Modified Majoron Model Overview

• The Modified Majoron Model extends the minimal singlet-majoron framework by embedding lepton-number breaking in enlarged symmetry sectors.
• It implements alternative neutrino mass generation schemes, including inverse-seesaw and radiative mechanisms, to align particle phenomenology with cosmological data.
• The framework yields diverse implications such as dark matter production, altered particle couplings, and gravitational wave signals, enriching experimental probes.

Modified Majoron Model denotes a non-minimal majoron framework in which the Goldstone mode associated with spontaneous lepton-number breaking is embedded in an enlarged symmetry, scalar, or fermion sector so that its mass, couplings, and cosmological function differ from those of the canonical singlet-majoron construction. In the literature, the label has been attached to several distinct realizations: a Higgs-triplet model with additional dimension-6 operators in the scalar potential (Barenboim et al., 2020), an axion-like majoron with inverse-seesaw structure and resonant photon conversion in the early universe (Cuesta et al., 2021), a gauged $U(1)_{B-L}\times U(1)$ setup motivated by protection against gravity-induced hard breaking (Ghosh et al., 6 Jul 2025), and an electromagnetic-anomalous variant designed to overlap the axion dark-matter band (Liang et al., 2024). This usage suggests a family of deformations of the minimal majoron paradigm rather than a unique universal Lagrangian.

1. Canonical baseline and the meaning of “modified”

The natural baseline is the minimal singlet-majoron extension of the Type-I seesaw, with

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$

where $\langle \sigma\rangle=f/\sqrt2$, $M_R=f\lambda/\sqrt2$, and the majoron $J$ is the phase of $\sigma$. In that setup the leading low-energy interaction is the diagonal neutrino coupling

$$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$

while couplings to gauge bosons and off-diagonal quark flavors first arise only at two loops (Heeck et al., 2019).

A modified majoron model changes this baseline in one or more of three ways. First, it alters the symmetry realization, for example by replacing a single global $U(1)$ with $U(1)\times U(1)$, a gauged $U(1)_{B-L}$ plus an approximate global symmetry, or an accidental $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$0 emerging from a finite modular symmetry (Ghosh et al., 6 Jul 2025, Jung et al., 2024). Second, it changes the neutrino-mass mechanism, moving from a standard Type-I seesaw to inverse-seesaw, radiative inverse-seesaw, or split-sector implementations (Cuesta et al., 2021, Bonilla et al., 2023, Giorgi et al., 2023). Third, it modifies the majoron’s phenomenological role by giving it a nonzero mass, an anomalous electromagnetic coupling, interactions strong enough to affect neutrino free streaming, or a topological-defect sector relevant for gravitational waves (Liang et al., 2024, Bari et al., 2023, Fu et al., 11 Jul 2025).

A concise comparison is given below.

Framework	Defining modification	Characteristic implication
Minimal singlet majoron (Heeck et al., 2019)	Type-I seesaw plus singlet $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$1	Gauge-boson and FC quark couplings first appear at two loops
Triplet modified majoron (Barenboim et al., 2020)	Higgs triplet $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$2 plus dimension-6 terms in $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$3	Either $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$4 or $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$5 can be light while the other is heavy enough to forbid $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$6
Axion-like majoron (Cuesta et al., 2021)	Singlet majoron with inverse seesaw and photon coupling	Resonant $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$7 conversion and late $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$8
$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$9 modified model (Ghosh et al., 6 Jul 2025)	Gauged $\langle \sigma\rangle=f/\sqrt2$0 plus approximate global $\langle \sigma\rangle=f/\sqrt2$1	Local and effectively global string networks with PTA-scale GW signals
Electromagnetic-anomalous majoron (Liang et al., 2024)	Charge assignment producing a QED anomaly	Majoron dark-matter band overlapping the QCD-axion band

2. Symmetry engineering and field content

The most visible structural change in modified models is the symmetry sector. In the triplet construction of Chikashige-type origin but with explicit higher-dimensional deformations, the Standard Model is supplemented by an $\langle \sigma\rangle=f/\sqrt2$2 triplet $\langle \sigma\rangle=f/\sqrt2$3 with hypercharge $\langle \sigma\rangle=f/\sqrt2$4 and lepton number $\langle \sigma\rangle=f/\sqrt2$5. The scalar potential contains the renormalizable terms for $\langle \sigma\rangle=f/\sqrt2$6 and $\langle \sigma\rangle=f/\sqrt2$7, together with

$\langle \sigma\rangle=f/\sqrt2$8

which are dimension-6 operators. Their role is to separate the masses of the neutral scalar and pseudoscalar states so that one can keep one state light and the other sufficiently heavy to evade the LEP-1 bound from $\langle \sigma\rangle=f/\sqrt2$9 (Barenboim et al., 2020).

In the $M_R=f\lambda/\sqrt2$0 construction, the gauge group is extended to

$M_R=f\lambda/\sqrt2$1

with three right-handed neutrinos and two singlet complex scalars $M_R=f\lambda/\sqrt2$2 and $M_R=f\lambda/\sqrt2$3. Sequential breaking at $M_R=f\lambda/\sqrt2$4 and $M_R=f\lambda/\sqrt2$5 yields one eaten CP-odd mode and one physical pseudo-Goldstone majoron $M_R=f\lambda/\sqrt2$6. The explicit breaking arises from a gravity-induced operator

$M_R=f\lambda/\sqrt2$7

with $M_R=f\lambda/\sqrt2$8 and $M_R=f\lambda/\sqrt2$9 (Ghosh et al., 6 Jul 2025).

More radical symmetry engineering appears in the finite modular majoron model. There the accidental $J$0 is not imposed as a fundamental continuous symmetry; it emerges from a finite modular symmetry $J$1, with only the residual $J$2 left exact after spontaneous breaking. In the limit $J$3, the Lagrangian develops an accidental continuous shift symmetry in $J$4, and the majoron is identified with the canonically normalized real part of the modulus (Jung et al., 2024).

Multi-scalar variants pursue yet another objective: hierarchical right-handed-neutrino masses and defect phenomenology. The Minimal Multi-Majoron Model introduces two complex scalar Majoron fields $J$5 and $J$6, controlled by an extra global $J$7, plus a flavon $J$8 and vector-like fermions $J$9. The intended outcome is a realistic ultraviolet complete framework with two “personal” Majoron couplings to two right-handed neutrinos, full $\sigma$0 neutrino mixing, global strings, domain walls, and a combined gravitational-wave spectrum (Fu et al., 11 Jul 2025).

3. Neutrino-mass mechanisms in modified majoron theories

Although the majoron is defined by lepton-number breaking, the modified literature is unified at least as much by neutrino-mass engineering as by bosonic phenomenology. In the triplet model, the Yukawa interaction

$\sigma$1

produces

$\sigma$2

so the triplet VEV directly controls the Majorana mass of $\sigma$3 (Barenboim et al., 2020).

Several later models replace this direct Type-II-like structure with Type-I or inverse-seesaw architectures. In the axion-like majoron model, the Standard Model is extended by singlet Weyl fermions $\sigma$4, $\sigma$5, and $\sigma$6, together with singlet scalars $\sigma$7 charged under a global $\sigma$8. After $\sigma$9 acquires a VEV $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$0 and a residual $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$1 is softly broken by a suppressed $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$2 VEV $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$3, the higher-dimensional operators generate inverse-seesaw parameters and the $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$4 mass matrix yields three light eigenvalues

$$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$5

(Cuesta et al., 2021).

The minimal massive majoron seesaw model is more constrained. It introduces two right-handed singlet neutrinos $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$6 and $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$7, one complex scalar singlet $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$8, and a single small spurion $$\mathcal L_{J\nu\nu}=\frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar n_i\gamma_5 n_i+\cdots,$$9 that explicitly breaks the global Abelian symmetry in the Dirac sector. After spontaneous breaking, the neutral-fermion mass matrix is

$U(1)$0

and the light-neutrino mass is

$U(1)$1

Its distinguishing feature is that the same spurion controlling $U(1)$2 also generates the majoron mass at one loop, so the neutrino and majoron mass sectors are explicitly correlated (Giorgi et al., 2023).

Radiative inverse-seesaw variants add a dark sector stabilized by a residual symmetry. In the model with global $U(1)$3, the singlet scalar $U(1)$4 yields a physical Majoron $U(1)$5, the one-loop radiative $U(1)$6-term is generated by the $U(1)$7 loop, and the inverse-seesaw relation

$U(1)$8

coexists with a Majorana-fermion dark-matter candidate $U(1)$9 (Bonilla et al., 2023).

4. Majoron mass generation and coupling patterns

A recurrent misconception is that the majoron must be exactly massless. That statement applies only to the strict Goldstone limit. Modified majoron models are largely defined by mechanisms that make the boson pseudo-Goldstone while keeping the neutrino sector technically natural.

Several distinct mass-generation mechanisms occur in the literature. Soft explicit breaking by a scalar bilinear,

$U(1)\times U(1)$0

gives $U(1)\times U(1)$1 in the electromagnetic-anomalous construction (Liang et al., 2024). Radiative breaking by a tiny lepton-number-violating Majorana mass $U(1)\times U(1)$2 in the canonical seesaw generates a one-loop potential

$U(1)\times U(1)$3

with

$U(1)\times U(1)$4

in the $U(1)\times U(1)$5 limit (Chao et al., 2023). Planck-suppressed explicit breaking produces

$U(1)\times U(1)$6

up to the CP-odd mixing denominator in the $U(1)\times U(1)$7 model (Ghosh et al., 6 Jul 2025). In the finite modular majoron model the mass is exponentially suppressed by powers of $U(1)\times U(1)$8, reflecting the fact that the continuous symmetry is only accidental and broken by higher-order $U(1)\times U(1)$9-effects (Jung et al., 2024).

The coupling structure is equally diverse. In the minimal singlet theory, couplings to Standard Model particles beyond neutrinos are loop-suppressed, and the diphoton coupling arises only at two loops (Heeck et al., 2019). Modified models change this in two opposite directions. Some deliberately enhance the photon channel. In the anomalous majoron model, a second Higgs doublet and the charge assignment

$U(1)_{B-L}$0

generate a QED anomaly coefficient $U(1)_{B-L}$1, yielding

$U(1)_{B-L}$2

(Liang et al., 2024). Others keep photon couplings tiny or indirect. In the axion-like majoron model, the induced one-loop coupling

$U(1)_{B-L}$3

is not used for laboratory axion searches but to trigger resonant $U(1)_{B-L}$4 conversion in the thermal plasma (Cuesta et al., 2021).

Another misconception is that modified majorons are necessarily photon-dominated axion-like particles. The opposite regime is equally important. The axion-like majoron of Cuesta et al. is constructed so that the boson decays into $U(1)_{B-L}$5 pairs near recombination rather than photons, and this is precisely why it avoids the strong bounds that affect other axion-like particles of similar mass and coupling to photons (Cuesta et al., 2021).

5. Cosmological implementations

Modified majoron models are often motivated not by collider phenomenology but by precision cosmology. The clearest example is the axion-like majoron cosmology of Cuesta et al. There, the majoron has $U(1)_{B-L}$6, decouples at $U(1)_{B-L}$7, contributes only $U(1)_{B-L}$8 before the resonance epoch, and then undergoes a resonant $U(1)_{B-L}$9 conversion at $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$00 induced by a primordial magnetic field. The conversion transfers roughly $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$01 of the photon energy into majorons, raises the post-BBN radiation density to $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$02, and after late $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$03 recoupling and $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$04 decay gives $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$05. A joint Planck 2018 + BAO fit yields

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$06

while $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$07-mediated $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$08-$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$09 scattering suppresses neutrino free streaming and preserves the CMB peak structure (Cuesta et al., 2021).

The triplet modified majoron model addresses a different set of anomalies. If the light neutral triplet state couples predominantly to the third-generation doublet $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$10, then the effective neutrino self-interaction strength

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$11

is favored in the range $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$12 at $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$13. The same parameter region modifies CMB fits in the $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$14 plane, allows inflationary potentials previously ruled out, and gives

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$15

at 95% C.L. when $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$16 and $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$17 are varied (Barenboim et al., 2020).

The split majoron model moves the cosmological focus to phase transitions below neutrino decoupling. A high-scale first-order transition in $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$18 creates the seesaw scale and the first majoron bath, while a later low-scale first-order transition in $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$19 at $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$20 produces extra radiation and nanohertz gravitational waves. For the minimal choice $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$21, $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$22, the low-scale reheating gives $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$23, marginally consistent with deuterium bounds, and one extra dark degree of freedom reduces this to $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$24. Majoron-mediated neutrino interactions with $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$25–$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$26 can then suppress free streaming into the eV era, mildly relieving Hubble and large-scale-structure tensions (Bari et al., 2023).

The finite modular majoron model addresses dark radiation differently. The heavy modulus $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$27 decays predominantly via $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$28, with

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$29

for $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$30, $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$31, and $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$32, a regime explicitly noted as potentially helpful for the Hubble tension (Jung et al., 2024).

6. Dark matter, topological defects, gravitational waves, and searches

Dark-matter realizations of the modified majoron idea span several regimes. In the anomalous-photon model, coherent oscillations from the misalignment mechanism lead to

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$33

so requiring the observed relic density fixes a one-to-one relation between $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$34 and $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$35. The predicted majoron band overlaps the standard QCD-axion target region,

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$36

making haloscope experiments such as ADMX directly relevant (Liang et al., 2024).

Radiatively generated or pseudo-Goldstone masses lead to other dark-matter mechanisms. The Majorana Majoron model combines ordinary misalignment with kinetic misalignment driven by a nonzero initial $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$37, and the same initial velocity acts as an effective chemical potential for lepton number so that the Weinberg operator can generate the observed baryon asymmetry, with numerical integration giving $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$38 for $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$39 GeV (Chao et al., 2023). In the $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$40 model, the majoron can be produced by thermal freeze-in, coherent oscillation, and string-induced radiation, but in the NANOGrav-compatible region $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$41 its relic abundance remains subdominant; saturating $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$42 instead requires higher $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$43, operator dimension $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$44, and parameter choices that begin to compete with cosmological bounds (Ghosh et al., 6 Jul 2025).

Topological defects are a particularly important signature of modified models with enlarged symmetry sectors. The multi-majoron framework predicts global cosmic strings from $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$45 or $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$46 breaking, residual $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$47 symmetries broken by the flavon $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$48, and “walls bounded by strings” that collapse rapidly. Its stochastic background is the sum

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$49

where the second Majoron field enhances the first-order phase transition and strengthens the signal (Fu et al., 11 Jul 2025). The gauged $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$50 model of Ghosh, Loho, and Manna has two independent string networks, one local and one effectively global. For $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$51, the combined spectrum can account for the NANOGrav 15-year signal with a “Type-C” best-fit range

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$52

at 95% C.L., though the paper explicitly notes that the fit is not as strong as that from supermassive black hole mergers (Ghosh et al., 6 Jul 2025).

The experimental profile of modified majoron models is correspondingly broad. Invisible or semi-invisible lepton-flavor violation remains a robust probe of singlet-derived constructions: in the minimal singlet-majoron effective theory,

$$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$53

with current limits $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$54, $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$55, and $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$56 (Heeck et al., 2019). Neutrino telescopes are relevant when $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$57 dominates, as in massive-majoron seesaw models (Giorgi et al., 2023). Haloscopes become relevant when an electromagnetic anomaly is engineered (Liang et al., 2024). Collider searches enter most directly in triplet realizations, where the singly and doubly charged Higgs bosons are lighter than $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$58 GeV and $$\mathcal L = -\,\bar L\,y\,N_R\,H - \frac12\,\bar N_R^c\,\lambda\,N_R\,\sigma - V(H,\sigma),$$59 GeV respectively (Barenboim et al., 2020).

Taken together, these constructions show that the modified majoron model is best understood as a research program: retain spontaneous lepton-number breaking and the associated Goldstone degree of freedom, then redesign the symmetry realization and infrared couplings so that the majoron also participates in neutrino-mass generation, cosmological anomaly relief, dark-matter production, baryogenesis, or gravitational-wave phenomenology (Barenboim et al., 2020, Cuesta et al., 2021, Ghosh et al., 6 Jul 2025).