Minimal Multi-Majoron Model (MMMM)

• MMMM is a UV-complete seesaw mechanism where distinct 'personal' Majoron fields generate hierarchical right-handed neutrino masses.
• It employs two Majoron fields, one flavon, and vector-like fermions to achieve realistic flavor mixing and effective Dirac neutrino couplings.
• The model predicts dual gravitational-wave signatures from cosmic strings and strong first-order phase transitions linked to its symmetry breaking scales.

Searching arXiv for the requested topic and closely related Majoron/MMMM papers. The Minimal Multi-Majoron Model (MMMM) is a realistic ultraviolet complete type-I seesaw construction in which hierarchical right-handed-neutrino masses arise from hierarchical vacuum expectation values of several “personal” Majoron fields, while generic lepton-flavour mixing is retained through a flavon-assisted Dirac sector. In its minimal form, the model contains two right-handed neutrinos, two complex scalar Majoron fields $\phi_1$ and $\phi_2$, one complex flavon $\theta$, and two vector-like fermions $\chi_1,\chi_2$, with global symmetry

$G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$

Its distinctive claim is that the same setup simultaneously realizes a flavour-realistic minimal seesaw and predicts a characteristic gravitational-wave spectrum sourced by both global cosmic strings and strong first-order phase transitions (Fu et al., 11 Jul 2025).

1. Conceptual position within Majoron model building

MMMM belongs to the class of Majoron models in which spontaneous breaking of a global lepton-number-like symmetry generates pseudo-Nambu–Goldstone bosons associated with neutrino-mass generation. Earlier minimal singlet-Majoron frameworks typically involved a single broken global $U(1)$ and therefore a single Majoron, whether in models of dark radiation and scalar dark matter, minimal Majoron dark matter, or the minimal massive Majoron seesaw (Chang et al., 2016, Akita et al., 13 May 2026, Giorgi et al., 2023). MMMM departs from that structure by introducing two global Abelian symmetries and two complex scalar Majoron fields, so that the Majoron sector is intrinsically multi-field rather than an extension of a single-Goldstone system (Fu et al., 11 Jul 2025).

The model’s stated purpose is to provide “a natural framework for hierarchical right-handed neutrinos” while preserving a realistic flavour structure. Its central organizing principle is the “personal Majoron” construction: each right-handed neutrino couples only to its own scalar $\phi_i$, and the hierarchy of right-handed-neutrino masses is traced to the hierarchy of the Majoron-sector vacuum expectation values. The price of this separation is that the direct Yukawa term $HL_\alpha N_i^c$ is forbidden by the extra $U(1)_N$, so a flavon field and heavy vector-like fermions are required to generate effective Dirac Yukawas (Fu et al., 11 Jul 2025).

A common misconception is to identify any model with several singlet phases as automatically “multi-Majoron.” In the single-Majoron literature, multiple singlets do not suffice if only one independent global $U(1)$ is spontaneously broken; in that case there is still only one true Goldstone mode (Chang et al., 2016, Giorgi et al., 2023). MMMM is explicitly multi-Majoron because its symmetry structure contains $\phi_2$0, and the spontaneous breaking pattern leaves more than one pseudo-Goldstone degree of freedom in the scalar phase sector (Fu et al., 11 Jul 2025).

2. Symmetry structure and field content

Beyond the Standard Model, the minimal field content consists of two Weyl right-handed neutrinos $\phi_2$1, two complex scalar Majoron fields $\phi_2$2, one complex scalar flavon $\phi_2$3, and two vector-like fermions $\phi_2$4. All Standard Model fields are neutral under $\phi_2$5, while carrying their usual $\phi_2$6 charges. With only two right-handed neutrinos, $\phi_2$7 cannot be gauged without anomalies; the model is therefore formulated with global $\phi_2$8 and global $\phi_2$9 (Fu et al., 11 Jul 2025).

Field	$\theta$0	Role
$\theta$1	$\theta$2	RHN coupled to $\theta$3
$\theta$4	$\theta$5	RHN coupled to $\theta$6
$\theta$7	$\theta$8	Majoron scalar for $\theta$9
$\chi_1,\chi_2$0	$\chi_1,\chi_2$1	Majoron scalar for $\chi_1,\chi_2$2
$\chi_1,\chi_2$3	$\chi_1,\chi_2$4	Flavon enabling effective Dirac Yukawas
$\chi_1,\chi_2$5	$\chi_1,\chi_2$6	Vector-like mediators for Dirac seesaw

The renormalizable interactions of the new sector are

$\chi_1,\chi_2$7

The charge assignment enforces the personal-Majoron structure: $\chi_1,\chi_2$8 and $\chi_1,\chi_2$9 are forbidden, and the usual $G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$0 terms are also forbidden (Fu et al., 11 Jul 2025).

Integrating out the heavy $G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$1 generates the dimension-five operators $G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$2 and $G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$3, with effective Yukawas

$G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$4

This UV completion is essential: the flavon is not an optional embellishment but the mechanism by which a generic $G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$5 Dirac matrix is recovered despite the $G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$6 selection rules (Fu et al., 11 Jul 2025).

3. Neutrino masses and hierarchical right-handed-neutrino scales

After symmetry breaking,

$G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$7

The full type-I seesaw matrix in the $G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$8 basis is

$G_{\rm SM}\times U(1)_{B-L}\times U(1)_N .$9

leading to the standard effective light-neutrino mass matrix

$U(1)$0

Since $U(1)$1 is a generic $U(1)$2 matrix, the low-energy theory below $U(1)$3 is effectively the familiar two-right-handed-neutrino seesaw, for which one light neutrino is massless or extremely light and the observed flavour mixing can be reproduced (Fu et al., 11 Jul 2025).

The hierarchy of heavy-neutrino masses is direct: $U(1)$4 Thus $U(1)$5 implies $U(1)$6. This mechanism is the model’s defining structural feature: the right-handed-neutrino hierarchy is encoded in a hierarchy of Majoron-sector scales rather than inserted as unrelated mass parameters (Fu et al., 11 Jul 2025).

The paper emphasizes that, at scales below $U(1)$7, the construction is effectively the same as the minimal two-right-handed-neutrino type-I seesaw. It also explicitly notes a limitation: it does not present a full numerical oscillation-data fit or benchmark Yukawa textures, but relies on the known generality of the two-right-handed-neutrino seesaw to accommodate the measured neutrino spectrum and mixing pattern (Fu et al., 11 Jul 2025).

4. Scalar potential, spontaneous symmetry breaking, and the Majoron sector

The renormalizable scalar potential is

$U(1)$8

For many purposes the $U(1)$9 subsector

$\phi_i$0

is singled out, because the $\phi_i$1 interaction controls both symmetry breaking and the phase-transition structure (Fu et al., 11 Jul 2025).

The assumed hierarchy is

$\phi_i$2

A single scalar VEV can break only one linear combination of $\phi_i$3 and $\phi_i$4, so it is convenient to define

$\phi_i$5

The symmetry-breaking chain is

$\phi_i$6

This sequence implies global cosmic strings from the first stage and domain walls when the residual $\phi_i$7 symmetries are later broken by $\phi_i$8 (Fu et al., 11 Jul 2025).

Before explicit breaking, two spontaneously broken global $\phi_i$9 factors imply two Goldstone modes. Because three complex scalars carry the relevant charges, there are three phase directions in total. The paper does not diagonalize the full mixed phase sector, but states that the pNGBs left from $HL_\alpha N_i^c$0 and $HL_\alpha N_i^c$1 are recognized as the Majorons, while one additional pNGB is left from $HL_\alpha N_i^c$2; their mass spectrum would be determined by soft symmetry-breaking terms such as $HL_\alpha N_i^c$3 (Fu et al., 11 Jul 2025).

This feature sharply contrasts with single-Majoron models. In the minimal majoron dark matter framework, for example, one complex scalar $HL_\alpha N_i^c$4 breaking one global $HL_\alpha N_i^c$5 produces one Majoron $HL_\alpha N_i^c$6 (Akita et al., 13 May 2026). Likewise, the minimal massive Majoron seesaw employs one complex scalar $HL_\alpha N_i^c$7 and one global $HL_\alpha N_i^c$8, yielding one pseudo-Goldstone mode whose mass is tied to the same explicit-breaking spurion that generates light-neutrino masses (Giorgi et al., 2023). MMMM replaces that single-breaking structure with a genuinely multi-Goldstone phase sector (Fu et al., 11 Jul 2025).

5. Topological defects and gravitational-wave phenomenology

The cosmological signature emphasized by MMMM is the coexistence of a broad gravitational-wave background from global cosmic strings and a peaked contribution from a strong first-order phase transition. For a global string formed at a symmetry-breaking scale $HL_\alpha N_i^c$9, the tension is

$U(1)_N$0

with core radius $U(1)_N$1 and inter-string separation $U(1)_N$2. Loops obey

$U(1)_N$3

where $U(1)_N$4 and

$U(1)_N$5

Because $U(1)_N$6, global-string loops are short-lived and lose energy efficiently through Goldstone emission (Fu et al., 11 Jul 2025).

When $U(1)_N$7 breaks the residual $U(1)_N$8 symmetries, domain walls form. Their surface energy density is parametrically

$U(1)_N$9

The paper argues that although walls bounded by global strings modify the network transiently, the gravitational-wave signal from the hybrid system is essentially that of a pure global-string network, with a possible late-time enhancement of the string-induced background by a factor $U(1)$0 due to string-wall dynamics. This part is presented as a qualitative picture; a quantitative simulation is left for future work (Fu et al., 11 Jul 2025).

The first-order phase transition arises in the lower-scale $U(1)$1 sector and is strengthened by the second Majoron field. After $U(1)$2 acquires a large VEV $U(1)$3, the radial modes mix as

$U(1)$4

with

$U(1)$5

Expanding the mixed quartic term produces an effective cubic interaction,

$U(1)$6

which supplies the barrier that enhances the strength of the $U(1)$7 transition. This mechanism is one of the model’s main novelties: without $U(1)$8, the enhancement would be absent or much weaker (Fu et al., 11 Jul 2025).

The total gravitational-wave spectrum is written as

$U(1)$9

where the three terms arise from bubble collisions, sound waves, and magnetohydrodynamic turbulence. Two benchmark FOPT parameter sets are given: $\phi_2$00 realized for $\phi_2$01, and

$\phi_2$02

realized for $\phi_2$03. In both cases, the string background is evaluated for

$\phi_2$04

The first benchmark places the FOPT peak in the deci-Hz band relevant to LISA, DECIGO, BBO, Taiji, and $\phi_2$05Ares, while the second moves it to higher frequencies potentially accessible to the Einstein Telescope. The resulting total spectrum combines a broad string-induced component extending from nano-Hz to kHz with a narrower FOPT peak whose position is controlled by the lower Majoron scale (Fu et al., 11 Jul 2025).

6. Determination of right-handed-neutrino scales, broader context, and open questions

Because

$\phi_2$06

the two right-handed-neutrino mass scales are tied directly to the two symmetry-breaking scales. The paper’s central phenomenological claim is therefore that the combined gravitational-wave spectrum can, in principle, determine both $\phi_2$07 and $\phi_2$08, and hence constrain both heavy-neutrino masses up to the unknown Yukawas $\phi_2$09. The broad string component probes the higher scale $\phi_2$10, while the FOPT peak probes the lower scale $\phi_2$11 through the finite-temperature potential (Fu et al., 11 Jul 2025).

This separates MMMM from recent single-Majoron cosmology, where the key observables are instead the dark-matter abundance, dark-radiation fraction, or decays of a single pseudo-Nambu–Goldstone boson. Minimal Majoron dark matter models analyze freeze-in and misalignment production for one Majoron and one complex scalar (Akita et al., 13 May 2026). The cosmological-window analysis for the singlet majoron ties dark-matter production, visible decays, and high-scale thermal leptogenesis to one effective decay constant $\phi_2$12 and one majoron mass $\phi_2$13 (Giorgi et al., 18 May 2026). MMMM does not perform an analogous dark-matter or leptogenesis scan; it instead identifies possible connections to Majoron dark matter and Majoron-driven leptogenesis as extensions of the framework (Fu et al., 11 Jul 2025).

The paper also states several limitations explicitly. It does not provide a full oscillation-data fit, does not diagonalize the complete mixed phase sector of $\phi_2$14, and does not present a full analysis of BBN, CMB, $\phi_2$15, laboratory Majoron searches, or astrophysical bounds (Fu et al., 11 Jul 2025). A plausible implication is that any such program would need the one-loop and derivative-coupling machinery developed for general Majoron models, including the model-independent charged-lepton coupling formalism in which each physical Majoron inherits couplings through its projection onto the underlying Goldstone directions (Herrero-Brocal et al., 2023).

Within Majoron model building, MMMM is therefore best understood as a structural and cosmological generalization of minimal singlet constructions. Its minimality is not the minimality of a single extra scalar, but the smallest UV-complete field content the authors propose for a realistic multi-Majoron seesaw: two right-handed neutrinos, two Majoron fields, one flavon, and two vector-like fermions (Fu et al., 11 Jul 2025). Its novelty lies in combining that field content with a controlled flavour mechanism and a two-scale gravitational-wave signature, so that the hierarchy of right-handed-neutrino masses is encoded in, and potentially testable through, the multi-Majoron symmetry-breaking sector itself (Fu et al., 11 Jul 2025).