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

TL;DR

This paper calculates two-photon exchange corrections in muonic and ordinary deuterium using pionless EFT up to N3LO, revealing larger elastic contributions than previous studies and refining deuteron charge radius extractions.

Contribution

It provides a comprehensive N3LO pionless EFT calculation of $2 ext{-} ext{γ}$-exchange effects in deuterium, including a novel extraction of the unknown low-energy constant from isotope shift data.

Findings

Elastic $2 ext{-} ext{γ}$-exchange in muonic deuterium is larger than recent estimates.

The study reduces the discrepancy between theory and experiment for $2 ext{-} ext{γ}$ effects.

Deuteron charge radius extractions from different methods are in perfect agreement.

Abstract

We present a study of the two-photon-exchange ($2\gamma$-exchange) corrections to the $S$-levels in muonic ($\mu$D) and ordinary (D) deuterium within the pionless effective field theory (pionless EFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the pionless 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 $2\gamma$-exchange contribution in $\mu$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 $2\gamma$-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 $2\gamma$-exchange contributions in $\mu$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 pionless EFT series and is quantified using a Bayesian approach. The resulting extractions of the deuteron charge radius from the $\mu$D Lamb shift, the $2S-1S$ transition in D, and the $2S-1S$ hydrogen-deuterium isotope shift, with the respective $2\gamma$-exchange effects evaluated in a unified pionless EFT approach, are in perfect agreement.