Leptogenesis via Resonant Sequential Dominance and TBC3 Mixing in a Type-I Seesaw Model

Michael Fodroci; Teruyuki Kitabayashi

Leptogenesis via Resonant Sequential Dominance and TBC3 Mixing in a Type-I Seesaw Model

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper argues that the observed baryon asymmetry of the Universe can be produced by resonant leptogenesis in a Type-I seesaw model, with the lightest right-handed neutrino mass fixed to between 0.13 and 230 TeV.

desk verdict A worthwhile model-building paper with a genuinely new PMNS ansatz and RSD framework, but the headline leptogenesis mass range rests on a dimensionally inconsistent splitting parameter that must be fixed before the numerical claim is credible. read the letter →

arxiv 2608.06983 v2 pith:AAL4FVCO submitted 2026-08-07 hep-ph

Michael Fodroci , Teruyuki Kitabayashi This is my paper · ORCID

classification hep-ph

keywords ResonantSequentialDominanceTBC3mixingansatzType-IseesawleptogenesisbaryonasymmetryoftheUniverseA4familysymmetryneutrinoright-handedmass

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

This paper tries to show that Sequential Dominance — the idea that each column of the Dirac neutrino mass matrix primarily generates one light neutrino mass — still works when two right-handed neutrinos are nearly degenerate, a variant it calls Resonant Sequential Dominance. It introduces a PMNS matrix ansatz, TBC3, whose deviations from tri-bimaximal mixing are set equal and proportional to the Cabibbo parameter, and shows that a simple Dirac texture reproduces current neutrino oscillation data within $1\sigma$. It then constructs an $A_4\times(Z_3)^4\times(Z_2)^2$-symmetric Lagrangian that realizes this texture and drives resonant leptogenesis. Solving the Boltzmann equations, it finds that matching the observed baryon asymmetry of the Universe requires the lightest right-handed neutrino mass to lie in $0.13\text{ TeV}\le m_{N_1}\le230\text{ TeV}$.

What carries the argument

The load-bearing structures are Resonant Sequential Dominance (RSD), the TBC3 PMNS ansatz, and the Master Formula that relates the Dirac matrix to light-neutrino observables. RSD replaces the conventional hierarchy of right-handed neutrino masses with a Yukawa hierarchy $y_{D1}<y_{D2}$ while keeping $M_1\approx M_2$, which is what makes resonant leptogenesis available. TBC3 fixes the deviations of the solar and atmospheric angles from tri-bimaximal mixing to be equal, $s=a=-(3\sqrt3)^{-1}\lambda$, with the reactor deviation $r=(2/3)\sqrt2\,\lambda$; this equality is what makes the low-energy fit compatible with 'simple' flavon vacuum expectation values. The Master Formula computes the ratios $Z_i$ of Dirac matrix elements and the two physical complex phases from the PMNS matrix, and the resulting Simple ($\sqrt2,1;1,2$) Alignment supplies the Dirac texture. Resonant leptogenesis then runs through the resummed Yukawa couplings of the resonant formalism and the coupled Boltzmann equations, with the small splitting $\Delta$ supplied by higher-dimensional operators.

What would settle it

A collider search that probes the $0.13\text{--}230$ TeV window and excludes the decay signatures of the Simple ($\sqrt2,1;1,2$) alignment would settle the central claim by falsifying the BAU mechanism. A precision measurement of $\theta_{12}$ or $\theta_{23}$ that moves outside the TBC3 $1\sigma$ predictions would also undercut the ansatz and the derived Dirac texture.

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Extended reading notes

Core claim

Within a Type-I seesaw model, the hierarchy of light neutrino masses need not come from the hierarchy of right-handed neutrino masses. In Resonant Sequential Dominance the two lightest right-handed neutrinos are nearly degenerate, $M_1\approx M_2$, and the sequential dominance of the Dirac columns is enforced purely by Yukawa couplings, $y_{D1}<y_{D2}$, while the third neutrino is ultra-heavy. Using the Master Formula and a $\chi^2$ fit, the paper identifies a Simple ($\sqrt2,1;1,2$) Alignment of the Dirac matrix that places all mixing angles, the Dirac phase, and $m_2/m_3$ inside the $1\sigma$ ranges of current oscillation data. The associated $A_4\times(Z_3)^4\times(Z_2)^2$ Lagrangian keeps the Majorana mass matrix diagonal up to dimension five, and a small off-diagonal splitting $\Delta$ triggers resonant enhancement of CP violation. Numerically solving the coupled Boltzmann equations at the four corners of the allowed $(y_{D1},\Lambda)$ space yields the observed baryon asymmetry for mass splittings $2.45\times10^{-9}\text{ GeV}\le x_{\Delta,\min}\le4.31\times10^{-6}\text{ GeV}$, which corresponds to $0.13\text{ TeV}\le m_{N_1}\le230\text{ TeV}$.

Load-bearing premise

The mass window is computed from hand-chosen ranges $0.1\le y_{D1}\le1$ and $100\text{ TeV}\le\Lambda\le419\text{ TeV}$; unless those ranges are justified by an underlying model, the headline window has no independent standing.

Editorial extensions

If this is right

If RSD is correct, the lightest right-handed neutrino can be as light as $0.13$ TeV, putting it within reach of current and future collider searches rather than at the usual $10^9$ GeV seesaw scale.
The TBC3 ansatz offers a parameter-reduced starting point for models that must accommodate a solar angle below $1/3$ together with a nonzero reactor angle.
This setup predicts a required range of mass splittings, $2.45\times10^{-9}\text{ GeV}\le x_{\Delta,\min}\le4.31\times10^{-6}\text{ GeV}$; constraining such a tiny splitting would directly test the mechanism.
The explicit $A_4\times(Z_3)^4\times(Z_2)^2$ Lagrangian shows that a flavor symmetry can produce the needed texture while keeping the heavy-neutrino mass matrix diagonal up to dimension five.

Reading between the lines

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

Editorial inference: the $0.13\text{--}230$ TeV window is not a fundamental prediction; it follows from the adopted ranges $y_{D1}\in[0.1,1]$ and $\Lambda\in[100,419]$ TeV, so a UV completion that fixes either constant could move the window.
The paper leaves the origin of the tiny splitting $\Delta$ to unspecified higher-dimensional operators; if those operators also mediate lepton-flavor-violating processes, their measured rates could indirectly test RSD.
The TBC3 equality of solar and atmospheric deviations pushes $\theta_{23}$ to the edge of its $1\sigma$ range; a future precise value away from the TBC3 prediction would favour relaxing that equality.
Because only normal mass ordering is considered, decisive evidence for inverted ordering would remove the paper's empirical basis.

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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

Colleague,

The genuinely new pieces here are RSD, TBC3, and the explicit A4 x (Z3)^4 x (Z2)^2 Lagrangian. RSD makes good sense: if you want resonant leptogenesis with sequential dominance, the hierarchy has to move from Majorana masses into the Yukawas, and the paper carries that through carefully. TBC3 is a clean parametrization—setting the solar and atmospheric deviations equal and both proportional to lambda—and it fits current NuFIT data, including JUNO, reasonably well. The Master Formula search and the chi-squared scan are standard but competently done, and the Simple (sqrt 2, 1; 1, 2) alignment is a concrete, useful result. A specialist can re-derive the main steps, and the Lagrangian section is explicit about the shaping symmetries. Credit where due: the framework and ansatz are worth having.

The soft spots are in the quantitative leptogenesis part. The headline m_N1 window is not a prediction in the strong sense; it follows from choosing y_D1 in [0.1,1] and Lambda in [100,419] TeV. The ranges have motivations (perturbativity, flavon VEVs, sphaleron freeze-out), but they are choices, so "requires" in the abstract oversells the result. More importantly, the x_Delta variable is dimensionally ambiguous. Eq. (75) defines m_N2 = m_N1 (1 + x_Delta), so x_Delta is dimensionless, but Table III and Eq. (77) label it in GeV. If x_Delta is relative, then the absolute splitting at the smallest quoted value is several orders of magnitude larger than the decay width, far from the resonance condition in Eq. (64) unless the numerical code is doing something else. If x_Delta is absolute, Eq. (75) is wrong. This is load-bearing because the BAU plots are tuned to x_Delta,min. It may be a units typo in the text; either way, the numerical claim needs a corrected, checked version before the m_N1 range is credible. The paper also admits the small splitting has no UV origin, which is fine for a model-building exercise but should be said more plainly.

Overall: the framework and ansatz deserve serious referee time. The BAU calculation needs fixing, not just polishing.

Referee Report

3 major / 4 minor

Summary. The paper introduces a framework called Resonant Sequential Dominance (RSD), applies it to a Type-I seesaw model with two nearly degenerate right-handed neutrinos, and proposes a new PMNS ansatz (TBC3) that modifies TBM mixing to accommodate the non-zero reactor angle and the JUNO measurement of sin^2(theta_12) < 1/3. Using King's Master Formula and chi^2 analyses, the authors identify a 'Simple (sqrt(2),1; 1,2) Alignment' that fits NuFIT 2025 data within 1 sigma. They construct an A4 x (Z3)^4 x (Z2)^2-symmetric Lagrangian that realizes this alignment, and numerically solve the resonant leptogenesis Boltzmann equations to claim that the observed baryon asymmetry can be produced for m_N1 in [0.13, 230] TeV.

Significance. If the leptogenesis calculation were correct, the paper would be of clear interest: the TBC3 ansatz is a simple alternative to TBM/TM2 that tracks current data, the RSD concept cleanly separates the light-neutrino mass hierarchy from the RH neutrino mass hierarchy, and the explicit A4 model provides a concrete starting point for UV completions. The paper's strengths include a transparent Master Formula treatment, reproducible chi^2 fits via the MPT package, and an explicit symmetry assignment with shaping symmetries. However, the headline quantitative claim (the m_N1 range) is currently not supported because it depends on hand-picked input ranges and on a mass-splitting parameter whose definition and numerical values are inconsistent.

major comments (3)

[Sec. VI.E, Eq. (75), Table III] The parameter x_Delta is defined by Eq. (75) as a dimensionless relative splitting, m_N2 = m_N1(1 + x_Delta), but Table III and Eq. (77) quote x_Delta,min in GeV. With the dimensionless reading, the Set 1 value x_Delta,min = 2.45e-9 gives a physical splitting m_N2 - m_N1 = m_N1 x_Delta ~ 3.2e-7 GeV for m_N1 = 0.13 TeV. Using the model's own Yukawa matrix in Eq. (63), one obtains (Y^dag Y)_11 ~ 0.9 y_D1^2/Lambda^2 ~ 5.2e-14 for Set 1, yielding a tree-level width Gamma_1 ~ 2.7e-13 GeV and Gamma_1/2 ~ 1.3e-13 GeV. The quoted splitting is therefore roughly 10^6 times Gamma_1/2, in plain contradiction of the resonant condition Eq. (64). If instead x_Delta is meant to be an absolute mass splitting in GeV, Eq. (75) is dimensionally wrong, and Set 1 still gives a splitting far below Gamma_1/2. The same factor of ~10^6 appears in all four rows under the dimensionless reading. Since x_Delta,min is defined as the minimum splitting needed to reach eta_B = 6.1e-10, the numerical leptogenesis results are not in the resonant regime and thus do not support the claimed resonant enhancement. Please clarify the definition, rerun the calculation with consistent units, and verify that the resonance condition is actually satisfied at the quoted splittings.
[Sec. VI.D, Eqs. (73)-(74)] The headline range 0.13 TeV <= m_N1 <= 230 TeV is not a derivation from the model but a consequence of the manually chosen intervals y_D1 in [0.1, 1] and Lambda in [100, 419] TeV. The lower bound Lambda_min = 100 TeV is 'conservatively selected' and the upper bound Lambda_max = 419 TeV is fixed by requiring z_crit >= 1 with y_D1 = 0.1. These choices are reasonable but ad hoc, and the abstract's wording 'accounting for the observed baryon asymmetry ... requires the lightest right-handed neutrino mass to lie in the range ...' overstates what has been shown. The correct statement is that within the chosen parameter box the model can reproduce the observed asymmetry. This distinction should be stated prominently in the abstract and conclusions.
[Sec. VI.B] The mass splitting Delta that breaks the M1 = M2 degeneracy is introduced from unspecified higher-dimensional operators, even though the shaping symmetries in Sec. V are specifically designed to keep the Majorana mass matrix diagonal up to dimension five. The leptogenesis result is therefore conditional on an assumption that is not part of the explicit Lagrangian. The paper acknowledges this in the conclusion, but the leptogenesis section should clearly flag Delta as a model input rather than a prediction, and should discuss how the required size of Delta compares with the scale of the higher-dimensional operators expected in a UV completion.

minor comments (4)

[Eq. (77) and Table III] The unit label [GeV] on x_Delta,min should be removed if x_Delta is dimensionless, or the definition in Eq. (75) should be changed if an absolute mass splitting is intended. The current inconsistency is confusing and should be fixed in any revision.
[Fig. 18 caption] The caption reads x_Delta = 4.36e-6 GeV, while Table III and the text quote 4.31e-6 GeV. Please correct the typo.
[Sec. IV.B, text near Eq. (48)] The sentence stating that 'the values predicted by this set of Dirac neutrino mass matrix elements are all in 1 sigma agreement' is immediately qualified by the marginal theta_23 value. Please rephrase to avoid the apparent contradiction.
[Sec. II.C] The acronym TBC3 is never expanded. Even if it follows the naming convention of Ref. [11], please spell it out on first use, or state explicitly that it denotes a particular Tri-Bi-Cabibbo-like ansatz.

Circularity Check

2 steps flagged · score 6.0 of 10

BAU agreement is a tuned consistency scan: the m_N1 window comes from hand-picked (y_D1, Λ) bounds, and x_Δ,min is defined to hit η_B = 6.1e-10.

fitted input called prediction [Section VI D, Eqs. (58), (73)-(74), and Abstract]
"So, we perform calculations at the extremes of the range 0.1≤y_{D1}≤1, 100 TeV≤Λ≤419 TeV, which via Eqs.(58) and (49) leads to the following range on the lightest Majorana mass 0.13 TeV≤m_{N1}≤230 TeV."

Eq. (58) is M = v^2/(2 m_a) (y_D1/Λ)^2. The parameter m_a = 0.01315 eV was fixed in Eq. (49) to reproduce Δm^2_21 and Δm^2_31, and the ranges 0.1≤y_D1≤1 and 100 TeV≤Λ≤419 TeV are selected by hand: Λ_min is 'conservatively selected' and Λ_max follows from z_crit≥1 with y_D1=0.1. Substituting these fitted and hand-picked inputs into Eq. (58) algebraically yields the quoted m_N1 interval. The BAU constraint does not appear in Eq. (58); it is imposed later through x_Δ,min. Therefore the abstract's statement that accounting for the observed baryon asymmetry 'requires' this mass range is a restatement of the chosen input box, not a prediction.
self definitional [Section VI E, Eqs. (75)-(77), and Section VI B; Conclusion]
"The value x_{Δ,min} is then defined as the minimum value for which η_B = 6.1×10^{-10}. Thus we have demonstrated that within the allowed parameter space there are a range of possible mass splittings that can produce sufficient resonant enhancement of the CP-violating processes that underpin leptogenesis and lead to baryogenesis."

x_Δ is a free parameter of unspecified higher-dimensional operators: 'it is assumed that the small off-diagonal corrections arise from higher-dimensional operators present at the scale of leptogenesis.' The paper defines x_Δ,min as the value that makes η_B equal to the observed target, then presents the resulting x_Δ,min values and the viability of the model as the numerical result. Since the splitting is not fixed by the Lagrangian, the successful leptogenesis is guaranteed by construction rather than predicted. The conclusion concedes the paper has 'not suggested ... a particular origin for the RH neutrino mass splittings.' The only genuine output is a mapping x_Δ,min(y_D1, Λ) that tunes every allowed mass point to the target asymmetry.

full rationale

The low-energy construction is not circular: the TBC3 ansatz and the A4 × (Z3)^4 × (Z2)^2 Lagrangian are explicit new structures, and the chi-squared fits of ε_ν and m_a to oscillation data are ordinary parameter fixing. The circularity enters when these fitted or hand-selected inputs are relabeled as BAU-derived constraints. Eq. (58) contains no BAU content; the 0.13–230 TeV window is the image of the chosen (y_D1, Λ) box under an algebraic identity. The leptogenesis step then tunes x_Δ,min to hit η_B = 6.1e-10 and reports that the model 'successfully drives resonant leptogenesis'; because x_Δ arises from unspecified higher-dimensional operators, this is a fit, not a prediction. There is no load-bearing self-citation: Ref. [11] is external and Ref. [32] is the authors' prior work used only as motivation. A separate correctness risk, independent of circularity: Eq. (75) defines x_Δ as a dimensionless relative splitting while Table III and Eq. (77) quote x_Δ,min in GeV; interpreted as dimensionless, the quoted minima lie orders of magnitude below the paper's own resonance condition Eq. (64). This makes the numerical BAU result additionally fragile. Overall, the headline m_N1 constraint reduces to the chosen parameter box and the BAU agreement reduces to the tuning of x_Δ; the independent content is the ansatz and Lagrangian construction.

Assumptions & free parameters 7 free parameters · 10 assumptions · 3 invented entities

The central claim rests on the type-I seesaw mechanism, King's Master Formula, assumed VEV alignments, an unspecified Froggatt-Nielsen mechanism for the Yukawa hierarchy, and an assumed higher-dimensional source for the mass splitting. The numerical m_N1 range is mostly determined by hand-chosen ranges for y_D1 and Lambda.

free parameters (7)

epsilon_nu (Yukawa squared ratio y_D2^2 / y_D1^2) = 0.2758 (Simple Alignment); 0.1393 and 0.1317 in chi-squared best fits
Controls the hierarchy between the A and B columns of the Dirac mass matrix; fixed by chi-squared fits to oscillation data (Eqs. 44, 57, Table I).
m_a (absolute light neutrino mass scale) = 0.01315 eV
Set by hand to reproduce Delta m^2_21 and Delta m^2_31 (Eq. 49).
y_D1 = 0.1 to 1
Chosen range from perturbativity; no theory predicts it; enters the m_N1 range via Eq. (58).
Lambda (flavon cutoff scale) = 100 to 419 TeV
Lower bound is conservatively selected, upper bound follows from z_crit >= 1 and y_D1 = 0.1 (Sec VI D); drives the m_N1 range.
x_Delta (mass splitting parameter) = 2.45e-9 to 4.31e-6 GeV
Tuned in Table III so that eta_B = 6.1e-10; no fundamental origin is given and the definition has unit ambiguity.
Flavon VEV alignments A = (0, sqrt(2), 1), B = (1, e^{-i pi/3}, 2) = Simple Alignment
Selected by simplicity and chi-squared fit (Eq. 47); the paper admits no symmetry produces this alignment.
CP phases (delta_CP, eta_2, eta_3, beta) = delta_CP = 192.2 deg, eta_2 = -pi/3, eta_3 = 0
Chosen for fit; the Majorana phase beta is not observable in oscillation experiments but is an input to the mass matrix construction (Sec IV B).

assumptions (10)

domain assumption Type-I seesaw mechanism with three right-handed Majorana neutrinos.
Used throughout (Eqs. 20-22) as the framework for generating light neutrino masses.
domain assumption King's Master Formula and Sequential Dominance texture assumptions.
Adopted from Ref. [11] and adapted to degenerate masses in Secs. III and IV.
ad hoc to paper Texture zero in the (1,1) entry of the Dirac mass matrix.
Assumed in Eq. (30) and connected to a non-zero reactor angle; no symmetry is shown to enforce it.
ad hoc to paper Yukawa hierarchy y_D1 < y_D2 is realized by an unspecified Froggatt-Nielsen-like mechanism.
Required for RSD (Sec III B) but no concrete FN model or charge assignment is given.
ad hoc to paper Flavon VEVs take the assumed alignments without potential minimization.
The Simple Alignment is chosen by hand in Sec IV B; the paper states no symmetry produces it.
ad hoc to paper Higher-dimensional operators generate the tiny mass splitting Delta without spoiling the Majorana mass matrix diagonality.
Assumed in Sec VI B (Eq. 60) but no explicit operators are written down.
domain assumption The third right-handed neutrino N3 is ultra-heavy and decoupled.
Standard in SD constructions; introduced in Eq. (21) and used throughout.
domain assumption Charged-lepton mass matrix is diagonal.
Basis choice in Eq. (20); standard in this type of model building.
standard math A4 group theory and tensor product rules.
Used in Sec V to build the Lagrangian (Eqs. 50-52).
domain assumption Pilaftsis-Underwood resonant leptogenesis formalism and Boltzmann equations.
The numerical BAU computation relies on the formalism of Ref. [41] (Secs. VI C-VI E).

invented entities (3)

Right-handed Majorana neutrinos N1, N2, N3
purpose: Generate light neutrino masses through the seesaw mechanism and produce lepton asymmetry via decays.
Canonical type-I seesaw fields, not yet observed; the paper gives a wide mass range but does not compute collider signatures or other falsifiable handles.
Flavon fields phi, phi_1, phi_2, phi_3, chi_1, chi_2, chi_3
purpose: Break the A4 family symmetry and generate the Dirac and Majorana mass textures.
No dynamical VEV derivation is provided; the alignments are inputs chosen for fit and simplicity.
A4 x (Z3)^4 x (Z2)^2 flavor symmetry
purpose: Restrict the allowed Yukawa and Majorana operators to the desired texture.
Chosen by hand in Sec V; no ultraviolet completion, anomaly check, or independent observable is presented.

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Cite this review

Pith. "Pith review of Leptogenesis via Resonant Sequential Dominance and TBC3 Mixing in a Type-I Seesaw Model." pith.science (2026). https://pith.science/paper/AAL4FVCO

@misc{pith2026260806983,
  author       = {Pith},
  title        = {Pith review of: Leptogenesis via Resonant Sequential Dominance and TBC3 Mixing in a Type-I Seesaw Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AAL4FVCO}},
  note         = {Machine review of arXiv:2608.06983}
}

abstract

We demonstrate how phenomenological model construction assuming Sequential Dominance can be extended to cases with degenerate right-handed neutrino masses. Because this framework accommodates resonant leptogenesis, we designate it as Resonant Sequential Dominance. We then investigate Dirac mass matrix textures compatible with current neutrino data within Resonant Sequential Dominance, utilizing a novel neutrino mixing matrix ansatz termed TBC3, proposed in this work. Additionally, we present an $A_4 \times (Z_3)^4 \times (Z_2)^2$-symmetric Lagrangian that is consistent with neutrino oscillation data and successfully drives resonant leptogenesis. Lastly, we find that accounting for the observed baryon asymmetry of the Universe requires the lightest right-handed neutrino mass to lie in the range $0.13\text{ TeV} \le m_{N_1} \le 230\text{ TeV}$.