Probing a Fifth Force in Muonic Atoms through Lamb Shifts and Hyperfine Structure

Abstract

Motivated by the ATOMKI anomalies in 8Be and 4He transitions, we study X17-induced Lamb shifts and hyperfine splittings in muonic atoms with stable nuclei up to Z <= 15. The bound-state problem is solved within the Gaussian Expansion Method using a unified Hamiltonian that includes the standard electromagnetic baseline together with vector and pseudoscalar X17 exchange. The spin-independent Lamb shift is described by a coherent vector muon-nucleus interaction, while the spin-dependent hyperfine sector is built isotope by isotope from shell-model spin fractions. We find a clear complementarity between mediator hypotheses: the vector Lamb-shift signal grows toward heavier nuclei, the vector hyperfine scenario favors odd-N nuclei, and the pseudoscalar scenario favors odd-Z nuclei. Using a signal-to-precision ratio, we identify muonic deuterium, muonic helium-3 ion, and muonic helium-4 ion as the most promising near-term Lamb-shift probes among systems with existing precision benchmarks. For future spectroscopy, the largest vector Lamb-shift signal is predicted in muonic silicon-29, while the leading 1S hyperfine targets are silicon-29 for the vector scenario and phosphorus-31 for the pseudoscalar scenario. The main theoretical uncertainty comes from the Schmidt-model treatment of nuclear spin content.

Figures (6)

• Figure 1: Schematic illustration of muon--nucleon interactions mediated by a light boson $X_{17}$. Panel (a) shows vector exchange, which contributes to both the Lamb shift and hyperfine structure. Panel (b) shows pseudoscalar exchange, which contributes only to the hyperfine sector at leading order.
• Figure 2: Single-particle occupancies in the $p$ shell for $^{13}$C, $^{14}$N, and $^{15}$N. The $1s_{1/2}$, $1p_{3/2}$, and $1p_{1/2}$ orbitals are shown; left (right) columns correspond to proton (neutron) states. Filled circles denote occupied states.
• Figure 3: Radial probability densities for the $2S_{1/2}$ and $2P_{1/2}$ states of muonic $^{31}\mathrm{P}$ under three QED treatments: no vacuum polarization, point-nucleus Uehling potential, and finite-size Uehling potential. Even for $^{31}\mathrm{P}$, the heaviest system considered in the present validation test, the three curves are nearly indistinguishable, showing that the finite-size Uehling correction does not affect the qualitative conclusions of the survey.
• Figure 4: Vector-$X17$ contribution to the $2S_{1/2}$--$2P_{1/2}$ Lamb shift for stable muonic atoms with $Z\le 15$. The signal grows systematically toward heavier nuclei and reaches its largest value in $\mu^{31}\mathrm{P}$ ($264$ meV).
• Figure 5: $X17$-induced $1S_{1/2}$ hyperfine corrections for stable muonic atoms with $Z\le 15$: (a) vector mediator and (b) pseudoscalar mediator. The vector case preferentially enhances odd-$N$ nuclei, while the pseudoscalar case favors odd-$Z$ nuclei. The largest signals are found in $\mu^{29}\mathrm{Si}$ ($0.643$ meV) and $\mu^{31}\mathrm{P}$ ($0.710$ meV), respectively.