Parity- and Time Reversal-Violating Pion Nucleon Couplings: Higher Order Chiral Matching Relations

Parity- and time reversal-violating (PVTV) pion-nucleon couplings govern the magnitude of long-range contributions to nucleon and atomic electric dipole moments. When these couplings arise from chiral symmetry-breaking CP-violating operators, such as the QCD $\theta$-term or quark chromoelectric dipole moments, one may relate hadronic matrix elements entering the PVTV couplings to nucleon and pion mass shifts by exploiting the corresponding chiral transformation properties at leading order (LO) in the chiral expansion. We compute the higher-order contributions to the lowest order relations arising from chiral loops and next-to-next-to leading order (NNLO) operators. We find that for the QCD $\theta$-term the higher order contributions are analytic in the quark masses, while for the quark chromoelectric dipole moments and chiral symmetry-breaking four-quark operators, the matching relations also receive non-analytic corrections. Numerical estimates suggest that for the isoscalar PVTV pion-nucleon coupling, the higher order corrections may be as large as $\sim 20\%$, while for the isovector coupling, more substantial corrections are possible.