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Critical behaviour of ($2+1$)-dimensional QED: $1/N_f$-corrections in an arbitrary non-local gauge

Dynamical chiral symmetry breaking (D$χ$SB) is studied within ($2+1$)-dimensional QED with $N$ four-component fermions. The leading and next-to-leading orders of the $1/N$ expansion are computed exactly. The analysis is carried out in an arbitrary non-local gauge. Resumming the wave-function renormalization constant at the level of the gap equation yields a strong suppression of the gauge dependence of the critical fermion flavour number, $N_c(ξ)$ where $ξ$ is the gauge fixing parameter, which is such that D$χ$SB takes place for $N<N_c(ξ)$. Neglecting the weak gauge-dependent terms yields $N_c = 2.8469$ while, in the general case, it is found that: $N_c(1) = 3.0084$ in the Feynman gauge, $N_c(0) = 3.0844$ in the Landau gauge and $N_c(2/3)= 3.0377$ in the $ξ=2/3$ gauge where the leading order fermion wave function is finite. These results suggest that D$χ$SB should take place for integer values $N \leq 3$.