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We investigate the impact of matter effects on $T$(time-reversal)-odd observables, making use of the quantum-mechanical formalism of neutrino-flavor evolution. We attempt to be comprehensive and pedagogical. Matter-induced $T$-invariance violation (TV) is qualitatively different from, and more subtle than, matter-induced $CP$(charge-parity)-invariance violation. If the matter distribution is symmetric relative to the neutrino production and detection points, matter effects will not introduce any new TV. However, if there is intrinsic TV, matter effects can modify the size of the $T$-odd observable. On the other hand, if the matter distribution is not symmetric, there is genuine matter-induced TV. For Earth-bound long-baseline oscillation experiments, these effects are small. This remains true for unrealistically-asymmetric matter potentials (for example, we investigate the effects of “hollowing out” 50% of the DUNE neutrino trajectory). More broadly, we explore consequences, or lack thereof, of asymmetric matter potentials on oscillation probabilities. While fascinating in their own right, $T$-odd observables are currently of limited practical use, due in no small part to a dearth of intense, well-characterized, high-energy electron-neutrino beams. Further in the future, however, intense, high-energy muon storage rings might become available and allow for realistic studies of $T$-invariance in neutrino oscillations. Published by the American Physical Society2025 

Abstract The existence of nonzero neutrino masses points to the likely existence of multiple Standard Model neutral fermions. When such states are heavy enough that they cannot be produced in oscillations, they are referred to as heavy neutral leptons (HNLs). In this white paper, we discuss the present experimental status of HNLs including colliders, beta decay, accelerators, as well as astrophysical and cosmological impacts. We discuss the importance of continuing to search for HNLs, and its potential impact on our understanding of key fundamental questions, and additionally we outline the future prospects for next-generation future experiments or upcoming accelerator run scenarios. 

Abstract Tau neutrinos are the least studied particle in the standard model. This whitepaper discusses the current and expected upcoming status of tau neutrino physics with attention to the broad experimental and theoretical landscape spanning long-baseline, beam-dump, collider, and astrophysical experiments. This whitepaper was prepared as a part of the NuTau2021 Workshop.