Direct detection of light dark matter charged under a $L_{\mu} - L_{\tau}$ symmetry

A possible extension of the Standard Model able to explain the recent measurement of the anomalous magnetic moment of the muon consists in adding a gauged $U(1)_{L_{\mu}-L_{\tau}}$ symmetry. If the dark matter particle is charged under this symmetry, the kinetic mixing between the new gauge boson and the photon induces dark matter-electron interactions. We derive direct detection constraints on light dark matter charged under a $U(1)_{L_{\mu}-L_{\tau}}$ symmetry with electron recoil experiments, and explore prospects with XLZD and OSCURA to close in the parameter space able to explain simultaneously the recent measurement on the anomalous magnetic moment of the muon and the observed relic density of dark matter. We further discuss the spin-dependent scattering contribution arising in this model, which was ignored previously in the literature.