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REVIEW 3 major objections 4 minor 45 references

Critical and super-critical scatterings in baryogenesis and leptogenesis

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A class of scatterings with rates that grow as the universe cools can generate the observed cosmic matter excess.

desk verdict New scaling route to asymmetry generation that deserves a referee, but the concrete model has two load-bearing gaps: the s/u topology proof and the Vµ IR cutoff. read the letter →

arxiv 2507.11532 v1 pith:A62ZQQF5 submitted 2025-07-15 hep-ph

Marcos M. Flores , Kalliopi Petraki , Anna Socha This is my paper
classification hep-ph
keywords baryogenesisleptogenesisCPviolationscatteringratessuper-criticalearlyuniversecosmologyU(1)B-LmodelBoltzmannequations
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

Baryogenesis and leptogenesis usually assume that scattering rates fall behind the Hubble expansion as the universe cools, leaving decays to do the work. The paper identifies a class of scatterings between non-relativistic heavy fermions and a relativistic scalar bath for which the opposite happens: the cross-section grows as $1/T^2$, so the scattering rate decreases only as $T$ while the Hubble parameter decreases as $T^2$. In a minimal $U(1)_{B-L}$ extension of the Standard Model, gauge-charge assignments forbid the $s$- and $u$-channel destructive interference that normally makes such amplitudes momentum-independent, and the CP-violating downscattering $X_2 \rho \to X_1 \rho$ consequently becomes super-critical. The authors solve the coupled Boltzmann equations and show that the asymmetry generated this way freezes out only when the heavier fermion is depleted, reproducing the observed baryon-to-entropy ratio with small couplings at a 10 PeV scale.

What carries the argument

The machinery is the interaction topology forced by $U(1)_{B-L}$ charge assignments: $\rho$ carries charge $-2$, and $X_1$ and $X_2$ carry charge $-1$. For a non-relativistic $X$ and a relativistic $\rho$, each of the $s$- and $u$-channel fermion-exchange diagrams in $X_i \rho \to X_j \rho$ contributes $\delta M \propto 1/k_{\mathrm{CM}}$ at leading order, but if both channels are present their leading terms cancel destructively, leaving a momentum-independent amplitude. The charge assignment admits only one channel per flavour-changing process, so no cancellation occurs and the cross-section scales as $1/k_{\mathrm{CM}}^2$, i.e. as $1/T^2$ in the bath; with $n_\rho \propto T^3$ the rate $R \propto T$, while $H \propto T^2/M_{\mathrm{Pl}}$. This 'super-critical' scaling is what lets the scattering rate outlive the Hubble rate. The companion processes $X_i V_\mu \leftrightarrow \bar X_j \rho$ have one gauge vertex, which is momentum suppressed, giving $\sigma \propto 1/k_{\mathrm{CM}}$ and a critical rate $\propto H$; these together with $X_2 \to \bar X_1 \rho$ deplete the heavier species and freeze out the asymmetry.

What would settle it

Compute the tree-level amplitude for $X_i \rho \to X_j \rho$ with generic $U(1)$ charges, allowing both $s$- and $u$-channel fermion exchanges: if the $1/k_{\mathrm{CM}}$ terms cancel and the thermally averaged cross-section stops growing as $1/T^2$, the claimed super-critical rate does not occur. A second check is to vary parameters until the condition of Eq. (6) is violated; the resulting O(1) thermal mixing would rotate the mass eigenstates and quench the asymmetry below the observed baryon-to-entropy ratio.

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Extended reading notes

Core claim

The central claim is that particle-number- or CP-violating scatterings with thermally averaged cross-sections scaling as $T^n$ with $n \le -1$ can drive asymmetry generation more efficiently than decays, because their rates track or overtake the Hubble rate at late times. For $n=-1$ the rate is Hubble-like ('critical'); for $n<-1$ it inevitably exceeds Hubble ('super-critical'), regardless of coupling strength. The paper realizes this in a two-flavour $U(1)_{B-L}$ model with heavy Dirac fermions $X_1, X_2$ at roughly 10 PeV and a complex scalar $\rho$: the charge assignments leave only one of the $s$- and $u$-channel fermion-exchange diagrams available for $X_i \rho \to X_j \rho$, so the leading $1/k_{\mathrm{CM}}$ terms do not cancel and $\sigma \propto 1/k_{\mathrm{CM}}^2$, or $1/T^2$ after thermal averaging. The CP-violating process $X_2 \rho \to X_1 \rho$ seeds a flavour asymmetry that becomes a global $B-L$ asymmetry, with freeze-out set by depletion of $X_2$; the benchmark numerical solution yields $|\Delta_\rho| \approx |\varepsilon| Y_1^{\mathrm{FO}}$, close to maximal efficiency, and matches the observed $Y_B$. Unlike super-critical dark-matter annihilation, the relativistic-target kinematics requires only $\gamma \ge 1$, consistent with unitarity at low momentum.

Load-bearing premise

The mechanism rests on the charge assignments under $U(1)_{B-L}$ leaving only one of the $s$- and $u$-channel fermion-exchange diagrams, so the leading $1/k_{\mathrm{CM}}$ terms in the amplitude do not cancel; if both channels contribute, or if thermal mixing restores the interfering topology, the cross-section becomes momentum-independent and the super-critical growth is lost.

Editorial extensions

If this is right

  • Asymmetry generation can remain efficient down to scales well below the heavy fermion mass, because the scattering rate grows rather than decays as temperature drops.
  • The required CP-violating phases and $B-L$-violating couplings can be much smaller than in ordinary decay-based leptogenesis, since the super-critical rate overtakes Hubble independently of the coupling strength.
  • Freeze-out is fixed not by the expansion rate but by the depletion of the heavier fermion population through decays and critical scatterings.
  • Models in which the CP-violating scalar mediator carries a gauge charge and participates in symmetry breaking should generically exhibit the same super-critical dynamics.
  • The same mechanism can serve asymmetric dark matter and low-scale leptogenesis scenarios, where only one relativistic target species is needed.

Reading between the lines

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

  • Beyond the paper: a purely group-theoretic diagnostic could classify $U(1)$ charge assignments by whether the $s$- and $u$-channel topologies interfere, turning the mechanism into a model-selection criterion.
  • Beyond the paper: if the mechanism is generic, low-scale searches for $B-L$ gauge bosons and CP-odd scalars become telling probes, because the couplings need not be small enough to suppress the scattering rate.
  • Beyond the paper: the same rate-scaling logic suggests that scatterings with $n<-1$ could dominate other rare processes at late times, so bounds on washout should be re-derived in models with scalar mediators carrying gauge charge.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper identifies a class of early-universe scattering processes in which the rate per non-relativistic particle can grow relative to the Hubble rate as the universe cools. Parametrizing the thermally averaged cross-section as <σv> ∝ T^n, the authors call n=-1 critical and n<-1 super-critical. The general argument is clean: a non-relativistic particle scattering off a relativistic bath with σ ∝ 1/k_CM^2 has Γ = n_T<σv> ∝ T, while H ∝ T^2. As a concrete realization, they construct a U(1)_{B-L} model with two heavy Dirac fermions X1,X2 (charge -1) and a complex scalar ρ (charge -2), in which CP-violating X2ρ→X1ρ scattering is claimed to have σ ∝ 1/k_CM^2, producing a rapidly growing asymmetry that freezes out when X2 is depleted. A Boltzmann calculation with a benchmark parameter set yields an asymmetry near the observed value. The paper also addresses thermal corrections to the X mass matrix and the IR-divergent t-channel V_μ exchange.

Significance. The general scaling argument is simple, correct, and potentially important: it shows that scattering processes need not become inefficient in an expanding universe if the target is relativistic and the cross-section grows fast enough, and it correctly notes that this evades the unitarity obstruction that applies to non-relativistic DM annihilation. If the model's topology claim holds, the mechanism offers a new and efficient route to baryogenesis/leptogenesis with possibly smaller couplings and lower scales. The manuscript is largely self-contained, with explicit appendices for decay rates, cross-sections, thermal mass corrections, and the Boltzmann decomposition. The main caveats are that the crucial topology assertion is not demonstrated and the numerical illustration is a benchmark fit rather than a predictive scan.

major comments (3)
  1. [Figs. 1-2 and the text between them] The entire super-critical behavior in the model rests on the statement that the U(1)_{B-L} charges of X_i (-1) and ρ (-2) leave only one of the s- and u-channel fermion-exchange diagrams for X2ρ→X1ρ, so that the two leading 1/k_CM contributions do not cancel and σ ∝ 1/k_CM^2. This claim is asserted but not demonstrated. Table III quotes cross-sections only in the Δm_X→0 limit, which obscures the propagator structure. Please provide an explicit charge-flow or amplitude-level demonstration that exactly one diagram contributes, and in particular rule out an antiparticle-exchange (crossed) diagram; if both diagrams contribute, the leading 1/k_CM terms cancel and the super-critical scaling disappears.
  2. [Appendix B, Eqs. (B5)-(B6)] The t-channel V_μ exchange divergences are regularized by introducing an angular cutoff θ_min and retaining only the finite parts (I1≈-2ln2, I2≈-1/3). No physical regulator (Debye screening, thermal width) is introduced, and no proof of scheme independence is given. Since the CP asymmetry coefficients in Table III contain terms proportional to g_{B-L}^2 ln2, the size of ε and hence the final asymmetry shown in Fig. 3 may depend on this ad hoc prescription. Please replace the cutoff by a physical regulator or demonstrate that the quoted results are regulator-independent.
  3. [Results, Table I and Fig. 3] The benchmark parameters in Table I are chosen so that the final asymmetry matches the observed value; the numerical agreement is therefore an illustration, not a prediction. The abstract and discussion claim that the mechanism is efficient 'even at low scales and with small couplings,' but the benchmark uses m1=30 PeV and no parameter scan is presented. A scan or at least a sensitivity estimate in (m1, Δm_X, α_ij) is needed to support the low-scale claim and to show that the freeze-out behavior in Fig. 3 is generic rather than selected.
minor comments (4)
  1. [Table III, definition of Y] The definition of Y just above Table III is garbled by the typesetting ('Y≡16· |y11y∗12+ y12y∗22|2'); it should read |y11 y12^* + y12 y22^*|^2 (or whichever combination is intended) so that the cross-sections can be checked.
  2. [Eq. (11)] Equation (11) quotes d ln ΔX/d ln x ≈ -α11 α_B-L M_Pl/m1 without a numerical coefficient or derivation; please provide the numerical factor or a reference.
  3. [Eq. (12)] The statement in Eq. (12) that |Δρ| ≃ |ε| Y_1^FO is presented as a numerical observation; it would be useful to state whether this is a fitted relation or a derived consequence of the Boltzmann system.
  4. [Discussion] There are several typographical artifacts in the text, including 'raison d'ˆetre' in the Discussion and the typesetting of T_{B-L} in the cosmology section; a careful proofreading pass would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the super-critical scaling is derived from the model and its cross-sections; the benchmark is an illustration, not a fitted prediction.

full rationale

The paper's central derivation chain is self-contained: Eq. (3) fixes the interactions, the U(1)_{B-L} charge assignments are used to argue that only one fermion-exchange channel contributes to X2 rho -> X1 rho, Table III lists the resulting sigma proportional to 1/k_CM^2, and Eqs. (7)-(10) convert this into scattering rates and asymmetry evolution. None of these steps redefines an output as an input or fits a parameter and then calls it a prediction. Table I is explicitly a benchmark ('which yield an asymmetry near the observed value'), not a fitted prediction, and the observed Y_B is not claimed as a derived prediction. The load-bearing topology assertion, given in the paragraph beginning 'However, the interaction topology dictated by...' around Figs. 1 and 2, is not displayed in detail, and the non-perturbative CP contribution is deferred to companion paper [24]; these are completeness or support issues, not circular reductions. Self-citations [19, 20] are used only to contrast with DM annihilation and do not carry the derivation. Therefore no significant circularity is identified.

Assumptions & free parameters 9 free parameters · 6 assumptions · 4 invented entities

The central mechanism depends on standard cosmological evolution, the Boltzmann framework, and the model's particle content. The benchmark parameters are chosen to reproduce the observed baryon asymmetry, so they count as fitted values rather than predictions. The most notable ad hoc input is the cutoff regularization of t-channel gauge-boson divergences, which introduces scheme dependence into the CP asymmetry coefficient.

free parameters (9)
  • m1 = 30 PeV
    Benchmark mass scale of the heavier X fermions; no scan over masses is presented.
  • Delta m_X / m_X = 0.2
    Mass splitting controls the freeze-out temperature and the kinematics of decays and downscatterings; chosen to produce the observed asymmetry.
  • alpha_B-L = 10^-5
    Small gauge coupling chosen to suppress the X1 V -> X1bar rho washout channel.
  • alpha_11 = 10^-4
    Yukawa coupling chosen small to prevent persistent washout of the X asymmetry.
  • alpha_22 = 10^-2
    Yukawa coupling for X2; chosen together with alpha_12 to match the final asymmetry.
  • alpha_12 = 10^-2
    Flavor-changing super-critical scattering coupling; central to the CP source and tuned to the benchmark.
  • Theta (CP phase) = pi/4
    CP phase chosen to maximize the CP-violating asymmetry.
  • w_ij = 0
    Left-right Yukawa differences set to zero in the benchmark, restricting CP violation to the super-critical scatterings and simplifying the calculation.
  • T_B-L = ~10 TeV (indicative)
    Scale of B-L breaking chosen to satisfy the LEP II bound and keep rho relativistic during asymmetry generation; not scanned.
assumptions (6)
  • standard math The universe is radiation-dominated with H proportional to T^2/M_Pl during the relevant epoch.
    Used throughout the scaling arguments and Boltzmann equations; standard cosmology.
  • domain assumption The Sakharov conditions hold and sphalerons convert the lepton asymmetry to a baryon asymmetry.
    Invoked in the intermediate-energy stage description; standard for leptogenesis/baryogenesis models.
  • domain assumption Rho remains in chemical equilibrium with the plasma via rapid rho rho* annihilations.
    Stated after Eq. (7b); this justifies relating rho and rho* chemical potentials and reducing the Boltzmann system.
  • domain assumption The X fermions are highly non-relativistic and their mass eigenstates are approximately the zero-temperature states, with thermal mixing negligible under Eq. (6).
    Appendix C derives the thermal mass matrix and the condition (6); the benchmark is chosen to satisfy it.
  • ad hoc to paper Infrared-divergent angular integrals from t-channel V_mu exchange are regularized by a cutoff angle and only the finite part is retained.
    Appendix B, Eqs. (B5)-(B6): the finite parts -2 ln 2 and -1/3 are taken without a physical regulator such as a Debye mass, affecting the CP asymmetry coefficient.
  • domain assumption The B-L phase transition at T ~ 10 TeV does not interfere with asymmetry generation at T ~ a few PeV.
    The benchmark mass splittings and couplings place asymmetry freeze-out well above the B-L breaking scale, so rho remains relativistic.
invented entities (4)
  • U(1)B-L gauge boson V_mu independent evidence
    purpose: Mediates long-range forces, Sommerfeld factors, and the critical Xi V_mu -> Xbar_j rho scatterings.
    The model predicts a gauged B-L boson with mV/g > 6 TeV from LEP II (refs. [29,30]); this is a falsifiable collider handle.
  • Complex scalar rho with B-L charge -2
    purpose: Target particle for super-critical scatterings and the field that breaks B-L at low temperature.
    No direct detection channel independent of the B-L gauge boson; it appears only through its VEV and Yukawa couplings in this model.
  • Heavy Dirac fermions X1 and X2
    purpose: Carry the global charge and generate the asymmetry through decays and super-critical scatterings.
    Masses around 30 PeV are far beyond collider reach and no direct signal is predicted.
  • Anomaly-canceling fermions chi independent evidence
    purpose: Cancel U(1)B-L and gravitational anomalies; Type A is a stable DM candidate, Type B contributes to Delta N_eff.
    Type A fermions could be dark matter and Type B affects the relativistic energy density, providing cosmological handles that can exclude parameter choices.

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Cite this review

Pith. "Pith review of Critical and super-critical scatterings in baryogenesis and leptogenesis." pith.science (2026). https://pith.science/paper/A62ZQQF5

@misc{pith2026250711532,
  author       = {Pith},
  title        = {Pith review of: Critical and super-critical scatterings in baryogenesis and leptogenesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A62ZQQF5}},
  note         = {Machine review of arXiv:2507.11532}
}
read the original abstract

In many theories, matter-antimatter asymmetries originate from out-of-equilibrium decays and scatterings of heavy particles. While decays remain efficient, scattering rates typically drop below the Hubble rate as the universe expands. We point out the possibility of scatterings between non-relativistic particles and the relativistic bath whose cross-sections grow with decreasing temperature, leading to scattering rates that track or exceed the Hubble rate at late times. This results in soaring asymmetry generation, even at low scales and with small CP- or baryon/lepton-violating couplings.

Figures

Figures reproduced from arXiv: 2507.11532 by the authors.

Figure 1
Figure 1. FIG. 1. Diagrams contributing to the CP asymmetry for the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Evolution of asymmetries for the model parameters [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. One-loop contribution of the self-energy of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

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