Hydrogen 1s-2s transition frequency: Comparison of experiment and theory

TL;DR

This paper calculates the hydrogen 1s-2s transition frequency using quantum electrodynamics and fundamental constants, compares it with experimental data, and explores how adjusting the proton charge radius can reconcile discrepancies.

Contribution

It provides a detailed theoretical calculation of the hydrogen 1s-2s transition frequency including uncertainties and demonstrates how fitting the proton charge radius aligns theory with experiment.

Findings

Experimental frequency exceeds theoretical uncertainty by 23.948 kHz.

Adjusting the proton radius to 0.830734 fm eliminates the discrepancy.

Theoretical uncertainty reduces to 6.4 kHz after fitting the proton radius.

Abstract

Using the Dirac equation, radiative corrections and finite nuclear size and mass corrections, we calculate the $1s$-$2s$ quantum transition frequency $f_{1s,2s}$ of hydrogen and its uncertainty due to the uncertainties $\delta m_e, \delta m_p, \delta \alpha, \delta r_p, \delta R_{\infty}$ of the electron mass $m_e$, proton mass $m_p$, fine structure constant $\alpha$, proton root mean squared charge radius $r_p$, and the Rydberg constant $R_{\infty}$. We use the 2018 CODATA [E. Tiesinga, P. J. Mohr, D. B. Newell, B. N. Taylor, Rev. Mod. Phys. {\bf 93}, 025010 (2021)] procedure for the calculation of $f_{1s,2s}$, and the fundamental constants given therein. We find that the value of the experimental frequency lies outside the theoretical uncertainty (the discrepancy between the theoretical and the experimental frequency is $\Delta f_{1s,2s}^{(2018)} = -23.948$~kHz). But, by fitting $r_p$ we obtain a vanishing discrepancy between the calculated and experimental frequencies and a 6.4 kHz theoretical uncertainty, with $r_p = 0.830734$~fm (and a theoretical uncertainty of $\delta r_p = 0.0022$ fm), consistent with a recent measurement~[W. Xiong, {\it{et al}}., Nature (London) {\bf 575}, 147 (2019)].