Spontaneous CP violation and leptogenesis in minimal flavour models

F. R. Joaquim; H. B. C\^amara; J. R. Rocha

REVIEW

Spontaneous CP violation and leptogenesis in minimal flavour models

H. B. Câmara, F. R. Joaquim, J. R. Rocha

High Energy Physics - Phenomenology · Submitted 29 Oct 2026 · 93 references

Pith's verdict: A genuinely predictive SCPV-flavour paper whose neutrino sector is solid; the leptogenesis leg is plausible but needs better documentation before the φ–ηB link is sold as quantitative.

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CitationF. R. Joaquim, H. B. C\^amara, J. R. Rocha. "Spontaneous CP violation and leptogenesis in minimal flavour models." arXiv:2610.39858 (2026). https://pith.science/paper/NDNP5DE3

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Two things to know. The neutrino-sector half is the real content and it holds up: the B5/B6 Z4 textures, with all flavour structure in the heavy Majorana mass matrix, give δ≈3π/2, fixed θ23 octants, m_lightest lower bounds around 60/50 meV, and nonzero mββ lower bounds. The ε-expansion in Sec. III.A agrees with the numerical χ² contours, and Eq. (35) is a clean way to reconstruct the vacuum phase from low-energy data. That part is healthy. What is genuinely new is the three-right-handed-neutrino, single-Higgs, single-singlet realization: one complex vacuum phase is the only CP source, and it feeds both the low-energy phase and the leptogenesis asymmetry. The paper also gives model-specific positive-asymmetry conditions, e.g. the sign relation (70) and bounds (71), which are useful. The citation pattern is normal: the B5/B6 textures come from Ref. [52], and the scalar-assisted leptogenesis machinery from the authors' own Ref. [80]; both are attributed properly. The soft spots are in the BAU leg. The abstract and conclusions quote an IO mββ lower bound of about 50 meV, while the body and Eq. (34) give about 70 meV; that discrepancy is real and should be fixed. More importantly, the uniqueness of the Appendix A inversion from the best-fit Mν to (y1,y2,y3) is asserted, not demonstrated. No algorithm, seed coverage, tolerances, or code is supplied, so a referee cannot independently check that the Fig. 4 regions are exhaustive. The perturbativity argument against identifying νR1 with N1,2 also has only about a 25% margin at the benchmark scales, which is thinner than the text implies. And the paper never explicitly states whether the Boltzmann equations were solved with the full numerically reconstructed couplings or with the approximate Eq. (69). I suspect the former, but the ambiguity should not be left to the reader. These issues do not make the central claim circular: the low-energy predictions stand independently of the leptogenesis analysis, and using the low-energy-fixed φ to compute ηB is a legitimate consistency test. But the φ–ηB link should be read as a benchmark-dependent consistency region, not as an independent high-energy prediction, until the inversion is documented. This paper deserves a serious referee. I would send it to review, with the expectation that the authors correct the mββ number and provide enough detail—ideally code or a reproducible search description—to verify the Appendix A inversion and the Fig. 4 computation.

Referee report

Significance

If it holds, the result is a genuinely economical link between laboratory and cosmological CP violation: one vacuum phase controls δ, the θ23 octant, the absolute mass scale and the baryon asymmetry, with no second Higgs doublet. Several features deserve credit independent of the leptogenesis leg. The ε-expansion of Sec. III A is a real derivation, not a fit: Eqs. (27), (30), (31) and (33) explain why δ ≃ 3π/2 and why the octant is anti-correlated with the ordering, and the closed-form m_lightest estimates of Eqs. (28) and (32) agree with the χ2 contours. The prediction of a non-zero mββ floor even for NO, escaping the usual cancellation region, is a falsifiable statement for LEGEND-200, CUPID and nEXO. The sign relation Eq. (70) and the resulting bound of Eq. (71) on M1/M2 are the kind of qualitative, checkable statement that makes a leptogenesis paper useful; so is the cross-sector constraint of Eq. (47) and Table IV, which ties neutrino data to the singlet scalar hierarchy. My reservations concern reproducibility of the heavy-spectrum inversion and the robustness of the Fig. 4 windows, not the neutrino-sector physics.

Major objections (4)

[Appendix A, Eqs. (A1)-(A3)] The uniqueness of the inversion is load-bearing for the entire second leg: every entry of the BEs (57), namely Y, U_R, Delta_a and Gamma_i, is computed from the (y1, y2, y3) obtained by inverting the fixed b.f. Mnu, and all of Fig. 4 rests on the branch nu_R1 ≃ N3. The paper deals with this in one sentence, 'By repeating the inversion without restricting y1,2,3, we find that every other solution violates at least one of the constraints of Fig. 4', with no algorithm, seeding window, number of solutions found, tolerances, or code. The stated argument for discarding nu_R1 ≃ N1,2 is that Eq. (A2) would then give y_R ≃ sqrt(2 M_{2,1} M_3)/v_sigma, which at M_3 = 10 v_sigma and M_{1,2} ≳ 10^12 GeV is about 4.5 against sqrt(4 pi) ≃ 3.54 — a 26% margin over what is itself a convention. Please describe the search (seed ranges, method, number of roots, convergence criteria), tabulate the actual y1,2,3 and max |(Y_R)_ij| at, say, three representative points of each panel of Fig. 4, and state which part of the plotted parameter space survives if the perturbativity ceiling is relaxed to 4 pi. As written, a reader cannot verify that Fig. 4 is exhaustive, and the 'unique viable solution' claim of Sec. V is asserted rather than demonstrated.
[Sec. IV B, Eqs. (64) and (69); Fig. 4] The manuscript does not state whether the eta_B contours of Fig. 4 are computed with the full CP asymmetry (64), using the couplings obtained from the numerical inversion of Appendix A, or with the approximate expression (69) obtained 'together with the approximate relations of Eq. (A2)'. This is not cosmetic: Eq. (68) makes the conventional type-I asymmetry vanish, so epsilon_2(wave) is the sole source of the asymmetry, and Eq. (69) drops precisely the small nu_R1 admixture that Appendix A says the numerical solution retains to all orders. Please state which expression enters the BEs (57) and, ideally, show eta_B at one benchmark point per panel (for instance M1/M2 = 0.87, M2/M3 = 0.15) computed both ways, so that the size of the neglected terms is visible. If Eq. (69) is used for the plots, the statement in Sec. V that the inversion 'includes (M_R)13,12 to all orders' is not reflected in the results.
[Sec. IV A, omission of Delta L = 2 scattering] The text states 'We include only two-body scattering processes, omitting LNV Delta L = 2 scatterings', with no quantitative justification, while the same section retains Delta N = 2 annihilations with full s-, t- and u-channel interference and the associated cubic scalar couplings (Eqs. (C8)-(C13)). At the benchmark M1 ~ 10^12 GeV and with the Yukawa couplings required by Eqs. (A2)-(A3), the heavy-neutrino-mediated Delta L = 2 rate is generically of comparable order to the Delta L = 1 rate that the paper keeps and then shows to matter (it shifts the windows of Fig. 4 by an appreciable fraction). Since the reported windows are narrow — for instance 0.865 <~ M1/M2 <~ 0.885 for B5 NO — even a factor-of-two change in efficiency moves them visibly. Please give a numerical estimate of the Delta L = 2 contribution to the washout at one benchmark point and either include it or state explicitly that the quoted windows carry that systematic uncertainty.
[Table IV and Fig. 4, choice of r_h] Fig. 4 fixes r_h = m_h1/m_h2 at the upper bound of Eq. (47), which for the two inverted-ordering cases means r_h <~ 0.03, i.e. a two-order-of-magnitude hierarchy in the singlet scalar sector. The wave asymmetry of Eq. (69) carries an explicit 1/(1 - r_h^2) and an r_h-dependent square root from the two branches of Eq. (46), so the BAU regions are being computed at a tuned corner of a sector otherwise unconstrained by the analysis (lambda_sigma, lambda'_sigma and m'^2_sigma). Please show, for one panel, how the eta_B = eta_B^0 band moves at r_h/2 and at the largest r_h consistent with Eq. (47), or state the sensitivity in the text. Without this it is not possible to tell whether the quoted windows are a property of the B5/B6 flavour structure or of the specific r_h choice.

Minor objections (9)

[Eqs. (A2), (A3), (69)-(71)] The compact notation y_{2,3} and y_{3,2} silently encodes the model: y_2 for B5 and y_3 for B6. Because this convention reverses the direction of the inequality, and hence of the bound on M1/M2 in Eq. (71) (M1/M2 > 0.864 for B5 NO versus M1/M2 < 0.881 for B6 IO), it should be stated explicitly in the text rather than left to the reader to decode from the subscripts.
[Eq. (35) and Table IV] Eq. (35), which is the linchpin connecting neutrino data to the phase used throughout Sec. V, is stated without derivation. It follows in one line from the entries of Table II — arg[(Mnu^{-1})23] = phi + const and arg[(Mnu^{-1})13,12] = -phi + const with real diagonal entries — and that derivation should be shown. It would also help to say explicitly, near Eq. (70), that only sin 4 phi enters the asymmetry, so the mod pi/2 ambiguity of Eq. (35) is physically harmless; this reassures the reader that the reconstructed phase is a one-valued input to Fig. 4.
[Table III and Ref. [4]] Table III and the surrounding text refer repeatedly to NuFIT 6.1 (IC24 with SK-atm), but Ref. [4] is the NuFIT 6.0 paper. Please reconcile the version label with the reference and confirm that the numerical values of delta and theta_23 actually used in the chi^2 are those of the stated data set.
[Abstract versus Eqs. (28) and (32); Sec. III bullets] The abstract and the text quote 3 sigma lower limits of 'approximately 60 meV (50 meV)' for m_lightest, whereas Eq. (28) gives 58 meV and 48 meV and Eq. (32) gives 60 meV and 58 meV. Please align the numbers and state clearly that these are 3 sigma lower limits on the model regions rather than best-fit values.
[Sec. III, B6 with NO] For B6 with NO the global minimum has chi^2_min = 7.1 and, as the text notes, a best-fit m_lightest beyond the KATRIN line. A short table giving (m_lightest, m_beta_beta, delta, theta_23) at chi^2_min for all four cases would make clear that the delta ≃ 3 pi/2 statement is a property of the 3 sigma region near the texture-zero solution rather than of the global minimum, and would justify the 'both models predict delta ≃ 3 pi/2' phrasing of the abstract.
[Abstract and Sec. III, cosmological bounds] The tension with the tightest cosmological bound is acknowledged in the text but not in the abstract, which advertises the m_lightest lower bound only as 'within reach of future neutrinoless double-beta decay searches'. Since the predicted scale is 60-100 meV, against sum m_nu < 0.062 eV from CMB+APS+BAO, the abstract should say that the same prediction is already in tension with the most stringent cosmological data sets, as the body of the paper correctly does.
[Eq. (52) and the lambda_Phi_sigma << 1 assumption] The neglect of the cubic couplings kappa_a and of the scatterings they induce rests on lambda_Phi_sigma << 1. Since kappa_a is proportional to v_sigma lambda_Phi_sigma with v_sigma = 10^12 GeV, please state the numerical value assumed for lambda_Phi_sigma and confirm that the neglected vertex correction to the asymmetry and the kappa-mediated scattering terms are indeed small at that value; otherwise a brief estimate of their size relative to the retained terms would be useful.
[Fig. 4 caption and reproducibility] The caption mixes filled regions with dashed contours ('blue regions (black dashed contours)') in a way that is hard to read; please state plainly which quantity is a fill and which is a contour. A small table of intermediate quantities at one benchmark per panel — epsilon_2(wave), K_i, Gamma_2/M2, N_{B-L} final, eta_B — would greatly improve reproducibility, especially as no public code accompanies the paper.
[Framing of the 'common origin' claim] Since phi is fixed to its neutrino best-fit value before the BAU scan, the leptogenesis panels of Fig. 4 are constraints on the heavy spectrum (M1/M2, M2/M3, m_h2/M1) given that phase, not a prediction of eta_B. The paper is internally consistent about this, but the abstract and the concluding paragraph could say it in one clause, which would also forestall a misreading of the 'common origin' as an over-determination.

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The paper

The paper claims

A single CP-violating phase — the vacuum expectation value of an extra scalar singlet — can simultaneously fix the leptonic Dirac phase, the atmospheric mixing octant, and generate the observed baryon asymmetry.

The reading

The paper asks whether one and the same CP-violating phase can explain both the CP violation that neutrino oscillations may soon pin down and the matter excess left over from the early universe. The authors build the smallest setting they can: the standard type-I seesaw with three right-handed neutrinos, one extra complex scalar singlet, and a $\mathcal{Z}_4$ flavour symmetry, with CP imposed exactly on the Lagrangian so that the only source of CP violation is the phase $\varphi$ of the singlet's vacuum expectation value. Because the charged-lepton and Dirac mass matrices are forced diagonal, all flavour information sits in the heavy Majorana mass matrix, and exactly two patterns survive, labelled $B_5$ and $B_6$. Those patterns turn out to be sharply predictive — $\delta\simeq3\pi/2$, a definite octant for $\theta_{23}$, and a lightest neutrino mass and effective Majorana mass above roughly 50–70 meV — and the same phase $\varphi$, fed into scalar-assisted leptogenesis, reproduces the observed baryon asymmetry in a calculable window of heavy-neutrino and scalar masses. If right, a vacuum phase inferred indirectly at low energy would be the same number that made the universe matter-dominated.

What carries the argument

Two ingredients carry the argument. The first is the pair of texture-zero conditions on $\widehat M_\nu^{-1}$ implied by the $\mathcal{Z}_4$ charge assignment — in the $B_5$ case $(\widehat M_\nu^{-1})_{12}=(\widehat M_\nu^{-1})_{33}=0$ and in the $B_6$ case $(\widehat M_\nu^{-1})_{13}=(\widehat M_\nu^{-1})_{22}=0$ — together with the observation that the two are related by $C^6_{1,2}(\theta_{23})=C^5_{1,2}(\theta_{23}+\pi/2)$, so one small-$\varepsilon$ expansion describes both. The second is the scalar-singlet-induced wave CP asymmetry $\varepsilon_2^{(\rm wave)}$, built from the couplings $\Delta_a$ between heavy neutrinos and the two real scalars $h_{1,2}$; it is the only non-zero asymmetry once the type-I piece cancels, and its overall sign, $\mathrm{sgn}\,\eta_B\simeq\mathrm{sgn}[(y_2^2-y_3^2)\sin4\varphi]$, is what converts the low-energy phase $\varphi$ into a definite direction for the baryon asymmetry.

Extended reading notes

Core claim

The claim is that the phase of a singlet vacuum expectation value can be the single CP-violating input behind both neutrino mixing and leptogenesis, in a model with no other source of CP violation. Imposing the $\mathcal{Z}_4$ symmetry on three generations gives the inverse light-neutrino mass matrix two texture zeros, in two inequivalent ways ($B_5$ and $B_6$); expanding in $\varepsilon=\Delta m^2_{21}/|\Delta m^2_{31}|\simeq0.03$ reduces these to a handful of leading-order relations, of which $(t^{(0)}_{23})^2=m_0/m_3$ fixes the atmospheric octant by the mass ordering and forces $m_{\rm lightest}\gtrsim58$ meV, while $\delta$ is left free at zeroth order and lands near $3\pi/2$ only at first order. On the leptogenesis side, the same structure makes the conventional type-I CP asymmetry vanish, since $H_{13}\simeq H_{23}\simeq0$ and $\mathrm{Im}[H_{12}^2]=0$, so the baryon asymmetry must come from the new scalar-mediated wave contribution to $\varepsilon_2$, proportional to $\sin4\varphi$. That ties the sign of $\eta_B$ to whether $y_2<y_3$, and solving the unflavoured Boltzmann equations shows that the observed $\eta_B^0=(6.12\pm0.04)\times10^{-10}$ is reproduced along narrow bands in $M_1/M_2$ versus $M_2/M_3$ and in $m_{h_2}/M_1$ versus $M_2/M_3$.

Load-bearing premise

The step from the measured light-neutrino mass matrix to a definite heavy-neutrino spectrum assumes that the numerical inversion for the three Yukawa couplings, started near the approximate relations of Appendix A, finds the only solution satisfying perturbativity and the kinematic constraints — if other viable solutions exist, the quoted baryon-asymmetry regions are incomplete rather than wrong.

Editorial extensions

If this is right

Both models place $\delta$ at $3\pi/2$, so an improved DUNE or Hyper-Kamiokande measurement away from that value would remove them, and the octant predictions ($B_5$ first for NO and second for IO; $B_6$ the reverse) give a second independent handle.
The predicted floors $m_{\beta\beta}\gtrsim60$ meV for normal and $\gtrsim70$ meV for inverted ordering lie inside the reach of KamLAND2-Zen, LEGEND, CUPID, nEXO and SNO+, so a null result at that sensitivity would exclude both models.
Because the phase is fixed by oscillation data, leptogenesis is not tuned: the best-fit values $\varphi/\pi\simeq0.26$–$0.28$ (NO) and $0.48$–$0.49$ (IO) for $B_5$, and $0.22$–$0.24$ (NO) and $0.01$–$0.02$ (IO) for $B_6$, are the same numbers that enter the Boltzmann equations.
Neutrino data also constrain the scalar sector through the reality of the mixing angle $\theta_\sigma$, bounding $r_h=m_{h_1}/m_{h_2}$ at $0.87$ ($B_5$ NO), $0.03$ ($B_5$ IO), $0.90$ ($B_6$ NO) and $0.03$ ($B_6$ IO).
Since the standard type-I asymmetry vanishes in these textures, any baryon asymmetry at all is a signature of the extra singlet: switching off the scalar-assisted contribution sets $\eta_B$ to zero rather than merely reducing it.

Reading between the lines

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

The $m_{\beta\beta}$ floor for normal ordering sits above the usual cancellation region where oscillation data alone allow near-zero values, so a signal in the 60–70 meV band would be suggestive but not decisive, while a well-motivated nearby variant of the charge assignment could sharpen the prediction enough to discriminate.
Because $\sin4\varphi$ controls both the magnitude and, through $y_2-y_3$, the sign of the asymmetry, the phase is doing two jobs at once; small perturbations of the $\mathcal{Z}_4$ assignment would plausibly either flip the sign of $\eta_B$ or kill the asymmetry entirely, which makes the construction fragile in an informative way.
The leptogenesis analysis is unflavoured and anchored at $M_1\gtrsim10^{12}$ GeV with $M_3=10^{13}$ GeV and $v_\sigma=10^{12}$ GeV; since flavour effects and the new scalar scatterings both grow as the mass scale falls, the natural next test is whether the same phase still works below the unflavoured regime.
The inverted-ordering branches, which fit the oscillation data best, are partly excluded by the stricter CMB+BAO bound on the sum of neutrino masses quoted here, so the model's viability is entangled with a cosmological dataset whose systematics are still under discussion.

Original abstract

We consider a type-I seesaw framework extended by a scalar singlet whose vacuum expectation value (VEV) breaks CP spontaneously. The singlet VEV phase provides a common origin for low-energy leptonic CP violation and the high energy CP violation needed to generate the observed baryon asymmetry of the Universe (BAU) through leptogenesis. We identify two minimal flavour models based on a discrete $\mathcal{Z}_4$ symmetry, whose neutrino flavour structure is encoded entirely in the heavy Majorana mass matrix. Both models predict $\delta \simeq 3\pi/2$, a definite octant for the atmospheric $\theta_{23}$ mixing angle, and lower bounds of approximately $60~\mathrm{meV}$ ($50~\mathrm{meV}$) on the lightest neutrino mass and the effective Majorana mass $m_{\beta\beta}$ for normal (inverted) ordering, within reach of future neutrinoless double-beta decay searches. We study scalar-singlet assisted leptogenesis in the non-resonant regime, including new decay, scattering, and washout processes involving the heavy scalars, as well as their contributions to the CP asymmetries. By solving the unflavoured Boltzmann equations, we identify the regions of parameter space in which the minimal flavour models successfully account for the observed BAU.

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, 2026-10-30: 1 request, 5,227 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.07 · 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.

We consider a type-I seesaw framework extended by a scalar singlet whose vacuum expectation value (VEV) breaks CP spontaneously. The singlet VEV phase provides a common origin for low-energy leptonic CP violation and the high energy CP violation needed to generate the observed baryon asymmetry of the Universe (BAU) through leptogenesis. We identify two minimal flavour models based on a discrete $\mathcal{Z}_4$ symmetry, whose neutrino flavour structure is encoded entirely in the heavy Majorana mass matrix. Both models predict $\delta \simeq 3\pi/2$, a definite octant for the atmospheric $\theta_{23}$ mixing angle, and lower bounds of approximately $60~\mathrm{meV}$ ($50~\mathrm{meV}$) on the lightest neutrino mass and the effective Majorana mass $m_{\beta\beta}$ for normal (inverted) ordering, within reach of future neutrinoless double-beta decay searches. We study scalar-singlet assisted leptogenesis in the non-resonant regime, including new decay, scattering, and washout processes involving the heavy scalars, as well as their contributions to the CP asymmetries. By solving the unflavoured Boltzmann equations, we identify the regions of parameter space in which the minimal flavour models successfully account for the observed BAU.

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	We consider a type-I seesaw framework extended by a scalar singlet whose vacuum expectation value (VEV) breaks CP spontaneously.	contribution 0.78	0.73	0.05	asserted 0.94
s1	The singlet VEV phase provides a common origin for low-energy leptonic CP violation and the high energy CP violation needed to generate the observed baryon asymmetry of the Universe (BAU) through leptogenesis.	contribution 0.60	0.43	0.38	asserted 1.00
s2	We identify two minimal flavour models based on a discrete $\mathcal{Z}_4$ symmetry, whose neutrino flavour structure is encoded entirely in the heavy Majorana mass matrix.	contribution 0.91	0.92	0.08	asserted 1.00
s3	Both models predict $\delta \simeq 3\pi/2$, a definite octant for the atmospheric $\theta_{23}$ mixing angle, and lower bounds of approximately $60~\mathrm{meV}$ ($50~\mathrm{meV}$) on the lightest neutrino mass and the effective Majorana mass $m_{\beta\beta}$ for normal (inverted) ordering, within reach of future neutrinoless double-beta decay searches.	result 1.00	0.93	0.36	asserted 0.99
s4	We study scalar-singlet assisted leptogenesis in the non-resonant regime, including new decay, scattering, and washout processes involving the heavy scalars, as well as their contributions to the CP asymmetries.	contribution 0.55	0.94	0.05	asserted 0.97
s5	By solving the unflavoured Boltzmann equations, we identify the regions of parameter space in which the minimal flavour models successfully account for the observed BAU.	contribution 0.44	0.96	0.22	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from simulation or computation 0.81.

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

not run

S sections
T body sentences
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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 3.** FIG. 3: Diagrams contributing to the unflavoured CP asymmetry in the LNV decay

Showing 3 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.39858 by the authors.

References

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Discussion

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