Unitarity and stability conditions in a 4-Higgs doublet model with an S3-family symmetry

In order to obtain a dark matter candidate we propose an extension of the S3 symmetric 3-Higgs doublet model, adding a new scalar doublet, occupying all the irreducible representations of the discrete symmetry. To ensure stability of the dark matter particle we impose an extra Z2 symmetry. We find the analytical masses of the scalar particles and constrain their values using stability and unitarity conditions.


Introduction.
Finding what dark matter (DM) is made of is one of the main challenges in particle physics and cosmology. It has been proposed in [1] that DM is conformed by neutral scalar particles, a very interesting hypothesis as these particles have the characteristics expected for DM: Neutral, cold and weakly interacting (see for example [2,3,4,5,6]).
Among the numerous proposals to extend the scalar sector of the standard model, the 3-Higgs Doublet Model with an S 3 -family symmetry (3H-S 3 ) 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 experimental results (see e.g. [7,8,9,10,11,12,13,14]) A natural generalization embracing these two ideas suggests itself: enlarge the 3H-S 3 model with an additional doublet representing a dark matter candidate. In this letter we present an analysis of the S 3 -symmetric 4-Higgs Doublet Model (4HDM) in which we occupy all irreducible representations of the S 3 symmetry: one symmetric singlet, one antisymmetric singlet and one doublet. The 3H-S 3 is constituted by the symmetric singlet and doublet representations, while for the fourth Higgs doublet (in the antisymmetric representation) we impose a Z 2 symmetry ensuring the stability of the potential dark matter candidates. In the following we present the analytical calculation of the masses of the scalar particles at tree level and constrain their values using unitarity and stability conditions. A complete analysis of the model with its dark matter phenomenology will be presented elsewhere [15] The most general SU (2) L ×U (1) Y invariant renormalizable scalar potential for a 4-Higgs doublet model with additional S 3 × Z 2 symmetry is given by: Here we have arranged the SU (2) L doublets H 1 and H 2 into a column vector transforming in the doublet representation of S 3 , while H s and H a are required to transform in the symmetric and antisymmetric singlet representations of S 3 respectively. In the following we will assume all the couplings λ i , i = 1...14 to be real, CP conserving, and |λ i | < 4π in order to fulfill perturbativity. Note that there are no terms with odd powers of H a , the only field assumed odd under Z 2 .
After electroweak symmetry breaking all Higgs doublets acquire a vacuum expectation value (vev) which we denote respectively by v 0 , v 1 , v 2 and v a . Nevertheless, to avoid the breaking of the Z 2 symmetry we fix v a identically to zero, and from the minimization conditions (a.k.a. tadpole equations) of the potential (1) the fourth equation ∂V /∂v a = 0 is therefore automatically satisfied. The three minimization equations left (∂V /∂v i = 0, i = 1, 2, 3) reduce to those of the 3H-S 3 , from which we reproduce the following conditions to the quadratic couplings: The self consistency of the above conditions require either λ 4 = 0 or the alignment of the vacuum expectation values v 1 and v 2 : Since the case λ 4 = 0 is phenomenologically unappealing in the context of the 3HDM, we will assume the alignment (3)  We parametrize the Higgs doublets by and similarly for H 1 , H 2 and H 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. The masses of the scalar particles are found by diagonalizing the corresponding mass matrices, e.g. for the neutral scalar fields the following matrix: which is block diagonal such that the primed fields 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 and the Z 2 odd field remains unmixed h n a = h n a . The expressions for the masses presented below are separated by neutral scalar, neutral pseudo scalar and charged particles.
• Masses of the neutral scalar particles: • Masses of the neutral pseudo scalar particles:

Stability conditions.
The stability conditions, ensure that the vacuum is stable i. e. that the potential has a minimum. They were found for the potential (1), following the procedure in [14] for the 3H-S 3 :

Unitarity constraints.
Unitarity constraints over the quartic parameters can be obtained from the elegant LQT method [16]. On account of renormalizability, scattering amplitudes cannot exhibit unphysical growth in the limit of high energies. Thus, one is led to impose (tree level) unitarity on different sets of scattering processes, in particular those of two particle states. Defining the matrix M ij = M i→j where the indices stand for all possible two particle processes, it suffices to consider processes involving Higgs scalars and longitudinal vector bosons. The eigenvalues of the matrix M ij , denoted a ± i and b i for charged and neutral two particle channels, will then be constrained according to | a ± i |, | b i |< 16π, reflecting the physical fact that the coefficient a 0 of the s-wave term in a partial wave expansion of the scattering amplitude is bounded (e.g. |a 0 | < 1/2) in the limit of high energy exchange.
Eigenvalues a ± i and b i where i = 1, 2, ..., 6. were found by Das and Dey for the 3HDM in [12], whereas a ± i where i = 7, 8, ..., 12 and b 7 correspond to 4H-S 3 interactions decoupled from the 3H-S 3 matrices, e.g. h n 1 h n a h n 1 h n a . The dispersion matrices involving fields from 4 Higgs Doublet, e.g. h n a h n a h n s h n s were solved numerically during the scan with pseudorandom values for the λ i couplings.

Numerical analysis.
To perform the analysis is convenient to parametrize the vevs in spherical coordinates, so  where θ ∈ (0, π) and φ ∈ (0, 2π). With this parametrization and the relation found in Equation 3 we get  Hence the vevs only depend on θ:

Conclusions.
The S3 flavor symmetry has been very successful accommodating fermion masses and mixings; in this work we use four SU (2) doublets, using all irreducible S3 representations, which provides a rich phenomenology where one or several dark matter candidates can be feasible. We found analytical expression for the masses of the scalar particles and constrained their numerical values applying stability and unitarity conditions. The mass spectra was computed for two cases: 1) the SM Higgs boson corresponds to h n s and 2) the SM Higgs boson corresponds to h n 2 . In the plots 1, 2 and 3 corresponding to the first case, we found that the masses of h n 1 , h p 2 and h c 2 are heavier than the rest for large tan θ. While, in the plots 4, 5 and 6 all masses have nearly the same range, with an upper limit close to 1TeV, the exception being m h c 2 and m h c a with a 500GeV upper limit. As a perspective of these results we are currently working in the complete analysis of the model and on the dark matter phenomenology in order to constrain the parameter space of the model by relic density measurements.