Confirming $U(1)_{L_\mu-L_{\tau}}$ as a solution for $(g-2)_\mu$ with neutrinos

The recent measurement of the muon anomalous magnetic moment by the Fermilab E989 experiment, when combined with the previous result at BNL, has confirmed the tension with the SM prediction at $4.2\,\sigma$ CL, strengthening the motivation for new physics in the leptonic sector. Among the different particle physics models that could account for such an excess, a gauged $U(1)_{L_\mu-L_{\tau}}$ stands out for its simplicity. In this article, we explore how the combination of data from different future probes can help identify the nature of the new physics behind the muon anomalous magnetic moment. In particular, we contrast $U(1)_{L_\mu-L_{\tau}}$ with an effective $U(1)_{L_\mu}$-type model. We first show that muon fixed target experiments (such as NA64$\mu$) will be able to measure the coupling of the hidden photon to the muon sector in the region compatible with $(g-2)_\mu$, and will have some sensitivity to the hidden photon's mass. We then study how experiments looking for coherent elastic neutrino-nucleus scattering (CE$\nu$NS) at spallation sources will provide crucial additional information on the kinetic mixing of the hidden photon. When combined with NA64$\mu$ results, the exclusion limits (or reconstructed regions) of future CE$\nu$NS detectors will also allow for a better measurement of the mediator mass. Finally, the observation of nuclear recoils from solar neutrinos in dark matter direct detection experiments will provide unique information about the coupling of the hidden photon to the tau sector. The signal expected for $U(1)_{L_\mu-L_{\tau}}$ is larger than for $U(1)_{L_\mu}$ with the same kinetic mixing, and future multi-ton liquid xenon proposals (such as DARWIN) have the potential to confirm the former over the latter.