Pith. sign in

REVIEW 3 major objections 4 minor 46 references

Neutrino Decoherence in Simple Open Quantum Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that neutrino coherence decays exponentially in any weakly coupled, memoryless environment, with a rate fixed by the coupling differences between mass eigenstates.

desk verdict The QM toy models are clean and the Lindblad dictionary is useful, but the QFT decoherence rate has a sign error and a missing derivation—worth a serious referee, not worth accepting as is. read the letter →

arxiv 2009.13471 v2 pith:2632RM67 submitted 2020-09-28 hep-ph hep-thquant-ph

Bin Xu This is my paper · ORCID
classification hep-phhep-thquant-ph PACS 03.65.Yz14.60.Pq
keywords neutrinodecoherenceopenquantumsystemsGKSLmasterequationLindbladoperatorsBorn-Markovapproximationfieldtheoryoscillationsenvironment-induced
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 argues that when neutrinos travel through an environment they interact with weakly, their quantum coherence—the off-diagonal entries of the neutrino density matrix that produce oscillations—does not decay in a model-dependent way: under weak couplings and a memoryless (Born-Markov) environment, it always decays exponentially, $F_{ij}(t)=e^{-\Gamma_{ij}t}$. The paper establishes this in three settings: an exactly solvable forced-oscillator bath, a collection of two-level systems delivering random phase kicks, and a quantum field theory of elastic scattering off environmental particles. In each case the decoherence rate is set by the squared difference of the couplings (or of the square roots of cross sections) between two mass eigenstates. If right, the result tells experimenters and model builders that environment-induced neutrino decoherence can be described by a standard GKSL master equation, with Lindblad operators that are simply the neutrino part of the interaction Hamiltonian.

What carries the argument

The load-bearing object is the decoherence form factor $F_{ij}(t)=\mathrm{Tr}(e^{iH_j t}e^{-iH_i t}\rho_E)$, the overlap of two environment states that have evolved under the two different mass-eigenstate interactions. With the assumption $[H_\nu,H_{\nu E}]=0$, the total evolution factorizes, and $F_{ij}(t)$ multiplies the off-diagonal neutrino density-matrix elements. Under a white-noise correlation function $C_f(\tau)=g\delta(\tau)$ and the van Hove weak-coupling limit, the oscillator model yields $F(t)=\exp[-\frac{1}{2}(\lambda_1-\lambda_2)^2Gt]$; the two-level kick model gives the same law via the central limit theorem; and the QFT calculation expresses $\mathrm{Re}\,F$ through the optical theorem, giving $\Gamma=-N_\phi v(\sqrt{\sigma_1}-\sqrt{\sigma_2})^2/2$. The GKSL master equation is derived from the same Born-Markov steps, so the Lindblad operator $L$ is shown to be the neutrino-space factor of $H_{\nu E}$.

What would settle it

A decisive check is to evaluate the commutator $[H_\nu,H_{\nu E}]$ for the standard matter potential expressed in the mass basis: if it is nonzero, the factorized form $F(t)=\mathrm{Tr}(e^{iH_2t}e^{-iH_1t}\rho_E)$ no longer holds, and a long-baseline experiment seeing decoherence with a rate incompatible with $(\sqrt{\sigma_i}-\sqrt{\sigma_j})^2$ would rule out the paper's universal formula.

Watch

Extended reading notes

Core claim

On the paper's own terms the central discovery is a universality result for weakly coupled, Markovian neutrino environments: the decoherence form factor $F_{ij}(t)=\mathrm{Tr}(e^{iH_j t}e^{-iH_i t}\rho_E)$ is universally $e^{-\Gamma_{ij}t}$, with $\Gamma_{ij}=\frac{1}{2}(\lambda_i-\lambda_j)^2 G$ in the two quantum-mechanical models and $\Gamma_{ij}=-\frac{N_\phi v}{2}(\sqrt{\sigma_i}-\sqrt{\sigma_j})^2$ in the QFT scattering model. The same exponential appears as the exact solution of the GKSL equation with a single Hermitian Lindblad operator $L=\mathrm{diag}\{l_1,l_2,l_3\}$, provided $l_i=\lambda_i\sqrt{G}$ (or $l_i=\sqrt{N_\phi v\sigma_i}$). This identifies the Lindblad operators with the neutrino part of the interaction Hamiltonian, and traces the universality to the Born-Markov approximation; when the environment has memory, as in the constant-force and constant-coupling cases, the decay becomes Gaussian or periodic rather than exponential.

Load-bearing premise

The derivation assumes the environment pulls on each neutrino mass state independently and never turns one mass state into another; real weak interactions in matter are tied to flavor, not mass, so this assumption may fail there.

Editorial extensions

If this is right

  • In any weak, memoryless environment, neutrino oscillation probabilities gain a factor $e^{-\Gamma_{ij}t}$ on the interference term, so coherence is lost on a time scale set by environment couplings, not by the mass splitting alone.
  • If all mass eigenstates couple to the environment with equal strength, no decoherence occurs; for unequal couplings, only two of the three rates are independent, with $\sqrt{\Gamma_{13}}=\sqrt{\Gamma_{12}}+\sqrt{\Gamma_{23}}$.
  • The GKSL master equation's Lindblad operators are no longer free phenomenological parameters: they are fixed by the interaction Hamiltonian, e.g. $L\propto\mathrm{diag}\{\lambda_1,\lambda_2,\lambda_3\}$ in the oscillator model.
  • Deviations from exponential decay signal non-Markovian environments; for constant driving forces the paper finds periodic revivals and Gaussian short-time decay, so a measured non-exponential decoherence would rule out the simple Born-Markov picture.
  • In the QFT treatment, decoherence rates follow from scattering cross sections, $\Gamma_{ij}=-N_\phi v(\sqrt{\sigma_i}-\sqrt{\sigma_j})^2/2$, so rates can be estimated from standard neutrino cross sections and medium density.

Reading between the lines

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

  • Because ordinary matter interactions are flavor-diagonal, the mass-basis diagonal assumption $[H_\nu,H_{\nu E}]=0$ is violated once lepton mixing is inserted; a natural extension is to derive a modified form factor and assess whether the exponential law and the squared-difference rate survive.
  • The two-level phase-kick model is essentially a quantum random walk in phase; one could test it against neutrino propagation in a fluctuating medium such as turbulent supernova matter, where the kick statistics are known from hydrodynamics.
  • The additivity $\sqrt{\Gamma_{13}}=\sqrt{\Gamma_{12}}+\sqrt{\Gamma_{23}}$ suggests decoherence rates behave like an $\ell^1$ metric on mass-basis couplings; this could be checked as a consistency condition in future global fits of decoherence parameters.
  • The simplified QFT Lagrangian ignores neutrino spin and treats the medium particle as a single scalar; upgrading to the full Standard Model neutral- and charged-current interactions is the direct path to realistic predictions for long-baseline and astrophysical neutrinos.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies environment-induced neutrino decoherence in open quantum systems. It first derives exact evolution in two quantum-mechanical toy models: a neutrino coupled to a bath of forced harmonic oscillators with white-noise forcing, and a neutrino coupled to a collection of two-level systems with random phase kicks. In both models, the decoherence form factor F(t) decays exponentially, F(t)=e^{-Γt}, with Γ∝(λ_i-λ_j)^2 in the van Hove/weak-coupling and Poisson-kick limits. The paper then attempts a quantum field-theoretic derivation in which the neutrino scatters elastically off environmental particles through a scalar interaction, claiming a rate Γ∝(√σ_1-√σ_2)^2. Finally, it connects the results to the GKSL master equation and identifies the Lindblad operator with the neutrino part of the interaction Hamiltonian, and argues that the Born-Markov approximation is the common basis of the exponential decay.

Significance. If the central claims were fully established, the paper would provide a useful, parameter-free bridge between microscopic interaction Hamiltonians and phenomenological Lindblad operators for neutrino decoherence. The two quantum-mechanical models are exactly solvable and are derived transparently: the white-noise harmonic-oscillator model and the two-level phase-kick model both lead to the advertised quadratic rate formula without fitted parameters. The GKSL connection in Appendix A is a valuable pedagogical demonstration that Lindblad operators can emerge from an explicit microscopic Hamiltonian. However, the QFT section, which is one of the three pillars of the claimed universality, contains a sign error and an inconsistent order-of-magnitude treatment. As printed, that section predicts growth rather than decay of the coherence factor, and the derivation mixes first- and second-order terms. The claimed universality of exponential decay across QM and QFT is therefore not established, and the paper needs substantive revision before the conclusion can be accepted.

major comments (3)
  1. [Section V, Eqs. (43)-(44) and Section VII] The QFT decoherence rate has the wrong sign. Equation (43) gives Re[F] = -T v/(2V)(√σ1 - √σ2)^2, so with F(t)=e^{-Γt+iα(t)} in Eq. (44) the correct rate is Γ = +N_φ v/2 (√σ1 - √σ2)^2, not the minus sign printed. With the printed sign, |F(t)| grows exponentially with time, which is the opposite of decoherence. The same negative sign is repeated in Section VII and is also inconsistent with Section VI, where l_i = √(N_φ v σ_i) implies the positive rate Γ_ij = N_φ v/2 (√σ_i - √σ_j)^2.
  2. [Section V, Eqs. (33)-(34), (41)-(43)] The derivation mixes first- and second-order terms in the coupling. The states |f1> and |f2> are constructed using S = 1 + iT truncated at first order in λ_i, and for the scalar interaction (25) the tree-level amplitude M_i is real at that order. The optical theorem in Eq. (42), Im M_i = 2|p| m_φ v σ_i, is a second-order unitarity statement with σ_i proportional to λ_i^2; it cannot be applied to the same first-order M_i that appears in the first two terms of Eq. (41). The third term in Eq. (41) is already O(λ1λ2), so the real part of F is not evaluated at a definite perturbative order. A consistent treatment must retain the O(λ^2) absorptive part of the forward amplitude, or else justify a resummation that makes the order counting valid.
  3. [Section II, Eq. (5)] The universal exponential decay derived in the two QM models relies on the assumption [H_ν, H_νE] = 0, i.e. the interaction is diagonal in the neutrino mass basis. In standard matter, weak interactions are flavor-diagonal before the PMNS rotation and therefore contain off-diagonal mass-basis terms after rotation. In that case the factorization e^{-iH_i t} used to obtain Eq. (6) does not hold, and the simple rate Γ ∝ (λ_i - λ_j)^2 requires modification. Since the concluding section proposes extending these models to realistic environments (e.g. matter propagation and MSW effects), this limitation should be stated explicitly and its quantitative consequences assessed.
minor comments (4)
  1. [Section V, Eq. (28)] The normalization reads N^{-1} = 2√(|p| m_φ V), but with the Lorentz-invariant single-particle normalization of Eq. (27) the volume should appear linearly, N^{-1} = 2√(|p| m_φ) V. Please check this formula and the subsequent flux factors.
  2. [Section III, text after Eq. (16)] The sentence 'coherence is maintained between |ν1⟩ and |ν1⟩' should read 'between |ν1⟩ and |ν2⟩'.
  3. [Appendix A, final equation] The equality L = (√G/2)(λ1+λ2)σ0 + (√G/2)(λ1-λ2)σ3 = √(2G)(λ1|ν1⟩⟨ν1| + λ2|ν2⟩⟨ν2|) is algebraically incorrect: the first expression equals √G(λ1|ν1⟩⟨ν1| + λ2|ν2⟩⟨ν2|). The √(2G) prefactor is inconsistent with the l_i = λ_i√G assignment used in Section VI to reproduce Γ_ij = (1/2)(l_i-l_j)^2, and the normalization should be corrected.
  4. [Section IV, text after Eq. (22)] The phrase 'replacing n with γt' is informal because n is an integer and γt is a real mean. Please clarify that this is an ensemble-average replacement in the Poisson process, not an exact identity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the decoherence rates are derived from explicit Hamiltonians with no fitted parameters or self-citation chain.

full rationale

The paper's central results are derived, not assumed. In Section III, the harmonic-oscillator environment is solved exactly via coherent states, and the form factor F(t) is computed as a product of overlaps yielding beta(t)=exp[-1/2 (lambda_1-lambda_2)^2 N g t], which becomes Eq. (15) in the van Hove limit. In Section IV, the two-level-system model computes F(t) from explicit stochastic phase kicks and a Gaussian ensemble average, giving the same exponential form in Eq. (22). In both cases the rate is obtained from the stated Hamiltonian and stochastic assumptions; no parameter is fitted to the quantity being predicted, and no result is imported from a self-citation. The GKSL connection in Section VI and Appendix A is also self-contained: the Lindblad operator L = sqrt(2G)(lambda_1 |nu_1><nu_1| + lambda_2 |nu_2><nu_2|) is obtained by diagonalizing the correlation matrix gamma_mn derived from the same interaction Hamiltonian, not imposed as an ansatz. The paper contains no load-bearing self-citations; references [38,39] are external model citations, and the author's own prior work is not used to force the conclusion. The QFT section contains an internal sign/order inconsistency in Eq. (44), where the printed formula yields a negative decoherence rate, and the use of the optical theorem with a first-order S-matrix is questionable. However, these are correctness defects, not circularity: Eq. (44) does not reduce to its input by definition or rename a fitted parameter as a prediction. Therefore the honest finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 7 assumptions · 1 invented entities

The paper's derivations rest on standard open quantum system approximations and on specially chosen interaction forms: mass-basis-diagonal couplings, factorized initial states, and Markovian (white-noise or Poisson-kick) environments. These assumptions are stated clearly. The exponential decay is a consequence of the Markovian and weak-coupling limits, not of a new physical mechanism. The QFT section additionally relies on a scalar toy medium and on an optical-theorem relation that is quoted rather than derived.

free parameters (3)
  • lambda1, lambda2 (and lambda3) mass-state couplings
    Coupling constants of neutrino mass eigenstates to the environment. The decoherence rate is proportional to (lambda_i - lambda_j)^2. They are introduced by hand and not fitted to data.
  • G in harmonic oscillator model (Ng in the van Hove limit)
    Effective white-noise strength defined by N to infinity, g to 0, with N g = G fixed. It sets the overall decoherence scale and is not predicted by the model.
  • G = sum_s gamma_s sigma_g_s^2 in the two-level model
    Product of kick rate and kick-strength variance for each environmental two-level system. The total G sets the exponential rate and is chosen rather than derived.
assumptions (7)
  • domain assumption Born approximation: the environment remains in its initial state rho_E and is not significantly entangled with the neutrino.
    Used in Appendix A (Eq. A.4) to derive the GKSL equation and in all models when tracing over the environment. This sets the weak-coupling regime.
  • domain assumption Markov approximation: the environment is memoryless, so rho_nu(t') in the integro-differential equation is replaced by rho_nu(t).
    Used in Appendix A and encoded as white-noise correlation C_f(tau)=g delta(tau) in Section III and Poisson kicks in Section IV. This is what produces time-local exponential decay.
  • domain assumption [H_nu, H_nuE] = 0, i.e., the interaction is diagonal in the neutrino mass basis and conserves neutrino energy.
    Stated in Section II after Eq. (5). It enables the factorization of the evolution and the form factor F(t)=Tr(e^{iH2t}e^{-iH1t}rho_E).
  • domain assumption The neutrino and environment are initially uncorrelated: rho(0)=rho_nu(x)rho_E.
    Assumed in Sections II, III, IV, and in the QFT initial state Eq. (26). If the system was created entangled, the decoherence dynamics would be different.
  • ad hoc to paper Van Hove weak-coupling limit: N to infinity, g to 0, with N g = G fixed.
    Used in Section III to obtain a finite exponential rate from many weakly coupled environmental modes. The value of G is not predicted.
  • domain assumption The random process f_s(t) is stationary and ergodic with white-noise correlation C_f(tau)=g delta(tau).
    Imposed in Section III. The delta correlation is a Markovian choice, not derived from a more fundamental environment.
  • domain assumption QFT model uses an effective scalar interaction L=(lambda1 nu1 nu1 + lambda2 nu2 nu2) phi phi with spinless particles and the optical theorem for elastic scattering.
    Section V. The scalar phi is a toy medium, and the optical-theorem relation is quoted to express the decoherence rate in terms of cross sections.
invented entities (1)
  • Scalar field phi (medium particle)
    purpose: Provides a simple elastic scattering environment for computing the QFT decoherence rate.
    Introduced as a toy model; no claim is made that this scalar exists. The QFT rate formula depends on this model device, so it has no independent experimental handle.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Neutrino Decoherence in Simple Open Quantum Systems." pith.science (2026). https://pith.science/paper/2632RM67

@misc{pith2026200913471,
  author       = {Pith},
  title        = {Pith review of: Neutrino Decoherence in Simple Open Quantum Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2632RM67}},
  note         = {Machine review of arXiv:2009.13471}
}
read the original abstract

Neutrinos lose coherence as they propagate, which leads to the fading away of oscillations. In this work, we model neutrino decoherence induced in open quantum systems from their interaction with the environment. We first present two different models in the quantum mechanical framework, in which the environment is modeled as forced harmonic oscillators with white noise interactions, or two-level systems with stochastic phase kicks. We then look at the decoherence process in the quantum field theoretic framework induced by elastic scatterings with environmental particles. The exponential decay is obtained as a common feature for all models, which shows the universality of the decoherence processes. We discuss connections to the GKSL master equation approach and give a clear physical meaning of the Lindblad operators. We demonstrate that the universality of exponential decay of coherence is based on the Born-Markov approximation. The models in this work are suitable to be extended to describe real physical processes that could be non-Markovian.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 40 canonical work pages

  1. [1]

    A. B. McDonald, Rev. Mod. Phys. 88, 030502 (2016)

    work page 2016

  2. [2]

    Kajita, Rev

    T. Kajita, Rev. Mod. Phys. 88, 030501 (2016)

    2016

  3. [3]

    2020 Global reassessment of the neutrino oscillation picture

    P. de Salas, D. Forero, S. Gariazzo, P. Mart ´ ınez-Mirav´e, O. Mena, C. Ternes, M. T´ ortola, and J. Valle, (2020), arXiv:2006.11237 [hep-p h]

    work page Pith review arXiv 2020

  4. [4]

    Pontecorvo, Sov

    B. Pontecorvo, Sov. Phys. JETP 7, 172 (1958)

    1958

  5. [5]

    Z. Maki, M. Nakagawa, and S. Sakata, Progress of Theoreti cal Physics 28, 870 (1962)

    work page 1962

  6. [6]

    Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory

    Q. Ahmad et al. (SNO), Phys. Rev. Lett. 89, 011301 (2002), arXiv:nucl-ex/0204008

    work page Pith review arXiv 2002

  7. [7]

    Nussinov, Phys

    S. Nussinov, Phys. Lett. B 63, 201 (1976)

    work page 1976

  8. [8]

    Giunti, C

    C. Giunti, C. Kim, and U. Lee, Phys. Lett. B 274, 87 (1992)

    work page 1992

Show all 46 references
  1. [9]

    Beuthe, Phys

    M. Beuthe, Phys. Rev. D 66, 013003 (2002), arXiv:hep-ph/0202068

    2002 arXiv

  2. [10]

    Kayser and J

    B. Kayser and J. Kopp, (2010), arXiv:1005.4081 [hep-ph ]

    2010 arXiv

  3. [11]

    F. P. An et al. (Daya Bay), Eur. Phys. J. C 77, 606 (2017), arXiv:1608.01661 [hep-ex]

    2017 arXiv

  4. [12]

    Akhmedov, J

    E. Akhmedov, J. Kopp, and M. Lindner, JCAP 09, 017 (2017), arXiv:1702.08338 [hep-ph]

    2017 arXiv

  5. [13]

    de Gouvea, V

    A. de Gouvea, V. de Romeri, and C. A. Ternes, JHEP 08, 018 (2020), arXiv:2005.03022 [hep-ph]

    2020 arXiv

  6. [14]

    W. H. Zurek, Phys. Today 44N10, 36 (1991)

    1991

  7. [15]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. Sudarshan, J. Math. Phys. 17, 821 (1976)

    1976

  8. [16]

    Lindblad, Commun

    G. Lindblad, Commun. Math. Phys. 48, 119 (1976)

    1976

  9. [17]

    Benatti and R

    F. Benatti and R. Floreanini, JHEP 02, 032 (2000), arXiv:hep-ph/0002221

    2000 arXiv

  10. [18]

    Benatti and R

    F. Benatti and R. Floreanini, Phys. Rev. D 64, 085015 (2001), arXiv:hep-ph/0105303

    2001 arXiv

  11. [19]

    E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000), arXiv:hep-ph/0002053

    2000 arXiv

  12. [20]

    A. Gago, E. Santos, W. Teves, and R. Zukanovich Funchal, Phys. Rev. D 63, 073001 (2001), arXiv:hep-ph/0009222

    2001 arXiv

  13. [21]

    Morgan, E

    D. Morgan, E. Winstanley, J. Brunner, and L. F. Thompson , Astropart. Phys. 25, 311 (2006), arXiv:astro-ph/0412618

    2006 arXiv

  14. [22]

    Farzan, T

    Y. Farzan, T. Schwetz, and A. Y. Smirnov, JHEP 07, 067 (2008), arXiv:0805.2098 [hep-ph]

    2008 arXiv

  15. [23]

    Bakhti, Y

    P. Bakhti, Y. Farzan, and T. Schwetz, JHEP 05, 007 (2015), arXiv:1503.05374 [hep-ph]

    2015 arXiv

  16. [24]

    Oliveira and M

    R. Oliveira and M. Guzzo, Eur. Phys. J. C 73, 2434 (2013)

    2013

  17. [25]

    M. M. Guzzo, P. C. de Holanda, and R. L. Oliveira, Nucl. Phys. B 908, 408 (2016), arXiv:1408.0823 [hep-ph]

    2016 arXiv

  18. [26]

    Oliveira and M

    R. Oliveira and M. Guzzo, Eur. Phys. J. C 69, 493 (2010)

    2010

  19. [27]

    R. L. N. Oliveira, Eur. Phys. J. C 76, 417 (2016), arXiv:1603.08065 [hep-ph]

    2016 arXiv

  20. [28]

    J. a. A. Coelho and W. A. Mann, Phys. Rev. D 96, 093009 (2017), arXiv:1708.05495 [hep-ph]. 20

    2017 arXiv

  21. [29]

    Carpio, E

    J. Carpio, E. Massoni, and A. Gago, Phys. Rev. D 97, 115017 (2018), arXiv:1711.03680 [hep-ph]

    2018 arXiv

  22. [30]

    Ohlsson, Phys

    T. Ohlsson, Phys. Lett. B 502, 159 (2001), arXiv:hep-ph/0012272

    2001 arXiv

  23. [31]

    Ohlsson and S

    T. Ohlsson and S. Zhou, (2020), arXiv:2006.02445 [quan t-ph]

    2020 arXiv

  24. [32]

    Fogli, E

    G. Fogli, E. Lisi, A. Marrone, D. Montanino, and A. Palaz zo, Phys. Rev. D 76, 033006 (2007), arXiv:0704.2568 [hep-ph]

    2007 arXiv

  25. [33]

    de Oliveira, M

    R. de Oliveira, M. Guzzo, and P. de Holanda, Phys. Rev. D 89, 053002 (2014), arXiv:1401.0033 [hep-ph]

    2014 arXiv

  26. [34]

    Balieiro Gomes, M

    G. Balieiro Gomes, M. Guzzo, P. de Holanda, and R. Olivei ra, Phys. Rev. D 95, 113005 (2017), arXiv:1603.04126 [hep-ph]

    2017 arXiv

  27. [35]

    J. A. B. Coelho, W. A. Mann, and S. S. Bashar, Phys. Rev. Lett. 118, 221801 (2017), arXiv:1702.04738 [hep-ph]

    2017 arXiv

  28. [36]

    Coloma, J

    P. Coloma, J. Lopez-Pavon, I. Martinez-Soler, and H. Nu nokawa, Eur. Phys. J. C 78, 614 (2018), arXiv:1803.04438 [hep-ph]

    2018 arXiv

  29. [37]

    Balieiro Gomes, D

    G. Balieiro Gomes, D. Forero, M. Guzzo, P. De Holanda, an d R. Oliveira, Phys. Rev. D 100, 055023 (2019), arXiv:1805.09818 [hep-ph]

    2019 arXiv

  30. [38]

    A. O. Caldeira and A. J. Leggett, Physica A: Statistical mechanics and its Applications 121, 587 (1983)

    1983

  31. [39]

    Zurek, Phys

    W. Zurek, Phys. Rev. D 26, 1862 (1982)

    1982

  32. [40]

    Sun and D

    C. Sun and D. Zhou, (1998), arXiv:hep-ph/9808334

    1998 arXiv

  33. [41]

    Benatti and H

    F. Benatti and H. Narnhofer, letters in mathematical ph ysics 15, 325 (1988)

    1988

  34. [42]

    Wolfenstein, Phys

    L. Wolfenstein, Phys. Rev. D 17, 2369 (1978)

    1978

  35. [43]

    Mikheev and A

    S. Mikheev and A. Smirnov, Nuovo Cim. C 9, 17 (1986)

    1986

  36. [44]

    Hawking, Commun

    S. Hawking, Commun. Math. Phys. 87, 395 (1982)

    1982

  37. [45]

    J. R. Ellis, N. Mavromatos, and D. V. Nanopoulos, Mod. Phys. Lett. A 12, 1759 (1997), arXiv:hep-th/9704169

    1997 arXiv

  38. [46]

    Benatti and R

    F. Benatti and R. Floreanini, Annals Phys. 273, 58 (1999), arXiv:hep-th/9811196. 21

    1999 arXiv

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.