Novel Soft-Pion Theorem for Long-Range Nuclear Parity Violation

The parity-odd effect in the Standard Model weak neutral current reveals itself in the long-range parity-violating nuclear potential generated by the pion exchanges in the $\Delta I=1$ channel with the parity-odd pion-nucleon coupling constant $h_\pi^1$. Despite decades of experimental and theoretical efforts, the size of this coupling constant is still not well-understood. In this Letter we derive a soft-pion theorem relating $h_\pi^1$ and the neutron-proton mass-splitting induced by an artificial parity-even counterpart of the $\Delta I=1$ weak Lagrangian, and demonstrate that the theorem still holds exact at the next-to-leading order in chiral perturbation theory. A considerable amount of simplification is expected in the study of $h_\pi^1$ by using either lattice or other QCD models following its reduction from a parity-odd proton-neutron-pion matrix element to a simpler spectroscopic quantity. The theorem paves the way to much more precise calculations of $h_\pi^1$, and thus a quantitative test of the strangeness-conserving neutral current interaction of the Standard Model is foreseen.