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Quantum Parity Conservation in Planar Quantum Electrodynamics

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Abstract

Quantum parity conservation is verified at all orders in perturbation theory for a massless parity-even U(1) × U(1) planar quantum electrodynamics (QED3) model. The presence of two massless fermions requires the Lowenstein-Zimmermann (LZ) subtraction scheme, in the framework of the Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein (BPHZL) renormalization method, in order to subtract the infrared divergences induced by the ultraviolet subtractions at 1- and 2-loops, however thanks to the superrenormalizability of the model the ultraviolet divergences are bounded up to 2-loops. Finally, it is proved that the BPHZL renormalization method preserves parity for the model taken into consideration, contrary to what happens to the ordinary massless parity-even U(1) QED3.

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Notes

  1. In perturbation theory, the proof on the absence of a parity anomaly in massless U(1) QED3 has also been performed by the Epstein-Glaser renormalization method [29].

  2. A quantum electrodynamics model describing electron-polaron–electron-polaron scattering and four-fold broken degeneracy of the Landau levels in pristine graphene.

  3. It is appropriated to stress that neither the ghosts (c and ξ) nor the antighosts (\(\overline {c}\) and \(\overline {\xi }\)) take part of vacuum-polarization tensor, self energy or vertex function Feynman diagrams at any perturbative order, since they are free quantum fields, thus they decouple.

  4. It should be pointed that, for the sake of subsequent renormalization, the symmetrical diagrams corresponding to \(\gamma _{11_{\pm }}\), \(\gamma _{12_{\pm }}\) and \(\gamma _{13_{\pm }}\) – those with the propagators \({\Delta }^{\mu \nu }_{AA}\), \({\Delta }^{\mu \nu }_{AA}\) or \({\Delta }^{\mu \nu }_{AA}\) inside the loop in its upper part – have to be taken into consideration.

  5. As already mentioned, since possible parity-even local counterterm of the type 𝜖μανAμpαaν has not been taken into consideration, it remains sixteen graphs that could generate parity-odd-like counterterms 𝜖μανAμpαAν and 𝜖μανaμpαaν.

  6. The explicitly BPHZL renormalization and the calculations of all counterterms at 1- and 2-loops, whether parity-even or -odd, are left to another work [44], since the purpose of this one was to verify if the LZ subtraction scheme in the framework of the BPHZ renormalization method would preserve or not parity symmetry.

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Acknowledgements

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 is acknowledged for invaluable financial help.

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  1. Departamento de Física - Campus Universitário, Universidade Federal de Viçosa (UFV), Avenida Peter Henry Rolfs s/n, 36570-900, Viçosa, MG, Brazil

    O. M. Del Cima, D. H. T. Franco, L. S. Lima & E. S. Miranda

  2. Ibitipoca Institute of Physics (IbitiPhys), 36140-000, Conceição do Ibitipoca, São João del Rei, MG, Brazil

    O. M. Del Cima, D. H. T. Franco, L. S. Lima & E. S. Miranda

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  1. O. M. Del Cima
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Del Cima, O.M., Franco, D.H.T., Lima, L.S. et al. Quantum Parity Conservation in Planar Quantum Electrodynamics. Int J Theor Phys 60, 3063–3075 (2021). https://doi.org/10.1007/s10773-021-04851-8

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