Scotogenic dark matter stability from gauged matter parity

We explore the idea that dark matter stability results from the presence of a matter-parity symmetry, arising naturally as a consequence of the spontaneous breaking of an extended $\mathrm{SU(3) \otimes SU(3)_L \otimes U(1)_X \otimes U(1)_{N}}$ electroweak gauge symmetry with fully gauged B-L. Using this framework we construct a theory for scotogenic dark matter and analyze its main features.


I. INTRODUCTION
Unveiling the nature of dark matter constitutes a big challenge in astroparticle physics, requiring the existence of new particles and also suggesting the presence of new symmetries capable of stabilizing the corresponding candidate particle on cosmological scales. A popular class of dark matter candidates in agreement with astrophysical and cosmological observations are the so-called Weakly Interacting Massive Particles, or WIMPs. For example, they are realized within supersymmetric extensions of the standard model [1]. In that case stability follows from a postulated Z 2 symmetry called R-parity which also avoids fast proton decay and neutrino masses.
WIMPS however, arise in many other ways including "low-scale" models of neutrino mass generation [2], such as scotogenic dark matter [3] scenarios in which the exchange of "dark sector particles" is responsible for the radiative origin of neutrino mass. In such attractive scenarios WIMP dark matter emerges as radiative neutrino mass messenger [4,5]. In Refs. [6,7] it was suggested that an extended gauge symmetry can provide a natural setting for a theory of cosmological dark matter. The associated electroweak extensions both involve the SU(3) L symmetry which has a long history. It is well-motivated due to its ability to "explain" the number of families to match that of colors, as a result of the anomaly cancellation requirement [8][9][10]. For recent papers see Refs. [11][12][13][14]. These theories can also be made consistent with unification [15] and/or with the understanding of parity as a spontaneously broken symmetry [16]. The two different models in [6,7] employ an extended electroweak gauge symmetry and the dark matter stability results from the presence of a matter-parity symmetry, M P , a non-supersymmetric version of R-parity, that arises naturally as a consequence of the spontaneous breaking of the extended gauge symmetry.
The purpose of this letter is to go a step further along this idea. For definiteness we set out to explore the SU(3) ⊗ SU(3) L ⊗ U(1) X ⊗ U(1) N model proposed in [6] as a possible template for a theory of scotogenic dark matter. To do so we consider an extension of the original model containing extra vector-like fermions as well as scalars. These naturally contain the new messenger dark sector particles required to implement the scotogenic scenario. In Sect. II we setup the stage for the theory, discussing the important issue of anomaly cancellation (details presented in appendix A). In Secs III and IV we study the loop-induced neutrino masses as well as the scalar boson and fermion mass spectra. In Sect V we briefly discuss the dark matter phenomenology, and conclude in the last section.

II. MODEL
We consider a variant of the model introduced in [6], based on the SU(3) ⊗ SU(3) L ⊗ U(1) X ⊗ U(1) N gauge symmetry (3-3-1-1 for short). This is an abelian extension of the class of models based on the SU(3) ⊗ SU(3) L ⊗ U(1) gauge symmetry and as such, it inherits many of the defining features of these models. The main motivation for the inclusion of the extra U(1) N symmetry is to allow for a fully gauged B − L symmetry within a 3-3-1 framework [17,18]. In the present model, electric charge and B − L are embedded into the gauge symmetry as with T i (i = 1, 2, 3, ..., 8), X and N as the respective generators of SU(3) L , U(1) X and U(1) N . Under suitable conditions, the spontaneous symmetry breaking (SSB) pattern is such that a residual discrete symmetry arises from the B − L symmetry breakdown. The role of the remnant symmetry is analogous to that of R-parity in supersymmetric theories, we call it matter parity, M P = (−1) 3(B−L)+2s . It follows that the stability of the lightest M P -odd particle leads to a potentially viable WIMP dark matter candidate. For recent related papers see Ref. [19][20][21].
Here we show how the natural M P symmetry described by the 3-3-1-1 models can be responsible for both the neutrino mass generation as well as for the stability of dark matter within a scotogenic scenario, without the need to impose any additional symmetry by hand [22].
The particle content of the model is shown in Table I. Anomaly cancellation requires that, if left-handed leptons l aL ; a = 1, 2, 3 transform as triplets under SU (3) L , i.e.
then two generations of quarks q iL ; i = 1, 2 must transform as anti-triplets and one as a triplet [8] This choice "explains" the number of generations as three (the same as the number of colors), an interesting feature of this class of models. The quark sector interactions are the same as in the original model [8].
The new ingredients of the model, with respect to [6], are the vector-like fermion triplets F aL,R 1 , and the extended scalar sector spanned by S, σ and Ω. These fields will be responsible for the neutrino mass generation mechanism 1 These are called vector-like insofar as their gauge charges are concerned, required for anomaly cancellation. However, as we will see later, they have Majorana as well as Dirac-type mass terms. I. 3311 model particle content (a = 1, 2, 3 and i = 1, 2 represent generation indices). Note the non-standard charges of "right handed neutrinos" νR.
described in the next section. The Yukawa terms involving leptons and vector-like fermions are thus given by where i, j = 1, 2, 3 are SU (3) L indices.
Notice the unconventional chiral charges of ν R fields, owing to which the tree level coupling between l aL and ν R is automatically avoided. Such chiral solutions were already known in context of B − L symmetry [23][24][25][26]. Here we show for the first time that they can also be embedded inside bigger gauge groups containing B − L symmetry. In appendix A we display explicitly the non-trivial way in which the anomalies involving the U(1) gauge symmetries cancel, despite the unconventional ν R charge assignments.
After the singlet scalar φ develops a vacuum expectation value (VEV), the gauged B −L symmetry is spontaneously broken by two units, leaving a discrete remnant symmetry M P = (−1) 3(B−L)+2s . The most general VEV alignment for the scalar triplets and φ, which is compatible with the preservation of M P symmetry, is given by Furthermore, if the VEV alignment for the scalars S, σ and Ω is then, M P is an exactly conserved symmetry.
Assuming w, Λ, w 2 , v s v 1 , v 2 , w 1 the spontaneous symmetry breaking (SSB) pattern of the model is given by and the phenomenology for quarks, charged leptons and gauge bosons of the model coincides largely with the analysis performed in [17].

III. NEUTRINO MASSES
First we notice that, thanks to the charges of the scalars in the model as well as the unusual assignments of ν R charges, tree level neutrino masses are absent in this model. These include a tree-level Dirac-like mass term coupling the electrically neutral isodoublet and isosinglet members of the lepton triplets [27]. Likewise, the absence of genuine right-handed neutrino fields implies no tree-level seesaw-type neutrino mass contributions, such as the type-I Majorana seesaw used in Ref. [6] or the type-II Dirac seesaw proposed in Ref. [28]. Matter-parity conservation also forbids seesaw-type neutrino Majorana masses mediated by the vector-like fermions.
As a result small neutrino masses are generated only at the one-loop level, mediated by the vector-like fermions F L,R , the singlet scalars S, σ and the scalar sextet Ω. The relevant interactions among these fields are where i, j represent SU (3) L indices. The resulting neutrino radiative mass is given as where the loop fuction Z(x) is defined as and mixing angles of (η3, σ) R,I and the odd component of F 0 L , F c0 R are obtained from Eqs. 29 and 34, respectively.
Notice that the fields νR remain massless after spontaneous symmetry breaking, and do not play a direct role in neutrino mass generation. They can contribute as extra degrees of freedom in primordial Big Bang nucleosyhthesis. However, this is not an issue since consistency with cosmological observations in such a case can be ensured by having the extra gauge bosons adequately heavy [29]. Alternatively, these extra fermions could be made massive trivially through the inclusion of extra scalar singlets with appropriate U(1)N charges.

IV. MASS SPECTRUM
The full scalar potential of the model is written as where the Vo piece consists of the following terms, The conditions for the minimization of the scalar potential are given as follows: Notice that, due to the assumed positivity of its squared mass, the field σ has zero vacuum expectation value, as required for the conservation of the matter parity symmetry.

A. Scalar masses
The physical scalars include the following particles, classified according to their electric charges and matter parities: • Q = ±2, MP = +: consists of only Ω ±2 22 complex scalar, its mass is given by • Q = ±1, MP = +: consists of two complex physical scalar eigenstates and one complex Nambu-Goldstone (NG) bo- ). Corresponding mass squared matrix is given by in the basis (η c 2 , ρ1, Ω12).
• Q = ±1, MP = −: consists of two complex physical scalar eigenstates and one complex Nambu-Goldstone (NG) boson ). Corresponding mass squared matrix is given by in the basis (ρ3, χ c 2 , Ω23). In order to calculate the neutrino masses we use the simplification w, Λ, w2, vs v1, v2, w1, then the mass squared matrices in Eqs. B3 and B4 become block diagonalized 2-by-2 matrices and are given by and 1 2 respectively. The part relevant for neutrino masses is the first 2-by-2 block of mass squared matrices 24 and 25 in the basis (η3R,I , σR,I ). The eigenvalues are given by and mixing is given by

B. Fermion masses
First note that the fermions NL,α = l3,α and ν c α,R do not mix with others at the tree level. The fermions which are relevant for neutrino mass generation are the MP odd components of (F 0 L , F c,0 R ), which lie in the sector with (Q = 0, MP = −). The corresponding mass matrix is given by in the basis of (F 0 L , F c,0 R ) of the MP -odd components. Corresponding eigenvalues and mass eigenstates are given by and θ f is given by

V. DARK MATTER PHENOMENOLOGY
Before concluding, let us briefly discuss the phenomenology of dark matter in our model. As can be seen from the discussion of previous sections, in our model dark matter is the mediator of neutrino mass generation. First notice that the stability of dark matter follows from the matter parity symmetry MP , which is a residual symmetry of the full SU(3) ⊗ SU(3)L ⊗ U(1)X ⊗ U(1)N gauge symmetry. All the particles odd under matter parity MP belong to the "dark sector", with the lightest amongst them being the dark matter candidate. As can be seen from Table I our model can have both fermionic or scalar dark matter, depending on which one is the lightest 2 . As an illustrative example, here we briefly discuss the phenomenological constraints for the case of scalar dark matter.
The direct detection and relic abundance constraints on the dark matter mass mσ R and its coupling λ σh to the Higgs boson. The red shaded region is ruled out by direct detection experiments, XENON1T [31] and LUX [32], while the blue shaded region is not compatible with the dark matter relic abundance [33]. The combination of relic density and direct detection constraints implies that, apart from a tiny region near half Higgs mass, the mass of dark matter must lie in the TeV range. The plot is obtained for a specific benchmark, see text for details.
As can be seen from Table I and Figure 1, the MP odd scalar σ takes part in the neutrino mass generation loop. Assuming that it is the dark matter particle we now analyze the associated phenomenology. For simplicity we assume that the mixing between σ and the other MP odd scalars is negligible. In order for σ to be dark matter it should also be the lightest particle amongst all dark sector particles. Due to its U (1)N charge, σ must be a complex field with real (σR) and imaginary (σI ) components. Owing to the µ2 coupling of Eq. 9, the two masses cannot be exactly degenerate once the S field get a vev. The vev of S field breaks U (1)B−L → Z2 subgroup, the symmetry protecting the small neutrino mass, see Fig. 1. In the limit of S → 0 the symmetry of the theory gets enhanced. The presence of µ2 term coupled with the S implies that σR and σI components cannot be degenerate in mass and the lighter of the two will be the dark matter candidate. In our analysis leading to Figure 2 we have assumed that σR is the lighter of the two components and is the dark matter.
Notice that in this model the new gauge bosons can lead to signatures at the LHC, as well as novel flavor violating effects in the neutral meson systems such as K-, D-and B-mesons. Current data already imply stringent limits. A recent phenomenological study [34] indicates that the masses of the new particles present in 3-3-1-1 models are expected to be high, so we take them heavy enough ( > ∼ O(10) TeV) so that they decouple at the electroweak scale. In this case, apart from standard model particles, the only particles that are not decoupled at the electroweak scale are the two components of σ which can be light, as they are SU(3) ⊗ SU(3)L ⊗ U(1)X singlets. Under these simplifying approximations, the relic density of our dark matter candidate σR is mainly controlled by its quartic coupling λ σh with the standard model Higgs boson. Furthermore, its "effective interaction strength" with the nuclei determining the dark matter-nucleus interaction cross section is also directly proportional to λ σh .
This simplified scenario has only one parameter, i.e. the λ σh coupling, responsible for both the relic abundance as well as the direct detection cross section. As a result, the constraints are rather tight, as can be seen from Figure 2. This Figure indicates the restrictions on the λ σh − mσ R plane obtained by requiring the correct dark matter relic abundance, as well as by imposing the dark matter direct detection constraints. The mass of the dark matter σR is confined to two distinct allowed regions. The first allowed region is near half the Higgs mass, where resonant annihilation of dark matter to the Higgs boson allows the relic density constraints to be satisfied for very small values of the coupling λ σh , well below the current direct detection bounds.
The second allowed region of mσ R starts at around 1 TeV, where the direct detection constraints on the coupling λ σh are weak.
Thus, even within this constrained scenario, the field σR can be a good dark matter candidate, provided its mass lies in one of these two allowed regions.
Before ending this section we wish to remark that Figure 2 is plotted for a very constrained scenario with all but one coupling of the dark matter field set to zero. This need not be the case. In the presence of other couplings, particularly the quartic coupling between σR and the other scalars, several additional channels for dark matter annihilation will open up. Thus the relic density constraints on the quartic coupling λ σh can be substantially weakened, opening up the allowed parameter space for λ σh and mσ R . Thus, Figure 2 should be taken as a kind of "worst case scenario" to illustrate consistency. Finally, as we have stated before, σR need not be the lightest MP -odd particle in our model. A complete phenemenological study of all possible dark matter candidates is not the main aim of our paper and hence we will not explore in detail other possibilities.

VI. DISCUSSION
Many general phenomenological features involving the weak SU(3) gauge group, such as present in SU(3) ⊗ SU(3)L ⊗ U (1) schemes, are common also to our model. These theories imply the existence of new Z gauge bosons that can be produced in proton-proton collisions through the Drell-Yan mechanism, leading to dilepton events at the LHC. In addition, and more distinctively when compared to other electroweak extensions, the anomaly cancellation solution based on having one of the quark families transforming differently from the others implies the existence of flavor changing neutral currents at the tree level [8]. As a result one can have effects in the neutral meson systems such as K-, D-and B-mesons. Current LHC, Belle and BaBar data already imply stringent limits, discussed in [34]. Several other phenomenological aspects of this SU(3) ⊗ SU(3)L ⊗ U(1)X ⊗ U(1)N models were already discussed in Refs. [6] and [17].
The main motivation of the current paper was the issue of dark matter. We note that in the present model all standard model fields have MP = 1, thus the lightest MP = −1 is automatically stable and constitutes a potential WIMP dark matter candidate particle. Among the electrically neutral fields with MP = −1, we have σ and the lighter of the F 's, i.e. (FaL,R)1.
Whichever is the lightest of these, can be a potential dark matter candidate. We discussed explicitly a benchmark for the scalar dark matter (σR) case.
As we noted, this model is characterized by the existence of extra fermions and scalar bosons needed for implementing the scotogenic scenario as well as for breaking the extended gauge symmetry. As a result one expects a plethora of possible collider signatures associated to the extra particles. Clearly, dedicated studies, similar to that in [35], would be required in order to scrutinize the associated detection potential at current and upcoming collider experiments, such as future runs of the LHC as well as future linear Colliders.