Dynamical Symmetry Breaking in QED3 from the Wilson RG Point of View

Ken-Ichi Kubota 0 Haruhiko Terao 0 0 Institute for Theoretical Physics, Kanazawa University Kanazawa 920-1192 , Japan Dynamical symmetry breaking in three dimensional QED with N flavors, which has been mostly analyzed by solving the Schwinger-Dyson equations, is investigated by means of the approximated Wilson, or non-perturbative, renormalization group (RG). We study the RG flows of the gauge coupling and the general four-fermi couplings allowed by the symmetry, concentrating our interest on study of the phase structure. The RG equations have no gauge parameter dependence in our approximation scheme. It is found that there exist chirally broken and unbroken phases for N > Ncr (3 < Ncr < 4) and that the unbroken phase disappears for N < Ncr. We also discuss spontaneous parity breaking in QED3 with four-fermi interactions. - incorporated. There the coupled SD equations for the photon self-energy and the mass function were examined. As a result, it was found that these more elaborate treatments of the SD equations also support the qualitative picture of the chiral symmetry breaking mentioned above. Even the critical flavor number has not been altered significantly in these later studies. Numerical simulations have been also performed to investigate non-compact QED3 defined on a lattice. 6) Interestingly, the results from these simulations are consistent with those obtained by solving the approximated SD equations, namely, 3 < Ncr < 4. QED3 is also invariant under parity transformations as long as the Chern-Simons term is absent. It has been shown by Redlich 7) that the Chern-Simons term should be generated through gauge invariant regularization with an odd number of twocomponent massless spinors. However, according to the Vafa-Witten argument, 8) parity symmetry is not spontaneously broken for QED3 with four-component spinors. Actually, it has been shown 9) that the results from the SD equation analyses are consistent with this expectation by considering the gap equations for chirally invariant but parity odd dynamical mass. In later works, these studies were extended to QED3 with fermion selfinteractions. 10) This was motivated by the hypothesis that the parity symmetry can be spontaneously broken in the presence of general four-fermi interactions. The SD analyses reveal the presence of a parity broken phase and also suggest that this phase is separated from the chiral symmetry broken phase. The Chern-Simons term is generated through radiative corrections in the parity broken phase even in the case of an even number of flavors. In other studies, the dynamics of QED3 with the Chern-Simons term have been also intensively investigated, 12), 13) in particular with regard to its applications in modelling (high-temperature) superconductors. 14), 15) In this paper, however, we consider only QED3 with an even number of flavors and do not include the ChernSimons term. Furthermore, the three-dimensional Thirring model has also been examined in numerical simulations as well as by solving the SD equations. 16) The exact renormalization group (ERG), 17) - 21) which is the continuous version of the Wilson RG transformation, has been known as an analytical method applicable to non-perturbative dynamics of field theories. Both the ERG and the SD equations are given in functional forms and have correlation functions as their solutions. Also in both cases, regularization is necessary and some approximations are inevitable in practical calculations, but it should be noted as a difference between the two that the ERG equations give RG flows for the effective couplings, while the SD gives order parameters in terms of bare parameters.) In analysis given in this paper, we make rather crude approximations. For this reason, we refer to the approximated ERG as the non-perturbative (NP) RG hereafter. The NPRG has not been applied to study the phase structures of dynamical symmetry breaking in QED3 (with four-fermi interactions) in spite of the great interest mentioned above.) The purpose of this paper is to show that using the NPRG, we can clarify the phase structures of these theories much more easily than in the SD approaches. It is clearly shown using the NPRG that the presence of the IR attractive fixed point is essential for the novel phase transition. Moreover, we are able to directly determine the boundaries between the chiral symmetry broken phase, the parity broken phase, and the unbroken phase by simply following the RG flows. This sharply contrasts the SD approach. Note that in the SD approach we must assume the order parameters needed for the symmetries to be broken a priori to derive the gap equations and to solve them for every theory. In the RG approach, contrastingly, we can treat any theory invariant under the symmetries on an equal footing, without a need to consider the order parameters. We aim to demonstrate such advantageous points of the NPRG in comparison with the SD analyses through explicit calculations for QED3, not to pursue further improvement of the approximations. Dynamical chiral symmetry breaking in (3+1)-dimensional gauge theories has been analyzed using the NPRG in the case of a single massless flavor. 24) - 26) In this analysis, four-fermi interactions are induced in the effective theories by the gauge interactions. It has been found that the RG flows of the four-fermi couplings reveal the phase structure. Also, if we truncate the corrections for the four-fermi interactions up to a set called the ladder-type, then the phase boundary obtained by solving the SD equation in the ladder approximation is found to be precisely reproduced. The anomalous dimensions of fermion composite operators can be immediately calculated from the RG equations. Moreover, the gauge-parameter dependence is remarkably improved by using the NPRG equations obtained through the derivative expansion. 24) The momentum cutoff introduced to define the Wilson RG breaks the gauge invariance. The remaining gauge dependence comes purely from this cutoff procedure, while the ladder approximation destroys the gauge invariance. We may remove the gauge dependence by substituting the anomalous dimension of the fermion with one evaluated in a gauge invariant way. We adopt such an approximation scheme in the RG analyses of QED3 as well. The remainder of the paper is organized as follows. In 2, we briefly review the NPRG for a simple four-fermi theory and the manner in which dynamical symmetry breaking is described in the RG framework. Section 3 is the main part of the paper. First we consider the general four-fermi operators allowed by the symmetries of QED . After explaining our approximation scheme to the NPRG, we explicitly 3 examine the phase structure by looking at the RG flows, while varying the number of flavors. In the Wilson RG framework we treat all the effective theories simultaneously. In our scheme, the phases of QED3 with general four-fermi interactions are explored automaticall (...truncated)