Measuring the $\mathit{P}$-Odd Pion-Nucleon Coupling ${\mathit{h}}_{\mathit{\pi }\mathit{NN}}^{\left(1\right)}$ in ${\mathit{\pi }}^{+}$-Photoproton Production near Threshold

We show that $\overrightarrow{\gamma }p\to {\pi }^{+}n$ in the threshold region is an excellent candidate for measuring the leading parity-violating pion-nucleon coupling $h_{\pi \mathrm{NN}}^{\left(1\right)}$ to an uncertainty of $20\%$ if it has a natural size from dimensional analysis. The conclusion is based on a large unpolarized cross section, a new low-energy theorem for the photon polarization asymmetry at the threshold $A_{\gamma }{|}_{\mathrm{th}}=\sqrt{2}{f}_{\pi }({\mu }_p-{\mu }_n)h_{\pi \mathrm{NN}}^{\left(1\right)}{/g}_A{m}_N\sim h_{\pi \mathrm{NN}}^{\left(1\right)}/2\mathrm{}$, and its strong dominance at forward and backward angles in the threshold region.