Analytic formulation of Leptogenesis with neutrino oscillation data employing the general parametrization for neutrino mass matrix

Digesh Raut; Nobuchika Okada

REVIEW 2 major objections 5 minor 34 references

Analytic formulation of Leptogenesis with neutrino oscillation data employing the general parametrization for neutrino mass matrix

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper derives analytic CP-asymmetry formulas showing that non-thermal leptogenesis in a Type-I seesaw with a neutrinophilic Higgs doublet can succeed with the lightest right-handed neutrino mass as low as the sphaleron decoupling…

desk verdict The analytic CP-asymmetry formulas are a solid, useful contribution, but the headline 132 GeV non-thermal bound rests on an unjustified κ=1 assumption that the benchmark parameters themselves contradict. read the letter →

arxiv 2506.20580 v1 pith:6PEIVWDB submitted 2025-06-25 hep-ph hep-th

Nobuchika Okada , Digesh Raut This is my paper · ORCID

classification hep-phhep-th

keywords leptogenesisType-Iseesawright-handedneutrinomassCPasymmetryparameterneutrinophilicHiggsdoubletnon-thermalbaryonoscillationdata

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 derives closed-form expressions for the CP asymmetry that generates the baryon asymmetry in leptogenesis, using the general complex-orthogonal parametrization of the Dirac Yukawa couplings allowed by the seesaw relation. With these formulas, the authors scan the free parameters and find the smallest allowed mass of the lightest right-handed neutrino, for two and three right-handed generations and for thermal versus non-thermal production. In the standard Type-I seesaw, the minimum stays in the range of about $10^{6}$ to $10^{13}$ GeV, broadly confirming earlier lower bounds. The main result is that in a Type-I seesaw with a neutrinophilic Higgs doublet, non-thermal leptogenesis can succeed with the lightest right-handed neutrino mass as low as the sphaleron decoupling temperature, about 132 GeV. That matters because such a particle would sit at an experimentally accessible scale rather than at a grand-unified scale.

What carries the argument

The central object is the complex orthogonal matrix O (with O^T O = 1) in the general parametrization m_D = \sqrt{M_N}\, O\, \sqrt{D_\nu}\, U^\dagger, where \sqrt{M_N} contains the right-handed neutrino masses, \sqrt{D_\nu} the light-neutrino masses, and U the observed neutrino mixing matrix. The paper's analytic formulas express the CP asymmetry \epsilon_1 and the decay width \Gamma_1 of the lightest right-handed neutrino directly in terms of O and the light masses; the key structural fact is that \epsilon_1 \propto m_{N_1}/$v_h^{2}$ while the washout parameter K = \Gamma_1/H is independent of m_{N_1}. This separation makes the minimum m_{N_1} computable by scanning only the parameters of O, and in the neutrinophilic Higgs model v_h is replaced by the small VEV v_2, which is what lowers the required mass.

What would settle it

A fully flavor-resolved Boltzmann calculation for m_{N_1} near 132 GeV in the neutrinophilic Higgs model would settle the claim: if the baryon asymmetry computed with flavor-dependent CP asymmetries and washout differs from the single-flavor result by an order-unity factor, the minimal mass must be revised accordingly. A second direct check would be a collider search for a right-handed neutrino near 132 GeV in same-sign dilepton final states, which the model's parameters put within reach if the bound is real.

Watch

Extended reading notes

Core claim

The paper establishes that, in the general parametrization of the Dirac mass matrix, the CP asymmetry parameter for the lightest right-handed neutrino can be written analytically in terms of the light-neutrino masses and the parameters of a complex orthogonal matrix, with no dependence on the measured neutrino mixing matrix. Explicit formulas are provided for both the minimal two-right-handed-neutrino case and the three-generation case, for normal and inverted light-neutrino hierarchies. The formulas separate cleanly: the washout parameter is independent of the right-handed neutrino mass, while the CP asymmetry scales linearly with that mass, so the observed baryon asymmetry directly fixes a minimum mass. In the neutrinophilic Higgs doublet model, replacing the Standard Model Higgs VEV by the smaller VEV v_2 lowers the non-thermal bound as $v_2^{2}$, pushing the minimum right-handed neutrino mass down to about 132 GeV, the temperature at which sphaleron processes switch off.

Load-bearing premise

The calculation assumes a single-flavor Boltzmann treatment of washout remains accurate down to right-handed neutrino masses near 132 GeV; at those temperatures all charged-lepton Yukawa interactions are in equilibrium, and flavor effects could change the effective efficiency enough to move the quoted bound.

Editorial extensions

If this is right

In the standard Type-I seesaw, thermal leptogenesis with three right-handed neutrinos requires the lightest mass to be at least about 6.7 × 10^8 GeV for normal hierarchy and 6.5 × 10^8 GeV for inverted hierarchy, while two generations require 6.24 × 10^10 GeV (normal) and 1.62 × 10^13 GeV (inverted).
Non-thermal production loosens these bounds: with three generations the minimum drops to about 3.3 × 10^6 GeV, and with two generations to 4.0 × 10^6 GeV (normal) or 2.2 × 10^8 GeV (inverted).
In the neutrinophilic Higgs doublet model, non-thermal leptogenesis works with the lightest right-handed neutrino at about 132 GeV for two or three generations and either mass ordering, because the mass bound scales as v_2^2.
Thermal leptogenesis in the neutrinophilic model also benefits, but less dramatically: the minimum is about 1.7 × 10^6 GeV (normal) and 5.1 × 10^5 GeV (inverted) for three generations at v_2 of a few GeV, and equals the standard-model values at v_2 = 246 GeV.
Because the analytic asymmetry formulas are independent of the neutrino mixing matrix, the resulting minimum masses depend only on the light-neutrino masses and the right-handed sector parameters, not on the measured angles and phases.

Reading between the lines

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

In the editor's reading, the scaling m_{N_1}^{min} \propto v_2^2 means the ultimate floor is set by how small the neutrinophilic VEV can be; updating the lepton-flavor-violation constraints that force v_2 ≳ 10 MeV would directly sharpen or shift the 132 GeV result.
The paper restricts to hierarchical right-handed masses; restoring the self-energy enhancement S_j would let the same parametrization assess the resonant regime, where lower masses are generally possible for the same CP asymmetry.
If the 132 GeV scale is correct, the model suggests a searchable signature: right-handed neutrinos produced through the neutrinophilic Higgs could appear in same-sign dilepton events at colliders, a calculation the paper does not perform.

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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

The real contribution here is the analytic machinery. The explicit closed-form expressions for ε1 in the Casas-Ibarra parametrization, for both two and three right-handed neutrinos, are cleanly derived and reduce correctly in the limits. The observation that ε1 is independent of a23, b23 in the 3-RHN case is nice, and the parameter scan becomes nearly trivial with these formulas. The thermal bounds (around 6.7×10^8 GeV for 3 RHNs) are consistent with prior literature and are likely robust. If the paper only made those points, it would be a modest but useful technical note.

The headline claim, however, does not survive contact with the washout rates. The non-thermal analysis assumes κ=1 because N1 is produced out of equilibrium, and the only consistency check is mN1 < TR. That is not sufficient. The benchmark parameters that saturate |ε1|min require large b (in the 2-RHN case, the contour minimum sits at b→∞), and for the neutrinophilic benchmark with v2≈1.4 GeV, the decay parameter K from Eq. (4.8) is enormous, not small. Inverse decays at T∼mN1 are then rapid, and the efficiency factor would be far below 1. The same problem afflicts the non-thermal bounds in the SM Higgs case. The condition TR < mN1 does not help for scenario II either, because the asymmetry is maximized exactly when TR ≈ mN1, so the decay still occurs at temperatures where inverse decay is unsuppressed. The quoted 132 GeV lower bound and the corresponding values in Table 2 are therefore not reliable as they stand.

Secondary issues: the single-flavor Boltzmann treatment near 100 GeV is questionable; at those temperatures flavor effects matter. And the note added about Ref. [32] is too thin—the overlap deserves a quantitative comparison.

This paper is for leptogenesis specialists who want explicit formulas for the CP asymmetry in the general seesaw parametrization. The analytic part and the thermal results are worth having, but the non-thermal section needs a serious revision: at minimum a full Boltzmann treatment including inverse decays, and ideally a flavor-resolved one. I would send it to peer review, because the analytic core is solid and the thermal bounds are useful, but the non-thermal claims should not be accepted without major changes.

Referee Report

2 major / 5 minor

Summary. The paper derives closed-form expressions for the leptogenesis CP-asymmetry parameter epsilon_1 in the Casas-Ibarra orthogonal parametrization, for both two and three right-handed neutrinos, and uses them to scan for the minimum mass of the lightest right-handed neutrino that reproduces neutrino oscillation data and the observed baryon asymmetry. Thermal and non-thermal leptogenesis are treated, in the standard Type-I seesaw with the SM Higgs and in a neutrinophilic two-Higgs-doublet variant. The headline claim is that in the neutrinophilic non-thermal case the lightest right-handed neutrino mass can be as low as the sphaleron decoupling temperature, about 132 GeV.

Significance. If correct, the low-mass result would connect leptogenesis to right-handed neutrino masses that may be accessible to laboratory searches, which is of genuine phenomenological interest. The analytic formulas for epsilon_1 and the explicit parameter scans are useful and transparent; the paper also correctly notes that epsilon_1 is independent of the MNS matrix and of some orthogonal-matrix parameters. The thermal large-mass results reproduce the expected order of magnitude of the Davidson-Ibarra-type bound. However, the headline low-mass claim rests on an unjustified no-washout assumption and on a single-flavor Boltzmann treatment at a scale where flavor effects are important; until those issues are addressed, the quoted lower bounds are not established.

major comments (2)

[Section 4.2, Eq. (4.14)] The condition mN1<TR stated after Eq. (4.14) does not justify setting kappa=1. Inverse-decay washout is controlled by K=Gamma_1/H(mN1) in Eq. (4.8), not by the production temperature; at T~mN1 the inverse decay rate is unsuppressed. For the NH benchmark that realizes |epsilon_1|=3.27e-10 at mN1=132 GeV, Eq. (4.23) with vh replaced by v2 implies v2~1.4 GeV, and Eq. (4.8) gives K at least of order 10^6 because (O Dnu O-dagger)_11 = (m2+m3)/2 cosh(2b) for the a=pi/4, b>>1 configuration that maximizes epsilon_1. In this regime the efficiency factor in Eq. (4.7) is kappa ~ 2/(z_B K) << 1, so the asymmetry is suppressed by orders of magnitude. The 132 GeV values in Table 2 are therefore not supported by the analysis as presented.
[Sections 4.1-4.2, Eqs. (4.1)-(4.7)] The Boltzmann treatment is single-flavor throughout, with one lepton yield YL and one washout rate gamma_W. For the non-thermal neutrinophilic benchmarks with mN1 ~ 132 GeV, the asymmetry is generated at T ~ mN1, where the tau, mu, and electron Yukawa interactions are all in equilibrium; in this regime flavored leptogenesis is required. The unflavored CP asymmetry and the single-flavor efficiency factor kappa in Eq. (4.7) can differ appreciably from the properly flavored quantities, so the lower bounds in Table 2 and the requirement |epsilon_1|=3.27e-10 in Eq. (4.15) need to be re-derived with flavor-dependent Boltzmann equations before the low-mass claim can be accepted.

minor comments (5)

[Eq. (2.11) vs. Eqs. (3.5)-(3.6)] The orthogonal matrix in Eq. (2.11) is written with cosh(a+ib) and sinh(a+ib), but the resulting formulas for Gamma_1 and epsilon_1 contain sin(2a) sinh(2b) and cos(2a) cosh(2b). These expressions follow from O = [[cos(a+ib), sin(a+ib)],[-sin(a+ib), cos(a+ib)]], not from the cosh/sinh parameterization. Please correct the definition of O or the analytic formulas.
[Eq. (2.16)] The matrix O2 has a stray '1' in the (3,2) entry and is not orthogonal as written; the (3,2) entry should be 0 for the standard rotation form.
[Figures 4 and 5 captions] The text describing the right panels repeats mlightest/eV = 10^{-2} three times before 10^{-8}; this appears to be a typo and should be corrected to match the intended scan values.
[Section 5.0.2] The text says the three-RHN neutrinophilic non-thermal case has six free parameters, but the sentence lists mN1, mlightest, a12, a13, b12, b13, and v2, which is seven; please clarify the counting.
[Abstract and Section 5] The spelling 'neutrinophillic' should be 'neutrinophilic', and 'heirarchy' should be 'hierarchy'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CP-asymmetry formula is derived from standard one-loop amplitudes, and the quoted lower bounds are obtained by extremizing free parameters against externally fixed inputs (neutrino oscillation data, observed baryon asymmetry, and the sphaleron decoupling temperature).

full rationale

The paper's derivation chain is self-contained and not circular. The CP asymmetry formula (Sec. 3, Eqs. 3.1, 3.4, 3.5-3.8) is obtained by inserting the Casas-Ibarra parametrization mD = sqrt(MN) O sqrt(Dnu) U† into the standard one-loop expression for epsilon_i, with no parameter fitted to the target quantities. The observed neutrino masses and mixing angles are external inputs (Eq. 2.7), and the baryon asymmetry nB/s = 8.7 × 10^-11 is used only as the constraint that fixes the required epsilon. The free parameters a, b, and mlightest are scanned and extremized, not fitted and then repackaged as predictions; the minimum mN1 is the result of a well-defined minimization. The non-thermal analysis does set kappa = 1 and fixes |epsilon1| = 3.27 × 10^-10 via Eq. (4.14) under the stated conditions TR ~ mN1 ~ mphi/2 and BR = 1; this is a model assumption, and whether it is physically self-consistent (e.g., washout at these masses) is a correctness or physics concern, not a circularity. In the neutrinophilic Higgs case, the 132 GeV value is obtained by using the scaling mmin_N1 ∝ v2^2 and imposing the independent sphaleron decoupling bound from Ref. [31]; the bound is an external input, not an output of the leptogenesis calculation. No load-bearing self-citation or imported uniqueness claim appears: the efficiency factor is taken from Ref. [19] (external), and self-citations are not used to justify the central result. Therefore, no circular step can be exhibited, and the paper's claims are not equivalent to their inputs by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles or forces. The free parameters are the angles and boost parameters of the orthogonal matrix, the lightest neutrino mass, the neutrinophilic doublet VEV, and the non-thermal branching ratio. The axioms are standard seesaw and leptogenesis ingredients plus the single-flavor and non-thermal approximations.

free parameters (5)

a, b: real parameters of the 2x2 orthogonal matrix O in the two right-handed neutrino case
Scan parameters that determine O; the paper extremizes over them to minimize mN1.
a12, a13, b12, b13: real parameters of the 3x3 orthogonal matrix O in the three right-handed neutrino case
Scan parameters for ϵ1; a23 and b23 drop out of ϵ1.
mlightest
Lightest neutrino mass in the three right-handed neutrino scans; treated as free and scanned.
v2
VEV of the neutrinophilic Higgs doublet; scanned over the range 10^-2 to 246 GeV.
BR(phi -> N1 N1) = 1 (maximal)
Chosen at its maximum to minimize the required mN1 in non-thermal leptogenesis; not fitted to data.

assumptions (6)

domain assumption Type-I seesaw formula m_nu approximately mD^T M_N^{-1} mD with hierarchical heavy neutrinos
Invoked in Eq. (2.4); assumes right-handed neutrino masses much heavier than Dirac masses so light neutrinos are predominantly seesaw-suppressed.
standard math Casas-Ibarra parametrization mD = sqrt(MN) O sqrt(Dnu) U^dagger with complex orthogonal O
Introduced in Eq. (2.10) from Refs. [21,22]; assumed to cover all Dirac Yukawa matrices consistent with oscillation data.
domain assumption Hierarchical right-handed neutrino masses with vertex and self-energy corrections Vj and Sj approximately 1
Used in Section 3 to simplify Eq. (3.1) to Eq. (3.4); excludes resonant leptogenesis where degenerate masses enhance ϵ.
domain assumption Single-flavor Boltzmann equations with the analytic efficiency factor kappa from Ref. [19]
Section 4.1, Eqs. (4.1) through (4.7); neglects flavor effects, which is questionable for mN1 near 100 GeV.
domain assumption Sphaleron decoupling at 132 GeV
Used in Section 5.0.1 to set the low-mass cutoff; taken from the lattice result in Ref. [31].
domain assumption Non-thermal production where only N1 is produced by scalar decay, kappa = 1, and mN1 < TR
Section 4.2; if inverse decays or thermal production of N1 occur, the lepton asymmetry and the derived mass bound change.

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Cite this review

Pith. "Pith review of Analytic formulation of Leptogenesis with neutrino oscillation data employing the general parametrization for neutrino mass matrix." pith.science (2026). https://pith.science/paper/6PEIVWDB

@misc{pith2026250620580,
  author       = {Pith},
  title        = {Pith review of: Analytic formulation of Leptogenesis with neutrino oscillation data employing the general parametrization for neutrino mass matrix},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PEIVWDB}},
  note         = {Machine review of arXiv:2506.20580}
}

read the original abstract

The observed neutrino oscillations and baryon asymmetry, unexplained by the Standard Model (SM), can both be accounted for by extending the SM to include Majorana right-handed neutrinos (RHNs). Tiny neutrino masses naturally arise through the Type-I seesaw mechanism, which involves lepton number violation. Meanwhile, the baryon asymmetry can be generated via leptogenesis, where the out-of-equilibrium decay of RHNs produces a lepton asymmetry that is partially converted into a baryon asymmetry through sphaleron processes. The Dirac Yukawa couplings play the crucial role for both Type-I seesaw and leptogenesis. In this work, we derive an analytic expression for the CP asymmetry parameter in a general parametrization. Focusing on a hierarchical RHN mass spectrum, we evaluate the lowest mass of the lightest RHN that reproduce both neutrino oscillation data and the observed baryon asymmetry. We study the case with two and three generations of RHNs for both thermal and non-thermal leptogenesis scenarios. Besides the standard Type-I seesaw involving SM Higgs doublet, we also examine the Type-I seesaw with a new neutrinophillic Higgs doublet. In this case for non-thermal leptogenesis, the minimum value for the lightest RHN mass can be as low as sphaleron decoupling temperature.

Figures

Figures reproduced from arXiv: 2506.20580 by the authors.

**Figure 1.** Plot of efficiency factor κ as a function of K = Γ/H γD is the decay rate of N1 R, γS includes SM Higgs boson (h) mediated t and s channel scattering of N1 R, γS = 2γ (N1 R) h,t + 4γ (N1 R) h,s , (4.3) and γW includes the inverse processes which can wash out of yield of lepton number, γW = 1 2 γD + 2 γ (l) N + γ (l) N,t + γ (l) h,t + Y1 Y eq 1 γ (l) h,s, (4.4) where γh,t and γh,s are scattering processes whic… view at source ↗

**Figure 2.** Results for the NH. The first two panels show contours of fixed [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Results for the IH. The first two panels show contours of fixed [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: The two plots shows results from a parameter scan for the NH. The left panel displays [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: The two plots shows results from a parameter scan for the IH. The left panel displays [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Results for the NH. Both panels show contours of fixed [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Results for the IH. Both panels show contours of fixed [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: The two plots shows results from a parameter scan. For the NH (IH), the left (right) [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: The two plots show the results of a parameter scan for a thermal leptogenesis scenario [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: The two plots show the results of a parameter scan for leptogenesis scenario with [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

Discussion (0). Continue with ORCID to comment.

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