One-loop matching of the $CP$-odd three-gluon operator to the gradient flow

Abstract: The calculation of the neutron electric dipole moment within effective field theories for physics beyond the Standard Model requires non-perturbative hadronic matrix elements of effective operators composed of quark and gluon fields. In order to use input from lattice computations, these matrix elements must be translated from a scheme suitable for lattice QCD to the minimal-subtraction scheme used in the effective-field-theory framework. The accuracy goal in the context of the neutron electric dipole moment necessitates at least a one-loop matching calculation. Here, we provide the one-loop matching coefficients for the $CP$-odd three-gluon operator between two different minimally subtracted 't Hooft-Veltman schemes and the gradient flow. This completes our program to obtain the one-loop gradient-flow matching coefficients for all $CP$-violating and flavor-conserving operators in the low-energy effective field theory up to dimension six.