Precision Rydberg State Spectroscopy with Slow Electrons and Proton Radius Puzzle

The so-called proton radius puzzle (the current discrepancy of proton radii determined from spectroscopic measurements in ordinary versus muonic hydrogen) could be addressed via an accurate measurement of the Rydberg constant, because the proton radius and the Rydberg constant values are linked through high-precision optical spectroscopy. We argue that, with manageable additional experimental effort, it might be possible to improve circular Rydberg state spectroscopy, potentially leading to an important contribution to the clarification of the puzzle. Our proposal involves circular and near-circular Rydberg states of hydrogen with principal quantum number around $n=18$, whose classical velocity on a Bohr orbit is slower than that of the fastest macroscopic man-made object, the Parker Solar Probe. We obtain improved estimates for the quality factor of pertinent transitions, and illustrate a few recent improvements in instrumentation which facilitate pertinent experiments.