Symmetry and Mass Degeneration in Multi-Higgs-Doublet Models

We investigate possible symmetry properties of the scalar sector of Multi-Higgs-Doublet Models, and, to some extent, the generalization of such models to gauge groups other than $SU(2)_L\times U(1)_Y$. In models where the ${\cal C}$ (charge conjugation) violating operator $\hat{C}$ is not present, the scalar potential is invariant under a group larger than the gauge group, O(4) when the Higgs fields are doublets. If the Higgs fields develop aligned vacuum expectation values, this symmetry will break to an O(3) subgroup, which in general is further broken by loop corrections involving the gauge bosons. Assuming such corrections are small, the physical properties of the Higgs sector will approximately organize into representations of SO(3). If the vacuum expectation values of the Higgs fields are aligned in the direction of the ${\cal C}$ even fields, the mass spectra of the charged and ${\cal C}$ odd sectors will be degenerate. Moreover, if the Higgs fields develop a pair of non-aligned vacuum expectations values, so that the charge conjugation symmetry is spontaneously broken (but not the U(1) electromagnetic gauge invariance), a pair of light charged Higgs bosons should appear.


Introduction
Future discoveries at the LHC may reveal a sector of scalar particles that is much richer than that of the Standard Model (SM). Some of the scalars may be responsible for generating the masses of fermions and the electroweak bosons [1], whereas others may be responsible for the dark matter [2,3,4,5,6,7,8].
It is natural to classify such scalars according to their properties under the SU (2) associated with the electroweak sector of the Standard Model. In order to be compatible with electroweak precision data, one usually considers only SU (2) doublets and singlets. Even these representations are severely constrained by the data [9,10,11,12].
Going beyond one or two doublets [13,14], one immediately has to face models having a large number of parameters. The structure of such potentials has been studied in [15].
Different doublets could be distinguished via their couplings to fermion fields. This idea is exploited in the so-called Model II version of the two-Higgs-doublet model (2HDM) [13], where one doublet couples to up-type fermions, and the other couples to down-type fermions. Another version of this idea is the one considered in Ref. [7], where each fermion or each family has its own Higgs field.
We shall here consider instead the case when the different doublets can not be distinguished (since we are not considering couplings to the fermions). Thus, we shall here study the symmetries of models with N doublets-it turns out that by setting a certain SO(4)violating parameter λ (3) to zero and assuming vacuum alignment, the spectrum simplifies considerably. In particular, a certain "custodial" SO(3) symmetry [16] leads to a degeneracy between the mass matrix of the C odd (or equivalently, CP odd) and the charged Higgs bosons.
This possibility of a symmetry group of the scalar polynomial which is larger than required by gauge invariance was pointed out by Weinberg [17] many years ago. In the theories considered the extra symmetry was assumed to be a symmetry of any quartic (i.e. renormalizable) potential of the scalar sector.
The Standard Model with its single Higgs doublet is an example of a theory were the most general scalar potential has an extra O(4) symmetry not contained in the SU (2)×U (1) gauge symmetry. On the other hand, an extension of the SM with an extra Higgs doublet (as required if we want to introduce supersymmetry) or more, and with C symmetry, induces a most general scalar potential where the extra O(4) symmetry is broken (before spontaneous symmetry breaking). In our notation, this extra symmetry is broken by the parameter(s) λ (3) , cf. eq. (2.8). Standard renormalizability instructs us to include into the Lagrangian, all terms allowed by the SU (2) × U (1) symmetry, hence it may be inconsistent to leave out terms proportional to λ (3) . The presence of such parameters will in general lead to an order λ (3) tree-level breaking of the additional symmetry. We don't think this in principle is any different from having the symmetry broken by loop corrections. It becomes in any case a question of the magnitude of the perturbation.
However, for renormalization it is only necessary to study the theory with unbroken symmetry, since the renormalization is not changed when v = 0 [26]. Hence, only quadratic counterterms have to be considered in the case of mass relations. Thus, in this case counterterms of the type λ (3) will be irrelevant.

The NHDM potential and Lagrangian
We define the N -Higgs-doublet model, abbreviated NHDM, to be a system of N two-component complex scalar fields Φ 1 , Φ 2 , . . . , Φ N , each with the same transformation property under SU (2) L ×U (1) Y as the Higgs field of the Standard Model, and with dynamics defined by the Lagrangian density where V (Φ 1 , Φ 2 , . . . , Φ N ) is a potential that-in its most general form-is given by (2.5) below. The covariant derivative D µ is defined as where W µ i and B µ are the SU (2) L and U (1) Y gauge fields, respectively, and T i = 1 2 σ i are the generators of SU (2) with σ 1 , σ 2 , σ 3 the Pauli matrices. Thus, our Higgs fields are labeled by two indices: The row index running from 1 to N is often written out explicitly as above, and an often hidden group index acted on by the gauge group. The latter are acted on by the matrices T i in (2.2) (whose indices are also hidden). When written explicitly we shall use Greek letters from the beginning of the alphabet.
To write the most general gauge-invariant potential in a renormalizable NHDM in a compact way, we introduce a set of linearly independent 1 hermitian operators invariant under local SU (2) L × U (1) Y transformations (this is a generalization of the approach for the 2HDM in [18]): 3) Due to (anti-)symmetry under interchange of m and n we may impose the restriction that 1 ≤ m < n ≤ N , and introduce indices a, b, . . . labeling such pairs. An explicit invertible encoding is We let m(a), n(a) denote the inverse of this encoding. We will use the summation convention that repeated indices from the beginning of the alphabet are summed from 1 to N , and repeated indices from the middle of the alphabet are summed from 1 to N . The most general potential in the NHDM thus becomes a linear combination of all different quadratic and quartic factors in the Φ m (and Φ † m ) which can be formed from A m , B a and C a : 2 V g (Φ 1 , . . . , Φ N ) = µ (1) m A m + µ (2) a B a + µ (3) a C a + λ (1) mn A m A n + λ (2) ab B a B b + λ (3) ab C a C b + λ (4) ma A m B a + λ (5) ma A m C a + λ (6) ab B a C b , (2.5) where the "g" in V g denotes "general". To avoid double counting we introduce the restriction m ≤ n in the term involving λ (1) mn , and the restriction a ≤ b in the terms involving λ (2) ab , λ (3) ab and λ (6) ab . We will not consider terms of degree higher than four, because these would destroy the renormalizability of the model [22]. From the hermiticity of the potential V g all parameters µ and λ in the expansion (2.5) must be real. Thus the number of free real parameters in (2.5) is which for N = 1 gives us the 2 parameters of the Standard Model (µ 2 and λ). N = 2 gives us the usual 14 parameters for the 2HDM. There are 54 parameters for N = 3 and 152 parameters for N = 4. This counting does not take into account the fact that we may make SU (N ) row transformations on the fields Φ m to eliminate some terms in (2.5). One possible choice is 1 There are no linear relations between the operators in (2.3). However, they are algebraically dependent when N ≥ 3, being restricted by (N − 2) 2 polynomial equations of 8'th order in the fields. 2 Since to transform the quadratic terms into a diagonal form, i.e. so that µ (2) a = µ (3) a = 0. This in general leaves a matrix of N − 1 independent diagonal phase transformations (such that the determinant is unity). We may for instance use it to transform all λ (5) 1a with n(a) = m(a)+1 to zero. This reduces the number of parameters by N 2 − 1, i.e. to N ′ tot = 1 2 N 4 + N 2 + 2 , yielding 11 parameters for N = 2 (in agreement with Barroso et. al. [14]), 46 parameters for N = 3, and 137 parameters for N = 4.

The most general C-invariant NHDM-potential
The charge conjugation operator C is a linear operator which leaves complex constants unaltered, but maps fields onto their hermitian conjugate; C(zΦ m ) = zΦ † m , where z is a complex number. 3 Then We obtain a C-invariant potential by leaving out all terms which are odd in C a , i.e., terms involving µ ma , and λ (6) ab . There are N + N N + N 2 = 1 4 N (N − 1)(N 2 + N + 2) such terms, leaving free parameters for the general renormalizable C-invariant potential, For N = 1 we get the usual 2 parameters of the standard model, the Higgs potential of which is automatically C-invariant. For N = 2 we get the usual (see e.g., [18]) 10 parameters. For N = 3 we get 33 parameters, and for N = 4 we get 86 parameters. This counting does not take into account that we may make O(N ) transformations on the row of fields Φ m to eliminate some terms in (2.8). A natural choice is to transform the quadratic terms to diagonal form, i.e. so that µ (2) a = 0. This reduces the number of parameters by 1 2 N (N − 1), i.e. to N ′ C = 1 4 N (N 3 + 3N + 4). This gives 9 parameters for N = 2 (in agreement with [14]), 30 parameters for N = 3, and 80 parameters for N = 4.
The difference N phases = N ′ tot − N ′ C = 1 4 N 2 (N 2 − 1) + 1 − N counts the number of genuine C-violating parameters in V g (in agreement with Branco et al. [19]).

Symmetries of A, B, C and C 2
For generality we here consider k (rather than 2)-component fields, i.e. with SU (k) × U (1) as gauge group. To make it simpler to explore all possible symmetries we express the field Φ m in terms of its independent real (hermitian) components, Φ m = Ψ m + iΘ m . Define 2k × 2k matrices where the subscript k indicates the linear dimension of the submatrix involved. The complex scalar product between two fields Φ m and Φ n , invariant under unitary (U (k)) transformations, can be expressed in terms of two real bilinear forms 4 The first is the Euclidean dot product between 2k-component real vectors, the second is the Poisson bracket (symplectic form) of the same quantities viewed as coordinates and momenta of 2k-dimensional phase space. The quantities in (2.10) are individually invariant under transformation groups larger than U (k). The first form B (with A as a special case) is invariant under the O(2k) group of real orthogonal transformations, the second form C is invariant under the Sp(k, R) group of real symplectic transformations, In this formulation the charge conjugation operator C discussed above can be represented as a particular O(2k) transformation when acting on the fields Ψ n and Θ n Considering infinitesimal transformations, O = I + ǫL + O(ǫ 2 ), S = I + ǫM + O(ǫ 2 ), the conditions (2.11) and (2.12) become Thus L must be a 2k × 2k antisymmetric real matrix; there is a set (Lie algebra) so(2k) of 2k 2 − k linearly independent such matrices. Writing out the condition for M in terms of k × k submatrices we find that it must have the form There is a set sp(k) of k 2 + k(k + 1) = 2k 2 + k linearly independent such matrices. The infinitesimal transformations of the original U (k) are the intersection of the sets so(2k) and sp(k). I.e., the matrices of the form A B −B A , with A = −A T and B T = B. There are 1 2 k(k − 1) + 1 2 k(k + 1) = k 2 such linearly independent matrices.
The symmetries of C 2 The form C 2 (or more precisely C mn C m ′ n ′ ) has a bigger symmetry group than the form C. Still, we will see that such operators (forms) will violate the full O(4) symmetry we can assign the rest of the Lagrangian. In analogy with (2.12), C 2 symmetries are given by which can be collected in a set which we in Appendix A show is a Lie group. The component consists of matrices with determinant as shown in appendix A. The group P (k, R) = Sp(k, R) ∪ P − (k, R) will have the same Lie algebra as Sp(k, R), since the new component P − (k, R) is not connected with the identity. This is manifested by the equation corresponding to eq. (2.14), not having any solution for the case of a −J on the right side, see appendix A for a proof.
The most general O(2k)-symmetric potential We can conclude that the most general O(2k)-invariant potential can be written since we have seen that operators not containing any factor C are O(2k)-invariant. We obtain a O(2k)-invariant potential by leaving out terms proportional to λ 2k) . For N = 1 we get the usual 2 parameters of the standard model, the Higgs potential being automatically O(2k)-invariant. For N = 2 we get the usual 9 parameters, one parameter less than for the C-invariant potential. For N = 3 we get 24 parameters, and for N = 4 we get 50 parameters. This counting does again not take into account that we may make O(N ) transformations on the row of fields Φ m to eliminate some terms in (2.21). We may transform the quadratic terms to diagonal form, so that µ (2) a = 0. This reduces the number of parameters by . This gives 8 parameters for N = 2, 21 parameters for N = 3, and 44 parameters for N = 4.

Symmetries of the NHDM potential
Since the NHDM-potential V g is constructed from the invariants (2.3) the symmetries of the latter are reflected in the symmetries of the former, but in a manner depending on details of the construction: 1. If V g depends only on the A m 's, i.e. if only the parameters µ (1) m and λ (1) mn are nonzero, then the symmetry group of V g is at least 5 N m=1 O(2k), since we can make independent transformations on each Φ m .
2. If V g depends only on the A m 's and the B a 's, i.e. for a C-invariant theory (2.8), where in addition the parameters λ  a and λ (2) ab vanish. To analyze this we partition the Φ m 's into sets: If a parameter µ (2) a is nonzero, then the fields Φ m(a) and Φ n(a) belong to the same set, with m(a) and n(a) denoting that m and n are contained in a. If a parameter λ (2) ab is nonzero, then the fields Φ m(a) and Φ n(a) belong to the same set, and the fields Φ m(b) and Φ n(b) belong to the same set. With this partitioning into a maximal number of sets we may make one independent O(2k) transformation for each set.
3. If V g depends only on the C a 's, i.e. with only the parameters µ ab being nonzero, then the symmetry group of V g is at least Sp(k, R). If we (in the same manner as above) can partition the fields into several sets, then we may make independent Sp(k, R) transformations on fields belonging to different sets. However, since the additional symmetries in this case fail to be symmetries of even the zero'th order kinetic term (2.30), their significance is uncertain. 4. With all parameters arbitrary the symmetry group of V g is just the original SU (k) × U (1) gauge symmetry.
In this work we will pay special attention to the second scenario, with k = 2.

Symmetries of the kinetic terms
We now turn to the (global) symmetries of the kinetic terms of the Lagrangian, Let K i denote the terms of the i'th order in the gauge fields. 5 It could possibly be larger, since there might be additional row symmetries transforming fields Φm with different m into each other; such symmetries would require special relations among the parameters µ Consider the transformation of the kinetic terms linear in the gauge fields, K 1 , under the map ρ defined in appendix B. We can then write 6 where the subscript k in (Φ n ) k indicates this is the usual complex Higgs k-plet (in the case k = 2 the usual complex Higgs doublet), while Φ n is the 2k dimensional real vector (2.26), where (Φ n ) k = Ψ n + iΘ n , and where we also use eqs. (B.2), (B.3) and (B.6). In eq. (2.25) we have applied the transformation ρ on the gauge terms G µ defined in eq. (2.24), where T then reads summed over the set of real symmetric generators T s i of SU (k), and W µ I = i ′ i W µ i T a i , summed over the set of imaginary antisymmetric generators T a i . For k = 2 the two sets are respectively 1 2 σ 1 , 1 2 σ 3 and 1 2 σ 2 . Third, we consider the kinetic terms quadratic in the gauge fields, The symmetries of K 0 When we first ignore couplings to the gauge fields the remaining terms can be written where the group index α labels the k components of Φ n = Ψ n + iΘ n . This term is invariant under rotation of all components {Ψ nα , Θ nα } into each other. I.e., the symmetry group of K 0 is O(2kN ). The connected part of this group is SO(2kN ), whose generators are all real antisymmetric matrices, L mn,αβ = −L nm,βα (i.e. L T = −L, where transposition refers to both sets of indices).
The symmetries of K 0 and K 1 : Next, consider the terms linear in the gauge fields again, cf. eq. (2.25), with T given in eq. (2.28).
Consider now an infinitesimal transformation δΦ m,α = L mn,αβ Φ n,β , and T denoting the 2k × 2k antisymmetric matrix in equation (2.28) (in group indices α, β -in addition it is proportional to the N × N unit matrix in row indices). The requirement that this is an infinitesimal symmetry transformation for K 1 is that L T T + T L = 0. Or, when we restrict L to be antisymmetric so that it also is an infinitesimal symmetry transformation for K 0 , L mn,αβ T βγ − T αβ L mn,βγ = 0. (2.32) In order to determine the allowed structure of L, we expand these matrices into terms of definite symmetries (L (s) symmetric, and L (a) antisymmetric) in the mn indices: with S (symmetric) and A (antisymmetric) restricted by the constraint (2.32). The sum runs over all possible combinations of allowed matrices 7 . We next note that the antisymmetric matrices T can be expanded in the set (2.36c) Using (2.36c) we find that X 11 and X 12 must commute with all matrices T a i , T s i (assumed to form an irreducible representation). By Schur's lemma they must then be proportional to the k × k unit matrix, so that S ∝ I and A ∝ J . Thus, the Lie algebra of the symmetry group of K 0 and K 1 consists of elements of the form This is the Lie algebra of U (N ) written in real variables.
The symmetries of K 0 and K 1 in the limit g ′ → 0: A more interesting situation arises if we remove J from the set T , as would apply to the limit g ′ → 0. Then we still find that X 11 + X 22 and X 12 − X 21 must commute with all matrices T a i , T s i , and hence must be proportional to the unit matrix. Further, X 11 − X 22 and X 12 + X 21 must commute with all T a i , but anticommute with all T s i . For k = 2, i.e. for the gauge group SU (2) L × U (1) Y in the limit g ′ → 0, we find that nonzero solutions of (2.32), where the 2 × 2 matrix ε was defined in eq. (2.38). The set G is a basis of generators for SU (2). Thus, the Lie algebra of symmetry generators for K 0 and K 1 in this case consists of elements of the form The results above were again found under the assumption that the fields W µ i are arbitrary, and kept constant under the transformation. Combined SU (k) transformations of the W µ i and the Φ m fields still remain a symmetry. For k = 2 this symmetry is enhanced to (at least) Sp(N ) × SO(4) as g ′ → 0. 8 (In the case g ′ = 0 it is U (N ) × SU (k).) The generators for SO(4) are the 3 generators in G plus the 3 generators for the SU (2) L gauge group (written in real form).
The symmetries of K 2 Next, consider the terms quadratic in the gauge fields cf. eq. (2.29), As in the symmetry analysis of K 1 we want to find all matrices X such that XT 2 = T 2 X.
All matrices X which commute with T will fulfill this criterion (since T 2 can be expanded in a set which consists of products of all possible pairs of matrices from T ). Therefore, the symmetries of K 1 are also symmetries of K 2 .

Spontaneous symmetry breakdown
In this section we return to the case of k = 2, i.e. with SU (2) L × U (1) Y as the gauge group and a row of N Higgs doublets Φ m . Note however that many of our considerations have straightforward generalizations to k > 2.
As for the Standard Model, the potential V g of equation (2.5), or V C of equation (2.8), may acquire its minimum at nonzero values of the scalar fields, Φ 0 = Φ (0) , where Φ (without a lower index) refers to the whole set of fields Φ m . This point, Φ (0) , will belong to one or more manifolds of equivalent minima related by the symmetries of the potential. One may use these symmetries to transform Φ (0) to a particular form. A possible one is to require for Φ with v 1 , u 2 , v 2 and δ real. The special case u 2 = δ = 0 is usually referred to as vacuum alignment, in which case we may also transform v 2 to zero by an orthogonal row transformation involving only Φ 1 and Φ 2 . This is known as the Higgs basis [24]. However, for other purposes it may be more convenient to adopt a "democratic" basis in which the lower component of all (or most) fields Φ m have a nonzero real expectation value. It is related to the Higgs basis by an orthogonal row transformation which preserves the form of C and U (1) electromagnetic gauge transformations (the latter preserving the definition of electric charge). 9 One may collect the quantities Φ (0) mα (α = 1, 2) into two N -component complex vectorsΦ (1) andΦ (2) . By a U (N ) row transformation one may first rotateΦ (1) m so that only the componentΦ real. There is a group of U (N − 1) transformations preserving this condition; this may be used to transformΦ (2) m so that only the componentsΦ 2 real. One cannot do better due to the existence of four real U (N ) invariant parameters in ||Φ (1) ||, ||Φ (2) ||, andΦ (1) †Φ (2) . But there remains a U (N −2) group of transformations preserving this condition which can be used for other purposes. For a SU (k) × U (1) gauge group one may generalize this procedure to k vectorsΦ (j) , j = 1 . . . k.
Assume now the case of vacuum alignment and a potential V O(4) which is O(4) invariant. Then the existence of the vacuum expectation values Φ (0) will break the (explicit) symmetry down to O(3), with the consequence that the Higgs boson particle spectra and other physical properties will organize themselves into multiplets of O(3) (broken by perturbative corrections in g ′ ). The number of broken symmetry generators is 3 whether we consider the symmetry broken from U (2) to U (1) or from O(4) to O(3); this leads to the existence of 3 Higgs ghosts and no extra Goldstone bosons 10 .

Mass-squared matrices
To make these statements slightly more explicit, as needed for calculation of the zero'th order (in g and g ′ ) particle masses, we expand the potential around Φ (0) to second order. There are no first order terms since we are expanding around a minimum. The matrix of second derivatives is the mass-squared matrix M 2 mnαβ . It is restricted by symmetries to have a block diagonal form in the group indices α, β. We use coordinates where Φ m = Ψ m + iΘ m is expressed in terms of four real fields, It is now convenient to represent these on real form as We have the expansion where Φ mρ denotes one of the four possibilities φ m1 , η m , φ m2 , φ m3 , and the subscript 0 indicates that a quantity is evaluated at Φ = Φ (0) . Now a set of generators for SO(4) 11 is where J 1 , J 2 , J 3 will transform the vacuum expectation value Φ (0) , while J 4 , J 5 , J 6 leave it unchanged, cf. Eq. (3.3). In terms of these, the broken generators of the SU (2) × U (1) Y gauge group, written in real form by the transformation ρ defined in (B.1), are and the unbroken U (1) (electromagnetic gauge) generator is there can be no terms in (3.4) mixing the sets {∆φ 1 , there can be no terms in (3.4) mixing ∆φ m1 and ∆φ m2 , and we must have We refer to this as the charged mass-squared matrix. The transformations considered generate a Z 4 subgroup of the U (1) gauge group generated by J 6 , assumed to be a symmetry of V . We have formulated it this way as a reminder that invariance under discrete subgroups may be sufficient to impose useful restrictions on the mass matrices.
(irrespective of how we define C to operate on the charged sector, f.i. ∆φ m1 → ∆φ m1 , ∆φ m2 → −∆φ m2 ) there can be no terms in (3.4) mixing ∆η m and ∆φ m3 . Thus the neutral mass-squared matrix decomposes into two more blocks, a C even and a C odd one, which generate a Z 4 subgroup of the SO(2) symmetry group generated by J 4 , we obtain the relation This explicitly demonstrates mass degeneracy between the charged and the C odd sectors [25]. Especially, if the potential is O(4)-invariant (2.21), that is, we have a C-invariant theory where also the parameters λ (3) ab = 0 [the latter implies (3.11)] 12 , the above symmetry criteria for mass degeneracy are valid. Moreover, since the the renormalization is not changed when the Higgs fields aquire a vacuum expectation value [26], we won't get any mass renormalization counterterms from the quartic operators. So even though O(4)-violating quartic terms proportional to λ (3) ab cannot alone be prohibited by any discrete symmetry imposed on the NHDM Lagrangian [27], they won't be showing up as counterterms when renormalizing the masses. Hence, the mass degeneration (3.12) will only be broken by loop corrections involving gauge bosons, since we get an exact SO(3) symmetry when g ′ = 0. With g ′ = 0 and hence with an approximate SO(3) symmetry, the mass differences of the charged and CP -odd sectors will be of order On the other hand, the SO(3) symmetry between CP -odd and charged sectors could also be broken by counterterms of the type λ (3) ab C 2 , even though these terms are set to zero in the original potential, if we are considering scattering processes and not mass relations.

The Higgs ghosts
Let ∆Φ be chosen so that Φ (0) +ǫ∆Φ+O(ǫ 2 ) is a family of minima related by the symmetry of the potential V , to first order in ǫ. By differentiating this relation with respect to ǫ and then setting ǫ = 0 we find which reflects the fact that the matrix M 2 mnαβ has zero eigenvalues with corresponding eigenvectors ∆Φ nβ . We may take the latter to be ∆Φ (i) ∝ J i Φ (0) for i = 1, 2, 3. Normalized, with a 2 = m v 2 m . The massless excitations in these directions correspond to a triplet of Higgs ghosts. There will be N − 1 additional SO(3) triplets of excitations in directions orthogonal to the ghosts. They correspond to physical particles. There will also be N SO(3) singlets, transforming evenly under C, corresponding to physical particles. In the case of N = 2, the triplet is (H + , H − , A), whereas the singlets are h and H [13].

Non-aligned vacuum expectation values
We have assumed vacuum alignment in much of the previous discussions of this section. The phenomenologically most realistic deviation from this case is that we have a situation with two (real) broken directions, as in (3.1) with u 2 = 0 but δ = 0. This corresponds to a situation which preserves the U (1) electromagnetic gauge symmetry, and its corresponding definition of electric charge, but where the C symmetry is spontaneously broken. In this situation the M 2 ch,mn mass-squared matrix remains in block form, but the C even and odd excitations may mix to give a 2N × 2N mass matrix for the neutral particles. One of the excitations will be massless, corresponding to a neutral Higgs ghost. We assume again that V is invariant under O(4) transformations. The explicit symmetry is now broken down to O(2) ≃ U (1), so that 5 generators are broken. As before, 3 of these will generate excitations which correspond to the Higgs ghosts; the remaining 2 will correspond to nearly massless charged pseudo-Goldstone 13 bosons (massless to zero'th order in g ′ ).
To analyze the situation we again write Φ = Φ (0) + Φ ′ in terms of real fields, J 4 and J 5 are now also broken by the vacuum expectation values. Acting with the broken generators on Φ (0) one finds five eigenvectors of the mass matrix with zero eigenvalues, where a 2 = m v 2 m and b 2 = m w 2 m . These eigenvectors are normalized, but they are not necessarily orthogonal to each other. Their nonvanishing inner products are Here | sin ϑ| > 0, since the vacuum expectation values by assumption are non-aligned. Thus, the orthonormalized eigenvectors corresponding to the Higgs ghosts can be written (2) generators as given by the map (3.6). The two eigenvectors corresponding to the Goldstone modes are orthogonal to those above, They have been orthonormalized. We note that the normalization constant becomes infinite in the limit of aligned vacuum expectation values, sin ϑ → 0. We recall that the set H (1) , H (2) , H (3) , G (1) , G (2) are just numerical eigenvectors of the mass-squared matrix. The corresponding zero mode fields are the quantum fields obtained by projecting Φ ′ on these eigenvectors, The field Φ H (3) is the neutral Higgs ghost field, while the fields Φ H (1) and Φ H (2) form the charged Higgs ghost field, and the fields G (1) and G (2) together form charged Goldstone boson fields. If the vacuum expectation values broke the symmetry in even more directions, as in (3.1) with both u 2 = 0 and δ = 0, the situation would be different: All 6 generators of SO(4) would be broken, 4 of them corresponding to the 4 broken generators of the U (2) gauge group. Thus, there would still be 2 pseudo-Goldstone bosons.

Concluding remarks
In this paper we have analyzed the additional (approximate) symmetries which may arise in multi-Higgs-doublet models, due to the fact that the scalar potential may have more symmetries than required by the imposed gauge invariance. Moreover, for the kinetic terms (as a whole) we found that the symmetry group was U (N ) × SU (k). In the case k = 2 (i.e. scalar doublets) we found that the symmetry group of the kinetic terms, in the limit g ′ → 0, is enhanced to Sp(N ) × SO(4). The most general C invariant Higgs potential (2.8) has the same SO(4) symmetry, only broken by the presence of the operator C 2 , that is, terms proportional to λ (3) ab . In the case where λ (3) ab is set to zero, we have a mass degeneration (3.12) (assuming vacuum alignment) between charged and C odd sectors in the limit g ′ → 0. When we don't have vacuum alignment, but rather two broken (real) directions with the electromagnetic generator left unbroken, a pair of light, charged Higgs bosons should emerge (cf. section 3.3).
A. P (k, R), the symmetry group of C 2 We will here show that the set given in eq. (2.17) is a Lie group: The associative law and the existence of the identity follow from GL 2k (R) (the set of all invertible, real 2k × 2k matrices) being a group. Define The other component of P (k, R) (what we could call P + (k, R)) is Sp(k, R). Then, if S − ∈ P − and S + ∈ Sp(k, R), then we easily see by the definition that S − S + , S + S − ∈ P − (k, R), So the set P (k, R) is closed under group multiplication. This set also includes the inverse of each element. We only have to show this for elements S ∈ X − , since we already know Sp(k, R) is a Lie group. Let S T J S = −J . Then and since we generally have that so S −1 ∈ P − too (still, P − is not a group considered isolated, since it is not closed under group multiplication, and does not include the identity).
We have now derived that P (k, R) is a group. To prove it is a Lie group, we must prove that it is a (topologically) closed subset of GL 2k (R): The determinant of P − (k, R) We will now show that the determinant of the matrices in the set P − (k, R), consisting of the real matrices with the property S T J S = −J , is (−1) k : First, we claim the set P − (k, R) is given by with C defined in eq. (2.13). This is so because if S ′ ∈ P − (k, R), then S ′ C ∈ Sp(k, R) since since all matrices in Sp(k, R) have determinant 1 [20]. The determinant of a n × n matrix A can be written (sum over repeated indices) det(A) = ǫ i 1 ,...,in A 1,i 1 · · · A n,in (A.10) (the Leibniz formula). Then there is only one non-zero term in this sum for the matrix C, so the determinant is given by (no sum over k) det(C) = ǫ 1,2,...,2k C 1,1 C 2,2 · · · C 2k, 2k = 1 k (−1) k = (−1) k . Sp(k, R) and P − (k, R) are not connected We want to show that Sp(k, R) and P − (k, R) are two components of P (k, R), i.e. they are not connected. Connected means the same as path connected for Lie groups. Assume that the two components are connected. Then there has to be a continuous path between e.g. I ∈ Sp(k, R) and R ∈ P − (k, R). Let X(t) be such a path, i.e. X(0) = I and X(1) = R, where X(t) is continuous. Consider the supremum We know that X(1) T J X(1) = −J . Moreover, consider the function

B. The map ρ
We will now introduce a map ρ which lets us easily translate between real and complex formulations of the kinetic terms we are studying. The map ρ preserves both matrix multiplication, addition and the identity. 14 We define ρ as a function from M k (C), the set of all k × k complex matrices, to M 2k (R), the set of all k × k complex matrices by With U a Lie group, ρ is a Lie group isomorphism from U ⊂ M k (C) to ρ[U ]. Now we want to show that the definition of ρ can be extended to vectors so that it preserve products of complex vectors and matrices: Let v be a complex k × 1 vector, let v = v R + iv I , with v R , v I real and define Moreover, let A be a complex k × k matrix and let A = (A R + iA I ), with A R , A I real, then Then,

Introduction
Future discoveries at the LHC may reveal a sector of scalar particles that is much richer than that of the Standard Model (SM). Some of the scalars may be responsible for generating the masses of fermions and the electroweak bosons [1], whereas others may be responsible for the dark matter [2,3,4,5,6,7,8]. Another interesting aspect of these theories is that CP violation is naturally accommodated, including its spontaneous breaking [9]. It is natural to classify such scalars according to their properties under the SU (2) associated with the electroweak sector of the Standard Model. In order to be compatible with electroweak precision data, one usually considers only SU (2) doublets and singlets. Even these representations are severely constrained by the data [10,11,12,13].
Going beyond one or two doublets [14,15,16], one immediately has to face models having a large number of parameters. The structure of such potentials has been studied in [17].
Different doublets could be distinguished via their couplings to fermion fields. This idea is exploited in the so-called Model II version of the two-Higgs-doublet model (2HDM) [14], where one doublet couples to up-type fermions, and the other couples to down-type fermions. Another version of this idea is the one considered in Ref. [7], where each fermion or each family has its own Higgs field.
We shall here consider instead the case when the different doublets can not be distinguished (since we are not considering couplings to the fermions). An introduction of Yukawa couplings would naturally have broken the symmetry among the different doublets [18]. Thus, we shall here study the symmetries of models with N doublets-it turns out that by setting a certain SO(4)-violating parameter λ (3) to zero and assuming vacuum alignment, the spectrum simplifies considerably. In particular, a certain "custodial" SO(3) symmetry [19,20] leads to a degeneracy between the mass matrix of the C odd (or equivalently, CP odd) and the charged Higgs bosons.
This possibility of a symmetry group of the scalar polynomial which is larger than required by gauge invariance was pointed out by Weinberg [21] many years ago. In the theories considered the extra symmetry was assumed to be a symmetry of any quartic (i.e. renormalizable) potential of the scalar sector.
The Standard Model with its single Higgs doublet is an example of a theory were the most general scalar potential has an extra O(4) symmetry not contained in the SU (2)×U (1) gauge symmetry. An extension of the SM with an extra Higgs doublet (as for instance required if we want to introduce supersymmetry) or more, and with C symmetry, also has a quadratic potential which is automatically O(4) symmetric. This is sufficient to enforce O(4) mass relations up to perturbative corrections in the parameter g ′ . (Hence these corrections involve the gauge bosons Z and γ.) For the quartic potential, the extra O(4) symmetry is broken by the parameter(s) λ (3) , cf. eq. (2.8). Standard renormalizability instructs us to include into the Lagrangian, all terms allowed by the SU (2)× U (1) symmetry, hence it may be inconsistent to leave out terms proportional to λ (3) . The presence of such parameters will in general lead to an order λ (3) tree-level breaking of the additional symmetry. This is in principle not different from having the symmetry broken by loop corrections. It becomes in any case a question of the magnitude of the perturbation.
We present a detailed classification of the possible terms in the potential, discussing charge conjugation and how the familiar custodial O(4) symmetry of the SM potential generalizes to an O(4) (or, more generally, O(2k)) symmetry for certain terms of the potential in the NHDM. The maximal O(2k) symmetric potential turns out to be a more constrained potential than the maximal charge-invariant one. The kinetic terms are in general subject to an independent classification, depending on the U (1) hypercharge coupling g ′ . We identify, in certain situations, a charged pair whose mass vanishes with g ′ .
The paper is organized as follows. In section 2 we discuss the general potential and classify the corresponding Lagrangian, including the kinetic terms. In section 3 we discuss spontaneous symmetry breakdown, and in section 4 we conclude. A couple of more matematical discussions are delegated to appendices.

The NHDM potential and Lagrangian
We define the N -Higgs-doublet model, abbreviated NHDM, to be a system of N two-component complex scalar fields Φ 1 , Φ 2 , . . . , Φ N , each with the same transformation property under SU (2) L ×U (1) Y as the Higgs field of the Standard Model, and with dynamics defined by the Lagrangian density is a potential that-in its most general form-is given by (2.5) below. The covariant derivative D µ is defined as where W µ i and B µ are the SU (2) L and U (1) Y gauge fields, respectively, and T i = 1 2 σ i are the generators of SU (2) with σ 1 , σ 2 , σ 3 the Pauli matrices. Thus, our Higgs fields are labeled by two indices: The row index running from 1 to N is often written out explicitly as above, and an often hidden group index acted on by the gauge group. The latter are acted on by the matrices T i in (2.2) (whose indices are also hidden). When written explicitly we shall use Greek letters from the beginning of the alphabet.
To write the most general gauge-invariant potential in a renormalizable NHDM in a compact way, we introduce a set of linearly independent 1 hermitian operators invariant under local SU (2) L × U (1) Y transformations (this is a generalization of the approach for the 2HDM in [22]): 3) Due to (anti-)symmetry under interchange of m and n we may impose the restriction that 1 ≤ m < n ≤ N , and introduce indices a, b, . . . labeling such pairs. An explicit invertible encoding is We let m(a), n(a) denote the inverse of this encoding. We will use the summation convention that repeated indices from the beginning of the alphabet are summed from 1 to N , and repeated indices from the middle of the alphabet are summed from 1 to N . The most general potential in the NHDM thus becomes a linear combination of all different quadratic and quartic factors in the Φ m (and Φ † m ) which can be formed from A m , B a and C a : 2 where the "g" in V g denotes "general". To avoid double counting we introduce the restriction m ≤ n in the term involving λ ab , 1 There are no linear relations between the operators in (2.3). However, they are algebraically dependent when N ≥ 3, being restricted by (N − 2) 2 polynomial equations of 8'th order in the fields.
other quartic invariants may be expressed by those chosen. λ (3) ab and λ (6) ab . We will not consider terms of degree higher than four, because these would destroy the renormalizability of the model [24]. From the hermiticity of the potential V g all parameters µ and λ in the expansion (2.5) must be real. Thus the number of free real parameters in (2.5) is This counting does not take into account the fact that we may make SU (N ) row transformations on the fields Φ m to eliminate some terms in (2.5). One possible choice is to transform the quadratic terms into a diagonal form, i.e. so that µ This in general leaves a matrix of N − 1 independent diagonal phase transformations (such that the determinant is unity). We may for instance use it to transform all λ (5) 1a with n(a) = m(a) + 1 to zero. This reduces the number of parameters by N 2 − 1, i.e. to N ′ tot = 1 2 N 4 + N 2 + 2 , yielding 11 parameters for N = 2 (in agreement with Davidson and Haber [15]), 46 parameters for N = 3, and 137 parameters for N = 4.

The most general C-invariant NHDM-potential
The charge conjugation operator C is a linear operator, multiplicative in the fields, which leaves complex constants unaltered, but maps fields onto their hermitian conjugate transposed; C(zΦ m ) = zΦ † T m , where z is a complex number. 3 Then C( C a ) = − C a , in contrast to C( A m ) = A m , and C( B a ) = B a . We obtain a C-invariant potential by leaving out all terms which are odd in C a , i.e., terms involving µ free parameters for the general renormalizable C-invariant potential, For N = 1 we get the usual 2 parameters of the standard model, the Higgs potential of which is automatically C-invariant. For N = 2 we get the usual (see e.g., [22]) 10 parameters. For N = 3 we get 33 parameters, and for N = 4 we get 86 parameters. This counting does not take into account that we may make O(N ) transformations on the row of fields Φ m to eliminate some terms in (2.8). A natural choice is to transform the quadratic terms to diagonal form, i.e. so that µ (2) a = 0. This reduces the number of parameters by 1 2 N (N − 1), i.e. to N ′ C = 1 4 N (N 3 + 3N + 4). This gives 9 parameters for N = 2 (in agreement with [16]), 30 parameters for N = 3, and 80 parameters for N = 4.
The difference N phases = N ′ tot − N ′ C = 1 4 N 2 (N 2 − 1) + 1 − N counts the number of genuine C-violating parameters in V g (in agreement with Branco et al. [23]).

Symmetries of A, B, C and C 2
For generality we here consider k (rather than 2)-component fields, i.e. with SU (k) × U (1) as gauge group. To make it simpler to explore all possible symmetries we express the field Φ m in terms of its independent real (hermitian) components, Φ m = Ψ m + iΘ m . Define 2k × 2k matrices where the subscript k indicates the linear dimension of the submatrix involved. The complex scalar product between two fields Φ m and Φ n , invariant under unitary (U (k)) transformations, can be expressed in terms of two real bilinear forms 4 The first is the Euclidean dot product between 2k-component real vectors, the second is the Poisson bracket (symplectic form) of the same quantities viewed as coordinates and momenta of 2k-dimensional phase space. The quantities in (2.10) are individually invariant under transformation groups larger than U (k). The first form B (with A as a special case) is invariant under the O(2k) group of real orthogonal transformations, the second form C is invariant under the Sp(k, R) group of real symplectic transformations, In this formulation the charge conjugation operator C discussed above can be represented as a particular O(2k) transformation when acting on the fields Ψ n and Θ n Thus L must be a 2k × 2k antisymmetric real matrix; there is a set (Lie algebra) so(2k) of 2k 2 − k linearly independent such matrices. Writing out the condition for M in terms of k × k submatrices we find that it must have the form There is a set sp(k) of k 2 + k(k + 1) = 2k 2 + k linearly independent such matrices. The infinitesimal transformations of the original U (k) are the intersection of the sets so(2k) and sp(k). I.e., the matrices of the form A B −B A , with A = −A T and B T = B. There are 1 2 k(k − 1) + 1 2 k(k + 1) = k 2 such linearly independent matrices.
The symmetries of C 2 The form C 2 (or more precisely C mn C m ′ n ′ ) has a bigger symmetry group than the form C. Still, we will see that such operators (forms) will violate the full O(4) symmetry we can assign the rest of the Lagrangian. In analogy with (2.12), C 2 symmetries are given by as shown in appendix A. The group P (k, R) = Sp(k, R) ∪ P − (k, R) will have the same Lie algebra as Sp(k, R), since the new component P − (k, R) is not connected with the identity. This is manifested by the equation corresponding to eq. (2.14), not having any solution for the case of a −J on the right side, see appendix A for a proof.
The most general O(2k)-symmetric potential We can conclude that the most general O(2k)-invariant potential can be written since we have seen that operators not containing any factor C are O(2k)-invariant. We obtain an O(2k)-invariant potential by leaving out terms proportional to λ free parameters for the general renormalizable O(2k)-invariant potential V O(2k) . For N = 1 we get the usual 2 parameters of the standard model, the Higgs potential being automatically O(2k)-invariant. For N = 2 we get the usual 9 parameters, one parameter less than for the C-invariant potential. For N = 3 we get 24 parameters, and for N = 4 we get 50 parameters. This counting does again not take into account that we may make O(N ) transformations on the row of fields Φ m to eliminate some terms in (2.21). We may transform the quadratic terms to diagonal form, so that µ

Symmetries of the NHDM potential
Since the NHDM-potential V g is constructed from the invariants (2.3) the symmetries of the latter are reflected in the symmetries of the former, but in a manner depending on details of the construction: ab vanish. To analyze this we partition the Φ m 's into sets: If a parameter µ (2) a is nonzero, then the fields Φ m(a) and Φ n(a) belong to the same set, with m(a) and n(a) denoting that m and n are contained in a. If a parameter λ (2) ab is nonzero, then the fields Φ m(a) and Φ n(a) belong to the same set, and the fields Φ m(b) and Φ n(b) belong to the same set. With this partitioning into a maximal number of sets we may make one independent O(2k) transformation for each set.
3. If V g depends only on the C a 's, i.e. with only the parameters µ ab being nonzero, then the symmetry group of V g is at least Sp(k, R). If we (in the same manner as above) can partition the fields into several sets, then we may make independent Sp(k, R) transformations on fields belonging to different sets. However, since the additional symmetries in this case fail to be symmetries of even the zero'th order kinetic term (2.30), their significance is uncertain. 5 It could possibly be larger, since there might be additional row symmetries transforming fields Φm with different m into each other; such symmetries would require special relations among the parameters µ 4. With all parameters arbitrary the symmetry group of V g is just the original SU (k) × U (1) gauge symmetry.
In this work we will pay special attention to the second scenario, with k = 2.

Symmetries of the kinetic terms
We now turn to the (global) symmetries of the kinetic terms of the Lagrangian, Let K i denote the terms of the i'th order in the gauge fields. Consider the transformation of the kinetic terms linear in the gauge fields, K 1 , under the map ρ defined in appendix B. We can then write 6 where the subscript k in (Φ n ) k indicates this is the usual complex Higgs k-plet (in the case k = 2 the usual complex Higgs doublet), while Φ n is the 2k dimensional real vector where (Φ n ) k = Ψ n + iΘ n , and where we also use eqs. (B.2), (B.3) and (B.6). In eq. (2.25) we have applied the transformation ρ on the gauge terms G µ defined in eq. (2.24), where T then reads with W µ R = ′ i W µ i T s i , summed over the set of real symmetric generators T s i of SU (k), and W µ I = i ′ i W µ i T a i , summed over the set of imaginary antisymmetric generators T a i . For k = 2 the two sets are respectively 1 2 σ 1 , 1 2 σ 3 and 1 2 σ 2 . Finally, we consider the kinetic terms quadratic in the gauge fields, The symmetries of K 0 When we first ignore couplings to the gauge fields the remaining terms can be written where the group index α labels the k components of Φ n = Ψ n + iΘ n . This term is invariant under rotation of all components {Ψ nα , Θ nα } into each other. I.e., the symmetry group of K 0 is O (2kN ). The connected part of this group is SO(2kN ), whose generators are all real antisymmetric matrices, L mn,αβ = −L nm,βα (i.e. L T = −L, where transposition refers to both sets of indices).
The symmetries of K 0 and K 1 : Next, consider the terms linear in the gauge fields again, cf. eq. (2.25), with T given in eq. (2.28). Consider now an infinitesimal transformation δΦ m,α = L mn,αβ Φ n,β , and T denoting the 2k × 2k antisymmetric matrix in equation (2.28) (in group indices α, β -in addition it is proportional to the N × N unit matrix in row indices). The requirement that this is an infinitesimal symmetry transformation for K 1 is that L T T + T L = 0. Or, when we restrict L to be antisymmetric so that it also is an infinitesimal symmetry transformation for K 0 , L mn,αβ T βγ − T αβ L mn,βγ = 0. (2.32) In order to determine the allowed structure of L, we expand these matrices into terms of definite symmetries (L (s) symmetric, and L (a) antisymmetric) in the mn indices: with S (symmetric) and A (antisymmetric) restricted by the constraint (2.32). The sum runs over all possible combinations of allowed matrices 7 . We next note that the antisymmetric matrices T can be expanded in the set By substituting (2.33) into (2.32) we are led to search for the set of 2k × 2k real matrices S and A which commute with T for arbitrary values of the fields W µ i and B µ . It is sufficient to verify that this property holds for all elements of the set T . Let Requiring commutativity [see eqs. (2.32) and (2.33)] with the three types of matrices in T we obtain the conditions (2.36c) Using (2.36c) we find that X 11 and X 12 must commute with all matrices T a i , T s i (assumed to form an irreducible representation). By Schur's lemma they must then be proportional to the k × k unit matrix, so that S ∝ I and A ∝ J . Thus, the Lie algebra of the symmetry group of K 0 and K 1 consists of elements of the form This is the Lie algebra of U (N ) written in real variables.
The symmetries of K 0 and K 1 in the limit g ′ → 0: A more interesting situation arises if we remove J from the set T , as would apply to the limit g ′ → 0. Then we still find that X 11 + X 22 and X 12 − X 21 must commute with all matrices T a i , T s i , and hence must be proportional to the unit matrix. Further, X 11 − X 22 and X 12 + X 21 must commute with all T a i , but anticommute with all T s i . For k = 2, i.e. for the gauge group SU (2) L × U (1) Y in the limit g ′ → 0, we find that nonzero solutions of (2.32), where the 2 × 2 matrix ε was defined in eq. (2.38). The set G is a basis of generators for SU (2). Thus, the Lie algebra of symmetry generators for K 0 and K 1 in this case consists of elements of the form In the more general case of an SU (k) × Sp(N ) symmetry of K 0 and K 1 in the limit g ′ → 0 there is, in the same manner as above, a "custodial" SU (k) × SU (2) R symmetry, which also will contain SU (2) × SU (2) R ∼ = SO(4) subgroups.
The symmetries of K 2 Next, consider the terms quadratic in the gauge fields cf. eq. (2.29), As in the symmetry analysis of K 1 we want to find all matrices X such that XT 2 = T 2 X.
All matrices X which commute with T will fulfill this criterion (since T 2 can be expanded in a set which consists of products of all possible pairs of matrices from T ). Therefore, the symmetries of K 1 are also symmetries of K 2 .

Spontaneous symmetry breakdown
In this section we return to the case of k = 2, i.e. with SU (2) L × U (1) Y as the gauge group and a row of N Higgs doublets Φ m . Note however that many of our considerations have straightforward generalizations to k > 2.
As for the Standard Model, the potential V g of equation (2.5), or V C of equation (2.8), may acquire its minimum at nonzero values of the scalar fields, Φ 0 = Φ (0) , where Φ (without a lower index) refers to the whole set of fields Φ m . This point, Φ (0) , will belong to one or more manifolds of equivalent minima related by the symmetries of the potential. One may use these symmetries to transform Φ (0) to a particular form. A possible one is to require for Φ with v 1 , u 2 , v 2 and δ real. The special case u 2 = δ = 0 is usually referred to as vacuum alignment, in which case we may also transform v 2 to zero by an orthogonal row transformation involving only Φ 1 and Φ 2 . This is known as the Higgs basis [26]. However, for other purposes it may be more convenient to adopt a "democratic" basis in which the lower component of all (or most) fields Φ m have a nonzero real expectation value. It is related to the Higgs basis by an orthogonal row transformation which preserves the form of C and U (1) electromagnetic gauge transformations (the latter preserving the definition of electric charge). Assume now the case of vacuum alignment and a potential V O(4) which is O(4) invariant. Then the existence of the vacuum expectation values Φ (0) will break the (explicit) symmetry down to O(3), with the consequence that the Higgs boson particle spectra and other physical properties will organize themselves into multiplets of O(3) (broken by perturbative corrections in g ′ ). The number of broken symmetry generators is 3 whether we consider the symmetry broken from U (2) to U (1) or from O(4) to O(3); this leads to the existence of 3 Higgs ghosts and no extra (pseudo-) Goldstone bosons 11 .

Mass-squared matrices
To make these statements slightly more explicit, as needed for calculation of the zero'th order (in g and g ′ ) particle masses, we expand the potential around Φ (0) to second order. There are no first order terms since we are expanding around a minimum. The matrix of second derivatives is the mass-squared matrix M 2 mnαβ . It is restricted by symmetries to have 10 One may collect the quantities Φ (0) mα (α = 1, 2) into two N -component complex vectorsΦ (1) andΦ (2) . By a U (N ) row transformation one may first rotateΦ (1) m so that only the componentΦ real. There is a group of U (N − 1) transformations preserving this condition; this may be used to transformΦ (2) m so that only the componentsΦ a block diagonal form in the group indices α, β. We use coordinates where Φ m = Ψ m + iΘ m is expressed in terms of four real fields, It is now convenient to represent these on real form as We have the expansion where Φ mρ denotes one of the four possibilities φ m1 , η m , φ m2 , φ m3 , and the subscript 0 indicates that a quantity is evaluated at Φ = Φ (0) . Now a set of generators for SO(4) 12 is where J 1 , J 2 , J 3 will transform the vacuum expectation value Φ (0) , while J 4 , J 5 , J 6 leave it unchanged, cf. Eq. (3.3). In terms of these, the broken generators of the SU (2) × U (1) Y gauge group, written in real form by the transformation ρ defined in (B.1), are and the unbroken U (1) (electromagnetic gauge) generator is there can be no terms in (3.4) mixing ∆φ m1 and ∆φ m2 , and we must have We refer to this as the charged mass-squared matrix. The transformations considered generate a Z 4 subgroup of the U (1) gauge group generated by J 6 , assumed to be a symmetry of V . We have formulated it this way as a reminder that invariance under discrete subgroups may be sufficient to impose useful restrictions on the mass matrices.
(irrespective of how we define C to operate on the charged sector, e.g. ∆φ m1 → ∆φ m1 , ∆φ m2 → −∆φ m2 ) there can be no terms in (3.4) mixing ∆η m and ∆φ m3 . Thus the neutral mass-squared matrix decomposes into two more blocks, a C even and a C odd one, If V in addition is invariant under the transformations which generate a Z 4 subgroup of the SO(2) symmetry group generated by J 4 , we obtain the relation This explicitly demonstrates mass degeneracy between the charged and the C odd sectors [27]. Especially, if the potential is O(4)-invariant (2.21), that is, we have a C-invariant theory where also the parameters λ ab = 0 [the latter implies (3.11)] 13 , the above symmetry criteria for mass degeneracy are valid. Moreover, since the renormalization is not changed when the Higgs fields acquire a vacuum expectation value [28], we do not get any mass renormalization counterterms from the quartic operators. So even though O(4)-violating quartic terms proportional to λ (3) ab cannot alone be prohibited by any discrete symmetry imposed on the NHDM Lagrangian [29], they will not show up as counterterms when renormalizing the masses. Hence, the mass degeneration (3.12) will only be broken by loop corrections involving gauge bosons, since we get an exact SO(3) symmetry when g ′ = 0. With g ′ = 0 and hence with an approximate SO(3) symmetry, the mass differences of the charged and CP -odd sectors will be of order O(g ′4 ) ∝ O(e 4 ).
On the other hand, the SO(3) symmetry between CP -odd and charged sectors could also be broken by counterterms of the type λ (3) ab C 2 , even though these terms are set to zero in the original potential, if we are considering scattering processes and not mass relations. 13 For supersymmetric theories we typically have λ

The Higgs ghosts
Let ∆Φ be chosen so that Φ (0) +ǫ∆Φ+O(ǫ 2 ) is a family of minima related by the symmetry of the potential V , to first order in ǫ. By differentiating this relation with respect to ǫ and then setting ǫ = 0 we find which reflects the fact that the matrix M 2 mnαβ has zero eigenvalues with corresponding eigenvectors ∆Φ nβ . We may take the latter to be ∆Φ (i) ∝ J i Φ (0) for i = 1, 2, 3. Normalized, with a 2 = m v 2 m . The massless excitations in these directions correspond to a triplet of Higgs ghosts. There will be N − 1 additional SO(3) triplets of excitations in directions orthogonal to the ghosts. They correspond to physical particles. There will also be N SO(3) singlets, transforming evenly under C, corresponding to physical particles. In the case of N = 2, the triplet is (H + , H − , A), whereas the singlets are h and H [14].

Non-aligned vacuum expectation values
We have assumed vacuum alignment in much of the previous discussions of this section. The phenomenologically most realistic deviation from this case is that we have a situation with two (real) broken directions, as in (3.1) with u 2 = 0 but δ = 0. This corresponds to a situation which preserves the U (1) electromagnetic gauge symmetry, generated by J 6 . Its corresponding definition of electric charge is preserved, but the C symmetry (or, here equivalently 14 , CP symmetry) is spontaneously broken. In this situation, assuming the potential only has the SU (2) × U (1) gauge symmetry, the three SU (2) generators are spontaneously broken, and we hence only get three Higgs ghosts and no (pseudo-) Goldstone bosons.
When we only have an SU (2) × U (1) symmetry in the potential, operators of the type C are present in the quadratic or in the quartic part of the potential. Two of the excitations of the charged mass squared matrix are still massless, corresponding to the usual charged Higgs ghosts. 15 In the case of non-aligned VEVs (or operators C present in the quadratic part of the potential) the C even and odd excitations generally mix to give a 2N × 2N mass squared matrix for the neutral particles. Here one of the excitations will be massless, corresponding to a neutral Higgs ghost.
On the other hand, assume that V is invariant under O(4) transformations. Then there are no operators of the type C present in the potential. The explicit O(4) symmetry is now broken down to O(2) ≃ U (1), so that 5 generators are broken. As before, 3 of these will generate excitations which correspond to the Higgs ghosts; the remaining 2 will correspond to nearly massless charged pseudo-Goldstone 16 bosons (massless to zero'th order in g ′ ).
To analyze the situation we again write Φ = Φ (0) + Φ ′ in terms of real fields, J 4 and J 5 are now also broken by the vacuum expectation values. Acting with the broken generators on Φ (0) one finds five eigenvectors of the mass matrix with zero eigenvalues, we can re-implement the matrix U C as U CP above, and re-implement P as the trivial transformation (3.16), and hence condition (1) of SCV is satisfied. Assume condition (2) of SCV does not hold. Then we, in a similar manner as for the case (SCV⇒SCPV), can let U CP = U C and define P as in (3.16), and hence SCPV does not hold either, which is a contradiction. Hence condition (2) of SCV is satisfied. 15 The relation between the complex and real formulations of the charged mass squared matrix is given by the map ρ of appendix B: Let M 2 c denote the N × N complex mass squared matrix, and let M 2 r be the corresponding 2N × 2N real matrix. They are related by as can be seen from eq. (B.6). Then ρ(M 2 c ) = M 2 r . The matrix M 2 c is Hermitian, and hence has only real eigenvalues. It then follows from the definition of ρ that λ is an eigenvalue of M 2 c if and only if λ is an eigenvalue of M 2 r . Moreover, if v is an eigenvector of M 2 c , (Re v, Im v) T and (Im v, −Re v) T will be eigenvectors of M 2 r with the same eigenvalue λ. Hence oppositely charged particles will have the same mass, although terms φ1m and φ2n mix by the presence of operators Cmn in the quadratic part of the potential or by complex (i.e. non-aligned) VEVs, and hence violate C. (This mass degeneration is a consequence of the remaining U (1) ∼ = SO(2) hypercharge symmetry, i.e. that the generator J6 is unbroken.) The identification (3.8) assumed that φ1m and φ2n did not mix, i.e. that Im(M 2 c ) = 0, cf. (B.1). 16 Pseudo-Goldstone bosons stem from broken generators of the extra O(4) symmetry of the potential, while Higgs ghosts by definition are generated by the broken generators of the gauge symmetry (which of course is a symmetry of the whole Lagrangian). The pseudo-Goldstone bosons acquire small masses from radiative corrections, and are hence not massless to all orders of perturbation theory, like true Goldstone bosons. True Goldstone bosons are, in contrast, generated by the spontaneous breaking of a symmetry of a total Lagrangian, not only a potential.

Concluding remarks
We have analyzed the additional (approximate) symmetries which may arise in multi-Higgsdoublet models, due to the fact that the scalar potential may have more symmetries than required by the imposed gauge invariance. Moreover, for the kinetic terms we found that the symmetry group is SU (k) × U (N ). In the limit g ′ → 0 the symmetry group of the kinetic terms is enhanced to SU (k) × Sp(N ), which has an SU (k) × SU (2) subgroup. In the case k = 2 the latter is the SU (2) L × SU (2) R ∼ = SO(4) custodial symmetry. The most general C invariant Higgs potential (2.8) has the same SO(4) symmetry, only broken by the presence of the operator C 2 , that is, terms proportional to λ (3) ab . In the case where λ (3) ab is set to zero, we have an exact mass degeneration (3.12) (assuming vacuum alignment) between charged and C odd sectors in the limit g ′ → 0. When there is no vacuum alignment, but rather two broken (real) directions with the electromagnetic generator left unbroken, a pair of light, charged Higgs bosons emerge (cf. section 3.3).
The introduction of Yukawa couplings could further constrain the theory. With N doublets, one could imagine "simplified" models analogous to Model I and Model II for the 2HDM, where only one doublet couples to all fermions, or where some doublets couple to up-type quarks, with others coupling to down-type fermions. Furthermore, with three or more doublets, one could arrange to let each fermion generation couple to its own doublet.
If n doublets are without any Yukawa couplings, for example due to a discrete Z 2 symmetry, Φ i → −Φ i , i = 1, . . . , n (4.1) then such a sector would provide a dark matter candidate [5]. Indeed, with n > 1, there would be a whole "family" of states in this "inert" sector, some of which would carry electric charge. Those would therefore be observable.
A. P (k, R), the symmetry group of C 2 We will here show that the set so S −1 ∈ P − too (still, P − is not a group considered isolated, since it is not closed under group multiplication, and does not include the identity).
We have now derived that P (k, R) is a group. To prove it is a Lie group, we must prove that it is a (topologically) closed subset of GL 2k (R): f (A) = A T J A is a continuous map, the set {±J } is closed in GL 2k (R), and hence P (k, R) = f −1 [{±J }] is closed in GL 2k (R).
The determinant of P − (k, R) We will now show that the determinant of the matrices in the set P − (k, R), consisting of the real matrices with the property S T J S = −J , is (−1) k : First, we claim the set P − (k, R) is given by with C defined in eq. (2.13). This is so because if S ′ ∈ P − (k, R), then S ′ C ∈ Sp(k, R) since since all matrices in Sp(k, R) have determinant 1 [32]. The determinant of an n × n matrix A can be written (sum over repeated indices) det(A) = ǫ i 1 ,...,in A 1,i 1 · · · A n,in (A.10) (the Leibniz formula). Then there is only one non-zero term in this sum for the matrix C, so the determinant is given by (no sum over k) det(C) = ǫ 1,2,...,2k C 1,1 C 2,2 · · · C 2k, 2k = 1 k (−1) k = (−1) k . Then, since the left hand side of eq. (B.6) equals its real part.