On the electron–polaron–electron–polaron scattering and Landau levels in pristine graphene-like quantum electrodynamics

Quantum parity conservation in planar quantum electrodynamics

“…that shows to be invariant under parity, thereby the BPHZL subtraction scheme does not break parity in the case of the 1-loop self-energy either. Finally, it has been finished the proof on the BPHZL parity invariance at 1-loop for the massless parity-even U (1) × U (1) Maxwell-Chern-Simons QED 3 model [10]. Nevertheless, thanks to divergent 2-loops vacuum polarization tensor diagrams (see Fig.…”

“…are related to the electromagnetic field (A µ ) and the pseudochiral gauge field (a µ ), respectively. The Dirac spinors ψ + and ψ − are two kinds of fermions where the ± subscripts are related to their pseudospin sign [10,13]. Also, the fields c and ξ are two kind of ghosts 3 and, c and ξ, the two antighosts, whereas b and π are the Lautrup-Nakanishi fields [14] playing the role of Lagrange multiplier fields for the gauge conditions.…”

“…The adopted gamma matrices are γ µ = (σ z , −iσ x , iσ y ). Finally, the Lowenstein-Zimmermann parameter s lies in the interval 0 ≤ s ≤ 1 and has the same status of an additional subtraction variable (as the external momentum) in the BPHZL renormalization scheme, in such a way that the massless model [10] is recovered by taking s = 1 at the end of calculations. Furthermore, some conventions and useful relations that shall be used in subsequent calculations follow:…”

“…Ultimately, based on the argumentations above, the 2-loops unsubtracted integrands associated to the vacuumpolarization tensors Π µν AA and Π µν aa do not produce parity-violating counterterms of the type, µαν A µ ∂ α A ν and µαν a µ ∂ α a ν , therefore it is concluded that parity is still preserved at 2-loops under the BPHZL renormalization procedures. Besides, due to the fact that the UV divergences are restricted up to 2-loops, thus for higher perturbative orders greater than two there is no need of UV subtractions, consequently it is definitely proved that the BPHZL renormalization method preserves parity for the massless parity-even U (1)×U ( 1) Maxwell-Chern-Simons QED 3 model [10].…”

“…In the meantime, a fundamental question arises if regardless of model the LZ subtraction scheme necessarily violates parity in three space-time dimensions, more specifically, if whether or not parity is broken at any order throughout the infrared subtraction in the Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein (BPHZL) renormalization procedure. Accordingly, the latter issue is dealt in this work by considering a massless parity-even U (1) × U (1) Maxwell-Chern-Simons QED 3 model [10], with two massless fermions, ψ + and ψ − , where the gauge mediating bosons, A µ (electromagnetic field) and a µ (pseudochiral field), associated to the both U (1) symmetries, are massive through a mixed Chern-Simons term.…”