⚛️ phenomenology

Majoron-driven spontaneous leptogenesis in type-II seesaw model

This paper proposes a minimal realization of spontaneous leptogenesis within the type-II seesaw model, demonstrating that a rotating Majoron background can generate lepton and Higgs asymmetries through decay and inverse-decay processes without requiring additional scalar triplets or explicit CP-violating interactions.

Original authors: Eung Jin Chun, Tae Hyun Jung, Jin Sun

Published 2026-09-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Eung Jin Chun, Tae Hyun Jung, Jin Sun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe we see today is made almost entirely of matter, with very little antimatter left over. This is a profound mystery because the Big Bang should have created equal amounts of both, which would have annihilated each other, leaving nothing but light. To explain why we exist, scientists look for a process that created a tiny imbalance between matter and antimatter in the early universe, a process known as baryogenesis. One leading theory suggests this imbalance started with leptons, the family of particles that includes electrons and neutrinos, before being converted into the matter we see today. However, standard versions of this theory face significant hurdles, often requiring complex setups with multiple heavy particles or specific, hard-to-find interactions to work.

A new study by researchers at the Korea Institute for Advanced Study and the Institute for Basic Science offers a simpler, more elegant solution. They investigated a scenario where the imbalance was driven not by a collision of particles, but by a rotating background field that permeated the early universe. This field is associated with a particle called the majoron, a ghostly messenger of a broken symmetry that governs the number of leptons. The researchers built a mathematical model based on a single type of heavy particle, known as a triplet, and showed how its decay could generate the necessary imbalance without needing extra ingredients or violating fundamental rules of physics in the usual way.

The core of their discovery lies in how this rotating background field changes the rules of the game. In the standard view, particles and their antimatter counterparts behave almost identically, making it difficult to create a preference for one over the other. The researchers found that in their model, the rotating field does not make the particles decay differently. Instead, it shifts the target state that the universe is trying to reach. Imagine a ball rolling down a hill; the field doesn't change the shape of the hill or the ball, but it tilts the ground itself, so the ball naturally rolls to a new, slightly different resting place. In this tilted state, the balance between particles and antiparticles is naturally shifted, creating an excess of matter as the universe evolves toward this new equilibrium.

To test this idea, the team wrote down a complete set of equations describing how the populations of these particles change over time as the universe cools. They tracked the heavy triplet particles and their decay products, which are either pairs of leptons or pairs of Higgs fields. A crucial finding was that this mechanism requires both decay paths to be active. If the heavy particle could only decay into leptons, or only into Higgs fields, the mechanism would fail completely. The asymmetry only arises when both channels are open, acting together to push the system toward the tilted equilibrium. The researchers showed that the final amount of matter created depends primarily on how often the particle chooses one path over the other, rather than on the mass of the particle itself.

The study also addressed a potential flaw in similar theories. In some models, the rotating background causes particles to decay faster than their antimatter counterparts, creating an imbalance directly. The researchers demonstrated that in their specific setup, this direct difference in decay rates is negligible. The asymmetry comes entirely from the shift in the equilibrium state, not from a difference in how fast the particles vanish. This distinction makes their model more robust and less dependent on fine-tuned details. They confirmed their results using computer simulations that tracked the evolution of the particle densities from the hot, dense early universe down to the point where these processes stopped.

The results indicate that this mechanism is a viable and efficient way to generate the matter we see today. The amount of matter produced scales with the square root of the branching ratios, meaning that even if one decay path is rare, the system can still produce a significant imbalance as long as the other path is active. The researchers found that to match the observed amount of matter in the universe, the rotation speed of the background field needs to be within a specific range, but this does not require any new, undiscovered particles beyond the single triplet and the majoron field. This approach avoids the need for multiple heavy triplets or complex interactions that have plagued previous attempts to explain the origin of matter.

Finally, the team considered the fate of the majoron field itself. After driving the creation of matter, the field would continue to rotate and eventually slow down, behaving like a form of dark matter. They calculated the constraints on how heavy this field could be and how much of the universe's dark matter it could constitute, finding that it could potentially account for a portion of the invisible mass in the cosmos without contradicting current observations. The study presents a self-contained, minimal explanation for the origin of matter, relying on a simple shift in the universe's equilibrium rather than a complex dance of particles, offering a fresh perspective on one of cosmology's deepest questions.

Technical Summary: Majoron-driven Spontaneous Leptogenesis in Type-II Seesaw Model

Problem Statement
The origin of the baryon asymmetry of the Universe (BAU) remains a central open question in particle cosmology. Standard thermal leptogenesis in the type-II seesaw framework faces significant theoretical obstructions. Specifically, with a single scalar triplet ( $\Delta$ ), tree-level and one-loop decay amplitudes cannot interfere to generate a CP asymmetry, necessitating at least two triplets or additional heavy states. Furthermore, the strong gauge interactions of the triplet tend to keep its abundance near thermal equilibrium, suppressing the efficiency of asymmetry generation. Spontaneous leptogenesis offers an alternative route where a rolling scalar field background acts as an effective chemical potential, biasing number-violating interactions without requiring explicit CP-violating phases or out-of-equilibrium decays. This work investigates the viability of this mechanism within the minimal type-II seesaw model extended by a Majoron.

Methodology
The authors construct a minimal model comprising an $SU(2)_L$ -triplet scalar $\Delta$ and a complex singlet $\Phi$ charged under a global $U(1)_{B-L}$ symmetry. The vacuum expectation value (vev) of $\Phi$ spontaneously breaks $B-L$ , generating the Majoron (the angular mode $\theta$ ) and inducing the trilinear coupling $\mu_H \Delta H^\dagger H^\dagger$ required for neutrino mass generation.

The analysis proceeds through the following steps:

Field Redefinition and Dispersion Relations: The Majoron phase is removed from interaction terms via field redefinitions ( $\Delta \to \Delta e^{-i\theta}$ , $\ell \to \ell e^{-i\theta/2}$ ). This shifts the dispersion relations of particles and antiparticles in the $\dot{\theta} \neq 0$ background, effectively introducing a chemical potential proportional to $\dot{\theta}$ .
Decay Rate Analysis: The authors calculate the partial decay widths for $\Delta \to \ell\ell$ and $\Delta \to HH$ . They demonstrate that in the massless final-state limit, the decay rates for particles and antiparticles are identical ( $\Gamma_{\Delta \to jj} = \Gamma_{\bar{\Delta} \to \bar{j}\bar{j}}$ ). The CP asymmetry in decay rates vanishes because the phase space shifts cancel out in the massless limit, contrasting with type-I seesaw scenarios where tree-level CP asymmetry exists.
Boltzmann Equations: A complete set of Boltzmann equations is derived for the asymmetric number densities ( $n_{i-} = n_i - n_{\bar{i}}$ $n_{i -} = n_{i} - n_{\overset{ˉ}{i}}$ ) of the triplet, leptons, and Higgs fields. The equations incorporate:
- Source terms driven by the shift in equilibrium distributions caused by $\dot{\theta}$ .
- Washout terms from inverse decays and gauge annihilations.
- Constraints from hypercharge neutrality and chemical equilibrium.
Analytical and Numerical Solutions: The system is solved analytically in limiting regimes (freeze-out and freeze-in) and numerically across the parameter space defined by the triplet mass ( $m_\Delta$ ) and branching fractions ( $Br_\ell, Br_H$ ).

Key Contributions and Results

Mechanism of Asymmetry Generation: The paper establishes that the asymmetry is not generated by CP-violating decay rates but by the system relaxing toward a $\dot{\theta}$ -shifted equilibrium state. The rolling Majoron background effectively violates CPT, creating a bias in number-violating interactions.
Independence from Triplet Mass: The final lepton asymmetry is found to be nearly independent of the triplet mass $m_\Delta$ , provided $m_\Delta$ is above the electroweak phase transition temperature. The efficiency is primarily controlled by the washout factor $K$ , which is determined by the branching ratios and the effective neutrino mass.
Scaling with Branching Fractions: A critical finding is the dependence of the final yield on the branching fractions $Br_\ell$ $B r_{ℓ}$ (decay to leptons) and $Br_H$ $B r_{H}$ (decay to Higgs).
- When $Br_\ell \sim Br_H$ , the system reaches thermal equilibrium, and the asymmetry freezes out.
- When either $Br_\ell$ or $Br_H$ is hierarchically small, the corresponding asymmetry undergoes a "freeze-in" process.
- In these limits, the final yield scales as $\sqrt{Br_\ell Br_H}$ . This reflects a symmetry argument: if either coupling vanishes, $B-L$ becomes an exact conserved charge, the Majoron phase becomes unphysical (removable by field redefinition), and no asymmetry can be generated.
Negligible CP Asymmetry in Decays: The authors explicitly show that the Majoron background does not induce a CP-asymmetric decay of the scalar triplet in the massless final-state limit. The difference $\Gamma_{\Delta \to HH} - \Gamma_{\bar{\Delta} \to \bar{H}\bar{H}}$ is suppressed by $O(m_H^2 \dot{\theta} / m_\Delta^3)$ , rendering it negligible for $m_\Delta \gg m_H$ .
Parameter Constraints: To reproduce the observed baryon asymmetry ( $Y_B^{obs} \approx 8.7 \times 10^{-11}$ ), the model requires a dimensionless parameter $\epsilon_1 \equiv (\dot{\theta}/T)|_{T=m_\Delta} \gtrsim 9 \times 10^{-8}$ . The relic Majoron abundance is estimated, yielding constraints on the symmetry-breaking scale $v_\sigma$ and the Majoron mass $m_J$ to avoid overclosing the universe or conflicting with dark matter observations.

Significance
The paper presents a minimal and robust realization of spontaneous leptogenesis in the type-II seesaw model. Unlike thermal type-II leptogenesis, this scenario:

Requires only a single scalar triplet and no additional heavy states.
Does not rely on explicit CP-violating phases in the Lagrangian; the CP violation is effectively provided by the time-dependent background.
Avoids the suppression issues associated with gauge interactions in thermal scenarios, as the strong washout factor actually aids the efficient production of asymmetry toward the shifted equilibrium.
Provides a simple scaling law ( $\propto \sqrt{Br_\ell Br_H}$ ) that highlights the necessity of both decay channels for the mechanism to operate.

The work establishes that Majoron-driven spontaneous leptogenesis is a viable alternative to standard thermal leptogenesis, capable of explaining the baryon asymmetry with a minimal particle content and without the need for fine-tuned CP-violating interactions.

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