Probing $L_\mu-L_\tau$ models with CE$\nu$NS: A new look at the combined COHERENT CsI and Ar data

The minimal gauged $U(1)_{L_\mu-L_\tau}$ model has long been known to be able to explain the tension between the theoretical and experimental values of the muon magnetic moment. It has been explored and tested extensively, pushing the viable parameter space into a very tight corner. Further, embedding the $U(1)_{L_\mu-L_\tau}$ model in a supersymmetric (SUSY) framework has been shown to relax some of these constraints and has recently been shown to explain the electron anomalous magnetic moment as well. In this model, the logarithm of the mass ratio of third to second generation (s)leptons control the non-negligible kinetic mixing and may crucially alter many of the constraints. We confront both the non-SUSY and SUSY versions of this class of models with the CsI(2017), the recently released CENNS10 data from the liquid Argon detector as well as the updated CsI(2020) data of the COHERENT experiment. We use the recoil energy and timing binned data from CsI(2017) and the energy, time, and Pulse Shape Discriminator binned data from CENNS10 to find estimates for the model parameters in a likelihood maximization test. We also show updated exclusions using all of the above data from the COHERENT Collaboration, as well as projected exclusions from the ongoing Coherent CAPTAIN-Mills experiment. The $(g-2)_\mu$ favored values of the $U(1)_{L_\mu-L_\tau}$ gauge coupling that are still unconstrained overlap with the estimates from COHERENT data within $1\sigma$. The combined COHERENT data is found to prefer the presence of the $U(1)_{L_\mu-L_\tau}$ gauge boson over the Standard Model at $\sim1.4\sigma$. The global minima of a chi-square deviation function using CsI(2020) as well as CENNS10 total counts has significant overlap with the $(g-2)_{\mu}$ favored parameter space in the context of the SUSY and non-SUSY $L_{\mu}-L_{\tau}$ models with a mediator mass in the $20-100$ MeV range.