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Analytic formulae for T violation in neutrino oscillations

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Pith's one-line read The paper derives analytic expressions for T violation in neutrino oscillations and shows that unitarity violation adds a distinctive sin(ΔE31L) term, making the energy spectrum a probe of new physics.

desk verdict A useful, clearly written analytic derivation of T-violation formulae, but the central unitarity-violating result is undercut by a dimensional inconsistency in Eq. (39) and a missing numerical check. read the letter →

arxiv 2502.04704 v1 pith:ZCW5MHBV submitted 2025-02-07 hep-ph

Osamu Yasuda This is my paper · ORCID
classification hep-ph
keywords TviolationneutrinooscillationsunitaritynonstandardinteractionsfactorymuTRISTANJarlskoginvariantmattereffects
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper derives closed-form expressions for T violation in neutrino oscillation—the difference between $P(\nu_\mu\to\nu_e)$ and $P(\nu_e\to\nu_\mu)$—under three scenarios: standard three-flavour mixing, flavor-dependent nonstandard interactions during propagation, and a nonunitary leptonic mixing matrix. In unitary scenarios the T-odd probability factors as the product of three sine functions of the matter-shifted energy splittings; the standard case reproduces the known Jarlskog-term formula, while nonstandard interactions add an energy-independent piece to the coefficient. In the unitarity-violating case, an additional term proportional to $\sin(\Delta\tilde E_{31}L)$ appears that has no counterpart in unitary evolution. The paper argues that measuring the energy spectrum of T violation, as envisioned with a polarized $\mu^+$ storage ring such as $\mu$TRISTAN, could therefore distinguish unitarity violation from the standard and nonstandard scenarios.

What carries the argument

The carrying object is the representation of the appearance amplitude as $A(\nu_\beta\to\nu_\alpha)=\sum_j \tilde X_j^{\alpha\beta} e^{-i\tilde E_j L}$, with $\tilde X_j^{\alpha\beta}=\tilde U_{\alpha j}\tilde U^*_{\beta j}$. The paper uses the identities $\sum_j \tilde E_j^m \tilde X_j^{\alpha\beta}=\bigl[(U E U^{-1}+A)^m\bigr]_{\alpha\beta}$ for $m=0,1,2$ to solve for the $\tilde X_j$ by inverting a Vandermonde matrix, reducing the problem to low moments of the matter Hamiltonian. Eigenvalues and moments are then expanded to first order in the small ratios $\Delta E_{21}/\Delta E_{31}$ and in the nonstandard-interaction or nonunitarity parameters. For the nonunitary case, the same Vandermonde inversion is applied to the modified amplitude built from $N^*W e^{-i\tilde E L}W^{-1}N^T$, producing the unitarity-violation formula.

What would settle it

Measure the energy spectrum of $P(\nu_\mu\to\nu_e)-P(\nu_e\to\nu_\mu)$ in a long-baseline neutrino factory using a polarized $\mu^+$ beam. If unitarity is exact, the spectrum should fit the factorized three-sine form with a coefficient following $1/E$ in the standard case or a constant offset under nonstandard interactions; a residual component with the functional form $\sin(\Delta\tilde E_{31}L)$ would confirm the paper's unitarity-violation signature, while its absence would falsify the distinguishability claim.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that T violation in matter has a scenario-dependent energy structure that can be written analytically. Starting from the Kimura-Takamura-Yokomakura procedure, it obtains the standard result that $P(\nu_\mu\to\nu_e)-P(\nu_e\to\nu_\mu)$ equals $16J$ times the product of three half-angle sine factors divided by the corresponding matter-shifted splittings, with $J$ the Jarlskog invariant. With flavour-dependent nonstandard interactions, the same factorized sine structure survives but the coefficient gains terms proportional to the matter potential $A$ that are independent of neutrino energy. In the unitarity-violating case, the paper finds an extra term proportional to $\sin(\Delta\tilde E_{31}L)$ multiplying $\operatorname{Im}[\eta_{\mu e}X_3^{\mu e*}]$, a contribution absent when time evolution is unitary. Because this term oscillates at a different frequency from the factorized three-sine term, the energy shape of T violation is qualitatively different.

Load-bearing premise

The unitarity-violating derivation assumes the atmospheric mass splitting and matter potential are much larger than the solar splitting and the nonunitarity parameters, and it evaluates one matter eigenvalue at zeroth order in those small quantities; if this hierarchy fails at the energies and baseline of a real experiment, the coefficient of the new $\sin(\Delta\tilde E_{31}L)$ term is not guaranteed.

Editorial extensions

If this is right

  • In every unitary scenario treated, the T-odd probability is proportional to $\sin(\Delta\tilde E_{32}L/2)\sin(\Delta\tilde E_{31}L/2)\sin(\Delta\tilde E_{21}L/2)$; only the coefficient carries scenario information.
  • Nonstandard interactions shift the coefficient by an energy-independent amount, so a constant offset in the T-violation energy spectrum is a signal of flavor-dependent nonstandard interactions.
  • Unitarity violation adds a separate term proportional to $\sin(\Delta\tilde E_{31}L)$ whose oscillation length is half that of the factorized term, making the two cases distinguishable in principle.
  • Because the T-violation asymmetry $P(\nu_\mu\to\nu_e)-P(\nu_e\to\nu_\mu)$ avoids the matter-potential complications of CP asymmetries, its analytic energy shape can be compared directly with data.
  • A polarized-muon neutrino factory such as $\mu$TRISTAN could access this channel and test whether the predicted unitarity-violating spectral feature is present.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The unitarity-violating extra term depends on the neutral-current matter potential $A_n$ through $2A_n(\{\eta,X_3\}_{\mu e}-\eta_{\mu e})$, so a measurement at two different baselines or matter densities could separate this contribution from the leading $4A\eta_{\mu e}$ term.
  • The distinct oscillation frequencies suggest that a Fourier analysis of the T-violation spectrum in $L/E$ could isolate the $\sin(\Delta\tilde E_{31}L)$ component even if its coefficient is small.
  • If the assumed hierarchy $|\Delta E_{31}|\sim A\gg|\Delta E_{21}|,A|\eta_{\alpha\beta}|$ fails at the energies and baseline of a real experiment, the analytic unitarity-violation formula would need to be replaced by a full numerical treatment; a sensitivity scan could map where the approximation breaks down.
  • The same Vandermonde moment technique could be applied to the $\nu_\mu\to\nu_\tau$ or $\nu_e\to\nu_\tau$ channels, where unitarity violation would enter through different combinations of the $\eta$ matrix elements.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper derives analytic expressions for T violation in neutrino oscillations, defined as P(νμ→νe) − P(νe→νμ), in three scenarios: standard three-flavor mixing, propagation with nonstandard interactions (NSI), and non-unitary mixing (unitarity violation). Using the Kimura-Takamura-Yokomakura formalism and first-order perturbation theory in ΔE21/ΔE31, the NSI parameters, and the non-unitarity parameter η, the author obtains: for the standard and NSI cases, T violation is proportional to sin(Δ~E31L/2) sin(Δ~E21L/2) sin(Δ~E32L/2), with NSI adding an energy-independent contribution to the coefficient; for unitarity violation, an additional term proportional to sin(Δ~E31L) appears, which is absent in unitary scenarios. The paper proposes that the energy spectrum of T violation at a future neutrino factory such as µTRISTAN could distinguish these scenarios.

Significance. If the central result holds, the paper provides useful analytic formulae and a clear qualitative signature: the sin(Δ~E31L) term in the unitarity-violation case offers a way to distinguish non-unitarity from the standard and NSI cases, which is a novel and falsifiable prediction. The derivation rests on established formalism (the KT-Y method and the non-unitary Hamiltonian from Ref. [48]), and the assumptions of constant matter density and a perturbative hierarchy are stated explicitly. However, the key new result currently rests on an intermediate equation with a dimensional inconsistency (Eq. (39)) and no numerical verification, so the significance is conditional on correcting and confirming that step.

major comments (2)
  1. [3.2, Eq. (39)] Eq. (39) is dimensionally inconsistent: the left-hand side Im[~X_2^{eμ} ~X_3^{eμ*}] is dimensionless, while the right-hand side has dimensions of inverse energy. Tracing from Eq. (38), the first two terms contribute 2A(ΔE31)^2 cos^2θ13 Im[η_{μe} X_3^{μe*}] and the third term contributes ΔE31 times an energy-squared bracket; after combining, the coefficient should be (ΔE31)^2/(Δ~E21 Δ~E31 Δ~E32), not ΔE31/(Δ~E21 Δ~E31 Δ~E32). Eq. (42) appears to use the dimensionally consistent coefficient, so this may be a typographical error, but as printed Eq. (39) does not establish the central result.
  2. [3.2, Eqs. (38)-(42)] The simplification of the η-dependent terms that leads from Eq. (38) to Eq. (39) (in particular the combination of 2(A−An)ΔE31 η_{μe}, (A+2An)ΔE31 {X3,η}_{μe}, the anticommutator terms, and the first-two-terms contribution 2A(ΔE31)^2 cos^2θ13 η_{μe} into the single term 4Aη_{μe}+2An({η,X3}_{μe}−η_{μe})) is not shown. Because Eq. (39) as printed is dimensionally wrong, this step is not verifiable from the text. The author should either display the intermediate algebra or provide a numerical check (e.g., comparison with exact diagonalization of Eq. (31) for a benchmark parameter point) to confirm the corrected Eq. (39) and the final result Eq. (42).
minor comments (3)
  1. [3.2, after Eq. (39)] The sentence 'terms of order O((ΔE21/ΔE31)^2), O((ǫαβ)^2) and O(ǫαβ ΔE21/ΔE31) have been neglected' appears in the section on unitarity violation; here ǫαβ should be ηαβ to match the notation of the section.
  2. [3.2, Eq. (33)] In the step [{(1+η)^2}^T]_{αβ} ≃ η_{βα}, the factor of 2 is not written explicitly because it is absorbed into the overall factor of 4 in the following line; a brief parenthetical remark would prevent confusion for readers.
  3. [3.1.2, Eq. (28)] The notation X3^{ττ}, X3^{eτ}, X3^{τμ} in Eq. (28) is compact; a reminder that X3 is the projector onto the third mass eigenstate (defined in Eq. (27)) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all three T-violation formulae are derived algebraically from the same Hamiltonian formalism, with the only self-citation (Ref. [48]) providing an independent nonunitary evolution equation rather than the target result.

full rationale

The paper starts from the standard Hamiltonian H = U E U^-1 + A and the Kimura-Takamura-Yokomakura inversion of the Vandermonde system (Eqs. (12)-(16)) to express Im[X~_2 X~_3*] in terms of Y_j. The standard three-flavour result Eq. (23) is checked against the known formula [10], not assumed. The NSI formula Eq. (28) follows from evaluating Y_2 and Y_3 with A + A_NP to first order in Delta E21/Delta E31 and epsilon; no NSI parameter is fitted to the final T violation. The unitarity-violation derivation uses Eq. (30), attributed to Ref. [48], for the nonunitary evolution Hamiltonian. That citation is a published, parameter-free derivation with stated assumptions and does not contain the paper's final sin(Delta~E31 L) term; the subsequent steps (Eqs. (33)-(42)) are explicit algebra and first-order perturbation theory. The new sin(Delta~E31 L) contribution arises from the |eta + oscillating amplitudes|^2 cross terms in Eq. (33), i.e., from the structure of the modified amplitude, not from any fitted input. There are no fitted parameters called predictions, no uniqueness claim imported from the authors, and no target result used as an input. The paper explicitly disclaims experimental sensitivity estimates, which further limits any risk of a result being forced by benchmark data. The dimensional inconsistency alleged in the skeptic note is not present: in Eq. (39) the factor Delta E31 times the energy-dimensional bracket divided by the three Delta~E's is dimensionless. Thus no circular step was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new entities or fitted constants are introduced. The inputs, such as mixing angles, mass splittings, epsilon, eta, and matter density, are standard phenomenological parameters from the cited literature. The paper-specific choices are the assumed perturbative hierarchy and the use of prior published Hamiltonians. The apparent factor error in Eq. (39) versus Eq. (42) is a typographical or algebraic inconsistency within the derivation, not a new physical entity.

assumptions (5)
  • domain assumption Constant matter density in the propagation Hamiltonian (Section 2, Eqs. (1)-(8)).
    All probabilities are derived for a uniform matter profile; Earth's varying density is neglected. This is standard in analytic oscillation papers but restricts direct applicability.
  • ad hoc to paper Perturbative hierarchy |ΔE31| ~ A >> |ΔE21| ~ A|epsilon| ~ A|eta|, retaining only first order (Sections 3.1.2 and 3.2).
    The NSI and unitarity-violating formulas are obtained only to first order in the small parameters. If higher orders or non-perturbative regions matter, Eqs. (28), (39)-(42) can fail.
  • domain assumption NSI Hamiltonian U E U^{-1} + A + A_NP with a hermitian epsilon matrix (Section 3.1.2, Eq. (24)).
    Flavor-dependent nonstandard interactions are parameterized with a hermitian epsilon matrix proportional to the standard matter potential A; this is a model assumption, not derived.
  • domain assumption Nonunitary mixing N = (1 + eta) U with hermitian eta and the evolution Hamiltonian from Ref. [48] (Section 3.2, Eqs. (29)-(31)).
    The unitarity-violating framework, including the neutral-current term proportional to A_n, is imported from the cited literature, one of whose authors is the present author. The target formula is not assumed in that Hamiltonian.
  • standard math Vandermonde matrix inversion identities (Eqs. (15)-(16), (35)-(37)).
    The derivation extracts the X matrices from three moments of the effective Hamiltonian using unitarity and invertibility of the Vandermonde matrix; this algebraic step is exact.

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Cite this review

Pith. "Pith review of Analytic formulae for T violation in neutrino oscillations." pith.science (2026). https://pith.science/paper/ZCW5MHBV

@misc{pith2026250204704,
  author       = {Pith},
  title        = {Pith review of: Analytic formulae for T violation in neutrino oscillations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZCW5MHBV}},
  note         = {Machine review of arXiv:2502.04704}
}
abstract

Recently, a concept known as $\mu$TRISTAN, which involves the acceleration of $\mu^+$, has been proposed. This initiative has led to considerations of a new design for a neutrino factory. Additionally, leveraging the polarization of $\mu^+$, measurements of T violation in neutrino oscillations are also being explored. In this paper, we present analytical expressions for T violation in neutrino oscillations within the framework of standard three flavor neutrino oscillations, a scenario involving nonstandard interactions, and a case of unitarity violation. We point out that examining the energy spectrum of T violation may be useful for probing new physics effects.

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Works this paper leans on

48 extracted references · 28 canonical work pages

  1. [48]

    CP- violation from non-unitary leptonic mixing,

    E. Fernandez-Martinez, M. B. Gavela, J. Lopez-Pavon and O. Yasuda, “CP- violation from non-unitary leptonic mixing,” Phys. Lett. B 649 (2007), 427-435 doi:10.1016/j.physletb.2007.03.069 [arXiv:hep-ph/0703098 [hep-ph]]. 15

    arXiv 2007

  2. [1]

    Review of Particle Physics,

    R. L. Workman et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2022 (2022), 083C01 doi:10.1093/ptep/ptac097

    doi:10.1093/ptep/ptac097 2022

  3. [2]

    µTRISTAN,

    Y. Hamada, R. Kitano, R. Matsudo, H. Takaura and M. Yoshida, “ µTRISTAN,” PTEP 2022 (2022) no.5, 053B02 doi:10.1093/ptep/ptac059 [arXiv:2201.06664 [hep-ph]]

    arXiv 2022

  4. [3]

    T-violation of neutrino oscillation at µTRISTAN,

    Sho Sugama, “T-violation of neutrino oscillation at µTRISTAN,” talk at Hokkaido Workshop on Particle Physics at Crossroads , Hokkaido University, Sapporo, Japan, 7 – 10 March, 2024. Available online: https://conference-indico.kek.jp/event/248/ (accessed on 24 April, 2024)

    work page 2024

  5. [4]

    Neutrino beams from muon storage rings: Character istics and physics poten- tial,

    S. Geer, “Neutrino beams from muon storage rings: Character istics and physics poten- tial,” Phys. Rev. D 57 (1998), 6989-6997 [erratum: Phys. Rev. D 59 (1999), 039903] doi:10.1103/PhysRevD.57.6989 [arXiv:hep-ph/9712290 [hep-ph]]

    arXiv 1998

  6. [5]

    Physics at a future Neutrino Factory and super-beam facility

    A. Bandyopadhyay et al. [ISS Physics Working Group], “Physics at a future Neutrino Factory and super-beam facility,” Rept. Prog. Phys. 72 (2009), 106201 doi:10.1088/0034- 4885/72/10/106201 [arXiv:0710.4947 [hep-ph]]

    work page Pith review arXiv 2009

  7. [6]

    Search for T-violation in Neutrino Oscillation with the Use of Muon Polarization at a Neutrino Factory

    T. Ota, J. Sato and Y. Kuno, “Search for T violation in neutrino os cillation with the use of muon polarization at a neutrino factory,” Phys. Lett. B 520 (2001), 289-297 doi:10.1016/S0370-2693(01)01135-2 [arXiv:hep-ph/0107007 [hep -ph]]

    work page Pith review arXiv 2001

  8. [7]

    Time Reversal Violation in Neutrino Oscillation,

    N. Cabibbo, “Time Reversal Violation in Neutrino Oscillation,” Phys. L ett. B 72 (1978), 333-335 doi:10.1016/0370-2693(78)90132-6

    work page doi:10.1016/0370-2693(78)90132-6 1978

Show all 48 references
  1. [8]

    Resonance Amplification and t V iolation Effects in Three Neutrino Oscillations in the Earth,

    P. I. Krastev and S. T. Petcov, “Resonance Amplification and t V iolation Effects in Three Neutrino Oscillations in the Earth,” Phys. Lett. B 205 (1988), 84-92 doi:10.1016/0370- 2693(88)90404-2

    1988 doi

  2. [9]

    On T violation in matter neutrino oscillations,

    S. Toshev, “On T violation in matter neutrino oscillations,” Mod. Phy s. Lett. A 6 (1991), 455-460 doi:10.1142/S0217732391000464

    1991 doi

  3. [10]

    Three neutrino oscillations in matter, CP violation and topological phases,

    V. A. Naumov, “Three neutrino oscillations in matter, CP violation and topological phases,” Int. J. Mod. Phys. D 1 (1992), 379-399 doi:10.1142/S0218271892000203

    1992 doi

  4. [11]

    CP and T violation test in neutrino oscilla tion,

    J. Arafune and J. Sato, “CP and T violation test in neutrino oscilla tion,” Phys. Rev. D 55 (1997), 1653-1658 doi:10.1103/PhysRevD.55.1653 [arXiv:hep-ph/ 9607437 [hep-ph]]

    1997 doi

  5. [12]

    Neutrino mixing, CP / T violation and textures in four neutrino models,

    V. D. Barger, Y. B. Dai, K. Whisnant and B. L. Young, “Neutrino mixing, CP / T violation and textures in four neutrino models,” Phys. Rev. D 59 (1999), 113010 doi:10.1103/PhysRevD.59.113010 [arXiv:hep-ph/9901388 [hep-ph]]

    1999 arXiv

  6. [13]

    CP and T violation in long baseline experiment s with low-energy neutrino from muon storage ring,

    M. Koike and J. Sato, “CP and T violation in long baseline experiment s with low-energy neutrino from muon storage ring,” Phys. Rev. D 61 (2000), 073012 [erratum: Phys. Rev. D 62 (2000), 079903] doi:10.1103/PhysRevD.61.073012 [arXiv:hep-ph/9 909469 [hep-ph]]

    2000 doi

  7. [14]

    T violation search with very long baseline ne utrino oscilla- tion experiments,

    M. Koike and J. Sato, “T violation search with very long baseline ne utrino oscilla- tion experiments,” Phys. Rev. D 62 (2000), 073006 doi:10.1103/PhysRevD.62.073006 [arXiv:hep-ph/9911258 [hep-ph]]

    2000 arXiv

  8. [15]

    CP and T violation in neutrino osc illations and in- variance of Jarlskog’s determinant to matter effects,

    P. F. Harrison and W. G. Scott, “CP and T violation in neutrino osc illations and in- variance of Jarlskog’s determinant to matter effects,” Phys. Lett . B 476 (2000), 349-355 doi:10.1016/S0370-2693(00)00153-2 [arXiv:hep-ph/9912435 [hep -ph]]

    2000 arXiv

  9. [16]

    Matter enhance ment of T violation in neu- trino oscillation,

    H. Yokomakura, K. Kimura and A. Takamura, “Matter enhance ment of T violation in neu- trino oscillation,” Phys. Lett. B 496 (2000), 175-184 doi:10.1016/S0370-2693(00)01288-0 [arXiv:hep-ph/0009141 [hep-ph]]. 13

    2000 arXiv

  10. [17]

    Optimizing T violating effects for neu trino oscilla- tions in matter,

    S. J. Parke and T. J. Weiler, “Optimizing T violating effects for neu trino oscilla- tions in matter,” Phys. Lett. B 501 (2001), 106-114 doi:10.1016/S0370-2693(01)00111-3 [arXiv:hep-ph/0011247 [hep-ph]]

    2001 arXiv

  11. [18]

    Ambiguities of theoretical param eters and CP/T violation in neutrino factories,

    M. Koike, T. Ota and J. Sato, “Ambiguities of theoretical param eters and CP/T violation in neutrino factories,” Phys. Rev. D 65 (2002), 053015 doi:10.1103/PhysRevD.65.053015 [arXiv:hep-ph/0011387 [hep-ph]]

    2002 arXiv

  12. [19]

    The Matter e ffect to T violation at a neutrino factory,

    T. Miura, E. Takasugi, Y. Kuno and M. Yoshimura, “The Matter e ffect to T violation at a neutrino factory,” Phys. Rev. D 64 (2001), 013002 doi:10.1103/PhysRevD.64.013002 [arXiv:hep-ph/0102111 [hep-ph]]

    2001 arXiv

  13. [20]

    T violatio n in neutrino oscil- lations in matter,

    E. K. Akhmedov, P. Huber, M. Lindner and T. Ohlsson, “T violatio n in neutrino oscil- lations in matter,” Nucl. Phys. B 608 (2001), 394-422 doi:10.1016/S0550-3213(01)00261-9 [arXiv:hep-ph/0105029 [hep-ph]]

    2001 arXiv

  14. [21]

    The Matt er fluctuation effect to T violation at a neutrino factory,

    T. Miura, T. Shindou, E. Takasugi and M. Yoshimura, “The Matt er fluctuation effect to T violation at a neutrino factory,” Phys. Rev. D 64 (2001), 073017 doi:10.1103/PhysRevD.64.073017 [arXiv:hep-ph/0106086 [hep-ph]]

    2001 arXiv

  15. [22]

    Rephasing invariants of CP and T violation in the four neu- trino mixing models,

    W. l. Guo and Z. z. Xing, “Rephasing invariants of CP and T violation in the four neu- trino mixing models,” Phys. Rev. D 65 (2002), 073020 doi:10.1103/PhysRevD.65.073020 [arXiv:hep-ph/0112121 [hep-ph]]

    2002 arXiv

  16. [23]

    On the energy and baseline optimization to study effects related to the delta phase (CP / T violation) in neutr ino oscillations at a neutrino factory,

    A. Bueno, M. Campanelli, S. Navas and A. Rubbia, “On the energy and baseline optimization to study effects related to the delta phase (CP / T violation) in neutr ino oscillations at a neutrino factory,” Nucl. Phys. B 631 (2002), 239-284 doi:10.1016/S0550-3213(02)00211-0 [arXiv:...

    2002 arXiv

  17. [24]

    Parameter Degen eracies in Neutrino Oscil- lation Measurement of Leptonic CP and T Violation,

    H. Minakata, H. Nunokawa and S. J. Parke, “Parameter Degen eracies in Neutrino Oscil- lation Measurement of Leptonic CP and T Violation,” Phys. Rev. D 66 (2002), 093012 doi:10.1103/PhysRevD.66.093012 [arXiv:hep-ph/0208163 [hep-ph]]

    2002 arXiv

  18. [25]

    Leptonic commutators and clean T violation in neutrin o oscillations,

    Z. z. Xing, “Leptonic commutators and clean T violation in neutrin o oscillations,” Phys. Rev. D 88 (2013) no.1, 017301 doi:10.1103/PhysRevD.88.017301 [arXiv:1304.76 06 [hep-ph]]

    2013 doi

  19. [26]

    On Neutrino Mixing in Matter and CP and T Violation Effects in Neutrino Oscillations,

    S. T. Petcov and Y. L. Zhou, “On Neutrino Mixing in Matter and CP and T Violation Effects in Neutrino Oscillations,” Phys. Lett. B 785 (2018), 95-104 doi:10.1016/j.physletb.2018.08.025 [arXiv:1806.09112 [hep-ph]]

    2018 arXiv

  20. [27]

    Model-Independent Test of T Vio lation in Neutrino Oscil- lations,

    T. Schwetz and A. Segarra, “Model-Independent Test of T Vio lation in Neutrino Oscil- lations,” Phys. Rev. Lett. 128 (2022) no.9, 091801 doi:10.1103/PhysRevLett.128.091801 [arXiv:2106.16099 [hep-ph]]

    2022 arXiv

  21. [28]

    T violation in nonstandard neutrin o oscillation scenarios,

    T. Schwetz and A. Segarra, “T violation in nonstandard neutrin o oscillation scenarios,” Phys. Rev. D 105 (2022) no.5, 055001 doi:10.1103/PhysRevD.105.055001 [arXiv:2112 .08801 [hep-ph]]

    2022 doi

  22. [29]

    Neutrino vs. Antineutrino Oscillat ion Parame- ters at DUNE and Hyper-Kamiokande,

    A. de Gouvˆ ea and K. J. Kelly, “Neutrino vs. Antineutrino Oscillat ion Parame- ters at DUNE and Hyper-Kamiokande,” Phys. Rev. D 96 (2017) no.9, 095018 doi:10.1103/PhysRevD.96.095018 [arXiv:1709.06090 [hep-ph]]

    2017 arXiv

  23. [30]

    Neutrinos, DUNE and the world best bound on CPT invariance,

    G. Barenboim, C. A. Ternes and M. T´ ortola, “Neutrinos, DUNE and the world best bound on CPT invariance,” Phys. Lett. B 780 (2018), 631-637 doi:10.1016/j.physletb.2018.03.060 [arXiv:1712.01714 [hep-ph]]

    2018 arXiv

  24. [31]

    CPT and CP, an entangled couple,

    M. A. T´ ortola, G. Barenboim and C. A. Ternes, “CPT and CP, an entangled couple,” JHEP 07 (2020), 155 doi:10.1007/JHEP07(2020)155 [arXiv:2005.05975 [hep- ph]]

    2020 arXiv

  25. [32]

    Stringent co nstraint on CPT violation with the synergy of T2K-II, NO νA extension, and JUNO,

    T. V. Ngoc, S. Cao, N. T. H. Van and P. T. Quyen, “Stringent co nstraint on CPT violation with the synergy of T2K-II, NO νA extension, and JUNO,” Phys. Rev. D 107 (2023) no.1, 016013 doi:10.1103/PhysRevD.107.016013 [arXiv:2210.13044 [hep-ph ]]. 14

    2023 arXiv

  26. [33]

    Neutrino CPT violation in the solar sector,

    G. Barenboim, P. Mart ´ ınez-Mirav´ e, C. A. Ternes and M. T´ ortola, “Neutrino CPT violation in the solar sector,” Phys. Rev. D 108 (2023) no.3, 035039 doi:10.1103/PhysRevD.108.035039 [arXiv:2305.06384 [hep-ph]]

    2023 arXiv

  27. [34]

    Exact formula of probability and CP violation for neutrino oscillations in matter,

    K. Kimura, A. Takamura and H. Yokomakura, “Exact formula of probability and CP violation for neutrino oscillations in matter,” Phys. Lett. B 537, 86 (2002) [arXiv:hep-ph/0203099]

    2002 arXiv

  28. [35]

    All you ever want to know about neutrino oscillation probabilities in constant matter,

    K. Kimura, A. Takamura and H. Yokomakura, “All you ever want to know about neutrino oscillation probabilities in constant matter,” Phys. Rev. D 66, 073005 (2002) [arXiv:hep-ph/0205295]

    2002 arXiv

  29. [36]

    Neutrino propagation in matter an d electromagnetic fields,

    W. Grimus and T. Scharnagl, “Neutrino propagation in matter an d electromagnetic fields,” Mod. Phys. Lett. A 8, 1943 (1993)

    1993

  30. [37]

    Neutrino Oscillations In Nonuniform Matter,

    A. Halprin, “Neutrino Oscillations In Nonuniform Matter,” Phys. R ev. D 34, 3462 (1986)

    1986

  31. [38]

    Derivation Of The Formalism For Neutrino Matte r Oscillations From The Neutrino Relativistic Field Equations,

    P. D. Mannheim, “Derivation Of The Formalism For Neutrino Matte r Oscillations From The Neutrino Relativistic Field Equations,” Phys. Rev. D 37, 1935 (1988)

    1988

  32. [39]

    Neutrino Oscillations In Inhomogeneous Matter ,

    R. F. Sawyer, “Neutrino Oscillations In Inhomogeneous Matter ,” Phys. Rev. D 42, 3908 (1990)

    1990

  33. [40]

    V. D. Barger, K. Whisnant, S. Pakvasa and R. J. N. Phillips, Phys . Rev. D 22, 2718 (1980)

    1980

  34. [41]

    Commutator of the Quark Mass Matrices in the St andard Electroweak Model and a Measure of Maximal CP Nonconservation,

    C. Jarlskog, “Commutator of the Quark Mass Matrices in the St andard Electroweak Model and a Measure of Maximal CP Nonconservation,” Phys. Rev. L ett. 55 (1985), 1039 doi:10.1103/PhysRevLett.55.1039

    1985 doi

  35. [42]

    On the MSW effect wit h massless neutrinos and no mixing in the vacuum,

    M. M. Guzzo, A. Masiero and S. T. Petcov, “On the MSW effect wit h massless neutrinos and no mixing in the vacuum,” Phys. Lett. B 260, 154 (1991)

    1991

  36. [43]

    Mikheyev-Smirnov-Wolfenstein effect with flavor-c hanging neutrino interac- tions,

    E. Roulet, “Mikheyev-Smirnov-Wolfenstein effect with flavor-c hanging neutrino interac- tions,” Phys. Rev. D 44, R935 (1991)

    1991

  37. [44]

    Davidson, C

    S. Davidson, C. Pena-Garay, N. Rius and A. Santamaria, JHEP 0303, 011 (2003) [arXiv:hep-ph/0302093]

    2003 arXiv

  38. [45]

    General bo unds on non- standard neutrino interactions,

    C. Biggio, M. Blennow and E. Fernandez-Martinez, “General bo unds on non- standard neutrino interactions,” JHEP 08 (2009), 090 doi:10.1088/1126-6708/2009/08/090 [arXiv:0907.0097 [hep-ph]]

    2009 arXiv

  39. [46]

    Global con- straints on non-standard neutrino interactions with quarks and e lectrons,

    P. Coloma, M. C. Gonzalez-Garcia, M. Maltoni, J. P. Pinheiro and S . Urrea, “Global con- straints on non-standard neutrino interactions with quarks and e lectrons,” JHEP 08 (2023), 032 doi:10.1007/JHEP08(2023)032 [arXiv:2305.07698 [hep-ph]]

    2023 arXiv

  40. [47]

    Unitarity of the Leptonic Mixing Matrix,

    S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela an d J. Lopez-Pavon, “Unitarity of the Leptonic Mixing Matrix,” JHEP 10 (2006), 084 doi:10.1088/1126-6708/2006/10/084 [arXiv:hep-ph/0607020 [hep-ph]]

    2006 arXiv

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