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 $\mathrm{\Lambda }$ determine the critical thresholds for chiral symmetry breaking in the reduced (2+1)D theory. We derive ${\alpha }_c(m,\mathrm{\Lambda })$ (quenched) and $N_c(m,\mathrm{\Lambda },g)$ (unquenched), with the gauge-field mass suppressing dynamical mass generation via Yukawa screening. For anisotropic fermions with Fermi velocity $v_F\mathrm{\ne }c$, the critical coupling scales as ${\alpha }_c^{\ast }\propto (1+v_F^2\mathrm{/}{c}^2{)}^{-1}$. These results clarify the non-perturbative dynamics of massive gauge fields in dimensionally reduced QED and connect to effective photon propagation in Proca metamaterials.