⚛️ phenomenology

How to Identify a Majoron: Effective Field Theories of Spontaneous Lepton Number Breaking

This paper constructs effective field theories for spontaneous lepton number breaking within Type I, II, and III Seesaw mechanisms coupled to a complex scalar, demonstrating that the resulting Majoron models are falsifiable through specific correlations between observables—such as the invisible Higgs width, non-unitarity in the leptonic mixing matrix, and a TeV-scale lepton-number breaking scale—rather than the absolute magnitude of individual couplings.

Original authors: Carla Biggio, Marta Fuentes Zamoro, Xu Li, Luca Merlo, Luca Ottonello

Published 2026-08-13

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CC BY 4.0

Original authors: Carla Biggio, Marta Fuentes Zamoro, Xu Li, Luca Merlo, Luca Ottonello

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

Imagine the universe as a giant, intricate machine. For decades, physicists have been trying to figure out how this machine works, specifically focusing on a tiny, ghostly part called the "neutrino." These particles are so light and shy that they barely interact with anything, yet we know they have mass, which is a puzzle because the standard rulebook of physics (the Standard Model) says they should be massless. To fix this, scientists proposed a "Seesaw" mechanism. Think of it like a playground seesaw: if one side (the heavy, hidden particles) goes down, the other side (the light neutrinos we see) goes up, explaining why they are so tiny.

But there's another mystery: why does the universe seem to have a hidden "leak" in its symmetry called Lepton Number? Usually, physicists think this leak is just a broken rule. However, some theories suggest it's not broken at all, but rather "spontaneously broken," like a pencil balanced on its tip that eventually falls over. When this happens, it doesn't just create a leak; it creates a new, invisible particle called a "Majoron." This particle is the "ghost" of the broken symmetry, similar to how a broken magnet creates a ripple in the magnetic field. The big question is: if these heavy Seesaw particles and the Majoron exist, how can we find them without building a collider the size of a galaxy?

This paper is a detective story written by a team of physicists who decided to look at the clues left behind after the heavy particles have vanished. They revisited three famous versions of the Seesaw mechanism (Type I, II, and III) and added the Majoron to the mix. Instead of trying to catch the heavy particles directly, they built a "low-energy map" (an Effective Field Theory) that shows how the heavy particles and the Majoron leave their fingerprints on the particles we can see, like the Higgs boson and electrons.

The authors found something fascinating: you don't need to see the heavy particles to know if the Majoron exists. Because everything in these models is tied together by a single "scale" (a specific energy level called $v_\phi$ ), the different parts of the theory are locked together like gears in a clock. If you measure one gear, you know exactly how the others must move. The paper proves that if the Majoron exists, it creates a very specific, unbreakable relationship between two things: how much the Higgs boson's strength is weakened, and how much the Higgs decays invisibly into Majorons. It's like saying, "If you hear a specific hum from a machine, you know exactly how fast its gears are spinning, even if you can't see the gears."

Furthermore, the paper rules out a popular idea: that we can find the Majoron by looking for it in "neutrinoless double beta decay" (a rare nuclear process often thought to be the best way to spot these particles). The authors show that in their models, this process is so suppressed by the tiny mass of the neutrino that it's effectively invisible—like trying to hear a whisper in a hurricane. Instead, the real clues are hiding in the precision measurements of the Higgs boson and in rare decays of muons (heavy cousins of electrons).

The team also discovered that the three different Seesaw models leave different "fingerprints." For example, one model predicts that the Majoron can turn into two photons (light particles), while the others cannot. Another model predicts a specific pattern of how the Higgs boson interacts with itself. By measuring these patterns, scientists can tell which version of the Seesaw is actually at work, even if the heavy particles are billions of times too heavy to ever be created in a lab.

In short, the paper argues that we don't need to wait for a super-powerful machine to solve the mystery of neutrino mass. We just need to look closely at the "echoes" the heavy particles leave behind. If the Higgs boson is slightly weaker than expected and decays invisibly in a specific way, and if muons decay into electrons and a Majoron with a strength tied to the same numbers, then we have found the Majoron. It's a clever, falsifiable plan: if the numbers don't match up perfectly, the whole theory falls apart. And right now, the numbers suggest that the scale of this new physics is likely between 1 and 10 TeV, a range that current and future experiments are just starting to explore.

Technical Summary: How to Identify a Majoron: Effective Field Theories of Spontaneous Lepton Number Breaking

Problem Statement
The discovery of neutrino oscillations necessitates physics beyond the Standard Model (SM), typically addressed by Type I, II, or III Seesaw mechanisms. These mechanisms generate small neutrino masses by introducing heavy degrees of freedom (fermionic singlets, scalar triplets, or fermionic triplets) that explicitly break lepton number (LN). Concurrently, the Peccei–Quinn (PQ) mechanism addresses the strong CP problem via a global $U(1)$ symmetry, the breaking of which yields an axion-like particle (ALP). A compelling theoretical possibility is that the global symmetry responsible for the ALP is identified with lepton number, leading to a "Majoron" ( $J$ ) that arises from the spontaneous breaking of LN.

While Majoron models have been studied individually, a systematic, model-independent Effective Field Theory (EFT) treatment of the interplay between the heavy Seesaw mediators, the radial mode of the LN-breaking scalar, and the resulting Majoron has been incomplete. Specifically, the order in which heavy fields are integrated out can obscure the physical origin of effective operators and complicate the interpretation of couplings. Furthermore, it remains unclear whether the Majoron sector can be identified experimentally if the heavy states lie far beyond the reach of current colliders.

Methodology
The authors revisit the three traditional Seesaw mechanisms (Type I, II, and III) augmented by a complex scalar singlet $\phi$ charged under a global $U(1)_L$ symmetry. Upon spontaneous symmetry breaking (SSB), $\phi$ acquires a vacuum expectation value (vev) $v_\phi$ , generating a massive radial mode $\rho$ and a massless angular mode identified as the Majoron $J$ .

The core methodology involves constructing the low-energy EFT (dubbed JSMEFT) by integrating out the heavy states in two complementary orders to ensure consistency:

Order A: Integrate out the radial mode $\rho$ first, followed by the Seesaw mediators.
Order B: Integrate out the Seesaw mediators first, followed by the radial mode $\rho$ .

The analysis covers all possible mass hierarchies between the radial mode ( $m_\rho$ ) and the Seesaw mediators ( $M_N$ , $M_\Sigma$ , or $M_\Delta$ ). This procedure yields seven intermediate high-energy EFTs (e.g., $N$ JSMEFT, $\rho$ JSMEFT-I) before reducing to the final low-energy Lagrangian containing only SM fields and the Majoron. The authors perform a complete matching up to dimension-7 operators, utilizing equations of motion (EOM) and perturbative solutions to derive Wilson coefficients. They verify that both matching orders yield identical low-energy Lagrangians, operator by operator, and reduce redundant structures to the Warsaw basis.

Key Contributions and Results

Complete EFT Construction: The paper provides the first systematic derivation of the JSMEFT for all three Seesaw types with spontaneous LN breaking. It identifies the complete set of effective operators up to dimension 7, including:
- Bosonic Operators: Generated by $\rho$ exchange (e.g., Higgs kinetic corrections, Higgs-Majoron derivative portals).
- Lepton-Number-Violating (LNV) Operators: Generated by Seesaw mediator exchange (e.g., Weinberg operator, unitarity-violating operators).
- Majoron Couplings: Derivative couplings to leptons and gauge bosons, and anomalous couplings arising from chiral rotations.
Matching Consistency: A rigorous check confirms that integrating out the radial mode before or after the Seesaw mediators yields the same physical predictions for the low-energy Wilson coefficients. This validates the EFT approach regardless of the specific mass hierarchy.
Discriminating Signatures: The analysis reveals that while individual observables depend on unknown couplings, the relations between them are fixed by the single scale $v_\phi$ . This allows for the identification of the specific Seesaw realization through correlation patterns:
- Type I vs. Type III: Distinguished by the ratio of Wilson coefficients in the Higgs-current operators ( $C^{(1)}_{Hl} : C^{(3)}_{Hl}$ is $-1:1$ for Type I and $3:1$ for Type III) and the presence of a two-photon Majoron coupling ( $J \to \gamma\gamma$ ) in Type III (due to charged triplet components) which is absent in Type I and II.
- Type II: Characterized by the absence of the Majoron-lepton derivative coupling ( $Q_{JHL}$ ) and the presence of a four-lepton contact interaction ( $Q_{ll}$ ) and custodial symmetry-breaking operators ( $Q_{HD}$ ).
Phenomenological Correlations:
- Higgs Sector: The invisible Higgs width ( $\Gamma(h \to JJ)$ ) is strictly correlated with the universal suppression of Higgs couplings to gauge bosons ( $\kappa_V$ ). The relation $\Gamma(h \to JJ) \propto (1-\kappa_V)$ is parameter-free. Crucially, the model predicts $\kappa_V < 1$ ; an observed enhancement ( $\kappa_V > 1$ ) would exclude the entire class of models.
- Leptonic Sector: The Majoron-lepton coupling is fixed by the measured non-unitarity of the PMNS matrix. Specifically, the coupling strength for processes like $\mu \to eJ$ is determined by the non-unitarity parameter $\eta$ and $v_\phi$ .
- Higgs Self-Couplings: The trilinear ( $\kappa_3$ ) and quartic ( $\kappa_4$ ) Higgs self-couplings are constrained to a specific half-line (slope 50/9) through the SM point for fermionic realizations (Type I/III), whereas the scalar realization (Type II) fills a band of slope 6.
- Neutrinoless Double Beta Decay: The paper demonstrates that neutrinoless double beta decay with Majoron emission ( $0\nu\beta\beta J$ ) has no sensitivity in this class of models. The derivative nature of the Goldstone coupling suppresses the rate by the active neutrino mass, rendering it nine orders of magnitude below current experimental reach.

Significance and Claims
The authors claim that their framework establishes a falsifiable foundation for identifying the origin of neutrino masses through the Majoron portal, even when the heavy states are inaccessible. The significance lies in shifting the experimental focus from searching for individual rare events to testing correlations among precision observables.

Falsifiability: The models predict that independent measurements of the Higgs invisible width and the non-unitarity of the leptonic mixing matrix must yield a consistent value for the lepton-number breaking scale $v_\phi$ . The paper finds that current bounds from both sectors independently suggest $v_\phi \gtrsim \mathcal{O}(1\text{--}10)$ TeV, a window where the consistency test is already being performed, albeit at low precision.
Experimental Strategy: The paper argues that the "flagship" signature of Majoron models ( $0\nu\beta\beta J$ ) is ineffective for these specific models. Instead, the most promising avenues for identification are Higgs precision physics (invisible width, self-couplings) and charged-lepton flavor violation ( $\mu \to eJ$ ).
Model Independence: By deriving the EFT for arbitrary hierarchies, the results provide a unified description valid for any Seesaw scale relative to the LN breaking scale, allowing for a model-independent interpretation of future data.

In summary, the paper posits that the spontaneous breaking of lepton number is not merely a theoretical decoration but imposes strict, testable relations between the Higgs sector, the leptonic mixing matrix, and Majoron interactions. These relations serve as the definitive diagnostic tool to distinguish between Seesaw realizations and confirm the spontaneous nature of lepton number breaking.

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