Strong Coupling Unquenched QED. II: Numerical Study

Abstract

Dynamical chiral-symmetry-breaking in massless QED with N fermion species is studied through the numerical solution of the coupled Schwinger-Dyson (SD) equation. We·have taken into account the fermion loop effect (at the I-loop level) in the SD equation for the photon propagator through the vacuum polarization function JI(k 2 ), with and without the standard approximation: JI((p_q)2) ,JI(max (p2, q2)). We have found that the scaling law is unchanged by this approximation and that, irrespective of the fermion flavor N, the dynamical fermion mass and chiral order parameter obey the same mean-field type scaling, while the quenched planar QED without the vacuum polarization (N =0 limit) obeys the Miransky scaling with the essential singularity. l ) we studied analytically the Schwinger-Dyson (SD) equation in massless QED with N fermion species. In this paper we confirm the analytical results by solving the SD equation numerically. The SD equation is the simultaneous integral equation among the fermion propagator S(p), the photon propagator Dpv(k) and the vertex function rp(p, q; k). The SD equation for the fermion propagator S(p) = [jfA(p2)- B(p2)]-l is decomposed into a pair of coupled integral equations for A and B, each of which is a non-linear integral equation containing multiple-integrals. In the previous analytical treatment,I) we avoided several difficulties appearing in solving the SD equation for the fermion propagator as follows. (1) Multiple-integral: The presence of the nontrivial vacuum polarization leads to the integral equation containing the double integral. This can be avoided by replacing the kernel by the separated (degenerate) form K(P, q)=K(p2)8(p2_q2)+K(q2)8(q2_p2) as a conse­ quence of the LAK l ) approximation a la Landau-Abrikosov-Khalatnikov for the vacuum polarization function: (1·1) Then we can carry out the angular integral exactly in the same manner as in the quenched planar case and the SD equation reduces to the integral equation containing the single integral only. (2) Simultaneousness: This has been evaded by taking the Landau gauge. In the Landau gauge A(p2) == 1 follows under the LAK approximation, and the SD equation for the fermion propagator reduces to the single integral equation for B(p2), as shown in the quenched planar approximation. 2 ) (3) Non-linearity: In order to study the scaling behavior in the neighborhood of the.critical point, we do not have to deal with the non-linear equation and it is sufficient to solve the linearized equation, as guaranteed by the bifurcation theory.3)