Minimally modified Frampton--Glashow--Yanagida ansatz and its leptogenesis consequences: in light of recent JUNO's result

Xing-Yu Wang; Zhen-hua Zhao

REVIEW

Minimally modified Frampton--Glashow--Yanagida ansatz and its leptogenesis consequences: in light of recent JUNO's result

Xing-Yu Wang, Zhen-hua Zhao

High Energy Physics - Phenomenology · Submitted 8 Oct 2026

Pith's verdict: Solid, honest, and thorough phenomenological study that clearly states its assumptions and provides a useful catalog of phase correlations and leptogenesis outcomes within a well-defined framework.

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arXiv ID2610.11789 (version 2)

Pith IDQ4HTDYXA, Pith's permanent record of this paper and its review

CitationXing-Yu Wang, Zhen-hua Zhao. "Minimally modified Frampton--Glashow--Yanagida ansatz and its leptogenesis consequences: in light of recent JUNO's result." arXiv:2610.11789 (2026). https://pith.science/paper/Q4HTDYXA

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Editor's note

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This is a well-executed piece of neutrino phenomenology. What's new: the explicit-CP MMFGY realization (twelve high-energy configurations reducing to three low-energy classes) and the systematic study of approximate second zeros and leptogenesis across multiple mass regimes for both realizations. The remnant-CP part extends Barreiros et al. (2019) with leptogenesis, which is genuinely new. The paper does what it says: it takes a minimal relaxation of the FGY texture-zero condition, imposes either a single CP phase or a remnant CP symmetry, works out the low-energy phase correlations, and then tests whether the resulting Yukawa structures can support leptogenesis.

What it does well: the mathematical derivation of the low-energy constraints (Eq. 21) is self-contained and correct. The leptogenesis scans use established numerical tools (ULYSSES) and cover three distinct mass regimes (non-degenerate, TeV-scale quasi-degenerate, and low-scale). The authors are honest about their assumptions—normal ordering, m1=0, minimal two-RHN seesaw—and they acknowledge the omission of off-shell ΔL=2 washout at high scales and the sensitivity to initial RHN abundance. They don't overclaim: they find parameter regions where successful leptogenesis is possible, not a unique prediction. The approximate second zero suppression reaches about a factor of ten, which is moderate but real.

Soft spots: the entire analysis assumes normal ordering with m1=0. That's a scope condition, stated upfront, but it means the framework has no predictive power if future data favor inverted ordering or nonzero m1. The approximate second zero is only about an order of magnitude suppression; the full-matrix hierarchy in the quasi-degenerate regime is 1:5, not extreme. These are not flaws, just results—the paper reports them accurately. The leptogenesis scan at the highest mass scales omits off-shell washout, which could strengthen the constraints, but the authors flag this.

This paper is for neutrino phenomenologists who work on texture zeros, CP phases, and leptogenesis within the minimal seesaw. It will inform future experimental searches for leptonic CP violation and neutrinoless double-beta decay. A serious referee should handle it.

Referee report

Minor objections (6)

[§2.1.1, Eq. (10)] The notation H_i;αβ is introduced but its connection to the texture patterns could be clarified. Consider adding a sentence immediately after Eq. (10) that explicitly states the column index i (1 or 2) and the flavor indices α (zero row), β (CP-source row), and γ (remaining row), since these are used throughout the derivation.
[§2.2.1, Fig. 1] The vertical axis label uses a tilde over sigma (˜σ) but the caption and text refer to 'continuous representative' of the Majorana phase. The description of the unwrapping convention (˜σ=0 at δ=π) is given in the text but could be repeated in the caption for clarity.
[§2.2.3, Figs. 4–10] The color scale used in multiple figures (e.g., Figs. 4 and 5) is very rich; consider adding a short note in the caption that the symmetric logarithmic scale is chosen to show both positive and negative values of R_B, and that black contours correspond to R_B=1. The dot-dashed curves in Fig. 6 are labelled in the caption but not in the legend inside the figure; adding a legend would improve readability.
[§3.2.2, Figs. 12–18] The notation 'No B = 1' in Fig. 13 panels (Y5-OI and Y5-OIII for thermal initial abundance) is helpful. However, the same absence occurs in some other panels (e.g., Y2-OII and Y2-OIII in Fig. 12 are missing R_B=1 contours) but is not flagged. Consider adding a note in the caption or a uniform marker for panels where no successful solution is found within the scanned region.
[§4, Conclusions] The summary of leptogenesis results states that the explicit-CP realization admits successful leptogenesis across all considered benchmarks. The text could briefly note that at M1=1 GeV with thermal initial abundance no resolved R_B=1 contour is found (as acknowledged in the body), to avoid potential overstatement in the abstract.
[General] Some references lack publication year or placeholders (e.g., Ref. [3] 'NuFIT webpage. Accessed 2026.'). The manuscript would benefit from consistent formatting of arXiv identifiers and, where available, journal references. Please also check that all references cited in the text appear in the bibliography (e.g., Ref. [32] is used but its full details are given only in the text).

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The paper

The paper claims

Minimally modified FGY texture works with normal ordering and predicts phase correlations.

The reading

The paper proposes a minimal modification of the Frampton–Glashow–Yanagida (FGY) texture in the type-I seesaw model with two right-handed neutrinos, motivated by the recent JUNO indication favoring normal neutrino mass ordering. Instead of two exact Yukawa zeros, the modified texture retains one exact zero and restricts the CP structure in one of two ways: either a single explicit CP-violating phase or a remnant CP symmetry. The authors show that under normal ordering with the lightest neutrino mass set to zero, the framework predicts correlations between the Dirac and Majorana CP phases and yields a definite range for the neutrinoless double-$\beta$ decay effective mass. Successful leptogenesis is found across a wide mass range from GeV to $10^{13}$ GeV, but the viable parameter space depends strongly on the chosen Yukawa configuration, the degree of right-handed neutrino degeneracy, and the initial abundance, demonstrating that different high-energy restrictions on the Yukawa matrix produce distinct baryogenesis predictions.

What carries the argument

The key machinery is the Casas–Ibarra parametrization, which expresses the neutrino Yukawa matrix in terms of the low-energy mixing matrix, the light-neutrino masses, a complex orthogonal angle, and the right-handed neutrino masses. Imposing one exact Yukawa texture zero fixes the complex Casas–Ibarra angle up to discrete choices, while the CP restriction (either a single explicit phase or a remnant CP symmetry) yields a relation between the Dirac and Majorana phases that depends only on the mixing angles and mass-squared differences. The two realizations differ in how the remaining Yukawa degrees of freedom are constrained: the explicit-CP realization requires that exactly one of the two physical phases in the Yukawa matrix be the sole source of CP violation, while the remnant-CP realization requires that the Casas–Ibarra matrix be real or purely imaginary (OI, OII, OIII branches) as a consequence of a generalized CP symmetry.

Extended reading notes

Core claim

The core discovery is that the minimally modified FGY (MMFGY) framework provides an economical setting in which low-energy neutrino observables can be connected with leptogenesis and characteristic correlations among the low-energy CP phases can be predicted, without imposing the full two-zero FGY texture. In the explicit-CP realization, twelve high-energy Yukawa configurations reduce to three independent low-energy equivalence classes, each giving a distinct relation between the Dirac phase δ and the Majorana phase σ and a corresponding prediction for the effective Majorana mass ⟨m_ee⟩ at the $10^{{-3}}$ eV level. In both realizations, a strongly suppressed second Yukawa entry can emerge at the level of about 10% of the other entries in the same right-handed neutrino column, although the full-matrix hierarchy remains moderate in the quasi-degenerate regime. The leptogenesis analysis demonstrates that the observed baryon asymmetry can be reproduced for non-degenerate, TeV-scale quasi-degenerate, and low-mass (100 GeV, 1 GeV) right-handed neutrinos, with successful parameter regions highly sensitive to the detailed high-energy Yukawa structure, the mass splitting, and the initial right-handed neutrino abundance.

Load-bearing premise

The entire analysis assumes normal neutrino mass ordering with the lightest neutrino mass exactly zero; if the true ordering is inverted or m1 is nonzero, the framework and all its predictions are invalidated.

Editorial extensions

If this is right

If the MMFGY framework is correct, the Dirac phase δ and Majorana phase σ must satisfy one of three distinct correlation patterns, which can be tested by future neutrino oscillation and neutrinoless double-beta decay experiments.
Successful leptogenesis in the framework restricts the allowed Dirac-phase intervals without removing the approximate second-zero behavior, implying that the approximate texture persists as a testable feature.
The two realizations (explicit-CP and remnant-CP) yield different leptogenesis predictions, meaning that future measurements of the baryon asymmetry and its relation to low-energy CP violation could distinguish the underlying mechanism that restricts the high-energy Yukawa structure.
The strong sensitivity of leptogenesis to the initial right-handed neutrino abundance at the 1 GeV scale provides a cosmology-dependent test: if the asymmetry is measured and the framework is assumed, the initial abundance can be inferred.
The predicted ⟨m_ee⟩ lies in the 10^{-3} eV range, which is within reach of upcoming neutrinoless double-beta decay experiments, so a positive signal in that range would provide a nontrivial test of the framework.

Reading between the lines

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

The MMFGY framework implicitly assumes that the exact texture zero arises from an underlying flavor symmetry; if future data confirm the predicted phase correlations, it would motivate a UV completion that realizes the one-zero texture and the specific CP restriction.
The fact that the explicit-CP and remnant-CP realizations produce distinct leptogenesis patterns suggests that the origin of CP violation in the seesaw sector could be inferred from a combined measurement of low-energy CP phases and the baryon asymmetry, even if the high-energy Yukawa entries are not directly accessible.
The approximate second-zero suppression at the 10% level is not accidental: it arises from the requirement that the framework reproduce the observed mixing angles and mass-squared differences, so any future experiment that measures the high-energy Yukawa couplings (e.g., through heavy neutrino searches) could verify this hierarchy.
If the mass ordering is eventually found to be inverted or if m1 is measured to be nonzero, the entire MMFGY analysis collapses, forcing model builders to seek analogous texture modifications that accommodate the inverted ordering while retaining predictive power.

Original abstract

Motivated by the recent JUNO indication favoring normal neutrino mass ordering, we study two realizations of the minimally modified Frampton--Glashow--Yanagida (MMFGY) framework in the minimal type-I seesaw with two right-handed neutrinos. Both retain one exact Yukawa texture zero, supplemented by either a single physical CP-violating phase or a remnant CP symmetry. The explicit-CP realization yields twelve high-energy Yukawa configurations that reduce to three low-energy classes, with correlated CP phases and predictions for neutrinoless double-beta decay. In both realizations, a strongly suppressed second Yukawa entry can emerge, although the full-matrix hierarchy remains moderate in the quasi-degenerate regime. We analyze leptogenesis for a non-degenerate benchmark, quasi-degenerate TeV-scale right-handed neutrinos, and lower masses where coherent production, oscillations, and electroweak sphaleron freeze-out become important. Successful leptogenesis is obtained across a broad range of the scanned benchmarks, with increased sensitivity to the initial right-handed-neutrino abundance at the lowest mass scale. Requiring successful leptogenesis restricts the allowed phases while preserving the possibility of an approximate second zero. The two realizations exhibit distinct approximate-zero patterns and leptogenesis predictions, demonstrating how different restrictions on the high-energy Yukawa structure affect baryogenesis.

The compiled record

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

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-09: 1 request, 6,417 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.65 · compares with prior results 0.23 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Motivated by the recent JUNO indication favoring normal neutrino mass ordering, we study two realizations of the minimally modified Frampton--Glashow--Yanagida (MMFGY) framework in the minimal type-I seesaw with two right-handed neutrinos. Both retain one exact Yukawa texture zero, supplemented by either a single physical CP-violating phase or a remnant CP symmetry. The explicit-CP realization yields twelve high-energy Yukawa configurations that reduce to three low-energy classes, with correlated CP phases and predictions for neutrinoless double-beta decay. In both realizations, a strongly suppressed second Yukawa entry can emerge, although the full-matrix hierarchy remains moderate in the quasi-degenerate regime. We analyze leptogenesis for a non-degenerate benchmark, quasi-degenerate TeV-scale right-handed neutrinos, and lower masses where coherent production, oscillations, and electroweak sphaleron freeze-out become important. Successful leptogenesis is obtained across a broad range of the scanned benchmarks, with increased sensitivity to the initial right-handed-neutrino abundance at the lowest mass scale. Requiring successful leptogenesis restricts the allowed phases while preserving the possibility of an approximate second zero. The two realizations exhibit distinct approximate-zero patterns and leptogenesis predictions, demonstrating how different restrictions on the high-energy Yukawa structure affect baryogenesis.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Motivated by the recent JUNO indication favoring normal neutrino mass ordering, we study two realizations of the minimally modified Frampton--Glashow--Yanagida (MMFGY) framework in the minimal type-I seesaw with two right-handed neutrinos.	contribution 0.89	0.81	0.05	asserted 0.83
s1	Both retain one exact Yukawa texture zero, supplemented by either a single physical CP-violating phase or a remnant CP symmetry.	method 0.58	0.53	0.05	asserted 1.00
s2	The explicit-CP realization yields twelve high-energy Yukawa configurations that reduce to three low-energy classes, with correlated CP phases and predictions for neutrinoless double-beta decay.	result 0.81	0.90	0.11	asserted 1.00
s3	In both realizations, a strongly suppressed second Yukawa entry can emerge, although the full-matrix hierarchy remains moderate in the quasi-degenerate regime.	result 1.00	0.84	0.06	hedged possibility 0.86
s4	We analyze leptogenesis for a non-degenerate benchmark, quasi-degenerate TeV-scale right-handed neutrinos, and lower masses where coherent production, oscillations, and electroweak sphaleron freeze-out become important.	method 0.76	0.89	0.05	asserted 0.89
s5	Successful leptogenesis is obtained across a broad range of the scanned benchmarks, with increased sensitivity to the initial right-handed-neutrino abundance at the lowest mass scale.	result 1.00	0.93	0.37	asserted 1.00
s6	Requiring successful leptogenesis restricts the allowed phases while preserving the possibility of an approximate second zero.	result 0.87	0.87	0.20	asserted 0.95
s7	The two realizations exhibit distinct approximate-zero patterns and leptogenesis predictions, demonstrating how different restrictions on the high-energy Yukawa structure affect baryogenesis.	interpretation 0.77	0.89	0.53	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.92 · evidence from simulation or computation 0.89.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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

levels run

P abstract

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

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

Figures

**Figure 1.** Correlations between the Dirac phase δ and the continuous representative σe of the Majorana phase for the three independent low-energy classes of the minimally modified FGY ansatz, evaluated using the NuFIT 6.1 best-fit inputs. Away from this point, the (e, τ ) and (e, µ) classes require σ to vary with δ, but with different slopes. This difference arises because the muon- and tau-flavor components enter the corresponding conditions Feτ = 0 and Feµ = 0 with different mixing-angle weights. By contrast, the (µ, e) trajectory remains close to σ = 0 throughout the scanned range of δ.…

**Figure 2.** Effective Majorana mass ⟨mee⟩ as a function of the Dirac phase δ for the three independent low-energy classes. The solid curves correspond to the NuFIT 6.1 best-fit inputs. The shaded bands show the ranges obtained from the endpoint combinations of the one-dimensional 3σ intervals of the three mixing angles and two mass-squared differences, with the phase relation re-solved for each input set. The widths of the shaded bands have two origins. Variations of the oscillation parameters in their 3σ intervals directly change the magnitudes of the two terms in Eq. (32). They also change…

**Figure 3.** Relative suppression ϵρj of the candidate second-zero entries for the six independent H1;αβ configurations, evaluated along the allowed phase correlations using the NuFIT 6.1 best-fit inputs. Both possible candidate flavor rows are shown for each configuration. The suppression factors are displayed without clipping on a logarithmic scale. Values ϵρj > 1 indicate that the specified entry is not the smallest nonzero element in its RHN column. The corresponding H2;αβ configurations follow by interchanging the two RHN columns. remaining row to the CP-sourcing flavor of each MMFGY class…

**Figure 4.** Non-degenerate leptogenesis in the explicit-CP realization with

**Figure 6.** Resonant leptogenesis at M1 = 1 TeV for the six H1;αβ configurations with the exact zero in the first RHN column. The colored regions show RB = ηB/ηobs B for vanishing initial RHN abundance in the (dM, δ) plane. The solid black contours mark RB = 1 for this initial condition, while the dashed black contours show the corresponding results for thermal initial RHN abundance where they are distinct. The dot-dashed curves mark ∆M/Γ = 1, and the horizontal dotted lines indicate δ = π. 15

**Figure 7.** Low-scale leptogenesis at M1 = 100 GeV for the six H1;αβ configurations of the explicit-CP realization. Within each pair, the left and right panels correspond to vanishing and thermal initial RHN abundances, respectively. Colors show RB = ηB/ηobs B , and black contours denote RB = 1 for the initial condition displayed in that panel. Both figures at this mass share the same color scale. 16

**Figure 9.** Low-scale leptogenesis at M1 = 1 GeV for the six H1;αβ configurations. The colored regions show RB = ηB/ηobs B in the (dM, δ) plane for vanishing initial RHN abundance, and the black contours denote RB = 1. 18

**Figure 11.** Fixed-position approximate-second-zero measure for the one-zero Yukawa textures

**Figure 12.** Non-degenerate leptogenesis in the remnant-CP framework with normal ordering and

**Figure 14.** Resonant leptogenesis results at M1 = 1 TeV for the six remnant-CP configurations Yi–OI , with i = 1, . . . , 6. Colors show RB = ηB/ηobs B for vanishing initial RHN abundance. Solid black contours mark RB = 1 for vanishing initial abundance; dashed black contours show the thermal-initial-abundance results where they differ from the solid contours. Dot-dashed curves mark ∆M/Γ = 1, and horizontal dotted lines indicate δ = 180◦ . The six OI configurations select different ranges of the Dirac phase for successful leptogenesis. For Y1, Y3, and Y5, the successful contours lie below δ = π

**Figure 15.** Low-scale leptogenesis at M1 = 100 GeV for the six remnant-CP configurations associated with Y1, Y2, and Y3. Within each pair, the left and right panels correspond to vanishing and thermal initial RHN abundances, respectively. Colors show RB = ηB/ηobs B , and black contours mark RB = 1 for the initial condition displayed in that panel. Both figures at this mass scale share the same color scale. The results show that all twelve configurations admit successful contours for both initial conditions, but the paired panels reveal substantial differences in their predictions. For…

**Figure 17.** Low-scale leptogenesis at M1 = 1 GeV for the six remnant-CP configurations associated with Y1, Y2, and Y3. The colored background shows RB = ηB/ηobs B for vanishing initial RHN abundance, and solid black contours mark RB = 1. Both figures at this mass scale share the same color scale. 27

Showing 12 of the paper's 18 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.11789 by the authors.

References

Discussion

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