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REVIEW 3 major objections 5 minor 98 references

Spontaneous Scoto-leptogenesis

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper shows that a rolling Majoron generated by spontaneous $U(1)_{B-L}$ breaking can produce the observed baryon asymmetry with right-handed neutrino masses as low as about 600 GeV, while the same sector explains radiative neutrino…

desk verdict A coherent and honest phenomenological study of spontaneous leptogenesis in the dynamical scotogenic model, held back only by the free initial Majoron yield that directly fixes the baryon asymmetry; worth refereeing. read the letter →

arxiv 2608.12497 v1 pith:X5YFCKV3 submitted 2026-08-12 hep-ph astro-ph.CO

Arghyajit Datta , Hyun Min Lee , Jun-Ho Song This is my paper · ORCID
classification hep-phastro-ph.CO
keywords spontaneousleptogenesisscotogenicmodelMajoronbaryonasymmetryinertdoubletdarkmatterkineticmisalignmentTeV-scaleright-handedneutrinosU(1)B-L
topics Dark MatterMatter-Antimatter Asymmetry
open problems Dark MatterMatter-Antimatter Asymmetry
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

The paper proposes that the observed baryon asymmetry of the Universe can be produced not by CP-violating right-handed neutrino decays but by a slowly rolling Majoron, the Goldstone boson of spontaneously broken global $U(1)_{B-L}$, which acts as a time-dependent chemical potential for $B-L$. In the dynamical scotogenic model with two hierarchical right-handed neutrinos and an inert Higgs doublet, this mechanism works precisely in the strong-washout regime where ordinary thermal leptogenesis fails, lowering the viable mass of the lightest right-handed neutrino to $M_1=\mathcal{O}(600)$ GeV. The same framework already generates radiative neutrino masses and dark matter, so if the mechanism works, one symmetry-breaking sector simultaneously explains neutrino masses, dark matter, and the observed matter-antimatter asymmetry in a way that TeV-scale experiments can test.

What carries the argument

The central object is the Majoron $\theta(x)=J(x)/v_\phi$, the pseudo-Nambu-Goldstone boson of the spontaneously broken global $U(1)_{B-L}$ symmetry. Its derivative coupling $-\frac{1}{2}\,\partial_\mu\theta\,J^\mu_{B-L}$ gives a background $\dot\theta$ that acts as an effective chemical potential for $B-L$; the $B-L$-violating decays and inverse decays $N_1\leftrightarrow\ell\eta$ keep the plasma in equilibrium so that a lepton asymmetry is washed in while the Majoron rolls. The final baryon yield is set by $Y_B\propto Y_\theta\,(M_1/v_\phi)^2/z_{\rm dec}^2$, with the coefficient $c_B$ fixed by the chemical potentials of the scotogenic plasma. The coupling $\lambda_5$ does double duty: through the mass splitting of $\eta_R$ and $\eta_I$ it controls the one-loop neutrino mass, and through $(\eta^\dagger\Phi)^2$ it determines whether an inert-doublet chemical potential participates in the spectator relations and whether a dark-matter asymmetry survives. The kinetic-misalignment condition $Y_\theta>Y_{\rm cr}$ selects the regime where the same rolling Majoron can supply the observed dark matter.

What would settle it

A full numerical solution of the coupled equations for the Majoron velocity and the lepton asymmetries, including thermal dissipation of $\dot\theta$ and the expansion history, would settle the claim: if $\dot\theta$ decays before inverse decays decouple, or if the Universe enters a kination phase before freeze-out, the predicted $Y_B$ falls below $8.7\times10^{-11}$. Observationally, a future CMB measurement that excludes $\Delta N_{\rm eff}\simeq0.025$ would rule out the sub-eV Majoron component of this scenario.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a rolling Majoron, through its derivative coupling to the $B-L$ current, induces an effective chemical potential that biases the $B-L$-violating decays and inverse decays of the lightest right-handed neutrino, and the resulting lepton asymmetry is converted by electroweak sphalerons into the observed baryon abundance $Y_B\simeq 8.7\times10^{-11}$. The key formula is $Y_B=(c_B/6)\,Y_\theta\,(M_1/(v_\phi z_{\rm dec}))^2$, where $Y_\theta=v_\phi^2\dot\theta/s$ is the conserved angular yield of the Majoron, $v_\phi$ is the $U(1)_{B-L}$ breaking scale, and $z_{\rm dec}=M_1/T_{\rm dec}$ is the inverse-decay decoupling point; the coefficient $c_B$ is obtained from chemical-equilibrium and spectator relations, including the role of the $\lambda_5$ interaction that couples the inert doublet to the Higgs. Because the asymmetry is sourced by the background motion rather than by CP-violating decay amplitudes, the strong washout that suppresses conventional thermal leptogenesis instead helps the plasma track the equilibrium asymmetry, and the lightest right-handed neutrino can be as light as about 600 GeV while still being hierarchical. The same rolling Majoron can later become dark matter through kinetic misalignment, with a sub-eV Majoron that is long-lived and contributes $\Delta N_{\rm eff}\simeq0.025$ as dark radiation.

Load-bearing premise

The mechanism needs the Majoron to start rolling with a sizable angular velocity $Y_\theta$ that stays roughly constant until inverse decays freeze out while the Universe is radiation dominated; if that initial motion is absent or is damped away first, the baryon asymmetry vanishes.

Editorial extensions

If this is right

  • TeV-scale hierarchical right-handed neutrinos ($M_1\simeq600$ GeV and above) become viable for leptogenesis, opening the right-handed neutrino sector to direct production and to missing-energy or displaced-vertex searches at colliders.
  • The strong-washout region $K_1\gg4$, which forbids conventional thermal leptogenesis in the two-right-handed-neutrino scotogenic model, is exactly where spontaneous leptogenesis is most efficient.
  • The $\lambda_5$ coupling ties neutrino mass, the inert-scalar dark-matter mass splitting, and the spectator conversion coefficient $c_B$ into one parameter, so measuring one of these quantities informs the others.
  • A sub-eV Majoron produced by kinetic misalignment can be the dominant dark matter while remaining cosmologically stable, and its thermal population contributes $\Delta N_{\rm eff}\simeq0.025$, which is within the projected sensitivity of future CMB experiments.
  • Collider searches for inert scalars above roughly 550 GeV and direct-detection bounds on the Higgs-portal coupling constrain the same parameter space that produces the baryon asymmetry.

Reading between the lines

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

  • Because the paper treats $Y_\theta$ as a free initial condition, the mechanism's ultimate viability depends on an explicit ultraviolet source for the Majoron motion; a concrete inflationary or phase-transition origin would complete the story.
  • The same derivative-coupling trick should work in other radiative neutrino-mass models with a pseudo-Goldstone boson, so the scotogenic structure may be one of several realizations of low-scale spontaneous leptogenesis.
  • The radiation-domination restriction means a kination phase before freeze-out would alter the final $Y_B$; a dedicated treatment of Majoron-domination could extend the allowed $(v_\phi, M_1)$ region or sharpen lower bounds on $v_\phi$.
  • If future CMB data resolve $\Delta N_{\rm eff}$ around 0.025, that would be a distinctive signature distinguishing this scenario from thermal leptogenesis, which predicts no such dark-radiation component from the Majoron.
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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 / 5 minor

Summary. The paper proposes a spontaneous leptogenesis mechanism in the dynamical minimal scotogenic model, where a rolling Majoron from global U(1)_{B-L} breaking induces an effective chemical potential for B-L, and RHN decay/inverse-decay processes convert this bias into a baryon asymmetry. The authors show that the strong-washout regime, which suppresses conventional thermal leptogenesis, becomes advantageous here, and they claim that the observed baryon asymmetry can be reproduced with hierarchical RHN masses as low as O(600) GeV while remaining consistent with radiative neutrino masses and inert-scalar or Majoron dark matter. The paper derives the final Y_B from chemical equilibrium conditions, computes the spectator coefficient c_B in the appendix, and explores parameter space in Y_theta versus M1 and v_phi, including constraints from DM relic density, direct detection, kination domination, Majoron overproduction, and Majoron decay lifetime.

Significance. If the central claim holds, the setup provides a testable low-scale alternative to high-scale thermal leptogenesis in a well-motivated radiative neutrino mass framework, linking the baryon asymmetry to neutrino masses and a multicomponent dark sector. The chemical equilibrium derivation in the Appendix is systematic, and the identification of the lambda_5-dependent spectator effects on the final asymmetry is a useful contribution. The paper also deserves credit for explicitly delineating several cosmological consistency conditions (kination bound, Majoron stability, dark radiation). However, the headline result is an existence proof contingent on the free initial Majoron yield Y_theta, and the paper does not currently demonstrate all conditions simultaneously in a concrete benchmark.

major comments (3)
  1. [Sec. V, Eq. (35)] The final baryon asymmetry, Y_B = (c_B/6) Y_theta (M1/(v_phi z_dec))^2, is directly proportional to the initial Majoron angular yield Y_theta, which the paper explicitly declares to be a free parameter in Sec. V. Figures 3 and 4 therefore show the values of Y_theta required to match the observed Y_B rather than predictions. The abstract's claim that the observed asymmetry is 'successfully generated' with M1 = O(600) GeV is accordingly an existence claim for a particular initial condition. I recommend that the authors either provide a quantitative estimate of Y_theta from the Planck-suppressed explicit breaking mechanism discussed in Sec. V, or state unambiguously that the TeV-scale result requires an ad hoc initial Majoron velocity and discuss the naturalness of the required values (e.g., Y_theta spanning roughly 10^-6 to 10^5 across the plotted range). Without this, the central claim is a parameter fit rather than a falsifiable prediction.
  2. [Sec. V, Eq. (39)] The kinetic misalignment condition Y_theta > Y_cr in Eq. (39) guarantees that the Majoron is not trapped at T_osc, but it does not guarantee that the Majoron is still rolling at the inverse-decay decoupling temperature T_dec. Because the kinetic energy density scales as rho_theta ∝ T^6 while T_dec can be much smaller than T_osc, one must also impose rho_theta(T_dec) > V_max, which gives an upper bound m_J < Y_theta s(T_dec)/(2 v_phi^2). This additional constraint is absent from Figs. 3-5 and from the text. For typical low-scale plateau parameters the required Y_theta appears large enough that the bound can be satisfied with sub-eV m_J, but the condition should be stated explicitly and imposed in the parameter-space analysis, especially for the higher-T_dec regime where Y_theta is smaller and the interval of allowed m_J can close.
  3. [Sec. VI] The paper's headline claim of M1 = O(600) GeV is not backed by an explicit benchmark point. Figures 3 and 4 show contours, but no single point is given with a full specification of v_phi, M1, M2, Y_theta, m_J, m_etaI, lambda_5, the Casas-Ibarra angle, and the resulting Yukawa couplings, together with checks of neutrino masses, DM relic abundance (including the subdominant eta_I contribution), direct detection limits, the kination bound, kinetic misalignment, the rolling condition of Eq. (39), and the Majoron stability bound. I request at least one or two benchmark tables demonstrating that all these constraints can be satisfied simultaneously.
minor comments (5)
  1. [Eq. (30)] Equation (30) contains a typesetting error in the chemical-potential term; the source term should be typeset as (2 mu_l_alpha + 2 mu_eta - theta_dot)/T, with the fraction unambiguous.
  2. [Sec. III] The phrase 'viable explodable region' should read 'viable explorable region'.
  3. [Eq. (33)] The derivation of the factor (M1/(v_phi z))^2 would be clearer if the text explicitly stated n_B = (T^2/6) mu_B and s = (2 pi^2/45) g_*s T^3 before presenting Y_B.
  4. [Fig. 5] Please clarify the meaning of the pink solid contours versus the pink shaded region; the text says the solid contours denote the Majoron yield required for the observed DM abundance while the shaded region is excluded, but the figure caption should be self-explanatory.
  5. [Sec. V] The statement that dissipation of the Majoron motion is suppressed is asserted by citations [24,84,87] without a parametric estimate; a one-line estimate of Gamma_diss/H at the benchmark temperatures would strengthen the argument.

Circularity Check

1 steps flagged · score 6.0 of 10

Central baryon-asymmetry claim reduces to a scan over the free initial Majoron yield Y_theta; DM and kination constraints supply partial but not full independence.

  1. fitted input called prediction [Sec. V, Eqs. (33)-(36); Sec. VI, Fig. 3 caption]
    "The final baryon asymmetry is then denoted as YB = cB/6 Yθ (M1/(vφzdec))^2 ... In the following, however, we remain agnostic about the underlying dynamics responsible for generating the initial condition for the Majoron motion ... Thus, we treat Yθ as a free parameter in our work."

    Equation (35) fixes Y_B = (c_B/6) Y_theta (M1/(v_phi z_dec))^2, so once Y_theta is declared a free parameter the equation can be inverted to force Y_B to equal the observed 8.7e-11. Figure 3 is explicitly described as the value of Y_theta required to reproduce the observed baryon asymmetry, and Fig. 4 draws contours for fixed Y_theta. The paper's central statement that the observed baryon asymmetry is 'successfully generated' at TeV-scale M1 is therefore an input-selection exercise rather than a first-principles prediction: for any point in the allowed (M1, v_phi) plane, Y_theta can be chosen so that Eq. (35) or Eq. (36) matches observation.

full rationale

The only substantive circular step is the treatment of Y_theta as a free input while Eq. (35) and Fig. 3 convert it directly into the observed Y_B. The inert-doublet DM analysis in Sec. III is independent: relic densities are computed with micrOMEGAs and confronted with LZ, LEP, and electroweak-precision constraints, and the neutrino-mass section uses the standard scotogenic loop. The c_B conversion factors are derived from chemical-equilibrium relations in the Appendix rather than imported from an author-specific uniqueness theorem. Self-citations [31,32,85,86] appear for Majoron dynamics and damping suppression, but they are accompanied by external references [24,84,87] and by the physical argument that inverse decays become active at M1 ~ T << v_phi, so they are not the load-bearing circularity. The paper itself flags the key limitation in Sec. V: 'we remain agnostic about the underlying dynamics responsible for generating the initial condition for the Majoron motion' and 'we treat Y_theta as a free parameter.' Because the central BAU claim is an inversion of Eq. (35), the low-scale plateau Eq. (36) is conditional on an unquantified initial-condition input. The DM relic, kinetic-misalignment, Majoron-stability, and radiation-domination constraints provide genuine independent restrictions, so the result is not fully tautological; but the quantitative baryon asymmetry itself is effectively fitted. Hence a score of 6, indicating partial circularity.

Assumptions & free parameters 5 free parameters · 8 assumptions · 1 invented entities

The central calculation rests on well-established Boltzmann and chemical equilibrium formalism plus the specific particle content of the scotogenic model. The most expensive input is Y_theta, an unconstrained initial condition that is scanned to match the baryon asymmetry; without it no asymmetry is generated. Additional scanned parameters (lambda_5, v_phi, m_J, theta) are model inputs rather than derived quantities. No genuinely new particle is invented beyond the Majoron that already follows from the global U(1)_{B-L} breaking, and the Majoron carries several falsifiable handles.

free parameters (5)
  • Y_theta (initial Majoron angular yield) = Not fixed; scanned in Figs. 3-5; benchmarks Y_theta=1e-3, 1e1, 1e5 appear in Fig. 4
    Treated as a free parameter in Sec. V and chosen in Fig. 3 to reproduce the observed baryon asymmetry. It sets the scale of the chemical potential and enters Y_B directly in Eq. (35).
  • Casas-Ibarra angle theta = pi/6 + i (benchmark)
    Chosen by hand in Sec. VI to fix the Yukawa structure via Eq. (10). The paper states changing Im(theta) mildly affects z_dec, but it remains an unconstrained input to the neutrino Yukawa matrix.
  • lambda_5 coupling = 1e-2 and 1e-4 benchmarks
    Chosen as representative values; controls eta_R-eta_I splitting, radiative neutrino mass, equilibration of the (eta†Phi)^2 interaction, and the spectator coefficient c_B.
  • v_phi (B-L breaking scale) = Scanned, e.g., 1e4 to 1e10 GeV
    Free symmetry-breaking scale of the model; sets RHN masses, Majoron couplings, and the size of Y_theta required. Constrained by the hierarchy M2 >= 10 M1 and by kination avoidance.
  • m_J (Majoron mass) = Sub-eV region, m_J <~ 0.017 eV for minimal v_phi
    Free parameter of the explicit U(1)_{B-L} breaking potential; scans in Fig. 5 use it to set kinetic misalignment and decay constraints.
assumptions (8)
  • standard math Standard Boltzmann and chemical equilibrium treatment of the early Universe plasma
    Used throughout Secs. IV-V and the Appendix to derive Y_B and c_B.
  • domain assumption Global U(1)_{B-L} is spontaneously broken by the vev of phi and is explicitly broken by Planck-scale effects to give the Majoron mass
    Defines the model in Sec. II and the Majoron potential in Eq. (38); the explicit breaking is assumed, not derived.
  • domain assumption The Majoron rolling is not damped by the thermal bath and Y_theta is conserved until inverse-decay freeze-out
    Assumed in Sec. V after Eq. (36), citing refs [24,84,87]; if false, the chemical potential disappears before freeze-out.
  • domain assumption Radiation domination holds until the baryon asymmetry freezes out
    Sec. VI restricts parameter space to below the kination boundary, the dot-dashed curves, so that the analytic expansion history applies.
  • ad hoc to paper The scalar portal couplings lambda_Phi_phi and lambda_eta_phi are negligible; lambda_eta = 0.2
    Assumed in Sec. III to simplify DM and Higgs-sector analysis; not derived from the symmetry.
  • ad hoc to paper eta_I is the lightest inert state and the only IHD DM candidate
    Sec. III imposes lambda_5 > 0 and lambda_3 > lambda_DM, and neglects the CP-even or RHN DM alternatives.
  • domain assumption Two hierarchical RHNs with M2 >= 10 M1 and Y_N^2 = 1 so that M2 = v_phi
    Sec. VI sets the RHN mass hierarchy and fixes M2, which determines the available (M1, v_phi) plane and excludes the red regions in Fig. 4.
  • domain assumption The initial asymmetries of U(1)_{eR1}, U(1)_{uR1-dR1}, and U(1)_{2B1-B2-B3} vanish
    Used in the Appendix to solve the chemical potential network; if non-zero initial charges existed, c_B would change.
invented entities (1)
  • Majoron J (theta = J/v_phi) independent evidence
    purpose: Rolling pseudo-Nambu-Goldstone boson that provides the B-L chemical potential for spontaneous leptogenesis and can constitute dark matter via kinetic misalignment.
    The model gives concrete observational handles: sub-eV mass range, decay to neutrino pairs bounded by tau_J > 250 Gyr in Eq. (43), Delta Neff near 0.025 from thermal decoupling, and a relic abundance link through Y_theta. These are falsifiable predictions, though the field itself is not yet observed.

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Pith. "Pith review of Spontaneous Scoto-leptogenesis." pith.science (2026). https://pith.science/paper/X5YFCKV3

@misc{pith2026260812497,
  author       = {Pith},
  title        = {Pith review of: Spontaneous Scoto-leptogenesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X5YFCKV3}},
  note         = {Machine review of arXiv:2608.12497}
}
abstract

We propose a low-scale spontaneous leptogenesis scenario within the dynamical minimal scotogenic model for accommodating neutrino masses and inert scalar dark matter simultaneously. Thus, we dub the mechanism Spontaneous Scoto-leptogenesis. In this setup, a rolling Majoron arising from the global $U(1)_{B-L}$ symmetry breaking induces an effective chemical potential for the $B-L$ charge in the presence of $B-L$ violating interactions that allow for the efficient decays and inverse decays of right handed neutrinos (RHN), so it gives rise to the observed baryon asymmetry of the Universe through the electroweak sphaleron conversion. The mechanism becomes effective in the strong washout regime and successfully lowers the viable mass scale of the lightest RHN to the range of TeV scales, thereby making the thermal scotogenic leptogenesis with two hierarchical RHNs accessible to direct tests. We identify the roles of the $\lambda_5$ coupling for spontaneous leptogenesis and inert scalar dark matter through the efficient erasure of the inert scalar asymmetry. We also explore the regime for Majoron dark matter from the kinetic misalignment, showing that a multicomponent dark sector comprising the inert scalar and the Majoron can be realized in the model. The resulting framework provides a unified origin for low-scale baryogenesis, neutrino masses, and multicomponent dark sector, so it can be tested by complementary experimental probes through direct detection experiments, collider searches for inert scalars, and future detection of Majoron dark matter or dark radiation.

Figures

Figures reproduced from arXiv: 2608.12497 by the authors.

Figure 1
Figure 1. FIG. 1. Relic density of the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The coupling λ5 nevertheless remains crucial for direct searches through the inelastic Z-mediated process. Since λ5 fixes the mass splitting between the CP-even and CP-odd neutral inert scalars, reducing λ5 makes the transition ηI + N → ηR + N kinematically easier. For a sufficiently small splitting, the halo DM kinetic energy can upscatter ηI into the heavier state ηR. For sub-TeV DM, splittings below roughly ∆M ≡ … view at source ↗
Figure 3
Figure 3. FIG. 3. Parameter space for [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Parameter space for successful scoto-leptogenesis in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Parameter space in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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