Measurement of the $2$S$_{1/2}-8$D$_{5/2}$ transition in hydrogen

We present a measurement of the hydrogen $2$S$_{1/2}-8$D$_{5/2}$ transition performed with a cryogenic atomic beam. The measured resonance frequency is $\nu=770649561570.9(2.0)$ kHz, which corresponds to a relative uncertainty of $2.6\times10^{-12}$. Combining our result with the most recent measurement of the $1$S$-2$S transition, we find a proton radius of $r_p=0.8584(51)$~fm and a Rydberg constant of $R_\infty=10973731.568332(52)$ m$^{-1}$. This result has a combined 3.1~$\sigma$ disagreement with the CODATA 2018 recommended value. Possible implications of the discrepancy are discussed.