**1. Introduction**

In this conference report, we present in a self-contained way the results of Refs. [1,2], where the critical behavior of Quantum Electrodynamics in 2 + 1 dimensions (QED3) have been studied. Contrary to previous reports [3–5], we here follow the Addendum of Ref. [2], which contains a strong upgrade of the exact results of [2] thereby proving the complete gauge-independence of the value of the critical fermion number, *Nc*, which is such that dynamical chiral symmetry breaking (D*χ*SB) in QED3 takes place only for *N* < *Nc*. Indeed, following Ref. [2] and after long discussions with Valery Gusynin, the expansion prescription used in Ref. [2] (and reported on in [3,4]) was modified in the Addendum. The expansion was initially based on (an NLO correction to) the gap equation and was modified to (an NLO correction to) the parameter *α* of its solution (see Equation (6) below). This subtle change in the interpretation of the NLO corrections does not affect at all the LO results of Appelquist et al. [6] but significantly modifies the NLO results (see below Section 4) leading to gauge-invariant *Nc* values after the so-called Nash resummation (see below for more).

The model is described by the Lagrangian:

$$L = \overline{\Psi} (i\hat{\partial} - e\hat{A}) \Psi - \frac{1}{4} F\_{\mu\nu\ \prime}^2 \tag{1}$$

where Ψ is taken to be a four component complex spinor. In the case of *N* fermion flavours, the QED3 has a *U*(2*N*) symmetry. The parity-invariant term *m*ΨΨ with fermion mass *m* breaks this symmetry up to *U*(*N*) × *<sup>U</sup>*(*N*). In the massless case, loop expansions suffer from infrared divergences. The latter are softened when analyzing the model in a 1/*N* expansion [7–9]. Since the theory is super-renormalizable, then the mass scale is given by the dimensional coupling constant: *a* = *Ne*2/8, which remains fixed as *N* → ∞. Early studies of this model (see Refs. [6,10]) showed that physics is damped rapidly at the momentum scales *p a* and that the fermion mass term, which violates the flavour symmetry, is dynamically generated at scales that are orders of magnitude smaller than the internal scale *a*. Since then, the D*χ*SB in QED3 and the *N* dependence of the mass of the dynamic fermion have been the subject of extensive research, see, e.g., [1–31].

One of the central problems is related to the critical number *Nc* of fermions, which is such that D*χ*SB takes place only for *N* < *Nc*. The exact definition of *Nc* is crucial for understanding the phase

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structure of QED3 with far-reaching consequences from particle physics to condensed matter physics systems with relativistic low-energy excitations [32–35]. It turns out that the values that can be found in the literature range from *Nc* → ∞ [10–15] corresponding to D*χ*SB for all *N* values, up to *Nc* → 0 in the case when the D*χ*SB sign is not found [16–18].

Central to our work is the approach of Appelquist et al. [6], which found that *Nc* = 32/*π*<sup>2</sup> ≈ 3.24 by solution of the Schwinger-Dyson (SD) gap equation using the 1/*N* expansion in leading order (LO) approximation. Lattice modeling in accordance with a finite nonzero value of *Nc* can be found in [22–25].

Shortly after [6], Nash approximately included the next-to-leading order (NLO) corrections and managed to partially resum the renormalization constant of the wave function at the level of the gap equation; he found [26]: *Nc* ≈ 3.28.

Recently, in [1], NLO corrections could be calculated exactly in the Landau gauge, obtaining *Nc* ≈ 3.17 (see Erratum to [1]), i.e., a value that is very close to the value of Nash in [26]. More recently, in Ref. [2] the results of [1] were generalized to an arbitrary nonlocal gauge [36,37]. In addition, Ref. [2] (see also its Appendix) showed that resumming the renormalization of the wave function gives a gauge-independent critical number of fermion flavors, *Nc* = 2.8469, the value of which coincides with the results obtained in [29].

The purpose of this paper is to present the main arguments of the papers [1,2] (and corresponding Addendum and Erratum, respectively) leading to exact D*χ*SB results in arbitrary nonlocal gauge [36,37]. This achievement represents a significant improvement in terms of the approximate Nash NLO results that were made mostly in the Feynman gauge. In this regard, considerable interest is currently being devoted to studying the gauge dependence of several models, see [31,38,39]. The use of the Landau gauge in [1] was motivated by recent results on QED3 [31] that revealed the gauge-independence of *Nc* upon using the Ball-Chiu vertex [40]. In fact, after resummation of the renormalization constant of the wave function, we find that the LO and NLO terms in the gap equation become gauge-invariant and match the results of [29].

## **2. SD Equations**

With the conventions of Ref. [1], the inverse fermion propagator has the following form: *<sup>S</sup>*−<sup>1</sup>(*p*) = [1 + *<sup>A</sup>*(*p*)] (*ip*ˆ + <sup>Σ</sup>(*p*)) where *<sup>A</sup>*(*p*) is the fermion wave function and <sup>Σ</sup>(*p*) is a dynamically generated parity-preserving mass, which is assumed to be the same for all fermions. The SD equation for the fermion propagator can be decomposed into scalar and vector components as follows:

$$\Sigma(p) = \frac{2a}{N} \text{Tr} \int \frac{d^3k}{(2\pi)^3} \frac{\gamma^\mu D\_{\mu\nu}(p-k)\Sigma(k)\Gamma^\nu(p,k)}{[1+A(k)]\left(k^2+\Sigma^2(k)\right)},\tag{2a}$$

$$A(p)p^2 = -\frac{2a}{N}\text{Tr}\int \frac{d^3k}{(2\pi)^3} \frac{D\_{\mu\nu}(p-k)\not p\gamma^\mu\not k\Gamma^\nu(p,k)}{[1+A(k)]\left(k^2+\Sigma^2(k)\right)}\,,\tag{2b}$$

where Σ ˜(*p*) = <sup>Σ</sup>(*p*)[<sup>1</sup> + *<sup>A</sup>*(*p*)], *<sup>D</sup>μν*(*p*) is the photon propagator in the non-local *ξ*-gauge:

$$D\_{\mu\nu}(p) = \frac{P\_{\mu\nu}^{\mathbb{Z}}(p)}{p^2 \left[1 + \Pi(p)\right]'}, \quad P\_{\mu\nu}^{\mathbb{Z}}(p) = \mathcal{g}\_{\mu\nu} - (1 - \mathbb{Z})\frac{p\_{\mu}p\_{\nu}}{p^2},\tag{3}$$

<sup>Π</sup>(*p*) is the polarization operator and <sup>Γ</sup>*<sup>ν</sup>*(*p*, *k*) is the vertex function. We shall study Equations (2) for arbitrary values of the gauge-fixing parameter *ξ*. All calculations will be performed using standard perturbation theory rules for massless Feynman diagrams, as in [41,42], see also recent reviews [43,44]. For the most complex diagrams, the Gegenbauer polynomial technique will be used following [45].
