Measuring the Low-Energy Weak Mixing Angle with Supernova Neutrinos

The weak mixing angle $\theta_W$ is a fundamental parameter in the electroweak theory with a value running according to the energy scale, and its precision measurement in the low-energy regime is still ongoing. We propose a method to measure the low-energy $\sin{^2\theta_W}$ by taking advantage of Argo, a future ton-scale liquid argon dark matter detector, and the neutrino flux from a nearby core-collapse supernova (CCSN). We evaluate the expected precision of this measurement through the coherent elastic neutrino-nucleus scattering (CE$\nu$NS) channel. We show that Argo is potentially capable of achieving a few percent determination of $\sin{^2\theta_W}$, at the momentum transfer of $q \sim 20$ MeV, in the observation of a CCSN within $\sim 3$ kpc from the Earth. Such a measurement is valuable for both the precision test of the electroweak theory and searching for new physics beyond the standard model in the neutrino sector.