Parity-violating and time-reversal-violating pion-nucleon couplings: Higher order chiral matching relations

Parity-violating 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 charge-parity-violating (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 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 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 nonanalytic 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.

Figure 1

Loop diagrams that give rise to PVTV pion-nucleon interactions. Each circular vertex denotes a PVTV interaction vertex defined in Sec. 3a. Diagram $\left(g\right)$ involves $O\left(E^2\right)NN\pi \pi$ coupling. The last diagram does not contribute to ${\overline{g}}_{\pi }^{\left(i\right)}$ due to the derivative nature of the chiral-invariant pion-nucleon coupling.

Figure 2

The nonvanishing one-loop amputated diagram contribution to the hadron mass shifts. Each square denotes a PCTC CSB interaction defined in Sec. 3b. Diagrams $\left(j\right)$ and $\left(k\right)$ involve $O\left(E^2\right)NN\pi \pi$ interaction vertexes.