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0704.0425

QED for fields obeying a square root operator equation

gleim200802 QED for fields obeying a square root operator equation Tobias Gleim Instead of using local field equations – like the Dirac equation for spin-1/2 and the Klein- Gordon equation for spin-0 particles – one could try to use non-local field equations in order to describe scattering processes. The latter equations can be obtained by means of the relativistic energy together with the correspondence principle, resulting in equations with a square root operator. By coupling them to an electromagnetic field and expanding the square root (and taking into account terms of quadratic order in the electromagnetic coupling constant e), it is possible to calculate scattering matrix elements within the framework of quantum electrodynamics, e.g. like those for Compton scattering or for the scattering of two identical particles. This will be done here for the scalar case. These results are then compared with the corresponding ones based on the Klein-Gordon equation. A proposal of how to transfer these reflections to the spin-1/2 case is also presented. Free scalar particles are usually described by means of the well-known Klein-Gordon equation (see e.g. [4,5,6,10]): ( ) ( ) 0,ˆ 222 =++∂ txmpt rr φ , (1) where we have used the momentum operator in configuration space ∇−= ip̂ (and set the velocity of light as well as Planck’s constant h to one). (1) can be regarded as an iteration of the following square root operator equation (see e.g. [1,2,4,5]): ( ) ( )txpmtxi t ,ˆ, 22  +=∂ φφ . (2) Introducing an electromagnetic field with a 4-vector potential ( ) ( )AAA r,0=µ and applying minimal coupling (with coupling constant e), i.e. replacing µµ xii ∂∂=∂ by ( )xeAxi µµ −∂∂ , the Klein- Gordon equation yields (see e.g. [6]) ( ) ( ) ( ) ( )( ) ( ) ( )[ ] ( )xxAxAexAxAiexmpt φφ µµµµµµ 2222 ˆ +∂+∂−=++∂ r , (3) where we have used the 4-vector notation ( ) ( ) ( )xtxxx rr ,,0 ==µ and Einstein’s summation convention. Here, the coupling terms on the right hand side of (3) could easily be separated from the term with the free particle Hamiltonian on the left hand side of (3). This is unfortunately no longer possible, if one couples the non-local equation (2) to the electromagnetic field: ( ) ( )( ) ( ) ( ) ( )xxeAxxAepmxi t φφφ 022 ˆ + , (4) because the vector potential A appears under the square root. But in a perturbation analysis of scattering processes, this property is useful, since such an analysis is based on the assumption that the coupling terms make only small contributions to the free particle solution due to the small value of the coupling constant e. By rewriting the Hamiltonian in (4), ( )( ) ( ) ( )22220220 ˆˆˆˆˆ AepAeApepmeAxAepmeAH rrrrrrrr +⋅−⋅−+++=−++=′ , (5) one can split off a factor with the free Hamiltonian ˆ pmH +=′ , (6) which yields ( )( ) ( )2122122220 ˆˆˆˆ1ˆ pmpmAepAeApeeAH rrrrrrr +++⋅−⋅−++=′ − . (7) With the above-mentioned assumption, it is now very tempting to expand the first square root factor. A very similar approach has already been proposed by [3]. We would like to restrict ourselves to a series expansion of the kind ...ˆˆ1ˆ1 2 1 +−+≈+ yyy (8) containing only constant, linear and quadratic terms, where ŷ denotes ( )( ) 12222 ˆˆˆˆ −++⋅−⋅−= pmAepAeApey rrrrrr . (9) Hamiltonian (7) is therefore approximated by ˆˆˆˆ HHHH ′+′+′≈′ (10) with ( )( )  ++⋅+⋅−=′ − 022 ˆˆˆˆ ApmpAApeH rrrrr , (11) ( ) ( )( ) ( )( )  +⋅+⋅+⋅+⋅−+=′ 22122 ˆˆˆˆˆˆˆˆ pmpAAppmpAAppmAeH rrrrrrrrrrrr , (12) where we have reordered the terms of expansion (9), retaining only terms to (and including the) quadratic order in e and recollected powers of ( )22 p̂m r+ . What we have won by (10) is a separation of the free Hamiltonian (6) from the coupling terms in (5) that approximately result in the sum of 1Ĥ ′ and 2Ĥ ′ (i.e. (11) and (12) respectively). For we are only interested in corrections to the free Hamiltonian anyway, this approximation might not hurt very much. But however, this separation seems not to be a true one, because of the multiple factors of powers of ( )22 p̂m r+ in (11) and (12). That is, we need an interpretation of these operators. To this end, it is useful to know that for the free square root operator equation (2), an integral representation can be given (see [1,2]): ( ) ( ) ( ) ( )( )txtxxxxdtxi t ,:,, 3 rrrrr φφφ Ω=′′−Ω′=∂ ∫ , (13) where Ω denotes an energy distribution ′−⋅−=′−Ω xxpip e rrrrr (14) with 22 pmpp r +== ωω . (15) (13) results from the fact, that one would expect to obtain the following momentum space representation of (2): ( ) ( )tptpi pt , φωφ =∂ with ( )tp,~ rφ denoting the Fourier-transformed ( )tx,rφ . If the operator 22 p̂m r+ corresponds to ( )∫ ′−Ω′ xxxd rr3 , the operator ( ) 2122 ˆ −+ pm r must correspond to ( )∫ ′−Ω′ − xxxd rr13 with ′−⋅−−− =′−Ω xxpip e rrrrr 1 , (16) because ( ) ( ) ( ) ( ) ( )xxxxxxxdxxxxxd ′−=′−′′Ω′′−Ω′′=′−′′Ω′′−Ω′′ −− ∫∫ rrrrrrrrrr 31313 δ (17) with the Dirac distribution ′−⋅−=′− xxpie rrrrr δ . (18) (17) should be an integral representation of the “symbolic equation” ( ) ( ) ( ) ( ) 1ˆˆˆˆ 21212121 22222222 =++=++ −− pmpmpmpm rrrr . Accordingly, terms with the nth power of ( )2122 p̂m r+ , ( )222 ˆ npm r+ , (19) correspond to integrals over “the nth power of Ω “: ′−⋅−=′−Ω xxpinp rrrrr . (20) By replacing the operators of type (19) by integrals over “powers of Ω ” as given in (20), 1Ĥ ′ and 2Ĥ ′ (see (11) and (12), respectively) can now be given a configuration space representation, too. With these preparations, we can now address to the quantisation of the scalar field with the aim to be able to calculate scattering matrix elements. Quantisation of the scalar field and the description of scattering processes Starting with Hamiltonian (10), it is now possible to describe scattering processes within the framework of quantum field theory. For free scalar particles, a quantum field theoretic ansatz is described e.g. in [2] and [4], using (2) and (13), respectively, as equations for a field operator ( )xφ . The latter one can be formulated with the help of creation and annihilation operators +pa rˆ and pa rˆ , respectively: ( ) p xip ae φ , (21) where as usual xptxpxp p ⋅−==⋅ ωµµ with the 4-vector ( )pp p r,ω= and the subsequent definitions are postulated: ,00ˆ =pa r (22) 0ˆ0 =+pa r , (23) [ ] ( )ppaa pp ′−=+′ rrrr 3ˆ,ˆ δ , (24 a) [ ] 0ˆ,ˆ =′pp aa rr , [ ] 0ˆ,ˆ =+′+ pp aa rr (24 b) with the vacuum state 0 . Since we are interested in a quantum theory for bosons, [ ]••, in (24) must be a commutator (for fermions we would use here an anti-commutator instead, cf. e.g. [4]). Equations (21) to (24) are identical to those that one would postulate within a non-relativistic quantum field theory for bosons. For the density of a Hamiltonian, we make the usual ansatz (see e.g. [7]): ( ) ( )xHxH φφ ′= + ˆˆ (25) which one can retrieve from a density of a Lagrangian (see [2]): ( ) ( ) ( ) φφφφφφφφ ++++ Ω−Ω−∂−∂= iL . (25 a) Substituting (10) into (25), we get ˆˆˆˆ HHHH ++≈ (26) with ( ) ( )xHxH φφ 00 ˆˆ ′= + , (27) ( ) ( )xHxH φφ 11 ˆˆ ′= + , (28) ( ) ( )xHxH φφ 22 ˆˆ ′= + . (29) (25) is (among other things) motivated by the fact that ppp aapdHxd rr ˆˆ ∫∫ = ω (30) reproduces the relativistic analogue of the free non-relativistic Hamiltonian: pp aa pd rr ∫ . (31) (28) together with (29) are the densities of the Hamiltonian to (and including the) quadratic order in e. (21) to (24) are valid for free particles, but can also be used for interacting ones, if Dirac’s representation is used instead of the so far applied Heisenberg representation. Then, with (28) and (29) combined to a Hamiltonian density ˆˆˆ HHH I += (32) for the interaction of scalar bosons with photons, we can now start to calculate scattering matrix elements. To this purpose, we need the serial expansion of the S-operator (see e.g. [7]) to the order of 2e : ( ) ( ) ...ˆˆ1ˆ 21 +++= SSS (33) with ( ) ( )( )∫−= xHTxdiS Iˆˆ 41 , (34) ( ) ( ) ( ) ( )( )∫ ∫−= 2124142212 ˆˆˆ xHxHTxdxdiS II , (35) where we have introduced a time ordering operator ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )1212010221020121 ˆˆˆˆˆˆˆˆ xHxHTxHxHxxxHxHxxxHxHT IIIIIIII =−+−= θθ (36) with tθ . (37) ( )1Ŝ does not only contribute to the expansion (33) with terms of order e, but also to order 2e . Therefore, we can split off ( )1Ŝ into a term 1Ŝ containing only terms in e, ( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )∫ +−+ +Ω⋅+⋅−−= xxAxxpxAxApxTxdieS φφφφ 012141 ˆˆˆ r (38) with ( )( ) ( ) ( )∫ −Ω=Ω −− txxxxdx ,111131 rrr φφ (39) and a part containing only terms in 2e : ( ) ( ) ( )( )( )( ( ) ( ) ( )( ) ( ) ( ) ( )( )( )( ) )∫ Ω⋅+⋅−Ω⋅+⋅− 21212221 111118 ,ˆ,,ˆˆˆ txptxAtxApxxxdpxAxApx xxAxTxdieS rrrrrrrrrrrrr (40) where the momentum operator 1p̂ contains a gradient acting on 1x and 2p̂ acting on 2x . In (38) and (40), we have already substituted (32) into (34) and replaced powers of ( )2122 p̂m r+ by integrals over “powers of Ω ” (see (20)) in (11). Thus we can rewrite (34) as 1 ˆˆˆ SSS += . (41) The time ordering operator appearing in ( )1Ŝ can be left out, because it contains only one time. In ( )2Ŝ we only want to retain terms of order 2e , therefore IĤ can be approximated by 1Ĥ : ( ) ( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( )( ( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ))22021101 222221 111114 111112 xxAxxxAx xpxAxApxxpxAxApx xxAxxpxAxApxTxdxdieS Ω⋅+⋅Ω⋅+⋅+ Ω⋅+⋅−−= ∫ ∫ rrrrrrrr . (42) Here, the time ordering operator must be taken into account, because there are two times: 1t and 2t . We have already quantised the field of scalar bosons, but must do now the corresponding with the electromagnetic field. Since the scalar field in (38), (40) and (42) couple in a different way to the vector potential A than to the scalar potential 0A , the choice of a Coulomb gauge seems to be appropriate: 0=⋅∇ A . (43) Then the field equations take on the form ( ) jAA tt r =∇∂+∇−∂ 022 , (44 a) ρ−=∇ 02 A (44 b) with charge and current densities ρ and j , respectively. From (44 b) we can see that in this gauge the scalar potential is just a c-number: ( ) ( )∫ ′− xdtxA rr , (45) whereas the vector potential A becomes an operator when being quantised. For a free field, A can be chosen like (see e.g. [4,7,10]) ( ) ( )λε λλ ,ˆˆ~22 kecec xA xik rr∫ ∑ ⋅+⋅− += (46) with the usual photon frequency 2~ kk and with creation and annihilation operators λkc rˆ and +λkc rˆ , respectively, for photons: [ ] ( )kkcc rr −′= ′ 3ˆ,ˆ δδ λλλλ , [ ] [ ] 0ˆ,ˆˆ,ˆ == ++ ′′′′ λλλλ kkkk cccc rrrr , (47) and 0A would even vanish. The polarisation vectors ( )λε ,krr fulfil the relation (see e.g. [4, 7]): ( ) ( ) δλελε . (48) Due to the Coulomb gauge condition (43), ( ) 0, =⋅ λε kk rrr (49) is valid, too. In the following sections, we are going to calculate ( )1Ŝ and ( )2Ŝ by substituting the field operators ( )xφ and ( )xA from (21) and (46) as well as the distributions (20). Since we are considering electromagnetic interactions between (charged) spin-0 bosons, we have to take 0A in (45) into account, too. Therefore, we first have to find out what the density of charge in (45) will be in this case. This can be done by coupling the Lagrangian density for free spin-0 bosons (25 a) to an electromagnetic field with the aid of the minimal coupling scheme µµµ eAii −∂→∂ . Then in that Lagragian density an extra term φφ +0eA (the only one with an 0A ) emerges. If one subsequently regards the sum of that spin-0 and the electromagnetic Lagrangian, it is possible to obtain from it the electromagnetic field equations by means of the Euler-Lagrange equations. The former equations then contain a charge density φφ +e . That is why we can set ρ in (45) to φφ + . With these results, one can then continue to calculate scattering matrix elements. The first term we calculate is ( ) ( ) ( )( )( )( )∫ −+ Ω⋅+⋅= xpxAxApxxdI φφ 141 ˆˆ:ˆ r (50) appearing in ( )1Ŝ (see (38)). To this end, it is useful to recognise that by means of (16) we get ( )( ) ⋅−−− =Ω p φ . (51) With (51) and integration by parts, (50) yields: ( ) ( ) ( ) ( )( )++ −−++−+⋅= ∫ ∑ λλ ωπ kk ckppckppaa pdpdkd I rrrr ˆˆˆˆ, , (52) where ( ) ( ) ( )kppkpp kpp ±−±−=±− 21 δωωωδδ (53) and, of course, the a operators commute with the c operators. In 12Ŝ (40), the first integral can be expressed in a similar way: ( ) ( )( )( ) ( ) ( ) ( ) ( )( ( ) ( ) ) 22112211 22112211 ~2~22 ccppkkccppkk ccppkkccppkk pdpdkdkd xxAxxdI −+−−+−++− +−+−+−++ (54) After a quite lengthy but straightforward calculation, the second integral in (40) yields ( ) ( ) ( )( ) ( ) ( ) ( )( )( )( ) =Ω⋅+⋅ ⋅−Ω⋅+⋅= 212122 111112 ,ˆ,,ˆ txptxAtxAp xxpxAxApxxdxdI rrrrrrr rrrrrr (55) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) −+−−+−++− +−+−+−++− 11221122 11221122 ~2~22 21222 ccppkkccppkk ccppkkccppkk ppkaa pdpdkdkd Finally, we want to determine the second term in ( )2Ŝ (see (42)) which is not just like the product of two operators 1̂I due to the time ordering operator as defined in (36). Unfortunately, we cannot use the famous Wick theorem, because the scalar field operator (21) contains only contributions to positive energy solutions: it does not consist of a sum of both positive and negative energy solutions as it would be the case for the field operator of the Klein-Gordon equation. Due to the symmetry of the time ordering operator (36) in its arguments, we may conclude ( ) ( )( ) ( ) ( ) ( )2102012414212414 ˆˆ2ˆˆ xHxHxxxdxdxHxHTxdxd IIII ∫∫∫∫ −= θ . (56) With this property, the calculation of the second term in (42) can be simplified a bit: ( ) ( ) ( )( )( )( )( ( ) ( ) ( )( )( )( ) =Ω⋅+⋅ ⋅Ω⋅+⋅= 22222 111112 xpxAxApx xpxAxApxTxdxdI (57 a) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) −−−−−−− ⋅−−′−′−′ ++−−−−−− ⋅+−′−′−′ +−−−+−−− ⋅−−′+′−′ ++−−+−−− ⋅+−′+′−′ 21212 21212 21212 21212 21212 ~exp~exp ~exp~exp ~exp~exp ~exp~exp ,,ˆˆˆˆ ~2~22 tititt kppkppcc tititt kppkppcc tititt kppkppcc tititt kppkppcc kaaaa pdpdpdpdkdkd kppkpp kppkpp kppkpp kppkpp ωωωωωωθ ωωωωωωθ ωωωωωωθ ωωωωωωθ rrrrrr rrrrrr rrrrrr rrrrrr The two integrals over the θ function can be performed by means of the introduction of the two variables 21: tt −=τ , 21: ttT += (58) with the Jacobian ( ) 2 A linear combination of these variables 21 tbtaTBA +=+τ (59 a) can be expressed by means of ( )baA −= 1 , (59 b) ( )baB += (57) contains four terms of the subsequent type that can be simplified with the help of (58) and (59): ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( )τωωτθ τωωτθ 21221 2122122 2211212 =+−−− titixpett (60 a) The function θ can be expressed by an integral in the complex plane (see e.g. [6,8]), ( ) ∫ 0 (61) with an ε approaching zero. Substituting this into (60 a), we get: ( ) ( ) ( )( ) ( )21 21221 τωωτθ ωωδ + =−−+ ∫ i id i . (60 b) With this result, (57) becomes: ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⋅−−′−′−′−−+−− ⋅+−′−′−′+−+−− ⋅−−′+′−′−−++− ⋅+−′+′−′+−++− ′′′′′ δδωωωωωωδ δδωωωωωωδ δδωωωωωωδ δδωωωωωωδ kppkppcc kppkppcc kppkppcc kppkppcc kaaaa pdpdpdpdkdkd kppkppkk kppkppkk kppkppkk kppkppkk 1~~ˆˆ 1~~ˆˆ 1~~ˆˆ 1~~ˆˆ ,,ˆˆˆˆ ~2~22 rrrrrr rrrrrr rrrrrr rrrrrr (57 b) So far, we have only considered terms of the scattering matrix containing the electromagnetic vector potential A . Now, we have to address to those terms containing the scalar potential 0A too. (38) does not only contain (50), but also a Coulomb potential term: += φφ 045ˆ AxdI . (62 a) By substituting ρ in (45) by φφ + , we obtain the following equation for (62 a), if we take into account that the Fourier transformed Coulomb potential looks like : (63) 43333 aaaakpkpkdkdpdpdI pkkp rr −−′+′′′−= +′ . (62 b) (42) contains two terms in 0A . The first term consists of a combination of (50) and (62 a), but taken at different times and therefore joined via the time ordering operator. That is why we have to use (60) as well as (45) and (63) again: ( ) ( ) ( )( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( ) ( ) ( ) −−+−−+−−+ −−+−−+−−+ +−+−−++−+ +−+−−++− −−′+′ =Ω⋅+⋅= ωωωωωωωδδ εωωωω ωωωωωωωδδ ωωωωωωωδδ εωωωω ωωωωωωωδδ 11121 11121 ˆˆˆˆˆˆ~ ˆˆˆˆˆˆ ˆˆˆˆˆˆ~ ˆˆˆˆˆˆ 111112 kppkkp kpppkpk kpkkpp kpppkpk kppkkp kpppkpk kpkkpp kpppkpk caaaaa caaaaa caaaaa caaaaa ppkkdkdpdpdpdpdkd xxAxxpxAxApxTxdxdI rrrrrr rrrrrr rrrrrr rrrrrr (64) The second term of (42) containing a scalar potential is even quadratic in 0A : ( ) ( ) ( ) ( ) ( ) ( )( )∫∫ ++= 2202110124147ˆ xxAxxxAxTxdxdI φφφφ . (65 a) (65 a) contains a factor of two integrands of the kind of (62 a), but taken at two different times. Thus the time ordering operator must be taken into account. With the same substitutions as in (62) and (64), we obtain the following result: ( ) ( ) ( ) ( ) ( ) εωωωω δδωωωωωωωωδ ikkkk kkppkkpp aaaaaaaapdkdkdpdpdkdkdpdiI kpkpkpkp pkkppkkp +−−+−′−′ ⋅−′+−′−′+−′−−++−−+ ⋅′′′′= 11112222 22221111 ˆˆˆˆˆˆˆˆ rrrrrrrr rrrrrrrr (65 b) The results (52), (54), (55), (57), (62), (64) and (65) substituted into (38) to (42) now enable us to evaluate scattering matrix elements for scattering processes to (and including) the order 2e . As two examples, we turn first to the scalar analogue of Compton scattering in order to address then to the scattering of two identical scalar bosons. These two scattering processes can be compared easily with the corresponding results of the well-known scalar QED dealing with the Klein-Gordon equation (3). Compton scattering For Compton scattering, we need one scalar boson and one photon each in the input and output channel. This means, we have to evaluate the element 0ˆˆˆˆˆ0:ˆ ++′′′= µµ hqqh caSacS rrrr , (66) with the Ŝ -operator (33). Firstly, we realise that the terms based on 1̂I (see (52)) as well as 6Î (see (64)) must vanish, because the subsequent two elements in the photon operators become zero: 00ˆ00ˆˆˆ0 == ′′ ′′ µλµµλµ δδ hhkhkh cccc rrrrrr , (67) 00ˆˆˆ0 =++′′ µλµ hkh ccc rrr . There, we have used the commutation relations (47), the properties of creation and annihilation operators corresponding to those of (22) and (23) as well as abbreviated the delta functional by ( )hk rr −= 3δδ . (68) The terms with 2Î and 3Î need qppqqppq aaaa rrrrrrrr 2121 0ˆˆˆˆ0 δδ ′ ′ = , (69) whereas a term qppppqqppppq aaaaaa rrrrrrrrrrrr 12121212 0ˆˆˆˆˆˆ0 δδδ ′′′ ′′ = belongs to 4Î . Here we have used again (22) to (24). For 2Î , 3Î and 4Î the following equations are necessary, too: 00ˆˆˆˆ0 =+′′′′ µλλµ hkkh cccc rrrr , 00ˆˆˆˆ0 =+++ ′′′′ µλλµ hkkh cccc , (70) µλλµµλλµ δδδδ ′′′′ ′′′′ = khkhhkkh cccc rrrrrrrr 0ˆˆˆˆ0 , λµλµµλλµ δδδδ ′′′′ ′′′′ = khkhhkkh cccc rrrrrrrr 0ˆˆˆˆ0 , where we have commutated the creation operators successively to the left and the annihilation operators to the right. Furthermore, we see that the term based on 5Î from (62) becomes zero, due to 00ˆˆˆˆˆˆ0 =++′ ′′ qpkkpq aaaaaa rrrrrr . (71) A similar result holds true for the term based on 7Î from (65). 00ˆˆˆˆˆˆˆˆˆˆ0 22221111 ′′ qpkkppkkpq aaaaaaaaaa rrrrrrrrrr . (72) Now, we can determine (66) by means of (33): ( )( ) ( ) ( )762144134122215121 ˆˆˆˆˆˆˆ1ˆ IIIIiIiieIIieS +−+−−++−−≈− . (73) For Compton scattering, 1̂I , 2Î , 3Î , 4Î , 5Î , 6Î and 7Î can be evaluated explicitly: 0ˆˆˆˆ 7651 ==== IIII , (74) ( ) ( ) ( )µεµε δ ′′⋅ ′−′−+ , (75) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ′−⋅′′+′−⋅ ++⋅′−+⋅ ′−′−+ hqhhhqh hqhhhqh hqqhh rrrrrrrrr rrrrrrrrr ωωωωπ (76) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +⋅′+′⋅′′ ′−⋅′′−′⋅′−′−+ ′−′′−′ ωωωωπ hqhhqh hqhhqhhqhq hqhqhq hqhqhqqhh rrrrrr rrrrrr rrrrrrrr rrrrrrrr 2,2,1 2,2,1 (77) The two terms 2Î and 4Î in (73) resemble the three terms in the corresponding formula of Compton scattering for scalar bosons, but this time being based on the Klein-Gordon equation (3) (see e.g. [6,8]): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )]µεµε ωωωωπ ′−⋅′′ ′+′⋅′′ ⋅′−′−+ 22~2~22 (78) where ε is the 4-dimensional generalisation of the three dimensional polarisation vector ε used so far. In (78), the first two terms correspond to 4Î . We realise that these terms in (78) look very much like those in (77), but also that especially the propagators of both theories are a bit different. (75) resembles the third term in (78). On the other hand, 3Î in (76) could also be regarded as the analogue of the first two terms in (78) – at least after having used the Coulomb gauge condition (49) in (76). The first two terms in (78) vanish, if we choose the incoming scalar boson to be at rest, ( ) ( )0,:,0 rr mqqq == , (79) and want to have transversally polarised photons in this laboratory system, ( ) ( ) 0,, =⋅′′=⋅ qhqh µεµε , (80) and use the Lorentz gauge condition ( ) ( ) 0,, =′⋅′′=⋅ hhhh µεµε . (81) Accordingly, 3Î and 4Î in (73) vanish too, if we adopt (79) again and use the analogue of (80), ( ) ( ) 0,, =⋅′′=⋅ qhqh rrrrrr µεµε , (82) as well as apply the Coulomb gauge condition (49). That is, under these conditions in both versions of scalar QED, only one term remains (i.e. (75) in (73) and, accordingly, the third term in (78)) which is a relativistically generalised version of the matrix element from which the well-known Thomson scattering cross section can be evaluated. Even though the propagators in both theories are rather different, for this example of scattering process, the results do not seem to differ very much from each other in the laboratory system chosen above. Therefore the question arises, whether this is also the case for further scattering processes. To this end, we are going to investigate what happens, if two identical scalar bosons interact with each other. Scattering of two identical scalar bosons If we want to calculate matrix elements of the S-operator (33) for the scattering of two identical scalar bosons, we need two scalar bosons in the input channel and two in the output channel: 0ˆˆˆˆ0:ˆ ′′= qqqq aaSaaS rrrr . (83) We can reuse (73), but have to evaluate 1̂I , 2Î , 3Î , 4Î , 5Î , 6Î and 7Î again. Firstly, we recognise that due to (52) and the analogue of (22) and (23) for the photon operators 01̂ =I , (84) is valid. Furthermore, we need 21212211 0ˆˆ0 λλλλ δδ kkkk cc rrrr =+ (85) for calculating 2Î with the help of (54), whereas all the other photon operator terms therein vanish. The same holds true for 3Î with (55) and 4Î with (57). As far as the scalar boson operators are concerned, for the evaluation of 2Î and 3Î 00ˆˆˆˆˆˆ0 121212 =+++′′ qqppqq aaaaaa rrrrrr . Hence, these two terms, 0ˆ2 =I , (86) 0ˆ3 =I , are zero, too. For the evaluation of the non-vanishing term 4Î , the following result is useful: 2122121121221211 212212112122121112121212 0ˆˆˆˆˆˆˆ0 pqpqpqpqpqpqpqpq pqpqpqpqpqpqpqpqqqppppqq aaaaaaaa ′′′′′′′′ ′′′′′′′′ rrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrr δδδδδδδδ δδδδδδδδ . (87) With (85), (87) and the invariance of (48) under the transformation kk −→ , we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 21212112 11111122 21212121 11111111 Ainterms qqqqqqqq qqqqqqqqqqqqie qqqqqqqq qqqqqqqq ′+⋅′−′−⋅′+  ′+⋅′−′−⋅′+′−′−+− ′−′′−′ ′−′′−′ εωωωεωωω εωωωεωωω rrrrrrrrrr rrrrrrrrrr (88 a) Here, we have omitted the terms 5Î , 6Î and 7Î containing terms in the scalar potential which are regarded later. (88 a) can be compared directly with the corresponding result of the Klein-Gordon equation (3) (see e.g. [9]): ( ) ( ) ( ) ( ) ( )  ′+⋅′+ ′+⋅′+′−′−+ qqqqqqqqie qqqq ωωωωπ , (89) if we use (48) and leave out the term ε approaching zero in (88) : ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 21212112 11111122 Ainterms qqqqqqqq qqqqqqqq qqqqqqqqie ′+⋅′−′−⋅′+ ′+⋅′−′−⋅′+ ′+⋅′+ ′+⋅′+′−′−+− rrvrvrrr rrvrvrrr rrrrrrvr (88 b) The first two terms in (88 b) correspond to (89) which contains the photon propagator in the Feynman gauge. But since we use a Coulomb gauge, only the space-like components of the linear 4-momenta appear in the numerators of the first two terms in (88 b). Moreover, in the second line of (88 b) two additional terms are present which can be reformulated by means of the delta distribution in (88 b): ( )( ) ( ) ( ) ( )( ) ( ) ( )221221 ′−′−′ − . (90) The structure of (88 b) is the same as that of (89). We have two terms: in the second term, the momenta of the two scalar bosons in the output channel have been exchanged compared to the first term. Now we address to the terms in the scalar potential 0A in (88). The Coulomb term 5Î contains a term pqkqqpqkkqpqqpqk pqkqqkqpkqpqqkqpqqpkkpqq aaaaaaaa ′′′′′′′′ ′′′′′′′′ rrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrr 21122112 211221121212 0ˆˆˆˆˆˆˆ0 δδδδδδδδ δδδδδδδδ Therefore, 5Î yields ( ) ( )  −−′+′= qqqqI rrrrδ . (91) The term 6Î vanishes, since the annihilation operators for photons in (64) act directly on the vacuum states. On the other hand, 7Î becomes a rather lengthy because of ( )( ) ( )( ) ( )( ) ( )( )11 12112121122222 21121121122222 121121211222 121121122122 122222111112 0ˆˆˆˆˆˆˆˆˆˆˆ0 aaaaaaaaaaaa kqpqkppkqpqkqp kkkqpqppqpqkqp kqpqkppkqkqp kqpqkkqkppqp qqpkkppkkpqq ′↔′−+ ′↔′−− ′↔′−− ′↔′−= ′′′′′′ ′′′′′′ ′′′′′′ ′′′′′′ rrrrrrrrrrrrrr rrrrrrrrrrrrrr rrrrrrrrrrrr rrrrrrrrrrrr rrrrrrrrrrrr δδδδδδδ δδδδδδδ δδδδδδ δδδδδδ where ( )11 kp ′↔′ denotes the same term as the immediately preceding one, but with exchanged momenta p′ and 1k ′ , respectively. Thus, 7Î gives −−′+′= −′−−′−+′−+′− 22222222 12112221 qpqpqpqpqpqpqpqp pqqpqq rrrrrrrrrrrrrrrr rrrrrr ωωωωωωωω ωωωωπ . (92) Hence in place of the time-like components (i.e. the energy terms) of the linear 4-momenta in the numerators of (89) derived from the Klein-Gordon equation, several terms arise: (90), (91) and (92). But (91) and (the non relativistic limit of) (92) would also have appeared, if we had started from the non-relativistic Schrödinger equation. Thus, these terms state the fact that the equation used for obtaining (88) (together with (90), (91) and (92)) is Schrödinger equation like. Conclusions and outlook For scalar bosons, we could see that it is possible to describe scattering processes by means of a square root operator equation being coupled to an electromagnetic field. We achieved this by splitting off a factor in the shape of the free square root operator from the equation and by a series expansion of a remaining square root factor containing terms in the electromagnetic vector potential and powers of the (inverse) free square root operator. The latter ones could be given an integral representation. Having quantised the fields involved, we could evaluate the scattering matrix elements for Compton scattering and for the scattering of two identical bosons (to – and including – the quadratic order of e) which, on the one hand, resembled the results derived with the help of the corresponding Klein- Gordon equation and, on the other hand, the results one would have obtained with a non-relativistic Schrödinger equation. Of course, now several questions arise, e.g.: • Can we formulate Feynman rules at all for our non-local scalar QED? • Do divergent terms appear and, if yes, can a renormalisation procedure be found? • Can the results of this non-local QED be confirmed (or refuted) by experiments? To the first question: if one tries to formulate Feynman rules, one must be aware that in each step of the approximation procedure, we have to expand the first square root factor (in the second term of) (7) to the desired order in e. Thus, the Hamiltonian we are using within that procedure must be adapted in each order of e of the approximation. Therefore we conclude that even if it were possible to formulate Feynman rules, they would be much more complicated than those for the Klein-Gordon theory. This is the price we have to pay for non-locality. But without Feynman rules, it is not very easy to analyse the renormalisability of that non-local scalar QED either. The answer to the last question listed above is negative due to a lack of elementary spinless bosons in nature. But that question would be sensible, if we had a corresponding non-local theory for spin-1/2 particles. We could even make a guess, how this theory would look like: for free spin-1/2 particles, the wave functions in (2) should be 2-spinors. It would also be possible to couple this equation to an electromagnetic field, but the usual minimal coupling scheme does not work. Instead we would have to postulate an equation: ˆ ′=∂ , (93) with the Hamilton operator ( ) ( ) 022 ˆˆ eAEiBeAepmH +⋅−−+=′ rmrrrr σ , (94) which contains an additional term with the Pauli matrices σ and the magnetic field B as well as the electric field E under the square root. For this equation, it has already been shown that it can reproduce the gyromagnetic factor of 2 for the electron as well as, when being applied to a hydrogen atom, that it can reproduce correct binding energies of the electron at least to (and including) the quadratic order in 2e (see [12]). We can apply the same approximation procedure to this equation as we have presented here for the scalar case. But of course, additional difficulties emerge: the Hamiltonian corresponding to the one shown in (25) should be Hermitian. And at least for the free spin-1/2 case, that Hamiltonian should be relativistically invariant. This means, that for the free case, (25) should be a Lorentz scalar with respect to the spin. To this end, we have to replace φ and +φ therein by combinations of a mixture of left and right handed 2-spinors Lφ and Rφ , respectively, and their Hermitian conjugates, because RLφφ + and LRφφ + are Lorentz scalars. Therefore, (25) could be replaced either by RLLR HHH φφφφ ± ++ ′+′= ˆˆˆ (95) or by RLLR HHH φφφφ mˆˆˆ 2 1 ′+′= +± + (96) so that the Hamiltonians (95) and (96) become Hermitian, because of the property ±′=′ HH ˆˆ m . (97) The results of a QED based on (95) or (96) could then be compared with the ones based on a corresponding Dirac equation. The author does not know, whether such an approach has already been performed for the spin-1/2 case or for the here presented spinless case. He does not know either, if an application for it can be found, where e.g. non-local properties are indispensable. But it seems to him that from the technical point of view, square root operator equations coupled to an electromagnetic field like (4) (or maybe even like (93) together with (94)) can be handled within the framework of a quantum field theory. Such equations were given up quite early in the history of quantum mechanics for several good reasons, e.g. due to their lack of relativistic invariance (see [13]), their non-local character accompanied by the difficulty of finding an appropriate mathematical interpretation and description, respectively (see e.g. [4,5,6]). While the first reason mentioned remains still valid, from today’s perspective, those non-local properties might not be refused as vehemently as in the past (e.g. when one looks out for approximations of the so called Bethe-Salpeter equation [11]). The author hopes to have shown, that at least answers to the question of possible descriptions of such non-local square root operator equations can be found. References [1] E. Trübenbacher, Z. Naturforschung 44a, 801-810 (1989). [2] C. Lämmerzahl, J. Math. Phys. 34 (9), 3918-3932 (1993). [3] T.L. Gill and W.W. Zachary, J. Phys. A: Math.Gen. 38 2479-2496 (2005). [4] W. R. Theis, Grundzüge der Quantentheorie (Teubner, Stuttgart, 1985). [5] A. Messiah, Quantenmechanik 2 (Walter de Gruyter, Berlin, 1990). [6] J.D. Bjorken and S.D. Drell, Relativistische Quantenmechanik (Bibliographisches Institut, Mannheim, 1966). [7] W. Greiner and J. Reinhardt, Feldquantisierung (Verlag Harri Deutsch, Thun/Frankfurt a.M., 1993). [8] W. Greiner and J. Reinhardt, Quantenelektrodynamik (Verlag Harri Deutsch, Thun/Frankfurt a.M., 1984. [9] W. Greiner and A. Schäfer, Quantenchromodynamik (Verlag Harri Deutsch, Thun/Frankfurt a.M., 1989. [10] L. H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1985). [11] W. Lucha and F.F. Schöberl, Int. J. Mod. Phys. A 14 2309 (1999). [12] T. Gleim, quant-ph/0601211; quant-ph/0602047. [13] J. Sucher, J. Math. Phys. 4 (1), 17-23 (1963).

0704.0426

Feedback from first radiation sources: H- photodissociation

Draft version November 4, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 FEEDBACK FROM FIRST RADIATION SOURCES: H− PHOTODISSOCIATION Leonid Chuzhoy , Michael Kuhlen and Paul R. Shapiro Draft version November 4, 2018 ABSTRACT During the epoch of reionization, the formation of radiation sources is accompanied by the growth of a H− photodissociating flux. We estimate the impact of this flux on the formation of molecular hydrogen and cooling in the first galaxies, assuming different types of radiation sources (e.g. Pop II and Pop III stars, miniquasars). We find that H− photodissociation reduces the formation of H2 molecules by a factor of Fs ∼ 1 + 10 3ksxf esc δ −1, where x is the mean ionized fraction in the IGM, fesc is the fraction of ionizing photons that escape from their progenitor halos, δ is the local gas overdensity and ks is an order unity constant which depends on the type of radiation source. By the time a significant fraction of the universe becomes ionized, H− photodissociation may significantly reduce the H2 abundance and, with it, the primordial star formation rate, delaying the progress of reionization. Subject headings: cosmology: theory – early universe – galaxies: formation – galaxies: high redshift 1. INTRODUCTION The first stars in the ΛCDM universe are believed to have formed inside dark-matter-dominated minihalos filled with mostly neutral, metal-free gas of virial tem- perature Tvir < 10 4 K, when H2 molecules formed in sufficient abundance to cool the gas radiatively to ∼ 102 K. If, as currently thought, these stars were massive, hot, and luminous, they may have contributed significantly to the reionization of the universe, which CMB polarization observations by WMAP indicate was highly ionized by z ∼ 10 (Spergel et al. 2006). The release of ionizing UV radiation by minihalos and other sources (e.g. stars in more massive halos, with Tvir > 10 4 K, or miniquasars), required to explain reionization, must have been accom- panied by radiation release at energies below the H Ly- man limit, as well, however. This may, in turn, have lim- ited the H2 abundance inside minihalos and their ability to form stars, thereby limiting their contribution to cos- mic reionization. In the absence of dust and at densities below the three- body formation regime (n . 1010 cm−3), the most im- portant reaction for the production of H2 is H− +H → H2 + e −, (1) (e.g., Shapiro & Kang 1987 and refs. therein) with re- action rate k− = 1.3× 10 −9 cm3 s−1 (Schmetekopf et al. 1967). Once formed, H2 can be destroyed by collisions with other species H2 +H + → H+2 +H, (2) H2 +H → H+H+H, (3) H2 + e − → H+H+ e−, (4) or by photodissociation via Lyman-Werner band photon absorption H2 + γ → H+H. (5) 1 McDonald Observatory and Department of Astronomy, The University of Texas at Austin, RLM 16.206, Austin, TX 78712, USA; [email protected] 2 Institute for Advanced Study, Princeton, NJ, 08540, USA; [email protected] The latter process becomes dominant once a substan- tial UV background is built up between 912 and 1110 Å, providing a feedback mechanism against the forma- tion of new radiation sources (e.g. Haiman et al. 1997; Haiman et al. 2000; Ciardi et al. 2000; Machacek et al. 2001; Mesinger et al. 2006). In this paper we explore the impact of another feedback mechanism, the photodissociation of H−, H− + γ → H+ e−. (6) The cross-section for photodissociation of H− is well fit- ted by (Wishart 1979) σ−(ǫ) = 2.1× 10 −16 (ǫ− 0.75) ǫ3.11 cm2, (7) where ǫ is the photon energy in eV. The cross section is zero below a threshold of ǫ < 0.755eV, the binding en- ergy of the second electron. In the absence of the UV background, the primary mode of H− destruction is the formation of H2 (Eq. [1]), 3 so introducing the H− pho- todissociating flux reduces the H2 formation rate by a factor Fs = 1 + , (8) where ζ− = nγ(ǫ)σ−(ǫ)cdǫ is the photodissociation rate per H− ion, nH is the hydrogen atom number density and nγ(ǫ) is the number density of photons with energy ǫ.4 Hence the importance of this mechanism depends pri- marily on the local density ratio of H− photodissociating photons and hydrogen atoms. 3 When gas fractional ionization is high (x & 0.01) mutual neu- tralization with H+ can provide another efficient channel for H− destruction. However, typically the fractional ionization of mini- halos is much lower. 4 This approximation for Fs breaks down when its value exceeds ∼ 50, since for such UV intensities H+ +H → H + γ reaction becomes a dominant channel of H2 production (assuming reaction rates given by Shapiro & Kang (1987)). Note also that k is still uncertain to within a factor of a few (see Glover et al. 2006), and this uncertainty carries over to Fs when Fs ≫ 1. http://arxiv.org/abs/0704.0426v2 The impact of H− photodissociation differs from that of H2 by two fundamental characteristics. First, the time required for H− abundance to approach equilibrium is very short (typically less than 10000 years), while for H2 the equilibration time can exceed the Hubble time. Therefore, when gas is exposed to a transient UV flux, produced by nearby Pop III stars, for example, H− pho- todissociation can generally be ignored, as it does not affect the subsequent thermal and chemical evolution. Secondly, photons that make up the H2 photodissociat- ing background are destroyed after a few percent of the Hubble time, as they redshift into one of the Lyman se- ries resonances, and must be replenished continuously. By contrast, photons that constitute the H− photodis- sociating background are very rarely destroyed, which allows them to accumulate over time. Consequently the importance of H− photodissociation increases over time, and as we show in this paper, by the time a significant (∼ 10 %) fraction of the Universe is ionized, H− pho- todissociation may result in a drastic reduction of the molecular hydrogen abundance. This in turn may lead to a reduced star formation rate and delay the progress of reionization. Recently, Glover (2007) considered the suppression of H2 formation due to the photodissociation of H − and H+2 . Whereas Glover (2007) focused on the local feed- back around and inside HII regions created by Pop III stars, we treat the problem globally and also consider long range effects due to the much lower optical depth of the universe below the Lyman limit. The paper is organized as following. In §2 and 3, we estimate the intensity of H− photodissociating flux pro- duced by UV and X-ray sources, respectively. In §4, we discuss the implication of our results for gas cooling in minihalos. 2. H− PHOTODISSOCIATING BACKGROUND - UV SOURCES 2.1. Recombination products Since the first radiation sources are expected to form within overdense gas clouds, only the escaping fraction of their ionizing photons, fesc, was available for ioniza- tion of the diffuse IGM. The rest was absorbed within the host halos and, via the process of radiative recombi- nation, converted into lower energy UV photons. Since the universe during that epoch is transparent to most non-ionizing UV photons, 5 almost all of them add to the H− photodissociation background. Neglecting recombinations in the diffuse IGM, the mean ionization is x = Nibfesc, where Nib is the to- tal number of ionizing photons per baryon produced up to this point. Inside halos, the recombination time is quite short, and so the number of ionizations taking place there, Nib(1− fesc) = x(1− fesc)/fesc, is almost equal to 5 An exception occurs for photons whose frequency is close to one of the high (n > 2) Lyman resonances, which, following their absorption by hydrogen atoms, are further split into two or more lower energy photons. For Lyα photons, the optical depth is also very high, but in their case the absorption in almost all cases is followed by reemission, with the destruction probability being ex- tremely low (e.g. Furlanetto & Pritchard 2006). Also, at the very early stage of reionization (x ≪ 1) the presence of H2 molecules makes the universe opaque in the Lyman-Werner range. However, since their initial abundance (∼ 10−6) is already very low, the number of photons they destroy is negligible. the number of electron recombinations to n ≥ 2 states (i.e., recombinations which do not result in emission of additional ionizing photons), Nrec. Therefore the average H− photodissociating rate is given by ζ− = Nrecnbarc〈σ−〉 = xnbarc〈σ−〉 1− fesc , (9) where nbar is the mean baryon density and 〈σ−〉 is the av- erage cross-section per recombination photon times the average number of photons per recombination, 〈σ−〉 = (jǫ/αrecnenpǫ)σ−(ǫ)dǫ. Note that since emissivity, jǫ, is proportional to nenp, 〈σ−〉 is in fact independent of ne and np. Using Osterbrock’s (1989, Sec. 4.3) calcula- tion of the recombination spectrum, (jǫ/αrecnenp), and assuming that the temperature of the recombining gas is close to 104 K, we find 〈σ−〉 = 3.4× 10 −17cm2. By combining equations (8) and (9), we can estimate the importance of the H− photodissociation due to re- combination radiation. Assuming that most of the re- combinations occurred recently, we find that, the recom- bination radiation alone will suppress the H2 formation rate by Fs = 1 + 800δ (1− fesc), (10) where δ = 1.08nH/nbar is the local overdensity. Here we have neglected recombinations in the diffuse intergalactic medium (IGM) and the associated H− dissociating pho- tons from these recombinations, but these would only further increase Fs. Cosmological redshift can affect the photodissociation rate by shifting the spectrum to longer wavelengths. Initially this leads to an increase in 〈σ−〉 due to the ǫ−3/2 dependence of the cross-section for ǫ ≫ 0.755 eV. Eventually, as more and more of the spectrum is shifted below the threshold, the cosmological redshift begins to decrease the dissociation rate. For recombi- nation photons this redshift effect is small, and the tran- sition to 〈σ−〉-depression occurs at a redshift factor of (1 + zi)/(1 + z) ≈ 2.5, see Figure 1. 2.2. Direct emission Unlike ionizing photons, whose intensity is heavily at- tenuated both in stellar atmospheres and in their host galaxies, most of the photons with frequencies below the Lyman limit escape freely into the IGM. From then on, photons with frequency below Lyβ undergo no evolution apart from cosmological redshift. By contrast, within a small fraction of the Hubble time, most photons with frequency between Lyβ and the Lyman limit are split by cascade into two or more photons after being redshifted into one of the hydrogen resonances. Most of the cas- cade products, which include lines such as Lyα, Hα, and Hβ, as well as a continuum spectrum produced by the two photon transition 2s → 1s, are above the 0.755 eV threshold for H− photodissociation. The relative importance of these directly emitted H− dissociating photons depends on the nature of the UV sources. Figure 2 shows the increase of the H− dissocia- tion rate due to inclusion of direct emission from metal- poor Pop III stars, which we calculated using the stellar atmosphere models of Schaerer (2002). Predictably, for 1 1.5 2 2.5 3 3.5 4 )/(1+z) Fig. 1.— Redshift evolution of the average H− photodissociation cross-section of the UV photons produced by recombination (dot- ted line), excitations by non-thermal electrons (dashed line) and massive Pop III stars (solid line). 4 6 8 10 12 14 Fig. 2.— The ratio between the total H− photodissociation rate and the photodissociation by recombination products alone for star with different effective surface temperature. very massive Pop III stars, with surface temperatures ∼ 105 K, adding the stellar continuum below the Lyman limit to the recombination spectrum increases the pho- todissociation rate by only ∼ 10%. If, on the other hand, most of the early ionizing flux was produced by stars with masses below 10M⊙, whose continuum emission is stronger at lower frequencies, then the total H− disso- ciation rate would be tripled at least. Likewise, direct emission may be important if most of the UV photons were produced by miniquasars. For example, assuming that their spectrum can be approximated by a power law, Lν ∝ ν −1.7, with a cutoff below 0.75 eV, adding the directly emitted photons to the recombination products increases the total photodissociation rate by a factor of 3. H− PHOTODISSOCIATING BACKGROUND - X-RAY SOURCES It has been suggested that X-ray photons could con- tribute a large fraction of the energy emitted by the first radiation sources (e.g. Ricotti & Ostriker 2004). By increasing the number of free electrons, X-rays can boost the production of H−, and thus of H2, provid- ing a positive feedback to the formation of new sources (Haiman et al. 2000; Kuhlen & Madau 2005). This ef- fect, however, would be at least partially offset by an in- crease of the H− photodissociating background, caused by conversion of X-rays into UV photons. The absorption of an X-ray photon is followed by re- lease of a non-thermal electron, which then loses some of its energy by inelastic collisions with atoms before it can thermalize its energy by elastic scattering with ions and other electrons. When the gas ionization fraction is low (x . 0.05), the photoelectron splits most of its energy evenly between collisional ionizations and excitations of hydrogen atoms (Shull & van Steenberg 1985). Using electron-hydrogen excitation cross-sections (Grafe et al. 2001; Stone et al. 2002), we find that around ∼ 5/6 of the excitations are to the 2p level, which are followed by emission of a Lyα photon. Most of the remaining excita- tions are to the 3p level, which decays via emission of one Hα photon and a subsequent two-photon decay from the 2s level. The Lyα, Hα and two-photon continuum each produce roughly equal contributions to H− photodisso- ciation. Per ionization, the average intensity-weighted cross-section for these photons is 〈σ−〉 = 1.6×10 −17cm2. Due to the low number of UV photons produced during this phase, the formation of H2 is not strongly affected Fs = 1 + 4δ . (11) After the ionized fraction climbs above x ∼ 0.05, most of the energy of the non-thermal electrons is converted to heat. However, simultaneously with the growth of the ionized fraction, the temperature of the gas rises, and as it crosses 104 K, the collisions between thermal electrons and atoms begin to dissipate the energy added by X- rays, mainly via emission of Lyα photons. Neglecting gas clumping, we find that the number of emitted Lyα photons per hydrogen atom is Nα = 4.6× 10 −8 cm3 s−1 x(1 − x)e−1.18×10 5/T nH dt. Assuming for simplicity that x and T are constants, we can rewrite the equation (12) as Nα = 11.2 (τe,X (1− x)e−1.18×10 , (13) where τe,X = xnσT dt is the Thompson optical depth from the epoch of partial ionization by X-rays. If X- ray preionization contributes at least half of the τe ∼ 0.1 measured by WMAP (i.e. τe,X ≈ 0.05), hydrogen atomic de-excitations in the diffuse IGM may produce & 30 Lyα photons per baryon. The suppression of H2 formation due to H − photodis- sociation by Lyα photons is Fs ≈ 1 + 100Nαδ −1. (14) Since the energy of Lyα photons (10.2 eV) is far above the H− photodissociation threshold (0.75 eV), the pho- todissociation rate grows roughly as (1+zi) 1.5/(1+z)1.5, where zi is the redshift at which the photon was emitted. In the case of an extended period of partial ionization, Fs may be increased by a factor of a few, possibly exceeding 104δ−1. Since, when the IGM temperature rises above 104 K, the formation of new minihalos is suppressed, the impact of H− photodissociating flux produced by X-ray conver- sion is relevant only for minihalos which have formed some time ago or for halos with Tvir > 10 4 K, which also rely on H2 cooling to form stars. 4. DISCUSSION As shown by our calculations, H− photodissociation reduces the formation of H2 molecules by a factor of Fs ∼ 1 + 10 3ksxf esc δ −1, (15) where ks is a constant of order a few, whose value de- pends on the type of radiation source and the growth history of the radiation background. Thus, by the time a significant fraction (& 0.1) of the universe becomes ion- ized, H− photodissociation can significantly reduce the H2 formation rate in regions with overdensities of up to a few thousands, i.e. in the interior regions of miniha- los. The equilibrium abundance of molecular hydrogen during this stage would be determined by the balance be- tween its formation and destruction rates (Eqs. [1] and nH2 = k− nH nH− , (16) where kLW is the H2 destruction rate by the Lyman- Werner photons. Thus a reduction of H− abundance by a factor Fs translates into the same reduction of the H2 abundance and, in minihalos, a comparable increase of the cooling time. Indirectly, H− photodissociation may affect the cool- ing in the central regions of minihalos even during the early stages of reionization. The maximum density that gas can reach in the core region of a minihalo is lim- ited by the amount of entropy it is able to radiate away during collapse. The lower density gas prevalent dur- ing the early collapse phase would be susceptible to H− dissociation from even a relatively low intensity H− dis- sociating flux, and the resulting lowered H2 abundance would limit its ability to radiate away entropy via H2 cooling. Furthermore, the density and H2 abundance at the center depend on the conditions in the low density outer regions, through their contributions to both the to- tal pressure and the self-shielding ability of the halo. We plan to investigate these effects further with numerical radiation-hydrodynamic simulations in the future. LC thanks the McDonald Observatory for the W.J. McDonald Fellowship. MK gratefully acknowledges sup- port from the Institute for Advanced Study. This work was partially supported by NASA Astrophysical Theory Program grants NAG5-10825 and NNG04G177G to P. R. S. REFERENCES Ciardi, B., Ferrara, A., & Abel, T. 2000, ApJ, 533, 594 Furlanetto, S. R., & Pritchard, J. R. 2006, MNRAS, 372, 1093 Grafe, A., Sweeney, C. J., & Shyn, T. W. 2001, Phys. Rev. A, 63, 052715 Glover, S. C. O. 2007, MNRAS, in press, astro-ph/0703716 Glover, S. C. O., Savin, D. W., & Jappsen, A. -K. 2006, ApJ, 640, Haiman, Z., Abel, T., & Rees, M. J. 2000, ApJ, 534, 11 Haiman, Z., Rees, M. J., & Loeb, A. 1997, ApJ, 484, 985 Kuhlen, M., & Madau, P. 2005, MNRAS, 363, 1069 Machacek, M. E., Bryan, G. L., & Abel, T. 2001, ApJ, 548, 509 Mesinger, A., Bryan, G.L., & Haiman, Z. 2006, ApJ, 648, 835 Osterbrock, D. E. 1989, “Astrophysics of gaseous nebulae and active galactic nuclei”, University Science Books, 1989, 422 p. 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0704.0427

The 3D +-J Ising model at the ferromagnetic transition line

arXiv:0704.0427v1 [cond-mat.dis-nn] 3 Apr 2007 The 3D ±J Ising model at the ferromagnetic transition line Martin Hasenbusch,1 Francesco Parisen Toldin,2 Andrea Pelissetto,3 and Ettore Vicari1 1 Dipartimento di Fisica dell’Università di Pisa and INFN, Pisa, Italy. 2 Scuola Normale Superiore and INFN, Pisa, Italy. 3Dipartimento di Fisica dell’Università di Roma “La Sapienza” and INFN, Roma, Italy. (Dated: November 1, 2018) Abstract We study the critical behavior of the three-dimensional ±J Ising model [with a random-exchange probability P (Jxy) = pδ(Jxy−J)+(1−p)δ(Jxy+J)] at the transition line between the paramagnetic and ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point at p = pN ≈ 0.767. By a finite-size scaling analysis of Monte Carlo simulations at various values of p in the region pN < p < 1, we provide strong numerical evidence that the critical behavior along the ferromagnetic transition line belongs to the same universality class as the three-dimensional randomly-dilute Ising model. We obtain the results ν = 0.682(3) and η = 0.036(2) for the critical exponents, which are consistent with the estimates ν = 0.683(2) and η = 0.036(1) at the transition of randomly-dilute Ising models. PACS numbers: 75.10.Nr, 75.40.Cx, 75.40.Mg, 64.60.Fr http://arxiv.org/abs/0704.0427v1 I. INTRODUCTION The ±J Ising model has played an important role in the study of the effects of quenched random disorder and frustration on Ising systems. It is defined by the lattice Hamiltonian H±J = − Jxyσxσy, (1) where σx = ±1, the sum is over the nearest-neighbor sites of a simple cubic lattice, and the exchange interactions Jxy are uncorrelated quenched random variables, taking values ±J with probability distribution P (Jxy) = pδ(Jxy − J) + (1− p)δ(Jxy + J). (2) For p = 1 we recover the standard Ising model, while for p = 1/2 we obtain the usual bimodal Ising spin-glass model. The phase diagram of the three-dimensional (3D) ±J Ising model is sketched in Fig. 1. The high-temperature phase is paramagnetic for any p. The low-temperature phase depends on the value of p: it is ferromagnetic for small values of 1 − p, while it is a spin-glass phase with vanishing magnetization for sufficiently large values of 1 − p. The different phases are separated by transition lines, which meet at a multicritical point N located along the so-called Nishimori line.1,2,3 The spin-glass transition has been mostly studied at the symmetric point p = 1/2, see, e.g., Refs. 3,4 and references therein. The spin-glass transition line extends up to the Nishimori multicritical point,2 located at5,6,7,8 pN ≈ 0.767. For larger values of p, the transition is ferromagnetic, up to p = 1 where one recovers the pure Ising model, and therefore a transition in the Ising universality class. At the ferromagnetic transition line, for pN < p < 1, the critical behavior is expected to belong to a different universality class. An interesting hypothesis, which has already been put forward in Refs. 3,9, is that the ferromagnetic transition of the ±J Ising model belongs to the 3D randomly-dilute Ising (RDIs) universality class (see, e.g., Refs. 10,11 for reviews on randomly-dilute spin mod- els). A representative of the RDIs universality class is the randomly site-dilute Ising model (RSIM) defined by the lattice Hamiltonian Hd = −J ρx ρy σxσy, (3) glass ferro 1 − p FIG. 1: Sketch of the phase diagram of the 3D ±J Ising model in the T -p plane. where ρx are uncorrelated quenched random variables, which are equal to 0, 1 with proba- bility P (ρx) = pδ(ρx − 1) + (1− p)δ(ρx). (4) For p < 1 and above the percolation threshold of the spins (pperc ≈ 0.3116081(13) on a cubic lattice12), the RSIM undergoes a continuous phase transition between a disordered and a ferromagnetic phase, whose nature is independent of p. This transition is definitely different from the usual Ising transition: for instance, the correlation-length critical exponent13,14,15,16 ν = 0.683(2) differs from the Ising value17,18 ν = 0.63012(16). The RDIs universality class is expected to describe the ferromagnetic transition in generic diluted ferromagnetic systems. For instance, it has been verified that also the randomly bond-diluted Ising model (RBIM) belongs to the RDIs universality class.13,19 These results do not necessarily imply that also the ±J Ising model has an RDIs ferromagnetic transition line. Indeed, while the RSIM (3) has only ferromagnetic exchange interactions, the ±J Ising model is frustrated for any value of p < 1. Therefore, the ferromagnetic transition in the ±J Ising model belongs to the RDIs universality class only if frustration is irrelevant, a fact that is not obvious and should be carefully investigated. Reference 9 investigated the issue by means of a Monte Carlo (MC) renormalization- group (RG) study, claiming that the ±J Ising model belongs to the same RDIs universality class as the RSIM and the RBIM. It should be noted however that the quoted estimate for the correlation-length exponent at the ferromagnetic transition, ν = 0.658(9), is close to but not fully consistent with the RDIs value ν = 0.683(2).13 Another numerical MC work20 investigated the nonequilibrium relaxation dynamics of the ±J Ising model and showed an apparent nonuniversal dynamical critical behavior along the ferromagnetic transition line. These results are not conclusive and further investigation is called for to clarify this issue. In this paper we focus on the transition line of the 3D ±J Ising model between the paramagnetic and the ferromagnetic phase. We investigate the critical behavior by means of MC simulations at various values of p in the region pN < p < 1. Our finite-size scaling (FSS) analysis provides a strong evidence that the critical behavior of the 3D ±J Ising along the ferromagnetic line belongs to the 3D RDIs universality class. For example, we obtain ν = 0.682(3) and η = 0.036(2), which are in good agreement with the presently most accurate estimates13 ν = 0.683(2) and η = 0.036(1) for the 3D RDIs universality class. The paper is organized as follows. In Sec. II we summarize some FSS results which are needed for the analysis of the MC data, and describe our strategy to check whether the transition belongs to the RDIs universality class. In Sec. III we describe the MC simula- tions. In Sec. IV we report the results of the FSS analysis. Finally, in Sec. V we draw our conclusions. In App. A we report the definitions of the quantities we compute. II. STRATEGY OF THE FINITE-SIZE SCALING ANALYSIS In this work we check whether the ferromagnetic transition line in the 3D ±J Ising models belongs to the RDIs universality class. For this purpose, we present a FSS analysis of MC data for various values of p in the region 1 > p > pN ≈ 0.767. We follow closely Ref. 13, which studied the ferromagnetic transition line in the 3D RSIM and RBIM and provided strong numerical evidence that these transitions belong to the same RDIs universality class. We refer to Ref. 13 for notations (a short summary is reported in App. A) and a detailed discussion of FSS in these disordered systems. According to the RG, in the case of periodic boundary conditions and for L → ∞, where L is the lattice size, a generic RG invariant quantity R at the critical temperature 1/βc behaves as R(L, β = βc) = R 1 + c11L −ω + c12L −2ω + · · ·+ c21L −ω2 + · · · , (5) where R∗ is the universal infinite-volume limit and ω and ω2 are the leading and next- to-leading correction-to-scaling exponents. In RDIs systems scaling corrections play an important role,16,21 since ω is quite small. Indeed we have ω = 0.33(3) and ω2 = 0.82(8) in the 3D RDIs universality class.13 These slowly-decaying scaling corrections make the accurate determination of the universal asymptotic behavior quite difficult. Instead of computing the various quantities at fixed Hamiltonian parameters, we keep a RG invariant quantity R fixed at a given value Rf . 22 This means that, for each L, we determine the pseudocritical inverse temperature βf(L) such that R(β = βf(L), L) = Rf . (6) All interesting thermodynamic quantities are then computed at β = βf (L). The pseudocrit- ical inverse temperature βf(L) converges to βc as L → ∞. The value Rf can be specified at will, as long as Rf is taken between the high- and low-temperature fixed-point values of R. The choice Rf = R ∗ (where R∗ is the critical-point value) improves the conver- gence of βf to βc for L → ∞; indeed βf − βc = O(L −1/ν) for generic values of Rf , while βf −βc = O(L −1/ν−ω) for Rf = R ∗. This FSS method has already been applied to the study of the critical behavior of N -vector spin models,22,23 and of randomly-dilute Ising models.13 As in Ref. 13, we perform a FSS analysis at fixed Rξ ≡ ξ/L = 0.5943, which is very close to the fixed-point value R∗ξ = 0.5944(7) of Rξ at βc. Given any RG invariant quantity R, such as the quartic cumulants U4 and U22, we consider its value at fixed Rξ, i.e., R̄(L) = R(L, βf(L)). For L → ∞, R̄(L) behaves as R(L, βc): R̄(L) = R̄∗ 1 + b11L −ω + b12L −2ω + · · ·+ b21L −ω2 + · · · , (7) where the coefficients bij depend on the Hamiltonian. The derivative R̄ ′ with respect to β of a generic RG invariant quantity R behaves as R̄′(L) = aL1/ν 1 + a11L −ω + a12L −2ω + · · ·+ a21L −ω2 + · · · . (8) Finally, the FSS of the magnetic susceptibility χ is given by13 χ̄(L) ≡ χ(L, β = βf (L)) = eL 1 + e11L −ω + e12L −2ω + · · ·+ e21L −ω2 + · · · + eb (9) where eb represents the background contribution. A standard RG analysis, see, e.g., Ref. 13, shows that the amplitudes of the O(L−kω) scaling corrections are proportional to uk3 (with a universal coefficient), where u3 is the leading irrelevant scaling field with RG dimension y3 = −ω. Hamiltonians such that u3 = 0— we call them improved Hamiltonians—have a faster approach to the universal asymptotic behavior, because the O(L−kω) scaling corrections vanish: b1k = a1k = e1k = 0 in Eqs. (7), (8), and (9). In this case the leading scaling corrections are proportional to u4L −ω2 , where u4 is the next-to-leading irrelevant scaling field and y4 = −ω2 is its RG dimension. In Ref. 13 it was shown that the RSIM for p = p∗ = 0.800(5) and the RBIM for p = p∗ = 0.56(2) are improved. Since scaling fields are analytic functions of the Hamiltonian parameters, u3 must be proportional to p − p∗ close to p = p∗, i.e. u3 ≈ c3(p − p ∗). Therefore, since the coefficients b1k, a1k, and e1k that appear in Eqs. (7), (8), and (9) are proportional to u 3, we b1k, a1k, e1k ∼ (p− p ∗)k. (10) Beside the quantities defined in App. A, we also consider observables—in analogy with the previous terminology, we call them improved quantities—characterized by the fact that the leading scaling correction proportional to L−ω (approximately) vanishes in any model belonging to the RDIs universality class.13 We consider the combination of quartic cumulants Ūim = Ū4 + 1.3Ū22, (11) and improved estimators of the critical exponent ν defined as R′ξ,im ≡ R̄ d , U 4,im ≡ Ū d (12) (Ūd is defined in App. A). In Ref. 13 we showed that, if the transition belongs to the RDIs universality class, the leading scaling correction proportional to L−ω of these improved ob- servables is suppressed. More precisely, we showed that the universal ratio of the amplitudes of the leading scaling correction in Ūim and Ū4 satisfies |b11,Ūim/b11,Ū4 | . , (13) while the one for the quantities R′ξ,im and R̄ ξ is bounded by |a11,R′ /a11,R̄′ . (14) The remaining scaling corrections are of order L−2ω and L−ω2. These improved observables are particular useful to check whether the transition in a given system belongs to the 3D RDIs universality class. 10 100 FIG. 2: Exponential autocorrelation time τ of the magnetic susceptibility for a mixture of Metropolis and cluster updates as discussed in the text, at p = 0.87. The dotted line shows the result of a fit to τ = cLz: this fit gives z ≈ 1.6. To summarize: in order to check whether the ferromagnetic transition of the 3D ±J Ising model belongs to the RDIs universality class, we perform a FSS analysis at fixed Rξ = 0.5943, and check if the results for the critical exponents and other universal quantities are consistent with those obtained for the RDIs universality class, which is characterized by13 critical exponents ν = 0.683(2) and η = 0.036(1), by the leading and next-to-leading scaling-correction exponents ω = 0.33(3) and ω2 = 0.82(8) and by the universal infinite- volume values of the quartic cumulants Ū∗22 = 0.148(1), Ū im = 1.840(4), and Ū d = 1.500(1). Notice that the fact that we fix Rξ = 0.5943 does not introduce any bias in our FSS analysis. III. MONTE CARLO SIMULATIONS We performed MC simulations of Hamiltonian (1) with J = 1 for p = 0.94, 0.90, 0.883, 0.87, 0.83, 0.80, close to the critical temperature on cubic lattices of size L3 with periodic boundary conditions, for a large range of lattice sizes: from L = 8 to L = 80 for p = 0.883, 0.87, to L = 64 for p = 0.94, 0.90, 0.83, and to L = 48 for p = 0.80. We chose values of p not too close to p = 1: indeed, as p → 1 we expect crossover effects due to the presence of the Ising transition for p = 1 and, therefore, that the asymptotic behavior sets in only for large values of L. We return to this point later. We used a Metropolis algorithm and multispin coding.24 In the simulation nbit systems evolve in parallel, where nbit = 32 or nbit = 64 depending on the computer that is used. For each of these nbit systems we use a different set of couplings Jxy. This allows us to perform 64 parallel simulations on a 64-bit machine, and therefore to gain a large factor in the efficiency of the MC simulations. We used high-quality random-number generators, such as the RANLUX25 or the twister26 generators.27 Using the twister random-number generator, we need about 1.2 × 10−9 seconds for one Metropolis update of a single spin on an Opteron processor running at 2 GHz. Our simulations took approximately 3 CPU years on an Opteron (2 GHz) processor. It is worth mentioning that cluster algorithms, such as the Swendsen-Wang cluster28 and the Wolff single-cluster29 algorithm, show significant slowing down in the ±J Ising model. At the earlier stage of this work we performed some simulations of the ±J Ising model at p = 0.87 using the algorithm used in Ref. 13 to simulate the RSIM and the RBIM. There we used a combination of Metropolis, Swendsen-Wang cluster,28 and Wolff single-cluster29 updates. More precisely, each updating step consisted of 1 Swendsen-Wang update, 1 Metropolis update, and L single-cluster updates. In all cases the exponential autocorrelation times τ of the magnetic susceptibility was small: τ . 1 in units of the above updating step, even for the largest lattice sizes considered, i.e. L = 192. In the ±J Ising model at p = 0.87 autocorrelation times are much larger. In Fig. 2 we plot estimates of τ as obtained from the magnetic susceptibility. They show a clear evidence of critical slowing down: τ ∼ Lz with z ≈ 1.6. Such a value of z should be compared with the dynamic exponent of Swendsen-Wang and Wolff cluster algorithms in the RSIM, which is much smaller:30 z . 0.5. These results show that cluster algorithms behave differently in the ±J Ising model, likely due to frustration. They suggest that frustration is relevant for the cluster dynamics. Taking also into account the computer time required by the cluster algorithms, we then turned to a multispin Metropolis algorithm. This turns out to be much more effective at the lattice sizes considered, although it has a larger dynamic exponent z & 2, see, e.g., Ref. 30 and references therein. We also mention that the autocorrelation time significantly increases with decreasing p (keeping L fixed). For example, for L = 48 it increases by approximately a factor of 10 from p = 0.90 to p = 0.80. This represents a major limitation to perform simulations for large lattices close to the multicritical point. For each lattice size we considered Ns disorder samples, with Ns decreasing with in- creasing L, from Ns & 10 6 for L = 8 to Ns & 2 × 10 4 for the largest lattices. For each disorder sample, we collected a few hundred independent measurements at equilibrium. The averages over disorder are affected by a bias due to the finite number of measures at fixed disorder.13,31 A bias correction is required whenever one considers the disorder average of combinations of thermal averages. We used the formulas reported in App. B of Ref. 13. Errors were computed from the sample-to-sample fluctuations and were determined by using the jackknife method.27 Our FSS analysis is performed at fixed Rξ ≡ ξ/L. In order to determine expectation values at fixed Rξ, one needs the values of the observables as a function of β in some neighborhood of the inverse temperature βrun used in the simulation. In Ref. 13 we used the reweighting method for this purpose. This requires that the observables and, in particular, the values of the energy are stored at each measurement. For the huge statistics like those we have for the smaller values of L, this becomes unpractical. Therefore, we used here a second- order Taylor expansion, determining O(β, L) from O(βrun, L)+aO(β−βrun)+ bO(β−βrun) The coefficients aO and bO are obtained from appropriate expectation values as in Ref. 23. Since their computation involves disorder averages of products of thermal averages, we have implemented in all cases an exact bias correction, using the formulas of Ref. 13. Derivatives with respect to β are then obtained as O′(β, L) = aO + 2bO(β − βrun). Of course, this method requires |βrun − βf | to be sufficiently small. We have carefully checked the results by performing, for each L and p, runs at different values of β. The MC estimates of the quantities introduced in Sec. II and in App. A at fixed Rξ ≡ ξ/L = 0.5943 are available on request. IV. FINITE-SIZE SCALING ANALYSIS In this section we present the results of our FSS analysis of the MC data at fixed Rξ = 0.5943. 0.0 0.1 0.2 0.3 0.4 0.5 -0.33 p=0.90 p=0.883 p=0.87 p=0.83 p=0.80 FIG. 3: MC estimates of Ū22 versus L −ω with ω = 0.33 for different values of p. The dotted lines show results of fits to c0 + c1L −ε1 + c2L −ε2 , fixing c0 = 0.148, ε1 = 0.33, and ε2 = 0.82. In the RDIs universality class Ū∗22 = 0.148(1). A. Renormalization-group invariant quantities In Fig. 3 we show the MC estimates of Ū22 versus L −ω with ω = 0.33(3), which is the leading scaling exponent of the RDIs universality class. The data vary significantly with p and L. This p and L dependence is always consistent with the existence of the expected next-to-leading scaling corrections, i.e. with a behavior of the form Ū22 = Ū 22 + c1L −ε1 + c2L −ε2 , (15) where Ū∗22, ε1 and ε2 are fixed to the RDIs values: 13 Ū∗22 = 0.148, ε1 = 0.33 and ε2 = 0.66, 0.82. The fits corresponding to ε2 = 0.82 are shown in Fig. 3. Note that in most of the cases it is crucial to include a next-to-leading correction. Only for p = 0.90 the data are well fitted by taking only the leading scaling correction. An unbiased estimate of ω can be obtained from the difference of data at different values of p, i.e. by considering Ū22(p1;L)− Ū22(p2;L) ≈ cL −ω. (16) Linear fits of the logarithm of these differences give results in reasonable agreement with 0.00 0.05 0.10 0.15 0.20 -0.82 p=0.90 p=0.883 p=0.87 p=0.83 p=0.80 FIG. 4: MC estimates of Ūim versus L −0.82. The filled square on the vertical axis corresponds to the RDIs estimate13 Ū∗im = 1.840(4). the RDIs estimate ω = 0.33(3), especially when only data corresponding to L ≥ Lmin = 24 are used. For Lmin = 24 [Lmin = 32], we obtain ω = 0.27(2) [ω = 0.27(3)] from the data at p1 = 0.83 and p2 = 0.90, ω = 0.19(5) [ω = 0.31(9)] from those at p1 = 0.883 and p2 = 0.90, and ω = 0.29(3) [ω = 0.25(4)] from the results at p1 = 0.83 and p2 = 0.883. We also fitted the difference Ū22− 0.148 at p = 0.90 to cL −ε (for this value of p next-to-leading corrections are apparently very small, see Fig. 3). We obtain ω = 0.35(3) [ω = 0.39(6)] for Lmin = 24 [Lmin = 32]. The results of the above-reported fits of Ū22 show that the leading scaling corrections proportional to L−ω vanish for p ≈ 0.883. Note that, close to p∗, the relevant next-to- leading scaling corrections should be those proportional to L−ω2 with ω2 ≈ 0.82. Indeed, according to Eq. (10), the coefficient of those proportional to L−2ω is of order (p − p∗)2, i.e. b12 ≈ b̄12(p − p ∗). Therefore, b12 is small if b̄12 = O(1) (we checked this numerically). This applies to the FSS at p = 0.87 and 0.90, where the L−2ω corrections can be neglected, although in these two cases we cannot neglect the leading L−ω correction whose coefficient is proportional to p−p∗. An analysis of the leading scaling corrections at p = 0.87, 0.883, 0.90, assuming the RDIs values Ū∗22 = 0.148(1) and ω = 0.33(3) (we perform combined fits to (15) 10 20 30 40 50 p=0.900, ε=0.82 p=0.900, ε=0.66 p=0.883, ε=0.82 p=0.870, ε=0.82 p=0.870, ε=0.66 p=0.830, ε=0.82 p=0.830, ε=0.66 p=0.800, ε=0.66 FIG. 5: Estimates of Ū∗im as obtained by fits to Ū im+ cL −ε, versus the minimum lattice size Lmin allowed in the fits. Some data are slightly shifted along the x-axis to make them visible. The dotted lines correspond to the RDIs estimate13 Ū∗im = 1.840(4). with ε1 = ω) gives the estimate p∗ = 0.883(3), (17) which is approximately in the middle of the ferromagnetic line, i.e. 1 − p∗ ≈ (1 − pN)/2. We performed a similar analysis for Ūd, obtaining a consistent estimate of p ∗. Thus, the ±J Ising model for p = 0.883 is approximately improved. Therefore, at p = 0.883, fits of the data assuming O(L−ω2) leading scaling corrections should provide reliable results. As discussed in Sec. II, a useful quantity to perform stringent checks of universality within the RDIs universality class is the combination Ūim of quartic cumulants reported in Eq. (11). For this quantity the scaling corrections proportional to L−ω are small, cf. Eq. (13), and thus the dominant corrections should behave as L−2ω, with 2ω ≈ 0.66. As already discussed, for values of p close to p∗, such as p = 0.87, 0.883, 0.90, also the L−2ω term is expected to be small and thus the dominant corrections should scale as L−ω2 with ω2 ≈ 0.82. In Fig. 4 we show the MC results for Ūim for various values of p. Fig. 5 shows results of fits to Ū∗ + cL−ε, (18) 0 20 40 60 80 L, L ε=0.82 ε=0.74 ε=0.90 FIG. 6: Estimates of Ū∗d as obtained by fits of Ūd at p = 0.883, to Ū d + cL −ε. Some data are slightly shifted along the x-axis to make them visible. The dotted lines correspond to the RDIs estimate13 Ū∗d = 1.500(1). with ε = 0.66, 0.82. We obtain Ū∗im = 1.840(3)[3], 1.842(2)[1], 1.845(2)[3] respectively for p = 0.90, 0.883, 0.87, fixing ε = ω2 = 0.82(8) (the error in brackets is related to the uncer- tainty of ω2) and using data with L ≥ 32; moreover we obtain Ū im = 1.847(3)[2], 1.840(7)[1] respectively for p = 0.83, 0.80, fixing ε = 0.66(6) and using data with L ≥ 24. For all values of p the results are in good agreement with RDIs estimate13 Ū∗im = 1.840(4). They provide strong support to a RDIs critical behavior along the ferromagnetic line. A further stringent check of universality comes from the analysis of the data Ūd at p = 0.883, because the data of Ūd are very precise due to a cancellation of the statisti- cal fluctuations.13 Since the model is improved, the L−kω scaling corrections are negligible and the large-L behavior is approached with corrections of order L−ω2 , ω2 = 0.82(8). We thus fit the data to Eq. (18) with ε = 0.74, 0.82, 0.90. In Fig. 6 we show the results. We obtain Ū∗d = 1.5001(1)[15], 1.5004(2)[9], 1.5003(10)[6] (the error in brackets is related to the uncertainty of ω2 = 0.82(8)) for Lmin = 12, 24, 48 respectively. Moreover, by fitting the data to Ū∗d +c1L −ε1+c2L −ε2 with ε1 = 0.33 and ε2 = 0.82, we obtain Ū d = 1.5006(7), 1.500(3) for Lmin = 12, 24 respectively. These results are in perfect agreement with the RDIs estimate 10 15 20 25 30 0.665 0.670 0.675 0.680 0.685 0.690 R’ξ,im , ε=0.82 R’ξ,im , ε=0.66 , ε=0.82 R’ξ , ε=0.82 R’ξ , ε=0.33 , ε=0.82 p=0.883 FIG. 7: Estimates of the critical exponent ν, as obtained by fits of R̄′ξ, Ū ξ,im, and U 4,im at p = 0.883. Lmin is the minimum lattice size allowed in the fits. Some data are slightly shifted along the x-axis to make them visible. The dotted lines correspond to the estimate ν = 0.682(3). Ū∗d = 1.500(1). Such an agreement is also confirmed by the analysis of the data of Ū22, for example a fit to Eq. (18) with ε = ω2 = 0.82(8) gives Ū 22 = 0.1486(8)[3] for Lmin = 32, to be compared with the RDIs estimate13 Ū∗22 = 0.148(1). We have not shown results for values of p too close to 1, for p > 0.90 say, because they are affected by crossover effects due to presence of the Ising transition for p = 1, as it also occurs in randomly dilute Ising models.16,21,32 For instance, for p = 0.94 the data are not compatible with a behavior of the form (15) with Ū∗22 fixed to the RDIs value. Our data that correspond to lattice sizes L ≤ 64 apparently converge to a smaller value, consistently with the expected crossover from pure to random behavior (in pure systems Ū∗22 = 0). The same quantitative differences are observed in the RSIM and in the RBIM close to the Ising transition. This suggests that in FSS analyses up to L ≈ 100 the asymptotic RDIs behavior can only be observed for p . 0.94. B. Critical exponents The correlation-length exponent ν can be estimated by fitting the derivative of Rξ and U4 to the expression (8). Accurate estimates are only obtained for improved Hamiltonians. For generic models, as shown in Ref. 13, good estimates are only obtained by using improved estimators, such as those reported in Eq. (12). We analyze the data at p = 0.883, which is a very good approximation of the improved value p∗ = 0.883(3). In Fig. 7 we report several results for the critical exponent ν, obtained by analyzing R̄′ξ, Ū 4, and their improved versions R ξ,im and U 4,im. We show results of fits of their logarithms to lnL+ a+ bL−ε, (19) fixing ε to several values. Since the Hamiltonian is approximately improved, scaling cor- rections are expected to decrease as L−ω2 with ω2 = 0.82(8). Since p = 0.883 is only approximately equal to p∗, one may be worried of the residual leading scaling corrections that are small but do not vanish exactly. Improved estimators should provide the most reliable results since the leading scaling corrections are additionally suppressed. As can be seen in Fig. 7, the results obtained by using R̄′ξ and Ū 4 and ε = 0.82 are perfectly consistent with those obtained from their improved versions. This confirms that the Hamiltonian is improved. Fits of R̄′ξ to (19) with ε = 0.33 do not provide stable results. The results approach the values obtained in the other fits only when increasing the minimum size Lmin allowed in the fit. This is expected, since the L −ω corrections should be negligible with respect to the L−ω2 ones. In conclusion, our final estimate of the correlation-length exponent is ν = 0.682(3), (20) which includes all results (with their errors) of the fits of R̄′ξ, Ū ξ,im, U 4,im to Eq. (19) with ε = 0.82(8) and Lmin = 16, 24. Estimate (20) is in perfect agreement with the most precise RDIs estimate ν = 0.683(2). Estimate (20) is also confirmed by the analysis of the data at the other values of p. Fig. 8 shows results obtained by fitting the logarithm of R′ξ,im to the function (19) for other values of p. They are definitely consistent with the result obtained at p = 0.883. Results for p = 0.80 are not shown because the available data are not sufficient to get reliable results. 10 15 20 25 30 p=0.900, R’ξ,im, ε=0.82 p=0.900, R’ξ,im, ε=0.66 p=0.900, U’ , ε=0.82 p=0.870, R’ξ,im, ε=0.82 p=0.870, R’ξ,im, ε=0.66 p=0.870, U’ , ε=0.82 p=0.830, R’ξ,im, ε=0.66 p=0.830, U’ , ε=0.66 FIG. 8: Results from ν obtained by fitting R′ξ,im to aL 1/ν(1+bL−ε). Some data are slightly shifted along the x-axis to make them visible. The dotted lines correspond to the estimate obtained at p = 0.883, i.e. ν = 0.682(3). In order to estimate the critical exponent η, we analyze the FSS of the magnetic suscep- tibility χ̄, cf. Eq. (9). We fit it to aL2−η + b (where b represents a constant background term), to aL2−η(1 + cL−ε), and to aL2−η(1 + cL−ε) + b (more precisely, we fit ln χ̄ to the logarithm of the previous expressions). The results at p = 0.883 are shown in Fig. 9, versus the minimum size Lmin allowed in the fits. We obtain the estimate η = 0.036(2), (21) which includes all results obtained for Lmin & 16. This estimate agrees with the most precise RDIs estimate η = 0.036(1). Fig. 10 shows results for the other values of p. Again, they are in good agreement. C. The critical temperature The critical temperature can be estimated by extrapolating the estimates of βf at Rξ = 0.5943, cf. Eq. (6). Since we have chosen Rξ = 0.5943 ≈ R ξ = 0.5944(7), 13 we expect 10 15 20 25 30 +cost ε=0.82 ε free +cost & ε=0.82 p=0.883 FIG. 9: Estimates of the critical exponent η, obtained by fitting ln χ̄ at p = 0.883, to a + (2 − η) lnL+bLη−2 (denoted by +cost), a+(2−η) ln L+a1L −ε, and to a+(2−η) ln L+a1L −ε+bLη−2. Some data are slightly shifted along the x-axis to make them visible. The dotted lines correspond to the final estimate η = 0.036(2). in general that βf − βc = O(L −1/ν−ω). For p = 0.883, since the model is approximately improved, the leading scaling corrections are related to the next-to-leading exponent ω2. Thus, in this case βf −βc = O(L −1/ν−ω2). This behavior is nicely observed in Fig. 11, which shows βf(L) at p = 0.883 vs L −1/ν−ω2 with 1/ν + ω2 ≈ 2.28. A fit to βc + aL −1/ν−ω2 gives βc = 0.300611(1). For the other values of p we expect βf − βc = O(L −1/ν−ω) with 1/ν + ω ≈ 1.79. Linear fits of βf (L) (for L ≥ Lmin with Lmin sufficiently large to give an acceptable χ 2) give the estimates βc = 0.25544(2) for p = 0.94, βc = 0.285285(5) for p = 0.90, βc = 0.313748(1) for p = 0.87, βc = 0.365459(5) for p = 0.83, βc = 0.42501(3) for p = 0.80. We finally recall that18 βc = 0.22165452(8) for p = 1 (the standard Ising model), and that 7 βc = 0.5967(11) at the multicritical Nishimori point at pN = 0.7673(3). In Fig. 12 we plot the available estimates of the critical temperature Tc ≡ 1/βc in the region 1 ≥ p ≥ pN . The estimates of Tc shown in Fig. 12 hint at a smooth linear behavior for small values of w ≡ 1− p, close to the Ising point at w = 0. This can be explained by some considerations 10 15 20 25 30 35 p=0.900, +cost p=0.900, ε free p=0.870, +cost p=0.870, ε free p=0.830, +cost p=0.830, ε free FIG. 10: Estimates of the critical exponent η, obtained by fitting χ̄ for various values of p. See the caption of Fig. 9 for an explanation of the fits. Some data are slightly shifted along the x-axis to make them visible. The dotted lines correspond to the result η = 0.036(2) obtained at p = 0.883. on the multicritical behavior around the Ising point at w = 0. The Ising critical behavior at w = 0 is unstable against the RG perturbation induced by quenched disorder at w > 0,33 which leads to the RDIs critical behavior. Indeed such a perturbation has a positive RG dimension yw at the Ising fixed point: 10,34 yw = αIs/νIs = 2/νIs − 3 where αIs and νIs are the Ising specific-heat and correlation-length critical exponents, and therefore17 yw = 0.1740(8). Thus, in the absence of an external magnetic field, beside the scaling field ut related to the temperature, there is another relevant scaling field uw associated with the quenched disorder parameter w ≡ 1− p. General RG scaling arguments21,35 show that the singular part of the free energy for w → 0 behaves as Fsing ∼ u 2−αIs t F (X), X = uwu t , (22) where φ = ywνIs = αIs = 0.1096(5) is the crossover exponent, and F (X) is a crossover scaling function which is universal (apart from normalizations). The scaling fields ut and uw depend on the parameters of the model. In general, we expect ut = t+ kw, (23) 0.000 0.002 0.004 0.006 -1/ν−ω2 0.3006 0.3007 FIG. 11: Estimates of βf (L) at p = 0.883 versus L −(1/ν+ω2) for 1/ν + ω2 ≈ 2.28. The dashed line corresponds to a linear fit of the data for L ≥ 12. where t ≡ T/TIs − 1, TIs is the critical temperature of the Ising model, and k is a constant. No such mixing between t and w occurs in uw, since uw vanishes for w = 0. Hence, we can take uw = w. The system has a critical transition for w > 0 at Tc(w). Since the singular part of the free energy close to a critical point behaves as (T −Tc) 2−α (α = −0.049(6) is the specific-heat exponent of the RDIs universality class), we must have F (Xc) = 0, where Xc is the value of X obtained by setting T = Tc(w) (see, e.g., Ref. 36 and references therein). Hence, we obtain w [Tc(w)/TIs − 1 + kw] = Xc, (24) and therefore Tc(w)/TIs − 1 = (w/Xc) 1/φ − kw + · · · , (25) where the dots indicate higher-order terms. This expression provides the w dependence of the critical temperature for w small. Note that the nonanalytic term in Eq. (25) is suppressed with respect to the analytic ones, because 1/φ ≈ 9.1. Thus, Tc(w) ≈ TIs(1 − kw + O(w Since Tc(w) < TIs, we can also infer that k > 0. From the results for Tc(w) we estimate k ≈ 2.2 for the ±J Ising model. 0.0 0.1 0.2 Ising multicritical point FIG. 12: The critical temperature Tc ≡ 1/βc vs 1− p. V. CONCLUSIONS In this paper we have studied the critical behavior of the 3D ±J Ising model at the transition line between the paramagnetic and the ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point at p = pN ≈ 0.767. We presented a FSS analysis of MC simulations at various values of p in the region pN < p < 1. The results for the critical exponents and other universal quantities are consistent with those of the RDIs universality class. For example, we obtained ν = 0.682(3) and η = 0.036(2), which are in good agreement with the presently most accurate estimates13 ν = 0.683(2) and η = 0.036(1) for the 3D RDIs universality class. Therefore, our FSS analysis provides a strong evidence that the critical behavior of the 3D ±J Ising along the ferromagnetic line belongs to the 3D RDIs universality class. We also note that the random-exchange interaction in the ±J Ising model gives rise to frustration, while the RDIs universality class describes transitions in generic diluted Ising systems with ferromagnetic exchange interactions. This implies that frustration is irrelevant at the ferromagnetic transition line of the 3D ±J Ising model. Moreover, the observed scaling corrections are consistent with the RDIs leading and next-to-leading scaling correction exponents ω = 0.33(3) and ω2 = 0.82(8). This indicates that frustration does not introduce new irrelevant perturbations at the RDIs fixed point with RG dimension yf & −1. APPENDIX A: NOTATIONS We define the two-point correlation function G(x) ≡ 〈σ0 σx〉, (A1) where the overline indicates the quenched average over the Jxy probability distribution. Then, we define the corresponding susceptibility χ ≡ xG(x) and the correlation length ξ G̃(0)− G̃(qmin) q̂2minG̃(qmin) , (A2) where qmin ≡ (2π/L, 0, 0), q̂ ≡ 2 sin q/2, and G̃(q) is the Fourier transform of G(x). We also consider quantities that are invariant under RG transformations in the critical limit. Beside the ratio Rξ ≡ ξ/L, (A3) we consider the quartic cumulants U4, U22 and Ud defined by , (A4) U22 ≡ µ22 − µ2 Ud ≡ U4 − U22, where µk ≡ 〈 ( k〉 . (A5) We also define corresponding quantities Ū4, Ū22, and Ūd at fixed Rξ = 0.5943. Finally, we consider the derivative R′ξ of Rξ, and U 4 of U4, with respect to β ≡ 1/T , which allow one to determine the critical exponent ν. 1 H. Nishimori, Prog. Theor. Phys. 66, 1169 (1981). 2 P. Le Doussal and A.B. Harris, Phys. Rev. Lett. 61, 625 (1988). 3 N. Kawashima and H. Rieger, in Frustrated Spin Systems, edited by H.T. Diep (World Scientific, Singapore, 2004); cond-mat/0312432. 4 H. Katzgraber, M. Körner, and A.P. Young, Phys. Rev. B 73, 224432 (2006). 5 Y. Ozeki and H. Nishimori, J. Phys. Soc. Japan 56, 3265 (1987). 6 R.R.P. Singh, Phys. Rev. Lett. 67, 899 (1991). 7 Y. Ozeki and N. Ito, J. Phys. A 31, 5451 (1998). 8 Refs. 5,6,7 report the estimates pN = 0.767(2), pN = 0.7656(20), and pN = 0.7673(3) respec- tively. 9 K. Hukushima, J. Phys. Soc. Japan 69, 631 (2000). 10 A. Pelissetto and E. Vicari, Phys. Rept. 368, 549 (2002). 11 R. Folk, Yu. Holovatch, and T. Yavors’kii, Uspekhi Fiz. Nauk 173, 175 (2003) [Phys. Usp. 46, 175 (2003)]. 12 H. G. Ballesteros, L. A. Fernández, V. Mart́ın-Mayor, A. Muñoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, J. Phys. A 32, 1 (1999). 13 M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, and E. Vicari, J. Stat. Mech.: Theory Exp. P02016 (2007). 14 P. Calabrese, V. Mart́ın-Mayor, A. Pelissetto, and E. Vicari, Phys. Rev. E 68, 036136 (2003). 15 A. Pelissetto and E. Vicari, Phys. Rev. B 62, 6393 (2000). 16 H.G. Ballesteros, L.A. Fernández, V. Mart́ın-Mayor, A. Muñoz Sudupe, G. Parisi, and J.J. Ruiz-Lorenzo, Phys. Rev. B 58, 2740 (1998). 17 M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. E 65, 066127 (2002). 18 Y. Deng and H.W.J. Blöte, Phys. Rev. E 68, 036125 (2003). 19 W. Janke, in Proceedings of the XXIII International Symposium on Lattice Field Theory, Dublin, July 2005, POS(LAT2005)018 20 N. Ito, Y. Ozeki, and H. Kitatani, J. Phys. Soc. Jpn. 68, 803 (1999). 21 P. Calabrese, P. Parruccini, A. Pelissetto, and E. Vicari, Phys. Rev. E 69, 036120 (2004). 22 M. Hasenbusch, J. Phys. A 32, 4851 (1999). 23 M. Campostrini, M. Hasenbusch, A. Pelissetto, and E. Vicari, Phys. Rev. B 74, 144506 (2006); Phys. Rev. B 63, 214503 (2001). 24 See, e.g., S. Wansleben, J.B. Zabolitzky, and C. Kalle, J. Stat. Phys. 37, 271 (1984); G. Bhanot, D. Duke, and R. Salvador, Phys. Rev. B 33, 7841 (1986). 25 M. Lüscher, Comput. Phys. Commun. 79, 100 (1994). 26 The SIMD-oriented fast Marsenne twister random number generator has been introduced by M. Matsumoto and M. Saito. Details can be found in M. Saito, Master Thesis (2007) and at http://www.math.sci.hiroshima-u.ac.jp/∼m-mat/MT/emt.html. 27 In order to make the use of these expensive (in terms of CPU-time) generators affordable, we employed the same sequence of random numbers for the update of all nbit systems (for the initialization of the configurations at the beginning of the simulation we used independent random numbers for each of the systems). This may give rise to a statistical correlation among the nbit systems. This effect is probably small and we have not detected it. Anyway, in order to ensure a correct estimate of the statistical error, all nbit systems that use the same sequence of random numbers have been put in the same bin in our jackknife analysis. 28 R.H. Swendsen and J-S. Wang, Phys. Rev. Lett. 58, 86 (1987). 29 U. Wolff, Phys. Rev. Lett. 62, 361 (1989). 30 D. Ivaneyko, J. Ilnytskyi, B. Berche, and Yu. Holovatch, Physica A 370, 163 (2006). 31 H.G. Ballesteros, L.A. Fernández, V. Mart́ın-Mayor, A. Muñoz Sudupe, G. Parisi, and J.J. Ruiz-Lorenzo, Nucl. Phys. B 512, 681 (1998). 32 The crossover exponent from pure Ising to RDIs critical behavior is the Ising specific-heat exponent34 αIs, see also Sec. IVC. This implies that the crossover scaling variable in the FSS at Tc is given by the combination X = cwL αIs/νIs , where w = 1 − p, αIs/νIs = 0.1740(8), and c is a normalization constant. When w → 0, strong crossover effects are expected for X . 1, which corresponds to L . (cw)−5.75. The RDIs asymptotic critical behavior is observed for X ≫ 1. 33 A. B. Harris, J. Phys. C 7, 1671 (1974). 34 A. Aharony, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic Press, New York, 1976), Vol. 6, p. 357. 35 I.D. Lawrie and S. Sarbach, in Phase Transitions and Critical Phenomena, Vol. 9, edited by C. Domb and J. Lebowitz (Academic, London, 1984). 36 A. Pelissetto and E. Vicari, cond-mat/0702273.

0704.0428

Star Formation in the Bok Globule CB54

Star Formation in the Bok Globule CB 54 David R. Ciardi Michelson Science Center/Caltech 770 South Wilson Avenue, M/S 100-22 Pasadena, CA 91125 [email protected] Cynthia Gómez Mart́ın University of Florida, Department of Astronomy 211 Bryant Space Sciences Building, Gainesville, FL 32611 [email protected] ABSTRACT We present mid-infrared (10.4 µm, 11.7 µm, and 18.3 µm) imaging intended to locate and characterize the suspected protostellar components within the Bok globule CB54. We detect and confirm the protostellar status for the near-infrared source CB54YC1-II. The mid-infrared luminosity for CB54YC1-II was found to be Lmidir ≈ 8 L⊙, and we estimate a central source mass of M∗ ≈ 0.8 M⊙ (for a mass accretion rate of Ṁ = 10−6 M⊙ yr −1). CB54 harbors another near-infrared source (CB54YC1-I), which was not detected by our observations. The non- detection is consistent with CB54YC1-I being a highly extinguished embedded young A or B star or a background G or F giant. An alternative explanation for CB54YC1-I is that the source is an embedded protostar viewed at an extremely high inclination angle, and the near-infrared detections are not of the central protostar, but of light scattered by the accretion disk into our line of sight. In addition, we have discovered three new mid-infrared sources, which are spatially coincident with the previously known dense core in CB54. The source tempera- tures (∼ 100K) and association of the mid-infrared sources with the dense core suggests that these mid-infrared objects may be embedded class 0 protostars. Subject headings: infrared: ISM — infrared: stars — ISM: globules — ISM: individual (CB54, LBN 1042) — stars: formation — stars: pre–main-sequence http://arxiv.org/abs/0704.0428v1 – 2 – 1. Introduction Protostars are young stellar objects that are still in the process of accreting the bulk of their material. Class 0 sources have been proposed as the evolutionary precursors to the class I protostars (André, Ward-Thompson, & Barsony 1993; André & Montmerle 1994). Work by Bontemps et al. (1996) and Saraceno et al. (1996) suggested a direct evolutionary se- quence from the class 0 stage to the class I stage. However, Jayawardhana, Hartmann, & Calvet (2001) have argued that class 0 protostars are located preferentially in higher density regions and that class I young stellar objects are located preferentially in lower density regions and, thus, may be at a comparable evolutionary states. To further complicate the distinction be- tween class 0 and class I objects, some class I objects, which are viewed at high inclination, may appear “class 0-like” because of the high optical depth associated with viewing the disk (nearly) edge-on (Masunaga & Inutsuka 2000). An example of this confusion may be seen in the binary system L1448N(A,B), where one component of the system has the spectral energy distribution (SED) of a class 0 protostar while the other has an SED of a class I protostar (Ciardi et al. 2003; O’Linger et al. 2006). This spread in apparent evolutionary status is also seen in larger clusters. In Perseus, Rebull et al. (2007) found that sources within clusters exhibited SEDs for a wide range in circumstellar environments, suggesting class 0 to class II protostars within the same aggregate. Either viewing geometry plays a significant role in our interpretation for each of the SEDs, the aggregates span a significant age spread, or if the sources are coeval, the disks/envelopes evolve faster than anticipated. To help address these issues, it would be beneficial to study a set of young stellar objects, located within the same environment, but isolated and free from the influence of other active star formation. The Bok Globule CB54, known to be a site of active star formation, may provide such an environment of isolated aggregate star formation. CB54 (LBN 1042; Clemens & Barvainis 1988) is a ∼ 100 M⊙ Bok globule associated with the Vela OB1 molecular cloud (d≈1500 pc; Launhardt & Henning 1997; Launhardt, Ward-Thompson, & Henning 1997). Bok globules are small (10−100 M⊙; Clemens, Yun, & Heyer 1991), isolated molecular clouds, most of which have been identified via opaque patches in optical images (Clemens & Barvainis 1988; Bourke, Hyland, & Robinson 1995). Globules have been found to be sites of star for- mation (e.g., Yun & Clemens 1990, 1994a,b; Alves & Yun 1995; Moreira & Yun 1995), of both single low-mass stars and multiple or binary stars (e.g., Yun 1996). Active star formation in CB54 was first identified by the association of a dense core with the IRAS point source PSC 07020-1618, which is located at the center of a collimated molec- ular outflow (Yun & Clemens 1994b). Near-infrared imaging revealed that CB54 actually – 3 – contains two bright near-infrared sources (CB54YC1-I, CB54YC1-II) and diffuse nebulosity of shocked H2 emission connecting the two sources (see Figure 1 and Yun 1996; Khanzadyan 2003). The positions of CB54YC1-I and -II are offset from the position of the IRAS point source (see Figure 1), although the IRAS beam size and PSC positional errors (20′′ × 4′′) do make it difficult to associate any one source with IRAS 07020-1618. Based upon its near-infrared colors (J−K = 5.29 mag, H−K = 2.58 mag), CB54YC1- II was classified as a class I young stellar object (Yun 1996). There is an MSX point source (G228.9946-04.6200), detected only in Band-A (8.3 µm), within a few arcsecs of CB54YC1-II (see Figure 1). CB54YC1-I (J −K = 4.34 mag, H −K = 1.63 mag) was also classified as a class I protostar (Yun 1996), but despite its similar near-infrared brightness to CB54YC1-II, it was not detected by MSX. Unlike CB54YC1-II, the near-infrared colors of CB54YC1-I could be explained with a highly extinguished (AV ∼ 20 mag) “bare” photosphere. VLA ob- servations detected a 3.6 cm and 6 cm source within a few arcseconds of CB54YC1-I (see Fig- ure 1), possibly indicating a stellar wind or accretion shock (Yun et al. 1996; Moreira et al. 1997). The position of the IRAS point source is spatially coincident with a dense core revealed in sub-mm, mm, and molecular line mapping; the position of which is offset from the positions of CB54YC1-I and CB54YC1-II but coincident with the IRAS point source (see Figure 1 and e.g., Wang et al. 1995; Zhou, Evans, & Wang 1996; Henning et al. 2001). Molecular line observations also indicated the presence of gravitational collapse in the core of CB54 (Wang et al. 1995; Afonso, Yun, & Clemens 1998). Water maser emission, almost exclusively associated with class 0 protostars in regions of low-mass star formation (e.g., Furuya et al. 2001), was also discovered in CB54 (de Gregorio-Monsalvo et al. 2006; Gómez et al. 2006). All of this suggests that the star formation in CB54 may be more substantial than revealed by the near-infrared imaging alone. We have observed the mid-infrared emission from the Bok globule CB54 at high spatial resolution (∼ 0.′′5) to clarify the evolutionary status of CB54YC1-I and CB54YC1-II and to search for additional protostars embedded in the globule core. Our work confirms the protostellar status of CB54YC1-II, but indicates that CB54YC1-I may be a more evolved young stellar object or a background giant star. In addition, we have discovered three new mid-infrared sources which are spatially coincident with the dense core and may be class 0 protostars. – 4 – 2. Observations and Data Reduction Mid-infrared imaging observations of CB54 were made on 2004 February 01 (UT) using the Thermal Region Camera and Spectrograph (T-ReCS; Telesco et al. 1998) on the Gemini South 8 m telescope. T-ReCS utilizes a 320×240 pixel Si:As blocked impurity band detector, with a spatial scale of 0.′′089 pixel−1 and a field of view of 28.′′8×21.′′6. The observations were centered on the J2000 coordinates (α, δ) = (α = 07h04m21s, δ = −16◦23′19′′). Imaging was obtained in three filters (N, Si-11.7 and Qa-18.3). The on-sky alignment of the T-ReCS field- of-view was chosen to cover the entire near-infrared nebulosity, the two known near-infrared sources, and the peak of the sub-millimeter (850 µm) core (see Figure 1). A standard off-chip 15′′ north-south chop-nod sequence was employed with total on- source integration times of 300 s per image. Three exposures in the Qa-18.3 filter were acquired for a total on-source integration time of 900 s. Flux calibration was obtained from imaging of the standard star HD 32887 (see the Gemini webpage for a compilation of mid-infrared standard stars and flux densities.)1. The weather quality was listed as the 50th-percentile, and the seeing at 11.7 µm was ≈ 0.′′4. A summary of the filters, frame times, total integration time per filter, and associated air masses is given in Table 1. The data were reduced with custom-written IDL routines for the T-ReCS data format. Four mid-infrared sources were detected by our observations, with no evidence of extended or diffuse mid-infrared emission (see Figure 2). Standard aperture photometry was performed using a 1′′ aperture radius. Detection limits were tested by inserting fake sources into the images and performing aperture photometry. A summary of the photometry (including estimated 1σ upper limits) and relative positional offsets with respect to CB54YC1-II is given in Table 2. 3. Discussion The near-infrared source CB54YC1-II is the brightest mid-infrared source and is de- tected in each of the mid-infrared filters. The other near-infrared source CB54YC1-I was not detected in any of the mid-infrared imaging. In addition, three new mid-infrared sources have been detected (MIR-a, MIR-b, and MIR-c). MIR-a and MIR-b were detected in each of three mid-infrared filters, while MIR-c was detected only at 18.3 µm. In the following sections, we evaluate the properties of these sources and discuss the possible star formation history of the globule. 1 http://www.gemini.edu/sciops/instruments/mir/MIRPhotStandards.html http://www.gemini.edu/sciops/instruments/mir/MIRPhotStandards.html – 5 – 3.1. CB54YC1-II CB54YC1-II, originally classified as a candidate class I protostar (Yun 1996), has a 2.2 − 10.3 µm spectral index [α = −d log(νFν)/d log(ν)] of α = 0.34 ± 0.01, which is consistent with the spectral index expected for a class I/flat spectrum young stellar object. The SED of CB54YC1-II in Figure 3. For comparison, the SED for a confirmed class I young stellar object (IRAS 04195+2251; Eisner, J. et al. 2006), with a similar spectral index α(2.2−10.6 µm) ≈ 0.3, has been scaled to the SED of CB54YC1-II (see Figure 3), exhibiting good agreement between the SEDs. If the mid-infrared emission is primarily the result of gravitational infall, (e.g., Ciardi et al. 2003), the mid-infrared luminosity provides a means of estimating the central protostellar mass. Integrating the SED from 1 − 20 µm, we estimate the mid-infrared luminosity to be Lmidir ≈ 8 ± 2 L⊙, for an assumed distance of 1500 pc. We estimate a central source mass from the relation L = (GṀM∗)/R∗, where Ṁ is the mass infall rate, R∗ is the source size, and M∗ is the source mass (Shu, Adams, & Lizano 1987). Using a standard R∗ = 3 R⊙ protostellar radius (Stahler, Shu, & Taam 1980) and typical mass accretion rates of Ṁ = 10−5 − 10−6 M⊙ yr −1 (Kenyon, Calvet, & Hartmann 1993), we estimate the central protostellar mass for CB54YC1-II to be M∗ = 0.08− 0.8 M⊙. A class II pre-main sequence star located behind a wall of extinction could also explain the observed SED for CB54YC1-II. In Figure 3, a median SED for T Tauri stars (TTS) (D’Alessio et al. 1999) has been scaled and convolved with an extinction model (R=3.1 assumed, Mathis 1990). In order to match both the mid-infrared flux densities and the slope of the near-infrared, the TTS SED must be extinguished by AV ≈ 25 magnitudes, and indeed, the position of CB54YC1-II in a JHK color-color (J −H = 2.71 mag, H −K = 2.58 mag) diagram is consistent with a heavily extinguished TTS (see Figure 4 in Haisch et al. 2000). The average volume density of the CB54 envelope (i.e., not including the central con- densation which is offset from the near-infrared sources), as derived from sub-mm (450 & 850 µm) imaging, is 〈nH〉 ≈ 5 × 10 4 cm−3 (Henning et al. 2001). At 1500 pc, the projected linear radius of the CB54 envelope is r ≈ 22500 AU (≈ 15′′). If we assume the globule is spherical, we can derive a peak column density of NH ≈ 4×10 22 cm−2, which corresponds to a visual extinction of AV ≈ 20 mag. The extinction estimatation does not take into account specific structure within the cloud including any dense envelope which may immediately surround the source, but does indicate the above derived extinction levels for CB54YC1-II are possible. Without a more complete SED or spectroscopy, especially at 3 − 8 µm, it is difficult to distinguish between the class I and class II models for CB54YC1-II – 6 – 3.2. CB54YC1-I CB54YC1-I was not detected in any of the three mid-infrared images, calling into ques- tion the original class I protostellar classification. Scaling the SED for the class I protostar (IRAS 04295+2251) to the JHK SED of CB54YC1-I, the predicted mid-infrared flux densi- ties for CB54YC1-I are Fν ∼ 80− 100 mJy at 11.7 µm and Fν ∼ 150 mJy at 18.3 µm. The predicted emission is ∼ 10σ above the detection limits (see Figure 4. The JHK slope for CB54YC1-I is too steep to match the near-infrared SED for a class II (TTS) pre-main sequence star. However, if TTS SED is modified with a screen of foreground extinction, the near-infrared SED for CB54YC1-I can be reproduced with a class II pre- main sequence model. In Figure 4, as was done for CB54YC1-II, the median TTS SED (D’Alessio et al. 1999) has been scaled and convolved with an extinction model and fitted to the JHK data for CB54YC1-I. For a best-fit extinction of AV = 15− 17 mag, the TTS SED can reproduce the near-infrared data. However, the model predicts mid-infrared densities (Fν ∼ 20−40 mJy at 11.7 µm and Fν ∼ 50−70 mJy at 18.3 µm). The predicted mid-infrared emission for the TTS SED is ∼ 7σ above the detection limits. It is possible that CB54YC-I is a more evolved star embedded in the globule or simply a star background to the globule. To explore these possibilities, we have fitted the JHK photometry with a blackbody function modified by a line-of-sight extinction curve: Sν = ΩBν(T ) exp (−Aν/1.086), where Bν(T ) is the Planck function, Aν is the frequency-dependent extinction, and Ω is the solid angle. At each extinction value in the range from AV = 0− 30 mag (∆AV = 0.1 mag), a range of temperatures (T = 500−50000 K in steps of 100 K) were tested. For a given extinction value, there is a unique blackbody temperature for which the chi- square is a minimum, but there is no global minimum representing a best fit the JHK data. The average temperature uncertainty for a given extinction value is ∼ 500 K. In Figure 5, the resulting reduced chi-squares and temperatures for each of the trial extinctions are plotted. The fitting uncertainty in the temperature for a given extinction value is approximately 10%. The reduced chi-square curve is relatively flat between 0 ≤ AV ≤ 26 mag. Beyond AV = 26 mag, the reduced chi-square climbs above χ ≈ 1 and begins to diverge rapidly. The best fit temperature at this boundary is T ≈ 30000 K. Because of the rapid change in the chi-square beyond this point, we regard this as the upper bound for the extinction and source temperature of CB54YC1-I. The lower bound to the extinction and temperature is constrained only by the detection limits of the mid-infrared observations. The combination of temperature and extinction must be such that CB54YC1-I is not detected in all three mid-infrared filters (N, Si-11.7, & Qa- – 7 – 18.3). For example, at zero extinction (AV = 0 mag), the best fit temperature is T ≈ 1100 K and the reduced chi-square is χ2 < 1, but the predicted mid-infrared flux densities violate the non-detections at both N and 11.7 µm (see Figure 4). The blackbody plus extinction models only predict mid-infrared flux densities below the detection limits in all three mid-infrared filters if AV & 23 mag. As an example, the predicted N-band flux densities (the most sensitive of the three observations) are plotted as a function of visual extinction in Figure 5 (bottom), showing that the predicted flux density drops below the detection limit at AV & 23 mag. We, therefore, regard AV ≈ 23 mag and the corresponding temperature (T ≈ 6000 K) as the lower bound for CB54YC1-I. The lower (AV = 23 mag, T = 6000 K) and upper (AV = 26 mag, T = 30000 K) bounds for the blackbody model fits to CB54YC1-I are shown in Figures 4. If the extinction is AV = 23 mag, the temperature (T ≈ 6000 K) suggests that the CB54YC1-I could be an early-G or late-F star. A main sequence dwarf of this spectral type has an absolute K magnitude ofMK ≈ 2.7 mag. With a measured K magnitude ofK = 11.76 (Yun 1996), the implied distance is only ∼ 250 pc, much too close to be extinguished by CB54. If, however, CB54YC1-I is a G or F giant star, the star would be approximately 4.0 magnitudes brighter and located at a distance of ≈ 1500 pc, sufficiently distant to place CB54YC1-I behind the globule. At the upper limit to the model fitting, (AV ≈ 26 mag, T ≈ 30000 K), the temperature corresponds to a B0 star. With an estimated absolute magnitude of MK = −3. CB54YC1-I would be at a distance of 2500 pc which is far enough to be behind the globule. The rarity and short lifespan (10 MYr) of B0 stars and the required chance alignment with the globule seem to make this a remote possibility. If, instead, the extinction is near the middle of the extinction range (AV = 24 − 25 mag), the best fit temperature (T ≈ 10000 − 15000 K) implies that CB54YC1-I may be a young A or B star. With an absolute K magnitude of MK ≈ −1.5− 0, this yields a distance of only 1200− 1500 pc placing CB54YC1-I at a distance consistent with being an embedded young star. An alternative explanation for CB54YC1-I is that the source is an embedded protostar viewed at an extremely high inclination angle, and the near-infrared detections are not of the central protostar, but of light scattered by the accretion disk into our line of sight. Unfortunately, near-infrared photometry alone can not distinguish these models. – 8 – 3.3. Mid-Infrared Sources Three new sources have been detected by our mid-infrared observations. These sources have no near-infrared counterparts. The mid-infrared sources lie just beyond the edge of the shocked H2 emission and do not correspond to any of the knots or condensations visible in the near-infrared diffuse emission (see Figure 1 and Yun 1996; Khanzadyan 2003). The mid-infrared sources, however, are located within the boundaries of the dense core in CB54 and clustered near the position of the IRAS point source (see Figure 1). The two brightest sources (MIR-a, MIR-b) were detected in all three filters, but MIR-c was detected only at 18.3 µm. A summary of the photometry is given in Table 2, and the SEDs for these sources are presented in Figure 6. To characterize and understand the relative temperatures of the mid-infrared sources, a single temperature blackbody was fit to both MIR-a and MIR-b. The blackbody fits did not include the broad-band N (10.3 µm) flux density which are potentially contaminated with an unknown amount of amorphous silicate, and were fit to the narrow-band 11.7 µm and 18.3 µm flux densities. The best-fit blackbody (Figure 6) temperatures for the two sources are quite similar (TA = 110 ± 10 K and TB = 100 ± 10 K) and are near what is expected for the bolometric temperatures of class 0 protostars (André, Ward-Thompson, & Barsony 2000). If the N-band photoometry is included in the fits, the resulting temperatures increase by 10− 20 K. Blackbody fits to the sub-mm and mm emission from the dense core yields a much colder envelope temperature of 25 K (Launhardt, Ward-Thompson, & Henning 1997). The summed flux densities predicted by the mid-infrared blackbody fits is not sufficient to explain the 100 µm flux density for IRAS PSC 07020-1618 (Fν ≈ 100 Jy), indicating that these class 0 protostars are harbored within the cold, dense envelope. If we assume that the mid-infrared emission is optically thin and little emission is con- tributed from the cold envelope, we can estimate the protostellar core mass associated with mid-infrared emission via (16/3)πaρD2 QνBν(Td) Fν (1) where Fν is the observed flux density at frequency ν, Qν is the grain emissivity at frequency ν, a is the grain radius, ρ is the grain mass density, D is the distance to CB54, and Bν is the Planck function at dust temperature Td. Assuming a = 0.5 µm, ρ = 1 g cm Qν = 0.1(λ/µm) −α, and α = 0.45 (e.g., Muthumariappan, Kwok, & Volk 2006), we estimate dust masses for MIR-a and MIR-b of Ma ≈ 0.014 M⊙ and Mb ≈ 0.043 M⊙, respectively. For an average gas-to-dust mass ratio of 100, the central mid-infrared cores have masses of Ma ≈ 1.4 M⊙ and Mb ≈ 4.3 M⊙. – 9 – MIR-c was detected only at 18.3 µm, but if use the 11.7µm uppper limit to restrict the blackbody fitting, we find an upper limit to the temperature of 120 K. (see Figure 6). Coupled with the 18.3 µm flux density, we estimate a limit to the total mass (gas+dust) of Mc & 0.2 M⊙. 4. Conclusions We have obtained high angular resolution 10 − 18 µm imaging of CB54, a ∼ 100 M⊙ Bok globule known to harbor a dense core and two near-infrared sources previously classified as class I young stellar objects. We have detected only one (CB54YC1-II) of the two near- infrared sources, confirming its protostellar evolutionary status. Based upon the mid-infrared luminosity, we estimate that the central protostellar mass for CB54YC1-II is M∗ = 0.08 − 0.8M⊙, depending on the mass transfer rate. The SED is also consistent with a more evolved T Tauri star behind a screen of extinction. Without a more complete SED, it is not possible to distinguish between these models. The other near-infrared source (CB54YC1-I) should have been detected if it were a class I protostar similar to that CB54YC1-II. We find that the near-infrared SED is consistent with the SED for a more evolved star extinguished by the globule itself. CB54YC1-I may be a background F- or G-giant or may be an embedded young A- or B-star. An alternative ex- planation for CB54YC1-I is that the source is an embedded protostar viewed at an extremely high inclination angle, and the near-infrared detections are not of the central protostar, but of light scattered by the accretion disk into our line of sight. High spatial resolution near- infrared polarimetry and/or mid-infrared spectroscopy could be used to ascertain the status CB54YC1-I. If CB54YC1-I is an embedded, young A or B star, its mass may be on the order of ∼ 2− 5 M⊙. Additionally, we have discovered three new mid-infrared sources (MIR-a, MIR-b, and MIR-c) which are spatially coincident with both the position of the associated IRAS point source and the center of the dense core in CB54. These sources are characterized with a 100 K blackbody, consistent the expected bolometric temperature of a class 0 protostar. Based upon the mid-infrared emission, we have estimated the masses for these sources to be ∼ 4 M⊙, ∼ 1.5 M⊙, and ∼ 0.2 M⊙. If CB54YC1-I is indeed an embedded A or B star, it is interesting to speculate that CB54YC1-I may have formed first and induced star formation further in the cloud, through the interaction of its outflow/winds with remainder of the globule. The total mass estimated for the sources within CB54 is about 10− 15 M⊙ or about 10− 15% of the total cloud mass. – 10 – Such a sequential process of star formation occurring in the Bok globule CB54 may be similar to what is observed in other globules (Huard, Weintraub, & Sandell 2000; Codella et al. 2006). Spectroscopy and a more complete SED in the mid-infrared and far-infrared is needed to disentangle the possible spectral types and evolutionary states for the sources in CB54. These observations were carried out during payback time to C. Telesco for develop- ment of T-ReCS. The authors would like to thank Charlie Telesco, Chris Packham, and Margaret Moerchen for collecting these data. Portions of this work were supported by the California Institute of Technology under contract with the National Aeronautics and Space Administration. C. G. M. acknowledges support from a University of Florida Graduate Mi- nority Fellowship, a SEAGEP Fellowship, and NSF grants AST97-3367 and AST 02-02976. C. G. M. would like to thank Eric McKenzie and Ana Matkovic for comments and sugges- tions. Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agree- ment with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Coun- cil (Australia), CNPq (Brazil) and CONICET (Argentina). This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Labora- tory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. REFERENCES Afonso, J. M., Yun, J. L., & Clemens, D. P. 1998, AJ, 115, 1111 Alves, J. F. & Yun, J. 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Summary of Observations λc ∆λ Frame Time On-Source Filter (µm) (µm) (ms) (seconds) Air Mass N 10.36 5.27 25.8 304 1.11 Si-11.66 11.7 1.13 25.8 304 1.84 Qa-18.30 18.3 1.51 25.8 912 1.18-1.34 Table 2. Source Positions and Flux Densities J H K MSX-A N Si-11.7 Qa-18.3 1.22 µm 1.65 µm 2.18 µm 8.28 µm 10.36 µm 11.66 µm 18.3 µm Source (∆α, ∆δ)a (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) References YCII (0.0, 0.0) 0.71± 0.07 5.56± 0.56 37.3± 3.7 278± 19 302± 5 217± 6 431± 11 1,2 YCI (−10.0,+8.3) 0.59± 0.06 4.63± 0.46 12.9± 1.3 · · · < 4 < 6 < 10 1,2 MIR-a (−15.8,+1.7) · · · · · · · · · · · · 16± 4 14± 6 219± 11 2 MIR-b (−13.6,−1.8) · · · · · · · · · · · · 12± 4 14± 6 322± 11 2 MIR-c (−12.6,−3.8) · · · · · · · · · · · · < 4 < 6 70± 12 2 aPositional offsets in arcsec from the source CB54YC1-II (α = 07h04m21.7s, δ = −16◦23′19′′ (J2000)). References. — 1. Yun (1996); 2. This Work – 15 – Fig. 1.— 2MASS KS image of CB 54. The dashed line delineates the area imaged with T-ReCS. The near-infrared sources CB54YC1-I and CB54YC1-II are annotated, and the positions of the detected mid-infrared sources are marked with the filled white circles (see Figure 2). The image (0,0) position is centered on CB54YC1-II. The circle (dotted line) is centered on the peak of 850 µm core; the size of the circle represents the approximate size of the 850 µm core (Henning et al. 2001). The cross marks the position of the IRAS source PSC 07020-1618. The open diamond marks the position of the 8 µm MSX source G228.9946-04.6200, and the open square marks the position for the cm source discovered with the VLA (Yun et al. 1996). The image pixel scale is 1′′ pix−1. – 16 – Fig. 2.— T-ReCS N-band, 11.7µm and 18.3µm images and contour plots of CB54. The (0,0) point of each image is centered on CB54YC1-II. The images have been stretched by an inverse hyperbolic sine to enhance the contrast. The detected mid-infrared sources are annotated. The position of CB54YC1-I (not detected by the mid-infrared observations) is marked in each image, and the position of MIR-c (detected only at 18.3µm) is marked in the N-band and 11.7µm images. The N-band contour levels are [0.03, 0.05, 0.15, 1.5, 3.5] mJy pix−1; the 11.7 µm contour levels are [0.05, 0.10, 0.15, 0.2, 0.4, 0.8, 1.6] mJy pix−1; the 18.3 µm contour levels are [0.4, 0.8, 1.6, 3.2] mJy pix−1. The pixel scale for each image is 0.′′089 pixel−1. – 17 – Fig. 3.— The SED of CB54YC1-II (solid points). The horizontal error bars represent the bandwidth of the filters. The dashed line represents the SED for the class I protostar IRAS 04295+2251. The dotted line represents a median TTS SED convolved with AV = 25 magnitudes of extinction. – 18 – Fig. 4.— The SED of CB54YC1-I (solid points). Upper limits for the mid-infrared obser- vations are represented by the downarrows. The long-dashed line represents the SED for the class I protostar IRAS 04295+2251. The dotted line represents a median TTS SED convolved with AV = 16 magnitudes of extinction. The solid line is a 1100 K blackbody; the dash-dot line is a 6000 K blackbody convolved with AV = 23 magnitudes of extinction, and the short-dashed line is a 30000 K blackbody convolved with AV = 26 magnitudes of extinction. – 19 – Fig. 5.— Reduced chi-squares (top), best-fit temperatures (middle), and predicted N- band flux densities (bottom), are plotted as a function of visual extinction for the black- body+extinction models fitted to the JHK flux densities of CB54YC1-I. The horizontal dashed line in the chi-square plot marks the sharp knee in the chi-square curve at AV = 26 mag (χ2 = 1.2). The horizontal dashed line in the flux density plot marks the 1σ detection limit (4 mJy) of the N-band observations. The vertical dashed lines delineate the extinction range of 23 ≤ AV ≤ 26 mag (see text for details). – 20 – Fig. 6.— The SEDs of the mid-infrared sources MIR-a (solid squares), MIR-b (solid circles), and MIR-c (open circles). The 110 K and 100 K best-fit blackbody curves are shown for MIR- a (dashed) and MIR-b (dot-dash). For MIR-c, the dotted line represents a 120 K blackbody which is fit to the 18.3µm flux density and the 11.7µm upper limit. The horizontal error bars represent the bandwidth of the filters. Introduction Observations and Data Reduction Discussion CB54YC1-II CB54YC1-I Mid-Infrared Sources Conclusions

0704.0429

Quantitative Resolution to some "Absolute Discrepancies" in Cancer Theories: a View from Phage lambda Genetic Switch

Quantitative Resolution to some ”Absolute Discrepancies” in Cancer Theories a View from Phage lambda Genetic Switch Ping Ao Department of Mechanical Engineering and Department of Physics University of Washington, Seattle, WA 98195, USA (Dated: April 3 (2007)) Abstract Is it possible to understand cancer? Or more specifically, is it possible to understand cancer from genetic side? There already many answers in literature. The most optimistic one has claimed that it is mission-possible. Duesberg and his colleagues reviewed the impressive amount of research results on cancer accumulated over 100 years. It confirms the a general opinion that considering all available experimental results and clinical observations there is no cancer theory without major difficulties, including the prevailing gene-based cancer theories. They have then listed 9 ”absolute discrepancies” for such cancer theory. In this letter the quantitative evidence against one of their major reasons for dismissing mutation cancer theory, by both in vivo experiment and a first prin- ciple computation, is explicitly pointed out. To cite: P. Ao, 2007, Orders of magnitude change in phenotype rate caused by mutation, Cellular Oncology 29: 67 - 69. http://arxiv.org/abs/0704.0429v1 In a forceful article Duesberg and his colleagues (2005) reviewed the impressive amount of research results on cancer accumulated over 100 years. It confirms the current conclu- sion that considering all available experimental results and clinical observations there is no cancer theory without major difficulties, including the prevailing gene-based cancer theories (Hanahan and Weinberg 2000; Prehn 2002; Vogelstein and Kinzler 2004; Beckman and Loeb 2005). Phrasing differently, any known cancer theory is refutable to a substantial degree. While the support for their own advocated chromosomal cancer theory appears strong, it seems that their wholesale criticism on mutation cancer theory is premature. In this letter the quantitative evidence against one of their major reasons for dismissing mutation cancer theory, by both in vivo experiment and a first principle computation, is explicitly pointed Duesberg et al (2005) listed 9 ”absolute discrepancies” or questions which they believe the mutation cancer theory cannot answer: ”(1) How would non-mutagenic carcinogens cause cancer? (2) What kind of mutation would cause cancer only after delays of several decades and many cell generations? (3) What kind of mutation would alter the phenotype of mutant cells perpetually, despite the absence of further mutagens? (4) What kind of mutation would be able to alter phenotypes at rates that exceed conven- tional gene mutations 4-11 orders of magnitude? (5) What kind of mutation would generate resistance against many more drugs than the one used to select it? (6) What kind of mutations would change the cellular and nuclear morphologies several-fold within the same ”clonal” cancer? (7) What kind of mutation would alter the expressions and metabolic activities of 1000s of genes, which is the hallmark of cancer cells? (8) What kind of mutation would consistently coincide with aneuploidy, although conven- tional gene mutations generate infinite numbers of new phenotypes without altering the karyotype? (9) Why would cancer not be heritable via conventional mutations by conventional Mendelian genetics?” Evidently those important questions should be considered seriously by any cancer re- searcher. There are, however, two general reasons that the temporally inability to address them is not enough to dismiss mutation cancer theory. First, in many situations whether the effects are the causes or the consequences, or mutual causes to each other, are still poorly understood, though their associations with cancers may be obvious. The clarification of such confusing requires further and more experimental and clinical studies. Such efforts have been carrying out, for example, by Weiss et al (2004), Hermsen et al (2005), Weber et al (2006); Bielas et al (2006), Levitus et al (2006), and Sjoblom et al (2006). Second, the absence of an explanation or of a theory is not a proof that it would not ever exist. If there were already enough amount of consistent experimental and clinical ob- servations, the emergence of a theory would be simply a matter of time. It is a test to our creativity and imagination. Therefore, those ”absolute discrepancies” are logically not nec- essarily against the mutation cancer theory. Having given a ”dodged” defense, here I would like to call the attention to one evidence specifically addressing above 4th discrepancy: ”(4) What kind of mutation would be able to alter phenotypes at rates that exceed conventional gene mutations 4-11 orders of magnitude?” The question (4) can be answered by mutation cancer theory in a quantitative and first principle manner. The quantitative evidence also suggests answers to questions (1) and (2) of Duesberg et al. In the phage lambda system, a bio-system arguably started the modern molecular biology (Cairns et al 1992), extensive and quantitative experiments have demonstrated that simple mutations can cause the change in phenotypes over many orders of magnitude (Ptashne 2004; Oppenheim et al 2005). The phenotype easily accessible to experimental study is the switching between lysogenic and lytic states. It is generally known that a mutation in the DNA binding sites can cause more than 100 and more folds change in the switching rate, controlled by a few base pairs in the genomic sequences (Revet et al 1999; Little et al 1999; Dodd et al 2005). Systematic study showed that at least over 8 orders of magnitude change can be observed among mutants. A quantitative study is summarized in Table I. Phage lambda genotype λ+ (λ+OR321, wild) λ OR3’23’ λ OR121 λ OR323 λ OR123 Switching rate (exper) 2× 10−9 5× 10−7 3× 10−6 2× 10−5 ∞ Switching rate (theor) 1× 10−9 1× 10−7 3× 10−6 7× 10−5 ∞ Table I: This table is based on Zhu et al (2004, 2006). There are 5 different phage lambda phenotypes, including the ”wild type”, which have been systematic studied experimentally (Little et al 1999). The switching rate is the probability to switch from lysogenic to lytic states per minute under normal lab condition. The symbol ? indicates that there is no stable lysogenic state, that is, the phage lambda would immediately switch to lytic state. The switching rate is then ”infinite”. Quantitative answer to ”absolute discrepancy” 4: The point should be emphasized is that the mathematical calculation is based on first prin- ciple modelling without ”free” parameters. What the ”first principle” means is that the interaction between involved proteins and the protein-DNA binding are based on carefully reasoned physical, chemical and biological principles during past 40 years. What the ”pa- rameter free” means is that all the kinetic parameters needed for the mathematical modelling have been fixed by other experiments. Thus, the remarkable consistency over at least 8 or- ders of magnitude between the experimental data and mathematical calculation shows that it is unlikely due to artifacts in experiment and/or in modelling. Because such effect can occur in phage lambda, there is no reason that same thing cannot occur in higher organisms (Ptashne and Gann, 2002): Numerous gene regulatory sites similar to that of phage lambda exist in our human genome and wrong switching in gene regulatory network is generally believed to contribute to cancer. Therefore, the answer to the question (4) of Duesberg et al based on mutation cancer theory is already positive. Quantitative answer to ”absolute discrepancy” 2: The viability of various mutants, some can live up to thousands of generations before going to lytic state to kill its host E. coli, suggests that there can be a long delay in the manifestation in phenotypes after a mutation. Such a gene regulatory example hence directly answers the question (2) of Duesberg et al. Quantitative answer to ”absolute discrepancy” 1: In addition, it is known that the stability of lambda genetic switch can be influenced both chemically and physically, without any mutagenic effect (Ptashne 2004; Zhu et al 2004), that is, non-mutagenic agents can cause the switch from lysogenic to lytic states, therefore changes the an otherwise robust phenotype. This fact suggests itself as an answer to the question (1) of Duesberg et al. To summarize, though whether chromosomal or mutation cancer theory, or both , are the candidates for the cancer theory is too early to call, Duesberg et al is premature to write out the mutation cancer theory. Even if neither were the final cancer theory, both already appear clinically relevant (Meijer, 2005) and should be studied thoroughly. Finally, I would like to venture a challenge to my fellow quantitative modelers: Is it possible to address all those ”absolute discrepancy” quantitatively? I believe you can do better than what presented here. I thank R. Prehn and L. Loeb for stimulating discussions. This work was supported in part by US National Institutes of Health under grant number HG002894. [1] Beckman RA, Loeb LA (2005) Genetic instability in cancer: Theory and experiment. Seminars in Cancer Biology 15: 423-435. [2] Bielas JH, Loeb KR, Rubin BP, True LD, Loeb LA (2006) Human cancers express a mutator phenotype. Proc Natl Acad Sci (USA) 103: 18238-42. [3] Cairns J, Stent GS, Watson JD (ed) (1992) Phage and the Origins of Molecular Biology (expanded edition, Cold Spring Harbor Laboratory Press, Cold Spring Harbor). [4] Dodd IB, Shearwin KE, Egan JB (2005) Revisted gene regulation in bacteriophage lambda. Current Opinion in Genetics and Development 15: 145-152. [5] Duesberg P, Li RH, Fabarius A, and Hehlmann R (2005) The chromosomal basis of cancer. Cellular Oncology 27: 293-318. [6] Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100: 57-70. [7] Hermsen M, Alonso GM, Meijer G, van Diest P, Suarez NC, Marcos CA, and Sampedro A (2005) Chromosomal changes in relation to clinical outcome in larynx and pharynx squamous cell carcinoma. Cell. Oncol. 27: 191-198. [8] Levitus M, Joenje H, and de Winter JP (2006) The Fanconi anemia pathway of genomic maintenance. Cell Oncol. 28: 3-29. [9] Little JW, Shepley DP, Wert DW (1999) Robustness of a gene regulatory circuit. EMBO J. 18: 4299-4307. [10] Meijer GA (2005) Chromosomes and cancer, Boveri revisited. Cell. Oncol. 27: 273-275. [11] Revet B, von Wilcken-Bergmann B, Bessert H, Barker A, Muller-Hill B (1999) Four dimers of repressor bound tp two suitably spaced pairs of operators form octamers and DNA loops over large distances. Current Biology 9: 151-154. [12] Oppenheim AB, Kobiler O, Stavans J, Court DL, Adhya S (2005) Switches in bacteriophage lambda development. Ann. Rev. Genetics 39: 409-429. [13] Prehn RT (2002) The epigenetic instruction manual for the operation of the genome. Med. Hypotheses 58: 177-181. [14] Ptashne M. (2004) A Genetic Switch: Phage λ revisited, 3rd edition (Cold Spring Harbor Laboratory Press, Cold Spring Harbor). [15] Ptashne M, Gann A (2002) Genes and Signals (Cold Spring Harbor Laboratory Press, Cold Spring Harbor). [16] Sjoblom T, Jones S, Wood LD, Parsons DW, Lin J, Barber TD, Mandelker D, Leary RJ, Ptak J, Silliman N, Szabo S, Buckhaults P, Farrell C, Meeh P, Markowitz SD, Willis J, Dawson D, Willson JKV, Gazdar AF, Hartigan J, Wu L, Liu CS, Parmigiani G, Park BH, Bachman KE, Papadopoulos N, Vogelstein B, Kinzler KW, Velculescu VE (2006) The consensus coding sequences of human breast and colorectal cancers. Science 314: 268-294. [17] Vogelstein B, Kinzler KW (2004) Cancer genes and the pathways they control. Nature Medicine 10: 789-799. [18] Weber A, Gutierrez MI, and Levens D (2006) The CT-element of the c-myc gene does not predispose to chromosomal breakpoints in Burkitt’s lymphoma. Cell. Oncol. 28: 31-35. [19] Weiss MM, Kuipers EJ, Postma C, Snijders AM, Pinkel D, Meuwissen SG, Albertson D, and Meijer GA (2004) Genomic alterations in primary gastric adenocarcinomas correlate with clinicopathological characteristics and survival. Cell. Oncol. 26: 307-317. [20] Zhu XM, Yin L, Hood L, Ao P. (2004) Robustness, stability and efficiency of phage lambda genetic switch: dynamical structure analysis. Journal of Bioinformatics and Computational Biology 2: 785-817. [21] Zhu XM, Yin L, Hood L, Galas D, and Ao P (2006) Efficiency, robustness and stochasticity of gene regulatory networks in systems biology: lambda switch as a working example. In Introduction to Systems Biology (Humana Press, 2007). ( http://arxiv.org/ftp/q-bio/papers/0512/0512007.pdf ) http://arxiv.org/ftp/q-bio/papers/0512/0512007.pdf [22] Ao P, 2007, Orders of magnitude change in phenotype rate caused by mutation, Cellular Oncology 29: 67 - 69 References

0704.0430

Reduced phase space and toric variety coordinatizations of Delzant spaces

Reduced phase space and toric variety coordinatizations of Delzant spaces J.J. Duistermaat∗ and A. Pelayo† November 4, 2018 Abstract In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained. 1 Introduction Let (M, σ) be a smooth compact and connected symplectic manifold of dimension 2n and let T be a torus which acts effectively on (M, σ) by means of symplectomorphisms. If the action of T on (M, σ) is moreover Hamiltonian, then dimT ≤ n, and the image of the momentum mapping µT : M → t ∗ is a convex polytope ∆ in the dual space t∗ of t, where t denotes the Lie algebra of T . In the maximal case when dimT = n, (M, σ) is called a Delzant space. Delzant [3, (*) on p. 323] proved that in this case the polytope ∆ is very special, a so-called Delzant polytope, of which we recall the definition in Section 2. Furthermore Delzant [3, Th. 2.1] proved that two Delzant spaces are T -equivariantly symplectomorphic if and only if their momentum mappings have the same image up to a translation by an element of t∗. Thirdly Delzant [3, pp. 328, 329] proved that for every Delzant polytope ∆ there exists a Delzant space such that µT (M) = ∆. This Delzant space is obtained as the reduced phase space for a linear Hamiltonian action of a torus N on a symplectic vector space E, at a value λN of the momentum mapping of the Hamiltonian N -action, where E, N and λN are determined by the Delzant polytope. Finally Delzant [3, Sec. 5] observed that the Delzant polytope gives rise to a fan (= éventail in French), and that the Delzant space with Delzant polytope ∆ is T -equivariantly diffeomorphic to the toric variety Mtoric defined by the fan. HereMtoric is a complex n-dimensional complex analytic manifold, and the action of the real torus T on Mtoric has an extension to a complex analytic action on Mtoric of the complexification ∗Research stimulated by a KNAW professorship †Partly supported by a Rackham Predoctoral fellowship http://arxiv.org/abs/0704.0430v1 TC of T . In our description in Section 5 of the toric variety M toric we do not use fans. The information, for each vertex v of ∆, which codimension one faces of ∆ contain v, already suffices to define Mtoric. In this note we show that the construction of the Delzant space M as a reduced phase space leads, for every vertex v of the Delzant polytope, to a natural coordinatization ϕv of a T -invariant open cell Mv in M , where Mv contains the unique fixed point mv in M of the T -action such that µT (mv) = v. We give an explicit construction of the inverse ψv of ϕv, which is a maximal diffeomorphism in the sense of Remark 3.10. The construction of ψv originated in an attempt to extend the equivariant symplectic ball embeddings from (B2nr , σ0) ⊂ (C n, σ0) into the Delzant space (M, σ) in Pelayo [10] by maximal equivariant symplec- tomorphisms from open neighborhoods of the origin in Cn into the Delzant space (M, σ). If v and w are two different vertices, then the coordinate transformation ϕw ◦ ϕv −1 is given by the explicit formulas (4.3), (4.4). Let Σ be the set of all strata of the orbit type stratification of M for the T -action. Then the domain of definition Mv of ϕv is equal to the union of all S ∈ Σ such that the fixed point mv belongs to the closure of S in M , see Corollary 5.5. The strata S ∈ Σ are also the orbits in the toric variety Mtoric ≃ M for the action of the complexification TC of the real torus T , and the domain of definition Mv of ϕv is equal to the domain of definition of a natural complex analytic TC-equivariant coordinatization Φv of a TC-invariant open cell. The diffeomorphism Φv ◦ ϕv −1, which sends the reduced phase space coordinates to the toric variety coordinates, maps Uv := ϕv(Mv) diffeomorphically onto a complex vector space, and is given by the explicit formulas (5.9). In the toric variety coordinates the complex structure is the standard one and the coordinate transfor- mations are relatively simple Laurent monomial transformations, whereas the symplectic form is generally given by quite complicated algebraic functions. On the other hand, in the reduced phase space coordinates the symplectic form is the standard one, but the coordinate transformations, and also the complex structure, have a more complicated appearance. Let F denote the set of all d codimension one faces of ∆ and, for every vertex v of ∆, let Fv denote the set of all f ∈ F such that v ∈ f . Note that #(Fv) = n for every vertex v of ∆. For any sets A and B, let AB denote the set of all A-valued functions on B. If A is a field and the set B is finite, then AB is a #(B)- -dimensional vector space over A. One of the technical points in this paper is the efficient organization of proofs and formulas made possible by viewing the Delzant space as a reduction of the vector space CF , and letting, for each vertex v, the coordinatizations ϕv and Φv take their values in C Fv . This leads to a natural projection ρv : C F → CFv obtained by the restriction of functions on F to Fv ⊂ F . For each vertex v the complex vector space CFv is isomorphic to Cn, but the isomorphism depends on an enumeration of Fv , the introduction of which would lead to an unnecessary complication of the combinatorics. Similarly our torus T is isomorphic to Rn/Zn, but the isomorphism depends on the choice of a Z-basis of the integral lattice tZ in the Lie algebra t of T . As for each vertex v a different Z-basis of tZ appears, we also avoid such a choice, keeping T in its abstract form. We hope and trust that this will not lead to confusion with our main references Delzant [3], Audin [2] and Guillemin [7] about Delzant spaces, where CF , each CFv , and T is denoted as Cd, Cn, and Rn/Zn, respectively. The organization of this manuscript is as follows. In Section 2 we review the definition of the reduced phase Delzant space, and introduce the notations which will be convenient for our purposes. In Section 3 we define the reduced phase space coordinatizations. In Section 4 we give explicit formulas for the coordinate transformations and describe the reduced phase space Delzant space as obtained by gluing together bounded open subsets of n-dimensional complex vector spaces with these coordinate transformations as the gluing maps. In Section 5 we review the definition of the toric variety defined by the Delzant polytope, prove that the natural mapping from the reduced phase space to the toric variety is a diffeomorphism, and compare the coordinatizations of Section 3 with the natural coordinatizations of the toric variety. In Section 6 we present these computations for the two simplest classes of examples, the complex projective spaces and the Hirzebruch surfaces. 2 The reduced phase space Let T be an n-dimensional torus, a compact, connected, commutative n-dimensional real Lie group, with Lie algebra t. It follows that the exponential mapping exp : t → T is a surjective homomorphism from the additive Lie group t onto T . Furthermore, tZ := ker(exp) is a discrete subgroup of (t, +) such that the exponential mapping induces an isomorphism from t/tZ onto T , which we also denote by exp. Note that tZ is defined in terms of the group T rather than only the Lie algebra t, but the notation tZ has the advantage over the more precise notation TZ that it reminds us of the fact it is a subgroup of the additive group t. Because t/tZ is compact, tZ has a Z-basis which at the same time is an R-basis of t, and each Z-basis of tZ is an R-basis of t. Using coordinates with respect to an ordered Z-basis of tZ, we obtain a linear isomorphism from t onto Rn which maps tZ onto Z n, and therefore induces an isomorphism from T onto n/Zn. For this reason, tZ is called the integral lattice in t. However, because we do not have a preferred Z-basis of tZ, we do not write T = R n/Zn. Let ∆ be an n-dimensional convex polytope in t∗. We denote by F and V the set of all codimension one faces and vertices of ∆, respectively. Note that, as a face is defined as the set of points of the closed convex set on which a given linear functional attains its minimum, see Rockafellar [11, p.162], every face of ∆ is compact. For every v ∈ V , we write Fv = {f ∈ F | v ∈ f}. ∆ is called a Delzant polytope if it has the following properties, see Guillemin [7, p. 8]. i) For each f ∈ F there is an Xf ∈ t and λf ∈ R such that the hyperplane which contains f is equal to the set of all ξ ∈ t∗ such that 〈Xf , ξ〉 + λf = 0, and ∆ is contained in the set of all ξ ∈ t ∗ such that 〈Xf , ξ〉+ λf ≥ 0. The vector Xf and constant λf are made unique by requiring that they are not an integral multiple of another such vector and constant, respectively. ii) For every v ∈ V , the Xf with f ∈ Fv form a Z-basis of the integral lattice tZ in t. It follows that ∆ = {ξ ∈ t∗ | 〈Xf , ξ〉+ λf ≥ 0 for every f ∈ F}. (2.1) Also, #(Fv) = n for every v ∈ V , which already makes the polytope ∆ quite special. In the sequel we assume that ∆ is a given Delzant polytope in t∗. For any z ∈ CF and f ∈ F we write z(f) = zf , which we view as the coordinate of the vector z with the index f . Let π be the real linear map from RF to t defined by π(t) := tf Xf , t ∈ R F . (2.2) Because, for any vertex v, the Xf with f ∈ Fv form a Z-basis of tZ which is also an R-basis of t, we have π(ZF ) = tZ and π(R F ) = t. It follows that π induces a surjective homomorphism of Lie groups π′ from the torus RF/ZF = (R/Z)F onto t/tZ, and we have the corresponding surjective homomorphism exp ◦π from RF /ZF onto T . Write n := ker π, a linear subspace of RF , and N = ker(exp ◦π′), a compact commutative subgroup of the torus RF /ZF . Actually, N is connected, see Lemma 3.1 below, and therefore isomorphic to n/nZ, where nZ := n ∩ Z F is the integral lattice in n of the torus N . 1 On the complex vector space CF of all complex-valued functions on F we have the action of the torus F/ZF , where t ∈ RF/ZF maps z ∈ CF to the element t · z ∈ CF defined by (t · z)f = e 2π i tf zf , f ∈ F. The infinitesimal action of Y ∈ RF =Lie(RF /ZF ) is given by (Y · z)f = 2π i Yf zf , which is a Hamiltonian vector field defined by the function z 7→ 〈Y, µ(z)〉 = Yf |zf | 2/2 = Yf (xf 2 + yf 2)/2, (2.3) and with respect to the symplectic form σ := (i /4π) dzf ∧dzf = (1/2π) dxf ∧dyf , (2.4) if zf = xf +i yf , with xf , yf ∈ R. Here the factor 1/2π is introduced in order to avoid an integral lattice (2π Z)F instead of our ZF . Because the right hand side of (2.3) depends linearly on Y , we can view µ(z) as an element of (RF )∗ ≃ F , with the coordinates µ(z)f = |zf | 2/2 = (xf 2 + yf 2)/2, f ∈ F. (2.5) In other words, the action of RF/ZF on CF is Hamiltonian, with respect to the symplectic form σ and with momentum mapping µ : CF → (Lie(RF /ZF ))∗ given by (2.3), or equivalently (2.5). It follows that the subtorus N of RF /ZF acts on CF in a Hamiltonian fashion, with momentum mapping µN := ι ◦ µ : CF → n∗, (2.6) where ιn : n → R F denotes the identity viewed as a linear mapping from n ⊂ RF to RF , and its transposed : (RF )∗ → n∗ is the map which assigns to each linear form on RF its restriction to n. Write λN = ι (λ), where λ denotes the element of (RF )∗ ≃ RF with the coordinates λf , f ∈ F . It follows from Guillemin [7, Th. 1.6 and Th. 1.4] that λN is a regular value of µN , hence the level set Z := µN −1({λN}) of µN for the level λN is a smooth submanifold of C F , and that the action of N on Z is proper and free. As a consequence the N -orbit space M = M∆ := Z/N is a smooth 2n-dimensional manifold such that the projection p : Z → M exhibits Z as a principal N -bundle over M . Moreover, there is a unique symplectic form σM on M such that p ∗σM = ιZ ∗σ, where ιZ is the identity viewed as a smooth mapping from Z to CF . Remark 2.1 Guillemin [7] used the momentum mapping µN − λN instead of µN , such that the reduction is taken at the zero level of his momentum mapping. We follow Audin [2, Ch. VI, Sec. 3.1] in that we use the momentum mapping µN for the N -action, which does not depend on λ, and do the reduction at the level λN . ⊘ 1We did not find a proof of the connectedness of N in [3], [2], or [7]. The symplectic manifold (M, σM ) is the Marsden-Weinstein reduction of the symplectic manifold (CF , σ) for the Hamiltonian N -action at the level λN of the momentum mapping, as defined in Abraham and Marsden [1, Sec. 4.3]. On the N -orbit space M , we still have the action of the torus (RF /ZF )/N ≃ T , with momentum mapping µT :M → t ∗ determined by π∗ ◦ µT ◦ p = (µ− λ)|Z . (2.7) The torus T acts effectively on M and µT (M) = ∆, see Guillemin [7, Th. 1.7]. Actually, all these properties of the reduction will also follow in a simple way from our description in Section 3 of Z in term of the coordinates zf , f ∈ F . The symplectic manifold M∆ together with this Hamiltonian T -action is called the Delzant space de- fined by ∆, see Guillemin, [7, p. 13]. This proves the existence part [3, pp. 328, 329] of Delzant’s theory. 3 The reduced phase space coordinatizations. For any v ∈ V , let ιv := ρv ∗ : (RFv)∗ → (RF )∗ denote the transposed of the restriction projection ρv : R F → RFv . If in the usual way we identify (RFv)∗ and (RF )∗ with RFv and RF , respectively, then ιv : R Fv → RF is the embedding defined by ιv(x)f = xf if f ∈ Fv and ιv(x)f ′ = 0 if f ′ ∈ F , f ′ /∈ Fv. Because ιv maps Z Fv into ZF and ιv(R Fv) ∩ ZF = ιv(Z Fv), it induces an embedding of the n-dimensional torus RFv/ZFv into RF/ZF , which we also denote by ιv. Lemma 3.1 With these notations, RF , ZF , and RF/ZF are the direct sum of n and ιv(RFv), n ∩ Zn and Fv), and N and ιv(R Fv/ZFv), respectively. It follows that N is connected, a torus, with integral lattice equal to n ∩ ZF . It also follows that π ◦ ιv is an isomorphism from the torus RFv/ZFv onto the torus T . Proof Let t ∈ RF . Because the Xf , f ∈ Fv, form an R-basis of t, there exists a unique t v ∈ RFv , such π(t) = (tv)f Xf = π(ιv(t that is, t− ιv(t v) ∈ n. Moreover, because the Xf , f ∈ Fv , also form a Z-basis of tZ, we have that t v ∈ ZFv , and therefore t− ιv(t v) ∈ n ∩ ZF , if t ∈ ZF . � Lemma 3.2 We have z ∈ Z if and only if µ(z) − λ ∈ π∗(t∗). More explicitly, if and only if there exists a ξ ∈ t∗ such that |zf | 2/2− λf = 〈Xf , ξ〉 for every f ∈ F. (3.1) When z ∈ Z , the ξ in (3.1) is uniquely determined. Furthermore, Z = (µ− λ)−1(π∗(∆)), (µ− λ)(Z) = π∗(∆), and Z is a compact subset of CF . Proof The kernel of ι∗ is equal to the space of all linear forms on RF which vanish on n := kerπ, and therefore ker ι∗ is equal to the image of π∗ : t∗ → (RF )∗. Because π is surjective, π∗ is injective, which proves the uniqueness of ξ. It follows from (3.1) that 〈Xf , ξ〉 + λf ≥ 0 for every f ∈ F , and therefore ξ ∈ ∆ in view of (2.1). Conversely, if ξ ∈ ∆, then there exists for every f ∈ F a complex number zf such that |zf | 2/2 = 〈Xf , ξ〉+ λf , which means that z ∈ Z and (µ− λ)(z) = π ∗(ξ). The set π∗(∆) is compact because ∆ is compact and π∗ is continuous. Because the mapping µ− λ is proper, it follows that Z = (µ− λ)−1(π∗(∆)) is compact. Let v ∈ V . The Xf , f ∈ Fv , form an R-basis of t, and therefore there exists for each z ∈ C Fv a unique ξ = µv(z) ∈ t ∗ such that (3.1) holds for every f ∈ Fv . That is, the mapping µv : C Fv → t∗ is defined by the equations |zf | 2/2 − λf = 〈Xf , µv(z)〉, z ∈ C Fv , f ∈ Fv. (3.2) In other words, µv is defined by the formula ρv ◦ π ∗ ◦ µv = ρv ◦ (µ− λ) ◦ ιv, (3.3) where ρv denotes the restriction projection from R F onto RFv . Lemma 3.3 If we let T act on CFv via RFv/ZFv by means of (t, z) 7→ (π ◦ ιv)−1(t) ·z, then µv : CFv → t∗ is a momentum mapping for this Hamiltonian action of T on CFv , with µv(0) = v. Here the symplectic form on CFv is equal to σ := (i /4π) dzf ∧dzf = (1/2π) dxf ∧dyf , (3.4) that is, (2.4) with F replaced by Fv . Let ρv denote the restriction projection from C F onto CFv , and let Uv be the interior of the subset ρv(Z) of CFv . Write ∆v := ∆ \ f ′∈F\Fv f ′. (3.5) Then ρv(Z) = µv −1(∆), µv(ρv(Z)) = ∆, Uv = µv −1(∆v), and µv(Uv) = ∆v. In particular ρv(Z) is a compact subset of CFv , and Uv is a bounded and connected open neighborhood of 0 in C Proof The first statement follows from (3.3), the fact that ρv◦µ◦ιv is a momentum mapping for the standard Fv/ZFv action on CFv , and the fact that a momentum mapping for a Hamiltonian action plus a constant is a momentum mapping for the same Hamiltonian action. It follows in view of (3.2) that 〈Xf , µv(0)〉+λf = 0 for every f ∈ Fv , hence µv(0) = v in view of i) in the definition of a Delzant polytope, and the fact that {v} is the intersection of all the f ∈ Fv . It follows from (3.2), Lemma 3.2, that z ∈ Z if and only if |zf | 2/2 = 〈Xf , µv(ρv(z))〉 + λf for every f ∈ F, (3.6) where we note that these equations are satisfied by definition for the f ∈ Fv. Therefore, if z ∈ Z , then (3.6) and (2.1) imply that µv(ρ(z)) ∈ ∆. Conversely, if ξ ∈ ∆, then it follows from Lemma 3.2 that there exists z ∈ Z such that π∗(ξ) = µ(z)− λ, of which the restriction to Fv yields ξ = µv(ρ(z)). If ξ ∈ ∆v, z v ∈ CFv , µv(z v) = ξ, then 〈Xf ′ , µv(z v)〉 + λf ′ > 0 for every f ′ ∈ F \ Fv, which will remain valid if we replace zv by z̃v in a sufficiently small neighborhood of zv in CFv . It follows that we can find z̃ ∈ CF such that ρv(z̃) = z̃ v and (3.6) holds with z replaced by z̃. That is, z̃ ∈ Z , and we have proved that zv ∈ Uv. Let conversely z ∈ Uv ⊂ C Fv . We have in view of (3.2) that |zf | 2/2 = 〈Xf , µv(z)− µv(0)〉 = 〈Xf , µv(z)− v〉 for every f ∈ Fv. Therefore µv(z) − v is multiplied by c 2 if we replace z by c z, c > 0. Because z is in the interior of ρv(Z), we have c z v ∈ ρv(Z), hence µv(c z) ∈ ∆ for c > 1, c sufficiently close to 1. On the other hand, if ξ belongs to a face of ∆ which is not adjacent to v, then v + τ (ξ − v) /∈ ∆ for any τ > 1. It follows that µv(z) does not belong to any f ′ ∈ F \ Fv, that is, µv(z) ∈ ∆v. � The equation (3.6) can be written in the form |zf | = rf (µv(ρv(z))), where, for each f ∈ F , the function rf : ∆ → R≥0 is defined by rf (ξ) := (2(〈Xf , ξ〉+ λf )) 1/2, f ∈ F, ξ ∈ ∆. (3.7) We now view the equations (3.6) for z ∈ Z as equations for the coordinates zf ′ , f ′ ∈ F \ Fv, with the zf , f ∈ Fv as parameters, where the latter constitute the vector z v = ρv(z). If z v ∈ Uv, then for each f ′ ∈ F \ Fv the coordinate zf ′ lies on the circle about the origin with strictly positive radius rf ′(µv(z Because Lemma 3.1 implies that the homomorphism which assigns to each element of N its projection to F\Fv/ZF\Fv is an isomorphism, and the latter torus is the group of the coordinatewise rotations of the zf ′ , f ′ ∈ F \ Fv, this leads to the following conclusions. Proposition 3.4 Let v be a vertex of ∆. The open subset Zv := ρv−1(Uv) ∩ Z of Z is a connected smooth submanfold of CF of real dimension 2n + (d − n), where d = #(F ) and d − n = dimN . The action of the torus N on Zv is free, and the projection ρv : Zv → Uv exhibits Zv as a principal N -bundle over Uv. It follows that we have a reduced phase space Mv := Zv/N , which is a connected smooth symplectic 2n-dimensional manifold, which carries an effective Hamiltonian T -action with momentum mapping as in (2.7), with Z replaced by Zv. There is a unique global section sv : Uv → Zv of ρv : Zv → Uv such that sv(z)f ′ ∈ R>0 for every z ∈ Uv and f ′ ∈ F \ Fv. Actually, sv(z)f ′ = rf ′(µv(z)) when z ∈ Uv and f ′ ∈ F \ Fv , and therefore the section sv is smooth. If pv : Zv → Mv = Zv/N denotes the canonical projection, then ψv := pv ◦ sv is a T -equivariant symplectomorphism from Uv onto Mv, where T acts on Uv via R Fv/ZFv , as in Lemma 3.3. Remark 3.5 When z belongs to the closure ρv(Z) = Uv of Uv in CFv , see Lemma 3.3, we can define sv(z) ∈ C F by sv(z)f = zf when f ∈ Fv and sv(z)f ′ = rf ′(µv(z)) when f ′ ∈ F \ Fv . This defines a continuous extension sv : Uv → C F of the mapping sv : Uv → Z . Therefore sv(Uv) ⊂ Z , and ψv := p ◦ sv : Uv →M is a continuous extension of the diffeomorphism ψv : Uv →Mv . The continuous mapping ψv : Uv → M is surjective, but the restriction of it to the boundary ∂Uv := Uv \Uv of Uv in C Fv is not injective. If zv ∈ ∂Uv, then the set G of all f ′ ∈ F \ Fv such that sv(z v)f ′ = 0, or equivalently µT (ψv(z v)) ∈ f ′, is not empty. The fiber of ψv over ψv(z v) is equal to the set of all tv · zv, where the tv ∈ RFv/ZFv are of the form tvf = − (v)fg tg, f ∈ Fv, where tg ∈ R/Z. It follows that each fiber is an orbit of some subtorus of R Fv/ZFv acting on CFv . ⊘ Recall the definition (3.5) of the open subset ∆v of the Delzant polytope ∆. Because the union over all vertices v of the ∆v is equal to ∆, we have the following corollary. Corollary 3.6 The sets Zv, v ∈ V , form a covering of Z . As a consequence, Z is a smooth submanifold of CF of real dimension n + d. The action of the torus N on Z is free, and we have a reduced phase space M := Z/N , which is a compact and connected smooth 2n-dimensional symplectic manifold, which carries an effective Hamiltonian T -action with momentum mapping µT :M → T as in (2.7). The sets Mv, v ∈ V , form an open covering of M and the ϕv := (ψv) −1 : Mv → Uv form an atlas of T -equivariant symplectic coordinatizations of the Hamiltonian T -space M . For each v ∈ V , we have Mv = µT −1(∆v), and µT |Mv = µv ◦ ϕv. For a characterization of Mv in terms of the orbit type stratification in M for the T -action, see Corollary 5.5, which also implies that Mv is an open cell in M . Corollary 3.7 For every f ∈ F the set µT−1(f) is a real codimension two smooth compact connected smooth symplectic submanifold of M . For each v ∈ V , the set Mv is dense in M , and the diffeomorphism ψv : Uv → Mv is maximal among all diffeomorphisms from open subsets of CFv onto open subsets of M . Proof If f ∈ F , then for each v ∈ V we have that −1(f) = {z ∈ Uv | zf = 0} (3.8) if v ∈ f , that is, f ∈ ∆v. This follows from (3.2) and i) in the description of ∆ in the beginning of Section 2. On the other hand, µv −1(f) = ∅ if f /∈ ∆v. Because µT −1(f) ∩Mv = ψv(µv −1(f)), and the Mv, v ∈ V , form an open covering of M , this proves the first statement. The second statement follows from the first one, because the complement of Mv in M is equal to the union of the sets µT −1(f ′) with f ′ ∈ F \ Fv. Remark 3.8 It follows from the proof of Corollary 3.7, that µT−1(f) is a connected component of the fixed point set in M of the of the circle subgroup exp(RXf ) of T . Actually, µT −1(f) is a Delzant space for the action of the (n− 1)-dimensional torus T/exp(RXf ), with Delzant polytope P ⊂ (t/(RXf )) ∗ such that the image of P in t∗ under the embedding (t/(RXf )) ∗ is equal to a translate of f . In a similar way, if g is a k-dimensional face of ∆, then µT −1(g) is a 2k-dimensional Delzant space for the quotient of T by the subtorus of T which acts trivially on µT −1(g). ⊘ Remark 3.9 Let, for each f ∈ F , cf ∈H 2(M, Z) ⊂H2(M, R) denote the Poincaré dual of the codimen- sion two Delzant subspace µT −1(f) of M , see Remark 3.8. Then, with our normalization of the symplectic form (2.4), the de Rham cohomology class [σM ] of the symplectic form σM of the Delzant space is equal to [σM ] = λf cf , (3.9) see Guillemin [8, Thm. 6.3]. In particular [σM ] ∈H 2(M, Z), and therefore [σM ] is equal to the Chern class of a complex line bundle over M , if all the coefficients λf , f ∈ F , are integers. If ∆ is a simplex, when M is isomorphic to the n-dimensional complex projective space, then the −1(f), f ∈ F , are complex projective hyperplanes, see Subsection 6.1, which are all homologous to each other. It follows that in this case [σM ] = γ c, where c is the Poincaré dual of a complex projective hyperplane and γ is equal to the sum of all the coefficients λf , f ∈ F . ⊘ Remark 3.10 Let ι : T → Rn/Zn be an isomorphism of tori, which allows us to let t ∈ T act on Cn via n/Zn by means of (t · z)j = e 2π i ι(t)j zj, 1 ≤ j ≤ n. Let U be a connected T -invariant open neighborhood of 0 in Cn, provided with the symplectic form (2.4) with F replaced by {1, . . . , n}. Let ψ : U → M be a T -equivariant symplectomorphism from U onto an open subset ψ(U) of M . Because 0 is the unique fixed point for the T -action in U , and the fixed points for the T -action in M are the pre-images under µT of the vertices of ∆, there is a unique v ∈ V such that µT (ψ(0)) = v. Let Iv : C Fv → Cn denote the complex linear extension of the tangent map of the torus isomorphism ι ◦ (π ◦ ιv). In terms of the notation of Lemma 3.3 and Proposition 3.4, we have that U ⊂ Iv(Uv) and ψv = ψ ◦ Iv on Iv −1(U), which leads to an identification of ψ with the restriction of ψv to the connected open subset Iv −1(U) of Uv, via the isomorphism Iv The ψ’s, with U equal to a ball in Cn centered at the origin, are the equivariant symplectic ball embed- dings in Pelayo [10], and the second statement in Corollary 3.7 shows that the diffeomorphisms ψv are the maximal extensions of these equivariant symplectic ball embeddings. ⊘ 4 The coordinate transformations Let v, w ∈ V . Then Uv, w := ϕv(Mv ∩Mw) = Uv ∩ ψv −1 ◦ ψw(Uw) = {zv ∈ Uv | (z v)f 6= 0 for every f ∈ Fv \ Fw}. (4.1) In this section we will give an explicit formula for the coordinate transformations ϕw ◦ ϕv −1 = ψw −1 ◦ ψv : Uv, w → Uw, v, which then leads to a description of the Delzant space M as obtained by gluing together the subsets Uv with the coordinate transformations as the gluing maps. Let f ∈ F . Because the Xg , g ∈ Fw, form a Z-basis of tZ, and Xf ∈ tZ, there exist unique integers f , g ∈ Fw, such that f Xg. (4.2) Note that if f ∈ Fw, then (w) f = 1 when g = f and (w) f = 0 otherwise. For the following lemma recall that rg is defined by expression (3.7). Lemma 4.1 Let v, w ∈ V , zv ∈ Uv, w. Then zw := ϕw ◦ ϕv−1(zv) ∈ Uw ⊂ CFw is given by zwg = (zvf ) f∈Fv\Fw |zvf | f (4.3) if g ∈ Fw ∩ Fv, and zwg = (zvf ) f rg(µv(z f∈Fv\Fw |zvf | f (4.4) if g ∈ Fw \ Fv. Proof The element zw ∈ Uw is determined by the condition that sw(zw) belongs to the N -orbit of sv(zv). That is, w)f = e i tf sv(z v)f for every f ∈ F for some t ∈ RF such that ∑ tf Xf = 0. (4.5) It follows from (4.5), (4.2) and the linear independence of the Xg, g ∈ Fw, that t ∈ n if and only if tg = − f∈F\Fw tf for every g ∈ Fw. (4.6) Note that µv(z v) = µT (m) = µw(z w), where m = ψv(z v) = ψw(z w). It follows from the definition of the sections sv and sw, see Proposition 3.4, that i) sv(z v)f = z f and sw(z w)f = z f if f ∈ Fv ∩ Fw, ii) sv(z v)f = z f and sw(z w)f = rf (µw(z w)) = rf (µv(z v)) if f ∈ Fv \ Fw, iii) sv(z v)f = rf (µv(z v)) and sw(z w)f = z f if f ∈ Fw \ Fv, and iv) sv(z v)f = rf (µv(z v)) = rf (µw(z w)) = sw(z w)f if f ∈ F \ (Fv ∪ Fw). It follows from ii) and iv) that tf = −arg z f and tf = 0 modulo 2π if f ∈ Fv \Fw and f ∈ F \ (Fv ∪Fw), respectively. Then (4.6) implies that, modulo 2π, f∈Fv\Fw f arg z f for every g ∈ Fw. It now follows from i) and iii) that if g ∈ Fw, then z g = sw(z w)g = e i tg sv(z v)g is equal to ei tg zvg = (zvf ) f∈Fv\Fw |zvf | if g ∈ Fv, and equal to ei tg |zvg | = (zvf ) f |zvg |/ f∈Fv\Fw |zvf | if g /∈ Fv , respectively. Here we have used that if g ∈ Fw, then (w) = 1 if f = g and (w)g = 0 if f ∈ Fw, f 6= g. Because |zvg | = rg(µv(z v)) if g /∈ Fv, see (3.6) and (3.7), this completes the proof of the lemma. � Remark 4.2 Note that zv ∈ Uv, w means that zv ∈ Uv and zvf 6= 0 if f ∈ Fv \ Fw. Furthermore, z v ∈ Uv implies that if g /∈ Fv , then µv(z v) /∈ g, and therefore rg is smooth on a neighborhood of µv(z v). Finally, note that if g ∈ Fw and f ∈ Fv ∩Fw, then (w) ∈ {0, 1}, and therefore each of the factors in the right hand sides of (4.3) and (4.4) is smooth on Uv, w. ⊘ Remark 4.3 In (4.3) and (4.4) only the integers (w)gf appear with f ∈ Fv and g ∈ Fw. Let (w v) denote the matrix (w)gf , where f ∈ Fv and g ∈ Fw. Then (w v) is invertible, with inverse equal to the integral matrix (v w). These integral matrices also satisfy the cocycle condition that (w v) (v u) = (w u), if u, v, w ∈ V . These properties follow from the fact that (4.2) shows that (w v) is the matrix which maps the Z-basis Xg, g ∈ Fw, onto the Z-basis Xf , f ∈ Fv, of tZ. It is no surprise that these base changes enter in the formulas which relate the models in the vector spaces CFv for the different choices of v ∈ V . ⊘ Corollary 4.4 Let, for each v ∈ V , the mapping µv : CFv → t∗ be defined by (3.2), which is a momentum mapping for a Hamiltonian T -action via RFv/ZFv on the symplectic vector space CFv as in Lemma 3.3. Define Uv := µv −1(∆v). If also w ∈ V , define Uv, w as the right hand side of (4.1), and, if z v ∈ Uv, w, define ϕw, v(z v) := zw, where zw ∈ CFw is given by (4.3) and (4.4). Then ϕw, v is a T -equivariant symplectomorphism from Uv, w onto Uw, v such that µw = µv ◦ ϕv, w on Uw, v. The ϕw, v satisfy the cocycle condition ϕw, v ◦ ϕv, u = ϕw, u where the left hand side is defined. Glueing together the Hamiltonian T -spaces Uv, v ∈ V , with the momentum maps µv, by means of the gluing maps ϕw, v, v, w ∈ V , we obtain a compact connected smooth symplectic manifold M̃ with an effective Hamiltonian T -action with a common momentum map µ̃ : M̃ → T such that µ̃(M̃ ) = ∆. In other words, M̃ is a Delzant space for the Delzant polytope ∆. The Delzant space M̃ is obviously isomorphic to the Delzant spaceM = µ−1({λ})/N introduced in Section 2, and actually the isomorphism is used in the proof that M̃ is a Delzant space for the Delzant polytope ∆. The only purpose of Corollary 4.4 is to exhibit the Delzant space as obtained from gluing together the Uv, v ∈ V , by means of the gluing maps ϕv, w, v, w ∈ V . 5 The toric variety Let T := {z ∈ C | |z| = 1} denote the unit circle in the complex plane. The mapping t 7→ u where uf = e 2π i tf for every f ∈ F is an isomorphism from the torus RF/ZF onto TF , where TF acts on CF by means of coordinatewise multiplication and RF/ZF acted on CF via the isomorphism from RF/ZF onto F . The complexification TC of the compact Lie group T is the multiplicative group C × of all nonzero complex numbers, and the complexification of TF is equal to TF := (TC) F = (C×)F , which also acts on F by means of coordinatewise multiplication. The complexification NC of N is the subgroup exp(nC) of U , where nC := n⊕ i n ⊂ C F denotes the complexification of n, viewed as a complex linear subspace, a complex Lie subalgebra, of the Lie algebra F of TF . In view of (4.6), we have, for every v ∈ V , that NC is equal to the set of all t ∈ T such that f∈F\Fv f , g ∈ Fv. (5.1) This implies that NC is a closed subgroup of T isomorphic to T , and therefore NC is a reductive complex algebraic group. If we define v := {z ∈ C F | zf 6= 0 for every f ∈ F \ Fv}, (5.2) then it follows from (5.1) that the action of NC on C v is free and proper. It follows that the action of NC on v (5.3) is free and proper, and therefore the NC-orbit space Mtoric := CF∆/NC (5.4) has a unique structure of a complex analytic manifold of complex dimension n such that the canonical pro- jection from CF∆ onto M toric exhibits CF∆ as a principal NC-bundle over M toric. On Mtoric we still have the complex analytic action of the complex Lie group group TF /NC, which is isomorphic to the complexifi- cation TC of our real torus T induced by the projection π. The complex analytic manifold M toric together with the complex analytic action of TC on it is the toric variety defined by the polytope ∆ in the title of this section. If v ∈ V and z ∈ CFv , then it follows from (5.1) that there is a unique t ∈ NC such that tf = zf for every f ∈ F \ Fv, or in other words, z = t · ζ , where ζ ∈ C F is such that ζf = 1 for every f ∈ F \ Fv. Let Sv : C Fv → CFv be defined by Sv(z v)f = z f when f ∈ F and Sv(z v) = 1 when f ∈ F \ Fv, as in Audin [2, p. 159]. If Pv : C v → C v /NC denotes the canonical projection from C v onto the open subset Mtoricv := C v /NC of M toric, then Ψv := Pv ◦ Sv is a complex analytic diffeomorphism from C Fv onto Mtoricv . It is TC-equivariant if we let TC act on C Fv via TFv as in Lemma 3.3. We use the diffeomorphism Φv := Ψv −1 from Mtoricv onto C Fv as a coordinatization of the open subset Mtoricv of M toric. If v, w ∈ V , then Utoricv, w := Φv(M toric toric w ) = C Fv ∩Ψv −1 ◦Ψw(C = {zv ∈ CFv | (zv)f 6= 0 for every f ∈ Fv \ Fw}. (5.5) Moreover, with a similar argument as for Lemma 4.1, actually much simpler, we have that for every zv ∈ Utoricv, w the element z w := Φw ◦Φv −1(zv) ∈ CFw is given by zwg = (zvf ) f , g ∈ Fw. (5.6) In this way the coordinate transformation Φw◦Φv −1 is a Laurent monomial mapping, much simpler than the coordinate transformation (4.3), (4.4). It follows that the toric variety Mtoric can be alternatively described as obtained by gluing the n-dimensional complex vector spaces CFv , v ∈ V , together, with the maps (5.6) as the gluing maps. This is the kind of toric varieties as introduced by Demazure [4, Sec. 4]. For later use we mention the following observation of Danilov [5, Th. 9.1], which is also of interest in itself. Lemma 5.1 Mtoric is simply connected. Proof Let w ∈ V . It follows from (5.5), for all v ∈ V , that the complement of Mtoricw in Mtoric is equal to the union of finitely closed complex analytic submanifolds of complex codimension one, whereas Mtoricw is contractible because it is diffeomorphic to the complex vector space CFw . Because complex codimension one is real codimension two, any loop in Mtoric with base point in Mtoricw can be slightly deformed to such a loop which avoids the complement of Mtoricw in Mtoric, that is, which is contained in M toric w , after which it can be contracted within Mtoricw to the base point in M toric w . � Recall the definition in Section 2 of the reduced phase space M = Z/N . Theorem 5.2 The identity mapping from Z into CF∆, followed by the canonical projection P from C Mtoric = CF∆/NC, induces a T -equivariant diffeomorphism ̟ from M = Z/N onto M toric. It follows that each NC-orbit in C ∆ intersects Z in an N -orbit in Z . Proof Because N is a closed Lie subgroup of NC, we have that the mapping P : Z → CF∆/NC induces a mapping ̟ : Z/N → CF∆/NC, which moreover is smooth. If v ∈ V , z ∈ Zv, then it follows from (5.1) that the tf , f ∈ F \ Fv , of an element t ∈ NC can take arbitrary values, and therefore the |zf |, f ∈ F \ Fv can be moved arbitrarily by means of infinitesimal NC-actions. Because Z is defined by prescribing the |zf |, f ∈ F \ Fv, as a smooth function of the zf , f ∈ Fv, and the Zv, v ∈ V , form an open covering of Z , this shows that at each point of Z the NC-orbit is transversal to Z , which implies that ̟ is a submersion. It follows that ̟(M) is an open subset of Mtoric. Because M is compact and ̟ is continuous, ̟(M) is compact, and therefore a closed subset of Mtoric. Because Mtoric is connected, the conclusion is that ̟(M) =Mtoric, that is, ̟ is surjective. Because ̟ is a surjective submersion, dimRM = 2n =dimRM toric, and M is connected, we conclude that ̟ is a covering map. Because Mtoric is simply connected, see Lemma 5.1, we conclude that ̟ is injective, that is, ̟ is a diffeomorphism. � Remark 5.3 Theorem 5.2 is the last statement in Delzant [3], with no further details of the proof. Audin [2, Prop. 3.1.1] gave a proof using gradient flows, whereas the injectivity has been proved in [7, Sec. A1.2] using the principle that the gradient of a strictly convex function defines an injective mapping. ⊘ Note that in the definition of the toric variety Mtoric, the real numbers λf , f ∈ F , did not enter, whereas these numbers certainly enter in the definition of M , the symplectic form on M , and the diffeomorphism ̟ :M →Mtoric. Therefore the symplectic form σtoricλ := (̟ −1)∗(σ) on Mtoric on Mtoric will depend on the choice of λ ∈ RF . On the symplectic manifold (Mtoric, σtoricλ ), the action of the maximal compact subgroup T of TC is Hamiltonian, with momentum mapping equal to µtoricλ := µ ◦̟ −1 :Mtoric → t∗, (5.7) where µtoricλ (M toric) = ∆, where we note that ∆ in (2.1) depends on λ. In the following lemma we compare the reduced phase space coordinatizations with the toric variety coordinatizations. Lemma 5.4 Let v ∈ V . Then Mtoricv = ̟(Mv), and θv := Ψv −1 ◦̟ ◦ ψv (5.8) is a TFv-equivariant diffeomorphism from Uv onto C For each zv ∈ Uv, the element ζ v := θv(z v) is given in terms of zv by ζvf = z f ′∈F\Fv rf ′(µv(z f ′ , f ∈ Fv , (5.9) where the functions rf ′ : ∆ → R≥0 are given by (3.7). We have µv(zv) = µT (ψv(z v)) = µtoricλ (Ψv(ζ v)), (5.10) and zv = θv −1(ζv) is given in terms of ζ zvf = ζ f ′∈F\Fv rf ′(ξ) f ′ , f ∈ Fv, (5.11) where ξ is the element of ∆ equal to the right hand side of (5.10). Proof It follows from Lemma 3.3 and the paragraph preceding Proposition 3.4 that if zv ∈ ρv(Z), then zv ∈ Uv if and only if z f ′ 6= 0 for every f ′ ∈ F \ Fv. That is, the set Zv in Proposition 3.4 is equal to Z ∩ CFv . It therefore follows from Theorem 5.2 that each NC-orbit in the NC-invariant subset C v of C intersects the N -invariant subset Zv of Z in an N -orbit in Zv, that is, Mtoricv = Pv(C v ) = ̟(pv(Zv)) = ̟(Mv). If zv ∈ Uv, then Proposition 3.4 implies that sv(z v)f = z f for every f ∈ Fv and v)f ′ = rf ′(µv(z v)), f ′ ∈ F \ Fv. If we define t ∈ TF tf ′ = rf ′(µv(z v))−1, f ′ ∈ F \ Fv, f ′∈F\Fv rf ′(µv(z f ′ , f ∈ Fv , then (t · sv(z v))t′ = 1 for every t ′ ∈ F \ Fv and, for every f ∈ Fv, ζ f := (t · sv)f is equal to the right hand side of (5.9). That is, t · sv(z v) = Sv(ζ v), see the definition of Sv in the paragraph preceding (5.5). On the other hand, it follows from (5.1) that t ∈ NC, and therefore Ψv(ζv) = Pv(t · sv(z v)) = Pv(sv(z v)) = ̟ ◦ pv(sv(z v)) = ̟ ◦ ψv(z that is, ζv = Ψv −1 ◦̟ ◦ ψv(z v). � Corollary 5.5 Let s be the relative interior of a face of ∆. Then µT−1(s) is equal to a stratum S of the orbit type stratification in M of the T -action, and also equal to the preimage under ̟ : M → Mtoric of a TC-orbit in M toric. If s = {v} for a vertex v, then µT −1(s) = {mv} for the unique fixed point mv in M for the T -action such that µT (mv) = v. The mapping s 7→ µT −1(s) is a bijection from the set Σ∆ of all relative interiors of faces of ∆ onto the set Σ of all strata of the orbit type stratificiation in M for the action of T . If s, s′ ∈ Σ∆ then s is contained in the closure of s′ in ∆ if and only if µT −1(s) is contained in the closure of µT −1(s′) in M . The domain of definition Mv of ϕv in M is equal to the union of the S ∈ Σ such that mv belongs to the closure of S in M . The domain of definition Mtoricv = ̟(Mv) of Φv is equal to the union of the corresponding strata of the T -action in Mtoric, each of which is a TC-orbit in M toric. Mv and M toric v are open cells in M and Mtoric, respectively. Proof There exists a vertex v of ∆ such that v belongs to the closure of s in t∗, which implies that s is disjoint from all f ′ ∈ F \ Fv. Let Fv, s denote the set of all f ∈ Fv such that s ⊂ f , where Fv, s = ∅ if and only if s is the interior of ∆. For any subset G of Fv, let C G denote the set of all z ∈ C Fv such that zf = 0 if f ∈ G and zf 6= 0 if f ∈ Fv \G. It follows from µv = µT ◦ ψv and (3.8) that ψ v (µT −1(s)) is equal to Uv ∩ C G with G = Fv, s. The diffeomorphism θv maps this set onto the set C G with G = Fv, s. Because the sets of the form CFvG with G ⊂ Fv are the strata of the orbit type stratification of the T Fv -action on CFv , and also equal to the (TC) Fv -orbits in CFv , the first statement of the corollary follows. The second statement follows from µv −1({v}) = {0} and the fact that 0 is the unique fixed point of the Fv-action in Uv. If s ∈ Σ∆ and v ∈ V , then mv belongs to the closure of µT −1(s) if and only if s is not contained in any f ′ ∈ F \ Fv. This proves the characterization of the domain of definition Mv := Zv/N = µT −1(∆v) of ϕv. The last statement follows from the fact that Φv is a diffeomorphism from M toric v onto the vector space Fv , and ̟ is a diffeomorphism from Mv onto M toric v . � Remark 5.6 If v, w ∈ V , then ϕw ◦ ϕv −1 = ψw −1 ◦ ψv = θw −1 ◦ (Ψw −1 ◦Ψv) ◦ θv = θw −1 ◦ (Φw ◦Φv −1) ◦ θv. Using the formula (5.6) for Φw ◦ Φv −1, this can be used in order to obtain the formulas (4.3), (4.4) as a consequence of (5.9). In the proof, it is used that ξ := µv(z v) = µw(z w), |zvf | = rf (ξ) if f ∈ Fv \ Fw, and = (w) if f ′ ∈ F \ Fv and g ∈ Fw. ⊘ In the following corollary we describe the symplectic form σtoricλ on the toric variety M toric in the toric variety coordinates. Corollary 5.7 For each v ∈ V , the symplectic form (Φv −1)∗(σtoricλ ) on C Fv is equal to (θv −1)∗(σv), where σv is the standard symplectic form on C Fv given by (3.4). Because rf ′(µv(z v))2 is an inhomogeneous linear function of the quantities |zvf | 2, it follows from (5.9) that the equations which determine the |zvf | 2 in terms of the quantities |ζvf | 2 are n polyomial equations for the n unknowns |zfv |2, f ∈ Fv , where the coefficients of the polyomials are inhomogeneous linear functions of the |ζvf |, f ∈ Fv. In this sense the |z 2, f ∈ Fv, are algebraic functions of the |ζ 2, f ∈ Fv, and substituting these in (5.9) we obtain that the diffeomorphism θv −1 from CFv onto Uv is an algebraic mapping. If ∆ is a simplex, when Mtoric is the n-dimensional complex projective space, we have an explicit formula for θv see Subsection 6.1. However, already in the case that ∆ is a planar quadrangle, when Mtoric is a complex two-dimensional Hirzebruch surface, we do not have an explicit formula for θv −1. See Subsection 6.2. Summarizing, we can say that in the toric variety coordinates the complex structure is the standard one and the coordinate transformations are the relatively simple Laurent monomial transformations (5.6). However, in the toric variety coordinates the λ-dependent symplectic form in general is given by quite complicated algebraic functions. On the other hand, in the reduced phase space coordinates the symplectic form is the standard one, but the coordinate transformations (4.3), (4.4) are more complicated. Also the complex structure in the reduced phase space coordinates, which depends on λ, is given by more complicated formulas. Remark 5.8 It is a challenge to compare the formula in Corollary 5.7 for the symplectic form in toric variety coordinates with Guillemin’s formula in [7, Th. 3.5 on p. 141] and [8, (1.3)]. Note that in the latter the pullback by means of the momentum mapping appears of a function on the interior of ∆, where in general we do not have a really explicit formula for the momentum mapping in toric variety coordinates. ⊘ 6 Examples 6.1 The complex projective space Let ∆ be an n-dimensional simplex in t∗. A little bit of puzzling shows that there is a Z-basis ei, 1 ≤ i ≤ n, of the integral lattice tZ in t, such that, with the notation e0 = − ei, (6.1) the Xf , f ∈ F , are the ei, 0 ≤ i ≤ n. That is, in the sequel we write F = {0, 1, . . . , n}. The Delzant simplex (2.1) is determined by the inequalities 〈ei, ξ〉+ λi ≥ 0, 0 ≤ i ≤ n, which has a non-empty interior if and only if λi > 0. (6.2) In the sequel we take for v the vertex determined by the equations 〈ei, ξ〉+ λi = 0 for all 1 ≤ i ≤ n, where Fv = {1, . . . , n}. If we write ξi = 〈ei, ξ〉, 1 ≤ i ≤ n, when ξ ∈ t ∗, then (3.2) yields that v)i = |zi| 2/2− λi, 1 ≤ i ≤ n. It follows from (3.7) that r0(ξ) = (2(− ξi + λ0)) and therefore (5.9) yields that ζvi = z i (2γ − ‖z v‖2)−1/2, 1 ≤ i ≤ n, (6.3) where we have written ‖zv‖2 = |zvi | Note that Uv is the open ball in C n with center at the origin and radius equal to (2c)1/2. The equations (6.3) imply that ‖ζv‖2 = ‖zv‖ 2/(2γ − ‖zv‖2), hence ‖zv‖2 = 2γ ‖ζv‖2/(1 + ‖ζv‖2). Therefore the mapping θv−1 : ζv 7→ zv is given by the explicit formulas zvi = ζ i (2γ/(1 + ‖ζ v‖2))1/2, 1 ≤ i ≤ n. (6.4) It can be verified that the symplectic form (θv −1)∗(σv), where σv = (1/2π) dxvi ∧dy is the standard symplectic form in (3.4), is equal to γ times the Fubini-Study form in Griffiths and Harris [6, p. 30, 31]. In view of Remark 3.9 this agrees with the fact that the de Rham cohomology class of the Fubini-Study form is Poincaré dual to the homology class of a complex projective hyperplane in the complex projective space, see Griffiths and Harris [6, p. 122]. 6.2 The Hirzebruch surface Let n = 2 and let ∆ be a quadrangle in the t∗ plane. A little bit of puzzling shows that there is an m ∈ Z≥0 and a Z-basis e1, e2 of the integral lattice tZ in t, such that the Xf , f ∈ F , are the ei, 1 ≤ i ≤ 4, with e3 = −e1 +me2, and e4 = −e2. We recognize the toric variety M toric as the Hirzebruch surface Σm, see Hirzebruch [9]. The Delzant polytope (2.1) is determined by the inequalities 〈ei, ξ〉 + λi ≥ 0, 1 ≤ i ≤ 4, which is a quadrangle if and only if γ± := λ1 + λ3 ±mλ4 > 0, (6.5) which inequalities imply that λ2 + λ4 = γ+ + γ− > 0. In the sequel we take for v the vertex determined by the equations 〈ei, ξ〉+ λi = 0 for i = 1, 2, where Fv = {1, 2}. If we write ξi = 〈ei, ξ〉, 1 ≤ i ≤ 2, when ξ ∈ t ∗, then (3.2) yields that v)i = |zi| 2/2 − λi, 1 ≤ i ≤ 2. It follows from (3.7) that r3(ξ) = (2(−ξ1 +mξ2 + λ3)) 1/2, r4(ξ) = (2(−ξ2 + λ4)) and therefore (5.9) yields that ζv1 = z 1 (2γ−|z 2 +m |zv2 | 2)−1/2, (6.6) ζv2 = z 2 (2γ−|z 2 +m |zv2 | 2)m/2 (2(γ+ + γ−)− |z 2)−1/2. (6.7) If we write ti = |z 2 and τi = |ζ 2, then this leads to the equations τ1 = t1/(2γ− − t1 +mt2), τ2 = t2 (2γ− + t1 +mt2) m/(2(γ+ + γ−)− t2) for t1, t2. If we solve t1 from the first equation, t1 = (2γ− +mt2) τ1/(1 + τ1), and substitute this into the second equation, then this leads to the polynomial equation (1 + τ1) m τ2 (2(γ+ + γ−)− t2) = t2 (2γ− +mt2) m (6.8) of degree m + 1 for t2. If we substract the left hand side from the right hand side then the derivative with respect to t2 is strictly positive, and one readily obtains that for every τ1, τ2 ∈ R≥0 there is a unique solution t2 ∈ R≥0, confirming the first statement in Lemma 5.4. On the other hand, if we work over C, and view both the parameter ε := (1 + τ1) m τ2 and the unknown t2 as elements of the complex projective line P 1, then the equation (6.8) defines a complex algebraic curve C in the (t2, ε)-plane P 1 × P1, where the restriction to C of the projection to the first variable t2 is a complex analytic diffeomorphism from C onto P1, as on C we have that ε is a complex analytic function of t2. In particular C is irreducible. The restriction to C of the projection to the second variable ε is an (m+ 1)-fold branched covering. Over ε = 0 and over ε = ∞ we have that m of the m+ 1 branches come together, whereas there are two more branch points on the ε-line over which only two of the branches come together. The fact that C is irreducible implies that the part of C over the complement of the branch points is connected, and therefore the analytic continuation of any solution t2 of (6.8), as a complex analytic function of ε in the complement of the branch points, will reach each other branch if ε runs over a suitable loop. In other words, the solution t2 is an algebraic function of ε of degree m+ 1, and no branch of a solution is of lower degree. This holds in particular for our solutions t2 ∈ R≥0 for ε ∈ R≥0. References [1] R. Abraham and J.E. Marsden: Foundations of Mechanics. Benjamin/Cummings Publ. Co., London, etc., 1978. [2] M. Audin: The Topology of Torus Actions on Symplectic Manifolds. Birkhäuser, Basel, Boston, Berlin, 1991. [3] T. Delzant: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France 116 (1988) 315–339. [4] M. Demazure: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. scient. Éc. Norm. Sup. 3 (1970) 507–588. [5] V.I. Danilov: The geometry of toric varieties. Russ. Math. Surveys 33:2 (1978) 97–154, translated from Uspekhi Mat. Nauk SSSR 33:2 (1978) 85–134. [6] P. Griffiths and J. Harris: Principles of Algebraic Geometry. J. Wiley & Sons, Inc., New York, etc., 1978. [7] V. Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces. Birkhäuser, Boston, etc., 1994. [8] V. Guillemin: Kaehler structures on toric varieties. J. Differential Geometry 40 (1994) 285–309. [9] F. Hirzebruch: Über eine Klasse von einfach-zusammenhn̈genden komplexen Mannigfaltigkeiten. Mathematische Annalen 124 (1951) 77–86. [10] A. Pelayo: Topology of spaces of equivariant symplectic embeddings. Proc. Amer. Math. Soc. 135 (2007) 277–288. [11] R.T. Rockafellar: Convex Analysis. Princeton University Press, princeton, N.J., 1970. J.J. Duistermaat Mathematisch Instituut, Universiteit Utrecht P.O. Box 80 010, 3508 TA Utrecht, The Netherlands e-mail: [email protected] A. Pelayo Department of Mathematics, University of Michigan 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA e-mail: [email protected] Introduction The reduced phase space The reduced phase space coordinatizations. The coordinate transformations The toric variety Examples The complex projective space The Hirzebruch surface

0704.0431

Fragmentation of general relativistic quasi-toroidal polytropes

FRAGMENTATION OF GENERAL RELATIVISTIC QUASI-TOROIDAL POLYTROPES Burkhard Zink,1, 2 Nikolaos Stergioulas,3 Ian Hawke,4 Christian D. Ott,5 Erik Schnetter,1, 6 and Ewald Müller7 1Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA 2Horace Hearne Jr. Institute for Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803, USA 3Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece 4School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK 5Department of Astronomy and Steward Observatory, The University of Arizona, Tucson, AZ, USA 6Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany 7Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching bei München, Germany How do black holes form from relativistic stars? This ques- tion is of great fundamental and practical importance in grav- itational physics and general relativistic astrophysics. On the fundamental level, black holes are genuinely relativistic ob- jects, and thus the study of their production involves ques- tions about horizon dynamics, global structure of spacetimes, and the nature of the singularities predicted as a consequence of the occurrence of trapped surfaces. On the level of as- trophysical applications, systems involving black holes are possible engines for highly energetic phenomena like AGNs or gamma-ray bursts, and also likely a comparatively strong source of gravitational radiation. The most simple model of black hole formation from, say, cold neutron stars, is a fluid in spherically symmetric poly- tropic equilibrium moving on a sequence of increasing mass due to accretion [1]. This assumes that (i) the stellar structure and dynamics are represented reasonably by the ideal fluid equation of state and the polytropic stratification, (ii) accre- tion processes are slow compared to the dynamical timescales of the star, and (iii) rotation is negligible. Our focus has been to study the effects of relaxing the third assumption. In spherical symmetry, the sequence of equilibrium poly- tropes has a maximum in the mass function M(ρ ), where ρ denotes the central rest-mass density of the polytrope. This maximum is connected to a change in the stability of the fun- damental mode of oscillation [1], and thus collapse sets in via a dynamical instability to radial deformations. During the subsequent evolution, a trapped tube forms at the center which traverses the stellar material entirely [2]. How much of this behaviour is preserved when rotation is taken into account? Rotation is known to change the equi- librium structure of the star, and, in consequence, its modes of oscillation and set of unstable perturbations. The collapse might also lead to the formation of a massive disk around the new-born black hole, and finally only systems without spher- ical symmetry can be a source of gravitational radiation. Numerical simulations have been used to study the collapse and black hole formation of general relativistic rotating poly- tropic stars [3]. For the uniformly and moderately differen- tially rotating models investigated in those studies, the dynam- ical process is described by the instability of a quasi-radial mode and subsequent collapse of the star up to the formation of an accreting Kerr black hole at the star’s center. Will strong differential rotation modify this picture? Even before our study, there was evidence that this should be the case. (i) Strong differential rotation can deform the high- FIG. 1: Development of the fragmentation instability in a model of a strongly differentially rotating supermassive star. The darker shades of grey indicate higher density. The closed white line in the last plot is a trapped surface. density regions of a star into a toroidal shape, thus chang- ing the equilibrium structure considerably. (ii) It admits stars of high normalized rotational energy T/|W | [1] which are stable to axisymmetric perturbations. (iii) It admits non- axisymmetric instabilities, for example by the occurrence of corotation points[4], at low values of T/|W |[5]. (iv) A bar- mode instability of the type found in Maclaurin spheroids[6] would likely express itself by the formation of two orbiting fragments if the initial high-density region has toroidal shape. This last property has motivated us to ask this question: Can a bar deformation transform a strongly differentially ro- tating star into a binary black hole merger with a massive accretion disk? If so, this process might occur in supermas- sive stars if the timescales associated with angular momentum transport are too large to enforce uniform rotation. We have investigated black hole formation in strongly dif- ferentially rotating, quasi-toroidal models of supermassive stars [7, 8], and found that a non-axisymmetric instability can lead to the off-center formation of a trapped surface (see fig- ure). An extensive parameter space study of this fragmen- tation instability [8] reveals that many quasi-toroidal stars of this kind are dynamically unstable in this manner, even for low values of T/|W |, and we have found evidence that the coro- tation mechanism observed by Watts et al. [4] might be active in these models. Since, on a sequence of increasing T/|W |, one of the low order m = 1 modes becomes dynamically un- http://arxiv.org/abs/0704.0431v1 stable before m = 2 and higher order modes, one would not expect a binary black hole system to form in many situations (although this may depend on the rotation law and details of the pre-collapse evolution as well). Rather, the off-center pro- duction of a single black hole with a massive accretion disk appears more likely. Since the normalized angular momentum J/M2 of the ini- tial model is greater than unity, there is another interesting consequence of this formation process: the resulting black hole, unless it is ejected from its shell, may very well be rapidly rotating, spun up by accretion of the material remain- ing outside the initial location of the trapped surface. Investi- gating the late time behaviour of this accretion process, esti- mating possible kick velocities of the resulting black hole, and finding the mass of the final accretion disk is, however, beyond our present-day capabilities and subject of future study. [1] S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs and Neutron Stars (Wiley 1983). [2] S. Shapiro and S. Teukolsky, Astrophys. J. 235, 199 (1980). [3] M. Shibata, T. Baumgarte and S. Shapiro, Phys. Rev. D 61, 044012 (2000). L. Baiotti, I. Hawke, P. Montero, F. Löffler, L. Rezzolla, N. Stergioulas, J. A. Font and E. Seidel, Phys. Rev. D 71, 024035 (2005), and references therein. [4] A. Watts, N. Andersson and D. Jones, Astrophys. J. L37 (2005). [5] J. Centrella, K. New, L. Lowe and J. Brown, Astrophys. J. 550 (2001). [6] S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale UP 1969). [7] B. Zink, N. Stergioulas, I. Hawke, C. D. Ott, E. Schnetter and E. Müller, Phys. Rev. Letters 96, 161101 (2006). [8] B. Zink, N. Stergioulas, I. Hawke, C. D. Ott, E. Schnetter and E. Müller, astro-ph/0611601 (2006). http://arxiv.org/abs/astro-ph/0611601

0704.0432

A Survey of Huebschmann and Stasheff's Paper: Formal Solution of the Master Equation via HPT and Deformation Theory

A Survey of Huebschmann and Stasheff’s Paper: Formal solution of the master equation via HPT and deformation theory Füsun Akman and Lucian M. Ionescu October 22, 2018 1 Introduction These notes, based on the paper [8] by Huebschmann and Stasheff, were pre- pared for a series of talks at Illinois State University with the intention of ap- plying Homological Perturbation Theory (HPT) to the construction of derived brackets [11, 16], and eventually writing Part II of the paper [1]. Derived brackets are obtained by deforming the initial bracket via a deriva- tion of the bracket. In [3] it was demonstrated that such deformations corre- spond to solutions of the Maurer-Cartan equation, and the role of an “almost contraction” was noted. This technique (see also [9]) is very similar to the itera- tive procedure of [8] for finding the most general solution of the Maurer-Cartan equation, i.e. the deformation of a given structure in a prescribed direction. The present article, besides providing additional details of the condensed article [8], forms a theoretical background for understanding and generalizing the current techniques that give rise to derived brackets. The generalization, which will be the subject matter of [2], will be achieved by using Stasheff and Huebschmann’s universal solution. A second application of the universal solu- tion will be in deformation quantization and will help us find the coefficients of star products in a combinatorial manner, rather than as a byproduct of string theory which underlies the original solution given by Kontsevich [10]. HPT is often used to replace given chain complexes by homotopic, smaller, and more readily computable chain complexes (to explore “small” or “minimal” models). This method may prove to be more efficient than “spectral sequences” in computing (co)homology. One useful tool in HPT is Lemma 1 (Basic Perturbation Lemma (BPL)). Given a contraction of N onto M and a perturbation ∂ of dN , under suitable conditions there exists a pertur- bation d∂ of dM such that H(M,dM + d∂) = H(N, dN + ∂). The main question is: under what conditions does the BPL allow the preser- vation of the data structures (DGA’s, DG coalgebras, DGLA’s etc.)? (We will use the self-explanatory abbreviations such as DG for “differential graded”, http://arxiv.org/abs/0704.0432v1 DGA for “differential graded (not necessarily associative) algebra”, and DGLA for “differential graded Lie algebra”.) Another prominent idea is that of a “(universal) twisting cochain” as a so- lution of the “master equation”: Proposition 1. Given a contraction of N onto M and a twisting cochain N → A (A some DGA), there exists a unique twisting cochain M → A that factors through the given one and which can be constructed inductively. The explicit formulas are reminiscent of the Kuranishi map [13] (p.17), and the relationship will be investigated elsewhere. Note: we will assume that the ground ring is a field F of characteristic zero. We will denote the end of an example with the symbol ♦ and the end of a proof by �. 2 Perturbations of (co)differentials 2.1 Derivations of the tensor algebra For any vector space V over F we have the isomorphismDer(TV ) ∼= Hom(V, TV ) where TV denotes the (augmented) tensor algebra on V . Namely, every linear map f from V into TV extends uniquely into a derivation of the algebra TV via the formula f̂(v1 ⊗ · · · vn) = v1 ⊗ · · · ⊗ f(vi)⊗ · · · vn. Equivalently, every derivation of TV is determined by its restriction to V . 2.2 Coderivations of the tensor coalgebra Similarly, we have the isomorphism Coder(T cV ) ∼= Hom(T cV, V ) where T cV is the (coaugmented) coassociative tensor coalgebra of V , with counit η : T cV → F (projection onto F ), and comultiplication ∆(v1 ⊗ · · · ⊗ vn) = (v1 ⊗ · · · ⊗ vi)⊗ (vi+1 ⊗ · · · ⊗ vn). Every linear map f = f1 + f2 + · · ·+ fn + · · · : T cV → V (where fi : V ⊗i → V ) factors uniquely through a coderivation f̂ of T cV defined via the formula f̂(v1 ⊗ · · · ⊗ vn) = v1 ⊗ · · · ⊗ f1(vi)⊗ · · · vn v1 ⊗ · · · ⊗ f2(vi ⊗ vi+1)⊗ · · · vn +fn(v1 ⊗ · · · ⊗ vn). That is, each coderivation on T cV is determined by itself followed by the projec- tion onto V . Recall that the condition for f̂ to be a coderivation can be written as ∆f̂ = (1⊗ f̂ + f̂ ⊗ 1)∆. 2.3 Coderivations of the symmetric coalgebra Let us consider the cofree cocommutative counital coassociative algebra ST cV on the vector space V as a subspace of T cV . The symmetric group Σn acts on the left on V ⊗n via σ(v1 ⊗ · · · ⊗ vn) = vσ−1(1) ⊗ · · · ⊗ vσ−1(n). Then ST cV = (V ⊗n)Σn is the space of invariants of this action. The action is compatible with the coproduct on ST cV , so ST cV is a subcoalgebra of T cV which is cocommutative. Note that ST (V ) is not a subalgebra with respect to the tensor multiplication in T (V ); the product has to be symmetrized so that it projects back onto this subspace (reminiscent of what T. Voronov does with derived brackets). The projection (symmetrization) map P : T cV → ST cV is given by P (v1 ⊗ · · · ⊗ vn) = σ(v1 ⊗ · · · ⊗ vn). This is not a coalgebra map, but is a retraction of the canonical inclusion ST cV →֒ T cV . Now a coderivation D : T cV → T cV induces a coderivation DS : ST cV → ST cV by the composition ST cV →֒ T cV → T cV → ST cV . In particular, a coderivation of T cV induces one of ΛcV = ST c[sV ] where V is thought of as living in degree zero: we introduce the graded symmetric coalgebra below. Once again, coderivations of ST cV are determined by their projections onto ST c1V ; a map f = f1 + f2 + · · · : ST c(V ) → V determines a coderivation f̂ as in the tensor coalgebra case. In the remaining part of this survey, we choose to identify ST cV with the abstract symmetric coalgebra ScV under the isomorphism v1 · · · vn 7→ P (v1 ⊗ · · · ⊗ vn). The coproduct in ScV is given by ∆(v1 · · · vn) = σ∈Σi,n−i vσ(1) · · · vσ(i) ⊗ vσ(i+1) · · · vσ(n). 2.4 DGLA’s and perturbations of the codifferential Definition 1. For any chain complex (X, d), and odd ∂, with (d + ∂)2 = 0, we say that ∂ is a perturbation of the differential d. We call d + ∂ the perturbed differential. Equivalently, we have [d, ∂] + ∂∂ = 0 in End(X). If ∂ is also compatible with an existing coalgebra structure on X , we say that it is a coalgebra perturbation. Let (g, d) be a graded chain complex (d lowers degrees) with a bracket [ , ] that is skew-symmetric (not necessarily Leibniz or a chain map). Consider the differential graded symmetric coalgebra Sc[sg], the differential d being induced by that on g. Also let ∂ be the coderivation on Sc[sg] of degree −1 induced by the bracket. Proposition 2. The bracket [ , ] turns (g, d) into a DGLA if and only if ∂ is a coalgebra perturbation of d. Also, any DGLA structure on g is determined by the coalgebra perturbation induced from the bracket. When g is an ordinary (degree-zero) Lie algebra over a field, Sc[sg] = Λcg with differential ∂ corresponding to the bracket is the ordinary Koszul or Chevalley-Eilenberg complex computing the homology of g with coefficients in the field. 2.5 Strongly homotopy Lie algebras Definition 2. Let (g, d) be a chain complex and let d also denote the codiffer- ential in Sc[sg] induced by d. A strongly homotopy Lie (sh-Lie, or L∞) structure on g is a perturbation ∂ = ∂2 + · · · + ∂n + · · · of d, i.e. an odd coderivation satisfying [d, ∂] + ∂∂ = 0 and ∂η = 0 (recall that η is the counit) so that the sum d+ ∂ endows Sc[sg] with a new coaugmented DG coalgebra structure. The corresponding mega-map ℓ2 + · · · + ℓn + · · · from S c(sg) to g extends the differential ℓ1 = d : sg → g, and the lower identities satisfied by ℓ = ℓ1 + ℓ2 + · · ·+ ℓn + · · · read as follows: ℓ21 = 0 ℓ1(ℓ2(a, b))± ℓ2(ℓ1(a), b)± ℓ2(ℓ1(b), a) = 0 ℓ1(ℓ3(a, b, c))± ℓ3(ℓ1(a), b, c)± ℓ3(ℓ1(b), a, c)± ℓ3(ℓ1(c), a, b) ±ℓ2(ℓ2(a, b), c)± ℓ2(ℓ2(a, c), b)± ℓ2(ℓ2(b, c), a) = 0. An sh-Lie morphism between two sh-Lie (or DGL) algebras (g, d + · · · ) and (g′, d′ + · · · ) is a collection of chain maps Fn : S n[sg] → S ′], satisfying ∆′F (u) = (F ⊗ F )(∆u). Then F is uniquely determined by its projection onto sg′, that is, we may assume Fn : S n[sg] → sg Definition 3. A quasi-isomorphism F between sh-Lie algebras g, g′ is an sh-Lie morphism such that F1 : sg → sg ′ induces an isomorphism between H(g, d) and H(g′, d′). Remark 1. Quasi-isomorphisms between DGLA’s are especially important in deformation theory. Such a map gives a one-to-one correspondence between moduli spaces of solutions to MC equations in ~g[[~]] and ~g′[[~]] (see [5]): given a quasi-isomorphism F : Sc[sg] → sg′, we define F̃ : ~g[[~]] → ~g′[[~]] by F̃ (r) = Fn(r, . . . , r) (also see [6]). 2.6 The Hochschild chain complex and DGA’s Let (A, µ) be a unital associative algebra (possibly graded), and T c[sA] denote the tensor coalgebra on the suspension of A. We recall that Coder(T c[sA]) ∼= Hom(T c[sA], A). In particular, the associative bilinear multiplication µ ∈ Hom(T c[sA], A) corre- sponds to a square-zero coderivation ∂ : T c[sA] → T c[sA] defined by ∂(a1 ⊗ · · · ⊗ an) (−1)i+1(a1 ⊗ · · · ⊗ µ(ai ⊗ ai+1)⊗ · · · ⊗ an) +(−1)n+1(µ(an ⊗ a1)⊗ a2 ⊗ · · · ⊗ an−1). The condition that ∂ is a codifferential is equivalent to the associativity con- dition m ◦ m = 0 where ◦ is the Gerstenhaber composition on multilinear maps (a right pre-Lie map). The complex (Hom(T c[sA], A), ∂) is known as the Hochschild chain complex. Now let (A, µ, d) be a DGA. Then d+µ ∈ Hom(T c[sA], A) corresponds to a perturbed codifferential d+ ∂ satisfying (d+ ∂)2 = 0, which is equivalent to the identities d2 = 0 and [d, ∂] + ∂∂ = 0. The latter can also be split into [d, ∂] = 0 and ∂∂ = 0. Proposition 3. The multiplication µ turns (A, d) into a DGA if and only if ∂ is a coalgebra perturbation of d. Also, any DGA structure on A is determined by the coalgebra perturbation induced from µ. 2.7 Strongly homotopy associative algebras Definition 4. Let (A, d) be a chain complex and let d also denote the codifferen- tial in T c[sA] induced by d. A strongly homotopy associative (or A∞) structure on A is a perturbation ∂ = ∂2 + · · · + ∂n + · · · of d, i.e. an odd coderivation satisfying [d, ∂] + ∂∂ = 0 and ∂η = 0 so that the sum d+ ∂ endows T c[sA] with a new coaugmented DG coalgebra structure. The corresponding mega-map m2+ · · ·+mn+ · · · from T c[sA] to A extends the differential m1 = d : sA → A, and the lower identities satisfied by m = m1 +m2 + · · ·+mn + · · · read as follows: m21 = 0 m1(m2(a, b))±m2(m1(a), b)±m2(m1(b), a) = 0 m1(m3(a, b, c))±m3(m1(a), b, c)±m3(a,m1(b), c)±m3(a, b,m1(c)) = 0. The mega-identity is m ◦ m = 0, sometimes written in the braces notation {m}{m} = 0. 3 Master equation If (A, d) is a differential graded associative algebra (DGA), then the equation dτ = ττ (1) is called the Master Equation (ME) (or Maurer-Cartan equation (MCE), etc.). Similarly if (g, d) is a DGLA, then the equation [τ, τ ] (2) is also called the Master Equation. Sometimes the sign convention is dτ + ττ = 0 (3) [τ, τ ] = 0. (4) Clearly any solution of such an equation must be an odd element of the algebra. Moreover, in case A is the graded universal enveloping algebra of g, or g is the Lie algebra obtained from A by the usual bracket, then solutions of the DGLA master equation are also solutions of the DGA master equation. Remark 2. If τ is a solution of Eq. (2) or Eq. (4) in a DGLA g, then the odd derivation dτ = d− adτ or dτ = d+ adτ respectively defines a new DGLA structure on g with respect to the old bracket. Example 1. If g is a DGLA or L∞ algebra, then the corresponding coal- gebra perturbation ∂ in Coder(Sc(sg)) is a solution of the ME in the DGA End(Sc(sg)), where the differential is D = add. ⋄ Example 2. Gauge Theory: Let ξ be a principal bundle with structure group G and Lie algebra g. There is a graded Lie algebra structure on the ad(ξ)-valued de Rham forms induced by g. Given a connection A and an ad(ξ)-valued 1-form η, the sum A+ η is again a connection, and its curvature is FA+η = FA + dAη + [η, η]. In particular, FA = FA+η if and only if dAη + [η, η] = 0 (the Maurer-Cartan equation). Here dA is the covariant derivative of the con- nection A. When A is a flat connection (zero curvature) then there exists a DGLA structure on the ad(ξ)-valued differential forms (d2A = 0) and FA+η is also flat iff the MCE is satisfied (then the covariant derivative for A+ η is dτ ). 4 Twisting cochain The notion of a twisting cochain generalizes that of a connection in differential geometry. 4.1 Differential on Hom If (C, dC) and (A, dA) are chain complexes, the following differential D makes Hom(C,A) into a chain complex: Dφ = dAφ± φdC . 4.2 Cup product and cup bracket Proposition 4. For any differential graded coalgebra C and a differential graded associative algebra (DGA) A, the chain complex (Hom(C,A), D) becomes a DGA via the cup (convolution) product a ⌣ b defined by the composition −→ C ⊗ C −→ A⊗A −→ A. The coaugmentation and augmentation maps η and ǫ on C and A respectively define an augmentation map on (Hom(C,A), D). Proposition 5. For any differential graded coalgebra C and a DGLA g, the chain complex (Hom(C, g), D) becomes a DGLA via the cup bracket [a, b] defined by the composition −→ C ⊗ C −→ g ⊗ g [ , ] −→ g. Example 3. If g is a Lie algebra, then the cup bracket on Hom(Sc[sg], g) is defined as above. For example, if τ and κ are maps Sc1[g] → g, then [τ, κ] may be nonzero only on Sc2[sg]. In this case, we compute ∆(xy) = 1⊗ xy + x⊗ y + y ⊗ x+ xy ⊗ 1 (x, y ∈ sg), [τ, κ](xy) = [τ(x), κ(y)] + [τ(y), κ(x)]. (5) Example 4. The Hochschild complex of an associative algebra (A, µ) (where µ2 = 0): Let C•(A) = Hom(T cA,A) = Hom( A⊗n, A), with differential D = adµ ∈ Der(C •(A)). The cup product x ⌣ y = {µ}{x, y} is the composition −→ T cA⊗ T cA −→ A⊗A −→ A; if x is an n-linear map and y is an m-linear map, then (x ⌣ y)(a1 ⊗ · · · ⊗ an+m) = x(a1 ⊗ · · · ⊗ an) · y(an+1 ⊗ · · · ⊗ an+m). Remark 3. The differential D above is an inner derivation and not derived from differentials on A and T cA. Still, it is a derivation of the cup product. 4.3 Twisting cochain Definition 5. Given a coaugmented DG coalgebra C and an augmented DGA A, a twisting cochain is a homogeneous morphism t : C → A of degree −1 such that ǫτ = 0 and τη = 0 , and which satisfies Dt = t ⌣ t. In other words, a twisting cochain is a solution of the master equation on Hom(C,A) with the usual differential D induced from those of C, A and the product is the cup product. Definition 6. Given a DG cocommutative coalgebra C and a DGLA g, a Lie algebra twisting cochain t : C → g is a homogeneous map of degree −1 whose composition with the coaugmentation map is zero, and which satisfies [t, t] (6) ([ , ] being the cup bracket). Recall that a DGLA structure on a graded chain complex (g, d) is given by a perturbation ∂ of the corresponding codifferential on Sc[sg]. Moreover, the piece d∂ + ∂d = 0 of (d+ ∂)2 = 0 says that the bracket is a chain map and the piece ∂2 = 0 says that the bracket satisfies the Jacobi identity. Let us denote the symmetric coalgebra with the codifferential ∂ by Sc [ , ] [sg]. Quillen’s notation C[g] for the same DG coalgebra reminds us that this is the Koszul or Chevalley- Eilenberg complex that computes the homology of g without any regard for the additional differential d on g. Example 5. For any DGLA g, its universal Lie algebra twisting cochain τg : S [ , ][sg] → g is given by τg(sx) = x for x ∈ g τg(y) = 0 for y ∈ S k[sg], k 6= 1. That is, an element with tensor degree one goes to its desuspension and ev- erything else goes to zero. Clearly, the composition τgη is zero, as τg = 0 on constants. Next, we show that τg satisfies the equation (6), but we note that in this construction the differential on g itself is taken to be zero. On the left-hand side, we have Dτg(x1 ∧ · · · ∧ xn) = τg∂(x1 ∧ · · · ∧ xn) which is zero if n 6= 2 and is equal to [x1, x2] if n = 2. Meanwhile, on the right-hand side, we have [τg, τg](x1 ∧ · · · ∧ xn) which is zero if n 6= 2 and is equal to {[τg(x1), τg(x2)]± [τg(x2), τg(x1)]} = [x1, x2] if n = 2. ⋄ Remark 4. The universal property of the universal LA twisting cochain is that every Lie algebra twisting cochain factors through this one: that is, whenever C is a coalgebra and τ : C → g is a twisting cochain, then τg ◦ c(τ) = τ where c(τ) : C → Sc [ , ] [sg] is the unique coalgebra map induced by τ . Using HPT, we will construct formal solutions τ ∈ Hom(ScD[sH(g)], g) of the master equation. Once we make the choice of a contraction, we will obtain explicit inductive formulas for D and τ . 5 Homological perturbation theory (HPT) “HPT is concerned with transferring various kinds of algebraic structure through a homotopy equivalence”. Also: “HPT is a set of techniques for the transference of structures from one object to another up to homotopy” (Real [14]). 5.1 Contraction Definition 7. Let (M,dM ) and (N, dN ) be chain complexes, π : N → M and ∇ : M → N be chain maps, and h ∈ End(N) be a morphism (possibly preserving some extra structure) of degree 1. Then a contraction N, h) (7) of N onto M is a collection of the above data satisfying π∇ = idM D(h) = addN (h) = ∇π − idN πh = 0, h∇ = 0, hh = 0. Another way to describe this structure is to say that M is a strong deformation retract (SDR) of N (also called Eilenberg-Zilber data). The properties on the last line are referred to as the annihilation properties or side conditions. Note that the first line makes π surjective (projection) and ∇, injective (inclusion). The map h is also known as the homotopy operator between ∇π and idN : ∇π − idN = D(h) = dNh+ hdN (D = addN is the induced differential on Hom(N,N)). Often filtered contractions are considered. Remark 5. Lambe and Stasheff [12] noticed that the side conditions on h are not restrictive: if πh = 0 and h∇ = 0 are not satisfied, then we can replace h by h′ = D(h)hD(h). Now if h2 = 0 is not satisfied either, we replace h′ by h′′ = h′dNh ′, which finally gives us an operator h′′ satisfying the side conditions. Lemma 2. Given a contraction (7), we have a (not necessarily direct) sum N = Im(∇) + Im(h) + Im(dN ). Proof. Each x ∈ N can be written as x = ∇π(x) − hdN (x) − dNh(x). (8) Lemma 3. [14] Given a contraction (7), we have Im(∇) + Im(h) = Im(∇)⊕ Im(h) = Ker(h). Proof. We have Im(∇) ⊂ Ker(h) and Im(h) ⊂ Ker(h) since h∇ = 0 and h2 = 0. Conversely, by (8), each x ∈ Ker(h) can be written x = ∇π(x) − hdN (x) ∈ Im(∇) + Im(h). That the sum is direct can be seen as follows: let x ∈ Im(∇) ∩ Im(h). Then we have x = ∇(y) = h(z) for some y ∈ M and z ∈ N . Rewriting the decomposition of x in Ker(h), we obtain x = ∇π(x) − hdN (x) = ∇πh(z)− hdN∇(y) = 0− hdN∇(y) (πh = 0) = h∇dM (y) (∇ chain map) = 0 (h∇ = 0). Corollary 1. For any contraction (7), we have H(N, h) ∼= Im(∇) ∼= M. Lemma 4. Given any contraction (7), we have Im(h) + Im(dN ) = Im(h)⊕ Im(dN ). Moreover, if dM ≡ 0, then Im(h)⊕ Im(dN ) = Ker(π). Proof. Say x ∈ Im(h) ∩ Im(dN ). Then x = h(y) = dN (z) for some y, z ∈ N , so that by (8) we obtain x = ∇πh(y) − hdNdN (z)− dNhh(y) = 0. It is always true that Im(h) ⊂ Ker(π) as πh = 0. If dM ≡ 0, we further have the result πdN (x) = −dMπx = 0, so that altogether Im(h)⊕ Im(dN ) ⊂ Ker(π). Conversely, for x ∈ Ker(π), we see that (even without the condition dM = 0) Ker(π) ⊂ Im(h) + Im(dN ) since x = −hdN (x)− dNh(x) ∀x ∈ Ker(π) due to (8). � Lemma 5. For any contraction (7) with dM = 0 we have Im(∇) + Im(dN ) = Im(∇) ⊕ Im(dN ) = Ker(dN ). Proof. The given sum is direct: let x ∈ Im(∇) ∩ Im(dN ). Then x = ∇(y) = dN (z) and by (8) x = ∇πdN (z)− hdNdN (z)− dNh∇(y) = ∇πdN (z) = −∇dMπ(z) = 0. Clearly, we have Im(dN ) ⊂ Ker(dN ). Also Im(∇) ⊂ Ker(dN ), because dN∇(x) = −∇dM (x) = 0. Conversely, by (8), Ker(dN ) ⊂ Im(∇) + Im(dN ) since we can write x = ∇π(x) − dNh(x) (no condition on dM ) for x ∈ Ker(dN ). � Corollary 2. For any contraction (7) with dM = 0 we have H(N, dN ) ∼= Im(∇) ∼= M. Proposition 6. For any contraction (7) with dM = 0 we have N = Im(∇)⊕ Im(h)⊕ Im(dN ) where Im(∇)⊕ Im(h) = Ker(h) Im(h)⊕ Im(dN ) = Ker(π) Im(∇)⊕ Im(dN ) = Ker(dN ), 2. Im(h) → Im(dN ) is an isomorphism with inverse Im(dN ) → Im(h), and 3. Im(∇) → M is an isomorphism with inverse M → Im(∇); we also have Im(∇) ∼= M = Im(π) ∼= H(N, dN ) ∼= H(N, h). Remark 6. This is a Hodge-type decomposition reminiscent of the case of a compact orientable Riemannian manifold M without boundary. If ∗ : Ωr(M) → Ωdim(M)−r is the “Hodge star operator” (an isomorphism) and d : Ωr−1(M) → Ωr(M) is the de Rham differential, then we define a “partial inverse” d† (the adjoint exterior derivative operator) to −d by d† = ± ∗ d∗. The commutator of d and d† is called the “Laplace-Beltrami operator”: ∆ = dd† + d†d. Then there exists a unique decomposition of the algebra of de Rham forms as follows: Ωr(M) = Harmr(M)⊕ d(Ωr−1(M))⊕ d†(Ωr+1(M)), where the “harmonic forms” are given by Harmr(M) = Ker(∆). In the case of our general contraction with dM = 0, the operators h and dN replace d and d respectively. What do we know about ∆ here? We have ∆ = D(h) = hdN + dNh = (h+ dN ) 2 = ∇π − idN . The kernel of this operator is equal to ∇(M), as we have (∇π − idN )(x) = 0 ⇔ ∇π(x) = x ⇔ x ∈ ∇(M), or Ker(∆) = Im(∇). So is there an analog of the Hodge star operator? If we define an isomorphism ∗ = h+ dN + Id∇(M) (where the last operator is zero on the remaining direct summands), then we have ∗−1 = −h− dN + Id∇(M), and −1 = (h+dN+Id∇(M))dN (−h−dN+Id∇(M)) = −hdNh = −hIdIm(dN ) = h. Remark 7. The operator d† is more like the BV operator than the (even) Laplacian, which is not square-zero. Another similar case is Q (BRST operator) and b0 (anti-ghost operator), for which we have Qb0 + b0Q = L0 (the degree operator which is zero on the cohomology). Proof. We only need to prove (2) and part of (3). First, we want to show that −dNh = idIm(dN ) But then for x = dN (y), we have − dNh(x) = [−dNh]dN (y) = [hdN −∇π + idN ]dN (y) = −∇[πdN ](y) + dN (y) = −∇[−dMπ]dN (y) + x Similarly, we would like to have −hdN = idIm(h). If x = h(y), then − hdN (x) = [−hdN ]h(y) = [dNh−∇π + idN ]h(y) = −∇πh(y) + h(y) Finally, we have π∇ = idM and ∇π∇ = ∇idM = ∇, which shows the isomor- phism between ∇(M) and π(N). � Example 6. Let (g, d) be a chain complex. Assume that the underlying ring is a field. Then there exists a contraction (H(g) g, h) (9) of chain complexes, where the differential on H(g) is zero: we can write g as a linear sum g = G⊕Ker(d) = G⊕ Im(d)⊕H(g) by choosing arbitrary representatives of the homology classes etc.; let us show the decomposition of an element x of g by x = xG + xIm(d) + xH(g). Then π is the projection of g onto H(g) and ∇ is the inclusion map of H(g) into g. Note that as vector spaces G and Im(d) are isomorphic via d: let x, y ∈ G. dx = dy ⇒ d(x − y) = 0 ⇒ x− y ∈ Ker(d) ∩G = {0} and d : G → Im(d) is one-to-one as well as onto. We define h to be the inverse of −d on Im(d) and zero on the rest of g. The linear map h is square-zero and increases degree by one. Moreover, (dh+ hd)(x) = (dh+ hd)(xG + xIm(d) + xH(g)) = dh(xIm(d)) + hd(xG) = −xIm(d) − xG (∇π − idg)(xG + xIm(d) + xH(g)) = xH(g) − (xG + xIm(d) + xH(g)) = −xIm(d) − xG. In comparison with the last corollary, we have (N, dN ) = (g, d) (M,dM ) = (H(g, d), 0) Im(∇) = H(g, d) Im(h) = G Im(dN ) = Im(d). Example 7. (The Tensor Trick) Any contraction (7) of chain complexes in- duces a filtered contraction (T c[M ] T c[N ], T ch) of coaugmented differential graded coalgebras. Here is how: the projection PN : T c[N ] → N followed by the surjective chain map π : N → M gives us a linear map π ◦ PN : T c[N ] → M (π ◦ PN )(x1 ⊗ · · · ⊗ xk) = π(x1) if k = 1 0 otherwise. which can then be made into a unique coalgebra map T cπ : T c[N ] → T c[M ] with the usual formula T cπ(x1 ⊗ · · · ⊗ xk) = x1 ⊗ · · · ⊗ π(xi)⊗ · · · ⊗ xk. Next, the morphisms T cπ and T c∇ pass to the corresponding morphisms on the coalgebras Sc[N ] and Sc[M ] respectively, and Sch is obtained from T ch by symmetrization, to yield a contraction (Sc[M ] Sc[N ], Sch). In particular, the contraction (9) induces (Sc[sH(g)] Sc[sg], Sch), (10) which is a filtered contraction of coaugmented DG coalgebras. (Warning: Scπ and Sc∇ are morphisms of coalgebras but Sch is not a coalgebra morphism, although it is somewhat compatible with the coalgebra structure, being a ho- motopy of coalgebra maps. One has to be careful when defining a homotopy of cocommutative coalgebras.) ⋄ 5.2 The first main theorem. Assume that ∂ is the codifferential corresponding to an sh-Lie algebra struc- ture on (g, d). Since the corresponding multilinear map on g has other compo- nents than the binary bracket, we will denote the symmetric coalgebra on sg with codifferential ∂ by Sc∂ [sg] and not by S [ , ][sg]. Given two sh-Lie algebras (g1, ∂1) and (g1, ∂2), an sh-morphism or sh-Lie map from g1 to g2 is a morphism Sc∂1 [sg1] → S [sg2] of DG coalgebras. Theorem 1. Given a DGLA g and a contraction of chain complexes such as (9), the data determine (i) a differential D on Sc[sH(g)] (a coalgebra perturbation of the zero differen- tial) turning the latter into a coaugmented DG coalgebra, hence endowing H(g) with an sh-Lie algebra structure, (ii) a Lie algebra twisting cochain τ : ScD[sH(g)] → g with adjoint τ̄ , written τ̄ = (Sc∇)∂ : S D[sH(g)] → C[g], that induces an isomorphism on the homology, and (iii) an extension of (Sc∇)∂ to a new contraction (ScD[sH(g)] (Scπ)∂ (Sc∇)∂ Sc∂ [sg], (S cd)∂) of filtered chain complexes (not necessarily of coalgebras). Notes on Notation. While the induced bracket on H(g) is a strict graded Lie bracket, the differential D may involve meaningful terms of higher order. Let us introduce a table for the notation used in [8] for different types of chain complexes and the corresponding symmetric coalgebras. Chain complex Bracket(s) Sym. coalgebra Coderivation Property (g, d) [ , ] (Sc[sg], d) ∂ graded generic DG coalgebra; coderivation chain complex bilinear bracket induced diff. d induced by [ , ] (g, d) [ , ] Sc[ , ][sg], d = C[g] ∂ (d+ ∂) DGLA Lie bracket on g; generalized Koszul or coderivation d derivation of it Chevalley-Eilenberg induced by [ , ] complex (g, 0) [ , ] (Sc[sg], 0) = Λ•(g) ∂ ∂2 = 0 Lie algebra Lie bracket Koszul complex codifferential for homology induced by [ , ] (g, d) ℓ2, ℓ3, . . . (S ∂ [sg], d) ∂ (d+ ∂) 2 = 0 L∞ algebra; higher brackets codiff. induced by d = ℓ1 ℓ2, ℓ3, . . . (H(g), 0) [ , ] (ScD[sH(g)], 0) D D 2 = 0 homology of induced Lie codiff. defined DGLA (g, d) bracket on H(g) in the proof with given contraction Sketch of Proof. We obtain the differential D and the twisting cochain τ on Sc[sH(g)] as infinite series by induction: for b ≥ 1, write Scb for the homogeneous degree-b component of Sc[sH(g)]. Then D, τ for b ≥ 2 are given by τ = τ1 + τ2 + · · · , τ1 = ∇τH(g), τ j : Scj → g, j ≥ 1, (11) h([τ1, τb−1] + · · ·+ [τb−1, τ1]) D = D1 +D2 + · · · (12) where Db−1 is the coderivation of Sc[sH(g)] determined by τH(g)D b−1 = π([τ1, τb−1] + · · ·+ [τb−1, τ1]) : Scb → H(g). That is, the coderivation followed by projection onto the degree-one subspace sH(g) of Sc[sH(g)] is given by the above formula. In the notation of Subsec- tion 2.3, we have fb = τH(g)D b−1 and D = f̂ . For example (dropping the symbol s for elements of sH(g)), we have τ1(x) = x ∈ H(g) ⊂ g, and τ2(xy) = h[x, y] by (5). Let us also compute two terms of D: τH(g)D 1(xy) = π[x, y] ∈ H(g), τH(g)D 2(xyz) = π( [x, h[y, z]] + [h[x, y], z] ), etc. We can see why τ is a LA twisting cochain: since τ satisfies τ = h [τ, τ ] we obtain −dτ = [τ, τ ] in case of the particular SDR we constructed, and the last equation is the mas- ter equation (the differential on H(g) being zero). The sums (11) and (12) are infinite, but when either one is applied to a specific element in some subspace of finite filtration degree, only finitely many terms will be nonzero. (The sum- mand D1 is the ordinary Cartan-Chevalley-Eilenberg differential for the classi- fying coalgebra of the graded Lie algebra H(g).) The proof that D is indeed a coalgebra differential and τ is a twisting cochain “will be given elsewhere”. A “spectral sequence argument” shows that τ̄ induces an isomorphism on the homology.� Remark 8. If ∇H(g) happens to be a Lie subalgebra of g, then [τ1, τ1] will have values in H(g) and τ2 = (1/2)h[τ1, τ1], as well as the remaining τ j , will be zero. Similarly, we will have D = D1. Corollary 3. Under the hypotheses of Theorem 1, τ : ScD[sH(g)] → g, (13) viewed as an element of degree −1 of the DGLA Hom(ScD[sH(g)], g), satisfies the master equation (2). The twisting cochain (13) is our most general solution of the master equation. The other solutions of the master equation can be derived from it. 6 Corollaries and the second main theorem 6.1 Other corollaries of Theorem 1. Corollary 4. Under the hypotheses of Theorem 1, suppose in addition that there is a differential D̃ on Sc[sH(g)] turning the latter into a coaugmented DG coalgebra in such a way that (Scπ)∂ = D̃(Scπ). Then D = D̃ and (Scπ)∂ may be taken to be Scπ. In particular, when (Scπ)∂ is zero, then the differential D on Sc[sH(g)] is necessarily zero, that is, the new contraction in Theorem 1 has the form (Sc[sH(g)] (Sc∇)∂ Sc∂ [sg], (S ch)∂). For example, this is the case when the composite g ⊗ g [ , ] → H(g) is zero. Corollary 5. Under the hypotheses of Theorem 1, suppose in addition that there is a differential D̃ on Sc[sH(g)] turning the latter into a coaugmented DG coalgebra in such a way that ∂(Sc∇) = (Sc∇)D̃. Then D = D̃ and (Sc∇)∂ = S∇. In particular, when ∂(Sc∇) is zero, then the differential D on Sc[sH(g)] is necessarily zero, that is, the new contraction in Theorem 1 has the form (Sc[sH(g)] (Scπ)∂ Sc∂ [sg], (S ch)∂). For example, this is the case when the composite H(g)⊗H(g) → → g ⊗ g [ , ] is zero. 6.2 The second main theorem Theorem 2. Given a DGLA g, a DGL subalgebra m of g, and a contraction (H(g) of chain complexes so that the composite [ , ] → H(g) is zero, then the induced bracket on H(g) is zero, that is, H(g) is abelian as a graded Lie algebra, and the data determine a solution τ ∈ Hom(Sc[sH(g)], g) of the master equation (2) in such a way that the following hold: (i) The composite πτ coincides with the universal twisting cochain Sc[sH(g)] → H(g) for the abelian Lie algebra H(g), and (ii) the values of τ lie in m. 7 Differential Gerstenhaber and BV algebras 7.1 Differential Gerstenhaber algebras Definition 8. A Gerstenhaber (or G-) algebra consists of • A graded commutative and associative algebra (A, µ) (µ suppressed), and • A graded Lie bracket (the Gerstenhaber or G− bracket) [ , ] : A⊗ A → A of degree −1, such that • For each homogeneous element a ∈ A, bracketing with a is a derivation of the Lie bracket of degree |a| − 1. That is, we want ad(a) to commute with the bracket for all a ∈ A. Definition 9. A differential G-algebra is a Gerstenhaber algebra (A, [ , ]) with a differential d of degree +1 on A which is a derivation of the multiplication on We want [d, µ] = 0 and [d, d] = 0 in Gerstenhaber’s composition bracket notation. Definition 10. A differential G-algebra is called strict if the differential d is a derivation of the G-bracket as well. We want d to commute with the bracket. Let (A, [ , ], d) be a strict differential G-algebra. We will for the moment ignore all the extra structure on A except for the G-bracket and the differential. As such, A is a DGLA, and we will change the notation to g to emphasize that. We will use the grading g1 = A 0, g0 = A 1, g−1 = A 2, . . . , g−n = A n+1, . . . so that the graded bracket and the differential on g are now “ordinary”: namely, [ , ] : gj ⊗ gk → gj+k, d : gj → gj−1. Consider a contraction of g onto H(g) as in (9). Let ∂ denote the operator (“perturbation of d”) on Sc∂ [sg] corresponding to the Lie bracket on g. By the Main Theorem (Theorem 1), we can transfer it to the symmetric coalgebra of the homology: there exists a new contraction (ScD[sH(g)] (Scπ)∂ (Sc∇)∂ Sc∂ [sg], (S cd)∂) of not only filtered chain complexes but of filtered differential graded coalgebras. The twisting cochain τ of (13) of Corollary 1 is now an element of Hom(ScD[sH(g)], s of degree −2, satisfying the master equation [τ, τ ], where D is the Hom-differential and the graded cup bracket on the right-hand side refers to the one induced by the graded coalgebra structure on Sc[sH(g)] and the graded Lie algebra structure on g. 7.2 Differential BV algebras Definition 11. Let (A, [ , ]) be a G-algebra with an additional operator ∆ on A of degree −1. If ∆ satisfies the condition [a, b] = (−1)|a| ∆(ab)− (∆a)b − (−1)|a|a(∆b) then it is said to be a generator of the G-algebra. In this case, (A,∆) is called a weak Batalin-Vilkovisky (BV-) algebra. If, moreover, ∆ is exact (i.e. ∆2 = 0), then (A,∆) is simply called a Batalin-Vilkovisky (BV-) algebra. Koszul has shown that ∆ behaves as a derivation for the G-bracket: ∆[x, y] = [∆x, y]− (−1)|x|[x,∆y] ∀x, y ∈ A. With respect to the original graded commutative and associative product on A, we can only say that ∆ is a second order differential operator, or Φ3∆(a, b) = 0, where Φr∆ are r-linear operators used to define higher order differential opera- tors. Now let us denote the bracket on a (weak) BV-algebra (A,∆) by [ , ]. Definition 12. If d is a differential of degree +1 that endows (A, [ , ]) with a differential G-algebra structure, such that ∆d+d∆ = 0, then the triple (A,∆, d) is called a (weak) differential BV-algebra. Proposition 7. For any weak differential BV-algebra (A,∆, d), the differential d behaves as a derivation of the G-bracket [ , ]: d[x, y] = [dx, y]− (−1)|x|[x, dy]. That is, (A, [ , ], d) is a differential G-algebra. Note: (A, [ , ],∆) is not a differential G-algebra unless ∆ is exact. Example 8. [7] Let V be a Z2-graded finite dimensional vector space and sV be its suspended-graded dual. If {xi} is a basis for V consisting of homogeneous elements, then the dual basis {x∗i } has the property that xi and x i always have opposite parities. Then the algebra C[[x1, . . . , xn, x 1, . . . , x n]] of formal power series has the following BV operator (Laplacian): Since the underlying algebra is graded commutative, the composition of two derivations is a second order differential operator by any definition. Moreover ∆2 = 0, which makes a BV-algebra out of this data. ♦ 7.3 Formality 7.3.1 Formality of differential graded P -algebras Recall that our ground ring is a field of characteristic zero. Let P be a differential graded operad and (A, d) be an algebra over P . We often want to know to which extent the cohomology of a space reflects the underlying topological or geometrical properties of that space. Definition 13. The P -algebra A is called formal if there exists a strongly homotopy P -algebra map (H(A), 0) → (A, d) which induces an isomorphism in homology. 7.3.2 Examples Example 9. (Commutative DG associative algebras.) A smooth mani- fold M is called formal if the commutative associative DG algebra of de Rham forms onM is formal in the sense of the above Definition. Examples are compact Kähler manifolds, Lie groups, and complete intersections. Poisson manifolds (proof by Sharygin and Talalaev). ⋄ Example 10. (DGLA’s.) The Hochschild complex for the algebra A = C∞(M) of smooth functions on a Poisson manifold M (Kontsevich). ⋄ 7.4 Differential BV algebras and formality Definition 14. We will say a differential BV-algebra (A,∆, d) (A=g as a Lie algebra) satisfies the statement of the Kählerian Formality Lemma (or the ∂∂̄ Lemma) if the maps Ker(∆), d |Ker(∆) →֒ (g, d), Ker(∆), d |Ker(∆) −→ H(g,∆) are isomorphisms on the homology, where H(g,∆) is endowed with the zero differential. Remark 9. If the statement of the K.F.L. is satisfied, then proj can be extended to a contraction (H(g, d), 0) Ker(∆), d |Ker(∆) Since we have H (Ker(∆), d) ∼= H(g, d) H (Ker(∆), d) ∼= H(g,∆), now we can have a contraction of Ker(∆) onto H(Ker(∆)) = H(g) as we did with g and H(g). Theorem 3. Let (A,∆, d) be a differential BV-algebra satisfying the statement of the Kählerian Formality Lemma and extend the projection proj to a contrac- (H(g, d), 0) where Ker(∆), d |Ker(∆) and π =proj. Then H(g) is abelian as a graded Lie algebra and the data deter- mine a solution τ ∈ Hom(Sc[sH(g)], g) of the master equation dτ = [τ, τ ] in such a way that the following hold: • The values of τ lie in m, that is, the composite ∆ ◦ τ : Sc[sH(g)] → g is zero; • The composite πτ coincides with the universal twisting cochain for the abelian graded Lie algebra H(g); so that • For k ≥ 2, the values of the component τk of τ on S k[sH(g)] lie in Im(∆). Proof. Follows from Theorem 2. � Let us add the following condition to the ones in Theorem 3: suppose that A consists of a single copy of the ground ring F (necessarily generated by the unit 1 of A) and that ∆(1) = 0. Then 1 generates a central copy of the ground ring in g (the ground ring commutes with all elements of A), and we may write g = F ⊕ g̃ as a direct sum of differential graded Lie algebras. Here g̃ is the uniquely determined complement of F . Why unique? We have g1 = A0 = F and we may g−n = An+1, where g̃ will be closed under the (degree-zero) Lie bracket: [ , ] : gj ⊗ gk → gj+k, with j + k ≤ 0 if j, k ≤ 0. Corollary 6. Assume that the hypotheses of Theorem 3, the abovementioned conditions (A0 = F , ∆(1) = 0), and the condition H1(g) 6= 0 hold. Then the contraction of the Theorem can be chosen in such a way that Im(τk) ⊂ g̃ for k ≥ 2. The statements of the main theorems have an interpretation in the context of deformation theory, as explained below. 8 Deformation theory Given a DGLA g, the construction of the universal solution τ from Theorems 2 and 3 relies on a chosen contraction. This provides a formal solution of the master equation (MCE), with a perturbed differential D on Sc[sHg] in the direction of (starting with) the Lie bracket induced on homology, endowing the former with a dg-coalgebra structure and a twisting cochain: τ : ScD[sHg] → g. The moduli space interpretation of the set of solutions is along the lines of Schlesinger-Stasheff [15]. Since our focus is on the construction of solutions of the MCE, the reader is referred to the original text [8]. Additional details in terms of deformation functors, tangent cohomology, and the Kuranishi functor can be found in [13]. The relation between the latter functor and the construc- tion of a twisting cochain corresponding to a contraction will be investigated elsewhere. References [1] F. Akman and L.M. Ionescu, Higher derived brackets and deformation the- ory I; arXiv:math.QA/0504541. [2] F. Akman and L.M. Ionescu, Higher derived brackets and deformation the- ory II, in preparation. [3] F. Akman, L.M. Ionescu, and P. Sissokho, On deformation theory and graph homology, J. Algebra 310 (2007), 730-741; arXiv:math.QA/0507077. [4] S. Barannikov and M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices 4 (1998), 201-215; arXiv:alg-geom/9710032. [5] P. Chen, On the formality theorem for the DGLA of Drinfeld; arXiv:math.QA/0601055. [6] V. Dolgushev, A formality theorem for Hochschild chains; arXiv:math.QA/ 0402248. [7] D. Fiorenza, An introduction to the Batalin-Vilkovisky formalism; arXiv: math.QA/0402057. [8] J. Huebschmann and J. Stasheff, Formal solution of the master equa- tion via HPT and deformation theory, Forum Math. 14 (2002), 847-868; arXiv:math.AG/9906036. [9] L.M. Ionescu, A combinatorial approach to coefficients in deformation quantization; arXiv:math.QA/0404389. [10] M. Kontsevich, Deformation quantization of Poisson manifolds I; arXiv:q-alg/9709040. [11] Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys. 69 (2004), 61-87. [12] L. Lambe and J.D. Stasheff, Applications of perturbation theory to iterated fibrations, Manuscripta Math. 58 (1987), 363-376. [13] M. Manetti, Deformation theory via differential graded Lie algebras; arXiv:math.AG/0507284. [14] P. Real, Homological perturbation theory and associativity, Homology, Ho- motopy, and Applications 2 (2000), 51-88. [15] M. Schlessinger and J. Stasheff, Deformation theory and rational homotopy type, Pub. Math. Sci. IHES (1998). [16] T. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005), 133-153. http://arxiv.org/abs/math/0504541 http://arxiv.org/abs/math/0507077 http://arxiv.org/abs/alg-geom/9710032 http://arxiv.org/abs/math/0601055 http://arxiv.org/abs/math/0402248 http://arxiv.org/abs/math/0402057 http://arxiv.org/abs/math/9906036 http://arxiv.org/abs/math/0404389 http://arxiv.org/abs/q-alg/9709040 http://arxiv.org/abs/math/0507284 Introduction Perturbations of (co)differentials Derivations of the tensor algebra Coderivations of the tensor coalgebra Coderivations of the symmetric coalgebra DGLA's and perturbations of the codifferential Strongly homotopy Lie algebras The Hochschild chain complex and DGA's Strongly homotopy associative algebras Master equation Twisting cochain Differential on Hom Cup product and cup bracket Twisting cochain Homological perturbation theory (HPT) Contraction The first main theorem. Corollaries and the second main theorem Other corollaries of Theorem ??. The second main theorem Differential Gerstenhaber and BV algebras Differential Gerstenhaber algebras Differential BV algebras Formality Formality of differential graded P-algebras Examples Differential BV algebras and formality Deformation theory

0704.0433

A variational formulation of electrodynamics with external sources

arXiv:0704.0433v3 [math-ph] 13 Oct 2008 A variational formulation of electrodynamics with external sources Antonio De Nicola CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal [email protected] W lodzimierz M. Tulczyjew Associated with Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Italy Valle San Benedetto, 2 – 62030 Montecavallo (MC), Italy [email protected] Abstract We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with external sources and permits the derivation of the constitutive relations. We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of different types of boundary problems in an unified way. In particular we obtain a smooth transition to the infinitesimal version by using a current with a one point support. 2000 Mathematics Subject Classification: 49S05, 70S05. Keywords: variational principles, electrodynamics. Introduction A general framework for variational formulations of physical theories was presented in [1]. Appli- cations to statics and dynamics of mechanical systems appear in [2, 3]. In this paper we present a variational formulation of electrodynamics based on that framework. Our work is related to the gen- eral formulation of linear field theories in a symplectic framework contained in [4] and to the earlier formulations of electrodynamics contained in [5, 6]. Some of the results contained in this paper were announced in [7]. We are presenting a variational formulation of electrodynamics in an intrinsic, frame independent fashion in the affine Minkowski space-time using de Rham odd and even differential forms ([8, 9, 10]) which permit the rigorous formulations of electrodynamics and the description of the transformation properties of electromagnetic fields relative to reflections (see [5, 6]). Observable quantities like the charge contained in a compact volume or the flux of the electromagnetic field through a surface require the integration of differential forms. The use of odd quantities is not just a matter of elegance. Even if the space-time is assumed to be orientable, the alternative approach of using standard differential forms and the Hodge star operator require the choice of a specific orientation, i. e. the addition of an extrinsic structure. Other recent treatments of classical electrodynamics using odd and even differential forms can be found in [11, 12, 13]. Our construction is an example of application of the general variational framework [1]. Similar constructions are needed for the variational formulation of a general field theory. The linearity of classical electrodynamics and the choice of formulating it on the affine Minkowski space makes our presentation simpler. Relying on a variational principle more complete than the Hamilton principle our formulation leads to field equations with external sources. This variational principle also permits http://arxiv.org/abs/0704.0433v3 the derivation of the constitutive relations which are usually postulated separately since the variations normally considered are not general enough to derive them from the variational principle. We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of different types of boundary problems in an unified way. As an example we obtain a smooth transition to the infinitesimal version by using a current with a one point support. De Rham currents are essentially objects dual to differential forms. The present paper is organized in the following way. In the first part we provide the geometric structures needed for the rigorous formulation of electrodynamics. Part of this material is based on [5] and is briefly reported here for the sake of completeness. We recall the Cartan calculus for odd and even differential forms and their integration theory over odd and even de Rham currents. The second part contains the main results. We start with the construction of a suitable space of fields for electrodynamics (not a differential manifold) and a construction of tangent and cotangent vectors. A convenient representation of these objects is introduced. The definition of the space of fields is inspired by a similar construction suited for the statics of continuous media which is contained in the final section of [1]. In Section 3 we formulate a variational principle for electrodynamics similar to the virtual action principle of analytical mechanics with external forces and boundary terms and derive the field equations which include the constitutive relations in addition to Maxwell’s equations. The boundary problem in a finite domain is treated in Section 4. Section 5 contains the Lagrangian formulation of electrodynamics. The Legendre transformation and the Hamiltonian formulation of electrodynamics in Section 6 conclude the paper. A. Preliminaries Here we provide the geometric structures needed for the rigorous formulation of electrodynamics in Part B. The material in Sections 1, 2, and 4 is based on [5] and is briefly recalled here for the sake of completeness. Nevertheless, we add a more explicit presentation of some useful details. 1.Orientations of vector spaces and vector subspaces. Let V be a vector space of dimension m 6= 0. We denote by F(V ) the space of linear iso- morphisms from V to Rm called frames. It is known that F(V ) is a homogeneous space with re- spect to the natural group action of the general linear group GL(m,R) in F(V ). Let GL+(m,R) and GL−(m,R) be the two connected components of the group GL(m,R). The set of orientations O(V ) = F(V ) GL+(m,R) has two elements. This set is a homogeneous space for the quotient group H(m,R) = GL(m,R) GL+(m,R). The sets E = GL+(m,R) and P = GL−(m,R) are the elements of the quotient group H(m,R) which is the group of permutations of the two elements of O(V ). There is an ordered base (e1, e2, . . . , em) of V associated with each frame ξ in a obvious way. Let W ⊂ V be a subspace of a vector space V . The subspace has the set O(W ) of orientations called inner orientations of W . Orientations of the quotient space V W are called outer orientations of W . An outer orientation o′′ of W can be determined by specifying an inner orientation o of W together with an orientation o′ of V . Let (e1, . . . , en) be the base of W associated with a frame ξ ∈ o. This base can be completed to a base (e′1, . . . , e m) of V with (e 1, . . . , e n) = (e1, . . . , en). The extended base can be chosen to be associated with a frame ξ′ ∈ o′. Let π:V → V W be the canonical projection. The sequence (e′′1 , . . . , e m−n) = (π(e n+1), . . . , π(e m)) (1) is a base of V W . It determines an orientation o′′ of V W . Hence an outer orientation of W . The outer orientation o′′ of W constructed from o ∈ O(W ) and o′ ∈ O(V ) is the same as the orientation constructed from Po and Po′. We have introduced inner orientation of subspaces of dimension different from zero and outer orien- tation of subspaces of codimension different from zero. Integration theory of differential forms requires the possibility of assigning inner orientations to the subspace W = {0} ⊂ V and outer orientations to the subspace W = V . Two possible orientations are assigned to the subspace W = {0} ⊂ V one of which is distinguished. The distinguished orientation is denoted by (+) and the other orientation is denoted by (−). In agreement with the conventions established for orientation the outer orientations of the subspace W = {0} ⊂ V are the orientations of V . An outer orientation of W = {0} can be specified in terms of an inner orientation and an orientation of V . If the inner orientation is (+) and the orientation of V is o, then o is the outer orientation of W . The orientation Po is the outer orientation derived from (−) and o. The subspace W = V has a distinguished outer orientation defined as an orientation of V 2.Multicovectors and multivectors. A q - covector in a vector space V is a mapping a :×qV ×O(V ) → R. This mapping is q - linear and totally antisymmetric in its vector arguments. A q - covector a is said to be even, if a(v1, v2, . . . , vq, Po) = a(v1, v2, . . . , vq, o). (2) It is said to be odd, if a(v1, v2, . . . , vq, Po) = −a(v1, v2, . . . , vq, o). (3) The vector space of even q - covectors will be denoted by ∧qeV ∗ and the space of odd q - covectors will be denoted by ∧qoV ∗. We will use the symbol ∧qpV ∗ to denote either of the spaces in constructions valid for both parities. The index p with the two possible values e and o will be used on other occasions. For the definition of exterior product of (even and odd) multicovectors we refer to [5]. Here we just recall that if two multicovectors a and a′ are even or both are odd, the product a∧a′ is even. In other cases the product is odd. The exterior product is commutative in the graded sense and associative. Let {eκ}κ=1,... ,m be a base of V and let {e κ}κ=1,... ,m be the dual base. Each element e κ defines an even covector eκe :V × O(V ) → R: (v, o) 7→ 〈e κ, v〉. (4) We choose an orientation o of V and introduce odd 0-covector eo defined by eo(o) = −eo(Po) = 1 and the even covector ee defined by ee(o) = ee(Po) = 1. We come now to multivectors. We denote by K(×qV × O(V )) the vector space of formal linear combinations of sequences (v1, v2, . . . , vq, o) ∈ × qV ×O(V ). In the space K(×qV ×O(V )) we introduce subspaces A pq (V ) = i=1 λi(v 2, . . . , v i) ∈ K(×qV × O(V )); i=1 λia(v 2, . . . , v i) = 0 for each a ∈ ∧qpV . (5) Subsequently we define quotient spaces ∧qpV = K(× qV × O(V )) A pq (V ). Elements of spaces ∧ eV and ∧qoV are called even q - vectors and odd q - vectors respectively. A multivector is said to be simple if it is represented by a single element of the space V q × O(V ) interpreted as a subspace of K(V q × O(V )). Evaluation of q - covectors on sequences (v1, v2, . . . , vq, o) ∈ × qV × O(V ) extends to linear combinations and their equivalence classes. If w is a q - vector represented by the linear combination∑n i=1 λi(v 2, . . . , v i) and a is a q - covector of the same parity as w, then 〈a, w〉 = i=1 λia(v 2, . . . , v i) (6) is the evaluation of a on w. We have constructed pairings 〈 , 〉:∧qpV ∗×∧qpV → R. The exterior product of multivectors can be easily defined using representatives. The parity of the exterior product is odd if the parity of one of the factors is odd. It is even otherwise. The exterior product is commutative in the graded sense and associative. The left interior multiplications are the operations :∧qpV × ∧ V ∗ → ∧ V ∗, defined for q 6 q′ by 〈w a,w′〉 = 〈a, w ∧w′〉. The parity pp′ which appears in this definition is constructed by assigning the numerical values +1 and −1 to e and p respectively. The parity of the multivector w′ must match the parity of the multicovector w a. The right interior multiplications are the operations :∧qpV×∧ ∗ → ∧ pp′ V, defined for q > q ′ by 〈a′, w a〉 = 〈a′∧a, w〉. The parity of the multicovector a′ in this definition must match the parity of the multivector w a. 3.The Weyl isomorphism and a useful formula. The space ∧mo V ∗ is one-dimensional. This makes it possible to define the tensor product ∧qeV ⊗∧ as the set of equivalence classes of pairs (w, e) ∈ ∧qeV ×∧ ∗. Pairs (w, e) and (w′, e′) are equivalent if there is a number λ such that w′ = λw and e = λe′ or w = λw′ and e′ = λe. The equivalence class of a pair (w, e) will be denoted by w ⊗ e. A tensor a ∈ ∧qeV ⊗ ∧ ∗ will always be presented as a product w ⊗ e. The set ∧qeV ⊗ ∧ ∗ is a vector space (see [5]). Proposition 1 ([5]). The linear mapping Weq:∧ eV ⊗ ∧ ∗ → ∧m−qo V ∗:w ⊗ e 7→ w e (7) is an isomorphism. The mapping Weq is called the Weyl isomorphism. We will show now that the values of any bilinear mapping b:∧qeV ∗ × ∧q ∗ → ∧mo V ∗ can be expressed using the exterior product. The following three technical lemmas will be used. Lemma 1. Let w ∈ ∧qeV . Then w (eo ∧ e e ∧ . . . ∧ e e ) = ν1<...<νq (νi−i)〈eν1e ∧ . . . ∧ e e , w〉eo ∧ e e ∧ . . . ∧ e e (8) where νq+1 < . . . < νm and (νq+1, . . . , νm) denotes the complementary (m− q)-tuple of (ν1, . . . , νq). Proof: It is enough to prove the claim for a simple q-vector. If w = w1 ∧ . . . ∧ wq, then (w1 ∧ . . . ∧ wq) (eo ∧ e e ∧ . . . ∧ e (−1)ν1−1〈eν1e , w1〉wq . . . eo ∧ e e ∧ . . . ∧ ê e ∧ . . . ∧ e = . . . = ν1 6=...6=νq (νi−i) sgn(ν1, . . . , νq)〈e e , w1〉 · · · 〈e e , wq〉 eo ∧ e 1 ∧ . . . ∧ êν1e ∧ . . . ∧ ê e ∧ . . . ∧ e ν1<...<νq σ∈S(q) (νi−i) sgnσ〈e νσ(1) e , w1〉 · · · 〈e νσ(q) e , wq〉 eo ∧ e 1 ∧ . . . ∧ êν1e ∧ . . . ∧ ê e ∧ . . . ∧ e ν1<...<νq σ∈S(q) (νi−i) sgnσ〈e νσ(1) e , w1〉 · · · 〈e νσ(q) e , wq〉eo ∧ e νq+1 ∧ . . . ∧ eνme ν1<...<νq (νi−i) det(〈eνre , ws〉)eo ∧ e νq+1 ∧ . . . ∧ eνme . (9) In the second line of this sequence of equalities we used the well known identity v (a1 ∧ . . . ∧ aq) = (−1)ν−1〈aν , v〉a1 ∧ . . . ∧ âν ∧ . . . ∧ aq (10) where v is a vector and a1, . . . , aq are covectors. In the third line we applied repeatedly this identity and we noted that when ν1 < . . . < νq each missing covector e e in the expression eo∧e 1∧. . .∧êν1e ∧. . .∧ e ∧ . . .∧ e e occupied the (νi− i+ 1)-th place in the exterior product, otherwise if νi−l−1 < νi < νi−l, then it occupied the (νi − i − l + 1)-th place, hence we get the factor (−1) (νi−i) sgn(ν1, . . . , νq) where the symbol sgn(ν1, . . . , νq) denotes the sign of the permutation (ν1, . . . , νq). Lemma 2. If a ∈ ∧qeV ∗ and w ⊗ e ∈ ∧qeV ⊗ ∧ ∗, then 〈a, w〉e = a ∧ Weq(w ⊗ e). (11) Proof: It is enough to prove the claim for e = eo ∧ e e ∧ . . . ∧ e e . By the Lemma 1 we get a ∧ Weq(w ⊗ e) = a ∧ (w (eo ∧ e e ∧ . . . ∧ e ν1<...<νq (νi−i)〈w, eν1e ∧ . . . ∧ e e 〉a ∧ eo ∧ e e ∧ . . . ∧ e ν1<...<νq (νi−i)w ν1...νq µ1<...<µq aµ1...µqee ∧ e e ∧ . . . ∧ e e ∧ eo ∧ e e ∧ . . . ∧ e e . (12) The right-hand side of (12) reduces to ν1<...<νq (νi−i)wν1...νqaν1...νqeo ∧ e e ∧ . . . ∧ e e , (13) since νq+1 < . . . < νm and (νq+1, . . . , νm) is the complementary (m− q)-tuple of (ν1, . . . , νq). Finally we note that moving each νi (for i = 1, . . . , q) to the νi-th place in (12) requires νi − i transpositions, since ν1 < . . . < νq. Then each of the remaining νi, i.e. those with i = q + 1, . . . ,m, will necessarily be at the νi-th place. Hence we obtain a ∧ Weq(w ⊗ e) = ν1<...<νq aν1...νqw ν1...νqeo ∧ e e ∧ . . . ∧ e e . (14) We denote by Hom(∧mo V ∗| ∧qe V ∗) the space of linear mappings from ∧qeV ∗ to ∧mo V ∗ and by iq: Hom(∧ ∗| ∧qe V ∗) → ∧qeV ⊗ ∧ ∗ (15) the isomorphism characterized by 〈iq(l), a ′ ⊗ u〉 = 〈l(a′), u〉 (16) for each l ∈ Hom(∧mo V ∗| ∧qe V ∗), a′ ∈ ∧qeV ∗ and u ∈ ∧mo V . The pairing 〈 , 〉: (∧qeV ⊗ ∧ ∗) × (∧qeV ∗ ⊗ ∧mo V ) → R: (w ⊗ e, a⊗ u) 7→ 〈a, w〉〈e, u〉 (17) is used. Lemma 3. If l ∈ Hom(∧mo V ∗| ∧qe V ∗), then l(a) = a ∧ Weq(iq(l)), (18) for each a ∈ ∧qeV Proof: Let iq(l) = wl ⊗ e with wl ∈ ∧ eV . Using Lemma 2 we obtain a ∧ Weq(i(l)) = a ∧ Weq(wl ⊗ e) = 〈a, wl〉e. (19) We will show now that l(a) = 〈a, wl〉e. Indeed if we denote by u ∈ ∧ o V the dual basis of e then we have l(a) = 〈l(a), u〉e and 〈l(a), u〉 = 〈i(l), a⊗ u〉 = 〈wl ⊗ e, a⊗ u〉 = 〈a, wl〉〈e, u〉 = 〈a, wl〉. (20) We can now finally show that a useful expression can be obtained for the values of any bilinear mapping b:∧qeV ∗ × ∧q ∗ → ∧mo V ∗. (21) We associate with b the linear mappings b:∧qeV ∗ → ∧q e V ⊗ ∧ ∗: a 7→ iq′(b(a, ·)) (22) ∗ → ∧qeV ⊗ ∧ ∗: a′ 7→ iq(b(·, a ′)). (23) Proposition 2. If b as in (21) is a bilinear mapping and b, b are the associated linear mappings (22) and (23), then b(a, a′) = a′ ∧ Weq′(b(a)) = a ∧ Weq(b(a ′)). (24) for each (a, a′) ∈ ∧qeV ∗ × ∧q Proof: Applying Lemma 3 to l = b(a, ·) we get b(a, a′) = b(a, ·)(a′) = a′ ∧ Weq′(iq′(b(a, ·))) = a ′ ∧ Weq′(b(a)). (25) The other equality is obtained in the same way by applying Lemma 3 to l = b(·, a′). 4. Integration of differential forms. Chains and currents. Let M be an affine space modelled on a vector space V . A differential q-form on M is a differentiable function A:M ××qV × O(V ) → R depending on a point, q vectors and an orientation. It is q-linear and totally antisymmetric in its vector arguments. A differential form A is said to be even or odd if for each point in x ∈ M it defines a even or odd multicovector, respectively. We note that a zero-form on M is a differentiable function f :M ×O(V ) → R. The vector space of even differential q-forms will be denoted by Φqe(M) and space of odd differential q-forms will be denoted by Φ o(M). The symbol Φqp(M) will be used to denote either of the two spaces when the distinction is of no importance. For the definition of the exterior product of two differential forms and for that of exterior differential of a differential form, we refer the reader to [5]. Here we just recall that if both forms A and A′ are even or both are odd, the product A ∧ A′ is even. In other cases the product is odd. The parity of the differential dA of a q-form A is the same as the parity of the original form A. A q-form A can be interpreted as a q-covector field Ã:M → ∧qpV ∗. The exterior product and the exterior differential are extended to this alternative interpretation of forms. The left and right interior multiplications of even and odd multivector fields with even and odd multicovector fields are defined point by point in an obvious manner. A cell of dimension q or a q - cell in M is a pair (χ, o), where χ is a differentiable mapping χ:Rq → M and o is an orientation of V . For q = 0, R0 is the vector space {0} with a single element 0. Hence a zero-cell in M is a pair of a point x ∈ M and an orientation of V . The integral of a q - form A on a cell (χ, o) is the Riemann integral (χ,o) · · · A (χ(s1, . . . , sq),D1χ(s1, . . . , sq), . . . ,Dqχ(s1, . . . , sq), o) ds1 · · · dsq. (26) The integral of a 0 - form f on a zero-cell (x, o) is the value (x,o) f = f(x, o). For each q we introduce the space K(Xq(M)) of formal linear combinations of q - cells. The formal linear combinations turn into real linear combinations if cells are identified with elements of K(Xq(M)). Integration of forms is extended to linear combinations by linearity. Subspaces N pq (M) ⊂ K(Xq(M)) are defined as the sets q (M) = C ∈ K(Xq(M)); A = 0 for each A ∈ Φqp(M) . (27) Elements of the quotient spaces Ξpq(M) = K(Xq(M)) N pq (M) are called even chains or odd chains of dimension q. We extend the sequence of even and odd chains to negative dimension q by defining the spaces Ξpq(M) = {0} for each q < 0. A chain is said to be simple if it has a single cell as one of its representatives. Integrals of q - forms on q - chains are well defined. The integral of a q - form A on the class C of C ∈ K(Xq(M)) is the integral of A on C. The boundary operator ∂ assigns to a chain C ∈ Ξ pq (M) its boundary ∂C ∈ Ξ q−1(M). The boundary of a simple chain represented by a q - cell (χ, o) is the chain represented by the combination i=1(−1) i−1((χ(i,1), o) − (χ(i,0), o)), (28) where the (q−1) - cells (χ(i,1), o) and (χ(i,0), o) are defined by (i,∗):Rq−1 → M : (s1, . . . , ŝi, . . . , sq) 7→ χ(s1, . . . , si−1, ∗, si+1 . . . , sq) (29) with ∗ replaced by 1 and 0, respectively. The cells introduced in (29) represent the faces of the simple chain. The construction of the boundary is extended to generic chains by linearity. The boundary of a boundary is the zero chain. It is known that Stokes theorem holds for chains and forms of the same parity. An even or odd de Rham current of dimension q on a manifold M is a linear function c: Φqp(M) → R. We will use the symbol A to denote the value c(A). The spaces of forms are given certain topologies and the continuity of the function is required. Chains will be treated as currents. They form a dense subspace in the space of currents. We will consider only very simple examples of currents other than chains. Topological considerations are of little importance for these examples. The boundary of a current is defined by assuming that Stokes theorem holds for currents. Thus if c is a current of dimension q, then the boundary of c is the mapping ∂c: Φq−1p (M) → R:A 7→ dA. (30) In addition to chains the odd de Rham current most frequently used is the Dirac current wδ(x) of dimension m derived from an odd m - vector w and a point x ∈ M . If A is an odd m - form, then wδ(x) A = 〈Ã(x), w〉. (31) B. Electrodynamics 1.The space of fields. In this section we will construct a suitable space of fields for electrodynamics which has the same role of the space of motions in mechanics (see [3]). The space of fields is not a differentiable manifolds. Nevertheless, we will introduce in this space enough structure to permit the construction of tangent and cotangent vectors which are the objects needed in in the formulation of the variational principle. A class of admissible functions defined on the space of fields has also to be specified. This class needs to be large enough to include the action, which for electrodynamics is generated (in a sense that will be specified) from a quadratic Lagrangian density, due to the linearity of the theory. Thus, only functions generated from quadratic mappings will be used in our variational formulation of electrodynamics. A larger class of admissible functions would be needed to deal with more general, non linear field theories, and the price of increased technical difficulties should be paid. Let M be the affine Minkowski space-time of special relativity with the 4-dimensional model space V and the non degenerate metric tensor g : V → V ∗ of signature (1, 3). The space of odd 4-currents with compact supports in M will be denoted by CM . Differential forms will always be presented as covector fields. We consider the set X(Φ1e(M);CM) of pairs (A, c), where c is an odd current of dimension 4 in M with a compact support Sup(c) and A is a local even 1 - form A:U → ∧1eV ∗ (32) defined in an open set U ⊂ M containing the support of c. The 1 - form A will represent the electro- magnetic potential. A mapping κ:M × ∧1eV ∗ × ∧2eV ∗ → ∧4oV ∗ (33) is said to be quadratic if for each x ∈ M there exists a symmetric bilinear mapping ∗ × ∧2eV ∗ × ∧2eV ∗) → ∧4oV ∗ (34) such that the mappings κx = κ(x, ·, ·) and δ 2κx satisfy the equality κx(a, f) = κx((a, f)(a, f)), (35) for each (a, f) ∈ ∧1eV ∗ × ∧2eV ∗. We are using the standard definition of quadratic mappings in terms of polarizations. The polarization of the mapping κx is the mapping δ 2κx and the equation (35) is the standard relation between a quadratic mapping defined on a vector space and its polarization. We will use the set of quadratic mappings (33) to introduce an equivalence relation in the set X(Φ1e(M);CM). In the next section they will be also used to define the admissible functions defined on the space of fields. Pairs (A, c) and (A′, c′) are equivalent if κ ◦ (x,A′, dA′) = κ ◦ (x,A, dA) (36) for each quadratic mapping κ:M × ∧1eV ∗ × ∧2eV ∗ → ∧4oV ∗. The symbol x is used to indicate the identity mapping of M and also a point of M . If µ is an arbitrary odd 4-form on M and κ is set to be the mapping κ:M × ∧1eV ∗ × ∧2eV ∗ → ∧4oV ∗: (x, a, f) 7→ µ(x), (37) then the equivalence condition (36) reduces to µ (38) and implies that c′ = c. Equivalence classes of elements of X(Φ1e(M);CM) will be called fields. Our fields are similar to those used by Freed in [14]. The space of fields will be denoted by Q(Φ1e(M);CM) or simply Q. The equivalence class of (A, c) will be denoted by q(A, c) or simply q. There is a natural projection ε:Q(Φ1e(M);CM) → CM : q(A, c) 7→ c (39) from the space of fields to the space CM of currents in M . It is not difficult to check that each fibre ε−1(c) of the projection ε is a vector space which will be denoted by the symbol Q(Φ1e(M); c) or Qc. Indeed, the sum of two pairs (A, c) and (A′, c) with A:U → ∧1eV ∗ and A′:U ′ → ∧1eV ∗ is the pair (A + A′, c) defined in U ∩ U ′ ⊃ Sup(c). Descending to the quotient with respect to the equivalence relation defined by (36) easily gives the sum in Qc. Therefore, the space of fields is the disjoint union of vector spaces Q = 2.Functions, vertical tangent vectors and covectors in the space of fields. With each quadratic mapping κ:M × ∧1eV ∗ × ∧2eV ∗ → ∧4oV ∗ we associate the function k:Q(Φ1e(M);CM) → R: q(A, c) 7→ κ ◦ (x,A, dA). (40) The above mentioned admissible functions defined on the space of fields are those constructed in this way. They will be called differentiable. The space of such functions will be denoted by K(Φ1e(M);CM). We note that from the definition of the space of fields Q = Q(Φ1e(M);CM) (as a quotient space with respect to the equivalence condition (36)) follows easily that the differentiable functions separate points of Q, i.e. if k(q′) = k(q) for each k ∈ K(Φ1e(M);CM), then q ′ = q. Indeed if k(q(A′, c′)) = k(q(A, c)) for each k ∈ K(Φ1e(M);CM), then equation (36) holds for each quadratic mapping κ:M×∧ ∗×∧2eV ∗. It follows that c′ = c and q(A′, c) = q(A, c). The tangent space to a vector space Qc (i.e. to a fibre of ε) coincides with Qc. The vertical tangent bundle to the vector bundle Q(Φ1e(M);CM) is the space VQ = Q × (ε,ε) Q = {(q, δq) ∈ Q×Q; ε(q) = ε(δq)} (41) where q = q(A, c) and δq = q(δA, c) and A:U → ∧1eV ∗, δA:U → ∧1eV ∗. The tangent projection is the canonical projection τQ:VQ → Q: (q, δq) 7→ q. (42) There is no obvious choice of the bundle dual to VQ. We will use the fibre derivatives of functions k ∈ K(Φ1e(M);CM) as models of covectors. The derivative Dk of a function k:Q → R is defined for each current c separately. It is evaluated on a pair of vectors q = q(A, c) ∈ Qc and δq = q(δA, c) ∈ Qc. The result is the expression Dk (q(A, c), q(δA, c)) = k (q(A + sδA, c)) (0) κ ◦ (x,A + sδA, F + sδF ) , (43) where F = dA and δF = dδA. The symbol dk(q(A, c)) will be used to denote the covector characterized by the pairing 〈dk (q (A, c)) , q(δA, c)〉 = Dk (q(A, c), q(δA, c)) (44) with vectors δq = q(δA, c) ∈ Qc. So we need to calculate the expression κ ◦ (x,A + sδA, F + sδF ) (x) = Dκx(A(x), F (x), δA(x) ⊕ δF (x)), (45) for each x ∈ U . We are using the following known definition. If V1, V2,W are vector spaces and m:V1 × V2 → W is a smooth mapping, then its derivative is the mapping Dm:V1 × V2 × (V1 ⊕ V2) → W : (a, f, δa⊕ δf) 7→ (m(a + sδa, f + sδf)) (0). (46) We define for each x ∈ U the bilinear mappings ∗ × ∧1eV ∗ → ∧4oV ∗: (a, a′) 7→ δ2κx((a, 0), (a , 0)), (47) ∗ × ∧2eV ∗ → ∧4oV ∗: (a, f) 7→ δ2κx((a, 0), (0, f)) (48) ∗ × ∧2eV ∗ → ∧4oV ∗: (f, f ′) 7→ δ2κx((0, f), (0, f ′)), (49) obtaining the equality κx(a, f) = λx(a, a) + µx(a, f) + νx(f, f), (50) for each (a, f) ∈ ∧1eV ∗ × ∧2eV ∗. The equality (50) in terms of the bilinear mappings (47), (48), and (49) will be useful to calculate the expression (45). We have the following lemmas. Lemma 4. If κx:∧ ∗ × ∧2eV ∗ → ∧4oV ∗ is a quadratic mapping and λx, µx, νx are the mappings defined above, then Dκx(a, f, δa⊕ δf) = Dλx(a, a, δa⊕ δa) + Dµx(a, f, δa⊕ δf) + Dνx(f, f, δf ⊕ δf), (51) for each (a, f, δa⊕ δf) ∈ ∧1eV ∗ × ∧2eV ∗ × ∧1eV ∗ ⊕ ∧2eV Proof: We have the equality λx ◦ ∆ ◦ pr1 + µx + νx ◦ ∆ ′ ◦ pr2, (52) where pr1:∧ ∗ × ∧2eV ∗ → ∧1eV ∗: (a, f) 7→ a, (53) ∆:∧1eV ∗ → ∧1eV ∗ × ∧1eV ∗: a 7→ (a, a), (54) pr2:∧ ∗ × ∧2eV ∗ → ∧2eV ∗: (a, f) 7→ f, (55) ∆′:∧2eV ∗ → ∧2eV ∗ × ∧2eV ∗: a 7→ (a, a). (56) The claim follows from the equality (52) by noting that Dκx(a, f, δa⊕ δf) = Dλx ◦ (∆,D∆) ◦ (pr1,Dpr1)(a, f, δa⊕ δf) + Dµx(a, f, δa⊕ δf) Dνx ◦ (∆ ,D∆′) ◦ (pr2,Dpr2)(a, f, δa⊕ δf) Dλx ◦ (∆,D∆)(a, δa) + Dµx(a, f, δa⊕ δf) Dνx ◦ (∆ ′,D∆′)(f, δf) Dλx(a, a, δa⊕ δa) + Dµx(a, f, δa⊕ δf) + Dνx(f, f, δf ⊕ δf). (57) Lemma 5. If V1, V2,W are vector spaces and b:V1 × V2 → W (58) is a bilinear mapping, then Db(a, f, δa⊕ δf) = b(a, δf) + b(δa, f), (59) for each (a, f, δa⊕ δf) ∈ V1 × V2 × (V1 ⊕ V2). Proof: From the definition (46) of the derivative it follows that Db(a, f, δa⊕ δf) = (b(a, f + sδf) + sb(δa, f + sδf)) (0) b(a, f) + sb(a, δf) + sb(δa, f) + s2b(δa, δf) = b(a, δf) + b(δa, f). (60) In view of Lemmas 4 and 5 the integrand in right hand side of the equality (43) reduces to Dκx(A(x), F (x), δA(x) ⊕ δF (x)) = Dλx(A(x), A(x), δA(x) ⊕ δA(x)) + Dµx(A(x), F (x), δA(x) ⊕ δF (x)) Dνx(F (x), F (x), δF (x) ⊕ δF (x)) (λx(A(x), δA(x)) + λx(δA(x), A(x))) + µx(A(x), δF (x)) + µx(δA(x), F (x)) (νx(F (x), δF (x)) + νx(δF (x), F (x))) , (61) Dκx(A(x), F (x), δA(x) ⊕ δF (x)) = λx(A(x), δA(x)) + µx(A(x), δF (x)) + µx(δA(x), F (x)) + νx(F (x), δF (x)), (62) since λx and νx are symmetric. By using the Proposition 2 contained in Section 6, this expression reduces to Dκx(A(x), F (x), δA(x) ⊕ δF (x)) =δA(x) ∧ We1(λx(A(x)) + δF (x) ∧ We2(µx(A(x)) + δA(x) ∧ We1(µx(F (x)) + δF (x) ∧ We2(νx(F (x)) =δA(x) ∧ We1(λx(A(x)) + µx(F (x))) + δF (x) ∧ (We2(µx(A(x)) + νx(F (x)))) . (63) where the linear mappings λx and λx are obtained from the bilinear mapping λx as prescribed in Section 6, i.e., ∗ → ∧1eV ⊗ ∧ ∗: a 7→ i1(λx(a, ·)), (64) ∗ → ∧1eV ⊗ ∧ ∗: a 7→ i1(λx(·, a)), (65) for each x ∈ U . The mappings µx, µx are obtained from µx, and νx, νx are obtained from νx in the same way. We define from λx, λx the mappings λ:M × ∧1eV ∗ → ∧1eV ⊗ ∧ ∗: (x, a) 7→ λx(a), (66) λ:M × ∧1eV ∗ → ∧1eV ⊗ ∧ ∗: (x, a) 7→ λx(a). (67) The mappings µ, µ, ν, ν are defined respectively from µx, µx, νx, νx in the same way. By using the equalities (47) and (63) which hold for each x ∈ U we obtain the equality κ ◦ (x,A + sδA, F + sδF ) We1 ◦ λ ◦ (x,A) + µ ◦ (x, F ) + (We2 ◦ (µ ◦ (x,A) + ν ◦ (x, F ))) ∧ δF. (68) We have also used the graded commutativity of the exterior product. Hence, Dk (q(A, c), q(δA, c)) = κ ◦ (x,A + sδA, F + sδF ) We1 ◦ λ ◦ (x,A) + µ ◦ (x, F ) (We2 ◦ (µ ◦ (x,A) + ν ◦ (x, F ))) ∧ δF. (69) Since F = dA and δF = dδA, this equality can be converted to the form Dk (q(A, c), q(δA, c)) = − We1 ◦ λ ◦ (x,A) + µ ◦ (x, dA) + d (We2 ◦ (µ ◦ (x,A) + ν ◦ (x, dA)))) ∧ δA d ((We2 ◦ (µ ◦ (x,A) + ν ◦ dA)) ∧ δA) . (70) We note that the first integral contains the exterior product of δA and an odd 3-form, while the second integral contains the exterior differential of the product of δA and an odd 2-form. The obtained expression suggests the representation of covectors as equivalence classes of elements of the sets X(Φ2o(M) × Φ o(M);CM) introduced below. An element of X(Φ2o(M)×Φ o(M);CM) is a triple (G, J, c) of a local odd differential 2-form G:U → ∗, a local odd differential 3-form J :U → ∧3oV ∗ and a current c with support contained in U . The odd 2-form G will be interpreted as the electromagnetic induction, while the odd 3-form J will be interpreted as the current. A covector p is an equivalence class p(G, J, c) of (G, J, c) ∈ X(Φ2o(M) × Φ3o(M);CM). The equivalence relations in X(Φ o(M) × Φ o(M);CM) are based on the expression J ∧ δA− d (G ∧ δA) . (71) Elements (G, J, c) and (G′, J ′, c′) are equivalent if c = c′ and J ′ ∧ δA− d (G′ ∧ δA) J ∧ δA− d (G ∧ δA) , (72) for each (δA, c) ∈ Qc. This equivalence relation is related to the equivalence relation (36). They can be used in the construction of dual objects. Indeed a (vertical) tangent vector might also (more generally) be defined as an equivalence class of pairs (δA, c) with respect to an equivalence relation similar to (36). A convenient representation of such relation is a relation similar to (72), but with (G, J, c) fixed, defining the equivalence between two pairs (δA, c) and (δA′, c). Such a construction is not needed in our case since each Qc is a vector space, so that its tangent space is canonically identified with Qc itself. The vector space Πc of covectors associated with the current c will be used as the dual space of the space Qc thought of as the space of vertical tangent vectors with the pairing 〈〈〈〈〈p, δq〉〉〉〉〉 J ∧ δA− d (G ∧ δA) , (73) where δq = q(δA, c) ∈ Qc and p = p(G, J, c) ∈ Πc. The space of all covectors is the union Πc. (74) There is a natural projection ε′: Π → CM : p(G, J, c) 7→ c. (75) from the space of fields to the space CM of currents in M . The phase space is the space Ph = Q × (ε,ε′) Π = {(q, p) ∈ Q× Π; ε(q) = ε′(a)}. (76) The symbol Phc will denote the set Qc × Πc ⊂ Ph. 3.A virtual action principle for electrodynamics. In this section a variational principle for electrodynamics similar to the virtual action principle of analytical mechanics (see [3]) will be formulated. The construction is an example of similar construc- tions needed for the variational formulation of a general field theory. The linearity of the theory and the choice of formulating it on the affine Minkowski space makes our presentation simpler though containing the core of the general framework (which can be found in [1]). The action is the differentiable function W :Q(Φ1e(M);CM) → R: q(A, c) 7→ L ◦ (A, dA) (77) derived from the quadratic Lagrangian density L:∧1eV ∗ × ∧2eV ∗ → ∧4oV ∗: (a, f) 7→ − 〈f,∧2eg −1(f)〉 |g|. (78) We are denoting by |g| the odd 4-covector defined by the Minkowski metric (see [5]). Later we will use the same symbol to denote the constant 4-covector field constructed from it. The 1-form A is called the potential, the 2-form F = dA is called the electromagnetic field. We note that the Lagrangian is a quadratic mapping which depends only on its second argument f and thus λ = µ = 0 and ν does not depend on x in the formula (50). A phase ph = (q(A, c), p(G, J, c)) satisfies the virtual action principle if the equality 〈dW (q), δq〉 − 〈〈〈〈〈p, δq〉〉〉〉〉 = 0 (79) holds for each virtual displacement δq = q(δA, c) ∈ Qc. For each current c the dynamics associated with the current c is the set Dc ⊂ Phc of phases which satisfy the virtual action principle. The dynamics is the subset Dc (80) of the phase space Ph. A phase space trajectory is a triple of local differential forms (A,G, J):U → ∧1eV ∗ × ∧2oV ∗ × ∧3oV . (81) The dynamics of a system can also be represented as a set DDDD of phase space trajectories (A,G, J) such that for each current c with support included in U the phase ph = (q(A, c), p(G, J, c)) is in Dc. The equation (79) is too abstract to be used directly. A more concrete expression is given in the following proposition. Proposition 3. A phase ph = (q(A, c), p(G, J, c)) satisfies the virtual action principle if and only if the equality −1 ◦ dA ∧ δA− d −1 ◦ dA J ∧ δA− d (G ∧ δA) is satisfied for each virtual displacement q(δA, c). Proof: By applying the formula (70) to the action W we obtain that its variation is 〈dW (q), δq〉 = (−d(We2 ◦ ν ◦ dA) ∧ δA + d ((We2 ◦ ν ◦ dA) ∧ δA)) , (83) since λ = µ = 0. Moreover νx(f, f) = − 〈f,∧2eg −1(f)〉 |g|, (84) from which it follows after some calculation that ν(x, f) = − ∧2e g −1(f) ⊗ |g|, (85) for each f ∈ ∧2eV ∗. Hence, We2 ◦ ν ◦ (x, dA) = − −1 ◦ dA and the variation of the action reduces to 〈dW (q), δq〉 = −1 ◦ dA ∧ δA− d −1 ◦ dA . (87) Recalling that, on the other hand 〈〈〈〈〈p, δq〉〉〉〉〉 J ∧ δA− d (G ∧ δA) , (88) the claim follows. A phase space trajectory belongs to the dynamics DDDD, if and only if it satisfies the virtual action principle for each current c with support included in its domain of definition. There is a characterization of the dynamics of phase space trajectories in terms of differential equations. This is shown in the following propositions. Theorem 4. A phase space trajectory (A,G, J) belongs to the dynamics DDDD if and only if it satisfies the Euler-Lagrange equation −1 ◦ dA J (89) and the constitutive relation −1 ◦ dA |g|. (90) Proof: If a phase space trajectory (A,G, J) satisfies the Euler-Lagrange equation and the constitutive relation, then by substituting the expressions (89) and (90) of J and G in terms of the electromagnetic field F in the virtual action principle (82) it follows that (A,G, J) belongs to the dynamics DDDD. The inverse implication will be proved in the next section. The constitutive relation (90) produced by our variational principle corresponds to the momentum- velocity relation of analytical mechanics. Proposition 5. A phase space trajectory (A,G, J) satisfies the Euler-Lagrange equation and the constitutive relation if and only if it satisfies the Maxwell’s equations J (91) and the constitutive relation −1 ◦ F |g|, (92) with F = dA. Proof: The constitutive relation (91) is satisfied if and only if the equation (90) is satisfied for F = dA. If a phase space trajectory (A,G, J) satisfies the Maxwell’s equations and the constitutive relation, then by substituting the expression (92) of the electromagnetic induction in the equation (91) we see that the Euler-Lagrange equation is satisfied, since F = dA. Conversely if the Euler-Lagrange equation and the constitutive relation are satisfied, then, again by substitution, we obtain that the Maxwell’s equations (91) are satisfied. We remark that since the virtual action principle (77) is more complete than the Hamilton principle, our formulation leads to the Maxwell’s equations (91) with external sources. The proposed variational principle also permits the derivation of the constitutive relations (92) which are usually postulated separately since the variations normally considered are not general enough to derive them from the variational principle. 4.The Dynamics in a compact domain. Let the current c consist in integrating an odd 4-form on a compact domain K ⊂ M with smooth boundary ∂K. Field configurations, tangent vectors and covectors are equivalence classes of equivalence relations based on the expressions ∫ κ ◦ (x,A, dA) (93) and ∫ J ∧ δA− d (G ∧ δA) J ∧ δA− G ∧ δA. (94) It follows that a field q = q(A,K) is represented by the restriction A|K:K → ∧1eV ∗ (95) of the potential A to the the domain K. A tangent vector δq = q(δA,K) is represented by the restriction (δA)|K:K → ∧1eV ∗ (96) of the variation δA to the domain K. A covector p = p(G, J,K) is represented by the pair of the restriction G|∂K: ∂K → ∧2oV ∗ (97) of the electromagnetic induction G to the boundary ∂K of the domain K and the restriction K → ∧3oV ∗ (98) of the current J to the interior K of the domain K. A phase is a pair (q, p) of a field q = q(A,K) and a covector p = p(G, J,K). The action is W :QK → R: q(A,K) 7→ L ◦ (A, dA). (99) The virtual action principle is the equality 〈dW (q), δq〉 − 〈〈〈〈〈p, δq〉〉〉〉〉 = 0 (100) and the dynamics in the domain K is the set DK ⊂ Ph of phases satisfying the virtual action principle. Proposition 6. A phase ph = (q(A,K), p(G, J,K)), defined in a compact domain K, belongs to the dynamics DK if and only if the Euler-Lagrange equation −1 ◦ dA K (101) and the constitutive relation G|∂K = −1 ◦ dA |∂K (102) are satisfied. Proof: If q = q(A,K), p = p(G, J,K) and δq = q(δA,K), then 〈dW (q), δq〉 = −1 ◦ dA −1 ◦ dA −1 ◦ dA −1 ◦ dA ∧ δA. (103) On the other hand 〈〈〈〈〈p, δq〉〉〉〉〉 J ∧ δA− G ∧ δA. (104) Thus the virtual action principle assumes the form −1 ◦ dA ∧ δA− −1 ◦ dA J ∧ δA− G ∧ δA. (105) By using variations with (δA)|∂K = 0 we derive the Euler-Lagrange equation −1 ◦ dA K. (106) Assuming that this equation is satisfied and using arbitrary variations, the constitutive relation G|∂K = −1 ◦ dA |∂K (107) follows. The following Proposition completes the proof of the Theorem 4. Proposition 7. If a phase space trajectory (A,G, J):U → ∧1eV ∗ × ∧2oV ∗ × ∧3oV . (108) belongs to the dynamics DDDD, then the Euler-Lagrange equation −1 ◦ dA J (109) and the constitutive relation −1 ◦ dA |g| (110) are satisfied in U . Proof: If (A,G, J) is a phase space trajectory, defined in the open set U ⊂ M , which belongs to the dynamics DDDD, then the equation −1 ◦ dA K (111) and the boundary relation G|∂K = −1 ◦ dA |∂K (112) are satisfied for each compact domain K ⊂ U . It follows that equations (109) and (110) are satisfied in every x ∈ U . 5.The Lagrangian formulation of electrodynamics. The Lagrangian formulation of dynamics is the infinitesimal limit of the formulation in a compact domain with the domain shrinking to a point. A formal method which greatly simplifies the passage to the infinitesimal limit is to replace the compact domain — which is used exclusively as domain of integration — with the current c = δ(x)w, where δ(x) is the Dirac delta function in x ∈ M and w ∈ ∧4oV is an odd 4-vector, with w 6= 0. The construction of infinitesimal fields, tangent vectors and covectors is based on the expressions δ(x)w κ ◦ (x,A, dA) = 〈κ (x,A(x), dA(x)) , w〉 (113) δ(x)w J ∧ δA− d (G ∧ δA) J(x) ∧ δA(x) − d(G ∧ δA)(x), w . (114) Since w 6= 0 and ∧4oV is one-dimensional, it follows from the first expression that a field q = q(A, c) is represented by the pair (A(x), F (x)) ∈ ∧1eV ∗ × ∧2eV ∗ (115) of an even 1-covector A(x) and an even 2-covector F (x). The second expression reduces to dG(x) − ∧ δA(x) + G(x) ∧ δF (x), w , (116) since dδA(x) = δF (x). It follows that a tangent vector δq = q(δA, c) is represented by the pair (δA(x), δF (x)) ∈ ∧1eV ∗ × ∧2eV ∗ (117) and a covector p = p(G, J, c) is represented by the pair G(x), dG(x) − ∈ ∧2oV ∗ × ∧3oV ∗. (118) The pairing 〈〈〈〈〈p, δq〉〉〉〉〉 J ∧ δA− d (G ∧ δA) (119) assumes the form 〈〈〈〈〈p, δq〉〉〉〉〉 dG(x) − ∧ δA(x) + G(x) ∧ δF (x), w . (120) We have constructed the space of infinitesimal fields Qδ = ∧ ∗×∧2eV ∗ and the space of infinitesimal covectors Πδ = ∧ ∗ × ∧3oV ∗. Hence, the infinitesimal phase space is Phδ = Qδ × Πδ = ∧ ∗ × ∧2eV ∗ × ∧2oV ∗ × ∧3oV ∗. (121) The infinitesimal action is W (q(A, δ(x)w)) = 〈L(A(x), F (x)), w〉 . (122) The infinitesimal dynamics is the set (q, δq) ∈ Phδ; ∀δq∈Qδ 〈dW (q), δq〉 = 〈〈〈〈〈p, δq〉〉〉〉〉 , (123) It is easy to verify that the infinitesimal dynamics Dδ admits also the following more explicit expression (a, f, g, h) ∈ Phδ; ∀ (δa,δf)∈∧1eV ∗×∧2eV ∗ DL(a, f, δa, δf) = − (h ∧ δa + g ∧ δf) . (124) The infinitesimal dynamics Dδ is characterized by the following Proposition. Proposition 8. An infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, belongs to the infinitesimal dynamics Dδ if and only if the equations G(x) = −1(F (x)) |g| (125) dG(x) = J(x) (126) are satisfied. Proof: If q = q(A, δ(x)w)), p = p(G, J, δ(x)w), and δq = q(δA, δ(x)w), with w 6= 0, then the variation of the action (87), which can also be expressed in the form 〈dW (q), δq〉 = − −1 ◦ F ∧ δF, (127) reduces to 〈dW (q), δq〉 = − −1(F (x)) ∧ δF (x), w . (128) Thus the virtual action principle 〈dW (q), δq〉 = 〈〈〈〈〈p, δq〉〉〉〉〉 (129) has the explicit form dG(x) − ∧ δA(x) + G(x) − −1(F (x)) ∧ δF (x), w = 0. (130) Therefore if ph satisfies the equations (125) and (126), then (130) is satisfied and ph ∈ Dδ. Conversely, if ph ∈ Dδ, then it satisfies the virtual action principle, which implies dG(x) − ∧ δA(x) + G(x) − −1(F (x)) ∧ δF (x) = 0, (131) since w 6= 0 and ∧4oV is one-dimensional. The equations (125) and (126) follow, since δA(x) and δF (x) are independent. In a preceding section we showed that the dynamics of phase space trajectories can also be charac- terized by the Maxwell’s equations and the constitutive relation. This fact can now be proved directly. Indeed, if a phase space trajectory (A,G, J) belongs to the dynamics DDDD and x is a point in the domain of definition of the trajectory, then the virtual action principle is satisfied for every infinitesimal current δ(x)w. Thus the Maxwell’s equations and the constitutive relation are satisfied in x, thanks to the previous proposition. It follows that they are satisfied in the whole domain of definition of the trajectory. To prove the inverse implication, we observe that the virtual action principle (82) can also be expressed in the form −1 ◦ F |g| ∧ δF = J ∧ δA− d (G ∧ δA) (132) −1 ◦ F ∧ δF + = 0, (133) for each virtual displacement q(δA, c). Thus if a phase space trajectory (A,G, J) satisfies the Maxwell’s equations, then it satisfies the virtual action principle for every current c with support contained in the domain of definition of the trajectory and hence it belongs to DDDD. 6.The Hamiltonian formulation of electrodynamics. We associate with the Lagrangian density L:∧1eV ∗ × ∧2eV ∗ → ∧4oV ∗: (a, f) 7→ − f,∧2eg −1(f) |g|, (134) the energy density E:∧1eV ∗ × ∧2oV ∗ × ∧2eV ∗ → ∧4oV ∗ (135) defined by E(a, g, f) = − g ∧ f − L(a, f) g ∧ f + f,∧2eg −1(f) g ∧ f + −1(f) 2g − ∧2eg −1(f) ∧ f (136) and treat this mapping as a family E(a, g, ·):∧2eV ∗ → ∧4oV ∗ (137) of mappings on the fibres of the projection prP :∧ ∗ × ∧2oV ∗ × ∧2eV ∗ → ∧1eV ∗ × ∧2oV ∗ (138) onto the field-momentum space P = ∧1eV ∗ × ∧2oV ∗. The set Cr(E, prP ) = (a, g, λ) ∈ ∧1eV ∗ × ∧2oV ∗ × ∧2eV δλ∈∧2 DE(a, g, λ, 0, 0, δλ) = 0 (139) is the critical set of the family. The equality DE(a, g, λ, δa, δg, δλ) = − (δg ∧ λ + g ∧ δλ) − DL(a, λ, δa, δλ) (δg ∧ λ + g ∧ δλ) + −1(λ) δg ∧ λ + −1(λ) (140) implies DE(a, g, λ, 0, 0, δλ) = − −1(λ) ∧ δλ. (141) Thus we obtain the expression Cr(E, prP ) = (a, g, λ) ∈ ∧1eV ∗ × ∧2oV ∗ × ∧2eV ∗; g = ∧2eg −1(λ) . (142) The critical set is the graph of the Legendre mapping Λ:∧1eV ∗ × ∧2eV ∗ → ∧2oV ∗: (a, λ) 7→ ∧2eg −1(λ) |g|. (143) For each a ∈ ∧1eV ∗, the mapping Λ(a, ·) is invertible. Its inverse is the mapping Λ(a, ·)−1:∧2oV ∗ → ∧2eV ∗: g 7→ ∧2eg |g−1| g , (144) where |g−1| denotes the odd 4-vector characterized by 〈 |g−1|〉 = 1. It follows that the critical set is the image of the section σ:∧1eV ∗ × ∧2oV ∗ → ∧1eV ∗ × ∧2oV ∗ × ∧2eV ∗: (a, g) 7→ a, g,∧2eg |g−1| g (145) of the projection prP . The Hamiltonian density is the mapping H = E ◦ σ:∧1eV ∗ × ∧2oV ∗ → ∧4oV ∗, (146) defined by the formula H(a, g) = − |g−1| g , (147) for each (a, g) ∈ ∧1eV ∗×∧2oV ∗. The passage from the Lagrangian density L to the Hamiltonian density H is the Legendre transformation of electrodynamics. We show that the energy density can be used to generate the infinitesimal dynamics Dδ. We consider the set (a, f, g, r) ∈ Phδ; ∃ V ∗ ∀(δa,δg,δλ)∈∧1 V ∗×∧2 V ∗×∧2 DE(a, g, λ, δa, δg, δλ) = (r ∧ δa− f ∧ δg) . (148) This set is obtained by projecting the set D̃E = (a, f, g, r, λ) ∈ Phδ × ∧ (δa,δg,δλ)∈∧1eV ∗×∧2oV ∗×∧2eV ∗ DE(a, g, λ, δa, δg, δλ) = (r ∧ δa− f ∧ δg) (149) onto the phase space Phδ = ∧ ∗ × ∧2eV ∗ × ∧2oV ∗ × ∧3oV It follows from (140) that λ = f and that the set D̃E reduces to D̃E = (a, f, g, r, λ) ∈ Phδ × ∧ ∗; λ = f, (δa,δg,δf)∈∧1eV ∗×∧2oV ∗×∧2eV ∗ DL(a, f, δa, δf) = (r ∧ δa + g ∧ δf) . (150) It projects onto the infinitesimal dynamics (a, f, g, r) ∈ Phδ; ∀ (δa,δf)∈∧1 V ∗×∧2 DL(a, f, δa, δf) = − (r ∧ δa + g ∧ δf) . (151) Hence, DE = Dδ. It is clear from the definition of the set D̃E that this set is included in the set (a, f, g, r, λ) ∈ Phδ × ∧ ∗; (a, g, λ) ∈ Cr(E, prP ) . (152) The use of the mapping σ results in D̃E = (a, f, g, r, λ) ∈ Phδ × ∧ ∗; λ = f, (δa,δg)∈∧1 V ∗×∧2 DH(a, g, δa, δg) = (r ∧ δa− f ∧ δg) . (153) The Hamiltonian description of the dynamics (a, f, g, r) ∈ Phδ; ∀ (δa,δg)∈∧1 V ∗×∧2 DH(a, g, δa, δg) = (r ∧ δa− f ∧ δg) (a, f, g, r) ∈ Phδ; ∧ |g−1| g = f , r = 0 . (154) is obtained by projecting onto the phase space Phδ. Proposition 9. An infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, belongs to the infinitesimal dynamics Dδ if and only if the equations F (x) = ∧2eg ◦ |g−1| G(x) (155) dG(x) = J(x) (156) are satisfied. Proof: We recall that an infinitesimal phase ph = (q(A, δ(x)w), p(G, J, δ(x)w)), with w 6= 0, is represented by ( A(x), F (x), dG(x) − J(x), G(x) (157) and use the Hamiltonian description (154) of infinitesimal dynamics Dδ. The claim easily follows. The resulting equations for the phase space trajectories (A,G, J) F = ∧2eg ◦ |g−1| G (158) J (159) are called Hamilton’s equations. The equations (159) are again Maxwell’s equations and the equation (158) is the inverse of constitutive relation (92). 7.References [1] W. M. Tulczyjew, The origin of variational principles, in the volume Classical and quantum inte- grability (Warsaw, 2001), 41–75, Banach Center Publ., 59, Polish Acad. Sci., Warsaw, 2003. [2] G. Marmo, W. M. Tulczyjew, P. Urbański, Dynamics of autonomous systems with external forces, Acta Physica Polonica B, 33 (2002), 1181–1240. [3] A. De Nicola, W. M. Tulczyjew, A variational formulation of analytical mechanics in an affine space, Rep. Math. Phys., 58 (2006), 335–350. [4] W. M. Tulczyjew, A symplectic framework of linear field theories, Annali di Matematica pura ed applicata, 130 (1982), 177–195. [5] G. Marmo, E. Parasecoli, W. M. Tulczyjew, Space-time orientations and Maxwell’s equations, Rep. Math. Phys. 56 (2005), 209–248. [6] G. Marmo, W. M. Tulczyjew, Time reflection and the dynamics of particles and antiparticles, Rep. Math. Phys. 58 (2006), 147–164. [7] A. De Nicola, W. M. Tulczyjew, A Note on a Variational Formulation of Electrodynamics, Proc. XV Int. Workshop on Geom. and Phys., Tenerife (Spain), 2006, Publ. de la RSME 11 (2007), 316–323. [8] J. A. Schouten, Ricci calculus, Springer, Berlin, 1955. [9] J. A. Schouten, Tensor Analysis for Physicists, Oxford University Press, London, 1951. [10] G. de Rham, Variétés Differentiables, Hermann, Paris, 1955. [11] T. Frankel, The Geometry of Physics: An Introduction, Cambridge University Press, Cambridge, 1997. [12] F. W. Hehl and Yu. N. Obukhov, Foundations of Classical Electrodynamics, Charge, Flux, and Metric, Birkhäuser, Boston, MA, 2003. [13] I. V. Lindell, Differential Forms in Electromagnetics, IEEE Press, Piscataway, NJ, and Wiley- Interscience, 2004. [14] D. Freed, Classical field theory and Supersimmetry, IAS Park City Mathematics Series Vol. 11, IAS, Princeton, 2001.

0704.0434

Protein and ionic surfactants - promoters and inhibitors of contact line pinning

RSC Communication Template (Version 2.1) Protein and ionic surfactants - promoters and inhibitors of contact line pinning Viatcheslav V. Berejnova We report a new effect of surfactants in pinning a drop contact line, specifically that lysozyme promotes while lauryl sulfate inhibits pinning. We explain the pinning disparity assuming difference in wetting: the protein-laden drop wets a "clean" surface and the surfactant-laden drop wets an auto-precursored surface. To date, the effect of surfactants on a drop's wetting and spreading has been well established 1-12. It was observed that low molecular weight surfactants extend spreading and decrease the contact line stability 13, 14. However, the effect of surfactants and proteins on pinning a drop contact line has not yet been thoroughly appreciated despite being a primary issue for several important methods in life science. Optimization of the pinning conditions could benefit the crystallization of globular and membrane proteins 15, 16, formulation of the pesticide sprays for protecting the plants 1, 17, and the influence of the pulmonary surfactants on physiological wettability of alveoli in the lungs 18. In this letter, we first study the quasi-static pinning of the protein and surfactant drops pinned by siliconized glass slides. We demonstrate that lysozyme (Lys) increases the hysteresis effect and stabilizes the drop contact line, enhancing the size of the completely pinned drops by a factor of 4-5 (compared to water). Conversely, the sodium dodecyl sulfate (SDS) reduces the apparent contact angles, decreasing the size of the completely pinned drops by a similar factor. We found the protein pinning to be similar to pinning induced by the geometrical and chemical corrugations of the contact line 15. In our experiments, DI water was purified by NANOpure II (Barnstead, Boston, MA). Lys protein, 6-x times crystallized hen egg white, was purchased from Seikagaku America (Mr~14 kD, lot: LF1121, Falmouth, MA). Lys was dissolved in a 50 mM sodium acetate buffer with pH = 5. pH of the SDS (0.29 kD, 99%, Sigma) solutions was 9. All solutions were filtered. Flat, siliconized 22 mm glass slides HR3-231 were purchased from Hampton Research (Laguna Niguel, CA). On a new siliconized slide, a water drop with a volume of ~20 μL formed a reproducible contact angle of ~(92±1)o. The siliconized material of the HR3-231 slides is similar to the organosilane-composed solution AquaSil (Hampton Research). We inspected the surface topography for some of the HR3-231 slides using a contact mode AFM (DI MultiMode III, Santa Barbara, CA) with NSC 1215 tips from MikroMasch. We found the manufacturer's coating to be homogeneous. We treated some hydrophobic slides chemically 15 in order to obtain the highly hydrophilic (contact angle <2°) circular area. Both the edge and hydrophilic-hydrophobic gradient provide a high threshold of pinning 15 that was taken as a reference in our experiments. Fig.1 Dimensionless diagram of the stability of inclined drops. The symbols: ■, □, and ∆ correspond to Lys (5 mg/mL), buffer (Bf), and SDS (33.3 mg/mL) solutions, respectively. Insert (a) is a geometrical sketch of an inclined drop. Insert (b) subtracts Vc from the raw-data diagrams (sin(α), V). The images 1, 2, and 3 represent three pinning zones: 1 - an absolutely stable drop; 2 - a drop is stable up to certain inclination < 90o, depending on V; and 3 - a drop is unstable and moves continuously. Each drop was dispensed manually onto initially horizontal glass slides, which were then slowly rotated in 2-4° steps (Fig. 1a). For the Lys drops the relaxation time between rotations was ~1 min to allow the transient disturbances of the drop to dissipate. For the SDS drops the initial spreading was similar to that described earlier 7-10, 12 and the relaxation time interval was different depending on concentration. We did not observe the autophobic 14 and Marangoni-induced 13 contact line displacements. The apparent advancing θa and receding θr contact angles were measured for some drops at quasi-static equilibrium using a horizontal goniometer 19. Dispensed drop volumes were accurate to 0.1-0.5%, and tilt and contact angle measurements were accurate to 1-2°. The surface tension γ for the Lys solutions was measured using the pendant drop counting method 15. We characterize pinning by measuring the critical tilt α corresponding to continuous motion of an inclined drop of volume V 19. We observed that the scaling law sin(α)~V-2/3 for large α may fit the (V, α) data for the different surfactant concentration C over a broad range of the supercriticality ratio V/Vc (Fig.1). Vc is a result of fitting, depends on C, and denotes the critical volume corresponding to a vertically (sin(α)=1) pinned drop (Fig.1b). The function (C, Vc) characterizes the surfactant pinning (Fig.2). a IESVic, University of Victoria, Victoria, BC V8W 3P6, Canada E-mail: [email protected] Two different regimes of pinning can be seen in Fig.2 depending on the chemical nature of the used surfactants. For very low concentrations close to point A, pinning of the Lys and SDS-laden drops behaves similarly. In the moderate and high concentration regions, adding Lys (triangles) and SDS (squares) increases and decreases the critical volumes of the pinned drops, respectively. The white and black circles represent the Lys drops initially dispensed on the circularly treated hydrophilic patterns with different radii 15 having edge profiles presented in Fig.2c. These patterns do not affect pinning of the SDS-laden drop. Both the similarities between the natural and pattern-induced pinning for the Lys drops and the differences in pinning between Lys and SDS are very remarkable. Summarized below are our results corresponding to the different pinning regimes presented in Fig.2. First, pinning is associated with the appearance of a trace of liquid film behind the drop 20-22 that can be treated as a sign of high adhesion. Verifying this, we mark intervals on the concentration axis in Fig.3a and Fig.3b where the trace behind the drop body appears. The buffer and pure water do not exhibit any traces on the hydrophobic slide (Fig.4a), but the treated slides show unstable traces covering the treated area (Fig.4c). The Lys and SDS-laden drops show similar traces (Fig.4b and Fig.4d). However, high pinning was observed only for the Lys drops (Fig3a and Fig3b). The SDS decreases pinning in the trace- region. Thus, the presence of the trace does not correlate with the strength of pinning for our choice of surfactants. Second, an equilibrium of an inclined drop yields the formula ρVcgsin(α)~DγΔ(cosθ)r,a 19, 20, 23, where ρ, g, and γ are the density of liquid, gravity, and the surface tension, respectively, and the contact angle difference Δ(cosθ)r,a equals cosθr-cosθa. Thus, the critical volume of an inclined drop (or pinning) is proportional to the wetted diameter D. Roughly, we can treat D as the diameter of an initially dispensed drop onto a horizontal substrate. This approximation holds in small drops when deformation of the drop surface describing by the Bond number Bo=ρgD2/γ (d1 for our case) is not too large. For the concentration range presented in Fig.3a we did not find a noticeable variation in D for the naturally dispensed Lys drops for either inclination or concentration. For the treated slides presented in Fig.2, increasing D qualitatively agrees with the above formula for low deviations of D in the Lys drops. The patterns with larger radii (open circle) provide higher drop stability (larger pinning). The diameters of the SDS drops on hydrophobic slides grow monotonically with concentration 7. However, unlike the Lys, the SDS decreases the drop pinning with respect to the concentration. Therefore, the D variation does not affect pinning for the SDS case, but it does for the Lys. Fig.3 Combined diagrams of critical volumes, contact angles, their difference, and surface tension versus the component concentrations. Black circles are γ, (a): Lys and (b): SDS 24. ∆, ▲, □, and ■ denote Vc and Δ(cosθ)r,a for the Lys and SDS drops, respectively. Dotted triangles (Lys) and squares (SDS) depict the contact angles: white - maximal θa and black - minimal θr. The zones of a liquid film trace appearing behind the quasi-statically displaced drop are marked on (a) and (b). Lines are not fits. Fig.2 Diagram of critical volumes versus concentration for vertically pinned drops. The inserts (a) and (b) represent drops on the untreated ∆ - Lys and □ - SDS, and treated ○ - Lys 8.0-diam-mm and ● - Lys 7.0-diam- mm glass slides, respectively. The point A depicts Vc for water and buffer. (c) is an AFM image of the etched-2 versus unetched-1 areas for the treated slide (bar is 1μ). Dashed lines are not fits. Third, we found that high pinning corresponds to high θa and Δ(cosθ)r,a, (Fig.3a and Fig.3c). For high concentration regions these parameters are related because the liquid-trace minimizes θr (Fig.3c dotted black triangles and squares). Keeping θa large is the only way to keep the difference Δ(cosθ)r,a also large. Adding Lys to the drops does not affect the initially high values of θa, and, consequently, Δ(cosθ)r,a (Fig.3a and Fig.3c). Conversely, adding the SDS decreases θa, controlling the overall low value of Δ(cosθ)r,a adequately, and provides the low pinning effect. Forth, the overall effect of concentration is noticeably different between Lys and SDS. Fig.3a demonstrates that pinning is tied to the Lys amount in the drop. Starting at 10-4 mM, the Lys-pinning increases as Δ(cosθ)r,a increases, reaches its maximum corresponding to the minimum of θr (appearance of the liquid- 2 trace), and finally saturates. Next, the high concentration region pinning decreases as γ(C) decreases (dγ/dC<0). The SDS-pinning, however, is much less sensitive to concentration. For the SDS drops, V c is close to the water drop, is stable up to 10mM, and decreases as θr decreases. Additional probing of the Lys and SDS solutions with the plain glass slides (HR3-227, Hampton Research) does not show a qualitatively different behavior of pinning. For plain glass, the Vc(C) curves generally follow the data presented in Fig.2, corresponding to the hydrophobic untreated slides for both the Lys and SDS surfactants. We see that the appearance of a trace indicating the high adhesion between solution and glass surface does not provide high pinning unambiguously: Lys increases and SDS decreases pinning. Neither does extending the drop's wetted diameter correlate with a pinning increase: for Lys, buffer, and water, a variation in D due to patterning increases pinning while it does not for SDS. Pinning and contact angle difference are correlated as well: an increase in Δ(cosθ)r,a corresponds to a pinning increase and decrease for the Lys and SDS solutions, respectively. We attribute such a dramatic difference in pinning to the different gas-solid surfaces near the contact line. We hypothesize that the Lys contact line advances the "clean" 10 solid surface, whereas the SDS contact line advances the pre-cursored zone existing in close proximity to the contact line. We expect that mobility of the SDS molecules through a liquid interface may affect the advancing zone. Indeed, such mobility of the small amphiphilic molecules is well documented 2-4, 7-10, 12. Though the mobility of proteins is not available, we argue it to be negligible compared to SDS. The globular proteins are weak surfactants with large, less amphiphilic, and therefore much less interfacially mobile molecules 25-27 and, in addition, they irreversibly adhere to the glass. We assume that the proteins adhere to the solid from the drop interior and provide high heterogeneity for the three-phase contact that results in the high pinning threshold similar to the corrugated case 15 (Fig.2, C>10-2mM). In contrast, SDS penetrates the drop- solid-air interface, increases affinity between the solid and the drop, and thus decreases the pinning threshold. Fig.4 Different regimes of displacement; (a): 40 μL drop of 50mM acetate buffer at pH=5 (water acts similarly); (b): 40 μL drop of 5mg/mL Lys; (c): 40 μl drop of water (bar is 5.5 mm); (d): side view of the 20 μL SDS 100 mg/mL drop (bar is 3.4 mm). Numbers are inclination in degrees; the slides (a), (b), and (d) are untreated, (c) is treated with D=5.5 The author thanks N. S. Husseini for proofreading the manuscript. References (1) Furmidge, C. G., J. Colloid Sci., 1962, 17, 309-&. (2) Eriksson, J.; Tiberg, F.; Zhmud, B., Langmuir, 2001, 17, 7274- 7279. (3) Kumar, N.; Varanasi, K.; Tilton, R. D.; Garoff, S., Langmuir, 2003, 19, 5366-5373. (4) Luokkala, B. B.; Garoff, S.; Tilton, R. D.; Suter, R. M., Langmuir, 2001, 17, 5917-5923. (5) Mohammadi, R.; Wassink, J.; Amirfazli, A., Langmuir, 2004, 20, 9657-9662. (6) Qu, D.; Suter, R.; Garoff, S., Langmuir, 2002, 18, 1649-1654. (7) Starov, V. M.; Kosvintsev, S. R.; Velarde, M. G., J. Colloid Interface Sci., 2000, 227, 185-190. (8) Stoebe, T.; Lin, Z. X.; Hill, R. M.; Ward, M. D.; Davis, H. T., Langmuir, 1996, 12, 337-344. (9) Stoebe, T.; Lin, Z. X.; Hill, R. M.; Ward, M. D.; Davis, H. T., Langmuir, 1997, 13, 7282-7286. (10) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davis, H. T., Langmuir, 1997, 13, 7276-7281. (11) Summ, B. D.; Soboleva, O. A.; Dolzhikova, V. D., Colloid J., 1998, 60, 598-605. (12) von Bahr, M.; Tiberg, F.; Zhmud, B. V., Langmuir, 1999, 15, 7069-7075. (13) Cachile, M.; Cazabat, A. M., Langmuir, 1999, 15, 1515-1521. (14) Frank, B.; Garoff, S., Langmuir, 1995, 11, 87-93. (15) Berejnov, V.; Thorne, R. E., Acta Cryst. B, 2005, 61, 1563-1567. (16) McPherson, A., Crystallization of Biological Macromolecules. Cold Spring Harbor Laboratory Press: New York, 1999. (17) Watanabe, T.; Yamaguchi, I., Pesticide Sci., 1992, 34, 273-279. (18) Hills, B. A., J. Appl. Phys., 1983, 54, 420-426. (19) Berejnov, V.; Thorne, R. E., Phys. Rev. E, 2007, submitted. http://lanl.arxiv.org/abs/physics/0609208 (20) Frenkel, Y. I., ZETF, 1948, 18, 659-667. http://lanl.arxiv.org/abs/physics/0503051 (21) Roura, P.; Fort, J., Phys. Rev. E, 2001, 64, 011601. (22) Tuck, E. O.; Schwartz, L. W., J. Fluid Mech., 1991, 223, 313-324. (23) Macdougall, G.; Ockrent, C., Proc. Roy. Soc. London Ser. A, 1942, 180, 0151-0173. (24) Persson, C. M.; Jonsson, A. P.; Bergstrom, M.; Eriksson, J. C., J. Colloid Interface Sci., 2003, 267, 151-154. (25) Mobius, D. M., R., Proteins at Liquid Interfaces. Elsevier: Amsterdam, 1998. (26) Vogler, E. A., Langmuir, 1992, 8, 2013-2020. (27) Krishnan, A.; Liu, Y. H.; Cha, P.; Allara, D.; Vogler, E. A., J. Biomed. Mater. Res., Part A, 2005, 75A, 445-457.

0704.0435

Type D Einstein spacetimes in higher dimensions

Type D Einstein spacetimes in higher dimensions V. Pravda1, A. Pravdová1 and M. Ortaggio2 ‡ 1 Mathematical Institute, Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic 2 Dipartimento di Fisica, Università degli Studi di Trento, and INFN, Gruppo Collegato di Trento, Via Sommarive 14, 38050 Povo (Trento), Italy E-mail: [email protected], [email protected], ortaggio‘AT’ffn.ub.es Abstract. We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, Ii, D or O. This applies also to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons, on which we briefly comment.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a one- dimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two- dimensional Lorentzian factor are restricted to the types D or O. By exploring algebraic consequences of the Bianchi identities, we then analyze the simplest non-trivial case from the above classes - type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for “generic” type D vacuum spacetimes (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact applies also to type II spacetimes). For n ≥ 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n ≥ 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 × 3 real matrix Φij . In the case with “twistfree” (Aij = 0) principal null geodesics we show that in a “generic” case Φij is symmetric and eigenvectors of Φij coincide with eigenvectors of the expansion matrix Sij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that Φij is symmetric. The five dimensional Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes. PACS numbers: 04.50.+h, 04.20.-q, 04.20.Cv 1. Introduction Algebraically special spacetimes play an essential role in the field of exact solutions of Einstein’s equations and many known exact solutions in four dimensions are indeed algebraically special [1]. Recently a generalization of the Petrov classification to higher dimensions was developed in [2, 3] and it turned out that many higher-dimensional solutions of Einstein’s equations are algebraically special as well (see e.g. [4]), in fact ‡ Now at: Departament de F́ısica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain http://arxiv.org/abs/0704.0435v2 Type D Einstein spacetimes in higher dimensions 2 so far there is only one known solution identified [5] as algebraically general - the static charged black ring [6]. There is, however, one important difference between four dimensional and n > 4 dimensional cases - the Goldberg-Sachs theorem does not hold in higher dimensions. Recall that for n = 4 the Goldberg-Sachs theorem implies that principal null directions of an algebraically special vacuum spacetime are necessarily geodetic and shearfree. It was stressed already in [7, 8] that the Goldberg-Sachs theorem cannot be straightforwardly extended to higher dimensions. Namely in [7] it was pointed out that principal null directions (or Weyl aligned null directions - WANDs [2]) of the n = 5 Myers-Perry black holes [9] are shearing though the spacetime is of type D. In [8] it was shown that in fact all vacuum, n > 4, type N and III expanding spacetimes are shearing. In [10] it was also shown that for n > 4, n odd, all geodetic WANDs with non-vanishing twist are again shearing. In this paper we study various properties of algebraically special vacuum spacetimes, such as geodeticity of multiple WANDs (not guaranteed in higher dimensions - another “violation” of the Goldberg-Sachs theorem) and relationships between optical matrices Sij and Aij and the Weyl tensor. Before approaching these problems, we study in the first part of the paper (sections 3 and 4) constraints on Weyl types of the spacetime following from various assumptions on the geometry. In section 3 we show that in arbitrary dimension (i.e., hereafter, n ≥ 4) the only Weyl types compatible with static spacetimes (and expanding stationary spacetimes with appropriate reflection symmetry) are types G, Ii, D and O. In section 4 we study direct or warped product spacetimes. It turns out that warped spacetimes with a one-dimensional Lorentzian factor are again of types G, Ii, D and O and that warped spacetimes with a two-dimensional Lorentzian factor are necessarily of type D or O. This also implies that spherically symmetric spacetimes are of type D or O. It follows that type D spacetimes play an important role as the simplest non-trivial case compatible with the above mentioned assumptions. Therefore, in the second part of the paper (sections 5 and 6) we focus on studying properties of type D Einstein spacetimes (i.e., vacuum with an arbitrary cosmological constant), dropping, however, the assumptions that the spacetime is static, stationary or warped. In section 5 we study type D spacetimes in arbitrary dimension and analyze geodeticity of WANDs. It turns out that in a “generic” case in vacuum the multiple WANDs are geodetic. Let us also point out that negative boost weight Weyl components do not enter relevant equations and thus the same results also hold for multiple WANDs in type II Einstein spacetimes. Surprisingly, it also turns out that explicit examples of special vacuum type D spacetimes not belonging to our “generic” class and admitting non-geodetic multiple WANDs can easily be constructed. Such examples for arbitrary dimension n ≥ 7 are given in section 5.4. This shows that there exist even more striking “violations” of the Goldberg-Sachs theorem in higher dimensions than the examples with non-zero shear discussed above. In section 5 we also study various properties of shearfree type D vacuum spacetimes. Perhaps not surprisingly, the situation in five dimensions is considerably simpler than for n > 5. In fact it turns out that for n = 5 the Weyl tensor of type D is fully determined by a 3 × 3 real matrix Φij . At the same time, five dimensional gravity is already an interesting arena where qualitatively new phenomena appear. We thus devote section 6 to five dimensional vacuum type D spacetimes. We study relationships between the Weyl tensor represented by Φij and optical matrices Sij and Aij . One Type D Einstein spacetimes in higher dimensions 3 of the results is that for “generic” spacetimes with non-twisting WANDs (Aij = 0) the antisymmetric part of Φij , Φ ij , vanishes and the symmetric part Φ ij is aligned with Sij (in the sense that the matrices Φ ij and Sij can be diagonalized together). Similarly, in the “generic” case the condition ΦAij = 0 implies vanishing of Aij . Again, there exist particular cases for which the “generic” proof does not hold, see section 6 for details. In this section a simple explicit example of a five-dimensional vacuum type D spacetime, the Myers-Perry metric, is also presented and Sij , Aij , Φ ij , and ΦAij are explicitly given. Finally in section 7 we concisely summarize main results and in the Appendix we briefly study geometric optics of type D Kerr-NUT-AdS metrics in arbitrary dimension. 2. Preliminaries Let us first briefly summarize our notation, further details can be found in [8]. In an n-dimensional spacetime let us introduce a frame of n real vectors m(a) (a, b . . . = 0, . . . , n − 1): two null vectors m(0) = m(1) = n, m(1) = m(0) = ℓ and n−2 orthonormal spacelike vectors m(i) = m(i) (i, j . . . = 2, . . . , n−1) satisfying ℓaℓa = n ana = ℓ am(i)a = n am(i)a = 0, ℓ ana = 1, m (i)am(j)a = δij . (1) The metric now reads gab = 2ℓ(anb) + δijm b . (2) We will use the following decomposition of the covariant derivative of the vector ℓ and the covariant derivative in the direction of ℓ ℓa;b = Lcdm b , D ≡ ℓ a∇a. (3) Note that ℓ is geodetic iff Li0 = 0 and for an affine parameterization also L10 = 0. We will often use the symmetric and antisymmetric parts of Lij , Sij ≡ L(ij) (its trace S ≡ Sii), Aij ≡ L[ij]. In case of geodetic ℓ, the trace of Sij represents expansion θ ≡ 1 n−2S, the tracefree part of Sij is shear σij ≡ Sij − θδij and the antisymmetric matrix Aij is twist.§ Optical scalars can be expressed in terms of ℓ (when Li0 = 0 = L10) σ2 ≡ σijσji = ℓ(a;b)ℓ(a;b) − 1n−2 , θ = 1 ;a, ω 2 ≡ AijAij = ℓ[a;b]ℓa;b. (4) The decomposition of the Weyl tensor in the frame (1) in full generality is given by [8] Cabcd = 4C0i0j n{am d} + 8C010i n{aℓbncm d} + 4C0ijk n{am + 4C0101 n{aℓbncℓd} + 4C01ij n{aℓbm d} + 8C0i1j n{am + Cijklm d} + 8C101i ℓ{anbℓcm + 4C1ijk ℓ{am d} + 4C1i1j ℓ{am where the operation { } is defined as w{axbyczd} ≡ 12 (w[axb]y[czd] + w[cxd]y[azb]). § For the sake of brevity, throughout the paper we shall refer to the corresponding quantities for non- geodetic congruences as “expansion”, “shear”, and “twist” (in inverted commas), bearing in mind that in that case expressions (4) do not hold. Type D Einstein spacetimes in higher dimensions 4 In the second part of this paper we will focus on type D spacetimes, possessing (in an adapted frame) only boost order zero components (see [8]) C0101, C01ij , C0i1j , Cijkl . For simplicity let us define the (n− 2)× (n− 2) real matrix Φij ≡ C0i1j , (5) with ΦSij , Φ ij , and Φ ≡ Φii being the symmetric and antisymmetric parts of Φij and its trace, respectively. Let us observe that for static spacetimes and for a large class of warped geometries one has ΦAij = 0 (see section 4). Note also that the above mentioned boost order zero components of the Weyl tensor are not completely independent. In fact from the symmetries and the tracelessness of the Weyl tensor (cf. eqs. (7) and (9) in [8]) it follows that C01ij = 2C0[i|1|j] = 2Φ ij , C0(i|1|j) = Φ ij = − 12Cikjk , C0101 = − Cijij = Φ. (6) The type D Weyl tensor is thus completely determined by m(m−1) independent components of ΦAij and m2(m2−1) independent components of Cijkl , where n = m−2.‖ 3. Static and stationary spacetimes 3.1. Static spacetimes Algebraically special spacetimes in higher dimensions are characterized by the existence of preferred null directions - Weyl aligned null directions (WANDs). A necessary and sufficient condition for a null vector ℓ being WAND in arbitrary dimension is [3, 11] ℓbℓcℓ[eCa]bc[dℓf ] = 0, (7) where Cabcd is the Weyl tensor. Let us now assume that a spacetime of interest is algebraically special and thus the equation (7) possesses a null solution ℓ = (ℓt, ℓA), A = 1 . . . n − 1 (note that necessarily ℓt 6= 0 and at least one of the remaining components is also non-zero). For static spacetimes the metric does not depend on the direction of time and consequently the form of the metric and of the Weyl tensor remains unchanged under the transformation t̃ = −t. Therefore, in these new coordinates equation (7) has the same form as in the original coordinates and admits a second solution ñ = (ℓt, ℓA). In the original coordinates n = (−ℓt, ℓA). Thus for static spacetimes the existence of a WAND ℓ implies the existence of a distinct WAND n which in fact has the same order of alignment. The only Weyl types compatible with this property are types G, Ii and D (or, trivially, O, i.e. conformally flat spacetimes). Therefore Proposition 1 All static spacetimes in arbitrary dimension are of Weyl types G, Ii or D, unless conformally flat. In fact explicit examples of static spacetimes of these Weyl types are known - charged static black ring (type G - [5]), vacuum static black ring (type Ii - [11]), the Schwarzschild-Tangherlini black hole (type D - [8]) and the Einstein universe R×Sn−1 (type O - cf. the results summarized in section 4). Cf. also the static examples given in [4]. ‖ In the standard n = 4 (i.e., m = 2) case these are essentially the imaginary and real part of Ψ2. More specifically, with the conventions of [1], one has Φ Φδij with Φ = −2Re(Ψ2), = Φ23 = −Im(Ψ2) as the only essential component of Φ , while the Cijks reduce to the only non-trivial component C2323 = −Φ. Type D Einstein spacetimes in higher dimensions 5 Note that in four dimensions there is no type G and type I is equivalent to type Ii [2, 3]. Thus for n = 4 only types I, D and O are compatible with static spacetimes. This was discussed already in [12] in the case of static, n = 4, vacuum spacetimes (see also additional comments in [13] and in section 6.2 of [1]). 3.2. Stationary spacetimes One can use the same arguments as above for stationary spacetimes with the metric remaining unchanged under reflection symmetry involving time and some other coordinates. E.g. in Boyer-Lindquist coordinates the Kerr metric remains unchanged under t̃ = −t, φ̃ = −φ and n = 5 Myers-Perry under t̃ = −t, φ̃ = −φ, ψ̃ = −ψ or, for general dimension, Myers-Perry under t̃ = −t, φ̃i = −φi. Note, however, that in contrast to the static case, in some special stationary cases one could in principle get from the original WAND ℓ a “new” WAND n = −ℓ which represents the same null direction. In order to deal with these special cases we note that the “divergence scalar” (or, loosely speaking, “expansion”, since it does coincide with the standard expansion scalar in the case of geodetic, affinely parameterized null directions) of both WANDs n and ℓ related by reflection symmetry is the same (as well as all the other optical scalars and the geodeticity parameters - this also applies to the static case), i.e. ℓa;a = n ;a while the “expansion” of −ℓ is equal to −ℓa;a. Therefore for all “expanding” spacetimes n 6= −ℓ. Thus Proposition 2 In arbitrary dimension, all stationary spacetimes with non-vanishing divergence scalar (“expansion”) and invariant under appropriate reflection symmetry are of Weyl types G, Ii or D, unless conformally flat. Note also that it is shown in [14] that Kerr-Schild spacetimes with the assumption R00 = 0 are of type II (or more special) in arbitrary dimension with the Kerr-Schild vector being the multiple WAND. Therefore all Kerr-Schild spacetimes that are either static or belong to the above mentioned class of stationary spacetimes are necessarily of type D. In particular, the Myers-Perry metric in arbitrary dimension is thus of type D.¶ In addition to the rotating Myers-Perry black holes for n ≥ 4, of type D, we can mention a number of physically relevant solutions as explicit examples of spacetimes subject to Proposition 2.+ First, rotating vacuum black rings [17], of type Ii [11]. To our knowledge, no stationary (non-static) type G solution has been so far explicitly identified. It is, however, plausible to expect that a rotating charged black ring (so far unknown in the standard Einstein-Maxwell theory) will be of type G as its static counterparts. Further interesting examples fulfilling our assumptions are expanding ¶ This was already known in the case n = 5 [4, 8]. Furthermore, it has been demonstrated recently in [15] by explicit computation of the full curvature tensor that the family [16] of higher dimensional rotating black holes with a cosmological constant and NUT parameter is of type D for any n. We observe in addition that, using the connection 1-forms given in [15], it is also straightforward to show (see the Appendix) that the mutiple WANDs (which are related by reflection symmetry) of all such solutions are twisting, expanding and shearing (except that the shear vanishes for n = 4). The fact that the WANDs found in [15] are complex is only due to the analytical continuation trick used in [16] to cast the line element in a nicely symmetric form - the WANDs of the associated “physical” spacetimes are thus real after Wick-rotating back one of the coordinates. + It is straightforward to verify the “reflexion symmetry” of the metric we mention in this context. The “expansion” condition, instead, has not been verified explicitly in all cases. However, it is plausible that these spacetimes are indeed “expanding” since they contain as special limits or subcases solutions with expansion, e.g. Myers-Perry black holes (cf. section 6.4, [8] and the preceding footnote). Type D Einstein spacetimes in higher dimensions 6 stationary axisymmetric spacetimes with n − 2 commuting Killing vector fields [18], which contain, apart from the (n = 5) black holes/rings mentioned above, also e.g. the recently obtained “black saturn” [19], doubly spinning black rings [20] and black di-rings [21]. In any dimension also rotating uniform black strings/branes satisfy the assumptions of Proposition 2 (see section 4), and so does the ansatz recently used in [22] for the numerical construction of corresponding n = 6 non-uniform solutions. Other examples are all the stationary solutions discussed in [4] and various black ring solutions reviewed in [23]. 3.3. Remarks and “limitations” of the results First, it is worth observing that we have not used any field equations for the gravitational field in the considerations presented above and the results are thus purely geometrical. Note that one can not relax the assumption ℓa;a 6= 0 in the case of stationary spacetimes. For example, the special pp -wave metric ds2 = gijdx idxj−2dudv−2Hdu2 such that H,u = 0 (note that it is always H,v = 0 by the definition of pp -waves) and ∂u · ∂u = −2H < 0 represents stationary spacetimes (cf., e.g., [24] for the n = 4 vacuum case) that are invariant under reflection symmetry (ũ = −u, ṽ = −v) and yet of type N [25]. In fact, the geodetic multiple WAND ℓ = ∂v is non-expanding (and n = −ℓ is not a new WAND). Furthermore, if we assume a null Killing vector field k instead of a timelike one we are led to different conclusions. Namely, it is easy to show that k must be geodetic, shearfree and non-expanding, which for Rabk akb = 0 implies that k is a twistfree WAND [10]. We thus end up with a subfamily of the Kundt class, for which (under the alignment requirement Rabk a ∝ kb, obeyed e.g. in vacuum) the algebraic type is II or more special [10] (cf. section 24.4 of [1] for n = 4). In particular, a similar argument applies locally at Killing horizons, where the type must thus be again II or more special (provided Rabk a ∝ kb).∗ This is in agreement with the result of [26] for generic isolated horizons. As an explicit example, vacuum black rings (which are of type Ii in the stationary region) become locally of type II on the horizon [11]. Finally, spacelike Killing vectors do not impose any constraint on the algebraic type of the Weyl tensor, in general, and all types are in fact possible. For example charged static black rings are of type G, vacuum black rings of type Ii, vacuum black holes of type D, and they all admit at least one spacelike Killing vector; Kundt spacetimes can be constructed that admit axial symmetry with all types II, D, III and N being possible (see, e.g., [1] for n = 4). 4. Direct/warped product spacetimes In this section we show that the algebraic types discussed above also characterize certain classes of direct/warped product geometries of physical relevance. In addition we discuss some optical properties of these spacetimes. ∗ The proof is a bit more tricky in this case since the Killing vector is null only at the horizon. Still, one can adapt techniques used in [26, 27] for related investigations. Note that the horizon of higher dimensional stationary black holes is indeed a Killing horizon (at least in the non-degenerate case) [27]. Type D Einstein spacetimes in higher dimensions 7 4.1. Weyl tensor Let us consider two (pseudo-)Riemannian spaces (M1, g(1)) and (M2, g(2)) of dimension n1 and n2 (n1, n2 ≥ 1 and n1 + n2 ≥ 4), parameterized by coordinates xA (A,B = 0, . . . , n1 − 1) and xI (I, J = n1, . . . , n1 + n2 − 1), respectively. Using adapted coordinates xµ (µ, ν = 0, . . . , n1 + n2 − 1) constructed from the coordinates xA of M1 and x I of M2, we define the direct product (M, g) to be the product manifold M = M1 ×M2, of dimension n = n1 + n2, equipped with the metric tensor g(xµ) = g(1)(x A) ⊕ g(2)(xI) defined (locally) by gAB = g(1)AB, gIJ = g(2)IJ , gAI = 0. For the sake of definiteness, we shall assume hereafter that (M1, g1) is Lorentzian and (M2, g2) is Riemannian. In general, any geometric quantity which can be split like the product metric (i.e., with no mixed components and with the A[I] components depending only on the xA[xI ] coordinates) is called a “product object” (or “decomposable”). Various interesting geometrical properties then follow [28] and, in particular, the Riemann and Ricci tensors and the Ricci scalar are all decomposable. As a consequence, a product space is an Einstein space iff each factor is an Einstein space and their Ricci scalars satisfy R(1)/n1 = R(2)/n2 [28]. Using the above coordinates it follows from the standard definition that the mixed components of the Weyl tensor are given by CABCI = CABIJ = CAIJK = 0, (8) CAIBJ = − g(1)ABR(2)IJ + g(2)IJR(1)AB R(1) +R(2) (n− 1)(n− 2) g(1)ABg(2)IJ , (9) where R(1)AB [R(2)IJ ] is the Ricci tensor of (M1, g1) [(M2, g2)]. For the non-mixed components one has to distinguish the special cases n1 = 1, 2 (and the “symmetric” cases n2 = 1, 2, which we omit for brevity). If n1 = 1 there are of course no non-mixed components CABCD since now the x A span a one-dimensional space. If n1 = 2 there is only one independent component, i.e. C0101 (notice that here, exceptionally, 0 and 1 are not frame indices but refer to the coordinates x0 and x1 in the factor space M1). For n1 ≥ 3, CABCD = C(1)ABCD + (n− 2)(n1 − 2) g(1)A[CR(1)D]B − g(1)B[CR(1)D]A (n− 1)(n− 2) R(2) −R(1) n2(n2 + 2n1 − 3) (n1 − 1)(n1 − 2) g(1)A[Cg(1)D]B (n1 ≥ 3), (10) where C(1)ABCD is the Weyl tensor of (M1, g1), whereas the remaining non-mixed components are given for any n1 ≥ 1 by CIJKL = C(2)IJKL + (n− 2)(n2 − 2) g(2)I[KR(2)L]J − g(2)J[KR(2)L]I (n− 1)(n− 2) R(1) −R(2) n1(n1 + 2n2 − 3) (n2 − 1)(n2 − 2) g(2)I[Kg(2)L]J (n2 ≥ 3), (11) where C(2)IJKL is the Weyl tensor of (M2, g2). It is thus obvious that the Weyl tensor is not decomposable, in general. It turns out that the Weyl tensor is decomposable iff both product spaces are Einstein spaces and n2(n2 − 1)R(1) + n1(n1 − 1)R(2) = 0 (the latter condition is identically satisfied whenever n1 = 1 or n2 = 1, while for n1 = 2 [n2 = 2] it implies that (M1, g1) [(M2, g2)] must be of constant curvature). When the Weyl tensor is decomposable the only non-vanishing components take the simple form CABCD = C(1)ABCD, CIJKL = C(2)IJKL. Therefore, in particular, the Type D Einstein spacetimes in higher dimensions 8 product space is conformally flat iff both product spaces are of constant curvature and n2(n2 − 1)R(1) + n1(n1 − 1)R(2) = 0. Determining the possible algebraic types of the Weyl tensor requires considering various possible choices for the dimension n1 of the Lorentzian factor. If n1 = 1, the full metric can always be cast in the special static form ds −dt2 + gIJdxIdxJ . Recalling the result of section 3, the Weyl tensor can thus only be of type G, Ii, D or O. In particular, one can show that C0i1j = C0j1i, so that for direct product spacetimes with n1 = 1 one has Φ ij = 0 identically. If n1 ≥ 2, it is convenient to adapt the null frame (1) to the natural product structure, so that gab = 2ℓ(anb) + δÂB̂m b + δÎĴm b (where Â, B̂ = 2, . . . , n1 − 1, Î , Ĵ = n1, . . . , n − 1 are now frame indices, and the frame vectors do not have mixed coordinate components, e.g. ℓI = 0 = nI etc.). From (10) and (11) it thus follows that CABCD and CIJKL do not give rise to mixed frame components, and from (9) that CAIBJ does not give rise to non-mixed frame components. Hence the only non-vanishing mixed components are (ordered by boost weight) C0Î0Ĵ = − R(1)00δÎĴ , C0ÎÂĴ = − R(1)0ÂδÎĴ , 0Î1Ĵ = − 1 (2)ÎĴ +R(1)01δÎĴ R(1) +R(2) (n− 1)(n− 2) ÂÎB̂Ĵ = − 1 R(2)ÎĴδÂB̂ +R(1)ÂB̂δÎĴ R(1) +R(2) (n− 1)(n− 2) , (12) C1ÎÂĴ = − n− 2R(1)1ÂδÎĴ , C1Î1Ĵ = − n− 2R(1)11δÎĴ . The non-mixed frame components are given for n1 = 2 by C0101 = − 2(n2 + 1) (n2 − 1)R(1) + 2R(2) (n1 = 2), (13) and for n1 ≥ 3 by 0Â0B̂ (1)0Â0B̂ (n− 2)(n1 − 2) R(1)00δÂB̂, 010 (1)010 (n− 2)(n1 − 2) (1)0 C0ÂB̂Ĉ = C(1)0ÂB̂Ĉ − (n− 2)(n1 − 2) R(1)0[ĈδB̂]Â, C0101 = C(1)0101 − (n− 2)(n1 − 2) R(1)01 (n− 1)(n− 2) R(2) −R(1) n2(n2 + 2n1 − 3) (n1 − 1)(n1 − 2) 01ÂB̂ (1)01ÂB̂ (n1 ≥ 3), (14) C0Â1B̂ = C(1)0Â1B̂ + (n− 2)(n1 − 2) R(1)ÂB̂ +R(1)01δÂB̂ (n− 1)(n− 2) R(2) −R(1) n2(n2 + 2n1 − 3) (n1 − 1)(n1 − 2) ÂB̂ĈD̂ = C(1)ÂB̂ĈD̂ + (n− 2)(n1 − 2) R(1)B̂[D̂δĈ] −R(1)Â[D̂δĈ]B̂ Type D Einstein spacetimes in higher dimensions 9 (n− 1)(n− 2) R(2) −R(1) n2(n2 + 2n1 − 3) (n1 − 1)(n1 − 2) B̂[D̂ Ĉ] ΠĴK̂L̂ (2)ÎĴK̂L̂ (n− 2)(n2 − 2) Î[K̂ (2)L̂]Ĵ Ĵ[K̂ (2)L̂]Î (n− 1)(n− 2) R(1) −R(2) n1(n1 + 2n2 − 3) (n2 − 1)(n2 − 2) Î[K̂ L̂]Ĵ 1ÂB̂Ĉ (1)1ÂB̂Ĉ − 2n2 (n− 2)(n1 − 2) (1)1[Ĉ B̂] 101 (1)101 (n− 2)(n1 − 2) (1)1 1Â1B̂ (1)1Â1B̂ (n− 2)(n1 − 2) R(1)11δÂB̂. (The expression for C ÎĴK̂L̂ holds only when n2 ≥ 3, while for n2 = 2 one gets only one component C2323 similar to (13).) For n1 = 2 the Weyl tensor of (M1, g1) of course vanishes, and in addition we have R(1)00 = 0 = R(1)11 identically (any 2-space satisfies 2R(1)AB = R(1)g(1)AB). Therefore among the above components (12) and (13) only the boost weight zero components C0Î1Ĵ and C0101 survive, so that the corresponding spacetime can be only of type D (or conformally flat), and ℓ and n, as chosen above, are multiple WANDs. Note also that Φij reduces to ΦÎĴ = C0Î1Ĵ = C0Ĵ1Î in this case, therefore ΦAij = 0. As an example, the higher dimensional electric Bertotti-Robinson solutions fall in this class, cf., e.g, [29, 30]. For n1 = 3, again the Weyl tensor of (M1, g1) vanishes. With the additional assumption that (M1, g1) is Einstein, we get R(1)00 = R(1)11 = R(1)0 = R(1)1 = 0 (here  = 2 only), and as above the Weyl tensor is of type D with ΦAij = 0. Similarly, for any n1 > 3, if (M1, g1) is an Einstein space the only non-zero mixed Weyl components (12) will have boost weight zero, and the non-mixed components (14) simplify considerably. As a particular consequence, if (M1, g1) is an Einstein space of type D, (M, g) will also be of type D (but now ΦAij 6= 0, in general) - this is the case, for example, of uniform black strings/branes (either static or rotating, see also the discussion concluding this section). If (M1, g1) is of constant curvature, (M, g) will be of type D with ΦAij = 0 (or O) - this includes the higher dimensional magnetic Bertotti-Robinson solutions [29]. One can consider other special cases using similar simple arguments. A spacetime conformal to a direct product spacetime is called a warped product spacetime if the conformal factor depends only on one of the two coordinate sets xA, xI (see e.g. [1]). Obviously, the algebraic type of two conformal spaces is the same.♯ Some of the results presented above can thus be straightforwardly generalized to the more general case of warped products. For example, Proposition 3 In arbitrary dimension, a warped spacetime with a one-dimensional Lorentzian (timelike) factor can be only of type G, Ii, D (with Φ ij = 0) or O. This case includes, in particular, the conclusion of section 3 for static spacetimes. As warped non-static/non-stationary examples we can mention the de Sitter universe (in global coordinates) and FRW cosmologies. For n = 4 Proposition 3 reduces to a result of [32]. ♯ This is true also for doubly warped product spacetimes discussed in [31], so that Propositions 3 and 4 hold also in that case. Type D Einstein spacetimes in higher dimensions 10 Furthermore, Proposition 4 In arbitrary dimension, a warped spacetime with a two-dimensional Lorentzian factor can be only of type D (with ΦAij = 0) or O. Cf. again [32] for n = 4. Notice that in this case the line element can always be cast in one of the two (conformally related) forms ds2 = 2A(u, v)dudv + f(u, v)hIJ(x)dx IdxJ or ds2 = 2f̃(x)A(u, v)dudv + gIJ(x)dx IdxJ (so that multiple WANDs are given by ∂u and ∂v), which include a number of known spacetimes. For example, the first possibility includes all spherically symmetric spacetimes, hence as a special case of Proposition 4 we have Proposition 5 In arbitrary dimension, a spherically symmetric spacetime is of type D (with ΦAij = 0) or O. For n = 4 this has been known for a long time (see e.g. [33] and sections 15.2, 15.3 of [1]), and in this case ΦAij = 0 means that Ψ2 is real (see the footnote on p. 4). For n > 4 this result has been proven in [34] in the static case. Other properties of decomposable Weyl tensors were discussed in [2]. 4.2. “Factorized” geodetic null vector fields Let us define an n-dimensional spacetime (M, g) as the warped product of an n1- dimensional Lorentzian space (M1, g(1)) and an n2-dimensional Riemannian space (M2, g(2)), with n = n1+n2 as in the preceding subsection. Hereafter we shall assume n1 ≥ 2. Using the adapted coordinates defined above, the metric can take one of the following two forms ds2 = gABdx AdxB + f(xA)hIJdx IdxJ , (15) ds2 = f̃(xI)hABdx AdxB + gIJdx IdxJ , (16) where gAB, hAB = g(1)AB depend only on the x A coordinates and gIJ , hIJ = g(2)IJ only on the xI coordinates. Given a null vector ℓ(1) = ℓ (1)∂A ofM1, this can be “lifted” to define a null vector ℓ of M with covariant components ℓA = ℓ(1)A (functions of the x A only) and ℓI = 0. From equations (15), (16) it follows that if ℓ(1) is geodetic (and affinely parameterized) in M1 then ℓ is automatically geodetic (and affinely parameterized) in M . We can thus “compare” the optical scalars of ℓ(1) in M1 with those of ℓ in M . For the warped metric (15), with the definitions (4) one finds σ2 = σ2(1) + (n1 − 2)n2 n1 + n2 − 2 θ(1) − (ln f),Aℓ n1 + n2 − 2 (n1 − 2)θ(1) + (ln f),Aℓ , (17) ω2 = ω2(1), where σ2 , θ(1) and ω are the optical scalars of ℓ(1) in (M1, g(1)). For the warped metric (16) one has σ2 = f̃−2 σ2(1) + (n1 − 2)n2 n1 + n2 − 2 θ2(1) n1 − 2 n1 + n2 − 2 f̃−1θ(1), (18) ω2 = f̃−2ω2(1). Type D Einstein spacetimes in higher dimensions 11 The special case of direct products is recovered for f, f̃ = const. (which can be rescaled to 1), in which case the shear of the full spacetime originates in the shear and expansion of the Lorentzian factor (while expansion and twist are essentially the same as in (M1, g(1))). Note that for n1 = 2 the definitions (4) for σ and θ(1) become formally singular because of the normalization, but for a Lorentzian 2-space (e.g., ds2 = 2A(u, v)dudv with the geodetic null vector ℓ = A−1∂v) one has ℓ(a;b)ℓ (a;b) = ℓa;a = ℓ[a;b]ℓ a;b = 0, so that we can essentially take σ2 = θ(1) = ω(1) = 0 and formulae (17), (18) still hold. The results of this section can be applied to several known solutions. For example, static [rotating] black strings and branes (i.e, direct products of Schwarzschild [Kerr] cross a flat space) are type D vacuum spacetimes with two shearing, expanding, twistfree [twisting] multiple WANDs. As such, they clearly “violate” the Golberg- Sachs theorem. In addition, spherically symmetric solutions in any dimensions (which necessarily take the metric form (15) with n1 = 2) are type D spacetimes with two shearfree, expanding, twistfree multiple WANDs (independently of any specific field equations; in the “exceptional case” (ln f),Aℓ A = 0 the vector ℓ is non-expanding, e.g. for Bertotti-Robinson/Nariai geometries, or for null generators of horizons). 5. Type D Einstein spacetimes in higher dimensions From the results of the previous sections it follows that type D spacetimes are the simplest non-trivial examples of static/stationary (“expanding” and with an appropriate reflection)/warped spacetimes. Therefore we will focus on type D spacetimes in general (without assuming staticity etc.). Recall that the quantities/symbols used below (e.g. Φij , Lij , D) are defined in section 2. 5.1. Algebraic conditions following from the Bianchi equations Various contractions of Bianchi identities Rabcd;e +Rabde;c +Rabec;d = 0 (19) lead to a set of first-order PDEs for frame components of the Riemann tensor given in Appendix B of [8]. In the following we shall concentrate on Einstein spaces (defined by Rab = gab), for which the same set of equations holds unchanged also for components of the Weyl tensor. In case of algebraically special spacetimes, some of these differential equations reduce to algebraical equations due to the vanishing of some components of the Weyl tensor. Here we derive algebraic conditions following from the Bianchi equations for type D Einstein spacetimes. These conditions will be employed in subsequent sections. In particular, by contracting (19) with m(i), ℓ, m(j), m(k) and ℓ (equation (B.8) in [8]) and assuming to have a type D Einstein space we get the first algebraic condition ΦijLk − ΦikLj + 2ΦAkjLi − CisjkLs = 0, (20) where we denoted Li0 by Li. We will also denote LiLi by L. The second algebraic equation follows from equation (B.15, [8]) 0 = 2 ΦAjkLim +Φ mjLik +Φ kmLij +ΦijAmk +ΦikAjm +ΦimAkj + CisjkLsm + CismjLsk + CiskmLsj (21) Type D Einstein spacetimes in higher dimensions 12 and contraction of k with i leads to 0 = SΦAmj +ΦAjm − (ΦSmi +ΦAmi)Sij + (ΦSji +ΦAji)Sim + 2(ΦAimAij − ΦAijAim) + 12CismjAsi. (22) By contracting m with j in equation (B.12) from [8] we get 2DΦSik = 4Φ ijAkj +ΦkjLij +ΦjiLjk − ΦkiS − ΦLik − 2ΦSisLsk − 2ΦSsk M i0 −2ΦSis Mk0 +CijksLsj , (23) where we employed Ciskj M j0 +Cijks M j0= 0 ( M j0 + M s0 = 0, cf. [8]). The symmetric part of equation (B.5, [8]) and equation (B.3) (that is equivalent to the antisymmetric part of (B.5)) give, respectively, 2DΦSik = −2ΦSik+(−2Φis+Φsi)Lsk+(−2Φks+Φsk)Lsi−2ΦSsk M i0 −2ΦSis Mk0, (24) 2DΦAik = −2ΦAik+(−2Φis+Φsi)Lsk−(−2Φks+Φsk)Lsi−2ΦAsk M i0 +2Φ Mk0 .(25) By subtracting (24) from (23) we finally obtain the third algebraic equation 0 = −ΦkiS +ΦLki +ΦkjLij + 4ΦAijAkj + (2Φkj − Φjk)Lji + 2ΦAijLjk + CijksLsj .(26) Its antisymmetric part is, thanks to CikjmAmj = 2CijksAsj , equal to equation (22) and its symmetric part reads 0 = −SΦSik +ΦSik +ΦSijSjk +ΦSkjSij + 3(ΦAijSjk +ΦAkjSji) + CijksSsj . (27) Equations (20), (22) and (27) will be extensively used in the following sections. In passing, let us observe here in what sense the n = 4 case is unique. Recalling the footnote on p. 4, from (20) we get Li = 0 (geodetic property) unless Φij = 0 (trivial case of zero Weyl tensor); equation (22) is identically satisfied (noting that necessarily ΦAij ∝ Aij when n = 4); equation (27) implies Sij ∝ δij (vanishing shear) again unless Φij = 0. Thus for n = 4 we correctly recover the standard Goldberg- Sachs result (here restricted to type D spacetimes) that multiple WANDs (PNDs) are geodetic and shearfree in vacuum (and Einstein) spaces [1]. The situation in higher dimensions, which is qualitatively different from the n = 4 case, is studied in the following sections. 5.2. WANDs in “generic” vacuum type D and II spacetimes in arbitrary dimension are geodetic In this section we study equation (20) in order to determine under which circumstances the multiple WAND ℓ is geodetic. By contracting i with k in (20) and using (6) we get 3ΦAij − ΦSij Li = ΦLj (28) and after multiplying (28) by Lj we obtain ΦSijLiLj = −ΦL. (29) By multiplying (20) by LiLj and using (29) we get 3ΦAikLi +Φ ikLi +ΦLk = 0. (30) Thus either L = 0 or −3ΦAij − ΦSij Li = ΦLj . (31) Type D Einstein spacetimes in higher dimensions 13 By adding and subtracting (28) and (31) we get ΦSijLi = −ΦLj, ΦAijLi = 0. (32) Finally multiplying (20) by Li and using (32) we get LΦAij = 0. (33) This implies that for a type D vacuum spacetime with non-vanishing ΦAij in arbitrary dimension corresponding WANDs are geodetic. In the case with vanishing ΦAij , let us choose a frame in which Φ ij is diagonal ΦSij = diag{p(2), p(3), . . . , p(n−1)}. Then from the first equation (32) it follows (p(i) +Φ)Li = 0, (34) where (from now on) we do not sum over indices in brackets. If p(i) 6= −Φ, ∀i, then Li = 0, ∀i, i.e. ℓ is geodetic. Note that so far we have employed only equation (20), which corresponds to equation (B.8) in [8] and which does not contain Weyl tensor components with negative boost order. Consequently, the same conclusions hold also for type II Einstein spacetimes. Proposition 6 In arbitrary dimension, multiple WANDs of type II and D Einstein spacetimes are geodetic if at least one of the following conditions is satisfied: i) ΦAij is non-vanishing; ii) for all eigenvalues of ΦSij: p(i) 6= −Φ. Note that the above argument can not be extended to more special algebraic classes of spacetimes since it relies on the fact that some Weyl components with boost weight zero are non-vanishing. However, it was already shown in [8] that multiple WANDs in type N and III vacuum spacetimes are geodetic (in that case with no need of extra assumptions). Therefore we can conclude that under most “generic” conditions multiple WANDs are geodetic. Note, however, that certain special type- D vacuum solutions with ΦAij = 0 and p(i) = −Φ (for some i) admit non-geodetic multiple WANDs. Explicit example of such spacetime is given in section 5.4. 5.3. Vacuum type D spacetimes with a “shearfree” WAND The algebraic equations (22) and (27) are quite complicated in general dimension and thus here we will limit ourselves to the “shearfree” case. This is of interest since it includes, for instance, the Robinson-Trautman solutions containing static black holes [35]. With the “shearfree” condition Sij = n−2δij , (35) equation (27) leads for S 6= 0 to ΦSij = n−2δij (S 6= 0), (36) whereas it is identically satisfied for S = 0. In the rest of this subsection we thus consider only the “expanding” case S 6= 0. For ΦSij in the form (36) with Φ 6= 0 the condition ii) of Proposition 6 is satisfied and thus the WAND ℓ is geodetic. Proposition 7 In arbitrary dimension, multiple “shearfree” and “expanding” WAND in a type D Einstein spacetime is geodetic whenever Φij 6= 0. Type D Einstein spacetimes in higher dimensions 14 Note that Φij has to be non-zero for type D spacetimes in four and five dimensions. Thus all such shearfree WANDs are geodetic.†† On the other hand, spacetimes with Φij = 0 are not necessarily conformally flat for n > 5 (Cijkl can be non-vanishing, and in that case equation (20) reduces to CisjkLs = 0) and in fact in section 5.4 we will present an example of such type D vacuum spacetime with a non-geodetic “shearfree” multiple WAND. Furthermore, using (35) and (36), equation (22) reads 0 = n−4 n−2SΦ ij +ΦAji + 2(Φ kiAkj +Φ jkAki) + CkmijAmk. (37) As mentioned above this is identically satisfied for n = 4. For n > 4 (and S 6= 0), if one assumes Aij = 0 it gives Φ ij = 0, while assuming Φ ij = 0 leads to CkmijAmk = 2ΦAij . On the other hand, from equation (25) with (35) and (36) we see that ΦAij = 0 implies Aij = 0, unless Φ = 0 (in which case the full Φij would be zero). We can thus summarize these results in Proposition 8 For a multiple “shearfree” and “expanding” WAND in a type D Einstein spacetime in n > 4 dimensions the following implications hold (i) Aij = 0 ⇒ ΦAij = 0. (ii) ΦAij = 0, Φ ij 6= 0 ⇒ Aij = 0. (iii) ΦAij = 0, Φ ij = 0 ⇒ CkmijAmk = 0. Note that for an arbitrary odd-dimensional spacetime with a geodetic and shearfree WAND one has Aij = 0 [10] and thus in the expanding case, θ 6= 0, by (i) ΦAij also necessarily vanish. Note also that the assumptions of (i) (i.e., σij = 0 = Aij , θ 6= 0) uniquely identify the Robinson-Trautman spacetimes (which are of type D for n > 4) in any dimensions and indeed ΦAij = 0 for the correspondingWeyl tensor [35]. In general ΦSij = n−2δij 6= 0 for Robinson-Trautman solutions [35] and by Proposition 7 the multiple WANDs are thus geodetic, however, in the next subsection we present a very special Robinson-Trautman solution with vanishing ΦSij and with a non-geodetic WAND. 5.4. An example of type D vacuum spacetimes with a non-geodetic WAND The conclusions in the preceding subsections about the geodetic character of multiple WANDs can not be (in contrast to the n = 4 case) extended to the most general case. In fact, here we point out that a special subclass of the Robinson-Trautman solutions [35] in n ≥ 7 dimensions represents type D vacuum spacetimes (with a possible cosmological constant) for which one of the multiple WANDs is non-geodetic. Namely, let us consider the vacuum family [35, 36] ds2 = r2hijdx idxj − 2dudr − 2Hdu2, 2H = K − 2r(lnP ),u − (n− 2)(n− 1) r2 (K = 0,±1), (38) where P 2 = (det hij) 1/(2−n) and hij represents an arbitrary (n − 2)-dimensional Einstein space (i, j = 2 . . . , n − 1 are, exceptionally, coordinate indices in this subsection). Using a suitable frame based on the null vectors ℓ = ∂r, n = −∂u +H∂r, (39) †† In fact, for n = 4 from the Goldberg-Sachs theorem we already knew that all multiple WANDs are automatically shearfree and geodetic. Type D Einstein spacetimes in higher dimensions 15 the only non-vanishing components of the Weyl tensor have boost weight zero and are given by [35] Cijkl = r 2(Rijkl − 2Khi[khl]j), (40) where Rijkl is the Riemann tensor associated to hij . This implies that the spacetime (38) is of type D, with Φij = 0, and that both ℓ and n are multiple WANDs. Now, the vector ℓ is geodetic, shearfree and twistfree by construction [35]. Next, one can easily show that ∇nn = −H,rn+H,idxi, (41) where, by (38), H,i = −r(lnP ),ui. Therefore n is geodetic if and only if (lnP ),ui = 0 ⇔ P = p1(u)p2(x2, x3, . . .). For a general (non-factorized) function P the multiple WAND n is thus non-geodetic (one can also easily check that it “shearfree”, “twistfree” and “expanding”). A simple explicit example of such spacetimes is obtained by extending to any n ≥ 7 the n = 7 dimensional solution discussed in [36], i.e. by taking in eq. (38) K = −1, P = f(u, z)−1/2 ρn−5(det ηαβ) ]1/(2−n) hijdx idxj = f(u, z) dz2 + V (ρ)dτ2 + V (ρ) dρ2 + ρ2ηαβdx , (42) f(u, z) = 4b(u)e2z/l l2[e2z/l − b(u)]2 , V (ρ) = where z ≡ x2, τ ≡ x3, ρ ≡ x4, ηαβ = ηαβ(x5, x6, . . .) is the metric of an (n − 5)- dimensional unit sphere (α, β = 5, . . . , n− 1), µ and l are constants and b(u) > 0 is an arbitrary function. The multiple WAND n is non-geodetic as long as db/du 6= 0. Note that there is not contradiction with the results of the previous subsections precisely because Φij = 0 here. 6. Type D vacuum spacetimes in five dimensions Let us now study the five-dimensional case. Note that the algebraic relation (6) between −2ΦSij and Cijkl is equivalent to the relation between the Ricci and the Riemann tensor of a m − 2 dimensional space. Therefore in five dimensions Cijkl is equivalent to ΦSij and thus a type D Weyl tensor in five dimesions is fully determined by Φij . In fact, for n = 5 it is possible to solve the second constraint from (6) for Cijkl : Cijkl (n=5) jk − δikΦSjl − δjlΦSik + δjkΦSil − Φ (δilδjk − δikδjl) . (43) Thus in the five dimensional case the algebraic equations we consider, (20), (21), (22), (27), can be expressed in terms of Φij , Li, and Lij . Plugging (43) into (20), recalling equation (32) and contracting with Lk one finds the equation LΦSij + 2ΦLiLj − ΦLδij = 0. (44) For n = 5 equation (21) takes the form 0 = ΦAjkLim + (Φ im + 3Φ im)Akj +Φ mjLik + (Φ ik + 3Φ ik)Ajm +Φ kmLij +(ΦAij + 3Φ ij)Amk + δij(Φ msLsk − ΦSksLsm) + δik(ΦSjsLsm − ΦSmsLsj) +δim(Φ ksLsj − ΦSjsLsk) + Φ[δijAkm + δikAmj + δimAjk]. (45) Type D Einstein spacetimes in higher dimensions 16 Equation (22) reduces to 0 = ΦAmjS + 2ΦAjm +Φ ji(Sim + 2Aim) + Φ im(Sij + 2Aij) +ΦSji(Sim − 2Aim) + ΦSmi(−Sij + 2Aij), (46) and equation (27) has the form 3[(ΦSij +Φ ij)Sjk + (Φ kj +Φ kj)Sji − SΦSki] = δik(2ΦSjsSjs − ΦS). (47) In the following sections we study (non-)geodecity of multiple WANDs (section 6.1), spacetimes admitting non-twisting WANDs Aij = 0 (section 6.2) and spacetimes with ΦAij = 0 (section 6.3). 6.1. Geodeticity of multiple WANDs It is interesting to return now to equation (20), which is related to the (non-)geodetic character of multiple WANDs and in five dimensions implies (44). Since we already know from Proposition 6 that WANDs are necessarily geodetic when ΦAij 6= 0, let us focus here on the case ΦAij = 0. If Φ = 0 we see that either L = 0 or Φ ij = 0, the latter case being now a conformally flat spacetime. Therefore an n = 5 type D Einstein spacetime requires (ΦAij = 0 and) Φ 6= 0 in order to admit a non-geodetic multiple WAND. In this case it follows from (44) that there exists an eigenframe of ΦSij such ΦSij = Φdiag(1, 1,−1), L2 = L3 = 0, (48) so that L4 6= 0 is responsible for the WAND ℓ being non-geodetic. Such spacetime is necessarily shearing since the “canonical” form of ΦSij given in equation (48) is not compatible with that of equation (36). It would be interesting to find such five dimensional vacuum type D spacetime with a non-geodetic WAND or prove that such spacetime does not exist. To summarize, Proposition 9 In five dimensions, the only type D spacetimes with non-geodetic multiple WAND ℓ are those satisfying ΦAik = 0 and Φ ik 6= 0, ΦSik = diag{Φ, Φ, −Φ}. 6.2. “Non-twisting” case - Aij = 0 In the non-twisting case Aij = 0, equation (46) reduce to ΦjiSim − ΦmiSij +ΦAmjS = 0. (49) Now we can, without loss of generality, choose a frame in which the symmetric matrix Sij is diagonal Sij = diag(s(2), s(3), s(4)). (50) Then equations (49) and (47) take the form (recall that we do not sum over indices in brackets) ΦSik(s(k) − s(i)) + ΦAik(s(k) + s(i) − S) = 0, ΦSik(s(k) + s(i) − S) + ΦAik(s(k) − s(i)) = 13δik(2Φ jsSjs − ΦS). (51) Now let us study components of the two above equations for i 6= k. By summing the two above equations we get (2s(k) − S)(ΦSik +ΦAik) = 0 (i 6= k). (52) Type D Einstein spacetimes in higher dimensions 17 In the “generic” case with 2s(i) 6= S ∀i, this implies ΦAik = 0 = Φ ik for i 6= k. (53) Consequently, ΦSij is also diagonal and from equation (51) ΦSij = diag(p(2), p(3), p(4)), p(i) = 2ΦSjsSjs − ΦS 3(2s(i) − S) . (54) Using (54), it is straightforward to express (two of) the p(i) in terms of the s(i) solving the linear relations (which are not all independent): (s(2) − s(3) − s(4))p(2) = (−s(2) + s(3) − s(4))p(3), (55) (s(2) − s(3) − s(4))p(2) = (−s(2) − s(3) + s(4))p(4), (56) (−s(2) + s(3) − s(4))p(3) = (−s(2) − s(3) + s(4))p(4). (57) Proposition 10 In five dimensions, in the “generic” (2s(i) 6= S ∀i) non-twisting (Aij = 0) type D spacetime, Φ ij also vanishes and Φ ij can be diagonalized together with Sij. Note that special cases with 2s(i) = S for some i have to be treated separately: 1) If one of s(i) = S/2, e.g. s(4) = S/2, and the others differ from S/2, 0 then only ΦS44 6= 0, all other component of ΦSij = 0 and ΦAij = 0. 2) If e.g. s(2) = s(3) = S/2, s(4) = 0 then Φ 24 = Φ 34 = Φ 44 = Φ 24 = Φ 34 = 0, the other components (ΦS22, Φ 33, Φ 23, Φ 23) are arbitrary. 6.3. Case ΦAij = 0 For ΦAij = 0 equations (46), (25) and (47) take the form 2(ΦSmiAij − ΦSjiAim +ΦAjm) + ΦSjiSim − ΦSmiSij = 0, (58) − ΦSimAij +ΦSjiAim + 2ΦAjm +ΦSjiSim − ΦSmiSij = 0, (59) 3(ΦSijSjk +Φ kjSji − SΦSki) = δik(2ΦSjlSjl − ΦS). (60) In previous section 6.2 it was efficient to choose a frame in which Sij was diagonal, however, now it is more efficient to choose a frame in which ΦSij is diagonal, ΦSij = diag{p(2), p(3), p(4)}. Then we obtain from (58)–(60) the following set of equations (2p(m) + 2p(j) − 2Φ)Amj + Smj(p(j) − p(m)) = 0, (61) (−p(m) − p(j) − 2Φ)Amj + Smj(p(j) − p(m)) = 0, (62) 3(p(i) + p(k))Sik = δik(3Sp(i) + 2Φ jlSjl − ΦS). (63) In the “generic” case p(i) + p(k) 6= 0, ∀i, k, from equation (63) Sik = diag{s(2), s(3), s(4)}, s(i) = 2ΦSjlSjl − ΦS 6p(i) . (64) From (64) we get the relations (which can be solved to fix two of the si, if desired): s(2)p(3)(p(2) + p(4)) = s(3)p(2)(p(3) + p(4)), (65) s(2)p(4)(p(2) + p(3)) = s(4)p(2)(p(3) + p(4)), (66) s(3)p(4)(p(2) + p(3)) = s(4)p(3)(p(2) + p(4)). (67) Type D Einstein spacetimes in higher dimensions 18 Subtracting (61) and (62) we obtain (p(m) + p(j))Amj = 0 and thus in the “generic” case p(m) + p(j) 6= 0, ∀m, j, Amj = 0. (68) Proposition 11 In five dimensions, the multiple WAND ℓ in a “generic” (p(i) + p(j) 6= 0, ∀i, j) type D spacetime with ΦAik = 0 and ΦSik 6= 0, is geodetic and non- twisting (Aij = 0) and Φ ik and Sij can be diagonalized together. There are some special cases to be treated: - Case a) one p(i) = 0 and Φ 6= 0: without loss of generality we choose p(2) = 0, then from (61)–(63) 2ΦSjlSjl − ΦS = 0, Sij = diag{0, S/2, S/2}, Amj = 0. - Case b) only one p(i) 6= 0: without loss of generality we choose p(4) 6= 0, p(2) = p(3) = 0 then from (61)–(63) 2Φ jlSjl − ΦS = 0, s(2) + s(3) = s(4) = S/2 and S23 is arbirary, Aij vanishes. - Case c) only one pair satisfies p(m)+ p(j) = 0, p(j) 6= 0: without loss of generality we choose p(3) + p(4) = 0, i.e. p(2) = Φ, then the diagonal components of Sij still satisfy (64), from (61)–(63) S34 is arbitrary and (p(m) + p(j))Amj = 0, 2ΦAmj = (p(j) − p(m))Smj (69) and thus if Φ 6= 0, A34 = − S34. If Φ = 0, then S34 = 0 and Sij is diagonal and A23 is arbitrary. - Case d) two pairs satisfy p(m) + p(j) = 0: without loss of generality we choose p(2) = p(3) = −p(4) = Φ. From (64) it follows that the diagonal components of Sij , s(2) and s(3), vanish and s(4) is arbitrary. Equation (63) implies that S24 and S34 are arbitrary and from equation (69) we get A23 = 0, A24 = −S24, A34 = −S34. This case is the non-geodetic case (48) from section 6.1. 6.4. An example - Myers-Perry black hole As an illustrative example we give Sij , Aij , Φ ij and Φ ij for the five-dimensional Myers-Perry black hole [9] ds2 = dx2 + ρ2dθ2 − dt2 + (x+ a2) sin2 θdφ2 + (x+ b2) cos2 θdψ2 (dt+ a sin2 θdφ+ b cos2 θdψ)2, where ρ2 = x+ a2 cos2 θ + b2 sin2 θ, ∆ = (x + a2)(x+ b2)− r02x. Two (multiple, geodetic) WANDs (related by reflection symmetry) are given by [7] (x+ a2)(x+ b2) x+ a2 x+ b2 x∂x, (70) n = α (x + a2)(x+ b2) x+ a2 x+ b2 , (71) where we chose α = −∆/2ρ2x in order to satisfy the normalization condition ℓ ·n = 1. Type D Einstein spacetimes in higher dimensions 19 As a basis of spacelike vectors we choose three eigenvectors of Sij (2) = (3) = (−ab∂t + b∂φ + a∂ψ) , (72) (4) = (a2 − b2) sin θ cos θ∂t − a tan−1 θ∂φ + b tan θ∂ψ with χ = a2 cos2 θ + b2 sin2 θ. In this frame Sij = , Aij = 0 0 −1 0 0 0 1 0 0  , (73) ΦSij = ρ2−2x 0 −1 0 0 0 ρ , ΦAij = 0 0 1 0 0 0 −1 0 0  .(74) Notice that in the static (Schwarzschild) limit (a = 0 = b so that ρ2 = x) one has Sij = δij/ x and σij = 0 = Aij , and indeed for Φij we recover the form discussed in subsection 5.3 in the shearfree expanding case and in subsection 6.2 in the “generic” non-twisting case (with p(2) = p(3) = p(4)). 7. Discussion Let us finally outline main results presented in the paper. In the first part of the paper (Sections 3 and 4) we study constraints on Weyl types of a spacetime following from various assumptions on geometry. It turns out that: - Static spacetimes are of types G, Ii, D or conformally flat (Proposition 1). - “Expanding” stationary spacetimes with appropriate reflection symmetry belong to these types as well (Proposition 2). - Warped spacetimes with one-dimensional Lorentzian factor are again of types G, Ii, D and O (Proposition 3). - Warped spacetimes with two-dimensional Lorentzian factor are necessarily of types D or O (Proposition 4), in particular this also applies to spherically symmetric spacetimes (Proposition 5). These results may have useful practical applications in determining the algebraic type of specific spacetimes (or at least in ruling out some types) just by “inspecting” the given metric and without performing any calculations. This is particularly important in higher dimensions, where it is more difficult to determine the algebraic class of a given metric. In the second part of the paper (sections 5 and 6) we study properties of type D vacuum spacetimes in general (without assuming that the spacetime is static, stationary or warped). In five dimensions a type D Weyl tensor is determined by a 3×3 matrix Φij with symmetric and antisymmetric parts being Φ ij and Φ ij , respectively. Type D Einstein spacetimes in higher dimensions 20 In general in the non-twisting case Φ is symmetric while in the twisting case antisymmetric part ΦAij appears. In higher dimensions n > 5 the (n−2)×(n−2) matrix Φij does not contain complete information about the Weyl tensor, but it still plays an important role. The matrix Φij can also be used for further classification of type D or II spacetimes, for example according to possible degeneracy of eigendirections of Φij . Special classes are also cases with Φij being symmetric or vanishing (such examples for n ≥ 7 are given in section 5.4) etc. First we focused on the geodeticity of multiple WANDs in type D vacuum space- times (these are always geodetic for n = 4). It was shown that: - The multiple WAND in a vacuum spacetime is geodetic in the “generic” case, i.e. if ΦAij 6= 0 or if all eigenvalues of ΦSij are distinct from minus the trace of Φij (Proposition - It is also geodetic in the type D, shearfree case whenever Φij 6= 0 (Proposition 7). - However, explicit examples of vacuum type D spacetimes with non-geodetic multiple WAND in n ≥ 7 dimensions are given in section 5.4. This provides us with the first examples of spacetimes “violating” the geodetic part of the Goldberg-Sachs theorem. - In five dimensions multiple WANDs are also geodetic when ΦAij = 0 and Φ ij 6= 0 has a “generic” form (Proposition 11), special cases are discussed in section 6.3. Properties of the matrix Φij , as well as the expansion and twist matrices Sij and Aij have been also studied: - For warped spacetimes with a one/two-dimensional Lorentzian factor (thus also for static spacetimes) the antisymmetric part of Φij , Φ ij , vanishes. - In vacuum type D spacetimes admitting a shearfree expanding WAND, ΦSij is pro- portional to δij and if Aij = 0 (this always holds in odd dimensions [10]) then Φ ij = 0 and in the case with ΦSij 6= 0 also vice versa (Proposition 8). - In five dimensions in a “generic” Einstein type D non-twisting spacetime, ΦAij van- ishes and eigendirections of Φij coincide with those of Sij (Proposition 10). - In five dimensions in a “generic” vacuum type D spacetime with symmetric Φij , the multiple WAND ℓ is non-twisting and eigendirections of Φij and Sij coincide (Propo- sition 11). These results provide interesting connections between geometric properties of principal null congruences and Weyl curvature. Hopefully, they can be also used for constructing exact type D solutions with particular properties. Acknowledgments V.P. and A.P. acknowledge support from research plan No AV0Z10190503 and research grant KJB100190702. Appendix A. Optics of WANDs in Kerr-NUT-AdS spacetimes in arbitrary dimension As discussed in Sec. 3.3, the assumption about non-zero “expansion” in Proposition 2 is essential. In this appendix we study optical properties of WANDs in Kerr- NUT-AdS spacetimes in arbitrary dimension [16] and show that the “expansion” in these cases is always non-vanishing. These metrics are thus subject to Proposition Type D Einstein spacetimes in higher dimensions 21 2. Indeed, it has been already shown in [15] that these spacetimes are of type D. In addition, since the expansion is non-zero, we can expect that possible (still stationary) generalizations of these spacetimes (such as charged black holes) with appropriate reflection symmetry are of types G, Ii or D (see also footnotes on page 5). This appendix also extends our example of five-dimensional Myers-Perry given in Section 6.4 to the case with NUT parameters and cosmological constant and to arbitrary dimension. Note, however, that now we use convenient but physically less “transparent” coordinates (x1, . . . , xm, ψ0, . . . ψm−1) in even dimensions n = 2m and (x1, . . . , xm, ψ0, . . . ψm) in odd dimensions n = 2m + 1, introduced in [16]. In our calculations, we employ results obtained in [15]. The metric of [16] for even and odd dimensions is, respectively, n = 2m: ds2 = A(k)µ dψk , (A.1) n = 2m+ 1: ds2 = A(k)µ dψk A(k)dψk .(A.2) The functions Qµ, A µ , A (k) and S̃ depend only on the coordinates (x1, . . . , xm) and their explicit expressions are given in [15, 16]. Appendix A.1. Even dimensions, n = 2m An orthonormal frame of 1-forms {e(A)} = {e(µ), e(m+µ)} with A = 1, 2, . . .2m, µ = 1, 2, . . . ,m, (µ) = , e(m+µ) = A(k)µ dψk (A.3) was intoduced in [15]. Denoting the duals of these forms with lower indices, let us here also define a null frame of vectors ℓ, n, m(i) by (e(m) + ie(2m)), n = −i (e(m) − ie(2m)), (A.4) with m(i) (i = 2 . . . n − 1) corresponding to e(µ), e(m+µ) (µ = 1 . . .m − 1 from now on). One can show [15] that the null vectors ℓ, n are multiple WANDs of the type D metric (A.1) and that they are geodetic (and affinely parametrized). Both WANDs are complex in the coordinates used above, but note that they become in fact real in “physical” coordinates since the metric (A.1) was obtained from a real Lorentzian metric by a Wick rotation with xm = ir in [16] and Qm < 0 in the outer stationary region, where ∂/∂r is spacelike. Thus Qm = i |Qm|, so that reintroducing r, both vectors ie(2m) and e(m) become real (e (r) = iδαm Let us now express the matrix Lij (defined in section 2) in terms of Ricci rotation coefficients, which can be easily obtained from the connection 1-forms given in [15] Lij = ℓa;bm (j) = (e(m)a;b+ie(2m)a;b)m (j) = − (γmij+iγ ij), (A.5) Type D Einstein spacetimes in higher dimensions 22 γmµµ = γ m+µ m+µ = − x2m − x2µ , (A.6) γ2mm+µ µ = − γ2mµ m+µ = − x2m − x2µ , (A.7) and with remaining Ricci rotation coefficients entering (A.5) being zero. Then Sij = r2+x2 0 δµν r2+x2µ  , Aij = 0 −δµν xµr2+x2 r2+x2 , (A.8) where terms proportional to δµν symbolically represent a (m− 1)× (m− 1) diagonal block. Note that Sij ∝ δij (that is, the shear is zero) iff n = 4. From this form of Sij it follows that shear is non-zero for arbitrary even dimension n > 4 and expansion r2 + x2µ (A.9) is non-zero in arbitrary even dimension n ≥ 4. Note indeed that the WANDs ℓ and n are related by reflection symmetry, in agreement with the discussion in section 3. The twist is also obviously non-zero for any n ≥ 4. Recall [16] finally that for n = 4 the metric (A.1) represents a subclass of the Plebański-Demiański family of type D spacetimes with two expanding, twisting and non-shearing principal null directions [1]. Appendix A.2. Odd dimensions, n = 2m+ 1 In odd dimensions, in addition to (A.3) we define (2m+1) = A(k)dψk . (A.10) Then the null frame consists of ℓ, n given in (A.4), m(i) (i = 2 . . . n−1) corresponding to e(µ), e(m+µ) (µ = 1 . . .m− 1), and e(2m+1). Again, the null vectors ℓ and n are geodetic multiple WANDs of the type D metric (A.2) [15]. Now together with (A.7) we have γm2m+1 2m+1 = − , (A.11) and thus Sij = r2+x2 0 δµν r2+x2µ 0 0 1 , Aij = 0 −δµν xµr2+x2 r2+x2 0 0 0 .(A.12) Shear, expansion and twist are thus non-zero for arbitrary odd dimension n > 4. Type D Einstein spacetimes in higher dimensions 23 References [1] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt. Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge, second edition, 2003. [2] A. Coley, R. Milson, V. Pravda, and A. Pravdová. Classification of the Weyl tensor in higher dimensions. Class. Quantum Grav., 21:L35–L41, 2004. [3] R. Milson, A. Coley, V. Pravda, and A. Pravdová. Alignment and algebraically special tensors in Lorentzian geometry. Int. J. Geom. Meth. Mod. Phys., 2:41–61, 2005. [4] A. Coley and N. Pelavas. Algebraic classification of higher dimensional spacetimes. Gen. Rel. Grav., 38:445–461, 2006. [5] M. Ortaggio and V. Pravda. Black rings with a small electric charge: gyromagnetic ratios and algebraic alignment. JHEP, 12:054, 2006 [gr-qc/0609049] [6] D. Ida and Y. Uchida. Stationary Einstein-Maxwell fields in arbitrary dimensions. Phys. Rev. D, 68:104014, 2003. [7] V. P. Frolov and D. Stojković. 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Pravdová, and R. Zalaletdinov. Generalizations of pp –wave spacetimes in higher dimensions. Phys. Rev. D, 67:104020, 2003. [26] J. Lewandowski and T. Pawlowski. Quasi-local rotating black holes in higher dimension: geometry. Class. Quantum Grav., 22:1573–1598, 2005. [27] S. Hollands, A. Ishibashi, and R. M. Wald. A higher dimensional stationary rotating black hole must be axisymmetric. Commun. Math. Phys., 271:699–722, 2007. [28] F. A. Ficken. The Riemannian and affine differential geometry of product-spaces. Ann. Math., 40:892–913, 1939. [29] P. G. O. Freund and M. A. Rubin. Dynamics of dimensional reduction. Phys. Lett. B, 97:233– 235, 1980. [30] V. Cardoso, Ó. J. C. Dias, and J. P. S. Lemos. Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions. Phys. Rev. D, 70:024002, 2004. [31] M. P. M. Ramos and E. G. L. R. Vaz. Double warped spacetimes. J. Math. Phys., 44:4839–4865, 2003. [32] J. Carot and J. da Costa. On the geometry of warped spacetimes. Class. Quantum Grav., 10:461–482, 1993. [33] J. Plebański and J. Stachel. Einstein tensor and spherical symmetry. J. Math. Phys., 9:269–283, http://arxiv.org/abs/gr-qc/0609049 http://arxiv.org/abs/hep-th/0701035 http://arxiv.org/abs/hep-th/0612005 http://arxiv.org/abs/hep-th/0702053 Type D Einstein spacetimes in higher dimensions 24 1979. [34] G. T. Horowitz and S. F. Ross. Properties of naked black holes. Phys. Rev. D, 57:1098–1107, 1998. [35] J. Podolský and M. Ortaggio. Robinson-Trautman spacetimes in higher dimensions. Class. Quantum Grav., 23:5785–5797, 2006. [36] M. Ortaggio. Higher dimensional spacetimes with a geodesic, shearfree, twistfree and expanding null congruence, gr-qc/0701036. http://arxiv.org/abs/gr-qc/0701036 Introduction Preliminaries Static and stationary spacetimes Static spacetimes Stationary spacetimes Remarks and ``limitations'' of the results Direct/warped product spacetimes Weyl tensor ``Factorized'' geodetic null vector fields Type D Einstein spacetimes in higher dimensions Algebraic conditions following from the Bianchi equations WANDs in ``generic'' vacuum type D and II spacetimes in arbitrary dimension are geodetic Vacuum type D spacetimes with a ``shearfree'' WAND An example of type D vacuum spacetimes with a non-geodetic WAND Type D vacuum spacetimes in five dimensions Geodeticity of multiple WANDs ``Non-twisting'' case - Aij=0 Case ijA=0 An example - Myers-Perry black hole Discussion Optics of WANDs in Kerr-NUT-AdS spacetimes in arbitrary dimension Even dimensions, n=2 m Odd dimensions, n=2 m + 1

0704.0436

Eigen Equation of the Nonlinear Spinor

arXiv:0704.0436v2 [physics.gen-ph] 4 Apr 2007 Eigen Equation of the Nonlinear Spinor Ying-Qiu Gu∗ Ta-tsien Li† Department of Mathematics, Fudan University, Shanghai, 200433, China Abstract How to effectively solve the eigen solutions of the nonlinear spinor field equa- tion coupling with some other interaction fields is important to understand the behavior of the elementary particles. In this paper, we derive a simplified form of the eigen equation of the nonlinear spinor, and then propose a scheme to solve their numerical solutions. This simplified equation has elegant and neat structure, which is more convenient for both theoretical analysis and numerical computation. PACS numbers: 11.10.Ef, 11.10.Lm, 11.10.-z Key Words: nonlinear Dirac equation, nonlinear spinor field, eigen equation, quaternion 1 Introduction Almost all the elementary fermions have spin-1 , which can be naturally described by spinors, so today spinors and spinor representations play a more and more important role in mathematical and theoretical physics. Noticing the limitations of the linear field equation, many physicists such as H. Weyl, W. Heisenberg, once proposed the nonlinear spinor equations[1, 2, 3, 4, 5, 6] to construct a unified field theory for elementary particles. However they have not gotten many definite results due to the mathematical difficulties. The rigorous solutions for some simple dark nonlinear spinor models were ∗email: [email protected] †email: [email protected] http://arxiv.org/abs/0704.0436v2 obtained in [7, 8, 9, 10], and we found it can provide negative pressure to guarantee a singularity-free and accelerating expanding universe[11, 12]. The theoretical proof about the existence of solitons was investigated in [13, 14, 15, 16, 17, 18]. The symmetries and many beautiful conditional exact solutions of the nonlinear spinor, vector and scalar differential equations are collected in [19]. However lots of these exact solutions seem to be non-physical. The spinor with its own electromagnetic potential was studied in [20, 21, 22, 23, 24], it was disclosed that the nonlinear spinor equations have particle like solution with anomalous magneton, and imply the exact classical mechanics and quantum mechanics for many-body[25, 26, 27]. In this paper we derive a simplified form of the eigen equation with general meaning for nonlinear Dirac equation, and then give a scheme to solve the solution. Denote the Minkowski metric by ηµν = diag[1,−1,−1,−1], Pauli matrices by ~σ = (σj) = . (1.1) Define 4× 4 Hermitian matrices as follows , γ = , β = 0 −iI . (1.2) In this paper, we adopt the Hermitian matrices (1.2) instead of Dirac matrices γµ, because this form is more convenient for calculation. For the system of a nonlinear spinor field φ in the 4−d potential Aµ, the Lagrangian describing the motion is generally given by L = φ+[αµ(~i∂µ − eAµ)− µcγ]φ+ F (γ̌, β̌), (1.3) where µ > 0 is a constant mass, F are the nonlinear coupling potential, which is usually the even polynomial of γ̌, β̌, and γ̌, β̌ are the quadratic scalars of φ defined by γ̌ = φ+γφ, β̌ = φ+βφ. (1.4) We can check γ̌ is a true-scalar, but β̌ a pseudo-scalar. The variation of (1.3) with respect to φ+ gives the dynamic equation αµ(~i∂µ − eAµ)φ = (µcγ − Fγγ − Fββ)φ, (1.5) where Fγ = , Fβ = . In the Hamiltonian form we have ~i∂tφ = Ĥφ, Ĥ ≡ ~α · (−~i∇− e ~A) + eA0 + (µc− Fγ)γ − Fββ. (1.6) In this paper we denote ~A = (A1, A2, A3) to be the spatial part of a contravariant vector Aµ. It is easy to check that the current conservation law holds ∂µq µ = 0, so we can take the normalizing condition as follows |φ|2d3x = 1. (1.7) The 4− d potential produced by spinor φ itself takes the following form µ = eα̌µ = eφ+αµφ. (1.8) 2 Simplification of the Equation Consider a spinor keeping motionless in an external magnetic field ~B = (0, 0, B). since the scale of the elementary particle is very small, we take external field B as a constant. By ∇× ~Aext = ~B, we have the general solution for external vector potential ~Aext = B(−y, x, 0) +∇Φ, (2.1) where Φ(~x) is any given smooth function. In the spherical coordinate system (r, θ, ϕ), we have ~σ · ∇ = σr∂r + (σθ∂θ + σϕ∂ϕ), (2.2) where (σr, σθ, σϕ) is given by cos θ sin θe−ϕi sin θeϕi − cos θ − sin θ cos θe−ϕi cos θeϕi sin θ r sin θ 0 −ie−ϕi ieϕi 0 . (2.3) Let Ĵ be the angular momentum operator for the spinor field Ĵ = ~r × (−~i∇) + ~~S, ~S = diag(~σ, ~σ), (2.4) then any eigenfunction of Ĵ3 = −~i∂ϕ + ~S3 takes the following form φ = (u1, u2e ,−iv1,−iv2e ϕi)T exp (2.5) with (κ = 0,±1,±2, · · · ), where uk, vk(k = 1, 2) are functions of r, θ but independent on ϕ and t. In this paper the index T stands for transposed matrix. For any spin-1 particle, it has a pole axis. If we set the pole axis as coordinate x3 = z, then Ĵ3 is commutative with the nonlinear Hamilton operator (1.6) by a U(1) gauge transformation for spinor as eΦiφ, which removes the uncertain function from external vector potential (2.1), thereby we have ~Aext = B(−y, x, 0) = Br sin θ(− sinϕ, cosϕ, 0). (2.6) For the above symmetric form of vector potential (2.6), substituting (2.5) into (1.6) we can check that all functions uk, vk can take real number. For the vector potential pro- duced by φ itself, we will find below it also takes the form of (2.6). This simplification of φ may be the essence of the gauge symmetry. In what follows, we set ~ = c = 1 as units for convenience. Making variable transformation u = u1(r, θ) + u2(r, θ)i, v = v1(r, θ) + v2(r, θ)i, (2.7) then we have { α̌0 = |u| 2 + |v|2, α̌ = (ūv − uv̄)i, γ̌ = |u|2 − |v|2, β̌ = ūv + uv̄, (2.8) with ~̌α = α̌(− sinϕ, cosϕ, 0). Substituting (2.7) and (2.8) into (1.3) we get the La- grangian of the eigen states as follows L = Re ū(∂r + ∂θ)v̄ − v̄(∂r + ∂θ)ū r sin θ (κ+ 1 )(ūv − uv̄) +(m− eA0)(|u| 2 + |v|2)− ie(ūv − uv̄)A− µ(|u|2 − |v|2) + F, (2.9) where (~r × ~A) · ez = (cosϕAy − sinϕAx) = A(r, sin θ) (2.10) including both external and inner vector potential. By variation with respect to ū, v̄, we get an elegant equation with double-helix structure eθi(∂r + ∂θ)ū = r sin θ [(κ+ 1 )u− 1 ū] + (µ+m− eA0 − Fγ)v + Fβu+ ieAu, eθi(∂r + ∂θ)v̄ = r sin θ [(κ + 1 )v − 1 v̄ ] + (µ−m+ eA0 − Fγ)u− Fβv + ieAv. (2.11) By (2.11) we find that, κ = 0 corresponds to spin 1 and κ = −1 corresponds to spin . Generally the coordinates r and θ can not be separable for nonlinear equation. The energy functional for (2.11) is given by E = 2π r2 sin θdrdθ ū(∂r + ∂θ)v̄ − v̄(∂r + ∂θ)ū r sin θ (κ+ 1 ) + eA i(ūv − uv̄) + eA0(|u| 2 + |v|2) + µ(|u|2 − |v|2) (2.12) The eigen solution of (2.11) is just the extreme point of E under the constraint of normalizing condition (|u|2 + |v|2)r2 sin θdrdθ = 1. (2.13) (2.13) is also the quantizing condition of the energy spectrum[9, 10]. 3 A Scheme for Solving Solution For general potential Aµ and F , the analytic solution of (2.11) u and v can not be solved. However they can be conveniently expressed by Fourier series of θ, and the equations of the radial functions can be derived by variation principle. For any given integer N ≥ 0, define 2N + 1 vectors Γ(θ) = (e−2Nθi, e−2(N−1)θi, · · · , e2(N−1)θi, e2Nθi), (3.1) U(r) = (U−N(r), U−(N−1)(r), · · · , U(N−1)(r), UN(r)) , (3.2) V (r) = (V−N(r), V−(N−1)(r), · · · , V(N−1)(r), VN(r)) T . (3.3) The eigen solution of (2.11) with even parity must take the form u = Γ · U = 2nθi, v = Γ̄ · V eθi = (−2n+1)θi, (3.4) and the eigen solution with odd parity takes u = Γ · Ueθi = (2n+1)θi, v = Γ̄ · V = −2nθi. (3.5) In what follows we only consider (3.4). For (3.5) we have similar results. For the cases κ 6= 0 and κ 6= −1, i.e. for the cases with nonzero magnetic quantum number, the solution must have consistent conditions at θ = 0, π as u(r, 0) = u(r, π) = Un(r) ≡ 0, v(r, 0) = v(r, π) = Vn(r) ≡ 0. (3.6) For this case, (3.4) minus (3.6) we get 2nθi − 1), v = −2nθi − 1)eθi. (3.7) By the form (3.7) we have e2nθi − 1 2i sin θ = (1 + e2θi + e4θi + · · ·+ e2(n−1)θi)eθi, (3.8) which removes the singularity of (2.11) at θ = 0, π. For the spin 1 state, i.e. for the case κ = 0, we can check from (2.11) that the solution U, V are all real functions. But for the spin −1 state, i.e. for the case κ = −1, the solution U, V are all pure imaginary functions. In what follows we only consider the real case. For the covariant quadratic forms (2.8), by (3.3) we have α̌0 = U TPU + V T P̄ V, α̌ = UT (Q+ −Q)V i, γ̌ = UTPU − V T P̄V, β̌ = UT (Q+ +Q)V, (3.9) where P = Γ+Γ and Q = ΓTΓeθi are (2N +1)× (2N +1) matrices with components as Pm,n = exp[2(n−m)θi], (−N ≤ n,m ≤ N), Qm,n = exp[(2(n+m) + 1)θi]. (3.10) By (3.9) and (3.10) we have α̌0 = n,m=−N (UnUm + VnVm)(e −2(n−m)θi + e2(n−m)θi), (3.11) n,m=−N (UnUm − VnVm)(e −2(n−m)θi + e2(n−m)θi), (3.12) n,m=−N UnVm(e −2(n−m)θi − e2(n−m−1)θi)eθii, (3.13) n,m=−N UnVm(e −2(n−m)θi + e2(n−m−1)θi)eθi. (3.14) The dynamic equation of the potential A is given by −∆A = eα̌ = e[a0(r) sin θ + a1(r) sin 3θ + · · · ], = e[a0(r)(e −2θi − 1) + a1(r)(e −4θi − e2θi) + · · · ]eθii. (3.15) Substituting (3.4), (3.7), (3.8) and (3.11)∼(3.15) into (2.11) and directly comparing the coefficients of all e2nθi, we can easily get a truncated ordinary differential equation of U(r), V (r). However the convergence this cut-off equation to the original solution needs proof. A more credible method to get the efficient equation of U(r), V (r) is via variational principle. The variational equation can be obtained by the following procedure. Define operators T̂u = ΓT (θ)e−θi sin θdθ, T̂v = Γ+(θ) sin θdθ, (3.16) then T̂u left multiplying the first equation of (2.11) and T̂v left multiplying the second give the variational equation of U(r), V (r). The coefficient matrix of U ′(r) and V ′(r) is the same positive definite symmetric matrix with components Mn,m = Pn,m sin θdθ = 1− 4(n−m)2 . (3.17) The other coefficient matrices can also be similarly obtained. The convergence of expansion (3.4) and the consistent condition (3.6) seem to have closely relation with the structure of nonlinear potential F (γ̌, β̌), which reflects the important properties of the elementary particles such as the exclusion princi- ple. The above procedure is valid for extensive models. The neat and elegant re- sults are profoundly rooted in the quaternionic structure of the physical variables and spacetime[28, 29, 30], so the 3 + 1 dimensional Universe is a miraculous masterpiece with unique feature. References [1] D. Iuanenko, Sov. Phys. 13, 141-149 (1938) [2] H. Weyl, Phys. Rev. 77, 699-701 (1950) [3] W. Heisenberg, Physica 19, 897-908 (1953) [4] K. Johnson, Phys. Lett. 78B, 259-262 (1978) [5] P. Mathieu, Phys. Rev. D29, 2879-2883 (1984) [6] A. F. Ranada, Classical nonlinear Dirac field models of extended particles, In:Quantum theory, groups, fields and particles, edited by A.O.Barut, Amster- dam, Reidel, 1983 [7] R. Finkelsten, et al, Phys. Rev. 83(2), 326-332(1951) [8] M. Soler, Phys. Rev. D1(10), 2766-2767(1970) [9] Y. Q. Gu, Some Properties of the Spinor Soliton, Adv in Appl. Clif. Alg. 8(1), 17-29(1998), http://www.clifford-algebras.org/v8/81/gu81.pdf [10] Y. Q. Gu, Characteristic Functions and Typical Values of the Nonlinear Dark Spinor, arXiv:hep-th/0611210 [11] Y. Q. Gu, A Cosmological Model with Dark Spinor Source, arXiv:gr-qc/0610147 [12] Y. Q. Gu, Accelerating Expansion of the Universe with Nonlinear Spinors, arXiv:gr-qc/0612176 [13] T. Cazenave, L. Vazquez, Comm. Math. Phys.105, 35-47 (1986) [14] F. Merle, J. Diff. Eq. 74, 50-68 (1988) [15] M. Balabane, et al, Comm. Math. Phys. 119, 153-176 (1988) [16] M. Balabane, et al, Comm. Math. Phys. 133, 53-74 (1990) [17] M. J. Esteban, E. Sere, C. R. Acad. Sci. Pavis, t. 319, Serie I, 1213-1218 (1994) [18] M. J. Esteban, E. Sere, Comm. Math. Phys. 171, 323-350(1995) [19] Wilhelm Fushchych, Renat Zhdanov, SYMMETRIES AND EXACT SOLUTIONS OF NONLINEAR DIRAC EQUATIONS, Kyiv Mathematical Ukraina Publisher, Ukraine (1997), arXiv:math-ph/0609052 [20] A. Garrett Lisi, A Solution of the Maxwell-Dirac Equations in 3+1 Dimensions, arXiv:hep-th/9410244 [21] M. Wakano, Prog. Theor. Phys. 35(6), 1117-1141(1996) [22] Y. Q. Gu, Spinor Soliton with Electromagnetic Field, Adv in Appl. Clif. Alg. V8(2), 271-282(1998), http://www.clifford-algebras.org/v8/82/gu82.pdf [23] A. O. Barut and J. Kraus, Found. Physics 13, 189 (1983) [24] A. O. Barut and J. F. Van Huele, Phys. Rev. A 32, 3187(1985) [25] Y. Q. Gu, The Electromagnetic Potential Among Nonrelativistic Electrons, Adv in Appl. Clif. Alg. V9(1), 55-60(1999), http://www.clifford-algebras.org/v9/91/gu91.pdf [26] Y. Q. Gu, New Approach to N-body Relativistic Quantum Mechanics, arXiv:hep-th/0610153 [27] Y. Q. Gu, Mass Energy Relation of the Nonlinear Spinor, arXiv:hep-th/0701030 [28] Y. Q. Gu, A Canonical Form For Relativistic Dynamic Equation, Adv in Appl. Clif. Alg. V7(1), 13-24(1997), http://www.clifford-algebras.org/v7/v71/GU71.pdf, arXiv:hep-th/0610189 [29] Y. Q. Gu, Green Functions of Relativistic Field Equations, arXiv:hep-th/0612214 [30] A. Gsponer, J.-P. Hurni, quaternions in mathematical physics (1): Alphabetical bibliography, arXiv:math-ph/0510059; (2): Analytical bibliography, arXiv:math-ph/0511092.

0704.0437

Measurement of B(D_S^+ --> ell^+ nu) and the Decay Constant f_D_{S^+}

CLNS 07/1990 CLEO 07-02 Measurement of B(D+s → ℓ +ν) and the Decay Constant fD+s T. K. Pedlar Luther College, Decorah, Iowa 52101 D. Cronin-Hennessy, K. Y. Gao, J. Hietala, Y. Kubota, T. Klein, B. W. Lang, R. Poling, A. W. Scott, A. Smith, and P. Zweber University of Minnesota, Minneapolis, Minnesota 55455 S. Dobbs, Z. Metreveli, K. K. Seth, and A. Tomaradze Northwestern University, Evanston, Illinois 60208 J. Ernst State University of New York at Albany, Albany, New York 12222 K. M. Ecklund State University of New York at Buffalo, Buffalo, New York 14260 H. Severini University of Oklahoma, Norman, Oklahoma 73019 W. Love and V. Savinov University of Pittsburgh, Pittsburgh, Pennsylvania 15260 O. Aquines, A. Lopez, S. Mehrabyan, H. Mendez, and J. Ramirez University of Puerto Rico, Mayaguez, Puerto Rico 00681 G. S. Huang, D. H. Miller, V. Pavlunin, B. Sanghi, I. P. J. Shipsey, and B. Xin Purdue University, West Lafayette, Indiana 47907 G. S. Adams, M. Anderson, J. P. Cummings, I. Danko, D. Hu, B. Moziak, and J. Napolitano Rensselaer Polytechnic Institute, Troy, New York 12180 Q. He, J. Insler, H. Muramatsu, C. S. Park, E. H. Thorndike, and F. Yang University of Rochester, Rochester, New York 14627 M. Artuso, S. Blusk, J. Butt, S. Khalil, J. Li, N. Menaa, R. Mountain, http://arxiv.org/abs/0704.0437v3 S. Nisar, K. Randrianarivony, R. Sia, T. Skwarnicki, S. Stone, and J. C. Wang Syracuse University, Syracuse, New York 13244 G. Bonvicini, D. Cinabro, M. Dubrovin, and A. Lincoln Wayne State University, Detroit, Michigan 48202 D. M. Asner, K. W. Edwards, and P. Naik Carleton University, Ottawa, Ontario, Canada K1S 5B6 R. A. Briere, T. Ferguson, G. Tatishvili, H. Vogel, and M. E. Watkins Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 J. L. Rosner Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 N. E. Adam, J. P. Alexander, D. G. Cassel, J. E. Duboscq, R. Ehrlich, L. Fields, L. Gibbons, R. Gray, S. W. Gray, D. L. Hartill, B. K. Heltsley, D. Hertz, C. D. Jones, J. Kandaswamy, D. L. Kreinick, V. E. Kuznetsov, H. Mahlke-Krüger, D. Mohapatra, P. U. E. Onyisi, J. R. Patterson, D. Peterson, J. Pivarski, D. Riley, A. Ryd, A. J. Sadoff, H. Schwarthoff, X. Shi, S. Stroiney, W. M. Sun, and T. Wilksen Cornell University, Ithaca, New York 14853 S. B. Athar, R. Patel, and J. Yelton University of Florida, Gainesville, Florida 32611 P. Rubin George Mason University, Fairfax, Virginia 22030 C. Cawlfield, B. I. Eisenstein, I. Karliner, D. Kim, N. Lowrey, M. Selen, E. J. White, and J. Wiss University of Illinois, Urbana-Champaign, Illinois 61801 R. E. Mitchell and M. R. Shepherd Indiana University, Bloomington, Indiana 47405 D. Besson University of Kansas, Lawrence, Kansas 66045 (CLEO Collaboration) (Dated: April 2, 2003) Abstract We examine e+e− → D−s D s andD s interactions at 4170 MeV using the CLEO-c detector in order to measure the decay constant f . We use the D+s → ℓ +ν channel, where the ℓ+ designates either a µ+ or a τ+, when the τ+ → π+ν. Analyzing both modes independently, we determine B(D+s → µ +ν) = (0.594±0.066±0.031)%, and B(D+s → τ +ν) = (8.0±1.3±0.4)%. We also analyze them simultaneously to find an effective value of Beff(D+s → µ +ν) = (0.638± 0.059± 0.033)% and extract f = 274 ± 13 ± 7 MeV. Combining with our previous determination of B(D+ → µ+ν), we also find the ratio f /fD+ = 1.23± 0.11± 0.04. We compare to current theoretical estimates. Finally, we find B(D+s → e +ν) < 1.3 × 10−4 at 90% confidence level. PACS numbers: 13.20.Fc, 13.66.Bc I. INTRODUCTION To extract precise information on the size of Cabibbo-Kobayashi-Maskawa matrix ele- ments from B − B mixing measurements, the “decay constants” for Bd and Bs mesons or their ratio, fBd/fBs , must be well known [1]. These factors are related to the overlap of the heavy and light quark wave-functions at zero spatial separation. Indeed, the recent measure- ment of B0s −B s mixing by CDF [2] that can now be compared to the very well measured B mixing [3] has pointed out the urgent need for precise values [4]. Decay constants have been calculated theoretically. The most promising of these calculations are based on lattice-gauge theory that includes light quark loops [5], often called “unquenched.” In order to ensure that these theories can adequately predict fBs/fBd it is useful to check the analogous ratio in charm decays f /fD+ . We have previously measured fD+ [6, 7]. Here we present the most precise measurement to date of f and the ratio f /fD+ . In the Standard Model (SM), the only way for a Ds meson to decay purely leptonically is via annihilation through a virtual W+, as depicted in Fig. 1. The decay rate is given by [8] Γ(D+s → ℓ +ν) = m2ℓMD+s |Vcs| , (1) where M is the D+s mass, mℓ is the mass of the charged final state lepton, GF is the Fermi coupling constant, and |Vcs| is a Cabibbo-Kobayashi-Maskawa matrix element with a value we take equal to 0.9738 [3]. FIG. 1: The decay diagram for D+s → ℓ In this paper we analyze both D+s → µ +ν and D+s → τ +ν, τ+ → π+ν. In both Ds decays the charged lepton must be produced with the wrong helicity because the Ds is a spin-0 particle, and the final state consists of a naturally left-handed spin-1/2 neutrino and a naturally right-handed spin-1/2 anti-lepton. Because the τ+ has a mass close to that of the D+s , the helicity suppression is broken with respect to the µ + decay, but there is an additional large phase space suppression. New physics can affect the expected widths; any undiscovered charged bosons would interfere with the SM W+. These effects may be difficult to ascertain, since they would simply change the value of the decay constants. The ratio f /fD+ is much better predicted in the SM than the values individually, so deviations from the the SM expectation are more easily seen. Any such discrepancies would point to beyond the SM charged bosons. For example, Akeroyd predicts that the presence of a charged Higgs boson would suppress this ratio significantly [9]. We can also measure the ratio of decay rates to different leptons, and the SM predictions then are fixed only by well-known masses. For example, for τ+ν to µ+ν: Γ(D+s → τ Γ(D+s → µ . (2) Using measured masses [3], this expression yields a value of 9.72 with a negligibly small error. Any deviation in R from the value predicted by Eq. 2 would be a manifestation of physics beyond the SM. This could occur if any other charged intermediate boson existed that affected the decay rate differently than mass-squared. Then the couplings would be different for muons and τ ’s. This would be a clear violation of lepton universality [10]. Previous measurements of fD+s have been hampered by a lack of statistical precision, and relatively large systematic errors [11, 12, 13, 14, 15, 16]. One large systematic error source has been the lack of knowledge of the absolute branching fraction of the normalization channel, usually D+s → φπ + [17]. The results we report here will not have this limitation. II. EXPERIMENTAL METHOD A. Selection of Ds Candidates The CLEO-c detector [18] is equipped to measure the momenta and directions of charged particles, identify them using specific ionization (dE/dx) and Cherenkov light (RICH) [19], detect photons and determine their directions and energies. In this study we use 314 pb−1 of data produced in e+e− collisions using the Cornell Elec- tron Storage Ring (CESR) and recorded near a center-of-mass energy (ECM) of 4.170 GeV. At this energy the e+e− annihilation cross-section into D∗+s D s is approximately 1 nb, while the cross-section for D+s D s is about a factor of 20 smaller. In addition, D mesons are produced mostly as D∗D∗, with a cross-section of ∼5 nb, and also in D∗D +DD∗ final states with a cross-section of ∼2 nb. The DD cross-section is a relatively small ∼0.2 nb [20]. There also appears to be DD π production. The underlying light quark “continuum” background is about 12 nb. The relatively large cross-sections, relatively large branching fractions and sufficient luminosities allow us to fully reconstruct one Ds as a “tag,” and ex- amine the properties of the other. In this paper we designate the tag as a D−s and examine the leptonic decays of the D+s , though, in reality, we use both charges for tags and signals. Track requirements, particle identification, π0, η, and K0S selection criteria are the same as those described in Ref. [6], except that we now require a minimum momentum of 700 MeV/c for a track to be identified using the RICH. We also use several resonances that decay via the strong interaction. Here we select intervals in invariant mass within ±10 MeV of the known mass for η′ → π+π−η, ±10 MeV for φ → K+K−, ±100 MeV for K∗0 → K−π+, and ±150 MeV for ρ− → π−π0. We reconstruct tags from either directly produced Ds mesons or those that result from the decay of a D∗s . The beam constrained mass, mBC, is formed by using the beam energy to construct the Ds candidate mass via the formula mBC = E2beam − ( −→p i) 2, (3) where i runs over all the final state particles. If we ignore the photon from the D∗s → γDs decay, and reconstruct the mBC distribution, we obtain the distribution from Monte Carlo simulation shown in Fig. 2. The narrow peak occurs when the reconstructed Ds does not come from the D∗s decay, but is directly produced. FIG. 2: The beam constrained mass mBC from Monte Carlo simulation of e +e− → D+s D s , D φπ± at an ECM of 4170 MeV. The narrow peak is from the D s and the wider one fromD s → γD (The distributions are not centered at the D+s or D s masses, because the reconstructed particles are assumed to have the energy of the beam.) Rather than selecting events based on only mBC, we first select an interval that accepts most of the events, 2.015 < mBC < 2.067 GeV, and examine the invariant mass. Distributions from data for the 8 tag decay modes we use in this analysis are shown in Fig. 3. Note that the resolution in invariant mass is excellent, and the backgrounds not abysmally large, at least in these modes. To determine the number of D−s events we fit the invariant mass distributions to the sum of two Gaussians centered at the D−s mass, a function we refer to as “two-Gaussian.” The r.m.s. resolution (σ) is defined as σ ≡ f1σ1 + (1− f1)σ2, (4) where σ1 and σ2 are the individual widths of each of the two Gaussians and f1 is the fractional area of the first Gaussian. The number of tags in each mode is listed in Table I. We will later use sidebands of the signal peaks shown in Fig. 3 for part of the background estimate. To select our sample of tag events, we require the invariant masses, shown in Fig. 3, to be within ± 2.5σ (±2σ for the ηρ− mode) of the known D−s mass. Then we look for an additional photon candidate in the event that satisfies our shower shape requirement. FIG. 3: Invariant mass of D−s candidates in the decay modes (a) K +K−π−, (b) KSK −, (c) ηπ−, (d) η′π−, (e) φρ−, (f) π+π−π−, (g) K∗−K∗0, and (h) ηρ−, after requiring the total energy of the D−s candidate to be consistent with the beam energy. The curves are fits to two-Gaussian signal functions plus a polynomial background. TABLE I: Tagging modes and numbers of signal and background events, within ±2.5σ of the D−s mass for all modes, except ηρ − (±2σ), determined from two-Gaussian fits to the invariant mass plots, and the number of photon tags in each mode, within ±2.5σ of the Ds mass-squared determined from fits of the MM∗2 distributions (see text) to a signal Crystal Ball function (see text) and a 5th order Chebychev background polynomial, and the associated backgrounds. Mode Invariant Mass MM∗2 Signal Background Signal Background K+K−π− 13871±262 10850 8053± 211 13538 − 3122±79 1609 1933±88 2224 ηπ−; η → γγ 1609 ± 112 4666 1024±97 3967 η′π−; η′ → π+π−η, η → γγ 1196±46 409 792±69 1052 φρ−; φ → K+K−, ρ− → π−π0 1678±74 1898 1050±113 3991 π+π−π− 3654±199 25208 2300±187 15723 K∗−K∗0; K∗− → K0Sπ −, K∗0 → K+π− 2030±98 4878 1298±130 5672 ηρ−; η → γγ, ρ− → π−π0 4142±281 20784 2195±225 17353 Sum 31302 ± 472 70302 18645±426 63520 Regardless of whether or not the photon forms a D∗s with the tag, for real D sDs events, the missing mass squared, MM∗2, recoiling against the photon and the D−s tag should peak at the D+s mass-squared. We calculate MM∗2 = (ECM − EDs −Eγ) −→pCM − −→pDs − −→p γ , (5) where ECM ( −→pCM) is the center-of-mass energy (momentum), EDs ( −→pDs) is the energy (mo- mentum) of the fully reconstructed D−s tag, and Eγ ( −→p γ) is the energy (momentum) of the additional photon. In performing this calculation we use a kinematic fit that constrains the decay products of the D−s to the known Ds mass and conserves overall momentum and en- ergy. All photon candidates in the event are used, except for those that are decay products of the D−s tag candidate. The MM∗2 distributions from the selected D−s event sample are shown in Fig. 4. We fit these distributions to determine the number of tag events. This procedure is enhanced by having information on the shape of the signal function. One possibility is to use the Monte Carlo simulation for this purpose, but that would introduce a relatively large systematic error. Instead, we use our relatively large sample of fully reconstructed DsD s events, where we use the same decay modes listed in Table I; we find these events and then examine the signal shape in data when one Ds is ignored. The MM ∗2 distribution from this sample is shown in Fig. 5. The signal is fit to a Crystal Ball function [21, 22]. The σ parameter, that represents the width of the distribution, is found to be 0.032±0.002 GeV2. We do expect this to vary somewhat depending on the final state, but we do not expect the parameters that fix the shape of the tail to change, since they depend mostly on beam radiation and the properties of photon detection. We fit the MM∗2 distributions for each mode using the Crystal Ball function with fixed tail parameters, but allowing σ to float, and a 5th order Chebyshev polynomial background. We find a total of 18645±426 events within a ±2.5σ interval defined by the fit to each FIG. 4: The MM∗2 distribution from events with a photon in addition to the D−s tag for the modes: (a) K+K−π−, (b) K0SK −, (c) ηπ−, (d) η′π−, (e) φρ−, (f) π+π−π−, (g) K∗−K∗0, and (h) ηρ−. The curves are fits to the Crystal Ball function and a 5th order Chebychev background function. mode. There is also a small enhancement of 4.8% on our ability to find tags in µ+ν (or τ+ν, τ+ → π+ν) events (tag bias) as compared with generic events, determined by Monte Carlo simulation, to which we assign a systematic error of 21% giving a correction of (4.8± 1.0)%. An overall systematic error of 5% on the number of tags is assigned by changing the fitting range, using 4th order and 6th order Chebychev background polynomials, and allowing the FIG. 5: The MM∗2 distribution from a sample of fully reconstructed D−s D s events where one Ds is ignored. The curve is a fit to the Crystal Ball function. parameters of the tail of the fitting function to float. B. Signal Reconstruction We next describe the search for D+s → µ +ν. Candidate events are selected that contain only a single extra track with opposite sign of charge to the tag. The track must make an angle >35.9◦ with respect to the beam line, and in addition we require that there not be any neutral cluster detected in the calorimeter with energy greater than 300 MeV. These cuts are highly effective in reducing backgrounds. The photon energy cut is especially useful to reject D+s → π +π0, should this mode be significant, and D+s → ηπ Since we are searching for events where there is a single missing neutrino, the missing mass squared, MM2, evaluated by taking into account the observed µ+, D−s , and γ should peak at zero; the MM2 is computed as MM2 = (ECM − EDs −Eγ − Eµ) −→pCM − −→pDs − −→p γ − −→p µ , (6) where Eµ ( −→p µ) are the energy (momentum) of the candidate muon track and all variables are the same as defined in Eq. 5. We also make use of a set of kinematical constraints and fit each event to two hypotheses one of which is that the D−s tag is the daughter of a D s and the other that the D s decays into γD+s , with the D s subsequently decaying into µ +ν. The kinematical constraints, in the center-of-mass frame, are −→pDs + −→pD∗s = 0 (7) ECM = EDs + ED∗s ED∗s = M2D∗s −M or EDs = M2D∗s −M MD∗s −MDs = 143.6 MeV. In addition, we constrain the invariant mass of the D−s tag to the known Ds mass. This gives us a total of 7 constraints. The missing neutrino four-vector needs to be determined, so we are left with a three-constraint fit. We perform a standard iterative fit minimizing χ2. As we do not want to be subject to systematic uncertainties that depend on understanding the absolute scale of the errors, we do not make a χ2 cut but simply choose the photon and the decay sequence in each event with the minimum χ2. In this analysis, we consider three separate cases: (i) the track deposits < 300 MeV in the calorimeter, characteristic of a non-interacting pion or a muon; (ii) the track deposits > 300 MeV in the calorimeter, characteristic of an interacting pion, and is not consistent with being an electron; (iii) the track satisfies our electron selection criteria defined below. Then we separately study the MM2 distributions for these three cases. The separation between muons and pions is not complete. Case (i) contains 99% of the muons but also 60% of the pions, while case (ii) includes 1% of the muons and 40% of the pions [7]. Case (iii) does not include any signal but is used later for background estimation. For cases (i) and (ii) we insist that the track not be identified as a kaon. For electron identification we require a match between the momentum measurement in the tracking system and the energy deposited in the CsI calorimeter and we also require that dE/dx and RICH information be consistent with expectations for an electron. C. The Expected MM2 Spectrum For the µ+ν final state the MM2 distribution can be modeled as the sum of two Gaussians centered at zero (see Eq. 4). A Monte Carlo simulation of the MM2 for the φπ− subset of K+K−π− tags is shown in Fig. 6 both before and after the fit. The fit changes the resolution from σ=0.032 GeV2 to σ=0.025 GeV2, a 22% improvement, without any loss of events. We check the resolution using data. The mode D+s → K K+ provides an excellent testing ground.1 We search for events with at least one additional track identified as a kaon using the RICH detector, in addition to a D−s tag. The MM 2 distribution is shown in Fig. 7. Fitting this distribution to a two-Gaussian shape gives a MM2 resolution of 0.025 GeV2 in agreement with Monte Carlo simulation. For the τ+ν, τ+ → π+ν final state a Monte Carlo simulation of the MM2 spectrum is shown in Fig. 8. The extra missing neutrino results in a smeared distribution. 1 In this paper the notation K K+ refers to the sum of K0K+ and K0K+ final states. FIG. 6: The MM2 resolution from Monte Carlo simulation for D+s → µ +ν utilizing a φπ− tag and a γ from either D∗s decay, both before the kinematic fit (a) and after (b). D. MM2 Spectra in Data The MM2 distributions from data are shown in Fig. 9. The overall signal region we consider is −0.05 < MM2 < 0.20 GeV2. The higher limit is imposed to exclude background from ηπ+ and K0π+ final states. There is a clear peak in Fig. 9(i) due to D+s → µ Furthermore, the region between the µ+ν peak and 0.20 GeV2 has events that we will show are dominantly due to the D+s → τ +ν decay. The events in Fig. 9(ii) below 0.20 GeV2 are also dominantly due to τ+ν decay. The specific signal regions are defined as follows: for µ+ν, −0.05 <MM2 < 0.05 GeV2, corresponding to ±2σ; for τν, τ+ → π+ν, in case (i) 0.05 <MM2 < 0.20 GeV2 and in case (ii) −0.05 <MM2 < 0.20 GeV2. In these regions we find 92, 31, and 25 events, respectively. E. Background Evaluations We consider the background arising from two sources: one from real D+s decays and the other from the background under the single-tag signal peaks. For the latter, we obtain the background from data. We define side-bands of the invariant mass signals shown in Fig. 2 starting at 4σ on the low and high sides of the invariant mass peaks for all modes. The intervals extend away from the signal peaks by approximately the same width used in selecting the signal, 5σ, so as to ensure that the number of background events in the sidebands accurately reflects the numbers under the signal peaks. Thus the amount of data corresponds to twice the number of background events under the signal peaks, except for the ηπ− and ηρ− modes, where the signal widths are so wide that we chose narrower side-bands only equaling the data. We analyze these events in exactly the same manner as those in the signal peak.2 2 The D mass used in the fit is chosen to be the middle of the relevant sideband interval. FIG. 7: The MM2 distribution for events with an identified K+ track. The kinematic fit has been applied. The curve is a fit to the sum of two Gaussians centered at the square of the K0 mass and a linear background. The backgrounds are given here as the sum of two numbers, the first being the number from all modes, except ηπ− and ηρ−, and the second being the number from these modes. For case (i) we find 2.5+1 background in the µ+ν signal region and 2.5+0 background in the τ+ν region. For case (ii) we find 2+1 events. Our total sideband background summing over all of these cases is 9.0±2.3. The numbers of signal and background events due to false D−s tags as evaluated from sidebands are given in Table II. TABLE II: Numbers of events in the signal region, and background events evaluated from sideband regions. Case Region (GeV2) Signal Background i -0.05<MM2 < 0.05 92 3.5±1.4 i 0.05<MM2 < 0.20 31 2.5±1.1 ii -0.05<MM2 < 0.20 25 3.0±1.3 Sum -0.05<MM2 < 0.20 148 9.0±2.3 This entire procedure was checked by performing the same study on a sample of Monte Carlo generated at an ECM of 4170 MeV that includes known charm and continuum produc- tion cross-sections. The Monte Carlo sample corresponds to an integrated luminosity that is four times larger than the data. We find the number of background events predicted directly by examining the decay generator of the simulation is 28 and the sideband method yields FIG. 8: The MM2 distribution from Monte-Carlo simulation for D+s → τ +ν, τ+ → π+ν at an ECM of 4170 MeV. 22. These are slightly smaller than found in the data, but consistent within errors. We note that the Monte Carlo is far from perfect as many branching fractions are unknown and so are estimated. The background from real D+s decays is studied by identifying each possible source mode by mode. For the µ+ν final state, the only possible background within the signal region is D+s → π +π0. This mode has not been studied previously. We show in Fig. 10 the π+π0 invariant mass spectrum from a 195 pb−1 subsample of our data. We do not see a signal and set an upper limit < 1.1 × 10−3 at 90% confidence level. Recall that any such events are also heavily suppressed by the extra photon energy cut of 300 MeV. There are also some D+s → τ +ν, τ+ → π+ν events that occur in the signal region. Using the SM expected ratio of decay rates from Eq. 2 we calculate a contribution of 7.4 π+νν events that we will treat as part of the signal. For the τ+ν, τ+ → π+ν final state the real D+s backgrounds include, in addition to the π+π0 background discussed above, semileptonic decays, possible π+π0π0 decays, and other τ+ decays. Semileptonic decays involving muons are equal to those involving electrons shown in Fig. 9(c). Since no electron events appear in the signal region, the background from muons is taken to be zero. The π+π0π0 background is estimated by considering the π+π+π− final state whose measured branching fraction is (1.02±0.12)% [17]. This mode has large contributions from f0(980)π + and other π+π− resonant structures at higher mass [23]. The π+π0π0 mode will also have these contributions, but the MM2 opposite to the π+ will be at large mass. The only component that can potentially cause background is the non-resonant component measured by FOCUS as (17±4)% [23]. This background has been evaluated by Monte Carlo simulation as have backgrounds from other τ+ decays, and each is listed in Table III. FIG. 9: The MM2 distributions from data for events with a D−s reconstructed in a tag mode, an additional positively charged track and no neutral energy clusters above 300 MeV. For case (i) when the single track deposits < 300 MeV of energy in the calorimeter. The peak near zero is from D+s → µ +ν events. Case (ii): track deposits > 300 MeV in the crystal calorimeter but is not consistent with being an electron. Case (iii): the track is identified as an electron. III. LEPTONIC BRANCHING FRACTIONS The sum of MM2 distributions for case (i) and case (ii), corresponding to the sum of D+s → µ +ν and D+s → τ +ν, τ+ → π+ν candidates, is compared in Fig. 11 with the expected shape, assuming the SM value of R as given in Eq. 2 for the ratio of τ+ν to µ+ν rates. The curve is normalized to the total number of events below MM2 <0.2 GeV2. Besides the prominent µ+ν peak and τ+ν; τ+ → π+ν shoulder, there is an enhancement between 0.25-0.35 GeV2, due to K0π+ and ηπ+ final states, where the decay products other than FIG. 10: The invariant π+π0 mass. The upper curve shows a fit using a background polynomial plus Gaussian signal functions, where the width of the Gaussian is fixed to a value determined by Monte Carlo simulation. The lower curve shows just the background polynomial. TABLE III: Backgrounds in the D+s → τ +ν, τ+ → π+ν sample for correctly reconstructed tags, case (i) for 0.05<MM2 < 0.20 GeV2 and case (ii) for -0.05<MM2 < 0.20 GeV2. Source B(%) # of events case (i) # of events case(ii) Sum D+s → Xµ +ν 8.2 0+1.8 −0 0 0 D+s → π +π0π0 1.0 0.03±0.04 0.08±0.03 0.11±0.04 D+s → τ +ν 6.4 τ+ → π+π0ν 1.5 0.55±0.22 0.64±0.24 1.20±0.33 τ+ → µ+νν 1.0 0.37±0.15 0 0.37±0.15 Sum 1.0+1.8 −0 0.7±0.2 1.7 the π+ escape detection. The data are consistent with our expectation that the region −0.05 <MM2 <0.2 GeV2 contains mostly signal. Recall there are 148 total events only 10.7 of which we estimate are background, 9.0 from fake D−s tags and 1.7 from real tags and D decays. Above 0.2 GeV2 other, larger backgrounds enter. The number of real µ+ν events Nµν is related to the number of events detected in the signal region Ndet (92), the estimated background Nbkgrd (3.5), the number of tags, Ntag, and the branching fractions as Nµν ≡ Ndet −Nbkgrd = Ntag · ǫ ǫ′B(D+s → µ +ν) + ǫ′′B(D+s → τ +ν; τ+ → π+ν) , (8) where ǫ (80.1%) includes the efficiency for reconstructing the single charged track including final state radiation (77.8%), the (98.3±0.2)% efficiency of not having another unmatched cluster in the event with energy greater than 300 MeV, and for the fact that it is easier to find tags in µ+ν events than in generic decays by 4.8%, as determined by Monte Carlo simulation. The efficiency labeled ǫ′ (91.4%) is the product of the 99.0% muon efficiency FIG. 11: The sum of case (i) and case (ii) MM2 distributions (histogram) compared to the predicted shape (curve) for the sum of D+s → µ +ν and τ+ν, τ+ → π+ν. The curve is normalized to the total number of events below MM2 <0.2 GeV2. for depositing less than 300 MeV in the calorimeter and 92.3% acceptance of the MM2 cut of |MM2| < 0.05 GeV2. The quantity ǫ′′ (7.9%) is the fraction of τ+ν; τ+ → π+ν events contained in the µ+ν signal window (13.2%) times the 60% acceptance for a pion to deposit less than 300 MeV in the calorimeter. The two D+s branching fractions in Eq. 8 are related as B(D+s → τ +ν; τ+ → π+ν) = R · B(τ+ → π+ν)B(D+s → µ +ν) = 1.059 · B(D+s → µ +ν) , (9) where we take the Standard Model ratio for R as given in Eq. 2 and B(τ+ → π+ν)=(10.90±0.07)% [3]. This allows us to solve Eq. 8. Since Ndet= 92, Nbkgrd=3.5±1.4, and Ntag = 18645± 426± 1081, we find B(D+s → µ +ν) = (0.594± 0.066± 0.031)%. (10) We can also sum the µ+ν and τ+ν contributions, where we restrict ourselves to the MM2 region below 0.20 GeV2 and above -0.05 GeV2. Eq. 8 still applies. The number of events in the signal region and the number of background events changes to 148 and 10.7+2.9 −2.3, respectively. The efficiency ǫ′ becomes 96.2%, and ǫ′′ increases to 45.2%. Using this method, we find an effective branching fraction of Beff(D+s → µ +ν) = (0.638± 0.059± 0.033)%. (11) The systematic errors on these branching fractions are given in Table IV. The error on track finding is determined from a detailed comparison of the simulation with double tag events where one track is ignored. “Minimum ionization” indicates the error due to the requirement that the charged track deposit no more than 300 MeV in the calorimeter; it is determined using two-body D0 → K−π+ decays (see Ref. [7]). The error on the photon veto efficiency, due to the 300 MeV/c extra shower energy cut, is determined using Monte Carlo simulation. The Monte Carlo was cross-checked using a sample of fully reconstructed D+D− events and comparing the inefficiency due to additional photons with energy above 300 MeV/c. These events have no real extra photons above 300 MeV/c; those that are present are due to interactions of the D± decay products in the detector material. The error on the number of tags of ±5% has been discussed earlier. In addition there is a small error of ±0.6% on the τ+ν branching fraction due to the uncertainty on the τ+ decay fraction to π+ν. Additional systematic errors arising from the background estimates are negligible. Note that the minimum ionization error does not apply to the summed branching fraction given in Eq. 11; in this case the total systematic error is 5.1%. TABLE IV: Systematic errors on determination of the D+s → µ +ν branching fraction. Error Source Size (%) Track finding 0.7 Photon veto 1 Minimum ionization∗ 1 Number of tags 5 Total 5.2 *-Not applicable for summed rate We also analyze the τ+ν final state independently. We use different MM2 regions for cases (i) and (ii) defined above. For case (i) we define the signal region to be the interval 0.05<MM2 <0.20 GeV2, while for case (ii) we define the signal region to be the interval -0.05<MM2 <0.20 GeV2. Case (i) includes the µ+ν signal, so we must exclude the region close to zero MM2, while for case (ii) we are specifically selecting pions so the signal region can be larger. The upper limit on MM2 is chosen to avoid background from the tail of the K0π+ peak. The fractions of the MM2 range accepted are 32% and 45% for case (i) and (ii), respectively. We find 31 signal and 3.5+1.7 −1.1 background events for case (i) and 25 signal and 5.1±1.6 background events for case (ii). The branching fraction, averaging the two cases is B(D+s → τ +ν) = (8.0± 1.3± 0.4)%. (12) Lepton universality in the Standard Model requires that the ratio R from Eq. 2 be equal to a value of 9.72. We measure Γ(D+s → τ Γ(D+s → µ = 13.4± 2.6± 0.2 . (13) Here the systematic error is dominated by the uncertainty on the minimum ionization cut that we use to separate the µ+ν and τ+ν regions at 300 MeV. We take this error as 2%, since a change here affects both the numerator and denominator. The ratio is consistent with the Standard Model prediction. Current results on D+ leptonic decays also show no deviations [26]. The absence of any detected electrons opposite to our tags allows us to set an upper limit of B(D+s → e +ν) < 1.3× 10−4 (14) at 90% confidence level; this is also consistent with Standard Model predictions and lepton universality. IV. CHECKS OF THE METHOD We perform an overall check of our procedures by measuring B(D+s → K K+). For this measurement we compute the MM2 (Eq. 6) using events with an additional charged track but here identified as a kaon. These track candidates have momenta of approximately 1 GeV/c; here our RICH detector has a pion to kaon fake rate of 1.1% with a kaon detection efficiency of 88.5% [19]. For this study, we do not veto events with extra charged tracks, or neutral energy deposits >300 MeV, because of the presence of the K0. The MM2 distribution is shown in Fig. 7. The peak near 0.25 GeV2 is due to the decay mode of interest. We fit this to a linear background from 0.02-0.50 GeV2 plus a two-Gaussian signal function. The fit yields 375±23±18 events. Events from the ηπ+ mode where the π+ fakes a K+ are very rare and would not peak at the proper MM2. Since ηK+ could in principle contribute a background in this region, we searched for this final state in a 195 pb−1 subsample of the data. Not finding any signal, we set an upper limit of B(D+s → ηK 2.8 × 10−3 at 90% confidence level, approximately a factor of ten below our measurement. This final state would peak at a MM2 of 0.30 GeV2 and would cause an asymmetric tail on the high side of the peak. Since we see no evidence for an asymmetry in the K K+ peak we ignore the ηK+ final state from here on. In order to compute the branching fraction we must include the efficiency of detecting the kaon track 76.2%, including radiation [24], the particle identification efficiency of 88.5%, and take into account that it is easier to detect tags in events containing a K K+ decay than in the average DsD s event due to the track and photon multiplicities, which gives a 3% correction.3 These rates are estimated by using Monte Carlo simulation. We determine B(D+s → K K+) = (2.90± 0.19± 0.18)%, (15) where the systematic errors are listed in Table V. We estimate the error from the signal shape by taking the change in the number of events when varying the signal width of the two-Gaussian function by ±1σ. The error on the background shapes is given by varying the shape of the background fit. The error on the particle identification efficiency is measured using two-body D0 decays [19]. The other errors are the same as described in Table IV. Again, the largest component of the systematic error arises from the number of tag events (5%). In fact, to use this result as a check on our procedures, we need only consider the 3 The tag bias is less here than in the µ+ν case because of the K0 decays and interactions in the detector. systematic errors that are different here than in the µ+ν case. Those are due only to the signal and background shapes and the particle identification cut. Those systematic errors amount to 3.7% or ±0.11 in the branching fraction. To determine absolute branching fractions of charm mesons, CLEO-c uses a method where both particles are fully reconstructed (so called “double tags”) and the rates are normalized using events where only one particle is fully reconstructed. Our preliminary result using this method for B(D+s → KSK +)=(1.50±0.09 ± 0.05)%, which when doubled becomes (3.00±0.19±0.10)% [25]. This is in excellent agreement with the number in Eq. 15. These results are not independent. TABLE V: Systematic errors on determination of the D+s → K K+ branching fraction. Error Source Size (%) Signal shape 3 Background shape 2 Track finding 0.7 PID cut 1.0 Number of tags 5 Total 6.3 We also performed the entire analysis on a Monte Carlo sample that corresponds to an integrated luminosity four 4 times larger than the data sample. The input branching fraction in the Monte Carlo is 0.5% for µ+ν and 6.57% for τ+ν, while our analysis measured B(D+s → µ +ν) =(0.514±0.027)% for the case (i) µ+ν signal and (0.521±0.024)% for µ+ν and τ+ν combined. We also find (6.6±0.6)% for the τ+ν rate. V. THE DECAY CONSTANT Using our most precise value for B(D+s → µ +ν) from Eq. 11, that is derived using both our µ+ν and τ+ν samples, and Eq. 1 with a Ds lifetime of (500±7)×10 −15 s [3], we extract fD+s = 274± 13± 7 MeV. (16) We combine with our previous result [6] fD+ = 222.6± 16.7 −3.4 MeV (17) and find a value for = 1.23± 0.11± 0.04, (18) where only a small part of the systematic error cancels in the ratio of our two measurements. VI. CONCLUSIONS Theoretical models that predict f and the ratio are listed in Table VI. Our result for fDs is slightly higher than most theoretical expectations. We are consistent with lattice gauge theory, and most other models, for the ratio of decay constants. There is no evidence at this level of precision for any suppression in the ratio due to the presence of a virtual charged Higgs [9]. TABLE VI: Theoretical predictions of f , fD+ , and fD+s /fD+ . QL indicates quenched lattice calculations. Model f (MeV) fD+ (MeV) fD+s Lattice (HPQCD+UKQCD) [27] 241± 3 208 ± 4 1.162 ± 0.009 Lattice (FNAL+MILC+HPQCD) [28] 249± 3± 16 201 ± 3± 17 1.24 ± 0.01± 0.07 QL (QCDSF) [29] 220 ± 6± 5± 11 206 ± 6± 3± 22 1.07 ± 0.02± 0.02 QL (Taiwan) [30] 266 ± 10 ± 18 235 ± 8± 14 1.13 ± 0.03± 0.05 QL (UKQCD) [31] 236 ± 8+17 −14 210± 10 −16 1.13 ± 0.02 +0.04 −0.02 QL [32] 231± 12+6 −1 211± 14 −12 1.10 ± 0.02 QCD Sum Rules [33] 205 ± 22 177± 21 1.16 ± 0.01± 0.03 QCD Sum Rules [34] 235 ± 24 203± 20 1.15 ± 0.04 Field Correlators [35] 210 ± 10 260± 10 1.24 ± 0.03 Quark Model [36] 268 234 1.15 Quark Model [37] 248±27 230±25 1.08±0.01 LFQM (Linear) [38] 211 248 1.18 LFQM (HO) [38] 194 233 1.20 LF-QCD [39] 253 241 1.05 Potential Model [40] 241 238 1.01 Isospin Splittings [41] 262± 29 By using a theoretical prediction for f /fD+ we can derive a value for the ratio of CKM elements |Vcd/Vcs|. Taking the value from Ref. [27] of 1.162± 0.009, we find |Vcd/Vcs| = 0.2171± 0.021± 0.0017 , (19) where the first error is due the statistical and systematic errors of the experiment and the second is due to the stated error on the theoretical prediction. This value is expected to be almost equal to the ratio of the CKM elements |Vus|/|Vud| We now compare with previous measurements. The branching fractions, modes, and derived values of f are listed in Table VII. Our values are shown first. We are generally consistent with previous measurements, although ours are more precise. Most measurements of D+s → ℓ +ν are normalized with respect to B(D+s → φπ +) ≡ Bφπ. An exception is the OPAL measurement which is normalized to the Ds fraction in Z 0 events that is derived from an overall fit to heavy flavor data at LEP [42]. It still, however, relies on absolute branching fractions that are hidden by this procedure, and the estimated error on the normalization is somewhat smaller than that indicated by the error on Bφπ available at TABLE VII: Our results for B(D+s → µ +ν), B(D+s → τ +ν), and f compared with previous measurements. Results have been updated for the new value of the Ds lifetime [3]. ALEPH combines both measurements to derive a value for the decay constant. Exp. Mode B Bφπ (%) fD+s (MeV) CLEO-c µ+ν (5.94 ± 0.66 ± 0.31) · 10−3 264 ± 15± 7 CLEO-c τ+ν (8.0± 1.3 ± 0.4) · 10−2 310 ± 25± 8 CLEO-c combined - 274 ± 13± 7 CLEO [11] µ+ν (6.2 ± 0.8± 1.3± 1.6) · 10−3 3.6±0.9 273± 19± 27± 33 BEATRICE [12] µ+ν (8.3 ± 2.3± 0.6± 2.1) · 10−3 3.6±0.9 312± 43± 12± 39 ALEPH [13] µ+ν (6.8± 1.1 ± 1.8) · 10−3 3.6±0.9 282± 19± 40 ALEPH [13] τ+ν (5.8± 0.8 ± 1.8) · 10−2 L3 [14] τ+ν (7.4 ± 2.8± 1.6± 1.8) · 10−2 299± 57± 32± 37 OPAL [15] τ+ν (7.0± 2.1 ± 2.0) · 10−2 283± 44± 41 BaBar [16] µ+ν (6.74 ± 0.83 ± 0.26 ± 0.66) · 10−3 4.71±0.46 283 ± 17± 7± 14 the time of their publication. The L3 measurement is normalized taking the fraction of Ds mesons produced in c quark fragmentation as 0.11±0.02, and the ratio of D∗s/Ds production of 0.65±0.10. The ALEPH results use Bφπ for their µ +ν results and a similar procedure as OPAL for their τ+ν results. We note that the recent BaBar result uses a larger Bφπ than the other results. The CLEO-c determination of fD+s is the most accurate to date. It also does not rely on the independent determination of any normalization mode. (We note that a preliminary CLEO-c result using D+s → τ +ν, τ+ → e+νν [43] is consistent with these results.) VII. ACKNOWLEDGMENTS We gratefully acknowledge the effort of the CESR staff in providing us with excellent luminosity and running conditions. D. Cronin-Hennessy and A. Ryd thank the A.P. Sloan Foundation. This work was supported by the National Science Foundation, the U.S. Depart- ment of Energy, and the Natural Sciences and Engineering Research Council of Canada. [1] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996) [hep-ph/9512380]. [2] A. Abulencia et al. (CDF), Phys. Rev. Lett. 97, 242003 (2006). See also V. Abazov et al. (D0), Phys. Rev. Lett. 97, 021802 (2006). [3] W.-M. Yao et al., J. Phys. G33, 1 (2006). 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[42] ALEPH, DELPHI, L3, and OPAL Collaborations, Nucl. Instr. and Meth. A378, 101 (1996). [43] S. Stone, ”Measurement Of D+s → ℓ +ν and the Decay Constant fDs,” to be published in Proceedings of XXXIII International Conference on High Energy Physics, Moscow, Russia, July 2006 [hep-ex/0610026]. http://arxiv.org/abs/hep-ph/0701263 http://arxiv.org/abs/hep-ex/0610026 Introduction Experimental Method Selection of Ds Candidates Signal Reconstruction The Expected MM2 Spectrum MM2 Spectra in Data Background Evaluations Leptonic Branching Fractions Checks of the Method The Decay Constant Conclusions Acknowledgments References

0704.0438

Broadening the Higgs Boson with Right-Handed Neutrinos and a Higher Dimension Operator at the Electroweak Scale

RUNHETC-04-2007 Broadening the Higgs Boson with Right-Handed Neutrinos and a Higher Dimension Operator at the Electroweak Scale Michael L. Graesser 1Department of Physics and NHETC, Rutgers University, Piscataway, NJ 08540 Abstract The existence of certain TeV suppressed higher-dimension operators may open up new decay channels for the Higgs boson to decay into lighter right-handed neutrinos. These channels may dominate over all other channels if the Higgs boson is light. For a Higgs boson mass larger than 2mW the new decays are subdominant yet still of interest. The right-handed neutrinos have macroscopic decay lengths and decay mostly into final states containing leptons and quarks. A distinguishing collider signature of this scenario is a pair of displaced vertices violating lepton number. A general operator analysis is performed using the minimal flavor violation hypothesis to illustrate that these novel decay processes can occur while remaining consistent with experimental constraints on lepton number violating processes. In this context the question of whether these new decay modes dominate is found to depend crucially on the approximate flavor symmetries of the right-handed neutrinos. http://arxiv.org/abs/0704.0438v2 1 Motivation Neutrino interactions with other Standard Model particles are well-described, forming a cornerstone of the Standard Model itself. But the origin of their masses remain unknown. If their masses are generated by a local quantum field theory, then other degrees of freedom must exist. These particles or “right-handed neutrinos” are either the missing Dirac part- ners of the neutrinos, or are much heavier than the O(eV) mass scale of the neutrinos and create a “see-saw” mechanism. Which of these scenarios is realized in Nature is dependent on the unknown scale of the Majorana mass parameters of the right-handed neutrinos. The existence of right-handed neutrinos may have other physical consequences, de- pending on the size of their Majorana masses. Right-handed neutrinos with Majorana masses violate overall lepton number, which may have consequences for the origin of the observed baryon asymmetry. Leptogenesis can occur from the out of equilibrium decay of a right-handed neutrino with mass larger than the TeV scale [1]. Interestingly, right-handed neutrinos with masses below the electroweak scale may also lead to baryogenesis [2]. But if right-handed neutrinos exist, where did their mass come from? The Majorana mass parameters are not protected by the gauge invariance of the Standard Model, so an understanding of the origin of their mass scale requires additional physics. The see-saw mechanism with order unity Yukawa couplings prefers a large scale, of order 1013−14 GeV. But in this case a new, intermediate scale must be postulated in addition to the four mass scales already observed in Nature. On the other hand, such a large scale might occur naturally within the context of a Grand Unified Theory. Here I explore the consequences of assuming that the Majorana neutrino mass scale is generated at the electroweak scale 1. To then obtain the correct mass scale for the left- handed neutrinos from the “see-saw” mechanism, the neutrino Yukawa couplings must be tiny, but not unreasonably small, since they would be comparable to the electron Yukawa coupling. It might be natural for Majorana masses much lighter than the Planck or Grand Unified scales to occur in specific Randall-Sundrum type models [7] or their CFT dual descriptions by the AdS/CFT correpsondance [8]. But as the intent of this paper is to be as model-independent as possible, I will instead assume that it is possible to engineer electroweak scale Majorana masses and use effective field theory to describe the low-energy theory of the Higgs boson and the right-handed and left-handed (electroweak) neutrinos. I will return to question of model-building in the concluding section and provide a few additional comments. With the assumption of a common dynamics generating both the Higgs and right- handed neutrino mass scales, one may then expect strong interactions between these par- ticles, in the form of higher dimension operators. However since generic flavor-violating higher dimension operators involving Standard Model fields and suppressed only by the TeV are excluded, I will use throughout the minimal flavor violation hypothesis [9, 10, 11] in 1For previous work on the phenomenology of electroweak scale right-handed neutrinos, see [3, 4, 5, 6]. None of these authors consider the effects of TeV-scale suppressed higher dimension operators. order to suppress these operators. The purpose of this paper is to show that the existence of operators involving the Higgs boson and the right-handed neutrinos can significantly modify the phenomenology of the Higgs boson by opening a new channel for it to decay into right-handed neutrinos. I show that the right-handed neutrinos are long-lived and generically have macroscopic decay lengths. For reasonable values of parameters their de- cay lengths are anywhere from fractions of a millimeter to tens of metres or longer if one of the left-handed neutrinos is extremely light or massless. As they decay predominantly into a quark pair and a charged lepton, a signature for this scenario at a collider would be the observation of two highly displaced vertices, each producing particles of this type. Further, by studying these decays all the CP -preserving parameters of the right-handed and left-handed neutrinos interactions could be measured, at least in principle. A number of scenarios for new physics at the electroweak scale predict long-lived parti- cles with striking collider features. Displaced vertices due to long-lived neutral particles or kinks appearing in charged tracks are predicted to occur in models of low energy gauge me- diation [12]. More recently models with a hidden sector super-Yang Mills coupled weakly through a Z ′ or by mass mixing with the Higgs boson can produce dramatic signatures with displaced jets or leptons and events with high multiplicity [13]. A distinguishing feature of the Higgs boson decay described here is the presence of two displaced vertices where the particles produced at each secondary vertex violate overall lepton number. That new light states or operators at the electroweak scale can drastically modify Higgs boson physics has also been recently emphasized. Larger neutrino couplings occur in a model with nearly degenerate right-handed neutrino masses and vanishing tree-level active neutrino masses, that are then generated radiatively at one-loop [3]. Decays of the Higgs boson into a right-handed and left-handed neutrino may then dominate over decays to bottom quarks if the right-handed neutrinos are heavy enough. Models of supersymmetry having pseudoscalars lighter than the neutral Higgs scalar may have exotic decay processes for the Higgs boson that can significantly affect limits and searches [14]. Supersymmetry without R-parity can have striking new signatures of the Higgs boson [15]. Two common features between that reference and the work presented here is that the Higgs boson decays into a 6-body final state and may be discovered through displaced vertices, although the signatures differ. Interesting phenomena can also occur without supersymmetry. Adding to the Standard Model higher dimension operators involving only Standard Model fields can modify the Higgs boson production cross-section and branching fractions [16]. Such an effect can occur in models with additional colored scalars coupled to top quarks [17]. The outline of the paper is the following. Section 2 discusses the new decay of the Higgs boson into right-handed neutrinos. Section 3 then discusses various naturalness issues that arise in connection with the relevant higher dimension operator. Section 4 discusses predictions for the coefficients of the new operator within the framework of minimal flavor violation [9, 10, 11]. It is found that the predicted size of the higher dimension operators depends crucially on the approximate flavor symmetries of the right-handed neutrinos. How this affects the branching ratio for the Higgs boson to decay into right-handed neutrinos is then discussed. Section 5 computes the lifetime of the right-handed neutrinos assuming minimal flavor violation and discusses its dependence on neutrino mass parameters and mixing angles. I conclude in Section 6 with some comments on model-building issues and summarize results. 2 Higgs Boson Decay The renormalizable Lagrangian describing interactions between the Higgs doubletH (1,2)−1/2, the lepton SU(2)W doublets Li (1,2)−1/2, and three right-handed neutrinos NI (1,1)0 is given by mRNN + λνH̃NL+ λlHLe c (1) where flavor indices have been suppressed and H̃ ≡ iτ2H∗ where H has a vacuum ex- pectation value (vev) 〈H〉 = v/ 2 and v ≃ 247 GeV. Two-component notation is used throughout this note. We can choose a basis where the Majorana mass matrix mR is di- agonal and real with elements MI . In general they will be non-universal. It will also be convenient to define the 3× 3 Dirac neutrino mass mD ≡ λνv/ 2. The standard see-saw mechanism introduces mass mixing between the right-handed and left-handed neutrinos which leads to the active neutrino mass matrix λTν m R λνv 2 = mTDm R mD . (2) This is diagonalized by the PMNS matrix UPMNS [18] to obtain the physical masses mI of the active neutrinos. At leading order in the Dirac masses the mass mixing between the left-handed neutrinos νI and right-handed neutrinos NJ is given by VIJ = [m R ]IJ = [m D]IJM J (3) and are important for the phenomenology of the right-handed neutrinos. For generic Dirac and Majorana neutrino masses no simple relation exists between the physical masses, left- right mixing angles and the PMNS matrix. An estimate for the neutrino couplings is fI ≃ 7× 10−7 0.5eV 30GeV . (4) where λν = URfUL has been expressed in terms of two unitary matrices UL/R and a diagonal matrix f with elements fI . In general UL 6= UPMNS. Similarly, an approximate relation for the left-right mixing angles is VIJ ≃ [UPMNS ]JI = 4× 10−6 0.5eV 30GeV [UPMNS]JI (5) which is valid for approximately universal right-handed neutrino masses MI ≃ M and UR ≃ 1. I note that these formulae for the masses and mixing angles are exact in the limit of universal Majorana masses and no CP violation in the Dirac masses [11]. For these fiducial values of the parameters no limits exist from the neutrinoless double β de- cay experiments or collider searches [5] because the mixing angles are too tiny. No limits from cosmology exist either since the right-handed neutrinos decay before big bang nucle- osynthesis if MI∼>O(GeV), which will be assumed throughout (see Section 5 for the decay length of the right-handed neutrinos). If a right-handed neutrino is lighter than the Higgs boson, MI < mh, where mh is the mass of the Higgs boson, then in principle there may be new decay channels h → NI +X (6) where X may be a Standard Model particle or another right-handed neutrino (in the latter case MI +MJ < mh). For instance, from the neutrino coupling one has h → NIνL. This decay is irrelevant, however, for practical purposes since the rate is too small. But if it is assumed that at the TeV scale there are new dynamics responsible for generating both the Higgs boson mass and the right-handed neutrino masses, then higher- dimension operators involving the two particles should exist and be suppressed by the TeV scale. These can be a source of new and relevant decay processes. Consider then δLeff = O(5)i + O(6)i + · · ·+ h.c. (7) where Λ ≃ O(TeV). Only dimension 5 operators are considered here, with dimension 6 operators discussed elsewhere [19]. The central dot ‘·’ denotes a contraction of flavor indices. At dimension 5 there are several operators involving right-handed neutrinos. However it is shown below that constraints from the observed scale of the left-handed neutrino masses implies that only one of them can be relevant. It is O(5)1 = H †HNN (8) where the flavor dependence is suppressed. The important point is that this operator is not necessarily suppressed by any small Yukawa couplings. After electroweak symmetry breaking the only effect of this operator at tree-level is to shift the masses of the right- handed neutrinos. Constraints on this operator are therefore weak (see below). This operator, however, can have a significant effect on the Higgs boson. For if MI +MJ < mh , (9) the decay h → NINJ (10) can occur. For instance, if only a single flavor is lighter than the Higgs boson, the decay rate is Γ(h → NINI) = 2β2I + (Imc where only half the phase space has been integrated over, c 1 /Λ is the coefficient of (8), and βI ≡ (1− 4M2I /m2h)1/2 is the velocity of the right-handed neutrino. The dependence of the decay rate on β may be understood from the following comments. The uninterested reader may skip this paragraph, since this particular dependence is only briefly referred to later in the next paragraph, and is not particularly crucial to any other discussion. Imagine a scattering experiment producing the two Majorana fermions only through an on-shell Higgs boson in the s-channel. The cross-section for this process is related to the decay rate into this channel, and in particular their dependence on the final state phase space are identical. Conservation of angular momentum, and when appropriate, conservation of CP in the scattering process then fixes the dependence of Γ on phase space. For example, note that the phase of c 1 is physical and cannot be rotated away. When Imc 1 = 0 the operator (8) conserves CP and the decay rate has the β 3 dependence typical for fermions. This dependence follows from the usual argument applied to Majorana fermions: a pair of Majorana fermions has an intrinsic CP parity of −1 [20], so conservation of CP and total angular momentum in the scattering process implies that the partial wave amplitude for the two fermions must be a relative p-wave state. If the phase of c non-vanishing, then CP is broken and the partial wave amplitude can have both p-wave and s-wave states while still conserving total angular momentum. The latter amplitude leads to only a βI phase space suppression. There is a large region of parameter space where this decay rate is larger than the rate for the Higgs boson to decay into bottom quarks, and, if kinematically allowed, not significantly smaller than the rate for the Higgs boson to decay into electroweak gauge bosons. For example, with Im(c 1 ) = 0 and no sum over I, Γ(h → NINI) Γ(h → bb) β3I (12) This ratio is larger than 1 for Λ∼<12|c I TeV . If all three right-handed neutrinos are lighter than the Higgs boson, then the total rate into these channels is larger than the rate into bottom quarks for Λ∼<20|c I TeV. If Im(c 1 ) 6= 0 the operator violates CP and the region of parameter space where decays to right-handed neutrinos dominate over decays to bottom quarks becomes larger, simply because now the decay rate has less of a phase space suppression, as described above. The reason for the sensitivity to large values of Λ is because the bottom Yukawa coupling is small. For mh > 2mW (13) the Higgs boson can decay into a pair of W bosons with a large rate and if kinematically allowed, into a pair of Z gauge bosons with a branching ratio of approximately 1/3. One finds that with Im(c 1 ) = 0 and no sum over I, Γ(h → NINI) Γ(h → WW ) f(βW ) where f(βW ) = 3/4 − β2W /2 + 3β4W /4 [21] and βW is the velocity of the W boson. Still, the decay of the Higgs boson into right-handed neutrinos is not insignificant. For example, with Λ ≃ 2 TeV, c(5)1 = 1 and βI ≃ 1, the branching ratio for a Higgs boson of mass 300 GeV to decay into a single right-handed neutrino flavor of mass 30 GeV is approximately 5%. Whether the decays of the Higgs boson into right-handed neutrinos are visible or not depends on the lifetime of the right-handed neutrino. That issue is discussed in Section 5. It is now shown that all the other operators at d = 5 involving right-handed neutrinos and Higgs bosons are irrelevant for the decay of the Higgs boson. Aside from (8), there is only one more linearly independent operator involving the Higgs boson and a neutrino, O(5)2 = LH̃LH̃ . (15) After electroweak symmetry breaking this operator contributes to the left-handed neutrino masses, so its coefficient must be tiny, c 2/Λ∼<O(mνL) . Consequently, the decay of the Higgs boson into active neutrinos from this operator is irrelevant. In Section 4 it is seen that under the minimal flavor violation hypothesis this operator is naturally suppressed to easily satisfy the condition above. It is then consistent to assume that the dominant contribution to the active neutrino masses comes from mass mixing with the right-handed neutrinos. Other operators involving the Higgs boson exist at dimension 5, but all of them can be reduced to (15) and dimension 4 operators by using the equations of motion. For instance, O(5)3 ≡ −i(∂ µN)σµLH̃ → mRNLH̃ + (H̃L)λTν (LH̃) , (16) where the equations of motion were used in the last step. As a result, this operator does not introduce any new dynamics. Still, its coefficients must be tiny enough to not generate too large of a neutrino mass. In particular, enough suppression occurs if its coefficients are less than or comparable to the neutrino couplings. Under the minimal flavor violation hypothesis it is seen that these coefficients are naturally suppressed to this level. Even if the operators O(5)2 and O 3 are not present at tree-level, they will be generated at the loop-level through operator mixing with O(5)1 . This is because the overall lepton number symmetry U(1)LN is broken with both the neutrino couplings and O 1 present. However, such mixing will always involve the neutrino couplings and be small enough to not generate too large of a neutrino mass. To understand this, it is useful to introduce a different lepton number under which the right-handed neutrinos are neutral and both the charged leptons and left-handed neutrinos are charged. Thus the neutrino couplings and the operators O(5)2 and O 3 violate this symmetry, but the operator O 1 preserves it. In the limit that λν → 0 this lepton number symmetry is perturbatively exact, so inserting O(5)1 into loops can only generate O 2 and O 3 with coefficients proportional to the neutrino couplings. Further, O(5)2 violates this symmetry by two units, so in generating it from loops of Standard Model particles and insertions of O(5)1 it will be proportional to at least two powers of the neutrino couplings. Likewise, in generating O(5)3 from such loops its coefficient is always proportional to at least one power of the neutrino coupling. In particular, O(5)2 is generated directly at two-loops, with c 2 ∝ λTν λνc 1 . It is also generated indirectly at one-loop, since O(5)3 is generated at one-loop, with c 3 ∝ c 1 λν . These operator mixings lead to corrections to the neutrino masses that are suppressed by loop factors and at least one power of mR/Λ compared to the tree-level result. As a result, no significant constraint can be applied to the operator O(5)1 .2 Instead the challenge is to explain why the coefficients of O(5)2 and O 3 in the effective theory are small to begin with. The preceding arguments show why it is technically natural for them to be small, even if O(5)1 is present. The minimal flavor violation hypothesis discussed below does provide a technically consistent framework in which this occurs. 3 Naturalness The operator H†HNN (17) violates chirality, so it contributes to the mass of the right-handed neutrino at both tree and loop level. At tree level δmR = c = 60c GeV . (18) There is also a one-loop diagram with an insertion of this operator. It has a quadratic divergence such that δmR ≃ 2c . (19) Similarly, at one-loop δm2h ≃ 1 mR]Λ . (20) 2This statement assumes c(5)∼<O(16π 2) and that the loop momentum cutoff Λloop ≃ Λ. Constraints might conceivably occur for very light right-handed neutrino masses, but that possibility is not explored here since MI∼>O(GeV) is assumed throughout in order that the right-handed neutrinos decay before big bang nucleosynthesis. 1 ∼ O(1) then a right-handed neutrino with mass MI ≃ 30 GeV requires O(1) tuning for TeV∼< Λ∼< 10 TeV, and mh ≃ 100 GeV is technically natural unless Λ∼> 10 TeV or mR is much larger than the range (MI∼<150 GeV ) considered here. Clearly, if Λ ∼> O(10 TeV) then a symmetry would be required to protect the right- handed neutrino and Higgs boson masses. One such example is supersymmetry. Then this operator can be generalized to involve both Higgs superfields and would appear in the superpotential. It would then be technically natural for the Higgs boson and right-handed neutrino masses to be protected, even for large values of Λ. As discussed previously, for such large values of Λ decays of the Higgs boson into right-handed neutrinos may still be of phenomenological interest. 4 Minimal Flavor Violation The higher dimension operators involving right-handed neutrinos and Standard Model leptons previously discussed can a priori have an arbitrary flavor structure and size. But as is well-known, higher dimension operators in the lepton and quark sector suppressed by only Λ ≃ TeV −10 TeV are grossly excluded by a host of searches for flavor changing neutral currents and overall lepton number violating decays. A predictive framework for the flavor structure of these operators is provided by the minimal flavor violation hypothesis [9, 10, 11]. This hypothesis postulates a flavor symme- try assumed to be broken by a minimal set of non-dynamical fields, whose vevs determine the renormalizable Yukawa couplings and masses that violate the flavor symmetry. Since a minimal field content is assumed, the flavor violation in higher dimension operators is com- pletely determined by the now irreducible flavor violation appearing in the right-handed neutrino masses and the neutrino, charged lepton and quark Yukawa couplings. Without the assumption of a minimal field content breaking the flavor symmetries, unacceptably large flavor violating four fermion operators occur. In practice, the flavor properties of a higher dimension operator is determined by inserting and contracting appropriate powers and combinations of Yukawa couplings to make the operator formally invariant under the flavor group. Limits on operators in the quark sector are 5− 10 TeV [10], but weak in the lepton sector unless the neutrinos couplings are not much less than order unity [11][22]. It is important to determine what this principle implies for the size and flavor structure of the operator 1 )IJH †HNINJ . (21) It is seen below that the size of its coefficients depends critically on the choice of the flavor group for the right-handed neutrinos. This has important physical consequences which are then discussed. In addition one would like to determine whether the operators O(5)2 and O 3 are suf- ficiently suppressed such that their contribution to the neutrinos masses is always sub- dominant. In Section 2 it was argued that if these operators are initially absent, radiative corrections involving O(5)1 and the neutrino couplings will never generate large coefficients (in the sense used above) for these operators. However, a separate argument is needed to explain why they are initially small to begin with. It is seen below that this is always the case assuming minimal flavor violation. To determine the flavor structure of the higher dimension operators using the minimal flavor violation hypothesis, the transformation properties of the particles and couplings are first defined. The flavor symmetry in the lepton sector is taken to be GN × SU(3)L × SU(3)ec × U(1) (22) where U(1) is the usual overall lepton number acting on the Standard Model leptons. With right-handed neutrinos present there is an ambiguity over what flavor group to choose for the right-handed neutrinos, and what charge to assign them under the U(1). In fact, since there is always an overall lepton number symmetry unless both the Majorana masses and the neutrino couplings are non-vanishing, there is a maximum of two such U(1) symmetries. Two possibilities are considered for the flavor group of the right-handed neutrinos: GN = SU(3)× U(1)′ or SO(3) . (23) The former choice corresponds to the maximal flavor group, whereas the latter is chosen to allow for a large coupling for the operator (21), shown below. The fields transform under the flavor group SU(3)× SU(3)L × SU(3)ec × U(1)′ × U(1) as N → (3,1,1)(1,0) (24) L → (1,3,1)(−1,1) (25) ec → (1,1,3)(1,−1) , (26) Thus U(1)′ is a lepton number acting on the right-handed neutrinos and Standard Model leptons and is broken only by the Majorana masses. U(1) is a lepton number acting only on the Standard Model leptons and is only broken by the neutrino couplings. Then the masses and Yukawa couplings of the theory are promoted to spurions transforming under the flavor symmetry. Their representations are chosen in order that the Lagrangian is formally invariant under the flavor group. Again for GN = SU(3) × U(1)′, λν → (3,3,1)(0,−1) (27) λl → (1,3,3)(0,0) (28) mR → (6,1,1)(−2,0) . (29) For GN = SO(3) there are several differences. First, the 3’s of SU(3) simply become 3’s of SO(3). Next, the U(1) charge assignments remain but there is no U(1)′ symmetry. Finally, a minimal field content is assumed throughout, implying that for GN = SO(3) mR ∼ 6 is real. With these charge assignments a spurion analysis can now be done to estimate the size of the coefficents of the dimension 5 operators introduced in Section 2. For either choice of GN one finds the following. An operator that violates the U(1) lepton number by n units is suppressed by n factors of the tiny neutrino couplings. In particular, the dangerous dimension 5 operators O(5)2 and O 3 are seen to appear with two and one neutrino couplings, which is enough to suppress their contributions to the neutrino masses. If GN = SO(3) such operators can also be made invariant under SO(3) by appropriate contractions. If however GN = SU(3)×U(1)′, then additional suppressions occur in order to construct GN invariants. For example, the coefficients of the dimension 5 operators O(5)2 and O 3 are at leading order λ Rλν/Λ and λνm R/Λ respectively and are sufficiently small. It is now seen that the flavor structure of the operator (8) depends on the choice of the flavor group GN . One finds GN = SU(3)× U(1)′ : c 1 ∼ a1 mRTr[m + · · · GN = SO(3) : c 1 ∼ 1+ d1 mR ·mR + · · ·+ e1λνλ†ν + · · · (30) where · · · denotes higher powers in mR and λνλ†ν . Comparing the expressions in (30), the only important difference between the two is that 1 is invariant under SO(3), but not under SU(3) or U(1)′. As we shall see shortly, this is a key difference that has important consequences for the decay rate of the Higgs boson into right-handed neutrinos. Next the physical consequences of the choice of flavor group are determined. First note that if we neglect the λνλ ν ∝ mL contribution to c 1 , then for either choice of flavor group the right-handed neutrino masses mR and couplings c 1 are simultaneously diagonalizable. For GN = SO(3) this follows from the assumption that mR ∼ 6 is a real representation. As a result, the couplings c 1 are flavor-diagonal in the right-handed neutrino mass basis. If GN = SO(3) the couplings c 1 are flavor-diagonal, universal at leading order, and not suppressed by any Yukawa couplings. It follows that Br(h → NINI) Br(h → NJNJ) ≃ 1 (31) up to small flavor-diagonal corrections of order mR/Λ from the next-to-leading-order terms in the couplings c 1 . βI is the velocity of NI and its appearance in the above ratio is simply from phase space. It is worth stressing that even if the right-handed neutrino masses are non-universal, the branching ratios of the Higgs boson into the right-handed neutrinos are approximately universal and equal to 1/3 up to phase space corrections. The calculations from Section 2 of the Higgs boson decay rate into right-handed neutrinos do not need to be rescaled by any small coupling, and the conclusion that these decay channels dominate over h → bb for Λ up to 20 TeV still holds. Theoretically though, the challenge is to understand why MI ≪ Λ. Similarly, if GN = SU(3) the couplings are flavor-diagonal and suppressed by at least a factor of mR/Λ but not by any Yukawa couplings. This suppression has two effects. First, it eliminates the naturalness constraints discussed in Section 3. The other is that it suppresses the decay rate of h → NINI by a predictable amount. In particular Γ(h → NINI) = I (32) where I have set a1 = 1, and Br(h → NINI) Br(h → NJNJ) up to flavor-diagonal corrections of order mR/Λ. In this case, the Higgs boson decays preferentially to the right-handed neutrino that is the heaviest. Still, even with this sup- pression these decays dominate over h → bb up to Λ ≃ 1 TeV if three flavors of right-handed neutrinos of mass MI ≃ O(50GeV) are lighter than the Higgs boson. For larger values of Λ these decays have a subdominant branching fraction. They are still interesting though, because they have a rich collider phenomenology and may still be an important channel in which to search for the Higgs boson. This scenario might be more natural theoretically, since an approximate SU(3) symmetry is protecting the mass of the fermions. 5 Right-handed Neutrino Decays I have discussed how the presence of a new operator at the TeV scale can introduce new decay modes of the Higgs boson into lighter right-handed neutrinos, and described the circumstances under which these new processes may be the dominant decay mode of the Higgs boson. In the previous section we have seen that whether that actually occurs or not depends critically on a few assumptions. In particular, on whether the Higgs boson is light, on the scale of the new operator, and key assumptions about the identity of the broken flavor symmetry of the right-handed neutrinos. Whether the decays of the Higgs boson into right-handed neutrinos are visible or not depends on the lifetime of the right-handed neutrinos. It is seen below that in the mini- mal flavor violation hypothesis their decays modes are determined by their renormalizable couplings to the electroweak neutrinos and leptons, rather than through higher-dimension operators. The dominant decay of a right-handed neutrinos is due to the gauge interactions with the electroweak gauge bosons it acquires through mass mixing with the left-handed neu- trinos. At leading order a right-handed neutrino NJ acquires couplings to WlI and ZνI which are identical to those of a left-handed neutrino, except that they are suppressed by the mixing angles VIJ = [m D]IJM J . (34) If the right-handed neutrino is heavier than the electroweak gauge bosons but lighter than the Higgs boson, it can decay as NJ → W+l−I and NJ → ZνI . Since it is a Majorana particle, decays to charge conjugated final states also occur. The rate for these decays is proportional to |VIJ |2M3J . If a right-handed neutrino is lighter than the electroweak gauge bosons, it decays through an off-shell gauge boson to a three-body final state. Its lifetime can be obtained by comparing it to the leptonic decay of the τ lepton, but after correcting for some additional differences described below. The total decay rate is 3 Γtotal(NI) Γ(τ → µνµντ ) = 2× 9 (cW + 0.40cZ ) . (35) The corrections are the following. The factor of “9” counts the number of decays available to the right-handed neutrino through charged current exchange, assuming it to be heavier than roughly few-10 GeV. The factor of “0.40” counts the neutral current contribution. It represents about 30% of the branching ratio, with the remaining 70% of the decays through the charged current. The factor of “2” is because the right-handed neutrino is a Majorana particle, so it can decay to both particle and anti-particle, e.g. W ∗l− and W ∗l+, or Z∗ν and Z∗ν. Another correction is due to the finite momentum transfer in the electroweak gauge boson propagators. This effect is described by the factors cW and cW where cG(xG, yG) = 2 dzz2(3− 2z) (1− (1− z)xG)2 + yG where xG = M G, yG = Γ G, cG(0, 0) = 1 and each propagator has been ap- proximated by the relativistic Breit-Wigner form. The non-vanishing momentum transfer enhances the decay rate by approximately 10% for mR masses around 30GeV and by ap- proximately 50% for masses around 50 GeV. This effect primarily affects the overall rate and is less important to the individual branching ratios. The formula (36) is also valid when the right-handed neutrino is more massive than the electroweak gauge bosons such that the previously mentioned on-shell decays occur. In that case (35) gives the inclusive decay rate of a right-handed neutrino into any electroweak gauge boson and a charged lepton or a left-handed neutrino. In this case the correction from the momentum transfer is obviously important to include! It enhances the decay rate by approximately a factor of 40 for masses around 100 GeV, but eventually scales as M−2I for a large enough mass. 3An ≈ 2 error in an earlier version has been corrected. An effect not included in the decay rate formula above is the quantum interference that occurs in the same flavor l+l−ν or ννν final states. Its largest significance is in affecting the branching ratio of these specific, subdominant decay channels and is presented elsewhere [19]. Using cττ = 87µm [23] and BR(→ µνµντ ) = 0.174 [23], (35) gives the following decay length for NI , cτI = 0.90m cW + 0.40cZ 30 GeV (120 keV)2 . (37) Care must be used in interpreting this formula, since the Dirac and Majorana masses are not completely independent because they must combine together to give the observed values of the active neutrino masses. This expression is both model-independent and model-dependent. Up to this point no assumptions have been made about the elements of the Dirac mass matrix or the right- handed neutrino masses, so the result above is completely general. Yet the actual value of the decay length clearly depends on the flavor structure of the Dirac mass matrix. In particular, the matrix elements [mDm D]II/MI are not the same as the active neutrino mass masses. This is fortunate, since it presents an opportunity to measure a different set of neutrino parameters from those measured in neutrino oscillations. The masses MI describe 3 real parameters, and a priori the Dirac matrix mD describes 18 real parameters. However, 3 of the phases in mD can be removed by individual lepton number phase rotations on the left-handed neutrinos and charged leptons, leaving 15 pa- rameters which I can think of as 6 mixing angles, 3 real Yukawa couplings and 6 phases. Including the three right-handed neutrino masses gives 18 parameters in total. Five con- straints on combinations of these 18 parameters already exist from neutrino oscillation experiments. In principle all of these parameters could be measured through detailed stud- ies of right-handed neutrino decays, since amplitudes for individual decays are proportional to the Dirac neutrino matrix elements. However, at tree-level these observables depend only on |[mD]IJ | and are therefore insensitive to the 6 phases. So by studying tree-level processes only the 3 right-handed neutrino masses, 3 Yukawa couplings, and 6 mixing angles could be measured in principle. In particular, the dominant decay is h → NINI → qqqqlJ lK which contains no miss- ing energy. Since the secondary events are highly displaced, there should be no confusion about which jets to combine with which charged leptons. In principle a measurement of the mass of the right-handed neutrino and the Higgs boson is possible by combining the in- variant momentum in each event. A subsequent measurement of a right-handed neutrino’s lifetime from the spatial distribution of its decays measures [mDm D]II . More information is acquired by measuring the nine branching ratios BR(NI → qq′lJ) ∝ |[mD]IJ |2. Such measurements provide 6 additional independent constraints. In total, 12 independent con- straints on the 18 parameters could in principle be obtained from studying right-handed neutrino decays at tree-level. To say anything more precise about the decay length would require a model of the neutrino couplings and right-handed neutrino mass parameters. Specific predictions could be done within the context of such a model. Of interest would be the branching ratios and the mean and relative decay lengths of the three right-handed neutrinos. The factor [mDm D]II/MI appearing in the decay length is not the active neutrino mass obtained by diagonalizing mTDm R mD, but it is close. If I approximate [mDm D]II/MI ≃ mI , then cτI ≃ 0.90m 30 GeV 0.48 eV cW + 0.4cZ A few comments are in order. First, the decay lengths are macroscopic, since by inspection they range from O(100µm) to O(10m) for a range of parameters, and for these values are therefore visible at colliders. Next, the decay length is evidently extremely sensitive to MI . Larger values of MI have shorter decays lengths. For instance, if MI = 100 GeV (which requiresmh > 200 GeV) andmI = 0.5 eV then cτI ≃ 0.2mm. Finally, if the active neutrino masses are hierarchical, then one would expect M4I cτI to be hierarchical as well, since this quantity is approximately proportional to m−1L . One or two right-handed neutrinos may therefore escape the detector if the masses of the lightest two active neutrinos are small enough. I have described decays of the right-handed neutrinos caused by its couplings to elec- troweak gauge bosons acquired through mass mixing with the left-handed neutrinos. How- ever, additional decay channels occur through exchange of an off-shell Higgs boson, higher dimension operators or loop effects generated from its gauge couplings. It turns out that these processes are subdominant, but may be of interest in searching for the Higgs bo- son. Exchange of an off-shell Higgs boson causes a decay NI → νJbb which is suppressed compared to the charged and neutral current decays by the tiny bottom Yukawa coupling. Similarly, the dimension 5 operator (8) with generic flavor couplings allows for the decay NI → NJh∗ → NJbb for NJ lighter than NI 4. However, using the minimal flavor vio- lation hypothesis it was shown in Section 4 that the couplings of that higher dimension operator are diagonal in the same basis as the right-handed neutrino mass basis, up to flavor-violating corrections that are at best O(λ2ν) (see (30)). As result, this decay is highly suppressed. At dimension 5 there is one more operator that I have not yet introduced which is the magnetic moment operator ·NσρσNBρσ (39) and it involves only two right-handed neutrinos. It causes a heavier right-handed neutrino to decay into a lighter one, NI → NJ+γ/Z for I 6= J . To estimate the size of this operator, first note that its coefficient must be anti-symmetric in flavor. Then in the context of minimal flavor violation with GR = SO(3), the leading order term is c 4 ≃ [λνλ ν ]AS where 4The author thanks Scott Thomas for this observation. “AS” denotes ‘anti-symmetric part’. This vanishes unless the neutrino couplings violate CP . In that case the amplitude for this decay is of order (λν) 2. If GR = SU(3)×U(1)′ the leading order term cannot be [mR]AS(Tr[mRm q)n/Λn+q, since they vanish in the right- handed neutrino mass basis. The next order involves λνλ ν and some number of mR’s, but there does not appear to be any invariant term. Thus for either choice of GR the amplitude for NI decays from this operator are O(λ ν) or smaller, which is much tinier than the amplitudes for the other right-handed neutrino decays already discussed which are of order λν . Subdominant decays N → ν + γ can occur from dimension 6 operators and also at also one-loop from electroweak interactions, but in both cases the branching ratio is tiny [19]. 6 Discussion In order for these new decays to occur at all requires that the right-handed neutrinos are lighter than the Higgs boson. But from a model building perspective, one may wonder why the right-handed neutrinos are not heavier than the scale Λ. A scenario in which the right-handed neutrinos are composite would naturally explain why these fermions are comparable or lighter than the compositeness scale Λ, assumed to be O(TeV). Since their interactions with the Higgs boson through the dimension 5 operator (8) are not small, the Higgs boson would be composite as well (but presumed to be light). These new decay channels of the Higgs boson will be the dominant decay modes if the right-handed neutrinos are also lighter than the electroweak gauge bosons, and if the coefficient of the higher dimension operator (8) is not too small. As discussed in Section 4, in the minimal flavor violation framework the predicted size of this operator depends on the choice of approximate flavor symmetries of the right-handed neutrinos. It may be O(1) or O(mR/Λ). In the former situation the new decays dominate over Higgs boson decays to bottom quarks for scales Λ∼<10 − 20 TeV, although only scales Λ ≃ 1 − 10 TeV are technically natural. This case however presents a challenge to model building, since the operator (8) breaks the chirality of the right-handed neutrinos. Although it may be technically natural for the right-handed neutrinos to be much lighter than the scale Λ (see Section 3), one might expect that any theory which generates a large coefficient for this operator to also generate Majorana masses mR ∼ O(Λ). In the case where the coefficient of (8) is O(mR/Λ) the new decays can still dominate over decays to bottom quarks provided that the scale Λ ≃ O(1 TeV). For larger values of Λ these decays are subdominant but have sizable branching fractions up to Λ ≃ O(10TeV). This situation might be more amendable to model building. For here an approximate SU(3) symmetry is protecting the mass of the right-handed neutrinos. In either case though one needs to understand why the right-handed neutrinos are para- metrically lighter than Λ. It would be extremely interesting to find non-QCD-type theories of strong dynamics where fermions with masses parametrically lighter than the scale of strong dynamics occur. Or using the AdS/CFT correspondence [8], to find a Randall- Sundrum type model [7] that engineers this outcome. The attitude adopted here has been to assume that such an accident or feature can occur and to explore the consequences. Assuming that these theoretical concerns can be naturally addresseed, the Higgs boson physics is quite rich. To summarize, in the new process the Higgs boson decays through a cascade into a six-body or four-body final state depending on the masses of the right-handed neutrinos. First, it promptly decays into a pair of right-handed neutrinos, which have a macroscopic decay length anywhere from O(100µm − 10m) depending on the parameters of the Majorana and Dirac neutrino masses. If one or two active neutrinos are very light, then the decay lengths could be larger. Decays occurring in the detector appear as a pair of displaced vertices. For most of the time each secondary vertex produces a quark pair and a charged lepton, dramatically violating lepton number. 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Dawson, The Higgs Hunter’s Guide; ibid., [arXiv:hep-ph/9302272]. http://arxiv.org/abs/hep-th/9711200 http://arxiv.org/abs/hep-th/9802109 http://arxiv.org/abs/hep-th/9802150 http://arxiv.org/abs/hep-ph/0207036 http://arxiv.org/abs/hep-ph/0507001 http://arxiv.org/abs/hep-ph/9601367 http://arxiv.org/abs/hep-ph/9607450 http://arxiv.org/abs/hep-ph/9609434 http://arxiv.org/abs/hep-ph/0604261 http://arxiv.org/abs/hep-ph/0605193 http://arxiv.org/abs/hep-ph/0607160 http://arxiv.org/abs/hep-ph/0502105 http://arxiv.org/abs/hep-ph/0510322 http://arxiv.org/abs/hep-ph/0511250 http://arxiv.org/abs/hep-ph/0608310 http://arxiv.org/abs/hep-ph/0607204 http://arxiv.org/abs/hep-ph/0601212 http://arxiv.org/abs/hep-ph/0606172 http://arxiv.org/abs/0705.2190 http://arxiv.org/abs/hep-ph/9302272 [22] V. Cirigliano and B. Grinstein, Nucl. Phys. B 752, 18 (2006) [arXiv:hep-ph/0601111]. [23] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006). [24] Michael L. Graesser, Amit Lath and Jessie Shelton, work in progress. http://arxiv.org/abs/hep-ph/0601111 Motivation Higgs Boson Decay Naturalness Minimal Flavor Violation Right-handed Neutrino Decays Discussion

0704.0439

Coulomb blockade of anyons

Coulomb blockade of anyons Dmitri V. Averin and James A. Nesteroff Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794-3800 (Dated: August 23, 2021) Coulomb interaction turns anyonic quasiparticles of a primary quantum Hall liquid with filling factor ν = 1/(2m + 1) into hard-core anyons. We have developed a model of coherent transport of such quasiparticles in systems of multiple antidots by extending the Wigner-Jordan description of 1D abelian anyons to tunneling problems. We show that the anyonic exchange statistics manifests itself in tunneling conductance even in the absence of quasiparticle exchanges. In particular, it can be seen as a non-vanishing resonant peak associated with quasiparticle tunneling through a line of three antidots. PACS numbers: 73.43.-f, 05.30.Pr, 71.10.Pm, 03.67.Lx Quasiparticles of two-dimensional (2D) electron liq- uids in the regime of the Fractional Quantum Hall effect (FQHE) have unusual properties of fractional charge [1] and fractional exchange statistics [2, 3]. The fractional quasiparticle charge was observed in experiments on an- tidot tunneling [4] and shot-noise measurements [5, 6]. The situation with fractional statistics is so far less cer- tain even in the case of the abelian statistics, which is the subject of this work. Although the recent experiments [7] demonstrating unusual flux periodicity of conductance of a quasiparticle interferometer can be interpreted as a manifestation of the fractional statistics [8, 9], this inter- pretation is not universally accepted [10, 11]. There is a number of theoretical proposals (see, e.g., [12, 13]) sug- gesting tunnel structures where the statistics manifests itself through noise properties. Partly due to complex- ity of noise measurements, such experiments have not been performed successfully up to now. In this work, we show that coherent quasiparticle dynamics in multi- antidot structures should provide clear signatures of the exchange statistics in dc transport. Most notably, in tun- neling through a line of three antidots, fractional statis- tics leads to a non-vanishing peak of the tunnel conduc- tance which would vanish for integer statistics. These effects rely on the ability of quantum anti- dots to localize individual quasiparticles of the QH liq- FIG. 1: Tunneling of anyonic quasiparticles between oppo- site edges of an FQHE liquid through quasi-1D triple-antidot systems: (a) loop, (b) open interval. Quasiparticles tunnel between the edges and the antidots with rates Γ1,2. The an- tidots are coupled coherently by tunnel amplitudes ∆. uids [4, 14, 15]. The resulting transport phenomena in antidots are very similar to those associated with the Coulomb blockade [16] in tunneling of individual elec- trons in dots. For instance, similarly to a quantum dot [19], the linear conductance of one antidot shows periodic oscillations with each period corresponding to the addi- tion of one quasiparticle [4, 14, 15, 17, 18]. Recently, we have developed a theory of such Coulomb-blockade-type tunneling for a double-antidot system [20], where quasi- particle exchange statistics does not affect the transport. The goal of this work is to extend this theory to antidot structures where the statistics does affect the conduc- tance. The two simplest structures with this property consist of three antidots and have quasi-1D geometries with either periodic or open boundary conditions (Fig. 1). A technical issue that needed to be resolved to calculate the tunnel conductance is that the anyonic field opera- tors defined through the Wigner-Jordan transformation [21, 22, 23, 24], are not fully sufficient in the situations of tunneling. As we show below, to obtain correct matrix elements for anyon tunneling, one needs to keep track of the appropriate boundary conditions of the wavefunc- tions which are not accounted for in the field operators. Specifically, we consider the antidots coupled by tun- neling among themselves and to two opposite edges of the quantum Hall liquid (Fig. 1). The edges play the role of the quasiparticle reservoirs with the transport voltage V applied between them. We assume that the antidot- edge coupling is weak and can be treated as a pertur- bation. Quasiparticle transport through the antidots is governed then by the kinetic equation similar to that for Coulomb-blockade transport through quantum dots with a discrete energy spectrum [25]. Coherent quasiparticle dynamics requires that the relaxation rate Γd created by direct Coulomb antidot-edge coupling is weak. This con- dition should be satisfied if the edge-state confinement is sufficiently strong [20]. The requirement on the con- finement is less stringent in the case of the antidot line (Fig. 1b), in which antidot quasiparticles move along the edge, suppressing the antidot-edge coupling at low fre- quencies. We also assume that all quasiparticle energies http://arxiv.org/abs/0704.0439v2 on the antidots, tunnel amplitudes ∆, temperature T , Coulomb interaction energies U between quasiparticles on different antidots, are much smaller than the energy gap ∆∗ for excitations on each antidot. This condition ensures that the state of each antidot is characterized completely by the occupation number n of its relevant quantized energy level. In any given range of the back- gate voltage or magnetic field (which produces the over- all shift of the antidot energies - see, e.g., [4, 14, 15]), there can be at most one quasiparticle on each antidot, n = 0, 1. This “hard-core” property of the quasipar- ticles means that they behave as fermions in terms of their occupation factors, despite the anyonic exchange statistics. All these assumptions can be summarized as: Γd,Γj ≪ ∆, U, T ≪ ∆∗. Under these conditions, the antidot tunneling is domi- nated by the antidot energies. The quasi-1D geometry of the antidot systems we consider makes it possible to in- troduce the quasiparticle “coordinate” x numbering suc- cessive antidots; e.g., x = −1, 0, 1 for systems in Fig. 1. The quasiparticle Hamiltonian can be the written as [ǫxnx − (∆xξ†x+1ξx + h.c.)] + Ux,ynxny , (1) where ǫx are the energies of the relevant localized states on the antidots (taken relative to the common chemical potential of the edges at V = 0), ∆x is the tunnel cou- pling between them, Ux,y is the quasiparticle Coulomb repulsion, and nx ≡ ξ†xξx. The quasiparticle operators ξ†x, ξx in (1) can be viewed as the Klein factors left in the standard operators for the edge-state quasiparticles when all the edge magneto-plasmon modes are suppressed by the gap ∆∗. Characteristics of such Klein factors depend on the geometry of a specific tunneling problem; non- trivial examples can be found in [12, 13, 26, 27]. In the Hamiltonian (1), ξx describe the hard-core anyons with exchange statistics πν. Wigner-Jordan transformation expresses them through the Fermi operators cx [21]: ξx = e iπ(ν−1) nzcx , ξyξx = ξxξye iπνsgn(x−y), (2) with similar relations for ξ†. Anyonic statistics creates an effective interaction be- tween the quasiparticles which can be understood as the Aharonov-Bohm (AB) interaction between a flux tube “attached” to one of the particles and the charge carried by another. In general, this interaction can be masked by the direct Coulomb interaction Ux,y. In the antidot loop (Fig. 1a), however, Ux,y is constant, Ux,y = U , and the interaction term in (1) reduces to Un(n− 1)/2, with n = x nx – the total number of the quasiparticles on the an- tidots. In this case, the Coulomb interaction contributes to the energy separation between the group of states with different n, but does not affect the level structure for given n. The hard-core property of quasiparticles limits n to the interval [0, 3]. For n = 0 and n = 3, the system has the “empty” and “completely filled” state with respective energies E0 = 0 , E3 = x ǫx+3U . The spectrum E1k of the three n = 1 states |1k〉 = x φk(x)ξ x|0〉, is obtained as usual from (1). In the uniform case ǫx = ǫ, ∆x = ∆, with an external AB phase ϕ, one has φk(x) = e ikx/L1/2 E1k = ǫ−∆cos k , k = (2πm+ ϕ)/L , (3) where m = 0, 1, 2, and the loop length is L = 3. Anyonic statistics can be seen in the n = 2 states, |2l〉 = (1/ xy ψl(x, y)ξ x|0〉. The fermion-anyon relation (2) suggests that the stationary two-quasiparticle wavefunctions should coincide up to the exchange phase with that for free fermions: ψl(x, y) = eiπ(1−ν)sgn(x−y)/2√ φq(x) φq(y) φp(x) φp(y) . (4) Here φs are the single-particle eigenstates of the Hamilto- nian (1). (The states (4) are numbered with the index l of the third “unoccupied” eigenstate of (1) complementary to the two occupied ones q, p.) The boundary conditions for the φs are affected by the exchange phase in Eq. (4). To find them, we temporarily assume for clarity that co- ordinates x, y are continuous and lie in the interval [0, L]. Subsequent discretization does not change anything sub- stantive in this discussion. The 1D hard-core particles are impenetrable and can be exchanged only by moving one of them, say x, around the loop from x = y + 0 to x = y − 0 (Fig. 2a). Since the loop is imbedded in the underlying 2D system, such an exchange means that the wavefunction acquires the phase factor eiπν , in which the sign of ν is fixed by the properties of the 2D system, e.g. the direction of magnetic field in the case of FQHE liq- uid. Next, if the second particle is moved similarly, from y = x+ 0 to y = x− 0, the wavefunction changes in the same way, for a total factor ei2πν . Equation (4) shows that only one of these changes can agree with the 1D form of the exchange phase. As a result, the wavefunction (4) satisfies different boundary conditions in x and y: ψl(L, y) = ψl(0, y)e iϕ, ψl(x, L) = ψl(x, 0)e i(ϕ+2πν). (5) Conditions (5) on the wavefunction (4) mean that the single-particle functions φ in (4) satisfy the boundary condition that correspond to the effective AB phase ϕ′ = ϕ+ π− πν, i.e. the addition of an extra quasiparti- cle to the loop changed the AB phase by π − πν, where −πν comes from the exchange statistics and π from the hard-core condition. This gives the energies of the two- quasiparticle states (4) as U + E1q + E1p, where, if the loop is uniform, the single-particle energies are given by Eq. (3) with ϕ → ϕ′. In this case, k E1k = 0, and the energies E2l of the two-quasiparticle states can be written as: E2l = 2ǫ+ U −∆cos l , l = (2πm′ + ϕ− πν)/3 , (6) (a) (b) FIG. 2: Exchanges of hard-core anyons on a 1D loop: (a) real exchanges by transfer along the loop embedded in a 2D sys- tem; (b) formal exchanges describing the assumed boundary conditions (5) of the wavefunction. where m′ = 0, 1, 2. One of the consequences of this discussion is that the sign of ν in the 1D exchange phases of Eqs. (2) and (4) can be chosen arbitrarily for a given fixed sign of the 2D exchange phase. Reversing this sign only exchanges the character of the boundary conditions (5) between x and y. This fact has simple interpretation. Although the 1D hard-core anyons can not be exchanged directly, formally, coordinates x and y in Eq. (4) are independent and one needs to define how they move past each other at the point x = y. Depending on whether the x-particle moves around y from below or (as in Fig. 2b) from above, its trajectory does or does not encircle the flux carried by the y particle, and the boundary condition for x is or is not affected by the statistical phase. The choice made for x immediately implies the opposite choice for y (Fig. 2b), accounting for different boundary conditions (5). This interpretation shows that in calculation of any matrix elements, the participating wavefunctions should be taken to have the same boundary conditions. While this requirement is natural for processes with the same number of anyons, it is less evident for tunneling that changes the number of anyons in the system. Indeed, the most basic, tunnel-Hamiltonian, description of tunneling into the point z of the system leads to the states ξ†z |1k〉 = (1/ ψk(x, y)ξ x|0〉 , (7) ψk(x, y) = [φk(x)δy,z − eiπ(1−ν)sgn(x−y)δx,zφk(y)]/ One can see that Eq. (7) automatically implies specific choice of the boundary conditions which corresponds to the tunneling anyon not being encircled by anyons al- ready in the system. This means that in the calculation of the tunnel matrix elements with the states (4), one should always pair the coordinate of the tunneling anyon with the discontinuous one in (5). Then, the tunnel ma- trix elements are obtained as 〈2l|ξ†z|1k〉 = ψ∗l (x, z)φk(x) . (8) For instance, in the case of uniform loop with states (3) and (6), we get up to an irrelevant phase factor 〈2l|ξ†z|1k〉 = (2/3) cos[(k − l)/2] . (9) Specific anyonic interaction between quasiparticles can be seen in the fact that the matrix elements (9) do not vanish for any pair of indices k, l. In the fermionic case ν = 1, one of the elements (9) always vanishes for any given k, since the two-particle state after tunneling neces- sarily has one particle in the original single-particle state. By contrast, the tunneling anyon can shift existing par- ticle out of its state. The matrix elements involving empty or fully occu- pied states coincide with those for fermions. Taken to- gether with Eqs. (8) and (9) for transitions between the partially filled states, they determine the rates Γj(E) = γjfν(E)|〈ξ†z〉|2 of tunneling between the jth edge and the antidots, where γj is the overall magnitude of the tun- neling rate, and fν(E) = (2πT/ωc) ν−1|Γ(ν/2+iE/2πT )|2e−E/2T /2πΓ(ν) is its energy dependence associated with the Luttinger- liquid correlations in the edges [28]. Here Γ(z) is the gamma-function and ωc is the cut-off energy of the edge excitations. The rates Γj(E) can be used in the stan- dard kinetic equation to calculate the conductance of the antidot system [20]. Anyonic statistics of quasiparticles affects the position and amplitude of the conductance peaks through the shift of the energy levels by quasiparti- cle tunneling (described, e.g., by Eq. (6)) and through the kinetic effects caused by the anyonic features in the ma- trix elements (8). In the case of the antidot loop (Fig. 1a), however, effects of statistics are masked by the external AB flux ϕ through the loop. Since the area of practical antidots is much larger than the internal area of the loop, ϕ is essentially random and can not be controlled by ex- ternal magnetic field on the relevant scale of one period of conductance oscillations. Below, we present the results for conductance for the similar case of a line of antidots (Fig. 1b), the conductance of which is insensitive to the AB flux, and shows effects of fractional statistics in the tunneling matrix elements. As before, the quasiparticle Hamiltonian is given by Eq. (1). In this geometry, the interaction energy U1 ≡ U1,0 = U0,−1 between the nearest-neighbor antidots is in general different from the interaction U2 ≡ U1,−1 between the quasiparticles at the ends. The localization energies on the antidots can be written as ǫj = ǫ+xδ+2λ|x|. We consider first the unbiased line, δ = 0. At low tempera- tures, T ≪ ∆, U , only the ground states of n quasipar- ticles with energies En participate in transport: E0 = 0, E1 = ǫ + λ − ω, E2 = 2ǫ + 3λ − ω̄ + (Ua + Ub)/2, and E3 = 3ǫ+ 2Ua + Ub + 4λ, where ω = (∆ 2 + λ 2)1/2 and ω̄ is given by the same expression with λ replaced by λ̄ = λ − (U1 − U2)/2. In this regime, the linear conduc- tanceG consists of three peaks, with each peak associated with addition of one more quasiparticle to the antidots, (eν)2 γ1 + γ2 anfν(En+1 − En) 1 + exp[−(En+1 − En)/T ] , (10) where an ≡ |〈n + 1|ξ†0|n〉|2. The amplitudes a0, a2 are effectively single-particle, and thus, independent of the −10 −5 0 5 = 2.0∆ = 1.5∆ T = 0.3∆ −10 −5 0 5 −2 0 2 FIG. 3: Linear conductance G of the antidot line in a ν = 1/3 FQHE liquid (Fig. 1b) as a function of the common anti- dot energy ǫ relative to the edges. In contrast to electrons (ν = 1, left inset), tunneling of quasiparticles with fractional exchange statistics produces non-vanishing conductance peak associated with transition between the ground states of one and two quasiparticles. The maximum of this peak is shown in the right inset (ν = 1/3 – solid, ν = 1 – dashed line) as a func- tion of the bias δ. The curves are plotted for ∆1 = ∆2, λ = 0, γ1 = γ2; conductance is normalized to G0 = (eν) 2Γ1(0)/∆1. exchange statistics: a0 = (ω + λ)/2ω, and a2 = (ω̄ − λ̄)/2ω̄. By contrast, the amplitude a1 of the transition from one to two quasiparticles is multi-particle, and is found from Eqs. (4) and (8) to be strongly statistics- dependent, (ω + λ)ω(ω̄ − λ̄)ω̄ cos2(πν/2) . (11) In particular, a1 vanishes for electron tunneling (ν = 1), but is non-vanishing in the case of fractional statistics, e.g., for ν = 1/3, when cos2(πν/2) = 3/4. This is illus- trated in Fig. 3 which shows the conductance G obtained by direct solution of the full kinetic equation for tunnel- ing through the antidots. Qualitatively, the vanishing amplitude a1 for electrons can be understood as a re- sult of destructive interference between the two terms in the two-particle wavefunction which correspond to dif- ferent ordering of the added/existing electron on the an- tidot line. The opposite signs of these two terms lead to vanishing overlap with the single-particle state in the tunnel matrix element. Fractional statistics of quasipar- ticles makes this destructive interference incomplete. Fi- nite bias δ 6= 0 along the line suppresses this interference making the effect of the statistics smaller. One can still distinguish the fractional statistics by looking at the de- pendence of the amplitude of the middle peak of conduc- tance on the bias δ shown in the right inset in Fig. 3. In conclusion, we have developed a model of coherent transport of anyonic quasiparticles in systems of mul- tiple antidots. In antidot loops, addition of individual quasiparticles shifts the quasiparticle energy spectrum by adding statistical flux to the loop. In the case without loops, energy levels are insensitive to quasiparticle statis- tics, but the statistics still manifests itself in the quasi- particle tunneling rates and hence dc tunnel conductance of the antidot system. The authors would like to thank F.E. Camino, V.J. Goldman, J.K. Jain, V.E. Korepin, Yu.V. Nazarov, O.I. Patu, V.V. Ponomarenko, and J.J.M. Verbaarschot for discussions. This work was supported in part by NSF grant # DMR-0325551 and by ARO grant # DAAD19- 03-1-0126. [1] R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983); Rev. Mod. Phys. 71, 863 (1999). [2] B.I. Halperin, Phys. Rev. Lett. 52, 1583 (1984). [3] D. Arovas, J.R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984). [4] V.J. Goldman and B. Su, Science 267, 1010 (1995); V.J. Goldman et al., Phys. Rev. B 64, 085319 (2001). [5] L. Saminadayar et al., Phys. Rev. Lett. 79, 2526 (1997). [6] R. de-Picciotto et al., Nature 389, 162 (1997). [7] F.E. Camino, W. Zhou, and V.J. Goldman, Phys. Rev. Lett. 95, 246802 (2005); Phys. Rev. B 72, 075342 (2005). [8] E.-A. Kim, Phys. Rev. Lett. 97, 216404 (2006). [9] V.J.Goldman, Phys. Rev. B 75, 045334 (2007). [10] J.K. Jain and C. Shi, Phys. Rev. Lett. 96, 136802 (2006). [11] B. Rosenow and B.I. Halperin, Phys. Rev. Lett. 98, 106801 (2007). [12] I. Safi, P. Devillard, and T. Martin, Phys. Rev. Lett. 86, 4628 (2001). [13] C.L. Kane, Phys. Rev. Lett. 90, 226802 (2003). [14] I.J. Maasilta and V.J. Goldman, Phys. Rev. B 57, R4273 (1998). [15] M. Kataoka et al., Phys. Rev. Lett. 83, 160 (1999). [16] D.V. Averin and K.K. Likharev, in: “Mesoscopic Phe- nomena in Solids” (eds. B.L. Altshuler, P.A. Lee, and R.B. Webb) p. 173 (Elsevier, Amsterdam, 1991). [17] M.R. Geller, D. Loss, and G. Kirczenow, Phys. Rev. Lett. 77, 5110 (1996). [18] A. Braggio et al., Phys. Rev. B 74, 041304R (2006). [19] L.P. Kouwenhoven et al. in: “Mesoscopic Electron Trans- port” (eds. L.L. Sohn, L.P. Kouwenhoven, and G. Schön) p. 105 (Kluwer, Dordrecht, 1997). [20] D.V. Averin and J.A. Nestroff, Physica E (2007); cond-mat/0702614. [21] E. Fradkin, “Field theories of condensed matter systems” (Addison-Wesley, 1991), Ch. 7. [22] J.-X. Zhu and Z.D. Wang, Phys. Rev. A 53, 600 (1996). [23] M.T. Batchelor, X.-W. Guan, and N. Oelkers, Phys. Rev. Lett. 96, 210402 (2006). [24] M. D. Girardeau, Phys. Rev. Lett. 97, 100402 (2006). [25] D.V. Averin and A.N. Korotkov, Sov. Phys. JETP 70, 937 (1990); C.W.J. Beenakker, Phys. Rev. B 44, 1646 (1991); D.V. Averin, A.N. Korotkov, and K.K. Likharev, Phys. Rev. B 44, 6199 (1991). [26] V.V. Ponomarenko and D.V. Averin, JETP Lett. 74, 87 (2001); Phys. Rev. B 70, 195316 (2004); Phys. Rev. B 71, 241308(R) (2005). [27] K.T. Law, D.E. Feldman, and Y. Gefen, Phys. Rev. B 74, 045319 (2006). [28] C. de C. Chamon and X.G. Wen, Phys. Rev. Lett. 70, 2605 (1993). http://arxiv.org/abs/cond-mat/0702614

0704.0440

Dynamics of a quantum phase transition in a ferromagnetic Bose-Einstein condensate

Dynamics of a quantum phase transition in a ferromagnetic Bose-Einstein condensate Bogdan Damski and Wojciech H. Zurek Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos, NM 87545, USA We discuss dynamics of a slow quantum phase transition in a spin-1 Bose-Einstein condensate. We determine analytically the scaling properties of the system magnetization and verify them with numerical simulations in a one dimensional model. Studies of phase transitions have traditionally focused on equilibrium scalings of various properties near the crit- ical point. Dynamics of the phase transition presents new challenges and there is a strong motivation for analyzing it. Nonequilibrium phase transitions may play a role in the evolution of the early Universe [1]. Their analogues can be studied in the condensed matter experiments. The latter observation led to development of the theory based on the universality of critical behavior [2], which in turn resulted in a series of beautiful experiments [3]. The re- cent progress in the cold atom experiments allows for time dependent realizations of different models under- going a quantum phase transition (QPT) [4, 5]. These experimental developments are only a proverbial tip of the iceberg, but they call for an in-depth theoretical un- derstanding of the QPT dynamics. A QPT is a fundamental change in ground state (GS) of the system as a result of small variations of an exter- nal parameter, e.g., a magnetic field [6]. It takes place ideally at zero absolute temperature, which is in strik- ing contrast to thermodynamical phase transitions. The most complete description of the QPT dynamics has been obtained so far in spin models [7, 8] that are exactly solv- able. In these systems the gap in the excitation spectrum goes to zero at the critical point, which precludes the adi- abatic evolution across the phase boundary. It leads to creation of excitations whose density and scaling with a quench rate follow from a quantum version [7, 9] of the Kibble-Zurek (KZ) theory [1, 2]. We study dynamics of a ferromagnetic condensate of spin-1 particles [10]. For simplicity, we consider 1D ho- mogeneous (untrapped) clouds: atoms in a box as in the experiment [11] with spinless bosons. We adopt the pa- rameters for our 1D model such that the length and time scales are comparable to experimental ones [12]. Assum- ing that the system is placed in a magnetic field B aligned in the z direction, one gets the following dimensionless mean-field energy functional [12] E[Ψ] = +Q〈Ψ|F 2z |Ψ〉 〈Ψ|Fα|Ψ〉2 (1) where ΨT = (ψ1, ψ0, ψ−1) describes the m = 0,±1 con- densate components, m |ψm|2 = 1, and Fx,y,z are spin-1 matrices [13]. The first term in (1) is the kinetic energy, the second and the fourth term describe spin- independent and spin-dependent atom interactions re- spectively, the third term is a quadratic Zeeman shift coming from atom interactions with a magnetic field. For 87Rb atoms considered here c1 < 0, which results in an interesting phase diagram due to the competition between the last two terms in (1). Restricting analysis to zero longitudinal magnetization case, and introducing q = Q/(n|c1|), n = Ψ†Ψ one finds a polar phase for q > 2, described by ΨTP ∼ (0, 1, 0), and the broken-symmetry phase where ΨTB ∼ ( 4− 2qeiχ1 , 8 + 4qei(χ1+χ−1)/2, 4− 2qeiχ−1) for 0 ≤ q < 2. The freedom of choosing the χ±1 results in rotational symmetry of the transverse magnetization on the (x, y) plane. The transition between these phases can be driven by the change of the magnetic field B im- posed on the atom cloud, q ∼ Q ∼ B2 [14], which was experimentally done in [5]. The dynamics of a QPT depends on the rate of quench driving the system across the phase boundary. For very fast “impulse” transition, the system has no time to ad- just to the changes of the Hamiltonian and arrives in a re- gion where a new phase is expected with the “old” wave- function untouched during the evolution. Slow transi- tions are different: the system has time to “probe” vari- ous broken symmetry “vacua” in the neighborhood of the critical point where it gets excited. We are interested in evolutions that are fast enough to produce macroscopic excitations of the system, but slow enough to reflect scal- ings of the critical region. By comparing analytical find- ings to numerical simulations for experimentally relevant parameters we provide the first complete description of QPT dynamics in a ferromagnetic condensate. Fast transitions were realized in the Berkeley experi- ment [5]. The 3D numerical simulations closely following this experiment were reported in [14]. Analytical studies of the evolution after “impulse” quench were presented in [15, 16]. The paper of Lamacraft [15] also discusses dy- namics of non instantaneous transitions in 2D spinor con- densates focusing on analytical predictions on the growth of the transverse magnetization correlation functions. We start with a qualitative discussion. Consider- ing small perturbations around the GS of the broken- symmetry phase one finds three Bogolubov modes as in http://arxiv.org/abs/0704.0440v2 [13] where quantum fluctuations are studied. In the long wavelength limit (important for slow transitions) there is only one nonzero eigenvalue mode: the gapped mode having eigenenergy ∆ ∼ 4− q2. Suppose now that we drive the system from polar to broken-symmetry phase. The system reaction time to Hamiltonian changes in the broken-symmetry phase is given by the inverse of the gap: τ ∼ 1 [7, 9]. For example, when τ is small enough the evolution becomes adiabatic so the system adjusts fast to parameter changes. Right after entering the broken- symmetry phase, the reaction time is large with respect to the transition time, ∆/ d∆ , and so the system under- goes the “impulse” evolution where its state is “frozen”. The gapped mode starts to be populated around the in- stant t̂ after entering the broken-symmetry phase: the system leaves the “impulse” regime to catch up with in- stantaneous GS solution. This happens when the two time scales become comparable: 1/∆(t̂) ∼ ∆/ d∆ |t=t̂. We consider here transitions driven by q(t) = 2− t/τQ, (2) where τQ is the quench time inversely proportional to the speed of driving the system through the phase transition. For slow transitions of interest here, τQ ≫ 1, we obtain t̂ ∼ τ1/3Q . (3) In the following we analyze dynamics induced by a linear decrease of q(t) (2). The evolution starts from t < 0, i.e., in the polar phase, and ends at t = 2τQ (q = 0). Such q(t) dependence is achieved by ramping down the magnetic field as ∼ 2− t/τQ. The initial state is chosen as a slightly perturbed GS in the polar phase, ΨT ∼ (δψ1, 1/ L+δψ0, δψ−1), where |δψm| ≪ 1/ L are random. We generate the real and imaginary part of δΨm at different grid points with the probability distribution p(x) = exp(−x2/2σ2)/ 2πσ. We take σ = 10−4 to start evolution closely to the polar phase GS. To find the full numerical solution within the mean- field approximation, we integrate three coupled nonlin- ear Schrödinger equations for the ψm condensates that can be easily obtained by the variation of (1). During evolution we look at the magnetization of the sample fα = 〈Ψ|Fα|Ψ〉, α = x, y, z. The transverse magnetization. A total transverse (to the magnetic field in the z direction) magnetization reads MT (t) = dz[f2x(z, t) + f y (z, t)] = dz mT , (4) and is experimentally measurable. It disappears in the GS of the polar phase and equals (1 − q2/4)/L in the broken-symmetry GS. Its typical evolution is depicted in Fig. 1. We see there that nothing happens in the po- lar phase. The system starts nontrivial evolution in the 0 0.5 1 1.5 2 2.5 3 q(t)=2-t/τ 1.5 1.75 2 0.4 1 4 16 64 FIG. 1: (color) Main plot: numerical solution (black solid line) vs. static prediction (red dashed line). The arrow depicts direction of evolution. Inset (a): the same as in the main plot plus a numerically obtained solution of the linearized problem (green divergent line). Inset (b): numerical data vs. fit to τQ ≥ 10 data only (see text for details). In the main plot τQ = 10 (see [12] for units). FIG. 2: The vectors represent (fx(z), fy(z)) × 103. Plot (a): snapshot at q(t = 2.81) = 1.72, i.e., at the first peak in MTL (see Fig. 1). Plot (b): snapshot by the end of time evolution: q(t = 20) = 0. The results come from the same numerical simulation as in Fig. 1 (see [12] for units). broken-symmetry phase at a distance t̂/τQ after the criti- cal point was passed. The magnetization grows fast from that point until it exceeds the static prediction and starts oscillations with the amplitude decreasing in time. We consider slow transitions. Therefore, by the end of time evolution, when q = 0, the system is in the slightly per- turbed ferromagnetic GS: globallyMTL ≈ 1 (Fig. 1) and locally L2mT (z) ≈ 1 (Fig. 3). We can now ask: Does the scaling (3) hold? To find out we define arbitrarily t̂ as the instant when MTL intersects 1%. A fit to numerics for τQ ≥ 10 yields ln t̂ = (0.056±0.01)+(0.332±0.002) lnτQ which confirms prediction (3). This fit is presented in Fig. 1a, where the gradual departure of the numerical data for τQ < 10 from t̂ ∼ τ1/3Q indicates that τQ ≫ 1 or 37ms has to be taken for the observation of 1/3 exponent: quench has to be slow enough to reflect the critical dynamics. 60 65 70 75 FIG. 3: (color) Magnetization of the system at t = 2τQ (q = 0). The dashed lines facilitate observation of extrema coincidences. Results come from the same simulation as in Figs. 1, 2; see [12] for units. In the GS configuration of the broken-symmetry phase the vector (fx, fy) can have arbitrary orientation, so in the dynamical problem considered here it is interesting to find out how is this rotational symmetry broken. When unstable evolution starts, spatial correlations in magne- tization appear (Fig. 2a). In the subsequent evolution these correlations evolve such that the correlation length increases: see Fig. 2b obtained by the end of time evolu- tion. This is a generic picture though the details depend on the quench time τQ and initial state of the system. This behavior suggests creation of spin textures [17, 18]. In our case, topological textures are spin configurations where the magnetization direction varies in space so that the kinetic energy term in (1) is not minimized, but mag- netization magnitude follows closely a GS result. Such structures appear in 1D when the first homotopy group of the vacuum manifold M is nontrivial, which happens here: π1(M) = Z [19]. These textures are characterized by the winding number, 1 Arg(fx + ify), which is not conserved. Indeed, it reads +1 in Fig. 2a, while by the end of that evolution (Fig. 2b) it equals 0. Are different stages of this evolution experimentally observable? Let’s look at τQ = 10 case presented in Figs. 1-3. The evolution from the phase boundary to the first peak in magnetizationMT (the q = 0 point) takes 2.81× 37ms ∼= 104ms (2τQ = 740ms). Both these time scales are well within the reach of the experiment [5]. The longitudinal magnetization. Initially, fz(z) ≈ 0 so dzfz ≈ 0. The conservation of the latter allows only for creation of a network of magnetic domains (non- topological structures with fixed fz sign) having opposite polarizations. The domains appear by the time when the system enters unstable evolution and the maxima of |fz| tend to move towards the minima of mT (Fig. 3). More quantitatively, we performed Nr evolutions starting from different initial conditions, but fixed σ. As in the exper- iment [5], we average over these runs to wash out shot- to-shot fluctuations. In Fig. 4 we plot the mean domain size: ξ = i ξz(i)/Nr, where i = 1, ..., Nr and ξz(i) is the mean domain size in the i-th run. As shown in Fig. 4a, for t . t̂ we observe ξ ≈ f(t/τ1/3Q ) as for MT (t). The domains are formed on a time scale of ∼ t̂. A simple anal- ysis based on KZ theory [1, 2] suggests that their charac- teristic post-transition size, ξ̂, should be roughly given by ∫ ∼t̂ dtvs(t), where vs(t) is a sound velocity. There are two sound modes in the broken-symmetry phase that prop- agate both spin and density fluctuations [13]: the faster (slower) one has velocity ∼ √c0 (∼ q|c1|). Putting any of these as vs into the integral, and assuming τQ ≫ 1 for the slower mode, we get ξ̂ ∼ τ1/3Q . (5) This result correctly predicts the scaling property of the size of post-transition “defects” as is evident from the overlap of different curves in Fig. 4, which shows up for τQ ≥ 25 or 0.9s. Quantitatively, we define ξ̂ as the value of ξ averaged over q ∈ [1/2, 1] to wash out post-transition fluctuations. A fit got us ln ξ̂ = (−0.38 ± 0.03) + (0.30 ± 0.01) ln τQ, in good agreement with (5). The fit was done to τQ ≥ 30 data and is pre- sented in Fig. 4b which illustrates that smaller τQ data gradually departs from 1/3 scaling law. Now we focus on the analytical calculations provid- ing predictions about early stages of time-evolution. We assume that the wave-function stays close to the polar phase GS, ΨT = (δψ1(t), 1/ δψ0(t), δψ−1(t)) exp(−iµt), where the chemical potential is µ = c0/L, |δψm| ≪ 1/ L, and dz(δΨ0 + δΨ 0) ≡ 0 to keep dzΨ†Ψ = 1 + O(δΨ2). Linearizing the cou- pled nonlinear-Schrödinger equations that describe the system we get fχ = ReGχ, where χ = x, y, Gx =√ 2(δΨ1 + δΨ−1)/ L, Gy = i 2(δΨ1 − δΨ−1)/ L, and Gχ = − qGχ − (Gχ +G where α = 2|c1|/L. To solve this equation we go to momentum space, aχ(k) = dzfχ exp(ikz) and bχ(k) = dzImGχ exp(ikz), getting 0 k2 + αq 2α− k2 − αq 0 . (6) Diagonalizing the matrix from Eq. (6) we see that there is instability for k2/α < 2 − q as in the Bogolubov spec- trum of this model [13]. Thus, the system is stable in the polar phase, and so small initial perturbations do not grow during the evolution towards broken-symmetry phase. The instability for q < 2 is responsible for the magnetization jump in Fig. 1 and the subsequent break- down of the linear approach (Fig. 1b). To solve Eq. (6) with q(t) given by (2) we derive the equation for d2aχ(t)/dt 2, keep leading order terms in the slow transition (τQ ≫ 1) and long-wavelength (k2/α ≪ 2) limits, and get that aχ(k, t) = αkχAi(s)+βkχBi(s), , (7) 0 0.5 1 1.5 2 q(t)=2-t/τ 0.75 1.5 4 16 64 FIG. 4: (color) Dynamics of magnetic domains in fz. Black line (τQ = 30), red line (τQ = 50), green line (τQ = 80). The arrow show direction of evolution on the main plot. Inset (a): early stages of ξ(t) evolution. Inset (b): dependence of the typical post-transition domain size, ξ̂, on quench time. The fit was done to τQ ≥ 30 data (see text for details). The figure shows results averaged over Nr = 44 runs; see [12] for units. where κ = (α2/2)1/3, αkχ and βkχ are constants given by initial conditions, while Ai and Bi are Airy func- tions. From (7) we see that the instability arises from unbounded increase of the Bi(s) function happening for s > 0, i.e., k2/α < 2−q(t), which is a dynamical manifes- tation of the static result for unstable modes. This solu- tion works till t ∼ t̂ ∼ τ1/3Q when a significant increase of fχ invalidates the linearized theory: this calculation rig- orously derives scaling (3). Additionally, the solution (7) can be reliably used as long as τQ ≫ 1 or 37ms, which is also supported by numerics (Fig. 1a). The quench time scale in the experiment [5] is much smaller than this bound. Finally, these results hold for any initial state spread over the k modes. The (re)scalings t/τ Q and ξ̂ ∼ t̂ ∼ τ Q derived above in a 1D system were also found by different means in a 2D spinor condensate [15]. A trivial extension of our mean- field analytical calculations to 2D and 3D systems shows that they hold for any number of spatial dimensions. To summarize, we have developed a theory of the dy- namics of symmetry-breaking in the quantum phase tran- sition inspired by the experiment [5], but for the range of quench rates that are sufficiently slow so that the critical scalings can determine phase transition dynamics. This regime should be accessible by a “slower” version of the quench [5]. Our analysis points to a Kibble-Zurek-like scenario, where the state of the system departs from the old symmetric vacuum with a delay ∼ t̂ after the critical point was crossed. This sets up an initial post-transition state with a characteristic length scale ξ̂ (5). This scale should determine the initial density of topological fea- tures. In our 1D simulations textures appear, but we predict that in real 3D experiments other topological de- fects are created (as they were in [5]), and the distance between them should be initially ∼ ξ̂. Such topological defects are more stable than textures so measurement of their density should be possible and would be a good test of the theory we have presented. [1] T.W.B. Kibble, J. Phys. A 9, 1387 (1976); Phys. Rep. 67, 183 (1980). [2] W.H. Zurek, Nature (London) 317, 505 (1985); Acta Phys. Pol. B 24, 1301 (1993); Phys. Rep. 276, 177 (1996). [3] I. Chuang et al., Science 251, 1336 (1991); M.J. Bowick et al., ibid. 263, 943 (1994); C. Bauerle et al., Nature (London) 382, 332 (1996); V.M.H. Ruutu et al., ibid. 382, 334 (1996); S. Ducci et al., ibid. 83, 5210 (1999); A. Maniv, E. Polturak, and G. Koren, Phys. Rev. Lett. 91, 197001 (2003); R. Monaco et al., ibid. 96, 180604 (2006). [4] M. Lewenstein et al., Adv. Phys. 56, 243 (2007). [5] L.E. Sadler, J.M. Higbie, S.R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, et al., Nature (London) 443, 312 (2006). [6] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge UK, 2001). [7] W.H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005). [8] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005); R.W. Cherng and L.S. Levitov, Phys. Rev. A 73, 043614 (2006). [9] B. Damski, Phys. Rev. Lett. 95, 035701 (2005). B. Damski and W.H. Zurek, Phys. Rev. A 73, 063405 (2006). [10] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998); T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). [11] T.P. Meyrath et al., Phys. Rev. A 71, 041604(R) (2005). [12] The experiment [5] is done in the harmonic potential mω2z(λ 2 + λ2yy 2 + z2)/2, where m is the 87Rb mass, ωz = 2π×4.3Hz, λx = 13, and λy = 81.4. We use the os- cillator units through the paper for time (1/ωz = 37ms), length ( ~/mωz = 5.1µm), and energy (~ωz), and as- sume that the system stays in the harmonic oscillator ground states in x and y directions. Then, after skip- ping constant terms, the 3D energy functional [10] re- duces to dimensionless (1) with the additional Ψ†Ψz2/2 term, and ci = 2Nαi mωz/~ λxλy/3 (N = 2× 106 is the atom number, α0 = 16nm, and α1 = −α0/216.1). To approximate such a system by a box we assume that the total density of atoms in the harmonic trap center is the same as in the box of size L. Neglecting in the Thomas-Fermi limit the first and the last term of (1) we get L = 2( c02/3) 2/3 ≈ 78, i.e., 78× 5.1µm ≈ 0.4mm. [13] K. Murata, H. Saito, and M. Ueda, Phys. Rev. A 75, 013607 (2007). [14] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. A 75, 013621 (2007). [15] A. Lamacraft, Phys. Rev. Lett. 98, 160404 (2007). [16] M. Uhlmann, R. Schützhold, and U.R. Fischer, cond-mat/0612664. [17] G.J. Stephens, Phys. Rev. D 61, 085002 (2000). [18] A. Vilenkin and E.P.S. Shellard, Cosmic strings and other topological defects (Cambridge University Press, Cambridge, 1994). [19] H. Mäkelä, Y. Zhang, and K.-A. Suominen, J. Phys. A 36, 8555 (2003). http://arxiv.org/abs/cond-mat/0612664

0704.0441

Evidence for a planetary companion around a nearby young star

Evidence for a planetary companion around a nearby young star J. Setiawan, P. Weise, Th. Henning, R. Launhardt, A. Müller Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany [email protected] J. Rodmann (European Space Agency, ESTEC/SCI-SA, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands) ABSTRACT We report evidence for a planetary companion around the nearby young star HD 70573. The star is a G type dwarf located at a distance of 46 pc with age estimation between 20 and 300 Myrs. We carried out spectroscopic observations of this star with FEROS at the 2.2 m MPG/ESO telescope at La Silla. Our spectroscopic analysis yields a spectral type of G1-1.5V and an age of about 100 Myrs. Variations in stellar radial velocity of HD 70573 have been monitored since December 2003 until January 2007. The velocity accuracy of FEROS within this period is about 10 m/s. HD 70573 shows a radial velocity variation with a period of 852(±12) days and a semi-amplitude of 149(±6) m/s. The period of this vari- ation is significantly longer than its rotational period, which is 3.3 days. Based on the analysis of the Ca II K emission line, Hα and Teff variation as stellar activity indicators as well as the lack of a correlation between the bisector veloc- ity span and the radial velocity, we can exclude the rotational modulation and non-radial pulsations as the source of the long-period radial velocity variation. Thus, the presence of a low-mass companion around the star provides the best explanation for the observed radial velocity variation. Assuming a primary mass m1 = 1.0 ± 0.1 MSun for the host star, we calculated a minimum mass of the companion m2 sin i of 6.1 MJup, which lies in the planetary mass regime, and an orbital semi-major axis of 1.76 AU. The orbit of the planet has an eccentricity of e = 0.4. The planet discovery around the young star HD 70573 gives an impor- tant input for the study of debris disks around young stars and their relation to the presence of planets. Subject headings: stars: general — stars: individual: HD 70573 — stars: plane- tary systems — techniques: radial velocities http://arxiv.org/abs/0704.0441v1 – 2 – 1. Introduction Precise radial velocity (RV) measurements are a well established technique in detecting extrasolar planets around non-active stars, like solar-type stars with similar masses and ages to our Sun (see e.g., Butler et al. 2006). This technique has been also applied in the late 1980’s for planet searches around cool evolved stars (Cochran & Hatzes 1989). However, the number of extrasolar planets around such non solar-type stars is still very small compared to planets around solar-like stars. The situation for young stars is similar, where practically no convincing case is known so far. Planet detections around young and active stars are indeed much more difficult than those around evolved and quiet solar-like stars. Many young stars possess high levels of stellar activity and are also known as fast rotators. Spectroscopically this is indicated by strong line broadening and the presence of emission line features, in particular Hα (λ6536 Å), Ca II H (λ3967 Å) and K (λ3934 Å). Within the same spectral class the stellar activity of young stars is considerably higher than for older stars. The rotational velocity of F-, G- and K-type young stars can be as high as a few hundreds km/s which can be observed by strong line broadening. This makes precise RV measurements very difficult. Intrinsic stellar activity, like non-radial pulsations and rotational modulation, manifests itself in RV variation. In order to distinguish the sources of RV variation in active stars, the stellar spectra have to be investigated carefully, for instance, via the bisector analysis (e.g., Hatzes 1996) and stellar activity indicators, like Ca II H & K emission lines and variation in Hα line, to avoid a misinterpretation of the observed RV variation. This kind of analysis is indispensable for planet searches around active young stars. The search for young planetary systems by the RV technique is indeed limited to young stars which do not show a high activity level. Such a high stellar activity affects the accuracy of the RVmethod, like in stars with high rotational velocity (v sin i > 20 km/s). Nevertheless, in comparison to other young planet search methods, like the direct imaging techniques, the RV method is more sensitive to planetary companions with closer orbits, i.e., less than 10 AU to the parent stars. A further advantage compared to direct imaging is, that the RV method is not strongly limited by distance. It can be applied to planet searches in nearby young moving groups (30–70 pc) and star-forming regions at > 100 pc (e.g., the Taurus-Auriga region at 140 pc), for which direct imaging methods are not possible. This work reports the discovery of a planetary companion around the nearby young star HD 70573. Our RV measurements of HD 70573 show a periodic variation on a time scale which is much longer than the stellar rotational period. This excludes rotational modulation as the source of RV variation. We will show that the bisector technique allows us to distinguish intrinsic stellar activity (non-radial pulsations or stellar rotational modulation – 3 – due to starspots) from variability due to companions. By measuring the bisector velocity spans we detected rotational modulation in other young stars of our sample (Setiawan et al., in preparation). The planet detection around HD 70573 is concluded by the lack of the correlation between the observed RVs and stellar activity indicators (Sect. 4). 2. HD 70573: A nearby young star HD 70573 was identified by Jeffries (1995) as a Lithium rich star. He predicted an age of this star to be substantially younger than 300 Myrs. In a study of young stellar kinematic groups by Montes et al. (2001a), HD 70573 has been classified as a member of the Local Association (Pleiades moving group) with an age range between 20 and 150 Myrs. Later, Lopéz-Santiago et al. (2006) classified HD 70573 as a member of the Hercules-Lyra association, a group of stars comoving in space towards the constellation of Hercules. This moving group has an estimated age of ∼200 Myrs. By comparing the equivalent width of Li λ6708 Å versus the spectral type diagram (Fig. 2 in Montes et al. 2001b), we derived an age within the Pleiades age regime (78–125 Myrs). The stellar parameters of HD 70573 are compiled in Table 1. We measured the equiv- alent widths (EW) of neutral and ionized lines as described in Gray (1992). By comparing our EW measurements with the EWs of standard stars adopted from Cayrel de Strobel (2001) and by using the relation between EWs and temperature we derive the spectral type of G1-1.5V for HD 70573. The stellar parameters Teff , [Fe/H], log g have been calculated by using the TGV model (Takeda et al. 2002), which computes the stellar parameters from the EW of FeI and Fe II. The absolute visual magnitude has been calculated from the visual brightnessmV = 8.70 mag and the distance d = 45.7 pc (Lopéz-Santiago et al. 2006). Henry et al. (2005) has measured photometric variations of HD 70573 and found a period of 3.296 days, which cor- responds to the rotational period of the star. We measured the projected rotational velocity v sin i from the spectral lines by using a cross-correlation method (Benz & Mayor 1981) with the instrumental calibration from Setiawan et al. (2004). Our measured value (see Table 1) is slightly higher than the value published by Henry et al. (1995), who derived v sin i = 11 km/s. – 4 – Fig. 1.— RV measurements of HD 70573. We observed a long-period RV variation of 852 days and short-period variation of few days (see text). 3. Observations and results We are carrying out a RV survey of a sample of young stars with FEROS at the 2.2 m MPG/ESO telescope located at ESO La Silla Observatory, Chile. The spectrograph has a resolution of R = 48 000 and a wavelength coverage of 3600–9200 Å (Kaufer & Pasquini 1998). The data reduction has been performed by using the online pipeline, which produces 39 orders of one-dimensional spectra. The RVs have been measured with the simultaneous calibration mode of FEROS and a cross-correlation technique (Baranne et al. 1996). During the period of three years we obtained a long-term stability of FEROS that is about 10 m/s. RV measurements of HD 70573 are shown in Fig. 1. We observed a long-term RV varia- tion with a period of 852±12 days, which is much longer than the period of the photometric – 5 – Fig. 2.— Lomb-Scargle Periodogram of the RV variation of HD 70573 variability. The semi-amplitude of the RV variation is 149±16 m/s. A Lomb-Scargle peri- odogram (Scargle 1982) of the RVs show the highest peak in the power, which corresponds to the long-period RV variation. On a smaller time scale of several days we also detected short-term RV variations. In the Lomb-Scargle periodogram we also found a lower peak in the power, which corresponds to a period of ∼ 2.6 days. This is comparable to the period in the photometric variation detected by Henry et al. (1995). The False Alarm Probability (FAP) of the peaks are 1.1 × 10−3 for the long-period RV variation and 3.5 × 10−2 for the short-period one. Additional RV measurements, taken with interval of few hours in several consecutive days, may increase the power in the frequency region that corresponds to the period of ∼3 days. – 6 – Fig. 3.— Bisector velocity span vs. RV of HD 70573. The figure shows no correlation between both quantities. This favors the presence of a low-mass companion rather than stellar activity as the source of RV variation. 4. Testing the stellar activity As detected in many surveys, young stars show high stellar activity, characterized by strong X-ray, Hα, Ca II H and K emission. In addition, they are also known as fast rotators. For example, large surveys of young stars in star-forming regions such as NGC 2264 (Lamm et al. 2004) show that the objects are often fast rotators with periods between 0.2 and 15 days. Stellar magnetic activity manifests itself by starspots and granulation, as observed in the Sun. Pulsations have also been observed in young stars (e.g., Marconi et al. 2000). To measure the stellar activity of HD 70573 we investigated the variation of the Ca II K emission line (λ3934 Å ) and Hα. We did not use of the Ca II H (λ3967 Å) to avoid the blend which can be caused by the Hǫ line of the Balmer series. Similar to the method used – 7 – by Santos et al. (2000), we computed an activity index by measuring the intensity of the Ca II K relative to the intensities of 2 Å windows located in the blue and red part of the spectra, which are close to the Ca II K region and do not have strong absorption features. Our measurements do not show any long period variation which might be correlated with the RV variation. The relative rms of the S-index variation is 4.5% of the mean value. In addition, we also measured the equivalent width (EW) variation of the Hα line and Teff variation by using the line-ratio technique (e.g., Catalano et al. 2002) to search for the stellar activity. The EW measurements of the Hα line give a value of 961±45 mÅ. The rms of 45 mÅ corresponds to 4.7% variation in the EW, that is similar to the variation observed in the Ca II K emission line. We observed a short-term Teff variation with a peak-to-peak value of ∼220 K and a period of few days, which is close to the stellar rotational period. This result means an approximately 4% variation in Teff (Table 1) and thus in good agreement with other stellar activity indicators. However, we did not find any long-term periodicity. The equivalent width variation of the Hα line also does not show any long period variation. The stellar activity will leave imprints on the spectral line profile. Another possibility to characterize the stellar activity in the spectra is by using the bisector or the bisector velocity span (Hatzes 1996), which measures the asymmetry of the spectral line profile. Equivalently, the bisector velocity span method can be applied to the cross-correlation function used for the RV computation (Queloz et al. 2001). A correlation between bisector velocity spans and RVs should be expected, if the activity is responsible for the RV variation. In contrast to non-active solar-like stars, the bisector velocity spans of active stars are not constant. The scatter in the velocity spans may provide information about the activity level of the star. In HD 70573 we found no correlation between the bisector velocity spans and RVs (Fig. 3). Thus, based on the results of our analysis of the Ca II K emission lines, Hα, temperature variation and bisector velocity spans as stellar activity indicators we conclude that the observed long-period RV variation of HD 70573 is most likely due to the presence of a low-mass (substellar) companion. 5. Discussion We computed an orbital solution for the RV data of HD 70573 by using a standard Keplerian fit with χ2 minimization. The orbital parameters are listed in Table 2. HD 70573 b is probably the youngest extrasolar planet detected so far with the RV technique (Fig. 4). Planet discoveries around young stars provide important constraints for theories of planet formation. An example is the migration process of planets occurring in the gas-rich – 8 – Fig. 4.— A histogram of the ages of exoplanets as of November 2006. HD 70573 b is the youngest planet detected so far by the RV method. phases of protoplanetary disks. The detection of young planets will also allow us to study the relation between extrasolar planets and the structure of debris disks (Moro-Mart́ın et al. 2006). Since HD 70573 is part of the young star sample of the SPITZER/FEPS legacy program (Meyer et al. 2004), the detection of a planetary companion around this star is of great interest for the study of the relation between debris disks and planets. With a spectral type of G1-1.5V and an age of only 3–6 % of the age of the Sun, the planetary system around HD 70573 could resemble the young Solar system. More planet discoveries around young stars will certainly improve our understanding of planetary systems in their early evolutionary stages. Since planet searches around young stars via the RV method are restricted to the visual wavelength region and are strongly af- fected by stellar activity, other detection techniques like, e.g., NIR direct imaging or astrom- etry, are gaining importance and will most likely soon deliver more discoveries. Astrometric – 9 – measurements with a precision level of few tens of µas, for example, will be able to detect the astrometric signal of the planet around HD 70573, which is ∼0.23 mas. Finally, with the detection of a planetary companion around the young star HD 70573 we have shown, that the RV technique is still potentially profitable for the planet search programs. We thank the La Silla Observatory team for the assistance during the observations at the 2.2 m MPG/ESO telescope. Facilities: FEROS, 2.2 m MPG/ESO. REFERENCES Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373 Benz, W. & Mayor, M. 1981, A&A, 93, 235 Butler, R. P., Wright, J. T., Marcy, G. W., et al. 2006, ApJ, 646, 505 Catalano, S., Biazzo, K., Frasca, A. et al. 2002, A&A, 394, 1009 Cayrel de Strobel, G., Soubiran, C., & Ralite, N. 2001, A&A, 373, 159 Cochran, W. D. & Hatzes, A. P. 1989, BAAS, 21, 114 Gray, D. F., 1992, The Observation and Analysis of Stellar Photosphere, Cambridge Uni- versity Press Hatzes, A. P. 1996, PASP, 108, 839 Henry, G. W., Fekel, F. C., & Hall, D. S. 1995, AJ, 110, 2926 Jeffries, R. D. 1995, MNRAS, 273, 559 Lamm, M., Mundt, R., Bailer-Jones, C. et al. 2004, A&A, 417, 557 Lopéz-Santiago, J., Montes, D., Crespo-Chacon, L. & Fernandez-Figueroa, M.J. 2006, ApJ, 643, 1160 Kaufer, A., & Pasquini, L. 1998, The Messenger, 95 Marconi, M., Ripepi, V., Alcalá, J.M. et al. 2000, A&A, 355, L35 – 10 – Meyer, M. R., Hillenbrandt, L. A., Bachman, D. E. et al. 2004, ApJS, 154, 422 Montes D., Lopéz-Santiago, J., Galvez, M. C. et al. 2001, MNRAS, 328, 45 Montes D., Lopéz-Santiago, J., Fernández-Figueroa, M. J. et al. 2001, A&A, 379, 976 Moro-Mart́ın, A., Carpenter, J. M., et al. 2006, astro-ph/0612242 Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279 Santos, N.C., Major, M., Naef, D. et al., 2000, A&A, 361, 265 Scargle, J. D. 1982, ApJ, 263, 835 Setiawan J., Pasquini, L., da Silva, L. et al., 2004, A&A, 421, 241 Takeda, Y., Ohkubo, M. & Sadakane, K. 2002, PASJ, 54, 451 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0612242 – 11 – Table 1: Stellar parameters of HD 70573. Spectral type G1-1.5V MV 0.4 mag distance 45.7 pc m 1.0 ±0.1 M⊙ Teff 5737 ±70 K [Fe/H ] -0.18 ±0.2 [Fe/H]⊙ log g 4.59 ±0.1 EW (Li) 156 ±20 mÅ Age 78–125 Myrs v sin i 14.7 ±1.0 km/s Prot 3.296 days Table 2: Orbital parameters of HD 70573 b P 851.8 ± 11.6 days K1 148.5 ± 16.5 m/s e 0.4 ± 0.1 ω 269.6 ± 14.3 deg JD0 − 2450000 2106.54 ± 25.72 days reduced χ2 1.34 O − C 18.7 m/s m1 1.0 ± 0.1 M⊙ m2sini 6.1 ± 0.4 MJup a 1.76 ± 0.05 AU Introduction HD 70573: A nearby young star Observations and results Testing the stellar activity Discussion

0704.0442

Quantum electromagnetic X-waves

Quantum electromagnetic X-waves Alessandro Ciattoni1, 2, ∗ and Claudio Conti3, † Consiglio Nazionale delle Ricerche, CASTI Regional Lab 67010 L’Aquila, Italy Dipartimento di Fisica, Universitá dell’Aquila, 67010 L’Aquila, Italy Research Center ”Enrico Fermi” and SOFT INFM-CNR, Universitá La Sapienza, Piazzale Aldo Moro 2, 00185 Rome, Italy (Dated: November 1, 2018) We show that two distinct quantum states of the electromagnetic field can be associated to a classical vector X wave or a propagation-invariant solution of Maxwell equations. The difference between the two states is of pure quantum mechanical origin since they are internally entangled and disentangled, respectively and can be generated by different linear or nonlinear processes. Detec- tion and generation of Schrödinger-cat states comprising two entangled X-waves and their possible applications are discussed. PACS numbers: 03.75.Lm, 05.30.Jp, 42.65.Jx INTRODUCTION Electromagnetic X waves and, more generally, “localized waves” [1] are propagation-invariant solutions of classical Maxwell equations in vacuum [2, 3, 4], dielectric media [5], plasmas [6], optically nonlinear materials [7, 8, 9, 10] and periodic structures [11, 12, 13, 14]. In recent years these waves have attracted a considerable interest since, despite at a first glance they are non-physical (for example some of them apparently travel superluminally and carry an infinite amount of energy), propagation-invariant fields support several applications like image transmission, reconstruction and telecommunications (for a recent review see Ref.[1]). Since the first observation of localized optical X waves [15], the fundamental implications of rigidly travelling spatio- temporal correlations have been recognized. Classically, these waves can be generated by means of linear devices (like axicons) [15] and, recently, their experimental feasibility has been also proved through nonlinear optical processes [16, 17]. Within the quantum framework, analytical techniques borrowed from the subject of localized waves have allowed some authors to consider highly localized states for the photon-wavefunction [18], whereas the role of quantum X waves in nonlinear optical process (limited to the paraxial regime) were analyzed in [19]. In this paper we consider the issue of “constructing a quantum state of the electromagnetic field whose classical counterpart is an X-wave satisfying the full set of Maxwell equations in vacuum”. More precisely, we investigate the possibility of introducing various quantum states on which the mean value of the electric field operator coincides with a prescribed classical vector X wave. We show that the solution of this problem is not-trivial by finding two different quantum states characterized by the above property. Even if from a classical perspective they describe the same entity, these two states essentially differ in the degree of entanglement of their internal modal structure so that both a “disentangled” and an “entangled” quantum description can be provided of an arbitrary classical X wave. In this respect one can investigate a method for discerning between the two possibilities and, correspondingly, for recognizing which kind of quantum X wave is produced by a given mechanism (e.g. axicons or nonlinear processes). In addition, we show that some X wave states can be considered as multi-mode, multi-dimensional Schrödinger cats, exhibiting the relevant feature of being propagation invariant; the corresponding macroscopic quantum content is controlled by the velocity of the X waves and can be maintained for long distances. QUANTUM DESCRIPTION OF X WAVES Within a classical formulation, the x-component of the electric field of a radially symmetric linearly polarized X wave can be written as (extension to more general X waves solutions [1] is trivial) Ex(r, t) = ℜ dkf(k)J0(sin θ0kr⊥)e ik cos θ0 cos θ0 where r⊥ = x2 + y2 and ℜ stands for the real part, whereas the longitudinal component is simply expressed in terms of the transversal components. (see e.g. [4]) This kind of classical waves propagates along the z−axis without distortion and it is fully characterized by its spectrum f(k) and its velocity c/ cos θ0. Physically, such an X wave http://arxiv.org/abs/0704.0442v1 arises as the superposition of various plane-waves at different frequencies whose wave-vectors are all inclined at the same angle θ0 with the propagation direction z. In order to give a quantum description of X waves, we construct a quantum state of the electromagnetic field through the requirement that the quantum mean value of the electric field operator on this state yields a propagation invariant wave of the kind of Eq.(1) and we show that this can be done in two different manners. In order to fix the notation, we start from the standard expression of the electric field operator in Heisenberg representation, Ê(r, t) = h̄c|k| 2ε0L3 iâk,s e i(k·r−c|k|t) ek,s + h.c., where L is the edge of the quantization cubic box (see e.g. [20, 21]), k = (2π/L)(nxex + nyey + nzez) (where nx, ny and nz are integers), ek,s are the pair (s = 1, 2) of polarization unit vectors associated to the mode k and âk,s are the standard photon annihilation operators. Bearing in mind that the mean value of the electric field operator on a coherent state is a plane wave, let us consider the quantum states |X〉 = D̂k,s[∆θ0(k)β(k, s)]|0〉 (2) where D̂k,s(α) is the displacement operator which, acting on the vacuum state |0〉, produces the coherent state |α〉k,s (associated to the mode k, s) according to the relation D̂k,s(α)|0〉 = |α〉k,s [20]. Here β(k, s) is a complex weight function to be determined below and, in order to deal with the discreteness of the k vectors, we have defined ∆θ0(k) = 1 if k ∈ C(θ0, δθ) and ∆θ0(k) = 0 elsewhere, where C(θ0, δθ) is the portion of space comprised between the two coaxial cones of aperture angles θ0 − δθ/2 and θ0 + δθ/2 and whose common axis coincides with the z−axis. The presence of the function ∆θ0(k) assures that the state |X〉 of Eq.(2) is the tensor product of coherent states whose modes have wave vectors globally lying, for δθ ≪ θ0, on a cone with aperture angle θ0 so that the relation kz = η|k⊥| with η = 1/ tan θ0 holds for each wave vector. The mean value of the electric field operator on the state |X〉 is given 〈X |Ê|X〉 = ℜ ∆θ0(k) 2h̄c|k| iβ(k, s) ei(k·r−c|k|t)ek,s  (3) from which we envisage that, for L → ∞ and δθ ≪ θ0, 〈X |Ê(r, t)|X〉 → E(x, y, z−V t), which is a genuine propagation- invariant vector field travelling along the z−axis. In order to prove this assertion we note that the limit is different from zero only if β is dependent on δθ and L and it is given by β(k, s) = (2π)3 2h̄c|k|L3 fs(k), (4) where k⊥ = kxex + kyey, fs(k) are two arbitrary function of k and the various coefficients are chosen for later convenience. Substituting Eq.(4) into Eq.(3) and performing the limits L → +∞ and δθ → 0 by means of the rule (2π)3 d3k and the relation limδθ→0 ∆θ0(k)/δθ = δ(θ − θ0) we obtain 〈X |Ê|X〉 = ℜ dφ [f1(k)ek1 + f2(k)ek2] e i(k·r−ckt) where, after defining k = k[sin θ0(cosφex + sinφey) + cos θ0ez], polar coordinates has been introduced for the k integration. Note that, inside the integral of Eq.(5), the relation k·r−ckt = k[(x cosφ sin θ0+y sinφ sin θ0)+cos θ0(z− ct/ cos θ0)] holds, so that the field in Eq.(5) describes a wave travelling undistorted along the z−axis with velocity V = c/ cos θ0, as expected. Note that, strictly speaking, this rigorous nondiffracting behavior is a consequence of the limit δθ → 0 which is an idealization of the realistic condition δθ ≪ θ0, δθ representing the experimental uncertainty of the wave vectors smeared around the selected conical surface. Since δθ can be experimentally chosen much smaller than θ0, the actual fields are nearly nondiffracting objects, their departure from ideal fields being experimentally tunable. A convenient choice for the modes polarization unit vectors is given by ek1 = cos θ0(cosφex + sinφey) − sin θ0ez and ek2 = − sinφex + cosφey so that a linearly polarized wave along the x−direction is obtained by letting f2 = −f1 tanφ cos θ0 with f1 = (cosφ/ cos θ0)[f(k)/k2] from which we get f1(k)ek1 + f2(k)ek2 = f(k) (ex − cosφ tan θ0ez) /k2 where the factor k−2 has been added for later convenience. Inserting this expression into Eq.(5), we obtain 〈X |Ê|X〉 = ℜ dkf(k) J0 (sin θ0kr⊥) ex − x tan θ0 J1 (sin θ0kr⊥) ez eik cos θ0(z−V t) (6) where Jn(ξ) is the Bessel function of the first kind of order n and the relation dφeih(x cosφ+y sinφ) = 2πJ0(hr⊥) has been exploited. The expectation value of the electric field in Eqs.(6) describes a linearly polarized X-waves (see Eq.(1)) which is an exact solution of Maxwell equations in vacuum (see Ref.[4]). Note that the longitudinal component Ez is due to the full non-paraxial character of the exact approach we are considering and it is negligible for small aperture angles θ0. Let us consider the state of the electromagnetic field given by |XE〉 = ∆θ0(k)Φ(k, s)D̂k,s [α(k, s)] |0〉 (7) where α(k, s) and Φ(k, s) are arbitrary complex function, the second one being constrained by the normalization conditions 〈XE |XE〉 = 1. Note that the very presence of the function ∆θ0(k) in Eq.(7) implies that modes are collected in such a way that their wave vectors almost lies, in the limit δθ ≪ θ0, on the surface of the cone characterizing the spectrum of an arbitrary classical X wave and, therefore |XE〉 is expected to describe a quantum X wave. In order to prove this assertion we note that 〈XE |âks|XE〉 = ∆θ0(k)α(k, s)Q(k, s) (8) where Q(k, s) = |Φ(k, s)|2 + Φ(k, s)e− 12 |α(k,s)|2 k′ 6=k,s′ 6=s ∆θ0(k ′)Φ∗(k′, s′)e− |α(k′,s′)|2 so that the mean value of the electric field on |XE〉 turns out to be 〈XE |Ê|XE〉 = ℜ ∆θ0(k)G(k, s)e i(k·r−c|k|t)  (9) where G(k, s) = i 2h̄c|k| α(k, s)Q(k, s). Comparing Eq.(9) with Eq.(3) and noting that their structures are identical, we deduce that, in the limit δθ → 0 and L → ∞, the state |XE〉 describes a quantum X wave. It is worth noting that this state is completely different from the |X〉 of Eq.(2) and it is obtained as a linear combination of the states D̂k,s[α(k, s)]|0〉 each corresponding to a coherent state in the mode (k, s), all other modes being in the vacuum state. This implies that the state |XE〉 arises from an entangled superposition of modes in coherent states, where with “entanglement” we mean that the state cannot be expressed as a product of kets containing different modes. We therefore conclude that a prescribed classical X wave can be represented by two different quantum states |X〉 and |XE〉 exhibiting an internal disentangled and entangled modal structure, respectively whose quantum difference is expected to play a fundamental role during the measurement process. As a matter of fact, a realistic measurement device sensible to a bounded set S of wave vectors k affects the states |X〉 and |XE〉 in a dramatically different way after the measurement since a measurement carried out on |X〉 leaves the modes k /∈ S unaltered whereas the same measurement on |X〉E profoundly changes the structure of the modes k /∈ S causing, as usual, an irreversible loss of information and notably of the propagation invariance. GENERATION AND DETECTION OF QUANTUM X WAVES The same difference between the two proposed quantum X waves also raises the problem of discerning which one (i.e. disentangled or entangled) is obtained from a prescribed mechanism capable of generating a classical X wave. Since the disentangled X wave is just the product of coherent states with different frequencies but with the same axicon angle, |X〉 is well expected to be produced by a simple linear device, namely an axicon (as in [15]), exposed to a coherent non-monochromatic source like a mode-locked laser. Conversely the entangled X wave is expected when considering nonlinear processes since it is well known that frequency-conversion or parametric processes are accompanied by tight phase-matching condition able to provide the spectral structure characterizing a classical X wave, as addressed by various authors with reference to various kind of nonlinearities (as in [22, 23, 24, 25]). A viable scheme for detecting entanglement on a generated X-waves is offered by homodyne detection that select a specific mode, eventually including a prism or a grating spatially separating the various angular frequencies. This cor- respond to consider the projection of the state |XE〉 onto a specific mode (at frequency ω1 and within a normalization constant) |XE〉projected = |α1〉+ΦE |0〉 (10) where ΦE will in general be dependent on the specific X-wave or on its axicon angle. Using homodyne detection (see [26]) and tuning the local field phase θ it is possible to unveil fringes in the probability distribution of the detected current. As an example, for θ = − arg(α1) + π/2, the probability of detecting the value x of the output current |〈x|α〉|2 is an oscillating function: |〈x|α〉|2 = e 1 + |ΦE |2 + 2|ΦE | cos( 2|α1|x− argΦE) . (11) Note that the same reasoning with the state |X〉 yields ΦE = 0 so that oscillations in |〈x|α〉|2 do not appear if the original state is not entangled. More general results can be obtained by considering beam-splitters and generic phases θ, but their mathematical description is cumbersome and will be reported elsewhere. In addition to the considered X waves, we discuss here the interesting possibility of turning a disentangled X wave into an entangled one by letting the former (appropriately generated by an axicon) to pass through a nonlinear Kerr medium. In order to avoid mathematical complications, we consider the typical spectrum of a broad band mode-locked laser containing M harmonics ωm = mω0 (m = 1, 2, ...,M) and we focus on the case where each mode is in a coherent state corresponding to a plane wave propagating with the conical angle θ0. The disentangled X wave generated by the axicon is formed by the modes whose wave vectors are such that km = mω0/c with kz = km/(1 + η 2) and reads |X〉 = |α1〉1|α2〉2...|αM 〉M where αm denotes the complex parameter of the coherent state with angular frequency ωm. When travelling through a Kerr medium the various modes induce cross and self-phase modulation and the relevant and well-known interaction Hamiltonian corresponds to that of a nonlinear oscillator (see [26]) and reads ĤI = χ(h̄/2) m,p n̂mn̂p, where n̂m = â mâm is the photon number operator of the mode m. If t is the interaction time, the output state from the crystal |Xout〉 = e− ĤI t|X〉 is given by |Xout〉 = e− |αr | n1,n2,...,nM=0 m,p=0 q=1,2,...,M |nq〉. (12) Note that, for t = 4π/χ the output state coincides with the incoming one whereas for t = 2π/χ the state |Xout〉 is obtained from |X〉 after the replacement αm → −αm and the resulting disentangled X wave state can be denoted as | − X〉. More interesting is the situation for t = π/χ since, defining N = m nm and exploiting the relation e−i(π/2)N = [e−iπ/4 + (−1)N eiπ/4]/ 2, we obtain obtain |Xout〉 = e−iπ/4|X〉+ eiπ/4| −X〉 . (13) which is an entangled superposition of two disentangled quantum X waves travelling at the same velocity. The key point is here that the interaction time t coincides with the time spent by the classical X wave to pass through the nonlinear medium or t = D cos θ0/c, D being the nonlinear medium length. As for coherent states, the states periodically evolve with temporal period 4π/χ and with spatial period 4πc/(cos θ0χ). As a consequence, by simply acting on the velocity of the X-wave (e.g. by varying the axicon angle) it is possible to fine-tune the degree of entanglement of the two macroscopic classical X-waves, and even to switching from classical states to Schrödinger-cat states. This is a not-trivial outcome of the interplay between the classical interference process supporting X waves and the entanglement of coherent states. In addition, X waves travel at a velocity which is different from the other linear waves so that, by using coincidences measurements, it is straightforward to distinguish these states from a statistical mixture, and hence to point out their purely quantum properties as described above. CONCLUSIONS In conclusion we have shown that classical electromagnetic X-waves may hide different degrees of quantum entan- glement, depending on their generation mechanism (exploiting a linear, a nonlinear medium or both of them). In this respect their velocity (i.e. the axicon angle) plays a prominent role. Other interesting consequences of the X-waves structure arise when dealing with interferometric setup, as for example that considered in [27]. If a nonlinear medium is placed in one or both arms of an interferometer, the output state is a “progressive undistorted squeezed vacuum” or entangled superposition of classical X waves so that any detection scheme is affected by the axicon angle, and tuning among various quantum states can be attained, as it will be detailed elsewhere. The quantum properties of X-waves can hence be exploited for free-space quantum communications, highly sensible interferometers or quantum computing. We acknowledge fruitful discussions with E. Del Re, S. Trillo, B. Crosignani and P. Di Porto. ∗ Electronic address: [email protected] † Electronic address: [email protected] [1] E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, eds., Localized Waves (Wiley-Interscience, 2007). [2] J. Lu and J. F. Greenleaf, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39, 19 (1992). [3] R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, Phys. Rev. Lett. 62, 147 (1989). [4] A. Ciattoni, C. Conti, and P. Di Porto, Phys. Rev. E 69, 036608 (2004). [5] H. Sonajalg and P. Saari, Opt. Lett. 21, 1162 (1996). [6] A. Ciattoni and P. Di Porto, Phys. Rev. E 70, 035601 (2004). [7] L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, Phys. Rev. Lett. 89, 153902 (2002). [8] S. Orlov, A. Piskarskas, and A. Stabinis, Opt. Lett. 27, 2103 (2002). [9] C. Conti, S. Trillo, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, and P. Di Trapani, Phys. Rev. Lett. 90, 170406 (2003). [10] K. Staliunas and M. Tlidi, Phys. Rev. Lett. 94, 133902 (pages 4) (2005). [11] C. Conti and S. Trillo, Phys. Rev. Lett. 92, 12040 (2004). [12] S. Longhi, Phys. Rev. E 68, 066612 (2004). [13] O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 30, 2611 (2005). [14] Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, Physical Review Letters 98, 023901 (2007). [15] P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997). [16] O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. D. Trapani, Physical Review Letters 97, 243903 (2006). [17] A. Picozzi and M. Haelterman, Phys. Rev. Lett. 88, 0839011 (2002). [18] P. Saari, M. Menert, and H. Valtna, Opt. Commun. 246, 445 (2005). [19] C. Conti, arXiv:quant-ph/0309069v3 (2003); arXiv.org:quant-ph/0409130 (2004). [20] L. Mandel and E. Wolf, Optical Coherece and Quantum Optics (Cambridge University Press, 1995). [21] R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000), 3rd ed. [22] R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevicius, and A. Stabinis, Opt. Commun. 244, 411 (2005). [23] S. Trillo, C. Conti, P. D. Trapani, O. Jedrkiewicz, J. Trull, G. Valiulis, and G. Bellanca, Opt. Lett. 27, 1451 (2002). [24] C. Conti and S. Trillo, Opt. Lett. 28 1251 (2003); C. Conti, Phys. Rev. E 68 016606 (2003); ibid. 70, 046613 (2004). [25] S. Longhi, Phys. Rev. E 69, 016606 (2004). [26] B. Yurke, Phys. Rev. A 32 311 (1985); B. Yurke and D. Stoler, Phys. Rev. Lett. 57 13 (1986); P. Tombesi and A. Mecozzi, J. Opt. Soc. Am. B 4 1700 (1987). [27] B. C. Sanders, Phys. Rev. A 45, 6811 (1992). mailto:[email protected] mailto:[email protected] http://arxiv.org/abs/quant-ph/0409130 Introduction Quantum Description of X waves Generation and Detection of quantum X Waves Conclusions Acknowledgments References

0704.0443

Neutron-neutron scattering length from the reaction gamma d --> pi^+ nn employing chiral perturbation theory

Neutron–neutron scattering length from the reaction γd → π+nn employing chiral perturbation theory V. Lenskya, V. Barub, E. Epelbauma,c, C. Hanharta, J. Haidenbauera, A. Kudryavtsevb, Ulf-G. Meißnerc,a a Institut für Kernphysik, Forschungszentrum Jülich GmbH, D–52425 Jülich, Germany b Institute of Theoretical and Experimental Physics, 117259, B. Cheremushkinskaya 25, Moscow, Russia c Helmholtz-Institut für Strahlen- und Kernphysik (Theorie), Universität Bonn, Nußallee 14-16, D–53115 Bonn, Germany November 4, 2018 Abstract We discuss the possibility to extract the neutron-neutron scattering length ann from experi- mental spectra on the reaction γd → π+nn. The transition operator is calculated to high accuracy from chiral perturbation theory. We argue that for properly chosen kinematics, the theoretical uncertainty of the method can be as low as 0.1 fm. FZJ-IKP-TH-2007-13, HISKP-TH-07/12 1 Introduction A precise knowledge of the neutron-neutron scattering length ann is, e.g., important for an understand- ing of the effects of charge symmetry breaking in nucleon–nucleon forces [1]. The scattering length ann characterizes scattering at low energies. It is related to the on–shell 1S0 scattering amplitude f fon(pr) = pr cot δ(pr)− ipr −a−1nn + 12 rnnp2r +O(p4r)− i pr , (1) where pr is the relative momentum between the two neutrons, δ(pr) the scattering phase shift in the 1S0 partial wave and rnn is the effective range. At low energies the terms of order p r can be neglected to very high accuracy. Obviously, a direct determination of ann in a scattering experiment is extremely difficult due to the absence of a free neutron target. For this reason, the value for ann is to be obtained from analyses of reactions where there are three particles in the final state, e.g. π−d → γnn [2, 3, 4] or nd → pnn [5, 6, 7]. There is some spread in the results for ann obtained by the various groups. In particular, two independent analyses of the reaction nd → pnn give significantly different values for ann, namely ann = −16.1± 0.4 fm [6] and ann = −18.7± 0.6 fm [7], whereas the latest value obtained from the reaction π−d → γnn is ann = −18.5 ± 0.3 fm [4]. At the same time, for the proton-proton scattering length, which is directly accessible, a very recent analysis reports app = −17.3 ± 0.4 fm [8] after correcting for electromagnetic effects. This means that even the sign of ∆a = app − ann is http://arxiv.org/abs/0704.0443v1 not fixed.1 It should be mentioned, however, that state of the art calculations for the binding energy difference of tritium and 3He suggest that ∆a > 0 [9, 10]. In the present work we discuss the possibility to determine ann from differential cross sections in the reaction γd → π+nn. Specifically, we show that one can extract the value of ann reliably by fitting the shape of a properly chosen momentum spectrum. In this case the main source of inaccuracies, caused by uncertainties in the single–nucleon photoproduction multipole E0+, is largely suppressed. Furthermore there is a suppression of the quasi-free pion production at specific angles. We show that at these angular configurations the extraction of ann can be done with minimal theoretical uncertainty. Our investigation is based on the recent work of Ref. [11] in which the transition operator for the reaction γd → π+nn was calculated up to order χ5/2 in chiral perturbation theory (ChPT) with χ = mπ/MN ≃ 1/7, where mπ (MN ) is the pion (nucleon) mass. Half-integer powers of χ in the expansion arise from the unitarity (two– and three–body) cuts (see also [12]). The results of Ref. [11] for the total cross section are in very good agreement with the experimental data. The only input parameter that entered the calculation was the leading single–nucleon photoproduction multipole E0+, which was fixed from a fourth-order one-loop calculation of Bernard et al. [13]. The uncertainty in E0+ is the main theoretical error in the calculation presented in Ref. [11]. Besides this transition operator, in the present study we use nucleon–nucleon (NN) wave functions constructed likewise in the framework of ChPT, namely those of the NNLO interaction of Ref. [14]. This allows us to estimate the theoretical uncertainty which arises from variations in the wave functions. In fact, as soon as we include consistently all terms up to order χ5/2, we expect the ambiguities due to different wave functions not to be larger than a χ3 correction, for only at this order the leading counter term which absorbs these effects enters. This expectation is indeed quantitatively confirmed in the concrete calculations. Since we work within chiral perturbation theory we can estimate the effect of higher orders in terms of established expansion parameters together with the standard assumption that additional short ranged operators, that enter at higher orders, behave in accordance with the power counting (the so-called naturalness assumption). This method was also applied in Refs. [15, 16], where the reaction π−d → γnn was investigated as a tool to extract ann. However, to know the effect of higher orders for sure, one has to calculate them. Therefore, to derive a reliable uncertainty estimate for the extraction of ann from the γd reaction, we use our leading order calculation as baseline result and estimate the theoretical uncertainty from the effects of the higher orders that we calculated completely. Based on this, we find a theoretical uncertainty δann . 0.1 fm. We therefore argue that the reaction γd → π+nn appears to be a good tool for the extraction of ann. To end this section, we remark that in Ref. [17] a method was proposed to extract scattering lengths from γd induced meson production. However, this approach should not be used here, since the momentum transfer is not sufficiently large to use this method and with our explicit calculation of the transition operator we can reach a significantly higher accuracy. 2 ChPT calculation for γd → π+nn The diagrams that contribute to the reaction γd → π+nn are shown on Fig. 1. The kinematical variables are defined in Fig. 2. Before going into the details some comments are necessary regarding the relevant scales of the problem. In the near threshold regime of interest here (excess energies of at most 20 MeV above the 1Note, that, in contrast to app, ann is not corrected for electromagnetic effects. However, since those are only of the order of 0.3 fm [1] they are not relevant for the sign of ∆a. But they ought to be taken into account for determining charge symmetry breaking effects quantitatively. χ , χ , χ χ , χ0 1 2 2 5/2 (d2)(b2) (b1) (c1) (d1) Figure 1: Diagrams for γd → π+nn. Shown are one–body terms (diagram (a) and (b) ), as well as the corresponding rescattering contribution (c)—all without and with final state interaction. Diagrams (d) shows the class of diagrams with intermediate NN interaction. Solid, wavy, and dashed lines denote nucleons, photons and pions, in order. Filled squares and ellipses stand for the various vertices (see Ref. [11] for the details), the hatched area shows the deuteron wave function and the filled circle denotes the nn scattering amplitude. Crossed terms (where the external lines are interchanged) are not shown explicitly. pion production threshold) the outgoing pion momenta are small compared even to the pion mass. Thus, in addition to the conventional expansion parameters of ChPT mπ/Λχ and qγ/Λχ, where Λχ denotes the chiral symmetry breaking scale of order of (and often identified with) the nucleon mass, and qγ denotes the photon momentum in the center–of–mass system which is of order of the pion mass, we can also regard kπ/mπ as small, where kπ denotes the momentum of the outgoing pion. In what follows we will perform an expansion in two parameters, namely χm = mπ/MN and χQ = kπ/mπ . Obviously, the value of the second parameter depends on the excess energy Q. The energy regime of interest to us corresponds to excess energies up to 20 MeV. The maximum value of χQ, χ 2Q/mπ, at the highest energy considered is thus about 1/2. Since this is numerically close to we use the following assignment for the expansion parameter: χ ∼ χm ∼ χ2Q . (2) The tree level γp → π+n vertex, as it appears in diagrams (a1) and (a2) in Fig 1 (the vertex is labeled as filled square), contributes at leading order (order χ0), and orders χ1 and χ2, depending on the one–body operator used. Note that the loop diagrams with πN rescattering (see diagrams (b), (c) and (d) in Fig. 1) contribute at order χ2m as well as at χ mχQ, χ m and at χ Q. The origin of the non–integer power of χ are the two–body (πN) and three–body (πNN) singularities. Thus, all terms up to χ5/2 are explicitly taken into account in our calculation of the transition operator. As already emphasized, we employ wave functions evaluated in the same framework in order to have a fully consistent calculation. In our work, we use the N2LO wave functions corresponding to the chiral NN forces introduced in Ref. [18] and based on the spectral function regularization (SFR) scheme [19]. At this order, the NN force receives contributions from one-pion exchange, two-pion p−q 2 Figure 2: Kinematical variables for γd → π+nn. The relative neutron–neutron momentum is defined as ~pr = (~p1 − ~p2). exchange at the subleading order as well as from all possible short-range contact interactions with up to two derivatives. In addition, the dominant isospin-breaking correction due to the charged-to-neutral pion mass difference in the one-pion exchange potential together with the two leading isospin-breaking S-wave contact interactions were taken into account [18]. The two corresponding low-energy constants were adjusted to reproduce the scattering lengths ann and app. The SFR cutoff Λ̃ is varied in the range 500 . . . 700 MeV. It was argued in Ref. [19] that such a choice for Λ̃ provides a natural separation of the long- and short-range parts of the nuclear force and allows to improve the convergence of the chiral expansion [19]. The cutoff Λ in the Lippmann-Schwinger equation is varied in the range 450 . . . 600 MeV. For an extensive discussion on the choice of Λ and Λ̃ the reader is referred to [14, 18]. 3 Differential cross sections: relevant features In this section we outline the features of the differential cross section for unpolarized particles that are important for our considerations. For later convenience let us consider the function F proportional to the square of the matrix element as well as the five–fold differential cross section F (pr, θr, φr, θπ, φπ) = C pr kπ(pr)|M(pr, θr, φr, θπ, φπ)|2 ∝ d5σ(pr, θr, φr, θπ, φπ) dΩ~prdΩ~kπ , (3) where ~pr (~kπ) stands for the relative momentum of the two final neutrons (momentum of the final pion) in the center–of–mass frame, θr, φr (θπ, φπ) for the corresponding polar and azimuthal angles, respectively, and |M|2 for the squared and averaged amplitude. In Eq. (3) C is an irrelevant dimen- sionful constant. In what follows we will consider only shapes of cross sections and therefore the value of C is not important for our considerations. The value of kπ at given pr and excess energy Q is fixed by energy conservation: , (4) hence we write kπ(pr) in Eq. (3). In the following we choose the momentum ~qγ of the initial photon to be along the z–axis. Then the cross sections at a certain excess energy Q depend on four variables, namely the magnitude of the relative momentum of the two final neutrons pr, the polar angles of the vectors ~pr and ~kπ, and the difference between the azimuthal angles of those two momenta. Unpolarized cross sections are invariant under rotations around the beam axis, which makes the dependence on the missing angle trivial. Typical differential cross sections F are shown in Fig. 3 as a function of pr at some fixed set of angles {φr, θπ, φπ} and Q = 5 MeV for two different values of θr. One can see from this figure that for the differential cross section F of Eq. (3) there are two characteristic regions: 1. The region of quasi-free production (QF) at large pr, which corresponds to the dominance of those diagrams of Fig. 1 that do not contain the NN interaction in the final or intermediate states. In the Appendix we give explicit expressions for the diagram a1 – the most significant diagram of this type. At large pr the pion momentum kπ is small (see Eq. (4)) and the arguments of the deuteron wave function in Eqs. (A.1) and (A.2) may become small for particular combinations of ± ~pr and ~qγ/2. This feature gives rise to a peak in the differential cross section at large pr. 2. The region with prominence of the strong nn final–state interaction (FSI) at small pr (in fact, we would have the strongest final state interaction at zero relative momentum, however the cross section goes to zero at pr = 0 due to the phase space, therefore we see a peak shape). One can see from Fig. 3 that the FSI peak depends on the value of θr only marginally, whereas the quasi-free peak shows significant dependence on this angle. In particular, the quasi-free production is largely suppressed at θr = 90 ◦ — at this angle the arguments of the wave functions in both terms in the r.h.s. of Eqs. (A.1) and (A.2) are large. It can also be seen from Fig. 3 (right panel) that the effect of higher orders is more important for the quasi-free production amplitude — the influence of higher-order effects on the FSI production is quite small. Another interesting observation is that the contributions of higher orders change the relative height of the two peaks – the FSI peak goes up whereas the QF peak goes down when we proceed from the LO calculation to the order χ5/2. In order to suppress the distortions of the spectrum due to higher orders in the chiral expansion, which is the condition for an extraction of ann with small theoretical uncertainty, configurations should be chosen where θr = 90 We now briefly discuss the dependence of the cross section on the remaining angles θπ, φπ (we always may choose φr to be zero). The dependence on θπ is illustrated in Fig. 4. One can see from this figure that the dependence on θπ is significant for both the quasi–free as well as the FSI peak. This can be easily understood from the explicit expressions for the matrix elements given in the Appendix keeping in mind that already at Q = 5 MeV the maximal value of kπ is about mπ/3 while qγ ≈ mπ. Thus, the momentum transfer to the nucleon pair, |~qγ −~kπ|, varies in the range 2mπ/3 to 4mπ/3 depending on θπ. Since the S-wave deuteron wave function is large only for very small arguments, the influence of the direction of ~kπ is significant. In addition, from Fig. 4 it follows that a variation of θπ not only changes the magnitude but also the shape of the cross section, even in the FSI region. This has to be taken into account in the analysis of any experiment. In contrast to the polar angles, the dependence of F on φπ is negligible for all configurations (there is no dependence at all for θr = 0 ◦ and at θr = 90 ◦, only the anyway small QF contribution changes by just 5 %). 4 Extraction of ann and estimate of the theoretical uncertainty In this section we discuss how to extract the scattering length from future data on γd → π+nn as well as the resulting theoretical uncertainty. Our focus is especially the latter point. As in the previous section we will only discuss results at excess energy Q = 5 MeV. However, the analysis can be repeated analogously at any excess energy within the range of applicability of the formalism, i.e. Q ≤ 20 MeV. We are interested in extracting the value of ann, which, in turn, is a low-energy characteristic of neutron-neutron scattering and manifests itself in the momentum dependence of the cross section at small values of the momentum pr. The influence of the value of ann on the cross section is illustrated in 0 10 20 30 40 50 60 70 [MeV] 0 10 20 30 40 50 60 70 [MeV] .] FSI QF 0 degrees QF 90 degrees Figure 3: Left panel: Differential cross section. The solid line corresponds to the configuration when the quasi–free peak is suppressed (θr = 90 ◦), whereas the dashed line corresponds to one of the configurations when the quasi–free production amplitude is maximal (θr = 0 ◦). The values of the remaining angles are θπ = 135 ◦, φr = φπ = 0 ◦; they are the same for both curves. Right panel: Differential cross section — relative strength of QF and FSI peaks. Here the dashed curves correspond to the calculation at LO, the solid ones to the calculation at χ5/2. Curves denoted by ”FSI” (”QF”) are obtained by retaining only those diagrams of Fig. 1 that contain (do not contain) the final or the intermediate nucleon–nucleon interaction. The labels ”0 degrees” and ”90 degrees” denote the corresponding values of θr for the ”QF” curves whereas the ”FSI” curves are almost insensitive to this angle. The values or the remaining angles are as on the left panel of this figure. The overall scale is arbitrary in both panels but the relative normalization is the same for all curves. Fig. 5, where the cross sections are shown for three different values of ann, namely −18, −19, −20 fm. For each value there are two curves, the dashed one corresponds to θr = 0 ◦, and the solid one to θr = 90 ◦. One can see from Fig. 5 that the influence of different values of ann is significant in the FSI peak and marginal in the quasi–free peak, as one would have expected. In the previous section we have shown (see right panel in Fig. 3) that the relative height of the quasi- free and the FSI peak changes if the effects of higher orders are included in the cross sections. Therefore those angular configurations are to be preferred, where the quasi-free production is suppressed. The central point of this study is to demonstrate that there is a large sensitivity of the momentum spectra to the scattering length and that this scattering length can be extracted with a small and controlled theoretical uncertainty. As outlined in the Introduction, we can estimate this uncertainty reliably, because the effect of the higher orders up to χ5/2 are calculated completely. In order to demonstrate the effect of those higher orders on the shape of the momentum distribution, in Fig. 6 we show as the light band the spread in the results for the calculation from LO to χ5/2. The results also include higher partial waves for the pion as well as the final nn system. There is some sensitivity to the behavior of the deuteron wave function at short distances. For the reaction π−d → γnn this sensitivity was identified as the largest effect at N3LO in Ref. [15] 2. Guided by that observation we include in the uncertainty estimate also the spread in the results due to the use of different wave functions. In order to remove the effect of the change in normalization when, e.g., changing the chiral order, all curves are normalized at pnorm = 30 MeV in Fig. 6. In the same 2Within the framework of ChPT with a consistent power counting scheme, the quantitative impact of the wave- function dependence is governed by the order at which a counter term appears that can absorb this model dependence. The corresponding counter term for the γd as well as the πd reaction arises at N3LO. 0 10 20 30 40 50 60 70 [MeV] 0 10 20 30 40 50 60 70 [MeV] Figure 4: Dependence of the differential cross section on θπ. The left panel corresponds to the suppressed quasi–free amplitude (θr = 90 ◦), the right panel to the maximal quasi–free amplitude (θr = 0 ◦). Solid, dashed, dotted, and dash-dotted lines correspond to θπ = 0 ◦, 45◦, 90◦, 135◦ respectively. The values of the remaining angles are φr = φπ = 0 ◦. The overall scale is arbitrary in both panels but the relative normalization is the same for all curves. Figure (with the same normalization) we also show the change in the shape that comes from different values of the scattering length: the dark band is generated by a variation of ann by ±1 fm around the central value of −18.9 fm. Clearly, the theoretical uncertainty is negligibly small compared to the signal of interest. One way to quantify the theoretical uncertainty is through the use of the function S, defined as S(ann,Φ) = F (pr|a(0)nn ,Φ(0))−N(ann,Φ) F (pr|ann,Φ) w(pr) , (5) where pmax = MNQ is the maximum value of pr, F (pr|ann,Φ) is proportional to the five-fold differential cross section as defined in Eq. (3). In the latter we refrained from showing the angular dependence in favor of the parametric dependence of the cross section on the nn scattering length ann as well as the multi–index Φ, which symbolizes the dependence of the cross section on the chosen chiral order and the wave functions used, as outlined above. The weight function w(pr) was introduced to allow us to suppress particular regions of momenta in the analysis — the role of w(pr) will be discussed in detail below. For simplicity we may assume that S is dimensionless; all dimensions can be absorbed into the constant C defined in Eq. (3). The value a nn denotes the central value of the scattering length (−18.9 fm) for which we perform the estimate of the theoretical uncertainty3 whereas Φ(0) corresponds to the baseline type of calcu- lation, namely leading order with chiral wave functions as specified in the Appendix. The relative normalization N(ann,Φ) is fixed by demanding that S gets minimized for any given pair of parameters ann,Φ (∂S/∂N = 0). This gives N(ann,Φ) = dpr F (pr|a nn ,Φ (0))F (pr|ann,Φ)w(pr) dpr F 2(pr|ann,Φ)w(pr) . (6) 3Note that the theoretical uncertainty practically does not change when the central value of the scattering length varies in the relevant interval ±1 fm. 0 10 20 30 40 50 60 70 [MeV] -20 fm -19 fm -18 fm Figure 5: The effect of varying the value of ann on the differential cross section. The solid and dashed lines correspond to the same angular configurations as in Fig. 3, left panel. The different values of ann are shown on the figure. The overall scale is arbitrary but the relative normalization is the same for all curves. Obviously S is the continuum version of the standard χ2 sum, i.e. it characterizes the mean-square deviation from the baseline cross section F (pr|a nn ,Φ (0)). In this way we determine the theoretical uncertainty in full analogy to the standard method of data analysis. In order to quantify the theoretical uncertainty we may define Φmax as that chiral order and choice of wave function, where S(a(0)nn ,Φmax) gets maximal: S(a(0)nn ,Φmax) = max S(a(0)nn ,Φ) . (7) Therefore S(a(0)nn ,Φmax) provides an integral measure of the theoretical uncertainty of the differential cross section. Demanding that the effect of a change in the scattering length by the amount ∆ann matches that by the inclusion of higher orders etc., we can identify ∆ann as an uncertainty in the scattering length. Expressed in terms of S, we may define ∆ann via S(a(0)nn +∆ann,Φ(0)) = S(a(0)nn ,Φmax) . (8) This relation is illustrated in Fig. 7. The dashed horizontal line corresponds to S(a(0)nn ,Φmax), where we use w(pr) = 1. The dashed parabolic line shows the corresponding S(a nn + ∆ann,Φ (0)) as a function of ∆ann. The calculation is performed for θr = 90 ◦, and θπ = 0 ◦. The crossing point of the curves corresponds to ∆ann = 0.16 fm, which can be identified as the theoretical uncertainty for the extraction of the scattering length. In the previous section we showed that the signal region is located at momenta lower than 30 MeV. On the other hand, the theoretical uncertainty of the differential cross section is largest for large values of pr due to the onset of the quasi–free contribution. In view of these two facts it seems reasonable to use such weight functions w(pr) that suppress the contribution of large momenta. For instance, we may use w(pr) = Θ(p cut−pr) for the weight function. If we choose, e.g., pcut = 30 MeV the theoretical uncertainty of the extraction of the scattering length reduces to 0.07 fm, as is demonstrated by the solid lines in Fig. 7. This figure nicely illustrates that the parabolic curve that represents the signal changes only very little when a restriction to small values of pr is applied. At the same time this procedure significantly reduces the value of the uncertainty S(a(0)nn ,Φmax). 0 10 20 30 40 50 60 70 [MeV] Figure 6: The light (white) band is the error band, and dark (blue) band correspond to ±1 fm shift in the scattering length from the central value −18.9 fm. The observation that the dependence of S(a(0)nn +∆ann,Φ(0)) on ∆ann is very well approximated by a parabola allows for a more systematic study of the pcut dependence of the theoretical uncertainty. We therefore define α(pcut) = S(a(0)nn +∆ann,Φ(0)|pcut) (∆ann)2 , (9) where the explicit pcut dependence is introduced into the function S through the weight function w as explained above. The dashed and the solid parabola in Fig. 7 can then be written as α(pcut) (∆ann) with α(pmax) = 41 fm −2 and α(30 MeV) = 33 fm−2. In the left panel of Fig. 8 we show α(pcut) as the solid line. In the same panel the dashed line represents the measure of the theoretical uncertainty given by S(a(0)nn ,Φmax|pcut), multiplied by a factor of 40. This figure makes more quantitative the statement made above: for very small values of pcut we cut into the signal region and therefore α shows a very rapid variation. However, as soon as pcut is larger than 30 MeV it goes to a plateau (in the figure indicated by the arrow). On the other hand, the theoretical uncertainty is monotonously growing once pcut is larger than 30 MeV. From this figure we deduce that the ideal value for pcut is between 25 and 40 MeV. This translates into a theoretical uncertainty between 0.05 and 0.1 fm, as illustrated in the right panel of the same figure. The value of θπ also has some impact on the theoretical uncertainty, however, in its whole parameter range the estimated uncertainty stays below 0.1 fm for pcut = 30 MeV. Clearly, also the experimental data, once they exist, should be analyzed using a procedure analogous to the one given above. This means that the scattering length is to be extracted from a χ2 fit of the theoretical curves to the data. In this work we used the calculation at LO as baseline result and the results at higher orders to estimate the theoretical uncertainty. Consequently, we propose to use the momentum spectrum calculated at LO in the fitting procedure of the experiment. The corresponding analytical expressions are given in the Appendix. The only parameter to be adjusted besides the scattering length is the overall normalization. In this fitting procedure only those data points should be included that are below a given pcut, in order to keep the theoretical uncertainty small. -0.2 -0.1 0 0.1 0.2 [fm] Figure 7: The functions S(a(0)nn ,Φmax) and S(a nn + ∆ann,Φ (0)) are shown by the horizontal and parabolic curves, respectively. The solid curves are obtained by adding the weight factor in Eq. (5) that cuts all momenta above 30 MeV in distinction from the dashed ones. The calculation is performed for the scattering length a nn = −18.9 fm, θr = 90◦, and θπ = 0◦. The value of ∆ann corresponding to the crossing point of the horizontal and parabolic curves determines the theoretical uncertainty of the calculation. 5 Discussion and conclusions In the previous section it was shown that for the angular configurations that suppress the quasi-free production the inclusion of higher order effects (NLO, N2LO, and χ5/2) as well as the use of different wave functions leads only to a minor change in the momentum dependence of the five-fold differential cross sections. Based on this observation we propose to use the momentum spectrum calculated at LO for the extraction of the neutron–neutron scattering length from the data. This procedure has the advantage that the corresponding matrix elements can be given in an analytic form (see Appendix) that could be used directly in the Monte Carlo codes for the experiment analysis. In this way the non–trivial dependence of the spectra on θπ, discussed above, can be easily controlled. The scattering length can then be extracted by a two parameter fit to the data where, simultaneously to a variation in ann, the normalization constant needs to be adjusted. Note that the leading order calculation basically agrees to the expression given in Ref. [20] long ago. However, a systematic and reliable study of the theoretical uncertainties of the extraction was possible only within our full calculation up to order χ5/2 in ChPT. In this way we could show that the reaction γd → π+nn is very well suited for a determination of the nn scattering length. The theoretical uncertainty of order 0.1 fm for the extracted scattering length, estimated in this paper, is of the same order as that claimed for π−d → γnn [16] and nd → pnn [4, 7]. We discussed in detail the theoretical uncertainty for a fixed excess energy of Q = 5 MeV only, however, it should be clear that the procedure can be easily repeated for any energy within the range of applicability of our approach (Q ≤ 20 MeV). For example, we checked that the theoretical uncertainty stays below 0.1 fm also at Q = 10 MeV. Note that the number of events in the signal region scales roughly with Q, the phase space available for the pion. It remains to be seen which energy is the best for the corresponding experiment. We showed that for a proper choice of both kinematics and weight function w, the theoretical 0 10 20 30 40 50 60 70 [MeV] 0 10 20 30 40 50 60 70 [MeV] Figure 8: Left panel: Comparison of the pcut dependence of functions S(a(0)nn ,Φmax|pcut) (dashed curve) and α(pcut) (solid curve). The calculation is performed for the scattering length a nn = −18.9 fm, θr = 90 ◦, and θπ = 0 ◦. Right panel: The corresponding theoretical uncertainty ∆ann as a function of pcut. uncertainty for the extraction of the neutron–neutron scattering length from γd → π+nn can be as low as 0.1 fm. It should be stressed, however, that this error was evaluated most conservatively – we use our LO calculation as baseline result and estimate the theoretical uncertainty from the effects of the higher orders that we calculated completely. This error can be significantly reduced by further studies. For example, if we include in the uncertainty estimate only the spread in the results due to the use of different wave functions, which is identified as the largest effect at N3LO for the reaction π−d → γnn [15], the theoretical uncertainty of the extracted scattering length reduces by one order of magnitude. This indicates that the theoretical uncertainty is indeed under control. However to put this N3LO estimation on more solid ground a complete calculation should be performed to this order. Most of the operators that are relevant at this order are the same as those of π−d → γnn, given explicitly in Ref. [21]. One counter term enters, which can be fixed from other processes [16], e.g., from nd scattering [22], the reaction NN → NNπ [23], or from weak decays [16]. Once this is done we may use our calculation to order χ5/2 as baseline result and estimate the theoretical uncertainty from the then available N3LO calculation. Although we have identified the angles θr = 90 ◦ as the preferred kinematics, also other configura- tions could be studied in order to control the systematics. However, then the spectra calculated at χ5/2 should be used in the analysis. Acknowledgments We thank A. Bernstein for useful discussions and interest in this work. We also thank D. R. Phillips and A. G̊ardestig for helpful discussions. This research is part of the EU Integrated Infrastructure Initiative Hadron Physics Project under contract number RII3-CT-2004-506078, and was supported also by the DFG-RFBR grant no. 05-02-04012 (436 RUS 113/820/0-1(R)) and the DFG SFB/TR 16 ”Subnuclear Structure of Matter”. A. K. and V. B. acknowledge the support of the Federal Program of the Russian Ministry of Industry, Science, and Technology No 02.434.11.7091. E. E. acknowledges the support of the Helmholtz Association (contract no. VH-NG-222). A Leading amplitudes In this appendix we give explicit expressions for the amplitudes that appear at leading order in the calculation for γd → π+nn. As outlined in the main text these expressions can be used directly in the analysis of the data, once available. In addition, they should also proof useful for the design of the experiment. Note, as outlined in the text, only near θr = 90 ◦ the leading order calculation gives a sufficiently accurate representation of the spectra. At all other angles one should use the complete calculation. At leading order only diagrams a1 and a2 of Fig. 1 contribute. Since only the momentum dependence of the amplitudes is relevant for the experimental analysis we drop an overall factor compared to Ref. [11]. The corresponding amplitudes read: M sa1 = u(~pr − ~kπ/2 + ~qγ/2) + u(−~pr − ~kπ/2 + ~qγ/2) (A.1) M ta1 = u(~pr − ~kπ/2 + ~qγ/2)− u(−~pr − ~kπ/2 + ~qγ/2) (A.2) Ma2 = 8π fon(pr) g(pr) (2π)3 u(~p− ~kπ/2 + ~qγ/2) g(p) p2 − p2r − i0 fon(pr) iqπγ g(pr) p2r + β αi − ipr + iqπγ αi − ipr − iqπγ αi + βj − iqπγ αi + βj + iqπγ (A.3) where u(~p) denotes the S–wave part of the deuteron wave function in momentum space. We checked by explicit calculations that the inclusion of the deuteron D-wave changes only the absolute scale of the differential cross sections but not its momentum dependence. Thus, the D-wave contribution is not taken into account in the parameterization. The quantity qπγ is defined as qπγ = |~kπ − ~qγ|/2. The labels s and t stand for spin singlet and triplet final two-nucleon states, respectively — we do not write out the corresponding spin structures. We take into account only the 1S0 partial wave in the final state interaction. For a discussion of the effect of nn P–waves see Ref. [11]. To derive the expression forMa2 we used the fact that the neutron–neutron scattering amplitude can be represented to high accuracy in separable form [11, 24]. The neutron–neutron scattering amplitude, f(p, k;E), can be written in half off–shell kinematics as f(p, k; k2/MN ) = 2π2MNg(p)g(k) g2(q) q2−k2−i0 = fon(k) , (A.4) where the corresponding on–shell amplitude fon(k) can then be expressed in terms of the scattering phase–shifts through fon(k) = f(k, k; k2/MN ) = k cot δ(k) − ik For small momenta one can use the effective range expansion for k cot δ = −1/ann + rnnk2/2+O(k4), in agreement with Eq. (1). Here ann is the parameter to be fitted to the data and rnn = 2.76 fm. We checked that changing the value of rnn within the bounds allowed (±0.1 fm [1]) leads to negligible effects on the extraction of the scattering length. In this way we expressed the matrix element explicitly in terms of the scattering length. We checked that the ratio g(p)/g(k) in Eq. (A.4) does not change when we vary the scattering length within acceptable range bounds. In order to evaluate the convolution of the deuteron wave function with the nn final state interaction analytically, we needed to employ the following parameterizations for the 1S0 nn form factor g(p) (see Eq. (A.4)) and the S-wave deuteron wave function g(p) = p2 + β2i ; u(p) = p2 + α2i ; (A.5) 1S0 form factor S-wave deuteron w.f. βi [MeV] Di [MeV] αi [MeV] Ci [MeV 1 164.53278 31.101228 45.334919 43.543212 2 246.85751 -1310.3056 242.66091 -35.643003 3 329.18224 9455.9603 439.98691 419.25214 4 411.50697 -9666.0268 637.31291 -1833.4708 5 493.83170 -55571.615 834.63891 -3710.8173 6 576.15643 64600.071 1031.9649 24903.150 7 658.48116 149128.85 1229.2909 -31673.576 8 740.80589 -84844.967 1426.6169 26476.636 9 823.13062 -295594.17 1623.9429 -118733.48 10 905.45536 -30332.710 1821.2689 259759.15 11 987.78009 560829.89 2018.5949 -223816.07 12 1070.1048 -307006.25 2215.9209 − i=1Ci Table 1: Parameters of the 1S0 form factor and the S-wave deuteron wave function for the separable representation of the N2LO chiral NN potential. where the parameters corresponding to the ChPT calculation at N2LOwith cut offs {Λ, Λ̃}= {550 MeV, 600 MeV} (see Ref [18] for details) are listed in Table 1. Note that the coefficients in the parameter- ization of the wave function have to fulfill the relation Ci = 0 in order to ensure the regularity of the deuteron wave function at the origin in coordinate space [25]. The squared and averaged amplitude to be used in the expression for the differential cross section, defined in Eq. (3) is |M(pr, θr, φr, θπ, φπ)fit|2 = |M sa1 +Ma2| ∣M ta1 . (A.6) In a fit to data two parameters are to be adjusted, namely the overall normalization C of Eq. (3) and the object of desire, ann. References [1] G. Miller, B. Nefkens and I. Šlaus. Phys. Rep. 194, 1 (1990). [2] B. Gabioud et al., Nucl. Phys. A 420, 496 (1984). [3] O. Schori et al., Phys. Rev. C 35, 2252 (1987). [4] C. R. Howell et al., Phys. Lett. B 444, 252 (1998). [5] D. E. Gonzáles Trotter et al., Phys. Rev. Lett. 83, 3788 (1999). [6] V. Huhn et al., Phys. Rev. Lett. 85, 1190 (2000); Phys. Rev. C 63, 014003 (2000). [7] D. E. Gonzáles Trotter et al., Phys. Rev. C 73, 034001 (2006). [8] R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C 51, 38 (1995) [arXiv: nucl-th/9408016]. http://arxiv.org/abs/nucl-th/9408016 [9] R. Machleidt and H. Müther, Phys. Rev. C 63, 034005 (2001) [arXiv:nucl-th/0011057]. [10] A. Nogga et al., Phys. Rev. C 67, 034004 (2003) [arXiv:nucl-th/0202037]. [11] V. Lensky, V. Baru, J. Haidenbauer, C. Hanhart, A. Kudryavtsev and U.-G. Meißner, Eur. Phys. J. A 26, 107 (2005) [arXiv: nucl-th/0505039]. [12] V. Baru, C. Hanhart, A. E. Kudryavtsev and U.-G. Meißner, Phys. Lett. B 589, 118 (2004) [arXiv: nucl-th/0402027]. [13] V. Bernard, N. Kaiser and U.-G. Meißner, Phys. Lett. B 383, 116 (1996) [arXiv:hep-ph/9603278]. [14] E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006) [arXiv: nucl-th/0505032]. [15] A. G̊ardestig and D. R. Phillips, Phys. Rev. C 73, 014002 (2006) [arXiv: nucl-th/0501049]. [16] A. G̊ardestig and D. R. Phillips, Phys. Rev. Lett. 96 (2006) 232301 [arXiv:nucl-th/0603045]. [17] A. Gasparyan, J. Haidenbauer, C. Hanhart and K. Miyagawa, arXiv:nucl-th/0701090; Eur. Phys. J. A in print. [18] E. Epelbaum, W. Glöckle and U.-G. Meißner, Nucl. Phys. A 747, 362 (2005) [arXiv: nucl-th/0405048] [19] E. Epelbaum, W. Glöckle and U.-G. Meißner, Eur. Phys. J. A 19, 125 (2004) 125 [arXiv:nucl-th/0304037]. [20] J. M. Laget, Phys. Rep. 69 (1981) 1. [21] A. G̊ardestig, Phys. Rev. C 74, 017001 (2006) [arXiv:nucl-th/0604035]. [22] E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner and H. Witala, Phys. Rev. C 66, 064001 (2002) [arXiv:nucl-th/0208023]. [23] C. Hanhart, U. van Kolck and G. A. Miller, Phys. Rev. Lett. 85 (2000) 2905 [arXiv:nucl-th/0004033]. [24] J. Haidenbauer and W. Plessas, Phys. Rev. C 30, 1822 (1984). [25] M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, P. Pires and R. de Tourreil, Phys. Lett. B 101, 139 (1981). http://arxiv.org/abs/nucl-th/0011057 http://arxiv.org/abs/nucl-th/0202037 http://arxiv.org/abs/nucl-th/0505039 http://arxiv.org/abs/nucl-th/0402027 http://arxiv.org/abs/hep-ph/9603278 http://arxiv.org/abs/nucl-th/0505032 http://arxiv.org/abs/nucl-th/0501049 http://arxiv.org/abs/nucl-th/0603045 http://arxiv.org/abs/nucl-th/0701090 http://arxiv.org/abs/nucl-th/0405048 http://arxiv.org/abs/nucl-th/0304037 http://arxiv.org/abs/nucl-th/0604035 http://arxiv.org/abs/nucl-th/0208023 http://arxiv.org/abs/nucl-th/0004033 Introduction ChPT calculation for d+nn Differential cross sections: relevant features Extraction of ann and estimate of the theoretical uncertainty Discussion and conclusions Leading amplitudes

0704.0444

Multiple Unfoldings of Orbifold Singularities: Engineering Geometric Analogies to Unification

Multiple Unfoldings of Orbifold Singularities: Engineering Geometric Analogies to Unification Jacob L. Bourjaily∗ Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Dated: 2nd April 2007) Katz and Vafa [1] showed how charged matter can arise geometrically by the deformation of ADE- type orbifold singularities in type IIa, M-theory, and F-theory compactifications. In this paper we use those same basic ingredients, used there to geometrically engineer specific matter representations, here to deform the compactification manifold itself in a way which naturally compliments many features of unified model building. We realize this idea explicitly by deforming a manifold engineered to give rise to an SU5 grand unified model into a one giving rise to the Standard Model. In this framework, the relative local positions of the singularities giving rise to Standard Model fields are specified in terms of the values of a small number of complex structure moduli which deform the original manifold, greatly reducing the arbitrariness of their relative positions. I. INTRODUCTION One of the ways in which a gauge theory with massless charged matter can arise in type IIa, M- theory, and F-theory is known as geometrical engi- neering. In this framework, gauge theory at low en- ergy arises from co-dimension four singular surfaces in the compactification manifold [2] and charged matter arises as isolated points (curves in F-theory) on these surfaces over which the singularity is en- hanced. Katz and Vafa [1] constructed explicit ex- amples of local geometry which would give rise to dif- ferent representations of various gauge groups. Their work was presented explicitly in the language of type IIa or F-theory, but the general results have been shown to apply much more broadly to M-theory as well [3, 4, 5, 6, 7]. The picture of matter and gauge theory arising from pure geometry via singular structures has been used very fruitfully in much of the progress of M- theory phenomenology. In [8] Witten engineered an interesting phenomenological model in M-theory which could possibly solve the Higgs doublet-triplet splitting problem; this model was explored in great detail together with Friedmann in [9]. There, the ex- plicit topology of the ADE-singular surface and the relative locations of all the isolated conical singulari- ties was motivated by phenomenology—the descrip- tion of the geometry of the singularities themselves was taken for granted. Unlike model building with D-branes, for exam- ple, geometrical engineering as it has been under- stood provides little information about the number, type, and relative locations of the many different sin- gularities needed for any phenomenological model. This information must either come a posteriori from phenomenological success or via duality to a con- crete string model. But recent successes in M-theory model building (for example, [10, 11]) motivate a new look at how to describe the relative structure ∗Electronic address: [email protected] of singularities—at least locally—within the frame- work of geometrical engineering itself. In this paper, we reduce the apparent arbitrari- ness of the number and relative positions of the sin- gularities required by low-energy phenomenology by showing how they can be obtained from deforming a smaller number of singularities in a more unified model. In section II we review the ingredients of geometrical engineering as described in [1]. The ba- sic framework is then interpreted in a novel way in section III to relate manifolds with matter singular- ities to those with more or less symmetry. The idea is used explicitly to deform an SU5 grand unified model into the Standard model. To be clear, as in [1] our results apply only strictly to N = 2 models from type IIa compactifications or N = 1 models from F-theory compactifications1; but we suspect that this framework has an M-theory analogue in the spirit of [7]. II. GEOMETRICAL ENGINEERING In the framework of geometrical engineering the compactification manifold is described as a fibration of (singular) K3 surfaces over a base space of appro- priate dimension. The collection of point-like (co- dimension four) singularities of the K3 fibres would then be a co-dimension four surface in the compact- ified manifold, giving rise to gauge theory of type corresponding to the singularities on each K3 fibre. Table I lists polynomials in C3 whose solutions can be (locally) taken to be the fibres for each corre- sponding gauge group. One of the strengths of this framework is its gen- erality: the local geometry is specified in terms of 1 We essentially describe non-compact Calabi-Yau three- folds which are K3-fibrations over C1. If the C1-base is fibred over CP1 as an O(−2) bundle, for example, then the total space will be a Calabi-Yau four-fold upon which F-theory can be compactified, giving rise to an N = 1 theory. http://arxiv.org/abs/0704.0444v1 mailto:[email protected] TABLE I: Hypersurfaces in C3 giving rise to the desired orbifold singularities. Gauge group Polynomial SUn (≡ An−1) xy = z SO2n (≡ Dn) x 2 + y2z = zn−1 2 = y3 + z4 2 + y3 = yz3 2 + y3 = z5 the K3 fibres, so that the description applies equally well to compactifications in type IIa, M-theory, and F-theory—the difference being the dimension of the space over which the surfaces in Table I are fibred. To obtain massless charged matter, however, addi- tional structure is necessary. Specifically, at isolated points (in type IIa or M-theory) on the co-dimension four singular surface, the type of singularity of the K3 fibres must be enhanced by one rank. Mathe- matically, this requires that one can describe how the various polynomials in Table I can be deformed into each other; and the possible two-dimensional deformations have been classified [12]. For example, to describe the embedding of a mass- less 5 of SU5 in type IIa, you would need to start with aK3-fibred Calabi-Yau where each of the fibres are of the type giving rise to SU5 gauge theory. From Table I we see that these four-dimensional fibres are locally the set of solutions to the equation xy = z5, (1) in C3. Now, to obtain matter in the 5 representa- tion, there would need to be an isolated point some- where on the two-dimensional base space where the fibre is enhanced to SU6, [1]. A description of the local geometry can be given by xy = (z + 5t)(z − t)5, (2) where t is a complex coordinate on the base over which the K3’s are fibred. Notice that when t = 0 the equation describes precisely the fibre which would have given rise to SU6 gauge theory if it were fibred over the entire base manifold. However, be- cause it is the fibre only over the origin in the com- plex t-plane, there is no SU6 gauge theory. Equation (2) is said to describe the ‘resolution’ SU6 → SU5, which is found to give rise to SU5 gauge theory at low energy with a single massless 5 at t = 0. This and many other explicit examples of such resolutions and the matter representations obtained are given in One subtlety which makes the description above not automatically apply to M-theory constructions, however, is that in equation (2) the complex pa- rameter t is two-dimensional: taken as a coor- dinate over which the K3 surfaces are fibred, it gives rise to a six-dimensional compactification man- ifold. In M-theory, co-dimension four singularities are three-dimensional and chiral matter would live at isolated points on these three dimensional orb- ifold singularities. So in M-theory the resolution SU6 → SU5 would need a three-dimensional defor- mation. Morally, the structure is identical to that described in equation (2), but the parameter t must be upgraded to describe three-dimensional deforma- tions. This can be done in terms of hyper-Kähler quotients. We suspect that all the resolutions de- scribed explicitly for type IIa here and in [1] can be upgraded to three-dimensional deformations needed in M-theory, and in many cases these generalizations have already been given [4, 5, 7]. III. ENGINEERING GEOMETRIC ANALOGIES TO UNIFICATION The main result of this paper is that distinct con- ical singularities on a surface with some gauge sym- metry can be deformed into each other in ways anal- ogous to unification; and conversely, that a descrip- tion of a single matter field in a unified theory can be ‘unfolded’ into distinct matter fields in a theory of lower gauge symmetry. Because the tools used to perform these unification-like deformations are pre- cisely the same as those used to describe the singu- larities themselves, some care must be taken to avoid unnecessary confusion. We will start by reinterpreting the tools used above to engineer charged matter, and then we will use both interpretations simultaneously to construct explicit examples of the geometric analogue to uni- fied model building. Consider again the resolution SU6 → SU5 de- scribed by xy = (z + 5s)(z − s)5, (3) where we have replaced t 7→ s from equation (2) to make a interpretative distinction that will soon become clear. We propose to momentarily discuss pure gauge theory and ignore any description of mat- ter. With this in mind, take a fixed (real) two- dimensional neighborhood over which every point is fibred by the solutions to equation (3) for any fixed value of s. Because the fibres are the same every- where on the manifold, there is no matter: for any s the geometry would give rise to pure gauge theory at low energy. For s 6= 0 solutions to equation (3) are SU5 fibres and so the compactification manifold would give rise to pure SU5. However, when s = 0 the fibres are all SU6 and so the low-energy theory would be pure SU6. Therefore s is a ‘global’ param- eter which deforms the gauge content of the theory such that for arbitrary values of s 6= 0 the theory is pure SU5 and for s = 0 it is pure SU6. That this deformation is ‘smooth’ is apparent at least when s 6= 0. An obvious question to ask is how this framework applies when conical singularities are present. We (1, 2)(3, 1) SU6 SU5 SU3×SU3SU4×SU2 FIG. 1: The plane describing the deformation of a theory with a single of SU into one of SU SU gauge theory with one ( and one ( as a function of as described by equation (4). For a fixed value of , the base space over which solutions to (4) are fibred are indicated by the black line. Notice that the relative positions of the two isolated (conical) singulari ties are fixed by will show that when the ADE-surface singularity changes because of some complex structure modu- lus such as above, the conical singularities giving rise to charged matter (often) behave as one would expect from unified model building intuition. This is best demonstrated with explicit examples. Suppose that the singular 3 surfaces are fibred over a two-dimensional base space with local com- plex coordinate . And say the four-dimensional fi- bre over the point is given by the solutions to xy = ( + 5 )( + 3 (4) for a given value of , which is now to be interpreted as a complex structure modulus deforming the entire local geometry near = 0. When = 0 the geometry is of course identical to our previous description of SU SU and so the theory would be SU with a single massless located at = 0. Consider now to be fixed at some non-zero value. The gauge theory is then SU SU : for generic values of , the fibres given by equation (4) have two singular points, at + 3 = 0 and = 0, and so the union of these points over the base manifold coordinatized by will be two distinct, two-dimensional singular surfaces: one giving rise to SU and the other SU . These surfaces become coincident as a single SU surface when = 0. Along the complex -plane, there are two iso- lated points over which the singularities are en- hanced: at s/2 the fibre is visibly SU SU and at s/3 the fibre is SU SU . There- fore the theory has two two charged, massless fields, in the ( and ( representations of SU SU at s/2 and s/3, respec- tively. Figure 1 indicates the singularity structure as a function of Notice how this description parallels unified model building: the = 0 theory of one of SU de- forms smoothly into one with ( of SU SU Similarly, we may ask how a 10 of SU would de- form into distinct singularities supporting Standard Model matter fields. The fibre structure giving rise to a massless 10 of SU is given as follows. Let be a local coordinate on the base space over which fibres are given by solutions to + 2yt 10 ; (5) at = 0, equation (5) describes an SO10 fibre, while for = 0 the fibres are SU —although in this case the result is harder to read off. This resolution, SO10 SU , gives rise to a 10 of SU [1]. Following the same idea as before, the deformation of this singularity into SU SU is given by + 2 + ( + ( (6) where is again interpreted as a complex structure modulus deforming the geometry near the singular- ity. Notice as before that = 0 describes an SU the- ory with one massless 10 located at = 0. However, for = 0 there are again two orbifold singularities corresponding to SU SU gauge theory. At three distinct points on the complex plane the rank of the fibre is enhanced: = 0, and give rise to matter in the ( , ( , and representations of SU SU , respec- tively. The structure of the deformation achieved by varying is shown in Figure 2. Again, our intuition from unified model building is realized naturally in this framework. FIG. 1: The t-s plane describing the deformation of a theory with a single 5 of SU5 into one of SU3 × SU2 × U1 gauge theory with one (3,1)1/3 and one (1, 2)−1/2 as a function of s as described by equation (4). For a fixed value of s, the base space over which solutions to (4) are fibred are indicated by the black line. Notice that the relative positions of the two isolated (conical) singulari- ties are fixed by s. will show that when the ADE-surface singularity changes because of some complex structure modu- lus such as s above, the conical singularities giving rise to charged matter (often) behave as one would expect from unified model building intuition. This is best demonstrated with explicit examples. Suppose that the singular K3 surfaces are fibred over a two-dimensional base space with local com- plex coordinate t. And say the four-dimensional fi- bre over the point t is given by the solutions to xy = (z + 5t)(z − t+ 3s)2(z − t− 2s)3, (4) for a given value of s, which is now to be interpreted as a complex structure modulus deforming the entire local geometry near t = 0. When s = 0 the geometry is of course identical to our previous description of SU6 → SU5 and so the theory would be SU5 with a single massless 5 located at t = 0. Consider now s to be fixed at some non-zero value. The gauge theory is then SU3×SU2×U1: for generic values of t, the fibres given by equation (4) have two singular points, at x = y = z − t+ 3s = 0 and x = y = z − t− 2s = 0, and so the union of these points over the base manifold coordinatized by t will be two distinct, two-dimensional singular surfaces: one giving rise to SU2 and the other SU3. These surfaces become coincident as a single SU5 surface when s = 0. Along the complex t-plane, there are two iso- lated points over which the singularities are en- hanced: at t = s/2 the fibre is visibly SU3 × SU3, and at t = −s/3 the fibre is SU4 × SU2. There- fore the theory has two two charged, massless fields, in the (1,2) −1/2 and (3,1)1/3 representations of SU3 × SU2 × U1 at t = s/2 and t = −s/3, respec- tively. Figure 1 indicates the singularity structure as a function of s. Notice how this description parallels unified model building: the s = 0 theory of one 5 of SU5 de- forms smoothly into one with (3,1)1/3 ⊕ (1,2)−1/2 of SU3 × SU2 × U1. Similarly, we may ask how a 10 of SU5 would de- form into distinct singularities supporting Standard Model matter fields. The fibre structure giving rise to a massless 10 of SU5 is given as follows. Let t be a local coordinate on the base space over which fibres are given by solutions to x2 + y2z + 2yt5 = z + t2 − t10 ; (5) at t = 0, equation (5) describes an SO10 fibre, while for t 6= 0 the fibres are SU5—although in this case the result is harder to read off. This resolution, SO10 → SU5, gives rise to a 10 of SU5 [1]. Following the same idea as before, the deformation of this singularity into SU3 × SU2 is given by x2 + y2z + 2y(t+ s)3(t− s)2 = z + (t− s)2 z + (t+ s)2 − (t− s)4(t+ s)6 , (6) where s is again interpreted as a complex structure modulus deforming the geometry near the singular- ity. Notice as before that s = 0 describes an SU5 the- ory with one massless 10 located at t = 0. However, for s 6= 0 there are again two orbifold singularities corresponding to SU3 × SU2 × U1 gauge theory. At three distinct points on the complex t plane the rank of the fibre is enhanced: t = −s, t = 0, and t = s give rise to matter in the (3,1) −2/3, (3,2)1/6, and (1,1)1 representations of SU3 × SU2 × U1, respec- tively. The structure of the deformation achieved by varying s is shown in Figure 2. Again, our intuition from unified model building is realized naturally in this framework. (3, 2) (3, 1) (1, 1) SO10 SU5 SU3×SU2×SU2SU4×SU2 FIG. 2: The plane describing the deformation of a theory with a single 10 of SU into one of SU SU gauge theory, with matter content ( , as a function of as described by equation (6). For a fixed value of , the base space over which solutions to (6) are fibred are indicated by the black line. Notice that the relative positions of the three isolated (conical) singularities are fixed by IV. DISCUSSION One of the primary reasons why geometrical engi- neering had not been more widely used phenomeno- logically is because the number, type, and relative lo- cations of the singularities giving rise to various mat- ter fields were explicitly ad hoc: the inherent local framework prevented relationships between distinct singularities from being discussed. In this paper, we have shown a framework in which these questions can be addressed concretely, systematically reduc- ing the arbitrariness of these models. Of course, the local nature of geometrical engi- neering is still inherent in this framework, and con- tinues to prevent us from addressing questions about the global structure such as stability, quantum grav- ity, and the quantization of seemingly continuous parameters like . However, in the spirit of [13], we think that local engineering is a good step toward re- alistic string phenomenology, and may perhaps offer new insights. In this paper we explicitly illustrated the geomet- ric unfolding of the matter content of an SU grand unified model into the Standard Model. But the pro- cedure can easily be generalized. It is not difficult to see how this will work for a more unified theory. For example, one can envision how an entire family could unfold out of a single SO10 resolution (which starts as a 16 of SO10), or how all three families of the Standard Mode could be unfolded out of a sin- gle SO10 SU or SU resolution. However, these examples require more sophisticated tools of analysis, and so we have chosen to describe these in a forthcoming work. V. ACKNOWLEDGEMENTS This work originated from discussions with Mal- colm Perry whose insights drove this work forward in its earliest steps. The author also appreciates helpful discussions, comments, and suggestions from Her- man Verlinde, Sergei Gukov, Gordon Kane, Edward Witten, Paul Langacker, Bobby Acharya, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov. This research was supported in part by the Michi- gan Center for Theoretical Physics and a Gradu- ate Research Fellowship from the National Science Foundation. [1] S. Katz and C. Vafa, “Matter from Geometry,” Nucl. Phys., vol. B497, pp. 146–154, 1997, hep- th/9606086. [2] A. Klemm, W. Lerche, and P. Mayr, “K3 Fibrations and Heterotic Type II String Duality,” Phys. Lett. vol. B357, pp. 313–322, 1995, hep-th/9506112. [3] M. Atiyah and E. Witten, “M-theory Dynamics on a Manifold of Holonomy,” Adv. Theor. Math. Phys., vol. 6, pp. 1–106, 2003, hep-th/0107177. [4] E. Witten, “Anomaly Cancellation on Mani- folds,” 2001, hep-th/0108165. [5] B. Acharya and E. Witten, “Chiral Fermions from Manifolds of Holonomy,” 2001, hep-th/0109152. [6] B. S. Acharya and S. Gukov, “M-theory and Singu- larities of Exceptional Holonomy Manifolds,” Phys. Rept., vol. 392, pp. 121–189, 2004, hep-th/0409191. [7] P. Berglund and A. Brandhuber, “Matter from Manifolds,” Nucl. Phys., vol. B641, pp. 351–375, 2002, hep-th/0205184. [8] E. Witten, “Deconstruction, Holonomy, and Doublet-Triplet Splitting,” 2001, hep-ph/0201018. [9] T. Friedmann and E. Witten, “Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy,” Adv. Theor. Math. Phys., vol. 7, pp. 577–617, 2003, hep-th/0211269. [10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, “An M Theory Solution to the Hierarchy Problem,” Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar, and J. Shao, “Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory,” 2007, hep- th/0701034. [12] S. Katz and D. Morrison, “Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups,” J. Algebraic Geometry vol. 1, pp. 449–530, 1992. [13] H. Verlinde and M. Wijnholt, “Building the Stan- dard Model on a D3-Brane,” JHEP, vol. 01, p. 106, 2007, hep-th/0508089. FIG. 2: The t-s plane describing the deformation of a theory with a single 10 of SU5 into one of SU3×SU2×U1 gauge theory, with matter content (3,1) −2/3⊕(3, 2)1/6⊕ (1,1)1, as a function of s as described by equation (6). For a fixed value of s, the base space over which solutions to (6) are fibred are indicated by the black line. Notice that the relative positions of the three isolated (conical) singularities are fixed by s. IV. DISCUSSION One of the primary reasons why geometrical engi- neering had not been more widely used phenomeno- logically is because the number, type, and relative lo- cations of the singularities giving rise to various mat- ter fields were explicitly ad hoc: the inherent local framework prevented relationships between distinct singularities from being discussed. In this paper, we have shown a framework in which these questions can be addressed concretely, systematically reduc- ing the arbitrariness of these models. Of course, the local nature of geometrical engi- neering is still inherent in this framework, and con- tinues to prevent us from addressing questions about the global structure such as stability, quantum grav- ity, and the quantization of seemingly continuous parameters like s. However, in the spirit of [13], we think that local engineering is a good step toward re- alistic string phenomenology, and may perhaps offer new insights. In this paper we explicitly illustrated the geomet- ric unfolding of the matter content of an SU5 grand unified model into the Standard Model. But the pro- cedure can easily be generalized. It is not difficult to see how this will work for a more unified theory. For example, one can envision how an entire family could unfold out of a single E6 → SO10 resolution (which starts as a 16 of SO10), or how all three families of the Standard Mode could be unfolded out of a sin- gle E8 → SO10×SU3 or E8 → E6 ×SU2 resolution. However, these examples require more sophisticated tools of analysis, and so we have chosen to describe these in a forthcoming work. V. ACKNOWLEDGEMENTS This work originated from discussions with Mal- colm Perry whose insights drove this work forward in its earliest steps. The author also appreciates helpful discussions, comments, and suggestions from Her- man Verlinde, Sergei Gukov, Gordon Kane, Edward Witten, Paul Langacker, Bobby Acharya, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov. This research was supported in part by the Michi- gan Center for Theoretical Physics and a Gradu- ate Research Fellowship from the National Science Foundation. [1] S. Katz and C. Vafa, “Matter from Geome- try,” Nucl. Phys., vol. B497, pp. 146–154, 1997, hep-th/9606086. [2] A. Klemm, W. Lerche, and P. Mayr, “K3 Fibrations and Heterotic Type II String Duality,” Phys. Lett., vol. B357, pp. 313–322, 1995, hep-th/9506112. [3] M. Atiyah and E. Witten, “M-theory Dynamics on a Manifold of G2 Holonomy,” Adv. Theor. Math. Phys., vol. 6, pp. 1–106, 2003, hep-th/0107177. [4] E. Witten, “Anomaly Cancellation on G2 Mani- folds,” 2001, hep-th/0108165. [5] B. Acharya and E. Witten, “Chiral Fermions from Manifolds of G2 Holonomy,” 2001, hep-th/0109152. [6] B. S. Acharya and S. Gukov, “M-theory and Singu- larities of Exceptional Holonomy Manifolds,” Phys. Rept., vol. 392, pp. 121–189, 2004, hep-th/0409191. [7] P. Berglund and A. Brandhuber, “Matter from G2 Manifolds,” Nucl. Phys., vol. B641, pp. 351–375, 2002, hep-th/0205184. [8] E. Witten, “Deconstruction, G2 Holonomy, and Doublet-Triplet Splitting,” 2001, hep-ph/0201018. [9] T. Friedmann and E. Witten, “Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy,” Adv. Theor. Math. Phys., vol. 7, pp. 577–617, 2003, hep-th/0211269. [10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, “An M Theory Solution to the Hierarchy Problem,” Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Ku- mar, and J. Shao, “Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory,” 2007, hep-th/0701034. [12] S. Katz and D. Morrison, “Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups,” J. Algebraic Geometry, vol. 1, pp. 449–530, 1992. [13] H. Verlinde and M. Wijnholt, “Building the Stan- dard Model on a D3-Brane,” JHEP, vol. 01, p. 106, 2007, hep-th/0508089. http://arxiv.org/abs/hep-th/9606086 http://arxiv.org/abs/hep-th/9506112 http://arxiv.org/abs/hep-th/0107177 http://arxiv.org/abs/hep-th/0108165 http://arxiv.org/abs/hep-th/0109152 http://arxiv.org/abs/hep-th/0409191 http://arxiv.org/abs/hep-th/0205184 http://arxiv.org/abs/hep-ph/0201018 http://arxiv.org/abs/hep-th/0211269 http://arxiv.org/abs/hep-th/0606262 http://arxiv.org/abs/hep-th/0701034 http://arxiv.org/abs/hep-th/0508089

0704.0445

Geometrically Engineering the Standard Model: Locally Unfolding Three Families out of E8

Geometrically Engineering the Standard Model: Locally Unfolding Three Families out of E8 Jacob L. Bourjaily∗ Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Dated: 3rd April 2007) This paper extends and builds upon the results of [1], in which we described how to use the tools of geometrical engineering to deform geometrically-engineered grand unified models into ones with lower symmetry. This top-down unfolding has the advantage that the relative positions of singular- ities giving rise to the many ‘low energy’ matter fields are related by only a few parameters which deform the geometry of the unified model. And because the relative positions of singularities are necessary to compute the superpotential, for example, this is a framework in which the arbitrariness of geometrically engineered models can be greatly reduced. In [1], this picture was made concrete for the case of deforming the representations of an SU5 model into their Standard Model content. In this paper we continue that discussion to show how a geometrically engineered 16 of SO10 can be unfolded into the Standard Model, and how the three families of the Standard Model uniquely emerge from the unfolding of a single, isolated E8 singularity. I. INTRODUCTION In [2], Katz and Vafa showed how to geometrically engineer matter representations in terms of the local singularity structure of type IIa, M-theory, and F- theory compactifications. In that framework matter and gauge theory both have purely geometrical ori- gins: SUn, SO2n and En gauge theories arise from the existence of co-dimension four singular curves of certain types in the compactification manifold [3]; and massless matter representations arise from iso- lated points (in type IIa or M-theory) or curves (in F-theory) along the singular surface over which the type of singularity is enhanced by one rank. Despite the extraordinary generality of this frame- work, it has not been widely used phenomenologi- cally. This is largely because the description of the isolated enhancements of singularities giving rise to various matter representations is inherently local: although the geometry near any particular enhance- ment could be described concretely, the framework had nothing to say about numbers, types, and rela- tive locations of different matter fields. This global data was either to be determined by duality to a concrete, global string theory model1, or suggested via the a posteriori success of a given set of relative positions (as in e.g. [5, 6]). Another way to relate the number and relative positions of (enhanced singularities giving rise to) matter fields was given in [1]: in that paper, we described for example how a local description of the geometry giving rise to a massless 5 of SU5 could be smoothly deformed into a local description of a (3,1) and a (1,2) of SU3×SU2 which live at distinct ∗Electronic address: [email protected] 1 Geometrically engineered models in M-theory are dual to intersecting brane models in type II, (see e.g. [4]). points—related by a single deformation parameter. A cartoon of what was described in that paper is shown in Figure 1. In this paper we describe pedagogically how to extend that idea to engineer analogies to SO10 and E6 × SU2 grand unified models 2. Although in [1] we were able to analyze explicit unfoldings of SO10 and SU6 singularities sufficiently well by sight, this will not be possible for our present examples. All of the examples in this paper involve the unfolding of isolated En singularities; and although algebraic descriptions of these are known and classified [7], it would be unnecessarily cumbersome and unenlight- ening to analyze them explicitly as we did in [1]. Therefore, in section II we describe a much more powerful and elegant language in which to study these resolutions. In section III we describe in detail how the un- folding of a 16 of SO10 into the Standard Model is derived in the language of section II. This is achieved in two stages: in the first stage, we unfold the 16 into 10⊕5⊕1 of SU5; we then unfold the resulting SU5 model into a single ‘family’ of the Standard Model. At the end, all the relative positions of the singu- larities of the family are set by the non-zero values of two complex structure moduli, thereby greatly re- ducing the arbitrariness of their relative positions. The next most obvious example would be a de- scription of how a 27 of E6 geometrically unfolds into the Standard Model. However, there are two reasons to leave this example to the reader: first, it is a most natural extension of the results of section III; secondly, it is a consequence of the E6 × SU2 grand unified model which we describe in section IV. Al- 2 As described in section IV, the resolution E8 → E6 × SU2 naturally starts as a theory with three 27’s of E6 related by an SU2 family symmetry. http://arxiv.org/abs/0704.0445v2 mailto:[email protected] FIG. 1: A cartoon of the geometric deformation of a 5 and 10 of SU5 into the Standard Model as described in [1]. The surface over which the singularities are enhanced is coordinatized by a complex parameter t and the geometry is deformed by changing the value of a complex parameter s. The relative locations of the ‘resolved’ singularities are given in terms of s. though not given as an example in [2], it is not hard to see3 that a single isolated E8 singularity at the intersection of a co-dimension four surfaces of types E6 and SU2 gives rise to matter in the representa- tion (27,2)⊕ (27,1) ⊕ (1,2). It is easy to see how this would unfold into the matter content of three families—one coming from each of the 27’s. That three families emerge from E8 is a general conse- quence of group theory and can be understood from the fact that E6 ×SU3 is a maximal subgroup of E8 into which the adjoint of E8 partially branches into an SU3 triplet of 27’s. As in the preceding paper [1], this work is pre- sented concretely in the language of Calabi-Yau compactifications of type IIa string theory, which can also be naturally extended to F-theory mod- els. Here, we engineer the explicit local geometry of (non-compact) Calabi-Yau three-folds which are K3-fibrations over C1. If type IIa string theory is compactified on this three-fold, a four-dimensional N = 2 theory with various massless hypermulti- plets will result. But if, for example, the C1 base of this three-fold were fibred as an O(−2) bundle over 1, the resulting total space would be a Calabi-Yau four-fold4 upon which F-theory would compactify to an N = 1 theory with chiral multiplets. However, because the manifold over which the singular K3’s are fibred in M-theory is a real, three-dimensional space, our fibrations over C1 do not have a direct application to M-theory. It would of course be desirable to have a similar description of geometric unfolding explicitly in the language of G2-manifolds so that this picture could be realized concretely in M-theory as well. This is particularly important in light of the recent advances in M-theory phenomenology (e.g. [8, 9]). By exten- 3 This is described in section II. 4 This is just one example of the ways in which these Calabi- Yau three-folds could be fibred over CP1 to result in a Calabi-Yau four-fold. sion of the work of Berglund and Brandhuber in [10], such a generalization should be relatively straight- forward, but we will not attempt to do this here. II. RESOLVING En-TYPE SINGULARITIES Recall that a gauge theory in type IIa string the- ory can arise from compactification to six dimen- sions over a singular K3 surface (similar statements apply to M-theory and F-theory) [3]. The complex structures of the singular compactification manifolds giving rise to SUn(≡ An−1), SO2n(≡ Dn), and En gauge theory are given in Table I—where the sur- faces are labelled conveniently by the name of the resulting gauge theory5. We can generalize this discussion by considering a complex, one-dimensional space B over which a smooth family of singular K3 surfaces are fibred. If almost everywhere over B the K3-fibres have singu- larities of a single type, then compactification of type IIa string theory over the total space will give rise Gauge group Polynomial SUn (≡ An−1) xy = z SO2n (≡ Dn) x 2 + y2z = zn−1 2 = y3 + z4 2 + y3 = 16yz3 2 + y3 = z5 TABLE I: Hypersurfaces in C3 giving rise to the desired orbifold singularities. 5 It is of curious historical interest that the equations listed in Table I were first identified by Fleix Klein in 1884 [11]. The reader may also be amused that the full resolutions of these surfaces were almost completely classified—up to a few computational errors—by Bramble in 1918 [12]. to gauge theory in four-dimensions of the type corre- sponding to the typical fibre. Massless charged mat- ter will arise if over isolated points in B the type of fibre is enhanced by one rank. The geometry about a single such isolated point where the singularity is enhanced was described in detail by Katz and Vafa in [2]. The representation of matter living at these ‘more- singular’ points was also given in [2]: suppose that G ⊃ H ×U1 and that the rank of H is one less than G; then, if there is an isolated G-type singularity over a surface of H-type singularities, the resulting massless representation is given by those parts of the decomposition of the adjoint of G into H×U1 which are charged under the U1. Because the question of how to (smoothly) deform the surfaces of Table I into ones of lower rank has in- trinsic mathematical interest, it is not too surprising that all possible two-dimensional deformations have been classified. Our discussion below will make use of the notation and results presented in [7]. In our present work, we are interested in defor- mations of En singularities into ones of lower rank. Unlike SUn singularities, the resolutions of which are easy enough to read off by sight, the algebraic com- plexity of En singularities is formidable. To appre- ciate what is meant by this, consider the resolution of E7. From Table I we know that an E7 singularity is locally isomorphic to the surface x2 + y3 = 16yz3 in C3. Its full resolution in terms of the seven defor- mation parameters ~t = (t1, t2, . . . , t7) is given by − x2 − y3 + 16yz3 + ǫ2(~t)y 2z + ǫ6(~t)y 2 + ǫ8(~t)yz + ǫ10(~t)z 2 + ǫ12(~t)y + ǫ14(~t)z + ǫ18(~t) = 0, (1) where the ǫn(~t) are n th order symmetric polynomi- als in the components of ~t which are tabulated over several pages of the appendix of [7]. A näıve way to determine the type of singularity found by resolving E7 “in the direction ~t” would be to expand equation (1) completely using the explicit functions ǫn(~t), find each of its singular points, and expand locally about each until an isomorphism with a singularity of lower rank in Table I was clear. This is the way, for example, that [2] demonstrated that the resolution of E7 in the direction (0, 0, 0, 0, 0, t, 0) gives rise to E6 for t 6= 0. All of the results in this paper could be verified in this way. Luckily, however, Katz and Morrison described a much more powerful and direct way to analyze the deformations of En singularities [7]. We would like a pragmatic answer to the follow- ing question: what is the type of fibre found by re- solving an En singularity in the direction ~t? That there is an easy answer to this question makes our work much simpler. Although an adequate treat- ment would take us well beyond the scope of our present discussion, the answer given in [7] is at least very easy to make use of6: for each of the equations in Table II satisfied by the components of ~t, the sin- gularity has the corresponding root. Given the list of roots, it is then a straight-forward exercise to con- struct the Dynkin diagram corresponding to the sin- gularity7. 6 Of course, this answer does depend on the parameterization used. As stated before, we are working with the conventions of [7]. 7 Misusing the notation of [7] in a way applicable only to In an admittedly bad notation, we consider each of the n deformation parameters ti(t) to be functions of t, the local coordinate on the base space B. A (non-Abelian) gauge theory will be present if there are roots implied by Table II which are preserved for generic values of t. And charged massless matter will exist if at isolated points {t∗} an additional root is added—or, in terms of Dynkin diagrams, if an additional node is added. At each isolated point we can therefore identify the resolution G → H and thereby determine the resulting representation. Equation Root ti − tj = 0 =⇒ ei − ej ti + tj + tk = 0 =⇒ e0 − ei − ej − ek tij = 0 =⇒ 2e0 − 2ti1 + tij = 0 =⇒ 3e0 − 2ei1 − TABLE II: The roots of the singularity resulting from the resolution of En in the direction~t. This is a reproduction of Table 4 of Ref. [7]. SUn, SO2n and En, one can think of the vectors ei as an or- thonormal basis in Minkowski space which is equipped with a mostly-plus metric. Then roots are vectors in this space of norm +2. Each (positive) root gives rise to a node in the resulting Dynkin diagram, and two nodes are connected by a line if their inner product is −1 and disconnected if they are orthogonal. III. GEOMETRIC ANALOGUE OF SO10 GRAND UNIFICATION A. The Description of a 16 of SO10 A necessary starting point to describe the unfold- ing of a 16 of SO10 into the Standard Model is a description of the initial geometry as was done in [2]. We will briefly review that construction in the lan- guage described above before we unfold it, first into an SU5 model, and later all the way into SU3×SU2. Let t be a local complex coordinate on the space B over which is fibred the resolution of E6 parame- terized by ~t = (t, t, t, t, t,−2t). To be clear, for each value of t, the vector~t describes an explicit surface in 3 given in reference [7] analogous to that of equa- tion (1) above. Considering the rules of Table II, we see that for an arbitrary value of t 6= 0 the root lattice of the fibre is (e0−e1−e2−e6) (e1 − e2) (e2 − e3) (e3 − e4) (e4 − e5) where we have displayed the roots suggestively so as to reproduce the SO10 Dynkin diagram. At t = 0, however, E6 is restored. So we have an isolated E6 fibre over the point t = 0, while for any t 6= 0 the fibre is SO10. This gives rise to SO10 gauge theory with a single massless 16 located at the origin in the t-plane. B. Unfolding the 16 of SO10 into SU5 We would like to unfold the manifold described above into one with SU5 gauge theory. It is not hard to guess in what ‘directions’ ~t we may deform the the geometry so that the fibre over a generic point is SU5. Let a denote a parameter independent of t which adjusts the whole geometry over the re- gion which is coordinatized by t. Then let the fibre over t be given by the resolution of E6 in the direc- tion (t, t, t, t, t+ a,−2t− a). Obviously when a = 0 the situation is the same as above and results in a single massless 16 of SO10. However, when a 6= 0 the situation is different: for generic values of t it is easy to see that the simple roots are (e0−e1−e5−e6) (e1−e2) (e2−e3) (e3−e4) which means that the generic fibre over t is just SU5—and so the resulting gauge theory is SU5. To find what matter representations exist, we must determine over which locations t the rank of the fibre is enhanced. This means we are seeking special values of t (determined by a) at which an ad- ditional equation in Table II is satisfied. For each of these points, we can draw the resulting Dynkin dia- TABLE III: The locations on the complex t-plane over which the singularity of the fibre is enhanced, and the representations of SU5 × U1 that result. Location Fibre Representation of SU5 × U1 3t + 2a = 0 SU5 × SU2 1−5 3t+ a = 0 SO10 10−1 t = 0 SU6 53 gram to determine the fibre over that point, thereby determining the representation which arises there. It is not hard to exhaustively find all these ‘more singular’ points. They are give in Table III. Notice that we have included the U1-charge assignments that result; these are normalized as in the appendix of [13]. C. Unfolding a 16 of SO10 into the Standard Model To complete our task and unfold the 16 of SO10 all the way to the Standard Model, we must deform the fibres by another ‘global’ parameter, which we will denote b. It is not hard to guess a direction over which the generic fibre will be SU3 × SU2: try for example (t, t, t, t + b, t + a,−2t− a − b). Again, we notice that for a general location t and generic fixed values a, b 6= 0, the singularity has the root structure (e1 − e2) (e2 − e3) (e0 − e4 − e5 − e6) , (4) which is visibly SU3 × SU2. Like above, it is a straight-forward exercise to de- termine all the locations over which the singularity is enhanced, and the resulting representation which arises. These points including their resulting rep- resentations (with U1-charges as normalized in [13]) are listed in Table IV. The entire unfolding is repro- duced graphically in Figure 2. Location Fibre Representationof SU3×SU2×U1 Name 3t + 2a+ b = 0 SU3 × SU2 × SU2 (1,1)0 ν 3t + a+ 2b = 0 SU3 × SU2 × SU2 (1,1)6 e 3t+ a+ b = 0 SU5 (3,2)1 Q 3t + a = 0 SU4 × SU2 (3,1)−4 u 3t+ b = 0 SU4 × SU2 (3,1)2 d t = 0 SU3 × SU3 (1,2)−3 L TABLE IV: The locations on the complex t-plane over which the singularity of the fibre is enhanced and the representations of SU3 × SU2 × U1 that result. FIG. 2: An illustration of the resolution of a geometrically engineered 16 of SO10 into the Standard Model as a function of two complex structure moduli a and b as described in section III. The coordinate along the base space is t and runs vertically in the diagram. When a = b = 0, along the left-hand side, there is just one isolated E6 singularity at t = 0. When b = 0 but a is allowed to vary, this single singularity splits into three, and any a =constant slice will have three isolated singularities in the complex t-plane as shown above. Moving rightward in the diagram, at the dashed line a is held fixed and b is allowed to grow, causing the three enhancements of SU5 to break apart into six total isolated singularities over SU3 ×SU2, which is shown on the right-hand-side. Also shown are the (appropriately normalized) U1 charges of fields obtained via this multiple unfolding. IV. GEOMETRIC ANALOGUE OF E6 × SU2 GRAND UNIFICATION After having completed the unfolding of a 16 of SO10 into the Standard Model, it is natural to ask if this idea can be extended to relate all the singulari- ties of the Standard Model as perhaps the unfolding of a single isolated singularity of higher-rank. The answer is in fact yes—and there is a sense in which precisely three families arise if the notion of ‘geomet- ric unification’ is saturated. Because a 16 of SO10 arises from the resolution E6 → SO10, it can only be unfolded out an ex- ceptional singularity. Clearly the highest level of unification one can achieve along this line would be to start with a resolution E8 → H where H is a rank-seven subgroup of E8 which contains SO10. The possible ‘top-level’ gauge groups are then E7, E6 × SU2, and SO10 × SU3. We choose to study E8 → E6 × SU2 as our example because it will nat- urally include a description of the unfolding of 27 of E6 into the Standard Model, which is interesting in its own right, and because it follows quite directly from our work in section III. The initial geometry which we will deform into the Standard Model is given as follows. Let t be a com- plex coordinate on the base space B over which is fi- bred the resolution (t, t, 0, 0, 0, 0, 0, 0) of E8. Clearly, when t = 0 we recover E8; when t 6= 0 we see that the roots of the fibre are (e0−e3−e4−e5) (e3 − e4) (e4 − e5) (e5 − e6) (e6 − e7) (e7 − e8) (e1 − e2) , (5) which is visibly E6 × SU2. Following the general rule to determine the representation resulting from a given resolution [2], we find that at t = 0 lives massless matter charged in the (27,2)1⊕(27,1)−2⊕ (1,2)3 representation of E6 × SU2 × U1ϕ . To avoid pedantic redundancy, in Figure 3 we have summarized in great detail the entire unfolding into SU3×SU2×U1Y ×U1χ×U1Ψ×U1ζ×U1ϕ . An outline of the steps involved in deriving this unfolding is given presently. First, the unfolding of the E6 × SU2 gauge the- ory into E6 gauge theory is obtained by defining the fibre over t to be given by the resolution of E8 in the direction (t + a, t − a, 0, 0, 0, 0, 0, 0) for some a 6= 0. This clearly kills the SU2 node of the fibre in equation (5). There are five locations at which the singularity is enhanced by one rank, giving rise to three 27’s and two singlets as shown in the left- most section of Figure 3. FIG. 3: An illustration of the resolution of a single isolated E8 into the Standard Model in terms of four deformation parameters a, b, c, d. Along the left hand side, for a = b = c = d = 0, the generic E6 × SU2 fibre is enhanced to E8 at t = 0. Moving from left to right, a, b, c, d are sequentially allowed to grow to some non-zero value—and between dashed lines all but one of the moduli are held fixed. Solid lines indicate the locations of enhanced singularities relative to the plane for as functions of a, b, c, d. The complete list of isolated singularities, their locations, and charge assignments are given on the right hand side of the diagram. The rest of the unfolding is a natural appli- cation of the work in section III. Let us now set the fibre over t to be given by the resolution (t+ a, t− a, b, b, b, b, b,−2b) of E8 for arbitrary com- plex deformation parameters a, b 6= 0. From section III we see immediately that the generic fibre is SO10. A thorough scanning for possible solutions to equa- tions in Table II shows that there are 11 isolated points on the complex t-plane over which the singu- larity is enhanced. These correspond to the ‘break- ing’ of each 27 of E6 into 16 ⊕ 10 ⊕ 1 of SO10, while the singlets remain singlets. This is seen in the second vertical strip (from the left) in Figure 3. Again, following our discussion above, it is easy to guess possibilities for the next two resolution di- rections. First, we set the fibre over t to be given by (t+ a, t− a, b, b, b, b, b+ c,−2b− c) which will result SU5 gauge theory with matter content correspond- ing to the ‘canonical’ decomposition of three 27’s of E6 with two singlets. And finally, the full resolution of the E6×SU2 grand unified model into SU3×SU2 can be given by letting the fibre over t be given by the (t+a, t−a, t, t, t, t+d, t+c,−2b−c−d) resolution of E8 for (generic) arbitrary fixed complex structure moduli a, b, c, d 6= 0. V. IMPLICATIONS Let us clarify what we have done. For a given set of fixed, nonzero complex structure moduli, the reso- lution given above describes the explicit, local geom- etry of a non-compact Calabi-Yau three-fold, which is a K3-fibration over C1. If type IIa string theory is compactified on this three-fold, the resulting four- dimensional theory will have SU3 × SU2 gauge the- ory with hypermultiplets at isolated points as given in Figure 3 which reproduce the spectrum of three families of the Standard Model with an extended Higgs sector and some exotics. Alternatively, if one takes this (non-compact) Calabi-Yau three-fold and fibres it over CP1 as described in section I so that the total space is Calabi-Yau, then F-theory on this space will give rise to N = 1 supersymmetry with SU3×SU2 gauge theory and chiral multiplets in the representations given in Figure 3. And although it does not follow directly from our construction above, considering the close similarities between two- and three-dimensional resolutions of the singularK3 sur- faces we have every reason to suspect an analogous geometry can be engineered for M-theory in terms of hyper-Kähler quotients by extension of the results in [10, 14, 15]. We are currently working on building this geometry in M-theory, and we expect to report on this work soon. Given these four complex structure moduli, all the relative positions of the 35 disparate singulari- ties giving rise to all three families of the (extended) Standard Model are then known8. Beyond the usual three families of the Standard Model, the manifold also gives rise to two Higgs doublets for each fam- ily, six Higgs colour triplets, three right-handed neu- trinos and five other Standard Model singlets. We should point out that this matter content (and their U1-charge assignments) is a consequence of group theory and algebraic geometry alone—it is simply what is found when unfolding E8 all the way to the Standard Model. And given the relative positions and local ge- ometry of the singularities together with the U1- structure, one can in principle compute the full su- perpotential coming from instantons wrapping dif- ferent singularities. Because these are fixed by the values of the complex structure moduli, there is a (complex) four-dimensional landscape9 of different, explicit SU3 × SU2 embeddings at the compactifi- cation scale. Although this large landscape may appear to have too much freedom, we remind the reader that in the traditional understanding of ge- ometrical engineering there would be hundreds of parameters describing the (independent) relative lo- cations of each of the isolated singularities. There are a few things to notice about the form of the superpotential that will emerge. First, because of the U1-charge assignments, each term in the su- perpotential must combine exactly one term arising from each of the 27’s. This greatly limits the form of the superpotential. And in particular, it implies that neither mass nor flavour eigenstates will arise from any single 27—that is, the ‘families’ in the col- loquial sense are necessarily linear combinations of fields resulting from different 27’s. Also notice that in general the terms in the su- perpotential will be proportional to e− dVol where dVol is the volume form of some cycle wrapping sin- gularities (the details of which depends on whether we are talking about type IIa, M-theory, or F-theory realizations), and are in principle calculable in terms of the deformation moldui. And because these coef- ficients are exponentially related to the volumes of cycles, we expect the high-scale Lagrangian will be generically hierarchical. This structure could be im- portant for solving problems in phenomenology—for example the µ problem in the Higgs potential, the Higgs doublet-triplet splitting problem, or avoiding 8 A subtlety, however, is that because our language has been explicitly that of N = 2 theory from type IIa, we are unable to distinguish the 5 from the 5 in the splitting of the 10’s of SO10. In Figure 3, a consistent choice was made—and although we do not justify this claim here, it is the choice that will be correct for the M-theory generalization of this work. 9 That it is continuous is a consequence of the fact that we are engineering non-compact Calabi-Yaus. If one matched this local geometry to a compact global structure, the landscape would of course be discrete. proton decay. We are in the process of studying the phenomenol- ogy of models on this landscape. At first glance, the U1-structure combined with high-scale hierar- chies could possibly be complex enough to be able to avoid some of the typical problems of E6-like grand unified models. We should point out that if there were no high-scale hierarchies, however, then the al- lowed terms in the superpotential would generically give rise to low-energy lepton and baryon number vi- olation, similar to any ‘generic’ E6 model—i.e. one which includes all types of terms allowed by the E6- mandated U1-structure [16]. We could always im- pose additional symmetries and add fields by hand to solve these problems, but this would not be very compelling. However, if viable models already ex- ist in the landscape which do not require additional fields or symmetries, these would be compelling even if we do not yet understand how they are selected. One of the most important phenomenological questions about these models is the fate of the additional U1 symmetries. Although we suspect that one can determine which of the U1 symmetries are dynamical below the compactification scale by studying the normalizability of their corresponding vector multiplets, we do not presently have have a complete understanding of this situation. Of course, if any additional U1’s survive to low en- ergy they could have very interesting—or damning— phenomenological consequences. VI. DISCUSSION An important point to bear in mind when consid- ering geometrically engineered models is that there generically exist10 moduli which can deform the ge- ometry into one which gives rise to a theory with less gauge symmetry. For example, if you are given a geometrically-engineered SO10 grand unified model, then our results show explicitly that the model can be locally deformed into an SU5 model, and this can be deformed further into the Standard Model; the original SO10 theory is seen to be a single point in a (complex) two-dimensional landscape of SU3×SU2 theories. And because larger symmetries always lie in lower dimensional surfaces of moduli space, it is very relevant to ask what physics pre- vents this unfolding from taking place. Indeed, this question applies to the Standard Model as well— our analysis could easily go further to unfold away SU3×SU2. We are not presently able to answer why this does not happen11; although this observation 10 There could be global obstructions which prevent such a deformation from taking place. But these are invisible to the non-compact, local constructions considered here. 11 Although, perhaps the unfolding of SU2 may provide an al- ternative to tuning in the usual Higgs sector [17]. It would be interesting to understand in greater detail the relation- suggests that perhaps theories with less symmetry, like SU3 × SU2, could be much more natural than grand unified theories. More generally, it is not presently understood what physics controls the values of the geometric moduli which deform the manifold—the parameters which deform the E8 → E6×SU2 complex structure, for example. We do not yet have a general mecha- nism which would fix these parameters; we simply observe that any non-zero values of the moduli will give rise to a geometrically engineered manifold with SU3×SU2 gauge theory ‘peppered’ with all the nec- essary singularities of the three families of the Stan- dard Model together with the usual E6-like exotics. And importantly, for any point in the complex four- dimensional ‘landscape,’ the relative locations of all the relevant singularities are known—and hence in principle so is the superpotential. This relationship between moduli-fixing and gauge symmetry breaking could be a novel feature of geometrically-unfolded models. It may allow one to apply the results in [8], for example, to single out theories on the landscape. However, a prerequisite to this type of analysis would be an identification of which moduli should be identified with the ones which deform the geometry as described here. Although the motivation in this paper and in [1] appears to be a top-down realization of grand unification, there is a sense in which we are re- ally engineering from the bottom-up. Specifically, because the local geometry we have described is non-compact, the resulting theory is decoupled from quantum gravity, and the parameters along the land- scape of deformations are continuous. This is not unlike the situation in [18]. But what we lose in global constraints we perhaps gain by concrete local structure. Not only do we have a framework which naturally predicts three families with a rather de- tailed phenomenological structure, but we have done so in a way that preserves all the information about the local geometry. And because this framework re- alizes the ‘physics from pure geometry’ paradigm in a potentially powerful way, it could prove impor- tant to concrete phenomenological constructions in M-theory, for example. Of course we envision these local geometries to be embedded within compact Calabi-Yau manifolds. It is an assumption of the framework that the pre- cise global topology of the compactification mani- fold can be ignored at least as a good first approx- imation. One may ask the extent to which these constructions can be glued into compact manifolds. Concretely: under what circumstances can a non- compact Calabi-Yau three-fold which is a fibration of K3 surfaces with asymptotically uniform ADE-type ship between unfolding and the Higgs mechanism. singularities be compactified? This is an important question for mathematicians, the answer to which would likely lead to important physical insight—e.g. quantization of the moduli space of deformations. A possible objection to this framework is that our constructions appear to depend on several seem- ingly arbitrary choices (the specific chain from E8 to the Standard Model, which roots were eliminated at each step, etc.). However, it is likely that the particle content, for example, which results is completely in- dependent of these choices. Furthermore, we suspect that different realizations of the unfolding merely re- sult in different parameterizations of the landscape, and do not reflect true additional arbitrariness. But this is still an area that deserves attention. Lastly, because in this picture the Standard Model is seen to unfold at the compactification scale, one may ask what has become of gauge coupling unifi- cation. Because the gauge coupling constants are functions of the volumes of their corresponding co- dimension four singular surfaces12 which depend on the deformation moduli, the traditional meaning of grand unification is more subtle here—as is typical in string phenomenology. For example, although we chose to unfold the Standard Model sequentially as a series of less unified models, there is no reason to suspect that that order has any physical importance. Surely, if as we parameterized the unfolding in sec- tion IV, setting d → 0 (or c → 0) would result in an SU5 grand unified theory; but setting a → 0 instead would result in a restoration of family symmetry. The four complex structure moduli tune different types of unification separately—and should simulta- neously be at play in the question of gauge coupling unification. It is interesting to note, however, that if one were to simultaneously scale the values of all the moduli to be very small, the spectrum would be more and more unified: the relative distances between singu- larities shrink, unifying the coefficients in the su- perpotential; and the volumes of the co-dimension four singularities (if realized in a compact manifold) would approach one another, resulting in a unifica- tion of their gauge couplings. What this may mean phenomenologically remains to be understood. In this paper we have described a local, purely ge- ometric framework in which gauge symmetry ‘break- ing’ can be re-cast as a problem of moduli fixing— and in which the same moduli which describe this geometric ‘unfolding’ also determine the physics of massless matter. And although we still do not un- derstand the mechanisms by which these moduli are fixed, the landscape of possibilities is already enor- mously reduced: what would have been the hun- dreds of parameters describing the relative positions on the compactification manifold of the Standard Model’s three families worth of matter fields, we specify them all in terms of only four complex struc- ture moduli which describe the unfolding of an iso- lated E8 singularity. And the fact that three families emerges is group-theoretic and not added by hand. VII. ACKNOWLEDGEMENTS It is a pleasure to thank helpful discussions with and insightful comments of Herman Verlinde, Sergei Gukov, Gordy Kane, Paul Langacker, Edward Wit- ten, Cumrun Vafa, Brent Nelson, Malcolm Perry, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov. 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0704.0446

The classification of surfaces with p_g=q=1 isogenous to a product of curves

THE CLASSIFICATION OF SURFACES WITH pg = q = 1 ISOGENOUS TO A PRODUCT OF CURVES GIOVANNA CARNOVALE, FRANCESCO POLIZZI Abstract. A smooth, projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C × F so that S = (C × F )/G. In this paper we classify all surfaces with pg = q = 1 which are isogenous to a product. 0. Introduction The classification of smooth, complex surfaces S of general type with small birational in- variants is quite a natural problem in the framework of algebraic geometry. For instance, one may want to understand the case where the Euler characteristic χ(OS) is 1, that is, when the geometric genus pg(S) is equal to the irregularity q(S). All surfaces of general type with these invariants satisfy pg ≤ 4. In addition, if pg = q = 4 then the self-intersection K S of the canon- ical class of S is equal to 8 and S is the product of two genus 2 curves, whereas if pg = q = 3 then K2S = 6 or 8 and both cases are completely described ([CCML98], [HP02], [Pir02]). On the other hand, surfaces of general type with pg = q = 0, 1, 2 are still far from being classified. We refer the reader to the survey paper [BaCaPi06] for a recent account on this topic and a comprehensive list of references. A natural way of producing interesting examples of algebraic surfaces is to construct them as quotients of known ones by the action of a finite group. For instance Godeaux constructed in [Go31] the first example of surface of general type with vanishing geometric genus taking the quotient of a general quintic surface of P3 by a free action of Z5. In line with this Beauville proposed in [Be96, p. 118] the construction of a surface of general type with pg = q = 0, K S = 8 as the quotient of a product of two curves C and F by the free action of a finite group G whose order is related to the genera g(C) and g(F ) by the equality |G| = (g(C)−1)(g(F )−1). Gener- alizing Beauville’s example we say that a surface S is isogenous to a product if S = (C ×F )/G, for C and F smooth curves and G a finite group acting freely on C ×F . A systematic study of these surfaces has been carried out in [Ca00]. They are of general type if and only if both g(C) and g(F ) are greater than or equal to 2 and in this case S admits a unique minimal realization where they are as small as possible. From now on, we tacitly assume that such a realization is chosen, so that the genera of the curves and the group G are invariants of S. The action of G can be seen to respect the product structure on C × F . This means that such actions fall in two cases: the mixed one, where there exists some element in G exchanging the two factors (in this situation C and F must be isomorphic) and the unmixed one, where G acts faithfully on both C and F and diagonally on their product. After [Be96], examples of surfaces isogenous to a product with pg = q = 0 appeared in [Par03] and [BaCa03], and their complete classification was obtained in [BaCaGr06]. The next natural step is therefore the analysis of the case pg = q = 1. Surfaces of general type with these invariants are the irregular ones with the lowest geometric genus and for this reason it would be important to provide their complete description. So far, this has been obtained only in the cases K2S = 2, 3 ([Ca81], [CaCi91], [CaCi93], [Pol05], [CaPi06]). The goal of the present paper is to give the full list of surfaces with pg = q = 1 that are isoge- nous to a product. Our work has to be seen as the sequel to the article [Pol07], which describe Date: November 4, 2018. 2000 Mathematics Subject Classification. 14J29 (primary), 14L30, 14Q99, 20F05. Key words and phrases. Surfaces of general type, isotrivial fibrations, actions of finite groups. http://arxiv.org/abs/0704.0446v2 all unmixed cases with G abelian and some unmixed examples with G nonabelian. Apart from the complete list of the genera and groups occurring, our paper contains the first examples of surfaces of mixed type with q = 1. The mixed cases turn out to be much less frequent than the unmixed ones and, as when pg = q = 0, they occur for only one value of the order of G. However, in contrast with what happens when pg = q = 0, the mixed cases do not correspond to the maximum value of |G| but appear for a rather small order, namely |G| = 16. Our classification procedure involves arguments from both geometry and computational group theory. We will give here a brief account on how the result is achieved. If S is any surface isogenous to a product and satisfying pg = q then |G|, g(C), g(F ) are related as in Beauville’s example and we have K2S = 8. Besides, if pg = q = 1 such surfaces are neces- sarily minimal and of general type (Lemma 2.1). If S = (C×F )/G is of unmixed type then the two projections πC : C×F −→ C, πF : C×F −→ F induce two morphisms α : S −→ C/G, β : S −→ F/G, whose smooth fibres are isomorphic to F and C, respectively. Moreover, the geometry of S is encoded in the geometry of the two cover- ings h : C −→ C/G, f : F −→ F/G and the invariants of S impose strong restrictions on g(C), g(F ) and |G|. Indeed we have 1 = q(S) = g(C/G) + g(F/G) so we may assume that E := C/G is an elliptic curve and F/G ∼= P1. Then α : S −→ E is the Albanese morphism of S and the genus galb of the general Albanese fibre equals g(F ). It is proven in [Pol07, Proposition 2.3] that 3 ≤ g(F ) ≤ 5; in particular this allows us to control |G|. The covers f and h are determined by two suitable systems of generators for G, that we call V and W, respectively. Besides, in order to obtain a free action of G on C×F and a quotient S with the desired invariants, V and W are subject to strict conditions of combinatorial nature (Proposition 2.2). The geometry imposes also strong restrictions on the possible W and the genus of C, so the existence of V and W and the compatibility conditions can be verified through a computer search. It is worth mentioning that the classification of finite groups of automorphisms acting on curves of genus lesser than or equal to 5 could have also been retrieved from the existing literature ([Br90], [Ki03], [KuKi90], [KuKu90]). If S = (C × C)/G is of mixed type then the index two subgroup G◦ of G corresponding to transformations that do not exchange the coordinates in C ×C acts faithfully on C. The quo- tient E = C/G◦ is isomorphic to the Albanese variety of S and galb = g(C) (Proposition 2.5). Moreover g(C) may only be 5, 7 or 9, hence |G| is at most 64 (Proposition 2.10). The cover h : C −→ E is determined by a suitable system of generators V for G◦ and since the action of G on C × C is required to be free, combinatorial restrictions involving the elements of V and those of G \ G◦ have to be imposed (Proposition 2.6). Our classification is obtained by first listing those groups G◦ for which V exists and then by looking at the admissible extensions G of G◦. We find that the only possibility occurring is for g(C) = 5 so that |G| is necessarily 16 (Propositions 4.1, 4.2, 4.3). In the last part of the paper we examine the structure of the subset of the moduli space corresponding to surfaces isogenous to a product with pg = q = 1. It can be explicitly described by calculating the number of orbits of the direct product of certain mapping class groups with Aut(G) acting on the set (of pairs) of systems of generators (Proposition 5.1). In particular it is possible to determine the number of irreducible connected components and their respective dimensions, see the forthcoming article [Pe08]. Our computations were carried out by using the computer algebra program GAP4, whose data- base includes all groups of order less than 2000, with the exception of 1024 (see [GAP4]). For the reader’s convenience we included the scripts in the Appendix. Now let us state the main result of this paper. Main Theorem. Let S = (C × F )/G be a surface with pg = q = 1, isogenous to a product of curves. Then S is minimal of general type and the occurrences for g(F ), g(C), G, the dimension D of the moduli space and the number N of its connected components are precisely those in the table below. IdSmall g(F ) = galb g(C) G Group(G) Type D N 3 3 (Z2) 2 G(4, 2) unmixed (∗) 5 1 3 5 (Z2) 3 G(8, 5) unmixed (∗) 4 1 3 5 Z2 × Z4 G(8, 2) unmixed (∗) 3 2 3 9 Z2 × Z8 G(16, 5) unmixed (∗) 2 1 3 5 D4 G(8, 3) unmixed 3 1 3 7 D6 G(12, 4) unmixed (∗∗) 3 1 3 9 Z2 ×D4 G(16, 11) unmixed 3 1 3 13 D2,12,5 G(24, 5) unmixed 2 1 3 13 Z2 ×A4 G(24, 13) unmixed 2 1 3 13 S4 G(24, 12) unmixed 2 1 3 17 Z2 ⋉ (Z2 × Z8) G(32, 9) unmixed 2 1 3 25 Z2 × S4 G(48, 48) unmixed 2 1 4 3 S3 G(6, 1) unmixed (∗∗) 4 1 4 5 D6 G(12, 4) unmixed 3 1 4 7 Z3 × S3 G(18, 3) unmixed 2 2 4 7 Z3 × S3 G(18, 3) unmixed 1 1 4 9 S4 G(24, 12) unmixed (∗∗) 2 1 4 13 S3 × S3 G(36, 10) unmixed 1 1 4 13 Z6 × S3 G(36, 12) unmixed 1 1 4 13 Z4 ⋉ (Z3) 2 G(36, 9) unmixed 1 2 4 21 A5 G(60, 5) unmixed (∗∗) 1 1 4 25 Z3 × S4 G(72, 42) unmixed 1 1 4 41 S5 G(120, 34) unmixed 1 1 5 3 D4 G(8, 3) unmixed (∗∗) 4 1 5 4 A4 G(12, 3) unmixed (∗∗) 2 2 5 5 Z4 ⋉ (Z2) 2 G(16, 3) unmixed 2 3 5 7 Z2 ×A4 G(24, 13) unmixed 2 2 5 7 Z2 ×A4 G(24, 13) unmixed 1 1 5 9 Z8 ⋉ (Z2) 2 G(32, 5) unmixed 1 1 5 9 Z2 ⋉D2,8,5 G(32, 7) unmixed 1 1 5 9 Z4 ⋉ (Z4 × Z2) G(32, 2) unmixed 1 1 5 9 Z4 ⋉ (Z2) 3 G(32, 6) unmixed 1 1 5 13 (Z2) 2 ×A4 G(48, 49) unmixed 1 1 5 17 Z4 ⋉ (Z2) 4 G(64, 32) unmixed 1 2 5 21 Z5 ⋉ (Z2) 4 G(80, 49) unmixed 1 2 5 5 D2,8,3 G(16, 8) mixed 2 1 5 5 D2,8,5 G(16, 6) mixed 2 3 5 5 Z4 ⋉ (Z2) 2 G(16, 3) mixed 2 1 Here IdSmallGroup(G) denotes the label of the group G in the GAP4 database of small groups. The calculation of N is due to Penegini and Rollenske, see [Pe08], except for the cases marked with (∗), which were already studied in [Pol07]. The cases marked with (∗∗) also appeared in [Pol07], but the computation of N was missing. This work is organized as follows. In Section 1 we collect the basic facts about surfaces isogenous to a product, following the treatment given by Catanese in [Ca00] and we fix the algebraic setup. In Section 2 we apply the structure theorems of Catanese to the case pg = q = 1 and this leads to Propositions 2.2 and 2.6, that provide the translation of our classification problem from ge- ometry to algebra. All these results are used in Sections 3 and 4, which are the core of the paper and give the complete lists of the occurring groups and genera in the unmixed and mixed cases, respectively. Finally, Section 5 is devoted to the description of the moduli spaces. Notations and conventions. All varieties, morphisms, etc. in this article are defined over C. By “surface” we mean a projective, non-singular surface S, and for such a surface KS denotes the canonical class, pg(S) = h 0(S, KS) is the geometric genus, q(S) = h 1(S, KS) is the irregularity and χ(OS) = 1 − q(S) + pg(S) is the Euler characteristic. Throughout the paper we use the following notation for groups: • Zn: cyclic group of order n. • Dp,q,r = Zp ⋉Zq = 〈x, y | x p = yq = 1, xyx−1 = yr〉: split metacyclic group of order pq. The group D2,n,−1 is the dihedral group of order 2n and it will be denoted by Dn. • Sn, An: symmetric, alternating group on n symbols. • If x, y ∈ G, their commutator is defined as [x, y] = xyx−1y−1. • If x ∈ G we denote by Intx the inner automorphism of G defined as Intx(g) = xgx • IdSmallGroup(G) indicates the label of the group G in the GAP4 database of small groups. For instance IdSmallGroup(D4) = G(8, 3) and this means that D4 is the third in the list of groups of order 8. Acknowledgements. The authors wish to thank M. Penegini and S. Rollenske for giving them a preliminary version of [Pe08] and for kindly allowing them to include their results in the Main Theorem. Moreover they are indebted with the referee for several valuable comments and suggestions to improve this article. 1. Basic on surfaces isogenous to a product In this section we collect for the reader’s convenience some basic results on groups acting on curves and surfaces isogenous to a product, referring to [Ca00] for further details. Definition 1.1. A complex surface S of general type is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C×F so that S = (C×F )/G. There are two cases: the unmixed one, where G acts diagonally, and the mixed one, where there exist elements of G exchanging the two factors (and then C, F are isomorphic). In both cases, since the action of G on C × F is free, we have K2S = K2C×F 8(g(C) − 1)(g(F ) − 1) χ(OS) = χ(OC×F ) (g(C)− 1)(g(F ) − 1) hence K2S = 8χ(OS). Let C, F be curves of genus ≥ 2. Then the inclusion Aut(C × F ) ⊃ Aut(C) × Aut(F ) is an equality if C and F are not isomorphic, whereas Aut(C × C) = Z2 ⋉ (Aut(C) × Aut(C)), the Z2 being generated by the involution exchanging the two coordinates. If S = (C × F )/G is a surface isogenous to a product, we will always consider its unique minimal realization. This means that • in the unmixed case, we have G ⊂ Aut(C) and G ⊂ Aut(F ) (i.e. G acts faithfully on both C and F ); • in the mixed case, where C ∼= F , we haveG◦ ⊂ Aut(C), for G◦ := G∩(Aut(C)×Aut(C)). (See [Ca00, Corollary 3.9 and Remark 3.10]). Definition 1.2. Let G be a finite group and let g′ ≥ 0, and mr ≥ mr−1 ≥ . . . ≥ m1 ≥ 2 be integers. A generating vector for G of type (g′ | m1, . . . ,mr) is a (2g ′ + r)-ple of elements V = {g1, . . . , gr; h1, . . . , h2g′} such that: the set V generates G; |gi| = mi and g1g2 · · · grΠ i=1[hi, hi+g′ ] = 1. If such a V exists, then G is said to be (g′ | m1, . . . ,mr)-generated. For convenience we make abbreviations such as (4 | 23, 32) for (4 | 2, 2, 2, 3, 3) when we write down the type of the generating vector V. By Riemann’s existence theorem a finite group G acts as a group of automorphisms of some compact Riemann surface X of genus g with quotient a Riemann surface Y of genus g′ if and only if there exist integers mr ≥ mr−1 ≥ . . . ≥ m1 ≥ 2 such that G is (g ′ | m1, . . . ,mr)- generated and g, g′, |G| and the mi are related by the Riemann-Hurwitz formula. Moreover, if V = {g1, . . . , gr; h1, . . . , h2g′} is a generating vector for G, the subgroups 〈gi〉 and their conjugates are precisely the nontrivial stabilizers of the G-action ([Br90, Section 2], [Bre00, Chapter 3], [H71]). The description of surfaces isogenous to a product can be therefore reduced to finding suitable generating vectors. Requiring that S has given invariants pg and q imposes numerical restrictions on the order of the group G and the genus of the curves C and F . Our goal is to classify all surfaces with pg = q = 1 isogenous to a product. The aim of the next section is to translate this classification problem from geometry to algebra. 2. The case pg = q = 1. Building data Lemma 2.1. Let S = (C × F )/G be a surface isogenous to a product with pg = q = 1. Then (i) K2S = 8. (ii) |G| = (g(C)− 1)(g(F ) − 1). (iii) S is a minimal surface of general type. Proof. Claims (i) and (ii) follow from (1). Now let us consider (iii). Since C×F is minimal and the cover C×F −→ S is étale, S is minimal as well. Moreover (ii) implies either g(C) = g(F ) = 0 or g(C) ≥ 2, g(F ) ≥ 2. The first case is impossible otherwise S = P1 × P1 and pg = q = 0; thus the second case occurs, hence S is of general type. � 2.1. Unmixed case. If S = (C ×F )/G is a surface with pg = q = 1, isogenous to an unmixed product, then g(C) ≥ 3, g(F ) ≥ 3 and up to exchanging F and C one may assume F/G ∼= P1 and C/G ∼= E, where E is an elliptic curve. Moreover α : S −→ C/G is the Albanese morphism of S and galb = g(F ), see [Pol07, Proposition 2.2]. This leads to Proposition 2.2. ([Pol07, Proposition 3.1]) Let G be a finite group which is both (0 | m1, . . . ,mr) and (1 | n1, . . . , ns)-generated, with generating vectors V = {g1, . . . , gr} and W = {ℓ1, . . . , ℓs; h1, h2}, respectively. Let g(F ), g(C) be the positive integers defined by the Riemann-Hurwitz relations 2g(F ) − 2 = |G| − 2 + 2g(C)− 2 = |G| Assume moreover that g(C) ≥ 3, g(F ) ≥ 3, |G| = (g(C) − 1)(g(F ) − 1) and 〈σgiσ 〈σℓjσ  = {1G}. Then there is a free, diagonal action of G on C × F such that the quotient S = (C × F )/G is a minimal surface of general type with pg = q = 1, K S = 8. Conversely, every surface with pg = q = 1, isogenous to an unmixed product, arises in this way. Here, condition (U) ensures that the G-action on C × F is free. Set m := (m1, . . . ,mr) and n := (n1, . . . , ns); if S = (C × F )/G is a surface with pg = q = 1 which is constructed by using the recipe in Proposition 2.2, it will be called an unmixed surface of type (G, m, n). Proposition 2.3. ([Pol07, Proposition 2.3]) Let S = (C ×F )/G be an unmixed surface of type (G, m, n). Then there are exactly the following possibilities: (a) g(F ) = 3, n = (22) (b) g(F ) = 4, n = (3) (c) g(F ) = 5, n = (2). The following lemma gives a restriction on m instead. Lemma 2.4. Let S = (C × F )/G be an unmixed surface of type (G, m, n). Then every mi divides (g(F )−1) Proof. Since 〈gi〉 is a stabilizer for the G-action on F and since G acts freely on (C × F ), the subgroup 〈gi〉 ∼= Zmi acts freely on C. By Riemann-Hurwitz formula applied to the cover C −→ C/〈gi〉 we have g(C)− 1 = mi(g(C/〈gi〉)− 1). Thus mi divides g(C)− 1 = (g(F )−1) 2.2. Mixed case. Proposition 2.5. Let S = (C × C)/G be a surface with pg = q = 1 isogenous to a mixed product. Then E := C/G◦ is an elliptic curve isomorphic to the Albanese variety of S. Proof. We have (see [Ca00, Proposition 3.15]) C = H0(Ω1S) = (H 0(Ω1C)⊕H 0(Ω1C)) G = (H0(Ω1C) G◦ ⊕H0(Ω1C) G◦)G/G = (H0(Ω1E)⊕H 0(Ω1E)) G/G◦ . Since S is of mixed type, the quotient Z2 = G/G ◦ exchanges the last two summands, whence h0(Ω1E) = 1. Thus E is an elliptic curve and there is a commutative diagram (3) C × C E × E K E(2) showing that the Albanese morphism α of S factors through the Abel-Jacobi map α̂ of the double symmetric product E(2) of E. � By Lemma 2.1 we have |G| = (g(C)− 1)2. In this case [Ca00, Proposition 3.16] becomes Proposition 2.6. Assume that G◦ is a (1 | n1, . . . , ns)-generated finite group with generating vector V = {ℓ1, . . . , ℓs; h1, h2} and that there is a nonsplit extension (4) 1 −→ G◦ −→ G −→ Z2 −→ 1 which gives an involution [ϕ] in Out(G◦). Let g(C) ∈ N be defined by the Riemann-Hurwitz relation 2g(C)−2 = |G◦| . Assume, in addition, that |G| = (g(C)−1)2 and that (M1) for all g ∈ G \G◦ we have {ℓ1, . . . , ℓs} ∩ {gℓ1g −1, . . . , gℓsg −1} = ∅; (M2) for all g ∈ G \G◦ we have g2 /∈ 〈σℓjσ Then there is a free, mixed action of G on C × C such that the quotient S = (C × C)/G is a minimal surface of general type with pg = q = 1, K S = 8. Conversely, every surface S with pg = q = 1, isogenous to a mixed product, arises in this way. Here, conditions (M1) and (M2) ensure that the G-action on C × C is free. Remark 2.7. The surface S is not covered by elliptic curves because it is of general type (Lemma 2.1), so the map C −→ C/G◦ = E is ramified. Therefore condition (M1) implies that G is not abelian. Remark 2.8. The exact sequence (4) is non split if and only if the number of elements of order 2 in G equals the number of elements of order 2 in G◦. Proposition 2.9. Let S = (C × C)/G be a surface with pg = q = 1, isogenous to a mixed product. Then galb = g(C). Proof. Let us look at diagram (3). The Abel-Jacobi map α̂ gives to E(2) the structure of a 1-bundle over E ([CaCi93]); let f be the generic fibre of this bundle and F ∗ := ρ∗ε∗(f). If Falb is the generic Albanese fibre of S we have Falb = π(F ∗). Let n = (n1, . . . , ns) be such that G◦ is (1 |n1, . . . ns)-generated and 2g(C)− 2 = |G . The (G◦ ×G◦)-cover ρ is branched exactly along the union of s “horizontal” copies of E and s “vertical” copies of E; moreover for each i there are one horizontal copy and one vertical copy whose branching number is ni. Since ε ∗(f) is an elliptic curve that intersects all these copies of E transversally in one point, by Riemann-Hurwitz formula applied to F ∗ −→ ε∗(f) we obtain 2g(F ∗)− 2 = |G◦|2 · On the other hand the G-cover π is étale, so we have 2g(Falb)− 2 = (2g(F ∗)− 2) = |G◦| = 2g(C)− 2, whence galb = g(C). � If S = (C × C)/G is a surface with pg = q = 1 which is constructed by using the recipe of Proposition 2.6, it will be called a mixed surface of type (G, n). The analogue of Proposition 2.3 in the mixed case is Proposition 2.10. Let S = (C × C)/G be a mixed surface of type (G, n). Then there are at most the following possibilities: • g(C) = 5, n = (22), |G| = 16; • g(C) = 7, n = (3), |G| = 36; • g(C) = 9, n = (2), |G| = 64. Proof. By Proposition 2.6 we have 2g(C) − 2 = |G◦| and |G◦| = 1 (g(C) − 1)2, so g(C) must be odd and we obtain 4 = (g(C)− 1) . Therefore 4 ≥ 1 (g(C)− 1) and the only possibilities are g(C) = 3, 5, 7, 9. The case g(C) = 3 is ruled out because G cannot be abelian by Remark 2.7. If g(C) = 5 then = 1, so n = (22) and |G| = 16. If g(C) = 7 then , so n = (3) and |G| = 36. If g(C) = 9 then , so n = (2) and |G| = 64. � We will see in Section 2.10 that only the case g(C) = 5 actually occurs. 3. The unmixed case The classification of surfaces of general type with pg = q = 1 isogenous to an unmixed prod- uct is carried out in [Pol07] when the group G is abelian. Therefore in this section we assume that G is nonabelian. Following [BaCaGr06, Section 1.2], for an r-ple m = (m1, . . . ,mr) ∈ N r we set Θ(m) := −2 + , α(m) := If S is an unmixed surface of type (G, m, n) then we necessarily have 2 ≤ m1 ≤ . . . ≤ mr and Θ(m) > 0. Besides, by Proposition 2.2 we have α(m) = g(F )−1 = g(C)− 1 ∈ N and by Lemma 2.4 each integer mi divides α(m). Then we get Proposition 3.1. Let S = (C × F )/G be a surface with pg = q = 1 isogenous to an unmixed product of type (G, m, n). Then the possibilities for m and α(m), written in the format mα(m), lie in the set T below: (2, 3, 7)84 , (2, 3, 8)48, (2, 4, 5)40, (2, 3, 9)36 , (2, 3, 10)30 , (2, 3, 12)24 , (2, 4, 6)24 , (3 2, 4)24, (2, 5 2)20, (2, 3, 18)18 , (2, 4, 8)16, (3 2, 5)15, (2, 4, 12)12 , (2, 6 2)12, (3 2, 6)12, (3, 4 2)12, (2, 5, 10)10 , (3 2, 9)9, (2, 82)8, (4 3)8, (3, 6 2)6, (5 3)5, (2 3, 3)12, (2 3, 4)8, (23, 6)6, (2 2, 32)6, (2 2, 42)4, (3 4)3, (2 5)4, (2 Proof. This follows combining [BaCaGr06, Proposition 1.4] with Lemma 2.4. � By abuse of notation, we write m ∈ T instead of mα(m) ∈ T. Now we analyze the three cases in Proposition 2.3 separately, according to the value of g(F ). Note that if g(F ) = 3, 4, 5 then |Aut(F )| ≤ 168, 120, 192, respectively ([Bre00, p. 91]). Proposition 3.2. If g(F ) = 3 we have precisely the following possibilities. IdSmall G Group(G) m D4 G(8, 3) (2 2, 42) D6 G(12, 4) (2 3, 6) Z2 ×D4 G(16, 11) (2 3, 4) D2,12,5 G(24, 5) (2, 4, 12) Z2 ×A4 G(24, 13) (2, 6 S4 G(24, 12) (3, 4 Z2 ⋉ (Z2 × Z8) G(32, 9) (2, 4, 8) Z2 × S4 G(48, 48) (2, 4, 6) Proof. Since n = (22) it follows that G is (1 | 22)-generated and by the second relation in (2) we have |G| = 2(g(C) − 1). So we must describe all unmixed surfaces of type (G,m,n) with m ∈ T, n = (22) and |G| = 2α(m). By a computer search through the r-tuples in Proposition 3.1 we can therefore list all possibilities, proving our statement. See the GAP4 script 1 in the Appendix to see how this procedure applies to an explicit example. Proposition 3.3. If g(F ) = 4 we have precisely the following possibilities. IdSmall G Group(G) m S3 G(6, 1) (2 D6 G(12, 4) (2 Z3 × S3 G(18, 3) (2 2, 32) Z3 × S3 G(18, 3) (3, 6 S4 G(24, 12) (2 3, 4) S3 × S3 G(36, 10) (2, 6 Z6 × S3 G(36, 12) (2, 6 Z4 ⋉ (Z3) 2 G(36, 9) (3, 42) A5 G(60, 5) (2, 5 Z3 × S4 G(72, 42) (2, 3, 12) S5 G(120, 34) (2, 4, 5) Proof. Since n = (3) it follows that G is (1 | 3)-generated and by the second relation in (2) we have |G| = 3(g(C) − 1). Therefore our statement can be proven searching by computer calculation all unmixed surfaces of type (G,m,n) with m ∈ T, n = (3), |G| = 3α(m) and α(m) ≤ 40. � Proposition 3.4. If g(F ) = 5 we have precisely the following possibilities. IdSmall G Group(G) m D4 G(8, 3) (2 A4 G(12, 3) (3 Z4 ⋉ (Z2) 2 G(16, 3) (22, 42) Z2 ×A4 G(24, 13) (2 2, 32) Z2 ×A4 G(24, 13) (3, 6 Z8 ⋉ (Z2) 2 G(32, 5) (2, 82) Z2 ⋉D2,8,5 G(32, 7) (2, 8 Z4 ⋉ (Z4 × Z2) G(32, 2) (4 Z4 ⋉ (Z2) 3 G(32, 6) (43) 2 ×A4 G(48, 49) (2, 6 Z4 ⋉ (Z2) 4 G(64, 32) (2, 4, 8) Z5 ⋉ (Z2) 4 G(80, 49) (2, 52) Proof. Since n = (2), it follows that G is (1 | 2)-generated and by the second relation in (2) we have |G| = 4(g(C) − 1). Therefore our statement can be proven searching by computer calculation all unmixed surfaces of type (G,m,n) with m ∈ T, n = (2), |G| = 4α(m) and α(m) ≤ 48. � 4. The mixed case In this section we use Proposition 2.6 in order to classify the surfaces with pg = q = 1 isogenous to a mixed product. By Proposition 2.10 we have g(C) = 5, 7 or 9. Let us consider the three cases separately. 4.1. The case g(C) = 5, |G| = 16. Proposition 4.1. If g(C) = 5, |G| = 16 we have precisely the following possibilities. IdSmall IdSmall G◦ Group(G◦) G Group(G) D4 G(8, 3) D2,8,3 G(16, 8) Z2 × Z4 G(8, 2) D2,8,5 G(16, 6) 3 G(8, 5) Z4 ⋉ (Z2) 2 G(16, 3) Proof. In this case n = (22), so our first task is to find all nonsplit sequences of type (4) for which G◦ is a (1 | 22)-generated group of order 8. The three abelian groups of order 8 and D4 are (1 | 22)-generated whereas the quaternion group Q8 is not. Since Z8 has only one element ℓ of order 2, condition (M1) in Proposition 2.6 cannot be satisfied for any choice of V. By Remark 2.7 we are left to analyze the possible embeddings of Z2 × Z4, D4 and (Z2) 3 in nonabelian groups of order 16. The groups Z2 × Z4, D4 and (Z2) 3 have 3, 5 and 7 elements of order 2, respectively. Therefore if n2 denotes the number of elements of order 2 in G, by Remark 2.8 we must consider only those groups G of order 16 with n2 ∈ {3, 5, 7}. The nonabelian groups of order 16 with n2 = 3 are D2,8,5, Z2 × Q8 and D4,4,−1 and they all contain a copy of Z2 × Z4. The only nonabelian group of order 16 with n2 = 5 is D2,8,3 and it contains a subgroup isomorphic to D4. The nonabelian groups of order 16 with n2 = 7 are Z4 ⋉ (Z2) 2 = G(16, 3) and Z2 ⋉ Q8, and only the former contains a subgroup isomorphic to 3 (cfr. [Wi05]). Summarizing, we are left with the following cases: D4 D2,8,3 Z2 × Z4 D2,8,5 Z2 × Z4 Z2 ×Q8 Z2 × Z4 D4,4,−1 Z4 ⋉ (Z2) Let us analyze them separately. • G◦ = D4, G = D2,8,3 = 〈x, y | x 2 = y8 = 1, xyx−1 = y3 〉 We consider the subgroup G◦ := 〈x, y2〉 ∼= D4. Set ℓ1 = ℓ2 = x and h1 = h2 = y 2. Condition (M1) holds because CG(x) = 〈x, y 4〉 ⊂ G◦. Condition (M2) is satisfied because the conjugacy class of x in G◦ is contained in the coset x〈y2〉 while for every g ∈ yG◦ we have g2 ∈ 〈y〉. Therefore this case occurs by Proposition 2.6. • G◦ = Z2 × Z4, G = D2,8,5 = 〈x, y | x 2 = y8 = 1, xyx−1 = y5〉 We consider the subgroup G◦ := 〈x, y2〉 ∼= Z2 × Z4. Set ℓ1 = ℓ2 = x and h1 = h2 = y Conditions (M1) and (M2) are verified as in the previous case, so this possibility occurs. • G◦ = Z2 × Z4, G = Z2 ×Q8 and G ◦ = Z2 × Z4, G = D4,4,−1. All elements of order 2 in G are central so condition (M1) cannot be satisfied and these cases do not occur. • G◦ = (Z2) 3, G = Z4 ⋉ (Z2) 2 = 〈x, y, z | x4 = y2 = z2 = 1, xyx−1 = yz, [x, z] = [y, z] = 1〉 We consider the subgroup G◦ := 〈y, z, x2〉 ∼= (Z2) 3. Set ℓ1 = ℓ2 = y and h1 = z, h2 = x Condition (M1) holds because G◦ is abelian and [x, y] 6= 1. Condition (M2) is satisfied because if g ∈ xG◦ then g2 ∈ 〈z, x2〉. Therefore this case occurs. � 4.2. The case g(C) = 7, |G| = 36. Proposition 4.2. The case g(C) = 7, |G| = 36 does not occur. Proof. In this case n = (3), so G◦ is a group of order 18 which is (1 | 3)-generated. There are five groups of order 18 up to isomorphism. By computer search or direct calculation we see that the only one which is (1 | 3)-generated is Z3 × S3 = G(18, 3). Thus G would fit into a short exact sequence (5) 1 −→ Z3 × S3 −→ G −→ Z2 −→ 1. A computer search shows that the only groups of order 36 containing a subgroup isomorphic to Z3 × S3 are G(36, 10) = S3 × S3 and G(36, 12) = Z6 × S3 (see GAP4 script 2 in the Appendix). They contain 15 and 7 elements of order 2, respectively. On the other hand Z3 × S3 contains 3 elements of order 2, so by Remark 2.8 all possible extensions of the form (5) are split and this case cannot occur. � 4.3. The case g(C) = 9, |G| = 64. Proposition 4.3. The case g(C) = 9, |G| = 64 does not occur. The proof will be the consequence of the results below. First notice that, since n = (2), the group G◦ must be (1 | 2)-generated. Computational Fact 4.4. There exist precisely 8 groups of order 32 which are (1 | 2)- generated, namely G(32, t) for t ∈ {2, 4, 5, 6, 7, 8, 12, 17}. The number n2 of their elements of order 2 is given in the table below: t 2 4 5 6 7 8 12 17 n2(G(32, t)) 7 3 7 11 11 3 3 3 Proof. Slightly modifying the first part of GAP4 script 1 in the Appendix we easily find that the groups of order 32 which are (1 | 2)-generated are exactly those in the statement. The number of elements of order 2 in each case are found by a quick computer search: see again the Appendix, GAP4 script 3. � Computational Fact 4.5. Let t ∈ {2, 4, 5, 6, 7, 8, 12, 17}. A nonsplit extension of the form (6) 1 −→ G(32, t) −→ G(64, s) −→ Z2 −→ 1 exists if and only if the pair (t, s) is one of the following: (2, 9), (2, 57), (2, 59), (2, 63), (2, 64), (2, 68), (2, 70), (2, 72), (2, 76), (2, 79), (2, 81), (2, 82), (4, 11), (4, 28), (4, 122), (4, 127), (4, 172), (4, 182), (5, 5), (5, 9), (5, 112), (5, 113), (5, 114), (5, 132), (5, 164), (5, 165), (5, 166), (6, 33), (6, 35), (7, 33), (8, 37), (12, 7), (12, 13), (12, 14), (12, 15), (12, 16), (12, 126), (12, 127), (12, 143), (12, 156), (12, 158), (12, 160), (17, 28), (17, 43), (17, 45), (17, 46). Proof. Assume t = 2. Using the GAP4 script 4 in the Appendix we find that the groups of order 64 containing a subgroup isomorphic toG(32, 2) areG(64, s) for s ∈ {8, 9, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82}. By Remark 2.8 and Computa- tional Fact 4.4, in order to detect all the groups G(64, s) fitting in some nonsplit extension of type (6) with t = 2, it is sufficient to select from the previous list the groups containing exactly n2 = 7 elements of order 2. This can be done with the GAP4 script 5 in the Appendix, proving the claim in the case t = 2. The proof for the other values of t may be carried out exactly in the same way. � Let us denote by [G, G]2 and [G ◦, G◦]2 the subsets of elements of order 2 in [G, G] and [G◦, G◦], respectively. Lemma 4.6. Assume g(C) = 9 and that one of the following situations occur: • [G,G]2 ⊆ Z(G); • there exists some element y ∈ G \G◦ commuting with all elements in [G◦, G◦]2. Then given any generating vector V = {ℓ1; h1, h2} of type (1 | 2) for G ◦, condition (M1) in Proposition 2.6 cannot be satisfied. Proof. Since ℓ1 ∈ [G ◦, G◦]2 ⊆ [G, G]2, in any of the above situations CG(ℓ1) is not contained in G◦, so (M1) cannot hold. � Computational Fact 4.7. Let G = G(64, s) be one of the groups appearing in the list of Computational Fact 4.5. Then [G,G]2 is not contained in Z(G) if and only if s = 5, 33, 35, 37. Proof. See the GAP4 script 6 in the Appendix. � Computational facts 4.5, 4.7 and Lemma 4.6 imply that we only need to analyze the following pairs (G◦, G): G(32, 5) G(64, 5) G(32, 6) G(64, 33) G(32, 7) G(64, 33) G(32, 6) G(64, 35) G(32, 8) G(64, 37) Proposition 4.8. The case G◦ = G(32, 5) does not occur. Proof. A presentation for the group G◦ is G◦ = 〈x, y, z | x8 = y2 = z2 = 1, [y, z] = [x, z] = 1, [x, y] = z〉. Its derived subgroup contains exactly one element of order 2, namely z. It follows that if {ℓ1; h1, h2} is any generating vector of type (1 | 2) for G ◦, then ℓ1 = z. Since [G ◦, G◦] is characteristic in G◦, condition (M1) cannot be satisfied for any embedding of G◦ into G. � By using the two instructions P:=PresentationViaCosetTable(G) and TzPrintRelators(P) and setting in the output x := f1, y := f2, z := f3, w := f4, v := f5, u := f6 one obtains the following presentations for G(64, 33), G(64, 35) and G(64, 37). G(64, 33) = 〈x, y, z, w, v, u | z2 = w2 = v2 = u2 = 1, x2 = w, y2 = u, [x, zy] = z, [x, vz] = v, [x, vu] = u, [y, z] = [y, v] = [z, v] = [w, v] = [x, u] = 1〉 G(64, 35) = 〈x, y, z, w, v, u | w2 = v2 = u2 = 1, z2 = y2 = u, x2 = w, [y, z] = [z, w] = u, [x, yz] = z, [x, z] = uv, [y, v] = [z, v] = [w, v] = [x, u] = 1〉 G(64, 37) = 〈x, y, z, w, v, u | v2 = u2 = 1, w2 = z2 = y2 = u, x2 = w, [y, z] = [z, w] = u, [x, yz] = z, [x, z] = uv, [y, v] = [z, v] = [w, v] = 1〉. Computational Fact 4.9. Referring to presentations (7), (8) and (9), we have the following facts. • The group G(64, 33) contains exactly one subgroup N1 isomorphic to G(32, 6) and one subgroup N2 isomorphic to G(32, 7), namely N1 := 〈x, z, w, v, u〉, N2 := 〈xy, z, w, v, u〉. • The group G(64, 35) contains exactly two subgroups N3, N4 isomorphic to G(32, 6), namely N3 := 〈x, z, w, v, u〉, N4 := 〈xy, z, w, v, u〉. • The group G(64, 37) contains exactly two subgroups N5, N6 isomorphic to G(32, 8), namely N5 := 〈x, z, w, v, u〉, N6 := 〈xy, z, w, v, u〉. In addition, for every i ∈ {1, . . . , 6} we have (a) [Ni, Ni] = 〈v, u〉 ∼= Z2 × Z2. (b) y /∈ Ni and y commutes with all elements in [Ni, Ni]. Proof. See the GAP4 script 7 in the Appendix. � Proposition 4.10. The cases G◦ = G(32, 6), G(32, 7), G(32, 8) do not occur. Proof. By Lemma 4.6 and Computational Fact 4.9 it follows that, given any nonsplit extension of type (6) with G◦ as above, condition (M1) in Proposition 2.6 cannot be satisfied. � Summing up, we finally obtain Proof of Proposition 4.3. It follows from Propositions 4.8 and 4.10. 5. Moduli spaces Let Ma,b be the moduli space of smooth minimal surfaces of general type with χ(OS) = a, K2S = b; by an important result of Gieseker, Ma,b is a quasiprojective variety for all a, b ∈ N (see [Gie77]). Obviously, our surfaces are contained in M1,8 and we want to describe their locus there. We denote by M(G,m,n) the moduli space of unmixed surfaces of type (G,m,n) and by M(G,n) the moduli space of mixed surfaces of type (G,n). We know that n = (22), (3) or (2) in the unmixed case, whereas n = (22) in the mixed one. By a general result of Catanese ([Ca00]), both M(G,m,n) and M(G,n) consist of finitely many irreducible connected components of M1,8, all of the same dimension. More precisely, we have dim M(G,m,n) = r + s− 3, dim M(G,n) = s. Consider the mapping class groups in genus zero and one: Mod0,[r] := 〈σ1, . . . , σr | σiσi+1σi = σi+1σiσi+1, σiσj = σjσi if |i− j| ≥ 2, σr−1σr−2 · · · σ 1 · · · σr−2σr−1 = 1〉, Mod1,1 := 〈tα, tβ , tγ | tαtβtα = tβtαtβ, (tαtβ) 3 = 1〉, Mod1,[2] := 〈tα, tβ , tγ , ρ | tαtβtα = tβtαtβ, tαtγtα = tγtαtγ , tβtγ = tγtβ, (tαtβtγ) 4 = 1, tαρ = ρtα, tβρ = ρtβ, tγρ = ρtγ〉. One can prove that Mod0,[r] : = π0 Diff +(P1 − {p1, . . . , pr}), Mod1,1 : = π0 Diff +(Σ1 − {p}), Mod1,[2] : = π0 Diff +(Σ1 − {p, q}), where Σ1 is the torus S 1×S1 ([Schn03], [CattMu04]). This implies that we can define actions of these groups on the set of generating vectors for G of type (0 | m1, . . . ,mr), (1 | n) and (1 | n respectively. If V := {g1, . . . , gr} is of type (0 | m1, . . . ,mr) then the action is given by gi −→ gi+1 gi+1 −→ g i+1gigi+1 gj −→ gj if j 6= i, i+ 1. If W := {ℓ1; h1, h2} is of type (1 | n) then ℓ1 −→ ℓ1 h1 −→ h1 h2 −→ h2h1 ℓ1 −→ ℓ1 h1 −→ h1h h2 −→ h2. If W := {ℓ1, ℓ2; h1, h2} is of type (1 | n 2) then ℓ1 −→ ℓ1 ℓ2 −→ ℓ2 h1 −→ h1 h2 −→ h2h1 ℓ1 −→ ℓ1 ℓ2 −→ ℓ2 h1 −→ h1h h2 −→ h2 ℓ1 −→ ℓ1 ℓ2 −→ h1h 1 ℓ2h1h2h h1 −→ h 2 ℓ1h1 h2 −→ h2 ℓ1 −→ h 1 ℓ2h1h2 ℓ2 −→ h 2 ℓ1h2h1 h1 −→ h h2 −→ h These are called Hurwitz moves and the induced equivalence relation on generating vectors is said Hurwitz equivalence (see [BaCa03], [BaCaGr06], [Pol07]). Now let B(G, m, n) be the set of pairs of generating vectors (V, W) such that the assumptions in Proposition 2.2 are satisfied; then we denote by R the equivalence relation on B(G, m, n) generated by Hurwitz moves on V, Hurwitz moves on W and the simultaneous action of Aut(G) on V and W. Similarly, let B(G, n) be the set of generating vectors V such that the assump- tions of Proposition 2.6 are satisfied; then we denote by R the equivalence relation on B(G, n) generated by the Hurwitz moves and the action of Aut(G) on V. Proposition 5.1. The number of irreducible components in M(G, m, n) equals the number of R-classes in B(G, m, n). Analogously, the number of irreducible components in M(G, n) equals the number of R-classes in B(G, n). Proof. We can repeat exactly the same argument used in [BaCaGr06, Propositions 5.2 and 5.5]; we must just replace, where it is necessary, the mapping class group of P1 with the mapping class group of the elliptic curve E. � Proposition 5.1 in principle allows us to compute the number of connected components of the moduli space in each case. In practice, this task may be too hard to be achieved by hand, but it is not out of reach if one uses the computer. Recently, M. Penegini and S. Rollenske developed a GAP4 script that solves this problem in a rather short time. We put the result of their calculations in the Main Theorem (see Introduction), referring the reader to the forthcoming paper [Pe08] for further details. 6. Appendix In this Appendix we include, for the reader’s convenience, some of the GAP4 scripts that we have used in our computations; all the others are similar and can be easily obtained modifying the ones below. Let us show how the procedure in the proof of Proposition 3.2 applies to an explicit exam- ple, namely mα(m) = (2, 4, 12)12 . First we find all the nonabelian groups of order 24 that are (0 | 2, 4, 12)-generated. This is done using GAP4 as below; the output tells us that there is only one such a group, namely G = G(24, 5). gap> # -------------- SCRIPT 1 ------------------ gap> s:=NumberSmallGroups(24);; set:=[1..s]; [1..15] gap> for t in set do > c:=0; G:= SmallGroup(24,t); > Ab:=IsAbelian(G); > for g1 in G do > for g2 in G do > g3:=(g1*g2)^-1; > H:= Subgroup(G, [g1,g2]); > if Order(g1)=2 and Order(g2)=4 and Order(g3)=12 and > Order(H)=Order(G) and > Ab=false then > c:=c+1; fi; > if Order(g1)=2 and Order(g2)=4 and Order(g3)=12 and > Order(H)=Order(G) and > Ab=false and c=1 then > Print(IdSmallGroup(G)," "); > fi; od; od; od; Print("\n"); [24,5] By using the two instructions P:=PresentationViaCosetTable(G) and TzPrintRelators(P) we see that G has the presentation 〈x, y | x2 = y12 = 1, xyx−1 = y5〉, hence it is isomorphic to the metacyclic group D2,12,5. In order to speed up further computations, we define the sets G2, G4 given by the elements of G having order 2 and 4, respectively. gap> G:=SmallGroup(24,5);; gap> G2:=[];; G4:=[];; gap> for g in G do > if Order(g)=2 then Add(G2,g); fi; > if Order(g)=4 then Add(G4,g); fi; od; Then we check whether G is actually (1 | 22)-generated; if not, it should be excluded. gap> c:=0;; gap> for l2 in G2 do > for h1 in G do > for h2 in G do > l1:=(l2*h1*h2*h1^-1*h2^-1)^-1; > K:=Subgroup(G, [l2, h1, h2]); > if Order(l1)=2 and Order(K)=Order(G) then > Print(IdSmallGroup(G), " is (1 | 2,2)-generated", "\n"); c:=1; fi; > if c=1 then break; fi; od; > if c=1 then break; fi; od; > if c=1 then break; fi; od; [24,5] is (1 | 2,2)-generated We finish the proof by checking whether the surface S actually exists; the procedure is to look for a pair (V, W) of generating vectors for G satisfying the assumptions of Proposition 2.2. gap> c:=0;; gap> for g1 in G2 do > for g2 in G4 do > g3:=(g1*g2)^-1; > H:=Subgroup(G, [g1, g2]); > for l2 in G2 do > for h1 in G do > for h2 in G do > l1:=(l2*h1*h2*h1^-1*h2^-1)^-1; > K:=Subgroup(G, [l2, h1, h2]); > Boole1:=l1 in ConjugacyClass(G, g1); > Boole2:=l1 in ConjugacyClass(G, g2^2); > Boole3:=l1 in ConjugacyClass(G, g3^6); > Boole4:=l2 in ConjugacyClass(G, g1); > Boole5:=l2 in ConjugacyClass(G, g2^2); > Boole6:=l2 in ConjugacyClass(G, g3^6); > if Order(g3)=12 and Order(l1)=2 and > Order(H)=Order(G) and Order(K)=Order(G) and > Boole1=false and Boole2=false and Boole3=false and > Boole4=false and Boole5=false and Boole6=false then > Print("The surface exists "); c:=1; fi; > if c=1 then break; fi; od; > if c=1 then break; fi; od; > if c=1 then break; fi; od; > if c=1 then break; fi; od; > if c=1 then break; fi; od; Print("\n"); The surface exists The script above can be easily modified in order to obtain the list of all admissible pairs (V, W); for instance, one of such pairs is given by g1 = x, g2 = xy −1, g3 = y ℓ1 = xy 2, ℓ2 = xy 2, h1 = y, h2 = y. Finally, here are the GAP4 scripts used in Section 4. gap> # -------------- SCRIPT 2 ------------------ gap> s:=NumberSmallGroups(36);; set:=[1..s]; [1..14] gap> for t in set do > c:=0; G:=SmallGroup(36,t); > N:=NormalSubgroups(G); > for G0 in N do > if IdSmallGroup(G0)=[18,3] then > c:=c+1; fi; > if IdSmallGroup(G0)=[18,3] and c=1 then > Print(IdSmallGroup(G), " "); > fi; od; od; Print("\n"); [36,10] [36,12] gap> # -------------- SCRIPT 3 ------------------ gap> set:=[2,4,5,6,7,8,12,17];; gap> for t in set do > n2:=0; > G0:=SmallGroup(32,t); > for g in G0 do > if Order(g)=2 then > n2:=n2+1; fi; od; > Print(IdSmallGroup(G0), " "); Print(n2, " "); > od; Print("\n"); [32,2] 7 [32,4] 3 [32,5] 7 [32,6] 11 [32,7] 11 [32,8] 3 [32,12] 3 [32,17] 3 gap> # -------------- SCRIPT 4 ------------------ gap> s:=NumberSmallGroups(64);; set:=[1..s]; [1..267] gap> for t in set do > c:=0; G:=SmallGroup(64,t); > N:=NormalSubgroups(G); > for G0 in N do > if IdSmallGroup(G0)=[32,2] then > c:=c+1; fi; > if IdSmallGroup(G0)=[32,2] and c=1 then > Print(IdSmallGroup(G), " "); > fi; od; od; Print("\n"); [64,8] [64,9] [64,56] [64,57] [64,58] [64,59] [64,61] [64,62] [64,63] [64,64] [64,66] [64,67] [64,68] [64,69] [64,70] [64,72] [64,73] [64,74] [64,75] [64,76] [64,77] [64,78] [64,79] [64,80] [64,81] [64,82] gap> # -------------- SCRIPT 5 ------------------ gap> set:=[8,9,56,57,58,59,61,62,63,64,66,67,68,69,70, >72,73,74,75,76,77,78,79,80,81,82];; gap> for t in set do > n2:=0; G:=SmallGroup(64,t); > for g in G do > if Order(g)=2 then n2:=n2+1; > fi; od; > if n2=7 then > Print(IdSmallGroup(G), " "); > fi; od; Print("\n"); [64,9] [64,57] [64,59] [64,63] [64,64] [64,68] [64,70] [64,72] [64,76] [64,79] [64,81] [64,82] gap> # -------------- SCRIPT 6 ------------------ gap> set:=[5,7,9,11,13,,14,15,16,28,33,35,37,43,45,46, >57,59,63,64,68,70,72,76,79,81,82,112,113,114, 122,126, >127,132,143,156,158,160,164,165,166,172,182];; gap> for t in set do > c:=0; G:=SmallGroup(64,t); > D:=DerivedSubgroup(G); > for d in D do > B:=d in Center(G); > if Order(d)=2 and B=false then > c:=c+1; fi; > if Order(d)=2 and B=false and c=1 then > Print(IdSmallGroup(G), " "); > fi; od; od; Print("\n"); [64,5] [64,33] [64,35] [64,37] gap> # -------------- SCRIPT 7 ------------------ gap> s:=[33, 35, 37];; I:=[1, 2, 3];; gap> r:=[ [[32,6], [32,7]], [[32,6]], [[32,8]] ];; > for i in I do > G:=SmallGroup(64, s[i]); Print(IdSmallGroup(G), "\n"); > for N in NormalSubgroups(G) do > if IdSmallGroup(N) in r[i] then > Print(N, "="); Print(IdSmallGroup(N), " "); > Print(DerivedSubgroup(N), "\n"); > fi; od; Print("\n"); od; [64,33] Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,7] Group( [ f5, f6 ] ) Group( [ f1, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] ) [64,35] Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] ) Group( [ f1, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] ) [64,37] Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,8] Group( [ f5, f6 ] ) Group( [ f1, f3, f4, f5, f6 ] )=[32,8] Group( [ f5, f6 ] ) References [BaCa03] I. Bauer, F. Catanese: Some new surfaces with pg = q = 0, Proceedings of the Fano Conference (Torino, 2002). [BaCaGr06] I. Bauer, F. Catanese, F. Grunewald: The classification of surfaces with pg = q = 0 isogenous to a product of curves, e-print math.AG/0610267 (2006) to appear in Pure Appl. Math Q., volume in honour of F. Bogomolov’s 60-th birthday. [BaCaPi06] I. Bauer, F. Catanese, R. Pignatelli: Complex surfaces of general type: some recent progress, (2006), to appear in Global methods in complex geometry, 1–58 Springer-Verlag. [Be96] A. Beauville: Complex algebraic surfaces, Cambridge University Press 1996. [Bre00] T. Breuer: Characters and Automorphism groups of Compact Riemann Surfaces, Cambridge University Press 2000. [Br90] S. A. Broughton: Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1990), 233-270. [Ca81] F. Catanese: On a class of surfaces of general type, in Algebraic Surfaces, CIME, Liguori (1981), 269-284. [Ca00] F. Catanese: Fibred surfaces, varieties isogenous to a product and related moduli spaces, American J. of Math. 122 (2000), 1-44. [CaCi91] F. Catanese and C. Ciliberto: Surfaces with pg = q = 1, Sympos. Math. XXXII (1991), 49-79. [CaCi93] F. Catanese and C. Ciliberto: Symmetric product of elliptic curves and surfaces of general type with pg = q = 1, J. Algebraic Geom. 2 (1993), 389-411. [CaPi06] F. Catanese, R. Pignatelli: Fibrations of low genus I, Ann. Sci. École Norm. Sup. (4)39 (2006), 1011- 1049. [CCML98] F. Catanese, C. Ciliberto and M. M. Lopes: Of the classification of irregular surfaces of general type with non birational bicanonical map, Trans. Amer. Math. Soc. 350 (1998), 275-308. [CattMu04] A. Cattabriga, M. Mulazzani: (1,1)-knots via the mapping class group of the twice punctured torus, Adv. Geom. 4 (2004), 263-277. [GAP4] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4 ; 2006, http : //www.gap− system.org. [Gie77] D. Gieseker: Global moduli for surfaces of general type, Invent. Math 43 (1977), 233-282. [Go31] L. Godeaux: Sur une surface algébrique de genre zero et bigenre deux, Atti Accad. Naz. Lincei 14 (1931), 479-481. [H71] W. J. Harvey: On the branch loci in Teichmüller space, Trans. Amer. Mat. Soc. 153 (1971), 387-399. [HP02] C. Hacon, R. Pardini: Surfaces with pg = q = 3, Trans. Amer. Math. Soc. 354 no. 7 (2002), 2631-2638. [Ki03] H. Kimura: Classification of automorphism groups, up to topological equivalence, of compact Riemann surfaces of genus 4, J. Algebra 264 (2003), 26-54. [KuKi90] A. Kuribayashi, H. Kimura: Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), no. 1, 80–103. [KuKu90] I. Kuribayashi, A. Kuribayashi: Automorphism groups of compact Riemann surfaces of genera three and four, J. Pure Appl. Algebra 65 (1990), no. 3, 277–292. [Par03] R. Pardini: The classification of double planes of general type with K2S = 8 and pg = 0, J. Algebra 259 (2003) no. 3, 95-118. [Pe08] M. Penegini: Surfaces with pg = q = 2 isogenous to a product of curves: a computational approach. With an appendix of S. Rollenske. Work in progress [Pir02] G. P. Pirola: Surfaces with pg = q = 3, Manuscripta Math. 108 no. 2 (2002), 163-170. [Pol05] F. Polizzi: On surfaces of general type with pg = q = 1, K S = 3, Collect. Math. 56, no. 2 (2005), 181-234. [Pol07] F. Polizzi: On surfaces of general type with pg = q = 1 isogenous to a product of curves, e-print math.AG/0601063, to appear in Comm. Algebra. [Schn03] L. Schneps: Special loci in moduli spaces of curves. Galois groups and fundamental groups, 217–275, Math. Sci. Res. Inst. Publ. 41, Cambridge Univ. Press, Cambridge, 2003. [Wi05] M. Wild: The groups of order sixteen made easy, Amer. Math. Monthly 112, Number 1 (2005), 20-31. Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy. E-mail address: [email protected] Dipartimento di Matematica, Università della Calabria, Via Pietro Bucci, 87036 Arcavacata di Rende (CS), Italy. E-mail address: [email protected] 0. Introduction 1. Basic on surfaces isogenous to a product 2. The case pg=q=1. Building data 2.1. Unmixed case 2.2. Mixed case 3. The unmixed case 4. The mixed case 4.1. The case g(C)=5, |G|=16 4.2. The case g(C)=7, |G|=36 4.3. The case g(C)=9, |G|=64 5. Moduli spaces 6. Appendix References

0704.0447

Manipulating the rotational properties of a two-component Bose gas

Manipulating the rotational properties of a two-component Bose gas J. Christensson1, S. Bargi1, K. Kärkkäinen1, Y. Yu1, G. M. Kavoulakis1, M. Manninen2, and S. M. Reimann1 Mathematical Physics, Lund Institute of Technology, P.O. Box 118, SE-22100 Lund, Sweden Nanoscience Center, Department of Physics, FIN-40014 University of Jyväskylä, Finland (Dated: October 25, 2018) A rotating, two-component Bose-Einstein condensate is shown to exhibit vortices of multiple quantization, which are possible due to the interatomic interactions between the two species. Also, persistent currents are absent in this system. Finally, the order parameter has a very simple structure for a range of angular momenta. PACS numbers: 05.30.Jp, 03.75.Lm, 67.40.-w When a superfluid is set into rotation, it demonstrates many fascinating phenomena, such as quantized vortex states and persistent flow [1]. The studies of rotational properties of superfluids originated some decades ago, mostly in connection with liquid Helium, nuclei, and neu- tron stars. More recently, similar properties have also been studied extensively in cold gases of trapped atoms. Quantum gases of atoms provide an ideal system for studying multi-component superfluids. At first sight, the rotational properties of a multi-component gas may look like a trivial generalization of the case of a single compo- nent. However, as long as the different components inter- act and exchange angular momentum, the extra degrees of freedom associated with the motion of each species is not at all a trivial effect. On the contrary, this coupled system may demonstrate some very different phenomena, see, e.g., Refs. [2, 3, 4]. Several experimental and theo- retical studies have been performed on this problem, see, e.g., Refs. [5, 6, 7, 8, 9, 10, 11], as well as the review article of Ref. [12]. In this Letter, the rotational properties of a super- fluid that consists of two distinguishable components are examined. Three new and surprising conclusions result from our study: Firstly, under appropriate conditions, one may achieve vortex states of multiple quantization. It is important to note that these states result from the interaction be- tween the different species, and not from the functional form of the external confinement. It is well known from older studies of single-component gases, that any external potential that increases more rapidly than quadratically gives rise to vortex states of multiple quantization, for sufficiently weak interactions [13]; on the contrary, in a harmonic potential, the vortex states are always singly- quantized. In the present study, vortex states of multiple quantization result purely because of the interaction be- tween the different components, even in a harmonic ex- ternal potential. Therefore, our study may serve as an alternative way to achieve such states [14]. Secondly, our simulations indicate that multi- component gases do not support persistent currents, in agreement with older studies of homogeneous superflu- ids [3]. Essentially, the energy barrier that separates the (metastable) state with circulation/flow from the non- rotating state, is absent in this case, as the numerical results, as well as the intuitive arguments presented be- low, suggest. Finally, we investigate the structure of the lowest state of the gas, in the range of the total angular momentum L between zero and Nmin = min(NA, NB), where NA and NB are the populations of the two species labelled as A and B. In this range of L, only the single-particle states with m = 0 and m = 1 are macroscopically occupied, as derived in Ref. [15] within the approximation of the low- est Landau level of weak interactions. Remarkably, our numerical simulations within the mean-field approxima- tion, which go well beyond the limit of weak interactions, show that this result is more general. For simplicity we assume equal masses for the atoms of the two components, MA = MB = M . Also, we model the elastic collisions between the atoms by a con- tact potential, with equal scattering lengths for colli- sions between the same species and different species, aAA = aBB = aAB = a (except in Fig. 4). Our results are not sensitive to the above equality and hold even if aAA ≈ aBB ≈ aAB, as in Rubidium, for example. For the atom populations we assume NA 6= NB, but NA/NB <∼ 1 (without loss of generality). The trapping potential is assumed to be harmonic, Vext(r) = M(ω 2ρ2 + ω2zz 2)/2. Our Hamiltonian is thus NA+NB + Vext(ri) + NA+NB i6=j=1 δ(ri − rj),(1) where U0 = 4πh̄ 2a/M . We consider rotation around the z axis, and also assume that h̄ωz ≫ h̄ω, and h̄ωz ≫ n0U0, where n0 is the typical atom density. With these assump- tions, our problem becomes effectively two-dimensional, as the atoms reside in the lowest harmonic oscillator state along the axis of rotation. Thus, there are only two quan- tum numbers that characterize the motion of the atoms, the number of radial nodes n, and the quantum number m associated with the angular momentum. The corre- sponding eigenstates of the harmonic oscillator in two dimensions are labelled as Φn,m. Within the mean-field approximation, the energy of the gas in the rest frame is i=A,B h̄2∇2 + Vext(r) http://arxiv.org/abs/0704.0447v1 (|ΨA| 4 + |ΨB| 4 + 2|ΨA| 2|ΨB| 2) d3r, (2) where ΨA and ΨB are the order parameters of the two components. By considering variations in Ψ∗A and Ψ we get the two coupled Gross-Pitaevskii-like equations, h̄2∇2 + Vext + U0|ΨB| ΨA + U0|ΨA| 2ΨA = µAΨA, h̄2∇2 + Vext + U0|ΨA| ΨB + U0|ΨB| 2ΨB = µBΨB, where µA and µB is the chemical potential of each com- ponent. We use the method of relaxation [16] to minimize the energy of Eq. (2) in the rotating frame, E′ = E−LΩ, where Ω is its angular velocity. For the diagonalization of the many-body Hamilto- nian, we further assume weak interactions, n0U0 ≪ h̄ω, and work within the subspace of the states of the lowest Landau level, with n = 0. This condition is not neces- sary, however it allows us to consider a relatively larger number of atoms and higher values of the angular mo- mentum. We consider all the Fock states which are eigen- states of the number operators N̂A, N̂B of each species, and of the operator of the total angular momentum L̂, and diagonalize the resulting matrix. Combination of the mean-field approximation and of numerical diagonalization of the many-body Hamiltonian allows us to examine both limits of weak as well as strong interactions. For obvious reasons we use the diagonal- ization in the limit of weak interactions, and the mean- field approximation (mostly) in the limit of strong in- teractions. The interaction energy is measured in units of v0 = U0 |Φ0,0(x, y)| 4|φ0(z)| 4 d3r = (2/π)1/2h̄ωa/az, where φ0(z) is the lowest state of the oscillator potential along the z axis, and az = (h̄/Mωz) 1/2 is the oscilla- tor length along this axis. For convenience we introduce the dimensionless constant γ = Nv0/h̄ω = 2/πNa/az, with N = NA + NB being the total number of atoms, which measures the strength of the interaction. We first study the limit of weak coupling, γ ≪ 1, and use numerical diagonalization. Considering NA = 4 and NB = 16 atoms, we use the conditional probability dis- tributions to plot the density of the two components, for L = 4, 16, 28, and 32, as shown in Fig. 1. When L = 4 = NA, and L = 16 = NB, the component whose population is equal to L forms a vortex state at the center of the trap, while the other component does not rotate, residing in the core of the vortex. This is a so-called “coreless” vortex state [5, 6, 17]. As L increases beyond L = NB = 16, a second vortex enters component B, and for L = 2NB = 32, this merges with the other vortex to form a doubly-quantized vortex state. For this value of L = 32, the smaller component A does not carry any an- gular momentum (apart from corrections of order 1/N). The fact that this is indeed a doubly-quantized vortex state, is confirmed by the occupancy of the single-particle FIG. 1: The conditional probability distribution of the two components, with NA = 4 (higher row), and NB = 16 (lower row). Each graph extends between −2.4a0 and 2.4a0 in both directions. The reference point is located at (x, y) = (1.25a0, 0) in the lower graphs (B component). The angular momentum L increases from left to right, L = 4(= NA), 16(= NB), 28, and 32(= 2NB). states. By increasing NA, NB, and L = 2NB proportion- ally, we observe that the occupancy of the single-particle state withm = 2 of componentB approaches unity, while the occupancy of all the other states are at most of order 1/NB. The same happens for the single-particle state with m = 0 of the non-rotating component A. A similar situation emerges for the case of stronger cou- pling, γ = 50, where we have minimized the mean-field energy of Eq. (2) in the rotating frame (in the absence of rotation the two clouds do not phase separate). For ex- ample, we get convergent solutions, shown in Fig. 2, for NB/NA = 2.777, and (i): LA = NA, LB = 0, for Ω/ω = 0.35 (top left), (ii) LA = 0, LB = NB, for Ω/ω = 0.45 (top middle), (iii) LA = 0.755NA, LB = 1.171NB, for Ω/ω = 0.555 (top right), (iv) LA = 0, LB = 2NB, for Ω/ω = 0.60 (bottom left), (v) LA = 0.876NA, LB = 2.057NB, for Ω/ω = 0.69 (bottom middle), and (vii) LA = 0, LB = 3NB, for Ω/ω = 0.73 (bottom right). Here, LA and LB is the angular momentum of each com- ponent, with L = LA + LB. Again, when L = 2NB, and L = 3NB the phase plots show clearly a doubly-quantized and a triply-quantized vortex state in component B, and a non-rotating cloud in component A. The picture that appears from these calculations is intriguing: as Ω increases, a multiply-quantized vortex state of multiplicity κ splits into κ singly-quantized ones, and on the same time, one more singly-quantized vortex state enters the cloud from infinity. Eventually all these vortices merge into a multiply-quantized one of multiplic- ity equal to κ + 1. Figure 2 shows the above results for various values of Ω. The mechanical stability of states which involve the gradual entry of the vortices from the periphery of the cloud is novel. This behavior is absent in one-component systems, in both harmonic, as well as anharmonic trap- ping potentials. In one component gases, only vortex phases of given rotational symmetry are mechanically stable [18, 19]. In the present problem, the mechanical stability of states with no rotational symmetry (shown in Fig. 2) is a consequence of the non-negative curva- ture of the dispersion relation (i.e., of the total energy) FIG. 2: The density (higher graphs of each panel) and the phase (lower graphs of each panel) of the order parameters ΨA (left graphs of each panel) and ΨB (right graphs of each panel), with NB/NA = 2.777 and a coupling γ = 50. Each graph extends between −4.41a0 and 4.41a0 in both directions. The values of the angular momentum per atom and of Ω in each panel are given in the text. E(L). This observation also connects with the (absence of) metastable, persistent currents (i.e., the second main result of our study), which we present below. In Ref. [15] we have given a simple argument for the presence of vortex states of multiple quantization within the mean-field approximation. At least when the ra- tio between NA and NB is of order unity (but NA 6= NB), there are self-consistent solutions of Eqs. (3) of vor- tex states of multiple quantization. Within these solu- tions, the smaller component (say component A), does not rotate, providing an “effective” external potential Veff,B(r) = Vext(r) + U0nA(r) for the other one (compo- nent B), which is anharmonic close to the center of the trap. This effectively anharmonic potential is responsible for the multiple quantization of the vortex states. There- fore, we conclude that for a relatively small population imbalance, the “coreless vortices” are vortices of multiple quantization. The second aspect of our study is the absence of metastable currents (in the laboratory frame, for Ω = 0). A convenient and physically-transparent way to think about persistent currents is that they correspond to metastable minima in the dispersion relation E = E(L) [20]. A non-negative curvature of E(L) for all values of L implies the absence of metastability. For all the cou- plings we have examined, both within the numerical diag- onalization, as well as the mean-field approximation, we have found a non-negative second derivative of the dis- persion relation. Figure 3 shows LA/NA, LB/NB, and L/N versus Ω, for γ = 50. These curves are calculated by minimizing the energy E(L) in the rotating frame for a fixed Ω, and plotting the angular momentum per par- ticle of the corresponding state for the given rotational frequency. Again, our argument for the effective anharmonic po- tential is consistent with this positive curvature. Let us 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 FIG. 3: Higher graph: The angular momentum LA/NA (crosses) and LB/NB (dots), as function of Ω. Lower graph: L/N as function of Ω. All curves result from the minimiza- tion of the energy in the rotating frame, within the mean-field approximation, for γ = 50. consider for simplicity aAA = aBB = 0, and aAB 6= 0. Then, the problem of solving Eqs. (3) becomes essen- tially a (coupled) eigenvalue problem. If E0,m are the (lowest) eigenvalues of the effective (anharmonic) po- tential felt by the rotating component for a given an- gular momentum mh̄, then ∂2E0,m/∂m 2 is always posi- tive. For example, if one considers a weakly-anharmonic effective potential, Veff(ρ) = Mω 2ρ2[1 + λ(ρ/a0) 2k]/2, where k = 1, 2, . . . is a positive integer, a0 = (h̄/Mω) is the oscillator length, and 0 < λ ≪ 1 is a small di- mensionless constant, according to perturbation theory, E0,m = h̄ω|m|+ λ(|m|+ 1) . . . (|m|+ k)/2, which clearly has a positive curvature. One may gain some physical insight into the absence of persistent currents by understanding the difference be- tween a gas with one and two components. In the case of a single component, for sufficiently strong (and repul- sive) interactions, an energy barrier that separates the state with circulation from the vortex-free state may de- velop. In the simplest model where the atoms rotate in a toroidal trap, in order for them to get rid of the circu- lation, they have to form a node in their density, which costs interaction energy, and this creates the energy bar- rier [20, 21]. On the other hand, in the presence of a second component, this node may be filled with atoms of the other species, and therefore the system may get rid of the circulation with no energy expense. This physical picture is also supported by the density plots in Figs. 2 and 4. For example, in the case of coreless vortices, the core of the vortex is filled by the other (non-rotating) component [12]. More generally, the density minima of FIG. 4: The density (higher graphs) and the phase (lower graphs) of the order parameters ΨA (left graphs of each panel) and ΨB (right graphs of each panel), with NB/NA = 2.777. Here Ω/ω = 0.6 and γ = 50. In the left panel LA = 0, and LB/NB = 2. In the right panel, the scattering length aBB is twice as large as in the left panel, aBB = 2a. In this case, LA/NA = 0.05, and LB/NB = 1.936. All graphs extend between −4.41a0 and 4.41a0. the one component coincide, roughly speaking, with the density maxima of the other component, resulting in a total density ntot = |ΨA| 2 + |ΨB| 2 which does not have any local minima or nodes. Our third result is based on the mean-field approxima- tion. For 0 ≤ L ≤ Nmin, where Nmin = min(NA, NB), the only components of the order parameters ΨA and ΨB are the single-particle states with m = 0 and m = 1, i.e., cn,0Φn,0 + cn,1Φn,1, dn,0Φn,0 + dn,1Φn,1, (4) where cn,0, cn,1, dn,0 and dn,1 are functions of L and of the coupling. The numerical simulations that we perform within a range of couplings γ ≤ 50 that extend well be- yond the lowest-Landau level approximation, reveal this very simple structure for the lowest state of both com- ponents. Also, the corresponding dispersion relation is numerically very close to a parabola, as in the case of weak interactions [15]. Again, one may attribute these facts to the effective potential that arises from the inter- action between the two species [15]. In the studies that have examined a single-component gas in an external anharmonic potential, it has been shown that as the strength of the interaction increases, there is a phase transition from the phase of multiple quantization to the phase of single quantization [19]. In the present case the situation is more complex, since the effective anharmonic potential is generated by the in- teraction between the two species as a result of a self- consistent solution. Still, a similar phase transition takes place here, as, for example, one keeps the scattering lengths aAA and aAB fixed, and increases aBB that cor- responds to the rotating component. Figure 4 shows the density and the phase of both species, for aAA = aBB = aAB = a (left panel), and aBB = 2aAA = 2aAB = 2a (right panel). Component B undergoes a phase transi- tion from a doubly-quantized vortex state to two singly- quantized vortices. To conclude, mixtures of bosons demonstrate numer- ous novel superfluid properties and provide a model sys- tem for studying them. Here we have given a flavor of the richness of this problem. Many of the results pre- sented in our study are worth investigating further, as, for example, one changes the ratio of the populations, the coupling constant between the same and different species, or the masses. We acknowledge financial support from the Euro- pean Community project ULTRA-1D (NMP4-CT-2003- 505457), the Swedish Research Council, the Swedish Foundation for Strategic Research, and the NordForsk Nordic Network on “Low-dimensional physics”. [1] A. J. Leggett, Rev. Mod. Phys. 71, S318 (1999). [2] N. D. Mermin and T.-L. Ho, Phys. Rev. Lett. 36, 594 (1976). [3] P. Bhattacharyya, T.-L. Ho, and N. D. Mermin, Phys. Rev. Lett. 39, 1290 (1977). [4] T.-L. Ho, Phys. Rev. Lett. 49, 1837 (1982). [5] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. 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Krotscheck, Phys. Rev. E 72 036705 (2005). [17] R. Blaauwgeers, V. B. Eltsov, M. Krusius, J. J. Ruohio, R. Schanen, and G. E.Volovik, Nature (London) 404, 471 (2000). [18] D. A. Butts, and D. S. Rokhsar, Nature (London) 397, 327 (1999). [19] See, e.g., G. M. Kavoulakis and G. Baym, New Journal of Phys. 5, 51.1 (2003), and references therein. [20] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). [21] D. Rokhsar, e-print cond-mat/9709212. http://arxiv.org/abs/cond-mat/9709212

0704.0448

Giant Planet Migration in Viscous Power-Law Discs

Giant Planet Migration in Viscous Power-Law Discs Richard .G. Edgar Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 [email protected] ABSTRACT Many extra-solar planets discovered over the past decade are gas giants in tight orbits around their host stars. Due to the difficulties of forming these ‘hot Jupiters’ in situ, they are generally assumed to have migrated to their present orbits through interactions with their nascent discs. In this paper, we present a systematic study of giant planet migration in power law discs. We find that the planetary migration rate is proportional to the disc surface density. This is inconsistent with the assumption that the migration rate is simply the viscous drift speed of the disc. However, this result can be obtained by balancing the angular momentum of the planet with the viscous torque in the disc. We have verified that this result is not affected by adjusting the resolution of the grid, the smoothing length used, or the time at which the planet is released to migrate. Subject headings: hydrodynamics – planetary systems: protoplanetary disks – planetary systems: formation 1. Introduction The past decade has seen the discovery of over one hundred extra-solar planets. Most of these planets (see, e.g. Marcy et al. 2005) are gas giants of Jupiter mass or greater, in orbits close to their host stars.1 Such planets are often called ‘hot Jupiters.’ Although detection methods (primarily radial velocity measurements) are strongly biased towards detecting such planets, a sufficient number have been discovered to require detailed explanation. The present orbits of the ‘hot Jupiters’ lie so close to their host stars that in situ forma- tion is almost impossible. The problem is two-fold. Firstly, the expected disc temperature is high, which suppresses gravitational capture of gas by a gas giant core. Secondly, there 1For the most up to date information, refer to http://exoplanet.eu/ and http://exoplanets.org/ http://arxiv.org/abs/0704.0448v1 http://exoplanet.eu/ http://exoplanets.org/ – 2 – simply isn’t much material at such small orbital radii. We are therefore led to the conclusion that the ‘hot Jupiters’ formed further from the star, and migrated to their present orbits. In this paper, we present a study of giant planet migration in power law discs. A planet in a circumstellar disc exerts a torque on the disc material at resonances (Goldreich and Tremaine 1978, 1979, 1980). By Newton’s Third Law, there is an opposite torque on the planet, which causes it to migrate. Planets of mass comparable to Jupiter (the exact threshold depends on the disc conditions, see below) exert torques comparable to the viscous torque in the disc. They can therefore open a gap in the disc. If the gap is perfectly clean, then the planet will act as a ‘relay station’ between the inner and outer discs. We then expect the planet’s orbital evolution to follow the viscous evolution of the disc – a process known as Type II migration (Ward and Hahn 2000). We shall compare our computational results to the predictions of Type II theory. We summarise the theory of Type II migration in Section 2. We describe our numerical method in Section 3, and demonstrate the formation of a gap in Section 4. Section 5 presents our main results. In Section 6 we discuss the implications of our findings, before presenting our conclusions in Section 7. 2. Type II Migration In this section, we shall briefly review the theory of Type II migration. For a more thorough analysis of the theory of planet–disc interactions, the reader should turn to one of the many published reviews (e.g. Lin and Papaloizou 1993; Ward and Hahn 2000; Lin et al. 2000). Planet–disc interactions are dominated by resonances. A planet embedded in a cir- cumstellar disc excites waves at its Lindblad resonances (LR). These waves carry angular momentum, and hence exert a torque on the disc material. The strength of the torque from the Lindblad resonances is proportional to TLR ∝ Σq 2, where Σ is the local disc surface density, and q is the planet-star mass ratio. These torques act to push material away from the planet. At the same time, the disc gas is expected to have an intrinsic viscosity, ν (al- though the precise origin and exact behaviour of such a viscosity are still much debated), which leads to a viscous torque Tν ∝ Σν. Since the Lindblad torques scale with q2, we can expect that as the planet accretes material, the Lindblad torques will eventually dominate the viscous torque in the disc. Balancing the two torques leads to the so-called gap opening criterion: q > 40R−1 (1) – 3 – where R = r2Ω/ν is the Reynolds number of the disc (Bryden et al. 1999; Nelson et al. 2000). There is a similar requirement that the planet’s Hill sphere exceed the local scale height of the disc, namely that q > 3 where h is the disc scale height. See Lin and Papaloizou (1993) for a further discussion. The tidal condition of Equation 2 leads to the gap width being at least twice the local scale height. If this condition is not satisfied, then the edge of the gap will be Rayleigh unstable. Consideration of the torque condition leads to the expectation that the gap will lie between the m = 2 Lindblad resonances of the planet (that is, between r = 0.6 and r = 1.3). For Jovian mass planets in circumstellar discs, these gap sizes are similar. Once the planet has opened a gap, it is assumed to isolate the inner and outer discs from each other. However, each part of the disc will still be undergoing viscous evolution. According to Type II migration theory, the planet will be locked to the viscous evolution of the disc (Ward and Hahn 2000). Setting the migration rate of the planet equal to the radial drift velocity of the gas in a thin accretion disc (see e.g. Pringle 1981), we find: ȧ = − so long as the disc is sufficiently massive. Note that Equation 3 is independent of the disc surface density. If the disc mass is too low, then the torques (also proportional to Σ above) will not be able to force the planet to migrate at this rate. Syer and Clarke (1995) and Ivanov et al. (1999) have examined this limit. Ida and Lin (2004) suggest an alternative prescription for determining the Type II mi- gration rate. They balance the angular momentum change of the planet, 1 MpΩpaȧ with the maximum viscous couple in the disc J̇ = 3 r2. Since we are using power law discs here, J̇ will not have a maximum, so we use the nominal value at the planet’s orbit. This leads to the prediction ȧ = −3 Since this equation was obtained by balancing angular momentum, we might expect it to be valid for a full range of disc masses. Surprisingly, giant planet migration does not seem to have been subjected to a system- atic test (this is in sharp contrast to the theory of Type I migration, which applies to low mass planets). Some curiosities in the behaviour of migrating giant planets have been seen (e.g. Schäfer et al. 2004), but these have not been explored in detail. The work of Nelson et al. – 4 – (2000) provides the most complete set of runs to date, but the physical parameters were not varied in a regular fashion. In this paper, we shall present a series of numerical experiments, following giant planets migrating in a variety of accretion discs. We will then compare our results to equations 3 and 4. Equation 3 makes particularly strong predictions about the expected migration rates - namely that the migration rate depends solely on the disc viscosity. This is the first time such a test has been performed. 3. Numerical Set up We use the Fargo code of Masset (2000a,b) to perform our calculations. Fargo is a simple 2d polar mesh code dedicated to disc planet interactions. It is based upon a standard, Zeus-like (Stone and Norman 1992) hydrodynamic solver, but owes its name to the Fargo algorithm upon which the azimuthal advection is based. This algorithm avoids the restrictive timestep typically imposed by the rapidly rotating inner regions of the disc, by permitting each annulus of cells to rotate at its local Keplerian velocity and stitching the results together again at the end of the timestep. The use of the Fargo algorithm typically lifts the timestep by an order of magnitude, and therefore speeds up the calculation accordingly. The mesh centre lies at the central star, so indirect terms coming from the planets and the disc are included in the potential calculation. We make use of an non-reflecting inner boundary, to prevent reflected waves from interfering with the calculations. The pitch angle of the wake is evaluated in the WKB approximation. The inner ring of active cells is then copied to the ghost cells, with an azimuthal shift appropriate to the pitch angle. Material which flows off the inner boundary is not added to the star (nor does the planet itself accrete). At the outer boundary, mass was added, to compensate for the viscous evolution of the system. We use units normalised such that G = M +Mp = 1, while the planet’s initial orbital radius is set at a = 1. References to times in terms of ‘orbits’ should be understood to mean “orbital times at the planet’s initial radius.” The grid extends between r = 0.4 and r = 2.5. Scaled to Jupiter’s orbit, this grid roughly covers the area between the asteroid belt and Saturn’s orbit. We assume a constant aspect ratio disc, with h/r = 0.05. We set the mass ratio q = Mp/M∗ to be 10 −3, approximately equal to the Jupiter-Sun value. In our parameter space search, we varied the disc surface density profile and viscosity. The surface density was initially a power law: Σ(r) = Σ0 and we take r0 = 1 (the planet’s initial orbital radius). Four values for δ are considered: – 5 – 0, 0.5, 1 and 1.5. These represent a reasonable range of alternatives, from a theoretically simple constant surface density disc to the canonical Minimum Mass Solar Nebula. We set Σ0 through qdisc = which provides a quick estimate of the disc’s mass within the planet’s orbit. This estimate is accurate for δ = 0 discs, but is an underestimate for larger δ values. We take four values for q : 5×10−4, 10−3, 2×10−3, and 3×10−3. The total disc mass lies between 1.9×10−3 (for q = 5× 10−4 and δ = 1.5) and 0.018 (for q = 3× 10−3 and δ = 0) in units of the stellar mass. By way of comparison, the Minimum Mass Solar Nebula (MMSN) requires at least 5MJ of gas in the vicinity of Jupiter’s orbit. 2 Thus, our lower mass discs are somewhat sub-Minimum. The viscosity is taken to be uniform, and has values ν = 10−4 and 10−5 in our units. With a uniform viscosity, δ = 0.5 yields a disc with an initially stationary surface density pro- file (cf equation 2.10 of Pringle 1981). These viscosities may be related to the α prescription for viscosity, ν = αcsh using ν = α Ωr2 (7) This implies that α varies with radius. With our aspect ratio, a viscosity of ν = 10−5 gives α ≈ 4 × 10−3 at the planet’s initial orbital radius. Note that for the highest viscosity, the gap opening criterion of Equation 1 is not satisfied. The tidal condition of Equation 2 is always satisfied in our numerical experiments. The gravitational effect of the planet on the disc is smoothed at 0.6 of the disc thickness at the planet’s orbital radius: φ = − r2 + ǫ2 where ǫ = 0.6h. There are two motivations for this, the first being the desire to avoid having a singularity wandering around the grid. The second is physical. The 2d approximation becomes poor close to the planet, where the vertical distribution of material becomes im- portant. The actual distance of material from the planet ceases to be the well approximated by the in-plane distance, which would lead to the gravitational effect being over-estimated. Accordingly, we soften the potential over distances comparable to the disc scale height. How- ever, this softening length is still substantially smaller than the expected gap size and the 2We calculate this by comparing Jupiter’s metal content to that of the Sun, and assuming that both condensed from the same gas cloud. Scaled to the Solar System, our grid roughly covers the region between the asteroid belt and Saturn – 6 – planet’s Hill sphere. When calculating the torque the disc exerts on the planet, material from within the Hill sphere is subject to an exponential cut off, for similar reasons. At the start of each run, the planet is introduced gradually (over about one orbit), and is not initially permitted to migrate. This is done to minimise the effect of transients caused by the sudden appearance of a planet in a smooth disc. We considered release times of approximately one orbit, and 100 & 1000 orbits. The computational grid is covered by 128 radial and 384 azimuthal cells (all uniformly spaced). So far as possible, this setup mirrors that used in the comparison project presented by de Val-Borro et al. (2006). In that comparison, the Fargo code was seen to give similar results to other codes used to study the disc–planet interaction problem. 4. Development of the Gap Since we introduce the planet into an initially unperturbed disc, there is a period of rapid evolution, as the planet clears a gap. In this section, we shall discuss the development of this gap. In Figure 1, we trace the evolution of the gap in a ν = 10−5 disc. The initial surface density profile had δ = 0.5 (cf equation 5). There are no surprises in this plot, when compared with the many other numerical calculations of gap formation. We see that the gap is mostly cleared in the first 100 orbits (note that the y-axis is logarithmic). The gap lies roughly between the m = 2 Lindblad resonances (located at r = 0.6 and r = 1.3). For a q = 10−3 planet, this distance is also comparable to the co-rotation region. This plot draws attention to the fact that the planet never completely clears the gap. Even after 1000 orbits, the surface density in the gap is around 3% of its initial value. The gap edge is covered by roughly ten grid cells. If we increase the viscosity to ν = 10−4, the gap becomes far less pronounced. We show the development of the gap in this case in Figure 2. Again, most of the depletion occurs during the first 100 orbits, but the total depletion is far less. The density has only dropped to around 50% of its initial value. This is not unexpected - according to Equation 1, a Jupiter mass planet should not be able to open a gap in such a viscous disc. Of course, the condition of Equation 2 is satisfied. Figures 1 and 2 both show that most of the gap clearing occurs withing the first 100 orbits. We shall therefore use this as our canonical release time below. However, we shall show the effect of varying the release time as well. – 7 – Gaps in protoplanetary discs are known to vary smoothly with q and ν (see, e.g. astro-ph/0608020), so what is taken to be a gap is somewhat arbitrary. We shall con- tinue with both viscosity values, but we must bear in mind that in the high viscosity case the gap is quite shallow. 5. Results We will now present the results of our numerical experiments of a Jupiter mass planet migrating in power law discs, grouped by viscosity. Such planets are conventionally assumed to undergo Type II migration. We have two predictions for the migration rate of giant planets in Equations 3 and 4. We shall compare our results to these predictions. 5.1. High Viscosity The higher viscosity runs had ν = 10−4. This viscosity means that for a Jupiter mass planet, the viscous gap opening criterion q > 40R−1 of Equation 1 is not quite satisfied. However, the tidal condition of Equation 2 is fulfilled. We shall discuss the results from the runs where the planet was released after 100 orbits. In Appendix A, we demonstrate that the release time is not significant. The orbital evolution of these planets is plotted in Figure 3. We cut the y-axis at 0.6, since at that point the m = 2 ILR of the planet encounters the edge of the grid. The migration rate of the planet therefore becomes unreliable. We can see that the migration rate is a strong function (or equivalently, Σ0). This is in direct contradiction to the prediction of Equation 3. Notice also how the migration rate varies with a. Equation 3 predicts that ȧ ∝ a−1, which we do not see. We see that the migration rate generally falls with a, which is consistent with equation 4. Figure 3 shows that the reduction of ȧ with a falls as δ increases (that is, there is a pronounced curve in the migrations for δ = 0, whereas those for δ = 1.5 are almost straight lines). This is broadly consistent with equation 4, were the migration rate is proportional to ȧ ∝ aΣ ≡ a1−δ. Complicating this is the viscous evolution of the disc itself, which is probably the reason why ȧ is not strictly proportional to a1−δ (which would predict accelerating migration for the δ = 1.5 case). We have seen that the migration rate of a Jupiter mass planet in these discs is strongly affected by the disc surface density. However, since the gap is not particularly deep, whether Type II behaviour should be expected is debatable. – 8 – 5.2. Medium Viscosity Here, we examine the results from runs with ν = 10−5. Again, we shall examine the case where the planet was released after 100 orbits first. In Figure 4, we show the orbital migration for planets embedded in a variety of discs. We see that planets embedded in higher surface density discs (parameterised by q Equation 6) consistently undergo faster migration. The migration rate is roughly propor- tional to the disc surface density. Note also the variation of ȧ with a. It is similar to that seen for figure 3 above. Eccentricities again remained low. 5.3. Summary of results In this section, we have presented a series of giant planet migration runs. Such planets have generally been thought to undergo Type II migration. One formulation of the theory predicts that the migration rate depends solely on the disc viscosity (Equation 3). However, we have found that the migration rates vary systematically with disc surface density. Higher disc surface densities give faster migration, which is consistent with Equation 4. There is a weaker variation with disc viscosity, which is inconsistent with both predictions. In Appendix A, we show that our conclusions are not affected by varying the time at which the planet was released to migrate. We demonstrate that the grid resolution does not affect our results in Appendix B. 6. Discussion In Section 5, we presented a series of runs designed to study how giant planets migrate. This migration of such planets is thought to be controlled by the viscous torque within the disc. Two different rates have been suggested. In the first (Equation 3), the planet is locked to the viscous evolution of the disc, and the migration rate depends solely on the disc viscosity. The second (Equation 4) computes an angular momentum balance between the planet and disc. In this theory, the migration rate also depends on the disc surface density and planet mass. We have seen that the migration rate we obtain varies strongly with disc surface density, indicating that Equation 3 in not appropriate. Although Equation 4 is more promising, we do not recover the same variation with viscosity. The higher viscosity runs underwent more rapid migration, but the difference in migration rates was not an order of magnitude. – 9 – However, this result is not as robust as the variation with surface density, since the high viscosity runs did not satify both gap opening criteria. Although Figure 2 shows that the high viscosity runs are definitely in the non-linear regime, the gap itself was not especially clean. We have shown (Appendix A) that our results are not simply ‘turn-on’ transients. The migration rates are not significantly affected by the time at which the planet is released from a fixed orbit. Our resolution tests (Appendix B) demonstrate that our results are not significantly affected by a doubling of the grid resolution. Our neglect of material within the Hill sphere when calculating the torque is a point of concern. D’Angelo et al. (2003) noted that most of the torque in their calculations came from within the Hill sphere.3 However, the theory of Type II migration takes no account of this material either. It is a simple 2d theory, which assumes that the planet is merely acting as a ‘relay station’ for the disc’s viscous torques. If the flow structure within the Hill sphere is of critical importance, then we should not expect giant planet migration to be as simple as Section 2 suggests. The accretion behaviour of the planet could also affect migration. This is directly linked to the previous point about flow within the Hill sphere. Kley (1999) showed that even in the presence of a gap, a planet could continue to accrete material from the disc.4 Similar results were reported by Lubow et al. (1999) and Kley et al. (2001). In this paper, we did not allow the planet to accrete, and this caused material to build up around the planet. Since we attentuated the torque from within the Hill sphere, this would not have affected our results directly. However, if accretion were allowed, then the planet could gain an appreciable amount of mass. This would both alter the gap structure, and make it more difficult for the disc to move the planet (due to the planet’s increased inertia). Related to this issue is the recent finding of Lubow and D’Angelo (2006) that the accretion rate through the gap could be over 10% of the viscous accretion rate in the main disc, despite the drop in gas density. Finally, there is the matter of viscosity. In our numerical experiments, we used a physi- cal viscosity in the Navier-Stokes equations. In reality, the ‘viscosity’ in protoplanetary discs probably originates fromMHD turbulence, and calculations have shown (Winters et al. 2003) that the gap structure obtained in an MHD calculation differs from that in a purely hydro- dynamic one. In particular, the gaps tend to be wider and shallower. If material in the 3However, they did not perform a parameter space search like we have done here 4Note that this finding in itself implicitly contradicted the usual assumption that the planet isolates the inner and outer discs – 10 – corotation region is important to determining the migration rate, then this alteration in gap structure will cause further changes to the migration rates. Nelson (2005) has already demonstrated that a magnetic turbulence strongly affects the migration of a low mass planet. Although our computations do not include MHD turbulence, the theory of Type II migration neglects it too, so this cannot be the reason for the differences we have observed. When might we expect migration to proceed according to equation 3? We believe this might be possible for a planet of moderate mass, in a cold, very low viscosity disc, which is more massive than the planet. Our reasoning is as follows: equation 3 is based on the assumption that the planet completely isolates the inner and outer discs. This is easiest to achieve in a very low viscosity disc (cf equation 1), which is also cold (cf equation 2). We also require the disc to be more massive than the planet, to ensure sufficient angular momentum reserves are available. The gap will lie roughly between the m = 2 Lindblad resonances, and we would want these particular resonances to be responsible for most of the gap clearing (i.e. the m = 2 resonances themselves must dominate the disc’s viscous torque). This is because we would need disc material to be kept well away from the corotation region of the planet. Masset and Papaloizou (2003) showed that corotation torques can give rise to extremely rapid migration - known as ‘runaway’ or Type III migration. A planet less massive than Jupiter will have its corotation region inside its m = 2 Lindblad resonances. However, Masset and Papaloizou found that such planets tended to undergo runaway migration. This reinforces the need for the disc itself to have a very low viscosity, so that the gap is as clean as possible. 6.1. Origin of the Torque We shall now discuss the radial origin of the torque. In figure 5, we show the torque profiles, Tz(r) acting on the planet after 100, 500 and 1000 orbits. The disc viscosity was ν = 10−5, and the surface density profile was initially flat (δ = 0). The planet was held on a fixed orbit for the entire calculation. In computing the torques, the same exponential cut off used with Fargo was applied. We see that most of the torque is generated within the range 0.8 < r < 1.2, and that the torques from the inner and outer discs have opposite signs. At later times, the torque peaks on either side of the planet lessen. This is due to the gap emptying further. The outer peak (which is pushing the planet inwards) also broadens, while the inner peak does not. This ultimately ensures that the planet migrates inwards. Figure 6 shows how the torque felt by the planet scales with q ∝ Σ0. These curves – 11 – are plotted for a δ = 0, ν = 10−5 disc after 1000 orbits. As we might expect from section 5, we see that the strength of the torques is directly proportional to the value of q . This leads to the migration rate varying strongly with the surface density of the disc. Finally, in figure 7, we show the effect of varying the initial disc power law, δ, on the torque profiles. Again, the torque profiles are for a ν = 10−5 disc, and are plotted after 1000 orbits of the (fixed) planet. We see that the torques are very similar, regardless of the initial δ value, indicating that the perturbations induced by the planet are not dependent on the background structure of the disc. Again, this is expected from section 5. Figures 5, 6 and 7 all show that the torque generation peaks at radii of r = 0.9 and r = 1.1 (roughly 1.5 Hill radii from the planet). Comparing to figure 1, we see that these locations lie deep within the gap, which is interesting for a number of reasons. The peaks are close to the cutoff radius generally applied to obtain the numerical factor in equation 1, namely the planet’s Hill sphere. They are also well within the corotation region, raising the possibility that corotation torques are affecting the orbital evolution of the planet. Unfortunately, at this resolution, the Hill radius is only covered by four or five grid cells, and the torque peak only lies seven grid cells from the planet itself. The smoothing lengths are also comparable to these distances. Our resolution tests (Appendix B) show that our resolution is adequate for the smooth- ing lengths used. However, with so much torque being generated close to the planet, it is likely that the smoothing is significantly affecting the torque. Reduction of the smoothing lengths is obviously desirable, but unfortunately not possible in a two dimensional calcula- tion. As noted in section 3, we must smooth the planet’s gravity at about the local scale height of the disc in order to make a 2d calculation valid. In Appendix C, we show the effect of reducing the exclusion radius for the calculation of the migration torque. The effect appears to be minimal, but these results should be treated with some caution, due to the issue noted above. With the structure of the flow close to the planet so obviously critical to determining the migration rate, a full 3d calculation would be required to determine a robust migration rate. However, we would be most surprised if such calculations undermined our main conclusion that migration rates of massive planets are proportional to the disc mass. 7. Conclusion In this paper, we have performed a systematic test of giant (Jupiter mass) planet migra- tion. The migration rates we obtained varied strongly with the initial disc surface density, and less strongly with the disc viscosity. We have shown that the simplest theory of Type – 12 – II migration, where the planet is locked to the viscous evolution of the disc (Equation 3), is incorrect. An alternative formulation, based on an angular momentum balance (Equation 4), looks more promising. However, we have not tested this second theory fully. We verified that our results were not simply ‘turn-on’ transients, or purely the effect of low resolution. Neither doubling the grid resolution, nor allowing the planet to clear its gap for 1000 orbits affected our central finding. Separate confirmation of our results, using a different code would be highly desir- able. Although we have no reason to believe that Fargo is misbehaving, the work of de Val-Borro et al. (2006) underlines how codes can give varying results, even for the ‘same’ physical scenario (it is for this reason that ‘simulations’ should properly be referred to as ‘numerical experiments’). The issues of accretion and gravitational softening (both of the planet’s effect on the disc, and the torque exerted on the planet) also merit closer consider- ation. Indeed, if the gap shape and flow through the gap are critical for migration, one is led to wonder if 2d calculations are sufficient. Two dimensional calculations have generally been thought adequate for Jovian mass planets because the gap would keep material away from the planet (where the 3d nature of the flow will become evident). If the flow within the Hill sphere is important, then 2d calculations cease to be convincing. A. Effect of Varying Release Time In this Appendix, we shall demonstrate that varying the time at which the planet is released does not affect our central conclusion. We start by showing that changing the release time only affects the migration rates slightly. We then show that, since the effect is consistent for all discs used, the conclusion that the planet migration rate varies with disc surface density is robust. In Figure 8, we show the effect of varying the release time on a planet in a high viscosity (ν = 10−4) disc with q = 0.002 and δ = 0.5. The effect is fairly minimal, and this plot is typical. The explanation for this lies in Figure 2, which shows that only a minimal gap is formed. Indeed, one can debate whether the surface density depression is a gap, since the usual criterion (Equation 1) is not satisfied. Figure 9 shows the effect of release time on the orbital migration of a planet in the ν = 10−5 case. The particular disc used in this comparison had q = 0.002 and δ = 0, but the behaviour was generically the same for all cases. We see that the time at which the planet is released does have an effect on the migration rate. However, the effect is not especially dramatic. – 13 – We demonstrate that the release time does not affect our central conclusions in Figure 10. This duplicates Figure 4, but with the release time increased to 1000 orbits (note that the x axis has the zero point shifted, to improve the use of space). Although the exact migration rates undoubtedly change, the main conclusion that these planets are not undergoing Type II migration is unaffected. The migration rates continue to be affected by the disc surface density. B. The Effect of Resolution In this appendix, we shall study the effect of increasing the grid resolution on our results. We re-ran four of our numerical experiments, but with the grid resolution doubled. We picked the set of four runs with ν = 10−5, and δ = 0.5. In Figure 11 we plot the results, compared to the top right panel of Figure 4. In this plot, we can see that doubling the grid resolution has a minimal effect on the migration rates obtained. From this, we see that our conclusions are not simply an artifact of low resolution. Of course, the smoothing we have used could be hiding some effects, but we would not expect decreasing the smoothing length to change the planet migration to Type II behaviour. The smoothing length, at 0.6h is already rather smaller than the gap width, so shrinking it further is unlikely to enable the planet to make the gap cleaner. Furthermore, this smoothing length is already as small as we can realistically make it. The flow close to the planet will really have a 3d structure, not resolved in these calculations. By forcing all material to lie in the disc plane (as required by a 2d calculation), we effectively make it closer to the planet – significantly so within the gap. By softening the potential over a distance comparable to the scale height, we approximate the true 3d strength of the planet’s gravity. C. The Effect of Hill Sphere Exclusion By excluding material within the planet’s Hill sphere when computing the migration torque, we potentially reduced the migration torque substantially. Although this has a sound physical motivation (cf section 3), it does potentially affect the migration rate of the planet. Figure 12 shows that this has negligible effect on our results. This shows the migration of eight planets, embedded in ν = 10−5, δ = 0 discs. The four standard q values were considered, and the planets were held for 1000 orbits before being released to migrate. The only difference between each pair of curves is whether the exclusion radius for the torque calculation was a full Hill radius, for the solid lines, or half the Hill radius, for the dotted – 14 – lines. The solid lines in figure 12 are equivalent to the δ = 0 (top left) panel of figure 4, up to the difference in release time. In every case, the migration of each pair of planets is almost identical. In figure 13, we show the effect of the exclusion radius reduction on the torque profiles. This figure should be directly compared to figure 5. We can see that the torque close to the planet is increased, particularly for the first curve, plotted after 100 orbits. There is also a stronger peak close to the planet for all the curves. Otherwise, the torque profiles are remarkably similar to figure 5. This is as expected, given the results shown in figure 12. The author acknowledges support from NSF grants AST-0406799, AST-0098442, AST- 0406823, and NASA grants ATP04-0000-0016 and NNG04GM12G (issued through the Ori- gins of Solar Systems Program). I would like thank Frederic Masset for use of the Fargo code. I am also very grateful to Eric Blackman and Alice Quillen, for reading early drafts of this manuscript. Some of the computations presented here used the resources of HPC2N, Ume̊a REFERENCES Bryden, G., Chen, X., Lin, D. N. C., Nelson, R. P., and Papaloizou, J. C. 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F.: 2003, ApJ 589, 543 This preprint was prepared with the AAS LATEX macros v5.2. – 16 – Fig. 1.— Development of the gap for a Jupiter-mass planet on a fixed circular orbit, em- bedded in a ν = 10−5 disc, with δ = 0.5 (cf equation 5). The azimuthally averaged surface density profile is show after 10, 100 and 1000 orbits. Note that the y-axis is logarithmic – 17 – Fig. 2.— Development of the gap for a Jupiter-mass planet on a fixed circular orbit, em- bedded in a ν = 10−4 disc, with δ = 0.5 (cf equation 5). The azimuthally averaged surface density profile is show after 10, 100 and 1000 orbits. For ease of comparison, the y-axis is identical to that in Figure 1 – 18 – Fig. 3.— Planetary migration in the ν = 10−4 case for disc power laws of δ = 0 (top left), 0.5 (top right), 1.0 (bottom left) and 1.5 (bottom right). The planets were released to migrate after 100 orbits. The lines are marked by the value of q (see Equation 6) – 19 – Fig. 4.— Planetary migration in the ν = 10−5 case for disc power laws of δ = 0 (top left), 0.5 (top right), 1.0 (bottom left) and 1.5 (bottom right). The planets were released to migrate after 100 orbits. The lines are marked by the value of q (see Equation 6) – 20 – Fig. 5.— Radial torque profile for a planet in a ν = 10−5 disc as a function of time. The disc initially had a flat (δ = 0) surface density profile, and the planet was on a fixed orbit for the entire time – 21 – Fig. 6.— Radial torque profiles for a planet in ν = 10−5 discs of differing surface density after 1000 orbits. The disc had a δ = 0 initial surface density profile, and each line is marked with its q value. The planet’s orbit was fixed – 22 – Fig. 7.— Radial torque profiles for a planet in ν = 10−5 discs of differing δ, plotted after 1000 orbits. All discs had the same q value, and the planet was held on a fixed orbit – 23 – Fig. 8.— The effect of release time on the ν = 10−4 case. The orbital evolution of a planet in a sample disc is shown for three different release times – 24 – Fig. 9.— The effect of release time on the ν = 10−5 case. The orbital evolution of a planet in a sample disc is shown for three different release times – 25 – Fig. 10.— Planetary migration in the ν = 10−5 case for planets released after 1000 orbits. The disc power laws are δ = 0 (top left), 0.5 (top right), 1.0 (bottom left) and 1.5 (bottom right). The lines are marked by the value of q (see Equation 6). This plot should be compared to Figure 4 – 26 – Fig. 11.— Resolution test for a ν = 10−5 disc with δ = 0.5, where the planet is released after 100 orbits. Four different q values are compared (refer to the top right panel of Figure 4) at two grid resolutions – 27 – Fig. 12.— Effect of reducing the exclusion radius to half the Hill radius when computing the migration torque on the planet. This figure is based on the δ = 0 (top left) panel of Figure 4, except that the planets were released after 1000 orbits. The standard four q migrations are plotted. The solid lines correspond to those in Figure 4 (up to the difference in release time), while the dotted lines trace planets where the torque exclusion was only half the Hill radius – 28 – Fig. 13.— Effect on the torque profiles of reducing exclusion radius to half the Hill radius. This figure should be compared to figure 5 Introduction Type II Migration Numerical Set up Development of the Gap Results High Viscosity Medium Viscosity Summary of results Discussion Origin of the Torque Conclusion Effect of Varying Release Time The Effect of Resolution The Effect of Hill Sphere Exclusion

0704.0449

Worldsheet Instantons and Torsion Curves, Part B: Mirror Symmetry

UPR 1178-T DISTA-2007 TUW-07-08 Worldsheet Instantons and Torsion Curves Part B: Mirror Symmetry Volker Braun1, Maximilian Kreuzer2, Burt A. Ovrut1, and Emanuel Scheidegger3 1 Department of Physics, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA 19104–6395, USA 2 Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria 3 Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale via Bellini 25/g, 15100 Alessandria, Italy, and INFN - Sezione di Torino, Italy Abstract We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z3 ⊕ Z3 Wilson lines. As we found in Part A [1], the integral homology group H2(X,Z) = Z Z3 ⊕ Z3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. We observe that X is self-mirror even at the quantum level. Using the self- mirror property, we derive the complete prepotential on X , going beyond the results of Part A. In particular, this yields the first example where the instanton number depends on the torsion part of its homology class. Another consequence is that the threefold X provides a non-toric example for the conjectured exchange of torsion subgroups in mirror manifolds. Email: vbraun, [email protected], [email protected], [email protected] http://arxiv.org/abs/0704.0449v1 Contents 1 Introduction 1 2 Calabi-Yau Threefolds 5 2.1 The Calabi-Yau Threefold X . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The Intermediate Quotient X . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Toric Geometry and Mirror Symmetry 7 3.1 Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 The Batyrev-Borisov Construction . . . . . . . . . . . . . . . . . . . . . 10 3.3 Toric Intersection Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Mori Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5 B-Model Prepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Quotienting the B-Model 20 4.1 The Quotient by G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 The Quotient by G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 B-Model on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Instanton Numbers of X . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 B-Model on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.6 Instanton Numbers of X∗ . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.7 Instanton Numbers Assuming The Self-Mirror Property . . . . . . . . . 33 5 The Self-Mirror Property 34 6 Factorization vs. The (3,1,0,0,0) Curve 38 7 Towards a Closed Formula 41 8 Conclusion 43 A Triangulation of ∇̄∗ and ∇∗ 44 B The Flop of X∗ 50 Bibliography 53 1 Introduction Counting world sheet instantons (that is, holomorphic curves) on a Calabi-Yau three- fold has had a large number of applications in mathematics and physics, ever since it was essentially solved by mirror symmetry several years ago [2]. The purpose of this paper is to take into account an important subtlety that does not appear in very sim- ple Calabi-Yau manifolds like hypersurfaces in smooth toric varieties. This subtlety is the appearance of torsion curve classes. That is, the homology1 group = Z3 ⊕ Z3 ⊕ Z3 (1) contains the torsion2 subgroup Z3⊕Z3. Here, the manifold of interest X is a quotient of one of Schoen’s Calabi-Yau manifolds [3, 4] by a freely acting symmetry group. There are already a few known examples of such Calabi-Yau manifolds with torsion curves [5, 6, 7, 8, 9], but the proper instanton counting has never been done before. The prime motivation for studying these curves is that one would like to compute the superpotential for the vector bundle moduli [10, 11, 12, 13, 14, 15, 16] in a heterotic MSSM [17, 18, 19, 20, 21, 22, 23, 24, 25]. Our main result will be that there exist smooth rigid rational curves in X that are alone in their homology class. This proves that, in general, no cancellation between contributions to the superpotential W from instantons in the same homology class can occur. Therefore we would like to count rational curves on X . In physical terms, we need to find the instanton correction F X,0 to the genus zero prepotential of the (A-model) topological string on X . This is usually written as a (convergent) power series in h11 variables qa = e 2πita . Each summand is the contribution of an instanton, and the (integer) coefficients are the multiplicities of instantons in each homology class. According to [26, 27, 1] the novel feature of the 3-torsion curves on X is that for each 3-torsion generator we need an additional variable bj such that b j = 1. The Fourier series of the prepotential on X becomes X,0(p, q, r, b1, b2) = n1,n2,n3∈Z m1,m2∈Z3 n(n1,n2,n3,m1,m2) Li3 pn1qn2rn3bm11 b , (2) where n(n1,n2,n3,m1,m2) is the instanton number in the curve class (n1, n2, n3, m1, m2). For the purpose of computing the prepotential, we can either use directly the A- model or start with the B-model and apply mirror symmetry. The A-model calculation was carried out in the companion paper [1], entitled Part A. The results were: • A set of powerful techniques to compute the torsion subgroups in the integral homology and cohomology groups of X . They are spectral sequences starting with the so-called group (co)homology of the group action on the universal cover 1In the following, Z3 = Z/3Z always denotes the integers mod 3. Similarly, we write (Z3) ⊕nZ3 = Z3 ⊕ · · · ⊕ Z3 for the Abelian group generated by n generators of order 3. 2Not to be confused with the torsion tensor of a connection. • A closed formula for the genus zero prepotential X,0(p, q, r, b1, b2) = i,j=0 pbi1b P (q)4P (r)4+O(p2) = i,j=0 Li3(pb 2)+· · · (3) to linear order in p, extending the one computed in [28] for the universal cover X̃ . Here, if p(k) is the number of partitions of k ∈ Z≥, then P (q) is the generating function for partitions, P (q) p(i)qi = ln q) . (4) • Expanding eq. (3) as an instanton series we find that the number of rational curves of degree (1, 0, 0, m1, m2) is: n(1,0,0,m1,m2) = 1, ∀ m1, m2 ∈ Z3. (5) Furthermore, these curves have normal bundle OP1(−1)⊕OP1(−1). Hence, there are indeed 9 smooth rigid rational curves which are alone in their homology class. Alternatively, one can start with the B-model topological string and apply mirror symmetry, which is what we will do in this paper, entitled Part B. This will allow us to obtain the higher order terms in p. The order in p up to which one wants to compute the instanton numbers is only limited by computer power. We will again find a closed formula at every order in p, however, this time by guessing it from the instanton calculation, and hence only up to the order given by this limitation. The way to arrive at this result is as follows: • The universal cover X̃ admits a simple realization as a complete intersection in a toric variety. In this situation, mirror symmetry boils down to an algorithm to compute instanton numbers. Unfortunately, there are many non-toric divisors which cannot be treated this way. It turns out that, after descending to X , precisely the torsion information is lost. In this approach one can only compute X,0(q1, q2, q3, 1, 1). • As a pleasant surprise we find strong evidence that the manifold X is self-mirror. In particular, we attempt to compute the instanton numbers on the mirror X∗ by descending from the covering space X̃∗. The embedding of X̃∗ into a toric variety is such that all 19 divisors are toric. In principle, this allows for a complete analysis including the full Z3 ⊕Z3 torsion information, but this is too demanding in view of current computer power. • Although the full quotient X = X̃/(Z3 × Z3) is not toric, it turns out that a certain partial quotient X = X̃/Z3 can be realized as a complete intersection in a toric variety. That way, one only has to deal with h11(X) = 7 parameters, which is manageable on a desktop computer. Assuming the self-mirror property, we work with the mirror X , for which again all divisors are toric, and we can compute the expansion of F X,0(p, q, r, 1, b2) to any desired degree. A symmetry argument then allows one to recover the b1 dependence as well. Finally, we can extract the instanton numbers n(n1,n2,n3,m1,m2) including the torsion information. • As can be seen from the A-model result eq. (3), we observe that the prepotential X,0 at order p factors into i,j=0 b 2 times a function of p, q, r only. This means that the instanton number does not depend on the torsion part of its homology class. We will explain the underlying reason for this factorization and show that it breaks down at order p3. This fits nicely with the B-model computation at order p3, where the instanton numbers do depend on the torsion part. • Another consequence of the self-mirror property is that X is a non-toric example for the conjecture of [6]. By this conjecture, certain torsion subgroups of the integral homology groups are exchanged under mirror symmetry. An easily readable overview and a discussion of the physical consequences of our findings for superpotentials and moduli stabilization of heterotic models was pre- sented in [27]. The present Part B is self-contained and can be read independently of Part A [1]. All necessary results from Part A are reproduced in this part. As a guide through this paper, we start in Section 2 with a brief overview of the topology of the various spaces involved as determined in Part A [1]. This is followed by a review of the Batyrev-Borisov construction of mirror pairs of complete intersections in toric varieties in Section 3. We illustrate this construction by means of the covering spaces X̃ and X̃∗ of our example. The review includes the techniques to compute the B-model prepotential and the mirror map. These are applied in Section 4 to the partial quotients X and X yielding the main results stated above. This assumes that X as well as X are self-mirror, and evidence for this property is recapitulated in Section 5. Moreover, we show how the torsion subgroups are exchanged. Section 6 contains an explanation for the breakdown of the factorization alluded to above. Putting all the information together we try to guess a closed form for the prepotential in Section 7. Finally, we present our conclusions in Section 8. In the course of this work we will notice that a certain flop of X is very natural from the toric point of view, and we will present it in Appendix B. 2 Calabi-Yau Threefolds 2.1 The Calabi-Yau Threefold X The Calabi-Yau manifold X of interest is constructed as a free G = Z3 × Z3 quo- tient of its universal covering space X̃ . The latter is one of Schoen’s Calabi-Yau threefolds [3]. It is simply connected and hence easier to study. Among its various descriptions are the fiber product of two dP9 surfaces, a resolution of a certain T orbifold [29], or a complete intersection in a toric variety. In the present Part B, we will mostly use the latter viewpoint. The simplest way is to introduce the toric ambient variety P2×P1×P2 with homogeneous coordinates [x0 : x1 : x2], [t0 : t1], [y0 : y1 : y2] ∈ P2×P1×P2 . (6) The embedded Calabi-Yau threefold X̃ is then obtained as the complete intersection of a degree (0, 1, 3) and a degree (3, 1, 0) hypersurface in P2×P1×P2. We restrict the coefficients of their defining equations Fi = 0 to a particular set of three complex parameters λ1, λ2, λ3, such that the polynomials Fi read x30 + x 1 + x x0x1x2 = F1 (7a) λ1t0 + t1 y30 + y 1 + y λ2t0 + λ3t1 y0y1y2 = F2. (7b) For the special complex structure parametrized by λ1, λ2, λ3 the complete intersection is invariant under the G = Z3 × Z3 action generated by (ζ [x0 : x1 : x2] 7→ [x0 : ζx1 : ζ [t0 : t1] 7→ [t0 : t1] (no action) [y0 : y1 : y2] 7→ [y0 : ζy1 : ζ [x0 : x1 : x2] 7→ [x1 : x2 : x0] [t0 : t1] 7→ [t0 : t1] (no action) [y0 : y1 : y2] 7→ [y1 : y2 : y0] One can show that the fixed points of this group action in P2 ×P1 ×P2 do not satisfy eqns. (7a) and (7b), hence the action on X̃ is free. 2.2 The Intermediate Quotient X The partial quotient G1 (9) will be of particular interest in this paper because this quotient is generated by phase symmetries, see eq. (8a), and hence is toric. In particular, we will need a basis of Kähler classes. As usual, we will not distinguish degree-2 cohomology and degree-4 homology classes but identify them via Poincaré duality. Part A [1] ?? shows that3 = H2(X̃,Z)G1 ⊕ Z3 = spanZ φ, τ1, υ1, ψ1, τ2, υ2, ψ2 ⊕ Z3. (10) Hence, by abuse of notation, we can identify the free generators on X with the G1- invariant generators on X̃ , see Part A eq. (??), via the pull back by the quotient map. The triple intersection numbers on X = X̃/Z3 are one-third of the corresponding intersection numbers on X̃ listed in Part A eq. (??). Hence, the intersection numbers on X are φτ1τ2 = 3 φτ1υ2 = 3 φτ1ψ2 = 6 φυ1τ2 = 3 φυ1υ2 = 3 φυ1ψ2 = 6 φψ1τ2 = 6 φψ1υ2 = 6 φψ1ψ2 = 12 τ 1 τ2 = 1 τ 21υ2 = 1 τ 1ψ2 = 2 τ1υ1τ2 = 3 τ1υ1υ2 = 3 τ1υ1ψ2 = 6 τ1ψ1τ2 = 3 τ1ψ1υ2 = 3 τ1ψ1ψ2 = 6 τ1τ 2 = 1 τ1τ2υ2 = 3 τ1τ2ψ2 = 3 τ1υ 2 = 3 τ1υ2ψ2 = 6 τ1ψ 2 = 6 υ 1τ2 = 3 υ21υ2 = 3 υ 1ψ2 = 6 υ1ψ1τ2 = 6 υ1ψ1υ2 = 6 υ1ψ1ψ2 = 12 2 = 1 υ1τ2υ2 = 3 υ1τ2ψ2 = 3 υ1υ 2 = 3 υ1υ2ψ2 = 6 2 = 6 ψ 1τ2 = 6 ψ 1υ2 = 6 ψ 1ψ2 = 12 ψ1τ 2 = 2 ψ1τ2υ2 = 6 ψ1τ2ψ2 = 6 ψ1υ 2 = 6 ψ1υ2ψ2 = 12 ψ1ψ 2 = 12. Clearly, G2 acts on the partial quotient X . From Part A eq. (??) it follows that, of the 7 non-torsion divisors above, only 3 are G2-invariant. This invariant part is particularly manageable and will be important in the following. We find = span φ, τ1, τ2 with products 3τ 2i = τiφ. In particular, the triple intersection numbers on X are τ 21 τ2 = 1, τ1φτ2 = 3, τ1τ 2 = 1, (13) and 0 otherwise. Finally, the second Chern class of X is c2(X) = 12(τ 1 + τ 2 ). There- fore, · τ1 = 12, c2 · φ = 0, c2 · τ2 = 12. (14) 2.3 Variables As we discussed in Part A ??, the instanton-generated superpotential should be thought of as a series with one variable for each generator in H2. In particular, we 3The torsion in H2 are just the Wilson lines, that is, first Chern classes of flat line bundles. They will play no role in the following. The torsion curves in H2, on the other hand, are the focus of this paper. Calabi-Yau threefold ) Free generators Torsion generators X̃ Z19 p0, q0, . . . , q8, r0, . . . , r8 X = X̃/G1 Z 7 ⊕ Z3 P,Q1, Q2, Q3, R1, R2, R3 X = X̃/G Z3 ⊕ Z3 ⊕ Z3 p, q, r b1, b2 Table 1: The different Calabi-Yau threefolds, curve classes, and variables used to expand the prepotential. will be interested in the Calabi-Yau threefolds X̃ , X , and X . For these, the degree-2 integral homology and the variables used (see Part A [1] for precise definitions) are in summarized Table 1. Pushing down the curves by the respective quotients lets us express the prepotential on the quotient in terms of the prepotential on the covering space. We found in Part A that P,Q1, Q2, Q3, R1, R2, R3, b1) PQ51Q 3 b1, Q 2 , Q 3b1, Q3, 1, b1, Q1Q 2 , R 1, R3, 1, b 1, R1R p, q, r, b1, b2) = p, q, b2, b2, r, b 2, b1 . (16) 3 Toric Geometry and Mirror Symmetry In this section we review mirror symmetry and the construction of the B-model for the mirror of the covering space X̃ . Since X̃ is a complete intersection in a toric variety, we can use the standard constructions. Because we expect the model to be self-mirror, we will analyze the B-model for X̃ and its mirror X̃∗. The toric geometry for X̃ is much simpler4 than for X̃∗, but contains less information. In this section we will start with the simpler model in order to review the Batyrev-Borisov construction for the mirror of a complete intersection in a toric variety. Then we will apply this construction to the more complicated model, now without going into too many details. We will see that, on the simpler side, not all parameters are toric and no torsion is visible. However, on the more complicated mirror side, all parameters are toric which will allow us, in principle, to perform the B-model computation of 4Meaning that X̃ is a complete intersection in the very simple toric variety P2 ×P1 ×P2, whereas X̃∗ is embedded in a complicated toric ambient variety. the complete prepotential. As X̃ ∼= X̃∗ is expected to be self-mirror, this determines the complete prepotential F eX∗,0 as well. In practice, however, the analysis is computationally too involved. Fortunately, the space X = X̃/G1 and its mirror will turn out to be both tractable with toric methods and sufficiently informative. This quotient will be the subject of Section 4. Finally, this is also the starting point for arguing in Section 5 that the self-mirror property persists at the level of instanton corrections. Recall that, in Subsection 2.1 we defined our Calabi-Yau manifold as the complete intersection F1 = 0, F2 = 0 ⊂ P2×P1×P2 (17) with the two polynomials F1, F2 as in eqns. (7a) and (7b), respectively. In order to construct the mirror manifold following Batyrev and Borisov, we need to reformulate this definition in terms of toric geometry. We review here some essential ingredients of toric geometry, for details we refer to [30, 31] and references therein. We will give the abstract definitions and concepts step by step, and at each step illustrate them with the example X̃ and its mirror manifold X̃∗. 3.1 Toric Varieties Given a lattice N of dimension d, a toric variety VΣ is defined in terms of a fan Σ which is a collection of rational polyhedral5 cones σ ⊂ N such that it contains all faces and intersections of its elements. VΣ is compact if the support of Σ covers all of the real extension NR of the lattice N . The resulting d-dimensional variety VΣ is smooth if all cones are simplicial and if all maximal cones are generated by a lattice basis. Let Σ(1) denote the set of one-dimensional cones (rays) with primitive generators ρi, i = 1, . . . , n. The simplest description of VΣ introduces n homogeneous coordinates zi corresponding to the generators ρi of the rays in Σ (1). These homogeneous coordinates are then subjected to weighted projective identifications z1 : · · · : zn 1 z1 : · · · : λ a = 1, . . . , h (18) for any nonzero complex number λ ∈ C×, where the integer n-vectors q i are genera- tors of the linear relations i ρi = 0 among the primitive lattice vectors 6 ρi. In or- der to obtain a well-behaved quotient, we must exclude an exceptional set Z(Σ) ⊂ Cn 5Here, the tip of the cone is always the origin of N . A cone is rational if it is spanned by rays which pass through lattice points (other than the origin), that is, have rational slopes. A cone is polyhedral if it is the cone over an (d − 1)-dimensional polytope. In other words, curved faces are not allowed. 6We will use the same symbol ρ∗ to denote the generators in Σ (1) and the corresponding primitive lattice vectors in N . that is defined in terms of the fan, as will be explained below. Hence, the quotient is Cn − Z(Σ) , (19) where Γ ≃ N/ span{ρi} is a finite abelian group. There are h = n−d independent C identifications, therefore the complex dimension of VΣ equals the rank d of the lattice N . The identifications by Γ are only non-trivial if the ρi do not span the lattice N . Refinements of the lattice N with fixed ρi can hence be used to construct quotients of toric varieties VΣ by discrete phase symmetries such as Z3. Such quotients will be discussed in Section 4. Note that the rays ρi are in 1–to–1 correspondence with the (C×)-invariant divisors Di on VΣ, which are defined as zi = 0 ⊂ VΣ. (20) Conversely, the homogeneous coordinate zi is a section of the line bundle O(Di). For example, consider the simplest compact toric variety, the projective space Pd. Its fan Σ = Σ(∆) is generated by the n = d+ 1 vectors ρ1 = e1, ρ2 = e2, . . . , ρn−1 = ed, ρn = − ei (21) of a d-dimensional simplex ∆. They satisfy a single linear relation, i=1 ρi = 0. Therefore qi = 1 for all i, and the homogeneous coordinates in eq. (18) are the usual homogeneous coordinates on Pd. For products of toric varieties we simply extend the relations for any single factor by zeros and take the union of them. Hence, the fan of the polyhedron ∆∗ describing the 5-dimensional toric variety P2×P1×P2 in eq. (17) is generated by the n = 5+3 = 8 vectors ρ1 = e1, ρ2 = e2, ρ3 = −e1 − e2, ρ4 = e3, ρ5 = −e3, ρ6 = e4, ρ7 = e5, ρ8 = −e4 − e5 satisfying the linear relations ρi = 0. (23) Except for the origin, there are no other lattice points in the interior of ∆∗. The corresponding homogeneous coordinates will be denoted by z1 = x0, z2 = x1, z3 = x2, z4 = t0, z5 = t1, z6 = y0, z7 = y1, z8 = y2. In more general situations, given a polytope ∆∗ ⊂ N we will denote the resulting toric variety by P∆∗ = VΣ(∆∗). 3.2 The Batyrev-Borisov Construction Batyrev showed that a generic section ofK−1 , the anticanonical bundle of P∆∗, defines a Calabi-Yau hypersurface if ∆∗ is reflexive, which means, by definition, that ∆∗ and its dual x ∈MR ∣∣∣ (x, y) ≥ −1 ∀y ∈ ∆∗ are both lattice polytopes. Here, M = Hom(N,Z) is the lattice dual to N and MR is its real extension. Mirror symmetry corresponds to the exchange of ∆ and ∆∗ [32]. The generalization of this construction to complete intersections of codimension r > 1 is due to Batyrev and Borisov [33, 34]. For that purpose, they introduced the notion of a nef partition. Consider a dual pair of d-dimensional reflexive polytopes ∆ ⊂ ∗ ⊂ NR. In that context, a partition E = E1 ∪ · · · ∪ Er of the set of vertices of ∆∗ into disjoint subsets E1, . . . , Er is called a nef-partition if there exist r integral upper convex Σ(∆∗)-piecewise linear support functions φl : NR → R, l = 1, . . . , r such φl(ρ) = 1 if ρ ∈ El, 0 otherwise. Each φl corresponds to a divisor D0,l = Dρ (27) on P∆∗ , and their intersection Y = D0,1 ∩ · · · ∩D0,r (28) defines a family Y of Calabi-Yau complete intersections of codimension r. Moreover, each φl corresponds to a lattice polyhedron ∆l defined as x ∈MR ∣∣∣ (x, y) ≥ −φl(y) ∀y ∈ NR . (29) The lattice points m ∈ ∆l correspond to monomials 〈m,ρi〉 i ∈ Γ (P∆∗ ,O(D0,l)) . (30) One can show that the sum of the functions φl is equal to the support function of K and, therefore, the corresponding Minkowski sum is ∆1 + · · · + ∆r = ∆. Moreover, the knowledge of the decomposition E = E1 ∪ · · · ∪Er is equivalent to that of the set of supporting polyhedra Π(∆) = {∆1, . . . ,∆r}, and therefore this data is often also called a nef partition. It can be shown that given a nef partition Π(∆) the polytopes7 {0} ∪ El ⊂ NR (31) define again a nef partition Π∗(∇) = {∇1, . . . ,∇r} such that the Minkowski sum ∇ = ∇1 + · · · + ∇r is a reflexive polytope. This is the combinatorial manifestation of mirror symmetry in terms of dual pairs of nef partitions of ∆∗ and ∇∗, which we summarize in the diagram ∆ = ∆1 + . . .+∆r ∆ ∇1, . . . ,∇r Mirror Symmetry uujjj MR NR ∆1, . . . ,∆r (∆l,∇l′) ≥ −δl l′ ∇ = ∇1 + . . .+∇r . (32) In the horizontal direction, we have the duality between the lattices M and N and mirror symmetry goes from the upper right to the lower left. The other diagonal has also a meaning in terms of mirror symmetry as we will explain below. The complete intersections Y ⊂ P∆∗ and Y ∗ ⊂ P∇∗ associated to the dual nef partitions are then mirror Calabi-Yau varieties. Let us now apply the Batyrev-Borisov construction to the complete intersection eq. (17), hence r = 2. There exist several nef-partitions of ∆∗. The one which has the correct degrees (3, 1, 0) and (0, 1, 3) is, up to exchange of t0 and t1, E1 = {ρi |i = 1, . . . , 4} and E2 = {ρi |i = 5, . . . , 8}. Adding the origin and taking the convex hull yields the polytopes ρ1, . . . , ρ4, 0 , ∇2 = ρ5, . . . , ρ8, 0 , (33) where the ρi are defined in eq. (22). The two divisors cutting out the Calabi-Yau threefold are, according to eq. (27), D0,1 = Di, D0,2 = Di ⇒ X̃ = D0,1 ∩D0,2 ⊂ P∆∗ (34) Note that, while ∆∗ has no further lattice points, its dual ∆ has 18 vertices and 300 lattice points. Using the computer package PALP [35], we determine the associated polytopes ∆1 and ∆2 of the global sections of O(D0,1) and O(D0,2), respectively. In an appropriate lattice basis there is, up to symmetry, a unique nef partition consisting ν1, . . . , ν6, 0 , ∆2 = ν7, . . . , ν12, 0 , (35) 7The brackets · · · denote the convex hull. where ν1 = 2e1 − e2, ν2 = −e1 + 2e2, ν3 = −e1 − e2, ν4 = 2e1 − e2 − e3, ν5 = −e1 + 2e2 − e3, ν6 = −e1 − e2 − e3, ν7 = 2e4 − e5, ν8 = −e4 + 2e5, ν9 = −e4 − e5, ν10 = e3 + 2e4 − e5, ν11 = e3 − e4 + 2e5, ν12 = e3 − e4 − e5. Among these 12 vectors there are the 7 independent linear relations 3ν3 + ν4 + ν5 − 2ν6 = 0, 3ν9 + ν10 + ν11 − 2ν12 = 0, ν1 − ν3 − ν4 + ν6 = 0, −ν1 + ν2 + ν4 − ν5 = 0, ν7 − ν9 − ν10 + ν12 = 0, −ν7 + ν8 + ν10 − ν11 = 0, −ν2 + ν5 − ν8 + ν11 = 0. The convex hull ∇∗ = 〈∆1,∆2〉 yields the fan Σ(∇ ∗) and, consequently, the toric variety P∇∗ . Let D i , i = 1, . . . , 12 be the divisors associated to the vertices νi. Then, by eq. (27), the nef partition eq. (35) defines the divisors D∗0,1 = D∗i , D 0,2 = D∗i , ⇒ X̃ ∗ = D∗0,1 ∩D 0,2 ⊂ P∇∗ (38) cutting out the mirror complete intersection X̃∗. In contrast to ∆∗, the polytope ∇∗ contains extra integral points. We find that it contains, in addition to the origin and the vertices in eq. (36), the 26 points ν13 = (ν4 + ν5 + ν6) = −e3, ν12+6k+i+j = (ν3k+i + 2ν3k+j), ν14 = (ν10 + ν11 + ν12) = e3, ν15+6k+i+j = (ν3k+j + 2ν3k+i) ∀ k ∈ {0, . . . , 3}, (i, j) ∈ (1, 2), (1, 3), (2, 3) For completeness, note that the dual polytope ∇ has 15 vertices and 24 lattice points. Running PALP to compute the Hodge numbers using the formula of [36], we obtain = h1,2 = h1,1 = h1,2 = 19, (40) in agreement with Part A [1], eq. (??). So far, we have mainly focused on the information contained in the reflexive poly- topes ∆∗ and ∇∗ and ignored their duals. We have already mentioned that in the reflexive case a generic section of K−1 defines a Calabi-Yau manifold, and that such sections are provided by the lattice points of ∆. In other words, ∆ and ∇ are the Newton polytopes of Y and Y ∗, respectively. That is, the complete intersection Y (Y ∗) is defined by r polynomial equations, and the exponents of the monomials in each equation are the lattice points in ∆ (∇). More precisely, the Minkowski sum for, say, ∆ = ∆1 + · · ·+∆r defines r homogeneous polynomials Fl(z) = ∇l′∩N 〈m,ρi〉+δl l′ i , l = 1, . . . , r (41) with coefficients al,m ∈ C. The simultaneous vanishing of F1, . . . , Fr then defines the complete intersection Calabi-Yau manifold Y ⊂ P∆∗ . Exchanging ∆l and ∇l′ in eq. (41) yields the equations F ∗l defining the mirror manifold Y ∗. It is in this sense that the map from the upper left to the lower right in eq. (32) is also a manifestation of mirror symmetry. Since we will not need the actual polynomials for X̃ and X̃∗, we refrain from writing them explicitly. Instead, we refer the reader to Section 4, where we determine the equations in a simpler situation. 3.3 Toric Intersection Ring Up to now we have only considered one of the ingredients in the fan Σ, namely, the generators ρ ∈ Σ(1) which defined the C× action in eq. (19). The second ingredient is the exceptional set Z(Σ). It corresponds to fixed loci of a continuous subgroup of for which the quotient eq. (19) is not well defined. Therefore, these loci have to be removed. In terms of the homogeneous coordinates zi, this happens precisely when a subset {zi |i ∈ I}, I ⊆ {1, . . . , n}, of the coordinates vanishes simultaneously such that there is no cone σ ∈ Σ containing all of the ρi ⊆ σ, i ∈ I. Hence, the set Z(Σ) is the union of the sets ZI = {[z1 : · · · : zn] |zi = 0 ∀i ∈ I}. Minimal index sets I with this property are called primitive collections [37]. In order to determine the index sets I we need a coherent8 triangulation T = T (∆∗) of the polytope ∆∗ for which all simplices contain the origin. Different triangulations will yield different exceptional sets and, hence, different toric varieties. However, for simplicity, we will mostly suppress the choice of a triangulation in the notation. In the case of complete intersections, only those triangulations of ∆∗ are compatible with a given nef partition that can be lifted to a triangulation of the corresponding Gorenstein cone, see [38]. The polytope defining projective space Pd admits a unique triangulation with the required properties, and this triangulation consists of n = d + 1 simplices. The only primitive collection is I = {1, . . . , n}. This is well-known from the definition of projective space, where we have to remove the origin z1 = · · · = zd+1 = 0 from Cd+1. Similarly, the polyhedron ∆∗ for the ambient space P∆∗ of X̃ admits a unique triangulation, and the primitive collections are those of its factors, that is, I1 = {1, 2, 3}, I2 = {4, 5}, I3 = {6, 7, 8}. (42) 8Coherent triangulations, sometimes also called regular triangulations, satisfy some technical property that is equivalent to the associated toric quotient being Kähler. The mirror polyhedron∇∗, on the other hand, admits a huge number of triangulations. We will discuss particularly interesting triangulations of the mirror polyhedron at the end of Appendix A. The primitive collections determine the cohomology ring of toric varieties and, together with the nef partition, complete intersections. Recall that if the collection ρi1 , . . . , ρik of rays is not contained in at least one cone, then the corresponding ho- mogeneous coordinates zil are not allowed to vanish simultaneously. Therefore, the corresponding divisors Dil have no common intersection. Hence, we obtain non-linear relations RI = Di1 · . . . ·Dik = 0 in the intersection ring. It can be shown that all such relations are generated by the primitive collections I = {i1, . . . , ik} defined above. The ideal generated by these RI is called Stanley-Reisner ideal ISR = RI , I primitive collection ⊂ Z[D1, . . . , Dn], (43) and Z[D1, . . . , Dn]/ISR is the Stanley-Reisner ring. The intersection ring of a non- singular compact toric variety PΣ is [39] = Z [D1, . . . , Dn] (m, ρi)Di . (44) In other words, the intersection ring can be obtained from the Stanley-Reisner ring by adding the linear relations (m, ρi)Di = 0, where it is sufficient to take a set of basis vectors for m ∈ M . In particular, the intersection number of the divisors spanning a maximal-dimensional simplicial cone σ = span R≥{ρi1 , . . . , ρid} is Di1 · . . . ·Did = Vol(σ) , (45) where Vol(σ) is the lattice-volume, that is, the geometric volume divided by the volume 1 of a basic simplex. For practical purposes it is sufficient to compute one of these volumes, the remaining intersections can be obtained using the linear and non-linear relations. Having found the intersection ring of the ambient toric variety, we now turn to the complete intersection Y ⊂ P∆∗ . The toric part of its even-degree intersection ring is [40] Hevtoric = Q [D1, . . . , Dn] IY , (46) where IY is the ideal quotient (m, ρi)Di D0,l. (47) Note that it can happen that some of the Di appear as generators of IY . This means that they can be set to zero in the intersection ring. Geometrically, this means that these divisors do not intersect a generic complete intersection Y . While the inter- section ring depends on the triangulation T (∆∗) through the primitive collections defining the Stanley-Reisner ideal, we conjecture that the divisors Di not intersecting Y are independent of the choice of triangulation. This conjecture is proven for r = 1 and supported by a large amount of empirical evidence for r > 1. We conclude that the dimension dimH2toric(Y ) is in general smaller than h 1,1(Y ) for the following two reasons: Only h = n − d = dimH2(P∆∗,Z) divisors are realized in the ambient toric variety P∆∗ , and some of them may not descend to the complete intersection Y . Using the adjunction formula we can compute the the Chern classes of Y by expanding c(Y ) = (1 +Di) (1 +D0,l) . (48) The intersection ring together with the second Chern class determine the diffeomor- phism type of a simply-connected Calabi-Yau manifold [41]. If we consider the coho- mology with integral coefficients there can be torsion and, in fact, this is what this paper is all about. Unfortunately, a combinatorial formula in terms of the fan Σ(∆) for the torsion in the integral cohomology of a toric Calabi-Yau manifold is only known in the hypersurface case [6]. We now illustrate these concepts in the example of the complete intersection X̃ ⊂ P∆∗ = P 2 × P1 × P2 and its mirror manifold X̃∗. In eq. (42) we already determined the primitive collections, hence the corresponding Stanley-Reisner ideal is ISR = D1D2D3, D4D5, D6D7D8 . (49) The linear equivalences are D1 = D2, D1 = D3, D4 = D5, D6 = D7, D6 = D8 and, hence, we can choose K1 = D4, K2 = D1, K3 = D6 as a basis for H 2(P∆∗). In terms of this basis, we obtain D0,1 = K1+3K2 and D0,2 = K1+3K3, see eq. (27). Therefore, the ideal I in eq. (34) is 2K2 −K2 2K3, K1K2 − 3K2 2, K1K3 − 3K3 2, K1 2, K2 3, K3 . (50) Next, we define the restriction of the Ki to X̃ to be the divisors J̃i = Ki · X̃ = Ki(K1 + 3K2)(K1 + 3K3). (51) We need to compute one of the intersection numbers directly from the volume of a cone, say, J̃1J̃2J̃3 = K1K2K3(K1 + 3K2)(K1 + 3K3) = 9K1K 3 , where we made use of the relations in I . Using eq. (45), this intersection can be evaluated to be 3 = 9D1D2D4D6D7 = 9/Vol 〈ρ1, ρ2, ρ4, ρ6, ρ7〉 = 9/Vol 〈e1, e2, e3, e4, e5〉 Then, again using eq. (50), we see that the only non-vanishing intersection numbers and the second Chern class are J̃22 J̃3 = 3, J̃1J̃2J̃3 = 9, J̃2J̃ 3 = 3, · J̃1 = 0, c2 · J̃2 = 36, c2 · J̃3 = 36. Note that only h toric(X̃) = 3 of the h 1,1(X̃) = 19 parameters are realized torically. Comparing the triple intersection numbers with eq. (13), it is clear that these 3 toric divisors are precisely the G-invariant divisors on X̃ . A similar, though much more complicated, calculation can be done for X̃∗ ⊂ P∇∗ . Using the results of Appendix A one can show that, among the points in eq. (39), the 14 divisors D∗13, D 14, D 12+6k+i+j , D 15+6k+i+j, k = 0, 2 appear as generators of eq. (47) and, therefore, do not intersect X̃∗. Subtracting from the remaining 24 divisors in eqns. (36) and (39) the remaining 5 linear relations in eq. (37), we find that all toric(X̃ ∗) = h1,1(X̃∗) = 19 moduli are realized torically. 3.4 Mori Cone As we have just seen, the cohomology classes Di span H 2(PΣ,R) = H 1,1(PΣ). The Kähler classes of a smooth projective toric variety PΣ form an open cone in H 1,1(PΣ) called the Kähler cone K(PΣ). This cone has a combinatorial description in terms of the fan Σ, which we now review. First, define a support function to be a continuous function ψ : NR → R given on each cone σ ∈ Σ by an mσ ∈MR via ψ(ρ) = (mσ, ρ) ∀ρ ∈ σ ⊂ NR. (54) A support function determines a divisor D = ψ(ρi)Di. We say that D is convex if ψ is a convex function on NR. The convex classes form a non-empty strongly convex polyhedral cone in H1,1(PΣ) whose interior is the Kähler cone K(PΣ). Such a support function is strictly convex if and only if ψ(ρi1 + · · ·+ ρik) > ψ(ρi1) + · · ·+ ψ(ρik) (55) for every primitive collection I = {i1, . . . , ik} [40]. The dual of the Kähler cone K(PΣ) is called the Mori cone or the cone of numerically effective curves NE(PΣ). Its generators can be described by vectors l(a) of the corresponding linear relations∑ i ρi = 0. Each face of the Kähler cone K(PΣ) is dual to an edge of NE(PΣ). These edges are generated by curves c(a), and the entries of the vector l(a) are = c(a) ·Di. (56) A practical algorithm to find the generators for l(a) in terms of the triangulation T (∆∗) is described in [42]. Of course, we are not interested in the ambient space but in a complete intersection Y ⊂ P∆∗ . The restriction of a Kähler class on the ambient space yields a Kähler class on Y , but not every Kähler class on Y arises that way. We define the toric part of the Kähler cone on Y as the restriction [43] K(Y )toric = K(PΣ) ⊂ K(Y ). (57) In the simplicial case, we can always take the basis Ji of H toric(Y,Q) to be edges of the Kähler cone. The dual of the toric Kähler cone of Y is the (toric) Mori cone NE(Y )toric. This is sufficient for mirror symmetry purposes, however, it can be larger than the actual cone of effective curves. Once the generators l(a) of NE(P∆∗) are determined, we need to add the information about the nef partition. For this purpose, we define = −D0,m · c (a) m = 1, . . . , r. (58) Finally, it is customary to write the generators of the Mori cone NE(Y )toric as l(a) = 0,1 , . . . , l 0,r ; l 1 , . . . , l , (59) which are, by abuse of notation, again denoted by l(a). The knowledge of the (toric) Mori cone is important for several reasons. It defines the local coordinates on the complex structure moduli space of the mirror Y ∗ near the point of maximal unipotent monodromy. Moreover, the generators enter the coefficients of the fundamental period which is a solution of the Picard-Fuchs equations as we will review in Subsection 3.5. For example, using the unique primitive collections in eq. (42), the Mori cone for P∆∗ is generated l(1) =(0, 0, 0, 1, 1, 0, 0, 0) l(2) =(1, 1, 1, 0, 0, 0, 0, 0) l(3) =(0, 0, 0, 0, 0, 1, 1, 1). Recalling the nef partition D0,1 = D1 + · · ·+D4, D0,2 = D5 + · · ·+D8, we prepend (−D0,1 · c (a),−D0,2 · c (a)) = (−3, 0), (−1,−1), (0,−3), a = 1, 2, 3, to obtain the gener- ators l(1) =(−1,−1; 0, 0, 0, 1, 1, 0, 0, 0) l(2) =(−3, 0; 1, 1, 1, 0, 0, 0, 0, 0) l(3) =( 0,−3; 0, 0, 0, 0, 0, 1, 1, 1) of the Mori cone NE(X̃)toric. Due to the large number of toric moduli, the calculation for the Mori cone NE(P∇∗) of the ambient toric variety of the mirror X̃ ∗ is much more complex. 9We sort the Mori cone generators such that the first one corresponds to the P1 of the ambient space, and the second and third generator are the hyperplane sections of the two P2. In other words, we have J̃a · c (b) = δb . This is the basis of curves that we used for the A-model computation. 3.5 B-Model Prepotential Mirror symmetry identifies the quantum corrected Kähler moduli space of Y with the classical complex structure moduli space of Y ∗, see the excellent treatise in [43] for details. The deformations of the complex structure of Y ∗ are encoded in the periods ̟ = Ω and the latter can be computed from the equations F ∗l that cut out Y ∗ ⊂ P∇∗ . Given the Mori cone eq. (59) and the classical intersections numbers κabc = Ja · Jb · Jc we follow [44, 45, 38, 43] to write down a local expansion of the periods, convergent near the large complex structure point, which is characterized by its maximal unipotent monodromy. In the following, we will review just the bare essentials. The coefficients ai in the polynomial constraints F l of the complete intersection Y see eq. (41), define the complex structure of Y ∗. A particular set of local coordinates ua on the complex structure moduli space on Y ∗ is defined by i b = 1, . . . , h (62) where h toric(Y ) and am,0 is the coefficient in (41) corresponding to the origin in ∇l. In these coordinates, the point of maximal unipotent monodromy is at ub = 0. We define the cohomology-valued period ̟(u, J) = {na≥=0} 0,mJa a=1 l 0,mna a=1 l una+Jaa . (63) where (x)n = Γ(x + n)/Γ(x) is the Pochhammer symbol. Note that the choice of triangulation is implicit in the generators l(a) of the Mori cone. Expanding ̟(u, J) by cohomology degree yields ̟(u, J) = ̟(0)(u) + ̟(1)a (u)Ja + ̟(2)a (u)κabcJbJc −̟ (3)(u) dVol, (64) where dVol is the volume form. The coefficients in eq. (64) are the fundamental period ̟(0)(u), that is, the unique solution to the Picard-Fuchs equations holomorphic at ua = 0, and ̟(1)a (u) = ∂Ja̟(u, J)|J=0, ̟ a (u) = κabc∂Jb∂Jc̟(u, J)|J=0, ̟(3)(u) = − κabc∂Ja∂Jb∂Jc̟(u, J)|J=0. These coefficients coincide with the basis of solutions of the Picard- Fuchs equations obtained from the Frobenius method in [46, 31]. The B-model prepotential FBY ∗,0 is Y ∗,0(u) = 2̟(0)(u)2 ̟(0)(u)̟(3)(u) + ̟(1)a (u)̟ a (u) . (66) At the large complex structure point the mirror map defines natural flat coordinates on the Kähler moduli space of the original manifold Y , which are i (u) ̟0(u) , i = 1, . . . , h. (67) We also define qj = e 2πitj = uj + O(u 2). One way to obtain the prepotential is to compute its third derivatives C∗abc = DaDbDcF Y ∗,0 = Ω ∧ ∂a∂b∂cΩ, (68) and apply the Picard-Fuchs operators. This leads to linear differential equations, which determine C∗abc up to a common constant, see again [46, 43] for details. The quantum corrected three point function Cijk(q) on Y follows from C abc(u) using the inverse mirror map eq. (67) u = u(t), and one obtains Cijk(q) = ̟(0)(u(q))2 C∗abc(u(q)). (69) In practice, we use the formula Cijk(q) = ∂ti∂tj k (u(q)) ̟(0)(u(q)) . (70) Integrating three times with respect to ti yields the prepotential F Y ∗,0(t) up to a polynomial of degree three in ti which can be determined partially by the topological data of Y . Mirror symmetry then ensures that the B-model prepotential, eq. (66), is equal to the A-model prepotential. That is, FY,0(q) = F Y ∗,0(u(q)). (71) This allows us to compute the instanton numbers nd. For the case of interest, X̃ ∈ P∆∗ = P 2 × P1 × P2, (72) we refer to [28] where this program been carried out in detail. The same calculation can in principle be done on the mirror X∗, but the large number of toric moduli again makes it highly extensive. Instead, we refer to the next section where a suitable quotient of X̃∗ will be treated in detail for which the computations are reasonably simple. 4 Quotienting the B-Model In this section we consider the quotientX = X̃/G in terms of toric geometry and study the mirror of X in this context. In order to achieve this, we first analyze the partial quotient X = X̃/G1. Using the techniques introduced in Section 3, we construct the mirror X . Using their toric realization, we perform the B-model computation for the non-perturbative prepotentials F and F , respectively. Finally, we explain how one can implement the quotient by G2 on both sides in order to obtain X and X 4.1 The Quotient by G1 We start with a review of the general discussion of free quotients of complete intersec- tions in toric geometry in [31]. Consider a fan Σ ⊂ NR and pick a lattice refinement N̄ such that Γ = N̄/N is a finite abelian group. Such a lattice refinement consists of a finite sequence of lattice refinements of the form N → N +wpZ which are described by a vector wp = αpiρi with αpi ∈ Z. The group Γ is then isomorphic to Zkp . Let Σ̄ be the fan obtained from Σ by relating everything to the lattice N̄ . In this context, we make some additional identifications in the toric quotient eq. (19) [47]. One finds that VΣ̄ = VΣ/Γ is the quotient of VΣ by the finite abelian group Γ. Its action on the homogeneous coordinates is by multiplication by phases z1 : · · · : zn ξα1z1 : · · · : ξ , ξ = e k , (73) for every cyclic subgroup of order k. We will denote such group actions by Zk : (α1, . . . , αn). If VΣ is a compact toric variety, then the quotient VΣ̄ is never free [39]. However, a hypersurface or complete intersection in VΣ need not intersect the set of fixed points, and in that case we get a smooth quotient manifold with nontrivial fundamental group. We now apply this to P∆∗ = P 2 × P1 × P2 defined in eq. (22). The first step in performing the quotient of P∆∗ by G1 thus amounts to a refinement N̄ = wZ+N of the lattice N with index |G1| = 3. From the definition eq. (8a) of the action of G1 on P∆∗ and eq. (24) we read off that the refinement is by a vector ρ2 + 2ρ3 + ρ7 + 2ρ8 + Z5. (74) The resulting polytope ∆̄∗ admits the same nef partition as ∆∗ in eq. (33), ∇̄1 = 〈ρ̄1, . . . , ρ̄4, 0〉, ∇̄2 = 〈ρ̄5, . . . , ρ̄8, 0〉. (75) where we express the generators ρ̄ in terms of ρ as ρ̄i = ρi, i = 1, . . . , 6 , ρ̄7 = ρ7 + e1 + 2e2 + e4 + 2e5, ρ̄8 = ρ8 − e1 − 2e2 − e4 − 2e5. It is easy to check that the ρ̄i satisfy the same linear relations eq. (23) as the ρi, and that w = 1 (ρ̄1 − ρ̄2 + ρ̄6 − ρ̄7) = −e2 − e5. The ρ̄i together with w therefore indeed generate the lattice N̄ . Note that, while all 8 non-zero lattice points of ∆̄∗ are vertices, the dual polytope ∆̄ has 18 vertices and 102 points. Using PALP [35] again, we compute the lattice points of the polytope ∇̄∗ = 〈∆̄1, ∆̄2〉 ⊂MR, which will describe the ambient space of the mirror X of X . We find ∆̄1 = 〈ν̄1, . . . , ν̄6, 0〉, ∆̄2 = 〈ν̄7, . . . , ν̄12, 0〉, (77) where we express the vertices ν̄i in terms of the vertices νi of ∇ ν̄3k+1 = ν3k+1, ν̄3k+2 = ν3k+2 − e5, ν̄3k+3 = ν3k+3 + e5, k = 0, . . . , 3. (78) Again, it is easy to check that the ν̄i satisfy the same linear relations eq. (37) as the νi. It turns out that the lattice points of ∇̄ ∗ generate a sublattice M̄ of index 3 in M , and the lattice refinement is generated by ν̄1 + 2ν̄2 + 2ν̄7 + ν̄8 = e2 + e4 − e5. (79) Among the points of ∇∗ listed in eq. (39) only ν13 and ν14 are also lattice points of the sublattice M̄ . In fact, we have ν̄13 = ν13 and ν̄14 = ν14. Hence, ∇̄ ∗ has 12 vertices and 15 lattice points; its dual ∇̄ = ∇̄1 + ∇̄2 has 42 lattice points among which 15 are vertices10. Once we have the polytopes ∆̄∗ and ∇̄∗, we can construct X and X as complete intersections entirely analogous to X̃ and X̃∗, see Section 3. That is, using eq. (27), we define X = D̄0,1 ∩ D̄0,2, X = D̄∗0,1 ∩ D̄ 0,2 (81) in terms of the nef partitions eq. (75) and (77), respectively. Here, D̄i and D̄ i denote the divisors associated to the generators ρ̄i and ν̄i, respectively. The absence of fixed points of the G1 action on the complete intersection X̃ is guaranteed by the fact that the resulting polytope ∆̄∗ ⊂ N̄R has no additional lattice points [31]. Hence, X = X̃/G1 has a non-trivial fundamental group π1(X) = Z3. Surprisingly, it turns out that the mirror X is a free quotient as well. To see this recall that, as noticed above, the lattice points of ∇̄∗ generate a sublattice M̄ of index 3 inM . Furthermore, ∇̄∗ also has no additional lattice points with respect to ∇∗. Therefore, there is a 10Note that all of our polytopes differ from the non-free Z3 × Z3 quotient of ∆ ∗ defined in [28], Proposition 7.1. In the notation of [31] their quotient is ∇∗ 6= P 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 Z3 : 0 1 2 0 0 0 0 0 Z3 : 0 0 0 0 1 2 0 0 and has 21 points and 8 vertices in the lattice N . group G∗1 ≃ Z3 acting torically on P∇∗ . On the homogeneous coordinates this action g∗1 : z1 : · · · : z12 ζz1 : ζ 2z2 : z3 : · · · : z6 : ζ 2z7 : ζz8 : z9 : · · · : z12 . (82) Hence, X = X̃∗/G∗1 also has a non-trivial fundamental group π1(X ) = Z3. Note that this never happens for hypersurfaces in toric varieties [6]. Having the toric representation of X and X , we can now compute their Hodge numbers. It turns out = h1,2 = h1,1 = h1,2 = 7, (83) in agreement with Part A [1], eq. (??). 4.2 The Quotient by G2 We now turn to the G2 action, which does not act torically. Hence, we cannot, in principle, find a toric variety containing X = X/G2 as we did for the G1 quotient above. However, at least we have to ensure that X and X are G2-symmetric. This can be achieved via suitable symmetries in the toric data. The easy part of the toric data for X is the polytope ∆̄∗. The G2 action on the ambient space permutes the homogeneous coordinates, see eq. (8b). In terms of toric geometry, this means that it permutes the corresponding points of the polytope. That is11, g2 : ρ̄i 7→ ρ̄1+(i mod 3) ∀i ∈ {1, 2, 3}, g2 : ρ̄4 7→ ρ̄4, ρ̄5 7→ ρ̄5, g2 : ρ̄5+i 7→ ρ̄6+(i mod 3) ∀i ∈ {1, 2, 3}. It induces a mirror group actionG∗2 onX which is geometrical, rather than a quantum symmetry as discussed in [48]. The action of G∗2 is obviously the dual group action on the dual lattice M , which again must be a symmetry of the relevant polytope ∇̄∗. We find that g∗2 : ν̄3k+i 7→ ν̄3k+1+(i mod 3) ∀k = 0, . . . , 3, i ∈ {1, 2, 3}. (85) As a check on the mirror group action, note that the matrix of scalar products, see eq. (87) below, is invariant. That is, g2(ρ̄l), g 2(ν̄l′) ρ̄l, ν̄l′ ∀ l, l′. (86) By abuse of notation, we denote the corresponding cyclic permutation of homogeneous coordinates by g∗2 as well. Using this action, we define the mirror of X to be X 11We define the modulus operation such that (i mod 3) ∈ {0, 1, 2}. /G∗2. This idea has already been used for the construction of mirrors of orbifolds of the quintic [49] soon after the discovery of the first mirror construction by Greene and Plesser. Following eq. (41), the equations for the Calabi-Yau complete intersections X and are defined by evaluating the matrix of scalar products 〈ρ̄i, ν̄j〉+ δl l′ , which are 〈 , 〉+ δl l′ ν̄1 ν̄2 ν̄3 ν̄4 ν̄5 ν̄6 ν̄13 ν̄7 ν̄8 ν̄9 ν̄10 ν̄11 ν̄12 ν̄14 ρ̄1 3 0 0 3 0 0 1 0 0 0 0 0 0 0 ρ̄2 0 3 0 0 3 0 1 0 0 0 0 0 0 0 ρ̄3 0 0 3 0 0 3 1 0 0 0 0 0 0 0 ρ̄4 1 1 1 0 0 0 0 0 0 0 1 1 1 1 ρ̄5 0 0 0 1 1 1 1 1 1 1 0 0 0 0 ρ̄6 0 0 0 0 0 0 0 3 0 0 3 0 0 1 ρ̄7 0 0 0 0 0 0 0 0 3 0 0 3 0 1 ρ̄8 0 0 0 0 0 0 0 0 0 3 0 0 3 1 The equations of X can now be read off from the columns of eq. (87), and one finds F1 = (λ5t0 + λ6t1)(x 0 + x 1 + x 2) + (λ7t0 + λ8t1)x0x1x2, (88a) F2 = (λ1t0 + λ4t1)(y 0 + y 1 + y 2) + (λ2t0 + λ3t1)y0y1y2, (88b) where the G2-symmetry has been imposed. Note that the last monomial in each equation corresponds to the vector 0 ∈ ∆̄l, l = 1, 2. Two of the eight coefficients λm can be fixed by normalizing the equations, say λ4 = λ5 = 1, and three correspond to the symmetries of P1, that is, SL(2) transformations of [t0 : t1]. Hence, we can, for example, set λ6 = λ7 = λ8 = 0. This leaves us with 3 complex structure deformations λ1, λ2, and λ3, see eqns. (7a) and (7b). The equations defining X∗ correspond to the rows of eq. (87), that is, F ∗1 = a1(z 4 + z 5 + z 6)z13 + (a2z10z11z12z14 + a3z4z5z6z13)z1z2z3, (89a) F ∗2 = a4(z 10 + z 11 + z 12)z14 + (a5z4z5z6z13 + a6z10z11z12z14)z7z8z9, (89b) where, again, invariance under G∗2 has been imposed and the last monomial of each equation comes from the lattice point 0 ∈ ∇̄l, , l = 1, 2. Both equations are homo- geneous with respect to all seven scaling degrees that follow from the linear relations eq. (37). Among the twelve scalings of the coordinates zi, six are compatible with the cyclic permutations g∗2, see eq. (85). Subtracting the three G2 symmetric indepen- dent scalings among the relations eq. (37), there remains one torus action that acts effectively on the parameters plus two normalizations of the equations. As expected, the six parameters am of the equations of X ∗ thus become the 3 complex structure moduli. So far, we only considered the polytopes ∆̄∗ and ∇̄∗. However, this is only part of the toric data defining the manifolds X and X , respectively. In addition, we need the triangulations and the corresponding exceptional sets. A change in the triangulation corresponds to a flop of the toric variety. The very real danger is that not all, and perhaps none, of the flopped Calabi-Yau manifolds are G2-symmetric. For X ⊂ P∆̄∗ this turns out to be unproblematic, but for X ⊂ P∇̄∗ we will find a condition for the choice of a triangulation. 4.3 B-Model on X We now return to the discussion of the triangulations and the intersection ring of X . The analogous, but technically much more involved discussion of X will be presented in Subsection 4.5. For X everything is straightforward since the G1-quotient did not introduce ad- ditional lattice points in the associated polytope ∆̄∗. Therefore, just like for the polytope ∆∗ of the covering space X̃ , there exists a unique triangulation. In particu- lar the primitive collections, the Stanley-Reisner ideal, and the ideal IX are identical to the ones in eqns. (42), (49), and (50) since they are derived from the same triangu- lation. Moreover, one can easily see that this triangulation is G2-invariant and, hence, X is G2 symmetric. The only change is in the normalization of the intersection ring in eq. (52), since the total volume has to be divided by 3 = |G1|. This can also be seen in eq. (76), where the volume of the cone is now 3 instead of 1. Hence, on X the intersection ring and the second Chern class are J̄22 J̄3 = 1, J̄1J̄2J̄3 = 3, J̄2J̄ 3 = 1, · J̄1 = 0, c2 · J̄2 = 12, c2 · J̄3 = 12. Comparing these intersection numbers with eq. (13), it is clear that the toric divisors should be identified with the G1-invariant divisors on X as J̄1 = φ, J̄2 = τ1, J̄3 = τ2. (91) The curves spanning the Mori cone on the cover turn out to be G1-invariant as well. Therefore, the Mori cones NE(P∆̄∗) and NE(X)toric are identical to those in eqns. (60) and (61), respectively. Following the steps given in Section 3 we now want to compute the B-model pre- potential FB , plug in the mirror map, and obtain the prepotential on X (P,Q1, Q2, Q3, R1, R2, R3, b1). (92) We immediately realize the following two caveats: • We do not know how to incorporate the torsion curves H2(X,Z)tors = Z3 into the toric mirror symmetry calculation. • Of the 7 Kähler classes on X , only 3 are toric. This means that only 3 out of the 7 + 1 variables in the prepotential are accessible, and the remaining ones are set to one. Looking at the intersection numbers eq. (90), it is clear that the 3 divisors are precisely the G2-invariant divisors on X , see eq. (13). Therefore, these 3 variables must be those that map to the variables p, q, and r on X . By comparing with eq. (16), we see that the corresponding variables on X are P , Q1, and R1. Hence, we actually only compute (P,Q1, 1, 1, R1, 1, 1, 1) = n1,n2,n3 nX(n1,n2,n3) Li3 P n1Qn21 R . (93) In effect, this means that the resulting instanton numbers are not just the instantons in a single integral homology class, but the instanton numbers in a whole set of integral homology classes. The instanton numbers sum over all curve classes that cannot be distinguished by P,Q1, R1 ∈ Hom H2(X,Z),C . Up to total degree 4 and the symmetry nX(n1,n2,n3) = n (n1,n3,n2) , (94) the resulting instanton numbers are nX(1,0,0) = 27 n (1,0,1) = 108 n (1,0,2) = 378 n (1,0,3) = 1080 nX(1,1,1) = 432 n (1,1,2) = 1512 n (2,0,1) = −54 n (2,0,2) = −756 nX(2,1,1) = 864 n (3,0,1) = 9. 4.4 Instanton Numbers of X Knowing the prepotential on X , we now want to divide out the free G2 action and arrive at the prepotential on X . Since we do not know the complete expansion but only eq. (93), we have to set b1 = b2 = 1 in the descent equation (16). This yields p, q, r, 1, 1) = p, q, 1, 1, r, 1, 1, 1 n1,n2,n3 nX(n1,n2,n3) Li3 pn1qn2rn3 Up to the symmetry nX (n1,n2,n3) (n1,n3,n2) , the non-vanishing instanton numbers for X up to total degree 5 are nX(1,0,0) = 9 n (1,0,1) = 36 n (1,0,2) = 126 n (1,0,3) = 360 nX(1,0,4) = 945 n (1,1,1) = 144 n (1,1,2) = 504 n (1,1,3) = 1440 nX(1,2,2) = 1764 n (2,0,1) = −18 n (2,0,2) = −252 n (2,0,3) = −1728 nX(2,1,1) = 288 n (2,1,2) = 3960 n (3,0,1) = 3 n (3,0,2) = 252 nX(3,1,1) = 756, Unfortunately, this direct calculation misses the torsion information and only yields the expansion F X,0(p, q, r, 1, 1). The b1 dependence was lost because the toric methods do not yield this part, and the b2 dependence was lost because the relevant divisor on X was not toric. Comparing with the full expansion of the prepotential p, q, r, b1, b2) = n1,n2,n3 m1,m2 nX(n1,n2,n3,m1,m2) Li3 pn1qn2rn3bm11 b , (98) see Part A eq. (??), this means we only obtain the sum of the instanton numbers over all torsion classes nX(n1,n2,n3) = m1,m2=0 nX(n1,n2,n3,m1,m2). (99) Clearly, this destroys the torsion information, that is, the instanton numbers nX (n1,n2,n3) do not depend on the torsion part of the integral homology. For comparison purposes, we list the instanton numbers nX(n1,n2,n3) for 0 ≤ n1, n2, n3 ≤ 5 in Table 2. 4.5 B-Model on X We now study the mirror X , which sits in a more complicated ambient toric variety. Consequently, the analysis is more involved. The big advantage, however, will turn out to be that all h11(X ) = 7 Kähler moduli are toric, which will enable us to obtain the full instanton expansion. Since the polytope ∇̄∗ in eq. (78) is not simplicial, we have to specify a resolution of the singularities, that is, a triangulation T (∇̄∗). Moreover, not any triangulation will do, but we have to make sure that it is compatible with the action of the permutation group G∗2. While a tedious technicality, the existence of such a resolution has to be shown in order to establish the existence of a geometrical mirror family of X . In particular, we show in Appendix A that there is no projective resolution of the ambient space among the 720 coherent star triangulations of ∇̄∗ that respects the permutation symmetry eq. (85). In other words, if one demands G∗2 symmetry then the ambient toric variety cannot be chosen to be Kähler, but only a complex manifold. Clearly, in that case there is no Kähler cone and the usual toric mirror symmetry algorithm does not work. What comes to the rescue is that there are two classes of non-symmetric projective resolutions for which the symmetry-violating exceptional sets do not intersect X . Hence the complete intersection is G2-symmetric, even though the ambient space is not. We conclude that the extended Kähler moduli space of X contains two sym- metric phases. We will denote these two classes of triangulations by T± = T±(∇̄ see Appendix A. In fact, the two phases are topologically distinct, and only the tri- angulation T+ describes the threefold X that we are interested in. In Appendix B, we will investigate the other triangulation T− which describes a flop of X (0,n2,n3) (3,n2,n3) 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 2 3 4 5 0 0 3 252 4158 40173 287415 1 3 756 15390 164280 1259685 7763364 2 252 15390 426708 5427684 46537092 310465062 3 4158 164280 5427684 73971360 657552966 4487097816 4 40173 1259685 46537092 657552966 5948103483 41016575313 5 287415 7763364 310465062 4487097816 41016575313 284581389204 nX(1,n2,n3) n (4,n2,n3) 0 1 2 3 4 5 0 9 36 126 360 945 2268 1 36 144 504 1440 3780 9072 2 126 504 1764 5040 13230 31752 3 360 1440 5040 14400 37800 90720 4 945 3780 13230 37800 99225 238140 5 2268 9072 31752 90720 238140 571536 0 1 2 3 4 5 0 0 0 −144 −6048 −107280 −1235520 1 0 −306 −12348 −207000 −2273400 −19066500 2 −144 −12348 348480 14609520 235219680 2505155400 3 −6048 −207000 14609520 520226784 8245864800 87989812560 4 −107280 −2273400 235219680 8245864800 131759049600 1417949658000 5 −1235520 −19066500 2505155400 87989812560 1417949658000 15365394415800 (2,n2,n3) (5,n2,n3) 0 1 2 3 4 5 0 0 −18 −252 −1728 −9000 −38808 1 −18 288 3960 27648 143748 620928 2 −252 3960 54432 380160 1976472 8537760 3 −1728 27648 380160 2654208 13799808 59609088 4 −9000 143748 1976472 13799808 71748000 309920688 5 −38808 620928 8537760 59609088 309920688 1338720768 0 1 2 3 4 5 0 0 0 45 5670 189990 3508920 1 0 36 13140 474840 8793648 111499020 2 45 13140 1112886 38961252 777759975 10723515300 3 5670 474840 38961252 1952428464 47357606430 732897531720 4 189990 8793648 777759975 47357606430 1237373786439 19911043749420 5 3508920 111499020 10723515300 732897531720 19911043749420 327006066948660 Table 2: Summed instanton numbers nX (n1,n2,n3) m1,m2 (n1,n2,n3,m1,m2) (hence not distinguishing torsion) com- puted by mirror symmetry. The table contains all non-vanishing instanton numbers for 0 ≤ n1, n2, n3 ≤ 6. Following Subsection 3.3, given the triangulation T+, we can determine the prim- itive collections. This immediately yields the Stanley-Reisner ideal ISR = D̄1D̄13, D̄2D̄4, D̄2D̄13, D̄3D̄4, D̄3D̄5, D̄3D̄13, D̄4D̄10, D̄4D̄11, D̄4D̄12, D4D̄14, D̄5D̄10, D̄5D̄11, D̄5D̄12, D̄5D̄14, D̄6D̄10, D̄6D̄11, D̄6D̄12, D̄6D̄14, D̄13D̄10, D̄13D̄11, D̄13D̄12, D̄13D̄14, D̄7D̄14, D̄8D̄10, D̄8D̄12, D̄8D̄14, D̄9D10, D̄9D̄14, D̄1D̄2D̄3, D̄1D̄2D̄6, D1D̄5D̄6, D̄4D̄5D̄6, D̄7D̄8D̄9, D̄7D̄9D̄11, D̄7D̄11D̄12, D̄10D̄11D̄12 (100) where we dropped the superscript ∗ on D̄ for ease of notation. From this, in turn, we obtain the generators l̄ + of the Mori cone NE(P∇̄∗): + =( 0, 0, 0, 0, 0, 0, 1, 0, 0,−1, 0, 0, 0, 1) + =( 1, 0, 0,−1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) + =(−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0, 0, 0) + =( 0,−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0, 0) + =( 0, 0,−1, 0, 0, 1, 0,−1, 0, 0, 1, 0, 0, 0) + =( 0, 0, 0, 0, 0, 0,−1, 0, 1, 1, 0,−1, 0, 0) + =( 0, 0, 0, 0, 0, 0, 0, 1,−1, 0,−1, 1, 0, 0) + =( 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0,−3, 0) + =( 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0,−3). (101) A dual basis for the generators of the Kähler cone K(P∇̄∗) is K̄1 = D̄13 + 2 D̄1 − D̄2 − D̄3 + D̄9 + D̄7 + D̄8 + 3 D̄4, K̄2 = 3 D̄1 + D̄13 + 3 D̄4, K̄3 = D̄13 + 2 D̄1 + 3 D̄4, K̄4 = D̄13 + 2 D̄1 − D̄2 + 3 D̄4, K̄5 = D̄13 + 2 D̄1 − D̄2 − D̄3 + 3 D̄4, K̄6 = D̄13 + 2 D̄1 − D̄2 − D̄3 + D̄9 + D̄8 + 3 D̄4, K̄7 = D̄8 + D̄13 + 2 D̄1 − D̄2 − D̄3 + 3 D̄4, K̄8 = D̄4 + D̄1, K̄9 = D̄10 + D̄7. (102) The Calabi-Yau complete intersection X is then defined by X = K̄1K̄2. It turns out that the divisors D̄13, D̄14 do not intersect X . Therefore, all toric = h1,1 = 7 (103) Kähler moduli are realized torically. Since there are two divisors that do not intersect, finding the Mori cone is somewhat subtle. First, we have to restrict the lattice of linear relations to the sublattice orthogonal to these two directions. For the generators of the toric Mori cone NE(X )toric, this means that l̄ + → 3l̄ + + l̄ + , l̄ + → 3l̄ + + l̄ and that we drop l̄ + , l̄ + as well as the entries corresponding to intersections with D̄13, D̄14. In addition, we prepend the intersection numbers with D̄0,1 and D̄0,2. This yields + =(−3, 0; 0, 0, 0, 0, 0, 0, 3, 0, 0,−2, 1, 1) + =( 0,−3; 3, 0, 0,−2, 1, 1, 0, 0, 0, 0, 0, 0) + =( 0, 0;−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0, 0) + =( 0, 0; 0,−1, 1, 0, 1,−1, 0, 0, 0, 0, 0, 0) + =( 0, 0; 0, 0,−1, 0, 0, 1, 0,−1, 0, 0, 1, 0) + =( 0, 0; 0, 0, 0, 0, 0, 0,−1, 0, 1, 1, 0,−1) + =( 0, 0; 0, 0, 0, 0, 0, 0, 0, 1,−1, 0,−1, 1). (104) The dual basis of divisors is J̄∗1 = K̄21K̄2, J̄ K̄1K̄ 2 , J̄ 5 = K̄1K̄2K̄5, J̄∗3 = K̄1K̄2K̄3, J̄ 4 = K̄1K̄2K̄4, J̄∗6 = K̄1K̄2K̄6, J̄ 7 = K̄1K̄2K̄7. (105) We now try to identify this basis J̄∗1 , . . . , J̄ 7 of divisors onX with the basis {φ, τ1, υ1, ψ1, τ2, υ2, ψ2} of divisors on X in eq. (10). It turns out that there is more than one way to identify the bases if one only wants to preserve the triple intersection numbers. To obtain a unique answer, we also need to identify the actions by G∗2 and G2 as well. First, the G∗2 action on H 2(P∇̄∗ ,Z) is defined by eq. (85). Using the linear equivalence relations 2D̄1 − D̄2 − D̄3 + 2D̄4 − D̄5 − D̄6 = 0 −D̄1 + 2D̄2 − D̄3 − D̄4 + 2D̄5 − D̄6 = 0 2D̄7 − D̄8 − D̄9 + 2D̄10 − D̄11 − D̄12 = 0 −D̄2 + D̄3 − D̄5 + D̄6 − D̄8 + D̄9 − D̄11 + D̄12 = 0 −D̄4 − D̄5 − D̄6 + D̄10 + D̄11 + D̄12 − D̄13 + D̄14 = 0 (106) and the definition eq. (105), one can compute the induced group action on H2(X We find    1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 −1 1 0 0 0 0 3 −1 0 1 0 0 0 0 0 0 1 0 0 3 0 0 0 1 0 −1 0 0 0 0 1 1 −1    . (107) Second, recall that the G2 action on the divisors of X    1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 3 0 −1 0 0 0 0 3 1 −1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 3 0 −1 0 0 0 0 3 1 −1    , (108) see Part A eq. (??). The essentially unique12 identification of divisors on X and X then turns out to J̄∗1 = τ1, J̄ 2 = τ2, J̄ 3 = ψ2, J̄ 4 = υ2, J̄∗5 = φ, J̄ 6 = 3τ1 + υ1 − ψ1 = g2(ψ1), J̄ 7 = υ1. (109) Note that we are identifying divisors on X with divisors on X in eq.(109), something that one would usually not do. However, in view of the anticipated self-mirror prop- erty, X ∗ ∼= X, this is a sensible thing to try to attempt. And, indeed, the identification above is an isomorphism of the intersection rings. Regardless of this identification, we now continue to apply mirror symmetry. First, the second Chern class is · J̄∗1 = 12, c2 · J̄∗5 = 0, c2 · J̄∗2 = 12, · J̄∗3 = c2 · J̄∗6 = 24, c2 · J̄∗4 = c2 · J̄∗7 = 12. (110) Using this information, we now compute the B-model prepotential (q1, . . . , q7) = 3q5 + 3q4q5 + q25 + 3q5q7 + 3q7q5q6 + 3q4q5q7 + + 3q3q4q5 + 7 + 3q3q4q5q7 + + 3q4q3q2q5 + 3q7q5q4q6 + 3q1q5q6q7 + q45 +O(q (111) Finally, we insert the mirror map and obtain the A-model prepotential on X . Since we already identified the bases J̄∗1 , . . . , J̄ 7 with the divisors on X , we will use the same names (but with an added ∗ superscript) for the Fourier-transformed variables to expand the prepotential. With this notation, we obtain (P ∗, Q∗1, Q 3, 1) = 3P ∗ + 3 P ∗2 + 1 P ∗3 + 3 P ∗4 + 3 + 3P ∗Q∗2 + P ∗2Q∗22 + 3P ∗Q∗2Q 3 + 3P ∗R∗2 + P ∗2R∗22 + 3P ∗R∗2Q + 3P ∗R∗2Q 3 + 3P ∗R∗2R 3 + 3P ∗R∗2R 2 + 3P ∗R∗2R + 3P ∗Q∗1R 3 + 3P ∗Q∗1R 2 + 9P ∗Q∗1R 3 + 3P ∗Q∗2Q + 3P ∗Q∗2Q 2 + 9P ∗Q∗2Q total degree ≥ 6 (112) 12Up to the ĝ2 and g 2 ĝ2 symmetry. see also Part A eq. (??). The instanton numbers on X are the expansion coefficients (P ∗, Q∗1, Q 3, 1) n1,...,n7 (n1,n2,n3,n4,n5,n6,n7) P ∗n1Q∗n21 Q . (113) We see that we almost get the complete instanton expansion eq. (15), we only miss the expansion in the b∗1 variable which is not computed by the toric mirror symmetry algorithm. Up to total degree 5, the instanton numbers are (1,0,0,0,0,0,0) =3 n (1,0,0,0,0,1,0) =3 n (1,0,0,0,0,1,1) =3 n (1,0,1,0,0,0,0) =3 (1,0,1,0,0,1,0) =3 n (1,0,1,0,0,1,1) =3 n (1,0,1,1,0,0,0) =3 n (1,0,1,1,0,1,0) =3 (1,0,1,1,0,1,1) =3 n (1,1,0,0,0,1,2) =9 n (1,0,1,2,1,0,0) =9 n (1,0,1,1,1,1,0) =3 (1,1,1,0,0,1,1) =3 n (1,1,0,0,0,1,1) =3 n (1,0,1,1,1,0,0) =3. (114) Finally, let us take a look at the G∗2 action, see eq. (85). Of the 7 generators of the toric Mori cone, eq. (104), only the 3 generators l̄ + , l̄ + and l̄ + are invariant. Not surprisingly, the dual G∗2-invariant divisors J̄∗5 = φ, J̄ 1 = τ1, J̄ 2 = τ2 (115) were identified with the G2-invariant divisors on X in eq. (109). Therefore, only 3 Kähler parameters survive to the quotient X∗ = X /G∗2, and we have = h1,2 = h1,1 = h1,2 = 3. (116) 4.6 Instanton Numbers of X∗ Now that we have the expression eq. (113) for the prepotential on X , we can again apply a suitable variable substitution P ∗, Q∗1, Q p∗, q∗, r∗, b∗1, b (117) and obtain the prepotential on the quotient X∗ = X /G∗2. The correct way to replace the variables is determined by the group action on the homology and cohomology as we explained in Part A. Having computed the G∗2-action in eq. (107), we determine the descent equation for the prepotential to be13 p∗, q∗, r∗, b∗1, b |G∗2| p∗, q∗, b∗2, b ∗, b∗22 , b 2 , b . (118) 13Interestingly, eq. (118) turns out to be exactly analogous to eq. (16), even though the identifica- tion of divisors on X and X is not just a relabeling of divisors. Using the series expansion of the prepotential for b∗1 = 1 on X from Subsection 4.5, we now find that X∗,0(p ∗, q∗, r∗, 1, b∗2) Li3(p 2 ) + 4 Li3(p 2 ) + 4 Li3(p + 14 Li3(p ∗q∗2b 2 ) + 16 Li3(p ∗q∗r∗b 2 ) + 14 Li3(p ∗r∗2b + 40 Li3(p ∗q∗3b 2 ) + 56 Li3(p ∗q∗2rb 2 ) + 56 Li3(p ∗q∗r2b + 40 Li3(p ∗r∗3b 2 ) + 105 Li3(p ∗q∗4b 2 ) + 160 Li3(p ∗q∗3r∗b − 2 Li3(p ∗2q∗b 2 )− 2 Li3(p ∗2r∗b − 28 Li3(p ∗2q∗2b 2 ) + 32 Li3(p ∗2q∗r∗b 2 )− 28 Li3(p ∗2r∗2b + 3Li3(p ∗3q∗) + 3 Li3(p ∗3r∗) total p∗, q∗, r∗-degree ≥ 5 (119) The corresponding instanton numbers X∗,0(p ∗, q∗, r∗, 1, b∗2) = n1,n2,n3,m2 (n1,n2,n3,m2) p∗n1q∗n2r∗n3b∗m22 (120) are listed in Table 3. For comparison purposes, we list the summed instanton numbers (n1, n2, n3) n (n1,n2,n3,0) (n1,n2,n3,1) (n1,n2,n3,2) (n1,n2,n3) (1, 0, 0) 3 3 3 9 (1, 0, 1) 12 12 12 36 (1, 0, 2) 42 42 42 126 (1, 0, 3) 120 120 120 360 (1, 1, 1) 48 48 48 144 (1, 1, 2) 168 168 168 504 (2, 0, 1) −6 −6 −6 −18 (2, 0, 2) −84 −84 −84 −252 (2, 1, 1) 96 96 96 288 (3, 0, 1) 3 0 0 3 Table 3: Instanton numbers nX (n1,n2,n3,m2) computed by toric mirror symmetry. They are invariant under the exchange n2 ↔ n3, so we only display them for n2 ≤ n3. on X as well, see eq. (99). One observes that the sum over the more refined instanton numbers on X∗ equals the summed instanton number on X , another clue towards X being self-mirror. 4.7 Instanton Numbers Assuming The Self-Mirror Property So far, we have alluded to X being possibly self-mirror, but not actually made use of this property. Now we are going to assume the self-mirror property and, hence, obtain the prepotential on X as X,0(p, q, r, b1, b2) = F X∗,0(p, q, r, b1, b2). (121) Note that at linear and quadratic order in p we can actually recover the b1, b2 expansion from the summed instanton numbers in Subsection 4.4 and the factorization which we will prove in Section 6. In contrast, for the prepotential terms at order p3 we have to use the X∗ pre- potential to obtain the b2 expansion from eq. (119). Since this is based on a toric computation on X , we do not directly obtain the b1 expansion. However, note that the fact that g1 acted torically, eq. (8a), and g2 non-torically, eq. (8b), is just a conse- quence of the choice of coordinate system on P2×P1×P2. By a suitable coordinate choice, we could have made any one of the four Z3 subgroups of G = Z3 × Z3 act torically. Therefore, any combination of b1, b2 other than 1 = b 2 has to occur in the same way in the complete series expansion of the prepotential. We conclude that the prepotential can only depend on b1 and b2 through the combinations i,j=0 2. (122) This observation lets us recover the full b1, b2 expansion of the prepotential. To summarize, we obtain X∗,0(p, q, r, b1, b2) i,j=0 Li3(pb 2) + 4 Li3(pqb 2) + 4 Li3(prb + 14 Li3(pq 2bi1b 2) + 16 Li3(pqrb 2) + 14 Li3(pr 2bi1b + 40 Li3(pq 3bi1b 2) + 56 Li3(pq 2rbi1b 2) + 56 Li3(pqr 2bi1b + 40 Li3(pr 3bi1b 2) + 105 Li3(pq 4bi1b 2) + 160 Li3(pq 3rbi1b + 196 Li3(pq 2r2bi1b 2) + 160 Li3(pqr 3bi1b 2) + 105 Li3(pr 4bi1b − 2 Li3(p 2qbi1b 2)− 2 Li3(p 2rbi1b 2)− 28 Li3(p 2q2bi1b + 32 Li3(p 2qrbi1b 2)− 28 Li3(p 2r2bi1b 2)− 192 Li3(p 2q3bi1b + 440 Li3(p 2q2rbi1b 2) + 440 Li3(p 2qr2bi1b 2)− 192 Li3(p 2r3bi1b + 3Li3(p 3q) + 3 Li3(p + 9 Li3(p 3q2) + 27 i,j=0 Li3(p 3q2bi1b + 9 Li3(p 3q2) + 27 i,j=0 Li3(p 3q2bi1b + 27 Li3(p 3qr) + 81 i,j=0 Li3(p 3qrbi1b total p, q, r-degree ≥ 6 (123) Obtaining all of these terms required a computation of FB in eq. (111) up to total degree 23 in the 7 variables, which is close to the limit of what can be done with current desktop computers. We list the instanton numbers in Table 4. Observe that the instanton numbers sometimes do depend on the torsion part of their homology class. 5 The Self-Mirror Property When one speaks of a Calabi-Yau manifold Y being self-mirror, one has to indicate which level of invariants one is referring to. In particular, one might think of four types of invariants that are natural from the point of view of string theory. The weakest level is just the Euler number. In general, exchanging complex structure and Kähler moduli changes the sign of χ(Y ) = 2h11(Y )− 2h21(Y ). Therefore, a necessary condition for Y and its mirror Y ∗ to be equal is obviously that χ(Y ) = −χ(Y ∗) = 0. (124) This level of invariants, however, is much too crude and therefore insufficient. A much stronger level is based on the fact that the cohomology groups of even degree come with (1,n2,n3,0,0) (1,n2,n3,m1,m2) , (m1, m2) 6= (0, 0) 0 1 2 3 4 0 1 4 14 40 105 1 4 16 56 160 2 14 56 196 3 40 160 4 105 0 1 2 3 4 0 1 4 14 40 105 1 4 16 56 160 2 14 56 196 3 40 160 4 105 (2,n2,n3,0,0) (2,n2,n3,m1,m2) , (m1, m2) 6= (0, 0) 0 1 2 3 0 0 −2 −28 −192 1 −2 32 440 2 −28 440 3 −192 0 1 2 3 0 0 −2 −28 −192 1 −2 32 440 2 −28 440 3 −192 (3,n2,n3,0,0) (3,n2,n3,m1,m2) , (m1, m2) 6= (0, 0) 0 1 2 0 0 3 36 1 3 108 0 1 2 0 0 0 27 1 0 81 Table 4: Instanton numbers nX (n1,n2,n3,m1,m2) computed by mirror symmetry. The table contains all non-vanishing instanton numbers for n1+n2+ n3 ≤ 5. The entries marked in bold depend non-trivially on the tor- sion part of their respective homology class. an integral lattice structure and form a ring, and therefore have a product. Because of Poincaré duality, that is, H2(Y ) = H4(Y )∨, it is sufficient to look at H2(Y ). There is a product H2(Y )×H2(Y ) → H2(Y ) whose structure constants κijk are the triple intersection numbers. These intersection numbers are finer invariants than just the dimensions of the cohomology groups, and a self-mirror Calabi-Yau threefold should satisfy κijk(Y ) = κijk(Y ∗). (125) For simply connected threefolds with torsion-free homology a theorem of Wall [41] states that the cohomology groups with the intersection product κijk(Y ) together with the second Chern class c2(Y ) determine the diffeomorphism type of Y . If, however, Y and Y ∗ have non-trivial fundamental groups then we cannot con- clude that easily that they are diffeomorphic. But the non-trivial fundamental group is often reflected in torsion in homology (for example if π1(Y ) is Abelian). In that case, the conjecture of [6] says that for any Calabi-Yau threefold Z . (126) Therefore, a self-mirror manifold Y = Y ∗ is expected to satisfy . (127) Of the many spaces Y satisfying eq. (124) there are only a few which also satisfy eq. (125). So far we only considered classical topology, but we know that the ring H2(Y ) experiences quantum corrections when going far away from the large volume limit. At small volume the intersection numbers are replaced by the three-point functions Cijk(q) of (topological) conformal field theory in eq. (69). In the large volume limit q goes to zero and the Cijk(q) go to κijk, as expected. The Cijk(q) are characterized by the genus zero instanton numbers n = nd. In mathematical terms, these are resummations of the Gromow-Witten invariants of Y and characterize the symplectic structure of Y . This level of invariants is even stronger than the cohomology ring, since there are examples of diffeomorphic manifolds which have different Calabi-Yau structures, i.e. different n [50, 51, 31]. Therefore, a self-mirror Calabi-Yau threefold Y must satisfy (Y ) = n (Y ∗). (128) One can go even further and couple the topological conformal field theory to topolog- ical gravity and define higher genus instanton numbers n , where now d (Y ) = n ∗), g > 0 (129) has to hold. These invariants are very difficult to compute, however see [52, 53] for recent progress. We do not know whether they contain more information about the symplectic structure than the genus zero invariants. In other words, there are presently no examples known whose n agree for g = 0 but differ for g > 0. Now, one can start with any Y and use some method to construct the mirror Y ∗. Among these are the Greene-Plesser construction in conformal field theory, or its geometric generalizations by Batyrev and Borisov for complete intersections in toric varieties. Then, to show that Y is self-mirror one proceeds to compute the various invariants. The simplest condition, eq. (124), can directly be checked in terms of the toric data. This concretely means that one starts with a mirror pair Y and Y ∗ satis- fying eq. (124) and checks whether eqns. (125), (127), (128), and (129) are satisfied. In fact, in Section 4 we collected a large amount of evidence in favor of the claim that X and its Batyrev-Borisov mirror threefold X∗ are the same. Indeed, eqns. (40), (83) and (116) show that X̃ , X , and X satisfy by construction the constraint eq. (124) on the Euler number. More interestingly, by the identifications found in eqns. (109) and (115) we observed that the condition on the intersection ring, eq. (125), is sat- isfied for X and X , respectively. Next, eq. (97) and Table 3 show that X also fulfils the requirement eq. (128) on the genus zero instanton numbers. It would be very interesting to see whether also the condition eq. (129) for higher genus curves can be Finally, we consider the torsion in cohomology. In Part A ?? we have shown that ≃ Z3 ⊕ Z3, (130) as we expect from a self-mirror threefold. Moreover, we can actually compute the fundamental group of the Batyrev-Borisov mirror independently. For that, first notice that the quotient X = X̃∗/G∗1 is fixed-point free, see Subsection 4.2. The mirror permutation G∗2 onX acts freely as well. Therefore, bothX andX∗ are free quotients by a group isomorphic to Z3 ⊕ Z3, thus their fundamental groups are ≃ Z3 ⊕ Z3. (131) Moreover, on can easily show that on a proper14 Calabi-Yau threefold Z one has H2(Z,Z)tors = π1(Z)ab, the Abelianization of the fundamental group. Hence, we see H3(X,Z)tors ≃ Z3 ⊕ Z3 ≃ H 2(X∗,Z)tors (132) and the first of eq. (126) is true. This provides the first evidence for the conjecture of [6] in a context other than toric hypersurfaces. Another point of view is that there is a geometrical or rather combinatorial reason for the self-mirror property in this case. From eqns. (36) and (39) one can easily see that the lattice points νi, ν6+i, ν13, ν14, i = 1, . . . , 3, span a sub-polytope of ∇ satisfying the same linear relations as all the lattice points ρi of ∆ ∗ in eq. (23). Hence, 14A proper Calabi-Yau threefold has holonomy group the full SU(3). In particular, this implies that the fundamental group is finite. this sub-polytope is isomorphic to ∆∗. The same is true for the polytopes ∇̄∗ and ∆̄∗. The toric variety P∇̄∗ which is the ambient space of X can therefore be regarded as a blow-up of a quotient of P∆̄∗ , the ambient space of X . Actually, this blow-up makes all 7 divisors of X toric. Similarly, P∇∗ can be regarded as a blow-up of a quotient of P∆∗ . As shown in Subsection 3.3 this entails that all 19 Kähler moduli of X̃∗ are realized torically. Note that it is possible that the mirror polytopes ∆∗ and ∇∗ are actually isomorphic. In fact, for toric hypersurfaces there are 41, 710 self-dual polytopes [54]. The novel feature in our case is that non-isomorphic polytopes lead to self-mirror complete intersections, consistent with the nef partitions. 6 Factorization vs. The (3,1,0,0,0) Curve One interesting observation is that the prepotential F X,0 at order p, see eq. (123) in this paper and eq. (??) in Part A [1], factors into i,j=0 b 2 times a function of p, q, r only. This means that the instanton number for any pseudo-section (curve contributing at order p) does not depend on the torsion part of its homology class. In other words, for any pseudo-section there are 8 other pseudo-sections with the same class in H2(X,Z)free and together filling up all of H2(X,Z)tors = Z3 ⊕Z3. In contrast, this factorization does not hold at order p3. For example, X,0(p, q, r, b1, b2) = · · ·+ 3p b1 + b 1 + b2 + b1b2 + b 1b2 + b 2 + b1b 2 + b + · · · . (133) The purpose of this subsection is to understand this behavior. First, the factorization of the prepotential at any order of p not divisible by 3 follows from an extra symmetry that we have not utilized so far. The covering space X̃ is, in addition to eqns. (8a) and (8b), also invariant under another Ĝ = Z3 × Z3 action generated by (ζ ĝ1 : [x0 : x1 : x2] 7→ [x0 : ζx1 : ζ [t0 : t1] 7→ [t0 : t1] (no action) [y0 : y1 : y2] 7→ [y0 : y1 : y2] (no action) (134a) ĝ2 : [x0 : x1 : x2] 7→ [x1 : x2 : x0] [t0 : t1] 7→ [t0 : t1] (no action) [y0 : y1 : y2] 7→ [y0 : y1 : y2] (no action) (134b) This symmetry has fixed points and, therefore, cannot be used if one is looking for a smooth quotient of X̃ . However, it commutes with G and hence descends to a Ĝ = Z3 × Z3 symmetry of X (with fixed points). Clearly, the instanton sum must observe this additional geometric symmetry. To make use of this symmetry, we have to express its action on the variables in F X,0(p, q, r, b1, b2). We can do so by first noting that the basic 81 curves s1×s2 ⊂ X̃, s1 ∈MW (B1), s2 ∈MW (B2) (135) are really one orbit under G × Ĝ. Recall that, after dividing out G, these curves became the 9 sections in MW (X) = Z3 ⊕ Z3, see Part A ??. We now observe that MW (X) = {sij} is one Ĝ-orbit; since each of these sections contributes pb i, j = 0, . . . , 2 the induced Ĝ action on the prepotential must be ĝ1 : F X,0(p, q, r, b1, b2) 7→ F X,0(b1p, q, r, b1, b2), ĝ2 : F X,0(p, q, r, b1, b2) 7→ F X,0(b2p, q, r, b1, b2). (136) Clearly, the prepotential must be invariant under the ĝ1, ĝ2 action. While imposing no constraint on the p3n terms in the prepotential, all other powers of p must appear in the combination i,j=0 , n 6≡ 0 mod 3. (137) This proves the factorization observed at the beginning of this subsection. Second, we would like to understand the p3q terms in eq. (133). These are the curves in the homology classes15 (3, 1, 0, ∗, ∗) ∈ Z3 ⊕ Z3 ⊕ Z3 = H2 . (138) We will show that the rational curves in this class come in a single family, that is, the moduli space of genus 0 curves on X in these homology classes X, (3, 1, 0, ∗, ∗) (139) is connected. In particular, all such curves have the same homology class (3, 1, 0, 0, 0) and only contribute to p3q in the prepotential eq. (133). As discussed in Part A ??, any such map CX : P 1 → X factors CX // C eX �� �������� . (140) 15Recall that the exponent of p is the degree along the base P1. This is why we pick a basis in H2(X,Z)free such that a curve in (n1, n2, n3,m1,m2) contributes at order p n1qn2rn3bm11 b 2 in the prepotential. The map C can be written in terms of homogeneous coordinates as a function : P1[z0:z1] 7→ P [x0:x1:x2] ×P1[t0:t1] ×P [y0:y1:y2] (141) satisfying the equations (7a) and (7b) defining X̃ , F1 ◦ C eX [z0 : z1] = 0 = F2 ◦ C eX [z0 : z1] ∀[z0 : z1] ∈ P 1 . (142) The curve CX ends up in the homology class (3, 1, 0, ∗, ∗) if and only if the defining equation (141) is of degree (3, 1, 0) in P2×P1×P2. Hence, eq. (141) is defined by complex constants αij, βij , γi (up to rescaling) such that xi = αi0 z0 + αi1 z1 i = 0, 1, 2 ti = βi0 z 0 + βi1 z 0z1 + βi2 z0z 1 + βi3 z 1 i = 0, 1 yi = γi i = 0, 1, 2. (143) These constants have to be picked such that the resulting curve lies on the complete intersection X̃, that is, they have to satisfy eq. (142). Inserting eq. (143), we find that F1 ◦ C eX [z0 : z1] is a homogeneous degree 6 polynomial in [z0 : z1]. Since the coefficients of zk0z 1 must vanish individually, this yields 7 constraints for the parameters αij , βij . What makes this system of constraint equations tractable is the fact that they are all linear in βij , F1 ◦ C eX = 0 ⇔  A1 0 0 0 A5 0 0 0 A2 A1 0 0 A6 A5 0 0 A3 A2 A1 0 A7 A6 A5 0 A4 A3 A2 A1 A8 A7 A6 A5 0 A4 A3 A2 0 A8 A7 A6 0 0 A4 A3 0 0 A8 A7 0 0 0 A4 0 0 0 A8    = 0 (144) where = α300 + α 10 + α 20 A5 = α00α10α20 = 3α01α 00 + 3α11α 10 + 3α21α 20 + α 20 A6 = (α01α10 + α00α11)α20 + α00α10α21 = 3α201α00 + 3α 11α10 + 3α 21α20 A7 = α01α11α20 + (α01α10 + α00α11)α21 = α301 + α 11 + α 21 A8 = α01α11α21. (145) Thinking of this as 7 linear equations for the 8 parameters βij, there is always a non- zero solution. The solution is generically unique up to an overall factor, and turns into an Pn for special values of the αij . Moreover, the parameter space of the αij is connected (essentially, the moduli space of lines in P2). Since we just identified the parameter space of the (αij, βij) as a blow-up thereof, it is therefore connected as well. It remains to satisfy F2 ◦ C eX = 0. One can easily see that the only way is to pick the γi to be simultaneous solutions of γ30 + γ 1 + γ 2 = 0 = γ1γ2γ3. (146) Since two cubics intersect in 9 points, there are 9 such solutions, permuted by G. Therefore, the parameter space of (αij , βij, γi) has 9 connected components, permuted by the G-action. The moduli space of curves CX on X is the G-quotient of the moduli space of curves C on X̃, and therefore has only a single connected component. By continuity, every curve CX in this connected family has the same homology class, explaining the piece of the prepotential given in eq. (133). 7 Towards a Closed Formula Putting all the information together we found out about the prepotential on X , one can try to divine a closed form for the prepotential. We guess that the order pn terms have the closed form X,0(p, q, r, b1, b2) i,j∈Z3 P (q)4P (r)4 M2n−2(q, r) (147) if n is not a multiple of 3 and, slightly weaker, that X,0(p, q, r, 1, 1) P (q)4P (r)4 M2n−2(q, r) (148) if n is a multiple of 3. Here, • P (q) is the usual generating function of partitions eq. (4). • The M2n−2 are polynomials in the Eisenstein series E2(q), E4(q), E6(q) and E2(r), E4(r), E6(r), starting with M−2(q, r) = 0 M0(q, r) = 1 M2(q, r) = E2(q)E2(r) M4(q, r) = E4(q)E4(r) + E4(q)E2(r) 2 + E2(q) 2E4(r) E2(q) 2E2(r) M6(q, r) = E6(q)E6(r) + E6(q)E4(r)E2(r) + E4(q)E2(q)E6(r) E6(q)E2(r) 3 + E2(q) 3E6(r) E4(q)E2(q)E4(r)E2(r) E2(q) 3E4(r)E2(r) + E4(q)E2(q)E2(r) E2(q) 3E2(r) M8(q, r) = E6(q)E2(q)E6(r)E2(r) + E4(q)E4(q)E4(r)E4(r) E6(q)E2(q)E4(r)E4(r) + E4(q)E4(q)E6(r)E2(r) E6(q)E2(q)E4(r)E2(r) 2 + E4(q)E2(q) 2E6(r)E2(r) E6(q)E2(q)E2(r) 4 + E2(q) 4E6(r)E2(r) + 137 E4(q)E4(q)E4(r)E2(r) 2 + E4(q)E2(q) 2E4(r)E4(r) E4(q)E4(q)E2(r) 4 + E2(q) 4E4(r)E4(r) E4(q)E2(q) 2E4(r)E2(r) 2 + 121 E2(q) 4E2(r) E4(q)E2(q) 2E2(r) 4 + E2(q) 4E4(r)E2(r) (149) They are symmetric under the exchange q ↔ r and of weight 2n in q and r separately. But, for example, M4 above does not factor into a function of q and a function of r. So theM2n−2 are not the products of the polynomials appearing in the dP9 prepotential. However, by setting q = 0 or r = 0 one recovers the corresponding polynomials in the dP9 prepotential [55]. • The E2i are the usual Eisenstein series E2(q) = 1− 24q − 72q 2 − 96q3 − 168q4 − 144q5 − 288q6 +O(q7) E4(q) = 1 + 240q + 2160q 2 + 6720q3 + 17520q4 + 30240q5 +O(q6) E6(q) = 1− 504q − 16632q 2 − 122976q3 − 532728q4 +O(q5). (150) Note that the naive Taylor series coefficients of the prepotential are fractional, but when expanding in terms of Li3’s (which account for the multicover contributions) one finds integral instanton numbers. These expressions for the prepotential agree with all instanton numbers computed in this paper. Unfortunately, we have not been able to guess a closed formula that includes the b1 and b2 dependence of the prepotential F X,0(p, q, r, b1, b2)|pn if n is divisible by 3. We expect that these involve extra functions beyond the Eisenstein series. 8 Conclusion In the initial paper Part A [1], we analyzed the topology of the Calabi-Yau manifold of interest and found that = Z3 ⊕ Z3 ⊕ Z3. (151) Although the presence of torsion curve classes complicates the counting of rational curves, we managed to derive the A-model prepotential to linear order in p. The goal of this paper is to go beyond the results of Part A using mirror symmetry. By carefully adapting methods designed for complete intersections in toric varieties, we can apply mirror symmetry to compute the instanton numbers onX , even thoughX is not toric. Using thatX is self-mirror, we completely solve this problem and are able to calculate the complete A-model prepotential to any desired precision (and for arbitrary degrees in p), limited only by computer power. Carrying out this computation, we find the first examples of instanton numbers that do depend on the torsion part of their integral homology class, see Table 4 on Page 35. Since the self-mirror property of X is important, we investigate it in detail. In doing so, we go far beyond just checking that the Hodge numbers are self-mirror. In particular, we find that the intersection rings are identical and that torsion in homology obeys the conjectured mirror relation [6]. Finally, going beyond classical geometry, we independently calculate certain instanton numbers onX and its Batyrev- Borisov mirror X∗. Again, we find that X and X∗ are indistinguishable, providing strong evidence for X being self-mirror. Both of these results extend those found in Part A [1]. Using these results, we are able to guess certain closed expressions for the pre- potential of X in terms of modular forms. In certain limits it specializes to the dP9 prepotential of [55]. There it is known that the coefficients in p of the dP9 prepoten- tial satisfy a recursion relation. Moreover, there is a gap condition, that is, a certain number of subsequent terms in a series expansion is absent. This condition provides sufficient data to determine the integration constants for the recursion and allows to determine the prepotential completely, even at higher genus. We expect a similar story to be valid for the prepotential of X . Acknowledgments The authors would like to thank Albrecht Klemm, Tony Pantev, and Masa-Hiko Saito for valuable discussions. We also thank Johanna Knapp for providing a Singular [56] code to compute the intersection ring of Calabi-Yau manifolds in toric varieties. This research was supported in part by the Department of Physics and the Math/Physics Research Group at the University of Pennsylvania under cooperative research agree- ment DE-FG02-95ER40893 with the U. S. Department of Energy and an NSF Focused Research Grant DMS0139799 for “The Geometry of Superstrings”, in part by the Austrian Research Funds FWF grant number P18679-N16, in part by the European Union RTN contract MRTN-CT-2004-005104, in part by the Italian Ministry of Uni- versity (MIUR) under the contract PRIN 2005-023102 “Superstringhe, brane e inter- azioni fondamentali”, and in part by the Marie Curie Grant MERG-2004-006374.E. S. thanks the Math/Physics Research group at the University of Pennsylvania for kind hospitality. A Triangulation of ∇̄∗ and ∇∗ In principle the coherent triangulations of the fan over ∇̄∗ can be computed with TOPCOM by finding the 720 star triangulations in the total of 230, 832 coherent tri- angulations of ∇̄∗. The discussion of the symmetry properties is greatly facilitated, however, by an explicit understanding of their structure. We will work out the trian- gulations by first triangulating the facets and then checking the compatibility of their maximal intersections and the coherence of the resulting star triangulations. We start with a couple of useful definitions. A circuit is a minimal collection of n affinely dependent points p1, . . . , pn, λ1p1 + . . . λnpn = 0 with λ1 + . . .+ λn = 0, λi 6= 0, (152) any proper subset of which is affinely independent. The coefficient vector λn hence has nonzero entries and is unique up to a prefactor. We indicate the unique separation into points with positive and negative coefficients with the notation 〈pi1 . . . pis|pis+1 . . . pin〉. Each circuit admits two different triangulations, which are obtained by dropping one of the points with positive coefficients and one of the remaining points, respectively. We indicate this with a hat over the relevant subset. The two resulting triangulations ̂pi1 . . . pis |pis+1 . . . pin pi1 . . . pis| ̂pis+1 . . . pin (153) hence consist of s and n− s simplices, respectively. If the first point is in the convex hull of the others, that is, s = 1, then only one of the triangulations is maximal (all points are vertices of at least one simplex). Furthermore, we introduce the notation: ai = ν̄i, bi = ν̄3+i, ci = ν̄6+i, di = ν̄12+i, i = 1, 2, 3, e = ν̄13, f = ν̄14. (154) Among these 14 vectors in eq. (154) there are 9 independent linear relations, see eq. (37), a1 + a2 + a3 = 0, c1 + c2 + c3 = 0, e+ f = 0, bi = ai + e, dl = cl + f, (155) which imply others like ai + bj = aj + bi and ai + cl = bi + dl or e = (b1 + b2 + b3) and f = 1 (d1 + d2 + d3). Lemma 1. ∇̄∗ has 15 facets, 6 of which are simplicial: aiajbibjclcmdldm] i<j aiajd1d2d3 b1b2b3clcm . (156) The nine non-simplicial facets form an orbit under the permutation symmetries Zab3 × Z 3 generated by gab : ( ( ai+1 and gcd : ( cl+1 . According to the linear relations eq. (155) the eight points on each non-simplicial facet form quadratic circuits ai+ bj = bi+aj , ai+ cl = bi+dl, and cl+dm = cm+dl, which we call mixed if they contain vertices of both elements of the nef partition 〈aicl|bidl〉, and pure circuits 〈aibj |biaj〉, 〈cldm|cmdl〉 otherwise. The coherent triangulations of the facets [aiajbibjclcmdldm] are most easily ob- tained from their Gale transform 1 −1 −1 1 0 0 0 0 1 0 −1 0 1 0 −1 0 0 0 0 0 1 −1 −1 1  , (157) which is the coefficient matrix of the basis ai−aj −bi+bj = 0, ai−bi+cl−dl = 0 and cl− cm−dl+dm = 0 of linear relations. The coherent triangulations are in one-to-one correspondence to chambers that are seperated by the facets of the cones generated by all linear bases µ = {v1, v2, v3} with vi selected among the 8 column vectors of the Gale transform [57, 58]. In the present case the cones over the faces of the parallel- epiped in Figure 1 are subdivided into 24 chambers, which are indicated by dashed lines. The triangulations, which we can label by the facet containing and the edge adjoining the chamber, are obtained as the sets of complements of those bases µ that span a cone containing the respective chamber. Hence, each non-simplicial facet has 24 coherent triangulations, which can be characterized by the triangulations of its 2 pure and of its 4 mixed circuits: Calling the triangulation 〈âicl|bidl〉 positive and the triangulation 〈aicl|b̂idl〉 negative, and arranging the cyclic permutations gab and gcd in the horizontal and vertical direction, respectively, we can assign one of 16 different types ± ±± ± to each triangulation, where Figure 1: Secondary fan of the non-simplicial facets. Chambers are indicated by dashed lines. the signs indicate the induced triangulations of the mixed circuits. The constraints that reduce the a priori 32 = 26 combinations to 24 all derive from the following rules: ai aj bi bj aicl|b̂idl âjcl|bjdl aibj |âjbi âicl|bidl ajcl|b̂jdl âibj |ajbi 〉 (158) i.e. a triangular prism can be triangulated in 6 different ways, which correlates the a priori 8 combinations of the triangulations of the 3 squares (with analogous constraints for the two “horizontal” prisms [aibiclcmdldm] contained in the facet [aiajbibjclcmdldm]). Putting the pieces together we obtain Lemma 2. The 24 triangulations of the non-simplicial facets can be assorted as fol- lows: • For + ++ + , − − the pure circuits are unconstrained, yielding 2 · 2 2 = 8 triangu- lations. • For + +− − , + + the pure ab-circuit is unconstrained; with the transposed types + − , − + this accounts for another 8 triangulations. • The final 8 triangulations come from the 8 types with an odd number of positive signs, for which the triangulation of the pure circuits is unique. • The two types + −− + and + − cannot occur because of contradictory implications for the triangulations of the pure circuits. The secondary fan and the induced triangulations for the codimension-two faces at which the non-simplical facets intersect can be obtained from Figure 1 by projection along the dropped vertices. The secondary fan of the prism of eq. (158), for example, which is shown in Figure 2, is obtained from Figure 1 by projection along the diagonal Taibj 〈aicl|bidl〉s Taicl 〈ajcl|bjdl〉 Tajcl 〈ajbi|aibj〉 Tajbi 〈bidl|aicl〉 s Tbidl ��〈ajcl|bjdl〉 bi Tbjdl 〈ajbi|aibj〉 Figure 2: Secondary fan of the codimension two face [aiajbibjcldl]. 〈cmdm〉. The wall crossings between the six cones in Figure 2 are labeled by the circuits whose flops relate the adjoining triangulations [57]. For the construction of the complete star triangulation we now observe that the non-simplicial intersections of the 9 non-simplicial facets [aiajbibjclcmdldm] are given by the 18 triangular prisms [aiajbibjcldl] and [aibiclcmdldm]. If we interpret the former as vertices and the latter as links then the resulting compatibility conditions corre- spond to a graph with the topology of a torus. The vertices of this graph are decorated by signs as shown in Table 5 and connected by horizontal and vertical links. The re- 1 · 26 9 · 22 18 · 22 6 · 24 36 · 20 36 · 21 9 · 22 Table 5: The 824 = 2 (64 + 36 + 72 + 96 + 36 + 72 + 36) star triangulations of ∇∗, including the 720 = 2 (36 + 36+ 72+ 72+ 36+ 72+ 36) coherent triangulations. striction on the compatible signs is due to the absence of the inconsistent types + −− + and − ++ − as subgraphs on the torus. The multiplicities µ · 2 n come from the number n of unconstrained pure circuits and from the order µ of the effective part of the symmetry group generated by transposition and permutations of lines and columns. We thus find a total of 824 triangulations. The cyclic permutation symmetry that we want to keep on the Calabi-Yau manifold X amounts to a diagonal shift, i.e. its induced action on the graph is generated by gabgcd. We are hence left with the types , and the shift symmetry furthermore aligns the triangulations of the pure circuits and thus reduced the multiplicities from 26 to 22, yielding a total of 8 triangulations for which P∇̄∗ is G 2 symmetric. The resulting triangulations of the facet [a2a3b2b3c2c3d2d3] are triangulation of [a2a3b2b3c2c3d2d3] 〈â2b3|a3b2〉, 〈ĉ2d3|c3d2〉 {[a3b2b3d2d3], [b2b3c3d2d3], [a2a3b2d2d3], [b2b3c2c3d2]} 〈â2b3|a3b2〉, 〈c2d3|ĉ3d2〉 {[a3b2b3d2d3], [b2b3c2d2d3], [a2a3b2d2d3], [b2b3c2c3d3]} 〈a2b3|â3b2〉, 〈ĉ2d3|c3d2〉 {[a2b2b3d2d3], [b2b3c3d2d3], [a2a3b3d2d3], [b2b3c2c3d2]} 〈a2b3|â3b2〉, 〈c2d3|ĉ3d2〉 {[a2b2b3d2d3], [b2b3c2d2d3], [a2a3b3d2d3], [b2b3c2c3d3]} (159) triangulation of [a2a3b2b3c2c3d2d3] 〈a2b3|â3b2〉, 〈c2d3|ĉ3d2〉 {[a2a3b3c2c3], [a2a3c2c3d3], [a2b2b3c2c3], [a2a3c2d2d3]} 〈a2b3|â3b2〉, 〈ĉ2d3|c3d2〉 {[a2a3b3c2c3], [a2a3c2c3d2], [a2b3b2c2c3], [a2a3c3d2d3]} 〈â2b3|a3b2〉, 〈c2d3|ĉ3d2〉 {[a2a3b2c2c3], [a2a3c2c3d3], [a3b2b3c2c3], [a2a3c2d2d3]} 〈â2b3|a3b2〉, 〈ĉ2d3|c3d2〉 {[a2a3b2c2c3], [a2a3c2c3d2], [a3b2b3c2c3], [a2a3c3d2d3]} (160) It can be checked that the triangulations listed in eqns. (159) and (160) come from the chambers contained in the cones over [a2a3c2c3] and [b2b3d2d3], respectively. For the first of these triangulations we consider the chamber adjoining the edge [a2c2], which is contained in the span of the four bases µ1 = {a2c2c3}, µ2 = {a2a3c2}, µ3 = {b3c2c3} and µ4 = {a2a3d3}, whose complements are [a3b2b3d2d3], [b2b3c3d2d3], [a2a3b2d2d3] and [b2b3c2c3d2] in agreement with the first triangulation in eq. (159). Unfortunately, coherent triangulations of the facets that induce the same trian- gulations on their common (maximal) intersections do not automatically combine to coherent star triangulations of the polytope, and indeed only 720 of the 824 triangu- lations in Table 5 turn out to be coherent. The non-coherent ones are easily isolated by observing that coherent triangulations (via their height functions) induce coherent triangulations of the prisms [a1a2a3b1b2b2] and [c1c2c3d1d2d3], which eliminates the triangulations for which Zab3 or Z 3 is not broken by the triangulation of the pure circuits. For the triangulation types this reduces the multiplicity from 82 to 62. The only other affected types are the ones in the middle column of Table 5, which have unbroken horizontal symmetry and for which the multiplicity is reduced from 12 · 8 to 12 · 6. This poses a problem for the eight Z3-symmetric triangulations, which are all non-coherent. Coherence of the remaining 720 triangulations can be established by checking that their Mori cones are all strictly convex [59]. What comes to our rescue is that, even if all projective ambient spaces break the diagonal Z3 permutation symmetry, it may be preserved on X if the obstructing ex- ceptional sets do not overlap with the complete intersection. In the present case these are the blow-ups of the singularities coming from the pure circuits, i.e. codimension two sets of the form ai · bj or cl · dm, where we use, for simplicity, the symbol of the vertex ν̄j for the corresponding divisor Dj . Recall from eq. (77) that X is given by the product D̄∗0,1 · D̄ 0,2 of the divisors D̄∗0,1 = a1 + a2+ a3 + b1 + b2 + b3 + e, D̄ 0,2 = c1 + c2 + c3+ c1+ d2+ d3+ f (161) defined by the nef partition. Taking into account the five linear equivalences, we observe that a1 + b1 = a2 + b2 = a3 + b3, c1 + d1 = c2 + d2 = c3 + d3, b1 + b2 + b3 + e = d1 + d2 + d3 + f, (162) for divisor classes in the intersection ring. We first show that e and f do not intersect : In any maximal triangulation e and f belong only to the simplices b̂1b2b3cmcle d̂1d2d3amalf , (163) respectively, so that e·ai = e·dl = e·f = 0 ⇒ e·D̄ 0,1 = e·(b1+b2+b3+e) = e·(d1+d2+d3+f) = 0 (164) and similarly f · D̄∗0,2 = 0. Putting everything together, we conclude that a1 · b2 · D̄ 0,1 = a1 · b2 · 3(a3 + b3) = 0 (165) because none of the facets, and hence no triangle in any of the triangulations contains {a1, b2, a3} or {a1, b2, b3} as a subset. Similarly cl · dm · D̄ 0,2 = 0 in the intersection ring for l 6= m. Consequently, all exceptional sets arising from triangulations of pure circuits do not intersect X and hence do not obstruct the cyclic permutation symmetry G∗2. We will denote any of the remaining 36 coherent triangulations of type by T+ = T+(∇̄ ∗) and T− = T−(∇̄ ∗), respectively. The polytope ∇∗ of the mirror X̃∗ of the universal cover has 39 lattice points, with the same 12 vertices as ∇̄∗ but living on the finer lattice M̄ . The 24 additional lattice points, see eq. (39), are aij = (ai + 2aj), bij = (bi + 2bj), (166) cij = (ci + 2cj), dij = (di + 2dj), (167) where i 6= j. These additional points are all located on edges of ∇∗. It is natural to consider triangulations that are refinements of the ones that we just discussed. Observing that the additional points turn all simplices in eqns. (159), (160) and (163) into pyramids over a tetrahedron with interior points on opposite edges it is easy to see that the maximal triangulations are unique and multiply the number 54 = 9 · 4+6 · 3 of triangles in the original triangulations by a factor of 9. The resulting triangulations have been used to show that the divisors corresponding to the vertices aij and cij do not intersect X̃∗. B The Flop of X∗ In Subsection 4.5 we have taken into account only one of the triangulations T+(∇̄ We can repeat the same calculation with one of the triangulations T−. We denote the resulting Calabi-Yau manifold by X −. Skipping the details, we find that the generators of the Mori cone NE(X −) can be expressed in terms of those of NE(X in eq. (104) as − = l̄ + + 3 + + l̄ + + l̄ + + l̄ − = l̄ + + 3 + + l̄ + + l̄ − = l̄ + + l̄ − = − l̄ − = − l̄ + − l̄ + − l̄ + − l̄ + − l̄ − = l̄ − = l̄ (168) One can also express the dual basis of divisors J̄ ′i on the flop in terms of the dual basis J̄∗i on X , see eq. (105). We find J̄ ′1 = J̄ 1 , J̄ 2 = J̄ 2 , J̄ 5 = 3J̄ 1 + 3J̄ 2 − J̄ J̄ ′3 = 3J̄ 1 + J̄ 3 − J̄ 5 , J̄ 4 = 3J̄ 1 + J̄ 3 − J̄ J̄ ′6 = 3J̄ 2 − J̄ 5 + J̄ 6 , J̄ 7 = 3J̄ 2 − J̄ 5 + J̄ (169) The intersection ring is J̄ ′21 J̄ 2 = 1, J̄ 3 = 2, J̄ 4 = 1, J̄ 5 = 3, J̄ 6 = 3, J̄ ′21 J̄ 7 = 3, J̄ 2 = 1, J̄ 3 = 3, J̄ 4 = 3, J̄ 5 = 3, J̄ ′1J̄ 6 = 3, J̄ 7 = 3, J̄ 3 = 6, J̄ 4 = 6, J̄ 5 = 9, J̄ ′1J̄ 6 = 9, J̄ 7 = 9, J̄ 4 = 3, J̄ 5 = 9, J̄ 6 = 9, J̄ ′1J̄ 7 = 9, J̄ 5 = 9, J̄ 6 = 9, J̄ 7 = 9, J̄ 6 = 9, J̄ ′1J̄ 7 = 9, J̄ 7 = 9, J̄ 3 = 3, J̄ 4 = 3, J̄ 5 = 3, J̄ ′22 J̄ 6 = 2, J̄ 7 = 1, J̄ 3 = 9, J̄ 4 = 9, J̄ 5 = 9, J̄ ′2J̄ 6 = 9, J̄ 7 = 9, J̄ 4 = 9, J̄ 5 = 9, J̄ 6 = 9, J̄ ′2J̄ 7 = 9, J̄ 5 = 9, J̄ 6 = 9, J̄ 7 = 9, J̄ 6 = 6, J̄ ′2J̄ 7 = 6, J̄ 7 = 3, J̄ 3 = 18, J̄ 4 = 18, J̄ 5 = 27, J̄ ′23 J̄ 6 = 27, J̄ 7 = 27, J̄ 4 = 18, J̄ 5 = 27, J̄ 6 = 27, J̄ ′3J̄ 7 = 27, J̄ 5 = 27, J̄ 6 = 27, J̄ 7 = 27, J̄ 6 = 27, J̄ ′3J̄ 7 = 27, J̄ 7 = 27, J̄ 4 = 9, J̄ 5 = 27, J̄ 6 = 27, J̄ ′24 J̄ 7 = 27, J̄ 5 = 27, J̄ 6 = 27, J̄ 7 = 27, J̄ 6 = 27, J̄ ′4J̄ 7 = 27, J̄ 7 = 27, J̄ 5 = 27, J̄ 6 = 27, J̄ 7 = 27, J̄5J̄ 6 = 27, J̄ 7 = 27, J̄5J̄ 7 = 27, J̄ 6 = 18, J̄ 7 = 18, J̄ ′6J̄ 7 = 18, J̄ 7 = 9. (170) The second Chern class is · J̄ ′1 = 12, c2 · J̄ ′5 = 18, c2 · J̄ ′2 = 12, · J̄ ′3 = c2 · J̄ ′6 = 24, c2 · J̄ ′4 = c2 · J̄ ′7 = 30. (171) We observe that both the intersection ring and the second Chern class cannot be brought into (11) and (110) by a linear transformation with integer coefficients, re- spectively. Hence, the second phase really is topologically distinct. We denote the Fourier-transformed variables in the B-model prepotential (71) by q′i, i = 1, . . . , 7. With this notation, we obtain (q′1, . . . , q 7) = 3 q 1 + 3 q 2 + 3 q + 3 q′3q 5 + 3 q + 3 q′3q 5 + 3 q 6 + 3 q 12333 12333 + 3 q′1q + 3 q′2q − 6 q′1q 5 − 6 q + 3 q′3q 6 + 3 q 7 +O(q (172) The instanton numbers on X − are the expansion coefficients in (q′1, . . . , q 7, 1) = n1,...,n7 (n1,n2,n3,n4,n5,n6,n7) . (173) Up to degree 4, they read 1,0,0,0,0,0,0 = 3 n 0,1,0,0,0,0,0 = 3 n 0,0,0,0,1,0,0 = 3 n 2,0,0,0,0,0,0 = −6 0,2,0,0,0,0,0 = −6 n 3,0,0,0,0,0,0 = 27 n 0,3,0,0,0,0,0 = 27 n 4,0,0,0,0,0,0 = −192 0,4,0,0,0,0,0 = −192 n 0,0,1,0,1,0,0 = 3 n 0,0,0,0,1,1,0 = 3 n 0,0,1,1,1,0,0 = 3 0,0,1,0,1,1,0 = 3 n 0,0,0,0,1,1,1 = 3 n 1,0,0,3,0,0,0 = 3 n 0,1,0,0,0,0,3 = 3 1,0,1,1,1,0,0 = −6 n 0,1,0,0,1,1,1 = −6 n 0,0,1,1,1,1,0 = 3 n 0,0,1,0,1,1,1 = 3 (174) It is easy to check that the symmetry G∗2 acts without fixed points on X − so that there are two phases of the quotient X∗, too, with h1,1(X∗) = h1,2(X∗) = 3 and fundamental group π1(X ∗) = Z3×Z3. They correspond to the two classes of triangulations T+ and T−. The first phase was studied in detail in Subsection 4.5 and 4.6. We denote the Calabi-Yau manifold in the second phase by X∗− = X 2. From the linear equivalence relations eq. (106) and the definition eq. (169) we can compute the induced group action on H2(X −,Z) and find  J̄ ′1 J̄ ′2 J̄ ′3 J̄ ′4 J̄ ′5 J̄ ′6 J̄ ′7   1 0 0 0 0 0 0 0 1 0 0 0 0 0 3 0 0 −1 1 0 0 3 0 1 −1 0 0 0 0 0 0 0 1 0 0 0 3 0 0 1 0 −1 0 3 0 0 0 1 −1   J̄ ′1 J̄ ′2 J̄ ′3 J̄ ′4 J̄ ′5 J̄ ′6 J̄ ′7  . (175) In terms of the three invariant divisors J ′1 = J̄ 2 = J̄ 3 = J̄ 2 the intersection ring and the second Chern class of X∗− then are J ′22 J 3 = 1, J 2 = 3, J 3 = 1, J 3 = 3, J ′21 J 2 = 9, J 3 = 3, J 3 = 9, J 1 = 27, 1 = 18, c2 J 2 = 12, c2 J 3 = 12. (176) Again, we observe that there is no linear basis transformation with integer coefficients that brings both the intersection ring and the second Chern class into (90). Hence, also the phase X∗− is topologically distinct from X (n1, n2, n3) n (n1,n2,n3,0) (n1,n2,n3,1) (n1,n2,n3,2) (n1,n2,n3) (1, 0, 0) 3 0 0 3 (2, 0, 0) −6 0 0 −6 (3, 0, 0) 18 0 0 18 (0, 1, 0) 3 3 3 9 (1, 1, 0) −6 −6 −6 −18 (1, 1, 1) 12 12 12 36 (2, 1, 0) 15 15 15 45 (1, 2, 0) 12 12 12 36 Table 6: Instanton numbers n (n1,n2,n3,m2) computed by toric mirror symmetry. They are invariant under the exchange n2 ↔ n3, so we only display them for n2 ≤ n3. To give a geometrical interpretation of what happens, we look at the induced action of G∗2 on the toric Mori cone NE(X −). Only the generators l̄ − , l̄ − and l̄ − in eq. (168) are invariant. This is exactly as in the first phase. Denoting the invariant generators ± , l̄ ± , l̄ ± by l ± , l ± , l ± , respectively, we observe that phase T− is obtained from the phase T+ as a flop by the curve corresponding to the generator l − = −l + , l − = l + + 3l + , l − = l + + 3l + . (177) If we use the realization of X in terms of the fiber product of two dP9 surfaces, the above result means that the base P1 of X has been flopped. Furthermore, having computed the G∗2-action in eq. (175), we determine the de- scent equation for the prepotential to be p′, q′, r′, b′1, b |G∗2| p′, q′, b′2, b ′, b′2 2, b′2 2, b′1 . (178) The corresponding instanton numbers ′, q′, r′, 1, b′2) = n1,n2,n3,m2 (n1,n2,n3,m2) p′n1q′n2r′n3b′2 (179) are listed in Table 6. Bibliography [1] V. Braun, M. Kreuzer, B. A. Ovrut, and E. Scheidegger, “Worldsheet instantons and torsion curves. Part A: Direct computation,” hep-th/0703182. (document), 1, 1, 1, 2.2, 2.3, 3.2, 4.1, 6, 8, 8 http://arXiv.org/abs/hep-th/0703182 [2] P. Candelas, X. C. De La Ossa, P. S. Green, and L. Parkes, “A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory,” Nucl. Phys. B359 (1991) 21–74. 1 [3] C. Schoen, “On fiber products of rational elliptic surfaces with section,” Math. 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A Introduction Calabi-Yau Threefolds The Calabi-Yau Threefold X The Intermediate Quotient Xbar Variables Toric Geometry and Mirror Symmetry Toric Varieties The Batyrev-Borisov Construction Toric Intersection Ring Mori Cone B-Model Prepotential Quotienting the B-Model The Quotient by G1 The Quotient by G2 B-Model on Xbar Instanton Numbers of X B-Model on the Mirror of Xbar Instanton Numbers of the Mirror of X Instanton Numbers Assuming The Self-Mirror Property The Self-Mirror Property Factorization vs. The (3,1,0,0,0) Curve Towards a Closed Formula Conclusion Triangulations The Flop of X* Bibliography

0704.0450

Disorder screening near the Mott-Anderson transition

Disorder screening near theMott-Anderson transition M. C. O. Aguiar a,b,∗, V. Dobrosavljević c, E. Abrahams b, G. Kotliar b aDepartamento de F́ısica, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Belo Horizonte, MG, Brazil bCenter for Materials Theory, Serin Physics Laboratory, Rutgers University, 136 Frelinghuysen Road, Piscataway, New Jersey 08854, USA cDepartment of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA Abstract Correlation-driven screening of disorder is studied within the typical-medium dynamical mean-field theory (TMT-DMFT) of the Mott-Anderson transition. In the strongly correlated regime, the site energies εiR characterizing the effective disorder potential are strongly renormalized due to the phenomenon of Kondo pinning. This effect produces very strong screening when the interaction U is stronger then disorder W , but applies only to a fraction of the sites in the opposite limit (U < W ). Key words: Strong correlation; disorder; metal-insulator transition; Hubbard model PACS: 71.27.+a, 72.15.Rn, 71.30.+h Introduction - Theories that are able to capture both the Mott [1] and the Anderson [2] mechanisms for electron lo- calization have remained elusive despite many years of ef- fort. An attractive approach to this difficult problem has recently been proposed by combining the dynamical mean- field theory (DMFT) [3] of the Mott transition, and the typical medium theory (TMT) [4] of Anderson localiza- tion. This new formulation of the Mott-Anderson problem has been explored in recent work by Vollhardt and collabo- rators [5] using numerical renormalization-group methods, but the precise mechanism for the critical behavior of this model remains to be elucidated. Here we examine the mech- anism for disorder screening within this theory, which ex- plains several aspects of the results found in Ref. [5] Within TMT-DMFT, a lattice problem is mapped onto an ensemble of single-impurities problems, which are em- bedded in a self-consistently determined bath. Recent work of Ref. [6] examined the behavior of a collection of single- impurity models in the situation where the bath seen by the impurities was chosen to mimic the approach to the Mott- Anderson transition. In this work, the impurity quasipar- ticle weight Zi was shown to present a scaling behavior as a function of the on-site energy εi and the distance t to the transition. These findings, however, are not sufficient to address the disorder screening behavior of the model, which requires the description of the renormalized disor- ∗ Corresponding author. Tel: +55 31 3499-5671 fax: +55 31 3499- Email address: [email protected] (M. C. O. Aguiar). -0.5 -0.2 0.0 0.2 0.5 t = 0.10 t = 0.08 t = 0.06 t = 0.04 t = 0.02 t = 0.01 t = 0.004 t = 0.001 Fig. 1. Renormalized energy εR as a function of the on-site energy ε for a collection of single-impurity problems close to the Mott-An- derson transition (t → 0). The parameters used were U = 1.75 and W = 2.8. der potential. In this paper, we demonstrate that a scaling procedure similar to that presented in Ref. [6] can also be carried on for the renormalized energy εiR. Renormalization of the disorder potential - We consider a collection of Anderson impurity models [6] with on-site re- pulsion U , on-site energies εi, and the total spectral weight t of the cavity field. Without loss of generality [6], we con- sider a featureless model bath with non-vanishing spectral weight for −t/2 < ω < t/2 and zero otherwise. Our goal is to describe the statistics of the renormalized site ener- gies as the metal-insulator transition is approached, corre- sponding to t → 0 within the TMT-DMFT scheme. Preprint submitted to Elsevier 1 November 2018 http://arxiv.org/abs/0704.0450v1 δε’=|ε/W-0.3125| ln(t*) = 0.58 + 2 ln(δε’) ln(t*) = 3.4 + 2 ln(δε’) Fig. 2. Scaled renormalized energy εR/t 1/2 as a function of t/t∗(δε) showing that the results for different (and positive) ε can be collapsed onto a single scaling function with two branches. Different symbols correspond to different ε; the upper (bottom) branch contains results for ε > U/2 (ε < U/2). The inset shows the scaling parameter t∗ as a function of |ε/W − 0.3125| for the upper (squares) and bottom (circles) branches. The parameters used were U = 1.75 and W = 2.8. The impurity models were solved at zero temperature us- ing the SB4 method [7,6], which provides a parametrization of the low-energy (quasiparticle) part of the local Green’s function, given by Gi(ωn) = iωn − ε R − Zi∆(ωn) . (1) Here Zi is the local quasiparticle weight and ε R is the renor- malized site energy. The details of the calculations mirror those of Ref. [6]. The results for the renormalized energy εiR as a func- tion of −W/2 < εi < W/2, in the vicinity of the Mott- Anderson transition (t → 0), are shown in Fig. 1. As in Ref. [6], we find two-fluid behavior, where sites with |εi| < U/2 turn into local magnetic moments, corresponding to “Kondo pinning” [8] εiR → 0. For the remaining sites, ε εi + U/2 or ε R → εi − U/2, as they become, respectively, doubly occupied (those with εi < −U/2) or empty (those with εi > U/2). We should emphasize that such two-fluid behavior thus emerges only for sufficiently strong disorder, such that U < W . Otherwise all sites turn into local mag- netic moments, and the Mott transition for moderate dis- order retains a character similar to that found within the standard DMFT approach [8]. Scaling analysis - These results can alternatively be pre- sented in a scaling form, as shown in Fig. 2. Here, we show that it is possible to collapse the family of curves εR(t, δε)/t 0.5, where δε = (εi − ε ∗) /ε∗ and ε∗ = U/2, onto a single universal scaling function εR(t, δε)/t 0.5 = f [t/t∗(δε)] with two branches, one for εi < ε ∗ and other for εi > ε ∗. In agreement with Ref. [6] (inset of Fig. 2), the crossover scale t∗(δε) ∼ |δε|φ, with exponent φ = 2. In the limit t → 0, we find that the branch corresponding to εi < ε ∗ has the asymptotic form f(x) ∼ x3/2 (here x = t/t∗(δε)), corresponding to εR(t) ∼ t 2. Similarly, for εi > ε∗, f(x) ∼ x−1/2 corresponding to εR(t) ∼ constant. For 0.0 0.5 1.0 1.5 2.0 2.5 clean value Fig. 3. Arithmetic and geometric density-of-states (ADOS and TDOS, respectively) at the Fermi level as a function of U , for W = 1.5, when the TMT-DMFT self-consistent loop is performed. x ≫ 1 the two branches merge, viz. f(x) ∼ A±B±x−0.5. Disorder screening - Within TMT-DMFT, the Anderson localization effects are manifested by the reduction of the typical density of states (TDOS), since the (algebraic) aver- age (ADOS) remains finite even in an Anderson insulator. When disorder is strongly screened due to the correlation effects, the two quantities should not differ much, as illus- trated by the results of Fig. 3. Here we present the results of the fully self-consistent solution, as the Mott-like transition is approached by increasing U for W = 1.5. Close to the transition, both averages approach the clean limit (dashed line), indicating a strong screening effect. These results are consistent with those found in the numerical renormaliza- tion group solution of the TMT-DMFT equation of Ref. [5]. As discussed above, strong disorder screening is expected near the Mott-like transition (U > W ), which indeed cor- responds to the mechanism responsible for the results in Fig. 3.When the transition is approached at strong disorder (U < W ) (not shown), strong screening effects are found only for a fractions of the sites (i.e of the volume of the sample), indicating different critical behavior at the Mott- Anderson transition. The details of the critical behavior in this case will be discussed elsewhere. Acknowledgements - This work was partially supported by NSF grants DMR-0312495 (M.C.O.A.), DMR-0234215 and DMR-0542026 (V.D.) and DMR-0096462 (G.K.). References [1] N.F. Mott, Metal-insulator Transitions (Taylor and Francis, London, 1974). [2] P.W. Anderson, Phys. Rev. 109 (1958) 1498. [3] A. Georges et al., Rev. Mod. Phys. 68 (1996) 13. [4] V. Dobrosavljević et al., Europhys. Lett. 62 (2003) 76. [5] K. Byczuk, W. Hofstetter, and D. Vollhardt, Phys. Rev. Lett. 94 (2005) 056404. [6] M.C.O. Aguiar et al., Phys. Rev. B 73 (2006) 115117. [7] G. Kotliar and A.E. Ruckenstein, Phys. Rev. Lett. 57 (1986) 1362. [8] D. Tanasković et al., Phys. Rev. Lett. 91 (2003) 066603. References

0704.0451

Electromigrated nanoscale gaps for surface-enhanced Raman spectroscopy

Electromigrated nanoscale gaps for surface-enhanced Raman spectroscopy Daniel R. Ward1, Nathaniel K. Grady2, Carly S. Levin3, Naomi J. Halas3,4, Yanpeng Wu2, Peter Nordlander1,4, Douglas Natelson1,4 1Department of Physics and Astronomy, 2Applied Physics Graduate Program, 3Department of Chemistry, 4Department of Electrical and Computer Engineering, and the Rice Quantum Institute, Rice University, 6100 Main St., Houston, TX 77005, USA (Dated: October 23, 2018) Abstract Single-molecule detection with chemical specificity is a powerful and much desired tool for bi- ology, chemistry, physics, and sensing technologies. Surface-enhanced spectroscopies enable single molecule studies, yet reliable substrates of adequate sensitivity are in short supply. We present a simple, scaleable substrate for surface-enhanced Raman spectroscopy (SERS) incorporating nanometer-scale electromigrated gaps between extended electrodes. Molecules in the nanogap active regions exhibit hallmarks of very high Raman sensitivity, including blinking and spectral diffusion. Electrodynamic simulations show plasmonic focusing, giving electromagnetic enhance- ments approaching those needed for single-molecule SERS. http://arxiv.org/abs/0704.0451v1 Multifunctional sensors with single-molecule sensitivity are greatly desired for a vari- ety of sensing applications, from biochemical analysis to explosives detection. Chemi- cal and electromagnetic interactions between molecules and metal substrates are used in surface-enhanced spectroscopies[1] to approach single molecule sensitivity. Electromagnetic enhancement in nanostructured conductors results when incident light excites local elec- tronic modes, producing large electric fields in a nanoscale region, known as a “hot spot”, that greatly exceed the strength of the incident field. Hot spots can lead to particularly large enhancements of Raman scattering, since the Raman scattering rate is proportional to |E(ω)|2|E(ω′)|2 at the location of the molecule, where E(ω) is the electric field component at the frequency of the incident radiation, and E(ω′) is the component at the scattered frequency. It has been an ongoing challenge to design and fabricate a substrate for systematic SERS at the single molecule level. Single-molecule SERS sensitivity was first clearly demon- strated using random aggregates of colloidal nanoparticles[2, 3, 4, 5]. Numerous other metal substrate configurations have been used for SERS, including chemically engineered nanoparticles[6, 7, 8], nanostructures defined by bottom-up patterning[9, 10], and those made by traditional lithographic approaches[11]. In the most sensitive substrate geome- tries, incident light excites adjacent subwavelength nanoparticles or nanostructures, result- ing in large field enhancements within the interparticle gap[12, 13]. Fractal aggregates of nanoparticles[14] can further increase field enhancements by focusing plasmon energy from larger length scales down to particular nanometer-scale hotspots[15]. However, precise and reproducible formation of such assemblies in predetermined locations has been extremely challenging. An alternative approach is tip-enhanced Raman spectroscopy (TERS), in which the incident light excites an interelectrode plasmon resonance localized between a sharp, metal scanned probe tip and an underlying metal substrate. Recent progress has been made in single-molecule TERS detection[16, 17, 18]. A similar approach was recently attempted using a mechanical break junction[19]. While useful for surface imaging, TERS requires feedback to control the tip-surface gap, and is not scalable or readily integrated with other sensing modalities. We demonstrate a scaleable and highly reliable method for producing planar extended electrodes with nanoscale spacings that exhibit very large SERS signals, with each electrode pair having one well-defined hot spot. Confocal scanning Raman microscopy demonstrates the localization of the enhanced Raman emission. The SERS response is consistent with a very small number of molecules in the hotspot, showing blinking and spectral diffusion of Raman lines. Sensitivity is sufficiently high that SERS from physisorbed atmospheric contaminants may be detected after minutes of exposure to ambient conditions. The Raman enhancement for para-mercaptoaniline (pMA) is estimated from experimental data to exceed 108. Finite-difference time-domain (FDTD) modeling of realistic structures reveals a rich collection of interelectrode plasmon modes that can readily lead to SERS enhancements as large as 5× 1010 over a broad range of illumination wavelengths. These structures hold the promise of integration of single-molecule SERS with electronic transport measurements, as well as other near-field optical devices. Our structures are fabricated on a Si wafer topped by 200 nm of thermal oxide. Electron beam lithography is used to pattern “multibowtie” structures as shown in Fig. 1A. The multibowties consist of two larger pads connected by multiple constrictions, as shown. The constriction widths are 80-100 nm, readily within the reach of modern photolithography. After evaporation of 1 nm Ti and 15 nm Au followed by liftoff in acetone, the electrode sets are cleaned of organic residue by exposure to O2 plasma for 1 minute. The multibowties are placed in a vacuum probe station, and electromigration[20] is used to form nanometer-scale gaps in the constrictions in parallel, as shown in Fig. 1B. Electromigration is a nonthermal process whereby momentum transfer from current-carrying electrons is transferred to the lattice, rearranging the atomic positions. Electromigration has been studied thoroughly[21, 22, 23, 24] as a means of producing electrodes for studies of single-molecule conduction. We have performed manual and automated electromigration at room temperature, with identical results. The number of parallel constrictions in a single multibowtie is limited by the output current capacity of our electromigration voltage source. A post-migration resistance of ∼ 10 kΩ for the structure in Fig. 1A appears optimal. Post-migration high resolution scanning electron microscopy (SEM) shows interelectrode gaps ranging from too small to resolve to several nanometers. There are no detectable nanoparticles in the gap region or along the electrode edges. Based on electromigration of 283 multibowties (1981 individual constrictions), 77% of multibowties have final resistances less than 100 kΩ, and 43% have final resistances less than 25 kΩ. We believe that this yield, already high, can be improved significantly with better process control, particularly of the lithography and liftoff. The optical properties of the resulting nanogaps are characterized using a WITec CRM 200 scanning confocal Raman microscope in reflection mode, with normal illumination from a 785 nm diode laser. Using a 100× objective, the resulting diffraction-limited spot is measured to be gaussian with a full-width at half-maximum of 575 nm. Fig. 1C shows a spatial map of the integrated emission from the 520 cm−1 Raman line of the Si substrate. The Au electrodes are clearly resolved. Polarization of the incident radiation is horizontal in this figure. Rayleigh scattered light from these structures shows significant changes upon polarization rotation, while SERS response is approximately independent of polarization. Freshly cleaned nanogaps show no Stokes-shifted Raman emission out to 3000 cm−1. However, in 65% of clean nanogaps examined, a broad continuum background (see Support- ing Information) is seen, decaying roughly linearly in wavenumber out to 1000 cm−1 before falling below detectability. This background is spatially localized to a diffraction-limited re- gion around the interelectrode gap and is entirely absent in unmigrated junctions. The origin of this continuum, similar to that seen in other strongly enhancing SERS substrates[5], is likely inelastic electronic effects in the gold electrodes[25]. In samples coated with molecules, this background correlates strongly with visibility of SERS. No junctions without this back- ground displayed SERS signals. Like the SERS signal, this background is approximately polarization independent. Temporal fluctuations of this background in clean junctions are minimal, strongly implying that fluctuations of the electrode geometry are not responsible for SERS blinking. The SERS enhancement of the junctions has been tested using various molecules. The bulk of testing utilized pMA, which self-assembles onto the Au electrodes via standard thiol chemistry. Particular modes of interest are carbon ring modes at 1077 cm−1 and 1590 cm−1. Fig. 1D shows a map of the Raman emission from the 1077 cm−1 line on the same junction as Fig. 1B,C, after self-assembly of pMA. This emission is strongly localized to the position of the nanogap. No Raman signal is detectable either on the metal films or at the edges of the metal electrodes. Fig. 1E shows the spatial localization of the continuum background mentioned above. Fig. 2 shows a more detailed examination of the SERS response of the gap region of a typical junction after self-assembly of pMA. Fig. 2A,B are time series of the SERS response, with known pMAmodes indicated. The modes visible are similar to those seen in other SERS measurements of pMA on lithographically fabricated Au structures[11]. Each spectrum was acquired with 1 s integration time, with the objective positioned over the center of the nanogap hotspot. Temporal fluctuations of both the Raman intensity (“blinking”) and Raman shift (spectral diffusion), generally regarded as hallmarks of few- or single-molecule SERS sensitivity[26], are readily apparent. Fig. 2C shows a comparison of the wandering of the 1077 cm−1 pMA line with that of the 520 cm−1 Raman line of the underlying Si substrate over the same time interval. This demonstrates that the spectral diffusion is due to changes in the molecular environment, rather than a variation in spectrometer response. Fig. 2D shows the variation in the peak amplitudes over that same time interval. This blinking and spectral diffusion are seen routinely in these junctions. We have ob- served such Raman response from several molecules, including self-assembled films of pMA, para-mercaptobenzoic acid (pMBA), a Co-containing transition metal complex[28], and spin- coated poly(3-hexylthiophene) (P3HT). These molecules all have distinct Raman responses and show blinking and wandering in the junction hotspots. Another indicator of very large enhancement factors in these structures is sensitivity to exogenous, physisorbed contamination. Carbon contamination has been discussed[29, 30, 31] in the context of both SERS and TERS. This substrate is sensitive enough to examine such contaminants (see Supporting Information). While clean junctions with no deliberately attached molecules initially show only the continuum background, gap-localized SERS sig- natures in the sp2 carbon region between 1000 cm−1 and 1600 cm−1 are readily detected after exposure to ambient lab conditions for tens of minutes. Nanojunctions that have been coated with a self-assembled monolayer (SAM) (for example, pMA) do not show this carbon signature, even after hours of ambient exposure. Presumably this has to do with the ex- tremely localized field enhancement in these structures, with the SAM sterically preventing physisorbed contaminants from entering the region of enhanced near field. Recently arrived contaminant SERS signatures abruptly disappear within tens of seconds at high incident powers (1.8 mW), presumably due to desorption. SERS from covalently bound molecules is considerably more robust, degrading slowly at high powers, and persisting indefinitely for incident powers below 700 µW. SEM examination of the nanogaps shows no optically induced damage after exposure to intensities that would significantly degrade nanoparticles[32]. The large extended pads likely aid in the thermal sinking of the nanogap region to the substrate. Estimating enhancement factors rigorously is notoriously difficult, particularly when the hotspot size is not known. Although SERS enhancement volume measurements are possible using molecular rulers[33], this is not feasible with such small nanogaps. Junctions made directly on Raman-active substrates (Si with no oxide; GaAs) show no clearly detectable enhancement of substrate modes in the gap region, suggesting that the electromagnetic enhancement is strongly confined to the thickness of the metal film electrodes. Figure 3 shows a comparison between a typical pMA SERS spectrum acquired on a junction with a 600 s integration time at 700 µW incident power, and a spectrum acquired over one of that device’s Au pads, for the same settings. The pad spectrum shows no detectable pMA features and is dark current limited. We use FDTD calculations to understand the strong SERS response in these structures and roughly estimate enhancement factors. It is important to note that the finite grid size (2 nm) required for practical computation times restricts the quantitative accuracy of these calculations. However, the main results regarding spatial mode structure (allowing assess- ment of the localization of the hotspot) and energy dependence are robust to these concerns, and the calculated electric field enhancement is an underestimate[34]. Fig. 4 shows a cal- culated extinction spectrum and map of |E|4 in the vicinity of the junction. Calculational details and additional plots are presented in the Supporting Information. These calculations predict that there should be large SERS enhancements across a broad bandwidth of excit- ing wavelengths because of the complicated mode structure possible in the interelectrode gap. Nanometer-scale asperities from the electromigration process break the interelectrode symmetry of the structure. The result is that optical excitations at a variety of polariza- tions can excite many interelectrode modes besides the simple dipolar plasmon commonly considered. For extended electrodes, a continuous band of plasmon resonances coupling to wavelengths from 500-1000 nm is expected[35]. This broken symmetry also leads to much less dependence of the calculated enhancement on polarization direction, as seen experi- mentally. The calculations confirm that the electromagnetic enhancement is confined in the normal direction to the film thickness. Laterally, the field enhancement is confined to a region comparable to the radius of curvature of the asperity. For gaps and asperities in the range of 2 nm, purely electromagnetic enhancements can exceed 1011, approaching that sufficient for single-molecule sensitivity. Using the data from the device in Fig. 3, we estimate the total enhancement in that device. To be conservative, we assume a hotspot effective radius of 2.5 nm with dense packing of pMA, giving N ≈ 100 molecules. Blinking and wandering suggest that the true N value is much closer to one. The integrated Raman signal over a gaussian fit to the 1077 cm−1 Raman line is 2.0 counts/sec/molecule when the incident power is 700 µW. For our apparatus the count rate from imaging a bulk crystal at the same equivalent power (see Supporting Information) is 4.2× 10−9 counts/sec/molecule, so that we estimate a total enhancement of at least 5× 108. We have demonstrated a SERS substrate capable of extremely high sensitivity for trace chemical detection. Unlike previous substrates, these nanojunctions may be mass fabricated in controlled positions with high yield using a combination of standard lithography and electromigration. The resulting hotspot geometry is predicted to allow large SERS enhance- ments over a broad band of illuminating wavelengths. Other nonlinear optical effects should be observable in these structures as well. The extended electrode geometry and underly- ing gate electrode are ideal for integration with other sensing modalities such as electronic transport. Tuning molecule/electrode charge transfer via the gate electrode may also enable the direct examination of the fundamental nature of chemical enhancement in SERS. DW acknowledges support from the NSF-funded Integrative Graduate Research and Ed- ucational Training (IGERT) program in Nanophotonics. NH, PN, and DN acknowledge support from Robert A. Welch Foundation grants C-1220, C-1222, and C-1636, respectively. DN also acknowledges the National Science Foundation, the David and Lucille Packard Foundation, the Sloan Foundation, and the Research Corporation. C.S.L. was supported by a training fellowship from the Keck Center Nanobiology Training Program of the Gulf Coast Consortia, NIH 1 T90 DK070121-01. YP and NKG are supported by US Army Research Office grant W911NF-04-1-0203. Supporting Information Available: Detailed examination of continuum background and adsorption of exogenous contaminants; extended discussion of FDTD calculations; more detailed discussion of SERS enhancement calculation. This material is available free of charge via the Internet at http://pubs.acs.org. http://pubs.acs.org FIG. 1: (A) Full multibowtie structure, with seven nanoconstrictions. (B) Close-up of an individual constriction after electromigration. Note that the resulting nanoscale gap (<∼ 5 nm at closest separation, as inferred from closer images) is toward the right edge of the indicated red square. (C) Map of Si Raman peak (integrated from 500-550 cm−1) in device from (B), with red corresponding to high total counts. The attenuation of the Si Raman line by the Au electrodes is clear. (D) Map of pMA SERS signal for this device based on one carbon ring mode (integrated from 1050-1110 cm−1). (E) Map of integrated low energy background (50-300 cm−1) for this device. FIG. 2: (A) Waterfall plot (1 s integration steps) of SERS spectrum at a single nanogap that had been soaked in 1 mM pMA in ethanol. Identified pMA peaks include the ring modes at 1077 cm−1 and 1590 cm−1, as well as an 1145 cm−1 δCH mode with b2 symmetry, an 1190 cm −1 mode identified as δCH of a1 symmetry, a mode near 1380 cm −1 identified as δCH+νCC of b2 symmetry, and a mode near 1425 cm−1 identified as νCC+δCH of b2 symmetry. Mode assignments are based on Ref. [27]. (B) Close-up of (A) to show correlated wandering and blinking of 1077 cm−1 and 1145 cm−1 modes. (C) Comparison of 1145 cm−1 mode position (blue) with the Si Raman peak (red), which shows no such wandering. The jitter in the Si peak position is 1 pixel in the detector. (D) Comparison of 1145 cm−1 peak height (green, found by a gaussian fit to the peak) fluctuations with those of the Si peak (blue). Figure 3: Comparison of pMA at hotspot (blue,left axis) and 5um over on Au pad (green,right axis). Prominent features include the 1077,1590 a1 symmetry mode peaks, and the less strong 1180 (1160 in moerner paper) b2 symmetry mode peak. Additionally the 520 Si peak is visible. The Si peak is significantly reduced in the Au film spectra because the film is only partially transparent at 785 nm. B2 symmetry modes appear weaker because of blinking on and off which reduces the measured intensity when averaged over long time periods. There are no pMA features in the Au pad spectrum dark current noise limited. Estimated enhancement factor is ~1e6. 0 500 1000 1500 2000 0 500 1000 1500 2000 Raman Shift (cm-1) FIG. 3: Blue curve (left scale): pMA SERS spectrum at hotspot center of one nanojunction densely covered by pMA, integrated for 10 minutes at incident power of 700 µW. Green curve (right scale): integrated signal under same conditions on middle of Au pad on the same nanojunction. The feature near 960 cm−1 is from the Si substrate. No Raman features are detectable on either the Au pads or their edges under these conditions. FIG. 4: (A) FDTD-calculated extinction spectrum from the model electrode configuration shown in (B). (B) Mock-up electrode tips capped with nanoscale hemispherical asperities, with |E|4 plotted for the 937 nm resonance of (A). Constriction transverse width at narrowest point is 100 nm. Gap size without asperities is 8 nm. Asperity on left (right) electrode has radius of 6 nm (4 nm). Au film thickness is 15 nm, SiO2 underlayer thickness is 50 nm. 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[26] Wang, Z.; Rothberg, L. J. J. Phys. Chem. B 2005, 109, 3387-3391. [27] Osawa, M.; Matsuda, N.; Yoshii, K.; Uchida, I. J. Phys. Chem. 1994, 98, 12702-12707. [28] Ciszek, J. W.; Keane, Z. K.; Cheng, L.; Stewart, M. P.; Yu, L. H.; Natelson, D.; Tour, J. M. J. Am. Chem. Soc. 2006, 128, 3179-3189. [29] Kudelski, A.; Pettinger, B. Chem. Phys. Lett. 2000, 321, 356-362. [30] Otto, A. J. Raman Spectrosc. 2002, 33, 593-598. [31] Richards, D.; Milner, R. G.; Huang, F.; Festy, F. J. Raman Spectrosc. 2003, 34, 663-667. [32] Schuck, P. J.; Fromm, D. P.; Sundaramurthy, A.; Kino, G. S.; Moerner, W. E. Phys. Rev. Lett. 2005, 94, 017402. [33] Lal, S.; Grady, N. K.; Goodrich, G. P.; Halas, N. J. Nano Lett. 2006, 6, 2338-2343. [34] Oubre, C.; Nordlander, P. J. Phys. Chem. B 2005, 109, 10042-10051. [35] Nordlander, P.; Le, F. Appl. Phys. B 2006, 84, 35-41. Supporting Information: Electromigrated nanoscale gaps for surface-enhanced Raman spectroscopy I. CONTINUUM BACKGROUND The strong continuum background observed at the nanogaps is shown in Fig. S1A. The continuum slopes down linearly in intensity from 0 cm−1 to almost 1500 cm−1. This continuum, seen only in the presence of the nanogap, is compared with the Au film and Si substrate spectra taken using the same microscope configuration. The 300 cm−1 and 520 cm−1 peaks are from the Si substrate. The spectrum shown in Fig. S1A also shows a small peak at 1345 cm−1 which is indicative of absorbates from the air settling at the nanogap. The continuum is localized to the nanogap as seen in Fig. S1B, where a spatial plot has been made by integrating the SERS spectrum from 600 cm−1 to 800 cm−1 at each point. This wavenumber range was chosen to avoid any of the Si substrate Raman active modes. Additionally a comparison of Fig. S1B with the spatial plot of the Si 520 cm−1 peak, Fig. S1C, shows that the continuum background of the nanogap is indeed located at the center of the bowtie, as expected. Although blinking of SERS at the nanogap has been observed for pMA and pMBA, the continuum itself does not blink in the absence of molecules. This is clear from the data in Fig S1D showing the time evolution of the SERS spectrum at the clean nanogap. No fluctuations are observed, and the continuum background remains constant. FIG. S5: (A) Raman spectra at hotspot (blue) of a clean bowtie, Au pad(green), and over Si substrate(red). The continuum is very strong at the hotspot and shows linear slope from 0 to ∼ 1500 cm−1. Also visible are the 300 cm−1 peak and 520 cm−1 peaks of the Si substrate and a weak peak at 1345 indicating the onset of atmospheric contamination after approximately 15 minutes of air exposure. Curves have been offset by 150 counts/s (green) and 200 counts/s (red) for clarity. (B) Spatial plot of integrated signal over 600-800 cm−1 showing the localization of continuum to the center of the device when compared to the Si plot (C). Yellow is strong signal; blue is no signal. (C) Spatial plot of integrated Si signal over 500-540 cm−1. Red indicates strong Si signal the blue areas show where the Au pads are. (D) Time spectra of clean bowtie. The intensity is reported in CCD counts/second. No blinking of the continuum is observed. The lowest wavenumbers are reported to have zero counts/second due to the low pass filter used to block out the laser. II. DEPENDENCE ON INCIDENT POLARIZATION The SERS signal from the nanogap does not have significant polarization dependence, as shown in Fig S2A. The two spectra are from the same nanogap with the polarization at 0 and 90 degrees to the gap. Although slightly different due to positioning and actual time variation of the spectrum, the two spectra do not show any strong differences in the intensities of the pMA signal or the continuum. The nanogap does exhibit a strong polarization dependence for Rayleigh scattering, as shown in Fig S2B-S2E. Figures S2B and S2D show a spatial map of the Rayleigh scattering for the polarization across the gap (B) and parallel with the gap FIG. S6: (A) Raman spectra at hotspot of bowtie with pMA assembled on surface. The blue spectrum is with polarization at 0 deg. (direction shown in (C)). The green spectrum has been shifted 50 counts/s for clarity and is at the same hotspot but with the polarization rotated 90 degrees relative to the substrate (direction shown in (D)). (B) Spatial plot of integrated signal over -40 to 40 cm−1 showing the Rayleigh scattering from the center of the device. Red is high intensity blue is low intensity. The large pads are at the left/right. Polarization direction indicated in (C). (C) Spatial plot of integrated Si signal over 500-540 cm−1. Red indicates Si, blue is Au pads. Polarization direction is indicated by the arrow. (D) Spatial plot of integrated signal over -40 to 40 cm−1 showing the Rayleigh scattering from the center of the device for the sample rotated 90 deg. relative to (B),(C). Red is high intensity blue is low intensity. Polarization direction is indicated in (E). There is a local maximum in the Rayleigh scattering at the center of the gap. (E) Spatial plot of integrated Si signal over 500-540 cm−1. Red indicates Si, blue is Au pads. Polarization direction is indicated by the arrow. III. DEPENDENCE ON SOLUTION CONCENTRATION We have examined SERS spectra for varying supernatant solution concentration during the assembly procedure. Ideally, successive dilutions of the solution should vary surface cov- erage of the assembled molecules. While molecular coverage on the edges of polycrystalline Au films is not readily assessed, we observe reproducible qualitative trends as concentration is reduced. For pMBA molecules assembled from solutions in nanopure water, we have var- ied concentrations from 1 mM down to 100 pM. The fraction of junctions showing SERS distinct from carbon contamination remains roughly constant down to concentrations as low as 1 µM. For our volumes and electrode areas, this is still expected to correspond to a dense coverage of 1 molecule per 0.19 nm2. At concentrations below 1 µM, SERS spectra change significantly, while remaining distinct from those of carbon contamination: blinking occurs more frequently; modes of b2 symmetry rather than a1 symmetry appear more frequently; and the molecular peaks can be more than 100× larger than the high coverage case for the same integration times. These observations are qualitatively consistent with the molecules exploring different surface orientations at low coverages, and charge transfer/chemical en- hancement varying with orientation. However, the actual coverage at the edges remains unknown. The concentration of the solution used for assembling molecules on the nanogap surface strongly influences the form of the observed Raman spectrum as well as the rate and intensity of the mode blinking. Raman spectra of pMBA were taken by soaking samples in 2 mL of different concentrations of pMBA. Although for all of these concentrations there are enough molecules in solution to form a monolayer over the bowtie surface, significant differences in the spectra were observed. Fig. S3A shows a representative Raman spectrum for pMBA at the nanogap for 1mM concentrations. The two carbon ring modes at 1077 cm−1 and 1590 cm−1 are clearly present along with a third peak at 1463 cm−1. The time spectra for this nanogap in Fig S3B. The 1077 cm−1 and 1590 cm−1 peaks are relatively stable and always present while other modes, such as the 1463 cm−1 mode, blink on and off for a few seconds at a time. As the concentration is decreased to 1 µM, the pMBA signal tends to be stronger with more intense blinking. Additionally the 1077 cm−1 mode is observed to disappear while the 1590 cm−1 mode remains. Additional modes begin to become more visible such as the 1265 cm−1 and 1480 cm−1 modes seen in Fig S3C. At even lower concentrations such as 1 nM, the pMBA signal is again more intense with even more blinking as seen in Fig. S3F (which has been plotted with intensity on a log scale). The 1077 cm−1 mode is again unseen while the 1590 cm−1 mode begins to fluctuate in intensity even more. The blinking becomes much more intense with the intensity of the signal periodically reaching close to ten times the maximum intensity observed for pMBA at 1 µM. We suggest that the increased blinking and larger amplitude signals are a result of the molecules not being as tightly packed on the surface in the 1 µM and 1 nM cases as in the 1 mM case. As a result of looser packing, the molecules are free to explore more surface conformations, including those with more and different charge transfer with the Au surface. We point out that these pMBA spectra are distinct from those seen in physisorbed carbon contamination on initally clean junctions. These data persist at high incident powers and do not show “arrival” phenomena as described in the subsequent section. Furthermore, they are unlikely to originate from photodecomposition of pMBA, since the illumination conditions are identical for all coverages. FIG. S7: (A) Raman from pMBA at 1 mM concentration taken at t = 10 s. (B) Corresponding time spectrum for 1 mM. (C) Raman from pMBA at 1 µM concentration taken at t = 251 s. (D) Corresponding time spectrum for 1 µM. (E) Raman from pMBA at 1 nM concentration taken at t = 24.5 s (F) Corresponding time spectrum for 1 nm, plotted on log intensity scale. IV. DETECTION OF ADSORBED CONTAMINANTS Due to the large enhancements possible with the nanogaps, contamination from airborne absorbates occurs readily in the absence of assembled molecules on the nanogap surface. We have observed the absorption of contaminants onto the surface of clean nanogaps in as little as 10 minutes. Collecting Raman spectra every 4 seconds, we can observe the appearance of contaminants on the surface as seen in Fig. S4A and S4B. It is difficult to identify the contaminants, as the spectra observed have large variations, although carbon ring modes are often observed in conjunction with other modes. Furthermore the Raman signal from contaminants often blinks very strongly, with periods of no or weak signal followed by several seconds of intense blinking, as seen in Fig S4C. The changes in intensity can be more than a factor of 100. Again we suggest that the strong blinking is a result of the weak attachment of the contaminants to the nanogap surface, allowing them to move considerably and explore many interactions with the Au surface. As previously mentioned, these contamination spec- tra are not observed when molecules of interest have been preassembled deliberately on the electrode surface. The likely explanation for this is that the self-assembled later sterically prevents contaminants from arriving at the nanogap region of maximum field enhancement. FIG. S8: (A) Raman spectra for clean bowtie (blue) and clean bowtie after a few minutes exposed to the air (green). This change in the Raman spectrum is indicative of contamination for surface absorbed molecules from the air. (B) Raman spectra for a clean bowtie showing the onset of a contaminant signal at 900 cm−1 as time progresses. (C) Waterfall plot showing the extremely strong blinking observed for adsorbed contamination. The fluctuations are much larger than the those observed for dense coverage of pMA, pMBA, or P3HT. Notice the scale relative to the 520 cm−1 Si peak seen at t = 340 s. V. FDTD CALCULATIONS The optical properties of the bowtie structure were calculated using the Finite-Difference Time-Domain method (FDTD) using a Drude dielectric function with parameters fitted to the experimental data for gold. This fit provides an accurate description of the optical properties of gold for wavelengths larger than 500 nm [S1]. These calculations do not account for reduced carrier mean free path due to surface scattering in the metal film, nor do they include interelectrode tunneling. However, such effects are unlikely to change the results significantly. The bowtie is modeled as a two finite triangular structures as illustrated in Fig. 4A of the manuscript. Our computational method requires the nanostructures to be modeled to be of finite extent. The plasmon modes of a finite system are standing modes with frequencies determined by the size of the sample and the number of nodes of the surface charge distribution associated with the plasmon. For an extended system such as the bowties manufactured in this study, the plasmon resonances can be characterized as traveling surface waves with a continuous distribution of wavevectors. A series of calculations of bowties with increasing length reveals that the optical spectrum is characterized by increasingly densely spaced plasmon resonances in the wavelength regime 500-1000 nm and a low energy finite-size induced split-off state involving plasmons localized on the outer surfaces of the bowtie. For a large bowtie, we expect the plasmon resonances in the 500-1000 nm wavelength interval to form a continuous band [S2]. The electric field enhancements across the bowtie junction for the plasmon modes within this band are relatively similar with large and uniform enhancements in the range of 50-150. The magnitudes of the field enhancements were found to increase with increasing size of the bowtie structure. For instance, the maximum field enhancement factor was found to be 115 for a 200 nm bowtie (Each half of the bowtie is modeled as a truncated triangle 200 nm long.) and 175 for a 400 nm bowtie. Our use of a finite gridsize also underestimates the electric field enhancements[S3]. Thus our calculated electric field enhancements are likely to significantly underestimate the actual electric field enhancements in the experimentally manufactured bowties. For a perfectly symmetric bowtie, significant field enhancements are only induced for incident light polarized across the junction. If the mirror symmetry is broken, for instance by making one of the structures thicker or triangular, large field enhancements are induced for all polarizations of incident light. To investigate the effects of nanoasperities, FDTD calculations were performed for a bowtie with two semi-spherical protrusions in the junction as shown in Fig. 4 of the main text, and Figs. S5-S7 of the Supporting Online Material. As expected, the presence of these protrusions does not influence the optical spectrum. However, the local field enhancements around the protrusions become very large, typically three or four times higher than for the corresponding structure without the defect. The physical mechanism for this increase is an antenna effect caused by the coupling of plasmons localized on the protrusion with the extended plasmons on the remaining bowtie structure [S4]. FIG. S9: Maps of FDTD-calculated |E| for the 1535 nm mode indicated in the main manuscript’s Fig. 4A. Color scale is logarithmic in |E|/|Einc|. Illumination direction is normal incidence, with electric field polarization oriented horizontally in (A)-(C). Maximum field enhancements are shown. (A) Overall view. (B) Close-up of interelectrode gap showing asperities. (C) Side-view of section indicated in (B) in red. (D) Side view of section indicated in (B) in blue. FIG. S10: Maps of FDTD-calculated |E| for the 937 nm mode indicated in the main manuscript’s Fig. 4A. Color scale is logarithmic in |E|/|Einc|. Illumination direction is normal incidence, with electric field polarization oriented horizontally in (A)-(C). Maximum field enhancements are shown. (A) Overall view. (B) Close-up of interelectrode gap showing asperities. (C) Side-view of section indicated in (B) in red. (D) Side view of section indicated in (B) in blue. FIG. S11: Maps of FDTD-calculated |E| for the 746 nm mode indicated in the main manuscript’s Fig. 4A. Color scale is logarithmic in |E|/|Einc|. Illumination direction is normal incidence, with electric field polarization oriented horizontally in (A)-(C). Maximum field enhancements are shown. (A) Overall view. (B) Close-up of interelectrode gap showing asperities. (C) Side-view of section indicated in (B) in red. (D) Side view of section indicated in (B) in blue. VI. ENHANCEMENT ESTIMATE To estimate an enhancement based on the data of Fig. 3 in the main text, it was necessary to understand the effective count rate per molecule of Raman scattering from bulk pMA in our measurement setup. This requires knowing the effective volume probed by the WITec system when the laser is focused on a bulk pMA crystal. The full-width-half-maximum (FWHM) of the laser spot size was found to be 575 nm. This was determined by measuring the count rate of the Rayleigh scattering peak (at zero wavenumbers) as a function of position as the beam was scanned over the edge of a Au film on a Si substrate. Averaging 16 such scans, the Rayleigh intensity was fit to the form of an integrated gaussian to determine the FWHM of the gaussian beam. The 575 nm figure is likely an overestimate due to systematic noise in the flat regions of the fit. For a gaussian beam with intensity of the form ∝ e− 2σ2 , the FWHM = 2 2 ln 2σ. The effective radius of an equivalent cylindrical beam is 2σ, or 346 nm in this case. The effective confocal depth [S5] was determined by measuring the 520 cm−1 Si Raman peak as a function of vertical displacement of a blank substrate. The effective depth profile was determined by numerical integration of the Si data using matlab. The effective volume probed by the beam is 1.92× 10−12 cm3. From the bulk properties of pMA, this corresponds to 1.09× 1010 molecules. The count rate for the bulk pMA 1077 cm−1 line, corrected by the ratio of (Si SERS rate/Si bulk rate) to accomodate for the difference in laser powers, is 46 counts/s, compared with 203 counts/s for the SERS data of Fig. 3. This leads to the enhancement estimate quoted in the main text of 5× 108. [S1] Oubre, C.; Nordlander, P. J. Phys. Chem. B 108, 108, 17740-17747. [S2] Nordlander, P.; Le, F. Appl. Phys. B 2006, 84, 35-41. [S3] Oubre, C.; Nordlander, P. J. Phys. Chem. B 2005, 109, 10042-10051. [S4] Hao, F.; Nehl, C. L.; Hafner, J. H.; Nordlander, P. Nano Lett. 2007, 7, 10.1021/nl062969c. [S5] Cai, W.B.; Ren, B.; Li, X. Q.; Shi, C. X.; Liu, F. M.; Cai, X. W.; Tian, Z. Q. Surf. Sci. 1998, 406, 9-22. References Continuum background Dependence on incident polarization Dependence on solution concentration Detection of adsorbed contaminants FDTD calculations Enhancement estimate

0704.0452

Dramatic Variability of X-ray Absorption Lines in the Black Hole Candidate Cygnus X-1

Dramatic Variability of X-ray Absorption Lines in the Black Hole Candidate Cygnus X-1 Chulhoon Chang1 and Wei Cui1 Department of Physics, Purdue University, West Lafayette, IN 47907 ABSTRACT We report results from a 30 ks observation of Cygnus X-1 with the High En- ergy Transmission Grating Spectrometer (HETGS) on board the Chandra X-ray Observatory. Numerous absorption lines were detected in the HETGS spectrum. The lines are associated with highly ionized Ne, Na, Mg, Al, Si, S, and Fe, some of which have been seen in earlier HETGS observations. Surprisingly, however, we discovered dramatic variability of the lines over the duration of the present observation. For instance, the flux of the Ne X line at 12.14 Å was about 5×10−3 photons cm−2 s−1 in the early part of the observation but became subsequently undetectable, with a 99% upper limit of 0.06 × 10−3 photons cm−2 s−1 on the flux of the line. This implies that the line weakened by nearly two orders of magnitude on a timescale of hours. The overall X-ray flux of the source did also vary during the observation but only by 20–30%. For Cyg X-1, the absorption lines are generally attributed to the absorption of X-rays by ionized stellar wind in the binary system. Therefore, they may provide valuable diagnostics on the physical condition of the wind. We discuss the implications of the results. Subject headings: binaries: general — black hole physics — stars: winds,outflows — stars: individual (Cygnus X-1) — X-rays: binaries 1. Introduction Cygnus X-1 is the first dynamically-determined black hole system (Webster & Murdin 1972; Bolton 1972). It is in a binary system with a massive O9.7 Iab supergiant, and the orbital period was determined optically to be 5.6 days. Cyg X-1 is thus intrinsically different from the majority of known black hole candidates (BHCs) whose companion stars are much 1Email: [email protected], [email protected] http://arxiv.org/abs/0704.0452v1 – 2 – less massive (see review by McClintock & Remillard 2006). Curiously, those that have a high-mass companion (including Cyg X-1, LMC X-1 and LMC X-3) are all persistent X- ray sources, while those that have a low-mass companion are exclusively transient sources. Perhaps, stellar wind from the companion star plays a significant role in this regard (Cui, Chen, & Zhang 1997). Unlike transient BHCs, in which mass accretion is mediated by the companion star overfilling its Roche-lobe, Cyg X-1 is thought to be a wind-fed system. In this case, however, the wind is thought to be highly focused toward the black hole, because the companion star is nearly filling its Roche lobe (Gies & Bolton 1986). The observed orbital modulation of the X-ray flux (Wen et al. 1999; Brocksopp et al. 1999a) has provided tentative evidence for wind accretion, because it is probably caused by varying amount of absorption through the wind (Wen et al. 1999). On the other hand, it is generally believed that an accretion disk is also present, based on the presence of an ultra-soft component, as well as Fe Kα emission line in the X-ray spectrum (e.g., Ebisawa et al. 1996; Cui et al. 1998). Cyg X-1 is probably still the most studied BHC. It is a fixture in the target list for all major X-ray missions. Much has been learned from modeling the X-ray continuum of the source, as well as from studying its X-ray variability. A recent development is the availability of high-resolution X-ray data, which may shed further light on the accretion process and the environment within the binary system. Cyg X-1 has been observed on many occasions with the High Energy Transmission Grating Spectrometer (HETGS) on board the Chandra X-ray Observatory and the Reflection Grating Spectrometer on board the XMM- Newton Observatory. The high-resolution spectra have revealed the presence of numerous absorption lines that are associated with highly ionized material (Marshall et al. 2001; Schulz et al. 2002; Feng et al. 2003; Miller et al. 2005). In this work, we report the detection of absorption lines, some of which have been seen previously but are much stronger here, and, more surprisingly, the discovery of dramatic variability of the lines, based on data from our HETGS/Chandra observation of Cyg X-1 during its 2001 state transition. The fluxes of some of the lines varied by nearly two orders of magnitude over the duration of the observation, while the overall flux of the source varied only mildly. 2. Observations and Data Reduction Cyg X-1 made a rare transition between the low-hard state and the high-soft state, as seen by the All-Sky Monitor (ASM) on the Rossi X-ray Timing Explorer (RXTE), about – 3 – five years after a similar episode in 1996 (Cui et al. 1997a and 1997b). Figure 1 shows the ASM light curve that covers the entire period. In this case, the flux of the source stayed high for ∼400 days, about twice as long as in 1996. Otherwise, the two episodes are very similar, including the flux levels of the two states, prominent X-ray flares in both states, and rapid transitions. Triggered by the ASM alert, the source was observed from 2001 October 28 16:13:52 to October 29 00:33:52 (UT) with the HETGS on Chandra (ObsID #3407). The HETGS consists of two gratings: Medium Energy Grating (MEG) and High Energy Grating (HEG). After passing the gratings, the photons are recorded and read out with the spectroscopic array of the Advanced CCD Imaging Spectrometer (ACIS). To avoid photon pile-up in the dispersed events, we chose to run the ACIS in continuous-clocking (CC) mode. We also applied a spatial window around the aim point to accept every 10th event in the zeroth order, to prevent telemetry saturation yet still have a handle on the position of the zeroth-order image for accurate wavelength calibration. The Chandra data were reduced and analyzed with the standard CIAO analysis package (version 3.2). Following the CIAO 3.2 Science Threads1, we prepared and filtered the data, produced the Level 2 event file from the Level 1 data products, constructed the light curves, and made the spectra and the corresponding response matrices and auxilliary response files (ARFs). We did have to work around a problem related to the use of maskfile for the CC- mode data, when we were making the ARFs. The solution is now included in the CIAO 3.3 Science Threads. To verify wavelength calibrations, we compared the plus and minus orders and found good agreement on the position of prominent lines. We did not subtract background from either the light curves or the spectra, because it is not obvious how to select background events from the CC-mode data. For a bright source like Cyg X-1, however, we expect the effects to be entirely negligible. In coordination with the Chandra observation, we also observed Cyg X-1 with the Pro- portional Counter Array (PCA) and High Energy X-ray Timing Experiment (HEXTE) de- tectors on board RXTE. The PCA and HEXTE covers roughly the energy ranges of 2–60 keV and 15-250 keV, respectively. In this work, we made use of the combined broad spec- tral coverage of the two detectors to constrain the X-ray continuum more reliably. The RXTE observation was carried out in six short segments, with exposure times of 1–3 ks. The data were reduced with the standard HEASOFT package (version 5.2), along with the associated calibration files and background models. We followed the usual procedures2 in 1See http://asc.harvard.edu/ciao3.2/threads/index.html 2See http://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook book.html http://asc.harvard.edu/ciao3.2/threads/index.html http://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook_book.html – 4 – preparing, filtering, and reducing the data, as well as in deriving light curves and spectra from the standard-mode data. While the HEXTE background was directly measured from off-source observations, the PCA background was estimated based on the background model appropriate for bright sources. 3. Analysis and Results 3.1. Light Curves Figure 2 shows the Chandra light curve of Cyg X-1 that is made from the MEG first- order data (with plus and minus one orders coadded). For comparison, we have over-plotted the average count rate of PCU #2 for each of the RXTE exposures. The agreement is quite good. Besides the rapid flares and other short-term variability that are expected in Cyg X-1, the source also varies significantly on a timescale of hours. The MEG count rate rises from a baseline level of roughly 170 counts s−1 to a peak of roughly 220 counts s−1 within 3–4 hours and quickly goes back down to the baseline level. We somewhat arbitrarily divided the light curve into two time periods (which are labeled as Period I and Period II in Fig. 2) for subsequent analyses. 3.2. High-Resolution Spectroscopy We analyzed and modeled the HETGS spectra in ISIS (version 1.2.9) (Houck & Denicola 2000)3. For this work, we focused only on the first-order spectra, because of the relatively poor signal-to-noise ratio of higher-order spectra. For both the MEG and HEG, we first co- added the plus and minus orders to produce an overall first-order spectrum, again following the appropriate CIAO 3.2 science threads. The resolution of the raw data is about 0.01 Å and 0.005 Å for the MEG and HEG, respectively, which represents a factor of 2 over-sampling of the instrumental resolution. We applied no further binning of the data. Each spectrum was broken into 3-Å segments for subsequent analyses. Each segment was fitted locally with a model that consists of a multi-color disk component and a power law for the continuum and negative or positive Gaussian functions for narrow absorption or emission features, with the interstellar absorption taken into account. Such a continuum model is typical of BHCs. However, the best-fit continuum differs among the segments or between the MEG and HEG, presumably due to remaining calibration uncertainties. Since we are only interested in using 3See also http://space.mit.edu/CXC/ISIS/ http://space.mit.edu/CXC/ISIS/ – 5 – the Chandra data to study lines, we think that the adopted procedure is justified. We use the RXTE data to more reliably constrain the continuum. One thing that one notices right away is the presence of many absorption lines in the high-resolution spectra. For this work, we consider a feature real if it is present both in the MEG and HEG data, with a significance of above 4σ. Figure 3 shows a portion of the MEG first-order spectrum for Period I, to highlight the lines detected. No absorption lines were found at wavelengths above 15 Å. We should point out that we also see the emission- like features at 6.74 Å and 7.96 Å, which most likely instrumental artifacts associated with calibration uncertainties around the Si K and Al K edges (Miller et al. 2005 and references therein). We have identified the absorption lines with highly ionized species of Ne, Na, Mg, Al, Si, S, and Fe, based mostly on the atomic data in Verner et al. (1996) and Behar et al. (2002), but also in ATOMDB4 1.3.1 for additional transitions. The line identification process involves three steps: (1) a transition is considered a candidate if the theoretical wavelength is within 0.03 Å of the measured value; (2) in cases where multiple candidates exist based on (1), the most probably one (based both on the oscillator strength of the transition and the relative abundance of the element) is chosen; and (3) consistency check is made to avoid the identification of a line when more probable transitions of the same ion are not seen. The last step is critical. For instance, we initially associated the lines at 10.051 Å and 11.029 Å with Fe XVIII 2s22p5–2s22p46d and Fe XVII 2s22p6–2s2p64p, respectively, because, assuming solar abundances, they are expected to be stronger than Na XI 1s–2p and Na X 1s2–1s1p, which were also viable candidates from Step 1. However, we did not detect other more probable transitions associated with Fe XVIII and Fe XVII, which made the identifications highly unlikely. We had to conclude that Fe is under-abundant by a factor of 2–3, at minimum, so that we could associate the lines with Na instead. If so, the line at 7.480 Å would more likely be associated with Mg XI 1s2–1s4p, as opposed to Fe XXIII 2s2–2s5p, which we initially identified. The problem with this is that we saw no hint of Mg XI 1s2–1s3p, which is more probable. Therefore, we had to conclude similarly that Mg is also under-abundant by at least a similar amount (but not by too much, as constrained by other Mg lines). Table 1 show all of the absorption lines that we detected and identified. The flux and equivalent width (EW) of each line shown were derived from the best-fit Gaussian for the line, as well as the local continuum around the line for the latter. Note that in a few cases our identifications are different from those in the literature. They include: Fe XXII at 8.718 Å and Fe XXI at 9.476 Å (cf. Marshall et al. 2001; Miller et al. 2005); Fe XX at 10.12 Å, 4See http://cxc.harvard.edu/atomdb http://cxc.harvard.edu/atomdb – 6 – Na X at 11.0027 Å, and Fe XX at 12.82 Å (cf. Marshall et al. 2001); and Fe XXI 11.975 Å (cf. Miller et al. 2005). Only about half of the absorption lines that we detected have been seen previously in Cyg X-1 (Marshall et al. 2001; Schulz et al. 2002; Feng et al. 2003; Miller et al. 2005). Many of these lines are much stronger in our case. On the other hand, we detected nearly all of the reported absorption lines. The exceptions include: S XVI at 4.72 Å (Marshall et al. 2001; Feng et al. 2003), which is detectable only at the 3σ level in our case (and is thus not included in the table), Fe XIX at 14.53 Å (Miller et al. 2005), which is actually detectable at about the 4σ level here (but just misses our threshold), and Fe XIX at 14.97 Å and Fe XVIII at 16.01 Å (Miller et al. 2005), whose presence is not apparent in our data (< 3σ). Miller et al. (2005) also reported a line at 7.85 Å (Mg XI) but only at the 3σ level. The line can also be seen in our data at the 4σ level (but also just misses our threshold). Therefore, we already see some indication that the absorption lines in Cyg X-1 may be variable from a comparison of our results with those published. Still, it is surprising that almost all of the absorption lines become undetectable in Period II. Figure 4 shows the MEG first-order spectrum for this time period, which can be directly compared to results in Fig. 3. The only exception is the line at 14.608 Å, which is detected with a significance of 5σ in Period II. We derived 99% upper limits on the flux and EW of each line seen in Period I but not in Period II. The results are also summarized in Table 1, for direct comparison. As an example, we examine the Ne X line at 12.1339 Å in the two periods. The integrated flux of the line is about 5× 10−3 photons cm2 s−1 in Period I, while its 99% upper limit for Period II is only 0.06×10−3 photons cm2 s−1. Therefore, the line weakened by nearly two orders of magnitude in flux over a timescale of merely several hours. This is the first time that such dramatic variability of the lines has been observed in any BHC. While this is the most extreme case, other lines also show large variability (see Table 1). To quantify the column density of each ion required to account for the corresponding absorption line detected and its variability, we carried out curve-of-growth analysis, following Kotani et al. (2000). The atomic data used in the analysis were again taken from Verner et al. (1996), Behar et al. (2002), and also ATOMDB 1.3.1 in some cases. The analysis assumes that the width of the lines is due entirely to thermal Doppler broadening. For resolved lines, we derived the characteristic temperature from the measured widths. For unresolved lines, on the other hand, we adopted a temperature that would lead to a line width equal to the resolution of the MEG (0.023 Å at FWHM). In these cases, therefore, the derived column density only represents a lower limit. The results are shown in Table 1. This explains why, e.g., the density of Ne X derived from the 12.144-Å line (which is unresolved) is significantly lower than that from the 10.245-Å line. Note, however, that the latter is much lower that that from the 9.727-Å line. We think that the inconsistency arises from the – 7 – fact that the 9.727-Å line is likely a mixture of the Ne line and the Fe XIX line at 9.700 Å. Similar inconsistency is also apparent in a few other cases (see Table 1), which may originate similarly in line blending. It is also worth noting that most lines that we have analyzed fall on the linear part of the curve of growth. All resolved lines at wavelengths λ > 11.7 Å are saturated; so is the unresolved line at 14.203 Å. To show the degree of variability, we also derived a 99% upper limit on the column density of each of the ions for Period II (assuming the same characteristic temperatures). 3.3. X-ray Continuum We now use the RXTE data to constrain the X-ray continuum of Cyg X-1. Both the PCA and HEXTE data were used. For the PCA, we used only data from the first xenon layer of each PCU, which is most accurately calibrated. Consequently, the PCA spectral coverage is limited to roughly 2.5–30 keV. We relied on the HEXTE data to extend spectral coverage to higher energies. The PCA consists of five detector units, known as PCUs. Not all PCUs were always operating. For simplicity, we used only data from PCU #0 and PCU #2, which stayed on throughout the observation, in the subsequent modeling. We chose to derive a spectrum for each PCU separately, as well as for each of the two HEXTE clusters. Residual calibration errors were accounted for by adding 1% systematic uncertainty to the data. We also rebinned the HEXTE spectra to a signal-to-noise ratio of at least 3 in each bin. The individual spectra were then jointly fitted with the same model that includes a multi-color disk component, a broken power-law that rolls over exponentially beyond a characteristic energy, and a Gaussian, taking into account the interstellar absorption. We also multiply the model by a normalization factor that is fixed at unity for PCU #2 but is allowed to vary for other detectors, in order to account for any uncalibrated difference in the overall throughput among the detectors. The model fits the data well for all six segments in the sense that the reduced χ2 is near unity (with 169 degrees of freedom). The spectral shape of Cyg X-1 varied little from one observation to the other. The best- fit photon indices are ∼2.1 and ∼1.7 below and above ∼10 keV. The roll-over energy stays at 20–21 keV and the e-folding energy roughly at 120–130 keV. Neither the disk component nor the absorption column density is well constrained by the data, due to the lack of sensitivity (and, to some extent, large calibration uncertainty) at low energies. The results can be compared directly with those of Cui et al. (1997a), who applied the same empirical model to the RXTE spectra of Cyg X-1 during the 1996 transition. It is quite apparent that the spectra here are significantly harder, implying that the source was certainly not yet in the true high-soft state (see Cui et al. 1997a and 1997b for discussions on the “settling period”). – 8 – From the long-term ASM monitoring data, we can see that Cyg X-1 was brighter during our observation than during any of the earlier Chandra observations (Marshall et al. 2001; Schulz et al. 2002; Miller et al. 2005), but not as bright as during a later observation (Feng et al. 2003), when the high-soft state appears to have been reached. 3.4. Photoionization Modeling To shed light on the physical properties of the absorber, we carried out a photo-ionization calculation with XSTAR version 2.1 kn35. The underlying assumption is that the absorber is photo-ionized by the X-ray radiation from the vicinity of the black hole. The input parameters include the 0.5-10 keV luminosity (Lx = 3.11× 10 37 erg s−1, for a distance of 2.5 kpc), power-law photon index (2.1), both of which are based on results from modeling one of the RXTE spectra with an assumed NH value of 5.5×10 21 cm−2 (Ebisawa et al. 1996; Cui et al. 1998). We should note that it is in general risky to extrapolating the assumed power-law spectrum to lower energies, because it could severely over-estimate the flux there. For the lines of interest here, however, only ionizing photons with energies > 1 keV contribute and the spectrum of those photons is described fairly well by a power law, because the effective temperature of the disk component is expected to very low (e.g., Ebisawa et al. 1996; Cui et al. 1998). One of the outputs of the calculation is abundances of the ions of interest, as a function of the ionization parameter, ξ = Lx/nr 2, where n is the density of the absorber and r is the distance of the absorber to the source of ionizing photons. Using these abundance curves, we could, in principle, constrain ξ to a range that is consistent with the ratio of the densities of any two ions of an element. The challenge in practice is, as already mentioned, that many of the lines are likely a blend of multiple transitions (of comparable probabilities), which makes it difficult to reliably determine the densities. Nevertheless, we made an attempt at deriving such constraints with the resolved, non-mixed lines. Figure 5 summarizes the results. The intervals do not all overlap, which implies that no single value of ξ could account for all the data. This is supported by the fact that we have detected all the lines that are expected for ionization parameters in the range of roughly 102.5–104.5. If one assumes that the “absorbers” are thin shells along the line of sight and that they all have the same density, e.g., n = 1011 cm−3 (see, e.g., Wen et al. 1999), one would have 1011 cm . r . 1012 cm. Compared with the estimated the distance between the compact object and the companion star (∼ 1.4 × 1012 cm; LaSala et al. 1998), this would put the “absorbers” within the 5See http://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html http://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html – 9 – binary system. One should, however, take the results with caution, because of, e.g., gross over-simplification regarding the geometry of the “absorbers”. 4. Discussion The observed dramatic variability of the absorption lines might be caused by a change in the degree of ionization in the wind. Since the overall X-ray luminosity varied only mildly, we speculate that it probably arises from a sudden change in the density of the wind. There is evidence that such a change could occur during a state transition or during flares (Gies et al. 2003). If the moderate decrease in the ionizing flux is accompanied by a more dramatic reduction in the density of the wind from Period I to Period II, the ionization parameter might increase sufficiently to cause a total ionization of the wind in Period II and thus the disappearance (or significant weakening) of the lines. It is worth noting that in Period I lighter elements seen are all H- or He-like but Fe is in an intermediate ionization state (as indicated by the absence of H- or He-like ions), suggesting a high but not extreme degree of ionization in the period. Conversely, a dramatic increase in the wind density could achieve the same effect. Numerical simulations of similar wind-accretion systems (e.g., Blondin, Stevens, & Kallman 1991) have revealed not only a significant jump in the column density at late orbital phases (& 0.6) that is associated with tidal streams but also large variability of the absorbing column. It is conceivable that Period II might coincide with a sudden increase in the column density. Since we found no apparent absorption lines that correspond to a lower degree of ionization in Period II, however, such lines must be outside the spectral range covered with our data, in order for the scenario to be viable. A quantitative assessment of these scenarios is beyond the scope of this work. Many of the absorption lines detected by Miller et al. (2005) are much stronger during Period I of our observation (see Table 1). Using only the lines that are detected with a significance above 5σ, we looked for a systematic red- or blue- shift of the lines, following up on the reported redshift of the lines by Marshall et al. (2001) based on data taken in the low-hard state. The results are summarized in Figure 6 (in the left panel). In this case, although the lines are still systematically redshifted on average, there is not an obvious single-velocity solution. Interestingly, if we limit the results only to those lines that were used by Marshall et al. (2001), as shown in the right panel of Figure 6, we would arrive at an average velocity that is very close to what Marshall et al. reported, although the scatter of data points is much larger in our case. On the other hand, our observation spans binary orbital phases from 0.85 to 0.92, according to the most updated ephemeris (Brocksopp et al. 1999b), while Marshall et al.’s covers a phase range of 0.83–0.86. If the redshift of the lines – 10 – is related to the focused-wind scenario advocated by Miller et al. (2005), we ought to see a larger (by about 30%) redshift. Given the large uncertainties, as well as the possibility that the wind geometry might be different for different states, it is difficult to draw any definitive conclusions. Feng et al. (2003) reported the detection of a number of absorption lines of asymmetric profile, when Cyg X-1 was in the high-soft state, which they interpreted as evidence for inflows. The same lines are also present in our data during Period I and are, in fact, much stronger (except for S XVI, as noted in § 3.2). Figure 7 shows an expanded view of the Si and Mg lines, which are the strongest in the group. As is apparent from the figure, the line profile can be fitted fairly well by a Gaussian function in all cases. Therefore, the lines show no apparent asymmetry here. This also seems to be the case for the S and Fe lines, although the statistics of the data are not as good. Taken together, our results and Feng et al.’s imply that that the phenomenon is either unique to the high-soft state (in which Feng et al. made the observation) or is intermittent in nature. We should also note that Feng et al’s observation was carried out around the superior conjunction (i.e., zero orbital phase), where absorption due to the wind is expected to be the strongest (e.g., Wen et al. 1999). It is not clear, however, how such additonal absorption would lead to an asymmetry in the profile of lines. No emission lines are apparent in our data. Evidence for weak emission lines has been presented (Schulz et al. 2002; Miller et al. 2005) but the significance is marginal in all cases. On the other hand, several absorption edges are easily detected in our data (see Figs. 3 and 4), as first reported and studied in detail by Schulz et al. (2002). The edges can almost certainly be attributed to the interstellar absorption. We thank Harvey Tananbaum for approving this DDT observation, Norbert Schulz and Herman Marshall for helpful discussion on the pros and cons of various observing configura- tions, John Houck for help with the use of ISIS and Tim Kallman for help on using XSTAR, and David Huenemoerder for looking into issues related to the Chandra data products. We acknowledge the use of the curve-of-growth analysis script that Taro Kotani has made pub- licly available. We also thank the anonymous referee for a number of useful comments that led to significant improvement of the manuscript. Support for this work was provided in part by NASA through the Chandra Award DD1-2011X issued by the Chandra X-ray Ob- servatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060, and through the LTSA grant NAG5-9998. – 11 – REFERENCES Behar, E., & Netzer, H., 2002, ApJ, 570, 165 Blondin, J. M., Stevens, I. R., & Kallman, T. R. 1991, ApJ, 371, 684 Bolton, C. T., 1972, Nature, 235, 271 Brocksopp, C., et al. 1999a, MNRAS, 309, 1063 Brocksopp, C., Tarasov, A. E., Lyuty, V. M., & Roche, P., 1999b, A&A, 343, 861 Cui, W., Heindl, W. A., Rothschild, R. E., Zhang, S. N., Jahoda, K., & Focke, W. 1997a, ApJ, 474, L57 Cui, W., Zhang, S. N., Focke, W., & Swank, J. 1997b, ApJ, 484, 383 Cui, W., Chen, W., & Zhang, S. N. 1997, in 1997 Pacific Rim Conference on Stellar Astro- physics, eds. K. L. Chan, et al., PASP Conf. Ser, vol 138, p. 75 Cui, W., Ebisawa, K., Dotani, T., & Kubota, A. 1998, ApJ, 493, L75 Ebisawa, K., et al. 1996, ApJ, 467, 419 Feng, Y. X., Tennant, A. F., & Zhang, S. N., 2003, ApJ, 597, 1017 Gies, D. R., & Bolton, C. T., 1986, ApJ, 304, 389 Gies, D. R., et al. 2003, ApJ, 583, 424 Houck, J. C., & Denicola, L. A. 2000, in Astronomical Data Analysis Software and Systems IX, eds. N. Manset, C. Veillet, and D. Crabtree, ASP Conf. Proc., Vol. 216, p.591 Kotani, T., Ebisawa, K., Dotani, T., Inoue, H., Nagase, F., Tanaka, Y., & Ueda, Y., 2000, ApJ, 539, 413 LaSala, J., Charles, P. A., Smith, R. A. D., Balucinska-Church, M., & Church, M. J. 1998, MNRAS, 301, 285 Marshall, H. L., Schulz, N. S., Fang, T,. Cui, W., Canizares, C. R., Miller, J. M., & Lewin, W. H. G., 2001, in X-Ray Emmission from Accretion onto Black Holes, eds. T. Yaqoob, & J. H. Krolik, 45 620, 398 McClintock, J. E., & Remillard, R. A. 2006, in Compact Stellar X-ray Sources, Eds. W. H. G. Lewin and M. van der Klis (Cambridge Univ. Press), p157 Miller, J. M., et al. 2005, ApJ, 620, 398 Schulz, N. S., Cui, W., Canizares, C. R., Marshall, H. L., Lee, J. C., Miller, J. M., & Lewin, W. H. G., 2002, ApJ, 565, 1141 Verner, D. A., Verner, E. M. & Ferland, G. J., 1996, At. Data Nucl. Data Tables, 64, 1 – 12 – Webster, B. L., & Murdin, P., 1972, Nature, 235, 37 Wen, L., Cui, W., Levine, A. M., & Bradt, H. V. 1999, ApJ, 525, 968 This preprint was prepared with the AAS LATEX macros v5.2. – 13 – Table 1: Detected Absorption Lines Theoretical Measured Shift Flux(I) Flux(II) EW(I) EW(II) Nz(I) Nz(II) Ion and Transition (Å) (Å) (km s−1) (10−3ph cm−2s−1) (mÅ) (mÅ) (10−16cm−2) (10−16cm−2) S XV 1s2-1s2p 5.039b 5.041(3) 120±180 1.8(3) <0.9 3.8(7) <1.5 2.5(5) ≤1.0 Si XIV 1s-2p 6.1822a 6.189(1) 330±50 3.0(2) <0.6 6.9(4) <1.0 5.4(3) ≤0.7 Si XIII 1s2-1s2p 6.648b 6.657(2) 410±90 2.6(2) <1.7 6.1(5) <2.9 2.4(2) ≤1.1 Mg XII 1s-3p 7.1062a 7.119(3) 540±130 1.2(2) <0.9 2.6(5) <1.4 8(2) ≤4.0 Al XIII 1s-2p 7.1727a 7.191( ) 760 1.3(2) <1.7 3.0(5) <2.7 1.6(3)d ≤1.5 Fe XXIII 1s22s2p-1s22s6d 7.2646c 7.268(5) 140±210 1.4(3) <2.0 3.1(6) <3.3 29(6) ≤31 Fe XXIII 1s22s2-1s22s5p 7.4722a 7.480( ) 310 0.9(2) <1.4 1.9(5) <2.3 5(1) ≤6.5 Fe XXIV 1s22s-1s24p 7.9893a 8.004(5) 550±190 2.2(3) <1.3 4.7(7) <2.2 9(1) ≤4.0 Fe XXII 1s22s22p-1s22s25d 8.0904c 8.096(3) 210±110 1.0(2) <0.6 2.1(5) <1.0 8(2)d ≤3.6 Fe XXII 1s22s22p-1s22s25d 8.1684c 8.166(3) -90±110 1.0(2) <1.1 2.3(5) <1.9 9(2)d ≤7.2 Fe XXIII 1s22s2-1s22s4p 8.3029a 8.319(2) 580±70 3.0(3) <0.6 6.6(6) <1.0 6.4(6) ≤0.9 Mg XII 1s-2p 8.4210a 8.428(1) 250±40 5.0(3) <0.4 10.8(6) <0.7 4.7(3) ≤0.3 Fe XXI 1s22s22p2-1s22s22p5d 8.573a 8.577(5) 140±170 1.3(3) <0.7 2.8(7) <1.2 6(2) ≤2.6 Fe XXII 1s22s22p-1s22s2p1/24p3/2 8.718 c 8.735( ) 580 2.2(3) <0.5 4.7( ) <0.9 7(1) ≤1.3 Fe XXI 1s22s22p2-1s22s2p1/22p3/24p3/2 8.8254 c 8.823( ) -80 1.4(3) <0.9 3.0( ) <1.4 21( ) ≤9.4 Fe XXII 1s22s22p-1s22s24d 8.98a 8.978( ) -70 1.8(3) <0.9 3.9(6) <1.6 4.6(8)d ≤1.8 Fe XXII 1s22s22p-1s22s24d 9.07a 9.083( ) 430 ) <0.6 4.0( ) <1.0 5.2 Mg XI 1s2-1s2p 9.170b 9.192( ) 720 6.2(4) − 13.5(9) − 2.9(2) − Fe XXI 1s22s22p2-1s22s22p4d 9.356a 9.378(5) 700±160 1.9(4) <2.8 4.3 <4.8 9(2) ≤10 Fe XXI 1s22s22p2-1s22s22p4d 9.476a 9.478(1) 60±30 3.8(3) <0.8 8.6( ) <1.5 6.6( )d ≤1.0 Fe XIX 1s22s22p4-1s22s22p3(2D)5d 9.68a 9.700(4) 620±120 4.4( ) <1.8 9(1) <3.1 30(3) ≤9.8 Ne X 1s-4p 9.7082a 9.727( ) 580 ) <0.7 5.3( ) <1.2 24(4) ≤5.0 Fe XX 1s22s22p3-1s22s22p2(3P)4d 9.991a 10.000(1) 270±30 3.7(4) <0.7 8.3(8) <1.3 5.6(6)d ≤0.8 Na XI 1s-2p 10.0250a 10.051(2) 780±60 5.6( ) <2.0 13(1) <3.5 4.0(3) ≤1.0 Fe XX 1s22s22p3-1s22s22p2(3P)4d 10.12a 10.127(3) 210±90 1.7(4) <0.3 3.8(9) <0.6 2.3(6)d ≤0.4 Ne X 1s-3p 10.2389a 10.245( ) 180 ) <0.5 6(1) <0.9 9(2) ≤1.2 Fe XXIV 1s22s-1s23p 10.619a 10.631( ) 340 9.1(6) <4.6 22(2) <8.9 10(1) ≤3.7 Fe XXIV 1s22s-1s23p 10.663a 10.674(3) 310±80 5.6(6) <3.2 14(2) <6.2 12(2) ≤5.0 Fe XIX 1s22s22p4-1s22s22p3(4S)4d 10.816c 10.818(5) 60±140 7.0(9) <3.0 18(2) <6.0 14(2) ≤4.3 Fe XXIII 1s22s2-1s22s3p 10.981a 10.990(1) 230±30 5.5(5) <2.7 14(1) <5.6 2.4(2)d ≤0.8 Na X 1s2-1s2p 11.0027a 11.029(2) 720±50 6.3( ) <4.4 17(2) <9.2 2.7(4) ≤1.3 Fe XVIII 1s22s22p5-1s22s22p4(1D)4d 11.326c 11.33(1) 100±260 8(1) <5.1 23( ) <11 24( ) ≤11 Fe XXII 1s22s22p-1s22s2p(3P0)3p 11.44a 11.431(1) -240±30 7.3(6) <2.5 21(2) <5.7 8(1)d ≤1.6 Fe XXII 1s22s22p-1s22s2p(3P0)3p 11.51a 11.500(3) -260±80 4.3( ) <0.6 12(2) <1.4 22( ) ≤2.3 Fe XXII 1s22s22p-1s22s23d 11.77a 11.781( ) 280 12.6(9) <3.8 39(3) <9.3 7.5 Fe XXI 1s22s22p2-1s22s2p23p 11.975c 11.982(2) 180±50 9.1(9) − 29(3) − 17( Ne X 1s-2p 12.1339a 12.144( ) 250 4.7(7) <0.06 16(2) <0.2 3.9( )d ≤0.04 Fe XXI 1s22s22p2-1s22s22p3d 12.259a 12.247( ) -290 ) <5.4 14(3) <14 10(3)d ≤10 Fe XXI 1s22s22p2-1s22s22p3d 12.285a 12.304(2) 460±50 22(1) <6.8 75(5) <18 12(2) ≤1.6 Fe XXI 1s22s22p2-1s22s22p3d 12.422c 12.438(2) 390±50 3.9(8) <2.9 14(3) <8.1 3.0( )d ≤1.6 Fe XX 1s22s22p3-1s22s2p33p 12.576c 12.583( ) 170 7(1) <3.6 26(4) <10 19( ) ≤5.2 Fe XX 1s22s22p3-1s22s22p2(3P)3d 12.82a 12.844(2) 560±50 26(2) <6.5 100(6) <20 34 Fe XX 1s22s22p3-1s22s22p1/22p3/23d 12.912 c 12.914(3) 50±70 12(2) <4.7 47(6) <14 33 Fe XX 1s22s22p3-1s22s22p1/22p3/23d 12.965 c 12.953( ) -280 12(1) − 48(6) − 56 Ne IX 1s2-1s2p 13.448b 13.448( 9(2) <8.5 38(9) <30 6( ) ≤4.1 Fe XIX 1s22s22p4-1s22s22p1/22p 3d 13.479c 13.482(3) 70±70 5(1) <3.3 20( ) <12 1.0( )d ≤0.5 Fe XIX 1s22s22p4-1s22s22p3(2D)3d 13.518c 13.523(3) 110±70 16(2) − 70( ) − 24 Fe XVIII 1s22s22p5-1s22s22p4(1D)3d 14.203a 14.220(3) 360±40 6(1) <4.0 32(7) <18 4( )d ≤1.5 Fe XIX 1s22s22p4-1s22s22p3(2P)3s 14.60a 14.608(5)e 100±100 19(3) 20(4) 81±13 73±14 200 −60Fe XVIII 1s22s22p5-1s22s22p4(3P)3d 14.610a Notes. — Results for Periods I and II are both shown for comparison. The errors in parentheses indicate uncertainty in the last digit of the measurement; 1σ errors are shown. Negative flux or EW upper limits (indicating emission) are not shown. a Verner et al. (1996); b Behar et al. (2002); c ATOMDB 1.3.3; d Unresolved; e The two transitions are equally probable. The average wavelength was used to derive the Doppler-shift of the line. – 14 – Fig. 1.— Daily-averaged ASM Light Curve of Cyg X-1 during the 2001 state transition. The vertical line indicates the time of the Chandra observation. – 15 – Fig. 2.— X-ray Light Curves of Cyg X-1. The solid curve shows data from the MEG first order, while the horizontal bars show the average count rates from PCU #2. The error bars are negligible in both cases. The dashed line defines the two time periods for subsequent analyses (see text). – 16 – Fig. 3.— MEG first-order spectrum of Cyg X-1 for Period I. No binning was applied. The presence of absorption lines are apparent. The identifications of the lines are shown. Note that the emission-like features at 6.74 Å and 7.96 Å are likely instrumental (see text). The solid line shows the best-fit to the (local) continuum. – 17 – Fig. 4.— As in Fig. 3, but for for Period II. Note the absence of nearly all the absorption lines seen in Fig. 3. – 18 – 2.5 3 3.5 4 Ionization Parameter (log ξ) Fe XX/Fe XXII Fe XX/Fe XXIV Fe XX/Fe XIX Fe XX/Fe XXI Fe XX/ Fe XXIII Si XIV/Si XIII Ma XII/Mg XI Ne X/Ne IX Fig. 5.— Allowed ranges of the ionization parameter, each of which is inferred from the ratio of the average densities of two ions of the same element. 4 6 8 10 12 14 Wavelength (Å) 4 6 8 10 12 14 Wavelength (Å) Fig. 6.— Inferred Doppler shift of the selected absorption lines, (left) all the lines with a significance above 5σ and (right) only the lines that were used by Marshall et al. (2001). The dotted line shows the average Doppler velocity in both cases. – 19 – Fig. 7.— Profiles of the selected absorption lines. In each case, the dot-dashed histogram shows a fit to the profile with a Gaussian function. Introduction Observations and Data Reduction Analysis and Results Light Curves High-Resolution Spectroscopy X-ray Continuum Photoionization Modeling Discussion

0704.0453

Antiferromagnetism-superconductivity competition in electron-doped cuprates triggered by oxygen reduction

Antiferromagnetism-superconductivity competition in electron-doped cuprates triggered by oxygen reduction P. Richard,1, ∗ M. Neupane,1 Y.-M. Xu,1 P. Fournier,2 S. Li,3 Pengcheng Dai,3, 4 Z. Wang,1 and H. Ding1 Department of Physics, Boston College, Chestnut Hill, MA 02467 Département de physique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1 Department of Physics and astronomy, The University of Tennessee, Knoxville, TN 37996 Neutron Scattering Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 (Dated: October 25, 2018) We have performed a systematic angle-resolved photoemission study of as-grown and oxygen- reduced Pr2−xCexCuO4 and Pr1−xLaCexCuO4 electron-doped cuprates. In contrast to the common belief, neither the band filling nor the band parameters are significantly affected by the oxygen reduction process. Instead, we show that the main electronic role of the reduction process is to remove an anisotropic leading edge gap around the Fermi surface. While the nodal leading edge gap is induced by long-range antiferomagnetic order, the origin of the antinodal one remains unclear. PACS numbers: 74.72.Jt, 74.25.Jb, 74.62.Dh, 79.60.-i Even though most of the work has been focused on hole-doped cuprates, the understanding of their electron- doped counterparts is essential for obtaining a univer- sal picture of high-Tc superconductivity. To achieve this goal, it is necessary to first solve the main mystery that holds since the discovery of the T’-structure electron- doped cuprates RE2−xCexCuO4 and RE1−xLaCexCuO4 (RE = Pr, Nd, Sm, Eu): why is superconductivity in these compounds achieved only when a tiny amount of oxygen (∼ 1%) is removed from the as-grown (AG) samples following a post-annealing process (reduction) [1, 2, 3, 4, 5]? In fact, the AG samples, even with suf- ficient electron-doping by adding Ce, are antiferromag- netic (AF) insulators at low temperature. Far from be- ing a simple materials issue, the understanding of the microscopic origin of the reduction process that triggers superconductivity may shed light on other related ques- tions in high-Tc superconductivity. Long considered as the microscopic explanation of the reduction process, the removal of extraneous oxygen atoms located above Cu (apical oxygen) has been ruled out by recent Raman and crystal-field infrared transmis- sion studies [6, 7]. Indeed, these studies revealed two main defects appearing with the oxygen reduction, which have been tentatively assigned to out-of-plane and in- plane oxygen vacancies, the latter being the only one observed at optimal doping. In parallel, a (RE,Ce)2O3 impurity phase epitaxial to the CuO2 planes appears in reduced superconducting samples but disappears in re- oxygenated nonsuperconducting samples [8, 9, 10, 11]. Based on this phenomenon, it has been proposed recently that the Cu excess released during the formation of the (RE,Ce)2O3 impurity phase fills Cu vacancies and makes the remaining structure more stoichiometric [11]. Which of these structural defects has the most significant impact on the electronic properties is still under intense debate ∗Electronic address: [email protected] and calls for a better characterization of the electronic band structure before and after the reduction process. In contrast to the widespread belief that the reduction process in the electron-doped cuprates can be consid- ered as an independent degree of freedom for carrier dop- ing, a recent systematic study of the Hall coefficient in Pr2−xCexCuO4 thin films with various oxygen contents showed that the carrier mobility rather than their con- centration is modified by the reduction process [12]. In particular, the annealing process leads to the delocaliza- tion of holelike carriers, most likely due to the suppres- sion of the long-range AF order [10, 13]. Understanding how the reduction process can tune the competition be- tween the AF and superconducting states and modify the electronic band structure is thus crucial. In this letter, we present the first systematic angle- resolved photoemission spectroscopy (ARPES) study of the impact of the reduction process in the electron-doped cuprates. We show that neither the electronic filling nor the band structure parameters are significantly changed by the reduction process. Instead, the reduction process suppresses long-range AF order and fills up a leading edge gap (LEG) which has two components. While the nodal LEG is of AF origin, the nature of the antinodal LEG remains unclear. High-quality Pr1.85Ce0.15CuO4 single crystals have been grown by the flux technique. Some of the non- superconducting AG samples have been annealed in an argon environment at temperatures between 850 and 925 oC for a typical period of 5 days encapsulated in a poly- crystalline matrix [14] and are referred in the text as reduced samples. The reduced samples exhibit super- conducting transitions around 24 K. Using the floating zone technique, high-quality Pr0.88LaCe0.12CuO4 single crystals have also been grown and have been annealed as described in Ref. [11]. The samples have been stud- ied by ARPES using the PGM and U1-NIM beamlines of the Synchrotron Radiation Center (Stoughton, WI) with photon energies of 73.5 and 22 eV. The data have been recorded at 40 K using a Scienta SES-2002 ana- http://arxiv.org/abs/0704.0453v1 mailto:[email protected] lyzer with a 30 meV energy resolution. The samples have been cleaved in situ in a vacuum better than 10−10 Torr. Although this letter focuses on the data obtained on Pr2−xCexCuO4, similar results have been obtained for the Pr1−xLaCexCuO4 samples. In order to transform AG samples into superconduc- tors, the reduction process must affect the electronic structure and especially the excitations near the Fermi energy (EF). Fig. 1 compares the constant energy in- tensity plots (CEIPs) of the reduced (a,b) and AG (d,e) samples in momentum space, as obtained by ARPES. Bright spots indicate regions with large photoemission intensity. The CEIPs centered at -100 meV with 20 meV energy integration window (b,e) are quite similar, and one can easily distinguish the X(±π,±π)-centered hole- like pockets. This contrasts with the CEIPs centered at EF (a,d). Contrary to the reduced sample (Fig. 1a), the intensity at EF is strongly suppressed in the AG sample (Fig. 1d). Nevertheless, underlying Fermi sur- face (FS) contours can be extracted in both cases and the results are reproduced in Figs. 1c and f for the re- duced and AG samples, respectively. Surprisingly, the data extracted for the reduced and the AG samples can be fitted, within uncertainties, with the same band pa- rameters µ = 0.05 eV, t = −1.1 eV and t′ = 0.32 eV, using the simple effective tight-binding (TB) model E − µ = t/2[cos(kx) + cos(ky)] + t ′ cos(kx) cos(ky). According to Luttinger theorem, the introduction of extra negative carriers following the reduction would lead to smaller X-centered holelike pockets and thus to at least an increase of µ, which is not observed. Actually, the un- derlying FS contours coincide with a doping of x ≈ 0.15 in both cases. This is a strong evidence that the re- duction process modifies neither the band filling nor the shape of the band dispersion sufficiently to induce the dramatic changes observed in the transport properties [12]. Instead, the CEIPs at EF indicate that the anneal- ing process removes a LEG that is present at EF in the AG samples. This assertion is confirmed by Fig. 1h, which compares the electron distribution curves (EDCs) of AG and reduced samples at different k-locations given in Fig. 1g. All the AG sample spectra have their leading edge shifted towards higher binding energies as compared with the corresponding reduced sample spectra and thus have much weaker intensities at EF. Now one asks the question: how can the reduction pro- cess suppress the LEG observed in the AG samples? We first checked that the samples were not charged by in- creasing the photon flux, which had no influence on the leading edge shift in our experiment. The most likely candidate to explain this mystery is the AF ordering, which exists in AG samples. It is known that AF is sup- pressed in the reduced samples [10, 13]. This effect is clearly observable by ARPES. We plotted in Figs. 2a and 2b the electronic dispersion as measured along the lines indicated in Fig. 2c for the reduced (dashed) and AG (solid) samples, respectively. In addition to the main band branches, indicated with dashed arrows, the spec- FIG. 1: (Color online) a-b and d-e) CEIPs (20 meV integra- tion) of the ARPES data obtained at 40 K using a 73.5 eV photons with a A||Γ-X polarization. c and f) Underlying FS contours associated with the reduced and AG samples. The experimental data are represented by circles while the points indicated by diamonds have been obtained by symmetry oper- ations. The data have been fitted by an effective tight-binding model (see the text). h) Comparison of the EDCs of reduced (solid) and AG (dashed) at the locations given in panel g. trum of the AG sample shows features that are not ob- served in the reduced one. As indicated directly by solid arrows on Fig. 2, these features correspond to the AF- induced folding (AIF) band. Such features, observed in the first as well as in the second Brillouin zones (BZs), are responsible for the M(-π,0)-centered electron FS pockets emphasized in Fig. 2d, which shows the AG CEIP at -50 meV obtained with 22 eV photons. In the presence of an AIF band, the hybridization of the main and AIF bands opens a gap at locations coin- ciding with the magnetic BZ boundary [15]. The energy position of the center of the gap between the upper and lower hybridized bands depends on the k-location along that boundary, defined by M(0,π) and equivalent points. Hence, along the (0,π)→(π,0) direction, it occurs below EF between the M points and the hot spots, which are defined as the k-locations where the intersection occurs exactly at EF. On the other hand, between the hot spots, the intersection occurs always above EF and the upper hybridized band can never cross EF and therefore cannot be observed by low temperature ARPES, whereas the top of the lower hybridized band is pushed down. When the AF gap is large enough, the small holelike FS pocket around (π/2,π/2) is gapped out. In order to check this scenario, we investigated the band dispersion of reduced and AG samples along the nodal (Γ-X) direction. The results are given in Fig. 3. Figs. 3a and b show the EDCs of the reduced and AG samples, respectively. Besides the clear leading edge shift FIG. 2: (Color online) Comparison of the reduced (a) and AG (b) spectra obtained along the lines given in c. MB (dashed arrows) and AIF (solid arrows) indicate the main and AIF band structures, respectively. The number next to MB or AFI indicates the zone in which the band is detected. d) CEIP at -50 meV (30 meV integration) of a Pr1.85Ce0.15CuO4 AG sample obtained at 22 eV. The suppression along the vertical Γ-M direction is due to ARPES selection rules. observed for the AG sample as compared to the reduced one, the EDC maxima of the AG sample exhibit a bend- ing back characteristic of hybridization: from the top to the bottom of Fig. 3b, the EDC maxima first move closer to EF, and then move away, with a decrease of intensity. A contrast in the shape of the momentum distribution curves (MDCs) of the reduced and AG samples, which are given respectively in Figs. 3c and d, is also observed. The asymmetric shape of the AGMDCs suggests the presence of a band folded along the AF boundary (vertical dashed line). This effect is clearly seen on the corresponding second momentum-derivative intensity (SMDI) plots dis- played in Figs. 3e and f for the reduced and AG samples, respectively. The position of the MDC peaks corresponds to the minimum in the SMDI plots (bright spots). In con- trast to the situation shown in Fig. 3e, Fig. 3f exhibits an additional band whose dispersion is the reflection of the main band with respect to the AF boundary. Using the position of the MDCs, we extracted the main band dispersion and reported it on Fig. 3f, along with its re- flection across the AF boundary. We also reported on Fig. 3f the position of the EDC maxima associated with the AG sample. These maxima, which coincide with the renormalized dispersion band, support the hybridization scenario and indicate that the portion of the FS around (π/2,π/2) is suppressed in the AG samples. Fig. 4a, which compares the k-dependence of the lead- ing edge shift for the AG and reduced samples, provides additional evidence that an AF hybridization gap is sup- pressed after the reduction process. While no clear lead- ing edge shift is observable for the reduced sample, an anisotropic LEG is observed for the AG sample. Hence, FIG. 3: (Color online) Comparison of the nodal dispersion (second zone) between reduced (top panels) and AG samples (bottom panels). a-b) EDCs from -0.87π/a (bottom) to - 0.62π/a (top). c-d) MDCs between - 80 meV (bottom) and 0 meV (top). e-f) SMDI plots. The vertical dashed line corre- sponds to the AF boundary, while the solid lines and the dots correspond to the unhybridized main and AF bands, and to the hybridized band, respectively. a maximum is observed around the hot spot, as expected from the hybridization scenario [15]. In order to illus- trate further the AF scenario, we plotted in Figs. 4b-e simulations of the nodal dispersion obtained using the fit parameters given above, with various AF gap sizes. We introduced a broadening to mimic realistic results and re- moved the Fermi function for sake of clarity. For a large gap, the band folding is clearly observable and the lower hybridized band never crosses EF. As a consequence, a large leading edge shift is recorded. This leading edge shift decreases as the gap becomes smaller, and it dis- appears when the lower band crosses EF, as illustrated in 4d. Our simulations indicate that a LEG of 20 meV along the nodal direction can be produced by a 60 meV AF LEG at the hot spot with a proper linewidth, in agreement with our observation in Fig. 4a. While the experimental results for the nodal region can easily be described by an AF hybridization gap, such a model alone fails to explain the LEG observed for the antinodal region, where the main and AIF bands inter- sect below EF. In particular, the original band position at M is ∼ 300 meV below EF, thus a small LEG (∼ 20 meV) cannot be produced by a simple AF hybridization effect. Nevertheless, AF order may still be responsible for the antinodal LEG. It has been predicted that the whole FS of the Nd1.85Ce0.15CuO4 superconducting sam- ples, including the antinodal region, is pseudogapped due to paramagnons in the semiclassical regime [16]. Even though this is in apparent contradiction with our exper- imental data, which indicate no antinodal LEG for the reduced samples, this idea may be valid for the AG sam- ples, for which the AF correlations are much stronger. FIG. 4: (Color online) a) k-dependece of the leading edge shift. Lines are guides for the eye. b-e) Simulations of the band dispersion in the presence of an AF gap. We used the TB parameters defined in the text and introduced a band broadening in order to mimic real data. We now turn to a critical question: how can a small amount of extra oxygen atoms induce AF order in the AG samples? Literature provides two opposite scenarios involving CuO2 plane defects and the competition be- tween the AF and superconducting states. It has been suggested that in-plane oxygen vacancies in the reduced samples can suppress the AF order, affect the band pa- rameters and induce superconductivity [6, 7]. However, the present study indicates that the band parameters are not modified significantly by the reduction process. Moreover, the antinodal LEG in this scenario would be more likely an indirect consequence of magnetic fluctu- ations such as paramagnons [16]. The lack of theoreti- cal study on the subject leaves open the possibility that charge disorder induced by oxygen vacancies can suppress the AF order and promote superconductivity. An opposite scenario is based on a recent neutron study suggesting a deficiency of Cu in AG Pr1−xLaCexCuO4 samples that is healed after the reduction process through the formation of a (RE,Ce)2O3 impurity phase [11]. It has been argued that, in hole-doped cuprates, a Cu va- cancy, like a nonmagnetic Zn impurity, would suppress local superconducting phase coherence and at the same time induce a staggered paramagnetic S=1/2 local mo- ment extending over a few unit cells [17]. Such impurity or Cu vacancy induced local magnetism has been recently observed by in-plane 17O NMR in the superconducting state of the optimally hole-doped YBa2Cu3O7 with dilute Zn impurities [18]. If the amount of Cu vacancies in the AG samples, although small, is sufficient, it is possible to establish AF long-range order by quantum percolation of the AF regions. In addition, the strong scattering of the Cu vacancies in CuO2 planes of AG samples may produce a localization gap (or Coulomb gap) that could explain the observed antinodal LEG. We note that the residual resistivity (∼ 500 µΩcm) at the superconductor-insulator transition, found in re-oxygenated Pr1.83Ce0.17CuO4 thin films [12], corresponds to the two-dimensional (2D) re- sistance ρ2D0 ≈ 8.3 kΩ/� per CuO2, close to the uni- versal 2D value h/4e2 ≃ 6.5kΩ/� [19, 20, 21]. Simi- lar results were also found in the Zn-substituted hole- doped cuprates [22]. Moreover, the 2%-4% of Zn sub- stitution needed to suppress completely superconductiv- ity in La2−xSrxCu1−zZnzO4 [22] is similar to the value of 1.2% to 2.3% Cu vacancies estimated in the AG and non-superconducting Pr0.88LaCe0.12CuO4 samples [11]. We caution that there are other possible explanations to account for the antinodal LEG, such as a charge den- sity wave induced by the nesting of two sides of the M- centered electron pockets. However, it would then be hard to explain why both the long-range AF and this charge density wave are suppressed in the reduced sam- ples. The unexpected presence of the antinodal LEG calls for further theoretical and experimental studies. We are indebted to A.-M. S. Tremblay and D. Sénéchal for useful discussions. We acknowledge support from NSF DMR-0353108, DOE DEFG02-99ER45747 and DE- FG02-05ER46202. This work is based upon research con- ducted at the Synchrotron Radiation Center supported by NSF DMR-0537588. P.F. acknowledges the support of NSERC (Canada), FQRNT (Québec), CFI and CIAR. [1] E. Moran et al., Physica C 160, 30 (1989). [2] J. S. Kim and D. R. Gaskell, Physica C 209, 381 (1993). [3] E. Wang et al., Phys. Rev. B 41, 6582 (1990). [4] E. Takayama-Muromachi et al., Physica C 159, 634 (1989). [5] K. Susuki et al., Physica C 166, 357 (1990). [6] G. Riou et al., Phys. Rev. B 69, 024511 (2004). [7] P. Richard et al., Phys. Rev. B 70, 064513 (2004). [8] K. Kurahashi et al., J. Phys. Soc. Jpn 71, 910 (2002). [9] M. Matsuura et al., Phys. Rev. B 68, 144503 (2003). [10] Pengcheng Dai et al., Phys. Rev. B 71, 100502 (2005). [11] H. J. Kang et al., Nature Materials 6, 224 (2007). [12] J. Gauthier et al. , Phys. Rev. B 75, 024424 (2007). [13] P. Richard et al., Phys. Rev. B 72, 184514 (2005). [14] M. Brinkmann et al., Physica C 269, 76 (1996). [15] H. Matsui et al., Phys. Rev. Lett. 94, 047005 (2005). [16] D. K. Sunko and S. Barǐsić, Phys. Rev. B 75, 060506 (2007). [17] Z. Wang and P. A. Lee, Phys. Rev. Lett. 89, 217002 (2002). [18] S. Ouazi et al., Phys. Rev. Lett. 96, 127005 (2006). [19] T. Pang, Phys. Rev. Lett. 62, 2176 (1989). [20] M. P. A. Fisher et al., Phys. Rev. Lett. 64, 587 (1990). [21] V. J. Emery and S. A. 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0704.0454

An Extrasolar Planet Census with a Space-based Microlensing Survey

Microsoft Word - Bennett_space_microlensing.doc An Extrasolar Planet Census with a Space-based Microlensing Survey D.P. Bennett , J. Anderson , J.-P. Beaulieu , I. Bond , E. Cheng , K. Cook , S. Friedman , B.S. Gaudi , A. Gould , J. Jenkins , R. Kimble , D. Lin , M. Rich , K. Sahu , D. Tenerelli , A. Udalski , and P. Yock University of Notre Dame, Notre Dame, IN, USA Rice University, Houston, TX, USA Institut d’Astrophysique, Paris, France Massey University, Auckland, New Zealand Conceptual Analytics, LLC, Glen Dale, MD, USA Lawrence Livermore National Laboratory, USA Space Telescope Science Institute, Baltimore, MD, USA Ohio State University, Columbus, OH, USA SETI Institute, Mountain View, CA, USA NASA/Goddard Space Flight Center, Greenbelt, MD, USA University of California, Santa Cruz, CA, USA University of California, Los Angeles, CA, USA Lockheed Martin Space Systems Co., Sunnyvale, CA, USA Warsaw University, Warsaw, Poland University of Auckland, Auckland, New Zealand ABSTRACT A space-based gravitational microlensing exoplanet survey will provide a statistical census of exoplanets with masses 0.1M and orbital separations ranging from 0.5AU to . This includes analogs to all the Solar System’s planets except for Mercury, as well as most types of planets predicted by planet formation theories. Such a survey will provide results on the frequency of planets around all types of stars except those with short lifetimes. Close-in planets with separations < 0.5 AU are invisible to a space-based microlensing survey, but these can be found by Kepler. Other methods, including ground-based microlensing, cannot approach the comprehensive statistics on the mass and semi-major axis distribution of extrasolar planets that a space-based microlensing survey will provide. The terrestrial planet sensitivity of a ground-based microlensing survey is limited to the vicinity of the Einstein radius at 2-3 AU, and space-based imaging is needed to identify and determine the mass of the planetary host stars for the vast majority of planets discovered by microlensing. Thus, a space-based microlensing survey is likely to be the only way to gain a comprehensive understanding of the nature of planetary systems, which is needed to understand planet formation and habitability. The proposed Microlensing Planet Finder (MPF) mission is an example of a space-based microlensing survey that can accomplish these objectives with proven technology and a cost that fits comfortably under the NASA Discovery Program cost cap. 1. Basics of the Gravitational Microlensing Method The physical basis of microlensing is the gravitational attraction of light rays by a star or planet. As illustrated in Fig. 1, if a “lens star” passes close to the line of sight to a more distant source star, the gravitational field of the lens star will deflect the light rays from the source star. The gravitational bending effect of the lens star “splits”, distorts, and magnifies the images of the source star. For Galactic microlensing, the image separation is 4 mas, so the observer sees a microlensing event as a transient brightening of the source as the lens star’s proper motion moves it across the line of sight. Gravitational microlensing events are characterized by the Einstein ring radius, = 2.0 AU (1 kpc) where ML is the lens star mass, and DL and DS are the distances to the lens and source, respectively. This is the radius of the ring image that is seen with perfect alignment between the lens and source stars. The lensing magnification is determined by the alignment of the lens and source stars measured in units of RE, so even low-mass lenses can give rise to high magnification microlensing events. The duration of a microlensing event is given by the Einstein ring crossing time, Fig. 1: The geometry of a microlensing planet search towards the Galactic bulge. Main sequence stars in the bulge are monitored for magnification due to gravitational lensing by foreground stars and planets in the Galactic disk and bulge. Fig. 2: MPF is sensitive to planets above the purple curve in the mass vs. semi-major axis plane. The gold, green and cyan regions indicates the sensitivities of radial velocity surveys, SIM and Kepler, respectively. The location of our Solar System’s planets and many extrasolar planets are indicated, with ground- based microlensing discoveries in red. which is typically 1-3 months for stellar lenses and a few days or less for a planet. Planets are detected via light curve deviations that differ from the normal stellar lens light curves (Mao & Paczynski 1991). Usually, the signal occurs when one of the two images due to lensing by the host star passes close to the location of the planet, as indicated in Fig. 1 (Gould & Loeb 1992), but planets can also be detected at very high magnification where the gravitational field of the planet destroys the symmetry of the Einstein ring (Griest & Safizadeh 1998). 2. Capabilities of the Microlensing Method Planets down to one tenth of an Earth mass can be detected. The probability of a detectable planetary signal and its duration both scale as RE ~M p , but given the optimum alignment, planetary signals from low-mass planets can be quite strong. The limiting mass for the microlensing method occurs when the planetary Einstein radius becomes smaller than the projected radius of the source star (Bennett & Rhie 1996). The ~5.5 M planet detected by Beaulieu et al. (2006) is near this limit for a giant source star, but most microlensing events have G or K-dwarf source stars with radii that are at least 10 times smaller than this. So, the sensitivity of the microlensing method extends down to < 0.1M , as the results of a detailed simulation of the MPF mission (Bennett & Rhie 2002) show in Fig. 2. Microlensing is sensitive to a wide range of planet-star separations and host star types. The host stars for planets detected by microlensing are a random sample of stars that happen to pass close to the line-of-sight to the source stars in the Galactic bulge, so all common types of stars are surveyed, including G, K, and M- dwarfs, as well as white dwarfs and brown dwarfs. Microlensing is most sensitive to planets at a separation of ~RE (usually 2-3 AU) due to the strong stellar lens magnification at this separation, but the sensitivity extends to arbitrarily large separations. It is only planets well inside RE that are missed because the stellar lens images that would be distorted by these inner planets have very low magnifications and a very small contribution to the total brightness. These features can be seen in Fig. 2, which compares the sensitivity of the MPF mission with expectations for other planned and current programs. Other ongoing and planned programs can detect, at most, analogs of two of the Solar System’s planets, while a space-based microlensing survey can detect seven—all but Mercury. The only method with comparable sensitivity to MPF is the Kepler space-based transit survey, which complements the microlensing method with sensitivity at semi-major axes, a 1. The sensitivities of MPF and Kepler overlap at separations of ~1 AU, which corresponds to the habitable zone for G and K stars. The red crosses in Fig. 2 indicate the two gas giant (Bond et al. 2004; Udalski et al. 2005) and two ~10M “super-earth” planets in orbits of ~3 AU discovered by ground-based microlensing (Beaulieu et al. 2006; Gould et al. 2006). A preliminary analysis suggests that about one third of all stars are likely to have a super-earth at 1.5-4AU whereas radial velocity surveys find that only about 3% of stars have gas giants in this region (Butler et al. 2006). Microlensing light curves yield unambiguous planet parameters. For the great majority of events, the basic planet parameters (planet:star mass ratio, planet-star separation) can be “read off” the planetary deviation (Gould & Loeb 1992; Bennett & Rhie 1996; Wambsganss 1997). Possible ambiguities in the interpretation of planetary microlensing events have been studied in detail (Gaudi & Gould 1997; Gaudi 1998), and these can be resolved with good quality, continuous light curves that will be routinely acquired with a space-based microlensing survey. A space-based survey will also detect most of the planetary host stars, which generally allows the host star mass, approximate spectral type, and the planetary mass and separation to be determined (Bennett et al. 2007). The distance to the planetary system is determined when the host star is identified, so a space-based microlensing survey will also measure how the properties of exoplanet systems change as a function of distance from the Galactic Center. There is usually some redundancy in the measurements that determine the properties of the host stars, and so the determination is robust to complicating factors, such as a binary companion to the background source star. Detailed simulations indicate a large number of planet detections. Bennett & Rhie (2002) and Gaudi (unpublished) have independently simulated space-based microlensing surveys. These simulations included variations in the assumed mission capabilities that allow us to explore how changes in the mission design will affect the scientific output, and they form the basis of our predictions in Figs. 2-4. In order to predict the number of planets that will be detected by a space-based microlensing survey, we must make assumptions regarding the frequency of exoplanets. Figs. 3 and 4 show the expected number of planets that MPF would detect at orbital separations 1-2.5 AU and (i.e. free-floating planets) assuming one such planet per star. The range 1-2.5 AU is presented because this is just outside the range of Kepler and inside the region of highest sensitivity for ground-based microlensing surveys. It also corresponds to the outer part of the habitable zone for G and K stars and contains the “snow-line” for part of the history of lower mass stars (Kennedy et al. 2006). Free-floating planets are expected to be a common by- Fig. 3: The expected number of MPF planet discoveries as a function of the planet mass if every star has a single planet at a separation of 1.0-2.5 AU. Fig. 4: The expected number of MPF free- floating planet discoveries. product of most planet formation scenarios (Levison et al. 1998; Goldreich et al. 2004), and only a space- based microlensing survey can detect free-floating planets of 1M . 3. A Space-based Microlensing Survey Is Needed Microlensing relies upon the high density of source and lens stars towards the Galactic bulge to generate the stellar alignments that are needed to generate microlensing events, but this high star density also means that the bulge main sequence source stars are not generally resolved in ground- based images, as Fig. 5 demonstrates. This means that the precise photometry needed to detect planets of 1M is not possible from the ground unless the magnification due to the stellar lens is moderately high. This, in turn, implies that ground-based microlensing is only sensitive to terrestrial planets located close to the Einstein ring (at ~2-3 AU). The full sensitivity to terrestrial planets in all orbits from 0.5 AU to comes only from a space-based survey. Planetary host star detection from space yields precise star and planet parameters. For all but a small fraction of planetary microlensing events, space-based imaging is needed to detect the planetary host stars, and the detection of the host stars allows the star and planet masses and separation in physical units to be determined. This can be accomplished with HST observations for a small number of planetary microlensing events (Bennett et al. 2006), but space-based survey data will be needed for the detection of host stars for hundreds or thousands of planetary microlensing events. Fig. 6 shows the distribution of planetary host star masses and the predicted uncertainties in the masses and separation of the planets and their host stars (Bennett et al. 2007) from simulations of the MPF mission. The host stars with masses determined to better than 20% are indicated by the red histogram in Fig. 6(a), and these are primarily the host stars that can be detected in MPF images. Ground-based microlensing surveys also suffer significant losses in data coverage and quality due to poor weather and seeing. As a result, a significant fraction of the planetary deviations seen Fig. 6: (a) The simulated distribution of stellar masses for stars with detected terrestrial planets. The red histogram indicates the subset of this distribution for which the masses can be determined to better than 20%. (b) The distribution of uncertainties in the projected star-planet separation. (c) The distribution of uncertainties in the star and planet masses. Fig. 5: A comparison between an image of the same star field in the Galactic bulge from CTIO in 1” seeing and a simulated MPF frame (based on an HST image). The indicated star is a microlensed main sequence source star. (a) (b) (c) in a ground-based microlensing survey will have poorly constrained planet parameters due to poor light curve coverage (Peale 2003). 4. A Space-Based Microlensing Survey Constrains Planet Formation Theories Rapid advancement in exoplanet research is driven by both extensive observational searches around mature stars as well as the construction of planet formation and evolution models. Perhaps the most surprising discovery so far is the great diversity in the planets' dynamical properties, but these results are largely confined to planets that are unusually massive or reside in very close orbits. The core accretion theory suggests most planets are much less massive than gas giants and that the critical region for understanding planet formation is the “snow-line”, located in the region (1.5-4 AU) of greatest microlensing sensitivity (Ida & Lin 2005; Kennedy et al. 2006). Early results from ground-based microlensing searches (Beaulieu et al. 2006; Gould et al. 2006) appear to confirm these expectations. A space-based microlensing survey would extend the current sensitivity of the microlensing method down to masses of ~0.1M over a large range (0.5AU- ) in separation, and in combination with Kepler, such a mission provides sensitivity to sub-Earth mass planets at all separations. The semi-major axis region probed by space- microlensing provides a cleaner test of planet formation theories than the close-in planets detected by other methods, because planets discovered at > 0.5 AU are more likely to have formed in situ than the close-in planets. The sensitivity region for space-microlensing includes the outer habitable zone for G and K stars through the “snow-line” and beyond, and the lower sensitivity limit reaches the regime of planetary embryos at ~0.1M . It may be that such planets are much more common than planets of 1M because their type-1 migration time is much longer. Space-microlensing tests core accretion. The space-microlensing census of low-mass planets should also provide direct evidence of features of the proto-planetary disk predicted by the core accretion theory. There are several physical processes that control the development of planetary embryos and planets in the proto-planetary disk. In the inner disk, the size of planetary embryos is controlled by the isolation mass, and the isolation mass is expected to jump by an order of magnitude across the “snow-line” because of the increased surface density of solids in the disk. But the number of gas giant and super-earth planets is also expected to increase beyond the snow line, while the planetary growth time increases. This means that it is more likely for the growth of outer planets to be terminated via gravitational scattering of planetesimals or the proto- planets themselves. Scattering would also result in the removal of lower mass planets into very distant orbits or even out of the gravitational influence of the host stars altogether, but space- based microlensing can still detect planets in these locations. The frequency of planets of different masses and separations that a space-based microlensing survey provides will yield a unique insight into the planetary formation process and will allow us to determine the importance of these processes. The habitability of a planet depends on its formation history. The suitability of a planet for life depends on a number of factors, such as the average surface temperature, which determines if the planet resides in the habitable zone. However, there are many other factors that also may be important, such as the presence of sufficient water and other volatile compounds necessary for life (Raymond et al. 2004; Lissauer 2007). Thus, a reasonable understanding of planet formation is an important foundation for the search for nearby habitable planets and life. 5. Overview of the Microlensing Planet Finder Mission Key requirements of the MPF mission are summarized in table 1. MPF continuously observes four 0.65 sq. deg. fields in the central Galactic Bulge using an inclined geostationary orbit to provide a continuous view of the Galactic Bulge fields and a continuous downlink. MPF will use a dedicated ground station co-located with other NASA facilities at White Sands, NM. Spacecraft commanding and on-board processing are minimized because of the simple observation plan and orbit design. MPF system. MPF uses a 1.1m Three-Mirror Anastigmat (TMA) telescope feeding a 145 Mpixel HgCdTe focal plane residing on a standard spacecraft bus as shown in Fig. 7. The MPF design leverages existing hardware and design concepts, many of which are already demonstrated on-orbit and/or flight qualified. The spacecraft bus is a near-identical copy of that used for Spitzer and has demonstrated performance that meets MPF requirements. The telescope system leverages Ikonos and NextView commercial Earth-observing telescope designs that provide extensive diffraction-limited images. The focal plane design taps proven technologies developed for JWST. All elements are at TRL 6 or better. The focal plane design uses common non-destructive readout CMOS multiplexers for two detector technologies that cover the visible and the near-IR. The MPF focal plane can track up to 35 guide stars in the field, providing a built-in fine guidance capability. The focal plane gains additional advantages from using the Teledyne SIDECAR™ application specific integrated circuit (ASIC) that condenses all the control and readout electronics into a “system-on-a-chip” implementation. This approach dramatically simplifies the support electronics while minimizing wire-count challenges. Cost and Schedule. The total cost for the MPF mission is (FY06) $390M including 30% contingency during development. The team that developed the costs included NASA GSFC, Lockheed Martin, ITT, STScI, Teledyne and University of Notre Dame. 6. Discussion and Summary A space-based microlensing survey provides a census of extrasolar planets that is complete (in a statistical sense) down to 0.1M at orbital separations 0.5 AU, and when combined with the results of the Kepler mission a space-based microlensing survey will give a comprehensive Property Value Units Launch Vehicle 7920-9.5 Delta II Orbit Inclined GEO 28.7 degrees Mission Lifetime 4.0 years Telescope Aperture 1.1 meters (diam.) Field of View 0.95x0.68 degrees Spatial Resolution 0.240 arcsec/pixel Pointing Stability 0.048 arcsec Focal Plane Format 145 Megapixels Spectral Range 600 – 1700 nm in 3 bands Quantum Efficiency > 75% > 55% 900-1400 nm 700-1600 nm Dark Current < 1 e-/pixel/sec Readout Noise < 30 e-/read Photometric Accuracy 1 or better % at J=20.5. Data Rate 50.1 Mbits/sec Table 1: Key MPF Mission Requirements Fig. 7: MPF On-Orbit Configuration picture of all types of extrasolar planets with masses down to well below an Earth mass. This complete coverage of planets at all separations can be used to calibrate the poorly understood theory of planetary migration. This fundamental exoplanet census data is needed to gain a comprehensive understanding of processes of planet formation and migration, and this understanding of planet formation is an important ingredient for the understanding of the requirements for habitable planets and the development of life on extrasolar planets. A subset of the science goals can be accomplished with an enhanced ground-based microlensing program (Gould et al. 2007), which would be sensitive to Earth-mass planets in the vicinity of the “snow-line”. But such a survey would have its sensitivity to Earth-like planets limited to a narrow range of semi-major axes, so it would not provide the complete picture of the frequency of exoplanets down to 0.1M that a space-based microlensing survey would provide. Furthermore, a ground-based survey would not be able to detect the planetary host stars for most of the events, and so it will not provide the systematic data on the variation of exoplanet properties as a function of host star type that a space-based survey will provide. The basic requirements for a space-based microlensing survey are a 1-m class wide field-of- view space telescope that can image the central Galactic bulge in the near-IR or optical almost continuously for periods of at least several months at a time. This can be accomplished as a NASA Discovery mission, as the example of the MPF mission shows, but there are a number of other proposed missions with similar requirements, such as a number of JDEM concept missions or a stare-mode astrometry mission (Johnston et al. 2007) that makes use of the same type of detectors as the MPF mission. Such an astrometry mission could complement the statistical planetary results from microlensing survey with data on nearby planets. Thus, a space-based microlensing survey could be accomplished with a standalone Discovery class mission or a joint mission with another project. As Fig. 2 shows, there is no other planned mission that can duplicate the science return of a space-based microlensing survey, and our knowledge of exoplanets and their formation will remain incomplete until such a mission is flown. References Basri, G., Borucki, W.J., & Koch, D. 2005, New Ast. Rev., 49, 478 Beaulieu, J.-P. et al. 2006, Nature, 439, 437 Bennett, D.P. et al. 2006, ApJL, 647, L171 Bennett, D.P., Anderson, J., & Gaudi, B.S. 2007, ApJ, in press (astro-ph/0611448) Bennett, D.P., & Rhie, S.H. 1996, ApJ, 472, Bennett, D.P., & Rhie, S.H. 2002, ApJ, 574, Bond, I. 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0704.0455

USco1606-1935: An Unusually Wide Low-Mass Triple System?

Draft version October 29, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 USCO1606-1935: AN UNUSUALLY WIDE LOW-MASS TRIPLE SYSTEM? Adam L. Kraus ([email protected]), Lynne A. Hillenbrand ([email protected]) California Institute of Technology, Department of Astrophysics, MC 105-24, Pasadena, CA 91125 Draft version October 29, 2018 ABSTRACT We present photometric, astrometric, and spectroscopic observations of USco160611.9-193532 AB, a candidate ultrawide (∼1600 AU), low-mass (Mtot ∼0.4 M⊙) multiple system in the nearby OB association Upper Scorpius. We conclude that both components are young, comoving members of the association; we also present high-resolution observations which show that the primary is itself a close binary system. If the Aab and B components are gravitationally bound, the system would fall into the small class of young multiple systems which have unusually wide separations as compared to field systems of similar mass. However, we demonstrate that physical association can not be assumed purely on probabilistic grounds for any individual candidate system in this separation range. Analysis of the association’s two-point correlation function shows that there is a significant probability (25%) that at least one pair of low-mass association members will be separated in projection by .15′′, so analysis of the wide binary population in Upper Sco will require a systematic search for all wide systems; the detection of another such pair would represent an excess at the 98% confidence level. Subject headings: stars:binaries:general; stars:low-mass,brown dwarfs;stars:pre-main se- quence;stars:individual([PBB2002] USco160611.9-193532) 1. INTRODUCTION The frequency and properties of multiple star systems are important diagnostics for placing constraints on star formation processes. This has prompted numerous at- tempts to characterize the properties of nearby binary systems in the field. These surveys (e.g. Duquennoy & Mayor 1991; Fischer & Marcy 1992; Close et al. 2003; Bouy et al. 2003; Burgasser et al. 2003) have found that binary frequencies and properties are very strongly dependent on mass. Solar-mass stars have high binary frequencies (&60%) and maximum separations of up to ∼104 AU. By contrast, M dwarfs have moderately high binary frequencies (30-40%) and few binary companions with separations of more than ∼500 AU, while brown dwarfs have low binary frequencies (∼15%) and few com- panions with separations >20 AU. The mass-dependent decline in the maximum observed binary separation has been described by Reid et al. (2001) and Burgasser et al. (2003) with an empirical function which is exponential at high masses (amax ∝ 103.3Mtot) and quadratic at low masses (amax ∝M tot). The mechanism that produces the mass dependence is currently unknown; N-body simulations show that the empirical limit is not a result of dynamical evolution in the field (e.g. Burgasser et al. 2003; Weinberg et al. 1987) since the rate of disruptive stellar encounters is far too low. This suggests that the limit must be set early in stellar lifetimes, either as a result of the binary formation process or during early dynamical evolution in relatively crowded natal environments. Surveys of nearby young stellar associations have identified several unusually wide systems (Chauvin et al. 2004; Caballero et al. 2006; ; Jayawardhana & Ivanov 2006; Luhman et al. 2006, 2007; Close et al. 2007), but not in sufficient numbers to study their properties in a statistically meaningful manner. We have addressed this problem by using archival 2MASS data to systematically search for candidate wide binary systems among all of the known members of three nearby young associations (Upper Sco, Taurus-Auriga, and Chamaeleon-I; Kraus & Hillenbrand 2007). Our re- sults broadly agree with the standard paradigm; there is a significant deficit of wide systems among very low- mass stars and brown dwarfs as compared to their more massive brethren. However, we did identify a small num- ber of candidate wide systems. One of these candidates is [PBB2002] USco160611.9-193532 (hereafter USco1606- 1935), a wide (10.87′′; 1600 AU) pair of stars with similar fluxes and colors. The brighter member of the pair was spectroscopically confirmed by Preibisch et al. (2002) to be a young M5 star. The fainter member fell just below the flux limit of their survey. In this paper, we describe our photometric, as- trometric, and spectroscopic followup observations for USco1606-1935 and evaluate the probability that the sys- tem is an unusually wide, low-mass binary. In Section 2, we describe our observations and data analysis methods. In Section 3, we use these results to establish that both members of the pair are young and co-moving, and that the primary is itself a close binary. Finally, in Section 4 we address the possibility that the pair is not bound, but a chance alignment of young stars, by analyzing the clustering of pre-main-sequence stars in Upper Sco. 2. OBSERVATIONS AND DATA ANALYSIS Most binary surveys, including our discovery survey, identify companions based on their proximity to the pri- mary star and argue for physical association based on the (usually very low) probability that an unbound star would have been observed in chance alignment. How- ever, the probability of contamination is much higher for very wide systems like USco1606-1935, so we de- cided to pursue additional information in order to con- firm its multiplicity and further characterize its system components. In this section, we describe our followup efforts: a search of publicly available databases to ob- http://arxiv.org/abs/0704.0455v1 tain additional photometry and astrometry, acquisition of intermediate-resolution spectra to measure the sec- ondary spectral type and test for signatures of youth, and acquisition of high-resolution images to determine if either component is itself a tighter binary and to test for common proper motion. 2.1. Archival Data We identified USco1606-1935 AB as a candidate bi- nary system using archival data from 2MASS (Skrut- skie et al. 2006). The binary components are bright and clearly resolved, so we were able to retrieve ad- ditional photometry and astrometry from several other wide-field imaging surveys. We collated results for the binary components themselves and for nearby field stars from 2MASS, the Deep Near Infrared Survey (DENIS; Epchtein et al. 1999), United States Naval Observatory B1.0 survey (USNO-B; Monet et al. 2003), and the Su- perCOSMOS Sky Survey (SSS; Hambly et al. 2001). The DENIS and 2MASS source catalogues are based on wide-field imaging surveys conducted in the optical/NIR (IJK and JHK, respectively) using infrared array de- tectors, while the USNO-B and SSS source catalogues are based on independent digitizations of photographic plates from the First Palomar Observatory Sky Survey and the ESO Southern-Sky Survey. 2.1.1. Photometry After evaluating the data, we decided to base our anal- ysis on the JHK magnitudes measured by 2MASS and the photographic I magnitude of USNO-B (hereafter de- noted I2, following the nomenclature of the USNO-B catalog, to distinguish it from Cousins IC). We chose these observations because their accuracy can be directly tested using the independent IJK magnitudes measured by DENIS; this comparison shows that the fluxes are con- sistent within the uncertainties. We do not directly use the DENIS observations because they are not as deep as the other surveys. We adopted the photometric uncer- tainties suggested in each survey’s technical reference. 2.1.2. Astrometry As we describe in Section 3.3, there appear to be large systematic differences in the astrometry reported by the USNO-B and SSS source catalogs. These surveys rep- resent digitizations of the same photographic plates, so these systematic discrepancies suggest that at least one survey introduces systematic biases in the digitization and calibration process. Given the uncertainty in which measurements to trust, we have chosen to disregard all available photographic astrometry and only use results from 2MASS and DENIS. Our discovery survey already measured 2MASS rel- ative astrometry for each filter directly from the pro- cessed atlas images, so we have adopted those values. We extracted DENIS astrometry from the source catalog, which contains the average positions for all three filters. Both surveys quote astrometric uncertainties of 70-100 mas for stars in the brightness range of our targets, but that value includes a significant systematic term result- ing from the transformation to an all-sky reference frame. We have conducted tests with standard binary systems of known separation which suggest that relative astrom- etry on angular scales of <1′ is accurate to ∼40 mas, so we adopt this value as the astrometric uncertainty for each survey. 2.2. Optical Spectroscopy We obtained an intermediate-resolution spectrum of USco1606-1935 B with the Double Spectrograph (Oke & Gunn 1982) on the Hale 5m telescope at Palomar Obser- vatory. The spectrum presented here was obtained with the red channel using a 316 l/mm grating and a 2.0′′ slit, yielding a spectral resolution of R ∼1250 over a wave- length range of 6400-8800 angstroms. Wavelength cali- bration was achieved by observing a standard lamp after the science target, and flux normalization was achieved by observation of the spectrophotometric standard star Feige 34 (Massey et al. 1988). The spectrum was pro- cessed using standard IRAF1 tasks. Our field and young spectral type standards were drawn from membership surveys of Upper Sco and Tau- rus by Slesnick et al. (2006a, 2006b) which used identical instrument settings for the spectroscopic confirmation of photometrically selected candidate members. 2.3. High-Resolution Imaging We observed USco1606-1935 A and B on February 7, 2006 (JD=2453773) using laser guide star adaptive op- tics (LGSAO; Wizinowich et al. 2006) on the Keck-II telescope with NIRC2 (K. Matthews, in prep), a high spatial resolution near-infrared camera. The seeing was average to poor (&1′′) for most of the observing run, but the system delivered nearly diffraction-limited correction in K ′ (60 mas FWHM) during the period of these ob- servations. The system performance was above average given the low elevation (34 degrees; 1.8 airmasses), most likely due to the proximity and brightness of the tip-tilt reference star (R = 14.2, d = 14′′). Images were obtained using the K ′ filter in both the narrow and wide camera modes. The pixel scales in these modes are 9.942 mas pix−1 (FOV=10.18′′) and 39.686 mas pix−1 (FOV=40.64′′). All wide-camera ob- servations were centered on the close Aab binary. The A and B components were too wide to fit reasonably into a single narrow-camera exposure, so we took separate exposure sequences centered on each. We obtained four wide-camera exposures of the AB system, seven narrow- camera exposures of A, and four narrow-camera expo- sures of B; the total integration times for each image set are 80s, 175s, and 100s, respectively. Each set was pro- duced with a 3-point box dither pattern that omitted the bottom-left position due to higher read-noise for the de- tector in that quadrant. Single exposures were also taken at the central position. Our science targets are relatively bright, so all obser- vations were taken in correlated double-sampling mode, for which the array read noise is 38 electrons/read. The read noise is the dominant noise term for identifying faint sources, yielding 10σ detection limits of K ∼ 19.2 for the wide camera observations, K ∼ 18.8 for the narrow-camera observations centered on component A, 1 IRAF is distributed by the National Optical Astronomy Ob- servatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. and K ∼ 18.3 for the narrow-camera observations cen- tered on component B; the detection limits for B are slightly shallower due to the shorter total integration time. The data were flat-fielded and dark- and bias- subtracted using standard IRAF procedures. The images were distortion-corrected using new high-order distortion solutions (P. Cameron, in prep) that deliver a significant performance increase as compared to the solutions pre- sented in the NIRC2 pre-ship manual2; the typical resid- uals are ∼4 mas in wide camera mode and ∼0.6 mas in narrow camera mode. We adopt these systematic limits as the uncertainty in astrometry for bright objects; all faint objects (K ∼16-18) have larger uncertainties (∼10 mas) due to photon statistics. We measured PSF-fitting photometry and astrome- try for our sources using the IRAF package DAOPHOT (Stetson 1987), and specifically with the ALLSTAR rou- tine. We analyzed each frame separately in order to esti- mate the uncertainty in individual measurements and to allow for the potential rejection of frames with inferior AO correction; our final results represent the mean value for all observations in a filter. In the wide-camera observations, we produced a tem- plate PSF based on the B component and the field star F1 (see Section 3.1 and Figure 1), both of which appear to be single sources. In the narrow-camera observations centered on A or B, the science target was the only bright object detected in our observations, so there was not a separate source from which to adopt a template PSF. We could have adopted a template PSF from another set of observations, but the AO correction usually varies sig- nificantly between targets since it is very sensitive to the seeing, elevation, laser return, and tip-tilt separation and brightness. We found that no other target in our survey provided a good PSF match. We addressed this issue for the Aab binary pair by developing a procedure to reconstruct the single-source PSF directly from the observations of the binary system. Our algorithm begins with a preliminary estimate of the single-source PSF, then iteratively fits both components of the binary system with the estimated PSF and uses the synthetic PSF to subtract the best-fit estimate of the secondary flux. This residual image (which is dominated by the primary flux distribution) is then used to fit an improved estimate of the single-source PSF. DAOPHOT characterizes an empirical PSF in terms of an analytical function and a lookup table of residuals, so we first iterated the procedure using a purely ana- lytical function until it converged, then added a lookup table to the estimated PSF and iterated until its con- tents also converged. Observations of single stars sug- gested that the penny2 function (a gaussian core with lorentzian wings) would provide the best analytic fit, so we chose it as our analytic function. Four iterations of the fitting process were required for the analytic func- tion to converge and 3 iterations were required for the lookup table to converge. Our algorithm does not work for the B component because it appears to be single, so we adopted the average synthetic single-source PSF from analysis of the Aab system to perform PSF fitting and verify that it is single. We calibrated our photometry using 2MASS K mag- 2 http://www2.keck.hawaii.edu/realpublic/inst/nirc2/ nitudes for the A and B components and the nearby field star F1 (Section 3). The 2MASS observations were con- ducted using the Ks filter rather than K ′, but the the- oretical isochrones computed by Kim et al. (2005) for the Ks and K ′ systems differ by .0.01 magnitudes for objects in this color range; this is much smaller than other uncertainties in the calibration. Carpenter (2001) found typical zero point shifts of .0.03 magnitudes be- tween 2MASS Ks and several standard K bandpasses, all of which are more distinctly different from Ks than K ′, which also demonstrates that the zero point shift between Ks and K ′ should be negligible. The calibration process could introduce systematic un- certainties if any of the three calibration sources are vari- able, but based on the small deviation in the individual calibration offsets for each source (0.03 mag), variability does not appear to be a significant factor. We tested the calibration using DENIS K magnitudes and found that the two methods agree to within 0.01 mag, albeit with a higher standard deviation (0.12 mag) for DENIS. 3. RESULTS 3.1. Images In Figure 1, we show a NIRC2 wide-camera image of the field surrounding USco1606-1935. The A and B com- ponents are labeled, as are 6 apparent field stars (named F1 through F6) which we use as astrometric comparison stars. We found counterparts for the first three field stars in existing survey catalogues: F1 was detected by all four sky surveys, F2 was detected by DENIS, USNO-B, and SSS, and F3 was detected only by USNO-B and SSS. In Figure 2, we show individual contour plots drawn from NIRC2 narrow-camera images of the A and B components. These high-resolution images show that USco1606-1935 A is itself composed of two sources; we designate these two components Aa and Ab. We do not possess any direct diagnostic information to determine if Aa and Ab are physically associated, but there are only two other bright sources in the field of view. If the source count is representative of the surface density of bright (K < 15) sources along the line of sight, the probability of finding an unbound bright source within <100 mas of the A component is only ∼ 10−5. Thus, we consider Aa and Ab to comprise a physically bound binary system. 3.2. Photometry Photometric data are generally sufficient to reject most nonmember interlopers because association members fol- low a bright, well-defined cluster sequence in color- magnitude diagrams and most field stars will fall be- low or bluer than the association sequence. In Table 2, we summarize the observed and archival photometry for each source in the NIRC2 wide-camera images. In Figure 3, we show three color-magnitude diagrams (K versus J − K, H − K, and I2 − K) for our observed sources and for all spectroscopically-confirmed members of Upper Sco (as summarized in Kraus & Hillenbrand 2007). The colors and magnitudes for USco1606-1935 B are consistent with the known members of Upper Sco, which supports the assertion that it is an association member. B is located marginally above and redward of the mean cluster sequence in the (K,J −K) and (K,H −K) dia- grams; if this result is genuine and not a consequence of Fig. 1.— The field surrounding USco1606-1935. The A and B components are labeled, as are 6 apparent field stars. The separation between the Aa and Ab components is too small to be apparent in this image. Fig. 2.— Contour plots showing our LGSAO observations of USco1606-1935. The first panel shows an original exposure for the Aab pair, the second and third panels show Aa and Ab after subtracting best-fit values for the other component, and the last panel shows an original exposure for B. The contours are drawn at 5% to 95% of the peak pixel values. Fig. 3.— Color-magnitude diagrams showing all spectroscopically-confirmed members of Upper Sco (black crosses), the A and B binary components (red), and the other six objects detected in our LGSAO images (blue). The NIR CMDs (top) demonstrate that F1 lies significantly below the association sequence, and therefore is an unrelated field star. The optical-NIR CMD (bottom) supports this identification and demonstrates that F2 and F3 are also field stars that lie below the association sequence. We measure formal upper limits only for stars F4-F6, but marginal R band detections in the POSS plates suggest that F4 and F6 are also field stars. Typical uncertainties are plotted on the left edge of each plot. TABLE 1 Coordinates and Photometry Name RAa DECa KLGS b K2MASS b Hb Jb I2b A 16 06 11.99 -19 35 33.1 11.04 11.02 11.35 12.01 14.1 Aa - - 11.71 - - - - Ab - - 11.88 - - - - B 16 06 11.44 -19 35 40.5 11.74 11.78 12.32 13.00 14.9 F1 16 06 12.09 -19 35 18.3 11.51 11.50 11.62 12.27 13.5 F2 16 06 12.90 -19 35 36.1 16.32 - - - 17.8 F3 16 06 13.23 -19 35 23.7 16.66 - - - 18.7 F4 16 06 11.75 -19 35 32.0 17.43 - - - - F5 16 06 12.40 -19 35 40.3 17.28 - - - - F6 16 06 12.94 -19 35 44.6 16.97 - - - - Note. — Photometry is drawn from our observations (KLGS), 2MASS (JHK2MASS), and the USNO-B1.0 catalogue (I2). a Coordinates are derived from the 2MASS position for USco1606-1935 A and the rel- ative separations we measure using LGSAO. The absolute uncertainty in the 2MASS position with respect to the International Coordinate Reference System (ICRS) is .0.1′′. b Photometric uncertainties are ∼0.03 mag for LGSAO and 2MASS photometry and ∼0.25 mag for USNO-B1.0 photometry. TABLE 2 Relative Astrometry LGSAO K 2MASS K 2MASS H 2MASS J DENIS IJK (JD=2453773) (JD=2451297) (JD=2451297) (JD=2451297) (JD=2451332) ∆α ∆δ ∆α ∆δ ∆α ∆δ ∆α ∆δ ∆α ∆δ Aa -0.0132 -0.0149 - - - - - - - - Ab +0.0201 +0.0266 - - - - - - - - B -7.825 -7.460 -7.757 -7.455 -7.749 -7.395 -7.834 -7.382 -7.865 -7.448 F1 +1.453 +14.844 +1.401 +14.762 +1.446 +14.732 +1.479 +14.735 +1.418 +14.728 F2 +12.839 -3.017 - - - - - - -a -a F3 +17.571 +9.370 - - - - - - - - F4 -3.438 +1.056 - - - - - - - - F5 +5.805 -7.224 - - - - - - - - F6 +13.385 -11.540 - - - - - - - - Note. — The zero-point for all coordinate offsets is the photocenter of the unresolved Aab system. The relative astrometric uncertainties for 2MASS and DENIS results are ∼40 mas; uncertainties for the LGSAO results are ∼5 mas for bright objects and ∼10 mas for faint objects. a F2 was marginally detected in i by DENIS, but the astrometry is not sufficiently precise to be useful in calculating its proper motion. the photometric uncertainties, it could be a consequence of differential reddening, aK band excess associated with a hot disk, or the presence of an unresolved tight binary companion. However, B does not appear to be as red in DENIS data (J −K = 0.98), which suggests that the 2MASS result may not be genuine. The three sources for which we have colors (F1, F2, and F3) all sit below the Upper Sco member sequence in the (K,I2 − K) color-magnitude diagram. Some USco members also fall marginally blueward of the associa- tion sequence in (K,I2−K); we can find no correlation with location, multiplicity, or other systematic factors, so this feature may be a result of intrinsic variability be- tween the epochs of K and I2. This result suggests that the (K,I2−K) CMD is not sufficient for ruling out the membership of F1. However, F1 also sits at the extreme blueward edge of the association sequence in (K,J −K) and is clearly distinct from the association sequence in (K,H − K). We therefore judge that all three sources are unassociated field star interlopers. We do not possess sufficient information to determine whether these three stars are field dwarfs in the Milky Way disk or background giants in the Milky Way bulge; the unknown nature of these sources could complicate fu- ture efforts to calculate absolute proper motions because comparison to nonmoving background giants is the best way to establish a nonmoving astrometric frame of refer- ence. As we will show in Section 3.3, F1 possesses a small total proper motion (<10 mas yr−1), so it may be a dis- tant background star. Its 2MASS colors (J −H = 0.65, H −K = 0.12) place it on the giant sequence in a color- color diagram, but reddened early-type stars with spec- tral type <M0 can also reproduce these colors. We are unable to measure colors for the stars F4, F5, and F6 because they were detected only in our LGSAO observations. However, visual inspection of the digitized POSS plates via Aladdin (Bonnarel et al. 2000) found possible R band counterparts to F4 and F6 that were not identified by USNO-B. If these detections are genuine and these two sources fall near the USNO-B survey limit (R ∼ 20 − 21), their colors (R − K ∼ 3 − 4 or I2 − K ∼ 2−3) are too blue to be consistent with association membership. 3.3. Astrometry Fig. 4.— Relative separations from the A component to the B component (left) and the field star F1 (right) for our LGSAO data and archival 2MASS/DENIS data. The blue circles denote LGSAO data, the red circles denote 2MASS data for each filter (J , H, and K), and the green circles denote the average DENIS values for all three filters (IJK). The black line shows the expected relative astrometry as a function of time for a stationary object, and the predicted archival astrometry values for the non-moving (background) case are shown on these curves with red asterisks. The results for component B are consistent with common proper motion; the results for F1 are inconsistent with common proper motion and suggest that the total proper motion is small, denoting a probable background star. Fig. 5.— The spectrum of USco1606-1935 B (red) as compared to a set of standard stars drawn from the field and from the young Taurus and Upper Sco associations. The overall continuum shape is best fit by a field standard with spectral type M5; the spec- trum around the Na doublet at 8189 angstroms is better fit by an intermediate-age (5 Myr) M5 than a young (1-2 Myr) or field M5, suggesting that the B component is also intermediate-aged. The standard method for confirming physical associa- tion of candidate binary companions is to test for com- mon proper motion. This test is not as useful for young stars in associations because other (gravitationally un- bound) association members have similar proper motions to within .2-3 mas yr−1. However, proper motion anal- ysis can still be used to eliminate nearby late-type field stars and background giants that coincidentally fall along the association color-magnitude sequence but possess dis- tinct kinematics. In Table 2, we summarize the relative astrometry for the three system components and for the field stars F1-F6 as measured with our LGSAO observations and archival data from 2MASS and DENIS. All offsets are given with respect to the photocenter of the unresolved Aab sys- tem; Aa and Ab have similar fluxes and do not appear to be variable in any of these measurements (Section 2.3), so this zero point should be consistent between different epochs. We evaluated the possibility of including astro- metric data from older photographic surveys like USNO- B and SSS, but rejected this idea after finding that the two surveys reported very large (up to 1′′) differences in the separation of the A-B system from digitization of the same photographic plates. We calculated relative proper motions in each dimension by averaging the four first-epoch values (2MASS and DENIS; Table 2), then comparing the result to our second-epoch observation ob- tained with LGSAO. We did not attempt a least-squares fit because the 2MASS values are coeval and the DENIS results were measured only 35 days after the 2MASS re- sults. In Figure 4, we plot the relative astrometry between A and B and between A and F1 as measured by 2MASS, DENIS, and our LGSAO survey. We also show the ex- pected relative motion curve if B or F1 are nonmoving background stars and A moves with the mean proper mo- tion and parallax of Upper Sco, (µα,µδ)=(-9.3,-20.2) mas yr−1 and π=7 mas (de Zeeuw et al. 1999; Kraus & Hil- lenbrand 2007). The total relative motion of B over the 6.8 year observation interval is (+24±25,-40±25) mas; the corresponding relative proper motion is (+3.5±3.7,- 5.9±3.7) mas yr−1, which is consistent with comovement to within <2σ. This result is inconsistent with the hy- pothesis that B is a nonmoving background star at the 8σ level. The relative motion of F1 is (+17±25,+105±25) mas or (+2.5±3.7,+15.4±3.7)mas yr−1, which is inconsistent with comovement at the 4σ level. The absolute proper motion of F1, assuming A moves with the mean proper motion of Upper Sco, is (-7±4,-5±4) mas yr−1, which is consistent with nonmovement to within <2σ. The impli- cation is that F1 is probably a distant background star, either a giant or a reddened early-type star. 3.4. Spectroscopy The least ambiguous method for identifying young stars is to observe spectroscopic signatures of youth like lithium or various gravity-sensitive features. Spectro- scopic confirmation is not strictly necessary in the case of USco1606-1935 since we confirmed common proper mo- tion for the A-B system, but a spectral type is also useful in constraining the physical properties of the secondary, so we decided to obtain an optical spectrum. In the top panel of Figure 5, we plot our spectrum for B in comparison to three standard field dwarfs with spectral types of M4V-M6V. We qualitatively find that the standard star which produces the best fit is GJ 866 (M5V). The M4V and M6V standards do not adequately fit either the overall continuum shape or the depths of the TiO features at 8000 and 8500 angstroms, so the corresponding uncertainty in the spectral type is .0.5 subclasses. In the bottom panel of Figure 5, we plot a restricted range of the spectrum (8170-8210 angstroms) centered on the Na-8189 absorption doublet. The depth of the doublet is sensitive to surface gravity (e.g. Slesnick et al. 2006a, 2006b); high-gravity dwarfs possess very deep absorption lines, while low-gravity giants show almost no absorption. We also plot standard stars of identi- cal spectral type (M5) spanning a range of ages. The depth of the B component’s Na 8189 doublet appears to be consistent with the depth for a member of USco (5 Myr), deeper than that of a Taurus member (1-2 Myr), and shallower than that of a field star, which confirms that the B component is a pre-main sequence member of Upper Sco. We have quantified our analysis by calculating the spectral indices TiO-7140, TiO-8465, and Na-8189, which measure the depth of key temperature- and gravity-sensitive features (Slesnick et al. 2006a). We find that T iO7140 = 2.28, T iO8465 = 1.23, and Na8189 = 0.92; all three indices are consistent with our assessment that B is a young M5 star which has not yet contracted to the zero-age main sequence. 3.5. Stellar and Binary Properties In Table 3, we list the inferred stellar and binary properties for the Aa-Ab and A-B systems, which we estimate using the methods described in Kraus & Hil- lenbrand (2007). This procedure calculates component masses by combining the 5 Myr isochrone of Baraffe et al. (1998) and the M dwarf temperature scale of Luhman TABLE 3 Binary Properties Property Aa-Ab A-B Measured Sep (mas) 53.2±1.0 10874±5 PA (deg) 38.7±1.0 226.45±0.03 ∆K (mag) 0.17±0.05 0.70±0.05 aproj (AU) 7.7±1.2 1600±200 Inferred q 0.88±0.05 0.53±0.08 SpTPrim M5±0.5 M5+M5.2(±0.5) SpTSec M5.2±0.5 M5±0.5 MPrim 0.14±0.02 0.26±0.04 MSec 0.12±0.02 0.14±0.02 Note. — The center of mass for the Aa-Ab pair is unknown, so we calculate all A-B separa- tions with respect to the K band photocenter. et al. (2003) to directly convert observed spectral types to masses. Relative properties (mass ratios q and relative spectral types) are calculated by combining the Baraffe isochrones and Luhman temperature scale with the em- pirical NIR colors of Bessell & Brett (1998) and the K- band bolometric corrections of Leggett et al. (1998) to estimate q and ∆SpT from the observed flux ratio ∆K. We have adopted the previously-measured spectral type for A (M5; Preibisch et al. 2002) as the type for component Aa, but the inferred spectral type for Ab is only 0.2 subclasses later, so this assumption should be ro- bust to within the uncertainties (∼0.5 subclasses). The projected spatial separations are calculated for the mean distance of Upper Sco, 145±2 pc (de Zeeuw et al. 1999). If the total radial depth of Upper Sco is equal to its angu- lar extent (∼15o or ∼40 pc), then the unknown depth of USco1606-1935 within Upper Sco implies an uncertainty in the projected spatial separation of ±15%. The sys- tematic uncertainty due to the uncertainty in the mean distance of Upper Sco is negligible (.2%). 4. IS USCO1606-1935 AB A BINARY SYSTEM? The unambiguous identification of pre-main sequence binaries is complicated by the difficulty of distinguishing gravitationally bound binary pairs from coeval, comov- ing association members which are aligned in projection. Most traditional methods used to confirm field binary companions do not work in the case of young binaries in clusters and associations because all association members share common distances and kinematics (to within cur- rent observational uncertainties), so the only remaining option is to assess the probability of chance alignment. We address this challenge by quantifying the clustering of PMS stars via calculation of the two-point correlation function (TPCF) across a wide range of angular scales (1′′ to >1 degree). This type of analysis has been at- tempted in the past (e.g. Gomez et al. 1993 for Taurus; Simon 1997 for Ophiuchus, Taurus, and the Trapezium), but these studies were conducted using samples that were significantly incomplete relative to today. The TPCF, w(θ), is defined as the number of excess pairs of objects with a given separation θ over the ex- pected number for a random distribution (Peebles 1980). The TPCF is linearly proportional to the surface density of companions per star, Σ(θ) = (N∗/A)[1 +w(θ)], where A is the survey area and N∗ is the total number of stars. Fig. 6.— The surface density of companions as a function of separation for young stars and brown dwarfs in Upper Sco. Red symbols denote results from our wide-binary survey using 2MASS (Kraus & Hillenbrand 2007) and blue symbols denote data for all spectroscopically- confirmed members in two fields surveyed by Preibisch et al. (2002). The data appear to be well-fit by two power laws (dashed lines) which most likely correspond to gravitationally bound binaries and unbound clusters of stars that have not yet completely dispersed from their formation environments. The data points which were used to fit these power laws are denoted with circles; other points are denoted with crosses. However, it is often easier to evaluate the TPCF via a Monte Carlo-based definition, w(θ) = Np(θ)/Nr(θ) − 1, where Np(θ) is the number of pairs in the survey area with separations in a bin centered on θ and Nr(θ) is the expected number of pairs for a random distribution of ob- jects over the same area (Hewett 1982). The advantage of this method is that it does not require edge correc- tions, unlike direct measurement of Σ(θ). We adopted this method due to its ease of implementation, but we report our subsequent results in terms of Σ(θ) since it is a more intuitive quantity. The current census of Upper Sco members across the full association is very incomplete, so we implemented our analysis for intermediate and large separations (θ > 6.4′′) using only members located in two heavily-studied fields originally observed by Preibisch et al. (2001, 2002; the 2df-East and 2df-West fields). The census of mem- bers in these fields may not be complete, but we ex- pect that it is the least incomplete. The census of com- panions at smaller separations (1.5-6.4′′) has been uni- formly studied for all spectroscopically-confirmed mem- bers (Kraus & Hillenbrand 2007), so we have maximized the sample size in this separation range by considering the immediate area around all known members, not just those within the Preibisch fields. Our survey was only complete for mass ratios q >0.25, so we do not include companions with mass ratios q < 0.25. These choices might lead to systematic biases if the Preibisch fields are still significantly incomplete or if the frequency and properties of binary systems show intra-association variations, but any such incompleteness would probably change the result by no more than a fac- tor of 2-3. As we will subsequently show, Σ(θ) varies by 4 orders of magnitude across the full range of θ. The well- established mass dependence of multiplicity should not affect our results since the mass function for the Preibisch fields is similar to that seen for the rest of the association. In Figure 6, we plot Σ(θ) for Upper Sco, spanning the separation range −3.5 < log(θ) < 0.25 (1.14′′ to 1.78 deg ). We have fit this relation with two power laws, one which dominates at small separations (.15- 30′′) and one at larger separations. We interpret the two segments, following Simon (1997), to be the result of gravitationally-bound binarity and gravitationally un- bound intra-association clustering, respectively. We fit the binary power law to the three lowest-separation bins (log(θ) < −2.75) because this is the separation range over which we possess uniform multiplicity data. The cluster power law was fit to the six highest-separation bins (log(θ) > −1.25) because those bins have the small- est uncertainties. Bins corresponding to intermediate separations seem to follow the two power laws. We found that the slope of the cluster power law (- 0.14±0.02) is very close to zero, which implies that there is very little clustering on scales of .1 deg. This re- sult is not unexpected for intermediate-age associations like Upper Sco; given the typical intra-association veloc- ity dispersion (∼1 km s−1) and the age (5 Myr), most association members have dispersed ∼5 pc (2 deg) rel- ative to their formation point, averaging out structure on smaller spatial scales. Simon (1997) found that the slopes for Taurus, Ophiuchus, and the ONC are steeper, suggesting that more structure is present on these small scales at young ages (∼1-2 Myr). The slope of the binary power law (-3.03±0.24) is much steeper than the clus- ter regime. The separation range represented is much larger than the peak of the binary separation distribu- tion (∼30 AU for field solar-mass stars; Duquennoy & Mayor 1991), so the steep negative slope corresponds to the large-separation tail of the separation distribution function. The two power laws seem to cross at separa- tions of ∼15-30′′ (aproj ∼ 2500− 5000 AU), though this result depends on the sample completeness in the binary and cluster regimes. We interpret this to be the maxi- mum separation range at which binaries can be identified. If we extrapolate the cluster power law into the separa- tion regime of the binary power law, we find that the ex- pected surface density of unbound coincidentally-aligned companions is ∼60 deg−2. Given this surface density, there should be ∼1 chance alignment within 15′′ among the 366 spectroscopically confirmed members of Upper Sco. Among the 173 known late-type stars and brown dwarfs (SpT≥M4) for which this separation range is un- usually wide, the expected number of chance alignments with any other member is 0.5. If the mass function of known members is similar to the total mass function, approximately half (∼0.25 chance alignments) are ex- pected to occur with another low-mass member. There- fore, we expect ∼0.25 chance alignments which might be mistaken for a low-mass binary pair. The probability that one or more such chance align- ments actually exists for a known low-mass USco member is 25% (based on Poisson statistics), which suggests that the nature of a single candidate wide pair like USco1606- 1935 AB can not be unambiguously determined. If any more pairs can be confirmed, then they would represent a statistically significant excess. The corresponding prob- ability of finding 2 chance alignments of low-mass mem- bers is only 2%. As we have described in our survey of wide multiplicity with 2MASS (Kraus & Hillenbrand 2007), we have identified at least three additional can- didate ultrawide systems in Upper Sco, so spectroscopic and astrometric followup of these candidate systems is a high priority. 5. SUMMARY We have presented photometric, astrometric, and spec- troscopic observations of USco1606-1935, a candidate ul- trawide (∼1600 AU), low-mass (Mtot ∼0.4 M⊙) hierar- chical triple system in the nearby OB association Upper Scorpius. We conclude that the ultrawide B component is a young, comoving member of the association, and show that the primary is itself a close binary system. If the Aab and B components are gravitationally bound, the system would join the growing class of young multiple systems which have unusually wide separations as compared to field systems of similar mass. However, we demonstrate that binarity can not be assumed purely on probabilistic grounds. Analysis of the association’s two-point correlation function shows that there is a sig- nificant probability (25%) that at least one pair of low- mass association members will be separated by .15′′, so analysis of the wide binary population requires a system- atic search for all wide binaries. The detection of another pair of low-mass members within 15′′ would represent an excess at the 98% confidence level. In principle, bina- rity could also be demonstrated by measuring common proper motion with precision higher than the internal velocity scatter of the association; given the astromet- ric precision currently attainable with LGSAO data (.1 mas), the test could be feasible within .5 years. The authors thank C. Slesnick for providing guidance in the analysis of young stellar spectra, P. Cameron for sharing his NIRC2 astrometric calibration results prior to publication, and the anonymous referee for returning a helpful and very prompt review. The authors also wish to thank the observatory staff, and particularly the Keck LGSAO team, for their tireless efforts in commissioning this valuable addition to the observatory. Finally, we recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to con- duct observations from this mountain. This work makes use of data products from the Two Micron All-Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Process- ing and Analysis Center/California Institute of Tech- nology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This work also makes use of data products from the DENIS project, which has been partly funded by the SCIENCE and the HCM plans of the European Com- mission under grants CT920791 and CT940627. It is supported by INSU, MEN and CNRS in France, by the State of Baden-Wrttemberg in Germany, by DG- ICYT in Spain, by CNR in Italy, by FFwFBWF in Austria, by FAPESP in Brazil, by OTKA grants F- 4239 and F-013990 in Hungary, and by the ESO C&EE grant A-04-046. Finally, our research has made use of the USNOFS Image and Catalogue Archive operated by the United States Naval Observatory, Flagstaff Station (http://www.nofs.navy.mil/data/fchpix/). 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0704.0456

Particle propagation in cosmological backgrounds

Particle propagation in cosmological backgrounds Daniel Arteaga Departament de F́ısica Fonamental. Facultat de F́ısica. Av. Diagonal 647, 08028 Barcelona (Spain) E-mail: [email protected] Abstract. We study the quantum propagation of particles in cosmological backgrounds, by considering a doublet of massive scalar fields propagating in an expanding universe, possibly filled with radiation. We focus on the dissipative effects related to the expansion rate. At first order, we recover the expected result that the decay rate is determined by the local temperature. Beyond linear order, the decay rate has an additional contribution governed by the expansion parameter. This latter contribution is present even for stable particles in the vacuum. Finally, we analyze the long time behaviour of the propagator and briefly discuss applications to the trans- Planckian question. 1. Introduction In this contribution we study the quantum propagation of particles in a cosmological background. We are particularly interested in understanding the dissipative phenomena related to the time dependence of the metric. To this end, we analyze the propagator of a massive particle which interacts with a massless radiation field in an expanding universe. Several points must be considered. First, we are dealing with an interacting field theory in a curved spacetime. In this situation, the asymptotic in and out vacua generally do not coincide. Being interested in expectation values, rather than in in-out matrix elements, we adopt the Keldysh-Schwinger formalism, or Closed Time Path (CTP) method [1–3] in curved spacetime [4–7]. Second, as it is well known, in a curved spacetime there is no single definition for the vacuum nor for the concept of particle. We face this issue by working within the adiabatic approximation [8]: the massive particles will have their Compton wavelengths much smaller than the typical curvature radius of the universe (in our case, the Hubble radius). Third, as explained in [9], in theories such as QED or perturbative quantum gravity, dissipative effects appear only at two loops, because the one-loop diagrams which could lead to dissipation vanish on the mass shell. Here, in order to keep the calculations simple, we have chosen a simple, yet physically meaningful, model which exhibits http://arxiv.org/abs/0704.0456v1 Particle propagation in cosmological backgrounds 2 dissipation at one loop. We expect the behaviour of QED or perturbative quantum gravity to be similar at two loops. In this contribution we compute the retarded self-energy of the lightest field in a massive doublet which propagates in a thermal bath of massless particles in an expanding universe, and from it we extract the decay rate. Notice that in the Minkowski vacuum the excitations of the lighter field are stable, hence the decay rate is zero. We concentrate on the physical insights and summarize the main results. A more detailed account will be given in separate publications [10, 11] The contribution is organized as follows. In section 2 we introduce the model and motivate the use of the adiabatic approximation for the massive fields. In section 3 we present the results for the imaginary part of the self-energy and the decay rate. In section 4 we study the time evolution of the interacting propagator. Finally, in section 5 we summarize the main points of the contribution and discuss its relevance to the trans-Planckian question. We use a system of natural units with ~ = c = 1, and the metric has the signature (−,+,+,+). 2. The model We consider spatially isotropic and homogeneous Friedmann-Lemâıtre-Robertson- Walker models with flat spatial sections: ds2 = −dt2 + a2(t)dx2. (1) The particle model is the following: two massive fields φm, and φM , interacting with a massless field, χ, via a trilinear coupling. The total action is S = Sm + SM + Sχ + Sint, where each term is given by dt d3x a3(t) (∂tφm) a2(t) (∂xφm) 2 −m2φ2m , (2a) dt d3x a3(t) (∂tφM) a2(t) (∂xφM) 2 −M2φ2M , (2b) dt d3x a3(t) (∂tχ) a2(t) (∂xχ) 2 − ξR(t)χ2 , (2c) Sint = gM dt d3x a3(t)φmφMχ, (2d) with R(t) being the Ricci scalar. We assume that the massless field is conformally coupled to gravity, so that ξ = 1/6. It is useful to work with rescaled massive fields defined by φ̄(t,x) := [−g(t,x)]1/4φ(t,x) = a3/2(t)φ(t,x). We consider the two massive fields having large masses but with a small mass difference ∆m := M − m ≪ M . As shown in [12], the model can be interpreted as a field-theory description of a relativistic two-level atom (of mass m and energy gap ∆m) interacting with a scalar radiation field χ. The radiation field χ is assumed to be at some conformal temperature θ (which can eventually be zero). The corresponding physical temperature, as well as the Hubble rate H(t) := ȧ(t)/a(t), are chosen to be Particle propagation in cosmological backgrounds 3 much smaller than the masses of the fields. These restrictions ensure that the number of massive particles is strictly conserved. The non-trivial dynamics concerns the transitions between the two massive fields accompanied by emission and absorption of massless quanta. In a curved spacetimes it is not a trivial task to compute even the free field vacuum propagators. For massless conformally coupled fields there is a natural vacuum state, the conformal vacuum. Propagators in this vacuum, when expressed in conformal time, essentially correspond to the flat spacetime propagators. For the massive fields, rather than attempting to find the exact free propagator, we will exploit the fact that their Compton wavelengths is much smaller than the Hubble length H−1. In this regime, the adiabatic (WKB) approximation is valid and explicit expressions for the free propagators can be computed [8, 10] —see for instance (19). 3. The self-energy and decay rates In this section we consider the interacting retarded Green function GR(t, t ′;p) := θ(t− t′)〈[φ̂mp(t), φ̂mp(t ′)]〉 (3) within the adiabatic approximation. It is related to the retarded self-energy ΣR via [12] GR(t, t ′;p) = G R (t, t ds ds′ −g(s) −g(s′)G R (t, s;p)ΣR(s, s ′;p)GR(s ′, t′;p) (4) where G R (t, t ′;p) is the free retarded propagator. In terms of the rescaled fields, φ̄(t;p) = a3/2φ(t;p), the above relation becomes ḠR(t, t ′;p) = Ḡ R (t, t ′;p)− i ds ds′ Ḡ R (t, s;p)Σ̄R(s, s ′;p)ḠR(s ′, t′;p). (5) We assume that the massive fields are in the adiabatic vacuum, and that the massless field χ is in a thermal state, characterized by a fixed conformal temperature θ. We will compute the imaginary part of the one-loop self energy to order g2 in the adiabatic approximation, evaluated at the mass shell. It will be evaluated in a a frequency representation around the average time coordinate T = (t1+t2)/2, by Fourier- transforming with respect to the difference coordinate ∆ = t1 − t2, which amounts to a local frequency representation (it is further analyzed in next section). As for the spatial part, we work in the momentum representation to exploit conservation of the conformal momentum. 3.1. Linear approximation to the scale factor As a first step, we approximate the evolution of the scale factor by a linear expansion: a(t) ≈ a(T )[1 +H(T )(t− T )]. (6) This approximation for the scale factor is appropriate when considering physical temperatures which are much larger than the expansion rate (but still much smaller than Particle propagation in cosmological backgrounds 4 the fields masses). On the mass shell, thermal corrections will dominate over curvature corrections, since the thermal energy scale is much larger than the curvature energy scale. Therefore, we expect the on-shell self-energy to be governed by the thermal bath at the instantaneous physical temperature at each moment of the expansion, θ/a(T ). The explicit calculation [10] confirms that the imaginary part of the on-shell self- energy is given by that of a thermal bath in Minkowski at a physical temperature θ/a(T ). In the limit in which the atoms are at rest this result is [10, 12] Im Σ̄R(m, T ; 0) = − M∆m nθ/a(T )(∆m), (7) where nθ/a(T )(∆m) is the Bose-Einstein function: nθ/a(T )(∆m) := e∆m a(T )/θ −1 . (8) As in Minkowski spacetime, the self-energy corresponds to a decay rate, Γ = − Im Σ̄R(m, T ; 0) = ∆m nθ/a(T )(∆m), (9) which amounts to the probability per unit time for the lightest state to absorb a massless particle from the thermal bath. 3.2. Beyond linear order: vacuum effects When the expansion rate of the universe is of the order of the temperature or larger, vacuum effects become relevant. Energy conservation does not hold for energy scales of the order of the expansion rate, and therefore we expect new channels for the particle decay which will contribute to the imaginary part of the self-energy. In order to study the vacuum effects we need to choose a explicit model for the evolution of the scale factor. For instance, in the case of de Sitter, a(t) = a(T ) eH(t−T ), (10) the vacuum contribution to the imaginary part of the retarded self-energy given by [11] Im Σ̄R(m, T ; 0) = − M∆m nH/(2π)(∆m), (11) which coicides with the self-energy in a Minkowski thermal bath at a temperature H/(2π). The result is not unexpected since the effective de Sitter temperature [13] is recovered. The corresponding decay rate Γ = − Im Σ̄R(m, T ; 0) = ∆m nH/(2π)(∆m), (12) amounts for the probability per unit time for the lightest field to emit a massless particle. Energy conservation forbids this process in Minkowski spacetime, but this restriction does not apply in an expanding universe. Particle propagation in cosmological backgrounds 5 4. Retarded propagator and self-energy in cosmology In expanding universes the propagators are no longer time-translation invariant. We can nevertheless always express the propagator in a frequency representation, ḠR(ω, T ;p) := d∆ eiω∆ ḠR(T +∆/2, T −∆/2;p) . (13) For short time differences as compared to the inverse expansion rate, i.e., |t−t′| ≪ H−1, (5) can be diagonalized: ḠR(ω, T ;p) = [−iḠ(0)(ω, T ;p)]−1 + Σ̄R(ω, T ;p) . (14) Fourier-transforming again we get the short-time behavior: ḠR(t, t ′;p) = Rp(T ) sin [Rp(T )(t− t ′)] e−Γp(T )(t−t ′)/2 θ(t− t′). (15) (T ) := E2 (T ) + Re Σ̄R(Ep, T ;p) := m a2(T ) + Re Σ̄R(Ep, T ;p) (16) Γp(T ) := − Rp(T ) Im Σ̄R(Ep, T ;p). (17) Therefore one recovers the usual interpretation, in which the real part of the self-energy corresponds to the energy shift, and in which the imaginary part corresponds to the decay rate. Notice that both quantities depend in general on time. One may also be interested in considering large time lapses, and in this case the frequency representation of the propagator around the average time does not make sense. Lifting the short-time requirement, and only imposing the adiabatic approximation, the following expression for the evolution of the retarded propagator is found [10]: ḠR(t1, t2;p) = Rp(t1)Rp(t2) dt′ Rp(t dt′ Γk(t θ(t1 − t2). (18) Notice that the long-time evolution of the propagator can be expressed in terms of integrals of quantities evaluated in the local frequency representation. Two time scales are clearely separated: the interaction timescale, in which the interaction process take place and in which the self-energy is evaluated, and the evolution timescale, which can be much longer and during which the propagators deviate significantly from the corresponding Minkowski expression. Equation (18) can be derived in a very similar way as the well-known adiabatic approximation for the free retarded propagator [8]: R (t1, t2;p) = Ep(t1)Ep(t2) dt′ Ep(t θ(t1 − t2). (19) Particle propagation in cosmological backgrounds 6 5. Summary and discussion The goal of this contribution is to analyze the quantum effects in the propagation of interacting fields in a cosmological background. This issue may play an important role in justifying the non-trivial dispersion relations which have been used when addressing the trans-Planckian question in the context of black holes [14–17] and cosmology [18–21]. Interactions could indeed significantly modify the field propagation when approaching the event horizon of a black hole [22–25] or at primordial stages of inflation [9]. In our model, the masses of the fields were assumed to be much larger than the expansion rate of the universe. This was a key assumption, because it allowed to introduce the adiabatic (WKB) approximation, which not only makes the problem solvable, but also allows having a well-defined particle concept even in absence of asymptotic regimes. Within this approximation, the time-evolution of the interacting propagators can be computed from the integral of the retarded self-energy, evaluated on-shell in a frequency representation around the mid time. The imaginary part of the self-energy determines the decay of the retarded propagator, and hence it is an expression of the dissipative properties. For temperatures higher than the expansion parameter the decay of the propagator is determined by the local temperature at each moment of expansion. For lower temperatures, the decay of the propagator is driven by the expansion rate of the universe. This second contribution, which is present even in the vacuum, can be interpreted as being a consequence of the absence of energy conservation at those energy scales comparable to the expansion rate. The decay rate, derived from the imaginary part of the self-energy, has a secular character. Even small decay rates could thus give an important effect when integrated over large periods of time. The exact significance of the generically dissipative properties of the propagator will be further analyzed elsewhere [11]. Acknowledgments I am very grateful with Renaud Parentani and Enric Verdaguer for a critical reading of the manuscript. This work is partially supported by the Research Projects MEC FPA2004-04582-C02-02 and DURSI 2005SGR-00082. References [1] J. S. Schwinger. J. Math. Phys., 2:407, 1961. [2] L. V. Keldysh. Zh. Eksp. Teor. Fiz, 47:1515, 1965. [Sov. Phys. JEPT 20:1018, 1965]. [3] K.-C. Chou, Z.-B. Su, B.-L. Hao, and L. Yu. Phys. Rept., 118:1–131, 1985. [4] R. D. Jordan. Phys. Rev. D, 33:444–454, 1986. [5] E. Calzetta and B. L. Hu. Phys. Rev. D, 35:495–509, 1987. [6] A. Campos and E. Verdaguer. Phys. Rev. D, 49:1861–1880, 1994. [7] S. Weinberg. Phys. Rev. D, 72:043514, 2005. [8] N. D. Birrell and P. C. W. Davies. Quantum fields in curved space. Cambridge University Press, Cambridge, England, 1982. Particle propagation in cosmological backgrounds 7 [9] D. Arteaga, R. Parentani, and E. Verdaguer. Phys. Rev. D, 70:044019, 2004. [10] D. Arteaga, R. Parentani, and E. Verdaguer. To appear in Int. J. Theor. Phys. [11] D. Arteaga, R. Parentani, and E. Verdaguer. In preparation. [12] D. Arteaga, R. Parentani, and E. Verdaguer. Int. J. Theor. Phys., 44:1665–1689, 2005. [13] E. Mottola, Phys. Rev. D, 31:754–766, 1985. [14] W. G. Unruh. Phys. Rev. Lett., 46:1351–1353, 1981. [15] T. Jacobson. Phys. Rev. D, 44:1731–1739, 1991. [16] W. G. Unruh. Phys. Rev. D, 51:2827–2838, 1995. [17] R. Balbinot, A. Fabbri, S. Fagnocchi, and R. Parentani. Riv. Nuovo Cim. 28:1–55, 2005 (gr-qc/0601079). [18] J. Martin and R. Brandenberger. Phys. Rev. D, 63:123501, 2001. [19] J. Martin and R. Brandenberger. Phys. Rev. D, 68:063513, 2003. [20] J. C. Niemeyer. Phys. Rev. D, 63:123502, 2001. [21] J. C. Niemeyer and R. Parentani. Phys. Rev. D, 64:101301, 2001. [22] C. Barrabès, V. Frolov, and R. Parentani. Phys. Rev. D, 62:044020, 2000. [23] R. Parentani. Phys. Rev. D, 63:041503, 2001. [24] R. Parentani. Int. J. Theor. Phys., 41:2175–2200, 2002. [25] R. Parentani. Int. J. Mod. Phys., A17:2721–2726, 2002. http://arxiv.org/abs/gr-qc/0601079 Introduction The model The self-energy and decay rates Linear approximation to the scale factor Beyond linear order: vacuum effects Retarded propagator and self-energy in cosmology Summary and discussion

0704.0457

Quantum analysis of a linear DC SQUID mechanical displacement detector

Quantum analysis of a linear DC SQUID mechanical displacement detector M. P. Blencowe Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755 E. Buks Department of Electrical Engineering, Technion, Haifa 32000 Israel (Dated: October 26, 2018) Abstract We provide a quantum analysis of a DC SQUID mechanical displacement detector within the sub- critical Josephson current regime. A segment of the SQUID loop forms the mechanical resonator and motion of the latter is transduced inductively through changes in the flux threading the loop. Expressions are derived for the detector signal response and noise, which are used to evaluate the position and force detection sensitivity. We also investigate cooling of the mechanical resonator due to detector back reaction. PACS numbers: 85.25.Dq; 85.85.+j; 03.65.Ta http://arxiv.org/abs/0704.0457v2 I. INTRODUCTION In a series of recent experiments1,2,3 and related theoretical work,4,5,6,7,8,9 it was demon- strated that a displacement detector based on either a normal or superconducting single electronic transistor (SSET) can resolve the motion of a micron-scale mechanical resonator close to the quantum limit as set by Heisenberg’s Uncertainty Principle.10,11,12 The displace- ment transduction was achieved by capacitively coupling the gated mechanical resonator to the SSET metallic island. When the resonator is voltage biased, motion of the latter changes the island charging energy and hence the Cooper pair tunnel rates. The resulting modula- tion in the source-drain tunnel current through the SSET is then read out as a signature of the mechanical motion. Given the success of this capacitive-based transduction method in approaching the quan- tum limit, it is natural to consider complementary, inductive-based transduction methods in which, for example, a superconducting quantum interference device (SQUID) is similarly used as an intermediate quantum-limited stage between the micron-scale mechanical res- onator and secondary amplification stages.13,14,15,16 Unavoidable, fundamental noise sources and how they affect the SSET and SQUID devices are not necessarily the same. Furthermore, achievable coupling strengths between each type of device and a micron-scale mechanical resonator may be different. Therefore, it would be interesting to address the merits of the SQUID in comparison with the established SSET for approaching the quantum limit of displacement detection. In the present paper, we analyze a DC SQUID-based displacement detector. The SQUID is integrated with a mechanical resonator in the form of a doubly-clamped beam, shown schematically in Fig. 1. Motion of the beam changes the magnetic flux Φ threading the SQUID loop, hence modulating the current circulating the loop. We shall address the operation of the SQUID displacement detector in the regime for which the loop current is smaller than the Josephson junction critical current Ic and at temperatures well below the superconducting critical temperature. We thus assume that resistive (normal) current flow through the junctions and accompanying current noise can be neglected. (See for example Ref. 17 for a quantum noise analysis of resistively shunted Josephson junctions and Ref. 18 for a related analysis of the DC SQUID.) Such an assumption cannot be made with the usual mode of operation for the SSET devices, where the tunnel current unavoidably involves the quasiparticle decay of Cooper pairs, resulting in shot noise. As noise source, we will consider the quantum electromagnetic fluctuations within the pump/probe feedline and also transmission line resonator that is connected to the SQUID. This noise is a consequence of the necessary dissipative coupling to the outside world and affects the mechanical signal output in two ways. First, the noise is added directly to the output in the probe line and, second, the noise acts back on the mechanical resonator via the SQUID, affecting the resonator’s motion. With the Josephson junction plasma frequencies assumed to be much larger than the other resonant modes of relevance for the device, the SQUID can be modeled to a good approximation as an effective inductance that depends on the external current I entering and exiting the loop, as well as on the applied flux. In this first of two papers, we shall make the further approximation of neglecting the I-dependence of the SQUID effective inductance, which requires the condition I ≪ Ic. In the sequel,19 we will relax this condition somewhat by including the next to leading O(I2) term in the inductance and address the consequences of this non-linear correction for quantum-limited displacement detection. Modeling the SQUID approximately as a passive inductance element, the transmission line resonator-mechanical resonator effective Hamiltonian is given by Eq. (24). This Hamil- tonian describes many other detector-oscillator systems that are modeled as two coupled harmonic oscillators, including the examples of an LC resonator capacitively coupled to a mechanical resonator20,21 and an optical cavity coupled to a mechanically compliant mirror via radiation pressure;22,23,24,25,26 the various systems are distinguished only by the depen- dences of the coupling strengths on the parameters particular to each system. Thus, many of the results of this paper are of more general relevance. The central results of the paper are Eqs. (69) and (70), giving the detector response to a mechanical resonator undergoing quantum Brownian motion and also subject to a classical driving force. In the derivation of these expressions, we do not approximate the response as a perturbation series in the coupling between the SQUID and mechanical resonator as is conventionally done, but rather find it more natural to base our approximations instead on assumed weak coupling between the mechanical resonator and its external heat bath and weak classical driving force. Thus, in the context of the linear response paradigm, our detector should properly be viewed as including the mechanical resonator degrees of freedom as well, with the weak perturbative signal instead consisting of the heat bath force noise and classical drive force acting on the mechanical resonator. Since the quality factors of actual, micron-scale mechanical resonators can be very large at sub-Kelvin temperatures (E.g., Q ∼ 105 in the experiments of Refs. 2,3), quantum electromagnetic noise in the transmission line part of the detector can have strong back reaction effects on the motion of the mechanical resonator, even when the coupling between the resonator and the SQUID is very weak. One consequence that we shall consider is cooling of the mechanical resonator fundamental mode, which requires strong back reaction damping combined with low noise. Nevertheless, as we will also show, one can still analyze the quantum-limited detector linear response to the mechanical resonator’s position signal using general expressions (69) and (70), under the appropriate conditions of small pump drive and weak coupling between the SQUID and mechanical resonator such that back reaction effects are small. The outline of the paper is as follows. In Sec. II, we write down the SQUID-mechanical resonator equations of motion corresponding to the circuit scheme shown in Fig. 1 and then derive the Heisenberg equations for the various mode raising and lowering operators, subject to the above-mentioned approximations. In Sec. III, we solve the equations within the linear response approximation to derive the detector signal response and noise. In Sec. IV, we analyze both the position and force detection sensitivity, and address also back reaction cooling of the mechanical resonator. Sec. V provides concluding remarks. II. EQUATIONS OF MOTION A. Transmission line-SQUID-mechanical oscillator Hamiltonian Fig. 1 shows the displacement detector scheme. The device consists of a stripline resonator (transmission line T ) made of two sections, each of length l/2, connected via a DC SQUID (see Refs. 27,28,29,30 for related, qubit detection schemes). The transmission line inductance and capacitance per unit length are LT and CT respectively. The Josephson junctions in each arm of the SQUID are assumed to have identical critical currents Ic and capacitances CJ . A length losc segment of the SQUID loop is free to vibrate as a doubly-clamped bar resonator and the fundamental flexural mode of interest (in the plane of the loop) is treated as a harmonic oscillator with mass m, frequency ωm and displacement coordinate y. The total external magnetic flux applied perpendicular to the SQUID loop is given by Φext+λBextloscy, where Φext is the flux corresponding to the case y = 0, Bext is the normal component of the magnetic field at the location of the vibrating loop segment (oscillator), and the dimensionless parameter λ < 1 is a geometrical correction factor accounting for the non- uniform displacement of the doubly-clamped resonator in the fundamental flexural mode. p T T FIG. 1: Scheme for the displacement detector showing the pump/probe line ‘p’, transmission line resonator ‘T ’, and DC SQUID with mechanically compliant loop segment having effective mass m and fundamental frequency ωm. Note that the scale of the DC SQUID is exaggerated relative to that of the stripline for clarity. The transmission line is weakly coupled to a pump/probe feedline (p), with inductance and capacitance per unit length Lp and Cp respectively, employed for delivering the input and output RF signals; the coupling can be characterized by a transmission line mode amplitude damping rate γpT (see section IIB below). Other possible damping mechanisms in the transmission line may be taken into account by adding a fictitious semi-infinite stripline environment (e), weakly coupled to the transmission line characterized by mode amplitude damping rate γeT . 31 While γeT can be made much smaller than γpT with suitable transmission line resonator design, we shall nevertheless include both sources of damping in our analysis so as to eventually be able to gauge their relative effects on the detector displacement sensitivity [see Eq. (90)]. The SQUID, on the other hand, is assumed to be dissipationless. The mechanical oscillator is also assumed to be coupled to an external heat bath (b), characterized by mode amplitude damping rate γbm. A convenient choice of dynamical coordinates for the SQUID are γ± = (φ1 ± φ2) /2, where φ1 and φ2 are the gauge invariant phases across each of the two Josephson junctions. 32 For the transmission line, we similarly use its phase field coordinate φ(x, t),30,33 where x describes the longitudinal location along the transmission line: −l/2 < x < l/2, with the SQUID located at x = 0. In terms of φ, the transmission line current and voltage are IT (x, t) = − ∂φ(x, t) VT (x, t) = ∂φ(x, t) , (2) where Φ0 = h/(2e) is the flux quantum. Neglecting for now the couplings to the feedline, stripline and mechanical oscillator environments, the equations of motion for the closed system comprising the superconducting transmission line-SQUID-mechanical oscillator are as follows (see, e.g., Ref. 14 for a derivation of related equations of motion for a mechanical rf-SQUID): = (LTCT ) , (3) ω−2J γ̈− + cos(γ+) sin(γ−) + 2β γ− − π (Φext + λBextloscy) = 0, (4) ω−2J γ̈+ + sin(γ+) cos(γ−)− = 0, (5) mÿ +mω2my − λBextloscγ− = 0, (6) where ωJ = 2πIc/(CJΦ0) is the plasma frequency of the SQUID Josephson junctions, the dimensionless parameter βL = 2πLIc/Φ0, L is the self inductance of the SQUID, n is an integer arising from the single-valuedness condition for the phase 2γ− around the loop, and IT is shorthand for IT (x = 0, t). Eq. (3) is simply the wave equation for the phase field coordinate φ(x, t) of the transmission line. Eq. (4) describes the current circulating the loop, which depends on the external flux threading the loop. Eq. (5) describes the average current threading the loop, which from current conservation is equal to one-half the transmission line current at x = 0. With the circulating SQUID current given by Φ0γ−/(πL) (up to a Φext dependent term), we recognize in Eq. (6) the Lorentz force acting on the mechanical oscillator. In addition to the equations of motion, we have the following current and voltage bound- ary conditions: IT (x = ±l/2, t) = 0 (7) ∂ (Leff [Φext(y), IT ]IT ) = VT (0 −, t)− VT (0+, t), (8) where the external flux and current-dependent, effective inductance Leff [Φext(y), IT ] of the SQUID as ‘seen’ by the transmission line is Leff [Φext(y), IT ] = , (9) with Φext(y) = Φext + λBextloscy. Note that we have set n = 0, since observable quantities do not depend on n. We now make the following assumptions and consequent approximations: (a) ωJ ≫ ωT ≫ ωm (where ωT is the relevant resonant mode of the transmission line); neglect the SQUID inertia terms ω−2J γ̈±. (b) βL ≪ 1; solve for γ± as series expansions to first order in βL. (c) |Bextloscy| /Φ0 ≪ 1; series expand the equations of motion to first order in y(t). (d) |IT/Ic| = 2πLT Ic ∂φ(0,t) ≪ 1; series expand the equations of motion to second order in IT . With ωJ ’s typically in the tens of GHz, assumption (a) is reasonable. From Eq. (4), we see that a small βL value prevents the γ− coordinate from getting trapped in its various potential minima, causing unwanted hysteresis. With the γ+ expansion in IT consisting of only odd powers, approximations (a) and (d) amount to describing the SQUID simply as a current independent, Φext-tunable passive inductance element Leff [Φext(y)] that also depends on the mechanical oscillator position coordinate y. Including the next-to-leading, I3T term in the γ+ expansion gives an I T -dependent, nonlinear correction to the SQUID effective induc- tance. The consequences of including this nonlinear correction term for the quantum-limited displacement detection sensitivity will be considered in a forthcoming paper.19 Solving for γ+ to order IT and substituting in Eq. (9), we obtain: Leff [Φext(y)] ≈ πΦext(y) , (10) where the self inductance L contribution has been neglected since it is of order βL ≪ 1. Solving for γ− to order I T and substituting into Eq. (6), we obtain for the mechanical oscillator equation of motion: mÿ +mω2my − πλBextloscI tan (πΦext/Φ0) sec (πΦext/Φ0) = 0, (11) where from (c), we have set y = 0 in the solution for γ− and have dropped an overall constant term. Since the γ− expansion in IT consists only of even powers, we must go to second order in IT so as to have a non-trivial transmission line-oscillator effective coupling. Thus, the SQUID phase coordinates γ± have been completely eliminated from the equations of motion, a consequence of approximation (a); the SQUID mediates the interaction between the mechanical oscillator coordinate y and transmission line coordinate φ without retardation effects. From Eq. (11), it might appear that the force on the mechanical oscillator due to the transmission line can be made arbitrarily large by tuning Φext close to Φ0/2. Note, however, that the proper conditions for the validity of the IT and βL expansions are: sec (πΦext/Φ0) ≪ 1 (12) |βL sec (πΦext/Φ0)| ≪ 1. (13) We now restrict ourselves to a single transmission line mode and derive approximate equations of motion for the mode amplitude. Suppose that the mechanical oscillator position coordinate is held fixed at y = 0. The following phase field satisfies the current boundary conditions (7): φ(x, t) = −φ(t) cos [k0 (x+ l/2)] ; x < 0 +φ(t) cos [k0 (x− l/2)] ; x > 0 , (14) with the wavenumber k0 determined by the voltage boundary condition (8): Leff (Φext) . (15) The wave equation (3) gives for the transmission mode frequency: ωT = k0/ LTCT . Substi- tuting the phase field (14) into the IT part of the oscillator equation of motion (11) further- more gives the transmission line force acting on the oscillator with fixed coordinate y = 0. Now release the mechanical oscillator coordinate and suppose that for small [condition (c)] , slow [condition (a)] displacements, the force is the same to a good approximation.Then the oscillator equation of motion becomes mÿ(t) +mω2my(t) + sin2 (k0l/2) −λBextlosc (Φ0/2π) 4πLT lIc tan (πΦext/Φ0) sec (πΦext/Φ0) 2(t) = 0, (16) From Eq. (16), we can determine the mechanical sector of the Lagrangian, along with the interaction potential involving y and the mode amplitude φ. The remaining transmission line sector follows from the wave equation (3) and we thus have for the total Lagrangian: φ, y, φ̇, ẏ mẏ2 − 1 mω2my sin2 (k0l/2) φ̇2 − 1 1− λBextloscy (Φ0/2π) 4πLT lIc tan (πΦext/Φ0) sec (πΦext/Φ0) . (17) From Eq. (17), we see that for motion occuring on the much longer timescale ω−1m ≫ ω−1T , the mechanical oscillator has the effect of modulating the frequency of the transmission line mode. The associated Hamiltonian is H (φ, y, pφ, py) = sin2 (k0l/2) p2φ + sin2 (k0l/2) 1− λBextloscy (Φ0/2π) 4πLT lIc tan (πΦext/Φ0) sec (πΦext/Φ0) mω2my 2. (18) Let us now quantize. For the transmission line mode coordinate, the raising(lowering) operator is defined as: â±T = CT l (Φ0/2π) sin2 (k0l/2) sin2 (k0l/2)ωT φ̂∓ ip̂φ and for the mechanical oscillator â±m = (mωŷ ∓ ip̂y) . (20) In terms of these operators, the Hamiltionian (18) becomes (for notational convenience we omit from now on the ‘hats’ on the operators and also the ‘minus’ superscript on the lowering operator): H = ~ωTa T aT + ~ωma mam + ~ωTKTm aT + a am + a , (21) where the dimensionless coupling parameter between the mechanical oscillator and trans- mission line mode is KTm = − λBextlosc∆xzp (Φ0/2π) 4πLT lIc tan (πΦext/Φ0) sec (πΦext/Φ0) , (22) with ∆xzp = ~/(2mωm) the zero-point uncertainty of the mechanical oscillator. From expression (10) for the effective inductance, another way to express the coupling parameter is as follows: KTm = − λBextlosc∆xzp (Φ0/2π) dLeff/dΦext . (23) From Eq. (23), we see that in order to increase the coupling between the mechanical oscillator and transmission line, the SQUID effective inductance-to-transmission line inductance ratio must be increased. The advantage of using a SQUID over an ordinary, geometrical mutual inductance between a transmission line and micron-sized mechanical oscillator is that the former can give a much larger effective inductance. As we shall see in Sec. IV, just requiring that the inductances be matched such that Φ0 dLeff/dΦext ∼ 1 is sufficient for strong back reaction effects with modest drive powers, even though the other term in KTm describing the flux induced for a zero-point displacement is typically very small. Assuming then that KTm ≪ 1 and making the rotating wave approximation (RWA) for the ‘T ’ part of the interaction term in the system Hamiltonian (21), i.e., neglecting the terms (aT ) 2 and (a+T ) 2, we have (up to an unimportant additive constant): H = ~ωTa T aT + ~ωma mam + ~ωTKTma am + a . (24) Many other systems are modeled by this form of Hamiltonian, a notable example being the single mode of an optical cavity interacting via radiation pressure with a mechanically compliant mirror.22,23,24,25,26 Thus, much of the subsequent analysis will be relevant to a broad class of coupled resonator devices–not to just the transmission line-SQUID-mechanical resonator system. B. Open system Heisenberg equations of motion So far, we have treated the transmission line and mechanical resonator as a closed system with SQUID-induced effective coupling . Of course, a real transmission line mode will expe- rience damping and accompanying fluctuations, not least because it must be coupled to the outside world in order for its state to be measured. Furthermore, the mechanical resonator mode will of course be damped even when decoupled from the SQUID. It is straightforward to incorporate the various baths and pump/probe feedline in terms of raising/lowering op- erators. Assuming weak system-bath couplings, which again justify the RWA, we have for the full Hamiltonian: H = ~ωTa T aT + ~ωma mam + ~ωTKTma am + a dωωa+p (ω)ap(ω) + ~ dωωa+e (ω)ae(ω) + ~ dωωa+b (ω)ab(ω) K∗pTa p (ω)aT +KpTa T ap(ω) K∗eTa e (ω)aT +KpTa T ae(ω) K∗bma b (ω)am +Kbma mab(ω) (am + a m)Fext(t), (25) where ap denotes the pump/probe (p) feed line operator, ae the transmission line bath (‘e’ for ‘environment’) operator, and ab the mechanical resonator bath (b) operator. These operators satisfy the usual canonical commutation relations: ai(ω), a = δijδ(ω − ω′). (26) The couplings between these baths and the transmission line and mechanical resonator systems are denoted as KpT , KeT , and Kbm. Note we have also included for generality a classical driving force Fext(t) acting on the mechanical resonator. This allows us the opportunity to later on analyze quantum limits on force detection in addition to displacement detection. Within the RWA, it is straightforward to solve the Heisenberg equations for the bath operators and substitute these solutions into the Heisenberg equations for the transmission line and mechanical oscillator to give = −iωmam + Fext(t)− iωTKTma+T aT dω |KTm|2 dt′e−iω(t−t ′)am(t ′)− i dωKbme −iω(t−t0)ab(ω, t0) (27) = −iωTaT − iωTKTmaT am + a dω |KpT |2 dt′e−iω(t−t ′)aT (t ′)− i dωKpTe −iω(t−t0)ap(ω, t0) dω |KeT |2 dt′e−iω(t−t ′)aT (t ′)− i dωKeTe −iω(t−t0)ae(ω, t0). (28) We now make the so-called ‘first Markov approximation’,34,35 in which the frequency dependences of the couplings to the baths are neglected: KpT (ω) = eiφpT KeT (ω) = eiφeT Kbm(ω) = eiφbm , (29) where the γ’s and φ’s are independent of ω as stated. The Heisenberg equations of motion (27) and (28) then simplify to = −iωmam + Fext(t)− iωTKTma+T aT −γbmam(t)− i 2γbme iφbmainb (t) (30) = −iωTaT − iωTKTmaT am + a −γpTaT (t)− i 2γpTe iφpT ainp (t) −γeTaT (t)− i 2γeTe iφeT aine (t), (31) where the γi’s are the various mode amplitude damping rates (assumed much smaller than their associated mode frequencies) and the ‘in’ operators10,31,34,35 are defined as aini (t) = dωe−iω(t−t0)ai(ω, t0), (32) with t > t0. The time t0 can be taken to be an instant in the distant past before the measurement commences and when the initial conditions are specified (see below). We can similarly define ‘out’ operators: aouti (t) = dωe−iω(t−t1)ai(ω, t1), (33) with t1 > t. The time t1 can be taken to be an instant in the distant future after the measure- ment has finished. From the Heisenberg equations for the bath operators and the definitions of the ‘in’ and ‘out’ operators, we obtain the following identities between them:34,35 aoutp (t)− ainp (t) = −i 2γpTe −iφpT aT (t) aoutb (t)− ainb (t) = −i 2γbme −iφbmam(t) aoute (t)− aine (t) = −i 2γeTe −iφeT aT (t). (34) In outline, the method of solution runs in principle as follows:31,34,35,36 (1) specify the ‘in’ operators. (2) Solve for the system operators am(t) and aT (t) in terms of the ‘in’ operators. (3) Use the relevant identity (34) to determine the ‘out’ operator aoutp (t), which yields the desired probe signal. It is more convenient to solve the Heisenberg equations in the frequency domain with the Fourier transformed operators O(t) = 1√ −∞ dωe −iωtO(ω). The equations for the system operators then become am(ω) = ω − ωm + iγbm 2γbme iφbmainb (ω)− 2m~ωm Fext(ω) ωTKTm aT (ω ′)a+T (ω ′ − ω) + a+T (ω ′)aT (ω + ω aT (ω) = ω − ωT + i(γpT + γeT ) 2γpTe iφpT ainp (ω) + 2γeTe iφeT aine (ω) ωTKTm√ dω′aT (ω am(ω − ω′) + a+m(ω′ − ω) , (36) while the relevant ‘in/out’ operator identity becomes aoutp (ω) = −i 2γpTe −iφpT aT (ω) + a p (ω). (37) C. Observables and ‘in’ states Before proceeding with the solution to Eqs. (35) and (36), let us first devote some time to deriving expressions for observables that we actually measure in terms of aoutp (ω). Model the pump/probe feedline as a semi-infinite transmission line −∞ < x < 0. Solving the wave equation for the decoupled transmission line and then using the expressions (1), (2) relating the current/voltage to the phase coordinate, we obtain Iout(x, t) = − sin (ωx/vp) e−iωtaoutp (ω) + e iωtaout+p (ω) V out(x, t) = i cos (ωx/vp) e−iωtaoutp (ω)− eiωtaout+p (ω) , (39) where the sinusoidal x dependence in the current expression follows from the vanishing of the current boundary condition at x = 0, the feedline impedance is Zp = Lp/Cp and the wave propagation velocity is vp = 1/ LpCp. Suppose the current/volt meter is at x → −∞, so that the actual observables correspond to measuring the left-propagating component of the current/voltage. Then decomposing the x-dependent trig terms into their real and imaginary parts, we can identify the left propagating current/voltage operators as Iout(x, t) = −i e−iω(x/vp+t) aoutp (ω)− aout+p (−ω) +eiω(x/vp+t) aoutp (−ω)− aout+p (ω) V out(x, t) = i e−iω(x/vp+t) aoutp (ω)− aout+p (−ω) +eiω(x/vp+t) aoutp (−ω)− aout+p (ω) . (41) The output signal of interest due to the mechanical oscillator signal input will lie within some bandwidth δω centered at ωs, the ‘signal’ frequency, and so we define the filtered output current Iout (x, t|ωs, δω) and voltage V out (x, t|ωs, δω) to be the same as the above, left-moving operators, but with the integration range instead restricted to the interval [ωs − δω/2, ωs + δω/2]. Since the motion of the mechanical resonator modulates the transmission line frequency, one way to transduce displacements is to measure the relative phase shift between the ‘in’ pump current and ‘out’ probe current using the homodyne detection procedure.35 Another common way is to measure the ‘out’ power relative to the ‘in’ power, or equivalently the mean-squared current/voltage (all three quantities differ by trivial factors of Zp). We will discuss the latter method of transduction; the former, homodyne method can be straight- forwardly addressed using similar techniques to those presented here. Thus, we consider the following expectation value: δIout (x, t|ωs, δω) Iout (x, t|ωs, δω) Iout (x, t|ωs, δω) , (42) where the angle brackets denote an ensemble average with respect to the ‘in’ states of the various baths and feedline (see below). If the mechanical oscillator is being driven by a classical external force whose fluctuations are invariant under time translations, i.e., 〈Fext(t)Fext(t′)〉 = C(t− t′), then the above, mean-squared current will be time-independent. Alternatively, if Fext(t) is, e.g., some deterministic, AC drive, then we must also time-average so as to get a time-independent measure of the detector response: [δIout (x, t|ωs, δω)]2 ∫ TM/2 −TM/2 Iout (x, t|ωs, δω) , (43) where TM is duration of the measurement, assumed much larger than all other timescales associated with the detector dynamics. We have also assumed that the time-averaged cur- rent vanishes in the signal bandwidth of interest: 〈Iout (ωs, δω)〉 = 0. Substituting in the expression (40) for Iout (x, t|ωs, δω) in terms of the aoutp operators, we obtain after some algebra: [δIout (ωs, δω)] ∫ ωs+δω/2 ωs−δω/2 dω1dω2 (ω1 − ω2) TM sin [(ω1 − ω2)TM/2] aoutp (ω1)a p (ω2) + a p (ω2)a p (ω1) . (44) As ‘in’ states, we suppose kBT ≪ ~ωT , such that the relevant transmission line ‘in’ bath modes (ωe ∼ ωT ) are assumed to be approximately in the vacuum state. On the other hand, with the mechanical mode typically at a much lower frequency ωm ≪ ωT , we assume that its relevant ‘in’ bath modes (ωb ∼ ωm) are in the proper, non-zero temperature thermal state. For the pump/probe feedline, we consider the following coherent state:30 |{α(ω)}〉p = exp dωα(ω) ain+p (ω)− ainp (ω) |0〉p , (45) where |0〉p is the vacuum state and α(ω) = −I0 e−(ω−ωp) , (46) normalized such that the amplitude of the expectation value of I in [the right propagating version of (40) with aoutp replaced by a p ] with respect to this state is just I0. Again, we suppose kBT ≪ ~ωp, so that thermal fluctuations of the feedline are neglected. The fre- quency width of this pump drive is assumed to be the inverse lifetime of the measurement. Below we shall see that the output mechanical signal will appear as two ‘satellite’ peaks on either side of the central peak at ωp due to the pump signal, i.e, the mechanical signal can be extracted by centering the filter at either of ωs = ωp ± ωm (up to a renormalization of the mechanical oscillator frequency), corresponding to the anti-Stokes and Stokes bands. Note that we do not have to specify the initial t0 states of the mechanical resonator and transmission line systems; aT (t0) and am(t0)-dependent initial transients have been dropped in the above equations for aT (ω) and am(ω), since they give a negligible contribution to the long-time, steady-state behavior of interest. III. SOLVING THE EQUATIONS OF MOTION A. Linear response approximation We are now ready to solve for [δIout] . Introduce the following shorthand notation: ST (ω) = 2γpTe iφpT ainp (ω) + 2γeTe iφeT aine (ω) Sm(ω) = 2γbme iφbmainb (ω)− 2m~ωm Fext(ω) ωTKTm√ , (47) and γT = γpT +γeT , the net transmission line mode amplitude dissipation rate due to loss via the probe line and the transmission line bath. Substituting Eq. (35) for am(ω) into Eq. (36) for aT (ω) yields the following, single equation in terms of aT (ω) only: aT (ω) = dω′aT (ω − ω′)A(ω, ω′) + dω′B(ω, ω′)aT (ω − ω′) aT (ω ′′)a+T (ω ′′ − ω′) + a+T (ω ′′)aT (ω ′′ + ω′) + C(ω), (48) where, for the convenience of subsequent calculations, we have made this equation as concise as possible with the following definitions: A(ω, ω′) = ω − ωT + iγT ω′ − ωm + iγbm S+m(−ω′) −ω′ − ωm − iγbm B(ω, ω′) = ω − ωT + iγT ω′ − ωm + iγbm −ω′ − ωm − iγbm C(ω) = ST (ω) ω − ωT + iγT . (49) We expand Eq. (48) for aT (ω) to first order in the mechanical oscillator bath operator ainb (ω) and external driving force Fext(ω) [equivalently expand in A(ω, ω ′)]: aT (ω) ≈ a(0)T (ω)+ T (ω), where T (ω) = dω′B(ω, ω′)a T (ω − ω ′′ − ω′) + a(0)+T (ω ′′ + ω′) + C(ω) (50) T (ω) = T (ω − ω ′)A(ω, ω′) + dω′B(ω, ω′)a T (ω − ω ′′ − ω′) + a(0)+T (ω ′′ + ω′) dω′B(ω, ω′)a T (ω − ω ′′ − ω′) ′′ + ω′) + a ′′ − ω′) ′′ + ω′) . (51) Eq. (50) then yields the detector noise, while (51) yields the detector response to the signal within the linear response approximation. Thus, our approach here is to treat the mechanical oscillator as part of the detector degrees of freedom, with the signal defined as the thermal bath fluctuations and classical external force acting on the oscillator. This is the appropriate viewpoint for force detection. On the other hand, if the focus is on measuring the quantum state of the mechanical oscillator itself, then the oscillator should not be included as part of the detector degrees of freedom. Nevertheless, as we shall later see, the latter viewpoint can be straightforwardly extracted from the former under not too strong coupling KTm and pump drive current amplitude I0 conditions. B. Semiclassical approximation The sequence of solution steps to Eqs. (50) and (51) are in principle as follows: (1) Solve first equation (50) for a T (ω) in terms of B(ω, ω ′) and C(ω); (2) Substitute the solution for T (ω) into Eq. (51) for a T (ω) and invert this Eq. (which is linear in a T (ω)) to obtain the solution for a T (ω) in terms of A(ω, ω ′), B(ω, ω′), and C(ω). It is not clear how to carry out these steps in practice, however, since the equations involve products of non- commuting operators. Thus, we must find some way to solve by further approximation. The key observation is that the feedline is in a coherent state, which is classical-like for sufficiently large current amplitude I0 so as to ensure signal amplification. We therefore decompose a T (ω) into a classical, expectation-valued part and quantum, operator-valued fluctuation part, a T (ω) = T (ω) T (ω), and subtitute into Eq. (50) for a T (ω), linearizing with respect to the quantum fluctuation δa T (ω). This gives two equations, one for the expectation value T (ω) dω′B(ω, ω′) T (ω − ω ′′ − ω′) ′′ + ω′) + 〈C(ω)〉 (52) and the other for the quantum fluctuation: T (ω) = dω′B(ω, ω′)δa T (ω − ω ′′ − ω′) ′′ + ω′) dω′B(ω, ω′) T (ω − ω ′′ − ω′) ′′ − ω′) ′′ + ω′) ′′ + ω′) + δC(ω). (53) Eq. (51) for a T (ω) is approximated by replacing a T (ω) with its expectation value T (ω) i.e., we drop the quantum fluctuation part δa T (ω). This is because Eq. (51) already depends linearly on the quantum fluctuating signal term A(ω, ω′), which we of course want to keep. Dropping the δa T (ω) contribution to Eq. (51) amounts to neglecting multiplicative detector noise, which is reasonable given that we are concerned with large signal amplification. C. Complete solution to detector signal response and noise The sequence of solutions steps are therefore in practice as follows: (1) Solve Eq. (52) first T (ω) ; (2) Substitute this solution into Eq. (51) for a T (ω) and invert; (3) Substitute the solution for T (ω) into the Eq. (53) for δa T (ω) and invert; (4) Use these solutions for T (ω) and δa T (ω) to determine the detector signal and noise terms, respectively. Beginning with step (1), we have 〈C(ω)〉 = − 2γpTe γT − i∆ω ainp (ω) 2γpTe γT − i∆ω e−(ω−ωp) /2, (54) where ∆ω = ωp−ωT is the detuning frequency (not to be confused with the bandwidth δω) and note aine (ω) = 0 (recall, we assume the transmission line resonant frequency ωT mode is in the vacuum state). Given that TM is the longest timescale in the system dynamics, 〈C(ω)〉 is sharply peaked about the frequency ωp and we will therefore approximate the exponential with a delta function: 〈C(ω)〉 = cδ(ω − ωp), where 2πeiφpT γT − i∆ω I20ZpγpT . (55) Considering for the moment an iterative solution to Eq. (52) for 〈a(0)T (ω)〉, we see that 〈a(0)T (ω)〉 must also have the form of a delta function peaked at ωp: 〈a T (ω)〉 = χδ(ω − ωp). Substituting this ansatz into Eq. (52), we obtain the following equation for χ: χ = 2χ |χ|2B(ωp, 0) + c. (56) This equation has a rather involved analytical solution. For sufficiently large |c|2 |B(ωp, 0)| the response can become bistable (i.e., two locally stable solutions for χ). This region will not be discussed in the present paper, however. When we consider actual device parameters later in Sec. IV, we will assume sufficiently small drive such that χ ≈ c, allowing much simpler analytical expressions to be written down for the detector response. Proceeding now to step (2), we substitute the expectation value T (ω) = χδ(ω − ωp) for the operator a T (ω) into Eq. (51) for a T (ω). Carrying out the integrals, we obtain 1− 2 |χ|2 [B(ω, 0) +B(ω, ω − ωp)] T (ω)− 2χ 2B(ω, ω − ωp)a(1)+T (2ωp − ω) = χA(ω, ω − ωp). (57) Before we can invert to obtain a T (ω), we require a second linearly independent equation also involving a T (2ωp−ω) and a T (ω). This equation can be obtained by replacing ω with 2ωp − ω in Eq. (57) and then taking the adjoint: 1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)] T (2ωp − ω) +2χ∗2B(ω − 2∆ω, ω − ωp)a(1)T (ω) = −χ ∗A(ω − 2∆ω, ω − ωp), (58) where we have used the identities A+(2ωp − ω, ωp − ω) = −A(ω − ∆ω, ω − ωp), B∗(2ωp − ω, ωp − ω) = −B(ω − 2∆ω, ω − ωp), and B∗(2ωp − ω, 0) = −B(ω − 2∆ω, 0). Inverting, we obtain T (ω) = α1(ω)A(ω, ω − ωp) + α2(ω)A(ω − 2∆ω, ω − ωp), (59) where α1(ω) = D(ω) −1{1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)] χ (60) α2(ω) = −2D(ω)−1 |χ|2B(ω, ω − ωp)χ, (61) with determinant D(ω) = 1− 2 |χ|2 [B(ω, 0) +B(ω, ω − ωp)] 1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)] +4 |χ|4B(ω, ω − ωp)B(ω − 2∆ω, ω − ωp). (62) Moving on now to step (3), we substitute the expectation value 〈a(0)T (ω)〉 = χδ(ω − ωp) into Eq. (53) for δa T (ω) and carry out the integrals to obtain: 1− 2 |χ|2 [B(ω, 0) +B(ω, ω − ωp)] T (ω)− 2χ 2B(ω, ω − ωp)δa(0)+T (2ωp − ω) = δC(ω). (63) Replacing ω with 2ωp − ω in Eq. (63) and then taking the adjoint: 1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)] T (2ωp − ω) +2χ∗2B(ω − 2∆ω, ω − ωp)δa(0)T (ω) = δC +(2ωp − ω). (64) Inverting Eqs. (63) and (64), we obtain T (ω) = β1(ω)δC(ω) + β2(ω)δC +(2ωp − ω), (65) where β1(ω) = D(ω) −1{1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)] β2(ω) = 2D(ω) −1χ2B(ω, ω − ωp) (67) We are now ready to carry out step (4). To obtain the detector response, we substi- tute into expression (44) for [δIout] the linear response approximation to the ‘out’ probe operator [see Eq. (37)]: aoutp (ω) = 2γpTe −iφpT a T (ω) 2γpTe −iφpT δa T (ω) + δa p (ω) . (68) The first square-bracketed term will give the signal contribution to the detector response, while the second bracketed term gives the noise contribution. Note that the average values T (ω) ainp (ω) are not required in the noise term since they give negligible contri- bution in the signal bandwidths of interest centered at ωs = ωp ± ωm. Substituting in the signal part of aoutp (ω), we obtain after some algebra: [δIout (ωs, δω)] signal I0KTmωT )2 γ2pT γ2T +∆ω ∫ ωs+δω/2 ωs−δω/2 (ω − ωp +∆ω)2 + γ2T α1(ω) α2(ω) ω − ωp +∆ω + iγT ω − ωp −∆ω + iγT (ω − ωp − ωm)2 + γ2bm [2n(ω − ωp) + 1] + (ωp − ω − ωm)2 + γ2bm [2n(ωp − ω) + 1] I0KTmωT )2 γ2pT γ2T +∆ω 2m~ωmγbm ∫ ωs+δω/2 ωs−δω/2 dωdω′ (ω − ωp +∆ω)2 + γ2T α1(ω) α2(ω) ω − ωp +∆ω + iγT ω − ωp −∆ω + iγT ×sin [(ω − ω ′)TM/2] (ω − ω′) TM/2 (ω − ωp − ωm)2 + γ2bm Fext(ω − ωp)F ∗ext(ω′ − ωp) (ωp − ω − ωm)2 + γ2bm Fext(ωp − ω)F ∗ext(ωp − ω′) , (69) where n(ω) = e~ω/kBT − 1 is the Bose-Einstein thermal occupation number average for bath mode ω. The signal part of the detector response comprises a thermal component and a classical force component. In the limit of weak coupling KTm → 0 and or small drive current amplitude I0 → 0, we have α1(ω)/c → 1, α2(ω)/c → 0 and we note that the frequency resolved detector response has the form of two Lorentzians centered at ωp ± ωm. The resulting expression for the detector response coincides with an O(K2Tm) perturbative solution to the detector response (44) via the linear response Eqs. (50) and (51) (but no semiclassical approximation). However, as shall be described in Sec. IV, when the current drive is not small and or coupling is not weak, then the αi terms will modify this simple form, at the next level of approximation renormalizing the Lorentzians, i.e., shifting their location and changing their width. Substituting in the noise part of aoutp (ω), we obtain after some algebra: [δIout (ωs, δω)] noise = Z−1p ∫ ωs+δω/2 ωs−δω/2 2γTγpT (ω − ωp +∆ω)2 + γ2T |β1(ω)|2 + (ω − ωp +∆ω)2 + γ2T (ω − ωp −∆ω)2 + γ2T |β2(ω)|2 − Re [β1(ω)] + (ω − ωp +∆ω) Im [β1(ω)] +Z−1p . (70) The noise part of the detector response comprises a back reaction component (the integral term) where transmission line noise drives the mechanical oscillator via the SQUID coupling, and a component that is added at the output due to zero-point fluctuations in the probe line. While not as obvious given the form of Eq. (70), one may again verify (see Sec. IV) that the detector back reaction on the mechanical oscillator takes the form of two Lorentzians centered at ωp ± ωm in the weak coupling and or weak current drive limit, coinciding with an O(K2Tm) perturbative calculation. Eqs. (69) and (70) are the main results of the paper, their sum giving the net output mean-squared current. D. Quantum bound on noise As articulated by Caves,10 the fact that the ‘in’ and ‘out’ operators satisfy canonical commutation relations places a lower, quantum limit on the noise contribution to the detector response, Eq. (70). We now derive this quantum limit. First write the ‘out’ operator (68) aoutp (ω) = −i 2γpTe −iφpT a T (ω) +N(ω), (71) where N(ω) = −i 2γpTe −iφpT δa T (ω) + δa p (ω) is the noise part. Taking commutators, we have the following identity relating the noise and signal operator terms: N(ω), N+(ω′) = δ(ω − ω′)− 2γpT T (ω), a . (72) Now, from the Heisenberg Uncertainty Principle, one can derive the following general in- equality: N [f ]N+[f ] +N+[f ]N [f ] N [f ], N+[f ] ∣ , (73) where N [f ] = dωf(ω)N(ω) and f(ω) is an arbitrary function. Inserting the commutator identity (72), Eq. (73) becomes N [f ]N+[f ] +N+[f ]N [f ] dω |f(ω)|2 − 2γpT T [f ], a T [f ] . (74) Choosing the ‘filter’ function f(ω) = ωΘ(ω − ωs + δω/2)Θ(ωs + δω/2 − ω) and evaluating the commutator, we obtain the following lower bound on the detector noise: [δIout (ωs, δω)] noise I0KTmωT )2 γ2pT γ2T +∆ω ∫ ωs+δω/2 ωs−δω/2 (ω − ωp +∆ω)2 + γ2T α1(ω) α2(ω) ω − ωp +∆ω + iγT ω − ωp −∆ω + iγT (ω − ωp − ωm)2 + γ2bm − 2γbm (ωp − ω − ωm)2 + γ2bm . (75) In the next section we will address the extent to which the detector noise can approach the quantum bound on the right hand side of Eq. (75), depending on the current drive amplitude I0 and other detector parameters. IV. RESULTS A. Analytical approximations To gain a better understanding of the detector response, we now provide analytical ap- proximations to Eqs. (69) and (70) that are valid under the condition |c|2 |B(ωp, 0)| ≪ 1 such that χ ≈ c [see Eq. (56)], i.e., the expectation value T (ω) for the transmission line depends approximately only on the pump/probe feedline state and not on the mechanical oscillator state. Explicitly, this condition reads: 2I20ZpK TmωTγpT ~ωm (γ T +∆ω ≪ 1, (76) placing an upper limit on I0 and KTm for the validity of this approximation. We also assume that the mechanical and transmission line mode frequencies are widely separated: ωm ≪ ωT , and with small damping rates: γbm ≪ ωm, γT ≪ ωT . We do not restrict the relative magnitudes of ωm and γT , however. A simple picture emerges in which the detector back reaction ‘renormalizes’ the mechanical oscillator frequency and damping rate: ωm → Rωωm and γbm → Rγγbm, where Rωωm = ωm + |c|2ω2TK2Tm |c|2ω2TK2Tm [γ2T +∆ω2 − ω2m] γ2T + (∆ω + ωm) γ2T + (∆ω − ωm) ] (77) Rγγbm = γbm − |c|2ω2TK2Tm 2|c|2ω2TK2TmωmγT γ2T + (∆ω + ωm) γ2T + (∆ω − ωm) ] , (78) where c is defined in Eq. (55). With the measurement filter bandwidth centered at either of ωs = ωp±Rωωm, the approximation to Eq. (69) for the signal response is (with the classical force term omitted): [δIout (ωs = ωp ±Rωωm, δω)]2 signal I0KTmωT )2 γ2pT γ2T +∆ω γ2T + (∆ω ± ωm) ∫ ωs+δω/2 ωs−δω/2 (ω − ωp ∓Rωωm)2 + (Rγγbm)2 [2n(Rωωm) + 1] . (79) When there is a classical force acting on the mechanical oscillator, we must add to Eq. (79) the term I0KTmωT )2 γ2pT γ2T +∆ω 2m~ωmγbm ∫ ωs+δω/2 ωs−δω/2 dωdω′ (ω − ωp +∆ω)2 + γ2T ×sin [(ω − ω ′)TM/2] (ω − ω′)TM/2 (ω − ωp − Rωωm)2 + (Rγγbm)2 Fext(ω − ωp)F ∗ext(ω′ − ωp) (ωp − ω − Rωωm)2 + (Rγγbm)2 Fext(ωp − ω)F ∗ext(ωp − ω′) . (80) The approximation to Eq. (70) for the detector noise is [δIout (ωs = ωp ±Rωωm, δω)]2 noise I0KTmωT )2 γ2pT γ2T +∆ω γ2T + (∆ω ± ωm) ∫ ωs+δω/2 ωs−δω/2 (ω − ωp ∓ Rωωm)2 + (Rγγbm)2 N± + Z , (81) where the back reaction noise parameter is |c|2K2Tmω2TγT γ2T + (∆ω ∓ ωm) ] ∓ 1 = 2I TmωTγTγpT ~γbm [γ T +∆ω γ2T + (∆ω ∓ ωm) ] ∓ 1. (82) The ∓1 term in the back reaction noise parameter depends on whether the filter is centered at ωs = ωp + ωm or ωs = ωp − ωm and corresponds respectively to ‘phase preserving’ or ‘phase conjugating’ detection as discussed in Caves.10 In the limit I0 → 0 and or KTm → 0, we see from Eqs. (79), (81), and (82) that the back reaction noise amounts to doubling the oscillator quantum zero-point motion signal in the phase conjugating case, while the back reaction noise exactly cancels the quantum zero-point motion signal in the phase preserving case. In both cases, the noise coincides with the lower quantum bound (75). However, in this small drive/coupling limit, we do not have a detector or amplifier but rather an attenuator, which is of only academic interest to us. Comparing the detector response (79) and back reaction part of Eq. (81), we see that the mechanical oscillator behaves in the steady state as if in contact with a thermal bath.8,9,12,26,37,38,39 The back reaction of the detector on the mechanical oscillator is effec- tively that of a thermal bath with damping rate γback = γbm(Rγ − 1) and effective thermal average occupation number nback defined as follows: γback(2n back + 1) = γbmN±. (83) Thus, n±back = (Rγ − 1) . (84) The failure to approach the lower quantum bound (75) when N± ≫ 1 then translates into having (2n±back + 1)γback/γbm ≫ 1. Thus, to get close to the bound, we necessarily require γback ≪ γbm;12 the back reaction occupation number n±back does not have to be small. With the mechanical oscillator also in thermal contact with its external bath, the net damping rate of the oscillator is γnet = γbm + γback = Rγγbm and the net, effective thermal average occupation number nnet of the oscillator is defined as follows: 2n±net + 1 = γbm [2n(Rωωm) + 1] + γback 2n±back + 1 . (85) Thus, n±net = R n(Rωωm) + . (86) From Eq. (78), we see that depending on the detuning parameter ∆ω = ωp − ωT , the damping rate of the oscillator due to the detector back reaction can be either negative or positive. Specifically, positive damping requires the following condition on the detuning parameter: ∆ω < − |c|2ω2TK2Tm 2I20ZpK TmωTγpT ~ωm (γ T +∆ω . (87) B. Displacement sensitivity In the absence of a classical force acting on the mechanical oscillator, from Eq. (79) the mechanical oscillator thermal noise displacement signal spectral density takes the familiar Lorentzian form: Sx(ω)|signal = 2Rγγbm (ω − ωp ∓Rωωm)2 + (Rγγbm)2 2mRωωm [2n (Rωωm) + 1] . (88) In order to be able to resolve this mechanical signal, the detector noise (81) referred to the mechanical oscillator input must be smaller than (88). The detector noise spectral density at the input is Sx (ω = ωp ±Rωωm)|noise = Rγγbm ∓1 + |c| 2K2Tmω γ2T + (∆ω ∓ ωm) γ2T + (∆ω ± ωm) |c|2K2Tmω2TγpT 2mRωωm , (89) where the first term on the right hand side is the back reaction noise acting on the mechanical oscillator and the second term is the output, probe line zero-point noise referred to the input. Note that the noise has been evaluated at ω = ωp±Rωωm, the maximum of the back reaction Lorentzian. If the detector output is to depend linearly on the mechanical oscillator signal input (i.e., function as a linear amplifier), then back reaction effects must be small. In particular, we require that γback ≪ γbm, i.e., Rγ ≈ 1. With |c| being proportional I0, we see from Eq. (89) that increasing the drive current amplitude I0 increases the back reaction noise, but decreases the probe line noise referred to the mechanical oscillator input. Thus, there is an optimum I0 such that the sum Sx|noise is a minimum. Making the approximation Rγ = 1 and Rω = 1 in Eq. (89) and optimizing with respect to |c|, we find Sx (ω = ωp ± Rωωm)|noise−optimum = mωmγbm ∓1 + 2 γ2T + (∆ω ± ωm) γ2T + (∆ω ∓ ωm) . (90) From Eq. (90), we see that the noise is further reduced if (i) the dominant source of trans- mission line mode dissipation is due to energy loss through the coupled probe (information gathering) line:12 γT ≈ γpT ; (ii) the detuning frequency is chosen to be ∆ω = ∓ γ2T + ω where the minus (plus) sign corresponds to phase preserving (conjugating) detection. With this detuning choice, the condition Rγ ≈ 1 requires (ωm/γT )2 ≪ 1 and so the minimum detector noise is Sx (ω = ωp ± Rωωm)|noise−optimum = mωmγbm 2∓ 1 +O (ωm/γT ) , (91) where in order to determine the O ((ωm/γT ) 2) term, the full form of Rγ given in Eq. (78) must be used in Eq. (89) when optimizing. Comparing with Eq. (88) for the signal noise, we see that to leading order the detector noise effectively doubles the zero-point signal in 1 1.5 2 2.5 3 3.5 4 I0 H10 -8 AL FIG. 2: Displacement detector noise spectral density (solid line) and lower bound (dashed line) versus drive current amplitude. The noise densities are evaluated at ω = ωp+Rωωm, corresponding to phase preserving detection. the phase preserving case. This exceeds the lower bound on the detector noise derived from Eq. (75), which is zero to leading order in the phase preserving case. We now numerically evaluate Eq. (89) for the detector noise. The feasible example parameter values we use are:14 Bext = 0.005 Tesla, Zp = 50 Ohms, ωT/2π = 3×109 s−1, QT = ωT/(2γT ) = 100, γT = 9.4×107 s−1, losc = 5 µm, λ = 1 (geometrical correction factor), m = 10−16 kg, ωm = 2.5× 107 s−1, and Qbm = ωm/(2γbm) = 103. These values give a mechanical oscillator zero-point uncertainty ∆xzp = 1.45 × 10−13 m, a zero-point displacement noise ~/(mωmγbm) = 3.4 × 10−30 m2/Hz, and a dimensionless coupling strength KTm = −1.1 × 10−5, where we assume that in the expression (22) for KTm, Φext can be chosen such that the dimensionless factor Φ0 4πLT lIc tan (πΦext/Φ0) sec (πΦext/Φ0) ≈ 1 (matching condition). We also suppose that γT ≈ γpT , i.e., the transmission line mode damping is largely due to the probe line coupling. Fig. 2 shows Sx (ω = ωp +Rωωm)|noise × mωmγbm/~ and also the lower bound on the detector noise that follows from Eq. (75) for phase preserving detection. Note that the minimum detector noise is approximately 0.8 ~/(mωmγbm). Thus, for this example, the next-to-leading O ((ωm/γT ) 2) term in Eq. (91) is approximately −0.2. Note also that the detector noise coincides with the lower bound in the small drive limit. C. Force sensitivity Consider a monochromatic classical driving force with frequency ω0 ∼ Rωωm acting on the oscillator: Fext(ω) = F0δ(ω−ω0). The force signal spectral density is then SF (ω)|signal = F 20 δ(ω − ω0). For force detection operation, the mechanical oscillator is included as part of the detector degrees of freedom. From Eqs. (79-82), the force noise spectral density evaluated at ω = ωp ± ω0 is SF (ω = ωp ± ω0)|noise = 2m~ωmγbm 2n (ω0) + 1∓ 1 + |c|2K2Tmω2TγT γ2T + (∆ω ∓ ωm) (ω0 −Rωωm)2 + (Rγγbm)2 γ2T + (∆ω ± ωm) γbm|c|2K2Tmω2TγpT . (92) Comparing the displacement noise (89) with the force noise (92), we see that the latter includes the additional 2m~ωmγbm [2n (ω0) + 1] mechanical quantum thermal displacement noise term. Since the mechanical oscillator forms part of the force detector, it need not necessarily be weakly driven and or weakly coupled to the transmission line; as explained in Sec. IIIA, the present analysis employs a linear response approximation for force detection, not displacement detection. Thus, in determining the optimum I0 (and or KTm) and ∆ω such that SF |noise is a minimum, we should not assume a priori the restrictions Rγ, Rω ≈ 1. 2 4 6 8 10 12 14 I0 H10 -8 AL FIG. 3: Force detector noise spectral density versus drive current amplitude for detuning ∆ω = 0 (solid line), ∆ω = −5ωm (dashed line), and ∆ω = −10ωm (dotted line) . The noise densities are evaluated at ω = ωp +Rωωm, corresponding to phase preserving detection. Fig. 3 shows the results of numerically evaluating the force noise spectral density given by Eq. (92) for phase preserving detection (ω = ωp + ω0) and a range of detuning values. The same example parameters are used as in the above displacement sensitivity analysis, with n(ω0) = 0 and ω0 = Rωωm. The force noise is expressed in units 2m~ωmγbm = 6.6 × 10−39 N2/Hz. Note that the minimum force noise is exactly 2 in these units, independently of the detuning, with the minimum occuring at larger I0 values as the detuning is made progressively more negative. D. Back reaction cooling From Eq. (86), we see that the net, thermal average occupation number nnet of the mechanical oscillator’s fundamental mode decreases as Rγ increases. Thus, by increasing the drive and or coupling strength such that γback ≫ γbm, the mechanical oscillator can be effectively cooled at the expense of increasing its damping rate.3,8,9,21,26,40,41,42,43,44,45,46,47,48 Consider sufficiently negative detuning such that −∆ω ≫ |c|2ω2TK2Tm/(πωm) [see Eq. (87)]. Substituting definition (78) for Rγ and definition (82) for N+ into Eq. (86) and supposing Rγ is large enough that we can neglect the external damping term γbm, we obtain approximately for the phase preserving case: n+net ≈ n (Rωωm) + n+back, (93) where n+back ≈ − γ2T + (∆ω + ωm) 4∆ωωm . (94) This expression agrees with that derived in Ref. 26, apart from the 1/2 which is simply due to a small difference in the way we define n±back in Eq. (83). Choosing optimum detuning ∆ω = − γ2T + ω m to minimize n back in Eq. (94), we therefore have n+net ≈ n (Rωωm) 1 + (γT/ωm) 2 − 1. (95) How much cooling can be achieved depends on (i) how large Rγ can be, subject to the above inequality on −∆ω; (ii) making the ratio γT/ωm as small as possible.26 Using the same example parameter values as above, but taking instead a larger but still realistic quality factor Qbm = 10 4 for the mechanical oscillator,6 the resulting numerically evaluated effective occupation number n+net [Eq. (86)] is given in Fig. 4 for a range of external 5 10 15 20 25 30 35 40 I0 H10 -8 AL FIG. 4: Net effective average occupation number of the mechanical oscillator versus drive current. The solid curve is for external bath temperature T = 100 mK [n(Rωω) = 523], the dashed curve is for T = 10 mK [n(Rωω) = 52], and the dotted curve is for T = 1 mK [n(Rωω) = 4.8]. bath occupation numbers n(ωm). Thus, even for small coupling strengths KTm and drive current amplitudes I0, significant cooling of the mechanical oscillator can be achieved. This is in part a consequence of the fact that the quality factor Qbm of the mechanical oscillator when decoupled from the detector is very large. V. CONCLUDING REMARKS In the present paper, we have attempted to give a reasonably comprehensive analysis of the quantum-limited detection sensitivity of a DC SQUID for drive currents well below the Josephson junction critical current Ic. In this regime, the SQUID functions effectively as a mechanical position-dependent inductance element to a good approximation and the resulting closed system Hamiltonian (24) takes the same form as that for several other types of coupled mechanical resonator-detector resonator systems. Thus, the key derived expressions (69) and (70) for the detector response and detector noise are of more general application. The main approximation made in analyzing the position and force detection sensitivity, as well as back reaction cooling, was to limit the drive current and or coupling strength according to Eq. (76). This allowed us to find much simpler, analytical approximations to the key expressions, in particular Eqs. (79) and (81). The regime of larger drive currents and or coupling strengths which exceed the limit (76) remains to be explored. However, with the SQUID in mind, it is more appropriate to consider larger drive currents in the context of including the non-linear I/Ic corrections to the SQUID effective inductance. This will be the subject of a forthcoming paper.19 Acknowledgements We thank A. Armour, A. Rimberg, and P. Nation for helpful discussions. This work was partly supported by the US-Israel Binational Science Foundation (BSF) and by the National Science Foundation (NSF) under NIRT grant CMS-0404031. One of the authors (M.P.B.) thanks the Aspen Center for Physics for their hospitality and support where part of this work was carried out. 1 R. Knobel and A. N. Cleland, Nature 424, 291 (2003). 2 M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, Science 304, 74 (2004). 3 A. 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Corbitt, Y. Chen, E. Innerhofer, H. Muller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, quant-ph/0612188 (unpublished). Introduction Equations of Motion Transmission line-SQUID-mechanical oscillator Hamiltonian Open system Heisenberg equations of motion Observables and `in' states Solving the equations of motion Linear response approximation Semiclassical approximation Complete solution to detector signal response and noise Quantum bound on noise Results Analytical approximations Displacement sensitivity Force sensitivity Back reaction cooling Concluding remarks Acknowledgements References

0704.0458

Rotation Measures of Extragalactic Sources Behind the Southern Galactic Plane: New Insights into the Large-Scale Magnetic Field of the Inner Milky Way

Rotation Measures of Extragalactic Sources Behind the Southern Galactic Plane: New Insights into the Large-Scale Magnetic Field of the Inner Milky Way J. C. Brown,1 M. Haverkorn,2,3 B. M. Gaensler,4,5,6 A. R. Taylor,1 N. S. Bizunok,5 N. M. McClure-Griffiths,7 J. M. Dickey,8 and A. J. Green4 ABSTRACT We present new Faraday rotation measures (RMs) for 148 extragalactic radio sources behind the southern Galactic plane (253◦ ≤ ℓ ≤ 356◦, |b| ≤ 1.5◦), and use these data in combination with published data to probe the large-scale structure of the Milky Way’s magnetic field. We show that the magnitudes of these RMs oscillate with longitude in a manner that correlates with the locations of the Galactic spiral arms. The observed pattern in RMs requires the presence of at least one large-scale magnetic reversal in the fourth Galactic quadrant, located between the Sagittarius-Carina and Scutum-Crux spiral arms. To quantitatively compare our measurements to other recent studies, we consider all available extragalactic and pulsar RMs in the region we have surveyed, and jointly fit these data to simple models in which the large-scale field follows the spiral arms. In the best-fitting model, the magnetic field in the fourth Galactic quadrant is directed clockwise in the Sagittarius-Carina spiral arm (as viewed from the North Galactic pole), but is oriented counter-clockwise in the Scutum-Crux arm. This contrasts with recent analyses of pulsar RMs alone, in which the fourth-quadrant field was presumed to be directed counter-clockwise in the Sagittarius-Carina arm. Also in contrast to recent pulsar RM studies, our joint modeling of pulsar and extragalactic RMs demonstrates that large numbers of large-scale magnetic field reversals are not required to account for observations. Subject headings: Galaxy: structure — ISM: magnetic fields — polarization 1Department of Physics and Astronomy, University of Calgary, T2N 1N4, Canada; [email protected]; [email protected] 2Jansky Fellow, National Radio Astronomy Observatory 3Astronomy Department, UC-Berkeley, 601 Campbell Hall, Berkeley, CA 94720; [email protected] 4School of Physics A29, The University of Syd- ney, NSW 2006, Australia; [email protected]; [email protected] 5Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 01238; [email protected] 6Alfred P. Sloan Research Fellow, Australian Research Council Federation Fellow 7Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia; Naomi.McClure- [email protected] 8Physics Department, University of Tasma- nia, Private Bag 21, Hobart TAS 7001, Australia; 1. Introduction The Galactic magnetic field is now recognized as a fundamental component of the interstellar medium, and plays a critical role in the formation and evolution of structures in the Milky Way. An important prediction in models of the large-scale magnetic field, in both the Milky Way and in other galaxies, is the existence of magnetic field reversals (regions of magnetic shear across which the field changes direction by roughly 180◦). Determining the number and location of these magnetic rever- sals is essential to understanding Galactic evolu- tion (Shukurov 2005). While the majority of the recent studies suggest several large-scale reversals in the Galaxy along its radius, it is interesting to [email protected] http://arxiv.org/abs/0704.0458v1 note that most other galaxies exhibit either one re- versal or none (Beck 2007). Is our Galaxy unique this way, or is it simply a difference in observing methods? The large-scale Galactic magnetic field is con- centrated in the disk, and is most often studied via observations of rotation measure (RM), the mea- surable consequence of Faraday rotation. For a source that emits linearly polarised radiation at an angle φ◦, the received radiation will have a po- larisation angle at a wavelength, λ [m] , given by: φ = φ◦ + λ 2 0.812 neB · dl = φ◦ + λ 2 RM (1) where ne [cm −3] is the electron density andB [µG] is the magnetic field along the propagation path dl [pc]. The RM integral is over the path from the source to the observer. For more than four decades, the RMs of both pulsars and extragalactic radio sources (EGS) have been used to probe the Galactic magnetic field. This has led to a series of clear conclu- sions. First, in the local spiral arm, the field is unquestionably directed clockwise (CW), as viewed from the North Galactic pole (Manchester 1972, 1974; Heiles 1996a). The field in the first quadrant (Q1; 0◦ ≤ ℓ ≤ 90◦; see Figure 1) of the Sagittarius-Carina spiral arm is reversed rel- ative to the local arm (Thomson & Nelson 1980; Simard-Normandin & Kronberg 1980; Lyne & Smith 1989), implying a large-scale field reversal be- tween the local arm and the Q1 component of the Sagittarius-Carina arm. Some evidence suggests that this reversed field extends into the fourth- quadrant (Q4; 270◦ ≤ ℓ ≤ 360◦) component of the Sagittarius-Carina arm (eg. Rand & Lyne 1994; Han et al. 1999; Frick et al. 2001). At larger distances, the existence of other large- scale reversals in the Galaxy remains unclear. Ideally, reconstruction of the Galactic magnetic field should utilize information from both pulsars and extragalactic sources. However, until recently there have been very few EGS RMs available at low-latitude that might be used to study the field in the disk. Studies that have utilized pulsar RMs alone are constrained by the comparatively sparse sampling of pulsars on the sky, which can make it difficult to map the field in complicated regions. Three recent pulsar RM studies are as follows. Using pulsar RMs, Weisberg et al. (2004) inves- tigated the existence and location of reversals in Q1, concluding that a reversal occurs between each arm so that the magnetic fields in adjacent arms are oppositely directed. While the evidence pre- sented by Weisberg et al. (2004) for a reversal between the local arm and the Sagittarius-Carina arm in Q1 is indisputable, their evidence for addi- tional reversals is based on limited data, and they acknowledge that the evidence for a reversal into the Q4-component of the Sagittarius-Carina arm is not well-defined. Vallée (2005) investigated azimuthal field con- figurations in the Galaxy using pulsar RMs aver- aged in concentric rings. He concluded that best- fit for this model was an overall clockwise mag- netic field with a 2 kpc wide counter-clockwise (CCW) ring, located between 4 and 6 kpc from the Galactic center (note that a non-standard Solar-circle radius of 7.2 kpc was assumed in this study). In this model, the Galactic field has two large-scale reversals, and the Q1 component of the Sagittarius-Carina arm is CCW, while the Q4 component is CW. Using 223 new pulsar RM observations primar- ily in the fourth quadrant, in conjunction with pre- vious pulsar RMs, Han et al. (2006) concluded there is a reversal at every arm-interarm boundary, so that the fields in the arms are directed CCW, and the interarm regions are directed CW. A po- tential inconsistency with this model is that the majority of the pulsars distributed along the Q4 component of the Sagittarius-Carina spiral arm have what Han et al. (2006) describe as ‘unex- pectedly positive’ RMs, which they suggest is the influence of HII regions along the lines-of-sight to the affected pulsars (see also Mitra et al. 2003). Recent surveys of the Galactic plane at high resolution and at multiple wavelengths have as- sisted greatly in the study of the Galactic mag- netic field by addressing the previous paucity of low-latitude EGS RM data (eg. Brown & Taylor 2001). One such survey is the Southern Galac- tic Plane Survey (SGPS; McClure-Griffiths et al. 2005; Haverkorn et al. 2006b). The SGPS images low latitudes in the third and fourth quadrants of the Galaxy, complementing the Canadian Galactic Plane Survey in the northern hemisphere (CGPS; Taylor et al. 2003). Rotation measures calculated from these data were used by Haverkorn et al. (2006a) to explore the structure of the small-scale field. Here, we present the tabulation of these RMs and use them to examine the structure of the large-scale field. 2. Rotation Measure Calculations The initial set of observations for the SGPS (ie. Phase I) spans an area of 253◦ ≤ ℓ ≤ 358◦ and |b| ≤ 1.5◦. The observations were done with the Australia Telescope Compact Ar- ray (ATCA) in New South Wales, Australia. For details about the SGPS observations and polari- metric data reduction, see Gaensler et al. (2001) and Haverkorn et al. (2006b). RMs were calculated using twelve separate 8 MHz bands centered on frequencies between 1336 MHz and 1432 MHz. The proximity of the 8 MHz bands allows for unambiguous RM calcula- tions using the algorithm designed by Brown et al. (2003b) for the CGPS, with appropriate modifica- tions for the ATCA. Specifically, an RM calcula- tion was considered reliable if the source was suf- ficiently polarized (>0.3%), had sufficient signal to noise (>2) across at least 3 pixels, was Fara- day thin (Vallée 1980), and had a consistent value across the source. Of the 215 polarised source candidates identi- fied, 148 sources had RMs that successfully passed the screening tests discussed above. These new data are given in Table 1. Three of our 148 sources had a RM determined by Gaensler et al. (2001), as part of the SGPS test region (325.5◦ ≤ ℓ ≤ 332.5◦,−0.5 ≤ b ≤ 3.5◦). The previously deter- mined values of these sources are within the er- rors of the new values quoted in Table 1. These sources are indicated by footnote marker d. Two additional test region RM sources fall within the latitude range of the SGPS but they lie within the noise perimeter of our data (see Gaensler et al. 2001) and were not observable. There is one other previously observed RM from an EGS, reported by Broten et al. (1988), that falls within our field at ℓ = 307.1◦, b = 1.2◦. This source is resolved in the SGPS and exhibits a large gradient across the source. Consequently, it failed the screening, though its calculated RM (+183± 23 rad m−2) is in agreement with the previously determined value (+185± 1 rad m−2). 3. Observations of the Galactic Magnetic Field Figure 2 shows the RMs of EGS and pulsars in the SGPS region. Most of the pulsars (99 of 120) are from Han et al. (2006). The remaining are from Taylor et al. (2000) and Han et al. (1999). The most striking feature of Figure 2 is the change in sign of RM from predominantly positive at low longitudes to predominantly negative at high lon- gitudes for both the pulsars and EGS at ℓ ∼ 304◦, though it is more prominent for the EGS. A change in sign of the overall trend in RM can only come from a change of direction in the dominant line- of-sight component of the magnetic field. The fact that the RM sign change is abrupt indicates that this directional change is the result of a physi- cal field reversal, rather than a viewing angle ef- fect such as that observed towards ℓ ∼ 180◦ (eg. Brown & Taylor 2001). Furthermore, the abrupt- ness indicates a thin current sheet and an associ- ated large gradient in the magnetic field (Heiles 1996b). By averaging the EGS RMs in ℓ across the sky to reduce small-scale variations (for example, due to the small-scale field or intrinsic effects from the EGS themselves), we can obtain more information about the large-scale field. The top panel in Fig- ure 3 shows the RM data from the SGPS, both raw and binned plotted as a function of Galac- tic longitude. The middle panel shows these data smoothed. In all three presentations of the EGS data, an oscillating pattern of RM with longitude is visible. The transition from positive to nega- tive RMs remains at ℓ ∼ 304◦, indicated by the solid vertical line. In the bottom panel of Figure 3, we plot the individual pulsar RMs as a function of Galactic longitude. These data were not aver- aged in ℓ because of the variation and uncertainty in the pulsar distances. In spite of this, there are features seen here similar to the oscillatory pattern observed in the EGS data. The dashed and dotted lines in the middle panel of Figure 3 are the approximate longitudes of |RM| maxima and minima respectively for the SGPS data. In Figure 1, we show these lines as lines-of- sight overlaid on a top-down view of the Galaxy where the grey scale is the Galactic electron den- sity model of Cordes & Lazio (2002), hereafter CL02. Interestingly, the blue dashed lines (|RM| maxima) tend to have the longest continuous frac- tion of their length along a spiral arm or through the central annulus put in the CL02 to correspond to the molecular ring. This is consistent with the expectation that EGS RMs should be dominated by the spiral arms as a result of the higher electron densities in these regions (Han et al. 2006). The strong positive RMs in the longitude range around ℓ ∼ 292◦, seen in both the EGS and pulsars, sug- gest the magnetic field in the Q4-component of the Sagittarius-Carina arm is directed CW. The same conclusion was reached by Caswell et al. (2004) using data from a distant supernova remnant. This CW field presents a simple explanation for the positive pulsar RMs identified by Han et al. (2006) as part of a ‘Carina anomaly’. Similarly, the strong negative RMs in the lon- gitude range around ℓ ∼ 312◦ suggest the mag- netic field in the Q4-component of the Scutum– Crux arm is directed CCW. Therefore, a magnetic field reversal must reside between the Sagittarius– Carina arm and the Scutum–Crux arm in Q4. Fur- thermore, the strong evidence for a field reversal between the local and Sagittarius–Carina arms in Q1 suggests the reversal must slice through the Sagittarius–Carina arm if this reversal is contin- uous between Q1 and Q4, such as in the model proposed by Vallée (2005). The subsequently im- plied lack of alignment of the magnetic field with the Sagittarius-Carina arm is consistent with ear- lier observations of at least a 5◦ offset in pitch an- gle between the orientation of the local field and that of the local spiral arm (Beck 2007). 4. Modeling the Large-Scale Magnetic Field As a separate approach to qualitative obser- vational analysis of the data, we can fit global magnetic field models to the RM data (with as- sumptions regarding the electron distribution in the Galaxy), to explore the structure of the uni- form field and the location or existence of mag- netic field reversals. Here we use the CL02 electron density model and the technique of Brown (2002) which uses linear inversion theory (Menke 1984) to obtain a least-squares fit to the data. With this method, the boundaries of magnetic field regions are fixed a priori but the strength and direction of the field within these regions are not. The model is fit to the observed RMs to derive the strength and direction within each region. This contrasts previ- ous analyses which also employed separate models for the electron density and magnetic field, but where the direction of the field was an input to the model (e.g. Indrani & Despande 1998). We present here a simple model where we con- sider nine magnetic field regions, eight of which are either arm or interarm regions, and the ninth corresponds to the molecular ring. This model does not include the differing pitch angle between the magnetic and spiral arms discussed in section 3, but is instead designed to be directly compa- rable with the results of Weisberg et al. (2004) and Han et al. (2006). The region boundaries are delineated by the green spirals in Figure 4. The constraints for the model field within each arm or interarm region are: 1) a log-spiral with a pitch angle of 11.5◦ as shown; 2) a field strength that is inversely proportional to Galactic radius (eg. Heiles 1996b; Beck 2001; Brown et al. 2003a); 3) a zero vertical component; 4) a coherent di- rection within each region. We set the magnetic field within the molecular ring to have the same constraints as in the arms, except that the field is assumed to be purely azimuthal. At Galactic radii less than 3 kpc or greater than 20 kpc, or at more than 1 kpc from the mid-plane, the field is set to zero. We constrain the model using RM data from individual sources only within the SGPS region (120 pulsars, 148 SGPS EGS, and 1 EGS from Broten et. al 1988; see sections 2 and 3).1 For modeling purposes, we assume that the inter- galactic contribution to the EGS RMs is neg- ligible (Simard-Normandin & Kronberg 1980; Gaensler et al. 2005), so that the EGS may be considered to reside at Galactocentric radii of 20 kpc. Most of the published pulsar distances used are based on the CL02 model, and are therefore consistent with this model. As a consequence of the limited sky coverage of the RM data used here, we confine our analysis of the result to within the SGPS longitudes. The best-fit output for this model is a CW field everywhere except for a CCW field in the Scutum- Crux arm and in the molecular ring, as shown in 1Global models using data from all quadrants is beyond the scope of this paper, but will be presented in a future paper. Figure 4. Figure 5 shows a plot of both the mea- sured and modeled RMs for the individual SGPS EGS data (top panel), these data averaged and smoothed as in the middle panel of Figure 3 (mid- dle panel), and the measured and modeled RMs for individual pulsars (bottom panel). This model is able to closely reproduce the RM structure seen in the smoothed SGPS EGS data (recall that the fit is to the individual EGS and pulsar data) . However, there are two (relatively small) discrepancies between the model and the observed data towards the outer Galaxy. The first is in the vicinity around ℓ ∼ 275 deg, where the measured data are more negative than the mod- eled data as also seen in the top panel Figure 5. In Figure 2, there is a contained region of small, neg- ative RM between 270◦ < l < 283◦ at b > 0◦, re- sembling a magnetic bubble like that discussed by Clegg et al. (1992) and Brown & Taylor (2001). RMs at these longitudes nominally should be dom- inated by the field in the local arm. Thus, if the negative RMs seen here were to be attributed to the large-scale field, this would imply that the field is directed counter-clockwise in the local arm. This is contrary to the many studies that show the field is clockwise in the local arm, as discussed in section 1. Interestingly, this region was previ- ously identified in the Parkes 2.4-GHz survey as containing a polarized feature of unknown origin (Duncan et al. 1997). Since the lines-of-sight in the outer Galaxy are considerably smaller through the interstellar medium, compared to lines-of-sight through the inner Galaxy, it is likely that this lo- calized feature is dominating the RM for these EGS. As a result, averaging the negative RMs through this localized feature with the otherwise positive RMs of the local arm creates the effect of RM ∼ 0 at l ∼ 275◦. The other discrepancy is around ℓ ∼ 265◦ where the measured RMs are more positive than the modeled RMs. Lines-of-sight at this longi- tude are again through the outer Galaxy where localized features including the Gum nebula (Chanot & Sivan 1983) could dominate the RM. As seen in the top panel of Figure 3 there are two EGS sources at ℓ = 263◦ that have RMs roughly double that of neighboring sources which biases the average RM near this longitude. Interestingly, there is also a localized peak in the pulsar data near this longitude, as seen in the bottom panel of Figure 3. The model tries to fit both the nega- tive region around ℓ ∼ 275◦ and the more positive region around ℓ ∼ 265◦, with the result being a compromise between the two. Although the individual pulsar data are noisier than the averaged EGS data, the trends of the pulsar data are also reproduced by this model. In particular, the model supports our observa- tional conclusion that the Q4-component of the Sagittarius-Carina arm is directed CW, while the Q4-component of the Scutum-Crux arm is directed Finally, we note that the direction of the Norma arm field is not well constrained by this model or by the data. Regardless of whether the field in the Norma arm is oriented CW or CCW, the results from this model contrast the previous suggestions of reversals with every arm (Weisberg et al. 2004) or at every arm-interarm boundary (Han et al. 2006). 5. Summary We present the rotation measures for 148 ex- tragalactic sources found in the southern Galactic plane survey. The oscillations of rotation mea- sure with longitude revealed by these sources, and as also seen in pulsar RM data, highlight the dominating effect of the spiral arms on rota- tion measure. Both empirically and with a di- rect fit to measurements, the new data show con- clusively that the field is directed clockwise in the fourth-quadrant component of the Sagittarius- Carina arm, and that a field reversal exists be- tween the Sagittarius-Carina arm and the Scutum- Crux arms in the fourth quadrant. A definitive measurement of the number of large-scale magnetic reversals in the Galaxy can only emerge from an analysis that includes pulsar and EGS RM data at all Galactic longitudes, and which considers a wide range of distinct field con- figurations. In addition, the technique presented here is constrained to geometries imposed by the CL02 model. With these caveats, the results from our study of southern RMs indicate that far fewer magnetic reversals are needed to explain the data than other recent studies have suggested. We thank the anonymous referee for the insight- ful comments that have improved this manuscript. The Australia Telescope is funded by the Com- monwealth of Australia for operation as a National Facility managed by CSIRO. This work was facili- tated in part by an associateship grant to JCB by the Alberta Ingenuity Fund. MH acknowledges support from the National Radio Astronomy Ob- servatory, which is operated by Associated Uni- versities, Inc., under cooperative agreement with the NSF. BMG and MH acknowledge the support of the NSF through grant AST-0307358. REFERENCES Beck, R. 2001, Space Sci. Rev., 99, 243 Beck, R. 2007 in Polarisation 2005, F. Boulanger & M. A. Miville-Deschenes (eds.), EAS Publi- cation Series, 19 (astro-ph/0603531) Brown, J. C. 2002, PhD Thesis, University of Cal- Brown, J. C., Taylor, A. 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Table 1 Rotation Measures of the SGPS ℓ a b a α δ I b m c RM (◦) (◦) (J2000) (J2000) (mJy) (%) (rad m−2) 253.30 0.89 08 19 46.2 -34 42 05 179 5.8 −8 ± 12 253.52 0.83 08 20 08.4 -34 55 20 21 6.4 −50 ± 26 253.68 -0.60 08 14 47.2 -35 51 08 72 3.5 −349 ± 27 254.16 -0.34 08 17 10.3 -36 06 01 64 5.5 −338 ± 19 254.60 -0.87 08 16 12.7 -36 46 05 62 4.3 −15 ± 24 254.81 0.93 08 24 08.9 -35 55 21 87 13.7 +84 ± 13 254.95 0.63 08 23 20.7 -36 11 56 41 4.9 +8 ± 32 255.16 0.24 08 22 21.9 -36 36 03 92 5.3 −35 ± 16 255.27 0.16 08 22 20.2 -36 43 53 84 3.4 +26 ± 23 255.36 -0.26 08 20 52.1 -37 02 48 136 9.0 +97 ± 10 255.36 0.50 08 23 59.5 -36 37 00 94 8.0 −115 ± 13 256.14 0.25 08 25 11.9 -37 24 00 92 5.3 +44 ± 17 256.64 -0.22 08 24 43.7 -38 04 40 104 4.7 +172 ± 15 257.47 0.54 08 30 20.4 -38 18 27 46 9.5 +23 ± 20 257.71 -0.66 08 26 02.2 -39 12 13 117 4.0 +144 ± 20 257.92 0.65 08 32 07.7 -38 36 33 383 2.3 +76 ± 14 258.52 1.02 08 35 30.1 -38 52 23 118 2.7 +196 ± 26 258.77 0.08 08 32 23.4 -39 37 38 131 3.4 +221 ± 16 259.05 -0.72 08 29 49.9 -40 19 35 123 5.7 +175 ± 12 259.05 -0.75 08 29 45.1 -40 20 52 103 10.0 +150 ± 12 259.77 1.22 08 40 15.2 -39 44 39 33 5.8 +250 ± 29 260.41 -0.43 08 35 21.8 -41 14 50 101 3.0 +221 ± 18 260.52 -0.55 08 35 11.4 -41 24 44 25 5.8 +247 ± 31 260.69 -0.23 08 37 05.4 -41 21 08 224 3.2 +204 ± 12 263.20 1.07 08 50 56.2 -42 30 54 165 5.6 +739 ± 14 263.22 1.08 08 51 02.6 -42 31 32 164 4.4 +826 ± 19 263.50 0.17 08 48 12.7 -43 19 14 139 0.9 +260 ± 28 264.24 0.88 08 53 48.3 -43 25 59 112 11.0 +406 ± 9 265.69 0.85 08 58 54.5 -44 33 41 106 1.8 +70 ± 28 266.14 1.08 09 01 33.6 -44 44 49 66 3.2 +211 ± 29 266.27 0.66 09 00 15.2 -45 07 17 95 3.4 +396 ± 18 267.03 0.04 09 00 27.1 -46 06 15 58 3.8 +298 ± 21 267.17 0.47 09 02 50.7 -45 55 36 598 0.3 +323 ± 28 268.62 0.58 09 08 56.5 -46 55 53 231 2.1 +256 ± 15 269.05 0.17 09 08 54.5 -47 31 15 118 3.6 +456 ± 16 269.55 0.45 09 12 09.7 -47 41 35 156 1.9 +137 ± 18 270.56 -0.85 09 10 32.9 -49 19 14 58 5.5 −152 ± 23 270.91 0.93 09 19 52.0 -48 20 10 57 6.2 −149 ± 22 271.22 -0.35 09 15 37.0 -49 27 04 66 2.5 +215 ± 29 271.30 -0.06 09 17 12.3 -49 18 17 83 5.0 +136 ± 16 Table 1—Continued ℓ a b a α δ I b m c RM (◦) (◦) (J2000) (J2000) (mJy) (%) (rad m−2) 271.52 -1.01 09 13 54.3 -50 07 34 120 1.7 +75 ± 28 271.70 -0.38 09 17 30.7 -49 49 16 116 3.5 −93 ± 18 272.36 0.62 09 24 45.8 -49 34 44 21 5.6 −296 ± 28 273.46 0.68 09 29 58.1 -50 17 41 18 13.6 −106 ± 19 273.57 1.28 09 33 00.8 -49 55 47 229 1.9 +20 ± 25 274.77 0.25 09 34 14.3 -51 30 18 51 3.9 −76 ± 22 275.02 0.82 09 37 51.7 -51 15 07 26 8.4 −101 ± 25 275.48 -0.68 09 33 32.0 -52 40 18 282 2.8 +248 ± 13 275.56 1.02 09 41 17.5 -51 27 26 281 6.3 +16 ± 7 275.56 -0.20 09 36 03.4 -52 22 03 46 2.9 +112 ± 30 275.83 0.16 09 38 55.3 -52 16 41 170 9.3 −107 ± 7 275.86 0.94 09 42 25.0 -51 42 48 29 11.0 0 ± 21 276.46 0.89 09 45 09.7 -52 08 29 153 2.5 −17 ± 20 277.44 0.86 09 49 58.5 -52 47 04 10 12.1 −71 ± 28 277.78 -0.73 09 44 52.1 -54 13 45 127 2.0 −36 ± 23 277.78 -0.81 09 44 32.3 -54 17 32 690 7.3 −77 ± 3 278.04 0.75 09 52 37.5 -53 15 08 635 7.8 −104 ± 3 278.37 0.15 09 51 47.9 -53 55 46 18 13.7 −4 ± 24 278.43 0.53 09 53 42.8 -53 40 12 42 4.4 +20 ± 27 278.47 -0.30 09 50 22.2 -54 20 21 22 6.6 −116 ± 24 279.04 -0.88 09 50 55.5 -55 09 04 242 2.4 +239 ± 17 279.09 0.89 09 58 46.2 -53 47 21 47 5.0 −120 ± 25 279.15 -0.63 09 52 36.5 -55 01 08 36 11.2 +341 ± 24 279.15 -0.65 09 52 32.6 -55 02 37 44 6.2 +329 ± 20 279.33 0.80 09 59 41.3 -54 00 11 35 7.8 −157 ± 18 279.80 1.21 10 03 53.7 -53 57 24 25 9.7 −145 ± 25 280.53 0.81 10 06 18.9 -54 42 57 37 5.4 −10 ± 23 280.62 -0.14 10 02 53.8 -55 32 09 134 3.7 −112 ± 15 282.07 -0.78 10 08 36.2 -56 54 09 723 0.9 +862 ± 16 282.46 0.24 10 15 09.3 -56 17 28 109 2.0 +256 ± 23 284.30 0.81 10 28 39.3 -56 48 09 80 2.4 −547 ± 25 285.15 0.96 10 34 33.2 -57 06 17 268 0.9 +168 ± 31 285.60 0.62 10 36 12.7 -57 37 38 44 3.3 +368 ± 35 286.04 -1.05 10 32 43.2 -59 17 54 2826 0.4 +809 ± 14 286.89 0.59 10 44 37.5 -58 16 43 58 3.0 +324 ± 28 288.27 -0.70 10 49 34.1 -60 03 21 255 0.6 +491 ± 25 290.81 0.74 11 12 45.4 -59 47 27 1858 0.4 +419 ± 23 292.90 -0.02 11 26 31.6 -61 13 37 52 6.6 +349 ± 28 293.39 0.73 11 32 19.4 -60 40 22 67 5.0 +121 ± 20 293.73 0.63 11 34 43.2 -60 52 06 30 3.4 +116 ± 23 Table 1—Continued ℓ a b a α δ I b m c RM (◦) (◦) (J2000) (J2000) (mJy) (%) (rad m−2) 294.29 -0.90 11 35 31.6 -62 29 13 32 4.2 +449 ± 26 294.38 -0.75 11 36 37.4 -62 22 07 265 0.7 +470 ± 24 295.17 0.01 11 44 55.1 -61 51 26 57 3.5 +363 ± 22 295.23 -1.05 11 43 02.3 -62 53 56 114 2.1 −207 ± 22 295.29 -1.23 11 43 05.7 -63 04 55 226 1.6 −43 ± 29 296.18 -0.59 11 52 05.2 -62 40 59 118 3.7 +752 ± 14 296.90 0.14 11 59 31.2 -62 07 15 452 1.6 +1113 ± 11 297.67 0.77 12 06 52.0 -61 38 54 283 2.6 +570 ± 11 299.42 -0.23 12 20 29.8 -62 53 37 55 4.0 +535 ± 30 299.51 -1.10 12 20 24.8 -63 46 10 192 2.9 +315 ± 21 300.25 -0.01 12 27 57.6 -62 45 42 109 4.1 +123 ± 14 300.47 -0.99 12 29 06.5 -63 45 08 112 4.9 +412 ± 18 300.65 -0.41 12 31 12.4 -63 11 40 809 0.9 +358 ± 19 301.14 -0.09 12 35 39.6 -62 54 30 375 1.3 +350 ± 18 301.70 0.25 12 40 45.4 -62 36 01 21 5.2 +296 ± 33 302.60 -1.17 12 48 26.2 -64 02 26 162 1.7 +159 ± 25 303.30 0.51 12 54 36.3 -62 21 25 149 3.5 −370 ± 14 304.53 1.00 13 05 00.1 -61 49 31 32 6.1 +40 ± 28 305.62 -1.16 13 15 52.5 -63 54 09 44 10.5 −61 ± 28 306.87 0.02 13 25 47.5 -62 35 15 215 1.0 −197 ± 27 306.92 -0.70 13 27 00.7 -63 17 46 275 2.0 +52 ± 16 307.20 -0.84 13 29 39.3 -63 23 47 41 8.1 +382 ± 18 308.64 -0.62 13 41 56.1 -62 55 38 40 2.8 −133 ± 27 308.73 0.07 13 41 33.5 -62 14 06 92 1.4 −661 ± 29 308.93 0.40 13 42 41.5 -61 52 38 152 2.5 −752 ± 17 309.06 0.84 13 43 00.9 -61 24 52 670 2.3 −504 ± 10 310.20 -1.04 13 56 09.7 -62 59 30 46 8.5 −584 ± 19 312.37 -0.04 14 11 37.1 -61 25 54 129 2.3 −438 ± 28 313.96 -0.76 14 26 14.9 -61 34 58 88 2.3 −480 ± 22 313.99 0.94 14 21 36.9 -59 59 01 507 1.8 −828 ± 17 314.02 1.01 14 21 40.3 -59 54 20 757 1.7 −579 ± 20 314.50 0.30 14 27 14.8 -60 24 07 86 5.3 −738 ± 19 314.82 0.89 14 27 57.3 -59 43 58 92 3.7 −507 ± 25 316.64 1.15 14 40 15.0 -58 47 30 64 4.6 −525 ± 27 317.54 -0.57 14 52 27.0 -59 58 11 152 2.1 +395 ± 27 318.53 0.30 14 56 18.1 -58 44 29 326 1.0 +53 ± 29 319.34 1.08 14 58 58.6 -57 40 31 1101 0.7 +241 ± 21 319.39 0.74 15 00 28.7 -57 57 04 85 1.4 +279 ± 30 320.83 0.88 15 09 19.7 -57 07 19 454 0.8 −8 ± 20 321.48 1.02 15 12 54.7 -56 40 20 933 2.0 −243 ± 8 Table 1—Continued ℓ a b a α δ I b m c RM (◦) (◦) (J2000) (J2000) (mJy) (%) (rad m−2) 321.58 -0.76 15 20 29.3 -58 07 41 1831 1.1 −138 ± 9 322.05 -0.95 15 24 17.5 -58 01 49 78 5.4 −397 ± 17 323.15 -0.52 15 29 19.5 -57 03 49 47 2.9 +83 ± 25 324.77 0.61 15 34 10.5 -55 12 30 161 3.9 −66 ± 14 325.81 1.08 15 38 05.1 -54 13 08 608 0.6 −15 ± 31 d 325.83 -0.30 15 43 57.7 -55 18 26 247 1.8 +356 ± 18 326.69 -1.16 15 52 28.2 -55 27 04 95 9.3 −142 ± 22 327.31 0.88 15 47 01.4 -53 28 29 256 1.2 −189 ± 26 d 328.36 -0.41 15 58 00.3 -53 48 30 493 0.3 −721 ± 35 329.48 0.22 16 00 57.1 -52 36 04 599 0.7 −100 ± 25 330.12 -1.08 16 09 49.5 -53 08 33 218 3.0 −931 ± 25 332.14 1.03 16 10 10.8 -50 13 21 151 2.8 −754 ± 22 d 333.72 -0.27 16 22 55.5 -50 03 31 108 2.5 +204 ± 29 335.32 0.60 16 26 00.4 -48 18 36 352 2.9 −138 ± 11 337.06 0.85 16 32 01.7 -46 52 39 100 2.5 −739 ± 32 339.65 -0.24 16 46 44.6 -45 40 00 222 2.2 −398 ± 19 342.16 -0.74 16 57 52.7 -44 02 49 264 2.2 +127 ± 15 342.62 -0.45 16 58 15.4 -43 30 19 50 3.9 −913 ± 30 343.29 0.60 16 56 01.8 -42 19 58 102 4.7 −1035 ± 16 345.22 0.68 17 02 02.9 -40 45 50 39 12.6 +183 ± 21 347.40 -1.04 17 16 09.8 -40 02 24 72 4.0 −524 ± 36 349.65 -0.36 17 19 58.3 -37 48 15 133 3.3 +110 ± 20 350.52 -0.73 17 23 58.9 -37 18 18 127 2.2 +277 ± 32 351.31 -0.53 17 25 22.9 -36 32 16 200 3.3 −247 ± 15 351.82 0.17 17 23 56.8 -35 43 34 118 6.1 +134 ± 16 352.13 1.15 17 20 51.7 -34 54 49 116 6.5 +76 ± 30 355.43 -0.81 17 37 32.3 -33 14 40 76 4.2 +601 ± 21 356.57 0.87 17 33 45.9 -31 22 37 116 1.6 +985 ± 30 aThe identified location is the peak of the gaussian fit to the source in polarised intensity. All sources were either unresolved or partially resolved. b I is the peak-pixel Stokes I value of the interferometric data. c m is the fractional polarisation (linear polarised intensity divided by Stokes I) averaged over the FWHM pixels used for RM calculation. dA RM for this source was calculated by Gaensler et al. (2001) using a simpler approach than the more rigorous method used here. These new values should replace the previously determined values. Norma Scutum-Crux Sagittarius- Carina Perseus local molecular l=253o l=358o Fig. 1.— View of the Galaxy from above. The grey scale is the CL02 electron density model, with labels indicating the spiral arms, and the asterisk indicating the location of the Sun. Quadrants 1, 2, 3, and 4 are labeled with Q1, Q2, Q3 and Q4, respectively. The bounding longitudes of the SGPS (Phase I) are indicated by black lines and are labeled. The circles represent the smoothed-averaged SGPS RMs as shown in the middle panel of Figure 3. Filled (open) circles indicate positive (negative) RM, with the size of the circles linearly proportional to |RM|, truncated between 59 and 592 rad m−2 (the |RM|max from the middle panel of Figure 3) so that sources with |RM| < 59 rad m−2 are set to 59 rad m−2. The blue dashed (dotted) lines are also from the middle panel of Figure 3, and indicate the approximate longitudes of |RM| maxima (minima) in the SGPS RM data. The solid blue line is the longitude where the RMs transition from primarily positive to primarily negative (ℓ ∼ 304◦). 350 330 310 304 290 270 250 PULSARS Galactic Longitude Fig. 2.— Rotation measures for sources in the SGPS. Top panel: extragalactic RM sources; Bottom panel: pulsar RM sources. Grey filled symbols indicate positive rotation measure, black open symbols indicate negative rotation measures; sizes of symbols are linearly proportional to the magnitude of RM truncated between 100-600 rad m−2, so that sources with |RM| < 100 rad m−2 are set to 100 rad m−2, and those with |RM| > 600 rad m−2 are set to 600 rad m−2. In the top panel, the square at ℓ = 307.1◦, b= 1.2◦ represents the EGS RM previously observed by Broten et al. (1988), while the squares at ℓ ∼ 330◦ represent the EGS RMs of the SGPS test region. The circles in the top panel represent the SGPS EGS as given in Table 1. The vertical line at ℓ = 304◦ is the approximate longitude where the EGS RMs change sign. The dashed boxes indicate boundaries of the SGPS region. -1200 SGPS EGS (raw, binned) -1200 SGPS EGS (smoothed) 360 350 340 330 320 310 300 290 280 270 260 250 -1200 PULSARS (raw) Galactic Longitude Fig. 3.— RM versus Galactic longitude for RM sources in the SGPS region. Top panel: circles represent the individual SGPS EGS sources (errors are smaller than the symbol size), while open red diamonds represent data averaged into into independent longitude bins containing 9 sources (the end bin at 255◦ contains 13 sources). Where symbol size permits, the error bars are the standard deviation in longitude and RM for each bin. Middle panel: purple diamonds represent boxcar-averaged SGPS EGS data over 9 degrees in longitude with a stepsize of 3 degrees. In contrast to the binned data in the top panel, the error bars are the standard error of the mean. The solid line marks the approximate longitude of transition from predominantly positive RMs to negative RMs (l ∼ 304◦) Dotted lines (dashed lines) mark approximate longitudes of minimum (maximum) |RM| in SGPS data. Bottom panel: squares represent the individual pulsars with known RM in the SGPS region (errors are smaller than the symbol sizes). Fig. 4.— A simple model of the magnetic field in the SGPS region, constrained using individual RMs of 149 EGS and 120 pulsars. Colored regions indicate the total strength for the model fit of magnetic field corresponding to the regions identified by the green lines as discussed in the text. Red (blue) shading indicates a clockwise (counter-clockwise) field direction as viewed from the Galactic north pole. The direction of the field within the arms is also indicated by arrows. The open-head arrow on the Norma arm indicates the field direction in this arm is not well-constrained in this model configuration with the data used. The bounding longitudes of the SGPS (Phase I) are indicated by black lines and are labeled. The grey scale is the CL02 electron density model with labels indicating the spiral arms. -1200 SGPS EGS (raw) Model -1200 SGPS EGS (smoothed) Model 360 350 340 330 320 310 300 290 280 270 260 250 -1200 Pulsars (raw) Model Galactic Longitude Fig. 5.— Comparison of modeled RMs (green symbols) and observed RMs. The modeled RMs are calculated using the CL02 electron density model and the magnetic field model shown in Figure 4. The format of this figure follows that of Figure 3, where the individual SGPS EGS are shown in the top panel, smoothed- averaged SGPS EGS data are shown in the middle panel, and the individual pulsars are presented in the bottom panel. Introduction Rotation Measure Calculations Observations of the Galactic Magnetic Field Modeling the Large-Scale Magnetic Field Summary

0704.0459

On iterated image size for point-symmetric relations

On iterated image size for point-symmetric relations Yahya Ould Hamidoune∗ Abstract Let Γ = (V,E) be a point-symmetric reflexive relation and let v ∈ V such that |Γ(v)| is finite (and hence |Γ(x)| is finite for all x , by the transitive action of the group of automorphisms). Let j ∈ N be an integer such that Γj(v)∩Γ−(v) = {v}. Our main result states that |Γj(v)| ≥ |Γj−1(v)|+ |Γ(v)| − 1. As an application we have |Γj(v)| ≥ 1+(|Γ(v)|− 1)j. The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta-Häggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson. 1 Introduction Let G be an abelian group and let A,S be finite subsets of G with 0 /∈ S. Shepherdson’s generalization of the Cauchy-Davenport Theorem states that |A ∪ (A + S)| ≥ |A| + |S| if A ∪ (A+ S) contains no subgroup generated by some element of S. As an application Shepherdson [14] proved that there are s1, · · · , sk ∈ S such that k ≤ ⌉ and 1≤i≤k si = 0, if G is finite. The paper of Shepherdson includes thanks to Heilbronn for suggesting this application together with a mention that Chowla obtained some related zero-sum results. Let D = (V,E) be a loopless finite digraph with minimal outdgree at least 1. It is well known that D contains a directed cycle. The smallest cardinality of such a cycle is called the girth of D and will be denoted by g(D). In 1970 Behzad, Chartrand and Wall [1] conjectured that |V | ≥ r(g(D) − 1) + 1, if d+(x) = d−(x) = r for all x ∈ V . In 1978, Caccetta and Häggkvist [3] made the stronger conjecture : |V | ≥ min(d+x : x ∈ V )(g(D) − 1) + 1. These conjectures are still largely open, even for the special case g(D) = 4. The reader may find references and results about this question in [2]. These conjectures were proved by the author for vertex-symmetric digraphs [6]. This result applied to Cayley graphs shows the validity of Shepherdson’s zero-sum result for all finite groups. Unfortunately we were not aware at that moment of Shepherdson’s result. Our Université Pierre et Marie Curie, Paris. [email protected] http://arxiv.org/abs/0704.0459v1 proof [6] is based on the properties of atoms of a finite digraph and Menger’s Theorem. A description of Cayley graphs on finite Abelian groups such that |V | = r(g(D)− 1) + 1, where r is the outdegree was obtained by the authors of [9] using Kemperman critical pair Theory [12]. A new proof of the Caccetta and Häggkvist conjecture for vertex-symmetric digraphs based on an additive result of Kemperman [11] and the representation of vertex symmetric digraphs as coset graphs is given in [10]. More recently Seymour proposed the following conjecture [13]: Let D be a loopless digraph and let r ≥ 1 be an integer. Then there is a vertex a such |Γ(a) ∪ Γ2(a) ∪ · · · ∪ Γg−2(a)| ≥ r(g − 2), where g = g(D). The case g = 4 of this conjecture is mentioned in [2]. Seymour’s Conjecture implies the conjecture of Behzad, Chartrand and Wall. Seymour’s Conjecture also implies that D contains a directed cycle C with |C| ≤ ⌈ |V |−1 ⌉ + 1. Notice that the Caccetta-Häggkvist Conjecture states that D contains a directed cycle C with |C| ≤ ⌈ We shall allow infinite relations. The classical strong connectivity of digraphs needs to be modified in this case in order to have a good lower bound of the size of the image of a set. Also the presence of loops will simplify the presentation of the connectivity method. Since this convention is unusual in this part of Graph Theory, we shall work with relations. Our terminology will be developed in the next section. Seymour’s conjecture may be formulated as follows : Conjecture 1 [13] Let Γ = (V,E) be a finite reflexive relation and let j be an integer. Then there is an x ∈ V such that one of the following conditions holds. • |Γj(x)| ≥ 1 + j(|Γ(x)| − 1). • Γ−1(x) ∩ Γj(x) 6= {x}. Our main result is the following one: Let Γ = (V,E) be a point-symmetric reflexive relation and let v ∈ V such that |Γ(v)| is finite. Let j ≥ 1 be such that Γj(v) ∩ Γ−(v) = {v}. Then |Γj(v)| ≥ |Γj−1(v)|+ |Γ(v)| − 1. This result implies the validity of the above conjectures for vertex-symmetric graphs. 2 Terminology Let V be a set. The diagonal of V is by definition ∆V = {(x, x) : x ∈ V }. Let E ⊂ V × V . The ordered pair Γ = (V,E) will be called a relation . The relation Γ is said to be reflexive if ∆V ⊂ E. Let a ∈ V and let A ⊂ V . The image of a is by definition Γ(a) = {x : (a, x) ∈ E}. The image of A is by definition Γ(A) = Γ(x). The cardinality of the image of x will be called the degree of x and will be denoted by d(x). The relation Γ will be called regular with degree r if the elements of V have the same degree r. We shall say that Γ is locally finite if d(x) is finite for all x. The reverse relation of Γ is by definition Γ− = (V,E−), where E− = {(x, y) (y, x) ∈ E}. The restriction of Γ = (V,E) to a subset W ⊂ V is defined as the relation Γ[W ] = (W,E ∩ (W ×W )). Let Φ = (W,F ) be a relation. A function f : V −→ W will be called a homomorphism if for all x, y ∈ V such that (x, y) ∈ E, we have (f(x), f(y)) ∈ F . The relation Γ will be called point-symmetric if for all x, y ∈ V , there is an automorphism f such that y = f(x). Clearly a point-symmetric relation is regular. We identify graphs and their relations. A loopless finite relation will be called a digraph. The reader may replace everywhere the term ”relation” by ”graph”. In this case we mention some differences between our terminology (which follows closely the standard notations of Set Theory) and the notations used in some text books of Graph Theory. We point out that our graphs are usually called directed graphs without multiple arcs or digraphs. Notice that the notion Γ(a) used here and in Set Theory is written Γ+(a) in some text books in Graph Theory. Also our notion of degree is called outdegree. We made the choice of Set Theory terminology since some parts of this paper could have some interest in Group Theory and Number Theory. We shall use the composition Γ1◦· · ·◦Γk of relations Γ1, · · · ,Γk on V . If all these relations are equal to Γ, we shall write Γ1 ◦ · · · ◦ Γk = Γ We shall write Γ0 for the identity relation IV = (V,∆V ). Also we shall write Γ −j instead of (Γ−)j . 3 Connectivity Let Γ = (V,E) be a relation. For X ⊂ V , we shall write ∂Γ(X) = Γ(X) \X. When the context is clear the reference to Γ will be omitted. Let Γ = (V,E) be a locally finite reflexive relation. The connectivity of Γ is by definition κ(Γ) = |V | − 1, if E = V × V. Otherwise κ(Γ) = min{|∂(X)| : 1 ≤ |X| < ∞ and Γ(X) 6= V }. (1) A subset X achieving the minimum in (1) is called a fragment of Γ. A fragment with minimum cardinality is called an atom. The cardinality of an atom of Γ will be denoted by a(Γ). It is not true that distinct atoms are always disjoint. But the author proved in [4] that, if V is finite, then distinct atoms of Γ are disjoint, or distinct atoms of Γ− are disjoint. In [7], it was observed that the same methods imply that distinct atoms of Γ are disjoint if V is infinite. One may find in [8] unified proofs and some applications to Group Theory and Additive Number Theory. As a consequence of this result we could obtain : Proposition 2 [4, 5, 7, 8] Let Γ = (V,E) be a locally-finite point-symmetric relation with E 6= V × V . Suppose that V is infinite or that a(Γ) ≤ a(Γ−). Let A be an atom of Γ. Then the subrelation Γ[A] induced on A is a point-symmetric relation. Moreover |A| ≤ κ(Γ). 4 Iterated image size Lemma 3 Let Γ = (V,E) be a point-symmetric relation. Then for all i, Γi is point-symmetric. Proof. Clearly any automorphism of Γ is an automorphism of Γi. Theorem 4 Let Γ = (V,E) be a point-symmetric reflexive locally finite relation and let v ∈ V. Let j ≥ 1 be an integer such that Γj(v) ∩ Γ−(v) = {v}. Then |Γj(v)| ≥ |Γj−1(v)|+ |Γ(v)| − 1. Proof. Set V0 = 0≤i Γ i(v). Clearly Γj(v) ⊂ V0. So we may assume that Γ = Γ[V0] and V = V0. In the finite case this means that we restrict ourselves to the connected component con- taining v. We shall assume j > 1, since the result is obvious for j = 1. With this hypothesis, clearly we have κ(Γ) ≥ 1. Clearly E 6= V × V. Set κ = κ(Γ). Let A be an atom of Γ containing v. The proof is by induction on |Γ(v)|. Put r = |Γ(v)|. Assume first κ = r − 1. Observe that Γj(v) 6= V . Then by the definition of κ, we have |Γj(v) \ Γj−1(v)| = |∂(Γj−1(v))| ≥ κ = r − 1. The result holds in this case. So we may assume κ ≤ r − 2, (2) and hence r ≥ 3. Then |A| ≥ 2, since otherwise κ = |∂(A)| = r − 1. Case 1. V is infinite or a(Γ) ≤ a(Γ−). By Proposition 2, Γ[A] is point-symmetric (and hence regular) and |A| ≤ κ. (3) Put r0 = |Γ(v) ∩A|. Put X = Γj−1(v) and Y = A ∪X. By the induction hypothesis, we have |∂(X) ∩A| = |Γj(v) ∩A| − |Γj−1(v) ∩A| ≥ r0 − 1. (4) Let us prove that ∂(Y ) ⊂ (∂(X) \ A) ∪ (∂(A) \ Γ(v)). (5) Since j ≥ 2, we have Γ(v) ⊂ X and hence Γ(v) ∩ ∂(X) = ∅. Then (5) clearly holds. It follows that ∂(Y ) \ ∂(X) ⊂ ∂(A) \ Γ(v), and hence we have |∂(Y ) \ ∂(X)| ≤ |∂(A)| − |∂(A) ∩ Γ(v)| = κ− |Γ(v)| + |A ∩ Γ(v)| = κ+ r0 − r. Hence |∂(Y ) \ ∂(X)| ≤ κ+ r0 − r. (6) Let us show that Γ(Y ) 6= V. This holds obviously if V is infinite. So we may assume V finite. In this case we have |Γ−(v)| = |Γ(v)|. Clearly we have Γ(Y ) = Γ(X) ∪ (A \ Γ(v)) ∪ (∂(Y ) \ ∂(X)). It follows using ( 3) and (6) that |Γ(Y )| ≤ |Γ(X)|+ |A \ Γ(v)|+ |∂(Y ) \ ∂(X)| ≤ |V \ (Γ−(v) \ {v})| + |A| − r0 + κ+ r0 − r = |V |+ |A|+ κ− 2r + 1 ≤ |V |+ 2κ− 2r + 1 ≤ |V | − 3. By the definition of κ, we have |∂(Y )| ≥ κ. By (4) and (6), |∂(X)| = |∂(X) ∩A|+ |∂(Y ) ∩ ∂(X)| ≥ r0 − 1 + |∂(Y )| − |∂(Y ) \ ∂(X)| ≥ r0 − 1 + κ− (κ+ r0 − r) = r − 1, and the result is proved since ∂(X) = Γj(v) \ Γj−1(v). Case 2. V is finite and a(Γ) > a(Γ−). The argument used in Case 1, shows that |Γ−j(v) \ Γ−(j−1)(v)| ≥ r − 1. By Lemma 3, Γj is point-symmetric. Since V is finite, Γj and its reverse Γ−j have the same degree. Therefore observing that these relations are reflexive r − 1 ≤ |Γ−j(v)| − |Γ−(j−1)(v)| = |Γj(v)| − |Γ(j−1)(v)|. The next result shows the validity of the conjecture of Seymour mentioned in the intro- duction in the case of relations with a symmetric group of automorphisms. Corollary 5 Let Γ = (V,E) be a point-symmetric reflexive relation with degree r and let v ∈ V . Let j ≥ 1 be an integer such that Γj(v) ∩ Γ−(v) = {v} then |Γj(v)| ≥ 1 + (r − 1)j. Proof. The proof follows by induction using Theorem 4 Corollary 6 [6] Let Γ = (V,E) be a point-symmetric digraph with degree r ≥ 1 and put g = g(Γ). Then |V | ≥ 1 + r(g − 1). Proof. Set Φ = (V,E ∪ ∆V ). Let v ∈ V . Clearly we have Φ g−2(v) ∩ Φ−(v) = {v}. By Corollary 5, |V | − r = |V \ (Φ−(a) \ {a})| ≥ |Φg−2(v)| ≥ 1 + (g − 2)r. This result, proved in [6], shows the validity of the Caccetta-Häggkvist Conjecture for point-symmetric graphs. But the proof obtained here is much easier. Corollary 7 [6] Let G be a group of order n and let S ⊂ G\{1} with cardinality = s. There are elements s1, s2, · · · , sk ∈ S such that k ≤ ⌈ ⌉ and 1≤i≤k si = 1. The proof follows by applying Corollary 6 to the Cayley graph defined by S on G. In particular the theorem of Shepherdson mentioned in the introduction holds for all finite groups. References [1] M. Behzad, G. Chartrand and C.E. Wall, On minimal regular digraphs with given girth, Fund. Math. 69 (1970), 227-231. [2] J. A. Bondy, Counting subgraphs: a new approach to the Caccetta-Häggkvist conjecture. Graphs and combinatorics (Marseille, 1995). Discrete Math. 165/166 (1997), 71-80. [3] L. Caccetta and R. Häggkvist, On minimal digraphs with given girth, Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978), (Winnipeg, Man.), Congress. Numer., XXI, Utilitas Math. (1978), 181-187. [4] Y.O. Hamidoune, Sur les atomes d’un graphe orienté, C.R. Acad. Sc. Paris A 284 (1977), 1253-1256. [5] Y.O. Hamidoune, Quelques problèmes de connexité dans les graphes orientés, J. Comb. Theory B 30 (1981), 1-10. [6] Y.O. Hamidoune, An application of connectivity theory in graphs to factorizations of elements in groups, Europ. J of Combinatorics 2 (1981), 349-355. [7] Y.O. Hamidoune, Sur les atomes d’un graphe de Cayley infini, Discrete Math., 73 (1989), 297-300. [8] Y.O. Hamidoune, On small subset product in a group. Structure Theory of set-addition, Astérisque. no. 258(1999), xiv-xv, 281-308. [9] Y.O. Hamidoune, A. Lladó and O. Serra, Vosperian and superconnected Abelian Cayley digraphs, Graphs and Combinatorics 7(1991), 143-152. [10] Melvyn B. Nathanson, The Caccetta-Häggkvist conjecture and Additive Number Theory, http://arxiv.org/archive/math: eprint arXiv:math/0603469. [11] J.H.B. Kempermann, On complexes in a semigroup, Nederl. Akad. Wetensch. Proc. Ser. A. 59= Indag. Math. 18(1956), 247-254. [12] J. H. B. Kemperman, On small sumsets in a Abelian group, Acta Math. 103 (1960), 63-88. [13] P. Seymour, Oral communication. [14] J. C. Shepherdson, On the addition of elements of a sequence, J. London Math Soc. 22(1947), 85-88. http://arxiv.org/archive/math: http://arxiv.org/abs/math/0603469 Introduction Terminology Connectivity Iterated image size

0704.0460

The Kilodegree Extremely Little Telescope (KELT): A Small Robotic Telescope for Large-Area Synoptic Surveys

Draft version October 3, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 THE KILODEGREE EXTREMELY LITTLE TELESCOPE (KELT): A SMALL ROBOTIC TELESCOPE FOR LARGE-AREA SYNOPTIC SURVEYS Joshua Pepper , Richard W. Pogge , D. L. DePoy , J. L. Marshall , K. Z. Stanek , Amelia M. Stutz , Shawn Poindexter , Robert Siverd , Thomas P. O’Brien , Mark Trueblood , & Patricia Trueblood Draft version October 3, 2018 ABSTRACT The Kilodegree Extremely Little Telescope (KELT) project is a survey for planetary transits of bright stars. It consists of a small-aperture, wide-field automated telescope located at Winer Obser- vatory near Sonoita, Arizona. The telescope surveys a set of 26◦×26◦ fields, together covering about 25% of the Northern sky, targeting stars in the range of 8 < V < 10 mag, searching for transits by close-in Jupiters. This paper describes the system hardware and software and discusses the quality of the observations. We show that KELT is able to achieve the necessary photometric precision to detect planetary transits around solar-type main sequence stars. Subject headings: Astronomical Instrumentation 1. INTRODUCTION The scientific value of planetary transits of bright stars is well known – for a comprehensive review see Charbonneau et al. (2007). These transits pro- vide the opportunity to study the internal structure of planets (Guillot 2005), their atmospheric compo- sition (Charbonneau et al. 2002), spin-orbit alignment (Gaudi & Winn 2007), and the presence of rings or moons (Barnes & Fortnoy 2004). Radial-velocity (RV) surveys have searched the brightest stars in the sky for planets and are probing increasingly fainter stars. Even with significant multiplexing, however, RV surveys are not able to search large numbers of stars fainter than V ∼ 8 mag. To find planets around fainter stars, transit surveys are more suitable, and a number of such surveys are underway. These surveys typically have wide fields of view and small apertures to simultaneously monitor tens or hundreds of thousands of stars. These surveys have so far discovered six planets transiting bright stars (Alonso et al. 2004; Bakos et al. 2007; Cameron et al. 2007; McCullough et al. 2006; O’Donovan et al. 2006). In order to discover more such scientifically valuable sys- tems, we have begun a survey to discover transiting plan- The Kilodegree Extremely Little Telescope (KELT) is designed to meet the objectives described in Pepper, Gould, & DePoy (2003) for a wide field, small- aperture survey for planetary transits of bright stars. That paper derives a model for the ability of a given transit survey to detect close-in giant planets (i.e. “Hot Jupiters”), and determines an optimal survey configura- tion for targeting 8 < V < 10 mag main-sequence stars. Based on the model of Pepper, Gould, & DePoy (2003), we expect to detect roughly four transiting planets with the KELT survey. The KELT system has two different observing config- Electronic address: [email protected] 1 The Ohio State University Department of Astronomy, 4055 McPherson Lab, 140 West 18th Avenue, Columbus, OH 43210- 2 Department of Astronomy, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721-0065 3 Winer Observatory, P. O. Box 797, Sonoita, AZ 85637-0797 urations. The primary configuration is a “Survey Mode” designed for wide area coverage, either in large strips or an all-sky survey, with the goal of covering broad sec- tions of the sky with a large field of view, at a cadence of a few minutes on a nightly basis throughout most of the observing year. This mode implements the primary scientific driver of KELT and gives this telescope a wider field of view and targets a brighter magnitude range than other transit surveys of its kind. The second configura- tion is a “Campaign Mode” that uses a smaller field of view and is designed to conduct short duration inten- sive observing campaigns on specific fields. In Campaign Mode we have undertaken a 74-day campaign towards the Praesepe open cluster. In this paper, we describe the instrumentation, deploy- ment, and operations of KELT (§2); we characterize the performance of the different components and the overall system in the field (§3); we show how precise our pho- tometry is (§4); and we provide examples of lightcurves for variable stars and transit-like events (§5). 2. KELT SYSTEM OVERVIEW 2.1. Instrument KELT consists of an optical assembly (CCD detector, medium-format camera lens, and filter) mounted on a robotic telescope mount. A dedicated computer is used to control the telescope, camera, observation scheduling, and data archiving system tasks. One goal in assembling KELT was to use as many off-the-shelf components and software packages as possible to speed the development. The KELT detector is an Apogee Instruments4 AP16E thermoelectrically cooled CCD camera. This camera uses the Kodak KAF-16801E front-side illuminated CCD with 4096 × 4096 9µm pixels (36.88 × 36.88mm detec- tor area) and has peak quantum efficiency of ∼65% at 600nm. The AP16E uses a PCI card and cable to control the camera and thermoelectric cooler (TEC). According to the camera specifications and confirmed by labora- tory testing, the camera is operated at a conversion gain of 3.6 electrons/ADU and delivers a measured readout 4 http://www.ccd.com http://arxiv.org/abs/0704.0460v2 mailto:[email protected] noise of ∼15 e−. The device is read out at 14-bit reso- lution at 1.3MHz, which gives a full-frame readout time of ∼30 seconds. The CCD specifications claim a full-well depth of ∼100,000e−, but the 14-bit ADC saturates at 16383 ADU (∼59,000 e−). The TEC can cool the device to ∼30◦C below the ambient air temperature. Nominal dark current is 0.1 − 0.2 e− pixel−1 sec−1 at an operat- ing temperature of −10◦C (typical for 20◦C ambient air temperature). We use two different lenses with KELT. For the wide- angle survey mode, we use a Mamiya 645 80mm f/1.9 medium-format manual-focus lens with a 42mm aper- ture. This lens provides a roughly 23′′ pix−1 image scale and a 26◦×26◦ field of view. To provide a narrow-angle campaign mode, we use a Mamiya 645 200mm f/2.8 APO manual-focus telephoto lens with a 71mm aper- ture. This provides a roughly 9.′′5 pix−1 image scale and effective 10.◦8×10.◦8 field of view. Both lenses have some vignetting at the corners, and the image quality declines toward the outer part of the field, so the effective field of view is circular (see §2.5 and §3.3 for details). To reject the mostly-blue background sky without greatly diminishing the sensitivity to stars (which are mostly redder than the night sky), we use a Kodak Wratten #8 red-pass filter with a 50% transmission point at ∼490nm (the filter looks yellow to the eye), mounted in front of the KELT lens. The calculated re- sponse function of the KELT CCD and filter is shown in Figure 1. The transmission function for the Wrat- ten #8 filter is taken from the Kodak Photographic Filters Handbook (Kodak 1998), and the quantum effi- ciency curve for the Kodak KAF-16801E CCD was pro- vided by the Eastman Kodak Company. The effect of atmospheric transmission on this bandpass is estimated for 1.2 airmasses at the altitude of Winer Observatory (1515m) using the Palomar monochromatic extinction coefficients (Hayes & Latham 1975), which are for an al- titude of 1700m. We did not estimate in detail the atmo- spheric water vapor or O2 extinction terms, as these are not important for our application. The effective wave- length of the combined Filter+CCD response function (excluding atmospheric effects) is 691nm, with an effec- tive width of 318nm, computed following the definition of Schneider, Gunn, & Hoessel (1983). This results in an effective bandpass that is equivalent to a very broad R-band filter. The optical assembly (camera+lens+filter) is mounted on a Paramount ME Robotic Telescope Mount manufac- tured by Software Bisque5. The Paramount is a research- grade German Equatorial Mount designed specifically for robotic operation with integrated telescope and cam- era control. According to manufacturer’s ratings the periodic tracking error of the mount before correction is ±5′′. That is smaller than the large pixels of the KELT camera and therefore does not affect our obser- vations. The mount can carry an instrument payload of up to 75 kg, more than sufficient for the KELT cam- era, which weighs approximately 9 kg. The Paramount is installed on a stock 91 cm high steel pier using a stock base adapter plate. The Paramount provides us with a robust, complete mounting solution for our telescope. The optical assembly is mated to the Paramount using 5 http://www.bisque.com a custom mounting bracket that mounts directly on the Paramount’s Versa-Plate mounting surface. The CCD camera and mount are controlled by a PC computer located at the observing site that runs the Win- dows XP Professional operating system and the Bisque Observatory Software Suite from Software Bisque. There are three main applications that we use for KELT: TheSky to operate the Paramount ME (pointing and tracking); CCDSoft CCD camera control software to op- erate the AP16E camera; and Orchestrate scripting and automation software to integrate the operation of TheSky and CCDSoft. Orchestrate provides a simple script- ing interface that lets us control all aspects of a night’s observing with a single command script. This software lets us prepare an entire night’s observing schedule and upload it to the KELT control computer during the af- ternoon. A scheduled task on the computer starts up the system at sunset and loads the observing script into Orchestrate, after which the system runs unattended for the entire night, weather permitting. The observing site provides AC power and Internet connectivity. To ensure clean AC power for the KELT telescope and control computer at the observing site (§2.2) we use a 1500VA Powerware 9125 Uninterrupt- able Power Supply (UPS). This filters the line power and protects the system against surges or brief power outages. The control computer is connected to the Internet through the observing site’s gateway, and its internal clock is synchronized with the network time servers at the Kitt Peak National Observatory using the Dimension 4 network time protocol (NTP) client6. This ensures sufficiently accurate timing for telescope point- ing and time-series photometry. Since we observe at few- minute cadences, we can tolerate few-second timing pre- cision, which is easily within the typical performance of Dimension 4 on the available T1 connection. 2.2. Observatory Site The KELT telescope has been installed at the Irvin M. Winer Memorial Mobile Observatory7 near Sonoita, Arizona. Located at N 31◦39′53′′, W 110◦36′03′′, ap- proximately 50 miles southeast of Tucson at an eleva- tion of 4970 feet (1515meters), Winer Observatory has a dedicated observing building with a 25×50-foot (7.6×15- meter) roll-off enclosure and provides site and mainte- nance services. Winer currently hosts four robotic tele- scopes, including KELT. The KELT telescope pier is bolted to the concrete floor, and its control cables are run into a nearby telescope control room that houses the control computer and UPS. Winer also provides Internet access (currently via a dedicated DS/T1 line, but early in the project the site used a slower ISDN link), that al- lows us to remotely login to the control computer via a secure gateway. The weather conditions at Winer Observatory are roughly as good as comparable sites in southern Arizona; about 60% of all observing time is usable, with half of that time being measurably photometric. Since our PSFs are between 2 and 3 pixels, and the pixel scales are 9.′′5 and 23′′ (for the 200mm and 80mm lenses, respectively), 6 v5.0 from Thinking Man Software (www.thinkman.com/dimension4/) 7 http://www.winer.org atmospheric seeing variations (which are on the scale of arcseconds) are not a factor in our observations. 2.3. Observing Operations Observations with KELT are carried out each non-clouded night using command scripts for pre- programmed, robotic operation; we do not undertake any remote real-time operations. The nightly observ- ing scripts are created at Ohio State University (OSU) using a script-generation program written in Perl, and then uploaded to Winer Observatory where they are used by Orchestrate to direct the telescope to observe the specified fields for each night. Each night has a differ- ent script, and we generally upload scripts in 3-4 week batches during the main survey season, or more fre- quently during pointed-target campaigns. The suite of software programs we use to control the telescope mount and camera works well, but has several limitations. Most importantly, the Orchestrate script- ing package does not provide the built-in ability to pro- gram control loops or conditional branching, which is why we use a Perl program to create the Orchestrate scripts we upload to the control computer. The scripts start the telescope each night based on the local clock time. On nights when the weather is judged to be good enough for observing, the observatory control computer automatically opens the roof of the observatory at nautical (12◦) twilight and the telescope begins obser- vations on schedule. If the weather is not suitable for observing, on-site personnel abort the command script. If the weather appears good at first but degrades dur- ing the night, the observatory computer closes the roof and the personnel abort the script. All data acquired are archived automatically at the end of the night. When the script is loaded into Orchestrate, it first waits until one hour before astronomical (18◦) twilight. At that point the telescope takes five dark images and five bias images, with the exposure times for the dark frames the same as the exposure times of the observations for the night. Once these calibration data are taken, the telescope goes back to sleep until astronomical twilight, at which point it slews to the first target field and begins the nightly observing sequence. Unless the weather turns bad, prompting the roof to close, observations continue until astronomical dawn. At this time the telescope is slewed to its stowed position, and five dark images and five bias images are acquired, ending observing for the night. (See §2.5 for a discussion of KELT flatfields.) We have so far used KELT in two distinct operating modes: campaign mode, using the 200mm lens, and sur- vey mode, using the 80mm lens. In campaign mode, the telescope intensively observes a single target field for an entire night. In survey mode, the telescope observes a number of fields that are equally spaced around the sky at 2h intervals of Right Ascension centered on Declina- tion +31◦39′ (the latitude of Winer). In campaign mode, the observing script instructs the telescope to wait until the target field is above 2 air- masses, and to then observe the field continuously until it sinks below 2 airmasses or astronomical dawn, whichever happens first. In survey mode, the telescope begins ob- serving after astronomical twilight, tiling between two fields at a time. New fields are observed as they rise above 1.4 airmasses. Provisions for Moon avoidance are built into the scripts to prevent observations of any sur- vey field when it is within 45◦ (two field widths from field center) of the Moon. An operational complication arises because the KELT mount is a German Equatorial design. This means that fields observed East of the Meridian are rotated by 180◦ relative to fields observed West of the Meridian. The practical effect is that we must separately reduce pho- tometry for fields taken in East and West orientations, especially when we use difference-imaging photometry. To avoid the complication of creating two separate data pipelines to reduce data taken in survey mode, we in- stead observe only fields in the Eastern part of the sky. We lose some observation time due to periods when the Moon is within 45◦ of all fields in the East above 1.4 airmasses, leading to downtime when no field is available that meets the observing criteria. In those situations, the scripts instruct the telescope to pause observing until the next field becomes available. Overall, the loss to Moon avoidance reduces the total amount of data acquired by ∼10%. 2.4. Data Handling and Archiving Data acquired by the camera are immediately writ- ten to a hard drive in the control computer, logged, and then copied to one of two 250GB external hard drives attached to the control computer by a USB interface. At the end of the night, all new images are automatically du- plicated onto the other external hard drive. During long winter nights, the telescope can take as many as 500-600 images per night, depending on the exposure time, fill- ing the drives every two weeks. Normally, however, bad weather and downtime due to Moon avoidance reduce the actual observing rate, so it typically takes 3-4 weeks for both storage drives to reach capacity. Data quality is monitored daily using automated scripts running on the Windows observing computer at Winer. At the end of the night, a Perl script selects three images from the beginning, middle, and end of the night and uploads them to a computer at OSU. These sample images are analyzed for basic statistics: mean, median, and modal sky value and the mean FWHM of stars mea- sured across the images, and then visually inspected to ensure that the camera and mount are operating cor- rectly. When the external drives approach full capacity, they are disconnected from the computer. One of the drives is a hardened drive made by Olixir Technologies (their Mobile DataVault) that serves as the transport drive. This drive is shipped via FedEx to the OSU Astronomy Department in Columbus, Ohio, in a cushioned trans- port box, as bandwidth limitations at Winer Observa- tory preclude online data transfer. The second drive is a conventional external USB drive without special mobile packaging made by Maxtor Corporation which serves as the backup drive and which is stored at Winer Obser- vatory. Until the transport drive arrives at OSU and its data have been successfully copied and verified, the backup drive at Winer is stored and left idle. In the event a transport drive arrives damaged, data from the backup drive will be copied to another transport drive and shipped. In the meantime, the removed drives are replaced with two others of each type and operations continue. At any given time, we have four external disk drives in use: two drives in operation, one in transit, and one stored as a backup. For transferring hundreds of gigabytes of images every few weeks during the prime observing season, this procedure has proven to be very reliable and efficient. To date, out of dozens of drive ship- ments, we have lost only two drives to damage in transit, with no loss of data (both arrived damaged at Winer af- ter their data were retrieved and copied at OSU). When a transport drive arrives at OSU, the drive is connected to our main data storage computer and all of the images are copied and run though a series of data quality checks. The image files are renamed, replacing the cumbersome default file name created by the CCDSoft application with a name indicating the field observed, the UTC date, and an image number. The images are then analyzed to measure image quality (modal sky and mean FWHM). If the modal sky value is above 800 ADU (due to moonlight, cloud cover, or other ambient light sources), the image is discarded. The cutoff at 800 ADU was determined using two months of representative data showing that images compromised by excessive light con- tamination consistently had sky values above that level and were unsuitable for photometry. The cutoff is high, with many poor images well below the cutoff, but we choose to be conservative about eliminating images early in the reduction process. The images that pass the initial filter on sky values, and the trimmed sections of the bad images, are stored on a multi-terabyte RAID storage ar- ray at OSU, providing data redundancy and fast access for data reduction and analysis. 2.5. Data Reduction Here we describe the data reduction process in brief, for the purposes of evaluating the performance of the KELT camera. Detailed descriptions of the reduction process will be included in an upcoming paper on the scientific results of KELT observing campaigns (J. Pep- per, in preparation). The data reduction pipeline consists of three steps. First, we process the images by subtracting dark frames and dividing by a flatfield. Second, we identify all the stars in the field and determine their instrumental mag- nitudes. Third, we obtain the photometry on all images using difference image analysis. Dark images are created for each night by median- combining 10 dark images – five from the beginning and end of each night. In early testing we determined that we can treat our dark frames as combined dark+bias. We take bias frames separately to monitor their stability, but do not incorporate them into the reduction process be- cause the bias has been extremely stable. In cases where dark frames were not taken or there were problems with the images, we use good dark frames from nights brack- eting the observations to create a substitute dark frame. We confirmed that using dark images from nearby nights did not significantly affect the statistics of the subtracted images – our dark images are quite stable from night to night. The KELT system is challenging to accurately flatfield. For the 200mm lens there is a combined decrease in flux of ∼18% between the center of the image and the edges, and a decrease of up to ∼26% between the center and the corners. For the 80mm lens, the decrease is ∼23% from the center to the edge, and ∼35% from the center to the corners. Because of the large KELT field of view (10.◦8 and 26◦ for the 200mm and 80mm lenses, respec- tively), twilight flats are not useful for flatfielding since the twilight sky is not uniform on those scales. Dome flats using a diffuse screen produced reasonable results with high signal-to-noise ratio. The flats are sufficiently repeatable that we do not need to regularly take dome flats. For relative photometry, the dome flats work ade- quately, and we are able to absolutely flatfield our images to ∼ 5% accuracy. Once images have been dark-subtracted and flatfielded, we can then create catalogs of images and measure the brightnesses of stars on the images. This photometric analysis is done in two basic steps. The first is to cre- ate a high-quality reference image for a field by com- bining a few dozen of the best images and then use the DAOPHOT software package (Stetson 1987) to identify all of the stars in the field down to a faint magnitude limit and measure their approximate instrumental mag- nitudes. See §4.1 for the details on how the instrumental magnitudes are calibrated to standard photometry. Once a template and DAOPHOT star catalog with baseline instrumental magnitudes have been created, the second step is to process the images with the ISIS image subtraction package (Alard & Lupton 1998; Alard 2000). Our reduction process is similar to that of Hartman et al. (2004). The ISIS package first spatially registers all of the images to align them with the reference image. The refer- ence image is convolved with a kernel for each image and subtracted, creating a difference image. The flux for each star identified on the reference image is then measured on each subtracted image using PSF-fitting photometry. Image subtraction has been shown in limited tests to be equal to or better than other photometric methods for the purposes of transit searches (Bakos 2006). In section §4 below, we provide additional information about the reduction process to obtain relative photometry. 3. INSTRUMENT PERFORMANCE In this section we quantify the performance of the KELT system by assessing in turn the telescope mount, the astrometric quality (geometric image quality), the image quality (position-dependent PSF), and photomet- ric sensitivity. 3.1. Telescope Performance Since the telescope was installed at Winer in October 2004, the hardware has performed up to specifications. There have been no significant problems with the mount or the control software. The pointing has not been per- fect: our fields are so large that minor pointing errors do not significantly affect our scientific results, but during normal operations the typical intra-night drift is ∼25′ in Declination and ∼9′ in Right Ascension. We believe the drift is due to a slight non-perpendicularity between the orientation of the camera and the axis of the mount. While the magnitude of the drift seems large, it repre- sents a movement of ∼65 pixels, less than 2% of the size of the field. It does not cause stars to move across large portions of the detector, and therefore does not lead to significant changes in the PSFs of individual stars. Since our reduction method utilizes image subtraction, we do lose the ability to take good photometry at the edges of a field. However, because of PSF distortions and other effects (see §3.3), we already have degraded sensitivity in those regions. Therefore the loss of coverage and sensi- tivity due to drift is quite small. Future alignment of the telescope will attempt to reduce or eliminate the drift. 3.2. Astrometric Performance Measurements of the positions of stars from the Tycho- 2 Catalog (Høg et al. 2000) are used to determine the conversion between pixel coordinates and celestial coor- dinates on the KELT images. We use the Astrometrix8 package to compute polynomial astrometric solutions for our images following the procedure described by Calabretta & Greisen (2002). To avoid stars that are sat- urated on the KELT images, we consider Tycho-2 stars with magnitudes 9.0 ≤ VTyc ≤ 10.0. From these we select up to 1000 stars per image. A first attempt to compute a global astrometric solution for the entire 4096×4096 im- age produced large residuals for most of the outer parts of the detector, with discrepancies between the predicted and actual positions of Tycho-2 stars of many tens of pix- els. Since our primary goal is to convert pixel coordinates into celestial coordinates on a star-by-star basis, a global solution is not required. We instead divide the image into 25 subimages on a 5×5 grid and perform a separate astrometric solution for each subimage. A third-order polynomial astrometric fit is computed for each subim- age using Astrometrix. The individual subimage fits give much better results, with offsets between predicted and measured positions of catalog stars at the subpixel level except at the extreme corners of the field. The subframes overlap by a few tens of pixels, and fits to stars common to adjacent subimages are consistent at the arcsecond level. For the 200mm lens, the typical RMS residuals are ±0.′′8, or < 10% of the average pixel size of ∼9.′′5. The 80mm lens has slightly worse RMS residuals, ±5′′or ∼20% of a pixel size of 22′′, but still well within tolerances for our purposes. Having a good astrometric fit to the images permits a quantitative assessment of the geometric performance of the KELT optics; specifically, variation in pixel size and shape across the field. Since this is a commercial lens with proprietary optical designs, there is no way to determine these propoerties a priori. This analysis will therefore be useful for anyone contemplating using ssim- ilar systems, and has implicatons for the potential use of theis setup - for example, such a camera/lens com- bination cannot be used to construct large-scale image mosaics. Furthermore, while sky subtraction deals with this effect, flatfielding does not. For the 200mm lens, the effective pixel size decreases by about 1% from center-to-edge from 9.′′537 near the center to 9.′′450 at the edges of the field (∼9.′′40 at the corners). Contours of constant effective pixel scale are circular and centered on the intersection of the optical axis of the 200mm camera lens and the CCD detector, as shown in Figure 2. The square CCD pixels do not perfectly project onto squares on the sky, but slowly dis- tort systematically away from the center, showing the characteristic signature of ∼0.5% pincushion distortion, expected for the manufacturer’s typical claims for their telephoto lenses. As Figure 3 shows, for the 80mm lens the effective pixel size decreases by about 3.5% from 8 http://www.na.astro.it/$\sim$radovich/wifix.htm center-to-edge, from 23.′′19 near the center to 22.′′40 at the edges of the field (and ∼21.′′8 at the corners). This effect is larger than the one seen with the 200mm lens, consistent with ∼2% pincushion distortion in this lens, typical of short focal-length wider-angle lenses. Contours of constant effective pixel scale are also circular and cen- tered on the CCD detector. The effect of the optical distortion is that pixels project onto smaller effective areas on the sky moving radially outward from the center of the CCD, making the sky appear non-flat (center-to-edge) at the ∼1.5% level for the 200mm lens and at the ∼6% level for the 80mm lens. There are two effects that act together to decrease the background sky level per pixel as you go radially outward from the center of the image: the decreasing pixel scale, and hence decreasing pixel area on the sky, with radius, and increasing vignetting with radius. 3.3. Image Quality As expected for such an optically fast system, the image PSF varies systematically as a function of posi- tion. The small physical pixel size (9µm) implies that the lens optics dominate the PSF, and we are insensitive to changes in atmospheric seeing. For both lenses, the systematic patterns in both the image full-width at half maximum (FWHM) and more refined measures of image quality (i.e. the 80% encircled energy diameter D80) can be used to quantitatively assess the position-dependent image quality. For the 200mm lens, typical image PSFs have FWHM of ∼1.8–2.9pixels, and 80% encircled energy diameters of D80 =4.7-9pixels. The 80mm lens has a similar range of FWHM for stellar image PSFs, and the 80% encircled energy diameters range from D80 =6-10pixels. There are significant changes in the detailed PSF shape across each image from the center to the extreme edges of the detector. Figure 4 shows representative stellar PSFs for a 5×5 grid across the CCD for the 200mm lens. The 80mm lens shows more pronounced distortions, as shown in Figure 5. Therefore, the 80mm lens has a roughly 24◦ diameter effective field of view with reasonably good images and little vignetting, whereas the 200mm lens works well over most of the CCD detector except at the extreme corners. Figures 6 and 7 show maps of the image FWHM as a function of position for the 200mm and 80mm lenses, respectively, derived from measurements of unsaturated, bright field stars in representative images. The most ob- vious feature in both is the strong vertical trend in in- creasing FWHM, with nearly no differences horizontally. Because this is seen with both lenses, which are of very different design, we believe this is because the CCD is tilted relative to the optical axis. Because the lens de- signs are proprietary, we do not know precisely how much the detector is tilted, nor the origin of the tilt at present. This effect could be due to how the detector is mounted inside the camera, or to the camera/lens mounting plate. This apparent field tilt also affects the maps of the 80% encircled-energy diameter D80, shown in Figures 8 and The 80mm lens has very stable imaging performance over time. The FWHM and D80 maps derived for im- ages of the same field over a 11-month period show no significant changes. We have, however, periodically ad- http://www.na.astro.it/$\sim $radovich/wifix.htm justed the focus when working with the telescope, which my cause some discontinuities in the data for the survey images. We will explore such effects in upcoming papers. Unfortunately, the PSF is not stable across the image over time for the 200mm lens. While intranight vari- ations in the FWHM maps are quite small, there are significant changes from night to night that have no ap- parent correlation with hour angle, CCD temperature, or any other physical or environmental parameter for which we have measurements. The effect of the changes we see is for the region of best FWHM (the base of the trough seen in the FWHM map in Figure 6) to move vertically on the CCD by many hundreds of pixels. We do not yet know the cause. The main effect is to complicate the difference-imaging reductions of the data. We will discuss these complications and their mitigation in the subsequent paper describing our results for the 200mm lens Praesepe cluster observing campaign. 3.4. Photometric Sensitivity Given the nature of the KELT bandpass (see Figure 1), we calibrate our instrumental magnitudes to the R band. We do so by rescaling our instrumental magnitudes by a constant, such that RK ≡ −2.5 log(ADU/sec) +RK,0 (1) where the instrumental ADU/sec is measured using aper- ture photometry with IRAF, RK is defined as an approx- imate KELT R magnitude, and RK,0 is the zero-point. We find that the RK magnitudes are within a few tenths of a magnitude of standard R band photometry, with the uncertainty dominated by the color term. Since we do not have V −I colors for all our stars, we quote observed magnitudes in RK , which can be considered equivalent to Johnson R, modulo a color term defined by V = RK + CV I(V − I) (2) where CV I is the (V − I) color coefficient, and V/I are in the Johnson/Cousins system. Since we do not have previously measured R magnitudes of large numbers of stars in our fields in our magnitude range, we relate RK to known magnitudes by matching stars from our ob- servations to the Hipparcos catalog, selecting only stars with measured V and I colors in Hipparcos. We take the mean instrumental magnitude from a set of high-quality images, and match the known magnitudes to the mean instrumental magnitudes, using Equations 1 and 2. For the cluster observations with the 200mm lens, we select 22 calibration stars, and measure their instrumen- tal magnitudes on 76 high-quality images, resulting in a magnitude zero-point of RK,0 = 16.38 ± 0.06 mag, and CV I = 0.55± 0.2. For the survey observations with the 80mm lens, we use 59 stars on 77 images, resulting in a magnitude zero-point of RK,0 = 15.15 ± 0.07 mag, and CV I = 0.5 ± 0.3. Since the Hipparcos stars we use to calibrate our data have (V − I) colors mostly between 0 and 1, we expect our calibrations to be less accurate for redder stars. These measured zero-point uncertainties suggest that the flatfield corrections are good to within ∼ 5% in absolute accuracy. Tying together the full calibration process, we find that a fiducial R = 10 mag star at the field center with (V − I) = 0 has a flux of 356 counts per second with the 200mm lens, and 115 counts per second with the 80mm lens. Scaling those numbers by the different aperture sizes of the lenses (71mm aperture for the 200mm lens and 42mm aperture for the 80mm lens), we find that the 200mm lens is about 8% more efficient than the 80mm lens. 4. RELATIVE PHOTOMETRY KELT was designed primarily for precision time-series relative photometry (see Everett & Howell (2001) for background). The crucial test for our instrument is the ability to obtain long-term lightcurves with low noise and minimal systematics. A simple test of KELT’s photo- metric performance is to examine the root-mean-squared (RMS) of the magnitudes of an ensemble of lightcurves as a function of magnitude. We apply the ISIS image sub- traction package to samples of our data to obtain relative photometry, and measure the statistics of the resulting lightcurves. Our criteria for the ability to detect plan- etary transits is the presence of substantial numbers of stars for which the RMS of the lightcurves are below the 2% and 1% levels. 4.1. Difference Imaging Performance The instrumental magnitudes for the KELT lightcurves are produced through a combination of ISIS and DAOPHOT photometry. This process involves some careful conversion between DAOPHOT and ISIS flux measurements – see Appendix B of Hartman et al. (2004) for the details of the conversion. First, we create a ref- erence image by combining a number of high-quality im- ages. DAOPHOT measures the instrumental magnitude of the stars on the reference image mi(ref) based on PSF fitting photometry, with the magnitude of each star i cal- culated from the flux by mi(ref) ≡ 25− 2.5 log[fi(ref)] + Cap, where fi is the flux measured by DAOPHOT and Cap is an aperture correction to ensure that mi(ref) = 25 − 2.5 log(ci), where ci is the counts per second from the star in ADU. ISIS then creates an ensemble of sub- tracted images for the whole data set using the refer- ence. To derive the full light curve, ISIS fits a PSF for each star on each subtracted image j, to obtain a flux fi(j). The DAOPHOT-reported instrumental mag- nitudes for the reference images serve as the magnitude baseline for the conversion of ISIS fluxes to magnitudes, where the magnitude of the ith star on the jth image is mi(j) = mi(ref)− 2.5 log[1− fi(j)/fi(ref)]. We then calculate the RMS variation of all the detected stars in both the Praesepe data set and for a sample of the survey data. Because of the night-to-night varia- tions in the position of the best image quality on the detector with the 200mm lens described at the end of §3.3, we calculate the RMS for the Praesepe data from a single night of observations, to better demonstrate the intrinsic instrumental performance. In Figure 10 we plot the distribution of RMS versus RK magnitude for 67,674 stars on 32 images with 60-second exposures from one night. With this lens and exposure time, we obtain pho- tometry of stars in the magnitude range RK = 8 − 16 mag. For stars brighter than about RK = 9.5mag, systematics begin to dominate the light curves, mostly due to saturation. Out of the 67,674 stars, 4,281 have RMS< 0.02mag, and 1,369 have RMS< 0.01mag. We perform the same analysis for one of the regu- lar survey fields observed with the 80mm lens using 239 observations over 8 nights with 150-second expo- sures. We obtain photometry on 49,376 stars in the range RK = 6 − 14 mag, and plot the data in Figure 11. We find 14,333 stars with RMS< 0.02mag, and 3,822 stars with RMS< 0.01mag, with systematics dominating for stars brighter than RK = 7.5mag. Overall, the RMS performance is mostly as expected from Poisson statistics except at the bright end. The best precision just reaches the theoretical noise limit, with a spread of RMS values above that limit due to real-world effects. For the brightest stars in our data we see a floor in which the RMS no longer decreases as the stars get brighter, and instead becomes roughly constant at RMS = 0.004 magnitudes. The RMS floor is indicative of a fixed pattern noise component, caused by intrapixel sen- sitivity on the CCD. The Kodak KAF-E series CCDs are 2-phase front-side illuminated devices in which the second poly-gate electrode on each pixel is a transparent gate made of Indium-Tin-Oxide (ITO) to boost the over- all quantum efficiency of the device (Meisenzahl et al. 2000). Over much of the wavelength regime of interest for KELT, the ITO material is ∼2 times more transparent than the regular silicon oxide material used on the first poly-gate. The result is significant pixel substructure in which the quantum efficiency varies stepwise across each 9µm pixel, which introduces a component of fixed- pattern noise that produces the observed RMS floor. We note that more recent models of commercial CCD cam- eras with the Kodak KAF-16801 detector are using a newer version of this device that incorporate a front- surface microlens array that mitigates the intrapixel step in transmission, but persons contemplating similar sys- tems to our own should be aware of the issue and take it into account. We do not expect to obtain high-precision photometry for the very brightest stars in our data, but the RMS floor is well below the 1% level and it should not significantly affect our ability to detect transits. We plot noise models in Figures 10 and 11, which include photon noise and sky noise, along with the RMS floor. In the future we will choose a lens focus that makes slightly larger FWHM images to minimize these effects. 5. REPRESENTATIVE RESULTS The RMS plots shown in Figures 10 and 11 demon- strate that our telescope can obtain precision relative photometry, with large numbers of stars measured at the 1% level. However, simply looking at the RMS informa- tion does not prove that the data set can yield the con- sistent quality with low systematics necessary for a tran- sit search. To illustrate that we can fulfill that require- ment, we show in Figure 12 the lightcurves of three sam- ple variable stars we have discovered using the 200mm lens while observing the field of Praesepe. Even with the night-to-night variations discussed in §3.3, we are able to achieve the consistent data quality that transit detec- tion requires, and are able to clearly see features in the phased lightcurves of a few percent or less. An even better example of our ability to detect tran- sits can be seen in Figure 13. Here we show two objects we detected in our data from the Praesepe field which exhibit transit-like dips in their lightcurves. These ef- fects can be clearly seen at the level of a few percent. These objects are not planetary: follow-up spectroscopy indicates that the top object is an F star with a tran- siting M dwarf companion, and the bottom object is a grazing eclipsing binary (Latham 2007). However, they demonstrate that we can confidently detect transit-like behavior at the 1%-2% level with our telescope. A full catalog of the variable stars and transit candidates from the KELT observations of the Praesepe field will appear in a forthcoming paper (J. Pepper, in preparation). 6. SUMMARY AND DISCUSSION The KELT project has been acquiring data since Oc- tober 2004. We observed the field of the Praesepe open cluster with the 200mm lens for two months in early 2005, and have spent the rest of the time using the 80mm lens for a survey of 13 fields around the Northern sky. The KELT telescope, used in “Survey Mode” with the 80mm lens, reflects the design specifications called for in the theoretical paper Pepper, Gould, & DePoy (2003). The performance of the telescope as described in this pa- per provides a real-world evaluation of the potential for this telescope to detect transiting planets. In addition to the lightcurves and RMS plots that show the tele- scope’s abilities, we find that the total number of pho- tons acquired from a fiducial V = 10 mag star over an entire observing run is well above the number assumed in Pepper, Gould, & DePoy (2003) with the parameter gamma0. This paper has described the KELT instrumentation and performance, with both the 200mm lens used for observing clusters and the 80mm lens used for conduct- ing the all-sky survey. It is the widest-field instrument that is currently being employed to search for transit- ing planets, and we observe brighter stars than other wide field surveys. The performance metrics demon- strate that it is capable of detecting signals at the ∼1% level needed to detect Jupiter size planets transiting solar-type stars. Further refinements of our reduction process promise to expand our sensitivity to transits, such as applying detrending algorithms of the sort devel- oped by Tamuz, Mazeh, & Zuker (2005). We also plan to conduct a full analysis of red noise (i.e. temporally correlated systematic noise, see Pont, Zuker, & Queloz (2006) for a full description) for all KELT data. To date, we have detected over 100 previously unknown variable stars in our observations towards Praesepe, and we have identified several lightcurves with transit-like be- havior, of which we show two in Figure 13. Future pa- pers will report on the success in discovering variable stars and searching for planets in Praesepe, along with the full transit search for the all-sky survey. We would like to thank the many people who have helped with this research, including Scott Gaudi, An- drew Gould, Christopher Burke, and Jerry Mason. We would also like to thank Apogee Instruments and Soft- ware Bisque for supplying the camera and mount for the telescope and for excellent service for the hardware. This work was supported by the National Aeronautics and Space Administration under Grant No. NNG04GO70G issued through the Origins of Solar Systems program. REFERENCES Alard, C. & Lupton, R.H. 1998, ApJ, 503, 325 Alard, C. 2000, A&A, 144, 363 Alonso, R., et al. 2004, ApJ, 613, L153 Bakos, G. A. 2006, private communication Bakos, G. A. 2006, ApJ, 656, 552 Barnes, J. W. & Fortnoy, J. J. 2004, ApJ, 616, 1193 Calabretta, M.R., & Greisen, E.W. 2002, å, 395, 1077 Cameron, A. C., et al. 2007, MNRAS, 375, 951 Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, R. L. 2002, ApJ, 568, 377 Charbonneau, D., Brown, T. M., Burrows, A., & Laughlin, G. 2007, in Protostars & Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil (Tucson: University of Arizona Press), in press (astro-ph/0603376) Everett, M. E. & Howell, S. 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K., Kubiak, M., Pietrzynski, G., Soszynski, I., Zebrun, K., Szewczyk, O., & Wyrzykowski, L. 2004, Acta, 54, 313 http://arxiv.org/abs/astro-ph/0603376 0.1.2.3.4.5.6 Response Fig. 1.— Calculated response function of the KELT CCD camera and Kodak #8 Wratten filter. The dashed curve is the response function including atmospheric transmission at Winer Observatory for 1.2 airmasses. This response function does not include the transmission of the camera lenses. −10 0 10 X (mm) 9.525 9.500 9.475 9.537"/pix Fig. 2.— Effective pixel scale in arcseconds pixel−1 for the KELT 200mm camera. Contours show curves of constant effective pixel scale. The cross (+) marks the optical center of the field, where the pixel scale is 9.′′537 pix−1. −10 0 10 X (mm) 21.68"/pix Fig. 3.— Effective pixel scale in arcseconds pixel−1 for the KELT 80mm camera. Contours show curves of constant effective pixel scale. The cross (+) marks the optical center of the field, where the pixel scale is 21.′′68 pix−1. −14,14 −7,14 0,14 7,14 14,14 −14,7 −7,7 0,7 7,7 14,7 −14,0 −7,0 0,0mm 7,0 14,0 −14,−7 −7,−7 0,−7 7,−7 14,−7 −14,−14 −7,−14 0,−14 7,−14 14,−14 Fig. 4.— Representative stellar PSFs from the 200mm telephoto lens, shown as intensity contours of bright, unsaturated stars taken with a 5×5 grid pattern on the CCD; the position of the center of each box, relative to the center of the image, is indicated in mm. Each box is 15 pixels (135mm) on a side. The scale-bar in the center panel indicates 50µm on the detector (5.5 pixels). Contours show levels of (5,10,15,20,25,30,40,50,60,70,80,90,100)% of the peak intensity. −14,14 −7,14 0,14 7,14 14,14 −14,7 −7,7 0,7 7,7 14,7 −14,0 −7,0 0,0mm 7,0 14,0 −14,−7 −7,−7 0,−7 7,−7 14,−7 −14,−14 −7,−14 0,−14 7,−14 14,−14 Fig. 5.— Representative stellar PSFs from the 80mm wide-angle lens, displayed in the same format as in Figure 4. The images are more severely distorted at the extreme edges of the field than with the 200mm lens. −10 0 10 X (mm) Fig. 6.— Contours of constant FWHM for stellar images in a representative 200mm lens KELT image, with FWHM given in pixels. Contours are based on a smooth polynomial surface fit to measurements of ∼1200 bright, unsaturated stars distributed across the image. Contour spacing is every 0.1 pixels, with particular contour level values in pixels as indicated. −10 0 10 X (mm) Fig. 7.— Contours of constant FWHM for stellar images in a representative 80mm lens KELT image, with FWHM given in pixels. Format is the same as in Figure 6. Contours are based on measurements of ∼2100 bright, unsaturated stars. −10 0 10 X (mm) D80=4.7pix Fig. 8.— Contours of constant 80% Encircled-Energy Diameter (D80) in pixels for stellar images in a representative 200mm lens KELT image. Contours are based on a smooth polynomial surface fit to measurements of ∼1200 bright, unsaturated stars distributed across the image. −10 0 10 X (mm) D80=6.12 pix Fig. 9.— Contours of constant 80% Encircled-Energy Diameter (D80) in pixels for stellar images in a representative 80mm lens KELT image. Contours are based on measurements of ∼2100 bright, unsaturated stars. Fig. 10.— RMS plot for one night of data from the 200mm lens. Data are shown for 67,674 stars. The dashed lines show the limits for 1% and 2% RMS. The solid line represents a noise model including photon noise and sky noise, along with an RMS floor of 0.004 magnitudes. Fig. 11.— RMS plot for eight nights of data from the 80mm lens. Data are shown for 49,376 stars. The dashed lines show the limits for 1% and 2% RMS. The solid line represents a noise model including photon noise and sky noise, along with an RMS floor of 0.004 magnitudes. Fig. 12.— Three variable stars discovered with the 200mm lens in the field of the Praesepe open cluster. Fig. 13.— Two transit candidates discovered with the 200mm lens in the field of the Praesepe open cluster. The lower panel of each plot shows the data binned in 10-minute bins. Follow-up spectroscopy indicates that object (a) is an F star with a transiting K dwarf companion, and object (b) is a grazing eclipsing binary.

0704.0461

Entanglement increase from local interactions with not-completely-positive maps

Entanglement increase from local interactions with not-completely-positive maps Thomas F. Jordan∗ Physics Department, University of Minnesota, Duluth, Minnesota 55812 Anil Shaji† The University of New Mexico, Department of Physics and Astronomy, 800 Yale Blvd. NE, Albuquerque, New Mexico 87131 E. C. G. Sudarshan‡ The University of Texas at Austin, Center for Statistical Mechanics, 1 University Station C1609, Austin Texas 78712 Simple examples are constructed that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. One of the two qubits interacts with a third qubit, a control, that is never entangled or correlated with either of the two entangled qubits and is never entangled, but becomes correlated, with the system of those two qubits. The two entangled qubits do not interact, but their state can change from maximally entangled to separable or from separable to maximally entangled. Similar changes for the two qubits are made with a swap operation between one of the qubits and a control; then there are compensating changes of entanglement that involve the control. When the entanglement increases, the map that describes the change of the state of the two entangled qubits is not completely positive. Combination of two independent interactions that individually give exponential decay of the entanglement can cause the entanglement to not decay exponentially but, instead, go to zero at a finite time. Keywords: Entanglement, Quantum information I. INTRODUCTION We construct simple examples here that show the entan- glement of two qubits being both increased and decreased by interactions on just one of them. In our first and basic step, taken in Sec. II, we have one of the two qubits interact with a third qubit, a control, that is never entangled or correlated with either of the two entangled qubits and is never entangled, but becomes correlated, with the system of those two qubits. In Sec. III, we do this for each of the two entangled qubits, and consider the combination of the two interactions, with sepa- rate control qubits that are not correlated and do not interact with each other. The two entangled qubits do not interact, but their state can change from maximally entangled to separable or from separable to maximally entangled. Similar changes for the two qubits are made with a swap operation between one of the qubits and a control; then there are compensating changes of entanglement that involve the control. This is de- scribed in Sec. II.A. Whenever the entanglement increases, and in some cases where the entanglement decreases, the map that describes the change of the state of the two entangled qubits is not com- pletely positive and does not apply to all states of two qubits. It all depends on whether there are correlations with the con- trols at the beginning of the interval for which the dynamics is considered. The maps are described in Sec. IV and discussed in Sec. V. The completely positive maps that decrease the en- tanglement have already been described [1]. When the interaction of each qubit with its control by itself ∗email: [email protected] †email: [email protected] ‡email: [email protected] gives exponential decay of the entanglement, the combination of the two interactions gives exponential decay at the rate that is the sum of the rates for the individual interactions, when the two interactions are made the same way. Making them differ- ently can cause the entanglement to not decay at that rate or at any single rate. Instead, the entanglement goes to zero at a finite time; the state becomes separable and remains separable at later times. This is described in Sec. III.A. Similar behavior has been observed in more physically interesting and mathe- matically complicated models [2, 3, 4]. These examples are built on the same framework, but to a very different design, from those we made for Lorentz trans- formations that entangle spins [5]. There the momenta that played the roles of controls were purposely correlated. Here the controls are kept independent. The framework makes the operations transparent by describing the qubit states with den- sity matrices written in terms of Pauli matrices, so you can see the Pauli matrices being rotated by the interactions. States are shown to be separable by writing out the density matrices explicitly as sums of products for pure states. For each in- teraction here, the map that makes the change of the density matrix for the entangled qubits is described by a simple rule that particular Pauli matrices in the density matrix are multi- plied by a number; equivalently, the map of the state of the entangled qubits is described by a rule that particular mean values are multiplied by a number. Our examples show that statements like “entanglement should not increase under local operations and classical com- munication” [6, 7] are not generally true outside the set of local operations considered in the original proof [6]. In our examples, each control qubit interacts with only one of the two entangled qubits. In this sense, the quantum operations are local. Correlation with a control at the beginning of the interval for which the dynamics is considered can give local operations that increase entanglement. http://arxiv.org/abs/0704.0461v2 mailto:[email protected] mailto:[email protected] mailto:[email protected] II. ONE INTERACTION We consider the entanglement of two qubits, A and B. We use Pauli matrices Σ1,Σ2,Σ3 for qubit A, and Pauli matrices Ξ1,Ξ2,Ξ3 for qubit B. We let qubit A interact with a third qubit, which we call C. We think of C as a control. By inter- acting with qubit A, it will control the entanglement of qubits A and B. We work with states represented by orthonormal vectors |α〉 and |β〉 for C. We consider a state of the three qubits represented by a density matrix Π = ρ⊗ 1 11C (2.1) with ρ the density matrix for the state of qubits A and B. We follow common physics practice and write a product of operators for separate systems, for example a product of Pauli matrices Σ andΞ for qubitsA andB, simply as ΣΞ, not Σ⊗Ξ. Occasionally we insert a ⊗ for emphasis or clarity. We write 11A, 11B , 11C , but we do not put labels A and B on the Σj and Ξk. The single statement that the Σj are for qubit A and the Ξk are for qubit B eliminates the need for continual use of both A and B lalels and ⊗ signs. Suppose ρ is one of the density matrices (11 ± Σ1Ξ1 ± Σ2Ξ2 − Σ3Ξ3). (2.2) Both ρ+ and ρ− represent maximally entangled pure states for the two qubits. They are Bell states. The state of zero total spin is represented by ρ− and the state obtained from that by rotating one of the spins by π around the z axis is represented by ρ+. For a rotation W , let DA(W ) be the 2 × 2 unitary rotation matrix made from the Σj so that DA(W ) ΣDA(W ) = W (Σ) (2.3) where W (Σ) is simply the vector Σ rotated by W . Let W (φ) be the rotation by φ around the z axis, and let DA(φ) be DA(W (φ)). We consider an interaction between qubits A and C de- scribed by the unitary transformation U = DA(φ)|α〉〈α| +DA(−φ)|β〉〈β| (2.4) or, in Hamiltonian form, U = e−iφH (2.5) H = Σ3 (|α〉〈α| − |β〉〈β|). (2.6) This changes the density matrix ρ for qubits A and B to ρ′ = TrC (U ⊗ 11B)Π(U ⊗ 11B)† DA(φ)ρDA(φ) DA(−φ)ρDA(−φ)†. (2.7) For ρ± this gives ρ′± = [11 ± (Σ1 cosφ+Σ2 sinφ)Ξ1 ± (−Σ1 sinφ+Σ2 cosφ)Ξ2 − Σ3Ξ3] [11 ± (Σ1 cosφ− Σ2 sinφ)Ξ1 ± (Σ1 sinφ+Σ2 cosφ)Ξ2 − Σ3Ξ3] [11 ± (Σ1Ξ1 +Σ2Ξ2) cosφ− Σ3Ξ3] = ρ± cos 2(φ/2) + ρ∓ sin 2(φ/2). (2.8) A. From maximally entangled to separable and back We focus first on the case where φ is π/2. Then both ρ+ and ρ− are changed to [11 − Σ3Ξ3] (11 − Σ3) (11 + Ξ3) (11 +Σ3) (11 − Ξ3). (2.9) The density matrix ρ for a maximally entangled state is changed to the density matrix ρ′ for a separable state that is a mixture of just two products of pure states. The inverse of the unitary dynamics of qubits A and R takes ρ′ back to ρ; it changes a separable state to a maximally entangled state. The dynamics continuing forward also changes this separa- ble state to a maximally entangled state. As φ goes from π/2 to π, the density matrix ρ′± changes from that of Eq. (2.9) to ρ′± = ρ∓. (2.10) There can be revivals of entanglement between two qubits when there is no interaction between them, as well as when [8] there is. Changes in the state of qubits A and B from maximally entangled to separable and back to maximally entangled can also be made very simply with a swap of states[9] between A and C. This can be done with a unitary operator U ⊗ 11B with U a unitary operator for qubits A and C that acts on a basis of product state vectors simply by interchanging the states of A and C. There is interaction between qubits A and C only; qubit B is not involved. Applied to an initial state described by Eqs. (2.1) and (2.2), where qubits A and B are maximally entangled, this swap op- eration gives a separable state for A and B. Applied a second time, it restores the initial state where A and B are maximally entangled. For qubits A and B, this is similar to what hap- pens when φ goes from 0 to π/2 to π. For the three qubits, it is different. The swap operation does not change the com- plete inventory of entanglements for the three qubits. It just moves the entanglements around. In particular, C becomes maximally entangled with B. We will see, in Secs. II.C and D, that the interaction described by Eqs. (2.4), (2.5) and (2.6) does change the complete inventory of entanglements for the three qubits. When the state of qubits A and B changes from maximally entangled to separable and back to maximally en- tangled, there are no compensating changes of other two-part entanglements. In particular, qubit C never becomes entan- gled with anything. B. Concurrence The change of entanglement is smaller when φ does not change by π/2. ¿From Eq. (2.8) we have ρ′± = [11±(Σ1Ξ1+Σ2Ξ2) cosφ+(Σ1Ξ1)(Σ2Ξ2)], (2.11) after rewriting the last term. This shows that for both ρ′ ρ′− the eigenvalues are (1 + cosφ), (1 − cosφ), 0, 0 (2.12) because Σ1Ξ1 and Σ2Ξ2 each have eigenvalues 1 and −1 and together they make a complete set of commuting operators: their four different pairs of eigenvalues label a basis of eigen- vectors for the space of states for the two qubits. The Wooters concurrence [10] is a measure of the entanglement in a state of two qubits. It is defined by C(ρ) ≡ max (2.13) where ρ is the density matrix that represents the state and λ1, λ2, λ3, λ4 are the eigenvalues, in decreasing order, of ρ Σ2Ξ2 ρ ⋆ Σ2Ξ2, with ρ ⋆ the complex conjugate that is ob- tained by changing Σ2 and Ξ2 to −Σ2 and −Ξ2. From Eq. (2.11) we have ρ′± Σ2Ξ2 (ρ ⋆ Σ2Ξ2 = ρ ⋆ (Σ2Ξ2) 2=(ρ′±) 2 (2.14) so for ρ′± the λi are the eigenvalues of ρ ± and the concur- rence is C(ρ′±) = | cosφ|. (2.15) We can consider the change of entanglement as φ changes through any interval. When | cosφ| decreases, the entangle- ment decreases. When | cosφ| increases, the entanglement increases. C. Two-part entanglements The only two-part entanglements are when qubit A is in one part and qubit B is in the other. There is entanglement between qubitA and the subsystem of two qubits B and C and between qubit B and the subsystem of two qubits A and C, as well as between qubits A and B. There is never entanglement between the state of qubit C and the state of the subsystem of two qubits A and B. The density matrix (U ⊗ 11B)Π(U ⊗ 11B)†= DA(φ)ρ±DA(φ) †|α〉〈α| DA(−φ)ρ±DA(−φ)†|β〉〈β| (2.16) is always a mixture of two products of pure states. The re- duced density matrix for the subsystem of qubits A and C, obtained by taking the trace over the states of qubit B, is just 11A ⊗ 11C/4, and the reduced density matrix for qubits B and C, obtained by taking the trace over the states of qubit A, is 11B ⊗ 11C/4. There is never entanglement or correlation between qubits A and C or between qubits B and C. The reduced density matrices for the individual single qubits are just 11A/2, 11B/2, and 11C/2. The only subsystem density ma- trix that carries any information is the density matrix ρ for the qubits A and B, which is changed by the interaction with qubit C. The entropy of the subsystem of qubits A and B can increase or decrease, but there is no change of entropy for any other subsystem or for the entire system of three qubits. D. Three-part entanglement There is three-part entanglement. The state represented by the density matrix (2.16) is called biseparable because it is separable as the state of a system of two parts, with C one part and the subsystem of two qubits A and B the other part. It is not separable as the state of a system of three parts A, B, and C. The density matrix (2.16) is not a mixture of products of density matrices for pure states of the individual qubits A, B, and C. If it were, its partial trace over the states of C, the reduced density matrix that represents the state of the subsys- tem of the two qubitsA andB, would be a mixture of products for pure states of A and B. That happens only when cosφ is 0. In that case, we can see that the density matrix (2.16) is not a mixture of products for pure states of the individual qubits A, B, and C because its partial transpose obtained by changing Ξ2 to −Ξ2 is not a positive matrix. In the classification of three-part entanglement for qubits, biseparable states are between separable states and states that involve W or GHZ entanglement [11, 12, 13]. Let Π1, Π2, Π3 be Pauli matrices for the qubit C such that |α〉〈α| is (1/2)(11 +Π3) and |β〉〈β| is (1/2)(11 −Π3). Bounds from Mermin witness operators say that for separable or bisepara- ble states −2 ≤ 〈ΣjΞjΠj−ΣjΞkΠk−ΣkΞjΠk−ΣkΞkΠj〉 ≤ 2 (2.17) for j, k = 1, 2, 3 and j 6= k; a mean value outside these bounds is a mark of W or GHZ entanglement [14]. In our examples, these mean values are always 0. A mean value 〈|GHZ〉〈GHZ|〉 larger than 3/4 for the projection operator onto the GHZ state, |GHZ〉 = 1√ |0〉|0〉|0〉+ 1√ |1〉|1〉|1〉, (2.18) is a mark of GHZ entanglement; it can not be larger than 3/4 for a W state [13]. A mean value 〈|GHZ〉〈GHZ|〉 larger than 1/2 is a mark of a W state; it can not be larger than 1/2 for a biseparable state [13]. In our examples, 〈|GHZ〉〈GHZ|〉 is always 0. A mean value 〈|W 〉〈W |〉 larger than 2/3 for the projection operator onto the W state, |W 〉 = 1√ |1〉|0〉|0〉+ 1√ |0〉|1〉|0〉+ 1√ |0〉|0〉|1〉, (2.19) is a mark of W entanglement; it can not be larger than 2/3 for a biseparable state [13]. In our examples, 〈|W 〉〈W |〉 = 1 (1 ± cosφ). (2.20) This mean value does not involve either entanglement or cor- relation of the qubitC; it would be the same if both |α〉〈α| and |β〉〈β| in the density matrix (2.16) were replaced by (1/2)C , the completely mixed density matrix for C. For any φ, the density matrices (2.16) for the two cases + and − are changed into each other by the local unitary trans- formation that changes the Pauli matrices for one of the qubits A or B by rotating its spin by π around the z axis. As a func- tion of φ, the mean value 〈|W 〉〈W |〉 changes in opposite di- rections for the + and − cases. So will any mean value for the states described by the density matrices (2.16), if it changes at For the states described by the density matrices (2.16), the only nonzero mean values that involve the qubit C are 〈Σ1Ξ2Π3〉 = ∓ sinφ 〈Σ2Ξ1Π3〉 = ± sinφ. (2.21) These would be the same if they were calculated with only the |α〉〈α| part or only the |β〉〈β| part of the density matrix (2.16). In fact, they are the same as 〈Σ1Ξ2〉〈Π3〉 and 〈Σ2Ξ1〉〈Π3〉 calculated for one of those parts. Their values do not require either entanglement or correlation of C. III. TWO INTERACTIONS If a control were coupled similarly to qubit B as well, then cosφ would be replaced by cosφA cosφB in the next to last line of Eq. (2.8) and in Eqs.(2.11) and (2.15). If the coupling of qubit B is made with a rotation around the x axis instead of the z axis, then the next to last line of Eq. (2.8) becomes ρ′± = [11 ± Σ1Ξ1 cosφA ± Σ2Ξ2 cosφA cosφB −Σ3Ξ3 cosφB ]. (3.1) Rewriting the last term and looking at eigenvalues in terms of Σ1Ξ1 and Σ2Ξ2 as before yields the concurrence C(ρ′±)= max[0, | cosφA|+| cosφA cosφB|+| cosφB|−1]. (3.2) When cosφA is 1, these Eqs.(3.1) and (3.2) describe the result obtained when there is only the interaction of qubit B made with a rotation around the x axis. If neither cosφA nor cosφB is 1, the concurrence becomes zero, the state separable, be- fore cosφA or cosφB is zero. The interactions of qubits A and B with their controls change maximally entangled states to separable states. The inverses change separable states to maximally entangled states. In the following subsection, we describe the density matrices that show explicitly that the sep- arable states are mixtures of products of pure states. A. Exponential decay To describe exponential decay of entanglement we let cosφA = e −ΓAt, cosφB = e −ΓBt (3.3) by letting each interaction be modulated by a time-dependent Hamiltonian H(t) that is related to the Hamiltonian H of Eqs.(2.4) and (2.5) by H(t) = H = HΓcotφ, (3.4) where φ and Γ are φA and ΓA or φB and ΓB . The same result could be produced in different ways. The interactions could be with large reservoirs instead of qubit controls [2, 3, 4]. Each qubit A or B could interact with a stream of reservoir qubits [15]. Here we are interested in the way the entanglement is changed by the combination of the two interactions. That de- pends only on the changes in the density matrix ρ for qubits A and B, not on the nature of the controls and the interactions. Maps that make the changes in ρ will be described in the next section. If there is only the interaction of qubit A with qubit C, the concurrence is e−ΓAt. If there is only interaction of qubit B with its control, the concurrence is e−ΓBt. If there are both and both are made with rotations around the z axis, the con- currence is e−ΓAte−ΓBt. If there are both and the interaction of qubit B with its control is made with a rotation around the x axis, the concurrence is C(ρ′±)= max[0, e−ΓAt+e−ΓAte−ΓBt+e−ΓBt−1]. (3.5) This concurrence (3.5) is zero when e−ΓAt + e−ΓAte−ΓBt + e−ΓBt = 1. (3.6) Then the state is separable; it is a mixture of six products of pure states: from Eqs.(3.1) and (3.3) ρ′± = e−ΓAt (11 +Σ1) (11 ± Ξ1) e−ΓAt (11 − Σ1) (11 ∓ Ξ1) e−ΓAte−ΓBt (11 +Σ2) (11 ± Ξ2) e−ΓAte−ΓBt (11 − Σ2) (11 ∓ Ξ2) e−ΓBt (11 +Σ3) (11 − Ξ3) e−ΓBt (11 − Σ3) (11 + Ξ3). (3.7) The state remains separable at later times; when the sum of the exponential decay factors is less than 1, the density ma- trix is a mixture in which just a multiple of the density ma- trix 1/4 for the completely mixed state is added to the terms of Eq. (3.7). This change of maximally entangled states to separable states can be described without reference to expo- nential decay by continuing to use cosφA and cosφB instead of e−ΓAt and e−ΓBt. Similar behavior involving exponential decay has been observed in more physically interesting and mathematically complicated models [2, 3, 4]. IV. MAPS The maps that make the changes in the density matrix ρ for qubits A and B could be described in different ways using various matrix forms. That is not needed here. Writing ρ in terms of Pauli matrices provides a very simple way to describe the maps. For any density matrix 〈Σj〉Σj + 〈Ξk〉Ξk + j,k=1 〈ΣjΞk〉ΣjΞk (4.1) for qubits A and B, the result of the interaction of qubit A with qubit C, described by Eq. (2.7), is that in ρ, in both the Σj and ΣjΞk terms, Σ1 −→ Σ1 cosφA, Σ2 −→ Σ2 cosφA; (4.2) the result of the interaction of qubit B with its control is that Ξ1 −→ Ξ1 cosφB, Ξ2 −→ Ξ2 cosφB (4.3) if the interaction is made with a rotation around the z axis; and the result of the interaction of qubit B with its control is that in ρ Ξ2 −→ Ξ2 cosφB, Ξ3 −→ Ξ3 cosφB (4.4) if the interaction is made with a rotation around the x axis. The terms with sin φ cancel out because there is an equal mix- ture of parts with φ and parts with −φ. The changes in the state of qubits A and B can be described equivalently by maps of mean values that describe the state: the result of the interaction of qubit A with qubit C, described by Eq. (2.7), is that 〈Σ1〉 −→ 〈Σ1〉 cosφA 〈Σ2〉 −→ 〈Σ2〉 cosφA 〈Σ1Ξk〉 −→ 〈Σ1Ξk〉 cosφA 〈Σ2Ξk〉 −→ 〈Σ2Ξk〉 cosφA (4.5) for k = 1, 2, 3; the result of the interaction of qubit B with its control is that 〈Ξ1〉 −→ 〈Ξ1〉 cosφB 〈Ξ2〉 −→ 〈Ξ2〉 cosφB 〈ΣjΞ1〉 −→ 〈ΣjΞ1〉 cosφB 〈ΣjΞ2〉 −→ 〈ΣjΞ2〉 cosφB (4.6) for j = 1, 2, 3 if the interaction is made with a rotation around the z axis; and the result of the interaction of qubit B with its control is that 〈Ξ2〉 −→ 〈Ξ2〉 cosφB 〈Ξ3〉 −→ 〈Ξ3〉 cosφB 〈ΣjΞ2〉 −→ 〈ΣjΞ2〉 cosφB 〈ΣjΞ3〉 −→ 〈ΣjΞ3〉 cosφB (4.7) for j = 1, 2, 3 if the interaction is made with a rotation around the x axis. When φA and φB change over intervals from initial values φAi and φBi to final values φAf and φBf , the cosφA and cosφB factors in the maps are replaced by cosφAf/ cosφAi and cosφBf/ cosφBi. If either of these factors is larger than 1, the map is not completely positive and does not apply to all density matrices ρ for qubitsA andB. This happens whenever the entanglement increases. It also happens in cases where the concurrence (3.2) decreases, when one of cosφA and cosφB increases and the other decreases and there is more decrease than increase. The completely positive maps that decrease the entanglement have already been described [1]. V. RECONCILIATION Entanglement being increased by local interactions may seem surprising from perspectives framed by experience in common situations where it is impossible. Entanglement is not increased by a completely positive map of the state of two qubits produced by an interaction on one of them. The in- teraction will produce a completely positive map if it is with a control whose state is initially not correlated with the state of the two qubits, as in Eq. (2.1). In our examples, that hap- pens only when the initial value of φ is 0 or a multiple of π. Otherwise, the state of the control is correlated with the state of the two qubits as in Eq. (2.16). When a subsystem is ini- tially correlated with the rest of a larger system that is being changed by unitary Hamiltonian dynamics, the map that de- scribes the change of the state of the subsystem generally is not completely positive and applies to a limited set of subsys- tem states [16, 17]. We see this in our examples whenever the entanglement increases and in some cases when the entangle- ment decreases. The map depends on both the dynamics and the initial cor- relations. It describes the effect of both in one step. Com- pletely positive maps are what you get in the simplest set-up where you bring a system and control together in independent states and consider the effect of the dynamics that begins then. Dynamics over intervals where the maps are not completely positive can be expected to play roles in more complex set- tings. We should not let expectations for completely positive maps prevent us from seeing things that can happen. Our perspective is enlarged when we look beyond the map and include the dynamics. We can see the dynamics and the initial preparation as two related but separate steps. We can consider the effect of the dynamics, whatever the preparation may be. Local interactions that increase entanglement are com- pletely outside a perspective that is limited to pure states. An interaction on one of the qubits can not change the entangle- ment at all if the state of the two qubits remains pure [18]. The entanglement of a pure state of two qubits depends only on the spectrum of the reduced density matrices that describe the states of the individual qubits, which is the same for the two qubits. If that could be changed by an interaction on one of the qubits, there could be a signal faster than light. In our examples, the state of the two qubits is pure only when it is maximally entangled. In our examples, the spectrum of the density matrices for the individual qubits never changes, and gives no measure of the entanglement. Acknowledgments We are grateful to a referee for very helpful suggestions, including the comparison with a swap operation. Anil Shaji acknowledges the support of the US Office of Naval Research through Contract No. N00014-03-1-0426. [1] M. Ziman and V. Buzek, Phys. Rev. A 72, 052325 (2005). [2] T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006). [3] X.-T. Liang, Phys. Lett. A 349, 98 (2006). [4] T. Yu and J. H. Eberly, arXiv:quant-ph/0703083 (2007). [5] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 75, 022101 (2007). [6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Woot- ters, Phys. Rev. A 54, 3824 (1996). [7] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, arXiv:quant-ph/0702225 (2007). [8] K. Zyczkowski, P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. A 65, 012101 (2001). [9] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys. Rev. Lett. 71, 4287 (1993). [10] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [11] W. Dür, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. 83, 3562 (1999). [12] W. Dür and J. I. Cirac, Phys. Rev. A 61, 042314 (2000). [13] A. Acı́n, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. 87, 040401 (2001). [14] G. Toth, O. Guhne, M. Seevinck, and J. Uffink, Phys. Revs A 72, 014101 (2005). [15] E. C. G. Sudarshan, Chaos, Solitons and Fractals 16, 369 (2003). [16] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A. 70, 52110 (2004). [17] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A. 73, 12106 (2006). [18] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996).

0704.0462

Extended solar emission - an analysis of the EGRET data

Microsoft Word - sun.doc Extended Solar Emission – an Analysis of the EGRET Data Elena Orlando, Dirk Petry and Andrew Strong Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, D-85741 Garching, Germany Abstract. The Sun was recently predicted to be an extended source of gamma-ray emission, produced by inverse- Compton scattering of cosmic-ray electrons with the solar radiation. The emission was predicted to contribute to the diffuse extragalactic background even at large angular distances from the Sun. While this emission is expected to be readily detectable in future by GLAST, the situation for available EGRET data is more challenging. We present a detailed study of the EGRET database, using a time dependent analysis, accounting for the effect of the emission from 3C 279, the moon, and other sources, which interfere with the solar signal. The technique has been tested on the moon signal, with results consistent with previous work. We find clear evidence for emission from the Sun and its vicinity. The observations are compared with our model for the extended emission. Keywords: Cosmic rays, gamma-ray emission, Sun, EGRET. PACS: 95.85, 96.50, 96.60 THE EXTENDED SOLAR EMISSION MODEL The heliosphere has been studied as an extended source of gamma-ray emission, produced by inverse-Compton scattering of cosmic-ray electrons with the solar photon field [1,2]. For this analysis our model [1] has been improved using the modulated electron spectrum at all distances following [2] instead of the measured local electron spectrum, and using the anisotropic inverse-Compton scattering formalism [3]. ANALYSIS OF THE EGRET DATA We analyzed the EGRET data using the code developed for the moving target Earth [4] and adding necessary features (solar and lunar ephemerides, occultation, background point source trace calculations). The diffuse background was reduced by excluding the Galactic plane. Otherwise all available exposure within mission phase 1-3 was used. When the Sun passed by other gamma-ray sources (moon, 3C 279 and several quasars), these sources were included in the analysis. Details will be given in [5]. We fitted the data in the Sun-centred system using a multi-parameter likelihood fitting technique, leaving as free parameters the solar extended inverse-Compton flux from the model, the solar disk flux from pion decay [7], a uniform background, and the flux of 3C279, the dominant background point source. The moon flux was determined from moon-centred fits and the 3EG source fluxes were fixed at their catalogue values. All components were convolved with the energy-dependent EGRET PSF. The region used for fitting is a circle of radius 10º centred on the Sun. Since the interesting parameters are solar disk source and extended emission, the likelihood is maximized over the other components. In order to verify our method, we checked that the fluxes of the Crab Nebula, 3C 279, and in particular the moon [6] were reproduced. Results The log-likelihood ratio for E >100 MeV is displayed in Fig.1 as a function of solar disk flux and extended flux, compared with the model prediction of solar inverse-Compton flux for modulation parameter 500 MV at 1 AU. The solar emission is detected with 5.3σ significance. There is evidence for the extension of the emission at a level of 2.7σ; the maximum log L indicates an extended component with a flux compatible with the IC model. The total flux from the Sun is more than expected for the disk source [7], so this is clear evidence for the IC emission even without the proof of extension. We find that the measured extended flux is fully consistent with the model. Figure 2 shows the fitted model counts of the main components and the total including uniform background. In future work we will perform a detailed spectral analysis, refine the analysis of systematic errors, and study different models for the modulated electron spectrum. This is important for future missions such as GLAST and for studying solar modulation. FIGURE 1. Log Likelihood above 100 MeV as function of the solar disk flux and extended solar flux, relative to point at (0,0). The level of our predicted IC model flux and the predicted disk flux [7] are shown. FIGURE 2. Fitted model counts of the main components centered on the Sun. From left to right: Sun disk, Sun IC, moon, 3C 279, and the total predicted counts including uniform background. The colors show the counts/pixel, for 0.5°µ 0.5° pixels. REFERENCES 1. E. Orlando and A. W. Strong, Ap&SS in press, astro-ph/0607563 (2006). 2. I. V. Moskalenko et al., ApJL 652 (2006) L65-L68. 3. I. V. Moskalenko and A. W. Strong, ApJ, 528, 357 (2000). 4. D. Petry, AIP Conf. Proc., 745, 709 (2005). 5. D. Petry, E. Orlando and A.W. Strong, in prep. (2007). 6. D.J. Thompson et al., Journal of Geophys. Res. 102 (A7), 14735 (1997). 7. D. Seckel et al., ApJ 382, 652 (1991).

0704.0463

Mass and Temperature of the TWA 7 Debris Disk

Draft version October 23, 2018 Preprint typeset using LATEX style emulateapj v. 08/13/06 MASS AND TEMPERATURE OF THE TWA 7 DEBRIS DISK Brenda C. Matthews Herzberg Institute of Astrophysics, National Research Council of Canada, 5071 West Saanich Road, Victoria, BC, V9E 2E7, Canada Paul G. Kalas Department of Astronomy, University of California, Berkeley, CA, 94720-3411, U.S.A. Mark C. Wyatt Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K. Draft version October 23, 2018 ABSTRACT We present photometric detections of dust emission at 850 and 450 µm around the pre-main sequence M1 dwarf TWA 7 using the SCUBA camera on the James Clerk Maxwell Telescope. These data confirm the presence of a cold dust disk around TWA 7, a member of the TW Hydrae Association. Based on the 850 µm flux, we estimate the mass of the disk to be 18 Mlunar (0.2 M⊕) assuming a mass opacity of 1.7 cm2 g−1 with a temperature of 45 K. This makes the TWA 7 disk (d = 55 pc) an order of magnitude more massive than the disk reported around AU Microscopii (GL 803), the closest (9.9 pc) debris disk detected around an M dwarf. This is consistent with TWA 7 being slightly younger than AU Mic. We find that the mid-IR and submillimeter data require the disk to be comprised of dust at a range of temperatures. A model in which the dust is at a single radius from the star, with a range of temperatures according to grain size, is as effective at fitting the emission spectrum as a model in which the dust is of uniform size, but has a range of temperatures according to distance. We discuss this disk in the context of known disks in the TW Hydrae Association and around low-mass stars; a comparison of masses of disks in the TWA reveals no trend in mass or evolutionary state (gas-rich vs. debris) as a function of spectral type. Subject headings: stars: circumstellar matter, pre-main sequence — stars: individual (TWA 7) — submillimeter 1. INTRODUCTION TWA 7 (2MASS J10423011-3340162, TWA 7A) is a weak-line T Tauri star identified as part of the TW Hydrae Association (TWA, Kastner et al. 1997) by Webb et al. (1999) based on proper motion studies in conjunction with youth indicators such as high lithium abundance, X-ray activity and evidence of strong chro- mospheric activity. Disk systems were inferred around four of the 18 association members (Zuckerman & Song 2004) from measurements of IR excess with the In- frared Astronomical Satellite (IRAS). TW Hydra itself (a K7 pre-main sequence star) hosts the nearest proto- stellar disk to the Sun. Another accreting disk is ob- served around one member of the triple system Hen 3- 600 (Muzerolle et al. 2000), and two debris disk systems have been detected, around the A0 star HR 4796A (Jura 1991; Schneider et al. 1999) and one of two spectroscopic binary components of the quadruple system HD 98800 (Jayawardhana et al. 1999; Gehrz et al. 1999), which is a K5 dwarf. TWA 7 was not detected by IRAS. Based on the width of the Li 6707 Ȧ line, Neuhäuser et al. (2000) deduced that TWA 7 is a pre-main sequence star. TWA 7 was not detected by Hipparcos, but its membership in the TWA sets its distance to be 55 ± 16 pc (Neuhäuser et al. 2000; Weinberger et al. 2004; Low et al. 2005). Its spectral Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] type is M1 based on LRIS spectra (Webb et al. 1999). Based on existing photometry, evolutionary tracks and isochrone fitting, Neuhäuser et al. (2000) derived an age of 1-6 Myr (i.e., roughly coeval with other TWA stars) and a mass of 0.55 ± 0.15 M⊙. The age of the association is generally taken to be ∼ 8 − 10 Myr (Stauffer, Hartmann & Barrado y Navascues 1995; Zuckerman & Song 2004). This is the age when planet formation is thought to be ongoing and when disk dissi- pation is occurring. Thus, the TWA is the ideal cluster in which to observe the transitions from pre-main sequence stars with proto-planetary (gas-rich) disks to main se- quence stars with debris (gas-poor) disks. TWA 7 has been observed for infrared excess emission several times. The presence of a disk around TWA 7 was first noted in submillimeter observations byWebb (2000). They measured a flux of 15.5± 2.4 mJy at 850 µm. Nei- ther Jayawardhana et al. (1999) nor Weinberger et al. (2004) detected any excess associated with TWA 7 in the mid-infrared. However, Low et al. (2005) report de- tections of IR excess at 24 and 70 µm toward TWA 7 with the Spitzer Space Telescope. Based on these data and ex- isting shorter wavelength data on the stellar photosphere, Low et al. (2005) derive a disk temperature of 80 K and a lower limit to the mass of 2.4 × 1023 g (3.3 × 10−3 Mlunar) for the disk. This is a lower limit because the 70 µm data are not sensitive to colder material in the outer disk and dust grains exceeding a few hundred micron in size. A search for substellar companions using the NIC- http://arxiv.org/abs/0704.0463v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] 2 Matthews et al. MOS chronograph on HST did not reveal any evidence of the disk (Lowrance et al. 2005); a nearby point source is identified as a background object. The study of debris disks around members of an as- sociation permits the study of the evolution of disks as a function of spectral type alone, since the disks likely formed coevally and with similar compositions. It is also possible to judge whether the presence and evolution of disks around multiple stellar systems is comparable to those around single stars. The discovery of disks around low-mass stars is relatively recent (Greaves et al. 1998; Liu et al. 2004; Kalas et al. 2004). The low radiation field of late-K and M dwarfs means that the disks are faint in scattered light compared to disks around more massive stars, and hence they were less frequently tar- geted by scattered light searches for disks. However, scattered light imaging is often a follow-up technique used after an infrared excess has been discovered (e.g., AU Mic, Kalas et al. 2004). In fact, the low radiation fields may favor long-lived and slowly evolving disks, since some disk material may be unchanged from the original proto-planetary disks (e.g., Graham et al. 2007). Spitzer has detected evidence of disks around a few K stars (Chen et al. 2005; Bryden et al. 2006; Uzpen et al. 2005; Gorlova et al. 2004; Beichman et al. 2005), but Beichman et al. (2006) note that in a sample of 61 K1 to M6 stars, no excess emission is detected at 70 µm. This is well below the expected detection rate if the disk fraction is at all comparable to the ∼ 15% observed around solar- type stars. The discovery of disks around K- and M-type dwarfs may be difficult at far-infrared wavelengths be- cause material at similar radii to disks around early-type stars will be cool and will not radiate sufficiently. How- ever, submillimeter observing sensitivities and the dust temperature conspire to allow these objects to be dis- covered at submillimeter wavelengths (Zuckerman 2001). Submillimeter observations are sensitive to colder, larger grains, which are more likely to be optically thin than the warmer far-infrared emitting dust (Hildebrand 1988). We report here detections of submillimeter excess emis- sion at 450 and 850 µm around TWA 7 using the James Clerk Maxwell Telescope (JCMT). In § 2, we summarize our observations. In § 3, we present our results. We discuss the relevance of these data to the TWA and the population of disks around low-mass stars in § 4; our results are summarized in § 5. 2. OBSERVATIONS AND DATA REDUCTION Observations were made in 2004 October 19 using the photometry mode on the Submillimeter Common User Bolometer Array (SCUBA) on the JCMT (Holland et al. 1999). The pointing center of the observation was α = 10h42m30.s3, δ = −33◦40′16.′′9 (J2000). The on-source in- tegration time was 1.6 hours. Flux calibration was done using Mars, yielding flux conversion factors (FCFs) of 289.2 ± 1.4 Jy beam−1 volt−1 at 850 µm and 367.5 +/- 15 Jy beam−1 volt−1 at 450 µm. The absolute flux cali- bration is accurate to ∼ 20− 30%. Pointing was checked on the source 1034 − 293. The weather was excellent during the observations, with a CSO tau measurment at 225 GHz of ∼ 0.04. The extinction was corrected using skydips to measure the tau at 850 and 450 µm. The mean tau value was 0.15 at 850 µm in four skydips and 0.59 at 450 µm in three skydips. The extinction values derived from the skydips were consistent with the values extrapolated from the CSO tau values according to the relations of Archibald et al. (2002). The data were reduced using the Starlink SURF pack- age (Jenness & Lightfoot 1998). After flatfielding and extinction correction, we flagged noisy bolometers rig- orously; eleven (of 37) bolometers were removed in the long wavelength array and 26 (of 91) were removed from the short wavelength array. The photometry data were then clipped at the 5 σ level to remove extreme values. Short timescale variations in the sky background were then removed using the mean of all bolometers except the central one in each array. The average and variance were then taken of each individual integration for the central bolometers of the long and short wavelength ar- rays after a clip of 3 σ was applied. In the case of the 450 µm data, the signal-to-noise ratio was improved by applying a subsequent 2σ clip to the remaining data. 3. RESULTS We have detected emission toward TWA 7 at both 850 µm and 450 µm. The fluxes measured are 9.7 ± 1.6 mJy (6.1σ) and 23.0 ± 7.2 mJy (3.2σ), respectively. Errors are statistical, and do not include the typical flux uncer- tainty of∼ 20−30% for submillimeter single-dish calibra- tion. Utilizing the measured fluxes of the star (Table 1) and recently published Spitzer data (Low et al. 2005), we construct the spectral energy distribution (SED) for this source (Figure 1). The submillimeter data clearly repre- sent an excess of emission over the photospheric emission from TWA 7, as detected at 70 µm by Low et al. (2005). Fig. 1.— The spectral energy distribution of TWA 7. The optical and near-infrared data (diamonds) are modeled with a NextGen model (Hauschildt et al. 1999) scaled to the mass and luminosity of TWA 7 (Low et al. 2005). The star has a temperature of ∼ 3500 K (best fit to stellar photometry, Low et al. 2005; Webb et al. 1999). Triangles mark detections from the Spitzer Space Telescope and submillimeter detections from the JCMT. These fluxes show clear excess when compared to the stellar photosphere. The flux values are reported in Table 1. Fits for two single temperature blackbodies (parameters are described in Table 2) to the TWA 7 data show that no single temperature fits all four measurements of excess emission. An 80 K blackbody, Model 1, (dotted line) fits the 24 and 70 µm data (Low et al. 2005), but underestimates the submillimeter fluxes. The 70, 450 and 850 µm data are well fit by a 45 K blackbody, Model 2, (dashed line), but a disk this cold cannot account for the observed 24 µm excess. The flux we measure at 850 µm is only about 70% that measured by Webb (2000). Taking into account the typical 20 − 30% uncertainty in absolute flux cali- bration between epochs, the fluxes become 15± 3.8 mJy (2000) and 9.7 ± 2.5 mJy (2004), which are consistent Properties of the TWA 7 Disk 3 TABLE 1 Fluxes of TWA 7 Wavelength Magnitude Flux Reference [µm] [mJy] 0.44 (B) 12.55 39.4 HST Guide Star Catalog (Lasker et al. 1996) 0.44 (B) 12.3 49.7 USNO-A2.0 (Monet et al. 1998) 0.54 (V) 11.06 142.4 reported in Low et al. (2005) 0.64 (R) 11.2 97.4 USNO-A2.0 (Monet et al. 1998) 1.25 (J) 7.78 1259.5 Webb et al. (1999) 1.65 (H) 7.13 1476.3 Webb et al. (1999) 2.16 (Ks) 6.90 1159.1 2MASS PSC 2.18 (K) 6.89 1148.8 Webb et al. (1999) 12 – 70.4 ± 8.6 Weinberger et al. (2004) 24 – 30.2 ± 3.0 Low et al. (2005) 70 – 85 ± 17 Low et al. (2005) 450 – 23 ± 7.2 this work 850 – 9.7 ± 1.6 this work Note. — Conversion from magnitudes to Janskys has been done using zero points from Allen’s Astrophysical Quantities (Cox 1999). The 12 µm estimate by Weinberger et al. (2004) agrees with that of Jayawardhana et al. (1999) to within 1 σ and is consistent with the photospheric flux expected from TWA 7. within the errors. The instability of flux calibration in the submillimeter is well known, and may be assuaged somewhat by the ability to flux calibrate more often with SCUBA-2, the next generation submillimeter camera on the JCMT. In the interests of consistency with the only 450 µm detection in this work, we adopt the 850 µm flux from the later epoch and do not attempt to combine the two datasets. 3.1. Submillimeter Excess and Temperature Low et al. (2005) derive a disk temperature of 80 K for TWA 7 based on its infrared excess values at 24 and 70 µm. The temperature of the star is derived to be 3500 K, which is consistent with the log(Teff) of 3.56 reported by Neuhäuser et al. (2000) based on the stellar SED. Figure 1 shows the flux density distribution toward TWA 7. The stellar fluxes and fit to the stellar pho- tosphere are taken from Low et al. (2005) based on NextGen models by Hauschildt et al. (1999) using a grid of Kurucz (1979). The submillimeter excesses are evi- dent when compared with the stellar spectra. Based on their detection of TWA 7 in the mid-infrared, Low et al. (2005) determine that the dust orbiting TWA 7 exists at radii ≥ 7 AU from the star and has a temperature of 80 K. We have constructed several models for dust emission around a 0.55M⊙ star (see Table 2 for their parameters). We present two single temperature models in Figure 1, neither of which is capable of fitting the measured ex- cess fluxes at all four wavelengths. Model 1 (dotted line in Fig. 1) is that of Low et al. (2005): a blackbody of 80 K, which fits the 24 and 70 µm data, but cannot re- produce the fluxes at longer wavelengths. Model 2 is a colder black body at a temperature of 45 K (dashed line of Fig. 1), which fits the three longest wavelength fluxes, but cannot reproduce the 24 µm flux. Thus we conclude that the dust in this disk cannot be at a single tempera- ture; rather there are a range of temperatures responsible for the observed emission. Two models that are able to fit the observed emission spectrum are shown in Figure 2. In both models the dust has a range of temperatures, however there are two different physical motivations behind the origin of this range. Fig. 2.— The measured SED for TWA 7. Symbols and fit to the stellar spectrum is as for Fig. 1. Two multi-temperature models are fit to the excess emission arising from the disk. These are described in detail in the text, and the parameters are summarized in Table 2. In Model 3 (dotted line), different temperatures arise due to grains of different sizes located at a common radius from the star; while Model 4 (dashed line) contains grains distributed at a range of distances from the star. In Model 3 (dotted line of Fig. 2), the dust is assumed to all lie at the same distance from the star, r0. It is assumed to have a range of sizes with a power law distribution defined by the relation n(D) ∝ D(2−3q) for grains of size D. This power law is assumed to be trun- cated below dust of size Dmin = 0.9 µm, the size for which β = Frad/Fgra = 0.5 for compact grains. The dust is assumed to be composed of compact spherical grains of a mixture of organic refractories and silicates (Li & Greenberg 1997), and interaction with stellar radi- ation determines the temperature that dust of different sizes attains. The same model was used in Wyatt & Dent (2002) to model the emission from the Fomalhaut disk, where more detail can be found on the modelling method. The emission spectrum could be fitted by dust at a ra- dius r0 = 100 AU with a size distribution described by q = 1.78. The temperatures of dust in this model range from 21 K for the largest grains (> 100 µm) to 65 K for the smallest grains (0.9 µm). The size distribution used in Model 3 is close to that expected in a collisional cascade wherein dust is replen- ished by collisions between larger grains, since this re- 4 Matthews et al. TABLE 2 Dust Model Parameters Model Plotted as Temperature Dmin Dmax q Radius Mass (M⊕) 1 dotted Fig. 1 80 K – – – – 0.025 2 dashed Fig. 1 45 K – – – – 0.2 3 dotted Fig. 2 21 - 65 K 0.9 µm 1 m 1.78 100 AU 6.0 4 dashed Fig. 2 > 38 K – – – ≤ 35 AU 0.2 sults in a size distribution with q ∼ 1.83 and would be truncated below the size of dust for which radiation pres- sure would place the dust on hyperbolic orbits as soon as they are created. The effect of radiation forces on small grains is quantified by the parameter β = Frad/Fgra (which is not to be confused with the index of dust emis- sivity of grains that moderates the Rayleigh-Jeans tail of the SED of cold dust), and it is dust with β > 0.5 which is unbound. However, due to the low luminosity of M stars, it is not clear whether radiation pressure is sufficient to remove dust grains from the disk system. Figure 3 shows β as a function of dust grain size for dust around TWA 7 (M∗ = 0.55 M⊙, L∗ = 0.31 L⊙). While β is larger than 0.5 for compact (< 0.9 µm) grains, this condition is only met for a narrow region of the size distribution. Furthermore, if the dust grains are porous, as around AU Mic (Graham et al. 2007), then no grains will have β > 0.5. An alternative origin for the small grain cut-off could be stellar wind forces, since these provide a pressure force similar to radiation pres- sure (Augereau & Beust 2006), and it is known that stel- lar wind forces can be significantly stronger than radia- tion forces for M stars (Plavchan, Jura & Lipscy 2005). While the smallest grain size in the distribution may dif- fer from our value of 0.9 µm, this does not affect the ability of the model to fit the observed emission spec- trum with suitable modifications to the slope in the size distribution and radius of the dust belt. In Model 4 (dashed line of Fig. 2), the dust grains are assumed to lie at a range of distances from the star, but they are all assumed to emit like black bodies, and so have temperatures T = 278.3 L0.25 r. The spatial distribution of the grains was taken from the model of Wyatt (2005b) in which dust is created in a planetesimal belt at r0, and then migrates inward due to Poynting- Robertson (P-R) drag, but with some fraction of the grains removed by mutual collisions on the way. In the model the dust ends up with a spatial distribution which can be described by the parameter η0, such that the sur- face density falls off ∝ 1/(1 + 4 η0 (1 − r/r0)). The emission spectrum could be fitted using the parameters r0 = 30 AU and η0 = 10, resulting in dust with tem- peratures upwards of 38 K. However, the density of the planetesimal belt required to scale the resulting emission spectrum showed that removal by collisions dominates over the P-R drag force in such a way that η0 should be closer to 1400 in this disk. Thus, if this model is to have a true physical motivation, then we need to invoke a drag force which is ∼ 140 times stronger than P-R drag. Such a force could come from the stellar wind, which causes a drag force similar to P-R drag and which can be incorporated into the model by reducing η0 by a fac- tor 1 + (dMwind/dt) c 2/L∗ (Jura 2004). Thus to achieve η0 = 10 in this way we would require the stellar wind to be ∼ 140 times stronger than that of the Sun. The high X-ray luminosity of TWA 7 (LX = 9.2±1.0×1029 erg/s, Stelzer & Neuhauser 2000) may be indicative of a strong stellar wind, since measured mass loss rates have been found to increase with X-ray flux (Wood et al. 2005). However the correlation found by (Wood et al. 2005, see their Fig. 3) breaks down at X-ray fluxes an order of mag- nitude lower than that of TWA 7 (for which FX ∼ 8×106 erg cm−2 s−1, and so it is not possible to use this flux to estimate the mass loss rate with any certainty, and we simply note that the mass loss rate required to achieve η0 = 10 is not incompatible with observations of mass loss rates of other stars. A stellar wind drag force has also been invoked to explain structure in the AU Mic disk (Strubbe & Chiang 2006; Augereau & Beust 2006). We note that we are not claiming that the radius, r, size distribution, q, or parameter η0 have been well con- strained by these fits. The SED does indicate that the disk around TWA 7 contains grains at a range of temper- atures. The fits of Figure 2 illustrate two ways in which multiple temperatures in the disk may arise from phys- ically motivated models: the dust could have a range of sizes, or it could be distributed over a range of distances. Other models may also fit the data, including those in which dust has a range of distances and sizes, and those in which the dust originates in not one but multiple dust zones, as has been inferred for AU Mic (Fitzgerald et al., in prep.). Fig. 3.— Ratio of the radiation force acting on dust grains of different size in the TWA 7 disk to the force of stellar gravity, β = Frad/Fgra. Two different models are shown: compact grains (p = 0) and porous grains (p = 0.9). Dust with β > 0.5 is unbound from the star as soon as it is released from a planetesimal. This figure demonstrates that radiation pressure is not a significant mechanism to remove dust grains from the TWA 7 system and is particularly ineffective if the grains are highly porous. The derived fits of Figure 2 do show differences in the mid-IR spectrum. While this suggests that knowledge of the mid-IR spectrum would enable us to distinguish between the two models, it is worth pointing out that Properties of the TWA 7 Disk 5 the exact shape of this spectrum is very sensitive to the size distributions of small grains (in Model 3) and to the radial distribution of grains interior to the planetesimal belt (in Model 4). Thus it is possible that different as- sumptions about these distributions could be made to provide a fit to the same observed spectrum with both models. The only way to definitively break the degen- eracy is through imaging of the thermal emission from the mid-IR to the submillimeter. The single radius disk must look the same at all wavelengths, while a radially distributed disk would look larger at longer wavelengths, as more of the cold dust further from the star is detected. 3.2. Mass of the Disk Low et al. (2005) estimate the minimum mass in dust of the TWA 7 disk to be 0.0033 Mlunar, under the as- sumption that the dust grain size is 2.8 µm. In deter- mining the mass of the disk from submillimeter measure- ments, a key parameter is the temperature of the dust. Where possible, we discuss the derived mass for each of the models discussed above. For submillimeter fluxes, we can estimate the mass of the disk directly for an assumed temperature and opacity from the relation: Mdisk = κν Bν(Td) where Bν(Td) is the Planck function at the dust temper- ature, Td, and κν is the absorption coefficient of the dust. The derived mass is a strongly dependent function of the value of κν . For debris disk studies, a value of 1.7 cm g−1 is appropriate at 850 µm (Dent et al. 2000). This is at the upper end of the values derived by Pollack et al. (1994). We can estimate the mass in Model 1 by using the submillimeter flux predicted from the Rayleigh-Jeans tail of the model (2.4 mJy) as well as the 80 K temperature and standard opacity. This gives a mass of 0.025 M⊕ (∼ 2 Mlunar). This can be interpreted as the mass of hot dust, with the proviso that the submillimeter observation shows that there is more mass in colder dust as well. Model 2 fits the submillimeter and 70 µm emission well. Under the estimate of 45 K for the dust temperature of the disk, the standard dust opacity and the measured 850 µm flux, the TWA 7 disk contains 0.2M⊕ of material (18 Mlunar). Based on the uncertainty in the flux (random and systematic), the uncertainty on the mass estimates are ∼ 30%. The mass of the TWA 7 disk is an order of magnitude greater than the mass of 0.011 M⊕ for the disk detected around AU Mic (Liu et al. 2004) that is also derived based on a single temperature fit to far-IR and submillimeter data with κν = 1.7 cm 2 g−1. Model 3 contains dust grains at different temperatures based on their size. In this case, the size distribution of dust is well defined (i.e., with known scaling), the total mass of dust is given by Mtot = 11 D max, where Mtot is the mass in Earth masses and Dmax is in meters. While it is impossible to observationally knowDmax (since large planetesimals are effectively invisible), in our model 95% of the 850 µm flux comes from grains < 0.4 m. This implies a mass of 6 M⊕, significantly higher than that of Model 2. In the TWA 7 system, Dmax could be even larger (or smaller) than this, so this discrepency carries little definitive weight. The size distribution is relatively shallow (i.e., there are lots of large grains) which explains why the mass is much larger than in Model 2. To derive a mass for Model 4 we assume an opacity of 1.7 cm2 g−1 and determine the mass of dust in the model required to reproduce the observed 850 µm flux, given that the dust has a range of temperatures. The derived mass is almost exactly the same as that at of Model 2 at 0.2M⊕. This is to be expected because the submillimeter flux in both models is dominated by the coldest dust at 38-45 K, and very little additional mass is required in Model 4 to explain the mid-IR emission (as illustrated by the low mass in Model 1). Given the inherent assumptions and unknowns in each of these models, we adopt the results of the highly sim- ple Model 2 as our most robust estimate of the mass of the dust disk in TWA 7. It depends on the tem- perature of the grains producing the observed 850 µm flux, which are well fit by the cold dust model of Fig- ure 1. The mass is also highly dependent on the value of κν , for which we have adopted a value in line with other disk modeling work (Dent et al. 2000), and so can be compared with the disk masses derived by other au- thors (e.g., Wyatt, Dent & Greaves 2003; Liu et al. 2004; Najita & Williams 2005). 4. DISCUSSION 4.1. Disks in the TW Hydrae Association Recent MIPS data from the Spitzer Space Telescope show that most stars in the TWA show no evidence of circumstellar dust out to 70 µm (Low et al. 2005). These results confirm the assertion of Weinberger et al. (2004) and Greaves & Wyatt (2003) that there is a bimodal dis- tribution in the TWA: either stars have strong excesses associated with warm dust emission, or they have very weak emission consistent with very cold disks. The ex- ceptions are the proto-planetary disks around TW Hya and Hen 3-600, which both show evidence of warm and cold disk components (Wilner et al. 2000; Zuckerman 2001). Lower limits on warm dust emission were set for all TWA members (except TW Hya) by Weinberger et al. (2004). The absence of dust near the star is often taken as a signature of a centrally depleted debris, rather than an accreting, proto-planetary disk. There are a few main sequence stars for which warm dust is present (i.e., HD 69830 and η Corvi), but this may be a transient phenomenon (Wyatt et al. 2007). Based on their data, Weinberger et al. (2004) deduced that the non-detections implied an absence of material in terrestrial planet region and that, except for TW Hya, there were no long-lived disks in the association which could still form planets. An outstanding question is whether the stars without warm disks have potential disk material locked up in undetected planets, which would imply that dusty sys- tems represent failed planetary systems, or whether non- detections represent disk systems in which the dust is too cold to be detected in the mid-IR. Only large-scale searches for cold dust around a statistically significant number of stars can clarify whether a sizable population of disks exist which are too cold to detect at mid-IR wavelengths. Such a survey is planned using the new SCUBA-2 camera at the JCMT (Matthews et al. 2007). The disks now known in the TWA each have very dif- ferent properties. The disk around TW Hya (K7) still 6 Matthews et al. TABLE 3 Fluxes and Masses of Disks in the TWA Star Sp. Type Gas? Optical Depth Flux λ Mass Temp Reference [mJy] [Mlunar] [K] HR 4796A A0 no thin 19.1 ± 3.4 850 µm 19 99 Sheret et al. (2004) HD 98800 K5 no thick 111.1 ± 0.01 800 µma 28 150 Prato et al. (2001) TW Hya K7 yes thick 8± 1 7 mm 8× 105 – Wilner et al. (2000), SED fit TWA 7 M1 no thin 9.7± 1.6 850 µm 18 45 this work TWA 13 M1 no thin 27.6 ± 5.9 70 µm > 0.0019 65 Low et al. (2005) Hen 3-600 M3 yes thick ∼ 65 850 µm 304b 20c Zuckerman (2001) Note. — a κ850µm adjusted by 1/λ. b mass estimate based on conservative value of κ850µm = 1.7 cm 2 g−1; c temperature constraint on cold dust component only. contains molecular gas (Kastner et al. 1997), meaning it is proto-planetary or in a transition from a proto- planetary to a debris disk. It is a broad, face-on disk which extends to 135 AU with evidence of a dip in flux at 85 AU (Krist et al. 2000; Wilner et al. 2000; Trilling et al. 2001; Weinberger et al. 2002; Qi et al. 2004). Hen 3-600 also shows evidence of hosting an accreting, gas-rich disk (Muzerolle et al. 2000). The HR 4796A (TWA 11) disk contains a narrow dust ring at 70 AU from the star (Jayawardhana et al. 1998; Koerner et al. 1998; Schneider et al. 1999; Telesco et al. 2000) and no detectable gas in emission line stud- ies (Greaves, Mannings & Holland 2000) or more recent searches for absorption due to circumstellar gas along the line of sight (Chen & Kamp 2004), a technique which is highly dependent on the temperature profile of the gas because the disk is not edge-on. In both the Hen 3-600 and HD 98800 systems, the disk orbits only one mem- ber of the binary (Jayawardhana et al. 1999; Gehrz et al. 1999) with evidence for cooler dust in circumbinary or- bits (Zuckerman 2001). In fact, HD 98800 is a quadruple system, so the dust is in a circumbinary orbit around a spectroscopic binary. The discussion of the TWA disk population can be il- luminated by the results of recent Spitzer study of the Upper Sco Association (with an estimated age of 3 − 5 Myr) by Carpenter et al. (2006). Their MIPS obser- vation found optically thin disks around A-type stars, no disks around solar-like stars, and optically thick, ac- creting disks around K- and M-type dwarfs, suggest- ing that disk evolution proceeds more quickly around higher mass stars. A similar result was found for the H and χ Persei double cluster at 13 Myr (Currie et al. 2007). The sample of stars in Upper Sco was 204 stars with 31 detections at 8 µm, well distributed across spectral types. In the TWA, we have disk detections around only a handful of members, and the associa- tion is much smaller, with only 18 members. How- ever, both the optically-thick, gas-rich disks in the TWA are hosted by K and M stars, whereas the only A star with a disk hosts an optically-thin debris disk. We note that HD 98800 is noted to be gas-poor, but with an optically-thick dust disk (Weintraub, Kastner & Bary 2000; Zuckerman & Becklin 1993). The masses and temperatures of the disks are com- pared for the TWA members with disks in Table 3. All six disks are detected and their SEDs modeled in the recent paper by Low et al. (2005). Where possible, the masses in Table 3 are derived from submillimeter fluxes or fits to SEDs, rather than infrared values which pro- vide lower limits only. This is only an issue for TWA 13 which has not yet been detected at submillimeter or millimeter wavelengths. Scaling κ850µm = 1.7 cm 2 g−1 to 70 µm implies a lower mass limit of 0.1 Mlunar based on cold grains. Of the debris disks, the most massive is the disk around the HD 98800 system, but its mass is comparable to that of the disk around the earlier star HR 4796A. Based on submillimeter masses, the TWA 7 disk is roughly compa- rable to that of HR 4796A. It is clear that we cannot yet identify a systematic trend in mass or evolutionary phase with spectral type in the TWA. We note that, given the presence of massive disks around components of the mul- tiple systems HD 98800 and HR 4796, their dust disk life- times do not appear to be any shorter than that around a single star. 4.2. Disks Around Late-type Stars Table 4 shows the compilation of the few known debris disks around late-type (K and M) stars. Although a de- bris disk has been historically claimed around HD 233517 (Sylvester et al. 1989), Jura et al. (2006) conclude that it is a giant, not a main sequence, star, and so we do not include it. Similarly, we exclude HD 23362, although it has a measured excess, because Kalas et al. (2002) at- tribute the emission to a uniform surrounding dust cloud, not a debris disk, around a K2III star at 187 pc dis- tance. All fluxes are measured at 850 µm with SCUBA, except where noted. For one of these stars, only an up- per limit is observed in the 850 µm flux. Of the eight solid detections, two (HD 92945, TWA 13) are at a sin- gle wavelength in the mid-infrared, and two (GJ 182 and GJ 842.2) are detected only in the submillimeter. For the four remaining disks (HD 53143, ǫ Eri, AU Mic and TWA 7), there is a trend of increasing mass with later spectral type, but it must be noted that the mass dependence could also be attributed to the known trend of declining dust masses around older stars (Rhee et al. 2006) in the cases of HD 53143 and ǫ Eri since they are significantly older than AUMic and TWA 7. The youngest star, TWA 7, has the most massive disk. We are obviously also bi- ased toward more massive disks at larger distances. AU Mic’s disk (9.9 pc) could not have been detected in the observation which detected TWA 7’s disk at 55 pc. Spitzer has made great progress in the last year de- tecting infrared excess from main sequence stars (see the review by Werner et al. 2006). However, not many of these have been around late-type stars, and for those Properties of the TWA 7 Disk 7 TABLE 4 Masses and Mass Limits of Debris Disks around Late-type Main Sequence Stars Star Spectral Distance Age Flux λ Mass Temp Mass Reference Type [pc] [Myr] [mJy] [Mlunar] [K] HD 69830a K0 12.6 600-2000 < 7 850 µm < 0.24 100 this work HD 92945 K1 22 20-150 271 70 µm > 0.002 40 Chen et al. (2005) HD 53143 K1 18 1000 82.0± 1.1 30-34 µm > 6.5× 10−6 120± 60 Chen et al. (2006) – optical > 0.0096 60b Kalas et al. (2006) ǫ Eri K2 3 730± 200 40 ± 1.5 850 µm 0.1 85 Sheret et al. (2004) GJ 842.2 M0.5 20.9 200 25 ± 4.6 850 µm 28± 5 13 Lestrade et al. (2006) GJ 182c M0.5 26.7 100 4.8± 1.2 850 µm > 2.1 40 + 150d Liu et al. (2004) AU Micc M1 9.9 10 14.4± 1.8 850 µm 0.89 40 Liu et al. (2004) TWA 7c M1 55 8 9.7± 1.6 850 µm 18 45 this work TWA 13c M1 55 8 27.6± 5.9 70 µm > 0.0019 65 Low et al. (2005) Note. — a We have used the adopted temperature of 100 K but revised downward the estimated flux and mass based on re-analysis of data originally presented in Sheret et al. (2004). Wyatt et al. (2007) suggest that the warm dust around this source must be transient. b Temperature from Zuckerman & Song (2004). c Ages derived from association membership. d two-component fit. detections which have been made (Gorlova et al. 2004; Beichman et al. 2005; Uzpen et al. 2005; Bryden et al. 2006; Smith et al. 2006), no estimates of mass in the disk exist. The detections are typically toward field stars or very distant targets in the galactic plane, although one (P922) is a member of the cluster M47 (Gorlova et al. 2004). We do not list these candidates in Table 4. One exception is the excess around HD 92945 (K1V), for which Chen et al. (2005) measure a minimum disk mass of 2× 10−3 Mlunar. As discussed in § 3.1, the degeneracies in the models are best broken with thermal imaging. Of the disks com- piled in Table 4, only ǫ Eri (3.3 pc) has been well resolved at submillimeter wavelengths. The distance of the TWA makes imaging of 100 AU (∼ 2′′) scale disks impossi- ble with single dish telescopes in the submillimeter. ǫ Eri would be exceedingly difficult to map if it were even three times more distant. We will have to rely on arrays with higher sensitivity to obtain maps like that of ǫ Eri around most low mass stars unless many more are dis- covered within 10 pc. In the long-term, mid-IR images will be possible with the planned MIRI instrument on the James Webb Space Telescope. In the short-term, submil- limeter imaging will be possible with the Atacama Large Millimeter-submillimeter Array (ALMA). Far-IR obser- vations will be possible with the Herschel Space Obser- vatory, although imaging of 100 AU disks will only be possible for stars within 10 pc. 5. SUMMARY We have detected submillimeter excess emission aris- ing from the dust disk around TWA 7 at 450 and 850 µm using SCUBA on the JCMT. Based on our photom- etry and recent data from Spitzer, we derive a disk mass of 0.2 M⊕ (18 Mlunar) for a temperature of 45 K. This model effectively fits the 70, 450 and 850 µm data with a blackbody. To fit these data and the 24 µm flux re- quires dust at a range of temperatures, and we show that this could arise from dust at one radius with a range of sizes, or from dust of one size at a range of distances from the star. Based on the SED alone, it is not possible to determine which physical model is dominant. While the multiple system HD 98800 appears to har- bour the most massive debris disk in the TWA, disks of relatively comparable masses are observed around the A0 star HR 4796A and the M1 star TWA 7. Therefore, the formation of debris disks does not appear to be solely a function of the mass of the parent star. A comparison of masses of disks in the TWA reveals no trend in mass or evolutionary state (gas-rich, proto-planetary vs. debris) as a function of spectral type, although the detection of proto-planetary disks around the latest stars is consis- tent with the results of Carpenter et al. (2006) toward the Upper Sco Association and Currie et al. (2007) in the double cluster H and χ Persei. Kalas et al. (2006) came to the same conclusion with regard to other debris disks. They surmise that nur- ture could explain the presence or absence of disks at later epochs. If the environment dynamically heats the disk such that the large planets fail to form, then dust remains for a longer timescale. The dynamically quiet systems then may quickly form planets, leaving no disk to be observed at later epochs, although there is as yet no evidence for any correlation between stars with debris and/or planets (Moro-Martin et al. 2006). The authors acknowledge our anonymous referee for an insightful and constructive report. As well, the au- thors thank B. Zuckerman for providing us with the pre- vious 850 µm flux measurement from the thesis of R. Webb, and P. Smith for providing the stellar fit to TWA 7 of Low et al. (2005) to maximize consistency with their analysis. We also acknowledge useful conversations with P. Hauschildt and R. Gray regarding the spectra of M dwarfs. We thank our telescope operator E. Lundin and the staff at the JCMT for their support. B.C.M ac- knowledges support of the National Research Council of Canada through a Plaskett Fellowship. P.K. acknowl- edges support from GO-10228 provided by STScI under NASA contract NAS5-26555. M.C.W. acknowledges sup- port of the Royal Society. 8 Matthews et al. 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0704.0464

Annealed importance sampling of dileucine peptide

7 Annealed importance sampling of dileucine peptide Edward Lyman∗, and Daniel M. Zuckerman† Department of Computational Biology, School of Medicine 3079 BST3 , 3501 Fifth Ave., University of Pittsburgh, Pittsburgh, PA 15261 October 28, 2018 Abstract Annealed importance sampling is a means to assign equilibrium weights to a nonequilibrium sample that was generated by a simu- lated annealing protocol[1]. The weights may then be used to calculate equilibrium averages, and also serve as an “adiabatic signature” of the chosen cooling schedule. In this paper we demonstrate the method on the 50-atom dileucine peptide, showing that equilibrium distributions are attained for manageable cooling schedules. For this system, as ∗[email protected] †[email protected] http://arxiv.org/abs/0704.0464v1 näıvely implemented here, the method is modestly more efficient than constant temperature simulation. However, the method is worth con- sidering whenever any simulated heating or cooling is performed (as is often done at the beginning of a simulation project, or during an NMR structure calculation), as it is simple to implement and requires mini- mal additional CPU expense. Furthermore, the näıve implementation presented here can be improved. 1 Introduction Simulated annealing (SA) is used in a wide variety of biomolecular calculations. Crystallographic refinement protocols[2] and standard NMR structure calculations[3, 4, 5] both rely on SA to optimize a “target function,” constructed so that the global minimum of the tar- get function corresponds to the native structure. Molecular dynamics calculations often begin by cooling a configuration from a high tem- perature ensemble to a lower temperature, at which the simulation is to be performed. In this paper, we consider a different use for SA calculations. Since a set of structures that is generated by a series of SA trajectories is a nonequilibrium sample, they may not be used to calculate equilibrium averages. However, Neal demonstrated a simple procedure, called “an- nealed importance sampling” (AIS) that allows the nonequilibrium sample to be reweighted into an equilibrium one[1]. AIS is closely connected with the Jarzynski relation[6]. To our knowledge, the al- gorithm has only appeared once in the chemical physics literature[7], where it was used (along with sophisticated Monte Carlo techniques) to sample a one-dimensional potential. Here, we demonstrate an ap- plication of the AIS algorithm to generate an equilibrium sample of an implicitly solvated peptide, and discuss other uses for AIS which may of interest to the molecular simulation community. The basic idea which underlies SA is also the motivation for other temperature based sampling methods, notably J-walking[8], simulated tempering[9, 10] and replica exchange/parallel tempering[11, 12]. By coupling a simulation to a high temperature reservoir, it is hoped that the low temperature simulation may explore the configuration space more thoroughly. This is achieved by thermally activated crossing of energetic barriers, which are large compared to the thermal en- ergy scale of the lower temperature simulation, but are crossed more frequently at higher temperature. Simulated and parallel tempering differ in the way that the different temperature simulations are cou- pled. Simulated tempering heats and then cools the system, in a way that maintains an equilibrium distribution. Parallel tempering couples simulations run in parallel at different temperatures by occasionally swapping configuartions between temperatures, again in such a way that canonical sampling is maintained. AIS offers yet another approach to utilizing a high temperature ensemble for equilibrium sampling at a lower temperature. A sam- ple of a high temperature ensemble is annealed to a lower temper- ature, by alternating constant temperature simulation with steps in which the tempertaure is jumped to a lower value. Each annealed structure is assigned a weight, which depends on the trajectory that was traced during the annealing process. Equilibrium averages over the lower temperature ensemble may then be calculated by a simple weighted average. Furthermore, the distribution of trajectory weights contains useful information about the statistics of the annealed sam- ple. Roughly, a schedule which quenches high temperature structures very rapidly to low temperature will result in a sample dominated by a few high weight structures, resulting in poor statistics. This con- nection between the distribution of weights and the extent to which the schedule is not adiabatic ought to be of interest to anyone who uses SA protocols—whether for equilibrium sampling or for structure calculation. We have used the AIS method to generate 298 K equilibrium en- sembles of the dileucine peptide, by annealing structures from a 500 K distribution with several different cooling schedules. For the most efficient schedule used, we found a modest gain (about a factor of 3) over constant temperature simulation. This result is consistent with earlier observations on the expected efficiency of temperature-based sampling methods[13]. 2 Theory Consider a standard simulated annealing (SA) trajectory, in which a protein is slowly cooled from a conformation x at a (high) temperature TM . The cooling is achieved by alternating constant temperature dynamics with “temperature jumps,” during which the temperature is lowered instantaneously. Usually, the system is cooled to a low temperature, since the aim of standard SA calculations is to find the global minimum on the energy landscape. But we can imagine instead ending the run at T0 = 300 K—in fact, we can think of many such runs, all ending at 300 K. We then have an ensemble of conformations, though clearly not distributed canonically at T0. We would like to know if there is a way to reweight this distribution, so that it can be used to compute equilibrium averages at T0. The affirmative answer is provided by the annealed importance sampling (AIS) method. To make the discussion more concrete, consider many independent annealing trajectories xj(t) which at time tM−1 have just been cooled from inverse temperature βM to βM−1. As usual, each temperature defines a distribution of conformations: πi(x) ∝ exp[−βiU(x)]. Imme- diately after tM−1, before the system is allowed to relax to πM−1(x), we can compute the equilibrium average of an arbitrary quantity A over πM−1(x) by using the weight w(x) = πM−1(x)/πM (x): 〈A〉M−1ZM−1 = dxA(x)πM−1(x) = dxA(x)πM (x)w(x), (1) where 〈A〉i denotes an average over πi, and Zi = dxπi(x). In other words, we may reweight the distribution πM(x) to calculate averages over πM−1(x), by multiplying by the ratio of Boltzmann factors. Generalizing the argument toM temperature steps is straightforward[1], by forming the product of weights for successive cooling steps: wj ≡ w(xj(t0)) = πi−1(xj(ti−1)) πi(xj(ti−1)) . (2) Equation 2 gives the weight for trajectory j, cooled at successive times tM−1, tM−2,... through inverse temperatures βM , βM−1,... to reach conformation xj(t0). At each temperature, reweighting ensures that averages may be calculated for the appropriate canonical distribution, even though the system has not yet relaxed. The AIS idea is easily turned into an algorithm for producing a canonical distribution from serially generated annealing trajectories: (i) Generate a sample of the distribution πM(x), by a sufficiently long simulation at TM . (ii) Pull a conformation from πM (x) at random and anneal down to β0, yielding conformation x1(t0). Keep track of the weight w(x1(t0)) for this trajectory by Eq. 2. (iii) Repeat steps (iii) and (iv) N times, yielding congiura- tions xj and weights w(xj) ≡ wj for j = 1, 1, ..., N . Equilibrium averages at temperature T0 are then calculated by a weighted average: 〈A〉0 = j=1wjAj∑N The cooling schedule is defined by the number and spacing of the temperature steps, as well as the duration of the constant temperature simulation at each step. As available resources necessarily limit the CPU time spent on each annealing trajectory, careful consideration of the schedule is in order. Clearly, a schedule in which high temperature configurations are quenched in one step to low temperature amounts to a single-step reweighting procedure[14]. We may expect that such a schedule would be quite ineffective for large temperature jumps, since very few configurations in the high temperature distribution have ap- preciable weight in the low temperature distribution. By introducing intermediate steps, the system is allowed to relax locally, bridging the high and low temperature distributions in a way that echoes replica exchange protocols[11, 12], simulated tempering[9, 10], and the multi- ple histogram method[15]. However, the “top-down” structure of the algorithm most closely resembles J-walking[8, 16]. 3 Results The dileucine peptide (ACE-[Leu]2-NME) is good choice for the vali- dation of new algorithms, as it is small enough (50 atoms, including nonpolar hydrogens) that exhaustive sampling by standard simulation methods is possible, yet more akin to protein systems than a one- or two-dimensional “toy” model. The high temperature ensemble was generated by 300 nsec of Langevin dynamics at TM = 500 K, as implemented in Tinker v. 4.2.2[17], with a timestep of 1.0 fsec, and a friction constant of 91 psec−1, and solvation was treated by the GB/SA method[18]. Frames were written every psec, resulting in a sample of 3× 104 frames in the high temperature sample. The 500 K sample was annealed down to 298 K using 4 different schedules, consisting of a total of 3, 5, 9, and 17 temperatures, includ- ing the endpoints. In each case, the temperatures were distributed geometrically. Following each temperature jump, the velocities were reinitialized by sampling randomly from the Maxwell-Boltzmann dis- tribution, and then allowed to relax at constant temperature for a time tR = 0.5 psec (except where noted) with the protocol described above. A total of N = 1.6× 104 annealing trajectories were generated for each schedule. The control of the integration routine to effect the annealing, as well as the calculation of the trajectory weights, were implemented in a Perl script. Figure 1 shows that the 298 K distribution of energy is recovered by the AIS procedure. It is noteworthy that the 500 K distribution (corresponding to the high T sample) overlaps very little with the 298 K distribution, and yet the 298 K distribution is reproduced well for the two slowest schedules. Equally interesting is how poorly the algo- rithm performs when the structures are cooled too rapidly, especially on the low E side of the distribution, where there is no overlap with the high T distribution. We conclude that the schedules with 3 or 5 T -steps quench the structures too rapidly, resulting in many of the tra- jectories becoming “stuck” in high-energy states that are metastable at 298 K. This last observation may be quantified by asking, “How many of the annealed structures contribute appreciable weight to averages calculated with Eq. 3?” To address this question, for each schedule we estimated the number of configurations n which contribute appreciable weight to the averages: ≡ fN, (4) where wmax is the largest weight observed (see Table 1). If this number is near 1, then a small number of trajectories dominate the average— see Eq. 3 —and poor results should be expected. The effective fraction of the annealing trajectories which generate “useful” or “successful” structures is denoteed by f . A more complete picture is provided by the full distribution of the (logarithm of) trajectory weights (Fig. 2). For each schedule, the weights which contribute the most to the T = 298 K sample are to the right, at large values of w. The trend is clear—as slower cooling is effected, the distribution narrows and shifts to the right. It has been shown that the accuracy of averages computed from this type of protocol is roughly related to the variance of the (adjusted) weights[1]. (The adjusted weight is the weight divided by the average weight.) This “rule of thumb” is borne out by the data in Fig. 2 and Table 1—as the cooling slows down the distribution of weights narrows, and the number of trajectories contributing to the equilibrium averages increases. This type of analysis may serve as a means of distinguishing between annealing schedules to decide on a cooling schedule which is slow enough to yield reasonable estimates of equilibrium averages. It is also essential for optimizing an AIS protocol for sampling efficiency, as discussed in the next few paragraphs. How much better than standard simulation (if at all) is equilib- rium sampling by AIS? In order to make a direct comparison between AIS and constant temperature simulation, we need to compare the CPU time invested per statistically independent configuration in each protocol. For the constant temperature simulation, this time may be estimated in several ways[19, 20], and is essentially the time needed for the simulation to “forget” where it has been. Following the con- vention for correlation times, we call this time τi = τ(Ti), where i labels the temperature: M for the high T distribution, and 0 for the low T distribution. For the system studied here, τM = 0.8 nsec and τ0 = 3.0 nsec, as estimated from timseries of the α → β backbone dihedral transition[13]. The total cost to generate a structure in an AIS simulation is the sum of the costs of generating a structure in the high T distribution plus that for the annealing phase. Of course, not every annealing trajectory contributes to thermodynamic averages(Eq. 3). What then is the total cost tcost of a “successful” annealed structure? The first part is from high temperature sampling—i.e., τM . The second part is the cost of all the annealing trajectories, divided by the number which contribute to equilibrium averages. The time tanneal is the time spent annealing each structure: tanneal = tR(M − 2) (5) Recall that tR is the duration of the constant temperature relaxation steps, and there is no relaxation phase at the highest and lowest tem- peratures. The total cost tcost is then the sum of τM and tanneal: tc = τM + tanneal/f. (6) The efficiency of an AIS protocol may then be computed by taking the ratio R ≡ τ0/tcost (see Table 1), which gives the factor by which an AIS protocol is more or less efficient than constant temperature simulation. The data in Table 1 show that the best schedule used here offer a modest speedup over constant temperature simulation, of a factor of about 3. These findings are in agreement with an analysis we have published of another temperature-based sampling protocol[13]. We note that an optimized AIS protocol would require tuningN based on (perhaps preliminary) estimates of f . It is instructive to compare the AIS results to simple reweighting— i.e., AIS with no intermediate temperature steps or relaxation. In this case, no computer time is spent annealing, and the efficiency gain is simply τ0/τM = 3.75. The fraction f is of course reduced compared to any AIS protocol—when reweighting our 500 K dileucine trajectory to 298 K distribution, f = 1.3 × 10−4—but this has no impact on the efficiency, provided a sufficient number of snapshots are available for reweighting. However, it is clear that f will be greatly reduced in systems which undergo a folding transition upon lowering the temperature. This is simply a reflection of the fact that there is negligible overlap between the folded and unfolded distributions. In such cases, a useful reweighting protocol would require the generation of astronomical numbers of structures in the TM distribution, and annealing is advised. 4 Conclusion We have demonstrated the application of Neal’s annealed importance sampling (AIS) algorithm for equilibrium sampling of the dileucine peptide. AIS allows the calculation of equilibrium averages from a nonequilibrium sample of strutures that results from a simulated an- nealing protocol. To our knowledge, AIS has not previously been applied to a molecular system. While the method, as näıvely im- plemented here, represents only a modest improvement over constant temperature simulation, it is interesting for several reasons beyond equilibrium sampling. First, in applications where simulated annealing is already in widespread use (most notably, NMR structure calculations[3, 4, 5]), the path weights may be used to calculate (perhaps noisy) equilibrium aver- ages, and perhaps ultimately Boltzmann-distributed ensembles. The path weights also contain information that can be used to discriminate between different schedules, which may provide a way to optimize the schedule, based on the analysis of tann, the cost of annealing to “good” structures. Second, it may be possible to improve considerably on the efficiency of the method by implementing a more sophisticated version, which uses a resampling procedure to prune the low weight paths at each cooling step. (For a detailed discussion of resampling methods, see the book by Liu[21].) In this approach, we first cool some number N of structures from the high temperature (TM ) ensemble, yielding N weighted structures at TM−1. We then resample N times from this TM−1 ensemble, according to the cumulative distribution function of the weights, pruning the low weight paths without biasing the sample. This type of approach was recently applied successfully to sampling near native protein configurations of a discretized and coarse-grained model[22]. Nevertheless, we emphasize that the ultimate efficiency of any AIS protocol limited by the intrinsic sampling rate of the highest temperature, which may be modest; see Ref. ??. Finally, the AIS procedure could be naturally combined with “an- nealing” in the parameters of the Hamiltonian. Such a hybrid of AIS and Hamiltonian switching might be used, for example, to transform an NMR target function into a molecular mechanics potential function, over the course of a structure calculation. The result of such a cal- culation would be an equilibrium ensemble of structures, distributed according to the molecular mechanics potential. Such ensembles would find wide application, for instance in docking or homology modeling. Acknowledgements The authors thank Gordon Rule for several enlightening discussions about NMR methodology. D. Z. thanks Chris Jarzynski for alerting him to Neal’s work on AIS. This research was supported by the NSF (MCB-0643456), the NIH (GM076569), and the Department of Computational Biology, University of Pittsburgh. References [1] Radford M. Neal. Annealed importance sampling. Stat. and Comp., 11:125–139, 2001. [2] Axel T. Brünger, Paul D. Adams, G. Marius Clore, Warren L. DeLano, Piet Gros, Ralf W. Grosse-Kuntsleve, Jian-Sheng Jiang, John Kuszewski, Michael Nilges, Navraj S. Pannu, Randy J. Read, Luke M. Rice, Thomas Simonson, and Gregory L. War- ren. Crystallograhy and NMR system: a new software suite for macromolecular structure determination. Acta. Cryst., D54:905– 921, 1998. [3] C.D. Schwieters, J.J. Kuszewski, N. Tjandra, and G.M. Clore. The Xplor-NIH NMR molecular structure determination package. J. Magn. Res., 160:66–74, 2003. [4] P. Güntert, W. Braun, and K. Wüthrich. Torsion angle dynamics for NMR structure calculation with the new program DYANA. J. Mol. Biol., 273:283–298, 1997. [5] Axel T. Brünger, Paul D. Adams, and Luke M. Rice. New ap- plications of simulated annealing in X-ray crystallography and solution NMR. Structure, 15:325–336, 1997. [6] C. Jarzynski. Nonequilibrium equality for free energy differences. Phys. Rev. Lett., 78:2690–2693, 1997. [7] S. Brown and T. Head-Gordon. Cool walking: A new Markov chain Monte Carlo method. J. Comp. Chem., 24:68–76, 2002. [8] D. D. Frantz, D. L. Freeman, and J. D. Doll. Reducing quasi- ergodic behavior in Monte Carlo simulations by J-walking: appli- cations to atomic clusters. J. Chem. Phys., 93:2768–2783, 1990. [9] A. P. Lyubartsev, A. A. Martsinovski, S. V. Shevkunov, and P. N. Vorontsov-Velyaminov. New approach to Monte Carlo calculation of the free energy: Method of expanded ensembles. J. Chem. Phys., 96:1776–1783, 1992. [10] E. Marinari and G. Parisi. Europhys. Lett., 19:451–458, 1992. [11] Charles J. Geyer. Markov chain Monte Carlo maximum likeli- hood. In E. M. Keramidas, editor, Proceedings of the 23rd sympo- sium on the interface, Computing science and statistics. Interface foundation of North America, 1991. [12] David J. Earl and Michael W. Deem. Parallel tempering: theory, applications, and new perspectives. Phys. Chem. Chem. Phys., 23:3910–3916, 2005. [13] Daniel M. Zuckerman and Edward Lyman. A second look at canonical sampling of biomolecules using replica exchange simu- lation. J. Chem. Th. and Comp., 4:1200–1202, 2006. [14] Alan M. Ferrenberg and Robert H. Swendsen. New Monte Carlo technique for studying phase transitions. Phys. Rev. Lett., 61:2635–2638, 1988. [15] S. Kumar, D. Bouzida, R. H. Swendsen, P. A. Kollman, and J. M. Rosenberg. The weighted histogram analysis method for free energy calculations in biomolecules. i. the method. J. Com- put. Chem., 13:1011–1021, 1992. [16] Alexander Matro, David L. Freeman, and Robert Q. Topper. Computational study of the structures and thermodynamic prop- erties of ammonium chloride clusters using a parallel jump- walking approach. J. Chem. Phys., 104, 1996. [17] http://dasher.wustl.edu/tinker/. [18] W. C. Still, A. Tempczyk, and R. C. Hawley. Semianalytical treatment of solvation for molecular mechanics and dynamics. J. Am. Chem. Soc., 112:6127–6129, 1990. [19] A. M. Ferrenberg, D. P. Landau, and K. Binder. Statistical and systematic errors in monte carlo sampling. J. Stat. Phys., 63:867– 882, 1991. [20] Edward Lyman and Daniel M. Zuckerman. On the convergence of biomolecular simulations by evaluation of the effective sample size. preprint: http://xxx.lanl.gov/abs/q-bio.QM/0607037. [21] Jun S. Liu. Monte Carlo strategies in scientific computing. Springer, New York, 2001. [22] Jinfeng Zhang, Ming Lin, Rong Chen, Jie Liang, and Jun S. Liu. Monte Carlo sampling of near-native structures of proteins with applications. PROTEINS, 66:61–68, 2007. T-steps Annealing time Successful Fractional Net cost Efficiency tanneal structures success rate gain M = (M − 2)tR n f ≡ n/N tcost (nsec) R 3† 0.5 psec 7.1 4.4× 10−4 1.94 1.5 5† 1.5 psec 43.7 2.7× 10−3 1.36 2.2 17† 1.5 psec 137.6 8.6× 10−3 0.97 3.1 33 1.5 psec 46.2 2.9× 10−3 1.32 2.3 9† 3.5 psec 163.2 1.0× 10−2 1.15 2.6 17 7.5 psec 205.3 1.3× 10−2 1.38 2.2 17 15.0 psec 237.2 1.5× 10−2 1.80 1.7 17 30.0 psec 353.8 2.2× 10−2 2.16 1.4 Table 1: Comparison of the efficiency of AIS between several cooling sched- ules. n is given by Eq. 4, tcost is given by Eq. 6. The efficiency gain is the total simulation time invested in each successful annealed structure tcost divided by the time needed to generate an indepenendent structure by constant tem- perature simulation. The † indicates schedules for which data are presented in Figs. 1 and 2. Figure Legends Figure 1. Distribution of energies, from standard, constant temperature simu- lation and AIS. The dashed line is the T = 500 K distribution that was used for the high T ensemble. The other data compare a 300 nsec, T = 298 K constant temperature simulation to 298 K ensembles generated by the AIS algorithm with different cooling schedules. The schedules are discussed in Table 1. Figure 2. Distribution of the logarithm of trajectory weights for the four cooling schedules used in Fig. 1 and discussed in Table 1. Figure 1: Figure 2: Introduction Theory Results Conclusion

0704.0465

A General Nonlinear Fokker-Planck Equation and its Associated Entropy

A General Nonlinear Fokker-Planck Equation and its Associated Entropy Veit Schwämmle, Evaldo M. F. Curado, Fernando D. Nobre Centro Brasileiro de Pesquisas F́ısicas, Rua Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180, Brazil (Dated: October 27, 2018) Abstract A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equa- tion, comes out as a very general tool to describe phenomenologically systems presenting complex behavior, like anomalous diffusion, in the presence of external forces. Such an equation is char- acterized by a nonlinear diffusion term that may present, in general, two distinct powers of the probability distribution. Herein, we calculate the stationary-state distributions of this equation in some special cases, and introduce associated classes of generalized entropies in order to satisfy the H-theorem. Within this approach, the parameters associated with the transition rates of the original master-equation are related to such generalized entropies, and are shown to obey some restrictions. Some particular cases are discussed. Keywords: Nonlinear Fokker-Planck Equation, Generalized Entropies, H-Theorem, Nonextensive Thermostatistics. PACS numbers: 05.40.Fb, 05.20.-y, 05.40.Jc, 66.10.Cb http://arxiv.org/abs/0704.0465v1 I. INTRODUCTION The standard statistical-mechanics formalism, as proposed originally by Boltzmann and Gibbs (BG), is considered as one of the most successful theories of physics, and it has en- abled physicists to propose theoretical models in order to derive thermodynamical properties for real systems, by approaching the problem from the microscopic scale. Such a prescrip- tion has led to an adequate description of a large diversity of physical systems, essentially those represented by linear equations and characterized by short-range interactions and/or short-time memories. Although BG statistical mechanics is well formulated (under certain restrictions) for systems at equilibrium, the same is not true for out-of-equilibrium systems, in such a way that most of this theory applies only near equilibrium [1, 2, 3]. One of the most important phenomenological equations of nonequilibrium statistical mechanics is the linear Fokker-Planck equation (FPE), that rules the time evolution of the probability distri- bution associated with a given physical system, in the presence of an external force field [4], provided that the states of the system can be expressed by a continuum. This equation deals satisfactorily with many physical situations, e.g., those associated with normal diffu- sion, and is essentially associated with the BG formalism, in the sense that the Boltzmann distribution, which is usually obtained through the maximization of the BG entropy under certain constraints (the so-called MaxEnt principle), also appears as the stationary solution of the linear FPE [4, 5]. Nevertheless, restrictions to the applicability of the BG statistical-mechanics formalism have been found in many systems, including for instance, those characterized by nonlinear- ities, long-range interactions and/or long-time memories, which may present several types of anomalous behavior, e.g., stationary states far from equilibrium [6, 7, 8]. These anoma- lous behaviors suggest that a more general theory is required; as a consequence of that, many attempts have been made, essentially by proposing generalizations of the BG entropy [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Among these, the most successful proposal, so far, appears to be the one suggested by Tsallis [9], through the introduction of a generalized entropy, characterized by an index q, in such a way that the BG entropy is recovered in the limit q → 1. The usual extensivity property of some thermodynamic quantities holds only for q = 1, and if q 6= 1 such quantities do not increase linearly with the size of the system; this has led to the so-called nonextensive statistical-mechanics formalism [6, 7, 8]. Among many systems that present unusual behavior, one should mention those char- acterized by anomalous diffusion, e.g., particle transport in disordered media. A possible alternative for describing anomalous-transport processes consists in introducing modifica- tions in the standard FPE. Within the most common procedure, one considers nonlinear FPEs [19], that in most of the cases come out as simple phenomenological generalizations of the usual linear FPE [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In these nonlinear systems, interesting new aspects appear, leading to a wide range of open problems, in such a way that their investigation has led to a new research area in physics, with a lot of applications in natural systems. It is very common to find the power-law-like probability distribution that maximizes Tsallis’ entropy as solutions of some nonlinear FPEs [19, 20, 21, 23, 24, 30]. It seems that the nonextensive statistical mechanics formalism appears to be intimately related to nonlinear FPEs, motivating an investigation for a better understanding of possible con- nections between generalized entropies and nonlinear FPEs [19, 20, 21, 25, 29, 31, 32, 33, 34]. Recently, a general nonlinear FPE has been derived directly from a standard master equation, by introducing nonlinear contributions in the associated transition probabilities, leading to [35, 36] ∂P (x, t) ∂(A(x)P (x, t)) Ω[P (x, t)] ∂P (x, t) ; Ω[P ] = aµP µ−1+b(2−ν)P ν−1 , where a and b are constants, whereas µ and ν are real exponents [38]. The system is in the presence of an external potential φ(x), associated with a dimensionless force A(x) = −dφ(x)/dx [φ(x) = − A(x′)dx′]; herein, we assume analyticity of the potential φ(x) and integrability of the force A(x) in all space. In what concerns the functional Ω[P (x, t)], we are assuming its differentiability and in- tegrability with respect to the probability distribution P (x, t), in such a way that at least its first derivative exists, i.e., that it should be at least Ω[P ] ∈ C1. Furthermore, this func- tional should be a positive finite quantity, as expected for a proper diffusion-like term; this property will be verified later on, as a direct consequence of the H-theorem. As usual, we assume that the probability distribution, together with its first derivative, as well as the product A(x)P (x, t), should all be zero at infinity, P (x, t)|x→±∞ = 0 ; ∂P (x, t) = 0 ; A(x)P (x, t)|x→±∞ = 0 , (∀t) . (2) The conditions above guarantee the preservation of the normalization for the probability distribution, i.e., if for a given time t0 one has that dx P (x, t0) = 1, then a simple integration of Eq. (1) with respect to the variable x yields, dx P (x, t) = − [A(x)P (x, t)] Ω[P (x, t)] ∂P (x, t) = 0 , (3) and so, dx P (x, t) = dx P (x, t0) = 1 (∀t) . (4) In the present work we investigate further properties of the nonlinear FPE of Eq. (1), finding stationary solutions in several particular cases, and discussing its associated entropies, that were introduced in order to satisfy the H-theorem. In the next section we present stationary solutions of this equation; in section III we prove the H-theorem by using Eq. (1), and show that the validity of this theorem can be directly related to the definition of a general entropic form associated with this nonlinear FPE. In section IV we discuss particular cases of this general entropic form and their associated nonlinear FPEs. Finally, in section V we present our conclusions. II. STATIONARY STATE The nonlinear FPE of Eq. (1) is very general and covers several particular cases, e.g., the one related to Tsallis’ thermostatistics [20, 21, 30]. In this section we will restrict ourselves to a stationary state, and will derive the corresponding solutions for particular values of the parameters associated with this equation. Let us then rewrite Eq. (1) in the form of a continuity equation, ∂P (x, t) ∂j(x, t) = 0 ; j(x, t) = A(x)P (x, t)− Ω[P (x, t)] ∂P (x, t) , (5) in such a way that a stationary solution of Eq. (1), Pst(x), is associated with a stationary probability flux, jst(x) = constant, which becomes jst(x) = 0, when one uses Eq. (2). Therefore, using the functional Ω[P ] of Eq. (1), the stationary-state solution satisfies, A(x) = st (x) + b(2 − ν)P st (x) ] ∂Pst(x) , (6) which, after integration, becomes φ0 − φ(x) = a st (x) + b 2 − ν ν − 1 P ν−1st (x) , (7) where φ0 represents a constant. The equation above may be solved easily in some particular cases, e.g., ν = µ, ν = 2, and µ = 0, Pst(x) = ; Z(1) = , (8) where α = µ, in the cases ν = µ and ν = 2, whereas α = ν, if µ = 0. In the equation above, [y]+ = y, for y > 0, and zero otherwise. Another type of solution applies for µ = 2ν − 1, or ν = 2µ− 1, Pst(x) = 1 +K (φ(x)− φ0) ; Z(2) = 1 +K (φ(x)− φ0) where (2ν − 1)(ν − 1) (2− ν)2 , if µ = 2ν − 1 (α = ν) , (3− 2µ)(µ− 1) , if ν = 2µ− 1 (α = µ) , and we are assuming that [1 + K(φ(x) − φ0)] ≥ 0. Some well-known particular cases of the stationary solutions presented above come out easily, e.g., from Eq. (8) one obtains, in all three situations that yielded this equation, the exponential solution associated with the linear FPE in the limit α → 1 [with φ0 ∝ (α− 1) −1], as well as the generalized exponential solution related to Tsallis thermostatistics, for α = 2− q, where q denotes Tsallis’ entropic index. III. H–THEOREM AND THE ASSOCIATED ENTROPY In this section we will demonstrate the H-theorem by making use of Eq. (1), and for that purpose, an entropic form related to this equation will be introduced. Let us therefore suppose a general entropic form satisfying the following conditions, dx g[P (x)] ; g[0] = 0 ; g[1] = 0 ; ≤ 0 , (10) where one should have g[P (x, t)] at least as g[P (x, t)] ∈ C2; in addition to that, let us also define the free-energy functional, F = U − S ; U = dx φ(x)P (x, t) , (11) where β represents a Lagrange multiplier, restricted to β ≥ 0. Furthermore, we will show that this free-energy functional is bounded from below; this condition, together with the H-theorem [(∂F/∂t) ≤ 0), leads, after a long time, the system towards a stationary state. A. H–Theorem The H-theorem for a system that exchanges energy with its surrounding, herein repre- sented by the potential φ(x), corresponds to a well-defined sign for the time derivative of the free-energy functional defined in Eq. (11). Using the definitions above, dx φ(x)P (x, t)− dx g[P ] φ(x)− ∂g[P ] . (12) Now, one may use the FPE of Eq. (1) for the time derivative of the probability distribution; carrying out an integration by parts, and using the conditions of Eq. (2), one obtains, dφ(x) P (x, t) + Ω[P ] dφ(x) ∂2g[P ] . (13) Usually, one is interested in verifying the H-theorem from a well-defined FPE, together with a particular entropic form, in such a way that the quantities Ω[P ] and ∂2g[P ]/∂P 2 are previously defined (see, e.g., Refs. [19, 37]). Herein, we follow a different approach, by assuming that the general Eqs. (1) and (10) should be satisfied; then, we impose the condition, ∂2g[P ] Ω[P ] P (x, t) , (14) in such way that, dx P (x, t) dφ(x) Ω[P ] P (x, t) ≤ 0 . (15) It should be noticed that Eq. (14), introduced in such a way to provide a well-defined sign for the time derivative of the free-energy functional, yields two important conditions, as described next. (i) Ω[P ] ≥ 0 [cf. Eq. (10)], which is expected for an appropriate diffusion-like term. (ii) It expresses a relation involving the FPE of Eq. (1) and an associated entropic form, allowing for the calculation of such an entropic form, given the FPE, and vice-versa. Since the FPE is a phenomenological equation that specifies the dynamical evolution associated with a given physical system, Eq. (14) may be useful in the identification of the entropic form associated with such a system. In particular, the present approach makes it possible to identify entropic forms associated with some anomalous systems, exhibiting unusual be- havior, that are appropriately described by nonlinear FPEs, like the one of Eq. (1). As an illustration of this point, let us consider the simple case of a linear FPE, that describes the dynamical evolution of many physical systems, essentially those characterized by normal diffusion. This equation may be recovered from Eq. (1) by choosing µ = ν = 1, in such a way that Ω[P ] = a+b = D, where D represents a positive constant diffusion coefficient with units (time)−1. One may now set the Lagrange multiplier β = kB/D, where kB represents the Boltzmann constant; integrating Eq. (14), and using the conditions of Eq. (10), one gets the well-known BG entropic form, g[P ] = −kBP lnP . (16) In the next section we will explore further the relation of Eq. (14), by analyzing other particular cases. The simplest situation for which condition (i) above is satisfied may be obtained by imposing both terms of the functional Ω[P ] to be positive, which leads to a ≥ 0 ; µ ≥ 0 , or a < 0 ; µ < 0 , (17) b ≥ 0 ; ν ≤ 2 , or b < 0 ; ν > 2 . (18) It should be stressed that it is possible to have Ω[P ] ≥ 0 with less restrictive ranges for the parameters above. However, an additional property for the free-energy functional of Eq. (11) to be discussed next, namely, the boundness from below, requires the conditions of Eqs. (17) and (18), with the additional restriction ν > 0. Now, integrating Eq. (14) for the general functional Ω[P ] of Eq. (1), and using the standard conditions for g[P ] defined in Eq. (10), one gets that g[P ] = −β P µ + b ν(ν − 1) P ν + aν(1 − ν) + b(2− ν)(1− µ) (1− µ)(1− ν)ν . (19) This entropic form recovers, as particular cases, the BG entropy (e.g., when µ, ν → 1) and several generalized entropies defined previously in the literature, like those introduced by Tsallis [9], Abe [10], Borges-Roditi [12], and Kaniadakis [16, 17]. Such particular cases, as well as their associated FPEs, will be discussed in the next section. For the simpler situation of an isolated system, i.e., φ(x) = constant, the H-theorem should be expressed in terms of the time derivative of the entropy, in such a way that Eq. (13) should be replaced by ∂S[P ] Ω[P ] ∂2g[P ] dx Ω[P ] ∂2g[P ] ≥ 0 . In this case all that one needs is the standard condition associated with the FPE [same condition (i) above], i.e., Ω[P ] ≥ 0, and the general restrictions of Eq. (10) for the entropy. A similar result may also be obtained by proving the H-theorem using the master equation from which Eq. (1) was derived, with the transition probabilities introduced in Ref. [35, 36] (see the Appendix). B. Boundness from Below Above, we have proven that the free-energy functional decreases in time, and so, for the existence of a stationary state at long times of an evolution process, characterized by a probability distribution Pst(x), one should have that F (P (x, t)) ≥ F (Pst(x)) (∀t). (21) In what follows, we will show this inequality and find the conditions for its validity. There- fore, using Eqs. (7), (10), and (11), we can write, F (P (x, t)) = P (x, t) φ0 − a st (x)− b ν − 1 P ν−1st (x) g[P ]dx , (22) and so, F (Pst)− F (P ) = (P − Pst) st + b ν − 1 P ν−1st (g[P ]− g[Pst]) dx , where we have used the normalization condition for the probabilities. Now, we insert the entropic form of Eq. (19) in the equation above to obtain, F (Pst)− F (P ) = st Γµ[P/Pst] + b(2− ν) ν(ν − 1) P νst Γν [P/Pst] dx , (24) where, Γα[z] = 1− α + αz − z α (α = µ, ν) . (25) By analyzing the extrema of the functional Γα[P/Pst], one can see that Γα[P/Pst] ≥ 0 for 0 < α < 1 and Γα[P/Pst] ≤ 0 for α > 1 and α < 0. Therefore, the inequality of Eq. (21) is satisfied for the range of parameters specified by Eqs. (17) and (18), if one considers additionally, ν > 0. For the case of an isolated system, the stationary solution turns out to be the equilibrium state, which is the one that maximizes the entropy. Therefore, one may use the concavity property of the entropy [cf. Eq. (10)] in order to get, S[Peq(x)]− S[P (x, t)] = (g[Peq]− g[P ])dx ≥ 0 (∀t). (26) IV. SOME PARTICULAR CASES In this section we analyze some particular cases of the entropic form of Eq. (19) and using Eq. (14), we find for each of them, the corresponding functional Ω[P ] of the associated FPE. In the examples that follow, we will set the Lagrange multiplier β = k/D, with k and D representing, respectively, a constant with dimensions of entropy and a constant diffusion coefficient. (a) Tsallis entropy [9]: This represents the most well-known generalization of the BG entropy, which has led to the development of the area of nonextensive statistical mechanics [6, 7, 8]. One may find easily that Eq. (19) recovers Tsallis entropy in several particular cases, e.g., {b = 0, a = D, µ = q}, {a = 0, b = Dν/(2−ν), ν = q}, and {a = D/2, b = Dν/[2(2−ν)], µ = ν = q}. For all these cases one may use Eq. (14), in order to get the corresponding functional Ω[P ], g[P ] = k P q − P , Ω[P ] = qDP q−1 . (27) With the functional Ω[P ] above, one identifies the nonlinear FPE that presents the well- known q-exponential, or Tsallis distribution (replacing q → 2 − q), as a time-dependent solution [20, 21, 25]. (b) Abe entropy [10]: This proposal was inspired in the area of quantum groups, where certain quantities, usually called q-deformed quantities, are submitted to deformations and are often required to possess the invariance q ↔ q−1. The Abe entropy may be obtained from Eq. (19) in the particular case {a = D(q−1)/(q−q−1), b = −Dq(q+1)/[(q−q−1)(q+2)], µ = −ν = q}, for which g[P ] = −k P q − P−q q − q−1 , Ω[P ] = D q(q − 1) q − q−1 P q−1 − q(q + 1) q − q−1 P−q−1 . (28) (c) Borges-Roditi entropy [12]: This consists in another generalization of Tsallis entropy, where now one has two distinct entropic indices, q and q′, with a more general invariance q ↔ q′; this case may be obtained from Eq. (19) by choosing {a = D(q − 1)/(q′ − q), b = Dq′(q′ − 1)/[(q − q′)(2− q′)], µ = q, ν = q′}. One gets, g[P ] = −k P q − P q q − q′ , Ω[P ] = D q − q′ q(q − 1)P q−1 − q′(q′ − 1)P q . (29) (d) Kaniadakis entropy [16, 17]: This is also a two-exponent entropic form, but slightly different from those presented in examples (b) and (c) above; it may be reproduced from Eq. (19) by choosing {a = b = D/[2(1 + q)], µ = 1 + q, ν = 1− q}, in such a way that g[P ] = − 1 + q P 1+q − P 1−q , Ω[P ] = (P q + P−q) . (30) Except for the well-known example (a), i.e., Tsallis entropy and its corresponding FPE, the other three particular cases presented herein were much less explored in the literature. Their associated FPEs, defined in terms of their respective functionals Ω[P ] above are, to our knowledge, presented herein for the first time. These equations, whose nonlinear terms depend essentially in two different powers of the probability distribution, may be appropriated for anomalous-diffusion phenomena where a crossover between two different diffusion regimes occurs [24]. V. CONCLUSION We have analyzed important aspects associated with a recently introduced nonlinear Fokker-Planck equation, that was derived directly from a master equation by setting non- linear effects on its transition rates. Such equation is characterized by a nonlinear diffusion term that may present two distinct powers of the probability distribution; for this reason, it may reproduce, as particular cases, a large range of nonlinear FPEs of the literature. We have obtained stationary solutions for this equation in several cases, and some of them recover the well-known Tsallis distribution. We have proven the H-theorem, and for that, an important relation involving the parameters of the FPE and an entropic form was intro- duced. Since the FPE is a phenomenological equation that specifies the dynamical evolution associated with a given physical system, such a relation may be useful for identifying en- tropic forms associated with real systems and, in particular, anomalous systems that exhibit unusual behavior and are appropriately described by nonlinear FPEs. It is shown that, in the simple case of a linear FPE, the Boltzmann-Gibbs entropy comes out straightforwardly from this relation. Considering the above-mentioned nonlinear diffusion term, this relation yields a very general entropic form, which similarly to its corresponding FPE, depends on two distinct powers of the probability distribution. Apart from Tsallis’ entropy, other en- tropic forms introduced in the literature are recovered as particular cases of the present one, essentially those characterized by two entropic indices. Nonlinear FPEs (as well as their associated entropic forms) whose nonlinear terms depend essentially on two different powers of the probability distribution, like the ones discussed in the present paper, are good candi- dates for describing anomalous-diffusion phenomena where a crossover between two different diffusion regimes may take place. As a typical example, one could have a particle trans- port in a system composed by two types of disordered media, characterized respectively, by significantly different grains (e.g., different average sizes), arranged in such a way that the diffusion process is dominated by one medium, at high densities, and by the other one, at low densities. [1] L. E. Reichl. A Modern Course in Statistical Physics. Wiley, New York, 2nd edition, 1998. [2] N. G. Van Kampen. Stochastic Processes in Physics and Chemistry. North-Holland, Amster- dam, 1981. [3] H. Haken. Synergetics. Springer–Verlag, Berlin, 1977. [4] H. Risken. 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Tsallis. J. Math. Phys., 39:6490, 1998. [31] A. Compte and D. Jou. J. Phys. A, 29:4321, 1996. [32] T. D. Frank. Phys. Lett. A, 267:298, 2000. [33] T. D. Frank. Physica A, 292:392, 2001. [34] S. Martinez, A. R. Plastino, and A. Plastino. Physica A, 259:183, 1998. [35] E.M.F. Curado and F.D. Nobre. Phys. Rev. E, 67:021107, 2003. [36] F. D. Nobre, E. M. F. Curado, and G. Rowlands. Physica A, 334:109, 2004. [37] M. Shiino. J. Math. Phys., 42:2540, 2001. [38] Even though one could use a simpler notation for the functional Ω[P ], e.g., Ω[P ] = a′Pµ ν′ , herein we will keep the notation of Eq. (1), as appeared naturally in the derivation of the above FPE, for a consistency with previous publications [35, 36]. Appendix In this appendix we will prove the H-theorem directly from the master equation, for an isolated system (i.e., no external forces). Let us consider a system described in terms of W discrete stochastic variables; then, Pn(t) represents the probability of finding this system in a state characterized by the variable n at time t. This probability evolves in time according to a master equation, (Pmwm,n − Pnwn,m) , (31) where wk,l(t) represents the transition probability rate from state k to state l. The nonlinear effects were introduced through the transition rates [35], wk,l(∆) = (δk,l+1 + δk,l−1)[aP (t) + bP ν−1 (t)] , (32) where ∆ represents the size of the step of the random walk. Herein, we shall consider a random walk characterized by ∆ = 1; substituting such a transition rate in the master equation one gets, n = 1 : 2 − aP 1 + bP2P 1 − bP1P 2 , (33) n = W : W−1 − aP W + bPW−1P W − bPWP W−1 , (34) n = 2, ..., (W − 1) : n+1 + P − 2aP µn + bP n (Pn+1 + Pn−1) − bPn P ν−1n+1 + P , (35) where we have treated the borders of the spectrum (n = 1 and n = W ) separately from the rest. Let us now consider the entropy, S = g[Pn], satisfying the same conditions specified in the text [see Eq. (10)]; the H-theorem, to be proven below, is expressed in terms of its time derivative, dg[Pn] ≥ 0 . (36) Then, using Eqs. (33)–(35) one can write this time derivative as, dg[Pn] n+1 − aP n − bPnP n+1 + bPn+1P dg[Pn] n−1 − aP n + bPn−1P n − bPnP The sum indices can be rearranged to yield all summations in the range , and thus the time derivative of the entropy becomes, dg[Pn] dg[Pn+1] dPn+1 n+1 − P − bPnPn+1 P ν−2n+1 − P . (38) The negative curvature of the entropic function [cf. Eq. (10)] implies that its first derivative decays monotonically with Pn. Hence, for Pn+1 > Pn, the condition of Eq. (36) is satisfied n+1 − P Pn+1 − Pn − bPnPn+1 P ν−2n+1 − P Pn+1 − Pn ≥ 0 , (39) where we divided the term inside the brackets in Eq.(38) by the difference ∆P = Pn+1−Pn. Now we consider the limit ∆P → 0 and obtain, aµP µ−1 + b(2− ν)P ν−1 ≥ 0 , (40) which corresponds to the condition Ω[P ] ≥ 0 found in the text, when proving the H-theorem by making use of the FPE of Eq. (1). It should be mentioned that the procedure above works also for Pn > Pn+1 ; therefore, in this appendix we have proven that the H-theorem holds for an arbitrary state n and all times t of an isolated system. Introduction Stationary State H–theorem and the Associated Entropy H–Theorem Boundness from Below Some Particular Cases Conclusion References Appendix

0704.0466

Conduction electron spin-lattice relaxation time in the MgB2 superconductor

Conduction electron spin-lattice relaxation time in the MgB2 superconductor F. Simon,∗ F. Murányi,† T. Fehér, A. Jánossy Budapest University of Technology and Economics, Institute of Physics and Condensed Matter Research Group of the Hungarian Academy of Sciences, H-1521 Budapest, PO BOX 91, Hungary L. Forró IPMC/SB Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne-EPFL, Switzerland C. Petrovic,‡ S.L. Bud’ko, P.C. Canfield Ames Laboratory, U.S. Department of Energy and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Dated: July 19, 2021) The spin-lattice relaxation time, T1, of conduction electrons is measured as a function of tem- perature and magnetic field in MgB2. The method is based on the detection of the z component of the conduction electron magnetization under electron spin resonance conditions with amplitude modulated microwave excitation. Measurement of T1 below Tc at 0.32 T allows to disentangle con- tributions from the two Fermi surfaces of MgB2 as this field restores normal state on the Fermi surface part with π symmetry only. INTRODUCTION The conduction electron spin-lattice relaxation time in metals, T1, is the characteristic time for the return to thermal equilibrium of a spin system driven out of equi- librium by e.g. a microwave field at electron-spin reso- nance (ESR) or a spin-polarized current. The applica- bility of metals in “spintronics” devices in which infor- mation is processed using electron spins [1] depends on a sufficiently long spin life-time. In pure metals T1 is lim- ited by the Elliott mechanism [2, 3], i.e. the scattering of conduction electrons by the random spin-orbit poten- tial of non-magnetic impurities or phonons. In supercon- ductors, the Elliott mechanism becomes ineffective and a long T1 is predicted well below Tc [3]. Here we report the direct measurement of the spin-lattice relaxation time of conduction electrons in MgB2 in the superconducting state. The motivation to study the magnetic field and temperature dependence of T1 is two-fold: i) to test the predicted lengthening of T1 to temperatures well below Tc, ii) to measure the contributions to T1 from different Fermi surface sheets and to compare with the correspond- ing momentum life-times, τ . The lengthening of T1 has been observed in a re- stricted temperature range below Tc in the fulleride superconductor, K3C60 by measuring the conduction electron-spin resonance (CESR) line-width, ∆H [4]. This method assumes 1/T1 = 1/T2 = γe∆H , where γe/2π = 28.0 GHz/T is the electron gyromagnetic ratio, and 1/T2 is the spin-spin or transversal relaxation rate. It is limited to cases where the homogeneous broadening of the CESR line due to a finite spin lifetime outweighs ∆Hinhom, the line broadening from inhomogeneities of the magnetic field. In a superconducting powder sam- ple, the CESR line is inhomogeneously broadened below the irreversibility line due to the distribution of vorte- ces, which is temperature and magnetic field dependent. This prevents the measurement of T1 from the line-width and calls for a method to directly measure T1. Electron spin echo techniques, which usually enable the measure- ment of T1, are not available for the required nanosec- ond time resolution range. The magnetic resonance tech- nique, termed longitudinally detected (LOD) ESR [5, 6] used in this work allows to measure T1’s as short as a few ns. The method is based on the measurement of the electron spin magnetization along the magnetic field, Mz, using modulated microwave excitation. Mz recovers to its equilibrium value with the T1 time-constant, thus the method allows the direct measurement of T1 independent of magnetic field inhomogeneities. MgB2 has a high superconducting transition tempera- ture of Tc = 39 K [7] and its unusual physical properties [8, 9, 10, 11] are attributed [12, 13] to its disconnected, weakly interacting Fermi surface (FS) parts. The Fermi surface sheets derived from B-B bonds with π and σ char- acters (π and σ FS) have smaller and higher electron- phonon couplings and superconductor gaps, respectively, and contribute roughly equally to the density of states (DOS). The strange band structure leads to unique ther- modynamic properties: magnetic fields of about 0.3-0.4 T restore the π FS well below Tc for all field orienta- tions in polycrystalline samples but the material remains superconducting to much higher fields. This is character- ized by a small and nearly isotropic upper critical field, c2 ∼ 0.3 − 0.4 T [10, 14] and a strongly anisotropic one, Hσ c2 = 2 − 16 T, [10, 15, 16] related to the π and σ Fermi surface sheets, respectively. Our measurements at low fields and low temperatures determine T1 from the π FS alone, while high field and high temperature experi- ments measure T1 averaged over the whole FS. We find http://arxiv.org/abs/0704.0466v1 that spin relaxation in high purity MgB2 is temperature independent in the high field normal state between 3 K and 50 K, indicating that it arises from non magnetic im- purities. Spin relaxation times for electrons on the π and σ Fermi surface sheets are widely different but are not proportional to the corresponding momentum relaxation times. EXPERIMENTAL The same MgB2 samples were used as in a previous study [17]. Thorough grinding, particle size selection and mixing with SnO2, an ESR silent oxide, produced a fine powder with well separated small metallic par- ticles. The nearly symmetric appearance of the CESR signal [18] proves that penetration of microwaves is ho- mogeneous and that the particles are smaller than the microwave penetration depth of ∼ 1µm. SQUID mag- netometry showed that grinding and particle selection do not affect the superconducting properties. The parti- cles are not single crystals but rather aggregates of small sized single crystals. Continuous wave (cw) and longitu- dinally detected ESR experiments were performed in a home-built apparatus [6] at 9.1 and 35.4 GHz microwave frequencies, corresponding to 0.32 and 1.27 T resonance magnetic fields. The 9.1 GHz apparatus is based on a loop-gap resonator with a low quality factor (Q ∼ 200) and the 35.4 GHz instrument does not employ a mi- crowave cavity at all. The cw-ESR was detected using an audio frequency magnetic field modulation. Line-widths are determined by Lorentzian fits to the cw-ESR data. For the LOD-ESR, the microwaves are amplitude modu- lated with f = Ω/2π of typically 10 MHz and the result- ing varying Mz component of the sample magnetization is detected with a coil which is parallel to the external magnetic field and is part of a resonant circuit that is tuned to f and is matched to 50 Ohms. cw-ESR at 420 GHz (centered at 15.0 T) was performed at EPFL using a quasi-optical microwave bridge with no resonant cavities. RESULTS Relaxation in the normal state The low temperature behavior of the spin-lattice re- laxation time in MgB2 in the normal state can be mea- sured using cw-ESR from the homogeneous line-width, ∆Hhom, using 1/T1 = γe∆Hhom at high fields, H > Hc2 that suppresses superconductivity. The maximum upper critical field is Hc2,max ∼ 16 T for particles with field in the (a, b) crystallographic plane in the polycrystalline sample at zero temperature [19]. We did not observe any effects of superconductivity on the CESR, at 15 T it is suppressed in the full sample above a temperature of a 0 10 20 30 40 50 60 T (K) FIG. 1: CESR line-width of MgB2 as a function of tempera- ture for the 15 T CESR measurement (�). Open circles (©) show the homogeneous line-width (∆Hhom) after correcting for the field dependent broadening as explained in the text. Representative error bars are shown at the lowest tempera- ture. 0 2 4 6 8 10 12 14 16 Magnetic field (T) FIG. 2: ESR line-width of MgB2 as a function of magnetic field measured at 40 K (�). Solid curve is a linear fit to the data with parameters given in the text. few K. Fig. 1 shows that the temperature dependence of the CESR line-width at 15 T is small below 40 K. The CESR line-width is magnetic field dependent as shown in Fig. 2 at 40 K: it is linear as function of magnetic field with ∆H = ∆H0 + b ∗ H , where ∆H0 = 0.90(1) mT is the residual line-width and b = 0.057(1) mT/T. The residual homogeneous line-width corresponds to T1 = 6.3 ns at 40 K. The linear relation can be used to correct the 15 T CESR line-width data to obtain the homogeneous contribution, ∆HHom(T ) = ∆H(15 T, T )− 15T · b as the magnetic field dependence is expected to be temperature independent. We show the homogeneous line-width in Fig. 1. We find that it is 0 10 20 30 40 50 60 32 34 36 38 T (K) T (K) FIG. 3: Inhomogeneous CESR line broadening in MgB2 below Tc at 0.32 T. Full and open symbols show the CESR line-width for up and down magnetic field sweeps, respectively. Inset shows the data near Tc. Note the line narrowing between Tc and Tirr and the field sweep direction dependent line-widths below the irreversibility temperature. temperature independent within experimental precision between 3 and 50 K. This means that the spin-lattice re- laxation time flattens to a residual value that is given by non-magnetic impurities. Relaxation in the superconducting state In type II superconductors, CESR arises from thermal excitations and from normal state vortex cores. The in- homogeneity of the magnetic field in the vortex lattice or glass states does not broaden the CESR line. The lo- cal magnetic field inhomogeneity is averaged since within the spin life-time itinerant electrons travel long distances compared to the inter-vortex distance [4]. This is in con- trast to the NMR case where the line-shape is affected: the nuclei are fixed to the crystal and nuclei inside and outside the vortex cores experience different local fields [20]. In other words, a superconducting a single crystal sample would display a narrow conduction electron ESR line if there were no irreversible effects. However, the CESR line is inhomogeneously broadened below the irre- versibility line for a superconducting powder sample: the vortex distribution depends on a number of factors such as the thermal and magnetic field history, grain size and, for an anisotropic superconductor such as MgB2, on the crystal orientation with respect to the magnetic field also. The resulting inhomogeneous broadening of the CESR line gives 1/γ∆Hinhom = T 2 ≪ T1,2, and T1 cannot be measured from the line-width. In Fig. 3 we show this effect: above Tc MgB2 has a relaxationally broadened line-width of ∆H = 0.9 mT. Between Tc and the irre- versibility temperature at the given field, Tirr, the CESR remains homogeneous and narrows with the lengthening of T1. However, below Tirr it broadens abruptly and the line-width depends on the direction of the magnetic field sweep: for up sweep it is broader than for down sweeps due to the irreversibility of vortex insertion and removal. To enable a direct measurement of the T1 spin lattice relaxation time, one has to resort to time resolved experi- ments. Conventional spin-echo ESR methods are limited to T1’s larger than a few 100 ns. To measure T1’s of a few nanoseconds, the so-called longitudinally detected ESR was invented in the 1960’s by Hervé and Pescia [21] and improved by several groups [22, 23]. The method is based on the deep amplitude modulation of the mi- crowave excitation with an angular frequency, Ω ∼ 1/T1. When the sample is irradiated with the amplitude mod- ulated microwaves at ESR resonance, the component of the magnetization along the static magnetic field, Mz, decays from the equilibrium value, M0, with a time con- stant T1. Mz relaxes back to M0 with a T1 relaxation time when the microwaves are turned off. The oscillat- ing Mz is detected using a coil which is part of a resonant rf circuit. The phase sensitive detection of the oscillat- ing voltage using lock-in detection allows the measure- ment of T1 using ΩT1 = v/u [5, 21], where u and v are the amplitudes of the in- and out-of-phase components of the oscillating magnetization after corrections for in- strument related phase shifts. The principal limitation of the LOD-ESR technique is its 3-4 orders of magnitude lower sensitivity compared to conventional cw-ESR. The LOD-ESR method and the experimental apparatus are detailed in Refs. [5, 6]. To prove that the LOD-ESR signal of the itinerant electrons is detected in the superconducting phase, we compare in Fig. 4 the LOD-ESR signal with that mea- sured with conventional continuous-wave CESR (referred to as CESR in the following) of MgB2 in the normal and superconducting states. The CESR signal is the deriva- tive of the absorption due to magnetic field modulation used for lock-in detection. This signal was previously identified as the ESR of conduction electrons in MgB2 in the superconducting and normal states [15, 17, 24] and its characteristics have been discussed in detail [15, 17]. Above Tc at 40 K, the CESR line is relaxationally broad- ened. Below Tc, it is inhomogeneously broadened and is diamagnetically shifted, i.e. to higher resonance fields. The irreversible effects also contribute to a non-linear baseline known as the non-resonant microwave absorp- tion [25]. The intensity of the CESR signal decreases below Tc as we discussed previously [17], due to the van- ishing of normal state electrons. The LOD-ESR signal shows the same characteristics as the CESR below Tc: it is broadened, shifted to higher fields and its intensity decreases. The values for the temperature dependent diamagnetic shifts and broad- ening and the relative intensity change agree for the two kinds of measurements within experimental precision (not shown). This unambiguously proves that the LOD- 0.30 0.32 0.340.30 0.32 0.34 ESR LOD-ESR Magnetic field (T) 15 Kb) 40 Ka) FIG. 4: ESR (a-b) and LOD-ESR (c-d) spectra of MgB2 at 9.1 GHz (0.32 T). a) and c) at 40 K in the normal state, and b) and d) in the superconducting state at 15 K. Solid and dashed curves are the in- and out-of-phase LOD signals, respectively and are offset for clarity. Vertical solid lines in- dicate the resonance field above Tc. Note the diamagnetic shift and broadening for for both kinds of spectra below Tc. Also note the rotated phase of the in-phase and out-of phase channels upon cooling. ESR signal originates from the conduction electrons. The change of the relaxation time T1 is visible in the LOD-ESR spectra in Fig. 4 as a change in the relative intensities of the in- and out-of-phase signals. At 40 K v/u = 0.47 and at 15 K v/u = 0.95, which together with Ω/2π = 11.4 MHz gives 6.3 and 13.3 ns relaxation times, respectively. In Fig. 5, we show the T1 data inferred from the LOD-ESR spectra at 0.32 and 1.27 T as a function of the reduced temperature T/Tc. DISCUSSION The observed lengthening of T1 below Tc ( Fig. 5) is expected from theory for non-magnetic scattering cen- ters and low magnetic fields where the susceptibility is dominated by excitations over the superconducting gap. On the other hand, the field independence between 0.32 and 1.27 T of T1 below Tc is surprising. The lengthen- ing of T1 below Tc in zero magnetic field for an isotropic, type I superconductor was calculated in the framework of weak-coupled BCS theory by Yafet [3]. He concluded that T1 lengthens as a result of the freezing of normal state excitations. However, no theory exists for a type II superconductor in finite fields with Hc2 anistotropy such as MgB2, thus here the T1 data are analyzed phenomeno- 0.0 0.5 1.0 1.5 2.0 FIG. 5: Spin-lattice relaxation time as a function of the re- duced temperature in MgB2 at 0.32 (�) and 1.27 T (©) mag- netic fields. Representative error bars are shown for some of the data. Dashed curve shows T1 corresponding to ∆HHom in the 15 T measurement such as in Fig. 1 with the reduced temperature normalized to 39 K. logically in the framework of the two-band/gap model of MgB2. In the following, we deduce the residual (low tem- perature), impurity related spin scattering contributions of the σ and π Fermi surface sheets. The DOS is distributed almost equally on the FS sheets of MgB2: Nπ/(Nπ + Nσ) = 0.56 [13], where Nπ and Nσ are the DOS of the two types of FS sheets. A magnetic field of ∼ 0.3 − 0.4 T closes the gap on the π FS sheets but leaves the gap on the σ sheet almost intact. [10, 17]. This suggests that well below Tc, our experiment at 0.32 T measures exclusively the relaxation of electrons on the fully closed π FS sheets. Since T1 at 0.32 T increases slowly with temperature between 10 and 20 K, we ex- trapolate T1π ≈ T1(10 K, 0.32 T) = 20(2) ns for the π In order to separate the contribution of the σ FS to the relaxation rate in the normal state, 1/T1n, we assume that inter-band relaxation is negligible and 1/T1n is equal to the average of the spin-lattice relaxation rates on the two FS’s weighted by the corresponding DOS: Nπ/T1π +Nσ/T1σ Nπ +Nσ Here T1σ is the spin-lattice relaxation time on the σ FS. The 15 T measurement shows that 1/T1n changes little with temperature between 3 K and 40 K. Thus we find T1σ = 3.4(5) ns for the contribution of the σ FS sheets using T1n = T1(Tc) = 6.3 ns, T1π = 20(2) ns and Eq. 1. For normal metals with a simple Fermi surface, the so- called Elliott relation [2, 3, 26, 27] holds, which states that for a given type of disorder (e.g. phonons or dislo- cations) T1 is proportional to the momentum relaxation time, τ . The proportionality constant depends on the spin orbit splitting of the conduction electron bands and has been estimated in a number of metals from the shift of the CESR from the free electron value. Metals with complicated Fermi surfaces i.e. with great variations of the electron-phonon coupling on the different FS parts are known to deviate from the Elliott relation [28] and calculation of T1 requires to take into account the de- tails of the band-structure [29, 30, 31]. Examples include polyvalent elemental metals such as Mg or Al. Clearly, a calculation of T1 is required for MgB2, which takes into account its band structure peculiarities. Comparison of spin scattering and momentum scattering times of the two types of Fermi surfaces is instructive. The relative values of τ for the two FS parts, τπ and τσ, and for in- terband scattering were estimated by Mazin et al. [32]. A very small impurity interband scattering and τπ < τσ, i.e. a larger π intraband scattering relative to σ intra- band scattering was required to explain the rather small depression of Tc in materials with widely different con- ductivities. De Haas-van Alphen [33] and magnetore- sistance [34] measurements of high purity samples yield τπ < τσ also. Such a behaviour could rise fromMg vacan- cies, which perturb more electrons of the π band relative to the σ band. However, our spin scattering data do not support this. In contrast to momentum scattering, spin scattering is stronger on the σ FS: T1π : T1σ = 6 : 1 in high purity samples and low temperatures. The rela- tive values of T1 and τ for the two FS do not necessarily follow the same trend, spin relaxation times at low tem- peratures depend on spin orbit relaxation on impurities while momentum relaxation is due to potential scatter- ing. However, a defect center such as a Mg vacancy with a strong modification of the electron-phonon coupling and an atomic number strongly differing from that of the reg- ular atoms constituting the compound would greatly af- fect T1 compared to τ . In the two gap model Mg defects are expected to shorten T1π more than T1σ and thus are unlikely to be the dominant scatterers. A final note concerns the validity of the above analysis of T1’s in the framework of the two-band/gapmodel. The field independence of the lowest temperature T1 for 0.32 and 1.27 T is unexpected within this model. The spin susceptibility increases strongly between these fields and more normal states are restored at 1.27 T than expected from the closing of the gap on the π FS sheets alone [17]. Based on this, one would expect to observe additional spin scattering from the restored σ FS parts, which is clearly not the case. This also indicates that a theoret- ical study, which takes into account the peculiarities of MgB2 is required to explain the anomalous spin-lattice relaxation times. In conclusion, we presented the measurement of the spin-lattice relaxation time, T1, of conduction electrons as a function of temperature and magnetic field in the MgB2 superconductor. We use a novel method based on the detection of the z component of the conduction elec- tron magnetization during electron spin resonance con- ditions with amplitude modulated microwave excitation. Lengthening of T1 below Tc is observed irrespective of the significant CESR line broadening due to irreversible dia- magnetism in the polycrystalline sample. The field inde- pendence of T1 for 0.32 T and 1.27 T allows to measure the separate contributions to T1 from the two distinct types of the Fermi surface. ACKNOWLEDGEMENTS The authors are grateful to Richárd Gaál for the de- velopment of the ESR instrument at the EPFL. F.S. and F.M. acknowledge the Zoltán Magyary postdoc- toral programme, the Bolyai fellowship of the Hungarian Academy of Sciences and the Alexander von Humboldt Foundation for support, respectively. Work supported by the Hungarian State Grants (OTKA) No. TS049881, F61733, PF63954 and NK60984 and by the Swiss NSF and its NCCR ”MaNEP”. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State Uni- versity under Contract No. W-7405-Eng-82. ∗ Corresponding author: [email protected] † Present address: Leibniz Institute for Solid State and Materials Research Dresden, PF 270116 D-01171 Dres- den, Germany ‡ Present address: Condensed Matter Physics and Ma- terials Science Department, Brookhaven National Labo- ratory, Upton, New York 11973-5000, USA [1] I. Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). [2] R. J. Elliott, Physical Review 96, 266279 (1954). [3] Y. Yafet, Phys. Lett. A 98, 287 (1983). [4] N. M. Nemes, J. E. Fischer, G. Baumgartner, L. Forró, T. Fehér, G. Oszlányi, F. Simon, and A. Jánossy, Phys. Rev. B 61, 7118 (2000). [5] F. Murányi, F. Simon, F. Fülöp, and A. Jánossy, J. Magn. Res. 167, 221 (2004). [6] F. Simon and F. Murányi, J. Magn. Res. 173, 288 (2005). [7] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, Nature 410, 63 (2001). [8] F. 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0704.0467

Modeling Accretion Disk X-ray Continuum of Black Hole Candidates

Modeling Accretion Disk X-ray Continuum of Black Hole Candidates Gabor Pszota and Wei Cui1 Department of Physics, Purdue University, West Lafayette, IN 47907 ABSTRACT We critically examine issues associated with determining the fundamental properties of the black hole and the surrounding accretion disk in an X-ray bi- nary based on modeling the disk X-ray continuum of the source. We base our work mainly on two XMM-Newton observations of GX 339−4, because they pro- vided high-quality data at low energies (below 1 keV), which are critical for reliably modeling the spectrum of the accretion disk. A key issue examined is the determination of the so-called “color correction factor”, which is often empir- ically introduced to account for the deviation of the local disk spectrum from a blackbody (due to electron scattering). This factor cannot be predetermined the- oretically, because it may vary with, e.g., mass accretion rate, among a number of important factors. We follow up on an earlier suggestion to estimate the color correction observationally by modeling the disk spectrum with saturated Comp- ton scattering. We show that the spectra can be fitted well, and the approach yields reasonable values for the color correction factor. For comparison, we have also attempted to fit the spectra with other models. We show that even the high-soft state continuum (which is dominated by the disk emission) cannot be satisfactorily fitted by state-of-the-art disk models. We discuss the implications of these results. Subject headings: accretion, accretion disks — black hole physics – stars: indi- vidual (GX 339–4) — X-rays: stars 1. Introduction The X-ray continuum of black hole candidates (BHCs) is roughly composed of two main elements (see review by Liang 1998), an ultra-soft component that is thought to be 1Email: [email protected],[email protected] http://arxiv.org/abs/0704.0467v3 – 2 – associated with emission from the accretion disk, and a hard component that is thought to be produced by inverse Compton scattering of soft photons by energetic electrons that can be either thermal or non-thermal in origin. Modeling the disk component could, in principle, allow one to determine the radius of the inner edge of the accretion disk in a BHC (review by Tanaka & Lewin 1995, and references therein). This has been tried and the results have provided evidence that the accretion disk extends all the way in to the last stable orbit under certain circumstances (Tanaka & Lewin 1995). Motivated by this observation, Zhang et al. (1997) suggested that modeling the X-ray continuum of a BHC could lead to a measurement of the spin of the black hole, if the mass of the black hole can be independently derived. In retrospect, we now know that the accretion disk reaches the last stable orbit probably only in the high-soft state (e.g., Narayan 1996) 1, so the proposed technique may only be applicable to data taken in such a state. Since the X-ray spectrum of BHCs is dominated by the disk component in the high-soft state, the determination of the disk parameters based on spectral modeling should, in principle, be quite accurate, even if one neglects the hard component whose physical origin is less well understood, particularly in the high-soft state. However, there are still serious issues associated with the exercise. First, the local spectrum of the X-ray emitting portion of the accretion disk is not a blackbody, because the opacity is dominated by electron scattering. Saturated Comp- tonization leads to a “diluted” blackbody spectrum, whose color temperature is given by Tcol = fcolTeff , where fcol is the color correction factor and Teff is the effective tempera- ture (Ebisuzaki et al. 1984). Much effort has gone into finding the values of fcol that are appropriate for BHCs (Shimura & Takahara 1995; Merloni et al. 2000; Davis et al. 2006). The situation is still uncertain, but it is clear that fcol depends on a number of important physical parameters, such as mass accretion rate, which can vary even for a given source. It is, therefore, not possible to know what value to use a priori. Cui et al. (2002) proposed an observational approach to derive fcol from the data (see also Shrader & Titarchuk 1999). Although the technique showed some promise with limited data, it needs to be tested further. Second, there is observational evidence (Zhang et al. 2000) that the surface layer of the accretion disk in BHCs might deviate from the standard α-disk structure (Shakura & Sunyaev 1 It has recently been argued, based on hard-state observations of BHCs (e.g., Miller et al. 2006), that the disk also reaches the last stable orbit in the low-hard state. We must, however, caution against drawing strong conclusions on the properties of the disk from modeling a hard-state spectrum, because it would require a reliable extraction of the weak disk component from the dominating hard component whose precise origin (e.g., the geometry of the emitting region and the nature of seed photons) is still being debated (see Cui et al. 2002 for an in-depth discussion). This is why we chose to focus on the soft-state observations in this work. – 3 – 1973). Such an effect is expected from X-ray heating of the disk by a central hard X-ray source (e.g., Nayakshin & Melia 1997; Mistra et al. 1998), but it is not clear why the effect is still significant even for the high-soft state, in which hard X-ray production is expected to be quite weak. The presence of such a “warm” layer would add further complication in modeling the observed X-ray spectrum (Zhang et al. 2000), because Compton scattering in the layer can further modify the spectrum. Third, some of the widely-used disk models (e.g., the multi-color disk; Mitsuda et al. 1984) do not take into account general relativistic effects that can affect the formation of the X-ray spectrum. Attempts have been made to incorporate the effects empirically in the analysis by introducing a number of correction factors (Zhang et al. 1997). Recently, two new disk models have been developed that account for the general relativistic effects (Li et al. 2005; Davis & Hubeny 2006). The models also consider spectral hardening due to scattering, with one treating fcol as a free parameter (Li et al. 2005) and the other carrying out radiative transfer in the disk (Davis & Hubeny 2006). The models have been applied to observations of a number of BHCs (Shafee et al. 2006; Davis et al. 2006; McClintock et al. 2006; Middleton et al. 2006). In this work, we examined some of the issues and also assessed the viability of the state- of-the-art disk models, making use of data of much improved quality that have recently become available. Specifically, we analyzed two XMM-Newton observations of GX 339−4 and attempted to fit the observed X-ray spectra with different models. With its large effective area and good sensitivity at low energies (< 1 keV), XMM-Newton offers distinct advantages over other X-ray observatories for our purposes. The low-energy sensitivity is often not appreciated as much as it should be; it is critical to reliable modeling of the disk spectrum, because the effective temperature of the disk is typically . 1 keV for BHCs. 2. Data 2.1. XMM-Newton Observations We analyzed data from two archival XMM-Newton observations (ObsIDs 0093562701 and 0148220201) of GX 339−4 during its 2002–2003 outburst. The first observation was taken near the peak of the outburst (on 2002 August 24), judging from the ASM/RXTE light curve 2, while the second one was taken at the tail end of the episode (on 2003 March 8). GX 339−4 was observed for about 61 and 20 ks during the two observations, respectively. 2See http://heasarc.gsfc.nasa.gov/xte weather http://heasarc.gsfc.nasa.gov/xte_weather – 4 – Since we are mainly interested in the X-ray continuum here, we focused on the EPIC data. The pn/EPIC detector was operated in the burst mode, with the thin optical blocking filter, during the first observation, and the MOS/EPIC detectors were not used. In the second observation, the pn and MOS detectors were both run in the timing mode with the medium blocking filter. Even with the timing mode, the MOS data still suffer from severe photon pile-up, due to the high count rate. In contrast, the pile-up effects are minimal in the pn data. This work is, therefore, based on the pn data. The data were reduced with the standard SAS package (version 7.0.0). We followed the procedures described in the XMM-Newton data analysis cookbook 3 in preparing and filtering the data, making light curves, extracting spectra, and generating the corresponding arf and rmf files for subsequent spectral modeling. We did need to turn off bad-pixel search in processing the first observation because of a bug in the searching routine for the burst mode. The effects should be negligible because the source was very bright then. The events of interest were extracted from a rectangular region, with RAWX 32–40 RAWY 3–179 and RAWX 34–42 RAWY 3–199 for the 2002 and 2003 observations, respectively. Filtering expressions “FLAG = 0” and “PATTERN ≤ 4” were applied to select good single and double events. Because the source was bright during both observations, a significant number of source events are present even near the edge of the CCD chip, which makes it impossible to cleanly extract background events. This should only affect the high-energy end of the spectrum (where the background counts may become comparable or exceed the source counts). Our choice of the central 9 columns of the chip was made to minimize the effect on the shape of the spectrum. However, it led to an underestimation of the overall normalization, which is also important here. To determine the normalization more accurately, we also made spectra with events from the whole chip. The difference amounts to roughly 8%. For spectral modeling, we added a 1% systematic error to the data and grouped the channels so that each bin contains at least 500 counts. 2.2. RXTE Observations To complement the soft-band coverage of XMM-Newton, we obtained simultaneous RXTE data from the public archive. GX 339−4 was observed with RXTE for about 4 and 16 ks, respectively, during the two XMM-Newton observing periods. The data were 3See http://wave.xray.mpe.mpg.de/xmm/cookbook. http://wave.xray.mpe.mpg.de/xmm/cookbook – 5 – reduced with FTOOLS 5.2. We followed the standard steps 4 in preparing and filtering the data, deriving PCA and HEXTE spectra from data taken in the standard modes, and generating the corresponding response files for spectral modeling. A PCA or HEXTE spectrum consists of separate spectra from the individual detector units that were in operation. In deriving the PCA spectra, we only used data from the first xenon layer of each detector unit (which is best calibrated) and combined spectra from all the live detectors into one, to maximize the signal-to-noise ratio (S/N). To estimate the PCA background, we used the background model for bright sources (pca bkgd cmbrightvle eMv20030330.mdl). As for the HEXTE data, we extracted a spectrum for each of the two clusters separately. For spectral modeling, we rebinned the HEXTE spectra so that each bin contains at least 5000 counts. We also added a 1% systematic error to both the PCA and HEXTE spectra. 3. Results We carried out spectral modeling in XSPEC (Arnaud 1996). The spectral bands of interest are 0.5–10 keV (pn/EPIC), 3–25 keV (PCA), and > 15 keV (HEXTE). The spectra are always jointly fitted with a common model, except for a normalization factor (fixed at unity for the pn data) that was introduced to account for any residual difference in the calibration of the throughput of the detectors. Strictly speaking, however, the XMM- Newton and RXTE coverages are not always simultaneous, due to the difference not only in the observing time but also in the orbits of the two satellites. To justify joint modeling, we broke each of the XMM-Newton observations into 8 segments and extracted a spectrum for each segment. We compared the individual spectra and observed no apparent variation in the shape of the spectrum in either case. We experimented with several models for the ultra-soft and hard components of the spectrum. The former is often modeled with a non-relativistic, multi-temperature blackbody model (“diskbb” in XSPEC; Mitsuda et al. 1984). For this work, we instead used the two relativistic disk models (“kerrbb” in XSPEC, Li et al. 2005; and “bhspec”, Davis & Hubeny 2006). To test the procedure of deriving the color correction factor from the data, as proposed by Cui et al. (2002), we also modeled the disk component with saturated Compton scattering (“comptt” in XSPEC, in a disk geometry; Titarchuk 1994). In all cases, the hard component of the spectrum was modeled with unsaturated Compton scattering (also “comptt” but in a spherical geometry). Interstellar absorption was taken into account (with “phabs” in XSPEC). 4see http://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook book.html http://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook_book.html – 6 – The best and only formally acceptable fit to the continuum was obtained with comptt+comptt. In this case, the residuals reveal the presence of discrete features, which include absorption edges at 863 eV and 880 eV for the 2002 and 2003 observations, respectively, and emission lines at 569 eV and 562 eV. We suspect that the edges are calibration artifacts, since we were not able to associate them with any elements. On the other hand, the emission features could be real, with the former being associated with O VIII and the latter with O VII (corre- sponding to transitions at rest-frame energies 569 eV and 561 eV, respectively), which would imply a plasma temperature of 0.1–0.2 keV. The lines are unresolved and are quite weak, with equivalent widths of 26 and 21 eV for the 2002 and 2003 observations, respectively. We will not discuss the discrete spectral features any further, since the main focus here is on the X-ray continuum. The 2002 data also show the presence of an emission feature at 2.2 keV, which is likely an artifact caused by calibration uncertainty around the M-edge of gold (in the mirror coating). However, the feature is not apparent in the 2003 data, which is a bit puzzling, because the statistics are comparable in the two cases. We consulted with the XMM-Newton Helpdesk about it, and were told that it had probably been corrected for by the calibration in the timing mode, but not so well in the burst mode. After accounting for the discrete spectral features (with “edge” and “gaussian” in XSPEC), we still saw, in the residuals, genuine inconsistency between the pn/EPIC and PCA data at low energies, which could be related to known PCA calibration uncertainties around the L-edge of xenon. For this work, we resolved the issue simply by excluding the PCA data below 9 keV in the joint fits. For the 2003 data, the continuum fit also shows significant structures in the residuals roughly in the range of 5–8 keV, which might be similar to those reported by Miller et al. (2004) based on an XMM-Newton observation taken several months earlier. They are most likely associated with the Kα emission of the iron and its associated absorption edge. The excess appears broad and asymmetric in shape, as illustrated in Figure 1. Therefore, we modeled it as a gravitationally redshifted disk line (“laor” in XSPEC; Laor 1991). Also, we included a smeared edge (“smedge” in XSPEC) in the fit. The results are: ELaor = 6.48 +0.07 −0.09 keV, i = 51◦ +2 −1, q = 5.2± 0.2, and Rin = 1.76 +0.10 −0.06 Rg (where Rg is the gravitational radius) for the line; Eedge = 8.5 ± 0.1 keV, W = 2.7 −0.4 keV, and τ = 0.59 +0.07 −0.05 for the edge. Note that we fixed Rout at 400 Rg in the “laor” model. The obtained value for the inclination angle (i) is consistent with those estimated for the system (e.g., Zdziarski et al. 2004). If this interpretation is correct, the results would require a very high value (a∗ & 0.97) for the black hole spin (cf. Miller et al. 2004). However, no such broad line (nor the edge) is apparent in the 2002 data. Adding the line (as a Gaussian component) to the model, we found that the data could accommodate it, but its equivalent width would be merely 14+12 eV, compared to 485+217 −130 eV based on the 2003 data. – 7 – Figure 2 shows the observed X-ray spectra of GX 339−4, along with the best-fit models and the associated residuals. The parameters of the continuum fits are summarized in Table 1. The source was clearly in the high-soft state during the 2002 observation, with the disk contributing about 96% of the 0.5–10 keV flux. The spectrum became harder during the 2003 observation, but the disk still contributed about 80% of the 0.5–10 keV flux. Following Cui et al. (2002), we attempted to derive the color correction factor from the continuum fits. Briefly, to account for the effects of scattering in a Shakura-Sunyaev disk (Shakura & Sunyaev 1973), one should, strictly speaking, start with a multitemperature blackbody spectrum for the seed photons. However, comptt assumes a Wien spectrum for the seed photons. Fitting the peak of diskbb with a Wien distribution leads to Tdiskbb = 2.7TWien. Based on spectral modeling with comptt, therefore, we can approximate the color correction factor as fcol = Te/2.7T0 (Cui et al. 2002; see also Zhang 2005). For the 2002 and 2003 observations, respectively, we have fcol = 1.48 +0.09 −0.08 and 1.35 +0.01 −0.01, which seem quite reasonable. This lends support to the viability of the observational approach in deriving fcol. We then replaced the saturated Compton component with a multicolor disk model, but failed to obtain any formally acceptable fits to the observed X-ray continua with either “kerrbb” or “bhspec”. In this case, we fixed the inclination angle at the value from relativistic line modeling (51◦), the mass of the black hole at 10 M⊙, and the distance at 8 kpc (Zdziarski et al. 2004). With “kerrbb”, we also adopted the default settings for torque-free inner boundary condition, returning radiation, and limb darkening, and fixed the normalization at unity and the color correction factors at the values that we derived. The best-fit models are shown in Figure 3. Neither one is formally acceptable, with χ2/dof = 2634/1203 and 2010/1079 for the 2002 and 2003 observations, respectively. The residuals show significant structures in both cases. Taken at its face value, the black hole spin would be about 0.7, after correcting for the loss of flux due to the use of the central nine columns of the pn chip (see § 2.1). The situation is hardly improved when the inclination angle and the color correction factor are allowed to vary. Figure 4 shows the best-fit models with “bhspec”. Again, significant features are no- ticeable in the residuals. The χ2 values of the fits are χ2/dof = 2246/1203 and 2505/1079 for the 2002 and 2003 observations, respectively. As already mentioned, in this model spectral hardening (due to electron scattering) is taken into account in modeling the disk atmosphere. Again, taken at its face value, the black hole spin is about 0.5. Relaxing the inclination angle does not improve the fits. – 8 – 4. Discussion The importance of accurately modeling the accretion disk X-ray continuum of BHCs goes beyond gaining insights into radiative processes associated with accretion flows. It also lies in the exciting prospect of deriving the spin of black holes from such spectral modeling. The technique is one of many that have been proposed for BHCs (Laor 1991; Bromley et al. 1997; Zhang at al. 1997; Nowak et al. 1997; Cui et al. 1998; Stella et al. 1999; Wagoner et al. 2001; Abramowicz & Kluzniak 2001). Although varying degrees of success have been achieved, it is fair to say that the techniques all have serious issues in their applications to the data. Further investigation, both theoretical and observational, is thus needed to examine the issues. We have demonstrated in this work that the high quality of the data is starting to demand a proper treatment of electron scattering in radiative transfer through the accre- tion disk around a stellar-mass black hole. Some of the effects that were not appreciated previously in fitting low S/N data are now becoming apparent. At present, this demanding situation fundamentally limits our ability to reliably derive the physical parameters of the accretion disk or the black hole in an X-ray binary, based on modeling the disk X-ray contin- uum. There are also observational issues that add additional uncertainties to the exercise. For instance, many key parameters (e.g., black hole mass, inclination angle, and distance) that characterize a source are often poorly determined but are needed to determine, e.g., the black hole spin. This is entirely independent of the quality of X-ray data. Also, perhaps less appreciated are the significant uncertainties in the absolute and cross calibrations of the detectors on different X-ray satellites. This issue is relevant, because the determination of the spin of a black hole in an X-ray binary depends critically on the overall normalization of the X-ray continuum. This is the reason why one must be very careful in comparing results based on data from different satellites. We have shown that neither of the two state-of-the-art disk models is capable of satis- factorily fitting the observed ultra-soft component of the spectra of GX 339−4. While this is perhaps not totally surprising for “kerrbb”, since it does not actually carry out radia- tive transfer calculations, it is for “bhspec”. These models have recently been applied to data to derive the spin of black holes in a number of systems, so our finding is somewhat disappointing. If we take the best-fit parameters at their face values, the models would suggest that GX 339−4 contains a moderately rotating black hole (with a∗ ∼ 0.5–0.6). On the other hand, if we attribute the asymmetry in the profile of the observed Fe Kα line to gravitational redshift, we would conclude that the source contains a rapidly rotating black hole (with a∗ ≈ 0.96). We should note, however, that the apparent inconsistency can be easily reconciled when we take into account the large uncertainties associated with, e.g., – 9 – black hole mass, inclination angle, and distance. For example, if we adopt 13.5 M⊙ for the black hole mass, 51◦ for the inclination, and 7.5 kpc for the distance, the “kerrbb’ model yields a∗ ≈ 0.93 and 0.96 when fitting the 2002 and 2003 data, respectively. We were able to fit the ultrasoft component quite satisfactorily with a simple saturated Compton scattering model. The results allowed us to test a procedure that was previously suggested by Cui et al. (2002) to empirically derive the color correction factor from the same X-ray data. The values obtained are very close to the theoretical expectation (e.g., Shimura & Takahara 1995), which is also often adopted in spectral modeling. Therefore, our results have provided further support for this observational approach. Although the use of a single color correction factor ignores possible radial dependence of spectral hardening in the disk, it does not seem unreasonable given that the X-ray emission from the disk originates from a relatively narrow region (closest to the black hole). 5. Conclusions Based on our joint spectral analysis of two simultaneous XMM-Newton/RXTE obser- vations of GX 339-4, we can draw following conclusions: • The empirical procedure to derive the color correction factor (fcol) observationally, as proposed by Cui et al. (2002), yields reasonable results. If confirmed by further investigations, this would eliminate a major (theoretical) uncertainty in deriving the parameters of the disk from modeling the X-ray continuum. • The observed X-ray continuum of GX 339-4 in the high-soft state, which is dominated by emission from the optically-thick accretion disk, cannot be satisfactorily fitted by any existing disk model. Therefore, one should excise caution in assessing quantitative results from such spectral modeling. We wish to thank Shuangnan Zhang for suggesting the derivation of the spectral hard- ening factor from modeling the disk X-ray continuum and for subsequently collaborating on the subject. This work is a follow-up to much of the initial discussions. We also thank Lev Titarchuk for candid discussions on the theoretical aspects of the subject. This research has made use of data obtained through the High Energy Astrophysics Science Archive Re- search Center Online Service, provided by the NASA/Goddard Space Flight Center. It was supported in part by NASA through the LTSA grant NAG5-9998. We also gratefully ac- knowledge financial support from the Purdue Research Foundation and from a Grodzins Summer Research Award from the Department of Physics at Purdue University (to G.P.). Table 1. Best X-ray Continuum Fitsa comptt comptt Obs NH kT0 kTe τ K kT0 kTe τ K χ 2/dof 1021 cm−2 keV keV keV keV 2002 4.5 (+1− 2) 0.20 (1) 0.793 (+3− 4) 13.4 (2) 25 (+2− 1) 1.7 (+2 − 1) 46 1.8 (+1− 2) 1.7 × 10−3 978/1201 2003 4.75 (1) 0.170 (1) 0.618 (1) 10.07 (2) 7.58 (2) 1.11 (1) 183 (2) 0.38 (2) 3.21 (3) × 10−3 920/1076 aThe numbers in parentheses indicate uncertainty in the last digit. For asymmetric errors, both the lower and upper bounds are shown, again for the last digit. The errors shown represent 90% confidence intervals for single parameter estimation. – 11 – REFERENCES Abramowicz, M. A., & Kluzniak, W. 2001, A&A, 374, L19 Arnaud, K. A. 1996, in ASP Conf. Ser. 101, Astronomical Data Analysis Software and Systems V, ed. G. Jacoby & J. Barnes (San Francisco: ASP), 17 Bromley, B. C., Chen, K., & Miller, W. A. 1997, ApJ, 475, 57 Cui, W., Feng, Y. X., Zhang, S. N., Bautz, M. W., Garmire, G. P., & Schulz, N. S. 2002, ApJ, 576, 357 Cui, W., Zhang, S. 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Shown are the residuals after the “laor” component is removed from the best-fit model (see text). – 14 – 1 10 100 Energy (keV) Energy (keV) Fig. 2.— Observed X-ray spectra of GX 339−4 from the 2002 (left) and 2003 (right) obser- vations. The best-fit models are shown in solid histograms. The bottom panels show the respective residuals of the fits. 1 10 100 Energy (keV) Energy (keV) Fig. 3.— Same as Fig. 2 but the disk emission was modeled with “kerrbb”. – 15 – Fig. 4.— Same as Fig. 2 but the disk emission was modeled with “bhspec”. Introduction Data XMM-Newton Observations RXTE Observations Results Discussion Conclusions

0704.0468

Inapproximability of Maximum Weighted Edge Biclique and Its Applications

Inapproximability of Maximum Weighted Edge Biclique and Its Applications Jinsong Tan Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania, Philadelphia, PA 19104, USA [email protected] Abstract. Given a bipartite graph G = (V1, V2, E) where edges take on both positive and negative weights from set S , the maximum weighted edge biclique problem, or S-MWEB for short, asks to find a bipartite sub- graph whose sum of edge weights is maximized. This problem has various applications in bioinformatics, machine learning and databases and its (in)approximability remains open. In this paper, we show that for a wide range of choices of S , specifically when ˛ ∈ Ω(ηδ−1/2)∩O(η1/2−δ) (where η = max{|V1|, |V2|}, and δ ∈ (0, 1/2]), no polynomial time algo- rithm can approximate S-MWEB within a factor of nǫ for some ǫ > 0 unless RP = NP. This hardness result gives justification of the heuristic approaches adopted for various applied problems in the aforementioned areas, and indicates that good approximation algorithms are unlikely to exist. Specifically, we give two applications by showing that: 1) finding statistically significant biclusters in the SAMBA model, proposed in [18] for the analysis of microarray data, is nǫ-inapproximable; and 2) no poly- nomial time algorithm exists for the Minimum Description Length with Holes problem [4] unless RP = NP. 1 Introduction Let G = (V1, V2, E) be an undirected bipartite graph. A biclique subgraph in G is a complete bipartite subgraph of G and maximum edge biclique (MEB) is the problem of finding a biclique subgraph with the most number of edges. MEB is a well-known problem and received much attention in recent years because of its wide range of applications in areas including machine learning [14], manage- ment science [16] and bioinformatics, where it is found particularly relevant in the formulation of numerous biclustering problems for biological data analysis [5,2,18,19,17], and we refer readers to the survey by Madeira and Oliveira [13] for a fairly extensive discussion on this. Maximum edge biclique is shown to be NP-hard by Peeters [15] via a reduction from 3SAT. Its approximability status, on the other hand, remains an open question despite considerable efforts [7,8,12] 1. In particular, Feige and Kogan [8] conjectured that maximum edge biclique 1 Note it might be easy to confuse the MEB problem with the Bipartite Clique problem discussed by Khot in [12]. Bipartite Clique, which also known as Balanced Complete http://arxiv.org/abs/0704.0468v2 2 Jinsong Tan is hard to approximate within a factor of nǫ for some ǫ > 0. In this paper, we consider a weighted formulation of this problem defined as follows Definition 1. S-Maximum Weighted Edge Biclique (S-MWEB) Instance: A complete bipartite graph G = (V1, V2, E) (throughout the paper, let η = max{|V1|, |V2|} and n = |V1|+ |V2|), a weight function wG : E → S, where S is a set consisting of both positive and negative integers. Question: Find a biclique subgraph of G where the sum of weights on edges is maximized. A few comments are in order. First note it is not a lose of generality but a technical convenience to require the graph be complete, one can always think of an incomplete bipartite graph as complete where non-edges are assigned weight 0. Also note we require that both positive and negative weights be in S at the same time because otherwise S-MWEB becomes a trivial problem. Our study of S-MWEB is motivated by the problem of finding statistically significant biclusters in microarray data analysis in the SAMBA model [18] and the Minimum Description Length with Holes (MDLH) problem [3,4,10]; detailed discussion of the two problems can be found in Sect. 4. Our main technical contribution of this paper is to show that if S satisfies the condition |minS | ∈ Ω(ηδ−1/2) ∩O(η1/2−δ), where δ > 0 is any arbitrarily small constant, then no polynomial time algorithm can approximate S-MWEB within a factor of nǫ for some ǫ > 0 unless RP = NP. This result enables us to answer open questions regarding the hardness of the SAMBA model and the MDLH prob- lem. Since maximum edge biclique can be characterized as a special case of S-MWEB with S = {−η, 1}, the nǫ-inapproximability result also provides inter- esting insights into the conjectured nǫ-inapproximability [8] of maximum edge biclique. The rest of the paper is organized in three sections. In Sect. 2, we present the main technical result by proving the aforementioned inapproximability of S- MWEB. We give applications of this by answering hardness questions regarding two applied problems in Sect. 3. We conclude this work by raising a few open problems in the last section. 2 Approximating S-Maximum Edge Biclique is Hard We start this section by giving two lemmas about CLIQUE, which will be used in establishing inapproximability for the biclique problems we consider later. Lemma 1 is a recent result by Zuckerman [20], obtained by a derandomization of results of H̊astad [11]; Lemma 2 follows immediately from Lemma 1. Lemma 1. ([20]) It is NP-hard to approximate CLIQUE within a factor of n1−ǫ, for any ǫ > 0. Bipartite Subgraph [8], aims to maximize the number of vertices of a balanced sub- graph whereas MEB aims to maximize the total weights on edges in a (not necessarily balanced) subgraph. Inapproximability of Maximum Weighted Edge Biclique and Its Applications 3 Lemma 2. For any constant ǫ > 0, no polynomial time algorithm can approx- imate CLIQUE within a factor of n1−ǫ with probability at least 1 poly(n) unless RP = NP. 2.1 A Technical Lemma We first describe the construction of a structure called {γ, {α, β}}-Product, which will be used in the proof of our main technical lemma. Definition 2. ({γ, {α, β}}-Product) Input: An instance of S-MWEB on complete bipartite graph G = V1×V2, where γ ∈ S and α < γ < β; an integer N . Output: Complete bipartite graph GN = V N1 × V N2 constructed as follows: V N1 and V N2 are N duplicates of V1 and V2, respectively. For each edge (i, j) ∈ GN , let (φ(i), φ(j)) be the corresponding edge in G. If wG(φ(i), φ(j)) = γ, assign weight α or β to (i, j) independently and identically at random with expectation being γ, denote the weight by random variable X. If wG(φ(i), φ(j)) 6= γ, then keep the weight unchanged. Call the weight function constructed this way w(·). For any subgraph H of GN , denote by wγ(H) (resp., w−γ(H)) the total weight of H contributed by former-γ-edges (resp., other edges). Clearly, w(H) = wγ(H) + w−γ(H). With a graph product constructed in this randomized fashion, we have the fol- lowing lemma. Lemma 3. Given an S-MWEB instance G = (V1, V2, E) where γ ∈ S, and a number δ ∈ (0, 1 ]; let η = max (|V1|, |V2|), N = η δ(3−2δ)+3 δ(1+2δ) , GN = (V N1 , V 2 , E) be the {γ, {α, β}}-product of G and S ′ = (S ∪ {α, β})− {γ}. If 1. |β − α| = O((Nη) 12−δ); and 2. there is a polynomial time algorithm that approximates the S ′-MWEB instance within a factor of λ, where λ is some arbitrary function in the size of the S ′-MWEB instance then there exists a polynomial time algorithm that approximates the S-MWEB instance within a factor of λ, with probability at least 1 poly(n) Proof. For notational convenience, we denote η −δ by f(η) throughout the proof. Define random variable Y = X − γ, clearly E[Y ] = 0. Suppose there is a poly- nomial time algorithm A that approximates S ′-MWEB within a factor of λ, we can then run A on GN , the output biclique G∗B corresponds to N 2 bicliques in G (not necessarily all distinct). Let G∗A be the most weighted among these N subgraphs of G, in the rest of the proof we show that with high probability, G∗A is a λ-approximation of S-MWEB on G. Denote by E1 the event that G B does not imply a λ-approximation on G. Let H be the set of subgraphs of GN that do not imply a λ-approximation on G, 4 Jinsong Tan clearly, |H| ≤ 4Nη. Let H ′ be an arbitrary element in H, we have the following inequalities Pr {E1} ≤ Pr at least one element in H is a λ-approximation of GN ≤ 4Nη · Pr H ′ is a λ-approximation of GN = 4Nη · Pr{E2} where E2 is the event that H ′ is a λ-approximation of GN . Let the weight of an optimal solution U1×U2 of G be K, denote by UN1 ×UN2 the correspondingN2-duplication in GN . Let x1 and x2 be the number of former- γ-edges in H ′ and UN1 × UN2 , respectively. Suppose E2 happens, then we must w−γ(H ′) + x1γ ≤ N2(Kλ − 1) w−γ(H ′) + wγ(H ′) ≥ 1 (w−γ(U 1 × UN2 ) + wγ(UN1 × UN2 )) where the first inequality follows from the fact that we only consider integer weights. Since w−γ(U 1 × UN2 ) = N2K − x2γ, it implies (wγ(H ′)− x1γ)− (wγ(U 1 × UN2 )− x2γ) ≥ N2 so we have the following statement on probability Pr{E2} ≤ Pr (wγ(H ′)− x1γ)− 1λ(wγ(U 1 × UN2 )− x2γ) ≥ N2 Let z1 (resp., z2 and z3) be the number of edges in E(H ′) − E(UN1 × UN2 ) ( resp., E(UN1 × UN2 ) − E(H ′) and E(UN1 × UN2 ) ∩ E(H ′) ) transformed from former-γ-edges in G. We have (wγ(H ′)− x1γ)− 1λ(wγ(U 1 × UN2 )− x2γ) ≥ N2 i=1 Yi − j=1 Yj + k=1 Yk ≥ N2 i=1 Yi + j=1 (−Yj) + k=1 Yk ≥ N2 i=1 Yi ≥ j=1 (−Yj) ≥ k=1 Yk ≥ i=1 Yi ≥ j=1 (−Yj) ≥ k=1 Yk ≥ i∈{1,2,3} 3zi(c1f(Nη)) (Hoeffding bound) ≤ 3 · exp −c2 · N η3−2δ (zi ≤ η2N2) where c1, c2 are constants (c2 > 0). Now if we set N = η +θ for some θ, we Pr {E1} ≤ 4Nη · Pr {E2} ≤ 3 · exp ln 4 · η (1+2δ) +θ − c2 · η(1+2δ)θ For this probability to be bounded by 1 as η is large enough, we need to have +θ < (1+2δ)θ. Solving this inequality gives θ > 2 δ(1+2δ) . Therefore, for any δ ∈ (0, 1 ], by setting N = η δ(3−2δ)+3 δ(1+2δ) , we have Pr{E1}, i.e. the probability that Inapproximability of Maximum Weighted Edge Biclique and Its Applications 5 the solution returned by A does not imply a λ-approximation of G, is bounded from above by 1 once input size is large enough. This gives a polynomial time algorithm that approximates S-MWEB within a factor of λ with probability at least 1 This lemma immediately leads to the following corollary. Corollary 1. Following the construction in Lemma 3, if S ′-MWEB can be ap- proximated within a factor of nǫ , for some ǫ′ > 0, then there exists a polyno- mial time algorithm that approximates S-MWEB within a factor of nǫ, where ǫ = (1 + δ(3−2δ)+3 δ(1+2δ) )ǫ′, with probability at least 1 poly(n) Proof. Let |G| and |GN | be the number of nodes in the S-MWEB and S ′-MWEB problem, respectively. Since λ = |GN |ǫ′ ≤ |G|(1+ δ(3−2δ)+3 δ(1+2δ) , our claim follows from Lemma 3. ⊓⊔ 2.2 {−1, 0, 1}-MWEB In this section, we prove inapproximability of {−1, 0, 1}-MWEB by giving a reduction from CLIQUE; in subsequence sections, we prove inapproximability results for more general S-MWEB by constructing randomized reduction from {−1, 0, 1}-MWEB. Lemma 4. The decision version of the {−1, 0, 1}-MWEB problem is NP-complete. Proof. We prove this by describing a reduction from CLIQUE. Given a CLIQUE instance G = (V,E), construct G′ = (V ′, E′) such that V ′ = V1∪V2 where V1, V2 are duplicates of V in that there exist bijections φ1 : V1 → V and φ2 : V2 → V . E′ = E1 ∪ E2 ∪E3 E1 = {(u, v) | u ∈ V1, v ∈ V2 and (φ1(u), φ2(v)) ∈ E} E2 = {(u, v) | u ∈ V1, v ∈ V2, φ1(u) 6= φ2(v) and (φ1(u), φ2(v)) /∈ E} E3 = {(u, v) | u ∈ V1, v ∈ V2, and φ1(u) = φ2(v)} Clearly, G′ is a biclique. Now assign weight 0 to edges in E1, −1 to edges in E2 and 1 to edges in E3. We then claim that there is a clique of size k in G if and only if there is a biclique of total edge weight k in G′. First consider the case where there is a clique of size k in G, let U be the set of vertices of the clique, then taking the subgraph induced by φ−11 (U)× φ 2 (U) in G′ gives us a biclique of total weight k. Now suppose that there is a biclique U1×U2 of total weight k in G′. Without loss of generality, assume U1 and U2 correspond to the same subset of vertices in 2 Note we are slightly abusing notation here by always representing the size of a given problem under discussion by n. Here n refers to the size of S ′-MWEB (resp. S- MWEB) when we are talking about approximation factor nǫ (resp. nǫ). We adopt the same convention in the sequel. 6 Jinsong Tan V because if (φ1(U1)−φ2(U2))∪ (φ2(U2)−φ1(U1)) is not empty, then removing (U1 −U2)∪ (U2 −U1) will never decrease the total weight of the solution. Given φ1(U1) = φ2(U2), we argue that there is no edge of weight −1 in biclique U1×U2; suppose otherwise there exists a weight −1 edge (i1, j2) (i1 ∈ U1, and j2 ∈ U2), then the corresponding edge (j1, i2) (j1 ∈ U1, and i2 ∈ U2) must be of weight −1 too and removing i1, i2 from the solution biclique will increase total weight by at least 1 because among all edges incident to i1 and i2, (i1, i2) is of weight 1, (i1, j2) and (i2, j1) are of weight −1 and the rest are of weights either 0 or −1. Therefore, we have shown that if there is a solution U1 × U2 of weight k in G′, U1 and U2 correspond to the same set of vertices U ∈ V and U is a clique of size k. It is clear that the reduction can be performed in polynomial time and the problem is NP, and thus NP-complete. ⊓⊔ Given Lemma 1, the following corollary follows immediately from the above reduction. Theorem 1. For any constant ǫ > 0, no polynomial time algorithm can approx- imate problem {−1, 0, 1}-MWEB within a factor of n1−ǫ unless P = NP. Proof. It is obvious that the reduction given in the proof of Lemma 4 preserves inapproximability exactly, and given that CLIQUE is hard to approximate within a factor of n1−ǫ unless P = NP, the theorem follows. ⊓⊔ Theorem 2. For any constant ǫ > 0, no polynomial time algorithm can approx- imate {−1, 0, 1}-MWEB within a factor of n1−ǫ with probability at least 1 poly(n) unless RP = NP. Proof. If there exists such a randomized algorithm for {−1, 0, 1}-MWEB, com- bining it with the reduction given in Lemma 4, we obtain an RP algorithm for CLIQUE. This is impossible unless RP = NP. ⊓⊔ 2.3 {−1, 1}-MWEB Lemma 5. If there exists a polynomial time algorithm that approximates {−1, 1}- MWEB within a factor of nǫ, then there exists a polynomial time algorithm that approximates {−1, 0, 1}-MWEB within a factor of n5ǫ with probability at least poly(n) Proof. We prove this by constructing a {γ, {α, β}}-Product from {−1, 0, 1}- MWEB to {−1, 1}-MWEB by setting γ = 0, α = −1 and β = 1. Since δ = 1 according to Corollary 1, it is sufficient to set N = η4 so that the probability of obtaining a n5ǫ-approximation for {−1, 0, 1}-MWEB is at least 1 poly(n) Theorem 3. For any constant ǫ > 0, no polynomial time algorithm can approx- imate {−1, 1}-MWEB within a factor of n 15−ǫ with probability at least 1 poly(n) unless RP = NP. Proof. This follows directly from Theorem 2 and Lemma 5. ⊓⊔ Inapproximability of Maximum Weighted Edge Biclique and Its Applications 7 2.4 {−η , 1}-MWEB and {−ηδ− 2 , 1}-MWEB In this section, we consider the generalized cases of the S-MWEB problem. Theorem 4. For any δ ∈ (0, 1 ], there exists some constant ǫ such that no poly- nomial time algorithm can approximate {−η 12−δ, 1}-MWEB within a factor of nǫ with probability at least 1 poly(n) unless RP = NP. The same statement holds for {−ηδ− 12 , 1}-MWEB. Proof. We prove this by first construct a {γ, {α, β}}-Product from {−1, 1}- MWEB to {−η 12−δ, 1}-MWEB by setting γ = −1, α = −(Nη) 12−δ and β = 1. By Corollary 1, we know that for any δ ∈ (0, 1 ], if there exists a polynomial time al- gorithm that approximates {−η 12−δ, 1}-MWEB within a factor of nǫ, then there exists a polynomial time algorithm that approximates {−1, 1}-MWEB within a factor of n δ(3−2δ)+3 δ(1+2δ) with probability at least 1 poly(n) . So invoking the hardness result in Theorem 3 gives the desired hardness result for {−η 12−δ, 1}-MWEB. The same conclusion applies to {−1, η 12−δ}-MWEB by setting γ = 1, α = −1 and β = (Nη) −δ. Since η is a constant for any given graph, we can simply divide each weight in {−1, η 12−δ} by η 12−δ. ⊓⊔ Theorem 4 leads to the following general statement. Theorem 5. For any small constant δ ∈ (0, 1 ], if ∣ ∈ Ω(ηδ−1/2)∩O(η1/2−δ), then there exists some constant ǫ such that no polynomial time algorithm can ap- proximate S-MWEB within a factor of nǫ with probability at least 1 poly(n) unless RP = NP. 3 Two Applications In this section, we describe two applications of the results establish in Sect. 3 by proving hardness and inapproximability of problems found in practice. 3.1 SAMBA Model is Hard Microarray technology has been the latest technological breakthrough in biolog- ical and biomedical research; in many applications, a key step in analyzing gene expression data obtained through microarray is the identification of a bicluster satisfying certain properties and with largest area (see the survey [13] for a fairly extensive discussion on this). In particular, Tanay et. al. [18] considered the Statistical-Algorithmic Method for Bicluster Analysis (SAMBA) model. In their formulation, a complete bipar- tite graph is given where one side corresponds to genes and the other size cor- responds to conditions. An edges (u, v) is assigned a real weight which could be either positive or negative, depending on the expression level of gene u in condi- tion v, in a way such that heavy subgraphs corresponds to statistically significant 8 Jinsong Tan biclusters. Two weight-assigning schemes are considered in their paper. In the first, or simple statistical model, a tight upper-bound on the probability of an observed biclusters in computed; in the second, or refined statistical model, the weights are assigned in a way such that a maximum weight biclique subgraph corresponds to a maximum likelihood bicluster. The Simple SAMBA Statistical Model: LetH = (V ′1 , V 2 , E ′) be a subgraph of G = (V1, V2, E), E′ = {V ′1 × V ′2} − E′ and p = |V1||V2| . The simple statistical model assumes that edges occur independently and identically at random with probability p. Denote by BT (k, p, n) the probability of observing k or more successes in n binomial trials, the probability of observing a graph at least as dense as H is thus p(H) = BT (|E′|, p, |V ′1 ||V ′2 |). This model assumes p < 12 and |V ′1 ||V ′2 | ≪ |V1||V2|, therefore p(H) is upper bounded by p∗(H) = 2|V 1 ||V 2 |p|E ′|(1− p)|V 1 ||V 2 |−|E The goal of this model is thus to find a subgraph H with the smallest p∗(H). This is equivalent to maximizing − log p∗(H) = |E′|(−1− log p) + (|V ′1 ||V ′2 | − |E′|)(−1− log (1− p)) which is essentially solving a S-MWEB problem that assigns either positive weight (−1 − log p) or negative weight (−1 − log (1 − p)) to an edge (u, v), de- pending on whether gene u express or not in condition v, respectively. The summation of edge weights over H is defined as the statistical significance of H . Since 1 ≤ p < 1 , asymptotically we have −1−log (1−p) −1−log p ∈ Ω( 1 log η ) ∩ O(1). Invoking Theorem 5 gives the following. Theorem 6. For the Simple SAMBA Statistical model, there exists some ǫ > 0 such that no polynomial time algorithm, possibly randomized, can find a bicluster whose statistical significance is within a factor of nǫ of optimal unless RP = NP. The Refined SAMBA Statistical Model: In the refined model, each edge (u, v) is assumed to take an independent Bernoulli trial with parameter pu,v, therefore p(H) = ( (u,v)∈E′ pu,v)( (u,v)∈E′(1 − pu,v)) is the probability of ob- serving a subgraph H . Since p(H) generally decreases as the size of H increases, Tanay et al. aims to find a bicluster with the largest (normalized) likelihood ra- tio L(H) = (u,v)∈E′ pc)( (u,v)∈E′(1− pc)) , where pc > max(u,v)∈E pu,v is a constant probability and chosen with biologically sound assumptions. Note this is equivalent to maximizing the log-likelihood ratio logL(H) = (u,v)∈E′ (u,v)∈E′ 1− pc 1− pu,v With this formulation, each edge is assigned weight either log pc > 0 or log 1−pc 1−pu,v < 0 and finding the most statistically significant bicluster is equiva- lent to solving S-MWEB with S = {log 1−pc 1−pu,v , log pc }. Since pc is a constant Inapproximability of Maximum Weighted Edge Biclique and Its Applications 9 and 1 ≤ pu,v < pc, we have log (1−pc)−log (1−pu,v)log pc−log pu,v ∈ Ω( log η ) ∩ O(1). Invoking Theorem 5 gives the following. Theorem 7. For the Refined SAMBA Statistical model, there exists some ǫ > 0 such that no polynomial time algorithm, possibly randomized, can find a bicluster whose log-likelihood is within a factor of nǫ of optimal unless RP = NP. 3.2 Minimum Description Length with Holes (MDLH) is Hard Bu et. al [4] considered the Minimum Description Length with Holes problem (defined in the following); the 2-dimensional case is claimed NP-hard in this paper and the proof is referred to [3]. However, the proof given in [3] suffers from an error in its reduction3, thus whether MDLH is NP-complete remains unsettled. In this section, by employing the results established in the previous sections, we show that no polynomial time algorithm exists for MDLH, under the slightly weaker (than P 6= NP) but widely believed assumption RP 6= NP. We first briefly describe the Minimum Description Length summarization with Holes problem; for a detailed discussion of the subject, we refer the readers to [3,4]. Suppose one is given a k-dimensional binary matrix M , where each entry is of value either 1, which is of interest, or of value 0, which is not of interest. Be- sides, there are also k hierarchies (trees) associated with each dimension, namely T1, T2, ..., Tk, each of height l1, l2, ..., lk respectively. Define level l = maxi(li). For each Ti, there is a bijection between its leafs and the ’hyperplanes’ in the ith dimension (e.g. in a 2-dimensional matrix, these hyperplanes corresponds to rows and columns). A region is a tuple (x1, x2, ..., xk), where xi is a leaf node or an internal node in hierarchy Ti. Region (x1, x2, ..., xk) is said to cover cell (c1, c2, ..., ck) if ci is a descendant of xi, for all 1 ≤ i ≤ k. A k-dimensional l-level MDLH summary is defined as two sets S and H , where 1) S is a set of regions covering all the 1-entries in M ; and 2) H is the set of 0-entries covered (unde- sirably) by S and to be excluded from the summary. The length of a summary is defined as |S|+ |H |, and the MDLH problem asks the question if there exists a MDLH summary of length at most K, for a given K > 0. In an effort to establish hardness of MDLH, we first define the following problem, which serves as an intermediate problem bridging {−1, 1}-MWEB and MDLH. Definition 3. (Problem P) Instance: A complete bipartite graph G = (V1, V2, E) where each edge takes on a value in {−1, 1}, and a positive integer k. Question: Does there exist an induced subgraph (a biclique U1 × U2) whose total weight of edges is ω, such that |U1|+ |U2|+ ω ≥ k. Lemma 6. No polynomial time algorithm exists for Problem P unless RP = NP. 3 In Lemma 3.2.1 of [3], the reduction from CLIQUE to CEW is incorrect. 10 Jinsong Tan Proof. We prove this by constructing a reduction from {−1, 1}-MWEB to Prob- lem P as follows: for the given input biclique G = (V1, V2, E), make N duplicates of V1 and N duplicates of V2, where N = (|V1| + |V2|)2. Connect each copy of V1 to each copy of V2 in a way that is identical to the input biclique, we then claim that there is a size k solution to {−1, 1}-MWEB if and only if there is a size N2k solution to Problem P . If there is a size k solution to {−1, 1}-MWEB, then it is straightforward that there is a solution to Problem P of size at leastN2k. For the reverse direction, we show that if no solution to {−1, 1}-MWEB is of size at least k, then the maximum solution to Problem P is strictly less than N2k. Note a solution UN1 × UN2 to Problem P consists of at most N2 (not necessarily all distinct) solutions to {−1, 1}-MWEB, and each of them can contribute at most (k − 1) in weight to UN1 ×UN2 , so the total weight gained from edges is at most N2(k− 1). And note the total weight gained from vertices is at most N(|V1|+ |V2|) = N N , therefore the weight is upper bounded by N N +N2(k − 1) < N2k and this completes the proof. As a conclusion, we have a polynomial time reduction from {−1, 1}-MWEB to Problem P . Since no polynomial time algorithm exists for {−1, 1}-MWEB unless RP = NP, the same holds for Problem P . ⊓⊔ Theorem 8. No polynomial time algorithm exists for MDLH summarization, even in the 2-dimension 2-level case, unless RP = NP. Proof. We prove this by showing that Problem P is a complementary problem of 2-dimensional 2-level MDLH. Let the input 2DmatrixM be of size n1×n2, with a tree of height 2 associated with each dimension. Without loss of generality, we only consider the ’sparse’ case where the number of 1-entries is less than the number of 0-entries by at least 2 so that the optimal solution will never contain the whole matrix as one of its regions. Let S be the set of regions in a solution. Let R and C be the set of rows and columns not included in S. Let Z be the set of all zero entries in M . Let z be the total number of zero entries in the R × C ’leftover’ matrix and let w be the total number of 1-entries in it. MDLH tries to minimize the following: (n1 − |R|) + (n2 − |C|) + (|Z| − z) + w = (n1 + n2 + |Z|)− (|R|+ |C|+ z − w) Since (n1 + n2 + |Z|) is a fixed quantity for any given input matrix, the 2- dimensional 2-level MDLH problem is equivalent to maximizing (|R|+|C|+z−w), which is precisely the definition of Problem P . Therefore, 2-dimensional 2-level MDLH is a complementary problem to Prob- lem P and by Lemma 6 we conclude that no polynomial time algorithm exists for 2-dimensional 2-level MDLH unless RP = NP. ⊓⊔ 4 Concluding Remarks Maximum weighted edge biclique and its variants have received much atten- tion in recently years because of it wide range of applications in various fields Inapproximability of Maximum Weighted Edge Biclique and Its Applications 11 including machine learning, database, and particularly bioinformatics and com- putational biology, where many computational problems for the analysis of mi- croarray data are closely related. To tackle these applied problems, various kinds of heuristics are proposed and experimented and it is not known whether these algorithms give provable approximations. In this work, we answer this question by showing that it is highly unlikely (under the assumption RP 6= NP) that good polynomial time approximation algorithm exists for maximum weighted edge biclique for a wide range of choices of weight; and we further give specific appli- cations of this result to two applied problems. We conclude our work by listing a few open questions. 1. We have shown that {Θ(−ηδ), 1}-MWEB is nǫ-inapproximable for δ ∈ ); also it is easy to see that (i) the problem is in P when δ ≤ −1, where the entire input graph is the optimal solution; (ii) for any δ ≥ 1, the problem is equivalent to MEB, which is conjectured to be nǫ-inapproximable [8]. Therefore it is natural to ask what is the approximability of the {−nδ, 1}-MWEB problem when δ ∈ (−1,− 1 ] and δ ∈ [ 1 , 1]. In particular, can this be answered by a better analysis of Lemma 3? 2. We are especially interested in {−1, 1}-MWEB, which is closely related to the formulations of many natural problems [1,3,4,18]. We have shown that no polynomial time algorithm exists for this problem unless RP = NP, and we believe this problem is NP-complete, however a proof has eluded us so far. References 1. N. Bansal, A. Blum, and S. Chawla. Correlation clustering, Machine Learning, 56:89-113, 2004. 2. A. Ben-Dor, B. Chor, R. Karp, and Z. Yakhini. Discovering local structure in gene expression data: The Order-Preserving Submatrix Problem. In Proceedings of RECOMB’02, 49-57, 2002. 3. S. Bu. The summarization of hierarchical data with exceptions. Master The- sis, Department of Computer Science, University of British Columbia, 2004. http://www.cs.ubc.ca/grads/resources/thesis/Nov04/Shaofeng Bu.pdf 4. S. Bu, L. V. S. Lakshmanan, R. T. Ng. MDL Summarization with Holes. In Pro- ceedings of VLDB’05, 433-444, 2005. 5. Y. Cheng, and G. Church. Biclustering of expression data. In Proceedings of ISMB’00, 93-103. AAAI Press, 2000. 6. M. Dawande, P. Keskinocak, J. M. Swaminathan, and S. Tayur. On Bipartite and multipartite clique problems. Journal of Algorithms, 41(2):388-403, 2001. 7. U. Feige. Relations between average case complexity and approximation complex- ity. In Proceedings of STOC’02, 534-543, 2002. 8. U. Feige and S. Kogan. Hardness of approximation of the Balanced Complete Bipartite Subgraph problem. Technical Report MCS04-04, The Weizmann Institute of Science, 2004. 9. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco, 1979. 10. P. Fontana, S. Guha and J. Tan. Recursive MDL Summarization and Approxima- tion Algorithms. Preprint, 2007. http://www.cs.ubc.ca/grads/resources/thesis/Nov04/Shaofeng_Bu.pdf 12 Jinsong Tan 11. J. H̊astad. Clique is hard to approximate within n1−ǫ. Acta Mathematica, 182:105- 142, 1999. 12. S. Khot. Ruling out PTAS for Graph Min-Bisection, Densest Subgraph and Bipar- tite Clique. In Proceedings of FOCS’04, 136-145, 2004. 13. S. C. Madeira, and A. L. Oliveira. Biclustering algorithms for biological data anal- ysis: a survey. IEEE/ACM Transactions on Computational Biology and Bioinfor- matics, 1:24-45, 2004. 14. N. Mishra, D. Ron, and R. Swaminathan. On finding large conjunctive clusters. In Proceedings of COLT’03, 448-462, 2003. 15. R. Peeters. The maximum edge biclique problem is NP-complete. Discrete Applied Mathematics, 131:651-654, 2003. 16. J. M. Swaminathan and S. Tayur. Managing Broader Product Lines Through De- layed Differentiation Using Vanilla Boxes. Management Science, 44:161-172, 1998. 17. J. Tan, K. Chua, L. Zhang, and S. Zhu. Algorithmic and Complexity Issues of Three Clustering Methods in Microarray Data Analysis Algorithmica, 48(2): 203- 219, 2007. 18. A. Tanay, R. Sharan, and R. Shamir. Discovering statistically significant biclusters in gene expression data. Bioinformatics, 18, Supplement 1:136-144, 2002. 19. L. Zhang, and S. Zhu. A New Clustering Method for Microarray Data Analysis. In Proceedings of CSB’02, 268-275, 2002. 20. D. Zuckerman. Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number. In Proceedings of STOC’06, 681-690, 2006. Inapproximability of Maximum Weighted Edge Biclique and Its Applications Jinsong Tan

0704.0469

On line arrangements with applications to 3-nets

ON LINE ARRANGEMENTS WITH APPLICATIONS TO 3-NETS GIANCARLO URZÚA Abstract. We show a one-to-one correspondence between arrangements of d lines in P2, and lines in Pd−2. We apply this correspondence to classify (3, q)-nets over C for all q ≤ 6. When q = 6, we have twelve possible combinatorial cases, but we prove that only nine of them are realizable over C. This new case shows several new properties for 3-nets: different dimensions for moduli, strict realization over certain fields, etc. We also construct a three dimensional family of (3, 8)-nets corresponding to the Quaternion group. 1. Introduction. We start by showing a one-to-one correspondence between arrangements of d lines in P2, and lines in Pd−2. This correspondence is a particular case of a more general one between arrangements of d sections on ruled surfaces, which generalize line arrangements (see Remark 2.1), and certain curves in Pd−2. This was developed by the author in [21]. In there, we consider arrangements as single curves via moduli spaces of pointed stable rational curves. One of the main ingredients is the description of these moduli spaces given by Kapranov in [14] and [15]. In the present paper, we only treat the case of line arrangements, giving a proof of this correspondence by means of quite elementary geometry. An important consequence of seeing an arrangement as a single curve is, as it turns out, to answer questions about realization of certain line arrangements. That is the second part of this paper. Through this correspondence, we are able to classify the so-called (3, q)-nets over C for all q ≤ 6, and the Quaternion nets. This classification shows various new properties for 3-nets, and opens the question about realization of Latin squares in P2 (they give the combinatorial data which defines a 3-net). The following is an outline of the paper. In Section 2, we prove a one-to-one correspondence between arrangements of d lines, and lines in Pd−2. An arrangement of d lines is a set A = {L1, L2, . . . , Ld} of d labeled lines in P2 such that i=1 Li = ∅. As it was pointed out above, our general correspondence is between arrangements of sections and curves. For this reason, instead of considering line arrangements, we consider pairs (A, P ) where A is an arrangement of d lines, and P ∈ P2 is a point outside of i=1 Li. In Proposition 2.1, we prove a one-to-one correspondence between these pairs, up to projective equivalence, and lines in Pd−2 outside of a fixed hyperplane arrangement Hd. We also sketch a second proof of it (see Remark 2.1) to hint the more general correspondence mentioned before (see [21] for details). Under this correspondence, for a fixed arrangement A, different choices of P may produce different lines in Pd−2. Thus, if one fixes the combinatorial type of A and wants to study http://arxiv.org/abs/0704.0469v4 its moduli space, the presence of this artificial point P introduces more parameters than needed. In practice, even the question of realization of A becomes hard with this extra point P . To eliminate this difficulty, we take P in A and consider the new pair (A′, P ), where the lines in A′ are the lines in A not containing P (with a certain labelling). By taking P as a point lying on several lines of A, one greatly simplifies computations to prove or disprove its realization over some field, and to find a moduli space for its combinatorial type. In Sections 3 and 4, we use our method to study a particular type of line arrangements which are called nets. There is a large body of literature about them (cf. [1], [3], [5], [6], [17], [22], [19], [20], [9]). Nowadays, they are of interest to topologists who study resonance varieties of complex line arrangements (see [17], [22], [4], [9]). In general, they can be thought as the geometric structures of finite quasigroups, which in turn are intimately related with Latin squares [6]. We define (p, q)-nets in Section 3. We exemplify our correspondence by computing the Hesse arrangement, which is the only (4, 3)-net over C, and by showing that (4, 4)-nets do not exist in characteristic different than 2 (they do in characteristic 2, see Example 3.3). In [22], Yuzvinsky proved that (p, q)-nets over C are only possible for p = 3, 4, 5 (not true in positive characteristic, where any p is possible, see Example 3.3). Examples of (5, q)-nets were unknown, and for (4, q)-nets the only example was the Hesse arrangement. In [19], Stipins proved that (5, q)-nets do not exist over C, leaving open the case p = 4. It is believed that the only (4, q)-net is the Hesse arrangement. In Section 4, we present a classification for (3, q)-nets over C with q ≤ 6. It is known that a q×q Latin square provides the combinatorial data which defines a (3, q)- net (see [6], [17], [16], [19]). Until very recently, the only known (3, q)-nets corresponded to Latin squares coming from multiplication tables of certain abelian groups. Yuzvinsky conjectured in [22] that this should always be the case. In [19], there was given a three dimensional family of (3, 5)-nets not coming from a group. For the case q = 6, we have twelve possible cases associated to the twelve main classes of 6×6 Latin squares. In Section 4, we show that only nine of them are realizable in P2 over C. These nine cases present new properties for 3-nets: we have four three dimensional and five two dimensional families, some of them define nets strictly over C, for others we have nets over R or even over Q, etc. After that, we construct a three dimensional family of (3, 8)-nets associated to the Quaternion group, which has members defined over Q. The new cases corresponding to the symmetric and Quaternion groups show that there are (3, q)-nets associated to non-abelian groups (see [22, Conj. 6.1]). Out of this, a natural question is: find a combinatorial characterization of the main classes of Latin squares (see Remark 3.1) which realize (3, q)-nets in P2 We denote the projective space of dimension n by Pn, and a point in it by [x1 : . . . : xn+1] = [xi] i=1 . If P1, . . . , Pr are r distinct points in P n, then 〈P1, . . . , Pr〉 is the projective linear space spanned by them. The points P1, . . . , Pn+2 in P n are said to be in general position if no n + 1 of them lie in a hyperplane. Acknowledgments: I am grateful to Dave Anderson, my thesis advisor Igor Dolgachev, Sean Keel, Finn Knudsen, Janis Stipins, and Jenia Tevelev for valuable discussions. I would also like to acknowledge the referee for helping me to improve the exposition of this paper. 2. Arrangements of d lines in P2, and lines in Pd−2. Definition 2.1. Let d ≥ 3 be an integer. An arrangement of d lines A is a set of d labeled lines {L1, . . . , Ld} in P2 such that i=1 Li = ∅. When the labelling is not relevant, we will consider A as the plane curve i=1 Li. We introduce ordered pairs (A, P ), where A is an arrangement of d lines in P2, and P is a point in P2 \A. If (A, P ) and (A′, P ′) are two such pairs, we say that they are isomorphic if there exists an automorphism T of P2 such that T (Li) = L i for every i, and T (P ) = P ′. Let Ld be the set of isomorphism classes of pairs (A, P ). For example, clearly L3 is a set with only one element, represented by the class of the pair {{x = 0}, {y = 0}, {z = 0}}, [1 : 1 : 1] On the other hand, let us fix d points in Pd−2 in general position. We precisely take P1 = [1 : 0 : . . . : 0], P2 = [0 : 1 : 0 : . . . : 0], . . . , Pd−1 = [0 : . . . : 0 : 1], Pd = [1 : . . . : 1]. Consider the projective linear spaces Λi1,...,ir = 〈Pj : j /∈ {i1, . . . , ir}〉, where 1 ≤ r ≤ d − 1 and i1, . . . , ir are distinct numbers, and let Hd be the union of all the hyperplanes Λi,j. Hence, Λi,j = {[x1 : . . . : xd−1] ∈ Pd−2 : xi = xj} for i, j 6= d, Λi,d = {[x1 : . . . : xd−1] ∈ Pd−2 : xi = 0}, and Hd = {[x1 : . . . : xd−1] ∈ Pd−2 : x1x2 · · ·xd−1 (xj − xi) = 0}. The proof of the following proposition is inspired by a particular case of the so-called Gelfand-MacPherson correspondence [15, Chap. 2]. Proposition 2.1. There is a one-to-one correspondence between Ld and the set of lines in Pd−2 not contained in Hd. Proof. Let us fix a pair (A, P ), where A is defined by the linear polynomials Li(x, y, z) = ai,1x+ ai,2y + ai,3z, 1 ≤ i ≤ d. Consider the embedding ι(A,P ) : P 2 →֒ Pd−1 given by [x : y : z] 7→ [L1(x, y, z) L1(P ) : . . . : Ld(x, y, z) Ld(P ) Then, ι(A,P )(P 2) is a projective plane, ι(A,P )(P ) = [1 : . . . : 1], and ι(A,P )(Li) = ι(A,P )(P {yi = 0} for every i ∈ {1, 2, . . . , d}. We now consider the projection ̺ : Pd−1 \ [1 : . . . : 1] → Pd−2, [y1 : y2 : . . . : yd] 7→ [y1 − yd : y2 − yd : . . . : yd−1 − yd]. In this way, if Σi,j = {[y1 : y2 : . . . : yd] : yi = yj}, we see that ̺(Σi,j) = Λi,j. Therefore, we have that ̺ ι(A,P )(P is a line in Pd−2 not contained in Hd. To show the one-to-one correspondence, we need to prove that (A, P ) 7→ ̺ ι(A,P )(P gives a well-defined bijection between Ld and the set of lines in P d−2 not contained in Hd. Clearly we have a bijection between projective planes in Pd−1 passing through [1 : . . . : 1] and not contained in Σi,j , and the set of lines in Pd−2 not contained in Hd. Let T : P2 → P2 be an automorphism of P2. Let B = be the 3× 3 invertible matrix corresponding to T−1. Consider the pair (A′, P ′) defined by A′ = {L′i = T (Li)}di=1 and P ′ = T (P ). Then, the equations defining the lines L′i are j=1 ai,jbj,1 j=1 ai,jbj,2 j=1 ai,jbj,3 z = 0. Hence, we obtain that ι(A,P ) = ι(A′,P ′) ◦ T , and so our map (A, P ) 7→ ̺ ι(A,P )(P is well- defined on Ld. It is clearly surjective, so we only need injectivity. Let ι(A,P ) and ι(A′,P ′) be the corre- sponding maps for the pairs (A, P ) and (A′, P ′) such that ι(A,P )(P2) = ι(A′,P ′)(P2). Let T = ι−1 (A′,P ′) ◦ ι(A,P ) : P2 → P2. Then, T is an automorphism of P2 such that T (Li) = L′i for every i and T (P ) = P ′. Hence they are isomorphic, and so we have the one-to-one correspondence. � Remark 2.1. The following is a sketch of how this one-to-one correspondence works for arrangements of sections on geometrically ruled surfaces (cf. [?, p. 369]) via the moduli spaces M0,d+1 (cf. [14]). We will do it only for line arrangements, and over C. For the general case see [21]. Let us fix a pair (A, P ) as before, and let BlP (P2) be the blow-up of P2 at the point P [?, p. 386]. Then, we have an induced genus zero fibration BlP (P 2) → P1. The pull-back of A in BlP (P 2) is an arrangement of d labeled sections, each of which belongs to the fix class E+F . Here E is the exceptional divisor of the blow-up, and F is any fiber. Conversely, given an arrangement of d sections A in BlP (P2) with members in the fix class E+F , we blow-down the exceptional divisor E to obtain a pair (A, P ) in P2. Isomorphic pairs (A, P ) correspond to isomorphic arrangements of sections (via automorphisms of the fibration BlP (P 2) → P1). Now the correspondence. We have fixed the pair (A, P ), and the fibration BlP (P2) → P1 as given above. Consider the genus zero fibration f : R → P1, where R is the blow-up at all the singular points of A in BlP (P2) except nodes. Then, f is a family of (d+ 1)-marked stable curves of genus zero. The markings are given by the labeled lines of A, which are now d labeled sections of f , and the (−1)-curve coming from the exceptional divisor E in BlP (P 2). Therefore, since M0,d+1 is a fine moduli space, we have the following commutative diagram coming from its universal family. // M 0,d+2 // M 0,d+1 Let B′ be the image of g in M 0,d+1. It is a projective curve, since f has singular and non-singular fibers, and so f is not isotrivial. Let us now consider the Kapranov map ψd+1 : M 0,d+1 → Pd−2 [15, p. 81], and let B = ψd+1(B′). Because of the geometry of the fibers of f and the Kapranov’s construction, one can prove (see [21]) that B intersects all the hyperplanes Λi,j transversally. Say B intersects Λi,j. This means that the lines Li and Lj of A intersect in P2. But since they are lines, they can only intersect at one point. Therefore, we must have B.Λi,j = 1, and so deg(B) = 1, that is, B is a line in P d−2. Observe that this line is outside of Hd. In particular, B′ is a smooth rational curve. It is not hard to see the converse, this is, how to obtain a pair (A, P ) from a line in Pd−2 outside of Hd (see [21]). Moreover, one can check that the pair we obtain is unique up to isomorphism of pairs. In this way, to prove the one-to-one correspondence, we have to show that the map g is an inclusion. Assume deg(g) > 1. Notice that g is totally ramified at the points corresponding to singular fibers of f , since again they come from intersections of lines in P2, and so all the singular fibers have distinct points as images in B′. Let sing(f) be the set of points in P1 corresponding to singular fibers of f . Then, since i=1 Li = ∅, we have | sing(f)| ≥ 3 (at least we have a triangle in A). Now, by the Riemann-Hurwitz formula, we have −2 = deg(g)(−2) + (deg(g)− 1)| sing(f)|+ ǫ where ǫ ≥ 0 stands for the contribution from ramification of f not in sing(f). But we re-write the equation as 0 = (deg(g) − 1)(| sing(f)| − 2) + ǫ, and since deg(g) > 1 and | sing(f)| ≥ 3, this is a contradiction. Therefore, deg(g) = 1 and we have proved the one- to-one correspondence. Again, we refer to [21, Ch. 2 and 3] for the general one-to-one correspondence involving arrangements of sections on geometrically ruled surfaces. In this way, for each pair (A, P ) ∈ Ld, we denote its corresponding line in Pd−2 by L(A, P ). We now want to describe more precisely how this one-to-one correspondence relates them. Definition 2.2. Let K be any field. The pair (A, P ) is said to be defined over K if the coefficients of the equations defining the lines in A, and the coordinates of P are in K. Hence, for arbitrary fields K, Proposition 2.1 gives a one-to-one correspondence between pairs (A, P ) defined over K, and lines L(A, P ) in Pd−2 defined over K. Definition 2.3. Let 1 < k < d be an integer. A point in P2 is said to be a k-point of A if it belongs to exactly k lines of A. If these lines are {Li1 , Li2 , . . . , Lik}, we denote this point by [[i1, i2, . . . , ik]]. The number of k-points of A is denoted by tk. Remark 2.2. The complexity of an arrangement relies on its k-points. There are more constraints for the existence of an arrangement, over some field, than the plane restriction: any two lines intersect at one point. Combinatorially there are possible line arrangements, with assigned k-points, which may not be realizable in P2 over C (we will return to this in the next sections, for the particular case of nets). For instance, we have the Fano arrangement (formed by seven lines with seven 3-points) which is not realizable in P2 over fields of characteristic 6= 2. A rather trivial restriction, which is purely combinatorial, is that the numbers tk must satisfy tk; this is the only linear relation they satisfy for a fix d. In [13], Hirzebruch proved the following inequality for an arrangement of d lines in the complex projective plane having td = td−1 = 0, t3 ≥ d+ (k − 4)tk. This is a non-trivial relation among the numbers tk, which comes from the Miyaoka-Yau inequality for complex algebraic surfaces (see [21] for more about this type of restrictions). This inequality is clearly not true in positive characteristic. Let us fix a pair (A, P ), and its line L(A, P ) in Pd−2. Let λ be a line in P2 passing through P . Notice that λ corresponds to a point in L(A, P ). Let K(λ) be the set of k-points of A in λ, for all 1 < k < d; it might be empty or consist of several points. We write K(λ) = {[[i1, i2, . . . , ik1 ]], [[j1, j2, . . . , jk2]], . . .}. Example 2.1. In Figure 1, we have the complete quadrilateral A, formed by the set of lines {L1, . . . , L6}, and a point P outside of A. Through P we have all the λ lines. In the figure, we have named two such lines: λ and λ′. Thus, K(λ) = {[[3, 6]], [[1, 4]]} and K(λ′) = {[[1, 2, 3]]}. [[1,4]][[3,6]] [[1,2,3]] Figure 1. Some K(λ) sets for the pair (Complete quadrilateral, P ). The set K(λ) imposes the following constraints for the the point [x1 : x2 : . . . : xd−1] in L(A, P ) corresponding to λ. For each k-point [[i1, i2, . . . , ik]] in K(λ), we have: • If for some j, ij = d, then xil = 0 for all il 6= d. • Otherwise, xi1 = xi2 = . . . = xik 6= 0. For [[i1, . . . , ik1]], [[j1, . . . , jk2]] in K(λ), we have that xia 6= xjb, otherwise [[i1, . . . , ik1 ]] and [[j1, . . . , jk2]] would not be distinct points in λ. We will work out various examples when we compute nets in the next sections. Assume we know the combinatorial data of (A, P ), but we do not know whether is realiz- able in P2 over some field K. Then, this realization question is equivalent to the realization question of L(A, P ) over K. If we are only interested in the line arrangement A, the point P introduces unnecessary dimensions which makes the realization question harder. Instead, we consider the new pair (A′, P ′), where P ′ ∈ A and the lines of A′ are the lines in A not containing P ′ in a certain order. Now, the line L(A′, P ′) corresponding to (A′, P ′) is in Pd ′−2, and d′ < d. So we have less dimensions to work with, and L(A′, P ′) completely represents our arrangement A, by keeping track of P ′. If we take P ′ as a k-point with k large, the previous observation will be important to simplify computations to prove or disprove the realization of A. In addition, we find a moduli space for the combinatorial type of A, forgetting the artificial point P . Again, by combinatorial type we mean the data given by some of the intersection of its lines. In the next two sections, we will compute some special configurations by means of the line L(A, P ). We make the following choices to write down equations for the lines in A: • The point P will be always [0 : 0 : 1]. • The arrangement A will be formed by {L1, . . . , Ld}, where Li are the lines of A, and also their linear polynomials Li(x, y, z) = (aix + biy + z) for every i 6= d, and Ld = (z). With these assumptions, it is easy to check that the corresponding line L(A, P ) in Pd−2 is [ait + biu] i=1 , where [t : u] ∈ P1. 3. (p, q)-nets in P2. We now introduce a specific type of line arrangements in P2 which are called nets. Our main references are [6], [17], [22], [16], [19], and [20]. We begin with the definition of a net taken from [19]. Definition 3.1. Let p ≥ 3 be an integer. A p-net in P2 is a (p + 1)-tuple (A1, ...,Ap,X ), where each Ai is a nonempty finite set of lines of P2 and X is a finite set of points of P2, satisfying the following conditions: (1) The Ai are pairwise disjoint. (2) The intersection point of any line in Ai with any line in Aj belongs to X for i 6= j. (3) Through every point in X there passes exactly one line of each Ai. One can prove that |Ai| = |Aj| for every i, j and |X | = |A1|2 (see [19], [22]). Let us denote |Aj| by q, this is the degree of the net. Thus, if we use classical notation (see for example [7] or [10]), a p-net of degree q is a (q2p, pqq) configuration. Following [19] and [22], we denote a p-net of degree q by (p, q)-net. We label the lines of Ai by {Lq(i−1)+j}qj=1 for all i, and define the arrangement A = {L1, L2, ..., Lpq}. We assume q ≥ 2 to get rid of the trivial arrangement, which is actually not considered in Definition 2.1. Assume for now that we work over an algebraically closed field K. A (p, q)-net A = (A1, ...,Ap,X ) defines a unique pencil of curves P(A) of degree q as follows. Take any two sets of lines Ai and Aj. Consider Ai and Aj as the equations which define them, i.e., the multiplication of its lines. Then, the pencil is defined as P(A) = {uAi + tAj : [u : t] ∈ P1}. This is well-defined. Take Ak with k 6= i, j, and a point Q in Ak \ X . Then, there exists [u : t] ∈ P1 such that uAi(Q) + tAj(Q) = 0. We write B = uAi + tAj, which is a curve of degree q containing X ∪ {Q}. If Ak and B do not have common factors, we have, by Bezout’s Theorem, that Ak belongs to P(A) (Ak is B times a non-zero constant). This proves the independence of the choice of i, j to define P(A). If Ak and B have a non-trivial common factor, then it has to be formed by the multiplication of 0 < q1 < q lines in Ak. In this way, this common factor C contains exactly qq1 points of X . Therefore, if B = CF and Ak = CG, the set {F = 0} ∩ {G = 0} has at least q(q − q1) > (q − q1)2 points, deg(F ) = deg(G) = q − q1, and gcd(F,G) = 1. This is impossible by Bezout’s Theorem. In addition, if the characteristic of K is zero, the general member of this pencil is smooth [?, p. 272], i.e., outside of finitely many points in P1, uAi + tAj is a smooth plane curve. Hence, after we blow up the q2 points in X we obtain a fibration of curves of genus (q−1)(q−2) with at least p completely reducible fibers. This fibration leads to the following restriction on nets defined over C, due to Yuzvinsky [22] (see [18] for the higher dimensional analogue). The proof is a simple topological argument which uses the topological Euler characteristic of the fibration. Proposition 3.1. For an arbitrary (p, q)-net in P2 defined over C, the only possible values for (p, q) are: (p = 3, q ≥ 2), (p = 4, q ≥ 3) and (p = 5, q ≥ 6). The combinatorial data which defines (p, q)-nets can be expressed using Latin squares. A Latin square is a q × q table filled with q different symbols (in our case numbers from 1 to q) in such a way that each symbol occurs exactly once in each row and exactly once in each column. They are the multiplication tables of finite quasigroups. Let A = (A1, ...,Ap,X ) be a (p, q)-net. The q2 p-points in X are determined by (p− 2) q × q Latin squares which form an orthogonal set, as explained for example in [19] or [16]. Although we have defined nets as arrangements of lines already in P2, we will first “think combinatorially” about the (p, q)-net through this orthogonal set of (p− 2) Latin squares, and then we will attempt to prove or disprove its realization on P2 over some field. This is the strategy from now on. Example 3.1. In this example we use our correspondence to reprove the existence of the famous Hesse arrangement. This (4, 3)-net has nice applications in algebraic geometry (see for example [13, 2, 21]). Let us denote this net by A = A1 ∪ A2 ∪ A3 ∪ A4, with Ai = {L3i−2, L3i−1, L3i}. By relabelling the lines of A, we may assume that the combinatorial data is given by the following set of orthogonal Latin squares. 1 2 3 2 3 1 3 1 2 1 2 3 3 1 2 2 3 1 These Latin squares give the intersections of A3 and A4 respectively with A1 (columns) and A2 (rows) (see [16] or [19]). For example, the left one tell us that L2, L6 and L7 (values) have a common point of incidence. The right one says L2, L6 and L12 have also non-empty intersection. Hence, [[2, 6, 7, 12]] ∈ X . In this way, we find X , which is completely described in the following tables. L1 L2 L3 L4 L7 L8 L9 L5 L8 L9 L7 L6 L9 L7 L8 L1 L2 L3 L4 L10 L11 L12 L5 L12 L10 L11 L6 L11 L12 L10 We now consider the new arrangement of lines A′ = A \ {L3, L4, L9, L12} together with the point P = [[3, 4, 9, 12]]. We rename the twelve lines in the following way: A′ = {L′1 = L1, L 2 = L2, L 3 = L5, L 4 = L6, L 5 = L7, L 6 = L8, L 7 = L10, L 8 = L11} and the lines passing through P , α = L3, β = L4, γ = L9, and δ = L12. By our correspondence, we have a line L(A′, P ) in P6 for the pair (A′, P ), and it passes through these distinguished four points α, β, γ, and δ (we abuse the notation, these lines correspond to points in L(A′, P )). Then, K(α) = {[[4, 6, 7]], [[3, 5, 8]]}, K(β) = {[[2, 6, 8]], [[1, 5, 7]]}, K(γ) = {[[2, 3, 7]], [[1, 4, 8]]}, and K(δ) = {[[2, 4, 5]], [[1, 3, 6]]}. Hence, we write: α = [a1 : a2 : 0 : 1 : 0 : 1 : 1], β = [1 : 0 : a3 : a4 : 1 : 0 : 1] γ = [0 : 1 : 1 : 0 : a5 : a6 : 1], δ = [1 : a7 : 1 : a7 : a7 : 1 : a8] for some numbers ai (with extra restrictions), and we take L(A′, P ) : αt+ βu, [t : u] ∈ P1. For some [t : u], we have the equation αt+ βu = γ, and from this we obtain: w − 1 1− w a4 = w − 1 a5 = 1− w a6 = w, where w is a parameter. For another pair [t : u], we have αt + βu = δ, and so w2 − w + 1 = 0, a7 = 1w , and . Therefore, our field of definition needs to have roots for the equation w2−w+1. For instance, over C, we take w = e 3 , and then L(A′, P ) is: [w − 1 : 1 : 0 : w : 0 : w : w]t+ [w − 1 : 0 : −1 : w : w − 1 : 0 : w − 1]u. According to our choices at the end of Section 2, we write down the lines of A as: {L1 = ((w−1)x+y+z), L2 = (x+z), L3 = (y)} {L4 = (x), L5 = (−y+z), L6 = (wx+wy+z)} {L7 = (y+w2z), L8 = (wx+z), L9 = (wx+y)} {L10 = (x+wy+ z), L11 = (z), L12 = (x−wy)}. Notice that the lines in A corresponding to α, β, γ, and δ are ux− ty = 0, where [t : u] is the corresponding point in P1 for each of them, as points in L(A′, P ). Example 3.2. In this example, we show that there are no (4, 4)-nets over fields of char- acteristic 6= 2. This fact has independently been shown in [8] over C. We start supposing their existence, let A = {Ai}4i=1 be such a net. Again, by relabelling the lines of A, we may assume that the orthogonal set of Latin squares is: 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 We consider (A′, P ) defined by the arrangement of twelve lines A′ = A\{L4, L5, L12, L16}, and the point P = [[4, 5, 12, 16]]. The lines of A′ are L′1 = L1, L′2 = L2, L′3 = L3, L′4 = L6, L′5 = L7, L 6 = L8, L 7 = L9, L 8 = L10, L 9 = L11, L 10 = L13, L 11 = L14, and L 12 = L15. The special lines are α = L4, β = L5, γ = L12, and δ = L16. Hence, we have that α = [a1 : a2 : a3 : 1 : a4 : 0 : 0 : a4 : 1 : a4 : 1] β = [1 : b4 : 0 : b1 : b2 : b3 : 1 : b4 : 0 : 1 : b4] γ = [1 : 0 : c4 : c4 : 0 : 1 : c1 : c2 : c3 : c4 : 1] δ = [d1 : 1 : d4 : 1 : d1 : d4, 1 : d4 : d1 : d2 : d3] as points in L(A′, P ), which we write as αt+ βu, [t : u] ∈ P1. Let c1 = a, c2 = b and c3 = c. Since γ ∈ L(A′, P ), we have: a+ b+ c− 1 b+ c− 1 c4 = a+ b+ c− 1 a + b− 1 1− b− c Also, since δ ∈ L(A′, P ), have ad4 = 1 and ad1 + b = 1, plus the following equations: (1) : d1(1 − c) = 1 − b − c, (2) : d1(1 − b)(c − 1) + d1c(1 − c) = (1 − b)c, (3) : (1 − b)(1 + d4(b + c − 1)) = d4c, and (4) : c2 = (1 − b)(b + c − 1) among others. These equations are enough to produce a contradiction. By isolating d1 in (1), replacing it in (2), and using (4), we get c3 = (1− b)3 which requires a 3rd primitive root of 1. Say w is such, so b = 1− wc. Then, by (3), we get w2(1 + 2c) = w − 1. Since the characteristic of our field is not 2, we have c = 1 , and so b = 0, and a = 1. This gives a1 = 0, which is a contradiction, because it would imply that L1 ∩ L4 ∩ L8 ∩ L9 ∩ L15 6= ∅. See next example for the char. 2 case. Example 3.3. Positive characteristic gives more freedom for the realization of nets com- pared to Proposition 3.1, and the previous examples. Let q be a prime number, and let K be a field with m = qn elements. In P2 , we have m2 + m + 1 points with coordinates in K, and there are m2 + m + 1 lines such that through each of these points passes exactly m + 1 of these lines, and each of these lines contains exactly m + 1 of these points [12, p. 65]. By eliminating one of these lines, we obtain a (m + 1, m)-net. Each of the m + 1 members of this net has m lines intersecting at one point, and so tm = m + 1, tm+1 = m and tk = 0 otherwise. Hence, in positive characteristic, there are p-nets for all p ≥ 3. If we want a (4, 4)-net, one takes q = 2 (necessary by Example 3.2) and n = 2, and considers the corresponding (5, 4)-net. We now eliminate one of its members to obtain a (4, 4)-net. In [20], Stipins proves that there are no 5-nets over C (see [23] for a generalization of his result). His proof does not use the combinatorics given by Latin squares. We will see that this issue matters for the realization of (3, 6)-nets, and so, it would be interesting to know if Latin squares are relevant or not for the possible realization of 4-nets over C. It is believed that, except for the Hesse arrangement, (4, q)-nets do not exist over C. In this way, by Proposition 3.1, the only cases left over C would be 3-nets. In [22], it is proved that for every finite subgroup H of a smooth elliptic curve, there exists a 3-net over C corresponding to the Latin square of the multiplication table of H . In the same paper, the author proves that there are no (3, 8)-nets associated to the group Z/2Z ⊕ Z/2Z ⊕ Z/2Z. In [19], it can be found a classification of (3, q)-nets for q ≤ 5. In the next section we classify (3, q)-nets for q ≤ 6, and the (3, 8)-nets associated to the Quaternion group. Remark 3.1. (Main classes of Latin squares) As we explained before, a q × q Latin square gives the set X for a (3, q)-net A = {A1,A2,A3}. What if we are interested only in the realization of A in P2 as a curve, i.e., without labelling lines? Then, we divide the set of all q × q Latin squares into the so-called main classes (see [6] or [16]). For a given q× q Latin square M corresponding to A, by rearranging rows, columns and symbols ofM , we obtain a new labelling for the lines in each Ai. If we writeM in its orthog- onal array representation, i.e. M = {(r, c, s) : r = row number, c = column number, s = symbol number}, we can perform six operations on M , each of them a permutation of (r, c, s) which translates into relabelling the members {A1,A2,A3}, and so we obtain the same curve in P2. We can partition the set of all q × q Latin squares in main classes (also called Species) which means: if M,N belong to the same class, then we can obtain N by applying a finite number of the above operations to M . In what follows, we will choose one member from each class. The following table shows the number of main classes for small q. q 1 2 3 4 5 6 7 8 9 10 # main classes 1 1 1 2 2 12 147 283 657 19 270 853 541 34 817 397 894 749 939 4. Classification of (3, q)-nets for q ≤ 6, and the Quaternion nets. In order to do this classification, we use again the trick of eliminating some lines passing through a k-point P , and considering the new pair (A′, P ). We work with (3, q)-nets, thus P is taken as a 3-point in X (and so, we eliminate three lines from A). If the (3, q)-net is given by A = {A1,A2,A3} such that Ai = {Lq(i−1)+j}qj=1, then the new pair (A′, P ) will be given by A′ = {L′1 = L2, L′2 = L3, . . . , L′q−1 = Lq, L′q = Lq+2, L′q+1 = Lq+3, . . . , L′2q−2 = L2q, L 2q−1 = L2q+2, L 2q = L2q+3, . . . , L 3q−3 = L3q}, P = L1 ∩ Lq+1 ∩ L2q+1, and α = L1, β = Lq+1, γ = L2q+1. The corresponding line L(A′, P ) is αt+ βu, [t, u] ∈ P1. We obtain X from a given Latin square. Then, we fix a point P in X , so the locus of the line L(A′, P ) is actually the moduli space of the (3, q)-nets with combinatorial data defined by that Latin square (or better its main class). We give in each case equations for the lines of the nets depending on parameters coming from L(A′, P ). (3, 2)-nets. Here we have one main class given by the multiplication table of Z/2Z: 1 2 . According to our set up, (A′, P ) is formed by an arrangement A′ of three lines and P = [[1, 3, 5]] ∈ X . The line L(A′, P ) is actually the whole P1. This tells us that there is only one (3, 2)-net, up to projective equivalence. The special points are α = [1 : 0], β = [0 : 1], and γ = [1 : 1]. This (3, 2)-net is represented by the singular members of the pencil λz(x−y)+µy(z−x) = 0 on P2, and it is called complete quadrilateral (see Figure 1). (3, 3)-nets. Again, there is one main class given by the multiplication table of Z/3Z. 1 2 3 3 1 2 2 3 1 For (A′, P ) we have an arrangement of six linesA′ and P = [[1, 4, 7]] ∈ X , the line L(A′, P ) is in P4. The special points can be taken as α = [a1 : a2 : 1 : 0 : 1], β = [1 : 0 : b1 : b2 : 1], and γ = [1 : c1 : c1 : 1 : c2]. Then, for some [t : u] ∈ P1, we have αt + βu = γ. Thus, if a2 = a, b2 = b and c1 = c, we have that α = a(b−1) : a : 1 : 0 : 1 and β = 1 : 0 : bc(a−1) : b : 1 . The rest of the points in X ′ (again, although A′ is not a net, we think of X ′ as the set of 3-points in A′ coming from X ) [[1, 3, 6]] and [[2, 4, 5]] give the same restriction (a − 1)(b − 1) = 1, i.e., a = b b−1 . Therefore, the line L(A ′, P ) has two parameters of freedom, and it is given by b−1 : 1 : 0 : 1 t + [1 : 0 : c : b : 1]u where c, b are numbers with some restrictions (for example, c, b 6= 0 or 1). Hence, we find that this family of (3, 3)-nets can be represented by: L1 = (y), L2 = ( x + y + z), L3 = ( b−1x+ z), L4 = (x), L5 = (x+ cy + z), L6 = (by + z), L7 = (x+ c(1− b)y), L8 = (x+ y + z), and L9 = (z). (3, 4)-nets. Here we have two main classes. We represent them by the following Latin squares. 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 They correspond to Z/4Z and Z/2Z ⊕ Z/2Z respectively. We first deal with M1. Then, we have α = [a1 : a2 : a3 : 1 : a4 : 0 : 1 : a4], β = [1 : b1 : 0 : b2 : b3 : b4 : 1 : b1] and γ = [1 : c1 : c2 : c2 : c1 : 1 : c3 : c4]. Let a3 = a, b4 = b and c2 = c. By imposing γ to L(A′, P ), one can find a1 = (−1+b)abc , a2 = (−b1+c1b)a , a4 = (−b1+c4b)a , b2 = −(c−c2a)b b3 = b1 − c4b + c1b, and c3 = 1b + . When we impose L(A′, P ) to pass through [[1, 5, 9]], [[2, 4, 9]] and [[1, 4, 8]], we obtain equations to solve for c4, c1, and b1 respectively. After that, the restrictions [[2, 6, 7]], [[3, 5, 7]], and [[3, 6, 8]] are trivially satisfied. The line L(A′, P ) is parametrized by (a, b, c) in a open set of A3, and it is given by: a1 = a(b−1) , a2 = abc+ab−a−bc , a3 = a, a4 = a2(b−1) abc+ab−a−bc , b1 = b2(a−1)c abc+ab−a−bc , b2 = bc(a−1) , b3 = abc+ab−a−bc and b4 = b. Similarly, for M2 we have α = [a1 : a2 : a3 : 1 : a4 : 0 : 1 : a4], β = [1 : b1 : 0 : b2 : b3 : b4 : 1 : b1], and γ = [1 : c1 : c2 : 1 : c1 : c2 : c3 : c4]. Of course, the only change with respect to the previous case is γ. By doing similar computations, we have that L(A′, P ) is parametrized by (a, b, c) in a open set of A3, and it is given by: a1 = (b−c)a , a2 = abc+ab−bc−ac , a3 = a, a2(b−c) abc+ab−bc−ac , b1 = b2(a−c) abc+ab−bc−ac , b2 = b(a−c) , b3 = abc+ab−bc−ac , and b4 = b (see [19, p. 11] for more information about this net). Hence, the lines for the corresponding (3, 4)-nets for Mr can be represented by: L1 = (y), L2 = (a1x + y + z), L3 = (a2x + b1y + z), L4 = (a3x + z), L5 = (x), L6 = (x + b2y + z), L7 = (a4x+b3y+z), L8 = (b4y+z), L9 = (ax−bc2−ry), L10 = (x+y+z), L11 = (a4x+b1y+z), and L12 = (z). For example, if we evaluate the equations for the cyclic type M1 at a = b = 1−i , and c = −i (where i = −1), we obtain the well-known net: A1 = {y, (1+i)x+2y+ 2z, (1+ i)x+y+2z, (1+ i)x+2z}, A2 = {x, 2x+(1− i)y+2z, x+(1− i)y+2z, (1− i)y+2z} and A3 = {x + y, x + y + z, x + y + 2z, z}. This net is projectively equivalent of the one given by the plane curve (x4 − y4)(y4 − z4)(x4 − z4) = 0, known as CEVA(4) [7, p. 435]. (3, 5)-nets. We have two main classes, and we represent them by the following Latin squares. 1 2 3 4 5 2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 5 1 2 3 4 1 2 3 4 5 2 1 4 5 3 3 5 1 2 4 4 3 5 1 2 5 4 2 3 1 The Latin square M1 corresponds to Z/5Z. As before, for M1 and M2 we have that α = [a1 : a2 : a3 : a4 : 1 : a5 : a6 : 0 : 1 : a5 : a6] and β = [1 : b1 : b2 : 0 : b3 : b4 : b5 : b6 : 1 : b1 : b2], but for M1, γ = [1 : c1 : c2 : c3 : c3 : c2 : c1 : 1 : c4 : c5 : c6], and for M2, γ = [1 : c1 : c2 : c3 : 1 : c1 : c2 : c3 : c4 : c5 : c6]. In the case ofM1, after we impose γ to L(A′, P ), we use the conditions [[2, 3, 5]], [[4, 8, 11]], [[2, 6, 12]], [[2, 8, 9]], and [[3, 8, 10]] to solve for b2, c6, c5, b1, and c2 respectively. After that we have four parameters left: a4 = a, b6 = b, c3 = c, and c1 = d, and we get the following constrain for them: b2(a−1)(d−c)(c−ad)+b(−d2a+dc+2d2a2−2da2c−da+ca−dc2+c2da)+ad(ca−da+1−c) = 0. Hence, the (3, 5)-nets for M1 are parametrized by an open set of the hypersurface in A defined by this equation. The values for the variables are: a(b− 1) ab(d− 1) a− ba + bc a3 = a(d− db+ bc) c2(a− 1)b a4 = a a2(d− 1)(d− db+ bc) (a− ba + bc)(a− 1)cd a6 = ad(b− 1) b(da− adb+ bc) a− ba + bc b2 = d− db+ bc bc(a− 1) (da− adb+ bc)(d− db+ bc)a (a− ba + bc)(a− 1)cd b5 = d b6 = b. In the case of M2, we obtain a three dimensional moduli space of (3, 5)-nets as well. It is parametrized by (a, b, c) in an open set of A3 such that a4 = a, b6 = b and c1 = c, and: a2(1− b) b(ab− a− b) a2 = c a3 = (−a2 + a2b+ cba− ab− cb)b (ab+ cb− a− b)(ab− a− b) a4 = a (a2 − a2b− cba + ab+ cb)a (ab− a− b)2 a6 = c(b− 1)a −a + ab+ cb− b cb2(1− a) a(ab− a− b) b2 = (a− ab+ b− c)b2 (−a + ab+ cb− b)(ab− a− b) b2(1− a) a(ab− a− b) b4 = ab(ab − a− b+ c) (ab− a− b)2 b5 = cb(a+ b− ab) a(ab− a+ bc− b) b6 = b. To obtain the lines for the nets corresponding to Mr, we just evaluate: L1 = (y), L2 = (a1x + y + z), L3 = (a2x+ b1y + z), L4 = (a3x+ b2y + z), L5 = (a4x+ z), L6 = (x), L7 = (x+b3y+z), L8 = (a5x+b4y+z), L9 = (a6x+b5y+z), L10 = (b6y+z), L11 = (ax−bc2−ry), L12 = (x + y + z), L13 = (a5x + b1y + z), L14 = (a6x + b2y + z), and L15 = (z). These two 3 dimensional families of (3, 5)-nets appear in [19]. We notice that both families of (3, 5)-nets have members defined over Q. For the case M1, we can make b 2 disappear from the equation by declaring c = ad (the relations a = 1 and d = c are not allowed). Then, b = 2da−1−da 2da−2da2−1+a−d2a+d2a2 , and it can be checked that for suitable a, d ∈ Z the conditions for being (3, 5)-net are satisfied. (3, 6)-nets. We have twelve main classes of Latin squares to check. The following is a list showing one member of each class. It was taken from [6, pp. 129-137]. 1 2 3 4 5 6 2 3 4 5 6 1 3 4 5 6 1 2 4 5 6 1 2 3 5 6 1 2 3 4 6 1 2 3 4 5 1 2 3 4 5 6 2 1 5 6 3 4 3 6 1 5 4 2 4 5 6 1 2 3 5 4 2 3 6 1 6 3 4 2 1 5 1 2 3 4 5 6 2 3 1 5 6 4 3 1 2 6 4 5 4 6 5 2 1 3 5 4 6 3 2 1 6 5 4 1 3 2 1 2 3 4 5 6 2 1 4 3 6 5 3 4 5 6 1 2 4 3 6 5 2 1 5 6 1 2 4 3 6 5 2 1 3 4 1 2 3 4 5 6 2 1 4 3 6 5 3 4 5 6 1 2 4 3 6 5 2 1 5 6 2 1 4 3 6 5 1 2 3 4 1 2 3 4 5 6 2 1 4 5 6 3 3 6 2 1 4 5 4 5 6 2 3 1 5 3 1 6 2 4 6 4 5 3 1 2 1 2 3 4 5 6 2 1 4 3 6 5 3 5 1 6 4 2 4 6 5 1 2 3 5 3 6 2 1 4 6 4 2 5 3 1 1 2 3 4 5 6 2 1 6 5 3 4 3 6 1 2 4 5 4 5 2 1 6 3 5 3 4 6 1 2 6 4 5 3 2 1 1 2 3 4 5 6 2 3 1 6 4 5 3 1 2 5 6 4 4 6 5 1 2 3 5 4 6 2 3 1 6 5 4 3 1 2 M10 = 1 2 3 4 5 6 2 1 6 5 4 3 3 5 1 2 6 4 4 6 2 1 3 5 5 3 4 6 2 1 6 4 5 3 1 2 M11 = 1 2 3 4 5 6 2 1 4 5 6 3 3 4 2 6 1 5 4 5 6 2 3 1 5 6 1 3 2 4 6 3 5 1 4 2 M12 = 1 2 3 4 5 6 2 1 5 6 4 3 3 5 4 2 6 1 4 6 2 3 1 5 5 4 6 1 3 2 6 3 1 5 2 4 The Latin squares M1 and M2 correspond to the multiplication table of the groups Z/6Z and S3, respectively. The following is the set up for the analysis of (3, 6)-nets. We first fix one Latin square M from the list above. Let A = {A1,A2,A3} be the corresponding (possible) (3, 6)-net, where A1 = {L1, . . . , L6}, A2 = {L7, . . . , L12},, and A3 = {L13, . . . , L18}. As before, we consider a new arrangement A′ together with a point P such that A′ = A \ {L1, L7, L13}, and P = [[1, 7, 13]] ∈ X . We label the lines of A′ from 1 to 15 following the order of A, i.e., L′1 = L2, . . . , L′5 = L6, L′6 = L8, etc, eliminating L1, L7, and L13. Let L(A′, P ) be the line in P13 for (A′, P ). The special lines (or points of L(A′, P )) α = L1, β = L7, and γ = L13 are as α = [a1 : a2 : a3 : a4 : a5 : 1 : a6 : a7 : a8 : 0 : 1 : a6 : a7 : a8], β = [1 : b1 : b2 : b3 : 0 : b4 : b5 : b6 : b7 : b8 : 1 : b1 : b2 : b3], and γ = γ(c1, c2, ..., c8) depending on M . Since there is [t, u] ∈ P1 satisfying αt + βu = γ, we can and do write a1, a2, a3, a4, a6, a7, a8, b4, b5, b6, and c5 with respect to the rest of the variables. After that, we start imposing the points in X ′ which translates, as before, into 2 × 2 determinants equal to zero. At this stage we have 20 equations given by these determinants, and 12 variables. We choose appropriately from them to isolate variables so that they appear with exponent 1. In the way of solving these equations, we prove or disprove realization for A. When the (3, 6)-net exists, i.e. A is realizable in P2 over some field, the equations for its lines can be taken as: L1 = (y), L2 = (a1x+y+z), L3 = (a2x+b1y+z), L4 = (a3x+b2y+z), L5 = (a4x + b3y + z), L6 = (a5x + z), L7 = (x), L8 = (x + b4y + z), L9 = (a6x + b5y + z), L10 = (a7x+b6y+z), L11 = (a8x+b7y+z), L12 = (b8y+z), L13 = (ux−ty), L14 = (x+y+z), L15 = (a6x + b1y + z), L16 = (a7x + b2y + z), L17 = (a8x + b3y + z), and L18 = (z), where [t, u] satisfies αt+ βu = γ. Now we apply this procedure case by case. We first give the result, after that we indicate the order we solve the equations coming from the points in X ′, and then we give a moduli parametrization whenever the net exits. For simplicity, we work always in characteristic zero. We often omit the final expressions for the variables, although they can be given explicitly. M1: (Z/6Z) This gives a three dimensional moduli space. We have that some of these nets can be defined over R. We solve the determinants in the following order: [[4, 6, 15]] solve for c3, [[5, 10, 14]] solve for c8, [[1, 9, 15]] solve for c1, [[5, 9, 13]] solve for c7, [[3, 10, 12]] solve for c6, [[2, 10, 11]] solve for b3, [[3, 9, 11]] solve for c2, and [[2, 8, 15]] solve for b2. If a5 = a, b1 = d, b8 = b, and c4 = c, then they must satisfy: c2(−1+ a)b4(a2− a2b+ cab+ ab− 2a+ ca− bc)− b2c(2c2b2+5cab+4a2b2c− 4ca2b− 2b2a3− a2c+3a2−2a3−5a2b+ca3+2a2b2−bc2a+4ba3−4ac2b2−3ab2c−a3b2c+c2ba2+2a2b2c2)d+ (bc+ a− ab)(a2b2c2 + c2b2 − 2ac2b2 + a2b2c− ab2c+ 2cab− ca2b+ a2b2 − 2a2b+ a2)d2 = 0. So, the moduli space for these nets is an open set of this hypersurface. M2: (S3) This gives a three dimensional moduli space parametrized by an open set of A3. It does not contains (3, 6)-nets defined over R. The reason is that we need the square root of −1 to define the nets. Moreover, all of them have extra 3-points, apart from the ones coming from X . The order we take is: [[5, 10, 14]] solve for c8, [[2, 6, 15]] solve for c1, [[1, 10, 13]] solve for c7, [[1, 9, 12]] solve for c6, [[2, 10, 11]] solve for b1, [[5, 6, 12]] solve for c3, [[1, 8, 15]] solve for b2, [[1, 7, 14]] solve for b8, and [[2, 8, 14]] solve for c2. If i = −1, a5 = a, b3 = e and c4 = c, then the expressions for the variables are: 3)(2c+e−i 2aec−ac−ce−ice 3+ae+iae 3+ica (−1+i 3)(ae−iae 3−2ce+2ac)a 2(2ae−2ce+2aec+ac+ica a4 = a a5 = a a6 = (−1+i 3)(ae−iae 3−2ce+2ac)a 2(2aec−ac−ce−ice 3+ae+iae 3+ica 3)(e−i 3e+2c)a2 2(2ae−2ce+2aec+ac+ica (−1+i 3)e2(a−c) 2aec−ac−ce−ice 3+ae+iae 3+ica 3)(−ce+ae+ac)e 2ae−2ce+2aec+ac+ica b3 = e b4 = (−1+i 3)(a−c)e 3)(−ce+ae+ac)e 2aec−ac−ce−ice 3+ae+iae 3+ica 2ae−2ce+2aec+ac+ica For instance, if we plug in a = c+ic 3−2c and e = c(1+i 2(c−1) , we get a one dimensional family of arrangements of 18 lines with t2 = 18, t3 = 39, t4 = 3, tk = 0 otherwise. M3: This gives a three dimensional moduli space which does not contains (3, 6)-nets defined over R. The reason again is that we need to have the square root of −1 to realize the nets. The order we solve is: [[5, 10, 11]] solve for b8, [[1, 9, 15]] solve for c8, [[5, 9, 12]] solve for c6, [[3, 6, 15]] solve for c1, [[1, 10, 13]] solve for c7, [[4, 9, 11]] solve for b3, [[1, 6, 12]] solve for b1, and [[3, 10, 12]] solve for b2. If a5 = a, c3 = d, c2 = e, and c4 = c, then they must satisfy: (e2a2 + e2 − e2a− 2a2de− de+ d2 + 3dea+ d2a2 − 2d2a) + (−ea− e+ ad− d)c+ c2 = 0 and so its moduli space is an open set of this hypersurface. Moreover, by solving for c, we have that: c = 1 (ea+ e− ad+ d± −3(a− 1)(d− e)). But, we cannot have a = 1 or d = e, and so this shows that the square root of −1 is necessary. M4: This case is not possible over C. To get the contradiction, we take: [[5, 10, 13]] solve for c7, [[3, 7, 15]] solve for c6, [[2, 8, 15]] solve for b2, [[4, 6, 15]] solve for c3, [[5, 6, 14]] solve for a5, [[1, 9, 15]] solve for c8, [[1, 10, 14]] solve for c1, [[3, 8, 14]] solve for c2, [[2, 10, 11]] solve for c4, and [[2, 6, 13]] solve for b1. At this stage, we obtain several possibilities from the equation given by [[2, 6, 13]], none of them possible (for example, a2 = a6). M5: This case is not possible over C. By solving [[5, 10, 13]] for c7, and then [[3, 7, 15]] for c6, we obtain a6 = a7 which is a contradiction. M6: This gives a two dimensional moduli space, and so this parameter space are not always three dimensional (see [19, p. 14]). Some of these nets can be defined over R. The order we take is: [[5, 10, 11]] solve for a5, [[1, 9, 15]] solve for c8, [[3, 7, 15]] solve for c6, [[2, 6, 15]] solve for b1, [[5, 6, 13]] solve for c7, [[4, 9, 11]] solve for b3, [[2, 9, 13]] solve for c1, [[1, 10, 12]] solve for c3, and [[3, 9, 12]] solve for b2. If b8 = b, c2 = d, and c4 = c, then they must satisfy: bc(1−c)(bc−c−b)+(bc3+b2−5bc2+3bc−2b2c+b2c2−c3+2c2)d+(−b+2bc−2c+c2)d2 = 0. Thus, its moduli space is an open set of this hypersurface. M7: This gives a two dimensional moduli space parametrized by an open set of A These nets can be defined over Q. The order we solve is the following: [[5, 6, 13]] solve for c7, [[3, 6, 15]] solve for b2, [[1, 9, 15]] solve for c8, [[5, 9, 12]] solve for c6, [[1, 10, 14]] solve for b3, [[3, 9, 11]] solve for b8, [[4, 8, 11]] solve for c3, [[4, 10, 13]] solve for c2, [[4, 7, 15]] solve for b1, and [[5, 7, 11]] solve for c1. If a5 = a and c4 = c, then we have: (c2−4c+2ac+4−2a)a c(a−2)(c−2) a2 = (c−1)(c−2)(a−2)a a2c2+a2−2a2c−2c2a+5ac−2a+c2−2c a3 = ac(a+c−2) −c2−ac+c2a+2c−2a+a2 (a−2)(a−ac+c−2)a −c2+c2a−3ac+2c+a2c+2a−a2 a5 = a a6 = (a+c−2)(−a+ac−c+2)a a2c2+a2−2a2c−2c2a+5ac−2a+c2−2c (a−2)a2(c−1) c2+ac−c2a−2c+2a−a2 a8 = a(c2−4c+2ac+4−2a) −c2+c2a−3ac+2c+a2c+2a−a2 (a−1)(a−2)(c−2)2c (a+c−2)(a2c2+a2−2a2c−2c2a+5ac−2a+c2−2c) b2 = (c−a)(a−2)(c−2) −c2−ac+c2a+2c−2a+a2 (a−2)2(c−1)a(c−2) (a+c−2)(c2−c2a+3ac−2c−a2c−2a+a2) b4 = (c−a)(a−2)(c−2) ac(a+c−2) b5 = (c−1)(c−2)(a−2)a a2c2+a2−2a2c−2c2a+5ac−2a+c2−2c c(a−2)(c−2)a(a−1) (c2+ac−c2a−2c+2a−a2)(a+c−2) b7 = c(a−2)(c−2) −c2+c2a−3ac+2c+a2c+2a−a2 b8 = (a−2)(c−2) 2−a−c M8: This also gives a two dimensional moduli space. Some of these nets can be defined over R. The order we solve is the following: [[2, 6, 15]] solve for b1, [[1, 10, 13]] solve for c7, [[1, 7, 15]] solve for c6, [[5, 7, 14]] solve for c8, [[4, 10, 11]] solve for c3, [[5, 6, 13]] solve for b2, [[2, 10, 14]] solve for b3, [[5, 9, 11]] solve for a5, and [[3, 7, 11]] solve for c1. If b8 = b, c4 = c, and c2 = e, then they have to satisfy: c2(c−b)(4c2−6cb−b3+3b2)+c(cb−2c+b)(6c2−9cb−b3+4b2)e+(bc−b+c)(cb−2c+b)2e2 = 0. Thus, its moduli space is an open set of this hypersurface. M9: This gives a three dimensional moduli space. Some of them can be defined over R. The order we solve is the following: [[5, 10, 11]] solve for a5, [[1, 10, 14]] solve for c8, [[4, 7, 15]] solve for c6, [[4, 9, 12]] solve for b3, [[1, 8, 15]] solve for c7, [[5, 8, 12]] solve for c2, [[5, 6, 14]] solve for c1, and [[3, 6, 15]] solve for b2. If b1 = e, b8 = b, c4 = c, and c3 = d, then they have to satisfy: (b2c2 + c2 + bc− b2c− 2bc2) + (−2c + 2bc+ ce− bec + e2b− eb)d+ (−e + 1)d2 = 0. Thus, its moduli space is an open set of this hypersurface. M10: This gives a two dimensional moduli space. Some of these nets can be defined over R. The order we solve is the following: [[5, 10, 11]] solve for a5, [[1, 7, 15]] solve for b1, [[1, 10, 12]] solve for c6, [[3, 6, 15]] solve for b2, [[5, 6, 13]] solve for c7, [[5, 7, 14]] solve for c8, [[4, 8, 15]] solve for b3, [[3, 7, 11]] solve for c3, and [[2, 8, 11]] solve for c2. If b8 = b, c4 = c, and c1 = e, then they have to satisfy: ce(c− 2e) + (2ce− c− e)(e− c)b+ c(1− e)(e− c)b2 = 0. Thus, its moduli space is an open set of this hypersurface. M11: This also gives a two dimensional moduli space. Some of these nets can be defined over R. The order we solve is the following: [[5, 10, 11]] solve for a5, [[1, 9, 15]] solve for c8, [[3, 8, 11]] solve for c7, [[3, 7, 15]] solve for c6, [[4, 6, 15]] solve for b3, [[2, 8, 15]] solve for b1, [[4, 9, 11]] solve for c2, [[5, 7, 14]] solve for c3, and [[1, 8, 14]] solve for c1. An extra property for this nets is that c7 has to be zero, and so L13, L16, and L18 have always a common point of incidence. If b2 = e, b8 = b, and c4 = c, then they must satisfy: c(b− 1)(bc− b− c) + (b2c− 2bc + c− b2 + 2b)e− e2 = 0. Thus, its moduli space is an open set of this hypersurface. M12: This case is not possible over C. To achieve contradiction, we take: [[2, 9, 15]] solve for c8, [[5, 10, 13]] solve for c7, [[3, 6, 15]] solve for b2, [[1, 8, 15]] solve for c3, [[1, 10, 12]] solve for c6, [[5, 9, 11]] solve for c2, and [[5, 6, 12]] solve for b1. Then, the equation induced by [[1, 9, 13]] gives six possibilities, none of them is possible. (3, 8)-nets corresponding to the Quaternion group. We now compute the (3, 8)-nets corresponding to the multiplication table of the Quater- nion group. 1 2 3 4 5 6 7 8 2 1 6 7 8 3 4 5 3 6 2 5 7 1 8 4 4 7 8 2 3 5 1 6 5 8 7 6 2 4 3 1 6 3 1 8 4 2 5 7 7 4 5 1 6 8 2 3 8 5 4 3 1 7 6 2 In this case, we have a three dimensional moduli space for them, given by an open set of A3. Also, these (3, 8)-nets can be defined over Q (so we can even draw them). This example shows again that non-abelian groups can also realize nets over C. The set up is similar to what we did before. In this case, A′ = A \ {L1, L9, L17} and P = [[1, 9, 17]]. Our distinguished points on L(A′, P ) ⊆ P19 are: α = [a1 : a2 : a3 : a4 : a5 : a6 : a7 : 1 : a8 : a9 : a10 : a11 : a12 : 0 : 1 : a8 : a9 : a10 : a11 : a12], β = [1 : b1 : b2 : b3 : b4 : b5 : 0 : b6 : b7 : b8 : b9 : b10 : b11 : b12 : 1 : b1 : b2 : b3 : b4 : b5], and γ = [1 : c1 : c2 : c3 : c4 : c5 : c6 : 1 : c4 : c5 : c6 : c1 : c2 : c3 : c7 : c8 : c9 : c10 : c11 : c12]. Let [t : u] ∈ P1 such that αt + βu = γ. We isolate first a1, a2, a3, a4, a5, a6, a8, a9, a10, a11, b6, b7, b8, b9, b10, b11, and c7 with respect to the other variables. The following is the order we solve (some of) the 2× 2 determinants given by the 3-points in X ′: [[1, 11, 21]] solve for c10, [[2, 10, 21]] solve for c9, [[3, 12, 21]] solve for c11, [[4, 8, 21]] solve for b3, [[5, 13, 21]] solve for c8, [[7, 14, 15]] solve for b12, [[5, 14, 20]] solve for c4, [[2, 14, 17]] solve for c1, [[4, 9, 20]] solve for c2, [[6, 13, 15]] solve for c5, [[3, 8, 20]] solve for b5, [[3, 10, 15]] solve for b4, [[3, 9, 18]] solve for c6, and [[3, 11, 19]] solve for b1. Then, if we write a7 = a, b2 = e, and c3 = d, the expressions for all the variables are: ad−a−d a−2 a2 = 2e2d−2ed+ed2−e2d2+(−2ed2+e2d2+2e+6ed−3e2d−4)a+(−4ed−e+4+ed2+e2d)a2 (ae−2a−2e+2)(ade−ed+d−a−da) e(ade−ed+d−a−da) ae−2a−2e+2 a4 = d 4d+2e2d−6ed+e2d2−ed2+(−2e2d2+2ed2−8d−e2d+10ed−2e)a+(4d+e+e2d2−ed2−4ed)a2 (ae−2a−2e+2)(a+d−ad−de) (a+d−ad−de)(ae+dae−4a−2e−ed+4) (ae−2a−2e+2)(ade−a−da−2ed+2+d) a7 = a a8 = (a+d−da−2)e ae−2a−2e+2 2e2d−2ed+ed2−e2d2+(−2ed2+e2d2+2e+6ed−3e2d−4)a+(−4ed−e+4+ed2+e2d)a2 (a+d−ad−2)(ae−2a−2e+2) a10 = a+ d− ad a11 = ade+ae−4a−2e−ed+4ae−2a−2e+2 a12 = 2e2d+4d−6ed+e2d2−ed2+(−2e2d2+2ed2−8d−e2d+10ed−2e)a+(4d+e+e2d2−ed2−4ed)a2 (ade−a−da−2ed+d+2)(ae−2a−2e+2) −2e+ed+ae−2a −ed+dae+d−a−da b2 = e b3 = 2 b4 = ae−2a−2e−ed+4 a+d−ad−de b5 = ade+ae−4a−2e−ed+4 ade−a−da−2ed+d+2 b6 = e(ad−d+2−a) ad+de−a−d b8 = − −2e+ed+ae−2a a+d−ad−2 b9 = 2− a b10 = ae+dae−4a−ed−2e+4 ade−ed+d−a−da b11 = ae−2a−2e+4−ed ade−a−da−2ed+d+2 b12 = a−1 , with [t : u] = [2− a : d(a− 1)] ∈ P1. Since b3 = 2, these (3, 8)-nets are not possible in characteristic 2. The lines for these (3, 8)- nets can be written as: L1 = (y), L2 = (a1x+y+z), L3 = (a2x+b1y+z), L4 = (a3x+b2y+z), L5 = (a4x+ b3y + z), L6 = (a5x+ b4y + z), L7 = (a6x+ b5y + z), L8 = (a7x+ z), L9 = (x), L10 = (x + b6y + z), L11 = (a8x + b7y + z), L12 = (a9x + b8y + z), L13 = (a10x + b9y + z), L14 = (a11x + b10y + z), L15 = (a12x + b11y + z), L16 = (b12y + z), L17 = (ux − ty), L18 = (x + y + z), L19 = (a8x + b1y + z), L20 = (a9x + b2y + z), L21 = (a10x + b3y + z), L22 = (a11x+ b4y + z), L23 = (a12x+ b5y + z), and L24 = (z). A natural question, which we leave open, is the following: Question 4.1. Is there a combinatorial characterization of the main classes of q × q Latin squares which realize (3, q)-nets in P2 References 1. J. Aczel. Quasigroups, nets, and nomograms, Adv. in Math. 1 (1965) 383-450. 2. M. Artebani and I. Dolgachev. 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Pereira and S. Yuzvinsky. Completely reducible hypersurfaces in a pencil, arXiv:math/0701312v2. 19. J. Stipins. Old and new examples of k-nets in P2, arXiv:math.AG/0701046. 20. J. Stipins. On finite k-nets in the complex projective plane, Ph.D. Thesis, University of Michigan (2007). 21. G. Urzúa. Arrangements of curves and algebraic surfaces, Ph.D. Thesis, University of Michigan (2008). 22. S. Yuzvinsky. Realization of finite abelian groups by nets in P2, Compos. Math. 140, no. 6, (2004) 1614–1624. 23. S. Yuzvinsky. A new bound on the number of special fibers in a pencil of curves, arXiv:0801.1521v2. Department of Mathematics and Statistics, University of Massachusetts at Amherst, USA. E-mail address : [email protected] http://arxiv.org/abs/math/0611590 http://arxiv.org/abs/math/0505435 http://arxiv.org/abs/math/0703142 http://arxiv.org/abs/math/0701312 http://arxiv.org/abs/math/0701046 http://arxiv.org/abs/0801.1521 1. Introduction. 2. 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0704.0470

Testing turbulence model at metric scales with mid-infrared VISIR images at the VLT

Mon. Not. R. Astron. Soc. 000, 1–8 (2005) Printed 4 November 2018 (MN LATEX style file v2.2) Testing turbulence model at metric scales with mid-infrared VISIR images at the VLT A. Tokovinin1⋆, M. Sarazin2, and A.Smette3 1Cerro-Tololo Inter American Observatory, Casilla 603, La Serena, Chile 2European Southern Observatory, Karl-Schwarzschild-Strasse, 2 D-85748 Garching bei München, Germany 3European Southern Observatory, Alonso de Cordova 3107, Casilla 19001, Vitacura, Santiago, Chile ABSTRACT We probe turbulence structure from centimetric to metric scales by simultaneous imagery at mid-infrared and visible wavelengths at the VLT telescope and show that it departs signifi- cantly from the commonly used Kolmogorov model. The data can be fitted by the von Kármán model with an outer scale of the order of 30 m and we see clear signs of the phase structure function saturation across the 8-m VLT aperture. The image quality improves in the infrared faster than the standard λ−1/5 scaling and may be diffraction-limited at 30-m apertures even without adaptive optics at wavelengths longer than 8 micron. Key words: site testing – atmospheric effects 1 INTRODUCTION As the ground-based telescopes become bigger, more emphasis is made on studying and modeling atmospheric optical distortions at large spatial scales. This knowledge is required, e.g. for specifying the stroke of deformable mirrors in adaptive optics or the range of fringe-trackers in interferometers. Moreover, the size of the atmo- spheric coherence length exceeds 1 m at infrared (IR) wavelengths. Thus, even classical long-exposure imagery in the IR is affected by the departures of turbulence statistics from the standard Kol- mogorov model (variance proportional to the 5/3 power of the base- line) which was so successfully used in the visible range. Optical path-length fluctuations were measured with long- baseline interferometers, but the published results are controver- sial. Saturation of the fringe motions at baselines from 8 m to 16 m was first observed by Mariotti & Di Benedetto (1984). Later, Davis et al. (1995) found a strong departure from the Kolmogorov law and a saturation of the path-length fluctuations at the 80- m baseline at a level of about 10µm rms. On the other hand, Colavita et al. (1987) have not found any departure from the Kol- mogorov law at baselines up to 12 m by doing a temporal analysis of the fringe motion, and Nightingale & Buscher (1991) reached the same conclusion from interferometric measurements of the fringe motion in a 4-m telescope aperture. The controversy could be caused by the use of the frozen-flow hypothesis in the inter- pretation of fringe motion on a single baseline. Another reason is the difficulty in separating fringe motion caused by the atmosphere from the mechanical noise due to instrument instabilities, tracking, ⋆ E-mail: [email protected] Turbulence outer scale L0 can be evaluated from the covari- ances of the image motion in small telescopes, as implemented in the GSM instrument (Ziad et al. 2000), or from the analysis of adaptive-optics data, e.g. by Fusco et al. (2004). These methods use the von Kármán (VK) turbulence spectrum (cf. Appendix A) and adjust its parameters (r0, L0) to fit the data. Measurements with the GSM at different sites show that typical L0 values are of the order of 20 m. Maire et al. (2006) compared directly fringe motion in a long-baseline interferometer with the L0 measured by GSM and found a good agreement. This short and non-exhaustive review demonstrates that fur- ther work on characterizing large-scale turbulence structure is needed. A direct comparison of mid-IR and optical images at a large telescope offers a new, independent way to probe turbulence models at spatial scales from centimeters to meters. To our knowl- edge, no such work has been done previously, so we made an exper- iment at the Very Large Telescope (VLT) located at the ESO obser- vatory Cerro Paranal in Chile. The main purpose of this experiment is to evaluate the atmospheric phase structure function (SF) and to check if the VK model is adequate. So far, the mathematically con- venient VK model has been used without being actually tested. The experiment is described in Sect. 2. The results are pre- sented in Sect. 3 and the conclusions are given in Sect. 4. Appen- dices contain some technical material. For reader’s convenience we reproduce the formulation of atmospheric models in Appendix A. c© 2005 RAS http://arxiv.org/abs/0704.0470v1 2 A. Tokovinin, M. Sarazin, & A. Smette Subtract backg. FFT −−> OTF Correct OTF(0) Compute SF Divide by T0* VISIR Guider S−H sensor Guide star Object Tilt errors Good spots Fit r0 Average, OTF Average A−B Sel. long axis Re−center DIMM, MASS, meteo Bright star Divide by Ref Baseline 3cm−15cm von Karman (r0,L0) 0.4m−4m Figure 1. Overview of the experiment and data interpretation. 2 EXPERIMENT DESCRIPTION 2.1 From PSF to structure function It is well known that the modulus of the long-exposure optical transfer function (OTF) T (f) in a perfect telescope is related to the atmospheric SF Dϕ(r) as (Tatarskii 1961; Roddier 1981) T (f) = T0(f) exp[−0.5Dϕ(λf)], (1) where T0(f) is the diffraction-limited OTF, f is the vector of spa- tial frequency on the sky in rad−1, f = |f |, λ is the imaging wave- length and r is the baseline vector. This relation can be inverted to reconstruct Dϕ(r) from the known image point spread function (PSF). However, it is only feasible at the baselines r = λf where Dϕ(r) is not very large or small, otherwise T (f) is close to either 1 or 0 and the sensitivity to the atmospheric turbulence is lost. In the visible range, the method probes centimetric and decametric scales, in the mid-IR it is sensitive to the metric scales because λ is much larger. The 8-m VLT telescope offers a unique platform for our exper- iment with access to metric baselines. The aberrations are removed by active optics and the turbulence inside the dome is low, ensuring that the image blur is dominated by the atmospheric seeing. Optical images as small as 0.18′′ FWHM have been recorded under excep- tional conditions, proving that the VLT intrinsic quality is nearly ideal even in the visible range.1 For our purpose, the mid-IR im- ager VISIR installed at the Cassegrain focus of the UT3 telescope is the best choice. A deeper analysis of the mid-IR images shows that some de- partures from the ideal-telescope model (1) are inevitable. Resid- ual image motion (tilt) is the largest source of uncertainty, as it can cause additional blur (e.g. wind shake), while, on the other hand, part of the atmospheric tilt is removed by guiding. Residual aber- rations in the VLT optics and instrument can add something to the atmospheric PSF, too. Therefore, opto-mechanical wave-front dis- tortions of instrumental nature cannot be separated cleanly from the large-scale atmospheric distortions. In this respect our new exper- iment is not fundamentally different from long-baseline interfer- ometers, but it was worth trying nevertheless because instrumental effects in both cases are different. 1 ESO Press Release, July 21, 2000. http://www.eso.org/outreach/press-rel/pr-2000/pr-16-00.html Table 1. Instrument parameters Instrument VISIR VISIR SH Filter PAH1 Q2 None Aperture diam., m 8.115 8.115 0.34 square λ/∆λ, µm 8.6/0.42 18.7/0.88 0.6/∼0.3 Pixel scale, arcsec 0.075 0.075 0.280 Detector format 256×256 256×256 592×573 Exposure time, s 30×2 90×1 ∼45 Chopping period, s 4 2 None 2.2 Overview The overall scheme of the experiment is presented in Fig. 1. Two different stars are observed through the VLT: the object with VISIR, the guide star with the Shack-Hartmann (SH) sensor of the VLT active optics (AO) system. The same guide star is used for the guiding, called field stabilization. The angular distance between the object and the guide star was from 3′ to 5′. Details of the optical and IR imagery and data reduction are given below and in Table 1. Data on the seeing and turbulence profile at the Paranal observatory are collected by the dedicated site monitor equipped with the Differential Image Motion Monitor (DIMM) (Sarazin & Roddier 1990) and Multi-Aperture Scintillation Sensor (MASS) (Kornilov et al. 2003). The monitor points to a bright star near zenith and measures the total seeing ǫ and the seeing in the free atmosphere ǫFA produced by turbulence above 500 m. Naturally, ǫFA < ǫ unless most turbulence is above 500 m. The VLT dome is higher than the DIMM tower, hence the seeing at VLT can be better than that measured by DIMM, but still worse or equal to ǫFA. The MASS also measures the adaptive-optics time constant τ0 and a crude turbulence profile. The effective wind speed in the free atmosphere V was evaluated from the relation V = 0.31r0,FA/τ0. The speed of the ground wind Vgr was taken from the Paranal ambient conditions database. 2.3 Conditions of the experiment The data for this study have been obtained by A.S. on June 19, 21, and 22, 2006. Table 2 lists relevant average parameters for each data set. On all nights the sky was clear, with stable air temperature and very low humidity. Individual (non-averaged) data from DIMM and MASS are plotted in Fig. 2 to characterize the variability of the turbulence. The conditions were rather stable on all 3 nights, with c© 2005 RAS, MNRAS 000, 1–8 http://www.eso.org/outreach/press-rel/pr-2000/pr-16-00.html Turbulence model with mid-IR images 3 Figure 2. Temporal evolution of the seeing ǫ and the seeing in the free atmosphere ǫFA as measured by DIMM and MASS respectively during the data acquisition periods on 3 nights. The seeing derived from the SH spots and reduced to zenith is over-plotted as large squares. Table 2. Night log Date Time Air ǫ, ǫFA, V Vgr Jun 2006 UT mass ′′ ′′ m/s m/s 19/20 23:38–23:44 1.7 1.38 0.63 30 5.6 21/22 1:40–1:49 1.5 1.21 0.68 13 2.5 22/23 23:07–23:16 1.1 0.54 0.27 16 6.4 A B BA Figure 3. The pairs of positive and negative PSFs in the PAH1 filter regis- tered with VISIR on June 21/22 (left, file 5) and June 22/23 (right, file 1). Only the central 20x20 pixels (1.5′′) of each image are displayed with a square-root intensity stretch. the DIMM seeing always dominated by the ground layer. However, the seeing derived from the SH spots indicates that the ground see- ing contribution at VLT was less than at DIMM on June 19/20 and 21/22. 2.4 VISIR data Mid-IR images of bright stars were obtained with VISIR in two filters called PAH1 (8.6µm) and Q2 (18.7µm), cf. Table 1. A stan- dard chopping-nodding technique was used. At each telescope po- sition (nod), the image was shifted on the detector back and forth by modulation (chop) of the VLT secondary mirror M2 with a period of 2-4 s. The chop throw was 10′′ in the North-South direction, with a little pause to stabilize the image after each chop. For the PAH1 filter, for example, a total of 30 images with 2 s exposure in each chop position are taken to produce the data cube with cumulative exposure of 60 s. Then the telescope is moved by 10′′ to the East and a second data cube is taken. Here we do not take advantage of the nodding and analyze only the average difference between images in two chopping positions A and B. This A−B difference suppresses the background and its slow drifts. It is averaged over all image pairs in the cube and contains positive (A) and negative (B) images of the same star, considered here separately as two in- dependent realizations of the PSF. The positive and negative PSFs are extracted as two 64x64 Figure 4. The cuts of the OTFs along fx axis for the VISIR images dis- played in Fig. 3 (top – June 21/22, bottom – June 22/23). Asterisks – positive images, plusses – negative images. The line shows the diffraction- limited OTF T0(λf). pixel (4.8′′) subsections and processed in parallel. The background is computed in the corners of these images (outside the radius of 32 pixels) as a median and subtracted. For reference only, these PSFs are approximated with 2-dimensional Gaussians to determine the Full Width at Half Maximum (FWHM) ǫl and ǫs in the long and short axes. The ellipticity is computed as e = (ǫl − ǫs)/(ǫl + ǫs) and the position angle θ of the long axis (counted from the x-axis counter-clockwise) is recorded as well. Figure 3 shows typical examples of the individual PSFs in the PAH1 filter on two nights. On June 21/22, the PSF is blurred c© 2005 RAS, MNRAS 000, 1–8 4 A. Tokovinin, M. Sarazin, & A. Smette Figure 5. The atmospheric OTF in the VISIR PAH1 filter obtained after correcting the observed OTFs for diffraction on June 22/23, file 1. All points are plotted without radial averaging (asterisks – image A, triangles – image B) to show the asymmetry. Compare with Fig. 4. by the turbulence. It is slightly elongated (e = 0.1...0.2) and has some structure, different between positive and negative images. On June 22/23, under a better seeing, the PSF is closer to a diffraction- limited one, but still elongated (e = 0.08...0.12). The elongation can be caused by a combination of several factors including some residual low-order aberrations, vibrations, etc. However, the domi- nant contribution to the image elongation is likely related to the tilt anisoplanatism between the guide star and the object, estimated in Appendix C. The direction of the elongation points approximately to the guide star. Each PSF is Fourier-transformed and the normalized modulus of the FT is identified with the observed T (f) (Fig. 4). The spa- tial frequencies f are translated to the baselines r = λf . A small correction is needed for the normalization (cf. Appendix B). The experimental OTF T (f) is divided by the calculated T0(f) to ob- tain the atmospheric OTF and then the SF. In calculating T0(f), we use the pupil diameter D = 8.115 m because it is not vignetted by the cold stop inside the VISIR instrument. Figure 5 shows the results of this division for the sharpest PAH1 image (compare with Fig. 4, bottom). The saturation of the OTF (hence SF) is obvious. The vertical spread is caused by the asymmetry and the differences between the images A and B. Note that at small baselines the derived atmospheric OTFs and SFs are sensitive to the normalization errors, while at large base- lines the experimental T (f) becomes noisy and its modulus is bi- ased by any image defects such as noise and bad pixels. For these reasons, we use for further analysis the baseline range from 0.4 m to 4 m, where T (f) is most reliable. In the following, the asym- metry is neglected and the atmospheric SFs are calculated from the radially-averaged OTFs. The data in the Q2 filter are rather noisy compared to the PAH1 filter. The PSFs are nearly diffraction-limited. At some nod- ding positions, the image is affected by bad detector pixels and/or horizontal stripes in the background. The OTFs registered on June 22/23 do not differ from the diffraction-limited ones within the er- rors. Therefore, no reliable estimates of the atmospheric SF are de- rived from the Q2 images on this night and we can only affirm that the phase fluctuations were much less than 1 rad at 18.7µm. Figure 6. The atmospheric OTF derived from the average SH spot de- convolved by the average reference spot (June 22/23, 23:07). The fitted Kolmogorov model is plotted in dashed line. 2.5 Shack-Hartmann data Long-exposure images from the Shack-Hartmann (SH) sensor were recorded quasi-simultaneously with the data. A typical exposure time is 45 s, but the SH “sees” the star only during 1/2 of the chop- ping cycle. On the other hand, the guiding was done continuously because the guiding “box” moved to compensate for the chopping. The SH images are “raw”, i.e. not corrected for bias, hot pix- els, flat field, etc. The geometry of the lenslet array is square, with 24 spots across pupil diameter, hence sub-aperture size d = 0.34m. The pixel scale 0.30′′ was calculated from the opto-mechanical data of the SH sensor and telescope. However, the real pixel scale depends, among other things, on the actual distance between the lenslet array and the detector. So we obtained images of the known double star HIP 73246 with a separation of 3.512′′ measured by Hipparcos and derived the pixel scale 0.280′′ ± 0.004′′ for the SH sensor at the Cassegrain focus of UT3. The images of individual spots show various distortions and are all elongated in one direction due to the un-corrected atmo- spheric dispersion (except for the data taken near zenith). Initially, we selected for the analysis only the sharpest spots in each frame. The spots were extracted, over-sampled, re-centered and averaged. The influence of local CCD defects is reduced by the averag- ing. The modulus of the Fourier Transform T (f) of the average spot was calculated and normalized so that T (0) = 1. Then it was divided by the diffraction-limited OTF for the square aperture T0(f) = (1− |λfx/d|) (1− |λfy/d|), assuming λ = 0.6µm. We presume that the elongation of the spots is caused primar- ily by the atmospheric dispersion. The long axis of T (f) is found by fitting a 2-dimensional Gaussian. The cut along this axis is fitted to the Kolmogorov atmospheric OTF (eq. A2), giving an estimate of the Fried’s parameter r0,Kolm and seeing ǫSH = 0.98λ/r0,Kolm (Fig. 6). The atmospheric SF at short baselines is derived from the same cut. We found that even the sharpest spots were distorted by resid- ual aberrations in the lenslets. A mosaic of selected sharp spots recorded under good seeing near zenith (Fig. 7) shows various de- grees of distortion and elongation. This became apparent on June 22/23, under excellent conditions, when the SH measured a seeing of about 0.75′′ , worse than the DIMM seeing. To overcome this problem, we used the image of the reference point source. A total of 306 sharpest reference spots were selected, re-centered and aver- aged in the same way as the star images. Then we re-processed the c© 2005 RAS, MNRAS 000, 1–8 Turbulence model with mid-IR images 5 Figure 7. Mosaic of 8 individual good SH spots (June 22/23, 23:15), each in a 3′′ × 3′′ field. The data are taken at air mass 1.1, with negligible atmospheric dispersion. The ellipticity of the average spot is e = 0.08. Figure 8. Structure functions on June 21/22, Q2 filter (files 1, top, and 3, bottom). The SFs derived from the VISIR images A and B are plotted as plusses and crosses, the SFs from the SH spots – as asterisks. Full lines – VK models, dashed lines – subtraction and addition of Dtilt, dotted lines – Kolmogorov SFs. data by selecting the same spots and de-convolving them by the av- erage reference spot instead of T0(f). A good agreement between DIMM and SH was reached (Fig. 2). 3 RESULTS The results of image processing are gathered in the Table 3. The time (to 1 min) refers to the end of each acquisition. The seeing at 0.6µm estimated from the SH spots ǫSH is listed as well (it is not reduced to zenith as in Fig. 2). The next columns give the parameters of the elliptical Gaussians approximating the positive (A) and negative (B) mid-IR images: the FWHM ǫs of the short axis (in arcseconds), the ellipticity e, and the position angle θ of the long axis in degrees. The SFs derived from the visible and mid-IR PSFs are con- verted to linear units (µm2) by multiplying them with (λ/2π)2 and combined on the same plots (Figs. 8,9,10). They are compared to Figure 9. Structure functions as in Fig. 8 for the PAH1 filter (June 21/22, files 5, top, and 6, bottom). the VK models (Appendix A). The model parameters r0 are derived from the SH spots, and the outer scale L0 is selected to match the data qualitatively. These parameters are also listed in Table 3. The exact degree of tip-tilt compensation by the field stabi- lization servo cannot be evaluated. A large part of the atmospheric tilt produced by the ground layer is compensated, but tilts from high layers are actually amplified (Appendix C). However, the total effect of the tilt compensation is not large and cannot explain the deviation of the SFs from the Kolmogorov model. In the figures, the dashed lines show the VK models with complete tilt compen- sation and tilt doubling, thus bracketing possible effects of the field stabilization system. Figure 8 shows the data from two consecutive acquisitions made with an interval of only 4 min. We see that L0 increased from 50 m to 200 m. Further data show that it decreased again in the next 6 min. (Fig. 9). Such “bursts” of L0 are typical (Ziad et al. 2000). On June 22/23, the images in the Q2 filter are so close to the diffraction limit that the SFs derived from them are uncertain (L0 values marked by colons in Table 3). The SFs derived from the PAH1 image shows saturation (Fig. 10). c© 2005 RAS, MNRAS 000, 1–8 6 A. Tokovinin, M. Sarazin, & A. Smette Table 3. Data log Time File Filt ǫSH , Image A Image B r0, L0, UT ′′ ǫs e θ ǫs e θ m m June 19/20 23:38 5 Q2 1.348 0.573 0.09 22 0.582 0.08 23 0.082 40 23:40 6 Q2 - 0.674 0.07 38 0.663 0.08 43 0.086 200 23:42 7 Q2 - 0.643 0.03 30 0.624 0.03 9 0.086 200 23:44 8 Q2 - 0.585 0.05 22 0.578 0.08 17 0.082 50 June 21/22 1:40 1 Q2 0.980 0.506 0.10 -25 0.513 0.09 -30 0.112 50 1:42 2 Q2 0.867 0.512 0.08 -19 0.530 0.08 -30 0.115 100 1:44 3 Q2 0.895 0.515 0.08 -19 0.529 0.06 -28 0.117 200 1:46 4 Q2 0.861 0.493 0.03 -20 0.505 0.02 -24 0.113 60 1:48 5 PAH1 0.719 0.379 0.10 -1 0.388 0.08 -9 0.110 30 1:49 6 PAH1 0.989 0.373 0.17 -7 0.400 0.19 3 0.111 40 June 22/23 23:07 1 PAH1 0.583 0.239 0.08 23 0.247 0.09 21 0.181 35 23:08 2 PAH1 0.575 0.245 0.12 32 0.251 0.09 22 0.182 40: 23:11 3 Q2 0.580 0.457 0.02 14 0.456 0.01 27 0.186 60: 23:12 4 Q2 0.614 0.455 0.02 20 0.454 0.02 8 0.186 60: 23:14 5 Q2 0.584 0.448 0.04 20 0.459 0.02 17 0.194 200: 23:16 6 Q2 0.584 0.451 0.02 4 0.448 0.02 14 0.194 200: Figure 10. Structure function as in Fig. 8 for June 22/23, PAH1 filter, file 4 CONCLUSIONS We were able to measure directly the structure function of atmo- spheric wave-front distortions at the VLT up to metric scales. The results show a broad agreement with the VK turbulence model. Hence, we can use this model with an increased degree of confi- dence for predicting the long-exposure PSF in the infrared or eval- uating the deformable-mirror stroke. Interpretation of the measured SFs in terms of atmospheric turbulence model cannot be done without reservations, however. Several instrumental effects bias these SFs. Instead of uncertain modeling of these effects, we simply present the results “as they are” and hope that new, deeper studies will be prompted by this work. It is preferable to use a good-quality optical imager rather than SH for continuing this study. The standard theory predicts an improvement of the image size in a very large telescope (neglecting diffraction) as λ−1/5, e.g. by 1.70 times between 0.6µm and 8.6µm. In fact, on a good night the VLT image quality at 8.6µm is limited by diffraction, and we observe a clear saturation of the SF. It means that the λ−1/5 scaling does not work. In a larger, 30-m telescope, the FWHM resolution at 8.6µm will be 0.067′′ (diffraction-limited) for the VK model with decametric outer scales because the SF saturates at large baselines. This example illustrates a dramatic effect of turbulence model for predicting the long-exposure image quality in the IR, demonstrated here experimentally. ACKNOWLEDGMENTS We thank Stephane Guisard for obtaining reference SH images on our request. A suggestion by anonymous Referee to de-convolve the SH spots with images of the reference source helped us to re- solve the problem of lenslet aberrations. APPENDIX A: MODELS OF TURBULENCE The von Kármán (VK) turbulence model describes the spatial power spectrum of atmospheric wave-front phase by the formula (Tatarskii 1961; Sasiela 1994; Ziad et al. 2000) Wϕ(κ) = 0.0229r −11/6 , (A1) where κ is the modulus of the spatial frequency (one over period). The model has two parameters, the coherence radius r0 (also called Fried radius) and the outer scale L0. The Kolmogorov turbulence model is a specific case of (A1) with L0 = ∞. The phase structure function (SF) Dϕ(r) is obtained from Wϕ(κ). An analytic expression for the SF in the VK model can be found in (Tatarskii 1961) and in (Tokovinin 2002). In the limit L0 = ∞ (Kolmogorov model) the SF is Dϕ(r) = 6.8839(r/r0) . (A2) If we try to approximate a VK SF (or PSF) with a simpler Kol- mogorov model, the derived r0,Kolm will be larger than the true r0. c© 2005 RAS, MNRAS 000, 1–8 Turbulence model with mid-IR images 7 An approximate relation between these parameters can be estab- lished numerically. Here we use a formula adapted from (Tokovinin 2002), r0 ≈ r0,Kolm 1− 1.5(r0,Kolm/L0)0.356. (A3) Thus we translate the r0,Kolm obtained by fitting the SH spots to the r0 parameter appropriate for the VK model by means of (A3). This ensures a good match between the Kolmogorov and VK SFs at short baselines (e.g. Fig. 8). APPENDIX B: CORRECTION FOR THE MISSING FLUX The wings of the PSF (essentially caused by diffraction) outside the selected field are cut off, hence T (0) (integral of the PSF) is under- estimated. The fraction of energy in the Airy PSF contained in the circle of angular radius a is (Born & Wolf 1965): E(a) = 1− J 0 (πad/λ)− J 1 (πad/λ). (B1) We take a = pNgrid = 2.4 ′′ , where p = 0.075′′ is the pixel size and Ngrid = 32 is the half-size of the PSF frame. A second correction is needed because we calculate the back- ground as the average intensity at the distance ≈ a from the center. Together with the background, we subtract some fraction of the PSF. This fraction ∆E, integrated over the field, can be estimated by differentiating (B1) as [E(a+∆a)− E(a)]. (B2) We take ∆a = λ/D to average out the “wiggles” of E(a). For PAH1 images, E(a) ≈ 0.98 and ∆E ≈ 0.01. We divide T (0) by E(a) − ∆E before the normalization, thus accounting for the missing flux. The correction in the Q2 filter is larger, reaching 5%. Without such correction, we over-estimate T (f) at small baselines and hence under-estimate the SF. We see in Fig. 9 that the first points of mid-IR SFs are still below the model curves, indicating that a larger correction for missing flux was probably needed for the PAH1 images. APPENDIX C: EFFECT OF THE TIP-TILT SERVO ON THE SF The variance of the image centroid motion in one coordinate σ2α (in square radians) can be computed by the known formula α = Ktilt(λ/D) (D/r0) , (C1) where Ktilt = 0.170 for the Kolmogorov turbulence, e.g. (Sasiela 1994). The image motion is achromatic because the right hand of Eq. C1 does not depend on λ. For the VK model, our numerical calculation leads to the approximation Ktilt ≈ −2.672 + 2.308x − 0.898x , (C2) where x = log (L0/D). The approximation is valid for L0/D < 300 with an accuracy of better than 1%. For a typical situation at VLT, L0 = 20m, Ktilt = 0.013, i.e. an order of magnitude smaller than for the Kolmogorov model. The field stabilization servo measures the tilt of the guide star α2, filters it with the closed-loop response h(t) and applies to com- pensate for the object tilt α1. The residual tilt error is then ∆α = α1 − α2 ⊙ h. (C3) It follows that the power spectrum of the residual tilt variance is Figure C1. Temporal power spectra of the tilt (full line) and residual tilts in longitudinal and transverse directions after compensation with a 1-Hz servo. Parameters: D = 8.115m, L0 = 20m, V = 10m/s, s = 0.72D, η = 0, ν0 = 1Hz. W∆α(ν) = Wα(ν)[1 + |h̃(ν)| ]− 2Re[W12(ν)h̃ (ν)], (C4) where Wα(ν) is the power spectrum of the tilt, W12(ν) is the cross- power spectrum between guide star and object, and Re stands for the real part. The residual tilt variance σ2∆α can be conveniently expressed as a fraction r of the un-corrected variance, W∆α(ν)dν = rσ α. (C5) The servo response is modeled here by a simple integrator with a 3-db cutoff frequency ν0, h̃(ν) = 1 + i(ν/ν0) . (C6) The power spectrum and cross-spectrum of tilt for VK turbu- lence model is computed by Avila et al. (1997). For a single layer moving with the speed V at direction η (η = 0 for the wind blow- ing from the object to the guide star), the cross-spectrum depends on the separation s of the beam footprints. For example, a layer at H = 4 km and a guide star at θ = 5′ from the object lead to s = θH = 5.8m. In these conditions, the 1-Hz servo actually in- creases the tilt variance: σ2∆α = [2.36, 1.03]×σ α for V = 10m/s, where the coefficients [rl, rt] = [2.36, 1.03] refer to the longitudi- nal and transverse directions. This situation is illustrated in Fig. C1. On the other hand, for the ground layer (s = 0) and V = 5.5m/s, we obtain rl = rt ≈ 0.1, i.e. a good tilt correction. Residual tilt errors of the VLT field stabilization system can be modeled. However, the input information for such a model must include the closed-loop servo response h̃(ν), altitude profiles of turbulence C2n(h) and L0(h), profiles of the wind speed and di- rection, and the geometry of the object and guide star. We do not have all this information for the data at hand. Instead, we evaluate the effects of the servo by assuming either rl = rt = 0 (perfect compensation) or rl = rt = 2 (un-correlated tilts). The first case is closer to reality when a large part of turbulence is near the ground. The wave-front structure function D(x, y) can be represented by the combination of the quadratic part caused by tilts Dtilt(x, y) and the tilt-removed part D0, D(x, y) = D (x, y)+D (x, y) = D (x, y)+σ ).(C7) When the field-stabilization servo is at work, the resulting SF Dg is c© 2005 RAS, MNRAS 000, 1–8 8 A. Tokovinin, M. Sarazin, & A. Smette Dg(x, y) = D0(x, y) + σ2α(rlx 2 + rty = D(x, y) + σ2α(rl − 1)x 2 + σ2α(rt − 1)y where it is assumed that the x-axis is directed to the guide star. With such orientation of the coordinates, we can neglect the cross- term proportional to xy, which otherwise would be required in the Eq. C8. Our two options (complete compensation, r = 0, or tilt doubling, r = 2) correspond to the subtraction or addition of Dtilt(x, y) from the VK model. The tilt variance σ2α is calculated with eqs. C1,C2. As an example, consider the case of file 5 (PAH1) on June 21/22. A good match between MASS and SH seeing (Fig. 2) and MASS profiles indicate that most of turbulence was located in a strong layer at 4 km. Our model leads to [rl, rt] = [2.4, 1.0] (as- sumed parameters: s = 5.8m, V = 10m/s, η = 0, L0 = 30m, seeing 1.0′′, ν0 = 1Hz). We compute the SF according to Eq. C8 and translate it to the PSF using Eq. 1. The resulting PSF at 8.6µm has a minimum FWHM of 0.37′′ and an ellipticity of 0.08, elon- gated towards the guide star at position angle −39◦. The actual FWHM was 0.38′′ with e = 0.1 elongated at −5◦. Thus, tilt aniso- plamatism can qualitatively explain the image elongation. REFERENCES Avila, R., Ziad, A., Borgnino, J. et al., 1997, JOSA(A), 14, 3070 Born, M. & Wolf, E., 1965, Principles of Optics. Pergamon Press: Oxford. Ch. 8, Eq. 17. Colavita, M.M., Shao, M., & Staehlin, D.H., 1987, Appl. Opt., 26, Davis, J., Lawson, P.R., Booth, A.J. et al. 1995, MNRAS, 273, Fusco, T. , Rousset, G., Rabaud, D. et al. 2004 J. of Opt. A, 6, 585 Kornilov, V., Tokovinin, A., Voziakova, O. et al., 2003, Proc. SPIE, 4839, 837 Maire, J., Ziad, A., Borgnino, J., Mourard, D. et al. 2006, A&A, 448, 1225 Mariotti, J.M. & Di Benedetto, G.P., 1984, in: Very Large Tele- scopes, Their Instrumentation and programs. Proc. IAU Coll. 79, Garching, 257. Nightingale, N.S. & Buscher, D.F., 1991, MNRAS, 251, 155 Roddier, F. in Progress in Optics, E. Wolf, ed., 19, p. 281, North- Holland, Amsterdam, 1981. Sasiela, R.J. Electromagnetic Wave Propagation in Turbulence. Springer-Verlag, Berlin, 1994 Sarazin, M. & Roddier, F., 1990, A&A, 227, 294 Tatarskii, V.I. Wave propagation in a turbulent medium. Dover Publ., Inc., New York: 1961. Tokovinin, A. 2002, PASP, 114, 1156 Ziad, A., Conan, R., Tokovinin, A. et al. 2000, Appl. Opt., 39, This paper has been typeset from a TEX/ LATEX file prepared by the author. c© 2005 RAS, MNRAS 000, 1–8 Introduction Experiment description From PSF to structure function Overview Conditions of the experiment VISIR data Shack-Hartmann data Results Conclusions Models of turbulence Correction for the missing flux Effect of the tip-tilt servo on the SF

0704.0471

Density dependence of the symmetry energy and the nuclear equation of state: A Dynamical and Statistical model perspective

Density dependence of the symmetry energy and the nuclear equation of state: A Dynamical and Statistical model perspective D.V. Shetty, S.J. Yennello, and G.A. Souliotis Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA (Dated: October 25, 2018) The density dependence of the symmetry energy in the equation of state of isospin asymmetric nuclear matter is of significant importance for studying the structure of systems as diverse as the neutron-rich nuclei and the neutron stars. A number of reactions using the dynamical and the statistical models of multifragmentation, and the experimental isoscaling observable, is studied to extract information on the density dependence of the symmetry energy. It is observed that the dynamical and the statistical model calculations give consistent results assuming the sequential decay effect in dynamical model to be small. A comparison with several other independent studies is also made to obtain important constraint on the form of the density dependence of the symmetry energy. The comparison rules out an extremely “ stiff ” and “ soft ” form of the density dependence of the symmetry energy with important implications for astrophysical and nuclear physics studies. PACS numbers: 21.30.Fe, 25.70.-z, 25.70.Lm, 25.70.Mn, 25.70.Pq I. INTRODUCTION The fundamental goal of nuclear physics is to under- stand the basic building blocks of nature - neutrons and protons - and the nature of interaction that binds them together into nuclear matter. Studying the nature of matter and the strength of nuclear interaction is key to understanding some of the fundamental problems such as, How are elements formed? How do stars explode into supernova? What kind of matter exists inside a neu- tron star? How are neutrons compressed inside a neutron star to density trillions of times greater than on earth ? What determines the density-pressure relation, the so- called equation of state? The key ingredient for constructing the nuclear equa- tion of state is the basic nucleon-nucleon interaction. Un- til now our understanding of the nucleon-nucleon inter- action has come from studying nuclear matter that is symmetric in isospin (neutron-to-proton ratio, N/Z ≈ 1) and matter found near normal nuclear density (ρo ≈ 0.16 fm−3). It is not known how far this understanding re- mains valid as one goes away from the normal nuclear density and symmetric nuclear matter. Various interac- tions used in “ ab initio ” microscopic calculations pre- dict different forms of the nuclear equation of state above and below the normal nuclear matter density, and away from the symmetric nuclear matter [1, 2, 3, 4, 5, 6]. As a result, the symmetry energy, which is the difference in en- ergy between the pure neutron matter and the symmetric nuclear matter, shows very different behavior above and below normal nuclear density [6] (see Fig. 1). In general, two different forms of the density depen- dence of the symmetry energy have been predicted. One, where the symmetry energy increases monotonically with increasing density (“ stiff ” dependence) and the other, where the symmetry energy increases initially up to nor- mal nuclear density and then decreases at higher densities (“ soft ” dependence). Constraining the form of the den- sity dependence of the symmetry energy is important not 0 0.5 1 ρ / ρ0 0 1 2 3 ρ / ρ0 SkLya var AV +δv+3-BF DD-TW DD-ρδ FIG. 1: (Color online) Symmetry energy as a function of den- sity predicted by microscopic “ ab initio ” calculations. The left panel shows the low-density region, while the right panel displays the high-density range. The figure is taken from Ref. only for a better understanding of the nucleon-nucleon interaction, and hence its extrapolation to the structure of neutron-rich nuclei [7, 8, 9, 10], but also for deter- mining the structure of compact stellar objects such as neutron stars [11, 12, 13, 14, 15, 16, 17]. For example, a “ stiff ” form of the density dependence of the symmetry energy is predicted to lead to a large neutron skin thick- ness compared to a “ soft ” dependence [8, 10, 18, 19, 20]. Similarly, a “ stiff ” dependence of the symmetry energy can result in rapid cooling of a neutron star, and a larger neutron star radius, compared to a “ soft ” density depen- dence of the symmetry energy [20, 21, 22]. The nuclear Equation Of State (EOS) is therefore a fundamental en- tity that determines the properties of systems as diverse as atomic nuclei and neutron stars, and the knowledge of which is of significant importance [16, 17, 23]. Experimentally, the best possible means of studying the nuclear equation of state at sub-normal nuclear den- http://arxiv.org/abs/0704.0471v1 sity is through intermediate-energy heavy-ion reactions [26, 27]. In this kind of reaction, an excited nucleus (the composite of the projectile and the target nucleus) expands to a sub-nuclear density and disintegrates into various light and heavy fragments in a process called multifragmentation. By studying the isotopic yield dis- tribution of these fragments one can extract important information about the symmetry energy and its density dependence. Current studies on the nuclear equation of state are limited to beams consisting of stable nuclei. It is hoped that in the future radioactive beam facilities such as, FAIR (GSI) [24], SPIRAL2 (GANIL) and FRIB (USA) [25] will provide tremendous opportunities for ex- ploring the nuclear EOS in regions never before studied (i.e., extreme isospin and away from normal nuclear den- sity). In this work, we have made an attempt to study the density dependence of the symmetry energy using two different theoretical approaches for studying multi- fragmentation, namely the dynamical and the statistical model approaches of multifragmentation. In section II, the isoscaling technique used to study the density de- pendence of the symmetry energy, and their different in- terpretations in terms of statistical and dynamical ap- proach, are presented. In section III and IV, a brief description of the experiment and the experimental re- sults are presented. The dynamical and the statistical approaches used to interpret the experimental results are presented in section V. A comparsion between the two approaches with other independent studies is presented in section VI. Finally, a discussion and summary, and conclusions are presented in section VII and VIII, re- spectively. II. SYMMETRY ENERGY AND THE ISOTOPIC YIELD DISTRIBUTION It has been shown from experimental measurements that the ratio of the fragment yields, R21(N ,Z), taken from two different multifragmentation reactions, 1 and 2, obeys an exponential dependence on the neutron num- ber (N) and the proton number (Z) of the fragments; an observation known as isoscaling [28, 29, 30]. The depen- dence is characterized by the relation, R21(N,Z) = Y2(N,Z)/Y1(N,Z) = Ce (αN+βZ) (1) Where, Y2 and Y1 are the fragment yields from the neutron-rich and the neutron-deficient systems, respec- tively. C is an overall normalization factor, and α and β are the parameters characterizing the isoscaling behavior. Isoscaling is also theoretically predicted by the dynam- ical [31, 32, 33, 34, 35] and statistical [36, 37, 38, 39] models of multifragmentation. In these models, the dif- ference in the chemical potential of systems with different neutron-to-proton ration (N/Z) is directly related to the isoscaling parameter α. The isoscaling parameter α, is related to the symmetry energy Csym, through the rela- tion, 4Csym where, Z1, A1 and Z2, A2 are the charge and the mass numbers from the two systems and T is the temperature. This relation provides a simple and straight-forward con- nection between the symmetry energy and the fragment isotopic yield distribution. It must be mentioned that although the above equation derived from the statistical and the dynamical models of multifragmentation appears similar in form, the physical meaning of the terms involved in this equation differ for each model. 1) In statistical models, the Z/A in Eq. (2) corresponds to the charge-to-mass ratio of the initial equilibrated frag- menting system. Whereas, in dynamical models, it cor- responds to the charge-to-mass ratio of the liquid phase at a certain time (≈ 300 fm/c) during the dynamical evo- lution of the colliding systems. 2) The interpretation of the symmetry energy Csym, in dynamical and statistical models also differs significantly. The dynamical models relate the symmetry energy in the above equation to that of the fragmenting source. The statistical models, on the other hand, relate Csym to that of the fragments formed at freeze-out. These conceptual differences between the statistical and the dynamical models are due to the radically differ- ent approaches taken in the interpretation of the multi- fragmentation process. The different interpretation has also lead to conflicting results from the use of Eq. 2, due to the different sequential decay effects predicted for the primary fragments by each model. The isoscaling parameter α, in Eq. 2 corresponds to the hot primary fragments which undergo sequential de- cay into cold secondary fragments. These secondary frag- ments are the ones that are eventually detected in exper- iments. The experimentally determined isoscaling pa- rameter must therefore be corrected for the sequential decay effect before comparing it to the theoretical mod- els. It has been observed that while statistical model calculations show no significant change in the isoscaling parameter after sequential decay [40], dynamical models give contrasting results; with some showing no significant changes [41], while others showing a change of as much as 50% [42]. In this work, we adopt both theoretical approaches with their respective interpretations to study the den- sity dependence of the symmetry energy. In particular, we use the Antisymmetrized Molecular Dynamics (AMD) model [31, 43] and the Statistical Multifragmentation Model (SMM) [36] for this study. A comparison between the two can provide useful insight into the physical mean- ing of the above equation in the two models. III. EXPERIMENT A. Experimental Setup The experiments were carried out at the Cyclotron Institute of Texas A&M University (TAMU) using the K500 Superconducting Cyclotron and the National Su- perconducting Cyclotron Laboratory (NSCL) at Michi- gan State University (MSU). Targets of 58Fe (2.3 mg/cm2) and 58Ni (1.75 mg/cm2) were bombarded with beams of 40Ar and 40Ca at 33 and 45 MeV/nucleon for the TAMU measurements [44], and targets of 58Fe (∼ 5 mg/cm2) and 58Ni (∼ 5 mg/cm2) were bombarded with beams of 40Ar and 40Ca at 25 and 53 MeV/nucleon for the NSCL measurements [45]. The various combinations of target and projectile nuclei allowed for a range of N/Z (neutron-to-proton ratio) (1.04 − 1.23) of the system to be studied, while keeping the total mass constant (A = 98). In a separate experiment at TAMU, beams of 58Ni and 58Fe at 30, 40, and 47 MeV/nucleon were also bom- barded on self-supporting 58Ni and 58Fe targets. The beams in the TAMU measurements were fully stripped by allowing them to pass through a thin alu- minum foil before being hit at the center of the target inside the TAMU 4π neutron ball [46]. Light charged particles (Z ≤ 2) and intermediate mass fragments (Z > 2) were detected using six discrete telescopes placed in- side the scattering chamber of the neutron ball at angles of 10◦, 44◦, 72◦, 100◦, 128◦ and 148◦. Each telescope con- sisted of a gas ionization chamber (IC) followed by a pair of silicon detectors (Si-Si) and a CsI scintillator detector, providing three distinct detector pairs (IC-Si, Si-Si, and Si-CsI) for fragment identification. The ionization cham- ber was of axial field design and was operated with CF4 gas at a pressure of 50 Torr. The gaseous medium was 6 cm thick with a typical threshold of ∼ 0.5 MeV/nucleon for intermediate mass fragments. The silicon detectors had an active area of 5 cm × 5 cm and were each sub- divided into four quadrants. The first and second silicon detectors in the stack were 0.14 mm and 1 mm thick, respectively. The dynamical energy range of the silicon pair was ∼ 16 - 50 MeV for 4He and ∼ 90 - 270 MeV for 12C. The CsI scintillator crystals that followed the silicon detector pair were 2.54 cm in thickness and were read out by photodiodes. Good elemental (Z) identifi- cation was achieved for fragments that punched through the IC detector and stopped in the first silicon detector. Fragments measured in the Si-Si detector pair also had good isotopic separation. Fragments that stopped in CsI detectors showed isotopic resolution up to Z = 7. The trigger for the data acquisition was generated by requir- ing a valid hit in one of the silicon detectors. The calibration of the IC-Si detectors were carried out using the standard alpha sources and by operating the IC at various gas pressures. The Si-Si detectors were cal- ibrated by measuring the energy deposited by the alpha particles in the thin silicon and the punch-through ener- gies of different isotopes in the thick silicon. The Si-CsI detectors were calibrated by selecting points along the different light charged isotopes and determining the en- ergy deposited in the CsI crystal from the energy loss in the calibrated Si detector. The setup for the NSCL experiment consisted of 13 silicon detector telescopes placed inside the MSU 4π Ar- ray. Four of which were placed at 14◦, each of which consisted of a 100 µm thick and a 1 mm thick silicon surface-barrier detector followed by a 20 cm thick plastic scintillator. Five telescopes were placed at 40◦, in front of the most forward detectors in the main ball of the 4π Array. They each consisted of a 100 µm surface-barrier detector followed by a 5 mm lithium drifted silicon de- tector. More details can be found in Ref. [45]. Good isotopic resolution was obtained as in TAMU measure- ments. B. Event Characterization The event characterization of the NSCL data was accomplished by detection of nearly all the coincident charged particles by the MSU 4π Array. Data were ac- quired using two different triggers; the bulk of which was obtained with the requirement of a valid event in one of the silicon telescopes. Additional data were taken with a minimum bias 4π Array trigger for normalization of the event characterization. The impact parameter of the event was determined by the mid-rapidity charge de- tected in the 4π Array as discussed in Ref. [47]. The effectiveness of the centrality cuts was tested by compar- ing the multiplicity of events from a minimum bias trigger with the multiplicity distribution when a valid fragment was detected at 40◦ [48]. The minimum bias trigger had a peak multiplicity of charged particles of one, whereas with the requirement of a fragment at 40◦, the peak of the multiplicity distribution increased to five. The event characterization for the TAMU data was accomplished by using the 4π neutron ball that sur- rounded the detector assembly. The neutron ball con- sisted of eleven scintillator tanks segmented in its me- dian plane and surrounding the vacuum chamber. The upper and the lower tank were 1.5 m diameter hemi- spheres. Nine wedge-shaped detectors were sandwiched between the hemispheres. All the wedges subtended 40◦ in the horizontal plane. The neutron ball was filled with a pseudocumene-based liquid scintillator mixed with 0.3 % (b.w.) of Gd salt (Gd 2-ethyl hexanoate). Scintilla- tions from thermal neutrons captured by Gd were de- tected by twenty 5-inch phototubes : five in each hemi- sphere, one on each of the identical 40◦ wedges and two on the forward edges. The efficiency with which the neu- trons could be detected is about 83%, as measured with a 252Cf source. The detected neutrons were used to differentiate be- tween the central and peripheral collisions. To un- derstand the effectiveness of neutron multiplicity as a centrality trigger, simulations were carried out us- ing a hybrid BUU-GEMINI calculations at various im- pact parameters for the 40Ca + 58Fe reaction at 33 MeV/nucleon. The simulated neutron multiplicity distri- bution was compared with the experimentally measured distribution. The multiplicity of neutron for the impact parameter b = 0 collisions was found to be higher than the b = 5 collision. By gating on the 10% highest neu- tron multiplicity events, one could clearly discriminate against the peripheral events. To determine the contributions from noncentral im- pact parameter collisions, neutrons emitted in coinci- dence with fragments at 44◦ and 152◦ were calculated at b = 0 fm and b = 5 fm. The number of events were ad- justed for geometrical cross sectional differences. A ratio was made between the number of events with a neutron multiplicity of at least six, calculated at b = 0 fm, and the number of events with the same neutron multiplicity at b = 5 fm. The ratios were observed to be 19.0 and 11.1 at 44◦ and 1.3 and 2.2 at 152◦ for 33 and 45 MeV/nucleon respectively. At intermediate angles, high neutron multi- plicities were observed to be outside the region in which b = 5 fm contributes significantly. At backward angles the collisions at b = 5 fm made a larger contribution to the neutron multiplicity. In addition to the neutron multiplicity distribution, the charge distribution of the fragments was also used to in- vestigate the contributions from central and mid-impact parameter collisions. The b = 5 collisions produced es- sentially no fragments with charge greater than three in the 44◦ telescope. In an earlier work [44], some analysis of the fragment kinetic energy and charge distributions were presented. It was shown that at a laboratory angle of 44◦ the ki- netic energy and the charge distributions are well repro- duced by the statistical model calculation. Using a mov- ing source analysis of the fragment energy spectra, it was also shown that the fragments emitted at backward an- gles originate from a target-like source, while those emit- ted at 44◦ originate primarily from a composite source. In this work, we will concentrate exclusively on data from the laboratory angle of 44◦, which corresponds to the cen- ter of mass angle ≈ 90◦, to study the symmetry energy and the isoscaling properties of the fragments produced. The choice of this angle enables one to select events which are predominantly central and undergo bulk multifrag- mentation. The contributions to the intermediate mass fragments from the projectile-like and target-like sources can therefore be assumed to be minimal. IV. EXPERIMENTAL RESULTS A. Fragment isotopic yield distribution The experimentally measured relative isotopic yield distributions for the Lithium (left) and Carbon (right) elements, in 58Ni + 58Ni (star symbols), 58Ni + 58Fe (square symbols), 58Fe + 58Ni (circle symbols) and 58Fe FIG. 2: Relative yield distribution of the fragments for the Lithium (left) and Carbon (right) isotopes in 58Ni + 58Ni (stars and solid lines), 58Fe + 58Ni (circles and dashed lines), 58Ni + 58Fe (squares and dashed lines), and 58Fe + 58Fe (tri- angles and dotted lines) reactions at various beam energies. + 58Fe (triangle symbols) reactions, are shown in Fig. 2 for beam energies of 30, 40 and 47 MeV/nucleon. Sim- ilarly, the isotopic yield distributions for Lithium (left), Berillium (center) and Carbon (right) elements, in 40Ca + 58Ni (star symbols), 40Ar + 58Ni (circle symbols) and 40Ar + 58Fe (square symbols) reactions, are shown in fig- ure 3 for beam energies of 25, 33 and 45 MeV/nucleon. The isotope distribution for each element in Fig. 3 shows higher fragment yield for the neutron rich iso- topes in 40Ar + 58Fe reaction (squares), which has the largest neutron-to-proton ratio (N/Z), in comparison to the 40Ca + 58Ni reaction (stars), which has the small- est neutron-to-proton ratio. The yields for the reaction, 40Ar + 58Fe (circles), which has an intermediate value of the neutron-to-proton ratio, are in between those of the other two reactions. A similar feature is also observed for the 58Ni + 58Fe, 58Fe + 58Ni and 58Fe + 58Fe reactions shown in Fig. 2. The fragment yield distributions there- fore show the isospin dependence of the composite system on the fragments produced in the multifragmentation re- action. One also observes that the relative difference in the yield distribution between the three reactions in each figure decreases with increasing beam energy. This is due to the secondary de-excitation of the primary fragments, a process that becomes important for systems with in- creasing neutron-to-proton ratio and excitation energy. FIG. 3: Relative yield distribution of the fragments for Lithium (left), Berillium (center) and Carbon (right) isotopes in 40Ca + 58Ni (stars and solid lines), 40Ar + 58Ni (circles and dashed lines), and 40Ar + 58Fe (squares and dotted lines) re- actions at various beam energies. B. Isotopic and Isotonic scaling As discussed in section II, the ratio of isotope yields in two different systems, 1 and 2, R21(N,Z) = Y2(N,Z)/Y1(N,Z), follows an exponential dependence on the neutron number (N) and the proton number (Z) of the isotopes in relation known as isoscaling. In Fig. 4, we show the isotopic yield ratio as a func- tion of neutron number N , for Ar + Fe, Ar + Ni and Ca + Ni systems at beam energies of 25, 33, 45 and 53 MeV/nucleon. The left column shows the ratio for the 40Ar + 58Fe and 40Ca + 58Ni pair of reaction and the right column shows the ratio for the 40Ar + 58Ni and 40Ca + 58Ni pair of reaction. One observes that the ratio for each element shows linear behavior in the logarithmic plot and aligns with the neighboring element quite well. This feature is observed for all the beam energies and both pairs of reactions studied. One also observes that the alignment of the data points varies with beam ener- gies as well as the pairs of reaction. To have a quantita- tive estimate of this variation, the ratio for each element (Z) was simultaneously fit using an exponential relation (shown by the solid lines) to obtain the slope parame- ter α. The values of the parameters are shown at the top of each panel in the figure. The value of the slope FIG. 4: Experimental isotopic yield ratios of the fragments as a function of neutron number N , for various beam energies. The left column correspond to 40Ar + 58Fe and 40Ca + 58Ni pair of reactions. The right column correspond to 40Ar + 58Ni and 40Ca + 58Ni pair of reactions. The different symbols correspond to Z = 3 (circles), Z = 4 (open stars), Z = 5 (triangles), Z = 6 (squares) and Z = 7 (filled stars) elements. The lines are the exponential fits to the data as explained in the text. parameter α is larger for the 40Ar + 58Fe and 40Ca + 58Ni reactions, which has a larger difference in the N/Z of the systems in the pair, compared to the 40Ar + 58Ni and 40Ca + 58Ni reactions, which has a smaller difference in the corresponding N/Z. The α value furthermore de- creases with increasing beam energy. A similar feature is also observed in Fe + Fe, Fe + Ni, Ni + Fe and Ni + Ni systems. Fig. 5 shows the isotope yield ratios and the isotone yield ratios for the Fe + Fe and Ni + Ni re- actions for the 30 MeV/nucleon beam energy. A relative comparison of how the isoscaling parameter α, evolves as a function of beam energy and the isospin of the system is shown in Fig. 6. The figure clearly shows that the α value decreases with beam energy from 25 MeV/nucleon to 53 MeV/nucleon. In addition, there is also a clear drop in the α values with the decrease of the N/Z values of the system. FIG. 5: Experimental isotope yield ratios (top) and isotone yield ratios (bottom) from 58Fe + 58Fe and 58Ni + 58Ni re- actions as a function of N and Z for 30 MeV/nucleon beam energy. The solid lines are fit to the data as discussed in the text. FIG. 6: Experimental isoscaling parameter α, as a function of the beam energy. The solid circles are for the 40Ar + 58Fe and 40Ca + 58Ni reactions. The open triangles are for 58Fe + 58Fe and 58Ni + 58Ni reactions. The solid stars are for 40Ar + 58Ni and 40Ca + 58Ni reactions. The open squares are for 58Fe + 58Ni and 58Ni + 58Ni reactions. V. THEORETICAL MODEL COMPARISON A. Dynamical AMD model The Antisymmetrized Molecular Dynamics (AMD) [31, 43] is a microscopic model that simulates the time evolution of a nuclear collision. The colliding system in this model is represented in terms of a fully antisym- metrized product of Gaussian wave packets. During the evolution, the wave packet centroids move according to the deterministic equation of motion. The followed state of the simulation branches stochastically and successively into a huge number of reaction channels. The interactions are parameterized in terms of an effective force acting be- tween nucleons and the nucleon-nucleon collision cross- sections. The advantage of using a dynamical model to study the nuclear equation of state is that it allows one to understand the functional form of the density depen- dence of the symmetry energy at a very fundamental level (i.e., from the basic nucleon-nucleon interaction). Recently [31], the fragment yields from heavy ion col- lisions simulated within the Antisymmetrized Molecular Dynamics (AMD) calculation were reported to follow a scaling behavior of the type shown in Eq. 1. A linear re- lation between the isoscaling parameter α and the differ- ence in the isospin asymmetry (Z/A)2 of the fragments (as given in Eq. 2), with appreciably different slopes, was predicted for two different forms of the density de- pendence of the symmetry energy; a “ stiff ” dependence (obtained from Gogny-AS interaction) and a “ soft ” de- pendence (obtained from Gogny interaction). In this section, we compare the experimentally de- termined isoscaling parameter with the predictions of the AMD model calculation. The isospin asymmetry of the fragments for the present systems was estimated at t = 300 fm/c of the dynamical evolution using the AMD calculation. The values for the fragment asym- metry (Z/A)2, were obtained by interpolating between those calculated for the 40Ca + 40Ca, 48Ca + 48Ca and 60Ca + 60Ca systems by Ono et al. [31]. These systems are symmetric and nearly similar in charge and mass as studied in the present work. Fig. 7 shows the AMD cal- culation of the fragment asymmetry, (Z/A)2 at t = 300 fm/c, as a function of initial asymmetry at time t = 0 fm/c, for two different choices of the nucleon-nucleon in- teraction, Gogny and Gogny-AS. The asymmetry values for the 40Ca + 40Ca, 48Ca + 48Ca and 60Ca + 60Ca sys- tems of Ref. [31] are shown by solid and hollow square symbols for the Gogny and Gogny-AS interaction, re- spectively. The lines are the linear fits to the calcula- tions. The interpolated values for the present systems are shown by the solid circles and triangles, and hollow circles and triangles for the Gogny and Gogny-AS inter- action, respectively. We note that the AMD calculations carried out in Ref. [31] and shown in Fig. 7 correspond to the beam energy of 35 MeV/nucleon. The interpolated values of the asym- metries for the present systems obtained from Fig. 7 are FIG. 7: AMD calculations of the fragment asymmetry (Z/A)2, at t = 300 fm/c for the Gogny (solid line and solid squares) and Gogny-AS (dotted line and hollow squares) in- teractions at 35 MeV/nucleon. The calculations are taken from Ref. [31] for the systems shown by the square symbols. The lines are linear fit to the square symbols. The other sym- bols are the interpolated values for the systems studied in this work. therefore for the beam energy of 35 MeV/nucleon. In or- der to compare the experimentally determined isoscaling parameter to that of the calculations, we therefore make use of the experimental isoscaling parameter for the beam energy of 35 MeV/nucleon using Fig. 6. Fig. 8 shows a comparison between the experimentally observed α and those from the AMD model calculations plotted as a function of the difference in the fragment asymmetry for the beam energy of 35 MeV/nucleon. The solid and the dotted lines are the AMD model predictions for the “ soft ” (Gogny) and the “ stiff ” (Gogny-AS) form of the density dependence of the symmetry energy, respectively. The solid and the hollow symbols (squares, stars, triangles and circles) are the results of the present study for the two different values of the fragment asym- metry, assuming Gogny and Gogny-AS interactions, re- spectively. Also shown in the figure are the scaling pa- rameters (asterisks, crosses, diamond and inverted tri- angle) taken from various other works in the literature [36, 49]. It is observed that the experimentally deter- mined α parameter increases linearly with increasing dif- ference in the fragment asymmetry of the two systems as predicted by the AMD calculation. Also, the data points are in closer agreement with those predicted by the Gogny-AS interaction (dotted line) than those from the usual Gogny force (solid line). In the above comparison between the data and the cal- FIG. 8: Isoscaling parameter α, as a function of the differ- ence in fragment asymmetry for 35 MeV/nucleon. The solid and the dotted lines are the AMD calculations for the Gogny and Gogny-AS interactions, respectively [31]. The solid and the hollow squares, stars, triangles and circles are from the present work as described in the text. The other symbols cor- responds to data taken from [49] (asterisks) and [36] (crosses, diamonds, inverted triangles). culation, the corrections for the isoscaling parameter α due to the sequential de-excitation of the fragments are not taken into account. The slightly lower values of the isoscaling parameters (symbols) from the present mea- surements with respect to the Gogny-AS values (dotted line) could be due to the small secondary de-excitation effect of the fragments not accounted for in this compar- ison. Recently, it has been reported by Ono et al. [42], that the sequential decay effect in the dynamical calcu- lations can affect the α value by as much as 50 %, and the ability to distinguish between the “ stiff ” and the “ soft ” form of the density dependence of the symmetry energy diminishes significantly. The calculations by Ono et al., were carried out for the above studied systems using the AMD model. However, dynamical calculation carried out by Tian et al. [41], using Isospin Quantum Molecular Dynamic (IQMD) model shows no significant difference between the primary and the secondary α. The sequential decay effect from the IQMD calculation was also carried out for the same systems and beam energy as studied by Ono et al. [42] using the AMD model. The contrasting results between the two dynamical calcula- tions for the same systems and energy currently present significant amount of uncertainty in reliably estimating the effect of sequential decay using dynamical models. One reason for this could be the large discrepancy that exists in the determination of the primary fragment exci- tation energy from the current dynamical models. It has been shown using another dynamical model (stochastic mean field calculation) (see Liu et al. [33]), that it re- quires a significantly lower value of the primary fragment excitation energy (by as much as 50%), to be able to reproduce the experimentally observed fragment isotope distribution. In the above comparison between the data and the cal- culation, we have therefore assumed the effect of the se- quential decay to be negligible. A correction of about 10 - 15 %, as determined and well established from various statistical model studies [28], results in a slight increase in the α values bringing them even closer to the dotted line. The observed agreement of the experimental data with the Gogny-AS type of interaction therefore appears to suggest a “ stiff ” form of the density dependence of the symmetry energy. Figure 9 shows various forms of the density dependence of the symmetry energy in isospin asymmetric nuclear matter used by Chen et al. [50], and those used in the present dynamical model analysis. The dot-dashed, dot- ted and the dashed curve corresponds to the momentum dependent Gogny interactions used by Chen et al., to ex- plain the NSCL-MSU isospin diffusion data. Assuming that the density dependence of the symmetry energy can be parametrized as, Csym(ρ) = C (MeV ) (3) where Cosym, is the value of the symmetry energy at sat- uration density and γ is the parameter that characterizes the stiffness of the symmetry energy, the above depen- dences used by Chen et al. can be written as, Esym ≈ 31.6 (ρ/ρ◦) γ , where, γ = 1.6, 1.05 and 0.69, respectively. The solid curve and the solid point in Fig. 9 correspond to those from the Gogny and Gogny-AS interactions used to study the isoscaling data in the present work. By parameterizing the density dependence of the sym- metry energy that explains the present isoscaling data, one obtains, Csym(ρ) ≈ 31.6 (ρ/ρ◦) γ , where γ = 0.69, from the dynamical model analysis. B. Statistical Multifragmentation Model The Statistical Multifragmentation Model (SMM) [51, 52] is based on the assumption of statistical equilibrium at a low density freeze-out stage. All breakup channels composed of nucleons and excited fragments are taken into account and considered as partitions. During each partition the conservation of mass, charge, energy, mo- mentum and angular momentum is taken into account, and the partitions are sampled uniformly in the phase space according to their statistical weights using Monte Carlo sampling. The Coulomb interaction between the fragments is treated in the Wigner-Seitz approximation. Light fragments with mass number A ≤ 4 are considered FIG. 9: Different forms of the density dependence of the nu- clear symmetry energy used in the dynamical analysis of the present measurements on isoscaling data and the isospin dif- fusion measurements of NSCL-MSU [50]. The curves are as described in the text. as elementary particles with only translational degrees of freedom (“nuclear gas”). Fragments with A > 4 are treated as heated nuclear liquid drops, and their individ- ual free energies FA,Z are parametrized as a sum of the volume, surface, Coulomb and symmetry energy. For the present study we make use of the SMM ver- sion adopted by Botvina et al. [36]. In this version, the secondary de-excitation of large fragments with A > 16 is described by Weisskopf-type evaporation and Bohr-Wheeler-type fission models [51, 53]. The decay of smaller fragments is treated with the Fermi-breakup model. All ground and nucleon-stable excited states of light fragments are taken into account and the popula- tion probabilities of these states are calculated according to the available phase space [53]. The sequential decay effect on the isoscaling parameter in this version of SMM has been established to be small and in good agreement with other versions of the statistical models. Unlike dynamical calculations, the form of the density dependence of the symmetry energy is not known a priori, but has to be deduced from the systematic correlations between the isoscaling parameter, temperature, symme- try energy and the density of the multifragmenting sys- tem. To build this correlation, we make use of the frag- ment yield distributions measured in 58Ni, 58Fe + 58Ni, 58Fe reactions at 30, 40 and 47 MeV/nucleon to study the isoscaling parameter α, as a function of the excita- tion energy of the fragmenting source. The parameter α was obtained from the ratio’s of the isotopic yields for two different pairs of reactions, 58Fe + 58Ni and 58Ni + 58Ni, and 58Fe + 58Fe and 58Ni + 58Ni as discussed in section 4.2. The excitation energy of the source for each beam energy was determined by simulating the initial stage of the collision dynamics using the Boltzmann-Nordheim- Vlasov (BNV) model calculation [54]. The results were obtained at a time around 40 - 50 fm/c after the projec- tile had fused with the target nuclei and the quadrupole moment of the nucleon coordinates (used for identifica- tion of the deformation of the system) approached zero. These excitation energies were also compared with those obtained from the systematic calorimetric measurements (see Ref. [55]) for systems with mass (A ∼ 100), and sim- ilar to those studied in the present work, and are in good agreement. Fig. 10 (a) shows the experimental isoscaling parameter α, as a function of the excitation energy for Fe + Fe and Ni + Ni, and Fe + Ni and Ni + Ni pairs of reactions. A systematic decrease in the absolute val- ues of the isoscaling parameter with increasing excitation energy is observed for both pairs. The α parameters for the 58Fe + 58Fe and 58Ni + 58Ni are about twice as large compared to those for the 58Fe + 58Ni and 58Ni + 58Ni pair of reactions. The experimental isoscaling parameter was compared with the predictions of the Statistical Multifragmentation Model (SMM) [51, 56] calculations to study their depen- dence on the excitation energy and the isospin content. The initial parameters such as, the mass, charge and ex- citation energy of the fragmenting source for the calcula- tion was obtained from the BNV calculations as discussed above. The possible uncertainties in the source param- eters due to the loss of nucleons during pre-equilibrium emission was accounted for by carrying out the calcula- tions for smaller source sizes. The break-up density in the calculation was taken to be multiplicity-dependent and was varied from approximately 1/2 to 1/3 the sat- uration density. This was achieved by varying the free volume with the excitation energy as shown in Ref. [51]. The form of the dependence was adopted from the work of Bondorf et al. [57, 58], (and shown by the solid curve in Fig. 10 (d)). It is known that the multiplicity-dependent break-up density, which corresponds to a fixed interfrag- ment spacing and constant pressure at break-up, leads to a pronounced plateau in the caloric curve [57, 58]. A constant break-up density would lead to a steeper tem- perature versus excitation energy dependence. The symmetry energy in the calculation was varied un- til a reasonable agreement between the calculated and the measured α was obtained. Fig. 10 (a) shows the com- parison between the SMM calculated and the measured α for both pairs of systems. The dashed curves corre- spond to the calculation for the primary fragments and the solid curves to the secondary fragments. The width in the curve is the measure of the uncertainty in the in- puts to the SMM calculation. It is observed that, within the given uncertainties, the decrease in the α values with increasing excitation energy and decreasing isospin dif- ference ∆(Z/A)2, of the systems is well reproduced by 4 6 8 10 0 2 4 6 8 10 E* (MeV/nucleon) 4 6 8 10 0 2 4 6 8 10 FIG. 10: (Color online) Isoscaling parameter α, temperature, symmetry energy and density as a function of excitation en- ergy for the Fe + Fe and Ni + Ni (inverted triangles), and Fe + Ni and Ni + Ni (solid circles) reactions at 30, 40 and 47 MeV/nucleon. a) Experimental isoscaling parameter as a function of excitation energy. The solid and the dashed curves are the SMM calculations as discussed in the text. b) Temperature as a function of excitation energy. The solid stars correspond to the measured values and are taken from Ref. [55]. The solid and the dashed curve corresponds to the Fermi-gas relation. The dotted curve corresponds to the one obtained from Eq. 4. c) Symmetry energy as a function of excitation energy. d) Density as a function of excitation en- ergy. The solid stars correspond to those from Ref. [69]. The open triangles are those from Ref. [70]. The solid curve is from Ref. [57]. the SMM calculation. One also notes that the effect of sequential decay on the isoscaling parameter is small as observed in several other studies [40, 59] using statistical models. We show in Fig. 10 (b), the temperature as a func- tion of excitation energy (caloric curve) obtained from the above SMM calculation that uses the excitation en- ergy dependence of the break-up density to explain the observed isoscaling parameters. These are shown by the solid and inverted triangle symbols. Also shown in the figure are the experimentally measured caloric curve data compiled by Natowitz et al. [55], from various measure- ments for this mass range. The data from these mea- surements are shown collectively by solid star symbols and no distinction is made among them. The Fermi-gas model predictions with inverse level density parameter Ko = 10 (solid and dashed curve), is also shown. It is evident from the figure that the temperatures obtained from the SMM calculations are in good agreement with the overall trend of the caloric curve. Somewhat lower value for the temperature is observed when the break-up density of the system is kept constant at 1/3 the nor- mal nuclear density. By allowing the break-up density to evolve with the excitation energy, a near plateau that agrees with the experimentally measured caloric curves is obtained. This assures that the input parameters used in the SMM calculation for comparing with the data are reasonable. The symmetry energies obtained from the statistical model comparison of the experimental isoscaling param- eter α, are as shown in Fig. 10 (c). A steady decrease in the symmetry energy with increasing excitation energy is observed for both pairs of systems. Such a decrease has also been observed in several other studies [60, 61, 62, 63]. We have also estimated the effect of the symmetry en- ergy evolving during the sequential de-excitation of the primary fragments [60, 64]. These are reflected in the large error bars shown in Fig. 10 (c). The phase diagram of the multifragmenting system is two dimensional and hence the excitation energy depen- dence of the temperature (the caloric curve) must take into account the density dependence too. Often this de- pendence is neglected while studying the caloric curve. In the following, we attempt to extract the density of the fragmenting system as a function of excitation energy. It has been shown by Sobotka et al. [65], that the plateau in the caloric curve could be a consequence of the ther- mal expansion of the system at higher excitation energy and decreasing density. By assuming that the decrease in the breakup density, as taken in the present statisti- cal multifragmentation calculation, can be approximated by the expanding Fermi gas model, and furthermore the temperature in Eq. 2 and the temperature in the Fermi- gas relation are related, one can extract the density as a function of excitation energy using the relation Ko(ρ/ρo)2/3E∗ (4) 0 0.5 1 1.5 FIG. 11: (Color online) Symmetry energy as a function of density for the Fe + Fe and Ni + Ni pair of reaction (inverted triangles), and Fe + Ni and Ni + Ni pair of reactions (solid circles) for the 30, 40 and 47 MeV/nucleon. The solid curve is the dependence obtained form the dynamical model analysis as explained in the text. In the above expression, the momentum and the fre- quency dependent factors in the effective mass ratio are taken to be one as expected at high excitation energies and low densities studied in this work [66, 67, 68]. The resulting densities for the two pairs of systems are shown in Fig. 10 (d) by the solid circles and inverted triangles. For comparison, we also show the break-up densities obtained from the analysis of the apparent level density parameters required to fit the measured caloric curve by Natowitz et al. [69], and those obtained by Viola et al. [70] from the Coulomb barrier systematics that are required to fit the measured intermediate mass fragment kinetic energy spectra. One observes that the present re- sults obtained by requiring to fit the measured isoscaling parameters and the caloric curve are in good agreement with those obtained by Natowitz et al. The figure also shows the fixed freeze-out density of 1/3 (dashed line) and 1/6 (dotted line) of the saturation density assumed in various statistical model comparisons. The caloric curve obtained using the above densities and excitation ener- gies (shown by solid stars, circles and the triangles) with Ko = 10 in Eq. 4, is shown by the dotted curve in Fig. 10 (b). The small discrepancy between the dotted curve and the data (solid stars) below 4 MeV/nucleon is due to the approximate nature of Eq. 4 being used. It is therefore evident from figure 10 (a), (b), (c) and (d) that the decrease in the experimental isoscaling pa- rameter α, symmetry energy, break-up density, and the flattening of the temperature with increasing excitation energy are all correlated. One can thus conclude that the expansion of the system during the multifragmentation process leads to a decrease in the isoscaling parameter, decrease in the symmetry energy and density, and the flattening of the caloric curve. TABLE I: Parameterized form of the density dependence of the symmetry energy obtained from various independent studies. Reference Parametrization Studies Fuchs et al. [85] 32.9(ρ/ρo) 0.59 Relativistic Dirac-Brueckner calculation Heiselberg et al. [82] 32.0(ρ/ρo) 0.60 Variational calculation Danielewicz et al. [81] 31(33)(ρ/ρo) 0.55(0.79) BE, skin, isospin analog states Tsang et al. [79] 12.125(ρ/ρo) 2 Isospin diffusion Chen et al. [50] 31.6(ρ/ρo) 1.05 Isospin diffusion Li et al. [80] 31.6(ρ/ρo) 0.69 Isospin diffusion Piekarewicz et al. [77, 78] 32.7(ρ/ρo) 0.64 Giant resonances Shetty et al. [73, 74, 75] 31.6(ρ/ρo) 0.69 Isotopic distribution Famiano et al. [87] 32.0(ρ/ρo) 0.55 neutron-proton emission ratio Tsang et al. [28] 23.4(ρ/ρo) 0.6 Isotopic distribution From the above correlation between the symmetry en- ergy as a function of excitation energy, and the density as a function of excitation energy, we obtain the sym- metry energy as a function of density. This is shown by the inverted triangles and solid circles in Fig. 11 for the Fe + Fe and Ni + Ni, and the Fe + Ni and Ni + Ni pair of reactions. The temperature in the present work remains nearly constant for the range of excitation ener- gies studied, the observed decrease in the symmetry en- ergy with increasing excitation energy is therefore a con- sequence of decreasing density. This is also supported by microscopic calculations which shows an extremely slow evolution of the symmetry energy with temperature [71, 72]. The evolution is practically negligible for the temperature range studied in this work. The solid curve in Fig. 11 corresponds to the dependence Csym(ρ) = 31.6 (ρ/ρ◦) 0.69 MeV, obtained from the dynamical Anti- symmetrized Molecular Dynamic (AMD) calculation, as discussed in the previous section. It is thus observed that the dynamical and statistical models lead to similar den- sity dependence of the symmetry energy. VI. COMPARISON WITH OTHER INDEPENDENT STUDIES In the following, we compare the form of the den- sity dependence of the symmetry energy obtained from the present experimentally measured isoscaling param- eter using the statistical and the dynamical multifrag- mentation models with several other recent independent studies. Fig. 12 shows this comparison. The green solid curve corresponds to the one obtained from Gogny-AS interaction in dynamical AMD model that explains the present results [73, 74], assuming the sequential decay effect to be small. The inverted triangle and the circle symbols also correspond to the present measurements ob- tained by comparing with the statistical multifragmenta- tion model [75]. The red dashed curve corresponds to the one obtained recently from an accurately calibrated rela- tivistic mean field interaction, used for describing the Gi- ant Monopole Resonance (GMR) in 90Zr and 208Pb, and the IVGDR in 208Pb by Piekarewicz et al. [76, 77, 78]. The pink dot-dashed curve correspond to the one used to explain the isospin diffusion results of NSCL-MSU us- ing the isospin dependent Boltzmann-Uehling-Uhlenbeck (IBUU) model by Tsang et al. [79]. The blue dot-dashed curve also corresponds to the one used for explaining the isospin diffusion data of NSCL-MSU by Chen et al. [50], but with the momentum dependence of the interaction included in the IBUU calculation. This dependence has been further modified to include the isospin dependence of the in-medium nucleon-nucleon cross-section by Li et al. [80], and is in good agreement with the present study. The shaded region in the figure corresponds to those ob- tained by constraining the binding energy, neutron skin thickness and isospin analogue state in finite nuclei us- ing the mass formula of Danielewicz [81]. The yellow solid curve correspond to the parameterization adopted by Heiselberg et al. [82] in their studies on neutron stars. By fitting earlier predictions of the variational calcula- tions by Akmal et al. [83, 84], where the many-body and special relativistic corrections are progressively incorpo- rated, Heiselberg and Hjorth-Jensen obtained a value of Cosym = 32 MeV and γ = 0.6, similar to those obtained from the present measurements. A similar result is also obtained from the relativistic Dirac-Brueckner calcula- tion, with Cosym = 32.9 MeV and γ = 0.59 [85]. The Dirac-Brueckner is an “ab-initio” calculation based on nucleon-nucleon interaction with Bonn A type potential instead of the AV18 potential used in the variational cal- culation of Ref. [84]. The density dependence of the sym- metry energy has also been studied in the framework of expanding emitting source (EES) model by Tsang et al. [28], where a power law dependence of the form Csym(ρ) = 23.4(ρ/ρo) γ , with γ = 0.6 was obtained. This depen- dence (shown by the black dotted curve) is significantly softer than other dependences shown in the figure. The solid square point in the figure correspond to the value of symmetry energy obtained by fitting the experimental differential cross-section data in a charge exchange reac- tion using the isospin dependent CDM3Y6 interaction of FIG. 12: (Color online) Comparison between the results on the density dependence of the symmetry energy obtained from various different studies. The various curves and the symbols are described in the text. the optical potential by Khoa et al. [86]. An alternate observable, the double neutron/proton ratio of nucleons taken from two reaction systems using four isotopes of the same element, has recently been pro- posed as a probe to study the density dependence of the symmetry energy [87]. This observable is expected to be more robust than the isoscaling observable. It was shown recently [87] that the experimentally determined double- ratio for the 124Sn + 124Sn reaction to that for the 112Sn + 112Sn reaction, results in a dependence with γ = 0.5 (shown by black dashed curve in Fig. 12), when com- pared to the predictions of the IBUU transport model calculations. This observation is in close agreement with other studies discussed above. However, this dependence has been obtained by using the momentum independent calculation of Ref. [88]. A more recent calculation [89] using a BUU transport model that includes momentum dependent interaction show significantly lower values for the double neutron/proton ratio of free nucleons com- pared to the one reported by Famiano et al. The parameterized forms of the density dependence of the symmetry energy obtained from all the above men- tioned studies are as given in Table I. The close agree- ment between various independent studies show that a constraint on the density dependence of the symmetry energy, given as Csym(ρ) = C sym(ρ/ρo) γ , where Cosym ≈ 31 - 33 MeV and γ ≈ 0.55 - 0.69 can be obtained. VII. DISCUSSION We make the following observations from the above comparison between the statistical and the dynamical model analysis : 1) Assuming a negligibly small sequential decay effect, the form of the density dependence of the symmetry en- ergy obtained from the dynamical model analysis is in good agreement with the one obtained from the statistical model analysis: As mentioned earlier, the sequential de- cay effect among various dynamical model calculations is still a subject of debate. The statistical models however consistently show small sequential decay effect. If the se- quential decay in both the dynamical and the statistical model is determined by the excitation energy, charge (Z) and mass (A) of the fragments, and not by the process that leads to these fragments, the de-excitation of the fragments must lead to a same amount of change in the isoscaling parameter (either a large change or no change at all). It is therefore unrealistic to assume that the se- quential decay effect is different in the dynamical and the statistical model calculations. One comparison by Hudan et al. [90], show good agreement between the experimen- tally determined primary fragment excitation energy and those calculated using the statistical multifragmentation model (see table II of Ref. [90]). Furthermore, if dy- namical and statistical models are merely two different ways of interpreting the same multifragmentation process (i.e., one simulating the entire process from the forma- tion to the breakup stage, and the other simulating only the later breakup stage), the isoscaling parameter from both interpretation must lead to consistent results. It is well known, and as discussed in section II, that both interpretations predict isoscaling in multifragmentation. As discussed in section V A, the apparent disagreement between the sequential decay effect in statistical and dy- namical models, could be due to the large discrepancy that exists in the determination of the primary fragment excitation energy from current dynamical models. It has been argued [91] that the effect of sequential decay on the isoscaling parameter α, in statistical multi- fragmentation model depends not only on the excitation energy but also on the value of the symmetry energy. The fragments in their primary stage are usually hot and the properties of hot nuclei (i.e., their binding energy and mass) differ from those of cold nuclei. If hot fragments in the freeze-out configuration have smaller symmetry energy, their mass at the beginning of the sequential de- excitation will be different and this effect should be taken into account. At smaller values of the symmetry energy the sequential decay effect can be large. In order to study the effect of symmetry energy evolution on isoscaling pa- rameter during sequential decay, we have adopted in this work a phenomenological approach of Buyukcizmeci et al. [64]. Fig. 13 and 14 shows the primary and the sec- ondary isoscaling parameter as a function of symmetry energy calculated from the statistical multifragmentation model (SMM) for the Ar + Ni and Ca + Ni, and Ar + Fe and Ca + Ni pair of reactions. The various panels from top to bottom correspond to different system excitation energies. Fig. 13 shows the result of the calculations where the symmetry energy is kept fixed, and Fig. 14 shows the result for the calculations where the symme- try energy is varied during the de-excitation process. The dashed lines in the figure correspond to the primary frag- ments (Eq. 2) and the solid lines to the secondary frag- ments. It is observed that there is no significant change in the primary and the secondary alpha. 2) The result of the statistical model analysis is in good agreement with other independent studies : A comparison between the density dependence of the symmetry energy obtained from the statistical model analysis (for which the sequential decay effect is known to be small) and other independent studies shows excellent agreement. 3) The isoscaling parameter probes the property of in- finite nuclear matter : The symmetry energy obtained from dynamical model analysis (shown by the solid curve in Fig. 11) relates to the volume part of the symmetry en- ergy as in infinite nuclear matter, whereas, the symmetry energy obtained from the statistical model analysis (solid circles and inverted triangles in Fig. 11) relates to the FIG. 13: SMM calculated isoscaling parameter α as a func- tion of symmetry energy for various excitation energies. The open circles joined by dotted lines correspond to the primary fragments and the open stars joined by solid lines to the sec- ondary fragments. The left column shows the calculation for 40Ar + 58Ni and 40Ca + 58Ni pair, and the right column for the 40Ar + 58Fe and 40Ca + 58Ni pair. fragment that is finite and has surface contribution. The similarity between the two can probably be understood in terms of the weakening of the surface symmetry free energy when the fragments are being formed. During the density fluctuation in uniform low density matter, the fragments are not completely isolated and continue to interact with each other, resulting in a decrease in the surface contribution as predicted by Ono et al. [43]. It must be mentioned that by fitting the nuclear masses with mass formula, a volume contribution to the symme- try energy of about 31 - 33 MeV and surface contribution of about 11 - 13 MeV was obtained by Danielewicz [92] for nuclei at normal density. Using the constraint ob- tained for the volume part of the symmetry energy from the present study, and following the expression for the symmetry energy of finite nuclei by Danielewicz, we write the general expression for the density dependence of the symmetry energy as, SA(ρ) = α(ρ/ρ◦) 1 + [α(ρ/ρ◦)γ/βA1/3] where, α ≡ Cosym = 31 - 33 MeV, γ = 0.55 - 0.69 and α/β = 2.6 - 3.0. The quantities α and β are the volume and the surface symmetry energy at normal nuclear den- sity. The above equation reduces to Eq. 3 for infinite FIG. 14: Same as in fig. 13, but with the modified secondary de-excitation with evolving symmetry energy. nuclear matter in the limit of A → ∞, and to the sym- metry energy of finite nuclei for ρ = ρ◦. The ratio of the volume symmetry energy to the surface symmetry energy (α/β) is closely related to the neutron skin thickness. De- pending upon how the nuclear surface and the Coulomb contribution is treated, two different correlations between the volume and the surface symmetry energy have been predicted [17] from fits to nuclear masses. Experimental masses and neutron skin thickness measurements for nu- clei with N/Z > 1 should provide tighter constraint on the surface-volume correlation. 4) The density dependence of the symmetry energy ob- tained using the statistical model approach is consistent with other experimentally determined observables : In the past, attempts have been made to study the density de- pendence of the symmetry energy by looking at specific observables and comparing them with the predictions of the dynamical models. Such an approach attempts to ex- plain the observable of interest without trying to simul- taneously explain other properties, such as, the temper- ature, density and excitation of the fragmenting system. This has lead to a variety of different dependences with- out any clue to what density is being probed. It might not be straightforward to distinguish different realistic EOS interactions using dynamical models, due to the large uncertainties that currently exist in the sequential decay effects for these models. But the idea of extracting infor- mation on the symmetry energy from the point of view of the basic nucleon-nucleon interaction is a very pow- erful approach. On the other hand, the determination of the density dependence of the symmetry energy from statistical model analysis by simultaneously explaining the isoscaling parameter, caloric curve and the density as a function of excitation energy is a reverse approach. This approach attempts to explain the experimental ob- servables without any prior knowledge of the governing interaction and arrives at a dependence which can then be compared with those predicted from the basic inter- actions. 5) Symmetry energy determined from the multifrag- mentation study is lower than that of normal nuclei : Theoretical many-body calculations [1, 93, 94, 95] and those from the empirical liquid-drop mass formula [96, 97] predict symmetry energy near normal nuclear density and temperature to be around 30 - 32 MeV. As- suming a negligible evolution of the symmetry energy as a function of temperature, as shown in Ref. [71, 72], the present statistical model analysis yields symmetry energy of the order of 18 - 20 MeV at half the normal nuclear density. 6) The above constraint on the density dependence of the symmetry energy has important implications for as- trophysical and nuclear physics studies : a) Neutron skin thickness: It has been shown [98] that an empirical fit to a large number of mean-field calcu- lations yield neutron skin thickness for 208Pb nucleus, Rn - Rp ≃ (0.22γ + 0.06) fm, where γ is the exponent that determines the stiffness of the density dependence of the symmetry energy. From the above comparison, as- suming only those density dependences of the symmetry energy which have symmetry energy at normal nuclear density 31 - 33 MeV, one obtains a neutron skin thick- ness of 0.18 - 0.21 fm. An accurate determination of the neutron skin thickness from the parity-violating elec- tron scattering measurement [18] will provide a unique observational constraint on the thickness of the neutron skin of a heavy nucleus. The above constraint also leads to symmetric matter nuclear compressibility of K ∼ 230 b) Neutron star mass and radius : The constraint also predicts a limiting neutron star mass of Mmax = 1.72 solar mass and a radius, R = 11 - 13 km for the “ canon- ical ” neutron star. Recent observations of pulsar-white dwarf binaries at the Arecibo observatory suggest a pul- sar mass for PSRJ0751+1807 of M = 2.1+0.4 −0.5 solar mass at a 95% confidence level [99]. c) Neutron star cooling : Furthermore, it predicts a direct URCA cooling for neutron stars above 1.4 times the solar mass. If such is the case, then the enhanced cooling of a M = 1.4 solar mass neutron star may provide strong evidence in favor of exotic matter in the core of a neutron star. These results have important implications for nuclear astrophysics and future experiments probing the prop- erties of nuclei using beams of neutron-rich nuclei. The above constraint was obtained by studying the low den- sity behavior of nuclear matter. Measurements at den- sities above normal nuclear matter should further con- straint the form of the symmetry energy. Such measure- ments should yield consistent results when extrapolated to low densities. VIII. SUMMARY AND CONCLUSIONS In summary, a number of reactions were studied to in- vestigate the density dependence of the symmetry energy in the equation of state of isospin asymmetric nuclear matter. The results were analyzed within the framework of the dynamical and the statistical models of multifrag- mentation. It is observed that a dependence of the form Csym(ρ) = 31.6 (ρ/ρo) 0.69 MeV, agrees reasonably with the experimental data indicating that a “ stiff ” form of the symmetry energy provides a better description of the nuclear matter EOS at sub-nuclear densities. A compar- ison with several other independent studies shows that a constraint on the density dependence of the symmetry energy given as Csym(ρ) = C sym(ρ/ρo) γ , where Cosym ≈ 31 - 33 MeV and γ ≈ 0.55 - 0.69, can thereby be obtained. The present observation has important implications for astrophysics, as well as, nuclear physics studies to be car- ried out at future radioactive beam facilities worldwide. IX. ACKNOWLEDGMENTS This work was supported in part by the Robert A. Welch Foundation through grant No. A-1266, and the Department of Energy through grant No. DE-FG03- 93ER40773. We also thank A. Botvina for the SMM code, and the Catania group for the BNV code. [1] A.E.L. Dieperink, Y. Dewulf, D. Van Neck, M. Waro- quier, and V. 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0704.0472

Competitive nucleation and the Ostwald rule in a generalized Potts model with multiple metastable phases

entropic.tex Competitive nucleation and the Ostwald rule in a generalized Potts model with multiple metastable phases David P. Sanders,∗ Hernán Larralde, and François Leyvraz Instituto de Ciencias Fı́sicas, Universidad Nacional Autónoma de México, Apartado postal 48-3, 62551 Cuernavaca, Morelos, Mexico (Dated: November 4, 2018) We introduce a simple nearest-neighbor spin model with multiple metastable phases, the number and decay pathways of which are explicitly controlled by the parameters of the system. With this model we can construct, for example, a system which evolves through an arbitrarily long succession of metastable phases. We also construct systems in which different phases may nucleate competitively from a single initial phase. For such a system, we present a general method to extract from numerical simulations the individual nucleation rates of the nucleating phases. The results show that the Ostwald rule, which predicts which phase will nucleate, must be modified probabilistically when the new phases are almost equally stable. Finally, we show that the nucleation rate of a phase depends, among other things, on the number of other phases accessible from it. PACS numbers: 64.60.My, 64.60.Qb, 05.10.Ln, 05.50.+q Metastability is a ubiquitous phenomenon in nature. Broadly speaking, it occurs when a system is “trapped” in a phase different from equilibrium. This non-equilibrium phase, the metastable state, can last for extremely long times. Thus, it is not surprising that metastable states play a cru- cial role in many physical processes and are at the center of much current research. For example, recently an intermedi- ate metastable phase was shown to provide an easier pathway for the growth of crystal nuclei from fluids (nucleation), with implications for the crystallization of proteins [1, 2]. Proteins themselves are known to get stuck in misfolded metastable structures [3], preventing them from reaching their equilib- rium configuration. The phenomenology observed in these and in many other systems can be thought of as arising from a complicated energy landscape, with several local “metastable minima” where the trapping occurs [4]. The extreme situation is that of glasses, in which the energy landscape can have ex- temely many local minima hindering relaxation of the system to a thermodynamically stable crystal [5]. The above systems present at least several metastable states. These states and the transitions between them usually arise from the microscopic interactions in a complicated way. When this is the case, the study of phenomena such as compe- tition between nucleating phases and specific nucleation path- ways may be obscured. In view of this, in this work we present a simple spin model with nearest-neighbour interac- tions, where the number of metastable phases and the decay pathways between them can be explicitly specified by varying the model parameters. It thus serves as a test-bed for theoreti- cal results relating to systems with multiple metastable phases [6, 7, 8], just as the kinetic Ising model, a special case of our model, has been central in the study of systems with a sin- gle metastable phase [9]. As discussed below, the model also describes the adsorption of multiple chemical species onto a surface, an interesting physical problem in its own right. After presenting the model, as an illustration of a possible application, we construct a system with arbitrarily long suc- cessions of metastable states. We then focus on competition between phases nucleating from a single initial metastable phase. An important question in this context is to understand which phases nucleate under which conditions. The Ostwald rule states that the nucleating phase is the one with the small- est free energy barrier from the initial phase: see Ref. [10] and references therein. Previous results have supported this prediction [11]. We show that in general the Ostwald rule must be modified probabilistically when the new phases are of similar stabil- ity, using an argument based on individual nucleation rates of each phase. We give a method by which these rates can be measured in simulations or experiments, and show that there is a parameter regime in which any of the new phases may nucleate—only the nucleation probability of each phase can be established, with the outcome in any given run being un- predictable. We finally show that the nucleation probability of a phase depends on the phases accessible from it. Model details:- Our model is based on the Potts model, in which each spin has one of q states [12] and each phase has a majority of spins in one state; the Ising model corresponds to q = 2. The relative stability of each phase is controlled by ex- ternal fields, and the interplay of these fields with interactions between different spin states allows us to obtain any desired transition pathways between phases. Viewing the fields as chemical potentials, we can recast the model as a multi-component lattice gas which describes ad- sorption on a lattice substrate (e.g., a crystal plane) of mul- tiple chemical species with lateral interactions [13]. Much experimental work has been done on the thermodynamics of such systems, but little on the kinetics—see [14] and refer- ences therein; nonetheless, our results should be testable in that context. A more complicated system where the kinetics has been characterized is a colloid–polymer system [15, 16], where possible pathways were found from considerations of the free energy landscape [17]. Our approach is complemen- tary in that specific pathways result from microscopic interac- tions. We work on an L×L square lattice with N := L2 spins and periodic boundary conditions, although the results are qualiti- tavely unaffected by lattice type. Each lattice site i has a spin http://arxiv.org/abs/0704.0472v3 σi taking values in {1, . . . ,q}, and the energy of a configura- tion ß is given by the Hamiltonian H(ß) :=− ∑ 〈i, j〉 Jσi,σ j − hαMα . (1) Here, Mα := ∑i δσi,α is the magnetisation (=number of spins) of the spin type α; δα ,γ = 1 if α = γ , and 0 otherwise. The first term is a sum over nearest-neighbor pairs of spins of a symmetric interaction energy Jα ,γ = Jγ,α , and the second de- scribes the effect of external fields hα acting on spin type α . We set the diagonal elements Jα ,α of the interaction ma- trix to unity (Jα ,α := 1 for all α), so that in the absence of non-diagonal interactions and fields, the model reduces to the standard Potts model [12]. This has q symmetrical phases coexisting below a critical temperature Tc = 1/(ln(1+ each phase has a majority of spins in one of the q spin states. Including fields breaks the symmetry between phases. If hα = hγ , then the α and γ phases coexist, with a first-order phase transition between them, for T < Tc; this is at the ori- gin of metastability. Weak non-diagonal interactions do not qualititavely affect this coexistence. To evolve the system we choose discrete-time Metropolis dynamics [18]: at each time step, a spin and its new value are chosen at random, the increment ∆H of the Hamiltonian (1) for this change is calculated, and the update is accepted with probability min{1,exp(−β ∆H)}, where β := 1/T is the inverse temperature. This gives a Markov chain on the space of all possible configurations. This Markov chain has a unique equilibrium distribution, concentrated on the phase(s) with the largest hα . The other phases are metastable, that is, when started in such a phase α , the system stays there for some time, before a transition to a more stable phase γ is nucleated by the appearance of a crit- ical droplet of the γ phase. At sufficiently low temperatures, the relative stability is determined by hγ > hα . The reverse transition is exponentially unlikely. In the standard Potts model, the equilibrium phase (almost) always nucleates. To obtain non-trivial transition pathways, nucleation of other phases must be promoted. This we achieve using non-diagonal interactions between distinct spin types α 6= γ: setting Jα ,γ > 0 favors nucleation of γ droplets inside the α phase by lowering the surface tension between α and γ regions, and hence decreasing the droplet free energy of for- mation (nucleation barrier), whereas formation of γ droplets in the α phase is suppressed if Jα ,γ < 0. We can now construct models whose phases obey arbitrary metastable transition graphs. These are directed graphs with the restriction that no loops returning to a previously visited phase are allowed. Each vertex corresponds to one phase, la- beled by its dominant spin state, and each arrow to a desired transition: α → γ means that phase γ can nucleate directly from phase α . Fig. 1 shows example transition graphs. To construct a model corresponding to a given transition graph, we proceed as follows. The number of spin types, q, is the number of vertices in the graph. To each spin type α we 2 3 4 FIG. 1: Metastable transition graphs: (a) kinetic Ising model; (b) suc- cession of 3 phases; (c) single metastable phase decaying to two com- peting phases; (d) as in (c), but such that one phase can decay further; (e) three competing phases. 0 10000 20000 30000 time (Monte Carlo steps per site) 1 2 3 4 5 FIG. 2: (Color online) Time dependence of magnetisation per site Mα/N of each phase α , in a single run of the model with transition graph 1 → 2 → 3 → 4 → 5 (a succession of phases). Parameters are L = 50, β = 1.25, hα = 0.1(α −1), K1 = 0.1, and K2 = 1.0. Labels denote the dominant phase. Configuration snapshots depict post- critical nuclei of each new phase embedded in the previous phase. These grow to fill the system, producing the next phase in sequence. assign a field hα , with hγ > hα if γ is below α in the graph. The off-diagonal interactions are given by Jα ,γ := K1 > 0 (at- tractive) if α → γ , and Jα ,γ :=−K2 < 0 (repulsive) otherwise. K2 must be large enough to inhibit immediate formation of non-adjacent phases with large fields. As an illustration, we construct a model exhibiting a lin- ear succession of metastable phases with transition graph 1 → 2 → ··· → q. We impose fields 0 = h1 < h2 < · · · < hq and attractive interactions Jα ,α±1 := K1 > 0 between neigh- bouring states, and set all other non-diagonal interactions to −K2 < 0. With suitable, moderately robust, parameters, we observe the desired behavior, shown for q = 5 in Fig. 2. A three-phase succession was previously observed in a ki- netic Blume–Capel model [19, 20], corresponding to a special case of our model with q = 3 [21]. The physical reason for the observed transitions is, however, much more transparent with the Hamiltonian in the form (1), with its intuitive interpreta- tion in terms of attractive and repulsive interactions. Competitive nucleation:- We now turn to the decay of one metastable phase into two competing phases (Fig. 1(c)). Sear studied competitive heterogeneous nucleation (occurring on impurities) in the 3-state standard Potts model [11]. In con- trast, all behaviors discussed in this work are endogenous: ob- 0 0.1 0.2 0.3 0.4 0.5 (h3 −h2)/h2 0 50 100 150 β = 1.25 β = 1.40 L = 150 L = 200 h3 = 0.105 h3 = 0.11 FIG. 3: (Color online) Probability p2(h3) that phase 2 nucleates be- fore phase 3 as a function of ∆h/h2, with h2 = 0.1, K1 = 0.1 and K2 = 1, and β = 1.25, L = 50 unless otherwise noted. Up to n = 104 trials were used for each data point; statistical errors are of the or- der of the symbol size. Inset: system-size dependence of p2 for two values of h3 with fixed parameter values. served transitions are not caused by external influences, but rather arise spontaneously from within the system itself. We fix 0 = h1 < h2 ≤ h3, J1,2 = J1,3 > 0 and J2,3 < 0. Let ∆h := h3 − h2 be the field difference between the new phases, 2 and 3. When ∆h = 0, these phases are symmetrical, each nucleating half of the time, while for ∆h > 0, we expect the 1–3 free energy barrier to be lower than the 1–2 one, so that according to the Ostwald rule, only phase 3 should nucleate. To test this, we perform n simulations starting from phase 1 for each ∆h, in n2 of which phase 2 nucleates before phase 3. The ratio n2/n is then an estimate of the probability p2(∆h) that phase 2 nucleates first. For efficiency, we use a rejection- free version of the Metropolis method [21, 22]. Fig. 3 plots p2 as a function of a non-dimensionalised ∆h. For ∆h sufficiently close to 0, phase 2 can still nucleate first, contrary to the simple Ostwald rule. The probability that it does so rapidly decreases for larger ∆h, until a point beyond which phase 3 effectively always nucleates. To explain these results in a general context, we assume, as in classical nucleation theory [23], that there are well-defined nucleation rates λi(L) of phases i = 2,3, giving the number of critical nuclei which form per unit time in a system of size L. The nucleation rates per site are µi(L) := λi(L)/N. A nucleation rate is the inverse of a mean nucleation time, which can be measured in experiments or simulations by aver- aging over many nucleation events in independent runs. In the case of competitive nucleation, however, we can only measure the mean time τ for the first phase to nucleate, after which this phase invades the entire system. The rate of this first nucleation is λ2 + λ3, since the total number of nucleation events per unit time is the sum of those for each type, so that τ = 1/(λ2 + λ3). For convenience, in simulations τ is taken to be the time until the new phase occupies half the system. Under the same assumptions, the probability that phase 2 0 0.1 0.2 0.3 0.4 (h3 −h2)/h2 λ2 direct λ2 forward-flux λ3 direct λ3 forward-flux -0.2 0 0.2 0.4 0.6 (h2 −h3)/h3 with phase 4 without phase 4 FIG. 4: (Color online) (a) Comparison of nucleation rates λi(h3) of phases i = 2,3 from phase 1 in the model of Fig. 1(c) with h2 fixed, calculated directly from simulations and using forward-flux sam- pling. (b) Nucleation probability p2(h2) for h3 = 0.1 and h4 = 0.2 varying h2, with and without phase 4, in the model of Fig. 1(d). Dashed lines show positions of equal nucleation probability and equal field of the two phases. In both subfigures parameters are as in Fig. 3, with L = 50 and β = 1.25. nucleates first is p2 = Prob{T2 ≤ T3}, where Ti, the time for phase i to nucleate, is an exponentially distributed random variable with mean 1/λi. This gives p2 = λ2/(λ2+λ3) = λ2τ . Individual nucleation rates of the two phases, which can- not be obtained directly, can now be calculated as: λ2 = p2/τ and λ3 = (1− p2)/τ . This generalizes to P compet- ing phases, where the measurable quantities are the mean nu- cleation time τ = 1/∑Pi=1 λi and the nucleation probabilities pi = λi/∑ j λ j = µi/∑ j µ j of each phase i. The nucleation rate of phase i is then λi = pi/τ . Figure 4(a) shows λ2 and λ3 calculated in this way. To con- firm the validity of such calculations, we use the “forward-flux sampling” method [24, 25], which directly calculates the tran- sition rate between two phases in a stochastic system. This has previously been used to study nucleation rates in the Ising model [26, 27]. In our case, the possibility of escape to an additional new phase must be taken into account [21]. Fig- ure 4(a) shows that the results indeed coincide with those of the direct method, within statistical errors. We remark that we are unaware of any analytical prediction giving the observed variation of nucleation rates. The above considerations used “in reverse” confirm that the Ostwald rule must in general be modified when the new phases have similar stabilities, as follows. Consider phases which are equally stable for given parameter values. We ex- pect nucleation barriers, and hence nucleation rates, to vary continuously with the parameters, so that the nucleation prob- abilities pi also vary continuously. Hence there is a region, where the phases have similar stabilities, in which all nucle- ation probabilities are non-zero—only the probability of each phase nucleating is well-defined, with the outcome in any given run being stochastic, as in Fig. 3. The definite prediction given by the Ostwald rule is thus invalid in this region. To see how our results depend on system size L, we note FIG. 5: (Color online) Configuration snapshots of the multidroplet regime in the model with transition graph Fig. 1(e), with L = 200, β = 1.25, and h1 = 0, h2 = h3 = h4 = 0.15. Nucleation of droplets of three equally stable phases from a single metastable phase is fol- lowed by droplet growth and then domain coarsening. The “coating” of the initial phase visible between domains in the final snapshot is due to repulsive interactions between the new phases. that in a broad region of L, the per-site nucleation rates µi, and hence also the pi, are independent of L, as confirmed by the plateaus in the inset of Fig. 3. This is valid when the nu- cleation process is mediated by growth of a single droplet [9]. Note that this regime may be of relevance for macroscopically large systems [9]. Above a certain system size, however, droplets of differ- ent phases may nucleate before any dominates the system (the ‘multidroplet’ regime) [9]. This results in coarsening, as shown in Fig. 5 for three competing phases of equal sta- bility. Even if a phase-α droplet nucleates first, droplets of a more-stable phase may then nucleate and grow to dominate the system before the α phase can do so, thus reducing pα , as seen in Fig. 3. Finally, a generic non-symmetric case can be obtained by adding a new phase, 4, and a decay path, 3→ 4, as in Fig. 1(d). Even when h2 = h3, phase 3 now nucleates more often, as shown by the horizontally displaced nucleation probability curve in Fig. 4(b): the presence of phase 4 reduces p2 by roughly half. This is due to an entropic effect: there are more 3-dominated critical droplets than 2-dominated ones, since spins of type 4 also appear in the former droplets, resulting in nucleation of a binary mixture [23]; a similar effect is vis- ible in Fig. 1. There is thus a lower free energy barrier to form phase 3, and yet nucleation of both phases 2 and 3 is still observed, again at odds with the simple Ostwald rule. In summary, we have introduced a generalized Potts model which can easily be tuned to have any given number of metastable phases and arbitrary transitions between them. We have shown generally that individual nucleation rates of com- petitively nucleating phases can be calculated from exper- imentally measurable quantities, and that the Ostwald rule must be modified when the nucleating phases have compa- rable stabilities. In future work [21], we will study the model in detail and compare its properties with theoretical results [8] on systems with multiple metastable phases. DPS thanks A. Huerta for useful discussions and the Uni- versidad Nacional Autónoma de México for financial sup- port. The financial support of DGAPA-UNAM project PA- PIIT IN112307 is also acknowledged. ∗ Electronic address: [email protected] [1] J. F. Lutsko and G. Nicolis, Phys. Rev. Lett. 96, 046102 (2006). [2] P. R. ten Wolde and D. Frenkel, Science 277, 1975 (1997). [3] S. Takada and P. G. Wolynes, Phys. Rev. E 55, 4562 (1997). [4] D. Wales, Energy Landscapes (Cambridge University Press, Cambridge, 2003). [5] G. Biroli and J. Kurchan, Phys. Rev. E 64, 016101 (2001). [6] A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, Comm. Math. Phys. 228, 219 (2002). [7] B. Gaveau and L. S. Schulman, Phys. Rev. E 73, 036124 (2006). [8] H. Larralde, F. Leyvraz, and D. P. Sanders, J. Stat. Mech. 2006, P08013 (2006). [9] P. A. Rikvold, H. Tomita, S. Miyashita, and S. W. Sides, Phys. Rev. E 49, 5080 (1994). [10] P. R. ten Wolde and D. Frenkel, Phys. Chem. Chem. Phys. 1, 2191 (1999). [11] R. P. Sear, J. Phys. Cond. Matt. 17, 3997 (2005). [12] F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982). [13] P. Rikvold, J. Collins, G. Hansen, and J. Gunton, Surf. Sci. 203 (1988). [14] S. Manzi, W. Mas, R. Belardinelli, and V. Pereyra, Langmuir 20, 499 (2004). [15] W. C. K. Poon, F. Renth, R. M. L. Evans, D. J. Fairhurst, M. E. Cates, and P. N. Pusey, Phys. Rev. Lett. 83, 1239 (1999). [16] F. Renth, W. C. K. Poon, and R. M. L. Evans, Phys. Rev. E 64, 031402 (2001). [17] R. M. L. Evans, W. C. K. Poon, and F. Renth, Phys. Rev. E 64, 031403 (2001). [18] M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, New York, 1999). [19] E. N. M. Cirillo and E. Olivieri, J. Stat. Phys. 83, 473 (1996). [20] T. Fiig, B. M. Gorman, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E 50, 1930 (1994). [21] D. P. Sanders, H. Larralde, and F. Leyvraz, (in preparation). [22] M. A. Novotny, Phys. Rev. Lett. 74, 1 (1995). [23] P. G. Debenedetti, Metastable Liquids (Princeton University Press, Princeton, 1996). [24] R. J. Allen, P. B. Warren, and P. R. ten Wolde, Phys. Rev. Lett. 94, 018104 (2005). [25] R. J. Allen, D. Frenkel, and P. R. ten Wolde, J. Chem. Phys. 124, 024102 (2006). [26] R. P. Sear, J. Phys. Chem. B 110, 4985 (2006). [27] A. J. Page and R. P. Sear, Phys. Rev. Lett. 97, 065701 (2006). mailto:[email protected]

0704.0473

Contrasting Two Transformation-Based Methods for Obtaining Absolute Extrema

Contrasting Two Transformation-Based Methods for Obtaining Absolute Extrema D. F. M. Torres and G. Leitmann Abstract. In this note we contrast two transformation-based methods to deduce absolute extrema and the corresponding extremizers. Unlike variation-based methods, the transformation-based ones of Carlson and Leit- mann and the recent one of Silva and Torres are direct in that they permit obtaining solutions by inspection. Mathematics Subject Classification 2000: 49J15, 49M30. Key Words. Optimal Control, direct methods, Calculus of Variations, absolute extrema, invariance, symmetry. 1 Introduction In the mid 1960’s a direct method for the problems of the calculus of vari- ations, which permits one to obtain absolute extremizers directly, without using variational methods, was introduced by Leitmann (Ref. 9). Since then, this direct method has been extended and applied to a variety of problems (see e.g. Refs. 1, 3, 4, 10). A different but related direct approach to prob- lems of optimal control, based on the variational symmetries of the problem (cf. Refs. 7, 8), was recently introduced by Silva and Torres (Ref. 11). The emphasis in Ref. 11 has been on showing the differences and similarities be- tween the proposed method and that suggested by Leitmann. In order to illustrate the relation between these two methods, only examples capable of 1Associate Professor in the Department of Mathematics, University of Aveiro, Aveiro, Portugal. 2Professor in the Graduate School, College of Engineering, University of California, Berkeley, California. http://arxiv.org/abs/0704.0473v1 treatment by both methods were presented in Ref. 11. In this note, we dis- cuss some differences between the method of Carlson and Leitmann (C/L) and Silva and Torres (S/T). In particular, we show how one succeeds when the other does not. 2 The Invariant Transformation Method of S/T3 Let us consider the problem of optimal control in Lagrange form: minimize an integral I [x(·), u(·)] = L (t, x(t), u(t)) dt (1) subject to a control system ẋ(t) = ϕ (t, x(t), u(t)) a.e. on [a, b] , (2) together with appropriate boundary conditions and constraint on the values of the control variables: x(a) = xa , x(b) = xb , u(t) ∈ Ω . (3) The Lagrangian L(·, ·, ·) is a real function assumed to be continuously dif- ferentiable in [a, b] × Rn × Rm; t ∈ R is the independent variable; x(·) : [a, b] → Rn the vector of state variables; u(·) : [a, b] → Ω ⊆ Rm the vector of controls, assumed to be a piecewise continuous function; and ϕ(·, ·, ·) : [a, b] × Rn × Rm → Rn the derivative function, assumed to be a continuously differentiable vector function. When Ω is an open set (it may be all Euclidean space Ω = Rm), problem (1)-(3) can be studied using the classical techniques of the Calculus of Variations. Optimal Control The- ory includes the classical Calculus of Variations and generalizes the theory by dealing with the cases where Ω is not an open set. The application of the invariant transformation method (Ref. 11) de- pends on the existence of a sufficiently rich family of invariance transfor- mations (variational symmetries). The reader interested on the study of variational symmetries is referred to Refs. 8, 12, 13 and references therein. Definition 2.1 Let hs be a s-parameter family of C1 mappings satisfying: hs(·, ·, ·) : [a, b] ×Rn × Ω −→ R× Rn × Rm , hs(t, x, u) = (ts(t, x, u), xs(t, x, u), us(t, x, u)) , h0(t, x, u) = (t, x, u) , ∀(t, x, u) ∈ [a, b] ×Rn × Ω . 3Throughout this Note, the notation conforms to that used in the references. If there exists a function Φs(t, x, u) ∈ C1 ([a, b],Rn,Ω;R) such that L ◦ hs(t, x(t), u(t)) ts (t, x(t), u(t)) = L (t, x(t), u(t)) + Φs (t, x(t), u(t)) xs (t, x(t), u(t)) = ϕ ◦ hs (t, x(t), u(t)) ts (t, x(t), u(t)) (5) for all admissible pairs (x(·), u(·)), then (1)–(2) is said to be invariant un- der the transformations hs(t, x, u) up to Φs(t, x, u); and the transformations hs(t, x, u) are said to be a variational symmetry of (1)–(2). The method proposed in Ref. 11 is based on a very simple idea. Given an optimal control problem, one begins by determining its invariance trans- formations according to Definition 2.1. With respect to this, the tools devel- oped in Refs. 7, 8 are useful. Applying the parameter-invariance transfor- mations, we embed our problem into a parameter-family of optimal control problems. Given the invariance properties, if we are able to solve one of the problems of this family, we also get the solution to our original problem (or to any other problem of the same family) from the invariant transformations. In section 4 we give an example which shows that the Invariant Transforma- tion Method (Ref. 11) is more general than the earlier C/L transformation method in the case of optimal control problems. 3 The Direct Solution Method of C/L Since this method is fully discussed in readily available references, e.g. Refs. 1, 3, 4, 9, 10, many in this journal, we shall only recall that the C/L transformation based method is applicable to problems in the Calculus of Variations format: minimize an integral I [x(·)] = F (t, x(t), ẋ(t)) dt (6) with given end conditions x(a) = xa , x(b) = xb . (7) If one wishes to solve an optimal control problem (1)-(3), the “elimination” of u(t) in favor of a function of t, x(t), ẋ(t) must be possible. As illustrated in section 4, this may fail even if the Implicit Function Theorem is satisfied. Both the S/T and the C/L methods are predicated on posing a problem “equivalent” to the original problems (1)-(3) and (6)-(7), respectively. Thus, these methods are useful only if the solution of the “equivalent” problem is directly obtainable, i.e., by inspection. There is, at present, no result assur- ing that this can be done in general for the S/T method. However, for the C/L method, at least in the scalar x case, it has been shown in Ref. 5 and generalized to open-loop differential games in Ref. 2, that the “equivalent” problem always has a minimizing solution obtained by inspection. The con- ditions sufficient for this result are convexity of integrand F (t, x(t), ẋ(t)) with respect to ẋ(t), and existence of a so-called “field of extremals”. In- deed, no matter what the integrand of the original problem is, provided the conditions above are met, the absolute minimizer of the equivalent problem is always a constant. 4 Example 1 The advantage of the invariant transformation method when compared with the earlier transformation method is that one can apply it directly to control systems whereas the method of C/L requires that the control u(t) can be expressed as a sufficiently smooth function of t, x(t), ẋ(t), e.g. such that the integrand be continuous in x(t) and ẋ(t). Here we use the invariant transformation method of S/T to solve a simple optimal control problem that is not covered by the classical theory of the Calculus of Variations and which can not be solved by the previous transformation method. Consider the global minimum problem I[u1(·), u2(·)] = u1(t) 2 + u2(t) dt −→ min ẋ1(t) = exp(u1(t)) + u1(t) + u2(t) , ẋ2(t) = u2(t) , x1(0) = 0 , x1(1) = 2 , x2(0) = 0 , x2(1) = 1 , u1(t) , u2(t) ∈ Ω = [−1, 1] . We apply the procedure introduced in Ref. 11 and briefly described in sec- tion 2. First we notice that problem (8) is variationally invariant according to Definition 2.1 under the one-parameter transformations4 = x1 + st , x = x2 + st , u = u2 + s (t s = t and us = u1) . (9) To prove this, we need to show that both the functional integral I[·] and the control system stay invariant under the s-parameter transformations (9). This is easily seen by direct calculations. We begin by showing (4): Is[us (·), us (·)] = + (us u1(t) 2 + (u2(t) + s) u1(t) 2 + u2(t) s2t+ 2sx2(t) = I[u1(·), u2(·)] + s 2 + 2s . We remark that Φs (t, x2) = s 2t+2sx2 and that I s[·] and I[·] have the same minimizers: adding a constant s2+2s to the functional I[·] does not change the minimizer of I[·]. It remains to prove (5): (t)) = (x1(t) + st) = ẋ1(t) + s = exp(u1(t)) + u1(t) + u2(t) + s = exp(us (t)) + us (t) + us (t) , (t)) = (x2(t) + st) = ẋ2(t) + s = u2(t) + s (t) . Conditions (10) and (11) prove that problem (8) is invariant under the s-parameter transformations (9) up to the gauge term Φs = s2t + 2sx2. Using the invariance transformations (9), we generalize problem (8) to a s-parameter family of problems, s ∈ R, which include the original problem 4A computer algebra package that can be used to find the invariance trans- formations (see Refs. 7, 8) is available from the Maple Application Center at http://www.maplesoft.com/applications/app center view.aspx?AID=1983 for s = 0: Is[us (·), us (·)] = (t))2 + (us (t))2dt −→ min (t) = exp(us (t)) + us (t) + us (t) , ẋ2(t) = u (t) , (0) = 0 , xs (1) = 2 + s , xs (0) = 0 , xs (1) = 1 + s , (t) ∈ [−1, 1] , us (t) ∈ [−1 + s, 1 + s] . It is clear that Is ≥ 0 and that Is = 0 if us (t) = us (t) ≡ 0. The con- trol equation, the boundary conditions and the constraints on the values of the controls, imply that us (t) = us (t) ≡ 0 is admissible only if s = −1: xs=−1 (t) = t, xs=−1 (t) ≡ 0. Hence, for s = −1 the global minimum to Is[·] is 0 and the minimizing trajectory is given by (t) ≡ 0 , ũs (t) ≡ 0 , x̃s (t) = t , x̃s (t) ≡ 0 . Since for any s one has by (10) that I[u1(·), u2(·)] = I (·), us (·)]−s2−2s, we conclude that the global minimum for problem I[u1(·), u2(·)] is 1. Thus, using the inverse functions of the variational symmetries (9), u1(t) = u (t) , u2(t) = u (t)−s , x1(t) = x (t)−st , x2(t) = x (t)−st , the absolute minimizer is ũ1(t) = 0 , ũ2(t) = 1 , x̃1(t) = 2t , x̃2(t) = t . This problem cannot be solved by the C/L method. While the existence of a function h(·) such that u1(t) = h (ẋ1(t)− ẋ2(t)) is assured by the satisfaction of the Implicit Function Theorem, this is not useful in “eliminating” the control in favor of t, x(t), ẋ(t) which is a re- quirement of the C/L method. This is so because h(·) is a solution of a transcendental equation. In addition, since the controls are bounded, even if “elimination” of the control were possible, this would lead to differential constraints of the form briefly discussed in Ref. 6. 5 Example 2 Consider the problem of attaining the absolute minimum of integral I[x(·)] = ẋ2(t) + tẋ(t) dt (12) with prescribed end conditions x(a) = xa , x(b) = xb . (13) This is a problem in the Calculus of Variations format for which the C/L or S/T transformation-based method applies. These methods can be applied ab initio to obtain the solution. However, here we shall employ the constructive sufficiency condition embodied in Theorem 7 of Ref. 5 towards that end. The Euler-Lagrange equation is ẍ(t) = − so that x(t) = − t2 + c1t+ c2 (15) is the extremal with the constants of integration given by end conditions (13), say ci = c , i = 1, 2, and the extremal satisfying (15) is x∗(t) = − t2 + c∗ t+ c∗ which, being the solution of necessary condition (14), is a candidate for the absolute minimizer of (12)-(13). Now we can readily show that the conditions of Theorem 7 of Ref. 5 are met. Consider the one-parameter family of extremals ξ(t, β) = − t2 + c∗ t+ c∗ + β . First of all, there exist a β∗, namely β∗ = 0, such that ξ(t, β∗) = x∗(t) . Secondly, ∂ξ(t, β) 6= 0 , and finally, the integrand of (12) is convex in ẋ(t). Thus, the extremal (16) is indeed the absolute minimizer of (12)-(13). Of course, this is precisely the solution obtained by employing the C/L method. Indeed, the more general method inherent in Theorem 7 of Ref. 5 was derived using the C/L transformation-based method. 6 Conclusion We have contrasted two transformation-based methods for obtaining abso- lutely extremizing solutions for two classes of problems. One, dubbed the Carlson/Leitmann method, is applicable to problems in the format of the Calculus of Variations. The other, due to Silva and Torres, is applicable to problems of Optimal Control. We have shown that it is not always possible to convert an Optimal Control problem into a Calculus of Variations one. Hence, the C/L method may fail to apply when the S/T succeeds. On the other hand, a classical constructive sufficiency condition, readily derived by the C/L method, ren- ders absolute extremizers for specific problems of the Calculus of Variations more directly and succinctly than the C/L and S/T methods. References 1. CARLSON, D. A., An Observation on Two Methods of Obtaining So- lutions to Variational Problems, J. Optim. Theory Appl., Vol. 114, pp. 345–361, 2002. 2. CARLSON, D. A., Fields of Extremals and Sufficient Conditions for a Class of Variational Games, Proceed. of 12th International Symposium on Dynamic Games and Applications, Sophia Antipolis, France, 2006, to appear. 3. CARLSON, D. A., and LEITMANN, G., Coordinate Transformation Method for the Extremization of Multiple Integrals, J. Optim. Theory Appl., Vol. 127, pp. 523–533, 2005. 4. CARLSON, D. A., and LEITMANN, G., A Direct Method for Open- Loop Dynamic Games for Affine Control Systems. Dynamic games: the- ory and applications, 37–55, GERAD 25th Anniv. Ser., 10, Springer, New York, 2005. 5. CARLSON, D. A., and LEITMANN, G., Fields of Extremals and Suffi- cient Conditions for the Simplest Problem of the Calculus of Variations, J. Global Optim., 2007, to appear. 6. CARTIGNY, P., and DEISSENBERG, C., An Extension of Leitmann’s Direct Method to Inequality Constraints, Int. Game Theory Rev., Vol. 6, pp. 15–20, 2004. 7. GOUVEIA, P. D. F., and TORRES, D. F. M., Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Con- trol, Computational Methods in Applied Mathematics, Vol. 5, pp. 387– 409, 2005. 8. GOUVEIA, P. D. F., TORRES, D. F. M., and ROCHA, E. A. M., Symbolic Computation of Variational Symmetries in Optimal Control, Control and Cybernetics, Vol. 35, pp. 831–849, 2006. 9. LEITMANN, G., A Note on Absolute Extrema of Certain Integrals, International Journal of Nonlinear Mechanics, Vol. 2, pp. 55–59, 1967. 10. LEITMANN, G., Some Extensions to a Direct Optimization Method, J. Optim. Theory Appl., Vol. 111, pp. 1–6, 2001. 11. SILVA, C. J., and TORRES, D. F. M., Absolute Extrema of Invariant Optimal Control Problems, Commun. Appl. Anal., Vol. 10, pp. 503–516, 2006. 12. TORRES, D. F. M., Quasi-Invariant Optimal Control Problems, Port. Math. (N.S.), Vol. 61, pp. 97–114, 2004. 13. TORRES, D. F. M., A Noether Theorem on Unimprovable Conserva- tion Laws for Vector-Valued Optimization Problems in Control Theory, Georgian Mathematical Journal, Vol. 13, pp. 173–182, 2006. Introduction The Invariant Transformation Method of S/T3 The Direct Solution Method of C/L Example 1 Example 2 Conclusion

0704.0476

Geometric phase of an atom inside an adiabatic radio frequency potential

Geometric phase of an atom inside an adiabatic radio frequency potential P. Zhang1 and L. You1, 2 School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China (Dated: November 4, 2018) We investigate the geometric phase of an atom inside an adiabatic radio frequency (rf) potential created from a static magnetic field (B-field) and a time dependent rf field. The spatial motion of the atomic center of mass is shown to give rise to a geometric phase, or Berry’s phase, to the adiabatically evolving atomic hyperfine spin along the local B-field. This phase is found to depend on both the static B-field along the semi-classical trajectory of the atomic center of mass and an “effective magnetic field” of the total B-field, including the oscillating rf field. Specific calculations are provided for several recent atom interferometry experiments and proposals utilizing adiabatic rf potentials. PACS numbers: 03.65.Vf, 39.20.+q, 03.75.-b, 39.25.+k I. INTRODUCTION Magnetic trapping is an important enabling technol- ogy for the active research field of neutral atomic quan- tum gases. A variety of trap potentials can be developed using magnetic (B-) fields with different spatial distri- butions and time variations. For instance, the widely used quadrupole trap and the Ioffe-Pritchard trap [1] are usually created with static B-fields, while the time aver- aged orbiting potential (TOP) [2] and time orbiting ring trap (TORT) [3, 4] are created using oscillating B-fields with frequencies larger than the effective trap frequen- cies. Atom chips [5] have brought further developments to magnetic trap technology, as they can provide larger B-fields and gradients at reduced power-consumptions or electric currents using micro-fabricated coils. Today, magnetic trapping is a versatile tool used in many labo- ratories around the world for controlling atomic spatial motion in regions of different scales and geometric shapes, e.g., 3D or 2D traps, double well traps, and storage ring traps [1, 2, 3, 4, 5, 6, 7]. Recently, magnetic traps based on adiabatic microwave [8, 9] and adiabatic radio frequency (rf) potentials (ARFP) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] have attracted considerable attention. An ARFP is typically created with the combination of a static B-field and an rf field. The idea for an ARFP has been around for some time [10, 11, 12], and experi- mental demonstrations recently have been carried out for confining both thermal [13] and Bose condensed atoms [14, 15, 16, 17]. Further development of improved ARFP with atom chip technology likely will assist in practi- cal applications of atom interferometry. For instance, a double well potential was constructed recently using low order multi-poles capable of atomic beam splitting while maintaining tight spatial confinement [21]. Sev- eral interesting recent proposals outline the construc- tion of small storage rings with radii of the order 1µm [18, 19, 20, 21, 22], which could become useful if imple- mented for atom Sagnac interferometry [27] setups on atom chips. When a neutral atom is confined in a magnetic poten- tial, its hyperfine spin is assumed to follow adiabatically the spatial variation of the B-field direction during its spatial translational motion. As a result of this adia- batic approximation, the center of mass motion for the atom experiences an induced gauge field [28], giving rise to a geometric phase (or Berry’s phase) to the atomic internal spin state [29, 30]. The effect of this geometric phase is widely known, and is first addressed carefully in a meaningful way for atomic quantum gases by an explicit calculation of the resulting geometric phase in a static or a time averaged magnetic trap in Ref. [31]. Sev- eral important consequences are predicted to occur for a magnetically trapped atomic condensate in a quadrupole trap, a Ioffe-Pritchard trap [31, 32], or a TORT based storage ring [33]. To our knowledge, this geometric phase effect has not been investigated in any detail for an atom inside an ARFP. In a recent paper, we show that this geometric phase causes an effective Aharonov-Bohm-type [34] phase shift in a magnetic storage ring based atom interferometer [33]. In addition, our studies imply that the spatial fluctuation of the geometric phase can lead to a reduction of the visibility of the interference contrast. In view of this, we decided to carry out this study as reported here for the atomic geometric phase in an ARFP in order to shed light on the proposed high precision atom Sagnac interference experiment [27]. Analytical derivations for this study at some places become rather tedious and complicated. We therefore first will summarize our major results here for readers who may not be interested in the intricate details. We find that the geometric phase in an ARFP generally takes a more complicated form in comparison to the case of a static trap or a time averaged trap. In an ARFP, this phase factor is found to be determined by the trajectory of the time independent component of the trap field as well as an “effective B-field” that depends on the total B-field. In contrast to the earlier result found for a static trap or a time averaged trap [31, 33], the final result turns out to be not expressible as a functional of the trajectory http://arxiv.org/abs/0704.0476v1 for the direction of the total B-field in the parameter space. This paper is organized as follows. In sec. II, we gener- alize the semi-classical approach as outlined in Ref. [21] for the operating principle of an ARFP to a form more convenient for discussing the geometric phase. Section III parallels that of sec. II by reformulating a full quan- tum theory for discussing the geometric phase inside an ARFP [22]. The explicit expression for the geometric phase inside an ARFP is given in a readily adaptable form for specifical calculations. In sec. IV, we discuss the effect of the geometric phase in several types of ARFP recently proposed for atomic splitters and storage rings [18, 19, 20, 21, 22] and beam splitters [14, 21]. Finally, concluding remarks are given in sec. V. II. A SEMI-CLASSICAL APPROACH In this section, we provide a semi-classical formula- tion for calculating the atomic geometric phase inside an ARFP. The semi-classical working principle for an ARFP is described in Ref. [21], although only for the special case when the atomic center of mass is assumed at a fixed location. In order to calculate the geometric phase, our formulation allows for the explicit consideration of atomic center of mass motion classically. In our approach, the geometric phase is obtained naturally, and the validity conditions for both the adiabatic and the rotating wave approximations are clearly shown for an ARFP. Inside an ARFP [21], the total B-field ~B(~r, t) is the sum of a static field component ~Bs(~r) and an oscillatory rf field ~Bo(~r, t), which conveniently is expressed as ~Bo(~r, t) = ~B rf (~r, t) cos(ωt) + rf (~r, t) cos(ωt+ η). (1) where ~r is the spatial position vector of the atom, ω is frequency of the rf field, and η is a relative phase factor. In this section, we will assume that the atomic spatial motion is pre-determined, i.e., ~r(t) is given (as a slowly varying function of time t). For weak B-fields, the system Hamiltonian is simply the linear Zemman interaction H(t) = gFµB ~F · ~B[~r(t), t], (2) where gF is the corresponding Lande g-factor and µB denotes the Bohr magneton. ~ = 1 is assumed. For a static or a time averaged magnetic trap, the Hamiltonian (2) varies slowly over time scales of the Lar- mor precession of the atomic spin in the total B-field. During the effectively slow trapped motion, the atomic hyperfine spin is assumed to be fixed at the instanta- neous eigenstate of the Hamiltonian (2). The geometric phase then can be calculated straightforwardly from the variation of the B-field direction in the parameter space [31, 33]. In an ARFP, the situation is more complicated. Al- though the variation of ~Bs[~r(t)] remains much slower than the Larmor precession, the rf frequency ω usually is assumed to be nearly resonant with the precession fre- quency. Thus, the Hamiltonian (2) contains both fast and slow time varying components, making the direct calculation of the geometric phase a more involved task. In the following, we will proceed step by step, clarifying the various approximations adopted along the way. To understand the working principle for an ARFP, we first decompose the Hamiltonian H(t) (2) into the fol- lowing form H(t) = Hs[~r(t)] +H+[~r(t)]e −iωt +H−[~r(t)]e iωt, (3) where Hs and H± are all slow varying functions of time and are given by Hs[~r(t)] = gFµB ~F · ~Bs[~r(t)], H+[~r(t)] = gFµB ~F · rf [~r(t)] + e −iη ~B rf [~r(t)] H−[~r(t)] = H +[~r(t)]. (4) Hs is diagonal in the spin angular momentum ba- sis defined along the local direction of the static B- field ~Bs[~r(t)]. The eigenstate takes the familiar form |mF [~r(t)]〉s, quantized along the direction of ~Bs[~r(t)], with the eigenvalue mF |Bs[~r(t)]| for ~Bs[~r(t)] · ~F and mF ∈ [−F, F ], in analogy with the usual case of the z- quantized representation result of Fz |mF 〉z = mF |mF 〉z. Next we introduce a unitary transformation U(t) = mF=−F |mF 〉z s〈mF [~r(t)]|eimF κωt, (5) with κ = sign(gF ) for the rotating wave approxima- tion. The quantum state in the interaction picture |Ψ(t)〉I = U(t)|Ψ(t)〉 defined by U(t) is governed by the Schroedinger equation i∂t|Ψ(t)〉I = HI(t)|Ψ(t)〉I , with the Hamiltonian in the interaction picture given by HI(t) = UHU † + i(∂tU)U mκ∆[~r(t)]|m〉z z〈m| − i m,n=−F |m〉z s〈m[~r(t)]| |n[~r(t)]〉s z〈n|ei(m−n)κωt m=−F+1 h(+)m [~r(t)]|m〉z z〈m− 1|+ h(−)m [~r(t)]|m〉z z〈m− 1|e2iκωt + h.c. hm[~r(t)]|m〉z z〈m|eiκωt + h.c. , (6) where the time dependent parameters are defined as ∆[~r(t)] = µB|gF ~B[~r(t)]| − ω, h(±)m [~r(t)] = s〈m[~r(t)]|H±[~r(t)]|(m− 1)[~r(t)]〉s, and(7) hm[~r(t)] = s〈m[~r(t)]|H±[~r(t)]|m[~r(t)]〉s. The above result is obtained easily if we note that the matrix element s〈m[~r(t)|H±(t)|m′[~r(t)〉s is non-zero only when m − m′ = 0,±1. So far, we have always as- sumed that |m[~r(t)]〉s is a single valued function of the atomic position ~r. A careful examination shows that the eigenstate |m[~r(t)]〉s cannot be determined uniquely be- cause of the presence of the U(1) gauge freedom for se- lecting a local phase factor exp{iφ[~r(t)]}, which conse- quently affects the resulting expressions for h m (t) and s〈m[~r(t)]|d/dt|m′[~r(t)]〉s. The rotating wave approximation neglects of the oscillating terms proportional to eimωt (m 6= 0) in the Hamiltonian HI (6). The error for this approximation is estimated easily from a time dependent perturbation calculation. The sufficient condition for its validity requires that all factors such as dt′hm(t ′) exp[iκωt′], ′)ξm,m−1(t ′) exp[iκ(2ω + ∆)t′], and dt′〈m[~r(t′)]|d/dt′|n[~r(t′)]〉ξmn(t′) exp[i(m − n)κ(ω + ∆)t′] are negligible, where ξmn(t) = exp dt′ s〈m[~r(t′)]| |m[~r(t′)]〉s dt′ s〈n[~r(t′)]| |n[~r(t′)]〉s . (8) Thus, the gauge independent factors h m ξm,m−1, 〈ms|d/dt|ns〉ξmn , hm, and ∆ should all vary slowly with time and with the modulus of their amplitudes much less than ω. The effective Hamiltonian in the interaction picture under the rotating wave approximation then becomes eff (t) = µBgF ~F · ~Beff [~r(t)] |m〉z s〈m[~r(t)]| |m[~r(t)]〉s z〈m|, (9) where the first term resembles a coupling between the atomic spin and an “effective B-field” ~Beff(~r), whose components in real space are given by Beffx (~r) = Re 2 s〈m(~r)|H±(~r)|(m− 1)(~r)〉s (F +m)(F −m− 1) Beffy (~r) = −Im 2 s〈m(~r)|H±(~r)|(m− 1)(~r)〉s (F +m)(F −m− 1) , and Beffz (~r) = ~Bs(~r) µB|gF | . (10) Clearly, the x- and y-components of the effective field ~Beff(~r) depend on the explicit form of the eigenstate |m(~r)〉s. In fact, it easily can be seen that different choices of the local phase factor for the |m(~r)〉s actually lead to different values of ~Beff(~r) related to each other through ~r-dependent rotations in the x-y plane. In practice, the eigenstate |n(~r)〉s and the effective field ~B eff can sometimes be constructed more simply, as in Ref. [21]. For any spatial position ~r, we first choose a rotation R[m̂(~r), χ(~r)] along the axis m̂(~r) with an angle χ(~r) that satisfies R[~n(~r), χ(~r)] ~Bs(~r) = | ~Bs(~r)|êz. It is then easy to show that the eigenstate |n[~r(t)]〉s can be chosen as |n(~r)〉s = exp i ~F · m̂(~r)χ(~r) |n〉z. (11) Unfortunately, the choice for R is not unique in a given static field ~Bs(~r), an analogous result to the U(1) gauge freedom for the the egienstate |n(~r)〉s. Corresponding to the choice (11) given above for |n(~r)〉s, the unitary transformation U defined in (5) would become U(t) = exp(−iFzωt) · exp −i ~F · m̂(~r)χ(~r) , (12) and the transverse components of the “effective B-field” given by B effx,y(~r) = Bx,y(~r)/2 [21] with ~B(~r) = R[m̂(~r), χ(~r)] ~B rf (~r) +R[êz,−κη]R[m̂(~r), χ(~r)] ~B(b)rf (~r). (13) In earlier discussions of an ARFP [21, 22], the atomic internal state is assumed uniformly to remain adiabati- cally in a certain eigenstate of the first term of H eff (t). To fully appreciate this adiabatic approximation and to calculate the geometric phase, we expand |Ψ(t)〉I into the instantaneous eigenstate basis |n[~r(t)]〉eff quantized along the direction of the effective B-field ~Beff accord- ing to |Ψ(t)〉I = n Cn(t)|n[~r(t)]〉eff . The first term of eff (t) is simply the effective Zemman interaction be- tween the atomic hyperfine spin and the effective B-field. The corresponding Schroedinger equation for the Hamil- tonian H eff (t) of (9) then becomes Cn(t) = [ǫ I (t) + νnn(t)]Cn(t) + m 6=n νnm(t)Cm(t), I (t) = nµBgF | ~B eff(t)|, νpq(t) = −i eff〈p[~r(t)]|l〉z s〈l[~r(t)]| |l[~r(t)]〉s z〈l|q[~r(t)]〉eff −i eff〈p[~r(t)]| |q[~r(t)]〉eff . (14) Under the adiabatic approximation, the atomic inter- nal state remains in a given eigenstate |n[~r(t)]〉eff with transitions to states |m[~r(t)]〉eff (m 6= n) being negligi- bly small. Thus, the transition probability, as estimated from the first order perturbation theory, dt′νnm(t dt′′[ǫ (t′′)+νnn(t ′′)−ǫm (t′′)−νmm(t should be much less than one. As before, we find the sufficient condition for the validity of the adiabatic ap- proximation is given by |νmn(t)| |ǫmI (t′)− ǫ ≪ 1, (15) provided that νmn(t) exp[i [νnn(t ′′) − νmm(t′′)]dt′], which is independent of the local phase factor for |n[~r(t)]〉eff and |n[~r(t)]〉s, remains a slowly varying func- tion of time. A straight forward calculation from the effective Hamiltonian (9) then gives the general expression for the geometric phase in an ARFP γn(t) = νnn(t ′)dt′ | eff〈n[~r(t′)|l〉z| s〈l[~r(t′)]| |l[~r(t′)]〉sdt′ +γ(I)n (t), (16) and γ n (t) = −i dt′ eff〈n[~r(t′)|d/dt′|n[~r(t′)〉eff . During the adiabatic motion in a given internal state, the time evolution of the coefficient Cn(t) takes the form Cn(t) = Cn(0)e (t′)dt′e−iγn(t). (17) Equation (16) is the central result of this work. The geometric phase of an atom inside an ARFP is shown to contain two parts. The second part, γ n (t), is clearly due to the interaction term µBgF ~F · ~Beff in H(I)eff (9), with its value determined by the trajectory of the direc- tion for the “effective B-field” ~Beff . The first part arises from the second term of H eff (9). It is determined by the trajectories of both the static field ~Bs and the ef- fective B-field ~Beff . The expression for γn in an ARFP is complicated because the internal quantum state in an ARFP is assumed to be adiabatically kept in an eigen- state of µBgF ~F · ~Beff , rather than an eigenstate of the total interaction Hamiltonian H eff (t). In section IV, we will perform explicit calculations for several examples of ARFP proposed for various applica- tions: e.g., as atomic storage rings or atomic beam split- ters. Most often we find that only the first part of Eq. (16) contributes a non-zero value to the geometric phase. Before proceeding to the next section for a quantal treatment of the geometric phase, we find the time evo- lution of the atomic spin state in the Schroedinger picture |Ψ(t)〉 = Cl(0) z〈m|l[~r(t)]〉eff × (t′)dt′e−iγl(t)e−imωt|m[~r(t)]〉s, (18) obtained directly from |Ψ(t)〉 = U †(t)|Ψ(t)〉I after the applications of the rotating wave and adiabatic approxi- mations. When the atom is prepared initially in a specific adiabatic state |n[~r(t)]〉eff of the interaction picture, we arrive at the simple case of Cl(0) = δln. III. A QUANTUM MECHANICAL TREATMENT In the previous section, we provided the result for the geometric phase γn(t) in an ARFP based on a semi- classical approach, where the atomic center of mass mo- tion is described classically. A clear physical picture ex- ists in this case for the appearance of the geometric phase in a certain parameter space. The validity conditions for the rotating wave and the adiabatic approximations as obtained above are all formulated in terms of gauge inde- pendent forms. However, if the influence of the geomet- ric phase on the atomic spatial motion is to be included, e.g., as in the Aharonov-Bohm-type, phase shift, inter- ference arrangement in an atomic Sagnac interferometer discussed earlier [33], we would need an improved de- scription where both the atomic spin and its center of mass motion are treated quantum mechanically. In a full quantum treatment of the atomic motion, the quantum state of an atom can be expressed as |Φ(t)〉 = φl(~r, t)|l〉z, where φl(~r, t) is the atomic spatial wave function for the internal state |l〉z of Fz. The state |Φ(t)〉 then satisfies the Schroedinger equation governed by the Hamiltonian + gFµB ~F · ~B(~r, t), (19) with ~P being the kinetic momentum and M the atomic mass. The rotating wave and adiabatic approximations can be introduced now by defining the interaction picture with the unitary transformation U(t) = |m〉z eff〈m(~r)| |n〉z s〈n(~r)|einκωt . (20) The state in the interaction picture |Φ(t)〉I = U(t)|Φ(t)〉 now is governed by the Schroedinger equation with the Hamiltonian Heff = UHU†. Under the rotating wave and adiabatic approximations, we neglect transitions be- tween states |m〉z and |n〉z (m 6= n) as well as the rapidly oscillating terms. We then obtain Heff ≈ |n〉z z〈n|Heff |n〉z z〈n| ad |n〉z z〈n|, (21) where the adiabatic Hamiltonian H ad for the n-th adia- batic branch is defined as ~P − ~An I (~r), (22) with the effective gauge potential ~An(~r) = −i | eff〈n(~r)|l〉z|2 s〈l(~r)|∇|l(~r)〉s −i eff〈n(~r)|∇|n(~r)〉eff . (23) In this form, it is well known that the geometric phase γn can be expressed as the integral of the gauge poten- tial ~An along the spatial trajectory for the atomic center of mass in an ARFP, i.e., one would expect generally that γn = ~An · d~r. Similar to the result of the semi- classical approach, the gauge potential ~An(~r) can be ex- pressed as the sum of two parts. The first part in Eq. (23) is the weighted sum of the atomic gauge potential −i s〈l(~r)|∇|l(~r)〉s from the static field ~Bs, while the sec- ond term is the atomic gauge potential from the “effective B-field” ~Beff . A full quantum treatment for atomic motion in an ARFP has been attempted earlier [22]. In fact, many of our formulations are identical to the results of Ref. [22]. For instance, it is easy to show that the unitary trans- formations US, UR, and UF in [22] are related directly to ours as U S = U . The only difference concerns the gauge potential ~An that was neglected in Ref. [22]. Thus, they did not give the expression for the gauge po- tential, and the result for the geometric phase was not obtained either [22]. Our study shows that the neglect of the adiabatic gauge potential potentially can give rise to a final result, dependent on the choice of the local phase factors for the internal eigenstate. IV. GEOMETRIC PHASES IN ARFP BASED APPLICATIONS In the above two sections, we obtain the expression for the atomic geometric phase in an ARFP. This section is devoted to the calculations of the geometric phases for several proposed applications of ARFP, such as storage rings or beam splitters for neutral atoms [18, 19, 20, 21, Before presenting our results for the more specific cases, we provide some general discussions of the geo- metric phases in several ARFP based storage rings. As was pointed out earlier, the geometric phase γn is given by the line integral of the gauge potential ~An along the trajectory for the atomic center of mass motion. For a closed path in the storage ring at a fixed ρ = ρc and z = zc, this can be further reduced to γn = q A(φ)n (ρ, φ, z)ρdφ, (24) where the integer q is the winding number of the path and A n is the component of ~An along the azimuthal direction êφ of the familiar cylindrical coordinate system (ρ, φ, z). Without loss of generality, we take q = 1 in this paper. For the storage rings proposed in Refs. [18, 19, 20, 21, 22], the gauge potentials A n (ρ, φ, z) are actually independent of the angle φ. Therefore, the geometric phase is simply given by γ(c)n = 2πρcA n (ρc, zc), (25) given out in explicit forms for different storage ring schemes [18, 19, 20, 21, 22]. In reality, because of thermal motion or when the atomic transverse motional state is considered, the center of mass for an atom can deviate from (ρc, zc) even for a closed trajectory. This uncertainty in the exact shape of the closed trajectory gives rise to a fluctuating geometric phase and is usually difficult to study. Assuming a simple closed path at fixed ρ and z, we have found previously that the subsequently fluctuations could decrease the vis- ibility of the interference pattern [33]. Quantum mechan- ically, such destructive interference can be explained as resulting from entanglement between the freedoms for φ and (ρ, z) because of the dependence of the gauge po- tential Aφn on ρ and z. Therefore, it is important to investigate this dependence near the trap center. For simplicity, our discussions below will focus on the closed loops where ρ and z are φ-independent constants. In this case, the geometric phase can be expressed as γn(ρ, z) = 2πρA n (ρ, z). We will show numerically the distributions for γn(ρ, z) obtained this way near the cen- tral region of (ρc, zc). If needed, a more rigorous ap- proach can be developed to investigate the fluctuations of the resulting geometric phase from the gauge potential n (ρ, z). A. The storage ring proposals of Refs. [18, 19, 20] This subsection is devoted to a detailed calculation of the geometric phases for the ARFP storage ring propos- als of Refs. [18, 19, 20]. We will derive the analyti- cal expressions for the azimuthal component A n of the gauge potential that arises in both cases from cylindri- cally symmetric static B-field and rf fields. Because of the cylindrical symmetry, the angle βs(ρ, z) between the local static B-field and the z-axis is required to be ana- lytical in the region near the storage ring. Therefore, the eigenstate |n(~r)〉s of ~F · ~Bs can be chosen as |n(~r)〉s = exp{−i[~F · êφβs(ρ, z) + nφ]}|n〉z . (26) Consequently, ~B eff(~r) is also cylindrically symmetric, which leads to the eigenstate |n(~r)〉eff of ~F · ~B eff as |n(~r)〉eff = exp{−i[~F · n̂ eff⊥ (~r)βeff(ρ, z) + nφ]}|n〉z, (27) with the unit vector n̂ eff⊥ (~r) in the x-y plane orthogonal to ~B eff(~r) and β eff(ρ, z) denoting the angle between ~B eff(~r) and the z-axis. We note that the unit vector field n̂ eff⊥ (~r) also possesses cylindrical symmetry, i.e., remains invari- ant under rotation around the z-axis. The expressions of (26) and (27) allow us to obtain the simple expression of the gauge potential A(φ)n (ρ, z) = − cosβeff(ρ, z) cosβs(ρ, z), (28) after straightforward calculations. In the scheme of Ref. [18], the static B-field is a “ring- shaped quadrupole field” that vanishes along a circle of a radius ρ0 in the x-y plane. Near ρ = ρ0, the B-field is given approximately by ~Bs(~r) = B ′(ρ− ρ0)êρ −B′zêz, (29) like a quadrupole field, while the rf-field takes a compli- cated form ~Bo(~r, t) = cos(ωt) + cos(ωt+ ϕ) sin(ωt) + sin(ωt+ ϕ) êz, (30) with constants a and b independent of ~r. From the expression of (26) for the eigenstate |n(~r)〉s, the “effective B-field” ~B eff becomes ~B eff(~r) = B′[ (ρ− ρ0)2 + z2 − r0]êz cos(θ + ϕ) + cos θ sin(θ + ϕ)− a√ sin θ êφ, (31) FIG. 1: (Color online) A cross-sectional view for the storage ring of Ref. [18]. The static field is zero in the ring at the fixed radius ρ0. The addition of rf-fields creates an ARFP centered at a ring through (ρc, zc). The distance from the trap center to the ring with radius ρ0 in the plane z = 0 is r0. where r0 and θ are given by |µBgFB′| cos θ(ρ, z) = ρ− ρ0 (ρ− ρ0)2 + z2 sin θ(ρ, z) = (ρ− ρ0)2 + z2 . (32) In an ARFP, as discussed here, the trap center at (ρc, zc) is determined by minimizing both the z- component and the transverse component of ~B eff . With- out loss of generality, we will assume a, b > 0. Then, (ρc, zc) is found to satisfy θ(ρc, zc) = −ϕ/2, (ρc − ρ0)2 + z2c = r0, (33) i.e., the trap center lies on the surface of the “resonance toroid” at ρ = ρ0 with a radius r0 as shown in Fig. 1. The relative angle of the trap center with respect to the center of the toroid cross-section is given by −ϕ/2. On this “resonance toroid,” the rf-field is resonant with the static field, i.e., B effz vanishes. As a result, the “effective B-field” lies again in the x-y plane on the “resonance toroid,” which gives cosβeff(ρc, zc) = 0 and leads to the result A n (ρc, zc) = γn = 0 as shown in the trap center for the storage ring considered before in Ref. [18]. From the expression (29) of the static field and the def- inition of the angle θ(ρ, z), we find a simple relationship βs(ρ, z) = π/2 + θ(ρ, z), with which the gauge potential n (ρ, φ) in (28) can be further simplified as A(φ)n (ρ, z) = cosβeff(ρ, z) sin θ(ρ, z) cosβeff(ρ, z) sin θ(ρc, zc), (34) near the trap center. Thus, the spatial fluctuation of the gauge potential A n (ρ, z) in the region around the trap center is closely related to the angle θ(ρc, zc) of the trap center, or the parameter ϕ of the oscillating field ~Bo. When ϕ = 0, the atom is trapped in the region with θ ≈ 0 or π, where the fluctuation of A(φ)n (ρ, z) is suppressed significantly due to the small value of sin θ. On the other hand, if the angle ϕ is set to π with the trap center located in the region with θ ≈ ±π/2, the fluctuation of the gauge potential becomes amplified. In Fig. 2, we illustrate numerical results for the dis- tribution of the geometric phase γ1(ρ, z) = 2πρA 1 (ρ, z) in the region near the trap center at ϕ = 0, π/2, π. We see clearly decreased fluctuations of γ1 when the absolute value of sin θ(ρc, zc) = − sin(ϕ/2) is decreased. Next we turn to the storage ring of Ref. [19] con- structed from a quadrupole static B-field ~Bs(~r) = B′(x, y,−2z) and an ~r-independent rf field ~Bo = Brf cos(ωt)êz along the z direction. The resulting ARFP provides a 2D ring shaped trap in the x-y plane. In ad- dition, a 1D optical potential along the z direction is em- ployed to confine atoms in the transverse plane at z = 0 [19]. The “effective B-field” takes the form ~B eff(~r) = B′(ρ− ρ0)êz − Brf êρ, (35) in the plane at z = 0, with ρ0 = ω/|µBgFB′|. Because the strength of ~B eff is near minimum at the ring ρ = ρ0, the trap center for this storage ring is located at ρc = ρ0 and zc = 0. At the trap center, the “effective B- field” is along the direction of êρ. Thus, according to Eq. (28), the geometric phase γ n at the trap center again vanishes. In Fig. 3, we show the distribution of the geometric phase γ1 in the region near the trap center for Brf = 0.05|B′|ρ0 and Brf = 0.15|B′|ρ0. We see that the fluctu- ation is relatively small when the strength of the rf-field is large. This can be explained by Eq. (28), which shows that A n is proportional to cosβeff and can be approx- imated as 2B effz /Brf near the trap center. When Brf is large, the gauge potential becomes a relatively slow vary- ing function of ρ and z. In this case, the presence of a 1D optical potential allows for the possibility of tuning the trap center position to a nonzero value of z, with the storage ring remaining in the x-y plane. Then cosβs is assumed to a nonzero value, leading to increased fluctu- ations for the geometric phase. Finally, we discuss the geometric phase in the “time averaged” ARFP storage ring proposed in Ref. [20]. Un- like previously considered ARFP based storage rings, the time dependence now exists in both the “static B-field” and the frequency of the rf field given by ~Bs(~r, t) = B ′ρêρ − 2B′zêz +Bm sin(ωmt)êz, −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 (ρ−ρc)/ρ0 −0.1 0 0.1 −0.1 0 0.1 FIG. 2: (Color online) The distribution of the geometric phase γ1 near the trap center (ρc, zc) of the storage ring proposed in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly displaying the sin(ϕ/2) dependence. ~Bo(t) = Brf sin[ω(t)t]êz, ω(t) = ω0 1 + (Bm/B′ρ0)2 sin 2(ωmt) . (36) The frequency ωm is assumed to be much smaller than ω0 but much larger than the trap frequency. The radius ρ0 is now defined as ρ0 = ω0/|µBgFB′|, and the “effective −0.05 0.05 −0.05 0.05 −0.05 0 0.05 −0.05 0.05 FIG. 3: (Color online) The geometric phase γ1 for the storage ring of Ref. [19] with (a) Brf = 0.15B ′ρ0 and (b) Brf = 0.05B′ρ0. B-field” takes the form ~B eff(~r, t) = ∆(~r, t)êz − |2 ~Bs(~r, t)| Brf êφ . (37) The operating principle for the time averaged storage ring of Ref. [20] is similar to the well-known TOP [2] and TORT traps [3, 4]. The effective trap potential ex- perienced by the atom is proportional to the time aver- aged value of the “effective B-field” ∫ 2π/ωm | ~B eff(~r, t)|dt. When Brf and Bm are much smaller than B ′ρ0, the center of the storage ring is located approximately at ρc = ρ0, zc = 0. Using the earlier result [33], we find that in the time averaged storage ring, the effective gauge po- tential à n (ρ, z) is reduced simply to the time averaged instantaneous gauge potential Ã(φ)n (ρ, z) = ∫ 2π/ωm A(φ)n (ρ, z, t)dt, (38) with A n (ρ, z, t) given in (28). The geometric phase then is given approximately by γn(ρ, z) = 2πà n (ρ, z). In this case, we find that the geometric phase always vanishes at −0.05 −0.05 −0.2 0 (ρ−ρ0)/ρ0 −0.05 −0.05 z/ρ0 FIG. 4: (Color online) The geometric phase γ1 for the storage ring of Ref. [20] at (a) Brf = 0.3B ′ρ0 and (b) Brf = 0.1B Bm = 0.05B the trap center (ρc, zc). Figure 4 illustrates the distribu- tion of the geometric phase in the region near the trap center for two different values of the rf-field amplitude Brf . Similar to the storage ring of Ref. [19], the fluctua- tion of the geometric phase is suppressed in this case for large Brf . B. The storage ring proposals of Refs. [21, 22] Next we consider the ARFP based storage ring pro- posed in Refs. [21, 22]. In this case, the static B-field is that of a Ioffe-Pritchard trap on an atom chip. In the Cartesian coordinate (x, y, z), it takes the form ~Bs = B ′xêx −B′yêy +B′Lêz, (39) where B′ is the B-field gradient and the bias field along the z-direction is denoted as B′L. The amplitudes ~B and ~B rf (z) of the rf field are rf = [Brf(z)/ 2]êx and rf = [Brf(z)/ 2]êy with Brf(z) = B rf +B ′′z2. (40) In the schemes of Ref. [21, 22] considered earlier, the phase η of the rf field is assumed to be κπ/2. The x- and y-components of the “effective B-field” ~B eff(~r) then become B effx (~r) = Brf(z) (1 + cosβs(ρ, z)), B effy (~r) = 0, (41) according to Eq. (10). Then the strength of the “effective B-field” ~B eff has its minimum along a circle with a non- zero radius ρc, provided a positive detuning ∆ exists at the origin (0, 0, 0) [21, 22]. The “effective B-field” ~B eff is easily shown to lie in the x-z plane along the trap bottom mapped out by the atomic center of mass motion. This gives rise to a vanishing γ F . With a proper choice for the local phase of |n(~r)〉eff , the gauge potential A(φ)n takes the form A(φ)n (ρ, z) = cosβeff(ρ, z) (1− cosβs(ρ, z)) . (42) Figure 5 displays the geometric phase along a closed path for a spin-1 atom as a function of ρc for the ARFP storage ring proposed in Refs. [21, 22]. The parameter λ is defined as ∆[~r = 0] |gFµBB(0)rf | . (43) To assure the validity of the rotating wave approx- imation, we find that the maximal values of ∆[~r = 0]/(|gF |µB) and B(0)rf / 2 must be restricted to the re- gion of λ ∈ [0, 0.15]. As shown in Fig. 1, the geometric phase remains much smaller than 2π in this situation. This fact can be appre- ciated easily if we look at the distribution of the “effec- tive B-field” ~B eff . According to Eq. (41), the component B effx has a nonzero minimal value Brf/(2 2), while |Beffz | can become arbitrarily small, although not necessarily zero in general. Therefore, at the trap center where | ~B eff | is a minimum, the value of cosβeff = B z /| ~B eff | can be- come very small, leading to small geometric phases. Yet, despite the relatively small geometric phase found here, our result remains important because it could represent a systematic error if not properly included in a Sagnac interference experiment. In Fig. 6, we show the spatial distribution of the ge- ometric phase γ1 around the trap center with λ = 1/3 and λ = 3. The fluctuation is found to be relatively small when λ is small or when the rf-field amplitude Brf is large. Although not discussed in Refs. [21, 22], a ring shaped trap also can be realized if we take η = −κπ/2. The “effective B-field” ~B eff still lies in the x-y plane ~B effx (~r) = − Brf(z) cos(2φ)(1 − cosβs(ρ, z)), ~B effy (~r) = Brf(z) sin(2φ)(1 − cosβs(ρ, z)), (44) 0 0.1 0.2 0.3 0.4 0.5 0.6 −0.1 −0.08 −0.06 −0.04 −0.02 0 λ=1/3 FIG. 5: (Color online) The geometric phase γ1 is plotted against the radius ρc for the ARFP storage ring of Ref. [21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3. To assure the validity of the rotating wave approximation, in the solid lines, the maximal value of ∆[~r = 0]/|gFµBB′ρ0| or Brf/( 2B′ρ0) are restricted to be smaller than 0.15. The extending dashed line is beyond the rotating wave approxi- mation for λ = 1/3 and Brf/( 2B′ρ0) ∈ [0.15, 0.3]. clearly giving rise to a non-zero solid angle with respect to a closed path along the storage ring. Therefore, the term (eff) F is non-zero in this case. We choose the eigenstates |n(~r)〉s and |n(~r)〉eff as |n(~r)〉s = exp[−i ~F · n̂s(~r)βs(ρ, z)]|n〉z, |n(~r)〉eff = exp[−i ~F · n̂ eff⊥ (~r)βeff(ρ, z)]|n〉z . (45) with the unit vector n̂s⊥(~r) in the x-y plane orthogonal to ~Bs(~r). In this case, the gauge potential A n can be expressed as A(φ)n (ρ, z) = cosβeff(ρ, z)[1 + cosβs(ρ, z)]. (46) In Figure 7, we show the fluctuation of the geometric phase γ1 for a closed path with a new parameter λ′ = 6 ∆[~r = 0] |gFµBB(0)rf | , (47) equal to 3 and 1/3. The fluctuation for γ1 is found to be much larger than the case of η = κπ/2, which can be explained by the transverse components B effx,y of the “ef- fective B-field.” Because cosβs is always close to unity. In the case of η = −κπ/2, B effx,y can take only small pos- itive values. Therefore, at the minimum of the ARFP trap ρ = ρ0 of | ~B eff |, both B effz and B effx,y have to be close to zero. In this case the value for cosβeff becomes a rapidly changing function of ρ in the region near ρc. Our above calculations have obtained analytical ex- pressions of the geometric phases in an ARFP based stor- age ring for η = ±κπ/2. We have further investigated the −0.05 0.05 −0.05 0.05 −0.06 −0.02 −0.05 0.05 −0.05 0.05 FIG. 6: (Color online) The spatial distribution of the geo- metric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ = 1/3 and (b) λ = 3. η = κπ/2. B rf = 0.08B ′L and B′′ = 10−12B′/L are assumed. fluctuations of the geometric phase for the two cases of η = ±κπ/2. It seems one benefits from implementing a Sagnac interferometer in the discussed ARFP storage ring with η = κπ/2 and operating at a relatively large λ. Before proceeding onto the concluding section, we will discuss the geometric phase in an ARFP based beam splitter created via a double potential [14, 21]. In such an implementation, the static field ~Bs is created from a Ioffe- Pritchard trap, while the oscillating rf field components are ~B rf = Brf [z]êx and rf = 0. By spatially tuning the amplitude of Brf from zero to a significant value, in the x-y plane, an ARFP can be tuned from a single well centered near the origin to a double well with two minimal points at the point with nonzero radius ρ0 and φ = 0, π. Therefore, a Y-shaped atom beam splitter can be accomplished when the Brf [z] initially is increased along the z-axis to a large value, and then decreased to zero. In such an arrangement, the atom beam moving along the z direction can be separated into two beams that move along the z-axis at φ = 0, π for a while, and then can be recombined again into a single beam. −0.05 0.05 −0.05 0.05 −0.05 0.05 −0.05 0 0.05 FIG. 7: (Color online) The spatial distribution of the geo- metric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ′ = 3 and (b)λ′ = 1/3. η = −κπ/2. B(0)rf = 0.08B ′L and B′′ = 10−12B′/L are assumed. In the atom interferometer considered above, both the static field ~Bs and the “effective B-field” ~B eff are limited to the x-z plane. Therefore, for motion along the closed path of the trap bottom, the solid angle enclosed by the trajectory of ~B eff is zero. Thus, the geometric phase in (16) can be expressed as γn(t) = −i | eff〈n(~r)|l〉z|2 s〈l(~r)|∇|l(~r)〉s · ~vdt′. (48) We can show that the product eff〈n(~r)|l〉z|2 s〈l(~r)|∇|l(~r)〉s is a function of ρc and is independent of z. Thus, the geometric phase can be expressed as an integral of this function with respect to ρc, from zero to a large value and then back to zero. Therefore, the value of the geometric phase would be zero in the end. V. CONCLUSION In this study, we develop theoretical formalisms for the calculation of the atomic geometric phase inside an ARFP. We show that, due to the complexity of the ARFP, the geometric phase depends on the spatial vari- ation of both the static field and an “effective B-field” ~B eff . We provide general expressions for the geometric phase and the corresponding adiabatic gauge potential in Eqs. (16) and (23), respectively. To shed light on actual applications of the atomic ge- ometric phase, we investigate the distribution of atomic geometric phases for several proposed or ongoing exper- iments with ARFP based storage rings and atom beam splitters. We prove rigorously that the geometric phase in the center of the storage rings proposed in Refs. [18, 19] is always zero. In addition, we find that in the storage ring of Ref. [18], the spatial fluctuation of the geometric phase sensitively depends on the position of the trap cen- ter on the “resonance toroid.” In the proposals of Refs. [19, 20, 21, 22], the fluctuation for the geometric phase becomes significantly suppressed when the amplitude Brf of the rf-field is large. In the proposals of [21, 22], the fluctuations of the geometric phase also is suppressed if the angle η is set to be κ2π. 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0704.0477

The Transition from the First Stars to the Second Stars in the Early Universe

The Transition from the First Stars to the Second Stars in the Early Universe Britton D. Smith and Steinn Sigurdsson 525 Davey Laboratory, Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA, 16802 [email protected],[email protected] ABSTRACT We observe a sharp transition from a singular, high-mass mode of star forma- tion, to a low-mass dominated mode, in numerical simulations, at a metallicity of 10−3 Z⊙. We incorporate a new method for including the radiative cooling from metals into adaptive mesh-refinement hydrodynamic simulations. Our re- sults illustrate how metals, produced by the first stars, led to a transition from the high-mass star formation mode of Pop III stars, to the low-mass mode that dominates today. We ran hydrodynamic simulations with cosmological initial conditions in the standard ΛCDM model, with metallicities, from zero to 10−2 Z⊙, beginnning at redshift, z = 99. The simulations were run until a dense core forms at the center of a 5 × 105 M⊙ dark matter halo, at z ∼ 18. Analysis of the central 1 M⊙ core reveals that the two simulations with the lowest metallicities, Z = 0 and 10−4 Z⊙, contain one clump with 99% of the mass, while the two with metallicities, Z = 10−3 and 10−2 Z⊙, each contain two clumps that share most of the mass. The Z = 10−3 Z⊙ simulation also produced two low-mass proto-stellar objects with masses between 10−2 and 10−1 M⊙. Gas with Z ≥ 10 −3 Z⊙ is able to cool to the temperature of the CMB, which sets a lower limit to the minimum fragmentation mass. This suggests that the second generation stars produced a spectrum of lower mass stars, but were still more massive on average than stars formed in the local universe. Subject headings: stars: formation 1. Introduction Numerical simulations have shown that the very first stars invariably formed in isolation and were much more massive than the sun, due mainly to the inability of primordial gas http://arxiv.org/abs/0704.0477v1 – 2 – to efficiently cool at low temperatures (Abel et al. 2002; Bromm et al. 2002; Yoshida et al. 2006). Tumlinson et al. (2004) have suggested that the Pop III IMF was not dominated by very massive stars (M > 140 M⊙), but instead by stars with M = 8–40 M⊙. Even this IMF, though, is still remarkably distinct from that observed for the local universe, which peaks at less than one solar mass (Miller & Scalo 1979; Kroupa 2002; Chabrier 2003). The deaths of the first stars produced and distributed copious amounts of metals into their surroundings, through either core-collapse (M ∼> 10 M⊙) or pair-instability (M ∼> 140 M⊙) supernovae (Heger & Woosley 2002). These metals provide additional avenues for ra- diative cooling of the ambient gas, through fine-structure and molecular transitions, as well as continuum emission from dust formed from the supernova ejecta, permitting the gas that will form the next generation of stars to reach temperatures lower than what is possible for metal-free gas. Fragmentation of collapsing gas will continue so long as the gas can keep decreasing in temperature as the density increases (Larson 2005), or until the gas becomes optically thick to its own emission (Low & Lynden-Bell 1976). The minimum fragment mass is determined by the local Jeans mass, MJ ≃ 700 M⊙(T/200K) 3/2(n/104cm−3)−1/2(µ/2)−2, (1) where T, n, and µ are the temperature, number density, and mean molecular weight, at the halt of fragmentation (Larson 2005). For metal-free gas, a minimum temperature of ∼ 200 K is reached at n ≃ 104 cm−3 when H2 cooling becomes inefficient, yielding a Jeans mass, MJ ≃ 10 3 M⊙ (Abel et al. 2002; Bromm et al. 2002). At some certain chemical abundance, it is conjectured that metals provide sufficient cooling, so that the temperature of the gas continues to decrease as the density increases past the stalling point for metal-free gas, allowing the collapsing gas-cloud to undergo fragmentation and form smaller and smaller clumps. The enrichment of gas to some critical metallicity, Zcr, will trigger the formation of the first low-mass (Pop II) stars in the universe, as the gas can cool to lower temperatures at higher metallicity, in general. The value of Zcr can be estimated by calculating the metallicity required to produce a cooling rate equal to the rate of adiabatic compression heating at a given temperature and density. This has been carried out for individual alpha elements, such as C and O, by Bromm & Loeb (2003), and C, O, Si, Fe, as well as solar abundance patterns by Santoro & Shull (2006), yielding roughly, 10−3.5 Z⊙ ∼< Zcr ∼< 10 −3 Z⊙. Aside from the minimum clump mass, however, not much more can be said about the spectrum of clump masses produced during fragmentation. Omukai et al. (2005) use one- zone models with very sophisticated chemical networks to follow the evolution of temperature and density in the center of a collapsing gas cloud, for a range of metallicities. The predic- tions of fragmentation from this work, though, are based solely on statistical arguments of elongation in prestellar cores and do not capture the complex processes of interaction and ac- – 3 – cretion associated with the formation of multiple stars (Bate et al. 2003). Tsuribe & Omukai (2006) simulate the high density (n ≥ 1010 cm−3) evolution of extremely low-metallicity gas (Z < 10−4 Z⊙), but the conclusions of this work are limited by the fact that the simulations are initialized at an extremely late phase in the evolution of the prestellar core. The nu- merical simulations by Bromm et al. (2001), which use cosmological initial conditions, show fragmentation in gas with Z = 10−3 Z⊙, but a mass resolution of 100 M⊙ prevents this study from saying anything conclusive about the formation of sub-stellar mass objects. In this paper, we present the results of three-dimensional hydrodynamic simulations of metal-enriched star-formation. These simulations are similar in nature to those of Bromm et al. (2001), but with vastly improved numerical methods and updated physics. We describe the setup of our simulations in §2, with the results in §3 and a discussion of the consequences of this work in §4. 2. Simulation Setup We perform a series of four simulations, with constant metallicities, Z = 0 (metal-free), 10−4 Z⊙, 10 −3 Z⊙, and 10 −2 Z⊙, using the Eulerian adaptive mesh refinement hydrodynamics/N- body code, Enzo (Bryan & Norman 1997; O’Shea et al. 2004). The metallicity is held con- stant throughout each simulation in order to isolate the role of heavy element concentration in altering the dynamics of collapse compared to the identical metal-free case. In reality, metals will be injected over time into star forming gas by Pop III supernova blast waves, and the mixing of those metals with the gas will not be completely uniform. Here we fo- cus on an idealized approximation in order to capture the essential physics of collapse and fragmentation. Each simulation begins at z = 99, in a cube, 300 h−1 kpc comoving per side, in a ΛCDM universe, with the following cosmological parameters: ΩM = 0.3, ΩΛ = 0.7, ΩB = 0.04, and Hubble constant, h = 0.7, in units of 100 km s−1 Mpc−1. We initialize all the simulations identically, with a power spectrum of density fluctuations given by Eisenstein & Hu (1999), with σ8 = 0.9 and n = 1. The computation box consists of a top grid, with 128 3 cells, and three static subgrids, refining by a factor of 2 each. This gives the central refined region, which is 1/64 the total computational volume, an effective top grid resolution of 10243 cells. The grid is centered on the location of a ∼ 5 × 105 M⊙ dark matter halo that is observed to form at z ∼ 18 in a prior dark-matter-only simulation, as is done similarly in Abel et al. (2002); O’Shea et al. (2005). Refinement occurs during the simulations whenever the gas, or dark matter, density is greater than the mean density by a factor of 4, or 8, respectively. We also require that the local Jeans length be resolved by at least 16 grid cells at all times – 4 – in order to avoid artificial fragmentation as prescribed by Truelove et al. (1997). To include the radiative cooling processes from the heavy elements, we use the method described in Smith, Sigurdsson, & Abel (2007), in preparation. The nonequilibrium abun- dances and cooling rates of H, H+, H−, He, He+, He++, H2, H 2 , and e − are calculated internally, as in Abel et al. (2002); Anninos et al. (1997). Meanwhile, the metal cooling rates are interpolated from large grids of values, precomputed with the photoionization soft- ware, CLOUDY (Ferland et al. 1998). We ignore the cooling from dust and focus only on the contribution of gas-phase metals in the optically-thin limit. Unlike other studies of the formation of the first metal-enriched structures, we do not assume the presence of an ionizing UV background. In our model, the singular pop III star that was associated with the dark matter halo in which our stars form has already died in a supernova. We also assume any other Pop III stars are too distant to affect the local star-forming region and that QSOs have yet to form. We use the coronoal equilibrium command when constructing the cooling data in CLOUDY to simulate a gas where all ionization is collisional. The metal cooling data was created using the Linux cluster, Lion-xo, run by the High Performance Computing Group at The Pennsylvania State University. As a consequence of our choice to ignore any external radiation, we do not observe the fine-structure emission of [C ii] (157.74 µm) that was reported by Santoro & Shull (2006) to be important. Instead, cooling from C comes in the form of fine-structure lines of [C i] (369.7 µm, 609.2 µm). The cooling from [C i] in our study dominates in the same range of densities and temperatures as the cooling from [C ii] in Santoro & Shull (2006). We observe the contributions of the other coolants studied by Santoro & Shull (2006), [O i], [Si ii], and [Fe ii], to be in agreement with their work. In addition, we find that emission from [S i] (25.19 µm) dominates the cooling from metals at n ∼ 107 cm−3 and T ∼ 1–3 × 103 K. The absence of UV radiation in our simulations also allows H2 to form, differentiating this study from Bromm et al. (2001). This allows for a more direct comparison between the metal-free and metal-enriched cases. The simulations are run until one or more dense cores form at the center of the dark matter halo and a maximum refinement level of 28 is reached for the first time, giving us a dynamic range of greater than 1010. Only the simulation with Z = 10−2 Z⊙ reached 28 levels of refinement. The three other simulations were stopped after reaching 27 refinement levels, since their central densities were already higher than the simulation with Z = 10−2 Z⊙. Table 1 summarizes the final state of each simulation, where zcol is the collapse redshift, lmax is the highest level of refinement, nmax is the maximum gas density within the box, and ∆tcol is the time difference to collapse from the metal-free simulation. – 5 – 3. Results As can be seen in Table 1, the runs with higher metallicities reach the runaway collapse phase faster. The relationship between metallicity relative to solar and ∆tcol is well fit by a power-law with index, n ≃ 0.22. Gas-clouds with more metals are able to radiate away their thermal energy more quickly, and thus, collapse faster. An inverse relation between metallicity and the number of grids and grid-cells exists because the low-density, background gas evolves at roughly the same rate in all simulations, yet has more time, in the runs with lower metallicities, with which to collapse to higher density, requiring additional refinement. Our simulations, shown in Figure 1, display a qualitative transition in behavior between metallicities of 10−4 Z⊙ and 10 −3 Z⊙. In the runs with the highest metallicities (Figure 1C and 1D), the central core is extremely asymmetric, and multiple density maxima are clearly visible. All four runs display similar large-scale density profiles (Figure 2A). Radiative cooling from H2 becomes extremely inefficient below T ∼ 200 K, creating the effective temperature floor, visible in Figure 2B for the metal-free case (Abel et al. 2002; Bromm et al. 2002). At n ≃ 104 cm−3, the rotational levels of H2 are populated according to LTE, reducing the cooling efficiency and causing the temperature to increase (Abel et al. 2002; Bromm et al. 2002). In the isothermal collapse model of Shu (1977), the accretion rate is proportional to the cube of the sound speed. The increase in temperature leads to an increase in the accretion rate, causing the density, and thus, the enclosed mass (Figure 2C), to be slightly higher inside the central ∼ 0.1 pc in the metal-free case. A similar situation occurs further within for the Z = 10−4 Z⊙ and, later, the 10 −3 Z⊙ cases, as the metal cooling is overwhelmed by adiabatic compression heating and the temperature begins to rise with density. The presence of metals at the level of 10−4 Z⊙ enhances the cooling enough to lower the gas temperature to ∼ 75 K. Metallicities greater than 10−3 Z⊙ provide sufficient cooling to bring the gas down to the temperature of the cosmic microwave background, where TCMB ≃ 2.7 K (1 + z). The gas temperatures are in general agreement with the calculations of Omukai et al. (2005) that include a CMB spectrum at z = 20. Fragmentation requires that the cooling time be less than the dynamical time. Figure 2D shows that this criterion is essentially never met in the zero metallicity case, and only marginally in the Z = 10−4 Z⊙ case. However, the fragmentation criterion is more than satisfied in the Z = 10−3 Z⊙ and 10 −2 Z⊙ cases over a wide mass-range. In order to locate fragments within our simulations, we employ an algorithm, based on Williams et al. (1994), that works by identifying isolated density countours. Before we search for clumps, we smooth the density field by assigning each grid-cell the mass-weighted mean density of the group of cells including itself and its neighbors within one cell-width. This serves to eliminate small density perturbations that would be misidentified as clumps by – 6 – the code. In order to directly compare the fragmentation from each simulation, we limit the search for clumps to the 1 M⊙ of gas surrounding the cell with the highest density. On larger scales, all of the runs display a filamentary structure that is qualitatively similar. No other region in any of the simulation boxes has collapsed to densities comparable to those found within the region where the clump search is performed. The results are shown in Figure 3. A single clump exists in the metal-free and 10−4 Z⊙ simulations, containing 99.7% of the total mass within the region of interest. In the simulation with Z = 10−3 Z⊙, 91% of the mass is shared between two clumps with 0.52 M⊙ and 0.39 M⊙. In the same simulation, we also find two smaller clumps 0.06 M⊙ and 0.02 M⊙. Finally, in the Z = 10 −2 Z⊙ simulation, we see two clumps with 0.79 M⊙ and 0.21 M⊙. 4. Discussion We have shown, through three-dimensional hydrodynamic simulations, that fragmen- tation occurs in collapsing gas with metallicities, Z ≥ 10−3 Z⊙. Our results indicate that star-formation occurs in exactly the same manner at metallicity, Z = 10−4 Z⊙, as it does at zero metallicity. The similarities between the simulations with metallicities, Z = 10−3 Z⊙ and 10−2 Z⊙, suggest that the transition to low-mass star-formation is complete by 10 −3 Z⊙, implying that the entire transition occurs over only one order of magnitude in metal abun- dance. More simulations, bracketing the metallicity range, 10−4 to 10−3 Z⊙, will test how abrupt the transition truly is. We will also explore the effect of non-solar abundances on the low metallicity IMF. It has been recently argued that dust cooling at high densities (n ≥ 1013 cm−3) can induce fragmentation for metallicities as low as 10−6 Z⊙ Schneider et al. (2006). In light of the work by Frebel et al. (2007), who note the absence of stars with Dtrans < -3.5, where Dtrans is a measure of the combined logarithmic abundance of C and O, it seems unlikely that Zcr is this low. While the fragmentation mode discussed in Schneider et al. (2006), and also Omukai et al. (2005), may truly exist, it is possible that metal yields from Pop III supernovae overshoot this metallicity, for realistic mixing scenarios, leaving almost no star-forming regions with such a low concentration of heavy elements. Similar to our results, Omukai et al. (2005) note that only high-mass fragments are produced when Z = 10−4 Z⊙. If Pop III supernovae are able to immediately enrich the local universe to Z = 10 Z⊙, the high-density dust cooling fragmentation mode would be skipped altogether, and the high-mass stars that formed via the mode observed at 10−4 Z⊙ would leave no record in the search for low-metallicity stars in the local universe. We have limited the search for fragments to the dense 1 M⊙ core at the center of each simulation. Within this region, it is unlikely that any more fragments will form in any of – 7 – the simulations. In all of the cases presented, the cooling has begun to be overwhelmed by compression heating such that the central temperature is now increasing with increasing density, which was indicated by Larson (2005) to be the end of hierarchical fragmentation. Fragmentation may continue in the surrounding lower density gas in the cases of Z = 10−3 Z⊙ and 10 −2 Z⊙. The final stellar masses of these objects will also be affected interaction and accretion that will occur in later stages of evolution. In the two lowest metallicity cases, the gas immediately surrounding the central core evolves slowly enough that it will not have sufficient time to reach high densities before the UV radiation from the central, massive star dissociates all of the H2. As was shown by Bromm et al. (2001), clouds with metallicities, Z ≤ 10−4 Z⊙ are unable to collapse without the aid of H2 cooling. In the two simulations in which significant fragmentation is observed, Z = 10−3 and 10−2 Z⊙, the gas is able to cool rapidly to the temperature of the CMB. Wise & Abel (2005) predict that the rate of Pop III supernovae peaks at a redshift, z ∼ 20, and then drops off sharply, implying that metal production from Pop III stars is effectively finished at this point. In this epoch, the characteristic mass-scale for metal-enriched star formation will be regulated by the CMB, as is predicted in Bromm & Loeb (2003). Thus, the first Pop II stars will be considerably more massive, on average, than stars observed today, as was suggested by Larson (1998). Observations of low-mass prestellar cores in the local universe reveal them to have temperatures of about 8.5 K (Evans 1999), implying that the IMF may not become completely ’normal’ until z < 3 when the CMB fell below this temperature. We thank Tom Abel, Greg Bryan, Mike Norman, Brian O’Shea, and Matt Turk for useful discussions. BDS also thanks Michael Kuhlen for providing an update to some useful analysis tools. We are also very grateful for insightful comments from an anonymous referee. This work was made possible by Hubble Space Telescope Theory Grant HST-AR-10978.01, and an allocation from the San Diego Supercomputing Center. REFERENCES Abel, T., Bryan, G. L., & Norman, M. L. 2002, Science, 295, 93 Anninos, P., Zhang, Y., Abel, T., & Norman, M. L. 1997, New Astronomy, 2, 209 Bate, M. R., Bonnell, I. A., & Bromm, V. 2003, MNRAS, 339, 577 Bromm, V., Coppi, P. S., & Larson, R. B. 2002, ApJ, 564, 23 Bromm, V., Ferrara, A., Coppi, P. S., & Larson, R. B. 2001, MNRAS, 328, 969 – 8 – Bromm, V. & Loeb, A. 2003, Nature, 425, 812 Bryan, G. & Norman, M. L. 1997, in Workshop on Structured Adaptive Mech Refinement Grid Methods, ed. N. Chrisochoides, IMA Volumes in Mathematics No. 117 (Springer- Verlag) Chabrier, G. 2003, PASP, 115, 763 Eisenstein, D. J. & Hu, W. 1999, ApJ, 511, 5 Evans, II, N. J. 1999, ARA&A, 37, 311 Ferland, G. J., Korista, K. T., Verner, D. A., Ferguson, J. W., Kingdon, J. B., & Verner, E. M. 1998, PASP, 110, 761 Frebel, A., Johnson, J. L., & Bromm, V. 2007, ArXiv Astrophysics e-prints Heger, A. & Woosley, S. E. 2002, ApJ, 567, 532 Kroupa, P. 2002, Science, 295, 82 Larson, R. B. 1998, MNRAS, 301, 569 —. 2005, MNRAS, 359, 211 Low, C. & Lynden-Bell, D. 1976, MNRAS, 176, 367 Miller, G. E. & Scalo, J. M. 1979, ApJS, 41, 513 Omukai, K., Tsuribe, T., Schneider, R., & Ferrara, A. 2005, ApJ, 626, 627 O’Shea, B. W., Abel, T., Whalen, D., & Norman, M. L. 2005, ApJ, 628, L5 O’Shea, B. W., G., B., Bordner, J., Norman, M. L., Abel, T., Harknes, R., & Kritsuk, A. 2004, in Lecture Notes in Computational Science and Engineering, Vol. 41, Adaptive Mesh Refinement - Theory and Applications, ed. T. Plewa, T. Linde, & V. G. Weirs Santoro, F. & Shull, J. M. 2006, ApJ, 643, 26 Schneider, R., Omukai, K., Inoue, A. K., & Ferrara, A. 2006, MNRAS, 369, 1437 Shu, F. H. 1977, ApJ, 214, 488 Truelove, J. K., Klein, R. I., McKee, C. F., Holliman, II, J. H., Howell, L. H., & Greenough, J. A. 1997, ApJ, 489, L179+ – 9 – Tsuribe, T. & Omukai, K. 2006, ApJ, 642, L61 Tumlinson, J., Venkatesan, A., & Shull, J. M. 2004, ApJ, 612, 602 Williams, J. P., de Geus, E. J., & Blitz, L. 1994, ApJ, 428, 693 Wise, J. H. & Abel, T. 2005, ApJ, 629, 615 Yoshida, N., Omukai, K., Hernquist, L., & Abel, T. 2006, ApJ, 652, 6 This preprint was prepared with the AAS LATEX macros v5.2. – 10 – Table 1 Simulation Final States Z (Z⊙) zcol lmax Grids Cells nmax (cm −3) ∆tcol (Myr) 0 18.231519 27 8469 4.82 ×107 4.11 ×1013 - 10−4 18.838816 27 8060 4.64 ×107 3.90 ×1013 9.19 10−3 19.336557 27 7911 4.56 ×107 1.65 ×1013 16.21 10−2 20.032518 28 7521 4.42 ×107 1.50 ×1013 25.33 – 11 – Fig. 1.— Slices through gas density for the final output of simulations with Z = 0 (A), 10−4 Z⊙ (B), 10 −3 Z⊙ (C), and 10 −2 Z⊙ (D). Each slice intersects the grid-cell with the highest gas density and has a width of 2 × 10−8 of the computation box, corresponding to a proper size of ∼4 × 10−4 pc (84 AU). The color-bar at bottom ascends logarithmically, from left to right, spanning exactly four orders of magnitude in density. – 12 – Fig. 2.— Radially averaged, mass-weighted quantities for the final output each simulation: Z = 0 (red), 10−4 Z⊙ (green), 10 −3 Z⊙ (blue), and 10 −2 Z⊙ (purple). A: Number density vs. radius. B: Temperature vs. enclosed mass. C: Enclosed gas mass vs. radius. D: Ratio of crossing time to cooling time vs. enclosed mass. The classical criterion for fragmentation is met when the ratio of the crossing time to the cooling time is greater than 1. – 13 – Fig. 3.— Masses of clumps found within the final output of each simulation. The location on the x and y axes corresponds to the log of the clump mass and the metallicity of the simulation. Colors are the same as in Figure 2. The radii of the circles are proportional to the masses of the clumps they represent. A factor of 10 in mass is equivalent to a factor of 2 in radius. The search for clumps is limited to the 1 M⊙ surrounding the grid cell with the highest gas density. Only clumps with at least 1000 cells are plotted. Introduction Simulation Setup Results Discussion

0704.0478

Super Star Cluster Velocity Dispersions and Virial Masses in the M82 Nuclear Starburst

Accepted by the Astrophysical Journal, March 7, 2007 Preprint typeset using LATEX style emulateapj v. 03/07/07 SUPER–STAR CLUSTER VELOCITY DISPERSIONS AND VIRIAL MASSES IN THE M82 NUCLEAR STARBURST1 Nate McCrady and James R. Graham Department of Astronomy, University of California, Berkeley, CA 94720-3411 Accepted by the Astrophysical Journal, March 7, 2007 ABSTRACT We use high-resolution near-infrared spectroscopy from Keck Observatory to measure the stellar velocity dispersions of 19 super star clusters (SSCs) in the nuclear starburst of M82. The clusters have ages on the order of 10 Myr, which is many times longer than the crossing times implied by their velocity dispersions and radii. We therefore apply the Virial Theorem to derive the kinematic mass for 15 of the SSCs. The SSCs have masses of 2× 105 to 4× 106 M⊙ , with a total population mass of 1.4 × 107 M⊙ . Comparison of the loci of the young M82 SSCs and old Milky Way globular clusters in a plot of radius versus velocity dispersion suggests that the SSCs are a population of potential globular clusters. We present the mass function for the SSCs, and find a power law fit with an index of γ = −1.91±0.06. This result is nearly identical to the mass function of young SSCs in the Antennae galaxies. Subject headings: galaxies: individual (M82) — galaxies: starburst — galaxies: star clusters — in- frared: galaxies 1. INTRODUCTION 1.1. Starburst Galaxies and Super Star Clusters Short-duration episodes of intense star formation known as “starbursts” are responsible for a significant portion of star formation activity in the present-day Uni- verse. Heckman (1998) estimates that the four most lu- minous circumnuclear starbursts (M82, NGC 253, M83 and NGC 4945) account for 25 percent of the high-mass (> 8 M⊙ ) star formation within 10 Mpc. The star- burst phenomenon is the present-day manifestation of the dominant mode of star formation in the early Uni- verse (Leitherer 2001). At z = 0, high-mass stars form predominantly in dense clusters and OB associations (Miller & Scalo 1978). Massive stellar clusters in nearby starburst galaxies thus provide a laboratory for studying intense star formation and related feedback processes, as well as physical conditions analogous to high-redshift star formation. Star formation in starbursts is resolved into young, dense, massive “super–star clusters” (SSCs) that rep- resent a substantial fraction of new stars formed in a burst event (Meurer et al. 1995; Zepf et al. 1999). Hubble Space Telescope (HST) observations with WFPC/WFPC2 in visible light (e.g., O’Connell et al. 1994; Whitmore & Schweizer 1995) have resolved SSCs in the nearest starburst galaxies, and SSCs appear ubiq- uitous in mergers and interacting galaxies. Of the roughly 30 gas-rich mergers observed by HST, all have young, massive, compact clusters (Whitmore 2001, and refs.). A spectacular example is the “Antennae” galax- 1 Based on observations made at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the Na- tional Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. 2 Now at Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547 3 [email protected] ies, NGC 4038/4039, with more than 103 optically-visible SSCs (Whitmore et al. 1999) and many other clusters deeply embedded in dust (Gilbert et al. 2000). The derived masses, radii and ages of SSCs suggest that they are young globular clusters, and the brightest and most massive may evolve into a population of glob- ular clusters similar to that of the Milky Way. The ques- tion of whether SSCs are in fact the progenitors of glob- ular clusters depends critically on their masses and their content of low-mass stars in particular (Meurer et al. 1995). If the stellar initial mass function (IMF) within the clusters is biased toward high-mass stars (or “top heavy”), for example by suppression of low mass star formation, the clusters may not survive the disruptive nature of mass loss resulting from both stellar evolu- tion and dynamical processes (e.g. Chernoff & Weinberg 1990; Takahashi & Portegies Zwart 2000). Local analogues of SSCs — massive, dense young clus- ters in the Galaxy and Large Magellanic Cloud — con- tain substantial populations of low-mass stars. The cen- tral ionizing star cluster in NGC 3603, the most massive H II region in the Galaxy, has 2000 M⊙ in OB stars alone. Brandl et al. (1999) find that the mass spectrum of the cluster is “well populated” down to 0.1 M⊙ . The clus- ter R136 at the center of the 30 Doradus nebula in the Large Magellanic Cloud is considered the closest example of a starburst region (Brandl et al. 1996). In addition to ∼ 103 OB stars, some with masses in excess of 100 M⊙ , Sirianni et al. (2000) find a “substantial population” of low-mass stars down to masses of 0.6 M⊙ in R136. With a cluster mass of ∼ 2 × 104 M⊙ (Walborn et al. 2002), R136 is at the lower end of the SSC mass range. 1.2. Mass Estimates Most mass estimates to date for SSCs beyond the Lo- cal Group are “photometric masses.” This technique in- volves measuring the luminosity and broadband color of a cluster, and comparing the results to the predictions of spectral synthesis models. Examples include optical http://arxiv.org/abs/0704.0478v1 2 McCRADY & GRAHAM studies of clusters in the Antennae galaxies (Whitmore 2000, and refs.) and the nuclear cluster in IC 4449 (Böker et al. 2001). The resulting values are highly de- pendent on the assumed IMF, age estimates and theo- retical stellar evolution models. Consequently, the pho- tometric method provides only very weak constraints on the IMF in a cluster. The question of how SSCs evolve or whether they can survive to become globular clusters cannot be directly addressed with confidence by photom- etry alone. Recently, however, several studies have obtained kine- matic masses for SSCs in starbursts directly from obser- vations of stellar velocity dispersions. Ho & Filippenko (1996b) use high-resolution optical spectra to measure the velocity dispersion of an SSC in the nearby amor- phous galaxy NGC 1705. They derive a cluster mass of (8.2± 2.1)× 104 M⊙ . The dwarf starburst galaxy NGC 1569 contains two prominent SSCs. Ho & Filippenko (1996a) used optical spectroscopy to determine the ve- locity dispersion and derive a mass of (2 − 6) × 105 M⊙ for SSC-A. Gilbert (2002) uses near-IR spectroscopy to identify two separate velocity components along the line of sight at the position of SSC-A, and finds masses of 3.0× 105 M⊙ for cluster A1 and 3.4× 10 5 M⊙ for clus- ter A2. Gilbert also finds a velocity dispersion for SSC-B and derives a mass of 1.8×105 M⊙ . Mengel et al. (2002) use high-resolution optical and near-IR spectroscopy to measure the velocity dispersions of six of the brightest clusters in the merging Antennae galaxies. They derive mass estimates ranging from 6.5× 105 to 4.7× 106 M⊙ . In addition to these clusters in starburst galaxies, the kinematic approach has been applied to several young, massive clusters in more quiescent galaxies. Böker et al. (1999) uses near-IR spectra to derive a mass of 6.6× 106 M⊙ for the nuclear star cluster in the giant spiral IC 342. Larsen et al. (2004) derives virial masses for two SSCs in the dwarf irregular NGC 4214 and two SSCs in NGC 4449 based on optical spectra, and one young SSC in the nearby spiral NGC 6946 based on near-IR spectra. Clusters in the Larsen study have masses between 2×105 and 1.8× 106 M⊙ , typical of SSCs. Mass estimates based on measured velocity dispersions are nearly independent of theoretical models, relying on simple application of the virial theorem. Armed with the virial mass, one can derive the light-to-mass ratio of an SSC, which may be compared to predictions of pop- ulation synthesis models to constrain the cluster’s IMF. Critical to understanding the IMF is detection of low- mass stars, the light of which is swamped by high lumi- nosity supergiants. Measurement of the kinematic mass provides the only means of detecting and quantifying the low-mass stellar population of a cluster based on its in- tegrated light. 1.3. Messier 82 M82 (NGC 3034) provides a useful laboratory. As one of the nearest starburst galaxies at 3.6 Mpc (Freedman et al. 1994), M82 presents an obvious reso- lution advantage: star-forming regions can be studied on small spatial scales (1′′ = 17.5 pc) and individual SSCs are resolved by HST. The galaxy’s high inclination of 81◦ (Achtermann & Lacy 1995) and prevalent dust lead to large, patchy extinction; infrared observations are re- quired to overcome this obstacle in characterizing the SSC population. The light of red supergiant stars (RSGs) dominates the near-IR continuum throughout the galaxy’s star- burst core. Satyapal et al. (1997) interpret the compact sources along the plane of M82 as young star clusters, the space density of which increase towards the nucleus. The smooth component of the near-IR emission is itself likely the integrated contribution from unresolved clus- ters of RSGs (Förster Schreiber 1998). Based on mid-IR observations, Lipscy & Plavchan (2004) find that at least 20 percent of the star formation in M82 is occurring in SSCs. HST/NICMOS images of the region (Figure 1) show many luminous SSCs within ∼ 300 pc of the nu- cleus. The nuclear starburst is “active” in the sense that the typical age for the starburst clusters is ∼ 107 years (Satyapal et al. 1997). Evolutionary synthesis models by Förster Schreiber (1998) suggest the nuclear starburst (i.e., the central 450 pc) consists of two distinct, short duration events with ages of about 5 Myr and 10 Myr. The most intense star formation took place parallel to the plane of the galaxy with a peak near the nucleus. O’Connell et al. (1995) image M82 in the V and I bands with the high-resolution Planetary Camera aboard HST, identifying over 100 SSCs within a few hundred parsecs of the nucleus. de Grijs et al. (2001) images a region in the disk of M82, 1 kpc from the nuclear starburst, with WFPC2 and NICMOS. They identify 113 SSC candi- dates which were part of a starburst episode ∼ 600 Myr ago (a “fossil starburst”), with little star formation in the past 300 Myr. The clusters in the fossil starburst have masses of 104−6 M⊙ . Smith & Gallagher (2001, SG01) estimate an age of 60±20 Myr for the SSC ‘M82-F’, inter- mediate between the ongoing nuclear burst and the fossil burst farther out in the disk. M82-F lies ∼ 500 pc west of the nucleus of M82. McCrady et al. (2005) measures a mass of 5.6× 105 M⊙ for the cluster based on near-IR observations, and found evidence for mass segregation. Early ground-based studies of M82 found evidence of an abnormal IMF. Rieke et al. (1993) uses population synthesis models to constrain the IMF based on the near- IR observations of McLeod et al. (1993). They conclude that the large K-band luminosity of the M82 starburst relative to its dynamical mass requires an IMF that is significantly deficient in low-mass stars (M < 3 M⊙ ). Doane & Mathews (1993) examine the supernova rate, molecular gas mass and total dynamic mass and con- clude that an IMF producing stars of mass > 3 M⊙ easily matches observations, whereas a power-law IMF (e.g., Salpeter 1955) would require an unreasonably small mass of stars in the region prior to the onset of the burst. In contrast to these global studies, Satyapal et al. (1997) use 1′′-resolution near-IR images to identify pointlike sources and find that at this scale starburst models can match observations without invoking a high-mass-biased IMF. High spatial-resolution studies are necessary to in- vestigate star formation in the cluster-rich M82 star- burst. In a pilot study for this article, McCrady et al. (2003) measure the kinematic mass of two clusters using near-IR spectra and imaging. Based on the light-to-mass ratios of the clusters, they find that one (MGG-11) ap- pears to have a top-heavy IMF, whereas the other (MGG- 9) appears consistent with a normal IMF. Measurement of the SSC mass independent of assumptions regarding SUPER–STAR CLUSTERS IN M82 3 the L/M ratio are required to further constrain the stel- lar IMF in the clusters. 1.4. Overview In this article, we measure the virial masses of the su- per star cluster population of the inner ∼ 500 pc of the M82 starburst, extending the work of McCrady et al. (2003). Our aim is to examine star formation in the starburst on the scale of individual super star clusters, regions only a few parsecs in extent, with an eye towards placing constraints on the IMF of individual SSCs. We use high-spectral-resolution near-IR spectroscopy from the W.M. Keck Observatory to measure the stellar velocity dispersions and dominant stellar spectral type of the SSCs. We then apply the virial theorem to derive their masses. Clusters for which the age may be deter- mined facilitate investigation of the IMF. In § 2 we de- scribe the kinematic mass, the method we use to measure the velocity dispersion of stars in a cluster, and discuss related systematic effects. In §3 we discuss the NIRSPEC observations, data reduction and spectral extraction. In § 4 we measure cluster velocity dispersions and derive the kinematic masses of the clusters, and present the cluster mass function for the nuclear starburst region. 2. APPROACH The mass of a gravitationally-bound star cluster may be determined by application of the virial theorem (Spitzer 1987). Specifically, the virial mass is a function of two observable quantities: M = 10 where rhp is the half-light radius in projection, σr is the one-dimensional line-of-sight velocity dispersion and G is Newton’s gravitational constant. Half-light radii for the M82 clusters were measured in McCrady et al. (2003) based on HST/NICMOS images. We assume the light profile of the cluster traces the mass distribution, and thus use the measured half-light radius as a proxy for the half-mass radius. In the case of mass segrega- tion, however, this assumption breaks down for near- IR light, and the resulting mass represents a lower-limit (McCrady et al. 2005). The M82 nuclear SSCs are la- belled in a NIRSPEC slit-viewing camera (SCAM) mo- saic shown in Figure 2. HST/NICMOSH-band (F160W) images of the clusters are shown in Figure 3. To measure σr for the clusters, we obtain high-spectral- resolution near-IR integrated light spectra and perform a cross-correlation analysis with template supergiant stars. The near-IR spectrum of a young SSC (i.e., ages < 100 Myr) is dominated by the light of cool, evolved super- giant stars. These highly luminous stars have a large number of molecular and atomic features in the H band. The integrated-light spectrum of an SSC resembles the spectrum of a red supergiant star, the features of which have been “washed out” by the velocity dispersion of stars in the cluster (Figure 4). Our cross correlation method, described in detail in McCrady et al. (2003), returns both the velocity dispersion of the cluster rel- ative to a particular template supergiant and a measure of the similarity of the cluster and supergiant as quan- tified by the peak value of the cross correlation function (CCF). We have prepared an atlas of 19 high-resolution (R ∼ 22, 000) template star spectra in the H band, rang- ing from spectral types G2 through M5 in luminosity class I (Kirian et al. 2006). Results of the cross correla- tion analysis are presented in §4. Determination of the velocity dispersion by cross cor- relation analysis is subject to several potential sources of systematic error. A detailed analysis is presented in McCrady (2005). We present an overview in the follow- ing paragraphs. One potential difficulty is metallicity differences be- tween the Galactic supergiants used as templates and the supergiants producing the cluster light. McLeod et al. (1993, and refs.) cited evidence from emission-line stud- ies of various elements and concluded that the present- day ISM in M82 has solar or slightly higher metallicity. Förster Schreiber et al. (2001) determined that near-IR stellar absorption features observed in the starburst core are consistent with the light from solar-metallicity red supergiants (RSG). Origlia et al. (2004) performed abun- dance analysis on the nuclear starburst region using spec- tral synthesis models for near-IR absorption and X-ray emission. They found an iron abundance roughly half of the solar value, but enhancement of α-elements to so- lar or slightly higher levels. This pattern is consistent with enrichment by recursive bursts of Type II super- novae. The template supergiants used in our analysis also have roughly solar metallicity, and thus we expect that metallicity effects are unlikely to significantly bias the measured cluster velocity dispersions. Cross-correlation with a mismatched template spec- trum can introduce systematic bias to the velocity dis- persion determination. Tests with supergiant spectra broadened with a Gaussian to simulate the effect of a cluster velocity distribution indicate that the cross cor- relation analysis correctly identifies the “best fit” based upon the peak value of the CCF. Increasing the level of added noise decreases the CCF peak, but does not lead to misidentification of the best template. We have elected to cross-correlate the cluster spectra with spectra of sin- gle RSG stars because the light of a young coeval cluster should be dominated by the light of the most massive stars. At an age of ∼ 107 years, this would be the light of evolved massive stars, i.e., the RSG stars. Mixing a composite spectrum (with the inclusion of intermediate mass stars) would generate a better match to the overall spectrum (particularly the depth of absorption features — see below). But for our purposes, it is more important to be able to isolate the width of the lines resulting from the velocity dispersion of cluster stars. In addition to the dominant light of the RSG stars, we expect the cluster spectra to contain a substantial con- tribution from intermediate mass stars still on the main sequence. In the H band, the spectra of A and late-B stars (with masses ∼ 2–6 M⊙ ) are essentially featureless, with the exception of the prominent (and wide) hydro- gen Brackett series absorption lines. The prevalent neb- ular emission in the disk of M82 requires us to avoid the wavelength ranges of these hydrogen lines in our analy- sis. Over the rest of the spectral range in our analysis, an admixture of intermediate mass star spectra would only change the slope of the spectrum (as the near-IR spectra of these stars are essentially thermal). One step in our analysis is the removal of any continuum slope, 4 McCRADY & GRAHAM as the cross-correlation technique is used to measure the velocity dispersion, information which is contained in the width of the absorption lines, not in the depth of the lines or in the continuum. As such, addition of intermediate mass star spectra would not affect the measured velocity dispersions or virial masses. Filtering of the spectra in Fourier space is a source of systematic error. Extracted spectra (§3) are cross- correlated with the spectra of template supergiant stars. The spectra are baseline-subtracted, apodized and both high- and low-pass filtered in Fourier space. At the high- frequency end, the cross correlation result is affected by random noise; high amplitude noise residuals from sky emission line subtraction are particularly noxious, as the effects are unpredictable and merely serve to increase uncertainty. Low frequencies contain information about spectrum-wide residual variations after baseline subtrac- tion. One likely source of such a variation is the presence of light from intermediate-mass main sequence stars in the integrated cluster light. Very broad hydrogen absorp- tion features typical of the otherwise largely featureless H-band spectra of A0V stars (Meyer et al. 1998), for ex- ample, would not be removed by the low-order baseline subtraction. Information pertaining to the velocity dis- persion of the cluster resides in the frequencies between the extremes. The frequency filtering applied to the NIRSPEC data leads to a systematic error of 0− 3 km s−1 , which varies between echelle orders. The offsetting correction applied to the results respresents a correction of no more than 20 percent, generally less. Noise in the input spectrum leads to uncertainty in the correction factor in the range of 0.1−0.5 km s−1 , setting the lower bound on the precision of the velocity dispersion measurements. 3. OBSERVATIONS 3.1. NIRSPEC Spectra The spectra used to determine the internal velocity dis- persions of the SSCs and template stars were taken with the facility near-infrared echelle spectrometer NIRSPEC (McLean et al. 1998) on the 10-m Keck II telescope on Mauna Kea, Hawaii. We used NIRSPEC in the echelle mode, which yields spectral resolution of R ∼ 22, 000. The integrated light of super star clusters aged 5 to 80 Myr is dominated by evolved supergiant and bright giant stars (Gilbert 2002). The near-IR spectrum of cool evolved stars is replete with atomic and molecular ab- sorption features — no “continuum” in the traditional sense (i.e., a Planck thermal spectrum) is evident. Both the H and K bands offer a large number of features that the cross correlation analysis effectively averages over in determination of the mean feature width. There is per- haps an advantage to the H band, in that warm cir- cumstellar dust may veil features in the K band. The NIRSPEC detector experiences significant “persistence” from exposure to large flux, for example bright sky OH emission lines or arc lamp lines. Operationally, this dis- courages changing of filters during an observing night as the persistent after-images of sky emission lines from a different filter add significantly to the noise in an echel- legram. In this analysis, we have opted to observe the clusters at the shorter wavelength only. The NIRSPEC-5 (N5) order-sorting filter covers the wavelength range 1.51–1.75 µm, corresponding approxi- mately to Johnson H . The N5 echelle data fall in seven echelle orders, ranging from 44 through 50. All obser- vations were taken with the echelle and cross-dispersion gratings set at their blaze angles. This position max- imizes signal-to-noise for a given exposure time. This advantage is mitigated by the fact that more than a sin- gle position is required to cover the free spectral range at 1.6 µm, and portions of the H band are not observed. Spectra used in this work were obtained over four ob- serving seasons, from February 2002 through January 2005. Table 1 presents a summary list of spectroscopic observations. NIRSPEC observations of evolved stars used as template spectra are discussed in Kirian et al. (2006) and McCrady et al. (2003). The minimum air- mass of M82 (declination +69◦40′) from Mauna Kea is 1.56, and efforts were made to observe at an airmass of no larger than 2.0 when possible. The slit used has a width of 0.432′′ (3 pixels) and length of 24′′. Use of the long slit improves background subtraction, and often al- lows multiple clusters to be observed simultaneously. Slit positions were chosen to include multiple objects where possible to increase observing efficiency. Certain pairs of targets are closely-separated and only resolvable in good seeing. Each individual cluster spectrum has an integration time of 600 seconds. Bright OH sky emission lines begin to saturate in longer exposures, increasing the difficulty of sky subtraction. Total integration time on a cluster is increased by repeating the observations. 3.2. Reduction and Extraction The spectra were dark-subtracted, flat-fielded and cor- rected for cosmic rays and bad pixels. The curved echelle orders were then rectified onto an orthogonal slit-position versus wavelength grid based on a wavelength solution from sky (OH) emission lines. Each pixel in the grid has a width of δλ = 0.019 nm. We sky-subtracted by fitting third-order polynomials to the 2D spectra column-by- column. The NIRSPEC echelle turret is jostled when the cryo- genic image rotator undergoes large slews, leading to stochastic shifts of the wavelength scale of up to sev- eral pixels. Doppler shift information is lost due to such shifts. Typically, the absolute wavelength scale is estab- lished using telluric OH emission. Throughout each data acquisition cycle the image rotator was either turned off, or only slow, small amplitude tracking motions were ex- ecuted. In either case the wavelength solution is stable to better than a few hundredths of a pixel. Thus the ve- locity broadening reported here is intrinsic to the source and is not an instrumental effect. The cluster spectra were extracted using Gaus- sian weighting functions matched to the wavelength- integrated profile of each cluster. To correct for atmo- spheric absorption in the cluster spectra, we observed a hot main sequence star at a similar airmass. This cali- bration star spectrum is divided by a spline function fit to remove photospheric absorption features (particularly Brackett series and helium lines) and continuum slope. The resulting atmospheric absorption spectrum is then divided into the cluster spectra. The adopted sky-subtraction method generally com- pletely removes sky emission lines. However, a high level of noise is left behind at the position of bright OH SUPER–STAR CLUSTERS IN M82 5 lines, particularly un- or barely-resolved doublets. These “noise spikes” must be removed to avoid introduction of systematic bias in the cross-correlation results. We smooth the spectrum with a broad (∼ 40 km s−1 ) step function, and subtract the original spectrum to obtain a residuals array. Data points greater than 5× the rms are replaced by the smoothed pixel. The step-function width and clipping level were chosen to limit replaced pixels to only those affected by strong sky emission lines. Cer- tain atmospheric OH emission lines were incompletely removed in the sky subtraction process. In these cases, we replaced the pixels affected by residual sky emission with the median value of the ∼ 5 pixels on either side of the contaminated range. The fraction of pixels replaced in a given echelle order typically amounts to a few per- cent. Tests on an OH emission-free echelle order indicate that the cross-correlation result is unaffected within the stated uncertainties. An atlas of the spectra for 19 SSCs is presented in Fig- ures 5 and 6. We have included here only the spectra for echelle orders 46 and 47; plots of all echelle orders for each cluster are available in McCrady (2005). Each cluster spectrum represents the summation of multiple observations (Table 1). The total is normalized by di- viding by the median value for that echelle order, such that the spectrum is centered about unity. The resulting scale is relative flux, which allows direct comparison of spectra of different clusters. The spectra are offset by an arbitrary integer amount for presentation of multi- ple clusters on a single set of axes, and labeled towards the right (long-wavelength) side. The clusters within a given echelle order are arranged in the atlas in order of increasing velocity dispersion. The signal-to-noise ratio (S/N) of the extracted spec- tra varies between clusters. The clusters vary in lumi- nosity over a range of four apparent magnitudes in H band (McCrady et al. 2003), which is a factor of ∼ 40. The luminosity differences carry through to the single- integration S/N as all clusters were observed for 600 sec- onds per integration. Light losses due to variable seeing and inefficiencies in fine guiding add additional varia- tion to the S/N for each cluster. Although faint clusters were observed more often, not all clusters were observed to the same S/N as a result of observing constraints. Differences in the total S/N are evident in the atlas of cluster spectra. For example, faint SSC-k was observed only twice and has total S/N ∼ 9 per pixel based on the CCD equation (Howell 2000, p. 54), which accounts for Poisson statistics, background, dark current and read noise. Bright SSC-1c was also observed just twice, but has total S/N ∼ 28 per pixel. Repeated observations (13 times) of SSC-r brought the total S/N up to ∼ 37 per pixel; a single observation of faint SSC-r results in S/N comparable to a single observation of SSC k. These examples provide a sense of the S/N range of the obser- vations. The spectra presented in the atlas have been shifted to rest wavelength. Across the top of each plot, we have identified the positions of certain prominent spectral fea- tures. The H-band spectra of the clusters resemble the spectra of supergiant stars, and have a large number of iron and OH absorption lines. Rovibrational ∆v = 3 bandheads of carbon monoxide are recognizable in or- ders 45 through 49. As seen in the spectra of supergiant stars (Kirian et al. 2006; Meyer et al. 1998), the strength of the CO bandheads and OH lines increase at cooler effective temperatures. Lines from other miscellaneous metals (Mn, Ti, Si, Ca, C, Ni) and molecules (CN) are also indicated. At the bottom of each atlas plot is a rep- resentative sky emission spectrum for that echelle order, arbitrarily scaled. The brightest OH emission lines leave a footprint of increased noise in the extracted cluster spectrum. The disk of M82 has substantial diffuse emission. The nuclear starburst displays mottled near-IR continuum emission (Figure 1). The cross-correlation analysis based upon the spectra of the clusters is more sensitive to line emission. Lynds & Sandage (1963) first noted the bipo- lar, filamentary network of Hα emission extending more than 1 kpc from the galactic center. Paschen α im- ages show patchy recombination emission throughout the nuclear starburst (Alonso-Herrero et al. 2003). K-band spectra show emission lines of hydrogen (Br γ), He I and H2. Prominent H-band emission lines are [Fe II] at 1.644 µm and the Br 6 line of hydrogen at 1.7367 µm. In sev- eral of the slit positions used, nebular emission varies along the slit and removal is difficult. Remnants of the lines are apparent in the spectra of certain clusters (e.g., SSC-1c in Figure 6). 3.3. Objects Observed The 20 objects of Table 3 of McCrady et al. (2003) and the 19 objects of Table 2 do not constitute equal sets. The intersection of the two tables contains the 15 SSCs for which we have herein derived a mass (see Table 2). The union of the two tables contains 24 objects. Moreover, Figure 2 identifies 26 objects. An accounting of the objects observed in this project is as follows. Five clusters for which we measured photometry and half-light radius had significant problems with their echelle spectroscopy. The practical single-exposure time limit of 600 seconds is set by the saturation point of atmospheric OH emission lines. Clusters fainter than [F160W] ∼ 15 mag have S/N < 2 in 600 seconds of typ- ical seeing. In poor seeing, these faint clusters are often too difficult to identify for positioning the spectrograph slit. SSC-1b is a frustrating case, as it is a bright clus- ter with good S/N in a 600-s exposure. In poor seeing, however, the light from SSC-1b is blended with the light of the nearby clusters SSC-1a and SSC-1c. Observations on 2003 Feb 6 were not used for SSC-1b or SSC-1c be- cause of inadequate seeing. On the night of 2005 Jan 24, the seeing was sufficient to resolve SSC-1c, however the NIRSPEC detector was contaminated by persistence due to sky emission from use in low-resolution mode by an unaffiliated observing team earlier in the night. While we were able to extract SSC-1c, SSC-1b was significantly contaminated and had to be rejected. Six other objects identified in Figure 2 have no derived virial mass. SSC-z is clearly a super star cluster, but lies just north of the edge of the HST/NICMOS field (McCrady et al. 2003). In the absence of a resolved im- age, we have no measurement of the half-light radius. Object “y” also lies off the edge of the NICMOS field, just south of SSC-L. Spectra of object “y” are inconclu- sive due to low S/N in any case. Object “10” is un- resolved by the NICMOS image, and we therefore have no measured half-light radius. Interestingly, object “10” 6 McCRADY & GRAHAM is coincident with a point source in Paschen α images, suggesting it may be a compact H II emission region sur- rounding one or several massive stars. SSC-j and SSC-a are not well fit by empirical King functions, and the half- light radii of these clusters are undetermined. SSC-h is likewise not well fit by the empirical King model, as it is clearly a collection of sources in NIC2 images (Figure 3). 4. ANALYSIS 4.1. Cross Correlation Results Each cluster was observed multiple times, with seven echelle orders in the N5 filter per observation. A single observation of the spectrum in a particular echelle order is treated as one “experiment” for the cluster. Each of the experiments for a cluster is cross correlated with the spectrum from the corresponding spectral order of each of the template evolved stars. The result of this analy- sis is an ensemble of cross correlation functions (CCFs) for each cluster/template star pair. The peak amplitude of the CCF measures the similarity of the cluster to the template spectral type (Table 2). For each cluster, we have identified the template supergiant spectrum which provides the best match. In most cases, several template stars match the cluster approximately equally well. The spectra of template stars with higher surface tempera- tures (e.g., G-type stars) proved to be poor matches for the cluster spectra. The CCFs for each cluster and its best-match template are presented in McCrady (2005). The velocity dispersions based upon cross correlation results for the best-match template star are listed in Ta- ble 2. The quoted uncertainties reflect the formal error based upon the standard deviation of the mean for the ensemble of experiments. Systematic errors are treated in McCrady (2005, see also §2); the stated uncertainties do not include any allowance for the applied correction of systematic offsets. In the course of the analysis, each cluster spectrum is cross correlated with each of the tem- plate spectra. We find that the velocity dispersions in- dicated by the best match template are consistent with velocity dispersions indicated by other templates of sim- ilar effective temperature to within the uncertainties. 4.2. Derived Virial Masses Armed with measurements of the cluster half-light radii and velocity dispersions, we are ready to derive the virial masses. Table 2 lists the derived mass for 15 SSCs in M82. Most of the SSCs have masses between 2 × 105 and 106 M⊙ ; clusters SSC-L, SSC-7 and SSC-9 haveM > 106 M⊙ . The median uncertainty on the virial mass measurements is 16 percent. A significant portion of the error budget is the 8 percent uncertainty in the adopted distance to M82. Figure 7 plots the half-light radii versus the velocity dispersions for the SSCs. We have plotted the locus of points for certain masses as dashed lines. These lines il- lustrate that the uncertainty in the velocity dispersion, σr, has a greater impact on the uncertainty in mass than does the uncertainty in the half-light radius, rhp. This is to be expected, as the virial mass (Eq. 1) is proportional to the square of σr. We have mitigated this effect by measuring σr to a precision sufficient to balance the er- ror budget roughly evenly between uncertainties on the velocity dispersions and the halflight radii. Application of the virial theorem to determine the mass of the clusters is based on the assumption that the clusters are at present bound (self-gravitating) en- tities. This assumption is supported by a comparison of the relevant timescales: the crossing time and the age of the clusters. The crossing time is the typical time re- quired for a star to cross the cluster, where tcr ≈ rhp/σr (Binney & Tremaine 1987, p. 190). The SSCs in Table 2 have crossing times in the range of 4 × 104 to 3 × 105 years, while their ages are on the order of ∼ 107 years (Satyapal et al. 1997). Thus, member stars have made tens of crossings of the clusters. After just a few cross- ing times, the stars of a cluster are well mixed (King 1981) and the virial theorem is well satisfied (Aarseth 1974). The assumption that the M82 clusters are cur- rently gravitationally bound is therefore valid. To provide context and a sense of scale for the derived cluster masses, we turn to virial mass measurements of SSCs and globular clusters in our own and other galax- ies from the literature. Pryor & Meylan (1993) use ve- locity dispersions and King-Michie model fits to derive virial masses of 56 Galactic globular clusters. Their ve- locity dispersions range from 1 − 19 km s−1 (σ = 6.8 km s−1 ) and masses of 104 to 4 × 106 M⊙ (M = 5.6 × 105 M⊙ ). The Milky Way globular clusters are plot- ted in Figure 7 for comparison with the M82 clusters. (Pryor & Meylan (1993) provide σr and M , from which we estimated rhp using Eq. 1.) In total, the Milky Way has about 180 globular clusters (Ashman & Zepf 1998, p. 31). If we assume an average cluster mass of 1.9 × 105 M⊙ (Mandushev et al. 1991), the total mass of the Galactic population is ∼ 3.4 × 107 M⊙ . The M82 SSCs in Table 2 have a total mass of ∼ 1.4 × 107 M⊙ , comparable to the aggregate mass of the much older Galactic globular clusters. (However, we do not mean to imply that all of the M82 SSCs will remain bound for 12 Gyr; see §5.) The old globular clusters of the Milky Way are spread widely in the σ − rh space of Figure 7, but in general the locus of points is below (lower velocity dispersion and to the right (larger radius) of the locus of points for the young M82 SSCs. What can we infer about the two populations from this plot? It is interesting to consider the time evolution of a cluster in this parameter space. Over time, mass loss from individual stars in the course of their evolution will cause a cluster to lose mass. As detailed in the Appendix, adiabatic mass loss by a viri- alized cluster progresses such that the product σr is con- served. An isolated cluster, evolving through adiabatic mass loss, would gradually move down and to the right in Figure 7 as indicated by the plotted vector, crossing the “isobaric” lines. Over the span of 15 Gyr, a cluster with a Kroupa IMF would lose around half of its initial mass (i.e., its mass at 10 Myr) as a result of stellar evolution (McCrady et al. 2003). Such adiabatic evolution of the young M82 SSCs over a Hubble Time would place the clusters in the same region as the bulk of the old globu- lar clusters in Figure 7. We discuss the implications of this plot further in §5. Additional context for our results comes from observed cluster systems in other galaxies. Dubath & Grillmair (1997) measured the masses of nine globular clusters in M31. They find velocity dispersions of 7−27 km s−1 (σ = SUPER–STAR CLUSTERS IN M82 7 14 km s−1 ), rhp = 2 − 5 pc (rhp = 3.6 pc) and masses of 4.3− 82× 105 M⊙ (M = 2.3× 10 6 M⊙ ). Larsen et al. (2002) measure virial masses for four globular clusters in M33. They find velocity dispersions of 4.4− 6.5 km s−1 , rhp = 2 − 8 pc and masses of 1.4 − 6.2 × 10 5 M⊙ . In general, the old globular clusters in these neighboring galaxies are larger with lower velocity dispersions. This pattern is consistent with the notion that the M82 SSCs are a population of young globular clusters, as adiabatic mass loss due to stellar evolution would cause the clus- ters to expand over time (see Appendix). A more direct comparison is provided by young SSCs in other galaxies. Mengel et al. (2002) examines five young (age ∼ 8 Myr) clusters in the merging Antennae galaxies. They find ve- locity dispersions of 9 − 21 km s−1 , rhp of 3.6 − 4.0 pc, and masses of 6.4− 47× 105 M⊙ . As discussed in McCrady et al. (2005), SSC-F shows evidence of mass segregation. While cluster-wide dynam- ical mass segregation in these young clusters is unlikely, it is possible that the most massive stars in a cluster have either rapidly sunk toward the core or simply formed nearer the core. In either case, the red supergiant ve- locity dispersions we measure in the near-IR would be smaller than the cluster mean and the masses we derive would represent lower limits. Gasdynamical modeling by Boily et al. (2005) indicates that inward migration of massive stars may cause the dimensionless geometric pa- rameter η in the virial mass formula (Equation 1) to in- crease by a factor of around two over a few ×107 yr. In our analysis, we have explicitly assumed that η = 10 as derived in (McCrady et al. 2003). If mass segrega- tion has in fact led to η > 10 in these M82 clusters, the masses we derive here would be underestimated by the corresponding factor. 4.3. Cluster Mass Function Armed with the virial masses of the SSCs, we can inves- tigate the cluster mass function for the M82 nuclear star- burst. The standard manner for making a cluster mass function is to prepare a histogram of the masses over log- arithmically spaced bins and fit a power law or lognor- mal distribution to the slope. Rosolowsky (2005) demon- strates the shortcomings of this method, citing specifi- cally the dependence of the power law index to the choice of bin size and spacing in cases involving a small number of data points. We choose instead to evaluate the cumu- lative mass function for the 15 M82 SSCs. Figure 8 plots the cumulative mass function as N(M ′ > M), which is the total number of clusters with mass greater than the reference mass M . In the common case of a power law, the slope of a standard mass function is dN/dM ∝ Mγ . For the cumulative mass function, the integration adds one to the exponent, such that N(M ′ > M) ∝ Mγ+1 (Rosolowsky 2005). The mass function for the M82 SSCs is well fit by a power law of index γ = −1.91±0.05. The uncertainty on the power law index is based on a Monte Carlo simulation of our mass data. We resampled the cluster masses by adding normal noise according to the uncertainties on the mass measurements, then fit for the index of the resulting cumulative mass function. The distribution of power law indices was Gaussian with a standard deviation of our quoted uncertainty. The power law index of γ ∼ −2 indicates that the stellar mass of the cluster population is divided rather equally between the high-mass clusters and low-mass clusters. Of the aggregate mass in our sample of 15 SSCs, roughly 60 percent of the mass is contained in the three most massive clusters (SSCs L, 9 and 7). Estimation of the completeness limit for our SSC mass function is somewhat difficult. Our mass measurements are based on measured cluster velocity dispersions. For us to measure a velocity dispersion, the cluster must be observable in the near-IR and be sufficiently bright and spatially resolved for us to obtain a usable spectrum. Limiting factors include the intrinsic mass of the cluster, the light-to-mass ratio for the cluster, confusion with ad- jacent clusters or background emission, and line-of-sight extinction. With the exception of SSC 1b, which suffers from source confusion, we are confident we have obtained the spectra of all clusters brighter than apparent mag- nitude H ∼ 13.8. At the distance of M82, this corre- sponds to a cluster luminosity of ∼ 1.8 × 105 L⊙ . To convert to a mass estimate, we can estimate the light- to-mass ratio for the clusters. For clusters in the age range of 7–13 Myr, suitable for the M82 nuclear clusters (McCrady et al. 2003), and a Kroupa (2001) field star mass function, Starburst99 models predict L/M ∼ 1 in units of L⊙ /M⊙ . The highly variable extinction in the dusty, inclined disk of the galaxy could be hiding ad- ditional bright clusters, on the far side of the disk for example. Assuming a typical extinction correction for the clusters of ∼ 0.5 mag in H band, we estimate that our mass function is largely complete for clusters more massive than ∼ 3× 105 M⊙ . To characterize the poten- tial effects of incompleteness, we added fake clusters of mass (3 ± 0.5)× 105M⊙ to the Monte Carlo simulation. Each additional undetected cluster with mass near this completeness limit would decrease the power law index by ∆γ ∼ −0.02 (i.e., the power law would become more steeply negative). 5. DISCUSSION In §4.2, we note that the evolution of an individual M82 SSC via adiabatic mass loss due to stellar evolution over a Hubble time would reposition the SSC in σ−rh param- eter space. Such a repositioning would leave any of the SSCs in our sample in a region consistent with the posi- tion of the old globular clusters in the Milky Way (Figure 7). In a sense, this represents a necessary but not suffi- cient condition for the hypothesis that these young SSCs represent the progenitors of globular clusters. If, instead, our analysis showed that stellar evolution would move the clusters to a point in σ− rh parameter space that would be inconsistent with the position of old globular clus- ters, it would represent strong evidence that these SSCs could not be progenitors of globular clusters. In fact, adiabatically evolving any individual M82 SSC from our sample for 15 Gyr would leave it solidly within the re- gion of σ − rh parameter space occupied by old globular clusters. But we hasten to note that this result is insufficient evidence that these SSCs are destined to become a population of old globular clusters. The M82 nuclear clusters are young, with ages on the order of 10 Myr (Förster Schreiber 1998; McCrady 2005). There is grow- ing evidence in the literature that a significant portion of 8 McCRADY & GRAHAM young clusters are disrupted on a timescale of approxi- mately 10 Myr from birth. Observations of massive clus- ters in the Antennae galaxies (Fall et al. 2005) and M51 (Bastian et al. 2005) and lower mass open clusters in the solar neighborhood (Lada & Lada 2003) find an excess of clusters with ages ∼ 10 Myr relative to what would be expected based on an assumption of constant cluster formation rate (Bastian & Gieles 2006). The naive in- terpretation of these findings is that there was a burst of star cluster formation in the past 10 Myr in each of these galaxies. But as noted by Fall et al. (2005), the age distribution of star clusters represents the combined histories of star cluster formation and disruption within a galaxy. The relative wealth of clusters aged 10 Myr in these various nearby galaxies suggests that we are fortu- nate to be observing them at a special time in their star formation histories, whereby we fall afoul of the cosmo- logical principle. An attractive alternative is that a high percentage of clusters are disrupted within approximately 10 Myr of formation, i.e., the e-folding survival time for a popula- tion of clusters is about 10 Myr (Mengel et al. 2005). This hypothesis goes by the morbid name of “infant mortality.” The energy and momentum output of mas- sive stars via stellar winds and supernovae could remove residual natal gas from a young massive cluster. If the gas removal were impulsive (i.e., occured over less than a crossing time), the cluster could become gravitation- ally unbound and begin expanding freely. Whether or not a cluster survives this phase depends largely on the star formation efficiency in the formation of the cluster from natal gas (see references in Bastian & Gieles 2006). Fall et al. (2005) posit that because a cluster with more mass has both more gas to remove and more massive stars to provide the energy, the fraction of clusters dis- rupted may be roughly independent of mass. They find this conjecture to be consistent with their observation that the shape of the cluster mass function in the Anten- nae galaxies is nearly independent of age. Zhang & Fall (1999) investigated the mass function of young star clusters in the Antennae galaxies (NGC 4038/9) based on photometric mass estimates, and found a power law mass function of γ = −2 over the range 104 ≤ M/M⊙ ≤ 10 6, a result confirmed by Fall et al. (2005). Mengel et al. (2005) found potential evidence for a turnover or change in slope of the mass function for the Antennae clusters, but cautions that the random and systematic uncertainties discourage overinterpretation of this result. They note that determination of the cluster mass function from photometric masses requires age de- terminations for individual clusters, which is particularly delicate work around ages of ∼ 107 yr when the cluster luminosity varies greatly with age. Our virial mass measurements obviate determination of the ages of individual clusters and assumptions re- garding the form and cutoff masses of the stellar IMF. The M82 nuclear clusters in our sample are also young, and follow a power law mass distribution very similar to the Antennae clusters. A power law mass distribution for young SSCs stands in contrast to the lognormal mass distribution for old globular clusters (Harris 1991), which imply a preferred mass scale at the peak of ∼ 2×105M⊙ for Milky Way globular clusters. Several processes, oper- ating on different timescales, have the ability to disrupt star clusters. As discussed above, infant mortality ap- pears to disrupt clusters independently of mass, thereby preserving the shape of the initial cluster mass function. On longer timescales (∼ 108−109 yrs), the strongly mass dependent processes of two-body relaxation and exter- nal perturbations (such as gravitational shocks and dy- namical friction) can disrupt the clusters (Bastian et al. 2005). Analytical models by Fall & Zhang (2001) find that the initial form of the high-mass end of the cluster mass function is preserved over time. Two-body relax- ation decreases the masses of clusters linearly over time, flattening the mass function at low masses but little af- fecting the shape at high masses. By 12 Gyr, the mass function develops a peak at a mass of about 2× 105M⊙ . Thus, an initial power law distribution of cluster masses will develop into a distribution resembling the lognormal mass function of old globular clusters over 12 Gyr, with disruption erasing all information regarding the initial shape of the mass function at the low-mass end. In a galactic environment, of course, the clusters are not isolated, and are additionally subject to external forces such as galactic tides and dynamical friction. The inner 1 kpc of a galaxy, with its strong tidal fields, is a particularly dangerous place for star clusters. But as noted by Fall & Zhang (2001), their models “support the suggestion that at least some of the star clusters formed in merging and interacting galaxies can be regarded as young globular clusters.” As shown in Figure 7, the adia- batic evolution of a cluster through stellar evolution and mass loss over a similar time frame will tend to move the M82 SSCs into the σ − rh parameter space occupied by old Galactic globular clusters. While we cannot predict the fate of any individual M82 SSC, our results suggest that any cluster which should happen to survive for a Hubble time could resemble the old globular clusters seen in the Milky Way today. 6. SUMMARY In this paper, we investigate the SSC population of the inner ∼ 500 pc of the M82 starburst. The nuclear star- burst in M82 contains roughly two dozen SSCs that are prominent in the near-IR. Based on high spectral reso- lution near-IR spectra, we measure line-of-sight velocity dispersions for 19 SSCs in the nuclear starburst. We find dispersions in the range of 7 − 35 km s−1 , comparable with values for older globular clusters. We apply the virial theorem to the measured velocity dispersions and halflight radii to derive the masses of 15 of the SSCs. The SSC masses lie in the range of 2.5 × 105 M⊙ to 4 × 10 M⊙ , placing them at the high end of the mass distribu- tion function for old Galactic globular clusters. The total mass of the 15 measured SSCs is 1.4× 107 M⊙ , which is of the same order of magnitude as the total mass of the globular cluster system of the Milky Way. Evolution of the clusters via gradual mass loss from stellar evolution would move them into the realm of σ − rh parameter space occupied by old Milky Way globular clusters. The cumulative mass function of the clusters follows a power law with an index of γ = −1.91 ± 0.06. This is very similar to the mass distribution of young SSCs in the Antennae galaxies, and lends credence to the suspicion that SSCs are potential future globular clusters. We would like to thank the staff of the Keck Ob- SUPER–STAR CLUSTERS IN M82 9 servatory for their assistance in our observations. We also thank the anonymous referee for helpful com- ments regarding implications of this work. NM thanks John Johnson for invaluable data wrangling advice and W. D. Vacca, L. Blitz and S. E. Boggs for helpful com- ments. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the in- digenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This material is based upon work sup- ported by the National Science Foundation under Grant No. 0502649, with additional support from NSF Grant AST–0205999. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. APPENDIX APPENDIX The consequences of mass loss from a virialized star cluster depend upon the rate of loss. We consider specifically two cases: (1) rapid mass loss, where a star cluster has a sufficiently long relaxation time that v and R are unable to readjust during the ejection of some quantity of mass, and (2) slow mass loss, where v and R for the cluster continually readjust to maintain equilibrium. In the case of rapid mass loss, Hills (1980) derives an expression for the relation between the initial mass and the amount of mass ejected: R = R0 M0 −∆M M0 − 2∆M . (A1) Evidently, as ∆M tends to M0/2 the cluster radius tends to infinity and the cluster becomes unbound. In the case of gradual mass loss, the cluster constantly adjusts to maintain equilibrium. The total energy at any instant, whether the cluster is in equilibrium or not, is Mv2 − η where v is the 3-d rms velocity and η is a non-dimensional form factor. The corresponding change in total energy of the system is v2 − 2η . (A3) After δm is lost, the cluster must readjust to the new equilibrium. 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M. 1999, ApJ, 527, L81 SUPER–STAR CLUSTERS IN M82 11 TABLE 1 NIRSPEC N5 Observations Date Objects texp Airmass Seeing Atm Star Remarks (UT) (min) (sec z) (′′) (SpT) 2002 Feb 23 3, 9, 11 40 1.6 0.5 HD 74604 (B8V) (1) 2003 Jan 19 F, L 70 1.9 0.8 HD 173087 (B5V) 2003 Feb 6 1b, 1c, r 60 1.8 0.7 HD 74604 (B8V) 2003 Feb 6 1a, 3, m 50 1.7 0.7 HD 82327 (B9V) (2) 2003 Feb 6 6, 7 60 1.6 0.7 HD 82327 (B9V) 2003 Feb 6 8, 10, c 120 1.6 0.6 HD 82327 (B9V) 2003 Feb 6 s, t 60 1.9 0.6 HD 146926 (B8V) 2003 Feb 7 a, b 90 2.0 1.5 HD 146926 (B8V) (3) 2003 Dec 5 6, 7 20 1.6 0.5 HD 63586 (A0V) 2003 Dec 5 3, 6, h 40 1.6 0.5 HD 63586 (A0V) (4) 2004 Feb 8 a, y 60 1.9 0.8 HD 146926 (B8V) 2004 Feb 8 r, z 30 2.2 1.0 HD 146926 (B8V) 2004 Feb 9 r, z 40 1.6 1.3+ HD 82327 (B9V) (3) 2004 Feb 9 a, y 30 1.7 1.3+ HD 82327 (B9V) (3) 2005 Jan 24 1a, 1b 20 1.6 0.6 HD 82327 (B9V) 2005 Jan 24 1a, 1c, q 20 1.5 0.6 HD 82327 (B9V) 2005 Jan 24 j, k, q 30 1.5 0.6 HD 82327 (B9V) Note. — Seeing values are estimates. Remarks. — (1) Only 30 min on object 3. (2) Only 40 min on object 1a. (3) Seeing very poor & variable. (4) Only 20 min on object 3. TABLE 2 Cross-Correlation Results Best Fit Mean CCF Dispersion Mass∗ tcr Object Template Peak Value (σ, km s−1 (105 M⊙ ) (10 5 yr) SSC L HR2289 0.70± 0.05 34.7± 0.4 40.± 6. 0.41± 0.05 SSC F HR2289 0.69± 0.1 12.4± 0.3 5.5± 0.8 1.2± 0.1 SSC a HD237008 0.53± 0.1 10.9± 0.4 SSC 11 HD237008 0.78± 0.07 12.1± 0.4 3.9± 0.6 0.9± 0.1 SSC 9 HD237008 0.70± 0.07 19.8± 0.5 23.± 4. 1.3± 0.2 SSC 8 HD14469 0.52± 0.1 10.5± 0.5 4.0± 0.7 1.5± 0.2 SSC 7 HD237008 0.53± 0.1 18.6± 1. 22.± 4. 1.4± 0.2 SSC 6 HD237008 0.76± 0.09 9.2± 0.3 2.7± 0.4 1.5± 0.2 SSC h HD237008 0.62± 0.07 33.2± 1.0 SSC j HD237008 0.55± 0.10 9.0± 0.8 SSC k HD237008 0.51± 0.1 9.± 1. 5.7± 2. 3.1± 0.5 SSC m HD14469 0.38± 0.1 15.2± 0.8 7.3± 1. 0.9± 0.1 SSC q HD237008 0.53± 0.1 7.9± 0.6 2.8± 0.6 2.5± 0.3 SSC 3 HD237008 0.76± 0.1 8.7± 0.3 2.7± 0.4 1.8± 0.2 SSC 1a HD237008 0.72± 0.07 13.4± 0.4 8.6± 1. 1.5± 0.2 SSC 1c HD237008 0.71± 0.08 12.2± 0.4 5.2± 0.8 1.2± 0.2 SSC r HD237008 0.65± 0.1 8.6± 0.3 3.0± 0.5 2.0± 0.2 SSC t HD237008 0.46± 0.1 7.9± 0.9 2.5± 0.7 2.2± 0.4 SSC z HD237008 0.69± 0.1 9.9± 0.3 ∗ Error in mass includes errors in distance to M82, half-light radius and ve- locity dispersion. † Crossing time, described in §4.2. 12 McCRADY & GRAHAM Fig. 1.— Color mosaic of HST ACS/WFC and NICMOS images of the nuclear region in M82. ACS F814W, NICMOS F160W and NICMOS F222M images are mapped to blue, green and red, respectively. The image is ∼ 25′′ × 65′′ (0.4 × 1.1 kpc) with north up and east to the left. About two dozen super star clusters are evident, many of which are spatially coincident with and reddened by the band of variable extinction running from upper left to lower right in the image. SUPER–STAR CLUSTERS IN M82 13 Fig. 2.— Mosaic of H-band (N5) NIRSPEC SCAM images of the nucleus of M82. Candidate SSCs are labeled for reference. Coordinates are J2000. Inset image is from HST/NICMOS. 14 McCRADY & GRAHAM Fig. 3.— HST/NICMOS F160W images of each cluster. Each image is 2.5′′ × 2.5′′, and the position angle of the y-axes is 349.4◦ (i.e., North is 10.6◦ left of straight up). The images are log-scaled, as the cores are substantially brighter than the halos. SUPER–STAR CLUSTERS IN M82 15 Fig. 4.— Comparison of the spectra of SSC-11 and several cool supergiants in echelle order 49. The cluster spectrum displays the same features as the supergiants, but appears washed out due to the velocity dispersion of its constituent stars. The supergiant stars are plotted in a temperature sequence, with the hottest star at the top. 16 McCRADY & GRAHAM Fig. 5.— Atlas of SSC spectra for echelle orders 47 & 46. SUPER–STAR CLUSTERS IN M82 17 Fig. 6.— Atlas of SSC spectra for echelle orders 47 & 46, continued. 18 McCRADY & GRAHAM Fig. 7.— Projected halflight radius (rhp) versus velocity dispersion (σr) for M82 SSCs (circles). Dashed lines indicate the locus of points for cluster mass as labeled. Error bars on the halflight radius do not include the uncertainty on the distance to M82. Galactic globular clusters (squares) from Pryor & Meylan (1993) are plotted for comparison. The vector indicates time evolution of a cluster due to adiabatic loss of half its mass (see Appendix). SUPER–STAR CLUSTERS IN M82 19 Fig. 8.— Cumulative mass function for the M82 SSCs. The dashed line indicates a power law fit where N(M ′ > M) ∝ Mγ+1. The best fit has a slope of γ = −1.91± 0.06. The estimated completeness point for cluster mass is marked ’C’ (see text). The fitted power law does not reflect any correction for completeness.

0704.0479

The affine part of the Picard scheme

THE AFFINE PART OF THE PICARD SCHEME (CORRECTED). THOMAS GEISSER Abstract. We describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field. (This is a corrected and improved version of the article originally published in Comp. Math. 145 (2009)). 1. Introduction For a proper scheme p : X → k over a perfect field, the Picard scheme PicX representing the functor T 7→ H0(Tet, R 1p∗Gm) exists, and its connected component Pic X is separated and of finite type [Mu64, II 15]. By Chevalley’s structure theorem [Chev60], the reduced connected component Pic 0,red X is an extension of an abelian variety AX by a linear algebraic group LX : (1) 0 → LX → Pic 0,red X → AX → 0. The commutative, smooth affine group scheme LX is the direct product of a torus TX and a unipotent group UX . The following theorem completely characterizes TX : Theorem 1. If X is proper over a perfect field, then the cocharactermodule Homk̄(Gm, TX) of the maximal torus of PicX is isomorphic to H et(X̄,Z) as a Galois-module. To analyze the unipotent part, we let Pic(X[t])[1] be the typical part, i.e. the subgroup of elements x of Pic(X[t]) such that the map X[t] → X[t], t 7→ nt sends x to nx. Theorem 2. Let X be proper over a perfect field. Then Pic(X[t])[1] is isomorphic to the group of morphisms of schemes f : Ga → UX satisfying f(nx) = nf(x) for every n ∈ Z. In particular, Homk(Ga, UX) ⊆ Pic(X[t])[1], and this is an equality in characteristic 0. To get another description of UX , we assume that X is reduced (the map on the Picard scheme induced by the map Xred → X is well understood by the work of Oort [Oort62]). The semi- normalization X+ → X is the largest scheme between X and its normalization which is strongly universally homeomorphic to X in the sense that the map X+ → X induces an isomorphism on all residue fields. A Theorem of Traverso [Tra70] implies that Pic(X[t])[1], hence UX , vanishes if X is reduced and seminormal. We use this to show Theorem 3. Let X be reduced and proper over a perfect field. a) We have a short exact sequence (2) 0 → KX → Pic 0,red X → Pic 0,red and inclusions of unipotent group schemes UX ⊆ KX ⊆ p∗(Gm,X+/Gm,X) with quotients finite p-primary group schemes. 2010 Mathematics Subject Classification. 14K30. Key words and phrases. Picard scheme, torus, unipotent subgroup, semi-normalization, etale cohomology. Supported in part by NSF grant No.0556263. http://arxiv.org/abs/0704.0479v4 2 THOMAS GEISSER b) The group scheme p∗(Gm,X+/Gm,X) represents the functor T 7→ {OX×T -line bundles L ⊆ OX+×T which are invertible in OX+×T }. Notation: For a field k, we denote by k̄ its algebraic closure, and for a scheme X over k we let X̄ = X ×k k̄. Unless specified otherwise, all extension and homomorphism groups are considered on the fpqc site. 1 Acknowledgements: This (original) paper was written while the author was visiting T. Saito at the University of Tokyo, whom we thank for his hospitality. We are indebted to G. Faltings for pointing out a mistake in a previous version, and the referee, whose comments helped to improve the exposition and to give more concise proofs. O. Gabber pointed out mistakes in the original version and suggested improvements. 2. The torus Proposition 4. If p : X → k is reduced, geometrically connected, and proper over a perfect field, then Gm,k → p∗Gm,X is an isomorphism. Moreover, if f : X ′ → X is a universal homeomorphism and X ′ is reduced as well, then f induces an isomorphism p∗Gm,X ∼= p∗Gm,X′ . Proof. Since any scheme T over k is flat, we have by flat base change Rjq∗OXT = H j(X,OX )⊗kOT , where q : XT → T is the projection. In particular, p∗Gm,X(T ) := Γ(X × T,OX×T ) × = (Γ(X,OX )⊗ Γ(T,OT )) and it suffices to show that Γ(X,OX) ∼= Γ(X ′,OX′) ∼= k. Since Γ(X̄,OX̄) Gal(k̄/k) = Γ(X,OX ), we can assume that k is algebraically closed and that X is connected, in which case the statement follows because X and X ′ are reduced, proper, connected, and have a k-rational point. ✷ Lemma 5. For any scheme X we have isomorphisms H1et(X,Z) fl(X,Z) ∼= Ext X(Gm,X ,Gm,X). Proof. The first isomorphism is [Mi80, III Rem. 3.11(b)]. To prove the second isomorphism, we note that HomX(Gm,X ,Gm,X) ∼= ZX by [SGA3, VIII Cor. 1.5], and that Ext X(Gm,X ,Gm,X) is isomorphic to the group of extensions of group schemes [Oort66, Cor. 17.5], which vanishes by [SGA7, VIII Prop. 3.3.1].3 Hence we obtain the isomorphism from the spectral sequence [Mi80, III Thm.1.22] 2 = H fl(X, Ext X(Gm,X ,GmX)) ⇒ Ext X (Gm,X ,Gm,X). Proof. (Theorem 1) Since the maps defined below are natural, we can assume that k is alge- braically closed and X is connected. We can also assume that X is reduced, because H1et(X,Z) H1et(X red,Z), and the map PicX → PicXred has unipotent kernel and cokernel [Oort62, Cor. page 9]. It suffices to calculate Homk(Gm,k,PicX), because there are no homomorphisms from Gm to commutative group schemes other than tori [Oort66, p. 81]. By Yoneda’s Lemma, the latter group 1In [Gei09] we used the étale topology 2This replaces [Gei09, Prop. 9 a)] which is incorrect as stated because the induction step in the proof does not preserve the hypothesis on reducedness. 3This was claimed without proof in [Gei09]. THE AFFINE PART OF THE PICARD SCHEME (CORRECTED). 3 is isomorphic to the group of homomorphisms of sheaves on the fpqc site Homk(Gm,k, R 1p∗Gm,X). The Leray spectral sequence (3) E 2 = Ext k(Gm,k, R tp∗Gm,X) ⇒ Ext X (Gm,X ,Gm,X). gives an exact sequence 0 → Ext1k(Gm,k, p∗GmX) → Ext X(Gm,X ,Gm,X) → Homk(Gm,k, R 1p∗Gm,X) −→ Ext2k(Gm,k, p∗Gm,X). By Proposition 4 the left term agrees with Ext1k(Gm,k,Gm,k), and this vanishes by [Oort66, Cor. 17.5]. Thus it suffices to show that δX is the zero map 4. Choose a closed point of X and let i : Z → X be the corresponding closed subscheme. Since p∗ ◦ i∗ = id we have R sp∗i∗ = R s(p◦ i)∗ = 0 for s > 0. Hence we obtain a diagram Homk(Gm,k, R 1p∗Gm,X) −−−−→ Homk(Gm,k, R 1p∗i∗Gm,Z) = 0 Ext2k(Gm,k, p∗Gm,X) −−−−→ Ext2k(Gm,k,Gm). By Proposition 4, the lower horizontal map is an isomorphism. ✷ Remark. The example in [Gei06, Prop. 8.2] shows that the map H iet(X̄,Z) → Ext (Gm,Gm) is not an isomorphism for i ≥ 2. One can ask if it is an isomorphism if one replaces H iet(X̄,Z) by the eh-cohomology group H ieh(X̄,Z) of [Gei06]. Example. If X is the node over an algebraically closed field, then H1et(X,Z) ∼= Z, and TX ∼= Gm. Let X be a node with non-rational tangent slopes at the singular point. Base changing to the algebraic closure, one sees that H1et(X̄,Z) ∼= Z, with Galois group acting as multiplication by −1, hence TX is an anisotropic torus. Using the theorem, we are able to recover the torsion of TX , AX and the diagonalizable part of NSX := PicX /Pic 0,red X in terms of etale cohomology: Corollary 6. Let X be proper over a perfect field k. Then we have canonical isomorphisms H1et(X̄,Z)⊗Q/Z ∼= colimHomk̄(µm, TX); Div(torH et(X̄,Z)) ∼= colimHomk̄(µm, AX); et(X̄,Z)/Div ∼= colimHomk̄(µm, NSX). Proof. Taking the colimit of the isomorphismH1et(X̄,Z/m) ∼= Homk̄(µm,PicX) of [Mi80, Prop.4.16] or [Ray70, §6.2], we obtain H1et(X̄,Q/Z) ∼= colimHomk̄(µm,PicX). Since Ext (Gm, TX) = 0, The- orem 1 implies that Homk̄(µm, TX) ∼= Homk̄(Gm, TX)/m ∼= H1et(X̄,Z)/m. Consider the commu- tative diagram: colimHomk̄(µm, TX) H et(X̄,Z)⊗Q/Z colimHomk̄(µm,Pic 0,red X ) −−−−→ colimHomk̄(µm,PicX) −−−−→ colimHomk̄(µm, NSX) colimHomk̄(µm, AX) −−−−→ torH et(X̄,Z) −−−−→ coker f. The middle column is the short exact coefficient sequence. The left column and middle row are short exact because Ext1 (µm, TX) = Ext (µm,Pic 0,red X ) = 0 by [Oort66, Cor. 17.5, II 14.2]. A dia- gram chase shows that f is injective, and the right vertical map is an isomorphism. The Corollary 4The remainder of the proof is a simplification suggested by O. Gabber. 4 THOMAS GEISSER follows because colimHomk̄(µm, AX) is divisible and colimHomk̄(µm, NSX) is finite. ✷ The above result should be compared to [Gei10, Prop.6.2], where we show that, for every proper scheme over an algebraically closed field, the higher Chow group of zero-cycles CH0(X, 1,Z/m) is the Pontrjagin dual of H1et(X,Z/m). This implies a short exact sequence 0 → torA X(k) → CH0(X, 1,Q/Z) → χ(TX)⊗Q/Z → 0, for AtX the dual abelian variety of AX , and χ(TX) the character module of TX . However, in this case the contribution from the torus and from the abelian variety are not compatible with the coefficient sequence 0 → CH0(X, 1) ⊗Q/Z → CH0(X, 1,Q/Z) → torCH0(X) → 0 as in Corollary 6. Looking at tangent spaces, the previous Corollary gives a dimension formula: Corollary 7. Let l be a prime different from char k. Then dimk H 1(X,OX ) = dimUX + dimk Lie(NS X) + rankH et(X,Z) + corankl H et(X̄,Ql/Zl). 3. The unipotent part Let N Pic(X) := ker Pic(X[t]) −→ Pic(X) . Since t 7→ 0t induces x 7→ 0x on the typical part, Pic(X[t])[1] is a subgroup of N Pic(X). In [Wei91], Weibel shows that for every scheme there is a direct sum decomposition Pic(X[t, t−1]) ∼= Pic(X) ⊕N Pic(X) ⊕N Pic(X)⊕H et(X,Z). Proof. (Theorem 2). We show first that N Pic(X) = ker 1) → UX(k) . Since there are no non-trivial morphisms of schemes from A1k to an abelian variety, a torus, an infinitesimal group, or a discrete group, we see that the kernel of UX(A k) → UX(k) agrees with the kernel of PicX(A k) → PicX(k). Let p : X → k and p ′ : X × A1k → A k be the structure morphisms. Then the Leray spectral sequence gives a commutative diagram 0 −−−−→ H1et(A Gm) −−−−→ Pic(X × A k) −−−−→ PicX(A k) −−−−→ H 0 −−−−→ H1et(k, p∗Gm) −−−−→ Pic(X) −−−−→ PicX(k) −−−−→ H et(k, p∗Gm), and it suffices to show that the outer vertical maps are isomorphisms. Let X −→ L → k be the Stein factorization of p, such that OL ∼= g∗OX and L is the spectrum of an Artinian k-algebra. Since A1k → k is flat, p OX×A1 = OA1 ⊗kp∗OX , and X×A −→ A1L −→ A k is the Stein factorization of p′. We obtain H iet(A Gm) ∼= H Gm) ∼= H L,Gm), and H iet(k, p∗Gm) ∼= H iet(L,Gm). Hence the terms on the left vanish because Pic(L) = Pic(A 0. To show that H2et(A L,Gm) → H et(L,Gm) is an isomorphism, we can assume that L is a local Artinian k-algebra with (perfect) residue field k′. By [Mi80, III Rem.3.11] we are reduced to showing that H2et(A k′ ,Gm) → H ′,Gm) is an isomorphism, and this can be found in [Mi80, IV Ex.2.20]. Given an element x of N Pic(X), the condition x ∈ Pic(X[t])[1] implies that the corresponding f ∈ HomSch(A 1, UX) satisfies f(nx) = nf(x) for all n. If k has characteristic 0, then UX ∼= G a for some THE AFFINE PART OF THE PICARD SCHEME (CORRECTED). 5 r, and the map f : Ga → UX corresponds to a morphism of Hopf algebras f ∗ : k[x1, · · · , xr] → k[t]. If f∗(xi) = j ajt j, then aj(nt) j = nf∗(xi) = f ∗(nxi) = n only if nj = n for all n, hence j = 1. ✷ Example. If k has characteristic p, then t 7→ t2p−1 induces a map Ga → Ga which is compatible with multiplication by n, but not a homomorphism of group schemes. Corollary 8. We have UX = 0 if and only if N Pic(X) = 0. Proof. This follows from N Pic(X) = ker 1) → UX(k) , because any unipotent, connected, smooth affine group is an affine space as a scheme, hence admits a non-trivial morphism from A1 which sends 0 to 0 if it is non-trivial. ✷ The kernel and cokernel of PicX → PicXred has been described in [Oort62], hence we will from now assume that X is reduced. If X+ is the semi-normalization of X, then the map OX → OX+ is an injection of sheaves on the same topological space. For X+ reduced and semi-normal, N Pic(X+) = 0 by Traverso’s theorem [Tra70] together with [Wei91, Thm. 4.7]. Hence the Corollary implies that UX+ = 0, and that UX = ker(Pic 0,red X → Pic 0,red (For curves, this recovers [BLR90, Prop.9.2/10].) Indeed, by Corollary 6, the map Pic 0,red 0,red induces an isomorphism on the torus and abelian variety part, because it induces an isomorphism on etale cohomology. Proof. (Theorem 3) a) We have isomorphims H iet(X,Z) ∼= H iet(X +,Z), which combined with Corollary 6 shows that the canonical map Pic 0,red X → Pic 0,red induces an isomorphism on the torus components, and is an isogeny with kernel a unipotent group scheme PX on the abelian variety parts. 5 Hence the map is surjective and the kernel KX is an extension of PX by UX . Applying the Proposition to the exact sequence of etale sheaves 0 → p∗Gm,X → p∗Gm,X+ → p∗(Gm,X+/Gm,X) → PicX → PicX+ on Spec k, we obtain the diagram with exact columns 0 −−−−→ KX −−−−→ Pic 0,red X −−−−→ Pic 0,red −−−−→ 0 0 −−−−→ p∗(Gm,X+/Gm,X) −−−−→ PicX −−−−→ PicX+ 0 −−−−→ coker u −−−−→ NSX −−−−→ NSX+ . Since v is injective, so is u. The Neron-Severi group schemes are extensions of finitely gener- ated étale group schemes by a finite connected group scheme. The isomorphism H2et(X,µm) H2et(X ′, µm) implies that PicX(k̄)/m → PicX+(k̄)/m is injective for any m prime to p, and since 0,red X (k̄) and Pic 0,red (k̄) are m-divisible, the same holds for NSX(k̄)/m → NSX+(k̄)/m, and consequently for NSX [ ] → NSX+ [ ]. Thus coker u is contained in the extension of the p-primary torsion subgroup NSX{p} by the finite connected group scheme NS X . Finally, the isomorphism 5O.Gabber [Gab20] showed that, conversely, any finite unipotent commutative group scheme can appear as PX . 6 THOMAS GEISSER Homk̄(µm,PicX) ∼= H1et(X̄,Z/m) from [Mi80, III Prop. 4.16] together with the isomorphism H iet(X,Z) ∼= H iet(X +,Z) and the result on KX shows that the three right maps in the diagram induce isomorphisms on Homk̄(µm,−) for all m, hence the three groups on the left are unipotent. b) Recall that q : XT → T , and consider the diagram 0 −−−−→ H1et(T, q∗Gm,X×T ) −−−−→ Pic(X × T ) −−−−→ PicX/k(T ) −−−−→ H et(T, q∗Gm,X×T ) 0 −−−−→ H1et(T, q∗Gm,X+×T ) −−−−→ Pic(X + × T ) −−−−→ PicX+/k(T ) −−−−→ H et(T, q∗Gm,X+×T ). Since q∗Ga,X×T = H 0(X,OX )⊗OT = H 0(X+,OX+)⊗OT = q∗Ga,X+×T is an isomorphism as in Proposition 4a), the outer maps are isomorphisms, and it suffices to calculate ker r. Let Y = X×T and Y ′ = X+ × T , and consider the tautological map f : {OY -line bundles L ⊆ OY ′ which are invertible in OY ′} → Pic(Y ). It suffices to show the following statements: a) The image of f is contained in ker Pic(Y ) → Pic(Y ′) b) f surjects onto ker Pic(Y ) → Pic(Y ′) c) f is injective. a) We claim that the map L ⊗OY OY ′ → OY ′ ⊗OY OY ′ −→ OY ′ is an isomorphism. We can check this on an affine covering, and in this case it is proved in [RS93, Lemma 2.2(4)]. b) Let L ∈ Pic(Y ) with L⊗OY OY ′ ∼= OY ′ . Since L is flat, we get an injection L = L⊗OY OY → L⊗OY OY ′ ∼= OY ′ . We claim that the inverse of L in OY ′ is the sheaf associated to the presheaf U 7→ {x ∈ OY ′(U)|xL(U) ⊆ OY (U)} ⊆ OY ′(U). This can be checked on an affine covering, and then it is [RS93, Lemma 2.2(2)]. c) Let L and L′ be subsheaves of OY ′ which are invertible in OY ′ and isomorphic as abstract invertible sheaves. Multiplying with the inverse of L′ inside OY ′ , it suffices to show that if L is a subsheaf of OY ′ , and f : OY → L an isomorphism, then L = OY ⊆ OY ′ . But f(1) is a global unit of OY ′(Y ), and by Proposition 4a), OY (Y ) × = OY ′(Y ) ×. Hence L = f(1)−1L = OY . ✷ References [BLR90] S. Bosch, W. Lutkebohmert, M. Raynaud, Neron models, Ergebnisse der Mathematik und ihrer Grenzge- biete (3), 21. Springer-Verlag. [Chev60] C. Chevalley, Une demonstration d’un theoreme sur les groupes algebriques, J. Math. Pures Appl. (9) 39 (1960), 307–317. [SGA3] M. Demazure, A. Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 2, Lecture notes in mathematics 152. Springer-Verlag. [Gab20] O. Gabber, Letter to the author Nov. 2020. [Gei06] T. Geisser, Arithmetic cohomology over finite fields and special values of ζ-functions, Duke Math. J. 133 (2006), no. 1, 27–57. [Gei09] T. Geisser, The affine part of the Picard scheme, Comp. Math. 145 (2009) 415–422. [Gei10] T. Geisser, Duality via cycle complexes, Ann. of Math. (2) 172 (2010), 1095–1126. [Gr62] A. Grothendieck, Technique de descente et theoremes d’existence en geometrie algebrique. VI. Les schemas de Picard. Proprietes generales, Seminaire Bourbaki, 1961/62, no. 236. [SGA7] A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1. Lecture Notes in Mathematics 288. Springer-Verlag. [Mi80] J. S. Milne, Etale cohomology, Princeton Math. Series 33. [Mu64] J. P. Murre, On contravariant functors from the category of pre-schemes over a field into the category of abelian groups (with an application to the Picard functor), Inst. Hautes Etudes Sci. Publ. Math. No. 23 (1964) 5–43. THE AFFINE PART OF THE PICARD SCHEME (CORRECTED). 7 [Oort62] F. Oort, Sur le schema de Picard, Bull. Soc. Math. France 90 (1962) 1–14. [Oort66] F. Oort, Commutative group schemes, Lecture Notes in Mathematics 15, Springer-Verlag, Berlin-New York (1966). [Ray70] M. Raynaud, Specialisation du foncteur de Picard, Inst. Hautes Etudes Sci. Publ. Math. No. 38 (1970) 27–76. [RS93] L. Roberts, B. Singh, Subintegrality, invertible modules and the Picard group, Compositio Math. 85 (1993), no. 3, 249–279. [Tra70] C. Traverso, Seminormality and Picard group, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), 585–595. [Wei91] C. Weibel, Pic is a contracted functor, Invent. Math. 103 (1991), no. 2, 351–377. Dep. of Math., Rikkyo University, Japan 1. Introduction 2. The torus 3. The unipotent part References

0704.0481

The few scales of nuclei and nuclear matter

The few scales of nuclei and nuclear matter A. Delfino a, T. Frederico b, V. S. Timóteo c, and Lauro Tomio d aInstituto de F́ısica, Universidade Federal Fluminense, 24210-900 Niterói, RJ, Brasil bDepartamento de F́ısica, Instituto Tecnológico de Aeronáutica, CTA, 12228-900, São José dos Campos, Brasil cCentro Superior de Educação Tecnológica, Universidade Estadual de Campinas, 13484-370, Limeira, SP, Brasil dInstituto de F́ısica Teórica, Universidade Estadual Paulista, 01405-900, São Paulo, Brasil Abstract The well known correlations of low-energy three and four-nucleon observables with a typical three-nucleon scale (e.g. the Tjon line) is extended to light nuclei and nuclear matter. Evidence for the scaling between light nuclei binding energies and the tri- ton one are pointed out. We show that the saturation energy and density of nuclear matter are correlated to the triton binding. From the available systematic nuclear matter calculations, we verify the existence of bands representing these correlations. PACS 21.45.+v, 21.65.+f, 21.30.Fe Key words: Scaling, nonrelativistic few-body systems, nonrelativistic nuclear matter Two-nucleon interactions are typically constructed to fit scattering data and deuteron properties. When such interactions are used to calculate three-nucleon observables, the results exhibit some discrepancies [1]. Basically, they are ex- plained as originated from different strengths of the two-nucleon tensor force and short-range repulsions, provided that all realistic two-nucleon interactions have the correct one-pion exchange tail. In four-nucleon bound state (4He ) calculations the discrepancies still remain. But, at least, they are correlated, as seen in the binding energies of 4He (Bα) and triton (Bt), which lie on a very narrow band [2], obtained when the short-range repulsion of the nucleon- nucleon interaction is varied while two-nucleon informations (deuteron and scattering) are kept fixed. This correlation is known as Tjon line [2]. Bα and Bt follows an almost straight line in the range of about 1-2 MeV of variation Preprint submitted to Elsevier 4 November 2018 http://arxiv.org/abs/0704.0481v1 of the triton binding energy around the experimental value. As the long-range two-nucleon scales we have the deuteron binding energy (Bd) and the singlet virtual-state energy (Bv). Two-body short-ranged interactions, supporting very low two-body binding energy and/or large scattering lengths, when used to calculate three-body sys- tems, approach what we call the universal Thomas-Efimov limit [3]. By trying to find the range r0 of the two-nucleon force, Thomas [4] showed that when r0 → 0, while the two-body binding energy B2 is kept fixed, the three-body binding energy goes to infinity (Thomas collapse). Much latter, Efimov [5] showed that, in the limit B2 = 0(r0 6= 0) the number of three-body bound states is infinite with an accumulation point at the common two- and three- body threshold. Note that both the Thomas and Efimov effects are claimed to be model independent, since they are due to a dynamically generated effective three-body potential acting at distances outside the range of the two-body potential. These apparently different effects are related to the same scaling mechanism, as shown in Ref. [3]. In other words, the Thomas effect appears when r0 is much smaller than the size of the two-body system (which is of the order of the scattering length |a|), while the Efimov effect arises for |a| >> r0. Therefore, what matters for both effects is the same condition: |a| >> r0 or the ratio |a|/r0 >> 1. In terms of the two-body energies this is translated m|B2|/h̄ r0 << 1 (m the boson mass). One would expect that the Thomas-Efimov effect is manifested in weakly-bound quantum few-body sys- tems which are much larger in size than the corresponding two-body effective range. Notice that a zero binding energy for a free two-body system is not known in nature. But, nowadays it was shown that, for trapped ultracold gases of certain atomic species, it is possible to adjust the two-body scattering length at very large values, using Feshbach resonance techniques, by tuning the external magnetic field [6]. In this case, it is expected that the Thomas-Efimov effect can be manifested [7]. The deuteron and triton may be viewed as low energy systems with large size scales in which the range of the potential is smaller than the corresponding healing distances of the wave functions, leading the nucleons to have a high probability to be outside of the interaction range. Then, the low-energy prop- erties of these systems can be studied with models that minimally includes the physics of the Thomas-Efimov effect, as in the case of a few-body model with renormalized pairwise s-wave zero-range force [8]. This approach shows that all the low-energy properties of the three-body system are well defined in the model, once one three-body scale and the two-body low-energy observables are given. As a consequence, correlations between two three-body s−wave observ- ables are expected to appear in model calculations with short-ranged interac- tions. Along this line, some previous works (see [9] and references therein) have studied weakly-bound halo states in exotic nuclei as well as possible Efimov states for He-He-Alkali molecules. The scaling of three-nucleon observables with the triton binding energy corre- sponds to universal behaviors found when a three-body scale is varied. For ex- ample, the Phillips plot [10] of the neutron-deuteron doublet scattering length, as a function of the triton energy is nowadays one of the universal scalings found in the three-nucleon system [11,12]. In general, it is observed for nu- clear and molecular weakly bound three-body systems [13,14,15] the scaling of observables with the three-body binding energy. The Thomas-collapse of the three-body energy in systems of maximum wave- function symmetry implies the existence of a three nucleon scale (identified with the triton binding energy) governing the short-range behavior of the wave-function. Four nucleons can also form a state of maximum symmetry, allowing in principle the collapse of such configuration, independently of the three-nucleon collapse [8]. This is under discussion and it is suggested in [16] that the four-body scale is not independent of the three-body one. However, in their work it was introduced a three-body force to stabilize the shallowest three-body state, against the variation of the cut-off. The three-body interac- tion can be attractive or repulsive and their conclusion lies on the repulsive sector. We note that the attractive part indicates a possible independent be- havior of the four body ground-state energy from the three-body one. Certainly this point merits further discussions and so far, we think, it is still open the possibility of a four-body scale. Anyway, as the nucleon-nucleon interaction is strongly repulsive at short range and therefore the probability of four nu- cleons to be simultaneously in a volume ∼ r30 is quite small, presumably the four-nucleon scale itself has much less opportunity to be evidenced in realistic nuclear models. Indeed, this is indicated by the existence of the Tjon line. Due to that, as we will see later, the four-nucleon binding energy is eliminated in favor of the triton binding energy. In this respect it is worthwhile to note that Platter et al. extended the effective field theory framework applied to four- bosons [16] to calculate the 4He binding energy by controlling the triton energy through a repulsive effective three-nucleon force. Within their approach [17] the Tjon line is reproduced. In a nuclear scenario dominated by an interaction with a range smaller than the nucleon-nucleon scattering lengths, and considering the triton and 4He nuclear sizes yet larger than the force range, the picture of nuclei would be of a many-body system with the wave-function being an eigenfunction of the free Hamiltonian almost everywhere. The Pauli principle allows only up to four nu- cleons at the same position, forbidding certain particular configurations with overlap of more particles. If more than four particles are allowed to overlap, it would imply that the asymptotic information from the interaction of the cluster would go beyond of those already fixed by the low-energy observables of two, three and four nucleons. By some unknown reason the parameters of Quantum Chromodynamics are close to this limit. It was conjectured in Ref. [18] that a small change in the light quark masses away from their physi- cal values could put the deuteron and the singlet virtual state at zero binding energy, and therefore the above idealized picture of the nuclear systems could not be far from reality. It is quite amazing thinking that nuclear wave func- tions could heal much beyond the interaction range. Therefore, the details of the long wavelength structure of nuclei are given by the free Hamiltonian and by few-nucleon scales, which determine the wave function at short distances. The universal behavior of the scaling functions are due to that. If one wonders about the neutron matter within a non-relativistic quantum framework, in the limit of a zero-range force, we could say that the only scale in this case is the neutron-neutron scattering length. Therefore, the binding energy of neutron droplets will be strongly correlated to that quantity, which is the only physical scale in this situation allowed by the Pauli principle. This discussion has been performed in the context of three neutron systems [19]. Moreover, it was concluded in Ref. [20] that stable tetra-neutron droplets would imply a major change in the neutron-neutron scattering length. Another example of the dominance of only two-body scale appears in three- boson systems in two dimensions, where the Thomas-Efimov effect is ab- sent [3]. In this case, only two-body low energy scales are enough to define the many-body properties in the limit of a zero-range interaction. The low-energy properties of a many-body system of spin-zero particles in two dimensions will be sensitive only to the two-boson binding energy. Even in the case where bosons are trapped, since the essential singularity of the point-like configura- tion is not affected by the confining force as the harmonic one. For the sake of generality, we start with the observables Bd, Bv, Bt and Bα as the scales determining the asymptotic properties of nuclei [9]. Then, in the limit of a zero-range interaction, we write the binding energy of a nucleus with mass number A and isospin projection Iz, considering isospin breaking effects, B(A, Iz) = A Bt B (βv, βd, βα, A, Iz) , (1) where βa = Ba/Bt with a = v, d and α. According to the Tjon line, βα remains approximately constant for a variety of two-nucleon potentials and the parametrization of the numerical results, given in MeV, for several two-nucleon potentials is Bα = 4.72 (Bt − 2.48) , (2) which for B t = 8.48 MeV gives B α = 28.32 MeV. Using (2) in (1), R(A, Iz) = B(A, Iz)/A = Bt R (Bt, A, Iz) , (3) where in the scaling function R(A, Iz) the values of Bd and Bv are fixed to the experimental values. The dependence of Bα with Bt for realistic nucleon- nucleon potentials is given by Eq. (2). Equation (3) generalizes the concept of the Tjon line to nuclei. Recent calcu- lations using the AV18 nucleon-nucleon potential plus three-body forces [21] show that there is a systematic improvement of the binding energy results for He, Li, Be and B isotopes simultaneously with the triton binding energy, when models are tuned to fit Bt. It is important to note that these AV18 calculations have at least two three-body parameters that are fitted to Bt and nuclear matter saturation properties. Consequently, one could argue that such calculations cannot provide evidence for one-parameter correlation. The fit- ting to nuclear matter calculation presumably is not that important for light nuclei in view of the dominance of the triton binding (or three-body correla- tions) in the four-nucleon bound state as given by the Tjon line. Therefore, it is reasonable to think that three-body correlations are quite important for light-nuclei, since that even for the alpha particle where the nucleons are in a very compact configuration this occurs. In our opinion, the fitting of nuclear matter saturation properties has more to do with the approximations done in nuclear matter calculations. The three-body potential should be somewhat tuned, which is, probably, not so important for light nuclei once the triton binding attains its physical value. For nuclear matter properties calculation using a variety of two-nucleon poten- tials, in which the tensor strength was varied but the deuteron binding energy was kept fixed, it was shown that these interactions cannot quantitatively ac- count for nuclear saturation [22,23]. Coester et al. [22] observed that, in an energy versus density plot, the saturation points of nuclear matter obtained by employing different realistic potentials are located along a band (Coester band). Also, in a relativistic framework it was observed such strong correla- tion [24]. The displayed nuclear matter binding energy (BA/A ≡ B(A,0)/A) versus saturation density [ ρo = (2/3)k 2, with kF the Fermi momentum] results are within a narrow band [22]. The observation given in Ref. [22] have been studied by many other authors that have used nuclear matter binding energies and saturation densities from different two-nucleon interactions. The main argumentation, as also in the case of three-nucleon calculations, is that this effect comes from different strengths of the two-nucleon tensor force and short range repulsion, which changes the triton binding energy, while keeping fixed the low-energy two-body scales. Basically, nuclear matter saturates due to the composed repulsive and attrac- tive short-range two-nucleon potential. Since, it may also be seen as a typical low-energy problem, it is natural to question whether any connection exists 30 40 50 60 70 [ MeV ] Fig. 1. Infinite nuclear matter binding energy as a function of EF extracted from Ref. [25] (solid circles and squares). The squares includes the single particle contri- bution in the continuum. The full triangle is given by the empirical values. between the proper few-body scales, Bd , Bv and Bt with those of the many- body problem, like the BA/A and the Fermi energy EF = h̄ 2k2F/(2mN) . For light nuclei there is strong evidences of scaling between Bd , Bv and Bt as expressed by Eq. (3). Here we are arguing that the scales of nuclear matter, BA/A and EF , are determined by Bd , Bv and Bt . Therefore, we suppose that going to the infinite isospin symmetrical nuclear matter, A → ∞ and Iz = 0, the limit B (βv, βd, βα, A, Iz = 0) =Bt G (βv, βd, βα) , (4) is well defined and expresses the correlation between the binding energy of the nucleon in nuclear matter with the few-nucleon scales. The Fermi energy EF = Bt EF (βv, βd, βα) , (5) will be correlated as well to the few-nucleon binding energies. The aim of this work is to study the possible correlation of the nuclear matter binding energy per nucleon with Bd , Bv and Bt , in order to improve our understanding of the general and important scaling of observables. Our inves- tigation is based on Eqs. (4) and (5), motivated by our previous discussion that leads to Eq. (1) and in the several works that recognize the role played by the low-energy few-body scales [8,9,13,12,14,15], in defining the observables of few-nucleon systems. In the present framework, the universal scaling functions connect the proper -9 -8.5 -8 -7.5 -7 -6.5 -6 [MeV] Fig. 2. BA/A as a function of Bt extracted from Ref. [25] (solid circles and squares). The squares includes the single particle contribution in the continuum. The full triangle is given by the empirical values. scales of the few-body system with those of the many-body system, as given by Eqs. (4) and (5). Different potentials, which describe the deuteron and the two- nucleon scattering properties give different values of Bt , Bα BA/A and EF . As we have done in deriving Eq. (3) for a class of changes in the short-range part of the nuclear force that keeps the deuteron and low energy scattering properties unchanged, and taking into account that for these variations of the potential the 4He and triton binding energies are strongly correlated as given by the Tjon line, one can rewrite Eqs. (4) and (5) in order to get a one parameter scaling: =Bt G (Bt) (6) for fixed Bd and Bv, where the only true dependence in the class of potential variations is dominated by Bt. The analogous expression for the Fermi energy EF = Bt EF (Bt) , (7) where EF scale with Bt. In the perspective of the one parameter functions of Eqs. (6) and (7), it is clear that one could express Et as a function of EF and immediately get = C(EF ), (8) -9 -8.5 -8 -7.5 -7 -6.5 -6 [MeV] Fig. 3. EF as a function of Bt extracted from Ref. [25] (solid circles and squares). The squares includes the single particle contribution in the continuum. The full triangle is given by the empirical values. the correlation implied by the Coester band. In order to enlighten our discussion we bring a variety of nuclear matter bind- ing energies BA/A at the corresponding saturation density, represented by the Fermi momenta kF , calculated from different two-nucleon potentials. In Fig. 1, we present the well known Coester band in which the results for BA/A and EF are showed. The two distinct bands represent the nuclear matter cal- culations with and without the single-particle continuum contributions. The empirical values are BA/A = 16 MeV and EF = 37.8 MeV from [27]. The correlation of BA/A with Bt expressed by (6) is plotted in Fig. 2, for the same set of results given in Fig. 1. The two-nucleon potentials present different values for the triton binding energy Bt , while the two-nucleon low- energy observables are fixed. We observe that the scaling function EtG(Et) is quite linear in the interval of about 2 MeV including the triton binding energy. We note that the ratio BA/A depends strongly on Bt. We understand this fact as a reminiscent manifestation of the three-body scale in the nuclear matter results obtained with only two-body correlations. In Fig. 3, we show the correlation between the Fermi energy and the triton binding energy. In general, we observe that the increase in the three-body scale leads to the increase of the Fermi energy, which is reasonable in view of the scaling function (7). However, we observe that the empirical values disagrees with the general trend of the correlation, a problem that could already be anticipated by looking at the Coester band in Fig. 1. The inclusion of three-body correlations, which carries the dynamics that sta- bilizes the Thomas collapse, presumably has a repulsive effect diminishing the saturation density and somewhat the nuclear matter binding. Due the short- range repulsion of the nucleon-nucleon interaction, the nuclear matter tends to saturate at large densities if only two-body correlations are considered and the empirical binding is achieved. The dynamically generated three-body sta- bilization mechanism carried only through the three-body correlations should appear in the nucleon-nucleon interaction range, however such repulsive con- tribution is absent if only two-body correlations are considered in the evalu- ation of nuclear matter properties. Therefore, we suspect that the inclusion of three-body correlations in nuclear matter calculations will bring the corre- lation curve of the BA/A with Bt in Fig. 2 toward the empirical values; and also in Fig. 3, where the saturation densities would be possibly found at lower values for a given triton binding energy. Recent sophisticated many-body calculations [26], where the triton binding en- ergy and nuclear matter saturation density were adjusted through two three- body parameters make subtle the clear appreciation of the real role of the three-body correlations, once the fit of these realistic forces mixes differences that come from the interaction and from the many-body approximations in nu- clear matter calculations. It is beyond our work the discussion of the quite in- volved many-body approximations needed to perform such calculations. Nev- ertheless, our conjecture is obviously based on qualitative arguments, which implies that just one three-body scale parameter is relevant for a systematic description of light nuclei and nuclear matter. The one parameter dependence in Eqs. (6) and (7) suggests to plot the dimen- sionless quantity BA/(ABt) as a function of the ratio EF/Bt, which should look as an almost linear correlation. We display in Fig. 4 the values for BA/(A Bt) versus EF/Bt . As we could anticipate, the results show a clear linear cor- relation. We are tempted to say that, if the correlation is extrapolated and assuming that the binding and saturation densities somewhat decreases when three-body correlations are considered, it looks to be possible that the empir- ical values would be consistent with the correlation band. In summary, we suggest for the first time a possible scaling of nuclei asymp- totic properties, in particular the nuclear binding energies with the triton binding energy, substantiated by recent realistic calculations of light nuclei. This observation generalizes to the many-nucleon context the correlations be- tween observables found in the three and four-nucleon systems. Beyond that, we found that the original correlation between the nuclear matter binding en- ergy per nucleon with the Fermi momentum described by the Coester band can now be seen as robustly represented by the scaling of nuclear matter prop- erties with the triton binding energy. The values of Bt carry different aspects of the used two- and three-nucleon potentials. To verify the extension of our conjecture, we propose that one could control the strength of the three-body force without a many-body parameter in order produce different triton binding 4 5 6 7 8 9 Fig. 4. Infinite nuclear matter binding energy as a function of EF , both in units of the triton binding energy. The calculation results are extracted from Ref. [25] (solid circles and squares). The squares includes the single particle contribution in the continuum. The full triangle represents the empirical values. energies and nuclear matter properties. We emphasizing that the nuclear mat- ter results should be obtained within the same approximations for all models. In this way, we expect that the plotted results in figures 2 and 3 should lie in a very narrow band. Our discussion may turn in a simple way to systematize results of possible forthcoming realistic calculations for many-nucleon systems. Consistent with our conclusions, it was argued in Ref. [18] that QCD implies that nuclear physics is close to the Thomas-Efimov limit. In this reference, it was conjec- tured that a small change in the light quark masses away from their physical values could be enough to move the deuteron and the singlet virtual state to zero binding energies. Therefore, nuclear physics could be dominated by long- range universal effective forces making the scalings not only a subtlety but also an evident reality in ab-initio non-relativistic nuclear model calculations. Acknowledgments. This work was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo and Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico. V. S. T. would like to thank FAEPEX/UNICAMP for partial financial support. References [1] W. Glöckle, H. Witala, D. Huber, H. Kamada, J. Golak, Phys. Rep. 274 (1996) [2] J.A. Tjon, Phys. Lett. B 56 (1975) 217; R.E. Perne and H. Kroeger, Phys. Rev. C 20 (1979) 340; J.A. Tjon, Nucl. Phys. A 353 (1981) 470. [3] S.K. Adhikari, A. Delfino, T. Frederico, I.D. Goldman and L. Tomio, Phys. Rev. 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0704.0482

Implementation of holonomic quantum computation through engineering and manipulating environment

Implementation of holonomic quantum computation through engineering and manipulating environment Zhang-qi Yin, Fu-li Li,∗ and Peng Peng Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China We consider an atom-field coupled system, in which two pairs of four-level atoms are respectively driven by laser fields and trapped in two distant cavities that are connected by an optical fiber. First, we show that an effective squeezing reservoir can be engineered under appropriate conditions. Then, we show that a two-qubit geometric CPHASE gate between the atoms in the two cavities can be implemented through adiabatically manipulating the engineered reservoir along a closed loop. This scheme that combines engineering environment with decoherence-free space and geometric phase quantum computation together has the remarkable feature: a CPHASE gate with arbitrary phase shift is implemented by simply changing the strength and relative phase of the driving fields. PACS numbers: 03.67Lx, 03.65.Vf, 03.65.Yz, 42.50.Dv I. INTRODUCTION Quantum computation, attracting much current inter- est since Shor’s algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this dif- ficulty, one proposed the decoherence-free space concept [2, 3]. It is found that when qubits involved in quantum computation collectively interact with a same environ- ment there exists a “protected” subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To per- form quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parame- ters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum compu- tation goal is implemented by changing the controlling parameters [4]. As well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may ac- quire a geometric phase when the time-dependent param- eters adiabatically undergo a closed loop in the parameter space [5]. The phase depends only on the swept solid an- gle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic con- trol errors [6, 7, 8]. On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9]. In the scheme, quantum logical bits are represented by degener- ate eigenstates of the parameterized Hamiltonian. These ∗Email: [email protected] states have the features: they belong to DFS, and unitar- ily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop. In the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broad- band squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. More- over, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum sys- tems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11]. These results strongly inspire us that in- stead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir. In this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a two- qubit CPHASE gate [15]. We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the charac- teristic relaxation time of an atom-reservoir coupled sys- tem. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment. This paper is organized as follows. In Sec. II, we in- troduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two dis- tant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity http://arxiv.org/abs/0704.0482v3 mailto:[email protected] modes through the double Raman transition configura- tion. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to re- alize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations. laserlaser fibre FIG. 1. Atom-field coupling scheme. |gjn〉 |rjn〉 |sjn〉 δjn|ejn〉 FIG. 2. Atomic level configuration for atom j in cavity II. ENGINEERING A SQUEEZING ENVIRONMENT AND GENERATING A DECOHERENCE-FREE SUBSPACE Our scheme is shown in Fig.1. A pair of four-level atoms are trapped in each of two distant cavities, respec- tively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18], only one fiber mode b is excited and coupled to cavity modes a1 and a2 with strength ν [19]. We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig.2. Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |gjn〉 and |ejn〉 of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |gjn〉 is taken to be zero as the energy reference point. The lower lying level |ejn〉, and upper levels |rjn〉 and |sjn〉 have the energy δjn, and ωrjn and ωsjn, respectively, in the unit with ~ = 1. Transitions |gjn〉 ↔ |sjn〉 and |ejn〉 ↔ |rjn〉 are driven by laser fields of frequencies ωLsjn and ω jn with Rabi frequencies Ω and Ωrjn and relative phase ϕ, respectively. Transitions |gjn〉 ↔ |rjn〉 and |ejn〉 ↔ |sjn〉 are coupled to the cav- ity mode an with the strengths g jn and g jn, respectively. Here, we set ∆rjn = ω jn − ω = ωrjn − ω jn − δjn, and ∆sjn = ω jn − ω − δjn = ωsjn − ω Under the Markovian approximation, the master equa- tion of the density matrix for the whole system under consideration can be written as [14] ρ̇T = −i[H, ρT ] + Lcav1ρT + Lcav2ρT + LfiberρT , (1) where H = H0 +Hd +Hac +Hcf with j,n=1 ωrjn|rjn〉〈rjn|+ ωsjn|sjn〉〈sjn|+ δjn|ejn〉〈ejn|) a†nan + b j,n=1 (Ωsjn t|sjn〉〈gjn| e−i(ω t+ϕ)|rjn〉〈ejn|+H.c. Hac = j,n=1 (grjn|rjn〉〈gjn|an + gsjn|sjn〉〈ejn|an +H.c.), Hcf =ν 1 + a 2) + H.c. Here, H0 is the free energy of atoms and cavity fields, Hd is the interaction energy between the atoms and laser fields, Hac is the interaction energy between the atoms and the cavity fields, and Hcf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms LcavnρT =κn(2anρTa n − a†nanρT − ρT a†nan), LfiberρT =κf (2bρT b † − b†bρT − ρT b†b), where κn is the leakage rate of photons from cavity n, and κf is the decay rate of the fiber mode. Let’s introduce collective basis: |a〉n = (|g1n〉|e2n〉 − |e1n〉|g2n〉)/ 2, | − 1〉n = |g1n〉|g2n〉, |0〉n = (|g1n〉|e2n〉+ |e1n〉|g2n〉)/ 2, |1〉n = |e1n〉|e2n〉. The states |a〉n and | − 1〉n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting = βrn and = βsn, from (2), we obtain the effective interaction Hamiltonian Heff = iϕS+n + β nSn) +H.c. +Hcf , (4) where S+n = |0〉nn〈−1|+|1〉nn〈0|. In the derivation of (4), we have assumed the resonant condition 〈a†nan〉 + 〈a†nan〉+ δ′jn. In order to satisfy the condition with the flexible choice of Ωrjn, Ω jn, ∆ jn and ∆sjn, we have introduced additional ac-Stark shifts δ jn to states |gjn〉, which can be generated by using a laser field to couple the level |gjn〉 to an ancillary level. We now introduce three normal modes c and c± with frequencies ω and ω± 2ν by use of the unitary transfor- mation a1 = (c+ + c− + 2c), a2 = (c+ + c− − b = 1√ (c+ − c−) [17, 18]. In the limit ν ≫ |βrj |, |βsj |, neglecting the far off-resonant modes c± and setting 1 = −β 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as Heff = (β reiϕS+ + βsS)c+H.c., (5) where S+ = S+1 + S Since the modes c± are nearly not excited and decou- pled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, LfiberρT can be neglected, and Lcav1ρT + Lcav2ρT can be approximated as LcavρT = κ(2cρT c † − c†cρT − ρT c†c), (6) where κ = (κ1 + κ2)/2. In the bad cavity limit, κ≫ β, adiabatically eliminat- ing the mode c [12, 14], from Eq. (1) with the replace- ment of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms ρ̇ = −Γ (R+Rρ+ ρR+R− 2RρR+), (7) where ρ = Trf (ρT ), R = S cosh r + e iϕS† sinh r, r = cosh−1(βr/ βr2 − βs2) and Γ = 2(βr2−βs2)/κ. Eq. (7) describes the collective interaction of two cascade three- level atoms with the effective squeezed vacuum reservoir [10]. The parameters βr, βs and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8]. We will show that a geometric phase gate can be realized through changing these parameters. The DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ)|ψDF(r, ϕ)〉 = 0 [10]. In terms of basis states |e1〉 = |a〉1|a〉2, |e2〉 = |a〉1| − 1〉2, |e3〉 = | − 1〉1|a〉2, |e4〉 = |a〉1|0〉2, |e5〉 = |0〉1|a〉2, |e6〉 = |a〉1|1〉2, |e7〉 = |1〉1|a〉2, |e8〉 = |1〉1|1〉2, |e9〉 = 1√2 (|1〉1|0〉2 + |0〉1|1〉2), |e10〉 = | − 1〉1| − 1〉2, |e11〉 = 1√2 (|0〉1| − 1〉2 + | − 1〉1|0〉2), |e12〉 = (|1〉1|−1〉2+ |−1〉1|1〉2)+ 2√6 |0〉1|0〉2), the DFS states can be written as |ψDF(r, ϕ)〉1 =|e1〉, |ψDF(r, ϕ)〉j = cosh r√ cosh 2r |ej〉 − eiϕ sinh r√ cosh 2r |ej+4〉, j = 2, 3, |ψDF(r, ϕ)〉4 = e2iϕ(tanh r)2|e8〉 − eiϕ tanh r|e12〉+ |e10〉 (tanh r)4 + 2 (tanh r)2 + 1 Let’s introduce a unitary transformation O(r, ϕ) by |φi〉 = i=1Oij(r, ϕ)|ej〉, where |φi〉 = |ψDF〉i for i = 1, 2, 3, 4. For the transformed density matrix ρ̄ = O†ρO, we have = i[G, ρ̄] +O† O, (9) where G(r, ϕ) = iO† dO = iO†[ṙ dO + ϕ̇dO ]. To solve Eq. (9) in the DFS, let’s define the time-independent projector Π(0) = O†Π(r, ϕ)O = †|φi〉〈φi|O = j=1 |ej〉〈ej | + |e10〉〈e10| onto the DFS. From (9), we obtain the equation of motion for ρ̄DF = Π(0)ρ̄Π(0) dρ̄DF =i[GDF, ρ̄DF] + iΠ(0)GΠ⊥(0)ρ̄Π(0) − iΠ(0)ρ̄Π⊥(0)GΠ(0) + Π(0)O† OΠ(0), where Π⊥(0) = 1− Π(0) and GDF = Π(0)GΠ(0). In the limit of ṙ, ϕ̇≪ Γ, the last three terms in Eq. (10) can be neglected [11]. In this way, Eq. (10) is reduced to dρ̄DF = i[GDF, ρ̄DF]. (11) Therefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time. III. REALIZING CONTROLLING PHASE GATES THROUGH MANIPULATING THE SQUEEZING ENVIRONMENT In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g〉 ↔ |s〉 is switched off but the laser field driving the transition |r〉 ↔ |e〉 is switched on and the atoms are in the DFS state |Ψ(0)〉a = (|a〉1|a〉2 + |a〉1| − 1〉2 + | − 1〉1|a〉2 + | − 1〉1| − 1〉2) = j=1 |ψDF(0, 0)〉j/2. To generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is di- vided into the following three steps: (1) From time 0 to T1, hold on ϕ = 0, and adiabatically increase the parame- ter r from 0 to r0; (2) From time T1 to T2, hold on r = r0, and adiabatically change the phase ϕ from 0 to ϕ0; (3) From time T2 to T3, hold on ϕ = ϕ0, and adiabatically decrease r from r0 to 0. When the cyclic evolution ends, the atomic state becomes |Ψ(T3)〉a = (|e1〉+ eiχ1 |e2〉+ eiχ1 |e3〉+ eiχ12 |e10〉), where geometric phases χ1 = −ν1ϕ0, χ12 = −ν12ϕ0 with sinh2 r0 sinh2 r0+cosh , ν12 = 2 tanh4 r0+ tanh2 r0 tanh4 r0+ tanh2 r0+1 . By per- forming local transformations U1 = e −iχ1 |−1〉11〈−1| and U2 = e −iχ1 | − 1〉22〈−1|, the state (12) can be written as |Ψ′(T3)〉a = U1U2|Ψ(T3)〉a = 12 (|a〉1|a〉2 + |a〉1| − 1〉2 + | − 1〉1|a〉2 + ei∆| − 1〉1| − 1〉2), where ∆ = χ12 − 2χ1 = (2ν1 − ν12)ϕ0. Thus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 “see” different environments, |χ12| must be equal to |2χ1| and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment. If r0 = atanh( 4/3− 1) ≃ 0.4157, |ν12| = |ν1|. Under this condition with ϕ0 = π/ν1, the state of the atoms at the time T3 is |Ψ′′(T3)〉a = − 12 (−|a〉1|a〉2 + |a〉1| − 1〉2 + | − 1〉1|a〉2 + | − 1〉1| − 1〉2). In this case, the Controlled- Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 T=100/Γ T=200/Γ T=400/Γ FIG. 3. Fidelity Fr of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.975 0.985 0.995 T = 200/Γ T = 400/Γ T = 1000/Γ FIG. 4. Fidelity Fp of the atomic state. The above results depend on the adiabatical approxi- mation. To check the adiabatical condition, we numer- ically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ1〉a = (|a〉1|a〉2+ |ψDF(0, 0)〉2)/ 2 and the laser field driving the transition |e〉 ↔ |r〉 are turned on. Then, by slowly switching the laser field driving the transition |g〉 ↔ |s〉, we increase the parameter r from 0 to r0 according to the linear function r(t) = r0t/T . In the adiabatical limit (T ≫ Γ−1), the atomic state becomes |Ψ′1〉a = (|a〉1|a〉2 + |ψDF(r0, 0)〉2)/ 2 at the time T . On the other hand, in the Hilbert space spanned by the ba- sis states {|ei〉} for i = 1, 2, · · · , 12 , we can numerically solve Eq. (7) and obtain the density matrix ρ1(T ) of the atoms. Let’s define Fr = a〈Ψ′1|ρ1(T )|Ψ′1〉a as the fidelity for this process. As shown in Fig. 3, if T > 100/Γ, Fr is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution. In the second example, we suppose that the atoms are initially in the state |Ψ2〉a = (|a〉1|a〉2+ |ψDF(r, 0)〉4)/ and all the driving fields are turned on to hold the pa- rameters r = r0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate ϕ̇ = 2π/T , the atomic state at the time T becomes |Ψ′2〉a = (|a〉1|a〉2 + eiχ12 |ψDF(r, 2π)〉4)/ 2. Let’s define the fidelity for this example as Fp = a〈Ψ′2|ρ(T )|Ψ′〉a, where ρ(T ) is the nu- merical solution of Eq. (7). As shown in Fig. 4, Fp increases as T increases but decreases as the parameter r0 increases. If T > 1000/Γ, Fp is larger than 0.992 for 0 < r0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3. A controlled-Z gate has been numerically simulated by directly solving Eq. (7) with r0 = 0.5, and ϕ0 = π/|2ν1 − ν12|. In the simulation, we set ṙ = r0/T1 in the steps 1 and 3, and ϕ̇ = ϕ0/(T2 − T1) in the step 2 with T1 = 0.05T3 and T2 − T1 = 0.90T3. If T3 > 1100/Γ, we find that the fidelity F = a〈Ψ(T3)|ρ(T3)|Ψ(T3)〉a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T3 must be longer than 6000/Γ. Now let’s briefly discuss the effects of the atomic spon- taneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γeff = γ(Ω 2/2∆2) [14, 16] and the effective decay rate of the fiber mode is κeff = κfΩ 2g2/(4∆2ν2). If κf ≤ γ and g2 ≪ ν2, κeff can be much smaller than γeff . Under this condition, the present scheme is feasible if Γ ≫ γeff . In the current cavity quantum dynamic (CQED) ex- periment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20]. If setting Ω/(2∆) = 1√ × 10−3, we have Γ ≃ 4 × 104γeff . The condition is held. In the present scheme, the large cavity decay rate is re- quired to ensure that the cavity modes are in a broad- band squeezed vacuum reservoir and then the atoms al- ways ”see” the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γeff for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the op- eration time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γeff and the single-atom trapping time in cavity [21]. On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22]. Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology. IV. CONCLUSIONS We propose a cavity-atom coupled scheme for the real- ization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving con- dition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be im- plemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength be- tween the environment and the atoms is, the more reli- able the realized CPHASE gate is. Acknowledgments We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01). [1] P. W. Shor, in Proceeings, 35th Annual Symposium on Foundations of Computer Science, edited by S. Gold- wasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006). [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003).

0704.0484

Search for Chaotic Behavior in a Flapping Flag

Search for chaotic behavior in a flapping flag J. O. McCaslin and P. R. Broussard Covenant College, Lookout Mountain, Georgia 30750 Abstract We measured the correlation of the times between successive flaps of a flag for a variety of wind speeds and found no evidence of low dimensional chaotic behavior in the return maps of these times. We instead observed what is best modeled as random times determined by an exponential distribution. This study was done as an undergraduate experiment and illustrates the differences between low dimensional chaotic and possibly higher dimensional chaotic systems. I. INTRODUCTION Low dimensional chaotic behavior has been seen in many dynamical systems. The study of a driven pendulum,1 dripping faucets,2 and many other systems have illustrated the transition from periodic to chaotic behavior. The dynamics of a flapping flag have also been claimed to be chaotic,3,4 but our attempt to find experimental data on flapping flags was not successful. There has been extensive work on the onset of flapping and flutter for paper and fabrics in air,5 as well as filaments in a two-dimensional soap film6 and heavy flags in water.7 However, none of these studies has characterized the dynamics of the systems studied as chaotic. The goal of this paper is to determine if low dimensional chaotic dynamics is revealed by looking at return maps of the time between successive flaps of a flag. II. EXPERIMENTAL SETUP AND DATA ANALYSIS A schematic of the experiment is shown in Fig. 1. In order to not deal with changes in the flag’s aspect ratio due to gravity, we let the flag hang vertically and have the wind directed vertically. The wind was produced from a square home box fan (approximately 0.5 m on a side) located approximately 0.75 m above the flag support rod, in order to have a more uniform distribution of wind. The flag was approximately 0.35 m long and 0.25 m wide, and the support for the flag was approximately 0.70 m above the supporting table so as to create a space for the air to move. Wind speed was controlled by using a variac to give a variable voltage to the fan. The voltage was varied from 30 to 120 V in steps of 10 V. The actual wind speeds were measured using a Skywatch handheld windspeed detector. Ten measurements of wind speed were taken for each voltage and then averaged. The relation between the wind speed and the voltage was approximately linear and is given in Table I. The flapping of the flag was measured using a tab glued to the edge of the flag, which would break the beam from a HeNe laser (shown in Fig. 2) as suggested in Ref. 4. The laser’s illumination was measured with a Vernier light sensor and Logger Pro software running on an inexpensive laptop. The rate of collecting illumination data was determined by making data collection runs at lower collection rates until the sensor began to “miss” the flapping. The data collection rate was set to 35 samples/s, and the data was collected for 15 min for each wind speed. An example of the illumination versus time is given in Fig. 3. To analyze the data, a program8 was used to examine the illumination versus time series to look for the time intervals between the flag flapping, defined in our case as the time between drops in intensity. The code outputs a list of time intervals, denoted as t1, t2, . . . , tN for each wind speed. These data were then used to create plots of the time intervals versus their order as well as return maps, where the correlation between time intervals are studied. III. EXPERIMENTAL RESULTS An example partial time series is shown in Fig. 4. The times are not uniformly distributed, but are more clustered around t = 0. If we plot the time intervals in a standard two- dimensional return map (tn+1 versus tn) as in Fig. 5, we see that there is no apparent pattern in the plot, as would be expected if the dynamics controlling the flapping exhibited low dimensional chaos, such as seen in dripping faucet times2 or the times between swings of a chaotic pendulum.1 Data was collected for voltages from 30 V to 120 V (in steps of 10 V) (wind speeds from just under 0.5 m/s to over 4 m/s). All the return maps showed similar behavior, as seen in Fig. 6 for the 30 V and 80 V data sets. The 30 V data exhibits a clustering of points not around t = 0, but instead near (tn, tn+1) = (1, 1). This behavior is partly due to the reduction of short time scales in the flapping of the flag. As the voltage decreases and the wind speed lowers, there are fewer and fewer rapid flapping events. In addition, there is a change in the distribution of times as we will discuss. The overall behavior of the return maps (higher density of points at low times, lower density for high times, no apparent structure) is unchanged. As mentioned, the lack of structure in the return maps is evidence for the lack of low dimensional chaotic dynamics in this experiment. There is a possibility that the chaotic behavior could be higher dimensional and not revealed in a simple two-dimensional re- turn map. To investigate this possibility, three-dimensional return maps were made with (x, y, z) = (tn, tn+1, tn+2) as shown in Fig. 7 for the 120 V data set. Again, there is no ap- parent structure observed in this return map. We observe similar three-dimensional return maps for the other wind speeds. The return maps for voltages above 50 V (or windspeed greater than 0.8 m/s) show strong clustering near the origin, implying that shorter time intervals are more probable than longer ones. (Even for the lower values of windspeed where there seems to be a minimum time between flaps, there is still clustering around the lowest times.) For the three-dimensional return maps, the points are also densest near the origin, and then next along the axes, with fewer points along the diagonals. Although there are differences in the time scaling, the lack of any apparent structure in the return maps motivated us to look at other possible expla- nations. Figure 8 shows that a histogram of the times follows an exponential distribution. The time constant is order 1 s, with no systematic dependence on the wind speed. For the lowest windspeed measured, the histogram shows a peak at times greater than zero (1.2 s for the 30 V data) which could be due to the natural period of the flag (estimated to be 0.8 s). For low wind speeds the dynamics could be due to the natural periodic motion of the flag. However, we did not observe any indication of periodic motion in the low wind speed return maps. As the windspeed is increased, the peak in the histograms decreases and for 50 V and higher the peak is located at time equal zero. To compare the observed histogram order, we generated a random distribution of time intervals with an exponential probability. A set of N uniform random numbers xi were generated in the interval (0,1), where N was equal to the number of time intervals in one of our data sets. Exponentially distributed numbers yi were generated from the xi by the relation y(i) = − ln(xi)9 and then scaled to match the range of collected times for the data set considered. This set was then used to produce a simulated return map that would be expected if the time intervals between flaps are uncorrelated and occur with an exponential probability. Such a return map is shown in Fig. 9 along with an actual return map. By comparing the randomly generated return map to the return map for 120 V, it is clear that the two show very similar behavior, namely, strong concentration of points for small times and lower probability for large time intervals. We did not find evidence of low dimensional chaotic behavior for the time intervals between flaps, but instead data that resembles uncorrelated times with an exponential probability distribution, which would be equivalent to an infinite dimensional dynamic system. It is possible that we did not take data over a wide enough range of wind speeds. Because we did not observe any periodic flutter, such as seen in the heavy flag in the water experiment,7 the system should be in a non-linear dominated regime (even though we did see behavior at low windspeeds that might have been due to the natural period of the flag). We must also consider that our data may need to be analyzed at a higher dimensional level (looking at projections into 3D space of higher dimensional return maps) to verify that there is no chaotic nature to the results. However, we can conclude that low dimensional chaos cannot be easily observed in a flapping flag experiment.4 Acknowledgments The authors would like to gratefully acknowledge Dr. John Hunt for assistance with Java programing and Dr. Donald Petcher for his help in the random number analysis. In addition, the authors express their thanks to the anonymous referees who made many helpful suggestions for improving this paper. 1 John P. Berdahl and Karel Vander Lugt, “Magnetically driven chaotic pendulum,” Am. J. Phys. 69, 1016–1019 (2001). 2 K. Dreyer and F. R. Hickey, “The route to chaos in a dripping water faucet,” Am. J. Phys. 59, 619-627(1991). 3 Edward N. Lorenz, The Essence of Chaos (University of Washington Press, Seattle, 1993), p. 5. 4 Chaos in the Laboratory, edited by David L. Vernier (Vernier Software, 1991, pp. 14–15. 5 Y. Watanabe, S. Suzuki, M. Sugihara, and Y. Sueoka, “A experimental study of paper flutter,” J. Fluids Struct. 16, 529–542 (2002). 6 J. Zhang, S. Childress, A. Libchaber, and M. Shelley, “Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind,” Nature 408, 835–839 (2000). 7 M. Shelley, N. Vandenberghe, and J. Zhang, “Heavy flags undergo spontaneous oscillations in flowing water,” Phys. Rev. Lett. 94, 094302-094501 (2005). 8 The Java program used for this analysis is available on our website, http://physics.kepler.covenant.edu. 9 Germund Dahlquist and Åke Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, 1974), p. 452. http://physics.kepler.covenant.edu Tables TABLE I: Voltage on fan and corresponding windspeed measured at the flag position. Voltage (V) Windspeed (m/s) 30± 0.5 0.40± 0.23 40± 0.5 0.52± 0.16 50± 0.5 0.83± 0.24 60± 0.5 1.33± 0.28 70± 0.5 2.36± 0.64 80± 0.5 2.98± 0.64 90± 0.5 3.16± 0.77 100± 0.5 3.52± 0.81 110± 0.5 3.87± 0.74 120± 0.5 4.07± 0.76 Figure Captions FIG. 1: Schematic of the experimental setup. FIG. 2: Flag breaking beam. FIG. 3: Example illumination change as the flag flaps. This plot is for a voltage setting of 30 V, or a wind speed of approximately 0.40 m/s. The plot shows two example time intervals between successive flaps. FIG. 4: The time between successive flaps versus the sequence iteration. This data was collected at 120 V. FIG. 5: Return map for the data collected at 120 V. FIG. 6: Return map for the data collected at (a) 30 V and (b) 80 V. FIG. 7: Three-dimensional return map for the data collected at 120 V. The tick marks on the axis correspond to 1 s. FIG. 8: Histogram of times between flaps for the 120 V run. FIG. 9: Comparison of the return maps for times generated by an exponential probability (a) with the figure of data collected at 120 V as shown in Fig. 5 (b). The random data was scaled to match the 120 V run in the number of points and range of times. Introduction Experimental Setup and Data Analysis Experimental results Acknowledgments References Tables Figure Captions

0704.0486

Kinematic Decoupling of Globular Clusters with Extended Horizontal-Branch

arXiv:0704.0486v1 [astro-ph] 4 Apr 2007 Accepted for Publication in ApJ Letters, April 2007 Preprint typeset using LATEX style emulateapj v. 08/13/06 KINEMATIC DECOUPLING OF GLOBULAR CLUSTERS WITH EXTENDED HORIZONTAL-BRANCH Young-Wook Lee , Hansung B. Gim , & Dana I. Casetti-Dinescu Accepted for Publication in ApJ Letters, April 2007 ABSTRACT About 25% of the Milky Way globular clusters (GCs) exhibit unusually extended color distribution of stars in the core helium-burning horizontal-branch (HB) phase. This phenomenon is now best understood as due to the presence of helium enhanced second generation subpopulations, which has raised a possibility that these peculiar GCs might have a unique origin. Here we show that these GCs with extended HB are clearly distinct from other normal GCs in kinematics and mass. The GCs with extended HB are more massive than normal GCs, and are dominated by random motion with no correlation between kinematics and metallicity. Surprisingly, however, when they are excluded, most normal GCs in the inner halo show clear signs of dissipational collapse that apparently led to the formation of the disk. Normal GCs in the outer halo share their kinematic properties with the extended HB GCs, which is consistent with the accretion origin. Our result further suggests heterogeneous origins of GCs, and we anticipate this to be a starting point for more detailed investigations of Milky Way formation, including early mergers, collapse, and later accretion. Subject headings: Galaxy: formation – globular clusters: general – stars: horizontal-branch 1. INTRODUCTION The discovery of multiple stellar populations in the most massive GC ωCen (Lee et al. 1999), together with the fact that the second most massive GC M54 is a core of the disrupting Sagittarius dwarf galaxy (Layden & Sarajedini 2000), have strengthen the view that some of the massive GCs might be remaining cores of disrupted nucleated dwarf galaxies (Freeman 1993). Among their several peculiar characteristics, both ωCen and M54 have extended horizontal-branch (EHB), with extremely hot horizontal-branch (HB) stars well sep- arated from redder HB (Lee et al. 1999; Rosenberg, Recio-Blanco, & Garcia-Marin 2004). High pre- cision Hubble Space T elescope (HST ) photometry (Bedin et al. 2004) has discovered that ωCen also has a curious double main-sequence (MS). Recent studies have shown that both these peculiar colour-magnitude diagram (CMD) characteristics are best understood as due to the presence of helium enhanced second gen- eration subpopulations (Norris 2004; Lee et al. 2005; Piotto et al. 2005; D’Antona et al. 2005). Further- more, the prediction of the models (Lee et al. 2005; D’Antona et al. 2005) that most of the GCs with EHB would have double or broadened MSs are now confirmed by HST/ACS (Advanced Camera for Survey) pho- tometry (Piotto et al. 2007). This ensures that EHBs are strong signature of the presence of multiple popula- tions in GCs. A significant fraction (∼30%) of the helium enriched subpopulation observed in these peculiar GCs is also best explained if the second generation stars were formed from enriched gas trapped in the deep gravita- tional potential well while these GCs were cores of the ancient dwarf galaxies (Bekki & Norris 2006). Despite the lack of apparently wide spread in iron-peak elements 1 Center for Space Astrophysics & Department of Astronomy, Yonsei University, Seoul 120-749, Korea ([email protected]) 2 Department of Astronomy, Yale University, New Haven, CT 06520, USA in most of these GCs, all of these recent developments suggest that GCs with EHB are probably not genuine GCs, but might have a unique origin in the formation history of the Galaxy. In order to test this working hypothesis further, we have carefully surveyed 114 GCs with reasonably good CMDs, and found that 28 (25%) of them have EHB (Lee et al., in preparation). Their NGC numbers are: 2419, 2808, 5139, 5986, 6093, 6205, 6266, 6273, 6388, 6441, 6656, 6715, 6752, 7078, and 7089 for the GCs with strongly extended HB; and 1851, 1904, 4833, 5824, 5904, 6229, 6402, 6522, 6626, 6681, 6712, 6723, and 6864 for the GCs with moderately extended HB, including those with bimodal HB distributions. We will collectively call all of them as “EHB GCs”. Our selection of EHB GCs were based on the reddening independent criteria on CMD in B&V passbands (∆VHB > 3.5 for strongly ex- tended HB; either 3.0 < ∆VHB < 3.5 or ∆(B − V )HB > 0.78 with clear bimodal colour distribution for mod- erately extended HB). But, since their appearances on CMDs are distinct enough from GCs with normal HB (Piotto et al. 2002), our selection agrees well with the re- sult based on smaller sample and other measures of HB temperature extension (e.g., Recio-Blanco et al. 2006). We have then investigated their properties compared to other normal GCs. 2. LUMINOSITY FUNCTION AND KINEMATICS First of all, from the luminosity function (Fig. 1), we found that EHB GCs are among the brightest GCs of the Milky Way, including 11 out of 12 brightest GCs (see also Recio-Blanco et al. 2006). It is surprising to see that not a single EHB GC is fainter than MV = - 7. Careful inspection of all CMDs confirms that this is not due to the smaller number of HB stars in fainter GCs. Because of significant fraction (18 - 51%) of the helium enriched bluer subpopulation observed in EHB GCs, its presence on the HB would be reliably detected (> 5 - 10 stars) even in a cluster of MV = -6 or -5 if it http://arxiv.org/abs/0704.0486v1 2 Lee, Gim, & Casetti-Dinescu -2 -4 -6 -8 -10 -12 40 EHB Normal Fig. 1.— The histogram of MV for 114 Milky Way GCs (data from Harris 1996). Blue and red are GCs with strongly and mod- erately extended HBs, respectively. EHB GCs are clearly brighter (more massive) than normal GCs. existed. This result is perhaps already suggesting that EHB GCs might have a peculiar origin, as their inferred current stellar mass, which might represent only a small fraction of their original mass, is comparable with that of low-luminosity dwarf galaxies in the Local Group. Motivated by this, we have investigated the kinemat- ics of EHB GCs, in order to see whether their kinematic properties are also distinct from other normal GCs. Fol- lowing previous investigation (Zinn 1993), we have first divided GCs into three subgroups (Fig. 2) based on the HB morphology and metallicity diagram (Lee, Demar- que, & Zinn 1994). Metal-poor ([Fe/H] < -0.8) GCs in the “Old halo (OH)” group have bluer HB morphol- ogy at a given metallicity, and those in the “Younger halo (YH)” group have redder HB morphology at fixed metallicity. The OH group, in the mean, is probably older than the YH group by ∼1 Gyr (Rey et al. 2001; Salaris & Weiss 2002). The metal-rich ([Fe/H] > -0.8) GCs are further classified as “disk/bulge (D/B)” group. EHB GCs belong in all three subgroups, although the majority of them are in the OH group. Note also that most (94%) GCs in YH group are in the outer halo (galactocentric distance, Rgc > 8 kpc), while the ma- jority (80%) of GCs in OH and D/B groups are in the inner halo (Rgc < 8 kpc). The result of the kinematic analysis based on the constant-rotational-velocity solutions (Zinn 1993; Frenk & White 1980) and the updated database of Har- ris (Harris 1996) is presented in Table 1. When all the GCs are considered, we are basically confirming the con- clusion of the previous work (Zinn 1993). YH group is dominated by random motion with no sign of significant rotation (Vrot), while OH group shows some prograde rotation and a smaller line-of-sight velocity dispersion (σlos). D/B group is mostly supported by rotation with a relatively small σlos. We find, however, EHB GCs, both belonging to YH and OH groups, are dominated by ran- dom motion and show no signs of rotation. Consequently, when they are excluded from the sample, normal GCs in OH group show increased rotation (from 1.5 to 1.8σ from zero Vrot) and higher value of Vrot/σlos. The same trend is also observed in the normal GCs in D/B group, but with much larger uncertainty. When only comparably bright (MV < -6) GCs are considered, the differences become significantly larger (2.5σ from zero Vrot). The TABLE 1 KINEMATICS OF GLOBULAR CLUSTERS BASED ON RADIAL VELOCITY ALONE* Group N Vrot σlos Vrot/σlos All GCs All Halo 71 25±27 124±10 0.20±0.22 YH 25 -18±66 153±22 -0.12±0.43 OH 46 40±27 104±11 0.38±0.26 D/B 14 168±28 65±12 2.57±0.65 EHB GCs All EHB 24 10±32 93±13 0.11±0.34 OH 18 4±35 91±15 0.05±0.38 Normal All Halo 48 32±39 137±14 0.24±0.29 YH 20 -42±80 162±26 -0.26±0.49 OH 28 70±39 111±15 0.63±0.36 D/B 13 188±22 48±9 3.94±0.89 Normal (MV < -6) All Halo 39 37±44 139±16 0.26±0.32 YH 17 -69±81 160±27 -0.44±0.51 OH 22 105±42 103±15 1.02±0.43 D/B 11 195±27 52±11 3.76±0.95 *For Rgc < 40 kpc and excluding GCs with (cosψ) > 0.2 above analysis, based only on the radial velocity data, provides good reason to suspect that EHB GCs are kine- matically decoupled from other normal GCs, especially in OH group. Below, we investigate this in more detail using the measurements of full spatial motions and or- bital parameters now available for 49 GCs in our sample (Dinescu et al. 2003). In Figure 3, we have plotted kinematic parameters ob- tained from full spatial motions as a function of metallic- -1.0 -0.5 0.0 0.5 1.0 EHB OH+D/B EHB YH Normal OH+D/B Normal YH Red Blue HB Type Fig. 2.— The subdivision of GCs in the HB morphology ver- sus metallicity diagram. The filled circles are GCs either in the “old halo (OH)” or metal-rich ([Fe/H] > -0.8) “disk/bulge (D/B)” groups, while open circles are those in the “younger halo (YH)” group. The EHB GCs belong in all three subgroups, but most of them are in OH group. The updated database (Lee et al. 1994) consisting 95 GCs in Rgc < 40 kpc zone are compared with model HB isochrones (Rey et al. 2001) in solid lines, the upper being older by 1.1 Gyr. Short dashed line has the same age as the upper solid line, but is for EHB GCs with 15 % of helium enhanced (Y = 0.33) subpopulation (Lee et al. 2005). Kinematic Decoupling of EHB GCs 3 -2000 -1500 -1000 EHB OH EHB YH Normal OH+D/B (P) Normal OH+D/B (R) Normal YH 47Tuc 200 M54 -2.5 -2.0 -1.5 -1.0 -0.5 47Tuc [Fe/H] 0.0 -2.5 -2.0 -1.5 -1.0 -0.5 [Fe/H] 0.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 -1000 47Tuc [Fe/H] Fig. 3.— The relationship between kinematics derived from full spatial motions and metallicity. From (a) to (f), total orbital energy (Etot), maximum distance perpendicular to the Galactic plane (Zmax), velocity component perpendicular to the plane (W ), rotational velocity (Θ), orbital eccentricity, and the angular momentum component associated with Θ (LZ) are plotted as a function of metallicity (Zinn 1993), respectively. Red filled circles are normal GCs with prograde rotation in OH and D/B groups. All of them are in the inner halo. Red open circles are normal GCs with retrograde rotation in OH and D/B groups. Only 3 of them are in the inner halo. ity, which shows more directly the systematic differences between EHB and normal GCs. In all panels of Fig- ure 3, EHB GCs have diversity in kinematics, and show no correlations with metallicity (correlation coefficient, r, of -0.02 to -0.31 with high p-values of 0.25 to 0.94). Normal YH GCs, mostly in the outer halo, show kine- matically hot signatures (high Etot, Zmax, & eccentricity, and large velocity dispersion) (Mackey & Gilmore 2004). To our surprise, however, when EHB GCs are excluded, most normal GCs with prograde rotation in OH and D/B groups (red filled circles) show clear signs of dissipational collapse. Etot, Zmax, W velocity, and perhaps orbital ec- centricity are all decreasing with increasing metallicity, among which the ‘chevron’ shape of W velocity distri- bution is most impressive. Rotational velocity, however, is increasing with metallicity, and LZ appears to be con- served. The orbital properties of NGC 6528 are known to be highly affected by the potential of the bar because of its proximity (Dinescu et al. 2003). Thus, excluding this one deviant point in panels (d) and (e), we obtain strong correlations for Etot, Zmax, |W |, Θ, and eccentricity (r = -0.58, -0.71, -0.95, 0.89, and -0.77, respectively) with small p-values (0.05, 0.01, 3.1× 10−6, 0.0002, and 0.005, respectively). In other words, the correlations are highly significant at the level of 95%, 99%, 99.999%, 99.98%, and 99.5%, respectively. As expected, however, corre- lation is low (r = 0.17) for LZ with a high p-value of 0.59. All of these trends observed for normal GCs with pro- grade rotation in OH and D/B groups are fully consistent with the model first envisioned by Eggen, Lynden-Bell, & Sandage (1962), where metal enrichment went on as dissipational collapse continued. Although these results are based on relatively modest numbers of GCs with full spatial motion information, their coherent behaviours in all panels of Figure 3, together with statistically signifi- cant correlations, confirm that we are detecting real sig- natures. Also, these results are consistent with the kine- matics solution obtained from radial velocity alone (Ta- ble 1), which is based on a larger sample of GCs. We argue, therefore, (1) EHB GCs in our sample are indeed kinematically decoupled from most of the normal GCs in OH group, and (2) when EHB GCs are excluded, we are detecting clearer signatures of dissipational collapse in the inner halo, which apparently led to the formation of the Galactic disk (Zinn 1993; Mackey & Gilmore 2004). The kinematics of EHB GCs, which are not following the dissipational collapse, are more consistent with what one would expect among the relicts of primeval star- forming subsystems that first formed the nucleus (EHB GCs with low Etot and Zmax) and halo (EHB GCs with high Etot and Zmax) of the Galaxy through both dissipa- tional and dissipationless mergers, as has been predicted by recent ΛCDM simulations for “high-σ peaks” (e.g., Diemand, Madau, & Moore 2005; Moore et al. 2006). As described above, a significant fraction of the helium enriched subpopulation also favours building block origin of EHB GCs. Normal YH GCs in the outer halo share 4 Lee, Gim, & Casetti-Dinescu their kinematic properties with the outlying EHB GCs, which is consistent with the view (Searle & Zinn 1978) that they were originally formed in the outskirts of iso- lated building blocks and later accreted to the outer halo of the Galaxy when their parent dwarf galaxies, like the Sagittarius, were merging with the Milky Way. The GCs with EHB also tend to show more extended Na-O and Mg-Al anticorrelations (Gratton 2007). Therefore, the suggested connection between some of these GCs with strong chemical inhomogenity and orbital parameters (Carretta 2006) might be due to the diversity of kine- matics among EHB GCs. According to the present picture, most of the normal GCs with retrograde rotation in OH and D/B groups could have also originated from the subsystems with ret- rograde rotation. Interestingly, their relatively confined distributions both in the angular momentum phase space (Helmi et al. 1999) and velocity space are not inconsis- tent with the possibility that some or most of them were former members of parent dwarf galaxies hosting two EHB GCs, ωCen and/or NGC 6723 (Lee et al., in prepa- ration). Their distribution in velocity space is also well consistent with the model prediction of the tidal debris from ωCen’s parent dwarf system (Mizutani, Chiba, & Sakamoto 2003), which was presumably formed in the outer halo and accreted to the inner halo. Note that a similar minor merging of subsystem with the thin disk (Quinn, Hernquist, & Fullagar 1993) could have also changed some of the original kinematic properties of two disk GCs in Figure 3 (47Tuc and M71). 3. DISCUSSION The clear differences in kinematics and mass be- tween GCs with and without EHB are strong evidence that they have different origins. Our results suggest present-day Galactic GCs are most likely an ensemble of heterogeneous objects originated from three distinct phases of the Milky Way formation: (1) remaining cores or central star clusters of building blocks that first assem- bled to form the nucleus and halo of the proto-Galaxy (Bromm & Clarke 2002; Santos 2003; Bekki 2005; Kravtsov & Gnedin 2005; Moore et al. 2006), (2) gen- uine GCs formed in the dissipational collapse of a transient gas-rich inner halo system that eventually formed the Galactic disk (Eggen et al. 1962), and (3) genuine GCs formed in the outskirts of outlying building blocks that later accreted to the outer halo of the Milky Way (Searle & Zinn 1978). In this picture, relicts of first building blocks that formed the flattened nucleus (Kravtsov & Gnedin 2005; Moore et al. 2006) are now observed as relatively metal-poor EHB GCs (e.g., NGC 6266, 6522, and 6626) having low Etot and Zmax near the centre. Formation of the slowly-rotating gas-rich inner halo system that later collapsed in phase (2) is still most unclear, but it is attractive to speculate that leftover gas from “rare peaks” (building blocks hosting EHB GCs) in the inner halo and gas from continuously falling “less rare peaks” (Moore et al. 2006) led to the formation of this structure, perhaps with the aids of some heating feedbacks (e.g., Schawinski et al. 2006) soon followed by cooling. Several lines of further study will certainly help to shed more light into the picture briefly sketched here. For example, search for the tidal streams that might be associated with EHB GCs, dark matter search in the outlying EHB GCs where preferential disruption of dark matter halo (Saitoh et al. 2006) might be less severe, kinematics analyses of extragalactic GC systems along with the ultraviolet survey for EHB GC candidates, together with more detailed high resolution ΛCDM simulations. We thank R. Zinn, R. Larson, and P. Demarque for helpful discussions, C. Chung for his assistance in HB isochrone construction, and H.-Y. Lee for her assistance in CMD compilation. Support for this work was provided by the Creative Research Initiatives Program of the Ko- rean Ministry of Science & Technology and KOSEF, for which we are grateful. REFERENCES Bedin, L. R., Piotto, G., Anderson, J., Cassisi, S., King, I. R., Momany, Y., & Carraro, G. 2004, ApJ, 605, L125 Bekki, K. 2005, ApJ, 626, L93 Bekki, K. & Norris, J. E. 2006, ApJ, 637, L109 Bromm, V. & Clarke, C. J. 2002, ApJ, 566, L1 Carretta, E. 2006, AJ, 131, 1766 D’Antona, F., Bellazzini, M., Caloi, V., Pecci, F. F., Galleti, S., & Rood, R. T. 2005, ApJ, 631, 868 Diemand, J., Madau, P., & Moore, B. 2005, MNRAS, 364, 367 Dinescu, D. I., Girard, T. M., van Altena, W. F., & Lopez, C. E. 2003, AJ, 125, 1373 Eggen, O. J., Lynden-Bell, D., & Sandage, A. R. 1962, ApJ, 136, Freeman, K. C. 1993, in ASP Conf. Ser. 48, The Globular Clusters- Galaxy Connection, ed. G. H. 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0704.0490

Long Distance Signaling Using Axion-like Particles

Microsoft Word - stancil_text+figs_prDR.doc Long Distance Signaling Using Axion-like Particles Daniel D. Stancil, Department of Electrical and Computer Engineering Carnegie Mellon University, Pittsburgh, PA 15213 Abstract The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5am < meV and 8 15 10 GeVag γγ − −> × . PACS: 14.80.Mz, 84.40.Ba, 84.40.Ua, 42.79.Sz The axion has been proposed as a solution to the strong-CP problem [1], and is also a candidate for the galactic dark matter [2]. Interest in axions has increased recently owing to the report by the PVLAS collaboration of optical rotation induced by a magnetic field in a vacuum [3], since the creation of axions or other similar particles was one possible explanation for this rotation. Subsequently a number of groups began experimental searches using axion generation and detection schemes that would more definitively point to these new particles as the explanation [4]. The PVLAS result was surprising because it suggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5]. Although mechanisms have been proposed to reconcile the reports [6], the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8]. There does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling. In particular, I would like to call attention to the observation that a new type of long- distance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked—even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites. Communication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10]. Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11]. The coupling was obtained by considering the Lagrangian density 1 1 1 1 4 4 2 2a a L F F g aF F a a m aμν μν μμν γγ μν μ= − − + ∂ ∂ − , (1) where F A Aμν μ ν ν μ= ∂ − ∂ is the electromagnetic field tensor, ( ),A V Aα = − where V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, ag γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by F F γδαβ αβγδε= . In these equations we have taken 1h c= = . As an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp( )i tω− , where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations 2 2 2 0 02 2, . a a a m a i g B A A i g B a z zγγ γγ ω ω ω ω + − = − + = (2) Thus the static magnetic field B0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a). This is sometimes referred to as an “invisible light through walls” experiment [4,12,13]. An electromagnetic wave with amplitude (0)A enters a region of magnetic field of strength BOT extending over a distance LT. If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R TP P p p= , (3) where inP is the optical input power, 0P is the optical output power, Tp , Rp are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11,12,14] 1 sin / 2 2 / 2aa p g B L k qLγγ ω ⎡ ⎤ = ⎢ ⎥ . (4) Here aq k kγ= − indicates the phase mismatch between the photon and axion fields. The efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1(b) [4,15]. In this figure a laser is used to generate the incident photons, and mirrors are used to cause the light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 /TF π , where FT is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor /TF π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 /RF π , where FR is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be /RF π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b). The conversion loss equation (3) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver. The general solution to Eq. (2) is given by [12] 3 0( ) ( ) ( )4 aik r r a r i g d r A r B r r rγγ ′ ′ ′= ′−∫ i . (5) If the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field 3 0( ) ( ) ( )4 a r i g d r e A r B r ′−′ ′ ′≈ ∫ i i . (6) Consider the case where the source 0( ) ( )A r B r′ ′i is only nonzero inside a cylinder of radius R and length L as shown in Figure 2. Further, we assume that within this cylinder 0 0ˆ TB xB= and 0ˆ / exp( )TA x F A ik zγπ= , where 0A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse TF . Using Eq. (6), the far-field potential is found to be sin / 2 2 / 2 aik r az T a TT a T T T a Taz T k k L J k RFe a r i g A B L s r k Rk k L ⎡ ⎤ ⎡ ⎤−⎣ ⎦≈ ⎢ ⎥ − ⎢ ⎥⎣ ⎦ , (7) where sin , cosa a az ak k k kρ θ θ= = , and T Ts Rπ= is the cross-sectional area of the source region at the transmitter. This expression has its maximum when 0θ = , for which , 0az a ak k k ρ= = . The factor containing the Bessel function in (7) has the limit ( )10lim / 1/ 2a a T a Tk J k R k Rρ ρ ρ→ ⎡ ⎤ =⎣ ⎦ . Consequently, the axion field on axis is sin / 2 4 / 2 aik r a T T T a r i g A B L s r qLγγ ≈ . (8) The time averaged transmitted power density is ( ) ( ) a a T T in S r k a r s p P = = , (9) where (0)in TP S sγ= , and ( ) (0) 0 S k Aγ γω= . If this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is 0 (2 / ) ( )R R a RP F p S r sπ= . (10) Substituting Eq. (9) for the power flux ( )aS r gives in aR T R R T T P kF F P p s p s rπ π π π = . (11) This expression can be understood in terms of antenna theory for electromagnetic waves. In this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as /(4 ) = , (12) where Prad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)a T T TS L F p Sγπ= , we have 2 ( ) (2 / )rad a T T T T inP S L s F p Pπ= = . (13) Substituting (13) for the total radiated power and (9) for the power flux, the directivity simplifies to 2 2(2 / ) (4 / )( / 2)T a T a TD s sπ λ π λ= = . (14) The relation between the directivity and physical area (14) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /rad inP Pη = , we also define the antenna gain as 2(2 / ) (2 / )T T T T T a TG D F p sη π π λ= = , (15) where (2 / )T T TF pη π= . Next suppose that at some distant location this transmitted field is incident upon a receive antenna with length LR, radius RR, and finesse FR. From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 , ( )e R aP s S r= , where we have defined the effective area of the receiving antenna as , ( / ) ,e R R c R Rs s n F pπ= (16) R Rs Rπ= is the physical cross-sectional area, and cn is the number of ends from which photons are collected (i.e., 1, 2cn = ). As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive: 2, ,/ / /(4 ).e R R e T T c as G s G n λ π= = (17) If photons were collected only from one end of the receive antenna ( 1cn = ), then Eq. (17) would be identical to conventional antenna theory. With these definitions, the expression for the received power (11) can be interpreted as a Friis-like equation: ( ) ( )( )2 2 20 , , ,/ / 4 /( ) /in c T R a e T c T a e RP P n G G r s n s rλ π λ= = . (18) Here ,c Tn is the value used to compute ,e Ts according to Eq. (16). The ratio , ,/e T c Ts n is independent of the choice of ,c Tn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna. It is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12]), then there is an optimum length for the conversion region of an antenna. This occurs when ( )sin / 2 1qL = , or qL π= . The optimum length is found to be 2( / )opt aL mγλ ω= . (19) In obtaining this expression, we have used 2 /(2 )aq m ω≈ , which is valid for am ω . The diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in (7): ( ) ( ) 214 sin /( sin )d a aP J k R k Rθ θ θ= ⎡ ⎤⎣ ⎦ . (20) The diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression 1.22 /dFWFN TRγθ λ≈ . (21) Similarly, the diffraction beam width at half maximum is determined by the roots of ( ) 1/ 2dP θ = , or (1.616 / ) / 0.514 /dFWHM a T TR Rγθ π λ λ= ≈ . (22) In contrast, the conversion-limited power pattern is given by sin cos / 2 2 cos / 2 k k L k k L ⎡ ⎤⎡ ⎤−⎣ ⎦⎢ ⎥= , (23) and depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by (19), the conversion beam widths are approximately given by 2( / ), 1.06( / ) c cFWFN a FWHM am mθ ω θ ω≈ ≈ . (24) For quantum-limited detection, the channel capacity is [16] 2log (1 / )d RC Nν η ν= Δ + Δ , (25) were dη is the quantum efficiency of the optical detectors, νΔ is the bandwidth of the resonator /(2 )c LFνΔ = [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus. As an example, consider transmitters and receivers with 1064 nm (1.17 eV)γλ = , 10 WinP = , 20.01 ms = , 0 10 TB = , 3L = m, 53.1 10F = × , 0.5dη = , and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves ( ) 1/a aM m g γγ= for distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the “4+4” experimental apparatus proposed in [15]. The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8]. The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12]. The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5am < meV and 72 10 GeVM < × ( 8 15 10 GeVag γγ − −> × ). Of particular interest, we note that the range 6 72 10 GeV 2 10 GeVM× < < × has not yet been excluded by photon regeneration experiments. For an area of 0.01 m2, the radius is 0.01/ 0.0564R π= = m. The half-power diffraction beam width for the example antennas is therefore 6 49.7 10 rad 5.56 10 deg 2dFWHMθ − −= × ≈ × ≈ arc-sec, while the half-power conversion beam width for 0.7am ≈ meV (optimum for L=3 m) is 4 26.3 10 rad 3.63 10 deg 131cFWHMθ − −= × ≈ × ≈ arc-sec. Consequently the total beam width is determined by the diffraction beam width in this example. While pointing with an accuracy of 2 arc-sec would be challenging, it is somewhat less stringent than that required for optical deep space communications [18]. The size of such axion transmitters and receivers would be roughly that of a medium size professional telescope. It is interesting to note that the size of the beam at a distance of the earth’s diameter is about 124 m. Consequently coordinates obtained with differential GPS at both sites should enable the computation of pointing directions to sufficient accuracy. Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal. In summary, for 3.5am < meV and 72 10 GeVM < × ( 8 15 10 GeVag γγ − −> × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the “4+4” experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3am < meV and 66 10 GeVM < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: [email protected] References 1. R. D. Peccei and H. R. Quinn, Phys. Rev. Lett., 38, 1440, (1977); S. Weinberg, Phys. Rev. Lett., 40, 223 (1978); F. Wilczek, Phys. Rev. Lett., 40, 279 (1978). 2. R. Bradley, et al., Rev. Mod. Phys. 75, 777 (2003). 3. E. Zavattini, et al. (PVLAS Collaboration), Phys. Rev. Lett., 96, 110406 (2006). 4. See, for example, R. Rabadan, A. Ringwald, and Kris Sigurdson, Phys. Rev. Lett., 96, 110407 (2006); and J. Jaeckel, E. Masso, J. Redondo, A. Ringwald, and F. Takahashi, arXiv:hep-ph/0605313. 5. S. Andriamonje et al. (CAST collaboration), J. Cosmol. Astropart. Phys. 04, 010 (2007). 6. See, for example, R.N. Mohapatra and S. Nasri, Phys. Rev. Lett., 98, 050402 (2007). 7. E. Zavattini et al., arXiv:0706.3419 [hep-ex]. 8. C. Robilliard, et al., arXiv:0707.1296v3 [hep-ex]. 9. See, for example, J. W. Eerkens, US Patent # 4,205,268, May 27, 1980; J. M. Pasachoff and M. L. Kutner, Cosmic Search Vol. 1 No. 3 p. 2 (1979), http://www.bigear.org/CSMO/PDF/CS03/cs03p02.pdf ; H. Xie, J. Gao, C. Liu, and Q. Zhai, 2006 IET Int. Conf. on Wireless Mobile and Multimedia Networks Proc., Hangzhou, China, 6-9 Nov., (2006). 10. D. G. Michael, et al., Phys. Rev. Lett., 97, 191801 (2006). 11. P. Sikivie, Phys. Rev. Lett., 51, 1415 (1983); Phys. Rev. D, 32, 2988 (1985). 12. K. Van Bibber, N.R. Dagdeviren, S.E. Koonin, A.K. Kerman, and H.N. Nelson, Phys. Rev. Lett., 59, 759 (1987). 13. R. Cameron, et al., Phys. Rev. D, 47 3707 (1993). 14. G. Raffelt and L. Stodolsky, Phys. Rev. D, 37, 1237 (1988). 15. P. Sikivie, D.B. Turner, and Karl van Bibber, Phys. Rev. Lett. 98, 172002 (2007). 16. C-C. Chen, in Deep Space Optical Communications, edited by H. Hemmati (John Wiley & Sons, Hoboken, NJ, 2006) p. 92. 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. Figure Captions Figure 1. (a) Basic system for the generation and detection of axions. (b) An axion communication system, with the axion generator and photon regenerator located remote from one another. Figure 2. A source with uniform amplitude over a cylindrical region. Figure 3. (Color Online) The ranges of inverse coupling parameter M and pseudoscalar mass ma that would permit communication at information bandwidths of at least 1 bps using the example system, and the 4+4 system proposed in [15]. The shaded regions are ruled out by [8] and [13]. LT LR Mod Laser Data In Data Out B0T Optical Detectors B0T B0R Photon input Photon output Axions penetrate barrier opaque to photons LT LR Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. Figure 2, Stancil, Phys. Rev. D. -LT /2 LT /2 ma (eV) earth diameter 1000 km moon, 4+4 Robilliard, et al. QPM extensions Figure 3, Stancil, Phys. Rev. D.

0704.0492

Refuting the Pseudo Attack on the REESSE1+ Cryptosystem

RefutePseudoAttack Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 Refuting the Pseudo Attack on the REESSE1+ Cryptosystem* Shenghui Su 1, 2, and Shuwang Lü 3 1 College of Computer Science, Beijing University of Technology, Beijing 100022, P.R.China 2 School of Info Engi, University of Science & Technology Beijing, Beijing 100083, P.R China [email protected] 3 School of Graduate, Chinese Academy of Sciences, Beijing 100039, P.R.China [email protected] Abstract: We illustrate through example 1 and 2 that the condition at theorem 1 in [8] dissatisfies necessity, and the converse proposition of fact 1.1 in [8] does not hold, namely the condition Z / M – L / Ak < 1 / (2 Ak 2) is not sufficient for f (i) + f (j) = f (k). Illuminate through an analysis and ex.3 that there is a logic error during deduction of fact 1.2, which causes each of fact 1.2, 1.3, 4 to be invalid. Demonstrate through ex.4 and 5 that each or the combination of qu+1 > qu ∆ at fact 4 and table 1 at fact 2.2 is not sufficient for f (i) + f (j) = f (k), property 1, 2, 3, 4, 5 each are invalid, and alg.1 based on fact 4 and alg.2 based on table 1 are disordered and wrong logically. Further, manifest through a repeated experiment and ex.5 that the data at table 2 is falsified, and the example in [8] is woven elaborately. We explain why Cx ≡ Ax W f(x) (% M) is changed to Cx ≡ (Ax W f(x))δ (% M) in REESSE1+ v2.1. To the signature fraud, we point out that [8] misunderstands the existence of T –1 and Q –1 % (M – 1), and forging of Q can be easily avoided through moving H. Therefore, the conclusion of [8] that REESSE1+ is not secure at all (which connotes that [8] can extract a related private key from any public key in REESSE1+) is fully incorrect, and as long as the parameter Ω is fitly selected, REESSE1+ with Cx ≡ Ax W f(x) (% M) is secure. Keywords: Public key cryptosystem; Security; Lever function; Continued fraction; Sufficient condition 1 Introduction In April 2001, we put forward the REESSE1 public-key encryption scheme [1]. In September 2003, we proposed the REESSE1 public-key cryptosystem which is an extension of the first version, and includes both encryption and signature [2]. In May 2005, it was argued that the lever function ℓ(.) is necessary and sufficient for the security of the REESSE1 encryption [3]. In [3], the continued fraction method of analyzing the key transforms Cx ≡ Ax W and Cx ≡ Ax W (x) (% M) with x ∈ [1, n] and ℓ (x) ∈ Ω was mentioned earlier than in any other publications. In November 2006, an abbreviation of the REESSE1+ cryptosystem was submitted to eprint.iacr.org [4]. As is pointed out in [4], the set Ω = {5δ, …, (n + 4)δ | δ ≥ 1} is not unique, and other Ω may be selected ― Ω = {n + 1, …, n + n} with ℓ (i) + ℓ (j) ≠ ℓ (k) ∀ i, j, k ∈ [1, n] for example. Clearly, Ω is a security dominant parameter, and just like p and q in the RSA cryptosystem. In May 2005, [5] pointed out that the REESSE1 signature scheme was insecure, which is right. In July 2005, [6] thought unreasoningly that the REESSE1 encryption scheme was insecure, which is wrong, and rebutted thoroughly by us in [7]. Moreover, [7] illuminated definitely that the idea of the continued fraction analysis of REESSE1 did not originate from [6] (naturally also not from [8]), ant the idea firstly formally appeared in our 2004 application for a national fund project [7]. What needs to be pointed out further is that the authors of [8] are the related reviewers of our 2004 application. In December 2006, [8] thought unreasoningly again that the REESSE1+ public-key cryptosystem is not secure at all, which connotes any private key in REESSE1+ can be extracted by [8]. It is of flubdub and gulf. * Received 05 April 2007, and revised 19 December 2009. Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 The ancients said ‘stop an advancing army with troops, and stop onrushing water with earth’. In what follows, the function f in [8] is namely the function ℓ in [4], namely f (i), f (j), f (k) in [8] are equivalent to ℓ (i), ℓ (j), ℓ (k), unless otherwise specified, the sign  represents ‘M – 1’, the sign % does ‘modulo’, and unattached (x) does x-th expression. In short, there exist 6 grave faults in [8]: The converse proposition of fact 1.1 does not hold. Clearly, fact 1.1 implies that if f (i) + f (j) = f (k), then Z / M – p u / q u < 1 / (2 qu 2) with L / Ak = p u / q u. We will prove by a counterexample that the former is only sufficient, but not necessary, namely if Z / M – p u / q u < 1 / (2 q u 2), then f (i) + f (j) = f (k) do not necessarily hold, and also namely Z / M – p u / q u < 1 / (2 q u for f (i) + f (j) = f (k) is only necessary, but not sufficient. Fact 1.2, 1.3 and 4 do not always hold. Even if they hold, fact 1.2, 1.3 and 4 each are insufficient for f (i) + f (j) = f (k), and further, property 1, 2, 3, 4 and 5 are invalid. Note fact 4 is essentially equivalent to each of fact 1.2 and 1.3. The converse proposition of fact 2.2 does not hold, namely table 1 is insufficient for f (i) + f (j) = f (k). Both algorithm 1 based on fact 4 and algorithm 2 based on table 1 are disordered & wrong logically. To achieve so-called “breaking”, the example in [8] was woven elaborately, and table 2 was falsified, namely its authors intendedly mutilated the two tuple data to cause indeterminacy. The inverse T –1 %  does not exist, and Q –1 %  not necessarily exist. Additionally, the case of Ω = {5 + δ, …, (n + 4) + δ | δ ≥ n – 4} with f (i) + f (j) ≠ f (k) ∀ i, j, k ∈ [1, n] is not analyzed at all. Therefore, the cryptanalysis of the REESSE1+ cryptosystem by [8] is a type of pseudo-attack and balderdash leading to which the most radical reason is that the authors of [8] are not aware of the indeterminacy of the lever function ℓ (.) namely f (.), as is mentioned in [4]: if the order of W is d < , then there is W f (x) ≡ W f (x) + d (% M), and when f (i) + f (j) = f (k), we see that f (i) + d + f (j) + d ≠ f (k) + d; when f (i) + f (j) ≠ f (k), there exist Ci ≡ Ai′ W ′ ), Cj ≡ Aj′ W ′ (j), and Ck ≡ Ak′ W ′ ) (% M) such that f ′(i) + f ′ (j) ≡ f ′ (k) (% ) with Ak′ ≤ ṕ , where ṕ is the maximal prime allowed. Another vital reason is that [8] always regarded necessary conditions for f (i) + f (j) = f (k) as sufficient and necessary conditions, and [8] did not consider the whole space of private keys or public keys. 2 Theorem 1 vs the REESSE1+ Cryptosystem 2.1 Condition at Theorem 1 in [8] Dissatisfies Necessity Theorem 1 in [8] is retailed as follows: Theorem 1: Let α be a real number, and let r / s be a rational with gcd(r, s) = 1 and |α – r / s| < 1 / (2s2). Then r / s is a convergent of the continued fraction expansion of α. Here, |α – r / s| represents the absolute value of (α – r / s). The proof of theorem 1 is referred to [9]. The condition |α – r / s| < 1 / (2s2) is only sufficient for r / s to be a convergent of the continued fraction of Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 α, but not necessary. Namely if r / s is a convergent of the continued fraction of α, |α – r / s| < 1 / (2s2) does not necessarily hold. An example is taken. Example 1. Let r / s = 2 / 13, and then 1 / 2s2 = 1 / (2 × 132) = 0.002958579882. Let α = 2039 / 13001, and then 2039 / 13001 – 2 / 13 = 0.002987935839 > 0.002958579882 = 1 / (2 × 132). On the other hand, the continued fraction of 2039 / 13001 is 1 / (6 + (1 / (2 + 1 / (1 + … 1 / 3)))). Thus, 2 / 13 is a convergent of the continued fraction of 2039 / 13001, which illustrates |α – r / s| < 1 / (2s2) is not necessary for r / s to be a convergent of the continued fraction of α. 2.2 Ak Will Emerge But Is Undecidable If f (i) + f (j) = f (k) Assume that ṕ is the maximum prime in the cryptosystem, {A1, …, An} is a coprime sequence with 0 < ∀Ax ≤ ṕ, M > ∏ x = 1 Ax is a prime, and Cx ≡ Ax W (x) (% M) for x = 1, …, n is a public key [4], where n ≥ 6, and f (x) ∈ Ω = {5δ, …, (n + 4)δ | δ = 1} = {5, …, n + 4}. Assume f (k) = f (i) + f (j) with i ≠ k, j ≠ k , and i, j, k ∈ [1, …, n]. Let Z ≡ Ci Cj Ck –1 (% M). Then Z ≡ Ai Aj (Ak) –1 (% M) Z (Ak) ≡ Ai Aj (% M) Z (Ak) – L M = Ai Aj, where L is a positive integer. Dividing the either side of the above equation by (M Ak) yields Z / M – L / Ak = Ai Aj / (M Ak). (1) Due to M > ∏ n x = 1 Ax and every Ax ≥ 2, we have Z / M – L / Ak < 1 / (2 2). (1′) Obviously, when n > 2 + 1, (1′) may have a variant, namely Z / M – L / Ak < 1 / (2 Ak 2). (1″) In terms of theorem 1, L / Ak is a convergent of the continued fraction of Z / M. Let p0 / q0, p1 / q1, …, pt / qt be the convergent sequence of continued fraction of Z / M, and L / Ak = p u / q u. Note that if pu / qu satisfies (1″), then pu + 1 / qu + 1, pu + 2 / qu + 2, …, pt / qt also likely satisfies (1″). Therefore, there likely exist multiple values of L / Ak by (1″), and Ak is undetermined. However, if we do not know in advance whether f (i) + f (j) = f (k), then even if Z / M – p u / q u < 1 / (2 q u we can not decide f (i) + f (j) = f (k). Namely Z / M – p u / q u < 1 / (2 q u 2) is only necessary for f (i) + f (j) = f (k), but not sufficient, which will be discussed further in what follows. 3 Conditions at Fact 1.1 and 4 Each Are Insufficient for f (i) + f (j) = f (k) Because fact 4 is essentially equivalent to each of fact 1.2 and 1.3, if the condition at fact 4 is Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 insufficient for f (i) + f (j) = f (k), the conditions at fact 1.2 and 1.3 each are also insufficient. The condition qu+1 > qu ∆ = qu (M / (2∏ x=n-2 prime 〈x〉)) 2 at fact 4 connotes (1″) at fact 1.1 because (1″) is the precondition of qu+1 > qu ∆ which is the dominant basis of alg.1 [8]. 3.1 Converse Proposition of Fact 1.1 does not Hold and (1″) Is Insufficient for f (i) + f (j) = f (k) Fact 1.1 in [8] is retailed as follows: Fact 1.1 [8]: If f (i) + f (j) = f (k), there exists a qu such that qu = Ak in {p0 / q0, p1 / q1, …, pt / qt}, the convergent sequence of continued fraction expansion of Z / M with Z ≡ Ci Cj Ck –1 % M. Due to f (i) + f (j) = f (k), Z / M = L / Ak + Ai Aj / (M Ak), M > ∏ x = 1 Ax and Ax ≥ 2, we have Z / M – L / Ak = Ai Aj / (M Ak) < Ai Aj / (Ak ∏ x = 1 Ax). Further, Z / M – L / Ak < 1 / (2 Ak 2). (1″) Let Z / M = [0; a1, a2, …, at] is the continued fraction expansion of Z / M. By theorem 1, ∃ u ∈ [1, t] makes Z / M – p u / q u < 1 / (2 q u Let L / Ak = pu / qu, where pu / qu = a0 + 1 / (a1 + 1 / (a2 + … + 1 / (au – 1 + 1 / au))). (2) Notice that it is possible that ∃ h > 0 makes Z / M – p u + h / q u + h < 1 / (2 q u + h 2), and moreover not fact 1.1 but its converse is the inner logical base of alg.1 in [8]. Through a counterexample, we will prove that the converse proposition of fact 1.1 does not hold, that is, the condition Z / M – p u / q u < 1 / (2 q u 2) is insufficient for f (i) + f (j) = f (k). Example 2. For convenience in computing, let n = 6, {Ax} = {11, 10, 3, 7, 17, 13}, δ = 1, and M = 510931. Arbitrarily select W = 17797, f(1) = 9, f(2) = 6, f(3) = 10, f(4) = 5, f(5) = 7, and f(6) = 8. From Cx ≡ Ax W f(x) (% M), we obtain {Cx} = {113101, 79182, 175066, 433093, 501150, 389033}, and its inverse sequence {Cx –1} = {266775, 236469, 435654, 149312, 434038, 425203}. Randomly select i = 1, j = 3, and k = 5. In this case, f(5) = 7 ≠ f(1) + f(3) = 9 + 10. Compute Z ≡ C1 C3 C5 ≡ 113101 × 175066 × 434038 ≡ 186640 (% 510931). Presume that W in C1 C3 is just neutralized by W –1 in C5 –1, then 186640 ≡ A1 A3 A5 –1 (% 510931). According to (1), 186640 / 510931 – L / A5 = A1 A3 / (510931 A5). By the Euclidean algorithm, a1, a2, a3, … are found out, and thus the continued fraction of 186640 / 510931 = 1 / (2 + 1 / (1 + 1 / (2 + 1 / (1 + 1 / (4 + … + 1 / 3))))). Heuristically let p 4 / q 4 = L / A5 = 1 / (2 + 1 / (1 + 1 / (2 + 1 / 1))) = 4 / 11, Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 which indicates that probably A5 = 11. On this occasion, there is 186640 / 510931 – 4 / 11 = 0.0016575801 < 1 / (2 A5 2) = 1 / (2 × 112) = 0.0041322314. The above expression satisfies (1″), namely the condition at theorem 1, and thereby A5 = 11 less than the maximum in {Ax} is deduced, which is in direct contradiction to factual A5 = 17. So the condition Z / M – p u / q u < 1 / (2 q u 2) is not sufficient for f (i) + f (j) = f (k), namely the converse proposition of fact 1.1 does not hold. 3.2 Each of Fact 1.2, 1.3 and 4 does not Hold Fact 1.2 in [8] is retailed as follows: Fact 1.2 [8]: There is sharp increase from qu to qu+1 since qu+1 ≥ (Ak M / (2Ai Aj)) 1/ 2. The derivation of fact 1.2 in [8] is retailed as follows: Let L / Ak be the u-th convergent, i.e., qu = Ak and pu = L, i.e., pu / qu = L / Ak. Then we know that |Z / M – pu+1 / qu+1| < Ai Aj / (Ak M) = 1 / (2 ((Ak M / (2Ai Aj)) 2)2). (2′) According to theorem 1 and convergence of sequence {p0 / q0, p1 / q1, …, pt / qt}, we obtain that qu+1 ≥ (Ak M / (2Ai Aj)) 2 = Ak (M / (2Ai Aj Ak)) 2. (3) Is the above derivation right? See the following analysis. Clearly, by the definition of a finite continued fraction, (2′) holds. In addition, in terms of [9], pu+1 are qu+1 are coprime, and there is qu+1 ≥ Ak = qu, which is a judgment foundation. If f (i) + f (j) = f (k), then there is |Z / M – pu / qu| < 1 / (2 qu 2) with L / Ak = pu / qu. Furthermore, through practical observations, in most cases, there is also |Z / M – pu+1 / qu+1| < 1 / (2 qu+1 2). (3′) According to (2′) and (3′), we have either |Z / M – pu+1 / qu+1| < 1 / (2 qu+1 2) < 1 / (2 ((Ak M / (2Ai Aj)) 2)2), (3″) |Z / M – pu+1 / qu+1| < 1 / (2 ((Ak M / (2Ai Aj)) 2)2) < 1 / (2 qu+1 2). (3′′′) If (3″) holds, there exists qu+1 ≥ Ak (M / (2Ai Aj Ak)) 2, which also indicates qu+1 ≥ Ak = qu. If (3′′′) holds, there exists Ak (M / (2Ai Aj Ak)) 2 ≥ qu+1. Notice that in this case, qu+1 ≥ Ak = qu is still possible. Therefore, qu+1 ≥ Ak (M / (2Ai Aj Ak)) 2, namely fact 1.2 does not necessarily hold, which indicates that there is a logic error during the derivation of (3) in [8]. Moreover, from (2′) and (3′) we can judge that when n is large enough ― 80 for example, the probability that (3′′′) holds is greater than one that (3″) holds. Now, we review fact 1.3 in [8]. It is retailed as follows: Fact 1.3 [8]: Due to fact 1.2, there is also a sharp increase from au to au+1, since qv+1 = av+1 qv + qv –1 for v = 1, 3, …, t. Here av s are items of Z / M determined by (2). Obviously, because fact 1.2 does not hold, fact 1.3 does not also hold. Further, because fact 1.2, namely (3) does not hold, naturally, fact 4 in [8] does not also hold, that is, qu+1 Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 > qu (M / (2∏ x=n-2 prime 〈x〉)) 2 is not always valid. Observe an example once more. In example 3, suppose that the bit-length of a plaintext block is 8, and two bits of a block correspond to three items of a coprime sequence {Ax}, which means that the encryption algorithm is optimized through a compact binary sequence. In practice, we do just so. Apparently, the length of {Ax} is 3 × (8 / 2) = 12. Example 3. Let {Ax} = {{23, 11, 17}, {41, 29, 26}, {15, 19, 37}, {31, 7, 43}}, and M = 2022169 > 31 × 37 × 41 × 43 = 2022161. Randomly select W = 1507351, f (1) = 6, f (2) = 14, f (3) = 9, f (4) = 11, f (5) = 12, f (6) = 10, f (7) = 8, f (8) = 16, f (9) = 5, f (10) = 13, f (11) = 15, and f (12) = 7. From Cx ≡ Ax W (x) (% M), we obtain {Cx} = {{572402, 1930240, 374715}, {25128, 265158, 350520}, {1674837, 1231458, 1448214}, {110225, 1198155, 757620}}, and {C6 –1, C7 –1} = {93176, 1591882}. Let Z ≡ (C4 C12) (C6 –1) ≡ (25128 × 757620) (93176 × 1591882) ≡ 776394 × 1123251 ≡ 689616 (% 2022169). Then, 689616 / 2022169 – L / (A6 A7) = (A4 A12) / (2022169 A6 A7). Further, the continued fraction of 689616 / 2022169 is 1 / (2 +1 / (1 + 1/ (13 + 1/ (1 + (1/ (3 + 1/ (2 + 1/ (2 + 1/ (2 + 1/ (97 + 4 / 9)))))))))). Heuristically let L / (A6 A7) = 1 / (2 +1 / (1 + 1/ (13 + 1/ (1 + (1/ (3 + 1 / 2))))) = 133 / 390, which indicates that probably A6 A7 = 390. Because the discriminant 689616 / 2022169 – 133 / 390 = 2.235477262e-6 < 1 / (2 × 3902) = 3.287310979e-6 satisfies the condition at theorem 1 in [8], A6 A7 = 390 is deduced out. The integer 390 may be factorized into the pairs (2, 195), (3, 130), (5, 78), (6, 65), (10, 39), (13, 30), or (15, 26), where the elements of (10, 39), (13, 30), and (15, 26) are less than maximal number in {Ax}. Thus, true (A6, A7) = (26, 15) is included in 6 potential cases. Here, au = 2 and also au + 1 = 2, and there is no sharp increase from au to au+1. Additionally, this example also illustrates that when one attempts to infer the suitable factors of the product Ak 1 Ak 2 by f (i) + f (j) = f (k1) + f (k2) with every f (x) ∈ Ω = {n + 1, …, 2n}, indeterminacy is increased remarkably. 3.3 Condition at Fact 4 Is Insufficient for f (i) + f (j) = f (k) In [6], the attackers attempted to seek Ak dominantly by the converse proposition of fact 1.1, and Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 however, disturbing values of Ak are too many to determine the original value of Ak. Therefore, in [8], the attackers attempted to diminish indeterminacy of Ak through fact 4 which connotes fact 1.1, and is equivalent to each of fact 1.2 and 1.3. To say the least, even if fact 4 is valid sometimes, we can prove by a counterexample that the condition at fact 4 is insufficient for f (i) + f (j) = f (k). Example 4. Still let n = 6, {Ax} = {11, 10, 3, 7, 17, 13}, and M = 510931 > 11 × 10 × 3 × 7 × 17 × 13 = 510510. Arbitrarily select W = 17797, f(1) = 9, f(2) = 6, f(3) = 10, f(4) = 5, f(5) = 7, and f(6) = 8. From Cx ≡ Ax W f(x) (% M), we obtain {Cx} = {113101, 79182, 175066, 433093, 501150, 389033}, and its inverse sequence {Cx –1} = {266775, 236469, 435654, 149312, 434038, 425203}. Randomly select i = 1, j = 3, and k = 6. In this case, f(6) = 8 ≠ f(1) + f(3) = 9 + 10. Compute Z ≡ C1 C3 C6 ≡ 113101 × 175066 × 425203 ≡ 425865 (% 510931). Presume that W in C1 C3 is just neutralized by W –1 in C6 –1, then 425865 ≡ A1 A3 A6 –1 (% 510931). According to alg.1 in [8], 425865 / 510931 – L / A6 = A1 A3 / (510931 A6). Compute the continued fraction of186640 / 510931 being 1 / (1 + 1 / (5 + 1 / (159 + 1 / 535))). Heuristically let L / A6 = 1 / (1 + 1 / 5) = 5 / 6, which indicates that probably A6 = 6. Further, can verify that 425865 / 510931 – 5 / 6 = 0.000174518 < 1 / (2 × 62) = 0.0138889 satisfies the condition at theorem 1 in [8]. Let u = 2, and qu = Ak = A6 = 6. Then pu+1/ qu+1 = p3 / q3 = 1 / (1 + 1 / (5 + 1 / 159)) = 796 / 955, and Ak (M / (2Ai Aj Ak)) 2 = 6 (510931 / (2 × 11 × 3 × 6))1/ 2 = 6 × 35.9197 = 215.5186. In addition, evidently prime〈1〉 = 2, prime〈2〉 = 3, prime〈3〉 = 5, prime〈4〉 = 7, prime〈5〉 = 11, prime〈6〉 = 13, prime〈7〉 = 17, and prime〈8〉 = 19 which are according to [8]. Then, by fact 4 in [8], m = 7, and ∆ = (M / (2∏ m x=n -2 prime〈x〉)) 2 = (15)1 / 2 = 3.8729. Thus, qu+1 = 955 > Ak (M / (2Ai Aj Ak)) 2 = 216 satisfies fact 1.2 namely (3), au+1 = 159 > au = 5 satisfies fact 1.3, and qu+1 = 955 > qu ∆ ≈ 24 satisfies fact 4 and alg.1. By the condition at fact 4, A6 = 6 < max A = 221 is deduced, namely alg.1 will output {1, 3, 6, 6}. However, it is in direct contradiction to true A6 = 13, which show the condition at fact 4 is not sufficient for Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 f (i) + f (j) = f (k), and every Ax will likely be evaluated to at least two eligible values (see example 5). Because the condition at fact 4 is insufficient for f (i) + f (j) = f (k), property 1, 2, 3, 4 and 5 are invalid. Further, the run result of alg.1 regarding an arbitrary public key {C1, …, Cn} as an input will contain enormous disturbing data as n ≥ 80, and it is infeasible that alg.2 find out the original coprime sequence {Ax} in polynomial time (see example 5), which manifests that alg.1 and 2 are invalid. 4 Example in [8] Is Woven Elaborately and Data at Table 2 Is Falsified 4.1 Example in [8] Illustrates Nothing about Breaking It is easily understood that according to fact 1.1 and 4, the authors of [8] can weave an example consistent with alg.1 and 2 since the lever function value {f (1), …, f (n)} may be known in advance; the coprime sequence {A1, …, An} may be selected elaborately in advance; the condition Z / M – L / Ak < 1 / (2 Ak 2) at fact 1.1 is necessary for f (i) + f (j) = f (k); the condition qu+1 > qu (M / (2∏ x=n-2 prime 〈x〉)) 2 at fact 4 is necessary for f (i) + f (j) = f (k) sometime. However, as is indicated in the above rebutment, a consistent example does not illustrates that a related {Ax} can be extracted accurately from an arbitrary public key {Cx} when {f (x)} and {Ax} are unknown in advance. The authors of [8] at most broke “their own REESSE1+”, which diverted themselves, but not our REESSE1+ with choice parameters. It is well understood that even though a cryptosystem is RSA or ECC, its parameter is must also selected; otherwise the cryptosystem is insecure. The example in [8] is neither readable nor verifiable in short time, and the proportion of n to log2 M is not also proper, which contravenes the optimization principle for the modulus M in the REESSE1+ cryptosystem. An obvious truth is that if M is too large, the length of a public key will increase rapidly. Therefore, M should be as small as possible while at least meets M > ∏ n x = 1 Ax meantime. Selection of the sequence {Ax} in [8] also contravenes the optimization principle. The intent for [8] to select such a large M that n is out of proportion to log2 M seems to want to increase the necessity of the conditions at fact 1.1 and 4 for f (i) + f (j) = f (k). However, it can not increase the sufficiency of the conditions. 4.2 Data at Table 2 Is Falsified for a Compatible Effect In above paragraphs, we illustrate that the condition Z / M – L / Ak < 1 / (2 Ak 2) at fact 1.1, namely (1″) is insufficient for f (i) + f (j) = f (k). Property I will make us better understand it. Property I: Let Cx ≡ Ax W (x) (% M), where every x ∈ [1, n], Ax ≤ ṕ, f (x) ∈ {5, …, n + 4}, M > ∏ x = 1 Ax is a prime. Then, ∀ i, j, k ∈ [1, n], even if f(i) + f(j) ≠ f(k), 1) there always exist Ci ≡ A′i W ′ f ′ (i), Cj ≡ A′j W ′ f ′ (j), and Ck ≡ A′k W ′ f ′ (k) (% M) such that f ′ (i) + f ′ (j) ≡ f ′ (k) (% ) with A′k ≤ ṕ. 2) Ci, Cj, Ck make (1″) hold with A′k ≤ ṕ in all probability. Proof: Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 Let Οd be an oracle for a discrete logarithm. Suppose that W ′ ∈ [1, ] is a generator of ( * M , ·). In terms of group theories, ∀ A′k ∈ {2, …, ṕ}, the equation Ck ≡ A′k W ′ (k) (% M) has a solution. f ′(k) may be taken through Οd. ∀ f ′(i) ∈ [1, ], and let f ′(j) ≡ f ′(k) – f ′(i) (% ). Then, from Ci ≡ A′i W ′ (i) and Cj ≡ A′j W ′ (j) (% M), we can obtain many distinct pairs (A′i, A′j), where A′i, A′j ∈ (1, M), and f ′(i) + f ′(j) ≡ f ′(k) (% ). 2) Let Z ≡ Ci Cj Ck ≡ A′i A′j W ′ (i) + f ′ (A′k W ′ (k))–1 (% M) with f ′(i) + f ′(j) ≡ f ′(k) (% ) but f(i) + f(j) ≠ f(k). Further, there is A′i A′j ≡ Ci Cj Ck A′k (% M). It is easily seen from the above equations the values of W′ and f ′(k) do not influence the value of A′i A′j. If A′k ∈ [2, ṕ] changes, A′i A′j also changes. Thus, ∀ i, j, k ∈ [1, n], the number of value of A′i A′j is ṕ – 1. Let M = 2 q ṕ 2 A′k, where q is a rational number. According to (1), Z / M – L / A′k = A′i A′j / (M A′k) = A′i A′j / (2q ṕ When A′i A′j ≤ q ṕ 2, there is Z / M – L / A′k ≤ q ṕ 2 / (2 q ṕ 2 A′k 2) = 1 / (2 A′k which satisfies (1″). Assume that the value of A′i A′j distributes uniformly on (1, M). Then, the probability that A′i A′j makes (1″) hold is P∀ i, j, k ∈ [1, n] = (q ṕ 2 / (2 q ṕ 2)) (1 / 2 + … + 1 / ṕ)) ≥ (1 / 2)(2 (ṕ – 1) / (ṕ + 2)) = 1 – 3 / (ṕ + 2). It is seen that the probability is very large. According to property I.2, for a certain Ck ∈ {C1, …, Cn} and ∀ Ci, Cj ∈ {C1, …, Cn}, Ak will have roughly n2 values by (1″) namely the condition at fact 1.1, including the repeated, and considering the symmetry, almost every value has at least one counterpart. Of course, if the condition at fact 4, namely qu+1 > qu ∆ which connotes (1″) is used as a constraint, the number of values of Ak = qu will decrease. Example 4 already shows that even though f(i) + f(j) ≠ f(k), an eligible Ak can still be found. Notice that when i, j, k all fix on, it is fully possible that L / Ak has multiple satisfactory values, which implies multiple convergents of the continued fraction of Z / M likely meet (1″) and even qu+1 > qu ∆ . To clarify the matter thoroughly, we program by alg.1 in MS Visual C++, make an executable file, repeat the experiment regarding the public key at the example in [8] as input, and obtain the following output which is classified the same as in [8]: Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 Ak Tuples (i, j, k) A1 = 9 (9, 9, 1) A2 = 253 (7, 5, 2), (9, 6, 2), (5, 7, 2), (6, 9, 2) A3 = 16127 (10, 7, 3), (7, 10, 3) A4 = 3 (8, 3, 4), (3, 8, 4) A4 = 205 (9, 3, 4), (6, 5, 4), (5, 6, 4), (7, 7, 4), (3, 9, 4) A4 = 152391460756 (8, 7, 4), (7, 8, 4) A6 = 53022327 (4, 3, 6), (3, 4, 6) A6 = 318461273008612 (4, 3, 6), (3, 4, 6) A6 = 4471789987666990 (5, 3, 6), (3, 5, 6) A6 = 1572955621791218 (5, 5, 6) A8 = 2809 (5, 5, 8), (9, 7, 8), (7, 9, 8) A10 = 49 (9, 5, 10), (5, 9, 10) A10 = 1894 (9, 6, 10), (6, 9, 10) A10 = 6957 (9, 7, 10), (7, 9, 10) Table I: Output of the program by alg. 1 given the public key at the example in [8] Obviously, table 2 in [8] misrepresented A3 = 16127 as A4 = 16127, and A6 = 53022327 as A10 = 53022327. What gets worse is that table 2 mutilated the two tuple data (4, 3, 6, 318461273008612) and (3, 4, 6, 318461273008612), which is a type of data falsification. These two tuple data illustrate that for fixed i, j, k, the L / Ak may have several satisfactory values, namely the several convergents of the continued fraction of Z / M meet fact 4 meantime, which reflects the insufficiency of the condition qu+1 > qu ∆ further, increases the indeterminacy of Ak greatly, and weakens the reliability of alg.1 in [8] greatly. 4.3 Example in [8] Is Woven Elaborately and Alg.2 in [8] Is Invalid In the above, it is mentioned that at most the authors of [8] broke “their own REESSE1+”, because the example in [8] is woven elaborately, and the parameters {Ax} and {f (x)} are selected deliberately. If we use another set of parameters for producing a public key as the input of the program by alg.1, the output result will contains so many disturbing data that the original sequence {A1, …, An} can not be distinguished in polynomial time. Example 5. Let n = 10, {Ax} = {437, 221, 77, 43, 37, 29, 41, 31, 15, 2}, and M = 13082761331670077 > ∏ n x = 1 Ax = 13082761331670030. Arbitrarily select W = 944516391, f(1) = 11, f(2) = 14, f(3) = 13, f(4) = 8, f(5) = 10, f(6) = 5, f(7) = 9, f(8) = 7, f(9) = 12, f(10) = 6. According to Cx ≡ Ax W f(x) (% M), we obtain {Cx} = {3534250731208421, 12235924019299910, 8726060645493642, 10110020851673707, 2328792308267710, 8425476748983036, 6187583147203887, 10200412235916586, 9359330740489342, 5977236088006743}. Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 Input the public key {Cx} into the program by alg.1, and obtain ∆ = 506, max A = 58642670, and the following tuples greater than 100: Ak Tuples (i, j, k) A1 = 187125 (1, 1, 1) A1 = 121089 (2, 1, 1), (1, 2, 1) A1 = 77 (5, 3, 1), (3, 5, 1) A1 = 23 (8, 6, 1), (6, 8, 1), (10, 10, 1) A1 = 437 (10, 6, 1), (6, 10, 1) A2 = 1251 (1, 1, 2) A2 = 187125 (2, 1, 2), (1, 2, 2) A2 = 121089 (2, 2, 2) A2 = 17 (8, 4, 2), (6, 5, 2), (5, 6, 2), (10, 7, 2), (4, 8, 2), (7, 10, 2) A2 = 221 (10, 4, 2), (7, 6, 2), (6, 7, 2), (8, 8, 2), (4, 10, 2) A2 = 77 (9, 8, 2), (8, 9, 2) A2 = 4204 (10, 10, 2) A3 = 187125 (3, 1, 3), (1, 3, 3) A3 = 12 (7, 1, 3), (1, 7, 3) A3 = 121089 (3, 2, 3), (2, 3, 3) A3 = 77 (6, 4, 3), (4, 6, 3), (10, 8, 3), (8, 10, 3) A3 = 11 (10, 4, 3), (7, 6, 3), (6, 7, 3), (8, 8, 3), (4, 10, 3) A3 = 2113 (8, 7, 3), (7, 8, 3) A3 = 769 (9, 8, 3), (8, 9, 3) A4 = 187125 (4, 1, 4), (1, 4, 4) A4 = 121089 (4, 2, 4), (2, 4, 4) A4 = 76 (10, 6, 4), (6, 10, 4) A4 = 56 (10, 9, 4), (9, 10, 4) A5 = 187125 (5, 1, 5), (1, 5, 5) A5 = 630269 (6, 1, 5), (1, 6, 5) A5 = 121089 (5, 2, 5), (2, 5, 5) A5 = 41 (8, 2, 5), (2, 8, 5) A5 = 97 (4, 3, 5), (3, 4, 5) A5 = 37 (6, 6, 5), (10, 6, 5), (6, 10, 5) A6 = 187125 (6, 1, 6), (1, 6, 6) A6 = 121089 (6, 2, 6), (2, 6, 6) A7 = 187125 (7, 1, 7), (1, 7, 7) A7 = 121089 (7, 2, 7), (2, 7, 7) A7 = 3 (9, 3, 7), (3, 9, 7) Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 A8 = 187125 (8, 1, 8), (1, 8, 8) A8 = 34945619 (6, 2, 8), (2, 6, 8) A8 = 121089 (8, 2, 8), (2, 8, 8) A9 = 187125 (9, 1, 9), (1, 9, 9) A9 = 121089 (9, 2, 9), (2, 9, 9) A9 = 5 (6, 4, 9), (4, 6, 9), (10, 8, 9), (8, 10, 9) A9 = 15 (8, 6, 9), (6, 8, 9), (10, 10, 9) A10 = 259970 (4, 1, 10), (1, 4, 10) A10 = 187125 (10, 1, 10), (1, 10, 10) A10 = 121089 (10, 2, 10), (2, 10, 10) A10 = 7629 (8, 3, 10), (3, 8, 10) Table II: Output of the program by alg. 1 given the public key at example 5 From table II, we observe that Ak from 5 tuples is A2 = 221 or A3 = 11, Ak from 4 tuples is A3 = 77 or A9 = 5, Ak from 3 tuples is A1 = 23, A5 = 37, or A9 = 15, Ak from 2 tuples is A1 = 77, A2 = 77, A3 = 12, A4 = 56, A5 = 41, or A7 = 3 etc, and Ak from 1 tuples is A1 = 187125, A2 = 1251, A2 = 121089, or A2 = 4204. Among these Ak′s, there exist at least 2 n – 5 compatible combinations. For instance, arbitrarily select compatible A3 = 11, A9 = 5, A1 = 23, A5 = 41, and A2 = 1251, and find out f(3) = 14, f(9) = 13, f(1) = 12, f(5) = 11, and f(2) = 10 by Table 1 in [8]. Again for instance, arbitrarily select compatible A3 = 11, A9 = 5, A5 = 37, A7 = 3, and A1 = 187125, and find out f(3) = 14, f(9) = 13, f(5) = 12, f(7) = 11, and f(1) = 10 by Table 1 in [8]. Therefore, if keep Ω = {5, .., n + 4} unvaried, we may select fit {Ax} and W so as to make the time complexity of the continued fraction attack by qu+1 > qu ∆ and table 1 get to at least O(2 n), which elucidates that the example woven elaborately in [8] has no practical meaning, and alg.2 in [8] is invalid. However, we had best select fit Ω while let {Ax} and W random so as to avoid attack by (1′) (see sect.5.1). 4.4 Distribution of Tuples Relating Ak does not Follow Table 1 in [8] In addition, from table II we also observe that A2 = 17 involves 6 tuples, and A5 = 37 involves 3 tuples (but in fact, 6 tuples is impossible, and f(5) = 10), which indicates that the distribution of tuples relating Ak does not follow table 1 in [8]. Besides, considering A3 = 11 from 5 tuples, A9 = 5 from 4 tuples etc, we see that table 1 is insufficient for f (i) + f (j) = f (k), that is, the converse proposition of fact 2.2 does not hold. 5 Why Is Cx ≡ Ax W f(x) (% M) Changed to Cx ≡ (Ax W f(x))δ (% M) in REESSE1+ v2.1 5.1 Lever Set Ω Needs to Be Complicated When Cx ≡ Ax W f(x) (% M) In REESSE1, Cx ≡ Ax W f(x) (% M) with f (x) ∈ Ω = {5, …, n + 4}. Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 In REESSE1+, Cx ≡ Ax W f(x) (% M) with f (x) ∈ Ω = {5δ, …, (n + 4)δ | δ ≥ 1}, {5 + δ, …, (n + 4) + δ | δ ≥ n − 4}, {5, 7, …, 19, 53, 55, …} etc. If let W′ = W δ (% M), we see that {5δ, …, (n + 4)δ | δ ≥ 1} is substantially the same as {5, …, n + 4}. Although [8] by Z / M – L / Ak < 1 / (2 Ak 2) and qu+1 > qu ∆ can not break REESSE1+ with Cx ≡ Ax W f(x) (% M) and Ω = {5δ, …, (n + 4)δ | δ ≥ 1}, attack by Z / M – L / Ak < 1 / (2 2), namely (1′) will filter out the most of disturbing data as n is large, which makes REESSE1+ be faced with danger. Therefore, in REESSE1+ with Cx ≡ Ax W f(x) (% M), Ω needs to be complicated, namely had best select Ω = {5, 7, …, 19, 53, 55, …} which is an odd set of 2n elements such that ∀ e1, e2 ∈ Ω, e1 ≠ e2, ∀ e1, e2, e3 ∈ Ω, e1 + e2 ≠ e3, ∀ e1, e2, e3, e4 ∈ Ω, e1 + e2 + e3 ≠ e4. 5.2 Key Transform Cx ≡ Ax W f(x) (% M) Needs to Be Strengthened When Still Ω = {5, …, n + 4} In REESSE1+ with Cx ≡ Ax W f(x) (% M) and f (x) ∈ Ω = {5, 7, …, 19, 53, 55, …}, because the elements of Ω are relatively large, decryption speed will decrease greatly. To keep Ω = {5, …, n + 4} unvaried, the key transform should be strengthened, so in REESSE1+ v2.1, we let Cx ≡ (Ax W f(x))δ (% M). In this way, REESSE1+ v2.1 is not only secure but also swift. 6 Attack on the Signature Is an Eisegesis 6.1 T –1 %  does not Exist and Q –1 %  not Necessarily Exist Section 4 of the original [8] deduces U ≡ ((Q / H) 1 / S  (GW) –1δ δ (δ + 1) – 1 / S) Q T (% M), which is right. However, (GW) –1δ δ (δ + 1) – 1 / S ≡ ((Q / H) – S – 1  – 1) U (Q T) – 1 (% M) further given in [8] is wrong because T –1 %  with T |  does not exist, and neither does Q –1 %  exist when gcd(Q, ) > 1. In the signature algorithm, it is easy to let gcd(Q, ) > 1. Denote x = (GW) –1δ δ (δ + 1) – 1 / S (% M). Then, the trivial solution to x Q T ≡ U ((Q / H) 1 / S) – Q T (% M) does not exist when gcd (T,  / T) > 1. Due to stipulating T ≥ 2 n in the key generation algorithm, the time complexity of finding out a random solution to x Q T ≡ U ((Q / H) 1 / S) – Q T is at least max (O(2 n – 1), O(M / (Q T))) through the probabilistic algorithm [10]. If a solution to x Q T ≡ U ((Q / H) 1 / S) – Q T is found through the discrete logarithm method, the probability that the solution is just equal to (GW) –1δ δ (δ + 1) – 1 / S (% M) is at most 1 / 2 n. If denote x = ((GW) –1δ δ (δ + 1) – 1 / S) T (% M), then x Q ≡ U ((Q / H) 1 / S) – Q T (% M). When gcd (Q, ) > 5 and M / Q > 2 n, seeking a solution to x Q ≡ U ((Q / H) 1 / S) – Q T is also at least the discrete logarithm problem. 6.2 Forging Attack in [8] May Be Easily Avoided through Turning D | (δ Q – W) to D | (δ Q – WH) In REESSE1+ [4], we definitely pointed out that Q ≠ Q1, where Q is produced currently, and Q1 is any of signature foreparts produced ever before. Of course, Q ≠ Q1 implied that the linear combination of Q1 with Q2 should be excluded from signature foreparts. However, such exclusion is infeasible in polynomial time. Therefore, in practical applications, it is suggested as a shortcut that users move the parameter H in Q ≡ (R G0) Hδ (% M) into D | (δ Q – W), and make D | (δ Q – W) become D | (δ Q – WH). In this wise, the Refuting the Pseudo Attack on the REESSE1+ Cryptosystem http://arxiv.org/pdf/0704.0492 forgery attack in [8] is easily avoided, namely Q′ can not be forged out at least in polynomial time. Notice that correspondingly, the λ S in the signature algorithm and the discriminant in the verification algorithm should also be adjusted. 7 Conclusion The above rebuttal shows that each or the combination of (1″), qu+1 > qu ∆, and table 1 is not sufficient for f (i) + f (j) = f (k), there exist logic errors in the deduction of (3), and alg.1 based on fact 4 and alg.2 based on table 1 are not valid. Additional, the signature forgery attack in [8] is easily avoided. Hence, the conclusion of [8] that REESSE1+ is not secure at all (which connotes that [8] can extract a related private key from any public key in REESSE1+) is completely incorrect, as long as Ω is fitly selected, REESSE1+ with Cx ≡ Ax W f(x) (% M) is secure, and the private key attack in [8] like [6] is a pseudo attack.. The authors of [8] attempt to convince people or credulous one of their opinion through an example woven elaborately, and their purpose is to want to suffocate REESSE1+, suppress us, and elevate themselves. Especially, [8] like [6] does not list the origin of idea of the continued fraction analysis of REESSE1, and falsifies the data at table 2, which violates scientific research ethics and honestness. We welcome unmalicious, co-promotive, and normal academic criticism which is utterly necessary. References [1] Shenghui Su, The REESSE1 Public-key Encryption Algorithms, Int. C1: H04L 9/14, ZL01110163.6, Chinese Patent, Apr. 2001. [2] Shenghui Su, The REESSE1 Public-key Cryptosystem, Computer Engineering & Science, Chinese, v25(5), 2003, pp.13-16. [3] Shenghui Su, Yixian Yang and Bingru Yang, The Necessity and Sufficiency Analysis of the Lever Function in the REESSE1 Encryption Scheme, Acta Electronica Sinica, Chinese, v34(10), 2006, pp.1892-1895. (Received May 13, 2005) [4] Shenghui Su and Shuwang Lü, The REESSE1+ Public-key Cryptosystem, http://eprint.iacr.org/2006/420.pdf. [5] Shengli Liu, Fangguo Zhang and Kefei Chen, Cryptanalysis of REESSE1 Digital Signature Algorithm, CCICS 2005, Xi’an, China, May 2005. [6] Shengli Liu, Fangguo Zhang and Kefei Chen, Cryptanalysis of REESSE1 Public Encryption Cryptosystem, Information Security, Chinese, n7, 2005, pp.121-124. [7] Shenghui Su, Refuting the Pseudo Attack on the REESSE1 Public-key Algorithms for Encryption, Computer Engineering and Applications, Chinese, v42(20), 2006, pp.129-133. [8] Shengli Liu and Fangguo Zhang, Cryptanalysis of REESSE1+ Public Key Cryptosystem, http://eprint.iacr.org/2006/ 480.pdf, Dec. 22, 2006. [9] Kenneth H. Rosen, Elementary Number Theory and Its Applications (5th ed.), Boston: Addison-Wesley, 2005, ch. 12. [10] Henri Cohen, A Course in Computational Algebraic Number Theory, Berlin: Springer-Verlag, 2000, ch. 1, 3. Remark The first version of this paper was sent to the authors of [8] via email on Mar. 6, 2007, and the draft of this revised version was sent to the authors of [8] via email between Oct. 23 and Nov. 12, 2009 repeatedly. The authors of [8] revised section 5 of [8] on Mar. 12, 2007 after they read this paper and the eprint.iacr.org′s demand that [8] should be withdrawn or modified, but the modification avoided the heavy and chose the light.

0704.0493

Phase structure of a surface model on dynamically triangulated spheres with elastic skeletons

Phase structure of a surface model on dynamically triangulated spheres with elastic skeletons Hiroshi Koibuchi∗ Department of Mechanical and Systems Engineering, Ibaraki National College of Technology, Nakane 866 Hitachinaka, Ibaraki 312-8508, Japan (Dated: August 12, 2021) We find three distinct phases; a tubular phase, a planar phase, and the spherical phase, in a triangulated fluid surface model. It is also found that these phases are separated by discontinu- ous transitions. The fluid surface model is investigated within the framework of the conventional curvature model by using the canonical Monte Carlo simulations with dynamical triangulations. The mechanical strength of the surface is given only by skeletons, and no two-dimensional bending energy is assumed in the Hamiltonian. The skeletons are composed of elastic linear-chains and rigid junctions and form a compartmentalized structure on the surface, and for this reason the vertices of triangles can diffuse freely only inside the compartments. As a consequence, an inhomogeneous structure is introduced in the model; the surface strength inside the compartments is different from the surface strength on the compartments. However, the rotational symmetry is not influenced by the elastic skeletons; there is no specific direction on the surface. In addition to the three phases mentioned above, a collapsed phase is expected to exist in the low bending rigidity regime that was not studied here. The inhomogeneous structure and the fluidity of vertices are considered to be the origin of such variety of phases. PACS numbers: 64.60.-i, 68.60.-p, 87.16.Dg I. INTRODUCTION A crumpling of surfaces has been investigated on the basis of the singularity analysis, and progress has been recently made on understanding the crumpling phenom- ena; the universal structure on the crumpled thin sheets was found in the formations of singularity of ridges and cones [1, 2]. A similar transition to this phenomena was also found experimentally between the smooth state and the crumpled state in an artificial membrane, which is partly polymerized [3]. Studies have also been focused on the transition in the surface model of Helfrich, Polyakov and Kleinert (HPK) [4, 5, 6] from the viewpoint of statistical mechanics [7, 8, 9, 10, 11, 12, 13]. The bending rigidity is known to be stiffened by the thermal fluctuation of the surface, and this was confirmed in the statistical mechanics of membranes [14, 15, 16, 17, 18]. Numerical studies were made to understand the transition in triangulated surface models [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. The transition was reported as first-order in recent numerical studies [31, 32]. On the other hand, the concern with inhomogeneous surfaces has been growing over the past decade [33, 34]. A homogeneous artificial membrane that is coated by elas- tic skeletons is also considered to be an inhomogeneous membrane. Some of the mechanical properties of such membranes were revealed experimentally [35]. The hop diffusion of membrane protein or lipids was observed, and as a consequence the compartment of cytoskeletons was ∗Electronic address: [email protected] confirmed to be in biological membranes [36]. It is also well known that the microtubule, which is an element of the cytoskeleton, gives a mechanical strength to the surface of the biological membranes. However, the surface collapsing phenomena and the surface fluctuation phenomena are almost unknown in such inhomogeneous models for membranes. Therefore, it is worthwhile to study an inhomogeneous fluid sur- face model within the framework of the conventional sur- face model of HPK. We note that the inhomogeneity in our model corresponds to the cytoskeletons in biologi- cal membranes as stated above. The fluidity realized by dynamical triangulations in the inhomogeneous model, as well as the fluidity in the homogeneous surface mod- els, corresponds to the lateral diffusion of lipids in mem- branes. In this paper we study a compartmentalized surface model by Monte Carlo (MC) simulations. The Hamil- tonian of the model includes no two-dimensional bend- ing energy but a one-dimensional bending energy. The model is defined on dynamically triangulated surfaces, where the free diffusion of vertices is confined inside the compartments. The mechanical strength of the surface is given only by the compartment boundary, which is com- posed of one-dimensional elastic chains and rigid junc- tions. Because the collapsed phase is expected to appear at sufficiently small bending rigidity b[kT ]→0(b 6=0), we concentrate on the phase structure at relatively large b in this paper. Consequently, information on the phase boundary at b→0 remains unanswered. We recently reported numerical results of three types of surface models [37, 38], which are similar to the model in this paper. Then, we should comment on the similar- ity/difference between the model in this paper and the http://arxiv.org/abs/0704.0493v1 mailto:[email protected] models in [37, 38]. Firstly, the lattice structure of the model in this paper is very similar to that of the first model in [37] and that of the model in [38], and is iden- tical to that of the second model in [37]. Secondly, the lattice in this paper and that of the first model in [37] are the dynamically triangulated one, while the lattice of the second model in [37] and that in [38] are the fixed- connectivity one. Thirdly, the Hamiltonian is different from the one in the first model in [37]. The Hamiltonian of the model of this paper includes only one-dimensional bending energy, which is defined on the compartment boundary, while the Hamiltonian of the first in [37] in- cludes only a two-dimensional bending energy, which is defined all over the surface, and no one-dimensional bending energy is given to the compartment boundary. Therefore, the model in this paper is different from the three models in [37, 38]. Our results obtained in this paper show that the model undergoes a first-order transition between the smooth phase and the crumpled phase. Moreover, the smooth phase can be divided into the spherical phase and the planar phase, and the crumpled phase can also be divided into the tubular phase and the collapsed phase, which is expected to appear at sufficiently small b because no self- avoiding property [39, 40, 41] is assumed in the model. It must be emphasized that such variety of phases can be seen neither in the conventional surface models nor in the compartmentalized models such as those in [37, 38]. One remarkable result is the appearance of planar sur- faces. The echinocytic shapes of erythrocytes were ex- tensively studied, and they are currently known to be described by many models such as the area difference bilayer model [34]. The shape of membranes is also sen- sitive to the flow fields [42]. Our model in this paper indi- cates that one possible origin of such planar shape comes from the inhomogeneity due to the cytoskeltal structure and the fluidity of lateral diffusion of vertices. II. MODEL Figure 1(a) shows a triangulated surface of size (N,NS , NJ , L) = (2322, 600, 42, 6), where N is the total number of vertices including the junctions, NS is the total number of vertices on the chains, NJ is the total number of junctions, and L is the length of chains between the two nearest-neighbor junctions. It should be noted again that NJ is included in N ; junctions are counted in the total number of vertices. The junctions are assumed as rigid plates; twelve of them are pentagon and all the oth- ers are hexagon. The junction size in Fig.1(a) is drawn many times larger than that of the lattices for the sim- ulations; and it will be discussed in the last part of this section. Thick lines on the surface in Fig.1(a) denote the chains, which are terminated at the junctions. The construction of the lattices is as follows: Let us start with the icosahedron. Every edge of the icosahe- dron is divided into ℓ pieces of uniform length, and then (a) (N,NS , NJ , L)= (2322, 600, 42, 6) (b) A rigid junction with the chains θ(ij)θ(ij) FIG. 1: (Color online) (a) Starting configuration of surfaces of size (N,NS, NJ , L)=(2322, 600, 42, 6), and (b) angles θ(ij) in the bending energy S2 of Eq.(4). Thick lines in (a) denote the compartment boundary composed of the linear chains and the rigid junctions of the hexagonal and the pentagonal plates, whose size is drawn many times larger than that of the lattices for the simulations. we have a triangulated surface of size N0 = 10ℓ 2+2 (= the total number of vertices on the surface). The com- partmentalized structures are constructed by dividing ℓ further into m pieces (m=1, 2, · · · ). Thus, we have the chains of uniform length L= (ℓ/m)−2 when m divides ℓ. The reason for the subtraction −2 is because of the junctions at the two end points of the chain. Because the compartmentalized structure is a sublattice, the to- tal number of junctions NJ is given by NJ = 10m The total number of bonds in the sublattice is 3NJ−6, and each bond contains L−1 vertices, then NS is given by NS =(3NJ−6)(L−1), which can be written as NS = 30m(ℓ−3m). The hexagonal (pentagonal) rigid junctions are composed of 7 (6) vertices, then NJ−12 hexagonal rigid junctions and 12 pentagonal rigid junctions reduce the total number of vertices N0 by (NJ−12)×6 and 12×5. Therefore, we have N=N0−6NJ+12, which can also be written as N=10ℓ2−60m2+2. The thermodynamic limit of our model is defined by N→∞, NS→∞, and NJ →∞ under the condition that L is finite. We have the thermo- dynamic limit at ℓ→∞ and m→∞. The lattice of size (N,NS , NJ , L)=(2322, 600, 42, 6) in Fig.1(a) is given by two independent integers (ℓ,m)=(16, 2). The surfaces can be characterized by the length L. In this paper, we assume three values for L such that L = 6, L = 8, L = 11. (1) The value of L has a one to one correspondence with the total number of vertices n in a compartment; in fact, the values of L in Eq.(1) correspond to n= 21, n= 36, and n=66, respectively [37]. We note that the effective phys- ical meaning of increasing (decreasing) L can be consid- ered as the increasing (decreasing) temperature. In fact, the surface fluctuation mainly comes from the thermal fluctuation of vertices inside the compartments. Because no bending energy is assumed inside the compartments, the fluctuation of vertices becomes large not only in the in-plane directions (free diffusion) but also in the direc- tion perpendicular to the surface. Thus, we consider that the fluctuations are expected to grow with increasing n, i.e., increasing L. We use the surfaces of size (N,NS , NJ) listed in Ta- ble I. Three different sizes (N,NS , NJ) are assumed for each L. The corresponding integers (ℓ,m) are as follows: (16, 2), (24, 3), and (32, 4) for the L=6 surfaces, (10, 1), (20, 2), and (30, 3) for the L = 8 surfaces, and (13, 1), (26, 2), and (39, 3) for the L=11 surfaces. TABLE I: The surface size assumed in the simulations. Three sizes (N,NS , NJ ) are assumed for each L. L (N,NS , NJ ) (N,NS , NJ ) (N,NS , NJ ) 6 (2322,600,42) (5222,1350,92) (9282,2400,162) 8 (942,210,12) (3762,840,42) (8462,1890,92) 11 (1632,300,12) (6522,1200,42) (14672,2700,92) The model is defined by the partition function dXi exp [−S(X, T )] , (2) S(X, T ) = S1 + bS2, where S1 is the Gaussian bond potential, which is de- fined all over the surface, and S2 is the one-dimensional bending energy, which is defined on the compartment boundary and will be given below. The parameter b is the bending rigidity. The integration symbol in Eq.(2) denotes that the center of mass of the surface is fixed. denotes the sum over all possible triangulations T , which are performed by the bond flip technique keeping the compartments unflipped. The bond flip procedure will be given in the following section. The integration measure i=1 dXi is given by the product dXi = qαj(i) (α = 3/2, 0), (3) where N ′ (=N−NJ) is the total number of vertices ex- cluding the junctions, i=1 dXiq i denotes the integra- tion over the 3D translational degrees of freedom (DOF) of the vertices i, and i=1 dXi j(i) q j(i) denotes those of the 3D translational DOF and the 3D rotational DOF of the junctions i. The co-ordination number qi is the total number of bonds meeting at the vertex i, and qj(i) is the total number of bonds meeting at the corner j(i) of the junction i. The parameter α was chosen to be α=3/2 in [43, 44], while α = 0 in many previous simulations on dynami- cally triangulated surfaces in the literatures. It is easy to understand that large positive α suppresses the configu- rations with large coordination number. Therefore, it is interesting to see the dependence of the phase structure on α. We chose both α = 3/2 and α = 0 for the weight qαi [43, 44], and see whether the phase structure of the model depends on α or not. If the parameter is chosen to α=3/2, then the coordination number qi serves as a weight of the integration dXi, while α=0 gives the uni- form weight. The weight i=1 q i can also be written i=1 q i = exp(α log qi), and therefore, i=1 q is considered to be the co-ordination dependent term log qi in the Hamiltonian; −α log qi changes its value only on dynamically triangulated surfaces. The Gaussian term S1 and the bending energy term S2 are defined by (Xi −Xj) , S2 = 1− cos θ(ij) , (4) where (ij) in S1 is the sum over bonds (ij) connecting the vertices i and j, and (ij) in S2 is also the sum over bonds (ij). θ(ij) in S2 is the angle between the bonds i and j, which include virtual bonds. The virtual bonds denote the lines between the center and the corners of the junction; the hexagonal (pentagonal) junction contains six (five) virtual bonds. Figure 1(b) is a junction and the chains linked to the junction on a fluctuating surface. Triangles are elimi- nated from the figure. One θ(ij) shown at a corner of the junction is defined by using a virtual bond and a real bond in a chain, and the other θ(ij) shown at a vertex is defined by real bonds on the same chain. The size of the junctions can be characterized by the edge length R, which is fixed to R = 0.1 (edge length of the junctions). (5) The value R = 0.1 is relatively smaller than the mean bond length 0.707, which corresponds to the relation S1/N = 1.5 satisfied in the equilibrium configuration of surfaces without the rigid junctions. As we will see later, the relation S1/N =1.5 is slightly violated in the model of this paper because of the rigid junctions. Here we comment on the unit of physical quantities. Let a be the length scale of the model, then the unit of physical quantity that has the length unit can be ex- pressed by a; the unit of S1 is [a 2]. The surface tension coefficient λ in λS1+bS2 has the unit [kT/a 2] and as- sumed to be λ=1[kT/a2], and the bending rigidity b has the unit of [kT ] as described above. Note that the bending rigidity b in the Hamiltonian is a microscopic quantity from the view point of statistical mechanical model, and therefore b is not always identical to the macroscopic bending rigidity of real physical mem- branes. However, the microscopic value b of real mem- branes can effectively be varied with the temperature, because b has the unit of kT . Therefore, it is possible to consider that the phase structure described in terms of b in the surface model corresponds to the phase structure described in terms of T in real physical membranes. The length scale a in the model is also a microscopic quantity and, we consider that a is sufficiently smaller than the membrane size. III. MONTE CARLO TECHNIQUE A sequence of random numbers called Mersenne Twister [45] is used in the canonical MC simulations. The Metropolis technique is applied to update X and T , where the variable X denotes the position of the ver- tices and that of the junctions. The vertex position X is shifted so that X ′ = X + δX , where δX is ran- domly chosen in a small sphere. The new position X ′ is accepted with the probability Min[1, exp(−∆S)], where ∆S = S(new)−S(old). The position X of a hexagonal (or pentagonal) junction, which is not a point but a rigid plate, is also integrated out by performing 3D random translations and 3D random rotations. Thus, the variable X is updated by a random N ′ (= N−NJ) shifts of vertices, a random NJ translations of junctions, and a randomNJ rotations of junctions. These updates are denoted by (N ′, NJ , NJ) updates of X . The N ′ shifts of X can be divided into NS shift of the vertices on the linear chains and N ′−NS shifts of all the other vertices, which are those inside the compartments. The radius of the small sphere for δX is fixed at the beginning of the MC simulations in order to maintain about 50% acceptance rate. The vertices on the linear chains carry the bending energy S2 in Eq.(4), while all the other vertices inside the compartments does not. There- fore, the acceptance rate is independently controlled in the two-groups of vertices. The radius for the random translation of the junctions and that for the random rota- tion are also independently chosen so that the acceptance rates are both about 50%. The summation over T in Z of Eq.(2) is performed by using the standard bond flip technique [22, 23]. The flip is accepted with the probability Min[1, exp(−∆S)]. The acceptance rate for the bond flip is not under control and is about 75%, which is almost independent of b. The bonds are labeled with sequential numbers. The total number of bonds is denoted by N ′B, which excludes the number of bonds on the linear chains because the bonds on the linear chains remain unflipped. The bond flip is performed as follows: Firstly, the odd- numbered bonds are sequentially chosen to be flipped for the N ′B/2 updates of T , and after that, the (N,NJ , NJ) updates of X are performed. Secondly, the remaining even-numbered bonds are chosen to be flipped for the N ′B/2 updates of T , and after that, the (N,NJ , NJ) up- dates ofX are performed. Thus, the (N,NJ , NJ) updates of X and the N ′B/2 updates of T are consecutively per- formed, and these make one MCS (Monte Carlo Sweep). We introduce the lower bound 1× 10−8 to the area of triangles. No lower bound is imposed on the bond length. IV. RESULTS OF SIMULATION A. α = 3/2 As mentioned in Section II, we assume the value of α in Eq.(3) as α = 3/2 and α = 0. In this subsection, we present the results obtained under α= 3/2 by using snapshots and figures, and in the next subsection we will show some of the results under α=0. The thermalization MCS is 1× 107 in almost all cases. However, more than 1 × 108 thermalization MCS were done close to the transition point in such cases that the surface is trapped in one phase at first and then changes its phase to a more stable one under a given condition. The total number of MCS for the production of samples is 0.8 × 108 ∼ 1.3 × 108. At the transition point, about 2 × 108 MCS was performed after the thermalization in some cases. (a) b= 21.2 (b) b= 21.8 (c) b= 22 (d) The section (e) The section (f) The section FIG. 2: (Color online) The snapshots of surfaces of size (N,NS , NJ , L) = (8462, 1890, 92, 8) obtained at (a) b = 21.2 (tubular phase), (b) b = 21.8 (planar phase), and (c) b = 22 (spherical phase), and (d),(e),(f) are the surface sections of (a),(b),(c), respectively. α=3/2. We show snapshots of the (N,NS , NJ , L) = (8462, 1890, 92, 8) surface in Figs.2(a)–2(c). They were obtained at (a) b = 21.2, (b) b = 21.8, and (c) b = 22, which respectively corresponds to the tubular phase, the planar phase, and the spherical phase. The snapshot of Fig.2(b) at b=21.8 was the final configuration produced after 2× 108 MCS including 1× 108 thermalizaion MCS; the planar surface was stable after the thermalization MCS. The surface sections are shown in Figs.2(d)–2(f); the sections in Figs.2(d) and 2(e) were obtained by slicing the surfaces perpendicular to the vertical axis, and the section in Figs.2(f) was obtained by slicing the surface perpendicular to a horizontal axis. All of the snapshots were drawn in the same scale. The axis of the tubular surface Fig.2(a) as well as the axis perpendicular to the planar surface Fig.2(b) is spontaneously chosen. The planar phase is stable only on the L=8 surfaces, while it seems unstable on the L=6 surfaces and on the L=11 surfaces. Even if the planar phase once appears on the surfaces of L=6 and L=11 of size at least N≤9282 and N ≤ 14672, respectively, it eventually collapses into the tubular phase. Therefore, we find that no planar phase can be seen on the L=6 and the L=11 surfaces; the tubular phase and the spherical phase are connected by a discontinuous transition on those surfaces. Thus, we understand that the planar phase appears depending on the size of the compartments. We should note that the planar surface may bend and fluctuate in the limit of N→∞, and the tubular surface may also bend and wind in the same limit. 20 21 22 23 S2/NS' N=8462 α=1.5 tubular planar spherical 9 9.5 10 S2/NS' N=9282 α=1.5 tubular spherical 46 48 50 52 spherical S2/NS' N=6522 α=1.5 tubular FIG. 3: The one-dimensional bending energy S2/N S against b obtained on the surfaces of (a) L = 6, (b) L = 8, and (c) L=11. N ′S(=NS+6NJ −12) is the total number of vertices where S2 is defined. Figures 3(a),3(b), and 3(c) show the bending energy S of Eq.(4) against b, which were obtained on the surfaces of L = 6, L = 8, and L = 11, respectively. N ′S(=NS+6NJ−12) is the total number of vertices where S2 is defined. 6NJ −12 is the total number of corners of the junctions, which include 12-pentagons. The solid lines on the data were drawn to guide the eyes. Dashed lines drawn vertically denote the phase boundary be- tween the tubular and the spherical phases, the bound- ary between the tubular and the planar phases, and the boundary between the planar and the spherical phases. The discontinuous change of S2/N S between the tubu- lar phase and the spherical (or the planar) phase is very clear in the figures and considered to be a sign of the first-order transition. In order to see the difference between S2/N S in those three phases, we plot in Figs.4(a),4(b), and 4(c) the varia- tion of S2/N S against MCS obtained at b=21.2, b=21.4, and b = 21.8 on the (N,NS , NJ , L) = (8462, 1890, 92, 8) surface. The thermalization MCS were not discarded; they were included only in those variations. S2/N b=21.2 in Fig.4(a) shows a jump from the spherical phase to the planar phase and a jump from the planar phase to the tubular phase; the corresponding MCS at the jumps were indicated with the dashed vertical lines. We also find in Fig.4(b) a jump from the spherical phase to the planar phase. A jump is also seen in S2/N S at b=21.8 0 0.5 1 planar N=8462 L=8b=21.4 spherical [x108]0 0.5 1 N=8462 b=21.2 tubularplanarspherical [x108] 0 1 2 N=8462 b=21.8 spherical planar 0.05 0.06 b=21.2 (d) 0.05 0.06 b=21.4 (e) 0.05 0.06 b=21.8 FIG. 4: The variation of S2/N S against MCS, which were obtained on the (N,NS , NJ , L)=(8462, 1890, 92, 8) surface at (a) b=21.2, (b) b=21.4, and (c) b=21.8. The dashed lines denote the MCS where the jumps occurred. The correspond- ing normalized histogram h(S2) obtained at (d) b=21.2, (e) b = 21.4, and (f) b = 21.8. The parameter α was fixed to α=3/2. in Fig.4(c) from the spherical phase to the planar phase. The value of b=21.2 corresponds to the tubular phase, whereas b = 21.4 and b= 21.8 correspond to the planar phase, because the final states are considered to be stable states. The surfaces at b=21.2 and b=21.8 can be seen in the snapshots in Figs.3(a) and 3(b). The distribution of S2/N S are shown as the normal- ized histograms h(S2) in Figs.4(d)–4(f), which respec- tively correspond to the variations in Figs.4(a)–4(c). We see that h(S2) in Fig.4(d) has three peaks; two of them are almost overlapping and the other one is distinctly separated from the previous two. Those three peaks in h(S2) correspond to the spherical phase, planar phase, and the tubular phase. Two almost overlapping peaks can also be seen in h(S2) in Figs.4(e) and 4(f), and they are corresponding to the spherical phase and the planar phase. We remark that the surfaces hardly change not only from the tubular phase to the smooth (= spherical or planar) phase but also from the planar phase to the spherical phase on the L = 8 and L = 11 surfaces. For this reason, we find in Figs.4(a)–4(c) no jump-back from a higher S2 state (such as the tubular state) to a lower S2 state (such as the planar state). The two-dimensional bending energy is defined by (1− ni · nj) , (6) where ni is the unit normal vector of the triangle i, and ni ·nj is defined on the common bond (ij) of the triangles i and j. S3 is not included in the Hamiltonian and is defined even on the edges of the rigid junctions. Figures 5(a)–5(c) show S3/NB against b obtained on the surfaces of L=6, L=8, and L=11, where NB is the total number of bonds including the edges of the junctions. The jump of S3/NB in Fig.5(b) is clearly seen between the tubular 20 21 22 23 S3/NB N=8462 α=1.5 tubular planar spherical 9 9.5 10 S3/NB N=9282 α=1.5 tubular spherical 46 48 50 52 spherical S3/NB N=6522 α=1.5 tubular FIG. 5: The two-dimensional bending energy S3/NB against b obtained on the surfaces of (a) L = 6, (b) L = 8, and (c) L=11. NB is the total number of bonds where S3 is defined. phase and the planar phase. On the contrary, S3/NB in the planar phase in Fig.5(b), as well as S2/N S in the planar phase in Fig.3(b), is not so clearly distinguishable from that in the spherical phase. 20 21 22 23 1.512 1.514 1.516 N=8462 α=1.5 tubular planar spherical 9 9.5 10 1.514 1.516 1.518 N=9282 α=1.5 sphericaltubular 46 48 50 52 1.508 1.512 N=6522 α=1.5 tubular spherical FIG. 6: The Gaussian bond potential S1/N against b obtained on the surfaces of (a) L=6, (b) L=8, and (c) L=11. It is expected that the Gaussian bond potential S1/N is influenced by the phase transitions. The potential S1/N should be S1/N ≃ 3/2, which is satisfied in the model without the rigid junctions because of the scale invariant property of the partition function in that case. However, the junction size R in Eq.(5) is finite in the model of this paper, and therefore S1/N can slightly de- viate from 3/2. Figures 6(a)–6(c) show S1/N against b obtained on the surfaces of (a) L=6, (b) L=8, and (c) L=11. Discontin- uous changes in S1/N shown in the figures are consistent with the discontinuous transitions of the model, although the changes are very small compared to the value of S1/N itself. We find also the expected deviation of S1/N from 3/2 in the figures. Figures 7(a)–7(c) show the mean square sizeX2, which is defined by Xi − X̄ , X̄ = Xi, (7) where X̄ is the center of mass of the surface. We see that the phase transition is not reflected in X2 on the L=6 surfaces in Fig.7(a), and the transition is also not reflected in X2 on the L= 8 surfaces in Fig.7(b) at the transition point between the planar phase and the spher- ical phase. To the contrary, X2 discontinuously changes 20 21 22 23 N=942 α=1.5 N=8462 N=3762 tubular spherical planar 9 9.5 10 α=1.5 N=2322 N=9282 N=5222 tubular spherical 46 48 50 52 N=6522 α=1.5 N=1632 N=14672 tubular spherical FIG. 7: The mean square size X2 against b obtained on the surfaces of (a) L=6, (b) L=8, and (c) L=11. in Fig.7(b) at the transition point between the tubular phase and the planar phase and also at the transition point in Fig.7(c). All of these behaviors ofX2 at the tran- sition points are consistent with those of S2/N S , S3/NB, and S1/N . B. α = 0 In this subsection, we present some of the results ob- tained under α=0. (a) b= 20.9 (b) b= 21.4 (c) b= 21.8 (d) The section (e) The section (f) The section FIG. 8: (Color online) The snapshots of surfaces of size (N,NS , NJ , L) = (8462, 1890, 92, 8) obtained at (a) b = 20.9 (tubular phase), (b) b=21.4 (planar phase), and (c) b=21.8 (spherical phase), and (d),(e),(f) are the surface sections of (a),(b),(c), respectively. α=0. Snapshots of surfaces of α=0 are shown in Figs.8(a), 8(b), 8(c), which respectively correspond to the tubu- lar phase (b = 20.9), the planar phase (b = 21.4), and the spherical phase (b = 21.8). The surface size is (N,NS , NJ , L)= (8462, 1890, 92, 8), which is identical to that in Fig.2. The snapshot in Fig.8(b) at b = 21.4 is the final configuration produced after 1.9× 108 MCS in- cluding 1 × 107 thermalizaion MCS; the planar surface was stable throughout the simulation. Thus, we find that three distinct phases are seen also in the surfaces of L = 8, and that the planar phase is unstable on the surfaces of L= 6 and L= 11 under the condition α= 0. Therefore, we consider that the phase structure of the model is independent of whether α=3/2 or α=0. 20 21 22 23 S2/NS' N=8462 tubular planar spherical 9 9.5 10 S2/NS' N=9282 tubular spherical 46 48 50 52 spherical S2/NS' N=14672 tubular FIG. 9: The one-dimensional bending energy S2/N S against b obtained on the surface of (a) L=6, (b) L=8, and (c) L=11. N ′S(=NS+6NJ−12) is the total number of vertices where S2 is defined. The one-dimensional bending energy S2/N S obtained under α=0 is shown in Figs.9(a)–9(c). A discontinuous change can be seen in S2/N S not only in Fig.9(b) at the phase boundary between the tubular phase and the pla- nar phase but also in Fig.9(c) at the phase boundary be- tween the tubular phase and the spherical phase. A jump of S2/N S in Fig.9(b) at the transition point between the planar phase and the spherical phase is very small, and hence is hardly seen just the same as in Fig.3(b) under α = 3/2 in the previous subsection. Thus, we find no difference between S2/N S of α=0 and that of α=3/2. 20 21 22 23 N=942 N=8462 N=3762 tubular spherical planar 9 9.5 10 N=2322 N=9282 N=5222 tubular spherical 46 48 50 52 N=14672 tubular spherical FIG. 10: The mean square size X2 against b obtained on the surfaces of (a) L=6, (b) L=8, and (c) L=11. The mean square size X2 are shown in Figs.10(a)– 10(c). A jump is also seen in X2 on the L = 8 and L=11 surfaces in Figs.10(b) and 10(c), and it is hardly seen on the L=6 surfaces of size up to (N,NS , NJ , L)= (9282, 2400, 162, 6). These results are identical to those observed in Figs.7(a)–7(c) under α=3/2. Finally, we comment on the planar phase appeared only on the L = 8 surface. The thermal fluctuation of vertices inside the compartments disorders the surface against the bending energy of the compartment bound- ary. Therefore, the strength to disorder the surface in- creases (decreases) with increasing (decreasing) L if N remains fixed, as stated in Section II. On the other hand, the mechanical strength of the surface increases (decreases) with decreasing (increasing) L, because the total number of junctions increases (decreases) with de- creasing (increasing) L. Therefore, the strength to or- der the surface increases (decreases) with decreasing (in- creasing) L. Then, we expect that the surface is ordered (disordered) at sufficiently small (large) L at given in- termediate value of b. Moreover, it seems possible that two competitive forces to order/disorder the surface are balanced with each other at intermediate values of L and consequently, some new phase appears depending on b at those L. Note also that the possibility of the appear- ance of planar phase is not completely eliminated on the surfaces of L=6 and L=11 of sufficiently large size. V. SUMMARY AND CONCLUSION We have shown that a dynamically triangulated spher- ical surface has three distinct phases; the tubular phase, the planar phase, and the spherical phase, and that they are separated by discontinuous transitions. The first- order nature was very clear from the discontinuity in the bending energies S2 and S3 not only at the transition point between the tubular phase and the planar phase but also at the transition point between the tubular phase and the spherical phase. We know that the model has the collapsed phase at sufficiently small b, since the self- avoiding property is not assumed at least. Therefore, we expect that the model has four different phases including the collapsed phase, although the order of the transition between the collapsed phase and the tubular phase is un- known. The mechanical strength of the surface is given only by elastic linear-chains with rigid junctions. The triangulated surfaces are characterized by the size (N,NS , NJ , L), where N is the total number of vertices including the junctions, NS is the total number of vertices on the chains, NJ is the total number of junctions, and L is the length of chains between the two nearest-neighbor junctions on the starting configurations. These four pa- rameters are not totally independent, because these are given by two independent integers (ℓ,m), where m di- vides ℓ. In fact, N =10ℓ2−60m2+2, NS =30m(ℓ−3m), NJ =10m 2+2, and L=(ℓ/m)−2. We assumed three different values for L such that L= 6, L=8, and L=11 in the simulations. The edge length R of the rigid junction was fixed to be R = 0.1. The parameter α, which represents a weight for the three- dimensional integrations of the partition function, was assumed as α=3/2 and α=0. It is remarkable that the model has the planar phase, which is stable only on the surfaces with a specific struc- ture. In fact, the planar phase can be seen on the surfaces of L=8, and it is unstable on the L=6 and L=11 sur- faces. The planar phase appears in a narrow region on the b-axis between the tubular phase and the spherical phase, and it is distinguishable from the spherical phase because a small but finite discontinuity can be seen in the bending energies S2/N S and S3/NB. The gap of the bending energy S2 at the transition point is very small, i.e., S2 in the planar phase is almost identical to that in the spherical phase; however, the double peak struc- ture was clearly seen in the histogram of S2, which is included in the Hamiltonian. From this, we confirmed that the transition between the planar phase and the spherical phase is of first order. 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0704.0495

The Veldkamp Space of Two-Qubits

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 075, 7 pages The Veldkamp Space of Two-Qubits Metod SANIGA †, Michel PLANAT ‡, Petr PRACNA § and Hans HAVLICEK ¶ † Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic E-mail: [email protected] URL: http://www.ta3.sk/~msaniga/ ‡ Institut FEMTO-ST, CNRS, Département LPMO, 32 Avenue de l’Observatoire, F-25044 Besançon Cedex, France E-mail: [email protected] § J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Doleǰskova 3, CZ-182 23 Prague 8, Czech Republic E-mail: [email protected] ¶ Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria E-mail: [email protected] Received April 13, 2007, in final form June 18, 2007; Published online June 29, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/075/ Abstract. Given a remarkable representation of the generalized Pauli operators of two- qubits in terms of the points of the generalized quadrangle of order two, W (2), it is shown that specific subsets of these operators can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements – the so-called Veldkamp space of W (2). An intriguing novelty is the recognition of (uni- and tri-centric) triads and specific pentads of the Pauli operators in addition to the “classical” subsets answering to geometric hyperplanes of W (2). Key words: generalized quadrangles; Veldkamp spaces; Pauli operators of two-qubits 2000 Mathematics Subject Classification: 51Exx; 81R99 1 Introduction A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3]. The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space – the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle. 2 Finite generalized quadrangles and Veldkamp spaces In this section we will briefly highlight the basics of the theory of finite generalized quadran- gles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. http://arxiv.org/abs/0704.0495v3 mailto:[email protected] http://www.ta3.sk/~msaniga/ mailto:[email protected] mailto:[email protected] mailto:[email protected] http://www.emis.de/journals/SIGMA/2007/075/ 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence struc- ture S = (P,B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1+ t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y,M) ∈ P ×B for which xIMIyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle SD of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick. Given two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x⊥ = {y ∈ P |y ∼ x} and note that x ∈ x⊥; obviously, x⊥ = 1 + s+ st. Given an arbitrary subset A of P , the perp(-set) of A, A⊥, is defined as A⊥ = {x⊥|x ∈ A} and A⊥⊥ := (A⊥)⊥. A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T⊥ is called its center and we say that T is acentric, centric or unicentric according as |T⊥| is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st+ 1 points. The concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P,B), which is a proper subset of P such that each line of Γ meets H in one or all points [6]. For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x⊥; (ii) a (full) subquadrangle of order (s, t′), t′ < t; and (iii) an ovoid. Finally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P,B), V(Γ) [5]. V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection H1H2 of all geometric hyperplanes H of Γ such that H1 H = H2 H or H = Hi (i = 1, 2), where H1 and H2 are distinct points of V(Γ). Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types. 3 The smallest thick GQ and its Veldkamp space The smallest thick GQ is obviously the one with s = t = 2, dubbed the “doily.” This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W3(2) or simply W (2), is in terms of the points of PG(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in PG(4, 2). By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x⊥, seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each – as depicted in Fig. 1. The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2, left); hence, 1It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper. The Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are repre- sented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points’ perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each picture – except that in the bottom right-hand corner – stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon. Figure 2. Left: – The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center’s perp-set. Right: – A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the “yellow” hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2). The Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in “complementary” pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graph- combinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our “doily”, V(W (2)).2 From the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of “core-sets”, i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type’s lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type – the one exhibiting all the three kinds of hyperplanes – being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1. We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as PG(4, 2), the four-dimensional projective space over the Galois field of two elements [8]; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5]. 4 Pauli operators of two-qubits in light of V(W (2)) As discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1). As already mentioned, the role of tricentric triads of the operators has 2As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as “the doily”, and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type. Type of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3]. PO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin’s square GQ ovoid perp-set\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2, left) and three pentads (Fig. 3, 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10], because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x⊥ and whose lines are the spans {u, v}⊥⊥, where u, v ∈ x⊥ with u 6= v [4]. Identifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen–Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11]. At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper. Concerning three-qubits, our preliminary study indicates that the corresponding finite geo- metry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12]. As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of two- qubits, which would imply the expected difference between entanglement properties of the two The Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits. 5 Conclusion By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3], distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13], in particular when the dimension of Hilbert space is not a power of a prime [14]. Acknowledgements This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT–51–012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ “Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases” (France), the CNRS–SAV Project # 20246 “Projective and Related Geometries for Quantum Information” (France/Slovakia) and by the 〈Action Austria–Slovakia〉 project # 58s2 “Finite Geometries Behind Hilbert Spaces”. References [1] Saniga M., Planat M., Pracna P., Projective ring line encompassing two-qubits, Theor. Math. Phys., to appear, quant-ph/0611063. [2] Saniga M., Planat M., Multiple qubits and symplectic polar spaces of order two, Adv. Studies Theor. Phys. 1 (2007) 1–4, quant-ph/0612179. [3] Planat M., Saniga M., On the Pauli graph of N-qudits, quant-ph/0701211. [4] Payne S.E., Thas J.A., Finite generalized quadrangles, Pitman, Boston – London – Melbourne, 1984. [5] Buekenhout F., Cohen A.M., Diagram geometry (Chapter 10.2), Springer, New York, to appear; preprints of separate chapters can be found at http://www.win.tue.nl/~amc/buek/. [6] Ronan M.A., Embeddings and hyperplanes of discrete geometries, European J. Combin. 8 (1987), 179–185. [7] Shult E., On Veldkamp lines, Bull. Belg. Math. Soc. 4 (1997), 299–316. [8] Hirschfeld J.W.P., Thas J.A., General Galois geometries, Oxford University Press, Oxford, 1991. [9] Lawrence J., Brukner Č., Zeilinger A., Mutually unbiased binary observable sets on N qubits, Phys. Rev. A 65 (2002), 032320, 5 pages, quant-ph/0104012. [10] Saniga M., Planat M., Rosu H., Mutually unbiased bases and finite projective planes, J. Opt. B: Quantum Semiclass. Opt. 6 (2004), L19–L20, math-ph/0403057. [11] Saniga M., Planat M., Minarovjech M., Projective line over the finite quotient ring GF (2)[x]/(x3 − x) and quantum entanglement: the Mermin “magic” square/pentagram, Theor. Math. Phys. 151 (2007), 625–631, quant-ph/0603206. [12] Planat M., Saniga M., Pauli graph and finite projective lines/geometries, Optics and Optoelectronics, to appear, quant-ph/0703154. [13] Thas K., Pauli operators of N-qubit Hilbert spaces and the Saniga–Planat conjecture, Chaos Solitons Frac- tals, to appear. [14] Thas K., The geometry of generalized Pauli operators of N-qudit Hilbert space, Quantum Information and Computation, submitted. http://arxiv.org/abs/quant-ph/0611063 http://arxiv.org/abs/quant-ph/0612179 http://arxiv.org/abs/quant-ph/0701211 http://www.win.tue.nl/~amc/buek/ http://arxiv.org/abs/quant-ph/0104012 http://arxiv.org/abs/math-ph/0403057 http://arxiv.org/abs/quant-ph/0603206 http://arxiv.org/abs/quant-ph/0703154 Introduction Finite generalized quadrangles and Veldkamp spaces The smallest thick GQ and its Veldkamp space Pauli operators of two-qubits in light of V(W(2)) Conclusion References

0704.0496

Fusion process studied with preequilibrium giant dipole resonance in time dependent Hartree-Fock theory

Fusion process studied with preequilibrium giant dipole resonance in time-dependent Hartree-Fock theory C. Simenel1,2, Ph. Chomaz2 and G. de France2 1 DSM/DAPNIA/SPhN, CEA/SACLAY, F-91191 Gif-sur-Yvette Cedex, France and 2 Grand Accélérateur National d’Ions Lourds (GANIL), CEA/DSM-CNRS/IN2P3, Bvd Henri Becquerel, BP 55027,F-14076 CAEN Cedex 5, France (Dated: November 1, 2018) The equilibration of macroscopic degrees of freedom during the fusion of heavy nuclei, like the charge and the shape, are studied in the Time-Dependent Hartree-Fock theory. The preequilibrium Giant Dipole Resonance (GDR) is used to probe the fusion path. It is shown that such isovector collective state is excited in N/Z asymmetric fusion and to a less extent in mass asymmetric systems. The characteristics of this GDR are governed by the structure of the fused system in its preequi- librium phase, like its deformation, rotation and vibration. In particular, we show that a lowering of the preequilibrium GDR energy is expected as compared to the statistical one. Revisiting ex- perimental data, we extract an evidence of this lowering for the first time. We also quantify the fusion-evaporation enhancement due to γ-ray emission from the preequilibrium GDR. This cooling mechanism along the fusion path may be suitable to synthesize in the future super heavy elements using radioactive beams with strong N/Z asymmetries in the entrance channel. PACS numbers: 24.30.Cz, 21.60.Jz, 25.70.Gh, 25.70.Jj I. INTRODUCTION The fusion of two nuclei occurs at small impact pa- rameters when the overlap between their wave functions is big enough to allow the strong interaction to overcome the Coulomb repulsion. Heavy-ion fusion reactions have numerous applications, like the study of high spin states in yrast and super-deformed bands [1] or the formation of Heavy and Super Heavy Elements (SHE) [2]. Induced by beams of unstable nuclei, this mechanism will also allow to produce very exotic species and allow for the study of isospin equilibration in the fused system. The fusion process can be schematically divided in three steps: (i) an approach phase during which each nu- cleus feels only the Coulomb field of its partner and which ends up when the nuclear interaction starts to dominate, (ii) a rapid equilibration of the energy and the angular momentum transfered from the relative motion to the internal degrees of freedom, leading to the formation of a Compound Nucleus (CN) and (iii) a statistical decay of the CN. Lots of theoretical and experimental efforts [3] are made to understand step (i). These studies focus on an energy range located around the fusion barrier. At these energies the fusion is controlled by quantum tunnel- ing which is strongly influenced by the couplings between the internal degrees of freedom and the relative motion of the two colliding partners. Although the cooling mecha- nisms involved in (iii) are well known and consist mainly in light particle and γ-ray emission in competition with fission for heavy systems, the initial conditions of the statistical decay depend on the equilibration process (ii) which is still subject to many debates nowadays. Indeed, step (ii) is characterized by an equilibration of several degrees of freedom like the shape [4] or the charge [5] which can be accompanied by the emission of preequi- librium particles. Such emission decreases the excitation energy and the angular momentum. The latter quanti- ties are crucial and must be determined precisely because they have a major influence on the CN survival probabil- ity and therefore on the synthesis of very exotic systems such as the SHE. In this paper we study the equilibration of the charges in fused systems, its interplay with other macroscopic de- grees of freedom like the shape and the rotation, and its implications on the statistical decay. To probe theoret- ically and experimentally this way to fusion, we use the preequilibrium isovector Giant Dipole Resonance (GDR) [6, 7, 8, 9, 10]. Giant Resonances are interpreted as the first quantum of collective vibrations involving protons and neutrons fluids. The Giant Monopole Resonance can be described as a breathing mode, an alternation of compression and dilatation of the whole nucleus. The GDR corresponds to a collective oscillation of the pro- tons against the neutrons. The Giant Quadrupole Res- onance consists in a nuclear shape oscillation between prolate and oblate deformations. Many other resonances have been discovered [11, 12]. In particular Giant Res- onances have been observed in hot nuclei formed by fu- sion [13, 14]. This demonstrates the survival of ordered vibrations in very excited systems, which are known to be chaotic, even if some Giant Resonance characteristics like the width are affected by the temperature [15, 16]. Moreover, the strong couplings between various collec- tive modes which occur for Giant Resonances built on the ground state [17, 18] are still present in fusion reac- tions [10, 19]. It might therefore be possible to use the Giant Resonances properties to probe the nuclear struc- ture of the composite system on its way to fusion. The choice of the preequilibrium GDR, that is, a GDR excited in step (ii) before the formation of a fully equi- librated CN, is motivated by the fact that its properties strongly depend on the structure of the state on which http://arxiv.org/abs/0704.0496v3 it is built, for instance the deformation [5]. The idea is to form a CN with two N/Z asymmetric reactants. Such a reaction may lead to the excitation of a dipole mode because of the presence of a net dipole moment in the en- trance channel. This dipole oscillation should occur be- fore the charges are fully equilibrated, that is, during the preequilibrium phase in which the system keeps a mem- ory of the entrance channel [5, 6, 7, 8, 9, 10, 20, 21]. In addition, for such N/Z asymmetric reactions, an enhance- ment of the fast GDR γ-ray emission is expected as com- pared to the ”slower” statistical γ-ray yield [7, 8, 9, 10]. This is of particular interest since the properties of these GDR γ-rays characterize the dinuclear system which pre- cedes the hot equilibrated CN. The first experimental in- dications on the existence of such new phenomenon have been reported in [22, 23, 24, 25, 26] for fusion reactions and in [26, 27, 28, 29, 30, 31, 32, 33] in the case of deep inelastic collisions. The paper is organized as follows: In Sec. II we study the properties of the preequilibrium GDR using the Time-Dependent Hartree-Fock (TDHF) formalism. In Sec. III we show how an N/Z asymmetric entrance chan- nel may increase the fusion-evaporation cross-sections. Finally, we conclude in section IV. II. TDHF STUDY OF THE PREEQUILIBRIUM GIANT DIPOLE RESONANCE At the early time of the fusion reaction, the system keeps the memory of the entrance channel. We call this stage of the collision the preequilibrium phase which ends when all the degrees of freedom are equilibrated in the compound system and when the statistical decay starts. One of these degrees of freedom is the isospin, which measures the asymmetry between protons and neutrons. When the two nuclei have different N/Z ratios, the pro- ton and neutron centers of mass of the total system do not coincide. As shown in [6, 21], there is a non zero force between the two kind of nucleons which tends to restore the initial isospin asymmetry. In such a case, an oscil- lation of protons against neutrons on the way to fusion might occur, that is, the so-called preequilibrium GDR [5, 6, 7, 8, 9, 10, 20]. In fusion reactions the shape of the system changes drastically during the preequilibrium phase. Studies of the dynamics in the fusion reaction mechanism requires sophisticated calculations to extract the preequilibrium GDR characteristics (energy, width...) and in turn, on the way to fusion. To achieve this goal, we choose to use, as in the pioneer work of Bonche and Ngô on charge equilibration [5], the TDHF approach because it is a fully microscopic theory which takes into account the quantal nature of the single particle dynamics. Moreover in the present study we will restrict ourself to the observation of one-body observables (e.g. the density ρ(r)) which are supposed to be well described by such a mean field approach. However it is clear that an important challenge is to develop methods going beyond mean field which is beyond the scope of this paper. In this section we present quantum calculations on preequilibrium giant collective vibrations using the TDHF theory. We shall start with a brief description of the TDHF theory in Sec. II A. Then we examine the role of various relevant symmetries in the entrance channel, namely the N/Z and mass symmetries (Sec. II B-IID). Finally, in Sec. II E we shall compare our results with the experimental data obtained by Flibotte et al. [22]. A. TDHF approach In the TDHF approach [34, 35, 36, 37, 38, 39, 40], each single particle wave function is propagated in the mean field generated by the ensemble of particles. The mean field approximation does not take into account the dissi- pation due to two-body interactions [41, 42, 43, 44]. How- ever TDHF takes care of one-body mechanisms such as Landau spreading and evaporation damping [45]. Quan- tum effects induced by the single particle dynamics like shell effects or modification of the moment of inertia [46] are accounted for properly. The main advantage of TDHF is its fully microscopic treatment of the N-body dynamics with the same effec- tive interaction as the one used for the calculation of the Hartree-Fock (HF) ground sates of the collision partners. The consistency of the method for the structure of nuclei and the nuclear reactions increases its prediction power and its availability to study the interplay between exotic structures and reaction mechanisms. Moreover the TDHF equation is strongly non linear which is of great importance for reactions around the bar- rier because it includes couplings between relative motion and internal degrees of freedom of the collision partners. Also TDHF provides a good description of collective mo- tion and can even exhibit couplings between collective modes [17]. In fact the TDHF theory is optimized for the prediction of expectation values of one-body observ- ables and gives their exact evolution in the extreme case where the residual interaction vanishes. However, the TDHF prediction of multipole moments in nuclear colli- sion, for instance, may differ from the correct evolution because of the omission of the residual interaction. An improvement of the description would be given by the inclusion of the effect of the residual interaction on the dynamics, which would increase considerably the compu- tational time and is beyond the scope of this paper. The TDHF theory describes the evolution of the one-body density matrix ρ(t) of matrix elements 〈rsq|ρ̂|r′s′q′〉 = ϕ∗i (r ′s′q′)ϕi(rsq), where ϕi(rsq) = 〈rsq|i〉 denotes the component with a spin s and isospin q of the occupied single particle wave-function ϕi. This evolution is determined by a non linear Liouville-von Neumann equation, ρ− [h(ρ), ρ] = 0 (1) where h(ρ) is the matrix associated to the self consistent mean-field Hamiltonian. We have used the code built by P. Bonche and coworkers [47] with an effective Skyrme interaction [48] and SLy4d parameters [47]. In its actual version, TDHF does not account for pairing interactions. B. N/Z asymmetric reactions As far as the dipole motion in the preequilibrium phase is concerned, it is obvious that the main relevant asym- metry responsible for such a motion is a difference in the charge-to-mass ratio between the collision partners [6]. The associated experimental signature is an enhancement of the γ-ray emission in the GDR energy region of the compound system [22, 23, 24, 25, 26] which is attributed to a dipole oscillation. Several informations about the fusion path can be extracted from such a dipole oscilla- tion and its corresponding γ-ray spectrum. For numerical tractability we start our study of the fusion process with a light system: 12Be+28S→40Ca. We first deduce the γ-ray spectrum from the dipole motion. Then we study the effects of the deformation of the compound system, and of the impact parameter on this motion. 1. The preequilibrium GDR γ-ray spectrum We first consider a central collision at an energy of 1 MeV/nucleon in the center of mass. The expectation value of the dipole moment Q̂D is defined by QD = 〈Q̂D〉 = (Xp −Xn) (2) whereXp = 〈x̂p〉 andXn = 〈x̂n〉 are the positions of the proton and neutron centers of mass respectively. The expectation value of the conjugated dipole moment P̂D is then associated to the relative velocity between protons and neutrons, and is defined by the relation PD = 〈P̂D〉 = (Pp − Pn) (3) where Pp = 〈p̂p〉 and Pn = 〈p̂n〉 are the to- tal proton and neutron moments respectively. These definitions ensure the canonical commutation relation Q̂D, P̂D = i~. The time evolutions of QD and PD are plotted in Figs. 1-c and 1-b respectively. The trajectories in both the (QD,t) and (PD,t) planes exhibit oscillations which we attribute to the preequilibrium GDR. We also note that PD(t) oscillates in phase quadrature with QD(t) and that those oscillations are damped due to the one-body dissipation. Consequently, the plot of PD as a function of QD shown in Fig. 1-a is a spiral. The GDR period extracted from these plots is around 107 fm/c, which corresponds to an energy of ∼ 11.6 MeV. FIG. 1: Time evolution of the expectation value of the dipole moment, QD, and its conjugated moment, PD, in the reaction 12Be+28S→40Ca at an energy of 1 MeV/nucleon in the center of mass and at zero impact parameter. During the collision and before the equilibrium is reached, a fast rearrangement of charges occurs within the composite system [5], generating the γ-ray emission. We extract the preequilibrium GDR γ-ray spectrum from the Fourier transform of the acceleration of the charges [9, 49] (Eγ) = |I(Eγ)|2 where α is the fine structure constant and I(Eγ) = The spectrum obtained from Eq. 4 is plotted in Fig. 2 (solid line). In order to have a spectrum without spu- rious peaks coming from the finite integration time, we multiply the quantity d by a gaussian function [50]. In addition, this function plays a role of a filter in the time domain. This filter prevents the sig- nal to be affected by the interaction between the nucleus and the emitted nucleons which have been reflected on the box [51]. We choose τ = 320 fm/c in our calculations. This ensures the fact that the spectra are free of spuri- ous effects coming from the echo. However this procedure adds a width Γ ∼ ~ ∼ 0.6 MeV. This is a drawback if one is concerned with detailed spectroscopy. However, in this paper, we are only interested by the gross properties of the preequilibrium GDR in order to study the fusion mechanisms. As we can see in Fig. 2, the preequilibrium GDR energy is E GDR = 11.64 MeV, which corresponds FIG. 2: preequilibrium GDR γ-ray spectrum calculated in the reaction 12Be+28S → 40Ca (solid line) at an energy of 1 MeV/nucleon in the center of mass and γ-ray spectrum of a GDR built on the ground state of 40Ca (dotted line). to the previous value deduced from the GDR oscillation period. The energy of the preequilibrium GDR is much lower than the one of the GDR built on the spherical ground state of the 40Ca. This situation will be now explored into more details. 2. Deformation effect To better characterize the preequilibrium GDR, it is necessary to compare it with the usual GDR built upon the CN ground state [20]. This GDR is generated by applying an isovector dipole boost with a velocity kD on the 40Ca HF ground state |ψ(t)〉 = exp −ikDQ̂D |HF 〉 yielding an oscillation of QD(t) and PD(t) in phase quadrature as we can see in Fig. 3. The period of the oscillation is around 80 fm/c which is lower than in the fusion case and corresponds to a higher energy (EGDR = 15.5 MeV) as it is shown in the associated GDR γ-ray spectrum in Fig. 2 (dotted line). The lower energy obtained for the fusing system reveals a strong prolate deformation [5, 9, 10, 20]. The two mechanisms (fusion reaction and dipole boost) are expected to gen- erate a GDR with quite different dynamical properties. This can be seen in the density plot projected in the re- action plane shown in Fig. 4, which shows that in the case of a fusion reaction, the CN relaxes its initial pro- late elongation along the collision axis with a time which is larger than the typical dipole oscillation period of the FIG. 3: GDR built upon the HF ground state in 40Ca and excited by an isovector dipole boost: evolution of the expec- tation value of the associated dipole moment, QD, and its conjugated moment, PD, as a function of time. FIG. 4: Density plots projected on the reaction plane for dif- ferent times in the case of the fusion reaction. Lines represent isodensities. Deformation effects can be studied all along the fusion path [4, 20]. The quadrupole deformation parameter ǫ is defined by a scaling of the axis from a spherical to a deformed shape along the x-axis Rx = R0(1 + α) Ryz = R0(1− ǫ) (5) where α is defined by the conservation of the volume of the nucleus RxRyRz = R 0, which leads to (2− ǫ)ǫ (1 − ǫ)2 . (6) If one neglects high order terms in ǫ, we get the usual value α ≃ 2ǫ. The deformation parameter is related to the expecta- tion values of the monopole and quadrupole moments Q̂0 FIG. 5: Time evolution of the deformation, ǫ, in 40Ca formed in the 12Be+28S fusion reaction at an energy of 1 MeV/nucleon in the center of mass. The time axis origin is chosen when the maximum of the fusion barrier is reached. The average preequilibrium deformation ǫp obtained from the GDR energy (see Eq. 13) is represented by a dashed line. and Q̂2 which are expressed by Q0 = 〈Q̂0〉 = dr ρ(r)r2 (7) Q2 = 〈Q̂2〉 = dr ρ(r)r2 . (8) We can write Q2 as a function of Q0 Q2 = − Q0 + 3 dr ρ(r)x2. (9) Eqs. 5 and 7 lead to dr ρ(r)x2 = (1 + α)2 Q0. (10) Using Eqs. 6, 9, 10 and ǫ < 1, we get ǫ(t) = 1− 2Q2(t)√ 5Q0(t) which, at first order in ǫ, becomes ǫ(t) = Q2(t) 5Q0(t) . (12) In Ref. [10] we used Eq. 12 to characterize the average deformation. In Fig. 5 we present the time evolution of the deformation, ǫ(t), obtained from the more general FIG. 6: a) Energy of the preequilibrium GDR obtained from the first oscillation of the dipole moment and b) the defor- mation parameter, ǫ, obtained from Eq. 13 (dashed line) and from Eq. 11 (solid line), as a function of the center of mass energy. expression of ε given in Eq. 11. We consider a 40Ca formed in the 12Be+28S fusion reaction at an energy of 1 MeV/nucleon in the center of mass. The important point here is that the deformation does not relax and strongly affects the frequency of the oscillations. A lower energy is expected for the longitudinal collective motion E GDR in the fused system as compared to the one simulated in a spherical 40Ca [5, 7, 8, 9, 10, 20]. Following a macroscopic model for the dipole oscillation, we expect the energy of the GDR to evolve with the deformation along the x-axis (collision axis) as = (1 − ǫp)2 (13) where ǫp is the average deformation during the preequi- librium stage. The frequency of the GDR along the defor- mation axis fulfills this relation with ǫp ≃ 0.13 in excel- lent agreement with the observed deformation in Fig. 5. We have also investigated the effect of the center of mass energy ECM on the preequilibrium GDR energy and on the deformation parameter (see Fig. 6). The GDR energy exhibits small variations (less than 1 MeV) with the center of mass energy (Fig. 6-a). For ECM < 40 MeV, the increase of E with ECM is attributed to the formation of a dinuclear system with a slow neck dynamics at low energy [9]. The presence of the neck is in fact expected to slow down the charge equilibration process, and then to increase the GDR period. For ECM > 40 MeV, Fig. 6-a a decrease of E when ECM increases. As illustrated in Fig. 6-b, this is associated to a larger quadrupole deformation when the collision is more violent. Consequently, the higher the center of mass energy, the more prolately deformed the CN. In Fig. 6-b, the deformation is estimated from Eq. 13 (dashed line) and from Eq. 11 (solid line) at the first max- imum after one oscillation of ǫ(t) (e.g. at t ∼ 225 fm/c in the case of ECM = 1 MeV/u as we can see in Fig. 5). We also observe in this energy domain a good agreement between the deformations calculated with both methods. This lowering of the GDR energy due to deformation is not specific to nuclear physics. Indeed, an energy split- ting of the isovector dipole mode has been observed in fissioning atomic clusters due to a strong prolate defor- mation of the fission phase [52]. In such systems, the use of LASERs with the ”pulse and probe” technique is ex- pected to give access to the deformation and also to the fission time [53]. 3. non central collisions To better mimic the situation of a fusion reaction, we extended our calculations to non-zero impact parameters. In fact, a non central collision may excite collective ro- tational states in the deformed preequilibrated CN. This rotation may be coupled to the preequilibrium GDR [20]. In particular, the interplay of dipole vibration and defor- mation can be affected by the rotation. In addition to the center of mass coordinates with x along the beam axis and y perpendicular to the reaction plane, we de- fine a new coordinate system x′, y′, z′, where x′ is the deformation axis, and y = y′ is the rotation axis (see Fig. 7). In the head-on collision example studied previ- ously, those two frames are the same. For symmetry rea- sons, the dipole oscillation cannot occur along the z = z′ and y = y′ axis. For non-central collisions, the oscillation is only for- bidden along the y = y′ axis [5, 20]. In this case the amplitude of the oscillation along x′ slightly decreases with the impact parameter. This decrease becomes sig- nificant at rather large impact parameters as we can see in Fig. 8 where we have plotted the amplitude of the first oscillation of the dipole moment along x′ (solid line) as a function of the impact parameter. This decrease is ac- companied by an oscillation of the dipole moment along the z′ axis with a smaller amplitude which increases with the impact parameter b. Both amplitudes are of the same order when b ∼ 5 fm. FIG. 7: Description of the two frames used in non central collisions. FIG. 8: Amplitude of the first oscillation of the dipole moment along x′ (solid line) and along z′ (dashed line) as a function of the impact parameter, b, in the 12Be+28S fusion reaction at an energy of 1 MeV/nucleon in the center of mass. The oscillation along the z′ axis results from a weak symmetry breaking due to the rotation of the system [10]. In order to demonstrate this, let us start with the time-dependent Schrödinger equation in the laboratory frame R: i~|ψ̇〉 = Ĥ |ψ〉. In the rotating frame R′, the expression of the wave function is |ψ′〉 = R̂(α)|ψ〉 where R̂(α) = e−iα(t)Ĵy is a rotation matrix, Ĵy is the generator of the rotations around y and α(t) is the angle between the two frames (see Fig. 7). We express the Schrödinger FIG. 9: Time evolution of the total dipole moment for the 8Be+32S→40Ca reaction at an energy of 1 MeV/nucleon in the center of mass. At time t = 0 fm/c, the distance between the centers of mass of the nuclei is 92.8 fm. The arrow indi- cates the time when the fusion barrier is reached. The dashed line gives the result of the adiabatic model (cf. Eq. 15). equation as −~α̇ĴyR̂−1|ψ′〉+i~R̂−1|ψ̇′〉 = ĤR̂−1|ψ′〉 and we get [10] i~|ψ̇′〉 = R̂ĤR̂−1 + ~α̇Ĵy |ψ′〉. (14) Eq. 14 is the Schrödinger equation expressed in the rotating frame R′ of the CN and Ĥ ′ = R̂ĤR̂−1+~α̇Ĵy is the Hamiltonian expressed in this frame. The last term induces a motion along the z′ axis from a dipole vibration along x′. It is quantified by the dipole moment along z′ which is plotted as a dashed line in Fig. 8. This is a clear manifestation of couplings between rotational and vibrational motions in nuclei. In this subsection we have shown that an N/Z asymme- try in the entrance channel generates a dipole oscillation during the preequilibrium phase of a fusion reaction. In the next one we will see that, due to polarization effects, such a motion also occurs in N/Z symmetric systems al- though with a smaller amplitude. C. N/Z symmetric reactions We now examine the situation of a central collision in- volving two N = Z nuclei using the example of 8Be+32S at ECM = 1 MeV/nucleon ( 8Be is bound with a strong prolate deformation in Hartree-Fock calculations with the SLy4d force). As we can see in Fig. 9, the amplitude FIG. 10: Schematic representation of the isovector polariza- tion due to Coulomb repulsion between protons that occurs before fusion. The protons are represented by a solid line and the neutrons by a dotted line. Xi is the position of the center of mass of the nucleus i. of the dipole oscillations is significantly reduced as com- pared to the N/Z asymmetric case (cf. Fig. 1-c). In this latter system (12Be+28S), the dipole oscillations are gen- erated by the N/Z asymmetry, whereas in the 8Be+32S reaction, they are only due to the mass asymmetry of the two collision partners. Indeed, a mass asymmetry induces a difference in the isovector polarization in the collision partners. This polarization is due to Coulomb repulsion between protons of the colliding nuclei before the fusion starts [5]. To show it, let us use an adiabatic approach in which we consider that the polarization of a nucleus at a dis- tance X = X2 −X1 between the centers of mass is gen- erated by the Coulomb field of its collision partner. Xi is the position of the center of mass of the nucleus i. The distance between the proton and neutron centers of mass in nucleus i is supposed to be small as compared to X (see Fig. 10). The equality between the external Coulomb field and the restoring force between protons and neutrons leads to a dipole moment in the nucleus i QDi(t) ≃ (−1)i NiZiZje AiEGDR imX(t) where i 6= j = 1 (for 32S) or 2 (for 8Be). The GDR energy is calculated in each collision partner from the dipole response frequency following a small amplitude dipole boost. We get EGDR = 23.0 MeV for 32S and EGDRx = 17.2 MeV for 8Be along its deformation axis which is chosen to be aligned with the collision axis. The dipole moment in the total system becomes QD(t) = N1Z2 −N2Z1 X(t) +QD1(t) +QD2(t). (15) The first term of the right hand side of Eq. 15 is usually dominant for a N/Z asymmetric reaction [8]. However, it vanishes for a N/Z symmetric one. In this case, one is left with the sum of the dipole moments of the partners. This simple adiabatic model (dashed line in Fig. 9) gives the good trend of the total dipole moment up to the vicinity of the contact point. After the fusion starts, the dipole moment increases and oscillates in the preequilibrium system. The adia- batic model is too simple to describe this phenomenon. In fact, due to the polarization, the nuclear interaction acts first on neutrons and then is expected to modify strongly the dipole moment at the initial stage of the fusion [5]. The consequence of this polarization in a mass asym- metric system is a dipole oscillation which can be in- terpreted, as previously, in term of an excitation of a preequilibrium GDR. However, the GDR excitation is very small as compared to the N/Z asymmetric case. Of course, for a mass and N/Z symmetric reaction no pree- quilibrium GDR are allowed for symmetry reason [10]. As we will see in the next section, the special case of an N/Z asymmetric and mass symmetric system exhibits some interesting behaviors as far as the collective motions are concerned. D. mass asymmetry and isoscalar vibrations In this subsection we study the couplings between the isovector dipole motion and isoscalar vibrations in the preequilibrium phase and their dependence on the mass asymmetry in the entrance channel. The dipole motion can be coupled to isoscalar vibrations through the non linearity of the TDHF equation [10, 17, 19]. The presence of such isoscalar vibrations in the preequilibrium system depends on the structure of the colliding partners and on their mass asymmetry. For instance a mass symmet- ric system has a stronger quadrupole deformation at the touching point than a mass asymmetric one. In such a system a quadrupole vibration might appear. Let us start this study with the time evolution of the instantaneous dipole period [10] which is very sensitive to couplings with isoscalar vibrations. We define this period as being twice the time to describe half a revolution in the spiral diagram representing the evolution of the system in the (PD, QD) space. The resulting evolution is plotted in Fig. 11 for two N/Z asymmetric central collisions: • the mass asymmetric 12Be+28S reaction at ECM = 1 MeV/nucleon. • the mass symmetric 20O + 20Mg reaction at ECM = 1.6 MeV/nucleon. The center of mass energy has been chosen to obtain the same ECM/VB ratio for both reactions (VB is the Coulomb barrier). The mean values of the GDR period obtained for the two reactions are different. For the mass symmetric re- action, this value is ≃ 170 fm/c, whereas in the mass asymmetric case it is ≃ 105 fm/c, in good agreement with the one obtained from Fig. 1 (107 fm/c). This dif- ference is attributed to a larger deformation of the CN FIG. 11: Time evolution of the GDR period for 20O+20Mg at 1.6 MeV/nucleon (solid line) and for 12Be+28S at 1 MeV/nucleon (dashed line). Both energies are in the center of mass. in the mass symmetric case which, in average, is ǫ ∼ 0.2 (from Eq. 12), as compared to the mass asymmetric sys- tem (ǫ ∼ 0.13). Note that it is not appropriate to use Eq. 13, to calculate the deformation from the observed GDR energy frequency for 20O+20Mg since it is valid only for small deformations. The dipole moment time evolution for those two re- actions (Figs. 1 and 12), shows that unlike 20O+20Mg, the oscillations in the 12Be+28S system are dominated by a single energy. This is consistent with the evolution of the GDR period in Fig. 11 which is rather constant in the mass asymmetric case whereas it exhibits strong oscillations in the mass symmetric one. This anharmonic- ity can also be seen in the GDR γ-ray spectrum of the 20O+20Mg reaction plotted in Fig. 13. Indeed, one can clearly identify two peaks in this spectrum at 7.7 MeV and 10.8 MeV. To better understand what is the origin of the dif- ferences between the two systems, we have calculated the evolutions of the monopole Q0 and quadrupole Q2 moments defined by Eqs. 7 and 8 respectively. Those evolutions are plotted in Fig. 14-a for 20O+20Mg. We first note that Q2 is always positive, that is, the com- pound system keeps a prolate deformation. In addition, Q0 and Q2 exhibit strong oscillations with the same pe- riod ∼ 165 fm/c. Therefore, we conclude that they have the same origin which is interpreted as a vibration of the density around a prolate shape [10]. This mode is only excited in the mass symmetric channel: the evolu- tions of Q0(t) and Q2(t) for the mass asymmetric reac- tion (12Be+28S) at 1 MeV/nucleon in the center of mass FIG. 12: Evolution of the expectation value of the dipole mo- ment, QD, and its conjugated moment, PD, in the reactions 20O+20Mg→40Ca at an energy of 1.6 MeV/nucleon in the center of mass. FIG. 13: GDR γ-ray spectrum calculated in the 20O+20Mg→40Ca reaction at an energy of 1.6 MeV/nucleon in the center of mass. (thick lines in Fig. 14-b) do not show any significant oscil- lation of these moments. Evolutions of Q0(t) and Q2(t) at 1.6 MeV/nucleon in the center of mass are also plot- ted (thin lines in Fig. 14-b). They do not exhibit any significant oscillation neither. Therefore, the vibrations observed in Fig. 14-a are not attributed to a difference in the collision energy but to the mass asymmetry in the entrance channel. The monopole and quadrupole oscillations modify the properties of the dipole mode in a time dependent way [8, 20]. Let us consider a harmonic oscillator for the dipole motion with a time dependent rigidity constant. This is a way to simulate the non linearities of TDHF. Indeed, the observed oscillation of the density modifies the restoring force between protons and neutrons. This is due to the fact that the density enters in the mean field potential of the TDHF equation (Eq. 1). This restor- ing force is lower along the deformation axis of a pro- lately deformed nucleus than in the perpendicular axis. Thus, variations of the density profile in the TDHF equa- tion can be modeled by a corresponding variation of the rigidity constant k(t). In such a model, the evolution of the dipole moment is given by the differential equa- tion Q̈D(t) + (k(t)/µ)QD(t) = 0 where µ = the reduced mass of the system. We note ω0 the av- erage pulsation related to the rigidity constant given by k(t)/µ = ω20(1 + η cosωt), where ω is the pulsation of the density oscillation deduced from Fig. 14-a and η is a dimensionless constant which quantifies the coupling be- tween the GDR and the other collective mode associated to the density vibration. We thus have Q̈D(t) + ω 0 [1 + η cosωt]QD(t) = 0. (16) This equation is the so called Mathieu’s equation [10]. It is interesting to show how we can get this equation from a more microscopic equation like the TDHF one (Eq. 1) in a one dimensional framework. Following the way of ref. [54], the Wigner transform of Eq. 1 for a local self consistent potential V is V f (17) where f(x, p, t) = ds exp(−ip.s/~) ρ(x+ s , x− s , t) is the Wigner transform of the density matrix ρ(x1, x2, t) = 〈x1|ρ̂(t)|x2〉. The upper indices on the derivative opera- tors in Eq. 17 stand for the function on which the oper- ator acts. We have of course f = fp + fn where fp and fn are the Wigner transforms of the proton and neutron density matrices respectively. We now apply the Wigner Function Moment (WFM) method to get a closed system of dynamical equations for the dipole and its conjugated moments. We calculate the integrals on the phase space of Eq. 17 with the weights xτ on the one hand, and pτ on the other hand (τ=1 for protons and −1 for neutrons). The distance D between proton and neutron centers of mass can be written as dx dp x (fp − fn) and we get (fp − fn) dx dp x sin V (fp − fn) where the time dependence has been omitted for simplic- ity. The right hand side term is the integral of multiple p-derivatives of f so it vanishes because fp, fn and all their p−derivatives vanish for |p| → ∞. With P being the relative momentum between protons and neutrons dp dx p (fp − fn) we get . (18) We now calculate the integral of Eq. 17 with the weight pτ . Noting the matter density n(x, t) = dp f(x, p, t) and the kinetic energy density A(x, t) = dp p2 f(x, p, t) we have (Ap −An) = − (np − nn) . Using A = 0 for |x| → ∞ we have Ṗ = − (np − nn) . (19) Eqs. 18 and 19 are the system of dynamical equations of motion we were looking for. It is important to stress that this system of equations is obtained without approx- imation for a local potential. To go further, we need an explicit form of the potential. If we consider for instance a harmonic oscillator V = kx2/2, we obtain the dipole moment evolution equation: mD̈ = −k D with the solu- tion D = D0 cosω0 t, where ω0 = If a breathing mode occurs at a pulsation ω, then the density n(x, t) oscillates with the pulsation ω: n(x, t) = n0(x) [1 + λ(x) cosωt]. Since the potential is self consis- tent, it also presents oscillations which are a function of cosωt: V (x, t) ≡ V (x, cosωt). We assume for this poten- tial the separable form V (x, t) = V0(x) (1 + F [cosωt]), where V0(x) is the potential when no breathing mode is excited. Using a harmonic picture for V0, that is, V0(x) = 2, we get from Eqs. 18 and 19 the equa- tion for the dipole moment QD = Q̈D(t) + ω 0 (1 + F [cosωt])QD(t) = 0. (20) We finally see that the Mathieu’s equation (Eq. 16) ap- pears to be an approximation of Eq. 20 where only the linear part of the function F(ξ) ≃ ηξ is conserved. We have solved the Mathieu’s equation numerically with a set of parameters suitable for our problem. The pulsation of the density oscillation is extracted from Fig. 14-a and we get ω ≃ 7.5 MeV/~. For the pulsation of the GDR we choose the main peak at ωGDR ≃ 7.7 MeV/~ (see Fig. 12). It is related to the pulsation ω0 by the relation ω0 = rωGDR. The constants r and η are tuned to reproduce approximatively the TDHF results period. The parameter r is expected to be close to 1 but not exactly 1 because of the presence of the oscillating term which may slightly change the mean value of the dipole pulsation. The solution of the Mathieu’s equation oscillates with a time-dependent period which reproduces the TDHF case quite well with r ≃ 1.1 and η ≃ 0.5 (see Fig. 15). FIG. 14: Evolution with time of the monopole (Q0, solid line) and quadrupole (Q2, dashed line) moments in the reactions 20O+20Mg→40Ca at 1.6 MeV/nucleon (a) and for 12Be+28S at 1 MeV/nucleon (thick lines) and 1.6 MeV/nucleon (thin lines) (b). Both energies are in the center of mass. FIG. 15: Time evolution of the GDR period calculated for the reaction 20O+20Mg→ 40Ca at an energy of 1.6 MeV/nucleon in the center of mass (solid line) and its modelization by the Mathieu’s equation (dashed line). FIG. 16: Evolution of the expectation value of the dipole moment, QD, and its conjugated moment, PD, in the case of the N/Z asymmetric reaction 40Ca+100Mo at a center of mass energy of 0.83 MeV/nucleon. In a recent paper [19], following the formalism devel- oped in a study of non linear vibrations [17], we related η to a matrix element of the residual interaction coupling collective states. As a consequence, the excitation of collective modes such as the quadrupole and monopole vibrations is cou- pled to the preequilibrium GDR. Such vibrations occur only in the mass symmetric reaction we studied. The ef- fects of this coupling are a reduction of the GDR energy (estimated around 10 per cent in this case) and an addi- tional spreading of the resonance line shape due to the modulation of the dipole frequency. E. comparison with experiments As a test case, we have performed TDHF calcula- tions of the reactions studied by Flibotte et al. [22]. In this paper, two systems have been investigated: an N/Z asymmetric one (40Ca+100Mo) and an N/Z quasi- symmetric one (36S+104Pd) at a center of mass energy of 0.83 MeV/nucleon. These systems have been cho- sen because they lead to the same composite system (140Sm). The corresponding dipole evolutions obtained from TDHF are plotted in Fig. 16 for the N/Z asymmet- ric reaction and in Fig. 17 for the N/Z quasi-symmetric one. A dipole oscillation is observed in both reactions but with a stronger amplitude in the N/Z asymmetric The preequilibrium GDR γ-ray spectra for those reac- tions are calculated using Eq. 4 and plotted in Fig. 18-a. FIG. 17: Evolution of the expectation value of the dipole moment, QD, and its conjugated moment, PD, in the case of the N/Z quasi-symmetric reaction 36S+104Pd at a center of mass energy of 0.83 MeV/nucleon. The area under the peak associated to the N/Z asymmet- ric reaction (solid line) is considerably larger than the one under the N/Z quasi-symmetric one (dashed line). To estimate the importance of the preequilibrium γ- ray emission with respect to the statistical decay and its role on the fusion process, we have calculated the spec- trum associated to the first chance statistical γ-ray decay. It is obtained from the γ-ray emission probability in all directions per energy unit assuming an equilibrated CN [9, 55, 56]. Its expression is E4γ e E2γ − E2GDR where m is the nucleon mass, ΓGDR and EGDR are the width and the energy of the statistical GDR respectively, and T is the temperature of the equilibrated CN. At first order, the energy of the GDR does not depend on the temperature [14]. We use the values EGDR = 15 MeV and ΓGDR = 7 MeV. Following the same method as the one employed in Ref. [9], we approximate the CN width ΓCN with the total neutron width ΓCN ≃ Γn = 2mr20A T 2e− T (22) where Bn = 8.5 MeV is the neutron binding energy and r0 = 1.2 fm. The temperature T is calculated from the equation where a ≃ 1/10 MeV−1 is the level density parameter and E∗ = 71 MeV is the excitation energy. The resulting spectrum is plotted in Fig. 18-a (dotted line). We note that the N/Z asymmetric preequilibrium spec- trum is comparable in intensity to the first step statistical one. This fact has already been pointed out by Baran et al. [9] who got a similar spectrum for the N/Z asymmet- ric reaction with a semiclassical approach. Another important conclusion which can be drawn from Fig. 18-a is the lowering of the GDR γ-ray energy for the non statistical part as compared to the statistical one which is attributed to the deformation of the nucleus (see sec. II B 2). This phenomenon is also reported by Baran et al. [9]. In fact we get from Fig. 18-a a position of the peak of about 7.5 MeV for the preequilibrium GDR while Baran et al. obtained ∼9 MeV. On the experimen- tal side, the γ-ray spectra are dominated by a statistical background decreasing exponentially. In addition to this background, the GDR creates a bump located around the GDR energy (Fig. 1 of Ref. [22]). To get rid of the statistical background, the authors of [22] linearized the γ-ray spectra by dividing them by a theoretical statistical background. The resulting spectra are plotted in Fig. 2 of Ref. [22]. This procedure is used by the authors to determine the preequilibrium to statistical ratio for the GDR component. However it cannot be used to deter- mine the positions in energy of the peaks because the division by an exponential background induces a shift in energy which is different for both contributions (statisti- cal and preequilibrium) if they are not centered around the same energy, as expected from Fig. 18-a. We modified the procedure as follows. First, we assume that no preequilibrium γ-ray is emitted in the N/Z quasi- symmetric reaction 36S+104Pd. We then subtract the total γ-ray spectrum associated to the quasi-symmetric reaction from the N/Z asymmetric one. These two spec- tra are plotted in Fig. 1 of Ref. [22]. The result of this subtraction is the preequilibrium component of the GDR in the reaction 40Ca+100Mo, and is plotted in Fig. 18-b. The error bars are both statistical and systematic due to the graphical extraction of the data. Below 5 MeV the systematic error is to high to get relevant data. Focusing on the energy position of the preequilibrium component, we note a good agreement between TDHF predictions and experimental data. To conclude, we extracted from existing data, for the first time, an experimental observation of the lowering of the preequilibrium GDR predicted by our TDHF cal- culations. This analysis shows that the preequilibrium GDR is, indeed, a powerful experimental tool to study the fusion path. Another application of N/Z asymmetric fusion reactions is proposed in the next section. FIG. 18: a) preequilibrium GDR γ-ray spectrum calcu- lated in the reactions 40Ca+100Mo (solid line) and 36S+104Pd (dashed line). The dotted line represents the first chance sta- tistical γ-ray decay spectrum. b) Experimental data resulting from the subtraction of the γ-ray spectra obtained by Flibotte et al. [22] in the reactions 40Ca+100Mo and 36S+104Pd. III. FUSION/EVAPORATION CROSS SECTIONS OF HEAVY NUCLEI As mentioned in [9], the emission of a preequilibrium GDR γ-ray decreases the excitation energy hence the ini- tial temperature of the nucleus reaching the statistical phase. The emission of preequilibrium particles, which can be controlled in our example by the N/Z asymmetry, is thus a new interesting cooling mechanism for the for- mation of Heavy and Super Heavy Elements. For such nuclei, the statistical fission considerably dominates the neutron emission and the survival probability of the CN becomes very small. SHE must be populated at low excitation energy. Firstly, because the smaller the excitation energy, the smaller the fission probability. Secondly, because the shell corrections decrease with excitation energy [57]. These corrections are responsible for the stability of the transfermiums nuclei (Z > 100) in their ground state. The quantum stabilization decreases quite rapidly with excitation energy until the fission barrier vanishes. Those two reasons are strong motivations to study the cooling mechanisms involved in the preequilibrium phase of the CN formation. In the following, we expose one cooling mechanism re- sponsible for the predicted enhancement of the survival probability in the case of a N/Z asymmetric reaction. As an illustration, we treat only the γ-emission part of the preequilibrium GDR decay. Although it may play an im- portant role, we do not treat the preequilibrium neutron emission for two reasons: • Only the direct neutron decay of giant resonances can be assessed in TDHF. Then, we would be able to describe only a small part of this neutron emis- sion, the other parts being the sequential and sta- tistical decays. Missing the sequential decay would be a strong limitation of the description. • We would need not only the number of emitted neu- trons, but also their energy. Consequently, huge spatial grid would have to be used in order to per- form a spatial Fourier transform of the single par- ticle wave functions, which is out of range of three dimensional TDHF codes because of computational limitations. Let us define PE∗ init. (E∗) the survival probability at an excitation energy E∗ of a CN which started its statistical decay at the energy E∗init. We also note P surv and P the final survival probabilities of the CN formed by N/Z symmetric and asymmetric reactions respectively. Fig. 19-a illustrates schematically the evolution of the survival probability (x-axis) when the excitation energy decreases (y-axis) in a case of an N/Z symmetry in the entrance channel. In this case, no γ-ray emission is ex- pected in the preequilibrium phase and the initial ex- citation energy is always maximum E∗init = E 0 , where E∗0 = Q + Ecm is the excitation energy when no pree- quilibrium particles are emitted, Ecm is the center of mass energy and Q = (M1 +M2 −MCN )c2. During the statistical decay, the excitation energy decreases mainly through neutron emission, but at the same time the survival probability of the compound nucleus decreases too. For instance, when the excitation energy reaches E∗1 = E 0−EGDR, the survival probability P1 = PE∗0 (E at this energy might be small. At the end of the decay, when the excitation energy is zero, the survival probabil- ity becomes PSsurv = PE∗0 (0) = P1PE Fig. 19-b shows the same for an N/Z asymmetric re- action. In this last case, the nucleus can emit a pree- quilibrium GDR γ-ray with a probability Pγ . The nu- clei which emit such a γ-ray begin the statistical decay at a lower energy Einit = E 1 , whereas those which did not emit a γ-ray still starts their decay at Einit = E The probability for the latter case is 1 − Pγ . The sur- vival probability at the end of the decay then reads PAsurv = [(1− Pγ)P1 + Pγ ]PE∗1 (0). The ratio of the sur- FIG. 19: Schematic representation of the CN population dur- ing the statistical decay in the case of an N/Z symmetric collision (a) and an N/Z asymmetric reaction (b). vival probabilities between the N/Z symmetric and asym- metric cases is PAsurv PSsurv = 1 + (1− P1) . (24) We now use a simple model to get an estimate of this quantity. It is clear that, to get a quantitative predictions of survival probabilities, the studied mechanism has to be included in more elaborated statistical models, which is beyond the scope of this paper. The probability Pγ can be calculated by integrating Eq. 4 over the energy range. This can be done for example with a TDHF cal- culation or using the classical electrodynamic formulae from Ref. [49]. Following these formulae, we approximate the probability to emit a preequilibrium GDR γ-ray per interval of energy by 2e2QD(0) 3π(~c)3 E21 + (E − E1)2 + ΓGDR (E + E1)2 + where E1 = − ΓGDR2 is the “shifted” energy of the damped harmonic motion and ΓGDR is the damping width of the preequilibrium GDR. The initial value of the dipole moment, QD(0), can be estimated from Eq. 15 at the touching point and neglecting the polarization of the collision partners [8]. We get QD(0) ≃ R1 +R2 (Z1N2 − Z2N1) where Ri is the radius of nucleus i. To determine PE∗0 (E 1 ), we need to solve a system of six equations: Eqs. 22, 23 and = −Γn(t) (Bn + T (t)) (25) = −Γf(t) P (t) (26) Γf (t) = ~ω0ωs Bf (t) T (t) (27) Bf (t) ≡ Bf [E∗(t)] = Bf (0)e− Ed (28) Eq. 25 gives the evolution of the excitation energy, assum- ing as in [9] that the CN width can be identified to the neutron width. This implies that we neglect the statisti- cal gamma emission. This choice is justified by the fact that the statistical neutron emission is much more prob- able than the gamma emission in the excitation energy domain of interest where the fission dominates, which is above the neutron emission threshold Bn. Eq. 26 gives the evolution of the survival probability against fission P . Eq. 27 gives the evolution of the fission width. The parameters ω0 and ωS are the oscillator frequencies of the two parabolas approximating the potential V (x) in the first minimum and at the saddle point respectively. The variable x is related to the distance between the mass centers of the nascent fission fragments (see [58]) and β = 5× 1021s−1 is the reduced friction. Eq. 28 gives the evolution of the fission barrier Bf . For SHE, this barrier has only a quantum nature and vanishes at high excita- tion energy. Ed ≃ 20 MeV is the shell damping energy [58]. We consider that a CN with an excitation energy between E∗1 and E 0 decays only by fission or neutron emission. We take here the example of the reac- tion 124Xe+141Xe→265Hs∗ at the fusion barrier (Ecm = Bfus), that is, an excitation energy E 0 = 54 MeV. With an energy and a width of the GDR of 13 MeV and 4 MeV respectively, the preequilibrium γ-ray emission probability is Pγ ≃ 0.05. For the statistical decay we take Bf [E ∗ = 0] ≃ 8.5 MeV, Bn = 6.5 MeV and ω0 ≃ ωS ≃ 1 MeV/~. We also get a survival probability PE∗0 (E 1 ) ≃ 0.01 which is small as compared to Pγ . Following Eq. 24, the enhancement of the total survival probability due solely to the N/Z asymmetry in the entrance channel becomes PAsurv/P surv ∼ 6. To conclude, we see that such an effect may be useful for the formation of Heavy and Super Heavy Elements. Indeed, based on our conclusions, very asymmetric N/Z collisions induced by radioactive ion beams that are com- ing online in several laboratories, should allow the syn- thesis SHE with a larger cross sections than are obtain- able with beams of stable isotopes. IV. CONCLUSION In this paper we have performed TDHF calculations to study in some details the properties of the preequilibrium GDR that can be excited before the formation of a fully equilibrated CN. We have shown that this probe can be used to better understand the early stage of the fusion path, and more precisely the charge equilibration. We have clarified the role of the N/Z and/or mass asymme- tries on the GDR excitation. The energy of the preequi- librium GDR is expected to decrease with excitation en- ergy, an effect attributed to a strong prolate shape asso- ciated to the fused system. We presented the first experi- mental indication of this shift in energy. The calculations for an N/Z asymmetric collisions at non zero impact pa- rameters have been performed and revealed couplings be- tween the dipole oscillations and the CN rotation. Other couplings between vibrational modes for mass symmetric reactions have also been studied. Finally we suggest that the use of N/Z asymmetric fu- sion reactions is a good choice to synthesize Heavy and Super Heavy Elements. In that case, the preequilibrium GDR γ-ray emission cooling mechanism might be well suited to reach the statistical phase with a low excita- tion energy yielding a larger survival probability against fission. 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0704.0497

Penalization approach for mixed hyperbolic systems with constant coefficients satisfying a Uniform Lopatinski Condition

Penalization approach for mixed hyperbolic systems with constant coefficients satisfying a Uniform Lopatinski Condition. B. Fornet October 24, 2018 Abstract In this paper, we describe a new, systematic and explicit way of approximating solutions of mixed hyperbolic systems with constant coefficients satisfying a Uniform Lopatinski Condition via different Pe- nalization approaches. LATP, Université de Provence,39 rue Joliot-Curie, 13453 Marseille cedex 13, France. LMRS, Université de Rouen,Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France. http://arxiv.org/abs/0704.0497v1 1 Introduction. In this paper, we describe a new, systematic and explicit way of approximat- ing solutions of mixed hyperbolic systems with constant coefficients satisfy- ing a Uniform Lopatinski Condition via different Penalization approaches. In applied Mathematics like, for instance, in the study of fluids dynamics, the method of penalization is used to treat boundary conditions in the case of complex geometries. By replacing the boundary condition by a singular perturbation of the PDE extended to a larger domain, this method allows the construction of an approximate, often more easily computable, solution. We consider mixed boundary value problems for hyperbolic systems: Aj∂j , on {xd ≥ 0}, with boundary conditions on {xd = 0}. The n× n real valued matrices Aj are assumed constant. Of course, we assume the coefficients to be constant as a first approach, aiming to generalize the results obtained here in future works. We assume that the boundary {xd = 0} is noncharacteristic, which means that detAd 6= 0. We denote by y := (x1, . . . , xd−1) and x := xd. The problem writes: (1.1) Hu = f, {x > 0}, Γu|x=0 = Γg, u|t<0 = 0 , where the unknown u(t, x) ∈ Rn, Γ : Rn → Rp is linear and such that rg Γ = p; which implies that Γ can be viewed as a p×n real valued constant matrix. Let us fix T > 0 once and for all for this paper. Let Ω+ denotes the set [0, T ] × Rd+ and ΥT denote the set [0, T ] × R d−1. f is a function in Hk(Ω+ ), g is a function in Hk(ΥT ), where k ≥ 3 or k = ∞, such that: f |t<0 = 0 and g|t<0 = 0. We make moreover the following Hyperbolicity assumption on H : Assumption 1.1. For all (η, ξ) ∈ Rd−1 × R− {0}, the eigenvalues of ηjAj + ξAd are real, semi-simple and of constant multiplicity. Let us introduce now the frequency variable ζ := (γ, τ, η), where iτ + γ, with γ ≥ 0, and τ ∈ R stands for the frequency variable dual to t and η = (η1, . . . , ηd−1) where ηj ∈ R is the frequency variable dual to xj. We note: A(ζ) := − (Ad) (iτ + γ)Id+ iηjAj Denote by M a N ×N, complex valued, matrix; E−(M)[resp E+(M)] is the linear subspace generated by the generalized eigenvectors associated to the eigenvalues of M with negative [resp positive] real part. If F and G denote two linear subspaces of CN such that dimF+dimG = N, det(F,G) denotes the determinant obtained by taking orthonormal bases in each space. Up to the sign, the result is independent of the choice of the bases. We shall now explicit the Uniform Lopatinski Condition assumption: Assumption 1.2. (H,Γ) satisfies the Uniform Lopatinski Condition i.e for all ζ such that γ > 0, there holds: (1.2) |det(E−(A), ker Γ)| ≥ C > 0. The mixed hyperbolic system (1.1) has a unique solution in Hk(Ω+ ), and, since H is hyperbolic with constant multiplicity, for all γ positive, the eigenvalues of A stay away from the imaginary axis. More- over, as emphasized for instance by Chazarain and Piriou in [3] and Mé- tivier in [8], there is a continuous extension of the linear subspace E−(A) to {γ = 0, (τ, η) 6= 0 } that we will denote by Ẽ−(A). Ẽ+(A) extends as well continuously to {γ = 0, (τ, η) 6= 0 } and we will denote Ẽ+(A) this extension. Moreover, there holds: Ẽ−(A) Ẽ+(A) = C We can refer the reader to [3], [6], [7], or [8] for detailed estimates concern- ing mixed hyperbolic problems satisfying a Uniform Lopatinski Condition. Moreover, we can refer to [10] for the proof of the continuous extension of the linear subspaces mentioned above in the hyperbolic-parabolic framework. Remark 1.3. As a consequence of the uniform Lopatinski condition, there holds, for all ζ 6= 0 : rg Γ = p = dim Ẽ−(A(ζ)). 1.1 A Kreiss Symmetrizer Approach. We will now describe a penalization method involving a Kreiss Symmetrizer and a matrix constructed by Rauch in [12], in the construction of our singular perturbation. Note well that we have some freedom in both the choice of the Kreiss Symmetrizer and of Rauch’s matrix. Let us denote respectively by û, f̂ , and ĝ the tangential Fourier-Laplace transform of u, f, and g. Since the Uniform Lopatinski Condition is holding for the mixed hyperbolic system (1.1), there is, see [9] a Kreiss symmetrizer S for the problem: (1.3) ∂xû = Aû+ f̂ , {x > 0}, Γû|x=0 = Γĝ, That is to say there exists a matrix S(ζ), homogeneous of order zero in ζ, C∞ in R+ × Rd − {0 } and there are λ > 0, δ > 0 and C1 such that: • S is hermitian symmetric. • ℜ (SA) ≥ λId. • S ≥ δId− C1Γ An algebraic result proved by Rauch in [12] can be reformulated as follow, and a proof is recalled in section 2.2: Lemma 1.4. There is a hermitian symmetric, uniformly definite positive, N ×N matrix B such that: ker Γ = E+((S) −1B). Moreover B depends smoothly of ζ. Remark 1.5. This result is proved by constructing explicit matrices satisfy- ing the desired properties. Thus, it is not merely an existence result and we can use the explicitly known matrix B in our construction of a penalization operator. Let us denote by R := B 2 and SR := R −1SR−1. We will denote by P− the projector on E−(SR) parallel to E+(SR) and by P + the projector on E+(SR) parallel to E−(SR); P − and P+ denoting the associated Fourier multiplier. We recall that, denoting by F the tangential Fourier transform, the Fourier multiplier P−(∂t, ∂y, γ) [resp P +(∂t, ∂y, γ)] is then defined, for all w ∈ Hk(Rd+1), and γ > 0, by: −(∂t, ∂y, γ)w = P−(ζ)F(w), [resp +(∂t, ∂y, γ)w = P+(ζ)F(w)], in the future we will rather write: ±(∂t, ∂y, γ)w = P±(ζ)F(w). We fix, once and for all, γ > 0 big enough. Let us consider then the solution uε of the well-posed Cauchy problem on the whole space (1.4): (1.4) Huε + Muε1x<0 = f1x>0 + θ1x<0, {x ∈ R}, uε|t<0 = 0, where M := −eγtAdS −1RP−Re−γt, θ := −eγtAdS −1RP−Γg̃, and S(∂t, ∂y) [resp R(∂t, ∂y)] denotes the Fourier multiplier associated to S(ζ) [resp R(ζ)]. Let us define g̃ by: g̃ := e−x In what follows, ĝ will denote the Fourier-Laplace transform of g̃. Let us denote by ũ := u−1x<0 + u1x≥0 = u 1x≤0 + u1x>0. u denotes the solution of (1.1), and thus belongs to Hk(Ω+ ). u− is a function belonging to Hk(Ω− ) and such that u−|x=0 = u|x=0. More precisely, u − can be computed by: eγtF−1 R−1(v̂− + P−Γĝ) , where v̂− is the solution of the problem: SR∂xv̂ − − P+SRARv̂ − = P+SRARP −Γĝ, {x < 0}, v̂−|x=0 = P +Rû|x=0, and û denotes the Fourier-Laplace transform of the solution u of (1.1). Theorem 1.6. For all k ≥ 3, if f ∈ Hk(Ω+ ) and g ∈ Hk(ΥT ), then there holds: ‖uε − u−‖ Hk−3(Ω + ‖uε − u‖ Hk−3(Ω = O(ε), where uε denotes the solution of the Cauchy problem (1.4) and u denotes the solution of the mixed hyperbolic problem (1.1). If g = 0 then: ‖uε − u−‖ + ‖uε − u‖ = O(ε). Of course, since uε is defined for all {x ∈ R}, its limit as ε → 0+, ũ is can be viewed as an ”extension” of u on the fictive domain {x < 0}. The ”extension” resulting from our method of penalization gives a continuous ũ across {x = 0}, while the method used in [2] gave simply: ũ|x<0 = 0. We have the following Corollaries: Corollary 1.7. Assume for example that f ∈ H∞(Ω+ ) and g ∈ H∞(ΥT ) then ‖uε − u‖ Hs(Ω+ = O(ε); ∀s > 0. Corollary 1.8. If f belongs to L2(Ω+ ) and g = 0 then: ‖uε − ũ‖L2(ΩT ) = 0. One of the interest of this first approach lies in the rate of convergence of uε towards u. Indeed, in general, a boundary layer will form near the boundary in this kind of singular perturbation problem. For example in the paper by Bardos and Rauch [2], as confirmed by Droniou [4], a boundary layer forms. It is also the case in [11], as analyzed in our Appendix. There are also boundary layers phenomena in the parabolic context: see the ap- proach proposed by Angot, Bruneau and Fabrie [1] for instance. However, surprisingly, and like in the penalization method proposed by Fornet and Guès in [5], our method allows the convergence to occur without formation of any boundary layer on the boundary. As a result, this leads to the kind of sharp stability estimate given in Theorem 1.6. These results concern the case where f and g are sufficiently regular. The reason is that we construct an approximate solution. In the case of g only in L2(ΥT ), such a simple treatment does not work. However, let δ > 0 be given. If we approximate f and g by smooth functions fν ∈ H ) and gν ∈ H ∞(ΥT ) such that ‖f − fν‖L2(Ω+ < δ and ‖g − gν‖L2(ΥT ) < δ, by the uniform Lopatinski condition, we get: ‖uν − u‖L2(Ω+ < Cν, where uν is the solution of the mixed hyperbolic problem (1.1) with data fν and gν . We can now apply Corollary 1.8 to uν , and obtain by penalization a sequence uεν in L 2(ΩT ) such that: limε→0+ u ν = uν in L ). Finally, by choosing, ε sufficiently small, we get ‖u − uεν‖L2(Ω+ < 2Cδ. By choosing ε and ν as functions of δ, and noting u(δ) = u , we have: (1.5) lim ‖u(δ) − u‖L2(Ω+ 1.2 A second Approach. In the first approach we have just introduced, it is necessary to compute a Kreiss’s Symmetrizer and a Rauch’s matrix. In view of future numerical applications, we will now introduce another method preventing the compu- tation of these matrices. The price to pay is that we need the preliminary computation of v, which is by definition the solution of the Cauchy problem on the free space: (1.6) Hv = f, (t, y, x) ∈ ΩT , v|t<0 = 0 ∀(y, x) ∈ R Let us denote P−(ζ) the spectral projector on Ẽ−(A(ζ)) parallel to Ẽ+(A(ζ)), andP +(ζ) the spectral projector on Ẽ+(A(ζ)) parallel to Ẽ−(A(ζ)). Let us introduce P±(∂t, ∂y, γ), the Fourier multiplier associated to P ±(ζ). Let us denote by Π the projector on Ẽ−(A(ζ)) parallel to KerΓ, which has a sense because of the Uniform Lopatinski Condition and denote Π the associated Fourier multiplier. We define then h̃ by: h̃ := e−x −(e−γtv|x=0) +Πe −γt(g − v|x=0) where g denotes the function involved in the boundary condition of the mixed hyperbolic problem (1.1). Now, let us consider the following singularly perturbed Cauchy problem on the whole space: (1.7) Huε + −e−γtuε1x<0 = f1x>0 + γth̃1x<0, uε|t<0 = 0 . Let us denote by ũ := u−1x<0 + u1x≥0 = u 1x≤0 + u1x>0. u denotes the solution of (1.1) thus belonging toHk(Ω+ ) and u− is a function belonging to Hk(Ω− ) and such that u−|x=0 = u|x=0. More precisely, u − can be computed by: eγtF−1(F(h̃)+v̂−), where v̂− is the solution of the problem: +v̂−)−A(P+v̂−) = 0, {x < 0}, +v̂−|x=0 = P +û|x=0. and û denotes the Fourier-Laplace transform of the solution u of (1.1). The problem (1.7) is well-posed and, for all ε > 0, there exists a unique uε ∈ Hk(ΩT ) solution. We will fix γ adequately big beforehand. We observe then the following result: Theorem 1.9. For all k ≥ 3, if f ∈ Hk(Ω+ ) and g ∈ Hk(ΥT ), then there holds: ‖uε − u−‖ Hk−3(Ω + ‖uε − u‖ Hk−3(Ω = O(ε), where uε denotes the solution of the Cauchy problem (1.7) and u denotes the solution of the mixed hyperbolic problem (1.1). The singular perturbation involved in the definition of uε does not de- pend either of Kreiss’s Symmetrizer or Rauch’s matrix. As a result, for this method of penalization far less computations are necessary in order to obtain our singular perturbation. Note well that the proof of the energy estimates in Theorem 1.9 is completely different from the proof of the en- ergy estimates in Theorem 1.6. Indeed, for our first approach our singularly perturbed problem was treated as a Cauchy problem, contrary to our second approach where it was interpreted as a transmission problem. Corollary 1.10. Assume for example that f ∈ H∞(Ω+ ) and g ∈ H∞(ΥT ) then ‖uε − u‖ = O(ε); ∀s > 0. Of course, we see that the same problem of regularity arises in Theorem 1.9 and Theorem 1.6. However, by a simple density argument, we can also prove here the exact analogous of (1.5). Remark 1.11. In the case where f = 0, then the solution v of (1.6) is v = 0 and thus, the perturbed cauchy problem (1.7) rewrites: Huε + −e−γtuε1x<0 = γte−x Πe−γtg 1x<0, {x ∈ R}, uε|t<0 = 0 . 2 Underlying approach leading to the proof of Theorem 1.6. 2.1 Some preliminaries. Since the Uniform Lopatinski Condition holds, there is S, homogeneous of order zero in ζ, and such that there are λ > 0, δ > 0 and C1 and there holds: • S is hermitian symmetric. • ℜ (SA) ≥ λId. • S ≥ δId− C1Γ S is then called a Kreiss Symmetrizer for the problem: (2.1) ∂xû = Aû+ f̂ , {x > 0}, Γû|x=0 = Γĝ, where f̂ and ĝ denotes respectively the Fourier-Laplace transforms of f and g̃; and û denotes the Fourier-Laplace transform of the solution u of the well- posed mixed hyperbolic problem (1.1). û is also solution, for all fixed ζ 6= 0 of the following equation: (2.2) S∂xû = SAû+ S(Ad) −1f̂ , {x > 0}, Γû|x=0 = Γĝ, Remark 2.1. Following our current assumptions, Γ is independent of ζ 6= 0, however, more general boundary conditions, of the form: Γ(ζ)û|x=0 = Γ(ζ)ĝ, can be treated. It would imply taking as boundary condition for (1.1): Γγu|x=0 = Γγg, with for γ big enough, Γγ := Γ(∂t, ∂y)e where, Γ(∂t, ∂y) denotes the Fourier multiplier associated to Γ(ζ), that is to say is defined by: F(Γ(∂t, ∂y)u) = Γ(ζ)F(u). Referring for example to [3] and [7], Kreiss has proved that the exis- tence of a Kreiss symmetrizer for the symbolic equation is sufficient to prove the well-posedness of the associated pseudodifferential equation (here (1.1)). Indeed, multiplying by û and integrating by parts the equation: S∂xû = SAû+ S(Ad) leads to the desired a priori estimates. For all ζ 6= 0, S(ζ) is hermitian symmetric and definite positive on ker Γ. Let us sum up the properties crucial in the proof of the well-posedness of our problem: Proposition 2.2. For all ζ = (τ, γ, η) such that τ2 + γ2 + j=1 η j = 1, there holds: • S(ζ) is hermitian symmetric. • ℜ (SA) (ζ) := 1 (SA+ (SA)∗)(ζ) is positive definite. • −S(ζ) is definite negative on ker Γ and ker Γ is of same dimension as the number of negative eigenvalues in −S(ζ). Note that, by homogeneity of S, it is equivalent for the properties in Propo- sition 2.2 to hold for |ζ| = 1 or for |ζ| > 0. As a consequence of the first point and third point of Proposition 2.2, the Lemma 1.4 applies and gives a matrix B such that: ker Γ = E+(S −1B). In the sequel, such a matrix B is fixed once for all. The following chapter contains a proof of Lemma 1.4 assorted of a detailed construction of B. 2.2 Detailed proof of Lemma 1.4: Construction of the ma- trices B solving Lemma 1.4. As we will emphasize in next chapter, Lemma 1.4 is a crucial feature in our first method of Penalization. The aim of this chapter is to give a more com- plete proof rather than simply recalling Rauch’s result and, in the process, to precise how the matrices B solving Lemma 1.4 are constructed. For all ζ 6= 0, S(ζ) is hermitian symmetric, uniformly definite positive on Ẽ+(A(ζ)), and uniformly definite negative on Ẽ−(A(ζ)); as a consequence, S(ζ) keeps exactly p positive eigenvalues and N − p negative eigenvalues for all ζ 6= 0. Basically, knowing that S is uniformly definite positive on ker Γ; we search to express ker Γ in a way involving S. Consider q ∈ ker Γ, since, for all ζ 6= 0, E−(S(ζ)) E+(S(ζ)) = C N , we can split q in: q := q+ + q− with q+ ∈ E+(S(ζ)) and q − ∈ E−(S(ζ)). Since dim kerΓ = dimE+(S(ζ)) = p, these two linear subspaces are in bijec- tion. Let us give the two main ideas behind this proof: one idea is to detail the bijection between q ∈ kerΓ and q+ ∈ E+(S(ζ)) as it satisfies some con- straints, the other is to come down to the model case where the eigenvalues of S are either 1 or −1. Let us denote: S̃−1 = −IdN−p 0 0 Idp In a first step, we will prove the following result: Proposition 2.3. If we assume that V is a linear subspace of CN of dimen- sion p, and that there is C > 0 such that, for all q ∈ V, there holds: 〈S̃−1q, q〉 ≥ C〈q, q〉, then the two following equivalent properties hold: • There is a hermitian symmetric, positive definite matrix R̃, such that: [q ∈ V] ⇔ q ∈ E+(R̃S̃R̃) which is equivalent to: V = E+(R̃ • There is a hermitian symmetric, positive matrix R̃, such that: [q ∈ V] ⇔ R̃q ∈ E+(R̃S̃ −1R̃) which is equivalent to: V = E+(S̃ −1R̃2). Moreover, we can link the two properties by taking: R̃2 = S̃R̃ Proof. In this proof, we will show how to construct some matrices R̃ satisfying the required properties. There is a (N − p)× p matrix ℵ of rank N − p such that ‖ℵ‖ ≤ 1 and: V = {q ∈ CN , q− = ℵq+}, where q+[resp q−] denotes the projector on E+((S̃) −1) [resp E−((S̃) −1)]parallel to E−((S̃) −1) [resp E+((S̃) −1)]. Indeed, dimV = p = dimE+((S̃) −1), and N = E−((S̃) E+((S̃) −1). Moreover, there ic C > 0 such that, for all q ∈ V, there holds: 〈(S̃)−1q, q〉 = −〈q−, q−〉+ 〈q+, q+〉 ≥ C〈q, q〉. and thus |q+|2 − |ℵq+|2 ≥ C|q|2, which implies that ‖ℵ‖ < 1. We will show now that, for R̃ constructed as follow: IdN−p −ℵ −ℵ∗ Idp there holds: [q ∈ V] ⇔ R̃q ∈ E+(R̃S̃ −1R̃) First,we see that the constructed R̃ is trivially hermitian symmetric and positive definite since ‖ℵ‖ < 1. First, we have: R̃S̃−1R̃ = −IdN−p +NN 0 Idp −N R̃q = q− − ℵq+ −ℵ∗q− + q+ Thus, since ‖ℵ‖ < 1, there holds: R̃q ∈ E+(R̃S̃ −1R̃) q− − ℵq+ = 0 ⇔ [q ∈ V] . We will now prove that we have: (R̃)−1E+(R̃S̃ −1R̃) = E+(S̃ −1R̃2). Since R̃S̃−1R̃ is hermitian symmetric, the linear subspace E+(R̃S̃ −1R̃) is generated by the eigenvectors of R̃S̃−1R̃ associated to positive eigenvalues. A basis of (R̃)−1E+(R̃S̃ −1R̃) is thus given by ((R̃)−1vj)j where vj denotes an eigenvector of R̃S̃−1R̃ associated to a positive eigenvalue λj . We have: R̃S̃−1R̃vj = λjvj. Let us denote wj = (R̃) −1vj, we have then: R̃S̃−1R̃2wj = λjR̃wj ⇔ S̃ −1R̃2wj = λjwj . As a result, wj is an eigenvector of S −1R̃2 associated to the eigenvalue λj hence we obtain that: (R̃)−1E+(R̃S̃ −1R̃) = E+(S̃ −1R̃2). We can also prove, the same way, that: R̃E+(R̃S̃R̃) = E+(R̃ Now, taking R̃2 = S̃R̃ we can check that: E+(S̃ −1R̃2) = E+(R̃ which concludes the proof. ✷ Lemma (??) is a Corollary of the following Proposition: Proposition 2.4. If S−1 denotes a smooth in ζ 6= 0, matrix-valued function in the space of hermitian symmetric matrices with p positive eigenvalues and N −p negative eigenvalues and ker Γ denotes a linear subspace of dimension p and there is C > 0 such that, for all q ∈ ker Γ, there holds: 〈S−1q, q〉 ≥ C〈q, q〉, then the two following equivalent properties hold: • There is a smooth in ζ 6= 0, matrix-valued function R, in the space of hermitian symmetric, positive matrices such that: [q ∈ KerΓ] ⇔ ∀ζ 6= 0, R−1(ζ)q ∈ E+(R(ζ)S(ζ)R(ζ)) which is equivalent to: ∀ζ 6= 0, KerΓ = E+(R 2(ζ)S(ζ)). • There is a smooth in ζ 6= 0, matrix-valued function R, in the space of hermitian symmetric, positive matrices such that: [q ∈ KerΓ] ⇔ ∀ζ 6= 0, R(ζ)q ∈ E+(R(ζ)S −1(ζ)R(ζ)) which is equivalent to: ∀ζ 6= 0, KerΓ = E+(S −1(ζ)R2(ζ)). Moreover, for all ζ 6= 0, these two properties can be linked by taking: (R(ζ))2 = S(ζ)(R(ζ))2S(ζ). Proof. We will show here that Proposition 2.4 can be deduced from Propo- sition 2.3. For all ζ 6= 0, S(ζ) is a hermitian symmetric matrix, moreover S depends smoothly of ζ. As a consequence S−1 is also a hermitian symmetric matrix depending smoothly of ζ, and as such, there is a nonsingular matrix V such that: S̃−1 = V ∗ Let us denote Λ the diagonalized version of S−1 with eigenvalues sorted by increasing order, then there is Z depending smoothly of ζ such that, for all ζ 6= 0, we have: Z∗(ζ) = Z−1(ζ), Λ(ζ) = Z∗(ζ) (ζ)Z(ζ). As a consequence, V depends smoothly of ζ since, for all ζ 6= 0: V (ζ) = (Λ(ζ))− 2Z(ζ), where Λ is the diagonal matrix obtained by taking the absolute value of each eigenvalue of Λ. For the sake of simplicity, let us omit the dependence in ζ. Now, for all q ∈ V −1 ker Γ, there is C > 0, such that: 〈S̃−1q, q〉 = 〈V ∗S−1V q, q〉 = 〈S−1(V q), (V q)〉 ≥ C〈(V q), (V q)〉. Moreover V is nonsingular, thus there is C ′ > 0, such that, for all q ∈ V −1 ker Γ, there holds: 〈S̃−1q, q〉 ≥ C ′〈q, q〉. Moreover dimV −1 ker Γ = p, using Proposition 2.3, for all fixed ζ 6= 0, there is a hermitian symmetric, positive definite matrix R̃(ζ), such that: V −1(ζ) ker Γ = E+((R̃(ζ)) 2S̃(ζ)) = R̃(ζ)E+(R̃(ζ)S̃(ζ)R̃)(ζ). We will now prove that we can construct R̃ depending smoothly of ζ. First there is a (N − p) × p matrix ℵ of rank N − p, depending smoothly of ζ, such that fore all ζ 6= 0 ‖ℵ(ζ)‖ ≤ 1 and: V −1(ζ) ker Γ = {q ∈ CN , q− = ℵ(ζ)q+}, where q+[resp q−] denotes the projector on E+((S̃) −1) [resp E−((S̃) −1)]parallel to E−((S̃) −1) [resp E+((S̃) −1)]. R̃ is given, for all ζ 6= 0, by: R̃(ζ) = S̃−1(ζ)R̃2(ζ)S̃−1(ζ), with R̃ given by: R̃(ζ) = IdN−p −ℵ(ζ) −ℵ∗(ζ) Idp Since S̃−1 = V ∗ V, there holds: S̃ = V ∗SV, and, as a consequence: (V R̃)−1 ker Γ = E+(R̃V ∗SV R̃). As R̃V ∗SV R̃ is hermitian symmetric, a basis of the linear subspace E+(R̃V ∗SV R̃) is given by the eigenvectors of R̃V ∗SV R̃ associated to positive eigenvalues. This leads us to consider vj = (V R̃) −1uj satisfying: R̃V ∗SV R̃vj = λjvj. We have: R̃V ∗SV R̃(V R̃)−1uj = λj(V R̃) −1uj. hence: (V R̃)R̃V ∗Suj = λjuj . Since (V R̃)R̃V ∗ = (R̃V ∗)∗(R̃V ∗) is hermitian symmetric and positive defi- nite, we can then define its square root. We define R by: (R̃V ∗)∗(R̃V ∗). Since both R̃ and V depends smoothly of ζ, so does R. Moreover, there holds: R2Suj = λjuj, which gives: ker Γ = V R̃E+(R̃V ∗SV R̃) = E+(R We have thus proved there is a smooth in ζ 6= 0, matrix-valued function R, in the space of hermitian symmetric, positive matrices such that: [q ∈ KerΓ] ⇔ ∀ζ 6= 0, R−1(ζ)q ∈ E+(R(ζ)S(ζ)R(ζ)) which is equivalent to: ∀ζ 6= 0, KerΓ = E+(R 2(ζ)S(ζ)). Now consider R defined, for all ζ 6= 0, by: R(ζ) = S(ζ)(R(ζ))2S(ζ), R(ζ) = (R̃(ζ)V ∗(ζ)S(ζ))∗(R̃(ζ)V ∗(ζ)S(ζ)). ζ 7→ R(ζ) is smooth and, for all ζ, R(ζ) is a hermitian symmetric, positive definite matrix. Moreover, there holds: [q ∈ KerΓ] ⇔ ∀ζ 6= 0, R(ζ)q ∈ E+(R(ζ)S −1(ζ)R(ζ)) which is equivalent to: ∀ζ 6= 0, KerΓ = E+(S −1(ζ)R2(ζ)). Let us detail the computation of R(ζ). R(ζ) = S(ζ)V (ζ)R̃ (ζ)V ∗(ζ)S(ζ). Moreover (ζ) = S̃−1(ζ)R̃2(ζ)S̃−1(ζ), we have thus: R(ζ) = R̃(ζ)S̃−1(ζ)V ∗(ζ)S(ζ) R̃(ζ)S̃−1(ζ)V ∗(ζ)S(ζ) which gives: B(ζ) = R̃(ζ)S̃−1(ζ)V ∗(ζ)S(ζ) R̃(ζ)S̃−1(ζ)V ∗(ζ)S(ζ) We recall that R̃ is given, for all ζ 6= 0, by: R̃(ζ) = IdN−p −ℵ(ζ) −ℵ∗(ζ) Idp and that for all ζ 6= 0, V (ζ) is given by: V (ζ) = (Λ(ζ))− 2Z(ζ), where Λ(ζ) = Z∗(ζ) (ζ)Z(ζ) with Λ is a diagonal matrix with real coefficients: (λ1, . . . , λN ), and Λ de- notes the diagonal matrix with diagonal coefficients (|λ1|, . . . , |λN |). Remark 2.5. In the construction of B the only freedom we have resides in the choice of ℵ. 2.3 A change of dependent variables. Let us denote by R := B 2 and v̂ := Rû. v̂ is hence solution of (2.3): (2.3) R−1SR−1∂xv̂ = R −1SAR−1v̂ +R−1S(Ad) −1f̂ , {x > 0}, ΓR−1v̂|x=0 = Γĝ, We will adopt the following notations: SR := R −1SR−1, AR := RAR and ΓR := ΓR −1. We first observe that: ker ΓR = R ker Γ = RE+((S) −1R2). but S−1 = RS−1R thus ker ΓR = RE+(R −1SRR) = E+(SR). This is where Lemma 1.4 is used in a crucial manner. Let us denote by − the projector on E−(SR) parallel to E+(SR) and by by P + the projector on E+(SR) parallel to E−(SR); P − and P+ denoting the associated Fourier multiplier. Since SR is hermitian symmetric, P − is in fact the orthogonal projector on E−(SR). The problem (2.3) can then be written: SR∂xv̂ = SRARv̂ +R −1S(Ad) −1f̂ , {x > 0}, −v̂|x=0 = P −Γĝ, This problem is well-posed because, as a direct Corollary of Proposition 2.2, we have: Proposition 2.6. For all ζ such that τ2 + γ2 + |η|2 = 1, there holds: • SR(ζ) is hermitian symmetric. • ℜ (SRAR) (ζ) is positive definite. • −SR(ζ) is definite negative on ker ΓR and the dimension of ker ΓR is the same as the number of negative eigenvalues of −SR(ζ). Proof. For the sake of simplicity, let us omit the dependence in ζ in our notations. • SR := R −1SR−1, and both S and R are hermitian thus SR is hermi- tian. • SRAR = R −1SAR−1, thus for all q ∈ CN , there holds: 2〈ℜ(SRAR)q, q〉 = 〈SRARq, q〉+〈q, SRARq〉 = 〈R −1SAR−1q, q〉+〈q,R−1SAR−1q〉, since R−1 is hermitian, we have then: = 〈SAR−1q,R−1q〉+ 〈R−1q, SAR−1q〉 = 2〈ℜ(SA)R−1q,R−1q〉. Since ℜ(SA) is positive definite and R is invertible, ℜ (SRAR) is thus positive definite. • By construction of R, it satisfies ker ΓR = E+(SR), with SR hermitian. As a consequence −SR is definite negative on ker ΓR and the dimension of ker ΓR is the same as the number of negative eigenvalues of −SR. ✷ Let us mention that, since R and S remains uniformly bounded in ζ 6= 0, f̂ and R−1S(Ad) −1f̂ belongs to the same space. In a same spirit as [5], this suggests the following singular perturbation of (2.3): SR∂xv̂ −v̂ε1x<0 = SRARv̂ −Γĝ1x<0 +R −1S(Ad) −1f̂ , {x ∈ R}, This is equivalent to perturb (2.2) as follow: S∂xû RP−Rûε1x<0 = SAû RP−Γĝ1x<0 + S(Ad) −1f̂ , {x ∈ R}, Finally, this induces the following perturbation for (1.1): (2.4) Huε + Muε1x<0 = f1x>0 + θ1x<0, {x ∈ R}, uε|t<0 = 0, where M := −eγtAdS −1RP−Re−γt, θ = −eγtAdS −1RP−Γg̃, and S(∂t, ∂y) [resp R(∂t, ∂y)] denotes the Fourier multiplier associated to S(ζ) [resp R(ζ)]. 3 Proof of Theorem 1.6. First, we construct an approximate solution of equation (2.4) (which is also equation (1.4)), then prove suitable energy estimates that ensures uε and its approximate solution both converges towards the same limit as ε → 0+. 3.1 Construction of the approximate solution. uε is the solution of the well-posed Cauchy problem: Huε + Muε1x<0 = f1x>0 + θ1x<0, {x ∈ R}, uε|t<0 = 0. uε is moreover the solution of the well-posed Cauchy problem: Huε + Muε1x<0 = SA f1x>0 + θ1x<0, {x ∈ R}, uε|t<0 = 0. The associated equation after tangential Fourier-Laplace transform writes : S∂xû RP−Rûε1x<0 − SAû ε = − RP−Γĝ1x<0 + S(Ad) −1f̂1x>0, {x ∈ R}. or alternatively: ûε = R−1v̂ε SR∂xv̂ −v̂ε1x<0 = SRARv̂ −Γĝ1x<0 +R −1S(Ad) −1f̂ , {x ∈ R}, We will use the following formulation as a transmission problem in our con- struction of an approximate solution: SR∂xv̂ ε+ = SRARv̂ ε+ +R−1S(Ad) −1f̂ , {x > 0}, SR∂xv̂ −v̂ε− = SRARv̂ −Γĝ, {x < 0}, v̂ε+|x=0+ = v̂ ε−|x=0− . For Ω an open regular subset of Rd+1, and ρ ∈ N, let us introduce the weighted spaces H γ (Ω) defined by: H̺γ (Ω) = {̟ ∈ e γtL2(Ω), ‖̟‖H̺γ (Ω) < ∞}; where γ (Ω) α,|α|≤̺ γρ−|α|‖e−γt∂α̟‖2 L2(Ω). We will construct an approximate solution uεapp of u ε. uεapp will be con- structed as follow: uεapp = u app1x>0 + u app1x<0, where uε±app is an approximate solution of u ε± satisfying the following ansatz: uε±app = U±j (ζ, x)ε where the profiles U±j belong toH ), where Ω± stands for [0, T ]×Rd±. Denote v̂εapp = RF(e −γtuεapp) := v̂ app1x>0 + v̂ app1x<0. v̂ε±app is then an approximate solution of v ε± and is of the form: v̂ε±app = j (ζ, x)ε where = RF(e−γtU± and conversely U±j = e γtF−1 R−1V ±j The profiles U±j can be constructed inductively at any order. Let us show how the first profiles are constructed: Identifying the terms in ε−1 gives: −V −0 = P −Γĝ. Hence, P+V −0 remains to be computed in order to obtain the profile 0 = e γtF−1 Identifying the terms in ε0 gives then that V +0 is solution of the well-posed problem: (3.1) SR∂xV 0 = SRARV −1S(Ad) −1f̂ , {x > 0}, −V +0 |x=0 = P −Γĝ. The associated profile U+0 = e γtF−1 R−1V +0 belongs then toHkγ (Ω ).Moreover, the problem (3.1) is Kreiss-Symmetrizable and thus the trace of the profile U+0 , see [3] for instance, satisfies: U+0 |x=0 ∈ H γ (ΥT ). Since V +0 has just be computed, V 0 |x=0 is given by: V 0 |x=0 − V 0 |x=0 = 0 and thus, there holds: −V +0 |x=0 = P −V −0 |x=0. Moreover SR∂xV 0 − P −V −1 = SRARV 0 , {x < 0}. Projecting this equation on E+(SR) collinearly to E−(SR) gives then: SR∂xP 0 − P +SRARV 0 = 0, {x < 0}, Since +SRARV 0 = P +SRARP +V −0 + P +SRARP −Γĝ, we have then: SR∂x(P +V −0 )− P +SRAR(P +V −0 ) = P +SRARP −Γĝ, {x < 0}, and as a consequence, P+V −0 is solution of the following problem: (3.2) SR∂x(P 0 )− P +SRAR(P 0 ) = P +SRARP −Γĝ {x < 0}, +V −0 |x=0 = P +V +0 |x=0. Let us precise how (3.2) has to be interpreted: we denote w = P+V −0 . w is then totally polarized on E+(SR), and satisfies the problem: (3.3) +w = w SR∂xw − P +SRARw = P +SRARP −Γĝ {x < 0}, w|x=0 = P +V +0 |x=0. As we will see, the problem (3.3) is Kreiss-Symmetrizable and thus well- posed. Indeed, for all ζ such that τ2 + γ2 + |η|2 = 1, we have, omitting the dependencies in ζ in our notations: • For all q ∈ CN , there holds: 〈SRq, q〉 = 〈q, SRq〉. • Since Re(SRAR) is positive definite and P + is an orthogonal projector, there is C > 0 such that, for all q ∈ E+(SR), there holds: 〈P+SRARP +q, q〉+ 〈q,P+SRARP +q〉 ≥ C〈q, q〉. Indeed, for all q ∈ E+(SR), there holds: 〈P+SRARP +q, q〉 = 〈P+SRARP +q,P+q〉 = 〈SRARP +q,P+q〉. • −SR is definite negative on kerP + that is to say, that there is c > 0 such that, for all q ∈ kerP+, there holds: 〈−SRq, q〉 ≤ −c〈q, q〉. Moreover kerP+ has the same dimension as the number of negative eigenvalues in −SR. The profile U−0 can then be computed by: U−0 := e γtF−1 R−1(w + P−Γĝ) belongs to Hkγ (Ω ), moreover its trace U−0 |x=0 belongs to Hkγ (ΥT ). Consider now the equation: 1 = SR∂xV 0 − SRARV 0 , {x < 0}. Since P−V −1 |x=0 = P −V +1 |x=0, V 1 is solution of the well-posed problem: SR∂xV 1 = SRARV 1 , {x > 0}, −V +1 |x=0 = SR∂xV 0 |x=0 − SRARV 0 |x=0. Due to the loss of regularity in the boundary condition, the associated profile U+1 = e γtF−1 R−1V +1 belongs to H ), moreover its trace U+1 |x=0 belongs γ (ΥT ). Moreover, applying P + to the equation: −V −2 + SRARV 1 = SR∂xV 1 , {x < 0}, we obtain: SR∂xP +V −1 = P +SRARP +V −1 + P +SRARP −V −1 , {x < 0}, +V −1 |x=0 = P +V +1 |x=0. As before, let us take P+V −1 as the unknown of the well-posed problem: SR∂x(P +V −1 )− P +SRAR(P +V −1 ) = P +SRAR SR∂xV 0 − SRARV , {x < 0}, (P+V −1 )|x=0 = P 1 |x=0. This problem is Kreiss-Symmetrizable since, for all ζ such that τ2 + γ2 + |η|2 = 1, there holds: • For all q ∈ CN , there holds: 〈SRq, q〉 = 〈q, SRq〉. • There is C > 0 such that for all q ∈ E+(SR), there holds: 〈P+SRARP +q, q〉+ 〈q,P+SRARP +q〉 ≥ C〈q, q〉. • −SR is definite negative on kerP + that is to say, that there is c > 0 such that, for all q ∈ kerP+, there holds: 〈−SRq, q〉 ≤ −c〈q, q〉. Moreover kerP+ has the same dimension as the number of negative eigenvalues in −SR. However, due to a loss of regularity in both the source term and the boundary condition, the associated profile U−1 = e γtF−1 +V −1 + SR∂xV 0 − SRARV belongs to H ). The construction of the following profiles can be pursued at any order the same way. In practice, we take: uε+app = U uε−app = U 0 + εU As a result, the approximate solution writes uεapp := u app1x>0 + u app1x<0; where uε+app belongs to H ) and uε−app belongs to H ). uεapp is then solution of a well-posed problem of the form: (3.4) Huεapp + Muεapp1x<0 = f1x>0 + θ1x<0 + εr ε, {x ∈ R}, uεapp|t<0 = 0 . Where rε := rε+1x>0 + r ε−1x<0, with r ε+ ∈ H ) and rε− ∈ Hk−3γ (Ω Remark 3.1. In the case where g = 0, the loss of regularity in the profiles is delayed by one step. As a result, in this case we obtain: uε+app ∈ H uε−app ∈ H rε+ ∈ Hkγ (Ω rε− ∈ H 3.2 Stability estimates We will begin by proving energy estimates on the following equation: (3.5) SRARê ε − SR∂xê −êε1x<0 = εr̂ ε, {x ∈ R}, where êε = R F(e−γtuε)−F(e−γtuεapp) := ŵε; with wε = uε − uεapp. Refering to (3.4), wε is the solution of the Cauchy problem: (3.6) Hwε + Mwε1x<0 = εr wε|t<0 = 0 . For a fixed positive ε, the perturbation is nonsingular and thus the principal part of the pseudodifferential operator H+ 1 M is the same as the principal part of H. Hence, there is a unique solution of the Cauchy problem (3.6): wε which belongs to Hk−3γ (ΩT ). In order to simplify the notations, in this chapter we shall denote by L2 and H γ the spaces: L 2(ΩT ) and H γ (ΩT ). We recall the definition of the weighted spaces: H γ (ΩT ) for ρ ∈ N. H̺γ (ΩT ) = {̟ ∈ e γtL2(ΩT ), ‖̟‖H̺γ (ΩT ) < ∞}; where γ (ΩT ) α,|α|≤̺ γρ−|α|‖e−γt∂α̟‖2 L2(ΩT ) For fixed positive ε, there holds: ∂x〈SRê ε, êε〉L2 dx = 0. 2Re〈SR∂xê ε, êε〉L2 dx = 0. Using the equation, we have then: Re〈SRARê −êε − εr̂ε, êε〉L2 dx = 0. which is equivalent to: Re〈SRARê ε, êε〉L2 dx = Re〈P−êεεr̂ε, êε〉L2 dx Re〈r̂ε, êε〉L2) dx. But Re〈SRARê ε, êε〉 = 〈Re (SRAR) ê ε, êε〉 and Re (SRAR) is positive defi- nite, hence there is C > 0, independent of ε such that: Cγ‖êε‖2 Re〈P−êε, êε〉 ≤ Re〈εr̂ε, êε〉L2 dx. Thus, because P− is an orthogonal projector, for all positive λ, there holds: Cγ‖êε‖2 ‖P−êε‖2 ‖êε‖2 ‖εr̂ε‖2 Choosing λ big enough we have C− ε > 0 and the following energy estimate: γ‖êε‖2 ‖P−êε‖2 ‖r̂ε‖2 This shows that êε converges towards zero in L2 when ε tends to zero, with a rate in O(ε). We recall that êε is given by: êε := RF e−γt(uεapp − u and r̂ε is given by: r̂ε := RF e−γtrε Moreover, since R and P− are two uniformly bounded, uniformly definite positive matrices, there are two positive real numbers α and β such that, for all ζ 6= 0 and x ∈ R, there holds: • α‖F e−γt(uεapp − u ≤ ‖êε‖2 • α‖P−F e−γt(uεapp − u ≤ ‖P−êε‖2 • ‖r̂ε‖2 ≤ β‖F e−γtrε Applying then Plancherel’s equality we obtain then: γ‖uεapp − u eγtL2 uεapp − u eγtL2 ‖rε‖2 eγtL2 We have thus proved there are two positive constants c and C such that: cγ‖uεapp − u eγtL2 uεapp − u eγtL2 ‖rε‖2 eγtL2 Let us denote by ‖.‖∗ + ‖.‖2 . More generally, when rε ∈ H̺, there is two positive constants cρ and Cρ such that: cργ‖u app − u uεapp − u ‖rε‖∗2 As we have seen during the construction of the profiles, ̺ = k− 3 in general and ρ = k − 3 in the case where g = 0. 3.3 End of the proof of Theorem 1.6. As a consequence of our stability estimate, there holds: ‖uεapp − u Hk−3(Ω + ‖uεapp − u Hk−3(Ω = O(ε2). Moreover, by construction of uεapp, there holds: ‖uεapp − u Hk−3(Ω + ‖uεapp − u‖ Hk−3(Ω = O(ε2). Hence, we have proved that: ‖uε − u−‖2 Hk−3(Ω + ‖uε − u‖2 Hk−3(Ω = O(ε2). By the same arguments, if g = 0, there holds: ‖uε − u−‖2 + ‖uε − u‖2 = O(ε2). This completes the proof of Theorem 1.6. 4 Proof of Theorem 1.9. Like in the proof of Theorem 1.6, we begin by constructing formally an approximate solution of equation (1.7). We prove then suitable energy esti- mates that ensures both uε and its approximate solution converges towards ũ as ε → 0+. 4.1 Construction of an approximate solution. The goal of this Lemma is to replace the boundary condition Γu|x=0 = Γg of problem (1.1) by a condition of the form P−(e−γtu)|x=0 = h with a suitable h ∈ Hk(ΥT ). Lemma 4.1. Let u denote the unique solution in Hk(Ω+ ) of the mixed hyperbolic problem (1.1), P+(∂t, ∂y, γ) e−γtu does not depend of the choice of the boundary operator Γ and of g. Let us introduce the function h of Hk(ΥT ) defined by: e−γtv|x=0 e−γt(g − v|x=0) The solution u of the mixed hyperbolic problem (1.1) is the unique solution of the following well-posed mixed hyperbolic problem (4.1): (4.1) Hu = f, {x > 0}, −(∂t, ∂y, γ) e−γtu|x=0 u|t<0 = 0 . In addition, the mapping (f, g) → h is linear continuous from Hk(Ω+ )×Hk(ΥT ) to H k(ΥT ). Proof. Let v denote a solution in Hk(ΩT ) of the equation: Hv = f, (t, y, x) ∈ ΩT , v|t<0 = 0 . We introduce then U which is, by definition, the solution of the following mixed hyperbolic problem: HU = 0, {x > 0}, Γ(∂t, ∂y, γ)U|x=0 = Γ(∂t, ∂y, γ)g − Γ(∂t, ∂y, γ)v|x=0, U|t<0 = 0 . The right hand side of the boundary condition is, a priori, in Hk− 2 (ΥT ). Hence the solution U belongs to H 2 (Ω+ ). By construction we have: (4.2) u = U+ v. Hence, since u ∈ Hk(Ω+ ) and v ∈ Hk(Ω+ ), in fact we have: U ∈ Hk(Ω+ Let Û denote the Fourier-Laplace transform in (t, y) of U (Fourier-Laplace transform tangential to the boundary) given by: F(e−γtU). It satisfies the following symbolic equation: ∂xÛ = A(ζ)Û, {x > 0}, Γ(ζ)Û|x=0 = Γ(ζ)ĝ − Γ(ζ)v̂|x=0, where ĝ and v̂ denotes respectively the tangential Fourier-Laplace transform of g and v. Since A(ζ) is independent of x, projecting the above equation on E+(A(ζ)) gives then: Û = A(ζ)P+Û. Moreover P+Û|x=0 ∈ E−(A(ζ)) E+(A(ζ)) since limx→∞P Û = 0. Hence, there holds: Û = 0, and thus Û = P−Û. The boundary condition: Γ(ζ)Û|x=0 = Γ(ζ)ĝ − Γ(ζ)v̂|x=0 is equivalent to: Û|x=0 ∈ ĝ − v̂|x=0 +KerΓ. We have thus: Û|x=0 ∈ ĝ − v̂|x=0 + ker Γ. Let us denote by Π the projector on Ẽ−(A) parallel to ker Γ, which has a sense because the Uniform Lopatinski Condition holds. Since Û|x=0 ∈ Ẽ−(A), and of the Uniform Lopatinski Condition, the above boundary condition is equivalent to: Û|x=0 = Π(ĝ − v̂|x=0), and thus, because P−Û|x=0 belongs to E−(A), we have: Û|x=0 = Π(ĝ − v̂|x=0). As a consequence, we obtain: −û|x=0 = P −v̂|x=0 +Π(ĝ − v̂|x=0). Hence, there holds: e−γtu|x=0 e−γtv|x=0 e−γt(g − v|x=0) := h. P+(∂t, ∂y, γ) e−γtu = P+(∂t, ∂y, γ) e−γtv , thus it does not depend of the choice of the boundary operator Γ and of g.Moreover, since u|x=0 ∈ H k(ΥT ), it follows that g ∈ Hk(ΥT ). Now, since the Uniform Lopatinski Condition holds, u satisfies the following energy estimate: eγtL2(Ω + ‖u|x=0‖ eγtL2(ΥT ) ≤ γ‖f‖2 eγtL2(Ω + ‖g‖eγtL2(ΥT ), More generally, we have: Hkγ (Ω + ‖u|x=0‖ Hkγ (ΥT ) ≤ γ‖f‖2 Hkγ (Ω + ‖g‖Hkγ (ΥT ). where ‖̟‖2 |α|=0 γ k−|α|‖∂α̟‖2 eγtL2 h = P−(e−γtu|x=0) hence L2(ΥT ) ≤ C‖e−γtu|x=0‖ L2(ΥT ) = C‖u|x=0‖ eγtL2(ΥT ) and for 0 ≤ j ≤ d− 1, there holds: ‖∂jh‖ L2(ΥT ) ≤ cj‖ηjP −F(e−γtu)|x=0‖ ≤ c j‖u|x=0‖H1γ(ΥT ). More generally, we have: Hkγ (ΥT ) ≤ Ckγ‖f‖ Hkγ (Ω + Ck‖g‖ Hkγ (ΥT ) But γ is a positive real number fixed once and for all at the beginning of the paper, hence this proves that the mapping (f, g) → h is continuous from Hk(Ω+ )×Hk(ΥT ) to H k(ΥT ). ✷ As we will see, Lemma 4.1 is central in our construction of an approximate solution. We will construct an approximate solution uεapp := u app1x>0 + u app1x<0, along the following ansatz: uε+app := εju+j (t, y, x), with u+j ∈ H ), u+j |x=0 ∈ H γ (ΥT ); and uε−app := j (t, y, x), with u−j ∈ H ), u−j |x=0 ∈ H γ (ΥT ). As usual, we will refer to the terms u± as profiles. We will rather work on the reformulation of problem (1.7) as the transmission problem (4.3): (4.3)   Huε+ = f, {x > 0}, Huε− + −e−γtuε− = γth̃, {x < 0}, uε+|x=0 − u ε−|x=0 = 0, uε±|t<0 = 0 . Plugging uε+app and u app in (4.3) and identifying the terms with same power in ε, we obtain the following profiles equations: • Identification of the terms of order ε−1 : (4.4) Ade −e−γtu 0 = Ade γth̃, {x < 0}. • Identification of the terms of order ε0 : (4.5) Hu−0 +Ade −e−γtu−1 = 0, {x < 0}. (4.6) Hu+0 = f, {x > 0}, • Identification of the terms of order εj for j ≥ 1 : (4.7) Hu j +Ade −e−γtu j+1 = 0, {x < 0}. (4.8) Hu j = 0, {x > 0}, • Translation of the continuity condition over the boundary on the pro- files: For all 1 ≤ j ≤ M, there holds: (4.9) u j |x=0 − u j |x=0 = 0. Denote by û±j := F(e −γtu±j ) . We have then: j := e γtF−1(û±j ). We will now give the equations satisfied by the Fourier-Laplace transform of the profiles: û±j . First, equation (4.4) is equivalent to: −û−0 = F(h̃), {x < 0}. We deduce from this equation that there holds: −û−0 |x=0 = ĥ. Then, using (4.9) for j = 0, and (4.6) gives that, for γ big enough, 0 = F(e −γtû where û+0 is the solution of the well-posed first order ODE in x: 0 −Aû 0 = F(e −γt(Ad) −1f), {x > 0}, −û+0 |x=0 = h, Thus u+0 is solution of: Hu+0 = f, {x > 0}, eγtP−e−γtu+0 |x=0 = h. Thanks to Lemma 4.1, we recognize u+0 as the solution of our starting mixed hyperbolic problem (1.1). Once u+0 is known, so is û 0 and thus û 0 |x=0 is given by: û−0 |x=0 = û 0 |x=0. Moreover, u+0 |x=0 = u 0 |x=0 ∈ H γ (ΥT ). By (4.5), there holds: 0 −Aû −û−1 = 0, {x < 0}. As a consequence, P+û−0 is given by the well-posed ODE: 0 )−A(P 0 ) = 0, {x < 0}, +û−0 |x=0 = P +û+0 |x=0. Indeed, since kerP+(ζ) = E−(A(ζ)), this problem satisfies the Uniform Lopatinski Condition: for all ζ 6= 0, there holds: E−(A(ζ)) E+(A(ζ)) = C For γ big enough, by linearity of the inverse Fourier transform, u−0 can then be computed by: u−0 := e γtF−1(P−û−0 ) + e γtF−1(P+û−0 ). Following up with that process of construction, we can go on with the con- struction of the profiles at any order. Indeed, assume that all the profiles (u+j , u j ) up to order j have been computed. Then consider the equation obtained through identification: j+1 = −∂xû j +Aû j , {x < 0}. We see there is a loss of regularity between û−j+1 and û Let us say that u±j ∈ H ). Considering the traces, we have then: j |x=0 ∈ H γ (ΥT ). We will show in this part how the Sobolev regularity of the profiles u±j+1, which is by definition mj+1, can be computed know- ing mj . To begin with P j+1 belongs to H ). P−u+j+1|x=0, which belongs to H γ (ΥT ), is known by P j+1|x=0 = e γtF−1(P−û+j+1|x=0), with: j+1|x=0 = P j+1|x=0. Hence, û+j+1 := F(e −γtu+j+1) is the solution of the first order ODE in x : j+1 −Aû j+1 = 0, {x > 0}, −û+j+1|x=0 = P −û−j+1|x=0. Since kerP−(ζ) = E+(A(ζ)), this problem satisfies the Uniform Lopatinski Condition: for all ζ 6= 0, there holds: E−(A(ζ)) E+(A(ζ)) = C As a consequence, this problem is well-posed and, u+j+1 ∈ H ).More- over, there holds: u+j+1|x=0 = u j+1|x=0 ∈ H γ (ΥT ). Indeed, P+û+j+1 ∈ H ∞(Rd+1+ ) hence P +u+j+1|x=0 ∈ H γ (ΥT ) and thus u+j+1|x=0 ∈ H γ (ΥT ). Furthermore, we have: u−j+1|x=0 = u j+1|x=0. Applying P+ on the following equation: −û−j+2 = −∂xû j+1 +Aû j+1, {x < 0}; we obtain then the equation: j+1)−AP j+1 = 0, {x < 0}. Remark 4.2. −û−j+2 = −∂xû j+1 +Aû j+1, {x < 0}. shows that the ”Fourier profile” û−j+1 must be so that −∂xû j+1 + Aû j+1 is polarized on E−(A). It is indeed the case because we search for û j+1 satisfy- +û−j+1)−AP +û−j+1 = 0, {x < 0}. j+1 is given by: j+1 := e γtF−1(P−û−j+1) + e γtF−1(P+û−j+1). with P+u−j+1 = e γtF−1(P+û−j+1) belongs to H ) and is the unique solution of the well-posed first order ODE: +û−j+1)−A(P +û−j+1) = 0, {x < 0}, +û−j+1|x=0 = P +û+j+1|x=0. The profile u−j+1 belongs to H ). This achieves to show that the knowledge of (u+j , u j ), allows us to compute (u j+1, u j+1). Moreover mj+1 = mj − , that is to say that a construction of each supple- mentary profile consummate 3 of Sobolev regularity. In practice, we take: uε+app = u uε−app = u 0 + εu As a result, the approximate solution writes uεapp := u app1x>0 + u app1x<0; where uε+app belongs to H ) and uε−app belongs to H ). The so de- fined uεapp is solution of a well-posed problem of the form: (4.10) Huεapp + −e−γtuεapp1x<0 = f1x>0 + γth̃1x<0 + εr uεapp|t<0 = 0 . Where rε := rε+1x>0 + r ε−1x<0, with r ε+ ∈ H ) and rε− ∈ Hk−3γ (Ω 4.2 Asymptotic Stability of the problem as ε tends towards zero. Denote by vε = uεapp − u ε. By construction of uεapp, v ε is solution of the following Cauchy problem: (4.11) Hvε + −e−γtvε1x<0 = εr vε|t<0 = 0 . For all positive ε, this problem is well-posed. In order to investigate the stability of this problem as ε goes to zero, we will reformulate it as a trans- mission problem. The restrictions of vε to {x > 0} and {x < 0}, respectively denoted by vε+ and vε− are solution the following transmission problem: (4.12)   Hvε+ = εrε+, {x > 0}, Hvε− + −e−γtvε− = εrε−, {x < 0}, vε+|x=0 − v ε−|x=0 = 0, vε±|t<0 = 0 . Let us denote by V ε the function, valued in R2N , defined for all {x > 0} and (t, y) ∈ [0, T ]× Rd−1 by: V ε(t, y, x) = V ε+(t, y, x) V ε−(t, y,−x) vε is solution of the Cauchy problem (4.11) iff V ε is solution of the mixed hyperbolic problem on a half space (4.13) given below: (4.13) H̃V ε +BεV ε = εRε, {x > 0}, Γ̃V ε|x=0 = 0, V ε|t<0 = 0 , where H̃ = ∂t + 0 −Ad γtP−e−γt Rε(t, y, x) = rε+(t, y, x) rε−(t, y,−x) Id −Id Returning to the construction of our approximate solution, we have Rε ∈ H )×Hk−3γ (Ω ) and is such that Rε|t<0 = 0. In fact Rε ∈ Hk ) with k′ = k−3. For all positive ε, there exists a unique solution V ε in Hkγ (Ω ) to the above problem. We will prove here that this solution converges, uniformly in ε, towards 0 in Hk ), as ε vanishes. As in the proof of Kreiss Theorem, see [3] for instance, existence of solution for mixed hyperbolic systems like (1.7) or (4.13), are obtained through the proof of both direct and ”dual”a priori estimates on an adjoint problem. This estimates results in the constant coefficient case of estimates on the Fourier- Laplace transform of the solution. Additionally, if this ”Fourier” estimate can be proved, both direct and ”dual” energy estimates are deduced from it. In a first step, let us recall formally how to conduct the Fourier-Laplace transform of a mixed hyperbolic problem: Hu = f, {x > 0}, Γu|x=0 = g, u|t<0 = 0 , Denote by u∗ := e −γtu, u∗ is in particular a solution of the following problem: Hu∗ + γu∗ = e −γtf, {x > 0}, Γu∗|x=0 = g . We take then the tangential (with respect to (t,y)) Fourier transform of this equation, which gives: Ad∂xû∗ + (γ + iτ)û∗ + iηj Aj û∗ = F e−γtf , {x > 0}, Γû∗|x=0 = ĝ . Multiplying this equation by A−1 , we obtain that u∗ is solution of the fol- lowing ODE in x: ∂xû∗ −Aû∗ = (Ad) e−γtf , {x > 0}, Γû∗|x=0 = ĝ . Note that, û∗ and u can be freely deduced from each other through the formulas: û∗ = F(e −γtu) u = eγtF−1(û∗). We shall now introduce a rescaled solution V ε of the solution V ε of (4.13) defined as follows: V ε(t, y, x) := V ε(t, y, εx), and the rescaled remain- der: Rε(t, y, x) := Rε(t, y, εx). Denoting by V̂ = F(e−γtV ), the associated equation writes then: − εÃV̂ = ε2R̂ε, {x > 0}, |x=0 = 0 . where M(ζ) = 0 P−(ζ) We remark that εÃ(ζ) = Ã(εζ) = Ã(ζ̂), with ζ̂ = (τ̂ , γ̂, η̂) := εζ. Moreover P− is homogeneous of order zero in ζ. Let us denote R̃ε(ζ̂ , x) := R̂ε(ζ, x) and Ṽ (ζ̂ , x) := V̂ (ζ, x). Hence Ṽ solution of the following problem: −Ã(ζ̂) +M(ζ̂) = ε2R̃ε(ζ̂ , x), {x > 0}, |x=0 = 0 . As a consequence, the Uniform Lopatinski Condition for problem (4.13) writes: For all γ̂ > 0, |det(E−(Ã(ζ̂)−M(ζ̂), ker Γ)| ≥ C > 0. In view of the proof of the Proposition (4.3), we recall that the spaces E±(A) have to be considered in the extended sense defined above. Proposition 4.3. Since H satisfies the hyperbolicity Assumption in As- sumption 1.1, the Uniform Lopatinski Condition is satisfied for our present problem; that is to say that, for all ζ̂ such that γ̂ > 0 there holds: |det(E−(Ã(ζ̂)−M(ζ̂), ker Γ)| ≥ C > 0. Proof. We will begin to show that the Uniform Lopatinski Condition writes as well that for all ζ̂ 6= 0 there holds: (4.14) E+(A(ζ̂)−P −(ζ̂)) E−(A(ζ̂)) = {0} . This notation keeps a sense for ζ̂ such that γ̂ = 0 because we will prove a posteriori that the involved linear subspaces continuously extends from {ζ̂ , γ̂ > 0} to {ζ̂ , γ̂ = 0}. Then we will prove that, for all ζ̂, the property 4.14 holds true. The Uniform Lopatinski Condition writes actually, for all ζ̂ 6= 0 : E−(Ã(ζ̂)−M(ζ̂)) ker Γ̃ = {0}. and thus, since we have: E−(Ã(ζ̂)−M(ζ̂)) = E−(A(ζ̂))× E+(A(ζ̂)−P −(ζ̂)), and by definition of Γ̃, the Uniform Lopatinski Condition writes then that, for all ζ̂ 6= 0, there holds: E+(A(ζ̂)−P −(ζ̂)) E−(A(ζ̂)) = {0}. Lemma 4.4. A(ζ̂)−P−(ζ̂) A(ζ̂) A(ζ̂)−P−(ζ̂) A(ζ̂) Proof. For all ζ̂ 6= 0, there is an invertible N × N matrix with complex coefficients P (ζ̂) such that: P−1AP is trigonal and the diagonal coefficients are sorted by increasing order of their real parts. Let us denote by ν the dimension of E− (A) . The above matrix P traduces the change of basis from the canonical basis of CN into (v1, . . . , vν , vν+1, . . . , vN ), where Span ((vk)1≤k≤ν) = E− (A) , Span ((vk)ν+1≤k≤N ) = E+ (A) . Moreover, there holds P−1P−P = D where D is the diagonal matrix whose ν first diagonal terms are equal to 1 and the N − ν last diagonal terms are null. P−1(A−P−)P = P−1AP −D. P−1AP−D is also trigonal, with the same eigenvalues with positive real part as P−1AP and the same eigenvalues with negative real part as P−1AP −Id. As a consequence, for all ζ̂ 6= 0, there holds: A(ζ̂)−P−(ζ̂) A(ζ̂) A(ζ̂)−P−(ζ̂) A(ζ̂) As a consequence of Lemma 4.4, the rescaled Uniform Lopatinski Con- dition for ε > 0, ε → 0 happens to be exactly the same as the one written for bigger positive ε. Indeed, it writes, for all ζ̂ 6= 0 : E+(A(ζ̂)) E−(A(ζ̂)) = {0}. ✷ The Lopatinski condition is satisfied, and, as a result, the following, uniform in ε, energy estimate holds for V ε, for all γ ≥ γk′ > 0 : γ‖V ε‖2 + ‖V ε|2x=0‖Hk′γ (ΥT ) ‖εRε‖2 which is equivalent to: (4.15) γ‖V ε‖2 + ‖V ε|x=0‖ γ (ΥT ) ‖εRε‖2 This proves the convergence of V ε towards zero in Hk ). The weight γ is fixed beforehand thus, in fact, the solution of (4.13) tends to zero in ) at a rate at least in O(ε). 5 End of proof of Theorem 1.9. Let us consider V ε defined by: V ε(t, y, x) := uε+app(t, y, x)− u ε+(t, y, x) uε−app(t, y,−x)− u ε−(t, y,−x) This notation is perfectly fine because the so-defined function is solution of an equation of the form (4.13). Moreover, thanks to the stability estimate (4.15), there is γk positive such that, for all γ > γk, we have: γ‖uεapp−u +γ‖uεapp−u +‖uεapp−u γ (ΥT ) ‖εRε+‖2 Hence, it follows that: ‖uεapp − u Hk−3(Ω + ‖uεapp − u Hk−3(Ω = O(ε2). Moreover, by construction of uεapp, we have: ‖uεapp − u‖ Hk−3(Ω + ‖uεapp − u Hk−3(Ω = O(ε2). As a result, we obtain that there holds: ‖uε − u‖2 Hk−3(Ω + ‖uε − u−‖2 Hk−3(Ω = O(ε2). This concludes the proof of Theorem 1.9. 6 Appendix: answer to a question asked in [11]. In this chapter, we will show that the loss of convergence observed numeri- cally in [11] in a neighborhood of the boundary is due to a boundary layer phenomenon. We consider the 1-D wave equation: (6.1) ∂ttU − c 2∂xxU = 0, (x, t) ∈]0, π[×R U |x=0 = U |x=π = 0, U |t=0(x) = sin(x), ∂tU |t=0 = 0. As in [11], we define then U ε = U ε+1x>0 + U ε−1x<0 by: (6.2)   ε+ − c2∂xxU ε+ = 0, (x, t) ∈]0, π[×R+, ε− − c2∂xxU U ε− = 0, (x, t) ∈]−∞, 0[×R+, U ε+|x=0 − U ε−|x=0 = 0 ε+|x=0 − ∂xU ε−|x=0 = 0 U ε+|x=π = 0. U ε±|t=0(x) = sin(x), {±x > 0}. ε±|t=0 = 0, {±x > 0}. We will now construct formally an approximate solution U ε±app of U ε± satis- fying the following ansatz: U ε+app = U+j (t, x)ε U ε−app = t, x, where the profiles U−j (t, x, z) := U j (t, x) + U j (t, z), with e−αzU∗−j = 0, for some α > 0. Since the stability estimates are trivial here, we will only focus on the construction of U εapp := U app1x>0 + U app1x<0. Plugging U ε±app into problem (6.2) and identifying the terms with same power of ε, we obtain the following equations: U−0 = 0, Moreover, U∗−0 = 0 as it is the only solution of the problem: U∗−0 − c 2∂zzU 0 = 0, {z < 0}, 0 |z=0 = 0, 0 = 0. U ε+app converges towards U 0 as ε → 0 +. As awaited U+0 is the solution of the well-posed 1-D wave equation:   0 − c 2∂xxU 0 = 0, (x, t) ∈]0, π[×R U+0 |x=0 = U 0 |x=0 + U 0 |z=0 = 0. U+0 |x=π = 0. 0 |t=0(x) = sin(x), {x > 0}. 0 |t=0 = 0, {x > 0}. Let us write the following profiles equations: First, we can see that, for all j ≥ 1, there holds: where U∗−1 is the solution of the well-posed profile equation: 1 − c 2∂zzU 1 = −∂ttU 0 = 0, {z < 0}, 1 |z=0 = ∂xU 0 |x=0, U∗−1 = 0. Hence U∗−1 is given by: U∗−1 = c∂xU 0 |x=0e We will show now that the profiles can be computed as any order. Assume that U∗−j has been computed, U j is solution of the well-posed 1-D wave equation:    − c2∂xxU = 0, (x, t) ∈]0, π[×R+, j |x=0 = U j |z=0. U+j |x=π = 0. U+j |t=0(x) = 0, {x > 0}. 0 |t=0 = 0, {x > 0}. U∗−j+1 is then solution of the well-posed profile equation: U∗−j+1 − c 2∂zzU j+1 = −∂ttU j , {z < 0}, j+1|z=0 = ∂xU |x=0, U∗−j+1 = 0. Let us answer the question asked in [11]: U ε− is bound to present boundary layer behavior in {x = 0−}, indeed its approximate solution is composed exclusively of boundary layer profiles, which describes quick transitions at the boundary using a fast scale in ε. As a result of the loss in convergence induced by the boundary layer, the following estimate holds: ‖U ε − U‖L2(]−∞,π[×R+) = O(ε In [11], their small parameter is µ = ε2, as a result, adopting the same notations as them, our estimate writes: ‖Uµ − U‖L2(]−∞,π[×R+) = O(µ which is in agreement with the estimates given in [11]. Like in the penaliza- tion approach proposed by Bardos and Rauch [2] and underlined by Droniou in [4], the boundary layer only forms on one side of the boundary. The ap- proximation U ε+ of U, is computed by taking U ε+|x=0 = U ε−|x=0, thus, in numerical applications, the boundary layer phenomenon also affects the rate of convergence of U ε+ towards U, as ε → 0+. References [1] Ph. Angot, Ch.H. Bruneau, P. Fabrie, A penalization method to take into account obstacles in viscous flows. Numerische Mathematik 1999; 81:497-520. [2] C. Bardos, J. Rauch, Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc.,270 (1982), pp 377-408. [3] J. Chazarain, A. Piriou, Introduction to the theory of linear partial dif- ferential equations. translated from the french , Studies in Mathematics and its Applications, 14 , North Holland Publishing Co., Amsterdam- New York,1982. [4] J. Droniou, Perturbation Singulière par Pénalisation d’un Système Hy- perbolique, Rapport de stage (1997). [5] B. Fornet, O.Guès, Penalization approach of semi-linear symmetric hy- perbolic problems with dissipative boundary conditions, preprint (2007). [6] O. Guès, G. Métivier, M. Williams, K.Zumbrun Uniform stability esti- mates for constant-coefficient symmetric hyperbolic boundary value prob- lems. preprint (2005). [7] H.O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. math 13 (1970), 277-298. [8] G. Métivier, Small Viscosity and Boundary Layer Methods : Theory, Stability Analysis, and Applications, Birkhauser (2003). [9] G. Métivier, K. Zumbrun Viscous Boundary Layers for Noncharacteris- tic Nonlinear Hyperbolic Problems, Preprint. [10] G. Métivier, K. Zumbrun Symmetrizers and Continuity of Stable Sub- spaces for Parabolic-Hyperbolic Boundary Value Problems, Preprint. [11] A. Paccou, G. Chiavassa, J. Liandrat and K. Schneider A penalization method applied to the wave equation. C. R. Acad. Sci. Paris Serie II, (2003) [12] J. Rauch, Boundary value problems as limits of problems in all space., In Séminaire Goulaouic-Schwartz (1978/1979), pages Exp. No. 3, 17. École Polytech., Palaiseau, 1979. Introduction. A Kreiss Symmetrizer Approach. A second Approach. Underlying approach leading to the proof ofTheorem ??. Some preliminaries. Detailed proof of Lemma ??: Construction of the matrices B solving Lemma ??. A change of dependent variables. Proof of Theorem ??. Construction of the approximate solution. Stability estimates End of the proof of Theorem ??. Proof of Theorem ??. Construction of an approximate solution. Asymptotic Stability of the problem as tends towards zero. End of proof of Theorem ??. Appendix: answer to a question asked in paccou.

0704.0498

A unified analysis of the reactor neutrino program towards the measurement of the theta_13 mixing angle

RN.eps A unified analysis of the reactor neutrino program towards the measurement of the θ13 mixing angle G. Mention (a), Th. Lasserre (a,b), D. Motta (a) (a)DAPNIA/SPP, CEA Saclay, 91191 Gif sur Yvette, France (b) Laboratoire Astroparticule et Cosmologie (APC), Paris, France November 9, 2018 Abstract We present in this article a detailed quantitative discussion of the measurement of the leptonic mixing angle θ13 through currently scheduled reactor neutrino oscillation experiments. We thus focus on Double Chooz (Phase I & II), Daya Bay (Phase I & II) and RENO experiments. We perform a unified analysis, including systematics, backgrounds and accurate experimental setup in each case. Each identified systematic error and background impact has been assessed on experimental setups following published data when available and extrapolating from Double Chooz acquired knowledge otherwise. After reviewing the experiments, we present a new analysis of their sensitivities to sin2(2θ13) and study the impact of the different systematics based on the pulls approach. Through this generic statistical analysis we discuss the advantages and drawbacks of each experimental setup. 1 Experimental context Over the last years the phenomenon of neutrino flavor conversion induced by nonzero neutrino masses has been demonstrated by experiments with solar [1, 2, 3], atmospheric [4], reactor [5, 6] and accelerator neutrinos [7, 8]. Neutrino oscillation, that can be described by the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix [11], is the current best mechanism to explain the data. Considering only the three known families, the neutrino mixing matrix is parameterized by the three mixing angles (θ12, θ23, θ13) and a possible δ CP violation phase. The angle θ12 has been measured to be large (sin 2(2θ12) ≃ 0.8), the angle θ23 has been measured to be close to maximum (sin2(2θ23) & 0.9); but the last angle θ13 has only been upper bounded sin2(2θ13) . 0.15 at 90 % C.L., by the CHOOZ reactor experiment [9, 12]. These achievements have now shifted the field of neutrino oscillation physics into a new era of precision measurements. Next generation experiments are underway all around the world to further pin down the values of the oscillation parameters of both solar and atmosheric driven oscillations. Currently the most important task, for the experimentalists, is the determination of the last oscillation through the measurement of the unknown mixing angle θ13. An improved sensitivity on θ13 is not only important for the understanding of neutrino oscillations, but also to open up the possibility of observing CP-violation in the lepton sector if the θ13 driven oscillation is discovered by the forthcoming experiments. In order to improve the CHOOZ constraint on θ13 at least two identical unsegmented liquid scintillator neutrino detectors close to a nuclear power plant (NPP) are required. The near detector(s) located a few hundred meters away from the reactor cores monitor the unoscillated νe flux. The far detector(s) is(are) located at a distance between 1 and 2 km, searching for a departure from the 1/L2 behavior induced by oscillations. Experimental errors are being partially cancelled when using identical detectors. The goal is to achieve an overall effective systematic error of less than 1 % [19, 34]. Three experiments have received partial or full approval to perform such a measurement in a near future: Double Chooz in France [24], Daya Bay in China [25] and RENO in Korea [26]. In addition, a project is http://arxiv.org/abs/0704.0498v2 under study at the Angra power plant in Brazil to further improve the sensitivity of the measurement on a longer time scale [29]. Furthermore the Japanese KASKA collaboration is promoting the reactor neutrino oscillation physics [28]. 2 Neutrino oscillation at reactor and θ13 Fission reactors are prodigious sources of electron antineutrinos which have a continuous energy spectrum up to about 10 MeV. For Eνe > Ethr = 1.806 MeV they can be detected though the νe+p → e++n reaction using the delayed coincidence technique, where an electron antineutrino interacts with a free proton in a tank containing a target volume filled with Gd loaded liquid scintillator. The positron and the resultant annihilation gamma-rays are detected as a prompt signal while the neutron slows down and then thermalizes in the liquid scintillator before being captured by a hydrogen or gadolinium nucleus. The excited nucleus then emits gamma rays which are detected as the delayed signal. Electron antineutrinos energy is derived from the measured visible energy from positron energy loss and annihilation, Evis = Ee+ +me ≃ Eν − Ethrν + 2me . (1) The νe spectrum is calculated from measurements of the beta decay spectra of 235U, 239Pu and 241Pu [15] after fissioning by thermal neutrons and theoretical 238U spectrum, since no data are available for 238U 1. As a nuclear reactor operates, the fission element proportions evolve in time (the so-called burn-up). Since we are interested here on long term interpretation of the data for oscillation search, we will use an average fuel composition for a reactor cycle corresponding to 235U (55.6 %), 239Pu (32.6 %), 238U (7.1 %) and 241Pu (4.7 %) . (2) The mean energy release per fission, 〈Ef 〉, is then 205 MeV and the energy weighted cross section for νep → e+n amounts to 〈σf 〉 = 6 10−43 cm2 per fission. Let us introduce a new luminosity unit, called the r.n.u. (for reactor neutrino unit) and defined as 1 r.n.u. = 0.197 1060 MeV. With this unit, an experiment taking data for T years with a total NPP (nuclear power plant) thermal power of P GW and with N 1030 free protons inside the target has a luminosity L = T P N r.n.u.. The event number, N(L), at a distance L from the source, assuming no - oscillation, can be quickly assessed with N(L) = 〈σf 〉 4π 〈Ef 〉 ≃ 4.6 109 1 GWth For the full antineutrino reactor energy spectra simulation, we follow Vogel’s analytical parameteriza- tion [16], based on second order Eν polynomials. Higher order parameterizations [18] give very comparable results and do not require a specific attention for our aim in this article. The antineutrino event rates per energy range is then computed according to the mean reactor core composition (2) and experimental site specifications (reactor and detector locations, average efficiencies and running time as described in section 6). Reactor neutrino experiments measure the survival probability Pee which does not depend on the Dirac δCP phase. In addition, the oscillation of MeV’s reactor neutrinos studied over a distance of a few kilome- ters is not affected by the modification of the coherent forward scattering from matter electrons [32, 33]. Expanding the full three flavors νe oscillation probability as a function of (∆m 21/∆m 2 ≃ 1/302 ratio and sin2(2θ13), Pee measurements from reactor experiments on the kilometer scale may be described by the simplest two flavors oscillation formula: Pee ≃ 1− sin2(2θ13) sin2 ∆m231L . (4) as long as sin2(2θ13) & 10 −3. We assume that in eq. (4), ∆m231 is measured by other experiments (MINOS [8], K2K [7] and super-K [4]). With a determination of ∆m231 better than 10 %, the impact on sin 2(2θ13) 1238U fissions only with fast neutrons. Theoretical predictions are computed by summing all known beta decay processes contributing. determination can be neglected [34]. All results within this study are, unless otherwise stated, computed for a representative value of ∆m231 [13]: ∆m231 = 2.5 +0.25 −0.25 10 −3 eV2 at 68 % C.L.. (5) 3 Generic analysis of θ13 sensitivity The calculation of event rates is a convolution of the νe flux spectrum, the cross section, the oscillation probability, the detector efficiency with the energy response function. The detector energy response simulation, as well as its correction through detector callibrations are specific to each experiment. The details of the corrections are thus beyond the scope of this article. We therefore assume in the following that the reconstructed energy is identical to the true deposited energy. We will implement, anyway, a simple energy scale systematic uncertainty (section 6.1). We based our event rates computations on an extended version of the numerical code developed for Dou- ble Chooz [34, 24]. These computations take into account the characteristics of each experimental setup as the number of reactors, detectors, locations, overburdens, efficiencies, operating time, and so on which will be described in section 6. The resulting event rates, then, form the basis for a χ2 analysis, where systematic uncertainties are properly included. Since event rates in the disappearance channel of reactor experiments are quite large, we can use a Gaussian χ2, which has the advantage of allowing a natural inclusion of systematic errors through the so-called “pull-approach” [14]: χ2 = min i=1,...,Nb D=D1,...,DN ∆Di − k,k′=1 k,k′αk′ . (6) This generic χ2 definition encompasses all the spectral information (i index) from each detector (D index) and systematics parameterization through αk and S i,k. For S i,k = 0, we recover the classical χ 2 definition, χ2no syst = , through ∆Di = i −NDi /UDi (7) where we assume that the simulated data event numbers, N i , are uncorrelated between bins and detectors. In the absence of real data, they are computed for fixed given values of ∆m231 ⋆ and sin2(2θ13) ⋆. On the other hand NDi , the theoretical model, relies on the searched sin 2(2θ13) value. We assume an uncorrelated weight error UDi which, in the absence of systematic uncertainty, is simply expressed as the statistical error: UDi = NDi . Systematic uncertainties are included in the χ2 definition (6) through αk and S i,k coefficients. The S coefficient represents the shift of the ith bin of detector D spectrum due to a 1 σ variation in the kth systematic uncertainty parameter αk. Following this definition, we introduce bin, detector and reactor correlations in the systematic errors through SDi,k definitions whereas systematic parameter correlations are gathered in Wk,k′ (we refer to the appendix for details). Eventually, some fully uncorrelated systematic uncertainties may be included through the UDi definition, by adding quadratically all their effects together with the statistical uncertainty. We will use this property to include background shape uncertainties in our analysis as described in section 5. We refer to the appendix for the proper inclusion and definition of systematic coefficients SDi,k and αk inside the χ 2 definition (6). We define the sensitivity or sensitivity limit sin2(2θ13)lim as the largest value of sin 2(2θ13) which fits the true value sin2(2θ13) ⋆ = 0 at the chosen confidence level. We therefore determine the sin2(2θ13) sensitivity at 90 % C.L. of an experiment as the value of sin2(2θ13) for which ∆χ2 = χ2(sin2(2θ13))− χ2min = 2.71 . (8) 4 Generic overview of systematic error inputs Systematic errors can be classified into three main categories: reactor, detector and data analysis induced uncertainties. In this section we provide a brief and generic description of the systematic uncertainties included in our modelization. Details concerning specific experimental cases are given in sections 6, 7. The dominant reactor induced systematic error comes from our limited knowledge of the physical processes which produce electron antineutrinos in nuclear reactors. This leads to an overall normalization error on the production rate of 1.9 % [9]. Similarily we include a 2.0 % uncertainty on the antineutrino spectral shape [15], with the conservative hypothesis that the energy bins are not correlated among themselves. Furthermore, even at a perfectly defined thermal power and with an absolute knowledge of the number of antineutrinos emitted for each fission, an underlying uncertainty remains since the nuclear energy released per fission is known to about 0.5 % [9]. In our model the last three uncertainties are taken to be fully correlated between the nuclear cores. We included another group of reactor induced systematics, taken to be uncorrelated between the reactor cores: the uncertainty on the determination of the thermal power of each nuclear core, within 0.6 – 3 % and the uncertainty on the isotopic composition of the nuclear core fuel elements, within 2 – 3 %. Also, finite size and solid angle effects, distances bias between reactors and detectors, as well as displacements of the neutrino production barycenter might affect the fluxes at the near detector(s) if they are close to the power plants (below 200 m) up to a level of 0.1 %. We did not implemented the uncertainty coming from neutrinos produced in the spent fuel pools, often located within tens of meters from the nuclear cores since we could not gather all the relevant information for the different sites. We thus neglect this effect in our simulation, whithout any justification except in the case of Double Chooz, since a detailed evaluation has shown that this uncertainty does not affect its sensitivity [24]. We point out that in particular cases this additional neutrino source slightly affects the antineutrino spectrum, and is thus relevant for experiments aiming at high sensitivities, e.g. sin2(2θ13) ∼ 0.01. We could also have introduced a specific error on the inverse-beta decay neutrino cross section of 0.1 % [9, 36]. However, being fully correlated between the detectors, the latter can be gathered into a global uncertainty, adding up to the overall neutrino rate knowledge. The basic principle of the multi-detector concept is the cancellation of the reactor induced systematics; additional contributions would not modify significantly the sensitivities of the forthcoming experiments. Let us now focus on the uncorrelated errors between detectors that could affect strongly the experimental results. The uncorrelated errors between detectors directly contribute to the relative normalization of the mea- sured antineutrino energy spectrum of each detector. One of the major improvement with respect to the CHOOZ experiment relies on the precise measurement of the number of free protons inside target volumes, proportional to the antineutrino rates. Experimentally the target mass will be determined at 0.2 %. An uncertainty on the fraction of free hydrogen per unit volume remains, at the level of 0.2 – 0.8 %, if different batches of liquid scintillator are used to fill the detectors of a given experiment. This complex case diserve a special treatment as described in section 7.2. We did not include any error associated with the measurement of the live time of the experiments. The last set of systematics concerns the selection cuts applied to extract the antineutrino signal and reject the backgrounds. Neutrino events are identified as positrons followed in time by a single neutron captured on a gadolinium nucleus. New detector designs have been proposed in order to simplify the analysis, reducing the systematic errors while keeping high statistics and high detection efficiency. We accounted for three uncertainties for both positron and neutron associated to a candidate event: the possibility of missing the particle as it escapes the target (escape), the uncertainty related to the particle interactions2 (capture) and 2Note that it affects essentially the neutrons, since the Gd concentration might differ between the detectors if they are not the identification cut based on the energy deposited in the detector active region (identification). Finally we take into account the uncertainty on the efficiency of the delay cut and an error on the neutron unicity of the event selections, whereas we do not consider any position vertex reconstruction. We provide in section 6.1 (Table 1) the detailed systematic inputs necessary for our simulations. The uncertainty on these efficiencies have been treated, otherwise stated, as uncorrelated between detectors. Uncertainties induced by the background subtractions are discussed in sections 5 and 6.3. 5 Backgrounds: description and modelization In this section we briefly review the three main kinds of background for the next generation of reactor neutrino experiments: accidental coincidences, fast neutrons and the long-lived muon induced isotopes 9Li/8He. We then describe our simplified background modelization. Naturally occurring radioactivity mostly creates accidental background, defined as a coincidence of a prompt energy deposition between 0.5 and 10 MeV, followed by a delayed neutron-like event in the fiducial volume of the detector within a few hundredths of a millisecond. Selection of high purity materials for detector construction (scintillator, mineral oil, PMT’s, steel, etc.) and passive shielding provide an efficient handle against this type of background. We assume that the accidental background rate can be measured in situ with a precision of 10 %. Cosmic ray muons will be the dominating trigger rate at the depth of all near detector sites. Even though the energy deposition corresponds to about 2 MeV per centimeter path length (providing a strong discrimination tool) they induce the main source of background. Muon induced production of the radioactive isotopes 8He, 9Li and 11Li can not be correlated to the primary muon interaction since their lifetimes are much longer than the characteristic time between two subsequent muon interactions. The characteristic signature of this last class of events consists in a four-fold coincidence (µ → n → β → n). The initial muon interaction is followed by the capture of spallation neutrons within about 1 ms. The time scale of the decay of the considered isotopes is on the order of a few 100 ms, again followed by a neutron capture. This background mimics the νe signal and is considered among the most serious difficulty to overcome for the next generation reactor neutrino experiments. In our simulation we will assume that it can be estimated to within 50 %. A further source of background are neutrons that are produced in the surrounding rocks by radioactivity and in cosmic ray muon induced hadronic cascades. In the latter case, which is dominant at shallow depth, the primary cosmic ray muon may not penetrate the detector, being thus invisible. Fast neutrons may then enter the detector and create recoil protons and be captured by hydrogen or gadolinium nuclei after thermalization. Such a sequence can mimick a νe event. In the case of Double Chooz far detector (depth of 300 m.w.e.), muon induced neutron production can be fairly well estimated from the results of the CHOOZ experiment, since it was the dominating backgroundmonitored during reactor off periods [9]. We will assume that this background rate can be estimated within a factor of two. Figure 1 illustrates the background spectra that we implemented in our modelization. We used the CHOOZ reactor off data [9] to estimate the fast neutron background spectral shape, measured to be flat. We used a simple approximate exponential shape for accidental backgrounds. Finally we implemented the spectra of 8He and 9Li based on nuclear data information; we weighted them in the ratio 0.2/0.8, respectively. We checked that slight modifications of the background shapes do not change the results of our simulations for sin2(2θ13) sensitivities above 0.01. Background rates are adjusted to match to the predictions at each detector location. Afterward they are subtracted from the total Evis spectrum in each detector. These three backgrounds, Bn, have rates and shapes known at σBD and σshp,BD , respectively. We take the conservative assumption that these shape uncertainties are fully uncorrelated between bins. Thus, the σshp,BD contributions may be directly included inside UDi (eq. (7)). These generic uncorrelated errors will then take into account the statistical and the background shape subtraction uncertainties: filled with a single batch of liquid scintillator. 0 1 2 3 4 5 6 7 8 9 10 Evis (MeV) Proton recoils Accidentals Cosmogenics Figure 1: Energy spectrum of the backgrounds from spallation neutrons, accidentals, and cosmogenic 20 % 8He and 80 % 9Li. Each curve is normalized to unity. UDi = NDi + BDn,i + σ shp,BD BDn,i As the background rate uncertainties are correlated between bins, we treat them with additional pull terms, , in eq. (6), with weights σBD included in SDi,k (see the appendix). 6 Comparison of the current proposals Several sites are currently being considered for new reactor experiments to search for θ13: Angra dos Reis (Angra, Brazil), Chooz (Double Chooz, France, and possibly Triple Chooz), Daya Bay (Daya Bay, China), Kashiwazaki (KASKA, Japan) and Yonggwang (RENO, Korea). All these experiments may be classified in two generations. The first aims to probe the value of sin2(2θ13) till 0.02 – 03, and the second to track sin2(2θ13) down to 0.01 (90 % C.L.). The first phase concerns Double Chooz, RENO, and possibly Daya Bay (with its phase I). This phase should end by 2013. Angra, Daya Bay (nominal setup with 8 detectors), KASKA and possibly Triple Chooz are focusing on the second phase. For these second generation experiments, a significant R&D effort is required since the effective Gd-scintillator mass will be increased by, at least, one order of magnitude, and systematics, as well as backgrounds uncertainties, have to be further reduced with respect to the first phase experiments. In the following comparisons we will not include Angra, KASKA and Triple Chooz. We will only focus on Double Chooz, Daya Bay and RENO. In the next sections we first introduce a generic discussion on systematics, scintillator composition and backgrounds (sections 6.1 to 6.3). We will then shortly describe each setup and compute the associated baseline sensitivity (sections 7.1 to 7.3). We then perform a comparative analysis of the setups in section 8 to highlight advantages and drawbacks of each setup. 6.1 Detailed systematics review The two Double Chooz [24] and Daya Bay [25] proposals take a careful inventory of systematics, compiled in Table 1 for comparison. Double Chooz and Daya Bay estimates for the case of no additional R&D are at hand. We decided to strictly use the systematic errors (Table 1) and background values quoted by the collaborations [24, 25, 26]. However, we could not find any detailed background estimate for the RENO and Daya Bay Mid site setups. In the latter cases we use a simple model to estimate the background subtraction uncertainties from the scaling of the Double Chooz far detector (see section 6.3). Error Description CHOOZ Double Chooz Daya Bay No R&D R&D Absolute Absolute Relative Absolute Relative Relative Reactor Production cross section 1.90 % 1.90 % 1.90 % Core powers 0.70 % 2.00 % 2.00 % Energy per fission 0.60 % 0.50 % 0.50 % Solid angle/Bary. displct. 0.07 % 0.08 % 0.08 % Detector Detection cross section 0.30 % 0.10 % 0.10 % Target mass 0.30 % 0.20 % 0.20 % 0.20 % 0.20 % 0.02 % Fiducial volume 0.20 % Target free H fraction 0.80 % 0.50 % ? 0.20 % 0.10 % Dead time (electronics) 0.25 % Analysis (paticle id.) e+ escape (D) 0.10 % e+ capture (C) e+ identification cut (E) 0.80 % 0.10 % 0.10 % n escape (D) 0.10 % n capture (% Gd) (C) 0.85 % 0.30 % 0.30 % 0.10 % 0.10 % 0.10 % n identification cut (E) 0.40 % 0.20 % 0.20 % 0.20 % 0.20 % 0.10 % νe time cut (T) 0.40 % 0.10 % 0.10 % 0.10 % 0.10 % 0.03 % νe distance cut (D) 0.30 % unicity (n multiplicity) 0.50 % 0.05 % 0.05 % Total 2.72 % 2.88 % 0.44 % 2.82 % 0.39 % 0.20 % Table 1: Breakdown of the systematic errors included in the computation of the sensitivity of Dou- ble Chooz [24] and Daya Bay [25]. Since no breakdown of the RENO systematic errors has been pub- lished we use the same systematic error budget as for Double Chooz. Double Chooz and Daya Bay relative systematics are almost comparable, the only main difference coming from the determination of the gadolin- ium concentration and the free proton fraction inside the target volume. The absolute determination of the free proton fraction can have some impact in Daya Bay since multiple batches will be used to fill all the 8 detectors (see section 7.2 for details). Nevertheless, there is no published value in the Daya Bay proposal [25]. Through all the available publications [24, 25, 26, 28], the differences between the systematics are found only for the relative normalization of the dectetor (σrel) and the subtraction uncertainties on background rates (σBD ). From section 4 and Table 1, we thus group systematics in two categories to be used in our χ2 analysis eq. (6): 1. generic systematics common to all the experiments (Table 2): – σabs, the theoretical uncertainty on reactor antineutrino spectrum prediction. We call it also the absolute normalization of event rates (since common to all the detectors), extracted from Table 1 without power uncertainty contributions. σabs is at the level of 2 %; – σshp, the theoretical reactor spectrum shape uncertainty, at the level of 2 %; – σabs , σrel , the absolute and relative energy scale uncertainties, roughly assessed at the level of 0.5 % each; – σpwr, the reactor thermal power uncertainty, at the level of 2.0 %; – σcmp, the reactor core specific fuel composition uncertainty which is roughly at the level of the power uncertainties (2 – 3 %) on each fuel element. σabs σshp σ σpwr σcmp 2.0 % 2.0 % 0.5 % 0.5 % 2.0 % 2 – 3 % Table 2: Generic systematic uncertainties as included in the χ2 analysis. For more details about the correlations between the systematics, we refer to the appendix, and more particularly to Table 15. 2. specific systematics: – σrel, the relative normalization of event rates between all the detectors. This uncertainty is uncorrelated between detectors; – σBD , the background subtraction unceratinties, described in section 6.3. 6.2 Impact of the scintillator composition In this section we stress the impact of the scintillator composition on the sensitivity of reactor neutrino experiments. All current projects will use a gadolinium doped liquid scintillator to enhance the neutron capture. The long-term stability of this scintillator is among the most difficult experimental challenges, since a degradation of the scintillator transparency would induce large systematic uncertainties. In the following we consider a stable scintillator for all experiments. The choice of the scintillating base has some importance since it defines the free proton number per unit volume, the νe rate and proton recoil background rate. Similarly the 12C number per volume drives the long-lived muon induced isotopes in the target scintillator. Different bases can be used as neutrino target scintillator, mixture of dodecane (DOD), pseudocumene (PC) or phenylxylylethane (PXE), or linear alkylbenzene (LAB), as described in details in Table 3. If we consider the Double Chooz scintillator (80 % DOD + 20 % PXE) as our reference, a pure LAB scintillator contains 4.9 % less free proton per volume, and 9.5 % more carbon atoms. In the following we will assume the Daya Bay [25] and RENO [26] experiments will use pure LAB as the target scintillator, and we will renormalize the neutrino and the backgound rates accordingly in section 7.2 and 7.3. 6.3 Estimation of the µ-induced backgrounds Our estimates of the backgrounds induced by cosmic muons is based on the Double Chooz proposal [24]. A modification of the MUSIC code [20] was used to compute the muon rates and energy spectra by propagating Liquid Formula density 1028H/m3 1028C/m3 Dodecane (DOD) C12H26 0.753 6.93 3.20 pseudocumene (PC) C9H12 0.88 5.30 3.97 phenylxylylethane (PXE) C16H18 0.985 5.08 4.52 90 % DOD+10 % PC mixture in vol. 0.77 6.77 3.32 80 % DOD+20 % PXE mixture in vol. 0.80 6.56 3.46 linear alkylbenzene (LAB) C16H30 0.86 6.24 3.79 Table 3: Impact of the scintillator composition on the target free proton number driving the νe rate as well as the recoil proton background, and on the carbon composition which drives the long-lived muon induced isotopes produced in the detector. In the following we consider that Double Chooz is using modules containing 8.26 tons of 80 % dodecane +20 % PXE (in volume) based target scintillator, and that Daya Bay and RENO are both using 20 ton modules of a LAB based target scintillator. surface muons through rock. The site topographies have been included, according to a digitized map of the Chooz hill profile [22] in the case of the far site (300 m.w.e. overburden), and according to a flat topography in the case of the near site (80 m.w.e. overburden). For the far site this full Monte-Carlo simulation predicts a muon flux Φµ=0.612± 0.007 m−2 s−1, slightly higher than the approximate measured value quoted in [9]. The mean muon energy computed according to this method is 〈Eµ〉 = 61 GeV. For the near site, we get Φµ = 5.9 m −2 s−1, and 〈Eµ〉 = 22 GeV. A similar detailed computation was performed for the three sites of the Daya Bay collaboration [25]. Thus, for both the cases of Double Chooz and Daya Bay we use only the values of the muon flux and mean energy computed by the collaborations. However, we could not find any published data for the case of RENO. We then use the underground muon fluxes and mean energies calculated analytically following [21]. In order to justify this approximation, Table 4 reports the muon flux and mean energy computed by the Double Chooz and the Daya Bay collaborations, from 300 m.w.e. to 923 m.w.e., as well as the analytical computations according to [21]. We found a reasonable agreement which bears out the use of the analytical model [21] for the case of the RENO sites (depths of 255 m.w.e. and 675 m.w.e.). Nevertheless, note that the analytical simulation assumes a flat topography. It is worth noting, however, that the mean muon energy predicted by the analytical computation is systematically ∼ 25 % lower in the depth range of interest. Therefore we arbitrarily renormalize the analytical calculation by 25 % to estimate the mean muon energy at the RENO sites. We also note a large discrepancy between the detailed computation and the analytical model of [21] for the Double Chooz near site, probably due to its very shallow depth. Cross sections of muon induced isotope production on liquid scintillator targets (12C) have been measured by the NA54 experiment at the CERN SPS muon beam at 100 GeV and 190 GeV muon energies [23]. The energy dependence was found to scale as σtot(Eµ) ∝ 〈Eµ〉α with α = 0.73± 0.10 averaged over the various isotopes produced. We consider in the following that both long-lived muon induced isotopes and muon induced fast neutron backgrounds scale as Φµ ×〈Eµ〉α, our reference being taken at the Chooz far site (full Monte-Carlo simulation). We define the depth scaling factor as DSF = (Φµ 〈Eµ〉α)/(Φµ 〈Eµ〉α)Double Chooz far , (10) which is illustrated in the last column of Table 4 for various detector sites. Daily background rates computed for Daya Bay, Double Chooz, and RENO detectors at the different sites are then given in Table 5. Two cases are considered and compared: the background rates taken from the literature when available, and the background rates extrapolated from the background computed for the Double Chooz far detector, scaled with the target mass, the scintillator free proton and carbon numbers, as well as the depth scaling factor (DSF). We note here the good agreement between the Daya Bay cosmogenic induced backgrounds (9Li/8He and fast neutrons) estimated from our Double Chooz extended model and the original estimates of the Daya Bay collaboration. This bears out the use of our model for the RENO and Daya Bay mid site configurations. For the case of the 9Li/8He background, we understand well this agreement since the background rate mainly depends on the mass of the neutrino target region. The agreement is more Site depth (m.w.e.), Detailed simulation Analytical model DSF topography Φµ 〈Eµ〉 Φµ 〈Eµ〉 m−2 s−1 GeV m−2 s−1 GeV DC near 80, flat 5.9 22 9.9 17 6.80 RENO near 230, hill — — 1.2 40 1.57 DB near 1 255, hill 1.2 55 0.9 44 1.32 DB near 2 291, hill 0.73 60 0.72 49 1.06 DC far 300, hill 0.61 61 0.67 50 1 DB mid 541, hill 0.17 97 0.15 71 0.32 RENO far 675, hill — — 0.084 94 0.20 DB far 923, hill 0.04 138 0.035 118 0.10 Table 4: Muon flux Φµ and mean energy 〈Eµ〉 for the underground site of the reactor neutrino experiments. We compare the values obtained from a full Monte-Carlo simulation for Double Chooz and Daya Bay to the analytical model of [21]. We use the latter model for RENO. The depth scaling factor (DSF) is defined by the product (Φµ×〈Eµ〉α)/(Φµ×〈Eµ〉α)Double Chooz far, the Double Chooz far site is taken as the reference. Backgrounds induced by cosmic muons are scaled according to this factor. Detector Accidental (d−1) µ-induced fast-n (d−1) µ-induced 9Li/8He (d−1) Site Original DC ext. Original DC ext. Original DC ext. Double Chooz near 13.60 ± 1.36 1.36 ± 1.36 9.52 ± 4.76 RENO near — 7.10 ± 0.71 — 0.68 ± 0.68 — 5.40 ± 2.70 Daya Bay DB 1.86± 0.19 5.98 ± 0.60 0.50 ± 0.50 0.57 ± 0.57 3.7 ± 1.85 4.55 ± 2.27 Daya Bay LA 1.52± 0.15 4.76 ± 0.48 0.35 ± 0.35 0.45 ± 0.45 2.5 ± 1.25 3.63 ± 1.81 Double Chooz far 2.00± 0.20 — 0.20 ± 0.20 — 1.40 ± 0.70 — Daya Bay mid — 1.45 ± 0.14 — 0.14 ± 0.14 — 1.10 ± 0.57 RENO far — 0.90 ± 0.09 — 0.09 ± 0.09 — 0.69 ± 0.35 Daya Bay far 0.12± 0.01 0.44 ± 0.04 0.03 ± 0.03 0.04 ± 0.04 0.26 ± 0.13 0.33 ± 0.17 Table 5: Daily background rates computed for Daya Bay, Double Chooz, and RENO detectors at the different sites. We consider the three main background sources: accidental events, µ-induced fast neutrons, and µ-induced 9Li/8He. The columns labelled “Original” quote the background rates taken from the literature when available. The columns labelled “DC ext.” (for extended) quote the background value extrapolated from the background computed for the Double Chooz far detector, scaled with the detector target mass, the scintillator free proton and carbon numbers, as well as the depth scaling factor (DSF). Background subtraction systematic errors (%) Detector Accidental µ-induced fast-n µ-induced 9Li/8He Site Original DC ext. Original DC ext. Original DC ext. Double Chooz near 0.123 0.123 0.043 RENO near — 0.019 — 0.019 — 0.074 Daya Bay DB 0.020 0.064 0.054 0.061 0.199 0.245 Daya Bay LA 0.020 0.063 0.046 0.060 0.164 0.239 Double Chooz far 0.292 — 0.292 — 1.020 — Daya Bay mid — 0.120 — 0.115 — 0.458 RENO far — 0.100 — 0.095 — 0.382 Daya Bay far 0.010 0.036 0.025 0.035 0.108 0.138 Table 6: Background subtraction systematic errors (in percent) computed for Daya Bay, Double Chooz, and RENO detectors at the different sites. We consider the three main background sources: accidental events, µ-induced fast neutrons, and µ-induced 9Li/8He. The columns labelled “Original” quote the systematic errors taken from the literature when available. The columns labelled “DC ext.” (for extended) quote the systematic errors extrapolated from the Double Chooz far detector, taking into account the estimated detector signal to background ratio as well as the background rate uncertainty. surprising, however, for the case of the fast neutron background, since the size of the liquid shielding around the detector active area is rather different between the Double Chooz and Daya Bay detectors (the RENO detector design is very close to the Double Chooz case). In addition, further detector differences such as the thickness of the buffer oil shielding the inner target region, as well as the different mechanical structure explain the discrepancy between the Daya Bay computation and the DC extended model for the case of the accidental background. Nevertheless, we found out that these differences influence only weakly the sensitivity computed for the three experiments since the accidental background energy spectrum is different enough from the expected oscillation signal, and it is supposed to be known with a precision of 10 %. In a similar way, Table 6 gives the background subtraction systematic errors (in percent) computed for Daya Bay, Double Chooz, and RENO detectors at the different sites, taking into account the estimated detector signal to noise ratios as well as the background rate uncertainties. 7 Reactor experiments baseline sensitivity 7.1 Double Chooz The Double Chooz collaboration is composed of institutes from Brazil, France, Germany, Japan, Russia, Spain, United Kingdom, and the United States. The experimental site is located close to the twin reactor cores of the Chooz nuclear power station (two PWR3 producing 8.5 GWth), operated by the French company Électricité de France (EDF). The two, almost identical, detectors will contain a 8.3 ton fiducial volume of liquid scintillator (density of 0.8) doped with 0.1 g/l of gadolinium (Gd). The far detector will be installed in the existing laboratory, 1.05 km from the cores barycenter, shielded by 300 m.w.e. of rock. This detector should be operating alone for 1.5 – 2 years (DC Phase I), starting data taking by the end of 2008. The second detector will be installed in the meantime about 280 m from the nuclear core barycenter, at the bottom of a 40 m shaft (80 m.w.e.) to be excavated. Distances between detectors and nuclear cores as well as site overburdens are given in Table 7. Since there are no more than two NPP cores, it is still possible to install the near detector at a suitable position where the ratio of reactor νe fluxes from each core is the same as for the far detector (the iso-ratio curve is plotted on figure 2). This allows reactor relative uncertainty cancellations (NPP core compositions). This detector should be operational by 2010, and will take data 3Pressurized Water Reactor Detector near far Distance from West reactor (m) 290.7 1114.6± 0.1 Distance from East reactor (m) 260.3 997.9± 0.1 Detector Efficiency 80 % 80 % Dead Time 25 % 2.5 % Rate without efficiency (d−1) 977 66 Rate with detector efficiency (d−1) 782 53 Integrated rate (y−1) 1.67 105 1.48 104 Table 7: Double Chooz antineutrinos rate expected in the near and far detectors, with and without reactor and detector efficiencies. The integrated rate in the last line includes detector efficiency, dead time, and reactor off periods averaged over a year. The averaged reactor global load factor is estimated at 79 % [30]. Figure 2: Double Chooz experiment site configuration. We show also on this figure the far flux iso-ratio line. Another detector located on this particular curve will receive the same reactor flux ratio as for the far detector: 44.5 % from West and 55.5 % from East reactor. The near detector of Double Chooz is foreseen to be placed on this line. for three years (DC Phase II). Other details concerning the experiment may be found in the collaboration proposals [24]. For the Double Chooz phase I (DC Phase I) analysis, we used systematics of Table 2, setting σrel and to 0 since only one detector will be present. For a data taking period of 1.5 years, the sensitivity is sin2(2θ13)lim = 0.0544. The second phase (DC Phase II) will then start and both detectors will take data for 3 years as scheduled in the proposal. The full experiment will then achieve a final sensitivity sin2(2θ13)lim = 0.0278, assuming systematics from Table 2 and σrel = 0.6 % [24]. The sensitivity worsens for ∆m231 < 2.5 10 −3 eV2, due to the close distance of the far detector4. If we take the lower bound [12, 13] on ∆m231 (2.0 10 −3 eV2), Double Chooz will lose 30 % in sensitivity whereas for the upper bound [12, 13] on ∆m231 (3.0 10 −3 eV2), Double Chooz will gain 15 % in sensitivity. 7.2 Daya Bay Daya Bay is an experiment proposed by institutes from China, the United States, and Russia. Daya Bay [25] will be located in the Guang-Dong Province, on the site of the Daya Bay nuclear power station. The site is made up of two pairs of twin reactors, Daya Bay (DB) and Ling Ao I (LA I). An additional pair of reactors, Ling Ao II (LA II), is currently under construction and should be operational by 2010 – 2011 [25]. Each core has a thermal power of 2.9 GW [30]. In the full installation setup 3.3 km of tunnel and 3 detector halls have to be excavated, in order to accommodate 8 detector modules [25]. Each module contains an effective volume of 20 tons of Gd-loaded LAB liquid scintillator (Table 3). Distances between detectors and nuclear cores as well as site overburdens are given in Table 8. This site yields a rather complex signal composition Detector near DB near LA mid far Distance from DB 1 (m) 350 1,356 1,153 1,970 Distance from DB 2 (m) 381 1,331 1,161 2,000 Distance from LA I 1 (m) 942 492 783 1,619 Distance from LA I 2 (m) 1,030 475 818 1,623 Distance from LA II 1 (m) 1,378 500 968 1,602 Distance from LA II 2 (m) 1,463 555 1,029 1,624 Detector eff. 80 % 80 % 80 % 80 % Dead Time 7.2 % 4.3 % 1 % 0.2 % Rate without eff. (d−1) 1,938 1,813 494 430 Rate with detector eff. (d−1) 1,550 1,450 395 344 Integrated rate (y−1) 4.10 105 3.95 105 1.11 105 9.77 104 Table 8: Daya Bay antineutrino rates expected in the near and far detectors, with and without reactor and detector efficiencies. The integrated rate in the last line includes detector efficiency, dead time, and reactor off periods averaged over a year. We assumed that LA II NPP will be operating for the time the far site will be fully installed, but for the Mid site installation we assumed that LA II will be off. in each detector coming from up to 6 different NPP cores, as shown in Table 9. According to the Day Bay proposal, our estimate of the sensitivity is sin2(2θ13) = 0.0085 very close to the Daya Bay quoted value (sin2(2θ13) = 0.008). However, in the Daya Bay proposal, the uncertainties on the reactor fuel composition, σcmp, as well as the energy scale associated uncertainties σ and σrel are neglected. Taking into account these systematics (Table 2) and the quoted value of σrel = 0.39 % with no R&D [25], our estimate of the Daya Bay final sensitivity is then sin2(2θ13) = 0.009, with the full installation (DB Phase II, Figure 3) after 3 years of data taking. We draw the attention of the reader on the point that these computations are based on the assumption that σrel = 0.39 % is fully uncorrelated between all the detectors, and in particular 4a bit too close to the NPP to get the maximum amount of information from first minimum of oscillation over the reactor spectrum distortion DB LA1 LA2 near 1 (DB) 83.1 % 11.4 % 5.5 % near 2 (LA) 6.5 % 50.6 % 42.8 % Mid (LA2 OFF) 32.3 % 67.7 % 0.0 % Mid (LA2 ON) 22.5 % 47.1 % 30.4 % far 24.9 % 37.4 % 37.7 % Table 9: Daya Bay rate contributions from each NPP set (2 cores by set) while assuming, except if otherwise noticed (3rd line), the new Ling Ao II NPP is operating at full power. between detectors on a same site. Nevertheless this hypothesis is not guarenteed. We discuss this point as well as the current filling scenario (multi-batches) just below. Figure 3: Daya Bay installation phases and site configuration. On the left: phase I, with 2 × (2 × 20 t) detectors, Daya Bay and Ling Ao I power plants are operating. On the right: phase II with 2× (2× 20 t) near sites, and 4× 20 t at far site, all three power plants are operational. A preliminary fast phase (DB Phase I) is proposed by the Daya Bay collaboration. This phase includes only 4 detectors, 2 of them located at the DB near site and the other 2 at the mid site (see Figure 3). Taking systematics from Table 2 and σrel = 0.39 % fully uncorrelated between detectors, we get a sensitivity after 1 year of data taking of sin2(2θ13) = 0.040. if LA II NPP is off. If DB Phase I starts after 2010, with LA II operational, we get a sensitivity of sin2(2θ13) = 0.038 still after 1 year of data taking. The reason for a better sensitivity in the second scenario is that the mid detectors get 44 % more νe events (Table 8, 9 and Figure 3), with a larger oscillation baseline with respect to LA I. Daya Bay site correlation The main concept of the Daya Bay experiment is based on the multi-inter-calibration of detectors. Since many detectors are installed on a same site, the total uncorrelated uncertainty of a site is decreased by a factor of 1/ NSd compared to the single detector uncorrelated uncertainty, where N d is the number of detectors on a given site. In the Daya Bay proposal, the full relative uncertainty (σrel = 0.39 %) is assumed to be uncorrelated between all the detectors. However, the fraction of correlated and non-correlated error between detectors of a same site is not trivial. It relies on many experimental assumptions on uncertainty correlations. The absence of correlations between detectors on a same site implies there will be no detector- to-detector correction applied. If detector responses happen to be different, any data correction from detector to detector on a same site would yield correlations between detector uncertainties. For DB Phase II, if we assume that σrel is fully uncorrelated between the detectors, we get a sensitivity of sin2(2θ13) = 0.009. On the contrary, if we assume that σrel is fully uncorrelated between detectors in different sites, but fully correlated between detectors on a same site, we get a sensitivity of sin2(2θ13) = 0.012. The real sensitivity should lay between these two extremes. For DB Phase I, if we assume σrel is fully correlated on a same site and fully uncorrelated on distant sites, we get sin2(2θ13) = 0.041 if LA II is off, and sin 2(2θ13) = 0.038 if LA II is on. We conlude from these results that DB Mid sensitivity does not strongly depend on the correlations in σrel. More generally, DB Mid setup weakly depends on σrel. This latter point will be discussed in section 8, together with a full description of the impact of each systematic on the forseen sensitivity. Daya Bay and the filling procedure A large amount of Gd-loaded liquid scintillator will have to be produced, stored and filled into the detectors (8×20 tons, which is 10 times more than in Double Chooz). Due to the large number of detectors and the large amount of liquid to manage, the Daya Bay collaboration plans to fill detectors with four different Gd-doped liquid scintillator batches. A single batch will be used to fill detectors by pairs [25] (Figure 4). The best installation scenario (which is the one chosen by the Daya Bay collaboration) is then to move one of the filled detector to a near site, the other one to the far site (scheme 1 of Figure 4 and Table 10). Another batch of Gd-loaded liquid scintillator will be used for the next pair of detectors and so on. With the adopted filling procedure [25], extra systematic uncertainties on the hydrogen content between different batches have to be included. DB LA Pair 1 Pair 2 Pair 3 Pair 4 Scheme 1 Scheme 2 Scheme 3 Scheme 4 Scheme 5 Scheme 6 Figure 4: Possible installation of detector pairs in the Daya Bay experiment according to the adopted filling procedure [25]. Due to the large number of detectors, and the large amount of liquid to be managed, the Daya Bay collaboration plans to fill detectors by pairs with four different Gd-doped liquid scintillator batches. The forseen detector installation scenario [25] corresponds to scheme 1. We illustrate here other installation possibilities (schemes 2 to 6). Although the relative uncertainty on the free proton content within a same batch could be kept at a very low level (0.2 % in the Daya Bay proposal [25], negligeable in the Double Chooz proposal [24]), it is not necessarily true in the case of different batches, in which the chemical composition may slightly change. The free proton content between different batches relies then on the measurement of this quantity. In the CHOOZ experiment [9], the free proton fraction inside the Gd-loaded liquid scintillator was known to 0.8 %. In Double Chooz, this uncertainty is assessed at the level of 0.5 %. Since there is no published value on the absolute determination of this quantity in the Daya Bay proposal, we assume here, as in Table 1, a 0.5 % uncertainty between different batches. The filling systematic coefficients are explained in Table 10 (refer to the appendix and to Table 15 for full description). The uncertainty on the free proton fraction of a single Scheme 1 Scheme 2 Scheme 3 Scheme 4 Scheme 5 Scheme 6 0.008 0.01 0.012 0.014 0.016 (2θ13) Single batch Four batches Figure 5: Sensitivity on sin2(2θ13) at 90 % C.L. after 3 years of data taking for the 6 different installation schemes illustrated on Figure 4, with σpair = 0.5 %. Left bounds are computed assuming uncorrelated errors between all the detectors and right bounds are for the assumption of full correlation of σrel between detectors of a same site. The real sensitivity should be somewhere in between these two bounds. For comparison, we also show on this graph the computations for the single batch hypothesis where we take σrel = 0.39 %. Note that obviously the first installation scheme provides the best sensitivity. In these results we do not include any detector swapping scenario. Error type k SD Filling (Ns = 5Nd +Nb + 5Nr + 2) of batch 1 Ns +1 σpair αpair,1 of batch 2 Ns +2 σpair αpair,2 of batch 3 Ns +3 σpair αpair,3 of batch 4 Ns +4 σpair αpair,4 Table 10: Daya Bay, phase II (8 detectors), specific filling systematic parameters table in addition to standard one (see the appendix for details). Here we adopt a fully uncorrelated uncertainty of the different batches, σpair = 0.5 %, but fully correlated in a same batch. batch is taken to σpair = 0.5 %. This uncertainty is taken to be fully correlated when detectors are filled with the same batch and fully uncorrelated otherwise. According to this filling procedure, the final sensitivity of DB Phase II would be sin2(2θ13) = 0.0093 instead of 0.0089 with the initial installation scenario (scheme 1 of Figure 4). In all the other configuration schemes (2 – 6), which allows comparing on a same site at least two detectors filled with the same batch, the sensitivity is more largely weakened (Figure 5), to an extent depending on σpair. Note that we did not include any detector swapping option in previous conclusions. In the Day Bay proposal, the retained installation scenario is the first scheme of Figure 4. The baseline swapping option is then the permutation of two detectors filled with the same batch, in 4 steps, 1 step per batch. On the one hand, the drawback of such a swapping scenario is that two detectors filled with the same batch will never be directly compared. On the other hand, configuration schemes 2 – 6 allow detector intercalibration within a pair. However, it should be noticed that the time spent in any configuration different from scheme 1 may decrease the combined final sin2(2θ13) sensitivity (Figure 5). 7.3 RENO The RENO experiment [26] will be located close to the Yonggwang nuclear power plant in Korea, about 400 km south of Seoul. The power plant is a complex of six PWR reactors, each of them producing a thermal power of 2.74 GW [30]. The Yonggwong power station is ranked number 4 in the world, with a total thermal power of 16.4 GW. Its power rating is often cited as an advantage of the RENO experiment. These six reactors are equally distributed on a straight segment spanning over 1.5 km. The average cumulative operating factors for the reactors are all above 80 %. Figure 6 shows the foreseen layout of the experimental site. The near and far detectors will be located 150 m and 1,500 m away from the center of the reactor Figure 6: RENO experiment site configuration. row (Table 11), and will be shielded by a 88 m hill (230 m.w.e.) and a 260 m “mountain” (675 m.w.e.) respectively. Two neutrino laboratories have to be excavated and equipped in order to host the detectors. They will be located at the bottom of two tunnels having a length of 100 m and 600 m for near and far detector, respectively. In this configuration, the flux contribution from R1 to R6 ranges from 3 % to 39 % for the near detector, and from 15 % to 18 % for the far detector (Table 12). Assuming the sytematics of Table 1 and, lacking of published data, fixing σrel = 0.6 % as for Double Chooz, we obtain a final sensitivity after 3 years of data taking of sin2(2θ13) = 0.021. This is calculated for 2× 20 t detectors. However, a recent talk [27], quotes a different detector size, with 15 t of Gd-doped liquid scintillator. In that case, the final sensitivity after 3 years, would be sin2(2θ13) = 0.023. As seen from Table 12, the near detector monitors mainly the two central NPP cores, and the experiment sensitivity is quite affected by the reactor power uncertainties. If we compute the sensitivity with only the 2 central NPP cores, we get even better results compared to the full NPP configuration. This means that four of the six cores are useless for the sin2(2θ13) measurement. This can be understood by the fact that the Detector near far Distance from R1 (m) 765 1677 Distance from R2 (m) 474 1566 Distance from R3 (m) 212 1507 Distance from R4 (m) 212 1507 Distance from R5 (m) 474 1566 Distance from R6 (m) 765 1677 Detector Efficiency 80 % 80 % Dead Time 7.2 % 0.5 % Rate without efficiency (d−1) 2859 121 Rate with detector efficiency (d−1) 2287 97 Integrated rate (y−1) 6.20 105 2.82 104 Table 11: RENO antineutrino rates expected in the near and far detectors, with and without reactor and detector efficiencies. The integrated rate in the last line includes detector efficiency, dead time, and reactor off periods averaged over a year. R1 R2 R3 R4 R5 R6 near 3.0 % 7.8 % 39.2 % 39.2 % 7.8 % 3.0 % far 14.8 % 16.9 % 18.3 % 18.3 % 16.9 % 14.8 % Table 12: RENO rate contributions from each reactor core. sensitivity gained by statistics is compensated by a loss of information on the νe rates and energy spectra. Thus the appeal of this site is diminished. The sensitivity is equivalent to a 5.8 GWth NPP reactor neutrino experiment for σpwr = 2.0 %. The RENO collaboration considers using 3 small very near detectors (200 – 300 kg) to monitor sub-groups of cores of the NPP. However, taking into account current knowledge on reactor spectra (σabs = 2.0 %, σshp = 2.0 % [15]), even dedicated detectors with 10 5 νe events will not improve the thermal power knowledge below ∼ 2 %. Thus, the overall sin2(2θ13) sensitivity will not improve. 8 Discussion We develop here a two step comparison of the experiments described before: Double Chooz phase I (single detector, DC I ), phase II (both detectors, DC II ), Daya Bay phase I (DB Mid) and phase II (DB Full) and, RENO (RN ). The first elements of comparison are based on a single core equivalent approach. Although purely hypothetical, this analysis provides a lot of information on the impact of the layout of the site (locations of NPP reactor cores and detectors). The second approach, giving far more information on the impact of systematics on each experiment, is based on the pulls-approach, presented in section 3 and detailed in the Appendix. Note that in the following discussion we adopt the approximation that all the NPP cores operate with the same average efficiency. However, this assumption is not guaranteed, and, especially for NPP with many reactors such as RENO and Daya Bay experiments, running time and procedure of each NPP core have to be taken into account in the final analysis. For the Double Chooz experiment, which places the near detector on the flux iso-ratio line of the far detector, the full running operation time and procedure of each core is not needed to perform the final analysis. 8.1 The single core equivalent approach First of all we may simplify each experiment to its roughest single core equivalent (SCE) with total matching power P = r=1 P r, with P r the power of the rth reactor and Nr the number of available reactors on site. In this case, we compute the sin2(2θ13) sensitivity for each experiment (see Table 13) for their baseline option [25, 24, 26] but also for a single core equivalent, and the averaged near and far detector locations computed with )−1/2 . (11) This average distance, L, yields the same event rates associated to near and far detectors of each experiment except for the Daya Bay Phase II experiment. For this particular case, we have to compute this average distance in two steps for the equivalent near detector. In this special case, since there are two near sites, we compute the L for the DB near site, LDB, and for the LA near site, LLA, and then compute the overall single near detector equivalent distance as )−1/2 . (12) Following this SCE simplification, numerical values of sin2(2θ13) sensitivity are gathered in Table 13. As DC DB Mid DB Full RN LA II OFF LA II ON L (in r.n.u.) 17 67 101 303 71 Lfar (in m) 1,051 931 951 1,716 1579 Lnear (in m) 274 484 576 441 ⋆ 325 sin2(2θ13)lim 0.0278 0.0410 0.0381 0.0110 0.0213 sin2(2θ13) lim 0.0274 0.0274 0.0289 0.0105 0.0176 Table 13: In this table we provide the luminosity (expressed in r.n.u., see eq. (3)), sin2(2θ13)lim at 90 % C.L. of Double Chooz (DC II ), Daya Bay phase I (DB Mid) and phase II (DB Full), and RENO (RN ). Also quoted in this table are the average distances, L, to a single equivalent core (SCE) with total power r=1 P r (see text for explanations, the star (⋆) indicates a particular treatement). The sensitivity in the SCE case, sin2(2θ13) lim , is then computed to highlight possible drawback on the site configuration by comparison with the baseline sensitivity sin2(2θ13)lim. a first remark, the biggest discrepancy between sin2(2θ13)lim and sin 2(2θ13) is as large as 40 % for the DB Mid experiment. With four to six times higher luminosity compared to DC, a farther near site from the cores and a closer far site, the sensitivity for the hypothetical DB Mid SCE experiment is not improved with respect to DC. The discrepancy between DB Mid and DB Mid SCE clearly comes from the wide repartition of NPP. LA cores can not be properly monitored at the DB near site. We conclude that the DB Mid experiment is half way between an experiment with a single far detector and an experiment with two identical detectors, one near and one far. One may notice that DC is an experiment where the reactor cores may be considered as a single equivalent core with double power since there is only a less than 1 % difference between DC and DC SCE. This is not the case for the RN experiment where the relative discrepancy between RN and RN SCE is at the level of 15 %. Daya Bay Phase II, thanks to its two near sites, is only marginally affected by a ∼ 5 % difference between DB Full and DB Full SCE (more on section 8.2). 8.2 The pulls analysis In this discussion we are interested in assessing how much a given experiment sensitivity relies on the knowldege of the systematics (e.g.the number of systematics and the impact on sensitivity). The pulls- approach is perfectly adapted to this analysis. The idea is to break down the total ∆χ2 ∆χ2(slim) = χ 2(slim)− χ2min (13) = min {α1,...,αK} χ2(slim, α1, . . . , αK)− min {s,α1,...,αK} χ2(s, α1, . . . , αK) , (where s is shorthand for sin2(2θ13)) (14) into sub-parts δχ2i which represent their relative contribution to the overall ∆χ 2 of eq. (13). Since the sensitivity limit on sin2(2θ13) is computed at 90 % C.L., ∆χ 2 has a common value for every experiment and is equal to 2.71. Thus, δχ2i is defined as δχ2i = ith pull term with the trivial induced normalization: i = 1. Table 14 shows the results of our computation for the baseline option of each experimental setup. We report, together with the final sensitivities discussed earlier, the relative contributions δχ2i where δχ N1,N2,F are the observables (the contribution from the first term in eq. (6) for the respective detector) and the other δχ2i are the pulls (the weight term contributions in eq. (6)). Since the role of the far detector is to determine δχ2i in % DC I DC II DB Mid DB Full RN δχ2N1 — 3.0 % 4.3 % 1.1 % 1.5 % δχ2N2 — — — 3.3 % — δχ2F 29.5 % 38.0 % 23.4 % 31.2 % 34.4 % 29.1 % 1.5 % 9.0 % 1.0 % 7.9 % 18.4 % 1.3 % 8.4 % 0.5 % 1.0 % δχ2rel — 48.3 % 6.6 % 56.8 % 28.5 % δχ2scl,abs 6.5 % 1.2 % 6.1 % 0.1 % 0.1 % δχ2scl,rel — 5.0 % 11.8 % 1.6 % 0.2 % 1.0 % 0.8 % 0.4 % 0.1 % 0.5 % δχ2pwr 14.7 % 0.8 % 27.3 % 3.9 % 16.9 % δχ2cmp 0.1 % 0.0 % 0.6 % 0.1 % 9.1 % δχ2ε 0.6 % 0.0 % 2.2 % 0.3 % 0.0 % sin2(2θ13)lim 0.054 0.028 0.041 0.011 0.021 Table 14: Relative contributions, δχ2i , to the global ∆χ 2. The higher is the δχ2i contribution, the more the sin2(2θ13) sensitivity depends on the considered parameter. In red we highlight the main contributions (20 – 60 %), in orange other significant terms (5 – 20 %). All the values are calculated for the base case of each experimental setup. For the DB Full setup, where correlations between detectors on a same site are possible, we take σrel half correlated and half uncorrelated between detectors of a same site, and completely uncorrelated between detectors of different sites (see section 7.2 for details). Note that for Daya Bay we do not include here batch-to-batch uncertainties. sin2(2θ13), it is obvious that the associated residual should significantly contribute to the sensitivity. How- ever what we see is that all computed sensitivities mainly depend on systematics, which contribute from 60 % to 70 % of the overall ∆χ2. In the concept of identical detectors, correlated uncertainties between detectors should weakly impact the sensitivity (at the level of near detector “precision”). This is automatically the case if the near detector successfully monitors the NPP cores. Two of the quoted experimental setups reach this goal: DC II and DB Full. Double Chooz, with its final two detector installation, has one dominant systematic: the relative normalization uncertainty. The second most important contribution comes from the relative energy scale uncertainty. Daya Bay full installation is mostly limited by the relative normalization uncertainty. The weaker impact of the relative energy scale uncertainties comes from the better far site distance to the NPP Cores. The uncertainty on the energy scale matches less the oscillation induced distortion. On the other hand, three of the described experimental setups still rely on theoretical knowledge of the spectrum and NPP cores associated uncertainties: DC I, DB Mid and RN. In particular, the DB Mid installation is sensitive to several systematics, especially the NPP power uncertainties. Taking data on a longer time scale (3 years, for instance) with such a detector configuration will not improve the sin2(2θ13) sensitivity as much. Single core power uncertainties do not weaken the sensitivity in two particular cases: – if the near and far detectors have the same NPP core flux ratio contributions (this is the case of DC II); – if the far detector distance to each NPP core is the same, even if the spectra ratios are not the same in near/far detectors. In this particular case, the oscillation pattern is not entangled with power uncer- tainties in the far detector, since the νe travelling distances are the same. Power uncertainties would only contribute, weakly, through the absolute normalization and correlations with other systematics in the near detector. The DB site configuration does not meet any of the above conditions. Moreover, for DB Mid, no near site monitors the LA I NPP. This makes the DB Mid setup an intermediate between a two identical near/far detector experiment and a single far detector configuration. Since the near site is farther away from the NPP cores (Table 13), theoretical uncertainties on the spectrum have a larger impact than in DC II. Also, since the average distance of DB Mid far site is closer, the oscillation pattern matches slightly better with the energy scale associated distortion (bigger contribution than in DC II). The RENO far site location is the best among the quoted setups in canceling the impact of the relative and absolute energy scale uncertainties. However, because the site configuration does not fill any of the two conditions for the cancellation of the NPP core power uncertainties, this experiment relies on the precision with which each core power can be determined. Moreover, the near detector is a bit farther away than in the DC II case, which explains why the global reactor νe rate is less effectively determined and have a larger contribution than in DC II to the final sin2(2θ13) sensitivity. 8.3 Touching the “right systematic chord” In the previous section we have determined the dominant systematics of each setup. In this section we focus on the comparison of all the reactor experiments under the assumption that systematics are known at the same level. Moreover, we want to illustrate the impact of the determination of the most significant systematics on the sensitivity of each experimental setup: • the single core power uncertainties; • the relative normalization between detectors; • the energy scale uncertainties (absolute and relative, between detectors). In the CHOOZ experiment, the power uncertainties were assessed at 0.6 % [9]. However, this estimate was uniquely based on the heat balance of the steam generators. Even if quite precise, this method could be inaccurate. Other methods, such as the external neutron flux measurements, more directly linked to the fission rates inside the cores, lead to a power evaluation less effectively determined with an assessed error around 1.5 %. This latter method allows continuous tracking of the NPP core power variations. The Double Chooz and Daya Bay proposals set their baseline estimates of these uncertainties at a conservative value of 2 %. We studied the impact of this knowledge on the sensitivity by assuming two extreme scenarios: in the worst case, a 3 % error, and the best case of 0.6 % precision. As a standard, we set central value σpwr = 2.0 % for all the experiments. The relative normalization between detectors is the most significant systematic in two identical detector setups, with a near detector successfully monitoring the whole NPP (DC II and DB Full setups). The Total ∆m231 0.4 0.6 0.8 1 1.2 1.4 1.6 Double Chooz Phase I sin2(2θ13)lim = 0.0539 (90 % C.L.) Total ∆m231 0.4 0.6 0.8 1 1.2 1.4 1.6 Double Chooz Phase II sin2(2θ13)lim = 0.0235 (90 % C.L.) Total ∆m231 0.4 0.6 0.8 1 1.2 1.4 1.6 Daya Bay Phase I sin2(2θ13)lim = 0.0402 (90 % C.L.) Total ∆m231 0.4 0.6 0.8 1 1.2 1.4 1.6 Daya Bay Phase II sin2(2θ13)lim = 0.0106 (90 % C.L.) Total ∆m231 0.4 0.6 0.8 1 1.2 1.4 1.6 sin2(2θ13)lim = 0.0180 (90 % C.L.) Common systematic framework σabs 2.0 % σshp 2.0 % σrel 0.4 % σpwr 2.0 % 0.5 % (eV2) 2.5 10−3 Best Worst σpwr 0.6 % 3.0 % σrel 0.2 % 0.6 % 0.0 % 1.0 % (eV2) 3 10−3 2 10−3 Figure 7: Double Chooz, Daya Bay and RENO sensitivities as a function of the size of the main systematics. The common systematic framework is what experimentalists believe to be achievable, without any further R&D. It is worth noting that the main difference between the common and the baseline cases comes from Double Chooz, which takes a conservative value of the relative normalization at 0.6 %. The common framework is used to compute the reference sin2(2θ13) sensitivity of each setup (value on top of each graph). Then each systematic (σpwr, σrel, σscl) impact on sensitivity is separately computed and illustrated as ratio R = sin2(2θ13)best or worst/ sin 2(2θ13)baseline on each graph. The overall impact changing all three systematics together is also illustrated with the “Total” label. Moreover we also provide a quick guess on sin2(2θ13) sensitivity behaviour as a function of ∆m 31 best fit value provided by other experiments. For the Daya Bay Phase II experiment, where possible correlation between detectors on a same site may happen, we take σrel half correlated and half uncorrelated between detectors of a same site, and completely uncorrelated between detectors of different sites (see section 7.2 for details). two available detailed quantifications of this uncertainty are the Double Chooz [24] and Daya Bay [25] proposals. Each detector is estimated to measure the νe rate to a relative accuracy with respect to each other of 0.39 % (DB Full) and 0.44 % (DC II). In the Double Chooz proposal, however, a conservative value has been preferred for the baseline sensitivity calculation: σrel = 0.6 %. We will take this number as the worst case. The Daya Bay collaboration plans, after some R&D, to reach a relative uncertainty of 0.2 %. This value has been taken as the best case. We set as a standard central value σrel = 0.4 % for all experiments. Even if we did not implement detector response in this simulations, we included energy scale uncertainties. The Double Chooz proposal quotes σabs = 0.5 % and σrel = 0.5 %. This is taken as the common central values for all the experiments. However, in the CHOOZ experiment [9], the energy scale uncertainty was estimated at the level of 1.1 %. We thus take a 1.0 % uncertainty on both relative and absolute energy scale determination as the worst case. As a best case, we switched off the impact of energy scale systematics on the sensitivity. Although the impact of the ∆m231 uncertainty on the sensitivity to sin 2(2θ13) is negligible with the current knowledge, the central best fit value on this parameter from other experiments such as MINOS [8], K2K [7] and SuperK [4], may have some influence on the reactor experiment sensitivities. We thus include the current ∆m231 bounds on the sensitivity computations: the “worst case” is taken to be ∆m 31 = 2.0 10 −3 eV2 and the “best” one ∆m231 = 3.0 10 −3 eV2 (2 σ bounds from [13]). The standard central value for all the experiments has been taken to be ∆m231 = 2.5 10 −3 eV2. We illustrate in Figure 7 these three systematic scenarii: best, central and worst cases. Each contribution is assessed separately, but we also show on this graph the total impact by summing the effect of the three discussed systematics (σpwr, σrel and σscl) to their respective best, central and worst values. We also illustrate the impact of the true central value of ∆m231 on the sensitivity. In this representation, we show the ratio of the computed sensitivity sin2(2θ13)b,w for the best (resp. worst) case over the sensitivity sin 2(2θ13)c for standard central systematic values. The “Total” bar shows that in Daya Bay and RENO the sensitivity can vary from 0.6 to 1.2 – 1.3 of the baseline case, for experimental systematics ranging from a “best” to a “worst” scenario. In the case of Double Chooz, the impact of systematics is less significant, at the level of 20 % on both sides. In the Double Chooz and DB Mid cases, the sensitivity could be worsened for best fit values of ∆m231 below 2.5 10 −3 eV2. In DB Full, the sensitivity is quite stable over the current allowed range for ∆m231 with only a 5 % effect on sensitivity. Double Chooz is an optimized experiment in the sense of robustness with respect to systematics for a goal sensitivity in the 0.02 – 0.03 range. Daya Bay phase II is adequate to reach a sensitivity at the level of 0.01. However, a simpler experiment for this class of sensitivity would be a scaled-up variant of Double Chooz, with a very close near site at 150 m and a 1.5 km baseline for the far site. At the Diablo Canyon power plant [19], where two 3.19 GWth cores are operational and modest civil engineering works would be required, four 20 t detectors (2 near, 2 far) would give a sensitivity of 0.013 after 3 years of data taking. 9 Conclusion In this work we have presented a detailed comparative analysis of the sensitivities to sin2(2θ13) of upcoming and proposed reactor experiments. We have first calculated the sensitivities using all available data published by the respective collaborations for the baseline of both the systematical uncertainties and the experimental setups. Our results are generally in good agreement with the sensitivities quoted by the collaborations: 0.054 for Double Chooz Phase I; 0.028 for Double Chooz Phase II; 0.041 for Daya Bay “mid”; 0.0089 for Daya Bay “full”; and 0.021 for RENO. In the case of Daya Bay, we have additionally evaluated the impact of the proposed filling procedure, with pairs of near-far detectors filled with the same scintillator batch and 4 different scintillator batches. If the hydrogen mass fraction is controlled to 0.5 % between different batches, the sensitivity worsens slightly to 0.0093. We also examined, still for Day Bay, the until-now implicit assumption that the errors of the relative normalizations between the 8 detectors are fully uncorrelated. In reality this is the most optimistic scenario, since a part of these uncertainties may come from site-dependent systematics. In the most pessimistic scenario (all relative normalizations fully correlated for the same site), the Daya Bay sensitivity would be 0.012. An important result of this work is that the total thermal power available for an experiment, a figure of merit that has been often used as a strong argument to “rank” different projects, has a modest impact on the success of an experiment. Large powers are only available in multi-core sites, which are very difficult to monitor. The associated systematics can be overwhelming with respect to the benefit from the statistics. This is very nicely exemplified by the case of RENO, which would reach the same sensitivity with just 2 of its 6 reactors on, and by Daya Bay “mid”, which results to be just half way between Double Chooz Phase I and Double Chooz Phase II. We have illustrated how optimal is the use of the available thermal power in each site through a comparison of the real experiments with ideal “single core equivalent” setups: Double Chooz and Daya Bay “full” are nearly optimal, Daya Bay “mid” and RENO do not make full advantage of their huge power. Daya Bay pays for the complexity of its nuclear power plant by the inevitable construction of two near sites. We have carried out a detailed and unified χ2−pull analysis for all the experiments, under common assump- tions for all systematic uncertainties. This study allowed us to compare all projects on an equal footing and evaluating the impact of each single systematic uncertainty on the sensitivity of each experiment. In this calculation the Double Chooz baseline sensitivity is 0.0235 and a single systematic dominates, which is the error of the relative normalization between near and far detector. This shows that taking into consideration its small mass, Double Chooz is an optimized experiment. With respect to the other projects, the Double Chooz sensitivity has a more pronounced dependence on the best fit ∆m213, with sin 2(2θ13)lim ranging from 0.020 to 0.031 for the currently 2 σ−allowed ∆m213 interval. Daya Bay “full” proves to be robust with respect to the size of the systematical errors and to variations of ∆m213 and is the only partly approved project with an achievable sensitivity potential below 0.01. The Daya Bay sensitivity is dominated by the accuracy of the relative normalization between detectors and the degree of correlation existing on a same site between the latter parameters. With the knowledge of the systematics we have today, the Daya Bay sensitivity would be on average between 0.0094 and 0.0123, depending on the degree of correlation of the systematics between detectors on the same site. Contrary to Double Chooz and Daya Bay, the sensitivity of RENO is largely degraded by the uncertainties of the reactor powers and fuel composition. This, again, shows that the site is not optimal. Nevertheless, due to the large target mass and optimal baseline, a very competitive sensitivity, around 0.02, is achievable. This pull analysis also shows that the impact of the backgrounds on the χ2 is minor with respect to other systematics. Backgrounds, at least in our simulation, are therefore not critical in any of the analyzed experiments. Taken into consideration all the above results, we come to a conclusion about the features of the optimal experiment to approach a sin2(2θ13) sensitivity of 0.01. Where by “optimal” we mean a robust sensitivity, the simplest configuration, the minimal amount of civil works and the smallest mass. Such an optimal experiment would have a nuclear power plant layout as simple and powerful as Double Chooz; a favorable topography for sufficiently deep underground laboratories, a very close, single near site, a ∼ 1.5 km baseline for the far site and a total target mass of about the half of Daya Bay. Diablo Canyon, California, is a good example of an suitable site for such an experiment. Acknowledgement We are greatful to A. Milsztajn for the careful proofreading of this article and his useful comments. We wish also to acknowledge M. Lindner for fruitful discussions on the comparison of the θ13 experiments. It is also a pleasure to thank M. Cribier and H. de Kerret for continuous helpful exchanges. Appendix In this section we first describe the way we computed event rates, and then the χ2 analysis implementation with detailed systematic inclusions. Event rates The visible energy inside the detector has a simple expression as a function of the positron energy or neutrino energy of the inverse β-decay reaction: Evis = Ee+ +me ≃ Eν − Ethrν + 2me , (16) where Ethrν = Mn−Mp+me is the νe threshold energy of the reaction. The event rates produced in reactor R and recorded in detector D per visible energy bin [Ei;Ei+1] may be written as i (θ13,∆m 31) = ∫ Ei+1−me Ei−me (Ee+ , Eν) (17) (Ee+ , Eν) = 4πL2R,D h(LR,D, L)σ(Eν)φ R,D(Eν) ǫ(Ee+) (18) ×R(Ee+ , Eν)Pee(Eν , L, θ13,∆m231) , where h(LR,D, L), σ(Eν ), ǫ(Ee+), φ R,D(Eν), R(Ee+ , Eν) stand for the finite size effect function the νe inverse β-decay reaction cross section, the e+ detection efficiency, the νe flux from reactor R in detector D and the energy response of the detector respectively. The generic normalization factor NR,D is the product of the experiment life time by the number of available free protons inside the target volume, the global load factor of reactor R and the dead time of detector D. The νe flux from reactor R in detector D is described in term of the isotope composition as: φR,D(Eν) = P Efisℓ ℓ (Eν) (19) where ℓ = 235U, 238U, 239Pu, 241Pu labels the most important isotopes contributing to the νe flux, C is the relative contribution of the isotope ℓ to the total reactor power (CRℓ = N R, and Nfisℓ is the number of fissions per second of isotope ℓ), Efisℓ is the energy release per fission for isotope ℓ and P R is the thermal power of reactor R. In eq. (19), φ ℓ (Eν) is the energy differential number of neutrinos emitted per fission by the isotope ℓ, and we adopt the parameterization for the φ ℓ (Eν) from ref. [16]. For the C we take a typical isotope composition in a nuclear reactor given in eq. (2). We take for Pee the oscillation probability expressed in eq. (4). We assume in this article a constant efficiency of ǫ(Ee+) = 80 % and a Gaussian energy response function R(Ee+ , Eν) = 2πρ(Ee+) Ee+ − Eν +Mn −Mp 2ρ(Ee+) . (20) with an energy resolution of ρ(Ee+) = 8 %/ Ee+ [MeV ]. The finite size effect function h(LR,D, L) is assumed to be a Dirac distribution (pointlike sources and detectors) in this paper. The total event rates in ith bin of detector D is then simply expressed as NDi (θ13,∆m 31) = R=R1,...,RNr i (θ13,∆m 31) . (21) χ2 analysis and systematics inclusion The computed event rates, N i , are then included in a χ 2 pull-approach analysis [14], where correlations between the systematic uncertainties are properly included: χ2 = min i=1,...,Nb D=D1,...,DN ∆Di − k,k′=1 k,k′αk′ . (22) with, ∆Di = i −NDi /UDi (23) and UDi are given by eq. (9). The αk and S i,k coefficients are described in Table 15. The SDi,k coefficient represents the shift of the i th bin of detector D spectrum due to a 1 σ variation in the kth systematic uncertainty parameter αk. Most of the systematics are expressed as function of N i of B quantities already described previously. The energy scale systematic coefficients in SDi,k are defined through MDi which follows the relation MDi = R=R1,...,RNr i , (24) where ∫ Ei+1−me Ei−me (Ee+ , Eν) . (25) It is often assumed in the pull-approach that Wk,k′ = δk,k′ . If we keep this definition of Wk,k′ then we are faced with the problem that the reactor spectrum shape uncertainties may contribute to the absolute normalization error and the fuel composition uncertainties may contribute to the reactor core power errors. If we want to get rid of these free contributions we could use two methods: 1. the first one consists to infer that ℓ αcmp,ℓC ℓ = 0 2. the second one is to introduce additional weight terms in the χ2 definition (22): ℓ αcmp,ℓC ℓ /εcmp The first method allows disentangling fuel composition from power uncertainties. However, this assumption is a bit too restrictive since in practice the sum ℓ αcmp,ℓC R = 0 is only constrained at the knowledge level of the power of reactor R. The second method has the advantage of allowing an estimate of the level of contribution of this systematic to the power uncertainties. Moreover this simply leads to redefining the Wk,k′ matrix as W−1k,k′ = δk,k′ + ℓ,ℓ′,R cmp,R ℓ,ℓ′,k,k′ CRℓ C ε2cmp , (26) cmp,R ℓ,ℓ′,k,k′ = δℓ,k−kR δℓ′,k′−kR if k, k′ ∈ {kR0 + 1, . . . , kR0 + 5}, kR0 = 5Nd +Nb +Nr + 2 + 4(R− 1), 0 otherwise. With these definitions, εcmp determines at which level the fuel composition uncertainties are allowed to contribute to the core power errors. One wants typically that fuel composition uncertainties contribute within the allowed region of power uncertainty. Thus, εcmp = σpwr/P R . (28) Regarding the fuel composition uncertainties, σcmp may be assessed roughly at the level of σpwr uncertainties: σ2cmp = 2 − 3 % . (29) Error type k SD i,k ×UDi αDi,k Absolute normalization 1 σabsN i αabs Relative normalization (Ns = 1) in D1 Ns + 1 σrelN in DNd Ns +Nd σrelN Absolute Energy scale Nd + 2 σsclN i αscl Relative Energy scale (Ns = Nd + 2) in D1 Ns + 1 σ MD1i α in DNd Ns +Nd σ Backgrounds (Ns = 2Nd + 2) accidentals in D1 Ns + 1 σ BD11,i α accidentals in DNd Ns +Nd σ 1,i α cosmogenics in D1 Ns +Nd + 1 σ BD12,i α cosmogenics in DNd Ns + 2Nd σ 2,i α proton recoils in D1 Ns + 2Nd + 1 σ BD13,i α proton recoils in DNd Ns + 3Nd σ 3,i α Reactor spectrum shape (Ns = 5Nd + 2) in bin 1 Ns + 1 σshpN 1 αshp,1 in bin Nb Ns +Nb σshpN αshp,Nb Reactor power (Ns = 5Nd +Nb + 2) from R1 Ns + 1 σ from RNr Ns +Nr σ pwr N RNr ,D Reactor R composition (Ns = 5Nd +Nb +Nr + 2 + 4(R− 1)) from 235U Ns + 1 σcmpN cmp,R from 239Pu Ns + 2 σcmpN cmp,R from 238U Ns + 3 σcmpN cmp,R from 241Pu Ns + 4 σcmpN cmp,R Table 15: Systematic parameters table. We used the following definitions: Nb is the number of bins in the reactor spectra, Nd is the number of detectors in the experiment, Nr is the number of reactor cores in the NPP, Ns is short for the number of previously defined systematic parameters. For specific values of σabs, σrel, σshp, σBD , σpwr, σcmp, we refer to the experiment comparison sections 6 – 8. References [1] B. T. Cleveland et al., Astrophys. J. 496 (1998) 505; J.N. Abdurashitov et al. [SAGE Collaboration], J. Exp. Theor. Phys. 95 (2002) 181, astro-ph/0204245; T. Kirsten et al. [GALLEX and GNO Collab- orations], Nucl. Phys. B (Proc. Suppl.) 118 (2003) 33; C. Cattadori, Talk given at Neutrino04, June 14-19, 2004, Paris, France. [2] Super-K Collaboration, S. Fukuda et al., Phys. Lett. B 539 (2002) 179, hep-ex/0205075; J. Hosaka et al. hep-ex/0508053. [3] SNO Collaboration, Q.R. Ahmad et al., Phys. Rev. Lett. 89, 011302 (2002), nucl-ex/0204009; B. Aharmim et al., Phys. Rev. C 72, 055502 (2005), nucl-ex/0502021. 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Guo [Daya Bay Collaboration], hep-ex/0701029. Yifang Wang, hep-ex/0610024. Seminar of D. Jaffe at CEA/Saclay on 0ctober 23 2006, www-dapnia.cea.fr/Phocea/file.php?class=std&file=Seminaires/1425/1425.pdf [26] http://neutrino.snu.ac.kr/RENO/ [27] http://www.ba.infn.it/~now/now2006/Program/program.html [28] M. Aoki et al., hep-ex/0607013. [29] J.C. Anjos et al., hep-ex/0511059. [30] CEA, Nuclear power plants in the world (2005). http://132.166.172.2/fr/Publications/trilogie/Elecnuc-2005.pdf CEA, Energy handbook (2005). http://132.166.172.2/fr/Publications/trilogie/Infos-Energie-2005.pdf [31] Achkar B. et al., Phys. Lett. B374, 243(1996). [32] H. Minakata, et al., Phys.Rev. D68 (2003) 033017, hep-ph/0211111. [33] P. Huber, et al., Nucl.Phys. B665 (2003) 487-519, hep-ph/0303232. [34] G. Mention, PhD. Thesis, http://doublechooz.in2p3.fr/ [35] M. Apollonio, et al., Eur. Phys. J. C27, (2003) 331, hep-ex/0301017. [36] S. Eidelman et al., Phys. Lett. B592 (2004). http://arxiv.org/abs/hep-ex/0405032 http://arxiv.org/abs/hep-ex/0410081 http://arxiv.org/abs/hep-ex/0606025 http://arxiv.org/abs/hep-ex/0701029 http://arxiv.org/abs/hep-ex/0610024 http://arxiv.org/abs/hep-ex/0607013 http://arxiv.org/abs/hep-ex/0511059 http://arxiv.org/abs/hep-ph/0211111 http://arxiv.org/abs/hep-ph/0303232 http://doublechooz.in2p3.fr/ http://arxiv.org/abs/hep-ex/0301017 Experimental context Neutrino oscillation at reactor and 13 Generic analysis of 13 sensitivity Generic overview of systematic error inputs Backgrounds: description and modelization Comparison of the current proposals Detailed systematics review Impact of the scintillator composition Estimation of the -induced backgrounds Reactor experiments baseline sensitivity Double Chooz Daya Bay RENO Discussion The single core equivalent approach The pulls analysis Touching the ``right systematic chord'' Conclusion

0704.0499

Optimal Routing for Decode-and-Forward based Cooperation in Wireless Networks

Optimal Routing for Decode-and-Forward based Cooperation in Wireless Networks Lawrence Ong and Mehul Motani Department of Electrical & Computer Engineering National University of Singapore Email: {lawrence.ong, motani}@nus.edu.sg Abstract We investigate cooperative wireless relay networks in which the nodes can help each other in data transmission. We study different coding strategies in the single-source single- destination network with many relay nodes. Given the myriad of ways in which nodes can cooperate, there is a natural routing problem, i.e., determining an ordered set of nodes to relay the data from the source to the destination. We find that for a given route, the decode- and-forward strategy, which is an information theoretic cooperative coding strategy, achieves rates significantly higher than that achievable by the usual multi-hop coding strategy, which is a point-to-point non-cooperative coding strategy. We construct an algorithm to find an optimal route (in terms of rate maximizing) for the decode-and-forward strategy. Since the algorithm runs in factorial time in the worst case, we propose a heuristic algorithm that runs in polynomial time. The heuristic algorithm outputs an optimal route when the nodes transmit independent codewords. We implement these coding strategies using practical low density parity check codes to compare the performance of the strategies on different routes. 1 Introduction Research in high data rate wireless systems has enabled applications to go wireless and become more interesting, e.g., wireless Internet access, mobile video conferencing and mobile TV on buses and trains. These applications would have been impossible without high rate wireless transmission links. As many wireless devices are battery operated, power constraint is often imposed on them to make sure that they maintain a certain desired lifespan. In this paper, we investigate optimal routing problem to maximize the transmission rate in the wireless network where there is a power constraint on each node. The wireless channel is inherently broadcast, in that messages sent out by a node are heard by all nodes listening in the same frequency band and in communication range. This opens up opportunities for richer forms of cooperation among the wireless users/nodes. For example, rather than using point-to-point multi-hop routing (a direct adaptation from wired networks), where a node only transmits to the next node in the “route”, cooperative strategies, such as information theoretic relaying [1, 2, 3] and opportunistic routing [4], could be used. These richer forms of cooperation can lead to efficient distributed algorithms and can increase the end-to-end data rates. The gain from cooperation has been shown in information theoretic analyses [5][6] and demonstrated in practical implementations [7, 8, 9]. We now briefly describe what we mean by these richer forms of cooperation, often using the term “coding” to highlight that our approach stems from information theory [10]. Figs. 1(a)–(c) depict wireless networks in which node 1 is the source, nodes 2 and 3 are relays, and node 4 is the destination. In Fig. 1(a), since every node can hear what node 1 transmits, the simplest strategy is for node 4 to directly decode from node 1, which we call the single-hop coding strategy (SH). However, when nodes 1 and 4 are situated far apart, signals from node 1 go through severe attenuation before they reach node 4. This is when relay nodes 2 and 3 can help. Referring to http://arxiv.org/abs/0704.0499v1 Figure 1: Different coding strategies for multiple relay channels. Fig. 1(b), node 1 transmits to node 2. Node 2 fully decodes the data and re-transmits to node 3. Node 3 does the same and relays the data to node 4. This is the well known multi-hop coding strategy (MH). Although we can view relay nodes helping the source to transmit data as a form of cooperation, as far as decoding is concerned, SH and MH are still point-to-point strategies (a node only decodes from one node) and we categorize them as non-cooperative coding strategies. Taking a closer look at MH, we see that node 3 can hear and decode node 1’s transmission (although it is intended for node 2). This suggests a cooperative coding strategy, depicted in Fig. 1(c), in which node 3 decodes transmissions from nodes 1 and 2, and node 4 decodes transmissions from nodes 1–3. This cooperative way of encoding and decoding stems from an information theoretic approach and is termed the decode-and-forward coding strategy (DF) [1, 2, 3]. Regardless of whether MH or DF is used for data transmission, there is a sequence of nodes through which data flows. Kurose and Ross [11] define a route as “the path taken by a datagram between source and destination”. The datagram “hops” from one node to the next node, capturing the scenario in which a node receives data only from a node behind and forwards data only to the node in front. However, in the cooperative coding paradigm, data does not flow from one node to another; rather it is from many to many with complex ways of cooperating. To describe the flow of information in these new modes of cooperation, we define a route as follows. Definition 1 The route taken by a packet from the source to the destination is an ordered set of nodes involved in encoding/transmitting and receiving/decoding of the packet. The sequence of the nodes in the route is determined by the order in which nodes’ transmit signals first depend on the packet. Remark 1 If a group of nodes transmits simultaneously, then they can be ordered arbitrarily within the group. For example, consider a four node network, in which node 1 first broadcasts the message, and then nodes 2 and 3 listen and simultaneously transmit to node 4. The route here can be described by {1, 2, 3, 4} or {1, 3, 2, 4}. Remark 2 Fig. 1 describes three coding strategies for the same four-node network. The route for SH in Fig. 1a is {1, 4}. The routes for MH and DF in Figs. 1b & 1c respectively are both {1, 2, 3, 4}. Given the myriad of ways in which nodes can cooperate, there is a natural routing problem in the cooperative coding paradigm. Furthermore, route selection directly affects the end-to-end data transmission rate. For DF, the current routing solutions for MH cannot be applied trivially. In this paper, we construct an algorithm to find an optimal route (in terms of maximizing rates) for DF. Our contributions in this paper are as follows. 1. We show how much gain one can expect using DF, a cooperative coding strategy, over MH, a non-cooperative coding strategy, on the same route. 2. We construct an algorithm that finds rate maximizing routes for DF. 3. We construct a heuristic algorithm that runs in polynomial time. We show that the heuristic algorithm finds an optimal route for DF when the nodes send independent codewords. 4. We implement DF using low-density parity-check (LDPC) codes [12][13]. Also, we show the performance of codes using different coding strategies and on different routes. This paper investigates cooperative coding and routing in the wireless network based on an information theoretic approach. A few idealized assumptions are made (e.g., infinite block length, unbounded communication range). Some of these assumptions are, however, relaxed in the simu- lations in Section 8. 1.1 Related Work Communications in wireless networks has been progressing from MH to that using cooperative strategies. More research is being directed toward designing codes that are based on information theoretic cooperative coding strategies to harvest the gain in transmission rates predicted by information theory. Examples of codes based on cooperative coding strategies include DF-based Turbo codes [14][15] and LDPC codes [16, 17, 18, 19] for the single relay channel. It has been mentioned that some of these codes can be extended to the multiple relay channel [2][3][5][6]. In the past, link optimization (i.e., maximizing the transmission rate between node pairs) and route optimization were done separately. Routing was optimized after the links between the nodes had been established. Algorithms such as Bellman-Ford [20, Section 24.1][21] and Dijkstra’s algorithm [22] that assign costs to all links were used to find a route with the lowest cost from source to the destination. These ways of separating routing and coding are not optimal for MH or DF as the rates of the links change depending on which route is chosen. Realizing the inter- dependency between links and routes, it has been suggested that links and routes be jointly optimized [23, 24, 25, 26]. This gives rise to cross-layering [27] in the OSI model. However, in these joint routing and coding work, data transmission from the source to destination is still based on MH. Routing algorithms that are optimized for MH might not be suitable for DF. In Ad hoc On-demand Distance Vector Routing (AODV) [28] and Dynamic Source Routing (DSR) [29], the source node broadcasts a route discovery packet. Neighboring nodes receive and re-broadcast the packet. When the destination receives the packet, a route is formed by tracing the path that the packet took. These routing algorithms minimize the transmission delay but might not optimize the transmission rate. In Extremely Opportunistic Routing (ExOR) [4], a node broadcasts its data to a set of potential relays. Nodes in this set transmit acknowledgments and then selected nodes forward the data. Though ExOR does not have predefined routes, MH is used on the effective route taken by a packet. As far as we know, routing algorithms for cooperative coding have not been investigated. In this paper, we propose algorithms to find optimal routes for DF-based codes in the multiple relay channel. Our work complements code design by finding the best route (rate maximizing) on which the codes can be used. As previous work focused on cooperative coding for the single relay channel, in this paper, we implement DF-based LDPC codes on the multiple relay channel. We then compare the transmission rate of different coding strategies on different routes. We focus on the multiple relay channel [2][3][5][6], which is a single-source single-destination network, as a first step towards understanding general multiple-source multiple-destination net- works. We study DF because it is one of the “more implementable” information theoretic coding strategies [19, 14, 15, 16, 17, 18]. 2 Motivating Cooperation 2.1 Network Model We consider a D-node network S = {1, 2, 3 . . . , D − 1, D} with one source (node 1) and one destination (node D). Node i, ∀i ∈ S, either transmits at fixed average power Pi or turns off. We use the standard path loss model for signal propagation. The received power at node t from node i is given by Pit = κd it Pi, where dit is the distance between nodes i and t, η is the path loss exponent (η ≥ 2 with equality for free space transmission), and κ is a positive constant. The receiver at node t is subject to thermal ambient noise of power Nt. We assume duplex nodes, i.e., nodes can transmit and receive simultaneously. We assume that all nodes have the same noise variance. Given this network model, we investigate how nodes can cooperatively send messages from the source to destination. We study and compare several coding strategies. Remark 3 We consider single-flow networks. This is the first step in understanding a more complicated problem of multiple flows. The relevance of our work in multiple-flow networks is as follows: 1. In a multiple-flow networks where each flow uses an allocated orthogonal channel, the rate of each flow can be optimized in respective channel using the algorithm derived in this paper. 2. In a multiple-flow network with existing flows, if we wish to add a new flow, the algorithm in this paper finds an optimal route for the new flow. Note that adding a new flow might affect existing flows. We can restrict the transmit power of nodes in the new flow to control the interference introduced. 2.2 Single-Hop Coding Strategy (SH) In SH, the source directly transmits data to the destination. The signal-to-noise ratio (SNR) at the destination, node D, is given by γSH(D) = P1DN . The Shannon capacity of this SH link is RSH = log (1 + γSH(D)) . This rate depends on the source-destination distance and can be poor if the source and destination are situated far away from each other (because of signal attenuation). Remark 4 We assume that nodes that do not participate in relaying the data for a source- destination pair do not cause interference. Another way to account for the external noise is to include it in the receiver noise. 2.3 Multi-Hop Coding Strategy (MH) In MH, we make use of the relays to aid the transmission from the source to the destination. The source simply transmits to the next relay. The first relay decodes the message and re-transmits it to the second relay, and so on until the destination. This can improve the transmission rate if the attenuation from the source/relay to the next relay is reduced as compared to that in SH. However, since all relays transmit simultaneously, there exists interference, beside noise, at the receiver. In the rest of this paper, we denote a route by M = {m1,m2, . . . ,m|M|}. We define the set of all possible routes from the source (node 1) to the destination (node D) by Π(S) = {m1,m2, . . . ,m|M|} : m2, . . . ,m|M|−1 are all possible selections and permutations of the relays (including the empty set), m1 = 1,m|M| = D Using the routeM, the SNR at node mt is γMH(mt,M) = Pmt−1mt |M|−1 i=1,i6=t Pmimt +Nmt . (1) Since all relays and the destination must fully decode the messages, the transmission rate from the source to the destination using routeM is RMH(M) = min m∈M\{m1} log (1 + γMH(m,M)) . (2) We term RMH(M) the rate supported by the route M using MH. The maximum rate using by MH, optimized over all possible routes, is RmaxMH = max M∈Π(S) RMH(M). (3) We note that there may exist more than one route that support this maximum rate. 2.4 Decode-and-Forward Cooperative Coding (DF) Using DF [1, 2, 3], each decoder decodes transmissions from all nodes behind. E.g., the third node in the route decodes the transmissions from the first and the second node. So, the third node decodes each source message using two blocks of received codewords. In addition, assuming that the nodes in front decode the messages correctly, a node knows what they transmit and hence it cancels the interference from these nodes. It has been shown that in order to maximize the DF rate on routeM, node mi transmits Xmi = j=i+1 αmimjPmiUmj , for 0 ≤ j=i+1 αmimj ≤ 1, ∀i = 1, . . . , |M|−1. Umj are independent Gaussian random variables with unit variance. {αij |j = i+1, . . . , |M|} are the power splits of node i, allocating portions of its transmit power to transmit independent sub-codewords Uj . Doing this the SNR of node mt in routeM is γDF(mt,M) = N αmimjPmimt . (4) Using the routeM, DF can achieve rates up to RDF(M) = max m∈M\{m1} log (1 + γDF(m,M)) , (5) and the maximum rate using by DF is RmaxDF = max M∈Π(S) RDF(M). (6) Definition 2 We define the reception rate of node m in routeM as Rm(M) = log (1 + γDF(m,M)) . It is the rate at which node m can fully decode the messages. The same concept applies to MH. Remark 5 In practice, a relay in the route might decode a message wrongly, and hence forward the wrong message. When this happens, the nodes behind, when trying to cancel the co-channel in- terference introduced by this relay, will introduce more noise at their decoders. While this scenario is not captured in (4), we allow imperfect interference cancellation in our simulations (Section 8). 2.5 Comparing the Strategies It is easy to see that, for any chosen routeM with four nodes or more, γDF(m,M) > γMH(m,M), ∀m ∈ M. Also, we can show that for anyM∈ Π(S), RDF(M) = RMH(M) = RSH, for |M| = 2, (7a) RDF(M) ≥ RMH(M), for |M| = 3, (7b) RDF(M) > RMH(M), for |M| ≥ 4, (7c) 1000 5 10 15 20 25 30 35 No. of nodes in route, |M| Samples taken for each |M| = 105 (1) DF rate/MH rate (1) DF rate/SH rate (2) DF rate/MH rate (2) DF rate/SH rate Figure 2: Ratio of average transmission rate SH, MH, and DF versus |M| for two cases: (1) d1D = 10 and (2) d1D = |M| − 1. and RmaxDF ≥ R MH ≥ RSH. However, it is not clear how much again, on average, we can expect using DF compared to MH and SH. Now, we compare the rates of SH, MH, and DF for randomly generated routes of different lengths in a line topology. We consider two cases: (1) d1D = 10 and (2) d1D = |M| − 1. Then |M|− 2 nodes are randomly placed along the straight line joining nodes 1 and D. Note that in case 1, node density increases with the number of nodes while in case 2, the average adjacent node spacing is constant for all |M|. We set Pi = Ni = 1 for all transmitters and receivers, and κ = 1, η = 2. For each randomly generated route, we calculate the transmission rate using SH, MH, and DF. Here we restrict the nodes to transmit independent codewords for easier optimization, i.e., we set αij = 1, ∀i, ∀j = i+ 1 and αij = 0, ∀j 6= i+ 1. Fig 2 shows the results for cases 1 and 2. For a route of 25 nodes, the DF rate is roughly two orders of magnitude higher than that of SH and MH for both cases. Moreover, as we increase the number of nodes in the route, the gain of DF over MH/SH increases for both cases. We note that if the nodes are allowed to send arbitrarily correlated codewords, the DF rate can be higher. For example, consider the route M = {(0, 0), (0.5, 0), (2, 0), (3, 0), (4, 0)}. With independent codewords, RDF(M)/RMH(M) = 2.95, but with arbitrarily correlated codewords, RDF(M)/RMH(M) = 4.40. 3 The Optimal Routing Problem We define the optimal route set for DF as QDF , {M ∈ Π(S) : RDF(M) = R DF } (8) where Π(S) is a set of all possible routes from the source to the destination. We define the optimal route set because the rate maximizing route may not be unique. Then the optimal DF routing problem is Find at least oneM DF ∈ QDF and RDF The optimal route set and routing problem for MH are similarly defined. Finding optimal routes for MH and DF by brute force is hard, as it involves testing all routes in Π(S). In Sections 4 and 5, we construct an algorithm that finds M DF, potentially without having to test all routes in Π(S). However, this algorithm runs in factorial time in the worst case. Hence, in Section 7, we proposed a heuristic algorithm, which runs in polynomial time. 4 The Nearest Neighbor Algorithm Now, we present an algorithm to find an optimal route for DF. In the section, we assume that nodes use independent codewords, i.e., we set αij = 1, ∀i, ∀j = i+ 1 and αij = 0, ∀j 6= i+ 1. Remark 6 Although the theorems in this section are proven assuming that the nodes send in- dependent codewords, we can also show that they hold even when the nodes transmit arbitrarily correlated codewords [30]. Remark 7 We consider independent codewords in this section because coherent combining is practically infeasible. When the nodes are operating in the GHz range, it is difficult, if not im- possible, to synchronize the carriers to nanosecond accuracy. Furthermore, even if we have very precise clocks, coherent combining is still unlikely in a multiple-node network. For example, in a four-node route, even if we manage to synchronize nodes 1 and 2 to allow coherent combining at node 3, we might not be able to ensure that they will coherently combine at node 4. First, we define the nearest neighbor with respect to a route. Definition 3 Node i /∈M is a nearest neighbor with respect to the routeM iff Pmi ≥ Pmj , ∀m ∈ M, ∀j ∈ S \ (M∪ {i}). (9) Note that nearest neighbor might not be unique. Now, we describe the nearest neighbor algorithm (NNA). Algorithm 1 (NNA) 1. Initialize M = {m1}, where m1 = 1. 2. If there exists a unique nearest neighbor i∗ with respect to the current routeM, we append i∗ to the current route: M←M∪ {i∗}. Else, the NNA terminates prematurely. Since M is an ordered set, the notation A ∪ B means appending ordered set B to the end of ordered set A. 3. Step 2 is repeated until the destination node, node D, is added intoM. The algorithm is said to terminate normally if node D is added to the route. Otherwise, the algorithm is said to terminate prematurely. If the NNA terminates normally, we have the following theorem. Theorem 1 Consider a multiple node wireless network with one source and one destination. If the NNA terminates normally, then the NNA route is optimal for DF. To prove Theorem 1, we need the following lemmas. Lemma 1 When we add the unique nearest neighbor, node a∗, to routeM, the rate supported by the new route M1 =M∪ {m|M1| = a ∗} is greater or equal than the rate supported by the route formed by adding any other node toM. Mathematically, RDF(M∪ {a ∗}) ≥ RDF(M∪ {b}), ∀b ∈ S \ (M∪{a ∗}). (10) Proof: [Proof for Lemma 1] ConsideringM2, the reception rate of node m|M2| = b is Rb(M∪ {b}) = 1 +Nb  , (11) and the reception rate of the node m|M1| = a ∗ in routeM1 is Ra∗(M∪ {a ∗}) = 1 +N−1a∗ Pmia∗  . (12) Clearly, if Pma∗ ≥ Pmb, ∀m ∈ M with at least one inequality, Ra∗(M1) > Rb(M2). Hence RDF(M1) ≥ RDF(M2). Hence we have Lemma 1. We have proven that at any point of time during route construction, in order to maximize the rate supported by the route, we must choose the nearest neighbor (assuming it exists). Next, we show that choosing the nearest neighbor will not harm the rate supported by the route even when more nodes are added. Lemma 2 Let M = {a∗1, a 2, . . . , a } be a route formed by adding the nearest neighbor one by one starting from the source. Now, arbitrarily add K nodes to M. The first node b1 is not a nearest neighbor and the rest may or may not be nearest neighbors. In other words, M1 = {a∗1, a 2, . . . , a , b1, b2, . . . , bK}, where b1 is not a nearest neighbor toM. We can always replace b1 by the nearest neighbor a |M|+1 (assuming it exists) to obtain {a∗1, . . . , a |M|, a |M|+1, b1, . . . , bK−1}, if a∗|M|+1 /∈ {b1, . . . , bK−1}, {a∗1, . . . , a |M|+1 , b1, . . . , bk−1, bk+1, . . . , bK}, if a∗ |M|+1 = bk, for some bk ∈ {b1, . . . , bK−1}, where RDF(M2) ≥ RDF(M1). Proof: [Proof for Lemma 2] For both cases in (13), the reception rates for the first |M| nodes in both M1 and M2 remain the same as each of them decodes from the same nodes behind the route. In equations, (M2) = Ra∗ (M1), ∀i = 2, 3, . . . , |M|. (14) From Lemma 1, Ra∗ |M|+1 (M2) > Rb1(M1). Now, we study the case when a∗ |M|+1 /∈ {b1, . . . , bK−1}. For nodes {b1, . . . , bK−1} inM2, with an additional node behind, i.e., a∗ |M|+1 , the reception rates of these nodes are higher than the same nodes inM1: Rbi(M2) > Rbi(M1), ∀i = 1, 2, . . . ,K − 1. (15) Now, we study the case when a∗ |M|+1 = bk for some bk ∈ {b1, . . . , bK−1}. Similar to the first case, with an additional transmitting node a∗ |M|+1 , the nodes {b1, . . . , bk−1} in M2 have higher reception rates compared to those inM1, i.e., Rbi(M2) > Rbi(M1), ∀i = 1, 2, . . . , k − 1. (16) For nodes {bk+1, . . . , bK}, there is no change in the reception rate because each of them has exactly the same nodes behind them in bothM2 andM1. So, Rbi(M2) = Rbi(M1), ∀i = k + 1, k + 2, . . . ,K. (17) Lemma 2 follows by noting thatRDF(M2) = mini∈M2\{a∗1} Ri(M2) ≥ mini∈M1\{a∗1} Ri(M1) = RDF(M1). Lemma 3 For a route that contains all nearest neighbors, the supported rate is always higher or equal to any route, of the same length, with one or more non-nearest neighbors in it. Proof: [Proof for Lemma 3] Lemma 3 can be proven by applying Lemma 2 recursively until the entire set is replaced by nearest neighbor nodes. Now we consider routes of different lengths but that end on the same node. Lemma 4 Consider a route M1, where node 1 is the source, and node m|M1| = D is the desti- nation, that is M1 = {m 1,m2, . . . ,m|M1|}. (18) Here, one or more nodes in {m2, . . . ,m|M1|} are not nearest neighbors. The following route where all nodes are added according to the NNA (assuming that it does not terminate prematurely) supports rate as good or higher than that supported byM1. M2 = {m 2, . . . ,m }, (19) where m∗|M2| = D and |M1| not necessarily equals |M2|. In other words, RDF(M2) ≥ RDF(M1). Proof: [Proof for Lemma 4] First of all, we consider the case |M1| = |M2|. The results follows immediately from Lemma 3. Second, we consider |M1| > |M2|. We consider first |M2| nodes in M1, i.e.,M 1 = {m 1,m2, . . . ,m|M2|}. Then, RDF(M2) ≥ RDF(M 1) ≥ RDF(M1). (20) The first inequality is obtained by applying Lemma 3. M2 andM 1 are of the same length. The former is formed using the NNA while the latter is not. The second inequality can be argued as follows. The first |M2| nodes in both routesM 1 andM1 are identical. Hence the reception rates are the same. However, there are additional nodes in M1 whose reception rate might be lower than RDF(M 1). Hence, the rate supported byM 1 can only be higher than that ofM1. Lastly, consider |M2| > |M1|. We replace the transmitting nodes inM1 with nearest neighbors and obtain M3 = {m 2, . . . ,m |M1|−1 ,m|M1| = D}. (21) Note that m|M1| might not be the nearest neighbor. Clearly, using Lemma 2, RDF(M3) ≥ RDF(M1). Now, RDF(M2) = min{Rm∗ (M2), . . . , Rm∗ |M1|−1 (M2), Rm∗ (M2), . . . , Rm∗ |M2|−1 (M2), Rm∗ (M2) = RD(M2)}, (22a) ≥ min{Rm∗ (M3), . . . , Rm∗ |M1|−1 (M3), RD(M3)}, (22b) = RDF(M3) ≥ RDF(M1). (22c) The inequality in (22b) is because inM2, {m , . . . ,m∗ |M2|−1 } are added to {m∗1, . . . ,m |M1|−1 before D. A necessary condition for this is Pmn ≥ PmD, ∀m ∈ {m 1, . . . ,m |M1|−1 ∀n ∈ {m∗|M1|, . . . ,m |M2|−1 }, (23) with at least one strict inequality for each n. Hence, Rn(M2) > RD(M3), ∀n ∈ {m , . . . ,m∗ |M2|−1 With additional nodes transmitting to D inM2, RD(M2) > RD(M3). Hence, we have Lemma 4. Proof: [Proof for Theorem 1] From Lemma 4, we know that if the NNA terminates normally, the route (from the source to the destination) formed using the NNA can support transmission rates as high as any other route. In other words, the NNA finds a route that supports the highest rate achievable by DF. Theorem 1 follows. Remark 8 We note the NNA terminates normally if and only if a unique nearest neighbor exists at each step. In the next section, we extend the NNA to an algorithm which terminates normally given any network topology. 5 The Nearest Neighbor Set Algorithm In this section, we modify the NNA so that it terminates normally in any multiple node wireless network with a single source and a single destination. We term this algorithm the nearest neighbor set algorithm (NNSA). First, we define the nearest neighbor set. Definition 4 The nearest neighbor set N = {n1, n2, . . . , n|N |} with respect to routeM = {m1,m2, . . . ,m|M|} is defined as the smallest set N where each n ∈ N ⊆ S \ M satisfies the following condition. Pmn ≥ Pma, ∀m ∈M, ∀a ∈ S \ (M∪N ), (24) with at least one strict inequality for every pair of (n, a) ∈ {(n, a) : n ∈ N , a ∈ S \ (M∪N )}. Now we describe the NNSA. Algorithm 2 (NNSA) 1. Starting with the source node, we have M = {1}. 2. Find the nearest neighbor set N . The original route M branches out to |N | new routes as follows: Mi ←M∪ {ni}, i = 1, . . . , |N |. (25) 3. For each new route in (25), step 2 is repeated until the destination is added to all routes. When the algorithm terminates, we end up with many routes from the source to the destination. We term these routes NNSA candidates. We calculate the supported rate of each candidate and choose the one which gives the highest supported rate. The following theorem says that any NNSA candidate that gives the highest supported rate is an optimal route for DF. Theorem 2 Consider a single-source single-destination multiple node wireless network. The NNSA candidate routes that give the highest supported rate are optimal for DF. Proof: [Sketch of proof for Theorem 2] Using the technique used in the proof of Theorem 1, we can show that adding a node that does not belong to the nearest neighbor set can only be suboptimal. We can always replace that node with one from the nearest neighbor set and obtain an equal or higher rate. In other words, we can show that the supported rate of M1 = {1,m2, . . . ,m|M1|−1, D}, with one or more nodes in {m2, . . . ,m|M1|} not from the nearest neighbor set, is lower or equal to the supported rate ofM2 = {1,m 2, . . . ,m |M2|−1 , D}, where all nodes in M2 are added according to the NNSA. In other words, RDF(M2) ≥ RDF(M1). The NNSA finds all possible routes for which every node is added from the nearest neighbor set. Hence one or more of the NNSA candidates must achieve the highest rate achievable by DF. This gives us Theorem 2. Remark 9 We can show that a shortest optimal route, defined as some MSOR ∈ QDF, s.t. |MSOR | ≤ |M|, ∀M ∈ QDF, is contained in one of the NNSA candidates that supports R Remark 10 The NNSA might output optimal routes that include more nodes from the network unnecessarily. In other words, shorter optimal routes exist. However, from Remark 9, we can find the shortest optimal route by pruning the optimal routes output by the NNSA. 6 Complexity of NNSA With the NNSA, we can now search for the optimal route in the NNSA candidate set, as compared to searching in Π(S) using brute force. The number of candidates determines the number of routes whose rate we need to calculate to find optimal routes for DF. We note that the size of the NNSA candidate set might still, in the worst case, equal |Π(S)|. Using brute force, the number of permutations we need to check is |Π(S)| = 1 + + · · · = O((D − 1)!), where D is the total number of nodes in the network and n×(n−1)×···×1 (n−k)×(n−k−1)×···×1 We ran the NNSA on 10000 randomly generated networks with a varying number of nodes uniformly distributed in a 1m×1m square area. The source, relays and destination were randomly assigned. On average, half of the NNSA candidate set sizes were less than 0.715% of the size of Π(S) for the 8-node channel and less than 0.253% for the 11-node channel. We note that the average size of the NNSA candidate set does grow factorially with the number of nodes. However this does increase the range of finite size networks for which we can find optimal routes. Furthermore, the NNSA provides insights for designing heuristic algorithms to find good routes for DF based codes. In the next section, we propose heuristic algorithms which find routes in polynomial time. Remark 11 The NNSA builds routes from the source. Relays are added to the route regardless of where the destination is. We will use this observation in designing heuristic algorithms. 7 Heuristic Algorithms In the NNSA, the optimal route is constructed by adding the “next hop” node one by one to the partial route. The node to be added is from the nearest neighbor set. If the nearest neighbor set contains more than one node, the current route branches to more than one route, leading to a possibly large NNSA candidate set size. To avoid this, we consider a heuristic approach that starts from the source node and repeatedly adds only one “good” candidate from the nearest neighbor set until the destination is reached. For the choice of the next hop node, we consider the node which receives the largest sum of received power from all the node in the existing route. We call this the maximum sum-of-received-power algorithm (MSPA). By choosing only one node to be added to the partial route, we prevent the algorithm from branching out to multiple routes. This heuristic approach yields only one route, regardless of the network size. We now explicitly describe the MSPA. Algorithm 3 (MSPA) 1. Start with the source node: M = {1}. 2. For every node t ∈ S\M, find the sum of received power from all nodes inM to t, i∈M Pit. 3. Let a∗ be any node with the highest sum of received power, i.e., i∈M Pia∗ ≥ j∈M Pjt, ∀t ∈ S \M. Append node a∗ to the route: M←M∪ {a∗}. 4. Repeat steps 2–3 until the destination is added to the route. Remark 12 Assuming that the value of the previous sum-of-received-power computations are cached, the complexity of step 2 in MSPA is O(D) because there are at most (D − 1) nodes not in the route. The complexity of the comparisons in step 3 is O(D). Steps 2–3 are repeated at most (D − 1) times, giving a worst case complexity of the MSPA of O(D2). Recall that D = |S|. It turns out that the MSPA is optimal if the nodes are restricted to sending independent codewords, as proven in the following theorem. Theorem 3 In a single-source single-destination multiple node wireless network in which the nodes send independent codewords, the MSPA route is optimal for DF. Proof: [Proof for Theorem 3] Consider an optimal route M1 = {m 2, . . . ,m k+1, . . . ,m }. Suppose that the first k nodes of the MSPA route are the same as this optimal route but the (k + 1)-th node is different, i.e., the MSPA route is M2 = {m 2, . . . ,m , a, . . . } where a 6= m∗ Since a is added to the route by MSPA, a necessary condition is i=1 Pm∗i a ≥ j=1 Pm∗jm . So, Ra(M2) ≥ Rm∗ (M1). (26) Now, consider the case where a 6= m∗i , ∀i = k + 2, . . . , |M|. We add a to M1 and obtain M3 = {m 2, . . . ,m , a,m∗ k+1, . . . ,m }. Then, (M3) = Rm∗ (M1), i = 2, . . . , k (27a) Ra(M3) ≥ Rm∗ (M1) (27b) (M3) > Rm∗ (M1), i = k + 1, . . . , |M|. (27c) So, RDF(M3) ≥ RDF(M1). Suppose a = m∗n, for some n ∈ {k + 2, . . . , |M|}. We swap the position of a and obtain M4 = {m 2, . . . ,m , a,m∗ k+1, . . . ,m n−1,m n+1,m }. Then, (M4) = Rm∗ (M1), i = 2, . . . , k, n+ 1, . . . , |M| (28a) Ra(M4) ≥ Rm∗ (M1) (28b) (M4) > Rm∗ (M1), i = k + 1, . . . , n− 1. (28c) So, RDF(M4) ≥ RDF(M1). In summary, we choose an optimal route. Starting from the second node, we compare the optimal route with the MSPA route. If the nodes are different, we insert (or swap, if the node is in the optimal route but at a different position) the node in the MSPA route to the optimal route to obtain a new optimal route. Repeat this by comparing the new optimal route to the MSPA route and changing the first node that differ until the MSPA is contained in an optimal route. Then, we have Theorem 3. Remark 13 However, unlike the NNA and the NNSA, the MSPA does not output an optimal route when the nodes are allowed to send arbitrarily correlated codewords. Consider a four-node network with node coordinates 1(0,0), 2(0.418,0), 3(0.209,0.6755), and 4(0.995,0). Assume Pi = 1, Ni = 1, κ = 1, η = 2. The MSPA route is M1 = {1, 2, 4}. The NNSA outputs M1 and M2 = {1, 2, 3, 4}. It is easy to compute that RDF(M1) = 1.30826 and RDF(M2) = 1.31576. 8 DF with LDPC Codes In the previous section, we computed achievable rates of different strategies and routes based on an information theoretic approach. In this section, we compare the different strategies and routes in a line network using practical low density parity check (LDPC) codes [12][13] with incremental redundancy. The aims of this section are: 1. To illustrate that DF on the multiple relay channel is implementable. 2. To demonstrate that DF performs better than MH under certain network topologies. 3. To show that routing backward (away from the destination) can be good in DF. 4. To demonstrate that the NNSA route performs better than other routes using DF. Figure 3: Network topology. 1e-05 1e-04 0.001 0.01 0 1 2 3 4 5 6 7 8 Eb/N0 (dB) LDPC in 4-node network. 1(0,0.5) 2(0,0.4) 3(0,1) 4(0,1.5) DF M=1234 DF M=134 DF M=1324 DF M=124 SH M=14 MH M=1234 MH M=134 Figure 4: Performance (information bit error rate (BER) versus transmit SNR) of different strate- gies on different routes in a 4-node network. From Fig. 4, we see that for a given route, DF performs better than MH. An interesting observation is that routing backward helps in DF but not MH. We find that the NNSA route (which is also the MSPA route), i.e., {1, 2, 3, 4}, achieves the lowest BER compared to other routes using DF. Remark 14 We note that the total transmit energy differs depending on the length of the route. One might argue that route {1, 4}, though having a higher BER, is better as only 1/3 power is consumed compared to route {1, 2, 3, 4}. However, we stress that this paper finds a route that max- imizes the transmission rate, given that each node must transmit within a given power constraint. Whether or not the node transmits, it is not important in the route comparison. Remark 15 We plotted BER versus SNR in Fig. 4. If the maximum raw channel data rate (in bps) is Ψmax, then the throughput is Ψ = (1 − PER) · Ψmax, where PER is the packet error rate and depends on the BER and the packet size. In simulations, we found that packet error rate (PER) had the same behavior as BER. 9 Concluding Remarks We first showed that DF gives a significant transmission rate gain over MH, for an arbitrary route in the wireless network. We presented an algorithm, the nearest neighbor set algorithm (NNSA), to find optimal routes, which maximize the rates achievable by DF. As this algorithm, in the worst case, runs in factorial time, we designed a heuristic algorithm, the maximum sum- of-received-power algorithm (MSPA), that runs in polynomial time. We showed that the MSPA finds an optimal route when the nodes can only send independent codewords. However, unlike the NNA and the NNSA, the MSPA does not find an optimal route when the nodes are allowed to send arbitrarily correlated codewords. We implemented DF on practical networks using LDPC codes with incremental redundancy to compare different routes. We would like to highlight that for a given route, the choice of coding strategies, be it MH or DF, does not affect the spatial re-use of the system. In both strategies, the nodes in the route (except the destination) transmit at the same power level. The difference lies in how the nodes decode the data. We also note that there are some practical problems in implementing DF in large networks. 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Motani, “Optimal routing for the decode-and-forward strategy in the gaus- sian multiple relay channel,” accepted and to be presented at the 2007 IEEE International Symposium on Information Theory (ISIT 2007), Acropolis Congress and Exhibition Center, Nice, France. Introduction Related Work Motivating Cooperation Network Model Single-Hop Coding Strategy (SH) Multi-Hop Coding Strategy (MH) Decode-and-Forward Cooperative Coding (DF) Comparing the Strategies The Optimal Routing Problem The Nearest Neighbor Algorithm The Nearest Neighbor Set Algorithm Complexity of NNSA Heuristic Algorithms DF with LDPC Codes Concluding Remarks

0704.0500

On the polynomial automorphisms of a group

ON THE POLYNOMIAL AUTOMORPHISMS OF A GROUP GÉRARD ENDIMIONI Abstract. Let A(G) denote the automorphism group of a group G. A polynomial automorphism of G is an automorphism of the form x 7→ (v−1 ǫ1v1) . . . (v ). We prove that if G is nilpo- tent (resp. metabelian), then so is the subgroup of A(G) generated by all polynomial automorphisms. 1. Introduction and main results Let G be a group. We shall write A(G) for the automorphism group of G. According to Schweigert [10], we say that an element f ∈ A(G) is a polynomial automorphism of G if there exist integers ǫ1, . . . , ǫm ∈ Z and elements u0, . . . , um ∈ G such that f(x) = u0x ǫ1u1 . . . um−1x for all x ∈ G. Since f(1) = 1, it is easy to see that f(x) can be expressed as a ‘product’ of inner automorphisms, that is f(x) = (v−1 xǫ1v1) . . . (v ǫmvm). We shall write P0(G) for the set of polynomial automorphisms of G. Actually, Schweigert defines a polynomial automorphism in the context of finite groups. In particular, in this context, the set P0(G) is clearly a subgroup of A(G). On the other hand, this is not necessarily the case when G is infinite. For instance, in the additive group of rational numbers, the set of polynomial automorphisms forms a monoid with respect to the operation of functional composition, which is isomorphic to the multiplicative monoid Z \ {0}. 1991 Mathematics Subject Classification. 20F28, 20F16, 20F18. Key words and phrases. polynomial automorphism, metabelian group, nilpotent group, IA-automorphism. http://arxiv.org/abs/0704.0500v1 2 GÉRARD ENDIMIONI In this paper, we shall consider the subgroup P(G) = 〈P0(G)〉 of A(G), generated by all polynomial automorphisms ofG. Hence P0(G) = P(G) when G is finite, but for example P(G) is distinct from P0(G) when G is the additive group of rational numbers (note that P(G) = A(G) in this last case). It is easy to verify that P0(G) is a normal subset of A(G). Thus P(G) is a normal subgroup of A(G); in addition, we have I(G)E P(G)E A(G), where I(G) is the group of inner automorphisms of G. Also P(G) contains the group of invertible elements of the monoid P0(G). It is worth noting that there exist finite groups G such that the quotient P(G)/I(G) is not soluble [7]. If G is abelian, each polynomial automorphism is of the form x 7→ xǫ, and so P(G) is abelian. When G is a finite nilpotent group of class k ≥ 2, it is proved in [4] that P(G) is nilpotent of class k − 1 (see also [10, Satz 3.5]). We show here that this result remains true when G is infinite. Theorem 1.1. Let G be a nilpotent group of class k ≥ 2. Then P(G) is nilpotent of class k − 1. Notice that conversely, if P(G) is nilpotent, then so is G since P(G) contains the group of inner automorphisms. When G is metabelian, it seems that nothing is known about P(G), even in the context of finite groups. In this paper, we shall prove the following. Theorem 1.2. Let G be a metabelian group. Then the group P(G) is itself metabelian. In Section 3, we shall interpret a result of C. K. Gupta as a very particular case of this theorem (see Corollary 3.1 below). 2. Proofs As usual, in a group G, the commutator of two elements x, y is de- fined by [x, y] = x−1y−1xy. Instead of [[x, y], z], we shall write [x, y, z]. We denote by [G,G] the derived subgroup of G. POLYNOMIAL AUTOMORPHISMS 3 Lemma 2.1. Let f, g be two functions over a group G, respectively defined by the relations f(x) = (v−1 xǫ1v1) . . . (v ǫmvm), g(x) = (w−1 xη1w1) . . . (w ηnwn) (we do not suppose that f and g are automorphisms). Let t be an element of G such that any two conjugates of t commute. Then we have the relation f(g(t)) = tǫiηj [tǫiηj , vi][t ǫiηj , wj][t ǫiηj , wj, vi] (notice that in this product, the order of the factors is of no conse- quence). Proof. Using the fact that any two conjugates of t commute, we can write f(g(t)) = w−1j t v−1i w ǫiηjwjvi tǫiηj [tǫiηj , wjvi]. We conclude thanks to the relation [x, yz] = [x, z][x, y][x, y, z]. � In a nilpotent group G of class ≤ 2, two conjugates of any element t ∈ G commute. Therefore, as an immediate consequence of Lemma 2.1, we observe that any two polynomial automorphisms of G commute. Since these automorphisms generate P(G), we obtain: Corollary 2.1. If G is a nilpotent group of class ≤ 2, then P(G) is abelian. We are now ready to prove our first theorem. Proof of Theorem 1.1. Since P(G) contains I(G) (which is nilpotent of class k − 1 exactly), it suffices to show that P(G) is nilpotent of class at most k − 1. We argue by induction on the nilpotency class k of 4 GÉRARD ENDIMIONI G. The case k = 2 follows from Corollary 2.1. Now suppose that our theorem is proved for an integer k ≥ 2 and consider a nilpotent group G of class k + 1. Denote by ζ(G) the centre of G. One can define a homomorphism Θ : P(G) → A(G/ζ(G)), where for each f ∈ P(G), Θ(f) is the automorphism induced by f in G/ζ(G). Clearly, if f is a polynomial automorphism of G, then Θ(f) is a polynomial automor- phism of G/ζ(G). Hence Θ(P(G)) is a subgroup of P(G/ζ(G)), and so, by induction, is nilpotent of class at most k − 1. Since Θ(P(G)) and P(G)/ kerΘ are isomorphic, it suffices to show that ker Θ is included in the centre of P(G) and the theorem is proved. For that, consider an element g ∈ ker Θ and put w(x) = x−1g(x) for any x in G. Thus g(x) = xw(x) and w(x) belongs to ζ(G) for all x ∈ G. Notice that w defines a homomorphism of G into ζ(G) since w(xy) = y−1x−1g(x)g(y) = y−1w(x)g(y) = w(x)w(y). In order to show that g belongs to the centre of P(G), it suffices to verify that g commutes with any polynomial automorphism f of G. Suppose that f is defined by the relation f(x) = (v−1 xǫ1v1) . . . (v ǫmvm). We have easily f(g(x)) = f(xw(x)) = f(x)f(w(x)) = f(x)w(x)ǫ, where ǫ = ǫ1 + · · ·+ ǫm. In the same way, by using the fact that w is a homomorphism, we can write g(f(x)) = f(x)w(f(x)) = f(x)(w(v1) −1w(x)ǫ1w(v1)) . . . (w(vm) −1w(x)ǫmw(vm)), whence g(f(x)) = f(x)w(x)ǫ. Thus g and f commute, as required, and the result follows. � Now we undertake the proof of our second theorem. First we need the following result, which is well known and easy to prove (see for example [8, Lemma 34.51] or [9, Part 2, p. 64]). Lemma 2.2. In a metabelian group G, if t is an element of the derived subgroup [G,G], we have the relation [t, x, y] = [t, y, x] for all x, y ∈ G. POLYNOMIAL AUTOMORPHISMS 5 We arrive to the key lemma in the proof of Theorem 1.2. This lemma shows that when G is metabelian, any element h ∈ [P(G),P(G)] operates trivially on [G,G] and on G/[G,G]. Lemma 2.3. Let G be a metabelian group. Suppose that h is an ele- ment of the derived subgroup [P(G),P(G)]. Then (i) h(t) = t for all t ∈ [G,G]; (ii) x−1h(x) belongs to [G,G] for all x ∈ G. Proof. (i) Consider the homomorphism Φ : P(G) → A([G,G]) defined like this: for any f ∈ P(G), Φ(f) is the restriction of f to [G,G]. We must show that ker Φ contains [P(G),P(G)]. For that, first notice that any two conjugates of t ∈ [G,G] commute since G is metabelian. Now we apply Lemma 2.1. If f and g are polynomial automorphisms of G defined as in this lemma, we obtain the equalities f(g(t)) = tǫiηj [tǫiηj , vi][t ǫiηj , wj][t ǫiηj , wj, vi], g(f(t)) = tǫiηj [tǫiηj , vi][t ǫiηj , wj][t ǫiηj , vj, wi], and so, by Lemma 2.2, f(g(t)) = g(f(t)) for all t ∈ [G,G]. It follows that [f, g] belongs to ker Φ. In other words, the images of f and g in P(G)/ ker Φ commute. Since P(G)/ ker Φ is generated by the images of the polynomial automorphisms, this quotient is abelian. It follows that ker Φ contains [P(G),P(G)], as desired. (ii) Here, we consider the homomorphism Ψ : P(G) → A(G/[G,G]), where for any f ∈ P(G), Ψ(f) is the automorphism induced inG/[G,G] by f . Since a polynomial automorphism of G induces in G/[G,G] a polynomial automorphism of G/[G,G], Ψ(P(G)) is a subgroup of P(G/[G,G]). But P(G/[G,G]) is abelian (see for instance Corollary 2.1 above) and Ψ(P(G)) is isomorphic to P(G)/ kerΨ. Hence P(G)/ ker Ψ is abelian. Consequently, kerΨ contains [P(G),P(G)] and the result follows. � Proof of Theorem 1.2. Let f, g be two elements of [P(G),P(G)]. For any x ∈ G, put v(x) = x−1f(x) and w(x) = x−1g(x). By Lemma 2.3, 6 GÉRARD ENDIMIONI v(x) and w(x) belong to [G,G]. Applying again Lemma 2.3, we can write f(g(x) = f(xw(x)) = f(x)f(w(x)) = xv(x)w(x). In the same way, we have g(f(x)) = xw(x)v(x) = xv(x)w(x). It follows that f and g commute. Thus [P(G),P(G)] is abelian, and so P(G) is metabelian. � 3. IA-automorphisms of two-generator metabelian groups By way of illustration, we apply Theorem 1.2 to IA-automorphisms of a two-generator metabelian group. We recall that an automorphism of a group G is said to be an IA-automorphism if it induces the iden- tity automorphism on G/[G,G]. In a free metabelian group of rank 2, each IA-automorphism is inner [1], and so is a polynomial automor- phism. It turns out that in any two-generator metabelian group, each IA-automorphism is polynomial. This result is implicit in [2] with a different terminology. For convenience, we give a proof since this one is short and elementary. Proposition 3.1. Each IA-automorphism of a two-generator metabelian group is polynomial. To prove this proposition, we shall use the following result. Lemma 3.1. In a metabelian group G, each function ϕ of the form x 7→ ϕ(x) = x[x, v1] η1 . . . [x, vn] ηn (vi ∈ G, ηi ∈ Z) is an endomorphism. Proof. Thanks to the relation [xy, z] = y−1[x, z]y[y, z], we get ϕ(xy) = xy y−1[x, vi]y[y, vi] POLYNOMIAL AUTOMORPHISMS 7 But since the derived subgroup of G is abelian, we can write ϕ(xy) = xy y−1[x, vi] [y, vi] [x, vi] [y, vi] = ϕ(x)ϕ(y), as required. � Proof of Proposition 3.1. Suppose thatG is a two-generator metabelian group generated by a and b. If f is an IA-automorphism of G, we have f(a) = av and f(b) = bw, where v and w belong to the derived sub- group [G,G]. Now notice that [G,G] is the normal closure of [a, b]. Therefore, [G,G] is generated by [a, b] and the elements of the form [a, b, u], with u ∈ G. Hence v and w can be written in the form v = [a, b]α [a, b, vi] w = [a, b]β [a, b, wi] where α, β, λ1, . . . , λn, µ1, . . . , µn are integers (possibly equal to 0). By using the relation [x, y, z] = [x, y]−1[x, z]−1[x, yz], we obtain v = [a, b]α−λ [a, vi] −λi [a, bvi] w = [a, b]β−µ [a, wi] −µi [a, bwi] where λ = λ1 + · · ·+ λn and µ = µ1 + · · ·+ µn. Now put ϕ(x) = x[x, b]α−λ[x, a]µ−β [x, vi] −λi[x, bvi] λi[x, wi] µi [x, awi] −µi . By Lemma 3.1, ϕ is an endomorphism of G. Moreover, we have ϕ(a) = a[a, b]α−λ [a, vi] −λi[a, bvi] λi = av = f(a) 8 GÉRARD ENDIMIONI since [a, wi] = [a, awi]. In the same way, we get ϕ(b) = b[a, b]β−µ [b, wi] µi [b, awi] −µi . By using the identity [a, wi] −1[a, bwi] = [b, awi] −1[b, wi] (valid in any group), we obtain ϕ(b) = b[a, b]β−µ [a, wi] −µi [a, bwi] µi = bw = f(b). Thus f = ϕ and the proof is complete. � We remark that Proposition 3.1 cannot be extended to three-generator metabelian groups. For example, in the free metabelian group of rank 3 freely generated by a, b, c, consider the IA-automorphism f defined by f(a) = a, f(b) = b and f(c) = c[a, b]. Suppose that f is polynomial. Since [a, b] = c−1f(c), the commutator [a, b] would be in the normal closure of c, hence would be a product of conjugates of c±1. Substi- tuting 1 for c in this expression gives then [a, b] = 1, a contradiction. Therefore f is an IA-automorphism which is not polynomial. As a consequence of Theorem 1.2 and Proposition 3.1, we obtain an alternative proof of a result due to C. K. Gupta [6] (see also [3]). Corollary 3.1 ([6]). In a two-generator metabelian group, the group of IA-automorphisms is metabelian. Let Md denote the free metabelian group of rank d. By a result of Bachmuth [1], if d ≥ 3, the group of IA-automorphisms of Md contains a subgroup which is (absolutely) free of rank d. Thus Corollary 3.1 fails in a d-generator metabelian group when d ≥ 3. Also Bachmuth’s result shows once again that the group of IA-automorphisms of Md is not included in P(Md) (if d ≥ 3), since P(Md) is metabelian. In conclusion we mention that the metabelian groups constitute an important source of polynomial endomorphisms and automorphisms. Indeed, by Lemma 3.1, each function of the form x 7→ x[x, v1] η1 . . . [x, vn] ηn (vi ∈ G, ηi ∈ Z) POLYNOMIAL AUTOMORPHISMS 9 is an endomorphism in a metabelian group G. Besides, when G is metabelian and nilpotent, such an endomorphism is an automorphism since in a nilpotent group, every function of the form x 7→ u0x ǫ1u1 . . . um−1x ǫmum (ui ∈ G, ǫi ∈ Z) is a bijection if ǫ1 + · · ·+ ǫm = ±1 (see [5, Theorem 1]). References [1] S. Bachmuth, Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965), 93–104. [2] A. Caranti and C. M. Scoppola, Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup, Arch. Math. 56 (1991), 218–227. [3] F. Catino and M. M. Miccoli, A note on IA-endomorphisms of two-generated metabelian groups, Rend. Sem. Mat. Univ. Padova 96 (1996), 99–104. [4] G. Corsi Tani and M. F. Rinaldi Bonafede, Polynomial automorphisms in nilpotent finite groups, Boll. U.M.I. 5 (1986), 285–292. [5] G. Endimioni, Applications rationnelles d’un groupe nilpotent, C. R. Acad. Sci. Paris 314 (1992), 431–434. [6] C. K. Gupta, IA-automorphisms of two-generator metabelian groups, Arch. Math. 37 (1981), 106–112. [7] G. Kowol, Polynomautomorphismen von Gruppen, Arch. Math. 57 (1991), 114–121. [8] H. Neumann, Varieties of Groups, Springer-Verlag, Berlin (1967). [9] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer-Verlag, Berlin (1972). [10] D. Schweigert, Polynomautomorphismen auf endlichen Gruppen, Arch. Math. 29 (1977), 34–38. C.M.I-Université de Provence, 39, rue F. Joliot-Curie, F-13453 Mar- seille Cedex 13 E-mail address : [email protected] 1. Introduction and main results 2. Proofs 3. IA-automorphisms of two-generator metabelian groups References

0704.0501

On universality of critical behaviour in the focusing nonlinear Schr\"odinger equation, elliptic umbilic catastrophe and the {\it tritronqu\'ee} solution to the Painlev\'e-I equation

On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation. B.Dubrovin∗, T.Grava∗, C.Klein∗∗ ∗ SISSA, Trieste; ∗∗ Max-Planck Institute, Leipzig November 7, 2018 Abstract We argue that the critical behaviour near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation i�Ψt + � Ψxx + |Ψ|2Ψ = 0, � � 1, with analytic initial data of the form Ψ(x, 0; �) = A(x) e S(x) is approximately described by a particular solution to the Painlevé-I equa- tion. 1 Introduction The focusing nonlinear Schrödinger (NLS) equation for the complex valued function Ψ = Ψ(x, t) iΨt + Ψxx + |Ψ|2Ψ = 0 (1.1) has numerous physical applications in the description of nonlinear waves (see, e.g., the books [47, 35, 36]). It can be considered as an infinite dimensional analogue of a completely integrable Hamiltonian system [48], where the Hamiltonian and the Poisson bracket is given Ψt + {Ψ(x),H} = 0 {Ψ(x),Ψ∗(y)} = i δ(x− y) (1.2) |Ψx|2 − |Ψ |4 (here Ψ∗ stands for the complex conjugate function). Properties of various classes of so- lutions to this equation have been extensively studied both analytically and numerically [5, 6, 7, 8, 16, 21, 25, 26, 29, 32, 34, 43, 44]. One of the striking features that distinguishes this equation from, say, the defocusing case iΨt + Ψxx − |Ψ|2Ψ = 0 is the phenomenon of modulation instability [1, 11, 16]. Namely, slow modulations of the plane wave solutions Ψ = Aei(kx−ωt), ω = k2 − A2 develop fast oscillations in finite time. The appropriate mathematical framework for studying these phenomena is the theory of the initial value problem Ψ(x, 0; �) = A(x) e S(x) (1.3) for the �-dependent NLS i �Ψt + Ψxx + |Ψ|2Ψ = 0. (1.4) Here � > 0 is a small parameter, A(x) and S(x) are real-valued smooth functions. Introduc- ing the slow variables u = |Ψ|2, v = (1.5) the equation can be recast into the following system: ut + (u v)x = 0 (1.6) vt + v vx − ux + The initial data for the system (1.6) coming from (1.3) do not depend on �: u(x, 0) = A2(x), v(x, 0) = S ′(x). (1.7) The simplest explanation of the modulation instability then comes from considering the so- called dispersionless limit �→ 0. In this limit one obtains the following first order quasilin- ear system ut +v ux + u vx = 0 vt − ux + v vx = 0  . (1.8) This is a system of elliptic type because of the condition u > 0. Indeed, the eigenvalues of the coefficient matrix ( are complex conjugate, λ = v ± i u. So, the Cauchy problem for the system (1.8) is ill- posed in the Hadamard sense (cf. [33, 7]). Even for analytic initial data the life span of a typical solution is finite, t < t0. The x- and t-derivatives explode at some point x = x0 when the time approaches t0. This phenomenon is similar to the gradient catastrophe of solutions to nonlinear hyperbolic PDEs [2]. For the full system (1.6) the Cauchy problem is well-posed for a suitable class of �- independent initial data (see details in [17, 46]). However, the well-posedness is not uniform in �. In practical terms that means that the solution to (1.6) behaves in a very irregular way in some regions of the (x, t)-plane when � → 0. Such an irregular behaviour begins near the points (x = x0, t = t0) of the “gradient catastrophe” of the solution to the dispersionless limit (1.8). The solutions to (1.8) and (1.6) are essentially indistinguishable for t < t0; the situation changes dramatically near x0 when approaching the critical point. Namely, when approaching t = t0 the peak near a local maximum1 of u becomes more and more narrow due to self-focusing; the solution develops a zone of rapid oscillations for t > t0. They have been studied both analytically and numerically in [8, 16, 21, 23, 25, 26, 34, 43, 44]. However, no results are available so far about the behaviour of the solutions to the focusing NLS at the critical point (x0, t0). The main subject of this work is the study of the behaviour of solutions to the Cauchy problem (1.6), (1.7) near the point of gradient catastrophe of the dispersionless system (1.8). In order to deal with the Cauchy problem for (1.8) we will assume analyticity2 of the initial data u(x, 0), v(x, 0). Then the Cauchy problem for (1.8) can be solved for t < t0 via a suitable version of the hodograph transform (see Section 2 below). An important feature of the gradient catastrophe for this system is that it happens at an isolated point of the (x, t)- plane, unlikely the case of KdV or defocusing NLS where the singularity of the hodograph solution takes place on a curve. We identify this singularity for a generic solution to (1.8) as the elliptic umbilic singularity (see Section 4 below) in the terminology of R.Thom [42]. This codimension 2 singularity is one of the real forms labeled by the root system of the D4 type in the terminology of V.Arnold et al. [3]. Our main goal is to find a replacement for the elliptic umbilic singularity when the disper- sive terms are added, i.e., we want to describe the leading term of the asymptotic behaviour for �→ 0 of the solution to (1.6) near the critical point (x0, t0) of a generic solution to (1.8). Thus, our study can be considered as a continuation of the programme initiated in [13] to study critical behaviour of Hamiltonian perturbations of nonlinear hyperbolic PDEs; the fundamental difference is that the non perturbed system (1.8) is not hyperbolic! However, many ideas and methods of [13] (see also [12]) play an important role in our considerations. The most important of these is the idea of universality of the critical behaviour. The 1Regarding initial data with local minima we did not observe cusps related to minima in numerical simula- tions. We believe they do not exist because of the focusing effect in the NLS that pushes maxima to cusps but seems to smoothen minima. 2We believe that the main conclusions of this paper must hold true also for non analytic initial data; the numerical experiments of [8] do not show much difference in the properties of solution between analytic and non analytic cases. However, the precise formulation of our Main Conjecture has to be refined in the non analytic case. general formulation of the universality suggested in [13] for the case of Hamiltonian pertur- bations of the scalar nonlinear transport equation says that the leading term of the multiscale asymptotics of the generic solution near the critical point does not depend on the choice of the solution, modulo Galilean transformations and rescalings. This leading term was identi- fied in [13] via a particular solution to the fourth order analogue of the Painlevé-I equation (the so-called P 21 equation). The existence of the needed solution to P 1 has been rigorously established in [9]. Moreover, it was argued in [13] that this behaviour is essentially indepen- dent on the choice of the Hamiltonian perturbation. Some of these universality conjectures have been partially confirmed by numerical analysis carried out in [21]. The main message of this paper is the formulation of the Universality Conjecture for the critical behaviour of the solutions to the focusing NLS. Our considerations suggest the description of the leading term in the asymptotic expansion of the solution to (1.6) near the critical point via a particular solution to the classical Painlevé-I equation (P-I) Ωζζ = 6 Ω 2 − ζ. The so-called tritronquée solution to P-I was discovered by P.Boutroux [4] as the unique solution having no poles in the sector | arg ζ| < 4π/5 for sufficiently large |ζ|. Remarkably, the very same solution3 arises in the critical behaviour of solutions to focusing NLS! The paper is organized as follows. In Section 2 we develop a version of the hodograph transform for integrating the dispersionless system (1.8) before the catastrophe t < t0. We also establish the shape of the singularity of the solution near the critical point; the latter is identified in Section 4 with the elliptic umbilic catastrophe of Thom. In Section 3 we develop a method of constructing formal perturbative solutions to the full system (1.6) before the critical time. In Section 5 we collect the necessary information about the tritronquée solution of P-I and formulate the Main Conjecture of this paper. Such a formulation relies on a much stronger property of the tritronquée solution: namely, we need this solution to be pole-free in the whole sector | arg ζ| < 4π/5. Numerical evidence for the absence of poles in this sector is given in Section 6. In Section 7 we analyze numerically the agreement between the critical behaviour of solutions to focusing NLS and its conjectural description in terms of the tritronquée solution restricted on certain lines in the complex ζ-plane. In the final Section 8 we give some additional remark and outline the programme of future research. Acknowledgments. The authors thank K.McLaughlin for a very instructive discussion. One of the authors (B.D.) thanks R.Conte for bringing his attention to the tritronquées solutions of P-I. The results of this paper have been presented by one of the authors (T.G.) at the Conference “The Theory of Highly Oscillatory Problems”, Newton Institute, Cambridge, March 26 - 30, 2007. T.G. thanks A.Fokas and S.Venakides for the stimulating discussion after the talk. The present work is partially supported by the European Science Foundation Programme “Methods of Integrable Systems, Geometry, Applied Mathematics” (MISGAM), 3It is interesting that the same tritronquée solution (for real ζ only) appeares also in the study of certain critical phenomena in plasma [41]. In the theory of random matrices and orthogonal polynomials a different solution to P-I arises; see, e.g., [14]. Marie Curie RTN “European Network in Geometry, Mathematical Physics and Applications” (ENIGMA). The work of B.D. and T.G. is also partially supported by Italian Ministry of Universities and Researches (MUR) research grant PRIN 2006 “Geometric methods in the theory of nonlinear waves and their applications”. 2 Dispersionless NLS, its solutions and critical behaviour The equations (1.6) are a Hamiltonian system ut + {u(x), H} = 0 vt + {v(x), H} = 0 with respect to the Poisson bracket originated in (1.2) {u(x), v(y)} = δ′(x− y), (2.1) other brackets vanish, with the Hamiltonian u v2 − u2 dx. (2.2) Let us first describe the general analytic solution to the dispersionless system (1.8). Lemma 2.1 Let uI(x), vI(x) be two real valued analytic functions of the real variable x satisfying (uI,x) + (vI,x) 2 6= 0. Then the solution u = u(x, t), v = v(x, t) to the Cauchy problem u(x, 0) = uI(x), v(x, 0) = vI(x) (2.3) for the system (1.8) for sufficiently small t can be determined from the following system x = v t+ fu 0 = u t+ fv  (2.4) where f = f(u, v) is an analytic solution to the following linear PDE: fvv + u fuu = 0. (2.5) Conversely, given any solution to (2.5) satisfying u2f 2uu + f uv 6= 0 at some point (u = ũ, v = ṽ) such that fv(ũ, ṽ) = 0, the system (2.4) determines a solution to (1.8) defined locally near the point x = x̃ := fu(ũ, ṽ) for sufficiently small t. Remark 2.2 The solutions to the linear PDE (2.5) correspond to the first integrals of dis- persionless NLS: f(u, v) dx, F = 0. (2.6) Taking them as the Hamiltonians us + {u(x), F} ≡ ut + (fv)x = 0 vs + {v(x), F} ≡ vt + (fu)x = 0  (2.7) yields infinitesimal symmetries of the dispersionless NLS: (ut)s = (us)t , (vt)s = (vs)t . (2.8) One of the first integrals will be extensively used in this paper. It corresponds to the Hamiltonian density g = − v2 + u(log u− 1). (2.9) The associated Hamiltonian flow reads us + vx = 0  (2.10) Eliminating the dependent variable v one arrives at the elliptic version of the long wave limit of Toda lattice: uss + (log u)xx = 0. Due to commutativity (2.8) the systems (1.8) and (2.10) admit a simultaneous solution u = u(x, t, s), v = v(x, t, s). Any such solution can be locally determined from a system similar to (2.4) x = v t+ fu s = u t+ fv  (2.11) where f = f(u, v), as above, solves the linear PDE (2.5). The system (2.11) determines a solution u = u(x, t, s), v = v(x, t, s) provided applica- bility of the implicit function theorem. The conditions of the latter fail to hold at the critical point (x0, s0, t0, u0, v0) such that x0 = v0t0 + fu(u0, v0) s0 = u0t0 + fv(u0, v0) fuu(u0, v0) = fvv(u0, v0) = 0, fuv(u0, v0) = −t0 (2.12) In sequel we adopt the following system of notations: the values of the function f and of its derivatives at the critical point will be denoted by f 0 etc. E.g., the last line of the conditions (2.12) will read f 0uu = f vv = 0, f uv = −t0. Definition 1. We say that the critical point is generic if at this point: f 0uuv 6= 0. Let us the introduce real parameters r, ψ determined by the third derivatives of the function f evaluated at the critical point, (cosψ − i sinψ) = f 0uuv + i uuu. (2.13) Due to the genericity assumption + πk. In order to describe the local behaviour of a solution to the dispersionless NLS/Toda equations we define a function R(X;S, ψ) of real variables X , S depending on the real parameter ψ satisfying (S + cosψ)2 + (X + sinψ)2 6= 0 (2.14) by the following formula R(X,S, ψ) = sign [cosψ] (2.15) 1 +X sinψ + S cosψ + 1 + 2(X sinψ + S cosψ) +X2 + S2 P0(X,S, ψ) = R(X,S, ψ) cosψ − (X cosψ − S sinψ) sinψ R(X,S, ψ) − cosψ (2.16) Q0(X,S, ψ) = (X cosψ − S sinψ) cosψ R(X,S, ψ) +R(X,S, ψ) sinψ − sinψ. Observe that P0(X,S, ψ) and Q0(X,S, ψ) are smooth functions of the real variable X pro- vided validity of the inequality (2.14). Lemma 2.3 Given an analytic solution u(x, s, t), v(x, s, t) to the dispersionless NLS/Toda equations with a generic critical point (x0, s0, t0, u0, v0), and arbitrary real numbers X , S satisfying (2.14), T < 0, then there exist the following limits λ−1/2 u(x0 + λ 1/2v0T + r X T 2, s0 + λ 1/2u0T + rS T 2, λ1/2T )− u0 = r T P0 (X,S, ψ) (2.17) λ−1/2 v(x0 + λ 1/2v0T + r X T 2, s0 + λ 1/2u0T + rS T 2, λ1/2T )− v0 T Q0 (X,S, ψ) where the parameters r, ψ are defined by (2.13). Proof From the linear PDE (2.5) it follows that fuvv = −ufuuu − fuu, fvvv = −ufuuv. Using these formulae we expand the implicit function equations (2.11) near the critical point in the form x̄− v0t̄ = v̄ t̄+ f 0uuu(ū 2 − u0v̄2) + 2f 0uuvū v̄ (|ū|2 + |v̄|2)3/2 (2.18) s̄− u0t̄ = ū t̄+ f 0uuv(ū 2 − u0v̄2)− 2u0f 0uuuū v̄ (|ū|2 + |v̄|2)3/2 where we introduce the shifted variables x̄ = x− x0, s̄ = s− s0, t̄ = t− t0 ū = u− u0, v̄ = v − v0. The rescaling x̄− v0t̄ 7→ λ(x̄− v0t̄) s̄− u0t̄ 7→ λ(s̄− u0t̄) t̄ 7→ λ1/2t̄ ū 7→ λ1/2ū v̄ 7→ λ1/2v̄ (2.19) in the limit λ→ 0 yields the quadratic equation z = t̄ w + aw2, t̄ 6= 0 (2.20) where the complex independent and dependent variables z and w read z = s̄+ i u0x̄− (u0 + i u0v0)t̄, w = ū+ i u0v̄ (2.21) and the complex constant a is defined by a = f 0uuv + i uuu, (2.22) therefore = r eiψ. The substitution X = 2 x̄− v0t̄ r t̄2 , S = 2 s̄− u0t̄ r t̄2 reduces the quadratic equation to( w + t̄ r eiψ = r2t̄2e2iψ 1 + e−iψ(S + iX) For t̄ < 0 we choose the following root w = r t̄ei ψ 1 + e−i ψ(S + iX)− 1 (2.23) where the branch of the square root is obtained by the analytic continuation of the one taking positive values on the positive real axis. Equivalently, w = t̄ r eiψ sign (cosψ) ∆ + 1 + S cosψ +X sinψ + i X cosψ − S sinψ ∆ + 1 + S cosψ +X sinψ where 1 + 2(S cosψ +X sinψ) +X2 + S2. This gives the formulae (2.15). The result of the lemma describes the local structure of generic solutions to the dis- persionless NLS/Toda equations near the critical point. It can also be represented in the following form u(x, s, t) ' u0 + r T P0(X,S, ψ) (2.24) v(x, s, t) ' v0 + r T Q0(X,S, ψ) where X = 2 x̄− v0t̄ r t̄2 , S = 2 s̄− u0t̄ r t̄2 , T = t̄. (2.25) We want to emphasize that the approximation (2.24) works only near the critical point. Indeed, for large x→ ±∞ the function u(x, s, t) and v(x, s, t) have the following behaviour u = − r |x|u1/40 1∓ sinψ + u0 − r t̄ cosψ +O (2.26) v = ∓ r |x| u01/4 sign (cosψ) 1± sinψ + v0 − t̄ sinψ +O . (2.27) So, for sufficiently large |x| the function u(x, s, t) defined by (2.24) becomes negative. The function u has a maximum at the point X = S tanψ, so locally u ≤ u0 + r T cosψ − r | cosψ| + r T 2 < u0. (2.28) At the critical point (x0, s0, t0, u0, v0) the function u develops a cusp. Let us consider only the particular case S = 0 in order to avoid complicated expressions. In this case the local behaviour of the function u near the critical point is given by t̄→−0 r |x̂| 1− sinψ, x̂ > 0 r |x̂| 1 + sinψ, x̂ < 0 (2.29) (here x̂ = u0(x̄ − v0t̄)). Thus the parameters r, ψ describe the shape of the cusp at the critical point. 3 First integrals and solutions of the NLS/Toda equations Let us first show that any first integral (2.6) of the dispersionless equations can be uniquely extended to a first integral of the full equations. Lemma 3.1 Given a solution f = f(u, v) to the linear PDE (2.5), there exists a unique, up to a total derivative, formal power series in � hf = f + f (u, v;ux, vx, . . . , u (2k), v(2k)) such that the integral hf dx commutes with the Hamiltonian of NLS equation: {H,Hf} = 0 at every order in �. Explicitly, hf = f − fuuu + u2x + 2fuuvuxvx − ufuuuv (3.1) fuuuu + u2xx + 2fuuuvuxxvxx − ufuuuuv fuuuuuxxv fuuuvvxxu 3456u3 30fuuu − 9ufuuuu + 12u2f5u + 4u3f6u 432u2 −3fuuuv + 6ufuuuuv + 2u2f5u v u3xvx + 9fuuuu + 9uf5u + 2u (9fuuuuv + 10uf5u v)uxv (18f5u + 5uf6u) v +O(�6) Here we use short notations f5u := , f6u := , f5u v := ∂u5∂v Example 1. Taking f = 1 (u v2 − u2) one obtains the Hamiltonian of the NLS equation (u v2 − u2) + In this case the infinite series truncates. It is easy to see that the series in � truncates if and only if f(u, v) is a polynomial in u. Polynomial in u solutions to the linear PDE (2.5) correspond to the standard first integrals of the NLS hierarchy. Example 2. Taking g = −1 v2 + u(log u− 1) (cf. (2.9)) one obtains the Hamiltonian of Toda equation hg = − v2 + u(log u− 1)− u2x + 2u v 240u3 144u5 360u3 +O(�6) (3.2) written in terms of the function φ = log u in the form �2φxx + e φ(s+�) − 2eφ(s) + eφ(s−�) = 0. Lemma 3.2 Any solution to the NLS/Toda equations in the class of formal power series in � can be obtained from the equations x = v t+ δu(x) (3.3) s = u t+ δv(x) where f = f(u, v; �) is an arbitrary admissible solution to the linear PDE (2.5) in the class of formal power series in �, hf dx. Now, we can apply to the system (3.3) the rescaling (2.19) accompanied by the transfor- mation � 7→ λ5/4�. (3.4) At the limit λ→ 0 we arrive at the following system of equations s̄− u0t̄ = ū t̄+ f 0uuv (ū2 − u0v̄2) + − u0f 0uuu ū v̄ + (3.5) x̄− v0t̄ = v̄ t̄+ f 0uuu (ū2 − u0v̄2) + + f 0uuv ū v̄ + Using the complex variables z, w defined in (2.21) we can rewrite the system in the following form: z reiψ = w t̄ reiψ + wxx. (3.6) The last observation is that the Toda equations generated by the Hamiltonian Hg = hg dx (see Example 2 above) after the scaling limit (2.19), (3.4) yield the Cauchy - Riemann equa- tions for the function w = w(z), ∂w/∂z̄ = 0. Therefore the system (3.5) can be recast into the form equivalent to the Painlevé-I (P-I) equation (see (5.1) below) z reiψ = w t̄ reiψ + u0wzz. (3.7) Choosing λ = �4/5 we eliminate � from the equation. In the Section 5 below we will write explicitly the reduction of (3.7) to the Painlevé-I equation and give a conjectural characterization of the particular solution of the latter. 4 Critical behaviour and elliptic umbilic catastrophe Separating again the real and complex parts of (3.6) one obtains a system of ODEs UXX + (U2 − V 2) + r t̄ (U cosψ − V sinψ)− r (S cosψ −X sinψ) = 0 (4.1) VXX + UV + r t̄ (U sinψ + V cosψ)− r (S sinψ +X cosψ) = 0 that can be identified with the Euler - Lagrange equations δS = 0, S = L(U, V, UX , VX) dX with the Lagrangian V 2X − U U3 − 3U V 2 (U2 − V 2) cosψ − 2U V sinψ (4.2) +r (X sinψ − S cosψ)U + r (S sinψ +X cosψ)V. In the “dispersionless limit” � → 0 the Euler - Lagrange equations reduce to the search of stationary points of a function (let us also set t̄ = 0) U3 − 3U V 2 + a+U + a−V (4.3) where we redenote a+ = r (X sinψ − S cosψ) , a− = r (S sinψ +X cosψ) . At a+ = a− = 0 the function F has an isolated singularity at the origin U = V = 0 of the type D4,− also called elliptic umbilic singularity, according to R.Thom [42]. This singular- ity appears in various physical problems; we mention here the caustics in the collisionless dark matter [40] to give just an example. The parameters a+ and a− define two particular directions on the base of the miniversal unfolding of the elliptic umbilic; the full unfolding depending on 4 parameters reads U3 − 3U V 2 b(U2 + V 2) + a+U + a−V + c. (4.4) It would be interesting to study the properties of the modified Euler - Lagrange equations for the Lagrangian L̂ = L+ b(U2 + V 2). This deformation does not seem to arrive from considering solutions to the NLS hierarchy. 5 The tritronquée solution to the Painlevé-I equation and the Main Conjecture In this section we will select a particular solution to the Painlevé-I (P-I) equation Ωζζ = 6Ω 2 − ζ (5.1) Recall [22] that an arbitrary solution to this equation is a meromorphic function on the com- plex ζ-plane. According to P. Boutroux [4] the poles of the solutions accumulate along the arg ζ = , n = 0, ±1, ±2. (5.2) Boutroux proved that, for each ray there is a one-parameter family of particular solutions called intégrales tronquées whose lines of poles truncate for large ζ . He proved that the intégrale tronquée has no poles for large |ζ| within two consecutive sectors of the angle 2π/5 around the ray, and, moreover it has the asymptotic behaviour of the form Ω = − )1/2 [ (1−ε) (5.3) for a suitable choice of the branch of the square root (see below) and a sufficiently small ε > 0. Furthemore, if a solution truncates along any two of the rays (5.2) then it truncates along three of them. These particular solutions to P-I are called tritronquées. They have no poles for large |ζ| in four consecutive sectors; their asymptotics for large ζ is given by (5.3). It suffices to know the tritronquée solution Ω0(ζ) for the sector | arg ζ| < . (5.4) In this case the branch of the square root in (5.3) is obtained by the analytic continuation of the principal branch taking positive values on the positive half axis ζ > 0. Other four tritronquées solutions are obtained by applying the symmetry Ωn(ζ) = e , n = ±1, ±2. (5.5) The properties of the tritronquées solutions in the finite part of the complex plane were studied in the important paper of N.Joshi and A.Kitaev [24]. A. Kapaev [27] obtained a complete characterization of the tritronquées solutions in terms of the Riemann - Hilbert problem associated with P-I. We will briefly sketch here the main steps of his construction. The equation (5.1) can be represented as the compatibility condition of the following system of linear differential equations for a two-component vector valued function Φ = Φ(λ, ζ)  Ωζ 2λ2 + 2Ωλ− ζ + 2Ω2 2(λ− Ω) −Ωζ  Φ (5.6) Φζ = −  0 λ+ 2Ω  Φ. (5.7) The canonical matrix solutions Φk(λ, ζ) to the system (5.6) - (5.7) are uniquely determined by their asymptotic behaviour Φk(λ, ζ) ∼ λ1/4 λ1/4 λ−1/4 −λ−1/4 (5.8) λ−3/2 eθ(λ,ζ)σ3 , |λ| → ∞, λ ∈ Σk in the sectors λ ∈ C | < arg λ < , k ∈ Z. (5.9) θ(λ, ζ) = λ5/2 − ζ λ1/2, σ3 = , H = Ω2ζ − 2 Ω 3 + ζ Ω, (5.10) the branch cut on the complex λ-plane for the fractional powers of λ is chosen along the negative real half-line. The Stokes matrices Sk are defined by Φk+1(λ, ζ) = Φk(λ, ζ)Sk, λ ∈ Σk ∩ Σk+1. (5.11) They have the triangular form S2k−1 = 1 s2k−1 , S2k = s2k 1 (5.12) and satisfy the constraints Sk+5 = σ1 Sk σ1, k ∈ Z; S1S2S3S4S5 = i σ1 (5.13) where Due to (5.13) two of the Stokes multipliers sk determine all others; they depend neither on λ nor on ζ provided Ω(ζ) satisfies (5.1). In order to obtain a parametrization of solutions to the P-I equation (5.1) by Stokes mul- tipliers of the linear differential equation (5.6) one has to reformulate the above definitions as a certain Riemann - Hilbert problem. The solution of the Riemann - Hilbert problem de- pends on ζ through the asymptotics (5.8). If the Riemann - Hilbert problem has a unique solution for the given ζ0 ∈ C then the canonical matrices Φk(λ, ζ) depend analytically on ζ for sufficiently small |ζ − ζ0|; the coefficient Ω = Ω(ζ) will then satisfy (5.1). The poles of the meromorphic function Ω(ζ) correspond to the forbidden values of the parameter ζ for which the Riemann - Hilbert problem admits no solution. We will now consider a particular solution to the P-I equation specified by the following Riemann - Hilbert problem. Denote four oriented rays γ0, γ±1, ρ in the complex λ-plane defined by γk = {λ ∈ C | arg λ = }, k = 0, ±1 (5.14) ρ = {λ ∈ C | arg λ = π} directed towards infinity. The rays divide the complex plane in four sectors. We are looking for a piecewise analytic function Φ(λ, ζ) on λ ∈ C \ (γ−1 ∪ γ0 ∪ γ1 ∪ ρ) depending on the parameter ζ continuous up to the boundary with the asymptotic behaviour at |λ| → ∞ of the form (5.8) satisfying the following jump conditions on the rays Φ+(λ, ζ) = Φ−(λ, ζ)Sk, λ ∈ γk (5.15) Φ+(λ, ζ) = Φ−(λ, ζ)Sρ, λ ∈ ρ. Here the plus/minus subscripts refer to the boundary values of Φ respectively on the left/right sides of the corresponding oriented ray, the jump matrices are given by , S±1 = , Sρ = . (5.16) The following result is due to A.Kapaev4 . Theorem 5.1 The solution to the above Riemann - Hilbert problem exists and it is unique | arg λ| < , |λ| > R (5.17) for a sufficiently large positive number R. The associated function Ω0(ζ) := dH(ζ) (5.18) H(ζ) := σ3Φ(λ, ζ) e−θ(λ,ζ)σ3 − 1 is analytic in the domain (5.17), it satisfies P-I and enjoys the asymptotic behaviour Ω0(ζ) ∼ − , |ζ| → ∞, | arg λ| < . (5.19) Moreover, any solution of P-I having no poles in the sector (5.17) for some large R > 0 coincides with Ω0(ζ). 4Our solution Ω0(ζ) coincides with y3(x) ≡ y−2(x), x = −ζ, of [27] (see eq. (2.73) of [27]; Kapaev uses a different normalization y′′ = 6y2 + x of the P-I equation). Joshi and Kitaev proved that the tritronquée solution has no poles on the positive real axis. They found a numerical estimate for the position of the first pole ζ0 of the tritronquée solution Ω0(ζ) on the negative real axis: ζ0 ' −2.3841687 (cf. also [10]). Besides this estimate very little is known about the location of poles of Ω0(ζ). Our numerical experiments (see below) suggest the following Main Conjecture. Part 1. The tritronquée solution Ω0(ζ) has no poles in the sector | arg λ| < . (5.20) We are now ready to describe the conjectural universal structure behind the critical be- haviour of generic solutions to the focusing NLS. For simplicity of the formulation let us assume cosψ > 0. Main Conjecture. Part 2. Any generic solution to the NLS/Toda equations near the critical point behaves as follows u(x, s, t; �) + i u0v(x, s, t; �) ' u0 + i u0v0 − t̄ reiψ + 2 �2/5(3r 5 Ω0(ζ) +O (5.21) s̄− u0t̄+ i u0(x̄− v0t̄) + 12re iψ t̄2 where Ω0(ζ) is the tritronquée solution to the Painlevé-I equation (5.1). The above considerations can actually be applied replacing the NLS time flow by any other flow of the NLS/Toda hierarchy. The local description of the critical behaviour remains unchanged. Remark 5.2 Note that the angle of the line ζ(x̄) in (5.21) for t̄ fixed is equal to ψ/5 + π/2, ψ ∈ [−π, π]. Thus the maximal value of argζ is equal to 7π/10 < 4π/5. The lines in (5.21) consequently do not get close to the critical lines of the tritronquée solution. 6 Numerical analysis of the tritronquée solution of P-I In this section we will numerically construct the tritronquée solution Ω0, i.e. the tritronquée solution with asymptotic behavior (5.3). We will drop the index 0 in the following. The solution will be first constructed on a straight line in the complex plane. In a second step we will then explore global properties of these solutions within the limitations imposed by a numerical approach5. 5Cf. [15] where a similar technique was applied to solve numerically the Painlevé-II equation in the complex domain. Let the straight line in the complex plane be given by ζ = ay + b with a, b ∈ C constant (we choose a to have a non-negative imaginary part) and y ∈ R. The asymptotic conditions Ω ∼ − , (6.1) for y → ±∞. The root is defined to have its cut along the negative real axis and to as- sume positive values on the positive real axis. This choice of the root implies the following symmetry for the solution: Ω(ζ∗) = Ω∗(ζ). (6.2) Thus Ω is real on the real axis, see [24]. Numerically it is not convenient to impose boundary conditions at infinity. We thus assume that the wanted solution can be expanded in a Laurent series in ζ around infinity. Such an asymptotic expansion is possible for the considered tritronquée solution in the sector | arg ζ| < 4π/5. The formal series can be written there (see [24]) in the form Ωf = − ζ5k/2 , (6.3) where a0 = 1, and where the remaining coefficients follow from the recurrence relation for k ≥ 0 ak+1 = 25k2 − 1 amak+1−m. (6.4) This formal series is divergent, the coefficients ak behave asymptotically as ((k − 1)!)2, see [24] for a detailed discussion. It is known that divergent series can be used to get numerically acceptable approxima- tions to the values of the sum by properly truncating the series. Generally the best approx- imations for the sum result from truncating the series where the terms take the smallest absolute values (see e.g. [18]). Since we work in Matlab with a precision of 16 digits and with values of |ζ| ≥ 10, we typically consider up to 10 terms in the series. In this case the terms corresponding to a10 are of the order of machine precision (10−14 and below). Thus we have constructed approximations to the numerical values of the tritronquée solution for large values of |ζ|. These can be used as in [24] to set up an initial value problem for the P-I equation and to solve this with a standard ODE solver. In fact the approach works well on the real axis starting from positive values until one reaches the first singularity on the negative real axis. It is straightforward to check the results of [24] with e.g. ode45, the Runge-Kutta solver in Matlab corresponding to the Maple solver used in [24]. If one solves P-I on a line that avoids the sector | arg ζ| > 4π/5, one could integrate until one reaches once more large values of |ζ| for which the asymptotic conditions are known. This would provide a control on the numerical accuracy of this so-called shooting approach. Shooting methods are problematic if the second solution to the initial value problem has poles as is the case for P-I. In this case the numerical errors in the initial data (here due to the asymptotic conditions) and in the time integration will lead to a large contribution of the unwanted solution close to its poles which will make the numerical solution useless. It is obvious that P-I has such poles from the numerical results in [24] and the property (5.5). In [24] the task was to locate poles in the tritronquée solution, and in this case the shooting approach seems to be the only available. Here we are studying, however, the solution on a line in the complex plane where we know the asymptotic conditions for the affine parameter tending to ±∞. Thus we use as in [15] the asymptotic conditions on lines avoiding the sector | arg ζ| > 4π/5 to set up a boundary value problem for y = ±y0, y0 ≥ 10. The solution in the interval [−y0, y0] is numerically obtained with a finite difference code based on a collocation method. The code bvp4c distributed with Matlab, see [39] for details, uses cubic polynomials in between the collocation points. The P-I equation is rewritten in the form of a first order system. With some initial guess (we use Ω = − ζ/6 as the initial guess), the differential equation is solved iteratively by linearization. The collocation points (we use up to 10000) are dynamically adjusted during the iteration. The iteration is stopped when the equation is satisfied at the collocation points with a prescribed relative accuracy, typically 10−10. The solution for a = i and b = 0 is shown in Fig. 1. The values of Ω in between the collocation !20 !15 !10 !5 0 5 10 15 20 Figure 1: Real (blue) and imaginary part (red) of the tritronquée solution to the Painlevé I equation for ζ = iy. points are obtained by interpolation via the cubic polynomials in terms of which the solution has been constructed. This interpolation leads to a loss in accuracy of roughly one order of magnitude with respect to the precision at the collocation points. To test this we determine the numerical solution via bvp4c for P-I on Chebychev collocation points and check the accuracy with which the equation is satisfied via Chebychev differentiation, see e.g. [45]. It is found that the numerical solution with a relative tolerance of 10−10 on the collocation points satisfies the ODE to roughly the same order except at the boundary points where it is of the order 10−8, see Fig. 2 where we show the residual ∆ by plugging the numerical solution into the differential equation for the above example. It is straightforward to achieve a prescribed accuracy by requiring a certain value for the relative tolerance. Notice that we are not interested in a high precision solution of P-I here, but in a comparison of solutions to the NLS equation close to the point of gradient catastrophe of the semiclassical system with an asymptotic solution in terms of P-I transcendents. For this purpose an accuracy of the solution of the order of 10−4 will be sufficient in all studied cases. !20 !15 !10 !5 0 5 10 15 20 Figure 2: Error in the solution of the Painlevé I equation. The quality of the used boundary conditions via the asymptotic behavior can be checked by computing the solution for different values of y0. One finds that the difference between the asymptotic square root and the tritronquée solution is only visible near the origin, see Fig. 3. For large x it can be seen that the difference between the square root asymptotics and the tritronquée solution reaches quickly values below the aimed at threshold of 10−4. It is interesting to note that this difference is actually smaller than the difference between the tritronquée solution and the truncated formal asymptotic series except at the boundary, where the latter condition is implemented (see Fig. 3). The dominant behavior of the square root changes if one approaches the critical lines a = exp(4πi/5), b = 0. As can be seen from Fig. 4, the solution shows oscillations on top of the square root. The closer one comes to the critical lines, the slower is the fall off of the amplitude of the oscillations. We conjecture that these oscillations will have on the critical 10 12 14 16 18 20 !20 !10 0 10 20 Figure 3: The plot on the left side shows the absolute value of the difference between the tritronquée solution and the asymptotic condition − ζ/6 for a = i and b = 0. The plot on the right side shows in blue the same difference for x > 10 and in red the difference between the tritronquée solution and the truncated asymptotic series. lines only a slow algebraic fall off towards infinity. The above approach thus allows the computation of the tritronquée solution for a line avoiding the sector | arg ζ| > 4π/5 with high accuracy. The picture one obtains by com- puting Ω along several such lines is that there are indeed no singularities in the sector | arg ζ| < 4π/5, and that the square root behavior is followed for large |ζ|. To obtain a more complete picture, we compute the tritronquée solution for | arg ζ| < 4π/5 − 0.05 and |ζ| < R ( we choose R = 20). The boundary data for |ζ| = R follow as before from the truncated asymptotic series, the data for arg ζ = ±4π/5 − 0.05 are obtained by computing the tritronquée solution on the respective lines as above. To solve the resulting boundary value problem for the P-I equation is, however, computa- tionally expensive since we have to solve an equation in 2 real dimensions iteratively. Since the solution we want to construct is holomorphic there, we can instead solve the harmonicity condition (the two dimensional Laplace equation) for the given boundary conditions. To this end we introduce polar coordinates r, φ and use a spectral grid as described in [45]: the main point is a doubling of the interval r ∈ [0, R] to [−R,R] to allow for a better distribution of the Chebychev collocation points. Since we work with values of φ < φ0, we cannot use the usual Fourier series approach for the azimuthal coordinate. Instead we use again a Cheby- chev collocation method. The found solution in the considered domain is shown in Fig. 5 !20 !15 !10 !5 0 5 10 15 20 Figure 4: Real (blue) and imaginary part (red) of the tritronquée solution close to the critical line (for a = exp(i(4π/5− .05))) with oscillations of slowly decreasing amplitude. and Fig. 6. The quality of the solution can be tested by plugging the found solution to the Laplace equation into the P-I equation. Due to the low resolution and problems at the bound- ary, the accuracy is considerably lower in the two dimensional case than on the lines. This is, however, not a problem since we need only the one dimensional solutions for quantitative comparisons with NLS solutions. The two dimensional solutions give nonetheless strong numerical evidence for the conjecture that the tritronquée solution has globally no poles in the sector | arg ζ| < 4π/5. 7 Critical behavior in NLS and the tritronquée solution of P-I: numerical results In this section we will compare the numerical solution of the focusing NLS equation for two examples of initial data for values of � between 0.1 and 0.025 with the asymptotic so- lutions discussed in the previous sections, the semiclassical solution up to the breakup and the tritronquée solution to the Painlevé I equation. The numerical approach to solve the NLS equation is discussed in detail in [29]. For values of � below 0.04 we have to use Krasny filtering [30] (Fourier coefficients with an absolute value below 10−13 are put equal to zero to avoid the excitation of unstable modes). With double precision arithmetic we could thus reach � = 0.025, but could not go below. Figure 5: Real part of the tritronquée solution in the sector r < 20 and |φ| < 4π/5− 0.05. 7.1 Initial data We consider initial data where u(x, 0) has a single positive hump, and where v(x, 0) is monotonously decreasing. For initial data of the form u(x, 0) = A2(x) and v(x, 0) = 0 where the function A(x) is analytic with a single positive hump with maximum value A0, the semiclassical solution of NLS follows from (2.11) with f(u, v) given by f(u, v) = 2= dη ρ(η) v)2 + u  (7.1) where ρ(η) = ∫ x+(η) x−(η) A2(x) + η2 and where x±(η) are defined by A(x±(η)) = iη. The formula (7.1) follows from results by S.Kamvissis, K.McLaughlin and P.Miller in [26]. From f(u, v) it is straightforward to recover the initial data from the equations x = fu, fv = 0. (7.2) Numerically we study the critical behavior of two classes of initial data, one symmetric with respect to x which were used in [34], and initial data without symmetry with respect to x Figure 6: Imaginary part of the tritronquée solution in the sector r < 20 and |φ| < 4π/5 − 0.05. which are built from the initial data studied in [43]. For the former class the corresponding exact solution of focusing NLS is known in terms of a determinant. Nonetheless we integrate the NLS equation for these initial data numerically since this approach is not limited to special cases, but can be used for general smooth Schwartzian initial data as in the latter case. 7.1.1 Symmetric initial data We consider the particular class of initial data data u(x, t = 0) = A20sech 2 x, v(x, t = 0) = −µ tanhx, µ ≥ 0. (7.3) Introducing the quantity − A20, we find that the semiclassical solution for these initial data follows from (2.11) with f(u, v) = (v − 2M)∆+ − (v + 2M)∆− − u log u u log v +M + ∆+)(− v −M + ∆−) ] (7.4) where v ±M)2 + u For µ = 0 we recover the Satsuma-Yajima [37] initial data that were studied numerically in [34]. The function f(u, v) takes the form f(u, v) = < + i A0 + i A0 + u log + i A0) + iA0)2 + u√ (7.5) which can also be recovered from (7.1) by setting ρ = i. The critical point is given by u0 = 2A 0, v0 = 0, x0 = 0, t0 = . (7.6) Furthermore we have f 0uuu = 0, f uuv = , r = 4A30, ψ = 0, (7.7) where r, ψ are defined in (2.13). For A0 = 1 the initial data (7.3) coincides with the one studied in [43] by A.Tovbis, S.Venakides and X.Zhou. In the particular case µ = 2, A0 = 1 the function f(u, v) in (7.1) simplifies to f(u, v) = v − v2 + u+ u log −12v + v2 + u  . (7.8) In this case the critical point is given by v0 = 0, u0 = 2 + µ, t0 = 2 + µ , x0 = 0. Furthermore, f 0uuu = 0, f uuv = (µ+ 2)3 , r = (µ+ 2)3 , ψ = 0. 7.1.2 Non-symmetric initial data Recall that we are interested here in Cauchy data in the Schwartz class of rapidly decreasing functions. The above initial data are symmetric with respect to x, u is an even and v an odd function in x. To obtain a situation which is manifestly not symmetric, we use the fact that if f is a solution to (2.5), the same holds for derivatives and anti-derivatives of f with respect to v and for any linear combination of those. If fv is an even function in v, this will obviously not be the case for a linear combination of f and fv. As a specific example, we consider the linear combination f = f1 + αf2, α = const, where f1 coincides with (7.8) and f2 = 2u v2 + u− v2 + u + u v log −12v + v2 + u  . (7.9) The function f2 is obtained from the integration f2,v = f1 − v. The critical point is given in this case by u0 = 4(1− 16α2), v0 = −16α, x0 = 1 + 4α 1− 4α , t0 = 1 + 4α 1− 4α thus we have |α| < 1/4. Furthermore, f 0uuv = − 4α2 − 1/8 1− 16α2 , f 0uuu = 1− 16α2 such that r = 8u0, ψ = − arctan 1− 16α2 1/8− 4α2 We determine the initial data corresponding to f for a given value of |α| < 1/4 by solving (7.2) for u, v in dependence of x. This is done numerically by using the algorithm of [31] which is implemented as the function fminsearch in Matlab. The algorithm provides an iterative approach which converges in our case rapidly if the starting values are close enough to the solution, which is achieved by choosing u and v corresponding to f0 as an initial guess. For α close to 1/4 we observe numerically a steepening of the initial pulse which will lead to a shock front in the limit α→ 1/4. For the computations presented here, we consider the case α = 0.1 which leads to the initial data shown in Fig. 7. The initial data are computed in the way described above to the order of the Krasny filter on the interval x ∈ [−15, 11] on Chebychev collocation points. Standard interpolation via Chebychev polynomials is then used to interpolate the resulting data to a Fourier grid. To avoid a Gibbs phenomenon at the interval ends due to the non-periodicity of the data, we use a Fourier grid on the interval [−10π, 10π] to ensure that the function u takes values of the order of the Krasny filter. For x < −15 and x > 11, the function u is exponentially small which implies the zero-finding algorithm will no longer provide the needed precision. Thus we determine the exponential tails of the solution to leading order analytically. We find for x→ −∞ u = v2+ exp 2(x− αv+) αv+ + 1 v = v+ − v+ exp 2(x− αv+) αv+ + 1 2 log(v+) + 2(x− αv+) αv+ + 1 , (7.10) !4 !3 !2 !1 0 1 2 3 4 !4 !3 !2 !1 0 1 2 3 4 Figure 7: Initial data for the NLS equations without symmetry with respect to x. and for x→ +∞ u = v2− exp 2(x+ α) αv− + 1 v = v− − v− exp 2(x+ α) αv− + 1 2 log(−v−)− 2(x+ α) αv− + 1 , (7.11) where v± = ( 1± α − 1)/α. The initial data for the NLS equation in the form Ψ =√ u exp(iS/�) are then found by integrating v on the Chebychev grid by standard integration of Chebychev polynomials. The exponential tails for S follow from (7.10) and (7.11). The matching of the tails to the Chebychev interpolant is not smooth and leads to a small Gibbs phenomenon. The Fourier coefficients decrease, however, to the order of the Krasny filter which is sufficient for our purposes. Thus we obtain the non-symmetric initial data with roughly the same precision as the analytic symmetric data. 7.2 Semiclassical solution For times t� tc, the semiclassical solution gives a very accurate asymptotic description for the NLS solution. The situation is similar to the Hopf and the KdV equation [19]. We find for the symmetric initial data for t = tc/2 that the L∞ norm of the difference between the solutions decreases as �2. More precisely a linear regression analysis in the case of symmetric initial data (for the values � = 0.03, 0.04, . . . , 0.1) for the logarithm of this norm leads to an error proportional to �a with a = 1.94, a correlation coefficient r = 0.9995 and standard error σa = 0.03. In the non-symmetric case, we find a = 1.98, r = 0.999996 and σa = 0.003. Close to the critical time the semiclassical solution only provides a satisfactory descrip- tion of the NLS solution for large values of |x − xc|. In the breakup region it fails to be accurate since it develops a cusp at xc whereas the NLS solution stays smooth. This behavior can be well seen in Fig. 8 for the symmetric initial data. The largest difference between the −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 8: The blue line is the function u of the solution to the focusing NLS equation for the initial data u(x, 0) = 2 sechx and � = 0.04 at the critical time, and the red line is the corresponding semiclassical solution given by formulas (2.4). The green line gives the multiscales solution via the tritronquée solution of the Painlevé I equation. semiclassical and the NLS solution is always at the critical point. We find that the L∞ norm of the difference scales roughly as �2/5 as suggested by the Main Conjecture. More precisely we find a scaling proportional to �a with a = 0.38 and r = 0.999997 and σa = 4.2 ∗ 10−4. For the non-symmetric initial data, we find a = 0.36, r = 0.9999 and σa = 0.002. The corresponding plot for u can be seen in Fig. 9. The function v for the same situation as in Fig. 8 is shown in Fig. 10. It can be seen that the semiclassical solution is again a satisfactory description for |x − xc| large, but fails to be accurate close to the breakup point. The phase for the non-symmetric initial data can be seen in Fig. 11. In the following we will always study the scaling for the function u without further notice. 0.2 0.3 0.4 0.5 0.6 0.7 Figure 9: The blue line is the function u of the solution to the focusing NLS equation for the non-symmetric initial data and � = 0.04 at the critical time, and the red line is the corre- sponding semiclassical solution given by formulas (2.4). The green line gives the multiscales solution via the tritronquée solution of the Painlevé I equation. 7.3 Multiscales solution It can be seen in Fig. 8 and Fig. 10 that the multiscales solution (5.21) in terms of the tritronquée solution to the Painlevé I equation gives a much better asymptotic description to the NLS solution at breakup close to the breakup point than the semiclassical solution for the symmetric initial data. For larger values of |x − xc|, the semiclassical solution pro- vides, however, the better approximation. The rescaling of the coordinates in (5.21) sug- gests to consider the difference between the NLS and the multiscales solution in an interval [−γ�4/5, γ�4/5] (we choose here γ = 1, but within numerical accuracy the result does not depend on varying γ around this value). These intervals can be seen in Fig. 12. We find that the L∞ norm of the difference between these solutions in this interval scales roughly like �4/5. More precisely we have a scaling �a with a = 0.76 (r = 0.998 and σa = 0.019). For the non-symmetric initial data, the situation at the critical point can be seen in Fig. 9 and Fig. 11. Again the multiscales solution (5.21) gives a much better description close to the critical point than the semiclassical solution. However, the approximation is here much better on the side with weak slope for u than on the side with strong slope. We consider again the L∞-norm of the difference between the multiscales and the NLS solution in the interval [−γ�4/5, γ�4/5]. The scaling behavior of the solution can be seen in Fig. 13. For γ = 1 we find a = 0.71, r = 0.998 and σa = 0.02. These values do not change much for −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 Figure 10: The blue line is the function v of the solution to the focusing NLS equation for the initial data u0(x) = 2 sechx and � = 0.04 at the critical time, and the red line is the corresponding semiclassical solution given by formulas (2.4). The green line gives the multiscales solution via the tritronquée solution of the Painlevé I equation. larger γ. For smaller γ there are not enough points to provide a valid statistics. The value of a smaller than the predicted 4/5 is seemingly due to the strong asymmetry in the quality of the approximation of NLS by the multiscales solution as can be seen from Fig. 9. In the considered interval, the deviation is already so big that the scaling no longer holds as in the symmetric case. To study the scaling with a reliable statistics would, however, require the use of a considerably higher resolution which would be computationally too expensive. Going beyond the critical time, one finds that the real part of the NLS solution continues to grow before the central hump breaks up into several humps. Notice that the multiscales solution always leads to a function u that is smaller than the corresponding function of the NLS solution at breakup and before. This changes for times after the breakup as can be inferred from Fig. 14 which shows the time dependence of the NLS and the corresponding multiscales solution for the non-symmetric initial data. The approximation is always best at the critical time. To study the quality of the approximation (5.21), we use rescaled times. The scaling of the coordinates in (5.21) suggests to consider the NLS solution close to breakup at the times t±(�) with t±(�) = tc + u0/r − (u0/r)2 ± �4/5β, (7.12) 0.2 0.3 0.4 0.5 0.6 0.7 Figure 11: The blue line is the function v of the solution to the focusing NLS equation for the non-symmetric initial data and � = 0.04 at the critical time, and the red line is the corresponding semiclassical solution given by formulas (2.4). The green line gives the multiscales solution via the tritronquée solution of the Painlevé I equation. where β is a constant (we consider β = 0.1). We will only study the symmetric initial data in this context. Before breakup we obtain the situation shown in Fig. 15. It can be seen that the multiscales solution always provides a better description close to xc than the semiclassical solution, and that the quality improves in this respect with decreasing �. We find that the L∞ norm of the difference scales in this case as �a with a = 0.55 (r = 0.994 and σa = 0.03). The situation for times after breakup can be inferred from Fig. 16. Close to the central region the multiscales solution shows a clear difference to the NLS solution. But it is inter- esting to note that the ripples next to the central hump are well approximated by the Painlevé I solution. The L∞ norm of the difference between the two solutions scales roughly like �. More precisely we find a scaling �a with a = 1.02 (r = 0.9999 and σa = 7.7 ∗ 10−3). 8 Concluding remarks In this paper we have started the study of the critical behavior of generic solutions of the fo- cusing nonlinear Schrödinger equation. We have formulated the conjectural analytic descrip- tion of this behavior in terms of the tritronquée solution to the Painlevé-I equation restricted to certain lines in the complex plane. We provided analytical as well as numerical evidence −0.05 0 0.05 ε=0.03 −0.1 0 0.1 ε=0.1 Figure 12: The blue line is the solution to the focusing NLS equation for the initial data u0(x) = 2 sechx at the critical time, and the green line gives the multiscales solution via the tritronquée solution of the Painlevé I equation. The plots are shown for two values of � at the critical time. supporting our conjecture. In subsequent publications we plan to further study the Main Conjecture of the present paper by applying techniques based, first of all, on the Riemann - Hilbert problem method [26, 43, 44] and the theory of Whitham equations (see [19] for the numerical implementation of the Whitham procedure in the analysis of oscillatory behavior of solutions to the KdV equations). The latter will also be applied to the asymptotic descrip- tion of solutions inside the oscillatory zone. Furthermore we plan to study the possibility of extending the Main Conjecture to the critical behavior of solutions to the Hamiltonian perturbations of more general first order quasilinear systems of elliptic type. Last but not least, it would be of interest to study the distribution of poles of the tritronquée solution in the sector | arg ζ| > 4π and to compare these poles with the peaks of solutions to NLS inside the oscillatory zone. The elliptic asymptotics obtained by Kitaev [28] might be useful for studying these poles for large |ζ|. In this paper we did not study the behaviour of solutions to NLS near the boundary u = 0. Such a study is postponed for a subsequent publication. 0.35 0.4 0.45 0.5 !=0.03 0.2 0.3 0.4 0.5 0.6 0.7 !=0.1 Figure 13: The blue line is the solution to the focusing NLS equation for the non-symmetric initial data at the critical time, and the green line gives the multiscales solution via the tritronquée solution of the Painlevé I equation. 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Introduction Dispersionless NLS, its solutions and critical behaviour First integrals and solutions of the NLS/Toda equations Critical behaviour and elliptic umbilic catastrophe The tritronquée solution to the Painlevé-I equation and the Main Conjecture Numerical analysis of the tritronquée solution of P-I Critical behavior in NLS and the tritronquée solution of P-I: numerical results Initial data Symmetric initial data Non-symmetric initial data Semiclassical solution Multiscales solution Concluding remarks

0704.0502

Generic representations of orthogonal groups: projective functors in the category Fquad

GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS IN THE CATEGORY Fquad CHRISTINE VESPA Abstract. In this paper, we continue the study of the category of functors Fquad, associated to F2-vector spaces equipped with a nondegenerate qua- dratic form, initiated in [?] and [?]. We define a filtration of the standard projective objects in Fquad; this refines to give a decomposition into inde- composable factors of the two first standard projective objects in Fquad: PH0 and PH1 . As an application of these two decompositions, we give a complete description of the polynomial functors of the category Fquad. Mathematics Subject Classification: 18A25, 16D90, 20C20. Keywords: functor categories; quadratic forms over F2; Mackey functors; rep- resentations of orthogonal groups over F2. Introduction In the paper [?] we defined the category of functors Fquad from a category having as objects the nondegenerate F2-quadratic spaces to the category E of F2-vector spaces, where F2 is the field with two elements. The motivation for the construction of this category is to obtain an analogous framework for the orthogonal groups over F2, to that which exists for the general linear groups. We recall that the category F of functors from the category Ef of finite dimensional F2-vector spaces to the category E of all F2-vector spaces is a very useful tool for the study of the stable cohomology of the general linear groups with suitable coefficients (see [?]). Another motivation, in topology, for the study of the category F is the connection which exists between this category and unstable modules over the Steenrod algebra (see [?]). In order to have a good understanding of the category Fquad, we seek to classify its simple objects. We constructed in [?] two families of simple objects in Fquad. The first one is obtained by the fully-faithful, exact functor ι : F → Fquad, defined in [?], which preserves simple objects. By [?], the simple objects in F are in one-to-one correspondence with the irreducible representations of finite general linear groups over F2. The second family is obtained by the fully-faithful, exact functor κ : Fiso → Fquad, which preserves simple objects, where Fiso is equivalent to the product of the categories of modules over the orthogonal groups of possibly degenerate quadratic forms. In [?], we constructed two families of simple objects in the category Fquad which are neither in the image of ι nor in the image of κ. These simple objects are subfunctors of the tensor product between an object in the image of ι and an object in the image of κ. We proved that these simple objects in Fquad are the composition factors of two particular mixed functors, defined in Date: November 11, 2018. http://arxiv.org/abs/0704.0502v1 2 CHRISTINE VESPA The aim of this paper is to begin a programme to obtain a complete classification of the simple objects in Fquad. Accordingly, we seek to decompose the projective generators of this category into indecomposable factors and to obtain the simple factors of these indecomposable factors. This paper begins the study of the standard projective objects in the category Fquad. Although explicit decompositions of all the projective generators are not provided in this paper, we give several useful tools, results and examples for the realization of this programme. Furthermore, we deduce from the results contained in this paper several interesting consequences for the structure of the category Fquad. In work in progress, we obtain a general decomposition of standard projective object PH of Fquad which is indexed by the subspaces of H . Here we present explicit decompositions of the standard projective objects associated to “small” quadratic spaces, since these decompositions play a fundamental rôle in the category Fquad (for example, for the description of the polynomial functors of Fquad). Furthermore, recall that the decompositions of the injective standard IF of the category F and thus, by duality, that of the projective standard PF , is fundamental for the comprehension of the other injective standards of F . Hence, the decompositions of the two smaller projective standard of Fquad represent an important step in the understanding of the category Fquad. We briefly summarize the contents of this paper. After some recollections on the category Fquad, where we recall the definitions of the isotropic functors and the mixed functors, we define a filtration of the standard projective objects PV in Fquad: 0 ⊂ P V ⊂ P V ⊂ . . . ⊂ P (dim(V )−1) V ⊂ P (dim(V )) V = PV . We obtain a general description of the two extremities of this filtration. Theorem. Let V be a nondegenerate F2-quadratic space. (1) There is a natural equivalence: P V ≃ ι(P ǫ(V ) ), where ι : F → Fquad, ǫ is the functor that forgets the quadratic form and PF ǫ(V ) is the standard projective object in F associated to the vector space ǫ(V ). (2) The functor P V is a direct summand of PV . Proposition. Let V be a nondegenerate F2-quadratic space, we have a natural equivalence: PV /P (dim(V )−1) V ≃ κ(isoV ) where κ : Fiso → Fquad and isoV is an isotropic functor in Fiso. An important consequence of the Theorem concerning the functor P V is given in following result. Theorem. The category ι(F) is a thick subcategory of Fquad. Then by an explicit study of the filtration of the functors PH0 and PH1 we obtain the following fundamental decompositions of these two standard projective functors. Theorem. (1) The standard projective object PH0 admits the following decom- position: PH0 = ι(P ⊕2)⊕ (Mix0,1 ⊕2 ⊕Mix1,1)⊕ κ(isoH0) where Mix0,1 and Mix1,1 are two mixed functors and isoH0 is an isotropic functor. GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 3 (2) The standard projective object PH1 admits the following decomposition: PH1 = ι(P ⊕2)⊕Mix1,1 ⊕3 ⊕ κ(isoH1) where Mix1,1 is a mixed functor and isoH1 is an isotropic functor. These decompositions have several interesting consequences. Firstly, thanks to this theorem we can complete the study of the functors Mix0,1 and Mix1,1 started in [?] by the following result. Proposition. The functors Mix0,1 and Mix1,1 are indecomposable. We want to emphasize that the complete structure of the direct summands of the decompositions of PH0 and PH1 is understood. The structure of the isotropic functors is given in [?], those of the mixed functors Mix0,1 and Mix1,1 is the main result of [?] and is completed by the previous proposition and those of PF ⊕2 follows from [?]. Then, these decompositions give rise to a classification of the simple functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0. Proposition. The isomorphism classes of non-constant simple functors of Fquad such that either S(H0) 6= 0 or S(H1) 6= 0 are: ι(Λ1), ι(Λ2), ι(S(2,1)), κ(iso(x,0)), κ(iso(x,1)), RH0 , RH1 , SH1 where RH0 , RH1 and SH1 are the simple functors introduced in Corollary 1.7. These decompositions also allow us to derive some homological calculations in the category Fquad. Proposition. For n a natural number, we have: ExtnFquad(RH0 , RH0) ≃ F2 and Ext Fquad (RH1 , RH1) ≃ F2 where RH0 and RH1 are the simple functors introduced in Corollary 1.7. Finally, after having introduced the notion of polynomial functor for the category Fquad, which generalizes that for F , we obtain the following result as an application of the classification of the simple functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0 and of the thickness of the subcategory ι(F) of Fquad. Theorem. The polynomial functors of Fquad are in the image of the functor ι : F → Fquad. Most of the results of this paper are contained in the Ph.D. thesis of the author 1. The category Fquad: some recollections We recall in this section some definitions and results about the category Fquad obtained in [?]. Let Eq be the category having as objects finite dimensional F2-vector spaces equipped with a non degenerate quadratic form and with morphisms linear maps that preserve the quadratic forms. By the classification of quadratic forms over the field F2 (see, for instance, [?]) we know that only spaces of even dimension can be nondegenerate and, for a fixed even dimension, there are two non-equivalent nondegenerate spaces, which are distinguished by the Arf invariant. We will denote by H0 (resp. H1) the nondegenerate quadratic space of dimension two such that Arf(H0) = 0 (resp. Arf(H1) = 1). The orthogonal sum of two nondegenerate 4 CHRISTINE VESPA quadratic spaces (V, qV ) and (W, qW ) is, by definition, the quadratic space (V ⊕ W, qV⊕W ) where qV ⊕W (v, w) = qV (v) + qW (w). Recall that the spaces H0⊥H0 and H1⊥H1 are isomorphic. Observe that the morphisms of Eq are injective linear maps and this category does not admit push-outs or pullbacks. There exists a pseudo push-out in Eq that allows us to generalize the construction of the category of co-spans of Bénabou [?] and thus to define the category Tq in which there exist retractions. Definition 1.1. The category Tq is the category having as objects those of Eq and, for V and W objects in Tq, HomTq (V,W ) is the set of equivalence classes of dia- grams in Eq of the form V ←− W for the equivalence relation generated by the relation R defined as follows: V −→ X1 ←− W R V −→ X2 ←− W if there exists a morphism α of Eq such that α ◦ f = u and α ◦ g = v. The composition is defined using the pseudo push-out. The morphism of HomTq (V,W ) represented by the diagram V ←−W will be denoted by [V ←−W ]. Remark 1.2. A morphism of HomTq (V,W ) is represented by a diagram of the form: V −→ W⊥W ′ ←−−W , where iW is the canonical inclusion. In the following, we will use this representation of a morphism, without further comment. By definition, the category Fquad is the category of functors from Tq to E . Hence Fquad is abelian and has enough projective objects. By the Yoneda lemma, for any object V of Tq, the functor PV = F2[HomTq (V,−)] is a projective object and there is a natural isomorphism: HomFquad(PV , F ) ≃ F (V ), for all objects F of Fquad. The set of functors {PV |V ∈ S}, named the standard projective objects in Fquad, is a set of projective generators of Fquad, where S is a set of representatives of isometry classes of nondegenerate quadratic spaces. There is a forgetful functor ǫ : Tq → E f in Fquad, defined by ǫ(V ) = O(V ) and −→W⊥W ′ ←−W ]) = pg ◦ O(f) where pg is the orthogonal projection from W⊥W ′ to W and O : Eq → E f is the functor which forgets the quadratic form. By the fullness of the functor ǫ and an argument of essential surjectivity, we obtain the following theorem. Theorem 1.3. [?] There is a functor ι : F → Fquad, which is exact, fully faithful and preserves simple objects. In order to define another subcategory of Fquad, we consider the category E having as objects finite dimensional F2-vector spaces equipped with a (possibly degenerate) quadratic form and with morphisms injective linear maps which pre- serve the quadratic forms. The category Edegq admits pullbacks; consequently the category of spans Sp(Edegq ) ([?]) is defined. By definition, the category Fiso is the category of functors from Sp(Edegq ) to E . As in the case of the category Fquad, the category Fiso is abelian and has enough projective objects: by the Yoneda lemma, for any object V of Sp(Edegq ), the functor QV = F2[HomSp(Edegq )(V,−)] is a projec- tive object in Fiso. We define a particular family of functors of Fiso, the isotropic functors, which form a set of projective generators and injective cogenerators of Fiso. The category Fiso is related to Fquad by the following theorem. Theorem 1.4. [?] There is a functor κ : Fiso → Fquad, which is exact, fully-faithful and preserves simple objects. GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 5 We obtain the classification of the simple objects of the category Fiso from the following theorem. Theorem 1.5. [?] There is a natural equivalence of categories Fiso ≃ F2[O(V )]−mod where S is a set of representatives of isometry classes of quadratic spaces (possibly degenerate) and O(V ) is the orthogonal group. The object ofFiso which corresponds, by this equivalence, to the module F2[O(V )] is the isotropic functor isoV , defined in [?]. Recall that, as a vector space, isoV (W ) is isomorphic to the subspace of QV (W ) generated by the elements [V ←− V →W ]. A straightforward consequence of the classification of simple objects of Fiso given in Theorem 1.5 is given in the following corollary. Recall that, by definition, an object F of Fquad is finite if it has a finite composition series with simple subquo- tients. Corollary 1.6. The isotropic functors κ(isoV ) are finite in the category Fquad. In section 3, we will require the composition series for the isotropic functors asso- ciated to some small quadratic spaces. For α ∈ {0, 1}, let (x, α) be the degenerate quadratic space of dimension one generated by x such that q(x) = α. Since the or- thogonal groups O(x, 0) and O(x, 1) are trivial and O(H0) ≃ S2 and O(H1) ≃ S3, we deduce from Theorem 1.5 and 1.4, the following corollary. Corollary 1.7. (1) The functors κ(iso(x,0)) and κ(iso(x,1)) are simple in Fquad. (2) The functor κ(isoH0) is indecomposable. We have the following non-split short exact sequence: 0→ RH0 → κ(isoH0)→ RH0 → 0 where RH0 is the functor obtained from the trivial representation of O(H0). (3) The functor κ(isoH1) admits the following decomposition: κ(isoH1) = FH1 ⊕ (SH1) where SH1 is the functor obtained from the natural representation of O(H1) and FH1 is an indecomposable functor for which we have the following non- split short exact sequence: 0→ RH1 → FH1 → RH1 → 0 where RH1 is the functor obtained from the trivial representation of O(H1). In [?], we define a new family of functors of Fquad, named the mixed functors and we decompose two particular functors of this family: the functors Mix0,1 and Mix1,1. We recall the following description of these functors. Proposition 1.8. [?] For α ∈ {0, 1}, the functors Mixα,1 : Tq → E are defined by Mixα,1(V ) = F2[SV ] where SV = {(v1, v2) |v1 ∈ V, v2 ∈ V, q(v1 + v2) = α, B(v1, v2) = 1} and Mixα,1([V −→W⊥L ←−−W ])[(v1, v2)] = [(pW ◦ f(v1), pW ◦ f(v2))] if f(v1 + v2) ∈W 0 otherwise where pW is the orthogonal projection. 6 CHRISTINE VESPA In [?], for a positive integer n, we defined subfunctors Lnα of ι(Λ n)⊗ κ(iso(x,α)), where Λn is the nth exterior power and we proved that these functors are simple. The functor L1α is equivalent to the functor κ(iso(x,α)). We obtain the following result. Theorem 1.9. [?] Let α be an element in {0, 1}. (1) The functor Mixα,1 is infinite. (2) There exists a subfunctor Σα,1 of Mixα,1 such that we have the following short exact sequence 0→ Σα,1 → Mixα,1 → Σα,1 → 0. (3) The functor Σα,1 is uniserial with unique composition series given by the decreasing filtration given by the subfunctors kdΣα,1 of Σα,1: . . . ⊂ kdΣα,1 ⊂ . . . ⊂ k1Σα,1 ⊂ k0Σα,1 = Σα,1. (a) The head of Σα,1 (i.e. Σα,1/k1Σα,1) is isomorphic to the functor κ(iso(x,α)) where iso(x,α) is a simple object in Fiso. (b) For d > 0 kdΣα,1/kd+1Σα,1 ≃ L where Ld+1α is a simple object of the category Fquad that is neither in the image of ι nor in the image of κ. The functor Ld+1α is a subfunctor of ι(Λ d+1)⊗κ(iso(x,α)), where Λ is the (d+ 1)st exterior power functor. 2. Filtration of the standard projective functors PV of Fquad In this section, we define a filtration of the standard projective functors PV of Fquad. This construction gives rise to an essential tool to obtain, in section 3, the direct decompositions of the projective objects PH0 and PH1 of Fquad, into indecomposable summands. After defining this filtration, we will deduce general results about the projective PV of Fquad. In Theorem 2.6 we prove that the rank zero part is a direct summand of PV and we identify this functor. This result allows us to prove that ι(F) is a thick subcategory of Fquad. We will also show that the top quotient of this filtration is isomorphic to κ(isoV ), where isoV is the isotropic functor. 2.1. Definition of the filtration. We recall that a morphism in Tq from V to W , where V and W are nondegenerate quadratic spaces, is represented by a diagram V → X ←W. Definition 2.1. A morphism [V → X ← W ] in Tq has rank equal to i if the pullback in Edegq of the diagram V → X ←W is a quadratic space of dimension i. Notation 2.2. We denote by Hom (V,W ) the subset of HomTq (V,W ) of mor- phisms of rank less than or equal to i. We have the following proposition: Proposition 2.3. For W an object in Tq, the following subvector space of PV (W ): V (W ) = F2[Hom (V,W )] defines a subfunctor P V of PV . GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 7 Proof. It is sufficient to verify that for all morphisms f = [W → Y ← Z] of Tq and g = [V → X ← W ] of Hom (V,W ), the composition f ◦ g has rank less than or equal to i. The composition f ◦ g is represented by the following commutative diagram: P ′ // V // X // X⊥W where P and P ′ are the pullbacks and X⊥ Y is the pseudo push-out defined in [?]. Consequently: f ◦ g = [V → X⊥ Y ← Z]. Since [V → X ← W ] is an element of V (W ), we know that the dimension of P is less than or equal to i. We deduce from the injectivity of the morphisms of Edegq , that P ′ has dimension smaller than or equal to i. � The following lemma is a straightforward consequence of Definition 2.1. Lemma 2.4. There exists a natural equivalence: P (dim(V )) V ≃ PV . We deduce the following proposition. Proposition 2.5. The functors P V , for i = 0, . . . , dim(V ), define an increasing filtration of the functor PV . Proof. The inclusion of vector spaces P V (W ) ⊂ P (i+1) V (W ) is clear, forW an object in Tq. Consequently, P V is a subfunctor of P (i+1) V by the proposition 2.3. � 2.2. Extremities of the filtration. In the previous section, we have obtained, for all objects V of Tq, the following filtration of the functor PV : 0 ⊂ P V ⊂ P V ⊂ . . . ⊂ P (dim(V )−1) V ⊂ P (dim(V )) V = PV . The aim of this section is to study the two extremities of this filtration, namely, the functor P V and the quotient PV /P (dim(V )−1) 2.2.1. The functor P V . For V an object in Tq, we recall that the functor P ǫ(V ) of F defined by PFǫ(V )(−) = F2[HomEf (ǫ(V ),−)], where ǫ : Tq → E f is the forgetful functor of Fquad, is projective, by the Yoneda lemma. The aim of this paragraph is to prove the following theorem: Theorem 2.6. Let V be an object of Tq. (1) There is a natural equivalence: P V ≃ ι(P ǫ(V ) ), where ι : F → Fquad is the functor given in Theorem 1.3. (2) The functor P V is a direct summand of PV . Before proving this result, we give the following useful characterization of the morphisms of rank zero, which is a straightforward consequence of the definition of the rank of a morphism. 8 CHRISTINE VESPA Lemma 2.7. Let V be an object in Tq. A morphism T = [V −→ W⊥W ′ ←−− W ] has rank zero if and only if pW ′ ◦ α is an injective linear map, where pW ′ is the orthogonal projection from W⊥W ′ to W ′. For V and W objects in Tq, the forgetful functor ǫ : Tq → E f gives rise to a map HomTq(V,W )→ HomEf (ǫ(V ), ǫ(W )). By passage to the vector spaces freely gener- ated by these sets and by functoriality of ǫ, we deduce the existence of a morphism from PV to ι(P ǫ(V ) ). As the functors P V are subfunctors of PV , we obtain a mor- phism f from P V to ι(P ǫ(V ) ). Consequently, to prove Theorem 2.6, it is sufficient to prove the following proposition. Proposition 2.8. The map P V (W ) −−→ PFǫ(V )(ǫ(W )) is an isomorphism for V and W objects in Tq. The surjectivity of fW relies on the following lemma, which is an improved version of the fullness of the forgetful functor ǫ given in [?]. Lemma 2.9. Let (V, qV ) and (W, qW ) be two objects of Tq and f ∈ HomEf (ǫ(V, qV ), ǫ(W, qW )) a linear map, then there exists a morphism T = [V −→W⊥Y ←−−W ] of rank zero such that ǫ(T ) = f . Proof. As the quadratic space V is nondegenerate, we know that it has even di- mension. We write dim(V ) = 2n. We prove the result by induction on n. To start the induction, let (V, qV ) be a nondegenerate quadratic space of di- mension two, with symplectic basis {a, b} and f : V → W be a linear map. The following linear map preserves the quadratic form: g1 : V → W⊥H1⊥H0⊥H0 ≃W⊥Span(a1, b1)⊥Span(a0, b0)⊥Span(a a 7−→ f(a) + (q(a) + q(f(a)))a1 + a0 b 7−→ f(b) + (q(b) + q(f(a)))a1 + (1 +B(f(a), f(b)))b0 + a Consequently, the morphism: T = V −→W⊥H1⊥H0 ←֓ W , is a morphism of rank zero of Tq such that ǫ(T ) = f . Let Vn be a nondegenerate quadratic space of dimension 2n, {a1, b1, . . . , an, bn} be a symplectic basis of Vn and fn : Vn →W be a linear map. By induction, there exists a map: gn : Vn → W⊥Y ai 7−→ fn(ai) + yi bi 7−→ fn(bi) + zi where yi and zi, for all integers i between 1 and n, are elements of Y . The map gn preserves the quadratic form and the morphism T = [Vn −→W⊥Y ←֓ W ] is of rank zero and verifies ǫ([Vn −→W⊥Y ←֓ W ]) = fn. Let Vn+1 be a nondegenerate quadratic space of dimension 2(n + 1), {a1, b1, . . . , an, bn, an+1, bn+1} a symplectic basis of Vn+1 and fn+1 : Vn+1 → W a linear map. To define the map gn+1, we will consider the restriction of fn+1 to Vn and extend the map gn given by the inductive assumption. For that, we need the following space: E ≃ W⊥W ′⊥H⊥n0 ⊥H 0 ⊥H1⊥H0⊥H0 for which we specify the notations for a basis: E ≃W⊥W ′⊥(⊥ni=1Span(a 0))⊥(⊥ i=1Span(A 0))⊥Span(A1, B1) ⊥Span(C0, D0)⊥Span(E0, F0). GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 9 The following map: gn+1 : V → W⊥W ′⊥H⊥n0 ⊥H 0 ⊥H1⊥H0⊥H0 ai 7−→ fn+1(ai) + yi + a 0 for i between 1 and n bi 7−→ fn+1(bi) + zi +A an+1 7−→ fn+1(an+1) + (q(an+1) + q(fn+1(an+1)))A1 + C0 i=1 B(fn+1(ai), fn+1(an+1))b i=1 B(fn+1(bi), fn+1(an+1))B bn+1 7−→ fn+1(bn+1) + (q(bn+1) + q(fn+1(bn+1)))A1 +(1 +B(fn+1(an+1), fn+1(bn+1)))D0 i=1 B(fn+1(ai), fn+1(bn+1))b i=1 B(fn+1(bi), fn+1(bn+1))B 0 + E0 preserves the quadratic form. Furthermore, the morphism T = [Vn+1 −−−→W⊥W ′⊥H⊥n0 ⊥H 0 ⊥H1⊥H0⊥H0 ←֓ W ] is of rank zero and satisfies: ǫ(T ) = fn+1, which completes the inductive step. The proof of the injectivity of fW relies on the following result, which can be regarded as Witt’s theorem for degenerate quadratic forms. Theorem 2.10. Let V be a nondegenerate quadratic space, D and D′ subquadratic spaces (possibly degenerate) of V and f : D → D′ an isometry between these two quadratic spaces. Then, there exists an isometry f : V → V such that the following diagram is commutative: f // V // D′. Proof. For a proof of this result, we refer the reader to [?] §4, theorem 1. � Proof of the injectivity of fW . The natural map f is induced by the natural map (V,−)→ HomEf (ǫ(V ), ǫ(−)) by passage to the vector spaces freely generated by these sets. So, f is injective if and only if this natural map is injective. Consequently, it is sufficient to verify that, for T = [V −→ W⊥W ′ ←−− W ] and T ′ = [V −→ W⊥W ′′ ←−−W ] two generators of V (W ) such that (2.10.1) pW ◦ O(α) = p W ◦ O(α we have T = T ′. Let {a1, b1, . . . , an, bn} be a symplectic basis of V . We deduce from 2.10.1 that, for all i ∈ {1, . . . , n}, we have: (2.10.2) α(ai) = wi + w i, α(bi) = xi + x (2.10.3) α′(ai) = wi + w i , α ′(bi) = xi + x 10 CHRISTINE VESPA where, for all i ∈ {1, . . . , n}, wi and xi are in W , w i and x i are in W ′ and x′′i and w i are in W ′′. By Lemma 2.7, since the morphisms are of rank zero, {w′1, x 1, . . . , w n} and {w 1 , x 1 , . . . , w n} are two linearly independent families of vectors. We will denote by W ′ = Span(w′1, x 1, . . . , w n) (respectively W ′′ = Span(w′′1 , x 1 , . . . , w n)) the subquadratic space (possibly degenerate), of W ′ (respectively W ′′) and we define the linear map f : W ′ → W ′′ by f(w′i) = w and f(x′i) = x i for all i ∈ {1, . . . , n}. Since α and α′ preserve the quadratic forms, we deduce from the relations 2.10.2 and 2.10.3 that f preserves the quadratic form. Hence, we can apply Theorem 2.10 to the nondegenerate space W ′⊥W ′′, which gives a morphism f : W ′⊥W ′′ → W ′⊥W ′′ of Eq, such that, the restriction of this morphism to W ′ coincides with f . We deduce the commutativity of the following diagram: α̃ // α̃′ ++WWW W W⊥(W ′⊥W ′′) W⊥(W ′⊥W ′′) where α̃ = iW⊥W ′ ◦ α and α̃′ = iW⊥W ′′ ◦ α ′. Consequently, we obtain the equality T = T ′ since, by inclusion, we have T = [V −→W⊥W ′ ←−−W ] = [V −→W⊥W ′⊥W ′′ ←−−W ] T ′ = [V −→W⊥W ′′ ←−−W ] = [V −→ W⊥W ′⊥W ′′ ←−−W ]. Notation 2.11. For V and W two objects of Eq, and f a morphism of HomEf (ǫ(V ), ǫ(W )), we denote by tf the morphism of Hom (V,W ) corresponding to f and by [tf ] the canonical generator of P V (W ) obtained from tf . To simplify the notation, we will denote the morphism tIdV of Hom (V, V ) by eV . We deduce from the first point of Theorem 2.6 the following corollary. Corollary 2.12. For V , W and X objects of Eq, f : ǫ(W )→ ǫ(X) and g : ǫ(V )→ ǫ(W ) morphisms of Ef , we have: tf ◦ tg = tf◦g where tf , tg and tf◦g are respectively the morphisms of Hom (W,X), Hom (V,W ) and Hom (V,X) associated to the linear maps f, g and f ◦ g. We can apply this result to the idempotents of the ring of endomorphisms End(PV ), to obtain the following proposition. Proposition 2.13. The canonical generator [eV ] of P V is an idempotent of the ring of endomorphisms End(PV ) such that PV .[eV ] ≃ P GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 11 Proof. The canonical generator [eV ] is an idempotent of End(PV ) by Corollary 2.12. By definition of the rank filtration PV .[eV ] ⊂ P V and, for a canonical generator [tf ] of P V , we have [tf ] = [tf ] · [eV ]. � The idempotent [eV ] plays a central rôle in the proof of the thickness of the subcategory ι(F) in Fquad, which is the subject of the following paragraph. For that, the following result is necessary. Lemma 2.14. Let V and W be objects of Tq, the functor ι induces an isomorphism: HomF (P ǫ(V ), P ǫ(W )) −→ HomFquad(P V , P where ǫ : Tq → E is the forgetful functor. Proof. By Proposition 2.13 and Theorem 2.6 we have the following equivalences: HomFquad(P V , P W ) ≃ P W (eV )P W (V ) ≃ P W (V ) ≃ ι(PFǫ(W ))(V ) ≃ P ǫ(W )(ǫ(V )) ≃ HomF (P ǫ(V ), P ǫ(W )). To conclude this paragraph, we give the following property of eV which will be useful in section 4 concerning the polynomial functors of Fquad. Lemma 2.15. For V and W two objects of Tq, we have: eV⊥W = eV⊥eW , where ⊥ : Tq × Tq → Tq is the functor induced by the orthogonal sum. Proof. This is a straightforward consequence of Proposition 2.8. � 2.2.2. The category ι(F) is a thick subcategory of Fquad. The aim of this paragraph is to prove the following result. Theorem 2.16. The category ι(F) is a thick subcategory of Fquad, where ι : F → Fquad is the functor defined in Theorem 1.3. To prove this theorem, we need the following general result about the precom- position functor which is proved in the Appendix of [?]. Proposition 2.17. Let C and D be two small categories, A be an abelian category, F : C → D be a functor and −◦F : Func(D,A)→ Func(C,A) be the precomposition functor, where Func(C,A) is the category of functors from C to A. If F is full and essentially surjective, then any subobject (respectively quotient) of an object in the image of the precomposition functor is isomorphic to an object in the image of the precomposition functor. Proof of Theorem 2.16. • The subcategory ι(F) of Fquad is full by Theorem • Let FF be an object in F and G a subobject of ι(FF ). Let F ′ be the category of functors from Ef−(even) to E , where Ef−(even) is the full sub- category of Ef having as objects the F2-vector spaces of even dimension. The categories F and F ′ are equivalent [?]. The functor ǫ : Eq → E factorizes through the inclusion Ef−(even) →֒ Ef . This induces a functor ǫ′ : Eq → E f−(even) which is full and essentially surjective. Consequently, we can use Proposition 2.17 to obtain: G ≃ ι(GF ). Similarly, we obtain the result for the quotient. 12 CHRISTINE VESPA • Let GF and HF be objects of F , we set G = ι(GF ) and H = ι(HF ). For a short exact sequence: 0 → G → F → H → 0, we have to prove that there exists a functor FF in F such that F = ι(FF ). Let P1 → P0 → G F → 0 and Q1 → Q0 → H F → 0 be projective presentations of GF and HF in F , we have the following commutative diagram 0 // ι(P1) // ι(P1)⊕ ι(Q1) // ι(Q1) // 0 // ι(P0) // ι(P0)⊕ ι(Q0) // ι(Q0) // 0 // G 0 0 0 where the columns are projective resolutions in Fquad, by the horseshoe lemma. By Lemma 2.14, the morphism ι(P1)⊕ ι(Q1)→ ι(P0)⊕ ι(Q0) is in- duced by a morphism of F denoted by f . Consequently, F ∼= ι(Coker(f)) ∈ ι(F). By Theorem 2.16, we deduce from Lemma 2.14, the following characterization of the simple functors of F in Fquad which will be used in section 4 of this paper concerning the polynomial functors of Fquad. Lemma 2.18. (1) Let F be a functor of Fquad, then F is in the image of the functor ι : F → Fquad if and only if, for all objects V in Tq, F (eV )F (V ) = F (V ). (2) Let S be a simple object in Fquad, then S is in the image of the functor ι : F → Fquad if and only if there exists an object W in Tq such that S(eW )S(W ) 6= 0. Proof. (1) The forward implication is a consequence of the following fact: for a functor F in the image of ι, HomFquad(P V , F ) = F (eV )F (V ) = F (V ). The reverse implication relies on the fact that the condition F (eV )F (V ) = F (V ) implies that F is a quotient of a sum of projective objects of the form P V . Since the category ι(F) is thick in Fquad by Theorem 2.16, we obtain the result. (2) Observe that, if S(eW )S(W ) 6= 0, we have HomFquad(P W , S) 6= 0, thus S is a quotient of P W by simplicity of S. Lemma 2.14 implies that there exists a one-to-one correspondance between the indecomposable factors of V and those of P ǫ(V ). We deduce that the simple quotients of P V arise from F . Consequently, S is in the image of the functor ι. GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 13 2.2.3. The quotient PV /P (dim(V )−1) V . The aim of this paragraph is to prove the following result: Proposition 2.19. Let V be an object in Tq, we have a natural equivalence: PV /P (dim(V )−1) V ≃ κ(isoV ) where isoV is an isotropic functor and κ : Fiso → Fquad is the functor given in Theorem 1.4. To prove this proposition, we need the following notation and result: Notation 2.20. Denote by σf the natural map PV −→ κ(isoV ) which corre- sponds to the canonical generator [V −→ V ] of isoV (V ) by the equivalence Hom(PV , κ(isoV )) ≃ isoV (V ) ≃ F2[O(V )] given by the Yoneda lemma. Lemma 2.21. The functor κ(isoV ) of Fquad is a quotient of the functor PV = F2[HomTq(V,−)]. Proof. The natural map PV −→ κ(isoV ) is surjective: a pre-image of the canonical generator [V −→ W ] of κ(isoV )(W ) by (σf )W , is the morphism g◦f−1 −−−−→W ←−W ]. � A formal consequence of the previous lemma is given in the following result. Lemma 2.22. The functor κ(isoV ) of Fquad is a quotient of the functor PV /P (dim(V )−1) Proof. By definition of the filtration and by the previous lemma, we have the dia- gram: 0 // P (dim(V )−1) i // PV // // PV /P (dim(V )−1) κ(isoV ) where i is the canonical inclusion of P (dim(V )−1) V in PV . By definition of σId, we have σId ◦ i = 0, from which we deduce the existence of the surjection τ : PV /P (dim(V )−1) V → κ(isoV ). � We will prove below that this natural map is an isomorphism. It is sufficient to prove the following result. Proposition 2.23. For V and W two objects of Tq, we have an isomorphism (PV /P (dim(V )−1) V )(W ) ≃ κ(isoV )(W ). The proof of this proposition relies on the following lemma. Lemma 2.24. For a non-zero canonical generator of (PV /P (dim(V )−1) V )(W ) repre- sented by the morphism T = [V −→ W⊥W ′ ←−− W ] of Tq, we have g(V ) ⊂ W and T = [V ←−W ], where g = iW ◦ f . 14 CHRISTINE VESPA Proof. By definition of the filtration, for V andW two objects of Tq, the vector space (PV /P (dim(V )−1) V )(W ) is generated by Hom [dim(V )] (V,W ) where Hom [dim(V )] (V,W ) is the set of morphisms from V to W whose the pullback D in Edegq is a quadratic space such that dim(D) = dim(V ). We deduce from the existence of a monomor- phism from D to V and from the equality of the dimensions, that D and V are isometric. Consequently, for the morphism T of the statement, we have, by defi- nition of the pullback, g(V ) ⊂ W . Thus, we have T = [V ←− W ], where g = iW ◦ f , by the equivalence relation defined over the morphisms of Tq in Defini- tion 1.1. Proof of Proposition 2.23. The natural map τ obtained in the proof of Lemma 2.22 defines, for W an object in Tq, the linear map τW : (PV /P (dim(V )−1) V )(W ) → κ(isoV )(W ) T = [V ←−W ] 7−→ [V −→ W ] which is clearly an isomorphism. � 3. Decomposition of the standard projective functors PH0 and PH1 On abelian categories, the decompositions into direct summands of a functor F of Fquad correspond to decompositions into orthogonal idempotents of 1 in the ring EndFquad(F ) (see for example [?]). One of the difficulties of the category Fquad lies in the fact that the rings of endomorphisms of projectives PV and their representations are not well-understood. The decompositions of projectives PV , obtained in work in preparation, using a refinement of the rank filtration will allow us to understand the structure of these rings better. In this section, we obtain the decompositions into indecomposable factors of the projective objects PH0 and PH1 by an explicit study of the filtration defined in section 2. This section concludes by several consequences of these decompositions. In particular, we give a classification of the “small” simple functors of Fquad, which is an essential ingredient in the following section about the polynomial functors of Fquad. 3.1. Decomposition of PH0 . To obtain the decomposition of the functor PH0 into indecomposable factors, we give an explicit description of the subquotients of the filtration; then, we prove that the filtration splits for this functor and we identify the factors of this decomposition. 3.1.1. Explicit description of the subquotients of the filtration. The aim of this para- graph is to give a basis of the vector spaces P (V ), P (V ) and PH0/P for V a given object in Tq. We deduce from Theorem 2.6 and Notation 2.11, the following result. Lemma 3.1. A basis B (V ) is given by the set: = {[tf ] for f ∈ HomEf (F2 ⊕2, ǫ(V ))}. By definition of the filtration, a canonical generator of P (V ), represented by the morphism T = [H0 −→ V⊥L ←− V ] of Tq, satisfies the following property: I = f(H0) ∩ i(V ) is a quadratic space of dimension one. GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 15 Lemma 3.2. Let T = [H0 −→ V⊥L ←− V ] be a morphism of Tq which represents a canonical generator of P (V ), and {a0, b0} be a symplectic basis of H0, then the map f in T has one of the three following forms. (1) If I = (f(a0), 0), the map f : H0 → V⊥L is defined by: f(a0) = v and f(b0) = w + l for v and w elements of V satisfying q(v) = 0 and B(v, w) = 1 and l a non-zero element of L. (2) If I = (f(b0), 0), the map f : H0 → V⊥L is defined by: f(a0) = v + l and f(b0) = w for v and w elements of V satisfying q(w) = 0 and B(v, w) = 1 and l a non-zero element of L. (3) If I = (f(a0 + b0), 1), the map f : H0 → V⊥L is defined by: f(a0) = v + l and f(b0) = w + l for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1 and l a non-zero element of L. Proof. The quadratic space H0 has three subspaces of dimension one which are: Span(a0) and Span(b0) isometric to (x, 0) and Span(a0 + b0) isometric to (x, 1). These three subspaces give rise to each one of the maps f defined in the statement. Notation 3.3. The morphisms [H0 −→ V⊥L ←− V ], where f is one of the mor- phisms described in the point (1) (respectively (2) and (3)) of the previous lemma, will be known as type A (respectively B and C) morphisms. We have the following proposition. Proposition 3.4. For T = [H0 −→ V⊥L ←− V ] and T ′ = [H0 −→ V⊥L′ ←− V ] morphisms of Tq which represent canonical generators of P (V ), the follow- ing properties are equivalent. (1) The morphisms T and T ′ of HomTq (H0, V ) have the same type and satisfy the relation pV ◦ f = p V ◦ f (2) The morphisms T and T ′ of HomTq (H0, V ) are equal. The proof of the implication (2)⇒ (1) relies on the following technical lemma. Lemma 3.5. Let T = [V ←− W ] and T ′ = [V −→ X ′ ←− W ′] be morphisms of HomTq (V,W ). If T = T ′, then g(V ) + h(W ) ≃ g′(V ) + h′(W ) in Edegq . Proof. By definition of the equivalence relation given in Definition 1.1, it is sufficient to prove that, for two morphisms T and T ′ such that TRT ′, we have g(V )+h(W ) ≃ g′(V ) + h′(W ). 16 CHRISTINE VESPA By definition, g(V )+h(W ) is the smallest, possibly degenerate, quadratic space such that we have a commutative diagram in Edegq , of the form: **UUU U g(V ) + h(W ) Similarly, g′(V ) + h′(W ) is the smallest quadratic space such that we have an analogous commutative diagram. By definition of the relation R, we have the existence of a morphism δ in Eq such that the following diagram is commutative: By the consideration of the following commutative diagram in Edegq where Y = g(V ) + h(W ), we deduce from the minimality of g′(V ) + h′(W ), the existence of a morphism in Edegq from g ′(V ) + h′(W ) to g(V ) + h(W ) such that the corresponding diagram is commutative. Then, by minimality of g(V ) + h(W ) for T , we obtain: g(V ) + h(W ) ≃ g′(V ) + h′(W ). � Proof of Proposition 3.4. Suppose that the morphisms T and T ′ of HomTq (H0, V ) are of type A such that pV ◦ f = p V ◦ f ′. We deduce that: f(a0) = v; f(b0) = w + l and f ′(a0) = v; f ′(b0) = w + l for v and w elements of V and l (resp. l′) a non-zero element of L (resp. L′). Since the maps f and f ′ preserve the quadratic forms, we have q(b0) = q(w) + q(l) = q(w) + q(l ′). We deduce that q(l) = q(l′), thus the map Span(l) −→ Span(l′), such that α(l) = l′, preserves the quadratic form. Conse- quently, we can apply Theorem 2.10 to obtain the existence of a map α : L⊥L′ → GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 17 L⊥L′ such that the following diagram is commutative: α // L⊥L′ Span(l) // Span(l′). We deduce the commutativity of the diagram: V � _ f ′ **VVV V⊥(L⊥L′) V⊥(L⊥L′). Since T = [H0 −→ V⊥L⊥L′ ←− V ] and T ′ = [H0 −→ V⊥L⊥L′ ←− V ], by inclusion, we deduce from the previous diagram that T = T ′. We reason in the same way, for the morphisms of type B and C. Conversely, if T = T ′, by Lemma 3.5 f(H0) + iV (V ) ≃ f ′(H0) + i V (V ). Conse- quently we deduce from Theorem 2.10 the existence of an isometry β : V⊥L⊥L′→ V⊥L⊥L′ making the following diagram commutative: V⊥L⊥L′ β // V⊥L⊥L′ f(H0) + iV (V ) // f ′(H0) + i V (V ). This yields the commutativity of the following diagram: f ′ **UUU U V⊥L⊥L′ V⊥L⊥L′ which implies that β ◦ iV = i V . Thus, β = IdV⊥β ′ where β′ : L⊥L′ → L⊥L′ is a morphism of HomEq (L⊥L ′, L⊥L′). Consequently, we have f(a0) = v + l; f(b0) = w + l ′ and f ′(a0) = v + β ′(l); f ′(b0) = w + β ′(l′) and we deduce that pV ◦ f = p V . Furthermore, since β ′ is inversible, for all x in L⊥L′ we have: x is non-zero if and only if β′(x) is non-zero. Consequently, T and T ′ have the same type. � This proposition justifies the following notation. Notation 3.6. We will denote by Av,w, Bv,w and Cv,w the morphisms of HomTq (H0, V ) respectively of type A, B and C and such that pV ◦ f(a0) = v and pV ◦ f(b0) = w. 18 CHRISTINE VESPA The following result is a straightforward consequence of Lemma 3.2 and Propo- sition 3.4. Lemma 3.7. A basis B (V ) is given by the set: = { [Av,w] for v and w elements of V satisfying q(v) = 0 and B(v, w) = 1, [Bv,w] for v and w elements of V satisfying q(w) = 0 and B(v, w) = 1, [Cv,w] for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1} By Proposition 2.19, we have (PH0/P )(V ) ≃ κ(isoH0)(V ). We deduce the following result. Lemma 3.8. A basis B of PH0/P (V ) is given by the set: = {[Df ] for f ∈ HomEq (H0, V )}, where Df is the morphism of Tq represented by the diagram: H0 ←− V . We end this paragraph by the rules of composition for the morphisms tf , Av,w, Bv,w, Cv,w and Df , summarized in the following proposition. This technical re- sult will be fundamental in the following paragraph, to prove the splitting of the filtration. Lemma 3.9. Let T = V −→ W⊥L ←−− W be a morphism of HomTq(V,W ). The following relations are satisfied: (1) For f a morphism of HomEf (F2 ⊕2, ǫ(V )) we have: T ◦ tf = tϕ◦f . (2) (a) For v and w elements of V satisfying q(v) = 0 and B(v, w) = 1, we have: T ◦Av,w = Aϕ(v),pW ◦ϕ(w) if ϕ(v) ∈W tpW ◦(ϕ⊥Id)◦α otherwise. (b) For v and w elements of V satisfying q(w) = 0 and B(v, w) = 1, we have: T ◦Bv,w = BpW ◦ϕ(v),ϕ(w) if ϕ(w) ∈W tpW ◦(ϕ⊥Id)◦α otherwise. (c) For v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1, we have: T ◦ Cv,w = CpW ◦ϕ(v),pW ◦ϕ(w) if ϕ(v + w) ∈W tpW ◦(ϕ⊥Id)◦α otherwise. (3) For f a morphism of HomEq (H0, V ), we have: T ◦Df = Dϕ◦f if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈ W Aϕ◦f(a0),pW ◦ϕ◦f(b0) if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W BpW ◦ϕ◦f(a0),ϕ◦f(b0) if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W CpW ◦ϕ◦f(a0),pW ◦ϕ◦f(b0) if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) ∈W tpW ◦(ϕ⊥Id)◦α if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) /∈W. GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 19 Proof. By definition of the composition in Tq, we have the following diagram: α // V⊥L′ // W⊥L⊥L′ (1) For a morphism tf , we have α(a0) = f(a0) + l and α(b0) = f(b0) + m. where {l,m} is a linearly independent family of L′. Consequently: (ϕ⊥Id) ◦ α(a0) = ϕ ◦ f(a0) + l et (ϕ⊥Id) ◦ α(b0) = ϕ ◦ f(b0) +m. We deduce that T ◦ tf = tϕ◦f . (2) For Av,w, we have α(a0) = v and α(b0) = w + l ′, where l′ is a non-zero element of L′. Consequently: (ϕ⊥Id) ◦ α(a0) = ϕ(v) et (ϕ⊥Id) ◦ α(b0) = ϕ(w) + l We have to distinguish two cases: • if ϕ(v) ∈W , since ϕ preserves quadratic forms, we have q(ϕ(v)) = q(v) and, since L′ is orthogonal to V , B(ϕ(v), pW ◦ϕ(w)) = B(ϕ(v), ϕ(w)) = B(v, w). Thus the morphism Aϕ(v),pW ◦ϕ(w) is defined and we have: T ◦Av,w = Aϕ(v),pW ◦ϕ(w); • otherwise, ϕ(v) = pW ◦ ϕ(v) + m where m is a non-zero element of L. Consequently, we obtain a morphism of nul rank and we have: T ◦Av,w = tpW ◦(ϕ⊥Id)◦α. The cases Bv,w and Cv,w are similar to the case of Av,w and are left to the reader. (3) For the morphism Df , where f is an element of HomEq (H0, V ), we have α(a0) = f(a0) = v and α(b0) = f(b0) = w where v and w are elements of V . Consequently: (ϕ⊥Id)◦α(a0) = ϕ(v) and (ϕ⊥Id)◦α(b0) = ϕ(w). Since ϕ◦f preserves the quadratic forms, we have: q(ϕ ◦ f(a0)) = q(ϕ ◦ f(b0)) = 0 and B(ϕ ◦ f(a0), ϕ ◦ f(b0)) = 1. Thus the morphisms Aϕ◦f(a0),ϕ◦f(b0), Bϕ◦f(a0),ϕ◦f(b0) and Cϕ◦f(a0),ϕ◦f(b0) are defined. We have to distinguish four cases: • if ϕ(v) ∈W and ϕ(w) ∈W then T ◦Df = Dϕ◦f ; • if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W , we have ϕ ◦ f(a0) = w ′ and ϕ ◦ f(b0) = w ′′ + l, where l is a non-zero element of L. Consequently, we obtain a morphism of type A and we have T ◦Df = Aϕ◦f(a0),ϕ◦f(b0); • if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W , we have ϕ ◦ f(a0) = w ′ + l and ϕ ◦ f(b0) = w ′′, where l is a non-zero element of L. Consequently, we obtain a morphism of type B and we have T ◦Df = Bϕ◦f(a0),ϕ◦f(b0); • if ϕ ◦ f(a0) /∈ W , ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) ∈ W , we have ϕ◦f(a0) = w ′+ l and ϕ◦f(b0) = w ′′+ l, where l is a non-zero element of L. Consequently, we obtain a morphism of type C and we have T ◦Df = Cϕ◦f(a0),ϕ◦f(b0); • if ϕ ◦ f(a0) /∈ W , ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) /∈ W , we have ϕ ◦ f(a0) = w ′ + l and ϕ ◦ f(b0) = w ′′ + l′, where l and l′ are non-zero 20 CHRISTINE VESPA elements of L. Consequently, we obtain a morphism of nul rank and we have T ◦Df = tpW ◦(ϕ⊥Id)◦α. 3.1.2. Splitting of the filtration for the functor PH0 . In this paragraph, we prove the following result. Proposition 3.10. The rank filtration splits for the functor PH0 , namely: PH0 = P ⊕ PH0/P Proof. By Theorem 2.6, we have: P . To prove the proposition, it is sufficient to prove that PH0 = P ⊕ PH0/P By definition of the filtration, we have the short exact sequence: (3.10.1) 0→ P → PH0 −→ PH0/P Let V be an object in Tq, we consider a morphism f of HomEq (H0, V ) and the generator [Df ] of PH0/P (V ) associated to f . Since the map f preserves the quadratic forms, we have: q(f(a0)) = q(f(b0)) = 0, q(f(a0 + b0)) = 1; thus B(f(a0), f(b0)) = 1. Consequently, the morphisms Af(a0),f(b0), Bf(a0),f(b0) and Cf(a0),f(b0) of HomTq (H0, V ) are defined. We define a map sV : PH0/P (V ) → PH0(V ) by: sV : PH0/P (V ) −→ PH0(V ) [Df ] 7→ [Df ] + [Af(a0),f(b0)] + [Bf(a0),f(b0)] + [Cf(a0),f(b0)]. We verify the two following statements. (1) pV ◦ sV = Id. For [Df ] a canonical generator of PH0/P (V ), we have pV ◦ sV ([Df ]) = [Df ] since the morphisms Af(a0),f(b0), Bf(a0),f(b0) and Cf(a0),f(b0) have a rank equal to one. (2) The maps sV define a natural map. One verifies that, for a morphism T = V −→W⊥L ←−−W of HomTq (V,W ), we have the commutativity of the following diagram: PH0/P sV // PH0/P PH0(V ) PH0 (T ) PH0/P sW // PH0(W ). In order to simplify notation, we will write: A′ = Aϕ◦f(a0),pW ◦ϕ◦f(b0), B′ = BpW ◦ϕ◦f(a0),ϕ◦f(b0), C ′ = CpW ◦ϕ◦f(a0),pW ◦ϕ◦f(b0) and t ′ = tpW ◦(ϕ⊥Id)◦α. On the one hand, by Lemma 3.9, we have: PH0 (T ) ◦ sV ([Df ]) = PH0(T )([Df ] + [Af(a0),f(b0)] + [Bf(a0),f(b0)] + [Cf(a0),f(b0)]) = [T ◦Df ] + [T ◦Af(a0),f(b0)] + [T ◦Bf(a0),f(b0)] + [T ◦ Cf(a0),f(b0)] GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 21 [Dϕ◦f ] +[A ′] +[B′] +[C′] if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈ W [A′] +[A′] +[t′] +[t′] = 0 if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W [B′] +[t′] +[B′] +[t′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W [C′] +[t′] +[t′] +[C′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) ∈W [t′] +[t′] +[t′] +[t′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) /∈W. [Dϕ◦f ] +[A ′] +[B′] +[C′] if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈W 0 otherwise. On the other hand, by Lemma 3.9, we have: PH0/P (T )([Df ]) = [Dϕ◦f ] if ϕ ◦ f(a0) ∈W and ϕ ◦ f(b0) ∈W 0 otherwise since the morphisms A, B, C and t are zero in the quotient PH0/P (W ). We deduce, sW ◦ PH0/P (T )([Df ]) = [Dϕ◦f ] + [A ′] + [B′] + [C′] if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈W 0 otherwise. Consequently, the maps sV define a natural map which is a section of p. This gives rise to the splitting of the exact sequence 3.10.1. � 3.1.3. Identification of the direct summands. The aim of this paragraph is to iden- tify the summands of the decomposition given in Proposition 3.10. We begin by proving that the morphisms of type A (respectively of type B and C) define a subfunctor of P which is a direct summand of this functor. Lemma 3.11. The functor P admits the following decomposition into di- rect summands: = FA ⊕ FB ⊕ FC where FA, FB and FC are subfunctors of P generated by, respectively, the morphisms of type A, B and C. Proof. By Lemma 3.7, we have an isomorphism of vector spaces (V ) = FA(V )⊕ FB(V )⊕ FC(V ), for all objects V in Tq. Consequently, it is sufficient to prove that FA, FB and FC are subfunctors of For FA, we have to verify the commutativity of the diagram FA(V ) iV // FA(T ) FA(W ) iW // P 22 CHRISTINE VESPA where T is a morphism of HomTq(V,W ). Let [Av,w] be a canonical generator of FA(V ), we have by Lemma 3.9 T ◦Av,w = Aϕ(v),pW ◦ϕ(w) if ϕ(v) ∈W tpW ◦(ϕ⊥Id)◦α otherwise. Consequently, (T ) ◦ iV ([Av,w]) = [Aϕ(v),pW ◦ϕ(w)] if ϕ(v) ∈ W 0 otherwise, since the morphism tpW ◦(ϕ⊥Id)◦α, has nul rank. We deduce that P (T ) ◦ iV ([Av,w ]) is in the vector space FA(W ); thus, FA is a subfunctor of P In the same way, by the use of values of T ◦Bv,w and T ◦ Cv,w given in Lemma 3.9, we prove that the functors FB and FC are subfunctors of P In the following lemma, we identify the functors FA, FB and FC with certain mixed functors defined in [?] and recalled in section 1. Lemma 3.12. (1) The functors FA and FB are isomorphic to the functor Mix0,1. (2) The functor FC is isomorphic to the functor Mix1,1. Proof. (1) The isomorphism FA ≃ Mix0,1. Let [Av,w] be a canonical generator of FA(V ), we have, by definition, B(v, w) = 1. Consequently, the following linear map exists: σ1V : FA(V ) → Mix0,1(V ) [Av,w] 7−→ [(w, v + w)]. The map σ1V is an isomorphism, whose inverse is given by (σ1V ) : Mix0,1(V ) → FA(V ) [(v, w)] 7−→ [Av+w,v]. We have to verify that the maps σ1V define a natural map; namely, for a morphism T = [V −→ W⊥L ←−− W ], that the following diagram is commutative: FA(V ) σ1V // FA(T ) Mix0,1(V ) Mix0,1(T ) FA(W ) σ1W // Mix0,1(W ). We have: Mix0,1(T ) ◦ σ V ([Av,w ]) = Mix0,1(T )[(w, v + w)] [(pW ◦ (ϕ(w)), pW ◦ (ϕ(v + w))] if ϕ(v) ∈W 0 otherwise GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 23 by the definition of the mixed functors, and σ1W ◦ FA(T )([Av,w]) = σ [Aϕ(v),pW ◦ϕ(w)] if ϕ(v) ∈W 0 otherwise [(pW ◦ (ϕ(w)), ϕ(v) + pW ◦ (ϕ(w))] if ϕ(v) ∈W 0 otherwise. When ϕ(v) ∈ W , we have: [(pW ◦ (ϕ(w)), ϕ(v) + pW ◦ (ϕ(w))] = [(pW ◦ (ϕ(w)), pW ◦ (ϕ(v + w))], what proves the naturality of σ1. Since the two following cases are very close to the previous one, we only give the definition of the isomorphism of vector spaces and we leave the reader to verify that they define natural equivalences. (2) The isomorphism FB ≃ Mix1,0. σ2V : FB(V ) → Mix0,1(V ) Bv,w 7−→ [(v, v + w)]. (3) The isomorphism FC ≃Mix1,1. σ3V : FC(V ) → Mix1,1(V ) Cv,w 7−→ [(v, w)] We deduce the following proposition. Proposition 3.13. The projective functor PH0 admits the following decomposition into direct summands: PH0 = ι(P ⊕2))⊕ (Mix0,1 ⊕2 ⊕Mix1,1)⊕ κ(isoH0) where Mix0,1 and Mix1,1 are mixed functors and isoH0 is an isotropic functor. Proof. This proposition is a straightforward consequence of Proposition 3.10, The- orem 2.6, Proposition 2.19 and Lemmas 3.11 and 3.12. � 3.2. Decomposition of PH1 . The study of the functor PH1 is analogous to that of the functor PH0 given in the previous section. Consequently, for the functor PH1 , we give only the principal results without proofs. 3.2.1. Explicit description of the subquotients of the filtration. In this paragraph, we give basis of the vector spaces P (V ), P (V ) and PH1/P (V ) for V an object in Tq. Lemma 3.14. A basis B (V ) is given by the set: = {tf for f ∈ HomEf (F2 ⊕2, ǫ(V ))}. Lemma 3.15. Let T = [H1 −→ V⊥L ←− V ] be a morphism of Tq which represents a canonical generator of P (V ), and {a1, b1} a symplectic basis of H1, then the map f in T has one of the three following forms. 24 CHRISTINE VESPA (1) If I = (f(a1), 0) the map f : H1 → V⊥L is defined by: f(a1) = v et f(b1) = w + l for v and w elements of V satisfying q(v) = 1 and B(v, w) = 1 and l a non-zero element of L. (2) If I = (f(b1), 0) the map f : H1 → V⊥L is defined by: f(a1) = v + l et f(b1) = w for v and w elements of V satisfying q(w) = 1 and B(v, w) = 1 and l a non-zero element of L. (3) If I = (f(a1 + b1), 1) the map f : H1 → V⊥L is defined by: f(a1) = v + l et f(b1) = w + l for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1 and l a non-zero element of L. Notation 3.16. The morphisms [H1 −→ V⊥L ←− V ], where f is one of the morphisms described in the point (1) (respectively (2) and (3)) of the previous lemma will be known as type E (respectively F and G) morphisms. The analogous proposition to Proposition 3.4 holds for H1. This justifies the following notation. Notation 3.17. Denote by Ev,w, Fv,w and Gv,w the morphisms of HomTq (H1, V ) respectively of type E, F and G and such that pV ◦ f(a1) = v and pV ◦ f(b1) = w. We deduce the following lemmas. Lemma 3.18. A basis B (V ) is given by the set: = { [Ev,w] for v and w elements of V satisfying q(v) = 1 and B(v, w) = 1, [Fv,w] for v and w elements of V satisfying q(w) = 1 and B(v, w) = 1, [Gv,w] for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1} Lemma 3.19. A basis B of PH1/P (V ) is given by the set: = {[Hf ] for f ∈ HomEq (H1, V )}, where Hf is the morphism of Tq represented by the diagram: H1 ←− V . The rules of composition of morphisms Ev,w, Fv,w , Gv,w and Hf are similar to those given for Av,w, Bv,w, Cv,w and Df in Lemma 3.9. The details can be provided by the reader. 3.2.2. Splitting of the filtration for the functor PH1 . Proposition 3.20. The rank filtration splits for the functor PH1 , namely: PH1 = P ⊕ PH1/P Proof. One verifies that the map sV : PH1/P (V )→ PH1(V ) given by: sV : PH1/P (V ) −→ PH1(V ) [Hf ] 7→ [Hf ] + [Ef(a1),f(b1)] + [Ff(a1),f(b1)] + [Gf(a1),f(b1)]. defines a natural map which is a section of the projection PH1 → PH1/P GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 25 3.2.3. Identification of the direct summands. We have the following lemma. Lemma 3.21. The functor P admits the following decomposition into di- rect summands = FE ⊕ FF ⊕ FG where FE, FF and FG are subfunctors of P generated by, respectively, the morphisms of type E, F and G. In the following lemma, we identify the functors FE , FF and FG with mixed functor. Lemma 3.22. The functors FE , FF and FG are equivalent to the functor Mix1,1. We deduce the following decomposition. Proposition 3.23. The projective functor PH1 admits the following decomposition into direct summands: PH1 = ι(P )⊕Mix1,1 ⊕3 ⊕ κ(isoH1) where Mix1,1 is a mixed functor and isoH1 is an isotropic functor. 3.3. Consequences of decompositions of functors PH0 and PH1 . In this sec- tion, we draw the conclusions of the decompositions of PH0 and PH1 given in Propo- sitions 3.13 and 3.23. We deduce the indecomposability of the functors Mix0,1 and Mix1,1, we study the projectivity of the first isotropic functors in Fquad and we give the classification of the “small” simple objects of Fquad. 3.3.1. Indecomposability of functors Mix0,1 and Mix1,1. The aim of this paragraph is to prove the following result: Proposition 3.24. The functors Mix0,1 and Mix1,1 are indecomposable. The proof of this proposition relies on the following obvious lemma. Lemma 3.25. If the functor F of Fquad decomposes as a direct sum: F1⊕ . . .⊕Fn, then the projections πi : F → Fi and the inclusions ji : Fi → F induce idempotents ei = ji ◦ πi in the ring End(F ). Proof of Proposition 3.24. (1) By the Yoneda lemma, we have: Hom(PH0 ,Mix0,1) = Mix0,1(H0). By a calculation, we obtain that the dimension of the space Mix0,1(H0) is equal to 4. According to Proposition 3.13, the order of multiplicity of the summand Mix0,1 in the decomposition of PH0 is equal to 2. Consequently, the dimension of the vector space E := Hom(Mix0,1,Mix0,1) is 2. We have the following basis: {Id, τ} where the map τ is given by: τ([(u, v)]) = ([(v, u)]). Consequently, E = ({0, Id, τ, Id + τ},+, ◦), as a ring, and it is easy to see that this ring does not admit a non-trivial idempotent. (2) Similarly, we have Hom(PH0 ,Mix1,1) = Mix1,1(H0) and dim(Mix1,1(H0)) = 2. The order of multiplicity of the summand Mix1,1 in the decomposition of PH0 is equal to 1. We deduce that the ring Hom(Mix1,1,Mix1,1) does not admit a non-trivial idempotent. 26 CHRISTINE VESPA We deduce from this proposition the following result, which complements The- orem 1.9, obtained in [?]: Corollary 3.26. The short exact sequence 0 → Σα,1 → Mixα,1 → Σα,1 → 0 does not split. 3.3.2. Projectivity of certain isotropic functors in Fquad. The decompositions given in Propositions 3.13 and 3.23 allow us to study the projectivity of isotropic functors in Fquad. Corollary 4.37 in [?] shows that the set of functors {isoV |V ∈ S} is a set of projective generators of Fiso, where S is a set of representatives of isometry classes of (possibly degenerate) quadratic spaces. Since the functor κ(isoH0) (respectively κ(isoH1) ) is a direct summand of the functor PH0 (respectively PH1) we have the following result. Proposition 3.27. The functors κ(isoH0) and κ(isoH1) are projective in the cat- egory Fquad. We deduce from Corollary 1.6 and the previous proposition, the following result. Corollary 3.28. The category Fquad contains non-constant finite, projective ob- jects. This corollary constitutes one of the new features of the category Fquad compared to F . Recall that, according to Corollary B7 in [?], due to Lionel Schwartz, the category F does not contain non-constant finite projective functors. Recall that, the functor κ(iso(x,0)) is the top composition factor of Mix0,1 and κ(iso(x,1)) is that of Mix1,1. We have the following result. Proposition 3.29. The projective cover of κ(iso(x,0)) (respectively κ(iso(x,1))) is the functor Mix0,1 (respectively Mix1,1). In particular, the functors κ(iso(x,0)) and κ(iso(x,1)) are not projective in Fquad. Proof. Since κ(iso(x,0))(H0) 6= {0} and κ(iso(x,1))(H0) 6= {0}, if these two functors were projective, they would be direct summands of the functor PH0 . We deduce from Proposition 3.13, that these functors are not projective. � Remark 3.30. Propositions 3.27 and 3.29 let us conjecture that, for a nondegen- erate F2-quadratic space H, κ(isoH) is a projective functor in Fquad and, for a degenerate quadratic space D, κ(isoD) is not a projective functor in Fquad and its projective cover is a generalized mixed functor. This result will be the subject of future work. 3.3.3. Classification of simple objects S of Fquad such that either S(H0) 6= 0 or S(H1) 6= 0. If S is a simple object in Fquad, such that S(H0) 6= 0, the Yoneda lemma implies that Hom(PH0 , S) = S(H0) 6= 0. Consequently, there exists a morphism of Fquad from PH0 to S which is an epimorphism, by simplicity of S. We deduce from the decompositions given in Proposition 3.13 and 3.23, from Corollary 1.7 concerning the functors isoH0 and isoH1 and from the study of the functors Mix0,1 and Mix1,1 done in [?] and recalled in section 1, the following result. Proposition 3.31. The isomorphism classes of non-constant simple functors of Fquad such that either S(H0) 6= 0 or S(H1) 6= 0 are: ι(Λ1), ι(Λ2), ι(S(2,1)), κ(iso(x,0)), κ(iso(x,1)), RH0 , RH1 , SH1 where RH0 , RH1 and SH1 are the simple functors introduced in Corollary 1.7. GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 27 3.3.4. Extension groups in Fquad. By Theorem 1.4 and 1.5 we obtain an exact, fully- faithful functor V ∈S F2[O(V )]−mod −→ Fquad, where S is a set of representatives of isometry classes of quadratic spaces (possibly degenerate). Consequently, for M and N two F2[O(V )]− modules, this functor induces a morphism of extension groups: F2[O(V )]−mod (M,N) (κ̃)∗ −−−→ Ext∗Fquad(κ̃(M), κ̃(N)). We have the following proposition. Proposition 3.32. For V ∈ {H0, H1}, the morphism (κ̃)∗ is an isomorphism. The proof of this proposition relies on the following lemma. Lemma 3.33. For V ∈ {H0, H1}, if P is a finite projective F2[O(V )]-module, κ̃(P ) is projective in Fquad. Proof. If P is a finite projective F2[O(V )]-module, there exists a F2[O(V )]-module Q such that P ⊕Q ≃ F2[O(V )] ⊕N . We deduce from the exactness of κ that κ̃(P ⊕ Q) ≃ κ̃(P ) ⊕ κ̃(Q). Since κ̃(F2[O(V )]) = κ(isoV ) and the functors κ(isoH0) and κ(isoH1) are projective, by Proposition 3.27, we obtain that κ̃(P ) is projective. � Proof of Proposition 3.32. Let M and N be F2[O(V )]-modules for V ∈ {H0, H1} and P• → M be a projective resolution of M . Lemma 3.33 implies that κ̃(P•) is a projective resolution of κ̃(M). The functor κ̃ induces a morphism of cochain complexes HomF2[O(V )]−mod(P•, N)→ HomFquad(κ̃(P•), κ̃(N)) which induces the morphism (κ̃)∗ in cohomology. Since the functor κ̃ is fully- faithful the previous morphism is an isomorphism and so induces an isomorphism in cohomology. We deduce the following corollary: Corollary 3.34. For n a natural number, we have: ExtnFquad(RH0 , RH0) ≃ F2 and Ext Fquad (RH1 , RH1) ≃ F2 where RH0 and RH1 are the simple functors introduced in Corollary 1.7. Proof. Let ǫ be an element in {0, 1}. For V = Hǫ, by Corollary 1.7 (1), we have κ̃(F2) = RHǫ . So, applying Proposition 3.32 to M = N = F2 we obtain: Ext∗Fquad(RHǫ , RHǫ) ≃ Ext F2[O(Hǫ)]−mod (F2,F2) = H ∗(O(Hǫ),F2). Since O(H0) ≃ S2 ≃ C2 and O(H1) ≃ S3 ≃ GL2(F2) we know by classical results of cohomology of groups that Hn(O(H0),F2) = H n(O(H1),F2) = F2. Remark 3.35. This corollary exhibits an important difference between the cate- gories F and Fquad; recall that in F Ext F (S, S) = 0 for all simple objects S of F (see [?]). 28 CHRISTINE VESPA 4. Application: the polynomial functors of Fquad In this section, having generalized the notion of polynomial functor to the cat- egory Fquad, we prove, by induction, that the polynomial functors of Fquad are in the image of the functor ι : F → Fquad. 4.1. Definition of polynomial functors of Fquad. 4.1.1. The difference functors of Fquad. We define the difference functors of Fquad which generalize the notion of difference functor of F . Recall that, according to [?], the difference functor ∆ : F → F is the functor given by ∆F (V ) := Ker(F (V ⊕ F2) F (p) −−−→ F (V )), for F an object in F , V an object in Ef and p : V ⊕ F2 → V the projection. Definition 4.1. The difference functors ∆H0 : Fquad → Fquad and ∆H1 : Fquad → Fquad are the functors defined by: ∆H0F (V ) := Ker(F (V⊥H0) F (T0) −−−−→ F (V )), ∆H1F (V ) := Ker(F (V⊥H1) F (T1) −−−−→ F (V )), for F an object in Fquad, V an object in Tq, and Ti = [V⊥Hi −→ V⊥Hi ←− V ] for i ∈ {0, 1} . We have the following result: Lemma 4.2. The functors ∆H0 and ∆H1 are exact. 4.1.2. Definition of polynomial functors. Before giving the definition of polynomial functors in Fquad, let us recall that of polynomial functors of F ( [?]). For an object F of F , F is a polynomial functor of degree 0 if and only if ∆F = 0 and, for an integer d, F is polynomial of degree at most d+ 1 if and only if ∆F is polynomial of degree at most d. Definition 4.3. Let F be an object in Fquad: (1) the functor F is polynomial of degree 0 if and only if ∆H0F = ∆H1F = 0; (2) for an integer d, the functor F is polynomial of degree at most d+1, if and only if ∆H0F and ∆H1F are polynomial of degree at most d. The following proposition allows us to simplify the definition of a polynomial functor of degree 0. Lemma 4.4. Let F be an object in Fquad. The functor ∆H0F is zero if and only if the functor ∆H1F is zero. Proof. If ∆H0F = 0, we have, for all objects V of Tq, F (V ) ≃ F (V⊥H0). Conse- quently, F (V ) ≃ F (V⊥H0) ≃ F (V⊥H0⊥H0) ≃ F (V⊥H1⊥H1), where the last isomorphism is obtained from the isomorphism H0⊥H0 ≃ H1⊥H1, recalled in section 1. We deduce from the existence of morphisms F (V ) →֒ F (V⊥H1) and F (V ⊥H1) →֒ F (V⊥H1⊥H1), induced by the inclusions and from the isomor- phism between F (V ) and F (V⊥H1⊥H1), that F (V ) ≃ F (V⊥H1) ≃ F (V⊥H1⊥H1). GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 29 Therefore ∆H1F = 0. The proof of the converse is similar. � 4.2. Study of polynomial functors of Fquad. The aim of this section is to prove the following result: Theorem 4.5. The polynomial functors of Fquad are in the image of the functor ι : F → Fquad. We will prove this theorem by induction over the degree of the polynomial func- tors. 4.2.1. Polynomial functors of degree zero of Fquad. In this paragraph, we start the induction. The proof of the following result relies, in an essential way, on the classification of simple functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0, obtained in Proposition 3.31. Lemma 4.6. Let S be a simple functor of Fquad, S is a polynomial functor of degree zero if and only if S is the constant functor F2. Proof. In order to prove the direct implication, we have to distinguish two cases. (1) If S(H0) = S(H1) = 0. By the classification of the nondegenerate quadratic spaces over F2, if W is a space of minimal dimension, satisfying S(W ) 6= 0, we have the existence of an element ǫ of {0, 1} and a nondegenerate quadratic space V which is non-zero, such that: W ≃ Hǫ⊥V. Since W is of minimal dimension, we have S(V ) = 0. This implies: ∆HǫS(V ) = S(Hǫ⊥V ) 6= 0. We deduce the result in this case. (2) If S(H0) 6= 0 or S(H1) 6= 0. In this case, we use the classification of simple functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0 obtained in Proposition 3.31. By an explicit calculation for all the functors S obtained in this classification, we obtain that the functors ∆H0S are non-zero except for the constant functor S = F2. The converse is trivial. 4.2.2. Proof of Theorem 4.5. To prove Theorem 4.5, we need the following result where the idempotents [eV ], obtained in Proposition 2.13, play a crucial rôle. Proposition 4.7. Let S be a non-trivial simple functor of Fquad which is not in the image of the functor ι : F → Fquad, then, one of the functors ∆H0S or ∆H1S is not in the image of the functor ι : F → Fquad. Proof. Let W be a nondegenerate quadratic space of minimal dimension, such that S(W ) 6= 0. We distinguish the two following cases. (1) If dim(W ) = 2. By an explicit calculation for all the functors S of the classification given in Proposition 3.31, we obtain the result. 30 CHRISTINE VESPA (2) If dim(W ) > 2. There exists a nondegenerate quadratic space V , possibly trivial, and an element ǫ of {0, 1}, such that: W ≃ H0⊥Hǫ⊥V. Suppose that ∆H0S and ∆H1S are in the image of the functor ι, we prove, below, that this implies that S is in the image of ι. By Lemma 2.18 it is sufficient to show the existence of an object W in Tq such that: S(eW )S(W ) 6= 0. By Lemma 2.15, we have: eW = eH0⊥eHǫ⊥eV . Since W is assumed to be a space of minimal dimension such that S(W ) 6= 0, we have S(H0⊥V ) = S(Hǫ⊥V ) = 0. This implies that ∆H0S(Hǫ⊥V ) ≃ S(W )(4.7.1) ∆HǫS(H0⊥V ) ≃ S(W ).(4.7.2) These isomorphisms are natural and, for (4.7.1), the action of EndTq(Hǫ⊥V ) on ∆H0S(Hǫ⊥V ) corresponds to the restriction of the action of EndTq (W ) on S(W ). In the same way, for (4.7.2), the action of EndTq (H0⊥V ) on ∆HǫS(H0⊥V ) corresponds to the restriction of the action of EndTq (W ) on S(W ). Suppose that ∆H0S and ∆H1S are in the image of ι. We deduce that: S(1H0⊥eHǫ⊥eV )S(W ) = ∆H0S(eHǫ⊥eV )∆H0S(Hǫ⊥V ) = ∆H0S(Hǫ⊥V ) = S(W ) where the first equality comes from the action described previously, the second is a consequence of Lemma 2.15 and the third is given by 4.7.1. In the same way, we obtain: S(eH0⊥1Hǫ⊥eV )S(W ) = ∆HǫS(eH0⊥eV )∆HǫS(H0⊥V ) = ∆HǫS(H0⊥V ) = S(W ). We deduce that: S(1H0⊥eHǫ⊥eV ) ◦ S(eH0⊥1Hǫ⊥eV )S(W ) = S(W ). Since S(1H0⊥eHǫ⊥eV ) ◦ S(eH0⊥1Hǫ⊥eV ) = S(eW ) by Lemma 2.15, we have: S(eW )S(W ) 6= 0, as required. The proof of the theorem relies, also, on the following lemmas. Lemma 4.8. (1) A functor F of Fquad which takes values in finite vector spaces and such that ∆HǫF is finite, is finite. (2) A polynomial functor F of Fquad which takes values in finite vector spaces is finite. Proof. (1) The functor Fquad −→ E × Fquad F 7→ (F (0),∆HǫF ) is exact and faithful by Lemma 4.2. Hence, if γ(F ) is finite then F is finite. (2) If F takes values in finite vector spaces, so does ∆HǫF . Consequently we can apply the first point recursively to obtain the result. Lemma 4.9. A finite object F of Fquad whose composition factors are in the image of the functor ι is in the image of the functor ι. GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 31 Proof. This result is a straightforward consequence of the thickness of the subcat- egory ι(F) in Fquad given in Theorem 2.16. � Proof of Theorem 4.5. Since a functor of Fquad is colimit of its subfunctors which take values in finite vector spaces, we deduce from Lemma 4.8, that it is sufficient to prove the result for the finite polynomial functors of Fquad. Furthermore, since the functors ∆H0 and ∆H1 are exact by Lemma 4.2, it is sufficient to consider the case of a simple functor S. We prove the theorem by induction over the polynomial degree. If S is polynomial of degree 0, according to Lemma 4.2, S is in the image of the functor ι. Suppose that all simple polynomial functors of Fquad and of degree d are in the image of the functor ι and consider a simple polynomial functor S of Fquad such that deg(S) = d + 1. By the definition of polynomial functor in Fquad given in 4.3, the functors ∆H0S and ∆H1S are polynomial of degree d. We deduce that all composition factors of ∆H0S and ∆H1S are polynomial of degree smaller than or equal to d and, by induction, we obtain that they are in the image of ι. Since S is a simple functor, it is a quotient of a standard projective functor PV . Consequently S takes its values in finite dimensional vector spaces. Therefore, ∆H0S and ∆H1S take their values in finite dimensional vector spaces. We deduce from Lemma 4.8 that the functors ∆H0S and ∆H1S are finite, and, by Lemma 4.9, we obtain that ∆H0S and ∆H1S are in the image of the functor ι. Consequently, by Proposition 4.7, S is in the image of the functor ι. � Ecole Polytechnique Fédérale de Lausanne, Institut de Géométrie, Algèbre et Topologie, Lausanne, Switzerland. E-mail address: [email protected] Introduction 1. The category Fquad: some recollections 2. Filtration of the standard projective functors PV of Fquad 2.1. Definition of the filtration 2.2. Extremities of the filtration 3. Decomposition of the standard projective functors PH0 and PH1 3.1. Decomposition of PH0 3.2. Decomposition of PH1 3.3. Consequences of decompositions of functors PH0 and PH1 4. Application: the polynomial functors of Fquad 4.1. Definition of polynomial functors of Fquad 4.2. Study of polynomial functors of Fquad

0704.0503

Manifolds admitting a $\tilde G_2$-structure

Manifolds admitting a G̃2-structure. Hông-Vân Lê Abstract We find a necessary and sufficient condition for a compact 7-manifold to admit a G̃2-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of G̃2-type. MSC: 55S35, 53C10 1 Introduction Recently a new class of geometries related with stable forms has been discovered [Hitchin2000], [Hitchin2001], [Witt2005], [Le2006], [LPV2007]. In some cases we can define easily a nec- essary and sufficient condition for a manifold M to admit a stable form of type ω in terms of topological invariants of M , for example if ω is a 3-form of G2-type [Gray1969]. But in general there is no method to solve the question how to find a necessary and sufficient condition for a manifold to admit a stable form. In a previous note [Le2006] we have wrongly stated a sufficient condition for an open manifold to admit a closed stable 3-form of G̃2-type. We recall that [Bryant1987] a 3-form on R 7 is called of G̃2-type, if it lies on the Gl(R7)-orbit of a 3-form 3 = θ1 ∧ θ2 ∧ θ3 + α1 ∧ θ1 + α2 ∧ θ2 + α3 ∧ θ3 Here αi are 2-forms on V 7 which can be written as α1 = y1 ∧ y2 + y3 ∧ y4, α2 = y1 ∧ y3 − y2 ∧ y4, α3 = y1 ∧ y4 + y2 ∧ y3 and (θ1, θ2, θ3, y1, y2, y3, y4) is an oriented basis of (V The group G̃2 can be defined as the isotropy group of ω 3 under the action of Gl(R7). Bryant proved that [Bryant1987] G̃2 coincides with the automorphism group of the split octonians. In this note we prove the following http://arxiv.org/abs/0704.0503v2 Main Theorem. Suppose that M7 is a compact 7-manifold. Then M7 admits a 3-form of G̃2-type, if and only if M 7 is orientable and spinnable. Equivalently the first and second Stiefel-Whitney classes of M7 vanish. Suppose that M7 is an open manifold which admits an embedding to a compact orientable and spinnable 7-manifold. Then M7 admits a closed 3-form of G̃2-type. 2 Proof of Main Theorem Our proof is based on the following simple fact on G̃2. 2.1. Lemma. We have π1(G̃2) = Z2. Hence its maximal compact Lie group is SO(4). This Lemma is well-known, (Bryant mentioned it but he omitted a proof in [Bryant1987]), but I did not find an explicit proof of it in popular lectures on Lie groups, though it could be given as an exercise. For a hint to a solution of this exercise we refer to [HL1982], p.115, for an explicit embedding of SO(4) into G2. The reader can also check that the image of this group is also a subgroup of G̃2 ⊂ Gl(R 7). We shall denote this image by SO(4)3,4. The Cartan theory on symmetric spaces implies that SO(4)3,4 is a maximal compact Lie subgroup of G̃2. Now let us return to proof of our Main theorem. Clearly if M7 admits a G̃2-structure, then it must be orientable and spinnable, since a maximal compact Lie subgroup SO(4)3,4 of G2 is also a compact subgroup of G2. 2.2. Lemma. Assume that M7 is compact, orientable and spinnable. Then M7 admits a G̃2-structure. Proof. SinceM7 is compact, orientable and spinable,M7 admits a SU(2)-structure [Friedrich1997]. Now it is easy to see that it admits a SO(4)3,4-structure, where SO(4)3,4 is a maximal com- pact Lie subgroup of G2. Hence M 7 admits a G̃2-structure. ✷ To prove the last statement of the Main Theorem we shall use the following theorem due to Eliashberg-Mishachev to deform the 3-form ω3 to a closed 3-form ω̄3 of G̃2-type on For a subspaceR ⊂ ΛpM we denote by CloaR a subspace of the space SecR which consists of closed p-forms ω : M → R in the cohomology class a ∈ Hp(M). Eliashberg-Mishashev Theorem [E-M2002,10.2.1] Let M be an open manifold, a ∈ Hp(M) a fixed cohomology class and R an open Diff M-invariant subset. Then the inclu- CloaR →֒ SecR is a homotopy equivalence. In particular, - any p-form ω : M → R is homotopic in R to a closed form ω̄. - any homotopy of p-form ωt : M :→ R which connects two closed forms ω0, ω1 ∈ a can be deformed in R into a homotopy of closed forms ω̄t connecting ω0 and ω1 ∈ a. Let R be the space of all 3-forms of G̃2-type on M = M 7. Clearly this space is an open DiffM7-invariant subset of Λ3M7. Now we apply the Eliashberg-Mishashev theorem to our 3-form ω3 of G̃2-type whose existence has been proved above. Hence M 7 admits a closed 3-form ω̄3 of G̃2-type. ✷ 2.3. Remark. It seems that we can drop the closedness condition in our Main Theorem and use the classical obstruction theory to prove the main Theorem. Acknowledgement. This note is partially supported by grant of ASCR Nr IAA100190701. References [Adams1996] J.F. Adams, Lectures on exceptional Lie groups, The Chicago University Press, 1996. [Bryant1987] R. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2), 126 (1987), 525-576. [E-M2002] Y. Eliashberg and N. Mishachev, Introduction to the h-Principle, AMS 2002. [Gray1969] A. Gray, Vector cross products on manifolds, TAMS 141, (1969), 465-504, (Errata in TAMS 148 (1970), 625). [Friedrich1997] Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, On nearly parallel G2-manifolds, Journal Geom. Phys. 23 (1997), 259-286. [HL1982] , R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. (182), 47-157. [Hitchin2000] N. Hitchin, The geometry of three-forms in 6 and 7 dimensions, J.D.G. 55 (2000), 547-576. [Hitchin2001] N. Hitchin, Stable forms and special metrics, Contemporean math., (2001), 288, 70-89. [Le2006] H. V. Le, The existence of symplectic 3-forms on 7-manifolds, arXiv:math.DG/0603182. [LPV2007] H.V.Le, M. Panak and J. Vanzura, Manifolds admitting stable forms, in preparation. [Witt2005] F. Witt, Special metric structures and closed forms, Ph.D. Thesis , arxiv:math.DG/0502443. Hong Van Le, Institute of Mathematics, Zitna 25, 11567 Praha 1, [email protected], http://arxiv.org/abs/math/0603182 http://arxiv.org/abs/math/0502443 Introduction Proof of Main Theorem

0704.0504

Compatibility of Exotic States with Neutron Star Observation

Compatibility of Exotic States with Neutron Star Observation Chang Ho Hyun∗) Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Korea Department of Physics, Seoul National University, Seoul 151-742, Korea We consider the effect of hard core repulsion in the baryon-baryon interaction at short distance to the properties of a neutron star. We obtain that, even with hyperons in the interior of a neutron star, the neutron star mass can be as large as ∼ 2M⊙. §1. Introduction Reports on recent observations of pulsars in various binary systems show that the maximum mass of a neutron star can be large as (1.7 ∼ 2.1)M⊙. 1), 2) Most of them still have large uncertainty, but a few are within the above range with relatively small error bars. The possibility of large mass of a neutron star thus has led to a claim that exotic states of matter at high densities are not necessary in the neutron star as far as its mass is concerned,3) but there also appeared a counter argument that the large mass does not necessarily rule out the exotic states.4) There are many sources of uncertainties at high densities, e.g. state of matter, constituent particles and their interactions, but the information available to reduce the uncertainties is not sufficient yet. We revisit the neutron star mass problem with a simple phenomenological ap- proach. One fixed point of nuclear matter physics is the nuclear saturation density; its properties such as density, binding energy, symmetry energy, and compression modulus are fairly well constrained. We describe these saturation properties in terms of quantum hadrodynamics (QHD).5) The other fixed point we choose is the hard core repulsion at short range. Though it is a kind of artifact adopted to describe the nucleon-nucleon data, its role is clear in many phenomena of nuclear physics. The effect of hard core can be parametrized with an excluded volume in the estimation of thermodynamic variables.6), 7), 8) It has been more frequently employed to explain the phase transition in the relativistic heavy ion collision environment, and could describe well the transition from hadronic to quark-gluon plasma phase.9) In this work, by including hard cores in the interaction of baryons, we explore the bulk properties of the neutron star, and compare the result with the recent observation of heavy neutron star masses. This paper is outlined as follows. In the next section, we briefly address the basic formalism of QHD with hard core. The next section comes up with numerical results, and brief concluding remarks are drawn in the following section. ∗) e-mail address: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.0504v1 2 C. H. Hyun §2. Formalism We employ the QHD Lagrangian, ψ̄B (i∂ · γ −mB + gσBσ − gωBγ0ω0 − gρBτ3γ0b30)ψB mN (gσNσ) (gσNσ) l=e, µ ψ̄l(i∂ · γ −ml)ψl, (2.1) where the baryon species B includes octet baryons, and σ, ω0 and b30 are non- vanishing meson fields in the mean field approximation. When we account for the forbidden region due to hard core, the baryon density is redefined as 1 + vevρ′ , (2.2) where ρ′ is the density in the case of point particle and vev the excluded volume. We assume vev = πr30 where r0 is the radius of hard core, which is treated as a free parameter in our consideration. Consistency with thermodynamic relations and self- consistency conditions alter the form of state variables (pressure, chemical potential, energy density and etc) and equation of motion of σ-meson field from those of point particle ones. The explicit formulas and equations can be found in old6), 7), 8) and recent10), 11) publications. Three meson-nucleon coupling constants gσN , gωN and gρN and two σ-meson self interaction coefficients b and c are fitted to five saturation properties, the sat- uration density (0.17 fm−3), binding energy (16.0 MeV), symmetry energy (32.5 MeV), compression modulus (300 MeV) and nucleon effective mass (0.75m∗N ), with a given hard core radius r0. Meson-hyperon coupling constants are determined by quark counting rules, gMY = gMN q=u,d nqY /3, where gMY is the meson-hyperon coupling constant, nqY is the number of u and d quarks in a hyperon species Y and gMN is the meson-nucleon coupling constant. As for the hard core radius of hyperons, we assume the same value as that of the nucleon for simplicity. Table I summarizes the parameters determined from the given saturation prop- erties and hard core radii. r0 (fm) gσN gωN gρN b (×10 3) c (×103) 0 8.44 8.92 7.76 3.97 4.00 0.2 8.43 8.91 7.72 3.80 4.37 0.3 8.39 8.89 7.64 3.38 5.26 0.4 8.30 8.85 7.47 2.51 7.11 0.5 8.16 8.78 7.19 0.93 10.47 Table I. Meson-nucleon coupling constants and coefficients b and c fitted to a set of saturation properties described in the text with a given r0 value. Compatibility of Exotic States with Neutron Star Observation 3 §3. Numerical result Fig. 1 shows the binding energy per a nucleon in the symmetric nuclear matter with different hard core radii. Though the saturation properties are the same re- gardless of r0 values, the equation of state becomes stiffer at high densities with a larger r0 value. 0 0.5 1 1.5 2 2.5 r0 = 0.0 0.2 0.4 0.5 Fig. 1. Binding energy of a nucleon in the symmetric nuclear matter with different hard core radii. The equation of state of neutron star matter is determined self-consistently by the baryon number conservation, charge neutrality, β-equilibrium of baryons and leptons, and equations of motion of meson fields. Once the equation of state is determined, the mass-radius relation of a neutron star can be obtained by solving Tolman-Oppenheimer-Volkoff (TOV) equation. Table II shows the maximum mass of a neutron star, corresponding radius and central density with nucleons only (columns of “np”) and with hyperons (columns of “npY ”). Consistent with the behavior of the equation of state at high densities in Fig. 1, the maximum mass becomes larger with a larger r0 value. For np case, when r0 = 0.5 fm, the increase amounts to 10% of the maximum mass without hard core. With hyperons, the maximum mass increases to the range of large mass in recent observations when r0 & 0.3 fm. n p n p Y r0 (fm) M (M⊙) R (km) ρcent (ρ0) M (M⊙) R (km) ρcent (ρ0) 0 2.10 10.9 6.4 1.53 11.3 6.1 0.2 - - - 1.58 11.4 6.1 0.3 2.14 11.1 6.2 1.70 11.5 5.9 0.4 2.20 11.4 5.9 1.97 12.2 5.2 0.5 2.34 11.7 5.4 - - - Table II. Maximum mass M in units of solar mass, and corresponding radius R in km and central density ρcent in unit of the saturation density ρ0. 4 C. H. Hyun §4. Conclusion We investigated the maximum mass of a neutron star in a simple phenomeno- logical approach where the hard-core repulsion is included in the QHD model. The hard core radius is treated as a free parameter, and the meson-nucleon coupling con- stants are fixed identical saturation properties. We obtained the equation of state of neutron star matter that satisfies thermodynamic equations and self-consistency conditions. Solving TOV equation, we obtained the mass-radius relation of a neu- tron star. Our result shows that the maximum mass with hyperons can be as large as observed masses with a hard core radius r0 & 0.3 fm. These values of r0 are in the range of hard core radius 0.3 ∼ 0.6 fm in well-known hard-core potential models such as Hamada-Johnston12) or Reid.13) More investigations are necessary to figure out the uncertainties. For instance, the hard core size of hyperons can matter. The effect of hard cores to the formation of other exotic states such as meson condensation or deconfined quark phases is also worthy to be studied. To conclude, the our result shows that the hyperon matter, which is known to give the biggest effect to the mass-radius relation of a neutron star among possible exotic states in the interior of a neutron star, is not necessarily incompatible with the observed mass. Acknowledgments The author thanks the Yukawa Institute for Theoretical Physics at Kyoto Uni- versity, where this work was initiated and developed during the YKIS2006 on ”New Frontiers on QCD”. Author is grateful to Shung-ichi Ando for reading the manuscript. This works was supported by the Basic Research Program of Korea Science & En- gineering Foundation (R01-2005-000-10050-0). References 1) D. J. Nice et al., Astrophys. J. 634 (2005), 1242 2) D. Page and S. Reddy, Ann. Rev. Nucl. Part. Sci. 65 (2006), 327 3) F. Özel, Nature 441 (2006), 1115 4) T. Klahn et al., nucl-th/0609067. 5) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16 (1986), 1 6) D. H. Rischke, M. I. Gorenstein, H. Stöcker and W. Greiner, Z. Phys. C 51 (1991), 485 7) S. Kagiyama, A. Nakamura and T. Omodaka, Z. Phys. C 53 (1992), 163 8) J. Cleymans, J. Stalnacke and E. Suhonen, Z. Phys. C 55 (1992), 317 9) S. Kagiyama et al., Eur. Phys. J. C 25 (2002), 453 10) P. K. Panda et al., Phys. Rev. C 65 (2002), 065206 11) R. M. Aguirre and A. L. De Paoli, Phys. Rev. C 68 (2003), 055804 12) T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962), 382 13) R. V. Reid, Ann. Phys. (N.Y.) 50 (1968), 411 http://arxiv.org/abs/nucl-th/0609067 Introduction Formalism Numerical result Conclusion

0704.0505

Exact Solutions of Einstein-Yang-Mills Theory with Higher-Derivative Coupling

OCU-PHYS 263 Exact Solutions of Einstein-Yang-Mills Theory with Higher-Derivative Coupling Hironobu Kihara∗ Osaka City University, Advanced Mathematical Institute (OCAMI), 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan Muneto Nitta† Department of Physics, Keio University, Hiyoshi, Yokohama, Kanagawa 223-8521, Japan Abstract We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton’s constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS4 × S6. ∗Electronic address: kihara(at)sci.osaka-cu.ac.jp †Electronic address: nitta(at)phys-h.keio.ac.jp http://arxiv.org/abs/0704.0505v3 mailto:kihara(at)sci.osaka-cu.ac.jp mailto:nitta(at)phys-h.keio.ac.jp Unification of fundamental forces with space-time and matter often requires higher- dimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S1 [1]. This old idea has been revis- ited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2]. String theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli. The moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4]. This flux compactification has been extensively studied in these years. We would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7]) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the ’t Hooft-Polyakov monopole [8] on S2 and the Yang-Mills instanton [9] on S4, both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S6. Higher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10]. Eight dimensional case is known as octonionic instantons [11]. Though several works have been done for generalized self-dual equations [12, 13], a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO(6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S6 [14]. In this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmo- logical constant is compactified to a four-dimensional Minkowski space M4 (anti de Sitter space AdS4) with a six-dimensional sphere S 6, where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol’nyi equations and solutions are BPS. We find for both M4 × S6 and AdS4 × S6 that certain relations exist between the radius of S6, the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling con- stant α vanishes in the case of M4 × S6, these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol’nyi equa- tions for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields. Let us consider that space-time is a ten-dimensional manifold. We consider an Einstein- Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields and a graviton field or the metric ĝµ̂ν̂ . Indices with a hat “̂ ” will refer to a ten- dimensional space-time (X0, X1, · · · , X9). Latin indices (a, b, · · · ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO(6) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γa} which satisfy the following anti-commutation relations, {Γa,Γb} = 2δab. These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so(6) are represented by Γab = [Γa,Γb]. We often abbreviate the Yang-Mills fields as Aµ̂ = Γab and we also use notations with differential forms. Thus the gauge fields are expressed as A = Aµ̂dX µ̂. In this notation, the corresponding field strength F is written as F = dA+eA∧A, where e is a gauge coupling. Covariant derivative Dµ̂ on an adjoint representation Y = Y [ab]Γab is defined as Dµ̂Y = ∂µ̂Y + e(Aµ̂Y − Y Aµ̂), where Y is a scalar multiplet. The action Stotal consists of two parts. One is the Einstein- Hilbert action SE and the other SYMT is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is: Stotal = SE + SYMT , SE = dvR , SYMT = −F ∧ ∗F + α2(F ∧ F ) ∧ ∗(F ∧ F )− V0dv . (1) Here the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = −ĝd10X in a local patch. The scalar curvature is denoted by R. The asterisk “∗” denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω∧∗ω is proportional to the invariant volume form dv.1 The parameters of this action are the Newton’s gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V0. We show the explicit form of the Yang-Mills part with components of A and F , SYMT = − µ̂ν̂ F µ̂ν̂,[ab] + [abcd] µ̂ν̂ρ̂σ̂ T µ̂ν̂ρ̂σ̂,[ab][cd] + 3 · 16 Sµ̂ν̂ρ̂σ̂S µ̂ν̂ρ̂σ̂ + , (2) dX µ̂ ∧ dX ν̂Γab , Sµ̂ν̂ρ̂σ̂ = F [ab]µ̂ν̂ F , (3) [ab][cd] µ̂ν̂ρ̂σ̂ [abcd] µ̂ν̂ρ̂σ̂ [ab][cd] µ̂ν̂ρ̂σ̂ [ac][db] µ̂ν̂ρ̂σ̂ [ad][bc] µ̂ν̂ρ̂σ̂ [cd][ab] µ̂ν̂ρ̂σ̂ [db][ac] µ̂ν̂ρ̂σ̂ [bc][ad] µ̂ν̂ρ̂σ̂ . (4) The Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields: Rµ̂ν̂ − ĝµ̂ν̂R = 8πGTµ̂ν̂ , Dµ̂ −gF µ̂ν̂ − 2α2 −gF [µ̂ν̂F ρ̂σ̂]Fρ̂σ̂ = 0 . (5) Here the energy-momentum tensor Tµ̂ν̂ is obtained by the variation of the Yang-Mills part with respect to the metric: Tµ̂ν̂ = ρ̂,[ab]F + α2T̃ [abcd] µ̂ρ̂σ̂τ̂ ρ̂σ̂τ̂ ,[ab][cd] + 3 · 2 Sµ̂ρ̂σ̂τ̂Sν̂ ρ̂σ̂τ̂ − 1 gµ̂ν̂χ F µ̂ν̂,[ab] + [abcd] µ̂ν̂ρ̂σ̂ T µ̂ν̂ρ̂σ̂,[ab][cd] + 3 · 8 Sµ̂ν̂ρ̂σ̂S µ̂ν̂ρ̂σ̂ + V0 . (6) To solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk. Our ansatz for the metric is the following: ds2 = ηµνdx µdxν + (1 + y2/4R20) dyIdyJ = ĝµ̂ν̂dX µ̂dX ν̂ , y2 = (yI)2 , (7) 1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation: (Fµνdx µν) ∧ ∗(Fρσdxρσ) = −FµνFµνdv. where the coordinates X are the total space-time coordinates. The metric ηµν = diag(− + ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat “̂ ”, for instance µ will refer to the first four variables. Capital indices (I, J, · · · ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R0. The Riemann tensor, Ricci tensor and scalar curvature are RIJKL = δIKgJL − δILgJK , RIJ = gIJ , R = . (8) The rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations, = 8πGTµν , 0 = TµI , − gIJ = 8πGTIJ . (9) We now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂µA = 0, and they have no four-dimensional components Aµ = 0. This implies that the field strengths are two forms on the six-dimensional sphere: A = AI(y)dy I , F = FIJdy I ∧ dyJ . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes −1 ηµνχ, and the equation reduces to 30/R 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition F = iβγ7 ∗6 (F ∧ F ), (10) where β is a real parameter. Here “∗6” means the Hodge dual on the six-dimensional sphere. Then the second part of the equations of motion (5) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF+e (A ∧ F − F ∧A). In fact we have an explicit solution to the self-dual equation: 4eR20 yaebΓab , F = 4eR20 ea ∧ ebΓab , β = . (11) Here we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes , χ = (1 + ζ) 4e2R40 + V0 , TIJ = − (1− ζ) 4e2R40 gIJ . (12) With these ansatzes we obtain algebraic equations from the Einstein equations: = 8πG (1 + ζ) 2e2R40 = 8πG (1− ζ) 5 8e2R40 , (13) From these we finally obtain e2R20 (2 + 4ζ) , V0 = 4e2R40 (1 + 3ζ) . (14) When the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5]. 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol’nyi bound and becomes a BPS state. The energy density is given by an integral over S6 as follows: YMT = −F ∧ ∗6F + α2(F ∧ F ) ∧ ∗6(F ∧ F ) Tr (iF ∓ αγ7 ∗6 (F ∧ F )) ∧ ∗6 (iF ∓ αγ7 ∗6 (F ∧ F ))± Trγ7F ∧ F ∧ F ≥ ± i Trγ7F ∧ F ∧ F = ∓ ǫabcdefF [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15) where the field strength F has only components along S6. If the coupling α is equal to β, the solution of eq. (10) satisfies the Bogomol’nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Qm transform as a representation of SO(6). The index m labels the representation space. Let us add an action SQ of the scalar fields Q with a Higgs potential dvDµ̂Q mDµ̂Qm + V (Q2) , Dµ̂Q m = ∂µ̂Q R(Γab)mm′Q m′ (16) to Stotal. The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition Dµ̂Q m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations (7), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions. Next we suppose that the four-dimensional part is an anti-de Sitter space AdS4 of radius RA. Our ansatz for the metric is the following: ds2 = ηµν(x)dx µdxν + gIJ(y)dy IdyJ = ĝµ̂ν̂dX µ̂dX ν̂ , (17) 2 We need to redefine e the half when we compare to the result of [5]. gIJ(y)dy IdyJ = (1 + y2/4R20) dyIdyJ y2 = (yI)2 , ηµν(x)dx µdxν = cos2 θ −dτ 2 + dθ2 + sin2 θdΩ2 , dΩ2 = |dz|2 (1 + |z|2/4)2 , (18) where z parametrizes a whole complex plane. The metric ηµν(x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are Rµνρσ = − δµρηνσ − δµσηνρ , Rµν = − ηµν , RIJKL = δIKgJL − δILgJK , RIJ = gIJ . (19) The total scalar curvature is obtained by a summation of those of two parts: R = − In this space, the Einstein equations are Rµν − ηµνR = 8πGTµν , RIJ − gIJR = 8πGTIJ . (20) The ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein equations as = −4πG (1 + ζ) 4e2R40 = −4πG (1− ζ) 5 4e2R40 We are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V0 = 0. In this case, the radii (RA, R0) are written by the couplings, R20 = (5 + 7ζ) , R2A = 5 + 7ζ 5 + 15ζ R20. (22) Thus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol’nyi equation again. Our solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional space- time [13]. Though the parameter ζ(= α2/β2) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15]. To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Qm are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16]. Finally our solution of AdS4 ×S6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17]. Acknowledgments We are grateful to D. H. Tchrakian for various comments. We would like to thank Y. Hosotani, H. Itoyama, Y. Yasui, M. Sakaguchi, T. Oota, T. Kimura, S. Shimasaki and E. Itou. We also thank M. Sheikh-Jabbari for an advice. This work is supported by the 21 COE program “Constitution of wide-angle mathematical basis focused on knots” from Japan Ministry of Education. [1] G. Nordström, Phys. Z. 15, 504 (1914) [arXiv:physics/0702221]; T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921, 966 (1921); O. Klein, Z. Phys. 37, 895 (1926) [Surveys High Energ. Phys. 5, 241 (1986)]. [2] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Phys. Rept. 130, 1 (1986). [3] K. Dasgupta, G. Rajesh and S. Sethi, JHEP 9908, 023 (1999) [arXiv:hep-th/9908088]. [4] S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240]. [5] E. Cremmer and J. Scherk, Nucl. Phys. B 108, 409 (1976); Nucl. Phys. B 118, 61 (1977). [6] Z. Horvath, L. Palla, E. Cremmer and J. Scherk, Nucl. Phys. B 127, 57 (1977). [7] R. Kerner and D. H. Tchrakian, Phys. Lett. B 215, 87 (1988). [8] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974); A. M. 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Rept. 323, 183 (2000) [arXiv:hep-th/9905111]. http://arxiv.org/abs/hep-th/0210037 http://arxiv.org/abs/hep-th/0408068 http://arxiv.org/abs/hep-th/0502025 http://arxiv.org/abs/hep-th/0703166 http://arxiv.org/abs/hep-th/0105047 http://arxiv.org/abs/hep-th/0612181 http://arxiv.org/abs/hep-th/0602170 http://arxiv.org/abs/hep-th/9711200 http://arxiv.org/abs/hep-th/9802150 http://arxiv.org/abs/hep-th/9905111 Acknowledgments References

0704.0506

Dimensional crossover of quantum critical behavior in CeCoIn$_5$

APS/123-QED Dimensional crossover of quantum critical behavior in CeCoIn5 J. G. Donath,1 P. Gegenwart,2 F. Steglich,1 E. D. Bauer,3 and J. L. Sarrao3 Max-Planck-Institute for Chemical Physics of Solids, D-01187 Dresden, Germany I. Physik. Institut, Georg-August-Universität Göttingen, D-37077 Göttingen Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: November 11, 2018) The nature of quantum criticality in CeCoIn5 is studied by low-temperature thermal expansion α(T ). At the field-induced quantum critical point at H = 5 T a crossover scale T ⋆ ≈ 0.3 K is observed, separating α(T )/T ∝ T−1 from a weaker T−1/2 divergence. We ascribe this change to a crossover in the dimensionality of the critical fluctuations which may be coupled to a change from unconventional to conventional quantum criticality. Disorder, whose effect on quantum criticality is studied in CeCoIn5−xSnx (0 ≤ x ≤ 0.18), shifts T ⋆ towards higher temperatures. PACS numbers: 71.10.Hf,71.27.+a,74.70.Tx Quantum criticality in heavy fermion (HF) systems continues to attract interest due to the occurrence of highly anomalous metallic states with severe deviations from Landau Fermi liquid (LFL) behavior [1, 2] and the emergence of unconventional superconductivity in close vicinity to antiferromagnetic (AF) quantum criti- cal points (QCPs) [3]. Neither the nature of the non- Fermi liquid (NFL) normal state related to quantum crit- icality, nor the superconducting (SC) pairing mechanism has been clarified up to now. It is thus of great inter- est to investigate whether quantum criticality in these systems can be described by conventional theory within the framework of a spin-density-wave (SDW) instability [4, 5], or whether unconventional scenarios in which the f-electrons localize at the magnetic QCP due to a de- struction of the Kondo resonance [6, 7, 8] may be more appropriate. For the formation of the latter, magnetic frustration leading to a reduced dimensionality of the critical fluctuations may be crucial. The CeMIn5 (M=Rh, Ir, Co) systems are prototyp- ical as they display a generic phase diagram with un- conventional HF superconductivity in close vicinity to an AF QCP [9]. They crystallize in a tetragonal struc- ture which can be viewed as an alternating series of CeIn3 and MIn2 layers. As a result of the layered crystal structure, the Fermi surface displays a strongly two-dimensional (2D) character with cylindrical sheets along the crystallographic c-axis [10]. Compared to cu- bic CeIn3, a HF superconductor with Tc = 0.2 K [3] in a very narrow pressure range close to the magnetic quantum phase transition at pc ≈ 2.6GPa, SC transition temperatures of about 2 K are observed over wide pres- sure ranges for the tetragonal CeRhIn5 (at p ≥ 1.6 GPa) and CeCoIn5 (at ambient pressure) [11, 12]. This Tc enhancement has been attributed to the layered crystal structure and, relatedly, strongly anisotropic magnetic fluctuations [12]. Indeed the nuclear magnetic relaxation rate 1/T1 of CeCoIn5 displays a weak T 1/4 dependence in the normal state between 2 and 40 K which signals strongly anisotropic quantum critical fluctuations [13]. ��� ��� ��� � � � � � �� :;<=>?@ABCDE J KLM TUVWXYZ Figure 1: (Color online) Phase diagram of CeCoIn5 for H ‖ c as determined from thermal expansion. Superconducting phase in gray with first-order boundary below 0.7 K indicated by thick black line. Regions where thermal expansion follows 2D- and 3D quantum critical behavior are marked in blue and yellow, respectively. The inset displays the evolution of the crossover with Sn-doping in CeCoIn5−xSnx at the respective Hc2(x). The aim of this Letter is a detailed investigation of the nature of quantum criticality in CeCoIn5. We focus in particular to the region very close to the upper critical field Hc2 for superconductivity (5 T for H ‖ c, cf. Figure 1) which previously has been studied by heat and charge transport [14, 15, 16] and specific heat measurements [17]. Diverging coefficients of the T 2 contributions to the electrical and the thermal resistivity prove the existence of a magnetic-field induced QCP at 5 T. NFL behavior in the temperature dependence of the electronic specific heat coefficient at 5 T has been described in the frame of the SDW theory [17]. However, below 0.3 K a large nu- clear contribution arising from the Zeeman splitting of In- nuclear moments needs to be subtracted. Therefore, the data do not allow to distinguish between a saturation or http://arxiv.org/abs/0704.0506v2 logarithmic divergence at lowest temperatures and thus further thermodynamic measurements are needed to de- termine the nature of quantum criticality in the system (transport data will be discussed later). Thermal expansion is ideally suited for this purpose. It probes the pressure dependence of the entropy which close to QCPs is accumulated at finite temperatures. Scaling arguments have revealed that thermal expansion α(T ) is far more singular than specific heat C(T ) in the approach of any pressure-sensitive QCP [18]. Within the SDW theory the leading contribution to α(T )/T di- verges like T−1/2 and T−1 for 3D and 2D AF QCPs, respectively [18]. Both can easily be distinguished from α(T )/T = const. expected for a LFL. Especially impor- tant in this context, thermal expansion, in contrast to specific heat, is not affected by nuclear hyperfine contri- butions. For our study, we have used high-quality single crys- tals of CeCoIn5−xSnx grown from In flux, whose low- temperature specific heat and electrical resistivity are discussed in [17, 19, 20]. For details on the sample char- acterization see [19, 20, 21]. Thermal expansion has been determined with the aid of high-resolution dilatometers at temperatures down to 0.04 K and in magnetic fields up to 10 T. We have measured the length change ∆Lc along the c-axis and determined the linear (c-axis) expansion coefficient α = ∂ lnLc/∂T . Figure 2 displays our thermal expansion data on un- doped CeCoIn5. At the upper critical field of 5 T, the thermal expansion coefficient α(T )/T (cf. inset a) grows much stronger upon cooling than the respective specific heat coefficient which diverges only logarithmically [17]. Over more than one decade in temperature, i.e. for 0.3 K ≤ T ≤ 6 K, the data follow a 1/T divergence which hints at 2D AF quantum critical fluctuations [18]. The latter may result from the layered crystal structure [12, 13]. At T ⋆ ≈ 0.3 K, the temperature dependence changes to α ∝ T 1/2 (see main part and inset b, which also displays data obtained on a second sample down to 40 mK). Note, that α/T does not show a saturation excluding the formation of a LFL above the lowest mea- sured temperature. The square-root behavior for α(T ) is compatible with a 3D AF QCP of itinerant nature [18] as observed for CeNi2Ge2 [2] and CeIn3−xSnx [22]. As H is increased above 5 T, T ⋆ increases and α(T ) becomes less singular, i.e. the coefficient of the square- root contribution decreases (cf. Fig.2, inset b). This suggests that the system is tuned away from the QCP, compatible with previous studies [14, 17], although LFL behavior is not yet fully established in thermal expansion. Our data on CeCoIn5 are summarized in the main part of Fig. 1. We have observed a crossover scale separating 2D from 3D quantum critical behavior. To provide fur- ther evidence for this crossover and to investigate how it is influenced by weak disorder, we now focus on the series CeCoIn5−xSnx where the Sn-atoms preferentially 0 1 2 3 4 5 6 0.1 1 7 0.0 0.5 1.0 1.5 T (K) 1/2 CeCoIn 5T // c T (K) α/T (10 H (T) 5 (S2) 8 (S2) 10 (S2) T (K) Figure 2: (Color online) Temperature dependence of the lin- ear thermal expansion coefficient of CeCoIn5 at H = 5 T (‖ c). Dotted line and arrow indicate α ∝ T and crossover temperature T ⋆, defined as upper limit for this T -dependence, respectively. Inset (a) displays data from main part as α/T vs T (on logarithmic scale). Solid line indicates T−1 depen- dence. Inset (b) compares data from main part in the low- temperature regime with 5, 8 and 10T data obtained from a second sample (S2). Lines display square-root behavior. x Tc (K) Hc2 (T) 0.00 (2.25 ± 0.05) (4.9± 0.1) 0.03 (1.80 ± 0.05) (4.5± 0.1) 0.06 (1.50 ± 0.05) (3.9± 0.1) 0.09 (1.15 ± 0.05) (3.4± 0.1) 0.12 (0.75 ± 0.05) (2.5± 0.1) 0.18 0 0 Table I: Values for the SC transition temperature Tc and up- per critical magnetic field Hc2 for CeCoIn5−xSnx [19]. occupy the In-1 position within in the tetragonal plane [21]. Sn doping weakens superconductivity, leading to a linear suppression of Tc towards zero for x = 0.18 [19]. The temperature-magnetic field phase diagram of var- ious CeCoIn5−xSnx single crystals has previously been studied by low-temperature electrical resistivity and spe- cific heat measurements [19, 20]. As Tc is reduced, a corresponding reduction of Hc2 is observed (for the x- dependence of Tc and Hc2, see Table I). For all differ- ent Sn concentrations the temperature dependence of the specific heat displays NFL behavior at the respective up- per critical field and the formation of a LFL state at fields exceeding Hc2(x) [19]. This suggests that field- induced quantum criticality is always pinned at the upper critical field Hc2 when the latter is reduced by Sn dop- ing. Furthermore, the low-T specific heat coefficient at 0.1 1 6 x 0.00 0.03 0.06 0.09 0.12 0.18 T (K) CeCoIn Figure 3: Linear thermal expansion coefficient as α vs. T on a logarithmic scale for CeCoIn5−xInx at H ≃ Hc2(x) (given in Table I). Note that data sets are shifted by 5× 10−6 K−2, subsequently. Arrows indicate lower limit of α/T ∝ T−1 be- havior. H = Hc2(x) remains unchanged within the scatter of the data for 0 ≤ x ≤ 0.12 [19]. On the other hand, the resid- ual resistivity ρ0 shows a tenfold increase for x ranging from 0 to 0.18, indicating the effect of disorder scatter- ing due to the random distribution of Sn-atoms on the in-plane In site [20]. The study of CeCoIn5−xSnx thus allows to systematically investigate the disorder depen- dence of NFL behavior without tuning the system away from the QCP. Figure 3 shows c-axis thermal expansion data for the various studied CeCoIn5−xSnx single crystals at their re- spective upper critical magnetic fields (for the zero-field data see [23]). In all these samples, 2D-like quantum crit- ical behavior α(T )/T ∝ T−1 is found from 6 K down to a lower bound which increases from about 0.3 K for x = 0 to about 1.4 K for x = 0.18. Like for x = 0, the low-T thermal expansion of all samples studied is well described by α(T ) ∝ T . For x = 0.18, this temperature dependence holds up to T ⋆ ≈ 1.4 K, i.e. over more than one decade (see Fig. 4), providing clear evidence for 3D AF quantum critical fluc- tuations in the latter system. Clearly, the temperature 0 1 2 3 4 T (K) CeCoIn x 0.18 Figure 4: Linear thermal expansion coefficient α(T ) of CeCoIn5 at H = 5 T (‖ c, open squares), as well as CeCoIn4.82Sn0.18 at H = 0 (open circles). Dotted lines and arrows indicate α ∝ T and crossover temperatures T ⋆, re- spectively. at which the dimensional crossover occurs is shifted with Sn doping in CeCoIn5−xSnx to values above 1 K, cf. the inset of Fig. 1. As stated above, the partial substitution of the In-(1) site by Sn-atoms enhances impurity scat- tering without tuning the system away from the QCP. Our observation of a shift of the crossover scale T ⋆ with x is then naturally attributed to the effect of isotropic impurity scattering, which ”smears out” the magnetic anisotropy. Crossovers have also been observed in the electrical and heat transport [15, 16] as well as in the Hall coefficient [24] for the current direction j ⊥ c. How- ever, transport experiments are influenced by electronic relaxational properties, which can give rise to compli- cated behavior for anisotropic and multiband systems like CeCoIn5. Indeed, for j ‖ c no crossover is visible in ρ(T ), and the Wiedemann-Franz law, which is obeyed for j ⊥ c, seems to be violated [16]. In order to clearly show that, at the QCP in CeCoIn5−xSnx, a finite energy scale kBT ⋆ exists which marks the crossover from 2D to 3D quantum critical behavior, measurements either of the fluctuation spec- trum in equilibrium, for example by inelastic neutron scattering (INS), or of thermodynamic properties are re- quired. q-scans of the INS over wide regions in recip- rocal space, required to decide on the dimensionality of the quantum critical fluctuations, are not possible at high fields. Therefore, our thermodynamic measurements pro- vide the only way to investigate this question and indeed prove such a crossover. We now address the nature of quantum criticality (SDW-type or unconventional) in the regime where 2D- like behavior is observed. Theory suggests that 2D fluctuations are necessary for the occurrence of locally- [\] ^ _ ������ ������ − ��  − ¢£¤¥ ª «¬­ Figure 5: (Color online) Critical Grüneisen ratio Γcr = Vm/κT × αcr/Ccr, where αcr and Ccr denote thermal expan- sion and specific heat after subtraction of non-critical back- ground contributions [18], of YbRh2(Si0.95Ge0.05)2 (left axis, [2]) and CeCoIn5−xInx (x = 0 at H = 5 T and x = 0.18 at 0 T, right axis). For the latter, the molar volume and isothermal compressibility equal Vm = 9.57 · 10−5 m3/mol and κT = (3.43 ± 0.16) × 10−3 GPa−1 [25], respectively. A small background term [0.35 × 10−6 K−2] has been sub- tracted from α/T for x = 0.12. No specific-heat background contributions have been subtracted since C(T )/T ∝ log T at T > T ⋆ (T ⋆ indicated by dotted arrows). The so-derived crit- ical Grüneisen ratio is invalid for T < T ⋆, where the specific heat is dominated by a non-critical contribution [26]. Lines indicate power-law behavior at T > T ⋆. critical quantum criticality [7]. The latter is well estab- lished for the magnetic-field tuned AF QCP in the heavy fermion system YbRh2Si2 and its slightly Ge-doped vari- ant YbRh2(Si0.95Ge0.05)2, for which the critical field is almost zero [2]. It is therefore very interesting to com- pare the low-T thermodynamics of CeCoIn5−xSnx with the latter system. Of particular importance is the tem- perature dependence of the critical Grüneisen ratio Γcr, i.e. the ratio of the critical components of thermal ex- pansion to specific heat. It has previously been shown, that Γcr(T ) ∝ T−ǫ with ǫ = 1 and 2/3 for conventional and unconventional quantum criticality, respectively [2]. Figure 5 shows striking similarities in the temper- ature dependence of the critical Grüneisen ratio of YbRh2(Si0.95Ge0.05)2 and CeCoIn5−xSnx at tempera- tures above T ⋆(x), i.e. in the 2D regime: A rather simi- lar fractional Grüneisen exponent is found which is close to the prediction for the locally-critical QCP scenario in the presence of xy anisotropy [7]. Theoretically, it has been shown that such behavior requires quasi-2D quan- tum critical fluctuations [7] supporting further the lat- ter at T > T ⋆ in CeCoIn5. In view of the lack of su- perconductivity in YbRh2Si2 (at least for T > 10 mK), it is highly desirable to check, whether or not a similar crossover towards conventional behavior at temperatures below the lower limit of previous studies (20 mK) takes place in the latter material. To summarize, we have found thermodynamic evidence for a finite crossover scale T ⋆ at the magnetic-field tuned QCP in CeCoIn5. We associate T ⋆ with a dimensional crossover from 2D (T > T ⋆) to 3D (T < T ⋆) quantum critical behavior. The introduction of disorder shifts the crossover scale towards higher temperatures. Stimulating discussions with M. Nicklas, Q. Si and S. Wirth are gratefully acknowledged. Work at Dresden and Göttingen was partially financed by the DFG Research unit 960 (Quantum phase transitions), while work at Los Alamos was carried out under the auspices of the U.S. [1] G.R. Stewart, Rev. Mod. Phys. 73, 797-855 (2001), 78, 743-753 (2006). [2] P. Gegenwart, Q. Si, F. Steglich, arXiv:0712.2045v2 and refs. therein. [3] N.D. Mathur et al., Nature 394, 39 (1998). [4] A.J. Millis, Phys. Rev. B 48, 7183 (1993). [5] T. Moriya and T. Takimoto, J. Phys. Soc. Jpn. 64, 90 (1995), and refs. therein. [6] P. Coleman et al., J. Phys. Cond. Matt. 13 R723 (2001). [7] Q. Si et al., Nature 413, 804 (2001). [8] T. Senthil, M. Vojta, S. Sachdev, Phys. Rev. B 69, 035111 (2004). [9] J.D. Thompson, et al., Physica B 329, 446 (2003) and refs. therein. [10] R. Settai et al., J. Phys. Condens. Matter 13 L627 (2001). [11] H. Hegger et al., Phys. Rev. Lett. 84, 4986 (2000). [12] C. Petrovic et al., J. Phys. Condens. Matter 13, L337 (2001). [13] Y. Kawasaki et al., J. Phys. Soc. Jpn. 72, 2308 (2003). [14] J. Paglione et al., Phys. Rev. Lett. 91, 246405 (2003). [15] J. Paglione et al., Phys. Rev. Lett. 97, 106606 (2006). [16] M.A. Tanatar, J. Paglione, C. Petrovich, L. Taillefer, Sci- ence 316, 1320 (2007). [17] A. Bianchi, R. Movshovich, I. Vekhter, P.G. Pagliuso, J.L. Sarrao, Phys. Rev. Lett. 91, 257001 (2003). [18] L. Zhu, M. Garst, A. Rosch, Q. Si, Phys. Rev. Lett. 91, 066404 (2003). [19] E.D. Bauer et al., Phys. Rev. Lett. 94, 047001 (2005). [20] E.D. Bauer et al., Phys. Rev. B 73, 245109 (2006). [21] M. Daniel et al., Phys. Rev. Lett. 95, 016406 (2005). [22] R. Küchler et al., Phys. Rev. Lett. 96, 256403 (2006). [23] J.G. Donath, et al., Physica B 378-380, 98 (2006). [24] S. Singh et al., Phys. Rev. Lett. 98, 057001 (2007). [25] R.S. Kumar, A.L. Cornelius, J.L. Sarrao, Phys. Rev. B 70, 214526 (2004). [26] For T < T ⋆, specific heat follows the predictions of the 3D SDW scenario [17], i.e. C(T )/T = γ′ − β′ T . Since αcr ∝ T , Γcr ∝ T−1 (as expected [18]). http://arxiv.org/abs/0712.2045

0704.0507

E_6 and the bipartite entanglement of three qutrits

arXiv:0704.0507v2 [hep-th] 9 Oct 2007 Imperial/TP/2007/mjd/1 CERN-PH-TH/2007-78 UCLA/07/TEP/7 E6 and the bipartite entanglement of three qutrits M. J. Duff 1† and S. Ferrara2‡ † The Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ ‡Physics Department,Theory Unit, CERN, CH1211, Geneva23, Switzerland Department of Physics & Astronomy, Universuty of California,Los Angeles, USA INFN-Laboratori Nazionale di Frascati, Via E. Fermi 40, 00044 Frascati, Italy ABSTRACT Recent investigations have established an analogy between the entropy of four-dimensional supersymmetric black holes in string theory and entanglement in quantum information the- ory. Examples include: (1) N = 2 STU black holes and the tripartite entanglement of three qubits (2-state systems), where the common symmetry is [SL(2)]3 and (2) N = 8 black holes and the tripartite entanglement of seven qubits where the common symmetry is E7 ⊃ [SL(2)] 7. Here we present another example: N = 8 black holes (or black strings) in five dimensions and the bipartite entanglement of three qutrits (3-state systems), where the common symmetry is E6 ⊃ [SL(3)] 3. Both the black hole (or black string) entropy and the entanglement measure are provided by the Cartan cubic E6 invariant. Similar analogies exist for “magic” N = 2 supergravity black holes in both four and five dimensions. [email protected] [email protected] http://arxiv.org/abs/0704.0507v2 Contents 1 D = 4 black holes and qubits 3 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 3 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 4 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 4 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Five-dimensional supergravity 8 3 D = 5 black holes and qutrits 10 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 10 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 11 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Conclusions 13 5 Acknowledgements 14 1 D = 4 black holes and qubits It sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual de- scriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string the- ory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples: 1.1 N = 2 black holes and the tripartite entanglement of three qubits The three qubit system (Alice, Bob, Charlie) is described by the state |Ψ〉 = aABC |ABC〉 (1.1) where A = 0, 1, so the Hilbert space has dimension 23 = 8. The complex numbers aABC transforms as a (2, 2, 2) under SL(2, C)A×SL(2, C)B ×SL(2, C)C . The tripartite entangle- ment is measured by the 3-tangle [7, 8] τ3(ABC) = 4|Det aABC |. (1.2) where Det aABC is Cayley’s hyperdeterminant [9]. Det a = − ǫA1A2ǫB1B2ǫA3A4ǫB3B4ǫC1C4ǫC2C3aA1B1C1aA2B2C2aA3B3C3aA4B4C4 (1.3) The hyperdeterminant is invariant under SL(2)A × SL(2)B × SL(2)C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 aABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The STU model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)]3. The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11]. The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley’s hyperdeterminant S = π |Det aABC |. (1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various “small” and “large” BPS and non-BPS black holes [1, 2, 3, 4, 5, 6]. For example, the GHZ state [12] |Ψ〉 ∼ |111〉+ |000〉 (1.5) with Det aABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ〉 ∼ |100〉+ |010〉+ |001〉 (1.6) with Det aABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state |Ψ〉 = −|000〉+ |011〉+ |101〉+ |110〉 (1.7) corresponds to a large BPS 4-charge black hole. 1.2 N = 2 black holes and the bipartite entanglement of two qubits An even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ〉 = aAB|AB〉 (1.8) where A = 0, 1, and the Hilbert space has dimension 22 = 4. The aAB transforms as a (2, 2) under SL(2, C)A × SL(2, C)B. The entanglement is measured by the 2-tangle τ2(AB) = C 2(AB) (1.9) where C(AB) = 2 |det aAB| (1.10) is the concurrence. The determinant is invariant under SL(2, C)A × SL(2, C)B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 aAB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det aAB| (1.11) For example, the Bell state |Ψ〉 ∼ |11〉+ |00〉 (1.12) with det aAB ≥ 0 corresponds to a large non-BPS 2-charge black hole. 1.3 N = 8 black holes and the tripartite entanglement of seven qubits We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E7(7). The black hole entropy is [15, 18] S = π |J4| (1.13) where J4 is Cartan’s quartic E7 invariant [13, 14]. It may be written J4 = P ijQjkP klQli − P ijQijP klQkl ǫijklmnopQijQklQmnQop + ǫijklmnop P ijP klPmnP op . (1.14) where P ij and Qjk are 8× 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E7(C) ⊃ [SL(2, C)] 7 (1.15) under which (2, 2, 1, 2, 1, 1, 1) +(1, 2, 2, 1, 2, 1, 1) +(1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) +(1, 2, 1, 1, 1, 2, 2) +(2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state. |Ψ〉 = aABD|ABD〉 +bBCE |BCE〉 +cCDF |CDF 〉 +dDEG|DEG〉 +eEFA|EFA〉 +fFGB|FGB〉 +gGAC |GAC〉 (1.17) where A = 0, 1, so the Hilbert space has dimension 7.23 = 56. The a, b, c, d, e, f, g transform as a 56 of E7(C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1. Alternatively, we can use the Fano plane. See Figure 2. The Fano plane also corresponds to the multiplication table of the octonions3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle τ3(ABCDEFG) = 4|J4| (1.18) J4 ∼ a 4 + b4 + c4 + d4 + e4 + f 4 + g4+ 3Not the “split” octonions as was incorrectly stated in the published version of [4]. Figure 1: The E7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G rep- resents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement. 2[a2b2 + b2c2 + c2d2 + d2e2 + e2f 2 + f 2g2 + g2a2+ a2c2 + b2d2 + c2e2 + d2f 2 + e2g2 + f 2a2 + g2b2+ a2d2 + b2e2 + c2f 2 + d2g2 + e2a2 + f 2b2 + g2c2] +8[bcdf + cdeg + defa+ efgb+ fgac+ gabd+ abce] (1.19) where products like a4 = (ABD)(ABD)(ABD)(ABD) = ǫA1A2ǫB1B2ǫD1D4ǫA3A4ǫB3B4ǫD2D3aA1B1D1aA2B2D2aA3B3D3aA4B4D4 (1.20) exclude four individuals (here Charlie, Emma, Fred and George), products like a2b2 = (ABD)(ABD)(FGB)(FGB) = ǫA1A2ǫB1B3ǫD1D2ǫF3F4ǫG3G4ǫB2B4aA1B1D1aA2B2D2bF3G3B3bF4G4B4 (1.21) exclude two individuals (here Charlie and Emma), and products like abce = (ABD)(BCE)(CDF )(EFA) = ǫA1A4ǫB1B2ǫC2C3ǫD1D3ǫE2E4ǫF3F4aA1B1D1bB2C2E2cC3D3F3eE4F4A4 (1.22) exclude one individual (here George)4. Once again large non-BPS, small BPS and large BPS black holes correspond to states with J4 > 0, J4 = 0 and J4 < 0, respectively. 4This corrects the corresponding equation in the published version of [4] which had the wrong index contraction. Figure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point. 1.4 Magic supergravities in D = 4 The black holes described by Cayley’s hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)]3. In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z)× SO(l − 1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic). 2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z)× SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 +m) representation (m+ 6 electric plus m+ 6 magnetic). 3) N = 8 supergravity where the symmetry is the non-compact exceptional group E7(7)(Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic). In all three case there exist quartic invariants akin to Cayley’s hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]). However, as suggested by Levay [5], one might also consider the “magic” supergravi- ties [22, 23, 24]. These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO∗(12) and E7(−25), respectively. Once again, as has been shown just recently [20], in all cases there are quartic invariants whose square root yields the corresponding black hole entropy. Here we demonstrate that the black-hole/qubit correspondence does indeed continue to hold for magic supergravities. The crucial observation is that, although the black hole charges aABC are real (integer) numbers and the entropy (1.13) is invariant under E7(7)(Z), the coefficients aABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E7(C) which contains both E7(7)(Z) and E7(−25)(Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact E7(7)(Z) ⊃ [SL(2)(Z)] 7 (1.23) E7(−25)(Z) 6⊃ [SL(2)(Z)] 7 (1.24) is irrelevant. All that matters is that E7(C) ⊃ [SL(2)(C)] 7 (1.25) The same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E7(C) ⊃ SL(2, C)× SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R)× SO(10, 2). Moreover, SO(l−1, 2) is contained in SO(l+1, C) and SO(6, m) is contained in SO(12+ m,C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)]2n. 2 Five-dimensional supergravity In five dimensions we might consider: 1)N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l+1) representation (all elec- tric) . 2)N = 4 supergravity coupled tom vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m+5) representation (all electric). 3) N = 8 supergravity where the symmetry is the non-compact exceptional group E6(6)(Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric). The electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy. In this section we briefly describe the salient properties of maximal N = 8 case, following [16]. We have 27 abelian gauge fields which transform in the fundamental representation of E6(6). The first invariant of E6(6) is the cubic invariant [13, 17, 16, 18, 19] J3 = qijΩ jlqlmΩ mnqnpΩ pi (2.1) where qij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges qij is then given by S = π |J3| (2.2) We will see that a configuration with J3 6= 0 preserves 1/8 of the supersymmetries. If J3 = 0 and ∂J3 6= 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J3 = 0 (and the charge vector qi is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality. In five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Zab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as eab = s1 + s2 − s3 0 0 0 0 s1 + s3 − s2 0 0 0 0 s2 + s3 − s1 0 0 0 0 −(s1 + s2 + s3) (2.3) we can order si so that s1 ≥ s2 ≥ |s3|. The cubic invariant, in this basis, becomes J3 = s1s2s3 (2.4) Even though the eigenvalues si might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t, where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities J3 6= 0 s1, s2, s3 6= 0 J3 = 0, 6= 0 s1, s2 6= 0, s3 = 0 J3 = 0, = 0 s1 6= 0, s2, s3 = 0 (2.5) Taking the case of type II on T 5 we can choose the rotation in such a way that, for example, s1 corresponds to solitonic five-brane charge, s2 to fundamental string winding charge along some direction and s3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymme- tries. The respective orbits are E6(6) F4(4) E6(6) SO(5, 4)×T16 E6(6) SO(5, 5)×T16 (2.6) This also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions. In five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction. It is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E6 → SO(5, 5)× SO(1, 1), leading to 27 → 161 + 10−2 + 14 (2.7) The last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27)3 → 10−2 10−2 14 + 161 161 10−2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some “perturbative” charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification. 3 D = 5 black holes and qutrits So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits. 3.1 N = 2 black holes and the bipartite entanglement of two qutrits The two qutrit system (Alice and Bob) is described by the state |Ψ〉 = aAB|AB〉 where A = 0, 1, 2, so the Hilbert space has dimension 32 = 9. The aAB transforms as a (3, 3) under SL(3)A × SL(3)B. The bipartite entanglement is measured by the concurrence [21] C(AB) = 33/2|det aAB|. (3.1) The determinant is invariant under SL(3, C)A × SL(3, C)B and under a duality that inter- changes A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by S = π|det aAB| (3.2) 3.2 N = 8 black holes and the bipartite entanglement of three qutrits As we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E6(6) and the entropy is given by (2.2). The qutrit interpretation now relies on the decomposition E6(C) ⊃ SL(3, C)A × SL(3, C)B × SL(3, C)C (3.3) under which 27 → (3, 3, 1) + (3′, 1, 3) + (1, 3′, 3′) (3.4) suggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two: |Ψ〉 = aAB|AB〉+ b C |BC〉+ c CA|CA〉 (3.5) where A = 0, 1, 2, so the Hilbert space has dimension 3.32 = 27. The three states trans- forms as a pair of triplets under two of the SL(3)’s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a com- plex 27 of E6(C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence C(ABC) = 33/2|J3| (3.6) J3 being the singlet in 27× 27× 27: J3 ∼ a 3 + b3 + c3 + 6abc (3.7) Figure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. where the products a3 = ǫA1A2A3ǫB1B2B3aA1B1aA2B2aA3B3 (3.8) b3 = ǫB1B2B3ǫ C1C2C3bB1C1b C3 (3.9) c3 = ǫC1C2C3ǫA1A2A3c C1A1cC2A2cC3A3 (3.10) exclude one individual (Charlie, Alice, and Bob respectively), and the product abc = aABb CA (3.11) excludes none. 3.3 Magic supergravities in D = 5 Just as in four dimensions, one might also consider the “magic” supergravities [22, 23, 24]. These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU∗(6) and E6(−26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20]. Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges aAB are real (integer) numbers and the entropy (2.2) is invariant under E6(6)(Z), the coefficients aAB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E6(C) which contains both E6(6)(Z) and E6(−26)(Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that E6(6)(Z) ⊃ [SL(3)(Z)] 3 (3.12) E6(−26)(Z) 6⊃ [SL(3)(Z)] 3 (3.13) is irrelevant. All that matters is that E6(C) ⊃ [SL(3)(C)] 3 (3.14) The same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 +m,C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)]n. 4 Conclusions We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 33-dimensional three qutrit Hilbert space given by (3, 3, 3), but rather a direct sum of three 32-dimensional Hilbert spaces. It is, however, a subspace of the 73-dimensional three 7-dit Hilbert space given by (7, 7, 7). Consider the decomposition SL(7)A × SL(7)B × SL(7)C → SL(3)A × SL(3)B × SL(3)C under which (7, 7, 7) → (3′, 3′, 3′) + (3′, 3′, 3) + (3′, 3, 3′) + (3, 3′, 3′) + (3′, 3, 3) + (3, 3′, 3) + (3, 3, 3′) + (3, 3, 3) +(3′, 3′, 1) + (3′, 1, 3′) + (1, 3′, 3′) + (3′, 1, 3) + (3′, 3, 1) + (1, 3, 3′) +(3, 3, 1) + (3, 1, 3) + (1, 3, 3) + (3, 1, 3′) + (3, 3′, 1) + (1, 3′, 3) +(3′, 1, 1) + (1, 3′, 1) + (1, 1, 3′) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely (3′, 3, 1) + (3, 1, 3) + (1, 3′, 3′) So the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them. Nevertheless, just as the exceptional group E7 describes the tripartite entanglement of seven qubits [4, 5], we have seen is this paper that the exceptional group E6 describes the bipartite entanglement of three qutrits. In the E7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, whereas in the E6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole. Moreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive. 5 Acknowledgements MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1. References [1] M. J. Duff, “String triality, black hole entropy and Cayley’s hyperdeterminant,” Phys. Rev. D 76, 025017 (2007) [arXiv:hep-th/0601134]. [2] R. Kallosh and A. Linde, “Strings, black holes, and quantum information,” Phys. Rev. D 73, 104033 (2006) [arXiv:hep-th/0602061]. [3] P. Levay, “Stringy black holes and the geometry of entanglement,” Phys. Rev. D 74, 024030 (2006) [arXiv:hep-th/0603136]. [4] M. J. Duff and S. Ferrara, “E7 and the tripartite entanglement of seven qubits,” Phys. Rev. D 76, 025018 (2007) [arXiv:quant-ph/0609227]. [5] P. Levay, “Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane,” Phys. Rev. D 75, 024024 (2007) [arXiv:hep-th/0610314]. [6] M. J. Duff and S. Ferrara, “Black hole entropy and quantum information,” arXiv:hep- th/0612036. [7] V. Coffman, J. Kundu and W. Wooters, “Distributed entanglement,” Phys. Rev. A61 (2000) 52306, [arXiv:quant-ph/9907047]. [8] A. Miyake and M. Wadati, “Multipartite entanglement and hyperdeterminants,” Quant. Info. Comp. 2 (Special), 540-555 (2002) [arXiv:quant-ph/0212146]. [9] A. Cayley, “On the theory of linear transformations,” Camb. Math. J. 4 193-209,1845. [10] M. J. Duff, J. T. Liu and J. Rahmfeld, “Four-dimensional string-string-string triality,” Nucl. Phys. B 459, 125 (1996) [arXiv:hep-th/9508094]. [11] K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W. K. Wong, “STU black holes and string triality,” Phys. Rev. D 54, 6293 (1996) [arXiv:hep-th/9608059]. [12] D. M. Greenberger, M. Horne and A. Zeilinger, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, ed. M. Kafatos (Kluwer, Dordrecht, 1989) [13] E. Cartan, “Oeuvres completes”, (Editions du Centre National de la Recherche Scien- tifique, Paris, 1984). [14] E. Cremmer and B. Julia, “The SO(8) Supergravity,” Nucl. Phys. B 159, 141 (1979). [15] R. Kallosh and B. Kol, “E(7) Symmetric Area of the Black Hole Horizon,” Phys. Rev. D 53, 5344 (1996) [arXiv:hep-th/9602014]. [16] S. Ferrara and J. M. Maldacena, “Branes, central charges and U -duality invariant BPS conditions,” Class. Quant. Grav. 15, 749 (1998) [arXiv:hep-th/9706097]. [17] S. Ferrara and R. Kallosh, “Universality of Supersymmetric Attractors,” Phys. Rev. D 54, 1525 (1996) [arXiv:hep-th/9603090]. [18] S. Ferrara and M. Gunaydin, “Orbits of exceptional groups, duality and BPS states in string theory,” Int. J. Mod. Phys. A 13, 2075 (1998) [arXiv:hep-th/9708025]. [19] L. Andrianopoli, R. D’Auria and S. Ferrara, “Five dimensional U-duality, black-hole en- tropy and topological invariants,” Phys. Lett. B 411, 39 (1997) [arXiv:hep-th/9705024]. [20] S. Ferrara, E. G. Gimon and R. Kallosh, “Magic supergravities, N = 8 and black hole composites,” Phys. Rev. D 74, 125018 (2006) [arXiv:hep-th/0606211]. [21] Heng Fan, Keiji Matsumoto, Hiroshi Imai, “Quantify entanglement by concurrence hierarchy” J.Phys.A:Math.Gen 36 022317 (2003), quant-ph/0205126 [22] M. Gunaydin, G. Sierra and P. K. Townsend, “Gauging The D = 5 Maxwell-Einstein Supergravity Theories: More On Jordan Algebras,” Nucl. Phys. B 253, 573 (1985). [23] M. Gunaydin, G. Sierra and P. K. Townsend, “The Geometry Of N=2 Maxwell-Einstein Supergravity And Jordan Algebras,” Nucl. Phys. B 242, 244 (1984). [24] M. Gunaydin, G. Sierra and P. K. Townsend, “Exceptional Supergravity Theories And The Magic Square,” Phys. Lett. B 133, 72 (1983).

0704.0508

Invariance principle for additive functionals of Markov chains

INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS YURI N.KARTASHOV, ALEXEY M.KULIK Abstract. We consider a sequence of additive functionals {φn}, set on a sequence of Markov chains {Xn} that weakly converges to a Markov process X. We give sufficient condition for such a sequence to converge in distribution, formulated in terms of the characteristics of the additive functionals, and related to the Dynkin’s theorem on the convergence of W -functionals. As an application of the main theorem, the general sufficient condition for convergence of additive functionals in terms of transition probabilities of the chains Xn is proved. 1. Introduction Let a sequence of processes Xn = Xn(·) be given, converging in distribution (in some sense, e.g., in a sense of convergence of finite-dimensional distributions, distributions in spaces C or D, etc.) to a limit process X = X(·). Also let the family of functionals φn of the processes Xn be given. Assume that they are additive in an appropriate sense with respect to time variable. The general question, considered in the present paper, is what an information about the limit behavior of the distributions of functionals φn can be obtained in a situation where the processesXn, X possess certain Markov properties. The starting point in our considerations is provided by the comparatively simple, but important particular case of the problem outlined above, in which all the processes Xn coincide. In this situation, φn are a functionals of the same process X , and if X is Markov process and φn are W -functionals (see [1], Chapter 6), then their limit behavior, according to the well known theorem by E.B.Dynkin ([1], Theorem 6.4), is determined by the limit behavior of their characteristics (that is, their expectations). In the present paper we consider the processes Xn that differ one from another. The class of sequences of processes Xn, considered in the framework of our approach, contains sequences of Markov chains with appropriately normalized time, embedded into C or D (for example, by means of standard operations of linearization or construction of graduated processes), and weakly convergent to Markov process X . Important partial case is provided by random broken lines (or random step functions) Xn, constructed by a random walk in Rd and weakly convergent to a homogenous stable process X (particularly, to the Brownian motion). We show that, under some structural assumption about processes Xn, X (the condition is that the sequence {Xn} provides Markov approximation for the process X), the full analogue of the Dynkin’s theorem takes place: if the characteristics of functionals φn converge weakly to the characteristics of W -functional φ of the limit process X , then the distributions of φn converge to the distribution of φ. Our method of proof is based on L2-estimates for the distance between additive functionals, similar to those given in Lemma 6.5 [1]. The proof of these estimates is concerned with a preliminary construction of processes Xn, X on one probability space in such a way, that the functionals φn, φ, associated initially with a different processes, are interpreted as a functionals of one two-component process. The (some kind of) Markov property of the two-component process is essential for the estimates, analogous to those given in Lemma 6.5 [1]; the structural assumption mentioned above is just the claim for such a property to hold true in an appropriate form. The method, proposed by authors, allows one to reduce the problem of studying of asymptotic behavior of the distributions of additive functionals to a priori more simple problem of studying of their means. In our opinion, it provides a good addition to the available methods of studying the limit behavior of additive functionals both for the important partial case of random walks (we do not give the detailed review here, 2000 Mathematics Subject Classification. Primary 60J55; Secondary 60F17. Key words and phrases. additive functional, characteristics of additive functional, Markov approximation. http://arxiv.org/abs/0704.0508v1 2 YURI N.KARTASHOV, ALEXEY M.KULIK referring the reader to monographs [2],[3],[4], papers [5],[6] and reviews there), and for general Markov chains. Among the latter, it is necessary to mention the method that is based on the passing to the limit in the difference equations that describe characteristic functions of additive functionals of Markov chains, and ascends to the works of I.I.Gikhman at 50-ies (see [7],[8], also [9] and the survey paper [10]). The structure of the article is following. In Chapter 2, we introduce the notion of Markov approximation and give examples that illustrate it. In Chapter 3, the main theorem of the article is introduced and proved. In Chapters 4,5, the two elementary examples of application of this theorem are given. In Chapter 6, the main theorem is applied to the proof of a general sufficient condition for weak convergence of additive functionals, set on the sequence of Markov chains, that is formulated in terms of transition probabilities of the chains. 2. Markov approximation. Further we assume that the processes Xn, X are defined on R + and have a locally compact metric phase space (X, ρ). We say that the process X possesses the Markov property at the time moment s ∈ R+ w.r.t. filtration {Gt, t ∈ R+}, if X is adapted to this filtration and for each k ∈ N, t1, . . . , tk > s there exists a stochastic kernel {Pst1...tk(x,A), x ∈ X, A ∈ B(Xk)} such that (2.1) E[1IA((X(t1), . . . , X(tk)))|Gs] = Pst1...tk(X(s), A) a.s., A ∈ B(Xk). The measure Pst1...tk(x, ·) has a natural interpretation as the finite-dimensional distribution of X at the points t1, . . . , tk, conditioned by {X(s) = x}; we denote below Pst1...tk(x, ·) = P ((X(t1), . . . , X(tk)) ∈ ·|X(s) = x). Remark 1. In some cases, (2.1) implies the following functional analogue of (2.1): (2.2) E[1I·(X |∞s )|Gs] = E[1I·(X |∞s )|X(s)], where X |∞s denotes the trajectory of the process X on the time interval [s,+∞), considered as an element of appropriate functional space. For instance, if the Kolmogorov’s sufficient condition for existence of continuous modification holds true both for unconditional and conditional distributions of X , then (2.2) holds with X |∞s considered as an element of C([s,+∞),X). Everywhere below we assume that the process X possesses the Markov property w.r.t. its canonic filtration at every point s ∈ R+ and for the processesXn the same property holds true at every point of the type in , i ∈ Z+ (the choice of the denominator here is quite arbitrary; it is possible to put any expression N(n) → ∞, n→ ∞ instead of n, but we avoid to do this in order to shorten the notation). The next definition is introduced in [11]. Definition 1. The sequence {Xn} provides Markov approximation for the process X , if for arbitrary γ > 0, T < +∞ there exists K(γ, T ) ∈ N and a sequence of two-componential processes {Ŷn = (X̂n, X̂n)}, defined on another probability space, such that (i) X̂n =Xn, X̂ n d=X ; (ii) the process Ŷn, together with the processes X̂n, X̂ n, possesses the Markov property at the points iK(γ,T ) , i ∈ N w.r.t. filtration {F̂nt = σ(Ŷn(s), s ≤ t)}; (iii) lim sup  sup i≤ Tn K(γ,T ) iK(γ,T ) , X̂n iK(γ,T )  < γ. Remark 2. Condition (ii) implies that, for i, k ∈ N, t1, . . . , tk > iK(γ,T )n , (x, y) ∈ X 2, the marginal distributions (Ŷn(t1), . . . , Ŷn(tk)) ∈ ·|Ŷn( iK(γ,T )n ) = (x, y) are equal to P (Xn(t1), . . . , Xn(tk)) ∈ ·|Xn( iK(γ,T )n ) = x and P (X(t1), . . . , X(tk)) ∈ ·|X iK(γ,T ) respectively. Let us give some examples that illustrate Definition 1. INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 3 Example 1. Let {ξk} be a sequence of i.i.d random vectors in Rd with E‖ξk‖2+δRd < +∞ for some δ > 0. Assume {ξk} to have zero mean and identity for their covariance matrix. Let us introduce the sequence of processes Xn (”random broken lines”) on R (2.3) Xn(t) = Sk−1√ + (nt− k + 1) − Sk−1√ , t ∈ k − 1 , k ∈ N, where Sn = k=1 ξk. Then Xn converge by distribution in C(R +,Rd) to the Brownian motion X in Rd. It is shown in [11] that the sequence {Xn} provides Markov approximation for the process X (part I. of Theorem 1 [11]). On the other hand, in the same paper (part II. of the same Theorem) the following effect is revealed. Let us denote by K(γ, T ) the minimal constant K(γ, T ) such that there exists a process Ŷn satisfying conditions (i)-(iii) of Definition 1. Then, in all the cases except one trivial case ξk ∼ N (0, I), for each fixed T > 0 the convergence K(γ, T ) → +∞, γ → 0+ takes place. In other words, while the accuracy of approximation of the Brownian motion X by the random walk Xn becomes better (this accuracy is described by the parameter γ), the Markov properties of the pair of processes (X,Xn) necessarily become worse (these properties are characterized by K(γ, T )). Example 2. Let {ξk} be i.i.d random variables, belonging to the normal domain of attraction for α-stable distribution L, α ∈ (0, 2). By the definition, this means that α [Sn − an] ⇒ L, an = 0, α ∈ (0, 1) nEξ1, α ∈ (1, 2) n2E sin ξ1 , α = 1 ([12], Chapter XVII.5). In order to shorten the notation, we assume that an ≡ 0 and consider processes Xn on R+ of the type (2.4) Xn(t) = n αSk−1 + (nt− k + 1) αSk − n− αSk−1 , t ∈ k − 1 , k ∈ N. Then Xn converge by distribution in D(R +) to the homogeneous process with independent increments X in R, for which X(1)−X(0) d=L (we call such process a process an α-stable one). It is shown in [11] (Theorem 2) that the sequence {Xn} provides Markov approximation for the process X . Furthermore, in this situation, on the contrary to the previous example, K(γ, T ) = 1 for all γ, T . This means that, in this case, the Markov properties do not become worse while accuracy of approximation improves. Remark 3. The last example shows that the property of Markov approximation does imply, in general, the convergence of distributions of the processes Xn to the distribution of X in C = C(R +,X) even if Xn has continuous trajectories. The same can be said about convergence in D = D(R+,X) (we omit the corresponding example). Let us remark that the approach, introduced in the present paper, is closely related to the Skorokhod’s method of embedding of random walk into Wiener process by means of of appropriate sequence of stopping moments ([13]), widely used in literature. The basic idea is the same: we have to construct two processes on the same probability space, with the pair keeping Markov or martingale properties. However, the Skorokhod’s method, while being quite efficient for one-dimensional random walks that approximate Wiener process, is much less appropriate in a multi-dimensional situation or for stable domain of attraction. Examples 1 and 2 show that the claim for the Markov approximation to hold true is not restrictive, at least for all basic classes of random walks with no regard to the dimension of the phase space or to the type of limit distribution. The following example shows that the property of Markov approximation is ”stable” in the following sense. This property is preserved under construction of a new pair (Zn, Z) from the pair (Xn, X), possessing this property, in some regular way (e.g., as a solution of a family of stochastic equations). 4 YURI N.KARTASHOV, ALEXEY M.KULIK Example 3. Let Xn, X be as in Example 1, functions a : R m → Rm, b : Rd → Rd×m be Lipschitz and b∗(x)b(x) > 0, x ∈ Rm (the sign ∗ denotes the operation of taking of the adjoint matrix). Define (2.5) Zn k + 1 , Zn(0) = z, ) ≡ [Xn(k+1n )−Xn( )]. Then ([14], [15]) Zn converge by distribution in C(R +,Rm) to the process Z, defined by SDE (2.6) dZ(t) = a(Z(t))dt + b(Z(t))dX(t), Z(0) = z, where X is the Brownian motion in Rd. It is natural to call the sequence Zn the difference approximation of the diffusion process Z. Let us show that the sequence {Zn} provides Markov approximation for the process Z. For arbitrary γ, T , we construct a pair (X̂n, X̂ n), corresponding to processes Xn, X and satisfying conditions of Definition 1 (such construction is possible due to Example 1). Let us construct the processes Ẑn, Ẑ n as the functionals of the processes X̂n, X̂ n by equalities (2.5),(2.6) with Xn replaced by X̂n and X replaced by X̂ n (note that (2.6) has unique strong solution, hence this procedure is correct). By the construction, the pair (Ẑn, Ẑ n) satisfies condition (i) of Definition 1. It is easy to verify that the Markov condition (ii) for the pair (X̂n, X̂ n) holds in the functional form (2.2) with Ŷn|∞s considered as an element of C([s,+∞),Rd × Rd) (see Remark 1). Hence, the pair (Ẑn, Ẑn) also satisfies condition (ii) of Definition 1. Let us write ∆(γ) = lim sup  sup i≤ Tn K(γ,T ) iK(γ, T ) , Ẑn iK(γ, T ) and show that (2.7) ∆(γ) → 0+, γ → 0 + . Note that (2.7) immediately implies Markov approximation: for arbitrary δ > 0 we chose, using (2.7), γ = γ(δ) such that inequalities γ < δ and ∆(γ) < δ hold. Then the pair (Ẑn, Ẑ n), constructed by the scheme described above, satisfy Definition 1 with the constant γ replaced by δ (note that, under this construction, the value K(δ, T ) ≡ KZ(δ, T ) for the pair (Ẑn, Ẑn) is expressed through the same value for the pair (X̂n, X̂n) by KZ(δ, T ) = KX(γ(δ), T )). Now assume that (2.7) does not hold, then there exist constant c > 0 and sequence γk → 0+, nk → +∞ such that (2.8) K(γk, T ) → 0, P  sup i≤ Tnk K(γk,T ) iK(γk, T ) iK(γk, T )  > c. Consider the sequence of four-component processes (X̂nk , X̂ nk , Ẑnk , Ẑ nk). Every component of this sequence is weakly compact in C(R+,Rd) or C(R+,Rm), hence the whole sequence is also weakly compact in C(R+,Rd× d ×Rm ×Rm). Consider arbitrary limit point (X̂∗, X̂∗, Ẑ∗, Ẑ∗) (in a sense of convergence by distribution) of this sequence. It follows from (2.8) that (2.9) P (Z∗ 6= Z∗) > 0. It follows from Theorem 2.2 [15] (see also Chapter 9.5 [14]) that the processes Z∗, Z ∗ satisfy SDE (2.6) with X replaced by X∗, X ∗. However, the SDE (2.6) possesses the property of pathwise uniqueness (see [16]), and the property (iii) of the pair (Xnk , X̂ nk) implies that the processes X∗, X ∗ coincide a.s. Therefore, the processes Z∗, Z ∗ also coincide a.s., that contradicts to (2.9) and show that our assumption that ∆(γ) 6→ 0+, γ → 0+ is false. INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 5 The examples given above show that the claim for the Markov approximation to hold is not very restrictive, and is provided in a typical situations. On the other hand, this claim is strong enough to provide one the opportunity to obtain an analog of the Dynkin’s theorem; this will be shown in the next chapter. 3. Main theorem We consider the functionals of the type (3.1) φs,tn (Y ) k:s≤k/n<t k + 1 , . . . , Y k + L− 1 , 0 ≤ s < t, where the functions Fn(·) are nonnegative, L is a fixed integer. Together with the functionals φn, that are ”stepwise” functions w.r.t. every time variable, we consider random broken lines, related to these functions: n = φ n + (ns− j + 1)φ n + (nt− k + 1)φ n , s ∈ j − 1 , t ∈ k − 1 We interpret the random broken lines ψn as a random elements in space C(T,R +), where T = {(s, t)|0 ≤ s ≤ If process Y possesses Markov property w.r.t. the filtration, associated with this process, at the points of the type s = i , i ∈ Z+, then, for functional φn, its characteristic fn is naturally defined by the formula (3.2) f s,tn (x) = E[φs,tn (Y )|Y (s) = x], s = , i ∈ Z+, t > s, x ∈ X. Note that the functional (3.1) is a function of values of Y at finite number of time moments, thus the mean value in (3.2) is well defined as the integral over the family {Pst1...tk(x, ·), t1, . . . , tk > s, k ∈ N} of conditional finite-dimensional distributions of the process Y . The main result of this chapter is given in the following theorem. Theorem 1. Assume that there exist the sequence Xn that provides Markov approximation for the homoge- neous Markov process X and the sequence {φn ≡ φn(Xn)} of the functionals of the type (3.1). Let the following conditions hold true: (1) The functions Fn(·) are bounded and uniformly tend to zero: δ(Fn) = sup{Fn(x1, . . . , xL)|x1, . . . , xL ∈ X} → 0, n→ ∞. (2) There exists a function f , that appears to be a characteristics (in a sense of Chapter 6 [1]) of some W -functional φ = φ(X) of the limiting Markov process X, such that, for each T , ,t∈(s,T ) ∥∥f s,tn (·)− f s,t(·) ∥∥→ 0, n→ ∞, where ‖g(·)‖ ≡ sup |g(x)|. (3) The limiting function f is uniformly continuous with respect to variable x, that is, for arbitrary T 0≤s≤t<T ∣∣f s,t(x′)− f s,t(x′′) ∣∣→ 0, |x′ − x′′| → 0. ψn(Xn) ⇒ φ(X) ≡ {φs,t(X), (s, t) ∈ T}, where ψn are the random broken lines corresponding to the functionals φn and convergence is understood in a sense of C(T,R+). Remark 4. Conditions 1,2 are analogous to those of the Dynkin’s theorem: condition 2 is exactly the condition for the characteristics to converge, condition 1 corresponds to the assumption that the prelimit functionals are W -functionals. In the present situation, of course, we can not say that φn are W -functionals, particulary, φn are not continuous with respect to temporary variable. Condition 1 means exactly that the values of jumps 6 YURI N.KARTASHOV, ALEXEY M.KULIK are negligible while n→ ∞. Condition 3, though not very restrictive, is specific, and is caused by necessity to consider functionals, set over different processes. Remark 5. If Xn ⇒ X in C or in D (this condition is not provided by the conditions of the Theorem, see Remark 3), then, as one can easily see from the proof, (Xn, ψn(Xn)) ⇒ (X,φ(X)) in C × C(T,R+) or in D× C(T,R+), respectively. Note that the result of the theorem also holds for the Markov process X that is not homogeneous w.r.t. time variable; the claim for the limit Markov process to be homogeneous is imposed in order to shorten the notation only. This remark concerns also the most of the results stated below. Proof of the theorem. The general scheme of the proof is close to the one, proposed in [17] in order to prove the analogue of the Dynkin’s theorem for the family of functionals of a single Markov process, for which the properties of additivity, continuity and homogeneity may fail, but the violations become negligible while n→ ∞. First let us show that the finite-dimensional distributions of φn converge to the corresponding distributions of φ. Let the constants γ, T be fixed and X̂n, X̂ n be processes satisfying conditions (i)-(iii) of Definition 1 with these constants. For these processes, one can consider the functionals φn(X̂n), φ(X̂ n); obviously, their distributions and characteristics coincide with those for φn(Xn), φ(X). In order to shorten notation, we denote further φn = φn(X̂n), φ = φ(X̂ n),K = K(γ, T ),Ft = F̂nt ≡ σ(X̂n(s), X̂n(s), s ≤ t). It follows from the condition (iii) and the definition of characteristics that, for arbitrary t ∈ (3.3) E ,t|FKi n |FKi almost surely. Lemma 1. For 0 ≤ s ≤ t ≤ T , the following estimate holds: lim sup n (X̂n)− φs,t(X̂) ∥∥f0,T ∥∥G(f, γ, T ) + 4 ∥∥f0,T ∥∥2 , where G(f, γ, T ) = sup 0≤s≤t≤T,|x′−x′′|<γ |f s,t(x′)− f s,t(x′′)|. Proof. We will prove the statement of lemma for s = 0, t = T ; in general case the proof is exactly the same. Consider the partition of the axis R+ by points of the type Ki , i ∈ N. Denote Mn = [nTK ] + 1, (i−1)K ,( iK n , ∆̃ (i−1)K T , i = 1,Mn. We have that φ0,Tn − φ0,T ∆ni − ∆̃ni ∆ni ∆̃ j = Σ 1 + 2Σ where (∆ni ) (∆̃ni ) 2 − 2 ∆ni ∆̃ 1≤i<l≤Mn ∆ni ∆ 1≤i<j≤Mn ∆ni ∆̃ 1≤j<k≤Mn ∆̃nj ∆̃ 1≤j<i≤Mn ∆ni ∆̃ Let us estimate the expectations Σn1 ,Σ 2 separately. Since the increments ∆ i , ∆̃ i are non-negative, the first sum can be estimated by the sum of the first two terms: (3.4) Σn1 ≤ (∆ni ) (∆̃ni ) INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 7 The expectation of the first term in (3.4) can be estimated via the definition of φn: (∆ni ) 2 ≤ E i=1,Mn ) Mn∑ ∆ni ≤ Kδnf0,Tn X̂n(0) ≤ Kδn ∥∥f0,Tn ∥∥→ 0, n→ +∞, where δn ≡ δ(Fn). Convergence to zero of the expectation of the second term in (3.4) is provided by the arguments, analogous to those used in [1] Chapter 6: on the one hand, by the continuity of functional φ,∑Mn i=1(∆̃ 2 → 0 by probability; on the other hand, i=1(∆̃ 2 is dominated by the variable (φ0,T )2; the expectation of this variable, due to Lemma 6.4 [1], does not exceed 2 ∥∥f0,T ∥∥2 <∞. Therefore, E i=1(∆̃ 0 due to the Lebesgue theorem on dominated convergence. Hence, lim sup EΣn1 ≤ 0. The expectation of Σn2 is equal EΣn2 = E 1≤i<l≤Mn ∆ni ∆ 1≤i<j≤Mn ∆ni ∆̃ 1≤j<k≤Mn ∆̃nj ∆̃ 1≤j<i≤Mn ∆ni ∆̃ (3.5) = E Mn−1∑ n − φ Mn−1∑ n − φ We estimate the second term in (3.5), using property (3.3). Since ∆̃ni is measurable w.r.t. FKi , the following estimate holds: Mn−1∑ n − φ Mn−1∑ ∆̃ni E n − φ Mn−1∑ − f Kin ,T Mn−1∑ n − f ∣∣∣+ E Mn−1∑ ∣∣∣∣f − f Kin ,T ))∣∣∣∣ ≤ (3.6) ≤ ‖f0,T‖ sup ,t∈(s,T ) ∥∥f s,tn (·)− f s,t(·) ∥∥+ E Mn−1∑ ∣∣∣∣f − f Kin ,T ))∣∣∣∣ (in the last inequality, we have used that ∑Mn−1 i=1 ∆̃ i ≤ φ0,T and Eφ0,T ≤ ‖f0,T‖). The first term in (3.6) tends to zero. In order to estimate the second term, we put Ωγ,T = i≤ Tn , X̂n (recall that P (Ωγ,T ) < γ due to the claim (iii) of Definition 1). We have Mn−1∑ ∣∣∣∣f − f Kin ,T ))∣∣∣∣ ≤ Eφ G(f, γ, T )1IΩ\Ωγ,T+ (3.7) + E Mn−1∑ ∣∣∣∣f − f Kin ,T ))∣∣∣∣ 1IΩγ,T . The first term in (3.7) can be estimated by ‖f0,T‖G(f, γ, T ). The second term is estimated by Cauchy inequality: Mn−1∑ ∣∣∣∣f − f Kin ,T ))∣∣∣∣ 1IΩγ,T ≤ ∥∥f0,T ∥∥Eφ0,T 1IΩγ,T ≤ ∥∥f0,T ∥∥ [E(φ0,T )2 2 [P (Ωγ,T )] ∥∥f0,T ∥∥2√2γ 8 YURI N.KARTASHOV, ALEXEY M.KULIK (here, the Lemma 6.4 [1] was applied). Summing up the above relations, we deduce that (3.8) lim sup Mn−1∑ n − φ ∥∥f0,T ∥∥G(f, γ, T ) + ∥∥f0,T ∥∥2√2γ. Now, let us proceed with the estimation of the first item in (3.5). Straightforward use of the property (3.3) is impossible here, since the variable ∆ni is a functional of values of the process X̂n at the points , Ki+1 , . . . Ki+L , that is, it is not measurable with respect to FKi . Without loss of generality, one can assume that K ≥ L (otherwise one can make the same procedure with the constant K replaced by K · L). Then the variable ∆ni is measurable with respect to FK(i+1) . The functionals φn, φ are additive at points of the type j . Applying (3.3) and condition 1 of the Theorem, we obtain the following relation Mn−1∑ n − φ Mn−1∑ K(i+1) n − φ K(i+1) Mn−1∑ K(i+1) K(i+ 1) K(i+1) K(i+ 1) (3.9) ≤ Kδn ∣∣f0,Tn ∣∣+ E Mn−1∑ K(i+1) K(i+ 1) K(i+1) K(i+ 1) The first term in (3.9) tends to zero. The second term in (3.9) is estimated in the same way with the second term in (3.5), with one necessary change. We cannot apply Lemma 6.4 [1] in order to estimate the second moment φ0,Tn , therefore this estimate must be obtained separately. This can be done in a following way: E(φ0,Tn ) 2 = E (∆ni ) 2 + 2E 1≤i<j≤Mn ∆ni ∆ j = E (∆ni ) 2 + 2E 1≤i≤Mn ∆ni φ (∆ni ) 2 + 2E 1≤i≤Mn ∆ni [φ iK/n,(i+1)K/n n + φ (i+1)K/n,T n ] ≤ (∆ni ) 2 + 2KδnE 1≤i≤Mn ∆ni + 2E 1≤i≤Mn ∥∥f0,Tn (3.10) ≤ (2K + 1)δn + 2‖f0,Tn ‖ n ≤ (2K + 1)δn ∥∥f0,Tn ∥∥+ 2 ∥∥f0,Tn ∥∥2 , all transitions here are analogous to those given above, and thus are not discussed in details. Repeating literally the estimates for the second term in (3.5), we obtain the estimate (3.11) lim sup Mn−1∑ n − φ ∥∥f0,T ∥∥G(f, γ, T ) + ∥∥f0,T ∥∥2√2γ. It follows from (3.8),(3.11) that lim sup [2Σn2 ] ≤ 4 ∥∥f0,T ∥∥G(f, γ, T ) + 4 ∥∥f0,T ∥∥2. This, combined with the estimate lim sup [Σn1 ] ≤ 0 proved before, provides the needed statement. The lemma is proved. Now, we can complete the proof of the convergence of finite-dimensional distributions of φn to those of φ. In order to shorten notation we consider the one-dimensional distributions only; in general case considerations are completely the same. Take arbitrary s, t, s < t. In order to prove weak convergence φs,tn (Xn) to φ s,t(X), it is sufficient to show that, for arbitrary bounded Lipschitz function g, (3.12) lim sup ∣∣Eg(φs,tn (Xn))− Eg(φs,t(X)) ∣∣ = 0. INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 9 Let g be fixed, consider a pair of processes X̂n, X̂ n, corresponding (in a sence of Definition 1) to T = t and given positive γ. By construction, φs,tn (Xn) =φs,tn (X̂n), φ s,t(X) =φs,t(X̂n). Applying Lemma 1, we obtain lim sup ∣∣Eg(φs,tn (Xn))− Eg(φs,t(X)) ∣∣ ≤ lim sup ∣∣∣g(φs,tn (X̂n))− φs,t(X̂n) ∣∣∣ ≤ ≤ Lip(g) lim sup ∣∣∣φs,tn − φs,t )∣∣∣ ≤ 2Lip(g) ‖f0,t‖G(f, γ, t) + 2γ ‖f0,t‖2, here Lip(g) denotes the Lipshits constant for g. Condition 3 of the Theorem provides that G(f, γ, t) → 0, γ → 0+. Therefore, since γ > 0 is arbitrary, (3.12) follows from the estimate given above. Since sups,t |ψs,tn − φs,tn | ≤ δn → 0, the finite-dimensional distributions of φn converge to corresponding distributions of φ. Thus, the only thing left to show in order to prove the Theorem, is that the family of distributions of ψn is dense in C(T,R +). The values of the functions ψn at the point s, t differ from the values at the closest knots of partition s∗, t∗ ∈ 1nZ+ at most on δn, and ψn are monotone as the functions of the time variables. Hence, in order to prove the required statement, it is sufficient to show that, for arbitrary sequence of partitions Sn = {sn0 = 0 < sn1 < · · · < snk < . . . } ⊂ 1nZ+, n ∈ N with σn ≡ maxk(snk −snk−1) → 0, n→ +∞ and arbitrary T ∈ R+, k:sk≤T snk−1,s → 0, n→ +∞. Set γn,T = sup0<t−s<σn,t<T ‖f n ‖, note, that γn,T → 0, n→ +∞ due to continuity of the limit characteristics f and uniform convergence of fn ⇒ f . In the same way with (3.10) we obtain the estimate (3.13) E snk−1,s ≤ {(2K + 1)δn + 2γn,T }Eφ snk−1,s Summing up the estimates (3.13) w.r.t. k (recall that φs,tn = ψ n when s, t ∈ 1nZ+), we obtain k:sk≤T snk−1,s ≤ {(2K + 1)δn + 2γn,T } ‖f0,Tn ‖ → 0, n→ +∞, what was to be proved. The theorem is proved. Let us make one remark. For the random walks, the Skorokhod’s method is well known, allowing one to reduce the investigation of the sums of the type (3.1) to the case L = 1. This method can be applied in the context of current paper, also. Namely, the reasoning, similar to the one used in the proof of Theorem 1, Chapter 5.3 [2], provides the following result (the proof is omitted). Proposition 1. Let the sequence of functionals {φn = φn(Xn)} of the type (3.1) be given, and, for every n, the process Xn possesses the Markov property at the time moments , i ∈ Z+. Consider the functionals n (Xn) k:s≤k/n<t , 0 ≤ s < t, where Ψn,k(x) ≡ E k + 1 , . . . , Xn k + L− 1 )) ∣∣∣Xn , x ∈ X. Let functions Fn(·) be non-negative and satisfy condition 1 of Theorem 1, then the functionals φs,tn have a limit distribution if and only if the functionals χs,tn have a limit distribution, and the limit distributions of the functionals φs,tn , χ n are equal as soon as they exist. It is worth to note that the Proposition 1 does not lead to simplification of the initial problem in the context of current paper. The number of values of process Xn, contained in a one summand for the functional φn (that is, number L), is not involved significantly into the proof of the main theorem. We will see later that the main problem in the application of the Theorem consists in verification of the condition 2 of uniform convergence of characteristics; the characteristics of the functionals φn and χn, obviously, coincide. In the following two chapters, the examples of application of Theorem 1 are given. 10 YURI N.KARTASHOV, ALEXEY M.KULIK 4. The local time of a random walk at a point. Let the processesXn be constructed w.r.t. one-dimensional random walk that belongs to the normal domain of attraction of an α-stable law, α ∈ (1, 2] (see Examples 1,2). We assume the centering sequence an to be equal to zero, and set the random broken lines Xn by equality (2.4). Consider, for arbitrary z∗ ∈ R, the functionals φn = φn(Xn) of the type (3.1) with L = 2, Fn(x, y) = |y−x| 1I(x−z∗)(y−z∗)<0 + (1Ix 6=z∗,y=z∗ + 1Ix=z∗,y 6=z∗) . For every s < t, s, t ∈ { j , j ∈ Z+}, with probability 1 the following equality takes place (4.1) φs,tn (Xn) = lim 1IXn(r)∈(z∗−ε,z∗+ε)\{z∗} dr, 0 ≤ s < t. Therefore the functionals φn can be naturally interpreted as the censored local times for the broken lines Xn at the point z∗ (the censoring operation consists in removing horizontal parts of the broken lines). Theorem 3.1 allows one to obtain the following limit result. Proposition 2. Let the distribution of the jump ξ1 of the random walk be concentrated on Z and aperiodic. Then the conditions of Theorem 1 hold true and φs,tn (Xn) converge by distribution to φ s,t(X) = P (ξ1 6= 0) · Ls,t(X, z∗), where L(X, z∗) is the local time of the limit α-stable process X at the point z∗. Proof. The condition for Xn to provide Markov approximation for X holds true (see Example 2). Condition 1 of the Theorem holds with δn = 2n −1 since either the increment of the process Xn in the neighboring knots is equal to zero or the absolute value of this increment is not less then n− α . Let us show that the characteristics of functionals φn converge uniformly to the function (4.2) f s,t(x) = P (ξ1 6= 0) ∫ t−s pr(z∗ − x) dr, where pr(·) is the density of distribution X(r) under condition X(0) = 0; this provides conditions 2,3 of the Theorem. In order to shorten notation we take z∗ = 0. Denote P i = P (Sk = i), Pj = P j = P (ξ1 = j), i, j ∈ Z. We have that f s,tn (x) = n j 6=0 i∈(xn α −j,xn P ki + + P k notation i ∈ (a, b) in the case a > b means that b < i < a. Using the appropriate version of the Gnedenko’s local limit theorem (see [18], Theorem 4.2.1), one can write (4.3) εk ≡ sup ∣∣∣∣k αP ki − p1 )∣∣∣∣→ 0, k → +∞. Hence f s,tn (x) = j 6=0 i∈(xn α −j,xn (4.4) + α − j + Ξn(x), where (4.5) |Ξn(x)| ≤ [nt]∑ and Ξn ⇒ 0, n→ +∞ via the Toeplitz’s theorem. INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 11 The density p1 is uniformly continuous over R, hence, using the same arguments, one can show that, up to a summand that uniformly converges to zero, the value of f s,tn (x) equals j 6=0 i∈(xn α −j,xn (4.6) = P (ξ1 6= 0) P (ξ1 6= 0) in the latter equality, we have used that the process X is self-similar, that is, pr(x) = r α p1(r αx), r > 0. The sum in the right hand part of (4.6) is exactly the integral sum for the integral in the right hand part of (4.2), the functions {pr(·), r ≥ r0} are uniformly continuous for arbitrary r0 > 0 and supx pr(x) ≤ Cr− α . This immediately provides the required uniform convergence of fn to f . The proposition is proved. The similar result can be proved for ξk with non-lattice distribution, for which there exists a bounded distribution density of Sn0 for some n0 (the proof is omitted). The result of Proposition 2 and its analog for non-lattice random walks is not essentially new; one can obtain it applying either Proposition 1 and the technique, exposed in §§III.2, III.3 [3], or the reasonings, similar to those used in the proof of Theorem 3 [9]. Our reason to give this example consists, on the one hand, in describing the way of application of Theorem 1 in a simple situation where an appropriate local limit theorem is available, and on the other hand, in emphasizing the following interesting fact, that is not reflected in a literature available for us. For a ”good” random walks (lattice or essentially non-lattice), their local times at the point, defined by the natural equality (4.1), converge by distribution exactly to the local time of the limit process at the same point, as soon as the broken lines corresponding to the random walk does not contain horizontal sections. 5. Difference approximations of diffusion processes. Consider the sequence {Zn} of difference approximations of diffusion process Z (see Example 3, equalities (2.5),(2.6)). The sequence {Zn} provides Markov approximation for Z, that allows one to apply Theorem 1 while considering the question on the limit behavior of the functionals of type (3.1) for {Zn}. One of possible way to proceed here is to apply the estimates based on an appropriate local limit theorem, like it was made in the previous chapter. In order to make this paper reasonably short, we do not give the detailed exposition of this subject here (see the separate paper [19]). In this chapter, we give a simple corollary of Theorem 1, that provides invariance principle for certain ”canonic” additive functionals, that are related to the Doob’s decomposition of |Zn(·)|. Let us consider the objects introduced in Example 3 with m = d = 1 and a, b, {ξn} satisfying conditions introduced there. Put (5.1) φs,tn (Zn) ≡ k∈(sn,tn] Zn( k−1n )Zn( Zn( k−1n )=0 ψn are corresponding broken lines. Proposition 3. The processes ψn converge by distribution in C(T,R) to the local time φs,t ≡ lim 1I|Z(r)|<εb 2(Z(r)) dr of the diffusion process Z at the point 0. Proof. Since the diffusion coefficient is non-degenerate, Z possesses continuous transition density pt(x, y) and the standard estimate supx pt(x, y) ≤ C(y)√ holds true. This implies existence of the local time of Z at 12 YURI N.KARTASHOV, ALEXEY M.KULIK the point 0. This local time is a W -functional with the characteristics f0,t(x) = b2(0) ps(x, 0) ds, that is, condition 3 of Theorem 1 holds. Straightforward calculations prove the equality (5.2) |Zn(t)| − |Zn(s)| = φ0,tn (Zn) + [nt]−1∑ where s ∈ 1 Z+, sign (0) = 0. This provides that f s,tn (x) = E [|Zn(t)||Zn(s) = x]− |x| − [nt]−1∑ sign (Zn ) ∣∣∣Zn(s) = x Processes Zn converge weakly to Z, function a(x)sign (x) has unique jump at point x = 0 and P (Z(r) = 0) = 0 for every r > 0. Hence the standard reasonings provide that (we omit the details) (5.3) f s,tn (x)⇒ E [|Z(t)||Z(s) = x]− |x| − E a(Zr)sign (Zr) dr ∣∣∣Z(s) = x This proves condition 2 of Theorem 1, since the right hand side of (5.3) is exactly the characteristics of the local time φ due to Ito-Tanaka formula. In order to provide condition 1, let us, for a while, suppose additionally that the coefficients a, b are bounded. We apply the standard ”cutting” procedure: on each step of approximation, together with the process Zn, we consider the process Z̃n, constructed by the same scheme from a sequence of i.i.d.r.v. {ξ̃n}, satisfying conditions ‖ξ̃n‖ ≤ n 2 and ξn = ξ̃n for ‖ξn‖ ≤ n 2 . For such Z̃n, condition 1 of theorem holds with δ(Fn) ≤ n−1 max |a(x)| + n |b(x)|, and the other conditions of theorem for Z̃n remain to hold true. This proves the statement of Proposition 3 for {Z̃n}. On the other hand, for arbitrary T ∈ R+ Zn|[0,T ] 6= Z̃n|[0,T ] 1− 2+δ = o(1), n→ +∞, and therefore the statement of Proposition 3 holds true for {Zn}. At last, the additional assumption that the coefficients a, b are bounded, can be removed via a standard localization procedure. The proposition is proved. Remark 6. Let a = 0, b = 1, P (ξk = ±1) = 12 (that is, Zn corresponds to the Bernoulli’s random walk), then functional (5.1) can be represented at the form (5.4) φ̃s,tn = # {k ∈ [sn, tn) : Zn(k) = 0} . The functional (5.4) is widely used in a literature as the difference analogue of the local time at the point zero for lattice random walks. Proposition 3 shows that the functional (5.1) is a natural difference analogue of the local time both for random walks and, more generally, for difference approximations of diffusion processes without any restrictions on the distribution of the sequence {ξk}. 6. Invariance principle for additive functionals of Markov chains In previous two chapters we have considered more or less particular examples illustrating possible ways to provide the main condition of Theorem 1 (condition 2). In this chapter we introduce general sufficient condition of weak convergence of additive functionals, constructed on the sequence of Markov chains, that is formulated in terms of the transition probabilities of these chains and the functions Fn involved in representation (3.1). This condition is obtained as an application of Theorem 1, and the main assumption here is that the local limit theorem (condition 4 of Theorem 2 below) takes place in an appropriate form. For recurrent Markov chains this condition, together with a natural condition of weak convergence of ”symbols” of additive functionals (exact formulation is given below), is sufficient for convergence of characteristics, and the estimates here are INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 13 similar to (4.4) – (4.6) (see Theorem 3 below). For transient chains these estimates are not powerful enough, since in this case the estimate (4.5) does not provide that Ξn is negligible. One possible way to overcome this difficultly is to apply a more strong version of local limit theorem, for instance, to claim explicitly the rate of convergence εk → 0 in (4.3). Such an approach restricts the range of possible applications, therefore we introduce another one, that is concerned with a uniform condition on the modulus of continuity of processes Xn (condition 5 of Theorem 2) and a ”dimensional” condition on the symbols of functionals (condition 6), adjusted one with another with an appropriate way (condition 7). We assume that a σ-finite measures ν, νn on X are given such that P (X(t) ∈ dy|X(s) = x) = pt−s(x, y)ν(dy), 0 ≤ s < t, x, y ∈ X, ∈ dy|Xn = pn,k(x, y)νn(dy), i ∈ Z+, k ∈ N, x, y ∈ X. The measurable functions pt, pn,k are interpreted as the transition probability densities for X,Xn w.r.t. mea- sures ν, νn. We assume the W -functional φ = φ(X) with the characteristics f to be given. It is known (see [1], Chapter 6) that s,t = L2 − lim 0,ε(X(r)) dr, and therefore f s,t(x) = lim pr(x, y) f0,ε(y)ν(dy) dr. We assume that, as ε→ 0+, the measures 1 f0,εdν converge weakly (i.e., on every bounded continuous function) to a finite measure µ, the characteristics f can be represented in the form (6.1) f s,t(x) = ∫ t−s pr(x, y)µ(dy) dr, and pr(x, y)µ(dy) dr < +∞, T ∈ R+. We also consider the sequence of the functionals φn = φn(Xn) of the type (3.1) with L = 1 and Fn = (the case L > 1 can be considered similarly and we omit it in order to shorten notation). The characteristics of φn has the form f s,tn (x) = pn,k(x, y)µn(dy), 0 ≤ s < t, x ∈ X, where µn(dy) ≡ gn(y)νn(dy) are the ”symbols” of the functionals φn. Theorem 2. Assume the following conditions to hold true. (1) Trajectories of the processes Xn are continuous, and the sequence {Xn} possesses Markov approxima- tion of X. (2) 1 supx gn(x) → 0, n→ +∞. (3) For arbitrary t0 > 0, the function (t, x, y) 7→ pt(x, y) is uniformly continuous on [t0,+∞) × X2, and for arbitrary y ∈ X x 6∈B(y,R) pt(x, y) → 0, R → +∞ (here and below B(x,R) ≡ {x ∈ X|ρ(x, y) < R}). Furthermore, there exist constants γ > 0, Cγ > 0 such that x,y∈X pt(x, y) ≤ Cγt−γ , t > 0. (4) There exist sequences {αn}, {βn} ⊂ R+ tending to zero, such that x,y∈X |pn,k(x, y)− p k (x, y)| ≤ (αn + βk) , n, k ∈ N. 14 YURI N.KARTASHOV, ALEXEY M.KULIK (5) There exist constants δ > 0, Cδ > 0 such that, for arbitrary T > 0, x∈X,n∈N t,s∈[0,T ],|t−s|≥ 1 ρ(Xn(t), Xn(s)) |t− s|δ ]Cδ ∣∣∣X(0) = x  < +∞. (6) Measures µn are finite and converge weakly to measure µ. There exist constants θ > 0, Cθ, cθ > 0 such µn(B(x,R)) ≤ CθRθ, x ∈ X, n ∈ N, R > cθn−δ (note that the latter condition provides that µ(B(x,R)) ≤ CθRθ, x ∈ X, R > 0). (7) The constants γ, δ, θ, Cδ satisfy the relations δθ + 1 > γ, Cδ > 2θ + 2. Then (Xn, ψn(Xn)) ⇒ (X,φ(X)) in a sense of convergence in distribution in C(R+,X)×C(T,R+) (ψn are the random broken lines corresponding to the functionals φn). Proof. In order to prove the Theorem, it is sufficient to show that, for every T ∈ R+, (6.2) f s,tn (x) ⇒ s≤t≤T,x∈X f s,t(x), n→ +∞. Indeed, the sequence {Xn} provides Markov approximation for X (condition 1), and condition 1 of Theorem 1 is provides by condition 2 of Theorem 2. Having (6.2) proved, we provide condition 2 of Theorem 1. Condition 3 of this theorem is provided by (6.1) and uniform continuity of the density p. At last, condition 5 of Theorem 2 provides weak convergence of Xn to X in C(R +,X), that allows one to apply Theorem 1 and Remark 5. Before proving (6.2), let us make some auxiliary estimates. Denote δ,n(Xn) = sup v,w∈[s,t],|v−w|≥ 1 ρ(Xn(v), Xn(w)) |v − w|δ , n,A = Xn(r) ∈ B(Xn(s), A(r − s)δ), r ∈ note that {Hs,tδ,n(Xn) < A} ⊂ D n,A. Also denote α = maxn αn, β = maxk βk, δn = supx |gn(x)| , B1 = maxn δn, B2(T ) = Cθ(Cγ+α+β) 1+δθ−γ T 1+δθ−γ. For arbitrary A > cθ, T ∈ R+, consider the functionals φs,tn,A = φs,tn 1IHs,t (Xn)<A , s ≤ t ≤ T . Lemma 2. 1. E n,A|Xn(s) = x ≤ B1 +B2(T )Aθ. |Xn(s) = x ≤ 3B1(B1 +B2(T )Aθ) + 2(B1 +B2(T )Aθ)2. 3. Let p ∈ 1, 2Cδ−2 Cδ+2θ (recall that 1 < 2Cδ−2 Cδ+2θ due to condition 7 of the Theorem). Then x∈X,n∈N,s≤t≤T )p |Xn(s) = x < +∞. Proof. Using condition 4 of the Theorem and then condition 6, we obtain, for t, s ∈ 1 Z+, the estimate n,A|Xn(s) = x φs,tn 1IDs,t |Xn(s) = x gn(x) n(t−s)−1∑ x,A( kn ) pn,k(x, y)µn(dy) ≤ ≤ δn + Cγ + α+ β n(t−s)−1∑ ≤ δn + Cγ + α+ β n(t−s)−1∑ )γ−δθ INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 15 that immediately proves the first statement of the Lemma. The second statement can be obtained from the first one via the estimate similar to (3.10) with the use of the inequality 1IHs,t (Xn)<A ≤ 1IHs,r (Xn)<A1IHr,t (Xn)<A that holds true for arbitrary r ∈ (s, t). Applying statement 2 and Hölder inequality we obtain φs,tn )p |Xn(s) = x φs,tn (Xn)∈[N−1,N)|Xn(s) = x (φs,tn ) 21IHT (Xn)<N |Xn(s) = x P (HTδ,n(Xn) ≥ N − 1) ] 2−p B3(T ) +B4(T )N 2 ·B5(T ) [(N − 1) ∨ 1]− here and below Bi(T ), i = 3, 4, . . . denotes a constant, that can be expressed explicitly through T and the constants introduced in the formulation of the Theorem, but an explicit expression is not needed in our consideration. Since θp− 2−p Cδ < −1 by the choice of p, this proves the statement 3. The lemma is proved. Let us proceed with the proof of (6.2). Choose non-increasing Lipschitz function Ψ : R+ → [0, 1] such that Ψ([0, 1]) = {1},Ψ([2,+∞)) = {0}, and set Ψr(x, y) = Ψ(r −1 · ρ(x, y)), r > 0, x, y ∈ X, Ψ0 ≡ 1. Note that, for arbitrary r0 > 0, the function (r, x, y) 7→ Ψr(x, y) is uniformly continuous on [r0,+∞)× X2. For fixed s ≤ t ≤ T,A ∈ R+ we decompose φs,tn as φs,tn = η n,A + ζ n,A, where n,A = A( kn−s) Xn(s), Xn We have that, on the set D n,A, for k such that s ≤ kn < t, Xn(0), Xn A( kn−s) Xn(s), Xn hence {φs,tn = η n,A} ⊃ D n,A and (6.3) {ζs,tn,A 6= 0} ⊂ Ω\D n,A ⊂ {H δ,n ≥ A}. Let p be the same as in statement 3 of Lemma 2. Then it follows from (6.3) and inequality 0 ≤ ζs,tn,A ≤ φs,tn (6.4) E n,A|Xn(s) = x (φs,tn ) p|Xn(s) = x δ,n ≥ A|Xn(s) = x) ] p−1 p ≤ B6(T )A−δ Similarly, one can write φs,t = η A + ζ A , where η ΨA(r−s)δ(X(s), X(r))dφ s,r , (6.5) E A |X(s) = x ≤ B6(T )A−δ We have ∣∣∣E n,A|Xn(s) = x A |X(s) = x ]∣∣∣ = ∣∣∣∣∣∣ gn(x) ]n(t−s)[−1∑ pk,n(x, y)ΨA( kn ) δ (x, y)µn(dy)− ∫ t−s pr(x, y)ΨArδ(x, y)µ(dy) dr ∣∣∣∣∣∣ ≤ δn +∆1n(x,A, s, t) + ∆2n(x,A, s, t) + ∆3n(x,A, s, t), 16 YURI N.KARTASHOV, ALEXEY M.KULIK where ]z[≡ min{N ∈ Z, N ≥ z}, ∆1n(x,A, s, t) = ∣∣∣∣∣∣ ]n(t−s)[−1∑ [pk,n(x, y)− p k (x, y)]Ψ A( kn ) δ (x, y)µn(dy) ∣∣∣∣∣∣ ∆2n(x,A, s, t) = ∣∣∣∣∣∣ ]n(t−s)[−1∑ (x, y)Ψ A( kn) δ(x, y)µn(dy)− ∫ t−s pr(x, y)ΨArδ(x, y)µn(dy) dr ∣∣∣∣∣∣ ∆3n(x,A, s, t) = ∫ t−s pr(x, y)ΨArδ (x, y)[µn(dy)− µ(dy)] dr ∣∣∣∣ . Denote ∆in(A, T ) = supx∈X,s≤t≤T ∆ n(x,A, s, t), i = 1, 2, 3. Since Ψr(x, y) ∈ [0, 1] and {Ψr(x, y) 6= 0} ⊂ {y ∈ B(x, 2r)}, ∆1n(A, T ) ≤ ]nT [−1∑ [αn + βk] x, 2A (6.6) ≤ Cθ(2A)θ · ]nT [−1∑ [αn + βk] )δθ−γ → 0, n→ +∞ by Toeplitz theorem. The function (r, x, y) 7→ pr(x, y)Ψr(x, y) is uniformly continuous over [r0,+∞)×X2 for any r0 > 0, therefore an estimate analogous to (6.6) provides that x∈X,s≤t≤T ∣∣∣∣∣∣ ]n(t−s)[−1∑ k=[r0n]+1 (x, y)Ψ A( kn) δ (x, y)µn(dy)− ∫ t−s pr(x, y)ΨArδ(x, y)µn(dy) dr ∣∣∣∣∣∣ (note that maxn µn(X) < +∞ since µn weakly converge to µ). The same arguments provide that lim sup ∆2n(A, T ) ≤ ≤ lim sup [r0n]∑  = B7(A, T )(r0)δθ−γ+1. Since r0 > 0 is arbitrary, this implies that (6.7) ∆2n(A, T ) → 0, n→ +∞. At last, the weak convergence of µn to µ and the first part of condition 3 provide that, for every t, In(A, t) ≡ sup pt(x, y)ΨArδ(x, y)[µn(dy)− µ(dy)] ∣∣∣∣→ 0, n→ +∞. Since In(A, t) ≤ Cγt−γ · Cθ(2Atδ)θ, the Lebesgue theorem of dominated convergence provides that (6.8) ∆3n(A, T ) → 0, n→ +∞. It follows from the estimates (6.4) – (6.8) that lim sup x∈X,s≤t≤T ∣∣f s,tn (x)− f s,t(x) ∣∣ ≤ 2B6(T )A−δ p , A > cθ. Taking A→ +∞ we obtain (6.2), that completes the proof. The theorem is proved. In order to make our exposition complete, let us formulate a version of Theorem 2 for the recurrent case. Theorem 3. Let conditions 1 – 4 of Theorem 2 hold true and γ < 1. Also let µn converge weakly to µ, and Xn converge to X by distribution in C(R +,X). Then (Xn, ψn(Xn)) ⇒ (X,φ(X)) in a sense of convergence in distribution in C(R+,X)× C(T,R+). INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS 17 The proof, with slight changes, repeats the proof of Theorem 2, and is omitted. Note that, under conditions of Theorem 3, the convergence of finite-dimensional distributions of φn can be provided with the use of the technique, mentioned in the Introduction, that was proposed by I.I.Gikhman and is based on studying of limit behavior of difference equations for characteristic functions of φs,tn (see for instance the proof of Theorem 3 [9]). In the transient case, treated in Theorem 2, this technique can not be applied since the uniform estimates, analogous to (4.4) – (4.6), are not available in this case. At last, let us give an example of application of Theorem 2. To shorten exposition we omit the proofs of some technical details. Example 4. Let X = Rd, d ≥ 2 and Xn, X be as in Example 1. Let K ⊂ Rd be a compact set, for which the surface measure λK is well defined by equality λK(·) ≡ w − lim λd(· ∩Kε) λd(Kε) where w − lim means the limit in the sense of weak convergence of measures, λd is Lebesgue measure on Rd, Kε ≡ {x|dist(x,K) ≤ ε}. Assume that the condition (6.9) λd(Kε) ≥ const · εβ, ε > 0 holds with some β < 2. In particular, the set K can be smooth (or, more generally, Lipschitz) surface of codimension 1 or fractal with its Haussdorf-Besikovich dimension greater then d− 2. It not hard to verify that µ ≡ λK is W -measure (see [1], Chapter 8.1 for the terminology), and therefore corresponds to some W -functional φ of the Wiener process X . This functional is naturally interpreted as the local time of Wiener process at the set K, and can be written as φs,t = λK(Xr) dr. We consider the functionals φn(Xn) of the form k∈[sn,tn) 1I{Xn( k )∈K 1√ and apply Theorem 2 in order to prove convergence of the distributions in C(R+,Rd)× C(T,R+) (6.10) (Xn, ψn(Xn)) ⇒ (X,φ(X)) (ψn are the broken lines corresponding to φn). Condition 1 holds true due to Example 1, condition 2 is provided by condition (6.9) (by this condition, supx gn(x) ≤ const · n 2 ). Condition 3 holds with pt(x, y) = (2πt) 2 exp{− 1 ‖y− x‖2 } and γ = d . Condition (6.9) implies condition 6 with θ = d− β. We assume that the random walk Sn is either aperiodic on some lattice hZ d or is strongly non-lattice (i.e., Sn0 has bounded distribution density for some n0). Under this assumption, condition 4 holds with αn ≡ 0, ν = λd and νn equal to counting measures on d in lattice case or λd in strongly non-lattice case. It remains to provide conditions 5, 7. We have γ−1 = d−2 2(d−β) < . Choose some δ ∈ and consider α > 0 such that > δ and α > 2θ + 2. Suppose that (6.11) E‖ξk‖αRd < +∞. Then applying Burkholder inequality we obtain that (6.12) E‖Xn(t)−Xn(s)‖αRd ≤ const · |t− s| 2 , |t− s| ≥ 1√ , x ∈ Rd. Repeating the standard proof of the Kolmogorov’s theorem on existence of continuous modification (see, for instance [20], p. 44,45), one can deduce from (6.12) that, for ς < α, ϑ < t,s∈[0,T ],|t−s|≥ 1 ‖Xn(t)−Xn(s)‖Rd |t− s|ϑ < +∞. 18 YURI N.KARTASHOV, ALEXEY M.KULIK Finally, choosing ϑ = δ, ς > 2θ + 2 we obtain that conditions 5,7 hold with Cθ = ς . Applying Theorem 2, we obtain weak convergence (6.10) under additional moment condition (6.11). One can remove this condition using the ”cutting” procedure, described in the proof of the Proposition 3. Let us remark that for the lattice random walks the result, exposed in Example 4, was obtained in [5] by a technique, essentially different from the one proposed here. Convergence (6.10) in continuous case, as far as it is known to authors, is a new result. References [1] Dynkin E.B. Markov processes, M.: Fizmatgiz, 1963 (in Russian). [2] Skorokhod A.V., Slobodeniuk M.P. Limit theorems for random walks, Kiev: Naukova dumka, 1970 (in Russian). [3] Borodin A.N., Ibragimov I.A. Limit theorems for the functionals of random walks, Proc. of the Mathematical Institute of R. Acad. Sci, vol. 195. St.-P.: Nauka, 1994 (in Russian). [4] Revesz P. Random walk in random and nonrandom environments, World Sci. Publ. Co., Inc., Teaneck, NJ, 1990. [5] Bass R.F., Khoshnevisan D. Local times on curves and uniform invariance principles, Prob. Theory Rel. Fields 92, 1992, p. 465 – 492. [6] Cherny A.S., Shiryaev A.N., Yor M. Limit behavior of the ”horizontal-vertical” random walk and some extensions of the Donsker-Prokhorov invariance principle. Probability theory and its applications, vol. 47, 3, 2002, p. 498 – 517. [7] Gikhman I.I. Some limit theorems for the number of intersections of a boundary of a given domain by a random function, Sci. notes of Kiev Un-ty, 1957, vol. 16, 10, p. 149 – 164 (in Ukrainian). [8] Gikhman I.I. Asymptotic distributions for the number of intersections of a boundary of a domain by a random function, Visnyk of Kiev Un-ty, serie astron., athem and mech., 1958, v. 1, 1, p. 25 – 46 (in Ukrainian). [9] Portenko N.I. Integral equations and limit theorems for additive functionals of Markov processes, robability theory and its applications, 1967, v. 12, 3, p. 551 – 558 (in Russian). [10] Portenko N.I. The development of I.I.Gikhman’s idea concerning the methods for investigating local behavior of diffusion processes and their weakly convergent sequences, Probab. Theory and Math. Stat., 1994, 50, p. 7 – 22. [11] Kulik A.M. Markov Approximation of stable processes by random walks, vol.12(28) 2006, .1-2, p. 87 – 93. [12] Feller W. An introduction to probability theory and its applications, Vol II, M.: Mir, 1984 (Russian, translated from W.Feller, An introduction to probability theory and its applications, John Wiley & Sons, New York, 1971). [13] Skorokhod A.V. Studies in theory of stochastic processes, Kiev, Kiev Univ-ty publishing house, 1961 (in Russian). [14] Jacod J., Shiryaev A. Limit theorems for stochastic processes,Springer, Berlin, 1987. [15] Kurtz T.G., Protter Ph. Weak limit theorems for stochastic integrals and SDE’s, Annals of Probability, 1991, vol. 19, 3, p. 1035 – 1070. [16] Yamada T., Watanabe S. On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 1971, vol. 11, p. 156 – 167. [17] Androshchuk T.O., Kulik A.M. Limit theorems for oscillatory functionals of a Markov process. Theory of stochastic proc- cesses, vol. 11(27), p. 3 – 13. [18] Ibragimov I.A., Linnik Yu.V. Linnik, Independent and stationary related variables, M.: Nauka, 1965 (in Russian). [19] Kulik A.M. Difference approximation for local times of multidimensional diffusions, arXiv:math/0702175 [20] Skorokhod A.V. Lections on theory of stochastic processes, Kyiv: Lybid, 1990 (in Ukrainian). E-mail address: [email protected] http://arxiv.org/abs/math/0702175 1. Introduction 2. Markov approximation. 3. Main theorem 4. The local time of a random walk at a point. 5. Difference approximations of diffusion processes. 6. Invariance principle for additive functionals of Markov chains References

0704.0509

Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity

arXiv:0704.0509v1 [math.PR] 4 Apr 2007 Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity. Fulvia Confortola Dipartimento di Matematica, Politecnico di Milano piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected] November 4, 2018 Abstract In this paper we study a class of backward stochastic differential equations (BSDEs) of the form dYt = −AYtdt−f0(t, Yt)dt−f1(t, Yt, Zt)dt+ZtdWt, 0 ≤ t ≤ T ; YT = ξ in an infinite dimensional Hilbert space H , where the unbounded op- erator A is sectorial and dissipative and the nonlinearity f0(t, y) is dissipative and defined for y only taking values in a subspace of H . A typical example is provided by the so-called polynomial nonlinearities. Applications are given to stochastic partial differential equations and spin systems. Key words Backward stochastic differential equations, stochastic evo- lution equations. MSC classification. Primary: 60H15 Secondary: 35R60 1 Introduction Let H,K be real separable Hilbert spaces with norms | · |H and | · |K . Let W be a cylindrical Wiener process in K defined on a probability space (Ω,F ,P) and let {Ft}t∈[0,T ] denote its natural augmented filtration. Let L 2(K,H) be the Hilbert space of Hilbert-Schmidt operators from K to H. http://arxiv.org/abs/0704.0509v1 We are interested in solving the following backward stochastic differential equation dYt = −AYtdt− f(t, Yt, Zt)dt+ ZtdWt, 0 ≤ t ≤ T, YT = ξ (1) where ξ is a random variable with values in H, f(t, Yt, Zt) = f0(t, Yt) + f1(t, Yt, Zt) and f0, f1 are given functions, and the operator A is an un- bounded operator with domain D(A) contained in H. The unknowns are the processes {Yt}t∈[0,T ] and {Zt}t∈[0,T ], which are required to be adapted with respect to the filtration of the Wiener process and take values in H, L2(K,H) respectively. In finite dimensional framework such type of equations has been solved by Pardoux and Peng [12] in the nonlinear case. They proved an existence and uniqueness result for the solution of the equation (1) when A = 0, the coefficient f(t, y, z) is Lipschitz continuous in both variables y and z, and the data ξ and the process {f(t, 0, 0)}t∈[0,T ] are square integrable. Since this first result, many papers were devoted to existence and uniqueness results under weaker assumptions. In finite dimension, when A = 0, the Lipschitz condition on the coefficient f with respect to the variable y is replaced by a monotonicity assumption; moreover, more general growth conditions in the variable y are formulated. Let us mention the contribution of Briand and Carmona [1], for a study of polynomial growth in Lp with p > 2, and the work of Pardoux [11] for an arbitrary growth. In [13] Pardoux and Rascanu deal with a BSDE involving the subdifferential of a convex function; in particular, one coefficient is not everywhere defined for y in Rk. In other works the existence of the solution is proved when the data, ξ and the process {f(t, 0, 0)}t∈[0,T ], are in L p for p ∈ (1, 2). El Karoui, Peng and Quenez [4] treat the case when f is Lipschitz continuous; in [2] this result is generalized to the case of a monotone coefficient f (both for equations on a fixed and on a random time interval) and is studied even the case p = 1. In the infinite-dimensional framework Hu and Peng [6], and Oksendal and Zhang [10] give an existence and uniqueness result for the equation with an operator A, infinitesimal generator of a strongly continous semigroup and the coefficient f Lipschitz in y and z. Pardoux and Rascanu [14] replace the operator A with the subdifferential of a convex function and assume that f is dissipative, everywhere defined and continuous with respect to y, Lipschitz with respect to z and with linear growth in y and z. Special results deal with stochastic backward partial differential equa- tions (BSPDEs): we recall in particular the works of Ma and Yong [8] and [9]. Earlier, Peng [16] studied a backward stochastic partial differential equation and regarded the classical Hamilton-Jacobi-Bellman equation of optimal stochastic control as special case of this problem. Our work extends these results in a special direction. We consider an operator A which is the generator of an analytic contraction semigroup on H and a coefficient f(t, y, z) of the form f0(t, y)+ f1(t, y, z). The coefficient f1(t, y, z) is assumed to be bounded and Lipschitz with respect to y and z. The term f0(t, y) is defined for y only taking values in a suitable subspaceHα of H and it satisfies the following growth condition for some 1 < γ < 1/α, S ≥ 0, P-a.s. |f0(t, y)|H ≤ S(1 + ||y|| ) ∀t ∈ [0, T ], ∀y ∈ Hα. Following [6], we understand the equation (1) in the following integral e(s−t)A[f0(s, Ys) + f1(s, Ys, Zs)]ds+ e(s−t)AZsdWs = e (T−t)Aξ, requiring, in particular, that Y takes values in Hα. This requires generally that the final condition also takes values in the smaller spaceHα. We take as Hα a real interpolation space which belongs to the class Jα between H and the domain of an operator A (see Section 2). Moreover f0(t, ·) is assumed to be locally Lipschitz from Hα into H and dissipative in H. We prove (Theorem 5) that if ξ takes its values in the closure of D(A) in Hα and is such that ||ξ||Hα is essentially bounded, then equation (2) has a unique mild solution, i.e. there exists a unique pair of progressively measurable processes Y : Ω×[0, T ] → Hα, Z : Ω×[0, T ] → L 2(K;H), satisfying P-a.s. equality (2) for every t in [0, T ] and such that E supt∈[0,T ] ||Yt|| ||Zt|| L2(K,H) This result extends former results concerning the deterministic case to the stochastic framework: see [7], where previous works of Fujita - Kato [5], Pazy [15] and others are collected. In these papers similar assumptions are made on the coefficients f0, f1 and on the operator A. The plan of the paper is as follows. In Section 2 some notations and definitions are fixed. In Section 3 existence and uniqueness of the solution of a simplified equation are proved, where f1 is a bounded progressively measurable process which does not depend on y and z. In Section 4, applying the previous result, a fixed point argument is used in order to prove our main result on existence and uniqueness of a mild solution of (2). Section 5 is devoted to applications. 2 Notations and setting The letters K and H will always denote two real separable Hilbert spaces. Scalar product is denoted by 〈·, ·〉; L2(K;H) is the separable Hilbert space of Hilbert-Schmidt operators from K to H endowed with the Hilbert-Schmidt norm. W = {Wt}t∈[0,T ] is a cylindrical Wiener process with values in K, defined on a complete probability space (Ω,F ,P). {Ft}t∈[0,T ] is the natural filtration of W , augmented with the family of P-null sets of F . Next we define several classes of stochastic processes with values in a Banach space X. • L2(Ω× [0, T ];X) denotes the space of measurable X-valued processes Y such that |Yτ | is finite, identified up to modification. • L2(Ω;C([0, T ];X)) denotes the space of continuousX-valued processes Y such that E sup τ∈[0,T ] |Yτ | is finite, identified up to indistinguisha- bility. • Cα([0, T ];X) denotes the space of α-Hölderian functions on [0, T ] with values in X such that [f ]α = sup 0≤x<y≤T |f(x)− f(y)| (y − x)α Now we need to recall several preliminaries on semigroup and interpo- lation spaces. We refer the reader to [7] for the proofs and other related results. A linear operator A in a Banach space X, with domain D(A) ⊂ X, is called sectorial if there are constants ω ∈ R, θ ∈ (π/2, π), M > 0 such that (i) ρ(A) ⊇ Sθ,ω = {λ ∈ C : λ 6= ω, |arg(λ− ω)| < θ}, (ii) ||(λI −A)−1||L(X) ≤ |λ−ω| ∀λ ∈ Sθ,ω where ρ(A) is the resolvent set of A. For every t > 0, (3) allows us to define a linear bounded operator etA in X, by means of the Dunford integral etA = ω+γr,η etλ(λI −A)−1dλ, t > 0, (4) where, r > 0, η ∈ (π/2, π) and γr,η is the curve {λ ∈ C : |argλ| = η, |λ| ≥ r} ∪ {λ ∈ C : |argλ| ≤ η, |λ| = r}, oriented counterclockwise. We also set e0Ax = x,∀x ∈ X. Since the function λ 7→ etλR(λ,A) is holomorphic in Sθ,ω, the definition of e tA is independent of the choice of r and η. If A is sectorial, the function [0,+∞) → L(X), t 7→ etA, with etA defined by (4) is called analytic semigroup generated by A in X. We note that for every x ∈ X the function t 7→ etAx is analytic (and hence continuous) for t > 0. etA is a strongly continuous semigroup if and only if D(A) is dense in X; in particular this holds if X is a reflexive space. We need to introduce suitable classes of subspaces of X. Definition 1. Let (α, p) be two numbers such that 0 < α < 1, 1 ≤ p ≤ ∞ or (α, p) = (1,∞). Then we denote with DA(α, p) the space DA(α, p) = {x ∈ X : t 7→ v(t) = ||t 1−α−1/pAetAx|| ∈ Lp(0, 1)} where ||x||DA(α,p) = ||x||X + [x]α = ||x||X + ||v||Lp(0,1). (We set as usual 1/∞ = 0). We recall here some estimates for the function t 7→ etA when t → 0, which we will use in the sequel. For convenience, in the next proposition we set DA(0, p) = X, p ∈ [1,∞]. Proposition 1. Let (α, p), (β, p) ∈ (0, 1)× [1,+∞]∪{(1,∞)}, α ≤ β. Then there exists C = C(p;α, β) such that ||t−α+βetA||L(DA(α,p),DA(β,p)) ≤ C, 0 < t ≤ 1. Definition 2. Let 0 ≤ α ≤ 1 and let D,X be Banach spaces, D ⊂ X. A Ba- nach space Y such that D ⊂ Y ⊂ X is said to belong to the class Jα between X and D if there is a constant C such that ||x||Y ≤ C||x|| X ||x|| D, ∀x ∈ D. In this case we write Y ∈ Jα(X,D). Now we give the definition of solution to the BSDE: e(s−t)A[f0(s, Ys) + f1(s, Ys, Zs)]ds+ e(s−t)AZsdWs = e (T−t)Aξ, Definition 3. A pair of progressively measurable processes (Y,Z) is called mild solution of (5) if it belongs to the space L2(Ω;C([0, T ];Hα))×L [0, T ];L2(K,H)) and P-a.s.solves the integral equation (5) on the interval [0, T ]. We finally state a lemma needed in the sequel. It is a generalization of the well known Gronwall’s lemma. Its proof is given in the Appendix. Lemma 1. Assume a, b, α, β are nonnegative constants, with α < 1, β > 0 and 0 < T < ∞. For any nonnegative process U ∈ L1(Ω× [0, T ]), satisfying P-a.s. Ut ≤ a(T − t) (s− t)β−1EFtUsds for almost every t ∈ [0, T ], it holds P-a.s. Ut ≤ aM(T − t) −α, for almost every t ∈ [0, T ]. M is a constant depending only on b, α, β, T . 3 A simplified equation As a preparation for the study of (2), in this section we consider the following simplified version of that equation: e(s−t)A[f0(s, Ys)ds+ f1(s)]ds + e(s−t)AZsdWs = e (T−t)Aξ, (6) for all t ∈ [0, T ]. We suppose that the following assumptions hold. Hypothesis 2. 1. A : D(A) ⊂ H → H is a sectorial operator. We also assume that A is dissipative, i.e. it satisfies < Ay, y >≤ 0,∀y ∈ D(A); 2. for some 0 < α < 1 there exists a Banach space Hα continuously embed- ded in H and such that (i) DA(α, 1) ⊂ Hα ⊂ DA(α,∞); (ii) the part of A in Hα is sectorial in Hα. 3. the final condition ξ is an FT -measurable random variable defined on Ω with values in the closure of D(A) with respect to Hα-norm. We denote this set D(A) . Moreover ξ belongs to L∞(Ω,FT ,P;Hα); 4. f0 : Ω× [0, T ]×Hα → H satisfies: i) {f0(t, y)}t∈[0,T ] is progressively measurable ∀y ∈ Hα; ii) there exist constants S > 0, 1 < γ < 1/α such that P-a.s. |f0(t, y)|H ≤ S(1 + ||y|| ) t ∈ [0, T ], y ∈ Hα; iii) for every R > 0 there is LR > 0 such that P-a.s. |f0(t, y1)− f0(t, y2)|H ≤ LR||y1 − y2||Hα for t ∈ [0, T ] and yi ∈ Hα with ||yi||Hα ≤ R; iv) there exists a number µ ∈ R such that P-a.s., ∀t ∈ [0, T ], y1, y2 ∈ < f0(t, y1)− f0(t, y2), y1 − y2 >H≤ µ|y1 − y2| H ; (7) 5. f1 : Ω × [0, T ] → H is progressively measurable and for some constant C > 0 it satisfies P-a.s. |f1(t)|H ≤ C, for t ∈ [0, T ]. Remark 1. We note that the pair (Y,Z) solves the BSDE (6) with final con- dition ξ and drift f = f0+f1 if and only if the pair (Ȳ , Z̄) := (e λtYt, e λtZt) is a solution of the same equation with final condition eλT ξ and drift f ′(t, y) := 0(t, y) + f 1(t) where f 0(t, y) = e λt(f0(t, e −λty)− λy), f 1(t) = e λtf1(t). If we choose µ = λ, then f 0 satisfies the same assumption as f0, but with (7) re- placed by < f0(t, y1)− f0(t, y2), y1 − y2 >H≤ 0. If this last condition holds, then f0 is called dissipative. Hence, without loss of generality, we shall assume until the end that f0 is dissipative, or equivalently that µ = 0 in (7). 3.1 A priori estimates We prove a basic estimate for the solution in the norm of H. Proposition 2. Suppose that Hypothesis 2 holds; if (Y,Z) is a mild solution of (6) on the interval [a, T ], 0 ≤ a ≤ T , then there exists a constant C1, which depends only on ||ξ||L∞(Ω;H) and on the constants S of 4.ii) and C of 5. such that P-a.s. supa≤t≤T ||Yt||H ≤ C1. In particular the constant C1 is independent of a. Proof. Let the pair (Y,Z) ∈ L2(Ω, C([a, T ];Hα)× L 2(Ω × [a, T ];L2(K;H)) satisfy (6). Let us introduce the operators Jn = n(nI − A) −1, n > 0. We note that the operators AJn are the Yosida approximations of A and they are bounded. Moreover |Jnx − x| → 0 as n → ∞, for every x ∈ H. We set Y nt = JnYt, Z t = JnZt. It is readily verified that Y n admits the Itô differential dY nt = −AY t dt− Jnf(t, Yt)dt− Jnf1(t)dt+ Z t dWt, and Y T = Jnξ. Applying the Ito formula to |Y nt | H , using the dissipativity of A, we obtain |Y nt | ||Zns || L2(K;H)ds ≤ |Jnξ| H + 2 < Jnf0(s, Ys), Y s >H ds+ < Jnf1(s), Y s >H ds− 2 < Y ns , Z s dWs >H . We note that < Jnf0(s, Ys) + Jnf1(s), Y s >H ds → < f0(s, Ys) + f1(s), Ys >H ds by dominated convergence, as n → ∞. Moreover by the dominated convergence theorem we have ||(Zns ) ∗Y ns − Z sYs|| Kds → 0 P-a.s. and it follows that < Y ns , Z s dWs >H→ < Ys, ZsdWs >H in probability. If we let n → ∞ in (8) we obtain ||Zs|| L2(K;H)ds ≤ |ξ| H + 2 < f0(s, Ys) + f1(s), Ys >H ds < Ys, ZsdWs >H . Recalling (7), that we assume to hold with µ = 0, it follows that ||Zs|| L2(K,H) ≤ ≤ |ξ|2H + 2 < f0(s, 0), Ys >H +2 < f1(s), Ys >H ds+ < Ys, ZsdWs >H ≤ |ξ|2H + |f(s, 0)|2Hds+ |f1(s)| Hds + 2 < Ys, ZsdWs >H . Now, since sup0≤t≤T |f(t, 0)| H ≤ S 2 and since the stochastic integral < Ys, ZsdWs >H , t ∈ [a, T ] is a martingale, if we take the conditional expectation given Ft we have H ≤ E Ft |ξ|2H + 2E |f(s, 0)|2Hds+ E |f1(s)| ≤ |ξ|2L∞(Ω,H) + (S 2 + C2)T + 2 Ft |Ys| Since Y belongs to L2(Ω;C([a, T ];Hα)) and, consequently, ||Y || L1(Ω× [0, T ]), we can apply Lemma 1 to |Y |2H and conclude that H ≤ (|ξ| L∞(Ω,H) + [S2 + C2]T )(1 + 2Te2T ). Now we will show that the result of Proposition 2, together with the growth condition satisfied by f0, yields an a priori estimate on the solution in the Hα-norm. Let 0 < α < 1 and let γ > 1 be given by 4.ii). We fix θ = αγ and consider the Banach space DA(θ,∞) introduced in Definition 1. It is easy to check (see [7]) that, if we take θ ∈ (0, 1), θ > α, then Hα contains DA(θ,∞) and belongs to the class Jα/θ between DA(θ,∞) and H, hence the following inequality is satisfied: |x|Hα ≤ c|x| DA(θ,∞) H , x ∈ DA(θ,∞). (9) Proposition 3. Suppose that Hypothesis 2 is satisfied. Let (Y,Z) be a mild solution of (6) in [a, T ], a ≥ 0 and assume that there exists two constants R > 0 and K > 0, possibly depending on a, such that, P-a.s., t∈[a,T ] ||Yt||Hα ≤ R, sup t∈[a,T ] |Yt|H ≤ K. (10) Then the following inequality holds P-a.s.: |Yt|L∞(Ω,DA(θ,∞)) ≤ C2 (T − t)θ−α , a ≤ t < T (11) with C2 depending on the operator A, ||ξ||L∞(Ω,Hα), θ, α, K, C of 5. and S of 4.ii) of Hypothesis 2. Proof. Taking the conditional expectation given Ft in equation (6) we find Yt = E e(T−t)Aξ + e(s−t)A[f0(s, Ys) + f1(s)]ds , a ≤ t ≤ T. Consequently, we have ||Yt||DA(θ,∞) ≤ E Ft ||e(T−t)Aξ||DA(θ,∞) ||e(s−t)A[f0(s, Ys) + f1(s)]||DA(θ,∞)ds, a ≤ t ≤ T . Since Hα ⊂ DA(α,∞), we have Ft ||e(T−t)Aξ||DA(θ,∞) ≤ ≤ EFt ||e(T−t)A||L(DA(α,∞),DA(θ,∞))||ξ||L∞(Ω,DA(α,∞)) (T − t)θ−α ||ξ||L∞(Ω,Hα), with C0 = C0(α, θ,∞), where in the last inequality we use Proposition 1. Moreover ||e(s−t)A[f0(s, Ys) + f1(s)]||DA(θ,∞)ds ≤ ≤ EFt ||e(s−t)A||L(H,DA(θ,∞))|f0(s, Ys) + f1(s)|Hds ≤ ≤ EFt (s− t)θ [|f0(s, Ys)|H + |f1(s)|H ]ds ≤ EFt (s− t)θ [S(1 + ||Ys|| ) + C]ds. In the inequality we used Hypotheses 4.ii) and 5. and Proposition 1. Re- calling (9), we conclude that the last term is dominated by (s − t)θ S(1 + c|Ys| γ(1−α)/θ H ||Ys|| DA(θ,∞) ) + C = EFt (s − t)θ S(1 + c|Ys| γ(1−α)/θ H ||Ys||DA(θ,∞)) + C by choosing θ = αγ. By the second inequality in (10) this can be estimated (s− t)θ S(1 + cKγ(1−α)/θEFt ||Ys||DA(θ,∞) + C)ds (s− t)θ (C + S)ds + (s− t)θ ScKγ(1−α)/θEFt ||Ys||DA(θ,∞)ds. Hence by (13) and (14) it follows ||Yt||DA(θ,∞) ≤ (T − t)θ−α ||ξ||L∞(Ω,Hα) + (s− t)θ (C + S)ds (s− t)θ ScKγ(1−α)/θEFt ||Ys||DA(θ,∞)ds, and (11) follows from Lemma 1. In order to justify the application of Lemma 1, we need to prove that ||Y ||DA(θ,∞) belongs to L 1(Ω × [a, T ]). This also follows from(13) and (14) since, for some constant K1, ||Yt||DA(θ,∞) ≤ (T − t)θ−α ||ξ||L∞(Ω,Hα) + E Ft [ sup s∈[a,T ] (1 + ||Ys|| (s − t)θ (T − t)θ−α ||ξ||L∞(Ω,Hα) + (1 +R (s − t)θ 3.2 Local existence and uniqueness We prove that, under Hypothesis 2, there exists a unique solution of (6) on an interval [T − δ, T ] with δ sufficiently small. To treat the ordinary integral in the left hand side of (6), we need the following result, whose proof can be found in [7], Proposition 4.2.1 and Lemma 7.1.1. Lemma 3. Let φ ∈ L∞((a, T );H), 0 < a < T and set v(t) = e(s−t)Aφ(s)ds, a ≤ t ≤ T. If 0 < α < 1, then v ∈ C1−α([a, T ];DA(α, 1)) and there is G0 > 0, not depending on a, such that ||v||C1−α([a,T ];DA(α,1)) ≤ G0||φ||L∞((a,T );H). Since DA(α, 1) ⊂ Hα, we also have v ∈ C 1−α([a, T ];Hα) and there is G > 0, not depending on a, such that ||v||C1−α([a,T ];Hα) ≤ G||φ||L∞((a,T );H). Theorem 4. Let us assume that Hypothesis 2 holds, except possibly 4.iv). Then there exists δ > 0 such that the equation (6) has a unique local mild solution (Y,Z) ∈ L2(Ω;C([T − δ, T ];Hα))× L 2(Ω× [T − δ, T ];L2(K;H)). Remark 2. The dissipativity condition 4.iv) only plays a role in obtaining the a priori estimate in H (Proposition 2) and consequently global existence, as we will see later. Proof. Let Mα := sup0≤t≤T ||e tA||L(Hα). We fix a positive number R such that R ≥ 2Mα||ξ||L∞(Ω;Hα). This implies that sup0≤t≤T ||e tAξ||Hα ≤ R/2 P-a.s. Moreover, let LR be such that |f0(t, y1)− f0(t, y2)|H ≤ LR||y1 − y2||Hα 0 ≤ t ≤ T, ||yi||Hα ≤ R We recall that the space L2(Ω;C([T − δ, T ];Hα)) is a Banach space en- dowed with the norm Y → E supt∈[T−δ,T ] ||Yt|| . We define K = {Y ∈ L2(Ω;C([T − δ, T ],Hα)) : sup t∈[T−δ,T ] ||Yt||Hα ≤ R, a.s.}. It easy to check that K is a closed subset of L2(Ω;C([T − δ, T ],Hα)), hence a complete metric space (with the inherited metrics). We look for a local mild solution (Y,Z) in the space K. We define a nonlinear operator Γ : K → K as follows: given U ∈ K, Y = Γ(U) is the first component of the mild solution (Y,Z) of the equation e(s−t)A[f0(s, Us)ds+ f1(s)]ds+ e(s−t)AZsdWs = e (T−t)Aξ (15) for t ∈ [T − δ, T ]. Since U ∈ K we have P-a.s. |f0(t, Ut) + f1(t)|H ≤ S(1 + ||Ut|| ) + C ≤ S(1 +Rγ) + C, (16) for all t in [T − δ, T ]. Hence f0(·, U·)+ f1(·) belongs to L 2(Ω× [T − δ, T ];H) and, by a result of Hu and Peng [6], there exists a unique pair (Y,Z) ∈ L2(Ω× [T − δ, T ];H)× L2(Ω× [T − δ, T ];L2(K;H)) satisfying (15). More- over, by taking the conditional expectation given Ft, Y has the following representation Yt = E e(T−t)Aξ + e(s−t)A[f0(s, Us) + f1(s)]ds We will show that Γ is a contraction for the norm of L2(Ω, C([T − δ, T ];Hα) and maps K into itself, if δ is sufficiently small; clearly, its unique fixed point is the required solution of the BSDE. We first check the contraction property. Let U1, U2 ∈ K. Then Γ(U1)t − Γ(U 2)t = Y t − Y t = E e(s−t)A(f0(s, U s )− f0(s, U Let v(t) = e(s−t)A f0(s, U s )− f0(s, U ds. Then, noting that v(T ) = 0 and recalling Lemma 3, for t ∈ [T − δ, T ] ||Y 1t − Y t ||Hα = = ||EFtv(t)||Hα ≤ E Ft ||v(t)||Hα ≤ δ1−αEFt ||v||C(1−α)([T−δ,T ],Hα) ≤ Gδ(1−α)EFt ||f0(·, U · )− f0(·, U · )||L∞([T−δ,T ],H) ≤ Gδ(1−α)LRE Ft sup t∈[T−δ,T ] ||U1t − U t ||Hα =: Mt, where {Mt, t ∈ [T − δ, T ]} is a martingale. Hence, by Doob’s inequality E sup t∈[T−δ,T ] ||Y 1t − Y ≤ E sup t∈[T−δ,T ] 2 ≤ 2E|MT | = 2G2L2Rδ 2(1−α) E sup t∈[T−δ,T ] ||U1t − U If δ ≤ δ0 = 2GLR (1−α) , then Γ is a contraction with constant 1/2. Next we check that Γ mapsK into itself. For each U ∈ K and t ∈ [T−δ, T ] with δ ≤ δ0 we have t∈[T−δ,T ] ||Γ(U)t||Hα = sup t∈[T−δ,T ] ||Yt||Hα ≤ sup t∈[T−δ,T ] Ft ||e(T−t)Aξ||Hα+ + sup t∈[T−δ,T ] Ft || e(s−t)A[f0(s, Us) + f1(s)]ds||Hα ≤ R/2 + sup t∈[T−δ,T ] ||e(s−t)A[f0(s, Us) + f1(s)]||Hαds ≤ R/2 + sup t∈[T−δ,T ] ||e(s−t)A[f0(s, Us) + f1(s)]||DA(α,1)ds, where in the last inequality we have used the fact that DA(α, 1) ⊂ Hα. Now, by Proposition 1, and from 4.ii) and 5., it follows that ||e(s−t)A[f0(s, Us) + f1(s)]||DA(α,1) ≤ ≤ ||e(s−t)A||L(H,DA(α,1))|f0(s, Us) + f1(s)|H (s− t)α [S(1 + ||Us|| ) + C]. Then, since U ∈ K, we arrive at t∈[T−δ,T ] ||Γ(U)t||Hα ≤ ≤ R/2 + sup t∈[T−δ,T ] (s− t)α [S(1 + ||Us|| ) + C]ds ≤ R/2 + sup t∈[T−δ,T ] (s− t)α [S(1 +Rγ) + C]ds ≤ R/2 + CαS [(1 +Rγ) + C] δ1−α, where Cα depends on A, α. Hence, if δ ≤ δ0 is such that CαS [(1+Rγ )+C] is less or equal to R/2, then sup t∈[T−δ,T ] ||Γ(U)t||Hα ≤ R. Due to Lemma 3, P- a.s. the function t 7→ Yt−E Fte(T−t)Aξ belongs to C[T−δ, T ];Hα); moreover, the map t 7→ EFte(T−t)Aξ belongs to C[T − δ, T ];Hα), since ξ is a random variable taking values in D(A) . Therefore, P-a.s. Y· ∈ C([T − δ, T ];Hα) and Γ maps K into itself and has a unique fixed point in K. Remark 3. By Lemma 3, using properties of analytic semigroups, it can be proved that for every fixed ω the range of the map Γ is contained in C1−β([T − δ, T − ǫ];DA(β, 1)) for every ǫ ∈ (0, δ), β ∈ [0, 1]. 3.3 Global existence Now we are able to prove a global existence theorem for the solution of the equation (6), using all the results presented above. Theorem 5. If Hypothesis 2 is satisfied, the equation (6) has a unique mild solution (Y,Z) ∈ L2(Ω;C([0, T ],Hα))× L 2(Ω× [0, T ]);L2(K;H)). Proof. By Theorem 4 equation (6) has a unique mild solution (Y 1, Z1) ∈ L2(Ω;C([T − δ1, T ],Hα)) × L 2(Ω × [T − δ1, T ]);L 2(K;H)) on the interval [T − δ1, T ], for some δ1 > 0. By Proposition 2 we know that there exists a constant C1 such that P-a.s. |YT−δ1 |H ≤ C1. (17) We recall that the constant C1 depends only on |ξ|L∞(Ω;H) and on the con- stants S of 4.ii) and C of 5. and is independent of δ1. Moreover, by Propo- sition 3, there exists a constant C2 such that P-a.s. ||YT−δ1 ||L∞(Ω,DA(θ,∞)) ≤ C2 δθ−α1 , (18) with C2 depending on the operator A, ||ξ||L∞(Ω,Hα), θ, α, C1. This implies that YT−δ1 belongs to L ∞(Ω;Hα) and it can be taken as final value for the problem ∫ T−δ1 e(s−t)A[f0(s, Ys)ds+ f1(s)]ds + ∫ T−δ1 e(s−t)AZsdWs = = e(T−δ1−t)AYT−δ1 on an interval [T − δ1 − δ2, T − δ1], for some δ2 > 0. As in the proof of Theorem 4, we fix a positive number R2 such that R2 = 2Mα ≥ 2Mα||YT−δ1 ||L∞(Ω,DA(θ,∞)). By Theorem 4 there exists a pair of progressively measurable processes (Y 2, Z2) in L2(Ω;C([T − δ1 − δ2, T − δ1];Hα)) × L 2(Ω × [T − δ1 − δ2, T − δ1];L 2(K,H)) which solves (19) on the interval [T − δ1 − δ2, T − δ1] where δ2 depends on the operator A, α, R2. We note that the continuity in T − δ1 of Y 2 follows from the fact that YT−δ1 takes values in DA(α, 1) (see Remark 3), so that YT−δ1 takes values in D(A) . Now, the process Yt defined by Y 1t on the interval [T − δ1, T ] and by Y t on [T − δ1 − δ2, T − δ1] belongs to L2(Ω;C([T − δ1 − δ2, T ];Hα)) and it easy to see that it satisfies (6) in the whole interval [T − δ1 − δ2, T ]. Consequently, by Proposition 2, P-a.s., |YT−δ1−δ2 |H ≤ C1 with C1 the constant in (17), and by (18) ||YT−δ1−δ2 ||L∞(Ω,DA(θ,∞)) ≤ (δ1 + δ2)θ−α , (20) where C2 is the same constant as in (18). Again, YT−δ1−δ2 can be taken as initial value for problem ∫ T−δ1−δ2 e(s−t)A[f0(s, Ys)ds+ f1(s)]ds + ∫ T−δ1−δ2 e(s−t)AZsdWs = e(T−δ1−δ2−t)AYT−δ1−δ2 on the interval [T − δ1 − δ2 − δ3, T − δ1 − δ2], where δ3 will be fixed later. In this case, by (20), we can choose R3 = R2 = 2Mα ≥ 2Mα||YT−δ1−δ2 ||L∞(Ω,DA(θ,∞)) and prove that there exists a unique mild solution (Y 3, Z3) of (21) on the interval [T−δ1−δ2−δ3, T−δ1−δ2], with δ3 = δ2 . So we extend the solution to [T − δ1 − 2δ2, T ]. Proceeding this way we prove the global existence to (6) on [0, T ]. 4 The general case We can now study the equation: e(s−t)A[f0(s, Ys) + f1(s, Ys, Zs)]ds + e(s−t)AZsdWs = e (T−t)Aξ We require that the function f1 satisfy the following assumptions: Hypothesis 6. 1. there exists K ≥ 0 such that P-a.s. |f1(t, y, z)− f1(t, y )|H ≤ K|y − y |H +K||z − z ||L2(K;H), for every t ∈ [0, T ], y, y ∈ H, z, z ∈ L2(K;H), 2. there exists C ≥ 0 such that P-a.s. |f1(t, y, z)|H ≤ C, for every t ∈ [0, T ], y ∈ H, z ∈ L2(K;H). Theorem 7. If Hypotheses 2 and 6 hold, then equation (22) has a unique solution in L2(Ω;C([0, T ];Hα))× L 2(Ω× [0, T ];L2(K;H)). Proof. LetM be the space of progressive processes (Y,Z) in the space L2(Ω× [0, T ];H) × L2(Ω× [0, T ];L2(K;H)) endowed with the norm |||(Y,Z)|||2β = E eβs(|Ys| H + ||Zs|| L2(K;H))ds, where β will be fixed later. We define Φ : M → M as follows: given (U, V ) ∈ M, (Y,Z) = Φ(U, V ) is the unique solution on the interval [0, T ] of the equation e(s−t)A[f0(s, Ys)ds+f1(s, Us, Vs)]ds+ e(s−t)AZsdWs = e (T−t)Aξ. By Theorem 5 the above equation has a unique mild solution (Y,Z) which belongs to L2(Ω;C([0, T ];Hα))×L 2(Ω×[0, T ];L2(K;H)). Therefore Φ(M) ⊂ M. We will show that Φ is a contraction for a suitable choice of β; clearly, its unique fixed point is the required solution of (22). We take another pair (U ) ∈ M and apply Proposition 3.1 in [3] to the difference of two equations. We obtain β|Y 1t − Y H + ‖Z s − Z L2(K;H)ds eβs < f0(s, Y s ) + f1(s, U s , V −f0(s, Y s )− f1(s, U s , V s ), Y s − Y s >H ds eβsK(|U1s − U s |H + ||V s − V s ||L2(K;H))|Y s − Y s |Hds eβs(|U1s − U H + ||V s − V L2(K;H))/2 + 4K 2|Y 1s − Y where we have used 4.iv) of Hypothesis 2 and 1. of Hypothesis 6. Choosing β = 4K2 + 1, we obtain the required contraction property. 5 Applications In this section we present some backward stochastic partial differential prob- lems which can be solved with our techniques. 5.1 The reaction-diffusion equation Let D be an open and bounded subset of Rn with a smooth boundary ∂D. We choose K = L2(D). This choice implies that dWt/dt is the so-called ”space-time white noise”. Moreover, since Hilbert-Schmidt op- erators on L2(D) are represented by square integrable kernels, the space L2(L2(D), L2(D)) can be identified with L2(D ×D). We are given a com- plete probability space (Ω,F ,P) with a filtration (Ft)t∈[0,T ] generated by W and augmented in the usual way. Let us consider a non symmetric bilinear, coercive continuous form a : H10 (D) × H 0 (D) → R defined by a(u, v) := − i,j aij(x)Diu(x)Djv(x)dx, where the coefficients aij are Lipschitz continuous and there exists α > 0 such that i,j=1 aij(x)ξiξj ≥ α|ξ|2 for every x ∈ D, ξ ∈ Rn. Let A be the operator associated with the bi- linear form a such that < Au, v >L2(D)= a(u, v), v ∈ H 0 (D) and u ∈ D(A). It is known that, in this case, D(A) = H2(D) ∩H10 (D), where H 2(D) and H10 (D) are the usual Sobolev spaces. We consider for t ∈ [0, T ] and x ∈ D the backward stochastic problem written formally ∂Y (t, x) = AY (t, x) + r(Y (t, x)) + g(t, Y (t, x), Z(t, x), x)+ + Z(t, x) ∂W (t, x) on Ω× [0, T ]× D̄ Y (T, x) = ξ(x) on Ω× D̄ Y (t, x) = 0 on Ω× [0, T ]× ∂D We suppose the following. Hypothesis 8. 1. r : R → R is a continuous, increasing and locally Lipschitz function; 2. r satisfies the following growth condition: |r(x)| ≤ S(1 + |x|γ) ∀x ∈ R for some γ > 1; 3. g is a measurable real function defined on [0, T ] × R × L2(D × D) ×D and there exists a constant K > 0 such that |g(t, y1, z1, x)− g(t, y2, z2, x)| ≤ K(|y1 − y2|+ ||z1 − z2||L2(D×D)) for all t ∈ [0, T ], y1, y2 ∈ R, z1, z2 ∈ L 2(D), x ∈ D; 4. there exists a real function h in L2(D×D) such that P-a.s. |g(t, y, z, x)| ≤ K1h(x) for all t ∈ [0, T ], y ∈ R, z ∈ L 2(D), x ∈ D; 5. ξ belongs to L∞(Ω;H2(D) ∩H10 (D)). We define the operator A by (Ay)(x) = Ay(x) with domain D(A) = H2(D) ∩ H10 (D). We set f0(t, y)(x) = −r(y(t, x)) for t ∈ [0, T ], x ∈ D and y in a suitable subspace of H which will be determined below. For t ∈ [0, T ], x ∈ D, y ∈ L2(D), z ∈ L2(D × D) we define f1 as the operator f1(t, y, z)(x) = −g(t, y(t, x), z(t, x), x). Then problem (23) can be written in abstract way as dYt = −AYtdt− f0(t, Yt)dt− f1(t, Yt, Zt)dt+ ZtdWt, YT = ξ. Under the conditions in Hypothesis 8, the assumptions in Hypotheses 2, 6 are satisfied. The operator A is a closed operator in L2(D) and it is the infinitesimal generator of an analytic semigroup in L2(D) satisfying ‖etA‖L(H) ≤ 1 (see [17], Chapter 3). In particular, by Lumer-Philips theo- rem, A is dissipative. The non linear function f0(t, ·) : L 2γ(D) → L2(D), y 7→ −r(y) is locally Lipschitz. We look for a space of class Jα between H and D(A) where f0 is well defined and locally Lipschitz. It is well known (see [18]) that the fractional order Sobolev space W β,2(D) is of class Jβ/2 between L2(D) and H2(D) for every β ∈ (0, 2). Hence the space Hα defined by Hα = W β,2(D) if β < 1, by W β,2(D) ∩H10 (D) if β ≥ 1 is of class Jβ/2 between H and D(A). Moreover the restriction of A on Hα is a sectorial operator ([18]). By the Sobolev embedding theorem, W β,2 is contained in Lq(D) for all q if β ≥ n , and in L2n/(n−2β)(D) if β < n . If we choose β ∈ (0, 2) we have W β,2(D) ⊂ L2γ(D) for n < 4 . It is clear that f0 is locally Lipschitz with respect to y from Hα into H. It is easy to verify that f0 satisfies 4.ii) of Hypothesis 2 with γ = 2n + 1 and that it is dis- sipative with constant µ = 0. The function f1 is Lipschitz uniformly with respect to y and z and it is bounded. The final condition ξ takes values in and belongs to L∞(Ω;Hα). Hence we can apply the global exis- tence theorem and state that the above problem has a unique mild solution (Y,Z) ∈ L2(Ω;C([0, T ];Hα))× L 2(Ω× [0, T ];L2(K,H)). 5.2 A spin system Let Z be the one-dimensional lattice of integers. Its elements will be inter- preted as atoms. A configuration is a real function y defined on Z. The value y(n) of the configuration y at the point n can be viewed as the state of the atom n. We consider an infinite system of equations dY nt = −anY t dt+ |n−j|≤1 V (Y nt − Y t )dt+ Z t n ∈ Z, 0 ≤ t ≤ T Yn(T ) = ξn n ∈ Z, where Y n and Zn are real processes, and V : R → R. Let l2(Z) be the usual Hilbert space of square summable sequences. To study system (24) we apply results of previous sections. To fit our assump- tion in Hypotheses 2 and 6, we suppose the following Hypothesis 9. 1. W n, n ∈ Z are independent standard real Wiener processes; 2. a = {an}n∈Z is a sequence of nonnegative real numbers; 3. ξ = {ξn}n∈Z is a random variable belonging to L ∞(Ω, l2(Z)); 4. the function V : R → R is defined by V (x) = x2k+1 k ∈ N. We will study system (24) regarded as a backward stochastic evolution equation for t ∈ [0, T ] dYt = (AYt + f0(t, Yt))dt+ ZtdWt, YT = ξ (25) on a properly chosen Hilbert space H of functions on Z. To reformulate problem (24) in the abstract form (25), we set K = H = l2(Z). We set Wt = {W t }n∈Z, t ∈ [0, T ]. By 1. of Hypothesis 9, W is a cylindrical Wiener process inH defined on (Ω,F , P ). We define the operator A(y) = (anyn)n, D(A) = {y ∈ l 2(Z) such that n∈Z a n < ∞}. It is easy to prove that A is a self-adjoint operator in l2(Z), hence the infinitesimal generator of a sectorial semigroup. The coefficient f0 is given by (f0(t, y))n = (V (yn+1−yn)+V (yn−1−yn)), t ∈ [0, T ], y ∈ D(f0) where D(f0) = {y ∈ l 2(Z) such that n∈Z |xn+1 − xn| 2(2k+1) < +∞}. Under Hy- pothesis 9, A, f0, ξ satisfy Hypotheses 2 and 6. We observe that in this case the domain of f0 is the whole space H: if y ∈ l 2(Z) then |yn+1 − yn| 2(2k+1)} 2(2k+1) ≤ { |yn+1 − yn| 2 ≤ 2||y||l2(Z). Consequently, we can take Hα with α = 0, i.e. H0 = H. The function f0 is dissipative. Namely < f0(t, y)− f0(t, y ′), y − y′ >l2(Z) = {[(yn+1 − yn) (2k+1) + (yn−1 − yn) (2k+1)]+ +[(y′n+1 − y (2k+1) + (y′n−1 − y (2k+1)])[yn − y [(yn+1 − yn) (2k+1) − (y′n+1 − y (2k+1)][(yn+1 − yn)− (y n+1 − y and the last term is negative. Moreover, f0 satisfies 4.ii) of Hypothesis 2 with γ = 2k + 1. The map f0 is also locally Lipschitz from H in to H. Then by Theorem 7, problem (25) has a unique mild solution (Y,Z) which belongs to L2(Ω, C([0, T ];H)) × L2(Ω× [0, T ];L2(K,H)). 6 Appendix This section is devoted to the proof of Lemma 1. Assume first that β = 1. Using recursively the inequality Ut ≤ a(T − t) −α + b FtUsds we can easily prove that ≤ a(T − t)−α + (r − t)k−1 (k − 1)! (T − r)α +bEFt (b(r − t))n−1 (n− 1)! Urdr. The last term in the above inequality tends to zero as n tends to infinity for each t in the interval [0, T ]. Thus Ut ≤ a(T − t) −α + a bk(T − t)k−1 (k − 1)! (T − r)α ≤ a(T − t)−α + abeb(T−t) (T − r)α ≤ a(T − t)−α + abeb(T−t) (T − t)1−α ≤ a(T − t)−αM where M = 1 + bebT 1 In the case β 6= 1 a similar proof can be given, based on recursive use of the inequality Ut ≤ a(T − t) −α + b (s − t)β−1EFtUsds. Acknowledgments: I wish to thank Giuseppe Da Prato for hospitality at the Scuola Normale Superiore in Pisa, suggestions and helpful discussions. I would like to express my gratitude to Marco Fuhrman: I am indebted to him for his precious help and encouragement. Special thanks are due to Alessandra Lunardi, who gave me valuable advice and support. References [1] Ph. Briand and R. Carmona. BSDEs with polynomial growth generators. J. Appl. Math. Stochastic Anal., 13(3):207–238, 2000. [2] Ph. Briand and B. Deylon and Y. Hu and E. Pardoux and L. Stoica. Lp solutions of backward stochastic differential equations. Stochastic Process. Appl., 108(1):109–129, 2003. [3] F. Confortola. Dissipative backward stochastic differential equations in infinite dimensions. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 9 (1):155–168, 2006. [4] N. El Karoui and S. G. Peng and M. C. Quenez. Backward Stochastic Differential equations in Finance. Math. Finance, 7(1):1–71, 1997. [5] H. Fujita and T. Kato. On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal. 16:269–315, 1964. [6] Y. Hu and S. G. Peng. Adapted solution of a backward semilinear stochastic evolution equation. Stochastic Anal. Appl., 9(4):445–459, 1991. [7] A. Lunardi. Analytic semigroups and optimal regularity in parabolic prob- lems volume 16 of Progress in Nonlinear Differential Equations and their Applications. Birkhser Verlag, Basel 1995. [8] J. Ma and J. Yong Adapted solution of a degenerate backward SPDE, with applications. Stochastic Process. Appl. 70:59–84, 1997. [9] J. Ma and J. Yong On linear, degenerate backward stochastic partial differential equations. Probab. theory Related Fields 113:135–170 1999. [10] B. Oksendal and T. Zhang. On backward stochastic partial differential equations, 2001. Preprint. [11] E. Pardoux. BSDEs, weak convergence and homogenization of semilin- ear PDEs. Nonlinear analysis, differential equations and control (Mon- treal, QC, 1998), 503–549, NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Acad. Publ., Dordrecht, 1999. [12] É. Pardoux and S. Peng. Adapted solution of a backward stochastic differential equation. Systems and Control Lett. 14:55–61, 1990. [13] E. Pardoux and A. Răşcanu. Backward stochastic differential equa- tions with subdifferential operator and related variational inequalities. Stochastic Process. Appl., 76(2):191–215, 1998. [14] E. Pardoux and A. Răşcanu. Backward stochastic variational inequali- ties. Stochastics Stochastics Rep., 67(3-4):159–167, 1999. [15] A. Pazy Semigroups of linear operators and applications to partial dif- ferential equations, Springer-Verlag, (1983). [16] S. Peng Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Con- trol Optim., 30:284–304, 1992. [17] H. Tanabe Equations of evolution. Monographs and Studies in Mathemat- ics, 6. 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0704.0510

Axino warm dark matter and $\Omega_b - \Omega_{DM}$ coincidence

IFT-UAM/CSIC-07-15 Axino warm dark matter and Ωb − ΩDM coincidence Osamu Seto Instituto de F́ısica Teórica, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain Masahide Yamaguchi Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara 229-8558, Japan Abstract We show that axinos, which are dominantly generated by the decay of the next-to-lightest super- symmetric particles produced from the leptonic Q-ball (L-ball), become warm dark matter suitable for the solution of the missing satellite problem and the cusp problem. In addition, Ωb − ΩDM coincidence is naturally explained in this scenario. PACS numbers: 95.35.+d, 12.60.Jv, 98.80.Cq, 04.65.+e http://arxiv.org/abs/0704.0510v2 I. INTRODUCTION Recent observations of cosmic microwave background anisotropies such as Wilkinson Mi- crowave Anisotropy Probe measured the abundance of components of the Universe very precisely. However, their origins are still one of the major mysteries of cosmology and parti- cle physics. The fact that the abundances of dark matter and baryon are of the same order may give us a great hint for their origins. In the minimal supersymmetric standard model (MSSM), flat directions consist of squarks/sleptons and produce a non-zero baryon or lepton number through the Affleck- Dine (AD) mechanism [1]. Then, Q-balls, which are non-topological solitons [2], can be produced due to the instability and absorb almost all the produced baryon or lepton num- bers [3]. In the gravity mediated supersymmetry breaking model, the mass per charge of a Q-ball is larger than that of a nucleon so that they are unstable against the decay into light fermions. Then, they can directly decay into baryons and the lightest supersymmetric particle (LSP). In case that the charge of the produced Q-balls is large enough, Q-balls can survive even after the freeze-out of weakly interacting massive particles (WIMPs). Thus, the reason why energy densities of dark matter and baryon are almost the same magnitude can be explained in this scenario [4, 5]. However, it was pointed out that LSPs are often overproduced by the decay of Q-balls if the LSP is the lightest neutralino in the MSSM [6], which gives the stringent conditions on the neutralino LSPs and AD fields. Only a few models free from this overproduction have been proposed [7, 8]. Instead, the supergravity models in which the LSPs is a stable gravitino are investigated [9]. In this scenario, Q-balls decay into the next-to-lightest supersymmetric particle (NLSP) directly instead of the LSP gravitino. The LSP gravitinos are produced by the decay of NLSP and becomes dominant over other gravitinos produced by thermal processed [10, 11] and by the decay of thermal produced NLSP [12, 13] in case that Q-balls can survive the evaporation. Then, it is found that such a gravitino dark matter scenario is still viable if the late decay of NLSP does not spoil the success of Big Bang Nucleosynthesis (BBN). Another interesting possibility is that the LSP is an axino, which is the fermionic superpartner of an axion. Axinos are also produced by the decay of NLSPs produced from the Q-ball [14] as well as by thermal processes and by the decay of thermally produced NLSP [15, 16, 17, 18, 19]. Such gravitinos and axinos often become an ideal candidate for cold dark matter. The models of cold dark matter (CDM) and dark energy combined with inflation-based scale- invariant primordial density fluctuations have succeeded at explaining many properties of the observed universe, especially the large scale structure of the universe. However, going into the smaller scales, some observations on galactic and subgalactic (. Mpc) seem to con- flict with predictions by high-resolution N-body simulations as well as analytic calculations based on the standard CDM model. The first discrepancy is called the missing satellite problem [20]. The CDM-based models predict an order of magnitude higher number of halos than those actually observed within the Local Group. The other is called the cusp problem [21]. The CDM-based models also predict overly cuspy mass profile for the CDM halos compared to actual observations within the Local Group. In order to reconcile such discrepancies, several authors proposed modifications to the standard CDM-based model though the photoionization mechanism may overcome such difficulties [22]. One method is to reduce the small-scale power of primordial density fluctuations, which can be realized in a specific model of inflation [23]. Another is to change the properties of dark matter. Spergel and Steinhardt introduced strong self-interaction among cold dark matter particles (collisional CDM), which enhances satellite destruction and suppress cusp formation [24]. The warm dark matter [25], which can have relatively large velocity dispersion at the epoch of the matter-radiation equality, can also reduce satellite production and cusp formation. In this paper, we consider axinos dominantly generated by the decay of NLSPs produced from the leptonic Q-ball (L-ball). Such axinos become warm dark matter suitable for the solution of the missing satellite problem and the cusp problem. In addition, Ωb − ΩDM coincidence is naturally explained through the Affleck-Dine mechanism and the subsequent L-ball formation in this scenario. In the next section, we discuss Ωb − ΩDM coincidence based on the Affleck-Dine mechanism and the subsequent L-ball formation. In section III, we show that axinos in our scenario become warm dark matter suitable for the solution of the missing satellite problem and the cusp problem. In the final section, we give concluding remarks. II. Ωb − ΩDM COINCIDENCE FROM AFFLECK-DINE LEPTOGENESIS We now discuss baryogenesis via Affleck-Dine leptogenesis and dark matter production from Q-ball decays within the framework of gravity mediated supersymmetry breaking. A. Lepton asymmetry The potential of the AD flat direction field is, in general, lifted by soft supersymmetric (SUSY) breaking terms and non-renormalizable terms [26, 27]. The full potential of the AD field is given by V (φ) = 1 +K ln − c1H2 c2H + Am3/2 nMn−3 +H.c. |φ|2n−2 M2n−6 . (1) Here, mφ is the soft SUSY breaking scalar mass for the AD field with radiative correction K ln |φ|2. A flat direction dependent constant, K, takes values from −0.01 to −0.1 [28]. Λ denotes a renormalization scale and −c1H2 represents the negative mass squared induced by the SUSY breaking effect which comes from the energy density of the inflaton, with an order unity coefficient c1 > 0 [27]. λ is the coupling of a nonrenormalizable term and M is some large scale acting as its cut-off. Terms proportional to A and c2 are the A-terms coming from the low energy SUSY breaking and the inflaton-induced SUSY breaking, respectively, where m3/2 denotes the gravitino mass. Here, we omitted possible terms which may appear by thermal effects [29, 30]. These terms are negligible as long as we consider a sufficient low reheating temperature after inflation, as we will. Moreover, the model would face with “gravitino problem” [31], if the reheating temperature after inflation is so high that these thermal effect become effective, unless gravitino is LSP [9]. The charge number density for the AD field φ is given by nq = iq(φ̇ ∗φ− φ∗φ̇) where q is the baryonic (or leptonic) charge for the AD field. By use of the equation of motion of the AD field, the charge density can be rewritten as nq(t) ≃ a(t)3 dt′a(t′)3 2qλm3/2 Im(Aφn), (2) with a(t) being the scale factor. When the AD field starts to oscillate around the origin, the charge number density is induced by the relative phase between A-terms. By taking into account s = 4π2g∗T 3/90, the charge to entropy ratio after reheating is estimated as 4M2PH q|A|λm3/2 TR|φos|n sin δ. (3) Here, MP ≃ 2.4 × 1018 GeV is the reduced Planck mass, tos is the time of the start of the oscillation and sin δ is the effective CP phase. In case that thermal corrections are ineffective, Hos ≃ mφ, which yields |φos| ≃ )1/(n−2) . (4) From now on, as a concrete example, we consider a LLē direction of n = 6 as the AD field for our scenario. Since this is a pure leptonic direction, the lepton asymmetry generated by the Affleck-Dine mechanism can be estimated as ≃ 1× 10−10 q|A| sin δ ( m3/2 100GeV 103GeV )3/2( 100GeV . (5) B. Baryon asymmetry and LSP production from Q-balls The produced lepton asymmetry is not directly released to thermal bath. Instead, L-balls are formed due to the instability and almost all produced lepton numbers are absorbed into L-balls [3]. First of all, we briefly summarize relevant properties of Q-balls in gravity mediated SUSY breaking models. The radius of a Q-ball, R, is estimated as R2 ≃ 2/(|K|m2φ) [5]. Numerical calculations provide a fitting formula for the Q-ball charge Q ≃ β̄ |φos| ǫ for ǫ & ǫc ǫc for ǫ < ǫc , (6) ≃ 2q|A| sin δ (7) where ǫc ≃ 10−2 and β̄ = 6× 10−3 [3]. The Q-ball charge can be evaluated as Q ∼ 2× 1020 4× 10−1 )3/2( , (8) where we assumed ǫ > ǫc because it looks to be more natural than the other which can be realized only for an accidental small sin δ . Furthermore, if ǫ < ǫc, additional “unnatural” parameters are required for our scenario, as we will show. A part of the charge of a Q-ball can evaporate by the interaction with particles in the thermal bath. The evaporation of charge of Q-ball is done by the evaporation with the rate Γevap ≡ = −4πRQDevneq ≃ −4πR2QDevµQT 2, (9) with Dev . 1 and by the diffusion with the rate Γdiff ≡ = −4πkRQDdiffneq ≃ −4πkRQDdiffµQT 2, (10) where µQ is the chemical potential of Q-balls and the numerical constant k is very close to unity so that we will drop it hereafter. Ddiff ≈ a/T is a diffusion constant [32] and a is a particle dependent coefficient given by [33, 34] 4 for squark 6 for quark 100 for left− handed (s)lepton 380 for right− handed (s)lepton . (11) Here, we see that both the evaporation and the diffusion are efficient for low temperature, from Eqs. (9) and (10) with the relation between the cosmic time and the temperature: for T & TR for T < TR . (12) Moreover, by comparing Γdiff and Γevap, Γdiff Γevap , (13) we can find that for low temperature T . a mφ ∼ 10mφ , (14) the diffusion is more crucial for estimation of the evaporated charge from Q-ball. Equation (10) is rewritten as ≃ −4πRQDdiffµQT 2 . (15) Integrating Eq. (15) from mφ, because the evaporation from Q-ball is suppressed by the Boltzmann factor, we can estimate the total evaporated charge as ∆Q ≃ 32kRQ T 2RMP ∼ 3.2× 2.4√ × 1019 for mφ & TR, ∆Q ∼ 3.6× 2.4√ × 1019 for mφ < TR. By taking Eq. (8) into account, we obtain × 10−1 4× 10−1 )( mφ for mφ < TR 1 and find that about 10% of Q-ball charge would be evaporated. Here, one can see why the case of ǫ < ǫc is irrelevant for us. If ǫ < ǫc ≃ 10−2, ǫ is replaced with ǫc in Eq. (18). Then these Q-balls cannot survive the evaporation unless the AD field mass is extremely small as mφ = O(10) GeV or (λ1/3MP/M)3/2 ≪ 1 . The evaporated charges are released into the thermal bath so that a part of them is transformed into baryonic charges through the sphaleron effects [35]. Then, the resultant baryon asymmetry is given as ∆Q× 10−30 103GeV 100GeV ≃ 10−10 formφ & TR formφ < TR . (19) Interestingly, the baryon asymmetry does not depend on the effective CP phase sin δ unlike usual Affleck-Dine baryogenesis, because the CP phase dependences in both lepton asym- metry nL/s and the charge of Q-ball Q cancel each other. In addition, for mφ < TR, the baryon asymmetry basically depends on only one free parameter, the reheating temperature TR, because other parameters are not free but known in a sense. When Q-balls decay, the supersymmetric particles are released from them. Since the Q-ball consists of scalar leptons, the number of the produced supersymmetric particles is given by YNLSP = NQ = 2× 10−9 4× 10−1 )1/2( , (20) where NQ is the number of produced NLSP particles per one leptonic charge. Such produced NLSPs decay into axino LSP with a typical lifetime of O(0.1− 1) second. Thus, NLSPs produced by the Q-ball decay become a source of axino production. Of course, like gravitinos, axinos can also be produced by other processes such as thermal processes (TP), namely the scatterings and decays in the thermal bath, and non-thermal processes (NTP), say the late decay of NLSPs produced thermally. The relevant Boltzmann equations can be written as ṅNLSP + 3HnNLSP = −〈σv〉(n2NLSP − n 2) + γQ−ball − ΓNLSPnNLSP, (21) ṅã + 3Hnã = 〈σv(i+ j → ã+ ...)〉ijninj + 〈σv(i → ã+ ...)〉ini + ΓNLSPnNLSP, (22) 1 For mφ & TR, the result is of the same magnitude but with different dependence on mφ and TR. where γQ−ball denotes the contribution to NLSP production by Q-balls decay, 〈σv〉ij and 〈σv〉i are the scattering cross section and the decay rate for the thermal production of axinos, and ΓNLSP is the decay rate of the NLSP. The total NLSP abundance, before its decay, is given by YNLSP = NQ + Y TPNLSP, (23) where NQnL/s denotes the NLSP produced by L-ball decay and Y NLSP is the abundance of NLSP produced thermally and given by Y TPNLSP ≃ T=mNLSP mNLSP/Tf 〈σv〉ann . (24) Here 〈σv〉ann is the annihilation cross section and Tf ∼ mNLSP/20 is the freeze-out temper- ature. The resultant total axino abundance is expressed as Yã = Y ã + Y ã . (25) Y NTPã = YNLSP = NQ + Y TPNLSP (26) is the nonthermally produced axino through the NLSP decay and Y TPã denotes the axi- nos produced by thermal processes. For nonthermally produced axinos, while the NLSP abundance produced by L-ball decay is = 2× 10−9 2× 10−9 , (27) the typical value of Y TPNLSP is given by Y TPNLSP ≃ 10−11 100GeV mNLSP 10−10GeV−2 〈σv〉ann . (28) Thus, nonthermal production of axinos due to the thermal relic NLSPs decay, Y TPNLSP, can be negligible compared to that from Q-ball produced NLSPs, NQnL/s. On the other hand, axino production by thermal processes is dominated by scattering processes for the case that the reheating temperature is larger than the masses of neutralinos and gluinos. In this case, the abundance of such axinos is proportional to the reheating temperature TR and the inverse square of Peccei-Quinn (PQ) scale fa and given by [18] Y TPã ≃ 10−8 1011GeV , (29) where N is the number of vacua and N = 1(6) for the KSVZ (DFSZ) model [36, 37]. Thus, for TR ≃ 1 TeV, if fa/N & several ×1011 GeV [38], Y TPã is subdominant compared with Y NTPã ≃ NQnL/s. If this is the case, the energy density of axino is given by ρã = mãnNLSP due to the R-parity conservation. Recalling ≃ 3.9× 10−10 GeV, (30) the density parameter of axinos is expressed as ( mã 0.2GeV 2× 10−9 . (31) Thus, axinos with the sub-GeV mass can be dark matter in our scenario. Now, one can see that the Ωb and ΩDM is related through the lepton asymmetry. In fact, from Eqs. (19) and (31), we obtain a relation between the abundances of dark matter and baryon asymmetry, 1× 10−1 mã/mp , (32) where mp(≃ 1 GeV) is the mass of proton. One may find the similar relation in the case of baryonic Q-ball (B-ball) [14]. The difference between the case of B-ball and L-ball is that the required mass of LSP from L-ball can be an order of magnitude smaller than that in B-ball where the mass of LSP dark mater must be ≃ 1 GeV, mainly because a part of lepton asymmetry produced by the Affleck-Dine mechanism, that is, only evaporated charges ∆Q/Q are converted to baryon asymmetry so that the number density of NLSPs produced by the L-ball decay become larger for a fixed baryon asymmetry. As shown in the next section, such a difference of axino masses is crucial for solving the missing satellite problem and the cusp problem. Equation (31) with TR ≃ 1 TeV to explain the observed baryon asymmetry yields a quite natural scale of the AD field mass mφ ≃ 1TeV 4× 10−1 ( mã 0.2GeV . (33) As mentioned above, in our scenario, NLSP decays into axino at late time. Such late decay is potentially constrained by BBN. The lifetime of NLSP is given as τχ ≡ τ(χ → ã+ γ) = 0.33sec C2aY Y Z 1/128 1011GeV 102GeV for the case that the lightest neutralino χ is NLSP in [18]. Here, CaY Y is the axion model dependent coupling coefficient between axion multiplet and U(1)Y gauge field, Z11 denotes the fraction of b-ino component in the lightest neutralino. According to Ref. [18], we can summerize the constraints as follows. First of all, for τχ ≤ 0.1 sec., there is no constraint. The corresponding mass of the NLSP neutralino is mχ = 320GeV 0.1sec )1/3( C2aY YZ )1/3( fa/N√ 10× 1011GeV from Eq. (34). Thus, if mχ & 320 GeV, this model is free from problems by the late decay of NLSP. This lower bound is a bit stringent than that in [18], because we need to take the PQ scale somewhat larger as we mentioned. On the other hand, for 0.1 sec. < τχ < 1 sec., the lower bound on axino mass exists and can be roughly expressed as 0.1GeV 102GeV (CaY Y Z11) 1011GeV + 6 ≃ −4 0.33sec. by reading Fig. 4 in Ref. [18]. In this case, axino must be heavier than a few hundred MeV. For τχ ≃ 1 sec., the corresponding mass of the NLSP neutralino and the lower bound of axino mass is given by mχ ≃ 150GeV, mã & 320MeV. (37) III. A SOLUTION TO THE MISSING SATELLITE PROBLEM AND THE CUSP PROBLEM An interesting consequence of such light axinos is their large velocity dispersion. There- fore, they can potentially solve the missing satellite problem and the cusp problem as stated in the introduction. For the scenario of dark matter particle produced by the late decay of long-lived particle, it is shown that the missing satellite problem and the cusp problem can be solved simultaneously if the lifetime of long-lived particle and the ratio of mass between dark matter particle and the mother particle satisfy the following relation [39]: 6.3× 102mã sec. . τχ . 1.0× 103mã sec., (38) where we identified dark matter with axino (LSP) and the long-lived particle with the lightest neutralino (NLSP), respectively. Combining Eq. (34) with Eq. (38), we have the following relation between the masses of the axino (LSP) and the lightest neutralino (NLSP), mχ ≃ (0.33− 0.83) ( mã C2aY Y Z 1/128 1011GeV GeV. (39) For mã ≃ 1 GeV in the B-ball case [14], mχ becomes a few GeV with its lifetime τχ ≫ 103 second even if we take fa/N to be several times 10 11 GeV, which is excluded. On the other hand, for mã = O(0.1) GeV in the L-ball case of this paper, mχ becomes O(100) GeV with its lifetime τχ . 1 second if we take fa/N to be several times 10 11 GeV. Thus, we find that axinos in this scenario can solve the missing satellite problem and the cusp problem simultaneously for natural mass scales with mã = O(0.1) GeV and mχ = O(100) GeV. IV. CONCLUDING REMARKS In this paper, we show that Affleck-Dine leptogenesis can explain baryon asymmetry and dark matter abundance simultaneously and that Ωb −ΩDM coincidence is explained for sub-GeV mass of the LSP axino. Though the basic idea is the same as Ref. [14], where B-balls are considered, the mass of LSP axino becomes an order of magnitude smaller in this scenario. On the other hand, the PQ scale is determined as fa/N = a few ×1011 GeV, which will be tested in the future if PQ scale can be measured by e.g., the manner proposed in [40]. The other attractive point is that axinos considered in this paper can potentially solve the missing satellite problem and the cusp problem simultaneously because they are relatively light and have large velocity dispersion. We have shown that axions in our scenario can solve both problems for natural mass scales with mã = O(0.1) GeV and mNLSP = O(100) GeV. For simplicity, we concentrate on the case of the lightest neutralino NLSP with the mass to be O(100) GeV. Such neutralinos are detectable in Large Hadron Collider. In addition, the corresponding lepton asymmetry is almost the maximal value under the assumption of (M3/λM3P ) ≃ 1. 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0704.0511

A unified approach to SIC-POVMs and MUBs

A UNIFIED APPROACH TO SIC-POVMs AND MUBs Olivier Albouy and Maurice R. Kibler Université de Lyon, Institut de Physique Nucléaire, Université Lyon 1 and CNRS/IN2P3, 43 bd du 11 novembre 1918, F–69622 Villeurbanne, France Electronic mail: [email protected], [email protected] Abstract A unified approach to (symmetric informationally complete) positive op- erator valued measures and mutually unbiased bases is developed in this arti- cle. The approach is based on the use of Racah unit tensors for the Wigner- Racah algebra of SU(2) ⊃ U(1). Emphasis is put on similarities and differ- ences between SIC-POVMs and MUBs. Keywords: finite–dimensional Hilbert spaces; mutually unbiased bases; positive op- erator valued measures; SU(2) ⊃ U(1) Wigner–Racah algebra 1 INTRODUCTION The importance of finite–dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum informa- tion theory, especially for quantum cryptography and quantum state tomography [4]-[27]. Along this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]-[27]. The introduction of POVMs goes back to the seventies [4]-[7]. The most general quan- tum measurement is represented by a POVM. In the present work, we will be interested http://arxiv.org/abs/0704.0511v3 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condi- tion (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15]-[18], has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one. The aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27]. Most of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A† denotes the adjoint of the operator A, δa,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality. 2 SIC-POVMs Let Cd be the standard Hilbert space of dimension d endowed with its usual inner product denoted by 〈 | 〉. As is usual, we will identify a POVM with a nonorthogonal decompo- sition of the identity. Thus, a discrete SIC-POVM is a set {Px : x = 1, 2, · · · , d2} of d2 nonnegative operators Px acting on C d, such that: • they satisfy the trace or symmetry condition Tr (PxPy) = , x 6= y; (1) moreover, we will assume the operators Px are normalized, thus completing this condition with = 1; (2) • they form a decomposition of the identity Px = I; (3) • they satisfy a completeness condition: the knowledge of the probabilities px defined by px = Tr(Pxρ) is sufficient to reconstruct the density matrix ρ. Now, let us develop each of the operators Px on an orthonormal (with respect to the Hilbert–Schmidt product) basis {ui : i = 1, 2, · · · , d2} of the space of linear operators on vi(x)ui, (4) where the operators ui satisfy Tr(u iuj) = δi,j . The operators Px are thus considered as vectors v(x) = (v1(x), v2(x), · · · , vd2(x)) (5) in the Hilbert space Cd of dimension d2 and the determination of the operators Px is equivalent to the determination of the components vi(x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give v(x) · v(y) = 1 (dδx,y + 1) , (6) where v(x) · v(y) = i=1 vi(x)vi(y) is the usual Hermitian product in C In order to compare Eq. (6) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators Px are rank-one operators. Therefore, by putting Px = |Φx〉〈Φx| (7) with |φx〉 ∈ Cd, the trace property (1, 2) reads |〈Φx|Φy〉|2 = (dδx,y + 1) . (8) From this point of view, to find d2 operators Px is equivalent to finding d 2 vectors |φx〉 in Cd satisfying Eq. (8). At the price of an increase in the number of components from d3 (for d2 vectors in Cd) to d4 (for d2 vectors in Cd ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators Px. In fact, there exists a lot of relations among these d4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {ui : i = 1, 2, · · · , d2} and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors. The cornerstone of this approach is to identify Cd with a subspace ε(j) of constant angular momentum j = (d− 1)/2. Such a subspace is spanned by the set {|j,m〉 : m = −j,−j + 1, · · · , j}, where |j,m〉 is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u(k) be the Racah unit tensor [29] of order k (with k = 0, 1, · · · , 2j) defined by its 2k + 1 components u(k)q (where q = −k,−k + 1, · · · , k) through u(k)q = m′=−j (−1)j−m j k j −m q m′ |j,m〉〈j,m′|, (9) where (· · ·) denotes a 3–jm Wigner symbol. For fixed j, the (2j + 1)2 operators u(k)q (with k = 0, 1, · · · , 2j and q = −k,−k + 1, · · · , k) act on ε(j) ∼ Cd and form a basis of the Hilbert space CN of dimension N = (2j + 1)2, the inner product in CN being the Hilbert–Schmidt product. The formulas (involving unit tensors, 3–jm and 6–j symbols) relevant for this work are given in Appendix (see also [29] to [31]). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining vi(x). Each operator Px can be developed as a linear combination of the operators u Hence, we have ckq(x)u q , (10) where the unknown expansion coefficients ckq(x) are a priori complex numbers. The determination of the operators Px is thus equivalent to the determination of the coefficients ckq(x), which are formally given by ckq(x) = (2k + 1)〈Φx|u(k)q |Φx〉, (11) as can be seen by multiplying each member of Eq. (10) by the adjoint of u p and then using Eq. (46) of Appendix. By defining the vector v(x) = (v1(x), v2(x), · · · , vN(x)), N = (2j + 1)2 (12) vi(x) = 2k + 1 ckq(x), i = k 2 + k + q + 1, (13) the following properties and relations are obtained. • The first component v1(x) of v(x) does not depend on x since c00(x) = 2j + 1 for all x ∈ {1, 2, · · · , (2j + 1)2}. Proof: Take the trace of Eq. (10) and use Eq. (48) of Appendix. • The components vi(x) of v(x) satisfy the complex conjugation property described ckq(x) = (−1)qck−q(x) (15) for all x ∈ {1, 2, · · · , (2j + 1)2}, k ∈ {0, 1, · · · , 2j} and q ∈ {−k,−k + 1, · · · , k}. Proof: Use the Hermitian property of Px and Eq. (43) of Appendix. • In terms of ckq, Eq. (6) reads 2k + 1 ckq(x)ckq(y) = 2(j + 1) [(2j + 1)δx,y + 1] (16) for all x, y ∈ {1, 2, · · · , (2j + 1)2}, where the sum over q is SO(3) rotationally invariant. Proof: The proof is trivial. • The coefficients ckq(x) are solutions of the nonlinear system given by 2K + 1 cKQ(x) = (−1)2j−Q k ℓ K −q −p Q k ℓ K j j j ckq(x)cℓp(x) (17) for all x∈ {1, 2, · · · , (2j+1)2}, K ∈ {0, 1, · · · , 2j} and Q∈ {−K,−K+1, · · · , K}. Proof: Consider P 2x = Px and use the coupling relation (51) of Appendix involving a 3–jm and a 6–j Wigner symbols. As a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation ‖v(x)‖2 = v(x) · v(x) = 1. • All coefficients ckq(x) are connected through the sum rule (2j+1)2 ckq(x) j k j −m q m′ = (−1)j−m(2j + 1)δm,m′ , (18) which turns out to be useful for global checking purposes. Proof: Take the jm–jm′ matrix element of the resolution of the identity in terms of the operators Px/(2j + 1). 3 MUBs A complete set of MUBs in the Hilbert space Cd is a set of d(d + 1) vectors |aα〉 ∈ Cd such that |〈aα|bβ〉|2 = δα,βδa,b + (1− δa,b), (19) where a = 0, 1, · · · , d and α = 0, 1, · · · , d − 1. The indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a. We know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]). The approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to find d+ 1 sets Sa (with a = 0, 1, · · · , d) of vectors in Cd, each set Sa = {|aα〉 : α = 0, 1, · · · , d − 1} containing d vectors |aα〉 such that Eq. (19) be satisfied. This amounts to finding d(d+ 1) projection operators Πaα = |aα〉〈aα| (20) satisfying the trace condition Tr (ΠaαΠbβ) = δα,βδa,b + (1− δa,b), (21) where the trace is taken on Cd. Therefore, they also form a nonorthogonal decomposition of the identity Πaα = I. (22) As in Sec. 2, we develop each operator Πaα on an orthonormal basis with expansion coefficients wi(aα). Thus we get vectors w(aα) in C w(aα) = (w1(aα), w2(aα), · · · , wd2(aα)) (23) such that w(aα) · w(bβ) = δα,βδa,b + (1− δa,b) (24) for all a, b ∈ {0, 1, · · · , d} and α, β ∈ {0, 1, · · · , d− 1}. Now we draw the same relations as for POVMs by choosing the Racah operators to be our basis in Cd . We assume once again that the Hilbert space Cd is realized by ε(j) with j = (d − 1)/2. Then, each operator Πaα can be developed on the basis of the (2j + 1)2 operators u Πaα = dkq(aα)u q , (25) to be compared with Eq. (10). The expansion coefficients are dkq(aα) = (2k + 1)〈aα|u(k)q |aα〉 (26) for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, k ∈ {0, 1, · · · , 2j} and q ∈ {−k,−k + 1, · · · , k}. For a and α fixed, the complex coefficients dkq(aα) define a vector w(aα) = (w1(aα), w2(aα), · · · , wN(aα)) , N = (2j + 1)2 (27) in the Hilbert space CN , the components of which are given by wi(aα) = 2k + 1 dkq(aα), i = k 2 + k + q + 1. (28) We are thus led to the following properties and relations. The proofs are similar to those in Sec. 2. • First component w1(aα) of w(aα): d00(aα) = 2j + 1 for all a ∈ {0, 1, · · · , 2j + 1} and α ∈ {0, 1, · · · , 2j}. • Complex conjugation property: dkq(aα) = (−1)qdk−q(aα) (30) for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, k ∈ {0, 1, · · · , 2j} and q ∈ {−k,−k + 1, · · · , k}. • Rotational invariance: 2k + 1 dkq(aα)dkq(bβ) = δα,βδa,b + 2j + 1 (1− δa,b) (31) for all a, b ∈ {0, 1, · · · , 2j + 1} and α, β ∈ {0, 1, · · · , 2j}. • Tensor product formula: 2K + 1 dKQ(aα) = (−1)2j−Q k ℓ K −q −p Q k ℓ K j j j dkq(aα)dℓp(aα) (32) for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, K ∈ {0, 1, · · · , 2j} and Q ∈ {−K,−K + 1, · · · , K}. • Sum rule: dkq(aα) j k j −m q m′ = (−1)j−m2(2j + 1)δm,m′ (33) which involves all coefficients dkq(aα). 4 CONCLUSIONS Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2–designs [8, 10, 19, 24]. On the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13]. Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22]. In dimension d, to find d2 operators Px of a SIC-POVM acting on the Hilbert space d amounts to find d2 vectors v(x) in the Hilbert space CN with N = d2 satisfying ‖vx‖ = 1, v(x) · v(y) = for x 6= y (34) (the norm ‖v(x)‖ of each vector v(x) is 1 and the angle ωxy of any pair of vectors v(x) and v(y) is ωxy = cos −1[1/(d+ 1)] for x 6= y). In a similar way, to find d + 1 MUBs of Cd is equivalent to find d + 1 sets Sa (with a = 0, 1, · · · , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in CN with N = d2 satisfying w(aα) · w(aβ) = δα,β, w(aα) · w(bβ) = for a 6= b (35) (each set Sa consists of d orthonormalized vectors and the angle ωaαbβ of any vector w(aα) of a set Sa with any vector w(bβ) of a set Sb is ωaαbβ = cos −1(1/d) for a 6= b). According to a well accepted conjecture [8, 10], SIC-POVMs should exist in any dimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d. The situation is different for MUBs. In dimension d, it is known that there exist d+ 1 sets of d vectors of type |aα〉 in Cd satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. (35) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. (19). In fact, we have [26, 27] |aα〉 = 2j + 1 ω(j+m)(j−m+1)a/2+(j+m)α|j,m〉, (36) ω = exp 2j + 1 , j = (d− 1) (37) for a, α ∈ {0, 1, · · · , 2j} while |aα〉 = |j,m〉 (38) for a = 2j + 1 and α = j +m = 0, 1, · · · , 2j. Then, Eq. (26) yields dkq(aα) = 2k + 1 2j + 1 m′=−j ωθ(m,m ′)(−1)j−m j k j −m q m′ , (39) θ(m,m′) = (m−m′) (1−m−m′)a+ α for a, α ∈ {0, 1, · · · , 2j} while dkq(aα) = δq,0(2k + 1)(−1)j−m j k j −m 0 m for a = 2j + 1 and α = j + m = 0, 1, · · · , 2j. It can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients dkq(aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. (35) has a solution only if d is a prime or the power of a prime. APPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ We limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers. The definition (9) of the components u q of the Racah unit tensor u(k) yields 〈j,m|u(k)q |j,m′〉 = (−1)j−m j k j −m q m′ , (42) from which we easily obtain the Hermitian conjugation property u(k)q = (−1)qu(k)−q . (43) The 3–jm Wigner symbol in Eq. (42) satisfies the orthogonality relations j j′ k m m′ q j j′ ℓ m m′ p 2k + 1 δk,ℓδq,p∆(j, j ′, k) (44) (2k + 1) j j′ k m m′ q j j′ k M M ′ q = δm,Mδm′,M ′. (45) The trace relation on the space ε(j) u(k)q u(ℓ)p 2k + 1 δk,ℓδq,p∆(j, j, k) (46) easily follows by combining Eqs. (42) and (44). Furthermore, by introducing j j′ 0 m −m′ 0 = δj,j′δm,m′(−1)j−m 2j + 1 in Eq. (44), we obtain the sum rule (−1)j−m j k j −m q m 2j + 1δk,0δq,0∆(j, k, j), (48) known in spectroscopy as the barycenter theorem. There are several relations involving 3–jm and 6–j symbols. In particular, we have (−1)j−M j k j −m q M j ℓ j −M p m′ j K j −m Q m′ = (−1)2j−Q k ℓ K −q −p Q k ℓ K j j j , (49) where {· · ·} denotes a 6–j Wigner symbol (or W Racah coefficient). Note that the intro- duction of k ℓ 0 j j J = δk,ℓ(−1)j+k+J (2k + 1)(2j + 1) in Eq. (49) gives back Eq. (44). Equation (49) is central in the derivation of the coupling relation u(k)q u (−1)2j−Q(2K + 1) k ℓ K −q −p Q k ℓ K j j j Q . (51) Equation (51) makes it possible to calculate the commutator [u q , u p ] which shows that the set {u(k)q : k = 0, 1, · · · , 2j; q = −k,−k + 1, · · · , k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32). Note added in version 3 After the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32]. The main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension. Acknowledgements This work was presented at the International Conference on Squeezed States and Un- certainty Relations, University of Bradford, England (ICSSUR’07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man’ko and M. Planat for interesting comments. References [1] A. Peres, “Quantum Theory: Concepts and Methods”, Dordrecht: Kluwer (1995) [2] A. Vourdas, J. Phys. A: Math. Gen. 38, 8453 (2005) [3] W. M. de Muynck, “Foundations of Quantum Mechanics, an Empiricist Approach”, Dordrecht: Kluwer (2002) [4] J. M. Jauch and C. Piron, Helv. Phys. Acta 40, 559 (1967) [5] E. B. Davies and J. T. Levis, Comm. Math. Phys. 17, 239 (1970) [6] E. B. Davies, IEEE Trans. Inform. Theory IT-24, 596 (1978) [7] K. Kraus, “States, Effects, and Operations”, Lect. Notes Phys. 190 (1983) [8] G. Zauner, Diploma Thesis, University of Wien (1999) [9] C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43, 4537 (2002) [10] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys. 45, 2171 (2004) [11] D. M. Appleby, J. Math. Phys. 46, 052107 (2005) [12] M. Grassl, Proc. ERATO Conf. Quant. Inf. Science (EQIS 2004) ed. J. Gruska, Tokyo (2005) [13] M. Grassl, Elec. Notes Discrete Math. 20, 151 (2005) [14] S. Weigert, Int. J. Mod. Phys. B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, “Irreducible Tensorial Sets”, New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) http://arxiv.org/abs/quant-ph/0205155 http://arxiv.org/abs/quant-ph/0502031 INTRODUCTION SIC-POVMs MUBs CONCLUSIONS

0704.0512

Stable oscillations of a predator-prey probabilistic cellular automaton: a mean-field approach

7 Stable oscillations of a predator-prey probabilistic cellular automaton: a mean-field approach. Tânia Tomé and Kelly C de Carvalho Instituto de F́ısica, Universidade de São Paulo Caixa Postal 66318 05315-970 São Paulo, São Paulo, Brazil E-mail: [email protected] Abstract. We analyze a probabilistic cellular automaton describing the dynamics of coexistence of a predator-prey system. The individuals of each species are localized over the sites of a lattice and the local stochastic updating rules are inspired on the processes of the Lotka-Volterra model. Two levels of mean-field approximations are set up. The simple approximation is equivalent to an extended patch model, a simple metapopulation model with patches colonized by prey, patches colonized by predators and empty patches. This approximation is capable of describing the limited available space for species occupancy. The pair approximation is moreover able to describe two types of coexistence of prey and predators: one where population densities are constant in time and another displaying self-sustained time-oscillations of the population densities. The oscillations are associated with limit cycles and arise through a Hopf bifurcation. They are stable against changes in the initial conditions and, in this sense, they differ from the Lotka-Volterra cycles which depend on initial conditions. In this respect, the present model is biologically more realistic than the Lotka-Volterra model. PACS numbers: 87.23.Cc, 05.65.+b, 05.70.Ln, 02.50.Ga http://arxiv.org/abs/0704.0512v1 Stable oscillations of a predator-prey probabilistic cellular automaton 2 1. Introduction The simplest model exhibiting time-oscillations in a two-component system is the model proposed independently by Lotka [1, 2, 3] and by Volterra [4]. In this model the individuals of two species are dispersed over an assumed homogeneous space. It is implicitly assumed in this approach that any individual can interact with any other one with equal intensity implying that their positions are not taken into account. The time evolution of the densities of the two species in the Lotka-Volterra model is given by a set of two ordinary differential equations [5, 6, 7, 8] and is set up in analogy with the laws of mass-action. Depending on the level of description wanted, the approach based on mass-action laws, contained on the Lotka-Volterra model, suffices. However, there are situations in which the coexistence takes place in a spatially heterogeneous habitat such that the population densities can be very low in some regions. In this case we need to proceed beyond the mass-law equations and consider the space structure of the habitat. In other words, it becomes necessary to analyze the coexistence by taking explicitly into account spatial structured models. In fact, the role of space in the description of population biology problems has been recognized by several authors in the last years [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In a very clear manner, Durrett and Levin [11] have pointed out that the modelling of population dynamics systems which are spatially distributed by interacting particle systems [11, 27, 28, 29] is the appropriate theoretical approach that is able to give the more complete description of the problem. We include in this approach probabilistic cellular automata (PCA) [29, 30, 31], which will concern us here. We refer to interacting particle systems and PCA as stochastic lattice models. They are both Markovian processes defined by discrete stochastic variables residing on the sites of a lattice; the former being a continuous time process and the latter a discrete time process. In the present work we study the coexistence and the emergence of stable self- sustained oscillations in a predator-prey system by considering a PCA previously studied by numerical simulations [24, 26]. This PCA is defined by local rules, similar to the ones of the contact process [27], that are capable of describing the interaction between prey and predator. Here, we focus on the analysis of the PCA by means of dynamic mean-field approximations [10, 28, 29, 30, 32, 33]. In this approach the equations for the time evolution of correlations of various orders are truncated at a certain level and high order correlations of sites are written in terms of small order correlations. The simplest approximation is the one in which all correlations are written in terms of one- site correlation, called simple approximation. In a more sophisticated approximation, called pair approximation [10, 29], any correlation is written in terms of one-site and two-site correlations. The simple mean-field approximation is capable of predicting coexistence of individuals in a stationary state where the densities of each species, and of empty sites, are constant. However, it is not capable of predicting possible time oscillating Stable oscillations of a predator-prey probabilistic cellular automaton 3 behavior of the population densities and we have proceed to the next order of mean- field approximation. The simple approximation, on the other hand, can be placed in an explicit correspondence with a patch model [7, 12, 34] where unoccupied patches can be colonized by prey and patches occupied by prey can be colonized by predators that in turn may become extinct. In this approximation the PCA can be seen as an extended version of the Lotka-Volterra model which includes an extra logistic term related to the empty sites. The pair-mean field approximation is able to predict possible time oscillating behavior of the population densities that are self-sustained and are attained thorough Hopf bifurcations. This is in contrast with the Lotka-Volterra model which presents no stable oscillations but exhibits instead infinite cycles that are associated to different initial conditions. However, from the biological point of view, one does not expect that a small variation in the initial densities of prey and predator result in different amplitudes of oscillations. Within our approach, a PCA treated in the pair-approximation, the oscillations are associated with limit cycles what mean to say that they are stable against the changes in the initial conditions. According to our point of view, the pair- approximation, in which the correlation between neighboring sites are treated exactly, provides a basic description of the predator-prey spatial interactions. For this reason, we will refer to the PCA in this approximation as a quasi-spatial-structured model. 2. Model 2.1. Probabilistic cellular automaton We consider interacting particles living on the sites of a lattice and evolving in time according to Markovian local rules. The lattice is the geometrical object that plays the role of the spatial region occupied by particles, in a general case, or by individuals of each species in the present case. The lattice sites are the possible locations for the individuals. Each site can be either empty or occupied by one individual of different species and a stochastic variable ηi is introduced to describe the state of each site at a given instant of time. The state of the entire system is denoted by η = (η1, . . . , ηi, . . . , ηN) where N is the total number of sites. The transition between the states is governed by the interactions between neighbor sites in the lattice and by a synchronous dynamics. The probability P (ℓ)(η) of configuration η at time step ℓ evolves according to the Markov chain equation P (ℓ+1)(η) = W (η|η′)P (ℓ)(η′), (1) where the summation is over all the microscopic configurations of the system, and W (η|η′) is the conditional transition probability from state η′ at time ℓ to state η at time ℓ + 1. This transition probability does not depend on time and contains all the information about the dynamics of the system. Taking into account that all the sites are simultaneously updated, which is the fundamental property of a PCA, the transition Stable oscillations of a predator-prey probabilistic cellular automaton 4 Figure 1. Transitions of the predator-prey model. The three states are: prey or herbivorous (H or 1), predator (P or 2) and empty or vegetable (V or 0). The allowed transitions obey the cyclic order shown. probability can be factorized and written in the form [29, 30] W (η|η′) = wi(ηi|η ′), (2) where wi(ηi|η ′) is the conditional transition probability that site i takes the state ηi given that the whole system is in state η′. Being a probability distribution, the quantity wi(ηi|η ′) must satisfy the following properties: wi(ηi, η ′) ≥ 0 and wi(ηi|η ′) = 1. (3) The average of any state function F (η) is evaluated by 〈F (η)〉ℓ = F (η)P (ℓ)(η). (4) The time evolution equation for 〈F (η)〉 is obtained from definition (4) and equation (1). For example, we can derive the equations for the time evolution of densities and two-site correlations. 2.2. Predator-prey probabilistic cellular automaton To model a predator-prey system by a PCA, the stochastic variable ηi associated to site i will represent the occupancy of the site by one prey, or the occupancy by one predator or the vacancy (a site devoid of any individual). The variable ηi is assumed to take the value 0, 1, or 2, according to whether the site is empty (V), occupied by a prey individual (H) or by a predator (P), respectively. That is, 0, empty (V), 1, prey (H), 2, predator (P), which defines a three-state per site PCA. The stochastic rules, embodied in the transition rate wi(ηi|η ′), are set up according to the following assumptions. (a) The space is homogeneous, which means to say that no region of the space will be privileged against the others, that is, in principle the individuals have the same conditions of surveillance in any space region. (b) The space is isotropic, which means to say that there is no preferential direction in this space for any interaction. (c) The allowed transitions between states are only the ones that obey Stable oscillations of a predator-prey probabilistic cellular automaton 5 the cyclic order shown in figure 1. Prey can only born in empty sites; prey can give place to a predator, in a process where a prey individual dies and a predator is instantaneously born; finally a predator can die leaving an empty site. The empty sites are places where prey can proliferate and can be seen as the resource for prey surveillance. The death of predators complete this cycle, reintegrating to the system the resources for prey. The predator-prey PCA has three parameters: a, the probability of birth of prey, b, the probability of birth of predator and death of prey, and c, the probability of predator death. Two of the process are catalytic: the occupancy of a site by prey or by a predator is conditioned, respectively, to the existence of prey or predator in the neighborhood of the site. The third reaction, where predator dies, is spontaneous, that is, it occurs, with probability c, independently of the neighbors of the site. We assume that a+ b+ c = 1, (6) with 0 ≤ a, b, c ≤ 1. The transition probabilities of the predator-prey PCA are described in what follows: (a) If a site i is empty, ηi = 0, and there is at least one prey in its first neighborhood there is a favorable condition for the birth of a new prey. The probability of site i being occupied in next time step by a prey is proportional to the parameter a and to the number of prey that are in the first neighborhood of the empty site. (b) If a site is occupied by a prey, ηi = 1, and there is at least one predator in its first neighborhood then the site has a probability of being occupied by a new predator in the next instant of time. In this process the prey dies instantaneously. The transition probability is proportional to the parameter b and the number of predators in first neighborhood of the site. (c) If site i is occupied by a predator, ηi = 2, it dies with probability c. The transition probabilities associated to the three processes above mentioned can be summarized as follows: wi(0|η) = cδ(ηi, 2) + [1− fi(η)]δ(ηi, 0), (7) wi(1|η) = fi(η)δ(ηi, 0) + [1 − gi(η)]δ(ηi, 1), (8) wi(2|η) = gi(η)δ(ηi, 1) + (1− c)δ(ηi, 2), (9) where fi(η) = δ(ηk, 1), gi(η) = δ(ηk, 2), (10) and the summation is over the four nearest neighbors of site i in a regular square lattice. The notation δ(x, y) stands for the Kronecker delta function. These stochastic local rules, when inserted in equation (2), define the dynamics of the PCA for a predator- prey system. The present stochastic dynamics predicts the existence of states, called absorbing states, in which the system becomes trapped. Once the system has entered such a state Stable oscillations of a predator-prey probabilistic cellular automaton 6 it cannot escape from it anymore remaining there forever. There are two absorbing states. One of them is the empty lattice. Since the predator death is spontaneous, a configuration where just predators are present is not stationary. This situation happens whenever the prey have been extinct. In this case the predator cannot reproduce anymore and also get extinct, leaving the entire lattice with empty sites. The other absorbing state is the lattice full of prey. This situation occurs if there are few predators and they become extinct. The remaining prey will then reproduce without predation filling up the whole lattice. The existence of absorbing stationary states is an evidence of the irreversible character of the model or, in other words, of the lack of detailed balance [29]. However, the most interesting states, the ones that we are concerned with in the present study, are the active states characterized by the coexistence of prey and predators. 2.3. Time evolution equations for state functions We start by defining the densities, which are the one-site correlations, and the two-site correlations. These quantities will be useful in our mean-field analysis to be developed below. The density of prey, predator, and empty sites at time step ℓ are defined thought the expressions i (1) = 〈δ(ηi, 1)〉ℓ, (11) i (2) = 〈δ(ηi, 2)〉ℓ, (12) i (0) = 〈δ(ηi, 0)〉ℓ. (13) The evolution equations for the above densities are obtained from their definitions as state functions, as given by equation (4), and by using the evolution equation for P (ℓ)(η), given by equation (1). The resulting equations can be formally written as (ℓ+1) i (1) = 〈wi(1|η)〉ℓ, (14) (ℓ+1) i (2) = 〈wi(2|η)〉ℓ, (15) (ℓ+1) i (0) = 〈wi(0|η)〉ℓ, (16) where the transition probabilities for this model are given in equations (7), (8) and (9). The correlation between a prey localized at site i and a predator localized at site j at time step ℓ is defined by ij (1, 2) = 〈δ(ηi, 1)δ(ηj, 2)〉ℓ. (17) The other two-site correlations are defined similarly. The time evolution equation for the correlation of two neighbor sites i and j, one being occupied by a prey and the other by a predator, is given by (ℓ+1) ij (1, 2) = 〈wi(1|η)wj(2|η)〉ℓ. (18) The other two-site evolution equations are given by similar formal expressions. We can also derive equations for three-site correlations. Since we are interested here on Stable oscillations of a predator-prey probabilistic cellular automaton 7 approximations in which only the one-site and two-site correlations should be treated exactly, the above equations suffice. We call the attention to the fact that equation (18) includes the product of two transition probabilities. This is a consequence of the synchronous update of the PCA which allows that both neighboring sites i and j have their states changed at same time step. This situation does not occur when we consider a continuous time one- site dynamics. Therefore, although local interaction in the present PCA and in the continuous time model considered in reference [10] are the same, the predator-prey system evolves according to different global dynamics which leads to different time evolution equations for the densities and the correlations. The exact evolution equations for the one-site correlations are P ′j (1) = Pji(01)− Pji(12) + Pj(1), (19) P ′j (2) = Pji(12) + (1− c)Pj(2), (20) where the summation in j is over the ζ nearest neighbors of site i. To simplify notation we are using unprimed and primed quantities to refer to quantities taken at time ℓ and ℓ+ 1, respectively. The exact evolution equations for the correlations of two nearest neighbor sites j and k are P ′jk(01) = n(6=j) Pjkn(001)− i(6=k) Pijkn(1001) + (1− Pjk(01)− n(6=j) Pjkn(012) i(6=k) Pijk(101)− n(6=j) Pijkn(1012) )Pjk(21)− n(6=j) Pjkn(212) n(6=j) Pjkn(201), (21) P ′jk(12) = n(6=j) Pjkn(012) + i(6=k) Pijkn(1012) n(6=j) Pjkn(112)− i(6=k) Pijkn(2112) + (1− c) )Pjk(12)− i(6=k) Pijk(212) (1− c) i(6=k) Pijk(102), (22) Stable oscillations of a predator-prey probabilistic cellular automaton 8 P ′jk(02) = n(6=j) )Pjkn(012)− i(6=k) Pijkn(1012) + (1− c) cPjk(22) + Pjk(02)− i(6=k) Pijk(102) Pjk(21) + n(6=j) Pjkn(212)  , (23) where the summation in i is over the nearest neighbors of j and the summation in n is over the nearest neighbors of k. 3. Mean-field approximation 3.1. One and two site approximations The evolution equation for a density in any interacting particle system which evolves in time according to local interaction rules always contains terms related to the correlations between neighbor sites in a lattice. The evolution equations for the correlations of two neighbor sites includes the correlation of clusters of three or more sites in the lattice and so on. In this way we can have an infinite set of coupled equations for the correlations which is equivalent to the evolution equation for the probability P (ℓ)(η), described in equation (1) for the automaton. The scope of the dynamic mean-field approximation consists in the truncation of this infinite set of coupled equations [30, 31, 32, 33]. The lowest order dynamic mean-field approximation is the one where the probability of a given cluster is written as the product of the probabilities of each site. That is, all the correlations between sites in the cluster are neglected. For example, let us consider the cluster constituted by a center (C) site and its first neighboring sites to the north (N), south (S), east (E) and west (W) as shown in figure 2. Within the one-site approximation the probability P (N,E,W, S, C) corresponding to the cluster shown in figure 2 is approximated by P (N,E,W, S, C) = P (N)P (E)P (W )P (S)P (C), (24) where P (X), X = N,E,W, S, C are the one-site probabilities corresponding to each site. For some stochastic dynamics models this approximation is able to give qualitative results that are in agreement with the expected results. In order to get a better approximation we must include fluctuations. The simplest mean-field approximation that includes correlations is the pair-mean field approximation. This approximation is better explained by taking again, as an example, the cluster constituted by a center site which and its four nearest neighbors, shown above. Within the pair-approximation the conditional probability P (N,E,W, S |C) is approximated by P (N,E,W, S |C) = P (N |C)P (E, |C)P (W |C)P (S |C), (25) Stable oscillations of a predator-prey probabilistic cellular automaton 9 Figure 2. A site (C) of the square lattice and its four nearest neighbor sites (N, E, W, S). that is, the conditional probability P (N,E,W, S |C) is written in terms of the product of the conditional probabilities P (X|C), X = N,E,W, S. Now using the definition of conditional probability we have P (N,E,W, S, C) P (C) P (N,C) P (C) P (E,C) P (C) P (W,C) P (C) P (S, C) P (C) , (26) P (N,E,W, S, C) = P (N,C)P (E,C)P (W,C)P (S, C) [P (C)]3 . (27) We see that the resulting probability is written as a function of two-site correlations P (X,C), and the one-site correlation P (C). 3.2. Patch model The simple mean-field approximation of the predator-prey PCA describes exactly the same properties of an extended Levins patch model [7, 34]. That is, the PCA with local rules similar to the contact process becomes, in the simple mean-field approximation, analogous to the Levins model for metapopulation with empty patches, patches colonized by prey and patches colonized by predators. In the one-site mean-field approximation we consider that the probability of any cluster of sites can be written as the product of the probabilities of each site, as in equation (24). Using this approach, and writing x = Pi(1), y = Pi(2), and z = Pi(0) it can be seen that the set of equations can be reduced to the following two-dimensional map [26] x′ = x+ axz − bxy, (28) which is an evolution equation for prey density x, and y′ = y + bxy − cy, (29) which is an evolution equation for predator density yℓ. Notice that z = 1− x− y. (30) The fixed point of this map are those that represent the stationary solutions x′ = x and y′ = y, and they correspond to the three following solutions x1 = 0, y1 = 0, and x2 = 1, y2 = 0, and x3 = a/b, y3 = (1 − c/b)/(1 + b/a). The first solution corresponds to an absorbing states where both species have been extinct. The second corresponds Stable oscillations of a predator-prey probabilistic cellular automaton 10 −0.5 0 0.5 Figure 3. Phase diagram of the patch model. The continuous line represents the transition, c1(p), between the prey absorbing (A) state and the active species coexistence (C) state. The dashed line separates the two asymptotic time behavior of the active state. to an absorbing state where predators have extinct. The third solution corresponds to an active state where prey and predator coexist. Due to the constraint (6), the parameters a, b and c are not all independent and only two can be chosen as independent. For this reason it is convenient to introduce the following parametrization [10] − p, b = + p, (31) and consider p and c as the independent variables. The parameter p is such that −1/2 ≤ p ≤ 1/2 and 0 ≤ c ≤ 1 as before. This parametrization will useful in the determination of the different phases displayed by the model. A linear stability analysis reveals that solution the (x1, y1) is a hyperbolic saddle point for any set of the parameters a, b and c and so it is always unstable. The empty absorbing state will never be reached. A linear stability analysis also shows that the solution (x2, y2) is a stable node in the following region of the phase diagram c > c1 where c1(p) = (1 + 2p). (32) The active solution is stable in the region c < c1 and is attained in two ways: by an asymptotic stable focus, where the successive interactions of the map show damped oscillations; or trough an asymptotic stable node. In the phase diagram of figure 3 we show the transition line between the prey absorbing state and the active state given by c = c1. In figure 4 it is shown the behavior of the densities against the parameter c, the probability of predators death, for the special case p = 0.2. In terms of phase transitions what happens is that in the phase diagram there is a transition line separating the absorbing prey phase and the active phase which is characterized by constant and nonzero densities of prey and predator. Stable oscillations of a predator-prey probabilistic cellular automaton 11 0 0.1 0.2 0.3 0.4 0.5 0.6 predator Figure 4. Densities of predator and prey as functions of the parameter c for p = 0.2, for the patch model. We may conclude that the mean-field approximation for the predator-prey probabilistic cellular automaton with rules (7), (8), and (9) is capable to show, under a robust set of control parameters, that prey and predators can coexist without extinction. However the map defined by equations (28) and (29) is not able to describe self-sustained oscillations of species population densities. 3.3. Quasi-spatial model In order to find if oscillations in the species populations can be described within a mean- field approach we consider a more sophisticated approximation, the pair-approximation, where correlations of two neighbor sites are included in the time evolution equations for the densities. This is the lowest order mean-field approximation which takes into account the spatial localization of neighboring individuals. In this analysis we will maintain the correlations of one site and the correlations of two-sites in the equations. Correlations of three and four neighbor sites will be approximated by means of equation (27). With these approximation the model is described by the following set of five coupled equations x′ = au− bv + x, (33) y′ = bv + (1− c)y, (34) u′ = αa[ ] + [(1− βa)− αa ][u− αb + αac + c[(1− βb)v − αb ], (35) v′ = αb[βa ] + αa(1− c) + αb[ ] + (1− c)[(1− βb)v − αb ], (36) Stable oscillations of a predator-prey probabilistic cellular automaton 12 −0.5 0 0.5 Figure 5. Phase diagram of the quasi-spatial model. The upper continuous line represents the transition, c1(p), between the prey absorbing (A) state and the nonoscillating coexistence (CNO) state. The lower continuous line represents the transition, c2(p), between the nonoscillating coexistence and the oscillating (COS) coexistence state. The dashed line separates the two asymptotic time behavior of the nonoscillating coexistence state. w′ = αb[(1− βa) ] + (1− c)[w − αa + c[βbv + αb ] + c(1− c)s, (37) where α and β are numerical fractions defined by α = (ζ − 1)/ζ and β = 1/ζ where ζ is the coordination number of the lattice. For the present case of a square lattice, ζ = 4 so that α = 3/4 and β = 1/4. We are using the following notation: u = P (0, 1), v = P (1, 2), and w = P (0, 2) and also r = P (1, 1), q = P (0, 0) and s = P (2, 2). The last three correlations are not independent but are related to others by r = x− u− v, (38) q = z − u− w, (39) s = y − v − w. (40) We used the properties P (1, 0) = P (0, 1), P (1, 2) = P (2, 1) and P (2, 0) = P (0, 2), that follows from the assumption that space is isotropic and homogeneous. We have analyzed numerically the five-dimensional map, described by the set of equations (33), (34), (35), (36) and (37), and we have obtained four types of solutions. Two solutions are trivial and are given by x = y = u = v = w = 0 and x = 1, y = u = v = w = 0. They correspond to the empty and prey absorbing states, respectively. The empty absorbing state, where both species have been extinct is an unstable solution and never occurs. However, the prey absorbing state is one of the possible stable stationary solutions and is stable above the critical transition line Stable oscillations of a predator-prey probabilistic cellular automaton 13 0 0.1 0.2 0.3 0.4 0.5 0.6 predator Figure 6. Densities of predator and prey as functions of c for the quasi-spatial model, for p = 0.2. c = c1(p) shown in figure 5. Below this line it becomes unstable giving rise to the active state. The other solutions correspond to the active states where both prey and predator coexist. These solutions are of two kinds: a stationary solution where there is a coexistence of the two species with densities constant in time, which we call the nonoscillating (NO) active state; and another solution where both population densities oscillate in time. This solution corresponds to a self-sustained oscillation of the predator- prey system and will be called the oscillating (O) active state. In the phase diagram of figure 5 there is a line c = c2(p) that separates the NO and O active phases. Figure 6 shows the behavior of the densities as a function of c for p = 0. 3.4. Oscillatory behavior In figure 7 we show an example of self-sustained oscillations of the densities of prey and predators as functions of time. The oscillating solutions are attained from the nonoscillating solutions by a Hopf bifurcation. The fixed point associated to this solution is an unstable center which produces a stable limit cycle as trajectories in the phase- space of the predator density versus prey density, as can be seen in figure 7. Notice that the oscillations are not damped and have a well defined period which is the same for the prey density and for the predator density, which implies that the oscillations are coupled. A maximum of predators always follow a maximum of prey. This means that the abundance of prey is a condition that favors the increase in the number of predators. As the predator number increases the prey population decays. The evanescence of prey is followed by a decrease in the predator number, giving conditions for the increase of prey population until the cycle starts again. A well defined oscillatory behavior is found for many biological population, the most famous being the one related to the time oscillations of the population of lynx and snowshoe hare in Canada for which data were collected for a long period of time Stable oscillations of a predator-prey probabilistic cellular automaton 14 1000 1200 1400 1600 1800 2000 predator 0 0.05 0.1 0.15 Prey population density Figure 7. (a) Densities of predator and prey as functions of time and (b) density of predator versus density of prey, for the quasi-spatial model, for p = 0 and c = 0.016. [7, 8]. If the hare population cycles are mainly governed by the lynx cycle then the oscillations shown by the present model reproduces qualitatively some of the features of this predator-prey dynamics. Next we analyze the behavior of the frequency and amplitude of oscillations. Fixing the parameter p and varying the parameter c, we verify that in all the oscillating region the frequency of oscillation is proportional to parameter c, ω ∼ c, (41) as can be seen in figure 8. Low frequencies are associated to low values of c; what means that, for small values of c, the greater the predator lifetime the greater will be period of the oscillation. As to the amplitude A of the oscillations, we have verified, that fixing the value of p and varying the parameter c, it increases as c decreases. Our results show that, A ∼ (c− c2) 1/2, (42) as expected for a Hopf bifurcation and shown in figure 8. The transition line c = c2 from the oscillating phase to the nonoscillating phase can either be obtained by using the criterion given by equation (42) or by analyzing the eigenvalues associated to the map given by the set of equations (33), (34), (35), (36) and (37). This last criterion means to find the points of phase diagram such that the real part of the dominant complex eigenvalue equals 1. 4. Discussion and conclusion The main result coming from the pair mean-field approximation applied to the predator- prey PCA is that it is possible to describe coexistence and self-sustained time oscillations. Moreover, these are stable oscillations. Given a set of parameters, just one limit cycle is achieved, no matter what the initial conditions are. This property is essential in describing a biological system since a small variation in the initial condition can not Stable oscillations of a predator-prey probabilistic cellular automaton 15 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 Figure 8. (a) Frequency of oscillations ω versus the parameter c. The frequency vanishes linearly as one approaches c = 0. (b) Amplitude A of oscillations versus c near the Hopf bifurcation point c2 = 0.019. The quantity A 2 vanishes linearly when c → c2 in accordance with a Hopf bifurcation. modify the amplitude, frequency and mean value of the time oscillation densities of a predator-prey system. Similar results were obtained from a continuous time version of the present model [10]. Although the simple mean-field equations are essentially the same in both versions this is not the case concerning the pair mean-field approximation. The time evolutions of the pair correlations for the PCA, presented here, depend on higher order correlations (up to fourth) when compared to the ones of the continuous version (up to third). The model studied here is a spatial structured model with individuals residing in sites of a lattice and described by discrete dynamic variables. When we perform simple mean-field approximation we neglect all the correlations of sites in the lattice. But we take into account that there are limited resources for the surveillance of each species. For example in the time evolution equation for the density of prey we have an explicit term relative to reaction of birth of prey which is the product of the density of prey x by the density of empty sites z = (1−x−y). This coincides with an extended patch model approach for predator-prey systems. The presence of this term is what differs the simple mean-field equations from the Lotka-Volterra equations. However, taking into account the limitation of space and resources the simple mean-field equations are not sufficient to get self-sustained oscillations although able to describe damped time oscillations of population densities. To get self-sustained time oscillations we had to proceed to the next level of approximation in which a pair of nearest neighbor sites is treated exactly. This approximation can be seen as representing a pair of nearest neighbor sites immersed in a mean field produced by the rest of the lattice. The most important feature being the fact that the two sites of this pair can be seen as localized in space. The set of five equations which results from the pair approximation for the PCA is indeed able to produce self-sustained oscillations of population densities. It presents an important Stable oscillations of a predator-prey probabilistic cellular automaton 16 property that the Lotka-Volterra model lacks, namely, the oscillating solutions are stable and are unique for a given set of the control parameters. Acknowledgements The authors have been supported by the Brazilian agency CNPq. References [1] Lotka A 1920 J. Am. Chem. Soc. 42 1595 [2] Lotka A 1920 Proc. Nat. Acad. of Sciences USA 6 410 [3] Lotka A 1924 Elements of Mathematical Biology (new York: Dover) [4] Volterra V 1931 Leçons sur la Théorie Mathématique de la Lutte pour la Vie Paris: Gauthier- Villars) [5] Haken H 1976 Synergetics, An Introduction (Berlin: Springer) [6] Renshaw E 1991 Modelling Biological Populations in Space and Time (Cambridge: Cambridge University Press) [7] Hastings A 1997 Population Biology: Concepts and Models (New York: Springer) [8] Ricklefs R E and Miller G L 2000 Ecology (New York: Freeman) [9] Tainaka K 1989 Phys. Rev. Lett. 63 2688 [10] Satulovsky J and Tomé T 1994 Phys. Rev. E 49 5073 [11] Durrett R and Levin S 1994 Theor. Popul. Biol. 46 363 [12] Hanski I and Gilpin M E (eds.) 1997 Metapopulation Biology: Ecology, Genetic and Evolution (San Diego: Academic Press) [13] Satulovsky J and Tomé T 1997 J. Math. Biol. 35 344 [14] Tilman D and Kareiva P 1997 Spatial Ecology: the Role of Space in Population Dynamics and Interactions (Princeton: Princeton University Press) [15] Fracheburg L and Krapvisky P 1998 J. Phys. A 31 L287 [16] Liu Y C, Durrett R and Milgroom M 2000 Ecol. Model. 127 291 [17] Antal T, Droz M, Lipowsky A and Odor G 2001 Phys. Rev. E 64 036118 [18] Ovaskanien O, Sato K, Bascompte J and Hanski I 2002 J. Theor. Biol. 215 95 [19] Aguiar M A M, Sayama H, Baranger M and Bar-Yam Y 2003 Braz. J. Phys. 33 514 [20] de Carvalho K C and Tomé T 2004 Mod. Phys. Lett. B 18 873 [21] Nakagiri N and Tainaka K 2004 Ecol. Model. 174 103 [22] Szabó G 2005 J. Phys. A 38 6689 [23] Stauffer D, Kunwar A and Chowdhury D 2005 Physica A 352 202 [24] de Carvalho K C and Tomé T 2006 Int. Mod. Phys. C 17 1647 [25] Mobilia M, Georgiev I T and Tauber U C 2006 Phys. Rev. E 73 040903 [26] Arashiro E and Tomé T 2007 J. Phys. A 40 887 [27] Liggett T M 1985 Interacting Particle Systems (New York: Springer) [28] Marro J and Dickman R 1999 Nonequilibrium Phase Transitions (Cambridge: Cambridge University Press) [29] Tomé T and de Oliveira M J 2001 Dinâmica Estocástica e Irreversibilidade (São Paulo: Editora da Universidade de São Paulo) [30] Tomé T 1994 Physica A 212 99 [31] Tomé T, Arashiro E, Drugowich de Feĺıcio J R and de Oliveira M J, 2003 Braz. J. Phys. 33 458 [32] Dickman R 1986 Phys. Rev. A 34 4246 [33] Tomé T and Drugowich de Feĺıcio J R 1996 Phys. Rev. E 53 3976 [34] Levins R 1969 Bull. Entomol. Soc. Am. 15 237 Introduction Model Probabilistic cellular automaton Predator-prey probabilistic cellular automaton Time evolution equations for state functions Mean-field approximation One and two site approximations Patch model Quasi-spatial model Oscillatory behavior Discussion and conclusion

0704.0513

SDSS J233325.92+152222.1 and the evolution of intermediate polars

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 4 November 2018 (MN LATEX style file v2.2) SDSSJ233325.92+152222.1 and the evolution of intermediate polars John Southworth1 ⋆, B. T. Gänsicke1, T. R. Marsh1, D. de Martino2, A. Aungwerojwit1,3 1 Department of Physics, University of Warwick, Coventry, CV4 7AL, UK 2 INAF – Osservatorio di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy 3 Department of Physics, Faculty of Science, Naresuan University, Phitsanulok, 60500, Thailand 4 November 2018 ABSTRACT Intermediate polars (IPs) are cataclysmic variables which contain magnetic white dwarfs with a rotational period shorter than the binary orbital period. Evolution- ary theory predicts that IPs with long orbital periods evolve through the 2–3hr pe- riod gap, but it is very uncertain what the properties of the resulting objects are. Whilst a relatively large number of long-period IPs are known, very few of these have short orbital periods. We present phase-resolved spectroscopy and photometry of SDSS J233325.92+152222.1 and classify it as the IP with the shortest known or- bital period (83.12 ± 0.09min), which contains a white dwarf with a relatively long spin period (41.66± 0.13min). We estimate the white dwarf’s magnetic moment to be µWD ≈ 2× 10 33 Gcm3, which is not only similar to three of the other four confirmed short-period IPs but also to those of many of the long-period IPs. We suggest that long-period IPs conserve their magnetic moment as they evolve towards shorter or- bital periods. Therefore the dominant population of long-period IPs, which have white dwarf spin periods roughly ten times shorter than their orbital periods, will likely end up as short-period IPs like SDSS J2333, with spin periods a large fraction of their orbital periods. Key words: stars: novae, cataclysmic variables – stars: binaries: close – stars: white dwarfs – stars: magnetic fields – stars: individual: SDSS J233325.92+152222.1 1 INTRODUCTION Cataclysmic variables (CVs) are interacting binary stars containing a white dwarf primary star and a low-mass sec- ondary star in a tight orbit (Warner 1995). In most of these systems the secondary star is unevolved and fills its Roche lobe, losing material to the white dwarf primary via an accretion disc. About a quarter of CVs contain magnetic white dwarfs, and these systems are split into two cate- gories: the AMHer stars (polars) and the intermediate po- lars (IPs). In polars, the strong magnetic field of the white dwarf causes its spin period, Pspin, to synchronise to the or- bital period, Porb. The evolution of these objects is expected to be strongly affected by interaction between the magnetic fields of the white dwarf and the late-type secondary star (Webbink & Wickramasinghe 2002). In IPs it is believed that the lower magnetic field of the white dwarf means that ⋆ E-mail: [email protected] (JS), [email protected] (BTG), [email protected] (TRM) its rotation has not become synchronised to the orbital mo- tion. The evolution of IPs and in particular their relation to polars is still a debated issue (Hellier 2001; Norton et al. 2004, hereafter NWS04). While the total number of confirmed IPs has been boosted substantially over the past few years (e.g. Woudt & Warner 2003; Araujo-Betancor et al. 2003; Rodŕıguez-Gil et al. 2004; Woudt et al. 2004; Schlegel 2005; Rodŕıguez-Gil et al. 2005; Gänsicke et al. 2005; Bonnet-Bidaud et al. 2006; Norton & Tanner 2006; de Martino et al. 2006), only a handful of these have short orbital periods (i.e., below the 2–3 hr gap which is prevalent in the observed population of CVs). Con- versely, the known population of polars mostly lies below Porb = 4hr (Webbink & Wickramasinghe 2002). This has led to suggestions that long-period IPs may evolve into short-period polars (see Patterson 1994). A detailed theoretical study by NWS04 found that in general this should be the case. However, those with a white dwarf magnetic moment of µWD . 5 × 10 33 Gcm3 and secondary stars with weak magnetic fields will remain IPs, because the c© 0000 RAS http://arxiv.org/abs/0704.0513v1 2 Southworth et al. magnetic interaction between the stars is not strong enough to synchronise the the rotation of the white dwarf to the orbital motion. This picture of the evolution of magnetic CVs has not yet been confirmed observationally because very few short-period IPs are known. We are conducting a research program (Gänsicke 2005) to measure the orbital periods of objects spectroscopi- cally identified as CVs (Szkody et al. 2002, 2003, 2004, 2005, 2006) by the Sloan Digital Sky Survey (SDSS). The main motivation of this work lies in the characterisa- tion of an homogeneously identified sample of CVs which can be used to investigate the properties of the intrinsic CV population. The relatively faint limiting magnitude of the SDSS spectroscopic observations means that this sur- vey is much less biased towards intrinsically brighter ob- jects than previous work. A very high proportion of the SDSS CVs which we have studied so far are faint short- period systems (Gänsicke et al. 2006; Southworth et al. 2006, hereafter Paper I; Littlefair et al. 2006). In Paper I we found that SDSSJ023322.61+005059.5 was a probable IP with Porb = 96.08 ± 0.09min and Pspin ≈ 60min. Here we present a detailed analysis of another system, SDSSJ233325.92+152222.1 (hereafter SDSSJ2333), which we find to be an IP with Porb = 83.12min and Pspin = 41.66min1. Szkody et al. (2005) presented the identification spec- trum of SDSSJ2333 alongside time-resolved photometry and spectroscopy. They found modulation at 21min in the light curve and at 82min in the Hα and Hβ emission line ra- dial velocities, and suggested that this object could be an IP with an orbital period close to the 80min period mini- mum for CVs (Knigge 2006). The SDSS spectrum shows the Balmer and He I emission lines characteristic of CVs as well emission at He II 4686 Å which is a hallmark of magnetic systems. Whilst the Balmer and He I lines have the double- peaked profile indicative of an accretion disc, the He II line is single-peaked. However, it is difficult to conclude anything from this because the line may be contaminated by emission from the C III and N III Bowen blend. X-ray emission from SDSSJ2333 was not detected by ROSAT. 2 OBSERVATIONS AND DATA REDUCTION 2.1 WHT spectroscopy Spectroscopic observations were obtained in 2006 September using the ISIS double-beam spectrograph on the William Herschel Telescope (WHT) at La Palma (Table 1). For the red arm we used the R316R grating and Marconi CCD binned by factors of 2 (spectral) and 4 (spatial), giving a wavelength range of 6300–9200 Å and a reciprocal disper- sion of 1.66 Å per binned pixel. For the blue arm we used the R600B grating and EEV12 CCD with the same bin- ning factors, giving a wavelength coverage of 3640–5270 Å at 0.88 Å per binned pixel. From measurements of the full 1 The reduced spectra and photometry presented in this work will be made available at the CDS (http://cdsweb.u-strasbg.fr/) and at http://www.astro.keele.ac.uk/∼jkt/ Figure 2. Greyscale plots of the continuum-normalised and phase-binned trailed spectra around the Hα and Hβ emission lines. Other Balmer lines give similar result, but trails for the helium emission lines are too noisy to be useful. widths at half maximum (FWHMs) of arc-lamp and night- sky spectral emission lines, we find that this gave resolutions of 3.5 Å (red arm) and 1.8 Å (blue arm). Data reduction was undertaken using optimal extrac- tion (Horne 1986) as implemented in the pamela2 code (Marsh 1989). The wavelength calibration was interpolated from copper-neon and copper-argon arc lamp exposures taken every hour. We removed the telluric lines and flux- calibrated the target spectra using observations of G191- B2B, treating each night separately. A total of 38 spectra were obtained over two nights and the average spectrum is shown in Fig. 1. They show strong double-peaked Balmer emission and double-peaked emission from the other usual suspects, He I, Fe II and Ca IIK. There is no hint of the Balmer lines of the under- lying white dwarf or of Ca II emission. However, there is significant single-peaked emission at He II 4686 Å, which is characteristic of systems containing magnetic white dwarfs (e.g., DWCnc, Rodŕıguez-Gil et al. 2004). The trailed spec- tra (Fig. 2) show that the emission lines have double-peaked emission with a strong S-wave component. The spectrum of SDSSJ2333 is reminiscent of those of the short-period IPs HTCam (Kemp et al. 2002), DWCnc (Rodŕıguez-Gil et al. 2004) and V1025Cen (Buckley et al. 1998). The equivalent widths of Hα, Hβ and He II 4686 Å are 78 ± 4, 56 ± 3 and 8±1 Å, respectively. The ratio EW (4686) EW (Hβ) = 0.14±0.02 is typ- ical for short-period IPs (e.g., DWCnc) and not unusual for short-period non-magnetic CVs (e.g. Thorstensen & Fenton 2003). 2.2 Calar Alto photometry Unfiltered differential photometry of SDSSJ2333 was ob- tained on the night of 2006 September 13 using the 2.2m telescope at Calar Alto Observatory equipped with the CAFOS focal reducer (Table 1). We used the blue- optimised SITe CCD without binning, giving a plate scale pamela and molly were written by TRM and can be found at http://www.warwick.ac.uk/go/trmarsh c© 0000 RAS, MNRAS 000, 000–000 http://cdsweb.u-strasbg.fr/ http://www.warwick.ac.uk/go/trmarsh The intermediate polar SDSSJ233325.92+152222.1 3 Figure 1. Flux-calibrated averaged blue-arm (left) and red-arm (right) WHT spectra of SDSS J2333. Table 1. Log of the observations of SDSS J2333. Start date Start time End time Telescope and Optical Number of Exposure (UT) (UT) (UT) instrument element observations time (s) 2006 09 13 19 57 23 58 CA 2.2 CAFOS (unfiltered) 293 40 2006 09 25 00 46 04 46 WHT ISIS R600B R316R 22 600 2006 09 25 23 07 01 39 WHT ISIS R600B R316R 15 600 of 0.53” px−1. The images were reduced and aperture pho- tometry was performed using the pipeline described by Gänsicke et al. (2004), which applies bias and flat-field cor- rections within midas3 and uses the sextractor package (Bertin & Arnouts 1996) to obtain aperture photometry for all objects in the field of view. The differential magnitudes were converted into apparent magnitudes using the SDSS g and r photometry of the main comparison star. 3 DATA ANALYSIS 3.1 Radial velocity analysis We measured radial velocities from the Balmer emission lines by cross-correlation with single and double Gaussian functions (Schneider & Young 1980). For each line we tried a range of different widths and separations for the Gaussians in order to verify the consistency of our results (see Paper I for further details). We found that the best results for period determination were obtained using the Hα line and a sin- gle Gaussian of FWHM 1000 kms−1, as expected given the prominence of the S-wave emission in the spectra (Fig. 2). For a double Gaussian the best parameters were FWHM 300 kms−1 and separation 2000 kms−1. The choice of emis- sion line, measurement technique and Gaussian parameters does not have a significant effect on the derived period. Ra- dial velocities derived for the He I 4471 Å line gave similar results but with much larger noise as the line is weaker. Periodograms were calculated from the measured Hα radial velocities using the Scargle (1982) method, analysis 3 http://www.eso.org/projects/esomidas/ of variance (AoV; Schwarzenberg-Czerny 1989) and orthog- onal polynomials (ORT; Schwarzenberg-Czerny 1996), as implemented within the tsa4 context in midas. All three types of periodogram contain a strong signal close to 83min, accompanied by one-day aliases, and a few small peaks at much shorter periods (Fig. 3). As the three periodogram techniques agree well we have restricted further analysis to the Scargle algorithm, which is the most appropriate for a simple sinusoidal variation. We obtained the orbital period by fitting a circular spectroscopic orbit (sine wave) to the radial velocities (Fig. 3) using the sbop5 program, which we have previously found to give reliable error estimates for the optimised parameters (Southworth et al. 2005). Whilst the radial velocities measured using a single Gaussian have a larger scatter than those from the double-Gaussian tech- nique, the velocity variation in the core of the emission lines is much larger. These radial velocities therefore give the best results, and we find a period of Porb = 83.12 ± 0.09min. To investigate the probability that our chosen alias is the correct one we have both fitted sine curves for each of the possible alias periods and conducted bootstrapping sim- ulations (see Paper I). For the sine curves we would expect the lowest errors and residuals, and the highest amplitude, for the correct period. The bootstrapping gives results which can be directly interpreted as a probability distribution, but there will be some bias away from the correct period because the process of resampling the observations leads to a loss of temporal resolution, degrading the resulting periodograms. Thus bootstrapping will underestimate the probability that http://www.eso.org/projects/esomidas/doc/user/98NOV/volb/node220.html 5 Spectroscopic Binary Orbit Program, written by P. B. Etzel, http://mintaka.sdsu.edu/faculty/etzel/ c© 0000 RAS, MNRAS 000, 000–000 http://www.eso.org/projects/esomidas/ http://www.eso.org/projects/esomidas/doc/user/98NOV/volb/node220.html 4 Southworth et al. Table 2. Results of the bootstrapping analysis and the best-fitting circular spectroscopic orbits for the Hα line, both radial velocity measurement techniques and for different possible periods. The orbit adopted as a final result is indicated in bold. The results of the bootstrapping analysis are given as the percentage of simulations returning a highest peak close to a particular period. Radial velocity Orbital period Velocity amplitude Systemic velocity σrms Scargle bootstrap measurement (day) ( km s−1) ( km s−1) ( km s−1) probability (%) Double Gaussian 0.051038± 0.00011 51.6± 7.8 −2.7± 5.6 34 Double Gaussian 0.054111± 0.00012 56.3± 6.4 −0.8± 4.6 28 11.5 Double Gaussian 0.057622± 0.00013 58.4± 6.0 2.6± 4.3 26 85.6 Double Gaussian 0.061608± 0.00016 57.1± 7.0 6.6± 4.9 30 2.9 Single Gaussian 0.051149± 0.00012 151.8± 17.6 21.3± 12.9 78 Single Gaussian 0.054222± 0.00008 168.2± 11.6 27.5± 8.5 52 17.4 Single Gaussian 0.057724±0.00006 176.3±8.8 38.9±6.4 39 82.6 Single Gaussian 0.061713± 0.00009 175.5± 12.9 52.5± 9.1 55 Figure 3. Scargle periodogram of the Hα radial velocities mea- sured using a single Gaussian of width 1000 km s−1 (upper panel) and the phased velocities with the best-fitting circular spectro- scopic orbit (lower panel). On the periodogram the chosen peak is indicated with an arrow. the best period is in fact the correct one. Both the sine curve fitting and the bootstrapping simulations clearly favour the 83.12min period. 3.2 Light curve analysis The light curve of SDSSJ2333 shows clear sinusoidal vari- ation with a large amplitude of about 0.2mag (Fig. 4). An ORT periodogram of these data has strong power at one half and one quarter of the orbital period (Fig. 4) but not at the orbital period itself. A visual inspection of the light curve suggested that it exhibits a double-humped variation so we fitted the function f(x) = a1 sin 2π(x+ φ1) + a2 sin 2π(2x+ φ2) Figure 4. Calar Alto 2.2m photometry of SDSS J2333. The top panel shows the observed light curve, which displays a clear sinu- soidal variation. The middle panel contains an ORT periodogram and the lower panel depicts the phased light curve with a double- sine fit. where the optimised parameters are a1 and a2 (the ampli- tudes of variation), P (the period) and φ1 and φ2 (the phase zeropoints). This process confirmed the double-humped na- ture of the variation (Fig. 4) and yielded a period of 41.66± 0.13min. Szkody et al. (2005) found a periodicity of 21min in their light curve of this system, which was obtained with a much smaller telescope (1.0m) than our data. Aside from the fact that the double-peaked nature of the variation was not apparent in their data, this periodicity is consistent with our own measurement and demonstrates that it is stable. We have found that SDSSJ2333 has Porb = 83.12 ± 0.09min, close to the minimum orbital period for CVs, and a light curve with a strong double-humped modulation of amplitude 0.2mag and period 41.66±0.13 min. We interpret c© 0000 RAS, MNRAS 000, 000–000 The intermediate polar SDSSJ233325.92+152222.1 5 the light variation to be due to accretion heated spots on the white dwarf primary, which has a largely bipolar magnetic field. Therefore we classify SDSSJ2333 as an IP with an orbital period below the period gap and white dwarf which rotates with twice the frequency of the orbital motion. 4 DISCUSSION In Table 3 we have collected the observed orbital and spin periods for the five confirmed IPs with Porb < 2 hr. These are plotted in Fig. 5. Classification as an IP re- quires the presence of coherent variation at the white dwarf spin period over a significant span of time (e.g. Buckley 2000). As the photometric observations for an additional two objects, RXJ1039.7−0507 (Woudt & Warner 2003) and SDSSJ023322.61+005059.5 (Paper I), have only a short baseline their IP nature needs additional confirmation. The white dwarf magnetic moments, µWD, of the objects in Ta- ble 3 have been obtained from NWS04 under the assumption that the systems are in rotational equilibrium, either taken from their Table 1 or estimated from their Fig. 2. SDSSJ2333 is joined by the objects DWCancri and V1025Centauri in having Pspin ≈ 0.5Porb and µWD ∼ 2 × 1033 Gcm3. Assuming a canonical white dwarf radius of 107 m, this gives a field strength of 2MG for the white dwarf. The properties of EXHya are also similar, whereas those of the fifth system, HTCam, are quite different. The majority of IPs with Porb > 3 hr have Pspin ≈ 0.1Porb (Barrett et al. 1988; Gänsicke et al. 2005) and µWD ∼ 2 × 10 33 Gcm3 (NWS04). This magnetic moment is strikingly similar to those of SDSSJ2333, DWCnc and V1025Cen. This is an empirical indication that long-period IPs with Pspin ≈ 0.1Porb conserve µWD as they evolve and become short-period IPs with Pspin ≈ 0.5Porb. This is consis- tent with the evolutionary picture of magnetic CVs drawn by Patterson (1994); Webbink & Wickramasinghe (2002) and NWS04. HTCam is a short-period IP with a lower µWD of 2.7 × 1032 Gcm3. In the scenario we have outlined, an ob- ject such as this will result from the evolution of systems which currently resemble SDSSJ223843.84+010820.7 (Aqr 1 in Woudt et al. 2004) for which µWD = 2.6×10 32 Gcm3 has been estimated (NWS04). A similar argument can be ad- vanced for RXJ1039.7−0507, which has µWD ≈ 10 33 which is very similar to that of several known IPs (NWS04), for example 1RXSJ062518.2+733433 (Staude et al. 2003). King & Wynn (1999) have found that there is a large continuum of equilibrium spin levels for short-period IPs, and that an IP with a specific mass ratio, mass transfer rate and orbital period can have any of a wide range of values of Pspin/Porb, depending on the value of µWD The theoretical calculations of NWS04 suggest that long-period IPs will evolve into polars unless µWD . 5 × 1033 Gcm3 and the magnetic field of the secondary star is weak. In this case they may become EXHya-like systems as the interaction between the magnetic fields is too weak to synchronise the rotation of the white dwarf to the orbital motion. Whilst there are twelve long-period IPs listed in NWS04 with µWD similar to that of SDSSJ2333, there are only four short-period systems with this property. As short-period Figure 5. Comparison of the orbital and spin periods of IPs with short orbital periods. Dotted lines indicate where the spin period is 0.1, 0.25 and 0.5 times the orbital period. Those sys- tems which are not confirmed IPs are plotted with open circles. The error bars are smaller than the points for all systems except SDSS J023322.61+005059.5. CVs are predicted to be intrinsically far more common than long-period ones (de Kool 1992; Kolb 1993; Politano 1996), this means either that the vast majority of the long-period IPs become polars or that the known population of short- period IPs is much less complete than for those with longer periods. The latter possibility could easily arise as short- period IPs are in general intrinsically fainter than long- period IPs (Warner 1995). 5 CONCLUSIONS SDSSJ2333 was identified as a cataclysmic variable from a spectrum taken by the SDSS, which shows strong double- peaked Balmer emission, double-peaked He I emission and the single-peaked He II emission which is often found in CVs containing a magnetic white dwarf. We have measured its orbital period to be 83.12 ± 0.09min from a radial veloc- ity analysis of the Balmer emission lines. It is therefore an- other SDSS-identified CV with an orbital period close to the minimum period for CVs containing unevolved mass donor stars. Its light curve shows a strong variability with a double- humped nature and a period of 41.66 ± 0.13min, which is precisely half that of its orbital period. We interpret this as arising from hot spots on the surface of the white dwarf pri- mary component caused by accretion of matter controlled by a mostly dipolar magnetic field. SDSSJ2333 is therefore the shortest-period example of a relatively rare class of short- period intermediate polars. An X-ray detection of its spin period is highly desirable to further investigate its IP nature. The fact that its spin period is precisely half of its orbital period suggests the presence of a physical mechanism which is maintaining this as an equilibrium state. Four out of a total of five confirmed short-period IPs, including SDSSJ2333, have a spin period of approximately half their orbital period. These systems contain a white c© 0000 RAS, MNRAS 000, 000–000 6 Southworth et al. Table 3. Orbital and spin periods of intermediate polar CVs with orbital periods shorter than the 2–3 hr period gap. The estimated magnetic moments of the systems are from NWS04, either taken from their Table 1 or traced from their Fig. 2. ∗ The IP nature of RXJ1039−0507 and SDSS J023322.61+005059.5 has not been confirmed. References: (1) Tovmassian et al. (1998); (2) Kemp et al. (2002); (3) de Martino et al. (2005); (4) Rodŕıguez-Gil et al. (2004); (5) Patterson et al. (2004); (6) Buckley et al. (1998); (7) Hellier et al. (1998); (8) Hellier et al. (2002); (9) Mumford (1967); (10) Vogt et al. (1980); (11) Hellier & Sproats (1992); (12) Woudt & Warner (2003). System Orbital period (min) Spin period Pspin Estimated magnetic References (min) (min) Porb moment (G cm HT Cam 85.9853± 0.0014 8.58430± 0.00000 0.100 0.3×1033 1, 2, 3 DW Cnc 86.1015± 0.0003 38.58377± 0.00006 0.448 1.5×1033 4, 5 V1025 Cen 84.62 35.73± 0.05 0.422 1.8×1033 6, 7, 8 EX Hya 98.256738± 0.000006 67.02688± 0.00001 0.682 ∼5×1033 9, 10, 11 SDSS J2333 83.12± 0.09 41.66± 0.13 0.501 ∼2×1033 This work RXJ1039.7−0507 ∗ 94.4597± 0.0001 24.062 ∗ 0.255 0.9×1033 12 SDSS J023322.61+005059.5 ∗ 96.08± 0.09 60± 5 ∗ 0.625 Paper I dwarf with a magnetic moment of µWD ∼ 2 × 10 33 Gcm3 (corresponding to a field strength of about 2MG). A rela- tively large number of long-period (Porb & 3 hr) IPs have spin periods close to a tenth of their orbital periods and µWD ∼ 2×10 33 Gcm3. From this we suggest that the popu- lation of long-period IPs with Pspin ∼ 0.1Porb will conserve µWD during their later evolution and become short-period IPs with Pspin ∼ 0.5Porb. ACKNOWLEDGEMENTS This work is based on observations made with the William Herschel Telescope, which is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observa- torio del Roque de los Muchachos (ORM) of the Instituto de Astrof́ısica de Canarias (IAC). JS acknowledges financial support from PPARC in the form of a postdoctoral research assistant position. BTG ac- knowledges financial support from PPARC in the form of an advanced fellowship. TRM was supported by a PPARC senior fellowship during the course of this work. DDM ac- knowledges financial support from the Italian Ministry of University and Research (MIUR). AA thanks the Royal Thai Government for a studentship. We thank the referee for a positive report. REFERENCES Araujo-Betancor, S., Gänsicke, B. T., Hagen, H.-J., Rodriguez-Gil, P., Engels, D., 2003, A&A, 406, 213 Barrett, P., O’Donoghue, D., Warner, B., 1988, MNRAS, 233, 759 Bertin, E., Arnouts, S., 1996, A&AS, 117, 393 Bonnet-Bidaud, J. 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0704.0514

Complexity Considerations, cSAT Lower Bound

Microsoft Word - Complexity_Considerations_FOL2.doc Radosław Hofman, cSAT problem lower bound, 2007 Abstract—This article deals with the lower bound that is considered as the worst case minimal amount of time required to calculate a problem result for cSAT (counted Boolean satisfiability problem). It uses the observation that Boolean algebra is a complete first-order theory where every sentence is decidable. Lower bound of this decidability is defined and shown. The article shows that deterministic calculation model made up of finite number of machines (algorithms), oracles, axioms, or predicates is incapable of solving considered NP-complete problem when its instance grows to infinity. This is a direct proof of the fact that P and NP complexity classes differ and oracle capable of solving NP-complete problems in polynomial time must consist of infinite number of objects (i.e., must be nondeterministic). Corollary of this article clears complexity hierarchy: P < NP Index terms—complexity class, P vs NP, Boolean algebra, first order theory, first order predicate calculus. I. INTRODUCTION Unknown relation between P and NP [5] complexity classes remains one of the significant unsolved problems in complexity theory. P complexity class consists of problems solvable by deterministic Turing machine (DTM) in polynomially bounded time, while NP complexity class consists of problem solvable by nondeterministic Turing machine (NDTM) in polynomially bounded time. This means that DTM can verify the solution of every NP problem in polynomially bounded time, even if polynomial algorithm for finding this solution is unknown [13]. All known attempts to prove whether these classes are or are not equal could not convince the community that arguments used there are final. Problem with attempts showing that P=NP is mainly with counter examples provided for methods described by solvers (see for example: [6], [9]), especially for large instances. Problem with proof attempts that P≠NP touches mainly the difference between a problem and an algorithm. Proving the inequality of these classes is equivalent to proving that “there is no such algorithm that solves a particular NP problem in polynomially bounded time.” Algorithm is an immaterial object, so proving that it does not exist is rather difficult. Can then the inequality of complexity classes be proved? One of the possible ways is to use the properties of first-order theory. Useful properties include every sentence ϕ in theory T is provable if there exists a set of axioms a, b, c, … such that ϕ can be obtained using these axioms and the inference Manuscript created December 29, 2006. Author is Ph. D. student of Department of Information Systems at The Poznan University of Economics, http://www.kie.ae.poznan.pl, email: [email protected]. rules “modus ponens” and “universal generalization” (a ∧ b ∧ c… → ϕ) [2]. II. BACKGROUND This section presents some background for the first-order theory and other rules used in the article. A. First-Order Theories First-order theory is a given set of axioms in some language. Language consists of logical symbols and set constants, functions, and relation symbols (predicates). Terms and formulas are built from language and give rise to sentences, which are formulas with no free variables in body. Theory is then a set of sentences which may be closed if it contains all consequences of its elements. Theory can be also complete (i.e. every sentence can be proved or disproved), consistent (not every sentence is provable), or decidable (every sentence can be proved or disproved and there exists a computational path (algorithm) showing which sentences are provable). An example of first-order theory that is complete and decidable is Boolean algebra [16] or Zermelo–Frænkel set theory. B. First-Order Logic First-order logic, also called first-order predicate calculus (FOPC), is a system of deductions extending propositional logic. Atomic sentences of first-order logic are called predicates and are written usually in the form P(t1, t2, …,tn). An important ingredient of the first-order logic not found in propositional logic is quantification. In 1929, Gödel [8] proved that every valid logical formula is valid in first-order logic. In other words, it is proved that for complete first-order theory, inference rules of FOPC are sufficient to prove any valid formula. First-order predicate calculus language consists of predicates, constants, functions, variables, logical operators (NOT, OR, AND), quantifiers, parentheses, and some types of equality symbol. There is also a set of rules for recognition of terms and well-formed formulas (wffs). There are four axioms for quantification: 1) PRED-1: (∀ x Z(x)) → Z(t) 2) PRED-2: Z(t) → (∃ x Z(x)) 3) PRED-3: (∀ x (W → Z(x))) → (W → ∀ x Z(x)) 4) PRED-4: (∀ x (Z(x) → W)) → (∃ x Z(x) → W) An important theorem for first-order logic is the outcome from Herbrand’s work (known as Herbrand’s theorem). It states that in predicate logic without equality, a formula A in prenex form (all quantifiers at the front) is provable if and only if a sequent S comprising substitution instances of the quantifier-free subformula of A is propositionally derivable, and A can be obtained from S by structural rules and quantifier rules only. In other words, it states that the formula cSAT problem lower bound Radosław Hofman, cSAT problem lower bound, 2007 is provable, if, and only if we can rewrite it without quantifier substituting values and obtain provable formula. For example: ∀ x Z(x) = Z(0) ∧ Z(1) ∃ x Z(x) = Z(0) ∨ Z(1) C. Boolean Algebra Boolean algebra (also called Boolean lattice) is an algebraic structure containing objects and operations upon them and set of axioms (see Section D). It consists of one unary operation ¬ (not) and two binary operations ∧ (and), ∨ (or) also with two distinct elements 0 (constant representing false), 1 (constant representing true). Language of Boolean algebra considered as language for first order logic also contains symbols: = (equality), ⇒ (implication), parentheses and quantifiers, ∀ (universal), and ∃ (existential). Boolean algebra has the essentials of logic properties as well as all set operations (union, intersection, complement). D. Axioms of Boolean Algebra Given below is a complete list of Boolean algebra axioms. This set is not a minimal set of axioms (especially staring from Ax13)) – some axioms can be derived from others, but it does not change the reasoning used in this article (the list is larger only for clearness and ensuring that it is complete): Ax1) a = b can be written as (a ∧ b) ∨ (¬a ∧ ¬b) Ax2) a ⇒ b = ¬a ∨ (a ∧ b) Ax3) a ∨ (b ∨ c) = a ∨ b ∨ c = (a ∨ b) ∨ c Ax4) a ∧ (b ∧ c) = a ∧ b ∧ c = (a ∧ b) ∧ c Ax5) a ∨ b = b ∨ a Ax6) a ∧ b = b ∧ a Ax7) a ∨ (a ∧ b) = a Ax8) a ∧ (a ∨ b) = a Ax9) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) Ax10) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) Ax11) a ∨ ¬a = 1 Ax12) a ∧ ¬a = 0 Ax13) a ∨ a = a Ax14) a ∧ a = a Ax15) a ∨ 0 = a Ax16) a ∧ 1 = a Ax17) a ∨ 1 = 1 Ax18) a ∧ 0 = 0 Ax19) ¬0 = 1 Ax20) ¬1 = 0 Ax21) ¬(a ∨ b) = ¬a ∧ ¬b Ax22) ¬(a ∧ b) = ¬a ∨ ¬b Ax23) ¬¬a=a Using universal generalization, one may add in every axiom definition, the universal quantifier stating “for all x axiom body” (). E. Computational Tree Corollary of the considerations stated earlier establishes that every formula in Boolean algebra is decidable. It is said to be proved (or called “tautology”) if there exists a transformation path from a set of axioms to a sentence that we are trying to prove. Until the authors are discussing FOPC, one may say that every sentence is provable, if, and only if we can start with axiom and repeatedly apply “modus ponens” or “universal generalization” and obtain this sentence [2]. One may then consider every possible (provable) sentence to be deducible from axioms, which may be presented as the graph shown in Fig. 1. Axiom 1 Axiom 2 Axiom 3 Axiom n Sentence 1.1 Sentence 1.2 Sentence 1.3 Sentence 1.m... Modus ponens General universalization Figure 1 Example of deducible tree Axioms may also be the result of computations, especially when they are not independent (e.g., ZF axioms) or when computations fall in a cycle. Usually, during computation one would skip deduction to already proven sentences because it does not introduce any new information, so deductions to axioms would have been omitted. F. Inference Rules and Deduction Modus ponens is an inference rule using the reasoning: if a and a → b are both proved, then b is also proved. Universal generalization is an inference rule using the reasoning: if P(a) is proved, and a is a free variable, then ∀ a P(a) is also proved. Deduction theorem (in fact deduction meta-theorem) states that if formula F can be deduced from E, then the implication P → Q can be directly shown to be deducible from the empty set. Using symbol “├” for deducible, one may write: if P ├ Q then ├ P → Q. One may generalize it to a finite sequence of assumption formulas P1, P2, P3, …, Pn ├ Q: P1, P2, P3… Pn-1 ├ Pn → Q and repeat it until we obtain the empty set on the left-hand side: ├ (P1 → (…(Pn-1 → (Pn → Q))…). Deduction follows three kinds of steps: setting up a set of assumptions (hypothesis), reiteration - calling hypothesis made previously to make it recent, and deduction, which is removing recent hypothesis. If one wants to convert proof done using deduction meta-theorem to axiomatic proof, then usually the following axioms would have been involved: 1) P → (Q → P) 2) (P → (Q → R)) → ((P → Q) → (P → R)) 3) Modus ponens: (P ∧ (P → Q)) → Q G. Corollaries Theorem 1––If formula expressible in FOPC language is deducible, then every possible transformation of this formula obtained by usage inference rules and axioms is also deducible, and can be expressed in the same language. Theorem 2––If every transformation of formula is Radosław Hofman, cSAT problem lower bound, 2007 expressible in FOPC, then the optimal for certain resource for chosen computational model is also expressible in the same language. Proofs of these theorems are provided in the appendix ( VI.A and VI.B). One needs to focus on Theorem 1 and have good understanding of significance of Gödels work. Let there be considered some formula ϕ which is intended to be proven or disproved. Assuming that there exists some deterministic transformation T1 which transform formula ϕ to ϕ1: ϕ1=T1(ϕ), there can also exist another transformation T2 taking ϕ1 as input and returning ϕ2 as output: ϕ2=T2(ϕ1)=T2(T1(ϕ)). Continuing this idea of transformations one reaches ϕTRUE or ϕFALSE formula allowing to prove or disprove ϕ. Theorem 1 is in fact summary of Gödels Theorem [8], stating that ϕ, ϕ1, ϕ2… ϕx can all be expressed in FOPC language. Above in fact causes statement of Theorem 2: if every possible transformation / reformulation of formula is expressible in FOPC language, then (by power of every quantification) also optimal transformation is expressible in FOPC language. It does not matter what is nature of this transformations. If they are deterministic (for certain input always returns same output in finite number of steps). Optimal way to solve the problem (decide on formula) can be then written as: ϕTRUE/FALSE=Tx(Tx-1(Tx-1(…(T2(T1(ϕ)))…)) III. CSAT LOWER BOUND A. Problem Definition In this work, the authors consider a problem called “Count of Satisfaction of Boolean Expression for formula ϕ.” This problem is almost the same as the classical SAT problem, but instead of the question “Is there an assignment to variables such that formula ϕ is satisfied?”, they ask the question “Are there at least L assignments such that formula ϕ is satisfied?”. L in problem instance is written unary, and the remaining part of the instance is exactly the same as in SAT problem (the authors assume that it is in conjunctive normal form (CNF)). It is easy to show that the problem is in NP – Guess & Check algorithm, for NDTM requires O(L*n) steps to check (certificate size is L*v where v is the number of Boolean variables used). It is also easy to show that the problem is NP-complete. One can show it using reduction from SAT problem and ask the question “Is there at least L=1 assignment such that formula ϕ is satisfied?” B. Measurable Predicate Problem question is easy to understand by a human, but it certainly extends to FOPC language defined in Section II. To express it in a defined language, one needs to define predicate “µ” – measure. This predicate is a representation of sigma-additive (countably additive) measurable function known as “set cardinality.” Definition of this predicate requires one constant variable n – number of different Boolean variables used. Predicate “µ” will measure number of assignments satisfying formula ϕ. 1: µ(∅) :- 0 2: µ(TRUE) :- 2n 3: µ(FALSE) :- 0 4: µ(¬ϕ1) :- 2n-µ(ϕ1) 5: µ(a1) :- 2n-1 6: µ(a1∧a2…∧ak) ∃ ai, aj: i≠j ∧ ai=¬aj :- µ(FALSE) ∃ ai, aj: i≠j ∧ ai=aj :- µ(a1∧a2…aj-1∧aj+1...∧ak) :- 2n-k µ(ϕ1∨ϕ2) :- µ(ϕ1)+µ(ϕ2)−µ(ϕ1∧ϕ2) One may think of adding some more conditions to this predicate, but the list given earlier is sufficient to calculate measure for every formula for a defined language (growth of number of axioms and definitions is discussed in Section H). It is also compliant with sigma-measurable function definition. One may also observe that usage of measure leads to exponential number of calculations required for CNF. This is a consequence of sigma-additivity property: for any sets a and b: µ(a∪b)=µ(a)+µ(b)−µ(a∩b). If one considers m sets, then this function transforms to: ⋅−= })..({ aaPS Sa aa IU µµ , where P({a1..am}) is power set over m sets, which means that it has 2m objects in it. If one is able to calculate the measure of a set or intersection of sets, then the calculation of union of m sets requires Ω(2m) intersections to be measured. Problem question using predicate “µ” is then: “µ(ϕ)≥L?”. Direct calculation may not be the only possible way for solving problems, and the authors now analyze the definition of lower bound, deterministic and nondeterministic computation models. C. Lower Bound Definition Lower bound in Big-O notation is denoted as Ω(g(n)), and for its use in this article, one may assume that it is used to express problem lower bound. Interpretation of lower bound is “minimum value of function in the worst case,” and is defined as f(n) ∈ Ω(g(n)) ⇔ 0 inflim > ∞→ ng In most of the complexity considerations, two types of resources are used in the expressions of problem lower bounds or algorithm upper bounds. These resources are time (number of steps required) and space (number of symbols/tape cells required). Theorem 3––Time complexity of problem/algorithm is always greater than or equal to space complexity. This theorem is proved in Section VI.C. Theorem 4––Minimal number of symbols required for unambiguous description of object is Ω(log(N)), where N represents the number of possible objects to be stored. In other words, this means that if one has N different objects that may occur in computations at a certain step and would want to store information on which one occurred, then Ω(log(N)) Radosław Hofman, cSAT problem lower bound, 2007 symbols are required. This theorem is proved in Section VI.D. In this work, the authors mainly consider time complexity using observation from Theorems 3 and 4. Theorem 5––Lower bound calculated to express a specific resource (time or space) usage for deciding formula expressed in FOPC for a chosen computational model is equal to the minimal usage of this resource for the best possible transformation of formula in this language. This theorem is proved in Section VI.E. Theorem 5 is consequence of Theorem 2. If one had set of deterministic transformations expressing optimal way to solve the problem: ϕTRUE/FALSE=Tx(Tx-1(Tx-1(…(T2(T1(ϕ)))…)) then by power of definitions it can be shown that lower bound for problem solution is exactly equal to time required by this optimal solution. This is consequence of lower bound definition – it is asymptotically minimal amount of resource required to solve the problem. Repeating most important observations till this point: a) formula ϕ can be expressed in FOPC language (from Gödels Theorem [8]) b) any possible transformation of formula can be expressed in FOPC language ϕ1=T1(ϕ) (Theorem 1) c) if every deterministic transformation can be expressed in FOPC language then also optimal deterministic transformation can be expressed in FOPC language (Theorem 2) ϕTRUE/FALSE=Tx(Tx-1(Tx-1(…(T2(T1(ϕ)))…)) d) resource cost of optimal transformation of formula is equal to deterministic lower bound of the problem Roughly speaking, lower bound should be considered as the minimal amount of resource used for computation for the worst case. In case of time, it is the minimal number of operations to perform. It is even intuitive to see that if one could express calculation in some “steps,” then lower bound is equivalent to minimal number of “steps” required in the worst case. D. Nondeterministic Calculation Model Nondeterministic calculation model may be considered as the “luckiest possible guesser.” Such an approach expresses that the role of NDTM to answer a problem question is to guess the certificate and check it. If the check can be performed in O(nc) for some constant c, then one considers the problem as part of NP complexity class. One has to remember that DTM is a “special case” of NDTM where from every machine state, only one possibility to choose the next state exists, regardless of the symbol in the cell where the tape read/write head is positioned. This means that every problem solvable by DTM in O(nc) steps is solvable also on NDTM in at most same number of steps (or may be less). A good example expressing the differences between DTM and NDTM is the 2SAT problem (classic satisfaction of Boolean expression in CNF problem, but where in each clause there are at most two literals). This problem is solvable by DTM in O(n3) steps, but NDTM may guess the correct assignment and verify it in O(n). In terms of first-order logic and Herbrand’s theorem, one can see that NDTM is a verifier of Herbrand’s subformulas. When the formula is expressed using existential quantifier: ∃ <a: assignment> F(a), then, according to Herbrand’s theorem, it is equivalent to: F(a1) ∨ F(a2)… ∨ F(ak). NDTM is able to check each of F(ai) simultaneously, even if the number of possible assignments is exponential, excepting when at least one of the computation paths led to an accepting state. One can see that for a nondeterministic calculation model problem, the number of steps of lower bound is equal to the minimal number of steps required to check the certificate. For example, for 2SAT problems one can have different approaches. Number of possible Herbrand’s subformulas Minimal number of steps to check each subformula Total calculatio n cost 1 2n N n (guessing only p variables) n*p3 N*p3 (without splitting) 3 n3 Table 1 Different approaches for nondeterministic calculation Table 1 presents different approaches differing mainly in the number of “guesses.” Calculation of problem lower bound for nondeterministic model of calculation returns the minimal number of steps required to check subformula. The last row presents the approach where the problem is not split, so it is calculated as in the deterministic model of calculations. E. Deterministic Calculation Model As mentioned in Section D, deterministic model of calculations follows a single computation path. It is obvious that despite direct calculations, DTM can also perform Guess & Check algorithm (simulating NDTM). This time, the authors do not assume that DTM is the “luckiest possible guesser” and for lower bound complexity calculation of this approach, they have to assume that DTM is the “worst possible guesser.” This is also a consequence of the slight change in computation goal - NDTM has to “accept” when there is computational path leading to accepting state, while DTM has to “decide” on input, which means that the answer “NO” can be produced only when there is no possible way of reaching the accepting state (NDTM can be defined without rejecting state). Additionally, DTM requires an iterator (space on tape where number of current “guess” can be stored), which according to Theorem 4 requires Ω(log(H)), where H represents the possible number of “guesses.” Table 2 shows what time complexity would look like. Radosław Hofman, cSAT problem lower bound, 2007 Number of possible Herbrand’s subformulas Minimal number of steps to check each subformula Total calculation cost 1 2n N 2n*n+log(2n) 2 2p n*p3 2n*n*p3+log(2p) K 1 n3 n3 Table 2 Different approaches for deterministic calculation In this table, the last row represents the minimal possible number of steps to calculate result. It is easy to show that for DTM, this row also presents deterministic problem lower bound because if any of the “guessing” approaches had been better, then it would have been used to present minimal deterministic calculation cost (we assume that values in the table are “best possible” not “best known” - see Theorem 5). F. cSAT Nondeterministic Algorithm Upper Bound Upper bound for algorithm solving cSAT problem is polynomial. It is a consequence of Herbrand’s theorem and ability of NDTM to: generate all possible subformulas in O(nc) verify each of them in O(nc) NDTM algorithm can be described using the following steps: 1) Guess sets of measure L consisting of assignments of variables (time O(L*v)) 2) Verify guessed set (time O(L*v)) This procedure leads to accepting the state (if at least one computation path is accepting) in at most O(L*v) steps and because instance size n∈Ω(L+v), the solution is provided in O(n2). G. cSAT Deterministic Lower Bound Now, using the observations described in the earlier sections, the authors calculate deterministic lower bound of cSAT problem (it is known that its nondeterministic upper bound is O(L*v)). First, one needs to write the problem in the FOPC language. One uses the predicate µ: µ(ϕ)≥L. This problem may be considered to be harder than the classic SAT problem. If one tries to guess all possible subsets, then we would have Ω( v22 ) subsets, so according to Theorem 4, it would require Ω(2v) symbols to store information about the considered subset, which, according to Theorem 3, leads to the conclusion that such a calculation requires at least Ω(2v) steps. “Guessing” only subsets of size L leads to Ω(2v) different subsets, so that it can be calculated by NDTM in polynomial time (Ω(v*L) steps), but requires Ω(2v*v*L) steps to calculate on DTM. In fact, following the assumption that DTM is the worst possible guesser, one may see that the number of hypotheses (“guesses”) used during computation can lead to an exponential usage of time if “depth” of hypothesis path is longer than O(log(n)) or any of the hypotheses has more than polynomial number of possible values. For example, if one states hypothesis A with possibilities, it is true or false (constant number of possibilities) and it is followed by hypothesis B (true or false), etc.; we need O(n) hypotheses before we can decide on formula, then in the worst case we require Ω(2n) steps to give the answer “NO.” Leaving then all Guess & Check approaches, the authors try to determine the minimal possible number of steps for DTM to decide on problem input. According to Theorem 5 and considerations from the earlier sections, the authors conclude that the shortest possible path consists of steps transforming input formula to axioms of theory. If one can show that every transformation requires exponential number of steps or usage of object using exponential number of symbols to store, then it will be direct proof that lower bound of cSAT problem is over-polynomial. When will one be able to observe exponential growth of minimum number of required steps? If after using an axiom or predicate, one will obtain a formula of multiplicative length by a factor greater than 1. For example, if for formula of size n1 (considered to be in CNF), the authors use Ax9) for one parenthesis, they obtain new formula in the format v1∧(n2,1)∨v2∧(n2,2)∨…∨vm∧(n2,m). In each of the m parts, one can use a variable from the beginning to remove all its negations from body, so |n2,*|<|n1|−m, but for very large n1, these parts of formula will still require further transformations, which if done only with Ax9) would lead to exponential growth. Concluding this paragraph, one may say that if transformation reduces size of formula substring by O(nc) and multiplies this shorter string in formula making string grow to n2, where n2∈Ω(n1*c), then this path leads to exponential growth of formula and thus its lower bound is Ω(2n). In Table 3, the authors present the effect obtained by usage of every possible transformation, but before this the authors define polynomial purifying function for formula. This function will use axioms Ax7), Ax8), Ax11), Ax12), Ax13), Ax14), Ax15), Ax16), Ax17), Ax18), Ax19), Ax20), Ax23), and two observations: µ(ϕ1)=µ(ϕ1∧(TRUE))=µ(ϕ1∧(v1∨v2∨…∨TRUE)); µ(v1∧ϕ1)=µ(ϕ2) where ϕ2 is obtained by replacing every occurrence of v1 in ϕ1 with constant TRUE. Roughly speaking, this function looks for variables that can be cleared out from formula and prepare it for the next step of calculation. The authors assume that at every step of calculation, formula is in a form not allowing the use of any of the above axioms or rules. It is also important to remember that the number of transformation rules does not matter - refer to Section H. Transfor mation used Length string used Result string length Lower bound for path Remarks for “worst case” Ax1) Ax2) These axioms cannot be used since input never contains these symbols Ax3) n1 n1 Ω(cSAT) Ax4) n1 n1 Ω(cSAT) These axioms do not change formula length Radosław Hofman, cSAT problem lower bound, 2007 Transfor mation used Length string used Result string length Lower bound for path Remarks for “worst case” Ax5) n1 n1 Ω(cSAT) Ax6) n1 n1 Ω(cSAT) Ax7) Ax8) These axioms cannot be used because formula is transformed by purifying function Ax9) m1+m2+ 2*m1* m2+nr- Ax10) m1+m2+ 2*m1* m2+nr- Used on two parenthesis replaces them with string of size 2*m1*m2, after using these axioms purifying function will reduce size but in the worst case only by 2 symbols Ax11), Ax12), Ax13), Ax14), Ax15), Ax16), Ax17), Ax18), Ax19), Ax20) These axioms cannot be used because formula is transformed by purifying function Ax21) n1 n1 Ω(cSAT) Ax22) n1 n1 Ω(cSAT) These axioms do not change formula length Ax23) As Ax20) above µ1 µ2 µ3 These rules cannot be used because formula is transformed by purifying function µ4 n1 n1 Ω(cSAT) Can be used with Ax21), Ax22) or Ax23), but does not change length of formula µ5 As µ3 above µ6 m1+m2 m1*m2 n Treating formula as consisting of two parts Table 3 Lower bounds for every possible transformation of cSAT formula The variables used in Table 3 are the following: n1 – Length of the formula nr – Length of the remaining part of the formula m1 – Length of first part/parenthesis of the formula m2 – Length of second part/parenthesis of the formula p1 – Number of parentheses in the formula – The authors consider asymptotic behavior of the function, so one may use kind of “mean” m1 – representing Ω(m1). It is clear that in the worst case, Ax9), Ax10), or µ6 have to be used several times before purifying function would make significant reduction of length. Lower bound is considered as the minimal worst case, so from this table it is clear that in the worst case it is Ω(mp)=Ω(2p*log(m)) and because p and m are both O(n), the whole lower bound is Ω(2n). H. More Conditions and More Axioms The above considerations prove clearly that deterministic lower bound for cSAT problem considered with FOPC language defined is exponential. But one needs to answer one additional question – Is it the result of too poor FOPC axioms set definition? Or are too few predicates defined? In [1] Baker–Gill–Solovay theorem, authors have shown that problem “Is P equal to NP?” can be relativized using oracles. Oracle is a machine (black box) that gives answers to certain type of problems in one step. One can then imagine that there are a very large number of oracles which can solve certain types of instances. DTM task is to pick up one of them (or use them sequentially because if the number of oracles is an attribute of the machine, then even if we have used millions of them, the complexity in terms of relation to instance size is O(1)). The authors presume then, that for cSAT problem, there exists some deterministic algorithm calculating answer in O(nc) steps. Following lower bound calculation, one knows that this algorithm calculates a result requiring Ω(2n) transformations. Reminding optimal transformation as described above: ϕTRUE/FALSE=Tx(Tx-1(Tx-1(…(T2(T1(ϕ)))…)) and x∈Ω(2n). The presumption made here can be presented as existence of some transformation TA≡Tx-k(Tx-k-1(…(Tx-k-m( ))…)), where m is exponential (TA is equivalent to exponential number of transformations in FOPC language, on optimal transformation path). The authors also assume that TA is deterministic, as deterministic lower bound is discussed in this section, and computable in polynomial number of steps. Now, one need to look on transformation path as on decision process, where at each step there is a decision to be made (decision which transformation is to be used). Each decision takes Ω(1) space to be stored. If m was dynamic and asymptotically equal to 2n, and also computable in polynomial number of steps then this would be equal to O(2n) decisions in O(nc) time what contradicts Theorem 3. Considering constant m (invented by algorithm designer) one may ask what is common in a large number of Turing machines (in the sense of defined algorithms), large number of axioms, large number of predicates, large number of oracles, or large m in above transformation TA? Their number is always a constant, even if very large. If then anyone defines multiple TMs, adds multiple axioms, predicates, defines large number of oracles, or finds one transformation TA equivalent to exponential number of other transformation, then in fact after defining them, one may have constant number of machines, axioms, predicates, transformations, and oracles. The authors then assume that there exists a machine denoted by LDTM in which implements are equivalent to large number of TMs, large number of axioms, predicates, implements TA and are connected to multitude of oracles. Such a defined machine is (by power of assumption) capable Radosław Hofman, cSAT problem lower bound, 2007 of answering cSAT questions for a finite number of differing input types (number of types is a consequence of maximal input size). In other words, the authors assume that there exists a machine LDTM able to answer cSAT questions for instance size less than or equal to nl. They may consider having ( )lncO different input types, and each type is covered at least by one combination of axioms, predicates, or oracles allowing LDTM to give answer in O(n) steps. One may assume that there are gl such combinations. Denoting |gi(nl)| number of instances solved by ith combination of axioms and oracles for instance nl symbols long, the authors have assumed that: |||)(| cng ≥∑ , where ( ) |||| cng <∀ Now the authors determine the ability of this machine to answer cSAT question where n=nl*y. Number of combinations of axioms and oracles remain constant (gl), but they assume that each combination covers more instances (considered to be “same type”) gl(n)=gl(nl*y)≤gl(nl) y. The number of possible types grows from ( )lncO to ( ) ( )( )ynyn ll cOcO =* . Calculating instances covered by gl definitions, we have: ngyng |)(||)*(| . If one proves that for y growing to infinity lcng |||)(| , it will be proof that not all instances of size O(nl*y) are solvable using LDTM definitions, so these large instances will require calculations using deterministic lower bound discussed in Section G. Proof is presented in VI.F, so corollary about impossibility to answer cSAT problems in polynomial time by LDTM holds. IV. COROLLARIES To summarize this article, the authors repeat the deduction path: a) formula ϕ can be expressed in FOPC language (from Gödels Theorem [8]) b) any possible transformation of formula can be expressed in FOPC language ϕ1=T1(ϕ) (Theorem 1) c) if every deterministic transformation can be expressed in FOPC language then also optimal deterministic transformation can be expressed in FOPC language (Theorem 2) ϕTRUE/FALSE=Tx(Tx-1(Tx-1(…(T2(T1(ϕ)))…)) d) resource cost of optimal transformation of formula is equal to deterministic lower bound of the problem e) TA equivalent to exponential number of transformations computable in polynomial time contradicts Theorem 3 f) large number of defined constant set of transformations, oracles, algorithms, machines ect. cannot cover all possible inputs for growing instance size (Theorem 6) g) optimal solution of problem requires Ω(2n) transformations h) deterministic lower bound for cSAT problem is then Ω(2n), then cSAT∉P (Theorem 5) i) NDTM solves cSAT in polynomial time, so cSAT∈NP j) this means that P≠NP If above considerations are correct then checking problem known to be in P has to show that it is in P using the same reasoning. Such check for 2SAT problem if presented in Section VI.G - lower bound for this problem is Ω(nc). In [1], there was presented an oracle A for which PA=NPA. Proof presented in Section VI.F and problem lower bound lead to corollary, and if A is able to solve cSAT in polynomial time, then A has to be nondeterministic - it has to consist of infinite number of objects: deterministic oracles, algorithms, DTMs, axioms, rules etc. (the authors also consider NDTM as an infinite set of DTM duplicates - each for one computational branch). This work discusses problem P=NP, as described in [5]. It may be said to relativize (see [1]) to deterministic model of computation showing that deterministic calculation model made up of finite number of machines (algorithms), oracles, axioms, or predicates is incapable of solving the considered problem when its instance grows to infinity. On the other hand, one may conclude that if restrictions on maximum input length problem are set, then the problem can be proved to be in P using a large number of machines, axioms, algorithms, predicates, or oracles. For deterministic model of computation, one knows then that P≠NP. Using Theorem 13 from [12], the authors also know that NP-complete≠(NP-P). In this theorem, the authors have proved that: if P≠NP and U is some NP-complete language then U=A∪B where neither A nor B language is NP-complete (at least one of them is also not equal to P: A≠P ∨ B≠P). Complexity classes can be put in a picture (Fig. 2): Figure 2 Relation between P, NP, and NP-complete classes V. REFERENCES [1] Baker T. P., Gill J., Solovay R., “Relativizations of the P =? NP question”, SIAM Journal on Computing, vol. 4, no. 4, 1975, pp. 431-442. [2] Barwise J., Etchemendy J., “Language Proof and Logic”, Seven Bridges Press, CSLI (University of Chicago Press) and New York, 2000. [3] Chandra A. K., Kozen D. C., Stockmeyer L. J., “Alternation”, Journal of the ACM, vol. 28, no. 1, 1981. [4] Cook S. A., “The complexity of theorem-proving procedures”, Proceedings of the Third Annual ACM Symposium on Theory of Computing, 1971, pp. 151-158. Radosław Hofman, cSAT problem lower bound, 2007 [5] Cook S. A., “P versus NP problem”, unpublished. Available at: http://www.claymath.org/millennium/P_vs_NP/Official _Problem_Description.pdf [6] Diaby M., “P = NP: Linear programming formulation of the traveling salesman problem”, 2006, unpublished. Available at: http://arxiv.org/abs/cs.CC/0609005 [7] Gallier J. H., "Logic for Computer Science: Foundations of Automatic Theorem Proving", Harper & Row Publishers, 1986. [8] Gödel K., "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme", I. Monatshefte für Mathematik und Physik, vol. 38, 1931, pp. 173-198. [9] Hofman R., “Report on article: P=NP linear programming formulation of the traveling salesman problem”, 2006, unpublished. Available at: http://arxiv.org/abs/cs.CC/0610125 [10] Jech T., “Set Theory: The Third Millennium Edition, Revised and Expanded”, ISBN 3-540-44085-2, 2003. [11] Karp R. M., “Reducibility among combinatorial problems”, In Complexity of Computer Computations, Proceedings of the Symposium of IBM Thomas J. Watson Research Center, Yorktown Heights, NY. Plenum, New York, 1972, pp. 85-103. [12] Landweber, Lipton, Robertson, “On the structure of sets in NP and other complexity classes”, Theoretical Computer Science, vol. 15, 1981, pp. 181-200. [13] Papadimitriou C.H., Steiglitz K., “Combinatorial Optimization: Algorithms and Complexity”, Prentice-Hall, Englewood Cliffs, 1982. [14] Razborov A., Rudich S., “Natural proofs”, Journal of Computer and System Sciences, vol. 55, no. 1, 1997, pp. 24-35. [15] Savitch W. J., “Relationships between nondeterministic and deterministic tape complexities”, Journal of Computation and System Science, vol. 4, 1970, pp. 177-192. [16] Tarski A., Givant S., “A Formalization of Set Theory Without Variables”, American Mathematical Society, Providence, RI, 1987. VI. APPENDIX A. Proof 1 - Proof of Theorem 1 Theorem 1 - If formula expressible in FOPC language is deducible, then every possible transformation of this formula obtained by usage inference rules and axioms is also deducible and can be expressed in the same language. This theorem is a direct consequence of FOPC definitions. If ϕ is deducible, then: • ϕ ∧ axiom • ϕ → axiom • ∀ x ϕ are also deducible. B. Proof 2 - Proof of Theorem 2 Theorem 2 - If every transformation of formula is expressible in FOPC, then the optimal for certain resource for chosen computational model is also expressible in the same language. This theorem is a consequence of Theorem 1 and FOPC definitions. If the goal of calculation is to decide on formula based on theory axioms, then it is required to obtain formula as a consequence of axioms (with empty left-hand side): ├ (P1 → (…(Pn-1 → (Pn → Q))…). The authors said that every possible transformation of formula is expressible in FOPC and this directly means that the optimal in the aspect of a certain resource (time or space) path is also expressible in FOPC. C. Proof 3 - Proof of Theorem 3 Theorem 3 - Time complexity of problem/algorithm is always greater than or equal to space complexity. This theorem is a consequence of Turing machine definition, which states that in one step, a machine can read or write one (or in general constant) number of symbols. If then f(n) symbols were written, then machine had used at least f(n) steps to write them. D. Proof 4 - Proof of Theorem 4 Theorem 4 - Minimal number of symbols required for unambiguous description of object is Ω(log(N)), where N represents the number of possible objects to be stored. In this section, the function log is considered to have ∑ in root, where ∑ represents the number of the symbols in the alphabet: log(∑)=1. The authors prove the theorem using contradiction. Suppose that one knows “compression” algorithm allowing to write each of N symbols using log(N)−f(N) symbols, where f(N) is a function such that: ∀ N: 0<f(N)<log(N). On log(N)−f(N),one can write at most ∑log(N)-f(N) different strings. )log( )log( )log( )()log( NfNfNf NfN NN Now the authors check whether this number is greater than the number of objects to be identified by checking limens: lim It is easy to see that if f(N)=0, then limens is equal to zero (which means that exactly N different objects can be described using a string of desired length), but when f(N)>1 (it is the smallest value making difference in the number of symbols used), it is negative which means that less than N objects can be represented using a string of this length. E. Proof 5 - Proof of Theorem 5 Theorem 5 - Lower bound calculated to express specific resource (time or space) usage for deciding formula expressed in FOPC for a chosen computational model is equal to the minimal usage of this resource for best possible transformation of formula in this language. Proof of this theorem is in fact a direct corollary of Radosław Hofman, cSAT problem lower bound, 2007 Theorems 1 and 2. If any transformation of formula is expressible in FOPC language, then the optimal in terms of chosen resource is also expressible in FOPC language and when the authors calculate lower bound for this resource for the whole transformation path (from input string to decidable form (to axioms)), then they obtain the value of lower bound for the considered problem. F. Proof 6 - Proof of Not Covering by Constant Set of Definitions All Possible Large Instances by LDTM Assumptions: ( ) |||| cng <∀ and |||)(| cng ≥∑ . Also gi(n) function operates on natural numbers and returns natural numbers, so ( ) |1||| cng . One want to solve yn lcng |||)(| for y growing to infinity. First one may observe that: ∑∑ |1||)(| if one can solve inequality yn yn ll cc |||1| , then it will be equivalent to prove that the proof is correct. The new equality presented by the authors is free from i variables, so it can be rewritten as: ynyn ll ccl |||1|* <− . Now the authors take logarithm on both sides to the base l: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )||log|1|log1 ||log*|1|log*1 ||log|1|loglog ||log|1|*log At this point, it is obvious that this inequality holds - 1/y when y grows to infinity may be omitted and one has inequality of two logarithms where this one on the left-hand side is the logarithm of lower value. More formally, one may calculate limens: ( ) ( ) 1loglim loglim ||log|1|log lim Proof is then correct. G. Proof 7 - Lower Bound for 2SAT Problem 2SAT problem is a special case of cSAT problem. Its special factors are: • L = 1 (problem question is “µ(ϕ)>=1” or “µ(ϕ)>0”) m1=m2=…=mp=2 The authors assume that the input string is in CNF and purifying function (defined in Section III.G) cannot be applied. They use Ax9) on parenthesis to select next parenthesis such that: • parenthesis has not been used yet • parenthesis contains negation of variable used in a previous step Every time the usage of Ax9) will be followed by purifying function usage. For example: (a∨b)1∧(a∨¬c)2∧(c∨d)3∧(¬b∨¬c)4∧(¬b∨¬a)5∧(¬d∨¬a) 6∧(e∨f)7 the authors would have used Ax9) for parenthesis: 1 and 6 (last variable is: ¬d), then 3 (lv: c), then 4 (lv: ¬b), then 1 (second time, lv: a), then 5 (lv: ¬b), then 1 (third time, lv: a), then 6 (second time, lv: ¬d), then 3 (second time, lv: c), and finally 2. Every parenthesis will be used at most on every path from any pair of parentheses. At every stage, calculation formula will contain at most p+1 conjunctions (where p is the number of parenthesis processed) and each conjunction will contain at most every variable once. Every parenthesis will be used at most p2 times, which means that at every stage of computation, formula length is O(p3). This may not be a time optimal solution. According to Theorem 2, optimal transformation path is expressible using axioms and predicates defined for FOPC, but to show that the problem is in P, one does not need to look for optimal transformation path - the authors have shown that there exists at least one transformation path polynomially bounded to instance size, and even if it is not the optimal one, it shows that 2SAT problem is in P.

0704.0515

Temperature dependence of Coulomb drag between finite-length quantum wires

arXiv:0704.0515v2 [cond-mat.mes-hall] 16 Jul 2007 Temperature dependence of Coulomb drag between finite-length quantum wires J. Peguiron,1 C. Bruder,1 and B. Trauzettel1 Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland (Dated: July 2007) We evaluate the Coulomb drag current in two finite-length Tomonaga-Luttinger-liquid wires cou- pled by an electrostatic backscattering interaction. The drag current in one wire shows oscillations as a function of the bias voltage applied to the other wire, reflecting interferences of the plasmon standing waves in the interacting wires. In agreement with this picture, the amplitude of the current oscillations is reduced with increasing temperature. This is a clear signature of non-Fermi-liquid physics because for coupled Fermi liquids the drag resistance is always expected to increase as the temperature is raised. PACS numbers: 71.10.Pm,72.10.-d,72.15.Nj Coulomb drag phenomena in coupled one- dimensional (1D) electron systems have been investigated quite extensively in the past [1–10]. The interest has mainly been driven by the fact that Coulomb drag, i.e. the electrical response of one wire as a finite bias is applied to the other wire, seems to be an ideal testing ground for Tomonaga-Luttinger- liquid (TLL) physics in nature. This is because both inter-wire and intra-wire Coulomb interactions substantially modify transport properties such as the average current and the current noise. On the experimental side, there have been a few works, some of which have claimed to have observed TLL behavior in the drag data [11–13]. Recently, Yamamoto and coworkers have measured Coulomb drag in coupled quantum wires of different lengths and found peculiar transport properties that depend, for instance, on the asymmetry of the two wires [14]. This experiment is the major motivation for our work. We analyze theoretically the Coulomb drag current of two electrostatically coupled quantum wires using the concept of the inhomogeneous TLL model [15–17]. This model is known to capture the essential physics of an interacting 1D wire of finite length coupled to non- interacting (Fermi liquid) electron reservoirs. Within this framework, we are able to study finite-length and finite-temperature effects and therefore to make quali- tative contact with the experimental setup of Ref. [14]. Since the Coulomb interaction varies between the wire regions and the lead regions, charge excitations feel the interaction difference at the boundaries and are known to exhibit Andreev-type reflections [17]. We show that these reflections play a crucial role in the Coulomb drag setup illustrated in Fig. 1. Further- more, we show that the quantum interference phe- nomena associated with the Andreev-type reflections considerably modify the drag current. This is particu- larly interesting as far as the temperature dependence of the drag current is concerned. For Fermi-liquid sys- tems, it is well known that the drag resistance should always increase as the temperature is raised [6]. In our setup instead, the drag current at a fixed drive bias can either increase or decrease as a function of temperature. It crucially depends on the interference pattern due to finite-length effects. This is a clear signature of non-Fermi-liquid physics which could be observed in the double-wire setup of Ref. [14]. The system considered consists of two in- teracting parallel wires (j = <,>) of fi- nite length L< (shorter wire) and L> (longer wire) connected to non-interacting semi-infinite 1D leads (Fig. 1) and is described by the Hamiltonian j=<,> H0j +H +HC. The intra-wire inter- action is modelled through a TLL description [15–17] H0j = Π2j + g2j (x) (∂xΦj) with the piecewise constant interaction parame- ter gj(x) = gj < 1 in the wire region |x| < Lj/2 and gj(x) = 1 in the non-interacting lead re- gions |x| > Lj/2. The Fermi velocity vFj , the in- teraction strength gj, and the wire length Lj set the frequency ωLj = vFj/gjLj of the collective plasmonic excitations hosted in each wire. A voltage eVj = interwire coupling wire < wire > FIG. 1: (color online). The system under consideration. Each interacting wire of length Lj [gray area, interaction parameter gj(|x| < Lj/2) = gj < 1] is connected to a pair of non-interacting leads [gj(|x| > Lj/2) = 1]. The region of backscattering inter-wire interaction (red dashed box) extends over the length of the shorter wire. A voltage V is applied between the leads connected to the drive wire (here the longer wire j = >) and the backscattering-induced current Idr in the drag wire (here the shorter wire j = <) is investigated. http://arxiv.org/abs/0704.0515v2 0 1 2 3 V / 2πV a) g=0.1 b) g=0.25 c) g=0.5 d) g=0.75 e) g=1 FIG. 2: Drag current as a function of drive voltage for identical wires at zero temperature (solid curves). The inter-wire interaction strength ranges from strongly inter- acting (g = 0.1) to non-interacting (g = 1) for the dif- ferent curves (with I dr = eλ 2α4g/~2ωL and VL = ~ωL/e). The dashed curves show the dominant contri- bution ∝ V 4g−2 [given in Eq. (10)] for g = 0.1, 0.25, 0.5. L − µ R applied to the leads is described by HVj = − dx µj(x)∂xΦj(x), (2) with the piecewise constant electro-chemical potential µj(x) = L for x < −Lj/2, 0 for |x| < Lj/2, R for x > Lj/2 with µ L = −µ R . This model is expected to cap- ture the essential physics of a quantum wire coupled smoothly to electron reservoirs (with typical smooth- ing length Ls) as long as Lj=<,> ≫ Ls ≫ λF , where λF is the electron Fermi wavelength [18, 19]. Fi- nally, we include an inter-wire backscattering inter- action over the length L< of the shorter wire, HC = λBS ∫ L</2 −L</2 dx cos{ 4π[Φ<(x)−Φ>(x)]}. (4) This term includes the contribution of the density- density interaction which is most relevant to Coulomb drag [1, 2] when the Fermi wave-vectors of both wires are similar in magnitude, i.e. kF< ≈ kF> [22]. In the following, we choose to apply a voltage V> = V to the longer wire (µ>L = −µ R = eV/2, drive wire) and none V< = 0 to the shorter wire (µ L = µ R = 0, drag wire). The average current in the wires may then be written as I< = Idr and I> = V − Idr. In our model, the two currents I< and I> always flow in the same direction, which is due to momentum conserva- tion. This is known as positive Coulomb drag. In order to get an expression for the drag cur- rent Idr, we follow the formalism used in [18] in the case of a single wire with an impurity. We consider the situation of weak inter-wire coupling. To second order in λBS, we obtain Idr = I ∫ 1−R drjdr(r, R), (5) with the normalization I dr = eλ 2ωL<, where α< = ωL</ωc is the ratio between the plas- mon frequency of the shorter wire and a cutoff fre- quency ωc, of the order of the wire bandwidth. The plasmon frequency defines a voltage VL = ~ωL</e and a temperature TL = ~ωL</kB. It is conve- nient to introduce corresponding dimensionless volt- age u = V/VL and temperature θ = T/TL. The inte- grand in (5), jdr(r, R) = eiuτ − e−iuτ × exp 4πC<(r, R; τ) + 4πC> , (6) involves the parameters l = L>/L<, p = g>/g<, and q = vF>/vF<. The correlation function Cj = CGSj + C j of each wire can be decomposed in a zero-temperature and a finite-temperature contribu- tion given by [18] CGSj (r, R; τ) = − αj + i(τ − sr − 2k) αj + i(−2k) |2k+1| αj + i(τ − sR− 2k − 1) [α2j + (r − sR− 2k − 1)2]1/2 , (7) CTFj (r, R; τ) = − sinch[πθ(τ − sr − 2k)] sinch[πθ(−2k)] |2k+1| sinch[πθ(τ − sR− 2k − 1)] sinch[πθ(r − sR− 2k − 1)] , (8) with γj = (1− gj)/(1 + gj) and sinchx = (sinhx)/x. It is to be noted that the expression for the drag cur- 3rent Idr does not depend on whether the drive wire is the longer wire or the shorter one due to the symme- try of our model. In the following, we present results obtained by numerical evaluation of the triple integral involved in (5) and (6) and discuss several analytical approximations. First we set the temperature to zero and consider identical wires (l = p = q = 1, thus we drop the wire index j). The drag current shows non-monotonous behavior and oscillations with period ∼ 2π~ωL/e as a function of the bias voltage (Fig. 2). It decays at large voltages for g < 1/2 whereas it increases for g > 1/2. Thus, we obtain qualitatively the same behavior as in a dual Coulomb drag setup where a drive current is applied and a drag voltage is measured [2]. Similar os- cillations as a function of voltage have been predicted in the context of two coupled fractional quantum Hall line junctions [20]. An analytic approximation can be derived in the limit u = eV/~ωL ≫ 1, that is for high voltages or long wires. In Eq. (7), the terms proportional to γ|m| account for contributions from plasmon exci- tations reflected |m| times inside the wire. When the wire length is much longer than other relevant length scales, the contribution without any reflection m = 0 becomes dominant and yields the integrand jdr(r, R) ∼ Γ(2g) )2g−1/2 J2g−1/2(ur), (9) where Γ(z) denotes the Gamma function and Jν(z) the Bessel function of order ν. The resulting expres- sion for the drag current, which involves hypergeo- metric functions, underestimates the amplitude of the oscillations with respect to the exact numerical re- sult. However, the behavior at large u, governed by the dominant contribution Idr ∼ 2Γ2(2g) )4g−2 , (10) shows good agreement in the appropriate parameter regime (dashed curves in Fig. 2). Since we do per- turbation theory in λBS, the relation Idr ≪ (e2/h)V has to hold. In the large u regime, this means (λBSL/~ωc) 2(ωL/ωc)(eV/~ωc) 4g−3 ≪ 1. Now we consider the situation where the two wires have different lengths. The qualitative behavior of the drag current does not change for increasing length ra- tio l = L>/L<, but the peak positions get shifted to lower voltages (Fig. 3). Here, neglecting plasmon re- flections in the correlation function of the wires leads again to the expression (9), which is independent of l. This fact explains why the drag current does not change appreciably as a function of l and indicates that the peak shifts observed result from plasmon re- flections inside the wires. Our studies are the first to analyze the effect of an asymmetry in the length on Coulomb drag phenomena in coupled quantum wires which is of recent experimental relevance [14]. 0 0.5 1 1.5 2 Idr/Idr L>/L<V/2πVL Idr/Idr FIG. 3: (color online). Drag current as a function of the bias voltage and of the length ratio of the wires (with dr = eλ 2ωL< and VL = ~ωL</e). We now discuss in detail the temperature depen- dence of the drag current. For clarity, we consider again symmetric wires (l = 1) [23]. The oscilla- tions of the drag current as a function of bias voltage get washed out with increasing temperature (Fig. 4). This behavior is consistent with the picture which at- tributes the oscillations to interferences of the plas- mon excitations of the wires. Thus, for bias voltages such that the drag current is close to a maximum at zero temperature, one observes a decrease of the drag current with increasing temperature, whereas the op- posite behavior can be observed close to a minimum of the zero-temperature drag current for strong interac- tions. This behavior is in stark contrast to the linear temperature dependence predicted for Coulomb drag between 1D Fermi-liquid conductors [4] and therefore a clear signature of TLL physics. Note that our Fig. 4 bears significant resemblance to Fig. 9 of Ref. [12]. A good approximation of the drag current can be obtained for temperatures much larger than the tem- perature associated with the plasmon frequency, θ = kBT/~ωL ≫ 1, by neglecting contributions from plas- mon reflections in the wires. Then, we obtain the dominant contribution Idr ∼ dr (2πθ) 4g−2 sinh g + iu 4Γ2(2g) Taking the limit of low bias voltage in this result, Idr ∼ θ≫1,u πΓ4(g) 4Γ2(2g) (2πθ)4g−3u, (12) we recover the power-law dependence T 4g−3 of the lin- ear conductance predicted by renormalization group analysis [5]. At large bias voltage u ≫ θ ≫ 1, we recover the zero-temperature result (10). In the case g = 1/2, Eq. (11) as well as the con- tribution to next order in θ−1 can be brought into a 0 1 2 3 V / 2πV 0 1 2 3 g=0.25 g=0.6 FIG. 4: Temperature dependence of the drag current for identical wires with interaction strength g = 0.25. The solid curves (labelled a - e) are evaluated for tempera- tures given by T/TL = kBT/~ωL = 0, 0.5, 1, 1.5, 5, respec- tively (with I dr = eλ 2α4g/~2ωL and VL = ~ωL/e). The dashed curve shows the high-temperature limit (11) for T/TL = 5. The inset shows a similar plot for weaker intra-wire interaction g = 0.6 where the oscillations are less pronounced. compact analytic form Idr ∼ , (13) where Ψ′(z) = d2 ln Γ(z)/dz2 denotes the trigamma function. The first term is Eq. (11) evaluated at g = 1/2 (dotted curve in Fig. 5), and the second one is the dominant correction, which takes values within [−I(0)dr /θ, 0] (included in the dashed curve in Fig. 5). This illustrates that the first-order approx- imation already yields a nice qualitative description for the full numerical result, which makes us confident that Eq. (11) is a good high-temperature approxima- tion also for g 6= 1/2. In summary, we have analyzed two coupled quan- tum wires that exhibit both a finite intra-wire interac- tion and a finite inter-wire interaction. We have taken into account finite-length effects within the inhomoge- neous TLL model that is known to capture the essen- tial physics of quantum wires coupled to Fermi-liquid reservoirs. We have investigated how an asymmetry in the lengths of the wires changes the drag current and we have predicted a rich temperature dependence of the drag current that shows clear signatures of non- Fermi-liquid physics. We would like to thank F. Dolcini, H. Grabert, M. Kindermann, Y. Nazarov, S. Tarucha, and M. Ya- mamoto for interesting discussions. This work was supported by the Swiss NSF and the NCCR Nanoscience. 0 1 2 3 4 5 6 7 8 V / 2πV exact numerical result [Eq. (5)] dominant term [Eq. (11)] dominant term + first correction [Eq. (13)] g=0.5, T/T =5g=0.5, T/T FIG. 5: Drag current at high temperature (T/TL = 5) for identical wires with interaction strength g = 0.5 (solid curve), where I dr = eλ 2α4g/~2ωL and VL = ~ωL/e. The dotted curve shows the dominant contribution in the high-temperature limit [first term in (13)], the dashed curve includes the first correction as well [both terms in (13)]. [1] K. Flensberg, Phys. Rev. Lett. 81, 184 (1998). [2] Y. V. Nazarov and D. V. Averin, Phys. Rev. Lett. 81, 653 (1998). [3] A. Komnik and R. Egger, Phys. Rev. Lett. 80, 2881 (1998). [4] V. L. Gurevich, V. B. Pevzner, and E. W. Fenton, J. Phys.: Condens. Matter 10, 2551 (1998). [5] R. Klesse and A. Stern, Phys. Rev. B 62, 16912 (2000). [6] V. V. Ponomarenko and D. V. Averin, Phys. Rev. Lett. 85, 4928 (2000). [7] B. Trauzettel, R. Egger, and H. Grabert, Phys. Rev. Lett. 88, 116401 (2002). [8] M. Pustilnik, E. G. Mishchenko, L. I. Glazman, and A. V. Andreev, Phys. Rev. Lett. 97, 126805 (2003). [9] G. A. Fiete, K. Le Hur, and L. Balents, Phys. Rev. B 73, 165104 (2006). [10] M. Pustilnik, E. G. Mishchenko, and O. A. Starykh, Phys. Rev. Lett. 97, 246803 (2006). [11] P. Debray et al., Physica E 6, 694 (2000). [12] P. Debray et al., J. Phys.: Condens. Matter 13, 3389 (2001). [13] M. Yamamoto, M. Stopa, Y. Tokura, Y. Hirayama, and S. Tarucha, Physica E 12, 726 (2002). [14] M. Yamamoto, M. Stopa, Y. Tokura, Y. Hirayama, and S. Tarucha, Science 313, 204 (2006). [15] D. L. Maslov and M. Stone, Phys. Rev. B 52, R5539 (1995). [16] V. V. Ponomarenko, Phys. Rev. B 52, R8666 (1995). [17] I. Safi and H. J. Schulz, Phys. Rev. B 52, R17040 (1995). [18] F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev. B 71, 165309 (2005). [19] The effect of a finite smoothing length Ls on the con- ductance in an equivalent model has been analyzed by K. Janzen, V. Meden, and K. Schönhammer, Phys. Rev. B 74, 085301 (2006). [20] U. Zülicke and E. Shimshoni, Phys. Rev. B 69, 085307 5(2004). [21] A. Komnik and R. Egger, Eur. Phys. J. B 19, 271 (2001). [22] Inter-wire forward-scattering is not explicitely in- cluded in our model as it can be recast in a mere renor- malization of the intra-wire interaction strength gj of each wire [5]. We also neglect electron tunneling be- tween the two wires for two reasons: (i) It is a less relevant process than HC in a renormalization group sense [21]. (ii) It can be tuned to zero in a Coulomb drag experiment [12]. We assume to be away from half-filling where Umklapp scattering is forbidden. [23] In view of the results shown in Fig. 3, we do not expect any qualitative changes of the predicted temperature dependence for the case l = 1 as we make the wires asymmetric in length (l 6= 1).

0704.0516

Effects of Imperfect Gate Operations in Shor's Prime Factorization Algorithm

Journal of the Chinese Chemical Society, 2001, 48: 449-454 Effects of Imperfect Gate Operations in Shor’s Prime Factorization Algorithm Hao Guo1,2, Gui-Lu Long1,2,3,4,5 and Yang Sun1,2,6,7 Department of Physics, Tsinghua University, Beijing 100084 Key Laboratory for Quantum Information and Measurements, MOE Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, P.R. China Centre for Nuclear Theory, Lanzhou National Laboratory of Heavy Ions Chinese Academy of Sciences, Lanzhou 740000, P.R. China Center of Atomic, Molecular and Nanosciences, Tsinghua University, Beijing 100084 Department of Physics, Xuzhou Normal University, Xuzhou, Jiangsu 221009 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, U.S.A. (Dated: 2001) The effects of imperfect gate operations in implementation of Shor’s prime factorization algorithm are investigated. The gate imperfections may be classified into three categories: the systematic error, the random error, and the one with combined errors. It is found that Shor’s algorithm is robust against the systematic errors but is vulnerable to the random errors. Error threshold is given to the algorithm for a given number N to be factorized. PACS numbers: PACS numbers: 03.67.Lx, 89.70.+c, 89.80.+h I. INTRODUCTION Shor’s factorization algorithm [1] is a very impor- tant quantum algorithm, through which one has demon- strated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3]. Environment influence on the system can cause de- coherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor’s an- other important work, in which he showed that quantum error correlation can be corrected [4]. With quantum error correction scheme, errors arising from both deco- herence and imperfection can be corrected. There have been several works on the effects of deco- herence on Shor’s algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5]. Palma studied the effects of both deco- herence and gate imperfection in ion trap quantum com- puters [6]. There have also been many other studies on the quantum algorithm [7, 8, 9, 10]. The error correction scheme uses available resources. Thus it is important to study the robustness of the algo- rithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor’s factoriza- tion algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algo- rithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available. The paper is organized as follows. Section II is devoted to an outline of Shor’s algorithm and different error’s modes. In Section III, we present the results. Finally, a short summary is given in Section IV. II. SHOR’S ALGORITHM AND ERROR’S MODES Shor’s algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ〉 = 1√ |a〉|0〉, where q = 2L and N2 ≤ q ≤ 2N2 with N being the number to be factorized; 2) implementing yamodN and putting the results into the 2nd register |ψ1〉 = |a〉|yamodN〉; 3) making a measument on the 2nd register; The state of the register is then |φ2〉 = |jr + l〉|z = yl = yjr+lmodN〉 where j ≤ 4) performing discrete Fourier transformation (DFT) on the first register |φ3〉 = f̃ (c) |c〉 |z〉, where f̃ (c) = 2πi(jr + l) 2πilc 2πijrc http://arxiv.org/abs/0704.0516v1 This term is nonzero only when c = k q , with k = 0, 1, 2...r − 1, which correspond to the peaks of the dis- tribution in the measured results, and thus this term be- comes f̃(c) = 1√ 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation Aj = , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation Bjk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 eiθjk with θjk = . The gate sequence for implementing DFT is (Aq−1)(Bq−2q−1Aq−2) . . . (B0q−1B0q−2 . . . B01A0). Errors can occur in both Aj and Bjk. Aj is actually a rotation about y-axis through π Aj(θ) = e Syθ = I cos( )−i sin(θ )σy = cos( θ ) − sin( θ sin( θ ) cos( θ If the gate operation is not perfect, the rotation is not exactly π . In this case, Aj is a rotation of Aj(δ) = cos(δ)− sin(δ) −(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ)− sin(δ) If δ is very small, we have: Aj(θ) = 1− δ −(1 + δ) 1 + δ+ 1− δ Similarly, errors in Bjk can be written as Bjk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ei(θjk+δ) With these errors, the DFT becomes |a〉 → i( 2π +δc)a(1 + δ′c)|c̃〉 = i( 2πc +δc)a(1 + δ′c)|c̃〉,(1) where δc and δ c denote the error of Aj and Bjk, respec- tively. Let us assume the following error modes: 1) system- atic errors, where δc or δ c in (1) can only have system- atic errors (EM1); 2) random errors (EM2), for which we assume that δc or δ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM3). In the next sec- tion, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor’s algorithm. III. INFLUENCE OF IMPERFECT GATE OPERATIONS We first discuss the influence of imperfect gate opera- tions in the initial preparation Al−1Al−2...A0|0...0〉 = 1√ (|0〉+ |1〉+ δ1(|0〉 − |1〉))⊗ (|0〉+ |1〉+ δ2(|0〉 − |1〉))⊗ . . .⊗ (|0〉+ |1〉+ δn(|0〉 − |1〉)) i1i2...in=0 |i1i2...in〉+ 1√ R=1 δn i1i2...in=0 (|i1..iR−10iR+1..in〉 − |i1..iR−11iR+1..in〉 If the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ1 = δ2 = . . . = δn = δ. In this case, we can write the 2nd term as |ψ〉 = 1√ i1i2...in=0 (2s− n)|i1i2...in〉, where s stands for the number of 1’s, and 2s− n = s − (n − s) is the difference in the number of 1’s and 0’s. Thus the results after the first procedure is (|a〉+ δ(2s− n)|a〉) = (1 + δa)|a〉. (2) This implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have (1 + δa)(1 + δc)e i( 2πc +δ′c)a = (1 + δ′′)ei( +δ′c)a, where δ′′c = δc + δa. In the DFT, we have |ψ〉 ⇒ (1 + δj)e i( 2πc +δ′j)(jr+l)|c̃〉, where we have rewrite δ′′ as δj here. Let Pc denote the probability of getting the state |c̃〉 after we perform a measurement, we have (1 + δm)(1 + δk)e i( 2πc +δ′m)(mr+l)×e−i( +δ′k)(kr+l) (1 + δm)(1 + δk) cos[ r(m − k) + (mr + l)δ′m − (kr + l)δ′k](3) From Eq. (3), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can’t be eliminated. Eq. (3) is very complicated, so we will make some predigestions to discuss different error modes for conve- nience. Generally speaking, the influence of exponential error δj is more remarkable than δj , so we can omit the error δj , thus DFTq |φ〉 = j=0 e i( 2πc +δ′j)(jr+l)|c〉 . A. Case 1 If only systematic errors (EM1) are considered, namely, all the δj ’s are equal, then f̃(c) can be given analytically f̃(c) = i( 2πc +δ)(jr+l) il( 2πc +δ) 1− ei( 1− ei( The relative probability of finding c is f̃(c) sin2( sin2(πcr and if c = k q , then r sin2( q2 sin2( δr It can be easily seen that limδ→0 Pc = , which is just the case that no error is considered. When δ takes certain values, say, δ = 2 (k− r )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 27 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q , then Pc = lim r sin2( δq q2 sin2( δr The probability distribution is almost identical to those without errors. (ii) Let us increase δ gradually, from Fig.2, we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is Pc = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly. B. Case 2 When both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the er- ror has the form δ = δ0 + s, where δ0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribu- tion. When δ0 = 0, we have only random errors which is our error mode 2. When δ0 6= 0, we have error mode 3. For the uniform distribution, s ∼ ±smax × u(0, 1) where u(0, 1) is evenly distributed in [0,1]. smax indicates the maximum deviation from δ0. For Gaussian distribution, s ∼ N(0, σ0). Through the figure, we see the following: (1) When only random errors are present (δ0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar re- sults. The results for uniform random error mode are shown in Fig.3. For the uniform distribution error mode, with increasing δmax, the final probability distribution of the final results become irregular. In particular, when δmax is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks ap- pear. For the Gaussian distribution error mode, as shown in Fig.4, the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their proba- bility is small. The influence of σ0 on the final results is also sensitive, because it determines the shape of the distribution. When σ0 increases, the final probability dis- tribution becomes very messy. A small change in σ0 can cause a big change in the final results. (2) When δ0 6= 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors. IV. SUMMARY To summarize, we have analyzed the errors in Shor’s factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover’s algorithm where systematic errors are disastrous while random errors are less harmful [10]. [1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society Press, Los Alami- tos, CA, 1994) p.124. -20 0 20 40 60 80 100 120 140 FIG. 1: Relative probability for finding state c in the absence of errors. [2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, ”Quantum error correction by codding” (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329. http://arxiv.org/abs/quant-ph/9511003 -20 0 20 40 60 80 100 120 140 -20 0 20 40 60 80 100 120 140 -20 0 20 40 60 80 100 120 140 -20 0 20 40 60 80 100 120 140 FIG. 2: The same as Fig.1. with systematic errors. In sub- figures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid cir- cles(with lower peaks) denotes the result with δ = 0.33. -20 0 20 40 60 80 100 120 140 c -20 0 20 40 60 80 100 120 140 -20 0 20 40 60 80 100 120 140 -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 FIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively. -20 0 20 40 60 80 100 120 140 -20 0 20 40 60 80 100 120 140 -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 -20 0 20 40 60 80 100 120 140 FIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02.

0704.0517

Using decomposed household food acquisitions as inputs of a Kinetic Dietary Exposure Model

Using decomposed household food acquisitions as inputs of a Kinetic Dietary Exposure Model. Olivier Allais and Jessica Tressou Abstract Foods naturally contain a number of contaminants that may have different and long term toxic effects. This paper introduces a novel approach for the assessment of such chronic food risk that integrates the pharmakokinetic properties of a given contaminant. The estimation of such a Kinetic Dietary Exposure Model (KDEM) should be based on long term consumption data which, for the moment, can only be provided by Household Budget Surveys such as the SECODIP panel in France. A semi parametric model is proposed to decompose a series of household quantities into individual quantities which are then used as inputs of the KDEM. As an illustration, the risk assessment related to the presence of methylmercury in seafoods is revisited using this novel approach. Keywords: household surveys, individualization, linear mixed model, risk assessment, spline-estimation. INRA-CORELA, Laboratoire de recherche sur la consommation, Ivry sur Seine, France. INRA-Mét@risk, Méthodologies d’analyse des risques alimentaires, Paris, France and Hong Kong University of Science and Technology, Department of Information and Systems Management, Hong Kong. The second author research is in part supported by Hong Kong RGC Grant #601906. Corresponding author: Dr Jessica Tressou; Mail address: Hong Kong University of Science and Technology, ISMT, Clear Water Bay, Kowloon, Hong Kong; Email: [email protected]. http://arxiv.org/abs/0704.0517v1 Introduction The quantitative assessment of dietary exposure to certain contaminants is of high priority to the Food and Agricultural Organization and the World Health Organization (FAO/WHO). For exam- ple, excessive exposure to methylmercury, a contaminant mainly found in fish and other seafood (mollusks and shellfish) may have neurotoxic effects such as neuronal loss, ataxia, visual disturbance, impaired hearing, and paralysis (WHO, 1990). Quantitative risk assessments for such chronic risk require the comparison between a tolerable dose of the contaminant called Provisional Tolerable Weekly Intake (PTWI) and the population’s usual intake. The usual intake distribution is gener- ally estimated from independent individual food consumption surveys (generally not exceeding 7 days) and food contamination data. Several models have been developed to estimate the distribu- tion of usual dietary intake from short-term measurements (see for example, Nusser et al., 1996; Hoffmann et al., 2002). The proportion of consumers whose usual weekly intake exceeds the PTWI can then be viewed as a risk indicator (see for example, Tressou et al., 2004). This kind of risk assessment does not account for the underlying dynamic process, i.e. for the fact that the contami- nant is ingested over time and naturally eliminated at a certain rate by the human body. Moreover, longer term measurements of consumption are available through household budget surveys (HBS). In this paper, we propose to use HBS data to quantify individual long term exposure to a contaminant. This data provides long time series of household food acquisitions which are first used in a decomposition model, similar to the one proposed by Chesher (1997, 1998) in the nutrition field, in order to obtain time series of individual intakes. Then, the pharmacokinetic properties of the contaminant are integrated into an autoregressive model in which the current body burden is defined as a fraction of the previous one plus the current intake. From a toxicological point of view, this approach is, to our knowledge, novel and hence requires the definition of an ad-hoc long term safe dose as proposed in the next section. We refer to this autoregressive model as Kinetic Dietary Exposure Model (KDEM). From a statistical point of view, such autoregressive models are well known in general time series analysis (see for example, Hamilton, 1994) and most of the paper is devoted to the description of the decomposition model. This statistical model aims at estimating individual quantities from total household quantities and structures. This problem is similar to that studied by Engle et al. (1986), Chesher (1997, 1998), and Vasdekis and Trichopoulou (2000), and is addressed in a slightly different way. In the present article, the individual contaminant intake is firstly viewed as a nonlinear function of age within each gender, with time and socioeconomic characteristics being secondly introduced in a linear way. The nonlinear function is represented by a truncated polynomial spline of order 1 that admits a mixed model spline representation (section 4.9 in Ruppert et al., 2003). These choices yield a simple linear mixed model which is estimated by REstricted Maximum Likelihood (REML, Patterson and Thompson, 1971). One major extension of the proposed model compared to Chesher (1997) is the introduction of dependence between the individual intakes of a given household. In the next section, focusing on the methylmercury example even though the method is much more general and could be applied to any chronic food risk, SECODIP data are described along with the construction of a household intake series and the individual cumulative and long term exposure concepts yielding the KDEM. Section 2 is devoted to the statistical methodology used to decompose the household intake series into individual intake series, namely the presentation of the model and its estimation and tests. Section 3 displays the results for the quantification of long term exposure to methylmercury of the French population using the 2001 SECODIP panel. Finally, a discussion on the use of household acquisition data, with the focus on the French SECODIP panel, is conducted in section 4 with respect to the proposed long term risk analysis. 1 Motivating example: risk related to methylmercury in seafoods in the French population In this section, the Kinetic Dietary Exposure Model (KDEM) and the concept of long term risk are defined. Then a brief panorama of consumption data in France is given and the way the SECODIP HBS data will be used as an input of the KDEM is described. 1.1 Cumulative exposure and long term risk: the Kinetic Dietary Exposure Model (KDEM) The main objective of the analysis is to assess individuals’ long term exposure to a contaminant to deduce whether these individuals are at risk or not. As mentioned in the introduction the only ”safe dose” reference is the PTWI expressed in terms of body weight (relative intake). Unfortunately, TNS SECODIP did not record the body weight of the individuals until 2001. The body weights are thus estimated from independent data sets; namely the French national survey on individual consumption (INCA, CREDOC-AFSSA-DGAL, 1999) for people older than 18, and the weekly body weight distribution available from French health records (Sempé et al. (1979)) for individuals under 18. In both cases, gender differentiation is introduced. Assume that estimations of the individual weekly intakes are available, that is yi,h,t denotes the intake of individual i belonging to household h for the tth week (with i = 1, . . . , nh,t; h = 1, . . . ,H and t = 1, . . . , T ), and Di,h,t denotes the same quantity expressed on a body weight basis. The cumulative exposure up to the tth week of this individual is then given by Si,h,t = exp(−η) · Si,h,t−1 +Di,h,t, (1.1) where η > 0 is the natural dissipation rate of the contaminant in the organism. This dissipation parameter is defined from the so called half life of the contaminant,which is the time required for the body burden to decrease by half in the absence of any new intake. For methylmercury, the half life, denoted by l1/2, is estimated to 6 weeks, so that η = ln(2)/l1/2 := ln(2)/6 (Smith and Farris, 1996). The autoregressive model defined by (1.1) and a given initial state Si,h,0 = Di,h,0 has a stationary solution since exp(−η) < 1. As a convention, Si,h,0 is set to the mean of all positive exposures (Di,h,t)t=1,. . . ,T . However, this convention has little impact on the level of an individual’s long term exposure since the contribution of the initial state Si,h,0 tends to zero as t increases. We call this autoregressive model ”KDEM” for Kinetic Dietary Exposure Model. The individual cumulative exposure Si,h,t can be considered to be the long term exposure of an individual for sufficiently large values of t. For methylmercury, the long term steady state of the individual exposure to a contaminant is reached after 5 or 6 half lives according to Dr P. Granjean, a methylmercury expert. Thus, the long term individual’s exposure to methylmercury is defined as the cumulative exposure reached after say 6l1/2 = 36 weeks. The risk assessment usually consists of comparing the exposure with the so called Provisional Tolerable Weekly Intake (PTWI). This tolerable dose, determined from animal experiments and extrapolated to humans, refers to the dose an individual can ingest throughout his entire life without appreciable risk. For methylmercury, the PTWI is set to 1.6 microgram per kilogram of body weight per week (1.6 µg/kg bw, see FAO/WHO, 2003). In our dynamic approach, the long term exposure is compared to a reference long term exposure denoted by Sref , and defined as the cumulative exposure of an individual whose weekly intake is equal to the PTWI, d, such as Sref = lim 1− exp(−η) , (1.2) where d exp(−η(t− s)) = dexp(−η(t+ 1))− 1 exp(−η)− 1 . (1.3) For methylmercury, the reference for long term exposure Sref is 14.6 µg/kg bw. An individual is then assumed to be at risk if his cumulative exposure Si,h,t exceeds the reference S t for any t > 6l1/2. This KDEM model requires some long surveys of individual intakes which are not monitored and can only be approximated from available consumption data and contamination data. 1.2 From household acquisition data to household intake series Two current major consumption data sources in France are the national survey on individual consumption (INCA, CREDOC-AFSSA-DGAL, 1999) and the SECODIP panel managed by the company TNS SECODIP. Most quantitative risk assessments conducted by the French agency for food safety (AFSSA) use the 7 day individual consumption data of the INCA survey jointly with contamination data collected by several French institutions. Regarding methylmercury, seafood contamination data have been collected through different analytical surveys (MAAPAR, 1998-2002; IFREMER, 1994-1998) and were used in Tressou et al. (2004) and Crépet et al. (2005) combined with the INCA survey. In this paper, a methodology using the SECODIP data is developed (see Boizot, 2005, for a full description of this database). The company TNS SECODIP has been collecting the weekly food acquisition data of about five thousand households since 1989. All participating households register grocery purchases through the use of EAN bar codes but other grocery purchases are registered differently: the fresh fruit and vegetable purchases are recorded by the FL sub-panel while fresh meat, fresh fish and wine purchases are recorded by the VP sub-panel. The households are selected by stratification according to several socioeconomic variables and stay in the survey for about 4 years. TNS SECODIP provides weights for each sub-panel and each period of 4 weeks to make sure of the representativeness of the results in terms of several socioeconomic variables. TNS SECODIP also defines the notion of household activity which refers to the correct and regular reporting of household purchases over a year. For each household, the age and gender of each member of the household are retained in our decomposition model with some socioeconomic variables: the region, the social class (from modest to well-to-do), the occupation category and level of education of the principal household earner. For methylmercury risk assessment, the households of the VP panel are considered; in the 2001 data set, there are H = 3229 active households (corresponding to 9288 individuals) and T = 53 weeks during which the households may or may not acquire seafood. The weekly purchases of seafood are clustered into two categories (”Fish” and ”Mollusks and Shellfish”) for which the mean contamination levels are calculated from the MAAPAR-IFREMER data and are given in table 1. Table 1 around here, see page 21 Household intake series ((yh,t)h=1,...,H;t=1,...,T) are computed as the cross product between weekly purchases of seafoods which are assimilated to weekly consumptions, and mean contamination levels. They are expressed in micrograms per week (µg/w). The food ”purchase-consumption” assimilation is of course arguable and will be the main subject of the final discussion (see section 4). An additional assumption concerns the household size, denoted by nh,t for the household h and the week t. This can indeed vary over time in the case of a birth or death of a household member. Since a new born baby will not consume fish in his first few months, we assume that food diversification (and hence consumption of seafoods) starts at one year of age, yielding a total sample of 8913 individuals for the 2001 panel. These household intake series are then decomposed into individual intake series using the model described in the next section. These individual intake series are then used as imputs of the KDEM. 2 Statistical methodology In this section, the decomposition model is described and compared to similar models described in the literature, namely Chesher (1997, 1998); Vasdekis and Trichopoulou (2000). Its estimation and some structure tests are then presented. 2.1 The decomposition model 2.1.1 General principle Consider a household composed of nh,t members, each member having unobserved weekly intakes yi,h,t, with i = 1,. . . , nh,t, h = 1,. . . ,H, and t = 1,. . . , T . The week t intake of a household h is simply the sum across household members of the individual weekly intakes, such as yh,t = nh,t∑ yi,h,t. (2.1) As detailed below, the individual weekly intake yi,h,t is assumed to depend on • the age and gender of the individual via a function f, • some socioeconomic characteristics of the household, • time (seasonal variations). There are obviously several ways to model the individual intake under these assumptions and this choice leads to more or less simple estimation procedures. In Chesher (1997, 1998); Vasdekis and Trichopoulou (2000), a discretization argument on age is used leading to a penalized least square estimation of a great number of parameters, that is one parameter for each year of age and gender. We propose to use a truncated polynomial spline of order 1 for each gender, which admits a mixed model spline representation for f. As far as socioeconomic characteristics are con- cerned, Chesher (1997) retained a multiplicative specification whereas Vasdekis and Trichopoulou (2000) chose the additive one. In the multiplicative model, a change in income for example would proportionally affect all the individual intakes whereas in the additive setting, they would be af- fected by the same value. Following Vasdekis and Trichopoulou (2000), we retained the additive specification since the difference between the two specifications may not be notable, and the addi- tive setting yields to a much simpler estimation procedure (linear model). Finally, time dependency is only introduced in Chesher (1998) to track changes with age within cohorts: this time depen- dency is directly introduced into the function f that is bivariately smoothed according to age and time (cf. Green and Silverman, 1994). Again, we adopt a simpler specification in which time is introduced as a dummy variable. All these assumptions yield an individual model of the form yi,h,t = xi,h,tβ + zi,h,tu+ wh,tγ + δtα+ εi,h,t, (2.2) where the terms xi,h,tβ + zi,h,tu stand for the mixed model spline representation of the function f, the term wh,tγ denotes the socioeconomic effects, the term δtα the time effect, and εi,h,t is the individual error term. Combining (2.1) and (2.2) , we obtain the final rescaled household model given by Yh,t = Xh,tβ + Zh,tu+ nh,twh,tγ + nh,tδtα+ εh,t, (2.3) where Yh,t ≡ ∑nh,t i=1 yi,h,t/ nh,t, Xh,t ≡ ∑nh,t i=1 xi,h,t/ nh,t, Zh,t ≡ ∑nh,t i=1 zi,h,t/ nh,t, and εh,t ≡ ∑nh,t i=1 εi,h,t/ nh,t. 2.1.2 Specification details Age-gender function specification Let ai,h,t and si,h denote the age and sex of individual i of household h for the tth week. Individual dietary intake is generally different according to the gender of individuals, so the function f takes the following form f(ai,h,t, si,h) = fM (ai,h,t)1l{si,h=M} + fF (ai,h,t)1l{si,h=F}, where fM(.) and fF (.) are age-intake relationships for males (M) and females (F) respectively, and 1l{A} is the indicator function of event A. The function fS(.) is approximated by a spline of order one with a truncated polynomial basis for either sex, such as fS(ai,h,t) = β 0 + β 1 ai,h,t + uSk (ai,h,t − κS,k)+ , (2.4) where the (κS,k)k=1,. . . ,KS are nodes chosen from an age list and (ai,h,t − κS,k)+ ≡ (ai,h,t − κS,k) 1l{ai,h,t−κS,k>0} denotes the positive part of the difference between the age of the individual ai,h,t and the node κS,k and the uSk are random effects assumed to be i.i.d. Gaussian with distribution N 0, σ2uS . This last assumption allows us to introduce some penalties into the model and to smooth the function fS yielding a mixed model representation for the spline as shown in Speed (1991); Verbyla (1999); Brumback et al. (1999); Ruppert et al. (2003). As in Ruppert et al. (2003), page 125, the total number of nodes KS is set to min {∣∣aS,d ∣∣ , 35 , where aS,d is the list of distinct ages for individuals of sex S, and the nodes κS,k are defined as the percentile of vector aS,d for k = 1,. . . ,KS . Defining xi,h,t as a line vector 1l{si,h=M} ai,h,t1l{si,h=M} 1l{si,h=F} ai,h,t1l{si,h=F} , and zi,h,t as the line vector (ai,h,t − κS,k)+ 1l{si,h=S} k=1,. . . ,KS ; S=M,F , we finally obtain the first terms of (2.2) , that is f(ai,h,t, si,h) = xi,h,tβ + zi,h,tu. Socioeconomic characteristics and time dependency In the application, all the socioe- conomic characterics are categorical variables. Consider the Q categorical variables W h,t , q = 1, . . . , Q, with mq modalities, and fix the m q modality as the reference modality, then the socioe- conomic effect term in (2.2) and (2.3) is wh,tγ = mq−1∑ γq,m1l where γq,m is the effect of the m th modality of the socioeconomic variable q. Similarly, time is only measured by weekly counts throughout the year so that the time effect in (2.2) and (2.3) is simply δtα = τ 6=τR ατ1l{τ=t}, where ατ is the effect of week τ and τR is the reference week. Error specification The error at the individual level εi,h,t is assumed to be Gaussian with zero mean, and the variance-covariance structure is such that • households are independent, i.e. ∀i, i′, t, t′ and ∀h 6= h′ cov(εi,h,t, εi′,h′,t′) = 0, • members of the same household are dependent, that is for ∀h, t and i 6= i′, cov(εi,h,t, εi′,h,t) = ρσ where ρ measures the dependence between individuals within the same household. • there is no time dependence, that is ∀i, i′ and ∀t 6= t′ cov(εi,h,t, εi′,h,t′) = 0. In the rescaled household model (2.3), the error εh,t ≡ ∑nh,t i=1 εi,h,t/ nh,t is i.i.d. Gaussian with a zero mean and a variance R such that ∀t, t′ and ∀h 6= h′, V(εh,t) = ρσ εnh,t + (1− ρ)σ2ε and cov(εh,t, εh′,t′) = 0. (2.5) 2.2 Estimation and tests The model (2.3) is a linear mixed model that can be estimated using restricted maximum likelihood (REML) techniques, see Ruppert et al. (2003) for details. An attractive consequence of the use of the mixed model representation of a penalized spline in (2.4) is that mixed model methodology and software can be used to estimate the parameters and predict the random effect in the resulting household model. The amount of smoothing of the underlying functions fS is estimated with the REML technique via the estimation of σ2uS . The estimation was conducted using R©SAS MIXED procedure. To get estimators for σ2ε and ρ, asymptotic least square techniques combined with the linear relationship between the variance given in (2.5) and the household size were used. More precisely, a residual variance σ2n is first estimated for each household size n = 1, . . . , N = maxnh,t using an option of the MIXED procedure (see the program for the detailed syntax). Then, ordinary least square regression and the delta method give estimators for σ2ε and ρ and their standard deviations. The individual intake is then predicted by ŷi,h,t = xi,h,tβ̂ + zi,h,tû+ wh,tγ̂ + δtα̂, (2.6) where β̂, γ̂, and α̂ are the estimators of β, γ, and α respectively and û is the best prediction of the random effect u in the model (2.3). Confidence and prediction intervals can be built for the prediction ŷi,h,t as proposed in Ruppert et al. (2003) and several tests can be conducted in this model: 1. Are the random effects different according to sex? In other words, is the assertion σ2uM = σ2uF = σ u true? 2. Another question is the necessity for such random effects. Is the assertion σ2u = 0 (resp. σ2uM = 0 or σ = 0) true? 3. More globally, is the function f the same for both sexes? Is the assertion fM = fS true? These tests can be conducted using classical likelihood (or restricted likelihood) ratio techniques. The likelihood ratio statistic is asymptotically distributed as a chi square with a degree of freedom being the number of tested equalities, except for point 2, where the limiting distribution is known to be a mixture of chi-square (Self and Liang, 1987; Crainiceanu et al., 2003) because the test concerns the frontier of the parameter definition (σ2u ∈ [0,+∞[). 3 Applying our methodology to the methylmercury risk assess- In this section, we illustrate our approach on our motivating example. Firstly, several tests are conducted on the decomposition model, and secondly, individual long term exposure is compared to the reference long term exposure described in section 1. 3.1 Estimation and tests on the structure of the model Table 2 shows the REML estimates for all socioeconomic variables (parameter γ) and the p-values of Student tests in the model (2.3). The socioeconomic variables used are household income, region of residence, occupation category and level of education of the principal household earner. For each socioeconomic variable, the reference modality is given in Table 2. We assume here that • the function f differs according to the gender but the random effect does not (fM 6= fF and σ2uM = σ • the maximum household size N is set to 6 for variance-covariance estimation. Indeed, the dependence between individuals within the same household depends on the household size nh in (2.5). For each household size, a variance is estimated, and estimates of ρ and σ are obtained using asymptotic least square techniques as mentioned in section 2.2. Since large households are not numerous in the database, the estimations are implemented with a maximum household size, N , set to 6; it is assumed that there is a common variance for all households with size greater than N . In this sub-section, we show the results of several tests we carried out to simplify the inter- pretation of our study. These tests have been implemented in a hierarchical way, starting with the highest-order interaction terms, combining to the reference modality the modality which does not differ significantly from the reference. All tests are performed on the 5% level of significance and each new hypothesis is tested, conditionally on the results of the previous tests. Each null hypothesis and the p-value resulting from the appropriate F-test are shown in Table 3. First of all, concerning the occupation category variable, the self-employed modality does not significantly differ from the reference modality blue collar workers (H1, Pval = 0.771). Refitting the model with the reference modality ”Blue collar workers and self employed”, all the socioeconomic variables are significantly different from the reference. Then, F-tests allow us to conclude that the resulting three groups are significantly different from each other (H2, H3, H4). Let us now consider the region of residence variable. First, there are some very substantial differences among the 4 regions of residence (H5, Pval =< 0.001). However, the modality ”North, Brittany, and Vendee coast” and the modality ”Paris and its suburbs” should be grouped (H6 c, Pvalc = 0.881). Then, the other tests implemented for the level of education and income variables suggest that no further simplification is possible (see p-values of null hypotheses H7, H8, H9 in Table 3). Finally, the overall F-test comparing our resulting final model to the original model (2.3) shows that no important variable has been left out of the model (Pval = 0.59). Table 4 shows the parameter estimates and p-values of the Student’s t-tests for all socioeco- nomic variables of the reduced final model. The income effects on individual exposure are those expected: the richer the households are, the higher their exposures are because seafoods are ex- pensive. Furthermore, living in a coastal region or in Paris and its suburbs brings about larger individual exposure relatively to living in a non coastal region because of the more ready supply of seafoods in these regions. Moreover, the more educated you are, the larger the individual exposure is. The occupation category of the principal household earner has an unexpected effect on the in- dividual exposure. Indeed a higher exposure is expected for white collar workers and retirees whan compared to blue collar workers but an opposite effect is observed. This may be explained by the fact that the reference modality for this variable is a very heterogeneous modality also comprising managers and self-employed persons (farmers and craftsmen). Another explanation could be that white collars workers have a higher propensity to eat out in restaurants whereas outside the home consumption is not included in the model. Table 2 around here, see page 21 Table 3 around here, see page 22 Table 4 around here, see page 22 Likelihood ratio tests are implemented to test the structure of the final model. First, the dependence of individual exposures to methylmercury within a household is tested. The null hypothesis ρ = 0 (cf. equation (2.5)) is rejected (null Pval) which confirms that individuals within the same household have correlated exposures. Then, we test if the function f is the same for both genders. The null hypothesis fM = fF is rejected (null Pval) but the null hypothesis σ = σ2uF is accepted. This means that individual exposure differs with gender but both functions need the same amount of smoothing. 3.2 The cumulative and the long term individual exposure The cumulative individual exposure Si,h,t is calculated from the estimated individual weekly intakes according to equation (1.1) and the resulting values for t > 35 are compared to the reference cumulative exposure defined by (1.3). Figure 1 shows the cumulative individual exposure over the 53 weeks of the year 2001 for different individuals. Only certain percentiles of the distribution of the individual cumulative exposures of the last week are displayed. For example, the curve Pmax represents the cumulative exposure of an individual whose last week’s cumulative exposure is the highest. This is the cumulative exposure of a girl who turned one year old during the 30th week of 2001, lives in Paris or its suburbs in a well to do household. Very few individuals have a cumulative individual exposure above the reference long term ex- posure. We estimate that only 0.186% of individuals are deemed at risk. This risk index should be compared to the more common one defined as the percentage of weekly intakes Di,h,t exceeding the PTWI, denoted R1.6, such as R1.6 = i=1 1l (Di,h,t > 1.6). R1.6 is equal to 0.45%, and is slightly higher since each occasional deviation above the PTWI increases the risk index whereas only long term deviations above this PTWI should be taken into account to assess the risk. A deeper analysis of at risk individuals shows that all these vulnerable individuals are children less than three years old. They represent 5.29% of the children aged between 1 and 3 in 2001. Further, no child of a modest households is found to be at risk. Figure 1 around here, see page 23 4 Discussion As mentioned in section 1, the use of household acquisition data in a food safety context, and in our case the use of the SECODIP database for assessing methylmercury dietary intakes, gives rise to some approximations: 1. Consumption outside of the home is out of the scope of household acquisition data. TNS SECODIP does not provide any information on the quantities of seafoods consumed out of the home or bought for outside consumption. Nevertheless, Serra-Majem et al. (2003) assert that these data are good estimates for the consumption of the whole household. Vasdekis and Trichopoulou (2000) avoid this question by using the term ”availaibility” in- stead of intake or consumption. However, as in Chesher (1997), auxiliary information about outdoor consumption could be introduced in the model as a correction factor accounting for the propensity to eat outside of the home according to age, sex or socioeconomic variables. The French INCA survey on individual consumptions gives details about inside / outside the home consumption for 3003 individuals people aged 3 and older. The mean outside the home consumption proportion is 20% for seafoods. Applying such a factor to all household intakes yields a long term risk of 0.226%, and R1.6 = 0.791%. Furthermore, in this case, a small proportion of consumers older than 3 years old are vulnerable. Nevertheless, children aged between 1 and 3 in 2001 still represent the most vulnerable consumer group, at 10% of the corresponding population. 2. The amount of food bought by a household can be different from the amount actually con- sumed. Indeed, namely for seafoods, a non negligible part is not edible: Favier et al. (1995) show than on average only 61% of fresh or frozen fish is edible. Besides, Maresca and Poquet (1994) also demonstrate some part of the purchased food is thrown away, which also reduces the actual amount of food consumed by a household. However, SECODIP does not specify whether the quantity of fresh or frozen fish bought is ready to be consumed or as a whole fish that needs some preparation. Applying such a factor to all household intakes yields a long term risk of 0.00%, and R1.6 = 0.043%. If both the 20% outside of the home consumption correction factor and the 61% edible proportion factor are applied to our series, the long term risk is equal to 0.021%, R1.6 = 0.13%, and 1.06% of the population of children aged between 1 and 3 are vulnerable. These results stress that applying such a correction factor to assess the actual quantity consumed is probably too strong and is certainly a crude approximation of the quantity of seafoods ingested. Thus, a more detailed database on fish and seafood is needed, to realize an accurate assessment of exposure to methylmercury, taking into account only the edible part of fish and other seafood. Body weight information is crucial in a food safety context and will be included in the future SECODIP data since it has now been added to the list of required individual characteristics. The measurement error afferent to this quantity will remain however, namely for children whose body weight changes a lot throughout a year. Nevertheless, approximating the weekly body weight of young children by the median of the weekly body weight distribution available in French health records is the best approximation possible. 3. The food nomenclature of the SECODIP database is not as detailed as the contamination database. Unfortunately, fish and seafood species are not well documented so it is not possible to consider more than two food categories when computing household intakes. This problem of nomenclature matching is ubiquitous of food risk assessments since contamination analysis are generally conducted independently from the food nomenclature of consumption data. These arguments mainly show the disadvantages of the use of household food acquisition data such as the SECODIP database. Nevertheless, they also present many advantages compared to the individual food record survey mainly used in France in the food safety context: • As mentioned before, households respond for a long period of time (the average is 4 years in the SECODIP panel) which allows us to observe long term behaviors and avoid some well known biases of individual food record surveys. For example, respondents might over- (under- ) declare certain foods with a good (bad) nutritional value either deliberately or just because they increased (reduced) their consumption for the short (7 days) period of the survey. • The individual surveys are expensive and very difficult to conduct. Highly trained interviewers are required and extraordinary cooperation is required from respondents. Household food acquisition data can serve many other applications (economics or marketing) and, at least for the SECODIP data, acquisition recording is simplified by optical scanning of food barcodes. Conclusion In this paper, we proposed a methodology to assess chronic risks related to food contamination using the example of methylmercury exposure through seafood consumption. This methodology includes the definition of a Kinetic Dietary Exposure Model (KDEM) that integrates the fact that contaminants are eliminated from the body at different rates, the rate being measured by the half life of the contaminant. In this paper, the estimation is based on the use of household food acqui- sition data which are first decomposed into individual intake data through a disaggregation model accounting for the dependence among household members. Several extensions of this methodology are currently studied. First, the disaggregation model could be improved by considering a prelim- inary step in which we determine what member is an actual consumer, in the spirit of the Tobit model. The KDEM idea is also currently being developed by studying the stability and ergodic properties of the underlying continuous time piecewise deterministic Markov process (Bertail et al., 2006). The parameters of this new model are the intake distribution, the inter intake time distri- bution and the dissipation rate distribution. In this framework, the dissipation parameter η of the KDEM model is random and the intake and inter-intake distributions can be estimated either from individual (INCA-type) data or household (SECODIP-type) data. References Bertail, P., S. Clémençon and J. Tressou (2006). A storage model with random release rate for modeling exposure to food contaminants. Submitted for publication. Boizot, C. (2005). Présentation du panel de données SECODIP. Technical report. INRA-CORELA. Brumback, B., D. Ruppert and M. P. Wand (1999). Comment on ”variable selection and function estimation in additive nonparametric regression using a data-based prior” by Shively, Kohn, and Wood. Journal of the American Statistical Association 94, 794–797. Chesher, A. (1997). Diet revealed?: Semiparametric estimation of nutrient intake-age relationships. Journal of the Royal Statistical Society A 160(3), 389–428. Chesher, A. (1998). Individual demands from household aggregates: Time and age variation in the quality of diet. Journal of Applied Econometrics 13(5), 505–524. Crainiceanu, C. M., D. Ruppert and T. J. Vogelsang (2003). Some properties of likelihood ratio tests in linear mixed models. (Working Paper). CREDOC-AFSSA-DGAL (1999). Enquête INCA (individuelle et nationale sur les consommations alimentaires). TEC&DOC ed.. Lavoisier, Paris. (Coordinateur : J.L. Volatier). Crépet, A., J. Tressou, P. Verger and J. Ch. Leblanc (2005). Management options to reduce ex- posure to methyl mercury through the consumption of fish and fishery products by the French population. Regulatory Toxicology and Pharmacology 42(2), 179–189. Engle, R. F., C. W. J. Granger, J. Rice and A. Weiss (1986). Non-parametric estimation of the rela- tionship between weather and electricity demand. Journal of the American Statistical Association 81, 310–320. FAO/WHO (2003). Evaluation of certain food additives and contaminants for methylmercury. Sixty first report of the Joint FAO/WHO Expert Committee on Food Additives, Technical Report Series. WHO. Geneva, Switzerland. Favier, C., J. Ireland-Ripert, C. Toque and M. Feinberg (1995). Rpertoire Gnral des Aliments, Table de composition, tome 1. TEC&DOC ed.. Lavoisier, Paris. Green, P.J. and B.W. Silverman (1994). Nonparametric Regression and Generalized Linear Models. Chapman & Hall. Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press. Hoffmann, K., H. Boeingand, A. Dufour, J. L. Volatier, J. Telman, M. Virtanen, W. Becker and S. De Henauw (2002). Estimating the distribution of usual dietary intake by short-term mea- surements. European Journal of Clinical Nutrition 56, 53–62. IFREMER (1994-1998). Résultat du réseau national d’observation de la qualité du milieu marin pour les mollusques (RNO). MAAPAR (1998-2002). Résultats des plans de surveillance pour les produits de la mer. Ministère de l’Agriculture, de l’Alimentation, de la Pêche et des Affaires Rurales. Maresca, B. and G. Poquet (1994). Collectes slectives des dchets et comportements des mnages. Technical Report R146. CREDOC. Nusser, S.M., A.L. A.L. Carriquiry, K.W. Dodd and W.A. Fuller (1996). A semiparametric trans- formation approach to estimating usual intake distributions. Journal of the American Statistical Association 91, 1440–1449. Patterson, H. D. and R. Thompson (1971). Recovery of inter-block information when block sizes are unequal. Biometrika 58, 545–554. Ruppert, D., M .P. Wand and R. J. Carroll (2003). Semiparametric regression. Cambridge Series in Statistical and Probabilistic Mathematics. Cambrige University Press. Self, S. G. and K.Y. Liang (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Associ- ation 82(398), 605–610. Sempé, M., G. Pédron and M. P. Roy-Pernot (1979). Auxologie, méthode et séquences. Théraplix. Paris. Serra-Majem, L., D. MacLean, L. Ribas, D. Brule, W. Sekula, R. Prattala, R. Garcia-Closas, A. Yngve and M. Lalondeand A. Petrasovits (2003). Comparative analysis of nutrition data from national, household, and individual levels: results from a WHO-CINDI collaborative project in Canada, Finland, Poland, and Spain. Journal of Epidemiology and Community Health 57, 74–80. Smith, J. C. and F. F. Farris (1996). Methyl mercury pharmacokinetics in man: A reevaluation. Toxicology And Applied Pharmacology 137, 245–252. Speed, T. (1991). Discussion of “that blup is a good thing: the estimation of random effects” by g. robinson. Statistical science 6, 42–44. Tressou, J., A. Crépet, P. Bertail, M. H. Feinberg and J. C. Leblanc (2004). Probabilistic exposure assessment to food chemicals based on extreme value theory. application to heavy metals from fish and sea products. Food and Chemical Toxicology 42(8), 1349–1358. Vasdekis, V.G.S. and A. Trichopoulou (2000). Non parametric estimation of individual food avail- ability along with bootstrap confidence intervals in household budget surveys. Statistics and Probability Letters 46, 337–345. Verbyla, A. (1999). Mixed Models for Practitioners. Biometrics SA, Adelaide. WHO (1990). Methylmercury, environmental health criteria 101. Technical report. Geneva, Switzer- land. Figures and Tables Table 1: Description of the contamination database (Unit: microgram per kilogram Mean Min Max Standard Deviation Number of analysis Fish 0.147 0.003 3.520 0.235 1350 Mollusk and Shellfish 0.014 0.001 0.172 0.011 1293 Table 2: Restricted maximum likelihood estimates (REML) for age and all socioeconomic variables and the p-value of the Student’s tests (Pval) Effect Parameter REML Pval Income (ref: Mean sup) Well to do γ1 6.027 <0.001 Mean inf γ2 2.686 <0.001 Modest γ3 -1.928 <0.001 Region of residence (ref: Noncoastal regions) North, Brittany, Vendee coast γ4 0.962 0.003 South West coast γ5 5.232 <0.001 Mediterranean coast γ6 2.303 <0.001 Paris and its suburbs γ7 1.023 0.009 Occupation category of the principal household earner (ref: Blue collar workers) self-employed persons γ8 -0.122 0.771 white collar workers γ9 -3.733 <0.001 retirees γ10 -5.261 <0.001 no activity γ11 -1.910 0.004 Level of Education of the principal household earner (ref: BAC and higher degree) student γ12 5.901 <0.001 no or weak diploma γ13 -1.281 <0.001 Table 3: The different steps performed in testing the socioeconomic part of our model. For each step, the null hypothesis tested and the p-value resulting from the appropriate F-test are shown. All tests are performed conditionally on the results of the previous tests (Pval) Null hypothesis Pval H1 : γ8 = 0 0.771 H2 : γ9 = γ10 0.030 H3 : γ9 = γ11 0.018 H4 : γ10 = γ11 <0.001 H5 : γ4 = γ5 = γ6 = γ7 <0.001 H6 : a : γ4 = γ5 <0.001 b : γ4 = γ6 <0.001 c : γ4 = γ7 0.881 d : γ5 = γ6 <0.001 e : γ5 = γ7 <0.001 f : γ6 = γ7 0.0103 H7 : γ12 = γ13 <0.001 H8 : γ1 = γ2 = γ3 <0.001 H9 : a : γ1 = γ2 <0.001 b : γ1 = γ3 <0.001 c : γ2 = γ3 <0.001 Table 4: Restricted maximum likelihood estimates (REML) for all age and socioeconomic variables of the reduced final model with all variance components and their standard errors (s.e) Effect Parameter REML Pval Income (ref: Mean sup) Well to do γ1 6.108 <0.001 Mean inf γ2 2.760 <0.001 Modest γ3 -1.915 <0.001 Region of residence (ref: Non coastal regions) Paris and North, Brittany, Vendee coast γ4= γ7 0.995 <0.001 South west coast γ5 5.156 <0.001 Mediterranean coast γ6 2.250 <0.001 Occupation category of the principal household earner (ref: Blue collar workers and self employed persons) white collar workers γ9 -3.745 <0.001 retirees γ10 -5.243 <0.001 no activity γ11 -1.871 0.005 Level of education of the principal household earner (ref: BAC and higher degree) student γ12 5.879 <0.001 no or weak diploma γ13 -1.279 <0.001 REML s.e Variance of the random effect σu 24.832 6.7316 Variance-covariance structure variance σ2 1260705 282309 correlation ρ -0.22 0.0434 Figure 1: Cumulative exposure to MeHg (unit: µg per kg of body weight) Motivating example: risk related to methylmercury in seafoods in the French population Cumulative exposure and long term risk: the Kinetic Dietary Exposure Model (KDEM) From household acquisition data to household intake series Statistical methodology The decomposition model General principle Specification details Estimation and tests Applying our methodology to the methylmercury risk assessment Estimation and tests on the structure of the model The cumulative and the long term individual exposure Discussion

0704.0518

Dust and gas emission in the prototypical hot core G29.96-0.02 at sub-arcsecond resolution

Astronomy & Astrophysics manuscript no. beuther˙g29 c© ESO 2018 November 4, 2018 Dust and gas emission in the prototypical hot core G29.96−0.02 at sub-arcsecond resolution H. Beuther1, Q. Zhang2, E.A. Bergin3, T.K. Sridharan2, T.R. Hunter4, S. Leurini5 1 Max-Planck-Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany e-mail: [email protected] 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA e-mail: [email protected] 3 University of Michigan, Dept. of Astronomy, Ann Arbor, MI 48109-1090 e-mail: [email protected] 4 NRAO, 520 Edgemont Rd, Charlottesville, VA 22903 e-mail: [email protected] 5 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany e-mail: [email protected] ABSTRACT Context. Hot molecular cores are an early manifestation of massive star formation where the molecular gas is heated to temperatures > 100 K undergoing a complex chemistry. Aims. One wants to better understand the physical and chemical processes in this early evolutionary stage. Methods. We selected the prototypical hot molecular core G29.96−0.02 being located at the head of the associated ultracompact Hii region. The 862 µm submm continuum and spectral line data were obtained with the Submillimeter Array (SMA) at sub-arcsecond spatial resolution. Results. The SMA resolved the hot molecular core into six submm continuum sources with the finest spatial resolution of 0.36′′ × 0.25′′ (∼1800 AU) achieved so far. Four of them located within 7800 (AU)2 comprise a proto-Trapezium system with estimated protostellar densities of 1.4× 105 protostars/pc3 . The plethora of ∼ 80 spectral lines allows us to study the molecular outflow(s), the core kinematics, the temperature structure of the region as well as chemical effects. The derived hot core temperatures are of the order 300 K. We find interesting chemical spatial differentiations, e.g., C34S is deficient toward the hot core and is enhanced at the UCHii/ hot core interface, which may be explained by temperature sensitive desorption from grains and following gas phase chemistry. The SiO(8–7) emission outlines likely two molecular outflows emanating from this hot core region. Emission from most other molecules peaks centrally on the hot core and is not dominated by any individual submm peak. Potential reasons for that are discussed. A few spectral lines that are associated with the main submm continuum source, show a velocity gradient perpendicular to the large-scale outflow. Since this velocity structure comprises three of the central protostellar sources, this is not a Keplerian disk. While the data are consistent with a gas core that may rotate and/or collapse, we cannot exclude the outflow(s) and/or nearby expanding UCHii region as possible alternative causes of this velocity pattern. Key words. stars: formation – ISM: jets and outflows – ISM: molecules – stars: early-type – stars: individual (G29.96−0.02) – (stars:) binaries (including multiple): close 1. Introduction Hot molecular cores represent an early evolutionary stage in massive star formation prior to the formation of an ultra- compact Hii region (UCHii). Single-dish line surveys toward hot cores have revealed high abundances of many molecu- lar species and temperatures usually exceeding 100 K (e.g., Schilke et al. 1997; Hatchell et al. 1998; McCutcheon et al. 2000). Unfortunately, most hot cores are relatively far away (a few kpc, Orion-KL being an important exception), and high- spatial resolution studies are important to disentangle the var- Send offprint requests to: H. Beuther ious components in the region, to resolve potential multiple heating sources, and to search for chemical variations through- out the regions. Here we present sub-arcsecond resolution submm spectral line and dust continuum observations of the hot core G29.96−0.02, characterizing the physical and chemi- cal properties of this prototypical region. The hot core/UCHii region G29.96−0.02 is a well studied source comprising a cometary UCHii region and approximately 2.6′′ to the west a hot molecular core (Wood & Churchwell 1989; Cesaroni et al. 1994, 1998). G29.96−0.02 is at a dis- tance of ∼6 kpc (Pratap et al. 1999), the bolometric luminos- ity measured with IRAS is very high with L ∼ 1.4 × 106 L⊙ http://arxiv.org/abs/0704.0518v1 2 Beuther et al.: SMA observations of G29.96−0.02 (Cesaroni et al. 1994). Since the region harbors at least two massive (proto)stars (within the UCHii region and the hot core) this luminosity must be distributed over various sources. Based on cm continuum free-free emission, Cesaroni et al. (1994) cal- culate a luminosity for the UCHii region of Lcm ∼ 4.4×10 5 L⊙. Furthermore, they try to estimate the luminosity of the hot core via a first order black-body approximation and get a value of Lbb ∼ 1.2×10 5 L⊙. Later, Olmi et al. (2003) derive a similar es- timate (∼ 9 × 104 L⊙) via integrating a much better determined SED. The exciting source of the UCHii region has been identi- fied in the near-infrared as an O5-O8 star (Watson & Hanson 1997). Furthermore, Pratap et al. (1999) identified two addi- tional sources toward the rim of the UCHii region and an en- hanced density of reddened sources indicative of an embedded cluster. A line survey toward a number of UCHii regions reveals that G29.96−0.02 is a strong molecular line emitter in nearly all observed species (Hatchell et al. 1998). High-angular res- olution studies show that many species (e.g., NH3, CH3CN, HNCO, HCOOCH3) peak toward the main H2O maser cluster ∼ 2.6′′ west of the UCHii region (e.g, Hofner & Churchwell 1996; Cesaroni et al. 1998; Olmi et al. 2003), whereas CH3OH peaks ∼ 4′′ further south-west associated with another iso- lated H2O maser feature (Pratap et al. 1999). Hoffman et al. (2003) detected one of the relatively rare H2CO masers toward the hot core position. These masers are proposed to trace the warm molecular gas in the vicinity of young forming massive stars (Araya et al. 2006). The signature of a CH3OH peak off- set from the other molecular lines is reminiscent of Orion-KL (e.g., Wright et al. 1996; Beuther et al. 2005b). Temperature estimates toward the hot core based on high-density trac- ers vary between 80 and 150 K (e.g., Cesaroni et al. 1994; Hatchell et al. 1998; Pratap et al. 1999; Olmi et al. 2003). While Gibb et al. (2004) detect a molecular outflow in H2S emanating from the hot core in approximately the south- east north-west direction, Cesaroni et al. (1998) and Olmi et al. (2003) detect a velocity gradient in the east-west direction in the high-density tracers NH3(4,4) and CH3CN, consistent with a rotating disk around an embedded protostar. However, Maxia et al. (2001) also report that their rather low-resolution 5.9′′ × 3.7′′ (≈ 0.15 pc) SiO(2–1) data are consistent with the disk scenario as well. This is a bit puzzling since SiO is usually found to trace shocked gas in outflows and not more quies- cent gas in disks. Inspecting their SiO image again (Fig. 6 in Maxia et al. 2001), this interpretation is not unambiguous, the data also appear to be consistent with the outflow observed in H2S (Gibb et al. 2004). It is possible that the spatial resolution of their SiO(2–1) observations is not sufficient to really disen- tangle the outflow in this distant region. Olmi et al. (2003) compiled the SED from cm to mid- infrared wavelengths. While the 3 mm data are still strongly dominated by the free-free emission (Olmi et al. 2003), at 1 mm the hot core becomes clearly distinguished from the ad- jacent UCHii region (Wyrowski et al. 2002). G29.96−0.02 is one of the few hot cores which is detected at mid-infrared wavelengths (De Buizer et al. 2002). Interestingly, the mid- infrared peak is ∼ 0.5′′ (∼3000 AU) offset from the NH3(4,4) hot core position. While Gibb et al. (2004) speculate that the mid-infrared peak might arise from the scattered light only, De Buizer et al. (2002) suggest that it could trace a second mas- sive source within the same core. This hypothesis can be tested via very-high-angular-resolution submm continuum studies. 2. Observations We have observed the hot core G29.96−0.02 with the Submillimeter Array (SMA1, Ho et al. 2004) during four nights between May and November 2005. We used all available ar- ray configurations (compact, extended, very extended, for de- tails see Table 1) with unprojected baselines between 16 and 500 m, resulting at 862 µm in a projected baseline range from 16.5 to 591 kλ. The chosen phase center was the peak position of the associated UCHii region R.A. [J2000.0]: 18h46m03.s99 and Decl. [J2000.0] −02◦39′21.′′47. The velocity of rest is vlsr ∼ +98 km s −1 (Churchwell et al. 1990). Table 1. Observing parameters Date Config. # ant. Source loop τ(225GHz) [hours] 28.May05 very ext. 6 7.0 0.13-0.16 18.Jul.05 comp. 7 7.5 0.06-0.09 4.Sep.05 ext. 6 4.5 0.06-0.08 5.Nov.05 very ext. 7 3.0 0.06 For bandpass calibration we used Ganymede in the com- pact configuration and 3C279 and 3C454.3 in the extended and very extended configuration. The flux scale was derived in the compact configuration again from observations of Ganymede. For two datasets of the more extended configurations, we used 3C454.3 for the relative scaling between the various baselines and then scaled that absolutely via observations of Uranus. For the fourth dataset we did the flux calibration using 3C279 only. The flux accuracy is estimated to be accurate within 20%. Phase and amplitude calibration was done via frequent observa- tions of the quasars 1743-038 and 1751+096, about 15.5◦ and 18.3◦ from the phase center of G29.96−0.02. The zenith opac- ity τ(348GHz), measured with the NRAO tipping radiometer located at the Caltech Submillimeter Observatory, varied dur- ing the different observation nights between ∼0.15 and ∼0.4 (scaled from the 225 GHz measurement). The receiver operated in a double-sideband mode with an IF band of 4-6 GHz so that the upper and lower sideband were separated by 10 GHz. The central frequencies of the upper and lower sideband were 348.2 and 338.2 GHz, respectively. The correlator had a bandwidth of 2 GHz and the channel spacing was 0.8125 MHz. Measured double-sideband system temperatures corrected to the top of the atmosphere were between 110 and 800 K, depending on the zenith opacity and the elevation of the source. Our sensitivity was dynamic-range limited by the side-lobes of the strongest 1 The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica. Beuther et al.: SMA observations of G29.96−0.02 3 emission peaks and thus varied between the line maps of dif- ferent molecules and molecular transitions. This limitation was mainly due to the incomplete sampling of short uv-spacings and the presence of extended structures. The 1σ rms for the velocity-integrated molecular line maps (the velocity ranges for the integrations were chosen for each line separately depend- ing on the line-widths and intensities) ranged between 36 and 76 mJy. The average synthesized beam of the spectral line maps was 0.65′′×0.48′′ (P.A. −83◦). The 862 µm submm continuum image was created by averaging the apparently line-free parts of the upper sideband. The 1σ rms of the submm continuum image was ∼ 21 mJy/beam, and the achieved synthesized beam was 0.36′′×0.25′′ (P.A. 18◦), the smallest beam obtained so far with the SMA. The different synthesized beams between line and continuum maps are due to different applied weightings in the imaging process (“robust” parameters set in MIRIAD to 0 and -2, respectively) because there was insufficient signal-to- noise in the line data obtained in the very extended configura- tion. The initial flagging and calibration was done with the IDL superset MIR originally developed for the Owens Valley Radio Observatory (Scoville et al. 1993) and adapted for the SMA2. The imaging and data analysis were conducted in MIRIAD (Sault et al. 1995). 3. Results 3.1. Submillimeter continuum emission Figure 1 presents the 862 µm continuum emission extracted from the line-free parts of the upper sideband spectrum (∼1.8 GHz in total used) shown in Figure 4. The very high spa- tial resolution of 0.36′′ × 0.25′′ corresponds to a linear res- olution of ∼ 1800 AU at the given distance of ∼6 kpc. The submm continuum emission peaks approximately 2′′ west of the UCHii region and is associated with the molecular line emission known from previous observations. We do not de- tect any submm continuum emission toward the UCHii re- gion itself. At the given spatial resolution, for the first time multiplicity within the G29.96−0.02 hot core is resolved and we identify 6 submm continuum emission peaks (submm1 to submm6) above the 3σ level of 63 mJy beam−1 (Fig. 1). We consider submm1 and submm2 to be separate sources instead of a dust ridge because we count compact spherical or ellip- tical sources and their emission peaks are separated by about one synthesized beam. The four strongest submm peaks, that are all > 6σ detections, are located within a region of (1.3′′)2 (7800 AU) in diameter. The submm peak submm1 is associated with H2O and H2CO maser emission (Hofner & Churchwell 1996; Hoffman et al. 2003), and we consider this to be probably the most luminous sub-source. The other H2O maser peaks are offset from the submm continuum emission. The mid-infrared source detected by De Buizer et al. (2002) is offset > 1′′ from the submm emission. This may either be due to uncertainties in the MIR astrometry or the MIR emission may trace another young source in the region. It should be noted that the class ii CH3OH masers detected by Minier et al. (2001) peak close to 2 The MIR cookbook by Charlie Qi can be found at http://cfa-www.harvard.edu/∼cqi/mircook.html. the MIR source as well, which indicates that the MIR offset from the hot core may well be real. Table 2 lists the absolute source positions, their 862µm peak intensities and the integrated flux densities approximately associated with each of the sub-sources. Calculating the bright- ness temperature Tb of the corresponding Planck-function for, e.g., submm1, we get Tb(Peak1) ∼ 27 K. Assuming hot core dust temperatures of ∼ 100 K, the usual assumption of opti- cally thin dust emission is not really valid anymore, and one gets an approximate beam-averaged optical depth τ of the dust emission of ∼0.3. To calculate the dust and gas masses, we can follow the mass determination outlined in Hildebrand (1983) and Beuther et al. (2002, 2005a), which assumes op- tically thin emission, and correct that for the increased dust opacity. Assuming constant emission along the line of sight, the opacity correction factor C is 1 − e−τ With τ ∼ 0.3, we get a correction factor C ∼ 1.16 still compa- rably small. Assuming a dust opacity index β = 1.5, the dust opacity per unit dust mass is κ(862µm) ∼ 1.5 cm2g−1 (with the reference value κ(250µm) ∼ 9.4 cm2g−1, see Hildebrand 1983), and we assume a gas-to-dust ratio of 100. Given the uncertainties in β and T , we estimate the masses to be accu- rate within a factor 4. Table 2 gives the derived masses and beam-averaged column densities. Each sub-peak has a mass of a few M⊙, and the main submm1 exhibits approximately 10 M⊙ of compact, warm gas and dust emission. The inte- grated 862 µm continuum flux density of the central region comprising the four main submm continuum sources amounts to 1.16 Jy. At an average dust temperature of 100 K, this cor- responds to a central core mass of 39.9 M⊙. In comparison to these flux density measurements, Thompson et al. (2006) ob- served with SCUBA 850µm peak and integrated flux densities of ∼ 11.5 ± 1.2 Jy/(14′′beam) and ∼19.2 Jy, respectively. The ratio between peak and integrated JCMT fluxes already indi- cates non-compact emission even on that scales. Furthermore, subtracting a typical line contamination of the continuum emis- sion in hot cores of the order 25% (e.g., NGC6334I, Hunter et al. in prep.), the total 850µm single-dish continuum flux den- sity should amount to ∼8.6 Jy. Compared with the integrated flux density in the SMA data of ∼1.74 Jy, this indicates that approximately 80% of the single-dish emission is filtered out by the missing short spacings in the interferometer data. The dust and gas in the central region have higher temperatures than the components filtered out on larger spatial scales, and since the dust and gas mass is inversely proportionally related to the temperature by MH2 ∝ (e hν/kT − 1) (e.g., Beuther et al. 2002), a greater proportion of the mass (> 80%) is filtered out in the SMA data. However, the SMA image reveals the most compact hot gas and dust cores at the center of the evolving massive star-forming region. The shortest baseline of the SMA obser- vations of ∼16.5 kλ correspond to scales > 12′′ which hence have to be filtered out entirely. However, even smaller scales are missing because the uv-spacings corresponding to scales ≥ 5′′ are still relatively poorly sampled and the image presented in Figure 1 is only sensitive to spatial scales of the order a few http://cfa-www.harvard.edu/~cqi/mircook.html 4 Beuther et al.: SMA observations of G29.96−0.02 Fig. 1. The hot core UCHii region G29.96−0.02. The grey-scale with contours shows the submm continuum emission with a spatial resolution of 0.36′′×0.25′′. The contour levels start at the 1σ level of 21 mJy beam−1 and continue at 63, 105 mJy beam−1 (black contours) to 147, 168 mJy beam−1 (white contours). The dashed contours outline the cm continuum emission from the UCHii region and the thick contours show the NH3 emission (Cesaroni et al. 1994). The contouring is done from 15 to 95% (step 10%) of the peak emission of each image, respectively (S peak(1.2cm) = 109mJy/beam, S peak(NH3) = 15mJy/beam). Triangles, circles and pentagons show the H2O (Hofner & Churchwell 1996), H2CO (Hoffman et al. 2003) and class ii CH3OH (Minier et al. 2001) maser positions. The star marks the peak of the MIR emission (De Buizer et al. 2002), which is not a point source but has a similar size as the NH3 emission. The squares mark the infrared sources by Pratap et al. (1999). arcseconds. The submm peaks detected by the SMA are much stronger than what would have been expected if the single-dish flux (∼8.6 Jy) were uniformly distributed over the SCUBA pri- mary beam of 14′′, even ignoring any spatial filtering and miss- ing flux effects (This would result in ∼ 4 mJy per synthesized SMA beam.). This shows that the emission measured on the small spatial scales sampled by the SMA represents the com- pact core emission much better than expected. However, it does not imply that the gas masses measured by the SMA are the only gas reservoir the embedded protostars have for their on- going accretion; they may also gain mass from the large-scale gas envelope that is filtered out by our observations (see also the competitive accretion scenario, e.g., Bonnell et al. 2004). The derived beam-averaged H2 column densities are of the or- der a few times 1024 cm−2, corresponding to visual extinctions Av of a few 1000 (Av = NH/0.94 × 10 21, Frerking et al. 1982). 3.2. Spectral line emission Figure 4 presents spectra extracted toward the main submm submm1 with an angular resolution of 0.64′′×0.47′′ compared to the submm continuum map (see §2). More than 80 spectral lines from 18 molecular species, isotopologues or vibrationally excited species have been identified with a minor fraction of ∼5% of unidentified lines (UL) (Tables 6 & 4). The range of up- per level excitation temperatures for the many lines varies be- tween approximately 40 and 750 K (Table 6). Therefore, with one set of observations we are able to trace various gaseous temperature components from the relatively colder gas sur- Beuther et al.: SMA observations of G29.96−0.02 5 Table 2. Submm continuum source parameters Source R.A. Dec. S peak S int M N [J2000] [J2000] [ mJy ] [mJy] [M⊙] [10 24cm−3] submm1 18:46:03.786 -02:39:22.19 173 288 11.5 5.7 submm2 18:46:03.789 -02:39:22.48 168 237 9.5 5.5 submm3 18:46:03.758 -02:39:22.16 138 178 7.1 4.5 submm4 18:46:03.736 -02:39:22.65 151 249 9.9 5.0 submm5 18:46:03.710 -02:39:23.33 68 106 4.2 2.2 submm6 18:46:03.665 -02:39:23.80 84 85 3.4 2.8 The Table shows the peak intensities S peak, the integrated intensities S int, the derived gas masses M as well as the H2 column densities N. rounding the hot core region to the densest and warmest gas best observed in some of the vibrationally excited lines. Table 3. Peak intensities, rms and velocity ranges for images in Figs. 2 & 3. Line S peak rms ∆v mJy/beam mJy/beam km/s 862µm cont., low res. 422 17 CH3OH(73,5 − 62,4) 878 64 [90,104] 13CH3OH(137,7 − 127,6) 752 51 [95,101] CH3OH(74,3 − 64,3), vt = 1 1419 69 [91,105] CH3OCH3(74,3 − 63,4) 669 46 [94,104] C2H5OH(157,9 − 156,10) 586 51 [95,100] SiO(8 − 7) 391 36 [75,105] C34S(7 − 6) 592 62 [92,104] H2CS(101,0 − 91,9) 933 69 [92,100] 34SO(88 − 77) 827 57 [95,103] SO2(144,14 − 183,15) 544 53 [94,100] HCOOCH3(275,22 − 265,21) 491 70 [96,100] CH3CN(198 − 188) 788 71 [94,100] CH3CH2CN(383,36 − 373,35) 791 56 [94,102] CH3CHCN(362,34 − 352,32) 655 68 [96,100] HC3N(37 − 36), v7 = 1 622 55 [94,102] HC3N(37 − 36), v7 = 2 416 57 [94,100] HN13C(4 − 3) 1149 76 [94,100] Figures 2 and 3 now present integrated images of the var- ious detected species, isotopologues and vibrationally excited lines. For comparison, Figure 2 also shows the submm con- tinuum emission reduced with the same degraded spatial res- olution as the line images. All images show emission in the vicinity of the hot molecular core and no emission toward the associated UCHii region. However, the morphology varies sig- nificantly between many of the observed molecular line maps. The molecular emission is largely confined to the central region of the main four submm continuum peaks, and we do not detect appreciable molecular emission toward the continuum peaks 5 and 6. Reducing the submm continuum data with the same spatial resolution as the line images, the four submm peaks are smoothed to a single elongated structure peaking close to the submm peak submm1 (Fig. 2, top-left panel). The ground state CH3OH emission is relatively broadly distributed with two peaks in east-west direction, and one may associate one with the submm peaks 1 and 2 and the other with the submm peak submm3, but most other maps show on average one spec- tral line peak somewhere in the middle of the 4 main submm continuum peaks, similar to the lower-resolution submm con- tinuum map. However, there are also a few species which significantly deviate from this picture and show a different spatial morphol- ogy. For example SiO is more extended in north-east south- west direction likely due to a molecular outflow (§3.3). Also interesting is the emission from C34S which lacks emission around the central four submm peaks but is stronger in the in- terface region between the hot molecular core and the UCHii region (§4.3). Furthermore, there are a few spectral line maps – mainly those from likely optically thin lines (HCOOCH3, HN13C), highly excited lines (CH3CHCN) and vibrationally excited lines (CH3OH vt = 1, 2, HC3N v7 = 1, 2) – which show their emission peaks concentrated toward the main submm peak submm1 (§4.5). Previous lower-resolution (∼ 10′′) molecular line observa- tions revealed strong CH3OH emission toward the H2O maser feature approximately 4′′ south-west of the hot core peak (Fig. 1, Hofner & Churchwell 1996; Pratap et al. 1999). A lit- tle bit surprising, we do not detect any CH3OH emission (nor any other species) toward that south-western position, even when imaged at low angular resolution using only the com- pact configuration data (therefore, we do not cover that posi- tion in Figures 2 and 3). Pratap et al. (1999) discuss mainly two possibilities to explain this discrepancy: Either their ob- served specific CH3OH(80 − 71) line is a weak maser and we do not cover any comparable CH3OH line, or the emis- sion covered by the lower-resolution data is relatively extended and filtered out by our observations. As discussed in the pre- vious section, the shortest baseline of our observations was ∼16 m, implying that we are not sensitive to any scales > 12′′. Since the CH3OH emission in Pratap et al. (1999) is slightly resolved by their synthesized beam of 12.6′′ × 9.8′′, it is un- likely that we would have filtered out all emission. However, among the many observed CH3OH lines (Table 6), some have similar excitation temperatures of the order 80 K as the line observed by Pratap et al. (1999), and we would expect to de- tect thermal emission from these lines as well. Therefore, our non-detection of CH3OH emission toward the south-western H2O maser position supports rather their suggested scenario of weak CH3OH maser emission in the previously reported obser- vations (Pratap et al. 1999). 6 Beuther et al.: SMA observations of G29.96−0.02 Fig. 4. Lower and upper sideband spectra extracted toward the submm1. The spatial resolution of these data is 0.64′′ × 0.47′′. The main line identifications are shown in both panels. Table 4. Detected molecular species Species Isotopologues Vibrational states CH3OH 13CH3OH CH3OH, vt = 1, 2 CH3OCH3 C2H5OH HCOOCH3 CH3CN CH3CH2CN CH3CHCN HC3N, v7 = 1, 2 HN13C a The detection of this CH3OH vt = 2 line is doubtful since other close vt = 2 lines with similar excitation temperatures were not detected. 3.3. Molecular outflow emission The SiO(8-7) spectrum spans a large range of velocities from ∼75 to ∼111 km s−1. Integrating the blue- and red-shifted emis- sion, one gets the outflow map presented in Fig. 5. The elon- gated north-west south-east structure is consistent with the pre- viously proposed outflow by Gibb et al. (2004). The additional red feature north-east of the central hot core region makes the interpretation ambiguous: If the north-west south-east outflow is a relatively highly collimated jet, then the north-eastern red feature could be attributed to an additional outflow leaving the core in north-east south-west direction. The blue wing of that potential second outflow would not detected in our data. However, since we are filtering out any larger-scale emission, it is also possible that the red SiO features south-east and north- east of the main core are part of the same wide-angle outflow tracing potentially the limb-brightened cavity walls. In this sce- nario, our observations would miss part of the blue-shifted wide-angle outflow lobe. With the current data, it is difficult to clearly distinguish between the two scenarios. However, com- paring the elongated blue-shifted SiO(8–7) data with the pre- vious north-west south-eastern outflow observed in H2S by Gibb et al. (2004), it appears that this is the most likely direc- tion of the main outflow of the region. Therefore, the multi- ple outflow scenario appears more likely for the hot core in G29.96−0.02. The lower resolution SiO(2–1) observation by Maxia et al. (2001) are also consistent with this scenario. Based on these data, we cannot conclusively say which of the submm continuum sources submm1 to submm4 contribute to driving the outflows. Fig. 5. SiO(8-7) outflow map. The full and dashed contours are integrated over the blue- and redshifted SiO emission as shown in the figure. The contouring starts at ±2σ and continues in ±1σ steps (thick contours positive, thin contours negative). The 1σ values for the blue- and red-shifted images are 48 and 46 mJy beam−1, respectively. The markers are the same as in the previous images, the synthesized beam of 0.68′′ × 0.49′′ is shown at the bottom right, and the arrows guide the eye for the potential directions of the two discussed outflows. The offsets on the axes are relative to the phase center. 4. Discussion 4.1. The formation of a proto-Trapezium system? The four main submm continuum peaks are located within a projected area of 7800 × 7800 (AU)2 on the sky. The projected separation ∆θ between individual sub-sources varies between 1800 AU (peaks 1 and 2) and 5400 AU (peaks 1 and 4, see Table 5). Could the four central submm peaks be the predecessors of a future Trapezium system? Trapezia are defined as non- hierarchical multiple systems of three or more stars where the Beuther et al.: SMA observations of G29.96−0.02 7 largest projected separation between Trapezia members should not exceed the smallest projected separation by a factor of 3 (Sharpless 1954; Ambartsumian 1955; Abt & Corbally 2000). This criterion is satisfied by the four submm peaks at the cen- ter of the G29.96−0.02 hot core. The 14 optically identified Trapezia discussed by Abt & Corbally (2000) have mean radii to the furthest outlying member of ∼ 4 × 104 AU, with the largest radius of ∼ 5.4 × 105 AU (∼2.6 pc), the approximate dimension of an open cluster. Therefore, the protostellar pro- jected separations of the tentative proto-Trapezium candidate in G29.96−0.02 are significantly smaller than in typical optically visible Trapezia systems. A similar small size for a candidate Trapezium system has recently been reported for the multiple system in W3IRS5 (Megeath et al. 2005). Table 5. Spatial separation Pair ∆θ ∆x [′′] [AU] 1-2 0.3 1800 1-3 0.5 3000 1-4 0.9 5400 2-3 0.6 3600 2-4 0.8 4800 3-4 0.6 3600 The numbers in column 1 correspond to the numbers of the submm peaks. The small sizes of the proto-Trapezia in G29.96−0.02 and W3IRS5 may be attributed to their youth. During their upcom- ing evolution, these young system will expel most of the sur- rounding gas and dust envelope via the protostellar outflows and strong uv-radiation. Therefore, the whole gravitational po- tential of the system will decrease and the kinetic energy may dominate. Systems with positive total energy will globally ex- pand and will eventually be observable as a larger-scale optical Trapezia systems (Ambartsumian 1955). With the given data it is hard to estimate how massive the expected Trapezia stars are and will finally be at the end of their formation processes. The integrated hot core luminosity is estimated to be ∼ 105 L⊙ (Cesaroni et al. 1994; Olmi et al. 2003), in contrast to the integrated luminosity of the whole re- gion measured by the large IRAS beam of ∼ 106 L⊙. Producing 105 L⊙ requires either an O7 star or a few stars of comparable but lower masses. Nevertheless, the numbers imply that this Trapezium system should form at least one or more massive stars. Although the gas masses we derived from our dust con- tinuum data (Table 2) are relatively low, that does not neces- sarily imply that their mass reservoir is restricted to these gas masses because it is possible that they may accrete additional gas from the larger-scale envelope that is filtered out by our observations. This scenario is predicted by the competitive ac- cretion model for massive star formation (e.g., Bonnell et al. 2004). The fact that the gas masses we find for the four strongest submm sources are all similar allows to speculate that they may form about similar mass stars in the end, however, this cannot be proven by these data in more detail. Assuming that the projected size of the potential proto-Trapezium system in G29.96−0.02 of approximately 7800 (AU)2 resembles a 3-dimensional sphere of radius ∼3900 AU, we can estimate the current protostellar volume density of the region to approximately 1.4 × 105 protostars per cubic pc. This number is larger than typical stellar den- sities in young clusters of the order 104 stars per cubic pc (Lada & Lada 2003), but it is still below the extremely high (proto)stellar densities required for protostellar merger models of the order 106 to 108 stars per cubic pc (Bonnell et al. 1998, 2004; Stahler et al. 2000; Bally & Zinnecker 2005). Although we have not yet observed the extremely high (proto)stellar densities predicted by the coalescence scenario, as soon as we observe massive star-forming regions with a spa- tial resolution ≤ 4000 AU, we begin to resolve multiplicity and potential proto-Trapezia (see also the recent observations of NGC6334I and I(N) by Hunter et al. 2006). Furthermore, this (proto)stellar density may even be a lower limit, since we ob- serve only a two-dimensional projection and are additionally sensitivity limited to masses ≥ 2.1 M⊙ (corresponding to the 3σ flux limit of 63 mJy beam−1 at the assumed temperature of 100 K). Higher spatial resolution has so far always increased the observed (proto)stellar densities, and it is possible that in the future we may reach the 106 requirement for merging to play a role. However, it is also important to get better theoret- ical predictions of potential merger signatures that observers could look for. 4.2. Various episodes of massive star formation? It is interesting to note that the previously identified mid- infrared source (De Buizer et al. 2002) is offset from the submm continuum peaks. Although the mid-infrared astrome- try is usually relatively uncertain, the association of the mid- infrared peak with class ii CH3OH maser emission with an absolute positional uncertainty of only 30 mas (Minier et al. 2001) is indicative that the offset may be real. Combining the facts that we find within a small region of only ∼20000 AU (∼0.1 pc) at least three different regions of massive star for- mation – the UCHii region, the mid-infrared source, and the submm continuum sources – indicates that not all massive stars within the same evolving cluster are coeval but that sequences of massive star formation may take place even on such small spatial scales. 4.3. Carbon mono-sulfide C34S One of the most striking spectral line maps is from the rare car- bon mono-sulfide isotopologue C34S(7–6). Its emission peak is not toward the hot core nor any of the submm continuum peaks, but largely east of it in the interface region between the submm continuum peaks and the UCHii region. Hence, one likes to understand why the C34S emission is that weak toward the hot core region and that strong at the hot core/UCHii region inter- face. CS usually desorbs from dust grains at moderate temper- atures of a few 10 K, hence it should be observable relatively 8 Beuther et al.: SMA observations of G29.96−0.02 early in the evolution of a growing hot molecular core (e.g., Viti et al. 2004). From 100 K upwards H2O is released from grains, then it forms OH molecules, and the OH can react with S to SO and SO2 (e.g., Charnley 1997). Therefore, the initial high CS abundances should decrease with time while the SO and SO2 are expected to increase with time (e.g., Wakelam et al. (2005)). As shown in Figure 2, 34SO peaks to- ward the hot core where the derived CH3OH temperatures ex- ceed the H2O evaporation temperature (see §4.4 and Fig. 7b, potentially validating this theoretical prediction. According to such chemical models, the hot core G29.96−0.02 should have a chemical age of at least a few times 104 years. The strong C34S emission in the hot core/UCHii interface region may be explained in the same framework. In the molec- ular evolution scheme outlined above, one would expect low C34S emission toward the hot core with a maybe symmetrical increase further-out. In the case of the G29.96−0.02 hot core, we have the decrease toward the center, but the emission rises only toward the east, north and west with the strongest increase in the eastern hot core/UCHii region interface. If one compares the C34S morphology in Figure 2 with the temperature distri- bution in Figure 7b, one finds the lowest CH3OH temperatures right in the vicinity of the C34S emission peaks, adding further support to the proposed chemical picture. Extrapolating this scenario to other molecules, it indicates that species which are destroyed by H2O, e.g., molecular ions such as HCO+ or N2H + (e.g., Bergin et al. 1998), are no good probes of the inner regions of hot molecular cores. 4.4. Temperature structure Leurini et al. (2004, 2007) investigated the diagnostic proper- ties of methanol over a range of physical parameters typical of high-mass star-forming regions. They found that the ground state lines of CH3OH are mainly tracers of the spatial density of the gas, although at submillimeter wavelengths high k tran- sitions are also sensitive to the kinetic temperature. However, in hot, dense regions such as hot cores, the effects of infrared pumping on the level populations due to the thermal heating of the dust is not negligible, but mimic the effect of collisional ex- citation. For the ground state line, Leurini et al. (2007) found that it is virtually impossible to distinguish between IR pump- ing and pumping by collisions, as both mechanisms equally populate the vt = 0 levels. On the other hand, the vibrationally or torsionally excited lines have very high critical densities (1010–1011 cm−3) and high level energies (T ≥ 200 K). They are difficult to be populated by collisions and trace the IR field instead. To study the physical conditions of the gas around the main continuum peaks in G29.96–0.02, we analyzed only the emis- sion coming from the vt = 1 lines, as their optical depth is lower than for the ground state, and their emission is confined to the gas around the dust condensations, while the vt = 0 transitions are more extended and can be affected by problems of missing flux. We first fitted the methanol emission of the vt = 1 lines (see Fig. 6) towards the peak position, using the method de- scribed by Leurini et al. (2004, 2007) that is based on an LVG analysis and includes radiative pumping (Leurini et al. 2007). The continuum emission derived in §3.1 was used in the calcu- lations to solve the equations for the level populations. The two main dust condensations submm1 and submm2 fall in the beam of the line data; however, we assumed that the emission is com- ing from only one component, which is more extended than our beam, and derived a CH3OH column density averaged over the beam of 4×1017 cm−2. The corresponding methanol abundance, relative to H2 is of the order of 10 −7, typical of hot core sources. Since the emission from the vt = 1 lines is optically thin for this column density, and also at higher values, we consider this approach valid. The temperature derived toward the line peak is 340 K. This corresponds to our best fit model, but from a χ2 analysis we can only infer a low limit of ∼220 K for the temper- ature of the gas. Since lines are optically thin, the degeneracy between kinetic temperature and column density is not solved, and the model delivers good fit to the vt = 1 lines for lower or higher temperatures by adjusting the methanol column den- sity. However, the low temperature solutions (Tkin=100–200 K) need high methanol abundances relative to H2(∼ 10 −6), which can be hardly found at these temperatures. Moreover, lines are optically thick for these column densities, and the assumption of our analysis is not valid anymore. We also investigated the line ratio between several vt = 1 lines at the column density derived for the main position, to find the best temperature diagnostic tool among the methanol lines and derive a temperature map of the region. We found that the line ratios with the blend of lines at ∼ 337.64 GHz increase with the temperature of the gas (Fig. 7a). However, the blending of several transitions together complicates the use of such diagnostic. In Fig. 7b, we show the map of the line ratio between the 71,6 → 61,5-E vt = 1 at 337.708 GHz and the blend between the 71,7 → 61,6-E vt = 1 at 337.642 GHz and 70,7 → 60,6-E vt = 1 at 337.644 GHz. Since line intensities do not simply add up, we did not correct for the overlapping between the two transitions. Two other lines, the 74,3 → 64,2-E vt = 1 and the 75,3 → 64,2-E vt = 1, are also very close in frequency. This is seen in the linewidth of the blending, which is wider than for the other lines. Therefore, we considered only half of the channels of the blending at 337.64 GHz in our line ratio analysis. From the ratio-map in Fig. 7b, submm1, submm2 and submm3 of Table 1 show high temperatures (T≥ 300 K), while relatively low temperature gas (T∼ 100 K) is found at R.A. [J2000]=18h46m03s.818 Dec. [J2000]= −02◦39′22′′.14, close to a secondary peak of many ground state lines of methanol (Fig. 2). The temperature then decreases towards submm4. The increase in the line ratio towards the south-east and north is probably not true, but due to the poor signal to noise ratio in these areas. Changes in the column densities along the area may affect our results. Fig. 6. Spectrum of the 7ka ,kb → 6ka ,kb−1 vt = 1 methanol band to- ward the main dust condensation. Overlaid in black is the synthetic spectrum resulting from the fit. Beuther et al.: SMA observations of G29.96−0.02 9 Fig. 7. a: Modeled line ratio between the 71,6 → 61,5-E vt = 1 line and the 71,7 → 61,6-E vt = 1 transitions, as function of the temperature. b: Map of the line ratio between the same transitions in the inner region around the peaks. The white stars mark the positions of the dust peaks; the white dashed contours show the values of the line ratio from ∼ 150 to ∼ 350 K, which correspond to levels from 0.3 to 0.7 in step of 0.1 in the map. The solid black contours show the continuum emission smoothed to the resolution of the line data (from 0.2 to 0.4 Jy/beam in step of 0.05). The offsets on the axes are relative to the phase center. 4.5. Tracing rotation toward the massive cores At the given lower spatial resolution of the spectral line data compared to the submm continuum, we cannot resolve the four submm peaks well. However, one of the aims of such multi-line studies is to identify spectral lines that trace the massive proto- stars and that are potentially associated with massive disk-like structures. Such lines may then be used for kinematic gas stud- ies of rotating gas envelopes, tori or accretion disks. Therefore, we analyzed the data-cubes searching for velocity structures indicative of any kind of rotation. In the large majority of spec- tral lines, this was not successful and we could mostly not identify coherent velocity structure. While chemical and tem- perature effects (§4.3 & 4.4) may be responsible for parts of that, the large column densities derived in §3.1 imply also large molecular line column densities and hence large optical depths. Therefore, many of the observed lines are likely optically thick tracing only outer gas layers of the hot molecular core not pen- etrating down to the deeply embedded protostars. Furthermore, many molecules would not only be excited in the central ro- tating disk-like structures but also in the surrounding envelope and maybe the outflow. Hence, disentangling the different com- ponents observationally remains a challenging task. Fig. 8. Moment 1 maps of HN13C(4–3) (top) and HC3N(37– 36)v7 = 1 (bottom). The markers are the same as in the previous images, and the synthesized beam of 0.68′′ × 0.49′′ is shown at the bottom left. The offsets on the axes are relative to the phase center. The major exceptions are the molecular lines of the rare isotopologue of hydrogen isocyanide HN13C(4–3) with a low excitation temperature of only 42 K, and the vibrationally ex- cited line of cyanoacetylene HC3N(37–36)v7 = 1 with a higher excitation temperature of 629 K (Fig. 8). In both cases we find a velocity gradient across the main submm peak submm1 with a position angle of ∼ 45◦ from north. This is approximately perpendicular to the molecular outflow discussed in §3.3 and by Gibb et al. (2004). Interestingly, Gibb et al. (2004) also find a similar velocity gradient in their central velocity channels of H2S. The previously reported NH3 and CH3CN velocity gradi- ents in approximately east-west direction (Cesaroni et al. 1998; Olmi et al. 2003) have been observed with slightly lower spa- tial resolution and are consistent with our data as well. Our observations as well as previous work in the liter- ature suggest that the G29.96−0.02 hot core exhibits a ve- locity gradient in the dense gas in approximately north-east south-west direction perpendicular to the molecular outflow observed at larger scales. Based on the HN13C(4–3) map, the diameter of this structure is ∼ 1.6′′ corresponding to radius of ∼4800 AU. Since this emission encompasses not only the submm peak submm1 but also the submm2 and submm3, it is not genuine protostellar disk as often observed in low- mass star-forming regions. The velocity structure does not re- semble Keplerian rotation and may hence be due to some larger-scale rotating envelope or torus that could transform into a genuine accretion disks at smaller still unresolved spa- tial scales (Cesaroni et al. 2007). Additional options to ex- plain such a velocity gradient may be (a) interaction with the 2nd outflow in north-east–south-western direction, (b) inter- action with the expanding UCHii region, and (c) global col- lapse like recently proposed for NGC2264 (Peretto et al. 2006). While we cannot exclude (a) and (b), option (c) of a globally collapsing core appears particularly interesting because com- bining rotation and collapse would result in an inward spi- raling kinematic structure, potentially similar to the models originally proposed for rotating low-mass cores (e.g., Ulrich 1976; Terebey et al. 1984). Recent hydrodynamic simulations by Dobbs et al. (2005) and Krumholz et al. (2006) as well as analytic studies by Kratter & Matzner (2006) find fragmenta- tion and star formation within the massive disks forming early in the collapse process of high-mass cores. This would be con- sistent with the found three sub-sources (submm1 to submm3) within the HN13C/HC3N structure. However, on a cautionary note it needs to be stressed that the collapse/rotation scenario is far from conclusive, and that the outflow and/or UCHii re- gion can potentially influence the observed velocity pattern as well. It remains puzzling that only these two lines exhibit the discussed signatures whereas all the other spectral lines in our setup do not. 5. Conclusions and Summary The new 862 µm submm continuum and spectral line data ob- tained with the SMA toward G29.96−0.02 at sub-arcsecond spatial resolution resolve the hot molecular core into sev- eral sub-sources. At an angular resolution of 0.36′′ × 0.25′′, corresponding to linear scales of ∼1800 AU, the central core contains four submm continuum peaks which resemble a Trapezium-like multiple system at a very early evolutionary stage. Assuming spherical symmetry for the hot core region, the protostellar densities are high of the order 1.4 × 105 pro- tostars per pc3. However, these protostellar densities are still below the required values between 106 to 108 protostars/pc3 to make coalescence of protostars a feasible process. Derived H2 column densities of the order a few 1024 cm−2 imply visual ex- tinctions of a few 1000. The existence of three sites of massive star formation in different evolutionary stages within a small region (the UCHii region, the mid-infrared source, and the submm continuum sources) indicates that sequences of mas- sive star formation may take place within the same evolving massive protocluster. The 4 GHz of observed bandpass reveal a plethora of ap- proximately 80 spectral lines from 18 molecular species, iso- topologues or vibrationally excited lines. Only about 5% of the 10 Beuther et al.: SMA observations of G29.96−0.02 spectral lines remain unidentified. Most spectral lines peak to- ward the hot molecular core, while a few species also show more extended emission, likely due to molecular outflows and chemical differentiation. The CH3OH line forest allows us to investigate the temperature structure in more detail. We find hot core temperatures≥ 300 K and decreasing temperature gra- dients to the core edges. The SiO(8-7) observations confirm a previously reported outflow Gibb et al. (2004) in north-west south-east direction with a potential identification of a second outflow emanating approximately in perpendicular direction. Furthermore, C34S exhibits a peculiar morphology being weak toward the hot molecular core and strong in its surroundings, particular in the UCHii/hot core interface region. The C34S de- ficiency toward the hot molecular core may be explained by time-dependent chemical desorption from grains, where the C34S desorbs early, and later-on after H2O desorbs from grains forming OH, the sulphur reacts with the OH to form SO and Furthermore, we were interested in identifying the best molecular line tracers to investigate the kinematics and po- tential disk-like structures in such dense and young massive star-forming regions. Most spectral lines do not exhibit any coherent velocity structure. A likely explanation for this un- correlation between molecular line peaks and submm contin- uum peaks is that many spectral lines may be optically thick in such high-column-density regions, and that additional chem- ical evolution and temperature effects complicate the picture. Furthermore, many molecules are excited in various gas com- ponents (e.g., disk, envelope, outflow), and it is often observa- tionally difficult to disentangle the different contributions prop- erly. There are a few exceptions of optically thin and vibra- tionally excited lines that apparently probe deeper into the core tracing submm1 better than other transitions. Investigating the velocity pattern of these spectral lines, we find for some of them a velocity gradient in the north-east south-west direc- tion perpendicular to the molecular outflow. Since the spatial scale of this structure is relatively large (∼4800 AU) compris- ing three of the central protostellar sources, and since the veloc- ity structure is not Keplerian, this is not a genuine Keplerian ac- cretion disk. While these data are consistent with a larger-scale toroid or envelope that may rotate and/or globally collapse, we cannot exclude other explanations, such as that the influence of the outflow(s) and/or expanding UCHII region produces the observed velocity pattern. In addition to this, these data con- firm previous findings that the high column densities, the large optical depths of the spectral lines, the chemical evolution, and the different spectral line contributions from various gas com- ponents make it very difficult to identify suitable massive ac- cretion disk tracers, and hence to study this phenomenon in a more statistical fashion. (e.g., Beuther et al. 2006) Acknowledgements. 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The offsets on the axes are relative to the phase center. Beuther et al.: SMA observations of G29.96−0.02 13 Fig. 3. Continued Figure 2. 14 Beuther et al.: SMA observations of G29.96−0.02 Table 6. Line parameters Freq. Line Eu Freq. Line Eu GHz K GHz K 337.279 CH3OH(72,5 − 62,4)E(vt=2) a 727 338.409 CH3OH(70,7 − 60,6)A 65 337.297 CH3OH(71,7 − 61,6)A(vt=1) 390 338.431 CH3OH(76,1 − 66,0)E 254 337.348 CH3CH2CN(383,36 − 373,35) 328 338.442 CH3OH(76,1 − 66,0)A 259 337.397 C34S(7–6) 65 CH3OH(76,2 − 66,1)A − 259 337.421 CH3OCH3(212,19 − 203,18) 220 338.457 CH3OH(75,2 − 65,1)E 189 337.446 CH3CH2CN(374,33 − 364,32) 322 338.475 CH3OH(75,3 − 65,2)E 201 337.464 CH3OH(76,1 − 60,0)A(vt=1) 533 338.486 CH3OH(75,3 − 65,2)A 203 337.474 UL CH3OH(75,2 − 65,1)A − 203 337.490 HCOOCH3(278,20 − 268,19)E 267 338.504 CH3OH(74,4 − 64,3)E 153 337.519 CH3OH(75,2 − 65,2)E(vt=1) 482 338.513 CH3OH(74,4 − 64,3)A − 145 337.546 CH3OH(75,3 − 65,2)A(vt=1) 485 CH3OH(74,3 − 64,2)A 145 CH3OH(75,2 − 65,1)A −(vt=1) 485 CH3OH(72,6 − 62,5)A − 103 337.582 34SO(88 − 77) 86 338.530 CH3OH(74,3 − 64,2)E 161 337.605 CH3OH(72,5 − 62,4)E(vt=1) 429 338.541 CH3OH(73,5 − 63,4)A + 115 337.611 CH3OH(76,1 − 66,0)E(vt=1) 657 338.543 CH3OH(73,4 − 63,3)A − 115 CH3OH(73,4 − 63,3)E(vt=1) 388 338.560 CH3OH(73,5 − 63,4)E 128 337.626 CH3OH(72,5 − 62,4)A(vt=1) 364 338.583 CH3OH(73,4 − 63,3)E 113 337.636 CH3OH(72,6 − 62,5)A −(vt=1) 364 338.612 SO2(201,19 − 192,18) 199 337.642 CH3OH(71,7 − 61,6)E(vt=1) 356 338.615 CH3OH(71,6 − 61,5)E 86 337.644 CH3OH(70,7 − 60,6)E(vt=1) 365 338.640 CH3OH(72,5 − 62,4)A 103 337.646 CH3OH(74,3 − 64,2)E(vt=1) 470 338.722 CH3OH(72,5 − 62,4)E 87 337.648 CH3OH(75,3 − 65,2)E(vt=1) 611 338.723 CH3OH(72,6 − 62,5)E 91 337.655 CH3OH(73,5 − 63,4)A(vt=1) 461 338.760 13CH3OH(137,7 − 127,6)A 206 CH3OH(73,4 − 63,3)A −(vt=1) 461 338.769 HC3N(37 − 36)v7 = 2 525 337.671 CH3OH(72,6 − 62,5)E(vt=1) 465 338.886 C2H5OH(157,8 − 156,19) 162 337.686 CH3OH(74,3 − 64,2)A(vt=1) 546 339.058 C2H5OH(147,7 − 146,8) 150 CH3OH(74,4 − 64,3)A −(vt=1) 546 347.232 CH2CHCN(381,38 − 371,37) 329 CH3OH(75,2 − 65,1)E(vt=1) 494 347.331 28SiO(8–7) 75 337.708 CH3OH(71,6 − 61,5)E(vt=1) 489 347.446 UL 337.722 CH3OCH3(74,4 − 63,3)EE 48 347.494 HCOOCH3(275,22 − 265,21)A 247 337.732 CH3OCH3(74,3 − 63,3)EE 48 347.759 CH2CHCN(362,34 − 352,32) 317 337.749 CH3OH(70,7 − 60,6)A(vt=1) 489 347.792 UL 337.778 CH3OCH3(74,4 − 63,4)EE 48 347.842 UL 337.787 CH3OCH3(74,3 − 63,4)AA 48 347.916 C2H5OH(204,17 − 194,16) 251 337.825 HC3N(37 − 36)v7 = 1 629 347.983 UL 337.838 CH3OH(206,14 − 215,16)E 676 348.261 CH3CH2CN(392,37 − 382,36) 344 337.878 CH3OH(71,6 − 61,5)A(vt=2) 748 348.340 HN 13C(4–3) 42 337.969 CH3OH(71,6 − 61,5)A(vt=1) 390 348.345 CH3CH2CN(402,39 − 392,38) 351 338.081 H2CS(101,10 − 91,9) 102 348.532 H2CS(101,9 − 91,8) 105 338.125 CH3OH(70,7 − 60,6)E 78 348.910 HCOOCH3(289,20 − 279,19)E 295 338.143 CH3CH2CN(373,34 − 363,33) 317 348.911 CH3CN(199 − 189) 745 338.306 SO2(144,14 − 183,15) 197 349.025 CH3CN(198 − 188) 624 338.345 CH3OH(71,7 − 61,6)E 71 349.107 CH3OH(141,13 − 140,14) 43 338.405 CH3OH(76,2 − 66,1)E 244 a The detection of this CH3OH vt = 2 line is doubtful since other close vt = 2 lines with similar excitation temperatures were not detected. This figure "ch3oh1.jpg" is available in "jpg" format from: http://arxiv.org/ps/0704.0518v1 http://arxiv.org/ps/0704.0518v1 This figure "hc3n_v1_mom1.jpg" is available in "jpg" format from: http://arxiv.org/ps/0704.0518v1 http://arxiv.org/ps/0704.0518v1 This figure "hn13c_mom1.jpg" is available in "jpg" format from: http://arxiv.org/ps/0704.0518v1 http://arxiv.org/ps/0704.0518v1 This figure "ch3oh2.jpg" is available in "jpg" format from: http://arxiv.org/ps/0704.0518v1 http://arxiv.org/ps/0704.0518v1 This figure "ch3oh3.jpg" is available in "jpg" format from: http://arxiv.org/ps/0704.0518v1 http://arxiv.org/ps/0704.0518v1 Introduction Observations Results Submillimeter continuum emission Spectral line emission Molecular outflow emission Discussion The formation of a proto-Trapezium system? Various episodes of massive star formation? Carbon mono-sulfide C34S Temperature structure Tracing rotation toward the massive cores Conclusions and Summary

0704.0519

Hamilton-Jacobi Fractional Sequential Mechanics

Microsoft Word - H-J Fractional Hamilton-Jacobi Fractional Sequential Mechanics Eqab M. RABEI* and Bashar S. ABABNEH Department of Physics, Mutah University, Al-Karak, Jordan Abstract As a continuation of Rabei et al. work [11], the Hamilton- Jacobi partial differential equation is generalized to be applicable for systems containing fractional derivatives. The Hamilton- Jacobi function in configuration space is obtained in a similar manner to the usual mechanics. Two problems are considered to demonstrate the application of the formalism. The result found to be in exact agreement with Agrawal's formalism. Keywords: Fractional derivative, Fractional systems, Hamilotonian formalisms, Hamilton-Jacobi treatment. *[email protected] 1-Introduction The Hamiltonian formulation of non-conservative systems has been developed by Riewe[1,2].He used the fractional derivative [3,4,5] to construct the Lagrangian and Hamiltonian for non-conservative systems. As a sequel to Riewe's work, Rabei et al. [6] used Laplace transforms of fractional integrals and fractional derivatives to develop a general formula for the potential of any arbitrary forces, conservative or non- conservative. This led directly to the consideration of the dissipative effects in Lagrangian and Hamiltonian formulations. Besides, the canonical quantization of non- conservative systems carried out by Rabei et al. [7]. Other investigations and further developments are given by Agrawal [8] .He presented the fractional variational problems and the resulting equations are found to be similar to those for variation problems containing integral order derivatives. This approach is extended to classical fields with fractional derivatives [9]. Besides, Kilmek [10] showed that the fractional Hamiltonian is usually not a constant of motion, even in the case when the Hamiltonian is not an explicit function of time. In addition, as a continuation of Agrawal’s work [8], Rabei et al. [11] achieved the passage from the Lagrangian containing fractional derivatives to the Hamiltonian. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In the present work, the Hamilton – Jacobi partial differential equation (HJPDE) is generalized to be applicable for systems containing fractional derivatives. The paper is organized as follows: In Sec. 2 Lagrangian and Hamiltonian formalisms with fractional derivatives are reviewed briefly. In Sec.3, Hamilton-Jacobi Partial differential equations with fractional derivatives is constructed, and two illustrative examples are given in Sec. 4. 2- Hamiltonian Formalism with Fractional Derivative Several definitions of a fractional derivative have been proposed. These definitions include Riemann–Liouville, Grünwald–Letnikov, Weyl, Caputo, Marchaud, and Riesz fractional derivatives. Here; the problem is formulated in terms of the left and the right Riemann–Liouville fractional derivatives. The left Riemann–Liouville fractional derivative defined as ∫ −−−⎟⎠ xa dfx xfD τττ αα )()( )( 1 (1) Which is denoted as the LRLFD and the right Riemann–Liouville fractional derivative reads as ∫ −−−⎟⎠ bx dfx xfD τττ αα )()( )( 1 (2) Which is denoted as the RRLFD. Here α is the order of the derivative such that nn ≤≤− α1 and Γ represents the Euler gamma function. If α is an integer, these derivatives are defined in the usual sense, i.e. ,....3,2,1,)()(,)()( =⎟ xfDxf xfD bxxa (3) The fractional operator xa D can be written as [13] D −= αα (4) Where the number of additional differentiations n is equal to [α] +1, where [α] is the whole part of α. The operator αxa D is a generalization of differential and integral operators and can be introduced as follows: ∫ − 0)Re()( 0)Re(1 0)Re( (5) Following to Agrawal [8], the Euler-Lgrange equations for fractional calculus of variations problem is obtained as (6) Where L is the genaralized Lagrangian function of the form ),,,( tqDqDqL btta The generalized momenta are introduced as ββαα ∂ = , (7) And the Hamiltonian depending on the fractional time derivatives reads as LqDpqDpH btta −+= α (8) In Ref [11], the Hamilton’s equations of motion are obtained in a similar manner to the usual mechanics. These equations read as, ; qD , ; β α pDpD tabt +=∂ It is observed that the fractional Hamiltonian is not a constant of motion even though the Lagrangian does not depend on the time explicitly. 3. Hamilton-Jacobi Partial Differential Equation with Fractional Derivatives In this section, the determination of the Hamilton-Jacobi partial differential equation for systems with fractional derivatives is discussed. According to Rabei et al. [11], the fractional Hamiltonian is written as ( ) ),,,(,,, tqDqDqLqDpqDptppqH bttabtta βαββααβα −+= (9) Consider the canonical transformation with a generating function ( )tPPqDqDF btta ,,,, 112 βαβα −− Then, the new Hamiltonian will take the form ( ) ),,,(,,, tQDQDQLQDPQDPtPPQK bttabtta βαββααβα ′−+= (10) The old canonical coordinates βα ppq ,, , satisfy the fractional Hamilton’s principle that can be put in the form ( ) 0 btta dtHqDpqDp αδ (11) At the same time the new canonical coordinates βα PPQ ,, , of course satisfy a similar principle. ( ) 0 btta dtKQDPQDP αδ (12) The simultaneous validity of Eq. (11) and Eq. (12) does not mean of course that the integrands in both expressions are equal. Since the general form of the Hamilton’s principle has zero variation at the end points, both statements will be satisfied if the integrands connected by a relation of the form [12] KQDPQDPHqDpqDp bttabtta +−+=−+ α (13) Where the function F is given as: ( ) QDPQDPtPPqDqDFF bttabtta 11112 ,,,, −−−− −−= ββααβαβα (14) The function F2 is called Hamilton’s principal function S for a contact transformation. ( )tPPqDqDSF btta ,,,, 112 βαβα −−= (15) Thus, btbttata 1111 −−−− −−−−= ββ By using definitions of fractional calculus given in Eq. (4) then we have QDPQD QDPQD btbttata αα −−−−= −− 11 (16) Substituting the values of the from Eq. (16) into the Eq. (13) we have KHqDpqDp bttabtta 11 −− −−+−=−+ ββααββ α (17) Again using definitions of fractional calculus given in Eq. (4) we have the following form α 11 (18) Substituting the values of the from Eq. (18) into the Eq. (17) we get = ββαα (19) http://scienceworld.wolfram.com/physics/ContactTransformation.html QD btta ∂ = −− 11 , (20) K + (21) We can automatically ensure that the new variables are constant in time by requiring that the transformed Hamiltonian K shall be identically zero, In other words, βα PPQ ,, are constants. We see by putting K = 0 that this generating function must satisfy the partial differential equation. 0= H (22) This equation is called the Hamilton –Jacobi equation. Let us assume that 21 , EPEP == βα Where 1E , 2E are constants. Then the action function (15), can be expressed as ( )tEEqDqDSS btta ,,,, 2111 −−= βα (23) Further insight into the physical significance of Hamilton’s principal function S is furnished by an examination of the total time derivative, which can be computed from the formula α 11 (24) By using Eq. (19) we have HqDpqDp btta −+= And using Eq. (9) we have L Thus ∫= LdtS (25) If we restrict our considerations to the time -independent Hamiltonians, then the Hamilton-Jacobi function can be written in the form ( ) ( ) ( )tEEfEqDWEqDWS btta ,,,, 21212111 ++= −− βα (26) Where W is called Hamilton’s characteristic function and the function, f takes the following form: ( ) EttEEf −=,, 21 Making use of equations (19) and (20) we obtain: = ββαα (27) 2 11 , λλ βα = QD btta (28) Here 1λ , 2λ are constants. The physical significance of W can be understood by writing its total time derivative qD = (29) Comparing this expression to the results of substituting Eq. (27) into Eq. (29) we see that ∫∫ −=⇒=⇒= qDdpWqdtDpWqDpdt tatata α (30) Again one may show that ∫ −= qDdpW bt 12 ββ (31) 4. Illustrative Examples To demonstrate the application of our formalism, let us discuss the following models: As a first model consider the lagrangian given by Agrawal [8] ( )20 qDL t The (HJPDE) for this Lagrangian is calculated as ( ) 0 1 2 = Using Eq. (27) we obtain 0 1 =−⎟⎟ − EqD Solving this equation we have qDEW t −= α Thus Ep 2=α Making use of Eq. (26) we obtain the function S as: EtqDES t −= Eq. (28) leads to 1 1 λαα =−= = −− tqD QD tt Thus ( )110 2 λα +=− tEqDt α α pEqDt == 20 This is the same result obtained by Rabei et al. [11], which is equivalent to Agrawal formalism [8]. As a second model consider the Lagrangian given by Rabei et al. [11] ( ) ( ) qDqDqDqDL tttt βαβα 102120 ++= The Hamiltonian is calculated as ( ) ( )22 βα ppH == Thus, the Hamilton-Jacobi partial differential equation reads as: ( ) 0 1 2 = Making use of Eq. (26) we have 0 1 =−⎟⎟ − EqD Thus, qDEW t −= α Again the (HJPDE) can be written as ( ) 0 1 2 = Then 0 2 =−⎟⎟ − EqD Which leads to qDEW t −= β Thus the Hamilton-Jacobi action function can be written as EtqDEqDES tt −+= Where E = −αα E = −ββ Using Eq. (28) we get 1 1 λβαα =−+= −−− tqD QD ttt Thus ( )11110 2 λβα +=+ −− tEqDqD tt EqDqD tt 210 =+ Then qDqDp tt α 10 += qDqDp tt β 10 += These Leads to 0))(( 1010 =++ qDqDDD tttt This result is in exact agreement with Rabei et al. [11]. 5- Conclusion In This work we have studied the Hamilton-Jacobi partial differential equation for systems containing fractional derivatives. A general theory to solve the Hamilton- Jacobi partial differential equation is proposed for systems containing fractional derivatives under the condition that this equation is separable. The Hamilton-Jacobi function is determined in the same manner as for usual systems. Finding this function enables us to get the solutions of the equations of motion. In order to test our formalism, and to get a somewhat deeper understanding, we have examined two examples of systems with fractional derivatives. The result found to be in exact agreement with Lagrangian formulation given by Agrawal [8] and with Hamiltonian formulation given by Rabei et al. [11]. 6- References [1]F. Riewe, Non Conservative Lgrangian and Hamiltonian mechanics. Physical Review E. 53:1890-1899, (1996). [2] F. Riewe, Mechanics with Fractional Derivatives. Physical Review E.55: 3581- 3592, (1997). [3] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, (2000). [4] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: theory and applications, Gordon and Breach, Amsterdam, (1993). [5] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999); Fractional Calculus and Applied Analysis 5: 367, (2002). [6] Eqab M. Rabei, T. Al-halholy, A. Rousan, Potentials of Arbitrary Forces with Fractional Derivatives, International Journal of Modern Physics A, 19: 3083-3092, (2004). [7] Eqab M. Rabei, Abdul-Wali Ajlouni, Humman B. Ghassib, Quantization of Non- Conservative Systems Using Fractional Calculus, WSEAS Transactions on Mathematics, 5: 853-863, (2006); Quantization of Brownian Motion, International Journal of Theoretical physics, (2006) in press. [8]Om P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl.272: 368-379, (2002). [9] Dumitru Baleanu, Sami I. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scripta (in press). [10] M. Klimek, Lagrangian and Hamiltonian fractional sequential mechanics; Czech J. Phys., 52: 1247-1253, (2002); Fractional sequential mechanics – models with symmetric fractional derivative, Czech J. Phys. 51: 1348-1354, (2001). [11] Eqab M. Rabei, Khaled I. Nawafleh, Raed S. Hijjawi , Sami I. Muslih, Dumitru Baleanu, The Hamilton Formalism With Fractional Derivatives, J. Math. Anal. Appl. (in press) [12] H. Goldstein, Classical Mechanics, Addison-Wesley Publishing Company, (1980). [13] Igor M. Sokolov,Joseph Klafter, Alexander Blumen, Fractional Kinetics, Physics Today(2002) American Institute of physics,S-0031-9228-0211-030-1. [14] B.N.N. Achar, J.W. Hanneken, T. Enck, T.Clarke, Dynamics of the fractional oscillator, Physica A, 297: 361-367, (2001).

0704.0520

A critical theory of quantum entanglement for the Hydrogen molecule

A critical theory of quantum entanglement for the Hydrogen molecule Tina A.C. Maiolo∗, Luigi Martina†, Giulio Soliani‡ Abstract In this paper we investigate some entanglement properties for the Hy- drogen molecule considered as a two interacting spin 1 (qubit) model. The entanglement related to the H2 molecule is evaluated both using the von Neumann entropy and the Concurrence and it is compared with the corresponding quantities for the two interacting spin system. Many as- pects of these functions are examinated employing in part analytical and, essentially, numerical techniques. We have compared analogous results obtained by Huang and Kais a few years ago. In this respect, some pos- sible controversial situations are presented and discussed. 1 Introduction and the model Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration. The concept of entanglement gives a physical meaning to the electron cor- relation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the ex- act ground state energy of the many electrons Schrödinger equation and the Hartree–Fock energy. In this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so–called correlation energy which is not directly observ- able. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian HH2 = − (1 + g)σ1 ⊗ σ1 − (1− g)σ2 ⊗ σ2 − B(σ3 ⊗ σ3 + σ0 ⊗ σ3), (1) ∗Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100 Lecce, Italy; e–mail: [email protected] †Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100 Lecce, Italy; e–mail: [email protected] ‡Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100 Lecce, Italy; e–mail: [email protected] http://arxiv.org/abs/0704.0520v2 where σi stand for the Pauli matrices (σ0 = I). Actually, this model was con- sidered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange in- teraction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree–Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model. We notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J . In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E0 = −2, which we may consider as the Hartree- Fock approximated fundamental level in molecular structure calculations. When J 6= 0, the fundamental energy eigenvalue is E= − 4 + g2λ2 in Re- gion I defined by 0 < λ ≤ 2√ , otherwise E = −λ (λ means the coupling constant) in Region II, which is the complement of I which respect to pos- itive real axis. The corresponding (non normalized) eigenstates are |ΨI〉 = g2λ2+4+2 , 0, 0, 1 and |ΨII〉 = 0, 1, 1, 0 , respectively. In both cases the state is entangled. Since we are dealing with pure states, the von Neumann entropy [2] SvN = −Tr ρ1log2ρ1 is chosen to be a measurement of the entanglement, where ρ1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has SvN,I = − g2λ2 + 4 log g2λ2 + 4 g2λ2 + 4 + 4 λ2 + 8 g2λ2 + 4 + 2 g2λ2 + 4 g2λ2 + 4 g2λ2 + 2 g2λ2 + 4 + 4 log(4) SvN,II = 1. (4) Scrutinizing Eq. (3) and Eq. (4) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin. Limiting all further considerations to the case of weak interaction, we observe that at the boundary point λb = a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5]. As it was pointed out in [6], for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7], Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement. The Concurrence reads C(ρ) = max(0, ν1 − ν2 − ν3 − ν4), (5) where the νi’s are the eigenvalues of the Hermitian matrix where ρ̃ = (σy ⊗ σy)ρ∗(σy ⊗ σy), ρ∗ being the complex conjugate of ρ taken in the standard basis [7]. Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence. To this aim, we compute the Concurrence CI and CII, i. e. CI = gλ g2λ2 + 4 , CII = 1. (6) where I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1−g2 , and E = −λ, respectively. In Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is pre- sented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 1 2 3 4 5 Conc. Figure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1. 2 A comparison between the entanglement and the correlation energy Now we look for a comparison between the entanglement with the energy cor- relation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by Ecorr = |E0| − 2 = 4 + λ2 − 2. (7) We observe that the entanglement measure is always bounded, while Ecorr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation ∆S2α dλ, with ∆Sα = Ecorr − αSvN . (8) Thus the minimizing parameter αmin will be given by αmin = EcorrSvN dλ ≈ −0.691217. (9) A formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have ∆C2α′ dλ, with ∆Cα′ = Ecorr − α′ C. (10) Now, in order to estimate the relative deviation of SvN with respect to Ecorr, let us report |∆Sαmin |/SvN and |∆Sαmin/Ecorr| as functions of λ at the optimal value αmin. The graphs of these functions are shown in Figure 2. 0.2 0.4 0.6 0.8 1 ÈDSmin�SvN È 0.2 0.4 0.6 0.8 1 ÈDSmin�EcorrÈ Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value αmin as a function of the coupling constant λ for g = 1. In Figure 3, the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α′min, is represented. 0.2 0.4 0.6 0.8 1 ÈDCminÈ�C 0.2 0.4 0.6 0.8 1 ÈDCmin�EcorrÈ Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α′min as a function of the coupling constant λ for g = 1. Remark 1 From these graphs, one can argue that the agreement between the two func- tions SvN and Ecorr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the corre- lation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1], i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a more concrete physical meaning, in particular whether the minimizing parame- ter αmin and the vanishing point of ∆Sαmin does possess any physical meaning (or α′min and the vanishing point of ∆Cα′min). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For in- stance one finds ∆Cα′ 0.383249 λ√ λ2 + 4 + 2 Remark 2 We note that in an interval of values around αmin, the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4). 0.2 0.4 0.6 0.8 1 -0.05 0.2 0.4 0.6 0.8 1 -0.05 d DSΑ � dΛ Figure 4: The deviation ∆Sα and its derivative with respect to λ are computed for values of −1.29(red) ≤ α ≤ −0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to αmin This behavior suggests to consider the function ∆Sαmin as a sort of ”free en- ergy” , where αmin mimics the ”temperature” specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆Sαmin . Similar considerations can be made looking at the graphs drawn for the function ∆Cα′ and its derivative with respect to λ (see Figure 5). The function ∆Sαmin or, alternatively, the minimum of ∆Cα′ can be ob- tained algebraically. Such a minimum is at the value of the coupling constant λSvNmin ≈ 0.485 and λCmin ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which con- sists in substituting the molecular binding with a position dependent exchange coupling: J(r) ≈ 1.641 r 2 e−2 r Ry, (11) where r is given in Bohr radius, see Figure 6. The maximum value taken by this function is at the point rmax = 1.25. Assuming B = 0.5 Ry, i.e. 12 of the funda- mental level of the Hydrogen atom, the maximum value λ′max = J(rmax)/B ≈ 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 dDCΑ'� dΛ Figure 5: The deviation ∆Cα′ and its derivative with respect to λ are computed for values of −0.98(red) ≤ α′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α′min 0.5 1 1.5 2 2.5 3 JHrLHRyL Figure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λSvNmin , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆Sαmin . Then, the equilibrium bal- ance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to rmax (see the first panel of Figure 7). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ′′max ≈ 0.628, which goes beyond λmin, even if it is always less than 1. Now, the deviation function ∆Sαmin has two minima as seen in the second panel of Figure 7, one of which is at r′′− ≈ 0.76 , the other one being at r′′+ ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is rexp ≈ 2.0. We point out that although the spin–model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆Sαmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value rexp. 0.5 1 1.5 2 2.5 3 -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 DS minHr; B= .5 RyL 0.5 1 1.5 2 2.5 3 -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 DS minHr; B= .375 RyL r''- r''+ Figure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by (11). The first question to answer is whether this draft works also for the Concur- rence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 r HBohrL B HRyL 0.2 0.4 0.6 0.8 1 B(Ry) r(Bohr) Figure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11). The range of values divided by the contour lines is [−0.038, 0, 04] for the left panel and [−0.03705, −0, 03000] for the right one that approximatively corresponding to the black area in the left panel. In correspondence of the same values considered above, for B = 0.5 Ry the function ∆Cα′ (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 Ry) for the function ∆Cα′ One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one. In other words, to conclude monitoring numerically B the equilibrium config- uration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆Cα′ 3 A quantum chemical framework to compare entanglement and correlation energy In this Section we represent the results produced in [1], where the electron entan- glement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ1, is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method: ρCISD1 = −Tr ρCISD1 log2ρ |c2i+11 |2 + |c2i+1,2i+21,2 |2 |c2i+11 |2 + |c2i+1,2i+21,2 |2 |c0|2 + |c2i+22 |2 |c0|2 + |c2i+22 |2 , (12) where c1 is the coefficient for a single excitation, and c1,2 is the double excitation (in Appendix A of [10] more details are shown). In this framework, entanglement (S) and correlation energy (Ecorr), as func- tions of nucleus – nucleus separation are those in Figure 9 0 1 2 3 4 5 R ( Å ) S ( ρ1 Figure 9: Comparison between the entanglement, calculated by the von Neu- mann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule. By the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve. In order to compare the entropy S with the electron correlation energy Ecorr, we rescale S with the parameter αmin calculated with some procedure illustrated in Eq. (8) and Eq. (9) replacing the integration variable λ with R; in this way we extract EcorrSvNdR SvNdR ≈ 0.009. (13) The corresponding ∆Sαmin = Ecorr−αS allows us to answer to the question ii); in fact, as it is shown in Figure 10, the vanishing point of ∆Sαmin is, according to the two –spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule. 0 1 2 3 4 5 −0.01 Figure 10: ∆Sαmin for theH2 molecule as a function of nucleus–nucleus distance. 4 Differences between the Configuration Inter- action approach and the two–spin Ising model The model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. (7), we can express λ in terms of corre- lation energy and substituting it in Eq. (3) we can obtain the variation of SvN in terms of Ecorr. SvN = − EcorrLog Ecorr 2(Ecorr+2) + (Ecorr + 4)Log Ecorr+4 2(Ecorr+2) (Ecorr + 2)Log4 . (14) In order to calculated the coefficient of proportionality among SvN and Ecorr we make an expansion of SvN for Ecorr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by mSvN (Ecorr) = ( )(1 + 1 ). Since this behavior is uncorrect to represent the logatithmic singularity of SvN in the origin, we make an expansion of Eq. (14), preserving the logarithmic deviation, and we obtain an expression of the SvN = AEcorr +BEcorrLog(Ecorr), (15) where A = 1/2 and B = −1/(4Log2). 0.02 0.04 0.06 0.08 0.1 0.025 0.075 0.125 Ecorr Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. (14) and its linear approx- imation and the logarithmic one, for the Ising model. In order to compare the behavior of SvN in Eq. (14), we have organized the numerical data, calculated with the method proposed in [1], by making a correspondence between each value of Ecorr and its respective value of SvN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 Figure 12: A correspondence of Ecorr and SvN calculated by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotoni- cally increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that Ecorr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,Trρ 6= Trρ2; as depicted in Figure 13. 0 1 2 3 4 R(Å ) Figure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ2. Probably, for this reason, the procedure adopted in [1] seems to be not cor- rect: the density matrix, in fact, is calculated starting by the excitation coeffi- cient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state. However, even if we consider only the first branch of the plot in Figure 12, i.e. , the numerical values of SvN corresponding with increasing values of Ecorr, and we fit the values around Ecorr → 0 with a F = AEcorr + BEcorrLog(Ecorr) we draw out numerical values of the coefficient different from the ones used in Eq. (15). This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of SvN as a function of Ecorr, around the origin, with a function of the form F = AEcorr +BEcorrLog(Ecorr) whose coefficients A and B assume the numerical values in Figure. In particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve. This fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1]. 5 Concluding remarks We have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0. The concept of entanglement gives a physical meaning to the electron cor- relation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7]. To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g. An interesting result is that for large coupling constants the entropy ap- proach 1, meaning that all levels are equiprobably visited by the considered spin. For weak interactions, at the boundary point λb = the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5]. In Sec. 2 a comparison between the entanglement and the correlation energy is performed. To quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concur- rence. The entanglement measure is always bounded, while the energy correlation, Ecorr = |E0| − 2 = 4 + λ2 − 2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ−axes has no sense. Thus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. (8). This procedure leads to the value αmin ≈ −0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1, for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 . In Figure 4 the deviation ∆Sα and its derivatives with respect to λ are computed and αmin is evaluated for α ranging in the interval −1.29 ≤ α ≤ −0.091. In Figure 5 the minimized Concurrence deviation ∆C for the four eigen- states of the 2-spin model is shown. We point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues. Formula (11), due to Heitler–London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the nucleus–nucleus separation in the Hydrogen molecule. To conclude, the magnetic field B has been monitored such that the equi- librium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 Ry and r = 2.25 for B = 0.375 Ry for the function ∆Cα′ We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%. However, when we organized the pairs of points (Ecorr, SvN ) calculated by following the procedure described by [1], it is clear that the von Neumann en- tropy cannot be considered a function of correlation energy. The principle cause is that the function Ecorr presents a maximum in the region where SvN is mono- tonically increasing. The reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function Ecorr in terms of the nucleus – nucleus distance R, increases till the state is pure, on the contrary, when Tr(ρ2) becomes discordant from Tr(ρ), the function Ecorr decreases. This fact suggests us that the numerical model based on the calculation of SvN starting by the excitation coefficients ci, isn’t completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12, the function obtained by the two spin Ising model, i. e., Eq. (14), is unsuitable for fitting these numerical data. On the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule. Acknowledgments The authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41. References [1] Z. Huang, S. Kais, Chem. Phys. Lett. 413, 1 (2005). [2] M. A. Nielsen and I. L. Chuang Quantum Computation and Quantum In- formation, Cambridge Univ. Press, Cambridge, 2000. [3] D. M. Collin, Z. Naturforsch A 48, 68 (1993). [4] P. Rungta and C. M. Caves Phys. Rev. A 67, 012307 (2003). [5] S. Sachdev Quantum Phase Transition, Cambridge University Press, 2001. [6] O. Osenda, Z. Huang and S. Kais Phys. Rev A 67, 062321 (2003). [7] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [8] C. Herring and M. Flicker, Phys. Rev. A 134, 362 (1964). [9] W. Heitler, F. London, Z. Physik 44, 455 (1927) [10] T. Maiolo, F. Della Sala, L. Martina, G. Soliani arXiv: quant–ph/ 0610238 (2006). http://arxiv.org/abs/quant--ph/0610238 Introduction and the model A comparison between the entanglement and the correlation energy A quantum chemical framework to compare entanglement and correlation energy Differences between the Configuration Interaction approach and the two–spin Ising model Concluding remarks

0704.0521

Fractionally charged excitations on frustrated lattices

FRACTIONALLY CHARGED EXCITATIONS ON FRUSTRATED LATTICES E. Runge1 and F. Pollmann and P. Fulde2 1Technische Universität Ilmemau, Fakultät für Mathematik und Naturwissenschaften, FG Theoretische Physik I, 98693 Ilmenau, Germany 2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany Systems of strongly correlated fermions on certain geometrically frustrated lattices at particular filling factors support excitations with fractional charges ±e/2. We calculate quantum mechanical ground states, low–lying excitations and spectral functions of finite lattices by means of numerical diagonalization. The ground state of the most thoroughfully studied case, the criss-crossed checker- board lattice, is degenerate and shows long–range order. Static fractional charges are confined by a weak linear force, most probably leading to bound states of large spatial extent. Consequently, the quasi-particle weight is reduced, which reflects the internal dynamics of the fractionally charged excitations. By using an additional parameter, we fine–tune the system to a special point at which fractional charges are manifestly deconfined—the so–called Rokhsar–Kivelson point. For a deeper understanding of the low–energy physics of these models and for numerical advantages, several conserved quantum numbers are identified. I. INTRODUCTION Quantization of charge is a very basic feature in the description of the physical world. Therefore, fractionally charged excitations came as a surprise to physicists. Already back in the year 1979, Su, Schrieffer, and Heeger1 showed that a model describing the one-dimensional (1D) chain molecule trans–polyacetylene (CH)n supports excitations with spin–charge separation. This is not yet charge fractionalization, but when the model is considered at different electron densities (corresponding to extremely high doping), it turns out that at certain filling factors elementary excitations with fractional charge exist [in the simplest case q = ±e/3 and q = ±2e/3]. A few years later, Laughlin2 interpreted the much celebrated fractional quantum Hall effect (FQHE) in terms of fractionally charged (quasi-) particles (fcp) and fractionally charged (quasi-) holes (fch). Thereby, he firmly established the idea of fractional charges in our understanding of solid–state physics. Direct electron– electron interactions are crucial in the FQHE: Correlations become strong in an applied magnetic field because the kinetic energy of the electrons is quenched. Of course, no one will ever extract a fraction of an electron from an FQHE sample. But thinking of an extra electron or an extra hole near fill factor ν = 1/3 as decaying into three separate entities of charge q = ±e/3 each proved enormously helpful for a qualitative and quantitative understanding of the FQHE—and was rewarded by the Nobel Prize in Physics 1998. The question was left open whether or not fractionally charged excitations exist in 2D or 3D systems without a magnetic field. In 2002, it was suggested by one of us3 that excitations with charge ±e/2 do exist in certain 2D and 3D lattices, e.g., the pyrochlore lattice, which is a prototype 3D structure with geometrical frustration. The original work was motivated by the transition metal compound LiV2O4, which surprisingly shows heavy–fermion behavior with, e.g., a large γ coefficient in the low-temperature specific heat C = γT .4 However, we will not address the issue whether the models discussed here apply to any particular material or artifical systems such as optical lattices, see e.g. Ref.5 and citations therein. Instead, we try to contribute to the very general question whether or not fractional charges can exist at all in truly 2D or 3D systems in the absence of magnetic fields. We would like to argue that general prerequisites for fractional charges are—as in the FQHE case— strong short–range correlations and certain band fillings. Furthermore, the short–range correlations should be somehow incompatible with the lattice structure in order to prevent the development of long–range order. Following the general usage, we will simply call the lattice “frustrated” even though calling the interaction “frustrated by the lattice” would be the more accurate terminology. In this contribution, we study the charge degrees of freedom on such lattices systematically and consider a class of models of strongly correlated spinless fermions. Most of our numerical calculations were done for the 2D checkerboard lattice, which is easier to deal with than the even more interesting 3D cases. For future work, one can hope to learn from comparison with spin systems where numerous studies exist for the pyrochlore lattice and its 2D relatives, the crisscrossed checkerboard lattice and http://arxiv.org/abs/0704.0521v1 (a) (b) (c) Figure 1: (a) Checkerboard lattice as 2D projection of the 3D pyrochlore lattice.11 (b,c) Two examples of allowed configurations on a checkerboard lattice at half filling. Occupied sites are connected by thick lines as guides to the eye. the kagome lattice.6,7,8,9,10 II. FRACTIONAL CHARGES ON FRUSTRATED LATTICES In order to illustrate the concept of fractional charges on frustrated lattices, we focus here on the (crisscrossed) 2D checkerboard lattice. Figure 1(a) illustrates that the checkerboard lattice can be thought of as a projection of the pyrochlore lattice onto a plane. In the following, we adopt the ideas of Ref.3 and consider a model Hamiltonian of spinless fermions H = −t 〈i,j〉 +H.c. 〈i,j〉 ninj (1) on a checkerboard lattice. The operators ci(c i ) annihilate (create) fermions on sites i. The density operators are ni = c ici. We assume half filling, for a system with N sites. Our main interest is the regime where the nearest–neighbor hopping t is much smaller than the nearest– neighbor repulsion V , i.e., |t|/V ≪ 1. Henceforth, we assume t > 0. In analogy to the tetrahedra in a pyrochlore lattice, all bonds in a crossed square are equivalent. a. Classical correlations and ground–state degeneracy. For a moment, let us set the hopping– matrix element t to zero. The ground–state manifold is then macroscopically degenerate: Every configuration that satisfies the so–called tetrahedron rule of having exactly two particles on each tetrahedron (crisscrossed square) is a (classical) ground state12 and will henceforth be referred to as an “allowed configuration.” Examples are shown in Fig. 1(b) and (c). Note that our coordinate system is rotated by 45◦ relative to that of, e.g., Ref.13. The resulting difference in boundary conditions can lead to noticeable numerical differences in particular for small cluster sizes. One can visualize the origin of the macroscopic degeneracy as follows: Take the set of the six allowed crisscrossed squares and construct row by row a larger allowed configuration. Whenever a crisscrossed square is added, we can choose between one, two, or three different possibilities, depend- ing on the neighboring crisscrossed squares. Since there often is a choice to make, an exponential number of different allowed configurations can be constructed. Thus, the system has a finite entropy at T = 0. The exact value of the ground-state degeneracy can be obtained from a mapping to the so–called six–vertex model.14 Its solution is highly non–trivial due to the existence of long–range correlations, which seem to be a generic feature seen in many frustrated lattice models.15 All classical (t = 0) ground states have the important property of being incompressible in the sense that no fermion can hop to another empty site without creating defects and thereby increasing the repulsion energy. In other words, we have to violate the tetrahedron rule in intermediate states if we want to connect via hopping processes one allowed configuration with another. b. Fractional charges. Placing one additional particle with charge e onto an empty site of an al- lowed configuration leads to a violation of the tetrahedron rule on two adjacent crisscrossed squares, see Fig. 2(a). The energy is increased by 4V since the added particle has four nearest neighbors (charge gap). There is no way to remove the violations of the tetrahedron rule by moving electrons. However, fermions on a crisscrossed square with three particles can hop to another neighboring criss- crossed square without creating additional violations of the tetrahedron rule, i.e., without increase of the repulsion energy [see Fig. 2(b,c)]. By these hopping processes, two local defects (violations of the tetrahedron rule) can separate and the added fermion with charge e breaks into two pieces. (a) (b) (c) Figure 2: (a) Adding one fermion to the half–filled checkerboard lattice leads to two defects (marked by black triangles) on adjacent crisscrossed squares. (b), (c) Two defects with charge e/2 can separate without creating additional defects. They are connected by a string consisting of an odd number of fermions. (a) (b) (c) (d) Figure 3: (a)–(c) Hopping of a fermion to a neighboring site in an allowed configuration generates a fluctu- ation: (a) A fractional charged particle (fcp) and fractional charged hole (fch) are generated. (b)–(c) The two defect (marked by triangles) with charge ±e/2 can separate without creating additional defects and are connected by a string consisting of an even number of fermions. (d) Example of an allowed configuration on a checkerboard lattice at half filling with possible low–order hopping processes. They carry a fractional charge of e/2 each. In the quantum mechanical case (t 6= 0), the separation leads to a lowering of the kinetic energy of order |t|. Energy and momentum must be conserved by the decay processes 1e → 2fcp′s. If we associate momentum k and energy E(k) with the inserted fermion, they must now be shared between the resulting two fcp’s into which it has decayed E (k) = 4V + ǫ (k1) + ǫ (k2) . (2) Here k = k1+k2 and ǫ (k) is the energy dispersion of a fcp. Even though ǫ (k) is at present completely unknown, Eq. (2) allows to predict that for deconfined fcp’s the electronic spectral function should not contain a Fermi–liquid peak, but should show a broad continuum instead. Figure 2 demonstrates that the two defects can be thought of as always being connected by a string of occupied sites consisting of an odd number of sites. The fractional charges can thus be alternatively interpreted as the ends of a string–like excitation. In this picture, the connection is not static, as two pairs of fcp’s can exchanges partners when the connecting strings come close. c. Quantum Fluctuations. If we relax the constraint of having two fermions on each crisscrossed square and consider a small but finite ratio t/V, quantum fluctuations come into play. The quan- tum fluctuations lead also to fractional charges, but do not change the net charge of the systems. Starting from an allowed configuration, the hopping of a fermion to a neighboring site increases the energy by V . One crisscrossed square contains three fermions while the other has only one fermion, see Fig. 3(a)–(c). These so-called vacuum fluctuations (virtual fcp–fch pairs) lead to two mobile fractional charges with opposite charges +e/2 and −e/2, which are connected by a string of an even number of fermions. The energy associated with a free fcp and a free fch is ∆Evac = V+ǫ (k)+ǭ (−k), where ǭ (−k) denotes the kinetic energy of a fch. Such virtual processes connect different allowed configurations, lower the total energy, and reduce the macroscopic degeneracy, as we will see more explicitely in the next section. d. Fractional charges in 3D. All arguments mentioned above for the existence of fcp’s on a 2D checkerboard lattice can be directly transferred to a 3D pyrochlore lattice at half filling, see Fig. 4. Fcp’s correspond to tetrahedra with three fermions and fch’s to tetrahedra with only one fermion. (a) (b) Figure 4: Adding one fermion to the half–filled three-dimensional pyrochlore lattice leads to two defects on adjacent tetrahedra. The two defects with charge e/2 can separate without creating additional defects and are connected by a string consisting of an odd number of fermions. III. EFFECTIVE HAMILTONIANS e. Ring exchange processes in the undoped case. The numerical and analytical work is greatly simplified, if a simpler Hamiltonian H can be derived that shows in the limit |t| ≪ V the same low– energy physics as the model Hamiltonian (1). A down–folding procedure can be used to defineH: Per definition, it acts only on the subspace of allowed configurations and includes all virtual processes up to the lowest non–trivial order, which is t3/V 2. The relevant processes are shown in see Fig. 3(d) and can be classified as self–energy contributions HΣ and ring–exchange processes Heff , H = HΣ +Heff . The former comprises the terms which are diagonal in the real space basis. The latter includes those which connect different allowed configurations. HΣ contains fermion hops to an empty neighboring site and back again as well as hops around an adjacent triangle. These contribute only a constant, configuration–independent energy shift. It does not lift the macroscopic degeneracy and will hence be ignored furtheron. The total amplitude of ring–exchange processes around empty squares is proportional to t2/V . It vanishes for spinless fermions because the amplitudes for clockwise and counter–clockwise ring–exchange cancel each other due to fermionic anti–commutation relations. Thus, the macroscopic ground–state degeneracy is first lifted by ring exchanges ∼ t3/V 2 around hexagons, and the looked–for effective Hamiltonian reads Heff = −g { , } ∣+H.c. , (3) with the effective hopping–matrix element g = 12 t3/V 2 > 0 and the sum taken over all vertical and horizontal oriented hexagons. The pictographic operators represent the hopping around hexagons which have either an empty or an occupied central site. The signs of the matrix elements depend on the representation and the sequence in which the fermions are ordered. When the sites are enumer- ated along diagonal rows, an exchange process commutes an odd number of fermionic operators if the site in the center of the hexagon is empty and an even number if the center is occupied. In Ref.13, it has been shown that the effective Hamiltonian (3) gives a good approximation of the low–energy excitations of the full Hamiltonian (1) in the limit considered. f. Propagation of defects and extra charges. We have derived Heff as effective Hamiltonian for local rearrangement processes as resulting from the (virtual/intermediate) generation of a defect pair which subsequently recombines, leaving behind a different allowed generation. For many questions of physical interest, it is necessary to consider the propagation of defects over large distances. Anal- ogously to conventional semiconductors, this refers to long–lived thermally generated defects (e–h pairs in the semiconductor analogy) as well as to the consequences of slight doping, i.e., addition of two fcp’s by addition of one extra fermion. The natural generalization of the effective Hamiltonian (3) to these cases includes besides the lowest order ring–exchange processes Heff a projected hopping term that moves the defects Htg = Heff − t 〈i,j〉 i cj +H.c. P. (4) The projector P ensures that Htg acts only on the subspace of configurations with the smallest pos- sible number of violations of the tetrahedron rule which is compatible with the number of particles, (a) (b) (c) Figure 5: (a) Height representation for examples of allowed configurations of a 32 checkerboard lattice with periodic boundary conditions at half filling. The height field h (numbers in the non–crossed squares) is uniquely defined for a given configuration up to an additive constant M . The field f = ∇h is indicated by small arrows on the lattice sites. Details of the mapping can be found in the text. The effect on the height fields of a ring–exchange process around a hexagon is shown explicitely. (b) The effective Hamiltonian conserves the number of fermions on each of the four sublattices of the checkerboard lattice which are labeled by “blue squares”, “red circles”, “yellow stars” and “green triangles.” (c) Ring–exchange processes change the number of fermions on sites marked by dark “purple squares” by two.16 e.g., two in the case of one added fermion. We refer to Htg as the t–g model and consider the param- eters t and g as independent (i.e., not restricted to the regime g ≪ t as enforced by t ≪ V ). This allows to study the effect of ring exchange onto the dynamics of fractionally charged excitations. In particular, the question of confinement can be studied even on rather small clusters by increasing the ring exchange strength g relative to t. IV. HEIGHT REPRESENTATION, CONSERVED QUANTITIES, AND GAUGE SYMMETRIES Conserved quantities of a Hamiltonian allow for a reduction of the numerical effort by exploiting the resulting block–diagonal form of the matrix representation. The real space configurations span- ning the subspaces corresponding to these blocks will be referred to as “subensembles.” Eigenstates can conveniently be classified by the eigenvalues of the conserved quantities as quantum numbers for our model. We will now identify several such quantum numbers in order to characterize different subensembles. A topological quantity, which is conserved by all local processes, i.e., ring–exchange processes, is the average tilt of a scalar height field which will be introduced in the next paragraph. Another useful set of quantum numbers are the number (NB, NY , NG , NR) of particles on the four sublattices referred to as blue, yellow, green and red as shown in Fig. 5(b). They are conserved by Heff . Any allowed configurations of the half–filled checkerboard can uniquely be represented by a vector field f for which the discretized lattice version of the (discrete) curl vanishes, i.e., a pure (discrete) gradient of a scalar field (height field) f = ∇h, see Fig. 5. The height field h, is derived from the local constraint expressed by the tetrahedron rule as follows: A clockwise or counter–clockwise orientation is assigned alternatingly to the crisscrossed squares. Arrows of unit length are placed on the lattice sites. The arrows point along (against) the orientation of the adjacent crisscrossed squares if the site is occupied (empty). It is easily checked that allowed configurations are those for which the discretized line integral of f around every closed loop vanishes. Thus, f = ∇h defines a height field h up to an arbitrary constant M . The height at the upper (right) and at the lower (left) boundary of a finite lattice of Nx × Ny squares with periodic boundary conditions can differ only by an integer −Ny(x) ≤ κy(x) ≤ Ny(x), which is the same for all columns (rows). This defines topological quantum numbers (κx, κy). They remain unchanged by all local processes that transform one allowed configuration into another, i.e., by ring–exchange processes along contractible loops. In particular, Heff merely lowers or raises the local height of two adjacent plain squares by ±2, as illustrated in Fig. 5(a). We refer to (κx/Nx, κy/Ny) as global slope. The lattice symmetry is broken for states with (κx, κy) 6= (0, 0): 13 E.g., a finite positive value of κx/Nx implies a charge modulation along diagonal stripes. Similarly, a charge density modulation is present if the condition NB = NY = NG = NR is violated. For the identification of exactly solvable points in parameter space and for future quantum Monte Figure 6: The three different actions of the effective Hamiltonian on the topology of loop configurations are shown in panel (a). Panels (b) and (c) show representations of two configurations by fully–packed directed loops.17 Carlo simulations, it would be very advantageous if all matrix elements had the same sign. As first steps in this direction, we describe a gauge transformation that changes the global sign of g in the effective Hamiltonian (3) and then show conditions under which it is possible to remove minus signs for the half-filled checkerboard lattice.17 Consider a “purple” sublattice P that contains the sites around every second plaquette as shown in Fig. 5(c). Define σP as the number of fermions on purple sites for a given configuration: σP = i∈P ni. One observes that ring–exchange processes change σP by two. Thus, if all configurations are multiplied by a factor of iσP , the sign of all ring–exchange matrix–elements are changed. The invariance with respect to this gauge transformation proves a global g ↔ (−g) symmetry. However, a fermionic sign problem remains: Ring–exchange processes around empty and occupied hexagons carry opposite signs in Eq. (3). We argue that it can be avoided in certain (but not all) cases.17 In order to do so, we represent the ground–state manifold by ensembles of fully–packed loops as exemplified in Fig. 1. We notice that a ring exchange around a hexagon with an occupied center site does not change the loop topology, whereas a ring exchange around an empty hexagon always does cause changes in one of the three topological different ways shown in Fig. 6(a). Let us consider allowed configurations with “fixed” boundary conditions with an even number of fermions on the four boundaries. They are represented by closed loops in the interior and loops terminating at a boundary. We orient the closed loops as follows: (i) Color the areas separated by the loops alternatively white and grey, with white being the outmost color; (ii) orient all loops so that the white regions are always to the right, see Fig. 6(b). If open loops are present [Fig. 6(c)], these are closed arbitrarily but intersection-free outside the sample and colored as described above. Let us assume without loss of generality that the color at infinity is white. We now notice by inspection of Fig. (6) that the relative signs resulting from the exchange processes around empty hexagons are consistent with multiplying each loop configuration by ir(−i)l, where r and l are the total number of the clockwise and counter–clockwise winding loops, respectively. Hence, by simultaneously changing the sign of the exchange–processes around empty hexagons and transforming the loop states |L〉 → il(L)(−i)r(L)|L〉, (5) we cure the sign problem, thus making the system effectively bosonic. This construction need not work for periodic boundary conditions: Firstly, only even–winding subensembles (sectors) on a torus allow for such a two–color coloring. Secondly, even then it might be possible to dynamically reverse the coloring while returning to the same loop configuration. However, the exact diagonalization results presented below (see Fig. 8) suggest that for periodic boundary conditions on even tori (preserving the bipartiteness of the lattice), the lowest–energy states belong to a sector where such a transformation works. We remark that the presented non–local loop–orienting construction is restricted to the effective Hamiltonian (3), i.e., to the ring–exchange processes of length six. (a) (b) (c) Figure 7: (a)–(b) Fragments of possible “frozen” ground states (no flippable hexagons). (c) One of several configurations of the half–filled checkerboard lattice that maximize the number of flippable hexagons. The unit cell contains 20 sites. V. GROUND STATES AND LOWEST EXCITATIONS IN THE UNDOPED CASE In order to discuss the possible confinement of fcp’s, we have to investigate the nature of the quantum–mechanical ground state of the undoped system. We do so in the approximation of an effective ring-exchange Hamiltonian (3) acting only on the subset of allowed configurations. Following Rokhsar and Kivelson,18 we add to Heff an extra term that counts the number of flippable hexagons. The extended Hamiltonian reads Hgµ = Heff + µ { , } bb 〉〈 bb |+ | b b 〉〈 b , (6) where the pictographic operators with grey–colored dots in the center are summed over all flippable hexagons, independent of the occupancy of the site in the center. Next, we discuss some limiting cases of the Hamiltonian (6): (i) µ→ +∞: All configurations which contain no flippable hexagons (frozen configurations) become ground states with E = 0. Some are shown as Fig. 7 (a,b). (ii) µ → −∞: Ground states are configurations with maximal number of flippable hexagons Nfl. Using a simple Metropolis–like Monte Carlo algorithm for lattices with up to 1000 sites, we always find that configurations of the type shown in Fig. 7(c) or slight variations thereof maximize Nfl. Such configurations will be referred to as “squiggle” configurations.19 Our numerical results suggest that in the thermodynamic limit the degenerate ground–state configurations all lie in the (κx, κy) = (0, 0) subspace. (iii) µ = g > 0: This is the exactly solvable Rokhsar–Kivelson (RK) point.18 Following the original RK construction, we rewrite the Hamiltonian (6) for µ = g in a way which explictly shows that some liquid–like ground states have energy E = 0 and, thus, become degenerate with the frozen states, i.e., the µ→+∞ solutions. We assume that we can use the gauge transformation (5) to change the sign in the second term and to rewrite the Hamiltonian as Hg=µ = g { , } bb 〉 − | b bb | − 〈 b . (7) Since this is a sum over projectors, all eigenvalues are non–negative. Furthermore, after re– gauging, all off–diagonal elements are non–positive and for each subensemble ℓ an exact ground– state wavefunction is given by the equally weighted superposition of its configurations |C i 〉, i.e., 0,RK〉 ∼ i 〉. These coherent superpositions are the analogs of the Resonating Valence Bond (RVB) state, originally discussed by L. Pauling20 and P. W. Anderson.21 Their energy is easily computed: Each flippable hexagon contributes −g and +µ, thus for g = µ: 〈ψ 0,RK |Hgµ|ψ 0,RK〉 = 0. Note that for fermionic systems, we find a well defined RK point only if a gauge transformation exists such that all off–diagonal matrix elements are non–positive. Otherwise the energy is most likely larger than zero and the subensembles consequently do not form a ground state. Next, we explore by means of numerically exact diagonalization the eigenstates of small clusters general µ values. For the actual calculations, we first generate all configurations that fulfill the tetra- hedron rule, then group them according to quantum numbers and generate a sparse block–diagonal matrix representation of the Hamiltonian (6). For a 72–site checkerboard cluster with periodic boundary conditions, the 16 448 400 dimensional low–energy Hilbert space of allowed configurations (a) (b) −2 0 2 −14.5 −13.5 −12.5 Fermi sign −2 0 2 no Fermi sign Figure 8: (a) Energies of the ground state and lowest excited states in each subensemble of a 72–site half– filled checkerboard cluster for different values of µ of the g–µ Hamiltonian. Level crossing of ground states occurs only at µ = g. The insets indicate different phases: Maximal flippable plus fluctuations for µ < g, a critical point µ = g where the ground state is an equally weighted superposition of all configurations, and frozen configurations as ground states for µ > g. (b) Left side: Ground–state energy and energies of the lowest excited states of the effective Hamiltonian Heff in subspaces with different global slopes (κx, κy) = (κx, 0) for 72–site cluster. Right side: Same system, but assuming same signs for all matrix elements (“bosonic calculation”).17 can be decomposed into a few hundred subspaces, where the largest one has 1 211 016 dimensions. The low–energy states of the sparse block–diagonal Hamiltonian matrix and physical properties such as charge density distribution and density–density correlations can easily be obtained on a 64–bit workstation. Figure 8(a) shows energies of the ground-state and the lowest excited states of all subensembles. (iv) For µ < g, two ground states are found in subensembles with (κx, κy) = (0, 0) and (NB, NY , NG , NR) = (6, 6, 12, 12) and (12, 12, 6, 6), respectively. These are superpositions of con- figurations with the maximal number of flippable hexagons. At the physical point µ = 0, the 72–site system is in a crystalline and confining phase. In the thermodynamic limit, we expect to recover the 10–fold degeneracy of the squiggle phase instead of the two–fold degeneracy. For small values µ > 0, the average weight of configurations in the ground state with the maximum number Nfl is large (not shown), but decreases with increasing µ until at the RK point, i.e., µ=g, the ground state is formed by an equally weighted superposition of all configurations of a certain subensemble. (v) For µ > g, we find essentially the same properties as described above in the limit µ→ ∞. In summary, we find a confining phase and a phase in which the ground states are given by static isolated configurations. The two phases are separated by a point with deconfined excitations, i.e., the Rokhsar–Kivelson (RK) point.18 The main finding is that the original effective Hamiltonian is in a confining phase with a long–range ordered ground state (squiggle phase). This phase maximizes the gain in kinetic energy and is stabilized by quantum fluctuations (order from disorder). The results of the exact diagonalization on small samples indicate that the symmetry remains broken all the way along the µ–axis up to the RK point. This observation is also strongly disfavoring a deconfining phase to the left of the RK point. Hence, the fermionic RK point is likely to be an isolated quantum critical point just as it is for the bosonic model.10 We come back to the fermionic sign problem. Figure 8(b) compares energies of a system in which the Fermi sign is taken into account to those from calculations which exclude the Fermi sign. The ground–state energy, the first excited states in the (κx, κy) = (0, 0) sector, and the weights of the different configurations in the corresponding eigenstates are the same in both case. In some subensembles with (κx, κy) 6= (0, 0) the energies of the ground states including the Fermi sign are higher than the ones for bosons (no Fermi sign). At the physical point µ = 0, the 72–site ground state is two-fold degenerate with quantum numbers (NB, NY , NG , NR) = (6, 6, 12, 12) or (12, 12, 6, 6). The resulting charge order with alternating stripes of average occupation 1/3 and 2/3 is shown in Fig. 9(a). The density–density correlation function in the quantum–mechanical ground states |ψ 0 〉 [Fig. 9(b)] 0 |nini0 |ψ 0 〉 − 〈ψ 0 |ni|ψ 0 〉〈ψ 0 |ni0|ψ 0 〉. (8) is best understood as reflecting the algebraic correlations present in the average over all classically (a) (b) (c) Figure 9: Half filling: (a) Charge density distribution for one of the two ground sates. (b) Corresponding density–density correlation function C . The site i0 with average density 2/3 shows up as the largest dot in the panels (b)–(c). The radius of the dots is proportional to the absolute value. A red or blue color represents a positive or negative value, respectively. (c) Classical density–density correlation function. (a) (b) (e) (c) (d) (f) Figure 10: (a), (b), and (e): Local loss of kinetic energy due to the separation of two fractionally (static) charged defects (fcp’s marked by light red squares or fch’s marked by dark blue squares). The radii of the circles are proportional to the local energy loss. (c), (d), and (f): Red (blue) circles show an increase (decrease) of the local density (vacuum polarization due to the two fcp’s or the fcp–fch pair). degenerate configurations, see Fig. 9(c) and Ref.15. Specific quantum–mechanical features resulting from ring hopping become visible in the difference of the actual correlations and have been discussed in Ref.13. VI. STATIC AND DYNAMICAL PROPERTIES OF THE DOPED SYSTEM Next, we turn to the important issue of confinement/deconfinement of defects generated either particle–hole excitations or by weak doping. Two different kinds of calculations can be done: In the first case—referred to as static charges—the defects are fixed to some lattice positions and ground–state calculations involving Heff are performed in the restricted subspace spanned by the configurations with exactly those defects at given positions. The total (free) energy is recorded as function of the defect positions. Its spatial derivative is interpreted as attractive or repulsive force. In the second case—referred to as dynamic charges—the defects propagate according to the Hamiltonian (4). The existence or non–existence of bound states is interpreted as confinement or deconfinement. g. Static charges. If an additional particle is added/doped to a half-filled checkerboard lattice, at least two crisscrossed squares are occupied by three particles. It is easy to see that, e.g., for the t–µ Hamiltonian in the limit of large positive µ (frozen configurations) the ground–state energy of two static fractional charges is independent of the distance between. Thus static charges are not attracted by a force, which suggests that dynamic fcp’s are deconfined at zero temperature in that limit. Also, it is obvious that at the RK point the total energy of a system with two static charges is independent of the distance between them. Thus dynamic fcp’s are expected to be deconfined. We will not discuss systems with a general values of µ further, but focus on the physical case µ = 0 and discuss the changes of the kinetic energy density in the presence of two static charges e/2.22 The energy change can be decomposed into local contributions ǫi from all ring–exchange processes involving a given site i. An increase in kinetic energy in the region between the two fractional charges, i.e., along the connecting string, is found and illustrated in Fig. 10(a,b). The total energy increase is approximately proportional to the length of the generated string and implies at large distances a constant confining force. The changes of the local energy density goes along with density changes, as illustrated in Fig. 10(c,d). Similar results are found when a particle is removed from the ground state or when a particle–hole excitation is generated out of the ground state, see Fig. 10(c,d,f). The calculations have been performed in a reduced Hilbert space in which only two static defects are present. In a calculation within the full Hilbert space, or a subspace allowing at least an additional fcp–fch pair, the energy would not increase linearly to infinity, but the connecting string is expected to break by creating additional pairs of defects when this energetically favorable. For the relevant parameters, e.g., g = 0.01t and V = 10t, this occurs when two fcp’s have separated over 1000 lattice sites. This effect is well known for the case of confined quarks where pair production (quark anti–quark pairs) occurs before the quarks have been separated to an observable distance. The attractive constant force acting between two fractional charges in the confining phase results from a reduction of vacuum fluctuations and a polarization of the vacuum in the vicinity of the connecting strings. These findings suggest that a number of features known from QCD are also expected to occur in a modified form in solid–state physics. Conversely, one would hope that by studying frustrated lattices or dimer models one might be able to obtain better insight into certain aspects of QCD. h. Dynamic charges. We turn now to the dynamical properties of fcp’s, in particular spectral functions and optical conductivity for the half–filled checkerboard lattice. Numerical studies of finite clusters are again the method of choice,17 because conventional approximation schemes such as mean–field theories or Green’s function decoupling schemes are unable to describe the strong local correlations expressed by the tetrahedron rule. Diagonalizations with up to 50 sites were performed within the minimal Hilbert space spanned by the configurations with the smallest possible number of violations of the tetrahedron rule [half–filled system: all allowed configurations plus those with one fcp–fch pair; doped case: 2 fcp’s or 2 fch’s]. The spectral function A(k, ω) = A−(k, ω) + A+(k, ω) of an interacting many–particle system is the sum of the probability amplitudes for adding (+) to the N–particle ground–state system |ψN0 〉 or removing (–) from it a particle with momentum k and energy ω (~ = 1) A+(k, ω) = lim Im〈ψN0 |ck ω + iη + E0 −H |ψN0 〉 (9) A−(k, ω) = lim Im〈ψN0 | c ω + iη − E0 +H 0 〉. (10) We use operators c eirjkc j in the extended Brillouin zone. The spectral functions yield direct insight into the dynamics of a many–body system, as seen, e.g., in angular–resolved photoemission spectroscopy (ARPES). Expectation values of the form G(z) = 〈ψ0|A (z − H) −1 A†|ψ0〉 can conveniently be calculated numerically by the Lanczos continued fraction method23 or kernel polynomial expansion24. We found essentially identical results for both algorithms.16 However, the implementation of the Lanczos method turned out to be slightly faster. Well converged results were obtained already after several hundred iterations. We checked numerically that in the limit of large V , it is sufficient to calculate the spectral functions within the minimal Hilbert space. This enables us to study rather large systems and to address the question whether or not the two defects created by injecting one particle are closely bound to one another or not. The concept of a spatial separation of fcp’s can be confirmed by comparing results within the minimal Hilbert space with those from an artificially restricted calculation in which a particle is prevented from decaying into two defects with charge e/2 each. In our finite–cluster calculations, a broad low–energy continuum is seen in the spectral functions for the unrestricted case, which is missing when the restriction is imposed.22 The respective bandwidths are about 13t and 8t. This suggests a simple interpretation: The dynamics of two separated fcp’s having a bandwidth of ≈ 6t −10 −5 0 5 10 −10 −5 0 5 10 (a) Spectral function A(k = π/2, ω) for V = 25t calculated for a 32 cluster for the effective t–g Hamiltonian with (a) g = 0.01 and (b) g = 1. A Lorentzian broadening η = 0.1t is used. each due to six nearest neighbors would explain the calculated 13t, while a confined added particle has a much smaller bandwidth 8t, which is close to 6t. Further insight can be obtained if the amplitude of the ring exchange g=12t3/V 2 in the effective Hamiltonian is considered an independent parameter which is no longer restricted to the regime g≪t as enforced by t≪V . Figure VI 0 h compares results for the “physical” regime corresponding to the previously considered parameters g ∼ t3/V 2 = 0.01t with those for g = t. In the latter case, the broad continuum at the bottom of the spectral density vanishes and a sharp δ–peak evolves instead. The latter should be viewed as a Landau quasi-particle peak. This suggests the following interpretation: The ring exchange term leads for the undoped case to charge order, which in finite– cluster calculations is destroyed when a particle is added by the separation of the two fcp’s. These are weakly bound to each other. If g≪t, the distance of the two fcp’s is larger than the system size considered and thus the excitations seem to be deconfined. An artificially increased g leads to much stronger confinement and the diameter of the bounded pair becomes smaller or comparable to the system size. The pair acts at low energies as one entity. The electron as a (however strongly modified) “particle” is recovered and a finite weight of the quasi-particle peak results. These findings suggest that for the physical regime V/t ≈ 10 quasi-particles with a spatial extent over more than hundred lattice sites are formed. The huge spatial extent is expected to lead to interesting effect. E.g., one may find with increasing doping concentration a transition from a “confined” phase to an “fcp plasma” phase with yet unknown properties when the average distance of bound fcp–fcp pairs falls below the diameter of a single bound fcp pair. Other open questions for future research include a thorough investigation of the 3D pyrochlore lattice. Even though the checkerboard lattice and pyrochlore lattice show many similarities, there exist certain differences between them, e.g., due to the higher spatial dimensionality.8,25 Here, differences of the two lattices arise in the U(1) gauge theory which describes the low energy excitations of the considered systems. The related compact electrodynamic is always confining in 2+1 dimensions while it allows for the existence of a deconfined phase in 3+1 dimension.26 Systematic exact diagonalization studies as well as the application of Monte Carlo techniques will provide a deeper insight and we hope to confirm the expected existence of a deconfined phase in a 3D pyrochlore lattice. In addition, a natural extension of the model considered in this paper is the inclusion of spin. This leads to a more realistic model and could provide a better link to experiments. 1 W. P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. Lett. 42, p. 1698 (1979). 2 R. B. Laughlin, Phys. Rev. Lett. 50, p. 1395 (1983). 3 P. Fulde, K. Penc and N. Shannon, Ann. Phys. (Leipzig) 11, p. 893 (2002). 4 S. Kondo, D. C. Johnston, C. A. Swenson, F. Borsa, A. V. Mahajan, L. L. Miller, T. Gu, A. I. Goldman, M. B. Maple, D. A. Gajewski, E. J. Freeman, N. R. Dilley, R. P. Dickey, J. Merrin, K. Kojima, G. M. Luke, Y. J. Uemura, O. Chmaissem and J. D. Jorgensen, Phys. Rev. Lett. 78, p. 3729 (1997). 5 W. Hofstetter, J. I. Chirac, P. Zoller, E. Demler and M. D. Lukin, Phys. Rev. Lett. 89, p. 220407 (2002). 6 G. Misguich and C. Lhuillier, cond-mat/0310405 (October 2003), in Ref.7, p. 229ff. 7 H. T. Diep (ed.), Frustrated Spin Systems (World Scientific, Singapore, 2005). 8 M. Hermele, M. P. A. Fisher and L. Balents, Phys. Rev. B 69, p. 064404 (2004). 9 A. Läuchli and D. Poilblanc, Phys. Rev. Lett. 92, p. 236404 (2004). 10 N. Shannon, G. Misguich and K. Penc, Phys. Rev. B 69, p. 220403(R) (2004). http://arxiv.org/abs/cond-mat/0310405 11 R. Moessner, O. Tchernyshyov and S. L. Sondhi, J. Stat. Phys. 116, p. 755 (2004). 12 P. W. Anderson, Phys. Rev. 102, p. 1008 (1956). 13 E. Runge and P. Fulde, Phys. Rev. B 70, p. 245113 (2004). 14 R. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, San Diego, 1982). 15 F. Pollmann, J. J. Betouras and E. Runge, Phys. Rev. B 73, p. 174417 (2006). 16 F. Pollmann, Charge Degrees of Freedom on Frustrated Lattices, Doctoral Dis- sertation, Technische Universität Ilmenau, (Ilmenau, Germany, 2006), http: //nbn-resolving.de/urn/resolver.pl?urn=nbn:de:gbv:ilm1-2006000140. 17 F. Pollmann, J. Betouras, K. Shtengel and P. Fulde, Phys. Rev. Lett. 97, p. 170407 (2006). 18 D. S. Rokhsar and A. A. Kivelson, Phys. Rev. Lett. 61, p. 2376 (1988). 19 K. Penc and N. Shannon, private communication and to be published (2006). 20 L. Pauling, Proc. Natl. Acad. Sci. 39, p. 551 (1953). 21 P. W. Anderson, Mater. Res. Bull. 8, p. 173 (1973). 22 F. Pollmann, E. Runge and P. Fulde, Phys. Rev. B 73, p. 125121 (2006). 23 E. R. Gagliano and C. A. Balseiro, Phys. Rev. Lett. 59, p. 2999 (1987). 24 R. N. Silver, H. Roeder, A. F. Voter and J. D. Kress, J. Comp. Phys. 124, p. 115 (1996). 25 F. Pollmann, J. J. Betouras, E. Runge and P. Fulde, Charge degrees in the quarter-filled checkerboard lattice, in Proceedings of ICM2006 (Kyoto, Aug. 20-25, 2006), (in print). 26 A. M. Polyakov, Nucl. Phys. B 120, p. 429 (1977). Introduction Fractional charges on frustrated lattices Effective Hamiltonians Height representation, conserved quantities, and gauge symmetries Ground states and lowest excitations in the undoped case Static and dynamical properties of the doped system References

0704.0522

Measurement of Decay Amplitudes of B -->(c cbar) Kstar with an Angular Analysis, for (c cbar)=J/psi, psi(2S) and chi_c1

BABAR-PUB-07/009 SLAC-PUB-12430 hep-ex/0704.0522 Measurement of Decay Amplitudes of B → (cc)K∗ with an Angular Analysis, for (cc) = J/ψ , ψ(2S) and χ B. Aubert,1 M. Bona,1 D. Boutigny,1 Y. Karyotakis,1 J. P. Lees,1 V. Poireau,1 X. Prudent,1 V. Tisserand,1 A. Zghiche,1 J. Garra Tico,2 E. Grauges,2 L. Lopez,3 A. Palano,3 G. Eigen,4 I. Ofte,4 B. Stugu,4 L. Sun,4 G. S. Abrams,5 M. Battaglia,5 D. N. Brown,5 J. Button-Shafer,5 R. N. Cahn,5 Y. Groysman,5 R. G. Jacobsen,5 J. A. Kadyk,5 L. T. Kerth,5 Yu. G. Kolomensky,5 G. Kukartsev,5 D. Lopes Pegna,5 G. Lynch,5 L. M. Mir,5 T. J. Orimoto,5 M. Pripstein,5 N. A. Roe,5 M. T. Ronan,5, ∗ K. Tackmann,5 W. A. Wenzel,5 P. del Amo Sanchez,6 C. M. Hawkes,6 A. T. Watson,6 T. Held,7 H. Koch,7 B. Lewandowski,7 M. Pelizaeus,7 T. Schroeder,7 M. Steinke,7 W. N. Cottingham,8 D. Walker,8 D. J. Asgeirsson,9 T. Cuhadar-Donszelmann,9 B. G. Fulsom,9 C. Hearty,9 N. S. Knecht,9 T. S. Mattison,9 J. A. McKenna,9 A. Khan,10 M. Saleem,10 L. Teodorescu,10 V. E. Blinov,11 A. D. Bukin,11 V. P. Druzhinin,11 V. B. Golubev,11 A. P. Onuchin,11 S. I. Serednyakov,11 Yu. I. Skovpen,11 E. P. Solodov,11 K. Yu Todyshev,11 M. Bondioli,12 S. Curry,12 I. Eschrich,12 D. Kirkby,12 A. J. Lankford,12 P. Lund,12 M. Mandelkern,12 E. C. Martin,12 D. P. Stoker,12 S. Abachi,13 C. Buchanan,13 S. D. Foulkes,14 J. W. Gary,14 F. Liu,14 O. Long,14 B. C. Shen,14 L. Zhang,14 H. P. Paar,15 S. Rahatlou,15 V. Sharma,15 J. W. Berryhill,16 C. Campagnari,16 A. Cunha,16 B. Dahmes,16 T. M. Hong,16 D. Kovalskyi,16 J. D. Richman,16 T. W. Beck,17 A. M. Eisner,17 C. J. Flacco,17 C. A. Heusch,17 J. Kroseberg,17 W. S. Lockman,17 T. Schalk,17 B. A. Schumm,17 A. Seiden,17 D. C. Williams,17 M. G. Wilson,17 L. O. Winstrom,17 E. Chen,18 C. H. Cheng,18 A. Dvoretskii,18 F. Fang,18 D. G. Hitlin,18 I. Narsky,18 T. Piatenko,18 F. C. Porter,18 G. Mancinelli,19 B. T. Meadows,19 K. Mishra,19 M. D. Sokoloff,19 F. Blanc,20 P. C. Bloom,20 S. Chen,20 W. T. Ford,20 J. F. Hirschauer,20 A. Kreisel,20 M. Nagel,20 U. Nauenberg,20 A. Olivas,20 J. G. Smith,20 K. A. Ulmer,20 S. R. Wagner,20 J. Zhang,20 A. M. Gabareen,21 A. Soffer,21 W. H. Toki,21 R. J. Wilson,21 F. Winklmeier,21 Q. Zeng,21 D. D. Altenburg,22 E. Feltresi,22 A. Hauke,22 H. Jasper,22 J. Merkel,22 A. Petzold,22 B. Spaan,22 K. Wacker,22 T. Brandt,23 V. Klose,23 H. M. Lacker,23 W. F. Mader,23 R. Nogowski,23 J. Schubert,23 K. R. Schubert,23 R. Schwierz,23 J. E. Sundermann,23 A. Volk,23 D. Bernard,24 G. R. Bonneaud,24 E. Latour,24 V. Lombardo,24 Ch. Thiebaux,24 M. Verderi,24 P. J. Clark,25 W. Gradl,25 F. Muheim,25 S. Playfer,25 A. I. Robertson,25 Y. Xie,25 M. Andreotti,26 D. Bettoni,26 C. Bozzi,26 R. Calabrese,26 A. Cecchi,26 G. Cibinetto,26 P. Franchini,26 E. Luppi,26 M. Negrini,26 A. Petrella,26 L. Piemontese,26 E. Prencipe,26 V. Santoro,26 F. Anulli,27 R. Baldini-Ferroli,27 A. Calcaterra,27 R. de Sangro,27 G. Finocchiaro,27 S. Pacetti,27 P. Patteri,27 I. M. Peruzzi,27, † M. Piccolo,27 M. Rama,27 A. Zallo,27 A. Buzzo,28 R. Contri,28 M. Lo Vetere,28 M. M. Macri,28 M. R. Monge,28 S. Passaggio,28 C. Patrignani,28 E. Robutti,28 A. Santroni,28 S. Tosi,28 K. S. Chaisanguanthum,29 M. Morii,29 J. Wu,29 R. S. Dubitzky,30 J. Marks,30 S. Schenk,30 U. Uwer,30 D. J. Bard,31 P. D. Dauncey,31 R. L. Flack,31 J. A. Nash,31 M. B. Nikolich,31 W. Panduro Vazquez,31 P. K. Behera,32 X. Chai,32 M. J. Charles,32 U. Mallik,32 N. T. Meyer,32 V. Ziegler,32 J. Cochran,33 H. B. Crawley,33 L. Dong,33 V. Eyges,33 W. T. Meyer,33 S. Prell,33 E. I. Rosenberg,33 A. E. Rubin,33 A. V. Gritsan,34 Z. J. Guo,34 C. K. Lae,34 A. G. Denig,35 M. Fritsch,35 G. Schott,35 N. Arnaud,36 J. Béquilleux,36 M. Davier,36 G. Grosdidier,36 A. Höcker,36 V. Lepeltier,36 F. Le Diberder,36 A. M. Lutz,36 S. Pruvot,36 S. Rodier,36 P. Roudeau,36 M. H. Schune,36 J. Serrano,36 V. Sordini,36 A. Stocchi,36 W. F. Wang,36 G. Wormser,36 D. J. Lange,37 D. M. Wright,37 C. A. Chavez,38 I. J. Forster,38 J. R. Fry,38 E. Gabathuler,38 R. Gamet,38 D. E. Hutchcroft,38 D. J. Payne,38 K. C. Schofield,38 C. Touramanis,38 A. J. Bevan,39 K. A. George,39 F. Di Lodovico,39 W. Menges,39 R. Sacco,39 G. Cowan,40 H. U. Flaecher,40 D. A. Hopkins,40 P. S. Jackson,40 T. R. McMahon,40 F. Salvatore,40 A. C. Wren,40 D. N. Brown,41 C. L. Davis,41 J. Allison,42 N. R. Barlow,42 R. J. Barlow,42 Y. M. Chia,42 C. L. Edgar,42 G. D. Lafferty,42 T. J. West,42 J. I. Yi,42 J. Anderson,43 C. Chen,43 A. Jawahery,43 D. A. Roberts,43 G. Simi,43 J. M. Tuggle,43 G. Blaylock,44 C. Dallapiccola,44 S. S. Hertzbach,44 X. Li,44 T. B. Moore,44 E. Salvati,44 S. Saremi,44 R. Cowan,45 P. H. Fisher,45 G. Sciolla,45 S. J. Sekula,45 M. Spitznagel,45 F. Taylor,45 R. K. Yamamoto,45 S. E. Mclachlin,46 P. M. Patel,46 S. H. Robertson,46 A. Lazzaro,47 F. Palombo,47 J. M. Bauer,48 L. Cremaldi,48 V. Eschenburg,48 R. Godang,48 R. Kroeger,48 D. A. Sanders,48 D. J. Summers,48 H. W. Zhao,48 S. Brunet,49 D. Côté,49 M. Simard,49 P. Taras,49 http://arxiv.org/abs/0704.0522v2 F. B. Viaud,49 H. Nicholson,50 G. De Nardo,51 F. Fabozzi,51, ‡ L. Lista,51 D. Monorchio,51 C. Sciacca,51 M. A. Baak,52 G. Raven,52 H. L. Snoek,52 C. P. Jessop,53 J. M. LoSecco,53 G. Benelli,54 L. A. Corwin,54 K. K. Gan,54 K. Honscheid,54 D. Hufnagel,54 H. Kagan,54 R. Kass,54 J. P. Morris,54 A. M. Rahimi,54 J. J. Regensburger,54 R. Ter-Antonyan,54 Q. K. Wong,54 N. L. Blount,55 J. Brau,55 R. Frey,55 O. Igonkina,55 J. A. Kolb,55 M. Lu,55 R. Rahmat,55 N. B. Sinev,55 D. Strom,55 J. Strube,55 E. Torrence,55 N. Gagliardi,56 A. Gaz,56 M. Margoni,56 M. Morandin,56 A. Pompili,56 M. Posocco,56 M. Rotondo,56 F. Simonetto,56 R. Stroili,56 C. Voci,56 E. Ben-Haim,57 H. Briand,57 J. Chauveau,57 P. David,57 L. Del Buono,57 Ch. de la Vaissière,57 O. Hamon,57 B. L. Hartfiel,57 Ph. Leruste,57 J. Malclès,57 J. Ocariz,57 A. Perez,57 L. Gladney,58 M. Biasini,59 R. Covarelli,59 E. Manoni,59 C. Angelini,60 G. Batignani,60 S. Bettarini,60 G. Calderini,60 M. Carpinelli,60 R. Cenci,60 A. Cervelli,60 F. Forti,60 M. A. Giorgi,60 A. Lusiani,60 G. Marchiori,60 M. A. Mazur,60 M. Morganti,60 N. Neri,60 E. Paoloni,60 G. Rizzo,60 J. J. Walsh,60 M. Haire,61 J. Biesiada,62 P. Elmer,62 Y. P. Lau,62 C. Lu,62 J. Olsen,62 A. J. S. Smith,62 A. V. Telnov,62 E. Baracchini,63 F. Bellini,63 G. Cavoto,63 A. D’Orazio,63 D. del Re,63 E. Di Marco,63 R. Faccini,63 F. Ferrarotto,63 F. Ferroni,63 M. Gaspero,63 P. D. Jackson,63 L. Li Gioi,63 M. A. Mazzoni,63 S. Morganti,63 G. Piredda,63 F. Polci,63 F. Renga,63 C. Voena,63 M. Ebert,64 H. Schröder,64 R. Waldi,64 T. Adye,65 G. Castelli,65 B. Franek,65 E. O. Olaiya,65 S. Ricciardi,65 W. Roethel,65 F. F. Wilson,65 R. Aleksan,66 S. Emery,66 M. Escalier,66 A. Gaidot,66 S. F. Ganzhur,66 G. Hamel de Monchenault,66 W. Kozanecki,66 M. Legendre,66 G. Vasseur,66 Ch. Yèche,66 M. Zito,66 X. R. Chen,67 H. Liu,67 W. Park,67 M. V. Purohit,67 J. R. Wilson,67 M. T. Allen,68 D. Aston,68 R. Bartoldus,68 P. Bechtle,68 N. Berger,68 R. Claus,68 J. P. Coleman,68 M. R. Convery,68 J. C. Dingfelder,68 J. Dorfan,68 G. P. Dubois-Felsmann,68 D. Dujmic,68 W. Dunwoodie,68 R. C. Field,68 T. Glanzman,68 S. J. Gowdy,68 M. T. Graham,68 P. Grenier,68 C. Hast,68 T. Hryn’ova,68 W. R. Innes,68 M. H. Kelsey,68 H. Kim,68 P. Kim,68 D. W. G. S. Leith,68 S. Li,68 S. Luitz,68 V. Luth,68 H. L. Lynch,68 D. B. MacFarlane,68 H. Marsiske,68 R. Messner,68 D. R. Muller,68 C. P. O’Grady,68 A. Perazzo,68 M. Perl,68 T. Pulliam,68 B. N. Ratcliff,68 A. Roodman,68 A. A. Salnikov,68 R. H. Schindler,68 J. Schwiening,68 A. Snyder,68 J. Stelzer,68 D. Su,68 M. K. Sullivan,68 K. Suzuki,68 S. K. Swain,68 J. M. Thompson,68 J. Va’vra,68 N. van Bakel,68 A. P. Wagner,68 M. Weaver,68 W. J. Wisniewski,68 M. Wittgen,68 D. H. Wright,68 A. K. Yarritu,68 K. Yi,68 C. C. Young,68 P. R. Burchat,69 A. J. Edwards,69 S. A. Majewski,69 B. A. Petersen,69 L. Wilden,69 S. Ahmed,70 M. S. Alam,70 R. Bula,70 J. A. Ernst,70 V. Jain,70 B. Pan,70 M. A. Saeed,70 F. R. Wappler,70 S. B. Zain,70 W. Bugg,71 M. Krishnamurthy,71 S. M. Spanier,71 R. Eckmann,72 J. L. Ritchie,72 A. M. Ruland,72 C. J. Schilling,72 R. F. Schwitters,72 J. M. Izen,73 X. C. Lou,73 S. Ye,73 F. Bianchi,74 F. Gallo,74 D. Gamba,74 M. Pelliccioni,74 M. Bomben,75 L. Bosisio,75 C. Cartaro,75 F. Cossutti,75 G. Della Ricca,75 L. Lanceri,75 L. Vitale,75 V. Azzolini,76 N. Lopez-March,76 F. Martinez-Vidal,76 D. A. Milanes,76 A. Oyanguren,76 J. Albert,77 Sw. Banerjee,77 B. Bhuyan,77 K. Hamano,77 R. Kowalewski,77 I. M. Nugent,77 J. M. Roney,77 R. J. Sobie,77 J. J. Back,78 P. F. Harrison,78 T. E. Latham,78 G. B. Mohanty,78 M. Pappagallo,78, § H. R. Band,79 X. Chen,79 S. Dasu,79 K. T. Flood,79 J. J. Hollar,79 P. E. Kutter,79 Y. Pan,79 M. Pierini,79 R. Prepost,79 S. L. Wu,79 Z. Yu,79 and H. Neal80 (The BABAR Collaboration) 1Laboratoire de Physique des Particules, IN2P3/CNRS et Université de Savoie, F-74941 Annecy-Le-Vieux, France 2Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain 3Università di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy 4University of Bergen, Institute of Physics, N-5007 Bergen, Norway 5Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 6University of Birmingham, Birmingham, B15 2TT, United Kingdom 7Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany 8University of Bristol, Bristol BS8 1TL, United Kingdom 9University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 10Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom 11Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia 12University of California at Irvine, Irvine, California 92697, USA 13University of California at Los Angeles, Los Angeles, California 90024, USA 14University of California at Riverside, Riverside, California 92521, USA 15University of California at San Diego, La Jolla, California 92093, USA 16University of California at Santa Barbara, Santa Barbara, California 93106, USA 17University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA 18California Institute of Technology, Pasadena, California 91125, USA 19University of Cincinnati, Cincinnati, Ohio 45221, USA 20University of Colorado, Boulder, Colorado 80309, USA 21Colorado State University, Fort Collins, Colorado 80523, USA 22Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany 23Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany 24Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France 25University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom 26Università di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy 27Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy 28Università di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy 29Harvard University, Cambridge, Massachusetts 02138, USA 30Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany 31Imperial College London, London, SW7 2AZ, United Kingdom 32University of Iowa, Iowa City, Iowa 52242, USA 33Iowa State University, Ames, Iowa 50011-3160, USA 34Johns Hopkins University, Baltimore, Maryland 21218, USA 35Universität Karlsruhe, Institut für Experimentelle Kernphysik, D-76021 Karlsruhe, Germany 36Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, B. P. 34, F-91898 ORSAY Cedex, France 37Lawrence Livermore National Laboratory, Livermore, California 94550, USA 38University of Liverpool, Liverpool L69 7ZE, United Kingdom 39Queen Mary, University of London, E1 4NS, United Kingdom 40University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom 41University of Louisville, Louisville, Kentucky 40292, USA 42University of Manchester, Manchester M13 9PL, United Kingdom 43University of Maryland, College Park, Maryland 20742, USA 44University of Massachusetts, Amherst, Massachusetts 01003, USA 45Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA 46McGill University, Montréal, Québec, Canada H3A 2T8 47Università di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy 48University of Mississippi, University, Mississippi 38677, USA 49Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7 50Mount Holyoke College, South Hadley, Massachusetts 01075, USA 51Università di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy 52NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands 53University of Notre Dame, Notre Dame, Indiana 46556, USA 54Ohio State University, Columbus, Ohio 43210, USA 55University of Oregon, Eugene, Oregon 97403, USA 56Università di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy 57Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France 58University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 59Università di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy 60Università di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy 61Prairie View A&M University, Prairie View, Texas 77446, USA 62Princeton University, Princeton, New Jersey 08544, USA 63Università di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy 64Universität Rostock, D-18051 Rostock, Germany 65Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom 66DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France 67University of South Carolina, Columbia, South Carolina 29208, USA 68Stanford Linear Accelerator Center, Stanford, California 94309, USA 69Stanford University, Stanford, California 94305-4060, USA 70State University of New York, Albany, New York 12222, USA 71University of Tennessee, Knoxville, Tennessee 37996, USA 72University of Texas at Austin, Austin, Texas 78712, USA 73University of Texas at Dallas, Richardson, Texas 75083, USA 74Università di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy 75Università di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy 76IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain 77University of Victoria, Victoria, British Columbia, Canada V8W 3P6 78Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 79University of Wisconsin, Madison, Wisconsin 53706, USA 80Yale University, New Haven, Connecticut 06511, USA (Dated: November 4, 2018) We perform the first three-dimensional measurement of the amplitudes of B → ψ(2S)K∗ and B → χc1K ∗ decays and update our previous measurement for B → J/ψK∗. We use a data sample collected with the BABAR detector at the PEP-II storage ring, corresponding to 232 million BB pairs. The longitudinal polarization of decays involving a JPC = 1++ χc1 meson is found to be larger than that with a 1−− J/ψ or ψ(2S) meson. No direct CP -violating charge asymmetry is observed. PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er In the context of measuring the parameters of the Unitarity Triangle of the CKM matrix, B0 decays to charmonium-containing final states (J/ψ , ψ(2S), χc1)K defined collectively here as B0 → (cc̄)K∗, are of in- terest for the precise measurement of sin 2β, where β ≡ arg[−VcdV ∗cb/VtdV ∗tb], in a similar way as for B0 → J/ψK0. Furthermore, the J/ψK∗ channel allows the measurement of cos 2β [1]. For the modes considered in this paper, the final state consists of two spin-1 mesons, leading to three possible values of the total angular momentum with different CP eigenvalues (L = 1 is odd, while L = 0, 2 are even). The different contributions must be taken into account in the measurement of sin 2β. The amplitude for longitudinal polarization of the two spin-1 mesons is A0. There are two amplitudes for polarizations of the mesons transverse to the decay axis, here expressed in the transversity basis [2]: A‖ for parallel polarization and A⊥ for their perpen- dicular polarization. Only the relative amplitudes are measured, so that |A0|2 + |A‖|2 + |A⊥|2 = 1. Previous measurements by the CLEO [3], CDF [4], BABAR [1] and Belle [5] collaborations for the B → J/ψK∗ channels are all compatible with each other, and with a CP -odd in- tensity fraction |A⊥|2 close to 0.2. Factorization predicts that the phases of the transver- sity decay amplitudes are the same. BABAR has observed [1, 6] a significant departure from this prediction. Precise measurements of the branching fractions of B → (cc̄)K∗ decays are now available [7] to test the theoretical description of the non-factorizable contribu- tions [8], but polarization measurements are also needed. In particular, measurements for ψ(2S) and χc1, com- pared to that of J/ψ , would discriminate the mass de- pendence from the quantum number dependence. CLEO has measured the longitudinal polarization of B → ψ(2S) K∗ decays to be |A0|2 = 0.45 ± 0.11 ± 0.04 [9]. Belle has studied B → χc1 K∗ decays and obtained |A0|2 = 0.87± 0.09± 0.07 [10]. B → (cc̄)K(∗) decays provide a clean environment for the measurement of the CKM angle β because one tree amplitude dominates the decay. Very small direct CP -violating charge asymmetries are expected in these decays, and no such signal has been found [7]. While more than one amplitude with different strong and weak phases are needed to create a charge asymmetry in a sim- ple branching fraction measurement, London et al. have suggested [11] that an angular analysis of vector-vector decays can detect charge asymmetries even in the case of vanishing strong phase difference. Belle has looked for, and not found, such a signal [5]. In this paper we present the amplitude measurement of charged and neutral B → (cc)K∗ using a selection simi- lar to that of Ref. [7], and a fitting method similar to that of Ref. [1]. We use the notation ψ for the 1−− states J/ψ and ψ(2S). ψ (χc1) candidates are reconstructed in their decays to ℓ+ℓ− (J/ψγ), where ℓ represents an electron or a muon. Decays to the flavor eigenstates K∗0 → K±π∓, K∗± → K0 π± and K∗± → K±π0 are used. The relative strong phases are known to have a two-fold ambiguity when measured in an angular analysis alone. In con- trast to earlier publications [3, 4, 6] we use here the set of phases predicted in Ref. [12], with arguments based on the conservation of the s-quark helicity in the decay of the b quark. We have confirmed experimentally this prediction through the study of the variation with Kπ in- variant mass of the phase difference between theK∗(892) amplitude and a non-resonant Kπ S-wave amplitude [1]. The data were collected with the BABAR detector at the PEP-II asymmetric e+e− storage ring, and correspond to an integrated luminosity of about 209 fb−1 at the center- of-mass energy near the Υ (4S) mass. The BABAR detec- tor is described in detail elsewhere [13]. Charged-particle tracking is provided by a five-layer silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH). For charged- particle identification (PID), ionization energy loss in the DCH and SVT, and Cherenkov radiation detected in a ring-imaging device (DIRC) are used. Photons are iden- tified by the electromagnetic calorimeter (EMC), which comprises 6580 thallium-doped CsI crystals. These sys- tems are mounted inside a 1.5-T solenoidal superconduct- ing magnet. Muons are identified in the instrumented flux return (IFR), composed of resistive plate chambers and layers of iron that return the magnetic flux of the solenoid. We use the GEANT4 [14] software to simulate interactions of particles traversing the detector, taking into account the varying accelerator and detector condi- tions. J/ψ → e+e− (µ+µ−) candidates must have a mass between 2.95 − 3.14 (3.06 − 3.14) GeV/c2. ψ(2S) can- didates are required to have invariant masses 3.44 < me+e− < 3.74 GeV/c 2 or 3.64 < mµ+µ− < 3.74 GeV/c Electron candidates are combined with photon candi- dates in order to recover some of the energy lost through Bremsstrahlung. J/ψ candidates and γ candidates with an energy larger than 150MeV, are combined to form χc1 candidates, which must satisfy 350 < mℓ+ℓ−γ −mℓ+ℓ− < 450 MeV/c2. π0 → γγ candidates must satisfy 113 < mγγ < 153 MeV/c 2. The energy of each photon has to be greater than 50MeV. K0 → π+π− candidates are required to satisfy 489 < mπ+π− < 507 MeV/c 2. In ad- dition, the K0 flight distance from the ψ vertex must be larger than three times its uncertainty. K∗0 and K∗+ candidates are required to satisfy 796 < mKπ < 996 MeV/c2 and 792 < mKπ < 992 MeV/c 2, respectively. In addition, due to the presence of a large background of low-energy non-genuine π0’s, the cosine of the angle θK∗ between the K momentum and the B momentum in the K∗ rest frame has to be less than 0.8 for K∗ → K±π0. In events where two B’s reconstruct to modes with the same cc̄ and K candidate, one with a π± and the other with a π0, the B candidate with a π0 is discarded due to the high background induced by fake π0’s. B candidates, reconstructed by combining cc̄ and K∗ candidates, are characterized by two kinematic variables: the difference between the reconstructed energy of the B candidate and the beam energy in the center-of-mass frame ∆E = E∗B − s/2, and the beam-energy substi- tuted mass mES ≡ (s/2 + p0 · pB)2/E20 − p2B, where subscript 0 and B correspond to Υ (4S) and the B can- didate in the laboratory frame. For a correctly recon- structed B meson, ∆E is expected to peak near zero and mES near the B-meson mass 5.279GeV/c 2. The analysis is performed in a region of the mES vs ∆E plane defined by 5.2 < mES < 5.3 GeV/c 2 and −120 < ∆E < 120 MeV. The signal region is defined asmES > 5.27 GeV/c and |∆E| smaller than 40 (30) MeV for channels with (without) a π0. For events that have multiple candi- dates, the candidate having the smallest |∆E| is chosen. mES distributions are available in Ref. [18]. The B decay amplitudes are measured from the dif- ferential decay distribution, expressed in the transversity basis [1, 6], Fig. 1, with conventions detailed in Ref. [15]. θK∗ is the helicity angle of the K ∗ decay. It is defined in FIG. 1: Definition of the transversity angles. Details are given in the text. the rest frame of the K∗ meson, and is the angle between the kaon and the opposite direction of the B meson in this frame. θtr and φtr are defined in the ψ (χc1) rest frame and are the polar and azimutal angle of the posi- tive lepton (J/ψ daughter of χc1) , with respect the axis defined by: • xtr: opposite direction of the B meson; • ytr: perpendicular to xtr, in the (xtr,pK∗) plane, with a direction such that pK∗ · ytr > 0; • ztr: to complete the frame, ie: ztr = xtr × ytr. In terms of the transversity angular variables ω ≡ (cos θK∗ , cos θtr, φtr), the time-integrated differential de- cay rate for the decay of the B meson is g(ω;A) ≡ 1 d cos θK∗d cos θtrdφtr Akfk(ω), (1) where the amplitude coefficientsAi and the angular func- tions fk(ω), k = 1 · · · 6 are listed in Table I. The ψ decays to two spin-1/2 particles, while the χc1 decays to two vector particles. The angular dependencies are therefore different [15]. The symbol A ≡ (A0, A‖, A⊥) denotes the transversity amplitudes for the decay of the B meson, and A for the B meson decay. In the absence of direct CP violation, we can choose a phase conven- tion in which these amplitudes are related by A0 = +A0, A‖ = +A‖, A⊥ = −A⊥, so that A⊥ is CP -odd and A0 and A‖ are CP -even. The phases δj of the amplitudes, where j = 0, ‖,⊥, are defined by Aj = |Aj |eiδj . Phases are defined relative to δ0 = 0. We perform an unbinned likelihood fit of the three- dimensional angle probability density function (PDF). The acceptance of the detector and the efficiency of the event reconstruction may vary as a function of the transversity angles, in particular as the angle θK∗ is strongly correlated with the momentum of the final kaon and pion. We use the acceptance correction method de- velopped in Ref. [1]. The PDF of the observed events, gobs, is : gobs(ω;A) = g(ω;A) 〈ε〉(A) , (2) where ε(ω) is the angle-dependent acceptance and 〈ε〉(A) ≡ g(ω;A)ε(ω)dω (3) is the average acceptance. We take into account the pres- ence of cross-feed from channels with the same cc̄ candi- date and a differentK∗ candidate that has (due to isospin symmetry) the same A dependence as the signal. The observed PDF for channel b (b = K±π∓,K0 π±,K±π0) is then gbobs(ω;A) = g(ω;A) εb(ω) k=1 Ak(A)Φbk , (4) TABLE I: Amplitude coefficients Ak and angular functions fk(ω) that contribute to the differential decay rate. An overall normalization factor 9/32π (for ψ) and 9/64π (for χc1) has been omitted. In the case of a B decay, the ℑm terms change sign. i Ak fk(ω) for ψ [1, 6] fk(ω) for χc1 [15] 1 |A0| 2 2 cos2 θK∗ 1− sin2 θtr cos 2 φtr 2 cos2 θK∗ 1 + sin2 θtr cos 2 φtr 2 |A‖| 2 sin2 θK∗ 1− sin2 θtr sin 2 φtr sin2 θK∗ 1 + sin2 θtr sin 2 φtr 3 |A⊥| 2 sin2 θK∗ sin 2 θtr sin 2 θK∗ 2 cos2 θtr + sin 2 θtr 4 ℑm(A∗‖A⊥) sin 2 θK∗ sin 2θtr sinφtr − sin 2 θK∗ sin 2θtr sinφtr 5 ℜe(A‖A sin 2θK∗ sin 2 θtr sin 2φtr sin 2θK∗ sin 2 θtr sin 2φtr 6 ℑm(A⊥A sin 2θK∗ sin 2θtr cos φtr − sin 2θK∗ sin 2θtr cos φtr where εb(ω) is the efficiency, defined as the ratio between the reconstructed and generated yield for the process (B → (cc̄)K∗, K∗ → b), and we do not distinguish be- tween correctly reconstructed signal and cross-feed in the numerator εb(ω) ≡ a→b(ω). (5) εa→b(ω) is the probability for an event generated in chan- nel a and with angle ω to be detected as an event in channel b. Fa, a = K π0,K±π∓,K±π0,K0 π± denotes the fraction of each channel in the total branching frac- tion B → ccK∗, a Fa = 1. The Φ k are the fk(ω) moments of the total efficiency εb, including cross-feed : Φbk ≡ fk(ω)ε a→b(ω)dω. (6) Under the approximations of neglecting the angular resolution for signal and cross-feed events, and the pos- sible mis-measurement of the B flavor such as in events where both daughters inK∗0 → K±π∓ are mis-identified (K-π swap), the PDF gobs can be expressed as in Eq. (2), and only the coefficients ΦbK are needed. The biases in- duced by these approximations have been estimated with Monte Carlo (MC) based studies and found to be negli- gible. The coefficients Φbk are computed with exclusive signal MC samples obtained using a full simulation of the ex- periment [14, 16]. PID efficiencies measured with data control samples are used to adjust the MC simulation to the observed performance of the detector. Separate co- efficients are used for different charges of the final state mesons, in particular to take into account the charge de- pendence of the interaction of charged kaons with matter, and a possible charge asymmetry of the detector. Writ- ing the expression for the log-likelihood Lb(A) for the PDF gbobs(ωi;A) for a pure signal sample of NS events, the relevant contribution is Lb(A) = ln (g(ωi;A))−NS ln Ak(A)Φbk , (7) since the remaining term i=1 ln εb(ωi) does not de- pend on the amplitudes. We use a background correction method [1] in which background events from a pure background sample of NB events are added with a negative weight to the log- likelihood that is maximized L′b(A) ≡ nB+NS L(ωi;A)− L(ωj ;A), (8) where L(ω;A) = ln(gbobs(ω;A)). The fit is performed within the mES signal region. Background events used here for subtraction are from generic (BB, qq) MC sam- ples. ñB is an estimate of the unknown number nB of background events that are present in the signal region in the data sample. As L′b is not a log-likelihood, the uncertainties yielded by the minimization program Minuit [17] are biased es- timates of the actual uncertainties. An unbiased esti- mation of the uncertainties is described and validated in Appendix A of Ref. [1]. With this pseudo-log-likelihood technique, we avoid parametrizing the acceptance as well as the background angular distributions. The measurement is affected by several systematic un- certainties. The branching fractions used in the cross- feed part of the acceptance cross section are varied by ±1σ, and the largest variation is retained. The uncer- tainty induced by the finite size of the MC sample used to compute the coefficients Φbk is estimated by the statis- tical uncertainty of the angular fit on that MC sample [6]. The uncertainty due to our limited understanding of the PID efficiency is estimated by using two different meth- ods to correct for the MC-vs-data differences. The back- ground uncertainty is obtained by comparing MC and data shapes of the mES distributions for the combinato- rial component and by using the corresponding branching errors for the peaking component. The uncertainty due to the presence of a Kπ S wave under the K∗(892) peak is estimated by a fit including it. The differential decay rate is described by Eqs. (6-9) of Ref. [1]. The results are summarized in Table II. The values of |A0|2, |A‖|2, |A⊥|2 are negatively correlated due to the constraint |A0|2+ |A‖|2+ |A⊥|2 = 1. In particular, |A‖|2, TABLE II: Summary of the measured amplitudes. For decays to χc1, as A⊥ is compatible with zero, its phase is not defined. Channel |A0| 2 |A‖| 2 |A⊥| 2 δ‖ δ⊥ J/ψK∗ 0.556 ± 0.009 ± 0.010 0.211 ± 0.010 ± 0.006 0.233 ± 0.010 ± 0.005 −2.93 ± 0.08± 0.04 2.91 ± 0.05± 0.03 ψ(2S)K∗ 0.48± 0.05 ± 0.02 0.22 ± 0.06 ± 0.02 0.30± 0.06 ± 0.02 −2.8± 0.4± 0.1 2.8± 0.3± 0.1 ∗ 0.77± 0.07 ± 0.04 0.20 ± 0.07 ± 0.04 0.03± 0.04 ± 0.02 0.0± 0.3± 0.1 – Ψ(2S) -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 cosθK* K+π- Ksπ + K+π0 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 cosθtr K+π- Ksπ + K+π0 0 2.5 5 0 2.5 5 0 2.5 5 0 2.5 5 0 2.5 5 0 2.5 5 0 2.5 5 0 2.5 5 0 2.5 5 K+π- Ksπ + K+π0 FIG. 2: Angular distributions with PDF from fit overlaid. The asymmetry of the cos θK∗ distributions induced by the S-wave interference is clearly visible. TABLE III: Difference between the interference terms mea- sured in B and B decays to J/ψ . δA4 δA6 (K+π−) 0.002 ± 0.025 ± 0.005 −0.011 ± 0.043 ± 0.016 (K+π0) −0.017 ± 0.047 ± 0.023 −0.051 ± 0.098 ± 0.064 (K0Sπ +) −0.008 ± 0.049 ± 0.011 0.075 ± 0.089 ± 0.009 which would be the least precisely measured parameter in separate one-dimensional fits, is strongly anti-correlated with |A0|2, which would be the best measured. The one-dimensional (1D) distributions, acceptance-corrected with an 1D Ansatz and background-subtracted, are over- laid with the fit results and shown on Figure 2. In con- trast with the dedicated method used in the fit, for the plots, we simply computed the 1D efficiency maps from the distributions of the accepted events divided by the 1D PDF. As in lower statistics studies, the cos θK∗ for- ward backward asymmetry due to the interference with the S wave is clearly visible. Our measurement of the amplitudes of B decays to J/ψ are compatible with, and of better precision than, previous measurements. A comparison of neutral and charged B decays (not shown) yields results consistent with isospin symmetry. The strong phase difference δ‖ − δ⊥ is obtained from a fit in which the phase origin is δ⊥ ≡ 0. We confirm our previous observation that the strong phase differences are significantly different from zero, in contrast with what is predicted by factorization. For B → J/ψK∗, it amounts to δ‖ − δ⊥ = 0.45± 0.05± 0.02. The presence of direct CP -violating triple-products in the amplitude would produce a B to B difference in the interference terms A4 and A6: δA4 and δA6. Our results (see Table III), with improved precision relative to Ref. [19], are consistent with no CP violation. In summary, we have performed the first three- dimensional analysis of the decays to ψ(2S) and χc1. The longitudinal polarization of the decay to ψ(2S) is lower than that to J/ψ , while the CP -odd intensity fraction is higher (by 1.4 and 1.0 standard deviations, respectively). This is compatible with the prediction of models of me- son decays in the framework of factorization. The lon- gitudinal polarization of the decay to χc1 is found to be larger than that to J/ψ , in contrast with the predictions of Ref. [8], which include non-factorizable contributions. The CP -odd intensity fraction of this decay is compatible with zero. The parallel and longitudinal amplitudes for χc1 seem to be aligned (|δ‖ − δ0| ∼ 0) while for ψ they are anti-aligned (|δ‖ − δ0| ∼ π). We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminos- ity and machine conditions that have made this work pos- sible. The success of this project also relies critically on the expertise and dedication of the computing organiza- tions that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospi- tality extended to them. This work is supported by the US Department of Energy and National Science Foun- dation, the Natural Sciences and Engineering Research Council (Canada), the Commissariat à l’Energie Atom- ique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsge- meinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Re- search on Matter (The Netherlands), the Research Coun- cil of Norway, the Ministry of Science and Technology of the Russian Federation, Ministerio de Educación y Cien- cia (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union) and the A. P. 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0704.0523

Quantum superpositions and entanglement of thermal states at high temperatures and their applications to quantum information processing

Quantum superpositions and entanglement of thermal states at high temperatures and their applications to quantum information processing Hyunseok Jeong and Timothy C. Ralph Centre for Quantum Computer Technology, Department of Physics, University of Queensland, St Lucia, Qld 4072, Australia (Dated: October 26, 2018) We study characteristics of superpositions and entanglement of thermal states at high tempera- tures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two single- photon detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits. I. INTRODUCTION In many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equiva- lently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open sys- tem which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle ob- jects whose properties are significantly more difficult to characterize than pure states. Schrödinger’s famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive〉 (or a wave function such as ψalive). The cat isolated from the environment is then assumed to inter- act with a microscopic superposition state, (|g〉+|e〉)/ where |g〉 and |e〉 are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e〉, while it will remain alive if other- wise. Thus in Schrödinger’s gedanken experiment the cat is entangled with the atom as (|g〉|alive〉+ |e〉|dead〉) where the alive and dead statuses of the cat are described by the state vectors |alive〉 and |dead〉. If one mea- sures out the atomic system on the superposed basis, (|g〉 ± |e〉)/ 2, the cat will be in a superposition of alive and dead states such as (|alive〉± |dead〉)/ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to ob- serve because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4]. However, this explanation is not fully satisfactory be- cause the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5]. In such a case, even though one can perfectly iso- late the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive〉 and |dead〉. We may need a more realistic assumption that the “cat” in Schrödinger’s paradox was in a significantly mixed clas- sical state. An intriguing question is then whether the quantum properties of the resulting state would still re- main or diminish under such an assumption. Recently, such an analogy of Schrödinger’s cat para- dox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6]. A thermal state with a high temperature is consid- ered a classical state in quantum optics. As the tem- perature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the tem- perature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coher- ent states with large amplitudes are known as the most classical pure states [7], and their superposition is of- ten regarded as a superposition of classical states [8]. However, coherent states are still pure states which may not well represent truly classical systems, and they dis- play some nonclassical features [9]. In Ref. [6], it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a sig- nificantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally http://arxiv.org/abs/0704.0523v2 feasible process. This result clarifies that unavoidable ini- tial mixedness of the cat does not preclude strong quan- tum phenomena. One of the results in Ref. [6] is that quantum entan- glement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10]. There is an interesting previous example shown by Filip et al. for the maximum violation of Bell’s inequal- ity when one of the modes is an extremely mixed thermal state [11]. Very recently, Ferreira et al. showed that en- tanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12]. However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite tem- perature limit [10, 12], which is obviously in contrast to the result presented in Ref. [6]. Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell’s inequality in our examples reaches up to Cirel’son’s bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures re- duce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6]. The purpose of this paper is twofold. Firstly, we review and further investigate various properties of superposi- tions and entanglement of thermal states at high tem- peratures [6]. In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum informa- tion processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlin- ear interactions without photon number resolving mea- surements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme. This paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en- tanglement of thermal states, i.e., Bell inequality viola- tions. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first de- fine the thermal-state qubit and the Bell-basis states us- ing thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V. II. SUPERPOSITIONS OF THERMAL STATES A. Generation of thermal-state superpositions Let us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as ρth(V, d) = d2αP th(V, d)|α〉〈α| (1) where |α〉 is a coherent state of amplitude α and P thα (V, d) = π(V − 1) exp[− 2|α− d|2 V − 1 ] (2) with variance V and displacement d in the phase space. The thermal temperature τ increases as V increases as e~ν/τ = (V + 1)/(V − 1), where ~ is Planck’s constant and ν is the frequency [15]. Suppose that a microscopic superposition state |ψ〉a = (|0〉a + |1〉a), (3) where |0〉 and |1〉 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρthb (V, d) and the interaction Hamiltonian is HK = λâ†âb̂†b̂ (4) which corresponds to the cross Kerr nonlinear interac- tion. The resulting state is then ρentab = d2αP th(V, d) |0〉〈0| ⊗ |α〉〈α| + |1〉〈0| ⊗ |αeiϕ〉〈α|+ |0〉〈1| ⊗ |α〉〈αeiϕ| + |1〉〈1| ⊗ |αeiϕ〉〈αeiϕ| and ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρentab is W entab (α, β) = e−2|α| W th(β; d) + 2αV c(β; d) + 2[αV c(β; d)]∗ + (4|α|2 − 1)W th(β; deiϕ) where α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and W th(α; d) = exp[−2|α− d| ], (7) V c(α; d) = exp[− 2 (1 − eiϕ)d2 − 1 (α− 2e )(α∗ − 2d )], (8) K = 2+ (V − 1)(1− eiϕ), J = (sinϕ/2 + iV cosϕ/2)/(2V sinϕ/2 + 2i cosϕ/2), and d has been assumed real without loss of generality. If one traces ρentab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the “microscopic part” on the superposed basis, i.e., (|0〉a ± |1〉a)/ 2, the “macroscopic part” for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to ρsup(±) = N±s d2αP th(V, d) |α〉〈α| ± |αeiϕ〉〈α| ± |α〉〈αeiϕ|+ |αeiϕ〉〈αeiϕ| , (9) where N±s are the normalization factors, and its Wigner function is W sup(±)(α) = N±s {W th(α; d)± V c(α; d) ± {V c(α; d)}∗ +W th(α; deiϕ)}. (10) The ± signs in Eqs. (8) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. (10) is a superposition of two thermal states. A feasible experimental setup to generate superposi- tions of thermal states is atom-field interactions in cavi- ties, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has al- ready been performed to produce a superposition of co- herent states [16]. In our cases, simply thermal states can be used instead of coherent states. Another pos- sible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard single- photon qubit could be used as the microscopic superpo- sition. Recently, there have been theoretical and experi- mental efforts to produce and observe giant Kerr nonlin- earities using electromagnetically induced transparency [17]. Furthermore, it was shown that a weak Kerr non- linearity can still be useful if a initially strong field is employed in this type of experiment [18]. We shall fur- ther explain this with examples in Sec. III. B. Negativity of the Wigner function The negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. (6) for V = 1 is −0.144 for d = 0 and −0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function (6) is −0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (− 1 , 0) and has negativity Wneg ≡W entab (− , 0) = 2(−2 + 1 exp[− 2d . (11) It can be shown that Wneg approaches −4/(π2 −0.246 when either d→ ∞ or V → ∞. This effect is obviously due to the interaction between the microscopic superposition and the macroscopic ther- mal state. If the initial microscopic state is not super- posed, e.g., |ψ〉a = |1〉a, the resulting state will be a simple direct product, (|1〉〈1|)a ⊗ ρthb (V,−d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0〉a instead of |1〉a, the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativ- ity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. − −0.04 100 0.15 0 0.040 0.005 0.015 FIG. 1: The probability distributions of x (left) and p (right) for a “superposition” of two distant thermal states. A thermal state with a large mixedness is converted to such a “thermal- state superposition” by interacting with a microscopic super- potion (see text). The variance V and displacement d for the thermal state are chosen as (a) V = 100 and d = 100, and (b) V = 1000 and d = 300. The fringe visibility is 1 regardless of V and the fringe spacing (the distance between the fringes) does not depend on the variance (i.e. mixedness) but only on the distance d between the two component thermal states. The Wigner functions of the single-mode states, W sup(±)(α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(−)(α) is W sup(−)(0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+)(α) approaches 2/π for d→ ∞ and disappears when d = 0. C. Quantum interference in the phase space When ϕ = π, the state (9) becomes ρ± = N(ρth(V, d)±σ(V, d)±σ(V,−d)+ρth(V,−d)), (12) where σ(V, d) = d2αP th(V, d)| − α〉〈α| and N = 2 exp[− 2d . (13) If the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as |Ψ̃±〉 = 1± e−2|α|2 (|α〉 ± | − α〉), (14) where α = d. The probability P± to obtain the state ρ± is obtained as [19] P± = 〈ψ±|Trb[ρentab ]|ψ±〉 = exp[− 2d2 ), (15) 1998 2000 −2002 5 5 FIG. 2: The probability distributions P for a “superposition” of thermal states where V = 5, d = 2000, ϕ = π/1000. The x′ (p′) axis in this figure has been rotated by π/2000 from the x (p) axis for clarity. where |ψ±〉 = (|0〉±|1〉)/ 2. The probability approaches P± = 1/2 when either d or V becomes large. As an analogy of Schrödinger’s cat paradox, the vari- ance V corresponds to the size the initial “cat”, and the distance d between the two thermal component states corresponds to distinguishability between the “alive cat” and the “dead cat”. Suppose that both V and d are very large for the initial thermal state. The two thermal states ρth(V,±d) become macroscopically distinguishable when V , and our example may become a more realis- tic analogy of the cat paradox in this limit. Both the states ρ± in this case show probability distributions with two Gaussian peaks and interference fringes [6]. Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ− (a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability dis- tribution of x (p) for ρ± can be obtained by integrating the Wigner function of ρ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quan- tum superposition between macroscopically distinguish- able states. The visibility v of the interference fringes is defined as [15] Imax − Imin Imax + Imin , (16) where I = dxW sup(−)(α) and the maximum should be taken over p. It can be simply shown that the visibil- ity v is always 1 regardless of the value of V . Note that d should increase proportionally to V to maintain the condition of classical distingushability between the two component thermal states ρth(V,±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not de- pend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are “superpositions” of severely mixed thermal states. An experimental realization of a nonlinear effect cor- responding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ≪ π) combined with a strong field (d≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference (a) (b) (c) 0.005 0.005 0.015 0.005 0.015 (d) (e) (f) FIG. 3: (Color online) The time dependent Wigner functions of the thermal state of V = 100 at the origin (d = 0) after an interaction with a microscopic superposition and a conditional measurement. The measurement result on the microscopic part was supposed to be (|0〉 + |1〉)/ 2. The interaction times are (a) θ = λt = 0, (b) θ = λt = π/32, (c) θ = π/16, (d) θ ≈ 3.102, (e) θ ≈ 3.122 and (f) θ = π. patterns. In Fig. 2, we have used experimentally acces- sible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18]. Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18]. D. Symmetric macroscopic quantum states Let us assume that d = 0, i.e., the initial state is the thermal state, ρth(V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ±, are produced with probabilities, P± = (1/2){1± (1/V )}, re- spectively. Figure 3 shows the Wigner functions of ρ+ dependent on the interaction time between the macro- scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3. In this figure, the initial state is a ther- mal state of V = 100 (Fig. 3(a)). In a relatively short time (θ = π/32 and θ = π/16), the state shows some in- terference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3. The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state. In the case of ρ−, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (−2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ− as shown in Fig. 4 . III. ENTANGLEMENT BETWEEN THERMAL STATES Entanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be gener- ated between high-temperature thermal states even when the temperature of each mode goes to infinity. −1 −0.5 0 0.5 1 FIG. 4: (Color online) The Wigner function of the thermal state of V = 100 at the origin (d = 0) after an interac- tion with a microscopic superposition and a conditional mea- surement. The measurement result on the microscopic part was supposed to be (|0〉 − |1〉)/ 2 with the interaction time θ = λt = π. 2001 400 600 800 1000 50 100 150 FIG. 5: (a) The optimized violation, B ≡ |B+|max, of Bell- CHSH inequality for the “thermal-state entenglement”, ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 2, when V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ+. V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve). A. Entanglement using two initial thermal states If the microscopic superposition interacts with two thermal states, ρthb (V, d) and ρ c (V, d), and the micro- scopic particle is measured out on the superposed basis, the resulting state will be ρtm(±) = Nt ρth(V, d)⊗ ρth(V, d)± σ(V, d) ⊗ σ(V, d) ± σ(V,−d)⊗ σ(V,−d) + ρth(V,−d)⊗ ρth(V,−d) where Nt = 2 exp[− 4d . (18) Such two-mode thermal-state entanglement can be gener- ated using two cavities and an atomic state detector [20]. Extending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24]. The parity mea- surements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25]. The Bell-CHSH inequality can be repre- sented in terms of the Winger function as [24] |B(±)| = π |W tm(±)(α, β) +W tm(±)(α, β′) +W tm(±)(α′, β)−W tm(±)(α′, β′)| ≤ 2, where W tm(±)(α, β) is the Wigner function of ρtm(±) in Eq. (17). As shown in Fig. 5, the Bell-violation ap- proaches the maximum bound for a bipartite measure- ment, 2 2 [13], when d ≫ V regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ+ and ρ− even though only the case of ρ+ has been plotted in Fig. 5(a). This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ V , and such “thermal-state entangle- ment” cannot be described by a local theory. B. Entanglement using a beam splitter A different type of macroscopic entanglement can be generated by applying the beam splitter operation exp[θ/2(eiφâ†sâd − e−iφâ dâs)], (20) on the “thermal-state superpositions” in Eq. (9). The state after passing through a 50:50 beam splitter can be represented as d2αP thα (V, d) ,− α√ 〉 ± | − α√ ,− α√ | ± 〈− α√ , (21) 200 400 600 800 1000 FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement gener- ated using a 50:50 beam splitter using ρ+ when d = 0. 3.13 3.14 3.15 3.16 3.13 3.14 3.15 3.16 FIG. 7: (a) The Bell-CHSH function B against θ (= λt) for V = 1 (solid curve), V = 10 (dashed curve) and V = 20 (dotted curve) for d = 30. (b) The Bell-CHSH function for d = 10 (solid curve), d = 20 (dashed curve) and d = 30 (dotted curve) for V = 10. The Bell violations are more sensitive to the interaction time as either V or d increases. where N is defined in Eq. (13). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 2 regardless of the level of mixedness V as shown in Fig. 5(b). Again, it is true for both of ρ+ and ρ− even though only the case of ρ+ has been plotted in Fig. 5(b). Furthermore, these states severely violate Bell’s inequality even when d = 0 as V increases as shown in Fig. 6. We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interest- ingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26]. Note that multilmode entangled states can be generated using multiple beam splitters. It should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate. IV. QUANTUM INFORMATION PROCESSING WITH THERMAL-STATE QUBITS In this section, we discuss the possibility of quan- tum information processing with thermal-state qubits and thermal-state entanglement. A. Qubits and Bell-state measurements We introduce a thermal-state qubit ρψ = |a|2ρth(V, d)± ab∗σ(V, d)± a∗bσ(V,−d) + |b|2ρth(V,−d), where a and b are arbitrary complex numbers. The ba- sis states, ρth(V, d) and ρth(V,−d), can be well discrimi- nated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as d2αP thα (V, d) a|α〉+ b| − α〉 a∗〈α| + b∗〈−α| which can be understood as a generalization of the co- herent state qubit, a|d〉+ b| − d〉, where |d〉 is a coherent state of amplitude d. The thermal-state qubit (23) be- comes identical to the coherent-state qubit when V = 1. We also define four thermal-Bell states as ρΦ(±) = Nt ρth(V, d) ⊗ ρth(V, d)± σ(V, d)⊗ σ(V, d)± σ(V,−d)⊗ σ(V,−d) + ρth(V,−d)⊗ ρth(V,−d) ρΨ(±) = Nt ρth(V, d)⊗ ρth(V,−d)± σ(V, d) ⊗ σ(V,−d)± σ(V,−d)⊗ σ(V, d) + ρth(V,−d)⊗ ρth(V, d) where Nt was defined in Eq. (18). The thermal-Bell states can be written as ρΦ(±) = Nt dα2dβ2P thα (V, d)P β (V, d) |α, β〉 ± | − α,−β〉 〈α, β| ± 〈−α,−β| , (26) ρΨ(±) = Nt dα2dβ2P thα (V, d)P β (V, d) |α,−β〉 ± | − α, β〉 〈α,−β| ± 〈−α, β| . (27) Homodyne detector C FIG. 8: A schematic of the thermal-Bell state measurement (a) using photon number resolving detection and (b) using ho- modyne measurements with cross-Kerr nonlinear interactions (NL). See text for details. For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discrimi- nate between the thermal-Bell states (25). We shall only briefly describe the first scheme using photon number re- solving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a). This scheme is basically the same as the Bell- state measurement scheme with pure entangled coherent states [27, 28]. Let us suppose that the amplitude, d, is large enough, i.e., d ≫ V . If the incident state was ρΦ(+) or ρΦ(−), most of the photons are detected on de- tector A in in Fig. 8(a). Meanwhile, most of the photons are detected on detector B when the incident state was ρΨ(+) or ρΨ(−). The average photon numbers between the “many-photon case” and the “few-photon case” are compared in Fig. 9. Furthermore, the states ρΨ(+) and ρΦ(+) contain only even numbers of photons while ρΨ(−) and ρΦ(−) contain only odd numbers of photons. There- fore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon num- ber detected by the two detectors are even, this means that state ρΦ(+) was measured by the thermal-Bell mea- surement. The nonzero failure probability can be made arbitrarily small by increasing d. However, the average photon numbers of the thermal- Bell states are high when V ≫ 1 and d≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 FIG. 9: The average photon number N for the “many-photon case” (solid line) and the “few-photon case” (dashed line) for V = 10 against d (a) when the input state is either ρΦ(+) or ρΨ(+) and (b) when the input state is either ρΦ(−) or ρΨ(−). these four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, in- stead of photon number resolving detection. Our alterna- tive scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b). Let us first sup- pose that the input field was ρΦ(+). The incident two- mode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is ρB = Nt d2αd2βP thα (V, d)P β (V, d) |η,−ξ〉〈η,−ξ| + |η,−ξ〉〈−η, ξ|+ | − η, ξ〉〈η,−ξ| + | − η, ξ〉〈−η, ξ| where η = (α+β)/ 2 and ξ = (α−β)/ 2. Two dual-rail single photon qubits, |ψ+〉ee′ and |ψ+〉ff ′ , where |ψ+〉 = (|0〉|1〉+ |1〉|0〉), (29) are prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then = UceUdfρ ff ′U df (30) where Uce = exp[iπHKce/λ~] and ρq = |ψq〉〈ψq |. An ex- plicit form of Eq. (30) can then be simply obtained using the identity Uce|α〉c|0〉e = |α〉c|0〉e, Uce|α〉c|1〉e = | − α〉c|1〉e where |α〉 is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e′, f and f ′, should be measured with the mea- surement basis {|++〉, |+−〉, | −+〉, | − −〉} (32) where |+ +〉 = |ψ+〉ee′ |ψ+〉ff ′ , | + −〉 = |ψ+〉ee′ |ψ−〉ff ′ , | − +〉 = |ψ−〉ee′ |ψ+〉ff ′ , | − −〉 = |ψ−〉ee′ |ψ−〉ff ′ , and |ψ−〉 = (|0〉|1〉 − |1〉|0〉)/ 2. This measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b). If detector A1 and B1 click, i.e., the mea- surement result is | + +〉, the resulting state at modes c and d is ρ++ = d2αd2βP thα (V, d)P β (V, d) (|η〉 + | − η〉)(〈η| + 〈−η|) (|ξ〉+ | − ξ〉)(〈ξ| + 〈−ξ|) Note that state ρ++ is not normalized, which implies that the probability of obtaining the corresponding measure- ment result is not unity. The probability of obtaining this result is P++ = (V + 1)(V + e− 2(V 2 + e− . (34) When the result is either |+−〉 or | −+〉, the result is 〈ψ2|ρB |ψ2〉 = 〈ψ3|ρB |ψ3〉 = 0, (35) which obviously means that the probability of the ob- taining this result is zero. When the result is | −−〉, i.e., detector A2 and B2 click, ρ−− = d2αd2βP thα (V, d)P β (V, d) (|η〉 − | − η〉)(〈η| − 〈−η|) (|ξ〉 − | − ξ〉)(〈ξ| − 〈−ξ|) which is not normalized. The probability of obtaining this result is P−− = (V − 1)(V − e− 4d 2(V 2 + e− , (37) and it can be simply verified that P+++P−− = 1. There- fore, only the measurement results |++〉 and | −−〉 can be obtained in the case of the input state ρΦ(+). This is exactly the same for the case of ρΨ(+). In the same way, it can be shown that if either the input state was ρΦ(−) or ρΨ(−), only the measurement results |+−〉 and | −+〉 −20 −10 10 20 Probability −20 −10 10 20 Probability FIG. 10: (a) The probability distributions, P++ (solid curve) and P++ (dashed curve), for homodyne measurements at de- tector C. (b) The probability distributions, P++ (solid curve) and P++ (dashed curve), for homodyne measurements at de- tector C. can be obtained. In other words, the parity of the to- tal incoming state is perfectly well discriminated by the measurements on single-photon qubits. Subsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b). We assume that ideal homodyne measurements are per- formed, i.e., when a homodyne measurement is per- formed the state is projected onto eigenstate |x〉 of oper- ator X with eigenvalue x, where (a+ a†). (38) Let us first consider the case when the measurement re- sult for the single photon qubits is | + +〉. In this case, the remaining state is ρ++ in Eq. (33). The probabil- ity distribution P++ for the homodyne measurement at detector C is = 〈x|Trd[ρ++]|x〉 = 2 (e−V x 2 (V + 1) . (39) Note that the superscript, ++, denotes that the qubit measurement result was |++〉, and the subscript, Φ(+), denotes that the input state was ρΦ . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | − −〉: = 〈x|Trd[ρ−−]|x〉 = 2 (e−V x 2 − e−x 2 (V − 1) . (40) In the same way, for another input state, ρΦ(−), it is straightforward to show: = P++ , P−+ = P−− , (41) and P++ = P−− = 0. On the other hand, if the input state was ρΨ(+), the probability distributions P++ at detector C are = 〈x|Trc[ρ++]|x〉 = x{4d+(2+V 2)x} (1+V 2)x2 V + 2e 2x(2d+x) V + e x(8d+x+V 2x) V V ) + 1 , (42) = 〈x|Trc[ρ−−]|x〉 = x{4d+(2+V 2)x} (1+V 2)x2 V − 2e 2x(2d+x) V + e x(8d+x+V 2x) V V )− 1 . (43) 20 4 6 8 10 d FIG. 11: The distinguishability Ps between states ρ Ψ(+) and ρΦ(+) by a homodyne measurement against for V = 10 (solid curve) and V = 20 (dashed curve) against distance d. See text for details. It is straightforward to show for the other input state ρΨ(−): = P−− , P−+ = P++ . (44) The probability distributions P++ and P++ are plotted in Fig. 10. Figure 10 shows that when the input state was ρΦ(+) or ρΦ(−), the homodyne measurement outcome by detector C, characterized by P++ and P−− , is located around the origin. However, when the input state was ρΨ(+) or ρΨ(−), the homodyne measurement outcome by detector C, characterized by P++ and P−− , is located far from the origin. Therefore, two of the Bell states, ρΦ(+) or ρΦ(−), can be well distinguished from the other two by the homodyne detector C for the case of the mea- surement outcome |++〉. Finally, by combining the ho- modyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement out- come of the single photon detectors was | + +〉 and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρΨ(−) has been mea- sured for the result of the thermal-Bell measurement. As implied in Fig. 10, the overlaps between the proba- bility distributions around the origin, P++ and P−− and the other distributions, P++ and P−− , are ex- tremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can cal- culate the distinguishability between the states ρΨ(+) and ρΦ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is |x|<d dxP++c (x) + |x|≥d dxP++d (x) which is plotted in Fig. 11. The distinguishability is Ps ≈ 0.99 for d = 5.5 (d = 7.8) when V = 10 (V = 20), and it becomes as high as Ps > 0.99999 for d = 10 (d = 15) when V = 10 (V = 20). If necessary, another homodyne measurement can be performed for mode d to enhance distinguishability of the Bell measurement. When the probability distribution at detector C is around the origin that of detector D is far from the origin and vice versa. Note also that the second scheme using homodyne de- tection is robust to detection inefficiency compared with the first scheme using photon number resolving measure- ments. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong mea- surement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be imme- diately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distribu- tions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ V , as shown in Fig. 10. On the other hand, loss in the Kerr medium will have a detrimental affect. B. Quantum teleportation and computation Quantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρψ, to Bob using a thermal-state entanglement, ρΨ(−), shared by the two parties. The total state can be represented as 1 ⊗ ρ 23 = Nt dα2dβ2dγ2P thα (V, d)P β (V, d)P γ (V, d) (a|α〉+ b| − α〉)1(|β,−γ〉 − | − β, γ〉)23 . (46) Alice first needs to perform the thermal-Bell measure- ment described in the previous subsection. To complete the teleportation process, Bob should perform an appro- priate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27]. When the mea- surement outcome is ρΨ(−), Bob obtains a perfect replica of the original unknown qubit without any operation. When the measurement outcome is ρΦ(−), Bob should perform |α〉 ↔ | − α〉 on his qubit in Eq. (23). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = eiϕa †a, where a and a† are the annihilation and creation operators. When the out- come is ρΨ(+), the transformation should be performed as |α〉 → |α〉 and | − α〉 → −| − α〉. It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29]. This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30]. When the outcome is ρΦ(+), σx and σz should be successively applied. V. CONCLUSION In this paper, we have studied characteristics of su- perpositions and entanglement of thermal states at high temperatures and discussed their applications to quan- tum information processing. The superpositions and en- tanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH in- equality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermal- state entanglement is larger. Therefore, in order to ob- serve the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, sym- metric in the phase space, can also be generated. Some of these states have neither squeezing properties nor neg- ative values in their Wigner functions but they are found to be highly nonclassical. We have introduced the thermal-state qubit and thermal-Bell states for applications to quantum informa- tion processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measure- ment enables one to perform quantum teleportation and gate operations for quantum computation with thermal- state qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discrim- inate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the de- tectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell mea- surement. Acknowledgments This work was supported by the DTO-funded U.S. Army Research Office Contract No. W911NF-05-0397, the Australian Research Council and Queensland State Government. [1] E. Schrödinger, Naturwissenschaften. 23, pp. 807-812; 823-828; 844-849 (1935). [2] A.J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985). [3] M.D. Reid, preprint quant-ph/0101052 and references therein. [4] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000). [5] H.M. Wiseman and J.A. Vaccaro, Phys. Rev. Lett. 87, 240402, (2001); See discussions in the introduction and references therein. [6] H. Jeong and T.C. Ralph, Phys. Rev. Lett. 97, 100401 (2006). [7] E. Schrödinger, Naturwissenschaften 14, 664 (1926). [8] W. Schleich, M. Pernigo, and F.L. Kien, Phys. Rev. A 44, 2172 (1991). [9] L.M. Johansen, Phys. Lett. A 329, 184. [10] S. Bose, I. Fuentes-Guridi, P.L. Knight, and V. Vedral, http://arxiv.org/abs/quant-ph/0101052 Phys. Rev. Lett. 87, 050401 (2001). [11] R. Filip, M. Dusek, J. Fiurasek, L. Mista, Phys. Rev. A 65, 043802 (2002). [12] A. Ferreira, A. Guerreiro, and V. Vedral, Phys. Rev. Lett. 96, 060407 (2006); We note that this work appeared on the Los Alamos archive (quant-ph/0504186) after we up- loaded the main results of our work (quant-ph/0410210). [13] B. S. Cirel’son, Lett. Math. Phys. 4, 93 (1980). [14] V. Vedral, New J. Phys. 6 102 (2004). [15] D. F. Walls and G. J. Milburn, Quantum Optics, Springer-Verlag (1994). [16] M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996); A. Auffeves et al., Phys. Rev. Lett. 91 230405 (2003). [17] H. Schmidt and A. Imamoglu, Opt. Lett. 21, 1936 (1996); L. V. Hau et al., Nature 397, 594 (1999). [18] H. Jeong, Phys. Rev. A 72, 034305 (2005) and references therein. [19] We note that denominator V was missing in the genera- tion probability P± in [6]. [20] M. S. Kim and J. Lee, Phys. Rev. A 61 042102 (2000). [21] D. M. Greenberger, M. Horne and A. Zeilinger, Bells theorem, Quantum theory, and Conceptions of the the Universe, ed. M. Kafatos, Kluwer, Dordrecht, 69 (1989); [22] S. Bell, Physics 1, 195 (1964). [23] J. F. Clauser et al., Phys. Rev. Lett. 23, 880 (1969). [24] K. Banaszek and K. Wódkiewicz, Phys. Rev. A 58, 4345 (1998); Phys. Rev. Lett. 82, 2009 (1999). [25] B. -G. Englert, N. Sterpi, and H.Walther, Opt. Commun. 100 526 (1993). [26] H. Jeong, W. Son, M. S. Kim, D. Ahn, and C. Brukner, Phys. Rev. A 67, 012106 (2003). [27] H. Jeong, M. S. Kim, and J. Lee, Phys. Rev. A. 64, 052308 (2001). [28] S. J. van Enk and O. Hirota, Phys. Rev. A. 64, 022313 (2001). [29] H. Jeong and M. S. Kim Phys. Rev. A 65, 042305 (2002). [30] T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, Phys. Rev. A 68, 042319 (2003). http://arxiv.org/abs/quant-ph/0504186 http://arxiv.org/abs/quant-ph/0410210

0704.0524

Optimal control of stochastic differential equations with dynamical boundary conditions

Optimal control of stochastic differential equations with dynamical boundary conditions Stefano BONACCORSI∗, Fulvia CONFORTOLA†, Elisa MASTROGIACOMO Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38050 Povo (Trento), Italia In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problem with non standard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with non standard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation. Keywords: Stochastic differential equations in infinite dimensions, dynamical bound- ary conditions, optimal control 1991 MSC : 1 Setting of the problem Our model is a one dimensional semilinear diffusion equation in a confined system, where interactions with extremal points cannot be disregarded. The extremal points have a mass and the boundary potential evolves with a specific dynamic. Stochas- ticity enters through fluctuations and random perturbations both in the inside as on the boundaries; in particular, in our model we assume that the control process is perturbed by a noisy term. There is a growing literature concerning such problems; we shall mention the paper [2] where a problem in a domain O ⊂ Rn is concerned; the authors cite as an example an SPDE with stochastic perturbations which appears in connection with random fluctuations of the atmospheric pressure field. As opposite to ours, however, that paper is not concerned with control problems. Quite recently, the authors became aware of the paper [1] where a different application to some generalized Lamb model is proposed. The internal dynamic is described by a stochastic evolution problem in the unit ∗[email protected] †Current address: [email protected] http://arxiv.org/abs/0704.0524v1 2 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo interval D = [0, 1] ∂tu(t, x) = ∂ xu(t, x) + f(t, x, u(t, x)) + g(t, x, u(t, x))Ẇ (t, x) (1) which we write as an abstract evolution problem on the space L2(0, 1) du(t) = Amu(t) + F (t, u(t)) dt+G(t, u(t)) dW (t), (2) where the leading operator is Am = ∂ x with domain D(Am) = H 2(0, 1). We assume that f and g are real valued mappings, defined on [0, T ]× [0, 1]× R, which verify some boundedness and Lipschitz continuity assumptions. The boundary dynamic is governed by a finite dimensional system which follows a (ordinary, two dimensional) stochastic differential equation ∂tvi(t) = −bivi(t) + ∂νu(t, i) + hi(t)V̇i(t), i = 0, 1 where bi are positive numbers and hi(t) are bounded, measurable functions; ∂ν is the normal derivative on the boundary, and coincides with (−1)i∂x for i = 0, 1. For notational semplicity, we introduce the 2× 2 diagonal matrices B = diag(−b−0, b1) and h(t) = diag(h0(t), h1(t)). There is a constraint Lu = v which we interpret as the operator evaluating boundary conditions; the system is coupled by the presence, in the second equation, of a feedback term C that is an unbounded operator ∂xu(0) −∂xu(1) The idea is to write the problem in abstract form for the vector u = the space X = L2(0, 1)× R2, that is du = Au(t) + F(t,u(t)) dt +G(t,u(t)) dW(t) u(0) = Our main concern is to study spectral properties of the matrix operator on the domain D(A) = {u ∈ D(Am)× R2 : Lu = v}. Theorem 1. A is the infinitesimal generator of a strongly continuous, analytic semigroup of contractions etA, self-adjoint and compact. Control of stochastic differential equations with dynamical boundary conditions 3 We shall prove the above theorem in Section 2. Further, we shall prove that A is a self-adjoint operator with compact resolvent, which implies that the gener- ated semigroup is Hilbert-Schmidt. Moreover, we can characterize the complete, orthonormal system of eigenfunctions associated to A. Let us fix a complete probability space (Ω,F, {Ft},P); on this space we de- fine W (t), that is a space-time Wiener process taking values in X and V (t) = (V1(t), V2(t)), that is a R 2-valued Wiener process, such that W (t, x) and V (t) are independent. As a corollary to Theorem 1, using standard results for infinite dimensional stochastic differential equations, compare [3, Theorem 7.4], we obtain the following existence result Theorem 2. For any initial condition ∈ X×R2 there exists a unique process u ∈ L2F (0, T ;X × R2) such that u(t) = etA e(t−s)AF(u(s)) ds+ e(t−s)AG(u(s)) dW(s) that is by definition a mild solution of (3). The abstract semigroup setting we propose in this paper allows to obtain an optimal control synthesis for the above evolution problem with boundary control and noise. This means that we assume a boundary dynamics of the form: ∂tv(t) = bv(t)− ∂νu(t, ·) + h(t)[z(t) + V̇ (t)] (4) where z(t) is the control process and takes values in a given subset of R2. As before, we can write the system – defined by the internal evolution problem (1) and the dynamical boundary conditions described by (4) – in the following abstract form duzt = Au t dt+ F(t,u t ) dt+G(t,u t )[Pzt dt+ dWt] ut0 = u0. P : R2 → X denote the immersion of the boundary space in the product space X = L2(0, 1)× R2. The aim is to choose a control process z, within a set of admissible controls, in such way to minimize a cost functional of the form J(t0, u0, z) = E λ(s,uzs , zs)) ds+ Eφ(u T ) (6) where λ and φ are given real functions. In our setting, altough the control lives in a finite dimensional space, we obtain an abstract optimal control problem in infinite dimensions. Such type of problems has been exhaustively studied by Fuhrman and Tessitore in [8]. The control problem is understood in the usual weak sense (see [7]). We prove that if f and g are sufficiently regular then the abstract control problem, under suitable assumptions on λ and φ, can be solved and we can characterize optimal controls by a feedback law (see Theorem 17 and compare Theorem 7.2 in [8]). 4 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo Theorem 3. In our assumptions, there exists an admissible control {z̄t, t ∈ [0, T ]} taking values in a bounded subset of R2, such that the closed loop equation: duτ = Auτ dτ +G(τ,uτ )PΓ(τ,uτ ,G(τ,uτ ) ∗∇xv(τ,uτ )) dτ + F(τ,uτ ) dτ +G(τ,uτ ) dWτ , τ ∈ [t0, T ], ut0 = u0 ∈ X. admits a solution and the couple (z,u) is optimal for the control problem. Stochastic boundary value problems are already present in the literature, see the paper [11] and the references therein; in those papers, the approach to the solution of the system is more similar to that in [2]. We also need to mention the paper [5] for a one dimensional case where the boundary values are set equal to a white noise mapping. 2 Generation properties Let X = L2(0, 1) be the Hilbert space of square integrable real valued functions defined on D = [0, 1] and X = X × R2. In this section we consider the following initial-boundary value problem on the space X u(t) = Amu(t) v(t) = Lu(t) v(t) = Bv(t)− Cu(t) u(0) = u0 ∈ X, v(0) = v0 ∈ R2. In the above equation, Am is an unbounded operator with maximal domain Am = ∂ x, D(Am) = H 2(0, 1); B is a diagonal matrix with negative entries (−b0,−b1). Let C : D(C) ⊂ X → ∂X the feedback operator, defined on D(C) = H1(0, 1) ∂xu(0) −∂xu(1) The boundary evaluation operator L is the mapping L : X → R2 given by Its inverse is the Dirichlet mapping D λ : R 2 → D(Am) λ φ = u(x) ∈ D(Am) : (λI −Am)u(x) = 0, Lu = φ. As proposed in [10], we define a mild solution of (8) a function u ∈ C([0, T ];X) such that u(t) = u0 +Am u(s) ds, t ∈ [0, T ] v(t) = v0 +B v(s) ds+ C u(s) ds. Control of stochastic differential equations with dynamical boundary conditions 5 In order to use semigroup theory to study equation (8), we consider a matrix operator describing the evolution with feedback on the boundary on the domain D(A) = {u ∈ D(Am)× R2 : Lu = v}. Then a mild solution for equation (8) exists if and only if A is the generator of a strongly continuous semigroup. The above definition of the domain D(A) puts in evidence the relation between the first and the second component of the vector u. There is a different characteri- zation that is sometimes useful in the applications. Let us define the operator A0 as A0 = Am on D(A0) = {u ∈ D(Am) : Lu = 0}. We can then write the domain of A as D(A) = {u ∈ D(Am)× ∂X : u−DA,L0 v ∈ D(A0)}. The operator A can be decomposed as the product I −DA,L0 Then, according to Engel [6], A is called a one-sided K-coupled matrix-valued operator. Proof of Theorem 1 In this section we apply form theory in order to prove generation property of the operator A, compare the monograph [13]. Proposition 4. A is the infinitesimal generator of a strongly continuous, analytic semigroup of contractions, self-adjoint and compact. We will give the proof in two steps. First of all we will consider the following form: a(u,v) = u′(x)v′(x) dx + b0 u(0) v(0) + b1 u(1) v(1) on the domain u = (u, α) ∈ H1(0, 1)× R2 | u(0) = α0, u(1) = α1 and we will show that it is densely defined, closed, positive, symmetric and continue. Moreover, the operator associated with the form a is (A, D(A)) defined above. According to [13], this implies that the operator A is self-adjoint and generates a contraction semigroup etA on X that is analytic of angle π . Then we will show the self-adjointness and the compactness of the semigroup etA. To see this, we will refer to [9]. Let us begin with the properties of the form a. 6 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo Lemma 5. The form a is densely defined, closed, positive, symmetric and continue. Proof. By assumption, since b0 and b1 are positive real numbers, it follows that in particular a is symmetric and positive. It is clear that V is a linear subspace of X. Observe that V is dense in X if any u ∈ X can be approximated with elements of V . Consider (u, α) ∈ L2[0, 1] × R2. Since C∞c [0, 1] is dense in L 2(0, 1) it follows that for all ε > 0 there exists v ∈ C∞c [0, 1] such that |u− v|L2[0,1] ≤ Now let ρ0(x) be a symmetric function in C c (R) with support in Bε(0), ρ0(0) = 1 ρ0(x) dx = ε/3. Finally, let ρ1(x) = ρ0(x−1). Then, if we define the function ρ = v + α0 ρ0 [0,1] + α1 ρ1 [0,1] , we have: |u− ρ|L2[0,1] ≤ |u− v|L2[0,1] + |α0ρ0|L2[0,1] + |α1ρ1|L2[0,1] ≤ ≤ max {1, α0, α1} ε. Morever, ρ(0) = α0 and ρ(1) = α1. Thus |(u, α)− (ρ, ρ(0), ρ(1))| for a suitable M . This shows that V is dense in X. In order to check closedness and continuity of a, observe first that the norm induced by a on the space V is equivalent to the norm given by the inner product (u,v)V = [u′(x)v′(x) + u(x)v(x)] dx+ u(1)v(1) + u(0)v(0). In fact, if we set b = b0 + b1, we have ‖u‖a = a(u,u) + ‖u‖2V so that ‖u‖2a ≤ 2 ‖u‖ H1(0,1) + 2b u(0)2 + u(1)2 ≤ max {2, 2b} ‖u‖2V . Now observe that V becomes a Hilbert space when equipped with the inner product defined above since V is a closed subspace of H1(0, 1)× R2. Then a is closed. Finally, a is continuous. To see this, take u,v ∈ V ; then |a(u,v)| ≤ |u′(x)v′(x)| dx+ b [|u(0)| |v(0)|+ |u(1)| |v(1)|] ≤ ‖u‖H1(0,1) ‖v‖H1(0,1) + b [|u(0)| |v(0)|+ |u(1)| |v(1)|] ≤ ‖u‖V ‖v‖V ≤M ‖u‖a ‖v‖a by the Cauchy-Schwartz inequality. Control of stochastic differential equations with dynamical boundary conditions 7 Lemma 6. The operator associated with a is (A, D(A)) defined above. Proof. Denote by (C, D(C)) the operator associated with a. By definition, C is given D(C) = {f ∈ V | ∃g ∈ X s.t. a(f ,g) = (g,h)X∀h ∈ V } Cf = −g. Let us first show that A ⊂ C. Take f ∈ D(A). Then for all h ∈ V a(f ,h) = f ′(x)h′(x) dx + b0f(0)h(0) + b1f(1)h(1) = f ′(x)h(x)|10 − f ′′(x)h(x) dx + b0f(0)h(0) + b1f(1)h(1) = f ′(1)h(1)− f ′(0)h(0)− f ′′(x)h(x) dx + b0f(0)h(0) + b1f(1)h(1). At the same time, if we set α = (f(0), f(1)), β = (h(0), h(1)), we have (Af ,h) = (Af, h)L2(0,1) + (Cf +Bα, β)R2 = f ′′(x)h(x) dx + f ′(0)h(0)− f ′(1)h(1) − b0f(0)h(0)− b1f(1)h(1) = −a(f ,g). The last equality shows that A ⊂ C. To check the converse inclusion C ⊂ A take f ∈ D(C). By definition, there exists g ∈ X such that a(f ,h) = (g,h)X, ∀h ∈ V that is, f ′(x)h′(x) dx = g(x)h(x) dx. Now choose h = (h, α) ∈ V such that the function h belongs to H10 (0, 1) (the existence of such a function is ensured by the continuous embedding of H10 (0, 1)in H1(0, 1)). Then by the last equality we cand derive that f ′ ∈ H1(0, 1) and g is the weak derivative of f ′: it follows that f ′ ∈ H1(0, 1) and we conclude that f ∈ H2(0, 1). Integrating by parts as in the proof of the first inclusion we see that a(f ,h) = f ′(x)h′(x) dx + b0f(0)h(0) + b1f(1)h(1) = f ′(x)h(x)|10 − f ′′(x)h(x) dx + b0f(0)h(0) + b1f(1)h(1) = (−Af ,h) = (g,h), ∀h ∈ V. This implies that Af = −g, and the proof is complete. 8 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo Corollary 7. The operator (A, D(A)) is self-adjoint and dissipative. Moreover it has compact resolvent. Proof. The self-adjointness of A follows by [13] (Proposition 1.24) and he dissipativ- ity is obsvious. Since D(A) ⊂ H2(0, 1)×R2, the operator A has compact resolvent and the claim follows. Taking into account the above corollary, it follows that A generates a contraction semigroup (etA)t≥0 on X that is analytic of angle π/2 and self-adjoint. Finally, by [9, Corollary XIX.6.3] we obtain that etA is compact for all t > 0. Thus we have just proved Proposition 4. Remark 1. By the Spectral Theorem [9, Chapter XIX, Corollary 6.3] it follows that there exists an orthonormal basis {en}n∈N of X and a sequence {λn}n∈N of real negative numbers λn ≤ 0, such that en ∈ D(A), Aen = λnen and lim λn = −∞. Moreover, A is given by λn(u, en)en, u ∈ D(A) etAu = eλnt(u, en)en, u ∈ X. 2.1 Spectral properties of the matrix operator We shall now apply Theorem 2.5 in Engel[6] in order to describe the spectrum of A. According to that result σ(A) ⊆ σ(A0) ∪ σ(B) ∪ S (9) where S = {λ ∈ ρ(A0) ∩ ρ(B) : Det(F (λ)) = 0}. (10) The matrix F (λ) is defined as F (λ) = I − (λ−B)LλKλR(λ,B) where the operators Lλ and Kλ are given by Lλ = −BR(λ,B)R(0, B)C, Kλ = −A0R(λ,A0)DA,L0 . Notice that the matrix F (λ) can also be written as F (λ) = I + CA0R(λ,A0)D 0 R(λ,B). Remark 2. In case when the feedback operator matrix C is identically zero, the above construction implies that S = ∅. Control of stochastic differential equations with dynamical boundary conditions 9 Determining the set S In the following, we construct explicitly the set S. The idea is to construct the matrix F (λ) and compute its determinant. We have to distinguish two cases. If λ < 0 we have Det(F (λ)) = 1 + −λcos( λ+ b0 λ+ b1 (λ+ b0)(λ + b1) We note that the equation Det(F (λ)) = 0 has infinite solutions {λj}j∈N and every λj belongs to the interval (−π2(j + 1)2,−π2j2). Each λj is eigenvalue of the operator A corresponding to the eigenfunction φj = (ej(x), ej(0), ej(1)) where ej(x) = −λjBj b0 + λj −λjx+Bj sin −λjx. for a normalizing constant 0 < Bj < If λ > 0 then Det(F (λ)) = 1 + 1 + e2 − 1 + e2 b0 + λ b1 + λ (b0 + λ) (b1 + λ) We note that Det(F (λ)) > 0 for every λ > 0. This means that there are not elements λ strictly positive in S. Moreover the eigenvalues of A in S are all negative. Remark 3. It is possible to verify directly with some computation that the eigen- values of A are not eigenvalues of A. Further, the same happens in general with the eigenvalues of B, except in case b0 and b1 satisfy an explicit relation. In any case, also if b0 and b1 happen to belong to σ(A), they are in a finite number and do not affect its behaviour. Therefore, with no loss of generality, in the following we may and do assume that all the eigenvalues of A are contained in S. Theorem 8. In the above assumptions the semigroup etA is Hilbert-Schmidt, that |etAφi|2L2(0,1)×R2 <∞ (11) for any orthonormal basis {φi} of L2(0, 1)× R2. Proof. In order to prove that the semigroup etA is Hilbert-Schmidt, it is enough verify the (11) for an orthonormal basis. Let {φi} the orthonormal sequence of eigenfunctions of the operator A described in Remark 1. Then |etAφi|2L2(0,1)×R2 = e2tλi 10 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo where λi are the eigenvalues of the operator A. By (9) it follows that e2tλi ≤ i:λi∈σ(A) e2tλi + i: λi∈σ(B) e2tλi + i:λi∈S e2tλi . But, by Remark 3 we have that e2tλi ≤ i: λi∈σ(B) e2tλi + i:λi∈S e2tλi and the first of the last two series is a finite sum and the second one converges since the eigenvalues λi in S are asymptotic to −π2i2. 3 The abstract problem In this section we are concerned with problem (3): we introduce the relevant assump- tions and we formulate the main existence and uniqueness result for its solution. Let W = (W,V ) be the Wiener process taking values in = L2(0, 1) × R2. We denote {Ft, t ∈ [0, T ]} the natural filtration of W, augmented with the family N of P-null sets of FT : Ft = σ(W(s) : s ∈ [0, t]) ∨N. The filtration {Ft} satisfies the usual conditions. Define F : [0, T ]× X → X for every u = F(t,u) = F F (t, u) where F (t, u)(ξ) = f(t, ξ, u(ξ)). Let G be the mapping [0, T ]×X → L(X,X) such that, for u = and y = in X, G1(t, u) y G2(t, v) η where (G1(t, u) y)(ξ) = g(t, ξ, u(ξ))y(ξ) and (G2(t, v) · η) = h(t) η; we stress that h is a diagonal matrix. Therefore, we are concerned with the following abstract problem dut = Aut dt+ F(t,ut) dt+G(t,ut)dWt ut0 = u0 on which we formulate the following assumptions. Control of stochastic differential equations with dynamical boundary conditions 11 Assumption 9. (i) f : [0, T ] × [0, 1] × R → R, is a measurable mapping, bounded and Lipschitz continuous in the last component |f(t, x, u)| ≤ K, |f(t, x, u)− f(t, x, v)| ≤ L|u− v|. for every t ∈ [0, T ], x ∈ [0, 1], u, v ∈ R. (ii) g : [0, T ]× [0, 1]× R → R, is a measurable mapping such that |g(t, x, u)| ≤ K, |g(t, x, u)− g(t, x, v)| ≤ L|u− v| for every t ∈ [0, T ], x ∈ [0, 1], u, v ∈ R. (iii) h : [0, T ] →M(2, 2) is a bounded measurable mapping verifying |h(t)| ≤ K for every t ∈ [0, T ]. The existence and uniqueness of the solution to (12) is a standard result in the literature, see for instance the monograph [3]. In order to apply the known results, we shall verify that the nonlinear coefficients F and G satisfy suitable Lipschitz continuous conditions. That will be enough to prove the existence of a mild solution which is a process ut adapted to the filtration Ft satisfying the following integral equation ut = e e(t−s)AF(s,us) ds+ e(t−s)AG(s,us) dWs. (13) Proposition 10. Under Assumptions 9(i)–(iii), the following hold: 1. the mapping F : X → X is measurable and satisfies, for some constant L > 0, |F(t,u)− F(t,v)|X ≤ L|u− v|X u,v ∈ X. 2. G is a mapping [0, T ]× X → L(X) such that a. for every v ∈ X the map G(·, ·)v : [0, T ]× X → X is measurable, b. esAG(t,u) ∈ L2(X) for every s > 0, t ∈ [0, T ] and u ∈ X, and c. for every s > 0, t ∈ [0, T ] and u.v ∈ X we have |esAG(t,u)|L2(X) ≤ L s −1/4 (1 + |u|X), (14) |esAG(t,u)− esAG(t,v)|L2(X) ≤ L s −1/4|u− v|X, (15) |G(t,u)|L(X) ≤ L (1 + |u|X), (16) for a constant L > 0. Proof. 1. We have, for u = and v = |F(t,u)− F(t,v)|X = |F (t, u)− F (t, v)|X ≤ L|u− v|X ≤ L|u− v|X. 12 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo 2. Condition (16) follows from the definition of G and the Assumptions 9 (ii)-(iii) on g and h. Now we prove condition (14). Let {φk}k∈N be an orthonormal basis in X. |esAG(t,u)|2L2(X) = | < esAG(t,u)φj , φk > |2X | < G(t,u)φj , esAφk > |2X ≤ |G(t,u)|2L(X) |esA|2L2(X) ≤ L 2(1 + |u|2 )|esA|2L2(X). Using Theorem 8, |esA|2L2(X) ≈ e−2sn where f(t) ≈ g(t) means that f(s)/g(s) = O(1) as s→ 0; this verifies (14). In order to prove the last statement (15), we take the orthonormal basis {φk}k∈N consisting of eigenvectors of A (see Remark 1). We recall that φk = (ek(x), ek(0), ek(1)) where ek(x) = Bk b0 + λk −λkx+Bk sin −λkx. We have |esAG(t,u)− esAG(t,v)|2L2(X) = | < esA[G(t,u)−G(t,v)]φj , φk > |2X | < G(t,u)−G(t,v)φj , esAφk > |2X = e2sλk |G(t,u)−G(t,v)φk|2. But, for u = and v = , by the definition of the operator G, we have |G(t,u)−G(t,v)φk|2X = |g(t, x, u(x))− g(t, x, v(x))|2|ek(x)|2dx K2|u(x)− v(x)|2dx ≤ K2|u− v|2 since the function g is Lipschitz and |ek(x)| ≤ Bk is uniformly bounded in k. Consequently |esAG(t,u)− esAG(t,v)|L2(X) ≤ { e2tλk}1/2K|u− v|X ≤ |esA|L2(X)K|u− v|X which concludes the proof. Control of stochastic differential equations with dynamical boundary conditions 13 Proposition 11. Under the assumptions 9 for every p ∈ [2,∞) there exists a unique process u ∈ Lp(Ω;C([0, T ];X)) solution of (12). Proof. We can apply Theorem 5.3.1 in [4]. In fact by Proposition 4 the operator A generates a strongly continuous semigroup {etA} of bounded linear operators in the Hilbert space X. Moreover, for this theorem to apply we need to verify that coefficients F and G satisfy conditions (14)—(16), which follows from Proposition 4 Stochastic control problem After some preliminaries, in this section we are concerned with an abstract control problem in infinite dimensions. We settle the problem in the framework of weak control problems (see [7]). We aim to control the evolution of the system by the boundary. This means that we assume a boundary dynamic of the form: ∂tv(t) = bv(t)− ∂νu(t, ·) + h(t)[z(t) + V̇ (t)] (17) where z(t) is the control process. We require that z ∈ L2(Ω× [0, T ];R2). As in the previous section we can write the system ∂tu(t, x) = ∂ xu(t, x) + f(t, x, u(t, x)) + g(t, x, u(t, x))Ẇ (t, x) ∂tv(t) = bv(t)− ∂νu(t, ·) + h(t)[z(t) + V̇ (t)] in the following abstract form duzt = Au t dt+ F(t,u t ) dt+G(t,u t )[Pzt dt+ dWt] ut0 = u0 (19) where P : R2 → X is the immersion of the boundary space in the product space X = X×R2. Equation (19), in the framework of stochastic optimal control problem, is called the controlled state equation associated to an admissible control system. We recall that, in general, fixed t0 ≥ 0 and u0 ∈ X, an admissible control system (a.c.s) is given by (Ω,F, {Ft}t≥0,P, {Wt}t≥0, z) where • (Ω,F,P) is a probability space, • {Ft}t≥0 is a filtration in it, satisfying the usual conditions, • {Wt}t≥0 is a Wiener process with values in X and adapted to the filtration {Ft}t≥0, • z is a process with values in a space K, predictable with respect to the fil- tration {Ft}t≥0 and satisfies the constraint: z(t) ∈ Z, P-a.s., for almost every t ∈ [t0, T ], where Z is a suitable domain of K. 14 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo In our case the space K coincide with R2. To each a.c.s. we associate the mild solution uz of state equation the mild so- lution uz ∈ C([t0, T ];L2(Ω;X)) of the state equation. We introduce the functional J(t0, u0, z) = E λ(s,uzs , zs)) ds+ Eφ(u T ) (20) We consider the problem of minimizing the functional J over all admissible control systems (which is known in the literature as the weak formulation of the control problem); any a.c.s. that minimize J -if it exsts- is called optimal for the control problem. We define in classical way the Hamiltonian function relative to the above problem ψ : [0, T ]× X× X → R setting ψ(t,u,w) = inf {λ(t,u, z)+ < w, P z >} (21) and we define he following set Γ(t,u,w) = {z ∈ Z : λ(t,u, z)+ < w, P z >= ψ(t,u, z)} We consider the Hamilton-Jacobi-Bellman equation associated to the control problem ∂v(t, x) + Lt[v(t, ·)](x) = ψ(t, x, v(t, x),G(t, x)∗∇xv(t, x)), t ∈ [0, T ], x ∈ X, v(T, x) = Φ(x). where the operator Lt is defined by Lt[φ](x) = Trace G(t, x)G(x) ∗∇2φ(x) + < Ax,∇φ(x) > . Under suitable assumptions, if we let v denote the unique solution of (22) then we have J(t, x, z) ≥ v(t, x) and the equality holds if and only if the following feedback law is verified by z and uzσ: z(σ) = Γ(σ,uzσ,G(σ,u ∗∇xv(σ,uzσ)). Thus, we can characterize optimal controls by a feedback law. This class of stochastic control problems, in infinite dimensional setting, has been studied by Fuhrman and Tessitore [8] (We refer to Theorem 7.2 in that paper for precise statements and additional results). In order to characterize optimal controls by a feedback law we have to require that the abstract operators F and G satisfy further regularity conditions. We will prove that, under suitable assumptions on the functions f and g in the problem (18), the abstract operators fit the required conditions. Control of stochastic differential equations with dynamical boundary conditions 15 We impose that the operators F and G are Gâteaux differentiable. This notion of differentiability is weaker than the differentiability in the Fréchet sense. We recall that for a mapping F : X → V , where X and V denote Banach spaces, the directional derivative at point x ∈ X in the direction h ∈ X is defined as ∇F (x;h) = lim F (x+ sh)− F (x) whenever the limit exists in the topology of V . F is called Gâteaux differentiable at point x if it has directional derivative in every direction at point x and there exists an element of L(X,V ), denoted ∇F (x) and called Gâteaux derivative, such that ∇F (x;h) = ∇F (x)h for every h ∈ X . Definition 12. We say that a mapping F : X → V belongs to the class G1(X ;V ) if it is continuous, Gâteaux differentiable on X, and ∇F : X → L(X,V ) is strongly continuous. The last requirement of the definition means that for every h ∈ X the map ∇F (·)h : X → V is continuous. Note that ∇F : X → L(X,V ) is not continuous in general if L(X,V ) is endowed with the norm operator topology; clearly, if this happens then F is Fréchet differentiable on X . Membership of a map in G1(X,V ) may be conveniently checked as shown in the following lemma. Lemma 13. A map F : X → V belongs to G1(X,V ) provided the following condi- tions hold: i) the directional derivatives ∇F (x;h) exist at every point x ∈ X and in every direction h ∈ X; ii) for every h, the mapping ∇F (·;h) : X → V is continuous; iii) for every x, the mapping h 7→ ∇F (x;h) is continuous from X to V . When F depends on additional arguments, the previous definitions and proper- ties have obvious generalizations. The following assumptions are necessary in order to provide Gâteaux differen- tiability for the coefficients of the abstract formulation. Assumption 14. For a.a. t ∈ [0, T ], ξ ∈ [0, 1] the functions f(t, ξ, ·) and g(t, ξ, ·) belong to the class C1(R). Proposition 15. Under assumptions 9 and 14, for every s > 0, t ∈ [0, T ], F(t, ·) ∈ G1(X,X), esAG(t, ·) ∈ G1(X, L2(X)). Proof. The first statement is an immediate consequence of the fact that f(t, ξ, ·) ∈ C1(R,R). In order to prove that esAG(t, ·) belongs to the class G1(X, L2(X)) we use the continuous differentiability of g and an argument similar to that used in the proof of Proposition 10. We note that, for u = and v = , the gradient operator∇ esAG(t,u) is an Hilbert Schmidt operator that maps 7→ esA gu(t, ·, u(·))w(·)v(·) = esA (∇u(G(t,u)v)(w)) 16 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo In fact, we have esAG(t,u+ rv) − esAG(t,u) −∇esAG(t,u)v L2(X) = lim esAG(t,u+ rv) − esAG(t,u) φj − esA (∇u(G(t,u)v)φj) , φk > = lim G(t,u+ rv) −G(t,u) −∇uG(t,u)v φj , e sAφk > = lim e2sλk G(t,u+ rv) −G(t,u) −∇uG(t,u)v = lim e2sλk g(t, u(ξ) + rv(ξ)) − g(t, u(ξ)) ek(ξ)− gu(t, u(ξ))v(ξ)ek(ξ) ≤ c lim e2sλk g(t, u(ξ) + rv(ξ)) − g(t, u(ξ)) − gu(t, u(ξ))v(ξ) = c lim e2sλk gu(t, u(ξ) + αrv(ξ)) − gu(t, u(ξ)) dα v(ξ) and, by dominated convergence, this limit is equal to zero. In similar way we can prove the points (ii)− (iii) of Lemma 13 to obtain the thesis. In order to prove the main result of this section we require the following hypoth- esis. Assumption 16. (i) λ is measurable and for a.e. t ∈ [0, T ], for all u,u′ ∈ X, z ∈ Z |λ(t,u, z)− λ(t,u′, z)| ≤ C|1 + u+ u′|m|u− u′| |λ(t, 0, z)| ≤ C for suitable C ∈ R+, m ∈ N; (ii) Z is a Borel and bounded subset of R2; (iii) Φ ∈ G1(X,R) and, for every σ ∈ [0, T ], ψ(σ, ·, ·) ∈ G1,1(X × X,R); (iv) for every t ∈ [0, T ], u,w,h ∈ X ψ(t,u,w)h| + |∇ φ(u)h| ≤ L|h|(1 + |u|)m; (v) for all t ∈ [0, T ], for all u ∈ X and w ∈ X there exists a unique Γ(t,u,w) ∈ Z that realizes the minimum in (21). Namely λ(t,u,Γ(t,u,w))+ < w, PΓ(t,u,w) >= ψ(t,u,w) Control of stochastic differential equations with dynamical boundary conditions 17 Theorem 17. Suppose that assumptions 9, 14 and 16 hold. For all a.c.s. we have J(t0, u0, z) ≥ v(t0, u0) and the equality holds if and only if the following feedback law is verified by z and uz: z(σ) = Γ(σ,uzσ, G(σ,u ∗∇xv(σ,uzσ)), P− a.s. for a.a. σ ∈ [t0, T ]. (23) Finally there exists at least an a.c.s. for which (23) holds. In such a system the closed loop equation: duτ = Auτ dτ +G(τ,uτ )PΓ(τ,uτ ,G(τ,uτ ) ∗∇xv(τ,uτ )) dτ + F(τ,uτ ) dτ +G(τ,uτ ) dWτ , τ ∈ [t0, T ], ut0 = u0 ∈ X. admits a solution and if z(σ) = Γ(σ,uσ, G(σ,uσ) ∗∇xv(σ,uσ)) then the couple (z,u) is optimal for the control problem. Proof. By Proposition 4 we know that A generates a strongly continuous semigroup of linear operators etA on X. The assumption 9 ensures that the statements in Proposition 10 hold. Moreover the assumption 14 guarantees that the results in Proposition 15 are true. Finally these conditions together with the assumption 16 allow us to apply Theorem 7.2 in [8] and to perform the synthesis of the optimal control. References 1. M. Bertini, D. Noja, A. Posilicano, Dynamics and Lax-Phillips scattering for gen- eralized Lamb models, J. Phys. A: Math. Gen. 39 (2006), 15173–15195 2. Igor Chueshov, Björn Schmalfuss, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations 17 (2004), no. 7-8, 751–780. 3. Giuseppe Da Prato, Jerzy Zabczyk, Stochastic equations in infinite dimen- sions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. 4. G. Da Prato, J. Zabczyk, Ergodicity for infinite-dimensional systems, Lon- don Mathematical Society Lecture Notes Series, 229, Cambridge University Press, 1996. 5. A. Debussche, M. Fuhrman, G. Tessitore, Optimal Control of a Stochastic Heat Equation with Boundary-noise and Boundary-control, to appear in ESAIM Con- trol, Optimisation and Calculus of Variations. 6. K.-J. Engel, Spectral theory and generator property for one-sided coupled operator matrices, Semigroup Forum 58 (1999), 267–295. 7. W. H. Fleming, H. M. Soner, Controlled Markov processes and viscosity solutions, Springer-Verlag, 1993. 8. M. Fuhrman, G. Tessitore, Non linear Kolmogorov equations in infinite dimen- sional spaces: the backward stochastic differential equations approach and appli- cations to optimal control, Ann. Probab. 30 (2002), no. 3: 1397-1465. 9. Israel Gohberg, Seymour Goldberg, Marinus A. Kaashoek, Classes of linear oper- ators, Vol. I, Birkhser Verlag, Basel, 1990. Operator Theory: Advances and Applica- tions, 49. 18 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo 10. Marjeta Kramar, Delio Mugnolo, Rainer Nagel, Semigroups for initial-boundary value problems.In: Evolution equations: applications to physics, industry, life sciences and economics (Levico Terme, 2000), 275–292, Progr. Nonlinear Differential Equations Appl., 55, Birkhuser, Basel, 2003. 11. Bohdan Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 55–93. 12. D. Mugnolo, Asymptotics of semigroups generated by operator matrices, Ulmer seminare 10 (2005), 299–311. 13. El Maati Ouhabaz, Analysis of heat equations on domains, London Math- ematical Society Monographs Series, 31. Princeton University Press, Princeton, NJ, 2005. Setting of the problem Generation properties Spectral properties of the matrix operator The abstract problem Stochastic control problem

0704.0526

Fractional WKB Approximation

Fractional WKB approximation FRACTIONAL WKB APPROXIMATION Eqab M. Rabei٭ and Ibrahim M.A.Altarazi Department of physics, Mu'tah University, Al-karak, Jordan Sami I. Muslih Department of physics, Al-Azhar University, Gaza, Palestine Dumitru Baleanu Department of Mathematics and Computer Science, Faculty of Arts and Science, Çankaya University Institute of Space Sciences, P.O.BOX, MG-23, R 76900, Magurele-Bucharest, Romania Abstract Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function "S". To demonstrate our proposed approach two examples are investigated in details. Keywords: fractional derivative, fractional WKB approximation Hamilton's principle function. [email protected]٭ I. Introduction Fractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number. Nowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems. Recently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references]. This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2], where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems. From these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4]. Starting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5]. In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6]. The equations of motion are obtained in a similar manner to the usual mechanics. All these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10,14]. In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB. The purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions. The plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions. II. Basic Tools The left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by αα dfx xa ∫ −−−⎟⎠ = )()( )( 1 (1) The right Riemann-Liouville fractional derivative has the form ββ dfx bx ∫ −−−⎟⎠ = )()( )( 1 (2) Here α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero. If α is an integer, these derivatives are defined in usual sense as )()( xf xfDxa ⎛= (3-a) )()( xf xfDbx ⎛−= (3-b) Hamilton formalism with fractional derivative was proposed in [5] namely ),,,(),,,( tqDqDqLqDpqDptppqH bttabtta αβα −+= , (4) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5]. Hamilton's equations of motion are obtained as follows [5] ; ;qDp ;qDp β pDpD btta +=∂ (5) In [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as StPPqDqDFF btta == −− ),,,,( 112 βα ),,,('),,,( tQDQDQLQDPQDPtPPQK bttabtta αβα −+= (7) It is concluded that, the following relation relates the two Hamiltonians KQDPQDPHqDpqDp bttabtta +−+=−+ α (8) According to reference [6] the function F is proposed as ),,,,( 11 tPPqDqDSF btta βα βα −−= QDPQDP Btta 11 −− −− ββ α , (9) The function S is called Hamilton's principle function. Therefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, Pα, Pβ are constants. H (10) Since Q, Pα, Pβ are constants, The Hamilton’s principle function is written ),,,,( 21 11 tEEqDqDSS btta −−= βα (11) where 1EP =α 2EP =β If the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ),,(),(),( 212 1 tEEfEqDWEqDWS btta ++= −− βα W represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as: αα qD ββ (13) 11 λα = QDta 2 21 λβ = QDbt (14) Here λ1 ,λ2 are constants. III. Fractional WKB approximation The outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13]. exp),( tqψ (15) The phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12). Thus we propose the fractional state function as: ⎛= −−−− tqDqDS tqDqD bttabtta ,,exp),,( 1111 βαβαψ (16) From the quantization using WKB approximation [7,8,9,10,14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as: exp)( ψψ (17) where iio qp (18) In our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus, (19) qDq ta −≡ α αPP ˆ1≡ (20) qDq bt −≡ β βPP ˆ2 ≡ And the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form = −−−− tEEqDqDS tqDqD bttabtta ,,,,exp ),,( 21 1111 βα h (21) and the momenta operators in the form −− qDi ˆ,ˆ ββαα (22) We conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum IV. Examples IV. a) Example 1: As a first model let us consider the following fractional Lagrangian, ( ) ( )21 qDqDL tt βα += (23) The fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as: ( ) ( ) .0 1 22 = PP βα (24) where qD = ; qD = Making use of equation (13), the fractional Hamilton-Jacobi equation (24) becomes: βα (25) Taking into account t −= (26) If we apply (26) on a wave function it gives: )( 21 EEEt +−≡−= (27) By using the fact that E is the total energy of the system and taking into account (27) we obtain βα (28) Thus, both sides of (28) should be zero, and we obtain qDEWqDEW tt 011 2,2 −− == βα (29) By using (12) and (21) we obtain −+= −−−− tEqDEqDE tqDqD tttt 0 22exp ),,( βα (30) Which represents the wave function of the following Hamiltonian: ( ) ( )22 βα PPH += (31) Let us deal now with the momenta as operators of the form (22), and applying these operators on the wave function, one obtain the following momenta eigenvalues ψψψψ βα 21 2ˆ2ˆ EPEP == (32) Then, 21 2ˆ2ˆ EPEP == βα (33) It’s the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues: ( ) ( ) ψψψ βα 22 ˆ2 PPH += ( )ψ21 EE += (34) as in the classical case. IVb.)Example 2: As a second example let us consider the following fractional Lagrangian ( ) ( ) 210 qqDqDqDqDL tttt ++++= (35) The corresponding fractional Hamilton is calculated as follows ( ) ( ) 222 qPPH −−+−= βα (36) Thus, the fractional Hamilton-Jacobi equation becomes ( ) ( ) 0 1 222 = +−−+− qPP βα (37) The fractional Hamilton's principle function is calculated as, ( ) ( ) tEEqDEqDEqS tt )(1212 211121012 +−++++= −− βα (38) As a result the wave function can be written in the form ( ) ( )( )⎟ −++++ tEqDEqDEq tqDqD 1212exp (39) To identify the influence of the operators let us test the effect of the momenta −− qDi (40) Using the characteristic equations, it can be shown that 12ˆ,12ˆ 21 2 +=++= EPEqP βα (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψψψψ βα 222 2 qPPH −−+−= (42) ( ) ( ) ψψψψ βαβα 222 2 1ˆˆˆˆ qPPPP −++−+= ( ) ψ} 222222 EEqEq −++++− +++++= Then we get ψψ EH = (43) which is exactly the total energy as the case for the classical systems. V. Conclusions We use the generating function "S" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach. Giving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions. References [1] Om P.Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J.Math. Anal. Appl. 272,(2002),368-379. [2] F. Riewe, Non-conservative Lagrangian and Hamiltonian mechanics, Phys. Rev.E 53: (1996), 1890-1899. [3] Eqab M.Rabei, Tareq S.Alhalholy and Akram A. Rousan, Potential of arbitrary forces with fractional derivatives, International journal of modern physics A.19: 17&18July(2004), 3083-3092. [4] Eqab M.Rabei, Abdul-wali Ajlouni, an Humam B.Ghassib. Quantization of Brownian motion. International Journal of theoretical physics, 45: (2005), 1613-1623. [5] Eqab M.Rabei and khaled I.Nawafleh, Raed S.Hijjawi, Sami I.Muslih, Dumitru Baleanu.The Hamilton Formalism with fractional derivatives, J. Math. Anal. Appl.: 327, (2007) ,891–897 [6] Eqab M.Rabei, Bashar S. Ababneh, Hamilton-Jaccobi fractional Sequential Mechanics , J. Math. Anal. Appl. (in press). [7] Eqab M. Rabei, Eyad H. Hasan, and Humam B. Ghassib, Hamilton- Jaccobi Treatment of Constrained Systems with Second-Order Lagrangians, International Journal of Theoretical Physics, Vol. 43, No. 4, April (2004), 1073-1096. [8] Eqab M.Rabei, Khaled I.Nawafleh, Y.S Abdelrahman, H.Y.R Omari. Hamilton-Jaccobi treatment of Lagrangians with linear velocities Modern physics letters A.18:(2003), 1591-1596 [9] Eqab M.Rabei, Eyad H. Hasan, Humam B.Ghassib, S Muslih, Quantization of Second-Order constrained Lagrangians systems using the WKB approximation, International Journal of geometric methods in modern Physics 2: (2005)1-20. [10] Khaled I.Nawafleh, Eqab M. Rabei and H. B. Ghassib, Hamilton- Jacobi Treatment of constrained systems, International Journal of Modern Physics A, 19: (2004), 347-354. [11] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [12] H. Goldstein, Classical Mechanics, second ed., Addison Wesley, 1980. [13] David J Griffiths, Introduction to quantum mechanics, prentice hall, New Jersey (1995). [14] Eqab M. Rabei, K.I.Nawafleh and H.B.Ghassib, Quantization of Constrained Systems Using the WKB Approximation, Phys. Rev. A 66, (2002), 24101. Abstract

0704.0527

Towards Skyrmion Stars: Large Baryon Configurations in the Einstein-Skyrme Model

Towards Skyrmion Stars: Large Baryon Configurations in the Einstein-Skyrme Model Bernard M. A. G. Piette∗ and Gavin I. Probert† November 4, 2018 Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham. DH1 3LE. UK Abstract We investigate the large baryon number sector of the Einstein-Skyrmemodel as a possible model for baryon stars. Gravitating hedgehog skyrmions have been investigated previously and the existence of stable solitonic stars excluded due to energy considerations[1]. However, in this paper we demonstrate that by generating gravitating skyrmions using rational maps, we can achieve multi-baryon bound states whilst recovering spherical symmetry in the limit where B becomes large. ∗email:[email protected] †email:[email protected] http://arxiv.org/abs/0704.0527v1 1 Introduction The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2], [3]. Skyrme’s initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number. The lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the struc- tures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5], [6], mainly with the motivation of a comparison of its features with those of other non- linear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced. Other reasons for studying the Einstein-Skyrme model are cosmological and astrophysi- cal ones. Various authors have studied black hole formation in the model, with the conclu- sion that the so-called no-hair conjecture may not hold [7], [8]. The purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1]. However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions[9] which can be well approximated by the so called rational map ansatz [10]. In this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers. To construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations. The paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU(2) hedgehogs obtained by Bizon and Chmaj [1]. We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration. 2 The Einstein-Skyrme Model The action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field. LSk − x. (1) Here LSk is the Lagrangian density for the Skyrme model defined on the manifold M : LSk = Tr(∇µU∇µU−1) + Tr[(∇µU)U−1, (∇νU)U−1]2, (2) where U belongs to SU(2). As we eventually wish to study baryon stars, we take a spheri- cally symmetric metric, such as associated with the line element = −A2(r) 1− 2m(r) 1− 2m(r) + sin ), (3) where A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem. From (3), it can be shown that the Ricci scalar is −A′′r2 − 2A′r + 2A′′rm+ A′m+ 3A′rm′ + Arm′′ + 2Am′ which, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = −m′(r) . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2πG. We note that taking Fπ = 186Mev and G = 6.72 × 10−45Mev−2, then the physical value of the coupling is α = 7.3× 10−40. As the Skyrme field is an SU(2) valued scalar field, at any given time one can think of it as a map from R3 to the SU(2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU(2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number. 3 Gravitating Hedgehog Skyrmions Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU(2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field U = exp(i−→σ .r̂F (r)) (6) subject to the boundary conditions F (r = 0) = Bπ (7) F (r = ∞) = 0 (8) where B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B2 . It appears that the existence of a critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 − 2µ(x) ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling. The major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N) > NMADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric[11] but are stable i.e. E(B = N) < N ∗E(B = 1). It was actually shown by Houghton et al [10], [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively. In the following sections we will generalise the construction of Houghton et al to approx- imate the solution of the Einstein-Skyrme model. 4 The Rational Map Ansatz The rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R3 and defining the stereographic coordinates z = tan(θ/2) expiφ the ansatz reads [10] U = exp (i~σ · n̂RF (r, t)) (9) where n̂R = 1 + |R|2 2ℜ(R), 2ℑ(R), 1− |R|2 is a unit vector where R is a rational function of z. It can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions F (r = 0) = π F (r = ∞) = 0. (11) Substituting the ansatz (9) into the action for the model and scaling to dimensionless variables as earlier, we obtain the following reduced Hamiltonian 16πFπ S(x)F (x) + Bsin2F (x)(1 + S(x)F (x)′2) Isin4F (x) where S(x) = 1− 2µ(x) From which one obtains the following field equations S(x)x F (x) + B sin2 F (x) + S(x)BF (x)′2 sin2 F (x) + I sin 4 F (x) F (x) S(x)V (x) sin 2F (x) B + S(x)BF (x)′2 + I sin 2 F (x) − αS(x)F (x) ′3V (x)2 − S(x)′F (x)′V (x)− S(x)F (x)′V (x)′ = αA(x)F (x) x+ 2B sin 2 F (x) where, for convenience, we have defined V (x) as V (x) = x + 2B sin2 F (x). (17) B is the baryon number and I = 1 1 + |z|2 1 + |R|2 2idzdz (1 + |z|2)2 Its value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I. This has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B2 . The value of I so obtained is then used as a parameter and one can solve equations (14) - (16) for the radial profiles F (x), A(x) and µ(x). We should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe[11], when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons. 5 Hollow Skyrmion Shells Using the rational map ansatz, we will now solves the field equations (14) - (16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells. The first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 2 (Fig. 2). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if Bhedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B4hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed. In Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1× 10−6. These values were obtained by direct numerical solution of equations (14) - (16), where we have used the boundary data as specified in (11). We didn’t didn’t use the physical value of α (7.3×10−40) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10−6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz. The major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars. Another interesting feature of the data is the observed change in the radius of the solu- tions with increasing baryon number. We note that the radius grows as approximately B However there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persis- tently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases. As we approach the maximum baryon charge that can exist at α = 1 × 10−6, we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion. This is a tantalising property when one considers that generally a neutron star’s radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 Figure 2: Plot of the decrease in αcrit with increasing baryon number, for configurations generated with the rational map ansatz. +: αcr for the minimum value of I; curve: αcr = 0.0404B −1/2. to support the star. To motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric. Such structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to B R( 2 ) MADM ( 2etopconv ) Smin 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1× 103 27.4314 1.0825 0.9997 1× 104 86.7192 1.0821 0.9989 1× 105 274.0397 1.0814 0.9963 1× 106 864.6968 1.0792 0.9883 1× 107 2715.0729 1.0722 0.9628 1× 108 8377.4601 1.0500 0.88192 1× 109 23585.5315 0.9743 0.6107 1.5× 109 26860.2040 0.9463 0.5020 1.8× 109 27470.2449 0.9302 0.4256 1.81× 109 27456.5804 0.9296 0.4225 1.85× 109 27357.9201 0.9274 0.4090 1.9× 109 27078.6014 0.9246 0.3886 1.95× 109 26126.5508 0.9217 0.3517 1.951× 109 26050.7695 0.9217 0.3495 1.952× 109 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1× 10−6 1210 1215 1220 1225 1230 1235 1210 1215 1220 1225 1230 1235 0.99 0.995 1.005 1.01 1210 1215 1220 1225 1230 1235 Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 106 and α = 1× 10−6. modify the boundary condition (11) to read F (r = 0) = Nπ (19) F (r = ∞) = 0 (20) whilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in[12] to construct two shell configurations for the pure Skyrme model. The baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N-layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 1200 1205 1210 1215 1220 1225 1230 1200 1205 1210 1215 1220 1225 1230 0.995 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2× 106 and α = 1× 10−6. size stacked next to each other. We can therefore think of this as a naive way of constructing a gravitating Skyrmion. Instead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the asso- ciated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius. B R( 2 ) MADM ( 2etopconv ) Smin 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1× 103 19.4157 1.0913 0.9996 1× 104 61.3207 1.0833 0.9985 1× 105 193.7006 1.0812 0.9949 1× 106 610.6271 1.0779 0.9835 1× 107 1911.3704 1.0680 0.9475 1× 108 5825.2626 1.0362 0.8325 9.0× 108 13736.9982 0.9302 0.4258 9.7× 108 13263.0853 0.9224 0.3644 9.76× 108 12998.2817 0.9217 0.3480 9.764× 108 12931.5189 0.9216 0.3444 9.7647× 108 12895.6984 0.9216 0.3425 9.76472× 108 12891.4247 0.9216 0.3423 9.764724× 108 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1× 10−6 For the remainder of this section we will restrict ourselves to the case where N = 2. Table 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10−6. Briefly, we note the main features. Firstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions. Of course the results of this section are not really the main regime of interest. We clearly need to study configurations of extremely high baryon number (of order 1058) relevant for baryon stars. We will now discuss this high baryon number regime. 6 The Ramp-profile Approximation Unfortunately, at very high baryon numbers, eqns. ((14) - (16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10−16radius, and so even double precision data types have insufficient precision. Moreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 1017. With such a large number of layers we won’t be able to solve the equation nu- merically as we will need at least 10 times as many sampling points for the profile functions. We must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers. We have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions F (x) = − (x− x0) , (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (21) µ(x) = + (x− x0) , (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (22) A(x) = (1 + A0) + (x− x0) (1−A0) , (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (23) In the above there are four free parameters, namely the central radius x0 of the shell over which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter. The picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness. The above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type G(x) sinp F (x) for any function G(x) that varies very little over the width of the shell by G(x0) sin p F (x). We then use the fact that Z x0+NW/2 x0−NW/2 F (x) = y dy. (24) This leads to the following expression for the energy: E = −16πFπ 1 +A0 1 + A0 0 −Mx0 + 1− A0 W 2x0 − MWx0 1 + A0 −W − π − 3IW 16x20 1 + A0 To find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations. First of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10−6. All the general features of the full numerical solutions are reproduced. In particular, the approximate B 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation. Quantitatively though, there are some differences. The approximation allows a signif- icant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 105 so as to ensure our approximation, B R( 2 ) W MADM ( ) Smin 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1× 103 26.313 3.1397 1.1195 0.9996 1× 104 83.206 3.1396 1.1254 0.9987 1× 105 262.94 3.1357 1.1266 0.9960 1× 106 829.60 3.1230 1.1251 0.9872 1× 107 2604.2 3.0825 1.1186 0.9595 1× 108 8032.8 2.9512 1.0972 0.8713 1× 109 22899 2.4837 1.0272 0.5772 2× 109 29121 2.1092 0.9818 0.3645 2.8× 109 29098 1.6623 0.9505 0.1380 2.83× 109 28514 1.6066 0.9495 0.1119 2.839× 109 28024 1.5671 0.94922 0.09373 2.8397× 109 27869.3 1.5556 0.94924 0.08845 2.83975× 109 27869.8 1.5524 0.94925 0.08699 2.839752× 109 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1× 10−6 B R( 2 ) W MADM ( ) Smin 100 5.7924 3.0428 1.0692 0.9999 1× 103 18.5788 3.1305 1.1047 0.9995 1× 104 58.8202 3.1380 1.1201 0.9983 1× 105 185.8420 3.1332 1.1246 0.9944 1× 106 585.7950 3.1153 1.1233 0.9820 1× 107 1833.0500 3.0578 1.1143 0.9428 1× 108 5587.3600 2.8688 1.0840 0.8172 9× 108 14147.1782 2.1859 0.9900 0.4065 9.764724× 108 14472.3851 2.1276 0.9837 0.3746 1× 109 14560.5000 2.1092 0.9818 0.3646 1.4× 109 14549.0000 1.6623 0.9505 0.1381 1.41963× 109 13994.0523 1.5644 0.9492 0.0926 1.419635134× 109 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1× 10−6 that the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%. In general then, the data seems to confirm the reliability of the ramp-profile approxi- mation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made. Moreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B− 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 1058 for α = 7.3× 10−40. Table. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the con- figuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy. The Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration. Transferring to standard units, the single layer B = 1058 gravitating Skyrmion has a radius of 2.42× 1010km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star. The results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5. We note that not only does the radius decrease significantly, but the added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close to zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km. We reemphasise here that our approach to embedding shells of baryons is quite crude. For few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. NShell R( ) W MADM/(6π 2B) Smin 1× 102 8.3236× 1027 3.1416 1.1285 1.0000 1× 103 2.6321× 1027 3.1416 1.1285 1.0000 1× 104 8.3236× 1026 3.1416 1.1285 1.0000 1× 105 2.6321× 1026 3.1416 1.1285 1.0000 1× 106 8.3236× 1025 3.1416 1.1285 1.0000 1× 107 2.6321× 1025 3.1416 1.1285 1.0000 1× 108 8.3236× 1024 3.1416 1.1285 1.0000 1× 109 2.6321× 1024 3.1415 1.1285 1.0000 1× 1010 8.3234× 1023 3.1415 1.1285 0.9999 1× 1011 2.6319× 1023 3.1412 1.1285 0.9997 1× 1012 8.3216× 1022 3.1402 1.1283 0.9991 1× 1013 2.6301× 1022 3.1373 1.1278 0.9971 1× 1014 8.3034× 1021 3.1280 1.1263 0.9907 1× 1015 2.6118× 1021 3.0986 1.1213 0.9705 1× 1016 8.1147× 1020 3.0037 1.1057 0.9063 1× 1017 2.4001× 1020 2.6810 1.0552 0.6977 5× 1017 8.2066× 1019 1.7888 0.9552 0.2036 5.3× 1017 7.4172× 1019 1.6227 0.9491 0.1247 5.33× 1017 7.1871× 1019 1.5625 0.94866 0.0971 5.3306× 1017 7.1597× 1019 1.5549 0.94868 0.0936 5.33065× 1017 7.1525× 1019 1.5528 0.948694 0.0927 5.330657× 1017 7.1506× 1019 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 1058 and α = 7.3× 10−40. 7 Conclusions Previous work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons. We also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars. Although the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale. Finally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 1058 and α = 7.3 × 10−40 we find a minimal energy single layer configuration with radius=2.42 × 1010km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling. It seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 1058 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km. We note however that this approach to shell embedding has only be done naively thus far. We have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations. 8 Acknowledgements GIP is supported by a PPARC studentship. References [1] P. Bizon & T. Chmaj “Gravitating Skyrmions” Phys. Lett. B 297 (1992), 55-62 [2] T. H. R. Skyrme, “A Non-Linear Field Theory” Proc. Roy. Soc. A 260 (1961), 127-138 [3] T. H. R. Skyrme, “A Unified Theory of Mesons and Baryons” Nucl. Phys.31 (1962) [4] E. Witten “Global Aspects of Current Algebra” Nucl. Phys. B223 (1983), 422-433 [5] N. K. Glendenning, T. Kodama & F. R. Klinkhamer “Skyrme Topological Soliton Coupled to Gravity” Phys. Rev. D 38 Number 10 (1988), 3226-3230 [6] M. S. Volkov & D. V. Gal’tsov “Gravitating Non-Abelian solitons and Black Holes with Yang-Mills Fields” Physics Reports, 319, Numbers 1-2, 1-83 (1999) [7] H. Luckock & I. Moss “Black Holes HAve Skyrmion Hair” Phys. Lett B176 (1986),341- [8] S. Droz, M. Heusler & N. Straumann “New Black Hole Solutions with Hair” Phys. Lett. B268 (1991), 371-376 [9] R.A. Battye, P.M. Sutcliffe “ MULTI - SOLITON DYNAMICS IN THE SKYRME MODEL.” Phys.Lett. B391 (1997), 150-156 [10] C. Houghton, N. Manton & P. Sutcliffe “Rational MAps, Monopoles and Skyrmions” Nucl. Phys. B510 (1998), 507-537 [11] R. A. Battye & P. M. Sutcliffe “Skyrmions, Fullerenes and Rational Maps” Rev. Math. Phys. 14 (2002), 29-86 [12] N. S. Manton & B. M. A. G. Piette “Understanding Skyrmions using Rational Maps” hep-th/0008110 Understanding Skyrmions Using Rational Maps: Proceedings of the European Congress of Mathematics, Barcelona 2000, eds. C.Casacuberta et al., Progress in Mathematics, Birkhauser, Basel Vol. 201 (2001) 469-479 http://arxiv.org/abs/hep-th/0008110 [13] V. B. Kopeliovich “The Bubbles of Matter from MultiSkyrmions” JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich “MultiSkyrmions and Baryonic Bags” J.Phys. G28 (2002), 103-120 Introduction The Einstein-Skyrme Model Gravitating Hedgehog Skyrmions The Rational Map Ansatz Hollow Skyrmion Shells The Ramp-profile Approximation Conclusions Acknowledgements

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Many-to-One Throughput Capacity of IEEE 802.11 Multi-hop Wireless Networks

Microsoft Word - Transaction _Mar 2, 2007__1.doc IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID 1 Many-to-One Throughput Capacity of IEEE 802.11 Multi-hop Wireless Networks Chi Pan Chan, Student Member, IEEE, Soung Chang Liew, Senior Member IEEE, and An Chan, Student Member, IEEE Abstract—This paper investigates the many-to-one throughput capacity (and by symmetry, one-to-many throughput capacity) of IEEE 802.11 multi-hop networks. It has generally been assumed in prior studies that the many-to-one throughput capacity is upper-bounded by the link capacity L. Throughput capacity L is not achievable under 802.11. This paper introduces the notion of “canonical networks”, which is a class of regularly-structured networks whose capacities can be analyzed more easily than unstructured networks. We show that the throughput capacity of canonical networks under 802.11 has an analytical upper bound of 3L/4 when the source nodes are two or more hops away from the sink; and simulated throughputs of 0.690L (0.740L) when the source nodes are many hops away. We conjecture that 3L/4 is also the upper bound for general networks. When all links have equal length, 2L/3 can be shown to be the upper bound for general networks. Our simulations show that 802.11 networks with random topologies operated with AODV routing can only achieve throughputs far below the upper bounds. Fortunately, by properly selecting routes near the gateway (or by properly positioning the relay nodes leading to the gateway) to fashion after the structure of canonical networks, the throughput can be improved significantly by more than 150%. Indeed, in a dense network, it is worthwhile to deactivate some of the relay nodes near the sink judiciously. Index Terms—wireless mesh networks, many-to-one, one-to-many, data-gathering networks, 802.11, sensor networks, throughput capacity, wireless multi-hop networks. —————————— —————————— 1 INTRODUCTION any-to-one communication is a common communi- cation mode in many multi-hop wireless networks. Two relevant applications are sensor networks and multi-hop wireless mesh networks. In sensor networks, there is often a “data processing center” to which data collected at distributed sensors are to be forwarded. In multi-hop wireless mesh networks, there is an Internet gateway connecting the mesh network to the core wired Internet – the client stations and the Internet gateway form a many-to-one relationship. This paper investigates the many-to-one throughput capacity of IEEE 802.11 multi-hop networks. In this set- ting, there are multiple source nodes generating traffic streams to be forwarded to a common sink node via relay nodes. The relay nodes could be sources themselves. By symmetry, the throughput capacity thus found is also the same as that in a one-to-many scenario in which a source node generates multiple distinct data streams to be for- warded to their respective sinks (note: this is not to be confused with the multicast scenario in which the same data is to be forwarded to multiple sinks). For conven- ience, we shall refer to the sink in the many-to-one sce- nario as the “center” of the network. There have been many related studies on the capacity of general wireless networks. Gupta and Kumar [1] ana- lyzed the capacity in many-to-many situation. It provides the basic model that can be adapted for use in the analysis of the many-to-one communication. As a loose bound, it is obvious that the many-to-one throughput capacity is upper-bounded by L [1]-[3], where L is the single-link throughput capacity, since this is the rate at which the sink can receive data. There is a high probability, how- ever, that the throughput capacity is lower than L for a random network [3]. This paper follows the approach used in [1]-[3] in characterizing which nodes can transmit together without packet collisions. The main difference is that here we are interested in the capacity throughput obtained under the IEEE 802.11 distributed MAC protocol [4]. Specifically, we integrate into our analysis the effects of carrier sensing, the existence of an ACK frame for each DATA frame transmission, and the distributed nature of the CSMA protocol, while [1]-[3] do not and their bounds are obtained with the implicit assumption of perfectly scheduled transmissions. There are three main contributions to this paper: 1. We introduce the notion of “canonical networks”, which is a class of regularly-structured networks whose capacities can be analyzed more easily than general unstructured networks. We find that the throughput capacity of canonical networks under 802.11 is upper bounded by 3L/4 when the source nodes are at least two hops away from the sink. We conjecture that this is also the upper bound for general networks. Indeed, when all the links in the network are of equal length, canonical networks and general networks have the same upper bound of 2L/3. xxxx-xxxx/0x/$xx.00 © 200x IEEE ———————————————— • All authors are with the Department of Information Engineering, The Chinese University of Hong Kong, New Territories, Hong Kong. E-mail: C.P. Chan : [email protected] , S. C. Liew : [email protected], A. Chan : [email protected]. Manuscript received (insert date of submission if desired). Please note that all acknowledgments should be placed at the end of the paper, before the bibliography. 2 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID 2. We find that canonical networks give much insight on how a many-to-one network should be designed in general. Our simulations show that 802.11 networks with random topologies operated with AODV routing can only achieve throughputs far below the upper bound of canonical networks. However, if we route the traffic in accordance to the optimized routes ob- tained from an optimization algorithm, the routes near the center have a structure similar to that of the opti- mal canonical network structure. In other words, as a principle, routing or network design near the center should be fashioned after the canonical network. Our further investigation shows that a “manifold” canonical network structure near the center may yield through- put improvement of more than 150% relative to that obtained by using AODV routing in a general network structure. Indeed, in a dense network, it is worthwhile to deactivate some of the relay nodes near the sink ju- diciously. 3. We find that ensuring the many-to-one network is hidden-node free (HNF) in our design leads to higher throughputs as compared to not doing so. This is in contrast to the many-to-many case, in which the large carrier-sensing range required to ensure the HNF property may lower the network throughput due to the increased exposed-node problem [5]. This observa- tion is used as a design principle in much of the study in 1 and 2 above. The rest of this paper is organized as follows. Section II provides the definitions and assumptions used in our analysis. Section III derives the throughput capacities of canonical networks, and presents simulation results to support our findings. In addition, we demonstrate the desirability of ensuring the HNF property in many-to-one networks. Section IV investigates general networks not restricted to the canonical network structure. We show that the optimal routing in general networks results in a subset of selected routes that form a structure near the center that resembles the optimal canonical network. We then apply this insight to demonstrate the desirability of designing the network according to a “manifold” canoni- cal-network structure near the center. Section V concludes this paper. 2 DEFINITIONS AND ASSUMPTIONS Let us first provide some definitions used in our analy- sis. Definition 1: The source nodes are nodes that generate data traffic. Definition 2: The sink node is the center to which the data collected at the source nodes are to be forwarded. Definition 3: The relay nodes relay data traffic from the source nodes to the sink node. Note that a node can be classified as one of the follow- ings: 1) a source node; 2) a sink node; 3) a relay node; or 4) both a source node and a relay node. Definition 4: Given a network topology, the uniform throu- ghput capacity uC with respect to a set of source nodes and a sink node is the maximum total rate at which the data can be forwarded to the sink node, with equal amount of traffic from each source node to the sink node. The throu- ghput capacity, mC , on the other hand, does not require equal amounts of traffic from sources to sink. Thus, in general, m uC C≥ . Fig. 1 shows a simple example of a network consisting of three nodes. Suppose that node 2 is the sole source node and node 1 is the relay node that forwards packets from node 2 to node 0. Node 1 does not generate traffic by itself. Then, / 2mC L= , where L is the capacity of one link. This is because node 1 cannot receive and transmit at the same time (typical assumption of half-duplexity of wireless links). Also, since there is only one source node, m uC C= . If node 1 is also a source node in addition to being a re- lay node, then mC L= (obtained when only node 1 is allowed to transmit), and 2 / 3uC L= , with nodes 1 and 2 having a throughput of L/3 each. Since node 1 needs to serve as the relay node for node 2, node 1 will need to transmit twice as often as node 2. So, proper scheduling is required. Now, if we generalize the above linear network [7] to the one consisting (n+1) nodes, in which there are n sources nodes with (n-1) of them also being relay nodes. Then, uC can be obtained as follows. Node 1 will trans- mit to node 0, the sink node, at rate uC . Node 2 will transmit to node 1 at rate ( 1) /uC n n− , and so on. In gen- eral, node (i+1) transmits to node i at rate ( ) /uC n i n− . We note that when node i transmits, nodes (i+1) and (i+2) cannot: node (i+2) cannot transmit because the reception at node (i+1) will be corrupted by the transmission by node i. So, considering transmissions of nodes 3, 2, and 1 (which form the bottleneck), we have ( 1) / .u C n i n L − + =∑ That is, /(3 3) / 3uC Ln n L= − ≈ for large n. We note that L/3 is also the mC if node n were the only source node. As a matter of fact, / 3u mC C L= = if the source nodes in the linear network were nodes i for 3i ≥ only. Thus, for reasonably large n, if the traffic from nodes 1 and 2 is only a small fraction of the total traffic to the sink, we could treat nodes 1 and 2 as pure relay, non- source, nodes. Once we do that, we then do not have to distinguish between uC and mC . We next consider a general many-to-one network, such as that in Fig. 2. For the study of many-to-one networks in this paper, we focus on the case where the source nodes are two or more hops away from the sink. This is a good approximation when the nodes within one hop to the sink only generate a small fraction of the total traffic. Definition 5: The throughput capacity with respect a multi-access protocol p (e.g., IEEE 802.11), pC , is the total rate at which the data can be forwarded to the sink nodeusing that protocol, assuming the source nodes are two or more hops away from the sink. The transmission CHAN ET AL.: MANY-TO-ONE THROUGHPUT CAPACITY OF IEEE 802.11 MULTI-HOP WIRELESS NETWORKS 3 schedule by the links is dictated by the protocol. This paper focuses on the throughput capacity under the 802.11 CSMA protocol, 802.11C . Henceforth, by throughput capacity, we mean 802.11C . For illustration, let us consider the two-chain linear topology shown in Fig. 3. Suppose that only nodes 2 and 2’ are the source nodes. Under “perfect scheduling”, nodes 1 and 2’ will transmit together; and nodes 1’ and 2 will transmit together. This results in a throughput capacity of L. Under 802.11, how- ever, the transmissions are usually not perfectly aligned in time. In addition, a DATA frame is followed by an ACK frame in the reverse direction. Suppose nodes 1 and 2’ transmit together. Say, the transmission of the DATA frame of node 1 completes first, while the transmission node 2’ is ongoing. When node 0 returns an ACK to node 1, this ACK also reaches node 1’, the receiver of the transmission from node 2’, causing a collision there. Thus, under 802.11, simultaneous transmissions by nodes 1 and 2’ will usually result in a collision unless the completion times of their DATA transmissions are perfectly aligned, which is rare. In this case, 802.11C is at best 2L/3, since at best node 2 and 2’ can transmit together, and nodes 1 and 1’ will need to transmit at separate times. For many-to-one networks, the capacity bottleneck is likely to be near the sink node because all traffic travels toward the sink node. Specifically, nodes near the sink node are responsible for forwarding more traffic, and these nodes contend for access of the wireless medium because they are close to each other. To obtain an idea on the upper limit of the throughput capacity under 802.11, we consider a class of networks referred to as the canoni- cal networks. An example of a canonical network is shown in Fig. 4. We show that 3L/4 is the upper bound of the throughput capacity of canonical networks, and conjecture that this is also the upper bound for networks with general structures. We will motivate the study of the canonical networks shortly. In the special case in which all links have equal length, then the throughput capacities of the canonical network as well as general networks are upper-bounded by 2L/3. We now define the canonical networks. Definition 6: A chain is formed by a sequence of at least three nodes leading to the center sink node. Traffic is for- warded from one node to the next node in the sequence on its way to the sink node. A linear chain is a chain which is a straight line. In Fig. 4, for example, there are eight linear chains. Definition 7: An i-hop node is a node that is i hops away from the sink node in a chain (see Fig. 4). Definition 8: A canonical network is formed by a number of linear chains leading to a common center sink node; the nodes in different chains are distinct except the sink node. In addition, the distance between an i-hop node and an (i- 1)-hop node, di, is the same for all the linear chains (see Fig. 4). Definition 9: A ring is a circle centered on the sink node. An i-hop ring consists of all the i-hop nodes of the differ- ent linear chains in a canonical network (see Fig. 4). Motivation for the Study of Canonical Networks Canonical networks have regular structures and can be analyzed more easily than general networks. We con- jecture that the upper bound of throughput capacity ob- tained for canonical networks is also the upper bound for general networks, because intuitively canonical networks model a rich class of networks the optimal of which may yield very good throughput performance. Consider the following intuitive argument. (i) In a densely populated network (say, infinitely dense), we may choose to form linear chains from the source nodes to the center sink node for routing purposes. Since the direction of traffic flow is pointed exactly to the center, there is no “wastage” Fig. 1. Simple network example. Fig. 2. A random many-to-one network. Fig. 4. A Canonical Network. Fig. 3. A two-chain many-to-one network with equal link length. Sink Source Node 0 Node 1 Node 2 Relay Sink Source Node 0 Node 1 Node 2 Relay Relay Source Node 1’ Node 2’ ring 3-hop node d0 d1 d2 2-hop node 1-hop node chain 4 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID with respect to the case in which the routing direction is at an angle to the center. (ii) We have defined the class of canonical networks to be quite general in that we do not restrict the number of linear chains in it. Neither do we limit the distance di. In deriving the capacity of the ca- nonical network later, we allow for the possibility of an infinite number of linear chains and arbitrarily small di. This provides us with a high degree of freedom in identi- fying the best-structured canonical networks. The above intuitive reasoning will be validated by simulation results later. In addition, we will show later that in a random network with many nodes (so that there is a high degree of freedom in forming routes), establishing a canonical- network-like structure near the center for routing pur- poses will generally lead to superior throughput per- formance. In this paper, unless otherwise stated, we further as- sume the following: Assumptions: (1) The nodes and links are homogenous. They are con- figured similarly, i.e., same transmission power, carrier- sensing range (CSRange), transmission rate, etc. (2) ACK is sent by the receiver when a packet is received successfully, as per the 802.11 DCF operation. (3) The following constraints apply to simultaneous transmissions [1][6]. Consider two links (T1 ,R1) and (T2 ,R2). For simultaneous transmissions without collisions, they must satisfy all the eight inequalities below: 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 T R T R R R T R T T T R R T T R T R T R R R T R T T T R R T T R X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X − > + ∆ − − > + ∆ − − > + ∆ − − > + ∆ − − > + ∆ − − > + ∆ − − > + ∆ − − > + ∆ − (1) where Xi is the location of node i, |Xi – Xj| is the distance between Xi and Xj, ∆ > 0 is the distance margin (see next paragraph). These are the physical constraints that pre- vent DATA-DATA, DATA-ACK and ACK-ACK colli- sions. The received power function can be expressed in the form of ( ) /tP d P d α∝ , (2) where Pt is the transmission power, d is the distance and α is the path-loss exponent, which typically ranges from 2 to 6 according to different environments [8]. By the as- sumptions that all the nodes have the same transmission power and α = 4, and Signal-to-Interference Ratio (SIR) requirement of 10dB. Then at R1 , we require (| - |) (| - |) P X X P X X > (3) giving | - | 10 1.78 | - | > = In other words, ∆ = 0.78. Unless otherwise stated, we as- sume ∆ = 0.78 throughout this work. (4) In 802.11 networks, there are two types of packet colli- sions: collisions due to hidden nodes (HN) (see explana- tion of assumption (5) below or [6]), and collisions due to simultaneous countdown to zero in the backoff period of the MAC of different transmitters. In much of our throu- ghput-capacity analysis, we will neglect the latter colli- sions and assume that they have only small effects toward throughput capacity, a fact which has been borne out by simulations and which can be understood through intui- tive reasoning, particularly for a network in which a node is surrounded by only a few other active nodes who may collide with it. As will be shown later in this paper, this is generally a characteristic of a network with good throu- ghput performance (see results of Fig. 14 and Fig. 18, for example). Also, an upper bound on throughput capacity obtained by neglecting the countdown collisions is still a valid upper bound. It is a good upper bound so long as it is tight. We will see later that the upper bounds we obtain are reasonably tight when verified against simulations results in which countdown collisions are taken into ac- count. In the remainder of this paper, unless otherwise stated, the term “collisions” refers to collisions due to HN (i.e., caused by the failure of carrier-sensing) rather than simultaneous countdown to zero. (5) In this paper, unless otherwise specified, we assume the so-called Hidden-Node Free Design (HFD) [6] in the network. That is, we design the network such that simul- taneous transmissions that will cause collisions can be carrier-sensed by transmitters and be avoided. A reason for this assumption is that for many-to-one communica- tion, eliminating hidden nodes is worthwhile (see simula- tion results in Section III-C). According to [6], HFD re- quires (i) Use of Receiver Restart (RS) Mode, and (ii) Sufficiently large CSRange. This paper assumes the 802.11 basic mode and RTS/CTS are not used. We briefly describe the HFD re- quirements for understanding of the analysis later. More details can be found in [6]. Fig. 5 is an example showing that no matter how large CSRange is, the hidden node (HN) phenomenon can still occur in the absence of RS. In the figure, T1 and T2 are more than CSRange apart, and so simultaneous transmissions can occur. Furthermore, the SIR is sufficient at R1 and R2 so that no “physical colli- sions” occur. But HN can still happen, as described below. Assume T1 starts first to transmit a DATA packet to R1. After the physical-layer preamble of the packet is re- ceived by R2, R2 will “capture” the packet and will not attempt to receive another new packet while T1’s DATA is ongoing. If at this time T2 starts to transmit a DATA to R2, CHAN ET AL.: MANY-TO-ONE THROUGHPUT CAPACITY OF IEEE 802.11 MULTI-HOP WIRELESS NETWORKS 5 R2 will not receive it and will not reply with an ACK to T2, causing a transmission failure on link (T2, R2). This is the default receiver mode assumed in the NS-2 simulator [10] and most 802.11 commercial products. Note that the ex- ample in Fig. 5 is independent of the size of CSRange. This HN problem can be solved with the Receiver Re- start Mode (RS) which can be enabled in some 802.11 products (e.g., Atheros Wi-Fi chips; however, the default is that this mode is not enabled). With RS, a receiver will switch to receive the stronger packet if its power is Ct times greater than the current packet (say, 10 dB higher). The example in Fig. 5 will not give rise to HN with RS if CSRange is sufficiently large. RS Mode alone, however, cannot prevent HN without sufficiently large CSRange. To see this, consider the ex- ample in Fig. 6. Assume T1 transmits a DATA to R1 first. During the DATA’s period, T2 starts to send a shorter DATA packet to R2. With RS Mode, R2 switches to receive T2’s DATA and sends an ACK after the reception. If T1’s DATA is still in progress, R2’s ACK will corrupt the DATA at R1, since the distance between R1 and R2 is within interference range ( max(1 )d+ ∆ ). To prevent T2 from transmission (hence the collision), the following must be satisfied: 1 2| - | T TX X CSRange≤ . (4) Reference [6] proved that in general if CSRange > (3+∆) dmax, where dmax is the maximum link length, then HN can be prevented in any network. However, for a specific network topology, e.g., the canonical network, the re- quired CSRange can be smaller. Throughout this work, we primarily focus on the pair- wise-interference model [1][6]. The concept of CSRange and the constraints in (1) rely on this assumption. An analysis which at the outset takes into account the simul- taneous interferences from more than one source will complicate things significantly. So, given a network to- pology, our approach is to first identify the capacity based on pair-wise interference analysis only, and then verify the capacity is still largely valid under multiple interferences (this verification is done in Section III-D). 3 CANONICAL NETWORKS In this section, we derive the throughput capacities of canonical networks. Section A analyzes two kinds of ca- nonical networks: equal link-length and variable link- length networks. Simulation results are presented and discussed in Section B. Section C compares the perform- ance of HFD and non-HFD networks, and Section D veri- fies the results under multiple interferences. 3.1 Theoretical Analysis (1) Equal Link-Length Networks We first consider the case where all links have the same length d, i.e., d0 = d1 =… =d. Theorem 1, which fol- lows from Lemma 1 and Corollary 1 below, proves that the throughput capacity in this network is upper- bounded by 2L/3, where L is the single-link throughput. Lemma 1: Given three nodes on the periphery of a circle of radius d, we can identify two nodes with distance smaller than (1+∆)d between them. Proof: The three nodes form the vertices of a triangle. Consider the equilateral triangle inscribed on the circle of radius d , and let t be the length of one side (see Fig. 7). Then 2 cos = 1.731 (1+ ) t d d d = < ∆ That is, it is not possible to inscribe a triangle with all sides no less than (1+∆)d on the circle. Corollary 1: At any time, at most two 2-hop nodes can transmit at the same time. Proof: With reference to Fig. 8, suppose that three 2-hop nodes can transmit together. In order that the ACK of any 1-hop node to not interfere with the reception of DATA packet of another transmission, the distances between the three 1-hop nodes must all be larger than (1+∆)d. By Lemma 1, this is not possible. Theorem 1: For equal-link-length canonical networks, 802.11 2 / 3C L≤ , where L is the link capacity. Proof: Define “airtime” usage of a node to include the transmission time of DATA packets as well as the ACK from the receiver [7]. Let Sij be the airtime occupied by the transmission of the i-hop node on the j-th chain over a Fig. 5. Lack of RS Mode leads to HN no matter how large CSRange and SIR are. Fig. 6. With RS Mode, CSRange not sufficiently large still leads to HN due to insufficient SIR . Fig. 7. Equilateral triangle inscribed in a circle. R1 T1 T2 R2 dmax dmax DATA DATA CSRange >(1+ ∆ )dmax T1 R2 R1 DATA ACK dmax dmax <(1+ ∆ )dmax CSRange 6 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID long time interval [0, Time]. Let S1 = the union of airtimes occupied by all 1-hop nodes S1j. Similarly, let S2 = the union of airtimes occupied by all 2-hop nodes S2j . That is, 1 11 12 1... NS S S S= ∪ ∪ ∪ and 2 21 22 2... NS S S S= ∪ ∪ ∪ . We further define xij = |Sij|/Time. By definition, 1 2| |S S Time∪ ≤ (5) According to assumption (3), when any 1-hop node trans- mits, none of the other 1-hop nodes or 2-hop nodes can transmit at the same time if collisions are not to happen. Thus, if carrier-sensing works perfectly and collisions due to simultaneous countdown to zero in the 802.11 backoff algorithm are negligible (see assumptions (4) and (5) in Section II), then 1 2S S∩ = ∅ (6) and 1 1i jS S∩ = ∅ for i ≠ j . (7) This implies 1 2 1 2| | | | | |S S S S Time+ = ∪ ≤ (8) 1 11 12 1| | | | | | ... | |NS S S S= + + + . (9) By Corollary 1, 21 22 2 | | | | ... | | NS S SS + + + ≥ . (10) Recall that we assume that the 1-hop nodes are relay nodes that do not generate data (see Definition 5 and the justification be- fore that in Section II). All traffic transmitted by 1-hop nodes must therefore come from 2-hop nodes. By the “no collision” assumption, the sum of the airtimes of 1-hop nodes must not be greater than the sum of airtimes of 2-hop nodes. We have 11 12 1 21 22 2| | | | ... | | | | | | ... | |N NS S S S S S+ + + ≤ + + + (11) From (8)-(10), we have 11 12 1 21 22 2| | | | ... | | (| | | | ... | |) / 2N NS S S S S S Time+ + + + + + + ≤ . Applying (11), we get 11 12 1 11 12 1 ( ... ) ( ... ) 1 x x x x x x + + + + + + + ≤ giving 11 12 1 x x x+ + + ≤ where 11 12 1( ... )Nx x x L+ + + is the throughput. We now show a specific schedule on a 2-chain network which achieves the capacity of 2L/3. Consider the topol- ogy shown in Fig. 9. There are two chains, having link distance d and CSRange = 2.9d, which removes HN. Recall that the general HFD has two requirements, (i) RS mode and (ii) CSRange > (3+∆) dmax [6]. For the topology in Fig. 9, it turns out that CSRange = 2.9d is enough. The numbers shown on the links in Fig. 9 represent a possible transmission schedule. Links with same number transmits at the same time. Following this pattern, the throughput capacity of 2L/3 is “potentially” achievable. Our simulation results in Subsection B below show that the 802.11 protocol throughput capacity is below but close to this upper bound. The reader may be curious as to why we did not use a “symmetric” 2-chain network (where the angle between the chains isπ ) as the illustrating example above. It turns out that the symmetric structure cannot achieve the throughput of 2L/3 if there are source nodes four or more hops away. To see this, first we note that for a symmetric 2-chain network, CSRange must be at least 3d to ensure HFD in the areas around the sink node (see discussion of the example in Fig. 3 in Section II). Given CSRange=3d, each of the chains (assuming a long chain with more than four hops (or five nodes)) cannot have throughput of L/3, as can be easily verified by analysis of one linear chain [7], [9]. Before going to the next subsection, we note that Theo- rem 1 actually applies not just to canonical networks (the proof does not require it), but general networks in which (i) all links are of the same length; and (ii) source nodes are two hops are more away from the center. In other words, the chains leading to the data center need not be straight-line linear chains. Thus, Theorem 1 can be stated more generally as Theorem 1’ below: Theorem 1’: For equal-link-length general networks, 802.11 2 / 3C L≤ , where L is the link capacity. Proof: Same as Theorem 1 since Lemma 1 and Corollary 1 apply to general networks with equal link length also. (2) Variable Link-Length Networks In this subsection, we consider canonical networks in which the distance between adjacent rings can be varied (i.e., d0 , d1 ,… may be distinct). With this assumption, the capacity is upper-bounded by 3L/4. This is proved in Fig. 8. At most two simultaneous transmissions from 2-hop nodes. Fig. 9. Example of equal-link-length topology, CSRange=2.9d. 1 2 3 O N11 N12 >(1+∆)d >(1+∆)d <(1+∆)d CHAN ET AL.: MANY-TO-ONE THROUGHPUT CAPACITY OF IEEE 802.11 MULTI-HOP WIRELESS NETWORKS 7 Theorem 2 after Lemma 2 in the following. Lemma 2: At any time, at most three 2-hop nodes can transmit at the same time. Proof: Assume the contrary that we can have four 2-hop nodes belonging to four different chains transmitting at the same time. With respect to Fig. 10, consider the four straight lines formed by the four nodes to the center (note: the network could have more chains, just that we are fo- cusing on the four chains of the four 2-hop nodes in focus here). Four angles are formed between adjacent lines. Let θ < / 2π be the minimum of the four angles. Four angles are also formed between non-adjacent lines. Let β π≤ be the angle encompassing θ (see Fig. 10). For simultaneous transmissions of 2-hop nodes, the transmitters should not be able to carrier-sense each other. This implies an upper bound for CSRange as follows: 0 12( )sin 2 CSRange d d < + . (12) In addition, by assumption (5), to prevent collisions of 1- hop nodes and 2-hop nodes, they should be able to car- rier-sense each other. This implies a lower bound for CSRange. By (4), 0 1 0 0 0 1( ) 2 ( )cosCSRange d d d d d d≥ + + − + β . (13) By assumption (3), the receivers of simultaneous transmis- sions should not violate the physical constraints. By (1), 1 0(1 ) 2 sin 2 + ∆ < . (14) Since there are four chains, / 2θ π≤ and β π≤ . From the definitions of θ and β, we have 2θ β π≤ ≤ . (15) From (13) and (15), 0 1 0 0 0 1( ) 2 ( )cos(2 )CSRange d d d d d d≥ + + − + θ . (16) Let d1= α d0. We can form two inequalities from (12), (14) and (16): 2(1 cos ) , (17) (2cos 1) 1 2cos1 2cos 1 2cos 1 2cos (2cos 1) 1 2cos1 2cos 1 2cos 1 2cos ⎧ θ − + − θ− θ α > + −⎪ − θ − θ⎪⎪ θ − + − θ− θ⎪α < − −⎪ − θ − θ⎩ . (18) Fig. 11 shows the plot of (17) and (18) when ∆ = 0.78. The shadowed region is the area of solution. From the plot, 1.73 / 2θ π> > . This leads to a contradiction. Thus, there can be at most three simultaneous 2-hop transmissions. Theorem 2: For variable-link-length canonical networks, 802.11 3 / 4C L≤ , where L is the link capacity. Proof: Similar to the proof of Theorem 1, from Lemma 2, 21 22 2 | | | | ... | | NS S SS + + + ≥ . (19) Hence, 11 12 1 11 12 1 ( ... ) ( ... ) 1 x x x x x x + + + + + + + ≤ or 11 12 1 x x x+ + + ≤ , where 11 12 1( ... )Nx x x L+ + + is the throughput Fig. 12 shows an example of a canonical network. The CSRange has to be set larger than 2.62d0 and smaller than 3.417d0. The numbers on the links show a possible trans- mission schedule that achieves capacity of 3L/4. 1 Our simulation results in Subsection B below show that 802.11 throughput capacity is below but close to this upper bound. In the analysis of canonical networks, we have as- sumed that the loss exponent is 4, corresponding to ∆ = 1 For the one-to-many network (i.e., the sink becomes the source, and the sources become the sinks with respect to the many-to-one case here), some parameters should be changed to attain the capacity of 3L/4. Spe- cifically, CSRange = 1.7d0, and di = 0.7d0 for i=1, 2, … The derivation method for the capacity of the one-to-many case is similar to that in the many-to-one case here. Fig. 11. Plot of Inequalities (17) and (18). Fig. 10. Example of 4-chain canonical network. d0 d1 d1 (12) (13) (14) 8 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID 0.78. In outdoor environment, the typical value of loss exponent is in the range 2 to 4. Similar analytical tech- nique can be used to find their throughput capacities. Since smaller loss exponent implies larger ∆ (larger inter- ference), the throughput capacity under the assumption of loss exponent 4 serves as an upper-bound for the throughput capacity in outdoor environment. 3.2 Simulation We use the network simulator NS2 [10] to simulate the canonical network shown in Fig. 12. As shown in Subsec- tion A, for the 3-chain canonical network, 802.11 3 / 4C L≤ . In the simulation, the RS Mode is enabled. Table I shows the details of the simulated configuration. Only the n-hop nodes at the boundary are source nodes that generate data. Offered load control is applied to prevent them from injecting too much traffic into the network. For the interested reader, it has been shown in [7] that offered- load control can yield higher throughput in multi-hop networks. Fig. 13 shows the simulation result assuming the set- up of Table I. The x-axis is the number of nodes per chain, including the sink. Given a number of nodes per chain, we vary the offered load in the simulation to identify an offered load that achieves the highest average throughput. When the number of nodes per chain is 3, i.e., the 2-hop nodes are the source nodes, the throughput is 4.62Mbps (0.740L), which is very close to the theoretical capacity 3L/4, where the link capacity L is around 6.24Mbps as obtained by simulating one single link. But when the number of nodes per chain increases, the throughput drops to 4.30Mbps (0.690L). An explanation for this phenomenon is that the sched- uling scheme of IEEE 802.11 does not result in the optimal transmission schedule presented in Fig. 12 needed to achieve the 3L/4 upper bound. That is, the incorporation of random backoff countdown time in 802.11 causes im- perfect scheduling. Consider Fig. 12, it is possible for 2- hop and 3-hop nodes of different chains to transmit at the same time in 802.11, since they are out of the carrier- sensing range of each other. To achieve capacity 3L/4, however, all the 2-hop nodes must transmit together. However, a 3-hop transmission may prevent this, result- ing in only some of the 2-hop nodes transmitting together. In other words, there are times when not all 2-hop nodes transmit together, meaning |S2| cannot reach the lower bound in (19). Meeting the lower bound, however, is es- sential to achieving the optimal throughput 3L/4. Fig. 14 shows the simulation results of canonical net- works with different numbers of chains but with equal link length. The simulated configuration is shown in Ta- bles II and III. For the 2-chain canonical network, we use the network structure in Fig. 9. The angle between two chains are slightly less than π . The reason of not using a symmetric structure has been given in Subsection A above. For other cases, the chains are evenly placed on the network. The CSRange for each topology is determined by minimizing its value while preventing HN. The throughput is obtained by varying the offered load and choosing the highest one. From the graph, the highest throughput is 3.86Mbps (0.619L), which is slightly smaller than the theoretical capacity of 2L/3. This is due to the imperfect scheduling by 802.11, which has been discussed in the previous paragraph. In Fig. 14, the throughput converges to around 2.0Mbps (0.321L) when the number of chains increases. The convergence can be explained as follows. From the analysis in Subsection A, we see that the bottleneck is around the center. When the number of chains is large, the area near the center will become dense. The possible transmission patterns are similar in this area, and thus the throughput converges. In addition, note that the con- verged value, 0.321L, is considerably smaller than the value achieved when the number of chains is three, 0.619L. This is again due to imperfect scheduling of 802.11 MAC protocol. An interesting insight is that when the number of chains is small, the possible transmission pat- terns arise from “random” 802.11 MAC scheduling is more limited. And by limiting this degree of freedom, higher throughput can actually be achieved because ran- dom transmission patterns that degrade throughputs are eliminated. The above observation has two implications: (i) For network design, we may want to design the network in such a way that the number of routes leading to the center is limited. (ii) Even for a general, non-canonical, network densely popu- lated with nodes and with many routes leading to the center, it is better to selectively turn on only a subset of the nodes to limit the routes to the center. This principle will be further dis- cussed in Section IV. TABLE I SIMULATION CONFIGURATION FOR VARIABLE-LINK-LENGTH CA- NONICAL NETWORKS Number of chains 3 d0 250m d1 242m di for i>1 250m Transmission Range 250m Carrier Sensing Range 675m Routing Protocol AODV Propagation Model Two Ray Ground Packet Data Size 1460 bytes Fig. 12. Example of 3-chain canonical network, CSRange=2.7d. 1.732d0 0.973d0 2.62d0 3.417d0 CHAN ET AL.: MANY-TO-ONE THROUGHPUT CAPACITY OF IEEE 802.11 MULTI-HOP WIRELESS NETWORKS 9 3 4 5 6 7 8 9 10 Number of nodes per chain Fig. 13. Simulated throughput of a 3-chain canonical network with offered load control. TABLE II SIMULATION CONFIGURATION FOR EQUAL-LINK-LENGTH CA- NONICAL NETWORKS Number of nodes per chain 8 di for all i 250m Transmission Range 250m Carrier Sensing Range Refer to Table III Routing Protocol AODV Propagation Model Two Ray Ground Packet Data Size 1460 bytes TABLE III CARRIER SENSING RANGE FOR EQUAL-LINK-LENGTH CANONICAL NETOWKRS Number of chains Carrier Sensing Range 2 725m 3 875m 4 750m 5 725m 6 875m 7 800m 8 750m 9 875m 10 825m >10 900m 2 6 10 14 18 Number of chains Fig. 14. Simulated throughput of equal-link-length canonical net- works with offered load control. 3.3 HFD versus Non-HFD Performance In the preceding sections, we have assumed HFD net- works to simplify the analysis by eliminating the effect of collision. We now investigate the performance of HFD versus that of non-HFD networks. As a reminder, HFD requires (i) Use of Receiver Restart (RS) Mode, and (ii) Sufficiently large CSRange. From [11], we know that increasing CSRange increases the number of exposed nodes (EN) and decrease the number of hidden nodes (HN), and vice versa. When HN is removed, say with HFD, the EN phenomenon will be more severe, which lowers the throughput. However, that is the case for many-to-many data delivery only. For this paper, we are interested in many-to-one data delivery. Table IV shows the simulation results with same configu- ration as in Table II with varying CSRange. The shaded entries correspond to HFD. From the table, when the number of chains is between 2 to 10, the highest through- put is achieved if we choose the smallest CSRange within HFD. This shows that the best HFD configuration gener- ally works better than non-HFD. TABLE IV SIMULATION RESULT FOR EQUAL-LINK-LENGTH CANONICAL NETWOKRS No. of Chains Through- (Mbps) 2 3 4 5 6 7 8 9 10 975 2.388 2.981 3.355 2.833 2.863 3.022 2.891 3.054 3.114 925 2.793 2.993 3.329 3.518 2.837 2.805 2.943 3.270 3.108 875 2.797 2.999 3.508 3.535 3.393 3.272 3.163 3.384 2.883 825 2.795 2.490 3.513 3.483 2.615 3.681 3.575 3.053 3.366 775 2.808 2.473 3.724 3.540 2.760 2.754 3.709 3.367 3.269 725 3.589 2.226 3.210 3.854 2.095 2.264 3.147 3.199 2.686 675 3.170 2.288 2.398 2.799 2.142 2.261 2.176 2.367 2.633 625 3.166 1.806 2.219 2.657 1.735 2.020 2.670 1.906 2.156 575 3.183 1.788 2.168 2.202 1.657 1.609 2.280 1.929 2.041 bold: highest throughput; shaded: HFD The better performance of HFD could be explained as follows. When CSRange is decreased, the number of HN increases and the number of EN decreases. More links could be active when there are fewer EN, thus the throughput in multiple-source-multiple-destination net- work could be higher in the non-HN free situation. In a many-to-one network, however, all the traffic is directed toward the same destination. With a non-HN free design, although the total throughput on a link basis (point-to- point throughput) may be increased, the many-to-one throughput (or the end-to-end throughput) could not benefit from the increase, because all the traffic in the end will flow toward the bottleneck and be dropped there due to HNs. We will see later that this observation suggests a design in which the area near the center should be made HN-free, while areas far away from the data center need not be HN-free. 3.4 Multiple Interference Thus far, we have considered pair-wise interferences only. The analysis of pair-wise interferences is appealing from the simplicity viewpoint. However, it may not have taken into account the fact that the interferences from several other simultaneously transmitting sources may add up to yield unacceptable SIR even though each of the interferences may not be detrimental. In this section, we extend our analysis to take into account the effect of mul- tiple interferences. For brevity, we refer to the throughput 10 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID capacity obtained by assuming pair-wise interferences as pair-wise-interference throughput capacity, and the throughput capacity with mul-tiple interferences taken into account as multiple-interference throughput capacity. The multiple-interference throughput capacity is in general less than or equal to that of the pair-wise throughput capacity. The question then is whether the pair-wise-inter-ference capacity is a tight bound for mul- tiple-interference capacity. We show in the following that this is indeed the case. In the following, we focus on the 3- chain network. The analytical argument and the qualita- tive results for the 2-chain network are similar. Consider the canonical network in the Fig. 15, where d0=d2=d3=d4, and d1=0.9d0. In some cases, the SIR may not satisfy the constraint 10dB. For example, when N11 is receiving DATA from N12, and at the same time N21 and N31 are replying ACK to N22 and N32, the SIR is at most 11 11 21 31 4 4 4 ( ) (0.9 ) 6.859 1 1( ) ( ) ( ) 1.7321 1.7321 P N d PP N P N where PX(Y) is the received power from node Y to node X, Pt is the transmission power. This situation, however, occurs only if multiple ACKs are transmitted simultaneous in nearby links near the center. The probability of this occurring is low, since the transmission time of ACK is much lower than that of DATA. If we ignore the simultaneous transmissions of ACKs in these nearby links, we can show that the SIR due to multiple interferences is still more than 10dB, given that the SIR due to pair-wise interferences is more than 10dB, as follows. 1. 1-hop node to sink node When the sink node is receiving DATA from N11, the nearest three active links that cause largest interference are: N23 to N22, N33 to N32 and N14 to N13. If no two ACKs are transmitted simultaneously by these three links, the “worst-case” interference power at N0 (which includes ACK from N22 DATAs from N33 and N14, and transmis- sions by other nodes) is at most Hence, the SIR is at least 1/0.09949=10.513 2. 2-hop node to 1-hop node Consider the link N12 to N11. The nearest three active links are: N22 to N21, N32 to N31, N15 to N14. Similar to above, the SIR is at least 3. 3-hop node to 2-hop node and others The interference is less than the above cases. This part is skipped because the analytical approach is similar. In the above, we have argued analytically the consid- eration of multiple interferences will not have substan- tially different performance than that of pair-wise inter- ference. We have focused on the 3-chain network with variable link distance because this structure provides the highest capacity bound among the canonical networks. We now present simulation results for general canoni- cal networks with arbitrary number of chains. We have modified the NS2 simulator to take into account the ef- fects of multiple interferences (the modified NS2 code can be downloaded from the website in [12]). The throughput results are shown in Fig. 16. The multiple-interference throughput is only lower than the pair-wise-interference throughput by a small margin, and therefore the pair- wise-interference throughput serves a good bound for multiple-interference throughput. 4 GENERAL NETWORKS In this section, we consider the throughput of general networks. Since general networks may not have the regu- lar structure of canonical networks, the throughput capac- Fig. 15. Example of 3-chain canonical network, CSRange=2.7d. Fig. 16. Simulated throughput of 3-chain canonical network with offered load control. 0 0 0 0 0 022 33 14 25 35 16 4 4 4 4 4 4 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ... 1 1 1 1 1 1 ( ...) 0.0995 1.9 2.9 3.9 4.9 4.9 5.9 N N N N N N P N P N P N P N P N P N + + + + + + = + + + + + + ≈ 11 11 11 11 11 11 N 21 N 32 N 15 N 24 N 34 N 17 4 4 4 4 4 4 4 (N ) (N ) (N ) (N ) (N ) (N ) ... (0.9 ) 1 1 1 1 1 1 ( ...) 1.7321 2.5515 3.9 4.4844 4.4844 5.9 10.5259 P P P P P P + + + + + + + + + + + + 1.732d0 0.9d0 2.55d0 3.29d0 N11 3 4 5 6 7 8 9 10 Number of nodes per chain multiple interference pairw ise interference CHAN ET AL.: MANY-TO-ONE THROUGHPUT CAPACITY OF IEEE 802.11 MULTI-HOP WIRELESS NETWORKS 11 ity could be lower than 3L/4. We propose a method to find the capacity by selecting Hidden-node Free Paths (HFP). 4.1 Discussion of HFP In Section III-C, we found that the network with HN- free outperforms that with HN in terms of throughput capacity. We could have three schemes which satisfy the HN-free condition for general network analysis. As one of the requirements of HFD, we assume RS Mode is used in all the analyses and experiments in the remaining of the paper. We assume that all nodes use a common fixed CSRange in each of the following schemes (assumption (1) in Section II); however, the schemes set the fixed CSRange differently. Scheme 1: CSRange is set to 3.78‧TxRange, where TxRange is the transmission range. This is a sufficient condition of HN free for any networks [6]. Scheme 2: CSRange is minimized according to the network topology so that no hidden node exists with respect to any two links in the network. This scheme, for example, was used in the analysis of canonical networks. Scheme 3: HFP - We select a subset of links to form paths to the center which are hidden-node free and achieve the highest possible throughput. Since some links are not used, the CSRange can be smaller than scheme 1 and 2 (i.e., only the links in the path are considered when fixing CSRange.) Based on Table IV, the highest throughput is achieved when we choose the smallest CSRange within HFD. So we have the following predictions for the throughputs of the different schemes above. The throughput of scheme 1 cannot be higher than that of scheme 2 (because the CSRange of some links are forced to adopt a higher value than necessary in scheme 1). Also, the throughput of scheme 2 cannot be higher than that of scheme 3 (because scheme 3 requires the HN property to be maintained only for links along the paths, and the paths that will be used are optimally chosen with regard to the throughput; whereas scheme 2 requires all links to be HN-free, even for links that are not used). For an example where HFP can achieve a higher throughput than scheme 2, we add two nodes to the 3-chain canonical network in Fig. 12 to yield the network in Fig. 17. In the network, link BB’ in- terfere with link AA’. If we set CSRange to be less than 3.417d0, node B will become a hidden node of link AA’. If we set CSRange larger than 3.417d0, the capacity upper- bound 3L/4 cannot be achieved. On the other hand, if we use HFP, we could select the links in the canonical net- work only. So node A could be “switched off” and there will not be hidden-node problem if we set CSRange to 2.7d0. 4.2 Experiments and Discussions To conserve space, this paper will not go into the de- tails of the formulation of the HFP problem, and the HFP experimental methodology. For the interested readers, such details can be found in the Appendix of our techni- cal report [12]. In a nutshell, our approach extends that of [13] by additionally taking into consideration the effects of carrier sensing and HFD requirements. We also pro- vide a branch-and-bound heuristic algorithm for the re- sulting integer linear program (ILP). Here we only pre- sent the performance results of experiments on schemes 1, 2, and 3 and their implications. Solving the ILP of scheme 3 is computationally intensive. The experimental results of scheme 3 in this subsection are therefore obtained us- ing our branch-and-bound heuristic. Schemes 1 and 2 are still solved in an optimal manner. As will be seen, even with a suboptimal heuristic, scheme 3 still yields better results. In our experiments, we put the nodes inside a disk of radius one. A sink node is placed at the center of the disk, and six source nodes are placed evenly at the boundary of the disk spaced evenly apart. For each source node, a node is randomly generated within the transmission range 0.4. More nodes are generated similarly with refer- ence to the newly created node until a node is within the transmission range from the sink node. In this way, we could ensure that there is a path from any source node to the sink node. By setting the transmission range to 0.4, the data from the source nodes will need at least three hops to reach the sink node. Table V shows the experiment results for five ran- domly generated networks, Net1, Net2, …, Net5 . T1, T2 and T3 are the throughputs of the three schemes. In ob- taining Ti, we vary the offered load at the source nodes until the highest throughput is obtained [7]. From Table V, scheme 3 has improvements of 4.8% to 43.8% over scheme 1, and 4.8% to 23.2% over scheme 2. As related earlier, we did not solve scheme 3 optimally, but rather used a heu- ristic. Therefore, the CSRange (CS) found for HFP in the experiments may not be the shortest possible CSRange. Nevertheless, the result shows that the solutions of scheme 3 exhibit some properties similar to the canonical network, as shown in Fig. 12. We discuss the similarities in the following paragraph. First, for scheme 3, CSRange/TxRange (CS3/TX) for Fig. 17. Example of HFP. 1.732d0 0.973d0 2.62d0 3.417d0 12 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID Net1 to Net5 is in the range of 2.62 to 3.417, which is the CSRange region we mentioned near the end of Section III- A for achieving the capacity of 3L/4 in a canonical net- work. Second, exactly three paths leading to the sink node are used, which is the same as the 3-chain canonical network (Fig. 18). This gives us an intuition that the ca- nonical network is in a sense optimal – that is, we may want to form a structure similar to the canonical network by turning on only some of the relay nodes. TABLE V RESULT FOR THROUGHPUT OF RANDOM NETWORKS T1 T2 T3 T3/T1 T3/T2 CS3 CS3/TX Net1 0.4 0.5 0.575 1.438 1.15 1.253 3.133 Net2 0.412 0.439 0.541 1.313 1.232 1.265 3.162 Net3 0.429 0.451 0.536 1.25 1.189 1.265 3.163 Net4 0.429 0.5 0.6 1.4 1.2 1.205 3.012 Net5 0.5 0.5 0.524 1.048 1.048 1.287 3.216 Neti: Network i TX: Transmission range, set to 0.4 in experiments T1: Throughput when CSRange=3.78 TX (Scheme 1) T2: Throughput when CSRange is minimized with respect to links in the network (Scheme 2). T3: Throughput when only some links in the network are activated (HFP) (Scheme 3) CS3: CSRange for Scheme 3 4.3 Applying Canonical Network to General Networks The preceding subsection shows that HFP outperforms other HN-free schemes in terms of throughput. We also observe from the results that (i) HFP solutions for a ran- dom network exhibit structures similar to that of the 3- chain canonical network near the center. Furthermore, from simulation results in Section III-B (see Fig. 13), we observe that (ii) IEEE 802.11 scheduling in the canonical network achieves throughput close to that of perfect scheduling. Observations (i) and (ii) lead to the following general engineering principle: Centric Canonical-Network Design Principle In a general multi-hop network densely populated with relay nodes, instead of solving the complex HFP optimization problem, as a heuristic, we may select routes near the center so that the structure looks like that of a 3-chain canonical network. If we have the freedom for node placement near the center during the network design process, then the nodes around the center should be structured like a 3-chain canonical network. Note that there is no restriction on nodes far away from the center, and that they can be randomly distributed (see Fig. 19 for illustration). This subsection investigates the application of the Cen- tric Canonical-Network Design Principle. For our simula- tions, we assume there is a disk with radius 2000m. Within the disk, there is an inner circle with radius 980m. As illustrated in Fig. 19, the inner circle is structured as a canonical network. The nodes outside the inner circle are placed randomly with the constraint that the smallest distance between any two of them is not shorter than 125m. The nodes outside the inner circle act as source nodes and relay nodes at the same time, while the nodes inside the inner circle act merely as relay nodes. We refer to the network structure in Fig. 19 as centric canonical net- work, alluding to the fact that only the vicinity of the cen- ter looks like a canonical network. Henceforth, we shall refer to vicinity of the center as the canonical network and the randomly-structured part beyond that as the random network. The number of nodes beyond the inner circle is 284. We use the default setting in NS2, CSRange of 550m and TXRange of 250m, for performing the simulations. AODV routing is assumed. For the canonical network, with respect to Fig.12, we set d0=200m. Since 550m/200m=2.75, which is within the range 2.62 to 3.417 (see Fig. 12), the canonical network is HN free. The ran- dom network, however, is not necessarily HN-free in our experiments. The assumption is reasonable, and corre- sponds to the real situation in which we only try to de- sign the network architecture near the center judiciously by careful node placement. As a benchmark, we have also conducted simulation experiments for a random network in which the inner circle is populated by 146 randomly placed nodes with no constraint on the node-to-node distance. In all our simula- tions below, the offered load to the source nodes are var- ied until we find the largest throughput for each network structure [7]. Simulation of 802.11 with AODV yields a Fig. 18. Random Networks and HFP.. CHAN ET AL.: MANY-TO-ONE THROUGHPUT CAPACITY OF IEEE 802.11 MULTI-HOP WIRELESS NETWORKS 13 throughput of 1.16 Mbps for the benchmark random net- work, and a throughput of 2.79Mbps for the centric ca- nonical network. That is, the throughput of the centric canonical network is more than 100% higher. This dem- onstrates that a carefully designed structured network around the data center yields superior performance. Although the improvement is significant, 2.79 Mbps is still a bit lower than the 4.30Mbps simulated throughput of the 3-chain canonical network in Section III. It turns out that the centric-canonical network actually fails to take another bottleneck into account. That is, in addition to the bottleneck around the center, there is also a bottleneck at the “confluence” of the random network and the canoni- cal network, where the canonical network may branch off to many paths in the random network, and the nodes on these branches may interfere with each other in a negative way to bring down the throughput. To mitigate the bottleneck at the confluence, we mod- ify the canonical network as in Fig. 20. As shown, each chain in the canonical network only branches out further into two chains before meeting the random network. We refer to this design as the manifold canonical network, in reference to the fact that there are actually two “layers” of canonical networks. The first one is at the center, with three more before meeting the random network. We refer to this design principle as the Manifold Canonical-Network Design Principle. In our simulations, the manifold canonical network is placed inside an inner circle of radius 1026s. The nodes beyond the manifold canonical network are randomly generated with the same constraints as the nodes gener- ated beyond the inner circle of the centric canonical net- work. As the inner circle is larger than previous networks and the number of nodes (which are relay nodes) in the manifold canonical network is 31, to keep the total num- ber of nodes in the network constant, the number of ran- domly generated nodes (which are also the source nodes) outside the inner circle is decreased from 284 to 269. We set CSRange 550m and d0=200m in the manifold canonical network in our simulation (see Fig. 12). Simulation of 802.11 with AODV routing yields a throughput of 3.34Mbps, which is 20% higher than that of the centric canonical network. For fair bench-marking, we again per- form the simulation with the inner circle replaced by ran- dom node placements, but this time with the inner circle having a radius of 1026m, as in the manifold canonical network. The simulation of the benchmark network yields a throughput of 1.31Mbps. We find that the throughput of the manifold canonical network is more than 150% over that by the pure random benchmark net- work. We have also investigated the robustness of the mani- fold canonical network with respect node positioning. Simulations show that 5% position error of the nodes in the two “layers” of the canonical network only decreases the throughput by 10% on average, as summarized in TABLE VI. TABLE VI COMPARISON OF THROUGHPUTS OF MANIFOLD CANONICAL NETWORKS WITH AND WITHOUT NODE POSITION ERROR Throughput without position error (Mbps) Throughput with position error (Mbps) Ratio 3.44 3.45 1.003 3.35 3.11 0.928 3.32 3.18 0.958 3.29 2.94 0.894 3.37 2.96 0.878 3.36 2.82 0.839 5 CONCLUSION In this paper, we have studied the throughput capacity of many-to-one multi-hop wireless networks based on the IEEE 802.11 MAC protocol. We have defined a class of canonical networks whose throughput capacity serves as a benchmark for general networks. Specifically, the throu- ghput capacity of canonical networks under 802.11 is up- per bounded by 3L/4, where L is the single-link capacity, when the source nodes are at least two hops away from the sink. If we restrict our attention to networks in which all links have the same length, the upper bound is further reduced to 2L/3. While the 3L/4 result in the previous paragraph has been established for canonical networks only, the 2L/3 result applies to general networks so long as (i) source nodes are at least two hops away from the data center; (ii) all links have the same length. Our 802.11 simulation results yield throughputs are Fig. 19. Example of a centric-canonical. Fig. 20. Example of a manifold canonical network. -2000 -1000 -2000 -1000 0 1000 2000 -2000 -1000 -2000 -1000 0 1000 2000 14 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID around 0.690L (for variable-link-length canonical net- works) and 0.619L (for equal-link-length canonical net- works) under the worse-case scenario when the source nodes are very far away and their traffic needs to go through many hops before reaching the sink node. That is, the simulated throughputs are reasonably close to the theoretical upper bounds of 3L/4 and 2L/3, respectively. This is a quite positive result considering the fact that 802.11 schedules transmissions in a rather random man- ner, while the examples we gave in Section III-A to achieve throughputs of 3L/4 and 2L/3 require very spe- cific transmission orders. The above results also imply that using variable link length is more desirable than using fixed link length. When the network is very dense (say, infinitely dense), if each node chooses a routing path with maximum hop distance in each hop, an equivalent network with fixed link length dmax, may result, where dmax is the maximum hop-distance governed by the transmit power and re- ceiver sensitivity. This max-hop-distance routing is not optimal for the many-to-one traffic pattern. This paper has considered both canonical networks with and without hidden nodes. Our results indicate that hidden-node free designs (HFD) yield higher throughput capacity. This is in contrast to the many-to-many case where HFD may not yield better throughputs [5] [6] and may actually decrease the overall system throughput. For general networks, we have used the concept of HFP (Hidden-node Free Path) to set up routes that yield optimal throughput. HFP routing, however, requires solving a complicated integer linear program, which may not be practical. Fortunately, our experimental results indicate that the routes selected by the HFP algorithm resemble the structure of the canonical network near the center. This gives rise to simple network design principles that attempt to approximate the canonical network struc- ture in the center. Specifically, we have shown that a manifold canonical network structure near the sink can yield superior throughput that is as much as 150% higher than that of a dense random network. A key insight is that in a network densely populated with nodes, deliberating turn- ing off some nodes in the area near the sink node so as to approximate the canonical network structure can actually give rise to better throughput performance. REFERENCES [1] P. Gupta and P. R. Kumar, “The Capacity of Wireless Net- works,” IEEE Transactions on Information Theory, vol. IT-46, March 2000. [2] D. Marco, E.J. Duarte-Melo, M. Liu, and D.L. Neuhoff, “On the Many-to-One Transport Capacity of a Dense Wireless Sensor Network and the Compressibility of Its Data,” IPSN 2003, pp. 1- 16, April 2003 [3] E.J. Duarte-Melo, M. Liu, “Data-Gathering Wireless Sensor Networks: Organization and Capacity,” Computer Networks, vol. 43, pp.519-537, Nov. 2003 [4] IEEE Computer Society LAN MAN Standards Committee, “Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications,” IEEE Std. 802.11, 1997 [5] L. Jiang, “Improving Capacity and Fairness by Elimination of Exposed and Hidden Nodes in 802.11 Networks,” M.Phil Thesis, The Chinese University of Hong Kong, Jun. 2005. [6] L. Jiang and S. C. Liew, “Removing Hidden Nodes in IEEE 802.11 Wireless Networks,” IEEE VTC, Sept. 2005. More com- prehensive version to appear as “Hidden-node Removal and Its Application in Cellular WiFi Networks” IEEE Trans. On Vehicu- lar Technology, Nov 2007. [7] P.C. Ng and S.C. Liew, “Offered Load Control in IEEE802.11 Multi-hop Ad-hoc Networks,” IEEE MASS, Oct. 2004. More comprehesive version to appear as “Throughput Analysis of IEEE 802.11 Multi-hop Ad hoc Networks,” IEEE/ACM Transac- tions on Networking, June 2007. [8] The Institute of Electrical and Electronics Engineers Inc. Press, “Wireless Communications Principles and Practice” [9] J. Li, C. Blake et al., “Capacity of Ad Hoc Wireless Networks,” ACM MobiCom, July 2001 [10] “The Network Simulator NS-2”, http://www.isi.edu /nsnam/ns [11] P. C. Ng, S. C. Liew, and L. Jiang, “Achieving Scalable Perform- ance in Large-Scale IEEE 802.11 Wireless Networks,” IEEE WCNC, March 2005 [12] http://www.ie.cuhk.edu.hk/soung/many_to_one, Technical Report with Appendix on HFP Algorithm and NS-2 code modeling multiple interferences. [13] K. Jain, J. Padhye et al, “Impact of Interference on Multi-hop Wire- less Network Performance”, MobiCom ’03, Sept. 2003 Chi Pan Chan received his B.Eng and M.Phil. degrees in Informa- tion Engineering from The Chinese University of Hong Kong in 2004 and 2006. His research was mainly related to capacity analysis in multi-hop wireless networks. He is now involved in the software in- dustry in the field of multimedia and networking. Soung Chang Liew received his S.B., S.M., E.E., and Ph.D. de- grees from the Massachusetts Institute of Technology. From March 1988 to July 1993, Soung was at Bellcore (now Telcordia), New Jer- sey, where he engaged in Broadband Network Research. Soung is currently Professor and Chairman of the Department of Information Engineering, the Chinese University of Hong Kong. Soung’s current research interests focus on wireless networking. Recently, Soung and his student won the best paper awards in the 1st IEEE Interna- tional Conference on Mobile Ad-hoc and Sensor Systems (IEEE MASS 2004) the 4th IEEE International Workshop on Wireless Local Network (IEEE WLN 2004). Separately, TCP Veno, a version of TCP to improve its performance over wireless networks proposed by Soung and his student, has been incorporated into a recent release of Linux OS. Publications of Soung can be found in www.ie.cuhk.edu.hk/soung. Besides academic activities, Soung is also active in the industry. He co-founded two technology start-ups in Internet Software and has been serving as consultant to many com- panies and industrial organizations. He is currently consultant for the Hong Kong Applied Science and Technology Research Institute (ASTRI), providing technical advice as well as helping to formulate R&D directions and strategies in the areas of Wireless Internetwork- ing, Applications, and Services. An Chan received the B.Eng degree in Information Engineering from The Chinese University of Hong Kong, Hong Kong in 2005. He is currently working toward a M.Phil degree in the same field at The Chinese University of Hong Kong. His research interests are in QoS over wireless network and advanced IEEE 802.11-like multi-access protocols. He is a graduate student member of IEEE.

0704.0529

Scanning Tunneling Spectroscopy in the Superconducting State and Vortex Cores of the beta-pyrochlore KOs2O6

Scanning Tunneling Spectroscopy in the Superconducting State and Vortex Cores of the β-pyrochlore KOs2O6 C. Dubois,∗ G. Santi, I. Cuttat, C. Berthod, N. Jenkins, A. P. Petrović, A. A. Manuel, and Ø. Fischer DPMC-MaNEP, Université de Genève, Quai Ernest-Ansermet 24, 1211 Genève 4, Switzerland S. M. Kazakov, Z. Bukowski, and J. Karpinski Laboratory for Solid State Physics ETHZ, CH-8093 Zürich, Switzerland (Dated: October 24, 2018) We performed the first scanning tunneling spectroscopy measurements on the pyrochlore super- conductor KOs2O6 (Tc = 9.6 K) in both zero magnetic field and the vortex state at several temper- atures above 1.95 K. This material presents atomically flat surfaces, yielding spatially homogeneous spectra which reveal fully-gapped superconductivity with a gap anisotropy of 30%. Measurements performed at fields of 2 and 6 T display a hexagonal Abrikosov flux line lattice. From the shape of the vortex cores, we extract a coherence length of 31–40 Å, in agreement with the value derived from the upper critical field Hc2. We observe a reduction in size of the vortex cores (and hence the coher- ence length) with increasing field which is consistent with the unexpectedly high and unsaturated upper critical field reported. PACS numbers: 74.70.Dd, 74.50.+r, 74.25.Qt The discovery of superconductivity in the β-pyrochlore osmate compounds AOs2O6 (A = K, Rb, Cs) [1] has high- lighted the question of the origin of superconductivity in classes of materials which possess geometrical frustra- tion [2, 3]. Interest has been predominantly focused on the highest-Tc compound KOs2O6 which presents many striking characteristics. In particular, the absence of in- version symmetry in its crystal structure [4] raises the question of its Cooper pair symmetry and the possibility of spin singlet-triplet mixing [5, 6]. The pyrochlore osmate compound KOs2O6 displays a critical temperature Tc = 9.6 K, the largest in its class of materials (CsOs2O6 and RbOs2O6 which differ only by the nature of the alkali ion have Tcs of 3.3 and 6.3 K re- spectively). Although band structure calculations show that the K ion does not influence the density of states (DOS) at the Fermi level [7, 8], it seems to affect sev- eral key properties [9]. In particular, the first order phase transition revealed by specific heat measurements in magnetic fields at the temperature Tp ≈ 7.5 K has been ascribed to a “freezing” of its rattling motion [10]. The negative curvature of the resistivity as a function of temperature also indicates a large electron-phonon scattering [11]. Specific heat measurements [12] sug- gest the coexistence of strong electron correlations and strong electron-phonon coupling, two generally antago- nistic phenomena with respect to the superconducting pairing symmetry. The nature of the symmetry remains a controversial subject in the literature. NMR [13] and µSR [14] data suggest anisotropic gap functions with nodes whereas thermal conductivity experiments [15] fa- vor a fully-gapped state. The peculiar behavior of KOs2O6 is demonstrated by its upper critical magnetic field Hc2, whose tem- perature dependence is linear down to sub-Kelvin tem- peratures and whose amplitude is above the Clogston limit [16]. One possible interpretation is the occur- rence of spin-triplet superconductivity driven by spin- orbit coupling [5, 6]. Alternatively, it has also been sug- gested that this behavior can be explained by the peculiar topology of the Fermi surface (FS) sheets of KOs2O6, assuming that superconductivity occurs mainly on the closed sheet [16]. The understanding of the physics of this compound would greatly benefit from a detailed knowledge of the local density of states (LDOS). Scanning Tunneling Spec- troscopy (STS) is an ideal tool for this, particularly since it allows one to map the vortices in real space and also access the normal state below Tc by probing their cores [17, 18, 19, 20]. In this Letter we present a detailed STS study of KOs2O6 single crystals, including the first vortex imaging in this material. The KOs2O6 single crystals were grown from Os and KO2 in oxygen-filled quartz ampoules. Their dimensions are around 0.3 × 0.3 × 0.3 mm3. The details of their chemical properties as well as their growth conditions can be found in Ref. 4. AC susceptibility measurements show a very sharp superconducting transition (∆Tc = 0.35 K). Our measurements are carried out using a home-built low temperature scanning tunneling microscope featuring a compact nanopositioning stage [21] to target the small- sized crystals. Electrochemically etched iridium tips are used for STS measurements on as-grown single crystal surfaces and the differential conductivity was measured using a standard AC lock-in technique. The surface topography of as-grown samples (Fig. 1a) reveals atomically flat regions speckled with small corru- gated islands a few Ångströms high whose spectroscopic characteristics are noisy and not superconducting (thus restraining our field of view for spectroscopic imaging). http://arxiv.org/abs/0704.0529v1 0 50 100 150 200 PSfrag replacements x (nm) Distance d (nm) Bias voltage V (mV) 100 (Å) Conductance (shifted, arb. units) −5 −4 −3 −2 −1 0 1 2 3 4 5 PSfrag replacements x (nm) y (nm) Distance d (nm) Height z (Å) Bias voltage V (mV) )(a) (b) FIG. 1: (a) Large-scale topography of KOs2O6 (T = 2 K, Rt = 60 MΩ); the box shows the measurement area for the vortex maps. (b) Spectroscopic trace along a 100 Å path taken on an atomically flat region with one spectrum every 1 Å. The spectra show raw data offset vertically for clarity (T = 2 K, Rt = 20 MΩ). The large flat regions display highly homogeneous super- conducting spectra (Fig. 1b), which were perfectly repro- ducible over the timescale of our experiments (4 months). We have checked that the spectra obtained by varying the tunnel resistance Rt all collapse onto a single curve, thus confirming true vacuum tunneling conditions. We have also verified that the numerical derivative of the tunnel current with respect to the voltage gives the same spectroscopic signature as the dI/dV lock-in signal. We stress that all measurements presented in this paper are raw data. The lack of inversion symmetry in this compound to- gether with several experimental findings raises the ques- tion of the symmetry of the gap function. In order to clarify this point, we have fitted our data to several sym- metry models, focusing on the question of the presence or absence of nodes and the amplitude of any possible gap anisotropy. We therefore considered three scenarii with an approximate angular dependence of the gap, i.e. an isotropic s-wave (∆0), a d-wave (∆ cos 2φ) with nodes and an “anisotropic” s-wave (∆0 + ∆sinφ) which has the same angular dependence as the s-p-wave singlet- triplet mixed state [6]. We do not take the real topol- ogy of the FS [7] into account, since it comprises two 3D Fermi sheets and is hence unlikely to have any sig- nificant effect on the gap structure. For an anisotropic gap, ∆(φ), the quasiparticle DOS is given by N(ω) ∝ |Re[〈(ω+iΓ)/ (ω + iΓ)2 − |∆(φ)|2〉φ]| where Γ is a phe- nomenological scattering rate. In addition, we included broadenings due to the experimental temperature and the lock-in in our fits. The results are presented in Fig. 2. The d-wave model can be rejected at this stage since its zero bias conductance (ZBC) is larger than in experi- ment (increasing Γ in the model can only increase the ZBC). The differences between symmetries appear much more clearly in the second derivative spectrum (d2I/dV 2, Fig. 2d) which is not surprising as it emphasizes the varia- tions of the DOS on a small energy scale and is very sensi- tive to the model parameters (in contrast with the dI/dV −4 −3 −2 −1 0 1 2 3 4 V (mV) Experiment anisotropic s−wave s−wave d−wave PSfrag replacements x (nm) y (nm) Distance d (nm) Height z (Å) Bias voltage V (mV) 100 (Å) Conductance (shifted, arb. units) −2 −1 0 1 2 V (mV) PSfrag replacements x (nm) y (nm) Distance d (nm) Height z (Å) Bias voltage V (mV) 100 (Å) Conductance (shifted, arb. units) −5 0 5 V (mV) 1.95 K 3.10 K 4.00 K 5.10 K 6.00 K 9.00 K 10.00 K PSfrag replacements x (nm) y (nm) Distance d (nm) Height z (Å) Bias voltage V (mV) 100 (Å) Conductance (shifted, arb. units) anisotropic (meV) s d s-wave ∆0 1.22 - 1.09 ∆ - 1.52 0.40 Γ 0.12 0 0.05 2.93 3.66 3.58 (a) (b) T = 1.95 K T = 1.95 K FIG. 2: Experimental and theoretical tunneling spectra. (a) Normalized dI/dV spectra at different temperatures from 1.95 to 10 K (spectra are offset vertically for clarity). (b) Pa- rameters for the different theoretical models. (c) Comparison of the experimental spectrum at low temperature and low en- ergy with the different theoretical models; the color codes are explained in (d). (d) Same as (c) for the second derivative d2I/dV 2. curve). The best fit is clearly given by the “anisotropic” s-wave model with an anisotropy of around 30%. With respect to the singlet-triplet mixed state, we note that we do not see any evidence in our data for a second co- herence peak arising from spin-orbit splitting. Since the 3D nature of both sheets implies that tunneling takes place in both of them, the absence of a second peak also rules out the possibility of two different isotropic gaps on separate FS sheets. Our results would however be compatible with multiband superconductivity with two (overlapping) anisotropic gaps. Finally, we see no signa- ture of a normal-normal tunneling channel in our junc- tion, suggesting that all electrons involved in the tunnel- ing process come from the superconducting condensate. To investigate the temperature evolution of the quasi- particle DOS, we acquired tunneling conductance spec- tra at different temperatures between 1.95 K and 10 K (Fig. 2a). The closure of the gap at the bulk Tc shows that we are probing the bulk properties of KOs2O6. This −6 −4 −2 0 2 4 6 bias voltage V (mV) −6 −4 −2 0 2 4 6 bias voltage V (mV) (a) (b)H = 2 T H = 6 T FIG. 3: Spectroscopic traces at T = 2 K across vortices for a field of 2 T (a) and 6 T (b). The spectra at the vortex centers are highlighted in red. The spatial variation of the conductance is shown in the corresponding insets. is further confirmed by the fact that similar spectra were also obtained on freshly cleaved surfaces. The totally flat conductance spectra at higher temperature show no support for a pseudogap in the DOS above Tc, imply- ing that the steep decrease in the 1/(T1T ) curve around 16 K in NMR data [13] must have a different origin. The spectra taken between 6 and 9 K (not shown) were very noisy. This could be explained by the proximity to the first order transition at Tp ≃ 7.5 K [10]. The BCS coupling ratio 2∆max/kBTc inferred from our measured gaps and critical temperature is about 3.6 for the anisotropic s-wave case, a value slightly smaller than that reported from specific heat measurements [12]. However, we stress that STS is a direct probe of the su- perconducting gap. Our findings therefore lead us to the conclusion that KOs2O6 is fully gapped with a significant anisotropy of around 30%. We now focus on measurements performed in an ap- plied magnetic field. In the vortex cores whose radial size is roughly given by the coherence length ξ, superconduc- tivity is suppressed leading to a drastic change in the LDOS which can be measured by STM. Our measure- ments were performed for two fields, 2 and 6 T, over the particularly flat region of about 60 × 60 nm2 (Fig. 1a). Each measurement was taken at 2 K with a typical ac- quisition time of 40 hours. The results are presented in Figs 3 and 4. The vor- tex maps (insets of Fig. 3 and Figs 4a and 4b) show the ZBC normalized to the conductance at 6 meV. Fig. 3 displays the spectra taken along traces passing through vortex cores for each of the two fields considered. The suppression of superconductivity and its effect on the conductance in a vortex core can clearly be seen. The vortex maps show a roughly hexagonal vortex lattice with vortex spacings d = 352 ± 17 Å and 216 ± 21 Å at 2 and 6 T respectively, in agreement with the spacings 2Φ0/H expected for an Abrikosov hexago- nal lattice [22], i.e. 345 Å and 199 Å. We ascribe the variations in the core shapes and the deviation from a perfectly hexagonal lattice to vortex pinning. In partic- ular, the vortex identified by the arrow in Fig. 4 appears to be split. We attribute this to the vortex oscillating between two pinning centers during the measurement, a situation which has been seen in other compounds [23]. One should also note that the islands (surface defects) at the border of the measurement area (Fig. 1) could influence the vortex core shapes and positions. In order to estimate the coherence length ξ from our measurements, we now consider the spatial dependence of the ZBC. Due to the proximity of the vortices, we model the LDOS as a superposition of isolated vortex LDOS which can be expressed as N(ω, r) = n |un(r)| δ(ω − En) + |vn(r)|2 δ(ω + En), where ψn(r) = (un(r), vn(r)) is the wave function of the nth vortex core state and En its energy. An approximate solution for the iso- lated vortex was given long ago [24] in which the ra- dial dependence of each ψn(r) consists of a rapidly os- cillating n-dependent Bessel function multiplied by a cosh−1/π(r/ξ) envelope common to all states. We there- fore construct a phenomenological model for our 2D ZBC maps, σ(ω = 0, r) ∝ N(ω = 0, r), by retaining the slowly varying parts of the wave functions alone, i.e. σ(ω = 0, r) = σ0 + Λ |r − ri| where σ0 = 0.13 is the residual normalized conductance at zero bias in the absence of field (Fig. 2c), Λ a scaling factor, ξ the coherence length and the sum runs over all the vortices with positions ri in the map. Using (1), we fitted ri and ξ over the entire map for each field, thus considering all imaged vortices to determine ξ. The results from the 2D fits are presented in Fig. 4c and d in map format and along traces selected to pass through vortex cores in Fig. 4e and f. The traces help to visualize the spatial extent of the vortices and assess the (extremely high) quality of the 2D fits. We first ob- serve that the normalized ZBC between the vortices is slightly enhanced at H = 2 T but increases strongly at H = 6 T with respect to the value at zero-field (Fig. 2c), indicating a significant core overlap. From our data taken at T = 2 K, we obtain ξ = 35 ± 3 Å and 45 ± 7 Å at H = 6 and 2 T respectively (the uncertainties are esti- mated from the spread of the results obtained on several maps: two for 6 T and three for 2 T). Using Ginzburg- Landau theory, we extrapolate the corresponding T = 0 values as ξ = 31± 3 and 40± 6 Å respectively, consistent with the value derived from Hc2. Furthermore, our re- sults indicate that the vortex size decreases with increas- ing field and, although at the limit of the error bars, we believe this trend to be genuine. In addition, this finding is consistent with the abnormally large Hc2: if the vor- tices become smaller as the field increases, the material can accommodate more vortices before the breakdown 0 20 40 60 Distance (nm) 0 20 40 60 Distance (nm) ) (e) x (nm) 0 10 20 30 40 50 x (nm) 0 10 20 30 40 50 60 60 (c) H = 2 T H = 6 T 0 0.2 0.4 0.6 0.8 1 FIG. 4: (a), (b) Experimental ZBC maps (T = 2 K) normal- ized to the background conductance at 2 and 6 T respectively with corresponding fits (c), (d); large values (red) correspond to normal regions (i.e. vortex cores) and low values (blue) to superconducting (gapped) regions. (e), (f) Experimental ZBC profiles across vortex centers together with the corresponding profiles from the 2D fits (red lines). of superconductivity, leading to a higher upper critical field. This correlates with the observed temperature de- pendence of the upper critical field. We find that the spectra at the vortex centers are flat for both fields (Fig. 3), showing the presence of localized quasiparticle states in the vortex cores. However, our spectra show no excess spectral weight at or close to zero bias and thus no ZBCP which is the generally expected signature of vortex core states. The absence of a ZBCP is at first glance striking considering the large mean free path ℓ ≈ 200 nm ≫ ξ in KOs2O6 [15]. In fact, this ab- sence is common to many non-cuprate superconductors, the only known exceptions being 2H-NbSe2 [17, 25, 26] and YNi2B2C [27]. Although no definitive theory cur- rently exists to explain such an absence, a possible ex- planation assumes that the scattering rate is strongly en- hanced in the vortex cores. This interpretation is sup- ported by our numerical solutions of the Bogoliubov-de Gennes equations for a single vortex with an r-dependent scattering rate Γ. Furthermore, these simulations show a radial dependence of the LDOS which is fully consistent with (1). In conclusion, we have presented the first scanning tun- neling spectroscopic measurements on superconducting KOs2O6. The fitted spectra demonstrate that KOs2O6 is a fully-gapped superconductor with an anisotropy of around 30%, possibly resulting from a s-p singlet-triplet mixed state allowed by the lack of inversion symme- try. We have imaged hexagonal vortex lattices matching Abrikosov’s prediction for 2 and 6 T fields. Using Caroli- de Gennes-Matricon theory we extract a field-dependent coherence length of 31–40 Å, in good agreement with the thermodynamic estimate fromHc2. The absence of a zero bias conductance peak, the apparent field dependence of ξ and the precise radial dependence of the LDOS all call for deeper exploration. We acknowledge T. Jarlborg, M. Decroux, I. Maggio- Aprile and P. Legendre for valuable discussions and thank P.E. Bisson, L. Stark and M. Lancon for technical sup- port. This work was supported by the Swiss National Science Foundation through the NCCR MaNEP. ∗ Electronic address: [email protected] [1] S. Yonezawa, Y. Muraoka, Y. Matsushita and Z. Hiroi, J. Phys.:Condens. Matter 16, L9 (2004); ibid, J. Phys. Soc. Jpn 73, 819 (2004); S. Yonezawa, Y. Muraoka and Z. Hiroi, J. Phys. Soc. Jpn 73, 1655 (2004). [2] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973). [3] H. Aoki, J. Phys.: Condens. Matter 16, V1 (2004). [4] G. Schuck, S. Kazakov, K. Rogacki, N. Zhigadlo, and J. Karpinski, Phys. Rev. B 73, 144506 (2006). [5] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004); ibid, Phys. Rev. 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[15] Y. Kasahara, Y. Shimono, T. Shibauchi, Y. Matsuda, S. Yonezawa, Y. Muraoka, and Z. Hiroi, Phys. Rev. Lett. 96, 247004 (2006). [16] T. Shibauchi, L. Krusin-Elbaum, Y. Kasahara, Y. Shi- mono, Y. Matsuda, R. D. McDonald, C. H. Mielke, S. Yonezawa, Z. Hiroi, M. Arai, et al., Phys. Rev. B 74, 220506 (2006). [17] H. Hess, R. Robinson, R. Dynes, J. J. Valles, , and J. Waszczak, Phys. Rev. Lett. 62, 214 (1989). [18] Y. DeWilde, M. Iavarone, U. Welp, V. Metlushko, A. Koshelev, I. Aranson, G. Crabtree, and P. Canfield, Phys. Rev. Lett. 78, 4273 (1997). [19] M. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. Kazakov, J. Karpinski, and Ø. Fischer, Phys. Rev. Lett. 89, 187003 (2002). [20] N. Bergeal, V. Dubost, Y. Noat, W. Sacks, D. Roditchev, N. Emery, C. Hérold, J.-F. Marêché, P. Lagrange, and G. Loupias, Phys. Rev. Lett. 97, 077003 (2006). [21] C. Dubois, P. E. Bisson, S. Reymond, A. A. Manuel, and Ø. Fischer, Rev. Sci. Instrum. 77, 043712 (2006). [22] A. A. Abrikosov, Sov. 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0704.0530

Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions

arXiv:0704.0530v2 [hep-th] 14 Jun 2007 ITP–UH–09/07 Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions Olaf Lechtenfeld1 and Alexander D. Popov1,2 1Institut für Theoretische Physik, Leibniz Universität Hannover Appelstraße 2, 30167 Hannover, Germany 2Bogoliubov Laboratory of Theoretical Physics, JINR 141980 Dubna, Moscow Region, Russia Email: lechtenf, popov @itp.uni-hannover.de Abstract We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncom- mutative R2,1 and analyze some of their properties. http://arxiv.org/abs/0704.0530v2 1 Introduction In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4]. In particular, if the open string has N=2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5]. Furthermore, open N=2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7]. The topological nature of N=2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable.1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R2. These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11]. In fact, Derrick’s theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13]! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16]. In [12, 13] and [17]–[20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21]. The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16]. Recall that the critical N=2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8], which gets deformed noncommutatively in the presence of a B-field [5]. Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N=2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N=4 topological string description [23]. It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space2 [26] CP 3 and the topological N=4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N=2 open SFT in [27, 28]. However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31]) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions? 1For discussing some other noncommutative integrable models see e.g. [9, 10] and references therein. 2For reviews of twistor theory see, e.g., the books [24, 25]. There are some proposals [33, 34, 35, 36] for extending N=2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N=4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space.3 Note that N<4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N=4 SDYM can be considered as a truncation of the full N=4 super Yang-Mills theory [37]. It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N=2 (and topological N=4) string theory which also leads to the N=4 SDYM model. A dimensional reduction of the above relations between twistor strings and N=4 super Yang- Mills and SDYM models was considered in [44, 45, 46, 47]. The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P2|4. In [47] it was shown that the 2N=8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N=8 supersymmetric modified U(n) chiral model on R2,1. The subject of the current paper is an 2N≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (− + +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory. 2 Supersymmetric Bogomolny model in 2+1 dimensions 2.1 N -extended SDYM equations in 2+2 dimensions Space R2,2. Let us consider the four-dimensional space R2,2 = (R4, g) with the metric ds2 = gµνdx µdxν = det(dxαα̇) = dx11̇dx22̇ − dx21̇dx12̇ (2.1) with (gµν) = diag(−1,+1,+1,−1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α̇ = 1̇, 2̇ are spinor indices. We choose the coordinates4 (xµ) = (xa, t̃) = (t, x, y, t̃) with a, b, . . . = 1, 2, 3 , (2.2) and the signature (− ++−) allows us to introduce real isotropic coordinates (cf. [19, 6]) x11̇ = 1 (t− y) , x12̇ = 1 (x+ t̃) , x21̇ = 1 (x− t̃) , x22̇ = 1 (t+ y) . (2.3) SDYM. Recall that the SDYM equations for a field strength tensor Fµν on R 2,2 read εµνρσF ρσ = Fµν , (2.4) 3For other variants of twistor string models see [38, 39, 40]. For recent reviews providing a twistor description of super Yang-Mills theory, see [41, 42] and references therein. 4Our conventions are chosen to match those of [12] after reduction to the space R2,1 with coordinates (t, x, y). where εµνρσ is a completely antisymmetric tensor on R 2,2 and ε1234 = 1. In the coordinates (2.3) we have the decomposition αα̇,ββ̇ = ∂αα̇Aββ̇ − ∂ββ̇Aαα̇ + [Aαα̇, Aββ̇ ] = εαβ Fα̇β̇ + εα̇β̇ Fαβ (2.5) := −1 αα̇,ββ̇ and Fαβ := −12ε α̇β̇F αα̇,ββ̇ , (2.6) where εαβ is antisymmetric, εαβε βγ = δ α, and similar for ε α̇β̇, with ε12 = ε1̇2̇ = 1. The gauge potential (Aαα̇) will appear in the covariant derivative , · ] . (2.7) In spinor notation, (2.4) is equivalently written as = 0 ⇔ F αα̇,ββ̇ Fαβ . (2.8) Solutions {Aαα̇} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R2,2. N -extended SDYM in component fields. The field content of N -extended super SDYM is5 N = 0 Aαα̇ (2.9a) N = 1 Aαα̇, χiα with i = 1 (2.9b) N = 2 Aαα̇, χiα, φ[ij] with i, j = 1, 2 (2.9c) N = 3 Aαα̇, χiα, φ[ij], χ̃ [ijk] with i, j, k = 1, 2, 3 (2.9d) N = 4 Aαα̇, χiα, φ[ij], χ̃ [ijk] [ijkl] with i, j, k, l = 1, 2, 3, 4 . (2.9e) Here (Aαα̇, χ [ij], χ̃ [ijk] [ijkl] ) are fields of helicities (+1,+1 , 0,−1 ,−1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37] = 0 , (2.10a) Dαα̇χ iα = 0 , (2.10b) Dαα̇D αα̇φij + 2{χiα, χjα} = 0 , (2.10c) Dαα̇χ̃ α̇[ijk] − 6[χ[iα, φjk]] = 0 , (2.10d) D γ̇α G [ijkl] + 12{χ[iα, χ̃ } − 18[φ[ij ,D φkl]] = 0 . (2.10e) Note that the N < 4 SDYM field equations are governed by the first N+1 equations of (2.10), where F = 0 is counted as one equation and so on. 5We use symmetrization (·) and antisymmetrization [·] of k indices with weight 1 , e.g. [ij] = 1 (ij − ji). 2.2 Superfield formulation of N -extended SDYM Superspace R4|4N . Recall that in the space R2,2 = (R4, g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors6 θα and ηα̇ of helicities +1 and −1 as anti- commuting (Grassmann-algebra) objects. Using 2N such spinors with components θiα and ηα̇i for i = 1, . . . ,N , one can define the N -extended superspace R4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators Pαα̇ = ∂αα̇ , Qiα = ∂iα − ηα̇i ∂αα̇ and Qiα̇ = ∂iα̇ − θiα∂αα̇ , (2.11) where ∂αα̇ := ∂xαα̇ , ∂iα := and ∂iα̇ := ∂ηα̇i . (2.12) The commutation relations for the generators (2.11) read {Qiα, Qjα̇} = −2δ iPαα̇ , [Pαα̇, Qiβ ] = 0 and [Pαα̇, Q ] = 0 . (2.13) To rewrite equations of motion in terms of R4|4N superfields one uses the additional operators Diα = ∂iα + η i ∂αα̇ and D α̇ = ∂ α̇ + θ iα∂αα̇ , (2.14) which (anti)commute with the operators (2.11) and satisfy {Diα,Dj } = 2δjiPαβ̇ , [Pαα̇,Diβ ] = 0 and [Pαα̇,D ] = 0 . (2.15) Antichiral superspace R4|2N . On the superspace R4|4N one may introduce tensor fields de- pending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives LX etc.. Furthermore, on any such superfield A one can impose the constraint equations LDiαA = 0, which for a scalar superfield f reduce to the so-called antichirality conditions Diαf = 0 . (2.16) These are easily solved by using a coordinate transformation on R4|4N , (xαα̇, ηα̇i , θ iα) → (x̃αα̇ = xαα̇−θiαηα̇i , ηα̇i , θiα) , (2.17) under which ∂αα̇,Diα and D α̇ transform to the operators ∂̃αα̇ = ∂αα̇ , D̃iα = ∂iα and D̃ α̇ = ∂ α̇ + 2θ iα∂αα̇ . (2.18) Then (2.16) simply means that f is defined on a sub-superspace R4|2N ⊂ R4|4N with coordinates x̃αα̇ and ηα̇i . (2.19) This space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace. 6Note that in Minkowski signature the Weyl spinor θα is complex and ηα̇ = εα̇β̇η β̇ = θα is complex conjugate to θα. For the Kleinian (split) signature 2 + 2, however, these spinors are real and independent of one another. N -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R4|2N [33, 48]. Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields Aαα̇ and Aiα̇ depending on xαα̇, ηα̇i ∈ R4|2N and giving rise to covariant derivatives ∇αα̇ := ∂αα̇ +Aαα̇ and ∇iα̇ := ∂iα̇ +Aiα̇ . (2.20) In such terms the N -extended SDYM equations (2.10) read [∇αα̇,∇ββ̇] + [∇αβ̇ ,∇βα̇] = 0 , [∇ α̇,∇ββ̇ ] + [∇ ,∇βα̇] = 0 , {∇iα̇,∇ }+ {∇i } = 0 , (2.21) which is equivalent to [∇αα̇,∇ββ̇] = εα̇β̇ Fαβ , [∇ α̇,∇ββ̇ ] = εα̇β̇ F β and {∇iα̇,∇ } = ε F ij , (2.22) where F ij is antisymmetric and Fαβ is symmetric in their indices. The above gauge potential superfields (Aαα̇, Aiα̇) as well as the gauge strength superfields (Fαβ , F iα, F ij) contain all physical component fields of theN -extended SDYMmodel. For instance, the lowest component of the triple (Fαβ , F iα, F ij) in an η-expansion is (Fαβ , χiα, φij), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields. It is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, D̃iα̇ = ∂ α̇ + 2 θ iα ∂αα̇ + + + Ãiα̇ := Aiα̇ + 2 θiαAαα̇ ‖ ‖ ‖ ∇̃iα̇ := ∇iα̇ + 2 θiα∇αα̇ (2.23) where ∇αα̇, ∇iα̇ and D̃iα̇ are given by (2.20) and (2.18), while Aiα̇ and Aαα̇ depend on xαα̇ and ηα̇i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single {∇̃iα̇, ∇̃ } + {∇̃i , ∇̃j } = 0 ⇔ {∇̃iα̇, ∇̃ } = ε F̃ ij , (2.24) with F̃ ij = F ij + 4 θ[iαF j]α + 4 θiαθjβFαβ . The concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21). Linear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations ζ α̇(∂αα̇ +Aαα̇)ψ = 0 and ζ α̇(∂iα̇ +Aiα̇)ψ = 0 , (2.25) where (ζ and ζ α̇ = εα̇β̇ζ . The extra (spectral) parameter7 ζ lies in the extended complex plane C∪∞ = CP 1. Here ψ is a matrix-valued function depending not only on xαα̇ and ηα̇i but also (meromorphically) on ζ ∈ CP 1. We subject the n×n matrix ψ to the following reality condition: ψ(xαα̇, ηα̇i , ζ) ψ(xαα̇, ηα̇i , ζ̄) = 1l , (2.26) 7The parameter ζ is related with λ used in [45] by the formula ζ = i 1−λ (cf. e.g. [31]). where “†” denotes hermitian conjugation and ζ̄ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as ζ α̇(∇iα̇ + 2θiα∇αα̇)ψ = 0 ⇔ ζ α̇(D̃iα̇ + Ãiα̇)ψ = 0 ⇔ ζ α̇ ∇̃iα̇ ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions The supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R2,2 → R2,1. In particular, for the N=0 sector we demand the components Aµ of a gauge potential to be independent of x4 and put A4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R2,2 by imposing the ∂4-invariance condition on all the fields (Aαα̇, χ [ij], χ̃ [ijk] [ijkl] ) from the N=4 supermultiplet or its truncation to N<4 and obtain supersymmetric Bogomolny-type equations on R2,1. Spinors in R2,1. Recall that on R2,2 both N=4 SDYM theory and full N=4 super Yang- Mills theory have an SL(4, R) ∼= Spin(3,3) R-symmetry group [33]. A dimensional reduction to 2,1 enlarges the supersymmetry and R-symmetry to 2N=8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2) ∼= Spin(2,1)L×Spin(2,1)R of R2,2 to its diagonal subgroup Spin(2,1)D as the local rotation group of R2,1, the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth. Coordinates and derivatives in R2,1. The above discussion implies that one can relabel the bosonic coordinates xαβ̇ from (2.3) by xαβ and split them as xαβ = 1 (xαβ + xβα) + 1 (xαβ − xβα) = x(αβ) + x[αβ] (2.28) into antisymmetric and symmetric parts, x[αβ] = 1 εαβx4 = 1 εαβ t̃ and x(αβ) =: yαβ , (2.29) respectively, with y11 = x11 = 1 (t− y) , y12 = 1 (x12 + x21) = 1 x , y22 = x22 = 1 (t+ y) . (2.30) We also have θiα 7→ θiα and ηα̇i 7→ ηαi for the fermionic coordinates on R4|4N reduced to R3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators ∂(αβ) = (∂αβ + ∂βα) (2.31) which read explicitly as ∂(11) = = ∂t−∂y , ∂(12) = ∂(21) = 12 = ∂x , ∂(22) = = ∂t+∂y . (2.32) We thus have = ∂(αβ) − εαβ∂4 = ∂(αβ) − εαβ∂t̃ , (2.33) where ε12 = −ε21 = −1, ∂4 = ∂/∂x4 and ∂t̃ = ∂/∂t̃. The operators Diα and D α̇ acting on t̃-independent superfields reduce to Diα = ∂iα + η i ∂(αβ) and D α = ∂ α + θ iβ∂(αβ) , (2.34) where ∂iα = ∂/∂θ iα and ∂iα = ∂/∂η i . Similarly, the antichiral operators D̃iα and D̃ α̇ in (2.18) become D̂iα = ∂iα and D̂ α = ∂ α + 2θ iβ∂(αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A of a gauge potential in four dimensions split into the components A(αβ) of a gauge potential in three dimensions and a Higgs field A[αβ] = −εαβ ϕ, i.e. Aαβ = A(αβ) +A[αβ] = A(αβ) − εαβ ϕ . (2.36) Then the covariant derivatives D reduced to three dimensions become the differential operators Dαβ − εαβ ϕ = ∂(αβ) + [A(αβ), · ]− εαβ [ϕ, · ] , (2.37) and the Yang-Mills field strength on R2,1 decomposes as Fαβ, γδ = [Dαβ , Dγδ] = εαγ fβδ + εβδ fαγ with fαβ = fβα . (2.38) Substituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x4 = t̃, we obtain the following supersymmetric Bogomolny-type equations on R2,1: fαβ +Dαβϕ = 0 , (2.39a) Dαβ χ iβ + εαβ [ϕ, χ iβ] = 0 , (2.39b) Dαβ D αβφij + 2[ϕ, [ϕ, φij ]] + 2{χiα, χjα} = 0 , (2.39c) Dαβ χ̃ β[ijk] − εαβ [ϕ, χ̃β[ijk]]− 6[χ[iα, φjk]] = 0 , (2.39d) [ijkl] + [ϕ,G [ijkl] ] + 12{χ[iα, χ̃jkl]β } − 18[φ[ij ,Dαβφkl]]− 18εαβ [φ[ij , [φkl], ϕ]] = 0 .(2.39e) Supersymmetric Bogomolny-type equations in terms of superfields. Translations gen- erated by the vector field ∂4 = ∂t̃ are isometries of superspaces R 4|4N and R4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂4, we obtain the reduced full superspace R3|4N ∼= R4|4N /G and the reduced antichiral superspace R3|2N ∼= R4|2N/G. In the following, we shall work on R3|2N and R3|2N × CP 1, since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space. The linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t̃-independence on all fields in the linear system (2.27), we arrive at the linear equations ζα ∇̂iα ψ ≡ ζα(D̂iα + Âiα)ψ = 0 (2.40) of the same form but with D̂iα = ∂ α + 2θ iβ∂(αβ) and Âiα = Aiα + 2θiβ(A(αβ) − εαβΞ) , (2.41) where Aiα, A(αβ) and Ξ are superfields depending on yαβ and ηαi only. These linear equations expand again to the pair (cf. (2.25)) ζβ(∂(αβ) +A(αβ) − εαβΞ)ψ = 0 and ζα(∂iα +Aiα)ψ = 0 . (2.42) The compatibility conditions for the linear system (2.40) read {∇̂iα, ∇̂ } + {∇̂iβ, ∇̂jα} = 0 ⇔ {∇̂iα, ∇̂ } = εαβ F̂ ij (2.43) and present a condensed form of (2.39) rewritten in terms of R3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent. 3 Noncommutative N -extended U(n) chiral model in 2+1 dimensions As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentz- invariant or integrable but not both [7, 11]. We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7]. Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9]–[20]) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R2,1 can always be restored by taking the commutative limit θ → 0. Star-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism8. The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product (f ⋆ g)(x) = f(x) exp{ i ab −→∂b} g(x) ⇒ xa ⋆ xb − xb ⋆ xa = iθab (3.1) with a constant antisymmetric tensor θab. Specializing to R2,1, we use real coordinates (xa) = (t, x, y) in which the Minkowski metric g on R3 reads (gab) = diag(−1,+1,+1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the “bosonic” Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θiα and ηαi . For later use we consider not only isotropic coordinates and vector fields u := 1 (t+y) = y22 , v := 1 (t−y) = y11 , ∂u = ∂t + ∂y = ∂(22) , ∂v = ∂t − ∂y = ∂(11) (3.2) 8See [50] for reviews on noncommutative field theories. introduced in Section 2, but also the complex combinations z := x+ iy , z̄ := x− iy , ∂z = 12 (∂x − i∂y) , ∂z̄ = (∂x + i∂y) . (3.3) Since the time coordinate t remains commutative, the only nonvanishing component of the non- commutativity tensor θab is θxy = −θyx =: θ > 0 ⇒ θzz̄ = −θz̄z = −2i θ . (3.4) Hence, we have z ⋆ z̄ = zz̄ + θ and z̄ ⋆ z = zz̄ − θ (3.5) as examples of the general formula (3.1). Operator formalism. The nonlocality of the star products renders explicit computation cum- bersome. We therefore pass to the operator formalism, which trades the star product for operator- valued spatial coordinates (x̂, ŷ) or their complex combinations (ẑ, ˆ̄z), subject to [t, x̂] = [t, ŷ] = 0 but [x̂, ŷ] = iθ ⇒ [ẑ, ˆ̄z] = 2 θ . (3.6) The latter equation suggests the introduction of annihilation and creation operators, ẑ and a† = ˆ̄z with [a , a†] = 1 , (3.7) which act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ〉, ℓ = 0, 1, 2, . . .} such that a |ℓ〉 = ℓ |ℓ−1〉 and a† |ℓ〉 = ℓ+1 |ℓ+1〉 . (3.8) Any superfield f(t, z, z̄, ηαi ) on R 3|2N can be related to an operator-valued superfield f̂(t, ηαi ) ≡ F (t, a, a†, ηαi ) on R 1|2N acting in H, with the help of the Moyal-Weyl map f(t, z, z̄, ηαi ) 7→ f̂(t, ηαi ) = Weyl-ordered f 2θa†, ηαi . (3.9) The inverse transformation recovers the ordinary superfield, f̂(t, ηαi ) ≡ F (t, a, a†, ηαi ) 7→ f(t, z, z̄, ηαi ) = F⋆ t, z√ , z̄√ , ηαi , (3.10) where F⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have f ⋆ g 7→ f̂ ĝ and dx dy f = 2π θTrf̂ = 2π θ 〈ℓ|f̂ |ℓ〉 , (3.11) and the spatial derivatives are mapped into commutators, ∂zf 7→ ∂̂z f̂ = − 1√ [a†, f̂ ] and ∂z̄f 7→ ∂̂z̄ f̂ = 1√ [a , f̂ ] . (3.12) For notational simplicity we will from now on omit the hats over the operators except when con- fusion may arise. Gauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation ψ 7→ ψ′ = g−1ψ , (3.13a) A 7→ A′ = g−1A g + g−1∂ g (with appropriate indices) , (3.13b) Ξ 7→ Ξ′ = g−1Ξ g , (3.13c) where g = g(xa, ηαi ) is a U(n)-valued superfield globally defined on the deformed superspace R CP 1. Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51]) ψ = Φ−1 + O(ζ) for ζ → 0 , (3.14a) ψ = 1l + ζ−1Υ + O(ζ−2) for ζ →∞ , (3.14b) where the U(n)-valued function Φ and u(n)-valued function Υ depend on xa and ηαi . This “unitary” gauge is compatible with the reality condition for ψ, ψ(xa, ηαi , ζ) ψ(xa, ηαi , ζ̄) = 1l , (3.15) obtained by reduction from (2.26). Gauge fixing for Âiα. After fixing the unitary gauge (3.14) for ψ and inserting (ζα) = the linear system (2.40), one can easily reconstruct the superfield given in (2.41) from Φ or Υ via Âi1 = 0 and Âi2 = Φ−1D̂i2Φ = D̂i1Υ (3.16) and thus fix a gauge for the superfields Âiα. The operators D̂iα were defined in (2.35). One can express (3.16) in terms of Aiα and A(αβ) − εαβΞ as Ai1 = 0 and Ai2 = Φ−1∂i2Φ = ∂i1Υ , (3.17) A(11) = 0 and A(12) + Ξ = Φ−1∂(12)Φ = ∂(11)Υ , (3.18) A(21) − Ξ = 0 and A(22) = Φ−1∂(22)Φ = ∂(12)Υ . (3.19) Using (2.32), we can rewrite the nonzero components as A := Φ−1∂uΦ = ∂xΥ , B := Φ−1∂xΦ = ∂vΥ , Ci := Φ−1∂i2Φ = ∂i1Υ . (3.20) Recall that the superfields Φ and Υ depend on xa and ηαi . Linear system. In the above-introduced unitary gauge the linear system (2.42) reads (ζ∂x − ∂u −A)ψ = 0 , (ζ∂v − ∂x − B)ψ = 0 , (ζ∂i1 − ∂i2 − Ci)ψ = 0 , (3.21) which adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(xa, ηαj ), B(xa, ηαj ) and Ci(xa, ηαj ). The concise form of (3.21) reads ζ D̂i1 − D̂i2 − Âi2 ψ = 0 (3.22) or, in more explicit form, ∂i1 + 2θ i1∂v + 2θ ∂i2 + Ci + 2θi1(∂x + B) + 2θi2(∂u +A) ψ = 0 . (3.23) N -extended sigma model. The compatibility conditions of this linear system are the N - extended noncommutative sigma model equations D̂i1(Φ −1D̂j2 Φ) + D̂ −1D̂i2 Φ) = 0 (3.24) which in expanded form reads (gab + vcε cab) ∂a(Φ −1∂bΦ) = 0 ⇔ ∂x(Φ−1∂xΦ) − ∂v(Φ−1∂uΦ) = 0 , (3.25a) ∂i1(Φ −1∂xΦ) − ∂v(Φ−1∂i2Φ) = 0 , ∂i1(Φ−1∂uΦ) − ∂x(Φ−1∂i2Φ) = 0 , (3.25b) ∂i1(Φ −1∂j2Φ) + ∂ −1∂i2Φ) = 0 . (3.25c) Here, the first line contains the Wess-Zumino-Witten term with a constant vector (vc) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimen- sions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on xa and 2N Grassmann variables ηαi . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolny- type equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A,B, Ci) given in (3.17)–(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to (∂2x − ∂u∂v)Υ + [∂vΥ , ∂xΥ] = 0 , (3.26a) (∂i2∂v − ∂i1∂x)Υ + [∂i1Υ , ∂vΥ] = 0 , (∂i2∂x − ∂i1∂u)Υ + [∂i1Υ , ∂xΥ] = 0 , (3.26b) (∂i2∂ 1 + ∂ 1)Υ + {∂i1Υ , ∂ 1Υ} = 0 , (3.26c) which in concise form read (D̂i2 D̂ 1 + D̂ 1)Υ + {D̂i1Υ , D̂ 1Υ} = 0 . (3.27) Recall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on xa and 2N Grassmann variables ηαi . For N=4, the commutative limit of (3.27) can be considered as Siegel’s equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via ∂i1Υ = A 2 , ∂ 1Υ = φ ij , ∂i1∂ 1Υ = χ̃ [ijk] 2 , ∂ 1Υ = G [ijkl] 22 , (3.28a) ∂(α1)Υ = A(α2) − εα2ϕ , ∂(α1)∂i1Υ = χiα , ∂(α1)∂(β1)Υ = fαβ , (3.28b) where one takes Υ and its derivatives at η2i = 0. The other components of the physical fields, i.e. χ̃ [ijk] 1 , G [ijkl] 11 , G [ijkl] 21 , A(11) and A(21)−ϕ, vanish in this light-cone gauge. Supersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimen- sions to the form Qiα = ∂iα − ηβi ∂(αβ) and Q α = ∂ α − θiβ∂(αβ) . (3.29) Their antichiral version, matching to D̂iα and D̂ of (2.35), reads Q̂iα = ∂iα − 2ηβi ∂(αβ) and Q̂ , (3.30) so that {Q̂iα , Q̂jβ} = −2 δ i ∂(αβ) . (3.31) On a (scalar) R3|2N superfield Σ these supersymmetry transformations act as δ̂Σ := εiαQ̂iαΣ + ε αΣ (3.32) and are induced by the coordinate shifts δ̂ yαβ = −2εi(αηβ)i and δ̂ η i = ε i , (3.33) where εiα and εαi are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators D̂iα and D̂ anticommute with the supersymmetry generators Q̂iα and Q̂ . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N ,N ) in 2+1 and Spin(N ,N )× Spin(N ,N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above “noncompact” 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53]. This may be of interest and deserves further study. Action functionals. In either formulation of the N -extended supersymmetric SDYM model on 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27), where only the operators D̂iα but not D̂iα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47]). One proposal for an action functional stems from Siegel’s idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂i2Υ enters only linearly into the last two lines in (3.26). Therefore, if we introduce Υ(1) := Υ|η2 =0 (3.34) then it must satisfy the first equation from (3.26), and the remaining equations iteratively define the dependence of Υ on η2i starting from Υ(1). Hence, all information is contained in Υ(1), as can also be seen from (3.28). In other words, the dependence of Υ on η2i is not ‘dynamical’. For an action one can then take (cf. [33]) d3x dN η1 Υ(1)∂(αβ)∂ (αβ)Υ(1) + Υ(1) ε αβ∂(α1)Υ(1) ∂(β1)Υ(1) . (3.35) Extremizing this functional yields the first line of (3.26) at η2i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55]. One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56]. 4 N -extended multi-soliton configurations via dressing The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of theN -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ† = ψ|ζ=0 and Υ = lim ζ (ψ−1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12]–[20], and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case. Multi-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form ψ(∂u − ζ∂x)ψ† = A , ψ(∂x − ζ∂v)ψ† = B , ψ(∂i2 − ζ∂i1)ψ† = Ci . (4.1) Recall that ψ† := (ψ(xa, ηαi , ζ̄)) † and (A,B, Ci) depend only on xa and ηαi . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1. Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µk ∈ CP 1. Here we will build a (multi-soliton) solution ψm featuringm simple poles at positions µ1, . . . , µm with9 Imµk < 0 by left-multiplying an (m−1)-pole solution ψm−1 with a single-pole factor of the µm − µ̄m ζ − µm a, ηαi ) , (4.2) where the n×n matrix function Pm is yet to be determined. Starting from the trivial (vacuum) solution ψ0 = 1l, the iteration ψ0 7→ ψ1 7→ . . . 7→ ψm yields a multiplicative ansatz for ψm, µm−ℓ − µ̄m−ℓ ζ − µm−ℓ , (4.3) which, via partial fraction decomposition, may be rewritten in the additive form ψm = 1l + ζ − µk , (4.4) 9This condition singles out solitons over anti-solitons, which appear for Imµk > 0. where Λmk and Sk are some n×rk matrices depending on xa and ηαi , with rk ≤ n. Equations for Sk. Let us first consider the additive parametrization (4.4) of ψm. This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = µ̄k on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that µ̄k − µℓ Sk = 0 , (4.5) while from (4.1) we obtain the differential equations µ̄k − µℓ A,B,i Sk = 0 , (4.6) where L̄ A,B,i stands for either L̄Ak = ∂u − µ̄k∂x , L̄Bk = µk(∂x − µ̄k∂v) or L̄ik = ∂i2 − µ̄k∂i1 . (4.7) Note that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . ,m if we consider poles at ζ = µ̄k. Because the L̄ A,B,i for k = 1, . . . ,m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η1i , η i ) to “co-moving coordinates” (wk, w̄k, sk; η , η̄i ). The precise relation for k = 1, . . . ,m is [12, 58] wk := x+ µ̄ku+ µ̄ v = x+ 1 (µ̄k−µ̄−1k )y + (µ̄k+µ̄ )t and ηik := η i + µ̄kη i , (4.8) with w̄k and η̄ obtained by complex conjugation and the co-moving time sk being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity (vx , vy)k = − ( µk + µ̄k µkµ̄k + 1 µkµ̄k − 1 µkµ̄k + 1 , (4.9) so that the static case wk=z is recovered for µk = −i. On functions of (wk, ηik, w̄k, η̄ik) alone the operators (4.7) act as L̄Ak = L̄ k = (µk−µ̄k) =: L̄k and L̄ k = (µk−µ̄k) . (4.10) By induction in k = 1, . . . ,m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is L̄kSk = SkZ̃k and L̄ kSk = SkZ̃ k (4.11) with some rk×rk matrices Z̃k and Z̃ik depending on (wk, w̄k, η Passing to the noncommutative bosonic coordinates we obtain ŵk , ˆ̄wk = 2θ νkν̄k with νkν̄k = µk−µ̄k−µ−1k +µ̄ . (4.12) Thus, we can introduce annihilation and creation operators and c so that [ck , c ] = 1 (4.13) for k = 1, . . . ,m. Naturally, this Heisenberg algebra is realized on a “co-moving” Fock space Hk, with basis states |ℓ〉k and a “co-moving” vacuum |0〉k subject to ck|0〉k = 0. Each co-moving vacuum |0〉k (annihilated by ck) is related to the static vacuum |0〉 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12]) which is time-dependent. The fermionic coordinates ηik and η̄ k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as 7→ −[c† , · ] and ν̄k 7→ [ck , · ] . (4.14) After the Moyal deformation, the n×rk matrices Sk have become operator-valued, but are still functions of the Grassmann coordinates ηi and η̄i . The noncommutative version of the BPS conditions (4.11) naturally reads ck Sk = Sk Zk and Sk = Sk Z k (4.15) where Zk and Z k are some operator-valued rk×rk matrix functions of η and η̄ Nonabelian solutions for Sk. For general data Zk and Z it is difficult to solve (4.15), but it is also unnecessary because the final expression ψm turns out not to depend on them. Therefore, we conveniently choose Zk = ck ⊗ 1lrk×rk and Zik = 0 ⇒ Sk = Rk(ck, ηik) , (4.16) where Rk is an arbitrary n×rk matrix function independent of c†k and η̄ik.10 It is known that nonabelian (multi-) solitons arise for algebraic functions Rk (cf. e.g. [7] for the commutative and [12] for the noncommutative N=0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the ηi dependence, i.e. Rk = Rk,0 + η kRk,i + η Rk,ij + η Rk,ijp + η Rk,ijpq . (4.17) Abelian solutions for Sk. It is useful to view Sk as a map from C rk⊗Hk to Cn⊗Hk (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of Cn ⊗Hk. In particular, we may choose it to be finite-dimensional, say Cqk , and represent the map by an n×qk array |Sk〉 of kets in H. In this situation, Zk and Zik in (4.15) are just number -valued qk×qk matrix functions of ηjk and η̄ . In case they do not depend on η̄ , we can write down the most general solution as |Sk〉 = Rk(ck, ηjk) |Zk〉 exp ) η̄ik with |Zk〉 := exp |0〉k . (4.18) 10Changing Zk or Z k multiplies Rk by an invertible factor from the right, which drops out later, except for the degenerate case Zk=0 which yields Sk = Rk |0〉k〈0|k. As before, we may put Zi = 0 without loss of generality, but now the choice of Zk does matter. For any given k generically there exists a qk-dimensional basis change which diagonalizes the ket-valued matrix |Zk〉 7→ diag c† , eα c†, . . . , eα |0〉k = diag |α1k〉 , |α2k〉 , . . . , |α , (4.19) where we defined coherent states |αlk〉 := eα c† |0〉k so that ck |αlk〉 = αlk |αlk〉 for l = 1, . . . , qk and αlk ∈ C . (4.20) Note that not only the entries of Rk but also the α k are holomorphic functions of the co-moving Grassmann parameters η and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×qk matrices |Sk〉 for all k, yielding rows |Sk〉 = R1k |α1k〉 , R2k |α2k〉 , . . . , R for k = 1, . . . ,m , (4.21) with functions αl ). Here, the Rl only affect the states’ normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |Sk〉 → |Sk(ηjk)〉. Explicit form of Pk. Let us now consider the multiplicative parametrization (4.3) of ψm which also allows us to solve (4.5). First of all, note that the reality condition (3.15) is satisfied if Pk = P = P 2k ⇔ Pk = Tk (T −1T † for k = 1, . . . ,m , (4.22) meaning that Pk is an operator-valued hermitian projector (of group-space rank rk ≤ n) built from an n×rk matrix function Tk (the abelian case of n=1 is included). The reality condition follows just because µk − µ̄k ζ − µk µ̄k − µk ζ − µ̄k = 1l for any ζ and k = 1, . . . ,m . (4.23) The rk columns of Tk span the image of Pk and obey Pk Tk = Tk ⇔ (1l−Pk)Tk = 0 . (4.24) Furthermore, the equation (4.5) with m = k (induction) rewritten in the form (1l−Pk) µk−ℓ − µ̄k−ℓ µ̄k − µk−ℓ Sk = 0 (4.25) reveals that (cf. (4.24)) T1 = S1 and Tk = 1l − µk−ℓ − µ̄k−ℓ µk−ℓ − µ̄k Sk for k ≥ 2 , (4.26) where the explicit form of Sk for k = 1, . . . ,m is given in (4.16) or (4.18). The final result reads µm−ℓ − µ̄m−ℓ ζ − µm−ℓ = 1l + ζ − µk (4.27) with hermitian projectors Pk given by (4.22), Tk given by (4.26) and Sk given by (4.16) or (4.18). The explicit form of Λmk (which we do not need) can be found in [12]. The corresponding superfields Φ and Υ are Φm = ψ m|ζ=0 = (1l− ρkPk) with ρk = 1− , (4.28a) Υm = lim ζ (ψm − 1l) = (µk−µ̄k)Pk . (4.28b) From (4.22) it is obvious that Pk is invariant under a similarity transformation Tk 7→ Tk Λk ⇔ Sk 7→ Sk Λk (4.29) for an invertible operator-valued rk×rk matrix Λk. This justifies putting Zik = 0 from the beginning and also the restriction to Zk = ck ⊗1lrk×rk in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {Rk}m1 of matrix- valued functions of ck and η k and the pole positions µk. The abelian moduli space, however, is larger by the set {Zk}m1 of matrix-values functions of ηik which generically contain the coherent- state parameter functions {αl )}. Restricting to ηi =0 reproduces the soliton configurations of the bosonic model [12]. Static solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ1 ≡ µ = −i implying w1 = z, so we drop the index k. Specializing (4.27), we have ψ = 1l − 2 i ζ + i P so that Φ = Φ† = 1l− 2P , (4.30) where a hermitian projector P of group-space rank r satisfies the BPS equations (1l−P ) aP = 0 ⇒ (1l−P ) aT = 0 , (4.31a) (1l−P ) ∂ P = 0 ⇒ (1l−P ) ∂ T = 0 , (4.31b) with P = T (T †T )−1T † and ηi = η1i + iη i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, ηi) as an n×1 column. For the simplest case of N=1 we just have (cf. [59]) T = Te(a) + η To(a) with η = η 1 + iη2 , (4.32) where Te(a) and To(a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N=1 projector (for n=1) is built from |T 〉 = |α1〉 , |α2〉 , . . . , |αq〉 . (4.33) At θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N=1 supersymmetric CPn−1 sigma model in two dimensions (see e.g. [59]).11 For 11In fact, Φ in (4.30) takes values in the Grassmannian Gr(r, n), and Gr(1, n) = CPn−1. this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge Q = 2πθ dη1dη2 Tr tr Φ D+Φ ,D−Φ (4.34) is proportional to the action (BPS bound) S = 2πθ dη1dη2 Tr tr D+Φ ,D−Φ (4.35) and is finite for algebraic functions Te and To. Here, the standard superderivatives D± are defined + iη ∂z and D− = + iη̄ ∂z̄ . (4.36) One-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain ψ1 = 1l + µ− µ̄ ζ − µ P with P = T (T †T )−1T † (4.37) Φ = 1l − ρP with ρ = 1− µ . (4.38) Now our n×r matrix T must satisfy (putting Zi = 0 and Z = c⊗ 1lr×r) [c , T ] = 0 and T = 0 with ηi = η1i + µ̄ η i , (4.39) where c is the moving-frame annihilation operator given by (4.13) for k=1. Recall that the operators c and c† and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N=1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e. T = Te(c) + η To(c) = T 1e (c) + η T o (c)... Tne (c) + η T o (c) (4.40) with polynomials T ae and T o of order q, say, analogously to the static case (4.32). Note that, for T ao to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation. For N=1 the moving frame was defined in (4.8) (dropping the index k) via w = x + 1 (µ̄−µ̄−1)y + 1 (µ̄+µ̄−1)t and η = η1 + µ̄η2 hence ∂tη = 0 . (4.41) Consider the moving frame with the coordinates (w, w̄, s; η, η̄) with the choice s = t and the related change of the derivatives (see [12, 58]) ∂x = ∂w + ∂w̄ , (4.42a) (µ̄−µ̄−1) ∂w + 12(µ−µ −1) ∂w̄ , (4.42b) (µ̄+µ̄−1) ∂w + (µ+µ−1) ∂w̄ + ∂s , (4.42c) ∂η1 = ∂η + ∂η̄ , (4.42d) ∂η2 = µ̄ ∂η + µ∂η̄ . (4.42e) In the moving frame our solution (4.38) is static, i.e. ∂sΦ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w̄; η1, η2) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case). Large-time asymptotics. Note that in the distinguished (z, z̄, t) coordinate frame (4.41) implies that at large times w→ κ t with κ = 1 (µ̄+µ̄−1). As a consequence, the tq term in each polynomial in (4.40) will dominate, i.e. T → tq a1 + η b1 an + η bn =: tq Γ , (4.43) where Γ is a fixed vector in Cn. It is easy to see that in the distinguished frame the large-time limit of Φ given by (4.38) is Φ = 1l − ρΠ with Π = Γ (Γ†Γ)−1Γ† (4.44) being the projector on the constant vector Γ. Consider now them-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have Φm = (1l− ρ1Π1) . . . (1l− ρℓ−1Πℓ−1)(1l− ρℓPℓ)(1l− ρℓ+1Πℓ+1) . . . (1l− ρmΠm) , (4.45) where the Πm are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times. 5 Conclusions In this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N=0 model, the supersymmetric extension was seen to promote the configurations’ building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here. The considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47]. In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60]. Here 2N appears due to fermionic worldvolume directions of our branes in the superspace de- scription [60]. Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions. Restricting to static configurations, the models can be specialized to Grassmannian supersym- metric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere. We derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25]) by suitable dimensional reduc- tions (see e.g. [61]–[65] and references therein). 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0704.0531

Gravitational Duality Transformations on (A)dS4

arXiv:0704.0531v3 [hep-th] 14 Aug 2007 Preprint typeset in JHEP style - HYPER VERSION hep-th/yymmnnn ILL-(TH)-07-02 Gravitational Duality Transformations on (A)dS4 Robert G. Leigh Department of Physics University of Illinois at Urbana-Champaign 1110 West Green Street, Urbana, IL 61801-3080, USA Email: [email protected] Anastasios C. Petkou Department of Physics University of Crete Heraklion 71003, Greece Email: [email protected] Abstract: We discuss the implementation of electric-magnetic duality transformations in four- dimensional gravity linearized around Minkowski or (A)dS4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynam- ics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. http://arxiv.org/abs/0704.0531v3 mailto:[email protected] mailto:[email protected] Contents 1. Introduction and Summary 1 2. Action and Hamiltonian 2 2.1 The 3 + 1 split 4 2.2 Shifted Variables 7 2.3 Linearization 7 3. Linearized Gravitational Duality and Holography 9 3.1 Duality and Holography 9 3.2 Linearized gravitational duality 10 3.3 Linearized Constraints and Bianchi Identities 11 3.4 Connection with other known dualities 12 4. The Effect on the Boundary Theory 13 5. Conclusions and Outlook 13 6. Appendix: other duality mappings 14 1. Introduction and Summary Duality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality trans- formation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS4 is responsible for a “natural” SL(2,Z) action on current two-point functions in three-dimensional CFTs.2 Shortly afterwards it was shown in [3] that such an SL(2,Z) action is intimately related to certain “double-trace” deformations in the boundary, 1For reviews of higher-spin theories see e.g. [1]. 2See [4] and [5] for more recent works. – 1 – assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain ”double-trace” deformations induce an SL(2,Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3]: linearized higher-spin theories on AdS4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2,Z) action on boundary two-point functions. Surprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture.3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12]. More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14]. The duality of linearized gravity around dS4 was later studied in [15]. In this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16]. Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17]). Our aim in this work is to formulate linearized first order gravity using suitable ”electric” and ”magnetic” variables, in close analogy with electromagnetism. We find that this is pos- sible only when the background geometry is Minkowski or (A)dS4. Then we implement the standard electric-magnetic duality rotations. We find that, up to ”boundary” terms, the lin- earized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The ”boundary” terms have important holographic consequences since they correspond to marginal ”double-trace” deforma- tions [3] that induce the boundary SL(2,Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces ”boundary” terms that correspond to relevant deformations as in [3]. 2. Action and Hamiltonian Having in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action4 IMM = ǫabcd Rab ∧ Rcd + 2Λea ∧ eb ∧ Rcd + Λ2ea ∧ eb ∧ ec ∧ ed , (2.1) 3An interesting formulation of first order duality for linearized gravity around flat space was presented in [7]. 4We note I = −16πGNS, where S is the usually normalized gravitational action. – 2 – where a, b, ... are Lorentz indices. In this formalism, the vierbein ea and the spin connection ωab are initially thought of as independent variables. The curvature 2-form is Rab = dω b + ω c ∧ ω Rabcde c ∧ ed. Varying the action with respect to ea and ωab, we find Rab + Λea ∧ eb = 0 , (2.2) T a = dea + ωab ∧ e b = 0 . (2.3) The relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables Rµν − Rgµν = +3Λgµν . (2.4) and the scalar curvature is R = −d(d − 1)Λ = −12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λcosm = −6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length)−2. In the SO(3, 2)-invariant formalism, IMM arises from IMM = ǫABCDEV ERAB ∧RCD , (2.5) where V E is a non-dynamical 0-form field (that we take to have value V −1 = 1 to gauge back to the SO(3, 1) formalism) and RAB is the curvature of Ω B ≡ {e a, ωab}. There are also quasi- topological terms of the form Itop = RAB ∧R RAB ∧ RACV BV C (2.6) that we could add to the action. In the stated gauge, this reduces to Itop = P2 + (θ + α)CNY + α Rab ∧ e a ∧ eb (2.7) where P2 = Rab∧R a is the Pontryagin class, CNY = (T a∧Ta−Rab∧e a∧eb) is the Nieh-Yan class and we also note the Euler class E2 = ǫabcdR ab∧Rcd. Note that in the presence of torsion, the action (2.7) contains the non-topological term Rab ∧ e a ∧ eb with “Immirzi parameter” γ = −2/α. In the absence of torsion, this term is a total derivative. The Hilbert-Palatini action is IHP = IMM − E2 . (2.8) It differs from IMM by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)- invariant. – 3 – 2.1 The 3 + 1 split Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a ‘time’ function t and a foliation of space-time Σt →֒ M . In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ⊥, equal to ±1 in dS(AdS). Proceeding as usual then, we get a vector field t that satisfies ∇tt = 1 ≡ t(t) (so t = and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as n = σ⊥Ndt , (2.9) which is normalized as (n, n) = σ⊥. The dual vector field n can be expanded as N , (2.10) where the shift N satisfies (N,n) = 0, and thus (t,n) = σ⊥N . Next, we will locally choose a basis of 1-forms e0 = σ⊥n = Ndt , (2.11) eα = ẽα +Nαdt . (2.12) The ẽα span T ∗Σt, and correspond to a 3-metric hij = ẽ j ηαβ . The quantities N α are the components of N: Nα = eαi N i. These basis 1-forms are dual to {e0 = n, eα = ẽα}, with b) = δba. We expand the spin connection in the same basis5 ωab = q bdt+ ω̃ b , (2.13) which leads to Rab = R̃ b + dt ∧ r b , (2.14) where R̃ is formed from ω̃ and d̃ only, and rab = ˙̃ω b − d̃q b − ω̃ b + q b . (2.15) Note that these quantities are merely decompositions along T ∗Σt in the 4-geometry; we will introduce the intrinsically defined objects shortly. We then find IHP = 2ǫαβγ N(R̃αβ + Λẽα ∧ ẽβ) ∧ ẽγ − 2Nα(R̃0β) ∧ ẽγ + r0α ∧ ẽβ ∧ ẽγ . (2.16) 5We have qab = Nω0 b and ω̃ b ≡ ωα – 4 – As is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS4. The final term in the action contains the real dynamics – r0α depends on the components R0α0β of the Riemann tensor. Note though that the tensors used here are 4-dimensional. Let us define the ”electric field” Kα = σ⊥ω̃ α = Kβαẽ β . (2.17) In the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find R̃αβ = (3)Rαβ − σ⊥K α ∧Kβ , (2.18) R̃0α = σ⊥(d̃Kα +Kβ ∧ ω̃ α) ≡ σ⊥(D̃K)α . (2.19) These equations amount to the Gauss-Codazzi relations. Furthermore, r0α contains time derivatives of ω̃0α as well as terms linear in components of q. We find 2ǫαβγr 0α ∧ ẽβ ∧ ẽγ = 2ǫαβγ α − (D̃q)0α ∧ ẽβ ∧ ẽγ , (2.20) = 2σ⊥ǫαβγ K̇α + qαδKδ ∧ ẽβ ∧ ẽγ + 4q0α ǫαβγ T̃ β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T̃ α = d̃ẽα + ω̃α β ∧ ẽ β. Since we wish to regard the ẽ as coordinate variables,6 we integrate the first term by parts to obtain (up to the total time-derivative ∂ α ∧ ẽβ ∧ ẽγǫαβγ 2ǫαβγr 0α ∧ ẽβ ∧ ẽγ = Πα ∧ ˙̃e α + 4q0αǫαβγ T̃ β ∧ ẽγ + 2σ⊥q αδǫαβγKδ ∧ ẽ β ∧ ẽγ . (2.21) where we have defined the momentum 2-form Πα = −4σ⊥ǫαβγK β ∧ ẽγ . (2.22) The qab appear as Lagrange multipliers. In particular, the qαβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K[αβ] to zero. Next, we define the ”magnetic field” σ⊥ǫαβγω̃ βγ, ωαβ = −ǫαβγBγ. (2.23) and we find that the q0α constraint ǫαβγ T̃ β ∧ ẽγ = ǫαβγ d̃ẽ β ∧ ẽγ − σ⊥Bβ ∧ ẽ β ∧ ẽα = 0 , (2.24) 6Without this integration by parts, we would be in the Ashtekar formalism. Here, our choice gives a formalism closely related to the metric variable formalism. Note that the induced boundary term may be written − 1 Πα∧ ẽ – 5 – involves only the antisymmetric part B[α,β] of the magnetic field Bα = Bαβ ẽ β . The antisymmetric part of Bα spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα, hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of Bαβ to derivatives of ẽ d̃ẽα + ǫαβγBβ ∧ ẽγ = 0 (2.25) At the end, one is left with the canonically conjugate variables ẽα and Πα. These results are familiar from the metric formalism. Dropping the torsional terms, we then arrive at the action IHP = ˙̃eα ∧ Πα + 2Nǫαβγ( (3)Rαβ − σ⊥K α ∧Kβ + Λẽα ∧ ẽβ) ∧ ẽγ −4σ⊥N αǫαβγ(D̃K) β ∧ ẽγ . (2.26) Furthermore, using ∗3ẽ α ∧ ẽβ = 1 αβγ ẽδ, we have Π̂α = ∗3Πα = −2(Kαβ − ηαβtrK)ẽ β , (2.27) where trK = ηαβKαβ . We can solve the above equation to get Kα = − (Π̂αβ − ηαβtrΠ̂)ẽ β . (2.28) As stated above, Kαβ (and Π̂αβ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find7 (3)Rαβ ∧ ẽγ = ǫαβγ d̃ω̃αβ + ω̃αδ ∧ ω̃ ∧ ẽγ = σ⊥ 2d̃Bγ + ǫαβγB α ∧ Bβ ∧ ẽγ . (2.29) Introducing Bα is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form IHP = ˙̃eα ∧Πα − 4σ⊥N αǫαβγ(D̃K) β ∧ ẽγ +2σ⊥N(2d̃Bγ + ǫαβγB α ∧ Bβ − ǫαβγK α ∧Kβ + σ⊥Λǫαβγ ẽ α ∧ ẽβ) ∧ ẽγ . (2.30) Note that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint. 7The spatial signature σ3 appears in ǫαβγǫφδρη αφ = σ3(ηβδηγρ − ηβρηγδ). We will always consider Lorentzian spacetime signature, so σ3 = −σ⊥. – 6 – 2.2 Shifted Variables It is possible to make a transformation of the canonical variables in order to absorb the cosmo- logical constant term in (2.30). This can be achieved by introducing the new variables K̂α = Kα − ρẽα , (2.31) and requiring that ρ2 = σ⊥Λ . (2.32) This is positive only when σ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS4 (Λ > 0) and dS4 (Λ < 0). We will often write Λ = σ⊥/L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes Πα → Pα − 4σ⊥ρǫαβγ ẽ β ∧ ẽγ . (2.33) The last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable Pα = −4σ⊥ǫαβγK̂ β ∧ ẽγ . Then, we get the action IHP = ˙̃eα ∧ Pα − 4σ⊥N αǫαβγ(D̃K̂ + ρT̃ ) β ∧ ẽγ − σ⊥ρǫαβγ (ẽα ∧ ẽβ ∧ ẽγ) +2σ⊥N 2d̃(Bα ∧ ẽ α) + 2Bγ ∧ T̃ γ − ǫαβγ Bα ∧Bβ + K̂α ∧ K̂β + 2ρK̂α ∧ ẽβ ∧ ẽγ .(2.34) Note that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K̂. Consistent with our previous discussion, we drop the terms involving the torsion T̃ , and disregard the boundary term to obtain IHP = ˙̃eα ∧ Pα − 4σ⊥N αǫαβγ(D̃K̂) β ∧ ẽγ +2σ⊥N 2d̃(Bα ∧ ẽ α)− ǫαβγ Bα ∧Bβ + K̂α ∧ K̂β + 2ρK̂α ∧ ẽβ ∧ ẽγ . (2.35) We note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion). 2.3 Linearization Next, we linearize the above action around an appropriate fixed background. We expand as ẽα = ẽα + Eα, N = 1 + n, Nα = nα, Bα = Bα + bα, K̂α = K̂ + kα . (2.36) The background values should satisfy the constraints. The simplest choice is the background where = 0 = Bα . (2.37) – 7 – In fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives IHP = Ėα ∧ pα − 4σ⊥n αǫαβγ d̃k β ∧ ẽγ + 4σ⊥n d̃(bα ∧ ẽ α)− ρǫαβγk α ∧ ẽβ ∧ ẽγ −2σ⊥ǫαβγ bα ∧ bβ + kα ∧ kβ + 2ρkα ∧ Eβ ∧ ẽγ ,(2.38) where pα = −4σ⊥ǫαβγk β ∧ ẽγ (2.39) are the linearized momentum variables conjugate to Eα. In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ˙̃eα + ρẽα = 0 . (2.40) Notice that we can also write the linearized action in the form IHP = (Ėα + ρEα) ∧ pα − 2σ⊥ǫαβγ bα ∧ bβ + kα ∧ kβ ∧ ẽγ −4σ⊥n αǫαβγ d̃k β ∧ ẽγ + n 4σ⊥d̃bγ + ρpγ ∧ ẽγ . (2.41) The form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of Eα is of the form e−ρt (correspondingly, the natural time dependence of pα is e +ρt). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS4 spacetimes. We can solve (2.40) to obtain e0 = dt, eα = e−ρtdxα . (2.42) With these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from −∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive ωα0 = −ρe −ρtdxα = −ρeα , (2.43) and so Rαβ = − eα ∧ eβ Rα0 = − eα ∧ e0 Rab = − ea ∧ eb . (2.44) Hence Ricab = − ηab and R = −12σ⊥/L 2 = −12Λ. We also evaluate Πα = −4σ⊥ρǫαβγ ẽ β ∧ ẽγ , Π̂ = 4ρẽα, trΠ̂ = 12ρ (2.45) Bα = 0, K α = ρẽα ⇒ K̂ = 0 (2.46) Note that in this gauge, (D̃K)α = 1 = 0, which solves the shift constraint, while the Hamil- tonian constraint is satisfied through a cancellation between the K2 term and the cosmological term. – 8 – 3. Linearized Gravitational Duality and Holography Let us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K̂ (the “electric field”) and B (the ”magnetic field”) vanish in such a background - as they do around flat space - we found that the background should be (A)dS4. Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS4 spacetimes. 3.1 Duality and Holography This is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p 7→ q and q 7→ −p to the action is (see e.g. [19]) dt[pq̇ −H(p, q)] 7→ ID = dt[−qṗ−H(q,−p)] . (3.1) Notice that ID involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q,−p) is in general not related to H(p, q). However, if H(q,−p) = H(p, q) we call the above transformation a duality. It then holds ID = I − qp . (3.2) The dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, ID is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS4 [2, 3]. Clearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization dt[pq̇ − (p2 + q2 + 2λpq)] (3.3) where λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p 7→ q and q 7→ −p – the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately. – 9 – We also note that the canonical transformation (implemented by a generating functional of the first kind) p 7→ q + 2λp , q 7→ −p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms8 S 7→ SD = S − pq . (3.5) 3.2 Linearized gravitational duality As a preamble to gravity we recall the duality properties of Maxwell theory IMax = A ∧ ∗3E − (E ∧ ∗3E +B ∧ ∗3B)−A0d̃ ∗3 E , (3.6) Under the duality E 7→ − ∗3 B, B 7→ ∗3E, Ã 7→ ÃD, we find IMax 7→ IMax,D = AD ∧ B − (E ∧ ∗3E +B ∧ ∗3B) + A0d̃B . (3.7) E and B in (3.7) should be expressed through ÃD. We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial ‘Bianchi’ identity dB = 0 for the dual magnetic field. Next we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40) kα 7→ −bα, bα 7→ kα . (3.8) To implement the map (3.8) we need to specify the mapping of Eα to a ‘dual 3-bein’ Eα. We do that using the linearized form of (2.25) as ǫαβγbβ ∧ ẽγ + d̃E α = 0 7→ ǫαβγkβ ∧ ẽγ + d̃E α = 0 = d̃Eα − pα (3.9) Since pα = 4σ⊥d̃Eα, it is natural to define pD,α = 4σ⊥d̃Eα = −4σ⊥ǫαβγb β ∧ ẽγ , (3.10) and thus the mapping (3.8) is supplemented by E 7→ E , E 7→ −E , p 7→ −pD , pD 7→ p (3.11) 8In holography, the latter terms correspond to the relevant ”multi-trace” boundary deformations discussed in – 10 – Now, let us see the effects of the above duality mapping. The action transforms to IHP 7→ IHP,D = −Ėα ∧ pD,α − ρE α ∧ pD,α − 2σ⊥ǫαβγ bα ∧ bβ + kα ∧ kβ ∧ ẽγ (3.12) +4σ⊥n αǫαβγ d̃b β ∧ ẽγ + n 4σ⊥d̃kγ + ρpD,α ∧ ẽγ where now kα and bα should be expressed in terms of the dual variables Eα and pD,α via (3.9) and (3.10). We notice that the ’kinetic’ part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix. 3.3 Linearized Constraints and Bianchi Identities By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as Cα ≡ ǫαβγ d̃k β ∧ ẽγ 7→ −ǫαβγ d̃b β ∧ ẽγ (3.13) C0 ≡ −σ⊥ d̃bγ − ρǫαβγk α ∧ ẽβ ∧ ẽγ 7→ −σ⊥ d̃kγ + ρǫαβγb α ∧ ẽβ ∧ ẽγ (3.14) To identify the right hand sides, we first note that the Bianchi identities are b = dR c ∧ ω b + ω b = 0 (3.15) BaT = dT a − Rab ∧ e b + ωab ∧ T b = 0 (3.16) which are obtained from the definitions of Rab and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that Rab ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is α = −((3)Rαβ − σ⊥K α ∧Kβ) ∧ ẽ β = 0 (3.17) which upon using the symmetry of Kα linearizes to α = −ǫαβγ d̃b β ∧ ẽγ + . . . (3.18) Note that this is the image under duality of the shift constraint as in (3.13). The second identity is 0 = −R̃0α ∧ ẽ α = −σ⊥(D̃K)α = −σ⊥ d̃kα + ρǫαβγb β ∧ ẽγ ∧ ẽα = 0 (3.19) – 11 – where to arrive in the second line we used (2.46). This is the image of the Hamiltonian constraint as in (3.14). Summarizing, the duality transformations between linearized constraints and Bianchi iden- tities are Cα 7→ BT,α C0 7→ B T (3.20) BT,α 7→ −Cα B T 7→ −C0 (3.21) 3.4 Connection with other known dualities The Maxwell-type duality operation (3.8) is closely related to the dualization of the first two indices of the Riemann tensor as9 Rab → S dRcd (3.22) at least at the linearized level. Let us investigate (3.22) by rewriting expressions in the 3+1 split. We have Rab = R̃ b + dt ∧ r Sab = S̃ b + dt ∧ s We begin with the spatial 2-forms when we have R̃αβ = −ǫαβγ d̃Bγ + σ⊥(B α ∧ Bβ −Kα ∧Kβ) (3.23) R̃0α = σ⊥(d̃Kα +Kβ ∧ ω̃ α) ≡ σ⊥(D̃K)α (3.24) S̃0γ = σ⊥ǫαβγR̃ αβ (3.25) S̃αβ = ǫαβγR̃ 0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) R̃ab 7→ −σ⊥S̃ ab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an ‘integrated form’ of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find rab 7→ −σ⊥s ab (3.28) To arrive at this result we have set to zero the Lagrange multiplier field q. 9For a discussion of the duality properties of gravity in terms of the Riemann tensor see [10]. – 12 – 4. The Effect on the Boundary Theory It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form Sbdy = pα ∧ E α (4.1) Applying the duality transformation to the bulk theory, although the bulk action is not invari- ant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign. Sdualbdy = − pD,α ∧ E α (4.2) This transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21].10 2 = −1 . (4.3) 5. Conclusions and Outlook Motivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable ”electric” and ”magnetic” field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nev- ertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have ex- hibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations. The main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach. Acknowledgments 10See also [22] for an interesting recent application of this formula. – 13 – The work of A. C. P. was partially supported by the research program ”PYTHAGORAS II” of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709. 6. Appendix: other duality mappings It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40) kα 7→ −bα − 2ρEα, bα 7→ kα . (6.1) The mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as IHP 7→ IHP,D = IHP + 4σ⊥ ǫαβγE α ∧ bβ + ρǫαβγE α ∧ Eβ ∧ ẽγ αǫαβγ d̃b β ∧ ẽγ − 8ρnαkβ ∧ ẽα ∧ ẽ +n(4σ⊥d̃kα + 4σ⊥ρǫαβγb β ∧ ẽγ + 8ΛǫαβγE β ∧ ẽγ) ∧ ẽα (6.2) The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total ”time” derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since kα is a symmetric one form kα = kαβ ẽ β with kαβ = kβα; see (2.21). The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for Eα after gauge fixing.11 One way to see this is in components. Write Eα = Eαβ ẽ β and (3.9) becomes γ − ∂γE α = ǫ γ − ǫ α (6.3) In the ”Lorentz gauge” where ∂αEβα = 0 = ∂ αkβα the above can be inverted as Eαβ = ǫαδγ∂ γkδβ (6.4) Using (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15]. References [1] D. Francia and A. Sagnotti, arXiv:hep-th/0601199. X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, arXiv:hep-th/0503128. 11This is the equivalent of inverting Ē = ∇× Ā in the discussion of duality in electromagnetism [16]. – 14 – [2] E. Witten, arXiv:hep-th/0307041. [3] R. G. Leigh and A. C. Petkou, JHEP 0312 (2003) 020 [arXiv:hep-th/0309177]. [4] R. Zucchini, Adv. Theor. Math. Phys. 8 (2005) 895 [arXiv:hep-th/0311143]. H. U. Yee, Phys. Lett. B 598 (2004) 139 [arXiv:hep-th/0402115]. [5] S. de Haro and P. Gao, arXiv:hep-th/0701144. [6] S. Hands, Phys. Rev. D 51 (1995) 5816 [arXiv:hep-th/9411016]. [7] P. C. West, Class. Quant. Grav. 18, 4443 (2001) [arXiv:hep-th/0104081]. [8] T. Curtright, Phys. Lett. B 165 (1985) 304. [9] J. A. Nieto, Phys. Lett. A 262 (1999) 274 [arXiv:hep-th/9910049]. [10] C. M. Hull, JHEP 0109 (2001) 027 [arXiv:hep-th/0107149]. [11] X. Bekaert, N. Boulanger and M. Henneaux, Phys. Rev. D 67 (2003) 044010 [arXiv:hep-th/0210278]. and [12] N. Boulanger, S. Cnockaert and M. Henneaux, JHEP 0306 (2003) 060 [arXiv:hep-th/0306023]. [13] M. Henneaux and C. Teitelboim, Phys. Rev. D 71, 024018 (2005) [arXiv:gr-qc/0408101]. [14] S. Deser and D. Seminara, Phys. Rev. D 71, 081502 (2005) [arXiv:hep-th/0503030]. S. Deser and D. Seminara, Phys. Lett. B 607, 317 (2005) [arXiv:hep-th/0411169]. [15] B. L. Julia, arXiv:hep-th/0512320. B. Julia, J. Levie and S. Ray, JHEP 0511, 025 (2005) [arXiv:hep-th/0507262]. [16] S. Deser and C. Teitelboim, Phys. Rev. D 13, 1592 (1976). [17] I. Papadimitriou and K. Skenderis, arXiv:hep-th/0404176. [18] S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38 (1977) 739 [Erratum-ibid. 38 (1977) 1376]. [19] H. Goldstein, ”Classical Mechanics”, Addison-Wesley Publishing Company Inc. (1980) [20] E. Witten, arXiv:hep-th/0112258. [21] A. C. Petkou, Fortsch. Phys. 53, 962 (2005). [22] C. P. Herzog, P. Kovtun, S. Sachdev and D. T. Son, arXiv:hep-th/0701036. – 15 –

0704.0532

Effect of transition-metal elements on the electronic properties of quasicrystals and complex aluminides

Publication in the honor of Prof. T. Fujiwara. Editors: Y. Hatsugai, M. Arai, S. Yamamoto, 2007, p. 128-144 Effect of transition-metal elements on the electronic properties of quasicrystals and complex aluminides Guy TRAMBLY de LAISSARDIÈRE 1, Didier MAYOU 2 1 Laboratoire de Physique Théorique et Modélisation, CNRS et Université de Cergy–Pontoise, France. [email protected] 2 Institut Néel, CNRS et Université Joseph Fourier, Grenoble, France. [email protected] 1 Introduction It is with great pleasure that we contribute to this book in honor of Prof. Takeo Fujiwara. GTL enjoyed eighteen months of Prof. Fujiwara’s hospi- tality at the University of Tokyo during the early 1990’s. At that time the work of Prof. Fujiwara in the field of electronic structure of quasicrystals had already made a major contribution to the literature (see for instance [1]). Since that time our research owes much to his work. Prof. Fujiwara was the first who performed realistic calculations of the electronic structure in quasicrystalline materials without adjustable param- eters (ab-initio calculations) [2]. Indeed these complex alloys [3] have very exotic physical properties (see Refs. [4, 5] and Refs therein), and it rapidly appeared that realistic calculations on the actual quasicrystalline materials are necessary to understand the physical mechanism that govern this prop- erties. In particular, these calculations allow to analyze numerically the role http://arxiv.org/abs/0704.0532v1 2 ELECTRONIC STRUCTURE 2 of transition-metal elements which is essential in those materials. In this paper, we briefly present our work on the role of transition-metal element in electronic structure and transport properties of quasicrystals and related complex phases. Several Parts of these works have been done or initiated in collaboration with Prof. T. Fujiwara. 2 Electronic structure 2.1 Ab-initio determination of the density of states A way to study the electronic structure of quasicrystal is to consider the case of approximants. Approximants are crystallines phases, with very large unit cell, which reproduce the atomic order of quasicrystals locally. Experiments indicate that approximant phases, like α-AlMnSi, α-AlCuFeSi, R-AlCuFe, etc., have transport properties similar to those of quasicrystals [4, 6]. In 1989 and 1991, Prof. Fujiwara performed the first numerical calculations of the electronic structure in realistic approximants of quasicrystals [2, 7, 8]. He showed that their density of states (DOS, see figure 1) is characterized by a depletion near the Fermi energy EF, called “pseudo-gap”, in agreement with experimental results (for review see Ref. [4, 9, 18]) and a Hume-Rothery stabilization [10, 11]. The electronic structure of simpler crystals such as orthorhombic Al6Mn, cubic Al12Mn, present also a pseudo-gap near EF which is less pronounced than in complex approximants phases (figure 1) [11]. 2.2 Models to analyze the role of transition-metal element sp–d hybridization model The role of the transition-metal (TM, TM= Ti, Cr, Mn, Fe, Co, Ni) elements in the pseudo-gap formation has been shown from experiments, ab-initio calculations and model analysis [4,13–19,11]. Indeed the formation of the pseudo-gap results from a strong sp–d coupling associated to an ordered 2 ELECTRONIC STRUCTURE 3 -12 -10 -8 -6 -4 -2 0 2 4 -12 -10 -8 -6 -4 -2 0 2 4 Energy (eV) Fα-Al Figure 1: Ab-initio total DOS of Al6Mn (simple crystal) and α- Al69.6Si13.0Mn17.4 (approximant of icosahedral quasicrystals) [11, 12]. sub-lattice of TM atoms [19, 11]. Consequently, the electronic structure, the magnetic properties and the stability, depend strongly on the TM positions, as was shown from ab-initio calculations [28–33,20,21]. How an effective TM–TM interaction induces stability? Just as for Hume-Rothery phases a description of the band energy can be made in terms of pair interactions (figure 2) [17, 19]. Indeed, it has been shown that an effective medium-range Mn–Mn interaction mediated by the sp(Al)–d(Mn) hybridization plays a determinant role in the occurrence of the pseudo-gap [19]. We have shown that this interaction, up to distances 10–20 Å, is essential in stabilizing these phases, since it can create a Hume- Rothery pseudo-gap close to EF. The band energy is then minimized as shown on figure 3 [20, 11]. 2 ELECTRONIC STRUCTURE 4 2 3 4 5 6 7 8 9 10 11 12 13 Mn−Mn distance r (A) -0.04 -0.02 with repulsive term without repulsive term repulsive term : b e − a r r = 4.8 A r = 6.7 A Figure 2: Effective medium-range Mn–Mn interaction between two non- magnetic manganese atoms in a free electron matrix which models aluminum atoms. [11] 0 4 8 12 16 20 24 28 32 36 40 L (A) α-AlMnSi β-AlMnSi Figure 3: Variation of the band energy due to the effective Mn–Mn interac- tion in o-Al6Mn, α-AlMnSi and β-Al9Mn3Si. [20] The effect of these effective Mn–Mn interactions has been also studied by several groups [17, 20, 21] (see also Refs in [11]). It has also explained the origin of large vacancies in the hexagonal β-Al9Mn3Si and ϕ-Al10Mn3 phases on some sites, whereas equivalent sites are occupied by Mn in µ-Al4.12Mn and λ-Al4Mn, and by Co in Al5Co2 [20]. On the other hand, an spin-polarized 2 ELECTRONIC STRUCTURE 5 effective Mn–Mn interaction is also determinant for the existence (or not) of magnetic moments in AlMn quasicrystals and approximants [21, 22, 32]. The analysis can be applied to any Al(rich)-Mn phases, where a small number of Mn atoms are embedded in the free electron like Al matrix. The studied effects are not specific to quasicrystals and their approximants, but they are more important for those alloys. Such a Hume-Rothery stabiliza- tion, governed by the effective medium-range Mn–Mn interaction, might therefore be intrinsically linked to the emergence of quasi-periodicity in Al(rich)-Mn system. Cluster Virtual Bound states One of the main results of the ab-initio calculations performed by Prof. Fujiwara for realistic approximant phases, is the small energy dispersion of electrons in the reciprocal space. Consequently, the density of states of approximants is characterized by “spiky” peaks [2, 7, 8, 28]. In order to analyze the origin of this spiky structure of the DOS, we developed a model that show a new kind of localization by atomic cluster [23]. As for the local atomic order, one of the characteristics of the quasicrys- tals and approximants is the occurrence of atomic clusters on a scale of 10–30 Å [25]. The role of clusters has been much debated in particular by C. Janot [24] and G. Trambly de Laissardière [23]. Our model is based on a standard description of inter-metallic alloys. Considering the cluster embedded in a metallic medium, the variation ∆n(E) of the DOS due to the cluster is cal- culated. For electrons, which have energy in the vicinity of the Fermi level, transition atoms (such as Mn and Fe) are strong scatters whereas Al atoms are weak scatters. In the figure 4 the variation, ∆n(E), of the density of states due to different clusters are shown. The Mn icosahedron is the actual Mn icosahedron of the α-AlMnSi approximant. As an example of a larger cluster, we consider one icosahedron of Mn icosahedra. ∆n(E) of clusters exhibits strong deviations from the Virtual Bound 2 ELECTRONIC STRUCTURE 6 6 7 8 9 10 11 12 13 14 15 Energy E (eV) )) 1 Mn atom 1 Mn icosahedron 1 icosahedron of 12 Mn icosahedra Figure 4: Variation ∆n(E) of the DOS due to Mn atoms. Mn atoms are embedded in a metallic medium (Al matrix). From [23]. States (1 Mn atom) [26]. Indeed several peaks and shoulders appear. The width of the most narrow peaks (50 − 100meV) are comparable to the fine peaks of the calculated DOS in the approximants (figure 1). Each peak indicates a resonance due to the scattering by the cluster. These peaks correspond to states “localized” by the icosahedron or the icosahedron of icosahedra. They are not eigenstate, they have finite lifetime of the order of ~/δE, where δE is the width of the peak. Therefore, the stronger the effect of the localization by cluster is, the narrower is the peak. A large lifetime is the proof of a localization, but in the real space these states have a quite large extension on length scale of the cluster. The physical origin of these states can be understood as follows. Elec- trons are scattered by the Mn atoms of a cluster. By an effect similar to that of a Faraday cage, electrons can by confined by the cluster provided that their wavelength λ satisfies λ & l, where l is the distance between two Mn spheres. Consequently, we expect to observe such a confinement by the 3 TRANSPORT PROPERTIES 7 cluster. This effect is a multiple scattering effect, and it is not due to an overlap between d-orbitals because Mn atoms are not first neighbor. 3 Transport properties Quasicrystals have many fascinating electronic properties, and in particular quasicrystals with high structural quality, such as the icosahedral AlCuFe and AlPdMn alloys, have unconventional conduction properties when com- pared with standard inter-metallic alloys. Their conductivities can be as low as 150–200 (Ω cm)−1 (see Refs. [4, 5, 27] and Refs. therein). Furthermore the conductivity increases with disorder and with temperature, a behavior just at the opposite of that of standard metal. In a sense the most striking property is the so-called “inverse Mathiessen rule” according to which the increases of conductivity due to different sources of disorder seems to be ad- ditive. This is just the opposite that happens with normal metals where the increases of resistivity due to several sources of scattering are additive. An important result is also that many approximants of these quasicrystalline phases have similar conduction properties. For example the crystalline α- AlMnSi phase with a unit cell size of about 12 Å and 138 atoms in the unit cell has a conductivity of about 300 (Ω cm)−1 at low temperature [4]. 3.1 Small Boltzmann velocity Prof. Fujiwara et al. was the first to show that the electronic structure of AlTM approximants and related phases is characterized by two energy scales [2, 7, 8, 28, 29] (see previous section). The largest energy scale, of about 0.5−1 eV, is the width of the pseudogap near the Fermi energy EF. It is related to the Hume–Rothery stabilization via the scattering of electrons by the TM sub-lattice because of a strong sp–d hybridization. The smallest energy scale, less than 0.1 eV, is characteristic of the small dispersion of the band energy E(k). This energy scale seems more specific to phases related to 3 TRANSPORT PROPERTIES 8 Temperature Metallic alloys "Perfect" stable quasicrystals Doped semi-conductors 4 K 300 K Metastable quasicrystals (i-AlMn), (i-AlCuFe and i- AlPdMn) "Imperfect" stable quasicrystals (i-AlLiCu) Amorphous alloys ρ Mott Figure 5: Schematic temperature dependencies of the experimental resistiv- ity of quasicrystals, amorphous and metallic crystals. 1e+14 1e+15 1/τ (s−1) Al (f.c.c.) Temperature Figure 6: Ab-initio elec- trical resistivity versus inverse scattering time, in cubic approximant α- Al69.6Si13.0Mn17.4, pure Al (f.c.c.), and cubic Al12Mn. the quasi-periodicity. The first consequence on transport is a small velocity at Fermi energy, Boltzmann velocity, VB = (∂E/∂k)E=EF . From numerical calculations, Prof. Fujiwara et al. evaluated the Bloch–Boltzmann dc con- ductivity σB in the relaxation time approximation. With a realistic value 3 TRANSPORT PROPERTIES 9 of scattering time, τ ∼ 10−14 s [27], one obtains σB ∼ 10 − 150 (Ωcm) −1 for a α-AlMn model [8] and 1/1-AlFeCu model [28]. This corresponds to the measured values [4, 6], which are anomalously low for metallic alloys. For decagonal approximant the anisotropy found experimentally in the conduc- tivity is also reproduced correctly [29]. 3.2 Quantum transport in Quasicrystals and approximants The semi-classical Bloch–Boltzmann description of transport gives inter- esting results for the intra-band conductivity in crystalline approximants, but it is insufficient to take into account many aspects due to the spe- cial localization of electrons by the quasi-periodicity (see Refs. [34–43] and Refs. therein). Some specific transport mechanisms like the temperature dependence of the conductivity (inverse Mathiessen rule, the defects influ- ence, the proximity of a metal / insulator transition), require to go beyond a Bloch–Boltzmann analysis. Thus, it appears that in quasicrystals and re- lated complex metallic alloys a new type of breakdown of the semi-classical Bloch-Boltzmann theory operates. In the literature, two different unconven- tional transport mechanisms have been proposed for these materials. Trans- port could be dominated, for short relaxation time τ by hopping between “critical localized states”, whereas for long time τ the regime could be dom- inated by non-ballistic propagation of wave packets between two scattering events. We develop a theory of quantum transport that applies to a normal bal- listic law but also to these specific diffusion laws. As we show phenomenolog- ical models based on this theory describe correctly the experimental trans- port properties [41, 42, 43] (compare figures 5 and 6). 3.3 Ab-initio calculations of quantum transport According to the Einstein relation the conductivity σ depends on the diffu- sivity D(E) of electrons of energy E and the density of states n(E) (summing 3 TRANSPORT PROPERTIES 10 the spin up and spin down contribution). We assume that n(E) and D(E) vary weakly on the thermal energy scale kT , which is justified here. In that case, the Einstein formula writes σ = e2n(EF)D(EF) (1) where EF is the chemical potential and e is the electronic charge. The tem- perature dependence of σ is due to the variation of the diffusivity D(EF ) with temperature. The central quantity is thus the diffusivity which is re- lated to quantum diffusion. Within the relaxation time approximation, the diffusivity is written [41] D(E) = C0(E, t) e −|t|/τ dt (2) where C0(E, t) = Vx(t)Vx(0) + Vx(0)Vx(t) it the velocity correlation functions without disorder, and τ is the relaxation time. Here, the effect of defects and temperature (scattering by phonons ...) is taken into account through the relaxation time τ . τ decreases as disorder increases. In the case of crystals phases (such as approximants of quasicrystals), one obtains [42, 43]: σ = σB + σNB (3) σB = e 2n(EF)V B τ and σNB = e 2n(EF) L2(τ) where σB is actual the Bolzmann contribution to the conductivity and σNB a non-Boltzmann contribution. L2(τ) is smaller than the square of the unit cell size L0. L 2(τ) can be calculated numerically for the ab-initio electronic structure [42]. From (3) and (4), it is clear that the Bolzmann term domi- nates when L0 ≪ VBτ : The diffusion of electrons is then ballistic, which is the case in normal metallic crystals. But, when L0 ≃ VBτ , i.e. when the Bolzmann velocity VB is very low, the non-Bolzmann term is essential. In the case of α-Al69.6Si13.0Mn17.4 approximant (figure 7) [42], with realistic value of τ (τ equals a few 10−14 s [27]), σNB dominates and σ increases when 3 TRANSPORT PROPERTIES 11 1e+14 2e+14 3e+14 4e+14 1/τ (s−1) Figure 7: Ab-initio dc-conductivity σ in cubic approximant α- Al69.6Si13.0Mn17.4 versus inverse scattering time. [42] 1e+14 2e+14 3e+14 4e+14 1/τ (s−1) 1e+05 2e+05 3e+05 4e+05 Figure 8: Ab-initio dc-conductivity σ in an hypothetical cubic approximant α-Al69.6Si13.0Cu17.4 versus inverse scattering time. [43] 1/τ increases, i.e. when defects or temperature increases, in agreement with experimental measurement (compare figures 5 and 6). To evaluate the effect of TM elements on the conductivity, we have considered an hypothetical α-Al69.6Si13.0Cu17.4 constructed by putting Cu atoms in place of Mn atoms in the actual α-Al69.6Si13.0Mn17.4 structure. Cu atoms have almost the same number of sp electrons as Mn atoms, but their d DOS is very small at EF. Therefore in α-Al69.6Si13.0Cu17.4, the effect of sp(Al)–d(TM) hybridization on electronic states with energy near EF is 4 CONCLUSION 12 very small. As a result, the pseudogap disappears in total DOS, and the conductivity is now ballistic (metallic), σ ≃ σB, as shown on figure 8. 4 Conclusion In this article we present the effect of transition-metal atoms on the physical properties of quasicrystals and related complex phases. These studies lead to consider these aluminides as spd electron phases [11], where a specific electronic structure governs stability, magnetism and quantum transport properties. The principal aspects of this new physics are now understood particularly thanks to seminal work of Prof. T. Fujiwara and subsequent developpements of his ideas. References [1] Fujiwara T, Tsunetsugu H. In: Di Vincenxo DP, Steinhart PJ, editors, Quasicrystals: The states of the art, Singapore: World Scientific, 1991. [2] Fujiwara T. Phys Rev 1989;B40:942. [3] Shechtman D, Blech I, Gratias D, Cahn JW. Phys Rev Lett 1984;53:1951. [4] Berger C. In: Hippert F, Gratias D, editors. Lecture on Quasicrystals. Les Ulis: Les Editions de Physique, 1994; p. 463. [5] Grenet T. In: Belin-Ferré E, Berger C, Quiquandon M, Sadoc A, edi- tors. Quasicrystals: Current Topics. Singapor: World Scientific, 2000; p. 455. [6] Quivy A, Quiquandon M, Calvayrac Y, Faudot F, Gratias D, Berger C, Brand RA, Simonet V, Hippert. J Phys Condens Matter 1996;8:4223. [7] Fujiwara T, Yokokawa T. Phys Rev Lett 1991;66:333. REFERENCES 13 [8] Fujiwara T, Yamamoto S, Trambly de Laissardière G. Phys Rev Lett 1993;71:4166. Mat Sci Forum 1994;150-151:387. [9] Mizutani U, Takeuchi T, Sato H. J Phys: Condens Matter 2002;14:R767. [10] Massalski TB, Mizutani U. Prog Mater Sci 1978;22:151. [11] Trambly de Laissardière G, Nguyen Manh D, Mayou D, Prog Mater Sci 2005;50:679. [12] Zijlstra ES, Bose SK. Phys Rev 2003;B67:224204. [13] Dankházi Z, Trambly de Laissardière G, Nguyen–Manh D, Belin E, Mayou D. J Phys: Condens Matter 1993;5:3339. [14] Trambly de Laissardière G, Mayou D, Nguyen Manh D. Europhys Lett 1993;21:25. J Non-Cryst Solids 1993;153-154:430. Trambly de Lais- sardière G, et al. Phys Rev 1995;B52:7920. [15] Berger C, Belin E, Mayou D. Annales de Chimie-Science des Matériaux 1993;18:485. [16] Mayou D, Cyrot–Lackmann F, Trambly de Laissardière G, Klein T. J Non-Cryst Solids 1993;153-154:412. [17] Zou J, Carlsson AE. Phys Rev Lett 1993;70:3748. [18] Belin-Ferré E. J Non-Cryst Solids 2004;334-335:323. [19] Trambly de Laissardière G, Nguyen Manh D, Mayou D. J Non-Cryst Solids 2004;334-335:347. [20] Trambly de Laissardière G. Phys Rev 2003;B68:045117. [21] Trambly de Laissardière G, Mayou D. Phys Rev Lett 2000;85:3273. [22] Simonet V, Hippert F, Audier M, Trambly de Laissardière G. Phys Rev 1998;B58:R8865. REFERENCES 14 [23] Trambly de Laissardière G, Mayou M. Phys Rev 1997;B55:2890. Tram- bly de Laissardière G, Roche S, Mayou D. Mat Sci Eng 1997;A226- 228:986. [24] Janot C, de Boissieu M. Phys Rev Lett 1994;72:1674. [25] Gratias D, Puyraimond F, Quiquandon M, Katz A. Phys Rev 2000;B63:24202. [26] Friedel J. Can J Phys 1956;34:1190. Anderson PW. Phys Rev 1961;124:41. [27] Mayou D, Berger C, Cyrot–Lackmann F, Klein T, Lanco P. Phys Rev Lett 1993;70:3915. [28] Trambly de Laissardière G, Fujiwara T. Phys Rev 1994;B50:5999. [29] Trambly de Laissardière G, Fujiwara T. Phys Rev 1994;B50:9843. Mat Sci Eng 1994;A181-182:722. [30] Hafner J, Krajč́ı M. Phys Rev 1998;B57:2849. [31] Krajč́ı M, Hafner J. Phys Rev 1998;B58:14110. [32] Nguyen–Manh D, Trambly de Laissardière G. J Mag Mag Mater 2003;262:496. [33] Zijlstra ES, Bose SK, Klanǰsek M, Jeglič P, Dolinšek J. Phys Rev 2005;B72:174206. [34] Tokihiro T, Fujiwara T, Arai M. Phys. Rev 1988;B38:5981. [35] Fujiwara T, Mitsui T, Yamamoto S. Phys Rev B 1996;53,R2910. [36] Roche S, Trambly de Laissardière G, Mayou D. J Math Phys 1996;38:1794. [37] Roche S, Mayou D. Phys Rev Lett 1997;79:2518. REFERENCES 15 [38] Mayou D. In: Belin-Ferré E, Berger C, Quiquandon M, Sadoc A, edi- tors. Quasicrystals: Current Topics. Singapor: World Scientific, 2000; p. 412. [39] Triozon F, Vidal J, Mosseri R, Mayou D. Phys Rev 2002;B65:220202. [40] Bellissard J. In: Garbaczeski P, Olkieicz R, editors. Dynamics of Dissi- pation, Lecture Notes in Physics. Berlin: Springer, 2003; p. 413. [41] Mayou D. Phys Rev Lett 2000;85:1290. [42] Trambly de Laissardière G, Julien JP, Mayou D. Phys Rev Lett 2006;97:026601. [43] Mayou D, Trambly de Laissardière G. In: Fujiwara T, Ishii Y, editors. Quasicrystals. Series “Handbook of Metal Physics”. Elsevier Science, 2007. to appear Introduction Electronic structure Ab-initio determination of the density of states Models to analyze the role of transition-metal element Transport properties Small Boltzmann velocity Quantum transport in Quasicrystals and approximants Ab-initio calculations of quantum transport Conclusion

0704.0533

Non-resonant and Resonant X-ray Scattering Studies on Multiferroic TbMn2O5

TbMn2O5 Non-resonant and Resonant X-ray Scattering Studies on Multiferroic TbMn2O5 J. Koo1, C. Song1, S. Ji1, J.-S. Lee1, J. Park1, T.-H. Jang1, C.-H. Yang1, J.-H. Park1,2, Y. H. Jeong1, K.-B. Lee1,2,∗ T.Y. Koo2, Y.J. Park2, J.-Y. Kim2, D. Wermeille3,† A.I. Goldman3, G. Srajer4, S. Park5, and S.-W. Cheong5,6 eSSC and Department of Physics, POSTECH, Pohang 790-784, Korea Pohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang 790-784, Korea Ames Laboratory, Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA Laboratory of Pohang Emergent Materials and Department of Physics, POSTECH, Pohang 790-784, Korea (Dated: November 1, 2018) Comprehensive x-ray scattering studies, including resonant scattering at Mn L-edge, Tb L- and M -edges, were performed on single crystals of TbMn2O5. X-ray intensities were observed at a forbidden Bragg position in the ferroelectric phases, in addition to the lattice and the magnetic modulation peaks. Temperature dependences of their intensities and the relation between the mod- ulation wave vectors provide direct evidences of exchange striction induced ferroelectricity. Resonant x-ray scattering results demonstrate the presence of multiple magnetic orders by exhibiting their different temperature dependences. The commensurate-to-incommensurate phase transition around 24 K is attributed to discommensuration through phase slipping of the magnetic orders in spin frustrated geometries. We proposed that the low temperature incommensurate phase consists of the commensurate magnetic domains separated by anti-phase domain walls which reduce spontaneous polarizations abruptly at the transition. PACS numbers: 77.80.e-, 75.25.+z, 64.70.Rh, 61.10.-i In recent years, much attention has been paid to mul- tiferroic materials, in which magnetic and ferroelectric orders coexist and are cross-correlated [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], due to theoretical interests and poten- tial application to magnetoelectric (ME) devices. Ma- nipulation of electric polarizations by external magnetic fields has been demonstrated in some of these materi- als [4, 5]. Orthorhombic TbMn2O5, one of the multi- ferroic materials, displays a rich phase diagram. Upon cooling through TN ∼ 41 K, TbMn2O5 becomes anti- ferromagnetic with an incommensurate magnetic (ICM) order which transits to a commensurate magnetic (CM) phase with spontaneous electric polarization at T c1 ∼ 36 K, and reenters a low temperature incommensurate magnetic (LT-ICM) phase at T c2 ∼ 24 K. Anomalies of ferroelectricity and dielectric properties were observed concurrently with these magnetic phase transitions [4, 9]. Especially, the reentrant LT-ICM phase is a phenomenon peculiar to RMn2O5 multiferroics while commensurate phases are more common as the low temperature ground states. Since the CM to LT-ICM phase transition is also accompanied with an abrupt loss of spontaneous polar- izations, it is critical to elucidate the natures of the in- commensurability of the material, including the mecha- nism of the CM to LT-ICM phase transition. The origin of the complex phases of the material is at- tributed to the coupling between magnetic moments of Mn ions and lattice [8, 9]. It is suggested that, when a magnetic order is modulated with a wave vector qm, the exchange striction affects inter-atomic bondings result- ing in a periodic lattice modulation with a wave vector qc = 2qm [5, 6, 7, 8, 9]. Recently, Chapon et al. pro- posed for RMn2O5 systems that ferroelectricity results from the exchange striction of acentric spin density waves for the CM phases [9]. Indeed, Kimura et al. insisted that CM modulations are indispensable to the ferroelec- tricity in the LT-ICM phase, from their neutron scat- tering results on HoMn2O5 under high magnetic fields [11]. However, lattice distortions derived from ICM spin structures turned out to describe well the spontaneous polarizations of YMn2O5 even in the ICM phase [12], implying that commensurability is not a necessary con- dition for the ferroelectricity. In order to understand the intriguing magnetoelectricity well, detailed information on the lattice and spin structure changes is necessary. However, only limited crystallographic data are available and even any direct evidence on the symmetry lowering has not been reported yet [9, 10, 11, 12, 13, 14]. In this letter, we present synchrotron x-ray scatter- ing results on single crystals of TbMn2O5. Since x-ray scattering is sensitive to both lattice and magnetic mod- ulations, x-ray scattering with intense undulator x-rays allowed simultaneous measurements for qm and qc. Non- resonant x-ray scattering results show the relationship of qc = 2qm, confirming lattice modulations are generated by the magnetic orders. A (3 0 0) forbidden Bragg peak, which is a direct evidence of the symmetry lowering to a non-centrosymmetry space group, was observed in the ferroelectric (FE) phases. Furthermore, the temperature dependence of the peak intensity, I (300), was found to coincide with those of the lattice modulation peak inten- sities, I c, and the spontaneous polarization square,P 2, in http://arxiv.org/abs/0704.0533v1 the CM phase. This indicates the ferroelectricity is gen- erated by the lattice modulations. In the LT-ICM phase, temperature dependences of I c cannot be described by a single order parameter, implying the presence of differ- ent magnetic orders. Resonant x-ray magnetic scattering results at Mn L-, Tb L3- and M5-edges show that each magnetic order has its own temperature dependence. It is proposed that CM to LT-ICM phase transition is in- duced by discommensuration through phase slipping due to competing magnetic orders under the frustrated ge- ometry. Moreover, the CM modulations with anti-phase domain walls are consistent with the temperature depen- dences of qm and I (300) in the LT-ICM phase, and explain well the abrupt loss of P at the transition. Single crystals of TbMn2O5 were grown by a flux method [4]. The specimen used for the hard x-ray scat- tering measurements has a plate-like shape with (1 1 0) as a surface normal direction. Its mosaicity was measured to be about 0.01◦ at (3 3 0) Bragg reflection. For soft x-ray scattering, a different sample was cut and polished to have (2 0 1) as a surface normal direction. Soft x-ray scattering measurements were performed at 2A beamline in the Pohang Light Source (PLS). Details of the soft x-ray scattering chamber were described elsewhere [15]. X-ray diffraction experiments were conducted at the 3C2 bending magnet beamline in the PLS and at the 6-ID undulator beamline in the Midwest Universities Collab- orative Access Team (MUCAT) Sector in the Advanced Photon Source. For non-resonant x-ray scattering exper- iments, 6.45 keV was selected as an incident x-ray energy below Mn K -edge (∼ 6.55 keV). All the incident x-rays were σ-polarized and PG(006) was used to have a σ-to-π channel at Tb L3-edge. Nonresonant x-ray scattering measurements were per- formed to investigate the temperature dependence of qm and qc simultaneously. The measured lattice modula- tion peak position of (2 5 -0.5) for the CM phase and those of its 4 split peaks for the ICM phases are pre- sented as solid and open circles, respectively, in Fig. 1 (a). For magnetic satellites, (2.5 5 -0.25) peak and its 2 split ones were measured for the CM and ICM phases. Their positions are presented as solid and open squares, respectively. The magnetic and lattice modulation satel- lites for ICM phases are linked with broken and solid lines to their corresponding main Bragg peaks. Temper- ature dependences of qm and qc are shown in Fig. 1 (b) and (c). From the results, it is obvious that relation, qc = 2qm, holds within experimental errors in the whole temperature range below TN . It is consistent with the magnetic order induced lattice modulations. The tem- perature dependence of qm shown here is qualitatively similar to the neutron scattering results by others [16]. Below TN , ICM magnetic peaks develop, and qm locks into a CM ordering at (1 ) via a first order transi- tion at T c1. On further cooling the sample below T c2, the CM to LT-ICM phase transition takes place. With FIG. 1: (Color online) Positions of the measured magnetic satellites (square) and lattice modulation peaks (circle) in the (h 5 l) reciprocal lattice plane are shown in (a). The temperature dependences of qxm (square) and q c (circle), and those of qzm (square) and q c (circle) are shown in (b) and (c), respectively. For direct comparisons with those of qm, the components of qc are divided by two. Vertical broken lines indicate TN ∼ 41 K, T c1 ∼ 36 K, T c2 ∼ 24 K and T c3 ∼ 13 K, respectively. further decreasing temperature, qm of the LT-ICM mod- ulations evolves and is eventually pinned around (0.486 0 0.308) which can be approximated to a CM value of (17 ) at T c3 ∼ 13 K. Such a long-period CM modulation can be interpreted as the CM modulations (qm = ( )) with domain walls, as is the case for ErNi2B2C [17]. As shown in Fig. 2 (a), measurable x-ray intensi- ties were observed, in the ferroelectric phase, at (3 0 0) Bragg position which is forbidden under a space group of the room temperature paraelectric phase, Pbam. Resid- ual intensities above T c1 are due to higher harmonic FIG. 2: (Color online) (a) Rocking curves of a (3 0 0) forbid- den Bragg peak measured below (open) and above T c1 (solid). (b) Temperature dependences of the integrated intensities of a (3 0 0) Bragg peak (circle), CM lattice modulation peak (square) and squared spontaneous polarization (broken line) taken from Ref. 4. All the data are properly scaled. contaminations. Values for full-width-at-half-maximum (FWHM) of the peak are about 0.01◦, close to those of (4 0 0) main Bragg peak in the LT-ICM phase. The results explicitly evidenced that inversion symmetry is broken concomitantly with the FE phase as speculated before. According to the models suggested by others [9, 10], dis- placements of Mn3+ are in ab-plane. While b-axis com- ponents of the atomic displacements mainly contribute to P, a-axis components enable the emergence of I (300). If the atomic displacements correspond to the periodic lat- tice modulations, it is expected that both P2 and I (300) are proportional to I c, as shown in Fig. 2 (b). (The spon- taneous polarization data are taken from Ref. 4 and are shifted in order to get the same values for T c1.) It con- firms that spontaneous polarization is due to the atomic displacements driven by magnetic orders: a direct crys- tallographic evidence of exchange striction as the origin of ferroelectricity in the material [8, 9, 10, 12]. Also it is noted that I (300) drops abruptly at T c2 and has a broad minimum around T c3. Though many interesting ME phenomena have been reported in the LT-ICM phases below T c2 [4, 11, 18, 19], their basic mechanisms still remain to be understood. Since the lattice modulations reflect basic ME natures, temperature dependences below T c2 of integrated inten- sities were measured at the four split ICM peak positions illustrated in Fig. 1 (a). From the results displayed in Fig. 3, it is clear that temperature dependences of all four peaks cannot be described by a single order parameter, implying the presence of various magnetic orders having the same qm’s but different temperature dependences. To investigate different magnetic orders, we performed resonant x-ray magnetic scattering measurements at Mn L-, Tb L3- and M5-edges. Figure 4 (a) shows energy pro- files around Mn L-edge of magnetic satellites at 10 K and FIG. 3: (Color online) Temperature dependences of the ICM lattice modulation peak intensities. x-ray absorption spectroscopy (XAS) at room tempera- ture. Magnetic peaks and XAS data clearly show reso- nances at both Mn L2- and L3-edges. XAS results show broad peaks containing contributions from the multiplet states of 3d electrons of Mn3+ and Mn4+ ions. Magnetic satellites show relatively sharp double peaks at both Mn L-edges. The sharp resonances represent different multi- plet states of Mn 3d electrons including charge transfer excitations, while Mn ions are expected to be in the high- spin configurations with all the 3d electron spins aligned FIG. 4: (a) Energy profiles of the ICM magnetic peaks (cir- cle)and XAS (solid line) around Mn L2,3-edges. Vertical bro- ken lines correspond to 640.8 eV and 644.2 eV, respectively. (b) Temperature dependences of the ICM (circle) and the CM (square) magnetic peaks. Open (Solid) symbols denote the data taken E = 640.8 eV (644.2 eV), respectively. (c) Temperature dependences of the ICM (open circle) and the CM (solid square) magnetic peak at Tb L3-edge. parallel. Therefore, although the resonances do not have one-to-one correspondences with the magnetic orders of Mn ions, changes in the resonances at magnetic satellites reflect the changes in spin ordering which are periodi- cally modulated with the wave vector qm. Temperature dependences of x-ray intensities at the ICM peak of Qm = (qxm 0 q m) were measured at the two resonances, 640.8 and 644.2 eV. The results are presented in Fig. 4 (b). Data for a CM peak of Qm = (0.5 0 0.25) at the reso- nance of 644.2 eV are presented together. It is clear that, above 15 K, intensities of each resonance have different temperature dependences from each other. Though the origin of the anomalous temperature dependences is not understood in detail, it reflects complicated natures of magnetic moments of Mn ions under the frustrated con- figuration. Magnetic ordering of Tb3+ ions was investigated with resonant x-ray scattering measurements at Tb L3-edge. Figure 4 (c) shows that ordering temperature of Tb mag- netic moments is the same with that of Mn, TN , which is consistent with neutron scattering results [9]. The modu- lation wave vector of Tb magnetic order is the same with the values of qm measured in nonresonant x-ray scatter- ing. Soft x-ray magnetic scattering measurements were also performed at Tb M 5-edge and the result not shown here confirms that observed x-ray intensities in Fig. 4 (c) reflect magnetic order of Tb 4f electrons which grows monotonically below TN . From the results shown in Fig. 4 (b) and (c), it is clear that there exist multiple magnetic order parameters hav- ing the same qm’s but different temperature dependences. The contributing portions of each magnetic order to scat- tering factors of magnetic satellites are different depend- ing on Qm(= QBragg + qm), and it results in different temperature dependence for each magnetic peak and its corresponding lattice modulation peak intensities, which explains the temperature dependences presented in Fig. Since the magnetic orders are located under the spin frustrated geometry, it is reasonable to suppose that phase-slips take place due to competitions between the magnetic orders, as their order parameters grow with different temperature dependences. The discommensu- ration results in the transition to the LT-ICM phase. Anti-phase domain walls for the phase slips are consis- tent with the aforementioned long-period CM modula- tions below T c3. Assuming the model suggested by oth- ers [9], atomic displacements are canted antiferroelectric type. Across an anti-phase domain wall, directions of the atomic displacements and the spontaneous polariza- tions are reversed. Therefore, not only the polarizations from domains separated by the domain wall cancel each other but also x-ray scattering amplitudes for the (3 0 0) Bragg peak are canceled due to the crystal symme- try. Then, only remnants resulting from unequal popu- lations of the domains contribute to P and I(300). Since a density of the domain walls determines qm, tempera- ture dependences of P, I(300) and qm down to T c3 can be explained consistently in terms of CM modulations with the anti-phase domain walls. This indicates that CM modulations are preferred as its low temperature ground state. Then, the low temperature phase seems to have a higher entropy due to the domain walls than the high temperature CM phase, violating the entropy rule. How- ever, due to the geometrical frustration and the presence of multiple magnetic orders many different energy scales can exist. The complicated temperature dependences of the magnetic orders in Fig. 4(c) reflect the presence of the different energy scales. Smaller energy scales become important at low temperatures and induce discommen- suration. Upturns of the electrical polarization and I(300) below T c3 are attributed to lattice modulations enhanced by increasing Tb magnetic moments, which is consistent with results of others demonstrating couplings between Tb moments and lattices [18, 19, 20]. In summary, we have shown that exchange striction is the driving mechanism for the magnetoelectricity in the material. The same temperature dependences of x- ray intensities at a (3 0 0) forbidden Bragg peak and a lattice modulation peak in the CM FE phase, together with observation of the relation, qc = 2qm, demonstrate that spontaneous electric polarization is due to atomic displacements driven by the exchange striction of mag- netic orders. Resonant x-ray magnetic scattering results confirm the presence of multiple magnetic orders hav- ing different temperature dependences. The CM to LT- ICM phase transition is attributed to discommensura- tion through phase slipping in the competing magnetic orders in the frustrated configurations. Temperature de- pendences of qm, P and I(300) in the LT-ICM phase are explained in terms of the CM modulations with anti- phase domain walls. We thank D.J. Huang for the useful discussions. This work was supported by the KOSEF through the eSSC at POSTECH, and by MOHRE through BK-21 pro- gram. The experiments at the PLS were supported by the POSTECH Foundation and MOST. Use of the Ad- vanced Photon Source (APS) was supported by the U.S. Department of Energy, Office of Science, Office of Ba- sic Energy Sciences, under Contract No. W-31-109-Eng- 38. The Midwest Universities Collaborative Access Team (MUCAT) sector at the APS is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, through the Ames Laboratory under contract No. DE-AC02-07CH11358. Work at Rutgers was supported by NSF-DMR-0520471. ∗ Electronic address: [email protected] † Present address: European Synchrotron Radiation Facil- mailto:[email protected] ity, BP 220, F-38043 Grenoble Cedex 9, France [1] M. Kenzelmann et al., Phys. Rev. Lett. 95, 87206 (2005). [2] G. Lawes et al., Phys. Rev. Lett. 95, 087205 (2005). [3] S. Kobayashi et al., J. Korean Phys. Soc. 46, 289 (2005). [4] N. Hur et al., Nature 429, 392 (2004). [5] T. Kimura et al., Nature 426, 55 (2003). [6] T. Goto et al., Phys. Rev. Lett. 92, 257201 (2004). [7] T. Kimura et al., Phys. Rev. B 68, 060403(R) (2003). [8] S.-W. Cheong et al., Nature Mater. 6, 13 (2007). [9] L. C. Chapon et al., Phys. Rev. Lett. 93 177402, (2004). G.R. Blake et al., Phys. Rev. B. 71, 214402 (2005). [10] I. Kagomiya et al., Ferroelectrics 286, 167 (2003). [11] H. Kimura et al., J. Phys. Soc. Jpn. 75, 113701 (2006). [12] L. C. Chapon et al., Phys. Rev. Lett. 96, 097601 (2006). [13] V. Polyakov et al., Physica B 297, 208 (2001). [14] D. Higashiyama et al., Phys. Rev. B 70, 174405 (2004). [15] J.-S. Lee, Ph. D. Thesis, POSTECH (2006). [16] S. Kobayashi et al., J. Phys. Soc. Jpn. 73, 3439 (2004). [17] H. Kawano-Furukawa et al., Phys. Rev. B. 65, 180508 (2002). [18] S. Y. Haam et al., Ferroelectrics 336, 153 (2006). [19] S.-H. Baek et al., Phys. Rev. B 74 140410(R) (2006). [20] R. Valdés Aguilar et al., Phys. Rev. B 74, 184404 (2006).

0704.0534

Self-diffusion and Interdiffusion in Al80Ni20 Melts: Simulation and Experiment

accepted for publication in Phys. Rev. B Self–diffusion and Interdiffusion in Al80Ni20 Melts: Simulation and Experiment J. Horbach1,2, S.K. Das1,3, A. Griesche4, M.-P. Macht4, G. Frohberg5, and A. Meyer2 (1)Institut für Physik, Johannes–Gutenberg–Universität Mainz, Staudinger Weg 7, 55099 Mainz, Germany (2)Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft– und Raumfahrt, 51170 Köln, Germany (3)Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA (4)Hahn–Meitner–Institut Berlin, Dept. Materials (SF3), Glienicker Str. 100, 14109 Berlin, Germany (5)Institut für Werkstoffwissenschaften und –technologien, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany (Dated: November 4, 2018) A combination of experimental techniques and molecular dynamics (MD) computer simulation is used to investigate the diffusion dynamics in Al80Ni20 melts. Experimentally, the self–diffusion coefficient of Ni is measured by the long–capillary (LC) method and by quasielastic neutron scat- tering. The LC method yields also the interdiffusion coefficient. Whereas the experiments were done in the normal liquid state, the simulations provided the determination of both self–diffusion and interdiffusion constants in the undercooled regime as well. The simulation results show good agreement with the experimental data. In the temperature range 3000K≥ T ≥ 715K, the interdif- fusion coefficient is larger than the self–diffusion constants. Furthermore the simulation shows that this difference becomes larger in the undercooled regime. This result can be refered to a relatively strong temperature dependence of the thermodynamic factor Φ, which describes the thermodynamic driving force for interdiffusion. The simulations also indicate that the Darken equation is a good approximation, even in the undercooled regime. This implies that dynamic cross correlations play a minor role for the temperature range under consideration. PACS numbers: 64.70.Pf, 61.20.Ja, 66.30.Hs I. INTRODUCTION Multicomponent liquids exhibit transport processes due to concentration fluctuations among the different components. In the hydrodynamic limit, these processes are described by interdiffusion coefficients [1, 2, 3]. In the simplest case of a binary AB mixture, there is one inter- diffusion coefficient DAB. This quantity plays an impor- tant role in many phenomena seen in metallic mixtures, such as solidification processes [4], the slowing down near the critical point of a liquid–liquid demixing transition [5] or glassy dynamics [6]. Many attempts have been undertaken for different bi- nary systems to relate DAB to the self–diffusion con- stants DA and DB via phenomenological formula (see e.g. [7, 8, 9, 10, 11, 12]). An example is the Darken equa- tion [13] that is widely used to estimate the interdiffusion constant of simple binary fluid mixtures. This equation expresses DAB as a simple linear combination of the self– diffusion coefficients, DAB = Φ(cBDA+ cADB) (with cA, cB the mole fractions of A and B particles, respectively). Here, the so–called thermodynamic factor Φ contains in- formation about static concentration fluctuations in the limit of long wavelength. The relationship between one–particle transport and collective transport properties is a fundamental question in undercooled liquids [6]. In the framework of the mode– coupling theory of the glass transition, Fuchs and Latz [14] have studied a binary 50:50 mixture of soft–spheres with a size ratio of 1.2. Their numerical data indicate that the Darken equation is a good approximation for the latter system in the undercooled regime. However, from experiments or computer simulations, not much is known about the validity of the Darken equation for undercooled liquids. This is due to the lack of experimental data for interdiffusion coefficients in this case. Moreover, most of the computer simulation studies on the relation between self–diffusion and interdiffusion have been only devoted to the normal liquid state. In this case, the Darken equa- tion often seems to work quite well [1, 10, 12, 15, 16]. In this work, a combination of experiment and molecu- lar dynamics (MD) simulation is used to study the diffu- sion dynamics in the metallic liquid Al80Ni20. In the MD simulation, the interactions between the atoms are mod- elled by an embedded atom potential proposed by Mishin et al. [17]. The present work is a continuation of a recent study [18], where a combination of quasielastic neutron scattering (QNS) and MD simulation was applied to in- vestigate chemical short–range order and self–diffusion in the system Al–Ni at different compositions. In the lat- ter study, we have shown that the MD simulation yields good agreement with the QNS data, both for structural quantities and the Ni self–diffusion constant, DNi. In the present work, an additional experimental method, the long–capillary (LC) technique, is used. This method al- lows to determine simultaneously the self–diffusion con- stant DNi and the interdiffusion coefficient DAB (see be- low). Above the liquidus temperature (i.e. in the normal liq- uid state), thermodynamic properties as well as structure and dynamics of Al80Ni20 have been studied by different approaches (see, e.g., [15, 19, 20, 21, 22, 23]). The Al– http://arxiv.org/abs/0704.0534v1 Ni system is an ordering system which is manifested in a negative enthalpy of mixing [24]. Thus, it does not exhibit a liquid–liquid miscibility gap where one would expect that the interdiffusion coefficient vanishes when approaching the critical point, whereas the self–diffusion constants are not affected by the critical slowing down (see [25] and references therein). Such a behavior is not expected for the system Al–Ni. In the computer simulation, the Al80Ni20 melt can be undercooled to an arbitrary extent avoiding the occur- rence of crystallization processes. Therefore, we were able to study a broad temperature range in our MD sim- ulations, ranging from the normal liquid state at high temperature to the undercooled liquid at low tempera- ture. In the experiments presented below crystallization occurs due to heterogeneous nucleation. Thus, the ex- periments were performed above the liquidus tempera- ture TL ≈ 1280K. The combination of experiment and simulation presented in this work allows for a test of the validity of the Darken equation in Al80Ni20. We will see below that this equation is indeed a good approximation, even in the undercooled regime. In the next section, we summarize the basic theory on self–diffusion and interdiffusion. The details of the exper- iments and simulation are given in Sec. III and Sec. IV, respectively. In Sec. V we present the results. Finally, we give a summary of the results in Sec. VI. II. SELF–DIFFUSION AND INTERDIFFUSION: BASIC THEORY Consider a three–dimensional, binary AB system of N = NA + NB particles (with NA, NB the number of A and B particles, respectively). The self–diffusion con- stant Ds,α (α = A,B) is related to the random–walk mo- tion of a tagged particle of species α on hydrodynamic scales. It can be calculated from the velocity autocorre- lation function [1], Cα(t) = (t) · v (0)〉 , (1) via a Green–Kubo integral: Ds,α = Cα(t)dt . (2) In Eq. (1), v j (t) is the velocity of particle j of species α at time t. The self–diffusion constant can be also expressed by long–time limit of the mean–squared displacement (MSD): Ds,α = lim (t)− r . (3) Here, r j (t) is the position of particle j of species α at time t. Note that Eq. (3) is equivalent to the Green– Kubo formula (2). Interdiffusion is related to the collective transport of mass driven by concentration gradients. The transport coefficient that describes this process is the interdiffusion constant DAB which can be also expressed by a Green– Kubo relation, i.e. by a time integral over an autocorre- lation function. The relevant variable in this case is the concentration or interdiffusion current [1] given by JAB(t) = i (t)− cA i (t) + i (t) where cA ≡ NA/N = 1 − cB is the total concentration (mole fraction) of A particles. As a matter of fact, the au- tocorrelation function of the variable JAB(t) depends on the reference frame and fluctuations of JAB(t) have to be adapted to the ensemble under consideration. Whereas experiments are usually done in the canonical ensemble, in a molecular dynamics simulation, the natural ensemble is the microcanonical ensemble with zero total momen- tum [26]. Thus, i = − i (5) follows, where mA and mB denote the masses of A and B particles, respectively. Introducing the “centre of mass velocity of component α (α = A,B)” by Vα(t) = i (t) , (6) we can use expression (5) to simplify the formula for the interdiffusion current, JAB(t) = NcBcA VA(t) . (7) Thus, we have to consider only the velocities of one species to compute JAB(t). Now, the autocorrelation function for the interdiffusion current is given by CAB(t) = 〈JAB(t) · JAB(0)〉 (8) = N2 (cBcA) 〈VA(t) ·VA(0)〉 . The Green–Kubo formula for DAB reads DAB = 3NScc(0) CAB(t) dt (9) where Scc(0) is the concentration–concentration struc- ture factor in the limit q → 0. The function Scc(q) is the static correlation function associated with concentration fluctuations. It can be expressed by a linear combination of partial static structure factors Sαβ(q) (α, β = A,B) as follows [1]: Scc(q) = c BSAA(q) + c ASBB(q)− 2cAcBSAB(q) (10) Sαβ(q) = 〈exp [iq · (rk − rl)]〉 . (11) Using elementary fluctuation theory [1], Scc(0) can be related to the second derivative of the molar Gibbs free energy g, ∂cA∂cB , (12) Scc(q = 0) . (13) In Eq. 12, kB is the Boltzmann constant and T the tem- perature. In the following, we will refer to Φ as the ther- modynamic factor. We note that the total structure factor for the number density, Snn(q), and cross correlation between number density and concentration, Snc(q), can also be written as a linear combinations of partial structure factors. These functions are given by [1] Snn(q) = SAA(q) + 2SAB(q) + SBB(q) , (14) Snc(q) = cBSAA(q)− cASBB(q) + (cB − cA)SAB(q) .(15) The typical behavior of these functions for a liquid mix- ture will be discussed in the result’s section. The func- tions Snn(q), Snc(q) and Scc(q) are often called Bhatia– Thornton structure factors [27]. In principle, these func- tions can be determined in neutron scattering experi- ments, either by using isotopic enrichment techniques (see, e.g., Ref. [21]) or by applying a combination of neu- tron scattering and X–ray diffraction [28]. With Eqs. (9) and (13), the interdiffusion constant can be written as DAB = N cAcBΦ )2 ∫ ∞ 〈VA(t) ·VA(0)〉 dt . Alternatively, DAB can be also easily related to the self– diffusion constants to yield DAB = Φ(cADB + cBDA + cAcB [ΛAA + ΛBB − 2ΛAB] dt) , (17) where the functions Λαβ(t) denote distinct velocity cor- relation functions, Λαβ(t) = 3Ncαcβ l 6=k if α=β (t) · v Note that the three functions Λαβ(t) can be expressed by the “centre–of–mass” correlation function CAB(t) and the velocity autocorrelation functions Cα(t) (the latter, multiplied by 1/cα, has to be subtracted in the case of ΛAA(t) and ΛBB(t)) [11]. Thus, the functions Λαβ(t) do not contain any additional information compared to CAB(t) and Cα(t) and so we do not consider them sepa- rately in the following. If one denotes the distinct part in (17) by ∆d = cAcB [ΛAA(t) + ΛBB(t)− 2ΛAB(t)] dt (19) one can rewrite Eq. (17), DAB = ΦS (cADB + cBDA) , (20) S = 1 + cADB + cBDA The quantity S measures the contribution of cross cor- relations to DAB. If S = 1 holds, the interdiffusion con- stant is determined by a linear combination of the self– diffusion constants. In this case, Eq. (20) leads to the Darken equation [13]. Note that, in the context of chem- ical diffusion in crystals, S is called Manning factor [29]. As in the case of self–diffusion, the interdiffusion con- stant can be also expressed via a mean–squared displace- ment which involves now the centre–of–mass coordinate of species A, RA(t) = j (t) . (22) Then, the “Einstein relation” for DAB reads DAB = lim NcAcBΦ [RA(t)−RA(0)] This formula can be used to determine DAB in a com- puter simulation, where the system is located in a simu- lation box with periodic boundary conditions. However, in this case one has to be careful because the difference RA(t) − RA(0) has to be calculated in an origin inde- pendent representation [30]. This can be achieved by computing this difference via the integral ′)dt′. III. EXPERIMENTAL METHODS A. Long–capillary technique The long–capillary technique (LC) has been used to measure interdiffusion and Ni self–diffusion in liquid Al80Ni20. The sample material production is similar to that of Al87Ni10Ce3, which is described in Ref. [31]. The -2.0 -1.0 0.0 1.0 2.0 x (10 FIG. 1: Typical concentration profiles of a combined interdif- fusion and self–diffusion experiment. The squares denote the Al and Ni concentrations measured by energy-dispersive X– ray spectrometry (EDS) and the dots denote the 62Ni concen- tration measured by inductively–coupled plasma mass spec- trometry (ICP–MS). The lines represent the best fit (least– square method) of the appropriate solution of Fick’s diffusion equations to the measured concentrations. experimental apparatus, the measurement of the concen- tration profiles and the evaluation of the concentration profiles, including the determination of Fick’s diffusion coefficients, are also described elsewhere [32, 33]. Thus, here the experimental technique is reported only briefly. In more detail we describe an improved diffusion cou- ple setup, which has been used in this work. This setup, with a vertical diffusion capillary of 1.5mm diameter, has an increased stabilization against natural convection and minimizes the systematic error of convective mass flow contributions to the total mass transport. The improvement of the diffusion couple setup implies the combination of interdiffusion and self–diffusion mea- surements in one experiment. An Al80Ni20 slice of 2mm thickness, containing the enriched stable 62Ni isotope, is placed between both rods of an interdiffusion couple. The interdiffusion couple consists of a 15mm long rod of Al85Ni15, placed above the slice, and a 15mm long rod of Al75Ni25, placed below the slice. This configuration al- lows the development of an error function shaped chem- ical interdiffusion profile simultaneously to the develop- ment of a Gauss function shaped self–diffusion profile. In a first approximation the diffusion of the enriched stable isotope takes place at the mean concentration Al80Ni20 without influence of the changing chemical composition of the melt in the diffusion zone. The only necessary cor- rection results from the mass spectrometric measurement of the self–diffusion profile. Here the measured isotope incidences i(62Ni) of 62Ni have to be corrected for the overlaying chemical concentration profile of natural Ni, cNi, by using the following formula: c(62Ni) = cNi i(62Ni)− i(62Ni0) with i(62Ni0) the natural incidence of 62Ni and c(62Ni) the concentration of this Ni isotope with respect to all Ni isotopes. Typical concentration profiles of a diffusion experiment are given in Fig. 1. The diffusion couple configuration minimizes the risk of convection compared to conventional self–diffusion ex- periments in pure melts because of the solutal stabilized density profile of the melt column. This stabilizing ef- fect has been described in Refs. [34, 35]. In a standard self–diffusion experiment without chemical gradient the solutal stabilization effect is only due to the enrichment of a tracer. As a test for other mass transport processes we mea- sured the mean–square penetration depth x̄2 of interdif- fusion as a function of time t. We found a deviation from the linear behavior x̄2 = 2DABt. This has been iden- tified as sedimentation of Al3Ni2 during solidification of the diffusion sample. This additional mass transport was simply corrected by subtracting this contribution as an off–set of the measured total mass transport. This pro- cedure adds a 5-10% error to the uncertainty of the dif- fusion coefficient. The total error in the long–capillary measurements of the self– and interdiffusion coefficients is about 30–40%. B. Neutron scattering experiments The second experimental technique used in this work is quasielastic neutron scattering. In this case, the Al80Ni20 alloy was prepared by arc melting of pure elements un- der a purified Argon atmosphere. The measurements were done at the time–of–flight spectrometer IN6 of the Institut Laue-Langevin. The standard Nb resistor high temperature vacuum furnace of the ILL exhibits a tem- perature gradient over the entire sample at 1800K that was less than five degrees and a temperature stability within one degree. For the scattering experiment we used a thin–walled Al2O3 container that provides a hol- low cylindrical sample geometry of 22mm in diameter and a sample wall thickness of 1.2mm. An incident neutron wavelength of λ = 5.1 Å yielded an energy resolution of δE ≃ 92µeV (FWHM) and an accessible wave number range at zero energy transfer of q = 0.4 − 2.0 Å . Measurements were performed at 1350K, 1525K, 1670K and 1795K in 2 hour runs each. A run at room temperature provided the instru- mental energy resolution function. The scattering law S(q, ω) was obtained by normalization to a vanadium standard, accompanied by a correction for self absorp- tion and container scattering, and interpolation to con- stant wave numbers q. Further, S(q, ω) was symmetrized with respect to the energy transfer ~ω by means of the detailed balance factor. -1.0 0.0 1.0 2.0 hω (meV) T=1525K, q=1.0Å T=300K, q=0.8Å FIG. 2: Normalized scattering law of liquid Al80Ni20. The data at 300K represent the instrumental energy resolution function. The line is a fit with a Lorentzian function that is convoluted with the instrumental energy resolution function. Diffusive motion of the atoms leads to a broadening of the quasielastic signal from which the Ni self–diffusivity can be obtained on an absolute scale. Figure 2 displays S(q, ω) at q = 1.0 Å of liquid Al80Ni20 at 1525K and the crystalline alloy at 300K at q = 0.8 Å . Diffusive motion in the liquid leads to a broadening of the quasielastic signal. The data were fit- ted with an Lorentzian function that is convoluted with the instrumental energy resolution function. From the full width at half maximum of the quasielastic line Γ a q– dependent diffusion coefficient D(q) can be computed via D(q) = Γ/(2~q2). Towards small q incoherent scattering on the Ni atoms dominates the signal and the diffusion coefficient D(q) becomes constant yielding an estimate of Ds,Ni. Thus, the self–diffusion constant Ds,Ni can be determined on an absolute scale [36, 37]. IV. DETAILS OF THE SIMULATION For the computer simulations of the binary system Al80Ni20, we used a potential of the embedded atom type that was recently derived by Mishin et al. [17]. In a re- cent publication [18], we have shown that this potential reproduces very well structural properties and the self– diffusion constant of Al–Ni melts at various compositions. The present simulations are performed in a similar way as the ones in the latter work: Systems of N = 1500 particles (NNi = 300, NAl = 1200) are put in a cubic simulation box with periodic boundary conditions. First, standard Monte–Carlo (MC) simulations in the NpT en- semble [38] were used to fully equilibrate the systems at zero pressure and to generate independent configurations for MD simulations in the microcanonical ensemble. In the latter case, Newton’s equations of motion were in- tegrated with the velocity Verlet algorithm using a time step of 1.0 fs at temperatures T ≥ 1500K and 2.5 fs at lower temperatures. The masses were set to 26.981539amu and 58.69 amu for aluminum and nickel, respectively. At each tempera- ture investigated, we made sure that the duration of the equilibration runs exceeded the typical relaxation times of the system. The temperatures considered were 4490K, 2994K, 2260K, 1996K, 1750K, 1496K, 1300K, 1100K, 998K, 940K, 893K, 847K, 810K, 777K, 754K, 735K, 715K, 700K, 680K, and 665K. In order to improve the statistics of the results we averaged at each temperature over eight independent runs. At the lowest tempera- ture, the duration of the microcanonical production runs were 40 million time steps, thus yielding a total simu- lation time of about 120ns. The latter production runs were used to study the tagged particle dynamics. For the calculation of the interdiffusion constant DAB addi- tional production runs were performed in the tempera- ture range 4490K ≥ T ≥ 715K that extended the pro- duction runs for the tagged particle dynamics by about a factor of ten. This was necessary in order to obtain a rea- sonable statistics for DAB. Note that DAB is a collective quantity that does not exhibit the self–averaging prop- erties of the self–diffusion constant and thus it is quite demanding to determine transport coefficients such as the interdiffusion constant or the shear viscosity from a MD simulation. V. RESULTS In Eq. (20), the interdiffusion constant DAB is ex- pressed as a linear combination of the self–diffusion con- stants. The prefactor in this formula is a product of the thermodynamic factor Φ and the Manning factor S. Whereas Φ can be computed from structural input, the Manning factor contains the collective dynamic correla- tions in the expression for DAB (see Sec. II). In the following, we compare the simulated diffusion constants for Al80Ni20 to those from experiments. Moreover, the simulations are used to disentangle differences between self–diffusion constants and the interdiffusion constants with respect to the thermodynamic quantity Φ and the dynamic quantity S. First, we discuss static structure factors at different temperatures, as obtained from the MD simulation. Fig- ure 3 displays the different partial structure factors at the temperatures T = 2000K and T = 750K. At both temperatures, a broad prepeak around the wavenumber q = 1.8 Å−1 emerges in the NiNi correlations, which indicates the presence of chemical short–ranged order (CSRO). This feature is absent in the AlAl correlations. In a recent work [18], we have found that the prepeak in SNiNi(q) is present in a broad variety of Al–Ni compo- sitions, ranging from xNi = 0.1 to xNi = 0.9. However, 0.0 2.0 4.0 6.0 8.0 10.0 12.0 , T=2000Ka) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 , T=750Kb) FIG. 3: Partial structure factors, as obtained from the MD simulation, for a) T = 2000K and b) T = 750K. The multipli- cation by 1/(cαcβ) 1/2 is introduced to increase the amplitude of SNiNi(q) relative to that of SAlAl(q). Note that the factor 1/(cαcβ) 1/2 leads also to the asymptotic value Sαα(q) = 1 for q → ∞. the width of the prepeak decreases significantly with in- creasing Ni concentration and, in melts with a high Ni concentration, it appears also in SAlAl(q). The prepeak in Sαβ(q) describes repeating structural units involving next–nearest αβ neighbors which are built in inhomoge- neously into the structure. Of course, for the Al rich sys- tem Al80Ni20 considered in this work, only next–nearest Ni–Ni units exhibit the CSRO that is reflected in the prepeak. From the partial static structure factors, the Bhatia– Thornton structure factors can be determined according to Eqs. (10), (14) and (15). These quantities are shown in Fig. 4, again at T = 2000K and at T = 750K. Although these structure factors look very different for q > 2 Å−1, they are essentially identical in the limit q → 0. As we have indicated before, the static susceptibility, asso- ciated with concentration fluctuations, can be extracted from the structure factor Scc(q) in the limit q → 0. As we can infer from Fig. 4, at the temperature T = 750K the value of this susceptibility is very small. The small 0.0 2.0 4.0 6.0 8.0 10.0 12.0 , T=2000Ka) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 , T=750Kb) FIG. 4: Bhatia–Thornton structure factors, as obtained from the MD simulation, at a) T = 2000K and b) T = 750K. value of Scc(q = 0) reveals that concentration fluctua- tions on large length scales are strongly suppressed. This is the typical behavior of a dense fluid that exhibits a strong ordering tendency. In contrast, at a critical point of a demixing transition a divergence of Scc(q = 0) is expected. As we have seen in Sec. II, the ratio DAB/Φ can be expressed as a linear combination of the self–diffusion constants, provided S = 1 holds. In order to quantify the temperature dependence of S, we first define the fol- lowing mean–squared displacements: 〈r2(t)〉int = NcAcB × ×〈[RA(t)−RA(0)] 〉 (25) 〈r2(t)〉self = cA j (t)− r j (0) j (t)− r j (0) 〉 (26) Whereas the interdiffusion constant can be calculated t (ps) interdiffusion selfdiffusion 3000K FIG. 5: Simulation results of mean squared displacements (MSD) for self–diffusion (dashed lines) and interdiffusion (solid lines) for the temperatures T = 3000K, 2000K, 1500K, 1000K, 850K, 750K, 715K, and 665K (corresponding to the curves from left to right. Note that for T = 665K only 〈r2(t)〉self was calculated. For the definitions of the MSD’s see Eqs. (25) and (26). via DAB = limt→∞Φ〈r 2(t)〉int/(6t), the equation DAB = limt→∞Φ〈r 2(t)〉self/(6t) is only correct for S = 1. Figure 5 shows the quantities 〈r2(t)〉int and 〈r 2(t)〉self for the dif- ferent temperatures. Both MSD’s show a very similar be- havior. At high temperature, a crossover from a ballistic regime (∝ t2) at short times to a diffusive regime (∝ t) at long times can be seen. At low temperature, a plateau– like region develops at intermediate times, i.e. between the ballistic and the diffusive regime. With decreasing temperature, the plateau becomes more pronounced. In 〈r2(t)〉self , the plateau indicates the so–called cage ef- fect [6]. The tagged particle is trapped by its neighbors on a time scale that increases with decreasing tempera- ture. Although the MSD for the interdiffusion, 〈r2(t)〉int, describes also collective particle transport, the plateau in this quantity has the same origin: The particles are “arrested” on intermediate time scales. Moreover, the differences between 〈r2(t)〉self and 〈r 2(t)〉int are anyway very small in the whole time and temperature range un- der consideration. This means that the cross correlations do not give a large contribution to 〈r2(t)〉int. From the MSD’s in Fig. 5, the Manning factor S can be extracted using Eq. (21). In Fig. 6 we see that the Manning factor varies only slightly over the whole tem- perature range, located around values between 0.8 and 1.0. Also shown in Fig. 6 is the thermodynamic factor Φ and the product ΦS. We have extracted Φ from the ex- trapolation of the structure factors Scc(q) toward q → 0 [see Eq. (13)]. In contrast to the Manning factor S, the thermodynamic factor Φ increases significantly with de- 750 1250 1750 2250 2750 T (K) S(T)φ(T) FIG. 6: Thermodynamic factor φ, “Manning” factor S(T ), and the product of both as obtained from the simulation. creasing temperature and thus, also the change in the product ΦS is dominated by the change in Φ. There- fore, differences in the qualitative behavior between the self–diffusion constants and the interdiffusion constant are dominated by the thermodynamic factor. An Arrhenius plot of the diffusion constants as ob- tained from simulation and experiment is shown in Fig. 7. The self–diffusion constants DNi and DAl from the simu- lation are very similar over the whole temperature range 4490K≥ T ≥ 665K. In a recent publication [18], we have found that, in the framework of our simulation model, this similarity of the self–diffusion constants occurs in Al rich compositions of the system Al–Ni, say for cAl > 0.7. Whether this is also true in real systems is an open ques- tion. However, the neutron scattering results for DNi as well as the single point obtained from the LC measure- ment is in very good agreement with the simulation data. Asta et al. [15] have computed the concentration de- pendence of the self–diffusion constants at T = 1900 K using two different embedded atom potentials, namely the one proposed by Voter and Chen [39] and the one proposed by Foiles and Daw [40]. For both potentials, they find very similar values forDNi andDAl in Al80Ni20, in agreement with our results. However, their results for the Ni diffusion constant are significantly higher than the ones found in our quasielastic neutron scattering ex- periment and our simulation. They report the values DNi ≈ 1.5 · 10 −8m2/s and DNi ≈ 1.9 · 10 −8m2/s for the Voter–Chen potential and the Foiles–Daw potential, re- spectively, whereas we obtainDNi ≈ 10 −8m2/s from sim- ulation and experiment. Thus, the potential proposed by Mishin et al. [17], which is used in this work, leads to a better agreement with the experiment, as far as self– diffusion in Al80Ni20 is concerned. We emphasize that the statistical error in both the neutron scattering data and the simulation data for the self–diffusion constants is relatively small. In both cases, the error bars for the corresponding data points in Fig. 7 0.3 0.6 0.9 1.2 1.5 1000/T (K sim. D sim. D sim. D QNS D 0.5 0.6 0.7 0.8 1000/T (K sim. D sim. D sim. D QNS D LC exp. D FIG. 7: Arrhenius plot of interdiffusion and self–diffusion con- stants, as obtained from experiment and simulation, as indi- cated. The experimental results are measured by quasielastic neutron scattering (QNS) and by the LC technique. The lines through the data points are guides to the eye. The vertical dotted line in a) marks the location of the experimental liq- uidus temperature, TL ≈ 1280K. The vertical dashed line is at the location of the critical temperature of mode coupling theory, Tc ≈ 700K, as estimated by the MD simulation [41]. Panel b) is an enlargement of the data of panel a) in a tem- perature range above TL. The error bars of simulation and QNS data are of the order of the size of the symbols. are smaller than the size of the symbols. Due to the lack of self–averaging, it is much more dif- ficult to yield accurate results for DAB from the simu- lation. Therefore, in this case we considered a smaller temperature range than for the self–diffusion constants to yield results with reasonable accuracy. As we can in- fer from Fig. 7, the interdiffusion constant is larger than the self–diffusion constants over the whole temperature range. The difference becomes more pronounced with decreasing temperature. At T = 715K, the diffusion co- efficient DAB is about a factor of 3 larger than DNi and DAl. This behavior is of course due to the increase of the thermodynamic factor Φ at low temperature. Also included in Fig. 7 are the results of the LC measurements of DAB and DNi. These results are much less accurate than those of the quasielastic neutron scattering experi- ments for the determination of DNi (see the error bars for the LC data in Fig. 7b). Nevertheless, the LC data show that DAB > DNi holds, in agreement with the simulation results. VI. CONCLUSION A combination of experiment and molecular dynamics (MD) simulation has been used to investigate the diffu- sion dynamics in liquid Al80Ni20. We find good agree- ment between simulation and experiment. Both in ex- periment and in simulation, the interdiffusion constant is higher than the self–diffusion constants. This is valid in the whole temperature range considered in this work, i.e. in the normal liquid state as well as in the undercooled regime. In the latter regime (which is only accessible by the simulation), the difference between the interdiffusion constant and the self–diffusion constants increases with decreasing temperature. All these observations can be clarified by the detailed information provided by the MD simulation. Both the thermodynamic factor Φ and the Manning factor S have been estimated directly and accurately over a wide tem- perature range, as well as self–diffusion and interdiffusion coefficients. The central result of this work is shown in Fig. 6. Whereas the thermodynamic factor Φ increases significantly by lowering the temperature, the Manning factor S shows only a weak temperature dependence. Moreover, the value of S is close to one which means that dynamic cross correlations are almost negligible and thus, even in the undercooled regime, the Darken equation is a good approximation. The temperature dependence of Φ is plausible for a dense binary mixture with a strong ordering tendency. The situation is similar to the case of the isothermal compressibility which normally decreases with temperature in a densely packed liquid leading to very low values in the undercooled regime. In the same sense, the response to a macroscopic concentration fluc- tuation described by Scc(q = 0) tends to become smaller and smaller towards the undercooled regime which corre- sponds to an increase of Φ with decreasing temperature (since Φ ∝ 1/Scc(q = 0)). We note that the data shown for DAB are all above the critical temperature Tc of mode coupling theory which is around 700K for our simulation model (see Fig. 7) [41]. Since it is expected that the transport mechanism changes below Tc [6], it would be interesting to see how such a change in the transport mechanism is reflected in the interdiffusion constants. This issue is the subject of forthcoming studies. Acknowledgments We are grateful to Kurt Binder for stimulating dis- cussions and a critical reading of the manuscript. We gratefully acknowledge financial support within the SPP 1120 of the Deutsche Forschungsgemeinschaft (DFG) un- der grants Bi314/18, Ma1832/3-2 and Me1958/2-3 and from DFG grant Gr2714/2-1. One of the authors was supported through the Emmy Noether program of the DFG, grants Ho2231/2-1/2 (J.H.). Computing time on the JUMP at the NIC Jülich is gratefully acknowledged. [1] J.–P. Hansen and I.R. McDonald, Theory of Simple Liq- uids (Academic Press, London, 1986). 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0704.0535

The Formation of Globular Cluster Systems in Massive Elliptical Galaxies: Globular Cluster Multimodality from Radial Variation of Stellar Populations

to be published in ApJ The Formation of Globular Cluster Systems in Massive Elliptical Galaxies: Globular Cluster Multimodality from Radial Variation of Stellar Populations Antonio Pipino1,3, Thomas H. Puzia2,4, & Francesca Matteucci3 ABSTRACT The most massive elliptical galaxies show a prominent multi-modality in their globular cluster system color distributions. Understanding the mechanisms which lead to multiple globular cluster sub-populations is essential for a complete pic- ture of massive galaxy formation. By assuming that globular cluster formation traces the total star formation and taking into account the radial variations in the composite stellar populations predicted by the Pipino & Matteucci (2004) multi-zone photo-chemical evolution code, we compute the distribution of glob- ular cluster properties as a function of galactocentric radius. We compare our results to the spectroscopic measurements of globular clusters in nearby early- type galaxies by Puzia et al. (2006) and show that the observed multi-modality in globular cluster systems of massive ellipticals can be, at least partly, ascribed to the radial variation in the mix of stellar populations. Our model predicts the presence of a super-metal-rich population of globular clusters in the most massive elliptical galaxies, which is in very good agreement with the spectroscopic obser- vations. The size of this high-metallicity population scales with galaxy mass, in the sense that more massive galaxies host larger such cluster populations. We predict an increase of mean metallicity of the globular cluster systems with host galaxy mass, and forecast that those clusters that were formed within the initial 1Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, U.K. email: [email protected]. 2Herzberg Institute of Astrophysics, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada, email: [email protected]. 3Dipartimento di Astronomia, Università di Trieste, Via G.B. Tiepolo 11, 34100 Trieste, Italy, email: [email protected]. 4Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA. http://arxiv.org/abs/0704.0535v1 – 2 – galaxy halo closely follow an age-metallicity and α/Fe-metallicity relation, where older clusters exhibit lower metallicities and higher α/Fe ratios. Furthermore, we investigate the impact of other non-linear mechanisms that shape the metallicity distribution of globular cluster systems, in particular the role of merger-induced globular cluster formation and a non-linear color-metallicity transformation, and discuss their influence in the context of our model. We find that a non-linear color-metallicity relation may be partly responsible for a color multi-modality. On the other hand, the formation of globular clusters from subsequently ac- creted gas, either with primordial abundances or solar metallicity, delivers model predictions which are at variance with the observations, and suggests that a sig- nificant fraction of metal-poor globular clusters is accreted from satellite halos which formed their globular clusters independently. Subject headings: galaxies: elliptical and lenticular, cD — galaxies: formation — galaxies: star cluster — globular clusters: general — galaxies: structure 1. Introduction 1.1. The Multimodality of Globular Cluster Systems One of the most significant developments in the study of extragalactic globular cluster systems (GCSs) was the discovery of bimodality in their color distributions (see Ashman & Zepf 1998; Harris 2001; West et al. 2004 and references therein). Today, we generally refer to glob- ular clusters (GCs) belonging to the blue peak of the color distribution as metal-poor GCs and to the red-peak members as the metal-rich sub-population. It is generally considered that the presence of multiple modes implies multiple distinct GC formation epochs and/or mechanisms and ties those directly into formation scenarios that have to describe the par- allel assembly histories of GCSs and the diffuse stellar populations in their host galaxies. In massive early-type galaxies the current GCS assembly paradigms view the origin of the two color peaks from the perspective of either episodic star-cluster formation bursts triggered by gas-rich galaxy mergers (e.g. Ashman & Zepf 1992), temporarily interrupted cluster forma- tion (so-called in-situ formation, e.g. Forbes et al. 1997; Harris et al. 1998), and star-cluster accretion as a result of the hierarchical assembly of galaxies (e.g. Côté et al. 1998). While the majority of GCSs in early-type galaxies show clearly bimodal color distri- butions, the general picture is much more complex, ranging from purely blue to purely red color distributions (e.g. Gebhardt & Kissler-Patig 1999; Kundu & Whitmore 2001a,b; Larsen et al. 2001; Peng et al. 2006). This complexity is exacerbated by the fact that color – 3 – bimodality is a function of galaxy mass and morphology, as less massive and later-type galaxies tend to have single-mode blue (i.e. metal-poor) GC populations (e.g. Lotz et al. 2004; Sharina et al. 2005; Peng et al. 2006). Furthermore, color bimodality is also a func- tion of galactocentric distance and is mainly due to the more extended spatial distribution of the metal-poor sub-population relative to metal-rich clusters (e.g. Harris & Harris 2002; Rhode & Zepf 2004; Dirsch et al. 2003, 2005). 1.2. Numerical Models of Globular Cluster System Formation The aspect of GCS formation and assembly entered recently the domain of numerical simulations of galaxy formation due to the increasing spatial resolution of these computa- tions. For instance, Li et al. (2004) model GC formation by identifying absorbing sink parti- cles in their smoothed particle hydrodynamics (SPH) high-resolution simulation of isolated gaseous disks and their mergers. They find a bimodal globular-cluster metallicity distribution in their merger remnant under the assumption of a particular age-metallicity relation. A key finding of their merger simulation is a more concentrated spatial distribution of metal-rich GCs with respect to the metal-poor sub-population in good agreement with observations. Since their models of isolated galaxies produce a smooth age distribution (implying a smooth metallicity and color distribution), Li et al. conclude that mergers are required to produce a bimodal metallicity (i.e. color) distribution. In a more detailed adaptive-grid cosmological simulation, Kravtsov & Gnedin (2005) followed the formation of a star-cluster system during the early evolution of a Milky Way-size disk galaxy to redshift z=3. Their model could reproduce the extended spatial distribution of metal-poor halo globular clusters as observed in M31 and the Milky Way. However, because their simulation does not follow the later evolution at z < 3 it is unclear whether it would produce a metallicity bimodality and any significant age-metallicity relation. An alternative, more statistical approach to modeling GCS assembly is to directly link the mode of GC formation to the star-formation rate in semi-analytic models. Beasley et al. (2002) were the first to explore this path by assuming that metal-poor GCs form in gaseous proto-galactic disks while metal-rich GCs are created during gaseous merger events. Their study showed that the observed globular-cluster color bimodality can only be reproduced by artificially stopping the formation of metal-poor GCs at redshifts z & 5. By construction, no spatial information on metal-rich and/or metal-poor GCs is provided in these models. – 4 – 1.3. A Spatially Resolved Chemical Evolution Model for Spheroid Galaxies Recently, Pipino & Matteucci (2004, hereafter PM04) presented a spatially resolved chemical evolution model for the formation of spheroids, which successfully reproduces a large number of photo-chemical properties that could be inferred from either the optical or from the X-ray spectra of the light coming from ellipticals. The model includes an initial gas infall and a subsequent galactic wind; it takes into account detailed nucleosynthesis prescriptions of both type-II and Ia supernovae as well as low and intermediate-mass stars. It has been extensively tested against the main photo-chemical properties of nearby ellipticals, including the observed increase of the α-enhancement in their stellar populations with galaxy mass (e.g. Worthey et al. 1992; Weiss et al. 1995). This is at variance with standard models based on the hierarchical merging paradigm, which do not reproduce this trend (Thomas et al. 2002). Since the PM04 model provides full radial information on the composite nature of stellar populations that make up elliptical galaxies, the observation of different GC sub- populations is, therefore, a new sanity check for the validity of this model. Moreover, we recall that PM04 and, more recently, Pipino et al. (2006, hereafter PMC06) suggested that elliptical galaxies should form outside-in, namely the outermost regions form faster as well as develop an earlier galactic wind with respect to the central parts (see also Martinelli et al. 1998). This mechanism implies that the stars in massive spheroids form a Composite Stellar Population (CSP), whose chemical properties, in particular their metallicity distribution, changes with galactocentric distance. Starting with the assumption that GC sub-populations trace the components of CSPs, we will show how the observed multi-modality in GCSs can be ascribed to the radial vari- ation in the underlying stellar populations. In particular, the observed GCSs are a linear combination of GC sub-populations inhabiting a given projected galactocentric radius. The paper is organized as follows: in Section 2 we briefly describe the adopted theo- retical model; in Section 3 we compare the predictions with observations and discuss the implications, while Section 4 presents the final conclusions. 2. The model 2.1. The Chemical Evolution Code The chemical evolution code for elliptical galaxies adopted here is described in PM04, where we refer the reader for more details. In this work, we present the results for a galaxy with Mlum ∼ 10 11M⊙, taken from PM04’s Model IIb. This model is characterized – 5 – by a Salpeter (1955) IMF, Thielemann et al. (1996) yields for massive stars, Nomoto et al. (1997) yields for Type-Ia SNe, and van den Hoek & Groenewegen (1997) yields for low- and intermediate-mass stars. An important feature of the PM04 model is its multi-zone nature, namely the model galaxy is divided into several non-interacting spherical shells of radius ri, which facilitate a detailed study of the radial variation of the photo-chemical properties of the GCS and its host galaxy. In each zone i, the equations for the chemical evolution of 21 chemical elements are solved (see PM04, Matteucci 2001). The model assumes that the galaxy assembles by merging of gaseous lumps on short timescales. The chemical composition of the lumps is assumed to be primordial. In fact, our model assumes that the accretion of primordial gas from the surroundings1 is more efficient in more massive systems, given their higher cross section per unit mass (see PM04). The model galaxy suffers a strong starburst which injects a large amount of energy into the interstellar medium, able to trigger a galactic wind, occurring at different times at different radii, mainly due to the radial variation of the potential well, which is shallower in the galactic outskirts. After the onset of wind activity the star formation is assumed to stop and the galaxy evolves passively with continuous mass loss. In order to correctly evaluate the amount of energy driving the wind, a detailed treatment of stellar feedback is included in the code (that takes into account the stellar lifetimes). In particular, the energy restored to the interstellar medium by both Type-Ia and Type-II supernovae has been calculated in a self-consistent manner according to the time of explosion of each supernova and the characteristics of the ambient medium (see PM04 for details). The potential well that keeps the gas bound to the galaxy is assumed to be dominated by a diffuse and massive halo of Dark Matter surrounding the galaxy. In the following we adopt the standard star formation rate ψ∗(t, ri) = ν · ρgas(ri, t) before the onset of the galactic wind (tgw), where ρgas is the gas density, ν the star-formation efficiency; otherwise we assume that ψ∗(t > tgw, ri) = 0. We recall here that the adopted star-formation efficiency is ν = 10Gyr−1, while the infall timescale is τ = 0.4Gyr in the galactic core and τ = 0.01Gyr at one effective radius (of the diffuse light, Reff), respectively. These values were chosen by PM04 in order to reproduce the majority of the chemical and photometric properties of ellipticals such as: the 〈[Mg/Fe]〉−σ (e.g. Faber et al. 1992), the Color-Magnitude (e.g. Bower et al. 1992), the Mass-Metallicity relation (e.g. Gallazzi et al. 2005) as well as the observed gradients in metallicity (e.g. Carollo et al. 1993), 〈[Mg/Fe]〉 1Since we lack a cosmological framework we cannot further specify the properties of the infalling primordial – 6 – (e.g. Mendez et al. 2005), and color (e.g. Peletier et al. 1990). Pipino et al. (2005) recently extended this model to explain also the properties of hot X-ray emitting halos surrounding more massive spheroids. 2.2. Globular Cluster Formation The formation rate of GCs, ψGC , in the i-th shell is assumed to be directly linked to its star formation rate ψ∗(t, ri, Zi) via a suitable function of time t, radius ri, and metallicity Z, which represents some scaling law between star formation rate ψ∗ and the star cluster formation ψGC and can be regarded as a GC formation efficiency. A similar relation be- tween the average star formation rate per surface area and the star cluster formation was recently found by Larsen & Richtler (2000) to hold in nearby spiral galaxies. In addition, the efficiency of cluster formation in massive ellipticals appears to be constant, where the mass ratio between the mass in star clusters and the baryons locked in field stars+gas is ǫGC≈ 0.25% (McLaughlin 1999). Here we extend this surface density relation to 3-D space. Moreover, PMC06 showed that at a given galactocentric radius model galaxies are made of a CSP, namely a mixture of several simple stellar populations (SSPs) each with a single age and chemical composition. The CSP reflects the chemical enrichment history of the entire system, weighted by the star formation rate. We define the stellar metallicity distribution Υ∗ as the distribution of stars belonging to a given CSP as a function of [Z/H]. We can then write the globular-cluster metallicity distribution ΥGC at a given radius ri and time t as: ΥGC(t, ri, Z) = f(t, ri, Z) ·Υ∗(t, ri, Z) , (1) where f includes all the information pertaining to the connection between ψGC and ψ∗. It is not trivial, and beyond the scope of the paper, to find an explicit definition for f(t, ri, Zi), which basically carries the information on the internal physics of gas clouds where GCs are expected to form. In the following we will show that, for a few and sensible choices of f , the observed multi-modality in the color distribution of globular clusters may be driven by the radial variations in the stellar population mix of ellipticals. We will first adopt a constant function f (see Sec. 3.1) and then allow f to mildly vary with Z (see Sec. 3.2). No absolute values for f will be given, since our formalism deals with normalized distributions. In particular, the total ΥGC summed over all radial shells can be written as: ΥGC,tot(t, Z) = f(t, ri, Z) ·Υ∗(t, ri, Z) . (2) – 7 – Similar equations hold for other GC distributions as a function of either [Mg/Fe] or [Fe/H]. At this stage it is useful to recall that Υ∗(t, ri, Z) can be represented in two following ways: i) as the fraction of mass of a CSP which is locked in stars at any given metallicity (Pagel & Patchett 1975; Matteucci 2001). In the following we refer to this stellar metal- licity distribution as Υ∗,m (MSMD: mass-weighted stellar metallicity distribution); ii) as a fraction of luminosity of the CSP in each metallicity bin. This definition is closer to the measurement as it can be directly compared to the luminosity-weighted mean Υ∗,l (LSMD: luminosity-weighted stellar metallicity distribution) at a given radius (see Arimoto & Yoshii 1987; Gibson 1996). This classification is important since PMC06 showed that the Υ∗,m and Υ∗,l might differ, especially at large radii, even for old stellar populations. The advantage of GCSs, for which accurate ages are known, is that they directly probe the mass-weighted distributions. At this point it is useful to recall that our the adopted chemical evolution model divides the galaxy in several non-interacting shells. In each shell the time at which the galactic wind occurs is self-consistently evaluated from the local condition. In particular, we follow Mar- tinelli et al. (1998) suggestion that gradients can arise as a consequence of a more prolonged SF, and thus stronger chemical enrichment, in the inner zones. In the galactic core, in fact, the potential well is deeper and the supernovae (SNe) driven wind develops later relative to the most external regions. This particular formation scenario leaves a characteristic imprint on the shape of both Υ∗,m and Υ∗,l and here we give some general considerations. In partic- ular, we can explain the slow rise in the low metallicity tail of the distributions as the effect of the initially infalling gas, whereas the onset of the galactic wind sets the maximum metal- licity of the Υ∗,m and Υ∗,l. In general, the suggested outside-in formation process reflects in a more asymmetric stellar metallicity distribution at larger radii, where the galactic wind occurs earlier (i.e. closer to the peak of the star formation rate), with respect to the galactic center. The qualitative agreement between these model predictions and the observed stellar metallicity distributions derived at different radii by Harris & Harris (2002, see their Fig. 18) for the stars in the elliptical galaxy NGC 5128 is remarkable. If confirmed from observations in other ellipticals, the expected sharp truncation of Υ∗,m at large radii might be the first direct evidence of a sudden and strong wind which stopped the star formation earlier in the galactic outskirts (see PMC06 and Pipino, D’Ercole, & Matteucci, in preparation). – 8 – 3. Results and discussion 3.1. The Multi-Modality of Globular Cluster Systems in Elliptical Galaxies The general presence of multi-modal GCSs implies that their host galaxies did not form in a single, isolated monolithic event, but experienced spatially and/or temporally separated star-formation bursts. In the recent past, both semi-analytic and hydrodynamic simula- tions of galaxy formation attempted to follow the process of GC formation (Beasley et al. 2002; Kravtsov & Gnedin 2005), but neither could produce a clearly bimodal MDF in their simulated GCSs. In this section we will show how to obtain a bimodal metallicity distribution function for GCs ΥGC,tot starting from single-mode stellar metallicity distribution functions Υ∗(t, ri, Z) (commonly known as G-dwarf-like diagrams) for the CSP inhabiting different radii of a prototypical elliptical galaxy according to Equation 2. 3.1.1. The Comparison Sample As stressed in the introduction, the multi-modality in GCSs varies as a function of host galaxy properties (e.g. mass, morphological type, etc.). Here we try to match the distri- butions resulting from the recent compilation of spectroscopic data by Puzia et al. (2006, hereafter P06), which samples the typical bimodal color distribution of GCs in nearby galax- ies (see also Puzia et al. 2004, 2005). This is illustrated in Figure 1, where we plot the (V−I)0 color distribution of the P06 sample of GCs in elliptical galaxies (top panel) together with those of GCs in NGC 4472 and the Milky Way (middle and bottom panel). NGC 4472 is the most luminous elliptical in the Virgo galaxy cluster and hosts a GC system with a prototypical color bimodality (e.g. Puzia et al. 1999). To allow direct comparison with the P06 sample, we use GCs in NGC 4472 that are brighter than V ≃ 22.5 since the P06 sample includes only the brightest GCs in nearby early-type galaxies in order to maximize the S/N of their spectra. The resulting color distribution is remarkably similar to the one of the P06 sample, which assures that the P06 sample includes a representative sampling of the GC color bimodality in massive elliptical galaxies. However, the comparison with the Milky Way GCs shows that the P06 sample covers few of the most metal-poor GCs. Therefore, the bimodality which we refer to in the fol- lowing may not be the same as the one observed in spirals or in some elliptical galaxies, where a substantial population of metal-poor clusters with [Z/H] . −1.5 is present (e.g. Gebhardt & Kissler-Patig 1999). – 9 – We do not include the dynamical evolution of GCs in our model, since we are considering only the brightest (most massive, i.e. > 105.5M⊙, see also Puzia et al. 2004, A&A 415, 123) clusters in nearby galaxies as comparison sample. The comparison sample includes GCs much brighter than the typical turnover magnitude of the globular cluster luminosity function and we, therefore, do not expect significant differential dynamical evolution for these massive systems (see Gnedin & Ostriker 1997, for details). In fact, it has been shown (e.g. Fall & Zhang, 2001) that the timescales for both evaporation by two body relaxation and tidal stripping of star clusters is longer than a Hubble time for GCs more massive than ∼ 105.5M⊙. In our model, the number of clusters formed in a star-formation burst of a given strength is adjusted to match the observations. Hence, the absolute scaling of GC numbers is arbi- trary, i.e., within physical limitations of the star formation rate any number of GCs can be reproduced by adjusting the function f in Equation 2. If, however, metal-poor and metal-rich GCs are on systematically different orbits and experience significantly different dynamical evolutions the effect of tidal disruption might be slowly changing relative GC numbers with time. Another complication is the variation of the initial star-cluster mass function, in particular as a function of metallicity. Modeling these effects requires detailed knowledge of the orbital characteristics and chemo-dynamical processes that lead to star cluster formation, and goes far beyond the scope of this work. We keep these potential systematics in mind, but expect negligible impact on our analysis. 3.1.2. A Simple Model In order to make a first-order comparison between our model predictions and the ob- served ΥGC,tot at t=13 Gyr, we first focus on the simple case in which: ΥGC,tot(Z) = fred ·Υ∗(t = 13Gyr, r1, Z) + fblue ·Υ∗(t = 13Gyr, r2, Z) , (3) with fred , fblue = const and r1 = 0.1Reff , r2 ≥ 1Reff . The first term (red) corresponds to a population typical of the galaxy core (well inside r < 1Reff). The second term (blue) represents Υ∗ in the outer regions. In order to take into account the different amounts of stars formed in each galactic region, we point out that the stellar metallicity distributions entering Equation 2 were not normalized. As a first step, we take several values for the weights fred , fblue in order to mimic different mixtures of the two GC populations. In particular, we used the relative numbers of – 10 – the red (here identified as the metal-rich core population) and the blue globular clusters (the halo metal-poor population) as a function of galactocentric radius for the elliptical galaxy NGC 1399 (Dirsch et al. 2003). Our particular choice is driven by observationally motivated values for the weights, although we realize that NGC 1399 is a quite peculiar, massive cD elliptical and might not be representative of less massive systems. In the context of this first step, the weights might reflect the effects of the projection on the sky of a three-dimensional structure. However, we show below that the results do not strongly depend on the weights. Therefore they might be interpreted as mean values and could be changed if one decides to model a particular galaxy, with a different ellipticity, inclination, and luminosity profile. In Figure 2 we show the globular-cluster metallicity distribution ΥGC by mass (compu- tation based on Υ∗,m) in two radial bins for three particular choices of weights. In particular in the following we will use the ratios fred = 0.77,fblue = 0.23, fred = 0.60,fblue = 0.40 and fred= fblue = 0.50 in order to define the theoretical innermost, intermediate and outermost sub-sample of the GCS, respectively. These ΥGC will be compared against subsamples of the P06 data, obtained by selecting GCs with either r < 1Reff (in the case of the innermost population) or r ≥ 1Reff (for the intermediate and the outemost cases, respectively), unless otherwise stated. 3.1.3. Globular Cluster Metallicity Distribution In order to plot the different cases on the same scale we normalize each ΥGC by its maximum value. In the left panel of Figure 2 the shaded histogram represents the innermost population. Our predictions match the data very well, especially in the metal-rich slope and the mean of the distribution. The same happens for the pure core populations, which shows how the GCS might be used to probe the CSP in ellipticals. It should be remarked that a second peak centered at super-solar metallicity appears in the distribution predicted by our models, although not evident in the data of the particular radial sub-sample. The right panel of Figure 2 illustrates model predictions which are more representative of the galaxy as a whole (either at 1Reff , i.e. the intermediate population, or at several effective radii, the outermost population), and we consider them as the fiducial case. These two cases look quite similar to each other and have clear signs of bimodality in remarkable agreement with the spectroscopic data (solid empty histogram, sub-sample of the P06 data with r ≥ Reff). A Kolmogorov-Smirnov test returns > 99% probability that both model predictions and observations are drawn from the same parent distribution in the left panel of Figure 2. The right panel statistics gives a lower likelihood of 98.4% that both distributions – 11 – have the same origin, which is mainly due to the observed excess of metal-poor GCs at large galactocentric radii compared to the model predictions. The prediction of a super- solar metallicity globular cluster sub-population is entirely new and a result of the radially varying and violent formation of the parent galaxy. Moving to the low-metallicity tail, we predict slightly fewer low metallicity objects than expected from observations. But we recall the systematics mentioned in Section 3.1.1. In Figure 3 (left panel) we show the results for a pure core GCs, namely one in which we adopt fred :fblue = 1:0. In this quite extreme case the observed GCs have been selected with radius r < 0.5Reff . The histogram reflects the shape of a G-dwarf-like diagram expected for a typical CSP inhabiting the galactic core. This finding is particularly important, because it might offer the opportunity to resolve the SSPs in ellipticals, at variance with data coming from the integrated spectra which deal with luminosity-weighted quantities. Whereas in Figure 3 (right panel), the intermediate population is compared to a sub-sample of P06 GCs with 0.5 < r < 0.5Reff . This is to show that the multimodality is not an artifact due to the particular radial binning adopted in this paper. Figure 4 shows the V -band luminosity-weighted ΥGC for which the computation is based on Υ∗,l. This metallicity distribution has been obtained by converting the mass in each [Z/H] bin of the previous figure into LV , by means of theM/LV ratio computed by Maraston (2005) as a function of [Z/H] for 13 Gyr old SSPs. Due to the well-known increase of the M/LV in the high metallicity tail of the distribution2, we notice in Figure 4 that now the second peak has a smaller intensity in all the cases. The corresponding diffuse-light population goes undetected in integrated-light studies. In any case, the conclusions reached by analyzing Figure 2 are not significantly altered. We conclude that our analysis is not significantly biased by some metallicity effect which may alter the shape of the observed ΥGC,tot by luminosity. We stress the power of GCSs in disentangling stellar sub-populations in massive ellipticals, due to their nature as simple stellar populations that can be directly compared to SSP model predictions, unlike diffuse light measurements. Even with this simple parametrization, where f in Equation 3 does not depend on metallicity, we suggest that the bimodality for the metal-rich GCs is the result of different shapes of Υ∗,m (and the Υ∗,l) at different galactocentric radii. 2See PMC06 for a comparison between G-dwarf like diagrams for Υ∗,m and Υ∗,l predicted for the same – 12 – 3.1.4. Globular Cluster [Mg/Fe] Distributions Finally, in Figure 5 we show the [Mg/Fe] distributions for GCs divided in radial bins as in Figure 2. According to PMC06, these [Mg/Fe] distributions are narrower, more symmetric, and exhibit a smaller radial variation with respect to the [Z/H] distributions. In any case, a small degree of bimodality is still present. We point out the impressive agreement with the spectroscopic observations by P06. There is a rather large discrepancy between data and models at the high-[Mg/Fe] end. Hence, the corresponding Kolmogorov-Smirnov likelihood tests return a probaility of 1% (for inner sample) and 95% (for the outer ones). If we limit the model predictions to [Mg/Fe] < 0.8 dex, the agreement slightly improves, reaching a 10% probablity in the inner region. However, the inner field data still does not reach the extreme [Mg/Fe] values of the models. In fact, due to the monotonic decrease of the [Mg/Fe] as a function of either metallicity or time (see PM04), the lack of low-metallicity GCs, evident from Figure 2, translates into a lack of α-enhanced clusters. The [Mg/Fe] bimodality of our model predictions and the match with the spectroscopic measurements strongly imply that globular clusters in massive elliptical galaxies form on two different timescales. Their chemical compositions are consistent with an early mode with a duration of ∆t . 100 Myr and a normal formation that lasted for ∆t . 500 Myr. In fact, according to the time-delay model (see Matteucci, 2001) and given the typical star formation history of our model ellipticals, the [Mg/Fe] ratio in the gas - out of which the GCs form - is quickly and continuously decreasing with time. We predict that the [Mg/Fe] ratio can be higher than 0.65 dex (i.e. in the bins in which our predictions exhibit a deficit of GCs with respect to the observed distribution) only in the first ∼ 100 Myr (see also Fig.3 in P06 and related discussion). In fact, such a high value for the [Mg/Fe] can be attained only if very massive type II SNe contribute to the chemical evolution, without any contribution from either lower-mass type II or type Ia SNe. The normal formation, instead, is the one already plotted in Fig. 5 and forms on a typical timescale of 0.5 − 0.7 Gyr. More quantitatively, our initial theoretical GCMD predicts that only ∼ 4% of the GCS forms at [Mg/Fe] larger than 0.65 dex. In order to improve the agreement with observations we require that the above fraction should be increased to ∼ 12− 15%. Since star and globular cluster formation are expected to be closely linked (e.g. Chandar et al. 2006) the same must be true for the diffuse stellar population of such galaxies. We therefore foresee the presence of a similar [Mg/Fe] bimodality in the diffuse light of massive elliptical galaxies. Unfortunately, there are not direct observations confirming our suggestions, until the metallicity distributions for the diffuse stellar component in ellipticals will become available for a number of galaxies. Indeed, the detection of bimodality in the [Mg/Fe]-distribution might be a benchmark test for our predictions. – 13 – We will still refer to multiple GC sub-populations. However, their differences ought to be ascribed only to the fact that they are created during an extended (and intense) star formation event during which the variation in chemical evolution is not negligible. The radial differences originate from the fact that the galactic wind epoch is tightly linked to the potential, occurring later in the innermost regions (e.g. Carollo et al. 1993; Martinelli et al. 1998). Moreover, PM04 and PMC06 found that also the infall timescale is linked to the galactocentric radius. In particular, it lasts longer in the more internal regions, owing to the continuous gas flows in the center of the galactic potential well. In particular, we recall that in our model the core experiences a longer (∼ 0.7 Gyr) star-formation history with respect to the outskirts where the typical star-formation timescale is ∼0.2 Gyr. According to our models the metal-rich population of GCs in massive elliptical galaxies may consist of multiple sub-populations which basically play the same role as the CSPs populating each galactocentric shell in our framework of the global galaxy evolution. At the same time, we point out the lack of our models to produce a significant fraction of metal- poor GCs similar to the halo GC population in the Milky Way, with the caveat that the star formation histories are very different. This, in turn, suggests that GCSs in giant elliptical galaxies were assembled by accretion of a significant number of metal-poor GCs. 3.2. Metallicity Dependent Globular Cluster Formation In this section we explore the approach outlined in Equation 2, by introducing the effect of metallicity in the function f. In particular, we start by assuming that f(t, ri, [Z/H]<−1) f(t, ri, [Z/H]>−1) = 2 , (4) roughly following what was found for the GCS of the most nearby giant elliptical galaxy NGC 5128 (Centaurus A) by Harris & Harris (2002, hereafter HH02) for their “inner field”, regardless of the radius of the i-th shell. Since the final distributions are normalized, the actual zeropoint of the function f is not relevant. Our model predictions are plotted in Figure 6. We notice a modest increase of the low-metalliticy tail of the distribution with respect to the simple picture sketched in Section 3.1 without metallicity dependence, as well as a lower fraction of globular clusters populating the high-metallicity peak. Including the metallicity dependence leads to an ambiguous change in agreement with the general observed trend. Hence, no firm conclusions on the real need for a metallicity dependence can be drawn. Similar results are obtained in the more realistic case in which f is a linearly decreasing function of [Z/H]. – 14 – At variance with HH02, we chose to adopt the same scaling irrespective of galacto- centric radius, for the following reason. Despite the fact that the HH02 stellar metallicity distributions Υ∗ as functions of [Z/H] confirm both the shape and the radial behavior of our model predictions for the mass-weighted stellar metallicity distribution Υ∗,m (compare their Fig. 7 with PMC06 Fig. 4), care should be taken when comparing their results for Υ∗ as a function of [Fe/H]. The latter, in fact, had been obtained by assuming a particular trend in the [α/Fe] as a function of galactocentric radius which disagrees with the results of our detailed chemical evolution model. In particular, we find an offset of at least 0.2 dex in the sense that [Fe/H] HH02 ∼ 0.2+ [Fe/H] PM04 at a given metallicity ([Z/H]). This disagreement becomes larger either at very low metallicity or at larger galactocentric radii, where we expect a stronger α-enhancement. Once the PM04 value for [Fe/H] is adopted in Fig. 18 of HH02, we find that: i) for the inner halo, the stellar Υ∗,m should be shifted by ∼ 0.2 dex toward lower metallicities, removing any metallicity effect, and ii) for the outer halo the discrepancy between the stellar Υ∗,m and the ΥGC,tot should be reduced. Nevertheless, we believe that some decrease with time of the function f could be mo- tivated by theoretical arguments. In fact, recent work (e.g. Elmegreen & Efremov 1997; Elmegreen 2004) shows that GCs of all ages preferentially form in turbulent high-pressure regions. If we interpret the decrease in the efficiency of star formation (inside the gas clouds that form GCs), as a function of the ambient pressure (Elmegreen & Efremov 1997) as a proxy for the temporal behaviour of our function f, we find again a reduction of a factor ∼2−3 from the early high-pressure epochs to a late, more quiescent evolutionary phase. 3.2.1. The Ratio of Metal-poor to Metal-rich Globular Clusters For our fiducial model we predict a ratio of metal-poor (namely with [Z/H] ≤ −1) to metal-rich GCs of ∼0.2. Previous photometric surveys found that the typical value for GC systems in elliptical galaxies is close to unity (Gebhardt & Kissler-Patig 1999; Kundu & Whitmore 2001a). Provided a linear color-metallicity transformation (see also Section 3.5), a possible explanation for the discrepancy between our models and the observations might be obtained by boosting the metal-poor population by a factor of f(t, ri, [Z/H]<−1)/f(t, ri, [Z/H]>−1) ≥ Another way to solve the discrepancy is to assume that all the missing globular clusters have been accreted from the surroundings, e.g. from dwarf satellites (e.g. Côté et al. 1998). We estimate the amount of the accreted metal-poor GCs, needed to achieve a ratio close to 1, as a factor of ∼4 of the number of globular clusters initially formed inside the galaxy. – 15 – 3.3. The Role of the Host Galaxy Mass A natural consequence of the scenario depicted in Sections 3.1 and 3.2 is that, at a given galactocentric radius, the mean metallicity and [α/Fe] ratios of a GCS coincide with the mass-weighted [〈Z/H〉∗] and [〈α/Fe〉∗] of the underlying stellar population, because the GC quantities are calculated either from Υ∗,m or the Υ∗,l (see Eqs. 1 and 2 of PMC06), unless the scaling function f is allowed to strongly vary with time. We expect this to happen at least in the innermost GC sub-populations, in which the effects of the accretion of GCs from the environment can be reasonably neglected. In particular, PM04 predict that more massive galaxies should show higher [〈α/Fe〉∗] and [〈Z/H〉∗]. If accretion plays a negligible role, we expect the same correlations for the total GC population with host galaxy mass for the most massive systems, in agreement with current observations (e.g. van den Bergh 1975; Brodie & Huchra 1991; Peng et al. 2006). In fact, if we perform the same study of the above sections for a 1012M⊙ galaxy (see Table 2 of PM04 for its properties), both peaks in ΥGC,tot shift their positions by about 0.2 dex to higher [Z/H]. This is in good agreement with the results of (Peng et al. 2006, see their Figure 13 and 14). This trend holds for smaller objects as well. If we apply the procedure to a 1010M⊙ galaxy (Model IIb of PM04), we find that the metal-rich peak shifts towards a lower metallicity by 0.3 dex (with respect to our fiducial model with Mlum = 10 11M⊙), while the other peak is now centered around [Z/H] = −0.8 dex. In particular, we find a faster decrease in the mean metalliticy of the metal-poor GCs than for the metal-rich ones, again in agreement with the Peng et al. results. Interestingly, the ratio of metal-poor to metal-rich cluster increases up to ∼ 0.5 for less massive halos. We recall that in the PM04 scenario, the low-mass galaxies are those forming on a longer timescales and with a slower infall rate. Therefore, we suggest that the combination of these factors is likely to at least partly explain the change of the GC distributions in different galaxy morphologies. This is especially the case in dwarf galaxies, where star formation is slow and still on-going, together with the fact that the probability for a substantial change in the pressure of the interstellar medium relative to its initial values is higher than in ellipticals, thus implying a much stronger variation of f with time. 3.4. Merger-Induced Globular Cluster Formation It has been suggested that GC populations are produced during major merger events which would lead to present-day ellipticals and their rich GCSs (e.g. Schweizer 1987; Ashman & Zepf 1992). Subsequent studies (e.g. Forbes et al. 1997; Kissler-Patig et al. 1998b) challenged this – 16 – view by pointing out the much higher SN and more metal-rich GCSs in early-type galaxies compared to those of spiral and irregular galaxies, which are thought to represent the early building blocks of massive ellipticals. In the following, we study the impact of the merger hypothesis on the predictions of our simulations. In order to do that, we extended the procedure sketched in the previous sections to the merger models presented by Pipino & Matteucci (2006, hereafter PM06). In this paper, the effects of late gas accretion episodes and subsequent merger-induced starbursts on the photo-chemical evolution of elliptical galaxies have been studied and compared to the picture of galaxy formation emerging from PM04; in particular the PM04 best model is taken here as a reference. By means of the comparison with the colour-magnitude relations and the [〈Mg/Fe〉V ]-σ relation observed in ellipticals (e.g. Renzini 2006), PM06 showed that either bursts involving a gas mass comparable to the mass already transformed into stars during the first episode of star formation and occurring at any redshift (major mergers), or bursts occurring at low redshift (i.e. z ≤ 0.2) and with a large range of accreted mass (minor mergers), are ruled out. The reason lies in the fact that the chemical abundances in the ISM after the galactic wind (and before the occurrence of the merger) are dominated by Type Ia SN explosions, which continuously enrich the gas with their ejecta (mainly Fe). When the merger-induced starburst occurs, most stars form out of this enriched gas (thus, e.g., lowering the total [〈Mg/Fe〉]); at the same time, we expect the metallicities of GCs formed out of this gas to be on average higher and their [Mg/Fe] ratios to be lower than those of the bulk of stars and GCs formed in the initial starburst. In this work we present the case in which the galaxy accretes a gas mass Macc = Mlum at tacc = 2 Gyr (i.e. ∼1 Gyr after the onset of the galactic wind). We make this choice for several reasons: i This model is quite similar to the PM06 models b and g, which were among those in good agreement with observations of the diffuse galaxy light properties. ii) The formation epoch of the bulk of these second generation GCs cannot occur &2 Gyr later than tgw, because the majority of GCs in the most massive elliptical galaxies studied today appear old within the age resolution of current photometric (∆t/t ≈ 0.4−0.5) and spectroscopic studies (∆t/t ≈ 0.2−0.3). Finally, the composition of the newly accreted gas is assumed to be primordial (see PM06 for a detailed discussion), but we remark that we reach roughly the same conclusion in the case of solar composition, in order to mimic some pre-enrichment for the newly accreted gas. We point out that, lacking dynamics, PM06 presented their results for one-zone models. Therefore, in this section we are considering Equation 2 limited to only one shell. In this way we can check whether the single merger hypothesis alone is enough to produce some bimodality in the total globular-cluster metallicity distribution ΥGC,tot, and if it is consistent with the predictions based on our fiducial model described in Section 3.2. – 17 – We show our results in Figure 7 and 8. We notice a clear change in the overall shape of the metallicity distribution ΥGC,tot with respect to the cases shown in the previous sections, in the sense that now ΥGC,tot is narrower and dominated by objects with super-solar metallicity (and sub-solar [Mg/Fe] ratios) with a dominant population at [Z/H] ≈ 0.1, which is not prominent in the observations of P06. The high-metallicity globular cluster populations dominate the metallicity distribution which is at variance with both the results from previous sections and the observations. Our merger model does not include the accretion of globular clusters that were already formed within the accreted satellite galaxies. The inclusion of this effect could remedy the match between models and observations at low metallicities, as the typical GC in a dwarf galaxy is metal-poor (e.g. Lotz et al. 2004; Sharina et al. 2005) and their addition to the initial GC population would enhance the total number of metal-poor GCs and improve the fit to the data. However, these clusters need to be α-enhanced to match the observations. The impact of GC accretion on our post-merger model predictions will be studied in detail in a future paper. Here, we remark that the time at which the purely gaseous subsequent merger event can occur (which does not import already formed GCs), is limited by the onset of the galactic wind, after which the type-Ia SNe dominates the nucleosynthesis, and needs to be completed at tmrg . 1 − 2 Gyr after the first starburst. However, this time constraint implies that such merger events would overlap with the initial starburst and be mostly indistinguishable from each other. Such a scenario closely resembles the Searle-Zinn scenario (Searle & Zinn 1978), in which galaxy halos are formed from the agglomeration of gaseous protogalactic fragments. Later merger events are excluded in our models, as they would produce GCs with sub-solar [Mg/Fe] ratios which is at variance with the observations. Note also that the fraction of GCs at [Z/H]< −1 can be recovered in our models only if the cluster formation at low metallicity is enhanced (e.g. using a value of 10 instead of 2 in eq.4). However, even in the case in which we adopt some f(Z) strongly declining with total metallicity, which may alter the shape of ΥGC,tot enhancing the low-metallicity tail and thus improving the agreement with observations, the position of both super-solar metallicity peaks will not change, remaining at variance with the data. 3.5. Other Mechanisms responsible for Multimodality The picture emerging from our analysis is far from being the general solution to ex- plaining the complexity of GC color distributions, and it suggests only a scheme in which multiple mechanisms could be at work together, either broadening or adding features to the observed distributions. – 18 – For instance, Yoon et al. (2006) suggested that the color bimodality could arise from the presence of hot horizontal-branch stars (so far not accounted for in SSP model predictions) that results in a non-linear color-metallicity transformation producing two color peaks from an originally single-peak metallicity distribution. We tested this scenario on our fiducial model GC metallicity distribution, by applying to each SSP the following transformation from [Fe/H] to the (g − z) color: (g − z) = α + β [Fe/H] + γ [Fe/H] +δ [Fe/H]3 + ǫ [Fe/H] The numerical values of the coefficients are given in Table 1 and the relation was adopted from Yoon et al. (2006) and is consistent with the best-fit relation presented in their Figure 1b. We show our results in Figure 9. Since we start from symmetric metallicity distributions, the non-linear transformation seems to work and produce a color bimodality for the CSP inhabiting the < 10Reff shell (Figure 9, solid line), although the bimodality is slightly exaggerated compared to real data (see Figure 1). In fact, a look at the color distribution which we obtained for the sole 0.1Reff shell reveals that it still has one peak and is roughly symmetric (Figure 9, dashed line). Obviously, since the (g − z)− [Fe/H] relationship is meant to explain the GCs color bimodality without invoking any other effect, we did not combine the two histograms, either according to Eq.2 or to Eq. 3 in our models, as we are comparing metallicity distributions to spectroscopic measurements. It is of great importance to investigate this transformation with large and homoge- neous data sets that cover a wide enough metallicity range to allow a robust analysis of the non-linear inflection point in the color-metallicity transformation. However, as a re- sult of Figure 9, we point out that the color bimodality typically found for GCSs in massive early-type galaxies might be only partly due to a non-linear color-metallicity transformation. Another effect put forward by, e.g.,Recchi & Danziger (2005) is the claim that GCs might have undergone a self-enrichment phase at the early stages of their formation, and Table 1. Numerical values of coefficients used in Equation 6. coefficient numerical value α 1.5033 β 0.172774 γ −0.623522 δ −0.453331 ǫ −0.089038 – 19 – therefore some GCs could have experienced a boost in metallicity which would be not rep- resentative of the metallicity of their parent gas cloud. Finally, as already mentioned above in Section 3.4, some GCs residing in the outermost regions of the galaxies (e.g. Lee et al. 2006) could have experienced entirely different chemical enrichment histories at the time of their formation and later been added to a more massive system through accretion (e.g. Côté et al. 1998). The inclusion of these effects goes far beyond the scope of this work, but we remind the reader that all the aforementioned effects might influence the interpretation of any globular cluster color and metallicity distribution. 4. Conclusions By means of the comparison between PM04’s best model predictions for the radial changes in the CSP chemical properties and the recent spectroscopic data on the metallicity distributions of extragalactic GCSs from Puzia et al. (2006), we are able to derive some conclusions on the GC metallicity distributions in massive elliptical galaxies. In particular, we focused on the main drivers of the multi-modality that is observed in the majority of GCSs in massive elliptical galaxies. Our main conclusions are: • We show that the observed multi-modality in the GC metallicity distributions can be ascribed to the radial variation in the underlying stellar populations in giant elliptical galaxies. In particular, the observed GCSs are consistent with a linear combination of the GC sub-populations inhabiting different galactocentric radii projected on the sky. • A new prediction of our models, which is in astonishing agreement with the spectro- scopic observations, is the presence of a super-solar metallicity mode that seems to emerge in the most massive elliptical galaxies. In smaller objects, instead, this mode disappears quickly with decreasing stellar mass of the host galaxy. • Our models successfully reproduce the observed [Mg/Fe] bimodality in GCSs of mas- sive elliptical galaxies. This, in turn, suggests a bimodality in formation timescales during the early formation epochs of GCs in massive galaxy halos. The two modes are consistent with an early (initial) and later (triggered) formation mode. • Since the GC populations trace the properties of galactic CSPs in our scenario, we predict an increase of the mean metallicity of the cluster system with the host galaxy mass, which closely follows the mass-metallicity relation for ellipticals. Moreover, we expect that a major fraction of the GCs (i.e. those born inside the galaxy) follows an age-metalliticity relationship, in the sense that the older ones are also more α-enhanced and more metal-poor. – 20 – • The role of host galaxy metallicity in shaping the observed GC metallicity distribution is non-negligible, although its effects have been estimated to change the function f ≃ ψGC/ψ∗ by a factor of ∼ 2 − 5, in order to match the sample of Puzia et al. (2006). Either a non-linear color-metallicity transformation, or a stronger metallicity effect, and/or accretion of GCs from the surrounding environment is needed to explain a ratio of metal-poor to metal-rich GCs close to unity, as reported for ellipticals based on results from photometric surveys. • Merger models which include the later accretion of primordial and/or solar-metallicity gas predict a shape for the GC metallicity distribution which is at variance with the spectroscopic observations. We thank the referee for a careful reading of the paper. A.P. warmly thanks S.Recchi for useful discussions. A.P. acknowledges support by the Italian Ministry for University under the COFIN03 prot. 2003028039 scheme. T.H.P. acknowledges support by NASA through grants GO-10129 and GO-10515 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA Contract NAS5-26555, and the support in form of a Plaskett Research Fellowship at the Herzberg Institute of Astrophysics. 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In order to allow a robust comparison between the P06 and NGC 4472 sample, only GCs in NGC 4472 with with luminosities brighter than V ≃ 22.5 mag are shown. The solid lines are Epanechnikov- kernel probability density estimates with their bootstrapped 90% confidence limits. – 25 – Fig. 2.— Predicted globular-cluster metallicity distribution ΥGC,tot by mass as a function of [Z/H] for three different radial compositions (i.e. fred/fblue). The left panel shows both model predictions and observations related to the central part of an elliptical galaxy. The right panel shows the same quantities for cluster populations residing at r ≥ Reff . Solid empty histograms: observational data taken as sub-samples of the P06 compilation, according to the galactic regions presented in each panel. – 26 – Fig. 3.— Predicted globular-cluster metallicity distribution ΥGC,tot by mass as a function of [Z/H] for two different projected galactocentric radii. The left panel shows both model predictions and observations related to the pure core of an elliptical galaxy (namely fred : fblue = 1 : 0). The right panel shows the same quantities for cluster populations residing either at 0.5Reff < r < 1.5Reff . Solid empty histograms: observational data taken as sub- samples of the P06 compilation, according to the galactic regions presented in each panel. – 27 – Fig. 4.— Predicted globular-cluster metallicity distribution ΥGC,tot by luminosity at three different projected galactocentric radii. Solid: innermost region; dotted: average galactic (intermediate population); dashed: outermost part. – 28 – Fig. 5.— Shaded histogram: Predicted distribution of globular-cluster [Mg/Fe] values at two different projected galactocentric radii (innermost and outermost regions). Solid empty histogram: observational data taken as sub-samples of the P06 compilation, according to the galactic regions presented in each panel (see text). – 29 – Fig. 6.— Predicted globular-cluster metallicity distribution ΥGC,tot by mass as a function of [Z/H] for three different radial compositions (i.e. different fred/fblue). In this case, the function f has an explicit dependence on [Z/H] (see text). The left panel shows both model predictions and observations related to the central part of an elliptical galaxy. The right panel shows the same quantities for cluster populations residing at r ≥ Reff . Solid empty histograms: observational data taken as sub-samples of the P06 compilation, according to the galactic regions presented in each panel. – 30 – Fig. 7.— Shaded histogram: predicted total GC metallicity distribution ΥGC,tot by mass for the < 1Reff shell, for a case in which a second episode of star formation, induced by a gaseous merger, is taken into account (see text). Solid histogram: observations from Puzia et al. (2006), their entire sample. – 31 – Fig. 8.— Shaded histogram: predicted total GC [Mg/Fe] distribution by mass for the < 1Reff shell, for a case in which a second episode of star formation, induced by a gaseous merger, is taken into account (see text). Solid histogram: observations by Puzia et al. (2006), their entire sample. – 32 – Fig. 9.— Predicted globular-cluster metallicity distribution by mass as a function of the (g−z) colour at two different projected galactocentric radii. Dashed: galactic core. Solid: galactic halo out to 10 Reff . Introduction The Multimodality of Globular Cluster Systems Numerical Models of Globular Cluster System Formation A Spatially Resolved Chemical Evolution Model for Spheroid Galaxies The model The Chemical Evolution Code Globular Cluster Formation Results and discussion The Multi-Modality of Globular Cluster Systems in Elliptical Galaxies The Comparison Sample A Simple Model Globular Cluster Metallicity Distribution Globular Cluster [Mg/Fe] Distributions Metallicity Dependent Globular Cluster Formation The Ratio of Metal-poor to Metal-rich Globular Clusters The Role of the Host Galaxy Mass Merger-Induced Globular Cluster Formation Other Mechanisms responsible for Multimodality Conclusions