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Models of gauged $U(1)_{L_\mu-L_\tau}$ can provide a solution to thelong-standing discrepancy between the theoretical prediction for the muonanomalous magnetic moment and its measured value. The extra contribution is dueto a new light vector mediator, which also helps to alleviate an existingtension in the determination of the Hubble parameter. In this article, weexplore ways to probe this solution via the scattering of solar neutrinos withelectrons and nuclei in a range of experiments and considering high and lowsolar metallicity scenarios. In particular, we reevaluate Borexino constraintson neutrino-electron scattering, finding them to be more stringent thanpreviously reported, and already excluding a part of the $(g-2)_\mu$explanation with mediator masses smaller than $2\times10^{-2}$ GeV. We thenshow that future direct dark matter detectors will be able to probe most of theremaining solution. Due to its large exposure, LUX-ZEPLIN will explore regionswith mediator masses up to $5\times10^{-2}$ GeV and DARWIN will be able toextend the search beyond $10^{-1}$ GeV, thereby covering most of the areacompatible with $(g-2)_\mu$. For completeness, we have also computed theconstraints derived from the recent XENON1T electron recoil search and from theCENNS-10 LAr detector, showing that none of them excludes new areas of theparameter space. Should the excess in the muon anomalous magnetic moment beconfirmed, our work suggests that direct detection experiments could providecrucial information with which to test the $U(1)_{L_\mu-L_\tau}$ solution,complementary to efforts in neutrino experiments and accelerators.

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