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

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)].