<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>T</mml:mi></mml:math>-invariance violation in neutrino oscillations and matter effects

We investigate the impact of matter effects on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>T</a:mi></a:math> (time-reversal)-odd observables, making use of the quantum-mechanical formalism of neutrino-flavor evolution. We attempt to be comprehensive and pedagogical. Matter-induced <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>T</c:mi></c:math>-invariance violation (TV) is qualitatively different from, and more subtle than, matter-induced <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>C</e:mi><e:mi>P</e:mi></e:math> (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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>T</g:mi></g:math>-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, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>T</i:mi></i:math>-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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>T</k:mi></k:math>-invariance in neutrino oscillations.