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.

• Received 7 September 2012

DOI:https://doi.org/10.1103/PhysRevD.87.013011