The symmetry S 3 in the dark matter

Experimental evidence so far suggests that there are only three generations of quarks and leptons. Before electroweak symmetry breaking, the three families of quarks and leptons are indistinguishable, so they are invariant under transformations of the S 3 group. Using the symmetry S 3 we have at our disposal 3 irreducible representations, 2 , 1 s , 1 a , where we can accommodate up to 4 Higgses doublets in a model that occupies all the irreducible representations of the group S 3 . This model with four Higgses doublets (4HDM) is of great interest, thanks to the fact that we can take the fourth Higgs doublet as a stable particle without interaction with fermions so that it becomes a candidate for dark matter, while with the remaining three Higgses the properties obtained are maintained. An important condition for having a viable dark matter candidate is its stability, i. e., it does not decay into Standard Model particles. The simplest


Introduction
The symmetric group Sn is the group of bijections of {1, 2, ..., n} to itself, also called the permutation group of n objects.It is a nite group of order n!, that is, there are n! ways to swap n objects.The group that interests us is S3.
The group S1, comprises the permutations of a single object, and its only element is the identity {E}.The group S2 comprises the permutations of two objects f1 and f2.This group has 2! = 2 elements {E, A}, where E is the identity that produces the trivial transformation, f2 −→ f2 and A produces the transformation A : f1 −→ f2, f2 −→ f1.Note that the symmetry groups are abelian.
Experimental evidence so far suggests that there are only three generations of quarks and leptons.Before electroweak symmetry breaking, the three families of quarks and leptons are indistinguishable, so they are invariant under transformations of the S3 group.This tells us that the S3 symmetry [117] is convenient.The fermions in the irreducible representation of the doublet are denoted as ψ D(L,R) , where and those found in the symmetric singlet representation as (2) where 1, 2, 3 represent the index of each family of the left (L) or right (R) fermionic eld.For quarks we have: ψ3,L = (bL, tL) , ψ3,R = tR, ψ3,R = bR (3) where (uL, dL) and (cL, sL) are doublets SU (2) L , while uR, cR, dR y sR are singlets SU (2) L .The Higgs elds are then denoted as: We accommodate four SU (2) doublets into the irreducible representations of the permutation group S3, denoting the symmetric and antisymmetric scalars by Hs and Ha respectively, while the remaining two doublet H1 and H2 are arranged in a column vector conforming the S3 doublet, i. e.
2 The symmetry group S 3 The group S3 is dened as the group of permutations of three objects [18].This is a non-abelian group and is formed by three even and three odd permutations of three objects, which can be labeled as follows (f1, f2, f3).
The elements of the group are where E is the identity element of the group and the Ai, with i = 1, 2, 3, 4, 5, label the permutations as follows The notation used refers to the exchange of the sub-scripts of f1, f2, f3, for example A1 acts as follows: In general, for the permutation group Sn the product between two of its elements is simply their successive application and this product is not commutative.Thus, for example, by multiplying A2 with A5 we obtain that: in terms of f1, f2 and f3 The products between the remaining elements of group S3 are carried out in an analogous manner to the previous example.Thus, the multiplication table of the group is given in table 2. Now in general, if where A −1 is the inverse element.Therefore, the condition is satised As a particular example in the case of S3, the inverse of A1 is analogously, we have the inverses of the other elements

Conjugate Classes
A conjugate class for a group G of order g is dened as The identity element E by itself forms a conjugate class, the remaining classes are determined with the help of the multiplication table of the group (see table 1).Thus, for the conjugate class whose representative is A1 and which is denoted as k1, we have: Denoting with k2 the conjugate class with representative A4, for this we have Therefore, the group S3 has three conjugate classes that are denoted as E, k1 and k2, and have the form: (10) The conjugate classes of the group are useful for constructing the so-called class operators and in turn constructing the projectors of the group.
(11) The way in which the element E acts on the base vectors is evident, so we are tempted to represent element in matrix form in the following form For a more illustrative example, we analyze the eect of applying A5 to the basis vectors, thus the matrix representation of A5 is analogously for the other elements of the group S3, we then obtain the so-called representation S3 The real representation is a three-dimensional representation.
We will now construct one more representation, taking advantage of the fact that the group S3 is isomorphic to the symmetry group of a rectangle triangle [19], elements A1, A2 and A3 are associated with reections on the indicated axes of symmetry and elements A4 and A5 with rotations corresponding to angles of 2π/3 and 4π/3 respectively, around the z axis.Let us consider again the three-dimensional vector space generated by the base vectors (11), as an example, let's see how A4 (rotation by 2π/3 around z) acts on the base vectors, A4 − → e 3 = − → e 3, so we represent element A4 as follows: Proceeding in a similar way with the other elements The matrices of this representation are unitary, therefore the representation we have just constructed is a unitary representation.From now on we will work specically on this representation.Now, from group theory, we know that the number of non-equivalent irreducible representations of a group G of order g is equal to the number of conjugate classes in the group, so in our case, because the number of conjugate classes of S3 is 3, we can say that the number of irreducible representations of S3 is 3. On the other hand, we can also determine the dimensions of the irreducible representations of S3 by the following relation by simple inspection we can notice that the values for which this equation holds are n1 = n2 = 1 and n3 = 2, so the irreducible representations of S3 have dimensions 1 and 2 respectively; that is, we can decompose the group into two singlets and a doublet.
We see that all the matrices in the representation (13) have a block structure, so it is evident that they are reducible and therefore we can decompose them into two matrices of dimensions (2 × 2) and (1 × 1), but we also know that there exists a trivial irreducible representation for each group, called the identical representation and which is characterized by D (1) All this is summarized in Table 2.
Using the equation to obtain the characters of a group [19] and using table 2 we can build the respective character table 3 for S3.
Table 2: Representations of the elements of S3.D (1) and D (2) correspond to onedimensional representations, while Table 3: Characters of S3, where k l i (i = 1, 2) corresponds to the ith class of the group and the superscript l indicates the number of elements in the class.
We can express the representations in terms of their irreducible components, that is where ⊕ denotes the direct sum and the aν can be determined according to the equation where g is the order of the group, ci is the number of elements of class Ki, χ (ν) i is the primitive character of class Ki, and χi is the composite character of the class Ki.We compute a1 a1 = 1 g c1χ (1) 2 χ2 + c3χ (1) consequently a1 = 0, similarly, we nd that which means that the representation D (2) and D (3) are included in D, but D (1) is not here.The matrix representation of equation ( 13) in terms of its irreducible components is expressed as the direct sum of a single singlet and a doublet, that is, We can obtain a representation of S3 by direct product of the irreducible representations, in particular we construct the direct product of each element of S3 with itself in the representation D (3) , this is: then we get which is the direct sum of a pair of singlets and a doublet.

Projectors
The projectors of the group we can obtain the functions adapted to the S3 symmetry (invariants).For the pro-jectors we use the equation where Ci are the class operators, obtained from the direct product of each representation element D (3) ; this is, Then, the form of the projection operator on the symmetric singlet is given by: Similarly, we nd P1 A , the projector on the antisymmetric singlet P 2 And nally we nd P2, the projector on the doublet P 3 , The projector P2 can be decomposed into two terms, each of which is a tensor of rank one, that is, where and The eigenvalues of the elementary projectors will allow the construction of the matrices U that diagonalize the product by blocks With the help of the projectors, the direct product X ⊗ Y is decomposed into a direct sum of singlets and doublets.Then, applying each of the projectors obtained previously to said direct product, we obtain: Or The coecients of the eigenvectors are the functions adapted to symmetry.If (x1, x2) and (y1, y2) are the components of two doublets of S3, the Kronecker product (x1, x2) ⊗ (y1, y2) contains the following components: A symmetrical singlet: which is invariant under the group S3.
An antisymmetric singlet: which is not invariant under the group S3.
A doublet whose components are:

Mass matrices
The Yukawa Lagrangian density for three families of quarks and leptons is written as follows where Qi and Li denote the weak isospin doublets of quarks and leptons, respectively; H is the Higgs eld.
The weak isospin doublets are expressed as follows: It is convenient to rearrange the terms of equation ( 28) and write the Lagrangian density LY as a function of the spinors ψ q and ψ l , whose components are dened in the space of families like: The Yukawa Lagrangian density in family notation, equation (28), is (31) Before the breaking of gauge symmetry, quarks and leptons have no mass and the theory is chiral.Therefore, the left and right spinors transform independently, that is: with q = u, d and l = ν l , l; where g ∈ S3L acts on the left spinors, and g ∈ S3R on the right spinors.
When the gauge symmetry is spontaneously broken, the fermions acquire mass.Therefore, the elds of quarks are transformed as follows: (36) In the same way, the elds of charged leptons and neutrinos transform as: (37) The left and right chirality components of the same eld are transformed with the same group element.The avor symmetry group of the bilinear forms in equations (36) and (37) is the group S diag 3 whose elements are the pairs (g, g), with g ∈ S3L and g ∈ S3R and g = g ′ .Clearly, S diag 3 ⊂ S3L × S3R.The mass term from the Yukawa coupling is transformed as follows: substituting the expressions for ψ ,q ,ψ ,q , ψ ,l ,ψ ,l , we obtain Therefore, under the action of the avor group S diag 3 , the mass matrices Mq and M l are transformed according to the following rule: If the Yukawa sector is invariant to the family group , it must be true that The Yukawa sector of the Standard Model has family symmetry if the mass matrix commutes with all elements of the group S3, that is 4 Model with 4 Higgs doublets and symmetry S 3 Using the symmetry S3 we have at our disposal 3 irreducible representations, 2, 1s, 1a, where we can accommodate up to 4 Higgses in a model that occupies all the irreducible representations of the group S3.This model S3 − 4H is of great interest, thanks to the fact that we can take the fourth Higgs as a stable particle without interaction with fermions so that it becomes a candidate for dark matter, while with the remaining three Higgses the properties obtained are maintained.in a model with four doublets.It is worth mentioning that the possibility of obtaining dark matter candidates with only three doublets was explored [2022], but no way was found without altering the results obtained for this model.
The Higgs potential has quadratic and quartic terms.This means that we need to nd the invariants of S3 made with two and four elds that are irreducible representations of S3.The invariants of S3 for the quadratic and quartic tensor product are, respectively, For the construction of the terms in the potential, we need to consider the weak index of the elds.We make the corresponding projections to generate the invariants of S3 and for this purpose we use the projections of the four-dimensional real basis of S3.

Invariants with two elds:
(a)

Invariants with four elds:
HD + Hs : (a) HD + Hs + Ha : (a) (g) To simplify the calculations, we introduce the following variables as in [23,24] Using the above and taking the assignment of the self-coupling coecients in the order in which the constructed invariant terms are listed, we obtain: where At the normal minimum, the VEV's of the Higgs elds are considered real.So, once the elds acquire VEV's, we can relate them to the new variables (43)
First of all, we calculate the dierent terms individually ∂v 1 = 0, for the rest.
Evaluating all partial derivatives at the minimum: Then, we can write the system of equations of the minimum: Using the potential (54), the equations of the minimum are Furthermore, we obtain the equation In this way we can describe the mass matrices in a simpler way. 4.2

4HDM Model
The scalar (symmetry invariant and renormalizable ) is a mixture of the potentials known from studies of the three Higgs model with permutation symmetry, and can be written as: The most general S3-invariant Yukawa Lagrangian density for the coupling of 4-Higgs coupled Dirac fermions (see table 4), where both components of the third family are assigned to the symmetric singlet of S3 is:  where Yi are complex Yukawa couplings.When writing the Yukawa Lagrangian density, for up-type quarks or Dirac neutrinos, the Higgs eld must be replaced by the conjugate Higgs eld → iσ2H * iW , i = 1, 2. After symmetry breaking [25], the Higgs doublets SU (2) L acquire expectation values in a vacuum, which we choose real.
The rest of the matrices (and in particular MY ν ) have the same structure, changing only the Yukawa coupling terms, i. e., with the mass matrices If we use equation ( 53), we obtain (69) Models with three Higgs doublets can be obtained as special cases of models with 4 Higgs doublets, e.g. to obtain the mass matrix of a 3HDM [2640] and the third fermion family in the singlet representation asymmetric, it is enough to take the limit when Ha → 0 (70) then, the mass matrices are given by: This model has a dark matter candidate from a model with S3 symmetry without interfering with the positive results obtained in [2632].An important condition for having a viable dark matter candidate is its stability.That is, it does not decay into Standard Model particles.The simplest way to establish the stability of a particle in a model beyond the standard, is imposing a discrete symmetry Z2, so that all the elds are transformed in the form Ψ −→ Ψ, while the two dark matter candidates are transformed as χ −→ −χ, this way we make sure we don't have terms denoting decays of χ.This method has been used in numerous models, such as the scotogenic [41] and the inert scalar doublet [42].It is worth mentioning that there are also models with more complex discrete symmetries, such as Z3 in [43].In general you can make models with symmetry Zn.
In this model with symmetry Z2, i. e. 4HDM [4446], dark matter candidate is the Higgs boson in the antisymmetric singlet representation Ha , so these transform under Z2 as Ha −→ −Ha.So, the Lagrangian density of Yukawa is given by: (75) and scalar potential with this new symmetry.The terms highlighted in bold, correspond to those that break the Z2 symmetry and are therefore omitted, note that Ha no longer appears in the Yukawa Lagrangian density.Another imposition required to pro-pose the candidacy of a eld of the doublet Ha, is that its corresponding VEV is equal to zero, va = 0 (For more details of this model, see [4446]).(77) Hence, the fourth doublet, Ha, contains four physical elds, two charged h c a and two neutral, the scalar h n a and the pseudoscalar h p a .Charged particles are restricted as dark matter candidates [47].Thus, viable candidates are the antisymmetric doublet neutral Higgs elds, h n a and h p a , with masses: the lightest neutral Higgs eld will be the dark matter candidate resulting from the fourth Higgs doublet.
There are theoretical and experimental constraints, which apply to the analysis of the candidate to constrain the mass range of this and the rest of the Higgs elds in the model.Using the IDM [48], we have the constraints: Theoretical restrictions 1.The potential must be bounded from below, so that it has a stable vacuum [44].
Experimental restrictions 1.The mass Higgs boson of the standard model is [49]: In this model, we consider the mass of the Standard Model Higgs m h = 125 GeV, taking the possibility of two of the scalar neutral elds (those corresponding to the Higgs doublets Hs and H2).

Concluding remarks
In this article, we study the 4HDM model in the theoretical framework of the minimum extension S3 of the standard model.We extend the Higgs sector by adding four Higgs doublets and making the theory invariant with respect to avor permutations.We impose Z2 symmetry on the fourth Higgs doublet, Ha.In this model, the dark matter candidate is the Higgs doublet in the antisymmetric singlet representation Ha, so these transform under Z2 as Ha −→ −Ha.
We accommodate four SU (2) doublets into the irreducible representations of the permutation group S3, denoting the symmetric and antisymmetric scalars by Hs and Ha respectively, while the remaining two doublet H1 and H2 are arranged how Additionally, the 4HDM model can continue to be studied and understood.Then future perspectives could be to calculate radiative corrections that show corrections to the mass values.Another additional perspective of the work would be the possibility of extending the dark matter sector of the model with additional particles.

5
Dark matter candidateThis model is based on S3 symmetry, which allows us to accommodate the four Higgs doublets:

Table 1 :
Multiplication table of group S3.

Table 4 :
Particle spectrum of SM group and S3 ⊗ Z2.