Two- and three-loop anomalous dimensions of Weinberg’s dimension-six C P -odd gluonic operator

We apply a fully automated extension of the ${R}^{*}$ operation capable of calculating higher-loop anomalous dimensions of $n$-point Green's functions of arbitrary, possibly nonrenormalizable, local quantum field theories. We focus on the case of the $CP$-violating Weinberg operator of the Standard Model effective field theory whose anomalous dimension is so far known only at one loop. We calculate the two-loop anomalous dimension in full QCD and the three-loop anomalous dimensions in the limit of pure Yang-Mills theory. We find sizable two-loop and large three-loop corrections, due to the appearance of a new quartic group invariant. We discuss phenomenological implications for electric dipole moments and future applications of the method.