Renormalization Group Running of the Parity Operator in Lorentz-Violating Quantum Field Theory

TL;DR

This work analyzes the renormalization-group running of discrete operators in a Lorentz-violating fermion theory within the Standard Model Extension, focusing on the parity operator. It shows that C, P, and T can exhibit one-loop, scheme-dependent RG running due to the redundancy between $f^{\mu}$ and $c^{\nu\mu}$, with a continuous family of renormalization schemes parameterized by $X$ and a physically observable combination $c_{\mathrm{eff}}^{\nu\mu}=c^{\nu\mu}-\tfrac{1}{2} f^{\nu} f^{\mu}$ whose RG flow is scheme-independent: $\partial_{\log(p/\mathcal{M})} c_{\mathrm{eff}}^{\nu\mu} = \frac{2 g^{2}}{3(4\pi)^{2}} c_{\mathrm{eff}}^{\nu\mu}$. The parity operator itself acquires scale dependence in general, $\partial_{\log(p/\mathcal{M})} S_{P} \propto (1-X) g^{2} f^{0}$, illustrating how RG effects on discrete symmetries can influence virtual states and high-energy observables, albeit with typically tiny numerical impact for electrons and potentially larger effects for heavier fermions. The results provide a framework for leveraging Lorentz-violating formulations in precision tests and for understanding how renormalization choices affect discrete symmetry properties in quantum field theory.

Abstract

In conventional relativistic quantum field theory, the discrete operators $\textbf{C}$, $\textbf{P}$, and $\textbf{T}$ are matrix operators with no renormalization scale dependence. However, in a Lorentz-violating theory with a fermion $f^μ$ term in the action, these operators may acquire nontrivial renormalization group behavior. Because the $f^μ$ term may actually be exchanged in the action for an equivalent $c^{νμ}$ term, the scale dependence depends explicitly on the renormalization scheme, even at one-loop order. The scheme dependence means it is always possible to set the scale dependence parameter $1-X$ to zero, but for analyses of some high-energy electron-photon processes, using a scheme with $X=0-$and thus definite scale dependences for $\textbf{C}$, $\textbf{P}$, and $\textbf{T}-$may nonetheless be more convenient.

Figures (3)

• Figure 1: Divergent one-loop diagram with two Lorentz-violating $f^{\mu}$ vertex insertions on the internal fermion line.
• Figure 2: Diagram with a single $\delta c^{\nu\mu}$ fermion insertion to cancel the $\mathcal{O}(f^{2})$ divergence in the diagram from figure \ref{['fig-one-loop-f']}.
• Figure 3: The two counterterm diagrams that together cancel the divergence with alternate $f^{\nu}$ and $\delta f^{\mu}$ insertions.