Constraints on Yukawa-type New Forces from the Lamb Shift in Muonic Hydrogen and Deuterium

We develop an analytical framework to constrain Yukawa-type fifth-force interactions using the Lamb shift in hydrogen and deuterium.

The sixfold coordinate-space integral for the energy shift is transformed into a one-dimensional momentum-space convolution of the atomic form factor and the Yukawa potential, eliminating the need for approximations.

This formulation directly incorporates empirical nuclear charge distributions through their form factors, making the implementation of experimental results straightforward.

Applying this method to the latest hydrogen and deuterium spectroscopy data, we derive constraints on the coupling constant $\alpha$ in the interaction range $\lambda \in [10^{-4}, 10^{6}]$~fm.

A resonance-like cancellation between the $2s$ and $2p$ orbital shifts is identified at a critical range $\lambda \sim 10^{5}$~fm, where the energy shift vanishes and the constraint on $\alpha$ diverges, marking a transition between attractive and repulsive regimes.

The constraints are shown to be independent of the nuclear charge distribution model.

Our results indicate that the transition frequency shifts induced by such new forces lie well below current spectroscopic resolution, establishing spectroscopic precision as the primary bottleneck for probing these interactions.