Resonant \tau-Leptogenesis with Observable Lepton Number Violation

The idea is worth your time. If sphalerons conserve B/3 − L_τ individually, you can build the BAU out of a single flavoured asymmetry, and that lets the heavy Majorana neutrinos sit at 0.5–1 TeV instead of 10⁹ GeV. That is the paper's actual contribution, and the phenomenological tail — |(m_ν)_ee| ≈ 0.013 eV, B(μ→eγ) reaching 10⁻¹³, μ-e conversion near 10⁻¹⁶ — is exactly the kind of falsifiable handle most leptogenesis papers don't bother to supply. The Yukawa texture in Eq. (2) is clean, the U(1)_l protection argument for vanishing light masses at ε_l = 0 is correct, and footnote [24] hands you a full parameter set someone could actually reproduce. Credit where due: this is a well-posed model, not a hand-wave, and the author is upfront that further study is needed. Now the soft spots. First, the arithmetic. The text quotes η_B ≈ −(15/4) η_Lτ for T ≪ T_c, but the figure caption calls η_Lτ ≈ −3×10⁻⁸ 'the value needed to obtain the observed BAU.' Combine those and you get η_B ≈ 10⁻⁷, roughly two hundred times the observed 6.1×10⁻¹⁰. The paper's own earlier statement, B = −28/51 L_τ above T_c, is a different factor again. So the two conversion relations quoted differ by about seven, and neither reproduces the figure's target. This reads like a normalization slip rather than a physics failure, but it does mean the claimed agreement between the benchmark and the BAU is not checkable as printed, and a referee should insist it be pinned down. Second, the washout bound |c| ≲ 10⁻⁴ is argued rather than computed, and setting γ_S/Lτ = Σ γ_NiLτΦ is a fudge the author flags as conservative. Fine, but one number is doing a lot of work. Third — and here I'd push back on the reader's report — the 10⁻⁸ 'conventional BEs get this wrong' claim is not load-bearing. Eq. (5) is already the flavoured Boltzmann equation; the result comes from Fig. 1, not from [25]. The deferral to [25], still a preprint here, is for detail, not for the central number. Verdict: worth a serious referee, mainly to force the normalization and a second parameter point. The mechanism is sound; the demonstration is thin.

The paper's central quantitative claim is that the benchmark point of [24] yields ηLτ(Tc) ≈ -3×10^-8, which, via the conversion ηB ≈ -(15/4)ηLτ, gives the observed BAU. But if ηLτ = -3×10^-8 and ηB = -(15/4)ηLτ, then ηB = 1.1×10^-7, which is ~180× the observed 6.1×10^-10. The figure's caption explicitly identifies -3×10^-8 as 'the value needed to obtain the observed BAU,' so either the conversion factor is not -(15/4) (perhaps a typo for -(15/4)×10^-2, which would give ηB ≈ 1.1×10^-9, still a factor ~2 high) or the figure's 'needed' value is wrong. Either way, the quantitative agreement between the benchmark and the observed BAU is not established as stated. This is separate from the reader's concern about the 10^-8 flavour factor: the paper's Eq. (5) is itself the flavoured computation, so the central claim is supported by Fig. 1, not by the 10^-8 factor. The conversion factor is the more concrete vulnerability.

The reader's concern about the 10^-8 flavour factor is reasonable but slightly mis-targeted: the paper's Eq. (5) is itself the flavoured Boltzmann equation, and Fig. 1 solves it, so the central claim is supported by a self-contained computation, not solely by the 10^-8 factor. The 10^-8 factor is a justification for why the unflavoured treatment fails, which is a separate point. The more concrete vulnerability is the conversion factor: the text's -(15/4) and the figure's -3×10^-8 are inconsistent by ~two orders of magnitude. This is a checkable numerical issue that directly affects the claimed agreement with the observed BAU. The paper's other features — the falsifiable predictions (µ→eγ ~10^-13, µ-e conversion ~10^-16, |(mν)ee| ~0.013 eV) and the honest acknowledgment of the benchmark nature of [24] — are strengths. The verdict should remain CONDITIONAL: the scenario is plausible and has independent handles, but the quantitative demonstration needs the conversion factor checked.

If correct, the scenario is significant because it lowers the leptogenesis scale to the electroweak scale, making it testable at colliders and in lepton-flavour-violation experiments. The individual-lepton-number mechanism is a novel variant of resonant leptogenesis, and the paper provides a concrete benchmark and falsifiable predictions. The paper also has a self-consistent radiative stability argument (Eq. 3) and demonstrates independence of initial conditions (Fig. 1). These are genuine strengths.

The paper asks whether the observed baryon asymmetry of the universe can be produced at the electroweak scale, rather than at the enormous energies usually assumed, by a minimal extension of the Standard Model with one right-handed singlet neutrino per generation. The proposal is that heavy Majorana neutrinos of mass 0.5 to 1 TeV decay out of equilibrium with a CP asymmetry stored in a single lepton flavour, the tau number, and that sphaleron processes then convert that tau excess into baryons while leaving the electron and muon numbers alone. The model is built so that a CP-violating Yukawa texture explains the solar and atmospheric neutrino data, with an inverted light-neutrino mass spectrum, even though the heavy states sit at the TeV scale. What makes the claim worth attention is that the same heavy neutrinos would be light enough to produce at an electron-positron linear collider and would generate lepton-flavour-violating signals such as $0.013$ eV in neutrinoless double $\beta$ decay and a muon-to-electron-gamma branching ratio near $10^{-13}$.

What carries the argument. The load-bearing mechanism is resonant enhancement of the CP asymmetry in heavy Majorana neutrino decays when two of the heavy states are nearly degenerate, with a mass splitting comparable to their decay widths. In this paper that resonance is combined with flavour bookkeeping: the tau asymmetry $delta_ta^{N_1}$ is computed for the three-generation case, and the baryon yield is estimated as $eta_B sim -10^{-2}, delta_ta^{N_1} K_{N_1} (arepsilon_ta/c)^2$, so that the answer is controlled by three dials: the out-of-equilibrium measure $K_{N_1}$, the resonant asymmetry, and the ratio of tau to electron/muon couplings. A second piece of machinery is the approximate $U(1)$ lepton symmetry of the degenerate limit, which is what allows TeV-scale heavy neutrinos to coexist with sub-$0.1$ eV light neutrinos.

Core claim. The central claim is that resonant leptogenesis can operate flavour by flavour: an asymmetry in a single individual lepton number, here $L_ta$, can be resonantly generated by the out-of-equilibrium decays of heavy Majorana neutrinos and then partially converted into the baryon asymmetry by sphalerons, which preserve $B - L_e$, $B - L_mu$ and $B - L_ta$ separately. Because the conversion in this regime is $B = -28/51, L_ta$, an excess of roughly $3 imes 10^{-8}$ in tau number per photon suffices for the observed baryon-to-photon ratio. The author exhibits a Yukawa texture with maximal CP violation, in which the two heavier singlet neutrinos sit in thermal equilibrium and decay mostly to electrons and muons while their couplings to taus are suppressed by $|c| < 10^{-4}$, so that the tau excess produced by the lightest heavy neutrino is not washed out. Solving a Boltzmann equation for the tau number density at a benchmark point with $m_{N_1} = 500$ GeV and $K_{N_1} approx 15$ gives a tau asymmetry independent of initial conditions and compatible with the observed baryon asymmetry. The same construction keeps light neutrino masses below $0.1$ eV: exact degeneracy plus vanishing $arepsilon_l$ leaves the light mass matrix zero, protected by a $U(1)$ lepton symmetry, and small breaking parameters of order $10^{-6}$ and $10^{-7} m_N$ supply the observed splittings and mixings without destabilising the spectrum radiatively.

The premise it rests on. The scenario works only if the standard flavoured Boltzmann treatment really misses single-lepton-flavour effects by the enormous factor of about $10^{-8}$ that the paper asserts, a claim justified only by deferring to a companion study rather than derived here.

Heavy Majorana neutrinos with masses of a few hundred GeV to about 1 TeV become the relevant cosmological actors, so the baryon asymmetry no longer requires a reheating temperature near $10^9$ GeV or heavier.

The two heavier heavy neutrinos decay predominantly into electrons and muons rather than taus, giving a distinctive collider signature at $e^+e^-$ linear colliders.

The inverted light-neutrino hierarchy implied by the benchmark point predicts an effective Majorana mass $|(m_ u)_{ee}| approx 0.013$ eV, testable in the next generation of neutrinoless double beta decay searches.

For Yukawa couplings $a, b sim 3 imes 10^{-3}$, the model predicts $B( u o egamma) sim 10^{-13}$ and a muon-to-electron conversion rate in $^{48}_{22}mathrm{Ti}$ at the $10^{-16}$ level, both within the reach of planned experiments.

Because the muon-to-electron-gamma rate scales as $a^2b^2v^4/m_N^4$ while non-decoupling contributions to muon-to-electron conversion scale as $a^2b^2v^2/m_N^2$, comparing the two channels would help fix the heavy neutrino mass scale.

The scenario ties cosmology to laboratory physics in a way that is unusual for leptogenesis: a single parameter set is simultaneously constrained by the baryon asymmetry, neutrino oscillation data, collider reach and lepton-flavour-violation limits, so an improvement in any one of these would tighten the others.

If the mechanism is right, the tau flavour plays a special role among the three lepton numbers, which suggests looking for correlated departures from flavour universality in tau decays and in the muon sector rather than treating them as independent anomalies.

The framework invites an extension to supersymmetric or unified completions, where the same near-degeneracy that drives the resonance would have to survive renormalisation-group running down from the unification scale.

A useful cross-check would be to redo the asymmetry calculation with an independent quantum kinetic treatment of flavour coherence, since the size of the predicted tau excess is the single number on which the whole scenario turns.

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