Single pion contribution to the hyperfine splitting in muonic hydrogen

TL;DR

This paper evaluates the long-range one-pion exchange contribution to the 2S hyperfine splitting in muonic hydrogen, providing a detailed calculation that was previously overlooked, and quantifies its small but non-negligible effect.

Contribution

It presents the first detailed calculation of the pion exchange contribution to muonic hydrogen hyperfine splitting using chiral expansions and experimental decay data.

Findings

Pion exchange contributes approximately -0.09 microelectronvolts to the hyperfine splitting.

The contribution has an uncertainty of about 0.06 microelectronvolts.

This effect is small but relevant for high-precision measurements.

Abstract

A detailed discussion of the long-range one-pion exchange (Yukawa potential) contribution to the 2S hyperfine splitting in muonic hydrogen which had, until recently, been disregarded is presented. We evaluate the relevant vertex amplitudes, in particular $\pi^0\mu^+\mu^-$, combining low energy chiral expansions together with experimental data on $\pi^0$ and $\eta$ decays into two leptons. A value of $\Delta{E}^\pi_{HFS}= -(0.09 \pm 0.06)\ \mu\rm{eV}$ is obtained for this contribution.