Elements of QED-NRQED effective field theory. II. Matching of contact interactions

In 2010 the first extraction of the proton charge radius from muonic hydrogen was found to be 5 standard deviations away from the regular hydrogen value. Eight years later, this proton radius puzzle is still unresolved. One of the most promising avenues to resolve the puzzle is by a muon-proton scattering experiment called MUSE. 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 nonrelativistic. The interaction between them can be described by QED–nonrelativistic QED (NRQED) effective field theory. In a previous paper we have shown 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 $\mathcal{O}(Z^2{\alpha }^2)$ and leading power in $M$. In this paper we determine the Wilson coefficients of the four-fermion contact interactions at $\mathcal{O}(Z^2{\alpha }^2)$ and power $1/M^2$. Surprisingly, we find that the coefficient of the spin-independent interaction vanishes, implying that MUSE will be sensitive mostly to the proton charge radius and not spin-independent two-photon exchange effects.

Figure 1

QED-NRQED two-photon exchange Feynman diagrams that contribute to forward lepton-proton scattering at $\mathcal{O}(Z^2{\alpha }^2)$ up to power $1/M^2$. The double line denotes the NRQED field. The dashed (curly) line represents the Coulomb (transverse) photon. The dot, circle, cross, circledot, and dotted curly line symbols are defined in Appendix pp3.