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Neutrino evolution reduces to a quadratic polynomial

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2026-07-09 17:02 UTC pith:5Y7Z2OGE

load-bearing objection Solid synthesis of known neutrino oscillation algebra; the resummation framing of DMP is the main new insight, but the Feldman-Cousins validation is claimed but never shown. the 2 major comments →

arxiv 2607.07213 v1 pith:5Y7Z2OGE submitted 2026-07-08 hep-ph hep-th

Algebraic Structure of Three-Flavor Neutrino Oscillations in Constant-Density Matter: Cayley--Hamilton Evolution, DMP Resummation, and Closed-Form Uncertainty Propagation

Aaryan Chaulagain , Anju Dhakal , Daya Nidhi Chhatkuli This is my paper
classification hep-ph hep-th PACS 14.60.Pq13.15.+g
keywords parametercayley--hamiltonclosed-formconstant-densityeigenvaluesevolutionexpressionsjacobians
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that three-flavor neutrino oscillation through constant-density matter has a simple but underexploited algebraic backbone. Because the Cayley-Hamilton theorem forces any function of a 3×3 matrix to be a polynomial of degree at most two, the full evolution operator — the thing that tells you how a neutrino changes flavor as it travels — is just c₀ + c₁H + c₂H², with three coefficients fixed by a Vandermonde system fed the three eigenvalues. Those eigenvalues are known exactly from a 16th-century trigonometric formula. The paper then shows that the well-known DMP approximation achieves its accuracy not by luck but because its key rotation step acts as a resummation: it absorbs a divergent factor that blows up near a particular energy-density resonance, replacing it with a bounded parameter no larger than 0.015 at any energy. The paper ties this together with a density-matrix treatment of decoherence, a decomposition of CP-violation signals into genuine and matter-faked parts, and closed-form derivative formulas that let one propagate parameter uncertainties through the probability expressions in a single matrix multiplication.

Core claim

The central mechanism is that the DMP 1-3 rotation is a degenerate-eigenvalue resummation. When the matter potential approaches a specific ratio with the mass-squared splitting (Â→1), two eigenvalues of the Hamiltonian become nearly degenerate, and the standard perturbative expansion has denominators like (1−Â) that blow up. The DMP rotation diagonalizes the problematic sector exactly first, absorbing the dangerous denominator into a mixing angle and leaving behind a residual expansion parameter ε₀ that is bounded by 0.015 uniformly across all energies. This is why the approximation stays accurate where naive expansions fail. The paper also makes explicit the Vandermonde system that converts

What carries the argument

The Cayley-Hamilton theorem reduces the 3×3 matrix exponential to a quadratic polynomial with coefficients solved via a 3×3 Vandermonde linear system using the three eigenvalues from Cardano's trigonometric formula. The DMP resummation replaces the unbounded (1−Â)⁻¹ with the bounded ε₀ = (Δm²₂₁/Δm²_ee)·sin(θ_M13−θ₁₃)·s₁₂c₁₂. Closed-form Jacobians ∂P/∂p_k propagate the NuFIT 6.0 covariance through Var[P] = J Σ Jᵀ. A Lindblad master equation extends the framework to open-system decoherence with matter-dressed coherence lengths L_coh ∼ 10⁴–10⁶ km.

Load-bearing premise

The entire algebraic framework rests on the matter density being constant along the neutrino's path. The paper cites prior work showing that a path-averaged constant density reproduces exact DUNE oscillation probabilities at the one-percent level, but if the true density profile sensitivity is larger than that estimate, the applicability of these analytic expressions to real long-baseline experiments weakens.

What would settle it

If the DMP expansion parameter ε₀ were found to exceed its claimed bound of 0.015 at some energy or density — for instance at mantle densities or near the atmospheric resonance — the resummation interpretation would fail to explain the approximation's accuracy. Separately, if the constant-density approximation breaks down for DUNE at better than percent-level precision, the Vandermonde-based evolution operator would not apply to the actual experimental configuration.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The resummation interpretation of DMP could guide similar denominator-absorbing rotations in other perturbative physics problems where eigenvalue near-degeneracy causes divergent series.
  • The closed-form Jacobians make real-time uncertainty propagation feasible in neutrino fit pipelines without requiring millions of Monte Carlo evaluations, complementing faster numerical solvers.
  • The framework extends naturally to four-flavor (sterile neutrino) scenarios where the Cayley-Hamilton polynomial rises to cubic order and the characteristic equation becomes quartic.
  • The density-matrix formulation with Lindblad terms provides a unified language for testing exotic decoherence or non-standard interaction scenarios against upcoming DUNE and Hyper-K data.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. This manuscript presents a unified algebraic treatment of three-flavor neutrino oscillations in constant-density matter. The core results are: (1) the explicit Vandermonde system (Eq. 15) that determines the Cayley-Hamilton polynomial coefficients of the evolution operator from the three eigenvalues; (2) a resummation interpretation of the DMP 1-3 rotation, showing that it replaces the unbounded $(1-Â)^{-1}$ divergence with the uniformly bounded parameter $ε_0 ≤ 0.015$ (Eq. 46); (3) a density-matrix framework with Lindblad terms for decoherence and wave-packet effects; (4) the decomposition of CP asymmetry into genuine and fake contributions; and (5) closed-form Jacobians (Eqs. 51-56) for linearized uncertainty propagation in the NuFIT 6.0 parameter basis. The paper is explicitly positioned as a theoretical companion to the NuFast-LBL numerical algorithm, not a replacement for it.

Significance. The paper's value lies in synthesis and pedagogical clarity rather than new physics. The resummation reading of the DMP rotation (Section 10) is a correct and useful conceptual framing: the bound $|ε_0| ≤ 0.015$ follows rigorously from $|sin(θ_M^{13} - θ_{13})| ≤ 1$ and the known parameter magnitudes, and the explanation of why the naive expansion diverges at $Â → 1$ (Section 10.1) is clearly stated. The closed-form Jacobians in Eqs. (51)-(56) are a genuine practical contribution for uncertainty propagation. The limiting-case checks in Section 14 (vacuum, high-energy, two-flavor) are appropriate and correct. The paper is honest about its scope and does not overclaim novelty for individual components.

major comments (2)
  1. [§13.3 and Abstract] The abstract states that the closed-form Jacobians are 'validated against a Feldman-Cousins profile-likelihood mapping near physical boundaries,' and Section 13.3 reiterates that 'All Jacobian-based bands reported here are therefore validated against a Feldman-Cousins unified profile-likelihood mapping [54] before use in fits.' However, no Feldman-Cousins construction is shown, no profile-likelihood map is displayed, and no quantitative comparison between the linearized Jacobian bands and the FC bands is provided anywhere in the manuscript or appendices. This is the bridge from pure algebra to the claimed practical utility of the Jacobian infrastructure, and it is currently an unsubstantiated assertion. The authors should either (a) include at least one quantitative comparison showing the Jacobian-based band against the FC band near a boundary (e.g., $δ_{CP} → 0$ or $θ_{13} → 0$), or (b)
  2. [§13.3 and Abstract] soften the language to accurately reflect that such validation is recommended or planned but has not been carried out in this work. As written, the claim is not supported by the content of the paper.
minor comments (7)
  1. [Table 3 caption] The caption states that accuracy figures are 'taken from [43, 47, 44]' and that the perturbative expressions 'were not independently benchmarked for speed.' This is fine, but the table header reads 'Fractional eigenvalue/probability accuracy |ΔP|/P' while several rows (e.g., Madrid, AJLOS, Freund) are known to have energy-dependent accuracy. A note specifying the energy or energy range at which these accuracies apply would improve clarity.
  2. [§5.5, Table 4] The Halley convergence results are described as 'indicative of the tested point and are not a parameter-space-wide guarantee.' This caveat is appropriate, but the text in Section 5.5 states that 'a single Halley step from the DMP seed reaches machine precision, matching two Newton-Raphson steps.' Given that only one energy point near $E = E_{Â=1}$ is tested, the claim should be more carefully scoped in the main text, not just the caption.
  3. [§8.1, Eq. (43)] The coherence length calculation uses $σ_x ∼ 10^{-13}$ m. The text notes that $σ_x$ ranges from $10^{-15}$ to $10^{-13}$ m, but Table 5 and all quantitative claims use only the upper end. The ratio $L/L_{coh}$ would be roughly 100 times larger if $σ_x = 10^{-15}$ m were used (though still small). A brief note on the sensitivity to this choice would strengthen the claim.
  4. [§13.1, Table 8] The fractional variance contributions are described as 'averaged over the appearance band $E ∈ [0.45, 0.85]$ GeV.' The averaging procedure is not specified — is it a flat average, flux-weighted, or something else? This matters for reproducibility. Also, the $f_i(P_{μμ})$ entry for $Δm^2_{31}$ is listed as '<0.01' with a note that 'its weight is larger in a spectral fit'; this caveat should perhaps be in the table note rather than buried in the caption.
  5. [§4.3] The claim that the explicit Vandermonde system (15) is 'not, to our knowledge, written out in this form in prior neutrino references' may be correct but should be stated with care. The Cayley-Hamilton reduction and Lagrange interpolation are standard in matrix analysis textbooks; the novelty is specifically in writing it out for the neutrino oscillation context.
  6. [§15.3] The sterile-neutrino extension mentions that the Cayley-Hamilton polynomial rises to cubic order for the 4×4 case, solvable by Ferrari's method. This is correct in principle but the one-sentence treatment is too brief to be useful; either expand slightly or note that this is deferred to future work.
  7. [Submission metadata] The arXiv identifier '2607.07213' implies a July 2026 submission date, which is in the future relative to the manuscript's references (which cite works through 2024). This may be a metadata issue that should be corrected.

Circularity Check

0 steps flagged

No significant circularity found; the derivation chain is self-contained with only minor self-citation to prior DMP work for accuracy benchmarks.

full rationale

The paper's central claim — that the DMP 1-3 rotation acts as a degenerate-eigenvalue resummation replacing the unbounded $(1-Â)^{-1}$ with the bounded $ε_0 ≤ 0.015$ (Eq. 46, §10.2) — follows directly from the mathematical structure of the rotation and the elementary bound $|sin(θ_M^{13} - θ_{13})| ≤ 1$. This is not circular: the bound is derived from first principles (the definition of the DMP rotation angle and the parameter magnitudes), not fitted to data or defined in terms of the result it claims to predict. The Cayley-Hamilton structure (§4.3, Eq. 15) is a standard algebraic identity applied to the evolution operator, with the Vandermonde system being explicit bookkeeping — no circularity. The Cardano eigenvalue solution (§5.1, Eq. 16) recovers the known Zaglauer-Schwarzer expressions, which is correctly attributed and not claimed as novel. The closed-form Jacobians (§13.2, Eqs. 51-56) are obtained by direct differentiation of the perturbative probability expressions — a mechanical procedure with no input-output equivalence. The paper does cite Denton, Parke, and collaborators heavily (e.g., [43,44,47,48]), but these citations provide independent numerical benchmarks (accuracy figures in Table 3, convergence rates in Table 4) that are externally falsifiable and not definitions of the present paper's results. The Feldman-Cousins validation claim (§13.3) is undemonstrated in the text — no FC construction is shown — but this is a gap in evidence (a correctness/completeness concern), not circularity: the Jacobian framework itself does not reduce to the FC validation by construction. The constant-density assumption (§2.3) is a stated physical premise, not a result disguised as an input. No step in the derivation chain reduces to its own inputs by definition, fit, or self-citation.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, or forces. It works within the standard three-flavor PMNS framework with known matter effects.

free parameters (2)
  • sigma_x (wave-packet size) = ~10^{-13} m
    Used in Eq. (42) for coherence length estimates; chosen as a representative value from prior literature, not fitted in this paper.
  • rho (matter density) = 2.6-2.84 g/cm^3
    Path-averaged crustal density used for benchmark calculations (Table 2).
axioms (5)
  • standard math Cayley-Hamilton theorem
    Invoked in Section 4.3 to reduce the evolution operator to a quadratic polynomial.
  • standard math Cardano's trigonometric formula for cubic roots
    Used in Section 5.1 to obtain exact eigenvalues.
  • domain assumption Lindblad master equation for Markovian open systems
    Used in Section 6.2 to model decoherence.
  • domain assumption Constant-density matter approximation along the neutrino path
    Invoked throughout the paper; justified for DUNE by citing Kelly and Parke [53] in Section 2.3.
  • domain assumption NuFIT 6.0 global fit parameter values
    Used as input parameters for all numerical estimates and uncertainty propagation (Table 1).

pith-pipeline@v1.1.0-glm · 26404 in / 2020 out tokens · 383911 ms · 2026-07-09T17:02:52.613241+00:00 · methodology

0 comments
read the original abstract

For three-flavor neutrino oscillations in constant-density matter, the Cayley--Hamilton theorem forces the evolution operator into a quadratic polynomial in $\hat{H}$, with coefficients determined by the three real eigenvalues through a Vandermonde system we write out explicitly. The eigenvalues follow from Cardano's trigonometric formula, recovering the Zaglauer--Schwarzer expressions. The Denton--Minakata--Parke (DMP) approximation achieves fractional accuracy better than $10^{-4}$ because its $1$--$3$ rotation is a resummation: it removes the near-degeneracy that makes the naive expansion diverge at $\hat{A}\to 1$, replacing the unbounded $(1-\hat{A})^{-1}$ with an effective parameter $\epsilon_0\lesssim 0.015$ bounded uniformly in energy. A density-matrix treatment with a Lindblad term handles open-system decoherence and wave-packet effects in the same language; matter-dressed coherence lengths satisfy $L/L^{ij}_{\rm coh}\sim 10^{-3}$--$10^{-2}$ for terrestrial baselines. The CP asymmetry $\mathcal{A}_{\rm CP}(\nu_\mu\to\nu_e)$ is split into genuine and matter-induced fake contributions. Closed-form Jacobians in the NuFIT~6.0 parameter basis feed Monte Carlo and linearized uncertainty-propagation schemes, the latter validated against a Feldman--Cousins profile-likelihood mapping near physical boundaries. The Denton--Parke NuFast-LBL algorithm [Phys.\ Rev.\ D {\bf 110}, 073005 (2024)] remains the tool of choice for production fits; the analytic expressions here supply what iterative solvers cannot -- parameter continuity, transparent limits, and Jacobians in closed form.

discussion (0)

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Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · 5 internal anchors

  1. [1]

    Mesonium and anti-mesonium,

    B. Pontecorvo, “Mesonium and anti-mesonium,”Sov. Phys. JETP6, 429 (1957). 20

    work page 1957

  2. [2]

    Inverse beta processes and nonconservation of lepton charge,

    B. Pontecorvo, “Inverse beta processes and nonconservation of lepton charge,”Sov. Phys. JETP7, 172 (1958)

    work page 1958

  3. [3]

    Remarks on the unified model of elementary particles,

    Z. Maki, M. Nakagawa, and S. Sakata, “Remarks on the unified model of elementary particles,”Prog. Theor. Phys.28, 870 (1962). doi:10.1143/PTP.28.870

    work page doi:10.1143/ptp.28.870 1962

  4. [4]

    Evidence for os- cillation of atmospheric neutrinos,

    Y. Fukuda et al. (Super-Kamiokande Collaboration), “Evidence for os- cillation of atmospheric neutrinos,”Phys. Rev. Lett.81, 1562 (1998). doi:10.1103/PhysRevLett.81.1562

    work page doi:10.1103/physrevlett.81.1562 1998

  5. [5]

    Direct evidence for neutrino flavor trans- formation from neutral-current interactions in SNO,

    Q. R. Ahmad et al. (SNO Collaboration), “Direct evidence for neutrino flavor trans- formation from neutral-current interactions in SNO,”Phys. Rev. Lett.89, 011301 (2002). doi:10.1103/PhysRevLett.89.011301

    work page doi:10.1103/physrevlett.89.011301 2002

  6. [6]

    First results from KamLAND,

    K. Eguchi et al. (KamLAND Collaboration), “First results from KamLAND,”Phys. Rev. Lett.90, 021802 (2003). doi:10.1103/PhysRevLett.90.021802

    work page doi:10.1103/physrevlett.90.021802 2003

  7. [7]

    Observation of electron- antineutrino disappearance at Daya Bay,

    F. P. An et al. (Daya Bay Collaboration), “Observation of electron- antineutrino disappearance at Daya Bay,”Phys. Rev. Lett.108, 171803 (2012). doi:10.1103/PhysRevLett.108.171803

    work page doi:10.1103/physrevlett.108.171803 2012

  8. [8]

    Observation of reactor electron antineutrino disappearance in RENO,

    J. K. Ahn et al. (RENO Collaboration), “Observation of reactor electron antineutrino disappearance in RENO,”Phys. Rev. Lett.108, 191802 (2012). doi:10.1103/PhysRevLett.108.191802

    work page doi:10.1103/physrevlett.108.191802 2012

  9. [9]

    Indication of reactor¯ν e disappearance in Double Chooz,

    Y. Abe et al. (Double Chooz Collaboration), “Indication of reactor¯ν e disappearance in Double Chooz,”Phys. Rev. Lett.108, 131801 (2012). doi:10.1103/PhysRevLett.108.131801

    work page doi:10.1103/physrevlett.108.131801 2012

  10. [10]

    Indication of electron neutrino appearance from an off-axis muon neutrino beam,

    K. Abe et al. (T2K Collaboration), “Indication of electron neutrino appearance from an off-axis muon neutrino beam,”Phys. Rev. Lett.107, 041801 (2011). doi:10.1103/PhysRevLett.107.041801

    work page doi:10.1103/physrevlett.107.041801 2011

  11. [11]

    First measurement of elec- tron neutrino appearance in NOvA,

    P. Adamson et al. (NOvA Collaboration), “First measurement of elec- tron neutrino appearance in NOvA,”Phys. Rev. Lett.116, 151806 (2016). doi:10.1103/PhysRevLett.116.151806

    work page doi:10.1103/physrevlett.116.151806 2016

  12. [12]

    Prospects for beyond the Standard Model physics searches at the Deep Underground Neutrino Experiment,

    B. Abi et al. (DUNE Collaboration), “Prospects for beyond the Standard Model physics searches at the Deep Underground Neutrino Experiment,”Eur. Phys. J. C 81, 322 (2021). doi:10.1140/epjc/s10052-021-09007-w

    work page doi:10.1140/epjc/s10052-021-09007-w 2021

  13. [13]

    Hyper-Kamiokande Design Report

    K. Abe et al. (Hyper-Kamiokande Collaboration), “Hyper-Kamiokande design re- port,” arXiv:1805.04163 [physics.ins-det] (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  14. [14]

    JUNO physics and detector,

    A. Abusleme et al. (JUNO Collaboration), “JUNO physics and detector,”Prog. Part. Nucl. Phys.123, 103927 (2022). doi:10.1016/j.ppnp.2021.103927

    work page doi:10.1016/j.ppnp.2021.103927 2022

  15. [15]

    NuFit-6.0: Updated global analysis of three-flavor neutrino oscillations,

    I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pinheiro, and T. Schwetz, “NuFit-6.0: Updated global analysis of three-flavor neutrino oscillations,” JHEP12, 216 (2024). doi:10.1007/JHEP12(2024)216

    work page doi:10.1007/jhep12(2024)216 2024

  16. [16]

    NuFIT 6.0 (2024),

    NuFIT collaboration, “NuFIT 6.0 (2024),”http://www.nu-fit.org, parameter ta- blev60.tbl-parameters(IC24 with-SK column), accessed 2026. 21

    work page 2024

  17. [17]

    Neutrino oscillations in the three flavor paradigm,

    P. B. Denton, “Neutrino oscillations in the three flavor paradigm,” arXiv:2501.08374 [hep-ph] (2025)

    work page arXiv 2025

  18. [18]

    Neutrino oscillations in matter,

    L. Wolfenstein, “Neutrino oscillations in matter,”Phys. Rev. D17, 2369 (1978). doi:10.1103/PhysRevD.17.2369

    work page doi:10.1103/physrevd.17.2369 1978

  19. [19]

    Resonance enhancement of oscillations in matter and solar neutrino spectroscopy,

    S. P. Mikheyev and A. Yu. Smirnov, “Resonance enhancement of oscillations in matter and solar neutrino spectroscopy,”Sov. J. Nucl. Phys.42, 913 (1985). doi:10.1007/BF02508049

    work page doi:10.1007/bf02508049 1985

  20. [20]

    Mattereffectsonthree- neutrino oscillations,

    V.D.Barger, K.Whisnant, S.Pakvasa, andR.J.N.Phillips, “Mattereffectsonthree- neutrino oscillations,”Phys. Rev. D22, 2718 (1980). doi:10.1103/PhysRevD.22.2718

    work page doi:10.1103/physrevd.22.2718 1980

  21. [21]

    The mixing angles in matter for three generations of neutrinos and the MSW mechanism,

    H. W. Zaglauer and K. H. Schwarzer, “The mixing angles in matter for three generations of neutrinos and the MSW mechanism,”Z. Phys. C40, 273 (1988). doi:10.1007/BF01555889

    work page doi:10.1007/bf01555889 1988

  22. [22]

    Review of particle physics,

    R. L. Workman et al. (Particle Data Group), “Review of particle physics,”Prog. Theor. Exp. Phys.2022, 083C01 (2022). doi:10.1093/ptep/ptac097

    work page doi:10.1093/ptep/ptac097 2022

  23. [23]

    Massive neutrinos and neutrino oscillations,

    S. M. Bilenky and S. T. Petcov, “Massive neutrinos and neutrino oscillations,”Rev. Mod. Phys.59, 671 (1987). doi:10.1103/RevModPhys.59.671

    work page doi:10.1103/revmodphys.59.671 1987

  24. [24]

    Another possible way to determine the neutrino mass hierarchy,

    H. Nunokawa, S. J. Parke, and R. Zukanovich Funchal, “Another possible way to determine the neutrino mass hierarchy,”Phys. Rev. D72, 013009 (2005). doi:10.1103/PhysRevD.72.013009

    work page doi:10.1103/physrevd.72.013009 2005

  25. [25]

    Unitarity and the three neutrino mixing matrix,

    S. J. Parke, “Unitarity and the three neutrino mixing matrix,”Phys. Rev. D93, 053008 (2016). doi:10.1103/PhysRevD.93.053008

    work page doi:10.1103/physrevd.93.053008 2016

  26. [26]

    Symmetries of baryons and mesons,

    M. Gell-Mann, “Symmetries of baryons and mesons,”Phys. Rev.125, 1067 (1962). doi:10.1103/PhysRev.125.1067

    work page doi:10.1103/physrev.125.1067 1962

  27. [27]

    Neutrino Oscillations in Matter using the Adjugate of the Hamiltonian

    A. M. Abdullahi and S. J. Parke, “Neutrino oscillations in matter using the adjugate of the Hamiltonian,” arXiv:2212.12565 [hep-ph] (2022);Eur. Phys. J. C84, 707 (2024). doi:10.1140/epjc/s10052-024-13078-8

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-024-13078-8 2022

  28. [28]

    Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra,

    P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, “Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra,”Bull. Am. Math. Soc.59, 31 (2022). doi:10.1090/bull/1722

    work page doi:10.1090/bull/1722 2022

  29. [29]

    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. D1, 379 (1992). doi:10.1142/S0218271892000203

    work page doi:10.1142/s0218271892000203 1992

  30. [30]

    CP and T violation in neutrino oscillations and invariance of Jarlskog’s determinant to matter effects,

    P. F. Harrison and W. G. Scott, “CP and T violation in neutrino oscillations and invariance of Jarlskog’s determinant to matter effects,”Phys. Lett. B535, 229 (2002). doi:10.1016/S0370-2693(02)01753-5

    work page doi:10.1016/s0370-2693(02)01753-5 2002

  31. [31]

    Onthegeneratorsofquantumdynamicalsemigroups,

    G.Lindblad, “Onthegeneratorsofquantumdynamicalsemigroups,”Commun. Math. Phys.48, 119 (1976). doi:10.1007/BF01608499

    work page doi:10.1007/bf01608499 1976

  32. [32]

    Completely positive dynamical semigroups ofN-level systems,

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups ofN-level systems,”J. Math. Phys.17, 821 (1976). doi:10.1063/1.522979 22

    work page doi:10.1063/1.522979 1976

  33. [33]

    Lisi , author A

    E. Lisi, A. Marrone, and D. Montanino, “Probing possible decoherence ef- fects in atmospheric neutrino oscillations,”Phys. Rev. Lett.85, 1166 (2000). doi:10.1103/PhysRevLett.85.1166

    work page internal anchor Pith review doi:10.1103/physrevlett.85.1166 2000

  34. [34]

    Status of non-standard neutrino interactions,

    T. Ohlsson, “Status of non-standard neutrino interactions,”Rept. Prog. Phys.76, 044201 (2013). doi:10.1088/0034-4885/76/4/044201

    work page internal anchor Pith review doi:10.1088/0034-4885/76/4/044201 2013

  35. [35]

    Solar neutrinos and neutrino mixing,

    S. Nussinov, “Solar neutrinos and neutrino mixing,”Phys. Lett. B63, 201 (1976). doi:10.1016/0370-2693(76)90648-1

    work page doi:10.1016/0370-2693(76)90648-1 1976

  36. [36]

    On the Quantum Mechanics of Neu- trino Oscillation,

    B. Kayser, “On the quantum mechanics of neutrino oscillation,”Phys. Rev. D24, 110 (1981). doi:10.1103/PhysRevD.24.110

    work page doi:10.1103/physrevd.24.110 1981

  37. [37]

    When Do Neu- trinos Really Oscillate? Quantum Mechanics of Neu- trino Oscillations,

    C. Giunti, C. W. Kim, and U. W. Lee, “When do neutrinos really oscillate?”Phys. Rev. D44, 3635 (1991). doi:10.1103/PhysRevD.44.3635

    work page doi:10.1103/physrevd.44.3635 1991

  38. [38]

    Real oscillations of virtual neutrinos,

    W. Grimus and P. Stockinger, “Real oscillations of virtual neutrinos,”Phys. Rev. D 54, 7414 (1996). doi:10.1103/PhysRevD.54.7414

    work page doi:10.1103/physrevd.54.7414 1996

  39. [39]

    Oscillations of neutrinos and mesons in quantum field theory,

    M. Beuthe, “Oscillations of neutrinos and mesons in quantum field theory,”Phys. Rept.375, 105 (2003). doi:10.1016/S0370-1573(02)00538-0

    work page doi:10.1016/s0370-1573(02)00538-0 2003

  40. [40]

    Paradoxes of neutrino oscillations,

    E. Kh. Akhmedov and A. Yu. Smirnov, “Paradoxes of neutrino oscillations,”Phys. Atom. Nucl.72, 1363 (2009). doi:10.1134/S1063778809080122

    work page doi:10.1134/s1063778809080122 2009

  41. [41]

    Quantum field theoretic approach to neutrino oscillations in matter,

    E. Kh. Akhmedov and J. Wilhelm, “Quantum field theoretic approach to neutrino oscillations in matter,”JHEP01, 165 (2013). doi:10.1007/JHEP01(2013)165

    work page doi:10.1007/jhep01(2013)165 2013

  42. [42]

    When the wavepacket is unnecessary,

    L. Stodolsky, “When the wavepacket is unnecessary,”Phys. Rev. D58, 036006 (1998). doi:10.1103/PhysRevD.58.036006

    work page doi:10.1103/physrevd.58.036006 1998

  43. [43]

    Neutrino oscil- lation probabilities through the looking glass,

    G. Barenboim, P. B. Denton, S. J. Parke, and C. A. Ternes, “Neutrino oscil- lation probabilities through the looking glass,”Phys. Lett. B791, 351 (2019). doi:10.1016/j.physletb.2019.03.002

    work page doi:10.1016/j.physletb.2019.03.002 2019

  44. [44]

    Fast and accurate algorithm for calculating long- baseline neutrino oscillation probabilities with matter effects: NuFast,

    P. B. Denton and S. J. Parke, “Fast and accurate algorithm for calculating long- baseline neutrino oscillation probabilities with matter effects: NuFast,”Phys. Rev. D110, 073005 (2024). doi:10.1103/PhysRevD.110.073005

    work page doi:10.1103/physrevd.110.073005 2024

  45. [45]

    Series expansions for three-flavor neutrino oscillation probabilities in matter,

    E. K. Akhmedov, R. Johansson, M. Lindner, T. Ohlsson, and T. Schwetz, “Series expansions for three-flavor neutrino oscillation probabilities in matter,”JHEP04, 078 (2004). doi:10.1088/1126-6708/2004/04/078

    work page internal anchor Pith review doi:10.1088/1126-6708/2004/04/078 2004

  46. [46]

    Simple and compact expressions for neutrino oscilla- tion probabilities in matter,

    H. Minakata and S. J. Parke, “Simple and compact expressions for neutrino oscilla- tion probabilities in matter,”JHEP01, 180 (2016). doi:10.1007/JHEP01(2016)180

    work page doi:10.1007/jhep01(2016)180 2016

  47. [47]

    Compact perturbative expressions for neutrino oscillations in matter,

    P. B. Denton, H. Minakata, and S. J. Parke, “Compact perturbative expressions for neutrino oscillations in matter,”JHEP06, 051 (2016). doi:10.1007/JHEP06(2016)051

    work page doi:10.1007/jhep06(2016)051 2016

  48. [48]

    Rotations versus perturbative expansions for calculating neutrino oscillation probabilities in matter,

    P. B. Denton, S. J. Parke, and X. Zhang, “Rotations versus perturbative expansions for calculating neutrino oscillation probabilities in matter,”Phys. Rev. D98, 033001 (2018). doi:10.1103/PhysRevD.98.033001 23

    work page doi:10.1103/physrevd.98.033001 2018

  49. [49]

    Golden measurements at a neutrino factory,

    A. Cervera, A. Donini, M. B. Gavela, J. J. Gomez Cádenas, P. Hernández, O. Mena, and S. Rigolin, “Golden measurements at a neutrino factory,”Nucl. Phys. B579, 17 (2000) [Erratum:Nucl. Phys. B593, 731 (2001)]. doi:10.1016/S0550-3213(00)00221- 2

    work page doi:10.1016/s0550-3213(00)00221- 2000

  50. [50]

    Analytic approximations for three neutrino oscillation param- eters and probabilities in matter,

    M. Freund, “Analytic approximations for three neutrino oscillation param- eters and probabilities in matter,”Phys. Rev. D64, 053003 (2001). doi:10.1103/PhysRevD.64.053003

    work page doi:10.1103/physrevd.64.053003 2001

  51. [51]

    Analytic approximation of the neutrino oscillation matter effects at largeθ 13,

    S. K. Agarwalla, Y. Kao, and T. Takeuchi, “Analytic approximation of the neutrino oscillation matter effects at largeθ 13,”JHEP04, 047 (2014). doi:10.1007/JHEP04(2014)047

    work page doi:10.1007/jhep04(2014)047 2014

  52. [52]

    Large-θ13 perturbation theory of neutrino oscillation for long-baseline experiments,

    K. Asano and H. Minakata, “Large-θ13 perturbation theory of neutrino oscillation for long-baseline experiments,”JHEP06, 022 (2011). doi:10.1007/JHEP06(2011)022

    work page doi:10.1007/jhep06(2011)022 2011

  53. [53]

    Matter density profile shape effects at DUNE,

    K. J. Kelly and S. J. Parke, “Matter density profile shape effects at DUNE,”Phys. Rev. D98, 015025 (2018). doi:10.1103/PhysRevD.98.015025

    work page doi:10.1103/physrevd.98.015025 2018

  54. [54]

    Unified approach to the classical statistical anal- ysis of small signals,

    G. J. Feldman and R. D. Cousins, “Unified approach to the classical statistical anal- ysis of small signals,”Phys. Rev. D57, 3873 (1998). doi:10.1103/PhysRevD.57.3873

    work page doi:10.1103/physrevd.57.3873 1998

  55. [55]

    Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation,

    C. Jarlskog, “Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation,”Phys. Rev. Lett.55, 1039 (1985). doi:10.1103/PhysRevLett.55.1039

    work page doi:10.1103/physrevlett.55.1039 1985

  56. [56]

    Cardano,Ars Magna

    G. Cardano,Ars Magna. Nürnberg, 1545

    work page

  57. [57]

    On the oscillations of neutrinos with Dirac and Majorana masses,

    S. M. Bilenky, J. Hosek, and S. T. Petcov, “On the oscillations of neutrinos with Dirac and Majorana masses,”Phys. Lett. B94, 495 (1980). doi:10.1016/0370- 2693(80)90927-2

    work page doi:10.1016/0370- 1980

  58. [58]

    Exploring neutrino mixing with low energy super- beams,

    H. Minakata and H. Nunokawa, “Exploring neutrino mixing with low energy super- beams,”JHEP10, 001 (2001). doi:10.1088/1126-6708/2001/10/001

    work page doi:10.1088/1126-6708/2001/10/001 2001

  59. [59]

    CP violation and neutrino oscilla- tions,

    H. Nunokawa, S. J. Parke, and J. W. F. Valle, “CP violation and neutrino oscilla- tions,”Prog. Part. Nucl. Phys.60, 338 (2008). doi:10.1016/j.ppnp.2007.10.001

    work page doi:10.1016/j.ppnp.2007.10.001 2008

  60. [60]

    CP violation versus matter effect in long-baseline neutrino oscillation experiments,

    H. Minakata and H. Nunokawa, “CP violation versus matter effect in long-baseline neutrino oscillation experiments,”Phys. Lett. B413, 369 (1997). doi:10.1016/S0370- 2693(97)01164-0

    work page doi:10.1016/s0370- 1997

  61. [61]

    Non-adiabatic crossing of energy levels,

    C. Zener, “Non-adiabatic crossing of energy levels,”Proc. R. Soc. Lond. A137, 696 (1932). doi:10.1098/rspa.1932.0165

    work page doi:10.1098/rspa.1932.0165 1932

  62. [62]

    Zur Theorie der Energieübertragung bei Stößen,

    L. D. Landau, “Zur Theorie der Energieübertragung bei Stößen,”Phys. Z. Sowjetu- nion2, 46 (1932)

    work page 1932

  63. [63]

    T violation in neu- trino oscillations in matter,

    E. Kh. Akhmedov, P. Huber, M. Lindner, and T. Ohlsson, “T violation in neu- trino oscillations in matter,”Nucl. Phys. B608, 394 (2001). doi:10.1016/S0550- 3213(01)00279-3 24

    work page doi:10.1016/s0550- 2001

  64. [64]

    Twomodesofsearchingfornewneutrinointeractions at MiniBooNE,

    A.FriedlandandC.Lunardini, “Twomodesofsearchingfornewneutrinointeractions at MiniBooNE,”Phys. Rev. D74, 033012 (2006). doi:10.1103/PhysRevD.74.033012

    work page doi:10.1103/physrevd.74.033012 2006

  65. [65]

    Are solar neutrino oscillations robust?

    O. G. Miranda, M. A. Tortola, and J. W. F. Valle, “Are solar neutrino oscillations robust?”JHEP10, 008 (2006). doi:10.1088/1126-6708/2006/10/008

    work page doi:10.1088/1126-6708/2006/10/008 2006

  66. [66]

    Observation of coherent elastic neutrino-nucleus scattering,

    D. Akimov et al. (COHERENT Collaboration), “Observation of coherent elastic neutrino-nucleus scattering,”Science357, 1123 (2017). doi:10.1126/science.aao0990

    work page doi:10.1126/science.aao0990 2017

  67. [67]

    Neutrino oscillations and non-standard interactions,

    Y. Farzan and M. Tortola, “Neutrino oscillations and non-standard interactions,” Front. Phys.6, 10 (2018). doi:10.3389/fphy.2018.00010

    work page doi:10.3389/fphy.2018.00010 2018

  68. [68]

    Evidence for neutrino oscillations from the observation of¯νe appearance in a¯νµ beam,

    A. Aguilar et al. (LSND Collaboration), “Evidence for neutrino oscillations from the observation of¯νe appearance in a¯νµ beam,”Phys. Rev. D64, 112007 (2001). doi:10.1103/PhysRevD.64.112007

    work page doi:10.1103/physrevd.64.112007 2001

  69. [69]

    Significant excess of electron-like events in MiniBooNE,

    A. A. Aguilar-Arevalo et al. (MiniBooNE Collaboration), “Significant excess of electron-like events in MiniBooNE,”Phys. Rev. Lett.121, 221801 (2018). doi:10.1103/PhysRevLett.121.221801

    work page doi:10.1103/physrevlett.121.221801 2018

  70. [70]

    Results from the Baksan ex- periment on sterile transitions (BEST),

    V. V. Barinov et al. (BEST Collaboration), “Results from the Baksan ex- periment on sterile transitions (BEST),”Phys. Rev. Lett.128, 232501 (2022). doi:10.1103/PhysRevLett.128.232501

    work page doi:10.1103/physrevlett.128.232501 2022

  71. [71]

    Solution of the quartic equation,

    L. Ferrari, “Solution of the quartic equation,” in G. Cardano,Ars Magna. Nürnberg, 1545

    work page

  72. [72]

    Addendum to compact perturbative expres- sions for neutrino oscillations in matter,

    P. B. Denton and S. J. Parke, “Addendum to compact perturbative expres- sions for neutrino oscillations in matter,”Phys. Rev. D98, 093001 (2018). doi:10.1103/PhysRevD.98.093001

    work page doi:10.1103/physrevd.98.093001 2018

  73. [73]

    Reactor antineutrino oscillations in matter,

    P. B. Denton and S. J. Parke, “Reactor antineutrino oscillations in matter,”Phys. Rev. D109, 053002 (2024). doi:10.1103/PhysRevD.109.053002

    work page doi:10.1103/physrevd.109.053002 2024

  74. [74]

    Terrestrial matter effects on reactor an- tineutrino oscillations at JUNO or RENO-50,

    Y.-F. Li, Y. Wang, and Z.-Z. Xing, “Terrestrial matter effects on reactor an- tineutrino oscillations at JUNO or RENO-50,”Chin. Phys. C40, 091001 (2016). doi:10.1088/1674-1137/40/9/091001

    work page doi:10.1088/1674-1137/40/9/091001 2016

  75. [75]

    Solar neutrinos before and after KamLAND,

    J. N. Bahcall, M. C. Gonzalez-Garcia, and C. Pena-Garay, “Solar neutrinos before and after KamLAND,”JHEP02, 009 (2003). doi:10.1088/1126-6708/2003/02/009

    work page doi:10.1088/1126-6708/2003/02/009 2003

  76. [76]

    Comprehensive measurement of pp- chain solar neutrinos,

    M. Agostini et al. (Borexino Collaboration), “Comprehensive measurement of pp- chain solar neutrinos,”Nature562, 505 (2018). doi:10.1038/s41586-018-0624-y

    work page doi:10.1038/s41586-018-0624-y 2018

  77. [77]

    IceCube-Gen2: The window to the extreme universe,

    M. G. Aartsen et al. (IceCube-Gen2 Collaboration), “IceCube-Gen2: The window to the extreme universe,”J. Phys. G48, 060501 (2021). doi:10.1088/1361-6471/abbd48

    work page doi:10.1088/1361-6471/abbd48 2021

  78. [78]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Numerical Recipes: The Art of Scientific Computing, 3rd ed. Cambridge University Press, 2007. ISBN 978-0-521-88068-8. 25 A Physical Constants and Reference Values Table 9: Physical constants and reference matter configuration. Quantity Value GF 1.1663787×10 −5GeV−2 mN 0.939GeV 1km5.06773×1...

    work page 2007