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The translation of experimental limits on the neutron electric dipole momentinto constraints on heavy $CP$-violating physics beyond the Standard Modelrequires knowledge about non-perturbative matrix elements of effectiveoperators, which ideally should be computed in lattice QCD. However, thisnecessitates a matching calculation as an interface to the effective fieldtheory framework, which is based on dimensional regularization andrenormalization by minimal subtraction. We calculate the one-loop matchingbetween the gradient-flow and minimal-subtraction schemes for the$CP$-violating four-quark operators contributing to the neutron electric dipolemoment. The gradient flow is a modern regularization-independent schemeamenable to lattice computations that promises, e.g., better control over powerdivergences than traditional momentum-subtraction schemes. Our results extendprevious work on dimension-five operators and provide a necessary ingredientfor future lattice-QCD computations of the contribution of four-quark operatorsto the neutron electric dipole moment.

One-loop matching of $CP$-odd four-quark operators to the gradient-flow scheme