Classifying Accidental Symmetries in Multi-Higgs Doublet Models

The potential of $n$-Higgs Doublet Models ($n$HDMs) contains a large number of SU(2)$_L$-preserving accidental symmetries as subgroups of the symplectic group Sp(2$n)$. To classify these, we introduce prime invariants and irreducible representations in bilinear field space that enable us to explicitly construct accidentally symmetric $n$HDM potentials. We showcase the classifications of symmetries and present the relationship among the theoretical parameters of the scalar potential for; (i) the Two Higgs Doublet Model (2HDM) and (ii) the Three Higgs Doublet Model (3HDM). We recover the maximum number of $13$ accidental symmetries for the 2HDM potential and for the first time, we present the complete list of $40$ accidental symmetries for the 3HDM potential.


I. INTRODUCTION
The discovery of the Higgs particle at the CERN Large Hadron Collider (LHC) [1,2], which was previously predicted by the Standard Model (SM) of Particle Physics [3][4][5][6][7], generated renewed interest in Beyond the SM (BSM) Higgs Physics. This is corroborated by the fact that the SM fails to address several key questions, such as the origin of the observed matter-antimatter asymmetry and the dark matter relic abundance in the Universe.
There is a plethora of well-motivated New Physics models with additional Higgs scalars that have been introduced to solve these problems [8][9][10][11][12][13]. To distinguish such models, one usually employs symmetry transformations that leave the particular model invariant. These symmetries impose constraints over the theoretical parameters of the models and thus enhance their predictability to be probed in future experiments.
Identifying the symplectic group Sp(2n) as the maximal symmetry group allows us to classify all SU(2) L -preserving accidental symmetries in nHDM potentials. We introduce prime invariants to construct accidentally symmetric potentials in terms of fundamental building blocks that respect the symmetries. In the same context, we use irreducible representations to derive all potentials that are invariant under non-Abelian discrete symmetries.
The layout of this paper is as follows. In Section II, we define the nHDMs in the bilinear scalar field formalism. Given that Sp(2n) is the maximal symmetry of the nHDM potential, we adopt the bi-adjoint representation of this group which becomes relevant to this bilinear formalism. In Section III, we start with classifying continuous symmetries for nHDM potentials. As illustrative examples, we classify all accidentally symmetric potentials for the 2HDM and the Three Higgs Doublet Model (3HDM). Then, we introduce prime invariants to build potentials that are invariant under SU(2) L -preserving continuous symmetries. In Section IV, we discuss possible and known discrete symmetries for nHDM potentials and recover the discrete symmetries for the 2HDM and the 3HDM cases [21,23,24]. This section also includes our approach to building 3HDM potentials with the help of irreducible representations of discrete symmetries. Having discussed the classifications of symmetries, we provide the list of all SU(2) L -preserving accidental symmetries for the 2HDM and the 3HDM potentials including the relationships among the theoretical parameters of the scalar potentials. Section V contains our conclusions. Finally, technical details are delegated to Appendices A, B, C, and D.
The matrices Σ A n have 4n × 4n elements and can be expressed in terms of double tensor products as, where t a S and t b A are the symmetric and anti-symmetric matrices of the SU(n) symmetry generators, respectively. Specifically, for the case of the 2HDM, the following 6 matrices may be defined: Correspondingly, for the 3HDM, we have the following 15 matrices: where G i are the standard Gell-Mann matrices of SU(3) [36]. Note that the Σ A n matrices satisfy the property, which means that Σ A n are C-even. Consequently, the vectors R A 2 and R A 3 for the 2HDM and the 3HDM cases are given by (II. 10) With the aid of the n(2n − 1)-dimensional vectors R A n , the potential V n for an nHDM can be written in the quadratic form as where M n A is the 1 × n(2n − 1)-dimensional mass matrix and L n AA is a quartic coupling matrix with n(2n − 1) × n(2n − 1) entries. Evidently, for a U(1) Y -invariant nHDM potential, the first n 2 elements of M n A and n 2 × n 2 elements of L n AA are only relevant, since the other U(1) Y -violating components vanish. The general 2HDM and 3HDM potentials with their corresponding M n A and L n AA are presented in Appendix A. The gauge-kinetic term of the Φ n -multiplet is given by where the covariant derivative in the Φ n space is In the limit g Y → 0, the gauge-kinetic term T n is invariant under Sp(2n)/Z 2 ⊗ SU(2) L transformations of the Φ n -multiplet. In general, the maximal symmetry group acting on the Φ n -space in the nHDM potentials is G Φn n-HDM = Sp(2n)/Z 2 ⊗ SU(2) L , which leaves the local SU(2) L gauge kinetic term of Φ n canonical. The local SU(2) L group generators can be represented as σ 0 ⊗ 1 n ⊗ (σ 1,2,3 /2), which commute with all generators of the Sp(2n) group.
Let us turn our attention to the Sp(2n) generators K B n , with B = 0, 1, . . . , n(2n + 1) − 1. They satisfy the important relation [25], which implies that K B n are C-odd. The Sp(2n) generators may conveniently be expressed in terms of double tensor products as [37] where t a S (t b A ) are the symmetric (anti-symmetric) generators of the SU(n) symmetry group. For instance, the 10 generators of Sp(4) are [25] K 0,1,3 It is interesting to state the Lie commutators between the Σ A n and K B n generators, where I, J = 1, . . . , n(2n − 1) − 1 and f BIJ n is a subset of the structure constants of the SU(2n) group. Employing (II.18), we may define the Sp(2n) generators in the bi-adjoint representation (i.e. the adjoint representation in the bilinear formalism) as Note that the dimensionality of the bi-adjoint representation differs from the standard adjoint representation. The former representation has (2n 2 − n − 1) × (2n 2 − n − 1) dimensions, whereas the latter has n(2n + 1) × n(2n + 1) dimensions [38][39][40][41]. The generators of T B n for Sp (4) and Sp (6) corresponding to the 2HDM and 3HDM are presented in Appendix B.
Knowing that Sp(2n) is the maximal symmetry group allows us to classify all SU(2) Lpreserving accidental symmetries of nHDM potentials. These symmetries can be grouped into two categories: (i) continuous symmetries and (ii) discrete symmetries (Abelian and non-Abelian symmetry groups). In the next section, we demonstrate the structure of continuous symmetries and prime invariants for building nHDM potentials.

III. CONTINUOUS SYMMETRIES AND PRIME INVARIANTS
The symplectic group Sp(2n) acts on the Φ n -space, such that the bilinear vector R A n transforms in the bi-adjoint representation of Sp(2n) defined in (II.18) and (II. 19). It is therefore essential to consider the maximal subgroups of Sp(2n). Then, the accidental maximal subgroups would be the combinations of smaller symplectic groups, such as [42] Sp where p + q = n. Note that local isomorphisms should also be taken into account, such as Following this procedure, it is straightforward to identify all accidental continuous symmetries for nHDM potentials. For the simplest case, i.e. that of the 2HDM, the above breaking chain gives rise to the following continuous symmetries [25], where HF indicates Higgs Family symmetries that only involve the elements of Φ = (φ 1 , φ 2 , . . . , φ n ) T and not their complex conjugates. In the case of 3HDM, the maximal symmetry is Sp (6), so in addition to all symmetries in (III.7) we find   We may now construct accidentally symmetric nHDM potentials in terms of fundamental building blocks that respect the symmetries. To this end, we introduce the invariants S n , D 2 n and T 2 n . In detail, S n is defined as which is invariant under both the SU(n) L ⊗U(1) Y gauge group and Sp(2n). Moreover, we define the SU(2) L -covariant quantity D a n in the HF space as Under an SU(2) L gauge transformation, D a n → D a n = O ab D b n , where O ∈ SO(3). Hence, the quadratic quantity D 2 n ≡ D a n D a n is both gauge and SU(n) invariant. Finally, we define the auxiliary quantity T n in the HF space as which transforms as a triplet under SU(2) L , i.e. T n → T n = U L T n U T L . As a consequence, a proper prime invariant may be defined as T 2 n ≡ Tr(T T * ), which is also both gauge and SO(n) invariant.
In addition to the above maximal prime invariants, it is useful to define minimal invariants. For instance, prime invariants that respect Sp(2) can be derived from the doublets where the identity σ 2 σ a σ 2 = −(σ a ) T has been used.
By analogy, to construct an SO(2)-invariant expression from the doublet φ i φ j , we may use quantities such as Moreover, extra prime invariants can be constructed from φ i iσ 2 φ * i and φ j iσ 2 φ * j , e.g.
Note that D a ij D a ij depends on S ij and S ii,jj , since Observe that T ij and S ij are not invariant under phase re-parametrizations, φ i → e iα i φ i , and so they need to be appropriately combined with their complex conjugates.
We are now able to build a symmetric scalar potential V sym in terms of prime invariants as follows: Obviously, the simplest form of the nHDM potentials obeys the maximal symmetry Sp(2n), which has the same form as the SM potential, with a single mass term and a single quartic coupling. For example, the 2HDM Sp(4)invariant potential, the so-called MS-2HDM is given by [29] with the obvious relations among the parameters, Note that the functional form of the potential in (III. 19 Similarly, the 3HDM potential invariant under Sp(6)/Z 2 will be a function of the symmetric block S 3 = S 11 + S 22 + S 33 , i.e.
where the non-zero parameters are Remarkably, the MS-nHDM potentials obey naturally the SM-alignment constraints and all quartic couplings of the MS-nHDM potential can vanish simultaneously [27,29].
Another example is the SU(3)⊗U(1)-invariant 3HDM potential. The corresponding symmetric blocks are S 3 = S 11 +S 22 +S 33 and D 2 3 = D 2 12 +D 2 13 +D 2 23 , given in (III.9) and (III.10), respectively. Therefore, the SU(3)⊗U(1)-invariant 3HDM potential takes on the form with the following relations between the parameters: This method can be applied to all continuous symmetries of nHDM potentials. We present all explicit symmetric blocks under all continuous symmetries for the 2HDM and the 3HDM potentials in Appendix C. The complete list of accidental symmetries for the 2HDM and 3HDM potentials, along with the relations among the non-zero parameters, are exhibited in Tables I and II.

IV. DISCRETE SYMMETRIES AND IRREDUCIBLE REPRESENTATIONS
As discussed in Section II, Sp(2n) is the maximal symmetry group for nHDM potentials. This will help us to classify all SU(2) L -preserving accidental symmetries of such potentials. In addition to continuous symmetries shown in Section III, there are also discrete symmetries as subgroups of continuous symmetries. Known examples of this type are the Standard CP symmetry, the Cyclic discrete group Z n , the Permutation group S n , or a product of them possibly combining with continuous symmetries. In general, these discrete symmetries can be grouped into Abelian and non-Abelian symmetry groups. In this section, we discuss all possible and known discrete symmetries for nHDM potentials, including our approach to build nHDM potentials by employing irreducible representations of discrete symmetry groups.

IV.I. Abelian Discrete Symmetries
To start with, let us first consider the Abelian discrete symmetry groups [44,45] Note that iff n and m have no common prime factor, the product Z n × Z m is identical to Z n×m . These discrete symmetries can be imposed to restrict the independent theoretical parameters of the model. For example, in the 2HDM, the Z 2 symmetry is invoked to avoid flavour changing neutral currents [46] or to ensure the stability of dark matter [47]. In addition to these discrete symmetries, there are Generalized CP (GCP) transformations defined as, (1), where U(1) can always be eliminated by U(1) Y transformation. The GCP transformations realize different types of CP symmetry. For example, in the case of the 2HDM (3HDM), there are two types of CP symmetries: (i) standard CP or CP1: In general, without continuous group factors, the discrete symmetries for the 2HDM are CP1, CP2 and Z 2 . The generators of these discrete symmetries can be expressed in terms of double tensor products as In the bilinear R A 2 -space, the transformation matrices (or the generating group elements) associated with CP1, CP2 and Z 2 discrete symmetries are given by where the U(1) Y -conserving elements are denoted in boldface.
Let us turn our attention to the 3HDM potential. In this case, the corresponding CP1 discrete symmetry can be represented as resulting in the following transformation matrix in the R A 3 -space, On the other hand, CP2 discrete symmetries may be given by where the phase e iα is an arbitrary phase factor. This results in the following transformation matrix in the bilinear R A 3 -space (dots stand for zero elements) Note that the CP2 transformation matrix D CP2 in the bilinear R A 3 -space is non-diagonal, contrary to the 2HDM case. We must remark here that in the case of 2HDM, ∆ 2 CP2 = −1 8 = 1 8 and ∆ 4 CP2 = 1 8 and in the bilinear space D 2 CP2 = 1 6 . However, in the case of the 3HDM, we have ∆ 2 CP2 = −1 12 and ∆ 4 CP2 = 1 12 and D 4 CP2 = 1 14 , in agreement with a property termed CP4 in [50]. Without loss of generality, we set α = 0.
In addition, there are several Abelian discrete symmetries for the 3HDM potential [51], i.e. (IV.14) The generators of these Abelian discrete symmetries are given by 16) with ω = e i2π/3 . In the bilinear R A 3 -space, as a result of flipping the sign of the one or two doublets, the following diagonal transformation matrices for the discrete symmetries Z 2 and Z 2 may be derived: In the same way, the transformation matrices for Z 3 and Z 4 may be represented by the non-diagonal matrices Now, with the help of the D-transformation matrices, we can construct accidentally symmetric nHDM potentials. For example, in the case of Z 4 symmetry, applying the transformation matrix D Z 4 on the U(1) Y -conserving elements of R A 3 yields Therefore, all possible combinations R i 3 R j 3 (i, j = 1, . . . , 8) that respect the Z 4 symmetry may be obtained as These combinations lead to the following Z 4 -invariant 3HDM potential: where the complex phase of λ 1212 can be rotated away by a field redefinition.
In the same fashion, we can use this procedure to construct nHDM potentials invariant under all Abelian discrete symmetries. Tables I and II give the parameter relations of the 2HDM and 3HDM potentials constrained by these symmetries.

IV.II. Non-Abelian Discrete Symmetries
Non-Abelian discrete symmetries constitute another class of discrete symmetries, which may be thought of as combinations of Abelian discrete symmetries. The most familiar non-Abelian groups can be summarized as follows [20,52]: (i) Permutation group S N . The best known non-Abelian discrete groups are the permutation groups. The order of this group is N !. An S 2 group is an Abelian symmetry group and consists of a permutation in the form (x 1 , x 2 ) → (x 2 , x 1 ). Thus, the lowest order non-Abelian group is S 3 .
(ii) Alternating group A N . This group consists of only even permutations of S N and thus, its order is N !/2. The smallest non-Abelian group of this class is A 4 since A 3 ∼ = Z 3 .
(iii) Dihedral group D N . This group is also denoted as ∆(2N ) and its order is 2N . The D N group is isomorphic to Z N Z 2 that consists of the cyclic rotation Z N and its reflections. Note that D 3 ∼ = S 3 .
(iv) Binary Dihedral group Q 2N . This group, which is also called Quaternion group, is a double cover of D N symmetry group and its order is 4N .
(v) Tetrahedral group T N . This group is of order 3N and isomorphic to Z N Z 3 , where N is any prime number. The smallest non-Abelian discrete symmetry of this type is T 7 . This would imply that a T N -symmetric nHDM potential will also be symmetric under Z 7 .
(vi) Dihedral-like groups. These generic groups obey the following isomorphisms: The simplest groups of this type are Σ(2) ∼ = Z 2 , ∆(6) ∼ = S 3 , ∆(24) ∼ = S 4 , ∆(12) ∼ = A 4 , and Σ(24) ∼ = Z 2 × ∆(12). The decomposition of tensor products of 3-dimensional irreducible representations of these groups are given in Appendix D. Note that imposing many of these symmetry groups lead to identical potentials or to potentials that are invariant under continuous symmetries. These non-Abelian discrete symmetries can be the symmetry of nHDM potentials for sufficiently large n, as discussed in Section III.
In the case of the 3HDM, the complete list of non-Abelian discrete symmetries has been reported in [20][21][22][23][24]. There are two non-Abelian discrete symmetries as subgroups of SO (3) where the symmetry groups stacked in curly brackets produce identical potentials. Here, we discuss the cases D 3 and A 4 , while the description of the rest of these types of symmetries for the 3HDM potential may be found in Appendix D.
Let us start with the smallest non-Abelian discrete group D 3 ∼ = S 3 . The irreducible representations of the D 3 symmetry group can be expressed by two singlets, 1 and 1 , and one doublet 2. The 2 ⊗ 2 tensor product of this group decomposes as Moreover, the generators of D 3 discrete symmetry group should satisfy the conditions: g 3 1 = 1, g 2 2 = 1 and g 1 · g 2 = g 2 · (g 1 · g 1 ). Thus, two generators of D 3 in terms of double tensor products are given by where Imposing D 3 on the U(1)-conserving part of the R A 3 vector will lead to the following linear decomposition: (IV.30) Note that in the bilinear space, there are three singlets 1, 1 and 1 , and two doublets, 2 and 2 . Thus, given the irreducible representations in (IV.30), we may parametrize the D 3 -invariant 3HDM potential as follows: This can be rewritten as where λ 1323 is complex while all other couplings are real.
Another example of a non-Abelian discrete symmetry for the 3HDM potential is A 4 , which is a subgroup of SU(3). This symmetry group consists of three singlets, 1, 1 and 1 , and one triplet 3. The 3 ⊗ 3 tensor product of A 4 decomposes: (IV.33) The generators of the A 4 discrete symmetry group in terms of double tensor products are where satisfy the conditions: g 3 1 = g 2 2 = (g 1 · g 2 ) 3 = 1. The A 4 -symmetric blocks in the bilinearspace can be represented as (IV. 36) Thus, an A 4 -invariant 3HDM potential may be written as Equivalently, the A 4 -symmetric potential can be rewritten as follows: In a similar way, the remaining 3HDM potentials that are invariant under non-Abelian discrete symmetries may be obtained. These are presented in Appendix D.
In Tables I and II, we present all SU(2) L -preserving accidental symmetries for the 2HDM and the 3HDM potentials. The 2HDM potential has a total number of 13 accidental symmetries [35], of which 6 preserve U(1) Y [43,48,49] and 7 are custodially symmetric [25]. Given the isomorphism of the Lie algebras: SO(5) ∼ Sp(4), the maximal symmetry group of the 2HDM in the original Φ-field space is G Φ 2HDM = [Sp(4)/Z 2 ] ⊗ SU(2) L [25]. For the case of the 3HDM potential, we find that there exists a total number of 40 SU(2) L -preserving accidental symmetries, of which 18 preserve U(1) Y and 22 are custodially symmetric. The maximal symmetry group of the 3HDM potential in the original Φ-field [25]. Note that the 40 accidental symmetries are subgroups of Sp(6).

V. CONCLUSIONS
The nHDM potentials may realize a large number of SU(2) L -preserving accidental symmetries as subgroups of the symplectic group Sp(2n). We have shown that there are two sets of symmetries: (i) continuous symmetries and (ii) discrete symmetries (Abelian and non-Abelian symmetry groups). For the continuous symmetries, we have offered an algorithmic method that provides the full list of proper, improper and semi-simple subgroups for any given integer n. We have also included all known discrete symmetries in nHDM potentials.
Having defined the bi-adjoint representation of the Sp(2n) symmetry group, we introduced prime invariants and irreducible representations in the bilinear field space to construct the scalar sector of nHDM potentials. These quantities have been systematically used to construct accidentally symmetric nHDM potentials by employing fundamental building blocks that respect the symmetries.
Using the method presented in this paper, we have been able to classify all symmetries and the relations among the theoretical parameters of the scalar potential for the following: (i) the 2HDM and (ii) the 3HDM. For the 2HDM potential, we recover the maximum number of 13 accidental symmetries. For the 3HDM potential, we derive for the first time the complete list of 40 accidental symmetries.
Our approach can be systematically applied to nHDM potentials, with n > 3, once all possible discrete symmetries have been identified. In Section II, we have shown that the potential V n for an nHDM can be written down in the quadratic form with the help of n(2n − 1)-vector R A n as A is the mass matrix and L n AA is a quartic coupling matrix. In the case of 2HDM, the general potential is given by Thus, in the bilinear formalism, the mass M 2 A and the quartic couplings L 2 AA matrices for the 2HDM potential assume the following forms [25]: and Evidently, for a U (1) Y -invariant 2HDM potential, not all the elements of M 2 A and L 2 AA are non-zero, but only those for which A, A = 0, 1, 2, 3.
Appendix B: The bi-adjoint representations of Sp(4) and Sp (6) In Section II, we introduced the Sp(2n) generators in the bi-adjoint representation as The maximal symmetry of the potential in the case of 2HDM is Sp (4). With the help of the above relation, we may derive the following 10 generators in the bi-adjoint representation of Note that these generators are identical to those of the SO(5) group in the fundamental representation [25], thereby establishing the local group isomorphism: Sp(4) ∼ = SO (5).
The other two-dimensional non-Abelian discrete symmetry of the 3HDM is D 4 . The 2 ⊗ 2 tensor product of D 4 decomposes as This group is generated by two generators, that satisfy the conditions: g 4 1 = 1, g 2 2 = 1 and g 2 · g 1 · g 2 = g −1 1 . The irreducible representations of D 4 in the bilinear-space may be given by . (D.5) Hence, the D 4 -invariant 3HDM potential takes on the form: This potential can also be written as where all parameters are real. Likewise, we find the S 4 invariant 3HDM potential. The S 4 group can be defined by the two generators [16][17][18]20], which obey the conditions: g 4 1 = g 3 2 = 1 and g 1 · g 2 2 · g 1 = g 2 . The 3 ⊗ 3 tensor product of S 4 consists of one singlet 1, one doublet 2 and two triplets, 3 and 3 (D.10) The S 4 -symmetric blocks 1, 2, 3 and 3 may be obtained as   .
These symmetries for the 3HDM potential, along with their non-zero theoretical parameters, are presented in Table II.