Two-photon exchange in (muonic) deuterium at N3LO in pionless effective field theory

We present a study of the two-photon-exchange (2[Formula: see text] -exchange) corrections to the S-levels in muonic ([Formula: see text] D) and ordinary (D) deuterium within the pionless effective field theory ([Image: see text]EFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the [Image: see text]EFT expansion. The only unknown low-energy constant entering the calculation at this order corresponds to the coupling of a longitudinal photon to the nucleon–nucleon system. To minimise its correlation with the deuteron charge radius, it is extracted using the information about the hydrogen–deuterium isotope shift. We find the elastic 2[Formula: see text] -exchange contribution in [Formula: see text] D larger by several standard deviations than obtained in other recent calculations. This discrepancy ameliorates the mismatch between theory and experiment on the size of 2[Formula: see text] -exchange effects, and is attributed to the properties of the deuteron elastic charge form factor parametrisation used to evaluate the elastic contribution. We identify a correlation between the deuteron charge and Friar radii, which can help one to judge how well a form factor parametrisation describes the low-virtuality properties of the deuteron. We also evaluate the higher-order 2[Formula: see text] -exchange contributions in [Formula: see text] D, generated by the single-nucleon structure and expected to be the most important terms beyond N3LO. The uncertainty of the theoretical result is dominated by the truncation of the [Image: see text]EFT series and is quantified using a Bayesian approach. The resulting extractions of the deuteron charge radius from the [Formula: see text] D Lamb shift, the [Formula: see text] transition in D, and the [Formula: see text] hydrogen–deuterium isotope shift, with the respective 2[Formula: see text] -exchange effects evaluated in a unified [Image: see text]EFT approach, are in perfect agreement.