Muon Capture on the Proton with Heavy-Light Currents
We construct a non-relativistic Lagrangian that describes muon-proton electroweak interactions. We determine the leading order coefficients by matching onto the theory with non-relativistic nucleons and relativistic leptons. The most impactful $\mathcal{O}(\alpha)$ corrections to those coefficients are determined by matching the non-relativistic amplitudes for capture of a free muon on a proton to the corresponding relativistic one. We use our non-relativistic effective field theory framework to calculate the capture rate in muonic hydrogen, thereby including radiative corrections of order $\alpha$ and up to order $1/m_\mu$. Using results for the Fermi coupling previously derived in EFT, we obtain singlet and triplet capture rates of $\Gamma({}^1 S_0) = 724.07 \pm 5.45~\rm{s}^{-1}$ and $\Gamma({}^3 S_1) = 11.55 \pm 0.18~\rm{s}^{-1}$, respectively.