Vacuum polarization and dynamical chiral symmetry breaking: Phase diagram of QED with four-fermion contact interaction

We study chiral symmetry breaking for fundamental charged fermions coupled electromagnetically to photons with the inclusion of a four-fermion contact self-interaction term, characterized by coupling strengths alpha and lambda, respectively. We employ multiplicatively renormalizable models for the photon dressing function and the electron-photon vertex that minimally ensures mass anomalous dimension gamma(m) = 1. Vacuum polarization screens the interaction strength. Consequently, the pattern of dynamical mass generation for fermions is characterized by a critical number of massless fermion flavors N-f = N-f(c) above which chiral symmetry is restored. This effect is in diametrical opposition to the existence of criticality for the minimum interaction strengths, alpha(c) and lambda(c), necessary to break chiral symmetry dynamically. The presence of virtual fermions dictates the nature of phase transition. Miransky scaling laws for the electromagnetic interaction strength alpha and the four-fermion coupling lambda, observed for quenched QED, are replaced by a mean field power law behavior corresponding to a second-order phase transition. These results are derived analytically by employing the bifurcation analysis and are later confirmed numerically by solving the original nonlinearized gap equation. A three-dimensional critical surface is drawn in the phase space of (alpha, lambda, N-f) to clearly depict the interplay of their relative strengths to separate the two phases. We also compute the beta functions (beta(alpha) and beta(lambda)) and observe that alpha(c) and lambda(c) are their respective ultraviolet fixed points. The power law part of the momentum dependence, describing the mass function, implies gamma(m) = 1 + s, which reproduces the quenched limit trivially. We also comment on the continuum limit and the triviality of QED. DOI: 10.1103/PhysRevD.87.013011