The $S3$ Symmetric Model with a Dark Scalar

We study the $S3$ symmetric extension of the Standard Model in which all the irreducible representations of the permutation group are occupied by $SU(2)$ scalar doublets, one of which is taken as inert and can lead to dark matter candidates. We perform a scan over parameter space probing points against physical constraints ranging from unitarity tests to experimental Higgs searches limits. We find that the latter constraints severely restrict the parameter space of the model. For acceptable points we compute the value of the relic density of the dark scalar candidates and find that it has a region for low dark matter masses which complies with the Higgs searches bounds and lies within the experimental Planck limit. For masses $\gtrsim 80$ GeV the value of the relic density is below the Planck bound, and it reaches values close to it for very heavy masses $\sim 5$ TeV. In this heavy mass region, this opens the interesting possibility of extending the dark sector of the model with additional particles.


Introduction
One of the main challenges in particle physics at present is to find the nature of dark matter (DM). It is generally accepted that it should be neutral, cold and weakly interacting (although other possibilities exist), and there are proposals for scalar, fermion and vector particles that satisfy the criteria [1]. Among the scalar candidates a very interesting proposal is to introduce inert Higgs scalars, i.e. that do not couple to matter, which is usually achieved by introducing an extra Z 2 discrete symmetry, and that do not acquire a vacuum expectation value (vev), thus guaranteeing the stability of the DM candidate [2][3][4][5].
Among the numerous proposals to extend the scalar sector of the standard model, the 3-Higgs Doublet Model with an S 3 -family symmetry (S3-3H) presents interesting phenomenology, such as the prediction of a non zero reactor neutrino mixing angle θ 13 and of a CKM matrix in accordance with the experimental results [6]. The 3HDM with S3 symmetry has been extensively studied in different contexts (see for instance, [6][7][8][9][10][11][12][13], and references therein). The aim of this project is to extend this model to a 4HDM in order to have dark matter candidates, without spoiling the good features of the S3-3H model in the quark and lepton sectors. To to do this we occupied all irreducible representations of the S 3 symmetry: one symmetric singlet, one antisymmetric singlet and one doublet. The S3-3H model is constituted by the singlet symmetric and doublet representations, with all these Higgs scalars acquiring vacuum expectation values. The fourth Higgs doublet is assigned to the antisymmetric singlet representation and assumed to be inert, it does not acquire a vev and we impose a Z 2 symmetry to ensure the stability of the potential dark matter candidates.
In this letter we present an analysis of the parameter space of the model focusing in the determination of the relic density of the dark scalar for physically acceptable points by requiring them to satisfy numerous physical constraints. In the study presented here we limit the calculations to include only tree level quantities, for example the values of the quartic couplings are approximated by on shell values. While higher order corrections can be of sizeable importance for non-supersymmetric models (see e.g. [14,15]), an analysis including full loop corrections is outside the scope of this work, and we leave it for future research.

The model
In the S3 symmetric model the scalar sector of the SM is extended with additional SU (2) scalar doublets Φ k with definite transformation properties with respect to the permutation symmetry. Whilst the matter sector content of the model remains the same as the SM, the Yukawa lagrangian is required to be invariant also with respect to S3. This, together with the mixing of the scalars after electroweak symmetry breaking, leads to Yukawa terms that can be very different from the SM, for example the proportionality of the fermion masses to single Yukawa couplings is in general lost in the extended model. Since our primary focus in this letter will be centered in the properties of the dark scalar, in particular a probe for the values of the relic density of this particle in the model parameter space, we'll make simplifying assumptions over the Yukawa lagrangian's explicit form which we shall argue not to have a strong impact in the conclusions driven from our results.

The scalar sector
We accommodate four SU (2) doublets into the irreducible representations of the permutation group S3, denoting the symmetric and antisymmetric scalars by Φ s and Φ a respectively, while the remaining two doublet scalars Φ 1 and Φ 2 are arranged in a column vector conforming the S3 doublet. The (invariant and renormalizable) scalar potential is a mixture of the potentials known from the studies of the three Higgs model with the permutation symmetry (e.g. [13]), and can be written as: where V 2 comprises the quadratic terms: V 4 contains quartic terms involving Φ 1 and Φ 2 only: while V 4s and V 4a represent the quartic terms involving Φ s and Φ a respectively: The expression for V 4a is very similar to eq. (4) with Φ a replacing Φ s and quartic couplings λ 9 , . . . , λ 13 , except that the λ 9 term analogous to the λ 4 term has Φ 1 and Φ 2 interchanged. Finally the mixed Φ s , Φ a term is given by: In the following we will assume no additional sources of CP violation other than the present in the SM and hence we shall take the quartic couplings λ i , i = 1 . . . 14 to be real and also restrict their absolute values with the usual perturbativity condition |λ i | < 4π. In order to force the scalar Φ a to be inert we introduce an additional discrete Z 2 symmetry with respect to which all fields are even except those with subindex a, taken as Z 2 -odd. This gets rid of the λ 9 and λ 15 term in the potential leaving only terms with even powers of Φ a . Incidentally this also leads to the appearance of an additional symmetry of the potential under the interchange Φ 1 → −Φ 1 , this fact explains the absence of vertices with odd number of fields with subindex 1 in the Feynman rules.
After electroweak symmetry breaking all the scalar doublets acquire a vacuum expectation value (vev) denoted by v s , v 1 , v 2 and v a respectively. However, in order to avoid the explicit breaking of the Z 2 symmetry we fix v a = 0 and henceforth from the minimization conditions for the scalar potential (tadpole equations) the fourth equation ∂V /∂v a = 0 is automatically satisfied. The three minimization equations (∂V /∂v i = 0, i = s, 1, 2) reduce to those of the three Higgs doublet model with S3 symmetry (e.g. [11]), whose self consistency compels us to consider only the case of aligned vevs v 1 = √ 3v 2 . Parametrizing the Higgs doublets as and similarly for Φ 1 , Φ 2 and Φ a (here the indexes n and p refer to neutral (scalar) and (neutral) pseudoscalar and we use primes to distinguish from the mass eigenstates denoted by the same letters), it is straightforward to obtain the mass and mixing matrices and mass eigenstates for the scalar fields. The mass matrices are block diagonal such that the primed fields, e.g. h n s , h n 1 and h n 2 , mix into the mass eigenstates h n s , h n 1 and h n 2 while the Z 2 odd field remains unmixed h n a = h n a . A detailed description of the above procedure with explicit expressions for the masses of the physical scalars and other details can be found in [16], here we just want to present the rotation matrix for the neutral scalars, so as to aid ourselves with an argument in the next section. The relation between the physical fields h j (j = s, 1, 2) and those appearing in (7) is through the expression: where the rotation matrix is given by: here A, B and C are complicated functions of the parameters of the model 1 .
The important thing to notice is that due to the zero value of the first element of the diagonal the field h s is a linear combination of h 1 and h 2 only, this fact will be used in the next section. Following the procedure outlined in [13], the stability constraints for this model can be obtained (see [16]). The unitarity conditions are calculated using the LQT prescription [22]; we consider all possible combination of two-particle scattering processes including some charged processes and the resulting S matrix is diagonalized numerically. The imposed condition is for the largest eigenvalue to be less than 16π.

The matter sector
In the matter sector invariance under the S3 symmetry implies a lagrangian of the form: here D denotes d-type quarks and the dots refer to similar expressions for utype quarks and leptons plus hermitian conjugate. Flavor indices are denoted by α and β, quark left doublets and right singlets are denoted by Q L and q R respectively, while unprimed quantities will refer to the mass eigenstate basis. The Yukawa matrices are given by: and and similar expressions for u-quarks and leptons. It is well known from the studies of multi-Higgs models that the coupling of the fermions to extra scalar doublets can induce flavor changing currents even at tree level. Nevertheless, it has been shown in previous research concerning the S3 model (e.g. [6,9]) that values of the parameters y d1 , . . . calculated from fits to the CKM and PMNS mixing matrices are naturally small, so that experimental bounds on flavor changing processes are not violated. For our purposes, we will assume in the following that the offdiagonal Yukawa couplings, given its small size, do not contribute noticeably to the physical processes calculated in the next section, and thus we simply take this couplings identically zero. Writing explicitly the quark fields for each family, with these assumptions the Yukawa lagrangian becomes: After rotating the scalars to the mass eigenstate basis using (8) and (9) we find that the S3-symmetric neutral scalar h s does not couple to the fermions in this limit. Thus either h 1 or h 2 or a linear combination of both must correspond to the SM Higgs doublet.

Numerical analysis and results
The implementation of the model is made through SARAH [17][18][19][20] taking advantage of this package's functionality to generate model files for other tools. We perform a random scan of the parameter space filtering points that do not satisfy all the conditions mentioned before; the generation of the scattering matrix to calculate unitary constraints is done using FeynArts [23] and FormCalc [24]; the generation of SLHA [25,26] input files for HiggsBounds [27][28][29][30][31] and MicrOMEGAS [32] is donde using the SARAH-SPheno [33][34][35] framework. We use HiggsBounds to further filter points that do not comply with current experimental limits from Higgs searches, and finally MicrOMEGAS is utilized to compute the value of the relic density of the dark particle for points that satisfy all the constraints. We present our findings in figure (1) which is a plot of the value of the relic density as a function of the mass of the dark scalar.
The first observation that we want to make is that from the entire sample of points probed (close to 10 6 ) only a small proportion (around 0.5%) turned out to be physically acceptable (in the sense of fulfilling all constraints). While the size of this sample 2 is rather small compared to the size of a parameter space of such dimensionality (15 total free parameters), we still think that the main conclusion drawn from our findings shows an important property of the model.
From figure (1) we draw attention to the fact that the value of the relic density transpires to be suboptimal for all the potential physical points probed. In other words the dark scalar doublet of the model contributes only a fraction of the experimentally measured value. This highly suggests the possibility of extending the dark sector of the model by including more dark particles, since there is "room" for additional contributions to the relic density.
This fact (the small value of the relic density of the dark scalar doublet) is in high dissimilarity to the case of the Inert Doublet Model [2,[37][38][39][40] wherein, in the simplest approximations, there are two regions of parameter space (high/small mass) for which the experimental value of the relic density is easily obtained and even surpassed. Of course the extra scalar doublets in the S3 model can account for this contrasting differences; namely the increase in the number of allowed channels, as our results confirm, transform  Figure 1: Computed relic density of the dark matter particle as a function of its mass; the current Planck value [36] is also shown.
the annihilating process of the dark scalar into an over-efficient one.

Conclusions
We have performed an analysis of the S3 symmetric model where the number of scalar doublets fill in all the irreducible representations of the permutation group; by taking one of the scalars as inert we are retaining convenient features of the Three Higgs Doublet Model with S3 symmetry whilst additionally extending the model with a dark sector. We have probed random samples of parameter space points obtaining a set of potentially physical points for which we calculated the value of the relic density of the dark scalar. The results obtained indicate that the dark scalar annihilates too efficiently to generate the observed experimental value of the relic density (as opposed to e.g. the Inert Doublet Model) due to the presence of the additional allowed channels. Our results suggest the possibility of extending the dark sector of the model with additional particles, potentially enriching the phenomenology of this type of models.