QED nuclear recoil effect in helium isotope shift

TL;DR

This paper derives the second-order in the electron–nucleus mass ratio ($m/M$) recoil correction to the leading $mα^5$ QED contribution and applies it to the helium atom, thereby reducing the dominant theoretical uncertainty in the $^3$He–$^4$He isotope shift. It extends the $mα^5$ QED framework to include recoil up to $(m/M)^2$ for three-body helium and provides detailed calculations of recoil, hyperfine mixing, and finite-nuclear-size effects, including the Bethe logarithm and delta-function contributions. The authors extract the mean-square charge radii difference $δr^2 = r_C^2(^3 ext{He}) - r_C^2(^4 ext{He})$ from high-precision electronic measurements, obtaining $δr^2 = 1.0679(13) ext{ fm}^2$, in agreement with the muonic-helium value $1.0636(6) ext{ fm}^2$ at $1.3σ$ and with about 2.4× improved precision. They conclude that electronic helium determinations are less sensitive to nuclear polarizability uncertainties, and highlight the need for future $mα^7$ recoil calculations and experimental improvements to push the precision frontier further.

Abstract

We present a detailed investigation of the leading-order $mα^5$ QED correction with inclusion of the finite-nuclear-mass effects. Previously, this correction had been calculated within an expansion in the electron-nucleus mass ratio $m/M$ up to the first order. In this work, we derive formulas for the $mα^5$ QED contribution that are valid up to the second order in $m/M$, and perform its calculation for the $^3$He$-$$^4$He isotope shift, leading to an improved determination of the nuclear charge-radius difference.