Two-loop corrections to QCD $\theta$ angle from evanescent operator in the BMHV scheme

We study the renormalization of the QCD $\theta$ angle at the two-loop level focusing on divergent and finite $CP$-violating contributions from evanescent operators, using dimensional regularization with the BMHV scheme. When one considers the Lagrangian in $d$-dimensional space-time instead of four dimensions in dimensional regularization, evanescent operators that break the chiral symmetry are induced. Consequently, $T$-odd and $P$-odd fermion loops in the BMHV scheme generate evanescent contributions to the QCD $\theta$ angle. We carefully classify the evanescent contributions into two types: one originating from the evanescent operators at the one-loop level and the other directly produced by two-loop calculations. We show that renormalization of the parameters in the BMHV scheme keeps removing those unphysical contributions to the QCD $\theta$ angle at any scale at the two-loop level. We also discuss how the rephasing invariance of the radiative correction to the QCD $\theta$ angle is realized in the BMHV scheme.