Muonic hydrogen and MeV forces

We explore the possibility that a new interaction between muons and protons is responsible for the discrepancy between the CODATA value of the proton-radius and the value deduced from the measurement of the Lamb shift in muonic hydrogen. We show that a new force carrier with roughly MeV-mass can account for the observed energy-shift as well as the discrepancy in the muon anomalous magnetic moment. However, measurements in other systems constrain the couplings to electrons and neutrons to be suppressed relative to the couplings to muons and protons, which seems challenging from a theoretical point of view. One can nevertheless make predictions for energy shifts in muonic deuterium, muonic helium, and true muonium under the assumption that the new particle couples dominantly to muons and protons.

Figure 1

The contribution to the energy-shift in muonic hydrogen, Eq. (2), plotted against the mass of the mediator. In the central solid-blue curve, we require the coupling to the muon, $g_{\mu }$, to be such that the scalar contribution to $(g-2{)}_{\mu }$ equals the theoretical deficit. In the upper/lower solid-blue curve, the scalar contribution to $(g-2{)}_{\mu }$ is determined to be $\pm 1\mathrm{s}.\mathrm{d}$. away from the theoretical deficit. The vector case is similarly given by the dashed-red curves. The coupling to the proton is fixed at $g_p=g_{\mu }$. The solid horizontal line is the discrepancy between the experimentally measured value of the energy and the theoretical prediction assuming the CODATA value [2] for the proton-radius, $r_p=0.8768\mathrm{fm}$. The dotted horizontal lines represent the $\pm 1\mathrm{s}.\mathrm{d}$. uncertainty about this value.

Figure 2

The contribution to the 2S-2P energy splitting in muonic helium, Eq. (9), plotted against the mediator’s mass. The curve is normalized to $\delta E_{\mu H}=-0.31\mathrm{meV}$, which is the discrepancy between the experimentally measured value and the theoretical prediction, assuming the CODATA value for the proton-radius, $r_p=0.8768\mathrm{fm}$. The dotted curves represent the $\pm 1\mathrm{s}.\mathrm{d}$. uncertainty about this value.

Figure 3

The contribution to the energy-shift in true muonium as a function of the force carrier mass. Curves are as in Fig. 1.