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We consider the two-photon exchange contribution to the $2P-2S$ Lamb shift inmuonic deuterium in the framework of forward dispersion relations. Thedispersion integrals are evaluated using experimental data on elastic deuteronform factors and inelastic electron-deuteron scattering, both in thequasielastic and hadronic range. The subtraction constant that is required toensure convergence of the dispersion relation for the forward Compton amplitude$T_1(\nu,Q^2)$ is related to the deuteron magnetic polarizability $\beta(Q^2)$.Based on phenomenological information, we obtain for the Lamb shift $\DeltaE_{2P-2S}=2.01\pm0.74$ meV. The main source of the uncertainty of thedispersion analysis is due to lack of quasielastic data at low energies andforward angles. We show that a targeted measurement of the deuteronelectrodesintegration in the kinematics of upcoming experiments A1 and MESA atMainz can help quenching this uncertainty significantly.

Nuclear structure contribution to the Lamb shift in muonic deuterium