Neutrino mass and dark matter from an approximate B − L symmetry

We argue that neutrino mass and dark matter can arise from an approximate B − L symmetry. This idea can be realized in a minimal setup of the flipped 3-3-1 model, which discriminates lepton families while keeping universal quark families and uses only two scalar triplets in order for symmetry breaking and mass generation. This proposal contains naturally an approximate non-Abelian B − L symmetry which consequently leads to an approximate matter parity. The approximate symmetries produce small neutrino masses in terms of type II and III seesaws and may make dark matter long lived. Additionally, dark matter candidate is either unified with the Higgs doublet by gauge symmetry or acted as an inert multiplet. The Peccei-Quinn symmetry is discussed. The gauge and scalar sectors are exactly diagonalized. The signals of the new physics at colliders are examined.


Introduction
The standard model is incomplete, since it cannot address neutrino oscillations. Additionally, since the early universe is a quantum system, we need to account for the cosmological challenges of particle physics, such as matter-antimatter asymmetry, dark matter, and cosmic inflation, which all lie beyond the standard model. On the theoretical side, the standard model cannot explain why there are just three fermion families, strong CP conservation, and electric charge quantization.
Among various approaches to the new physics, the model based upon the gauge symmetry SU(3) C ⊗ SU(3) L ⊗ U(1) X (3-3-1) [1][2][3][4][5][6] reveals as one of the strong candidates. Indeed, the number of fermion families matches that of colors [2,4] and the quantization JHEP05(2020)090 JHEP05(2020)090 2 Minimal flipped, approximate B − L and matter parity As mentioned, the gauge symmetry is where the last two factors are a nontrivial extension of the standard model electroweak group, while the first factor is the ordinary color group. Additionally, the electric charge and hypercharge are correspondingly embedded in the 3-3-1 symmetry as where T n (n = 1, 2, . . . , 8) and X are SU(3) L and U(1) X generators, respectively. The T 8 coefficient is fixed to make exotic fermion spectrum phenomenologically viable [50,63].
The key observation is that the [SU(3) L ] 3 anomaly of a sextet equals seven times that of a triplet, A(6) = 7A(3) [50,63]. This leads to a flipped fermion content and solution for family number, in contrast with the usual 3-3-1 approach, such that where a = 1, 2, 3 and α = 2, 3 are family indices. The new fields E, U take the same electric charges as e and u, respectively. This fermion content is free from all the anomalies, including the gravitational one.
The previous studies used one scalar sextet and three scalar triplets [50,63]. In this work, only two of them are needed,

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Multiplet ψ 1L ψ αL Q aL e aR E aR u aR d aR U aR ρ χ N −2/3 −4/3 2/3 −1 −2 1/3 1/3 4/3 −1/3 2/3 Table 1. N -charge of particle multiplets. which would make the model phenomenologically viable. They develop vacuum expectation values (VEVs), such as The 3-3-1 symmetry is broken down to the standard model due to w. The standard model gauge symmetry is broken down to SU(3) C ⊗ U(1) Q due to v. To keep consistency with the standard model, we impose v w. Although ρ and χ have the same gauge charges, they differ in B − L charge, as shown below. Additionally, χ 2 and ρ 3 have nonzero B − L charges which lead to the corresponding VEVs to be small, i.e. v , w v, set by the B − L violating scalar potential. This leads to v , w v w. On the other hand, if one chooses another scalar triplet, i.e. η = (η 0 1 η − 2 η − 3 ) T , instead of ρ, it is not naturally to induce 2 a large mass for the top quark, since this mass vanishes at tree-level.
In this work, to avoid such a 3-3-1-1 extension, i.e. keeping consistency for the 3-3-1 model, we interpret that B − L, thus 3-3-1-1 and matter parity, is an approximate symmetry, by contrast. The approximation is naturally recognized by a minimal scalar content, such that (i) the VEVs associate to odd scalar fields spontaneously break B − L and matter parity and (ii) the Yukawa and scalar interactions explicitly violate B − L and matter parity. In addition to the condition of B − L violating scalar potential that sets JHEP05(2020)090 u , w , the approximate symmetry obviously implies v , w v, since otherwise the B − L conservation ensures u = w = 0 [63].
The total Lagrangian consists of (2.10) The first part contains kinetic terms and gauge interactions, where F and S run over fermion and scalar multiplets, respectively. The covariant derivative and field strength tensors take the forms, Here (g s , g, g X ), (t n , T n , X), and (G n , A n , B) are coupling constants, generators, and gauge bosons according to the 3-3-1 subgroups, respectively, and f nmk define SU(3) structure constants. The Yukawa interactions are given, up to six dimensions, by +H.c., (2.12) where the couplings h conserve B − L while the couplings s (or s) and s violate B − L by one and two units, respectively. Λ is a new physics or cutoff scale that defines (induces) the effective interactions. The Lagrangian might include other six-dimensional operators that directly contribute to the existing renormalizable interactions of the same fermion types, e.g.Q aL χ * χρ * u bR ,ψ αL χ * χρe bR , andψ αL χ * χρE bR . However, concerning mass generation such contributions are radically smaller than the renormalizable ones, which should be neglected. It is clear that the first and second rows includes renormalizable interactions, giving leading tree-level masses for heavy quarks and leptons. The remainders are nonrenormalizable interactions, providing subleading masses for lighter particles. The interactions that violate B − L and/or matter parity must satisfy the condition s, s h, respectively.
The scalar potential takes the form,

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Like the Yukawa interactions, the potential parametersλ andμ 3 violate B − L by one or two units. These violating parameters should be small, i.e.λ λ andμ 3 µ 1,2 , since B − L is approximate and the fact that the B − L conservation impliesλ = 0 =μ 3 .

Quark mass
Quarks gain a mass from the Lagrangian, Substituting the VEVs, we obtain a mass matrix for down quarks, where we use the condition v , w v w and set Λ w without loss of generality. The up quark mass matrix in basis (u U ), where u = (u 1 u 2 u 3 ) and U = ( Using the above condition for VEVs and s u , s U h u , h U , the ordinary quarks u a and exotic quarks U a are decoupled, leading to Hence, the ordinary up and down quarks get appropriate masses proportional to the weak scale v, while the exotic quarks are heavy at the 3-3-1 breaking scale w.

Lepton mass
The Yukawa interactions of leptons can be divided into L Yukawa ⊃ L l + L ν , where the first term provides masses for the ordinary and new charged leptons (e a , E a ), while the second term gives masses for neutrinos and lepton triplet (ν a , ξ), given respectively by

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Note that there slightly mix between (e a , E a ) and (ξ ± ), which are strongly suppressed by the conditions v , w v w and s, s h, as neglected. Substituting the VEVs for scalar fields in (4.1), we get the mass matrix for the charged leptons which yields The ordinary charged leptons have appropriate masses as in the standard model, while the new leptons E a have large masses in w scale, as expected.
For the neutrinos and lepton triplet, we derive their mass Lagrangian from (4.2) as where m ξ ± −h ξ 11 w and M is given by Due to the conditions v v w and s, s h, the new neutral lepton ξ 0 is heavy and decoupled with a mass, m ξ 0 h ξ 11 w. Hence, the components of the lepton triplet ξ gain nearly degenerate masses at w scale, |m ξ 0 | |m ξ ± | ∼ w, set by the first term of L ν when ξ interacts with χ 3 .
By virtue of the seesaw approximation, we have  an approximate B − L symmetry, governed solely by the B − L violating couplings as seen in (4.6). That said, the neutrino mass matrix is generically small and able to fit the data. In addition, the neutrino masses are typically proportional to Taking v = 246 GeV, w ∼ 10 TeV and m ν ∼ 0.1 eV, we derive the B − L violating couplings to be s ∼ 10 −10 and s ∼ 10 −5 √ h ∼ 10 −6 , given that m ξ ∼ 1 TeV. Thus, the violating couplings (s, s ) are smaller than all B − L conserving couplings of charged leptons and quarks, for instance the smallest one h e ∼ 10 −5 associate to electron and h µ ∼ 10 −3 and h τ ∼ 10 −2 for muon and tau, respectively. To conclude, the small neutrino masses are suitably generated by an approximate B − L symmetry, characterized by the strength = (s, s )/h 1, as expected. To make sure of this conclusion, the nature size and stability of approximate symmetry violating strengths responsible for the seesaws and neutrino masses are further determined in the last appendix of this paper.
It is noted that the triplet lepton ξ 0 is W P -odd, playing the main role for neutrino mass generation. However, as shown below, it would fast decay and does not contribute to dark matter, in contrast to [63].

PQ symmetry
We now discuss the existence of Peccei-Quinn symmetry in the minimal flipped 3-3-1 model, which subsequently solve the strong CP problem. Let us recall the reader's attention to the original proposal [67,68] and in kind of the 3-3-1 model [12,13].
In addition to the 3-3-1 group, we introduce a global Abelian group, called U(1) H . In order for U(1) H to play the role of Peccei-Quinn symmetry, we require i) the renormalizable Lagrangian is invariant under U(1) H and ii) the color anomaly [SU H χ , and H ρ corresponding to the matter multiplets, Q aL , U aR , u aR , d aR , ψ 1L , ψ αL , E aR , e aR , χ, and ρ, respectively. Here note that the repeated flavors should have the same H charge due to the invariance of renormalizable Yukawa Lagrangian.
Applying the first condition to the scalar potential in (2.13) and to the first and second lines of the Yukawa Lagrangian in (2.12), we obtain the following relations, The second condition implies provided that H u = H d to simplify the problem. Observe that we have 6 equations with 9 variables (except for H ψ 1 that is arbitrary). Thus, there is an infinite number of different solutions that satisfy the above conditions. Correspondingly, there is an infinite number of Peccei-Quinn symmetries. Note that this property is actually valid for every 3-3-1 model [13]. We list, for instance, Since the VEVs χ, ρ are simultaneously charged under SU(3) L ⊗ U(1) X ⊗ U(1) H , a residual Peccei-Quinn charge that conserves the vacuum must take the form, in which d = 0. The vacuum annihilation conditions P Q χ = 0 and P Q ρ = 0 lead to We deduce after substituting the solution (a, b, c, d) and rescaling the U(1) charge by a where δ = d/a. Obviously the Peccei-Quinn symmetry is present after gauge symmetry breaking. This coincides with the fact that the three down quarks have vanishing masses in the renormalizable theory. The strong CP question is solved and there is no axion. Also, there is no Majoron although B − L is broken. As we will see in the following sections, all the Goldstone bosons are eaten by the corresponding gauge bosons. As shown in the quark mass section, the down quarks can get consistent masses via the effective interaction,Q aL χρd bR , which explicitly violates U(1) H , since −H Q + H χ + H ρ + H d = 3H χ = 0. It is noteworthy that the neutrino masses can also come from the violation of the Peccei-Quinn symmetry, since L ν in (4.2) is generally not invariant under U(1) H for a generic H ψ 1 charge.

JHEP05(2020)090 6 Scalar sector
Let us expand the scalar fields around their VEVs, The scalar potential is correspondingly summed of V = V min +V linear +V mass +V int . The first and last terms define vacuum energy and scalar self-interactions, which will be skipped.

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Of course, necessary conditions for the potential parameters are which are required in order for the potential to be bounded from below and to yield the desirable vacuum structure. The third term in the potential expansion yields scalar mass spectrum. Let us rewrite V mass = V S mass + V A mass , since all charged scalars have vanishing mass to be identified as the Goldstone bosons of W and X gauge fields, i.e. G ± The real parts obey a 4 × 4 mass matrix, which provides a Goldstone boson (G 1 ) and a new Higgs field (H) orthogonal to it, There remains a slight mixing between S 2 and S 3 , which yields physical states where the mixing parameter and masses are given by The imaginary parts mix via a 4 × 4 mass matrix, which leads to three Goldstone bosons and a massive pseudo-scalar, Note that H is light, identical to the standard model Higgs boson. Additionally, we can define H = (H + iA)/ √ 2 since H and A have the same mass, similarly for G 0 Y = (G 1 + iG 2 )/ √ 2 to be the Goldstone boson of Y gauge field. In the effective limit, v w, the Higgs spectrum is Apart from the standard model Higgs boson, there are two new Higgs fields: H 1 is W P -even, responsible for the 3-3-1 breaking, while H is W P -odd, unified with the Higgs doublet by the 3-3-1 symmetry [63]. Even if H is the lightest W P -particle, it may decay JHEP05(2020)090 due to the approximate matter parity. Also due to this approximate nature, the candidate may reveal a lifetime longer than our universe's age responsible for dark matter (see below).
For completion, we supply appendix A to prove that the VEVs as given obey stable minimum. Additionally, we include appendix B to present the nature of the VEV alignments under the gauge symmetry. Last, it is noteworthy that the small values of v , w as well as the hierarchies v , w v can always be maintained against radiative corrections, as shown in appendix C.

Gauge sector
The presence of v , w leads to a small mixing between the charged gauge bosons, as well as between the neutral gauge bosons A 3 , A 8 , B with A 6 -the real part of non-Hermitian gauge boson, However, as proved, v and w are strongly suppressed, hence such mixings can be neglected. This leads to the physical gauge bosons W, X and Y by themselves with corresponding masses, obtained by Here W is identical to the standard model weak boson, implying v = 246 GeV. The neutral gauge boson Y is W P -odd and may be long-lived. However, it does not contribute to dark matter, since it annihilates completely, before freeze-out, into W bosons via gauge self-interactions (cf. [41]). Similarly, the neutral gauge bosons are identified by with the corresponding masses where the sine of the Weinberg angle is defined by s W = √ 3t X / 3 + 4t 2 X with t X = g X /g. JHEP05(2020)090 The photon has zero mass and decoupled as a physical field, while there is a small mixing between Z and Z , determined by m 2 . Therefore, this defines the Z-Z mixing angle, 8) and the physical neutral fields Z 1 = c ϕ Z − s ϕ Z and Z 2 = s ϕ Z + c ϕ Z . Additionally, this shift in Z mass modifies the ρ parameter to be From the global fit ∆ρ < 0.00058 [69], we deduce w > 3.6 TeV. This implies m Z > 1.43 TeV.
Using the above results, we have the interactions of Z and Z with fermions. The vector and axial-vector couplings are listed in tables 3 and 4.

Dark matter
As investigated in [63], if the B − L symmetry was exact, it would be broken down to a matter parity as the residual gauge symmetry, which subsequently stabilized the dark matter candidates, H and ξ 0 .
In the model under consideration B−L, thus resultant matter parity, is an approximate symmetry, implying dark matter instability. However, we expect that the approximate symmetry would suppress H , ξ 0 decays or provide other candidates. Let us evaluate their lifetimes, in order to interpret a possible candidate for dark matter.
The scalar candidate interacts with the standard model Higgs boson via where (H ) = H, while (H ) = A only appears in pair in interactions, including the Yukawa and gauge interactions. Hence, A is stabilized. For the real part, we obtain the decay rate, Γ(H → HH) |2λ 1 w +λ 6 w| 2 /(32πm H ), which leads to a lifetime, provided that w = 10 TeV and λ 1 w ∼λ 6 w.
The fermion candidate interacts with the standard model lepton and Higgs boson via which leads to ξ 0 → ν a H decay, yielding Remarks are in order 1. The neutrino mass generation requires s 1a ∼ 10 −6 . Thus, ξ 0 fast decays, not contributing to dark matter.

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2. Note thatλ 6 and w are relaxed from the neutrino mass constraint. We requirē λ 6 ∼ w /w ∼ 10 −18 , such that H has a lifetime comparable to the universe's age.
In fact, the current bound of dark matter lifetime is a billion times longer than the universe's age [70], which leads toλ 6 ∼ w /w ∼ 10 −23 . Such tiny values should be strongly suppressed by the approximate B − L symmetry, which is unlike that from the neutrino mass bound. An extra analysis supplied in appendix D show that the stability ofλ 6 under radiative corrections may only be kept at the violating strength level for generating neutrino mass, i.e.λ 6 10 −10 , which subsequently rules out the H candidate.
3. The pseudo-scalar A is a natural candidate for dark matter. It is a standard model singlet, hence interacting with ordinary particles only via the Higgs, H and H 1 , portals. In the early universe, they annihilate to the standard model particles via s-channels that set the density, Comparing to the data, Ωh 2 0.1 [69], we obtain m A = λ × 2.7 TeV ∼ O(1) TeV. The dark matter A can scatter off ordinary quarks via t-channel H-exchange that sets its direct detection cross-section, Note that λ 1 is proportional to the standard model Higgs coupling. When A has a correct density, the model predicts σ A−p,n ∼ 10 −45 cm 2 , coinciding with the direct detection experiment [71] for the dark matter mass in TeV regime. [38,39,62], the minimal flipped 3-3-1 model with approximate B − L symmetry can have naturally a room for extra inert scalars that are odd under a Z 2 symmetry. Indeed, the inert scalar multiplets may be a hidden triplet η = (η 0 1 η − 2 η − 3 ) or hidden sextet S = (S ++ 11 S + 12 S + 13 S 0 22 S 0 23 S 0 33 ), which were presented in the complete version [63], but now Z 2 odd. The dark matter candidate is the lightest inert particle, which may be a singlet or doublet or triplet scalar as resided in η, S. The phenomenology of dark matter is analogous to [38,39,62], which will be skipped as out of the scope of this work.

Collider search
The LEPII experiment searches for new neutral gauge bosons Z through the channel e + e − → ff , where f is some fermion, e.g. f = µ. Because the collision energy is much smaller than the Z mass, we can integrate Z out of the Lagrangian and obtain corresponding effective interactions, (ēγ µ P L e)(μγ µ P L µ) + (LR) + (RL) + (RR), (9.1)

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where the last terms differ from the first one by chiral structures. Such chiral couplings were extensively studied [69]. Let us take a typical bound, which translates to m Z > 1.13 TeV. Our previous study for the LHC dilepton and dijet signals in the general flipped 3-3-1 model implied a bound m Z > 2.8 TeV [63]. Such bound can be applied for the considering model, since in the effective limit (w v) the Z couplings with leptons and quarks are identical to the previous version.
The new Higgs H 1 can be produced at LHC by gluon fusion and then decays to diphoton and/or diboson signals. A naive estimation shows that the diboson signals are negligible, while the remaining productions are σ(pp → Z → γγ) ∼ σ(pp → Z → γZ) ∼ λ 2 fb, which are below the current bound ∼ 1 fb [69] for λ 2 below the perturbative limit.
With the dark matter masses (A, H) in TeV regime, they can be created at the LHC due to exchanges of H 1 and Z via couplings as well as those of H 1 , Z to ordinary particles. The mono-X signature is a jet via H 1exchange gg → gAA, qq → gAA, and gq → qAA as well as via Z exchange qq → gHA and gq → qHA. Since the mediators H 1 , Z are heavy, such processes can be induced by effective interactions after integrating them out, Generalizing the result in [72], we obtain the bounds, The first condition leads to m H 1 > 200 |λ 3 + λ 4 |, which is below 1 TeV due to perturbative limit λ 3,4 < 4π, while the second condition implies m Z > 74 GeV. Indeed, since the dark matter and mediators are in TeV regime, the monojet signals are negligible.

JHEP05(2020)090 10 Conclusion
We have shown that the neutrino masses and dark matter can be addressed in a common framework that includes approximate non-Abelian B − L symmetry.
Interpreted in the minimal flipped 3-3-1 model, the neutrino masses automatically come from a combination of type II and III seesaws, while the natural smallness of masses are protected by the approximate symmetry.
The generation of neutrino and quark masses violates the Peccei-Quinn symmetry, while such symmetry preserves strong CP symmetry and has no Axion.
In the present model, dark matter arises from a B −L-charged pseudo-scalar (A) which is unified with the Higgs doublet in a gauge triplet. This candidate has the appropriate density and detection cross-sections. The existence possibility of other candidates such as its real part H and inert scalars have been examined. The

A Stable minimum
For simplicity, as appropriate to the results/conclusions, we assume that the scalar potential and vacuum conserve CP, i.e.μ 2 3 ,λ 5,6,7 , and the VEVs ρ = (0, v, w ) T / √ 2 and χ = (0, v , w) T / √ 2 are all real. For brevity, we define new variables a = ρ † ρ , b = χ † χ , and t = cos( ρ , χ ) -the cosine of the angle between two vectors ρ and χ , hence χ † ρ = ρ † χ = abt. 3 The corresponding potential value takes the form, where we also define µ 2 3 = 2μ 2 3 , λ 5,6,7 = 2λ 5,6,7 , and λ 45 = λ 4 + λ 5 . The extremum conditions read The last equation leads to a solution, ab = 0 or 2λ 45 abt + µ 2 3 + λ 6 a 2 + λ 7 b 2 = 0. However, the first solution ab = 0 implies a = b = 0 with the aid of (A.2), (A.3) and the potential parameter conditions in the body text. This defines a local maximum, i.e. vacuum JHEP05(2020)090 instability, which should be neglected. We consider the second solution, which obeys These equations always give a general solution for a, b as supplied in terms of v, w, v , w from the outset in (2.9). Additionally, since the B − L violating potential parameters are small,λ λ andμ µ, the equation (A.5) reveals t = (vv +ww )/ (v 2 +w 2 )(v 2 +w 2 ) 1, in agreement with the constraints v , w w, v in the body text. Now we show that ρ , χ given above is a stable vacuum. Calculate The deviation of the potential from the extremum value is given by the Taylor's expansion up to second orders, where note that the first-order terms vanish due to the extremum conditions and that the last inequality (A.14) results due to the conditions, t 1,λ λ, and λ 4 > 0 due to m 2 H,A > 0. The conditions for the potential bounded from below, i.e. λ 1,2 > 0, and positive Higgs squared-masses, i.e. |λ 3 | < 2 √ λ 1 λ 2 and λ 4 > 0, lead to which yields a minimum at ρ, χ . This minimum is global, i.e. vacuum stability, since the potential once bounded from below tends to V → +∞ for a, b → +∞.

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Last, but not least, expanding the scalar fields around the VEVs ρ, χ as small fluctuations, the scalar potential takes the form V (ρ, χ) V ( ρ , χ )+V mass , where the scalar self interactions negligibly contribute as perturbative, and V ( ρ , χ ) is the potential value as in (A.1). Since V mass is definitely positive, it follows that V (ρ, χ) > V ( ρ , χ ), confirming again the stable minimum, which is also the true vacuum since V → +∞ for χ, ρ → ∞.

B Gauge rotation
Usually, an SU(3) potential with two triplets φ 1 and φ 2 , the VEVs can be brought to the form T by a gauge rotation. And, the question that arises is how this is avoided here? The answer is simple: all the results and conclusions remain unchanged, since the theory is invariant under such gauge transformation. The difference in physics is only if which vacuum alignment, , is chosen/considered from the beginning, where the basis of representation space has already been fixed by the quark and lepton arrangements.
For details of the above judgement, the relevant gauge transformation is where we use the condition κ v 1 which is equivalent to v w, as expected. Correspondingly, the second triplet VEV is transformed as In the new basis by the gauge transformation, U = e −i(κ/v 1 )λ 7 , all the fields must be shifted correspondingly. For instance, the quark and lepton multiplets are transformed to and ψ 1L = U ψ 1L U T , whereas the right-handed fermion singlets remain unchanged. Since the Lagrangian is invariant under U , it takes the same form after transformation with multiplets primed. Consider, for instance, the Yukawa interactions of quarks,

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where note that χ ≡ φ 1 and ρ ≡ φ 2 . Substituting the VEVs, we obtain which yields quark masses coinciding with those obtained in the body text (3.2) and (3.3). Note that we approximate only up to the first orders in (v , w )/(v, w), so the second order contribution, say v w /vw, disappears in the above down-quark masses. Similarly, we can obtain those masses and mixings for leptons, scalars, and gauge bosons. Even if we work in the new basis, the conditions (5.15) for the P Q operator still retain. Indeed, the P Q operator now becomes P Q = U P QU † , which annihilates the χ and ρ vacua. We deduce, The equations (B.8) and (B.9) lead to (5.15), while (B.7) is a difference of these conditions. To conclude, all the results and conclusions that are obtained are independent of gauge rotation of a definite vacuum alignment. The choice of different vacuum alignments in a given basis would lead to different consequences, respectively. The mixing effects between ordinary and exotic quarks, leptons, and bosons in the current model due to χ, ρ make it distinct from the ordinary version with v , w = 0 where such physical effects disappear.

C Stability of hierarchies
Let us discuss the hierarchies: (i) v w within the vacuum χ = 1 √ 2 (0, v , w) T , even stronger v v w, and (ii) w v within the vacuum ρ = 1 √ 2 (0, v, w ) T , against quantum corrections. Note that these hierarchies have been determined from the physical points of view for the model consistency. (For instance, the u-U mixing may otherwise lead to a violation of CKM unitarity and dangerous tree-level FCNC [42].) Concerning the first hierarchy, note that χ 0 2 = 1 √ 2 v + G 0 Y carries a B − L charge by one unit and is W P -odd. If B − L is exact symmetry, but not spontaneously broken, its VEV, i.e. v , vanishes. Even if B − L is spontaneously broken, but preserving the matter parity W P , then v = 0. Hence, when v = 0, it breaks both B − L and W P . We expect that the approximate B − L (and W P ) symmetry prevents v naturally small, as constrained by the B − L violating parameters in the body text.

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The stability mechanism for such tiny v is that we do not have any physical (massive) Higgs field associate to the v scale, since G 0 Y is already a Goldstone boson eaten by the non-Hermitian Y 0 gauge boson, as seen from the equation (6.15). The quantum corrections to the mass of G Y , thus the shift ∆v , should vanish, up to any order of the perturbative theory, which is the consequence of the gauge symmetry or the Goldstone theorem. The stabilization of v is therefore followed, a direct result of the Goldstone theorem.
Indeed, apart from the vanishing mass term, the quantum contributions to the quartic coupling can be evaluated as [73] V ⊃ λ eff (χ * 2 χ 2 ) 2 + violating couplings, summarizes the tree value λ 2 and the one-loop corrections given in terms of A = 16λ 2 2 +λ 2 3 − 2(h E ) 4 −2(h U ) 4 +3(1+1/4c 4 W )g 4 , which comes from the mediation of scalars, fermions, and gauge bosons that couple to χ 2 , respectively. Here Λ is an arbitrary renormalization scale at which d 4 V /dχ 2 2 dχ * 2 2 = 4λ 2 . Further, the dots represent higher order corrections which we expect they are in higher powers of ln χ 2 2 /Λ 2 . Note that the violating potential slightly contribute to the quartic coupling, as neglected. All the log-terms are very insensitive to χ 2 2 /Λ 2 , approximately preserving a scale-symmetry at the regime of interest. These contributions should be radically smaller than the tree value λ 2 and we require λ eff > 0 in order for the resulting potential bounded from below. The dominant part of (C.1) yields a minimum at v = 0, while the small value of v is lifted by the violating potential.
Concerning the second hierarchy, note that ρ 0 3 = 1 √ 2 w + H carries a B − L charge by minus-one unit and is W P -odd, analogous to χ 0 2 . The B − L and W P symmetries yield w = 0, while the approximate B − L and W P symmetry, i.e. the violating potential, prevents w to be appropriately small. But, how does it work even at quantum levels?
Observe that the tree-level potential of ρ 3 takes the form, where note that λ 1 > 0 and m 2 H λ 4 2 (w 2 + v 2 ) > 0 unlike the standard model scalar potential with a negative squared mass. The quantum corrections to m 2 H and λ 1 should be positive too. The reason for the positive effective coupling of ρ 3 as radiatively induced λ eff = λ 1 + ∆λ 1 > 0 is analogous to that of χ 2 (λ eff ). Whereas, the contributions to the H mass depend on the cutoff scale, which comes from the couplings of ρ 1,2 and χ 1,2,3 to ρ 3 , where the contribution of fermions and gauge bosons is small, as omitted. Since this theory is renormalizable, it demands that the radiatively-induced effective mass m 2 eff = m 2 H + ∆m 2 H fixed at a renormalization scale should be finitely positive.

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Hence, we obtain where m 2 eff > 0 and λ eff > 0. The dominant potential, i.e. the first two terms, does not lead to spontaneous symmetry breaking, giving a minimum at w = 0. The small value of w is lifted, proportional to the violating parameters, but divided by the effective mass and coupling, as explicitly shown in the body text. The stability mechanism for w is just behind the scalar theory with positive squared-mass parameter and large quartic coupling, which ensures a stable minimum at w = 0, in agreement to [73]. The large radiative modifications to w directly translating to the scalar effective mass and coupling term as inversely proportional to w , making this w suitably tiny when including the violating potential.
Last, the violating potential is important to set the size of w , v as partly discussed below.

D.1 Implication from top-down approach
As shown in the body text, for the 3-3-1 model, if B − L is conserved it must be a gauged charge, requiring a 3-3-1-1 extension, since B − L neither commutes nor closes algebraically with SU(3) L . In such case, all the interactions and mass terms that explicitly violate B − L must be absent. In the considering model, the B − L violating couplings existing in the Yukawa and scalar sectors, say s, s ,μ, andλ, measure an approximate B − L symmetry, required in order for the 3-3-1 model to be self-consistent. We expect that a number of such B − L violating couplings are imprinted from the 3-3-1-1 symmetry breaking down to 3-3-1, after integrating out the heavy particles of extended symmetry such as the U(1) N gauge boson, right-handed neutrinos and Higgs scalars [41][42][43][44][45][46][47]. This would provide an appropriate estimation of the violating strength of the approximate symmetry.

D.2 Stability of violating couplings
The stability of violating effective Yukawa couplings is obvious, as can be seen from the realm of renormalization theory as well as the above discussion. Indeed, these effective interactions do not have any tree-level origin at renormalizable level (in other words, their bare coupling constant or corresponding counterterm vanish). Hence, if these effective couplings are radiatively induced by the model particles, they should be finite, but strongly suppressed by a loop factor and mass scale, e.g. 1/16π 2 Λ, multiplied by the contribution of the B − L violating renormalizable Lagrangian (since the B − L conservation suppresses them up to any order of perturbation theory  Figure 1. Dominant one-loop contributions to theλ 6 coupling, where each diagram-type has an additional contribution with parenthesized fields and couplings. neutrino mass constraint, s ∼ 10 −6 and s ∼ 10 −10 , one must take v , w ∼ 1 GeV which are unlikely in the current model. Thus, the radiative corrections by the model particles do not destroy the violating effective couplings implied by approximate B − L symmetry. On the other hand, the running of such couplings up to a high energy scale, the new physics enters, inducing them as suppressed by a seesaw scale -the B − L breaking energywith suitable values as determined above from the top-down approach. To investigate the stability of B − L violating renormalizable Yukawa and scalar self couplings, we consider onlyλ 6 . The other couplings of such kinds can be achieved similarly. The one-loop corrections toλ 6 are given in figure 1. Recall that although the neutrino masses are generated, the low energy theory obeys a good residual symmetry, the matter parity W P , where the violating renormalizable couplings are W P odd. Such couplings vanish up to any order of perturbation theory, if W P is preserved. Hence, the one-loop corrections toλ 6 are necessarily odd under W P asλ 6 is. This can be verified directly from the above diagrams, where s U,u , s E,e andλ 7 that enter are all W P odd. Since the ultraviolet divergences can be absorbed into the bare coupling by renormalization condition d 4 V /dχ † dρdρ † dρ = 2λ 6 , the one-loop coupling strength takes the form, λ eff 6 =λ 6 + 1 16π 2 h U s U † h u h u † F 1 + s u h u † h u h u † F 2 +h E † s E h e † h e F 3 + s e † h e h e † h e F 4 +λ 7 λ 3 F 5 +λ 6 λ 1 F 6 , (D. 6) where F 1,2 , F 3,4 and F 5,6 are finite functions of quark, lepton, and scalar masses divided by renormalization mass, respectively, which need not necessarily be determined. It is clear that the radiative corrections toλ 6 are strongly suppressed by the corresponding violating couplings, if one takes s U ∼ s u ∼ s E ∼ s e ∼λ 7 ∼λ 6 . From this view, we cannot retain some violating coupling, e.g.λ 6 , to be much smaller (i.e. large hierarchical) than the other couplings of this type. However, all of them must be as small as the approximate matter parity is allowed. Again using the top-down approach, if B − L is completely broken, we expect thatλ 6 is similar to that for neutrino mass generation. A detailed study on all hierarchies including the couplings and VEVs is interesting, but it is out of the scope of this work, to be published elsewhere.
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