Quantum field theory treatment of the spin-flavor precession of Dirac neutrinos with diagonal magnetic moments

Maxim Dvornikov (IZMIRAN)

REVIEW

Quantum field theory treatment of the spin-flavor precession of Dirac neutrinos with diagonal magnetic moments

Maxim Dvornikov (IZMIRAN)

High Energy Physics - Phenomenology · Submitted 4 Nov 2026 · 18 references

Pith's verdict: A genuinely new QFT computation of electron-to-muon neutrino spin-flavor precession, but the claimed match with the QM result rests on an unpublished propagator and a magnetic phase dropped without a bound.

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arXiv ID2611.06643 (version 1)

Pith ID746FGQTF, Pith's permanent record of this paper and its review

CitationMaxim Dvornikov (IZMIRAN). "Quantum field theory treatment of the spin-flavor precession of Dirac neutrinos with diagonal magnetic moments." arXiv:2611.06643 (2026). https://pith.science/paper/746FGQTF

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Editor's note

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The computation is new, and it deserves a referee's time, but the headline claim is shakier than the paper lets on. What is genuinely new: a QFT treatment of the electron-to-muon left-handed channel of spin-flavor precession with diagonal magnetic moments, using the field-modified propagator, ending in Eqs. (2.24) and (2.25). Ref. [14] used standard propagators, so this closes a real formalism gap, if it holds. The B goes to zero limit of (2.25) collapses to the standard oscillation formula, which is a real sanity check, and the zeta-sum in (2.15) reconstructs (2.24) up to an overall phase.

Two soft spots matter, one internal and one about provenance.

Internal: a zeta-even magnetic phase appears in Eq. (2.23), the plus mu-squared-B term in the exponent, which by dimensions is probably mu-squared-B-squared over 2E. It is common to both zeta projections, so it does not cancel in the zeta-sum. Each mass-eigenstate amplitude then carries an extra phase of that order, and since mu_1 differs from mu_2 those do not cancel in the squared matrix element: the interference term picks up an additional phase of order (mu_2^2 - mu_1^2) B^2 L / 2E that appears in neither (2.25) nor the QM target (3.3). The paper drops it between (2.23) and (2.24) without a bound. The size is probably benign in the regime where the cosine factors actually oscillate, and a cleaner expansion of the pole energies might partly cancel it, but as written the equality is assumed rather than shown. A referee has to demand an explicit bound or a cancellation.

Provenance: every magnetic factor descends from the propagator (2.8), imported from Ref. [15], an unpublished self-citation. The Dirac solution behind it goes back to Ternov and collaborators in 1965, which is external, but the 4x4 propagator itself cannot be checked from the cited literature. The reader is right that this is the weakest link.

Minor points: the antiparticle-pole truncation and the smallness claim about the f_1 integral are asserted with self-citation rather than shown; the mu-squared-B term as printed has wrong dimensions; and the comparison target (3.3) is the author's own 2007 result, so the headline agreement is partly self-consistency.

Send it out. A referee's job is to force the propagator into the open and get a quantitative account of the dropped phase. The fix may be easy, but right now the central equality is not established, so I would not build on it yet.

Referee report

Significance

If the result holds, the paper establishes that the standard QM two-level description of spin-flavor precession with diagonal magnetic moments is the leading-order limit of the field-theoretic treatment, and that the QFT expression is available as a starting point for sub-leading corrections — a meaningful, if incremental, statement in a literature (Refs. [6–11], [14]) where the QFT/QM correspondence for magnetic-moment effects has not been written out in this channel. Credit is due for the analytic work: the momentum integration is real work rather than a restatement of the input, and several of its steps check out — Eq. (2.21)'s I_0 and Eq. (2.22) agree with the standard Bessel identities, the ζ-sum in Eq. (2.24) closes to the cos(μ_aBL) factors, and at B = 0 Eq. (2.25) reduces correctly to sin²2θ sin²(Δm²L/4E). The paper is also explicit about several of its approximations (fixed nuclei, ultrarelativistic and left-handed leptons, transverse field, L >> E⁻¹) and has no free parameters beyond the physical ones. What tempers the significance is that both sides of the comparison descend from the same level structure of Eq.

Major objections (4)

[§II, Eqs. (2.19)–(2.24)] The fate of the ζ-even magnetic phase is not resolved, and the claimed equality with Eq. (3.3) depends on it. Expanding Eq. (2.11) in the stated regime (m_a << E, μ_aB << E) gives E_{aζ} ≃ E − ζμ_aB(S_a/E) − m_a²/(2E) + (μ_aB)²/(2E). The last term is even in ζ, so it does not cancel in the ζ-sum; each mass-eigenstate amplitude would carry a phase exp[i(μ_aB)²L/(2E)], and since μ₁ ≠ μ₂ the interference term of |M_{e→μ}|² would acquire an extra phase of order (μ₂² − μ₁²)B²L/(2E). That phase is linear in L, exactly like the oscillation phase Δm²L/(2E), so it is not small by virtue of the stated assumptions alone. Equation (2.24) instead carries only exp(−im_a²L/(2E)) next to cos(μ_aBL), so the term appears to have been dropped between Eqs. (2.23) and (2.24); the only text supporting this is the summary line before Eq. (2.24) and 'The remaining integrals in Eq. (2.15) can be calculated similarly.' Please either show explicitly that the ζ-even piece cancels when I_{2+} and I_{2−} are added, or state the condition under which it is negligible, e.g. |μ₂² − μ₁²|B² << Δm² (equivalently (μ₂² − μ₁²)B²L/(2E) << 1). The notation compounds the difficulty: Eqs. (2.19) and (2.21) print 'μ²B' where an energy is expected (in ρ̃₀ ≃ E + μ²B − m₂²/(2E) and in the prefactor (1/L)(μ²B/E) f₁(y)), and the reader cannot determine whether this denotes μ₂B, a mistyped (μ₂B)²/(2E), or a misprint. Clarifying this notation is necessary before the drop in Eq. (2.24) can be judged.
[§II, Eq. (2.8); Ref. [15]] Every magnetic factor in the headline result descends from the propagator of Eq. (2.8), whose construction, normalization and conditions of validity are not reproduced here, and which is cited to Ref. [15], a manuscript listed as 'to be published in Moscow Univ. Phys. Bull. (2026)'. The underlying exact solution is the published Ref. [16], so this is not a demand for a new derivation; it is a request that the present manuscript be verifiable on its own terms. Concretely: (i) state the conditions under which Eq. (2.8) holds (constant field filling all space, Dirac representation, any restriction on the momentum range or on the treatment of the field at the source and detector), and state whether the propagator neglects radiative corrections and the anomalous magnetic moment; (ii) supply one internal check of Eq. (2.8) that a reader can perform from this paper alone — the natural one is that the B → 0 limit of Eq. (2.8) reproduces the free Dirac propagator i(p̸ + m)/(p² − m² + i0), which would simultaneously fix the overall 1/8 prefactor and the A_±, Δ, z_±, f_± normalizations that feed Eq. (2.16). Without such a check the reader cannot distinguish a correct reduction of Eq. (2.8) from a self-consistent but differently normalized one, and the claimed agreement with Eq. (3.3) would then be a statement about that unverified normalization.
[§II Eq. (2.25) vs §III Eq. (3.3); sentences [b93] and [b112]] The sense in which the two expressions 'coincide' needs to be made precise. Equation (2.25) is derived from P ∝ |M_{e→μ}|² with an overall factor E²/(4π²L²) (visible in the prefactor of Eq. (2.24)) and the spinor normalization of Eq. (2.12) dropped, whereas Eq. (3.3) is a normalized probability bounded by sin²2θ. What actually agrees is the bracket in the two formulas: the L- and E-dependent interference structure, not the full probability. As written, [b93] and [b112] ('coincides with the QFT result in Eq. (2.25)', 'coincide for ultrarelativistic particles') overstate this, and this is also stated in the Conclusion as 'the leading contribution ∝ L⁻²' in the same paragraph, which sits uneasily with a probability that is O(1) in the QM treatment. Please state the normalization convention explicitly (including that u†u = 1 rather than 2E in Eq. (2.12)) and give the proportionality factor relating Eq. (2.24) to a physical rate, so that the L⁻² prefactor is unambiguously separated from the interference bracket. This matters for the program announced at the end of the Conclusion: sub-leading QFT corrections can only be compared with the QM probability once the normalization is tracked.
[§III, Eqs. (3.1)–(3.3), and Eq. (2.11)] The manuscript should state what the QFT–QM comparison actually tests. The QM Hamiltonian of Eq. (3.1) is, at the order retained, the same physics as Eq. (2.11): the diagonal entries ±μ_aB of the two 2x2 blocks are the leading term in the expansion of E_{aζ} in μ_aB/E, which is why both sides necessarily produce cos(μ_aBL). The comparison therefore validates the QFT reduction — the e^{iqL} phase, the residue at the particle pole, the Bessel asymptotics, the neglect of the antiparticle pole — rather than the spin-precession dynamics, which is common input. That is a useful but narrower statement than the abstract and Conclusion suggest, and it would strengthen the paper to say so and to identify what the field-theoretic treatment supplies that the QM treatment cannot. A related observation that would help the reader calibrate the size of the effect: since the ν_eL → ν_μL channel requires two L–R transitions, the magnetic correction to Eq. (2.25) is O((μ_aBL)²) relative to the magnetic-free oscillation term, and vanishes identically in the degenerate case μ₁ = μ₂ (where the bracket reduces to 2(1 − cos(Δm²L/2E))cos²(μBL)). It would be worth stating this explicitly, and connecting it to the 'next to leading QFT corrections' discussed at the end of the Conclusion.

Minor objections (7)

[Eq. (2.17)] The two lines of Eq. (2.17) have identical left-hand sides but different right-hand sides (J₀ and iJ₁); the second integrand must contain an additional angular factor such as cosφ. As printed, the two lines contradict each other, and the factor J₁ is used immediately in Eq. (2.18) through the ρcosφ term, so this should be corrected.
[Eq. (2.15)] The last term is printed as I_{1,ζ=1}; from the context (the four combinations a = 1,2 and ζ = ±) it should presumably read I_{1,ζ=−1}.
[§II, text between Eqs. (2.16) and (2.18)] 'The rest rest of the integrals' contains a duplicated word, and the sentence 'The remaining integrals in Eq. (2.15) can be calculated similarly' would read better if at least the leading-order results for I_{1±} were quoted, since these are precisely what enters Eq. (2.24).
[Eqs. (2.23) and (2.24)] The overall phase of Eq. (2.23) is exp[+i(E + μ₂B − m₂²/(2E))L] while the prefactor of Eq. (2.24) is exp(−iEL); please check the relative sign conventions of the two expressions. The overall phase does not affect |M|², but as written the two equations appear inconsistent.
[§II, Eq. (2.12) and the justification citing Ref. [17]] The neglect of the antiparticle pole is justified by citation to Ref. [17], which is appropriate; a single sentence quoting the resulting parametric accuracy in the kinematics of this paper (m_a << E, μ_aB << E) would make the error bar on Eq. (2.25) explicit for the reader.
[Eq. (2.10)] The subscript 3 in the operator Π, printed outside the parenthesis as −iγ⁰γ⁵(Σ × p)_3 − μ_aB, is ambiguous; please state that it denotes the component along the magnetic field direction, i.e. along z, and that the eigenvalue equation is written for B = B e_z.
[Fig. 1 caption and Conclusion] Typographical: the caption reads 'theS-matrix element' and the Conclusion contains 'spin-flavor pressesion'; the Conclusion also reads 'However, is it the subject of a separate study', which should presumably be 'it is the subject'. The S-matrix in Eq. (2.3) is printed as S = −(1/2)(√2 G_int)² …, and the powers of the coupling relative to Eq. (2.5) are worth double-checking.

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The paper

The paper claims

This paper claims that the standard quantum-mechanical probability for neutrino spin-flavor precession is recovered exactly from quantum field theory in the ultrarelativistic limit, with the field-theoretic amplitude ready for…

The reading

The paper aims to show that the familiar two-level quantum-mechanical description of spin-flavor precession for Dirac neutrinos with diagonal magnetic moments is not an approximation in need of separate justification: it is the leading-order limit of a genuine quantum field theory calculation. Neutrino mass eigenstates are treated as virtual particles, and the whole magnetic-field interaction is put exactly into their propagators. Integrating the exact propagator against the mixing matrix yields a matrix element for $\nu_{eL}\to\nu_{\mu L}$, and in the ultrarelativistic limit its square reproduces the standard QM probability term by term once the distance is identified with the flight time and the neutrino energy with the mean charged-lepton energy. If the derivation is correct, the widely used QM formula is grounded in the field theory, and the exact amplitude in Eq. (2.24) becomes the concrete starting point for computing the sub-leading QFT corrections that the QM treatment cannot capture.

What carries the argument

The load-bearing object is the exact $4\times4$ propagator of a massive Dirac neutrino in a uniform magnetic field, Eq. (2.8), taken from the paper's Ref. [15]. Its entries are built from the field-shifted energy levels $E_{a\zeta}=\sqrt{(S_a-\zeta\mu_a B)^2+p_3^2}$ and from the solution of the Dirac equation in the field, and because the magnetic-moment matrix is diagonal in the mass basis the propagator stays diagonal in mass eigenstates. The full $\nu_{eL}\to\nu_{\mu L}$ amplitude is then the mixing-weighted Fourier transform of that propagator, Eqs. (2.6) and (2.13)--(2.16); the calculation drops the antiparticle pole as justified by Ref. [17], evaluates the azimuthal integrals with Bessel functions, and keeps only the leading $L^{-2}$ term at large source–detector distance. What survives is the compact amplitude of Eq. (2.24), whose phase carries the mass splittings while the real $\cos(\mu_a B L)$ factors carry the magnetic-field dependence inherited from the field-shifted energy levels, and this is the exact structural counterpart of the magnetic sub-block of the effective QM Hamiltonian in Eq. (3.1).

Extended reading notes

Core claim

The central claim is that the QFT transition probability for $\nu_{eL}\to\nu_{\mu L}$ in a transverse magnetic field, computed from the exact propagator of a massive Dirac neutrino with a diagonal magnetic moment, is $$P(L)=\frac{\$sin^{2}$ 2\$\theta$}{4}\left[\$cos^{2}$(\mu_1 B L)+\$cos^{2}$(\mu_2 B L)-2\cos(\mu_1 B L)\cos(\mu_2 B L)\cos\left(\frac{\$\Delta$ $m^{2}$}{2E_\nu}L\right)\right],$$ Eq. (2.25), and that this coincides exactly with the quantum-mechanical probability of Eq. (3.3) obtained from the four-level Schrödinger equation, provided one sets $L\approx t$ and identifies $E_\nu$ with the mean energy $E=(E_e+E_\mu)/2$ of the charged leptons. The magnetic field enters only through the factors $\cos(\mu_a B L)$, which descend from the field-shifted energy levels $E_{a\zeta}=\sqrt{(S_a-\zeta\mu_a B)^2+p_3^2}$ on which the exact propagator of Eq. (2.8) is built. The author presents this coincidence as establishing that the standard QM treatment is the correct leading ultrarelativistic limit of the field theory, with the matrix element of Eq. (2.24) available as the starting point for the next-to-leading QFT corrections that become sizable at short macroscopic distances.

Load-bearing premise

Everything magnetic in the final probability descends from the exact propagator of Eq. (2.8), which the paper takes from an unpublished manuscript by its own author, and the calculation further drops antiparticle contributions on the strength of another short self-cited work; if that propagator is mis-normalized or holds only under conditions the paper does not state, the claimed identity between the QFT probability (2.25) and the QM probability (3.3) collapses.

Editorial extensions

If this is right

The standard quantum-mechanical probability for spin-flavor precession is confirmed as the leading ultrarelativistic limit of the field theory, closing the gap between the two descriptions at this order.
The matrix element of Eq. (2.24) is a ready-made starting point for the next-to-leading QFT corrections, which the author states are sizable at short macroscopic distances and analogous to terms that have been used to discuss the reactor antineutrino anomaly.
The dictionary between the two descriptions is fixed: the source–detector distance replaces the flight time ($L\approx t$), and the neutrino energy is the mean charged-lepton energy $(E_e+E_\mu)/2$, not an independent beam parameter.
In the equal-moment limit $\mu_1=\mu_2$, Eq. (2.25) reduces to $\sin^2 2\theta\,\cos^2(\mu B L)\,\sin^2(\Delta m^2 L/4E_\nu)$, so a common diagonal moment suppresses the vacuum oscillation uniformly instead of distorting its frequency.

Reading between the lines

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

Extending the same propagator machinery to the next order in $1/(EL)$ and $\mu_aB/E$ would turn the claimed leading-order identity into a quantitative prediction of where the QM treatment breaks down; computing that term and comparing with short-baseline data would be a natural test of the approach.
Generalizing the propagator from diagonal to a full $2\times2$ magnetic-moment matrix would bring transition-moment channels into the same QFT treatment and connect directly to the Majorana-neutrino case, where diagonal moments vanish by construction.
The $\cos(\mu_a B L)$ modulation is a distinctive field-strength-dependent signature: it grows with $BL$, so strong-field environments such as supernova shock regions or neutron-star magnetospheres would be the natural places to look for it, since standard-model diagonal neutrino moments are extremely small.

Original abstract

We apply the quantum field theory (QFT) approach to describe the spin-flavor precession of Dirac neutrinos. The magnetic moments matrix of these neutrinos is taken to be diagonal in the mass basis. Neutrino mass eigenstates are considered to be virtual particles in the QFT approach. The neutrino contribution to the matrix element of the overall process, which includes the emission, the propagation and the detection of neutrinos, is contained in the propagators of massive neutrinos. Using the recently derived propagator of a massive neutrino, which exactly accounts for the magnetic moment interaction, we derive the transition probability of the process $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{L}}$ under the influence of a magnetic field. The result is compared with the prediction of the standard quantum mechanical (QM) approach. We briefly discuss the possibility to derive the QFT corrections to the QM expressions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-05: 1 request, 5,565 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.34 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We apply the quantum field theory (QFT) approach to describe the spin-flavor precession of Dirac neutrinos. The magnetic moments matrix of these neutrinos is taken to be diagonal in the mass basis. Neutrino mass eigenstates are considered to be virtual particles in the QFT approach. The neutrino contribution to the matrix element of the overall process, which includes the emission, the propagation and the detection of neutrinos, is contained in the propagators of massive neutrinos. Using the recently derived propagator of a massive neutrino, which exactly accounts for the magnetic moment interaction, we derive the transition probability of the process $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{L}}$ under the influence of a magnetic field. The result is compared with the prediction of the standard quantum mechanical (QM) approach. We briefly discuss the possibility to derive the QFT corrections to the QM expressions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We apply the quantum field theory (QFT) approach to describe the spin-flavor precession of Dirac neutrinos.	contribution 0.66	0.94	0.05	asserted 0.98
s1	The magnetic moments matrix of these neutrinos is taken to be diagonal in the mass basis.	method 0.92	0.47	0.03	asserted 0.99
s2	Neutrino mass eigenstates are considered to be virtual particles in the QFT approach.	method 0.96	0.43	0.03	asserted 1.00
s3	The neutrino contribution to the matrix element of the overall process, which includes the emission, the propagation and the detection of neutrinos, is contained in the propagators of massive neutrinos.	method 0.91	0.38	0.07	asserted 1.00
s4	Using the recently derived propagator of a massive neutrino, which exactly accounts for the magnetic moment interaction, we derive the transition probability of the process $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{L}}$ under the influence of a magnetic field.	contribution 0.50	0.95	0.36	asserted 1.00
s5	The result is compared with the prediction of the standard quantum mechanical (QM) approach.	method 0.48	0.81	0.06	asserted 0.81
s6	We briefly discuss the possibility to derive the QFT corrections to the QM expressions.	future work 0.86	0.38	0.04	hedged possibility 0.95

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. The Feynman diagram corresponding to the

Figures reproduced from arXiv:2611.06643 by the authors.

References

Works cited (18)

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V. Egorov and I. Volobuev, J. Exp. Theor. Phys.135, 197–208 (2022) [arxiv:2107.11570]

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M. Dvornikov, to be published in Moscow Univ. Phys. Bull. (2026) [arxiv:2512.10587]

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[16]

I. M. Ternov, V. G. Bagrov, and A. M. Khapaev, Sov. Phys. JETP21, 613–616 (1965)

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M. Dvornikov, Phys. Part. Nucl.57, 922–925 (2026) [arxiv:2512.08361]

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[12]

M. Dvornikov and J. Maalampi, Phys. Lett. B657, 217–227 (2007) [hep-ph/0701209]

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[1]

C.-S. Lim and W. J. Marciano, Phys. Rev. D37, 1368–1373 (1988)

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W. Grimus and P. Stockinger, Phys. Rev. D54, 3414–3419 (1996) [hep-ph/9603430]

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[9]

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M. Dvornikov, Annals Phys.478, 170361 (2026) [arxiv:2510.23760]

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[13]

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View on Pith Pith review arXiv 2019
[18]

V. A. Naumov and D. S. Shkirmanov, Universe7, 246 (2021)

View on Pith 2021

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [14]

V. Egorov and I. Volobuev, J. Exp. Theor. Phys.135, 197–208 (2022) [arxiv:2107.11570]

View on Pith Pith review arXiv 2022

[2] [15]

M. Dvornikov, to be published in Moscow Univ. Phys. Bull. (2026) [arxiv:2512.10587]

View on Pith arXiv 2026

[3] [16]

I. M. Ternov, V. G. Bagrov, and A. M. Khapaev, Sov. Phys. JETP21, 613–616 (1965)

View on Pith 1965

[4] [17]

M. Dvornikov, Phys. Part. Nucl.57, 922–925 (2026) [arxiv:2512.08361]

View on Pith Pith review arXiv 2026

[5] [12]

M. Dvornikov and J. Maalampi, Phys. Lett. B657, 217–227 (2007) [hep-ph/0701209]

View on Pith Pith review arXiv 2007

[6] [1]

C.-S. Lim and W. J. Marciano, Phys. Rev. D37, 1368–1373 (1988)

View on Pith 1988

[7] [2]

E. Kh. Akhmedov, Phys. Lett. B213, 64–68 (1988)

View on Pith 1988

[8] [3]

M. Fukugita and T. Yanagida,Physics of Neutrinos and Applications to Astrophysics (Springer, Berlin, 2003), p. 462

View on Pith 2003

[9] [4]

C. Giuntiet al., Ann. Rev. Nucl. Part. Sci.75, 1–33 (2025) [arXiv:2411.03122]

View on Pith Pith review arXiv 2025

[10] [5]

D. V. Naumov and V. A. Naumov, Phys. Part. Nucl.51, 1–106 (2020)

View on Pith 2020

[11] [6]

I. Yu. Kobzarevet al., Sov. J. Nucl. Phys.35, 708–712 (1982)

View on Pith 1982

[12] [7]

C. Giunti, C. W. Kim, and J. A. Lee, Phys. Rev. D48, 4310–4317 (1993) [hep-ph/9305276]

View on Pith Pith review arXiv 1993

[13] [8]

W. Grimus and P. Stockinger, Phys. Rev. D54, 3414–3419 (1996) [hep-ph/9603430]

View on Pith Pith review arXiv 1996

[14] [9]

M. Dvornikov, Phys. Rev. D111, 056009 (2025) [arxiv:2411.19120]

View on Pith Pith review arXiv 2025

[15] [10]

M. Dvornikov, Nucl. Phys. B1025, 117419 (2026) [arxiv:2504.14726]

View on Pith arXiv 2026

[16] [11]

M. Dvornikov, Annals Phys.478, 170361 (2026) [arxiv:2510.23760]

View on Pith arXiv 2026

[17] [13]

A. Popov and A. Studenikin, Eur. Phys. J. C79, 144 (2019) [arXiv:1902.08195]

View on Pith Pith review arXiv 2019

[18] [18]

V. A. Naumov and D. S. Shkirmanov, Universe7, 246 (2021)

View on Pith 2021