T versus CP effects in DUNE and T2HK

Time reversal (T) symmetry violations in neutrino oscillations imply the presence of an $L$-odd component in the transition probability at fixed neutrino energy, with $L$ denoting the distance between neutrino source and detector. Within the standard three-flavour framework, we show that the combination of the transition probabilities determined at the DUNE and T2HK experiments can establish the presence of an $L$-odd component, and therefore provide sensitivity to T violation, up to $4\sigma$ significance. The optimal neutrino energy window is from 0.68 to 0.92 GeV, and therefore a crucial role is played by the low-energy part of the DUNE event spectrum covering the second oscillation maximum. We compare the sensitivity to T violation based on this energy range using neutrino data only with the more traditional search for charge-parity (CP) violation based on the comparison of neutrino versus anti-neutrino beam data. We show that for DUNE it is advantageous to run in neutrino mode only, i.e., searching for T violating effects, whereas T2HK is more sensitive to CP violation, comparing neutrino and anti-neutrino data. Hence, the two experiments offer complementary methods to determine the complex phase in the PMNS mixing matrix.