Elsevier

Annals of Physics

Volume 430, July 2021, 168504
Annals of Physics

On the ultraviolet finiteness of parity-preserving U(1)×U(1) massive QED3

https://doi.org/10.1016/j.aop.2021.168504Get rights and content

Highlights

  • The parity-preserving U(1)×U(1) massive QED3 is ultraviolet finiteness.

  • It exhibits vanishing β-functions and anomalous dimensions of the fields.

  • It is parity and gauge anomaly free.

  • The proof is independent of any regularization scheme and valid at all orders in perturbation theory.

Abstract

The parity-preserving UA(1)×Ua(1) massive QED3 is ultraviolet finiteness – exhibits vanishing β-functions, associated to the gauge coupling constants (electric and pseudochiral charges) and the Chern–Simons mass parameter, and all the anomalous dimensions of the fields – as well as is parity and gauge anomaly free at all orders in perturbation theory. The proof is independent of any regularization scheme and it is based on the quantum action principle in combination with general theorems of perturbative quantum field theory by adopting the Becchi–Rouet–Stora (BRS) algebraic renormalization method in the framework of Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) subtraction scheme.

Introduction

The perturbative finiteness in quantum field theory, particularly in Chern–Simons models [1] in three space–time dimensions, has drawn attention since the preliminary results at 1-loop order [2], and afterwards at 2-loops [3]. At all orders in perturbation theory, pure non-Abelian Chern–Simons model in the Landau gauge exhibits ultraviolet finiteness [4]. However, even though coupled to bosonic and fermionic matter fields, non-Abelian Chern–Simons model in three-dimensional Riemannian manifolds still manifests at all radiative order vanishing β-function associated to Chern–Simons coupling constant [5]. The massless U(1) QED3 exhibits ultraviolet and infrared perturbative finiteness, parity and infrared anomaly free at all orders [6]. Moreover, in opposition to some claims in the literature still now defending that parity could spontaneously be broken, even perturbatively, in massless U(1) QED3, known as parity anomaly, has already been discarded by the consistent and correct use of dimension regularization [7], Pauli–Villars regularization [8], algebraic renormalization in the framework of Bogoliubov–Parasiuk–Hepp–Zimmermann–Lowenstein (BPHZL) subtraction method [6], and more recently through the Epstein–Glaser method [9]. The exact quantum scale invariance in dimensional reduced to three dimensional space–time massless QED4 models was investigated in [10], and the gauge covariance of the massless fermion propagator was studied in quenched QED3 [11]. The massive U(1) QED3 can be odd (odd fermion families number) or even (even fermion families number) under parity symmetry. The parity-even massive U(1) QED3 is ultraviolet finite, the gauge coupling β-function and the anomalous dimensions of all the fields vanish, furthermore, is infrared and parity anomaly free at all orders [12]. Besides all the latter quantum field theory formal aspects, planar quantum electrodynamics (QED3) has been demonstrated potential applications in condensed matter phenomena and low energy physics, on the other hand in early universe models and high energy physics as well.

The main purpose in this work is to show the ultraviolet finiteness – vanishing β-functions of both gauge couplings and all field anomalous dimensions – at all orders in perturbation theory of the parity-even UA(1)×Ua(1) massive QED3 [13], and the absence of any kind of anomaly, e.g. gauge and parity, as well. The proof is done by using the BRS (Becchi–Rouet–Stora) algebraic renormalization method in the framework of Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) subtraction scheme, which is based on general theorems of perturbative quantum field theory [14], [15], [16], [17], [18], thus independent of any regularization scheme. Accordingly, the action of the model and its symmetries, the action for the gauge-fixing and the one which couples antifields to the nonlinear BRS transformations of the fields are established in Section 2. The extension of parity-even UA(1)×Ua(1) massive QED3 at the classical level to all orders in perturbation theory – its perturbative quantization – is arranged as follows. Prior the stability analysis of the classical action – if the radiative corrections can be reabsorbed by a redefinition of the initial parameters of the model – which is presented in Section 4, in Section 3 all potential anomalies are identified by means of the analysis of the Wess–Zumino consistency condition, in other words, solving the Slavnov–Taylor cohomology problem in the sector of ghost number one, in addition to, it is checked if the radiatively induced breakings might be fine-tuned by an appropriate choice of local non-invariant counterterms. Final comments and conclusions are left to Section 5.

Section snippets

The model and its symmetries

The action for the parity-even UA(1)×Ua(1) massive QED3 [13] is defined by:

where
, e (electric charge) and g (pseudochiral charge) are the coupling constants with mass dimension 12, and, μ and mμ are mass parameters with mass dimension 1. The field strengths, Fμν=μAννAμ and fμν=μaννaμ, correspond to the electromagnetic field (Aμ) and the pseudochiral gauge field (aμ), respectively. ψ+ and ψ are two kinds of Dirac spinors where the ± subscripts are associated to their spin sign [19],

The unitarity condition: in search for anomalies

The multiplicative renormalizability, more precisely the stability condition, does not assure the extension of the classical model to quantum level, it still remains to guarantee the non existence of any gauge anomaly, i.e. electromagnetic and pseudochiral anomalies, and also the parity anomaly, once the latter is sometimes claimed in the literature as a typical anomaly of three dimensional space–times.

The quantum vertex functional (Γ) matches the tree-level action (Γ(0)) at zeroth-order in ħ, Γ

The stability condition: in search for counterterms

The stability condition, i.e. the multiplicative renormalizability, is ensured if perturbative quantum corrections do not produce local counterterms corresponding to renormalization of parameters which are not already present in the classical theory, therefore those radiative corrections can be reabsorbed order by order through redefinitions of the initial physical quantities – fields, coupling constants and masses – of the theory. Consequently, so as to verify if the classical action Γ(0) (6)

Conclusion

In conclusion, the parity-even UA(1)×Ua(1) massive QED3 [13] is free from any gauge anomaly and parity anomaly at all orders in perturbation theory. Beyond that, it exhibits vanishing β-functions associated to the gauge coupling constants (e and g) and the Chern–Simons mass parameter (μ), and all the anomalous dimensions (γ) of the fields as well. The proof is independent of particular diagrammatic calculations or regularization schemes, since the BRS (Becchi–Rouet–Stora) algebraic

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors thank the anonymous referee for helpful comments. O.M.D.C. dedicates this work to his father (Oswaldo Del Cima, in memoriam), mother (Victoria M. Del Cima, in memoriam), daughter (Vittoria), son (Enzo) and Glaura Bensabat. CAPES-Brazil and CNPq -Brazil are acknowledged for invaluable financial help.

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