Nonperturbative renormalization of the quark chromoelectric dipole moment with the gradient flow: Power divergences

The $CP$-violating quark chromoelectric dipole moment (qCEDM) operator, contributing to the electric dipole moment (EDM), mixes under renormalization and---particularly on the lattice---with the pseudoscalar density. The mixing coefficient is power-divergent with the inverse lattice spacing squared, $1/{a}^{2}$, regardless of the lattice action used. We use the gradient flow to define a multiplicatively renormalized qCEDM operator and study its behavior at small flow time. We determine nonperturbatively the linearly divergent coefficient with the flow time, $1/t$, up to subleading logarithmic corrections, and compare it with the 1-loop perturbative expansion in the bare and renormalized strong coupling. We also discuss the $\mathrm{O}(a)$ improvement of the qCEDM defined at positive flow time.