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

We present a study of the two-photon-exchange (2γ-exchange) corrections to the S-levels in muonic (µD) and ordinary (D) deuterium within the pionless effective field theory (/πEFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the /πEFT 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 2γ-exchange contribution in µD larger by several standard deviations than obtained in other recent calculations. This discrepancy ameliorates the mismatch between theory and experiment on the size of 2γ-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 2γ-exchange contributions in µD, 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 /πEFT series and is quantified using a Bayesian approach. The resulting extractions of the deuteron charge radius from the µD Lamb shift, the 2S-1S transition in D, and the 2S-1S hydrogen-deuterium isotope shift, with the respective 2γ-exchange effects evaluated in a unified /πEFT approach, are in perfect agreement.