Spontaneous Leptogenesis: Mechanisms & Models

• Spontaneous leptogenesis is a mechanism where CP violation arises dynamically through the spontaneous breaking of symmetry by scalar field vacuum expectation values.
• This framework utilizes complex VEVs from fields such as singlets, Majorons, or axion-like particles to induce a CP phase that drives lepton-number violating processes.
• Models are classified by their CP mediation methods, neutrino mass generation techniques, and phenomenological implications linking cosmology, neutrino oscillations, and collider experiments.

Spontaneous leptogenesis is a class of baryogenesis mechanisms in which the violation of charge-parity (CP) symmetry—essential for generating the cosmological matter-antimatter asymmetry—arises dynamically through spontaneous symmetry breaking, rather than being put in by hand at the Lagrangian level. The spontaneous generation of CP violation typically proceeds via the vacuum expectation value (VEV) of new scalar fields, which break a global or discrete symmetry and induce a phase that is communicated coherently to the relevant high-energy processes. This article surveys the core concepts, model structures, and phenomenological implications of spontaneous leptogenesis, with representative frameworks ranging from models with triplet scalars and flavor symmetries to those involving axion-like particles or Majorons.

1. General Principles and Model Taxonomy

Spontaneous leptogenesis builds on the Sakharov criteria, with a distinctive focus on the dynamical origin of CP violation. The spontaneous phase—arising, for instance, from the VEV of a complex singlet (as in many $A_4$-flavored models), a Majoron field (from spontaneous $B-L$ breaking), or an axion-like particle—acts as a universal source of CP violation. This phase is then transmitted to the lepton sector through couplings that generically affect both low-energy observables (e.g., the Dirac and Majorana phases in the PMNS matrix) and high-energy CP-violating processes relevant for leptogenesis.

Spontaneous leptogenesis models can be grouped by:

• The mediator of CPV: scalar singlets (e.g., $S$), Majorons, ALPs.
• Neutrino mass origin: type-I, type-II, or radiative seesaw.
• Lepton-number violation mechanism: decays of heavy right-handed (RH) neutrinos, scalar triplets, or dimension-5 operators.

Common frameworks include type-II seesaw with flavored triplets and SCPV (Chongdar et al., 11 Jan 2024), radiative seesaw with $A_4$ flavor (Ahn et al., 2013), minimal $SO(10)$ GUTs with a single high-scale spontaneous phase (Babu et al., 20 Aug 2025), spontaneous Majoron-driven type-I seesaw leptogenesis (Chun et al., 6 Dec 2025), and derivative-coupled ALP scenarios (Datta et al., 11 May 2024).

2. Structure of Scalar Potentials and Vacuum Alignment

The scalar potential is engineered both to preserve CP symmetry at high scales and to permit spontaneous breaking at a lower scale via a complex VEV. A canonical illustration is:

$$V(S) = m_S^2 S^* S + \mu_S^2(S^2 + S^{*2}) + \lambda_S (S^* S)^2 + \lambda'_S (S^4 + S^{*4}) + \lambda''_S S^* S (S^2 + S^{*2})$$

Minimization leads to a VEV $\langle S\rangle = v_S e^{i\alpha}$, where $\alpha$ is a nontrivial phase (often at $\pm\pi/4$), spontaneously breaking CP (Chongdar et al., 11 Jan 2024, Karmakar et al., 2015).

In $SO(10)$ GUTs, a similar role is played by a $CP$-odd Higgs field $54_H$, whose VEV injects a phase into the low-energy sector via doublet mixing in the Higgs sector (Babu et al., 20 Aug 2025). In Majoron models, a singlet $\sigma$ with $U(1)_{B-L}$ charge breaks the symmetry, yielding a massless Goldstone (the Majoron) with cosmologically relevant kinetic backgrounds (Ibe et al., 2015, Chun et al., 6 Dec 2025).

3. Transmission of the Spontaneous Phase to Lepton-Number Violation

The acquired phase feeds into lepton-number violating processes via several routes:

• Triplet Scalar Models: SCPV in $S$ feeds the triplet scalar lepton-number violating couplings (e.g., $$\mu_1 = (\lambda_1 e^{i\alpha} + \lambda'_1 e^{-i\alpha}) v_S$$), controlling the CP asymmetry in $\Delta_a \to \ell_\alpha \ell_\beta$ (Chongdar et al., 11 Jan 2024). The A$_4 \times$Z$_4$ flavor structure fixes Yukawa textures, enforcing real couplings at high scale and dynamically generating all CP violation.
• Type-I Seesaw/Majoron Models: The Majoron derivative coupling acts as a chemical potential for B-L, so that a rotating background $\dot{\sigma}$ sources a CP-violating bias in heavy neutrino decays and inversely in the plasma (Chun et al., 6 Dec 2025). The effect is present both in out-of-equilibrium decay and in equilibrium via inverse-decay wash-in.
• ALP/Derivative Coupling Models: The ALP background, via $\frac{c_{B-L}}{f_a} \partial_\mu \phi J_{B-L}^\mu$, induces a dynamical chemical potential $\mu_{B-L} \sim \dot{\phi}/f_a$. This structure enables baryogenesis at low reheating temperatures even with very light ALPs, if assisted by inert-doublet mediated $L$-violating operators (Datta et al., 11 May 2024).
• Higgs-Relaxation Scenarios: A time-dependent Higgs VEV following inflation (driven by non-minimal gravity coupling or derivative kinetic terms) introduces a chemical potential via a dimension-6 operator; lepton-number violation arises via the Weinberg operator (Lee et al., 2020, Wu et al., 2019).

Spontaneous phases are therefore common sources for both high-scale (leptogenesis) and low-scale (neutrino mass, PMNS phases) CP violation.

4. Flavor Structures, Boltzmann Equations, and Parametric Control

The rich flavor structures strongly influence the efficiency and viability of leptogenesis:

• In $A_4 \times Z_4$ models, the vanishing of unflavored asymmetry due to symmetry constraints ($\mathrm{Tr}(m_\nu^{(1)} m_\nu^{(2)†}) = 0$) means that successful baryogenesis relies on resolving the $(\alpha,\beta)$ parameter space within flavor-sensitive Boltzmann or density-matrix equations (Chongdar et al., 11 Jan 2024).
• In Majoron/ALP backgrounds, the equations take the form of modified kinetic evolution equations with source ($\propto \dot{\sigma}$ or $\dot{\phi}$) and washout (inverse decays/scatterings) terms. Analytical limits reveal "freeze-in" and "freeze-out" regimes for the asymmetry, with the final yield sensitive to the equilibrium maintenance of B-L violation and the initial conditions of the CP-odd background (Ibe et al., 2015, Chun et al., 6 Dec 2025, Datta et al., 11 May 2024).
• In type-II wash-in scenarios, the "chemical potential" is injected at the Higgs-to-triplet inverse decay stage, then transferred to leptons via $T \to \ell \ell$ (Berbig, 29 Jun 2025). The final asymmetry is directly linked to the kinetic background parameter $\dot\theta$ (with $\theta = j/v_\sigma$ for the Majoron) and insensitive to initial triplet/ALP abundance in strong wash-in.

A salient feature in these frameworks is that the magnitude and phase of the CP-odd VEVs or dynamical backgrounds determine not only whether the observed baryon asymmetry $\eta_B \sim 6 \times 10^{-10}$ can be achieved, but also fix low-energy leptonic parameters (e.g., the Dirac CP phase $\delta$, the reactor angle $\theta_{13}$) (Karmakar et al., 2015, Ahn et al., 2013, Chongdar et al., 11 Jan 2024).

5. Phenomenological Implications and Experimental Constraints

Spontaneous leptogenesis frameworks yield multiple, tightly correlated predictions:

• Neutrino Mass and Mixing: The texture and absolute scale of $m_\nu$ is tied to the same spontaneous phase that sources leptogenesis. Quantitative fits in models such as $A_4$ flavor symmetry (Chongdar et al., 11 Jan 2024, Ahn et al., 2013) and minimal $SO(10)$ (Babu et al., 20 Aug 2025) ensure compatibility with current oscillation data and predict restricted windows for $\theta_{23}$ and $\delta_{\rm PMNS}$.
• Baryon Asymmetry: For realistic parameter values (e.g., triplet mass $M_a \gtrsim 10^{10}$ GeV, Yukawa $y_\Delta \sim 10^{-3}-10^{-2}$, VEV $v_S \gtrsim 10^{12}$ GeV), the observed baryon asymmetry is readily reproduced, with branching ratios and SCPV phase $\alpha$ finely tuning the output (Chongdar et al., 11 Jan 2024).
• 0$\nu\beta\beta$ Decay: The effective Majorana mass $|m_{ee}|$ is bounded by the allowed region of spontaneous phase, with constraints $|m_{ee}| \lesssim 0.15$ eV in triplet SCPV scenarios and values often within reach of next-generation experiments (Chongdar et al., 11 Jan 2024, Karmakar et al., 2015, Ahn et al., 2013).
• Lepton Number Breaking Scale: Strong bounds are placed on the symmetry-breaking scale (e.g., $v_S \gtrsim 10^{12}$ GeV to avoid domain wall problems, $f \gtrsim 10^{10}$ GeV for Majoron-induced spontaneous leptogenesis (Chun et al., 6 Dec 2025)).
• Axion-Like Particle Mass/Decay Constant: Experimental searches for sub-GeV ALPs constrain allowed $(m_a, f_a)$ regions, with viable leptogenesis requiring appropriately tuned initial velocities and consistency with BBN and beam-dump bounds (Datta et al., 11 May 2024).
• Collider Signatures: In type-II seesaw with a Majoron, characteristic decays of doubly-charged scalars (same-sign dileptons vs same-sign $W$ bosons) can distinguish this scenario experimentally. The viability window for triplet VEVs ($v_T \sim$ keV–MeV) allows measurement via branching ratios at colliders (Berbig, 29 Jun 2025).
• Cosmology: Predictions include negligible isocurvature for models without new light fields during baryogenesis, and robust compatibility with CMB data (e.g., in Higgs-relaxation scenarios (Lee et al., 2020, Wu et al., 2019)).

6. Predictivity and Correlations with Low-Energy Observables

Spontaneous leptogenesis frameworks generically correlate the high-energy baryogenesis epoch with low-energy observables:

• In A$_4$-flavored SCPV models, the same phase $\alpha$ entering $\langle S \rangle$ simultaneously fixes the reactor angle $\theta_{13}$, the Dirac phase $\delta$, and the magnitude of the produced $CP$ asymmetry $\varepsilon$, tightly constraining the range of viable parameter space (Chongdar et al., 11 Jan 2024, Karmakar et al., 2015, Ahn et al., 2013).
• Minimal $SO(10)$ with a single SCPV phase yields all CP-violating phases (CKM, PMNS, Majorana, and those in leptogenesis) as linear combinations of one high-scale input, enabling global fits to masses, mixings, and the baryon asymmetry with just 19 real parameters (Babu et al., 20 Aug 2025).
• In Majoron-based scenarios, the baryon asymmetry is a direct function of $\dot{\sigma}/(f M)$, so that the required value of $\eta_B$ selects a scale for $f$ and constrains the allowed Yukawa sector and RH neutrino mass scale (Chun et al., 6 Dec 2025, Ibe et al., 2015).

These cross-connections enable powerful experimental tests, including neutrino oscillation measurements, $0\nu\beta\beta$ searches, cosmological observations, and in some cases collider searches for exotic scalars or ALPs.

7. Summary Table: Core Mechanisms and Features

Mechanism	Spontaneous Source	L-violation Process	CPV Transmission	Key Parameter(s)	Phenomenological Handle	References
$A_4$ triplet + SCPV	$\langle S\rangle = v_S e^{i\alpha}$	Triplet decay $\Delta\to\ell\ell$	$\alpha$ feeds $\mu_1$, $m_\nu$	$(v_S,\,\alpha,\,M_\Delta,\,y_\Delta)$	Neutrino mixing, $0\nu\beta\beta$, $\eta_B$	(Chongdar et al., 11 Jan 2024, Karmakar et al., 2015, Ahn et al., 2013)
Majoron (Type-I seesaw)	$\dot\sigma$ background	$N$ decay + inverse decay	$\mu_{B-L} = \dot\sigma/(2f)$	$(f,\,M_N,\,\dot\sigma)$	$\eta_B$, neutrino mass, $0\nu\beta\beta$	(Ibe et al., 2015, Chun et al., 6 Dec 2025)
ALP + inert doublet	$\dot\phi$ background, ALP	$\ell\ell\to\Phi\Phi$ ($\Phi$ inert)	$\mu_{B-L} = \dot\phi/f_a$	$(f_a,\,\Lambda,\,m_a,\,T_R,\,\dot\phi)$	ALP searches, low $T_R$ baryogenesis	(Datta et al., 11 May 2024)
Higgs relaxation (inflation)	Time-dependent $\langle H\rangle$	$\Delta L=2$ via Weinberg operator	$\mu_L = -\phi\dot\phi/\Lambda_6^2$	$(\Lambda_5,\,\Lambda_6,\,T_{\rm reh})$	CMB parameters, neutrino mass	(Lee et al., 2020, Wu et al., 2019)
Type-II seesaw Majoron wash-in	$\dot j$ (Majoron)	$H^TH\leftrightarrow T$, $T\to\ell\ell$	$\mu_T^{\rm eff}=\mu_T+\dot\theta$	$(v_\sigma,\,v_T,\,\dot\theta,\,m_T)$	Collider branching ratios, DM-Majoron link	(Berbig, 29 Jun 2025)
$SO(10)$ minimal SCPV	Phase from $54_H$	$N_i\to\ell H$ decay	Down-mixing of SCPV phase	Single phase, GUT-scale vev	Global fit: CKM, PMNS, $\eta_B$, $m_{ee}$	(Babu et al., 20 Aug 2025)

• "Triplet scalar flavored leptogenesis with spontaneous CP violation" (Chongdar et al., 11 Jan 2024)
• "One Phase to Rule Them All: Spontaneous CP Violation and Leptogenesis in SO(10)" (Babu et al., 20 Aug 2025)
• "Spontaneous CP Violation in Lepton-sector: a common origin for $θ_{13}$, Dirac CP phase and leptogenesis" (Karmakar et al., 2015)
• "Spontaneous Leptogenesis with sub-GeV Axion Like Particles" (Datta et al., 11 May 2024)
• "Spontaneous Leptogenesis in Type I Seesaw" (Chun et al., 6 Dec 2025)
• "Spontaneous thermal Leptogenesis via Majoron oscillation" (Ibe et al., 2015)
• "Type II Seesaw Leptogenesis in a Majoron background" (Berbig, 29 Jun 2025)
• "Leptogenesis from spontaneous symmetry breaking during inflation" (Wu et al., 2019)
• "Spontaneous Leptogenesis in Higgs Inflation" (Lee et al., 2020)
• "Spontaneous CP Violation in $A_4$ Flavor Symmetry and Leptogenesis" (Ahn et al., 2013)