The BPHZL renormalization of a planar quantum electrodynamics up to 2-loops and beyond

Technical Summary: The BPHZL Renormalization of a Planar Quantum Electrodynamics up to 2-loops and Beyond

Problem Statement
The paper addresses the renormalization of a parity-even, massless $U(1) \times U(1)$ Quantum Electrodynamics in three space-time dimensions (QED$_3$), specifically a Maxwell-Chern-Simons model. This model is relevant for describing quasi-planar condensed matter phenomena, including the Quantum Hall Effect, high-$T_c$ superconductivity, topological insulators, and graphene. A central issue in the renormalization of massless theories in three dimensions is the interplay between ultraviolet (UV) and infrared (IR) divergences. While the standard Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) scheme handles UV divergences, it often fails for massless theories because UV subtractions can induce IR divergences. The Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein (BPHZL) scheme, which incorporates the Lowenstein-Zimmermann (LZ) subtraction method, is required to handle these induced IR divergences.

A specific theoretical question motivates this work: Does the LZ subtraction scheme necessarily violate parity symmetry in three-dimensional space-time? Previous studies on ordinary massless $U(1)$ QED$_3$ indicated that parity is explicitly broken at the 1-loop level due to LZ IR subtractions, which generate a parity-odd Chern-Simons term. The authors investigate whether this parity violation is a universal feature of the BPHZL scheme or if it depends on the specific model. They focus on the parity-preserving extended QED$_3$ model, which features two massless fermions and two gauge fields, to determine if parity is preserved under BPHZL renormalization up to 2-loops and beyond.

Methodology
The authors employ the BPHZL renormalization method combined with the LZ subtraction scheme. The procedure involves the following steps:

1. Model Definition: The action for the parity-even massless $U(1) \times U(1)$ Maxwell-Chern-Simons QED$_3$ is defined, incorporating a Lowenstein-Zimmermann mass term $m(s-1)$ to regulate IR divergences, where $s$ is a subtraction parameter ($0 \le s \le 1$). The model includes two fermion species ($\psi_+, \psi_-$) associated with graphene sublattices and two gauge fields ($A_\mu, a_\mu$).
2. Power Counting: The UV ($d$) and IR ($r$) dimensions of all fields are established to determine the superficial degrees of divergence for Feynman diagrams. The model is identified as superrenormalizable, meaning UV divergences are bounded to a finite number of loops. Specifically, the power counting indicates that divergences cease to exist beyond the 2-loop order ($d < 0$ for 3-loops and higher).
3. Diagram Analysis: The authors analyze all 1-loop and 2-loop Feynman diagrams.
   • At 1-loop, vacuum polarization and self-energy diagrams are identified as superficially divergent.
   • At 2-loops, four vacuum polarization diagrams ($\gamma_{8\pm}, \gamma_{9\pm}, \gamma_{11\pm}, \gamma_{12\pm}$) are identified as potentially logarithmically divergent ($d=0$).
4. Zimmermann's Forests: For the 2-loop renormalization, the authors construct Zimmermann's forests to identify subdiagrams and their overlapping structures.
   • Diagrams $\gamma_{8\pm}$ and $\gamma_{9\pm}$ are found to have no proper divergent subdiagrams (their subdiagrams are convergent).
   • Diagrams $\gamma_{11\pm}$ and $\gamma_{12\pm}$ contain divergent 1-loop self-energy subdiagrams.
5. Subtraction Procedure: The renormalized integrand $R_\Lambda$ is constructed using Zimmermann's forest formula. This involves applying subtraction operators $\tau_\lambda$ (Taylor operators in external momenta and the parameter $s$) to remove UV and IR divergences. The subtraction degrees $\delta$ (UV) and $\rho$ (IR) are determined for each diagram and subdiagram.
6. Explicit Calculation: The authors perform explicit calculations for the 2-loop diagrams. They utilize Feynman parametrization and a new general expression for $J_r$-integrals (loop integrals with arbitrary powers of momenta in the numerator) to evaluate the renormalized integrands. The limit $s \to 1$ is taken at the end to recover the massless theory.

Key Contributions and Results

• Proof of Finiteness: The paper provides a rigorous proof that the parity-preserving massless $U(1) \times U(1)$ Maxwell-Chern-Simons QED$_3$ is both UV and IR finite up to 2-loops and beyond. Due to the superrenormalizable nature of the model, no further renormalization is required for loop orders $L \ge 3$.
• Parity Preservation: A primary result is the demonstration that parity is not broken at the 1-loop or 2-loop levels in this specific model. The authors show that the BPHZL subtraction operators, when applied to the parity-even model, do not generate parity-odd counterterms (such as the Chern-Simons term). This contrasts with the ordinary $U(1)$ QED$_3$ where parity is broken by the LZ scheme.
• Non-Locality of Renormalized Terms: The explicit calculation of the 2-loop diagrams ($\gamma_{11\pm}, \gamma_{12\pm}$ and $\gamma_{8\pm}, \gamma_{9\pm}$) reveals that the resulting renormalized contributions are non-local. Specifically, the terms involve denominators containing powers of the external momentum $p^2$ (e.g., $(c' + p'^2)^{-1}$), which correspond to non-local operators in configuration space. Consequently, these diagrams do not contribute to the renormalization of the model's parameters (coupling constants or masses).
• Technical Developments: The authors derive a new general expression for $J_r$-integrals (Equation B.7) applicable to any number of momenta in the integrand numerator, which facilitates the evaluation of complex multi-loop integrals. They also explicitly compute the $X'_l$ terms associated with the 2-loop diagrams.

Significance
The paper claims that this work resolves a fundamental question regarding the behavior of discrete symmetries under the BPHZL renormalization scheme in three dimensions. By demonstrating that parity is preserved in the $U(1) \times U(1)$ model but broken in the ordinary $U(1)$ model, the authors conclude that the induction of parity violation by LZ infrared subtractions is model-dependent, not an inherent flaw of the renormalization scheme itself.

The authors state that, to the best of their knowledge, this is the first demonstration in quantum field theory that a discrete symmetry (parity) can be preserved in a massless 3D theory renormalized via the BPHZL method, despite the presence of two massless fermions requiring the LZ subtraction scheme. This result supports the consistency of the parity-even extended QED$_3$ model as a viable framework for describing phenomena in graphene and other planar systems where scale invariance and parity preservation are observed. The work confirms that the model's UV and IR finiteness emulates the scale invariance observed in graphene without requiring the introduction of parity-violating counterterms.