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

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