Low-scale seesaw with flavour and CP symmetries $\unicode{x2013}$ from colliders to leptogenesis

Claudia Hagedorn; Juraj Klaric; Marco Drewes; Yannis Georis

REVIEW 3 major objections 4 minor 2 cited by

Low-scale seesaw with flavour and CP symmetries $\unicode{x2013}$ from colliders to leptogenesis

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper shows that discrete flavour and CP symmetries make a large slice of leptogenesis-viable parameter space testable at planned colliders, with heavy-neutrino decay flavours and lifetimes identifying the symmetry case.

desk verdict The paper is a solid, honest extension of the authors' earlier low-scale seesaw framework, with real new results in lifetime ratios and full scans; the main caveat is the ad hoc ΔMR structure, which the authors flag but whose impact on the quantitative maps they do not quantify. read the letter →

arxiv 2412.10254 v1 pith:ZNKXRNLN submitted 2024-12-13 hep-ph

Marco Drewes , Yannis Georis , Claudia Hagedorn , Juraj Klaric This is my paper · ORCID

classification hep-ph

keywords low-scaleseesawresonantleptogenesisflavourandCPsymmetriesheavyneutralleptonscollidersearchesleptonratiosmixingpatternsbaryonasymmetryoftheUniverse

topics Matter-Antimatter Asymmetry

open problems Matter-Antimatter Asymmetry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a low-scale type-I seesaw with three right-handed neutrinos, whose flavour structure is dictated by a discrete symmetry $\Delta(3 n^2)$ or $\Delta(6 n^2)$ combined with CP, is not just a neutrino-mass mechanism but a testable origin of the baryon asymmetry of the Universe. For four symmetry cases that produce distinct lepton mixing patterns, it maps the region of the heavy-neutrino mass–mixing plane where resonant leptogenesis can generate the observed baryon asymmetry, and overlays the projected reach of SHiP, MATHUSLA, (HL-)LHC, and FCC-ee/CEPC. The paper also shows that the lifetimes and flavour ratios of heavy-neutrino decays are predicted differently in each case, so a future collider that observes these decays could identify which symmetry case is realised and, in several cases, the neutrino mass ordering and the lightest neutrino mass. A sympathetic reader should care because the conclusion is that a large part of the leptogenesis-viable parameter space is experimentally accessible whenever at least one of the small mass splittings $\kappa$ or $\lambda$ is non-zero.

What carries the argument

The load-bearing object is the factorised Yukawa parametrisation $Y_D = \Omega^{(3)} R_{ij}(\theta_L) \operatorname{diag}(y_1,y_2,y_3) P^{ij}_{kl} R_{kl}(-\theta_R) \Omega^{(3')\dagger}$, which encodes the residual flavour and CP symmetries in fixed matrices $\Omega^{(3)}$, $\Omega^{(3')}$ and in the fixed rotation planes, leaving three couplings, two angles and the mass scale $M$ as free parameters. On top of the degenerate Majorana mass matrix $M_R^0$ the symmetry-breaking splittings $\delta M_R = \kappa M \operatorname{diag}(2,0,-1;\,0,-1,0)$ and $\Delta M_R = \lambda M \operatorname{diag}(0,1,1)$ control the resonance condition for leptogenesis and, through the ratios $U_1^2:U_2^2:U_3^2$, the pattern of heavy-neutrino lifetimes. The argument is carried by the CP-violating combinations $C_{\mathrm{LFV},\alpha}$, $C_{\mathrm{LNV},\alpha}$, $C_{\mathrm{DEG},\alpha}$ together with the flavoured washout parameter $f_\alpha$, which decide whether the asymmetry is generated by lepton-number violation or by flavoured washout. The comparison with experiments uses analytic Z-pole event numbers and projected displaced-vertex and beam-dump sensitivities, with branching ratios fixed by the flavour ratios $U_\alpha^2/U^2$ shown in ternary plots.

What would settle it

At a Z-pole machine such as FCC-ee/CEPC with roughly $10^5$ reconstructed heavy-neutrino decays, measure the flavour ratios $U_e^2/U^2$, $U_\mu^2/U^2$, $U_\tau^2/U^2$ and the decay-length distribution: detecting $U_e^2/U^2 > 0.35$ would falsify Case 3 b.1) as analysed here, and a single-exponential distribution instead of the predicted mixture with ratios such as $2:1:3$ or $1:0:1$ would falsify the lifetime predictions. In the $\kappa$-$\lambda$ plane, observing successful leptogenesis with splittings that violate the consistency conditions of Eqs. (47)–(49) would break the framework's central assumption.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the symmetry-fixed structure of the heavy neutrino sector remains predictive once symmetry breaking is switched on. In the limit of large active-sterile mixing, realised when the angle $\theta_R$ sits near a special value so that one Yukawa coupling dominates, the physical heavy-neutrino masses are set by the degenerate scale $M$, the Higgs-induced contribution $\Delta \hat{M}_{\theta\theta}$, and two small splittings: $\delta M_R$ parametrised by $\kappa$ (motivated by the residual charged-lepton symmetry) and $\Delta M_R$ parametrised by $\lambda$ (admitted as a generic perturbation). Depending on which splitting dominates, the mixing ratios $U_1^2:U_2^2:U_3^2$ collapse to discrete patterns such as $2:1:3$, $4:1:3$, $1:0:1$ or $4:11:9$, producing decay-length distributions that deviate from a single exponential and can be fitted to recover the splittings. In the same regime the flavour ratios $U_\alpha^2/U^2$ equal the moduli squared of columns of the symmetry-determined PMNS matrix, up to case-specific permutations, giving sharp bounds such as $U_e^2/U^2 \leq 0.35$ for Case 3 b.1). Solving the quantum kinetic equations for flavoured resonant leptogenesis, the paper finds viable parameter space in all four cases, and shows that a sizeable portion of it lies within the projected reach of current and future accelerator experiments, especially for vanishing initial heavy-neutrino abundances and non-zero $\kappa$ or $\lambda$; for Case 2 and Case 3 b.1) the baryon asymmetry can even be generated at exact degeneracy through flavoured washout.

Load-bearing premise

The assumed flavour structure of the tiny right-handed-neutrino mass splittings, namely the $\kappa$ term tied to the residual charged-lepton symmetry and the more ad hoc $\lambda$ term that separates the second and third masses, carries the quantitative predictions for lifetime ratios, the leptogenesis resonance, and the borders of the testable region; if the actual symmetry-breaking terms had a different flavour structure, those predictions would change.

Editorial extensions

If this is right

If the central claim is right, a large fraction of the leptogenesis-viable region in the mass–mixing plane is within the projected reach of SHiP, MATHUSLA, (HL-)LHC, and FCC-ee/CEPC, especially for vanishing initial heavy-neutrino abundances and non-zero $\kappa$ or $\lambda$.
Heavy-neutrino decay-length distributions will not be simple exponentials: predicted ratios like $2:1:3$ or $1:0:1$ mean that the distribution of displacements carries information about the combination $3\kappa-\lambda$ even when the three masses cannot be resolved.
Measuring the flavour ratios $U_e^2/U^2$, $U_\mu^2/U^2$, $U_\tau^2/U^2$ at the one-percent level, plausible with roughly $10^5$ Z-pole events, can distinguish the four symmetry cases and, for Case 1 and parts of Case 2, the neutrino mass ordering and the value of the lightest neutrino mass $m_0$.
Even with exactly degenerate heavy neutrinos ($\kappa = \lambda = 0$), Case 2 and Case 3 b.1) can still generate the baryon asymmetry through flavoured washout, leaving a smaller but partly FCC-ee-testable region.
In the strong-inverted-ordering version of Case 3 a) and the strong-normal-ordering version of Case 3 b.1), the model effectively reduces to a two-heavy-neutrino framework for both collider searches and leptogenesis.

Reading between the lines

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

(Beyond the paper) A precise measurement of the decay-length distribution would effectively measure a combination of the two splitting parameters, turning the ad hoc $\lambda$ correction into an observable.
(Beyond the paper) The same ternary-plot logic could be applied to seesaw variants with two heavy neutrinos or non-degenerate masses: because the flavour ratios fix columns of the PMNS matrix, any measured ratio lying outside every predicted region would point to a different symmetry structure.
(Beyond the paper) A determination of $U_e^2/U^2$ at a Z-pole collider would act as a complementary probe of the neutrino mass ordering and of $m_0$, cross-checking cosmological and oscillation bounds, since Case 1 correlates these quantities tightly.
(Beyond the paper) The sensitivity forecasts assume idealised reconstruction efficiencies; realistic efficiencies will shrink the absolute reach, but the relative ordering of the cases and the qualitative distinction between single-exponential and multi-exponential decay distributions should survive.

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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 paper does what it says: it takes the flavour/CP-symmetric low-scale seesaw framework from the authors' earlier JHEP paper and runs it through lifetime, branching-ratio, flavour-ratio, and full leptogenesis scans for all four symmetry cases, including Case 3 a) which had not been studied numerically. The new content is real: the heavy-neutrino lifetime ratios in Table 1, the flavour-ratio ternary plots for all cases, the mu-e conversion constraint study, and the marginalised scans over the symmetry parameters. It is also unusually honest about its own limitations, flagging the ad hoc nature of the λ splitting and noting where a supplementary scan has not fully converged.

The central results look plausible. The mass matrices in Eqs. (62)–(71) are stated without derivation, which is a readability cost rather than a correctness problem—a patient reader can reproduce them from Eq. (61) and the given Ω(3') forms. The flavour-ratio predictions in Section 4 do reduce, in the generic case, to mod-squares of PMNS elements, so they are not independent of the global fit; the sharper value is in the case-specific regions, where the symmetry constrains the mixing enough to produce distinct ternary locations. That distinction is a genuine result.

The one soft spot that deserves more attention—and it is partly self-inflicted—is the sensitivity to the assumed form of ΔMR. λ enters as an ad hoc diag(0,1,1) perturbation, and Table 1, the resonance lines, and the grey-shaded excluded regions all follow from diagonalising the combined mass matrix with that specific structure. If the symmetry-breaking term that splits the three RH neutrinos had off-diagonal components, the lifetime ratios and the testability boundaries would shift. The paper acknowledges this in passing, but it does not explore how robust the quantitative maps are to that choice. I do not think this is fatal: κ alone already breaks the degeneracy and is motivated by the residual charged-lepton symmetry, and the qualitative conclusion that sizeable testable regions exist for all cases will likely survive. But readers should treat the specific numbers as benchmark predictions of a particular ansatz, not as robust model-independent statements.

The citation pattern is appropriate—this builds on the authors' own earlier work and cites the relevant experimental and global-fit literature. Who is this for? Anyone working on heavy neutral lepton searches, low-scale seesaw models, or leptogenesis will get value from the scans and the analytic limits. It deserves a serious referee: the conditions raised—derivation of the mass matrices, clarifying the PMNS-element dependence, and a sensitivity study for λ—are addressable in revision. I would send it to review and, after those points are handled, take the maps as useful reference points.

Referee Report

3 major / 4 minor

Summary. The paper studies the low-scale type-I seesaw with three right-handed neutrinos, a flavour symmetry Δ(3n²) or Δ(6n²), and CP, extending the earlier work of ref. [39] to a comprehensive phenomenological analysis. For the four symmetry-defined mixing cases (Case 1 through Case 3 b.1), it derives the heavy-neutrino mass matrices, the ratios of active-sterile mixings U_i²/U², the lifetimes, and the branching ratios U_α²/U², and then scans the parameter space for resonant leptogenesis with both vanishing and thermal initial conditions. The central claim is that a sizeable portion of the leptogenesis-viable parameter space is testable at SHiP, MATHUSLA, (HL-)LHC and FCC-ee/CEPC, especially for vanishing initial conditions and non-zero mass splittings κ or λ, and that lifetime and flavour ratios can distinguish the symmetry cases.

Significance. If correct, the paper provides concrete, falsifiable targets for a well-motivated flavour-symmetric low-scale seesaw: specific lifetime ratios, restricted flavour branching ratios, and experimentally accessible leptogenesis regions. The strengths are the comprehensive treatment of four symmetry cases, the inclusion of both vanishing and thermal initial conditions, analytic formulae for the CP-violating combinations (Appendix B.2), and the frank admission of limitations, including the ad hoc nature of the λ splitting and a non-converged supplementary scan. The flavour-ratio predictions, while partly inherited from fits to NuFIT data, are still useful discriminants between the cases. However, the quantitative lifetime and leptogenesis predictions in the λ-sensitive regimes rest on an assumed, unconstrained perturbation structure, which limits the robustness of the claimed discriminating power without further analysis.

major comments (3)

[Section 2.1, Eq. (23); Section 5.2, 'Impact of the splitting λ'] The form ΔMR = λ M diag(0,1,1) is introduced in Eq. (23) and the paper later states explicitly that the splitting λ is 'ad hoc'. Despite this, the quantitative predictions in the λ-sensitive regimes—the exact mixing-ratio formulae in Eqs. (64) and (68), the rows of Table 1 in the |λ| ≫ U² limit, the resonance lines in Figs. 14 and 24, and the λ-scan regions in Figs. 23, 24, and 26—all depend on this specific diagonal structure. The paper varies the magnitude of λ but never the flavour structure; a different symmetry-breaking perturbation, e.g. one mixing the first and second heavy-neutrino states, would change the diagonalisation of M_R and hence the lifetime ratios and leptogenesis parameter space. Since the summary claims that lifetime ratios distinguish the cases and that the testable regions are sizeable, the authors should either demonstrate robustness under variations of the λ structure or explicitly qualify these claims as benchmark-dependent.
[Section 3.1, Eqs. (62)-(71) and Table 1] The central heavy-neutrino mass matrices (Eqs. (62), (66), (71)) and the resulting mixing ratios in Table 1 are stated without derivation; the text simply says 'Its form reads' and 'we get'. These matrices underlie the lifetime-ratio predictions that are a principal novelty of the paper, and the limiting ratios (e.g. 2:1:3 vs 1:0:1 vs 4:11:9) are used in Fig. 1 to claim experimental distinguishability. A derivation, or at least a sketch of the diagonalisation procedure, should be provided in an appendix so that the reader can verify the structure and the limits; if this is already available in the companion paper [39], a precise pointer is needed.
[Appendix B.1, Fig. 25 (right plot)] The caption of Fig. 25 (right plot) states that 'the angular line shapes ... are due to a reduced convergence, since the scan has not been optimised, unlike for the other results shown in this work.' In the main text this plot is used to support the conclusion that for Case 3 b.1) with IO there is 'only a mild enlargement of the allowed parameter space' for m0 = 0.015 eV. A non-converged scan cannot reliably support this quantitative statement; either rerun the scan to converged results or explicitly soften the conclusion.

minor comments (4)

[Fig. 2 caption and Section 2.2.2] The caption of Fig. 2 says 'both κ and λ do not impact the value of the ratios U_α²/U² as long as the condition in Eq. (48) is fulfilled', but Eq. (48) is specific to Case 1) with cos 2θ_R ≈ 0; the analogous conditions for other cases are only 'alike' (as stated in the text). Please rephrase to refer to 'conditions such as Eq. (48)'.
[Section 4.1, Eqs. (84)-(91)] The flavour-ratio predictions in this section are mod-squares of PMNS matrix elements whose inputs are taken from the NuFIT global fit. The paper does note that the mixing angle is fitted, but it would be useful to state explicitly that these are not parameter-free predictions; their discriminating power is conditional on the fitted values of θL (or eθL) and the group parameters.
[Table 3] The row 'Case 3 b.1), κ = 0 ... 6.1 / 2.5 /' is ambiguous: the caption says '/' marks situations where the BAU always vanishes, but the row contains only three entries for a four-column table. Please clarify the formatting so that each of the NO/IO and VIC/TIC entries is unambiguous.
[Section 3.3, discussion of mean lifetime] The statement that 'the mean lifetime is 1/3 of the quantity Γ_N expected from Eq. (52)' appears to assume equal production of three heavy-neutrino species; as written it could be read as a general result. Please add the qualifying assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model's inputs (fitted lepton mixing angles, ad hoc MR corrections) are transparent, and the central new computations (lifetime ratios, leptogenesis testability maps) are independent numerical outputs.

full rationale

The paper's derivation chain is self-contained. The heavy-neutrino mass matrices in Eqs. (62)-(71) are diagonalized to obtain lifetime ratios (Table 1), and the leptogenesis parameter space is obtained by solving the quantum kinetic equations (Eqs. (140)) with rates taken from independent literature; neither step is equivalent to fitting the inputs. The flavour ratios in Sec. 4 are derived from the model's PMNS structure: e.g. Eq. (93) states U_e^2/U^2 = sin^2(theta13), but this is a prediction for the heavy-neutrino decay flavour ratio, a distinct observable from the neutrino oscillation measurement of theta13, so it is a model mapping from a known quantity to a different one rather than a fit of the same quantity. The paper transparently labels the data-only case as 'generic' and the symmetry cases as consequences of the fitted angle theta_L. The splitting DeltaMR in Eq. (23) is admitted to be ad hoc ('the splitting lambda is introduced ad hoc'), which is a model-assumption and robustness limitation rather than circularity; the impact of lambda and the grey-shaded excluded regions are explicitly discussed with their assumptions stated. Self-citations to [39] provide the model framework, but the present numerical scans, marginalisations, and experimental comparisons are new and do not reduce to those citations.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The model rests on the type-I seesaw with three right-handed neutrinos and on the discrete flavour/CP symmetry framework from earlier work. The central quantitative claims depend on the assumed mass-splitting corrections deltaMR and DeltaMR (Eqs. 21, 23), on the fitted angle thetaL, and on the scanned parameters M, kappa, lambda, thetaR and the group parameters. No new particles are introduced beyond the standard seesaw RH neutrinos.

free parameters (8)

M (RH neutrino mass scale)
Mass scale of the three right-handed neutrinos; scanned with log prior over [50 MeV, 70 TeV], with benchmark values 10 GeV, 100 GeV, 1 TeV.
kappa (splitting in deltaMR)
Gl-invariant correction to MR, Eq. (21); scanned over [10^-20, 10^-1] with log prior; controls the MR1 versus MR2/MR3 splitting.
lambda (splitting in DeltaMR) = 0 (default); benchmarks 10^-10, 10^-4
Ad hoc splitting separating all three RH masses, Eq. (23); scanned when its effect is studied; the paper calls it ad hoc in section 5.2.
thetaR
Free angle in YD, Eq. (14); scanned over 0 to 2pi with log-prior on |thetaR - k pi/4|; special values give large active-sterile mixing U2.
thetaL (or effective thetaL) = e.g. 0.183 (Case 1, NO), see Table 1 in [39]
Angle in the PMNS mixing matrix; fitted to reproduce observed lepton mixing angles; the flavour-ratio predictions in section 4 depend on it.
y1, y2, y3 (Yukawa couplings) = fixed by m0 and light-neutrino mass spectrum
Real couplings in YD; chosen to reproduce NO or IO spectra with the given m0; in the large-U2 regime one coupling is large and the others neglected.
s/n, u/n, v/n, m/n (discrete group parameters)
Ratios specifying the CP and residual Z2 symmetries; scanned linearly in the ranges of Table 2, which are constrained by fits to lepton mixing angles.
m0 (lightest neutrino mass) = 0 or 0.03 eV (NO) / 0.015 eV (IO)
Benchmark values at the extrema allowed by cosmology [117]; chosen rather than scanned.

assumptions (7)

standard math Type-I seesaw with three right-handed neutrinos gives light neutrino masses via mnu = -mD MR^-1 mD^T (Eq. 17).
The framework on which all neutrino mass and mixing predictions rest; standard seesaw mechanism from [1-5].
domain assumption The flavour group is Delta(3n^2) or Delta(6n^2) with residual symmetries Gl=Z3 and Gnu=Z2 x CP, plus an auxiliary Z3 to separate charged-lepton masses.
The whole case structure (Cases 1 to 3 b.1) follows from this choice; motivated by lepton-mixing fits from [23-25,45].
domain assumption In the unbroken limit MR = M diag(1,0,1; 0,1,0) (Eq. 11) and YD has the form of Eq. (14) with five real parameters.
Form fixed by the symmetry representation content and residual groups; taken from [39,44].
ad hoc to paper The Majorana mass corrections are deltaMR = kappa M diag(2,0,-1; 0,-1,0) (Eq. 21) and DeltaMR = lambda M diag(0,1,1) (Eq. 23), with |kappa|,|lambda| <= 10^-1.
deltaMR is Gl-invariant, but lambda is explicitly called ad hoc in section 5.2; the form of these corrections drives the lifetime ratios and the leptogenesis resonance structure.
domain assumption Heavy neutrino production and decay at colliders are governed only by the mixing Theta through the SM weak interactions (Eq. 50).
Standard assumption in HNL searches; footnote 6 notes that additional TeV-scale gauge interactions could enhance production, so collider reach is conservative.
domain assumption The quantum kinetic equations (Eqs. 140) with rates extrapolated from [122] to the non-relativistic regime reproduce the BAU.
Used to compute the leptogenesis parameter space; the extrapolation procedure follows [123].
ad hoc to paper The splittings kappa and lambda must be small enough not to destabilise the light neutrino masses (conditions such as Eqs. 47-49).
Imposed in the scans and used to shade grey regions; the paper notes these regions might still allow viable leptogenesis.

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Cite this review

Pith. "Pith review of Low-scale seesaw with flavour and CP symmetries $\unicode{x2013}$ from colliders to leptogenesis." pith.science (2026). https://pith.science/paper/ZNKXRNLN

@misc{pith2026241210254,
  author       = {Pith},
  title        = {Pith review of: Low-scale seesaw with flavour and CP symmetries $\unicodex2013$ from colliders to leptogenesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZNKXRNLN}},
  note         = {Machine review of arXiv:2412.10254}
}

abstract

We consider an extension of the Standard Model with three right-handed neutrinos, endowed with a flavour symmetry $G_f$, $G_f=\Delta (3 \, n^2)$ or $G_f=\Delta (6 \, n^2)$, $n \geq 2$, and CP. For large active-sterile mixing, we study the properties of the (nearly mass-degenerate) heavy neutrinos, such as their lifetimes and branching ratios. In doing so, we examine the four different cases, called Case 1) through Case 3 b.1), that lead to distinct lepton mixing patterns, all potentially compatible with current data. Furthermore, we comprehensively explore for each case the region of parameter space in which a sufficient amount of baryon asymmetry of the Universe can be generated via leptogenesis, while being testable at accelerator-based and potentially also precision flavour experiments.

Figures

Figures reproduced from arXiv: 2412.10254 by the authors.

**Figure 1.** Expected differential distribution of heavy neutrino decays as a function of l/λN assuming the different ratios from Tab. 1. We show the distribution both on a log scale (left) and on a linear scale (right). The following ratios of heavy neutrino mixing are considered: 2 : 1 : 3 (orange, dashed), 4 : 11 : 9 (green, dotted), 4 : 1 : 3 (red, dot-dashed) as well as 1 : 0 : 1 (blue, solid), all found in Tab. 1, together… view at source ↗

**Figure 2.** Event numbers displayed in the U2 e U2 − U2 -plane for semi-leptonic decays expected at FCC-ee/CEPC for 1012 produced Z bosons. The Majorana mass M is set to M = 10 GeV, the splitting λ to λ = 0 and we assume light neutrino masses follow NO, while marginalising over κ. The left plot refers to Case 1) with the lightest neutrino mass being zero, m0 = 0, while in the right plot results for Case 3 b.1) and m0 = 0.03 eV … view at source ↗

**Figure 3.** Generic lepton mixing matrix Results for the ratios U2 α U2 in case the lepton mixing matrix is not constrained by a symmetry, but by experimental data only [46]. We marginalise over the three lepton mixing angles θ12, θ13, θ23, as well as the CP phase δ. We use the tabulated χ 2 values for light neutrino masses with NO. using the results of the NuFIT collaboration [46]. This is shown as orange area in the ternary p… view at source ↗

Figures from the paper (26 more)

**Figure 4.** Figure 4: Case 1) Results for ratios U2 α U2 assuming light neutrino masses with NO (left plot) and IO (right plot), respectively. The different colours indicate different values of the lightest neutrino mass m0. The point shown in orange corresponds to the choice m0 = 0, while …

**Figure 4.** Figure 4: Including the effect of non-zero m0 requires taking into account that the angles θe L and θL do not coincide. We first note that the special values of θR which allow for large active-sterile mixing U 2 fulfil cos 2 θR ≈ 0. Thus, for large m0 the relation between the tw…

**Figure 5.** Figure 5: Case 2) Results for ratios U2 α U2 assuming light neutrino masses with NO (left plot) and IO (right plot), respectively. The colour-coding is the same as in [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

**Figure 6.** Figure 6: Case 3 a) Results for ratios U2 α U2 , assuming that light neutrino masses follow strong NO. In case we only consider results with the minimum value of χ 2 for the lepton mixing angles, we obtain the two red lines, while the pink area represents the results, if we dema…

**Figure 7.** Figure 7: Case 3 a) Results for ratios U2 α U2 assuming light neutrino masses with NO (left plot) and IO (right plot), respectively, for two different values of the lightest neutrino mass m0: m0 = 0 (darker red area) and m0 = 0.03 eV (lighter red) for NO and m0 = 0 (darker blue …

**Figure 6.** Figure 6: This area is also shown in darker red in the ternary plot in the left of Fig. 7. [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]

**Figure 8.** Figure 8: Case 3 b.1) Results for ratios U2 α U2 assuming light neutrino masses with NO (left plots) and IO (right plots), respectively, for a fixed value of ϕm, ϕm = 10 π 20 (upper plots) and ϕm = 9 π 20 (lower plots), while it is marginalised over the allowed values of ϕs. The…

**Figure 9.** Figure 9: Case 3 b.1) Results for ratios U2 α U2 assuming light neutrino masses with NO (left plots) and IO (right plots), respectively, for a fixed value of ϕs, ϕs = 10 π 20 (upper plots) and ϕs = 7 π 20 (lower plots), while it is marginalised over the allowed values of ϕm. The…

**Figure 10.** Figure 10: Case 3 b.1) Results for ratios U2 α U2 assuming light neutrino masses with NO (left plot) and IO (right plot), respectively, for two different values of the lightest neutrino mass m0: m0 = 0 and m0 = 0.03 eV for NO as well as m0 = 0 and m0 = 0.015 eV for IO. We margin…

**Figure 11.** Figure 11: Case 1) Comparison of parameter space consistent with leptogenesis in the M − U 2 - plane in case of fully marginalising over the splitting κ and the ratio s n (green- and blue-bordered regions) and for fixed κ and s (κ = 10−11 and s n = 1 10 , see [39], yellow- and o…

**Figure 12.** Figure 12: Case 1) Comparison of parameter space viable for leptogenesis in the M − U 2 µ -plane for vanishing lightest neutrino mass, m0 = 0, and its value, maximally allowed by cosmology, m0 = 0.03 (0.015) eV for NO (IO). Green and blue lines refer to m0 = 0, while yellow and …

**Figure 13.** Figure 13: Case 1) Viable parameter space for leptogenesis displayed in the different planes M − U 2 α, α = e, µ, τ (top, middle, bottom plots) for light neutrino masses with strong NO (left plots) and strong IO (right plots). Results for VIC and TIC are shown. Expected sensitiv…

**Figure 14.** Figure 14: Case 1) Parameter space leading to the successful generation of the BAU in the |κ|(∆M) − U 2 · M-plane for three different values of the Majorana mass M, M = 10 GeV (upper left plot), M = 100 GeV (upper right plot) and M = 1 TeV (bottom plot). Both types of initial co…

**Figure 15.** Figure 15: Case 2) Viable parameter space for leptogenesis displayed in the M − U 2 µ -plane in the case of fully marginalising over the splitting κ and the ratios u n and v n in their allowed ranges, see Tab. 2. The left (right) plot refers to light neutrino masses with strong …

**Figure 14.** Figure 14: Clearly, the largest attainable value, U 2 · M ∼ 105 eV, requires small M, |κ| of the order of 10−11 and VIC. For larger values of M this maximum value is of the order of 104 eV and can be obtained for both types of initial conditions. At the same time, we confirm tha…

**Figure 16.** Figure 16: Case 2) Parameter space consistent with leptogenesis shown in the M − U 2 µ -plane for vanishing splittings, κ = 0 and λ = 0, and light neutrino masses with NO (IO) and m0 = 0.03 (0.015) eV in the left (right) plot. Results for VIC and TIC are displayed. For details a…

**Figure 17.** Figure 17: Case 3 a) Viable parameter space for leptogenesis in the M − U 2 µ -plane in the case of fully marginalising over the splitting κ and the ratios m n and s n in their allowed ranges, see Tab. 2. The left (right) plot shows the results for light neutrino masses with str…

**Figure 18.** Figure 18: Case 3 b.1) Parameter space consistent with leptogenesis shown in the M − U 2 µ -plane, resulting from fully marginalising over the splitting κ and the ratios m n and s n according to the ranges in Tab. 2. Light neutrino masses with strong NO (left plot) and strong IO…

**Figure 19.** Figure 19: Case 3 b.1) Left plot: Comparison of the viable parameter space in the M − U 2 µ -plane resulting from fully marginalising over the splitting κ and the ratios m n and s n in the ranges found in Tab. 2 and for fixed values of the ratios m n and s n (corresponding to m …

**Figure 20.** Figure 20: Case 3 b.1) Left plot: Parameter space consistent with leptogenesis in the M−U 2 µ -plane for vanishing splittings, κ = 0 and λ = 0, and light neutrino masses with strong NO. Both VIC and TIC are considered. The different coloured dashed lines refer to various experim…

**Figure 21.** Figure 21: Case 3 b.1) Parameter space consistent with leptogenesis highlighted in the ternary plot for vanishing splittings, κ = 0 and λ = 0, and light neutrino masses with strong NO. The different colours indicate the size of the active-sterile mixing U 2 . mass M around 4 GeV…

**Figure 22.** Figure 22: Constraints from searches for charged lepton flavour violation in µ − e transitions on the parameter space consistent with leptogenesis, shown in the M − q U2 e U2 µ -plane, for Case 1) (fully marginalised over the splitting κ and the ratio s n ) and light neutrino ma…

**Figure 23.** Figure 23: Impact of splitting λ on the parameter space available for leptogenesis, shown in the M − U 2 µ -plane, for Case 1) (fully marginalising over the splitting κ and the ratio s n ), light neutrino masses with strong NO and VIC. We compare the parameter space for λ = 0 (g…

**Figure 24.** Figure 24: Impact of the splitting λ on the parameter space consistent with leptogenesis, shown in the |λ|(∆M) − U 2 · M-plane, for two different benchmark values of the Majorana mass M, M = 10 GeV (left plot) and M = 1 TeV (right plot). We take Case 1), assume light neutrino ma…

**Figure 25.** Figure 25: Case 3 b.1) Plots similar to those given in [PITH_FULL_IMAGE:figures/full_fig_p055_25.png]

**Figure 26.** Figure 26: Impact of splitting λ on the parameter space available for leptogenesis, shown in the M − U 2 µ -plane, for Case 1) (fully marginalising over the splitting κ and the ratio s n ), light neutrino masses with strong NO and VIC. Here, we display the results for λ = 10−4 w…

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A two-step diagonalization plus a dummy-particle trick lets standard event generators simulate GeV-scale sterile neutrino oscillations with crossing-width terms, and a QFT derivation gives displaced-vertex distances.
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