Two-loop mass anomalous dimension in reduced quantum electrodynamics and application to dynamical fermion mass generation

We consider reduced quantum electrodynamics (RQED${}_{d_{\gamma },d_e}$) a model describing fermions in a $d_e$-dimensional space-time and interacting via the exchange of massless bosons in $d_{\gamma }$-dimensions ($d_e\le d_{\gamma }$). We compute the two-loop mass anomalous dimension, ${\gamma }_m$, in general RQED${}_{4,d_e}$ with applications to RQED${}_{4,3}$ and QED${}_4$. We then proceed on studying dynamical (parity-even) fermion mass generation in RQED${}_{4,d_e}$ by constructing a fully gauge-invariant gap equation for RQED${}_{4,d_e}$ with ${\gamma }_m$ as the only input. This equation allows for a straightforward analytic computation of the gauge-invariant critical coupling constant, ${\alpha }_c$, which is such that a dynamical mass is generated for $\alpha_r > \alpha_c$,where$\alpha_r$ is the renormalized coupling constant, as well as the gauge-invariant critical number of fermion flavours, $N_c$, which is such that ${\alpha }_c\to \mathrm{\infty }$ and a dynamical mass is generated for $N < N_c$\mathrm{.}ForRQED$_{4,3}$, our results are in perfect agreement with the more elaborate analysis based on the resolution of truncated Schwinger-Dyson equations at two-loop order. In the case of QED${}_4$, our analytical results (that use state of the art five-loop expression for ${\gamma }_m$) are in good quantitative agreement with those obtained from numerical approaches.