Testing MSW effect in supernova explosion with neutrino event rates

Flavor transition mechanisms of supernova (SN) neutrinos during their propagation deserve a close scrutiny. We present a method to verify Mikheyev-Smirnov-Wolfenstein (MSW) effect during the propagation of SN neutrinos from the SN core to the Earth. The non-MSW scenarios to be distinguished from the MSW one are the incoherent flavor transition probability for neutrino propagation in the vacuum and the flavor equalization induced by fast flavor conversions. Our approach involves studying the time evolution of neutrino event rates in liquid argon, liquid scintillation, and water Cherenkov detectors. The liquid argon detector is sensitive to $\nu_e$ flux while liquid scintillation and water Cherenkov detectors can measure $\bar{\nu}_e$ flux through inverse $\beta$ decay process. The flux of $\nu_e$ ($\bar{\nu}_e$) is a linear combination of $\nu_e$ ($\bar{\nu}_e$) and $\nu_{\mu,\tau}$ ($\bar{\nu}_{\mu,\tau}$) fluxes from the source with the weighting of each component dictated by the flavor transition mechanism. Using currently available simulations for SN neutrino emissions, the time evolution of $\nu_e{\rm Ar}$ and $\bar{\nu}_e$ inverse $\beta$ decay event rates and the corresponding cumulative event fractions are calculated up to $t=100~{\rm ms}$ in DUNE, JUNO, and Hyper-Kamiokande detectors, respectively. It is shown that the area under the cumulative time distribution curve from $t=0$ to $t=100~{\rm ms}$ in each detector and their ratio can be used to discriminate different flavor transition scenarios of SN neutrinos.