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We present our studies of the forward unpolarised doubly-virtual Comptonscattering (VVCS) off the deuteron and the closely related two-photon-exchange($2\gamma$-exchange) corrections to the Lamb shift of muonic deuterium. Thedeuteron VVCS amplitude is calculated in the framework of pionless effectivefield theory, up to next-to-next-to-next-to-leading order (N3LO) for thelongitudinal and next-to-leading order (NLO) for the transverse amplitude. Thecharge elastic form factor of the deuteron, obtained from the residue of thelongitudinal VVCS amplitude, is used to extract the value of the single unknowntwo-nucleon one-photon contact coupling that enters the longitudinal amplitudeat N3LO. The obtained deuteron VVCS amplitude serves as a high-precisionmodel-independent input to examine the $2\gamma$-exchange corrections.Substantial differences with the recent dispersive evaluations are identified,namely, the elastic contribution appears to be larger by several standarddeviations, thus ameliorating the current discrepancy between theory andexperiment on the size of $2\gamma$-exchange effects. A correlation between thevalues of the deuteron charge and Friar radii is found that can be used tojudge on the quality of a parametrisation of the deuteron charge elastic formfactor. The discrepancy between the theory and the empirical result for the$2\gamma$-exchange correction in muonic deuterium appears to be completelyeliminated. To further confirm this, we revisit the hydrogen-deuterium isotopeshift in the same framework. Our work provides an alternative self-consistentand high-precision evaluation of the $2\gamma$-exchange correction in (muonic)deuterium.

Deuteron VVCS and nuclear structure effects in muonic deuterium at N3LO in pionless EFT