Elements of QED-NRQED effective field theory: NLO scattering at leading power

The proton radius puzzle, i.e. the large discrepancy in the extraction of the proton charge radius between regular and muonic hydrogen, challenges our understanding of the structure of the proton. It can also be an indication of a new force that couples to muons, but not to electrons. An effective field theory analysis using nonrelativistic quantum electrodynamics (NRQED) indicates that the muonic hydrogen result can be interpreted as a large, compared to some model estimates, muon-proton spin-independent contact interaction. The muonic hydrogen result can be tested by a muon-proton scattering experiment, MUSE, that is planned at the Paul Scherrer Institute in Switzerland. The typical momenta of the muons in this experiment are of the order of the muon mass. In this energy regime the muons are relativistic but the protons are still nonrelativistic. The interaction between the muons and protons can be described by a hybrid QED-NRQED effective field theory. We present some elements of this effective field theory. In particular we consider $\mathcal{O}(Z\ensuremath{\alpha})$ scattering up to power ${m}^{2}/{M}^{2}$, where $m$ ($M$) is the muon (proton) mass and $Z=1$ for a proton, and $\mathcal{O}({Z}^{2}{\ensuremath{\alpha}}^{2})$ scattering at leading power. We show how the former reproduces Rosenbluth scattering up to power ${m}^{2}/{M}^{2}$ and the latter the relativistic scattering off a static potential. Proton structure corrections at $\mathcal{O}({Z}^{2}{\ensuremath{\alpha}}^{2})$ will be considered in a subsequent paper.