Flavoured leptogenesis and ${\rm CP}^{\mu\tau}$ symmetry

Manibrata Sen; Rome Samanta

REVIEW 3 major objections 3 minor 134 references

Flavoured leptogenesis and ${\rm CP}^{\mu\tau}$ symmetry

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In a CP$\mu\tau$-symmetric minimal seesaw model, the observed baryon asymmetry can be produced with a right-handed neutrino mass hierarchy as mild as $M_2/M_1 \simeq 4.7$, lowering the lightest mass bound to about $7.5\times 10^9$ GeV.

desk verdict Relaxing the strong hierarchy in CPμτ leptogenesis is a genuine extension, and the numbers are usable as estimates, but the decoherence-derived upper bound and scan fractions need verification before being treated as precise. read the letter →

arxiv 1908.08126 v2 pith:2VFRYY5L submitted 2019-08-21 hep-ph

Rome Samanta , Manibrata Sen This is my paper · ORCID

classification hep-ph

keywords leptogenesisCPmu-tausymmetryreflectionminimalseesawtworight-handedneutrinosflavouredN2baryonasymmetry

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 asks whether the CP$\mu\tau$-symmetric minimal seesaw model, a neutrino mass model with a $\mu$-$\tau$ exchanging CP symmetry, can account for the observed baryon asymmetry using only two right-handed neutrinos once realistic flavour effects are included. The authors establish that the heavy neutrino mass hierarchy need not be extreme: a ratio as mild as $M_2/M_1 \simeq 4.7$ still supports a standard $N_1$-dominated leptogenesis scenario, lowering the lower bound on the lightest right-handed mass from about $6\times 10^{10}$ GeV to about $7.5\times 10^{9}$ GeV. They further show that $N_1$-domination is confined to a window $7.5\times 10^{9}\,\mathrm{GeV} < M_1 < 4\times 10^{10}\,\mathrm{GeV}$, fixed by tau-lepton flavoured decoherence, and that beyond this window the second right-handed neutrino $N_2$ can dominate the final asymmetry over a substantial fraction of the parameter space. When $N_2$ produces the asymmetry in the two-flavour regime and $N_1$ washes it out in the three-flavour regime, the surviving asymmetry is essentially electron-flavoured, and roughly 33% of the parameter space is consistent with pure $N_2$ leptogenesis.

What carries the argument

The argument is carried by the CP$\mu\tau$ ($\mu\tau$-reflection) symmetry acting on the neutrino mass Lagrangian, which forces the flavoured CP asymmetries to satisfy $\varepsilon_{ie} = 0$ and $\varepsilon_{i\mu} = -\varepsilon_{i\tau}$ while the total unflavoured asymmetry vanishes. Within the minimal two-right-handed-neutrino seesaw, three ingredients do the quantitative work: the flavoured decay parameters $K_{i\alpha}$, which set the washout strength in each flavour; the overlap probability $p_{12}$ (the squared projection of the $N_2$ lepton state onto the $N_1$ lepton state in the $\tau$-perpendicular plane, never equal to unity in this model because the two heavy states define different $\tau^\perp$ directions); and the decoherence condition $F_\tau = \Gamma_\tau/(2Hz) > W(z)$, which fixes the upper end of the $N_1$ mass window.

What would settle it

Compute the full density-matrix (quantum-kinetic) leptogenesis equations, including off-diagonal flavour coherence, for a point inside the claimed window (e.g., $M_1 = 10^{10}$ GeV with $M_2/M_1 = 4.7$ and $K_{1\tau} \approx K_{2\tau} \approx 5$); if the resulting baryon-to-photon ratio differs from the flavour-projected Boltzmann result by more than an order-one factor, the window is invalid. Alternatively, a future neutrinoless double-$\beta$ decay measurement finding $|(M_\nu)_{\beta\beta}|$ outside the model's required 3–5 meV range would exclude the parameter space that underlies all of these results.

Watch

Extended reading notes

Core claim

This paper's central claim is that in the CP$\mu\tau$ ($\mu\tau$-reflection) symmetric two-right-handed-neutrino seesaw model, flavoured leptogenesis works across a much wider region of parameter space than earlier analyses concluded. In the two-flavour regime, the heavy neutrino masses can be mildly hierarchical, $M_2 \simeq 4.7 M_1$, and still realize a perfectly hierarchical $N_1$-dominated scenario; the loop-function enhancement of the CP asymmetry at this mild hierarchy lowers the minimal allowed $M_1$ to about $7.5\times 10^{9}$ GeV. $N_1$-domination is valid only for $7.5\times 10^{9}\,\mathrm{GeV} \lesssim M_1 \lesssim 4\times 10^{10}\,\mathrm{GeV}$, with the upper bound set by the requirement that $\tau$ charged-lepton interactions decohere the lepton flavour states before the washout rates peak; beyond this window, about 26% of the parameter space favours $N_2$-domination while 37% favours $N_1$-domination. When $M_2$ lies in the two-flavour regime and $M_1$ in the three-flavour regime, the asymmetry produced by $N_2$ is essentially erased by $N_1$ in the $\mu$ and $\tau$ flavours, leaving the electron flavour as the only survivor, with the probability of survival (quantified by $K_{1e} < 1$) around 33%.

Load-bearing premise

The upper bound on $M_1$ (and hence the quoted domination fractions) rests on the condition that the $\tau$ charged-lepton interaction rate exceeds every relevant washout rate throughout the asymmetry epoch ($F_\tau > W$ at the relevant $z$); if this decoherence condition fails, the full quantum-kinetic density-matrix treatment is needed and the mass window and fractions could shift.

Editorial extensions

If this is right

The lower bound on $M_1$ drops by roughly an order of magnitude, so CP$\mu\tau$ models can be compatible with reheating temperatures around $10^{10}$ GeV rather than requiring $M_1 \gtrsim 6\times 10^{10}$ GeV.
A pure $N_1$-dominated analysis that assumes strong hierarchy ($M_2/M_1 = 10^3$) is not generally valid inside the allowed window, since $1-p_{12}$ never vanishes and $N_2$ dominates the final asymmetry for about 26% of the parameter space.
When $M_1 < 10^9$ GeV and $M_2$ is in the two-flavour regime, the surviving asymmetry is electron-flavoured; a pure $N_2$ leptogenesis signal would then be a single-flavour asymmetry, with the other flavours erased by $N_1$ washout.
The decoherence condition $F_\tau > W$ pins down a concrete scale, $M_1 \lesssim 4\times 10^{10}$ GeV, above which either $N_2$ contributions become relevant or the flavour-projected Boltzmann approach must be replaced by a density-matrix treatment.

Reading between the lines

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

The same flavour-survival logic applies to any pre-existing electron-flavour asymmetry: in the low-$M_1$ spectrum, a high-scale source of $e$-flavour lepton asymmetry would also evade $N_1$ washout, a possibility the paper sets aside by assuming no pre-existing asymmetry.
The claimed window is computed in the rate-equation (flavour-projected) approximation; a full quantum-kinetic computation could shift $M_{\rm max}$ and the 26%/37% fractions, so those percentages should be read as the rate-approximation estimate rather than exact model predictions.
The mild-hierarchy result depends on the loop-function enhancement in $g_1(x_{12})$; variants of CP$\mu\tau$ with additional right-handed neutrinos (or with a differently charged-lepton sector) would modify the allowed mass window, making the 33% electron-flavour survival fraction a useful discriminator between symmetry variants.

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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 is a serious extension of flavoured leptogenesis in CPμτ-symmetric seesaw models. The genuinely new result is that the heavy neutrino hierarchy need not be very strong: M2/M1 ~ 4.7 is enough for an N1-dominated hierarchical leptogenesis in the minimal seesaw, which lowers the lower bound on M1 to around 7.5e9 GeV. That directly contradicts the implicit strong-hierarchy assumption in earlier CPμτ leptogenesis papers, and the authors show it with a reasonable combination of analytic efficiency factors and explicit one-flavour Boltzmann checks. The second new piece, that N2 can dominate for a sizable part of parameter space in the two-flavour regime, is also well motivated by the structure of the model, and the claim that N2-generated asymmetry survives mainly in the electron flavour in the M1<1e9 GeV scenario is a clean symmetry-based result.

The central arguments are not circular. The observed baryon asymmetry is not used to fit any parameter; the M2/M1 threshold and the N1/N2 fractions are computed from the model. Self-citations are to the authors' own earlier work on the same symmetry, which is appropriate here rather than a red flag.

The soft spots are real but, in my view, not fatal. The most load-bearing one is Eq. V.10, the decoherence criterion that sets the upper bound M_max ~ 4e10 GeV. It compares the tau charged-lepton interaction rate with a scalar washout rate, which is a plausible sufficient condition, but it is not derived from the density-matrix equations, and the stress-test note is correct to point out that flavour off-diagonal coherence terms are not directly controlled by this inequality. The paper itself acknowledges the need for a full quantum-kinetic treatment beyond M_max, which is honest, but it also means the quoted mass window is only as good as that criterion. If the exact decoherence condition is stricter, the window shrinks; if the criterion is conservative, the window could be larger. The authors should either justify the criterion more carefully or run a density-matrix check for representative points.

A second, lesser issue is reproducibility. The scan fractions (37% N1-dominated, 26% N2-dominated, 33% electron-flavour survival) are not backed by public code or detailed scan specifications. The numbers are plausible as order-of-magnitude statements, but I would not quote them as precise predictions. The 33% probability in the three-flavour scenario also relies on single-flavour exponential washout and neglects flavour couplings, which the authors explicitly state; the ordering of flavour survival is likely right, but the exact percentage should be treated cautiously.

Overall, the paper is a solid, useful contribution for model builders working on flavour symmetries and leptogenesis. It deserves a serious referee, not a desk reject, and I would send it to review with a request to address the decoherence criterion and make the numerical scan more transparent.

Referee Report

3 major / 3 minor

Summary. This paper studies thermal flavoured leptogenesis in the minimal two-right-handed-neutrino seesaw with CP^{μτ} symmetry. The authors use analytic efficiency factors, validated against explicit one-flavour Boltzmann solutions, and scan the CP^{μτ} parameter space constrained by neutrino oscillation data. They claim three main results: (i) a mild hierarchy M2/M1 ≈ 4.7 is sufficient for a hierarchical N1-dominated scenario, lowering the lower bound on M1 to about 7.5 × 10^9 GeV; (ii) in the two-flavour regime the N1-dominated scenario is valid only for 7.5 × 10^9 GeV ≲ M1 ≲ 4 × 10^10 GeV, with N2 domination possible for a substantial part of the parameter space above the upper bound; and (iii) if M1 lies in the three-flavour regime, an asymmetry produced by N2 in the two-flavour regime survives N1 washout mainly in the electron flavour, with about 33% probability for a hierarchical light neutrino spectrum.

Significance. If the claims are correct, the paper is significant: it extends leptogenesis studies of CP^{μτ} models beyond the strong-hierarchy limit, identifies a specific RH neutrino mass window, and quantifies the conditions under which N2 contributions matter. The paper is commendable for the explicit one-flavour Boltzmann check of the analytic efficiency factors and for the transparent parameter scan tied to oscillation data. The main caveats are the semi-classical decoherence criterion used to set the upper bound M_max and the ad hoc introduction of N3 in the three-flavour-regime scenario without a corresponding refit of the light-neutrino parameter space; these points need to be addressed before the central quantitative claims can be considered fully established.

major comments (3)

[Sec. V.A, Eq. (V.10)] The upper bound M_max ≈ 4 × 10^10 GeV is set by the condition F_τ = Γ_τ/(2Hz_i) > W(z_i^max), where W is a single scalar washout rate. The two-flavour regime is a density-matrix problem with flavour off-diagonal coherence terms, as the paper itself notes in footnote 11, and the rate comparison in Eq. (V.10) does not by itself establish the decoherence of those off-diagonal terms. Because the quoted mass window and the N1/N2 dominance percentages in Section V.A rest on this criterion, the upper bound should be checked with a full quantum-kinetic calculation for representative parameter points or with a quantitative estimate of the error incurred by the rate comparison; otherwise the window 7.5 × 10^9 GeV < M1 < 4 × 10^10 GeV is not rigorously delimited.
[Sec. V.B, footnote 14] The M1 < 10^9 GeV scenario requires adding a third right-handed neutrino N3 with M3 > 10^12 GeV and Yukawa couplings 'of similar order of magnitude' to N1 and N2, but the light-neutrino parameter scan and the K_{1α} distributions in Fig. 13 are computed from the two-RHN mass matrix without N3. If y3 ~ y2 and M3 ~ 10^12 GeV, the N3 contribution to M_ν is of order 10^-4 eV for K ~ 20 and M2 ~ 10^10 GeV, which is a few percent of the solar mass scale and not manifestly negligible for the 3σ fit. The authors should demonstrate that the N3 contribution is below the fit sensitivity or repeat the scan with N3 included; otherwise the central 33% survival probability is not the probability in the stated model.
[Sec. V.A, Fig. 9 and Eq. (IV.13)] The critical mild hierarchy M2/M1 = 4.7 is determined from the one-flavour efficiency factors of Eqs. (III.13)/(III.18) applied to the tau-flavour decay parameters, not from a numerical solution of the flavoured Boltzmann equations (IV.10)-(IV.11). Since the claim is specifically about flavour-dependent N1 dominance, a flavoured Boltzmann check analogous to Fig. 2 for the representative points of Fig. 9 would strengthen the identification of δ12 = 3.7 as the critical point; as it stands, the conclusion relies on the analytic hierarchy-exponential approximation without direct verification in the flavoured case.

minor comments (3)

[Sec. II, Eq. (II.11)] The symbol θ is reused for the relative phase after the redefinition θ1 = 0, θ2 = θ; the allowed range of θ should be stated explicitly next to Eq. (II.11) rather than only in the discussion around Fig. 6.
[Sec. V.A, Eq. (V.6)] The expression for ε1^μ in the 'strong hierarchical limit' is used in a context where x12 = (4.7)^2 ≈ 22, which is not strongly hierarchical; please clarify whether the full g'(x12) expression of Eq. (V.5) is used in the numerical analysis.
[Fig. 12 caption] The caption does not state that parameter points with large decay parameters are discarded for M1 = 7 × 10^10 GeV, although the text explains this; the truncation should be stated in the caption so that the plot is not misread as covering the full allowed parameter space.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central mass-window and N2-survival claims are computed from neutrino-oscillation constraints and model dynamics, not fitted to the observed baryon asymmetry.

full rationale

The paper's central quantitative claims are (i) the mild hierarchy M2/M1 ~ 4.7 required for N1-dominated leptogenesis, (ii) the resulting window 7.5e9 GeV < M1 < 4e10 GeV, and (iii) the ~33% probability that N2-generated asymmetry survives in the electron flavour after N1 washout in the three-flavour regime. All three are derived from the CPmu-tau parametrization of the Dirac mass matrix (Eq. II.11), constrained by neutrino oscillation data, and then evolved through standard Boltzmann or analytic efficiency-factor equations. The observed baryon-to-photon ratio etaB is used as a target to compare against, but no parameter is fitted to it. The threshold M2/M1 = 4.7 is obtained by comparing efficiency factors kappa1 with kappa1^infinity and by scanning the model's allowed flavoured decay parameters; it is not imposed by construction. The upper bound M_max ~ 4e10 GeV follows from the explicit decoherence condition F_tau > W in Eq. V.10, again a computed consistency requirement rather than an input. The 33% electron-flavour survival probability is computed from the distribution of K1e in the model parameter space, with the washout condition K1e < 1 taken from standard leptogenesis literature; this is a model prediction, not a fitted output. The paper's self-citations supply the model texture (Ref. [35]) and some analytic tools (Ref. [63]), but these are external premises and standard machinery, not results that already contain the quoted mass window or survival probability. The limitations acknowledged by the authors, such as the neglect of flavour couplings and the need for a full density-matrix treatment beyond M_max, are correctness or robustness concerns, not circularity. The derivation chain is therefore self-contained with respect to the claims that are actually presented as predictions.

Assumptions & free parameters 5 free parameters · 7 assumptions · 1 invented entities

The central claims rest on a specific CPmu-tau-antisymmetric seesaw texture with six scanning parameters {x1,x2,y1,y2,theta,M1}, plus standard strong-washout and thermal-production assumptions. The second scenario adds an unobserved N3 with assumed Yukawa couplings. These are model assumptions rather than fitted constants for eta_B, which keeps the circularity burden low.

free parameters (5)

x1, x2, y1, y2 (Dirac mass texture parameters) = ranges shown in Fig. 6, in sqrt(eV)
Free parameters of the CPmu-tau-antisymmetric m_D texture of Eq. II.11-II.13, scanned and constrained by NuFIT 3-sigma oscillation data.
theta (relative phase) = -150 deg < theta < 150 deg
The phase difference theta2 - theta1 after phase redefinition in Sec. II, constrained by the neutrino oscillation fit.
M1 (lightest right-handed neutrino mass) = allowed window roughly 7.5e9 to 4e10 GeV
Input mass scale; the paper derives an allowed window but the value of M1 itself is not predicted.
M2/M1 hierarchy ratio = 4.7, corresponding to delta12 = 3.7
Chosen as the critical mild hierarchy for a valid N1-dominated scenario, derived from efficiency-factor comparisons rather than fitted to the baryon asymmetry.
M3 (third right-handed neutrino mass, second scenario) = greater than 1e12 GeV
Introduced in Sec. V B to make the N2 CP asymmetry nonvanishing when M1 is below 1e9 GeV.

assumptions (7)

domain assumption Type-I seesaw with diagonal charged-lepton and right-handed neutrino mass matrices
Eqs. II.2-II.3 establish the minimal seesaw framework and the basis used throughout.
domain assumption CPmu-tau-antisymmetric invariance of the Dirac mass matrix: m_D G = -i m_D^*
Eq. II.10 is the defining symmetry of the model, taken from prior work by some of the same authors.
domain assumption Two right-handed neutrino minimal seesaw with det(M_nu) = 0 and massless lightest neutrino
Sec. II after Eq. II.12; this texture is required by the CPmu-tau structure and sets a hierarchical light neutrino spectrum.
domain assumption Thermal production of right-handed neutrinos with TRH > M2 and no pre-existing asymmetry
Sec. III opening states both RH neutrinos are thermally produced and pre-existing asymmetries are not considered.
domain assumption Strong washout regime with neglect of Delta L = 1 and Delta L = 2 scatterings in the efficiency equations
Sec. III after Eq. III.7; this standard approximation is used for the strong-washout parameter region.
domain assumption Full charged-lepton flavour decoherence, valid when F_tau = Gamma_tau/(2Hz) dominates the washout rates
Eq. V.10 and the surrounding text define the condition used to set M_max ~ 4e10 GeV; violation requires density-matrix equations.
ad hoc to paper A third right-handed neutrino N3 with M3 > 1e12 GeV and Yukawa couplings of similar order to N1 and N2
Sec. V B introduces N3 to give N2 a nonvanishing CP asymmetry in the second scenario; this goes beyond the minimal two-RH seesaw used in the first part.

invented entities (1)

Right-handed neutrino N3 with M3 > 1e12 GeV
purpose: Provides a nonvanishing N2 CP asymmetry when M1 is in the three-flavour regime; its seesaw contribution is suppressed by M3.
Assumed in Sec. V B with Yukawa couplings of similar order to N1 and N2; no direct observable signature is identified in the paper.

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Cite this review

Pith. "Pith review of Flavoured leptogenesis and ${\rm CP}^{\mu\tau}$ symmetry." pith.science (2026). https://pith.science/paper/2VFRYY5L

@misc{pith2026190808126,
  author       = {Pith},
  title        = {Pith review of: Flavoured leptogenesis and $\rm CP^\mu\tau$ symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VFRYY5L}},
  note         = {Machine review of arXiv:1908.08126}
}

abstract

We present a systematic study of leptogenesis in neutrino mass models with $\mu\tau$-flavoured CP symmetry. In addition to the strong hierarchical $N_1$-dominated scenario ($N_1$DS) in the `two flavour regime' of leptogenesis, we show that one may choose the right-handed (RH) neutrino mass hierarchy as mild as $M_2\simeq 4.7 M_1$ for a perfectly valid hierarchical $N_1$DS. This in turn reduces the lower bound on the allowed values of $M_1$, compared to what is stated in the literature. The consideration of flavour effects due to the heavy neutrinos also translate into an upper bound on $M_1$. It is only below this bound that the observed baryon-to-photon ratio can be realized for a standard ${ N_1}$ domination, else a substantial part of the parameter space is also compatible with $N_2$DS. We deduce conditions under which the baryon asymmetry produced by the second RH neutrino plays an important role. Finally, we discuss another interesting scenario where lepton asymmetry generated by $N_2$ in the two flavour regime faces washout by $N_1$ in the three flavour regime. Considering a hierarchical light neutrino mass spectrum, which is now favoured by cosmological observations, we show that at the end of $N_1$-leptogenesis, the asymmetry generated by $N_2$ survives only in the electron flavour and around $33\%$ of the parameter space is consistent with a pure $N_2$-leptogenesis.

Figures

Figures reproduced from arXiv: 1908.08126 by the authors.

**Figure 2.** FIG. 2. Efficiency factor with [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3. Left: Possible error due to [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4. Ilustration of the two flavour regime for two RH neutrino model. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. Various mass pattern in a leptogenesis scenario dominated by two right handed neutrinos. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6. Parameter space of CP [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7. Flavoured decay parameters for both the RH neutrinos. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: FIG. 8. Plot showing the quantity (1 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: FIG. 9. Upper panel: (Colour codes for the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: FIG. 10. Comparison among rates of various processes involved in the leptogenesis. The horizontal [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: FIG. 11. Top panel [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: FIG. 12. Normalised baryon to photon ratio with the [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

**Figure 13.** Figure 13: FIG. 13. Distributions of the flavoured decay parameters. The probability for the electron decay [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

**Figure 14.** Figure 14: FIG. 14. Left: Possible range of the orientation of the state [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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(V.6) Similarly for i = 2 the CP asymmetry parameter can be calculated as εµ 2 = g′(x21)M1 4πv2 [(x1x2 +y1y2 cosθ)y1y2 sinθ x2 2 +y2 2 ] =−ετ
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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

(V.6) Similarly for i = 2 the CP asymmetry parameter can be calculated as εµ 2 = g′(x21)M1 4πv2 [(x1x2 +y1y2 cosθ)y1y2 sinθ x2 2 +y2 2 ] =−ετ

[2] [2]

− √ 2eiπ/4(x1y1eiθ1 +x2y2eiθ2) −i √ 2eiπ/4(x1y1e−iθ1 +x2y2e−iθ2) − √ 2eiπ/4(x1y1eiθ1 +x2y2eiθ2) −(e2iθ1y2 1 +e2iθ2y2

[3] [3]

−i(y2 1 +y2 2) −i √ 2eiπ/4(x1y1e−iθ1 +x2y2e−iθ2) −i(y2 1 +y2

[4] [4]

pre-existing asymmetry

e−2iθ1y2 1 +e−2iθ2y2 2  . (II.12) In (II.12), new real parameters x1,2 and y1,2 are deﬁned by scaling a1,2 and b1,2 with the square roots of the respective RH neutrino masses M1,2, i.e. a1,2√ M1,2 =x1,2, b1,2√ M1,2 =y1,2. (II.13) A few comments on the matrix MCPµτA ν are in order. Since det ( MCPµτA ν ) = 0, the lightest neutrino mass (either m1 for a n...

[5] [5]

(V.7) Armed with these equations, we can proceed toward a systematic discussion of leptogenesis for the relevant cases. A. Two ﬂavour regime: 109 GeV<M 1,2< 1012 GeV The N1-decay parameters in the muon and tau ﬂavour are strong enough [63, 64] to washout any pre-existing asymmetry (see right panel of Fig. 7). Thus, all the terms which contain the exponent...

[6] [6]

(V.9) 19 �-�� ⊥ �� FIG. 8. Plot showing the quantity (1 −p⊥

[7] [7]

Esteban, M

I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, JHEP 1901, 106 (2019) doi:10.1007/JHEP01(2019)106 [arXiv:1811.05487 [hep-ph]]

arXiv 2019

[8] [8]

An inter- esting fact is that (1−p⊥

with the model parameterx1 andx2. An inter- esting fact is that (1−p⊥