Toward resolving the discrepancy in helium-3 and helium-4 nuclear charge radii

Abstract

The discrepancy in the squared nuclear charge radius difference, ΔR^{2}, between ^{3}He and ^{4}He, as determined from electronic and muonic atom energy levels, presents an unresolved puzzle. This paper shows that accounting for off-diagonal hyperfine mixing effects can substantially reduce this discrepancy. We find that hyperfine mixing with the n^{3}S and n^{1}S states (n>2) in ^{3}He introduces a correction of −1.37 kHz to the isotope shift of the 2^{1}S−2^{3}S transition, a factor of seven times larger than the current uncertainty. This correction modifies ΔR^{2} by −0.0064fm^{2}, shifting it from 1.0757(15)fm^{2} to 1.0693(15)fm^{2}, as initially reported by Werf et al. [arXiv:2306.02333]. This brings ΔR^{2} closer to the value of 1.0636(31)fm^{2} obtained from muonic helium μHe^{+} by Schuhmann et al. [arXiv:2305.11679], narrowing the existing discrepancy from 3.6σ to 1.7σ. The adjusted value ΔR^{2} also agrees well with the result of 1.069(3) fm^{2} derived from the helium 2^{3}S−2^{3}P transitions, as analyzed by Patkóš et al. [Phys. Rev. A 94, 052508 (2016)10.1103/PhysRevA.94.052508]. Our results provide key insights for resolving the discrepancy in ΔR^{2}.