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

We consider reduced quantum electrodynamics ($$ {\mathrm{RQED}}_{d_{\gamma },{d}_e} $$ RQED d γ , d e ) a model describing fermions in a de-dimensional space-time and interacting via the exchange of massless bosons in dγ-dimensions (de ≤ dγ). We compute the two-loop mass anomalous dimension, γm, in general $$ {\mathrm{RQED}}_{4,{d}_e} $$ RQED 4 , d e with applications to RQED4,3 and QED4. We then proceed on studying dynamical (parity-even) fermion mass generation in $$ {\mathrm{RQED}}_{4,{d}_e} $$ RQED 4 , d e by constructing a fully gauge-invariant gap equation for $$ {\mathrm{RQED}}_{4,{d}_e} $$ RQED 4 , d e with γm as the only input. This equation allows for a straightforward analytic computation of the gauge-invariant critical coupling constant, αc, which is such that a dynamical mass is generated for αr> αc, where αr is the renormalized coupling constant, as well as the gauge-invariant critical number of fermion flavours, Nc, which is such that αc → ∞ and a dynamical mass is generated for N < Nc. For RQED4,3, 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 QED4, our analytical results (that use state of the art five-loop expression for γm) are in good quantitative agreement with those obtained from numerical approaches.