Muonic bound systems, virtual particles, and proton radius

The proton radius puzzle questions the self-consistency of theory and experiment in light muonic and electronic bound systems. Here we summarize the current status of virtual particle models as well as Lorentz-violating models that have been proposed in order to explain the discrepancy. Highly charged one-electron ions and muonic bound systems have been used as probes of the strongest electromagnetic fields achievable in the laboratory. The average electric field seen by a muon orbiting a proton is comparable to hydrogenlike uranium and, notably, larger than the electric field in the most advanced strong-laser facilities. Effective interactions due to virtual annihilation inside the proton (lepton pairs) and process-dependent corrections (nonresonant effects) are discussed as possible explanations of the proton size puzzle. The need for more experimental data on related transitions is emphasized.

Figure 1

The ALP-photon conversion in a strong magnetic field according to the interaction term in the Lagrangian given in Eq. (2). The large encircled cross denotes the interaction with an external magnetic field.

Figure 2

The leading (tree-level) correction to the Coulomb potential due to the ALP-photon interaction is given by the tree-level diagram shown. The upper fermion line corresponds to an electron $e$ (ordinary hydrogen) or a muon $\mu$ (muonic hydrogen).

Figure 3

Plot of the average field strength $E\equiv \langle E\rangle$ [see Eq. (13a)] experienced by a bound electron or muon in a one-muon ion (red line, $1\le Z\le 5)$ and for hydrogenlike (electronic) ions in the range $1\le Z\le 92$. For comparison, the average field strength in a laser field of intensity ${10}^{24}\mathrm{W}{\mathrm{cm}}^{-2}$ is given [39]. The Schwinger-critical-field strength is denoted by $E_{\mathrm{cr}}$.