Elements of QED-NRQED Effective Field Theory: I. NLO scattering at leading power

TL;DR

This paper develops elements of a hybrid QED-NRQED effective field theory to analyze muon-proton scattering at energies relevant to the MUSE experiment, aiming to shed light on the proton radius puzzle and potential new muon-specific forces.

Contribution

It introduces a hybrid QED-NRQED effective field theory framework for muon-proton interactions at intermediate energies, including calculations up to specific orders in coupling and power expansion.

Findings

Reproduces Rosenbluth scattering up to order m^2/M^2.

Describes relativistic scattering off a static potential at leading power.

Sets the stage for future proton structure correction analyses.

Abstract

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 Non Relativistic 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 non-relativistic. 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 ${\cal O}(Z\alpha)$ scattering up to power $m^2/M^2$, where $m$ ($M$) is the muon (proton) mass and $Z=1$ for a proton, and ${\cal O}(Z^2\alpha^2)$ 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 ${\cal O}(Z^2\alpha^2)$ will be considered in a subsequent paper.