Neutrino–Majoron Couplings

• Neutrino–majoron couplings are interactions between neutrinos and a Goldstone boson from spontaneous lepton-number breaking that naturally scale with neutrino masses.
• In seesaw frameworks, these couplings provide a direct probe of the neutrino mass mechanism and link to observable phenomena like neutrino decay and lepton-flavor violation.
• Their rich phenomenology extends to astrophysical and cosmological tests, including supernova neutrino transport and majoron dark matter decay signatures.

Searching arXiv for recent and foundational papers on neutrino–majoron couplings to ground the article in the provided literature. {"query":"majoron neutrino couplings supernova cosmology singlet seesaw arXiv", "max_results": 10} Neutrino–majoron couplings are the interactions between neutrinos and the Goldstone boson associated with spontaneous breaking of global lepton number, or equivalently of an anomaly-free $U(1)_{B-L}$-type symmetry in the standard singlet realization. In seesaw frameworks these couplings are not auxiliary additions: they are generated by the same symmetry-breaking sector that gives Majorana masses to right-handed neutrinos and, after integrating out the heavy states, to the light neutrinos. The resulting low-energy interaction is therefore a direct probe of the neutrino mass mechanism, while its phenomenology extends from neutrino decay and lepton-flavor violation to supernova transport, cosmic microwave background signatures, and long-lived dark matter based on a pseudo-Goldstone majoron (Heeck, 2017, Escudero et al., 2019, Iváñez-Ballesteros et al., 2024).

1. Symmetry origin and tree-level coupling structure

In the standard singlet-majoron construction, the Standard Model is extended by a complex scalar singlet and right-handed neutrinos. With

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$

the vacuum expectation value $f$ spontaneously breaks global lepton number or $U(1)_{B-L}$, and the phase field $J$ is the majoron. The relevant Yukawa sector is

$$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$

which yields the type-I seesaw relation

$$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$

with $M_R\gg m_D$ (Heeck, 2017).

Because the majoron is the phase of the scalar responsible for the Majorana mass term, its coupling to light neutrinos is proportional to the light-neutrino masses themselves. In the mass basis, the tree-level interaction can be written either in derivative form,

$$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$

or on shell as

$$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$

equivalently

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$0

This coupling is diagonal in the light-neutrino mass basis and has no flavor-changing structure at tree level (Jodłowski et al., 28 Feb 2026, Heeck, 2017).

The same logic persists across a broad class of majoron models. In the “minimal massive majoron seesaw model,” the active-neutrino coupling is

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$1

again tying the coupling matrix directly to the light-neutrino mass matrix (Giorgi et al., 2023). In inverse-seesaw-based constructions the low-energy parametrization is likewise mass proportional,

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$2

with the heaviest neutrino state typically dominating the late-time phenomenology (Cuesta et al., 2021).

A common source of confusion is whether a majoron must be exactly massless. In the exact Goldstone limit it is massless, but several papers treat it as a pseudo-Goldstone boson once there is small explicit breaking from scalar-potential terms, quantum gravity, or related effects. That explicit breaking generates a nonzero mass $\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$3 without changing the basic origin of the neutrino coupling (Heeck, 2017, Heeck, 2018, Escudero et al., 2019).

2. Seesaw dependence beyond the light-neutrino mass matrix

Although the irreducible tree-level coupling is fixed by $\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$4, much of the broader majoron phenomenology depends on seesaw data not reconstructible from low-energy oscillation observables. In the singlet-majoron model, the central object is

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$5

or, equivalently,

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$6

a Hermitian positive semidefinite matrix that encodes the combination $\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$7 rather than the oscillation-controlled combination $\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$8 (Heeck, 2017, Jodłowski et al., 28 Feb 2026).

At one loop, this matrix controls the majoron couplings to charged fermions. The effective interaction is written as

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$9

For quarks,

$f$0

whereas for charged leptons

$f$1

$f$2

The diagonal couplings are thus loop suppressed, while off-diagonal lepton couplings are generically nonzero and governed by $f$3 (Heeck, 2017).

This distinction is structurally important. The invisible decay $f$4 depends on low-energy neutrino masses and the breaking scale $f$5, whereas visible decays and charged-lepton flavor violation probe $f$6. The two sets of observables are therefore complementary rather than redundant (Garcia-Cely et al., 2017, Heeck, 2017).

The matrix $f$7 is constrained by positivity and perturbativity. The singlet-majoron analysis gives

$f$8

together with the perturbative estimate

$f$9

A recent intensity-frontier study formalizes this structure through benchmark textures, including positive semidefinite “anarchical,” single-flavor, and CP-violating cases, and uses them to map displaced-decay and lepton-flavor-violating signatures in the $U(1)_{B-L}$0 plane (Heeck, 2017, Jodłowski et al., 28 Feb 2026).

General one-loop formulas for majoron couplings to charged leptons have also been derived in a model-independent expansion valid for Majorana neutrino mass models with spontaneous global lepton-number violation and a clear scale hierarchy. Applied to explicit realizations, these formulas reproduce the suppression in minimal type-I seesaw and inverse seesaw models, while allowing significantly larger couplings in the Scotogenic model or in a type-I seesaw plus 2HDM (Herrero-Brocal et al., 2023). This suggests that the neutrino–majoron coupling $U(1)_{B-L}$1 is universal at tree level, but the experimentally dominant charged-sector manifestations are highly UV sensitive.

3. Vacuum diagonality, off-diagonal decay, and matter-induced couplings

In standard majoron models the vacuum coupling matrix is approximately diagonal in the neutrino mass basis,

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

so vacuum decay is suppressed (Iváñez-Ballesteros et al., 2024). That statement, however, is not universally sufficient for phenomenology.

A separate line of work studies visible neutrino decay through effective scalar and pseudoscalar interactions,

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

allowing heavier mass eigenstates to decay into lighter active neutrinos plus a massless majoron. In reactor analyses the relevant widths depend on the parent mass, the daughter-to-parent mass ratio $U(1)_{B-L}$4, and the functions $U(1)_{B-L}$5, $U(1)_{B-L}$6, and $U(1)_{B-L}$7, leading to constraints on $U(1)_{B-L}$8 and on the couplings $U(1)_{B-L}$9 (Porto-Silva et al., 2020).

Dense matter changes the problem qualitatively. In supernova matter, neutrinos experience flavor-dependent effective potentials,

$J$0

and the medium eigenstates differ from the vacuum mass eigenstates. In this environment, matter-induced effective masses and potentials can induce off-diagonal effective couplings and open decay channels that are absent or negligible in vacuum (Iváñez-Ballesteros et al., 2024).

The in-medium two-body non-radiative decay rate is written as

$J$1

with the kinematic requirement $J$2. For the supernova profiles analyzed in recent work, this condition is fulfilled only for antineutrino-to-neutrino decays, so neutrinos are treated as stable while antineutrino spectral distortions carry the signal (Iváñez-Ballesteros et al., 2024, Iváñez-Ballesteros et al., 29 Jul 2025).

This is one of the main conceptual corrections to the vacuum intuition. Tree-level diagonality in the light-neutrino mass basis does not imply the absence of phenomenologically relevant off-diagonal decay in matter. The full three-neutrino treatment therefore tracks propagation from the neutrinosphere, energysphere, and transportsphere, then MSW conversion in the envelope, and finally the distorted $J$3 spectrum at Earth (Iváñez-Ballesteros et al., 2024, Iváñez-Ballesteros et al., 29 Jul 2025).

4. Majoron dark matter and monochromatic neutrino lines

If the majoron is a pseudo-Goldstone boson with a small mass, its couplings are so suppressed that it can be long lived on cosmological timescales and become a dark-matter candidate. In the singlet-majoron model, the dominant tree-level decay is

$J$4

with total width

$J$5

The final state consists of two monochromatic neutrinos at

$J$6

Because the decay products are neutrino mass eigenstates, the flavor composition at Earth is fixed already at production and does not oscillate afterward (Garcia-Cely et al., 2017).

The resulting flavor ratios are a direct function of the neutrino hierarchy: $J$7 Benchmark ratios quoted for singlet-majoron dark matter are:

• normal hierarchy: $J$8,
• inverted hierarchy: $J$9,
• quasi-degenerate spectrum: $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$0 (Garcia-Cely et al., 2017).

The hierarchy sensitivity is physically distinctive. In normal ordering the electron-flavor fraction is strongly suppressed, approximately $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$1, whereas inverted ordering yields approximately $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$2 (Garcia-Cely et al., 2017). A related treatment emphasizes the same pattern qualitatively: only a small $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$3 component in normal hierarchy, roughly $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$4 electron flavor in inverted hierarchy, and approximately democratic flavor in the quasi-degenerate regime (Heeck, 2017).

For $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$5, the neutrino line can be sought in Borexino, KamLAND, and Super-Kamiokande; above the MeV scale inverse beta decay becomes effective, while Super-K is more sensitive at higher masses and can exploit the $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$6 component (Heeck, 2017). For $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$7, the neutrino line becomes too soft for standard direct searches, and the relevant indirect channels shift toward $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$8 and cosmological probes (Heeck, 2018).

An important model-building point is that visible decays do not directly fix the neutrino-line signal. In the singlet-majoron model, $$\mathcal{L}=\mathcal{L}_{\rm SM}+ i\overline{N}_R \gamma^\mu\partial_\mu N_R + (\partial_\mu \sigma)^\dagger (\partial^\mu \sigma) - V(\sigma)-\left(\overline{L} y N_R H +\tfrac12\overline{N}_R^c\lambda N_R \sigma +\text{h.c.}\right),$$9 depends on $$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$0, whereas $$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$1, $$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$2, and $$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$3 depend on the loop-induced structure governed by $$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$4. Visible-decay bounds therefore constrain different combinations of seesaw parameters and do not automatically invalidate the neutrino-line channel (Heeck, 2017, Heeck, 2018).

5. Astrophysical and cosmological constraints

Cosmology probes neutrino–majoron couplings at exceptionally small values. A widely used low-energy interaction is

$$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$5

with the seesaw-motivated relation

$$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$6

The majoron decay width is

$$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$7

and inverse decays $$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$8 can thermalize the majoron after Big Bang Nucleosynthesis but before recombination. Planck2018 is reported to be sensitive to couplings as small as $$M_\nu \simeq - m_D M_R^{-1} m_D^T,\qquad m_D=\frac{yv}{\sqrt{2}},\qquad M_R=\frac{\lambda f}{\sqrt{2}},$$9, and the same framework identifies parameter space in which neutrino–majoron interactions together with extra $M_R\gg m_D$0 can reduce the $M_R\gg m_D$1 tension from about $M_R\gg m_D$2 to $M_R\gg m_D$3 (Escudero et al., 2019).

Supernovae provide a complementary probe. One SN 1987A analysis considers universal couplings

$M_R\gg m_D$4

with production by

$M_R\gg m_D$5

followed by

$M_R\gg m_D$6

The non-observation of $M_R\gg m_D$7 MeV-range events in Kamiokande-II and IMB implies that less than about $M_R\gg m_D$8 of the total supernova energy could have been emitted this way, leading to

$M_R\gg m_D$9

(Fiorillo et al., 2022).

A different SN 1987A strategy uses spectral distortions induced by in-medium non-radiative decay. In a full three-neutrino likelihood analysis of the 24 observed events, the quoted $$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$0 bounds are

$$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$1

$$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$2

at $$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$3 CL for the reference supernova model. The same analysis reports new limits on $$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$4 and stronger bounds on $$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$5 and $$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$6 than earlier spectral studies (Iváñez-Ballesteros et al., 29 Jul 2025). An earlier likelihood study already emphasized that the resulting supernova constraints are complementary or competitive with laboratory limits and are especially valuable because matter effects activate decay channels suppressed in vacuum (Iváñez-Ballesteros et al., 2024).

Laboratory visible-decay searches offer an additional, lower-energy handle. Under the reactor-neutrino visible-decay formalism, KamLAND and JUNO constrain

$$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$7

and

$$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$8

at $$\mathcal{L}^{\rm tree}_{\rm int}= -\frac{\partial_\mu J}{4f}\sum_{i=1}^3\bar{\nu}_i\gamma^\mu\gamma_5\nu_i,$$9 CL, with the detailed coupling constraints depending strongly on the lightest neutrino mass and on whether the coupling is scalar or pseudoscalar (Porto-Silva et al., 2020).

6. Gauge-boson couplings, model variants, and open directions

Because $$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$0 is anomaly free, a massless majoron has no true anomalous $$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$1 coupling. Yet this does not imply that all gauge-boson amplitudes vanish. A one-loop study of the matrix element $$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$2 shows that the anomaly itself vanishes, but the amplitude is nonzero for nontrivial left–right neutrino mixing because of a UV-finite, non-local contribution (Latosinski et al., 2012). The effect disappears in the pure Dirac limit and in the decoupling limit of infinitely heavy right-handed neutrinos, which clarifies that it is generated by seesaw mixing rather than by a genuine anomaly.

This anomaly-like mechanism underlies a broader proposal identifying the majoron with the axion. In that construction the relevant low-energy gauge couplings arise through neutrino-mediated loop effects rather than through an anomalous Peccei–Quinn symmetry, and the effective $$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$3 vertex appears only at three loops after the neutrino triangle is embedded into electroweak and QCD subdiagrams (Latosinski et al., 2012). A plausible implication is that neutrino–majoron couplings may control a wider infrared effective theory than the minimal tree-level $$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$4 interaction alone suggests.

Recent model building has also diversified the neutrino–majoron sector itself. The “Minimal Multi-Majoron Model” introduces two complex scalar majoron fields $$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$5, two right-handed neutrinos, a flavon $$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$6, and vector-like fermions, with diagonal couplings

$$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$7

and effective Yukawas

$$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$8

This realizes a type-I seesaw in which each right-handed neutrino couples to its own majoron field, and the hierarchy of scalar vacuum expectation values controls the hierarchy of heavy-neutrino masses (Fu et al., 11 Jul 2025).

In the “minimal massive majoron seesaw model,” the same explicit breaking term that renders the majoron massive also participates in generating realistic neutrino masses. The low-energy coupling remains

$$\mathcal{L}^{\rm tree}_{\rm int}= \frac{iJ}{2f}\sum_{i=1}^3 m_i\,\bar{\nu}_i\gamma_5\nu_i,$$9

but the majoron mass is no longer arbitrary: it is radiatively generated and correlated with neutrino-sector parameters (Giorgi et al., 2023).

On the experimental side, displaced-decay and intensity-frontier searches have become a new arena. In the singlet-majoron model, a 2026 study finds that realistic seesaw-induced textures often make the lepton-flavor-violating decays

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$00

the dominant production mode at DUNE, NA62, FASER/FASER2, MATHUSLA, and SHiP, extending sensitivity to the intermediate-mass window

$\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$01

(Jodłowski et al., 28 Feb 2026). This does not alter the theoretical hallmark $\sigma=\frac{1}{\sqrt{2}}(f+\sigma^0+iJ),$02; rather, it reinforces the separation between the irreducible neutrino coupling and the loop-induced flavor-changing observables that make the majoron experimentally accessible.

Overall, neutrino–majoron couplings occupy a distinctive position in beyond-the-Standard-Model phenomenology. Their minimal tree-level form is fixed by the neutrino mass mechanism, their charged-sector and gauge-sector manifestations expose otherwise hidden seesaw parameters, and their observable consequences span cosmology, supernova neutrino transport, dark-matter decay, and flavor physics. The subject therefore remains a direct interface between neutrino mass generation and precision probes of weakly coupled new bosons (Heeck, 2017, Giorgi et al., 18 May 2026).