Solar neutrino probes of the muon anomalous magnetic moment in the gauged $U(1)_{L_\mu-L_\tau}$

Models of gauged $U(1)_{L_\mu-L_\tau}$ can provide a solution to the long-standing discrepancy between the theoretical prediction for the muon anomalous magnetic moment and its measured value. The extra contribution is due to a new light vector mediator, which also helps to alleviate an existing tension in the determination of the Hubble parameter. In this article, we explore ways to probe this solution via the scattering of solar neutrinos with electrons and nuclei in a range of experiments and considering high and low solar metallicity scenarios. In particular, we reevaluate Borexino constraints on neutrino-electron scattering, finding them to be more stringent than previously reported, and already excluding a part of the $(g-2)_\mu$ explanation with mediator masses smaller than $2\times10^{-2}$ GeV. We then show that future direct dark matter detectors will be able to probe most of the remaining solution. Due to its large exposure, LUX-ZEPLIN will explore regions with mediator masses up to $5\times10^{-2}$ GeV and DARWIN will be able to extend the search beyond $10^{-1}$ GeV, thereby covering most of the area compatible with $(g-2)_\mu$. For completeness, we have also computed the constraints derived from the recent XENON1T electron recoil search and from the CENNS-10 LAr detector, showing that none of them excludes new areas of the parameter space. Should the excess in the muon anomalous magnetic moment be confirmed, our work suggests that direct detection experiments could provide crucial information with which to test the $U(1)_{L_\mu-L_\tau}$ solution, complementary to efforts in neutrino experiments and accelerators.