Neutrino Non-standard Interactions with arbitrary couplings to u and d quarks

We introduce a model for Non-Standard neutral current Interaction (NSI) between neutrinos and the matter fields, with an arbitrary coupling to the up and down quarks. The model is based on a new $U(1)$ gauge symmetry with a light gauge boson that mixes with the photon. We show that the couplings to the $u$ and $d$ quarks can have a ratio such that the contribution from NSI to the Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) amplitude vanishes, relaxing the bound on the NSI from the CE$\nu$NS experiments. Additionally, the deviation of the measured value of the anomalous magnetic dipole moment of the muon from the standard-model prediction can be fitted. The most limiting constraints on our model come from the search for the decay of the new gauge boson to $e^-e^+$ and invisible particles, carried out by NA48/2 and NA64, respectively. We show that these bounds can be relaxed by opening up the decay of the new gauge boson to new light scalars that eventually decay into the $e^- e^+$ pairs. We show that there are ranges that can lead to both a solution to the $(g - 2)_\mu$ anomaly and values of $\epsilon_{\mu \mu} = \epsilon_{\tau \tau}$ large enough to be probed by future solar neutrino experiments.