Constraints on the neutrino extension of the Standard Model and baryon asymmetry of the Universe

Mariia Tsarenkova; Oleg Ruchayskiy; Oleksandr Khasai; Volodymyr Gorkavenko

REVIEW 4 major objections 5 minor 26 references

Constraints on the neutrino extension of the Standard Model and baryon asymmetry of the Universe

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read In a two-heavy-neutrino seesaw with unequal masses, the flavour-violating couplings obey $|S_{\alpha\beta}|^2 = S_{\alpha\alpha} S_{\beta\beta}$ for large complex mixing angle, tightening $\tau$–$e$ and $\tau$–$\mu$ bounds tenfold and…

desk verdict The saturation equalities are a real generalization, but the claimed numerical improvement is just positivity and the BAU link is a black box. read the letter →

arxiv 2502.07988 v1 pith:LDTZV7F6 submitted 2025-02-11 hep-ph hep-ex

Volodymyr Gorkavenko , Oleksandr Khasai , Oleg Ruchayskiy , Mariia Tsarenkova This is my paper · ORCID

classification hep-phhep-ex

keywords heavyneutralleptonschargedleptonflavourviolationseesawmechanismCasas-IbarraparameterizationneutrinoMinimalStandardModelbaryonasymmetryleptogenesiscLFVoperators

topics Matter-Antimatter Asymmetry

open problems Matter-Antimatter Asymmetry

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

The paper asks how tightly experiments can bound the flavour-violating couplings that heavy sterile neutrinos induce, and what those couplings imply for the origin of matter. It claims that in a seesaw extension with two heavy neutral leptons of different masses and massive active neutrinos, the effective couplings $S_{\alpha\beta}$ and $R_{\alpha\beta}$ obey $|S_{\alpha\beta}|^2 = S_{\alpha\alpha} S_{\beta\beta}$ (and likewise for $R$) whenever the Casas-Ibarra mixing angle has a large imaginary part. That saturation — previously known only for massless active neutrinos or degenerate heavy masses — strengthens the experimental upper limits on $|\hat{S}_{e\tau}|$ and $|\hat{S}_{\mu\tau}|$ to $0.58\times10^{-3}$. The same operator language turns the $\nu$MSM baryon asymmetry into a bound on the $S$ and $R$ entries, and shows that the saturation regime makes the asymmetry vanish. The result matters because it tells experimenters where to look and exposes a tension between the parameter region that gives the stronger collider bounds and the region that can generate the observed matter–antimatter asymmetry.

What carries the argument

The load-bearing object is the Casas-Ibarra parameterization of the Yukawa matrix, packaged here as $X = (i/v)\,U_\nu\sqrt{m^{\rm diag}_\nu}$. Under the large-$\mathrm{Im}\,\omega$ assumption (4), the sums over the two heavy neutrinos in (2)--(3) collapse into a single rank-one factor $(X_{\alpha 2}-iX_{\alpha 3})(X^*_{\beta 2}+iX^*_{\beta 3})$ times a mass-dependent coefficient. This factorization is what turns the Schwarz inequality into the saturation equality for both $S_{\alpha\beta}$ and $R_{\alpha\beta}$, and it is also why $\mathrm{Im}[S^*_{\alpha\beta}R_{\alpha\beta}]$, the combination controlling baryon asymmetry, vanishes identically in the same limit.

What would settle it

Evaluate the exact ratio $|S_{\alpha\beta}|^2/(S_{\alpha\alpha}S_{\beta\beta})$ from equations (5)--(6) without imposing (4), scanning over $\mathrm{Im}\,\omega$ and the heavy-mass splitting $\Delta M/M$ with realistic active-neutrino masses: if the ratio stays well below 1 for the values of $\mathrm{Im}\,\omega$ that still allow the observed baryon asymmetry, the improved bounds (9) do not hold in the baryogenesis-viable region, and a future measurement of $\tau\to\mu\gamma$ finding $|\hat{S}_{\mu\tau}| > 0.58\times10^{-3}$ would directly contradict the saturation bound.

Watch

Extended reading notes

Core claim

The central claim is that the seesaw observables $S_{\alpha\beta}$ and $R_{\alpha\beta}$ — the effective couplings generating charged lepton flavour violation — saturate the Schwarz inequality, $$|S_{\$\alpha$\$\beta$}|^2 = S_{\$\alpha$\$\alpha$} S_{\$\beta$\$\beta$}, \qquad |R_{\$\alpha$\$\beta$}|^2 = R_{\$\alpha$\$\alpha$} R_{\$\beta$\$\beta$},$$ not only in the previously studied limits (massless active neutrinos and degenerate heavy masses) but also when the active neutrinos are massive and the two heavy neutral leptons have different masses, provided the complex Casas-Ibarra angle $\omega$ obeys (4), i.e. $\cosh 2\,\mathrm{Im}\,\omega \simeq \sinh 2\,\mathrm{Im}\,\omega \simeq e^{2\,\mathrm{Im}\,\omega}/2 \gg 1$. From these equalities the paper derives the improved bounds $|\hat{S}_{e\tau}| \le 0.58\times10^{-3}$ and $|\hat{S}_{\mu\tau}| \le 0.58\times10^{-3}$. Re-expressing the $\nu$MSM baryon asymmetry in the same language yields Eq. (10), and shows that under (4) the asymmetry, being proportional to $\mathrm{Im}[S^*_{\alpha\beta}R_{\alpha\beta}]$, vanishes, so generating the observed asymmetry requires leaving the saturation regime.

Load-bearing premise

The load-bearing premise is that the imaginary part of the complex neutrino mixing angle is very large, so that $\cosh 2\,\mathrm{Im}\,\omega$, $\sinh 2\,\mathrm{Im}\,\omega$ and $e^{2\,\mathrm{Im}\,\omega}/2$ are all interchangeable and much larger than one; if that limit is not satisfied, the saturation equalities break, and the parameter region that yields a non-zero baryon asymmetry is exactly where it breaks.

Editorial extensions

If this is right

The current upper limits on the tau–electron and tau–muon seesaw couplings tighten from the $10^{-3}$–$10^{-2}$ level in Table 1 to $|\hat{S}_{e\tau}|, |\hat{S}_{\mu\tau}| \le 0.58\times10^{-3}$.
A future observation of charged lepton flavour violation in any one channel would, within this model, fix the values in the other channels through the saturation equalities.
The baryon asymmetry formula (10) expresses a cosmological observable in terms of the same $S$ and $R$ operators that control lepton flavour violation, making collider and cosmology constraints comparable.
In the large-$\mathrm{Im}\,\omega$ regime that produces the improved bounds, the baryon asymmetry vanishes, so the parameter region able to explain the matter–antimatter asymmetry is disjoint from the region with the strongest collider constraints.
Because the baryogenesis lower bound and the experimental upper bounds differ by many orders of magnitude, successful $\nu$MSM baryogenesis implies the actual $S$ and $R$ entries lie far below the limits in Table 1.

Reading between the lines

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

If the same large-parameter limit is realized with more than two heavy neutrinos, an analogous rank-one factorization should appear, so the saturation relations and the resulting bound improvement are likely not special to the two-flavour case.
Since the saturation regime forces $\mathrm{Im}[S^*_{\alpha\beta}R_{\alpha\beta}]$ to vanish, the goals of maximal cLFV sensitivity and successful baryogenesis pull toward opposite corners of parameter space; a positive cLFV signal near the improved bounds would count against the $\nu$MSM leptogenesis explanation of the baryon asymmetry.
A numerical scan of $|S_{\alpha\beta}|^2/(S_{\alpha\alpha}S_{\beta\beta})$ as a function of $\mathrm{Im}\,\omega$ and $\Delta M/M$, with finite active-neutrino masses, would turn the qualitative condition (4) into a quantitative statement of how large the large-parameter limit must be.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper has a genuinely new algebraic observation—the S-R linear relation (7) and the saturation equalities (8) extended to non-degenerate HNL masses with massive active neutrinos. But the advertised payoff, the improved bounds (9), does not actually test that result: it follows from trivial positivity of S and the diagonal limits already in Table 1. The BAU part is a black box imported from [11] and, as the paper admits, is not even valid under the same assumption (4) that makes (8) hold.

The good parts: extending the equality |S_αβ|^2 = S_αα S_ββ beyond the degenerate-mass/massless-active-neutrino cases is a useful contribution. If (4) holds, (7)-(8) give a compact description of the cLFV parameter space of the minimal seesaw. The paper is also honest in noting that the baryon asymmetry bound has limited practical value (right side >> 1) and that under (4) the asymmetry vanishes; the nonzero asymmetry requires going beyond the approximation. That is a welcome admission.

The soft spots are real. First, the derivation of (5)-(8) is not shown despite the 'effortlessly see' comment, and assumption (4) is left unquantified. Second, the central numerical claim (9) is an overreach: with Ŝ_ee + Ŝ_μμ ≤ 0.53·10^-3 and Ŝ_ττ ≤ 0.64·10^-3, positivity alone gives |Ŝ_eτ|^2 ≤ (Ŝ_ee + Ŝ_μμ) Ŝ_ττ, so |Ŝ_eτ| ≤ 0.58·10^-3; no saturation equalities needed. Same for μτ. Thus (9) is a corollary of the table, not a test of (8). That should be stated clearly. Third, the baryon asymmetry expression (10) uses symbols (sin φ, M0, TW?) that are never defined and is simply quoted from [11]. Since the whole BAU link is built on it, the paper needs at least a derivation or a precise reference with definitions.

Fourth, and maybe most interesting: the regime where (8) saturates (Im ω large) is exactly where baryon asymmetry vanishes, whereas the BAU formula requires the deviation from (4). The paper notes this but does not discuss what it means for combining the two constraints. That is a real physics point a referee should push on.

Verdict: worth sending to peer review. The algebraic generalization is plausible and deserves a careful check. But the referee should insist on a step-by-step derivation, quantification of the approximation, and a rewriting that separates the positivity bound from the saturation bound.

Referee Report

4 major / 5 minor

Summary. This paper studies the νMSM with two heavy neutral leptons and derives approximate algebraic forms for the effective cLFV matrices Sαβ and Rαβ under the large-Imω condition (4). The authors obtain the proportionality relation (7), the saturation equalities (8), and then claim improved off-diagonal bounds (9). In the second part, they import a baryon asymmetry expression from Ref. [11], translate it into an upper bound (10) on nB/s in terms of Ŝ entries, evaluate it with Table 1, and conclude that baryogenesis and accelerator constraints differ by many orders of magnitude. The paper contains no free parameters; the inputs are external experimental limits and standard seesaw relations.

Significance. The algebraic observation that non-degenerate heavy neutrino masses still produce a rank-one form for S and R in the large-Imω limit is potentially useful and would extend earlier degenerate/massless results. The derivation is not shown, however, and the advertised numerical improvement (9) is not actually powered by the new relations. The baryon-asymmetry part does not yield the claimed restrictions and contains internal logical inconsistencies. With the missing derivation supplied and the overclaims corrected, the paper could be a modest but acceptable contribution; in its present form its significance is limited.

major comments (4)

[§2, Eq. (9) and Table 1] The bounds |Ŝeτ| ≤ 0.58×10^{-3} and |Ŝμτ| ≤ 0.58×10^{-3} do not test or use the new saturation equalities (8). They follow directly from the diagonal bounds already quoted in Table 1 together with positive semidefiniteness of Ŝ: |Ŝeτ|² ≤ ŜeeŜττ ≤ (Ŝee+Ŝμμ)Ŝττ ≤ 0.53×10^{-3} × 0.64×10^{-3}, and likewise |Ŝμτ|² ≤ ŜμμŜττ ≤ (Ŝee+Ŝμμ)Ŝττ. Thus Eq. (9) is a corollary of the input bounds and of the ordinary Cauchy–Schwarz inequality, not of the rank-one limit (4). The paper should either demonstrate a quantitative consequence of Eq. (8) that goes beyond positivity or explicitly present Eq. (9) as the standard positivity bound and reframe the novelty claim.
[§2, Eqs. (5)–(8)] The central derivation is not shown. The paper does not give the explicit Casas-Ibarra R matrix used, nor the asymptotic expansion that leads to the factorized forms (5)–(6), nor an estimate of the error made when assumption (4) is imposed. Equation (8) is an exact equality only in the limit e^{2 Imω} → ∞; for finite Imω the ratio |Sαβ|²/(SααSββ) differs from unity by terms of relative order e^{-2 Imω}, with coefficients that depend on the other parameters and on the mass ratio. Since the experimental region 'above the seesaw line' may not satisfy (4) with the required accuracy, the practical validity of Eq. (8) is not established. A short appendix giving the R-matrix parameterization, the expansion, and a numerical example of the deviation would be necessary to support the claim that Eq. (8) 'holds true with sufficient accuracy'.
[§3, Eq. (10)] Equation (10) is quoted from Ref. [11] with no derivation and with undefined symbols (φ, M0, TW, and ΔM21 in Eqs. (11)–(12)). The accompanying statement that the baryon asymmetry is proportional to Im[S∗αβRαβ] needs clarification: since S and R defined in Eqs. (2)–(3) are Hermitian matrices, the sum over α≠β of Im[S∗αβRαβ] vanishes identically; if the relevant CP-violating combination is a fixed off-diagonal term or a different contraction, this must be stated and derived. The text also says that Eq. (10) is obtained 'without assumption (4)', although the preceding derivation of Eq. (7), which is the only visible link between S and R, relies on (4); this tension needs to be resolved.
[§3 and Conclusions] The estimates (11)–(12) are upper bounds on nB/s whose right-hand sides are much larger than the observed value for M ≳ 1 GeV and M/ΔM21 ≫ 1. An upper bound that is far above the observed asymmetry imposes no restriction on Ŝ or R̂, and it does not imply that 'the actual values of the observed elements of the Ŝ and R matrices are much lower than the experimental limits'. The Conclusions similarly refer to 'lower limits (baryon asymmetry)', but no lower limit on the seesaw parameters is derived in the paper. This part should be rewritten to state only what the inequality actually shows, and the abstract's claim of 'new restrictions' should be withdrawn or replaced by a correct statement.

minor comments (5)

[§2, Eq. (4)] The approximation in Eq. (4) implicitly requires Im ω > 0; for Im ω < 0, sinh 2 Im ω is negative and the stated chain of approximations fails. Please state the sign condition explicitly.
[Table 1] The 'Future experiments' column is empty for Ŝee+Ŝμμ and Ŝττ; please clarify whether no future projections exist or whether the entries are simply omitted.
[§3, Eqs. (11)–(12)] The notation '(M/1GeV)' should be typeset as '(M/GeV)' or 'M/(1 GeV)', and 'ΔM21' should be defined explicitly, presumably as M2−M1.
[§2, before Eq. (5)] The phrase 'as one can effortlessly see' is informal; the reality and positivity of the diagonal elements follow from the displayed factorized form and should simply be stated.
[Conclusions] The statement that Eq. (8) is 'independent of the mass difference between sterile neutrinos' should be qualified as holding only to leading order in the asymptotic expansion (4), not as an exact statement for arbitrary heavy neutrino masses.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations are algebraic consequences of stated assumptions and external inputs, with no fitted parameters renamed as predictions.

full rationale

The paper's central derivation (Eqs. (5)-(8)) is self-contained: starting from the Casas-Ibarra parametrization and the stated large-Im-omega assumption (4), it expresses S_alpha_beta and R_alpha_beta as rank-one products, from which the saturation equalities (8) follow algebraically. The numerical bounds (9) use experimental upper limits from Table 1 (cited to Ref. [25]) and the algebraic structure of the seesaw parameters; even though the skeptic correctly notes that |S_e_tau| <= 0.58e-3 and |S_mu_tau| <= 0.58e-3 also follow from positivity (Cauchy-Schwarz) applied to the tabulated diagonal bounds, this observation concerns the novelty or necessity of Eq. (8), not circularity. No parameter is fitted to data and then predicted; no result is defined in terms of the target conclusion; and the baryon asymmetry formula (10) is imported from Ref. [11] as external support, not from the present authors' prior work. The paper itself flags limitations: the practical value of the baryon bounds is explicitly called limited (Section 3), and the accuracy of assumption (4) is not quantified, but a missing quantification is a correctness or precision concern, not a circularity. No self-citation is load-bearing, and no uniqueness theorem is invoked to forbid alternatives. The finding is therefore no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the Casas-Ibarra parametrization for a two-HNL seesaw, the large-Im-omega approximation (4), the standard seesaw limit, and the assumed validity of the leptogenesis expression (10) translated from Ref. [11]. No new entities or fitted parameters are introduced.

assumptions (4)

standard math Casas-Ibarra parameterization with a single complex angle omega for two heavy HNLs and one massless active neutrino
Used to write Yukawa couplings in terms of light neutrino data; standard for the minimal two-HNL seesaw with normal ordering (m1=0) or inverted ordering (m3=0).
domain assumption Assumption (4): cosh 2 Im omega is approximately sinh 2 Im omega is approximately e^{2 Im omega}/2, all much greater than 1
Central premise for Eqs. (5)-(8); defines the 'above the seesaw threshold' parameter region relevant for HNL collider searches.
domain assumption Active neutrino masses are much smaller than heavy neutrino masses
Standard seesaw limit invoked before Eq. (8) and stated as a necessary condition for the derived relations.
domain assumption The baryon asymmetry formula from Ref. [11] and its translation into S and R variables, Eq. (10)
The paper states Eq. (10) is based on Ref. [11] with no derivation; correctness of this translation is assumed, and it is valid for M1 approximately M2 approximately M.

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Cite this review

Pith. "Pith review of Constraints on the neutrino extension of the Standard Model and baryon asymmetry of the Universe." pith.science (2026). https://pith.science/paper/LDTZV7F6

@misc{pith2026250207988,
  author       = {Pith},
  title        = {Pith review of: Constraints on the neutrino extension of the Standard Model and baryon asymmetry of the Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LDTZV7F6}},
  note         = {Machine review of arXiv:2502.07988}
}

read the original abstract

Heavy neutral leptons (HNLs) can cause new effective interactions of Standard Model particles, particularly charged lepton flavour violation (cLFV) processes. The non-observation of cLFV processes, therefore, puts constraints on the parameters of the HNLs. We find the relations between the cLFV effective operators in the realistic case when active neutrino masses are non-zero and masses of the HNLs are non-degenerate. This allows us to strengthen the existing cLFV constraints. We also link the baryon asymmetry of the Universe to the same cLFV effective operators, which imposes new restrictions on their values.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [11]

Asaka, M

T. Asaka, M. Shaposhnikov. The νMSM, dark matter and baryon asymmetry of the universe. Phys. Lett. B 620, 17 (2005) [DOI: 10.1016/j.physletb.2005.06.020]

doi:10.1016/j.physletb.2005.06.020 2005

[2] [1]

Cottingham, D.A

W.N. Cottingham, D.A. Greenwood. An Introduction to the Stand ard Model of Particle Physics, (Cambridge University Press, 2023) [ISBN: 978-1 -00-940168-5, 978- 1-00-940172-2, 978-1-00-940170-8, 978-0-511-27136-6, 9 78-0-521-85249-4]

work page 2023

[3] [2]

Bilenky, S.T

S.M. Bilenky, S.T. Petcov. Massive Neutrinos and Neutrino Oscillatio ns. Rev. Mod. Phys. 59, 671 (1987) [DOI: 10.1103/RevModPhys.59.671] [Erratum: Rev.Mod .Phys. 61, 169 (1989), Erratum: Rev.Mod.Phys. 60, 575–575 (1988)]

doi:10.1103/revmodphys.59.671 1987

[4] [3]

Strumia, F

A. Strumia, F. Vissani. Neutrino masses and mixings and... arXiv:hep -ph/0606054 (2006)

work page arXiv 2006

[5] [4]

de Salas, D.V

P.F. de Salas, D.V. Forero, C.A. Ternes, M. Tortola, J.W.F. Valle. St atus of neutrino oscillations 2018: 3 σ hint for normal mass ordering and improved CP sensitivity. Phys. Lett. B 782, 633 (2018) [DOI: 10.1016/j.physletb.2018.06.019]

work page doi:10.1016/j.physletb.2018.06.019 2018

[6] [5]

P.J.E. Peebles. Dark Matter. Proc. Nat. Acad. Sci. 112, 2246 (2015) [DOI: 10.1073/pnas.1308786111]

work page doi:10.1073/pnas.1308786111 2015

[7] [6]

Lukovic, P

V. Lukovic, P. Cabella, N. Vittorio. Dark matter in cosmology. Int. J. Mod. Phys. A 29, 1443001 (2014) [DOI: 10.1142/S0217751X14430015]

work page doi:10.1142/s0217751x14430015 2014

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Bertone, D

G. Bertone, D. Hooper. History of dark matter. Rev. Mod. Phys. 90, 045002 (2018) [DOI: 10.1103/RevModPhys.90.045002]

doi:10.1103/revmodphys.90.045002 2018