3-3-1 Model with Right-Handed Neutrinos

• 3-3-1 Model with Right-Handed Neutrinos is a gauge extension based on SU(3)_C × SU(3)_L × U(1)_X that unifies neutrino mass generation, family replication, and extended flavor dynamics.
• It achieves anomaly cancellation by assigning one quark family as a triplet and two as antitriplets, linking the structure of fermion families to the gauge symmetry.
• The framework supports varied neutrino mass mechanisms—including inverse, double, type-I, type-I+II, and radiative seesaws—with significant implications for collider signatures, lepton-flavor violation, and dark matter.

The 3-3-1 model with right-handed neutrinos is a class of gauge extensions based on $SU(3)_C\times SU(3)_L\times U(1)_X$ in which right-handed neutrino degrees of freedom are incorporated into the electroweak structure, typically through lepton multiplets of the form $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$. In its standard realization the electric charge is embedded as $Q=T_3-\frac1{\sqrt3}T_8+X$, one quark family transforms as an $SU(3)_L$ triplet and two as antitriplets, and anomaly cancellation links the construction to the existence of three fermion families. The framework is notable because neutrino mass generation, family replication, extended gauge dynamics, and flavor structure are treated within a single non-minimal gauge sector rather than as disconnected additions to the Standard Model (Pires, 2014, Huitu et al., 2019).

1. Gauge structure and defining matter content

A standard 3-3-1 right-handed-neutrino realization uses the gauge group $SU(3)_C\times SU(3)_L\times U(1)_X$ with

$Q=T_3-\frac1{\sqrt3}T_8+X.$

The characteristic lepton assignment places the would-be right-handed neutrino inside the third component of each $SU(3)_L$ multiplet,

$$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$

while the quark sector is family non-universal: $$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$ This non-universality is not optional in the canonical construction; it is required by anomaly cancellation and is the origin of the well-known connection between 3-3-1 models and the number of fermion families (Pires, 2014, Escalona et al., 20 Oct 2025).

The literature also includes broadened versions in which additional neutral singlets are introduced. In inverse-, double-, and linear-seesaw realizations, one supplements the canonical lepton triplets with gauge-singlet neutral fermions such as $N_{aL}$ or $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$0 in order to generate $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$1 neutral-fermion mass matrices with hierarchical subblocks (Pires, 2014, Cataño et al., 2012). Other variants alter the embedding while retaining right-handed neutrinos as explicit degrees of freedom; for example, models with exotic charged leptons use $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$2 together with $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$3, and some benchmark implementations adopt a different $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$4 embedding convention for the charge operator (Ahmad et al., 2024, Benavides et al., 18 Sep 2025). This suggests that “3-3-1 model with right-handed neutrinos” denotes a family of constructions rather than a single rigid Lagrangian.

A closely related notation uses $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$5 instead of $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$6 and parameterizes the embedding with $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$7. In that convention the same family-non-universal quark structure persists, and the new neutral gauge boson $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$8 remains a defining feature of the low-energy spectrum (Sánchez-Vega et al., 2016, Escalona et al., 20 Oct 2025).

2. Scalar sector, symmetry breaking, and gauge spectrum

The minimal scalar content sufficient for the canonical 3-3-1 breaking chain consists of three $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$9 triplets,

$Q=T_3-\frac1{\sqrt3}T_8+X$0

with neutral vacuum expectation values arranged so that

$Q=T_3-\frac1{\sqrt3}T_8+X$1

In the canonical basis one commonly takes

$Q=T_3-\frac1{\sqrt3}T_8+X$2

with $Q=T_3-\frac1{\sqrt3}T_8+X$3 and $Q=T_3-\frac1{\sqrt3}T_8+X$4 (Pires, 2014, Sánchez-Vega et al., 2016). The Froggatt-Nielsen realization uses a rotated but equivalent triplet set,

$Q=T_3-\frac1{\sqrt3}T_8+X$5

with VEVs $Q=T_3-\frac1{\sqrt3}T_8+X$6, $Q=T_3-\frac1{\sqrt3}T_8+X$7, and $Q=T_3-\frac1{\sqrt3}T_8+X$8 (Huitu et al., 2019).

The extended gauge spectrum contains the Standard Model $Q=T_3-\frac1{\sqrt3}T_8+X$9, $SU(3)_L$0, and photon together with extra charged and neutral bosons conventionally denoted $SU(3)_L$1, $SU(3)_L$2, $SU(3)_L$3, $SU(3)_L$4, and $SU(3)_L$5, depending on basis and convention. Their masses are governed primarily by the large 3-3-1 breaking scale $SU(3)_L$6, $SU(3)_L$7, or $SU(3)_L$8, and therefore lie in the TeV regime in phenomenologically viable realizations (Huitu et al., 2019, Escalona et al., 20 Oct 2025).

Several neutrino-mass implementations enlarge the scalar sector beyond the three triplets. A symmetric sextet

$SU(3)_L$9

is used in low-scale type-I constructions; in that case only the neutral components $SU(3)_C\times SU(3)_L\times U(1)_X$0 and $SU(3)_C\times SU(3)_L\times U(1)_X$1 are arranged to acquire VEVs, while the usual triplet VEVs still perform the gauge breaking (Cabrera et al., 2023). Type-I+II models based on $SU(3)_C\times SU(3)_L\times U(1)_X$2 employ three antisextets $SU(3)_C\times SU(3)_L\times U(1)_X$3 together with additional triplets $SU(3)_C\times SU(3)_L\times U(1)_X$4 and flavor-breaking vacuum alignments (Vien et al., 2016). Flavor models based on $SU(3)_C\times SU(3)_L\times U(1)_X$5, $SU(3)_C\times SU(3)_L\times U(1)_X$6, or $SU(3)_C\times SU(3)_L\times U(1)_X$7 further introduce numerous $SU(3)_C\times SU(3)_L\times U(1)_X$8 singlets whose large VEVs break discrete symmetries and generate hierarchical Yukawa textures (Hernández et al., 2013, Hernández et al., 2016, Hernández et al., 2017).

The scalar sector is therefore not merely a symmetry-breaking device. It determines whether the neutrino sector is type-I, inverse, double, linear, radiative, or mixed type-I+II, and it also controls FCNC alignment, accidental symmetries, and lepton-number violation.

3. Neutrino-mass mechanisms

A distinctive feature of the 3-3-1 right-handed-neutrino framework is that it supports several inequivalent neutrino-mass mechanisms without abandoning the same gauge backbone.

Realization	Characteristic light-neutrino structure	Representative feature
Inverse seesaw	$SU(3)_C\times SU(3)_L\times U(1)_X$9	TeV quasi-Dirac heavy states (Pires, 2014)
Double seesaw / FN331	$Q=T_3-\frac1{\sqrt3}T_8+X.$0	light, medium, and heavy neutral sectors (Huitu et al., 2019)
Type-I with sextet	$Q=T_3-\frac1{\sqrt3}T_8+X.$1	$Q=T_3-\frac1{\sqrt3}T_8+X.$2 from $Q=T_3-\frac1{\sqrt3}T_8+X.$3 (Cabrera et al., 2023)
Type-I + II	$Q=T_3-\frac1{\sqrt3}T_8+X.$4	predictive mixing textures (Vien et al., 2016, Ahmad et al., 2024)
Radiative linear / loop-induced	loop-generated $Q=T_3-\frac1{\sqrt3}T_8+X.$5 block or Majorana masses	TeV vector-like states or leptoquarks (Hernández et al., 2021, Doff et al., 2024)

In the inverse-seesaw version, three gauge-singlet neutral fermions $Q=T_3-\frac1{\sqrt3}T_8+X.$6 are added and the neutral mass matrix takes the form

$Q=T_3-\frac1{\sqrt3}T_8+X.$7

with $Q=T_3-\frac1{\sqrt3}T_8+X.$8. The smallness of the active masses is then controlled by the lepton-number-violating scale $Q=T_3-\frac1{\sqrt3}T_8+X.$9, while the six heavier states form quasi-Dirac pairs around the TeV scale (Pires, 2014). A related construction using only three scalar triplets and neutral singlet fermions shows that double, inverse, and linear seesaw limits can all be realized without scalar sextets or doubly charged Higgs fields (Cataño et al., 2012).

The Froggatt-Nielsen 3-3-1 model with three right-handed neutrino singlets implements a double seesaw in a $SU(3)_L$0 basis $SU(3)_L$1. Its hierarchical structure,

$SU(3)_L$2

produces heavy eigenstates of order $SU(3)_L$3, “medium” eigenstates of order $SU(3)_L$4, and light eigenstates of order $SU(3)_L$5. For $SU(3)_L$6–$SU(3)_L$7, $SU(3)_L$8–$SU(3)_L$9, and $$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$0, the resulting scales are $$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$1–$$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$2, $$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$3–$$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$4, and $$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$5–$$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$6 (Huitu et al., 2019).

In low-scale type-I implementations with a scalar sextet, the same Yukawa matrix controls both the Dirac and Majorana blocks: $$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$7 Because $$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$8 contributes to the charged-gauge-boson mass matrix, the recent $$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3),\qquad e_{aR}\sim(1,1,-1),$$9 discussion leads to the bound $$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$0 for $$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$1–$$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$2, and fixing $$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$3 then forces $$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$4–$$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$5 (Cabrera et al., 2023).

Mixed type-I+II realizations arise when left-handed Majorana masses are generated alongside the usual seesaw term. In the $$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$6 model with neutral leptons,

$$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$7

and both the mass eigenvalues and the PMNS structure are fixed by a small set of parameters (Vien et al., 2016). In the exotic-charge model with a $$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$8 symmetry, the light-neutrino matrix is

$$Q_{3L}\sim(3,3,1/3),\qquad Q_{iL}\sim(3,\bar3,0),\quad i=1,2.$$9

with $N_{aL}$0, $N_{aL}$1, and a small induced $N_{aL}$2 generated by the trilinear scalar coupling (Ahmad et al., 2024).

Radiative options are also available. In the extended linear-seesaw model, the neutral matrix is

$N_{aL}$3

where the $N_{aL}$4 block $N_{aL}$5 is generated at one loop through $N_{aL}$6–$N_{aL}$7–$N_{aL}$8–$N_{aL}$9–$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$00 exchange, leading to

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$01

for $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$02 (Hernández et al., 2021). A different loop realization introduces a minimal set of scalar leptoquarks, with active Majorana masses generated at one loop and sterile Majorana masses at two loops, yielding sub-eV active masses and eV-scale sterile masses from the same leptoquark Yukawa sector (Doff et al., 2024).

4. Flavor structures and predictive realizations

The 3-3-1 right-handed-neutrino framework is frequently combined with flavor symmetries in order to address the observed hierarchies of quark and lepton masses and the large size of leptonic mixing.

In the Froggatt-Nielsen realization, a global $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$03 is introduced with the gauge-singlet combination $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$04 playing the role of the flavon,

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$05

All Standard-Model-like Yukawa couplings arise from higher-dimensional operators, so charged-fermion hierarchies and the neutrino sector are generated by the same suppression mechanism. With equal left-handed lepton FN charges, both $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$06 and $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$07 are anarchical, and

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$08

naturally yields $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$09 (Huitu et al., 2019).

An $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$10 construction realizes an inverse seesaw with a $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$11 neutral-fermion matrix and fits the observed fermion masses and mixings. Its representative predictions are

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$12

together with $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$13, $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$14 in the normal hierarchy and $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$15, $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$16 in the inverted hierarchy (Hernández et al., 2013).

The $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$17 model with neutral leptons combines type-I and type-II seesaw contributions and obtains

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$18

with a fitted Dirac phase $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$19 and

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$20

Its Majorana phases vanish in the fitted solution (Vien et al., 2016).

A second $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$21 model implements a double seesaw and uses 14 effective free parameters to fit 18 observables. In that construction only the inverted hierarchy yields a good fit, with

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$22

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$23

and

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$24

The normal hierarchy is disfavored by the oscillation data in that setup (Hernández et al., 2016).

The $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$25 realization with a radiative inverse seesaw likewise selects only the inverted hierarchy and predicts

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$26

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$27

while an unbroken $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$28 stabilizes a scalar dark-matter candidate (Hernández et al., 2017).

These constructions show that the 3-3-1 gauge structure does not fix the flavor sector by itself. Instead, it provides a scaffold in which discrete symmetries, high-scale singlet VEVs, and texture-enforcing charge assignments can be overlaid to obtain predictive PMNS patterns and neutrinoless-double-beta targets.

5. Phenomenology: heavy neutrinos, LFV, dark matter, and collider signatures

The phenomenology of 3-3-1 models with right-handed neutrinos is driven by the coexistence of extended gauge bosons, non-minimal neutral-fermion sectors, and, in many realizations, extra charged or doubly charged scalars.

A particularly studied collider channel is resonant heavy-neutrino production through the extra neutral gauge boson. In a TeV-scale type-I seesaw realization with a sextet, one has

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$29

with heavy-neutrino masses such as $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$30 and active-sterile mixings $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$31 in a representative benchmark (Cabrera et al., 2023). A dedicated collider study of resonant $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$32 production finds

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$33

per generation, same-sign dilepton plus jets cross sections of order $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$34 for $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$35 and $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$36, and a $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$37 reach up to $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$38 with right-handed-neutrino masses up to about $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$39 at $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$40 and $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$41 (Zhang et al., 2018).

Lepton-flavor violation is typically suppressed, but mixed type-I+II realizations generate calculable one-loop amplitudes. In the $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$42-protected exotic-charge model the numerical outputs are

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$43

and

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$44

all below current bounds (Ahmad et al., 2024).

Dark-matter candidates appear in several distinct ways. In the low-scale type-I seesaw, if the lightest right-handed neutrino lies in the MeV-keV range, viability requires it to be effectively stable with $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$45, making it a natural warm-dark-matter candidate; in that case the lightest active neutrino must satisfy

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$46

for $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$47–keV (Cabrera et al., 2023). In the supersymmetric 3-3-1 model, the right-handed sneutrino emerges as a thermal cold-dark-matter candidate, with relic-density and direct-detection constraints implying

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$48

(Pires et al., 2016). In the $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$49 radiative inverse-seesaw model, the inert scalar $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$50 is the lightest $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$51-odd state and yields $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$52 for a WIMP mass in the $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$53–$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$54 interval and quartic coupling $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$55–$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$56 (Hernández et al., 2017).

Loop-induced flavor observables extend beyond the lepton sector. In the leptoquark-augmented model, the same couplings that generate active and sterile neutrino masses induce $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$57-physics effects and rare Higgs decays. For $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$58 one finds the constraint

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$59

from $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$60 and a representative prediction

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$61

below the quoted ATLAS bound $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$62 (Doff et al., 2024).

6. Consistency conditions, accidental symmetries, and viable parameter space

The same non-minimal structure that makes the 3-3-1 right-handed-neutrino framework versatile also creates stringent consistency conditions.

A central issue is the appearance of accidental global symmetries in the simplest three-triplet scalar sector. In the conventional $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$63-symmetric model with $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$64, $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$65, and $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$66, spontaneous symmetry breaking leaves a physical Nambu-Goldstone boson associated with an accidental Peccei-Quinn-type symmetry. Its couplings to electrons and nucleons are strong enough to violate astrophysical bounds, so the $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$67-symmetric model is excluded for all the studied symmetry-breaking patterns. The analysis finds that adding only the soft cubic term

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$68

removes the problematic massless boson while preserving a simple scalar sector (Sánchez-Vega et al., 2016).

Flavor-changing neutral currents are intrinsic because one quark family transforms differently under $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$69. Tree-level FCNC are mediated not only by $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$70 but also by the neutral scalars $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$71, $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$72, and $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$73. A recent synthesis of the subject emphasizes the role of the alignment limit

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$74

which removes the SM-like Higgs contribution to tree-level FCNC. Once scalar alignment and the quark-diagonalization pattern are specified, the allowed $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$75 scale varies widely: from a few hundred GeV when FCNC are effectively confined to the up sector, to $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$76 when down-sector constraints dominate (Escalona et al., 20 Oct 2025). In a benchmark implementation with three triplets and a sextet, global fits to weak charges, CKM unitarity, neutrino trident production, and $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$77 mixing exclude $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$78 except for finely tuned mixing angles, while direct dilepton and dijet searches push the preferred region to $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$79–$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$80 (Benavides et al., 18 Sep 2025).

Electroweak precision observables can constrain the seesaw-breaking sector directly. In the low-scale type-I model the sextet VEV $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$81 enters the charged-gauge-boson mass matrix and shifts the $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$82 mass by

$L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$83

which is why the mechanism is forced to operate with $L_a=(\nu_{aL},e_{aL},(\nu_{aR})^c)^T\sim(1,3,-1/3)$84 at the tens-of-GeV scale rather than at a conventional high scale (Cabrera et al., 2023). This is a structural difference from ordinary high-scale seesaws: the neutrino sector is not decoupled from gauge-boson precision physics.

Taken together, these results indicate that the viable 3-3-1 right-handed-neutrino parameter space is controlled by four intertwined requirements: anomaly cancellation and family assignment, consistent scalar symmetry breaking without accidental massless bosons, neutrino-mass generation compatible with oscillation data, and FCNC suppression through either large mediator masses or scalar/quark alignment. Within those conditions, the framework remains broad enough to support inverse, double, type-I, type-I+II, linear, and radiative neutrino-mass realizations, as well as collider, LFV, and dark-matter phenomenology that is qualitatively different from that of more minimal neutrino extensions.