Dynamical mass generation and critical behavior in pseudo-Proca quantum electrodynamics

We investigate dynamical mass generation in pseudo-Proca quantum electrodynamics (PPQED) using Schwinger--Dyson equations in rainbow-quenched and unquenched approximations. The (3+1)D gauge-field mass $m$, fine-structure constant $\alpha$, flavor number $N$, and ultraviolet cutoff $\Lambda$ determine the critical thresholds for chiral symmetry breaking in the reduced (2+1)D theory. We derive $\alpha_c(m,\Lambda)$ (quenched) and $N_c(m,\Lambda,g)$ (unquenched), with the gauge-field mass suppressing dynamical mass generation via Yukawa screening. For anisotropic fermions with Fermi velocity $v_F \neq c$, the critical coupling scales as $\alpha^*_c \propto (1 + v_F^2/c^2)^{-1}$. These results clarify the non-perturbative dynamics of massive gauge fields in dimensionally reduced QED and connect to effective photon propagation in Proca metamaterials.