Sensitivity to Supernovae Average $\nu_x$ Temperature with Neutral Current Interactions in DUNE

We explore a novel method for measuring the average temperature of the $\nu_x$ component in Type-II core-collapse supernovae. By measuring neutral current incoherent neutrino-Argon interactions in DUNE we can obtain spectral information for the combination of all active neutrino species. Combining this all-neutrino spectral information with detailed charged current measurements of the electron neutrino and electron anti-neutrino fluxes from DUNE and Hyper-Kamiokande, we can infer the average temperature for the remaining neutrino species in the $\nu_x$ component to within a factor two for most cases and to 30% for a small range of average $\nu_x$ temperatures. Due to the limited energy range of the emitted photons from incoherent neutral current interactions on Argon, the $\nu_x$ temperature reconstruction demonstrates a degeneracy in the one and two sigma credible regions. Furthermore, while large uncertainties on the NC cross-section penalize this measurement, we examined the efficacy of constraining NC cross-section uncertainties on improving $\nu_x$ measurements. We found that if additional measurements of B(M1$\uparrow$) 1$^+$ excited state transitions in Argon are able to reduce correlated cross section uncertainties from 15% to 7%, the size of the $1\sigma$ allowed regions for $T_{\nu_x}$ becomes sample size limited, and approaches the case where there are no uncertainties on the cross-section.