Primordial Lepton Family Asymmtries in Seesaw Model

Bottom line: the family-resolved asymmetries here are a genuine extension of the authors' own PRL work, and the headline observation — Y_mu and Y_tau each around 10⁻⁸, opposite in sign, cancelling to Y_L around 10⁻¹⁰, with the dominant family set by gamma_L — is a real effect. But the chemical-potential bookkeeping assumes all three charged-lepton Yukawas are in equilibrium at M1 = 2×10¹¹ GeV, and they are not. That is load-bearing for the abstract's claim and needs to be fixed before anyone treats this as a constraint on Yukawa textures. The stress-test note is right on this point, and I would not soften it. Eq. (3.25) sets mu_phi0 = mu_{l-iL} - mu_{l-iR} for i = e, mu, tau with no temperature qualification. At 2×10¹¹ GeV only the tau Yukawa is in equilibrium (boundaries roughly 10¹², 10⁹, 10⁵ GeV for tau, mu, e). So Y_e and Y_mu are not separately conserved; the resolved basis is Y_tau and Y_e+mu. That changes the A_ij coefficients in (3.37) and the reading of Fig. 8. To be fair to the authors, the mechanism itself — the flavor-dependent epsilon_ki from (4.9)-(4.10) — does not depend on the chemical-potential bookkeeping, so the effect is not an artifact. It is the O(1) coefficients and the interpretation of Y_mu as a separate observable that are at stake. Worth noting they cite Barbieri et al. (ref. 10), which is exactly the flavor-in-equilibrium paper, so the reference was in hand and the temperature boundaries were not applied. Second soft spot, smaller: the cancellation is shown at one hand-chosen point (x1, x2) = (0.0523, 0.0100) with only gamma_L varied. The 'we find the case' language is honest, but the abstract reads broader than an existence demonstration. What the paper does well: the Boltzmann equations, source terms, and reduced cross sections are laid out in enough detail to re-implement, and including both heavy neutrinos rather than just NR1 is the right call. This is for leptogenesis people working on flavor and for anyone building neutrino Yukawa textures who wants a cosmological handle. Send it to referees, but the chemical-potential assumption should be the first thing they are asked about.

The derivation of the chemical-potential relations in §3.2 assumes all three charged-lepton Yukawa interactions are in equilibrium. Eq. (3.25) sets μφ0 = μ_{l−iL} − μ_{l−iR} for i = e, μ, τ, and this feeds Eqs. (3.31) and (3.33), which in turn fix the A_ij matrix (3.37) and the conversion Y_B = −(28/51)Y_L (3.34). The standard flavor-regime boundaries are τ ~ 10^12 GeV, μ ~ 10^9 GeV, e ~ 10^5 GeV. At the paper's fixed M1 = 2×10^11 GeV (Section 4), only the tau Yukawa is in equilibrium. Therefore the relations for i = e, μ do not hold, and the mapping from μ to Y_Li (3.33) and the A_ij coefficients (3.37) are not valid for the electron and muon flavors. The paper's central claim is precisely that Y_mu and Y_tau are separately large and opposite (abstract:s5–s6, Figs. 6–8); if Y_mu is not a separately conserved quantity at this temperature, the claim needs restatement in the resolved basis (Y_tau and Y_{e+μ}). §3.2 simply states 'We assume that gauge and Yukawa interactions are in equilibrium' without temperature qualification, so the assumption is unexamined. I want to be fair: the qualitative mechanism — flavor-dependent ε_ki from Eq. (4.9)–(4.10) — is independent of the chemical-potential bookkeeping, so the effect itself is not an artifact. The concern is about the O(1) coefficients and the interpretation of Y_mu as a separate observable, not about the existence of flavor-dependent asymmetries. That is why I would keep the verdict at CONDITIONAL rather than REJECT.

The reader's weakest_assumption identifies the right load-bearing issue. The paper's family-resolved result depends on assigning independent chemical potentials to all three lepton doublets, which requires all three charged-lepton Yukawas to be in equilibrium; at M1 = 2×10^11 GeV only the tau is. The paper does not discuss this, and §3.2's blanket 'We assume that gauge and Yukawa interactions are in equilibrium' is exactly the unexamined step. I agree with the reader that this is the weakest link. I also agree with the reader's framing that the paper establishes an existence case at a single parameter point rather than a generic pattern; the 'we find the case' language in abstract:s4–s5 is honest about this, but the abstract's broader framing (abstract:s6) invites over-reading. A secondary point worth noting: the sphaleron conversion factor Y_B = −(28/51)Y_L (Eq. 3.34) is the one-flavor result and would also change in the two-flavor regime, compounding the same concern. Neither issue breaks the central claim as an existence demonstration, but both should be addressed before the result is treated as a generic constraint on seesaw textures. The paper's own acknowledged limitation (broken-phase Boltzmann equations beyond scope) is appropriately flagged and does not affect the symmetric-phase result. I therefore keep the reader's CONDITIONAL verdict and recommend no change: the concern is real, technically correct, and worth a quantitative check, but the qualitative mechanism survives it, so it does not warrant a downgrade to REJECT or a promotion to ACCEPT.

The paper provides a detailed and technically careful derivation of the Boltzmann equations for flavor-dependent lepton asymmetries, including the chemical-potential bookkeeping for all SM particles and a comprehensive set of scattering processes. The observation that family asymmetries can be much larger than the total, with the baryon asymmetry arising from a cancellation, is interesting and potentially relevant for leptogenesis phenomenology. The result that the dominant family is controlled by γL, while Y_L is controlled by γR, is a useful addition to the literature on flavor effects in leptogenesis. The main caveats are that the central numerical claim is demonstrated at a single hand-chosen parameter point and that the chemical-potential treatment assumes all three charged-lepton Yukawa interactions are in equilibrium, which is not the case at the chosen temperature.

The paper asks whether the decays of heavy right-handed neutrinos in the early universe produce not one net lepton excess but three separate family excesses, one each for electrons, muons and taus, and whether those family excesses can be far larger than the total. To answer it the authors write Boltzmann equations that track each lepton doublet with its own chemical potential, rather than assuming a single shared lepton chemical potential as earlier treatments did, and then convert the resulting number densities into family asymmetries $Y_e$, $Y_μ$, $Y_τ$ using the standard sphaleron and Yukawa equilibrium relations. They solve these equations in the minimal seesaw model with two heavy Majorana neutrinos, taking neutrino oscillation data to fix most of the input. The payoff is that the CP-violating phase $γ_L$ of the left-handed rotation $U_L$ steers which family carries the asymmetry: at $γ_L = 0$ the muon asymmetry dominates, at $γ_L = π$ the tau asymmetry does, and at $γ_L = π/2$ both reach order $10^{-8}$ with opposite signs and nearly cancel, leaving a total $Y_L$ of order $10^{-10}$, the size the observed baryon-to-photon ratio requires. The wider point is that the total lepton asymmetry, which depends mainly on the other CP phase $γ_R$, is a poor guide to what is actually happening family by family.

What carries the argument. The machinery is a set of Boltzmann equations in which the three lepton doublets carry independent chemical potentials, so that $Y_e$, $Y_μ$ and $Y_τ$ evolve separately instead of collapsing into a single lepton number. The source terms combine decays and inverse decays of the heavy Majorana neutrinos with two-body scatterings, including Higgs-exchange processes with the top quark and the $|ΔL| = 2$ and $ΔL = 0$ family-changing scatterings mediated by right-handed neutrino exchange. Chemical-potential relations from sphaleron equilibrium, charge neutrality and the gauge and Yukawa interactions then express every chemical potential in terms of the family asymmetries, giving $Y_{L_i} = (1/2)(μ_{l^-_iL}/T)T^3/s - (4/45)Y_L$ and the conversion $Y_B = -(28/51)Y_L$. The carrier of the family-dependent effect is the CP asymmetry $ε_{ki}$, whose second term comes from the self-energy diagram that contributes to family asymmetries but not to the total.

Core claim. The central claim is that in the minimal two-heavy-neutrino seesaw model the lepton family asymmetries $Y_μ$ and $Y_τ$ can each be of order $10^{-8}$ while the total lepton asymmetry $Y_L$ is of order $10^{-10}$, because the two large family asymmetries carry opposite signs and almost cancel. Which family dominates is controlled by the CP-violating phase $γ_L$ appearing in the left-handed mixing matrix $U_L$: $Y_μ$ dominates for $γ_L = 0$, $Y_τ$ for $γ_L = π$, and for $γ_L = π/2$ both are large and opposite. The total lepton asymmetry is nearly flat in $γ_L$, so a measurement of the baryon asymmetry alone would not reveal this structure. The authors trace the effect to the term $(y_ν)^*_{i1}(y_ν)_{i2}$ in the family CP asymmetry $ε_{ki}$, whose $γ_L$-dependent part is proportional to $(U_L^*)_{i2}(U_L)_{i3}$ and flips sign between $γ_L = 0$ and $γ_L = π$. They also stress that the heavier right-handed neutrino $N_{R2}$ contributes importantly and can dominate a given family asymmetry, in contrast to treatments that keep only the lightest one.

The premise it rests on. The derivation assumes that each of the three lepton doublets carries its own separately conserved chemical potential, which requires the electron and muon Yukawa interactions to be in equilibrium at the temperature where the asymmetry is generated.

The total lepton asymmetry and the baryon asymmetry can understate what is happening in any one family by two orders of magnitude, so a successful leptogenesis calculation that matches $η$ may still be hiding large, cancelling family asymmetries.

The phase $γ_L$ becomes an indirectly constrained quantity: different values of it give qualitatively different family patterns ($Y_μ$-dominated, $Y_τ$-dominated, or a large opposite-sign pair) while leaving $Y_L$ essentially unchanged.

Because the two heavy Majorana neutrinos contribute with opposite signs and different time profiles, both must be kept in the evolution; the heavier one can dominate a family asymmetry even when it is subdominant in the total.

The family asymmetries provide a handle for constraining the form of the neutrino Yukawa matrix, since different Yukawa textures give different family patterns at the same total $Y_L$.

The family-resolved treatment is only as good as the assumption that all three charged-lepton Yukawa interactions are in equilibrium at the relevant temperature; at the $M_1 = 2 \times 10^{11}$ GeV used here, only the tau Yukawa is comfortably fast, so a calculation that resolves just the tau flavor would be the natural next check on whether the $Y_μ$–$Y_τ$ cancellation survives.

If the cancellation pattern is real, it suggests a broader lesson for flavored leptogenesis: agreement with the observed baryon asymmetry is a weak test, and models should be compared on their family asymmetry vectors rather than on $Y_L$ alone.

The sensitivity of $Y_μ$ and $Y_τ$ to $γ_L$, combined with the weak sensitivity of $Y_L$ to it, implies that any future cosmological probe able to distinguish lepton flavors would constrain a CP phase that the baryon asymmetry cannot.

The paper's own framing invites a follow-up connecting $γ_L$-dependence of family asymmetries to low-energy leptonic CP violation, which the authors list as an open extension rather than a result.

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