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.


I. INTRODUCTION
There is no new Higgs doublet presenting in the model. The scalar sector of the model is now calculable and predictive.
In spite of a minimal scalar content, we argue that all the fermions including neutrinos get appropriate masses. Indeed, the Peccei-Quinn like symmetries are automatically realized in this framework which ensure the strong CP conservation. There is no axion since it is already eaten by a new gauge boson. All ordinary quarks get a mass since the Peccei-Quinn symmetries are completely broken or violated. In addition, the model naturally recognizes an approximate B − L symmetry. Hence, neutrino mass generation schemes arise from such approximate B − L and Peccei-Quinn symmetries. All the interactions are obtained and we make necessary bounds on the model under the current experimental data.
The rest of this work is organized as follows. In Sec. II, we propose the flipped 3-3-1 model with minimal scalar content. Secs. III and IV are devoted to determine fermion masses. In Sec.
V, we give a brief comment on Peccei-Quinn symmetry. In Secs. VI and VII, we study the mass spectra of the scalar and gauge bosons, respectively. Here, the gauge interactions of fermions are also investigated. In Sec. VIII we interpret dark matter candidates and obtain their observables.
The collider phenomena are given in Sec. IX. We summarize our results and conclude this work in Sec. X.

II. 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 e aR ∼ (1, 1, −1), E aR ∼ (1, 1, −1), 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, 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.
it is not naturally to induce 2 a large mass for the top quark, since this mass vanishes at tree-level.
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 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 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 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, Like the Yukawa interactions, the potential parametersλ andμ 3 violate B − L by one or two units.

III. 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 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.

IV. 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 Note that there slightly mix between (e a , E a ) and (ξ ± ), which are strongly suppressed by the Substituting the VEVs for scalar fields in (18), 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 (19) 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 The neutrinos obtain masses via a combination of type II and III seesaw mechanisms. Indeed, the type III seesaw comes from the interactions of heavy triplet (ξ + ξ 0 ξ − ) with (ν a e a )(ρ + 1 ρ 0 2 ) and with (ν 1 e 1 )(χ + 1 χ 0 2 ), contained in the Yukawa couplings s 1a and h ξ 11 in L ν , respectively. Whist the type II seesaw arises from the interactions of two lepton doublets with tensor products, (ρ + governed by the Yukawa couplings s 1a , s ′ 1a , and h ξ 11 in L ν . It is clear that such two mechanisms generate appropriate mixing angles between ν 1 and ν 2,3 in order to recover the neutrino data, which did not happen in the original model at renormalizable level [63].
By virtue of the seesaw approximation, we have If B −L is conserved, by contrast, we have s = s ′ = 0 (in this case, v ′ = w ′ = 0 as proved in the next section), thus m ν = 0. Hence, the nonzero masses of neutrinos indeed measure an approximate B − L symmetry, governed solely by the B − L violating couplings as seen in (23). 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 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].

V. 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]. Applying the first condition to the scalar potential in (13) and to the first and second lines of the Yukawa Lagrangian in (12), we obtain the following relations, which are equivalently given by 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 where H ψ 1 is left arbitrary. 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 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 It is noteworthy that the neutrino masses can also come from the violation of the Peccei-Quinn symmetry, since L ν in (19) is generally not invariant under U (1) H for a generic H ψ 1 charge.

VI. 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.
The second term provides conditions of potential minimization, Generally under an approximate B − L symmetry, the violating parameters should be radically smaller than the corresponding conserving ones,μ ≪ µ andλ ≪ λ, since otherwise the B − L conservation demandsμ = 0 =λ. From (44) and (45), we derive where t ≡ w ′ /v ′ is finite and we assumeμ 2 3 ∼λ 6 v 2 ∼λ 7 w 2 . Sinceμ 3 ≪ µ 1,2 ∼ (v, w), we get u ′ , w ′ ≪μ 3 . In practice, takingμ 3 = 1 MeV to 1 GeV, it implies u ′ ∼ w ′ ∼ 0.1 eV to 0.1 MeV, respectively. Such VEVs are responsible for the mentioned neutrino mass generation scheme. Since (42) and (43), we deduce, 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 ± W = ρ ± 1 , G ± X = χ ± 1 . 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 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 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).

VII. 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. 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, 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 III and IV.

VIII. 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 ℜ(H ′ ) = H, while ℑ(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.
2. Note thatλ 6 and w ′ are relaxed from the neutrino mass constraint. We requireλ 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 tō Such tiny values should be strongly suppressed by the approximate B − L symmetry, which is unlike that from the neutrino mass bound.
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, where λ ′ ≡ λ 1 + λ 3 (λ 3 + λ 4 )/[4(λ 4 − λ 2 )]. Comparing to the data, Ωh 2 ≃ 0.1 [69], we obtain 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.

IX. 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, 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 1 -exchange 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, where g m = max{|g Z ′ V (q)|, |g Z ′ A (q)|} = (3 + 2s 2 W )/[6 3 − 4s 2 W ]. 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.

X. 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 of other candidates such as its real part H and inert scalars have been examined.
The new fermions, gauge and Higgs bosons are obtained, having masses at TeV scale. The new physics effects at colliders are discussed, showing the viability of the model.