Thirring Model as a Gauge Theory

Abstract

We reformulate the Thirring model in $D$ $(2 \le D < 4)$ dimensions as a gauge theory by introducing $U(1)$ hidden local symmetry (HLS) and study the dynamical mass generation of the fermion through the Schwinger-Dyson (SD) equation. By virtue of such a gauge symmetry we can greatly simplify the analysis of the SD equation by taking the most appropriate gauge (``nonlocal gauge'') for the HLS. In the case of even-number of (2-component) fermions, we find the dynamical fermion mass generation as the second order phase transition at certain fermion number, which breaks the chiral symmetry but preserves the parity in (2+1) dimensions ($D=3$). In the infinite four-fermion coupling (massless gauge boson) limit in (2+1) dimensions, the result coincides with that of the (2+1)-dimensional QED, with the critical number of the 4-component fermion being $N_{\rm cr} = \frac{128}{3\pi^{2}}$. As to the case of odd-number (2-component) fermion in (2+1) dimensions, the regularization ambiguity on the induced Chern-Simons term may be resolved by specifying the regularization so as to preserve the HLS. Our method also applies to the (1+1) dimensions, the result being consistent with the exact solution. The bosonization mechanism in (1+1) dimensional Thirring model is also reproduced in the context of dual-transformed theory for the HLS.Comment: 33 page

Causal theory for the gauged Thirring model

“…(29) and (83) [6] [7]. Thus we verify that there is a solution of the equation (85) for all G for G > 0.…”

“…One can see that gauge invariance plays a central role in the derivation of this result and, again, we see the relevance of the construction outlined in section 1 [6] [7]. In addition, we must observe that in this theory there is no infrared difficulties because the gauge bosons are massive.…”

“…[6], the fact that the Lagrangian (11) has a gauge symmetry restricts the choice of the regularization schemes to be used, i.e., we only can employ the regularization schemes which preserve the gauge symmetry. This is the merit of the above construction in comparison with the naïve use of the Lagrangian (4) with a gauge fixing term, without the prior introduction of a gauge symmetry (see [3][4]).…”

“…This paper is organized as follows. In section 2 we introduce the gauged version of the Thirring model [6] [7], for one fermion flavor, using the Stückelberg formalism. In section 3 we give a brief presentation of the method of Epstein and Glaser.…”

Critical behavior of the (2+1)-dimensional Thirring model

“…Moreover, they are likewise intrinsically interesting: in these models the number of fermion flavors N f serves as a control parameter for a quantum phase transition at a critical value N cr f . Several previous works provide a substantial amount of evidence that chiral symmetry breaking (χSB) may be prohibited even for arbitrarily large coupling if N f > N cr f [22,[24][25][26][27][28][29][30][31]. This is a similarity to many-flavor nonabelian gauge theories in four dimensions which are used for particle physics models for dynamical electroweak symmetry breaking [42][43][44].…”