  A non-zero signal $A_\gamma^\mathrm{np}=(-3.0\pm1.4\pm0.2)\times 10^{-8}$ of
the gamma-ray asymmetry in the neutron-proton capture was recently reported by
the NPDGamma Collaboration which provides the first determination of the
$\Delta I=1$ parity-odd pion-nucleon coupling constant $h_\pi^1=(2.6\pm 1.2\pm
0.2)\times 10^{-7}$. The ability to reproduce this value from first principles
serves as a direct test of our current understanding of the interplay between
the strong and weak interaction at low energy. To motivate new lattice studies
of $h_\pi^1$, we review the current status of the theoretical understanding of
this coupling, which includes our recent work that relates it to a nucleon
mass-splitting by a soft-pion theorem. We further investigate the possibility
of calculating the mass-splitting on the lattice by providing effective field
theory parameterizations of all the involved quark contraction diagrams. We
show that the lattice calculations of the easier connected diagrams will
provide information of the chiral logarithms in the much harder quark loop
diagrams and thus help in the chiral extrapolation of the latter.

Nuclear Theory;High Energy Physics - Lattice;High Energy Physics - Phenomenology;Nuclear Experiment