PHYSICAL REVIEW D, VOLUME 62, 094014 Critical coupling for chiral symmetry breaking in QCD motivated models A. C. Aguilar, A. A. Natale, and R. Rosenfeld Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sa˜o Paulo, SP, Brazil ~Received 21 January 2000; revised manuscript received 8 June 2000; published 3 October 2000! We determine the critical coupling constant above which dynamical chiral symmetry breaking occurs in a class of QCD motivated models where the gluon propagator has an enhanced infrared behavior. Using methods of bifurcation theory we find that the critical value of the coupling constant is always smaller than the one obtained for QCD. PACS number~s!: 12.38.Lg, 12.38.Aw, 24.85.1p de m ce ta al ol d e b - ns t de th b th we ra te o de D i it o e tr o d e- av r ge rba- f. nt ice ean r in lf- he on ical tors na- ua- in f ef- - I. INTRODUCTION Hadronic physics at low energy is expected to be scribed by the infrared properties of quantum chromodyna ics ~QCD!, whose nonperturbative character in general for us to make use of approximate models in order to unders the strong interaction effects at a small momentum sc One of these models became known as the global c model ~GCM! @1–3#, which is a quark-gluon quantum fiel theory that describes QCD for low-energy processes. Th are many recent calculations exemplifying the remarka success of this procedure@4,5#. It relates the hadronic prop erties to the Schwinger functions of quarks and gluo Therefore, when comparing the theoretical calculations some low-energy data, e.g., pseudoscalars masses and constants or other chiral parameters, we learn about the frared behavior of the quark and gluon propagators. In near future, this semiphenomenological tool may reveal to even more successful than the relativistic quark model or bag model. In order to have an idea of what is behind the GCM, can recall that its action is obtained from the QCD gene ing functional in the standard way@1#, with the main differ- ence being that in the functional generator of the connec gluon n-point functions we neglect the higher than tw n-point functions, expecting that a phenomenological scription of the gluon propagatorg2Dmn(x2y) retains most of the information about the non-Abelian character of QC The effect of this approximation can only be measured comparison with experiments, and in fact it does work qu well once we model appropriately the infrared behavior the gluon propagator. Some of the gluon propagators used in the GCM hav quite enhanced infrared behavior. One example is the in duction of a delta functiond(k) as prescribed in Ref.@6#, which is a confining propagator according to the criterion absence of realk2 poles for the quark propagators@6,7#. An- otherAnsatzfor the gluon two-point function in the infrare is g2D(k2)53p2(x/D)2 exp(2k2/D) @8#, which was in- spired by and approaches thed function Ansatzof Ref. @6# for D→0, wherex andD are adjustable parameters. In d tailed calculations of chiral parameters more completeAn- sätze for the gluon propagators than the above ones h been used, in general including the asymptotic behavio the gluon propagator as predicted by QCD. In Ref.@5# the 0556-2821/2000/62~9!/094014~7!/$15.00 62 0940 - - s nd e. or re le . o cay in- e e e t- d - . n e f a o- f e of following form for the gluon propagator in the Landau gau was introduced: g2Dmn~k!5H dmn2 kmkn k2 J D~k2!, ~1! where D~k2![ g2 k2@11P~k2!# 54p2dF4p2mt 2d4~k!1 12e(2k2/[4mt 2]) k2 G , ~2! with d512/(3322nf), and nf53 ~considering only three quark flavors!. The mass scalemt determined in Ref.@5# was interpreted as marking the transition between the pertu tive and nonperturbative domains. AnotherAnsatzis @9# g2D~k2!53p2 x2 D2 exp~2k2!1 as~k2! k2 F~k2!, ~3! where as~k2!5 4p2dg2 ln~k2/L21t! , ~4! F(k2) is a function chosen differently in the papers of Re @9#, as(k 2) describes the QCD running coupling consta wheret is a parameter adjusted phenomenologically. Not that in the above expressions the momenta are in Euclid space. Of course, there are still other variations of theseAn- sätze @10# and attempts to explain the enhanced behavio the infrared@11#. In this work we study the bifurcation of the quark se energy within the context of the GCM, i.e., we determine t critical coupling constant of the truncated Schwinger-Dys equation for the quark propagator above which dynam chiral symmetry breaking occurs using the gluon propaga discussed in the previous paragraph. It is known from a lytical and numerical studies of the Schwinger-Dyson eq tions that dynamical chiral symmetry breaking takes place QCD when the coupling constant (as) is of O(1) @12–14#. These studies have also been performed with the use o fective potentials@15# and corroborated by lattice simula ©2000 The American Physical Society14-1 al n n on er e in u to e in g lu a w on ha f he ne d ho s ns al- a- nt. a- ith e ng. not fre- e r ou- inor A. C. AGUILAR, A. A. NATALE, AND R. ROSENFELD PHYSICAL REVIEW D62 094014 tions @16#. Therefore, it is natural to ask what is the critic coupling in the GCM with the phenomenological gluo propagators proposed in the literature. We will apply the standard techniques of bifurcatio theory as used by Atkinson and collaborators@13,17–19# and verify that for the gluon propagator given by a delta functi the chiral symmetry is always broken no matter what ‘‘p turbative’’ propagator is added to thed. For a propagator of the form given by Eq.~3! we obtain a lower bound on th critical coupling smaller than the one obtained for QCD@17#, and evaluate the smallest characteristic number confirm the result obtained with thed function propagator in the limit that the gaussian width goes to zero. In Sec. II we disc chiral symmetry breaking for an infrared gluon propaga similar to Eq. ~2! using a bifurcation analysis of th Schwinger-Dyson equations. In Sec. III we analyze the frared gluon propagator given by a Gaussian form usin different technique. The last section contains our conc sions. II. THE CRITICAL COUPLING FOR A CONFINING PROPAGATOR The Schwinger-Dyson equation for a massless qu propagator in Minkowski space can be written in the form S21~p!5p”2ıg2E d4q ~2p!4 Dmn ab 3~p2q!Gm~q,p! lc a 2 S~q!gn lc b 2 , ~5! where the gluon propagator in the Landau gauge, which be used throughout the paper, is given by Dmn ab~k!5dabF2gmn1 kmkn k2 G 1 k21 i e . ~6! In the case that the gluon propagator is given by the c fining delta function@6# Dmn ab~k!5dabFgmn2 kmkn k2 Gbd4~k!, ~7! whereb is an adjustable dimensional parameter. Note t the termkmknd(k)/k2 is undefined from the point of view o generalized functions. Of course, we could work with t Feynman gauge where this unwanted behavior is softe Fortunately thed function gives an integrable singularity an only because of this peculiarity we have not a strong pat logical behavior. With Eq.~7! it is quite easy to verify that the quark self-energy has a nontrivial solution for any po tive value of g2, in the rainbow approximation@Gm(q,p) 5gm#. With the quark propagator given by S21~p!5A~p2!p”2B~p2!, ~8! and the gluon propagator of Eq.~7! it follows from Eq. ~5!, in Euclidean space, the set of coupled integral equations 09401 - g ss r - a - rk ill - t d. - i- @A~p2!21#p25 4 3 g2bE d4q ~2p!4 d4~p2q! 3 A~q2! q2A2~q2!1B2~q2! 3Fpq12 q~p2q!~p2q!p ~p2q!2 G , B~p2!54g2bE d4q ~2p!4 d4~p2q! 3 B~q2! q2A2~q2!1B2~q2! . ~9! The solutions of the nonlinear coupled integral equatio were determined in Refs.@1,2#, showing thatA(p2) is a con- stant. Without loss of generality we can setA(p2)51 and verify the behavior of the self-energy apart from a norm ization constant. In this case we obtain the following equ tion for B(p2) B~p2!5 4g2b ~2p!4E d4q B~q2!d4~q2p! q21B2~q2! , ~10! whose integration leads to B~p2!5 4g2b ~2p!4 B~p2! p21B2~p2! . ~11! Equation~11! has the solution B~p2!5HA4g2b ~2p!4 2p2, p2< 4g2b ~2p!4 , 0, p2. 4g2b ~2p!4 , for any nonzero and positive value of the coupling consta If we consider only this infrared behavior of the gluon prop gator we would conclude that this model is inconsistent w previous work on QCD, which predicts a critical value of th coupling constant for the onset of chiral symmetry breaki However, the ultraviolet part of the gluon propagator can be neglected and the use of the following propagator is quent@2,5# Dmn ab~k!5dabFdmn2 kmkn k2 Gbd4~k!1dabFdmn2 kmkn k2 G 1 k2 . ~12! The following comments are in order:~a! In some papers the coupling constantg2 is absorbed in the parameterb @5#. Here we assume thatg2 can always be factorized, and th critical behavior is going to be related to it.~b! We assume a constantg2. In the next section it will be shown that ou result does not change with the inclusion of the running c pling in the ultraviolet part of the gluon propagator.~c! This propagator has been used in the literature with some m 4-2 su ic ts e a- tio ar a- a l - ia h s t e th is ic y e ng CRITICAL COUPLING FOR CHIRAL SYMMETRY . . . PHYSICAL REVIEW D 62 094014 differences; these variations are not important for our re as will become clear in the next section. When the gluon propagator is given by Eq.~12! we need a thorough analysis of the quark self-energy to find for wh value of the coupling constant the quark self-energy admi nontrivial solution. The substitution of Eq.~12! into Eq. ~5! yields B~x!5 g2 4p2 S kB~x! x1B2~x! 1E m2 L2 dy xmax yB~y! y1B2~y! D , ~13! wherex5p2, y5q2, xmax5max(x,y), andk5b/p2. We will be looking for solutions of Eq.~13! only in the interval @m2,L2#, although thed function has been integrated in th full range of momenta. A rigorous handling of this integr tion does not change the conclusions about the bifurca point in the limit of very small~large! infrared ~ultraviolet! cutoff. It is important to verify that Eq.~13! is a very peculiar one, in the sense that for very small values ofg2, keeping the productg2k fixed, we do have the solution of the nonline equation, which, at low momenta, approaches B~x→0!5SAg2k 4p2D g2→0, g2k fixed . ~14! To find nontrivial small solutions of the nonlinear equ tion ~13! we study the linearized equation, i.e., the function derivative of Eq. ~13! evaluated atB(x)50 @13,17–19#. Writing dB5 f , we obtain f ~x!5 g2 4p2 S k f ~x! x 1E m2 L2 dy f ~y! xmax D . ~15! Note that we are not considering the equation forA(p2) because this equation is of second order indB @19#. The existence of a solution for Eq.~15! is a sufficient condition for the onset of chiral symmetry breaking@13#. Defining a [g2/4p25as /p, Eq. ~15! is equivalent to the differentia equation x~x2ak! f 9~x!12x f8~x!1a f ~x!50, ~16! together with the ultraviolet boundary condition@(x 2ak) f (x)#8ux→L250 and another condition valid in the in frared region. The choice of the infrared boundary condition is a cruc one. Note that if we neglect the second term of the rig hand side of Eq.~13! we only obtain a trivial result, no matter what is the boundary condition coming from Eq.~13!. However, we know that the nonlinear equation always ha nontrivial solution for thed function propagator. The mos suitable infrared boundary condition is the one that com from the nonlinear integral equation~12!, since when applied to the linear equation the result must be consistent with known solution of the nonlinear equation for smallg2 @see Eq. ~14!#. Therefore, our infrared boundary condition given by 09401 lt h a n l l t- a s e f ~x!ux→m25Aak. ~17! Equation~16! can be put in the form of a hypergeometr equation performing a shiftx→z1ak and definingy5 2z/ak leading to y~12y! f 9~2aky1ak! 12~12y! f 8~2aky1ak!1a f ~2aky1ak!50, ~18! The solution of Eq.~18! that obeys the infrared boundar condition is f ~x!5AFS a,b;2;12 x ak D , ~19! where A5AakG~3/22s!G~3/21s!, ~20! is a normalization constant,s is defined ass5( 1 4 2a)1/2, and a5 1 2 1A1 4 2a, b5 1 2 2A1 4 2a. ~21! To apply the ultraviolet boundary condition we recall th following relation involving hypergeometric functions: d dz @zc21F~a,b;c;z!#5~c21!zc22F~a,b;c21;z!, ~22! which lead us to the analysis of the zeros of the followi equation: @~x2ak! f ~x!#8ux→L2 52akAFS 1/21s,1/22s;1,12 x ak D U x→L2 . ~23! We need to consider the solution of Eq.~23! in three different regions of the parametera, namely, 0,a, 1 4 , a 5 1 4 , anda. 1 4 , and study the asymptotic behavior off (x) in each one of these cases. Whena, 1 4 the relation 4-3 an a th e in y a ve ult er- i- t by - te f gy r- u- ba- f the r can r - led A. C. AGUILAR, A. A. NATALE, AND R. ROSENFELD PHYSICAL REVIEW D62 094014 F~a,b;c;z! 5 G~c!G~b2a! G~b!G~c2a! ~2z!2aF~a,11a2c;11a2b;z21! 1 G~c!G~a2b! G~a!G~c2b! ~2z!2bF~11b2c,b;11b2a;z21!, ~24! can be used together with F~a,b;c;0!51 ~25! to give @~x2ak! f ~x!#8ux→L2 5A8S x ak 21D 21/21s 1B8S x ak 21D 21/22sU x→L2 , ~26! whereA8 andB8 depend ons andak. In this cases is a real and positive number smaller th 1 2 . For large values ofx we see that Eq.~26! condition is satisfied. Therefore, Eq.~19! is a solution when 0,a, 1 4 . Note that whena5 1 4 we haves50, the constantsa and b are identical, implying that Eq.~24! cannot be used in this case. However, we can perform the limitx→L2 already in the differential equation~16! and study its behavior for this particular value ofa. In the asymptotic region~UV! we ob- tain f ~x!UV5x21/2~C1D ln x!. ~27! We easily see that the ultraviolet boundary condition is s isfied for largex and Eq.~19! is also a solution whena5 1 4 . When a. 1 4 in the parametersa and b of Eq. ~19! we make the substitutions→ ir with r defined as (a2 1 4 )1/2. The asymptotic behavior in this case is obtained with help of Eqs.~25! and ~24! leading to @~x2ak! f ~x!#8ux→L2 522~ak!3/2Re FG~2ir!G~3/22s!G~3/21s! G2~1/21 ir! G 3S x ak 21D 21/21 irU x→L2 , ~28! which is valid forx@ak. Equation~28! has an infinite set of zeros located atx5ak(xn11) for integern, which can be determined with the procedure of the second paper of R @13#. The real part of Eq.~28! can be written as expH lnFG~2ir!G~3/22s!G~3/21s! G2~1/21 ir! xn 21/21 irG J , ~29! as lnz5lnuzu1i(arg z62np), wherez is a complex number andn50,1,2, . . . , thezeros of this function will occur for 09401 t- e f. ln~xn!; 1 r F2np2argS G~2ir!G~3/22s!G~3/21s! G2~1/21 ir! D G ~30! with n51,2, . . . .This relation was obtained forx@ak, and n50 was excluded in order to obtain only positive values Eq. ~30!. Therefore, also fora. 1 4 we do have a nontrivial solution. In summary, when the gluon propagator is modeled b delta function plus a ‘‘perturbative’’ 1/k2 propagator we verified that the chiral symmetry is broken for any positi value of the coupling constant. One may ask if this res changes if we modify the nonperturbative as well as the p turbative propagator, as in the case of Eq.~2!, or with the inclusion of the running coupling or the effect of a dynam cal gluon mass@13,20#. We will show in the next section tha this is not the case. III. AN INFRARED MODEL IN THE GAUSSIAN FORM As discussed in the introduction a differentAnsatzfor the gluon propagator, exemplified by Eq.~3!, has frequently been used. Its nonperturbative infrared behavior is given the first term of Eq.~3!: Dmn ab~k!5dabFdmn2 kmkn k2 G3p2 x2 D2 exp 2k2 D , ~31! where the parameterx controls the intensity of the interac tion, andD gives the Gaussian width. It is important to no that in the limit D→0 we recover the infrared behavior o the propagator discussed in the previous section. In this section we will show that the quark self-ener calculated with the propagator of Eq.~31! has a well defined bifurcation point. Afterwards, we verify that adding a pertu bative tail to this propagator results in a smaller critical co pling constant than the one obtained only with the pertur tive part. Finally, in the limitD→0, we recover the result o the previous section showing that it is independent of perturbative part that is added to the delta function. To find nontrivial small solutions of the nonlinea Schwinger-Dyson equation for the quark self-energy we study the linearized equation. The substitution of Eq.~31! into Eq. ~5! gives the following linear integral equation fo f (x) @recalling thatf (x)5dB(x)]: f ~x!5 3 4 g2 x2 D2 F E 0 x dy f~y!exp 2x D 1E x ` dy f~y!exp 2y D G , ~32! where we consideredA(p2)51, since, according to bifurca tion theory, the equation for@A(p2)21# is of higher order in the functional derivative ofB(p2), and the critical coupling constant is determined only through Eq.~32!. To obtain Eq. ~32! we performed in the gluon propagator the so-cal angle approximation, which is given by 4-4 ith o e e to o he s uss- u- l ng. he on r- n- lu- ., a - vial e nly , ed, to de- CRITICAL COUPLING FOR CHIRAL SYMMETRY . . . PHYSICAL REVIEW D 62 094014 D@~p2q!2#'u~p22q2!D~p2!1u~q22p2!D~q2!. ~33! The Eq.~32! is a homogeneous Fredholm equation w the kernel KI~x,y!5expS 2 x D D u~x2y!1expS 2 y D D u~y2x!. ~34! The norm of Eq.~34! is easily calculated iKI i25E 0 ` dxE 0 ` dyKI 2~x,y!5 D2 2 , ~35! and we find nontrivialL2 solutions ofB(x) for g2 on a point set whose smallest positive point satisfies@17# g2> 4D2 3x2 1 iKI i 5S 4A2 3 D D x2 '1.88 D x2 . ~36! Equation~36! gives a lower bound for the critical pointgc 2 . However, we can do better than this. Using the method traces we can show that the approximate critical value indeed of the order of the smaller value of the bound giv by Eq. ~36!. The following approximate formula holds true for th smallest characteristic numbergc 2 @21#: ugc 2u'AA2 A4 , ~37! where, for a symmetric kernel, A2m52E 0 `E 0 x Km 2 ~x,y! dy dx, ~38! with m running over 1,2.K2(x,y) is given by K2~x,y!5E 0 ` K1~x,z!K1~z,y! dz, ~39! and, in the case of the infrared part of the propaga K1(x,y)5(3x2/4D2)g2KI(x,y). The calculation of Eq.~37! entails gc 2'A 9x4/32D2 297x8/4096D4 '1.97 D x2 . ~40! It is known that this method overestimatesgc 2 . In this case we verify that the smallest value of our lower bound Eq.~36! is a good approximation for the critical point. The value the critical coupling can be obtained substituting the p nomenological values ofx andD into Eq.~36!. It is obvious that in the limitD→0 we recover the result of the previou section, i.e., with the gluon propagator given by ad function the chiral symmetry is broken for any coupling constantg2 .0. 09401 f is n r, f - Let us now consider the case where we add to the Ga ian form of the infrared propagator a perturbative contrib tion. If this new contribution is of the form given by Eq.~2! or ~3!, it is not difficult to verify the existence of a critica coupling constant for the onset of chiral symmetry breaki The reason for this is that the ultraviolet behavior of t propagator is softened by the factors 12exp(2k2/@4mt 2#) and F(k2), producing an effect equivalent to a dynamical glu mass@10# for which it was shown the existence of a bifu cation point@13,20#. Therefore, we proceed to the most i triguing case where we add to theAnsatzfor the infrared gluon propagator the contribution of QCD with massive g ons and the effect of the running coupling constant, i.e propagator proportional toas(k 2)/(k21mg 2). If we denominate byKU the kernel related to this ultra violet propagator, we assume that we do have a nontri solution for g2> 1 iKUi . ~41! A lower bound for the bifurcation point in the case that w consider the sum of propagators will be given by gc 2I 3x2 4D2 KI1KUI>1, ~42! whereiKI i is given by Eq.~35!. With the triangle inequality iKA1KBi1. ~44! The solutions that break the chiral symmetry appear o for values of the coupling constant obeying Eq.~44!. Note that as long as we can factor outg2 from both propagators and as long as the kernel of the perturbative tail is bound we obtain a condition for values ofg2 above which the self- energy bifurcates. The lower bound for the critical value gc 2> 1 @~3A2/8!~x2/D!1iKUi # , ~45! approaches zero as we take the limitD→0, and is compat- ible with the result of the previous section. Considering the infrared and ultraviolet contributions the propagator, we apply again the method of traces to termine the critical coupling constant through Eqs.~37! and ~38!. In this case the kernelK1(x,y) in Eq. ~38! is given by K1~x,y!5H~x!u~x2y!1H~y!u~y2x!, ~46! where H~x!5 3x2 4D2 expS 2x D D1 1 x1mg 2 4p2d ln~x/L21t! . ~47! 4-5 h ri h ly e he f the e iral ob- f nt ral on to n cal en n tor ny lt if ribe t he ou- an in a g re, - he ller s. l de o o A. C. AGUILAR, A. A. NATALE, AND R. ROSENFELD PHYSICAL REVIEW D62 094014 Note that we have already factored out from the kernel t coupling g2. To determineA4 we needK2(x,y) which is equal to K2~x,y!5H~x!H~y!E 0 y dz1H~x!E y x dz H~z! 1E x ` dz H2~z!. ~48! The critical coupling constant was determined nume cally for four quark flavors,x252.0 GeV2, mg5600 MeV, L5300 MeV, and we assumedt516, where all these values are phenomenologically acceptable, and, in particular, t value of t is compatible with the infrared behavior of the running coupling constant in a theory admitting dynamical generated gluon masses@22# @in these papers we havet '4(mg 2/L2)]. The critical coupling is shown in Fig. 1 as a function of the parameterD. Note that as we decreaseD the value of the critical coupling goes to zero, confirming th result of the previous section. In this region also the integra FIG. 1. Critical coupling constant,gc 2 , as a function of the pa- rameterD, considering the infrared and ultraviolet contributions t the gluon propagator. We see that decreasing the value ofD the critical coupling goes to zero. The evaluation was performed f nf54, x252.0 GeV2, mg5600 MeV,L5300 MeV, andt516. s - re T. 09401 e - e ls appearing in Eq.~48! become more problematic because t Gaussian becomes very peaked. Changes in the form o ultraviolet propagator barely affect this result. Finally, w confirm that for this class of infrared propagators the ch phase transition happens at a smaller value than the one tained with perturbative QCD. IV. CONCLUSIONS An Ansatzfor the infrared gluon propagator in the form o a d4(q) function ~or a variation thereof!, frequently used in applications of the global color model, gives an excelle description of hadronic properties connected with chi symmetry breaking. In this work we studied the bifurcati of the quark self-energy within this model. The idea was compare GCM to QCD with just the perturbative gluo propagator, where it is known that above a certain criti value of the coupling constant the chiral symmetry is brok @13–15#. We verified that the introduction of a delta functio to describe the infrared behavior of the gluon propaga implies that the chiral symmetry is always broken for a value of the coupling constant. This is an interesting resu we remember that this propagator is considered to desc confined quarks@6#. However, this is certainly in contras with what is known to happen in QCD. With a model for the gluon propagator inspired by t delta function but softer at the origin~in a Gaussian form!, we verified, using the method of traces, that the critical c pling for the onset of chiral symmetry breaking is lower th the one expected for perturbative QCD, and recovered particular limit the result obtained with the model involvin a delta function for the infrared gluon propagator. Therefo the critical point for chiral symmetry breaking may distin guish among different QCD motivated models, and in t cases we have studied the critical coupling is always sma than the one known for QCD@14,15#. ACKNOWLEDGMENTS We would like to thank E.V. Gorbar for valuable remark This research was supported by the Conselho Naciona Desenvolvimento Cientı´fico e Tecnolo´gico ~CNPq! ~A.A.N., R.R.!, by Fundaca˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo~FAPESP! ~A.C.A., A.A.N., R.R.! and by Programa de Apoio a Núcleos de Exceleˆncia ~PRONEX!. r . J. @1# P. C. Tandy, Prog. Part. Nucl. Phys.39, 117 ~1997!. @2# C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys.33, 477 ~1994!. @3# R. T. Cahill and S. M. Gunner, Fiz. B7, 171 ~1998!. @4# C. D. Roberts, R. T. Cahill, M. Sevior, and N. Iannella, Phy Rev. D49, 125~1994!; C. D. Roberts, ANL Report No. ANL- PHY-7842-TH-94, 1994; inChiral Dynamics: Theory and Ex periment, Proceedings of the Workshop, MIT, 1994, Lectu Notes in Physics No. 452~Springer-Verlag, New York, 1995!, p. 68; K. L. Mitchell, P. C. Tandy, C. D. Roberts, and R. Cahill, Phys. Lett. B335, 282 ~1994!; M. R. Frank, K. L. . Mitchell, C. D. Roberts, and P. C. 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