General Kinetic Mixing in Gauged $U(1)_{L_\mu-L_\tau}$ Model for Muon $g-2$ and Dark Matter

The gauged $U(1)_{L_\mu-L_\tau}$ extension of the Standard Model is a very simple framework that can alleviate the tension in muon anomalous magnetic dipole moment, reinforced by the recent Fermilab measurement. We explore experimental probes of the $(g-2)_\mu$ target with a general treatment of kinetic mixing between the $Z'$ gauge boson and the photon. The physical value of the kinetic mixing depends on a free parameter of the model and energy scale of a process. We find neutrino constraints on the $(g-2)_\mu$ target including Borexino, CE$\nu$NS, and white dwarfs are sensitive to this freedom and can be lifted if the kinetic mixing lies in proximity of zero at low momentum transfer. As a further step, we explore $L_\mu-L_\tau$ charged dark matter with a thermal origin and show that the same scenario of kinetic mixing can relax existing direct detection constraints and predict novel recoil energy dependence in the upcoming searches. Future joint effort of neutrino and dark matter experiments and precision spectral measurement will be the key to test such a theory.