Muonic-hydrogen Lamb shift: Dispersing the nucleon-excitation uncertainty with a finite-energy sum rule

We assess the two-photon exchange contribution to the Lamb shift in muonic hydrogen with forward dispersion relations. The subtraction constant (T) over bar (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 +/- 5) 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.