Gauge invariance of the phase diagram in quenched QED

We incorporate an approximation to the full QED vertex containing both longitudinal and transverse pieces into the Schwinger-Dyson equation for the electron propagator in quenched QED, from which we find solutions to the dynamical electron mass $M\left(p\right)\mathrm{}$. Throughout we use a general covariant gauge, and find that the resulting four-Fermi coupling versus gauge-coupling diagram separating the chiral-symmetric and dynamically broken phases of quenched massless QED is gauge invariant.