$\mu-H$ Lamb shift: dispersing the nucleon-excitation uncertainty with a finite energy sum rule

TL;DR

This paper evaluates the two-photon exchange contribution to the muonic hydrogen Lamb shift using a finite energy sum rule, finding a small proton polarizability correction and suggesting nucleon structure uncertainties are unlikely to explain the observed discrepancy.

Contribution

It introduces a dispersive approach with a finite energy sum rule to assess nucleon structure effects on the Lamb shift in muonic hydrogen.

Findings

Proton polarizability correction is approximately -40 μeV.

Nucleon structure uncertainties are insufficient to explain the 300 μeV Lamb shift discrepancy.

The sum rule evaluation uses high-quality virtual photoabsorption data.

Abstract

We assess the two-photon exchange contribution to the Lamb shift in muonic hydrogen with forward dispersion relations. The subtraction constant $\bar T(0,Q^2)$ that is necessary for a dispersive evaluation of the forward doubly-virtual Compton amplitude, through a finite energy sum rule, is related to the fixed J=0 pole generalized to the case of virtual photons. We evaluated this sum rule using excellent virtual photoabsorption data that are available. We find that the "proton polarizability correction" to the Lamb shift in muonic hydrogen is $-(40\pm5)\mu$eV. We conclude that nucleon structure-dependent uncertainty by itself is unlikely to resolve the large (300$\mu$eV) discrepancy between direct measurement of the Lamb shift in $\mu H$ and expectations based on conventional Hydrogen measurements.