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

## TL;DR

This paper demonstrates that resumming the wave-function renormalization in ($2+1$)-dimensional QED removes gauge dependence in the critical fermion number, confirming the occurrence of dynamical chiral symmetry breaking below this threshold.

## Contribution

It shows that wave-function renormalization resummation leads to a gauge-independent critical fermion number in ($2+1$)-dimensional QED, confirming previous results.

## Key findings

- Critical fermion number $N_c=2.8469$ where D$ ext{chi}$SB occurs.
- Complete cancellation of gauge dependence after resummation.
- Agreement with previous independent calculations.

## Abstract

Dynamical chiral symmetry breaking (D$\chi$SB) is studied within ($2+1$)-dimensional QED with $N$ four-component fermions. The leading and next-to-leading orders of the $N$ expansion were computed exactly in Refs.~[\onlinecite{Gusynin:2016som,Kotikov:2016prf}] in an arbitrary non-local gauge. In this addendum to [\onlinecite{Kotikov:2016prf}], we show that the resummation of the wave-function renormalization constant at the level of the gap equation yields a {\it complete} cancellation of the gauge dependence of the critical fermion flavour number resulting in: $N_c= 2.8469$, which is such that D$\chi$SB takes place for $N<N_c$. The result is in full agreement with one of Ref.~[\onlinecite{Gusynin:2016som}].