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

Abstract

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 (EFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the 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 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 EFT 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 EFT approach, are in perfect agreement.