Two-loop electron self-energy for low nuclear charges

Calculations of the two-loop electron self-energy for the $1S$ Lamb shift are reported, performed to all orders in the nuclear binding strength parameter $Z\alpha$ (where $Z$ is the nuclear charge number and $\alpha$ is the fine structure constant). Our approach allows calculations to be extended to nuclear charges lower than previously possible and improves the numerical accuracy by more than an order of magnitude. Extrapolation of our all-order results to hydrogen yields a result twice as precise as the previously accepted value [E. Tiesinga et al. Rev. Mod. Phys. 93, 025010 (2021)], differing from it by 2.8 standard deviations. The resulting shift in the theoretical prediction for the $1S$-$2S$ transition frequency in hydrogen decreases the value of the Rydberg constant by one standard deviation.