Theorem for the Photon Asymmetry in Radiative Muon Capture

I show, for the standard theory of radiative muon capture including all couplings except $g_S$ and all usual diagrams, that the photon asymmetry $\alpha$ and circular polarization $\beta$ satisfy $\alpha =+1+O\left(\frac{1}{m^2}\right)$ and $\beta =+1+O\left(\frac{1}{m^2}\right)$, where $m$ is the nucleon mass. This may bear on current disagreement between ${\alpha }^{\mathrm{theor}}$ and ${\alpha }^{\mathrm{expt}}$ since $O\left(\frac{1}{m^2}\right)$ terms have never been calculated consistently and since other uncalculated corrections, some of which are discussed, may contribute in the same order.