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Vacuum Polarization and Dynamical Chiral Symmetry Breaking in Quantum Electrodynamics

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Vacuum Polarization and Dynamical Chiral Symmetry Breaking in Quantum Electrodynamics '••fA9Z00tl{ АКАДЕМИЯ НАУК УКРАИНСКОЙ ССР ИНСТИТУТ ТЕОРЕТИЧЕСКОЙ ФИЗИКИ ITP-89-45B V. P.Gusynin VACUUM POLARIZATION AND DYNAMICAL CHIRAL SYMMETRY BREAKING IN QUANTUM ELECTRODYNAMICS :шшттт *4 Academy of Soienoes o:t the Ukrainian SSR Institute for Theoretical Phyaioe Preprint MP-89-45E V.P.Guaynin VACUUM POLARIZATION AND DYNAMICAL CHIRAL SYMMETRY BREAKING IN QUANTUM ELECTRODYNAMICS Ki«Y - 1989 УДК 539.12; 530.145 В.П.Гусынин Поляризация вакуума и динамическое нарушение киральной симметрии в квантовой электродинамике Дяя массовой функции фермиона рассмотрено уравнение Швингера- Дайсона в лестничном приближении с учетом эффектов поляризации вакуума. Показано, что даже в ситуации "нуль-заряда" существует при достаточно большой константе связи ( о(. > о<с >0 ) решение со спонтанно нарушенной киральной симметрией. Обсуждается су- ществование локального предела в рассматриваемой модели. Y.P.Gusynin VAOUUB Polarisation and Dynamical Chiral Syanetry Breaking in QuaatiM BleatrodynaMios The Sohwinger-Dyson equation in the ladder approximation is considered for the feraion aaes funotlon taking into account the raouua polarisation effects. It la shown that even in the "мго-oharge" situation there exists, at rather large coupling oonetant ( Ы > oie > 0 ), a solution with apontaneously broken ohiral syaaetry. The existence of the looal limit in the model oonoemed is dlaoussed. 1969 Институт теоретической физики АН УССР -3- Reoent years have witnessed an increasing Interest in pos- sible existence of a new nonperturbative phase in quantum eleot- rodynamios (QBD) (for a review aee Ref.[i]) in oonneotion with application of this idea to describe the narrow e+e~ and ytf peaks [2-5] in heavy - ion collisions. Besides, the dynamics of supercritical QBD phase foras the basis of the dynamios with "walking** ooupling oonstant in eleotroweak models with teohnloo- lour [6-a]. For a long time the problem of the existence of a new QBD phase was associated with the searoh for a nontrivial zero of the renormalisation-group jS -funotion [9]. The existence of suoa a zero would allow one to solve the problem of the looal limit in QBD (the out off Д-900 ) that has triggered a keen interest since the papers of Landau, Pomeranohuk and Pradkin appeared [idj. essential progress ia the solution of this problem was made after the existenoe in QBD of a phase with spontaneously broken ohiral symmetry for a sufficiently strong bars ooupling oonstant (o( > d e ) has been shown. This faot was established by means of general methods of the renoxmalisation group [ill and also by a direct solution of the Sohwinger-Dyson (SD) [i2^J and Bethe-Sal- = < peter equations [13I in the ladder approximation (where o£e % ) with a finite (Л < <**) outoff. A physical interpretation of the critical oonstant ctc was given in Refs.[i4,15] where a olose relation between spontaneous ohiral symmetry breaking in QBD and the "fall into the centre" phenomenon ("the wave funotion oollap- se" meohanism) in quantum mechanics was also mentioned. The dyna- mical fermion mass in the ladder approximation is [t3,15l In the looal limit (/\ -=? 00 ) the mass 1П remains finite if the following renormalisation of the ooupling oonstant is perfor- med [13,15-17] The oharge renormalisation (2) corresponds to the в -funotion -4- which poaseaees the ultraviolet stable zero at the point oL = • oLc , The oritioal ooupling constant Ы c corresponds to the aeoond order phase transition |_17j an^ separates maasleaa and massive QED phases. The existence of auch a point has been sup- ported by ourrent Monte-Carlo computer simulations in lattioe nonoompaot QED both, in the approximation with quenched fermions and with aooount of dynamical fermions \)&]. An important feature of the ladder approximation is the lar- ge anomalous dimension of the oomposite operator ^ У (^т- 2) jjl9,2O] which ie reflected in a slower (as compared, for instan- ce, to QCD) asymptotic decrease of the dynamical masa function S(pj- -**o /- '™ ' (4) The latter circumstance turns out to be very important for BOI- ving the flavour-changing neutral-ourrent problem in technloo- lour models j_7,2ij. Also, the large anomalous dimension Ут - 1 makes it necessary to consider the four-fermion operators \Z0~[ since in the looal limit the dynamical dimension of auoh operators equals four and they are the relevant (a la Wilson) operators in this limit. All these results were obtained in the ladder approximation (with bare vertex ^iu and Ъаге photon propagator) for the fer- mion propagator in the Landau gauge (z =o)> It ie olear that the eearoh of the possibility of going beyond the ladder appro- ximation is a very important and nontrivial problem. In Refs. [22^] the approximation with "frozen" fermions (quenched, approxi- mation) was considered. Here all planar and nonplanar diagrams with photons exchange were taken into account. It was shown that the prinoipal peculiarities of the ladder approximation (the existence of a oritioal coupling constant oic , the essentially nonanalytlo dynamical mass dependence on the interaction constant of the form (1), large anomalous dimension of the operator *pf with a oharaoteristio mass function decrease (4)) are oonserred in this approximation too. In Refe. [23,24! the dependence on the ohoioe of gauge (Z ф 0)was studied both in the ladder approxi- mation [23З and with aooount of the more complete expression for the vertex /~V* satisfying the Ward-Takahashi identity [24]. -5- In Ref. [17^ the hypothesis about the existenoe in QSD of fixed ultraviolet stable point (ole^ l) connected with вроп- taneous ohiral symmetry breaking even in the "zero-oharge" situ- ation was put forward. This possibility wae illustrated in a model imitating the "zero-oharge" situation by introducing the infrared cutoff JA ~ A/p into the SD equation for the masa funo- tion (the numerical parameter y° > 1 ). In this paper we consi- der the immediate influence of the vacuum polarization effeots (in particular, the "zero-charge" situation) on the dynamical maas generation. At the end of the paper we'll discuss how the results of our paper are related to the results of Hef.[25] where the similar problems were discussed. The SD equation for the mass funotion &(pz) in the ladder approximation taking into acoount the vacuum polarization effeots haa the form (in the Landau gauge) The integration in (5) is carried out in Euolidean region. Having used the standard substitution f][(р-К)г~\ ~ ПС we can integrate over the angles in Bq.(5) and obtain о £ where the kernel X), (7) (в) ) For /7(x) we'll make use of the one-loop expression of vaouun polarization by maesless fermiona Her* /Vf is the number of fermiona. The effeotiv* ooupling oonetant Q(X) behaves qualitati- vely in the earn* manner both for maaelese and aaeeiTe fermiona (some difference la obaerved in the infrared region X ~-0 -6- re for the massive feimions and д(о~)~ _ л •— 0 (total screening) for messleas fermiona). To study the posaibillty for the appearanoe of a nontrivial solution of the nonlinear integral equation (6) it suffloes to oonaider the bifurcation equation [26J (9) m where fll ia dynamioal mase, C)(ffl2) - WL . The equation (9) is equivalent to the differential equation (10) with the boundary conditions 1 = 0. (и) It ie convenient to introduoe the variable Bqs. (10),(11) will be written as (12) We note that all the momentum dependence of the function В is expressed through the function 8 (X) » i.e., through the standard effective charge (in the approximation oonoerned the re- normallnation oonatant ^ = 1 and the function & Cx ) ia a renormalisation-group invariant). Moreover, if one aesuaes that the solution to Bq.(12) is a slowly varying funotion of i? and hence the term with the second derivative in this equation can be omitted, then we obtain (C is the integration oonatant). For >>1 th« funotion haves as follow* -7- The expression (15) agreea exactly with the aaymptotio expres- sion for the mass funotion obtained in the Landau, Abrikosov and Khalatnikov papers [27^). Returning to Bq.(12) we note that it ia a linear second or- der differential equation with three singular points £ -O^l^oo 6 ol which г = D, 1 a*" regular singular points} ? = ©o ia an irregular one. In virtue of this the solutions of Eqs.(12) can- not Ъе expressed in general oaee through the known special func- ЗЗГ tions. Since the variable i? changes within Zo ^-? / then at Ы.(л)г< 33f/tfi the variable F >> 1 and Eq.(i2) can be simplified by keeping at the derivative °f&/clz tlle expansion terms at large i? . In this case The general solution of Eq.(i6) has the form oft) -zr[C<P(**4) CSrc*c4->[ lm 0,С; Z) and 1^7«,C;?) are the confluent hypergeomet- ric functions [Щ,^~^Щ* а =f~f!i> C ~1+'Z{'- The boundary conditions (13) lead to the following transce- ndental equation for mass m : (18) г = In deriving Eqa.(18) use wae made of the formulae for derivati- ves of the oonfluent hypergeometrio funotiona [28J : -8- (19) For 'y/ft ""> oo^?o-»eo)we find using the asymptotice of funo- tione ф and ^ [28] с», (21) that Bq.(18) takes the form r irtiere the functions ^(f P ) aDd -f(y g ) are equal to jt In order to eatiafy Bq.(22) at Л/т -ъ &°, ?= %^ahould be oloee to zero of the function f(f,F) , thua, Bq. /^?)=£> determines the oritioal ooupling oonetant ci,, , From the asymp- totio behaviour of the function Ijr at infinity (21) and at «его s P(c-i) f-c м find -9- га+у-у*)?^^** (26) (27) Aa it follows from Bqs. (26), (27) the funotion f(f,Z) oan have only a finite number of zeros. We need the extreme right ze- ro of the function -/^^(corresponding to the lowest critioal value oic ) which should be euoh that •ХИ/е(О,Щ'$?1 .
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