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

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 $f^{\mu}$ term in the action, these operators may acquire nontrivial renormalization group behavior. Because the $f^{\mu}$ term may actually be exchanged in the action for an equivalent $c^{\nu\mu}$ 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.