Correlating neutrino millicharge and muon $(g-2)$ in an abelian $L_\mu-L_\tau$ model

The inclusion of an additional $U(1)$ gauge symmetry is a common feature in many extensions of the Standard Model, revealing the intricate connections between particle physics and cosmology. The $L_{\mu} - L_{\tau}$ model stands as a prominent member of this distinguished family, characterized by its anomaly-free nature and resilience in the face of collider constraints. This framework provides a unique vantage point for investigating both the intriguing mystery of the muon $(g-2)$ anomaly and the puzzling issue of the Hubble tension. However, due to the presence of kinetic mixing between the photon and $Z'$ in this model, the neutrinos have the potential to acquire minuscule electric charges, often referred to as millicharges ($q_{\nu}$) which is directly related to the strength of the new gauge couplings. A crucial question emerges: how does the model's inclusion of millicharges, while adhering to the stringent constraints imposed by experimental observations, influence its inherent ability to address the muon $(g-2)$ anomaly and the Hubble tension? We find the current upper bounds on $q_{\nu}$ derived from experiments such as the beam dump, XENONnT and LUX-ZEPLIN experiments can impose strong constraints on the $U(1)_{L_{\mu} - L_{\tau}}$ coupling. Consequently, these constraints may limit the ability of the model to fully accommodate the current measurement of $(g-2)_{\mu}$ while having a relatively minor impact on the resolution of the Hubble tension.