Insights on the Scale of Leptogenesis from Neutrino Masses and Neutrinoless Double-Beta Decay

Alessandro Granelli; Juntaro Wada; Kengo Shimada; Koichi Hamaguchi; Maura E. Ramirez-Quezada; Tatsuya Yokoyama

REVIEW 2 major objections 4 minor 1 cited by

Insights on the Scale of Leptogenesis from Neutrino Masses and Neutrinoless Double-Beta Decay

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that future neutrinoless double-beta decay measurements can constrain the minimal lightest-heavy-neutrino mass required for thermal leptogenesis to $(0.7-6)\times10^9$ GeV under mild fine-tuning.

desk verdict A credible, honest numerical map of the minimal leptogenesis scale onto (m_lightest, m_bb^eff), but the headline range is conditional on a zero-initial-N1 assumption that the paper does not stress-test. read the letter →

arxiv 2502.10093 v2 pith:RABY5H56 submitted 2025-02-14 hep-ph hep-ex

Alessandro Granelli , Koichi Hamaguchi , Maura E. Ramirez-Quezada , Kengo Shimada , Juntaro Wada , Tatsuya Yokoyama This is my paper

classification hep-phhep-ex

keywords leptogenesistype-IseesawheavyMajorananeutrinosneutrinolessdouble-betadecayneutrinomassorderingbaryonasymmetryflavoureffectsfine-tuning

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

Successful thermal leptogenesis in the type-I seesaw needs a sufficiently heavy lightest heavy neutrino, and this paper computes that minimum mass as a function of two quantities we may soon measure: the lightest ordinary neutrino mass and the neutrinoless double-$\beta$ decay effective mass. The calculation uses the flavoured density matrix equations, so flavour effects are included, and it covers both normal and inverted light-neutrino orderings. The central quantitative result is that, with mild fine-tuning in the seesaw, the minimum mass stays within $(0.7-6)\times10^9$ GeV for the sensitivity reach of planned neutrinoless double-$\beta$ decay experiments. If this is right, upcoming experiments would not just measure neutrinos; they would delimit the scale of baryogenesis and the reheating temperature of the early Universe.

What carries the argument

The argument is carried by the flavoured density matrix equations for the lepton-flavour asymmetries, together with the Boltzmann equation for the lightest heavy neutrino: the diagonal entries of the density matrix track the $B/3 - L_\alpha$ number densities of each flavour, while the off-diagonal entries track flavour coherence and are damped by charged-lepton Yukawa interactions. The heavy-neutrino Yukawa couplings are parameterised through the seesaw in terms of the light-neutrino masses, the neutrino mixing matrix, and a complex orthogonal matrix; the parameter $\Delta = \sum_a \sum_j |m_a^{(j)}|/m_a$ quantifies how much the individual heavy-neutrino contributions cancel to produce the light neutrino masses, with $\Delta=3$ corresponding to no cancellations. The machinery converts the requirement that the final baryon-to-photon ratio equal $6.1\times10^{-10}$ into a minimisation of $M_1$ over the remaining seesaw parameters at each point of the $(m_\nu^{\rm lightest}, m_{\beta\beta}^{\rm eff})$ plane.

What would settle it

If future measurements of the lightest neutrino mass and the neutrinoless double-$\beta$ decay effective mass land at a point where the numerical scan finds no value of the lightest heavy-neutrino mass that reproduces the observed baryon-to-photon ratio, the hierarchical thermal-leptogenesis assumption would be ruled out. A cheaper numerical test is to rerun the same scan with the assumptions $M_1\times10^3<M_2<M_3/3$ and zero initial heavy-neutrino population relaxed; if the quoted minimal mass moves outside the stated spread, the headline range depends on those assumptions.

Watch

Extended reading notes

Core claim

For thermal leptogenesis from the type-I seesaw with three hierarchical heavy Majorana neutrinos, the minimal lightest-heavy-neutrino mass $M_1^{\rm min}$ needed to match the observed baryon-to-photon ratio $\eta_B = 6.1\times10^{-10}$ is a computable function of the lightest neutrino mass $m_\nu^{\rm lightest}$ and the neutrinoless double-$\beta$ decay effective mass $m_{\beta\beta}^{\rm eff}$, with flavour effects included through the flavoured density matrix equations. Under mild (10%) fine-tuning in the seesaw relation, and within the sensitivity of planned neutrinoless double-$\beta$ decay experiments ($m_{\beta\beta}^{\rm eff}\gtrsim 0.0047$ eV), the scan finds $M_1^{\rm min}\sim(0.7-6)\times10^9$ GeV, with the spread driven mostly by $m_\nu^{\rm lightest}$ rather than by the Majorana-phase-dependent effective mass. In the least-tuned case of a real orthogonal matrix ($\Delta=3$), the minimal mass is larger, $M_1^{\rm min}\sim(1-2)\times10^{10}$ GeV; allowing stronger cancellations lowers it, reaching $\sim10^6$ GeV for $\Delta\sim10^5$.

Load-bearing premise

The result assumes that only the lightest of the three heavy neutrinos is produced in the early Universe and contributes to leptogenesis, enforced by a large hierarchy between the heavy masses and by starting the calculation at ten times the lightest heavy-neutrino mass with no initial heavy-neutrino population; if the heavier neutrinos or an initial population contributed, the required minimal mass could shift.

Editorial extensions

If this is right

Within the reach of planned neutrinoless double-beta decay searches, the minimal lightest-heavy-neutrino mass ranges over $(0.7-6)\times10^9$ GeV, so a non-observation at that sensitivity would exclude only a part of the allowed plane while an observation would fix a much narrower band.
A cosmological measurement of $m_\nu^{\rm lightest}$ near $2.5\times10^{-2}$ eV would point to the largest $M_1^{\rm min}\sim6\times10^9$ GeV in the mild-tuning case, whereas smaller values lower $M_1^{\rm min}$ toward $\sim7\times10^8$ GeV.
In the least-tuned scenario with a real orthogonal matrix, $M_1^{\rm min}\sim(1-2)\times10^{10}$ GeV is required, so a future preference for a lower mass scale would imply non-negligible cancellations among the heavy-neutrino contributions to the light neutrino masses.
If the fine-tuning can be as large as $\Delta\sim10^5$, thermal leptogenesis can operate with $M_1$ as low as $\sim10^6$ GeV, bringing the required reheating temperature into a range that lower-scale cosmological and collider probes could start to address.
The combined future sensitivities of neutrinoless double-beta decay experiments and cosmological neutrino-mass surveys define a concrete target region in the $(m_\nu^{\rm lightest}, m_{\beta\beta}^{\rm eff})$ plane where the minimal hierarchical thermal-leptogenesis picture can be confirmed or excluded.

Reading between the lines

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

Beyond the paper: if $M_1^{\rm min}$ stays near $10^9$ GeV, the required reheating temperature is near $10^{10}$ GeV, which would disfavour low-scale inflation models while fitting naturally into high-scale unification or $U(1)_{B-L}$ frameworks.
Beyond the paper: the same numerical machinery could rerun the scan without the $M_1\times10^3<M_2<M_3/3$ hierarchy to test whether heavier-neutrino contributions lower $M_1^{\rm min}$; the authors defer this, so treating the quoted range as robust requires that check.
Beyond the paper: a future measurement of $m_{\beta\beta}^{\rm eff}$ at the planned sensitivity, combined with a cosmological determination of $m_\nu^{\rm lightest}$, would locate the scenario on the authors' Figures 1 and 3 and, in the plateau region, single out $M_1^{\rm min}\sim(7-8)\times10^8$ GeV.
Beyond the paper: the steep drop of $M_1^{\rm min}$ with $\Delta$ offers a naturalness diagnostic, since evidence for $M_1$ around $10^6$ GeV would imply cancellations at the $10^5$ level among the heavy-neutrino contributions, a structural constraint on the seesaw model itself.

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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 content here is the systematic projection: for hierarchical type-I seesaw thermal leptogenesis with flavored density matrix equations, the paper maps M1^min onto the low-energy plane (m_lightest, m_bb^eff), for both mass orderings and for different fine-tuning levels. That specific map is new, even though the individual ingredients—flavored leptogenesis bounds, Casas-Ibarra scans, the dependence on m_lightest—exist in the literature. The numerics are credible: ULYSSES is a public, tested solver, the scan strategy is described well enough to re-implement, and the results line up with known bounds, including the drop toward ~10^6 GeV at large fine-tuning. The paper is also honest about what it does not do: N2/N3 contributions, scatterings, non-hierarchical spectra, and the initial-abundance question are all explicitly deferred.

The main soft spot is exactly that deferred initial-abundance question. The headline numbers assume zero initial N1 population at T_init = 10 M1, justified only by a footnote about an inflaton that does not couple to heavy neutrinos. That is a model-building carve-out, not a derived property. The paper never reports the decay parameter K1 at the found minima, nor any sensitivity to T_init. In the weak-washout regime (K1 ≲ 3), a thermal initial abundance would change the efficiency and could lower M1^min by an O(1) factor; in strong washout it would not matter. Without that information, the quoted range (0.7–6)×10^9 GeV should be read as 'under the zero-initial-abundance scenario,' not as a robust lower bound on M1 in the full framework. This is a real limitation, but not a fatal one: the map is still a valid statement about that scenario, and the qualitative shape of the contours is unlikely to change.

Two smaller issues. First, oscillation parameters are fixed at NuFit best fit with no uncertainty propagation; the plotted bands are therefore artificially sharp, especially near the NO lower boundary at m_bb^eff ~ 1.5×10^-3 eV, which is a delicate difference of two terms. Second, the exact scripts, parameter files, and output data are not provided. ULYSSES is public, but reproducing this 60×60 grid with differential evolution would take real effort; the authors should release their scan data. The citation pattern looks fine; the heavy use of ULYSSES is justified because it is their own publicly available code, and the relevant prior work is cited.

This paper deserves a serious referee. It is a useful phenomenological tool, not a breakthrough, and the central claim is conditional but clearly presented. I would send it to review, asking the authors to propagate oscillation uncertainties, report K1 and T_init sensitivity at the minima, and release scripts or data. If those requests are met, it becomes a solid reference for the leptogenesis scale.

For the record: I would not place the full weight of the quantitative range on the zero-initial-N1 assumption, but the paper itself flags that limitation clearly enough that a careful reader will not be misled.

Referee Report

2 major / 4 minor

Summary. The paper computes, within hierarchical thermal type-I seesaw leptogenesis with three heavy Majorana neutrinos, the minimum mass M1^min of the lightest heavy neutrino required to reproduce the observed baryon asymmetry, as a function of the lightest neutrino mass m_lightest^nu and the 0nu-beta-beta effective mass m_eff_beta_beta, for both normal and inverted light-neutrino orderings. The calculation uses the public ULYSSES flavoured density-matrix solver, with a Casas-Ibarra parameter scan and differential-evolution minimization at fixed values of a fine-tuning parameter Delta (Delta = 3, 10, and up to 10^5). The main quantitative results are: for Delta = 10 and within the sensitivity of future 0nu-beta-beta experiments (m_eff_beta_beta >= 0.0047 eV), M1^min is in the range (0.7-6) x 10^9 GeV; for Delta = 3 it is around (1-2) x 10^10 GeV; and increasing fine-tuning can lower M1^min toward 10^6 GeV. The calculation assumes a strongly hierarchical heavy spectrum (M1 x 10^3 < M2 < M3/3), an initial temperature T_init = 10 M1, and a vanishing initial N1 population.

Significance. This is a useful quantitative update that maps two potentially measurable low-energy quantities, m_lightest^nu and m_eff_beta_beta, onto the scale of thermal leptogenesis. Its main virtue is that it makes falsifiable predictions: if future 0nu-beta-beta and cosmological neutrino-mass measurements select a particular region of the (m_lightest^nu, m_eff_beta_beta) plane, the quoted M1^min intervals provide a definite target for the leptogenesis scale. The use of the established ULYSSES density-matrix solver with flavour effects is a strength, as is the explicit enumeration of the input assumptions and the introduction of a practical fine-tuning measure Delta. The computation is not circular: the observed baryon asymmetry enters as a constraint, while the low-energy observables are scanned inputs. If the robustness checks requested below confirm the initial-condition and minimization dependence, this will be a valuable reference for the community.

major comments (2)

[Sec. 4.1, Eqs. (24)-(25) and footnote 6] The headline result, exemplified by the Delta = 10 contours in Figs. 1-2 and the range (0.7-6) x 10^9 GeV quoted in Sec. 4.2.1, is computed under a vanishing initial N1 population with T_init = 10 M1. This is not an innocuous specification. In the weak-washout regime, K1 = (Y^dagger Y)_11 v^2 / (2 M1 m*) <~ 3, the final efficiency for a zero-initial-abundance run is smaller than for a thermal-initial-abundance run, so a thermal initial population would generally lower M1^min, while for K1 >~ 3 the two initial conditions coincide. The paper does not report K1 at the found minima, nor does it test the dependence on T_init. The robustness of the quoted 'required' M1 range to the initial-condition scenario is therefore not established. I request either a quantitative check (e.g., re-running selected grid points with a thermal initial N1 abundance and with T_init/M1 varied, say 1, 10, 100) or a prominent restatement that the results are bounds for the zero-initial-abundance scenario only, together with an estimate of the possible downward shift.
[Sec. 4.2, minimization procedure] The paper identifies its results as M1^min obtained by scanning eight or nine parameters with SciPy's differential evolution, but it does not provide convergence diagnostics, replicate runs with different random seeds, or a test of the 3x3 coarse-graining procedure. Since every plotted contour is an extremal statistic, local-minimum artifacts cannot be excluded without such checks. I request a short validation appendix, or statements in the text, reporting for representative grid points the spread of optimization outcomes across seeds and population sizes, and the sensitivity of the coarse-grained contours to the neighborhood size.

minor comments (4)

[Sec. 4.2] The success condition is printed as 'eta_B >= 6.1 x 10^10' but the observed baryon-to-photon ratio is 6.1 x 10^-10; this exponent error should be corrected.
[Figs. 2 and 4] The legend label 'Iiverted Ordering' is a typo for 'Inverted Ordering'.
[Sec. 4.2.3 and Fig. 5] The text states that for sufficiently large Delta M1^min can be reduced to 10^6 GeV, but the IO panel of Fig. 5 appears to reach values below 10^6 GeV; please align the statement with the figure or clarify that 10^6 GeV is an order-of-magnitude summary.
[Sec. 2.2 and Table 1] Freezing the NuFit 6.0 oscillation parameters at their best-fit values is a limitation that should be stated more prominently, because the NO boundary m_eff^-_{beta beta,N}(0) ~ 1.49 x 10^-3 eV that controls the rise in Fig. 2 is a small difference of two comparable terms and is sensitive to the input parameters at the ~10% level.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the M1^min scan is a forward-model inversion constrained by the observed baryon asymmetry, not a fitted input renamed as a prediction.

full rationale

The paper's central quantity, M1^min, is obtained by solving the flavoured density-matrix equations (Eqs. 12 and 13, with the CP-asymmetry tensor Eq. 18) over the Casas-Ibarra parameter space and then minimizing M1 subject to the successful-leptogenesis constraint eta_B >= 6.1 x 10^-10 (Sec. 4.2). This is a forward calculation with an inversion step: the observed baryon asymmetry is used as an inequality constraint, and the low-energy observables m_lightest and m_eff_beta_beta are scanned inputs, not fitted outputs. The relation between m_eff_beta_beta and the Majorana phases via Eq. (7) is a definitional constraint on the input grid, not a definition of M1^min, so no self-definitional reduction occurs. The self-citations to the ULYSSES code [34,35] and to the authors' earlier work [30] are not load-bearing in a circular way: the governing differential equations are written out in the paper itself, and ULYSSES is a public, independently usable code that solves those stated equations. The scenario assumptions — the hierarchy M1 x 10^3 < M2 < M3/3, T_init = 10 M1, vanishing initial N1 density, and fixed fine-tuning parameter Delta — are explicit modelling conditions whose numerical impact the authors also flag as future work; they affect the value of the extremal statistic M1^min but do not make the derivation equivalent to its inputs. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no known result is simply renamed. The paper is therefore self-contained as a numerical lower-bound study, with only scenario-dependence as a caveat rather than circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard type-I seesaw plus thermal leptogenesis framework. No new particles or forces are introduced. The numerical output depends on scenario choices (fine-tuning parameter Δ, heavy-mass hierarchy, initial temperature) that the authors vary or fix by hand; these are not fitted to data. The flavoured density matrix equations and the Casas-Ibarra parameterisation are taken from the prior literature.

free parameters (3)

Fine-tuning parameter Δ = 3, 10, ..., 10^5 (benchmarks)
Chosen to quantify the allowed cancellation between different heavy-neutrino contributions to light neutrino masses; not fitted to data, but a scenario parameter that controls the scan.
Heavy mass hierarchy ratio = M2 > 10^3 M1, M3 > 3 M2
Chosen to ensure only N1 contributes; without this choice, the numerical problem changes and N2/N3 contributions could alter M1^min.
Initial temperature T_init = 10 * M1
Chosen so that the heavy neutrino population starts from zero; affects the amount of washout and the minimal M1.

assumptions (5)

domain assumption Type-I seesaw with three heavy Majorana neutrinos generates light neutrino masses and provides the CP violation needed for leptogenesis
The entire analysis is built on this model (Sec. 2.1).
domain assumption The flavored density matrix equations (DMEs) correctly describe the asymmetry evolution, including flavor decoherence from charged-lepton Yukawa interactions
Adopted from prior literature [19-21] and implemented via ULYSSES; this is the core evolution framework (Sec. 3).
ad hoc to paper Only the lightest heavy neutrino contributes to leptogenesis because of the imposed hierarchy and initial conditions
This is a modeling restriction in Sec. 4.1 that makes the computation tractable and is a stated limitation.
domain assumption Standard cosmology: radiation domination, Hubble rate, sphaleron conversion, and dilution factor
Used in Eqs. (12)-(20) to convert the B-L asymmetry to the baryon-to-photon ratio.
standard math Casas-Ibarra parameterization covers all valid Yukawa matrices for given low-energy observables and heavy masses
Used to scan the model; relies on the orthogonality of O (Sec. 2.4).

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Cite this review

Pith. "Pith review of Insights on the Scale of Leptogenesis from Neutrino Masses and Neutrinoless Double-Beta Decay." pith.science (2026). https://pith.science/paper/RABY5H56

@misc{pith2026250210093,
  author       = {Pith},
  title        = {Pith review of: Insights on the Scale of Leptogenesis from Neutrino Masses and Neutrinoless Double-Beta Decay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RABY5H56}},
  note         = {Machine review of arXiv:2502.10093}
}

abstract

We revisit the thermal leptogenesis scenario in the type-I seesaw framework featuring three heavy Majorana neutrinos with a hierarchical mass spectrum. We focus on low energy observables, specifically the lightest neutrino mass $m_{\nu}^{\rm lightest}$ and the neutrinoless double-beta decay effective mass parameter $m^{\rm eff}_{\beta\beta}$. In particular, we numerically calculate the minimum mass of the lightest heavy Majorana neutrino, $M_1^{\rm min}$, required for successful leptogenesis as a function of $m_{\nu}^{\rm lightest}$ and $m_{\beta\beta}^{\rm eff}$, considering both normal and inverted light neutrino mass orderings. Flavour effects are taken into account within the flavoured density matrix formalism. We also examine the interplay between fine-tuned cancellations in the seesaw relation and $M_1^{\rm min}$. Recent and forthcoming searches for neutrinoless double-beta decay, along with cosmological probes of the sum of neutrino masses, motivate this analysis, as they can provide key insights into the minimal scale of thermal leptogenesis and its broader implications.

Figures

Figures reproduced from arXiv: 2502.10093 by the authors.

**Figure 1.** Contour plot of Mmin 1 on the (m lightest ν , meff ββ)-plane for the case of ∆ = 10. The left (right) panel is obtained for a light neutrino mass spectrum with NO (IO). Orange lines are the contours of Mmin 1 in units of 109 GeV. The current limits and future sensitivities to meff ββ and m lightest ν are as in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** The minimal lightest heavy neutrino mass Mmin 1 versus m lightest ν (left) and meff ββ (right), for NO (red) and IO (green). The red and green lines are for the cases of normal and inverted orderings, respectively. In both figures, the dashed curves are obtained without imposing the constraints on m lightest ν . Experimental constraints and future sensitivities on meff ββ and m lightest ν from cosmological and (ββ)0… view at source ↗

**Figure 3.** Contour plot of Mmin 1 on the (m lightest ν , meff ββ)-plane, for the cases of light neutrino mass spectrum with NO (left) and IO (right), and ∆ = 3 (real CI matrix case). As in [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Lower bound on M1 as a function of m lightest ν (left panel) and meff ββ (right panel), under the condition ∆=3. The red and green lines are for the cases of normal and inverted orderings, respectively. Experimental constraints on meff ββ and m lightest ν from cosmolog…

**Figure 5.** Figure 5: Dependence of Mmin 1 required by successful LG against the fine-tuning parameter ∆. The left (right) panel is for the NO (IO) case. In the NO case, we considered the benchmarks (m lightest ν /eV, meff ββ/eV) = (10−2.5 , 10−3.5 ), (10−3 , 10−2.5 ) and (10−4 , 10−2.8 ), …

**Figure 6.** Figure 6: The parameter space of neutrinoless double-beta decay searches based on Eq. (7), for either IO (blue) and NO (yellow). The neutrino oscillation parameter are fixed according to [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: the relations between F.T. and ∆ with a scatter plot. We find it more practical to quantify the amount of fine-tuning in the considered scenario with ∆, as it is more straightforwardly related to the Casas-Ibarra parameters. 101 102 103 104 105 106 107 108 ∆ 10−1 100 1…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Analytic formulation of Leptogenesis with neutrino oscillation data employing the general parametrization for neutrino mass matrix
hep-ph 2025-06 conditional novelty 5.0 of 10

Analytic CP asymmetry formulas from the Casas-Ibarra parametrization give minimum right-handed neutrino masses for successful leptogenesis, down to about 132 GeV with a neutrinophilic Higgs doublet and non-thermal production.

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