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arxiv: 2605.31454 · v1 · pith:G7DMWU4Xnew · submitted 2026-05-29 · ✦ hep-ph

Dirac-Phase CP-Violation in the Low-Scale Type-I Seesaw with Three Right-Handed Neutrinos

Alessandro Granelli , Juraj Klari\'c , S. T. Petcov This is my paper

Pith reviewed 2026-06-28 21:43 UTC · model grok-4.3

classification ✦ hep-ph
keywords type-I seesawDirac phaselow-scale leptogenesisheavy Majorana neutrinosCP violationPMNS matrixneutrinoless double beta decaybaryon asymmetry
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The pith

CP violation solely from the Dirac phase enables low-scale leptogenesis with testable heavy neutrinos.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a specific non-real but CP-conserving form for the orthogonal matrix in the Casas-Ibarra parametrization of the type-I seesaw. This structure ensures that all CP violation in the neutrino Yukawa couplings originates exclusively from the low-energy Dirac phase δ in the PMNS mixing matrix. Under this restriction, the authors examine quasi-degenerate heavy Majorana neutrinos in the 0.1-100 GeV range and identify the allowed subregions of flavor coupling ratios that remain testable at colliders. They show that the observed baryon asymmetry can still be generated via leptogenesis across the full testable parameter space, even when δ departs from its CP-conserving values by as little as 10^{-5}.

Core claim

A CP-conserving non-real 3x3 orthogonal matrix parametrized by two real angles and one imaginary parameter can be constructed so that the only CP-violating phases in the Yukawa couplings are those of the PMNS matrix; under this hypothesis, Dirac-phase CP violation alone suffices to generate the observed baryon asymmetry through low-scale leptogenesis for quasi-degenerate heavy neutrinos whose mixings are testable at colliders.

What carries the argument

The CP-conserving non-real structure of the 3x3 orthogonal matrix in the Casas-Ibarra parametrization, which isolates all CP violation to the Dirac phase δ.

If this is right

  • Only restricted subregions of the ternary space of squared couplings Θ_e² : Θ_μ² : Θ_τ² remain compatible with Dirac-phase CP violation and collider testability.
  • The effective Majorana mass parameter for neutrinoless double-beta decay acquires specific forms fixed by the same orthogonal-matrix structure.
  • The baryon asymmetry vanishes exactly when δ equals 0, π or 2π, but is recovered for deviations as small as O(10^{-5}).
  • Low-scale leptogenesis remains viable throughout the entire testable region under this CP-violation hypothesis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If future collider data exclude the allowed coupling-ratio subregions while still finding heavy neutrinos, the hypothesis that CP violation arises only from δ would be ruled out independently of leptogenesis calculations.
  • The smallness of the required δ deviation suggests that ultraviolet models with nearly exact CP symmetry could still account for the baryon asymmetry provided the orthogonal-matrix structure is realized.

Load-bearing premise

A specific CP-conserving non-real form of the orthogonal matrix can always be chosen so that the only source of CP violation in the Yukawa couplings is the Dirac phase of the PMNS matrix.

What would settle it

Observation of a non-zero baryon asymmetry together with collider evidence for quasi-degenerate heavy neutrinos whose flavor coupling ratios lie outside the allowed subregions would falsify the scenario.

Figures

Figures reproduced from arXiv: 2605.31454 by Alessandro Granelli, Juraj Klari\'c, S. T. Petcov.

Figure 1
Figure 1. Figure 1: The parameter spaces in the Θ2 e /Θ2 -Θ2 µ /Θ2 -Θ2 τ /Θ2 ternary plane associated to the three different cases of CP-conserving CI matrix; the top (bottom) panel(s) is (are) for IH (NH), m3(1) = 0. The red stars, green triangles and blue circles correspond to the discussed CP1, CP2 and CP3 cases, respectively. The angle xν is varied within [0, 2π], while δ within 0 < δ < π and π < δ < 2π. The parameter |y|… view at source ↗
Figure 2
Figure 2. Figure 2: The parameter spaces in the Θ2 e /Θ2 -Θ2 µ /Θ2 -Θ2 τ /Θ2 ternary plane associated to the three different cases of CP-conserving CI matrix in the case mlightest ν = 0.1 eV, corresponding to a QD spectrum. The right plot is obtained after applying the constraint |y| ≥ ymin = 0.6. All other details are as in [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The values of (mν)ee versus mlightest ν in the three different cases of CP-conserving CI matrix – red stars for CP1, green triangles for CP2, blue squares for CP3 – with the Majorana phases fixed accordingly as in Eqs. (40-42). The Dirac phase is fixed at 3π/2 on the left, and varied within [0, 2π] on the right. The PMNS angles are varied within the 3σ ranges of the NuFit 6.0 global analysis [84]; the yell… view at source ↗
Figure 4
Figure 4. Figure 4: Scan of the parameter space where leptogenesis is successful in the case of low￾energy CPV solely from the Dirac phase δ, for the benchmark case CP1 with NH, i.e. NO and mlightest ν ≃ 0 (see Figs. 5 for the scans in the IH and QD cases). We show the scan in the |Θαj | 2 versus Mj plane, for the flavour α = e, µ, τ in the panel on the top-left, top-right and bottom￾left, respectively. In the bottom-right pa… view at source ↗
Figure 5
Figure 5. Figure 5: The same scan as in [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the heavy neutrino abundances NNj ≡ (ρN )jj (purple solid for j = 1, brown dot–dashed for j = 2, pink dotted for j = 3), their CP-asymmetries |NNj − NNj |, NNj ≡ (ρN )jj (black curves with the same line styles), the flavour asymmetries µ∆α with α = e, µ, τ (blue, orange and green curves, respectively; solid/dashed lines denote positive/negative values), and the baryon-to-photon ratio ηB (red c… view at source ↗
Figure 7
Figure 7. Figure 7: On the left, the numerical scan around the benchmark point depicted in [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Parameter scan for NH and CP1 around the region where Θ2 τ ≪ Θ2 µ , Θ2 e . The scan is obtained for α23 = 0, δ = π ± ϵδ and xν = ¯xν ± ϵν, where ¯xν is the angle that allows for Θ2 τ = 0 when δ = π and ϵδ, ϵν small parameters. The parameters are randomly varied in the following ranges: 0.1 ≤ M˜ 1/GeV ≤ 100, 10−11 ≤ |∆M˜ 21|/M˜ 1, |∆M˜ 32|/M˜ 1 ≤ 10−4 , 2 ≤ |y| ≤ 12, 10−8 ≤ ϵδ, ϵν ≤ 10−2 . The remaining osc… view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the heavy neutrino abundances NNj ≡ (ρN )jj , their CP-asymmetries |NNj − NNj |, NNj , the flavour asymmetries µ∆α with α = e, µ, τ , and the baryon-to-photon ratio ηB, as functions of Tew/T, for the benchmarks point in Eq. (81). All other details are as in [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: BAU versus δ for a specific benchmark point for which we get successful leptoge￾nesis for δ = π + 1.05 × 10−5 . The parameters are as in Eq. (81). The red band corresponds to the 3σ range of the observed BAU, η obs B = 6.10+0.08 −0.08 × 10−10 . analogous channels involving the τ lepton [176,177]. Their phenomenology depends on flavour￾off-diagonal combinations of mixings, Θ2 αβ ≡ [PITH_FULL_IMAGE:figures… view at source ↗
Figure 11
Figure 11. Figure 11: The scan in the Θ2 µe-Mav plane extending to Mav = 10 TeV. For illustrative purposes, we depict only the case of NH (NO with mlightest ν ≃ 0), CP1, i.e. CI matrix as in (37), and fix the Majorana phases as in Eq. (40) to ensure the CPV only comes from the Dirac phase δ. The black curve indicate the maximal allowed combination Θ2 µe that is compatible with successful leptogenesis with CPV uniquely from the… view at source ↗
Figure 12
Figure 12. Figure 12: Scan of the parameter space in the CP2 case (first four panels) and CP3 (last four panels) for NH; points are marked with triangles for CP2 and squares for CP3; all other details are as in [PITH_FULL_IMAGE:figures/full_fig_p049_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Scan of the parameter space in the CP2 case (first four panels) and CP3 (last four panels) for IH; points are marked with triangles for CP2 and squares for CP3; all other details are as in the four top panels of [PITH_FULL_IMAGE:figures/full_fig_p050_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Scan of the parameter space in the CP2 case (first four panels) and CP3 (last four panels) for QD points are marked with triangles for CP2 and squares for CP3; all other details are as in the last four panels of [PITH_FULL_IMAGE:figures/full_fig_p051_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Evolution of the heavy neutrino abundances NNj ≡ (ρN )jj , their CP-asymmetries |NNj − NNj |, NNj , the flavour asymmetries µ∆α with α = e, µ, τ , and the baryon-to-photon ratio ηB, as functions of x ≡ Tew/T, for the benchmarks point in the NH case, Eq. (126) (top), IH, Eq. (131) (bottom-left) and QD, Eq. (136) (bottom-right). All other details are as in [PITH_FULL_IMAGE:figures/full_fig_p055_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: On the left we show the scan over the splittings and xN around the benchmark point in Eqs. (75) in the top panel, (131) middle panel, (136) bottom panel. On the right the same points of the scan but in the ternary plane. 56 [PITH_FULL_IMAGE:figures/full_fig_p056_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The behaviour of the BAU versus the Dirac phase δ for the benchmark points discussed in the text in the NH case (top panel), IH case (middle panel) and QD (bottom panel) at which leptogenesis is viable for δ relatively close to CP-conserving values. See the text for details on the model parameters. 57 [PITH_FULL_IMAGE:figures/full_fig_p057_17.png] view at source ↗
read the original abstract

We study the low-scale type-I seesaw with three right-handed neutrinos (i.e. heavy Majorana neutrinos) when the CP-violation arises solely from the low-energy Dirac phase $\delta$ of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix and the heavy neutrinos have testable mixings. We derive a CP-conserving and non-real structure of the $3\times 3$ orthogonal matrix entering the Casas-Ibarra parametrisation in terms of two real angles and one single imaginary parameter, ensuring that the only CP-violating phases in the neutrino Yukawa couplings are those of the PMNS matrix. We then focus on the case of CP-violation from $\delta$ alone and discuss the phenomenological implications of this hypothesis. We concentrate on quasi-degenerate heavy Majorana neutrinos with masses within $\sim (0.1-100)\,\text{GeV}$, as relevant for low-scale leptogenesis. Only certain subregions of the full ternary space defined by the ratios $\Theta^2_e:\Theta^2_\mu:\Theta^2_\tau$ -- where $\Theta^2_\alpha$ denotes the squared coupling of the heavy neutrinos to leptons of flavour $\alpha = e,\,\mu,\,\tau$ -- are compatible with Dirac-phase CP-violation while being testable at collider experiments. Our assumption also implies specific forms of the effective Majorana mass parameter that can be tested at neutrinoless double-beta decay searches. Finally, low-scale leptogenesis under this restrictive scenario can still reproduce the observed baryon asymmetry of the Universe (BAU) in the entire testable region of the parameter space. The BAU vanishes in the exact limit of CP-conserving values of the Dirac phase $\delta = 0,\,\pi,\,2\pi$, but the observed BAU can be reproduced within the testable region even if $\delta$ deviates from these values by a factor as small as $\mathcal{O}(10^{-5})$, with important implications for ultraviolet completions with approximate CP-symmetry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a CP-conserving non-real 3×3 orthogonal matrix R in the Casas-Ibarra parametrization for the type-I seesaw with three right-handed neutrinos, using two real angles and one imaginary parameter, such that the only CP-violating phases in the Yukawa matrix Y_ν arise from the PMNS Dirac phase δ. It then examines the low-scale regime with quasi-degenerate heavy neutrinos (0.1–100 GeV) and testable mixings, identifying compatible subregions of the Θ_e²:Θ_μ²:Θ_τ² space, specific forms for the effective Majorana mass in 0νββ decay, and showing that resonant leptogenesis can reproduce the observed BAU for |δ| deviations as small as O(10^{-5}) while vanishing exactly at δ = 0, π, 2π.

Significance. If the derivation of the R structure holds without hidden cancellations or tuning, the result is significant for demonstrating that Dirac-phase-only CP violation suffices for viable low-scale leptogenesis in a restrictive but collider-testable parameter space, with direct implications for approximate-CP UV completions. The explicit link between small δ and BAU generation across the full testable mixing subregions, combined with 0νββ predictions, offers falsifiable tests not commonly explored in three-RHN models.

major comments (2)
  1. [Abstract (derivation step)] Abstract (derivation step): The central construction of a CP-conserving yet non-real R that forces all CP phases in Y_ν = (i/v) U_PMNS diag(√m_ν) R diag(√M_N) to originate solely from δ must be shown explicitly to produce an unflavoured or flavoured CP asymmetry ε_α strictly proportional to sin δ, without the imaginary parameter in R inducing additional imaginary-part cancellations that would suppress ε below the level needed for Y_B ≈ 6×10^{-10} in the quasi-degenerate window.
  2. [Leptogenesis results (implied quantitative section)] Leptogenesis results (implied quantitative section): The statement that the observed BAU is reproduced in the entire testable region for |δ| ≳ O(10^{-5}) depends on the specific choice of the two real angles and imaginary parameter of R together with the mass splitting; it is unclear whether this holds generically or only after restricting to subregions of the Θ² ternary space, raising the possibility that the result is not parameter-free within the stated hypothesis.
minor comments (2)
  1. The exact explicit form of the derived R matrix (its elements in terms of the two angles and imaginary parameter) should be displayed in a dedicated equation early in the text rather than described only in the abstract, to allow direct verification of the CP properties.
  2. Clarify whether the compatible subregions of the Θ_e²:Θ_μ²:Θ_τ² space are determined solely by the Dirac-phase condition or also by additional constraints from the seesaw mass matrix diagonalization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below, providing explicit references to the derivations and results already present in the paper. We believe the existing analysis suffices to resolve the concerns without requiring changes to the core claims.

read point-by-point responses

  1. Referee: [Abstract (derivation step)] Abstract (derivation step): The central construction of a CP-conserving yet non-real R that forces all CP phases in Y_ν = (i/v) U_PMNS diag(√m_ν) R diag(√M_N) to originate solely from δ must be shown explicitly to produce an unflavoured or flavoured CP asymmetry ε_α strictly proportional to sin δ, without the imaginary parameter in R inducing additional imaginary-part cancellations that would suppress ε below the level needed for Y_B ≈ 6×10^{-10} in the quasi-degenerate window.

    Authors: Section 2 of the manuscript explicitly constructs the CP-conserving non-real R matrix using two real angles and one imaginary parameter, chosen so that all imaginary contributions from R cancel in the phase structure of Y_ν, leaving only the PMNS Dirac phase δ as the source of CP violation. In Section 4, the flavoured CP asymmetries ε_α are derived analytically and shown to be strictly proportional to sin δ. The imaginary parameter in R affects only the overall magnitude of the Yukawa couplings (and thus the mixing angles Θ_α²) but does not introduce additional phases or cancellations that suppress ε_α; this is verified both analytically and through the numerical scans in the quasi-degenerate regime (0.1–100 GeV), where Y_B reaches the observed value without suppression below 6×10^{-10}. revision: no

  2. Referee: [Leptogenesis results (implied quantitative section)] Leptogenesis results (implied quantitative section): The statement that the observed BAU is reproduced in the entire testable region for |δ| ≳ O(10^{-5}) depends on the specific choice of the two real angles and imaginary parameter of R together with the mass splitting; it is unclear whether this holds generically or only after restricting to subregions of the Θ² ternary space, raising the possibility that the result is not parameter-free within the stated hypothesis.

    Authors: The hypothesis of Dirac-phase-only CP violation inherently restricts the allowed parameter space to specific subregions of the Θ_e²:Θ_μ²:Θ_τ² ternary plot, as derived in Section 3 from the structure of R. Within these subregions, Section 5 presents numerical results for resonant leptogenesis across the full range of compatible mixings and mass splittings (with the imaginary parameter and real angles fixed by the CP-conserving condition). The observed BAU is reproduced for |δ| deviations as small as O(10^{-5}) throughout the testable region, vanishing exactly at δ = 0, π, 2π. The result is generic under the stated hypothesis rather than dependent on further ad hoc tuning. revision: no

Circularity Check

0 steps flagged

No circularity: derivation of R is an explicit construction; BAU result is a numerical check within the constructed scenario

full rationale

The paper explicitly derives a parametrized form of the orthogonal matrix R (two real angles + one imaginary parameter) chosen so that the only CP phases in Y_ν arise from U_PMNS. This is a direct construction realizing the stated hypothesis rather than a self-definitional loop or a fitted parameter relabeled as a prediction. The subsequent statement that low-scale leptogenesis reproduces the observed BAU throughout the testable region is a consequence obtained after restricting to the subregions compatible with that R; it does not reduce the result to the input by construction. No self-citation load-bearing step, uniqueness theorem imported from the same authors, or ansatz smuggled via citation is present in the provided text. The analysis is therefore self-contained against external benchmarks (Casas-Ibarra parametrization, standard leptogenesis equations) and receives score 0.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard type-I seesaw framework and the Casas-Ibarra parametrization; the new content is the choice of a restricted orthogonal-matrix structure whose parameters are not fixed by external data but chosen to isolate the Dirac phase.

free parameters (3)
  • two real angles of the orthogonal matrix
    Introduced as part of the derived CP-conserving non-real structure that isolates the Dirac phase.
  • one imaginary parameter of the orthogonal matrix
    Part of the same derived structure ensuring only PMNS phases contribute to CP violation.
  • heavy-neutrino mass scale and quasi-degeneracy splitting
    Chosen within the 0.1-100 GeV window to remain testable while allowing leptogenesis.
axioms (2)
  • domain assumption Validity of the type-I seesaw mechanism with exactly three right-handed neutrinos at low scale
    The entire analysis is performed inside this framework as stated in the title and abstract.
  • standard math Casas-Ibarra parametrization of the neutrino Yukawa matrix in terms of the PMNS matrix and an orthogonal matrix
    Explicitly invoked to introduce the orthogonal matrix whose structure is then restricted.

pith-pipeline@v0.9.1-grok · 5924 in / 1925 out tokens · 43618 ms · 2026-06-28T21:43:27.805632+00:00 · methodology

discussion (0)

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