Abstract
In 2010 the first extraction of the proton charge radius from muonic hydrogen was found be be five standard deviations away form 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 non-relativistic. The interaction between them can be described by 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 ${\cal O}(Z^2\alpha^2)$ and leading power in $M$. In this paper we determine the Wilson coefficients of the four-fermion contact interactions at ${\cal O}(Z^2\alpha^2)$ 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.
Highlights
The response of an on-shell proton to a one-photon electromagnetic probe can be parametrized in terms of two nonperturbative form factors, the so-called “Dirac” [F1ðq2Þ] and “Pauli” [F2ðq2Þ] form factors; see (2).Alternatively, one can define a different linear combination as the “electric” (GE 1⁄4 F1 þ q2F2=4M2, where M is the proton mass) and “magnetic” (GM 1⁄4 F1 þ F2) form factors
In a previous paper we studied some aspects of this effective field theory (EFT) [44]
Denoting by mðMÞ the muon mass and using Z 1⁄4 1 for a proton, we showed that one-photon exchange OðZαÞ QED-nonrelativistic QED (NRQED) scattering at power 1=M2 reproduces Rosenbluth scattering [45], and the two-photon exchange OðZ2α2Þ QED-NRQED scattering at leading power reproduces the scattering of a relativistic fermion off a static potential [46,47]
Summary
The response of an on-shell proton to a one-photon electromagnetic probe can be parametrized in terms of two nonperturbative form factors, the so-called “Dirac” [F1ðq2Þ] and “Pauli” [F2ðq2Þ] form factors; see (2). Other recent extractions use dipole [30], polynomial [26], continued fraction [26], modified z expansion [31], or other parametrizations of the form factors, as well as include inputs from chiral effective field theory (EFT) [32,33]. Most of these [26,30,33] favor the muonic hydrogen result. The Appendixes describe the matching in Coulomb gauge and the properties of the hadronic tensor, as well as list the NRQED Feynman rules
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