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In 2010 the first extraction of the proton charge radius from muonic hydrogenwas found be be five standard deviations away form the regular hydrogen value.Eight years later, this proton radius puzzle is still unresolved. One of themost promising avenues to resolve the puzzle is by a muon-proton scatteringexperiment called MUSE. The typical momenta of the muons in this experiment areof the order of the muon mass. In this energy regime the muons are relativisticbut the protons are non-relativistic. The interaction between them can bedescribed by QED-NRQED effective field theory. In a previous paper we haveshown how QED-NRQED reproduces Rosenbluth scattering up to $1/M^2$, where $M$is the proton mass, and relativistic scattering off a static potential at${\cal O}(Z^2\alpha^2)$ and leading power in $M$. In this paper we determinethe Wilson coefficients of the four-fermion contact interactions at ${\calO}(Z^2\alpha^2)$ and power $1/M^2$. Surprisingly, we find that the coefficientof the spin-independent interaction vanishes, implying that MUSE will besensitive mostly to the proton charge radius and not spin-independenttwo-photon exchange effects.

Elements of QED-NRQED Effective Field Theory: II. Matching of Contact Interactions