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Electroweak-Scale Resonant Leptogenesis

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Apostolos Pilaftsis, Thomas E.J. Underwood

High Energy Physics - Phenomenology · Submitted 10 Jun 2005 · 81 references

Pith's verdict: The reusable result is the flavour-resolved Boltzmann network, not the tuned η_B numbers; the sphaleron rate near Tc is the one soft spot that actually matters.

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arXiv IDhep-ph/0506107 (version 2)
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CitationApostolos Pilaftsis, Thomas E.J. Underwood. "Electroweak-Scale Resonant Leptogenesis." arXiv:hep-ph/0506107 (2005). https://pith.science/paper/S3IL3TVT
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The paper

The paper claims

Heavy neutrinos at the electroweak scale could explain why matter survived, and leave traces in muon decays.

The reading

The paper asks whether the observed excess of matter over antimatter can be generated by leptogenesis at energies as low as a few hundred GeV, rather than at the enormous scales usually assumed. The authors build a minimal three-generation model in which three heavy Majorana neutrinos sit near the electroweak scale and are almost degenerate in mass, so that the CP asymmetry in their decays is resonantly enhanced. Their central technical claim is that when the asymmetry lives in a single lepton flavour, here the tau number, the standard treatment that lumps all lepton flavours together fails badly: it can miss the answer by up to six orders of magnitude and even get the sign wrong. Once flavour is tracked properly, and once the shutting-off of sphaleron transitions near the electroweak phase transition is included, the model reproduces the measured baryon-to-photon ratio of about $6.1×10^{-10}$ and is nearly insensitive to whatever baryon, lepton, or heavy-neutrino abundance the universe started with. The payoff is a set of concrete low-energy and collider predictions, which is what makes the scenario worth taking seriously as a testable rather than decorative idea.

What carries the argument

The load-bearing object is the flavour-resolved network of Boltzmann equations, Eqs. (3.15) and (3.16), coupled to the sphaleron conversion equations (4.14) and (4.15). Three ingredients make it work. First, the resonant enhancement of the CP asymmetry: when two heavy Majorana neutrinos have a mass splitting comparable to their decay widths, the self-energy contribution to the asymmetry is amplified far beyond the usual one-loop estimate. Second, the flavour resolution itself: sphalerons conserve $B/3 - L_e$, $B/3 - L_μ$, and $B/3 - L_τ$ separately, so an asymmetry stored in one flavour can survive while the others are washed out, and the individual $K$-factors $K^l_{N_α}$ in Eq. (5.6) quantify which flavour is protected. Third, the sphaleron freeze-out: the rate of Eq. (4.4), scaled by the temperature-dependent Higgs expectation value of Eq. (4.2), decouples the baryon asymmetry from the lepton asymmetry at $T$ near $T_c$, freezing in whatever baryon number has accumulated. A compact order-of-magnitude estimate, Eq. (5.7), reproduces the numerical result and makes the flavour dependence transparent.

Extended reading notes

Core claim

The paper's claim is that resonant tau-leptogenesis works at the electroweak scale, and that seeing it requires resolving individual lepton flavours in the Boltzmann equations. In the model, an approximate $SO(3)$ symmetry makes the heavy neutrinos nearly degenerate, while an approximate $U(1)_l$ lepton symmetry keeps the tau Yukawa couplings small and the electron and muon couplings comparatively large. That texture lets the heavy states decay out of equilibrium and generate a large tau asymmetry while protecting the tau number from washout. The authors extend the standard network of Boltzmann equations to follow $L_e$, $L_μ$, and $L_τ$ separately, include standard-model chemical potentials, and add a temperature-dependent sphaleron rate so that lepton-to-baryon conversion switches off as the Higgs expectation value turns on near $T_c$. Solving these equations for heavy masses of 100, 250, 500, and 1000 GeV, they find final baryon asymmetries matching the observed value, with the low-mass cases depending on initial conditions only weakly and the $m_N ≥ 500$ GeV cases not at all. They contrast this with the flavour-blind treatment, which in their 250 GeV example is suppressed by roughly $10^{-6}$ and carries the wrong sign.

Load-bearing premise

The claim that leptogenesis can happen as low as 100 to 250 GeV depends on an approximate treatment of how quickly sphaleron transitions shut off near the electroweak phase transition, which assumes a smooth, spatially homogeneous transition and uses a rate formula valid only near or below the critical temperature.

Editorial extensions

If this is right

  • Electroweak-scale heavy Majorana neutrinos with couplings to electrons and muons of order $10^{-2}$ of the standard-model $W$ coupling would be producible at future $e^+e^-$ and $μ^+μ^-$ colliders running at 0.5 to 1 TeV, with of order 100 signal events expected at 100 fb$^{-1}$ before background reduction.
  • The same couplings generate lepton-flavour violation: the models predict $B(μ → eγ)$ in the range $6×10^{-14}$ to $6×10^{-12}$ and coherent $μ → e$ conversion in $^{48}_{22}$Ti around $10^{-13}$, both within the reach of the proposed MEG and MECO experiments.
  • Because the light neutrino spectrum comes out inverted, the effective Majorana mass $|\langle m\rangle| = |(m_ν)_{ee}|$ is predicted near $0.013$ eV, inside the target range of next-generation neutrinoless double beta decay experiments.
  • The final baryon asymmetry is essentially independent of the primordial baryon, lepton, and heavy-neutrino abundances for heavy masses at or above 250 GeV, so the scenario does not need to assume a particular reheating history.
  • The same flavour-resolved treatment should change predictions in ordinary hierarchical leptogenesis as well, where the authors show an order-of-magnitude suppression from neglecting flavour in a mildly hierarchical example.

Reading between the lines

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

  • If flavour-resolved Boltzmann equations shift hierarchical leptogenesis predictions by an order of magnitude, then some existing constraints on heavy-neutrino mass and reheating temperature derived with the flavour-blind treatment may need revisiting, not just the resonant scenarios.
  • The prediction of an inverted light-neutrino hierarchy with a massless lightest state is a sharp, independent handle: a confirmed normal hierarchy from oscillation data would remove the version of the model studied here, though not necessarily other resonant variants.
  • The near-degeneracy required in the heavy sector, at the $10^{-7}$ level relative to the mass, is the kind of tuning that a flavour symmetry could supply; the paper leaves the origin of the symmetry-breaking parameters open, which is where a ultraviolet completion would have to do real work.
  • Because the collider signature is a heavy neutrino that decays to electrons and muons but not taus, a null result at a 0.5 TeV collider would constrain the same couplings that control $μ → eγ$, giving two independent handles on one parameter pair.
Original abstract

We study minimal scenarios of resonant leptogenesis near the electroweak phase transition. These models offer a number of testable phenomenological signatures for low-energy experiments and future high-energy colliders. Our study extends previous analyses of the relevant network of Boltzmann equations, consistently taking into account effects from out of equilibrium sphalerons and single lepton flavours. We show that the effects from single lepton flavours become very important in variants of resonant leptogenesis, where the observed baryon asymmetry in the Universe is created by lepton-to-baryon conversion of an individual lepton number, for example that of the tau lepton. The predictions of such resonant tau-leptogenesis models for the final baryon asymmetry are almost independent of the initial lepton-number and heavy neutrino abundances. These models accommodate the current neutrino data and have a number of testable phenomenological implications. They contain electroweak-scale heavy Majorana neutrinos with appreciable couplings to electrons and muons, which can be probed at future e+ e- and mu+ mu- high-energy colliders. In particular, resonant tau-leptogenesis models predict sizeable 0\nu\beta\beta decay, as well as e- and mu-number-violating processes, such as mu -> e gamma and mu -> e conversion in nuclei, with rates that are within reach of the experiments proposed by the MEG and MECO collaborations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,461 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.10 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.53 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study minimal scenarios of resonant leptogenesis near the electroweak phase transition. These models offer a number of testable phenomenological signatures for low-energy experiments and future high-energy colliders. Our study extends previous analyses of the relevant network of Boltzmann equations, consistently taking into account effects from out of equilibrium sphalerons and single lepton flavours. We show that the effects from single lepton flavours become very important in variants of resonant leptogenesis, where the observed baryon asymmetry in the Universe is created by lepton-to-baryon conversion of an individual lepton number, for example that of the tau lepton. The predictions of such resonant tau-leptogenesis models for the final baryon asymmetry are almost independent of the initial lepton-number and heavy neutrino abundances. These models accommodate the current neutrino data and have a number of testable phenomenological implications. They contain electroweak-scale heavy Majorana neutrinos with appreciable couplings to electrons and muons, which can be probed at future e+ e- and mu+ mu- high-energy colliders. In particular, resonant tau-leptogenesis models predict sizeable 0\nu\beta\beta decay, as well as e- and mu-number-violating processes, such as mu -> e gamma and mu -> e conversion in nuclei, with rates that are within reach of the experiments proposed by the MEG and MECO collaborations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study minimal scenarios of resonant leptogenesis near the electroweak phase transition. contribution 0.99 0.89 0.04 asserted 0.69
s1 These models offer a number of testable phenomenological signatures for low-energy experiments and future high-energy colliders. interpretation 0.75 0.15 0.06 asserted 0.84
s2 Our study extends previous analyses of the relevant network of Boltzmann equations, consistently taking into account effects from out of equilibrium sphalerons and single lepton flavours. contribution 0.62 0.89 0.05 asserted 0.97
s3 We show that the effects from single lepton flavours become very important in variants of resonant leptogenesis, where the observed baryon asymmetry in the Universe is created by lepton-to-baryon conversion of an individual lepton number, for example that of the tau lepton. result 0.93 0.85 0.34 asserted 1.00
s4 The predictions of such resonant tau-leptogenesis models for the final baryon asymmetry are almost independent of the initial lepton-number and heavy neutrino abundances. result 0.99 0.89 0.25 asserted 1.00
s5 These models accommodate the current neutrino data and have a number of testable phenomenological implications. result 0.50 0.42 0.09 asserted 0.98
s6 They contain electroweak-scale heavy Majorana neutrinos with appreciable couplings to electrons and muons, which can be probed at future e+ e- and mu+ mu- high-energy colliders. interpretation 0.44 0.39 0.09 asserted 0.87
s7 In particular, resonant tau-leptogenesis models predict sizeable 0\nu\beta\beta decay, as well as e- and mu-number-violating processes, such as mu -> e gamma and mu -> e conversion in nuclei, with rates that are within reach of the experiments proposed by the MEG and MECO collaborations. result 0.69 0.74 0.25 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from simulation or computation 0.83.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Finite radiative effects contributing to the light-neutrino mass matrix. mN1 ≤ mN2 ≤ mN3 , and h νR ≡ h ν , these finite radiative corrections may conveniently be expressed as follows [40]: (δmν )ll′ = − αw 32π X α=1,2,3 h ν lα h ν l ′α v 2 mNα  3M2 Z M2 W  B0(0, m2 Nα , M2 Z ) − B0(0, 0, M2 Z )  + m2 Nα M2 Z  B0(0, m2 Nα , M2 H ) − B0(0, m2 Nα , m2 Nα )   , (2.9) where αw = g 2 w/(4π) and MH is the SM Higgs boson mass. In (2.9), B0(0, m2 1 , m2 2 ) is the usual Pasarino–Veltman one-loop function [41], i.e. B0(0, m2 1 , m2 2
Figure 2. The predicted evolution of ηLl and ηB, for models with mN = 100, 250, 500 and 1000 GeV, and η in Nα = 1. The model parameters are given in (5.1), (5.2), and Tables 2 and 3. The horizontal grey dashed line shows the baryon asymmetry needed to agree with observational data. 25
Figure 3. The (in)dependence of the final baryon asymmetry on the initial lepton, baryon and heavy Majorana neutrino abundances for (a) mN = 250 GeV and (b) mN = 100 GeV. The model parameters and the meaning of the horizontal grey line are the same as in
Figure 4. The complete independence of the final BAU on the initial lepton and baryon abundances for the mN = 500 GeV scenario, with model parameters the same as in
Figure 5. The ratio of various collision terms to the Hubble parameter plotted for the mN = 250 GeV scenario presented in
Figure 6. The predicted evolution of ηLtotal and ηNα for a model with a heavy neutrino spectrum: mN3 = 3 mN1 , mN2 = 2 mN1 and mN1 = 1010 GeV. where K = P l=e,µ τ Kl . In the RτL scenario with mN = 250 GeV, the dominant contribu￾tion to this estimate will come from N3, with a total CP asymmetry δN3 ∼ 10−3 . Taking the ratio of the two estimates yields η univ. B ηB ∼ δN3 δ τ N3 KN3 Kτ Kτ N3 K ≈ δN3 δ τ N3 |c| 2 |a| 2 + |b| 2 ≈ − 10−6 . (5.11) Thus, without considering single lepton flavour effects in this particular RτL model, one obtains an erroneous prediction for th
Figure 7. Feynman graphs pertaining to the effective γeµ and Zeµ couplings. 6.2 µ → eγ As shown in
Figure 8. Feynman graphs responsible for µ − → e −e +e − (µ → e conversion in nuclei). where αem = 1/137 is the electromagnetic fine structure constant, Zeff is the effective atomic number of coherence and Γcapt. is the muon nuclear capture rate. For 48 22Ti, ex￾perimental measurements give Zeff ≈ 17.6 for 48 22Ti [72] and Γ[µ 48 22Ti → capture] ≈ 1.705×10−18 GeV [73]. Moreover, |F(−m2 µ )| ≈ 0.54 is the nuclear form factor [74]. Finally, QW = Vu(2Z + N) + Vd(Z + 2N) is the coherent charge of the nucleus, which is associated with the vector current. Its explicit form is given by Vu =
Figure 9. Diagrams related to the production of the heavy Majorana neutrinos N2,3 at future e +e −(µ +µ −) high-energy colliders. As a final general remark, we should mention that RτL models, and leptogenesis models in general, do not suffer from too large contributions to the electron electric dipole moment (EDM) [12, 76], which first arises at two loops. The reason is that EDM effects are suppressed either by higher powers of small Yukawa couplings of order 10−4 and less, or by very small factors, such as (mN1 − mN2,3 )/mN ∼ 10−7 . The latter is the case in RτL models, which leads to…
Figure 10. ∆L = 0 interactions between leptons of different flavours mediated by heavy Majorana neutrinos. (a) corresponds to the process LkΦ ↔ LlΦ, (b) corresponds to the process LkΦ † ↔ LlΦ † and (c) corresponds to the process LkL C l ↔ Φ †Φ. In (A.21), P −1 α (x) is the Breit–Wigner s-channel propagator P −1 α (x) = 1 x − aα + i √ aαcα . (A.22) Therefore, following the procedure in [17], the RIS-subtracted propagator is determined by |D −1 α (x)| 2 = |P −1 α (x)| 2 − π √ aαcα δ(x − aα) → 0. (A.23) Processes (b) and (c) in

Figures reproduced from arXiv:hep-ph/0506107 by the authors.

References
How later papers used it

122 later papers cite it

Evidence: substantial. 114 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 209 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 4 of 11 papers with this label
How the literature uses it

116 arXiv papers cite this in 201 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 144 sentences without a confident label

Uses its method or tool 11 papers
  • These, in turn, are based on the Boltzmann equations (BEs) derived in [ref] specifically for flavoured leptogenesis in the resonant regime.

    Low-scale Leptogenesis with Minimal Lepton Flavour Violation · Baryon Asymmetry · 0.91

  • The set of Boltzmann equations that governs the dynamics of decay of the heavy pseudo-Dirac states and yield of lepton asymmetries are the following [ref] : [equation] [equation] where we have defined $z=M_N_1/T$.

    Rescuing leptogenesis parameter space of inverse seesaw · Leptogenesis · 0.93

  • The relevant Boltzmann equations, taking into account decays and inverse decays with proper subtraction of the resonant contribution from $ΔL=2$ [ref] and $ΔL=0$ [ref] processes as well as $ΔL=1$ scatterings [ref] , are given by [equation] where $z ≡M_ρ_3/T$, $Δ_α=B/3 -L_α$, and we indicated with $N_X$ any particle number or asymmetry $X$ calculated in a portion of co-moving volume containing one $ρ_3$ (i.e. three components) in ultra-relativistic thermal equilibrium, so that $N^eq_ρ_3(T≫M_ρ_3)=1$.

    Baryogenesis via Leptogenesis in Adjoint SU(5) · Flavored Regime · 0.90

Uses its data or values 1 paper
  • For non-unitarity induced through heavy right handed neutrinos the bound quoted in Ref. [ref] is |∑_i=1^3 W_e i W_μi^* | ≡ϵ_e ϵ_μ1.2 ·10^-4 The bound on the $μ$-$τ$ sector is much weaker |∑_i=1^3 W_μi W_τi^* | ≡ϵ_μϵ_τ2·10^-2 The $ϵ_α$'s are also constrained by electroweak measurements individually as [ref] ϵ_e^2< 0.012, ϵ_μ^2< 0.0096, ϵ_τ^2 < 0.016.

    Testing non-unitarity of neutrino mixing matrices at neutrino factories · Non-unitary mixing matrices and current constraints · 0.98

Compares with it 4 papers
  • Recall that in our framework this degeneration is not a consequence of the small breaking of total lepton number, as in other low scale leptogenesis scenarios [ref] , although it may be due to an additional symmetry of the heavy sector.

    Leptogenesis without violation of B-L · Summary · 0.85

  • In contrast to resonant leptogenesis via vacuum mass degeneracy [ref] , the appearance of soft-lepton resonance is itself a SM prediction and the resonant enhancement is protected from finite width under perturbative finite-temperature field theory.

    Forbidden neutrinogenesis · Introduction · 0.98

  • The enhancement mimics the mass degeneracy between two neutrinos or scalars in resonant leptogenesis [ref] .

    Resonant Forbidden CP Asymmetry from Soft Leptons · Introduction · 0.85

Cites it as support 2 papers
Cites it as background 34 papers
  • The asymmetries in lepton flavour $α$ associated with Higgs doublet $ϕ_k$ are [equation] The CP violation in scatterings was first taken into account in the early works on resonant leptogenesis [ref] , assuming factorisation, see also [ref] .

    Leptogenesis in Type Ib seesaw models · CP asymmetry · 0.88

  • For an almost degenerate heavy Majorana neutrino mass spectrum, leptogenesis can be naturally implemented in the so-called resonant leptogenesis framework [ref] .

    Resonant leptogenesis and tribimaximal leptonic mixing with A4 symmetry · Leptogenesis · 0.80

  • Low-scale leptogenesis resonantly enhanced due to quasi-degenerate heavy neutrino mixing, known as resonant lepto­genesis (RL) [ref] , and via sterile neutrino oscillations (the ARS mechanism) [ref] long stand out as being the leading paradigms that have attracted much attention in the literature, e.g. see Ref. [ref] for a review.

    Dominant Thermal Resonant Mechanism for Low-Scale Leptogenesis · Introduction · 1.00

Discussion

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