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 $1\mathrm{/}{m}_{\mu }$. Using results for the Fermi coupling previously derived in EFT, we obtain singlet and triplet capture rates of $\mathrm{\Gamma }({}^1{S}_0)=724.07\pm 5.45{\mathrm{s}}^{-1}$ and $\mathrm{\Gamma }({}^3{S}_1)=11.55\pm 0.18{\mathrm{s}}^{-1}$, respectively.