Electroweak gauge invariant Higgs multiplets
M. Maniatis
 
Black hole singularity resolution from unitarity
    Lucía Menéndez-Pidal
    September 9, 2024
================================================






§ INTRODUCTION


T.D. Lee originally proposed extending the Higgs sector to two Higgs-boson doublets to provide a new source of CP violation <cit.>. This motivation is still today valid. For a review of the two-Higgs doublet model (2HDM) see for instance <cit.>. The 2HDM has been studied with respect to stability, elecroweak symmetry breaking, CP symmetries, as well as general symmetries; see for instance <cit.>. 






Most extensions of Higgs multiplets in the literature focus on singlet and doublet copies.
In supersymmetric models, for instance, additional Higgs-boson singlets and doublets appear in a rather natural way.
As an example let us mention the next-to minimal sypersymmetric model - see the reviews <cit.> - which besides two Higgs-boson doublets contains one complex singlet. 


What about Higgs-boson triplets, quadruplets, and so on? With respect to general multiplets there is a severe restriction arising from the parameter defined as

    ρ = m_W^2/cos^2 (θ_W) m_Z^2 .

Here the masses of the electroweak gauge bosons are denoted by m_W and m_Z and the electroweak mixing angle by θ_W. This parameter is measured to be close to one <cit.>.

As usual, from the kinetic terms of the Higgs multiplets, written gauge covariantly, we compute the masses of the electroweak gauge bosons. From the neutral components of the Higgs multiplets we get contributions to the masses of the electroweak gauge bosons. Numbering the Higgs multiplets with i, denoting their weak isospin by T(i), its 3 component by T_3(i), and the vacuum-expectation value of its neutral component by v_i, the ρ parameter at tree level is given by <cit.>,

    ρ =
    ∑_i ( T(i) (T(i)+1) - T_3^2(i) ) |v_i|^2 /2 ∑_i T_3^2(i)  |v_i|^2 .

From this computation of the ρ parameter follows that 
for Higgs-boson singlet models with T=T_3=0 and Higgs-boson doublet models with T=1/2 and T_3=±1/2 we find 
ρ=1, consistent with experimental observation. One special case is the Standard Model which, since it employs one Higgs-boson doublet, predicts also ρ=1 at tree level.
 

An example of a higher multiplet resulting in ρ=1, as follows from (<ref>), are Higgs septuplets corresponding to T=3 and T_3= ± 2. Another possibility to comply with this constraint is to add multiplets with vanishing vacuum-expectation values of their neutral component. Also it is possible to have a potential with several multiplets and with the vacuum-expectation values arranged by an appropriate potential to eventually give ρ =1.
Examples are the Georgi-Machacek models <cit.> with one Higgs-boson doublet besides Higgs-boson triplets. The potential is arranged in these type of models such that the vacuum-expectation values satisfy at tree level the condition ρ=1. Let us also mention the recent work <cit.>, where a Higgs-boson quadruplet is combined with a doublet.
Eventually, even with (<ref>) in mind, we see that it is at least in principle possible to consider models with more general Higgs-boson multiplets. In this sense, we here would like to study Higgs potentials of arbitrary Higgs multiplets. 


Stability and electroweak symmetry breaking of Higgs-multiplet models have been studied in <cit.> and <cit.>; for the case of an arbitrary number of doublets in <cit.>. Couplings of Higgs multiplets to the electroweak gauge bosons have been
investigated in <cit.>. 
 

Let us briefly return to the Standard Model, where a single Higgs-boson doublet field φ^(2)(x) belongs to 
C^2. This field transforms as a doublet under electroweak  transformations as indicated by the superscript. It is straightforward to introduce a bilinear field,

    K^(2)(x) = φ^(2)^†(x) φ^(2)(x)

which is bilinear in the doublet fields, real, and it is a singlet under electroweak gauge transformations. Bilinear fields have been studied in detail for the case of an arbitrary number of doublets <cit.>. Here we will generalize the bilinear approach to arbitrary multiplets. 
Specifically, the bilinear representation enables us to determine the domain of these multiplets.
This domain can be split into distinct parts with respect to  the behavior of the multiplets with respect to electroweak symmetry breaking, that is, unbroken, fully, and partially broken electroweak symmetry →.

Having found the domain of the multiplets with respect to electroweak-symmetry breaking, we proceed and study the corresponding Higgs potential. 
In the potential we shall consider multiplets in the adjoint, that is, bilinear representation. 
This has many advantages, for instance with respect to the study of symmetries of the potential. For instance, for the case of two Higgs-boson doublets, it has been shown that CP transformations are given by reflections in terms of bilinears <cit.>. Let us mention in this context that for the case of the 2HDM, a description free of gauge redundancies has been given for the complete model <cit.>, including the Yukawa and gauge couplings. 

However, considering potentials of different multiplets it turns out to be possible to form gauge-invariant terms which are not bilinear in all the multiplets. One example is a gauge-invariant tensor product of a pair of Higgs-boson doublets with one Higgs-boson triplet. Motivated by a recent study of Higgs-boson quadruplets and doublets <cit.>, 
we show how these gauge-invariant terms can be formed systematically. The essential idea is to express the multiplets in terms of irreducible symmetric doublet representations.

 
Finally, we use the description of Higgs-boson multiplets in both the adjoint and symmetric representations to examine the general symmetries of the potential.
As an example we study standard CP symmetries for the case of general Higgs multiplets.




§ HIGGS MULTIPLETS


We are considering Higgs-boson multiplets with respect to electroweak gauge symmetry. In analogy with ordinary spin with respect to the group , the Higgs multiplets have to have isospin T=0, 1/2, 1, 3/2, 2, … with the 3 component running from T_3 = -T, -T+1, …, +T. Considering the group U(1)_Y,  each Higgs multiplet is assigned a hypercharge Y. 


The Higgs multiplets corresponding to isospin T have multiplicity m= 2T+1 in analogy with the spin. The charges of the components of the Higgs fields in the fundamental representation follow from the Gell-Mann-Nishijima formula Q = Y + T_3 <cit.>. If we choose, for instance, a hypercharge Y= -T_3 we find the lowest component of the Higgs-multiplet in the fundamental representation, corresponding to -T_3, to be electrically neutral. This of course is only one example and any hypercharge may be assigned to the multiplets. Independent of the assignment of the hypercharge the electric charge decreases from top to bottom in each component by one elementary charge unit.

In general, the tensor product of m-1 copies of doublets of , that is, 2 ⊗ 2 ⊗…⊗ 2 has one totally symmetric irreducible representation of dimension m.  It is very convenient to represent all multiplets in the symmetric representation in order to form gauge-invariant tensor products among them. 

Next, we examine the totally symmetric irreducible representations
 Δ_(a_1, …, a_m-1) in detail. As usual, the round brackets indicate that the tensor is totally symmetric under any permutation of the indices. 
For instance, the  Higgs triplet corresponding to multiplicity m=3 can be represented as,

    Δ_11 = ϕ^1,     Δ_12 = Δ_21 = 1/√(2)ϕ^2,     Δ_22 = ϕ^3

If we assign to the triplet the hypercharge Y=1
we get with the help of the Gell-Mann-Nishijima formula the expressions

    Δ_11 = ϕ^++,     Δ_12 = Δ_21 = 1/√(2)ϕ^+,     Δ_22 = ϕ^(0) ,

that is,

    (Δ_ij) = 
    [      ϕ^++ 1/√(2)ϕ^+; 1/√(2)ϕ^+     ϕ^(0) ].

Similar, we find for a quadruplet in terms of the 
symmetric representation the explicit expressions,

    Δ_111 = ϕ^1,     Δ_112 = Δ_121 = Δ_211  = 
    1/√(3)ϕ^2, 
    Δ_122 = Δ_212 = Δ_221  = 
    1/√(3)ϕ^3,     Δ_222 = ϕ^4 .

In general, the totally symmetric doublet product representation for a multiplet of multiplicity m in terms of the fundamental components reads

    Δ_(1, …, 1_n_1, 
    2, …, 2_m-n_1-1)
    =
    1/√(m-1n_1)ϕ^m-n_1.

Since the doublet representation is totally symmetric as indicated by the round brackets enclosing the indices, this expression holds for all permutations of the indices 1 and 2. 
In Tab. <ref> we summarize the properties of the Higgs multiplets. 





Now we seek to determine the domain of the Higgs multiplets. 












To achieve this we construct the Higgs multiplets in the adjoint representation in an analogous way as outlined in <cit.> for the case of Higgs-boson doublets. The case of three Higgs-boson doublets has been studied in detail in <cit.> and in <cit.> the case of an arbitrary number of doublets is discussed. 
Let us denote by n_m the present number of copies of Higgs-boson multiplets with multiplicity m. In particular n_1  denotes the number of singlets, which carry electroweak isospin T=0, that is, T=0=T_3=0. Of course, singlets are by definition gauge invariant. 
More generally, for the case of n_m copies of Higgs bosons of multiplicity m we build the matrix 

    .ψ^(m) = [   φ_1^(m)^;          ⋮; φ_n_m^(m)^ ]
    =
    [   ϕ^1_1       ⋯   ϕ^m_1;       ⋮       ⋮; ϕ^1_n_m       ⋯ ϕ^m_n_m ]∈ (n_m × m) .

Note that for the entries of this matrix, for instance ϕ^2_3, the subscript labels the number of the copy, whereas the superscript gives the component of this copy of the multiplet.

For arbitrary multiplets the next step is to form from ψ^(m) the gauge-invariant matrix: 

    K^(m) = ψ^(m)ψ^(m)^† =
    [     φ_1^(m)^†φ_1^(m)                    ⋯   φ^(m)_n_m^†φ^(m)_1;                                         ⋱                     ;     φ_1^(m)^†φ^(m)_n                    ⋯ φ^(m)_n_m^†φ^(m)_n_m ]∈ (n_m × n_m) .

By construction, the quadratic, gauge-invariant matrix K^(m) is hermitian, has rank smaller than or equal to m and is positive semidefinite. The rank condition follows from the fact that the (n_m × m) matrix ψ^(m) has trivially a rank smaller than or equal to m, whereas hermiticity and semi definiteness follow directly from the definition of the matrix K^(m).

As an example let us consider the two-Higgs triplet model. The two triplets correspond to T=1 with multiplicity m=3. We have in this case

    ψ^(3) = [ φ^(3)_1^; φ^(3)_2^ ] =
    [ ϕ^1_1 ϕ^2_1 ϕ^3_1; ϕ^1_2 ϕ^2_2 ϕ^3_2 ].

If we assign the hypercharge Y=1 to these two triplets we will have the charges of the components,

    ψ^(3) = [ φ^(3)_1^; φ^(3)_2^ ] =
    [ ϕ^++_1  ϕ^+_1  ϕ^0_1; ϕ^++_2  ϕ^+_2  ϕ^0_2 ].

The matrix K^(3) defined in (<ref>) for the case ψ^(3) in (<ref>) is of dimension 2 × 2 and reads

    K^(3) = ψ^(3)ψ^(3)^† =
    [ φ^(3)_1^†φ^(3)_1 φ^(3)_2^†φ^(3)_1; φ^(3)_1^†φ^(3)_2 φ^(3)_2^†φ^(3)_2 ].


The next step is to write the gauge-invariant bilinear matrix K^(m) for each set of the n_m multiplets in a basis of the unit matrix and the (generalized) Pauli matrices,

    K^(m) = 1/2 K^(m)_αλ_α,     α∈{0, …, n_m^2-1} .

Here we have added to the generalized Pauli matrices the conveniently scaled identity matrix written with an index zero,
 
    λ_α with α∈{0, …, n_m^2-1},     λ_0 = √(2/n_m)_n_m .

 An explicit scheme to construct the generalized Pauli matrices can for instance be found in <cit.>. 
 The matrices λ_α satisfy the equations,
 
    (λ_αλ_β) = 2 δ_αβ,     (λ_α) = √(2 n_m)δ_α 0  .

 Multiplying (<ref>) on both sides with (generalized) Pauli matrices and taking traces we can express the bilinears in terms of the Higgs multiplets,
 
    K^(m)_α =  ( K^(m)λ_α) .

 These bilinears are real, gauge invariant expressions. 

 In our example of two Higgs triplets we find for the real gauge-invariant bilinears explicitly
 
    K^(3)_0 = φ^(3)_1^†φ^(3)_1 + φ^(3)_2^†φ^(3)_2, 
       K^(3)_1 = φ^(3)_1^†φ^(3)_2 + φ^(3)_2^†φ^(3)_1, 
       K^(3)_2 = i φ^(3)_2^†φ^(3)_1 - i φ^(3)_1^†φ^(3)_2, 
       K^(3)_3 = φ^(3)_1^†φ^(3)_1 - φ^(3)_2^†φ^(3)_2 .


Let us determine the domain of the real bilinears K^(m)_α, α∈{0, …, n_m^2-1}. 
From the explicit construction we get for the zero component

    K^(m)_0 =   ( K^(m)λ_0) = √(2)/n_m(
     φ_1^(m)^†φ^(m)_1 + … +  φ^(m)_n_m^†φ^(m)_n_m)

 and we find immediately 
 
    K^(m)_0 ≥ 0 .

 The rank condition (K^(m)) ≤ m translates to the bilinears as follows (here we follow closely <cit.>):
 Since ψ^(m) has trivially a rank smaller than or equal to m, this holds also for the bilinear matrix
 K^(m)=ψ^(m)ψ^(m)^†. We can diagonalize K^(m) by a unitary similarity transformation,
 
    U K^(m) U^† = (κ_1, …, κ_m, 0, …, 0) .

 Positive semidefiniteness translates for the eigenvalues to
 
    κ_i ≥ 0,      i ∈{1, …, m} .

 For any hermitean matrix K^(m)∈ (n_m × n_m) with eigenvalues κ_i we can introduce the symmetric sums,
 
    s_0 = 1,      s_k = ∑_1 ≤ i_1 < i_2 < … < i_k ≤ n_mκ_i_1κ_i_2⋯κ_i_k,      k ∈{1, …, n_m} .

 In particular we have s_0 = 1, s_1 = κ_1 + ⋯ + κ_n_m and
 s_n_m = κ_1 ·…·κ_n_m = (K^(m)). 
 The symmetric sums can be expressed recursively in terms of basis-independent traces of powers of the bilinear matrix <cit.>,
 
    s_0 = 1,     
     s_k = 1/k∑_i=1^k (-1)^i-1 s_k-i( (K^(m))^i),    k ∈{1, …, n_m} .

 Explicitely, the first symmetric sums read
 
    s_0 = 1,
     
       s_1 = (K^(m)) = √(n_m/2) K^(m)_0, 
     
       s_2 = 1/2( ^2(K^(m)) - ((K^(m))^2) ),
     
       s_3 = 1/6( ^3(K^(m)) - 3 ((K^(m))^2)(K^(m)) + 2 ((K^(m))^3) ) .

As stated before, the matrix K^(m) has rank smaller than or equal to m.  Having K^(m) of rank
 r, with 0 ≤ r ≤ m, this is equivalent to the first r symmetric sums positive and the remaining ones vanishing. This can be shown as follows:
Suppose we have
a matrix K^(m) of rank r, 0 ≤ r ≤ m. The number of positive, non-vanishing eigenvalues is therefore r and without loss of generality we set

    κ_i > 0   for  i ∈{1, …, r},     κ_r+1 = κ_r+2= … = κ_n_m = 0 .

From the definition of the symmetric sums in (<ref>) we see that s_k = 0 for k>r since each summand in the sum is formed from a product of k different eigenvalues, where at least one is vanishing. 
Therefore, for a matrix K^(m) with rank 0 ≤ r ≤ m we have

    s_i  >  0   for  i ∈{1, …, r},     
    s_r+1 = s_r+2= … = s_n_m = 0 .

 
Vice versa, suppose we have the conditions for the symmetric sums as given in (<ref>). 
From the definition of the sums we see that starting with the last symmetric sum condition, s_n_m=0, that at least one eigenvalue, say κ_n_m, without loss of generality, has to vanish. Recursively from the preceding condition we find that succesively one more eigenvalue has to vanish. Eventually we find all the equality conditions in (<ref>) to be satisfied.
It remains to be shown that for s_i > 0 with i ∈{1, …, r} follows κ_i >0. This can be proven by induction.
The induction base follows from considering r=1: For s_1>0 we have κ_1 = s_1>0 and this forms the base of induction. Now we assume that for s_i >0, i ∈{1, …, r} it follows that κ_i >0. Under this assumption we have to show that this holds for r+1. We can see this by considering the last condition, that is, s_r+1 = κ_1 ·…·κ_r ·κ_r+1 > 0. Under the assumption κ_1 ·…·κ_r >0 follows κ_r+1 > 0, which completes the proof.

 With these preparations we can now find the domain of the bilinears. Each Higgs multiplet of multiplicity m consists of m complex components, therefore, the domain is φ^(m)∈C^m. 
 Of course, the domain is not unique with respect to gauge symmetries.
 For the general case, where we have n_m Higgs multiplets of multiplicity m, we have the domain ψ^(m)∈C^n_m × m. 
 The bilinears on the other hand are constructed from the bilinear matrix, that is, K^(m) = ψ^(m)ψ^(m)^† of dimension n_m × n_m with rank smaller than or equal to m. Therefore, the domain of the bilinears with respect to n_m Higgs multiplets of multiplicity m is given in terms of the symmetric sums s_i from (<ref>),
 
    s_i ≥ 0   for  i ∈{1, …, m} , 
         
     s_i = 0   for  i ∈{m+1, …, n_m} .

 The expressions for the symmetric sums in terms of bilinears are given in (<ref>). In this way we have found the domain of the multiplets in a gauge-invariant way.
 Let us see this in some examples: For n_1 Higgs complex singlets corresponding to multiplicity m=1, that is,
 φ^(1)_1, …,  φ^(1)_n_1 we have ψ^(1) of rank smaller than or equal to one. Therefore, in terms of bilinears, we have s_1 ≥ 0 that is K^(1)_0 ≥ 0 as well as s_2=0, …, s_n_1=0. 
 The case of doublets has been discussed in terms of bilinears in <cit.>. Let us recall the results. In case of n_2 Higgs-boson doublets, we have to have K^(2) of rank smaller than or equal to two. This gives s_1= K_0 ≥ 0 as well as s_2 ≥ 0 and s_3= … = s_n_2 =0. For the special case n_2=2, that is, the 2HDM we have, for instance, the domain of the bilinears K^(2)_0 ≥ 0, 
 s_2= 1/2 (^2(K^(2)) - ((K^(2))^2)) = 1/4[
 (K^(2)_0)^2 - (K^(2)_1)^2 -(K^(2)_2)^2-(K_3^(2))^2 ] ≥ 0. Note that we can always express the entries of the bilinear matrix K^(m) in terms of bilinear coefficients K^(m)_α. 

Let us recall that the Higgs-boson multiplets play a special role in particle physics since they provide in particular the masses to the fermions. In the Standard Model, for instance, one Higgs-boson doublet provides the masses to all fermions. In the two-Higgs doublet model, 2HDM, where the matrix K^(2) is of rank smaller than or equal to two, the observed electroweak symmetry breaking with a neutral vacuum of the doublets singles out the rank one case. The rank two case correspond to a completely broken electroweak symmetry and the rank zero case corresponds to an unbroken electroweak symmetry. With view at the Gell-Mann-Nishijima formula, we see that 
 for any assignment of hypercharge to a multiplet the electric charge drops by one unit from top to down of the components. This means that we can have at most one neutral component of the Higgs-boson multiplets. In case only the neutral component acquires a non-vanishing vacuum-expectation value at a minimum of the potential, this therefore corresponds to the case of rank one of the matrix K^(m). 
 
It is convenient to separate these different cases corresponding to different rank conditions. We refer to the studies mentioned above for a detailed discussion of this subject in the 2HDM and the nHDM.  In analogy to the nHDM this can be done by imposing Lagrange multipliers ensuring the rank conditions. 
 



§ THE HIGGS MULTIPLET POTENTIAL

 We now consider the general case of a multi-Higgs potential, where we have corresponding to each multiplicity m with m=1, 2, 3, … the number of n_m copies of Higgs-boson multiplets.

In the potential we typically would like to consider only terms up to the fourth power of Higgs-boson fields, since otherwise, we would spoil renormalizability. However, with respect to the analysis here, we are not restricting the order of the potential.
Starting with n_1 copies of Higgs-boson singlets, there are, due to the singlet nature, no restrictions with respect to  symmetry for any term which may appear in the potential. 
As singlets have isospin zero, their hypercharge equals their electric charge. Therefore, gauge invariance imposes the restriction that each potential term must have a total electric charge of zero.
Concerning multiplets with multiplicity larger than one, we consider their irreducible totally symmetric tensor product representations. As we have seen, a multiplet of multiplicity m can be written in the symmetric representation with m-1  indices, 
 Δ_(i_1… i_m-1). Writing all multiplets in this form
 allows us to build in a convenient way all gauge-invariant expressions which may appear in the potential. These multiplets transform in each index in the fundamental representation of . With the index notation borrowed from <cit.> we write conjugated multiplets with upper indices,
 
    Δ^*i_1, …, i_m-1≡  (Δ_i_1, … i_m-1)^*    with  i_k ∈{1,2} .

For instance, we write the anti quadruplet in the form
 Δ^*(ijk). The conjugated multiplets transform under  in every upper index in the anti-fundamental representation. Having the notation with upper and lower indices available, any product of multiplets with paired, that is, contracted indices, one above and one below, ensures invariance under  transformations. The hypercharge of the conjugated multiplets is also opposite to the original multiplets. An example, where we have one copy of the Higgs-boson doublet is the potential of the Standard Model, where a quadratic term is conventionally written as φ^†φ. In the symmetric representation we write this term Δ^*iΔ_i.
This term is manifestly gauge-invariant since the index i is contracted and any hypercharge assigned to the doublet appears once with positive and once with negative sign. Another example is the product Δ_i Δ_j Δ^* ij denoting an -invariant product of two doublets with one triplet. 
It general we have to take care that the assignments of the hypercharges results in total hypercharge zero in every tensor product forming a term of the potential. 
For instance if we assign to the doublets 
Δ_i the hypercharge 1/2 and to the triplet Δ_(ij) the hypercharge +1 we find the expression Δ_i Δ_j Δ^* ij to be gauge-invariant. 
We see at this point that we can form terms exclusively built from an odd number of copies of multiplets of the same multiplicity only if we assign to them a vanishing hypercharge. 

On the other hand, forming tensor products of different multiplicities, we can form odd  terms with non-vanishing hypercharges. An example is the term Δ_i Δ_j Δ^* ij with the appropriate assignments of hypercharge given in the last paragraph.
We can also form fundamental representations from anti-fundamental representations, that is, we can lower indices by applying the epsilon tensor,

    ϵ_ijΔ^* … j …  with ϵ = i σ_2 = [ +0  1; -1  0 ] .

 An expression of this form transforms with respect to the index i in the fundamental representation of .
An example is a term
Δ_j Δ_k Δ^*mϵ_miΔ^* ijk which is invariant under .

Eventually we arrive at the general recipe to construct a gauge invariant potential:



  * Write all multiplets beyond singlets as totally symmetric tensors, Δ_i, Δ_(ij), Δ_(ijk), ….  

  * Form conjugated multiplets with upper indices Δ^*i, Δ^* (ij), 
Δ^*(ijk), ….

  * Build products of the multiplet terms only with contracted indices. Here we consider that indices of the conjugated multiplets can be lowered by multiplying with ϵ = i σ. Singlets have no further restriction in this respect since they do not carry an  index. 

  * We ensure that the assigned hypercharge in every tensor product adds to zero. 


We now would like to illustrate this approach in some examples.
Let us again consider the potential of the Standard Model employing one copy of a Higgs-boson doublet 
φ^(2). In the symmetric representation we write the doublet as Δ_i. We form the conjugated doublet 
Δ^* i and find the well-known, most general potential up to order four,

    V_SM = μ^2  Δ^* iΔ_i + λ (Δ^* iΔ_i)^2 .

Next, let us consider the two-Higgs doublet model: In this case we have two copies of
Higgs-boson doublets which we write as Δ_1 i and Δ_2 i with the additional lower index labelling the number of the copy. The most general potential reads 

    V_2HDM = 
    μ^2_abΔ_a^* iΔ_b  i + 
    λ_abcdΔ_a^* iΔ_b  iΔ_c^* jΔ_d  j .
      with  a,b,c,d ∈{1, 2} .

Note that we have to sum over pairs of  indices i, j, as well as over pairs of the copy labels a, b, c, d.
Extending the range of the indices labelling the copies of the doublets, we can generalize the potential to cases with an arbitrary number of doublets. 

Eventually let us consider the case of one doublet corresponding to m=2 with the assignment of hypercharge +1/2 and one triplet, that is m=3 with hypercharge +1. We find the most general gauge-invariant potential up to order four,

    V_23 = 
    μ^2 Δ^* iΔ_i +  M^2 Δ^* ijΔ_ij + 
    c_1 Δ_ijΔ^* iΔ^* j +
    c_1^*  Δ^* ijΔ_iΔ_j + 
    λ_1 Δ^* iΔ_iΔ^* jΔ_j +
    λ_2 Δ^* ijΔ_ijΔ^* klΔ_kl +
    λ_3 Δ^* ijΔ_jkΔ^* klΔ_li +
    λ_4 Δ^* ijΔ_ijΔ^* kΔ_k .





§ CHANGE OF BASIS


Let us consider now a change of basis. Basis changes are defined as a unitary mixing of the n_m copies of Higgs multiplets in the fundamental representation,
φ^(m)_1,…, φ^(m)_n_m with the same multiplicity m,

    ψ'^(m) = [   φ_1^'(m)^;           ⋮; φ_n_m^'(m)^ ]
    =
    U^(m)[   φ_1^(m)^;          ⋮; φ_n_m^(m)^ ]
    =
    U^(m)ψ^(m) ,     with  U^(m)∈ (n_m × n_m) .

Under this unitary transformation the bilinear matrix transforms, as can be seen directly from its definition (<ref>),

    K'^(m) = U^(m)K^(m)U^(m)^† .

This in turn means for the bilinears; see (<ref>),

    K'^(m)_α =  ( K'^(m)λ_α ) =
     (  K^(m)U^(m)^†λ_α U^(m))
    =
    R_αβ ( K^(m)λ_β ) = R_αβ K^(m)_β ,
    α, β∈{0, …, n_m^2-1} ,

where we have written

    U^(m)^†λ_α U^(m) = R^(m)_αβλ_β .

Since λ_0 is the scaled identity matrix this reads,
 
    K'^(m)_0 = K^(m)_0,      
     K'^(m)_a = R_ab K^(m)_b, 
          a, b ∈{1, …, n_m^2-1}.

 The matrix R^(m) = (R^(m)_ab) has the properties
 
    R^(m)^* = R^(m),     R^(m)^ R^(m) = _n_m^2-1,       (R^(m)) = 1 .

 We see that R^(m)(U^(m)) ⊂ SO(n_m^2-1) but only forms a 
 subset. Similar to the case of doublets we find for any type of mulitplets, that a basis change corresponds in terms of bilinears to a proper rotation.
 
 Under basis changes, the symmetric representations of the multiplets Δ, where we suppress the symmetric  indices, transform in the same way as the multiplets in the (anti) fundamental representation, that is,
 
    Δ^(m)_a → 
     U^(m)_abΔ^(m)_b ,     Δ^(m)*_c →Δ^(m)*_d U^(m)†_dc.

 The indices a, b, c, d label the copies of the multiplets.
 We may consider basis changes of only specific multiplets, or all multiplets. In a model of two Higgs-boson doublets and two Higgs-boson triplets we could for instance be interested in a basis change of the two doublets and/or of the two triplets. 

A Higgs potential is invariant under a change of basis if every gauge-invariant term is invariant. Let us consider a general term built from arbitrary multiplets, 

    λ^a_1… a_n c_1 … c_mΔ^(m_1)_a_1…Δ^(m_n)_a_nΔ^*(m_n+1)_c_1…Δ^*(m_n+m)_c_m.

We emphasize that the multiplets can have different multiplicities indicated by the superscript index, whereas the lower indices a_k and c_k label the copy.   
This potential term is invariant under a changes of bases, (<ref>), if we simultaneously transform the parameter as

    λ^a_1… a_n c_1 … c_m→
    U^(m_n+1)_c_1 d_1… U^(m_n+m)_c_m d_mλ^b_1… b_n d_1 … d_m 
    U^(m_1) †_b_1 a_1… U^(m_n) †_b_n a_n .

 The unitary transformations correspond to the copies present with respect to the multiplets. 

Let us also consider terms of the potential written in terms of the adjoint representation, that is in terms of bilinears, which transform as shown in (<ref>). Let us consider an arbitrary term written in terms of bilinears,

    ξ^μ_1 …μ_n K^(m_1)_μ_1… K^(m_n)_μ_n .

This term is invariant under changes of bases, if we simultaneously with the changes of bases transform the parameter as

    ξ^μ_1 …μ_n→ξ^ν_1 …ν_n 
      R^(m_1) _ν_1 μ_1…
      R^(m_n) _ν_n μ_n
 
Also, we may have mixed terms in the potential written in terms of bilinears and symmetric representations. Then the indices of the parameters have to transform accordingly, that is, every copy index corresponding to a multiplet in the symmetric representation with respect to the unitary transformations, and every index with respect to a bilinear with respect to an orthogonal rotation. 













 

§ SYMMETRIES OF THE HIGGS-MULTIPLET POTENTIAL

 
 Similar to the study of basis transformations we can find an appropriate treatment of symmetries of the Higgs-multiplet potential. Let us mention that this is a generalization of the study of symmetries in the 2HDM potential; see for instance <cit.>. First of all we note that in a Higgs-multiplet model we have in particular to consider the kinetic terms of the Higgs multiplets. Therefore we have the condition that any symmetry transformation of the multiplets must be respected by the corresponding kinetic terms. Apart from a transformation of the space-time argument of the Higss-multiplet fields, any symmetry transformation must therefore be a subgroup of an unitarity transformation. 
 
 For the n_m copies of multiplets with multiplicity m in the fundamental representation, the symmetry transformation extends the basis change transformations (<ref>) by transformation of the argument, that is,
 
    ψ^(m)(x) →ψ'^(m)(x) = 
    U^(m)ψ^(m)(x') ,     with  U^(m)∈ (n_m × n_m) .

For the bilinears, the unitary transformation corresponds to an orthogonal transformation besides the transformation of the space-time argument x → x'; compare with (<ref>),

    K'^(m)_α(x) = 
    X^(m)_αβ K^(m)_β(x')
        α, β∈{0, …, n_m^2-1} .

Generalizing (<ref>), any symmetry transformation should be of the form
 
    K^(m)_0(x)  → K'^(m)_0(x) = K^(m)_0(x'), 
           
        K^(m)_a(x) → K^'(m)_a(x) = X^(m)_ab K^(m)_b(x'),
    
     X^(m) = (X^(m)_ab) ∈ O(n_m^2-1) ,      
     a, b ∈{1, …, n_m^2-1} .

 Note that for any symmetry transformation matrix X^(m) of the bilinears, there is, except for gauge transformation, a unique unitary transformation U; compare with (<ref>) and appendix <ref>, 
 
    U^(m)^†λ_a U^(m) = X^(m)_a bλ_b .

 
 Eventually, we consider the irreducible symmetric representations. Taking the transformation of the argument into account we want to consider the symmetry transformations; compare with (<ref>),

    Δ^(m)_a(x) → 
     U^(m)_abΔ^(m)_b(x') , 
         Δ^(m)*_c(x) →Δ^(m)*_d(x') U^(m)†_dc.

 Note that the indices a, b, c, d label the copies of the multiplets.

Now, a Higgs potential is invariant under a symmetry transformation,
if every gauge-invariant term is invariant. Let us consider a general term build from arbitrary copies of multiplets in the symmetric representation, 

    λ^a_1… a_n c_1 … c_mΔ^(m_1)_a_1(x) …Δ^(m_n)_a_n(x) 
    Δ^*(m_n+1)_c_1(x) …Δ^*(m_n+m)_c_m(x).

We emphasize again that the multiplets can have different multiplicities written as an additional upper index. We consider here the general case of n multiplets in the fundamental and m multiplets in the anti-fundamental representation.
The term (<ref>) is invariant under the symmetry transformation (<ref>) if the parameter fulfills the condition

    λ^a_1… a_n c_1 … c_m 
    =
    U^(m_n+1)_c_1 d_1… U^(m_n+m)_c_m d_mλ^b_1… b_n d_1 … d_m 
    U^(m_1) †_b_1 a_1… U^(m_n) †_b_n a_n .


Similar, we find the condition for a term of the potential written in terms of the adjoint representation, that is in terms of bilinears. Let us consider an arbitrary term written in terms of bilinears,

    ξ^μ_1 …μ_n K^(m_1)_μ_1(x) … K^(m_n)_μ_n(x) .

This term is invariant under the transformation (<ref>), if the 
bilinear parameter fulfills the condition

    ξ^μ_1 …μ_n =
      ξ^ν_1 …ν_n 
      X^(m_1) _ν_1 μ_1…
      X^(m_n) _ν_n μ_n .
 
Also, we may have mixed terms in the potential written in terms of bilinears and symmetric representations. Then the condition of invariance follow from the parameters: Under the transformation with respect to every index corresponding to a symmetric representation, and every index with respect to a bilinear the parameter have to stay invariant. 




















Let us turn to spontaneous symmetry breaking. This refers to the case of a symmetry of the potential, that is, we have a symmetry respected by the potential but not by the vacuum. In the case of a symmetric representation of the Higgs multiplets let us suppose we have a symmetry transformation (<ref>) such that (<ref>) holds. Similar, for the case of potential terms written in bilinear form, we assume to have a transformation (<ref>) such that (<ref>) holds (and similar for the mixed case). The symmetry is often called explicit in this context. For a vacuum of the potential we can then check whether this symmetry is respected. This requires for the multiplets in the symmetric representation,

    ⟨Δ^(m)_a ⟩ = 
     U^(m)_ab⟨Δ^(m)_b ⟩
     ,
       ⟨Δ^(m)*_c ⟩ =
     ⟨Δ^(m)*_d ⟩  U^(m)†_dc,
         
     a,b,c,d ∈{1, …, n_m} .

For the case of bilinear representations we have to check at the vacuum,
 
    ⟨ K^(m)_a ⟩ = X^(m)_ab⟨ K^(m)_b ⟩ ,
        
     a,b ∈{1, …, n_m^2-1} .

In the case that also (<ref>), respectively (<ref>) holds,
the vacuum respects the symmetry, that is, the symmetry is not spontaneously broken. 




§ CP TRANSFORMATIONS IN THE HIGGS MULTIPLET MODEL

As an application of the symmetry study in the last section let us consider the standard CP transformation of the multiplets. The CP transformation is defined by charge conjugation of the multiplets simultaneously with a parity transformation of the space-time argument,

    ψ^(m)(x) →ψ'^(m)(x) = ψ^(m)^*(x'),     with 
    x=(t, x),    x'=(t, -x) .

Similar to the terminology in work <cit.> we call these standard CP transformations in contrast to generalized CP transformations where simultaneously the multiplets are unitarily mixed. 
Indeed it has been shown that the simultaneous mixing of the multiplets can give new types of transformations which do not correspond to standard CP transformations in a different basis.

The bilinear matrix K^(m)(x) transforms under the standard CP transformation as

    K^(m)(x) →K^'(m)(x) = ψ'^(m)(x) ψ'^(m)^†(x) =
    (ψ^(m)(x') ψ^(m)^†(x') )^ 
    = K^(m)^(x') .

For the bilinears we get

    K^(m)_α(x)  → K^'(m)_α(x) =  (K^'(m)(x) λ_α)
    =  (K^(m)(x') λ_α^) =
    C̃^(m)_αβ K^(m)_β(x'), 
    
    α, β∈{0, …, n_m^2-1} ,

where we define the matrix C̃^(m) by

    λ_α^ = C̃^(m)_αβλ_β .

From the properties of the generalized Pauli matrices we find in particular,

    C̃^(m)_00 = 1 ,

and all off-diagonal terms of C̃^(m) vanishing. Therefore, we drop the trivial first row and the first column and define the matrix 
C^(m)=(C^(m)_a b) with a, b ∈{1, …, n_m^2-1}. 
Since the generalized Pauli matrices can be decomposed into symmetric and antisymmetric matrices, we get for the matrices C^(m),

    C^(m) = (± 1, …, ± 1) .

The transformations of the bilinears are then,

    K^(m)_0(x) → K^(m)_0(x’),     
    K^(m)_a(x) → C^(m)_ab K^(m)_b(x’)
    with λ_a^ = C̃^(m)_abλ_b, 
        
    a, b ∈{1, …, n_m^2-1} .

We see that, besides the space-time transformation of the argument of the multiplet fields, CP transformations give proper or improper rotations, depending on the number of symmetric and antisymmetric generalized Pauli matrices. For example, for the cases of two or three copies of multiplets of the same multiplicity we find reflection symmetry transformations, that is, matrices C^(m) with determinant minus one. However, for instance in the case of four or five multiplets of the same multiplicity we can see that the standard CP transformations correspond to proper rotations in bilinear space.
 
 Eventually, the irreducible symmetric representations transform under standard CP transformations as
 
    Δ^(m)_a(x) →Δ^*(m)_a(x') ,     Δ^*(m)_c(x) →Δ^(m)_c(x').

 Note that we have suppressed again the  indices.
Having formulated the standard CP transformation in terms of the bilinears, (<ref>), and in terms of the symmetric representation, (<ref>) we can verify the potential with respect to this symmetry. As discussed in the last section this requires to check the conditions (<ref>) in the case of the symmetric representation, 
respectively (<ref>) in the case of bilinear representations. Note that this study of standard CP transformations can directly be generalized to non-standard CP transformations, where an arbitrary unitary mixing is permitted - in addition to the complex conjugation and the parity transformation of the field argument. 

In the case we have a potential explicitly invariant under standard CP transformations, the conditions (<ref>), respectively (<ref>) determine whether this symmetry is respected by the vacuum.
Explicitly, these conditions for the standard CP symmetry considered here read

    ⟨Δ^(m)_a ⟩ = 
     ⟨Δ^*(m)_a ⟩ .

for the symmetric representations. In the case of bilinears we have the conditions

    ⟨ K^(m)_a ⟩ = C^(m)_a b⟨ K^(m)_b ⟩ .

Of course, (<ref>) and (<ref>) require to minimize the potential.
As an example let us consider the standard CP transformations in the case of n_2 copies of Higgs-boson doublets.
We write the potential in the symmetric representation,

    V_NHDM = 
    μ^2_abΔ_a^* iΔ_b  i + 
    λ_abcdΔ_a^* iΔ_b  iΔ_c^* jΔ_d  j .
      with  a,b,c,d ∈{1, … n_2} .

The potential has to be real, requiring for the quadratic term that we must have μ^2^*_abΔ_a  iΔ_b^* i = 
μ^2_abΔ_a^* iΔ_b  i = μ^2^*_baΔ_a^* iΔ_b  i, that is, 
μ^2_ab = μ^2^*_ba. Similar we have 
 λ_abcd = λ_badc^*. 
The CP transformation (<ref>) is given by Δ_a  i→Δ_a^* i and
Δ_a^* i→Δ_a  i. Using the relation μ^2_ab = μ^2^*_ba we find 
 μ^2_abΔ_a  iΔ_b^* i =
 μ^2^*_abΔ_a  iΔ_b^* i =
 μ^2_abΔ_a  iΔ_b^* i, that is,
 the condition μ^2_ab = μ^2^*_ab as expected. Similar we get the condition
λ_abcd = λ_abcd^*. 
 
 

§ EXAMPLE POTENTIAL OF TWO HIGGS-BOSON DOUBLETS AND THREE TRIPLETS


Let us illustrate the formalism in an explicit example of a potential with two Higgs-boson doublets, 

    φ_1^(2) = 
    [ ϕ_1^1; ϕ_1^2 ],     φ_2^(2) = 
    [ ϕ_2^1; ϕ_2^2 ] .

and three Higgs-boson triplets,

    φ_i^(3) = 
    [ ϕ_i^1; ϕ_i^2; ϕ_i^3 ],      i ∈{1,2, 3} .

This corresponds to n_2=2 and n_3=3. 
Usually we want the potential, as well as the complete Lagrangian of the model to be restricted by a symmetry. For a physically viable model this is typically required since, for instance, for two Higgs-boson doublets, the most general Yukawa couplings would lead to large flavor-changing neutral currents - which have never been observed. 
Here we will assume a specific potential of the form

    V_example = 
    m_2^2 
    (φ_1^(2)^†φ_1^(2)
    + φ_2^(2)^†φ_2^(2)) 
    +
    m_3^2 
    (φ_1^(3)^†φ_1^(3)
    + φ_2^(3)^†φ_2^(3)  
    + φ_3^(3)^†φ_3^(3)) 
    
    
    + λ_2 
    (φ_1^(2)^†φ_1^(2)
    + φ_2^(2)^†φ_2^(2))^2
    +
    λ_3
    (
    φ_1^(3)^†φ_1^(3)
    + φ_2^(3)^†φ_2^(3) 
    + φ_3^(3)^†φ_3^(3))^2
    
    
    + λ_23 
    (φ_1^(2)^†φ_2^(2) + φ_2^(2)^†φ_1^(2))
    · 
    (φ_1^(3)^†φ_2^(3) + φ_2^(3)^†φ_1^(3))

We translate the Higgs-doublet and triplet fields to the bilinear formalism. We form the 2 × 2 matrices ψ^(2) and the 3 × 3 matrices ψ^(3),

    ψ^(2) = [ φ_1^(2)^; φ_2^(2)^ ],     ψ^(3) = [ φ_1^(3)^; φ_2^(3)^; φ_3^(3)^ ] .

Then we can built the bilinear 2 × 2, respectively, 3 × 3 matrices,

    K^(2) = ψ^(2)ψ^(2)^† =
    [ φ_1^(2)^†φ_1^(2) φ_2^(2)^†φ_1^(2); φ_1^(2)^†φ_2^(2) φ_2^(2)^†φ_2^(2) ],
       K^(3) = ψ^(3)ψ^(3)^† =
    [ φ_1^(3)^†φ_1^(3) φ_2^(3)^†φ_1^(3) φ_3^(3)^†φ_1^(3); φ_1^(3)^†φ_2^(3) φ_2^(3)^†φ_2^(3) φ_3^(3)^†φ_2^(3); φ_1^(3)^†φ_3^(3) φ_1^(3)^†φ_2^(3) φ_3^(3)^†φ_3^(3) ] .

Now the bilinears follow in the basis of the Pauli and the Gell-Mann matrices together with the identity matrices σ_0 = _2, and λ_0 = √(2/3)_3,

    K^(2)_α =  ( K^(2)σ_α), 
       α∈{0, …, 3},
       
     K^(3)_β =  ( K^(3)λ_β), 
       β∈{0, …, 8} .

 Explicitly, we find 
 
    K^(2)_0 = φ^(2)_1^†φ^(2)_1 + φ^(2)_2^†φ^(2)_2,           
 K^(2)_1 = φ^(2)_1^†φ^(2)_2 + φ^(2)_2^†φ^(2)_1, 

    K^(2)_2 = i φ_2^(2)^†φ^(2)_1 - i φ^(2)_1^†φ^(2)_2,
       
 K^(2)_3 = φ^(2)_1^†φ^(2)_1 - φ^(2)_2^†φ^(2)_2 ,
 

and
    K^(3)_0 = √(2/3) (
 φ^(3)_1^†φ^(3)_1 + φ^(3)_2^†φ^(3)_2 + φ^(3)_3^†φ^(3)_3 ),
        
 K^(3)_1 = φ^(3)_1^†φ^(3)_2 + φ^(3)_2^†φ^(3)_1, 

    K^(3)_2 = φ^(3)_1^†φ^(3)_3 + φ^(3)_3^†φ^(3)_1, 
      
 K^(3)_3 = φ^(3)_2^†φ^(3)_3 + φ^(3)_3^†φ^(3)_2, 

   K^(3)_4 = i φ^(3)_2^†φ^(3)_1 - i φ^(3)_1^†φ^(3)_2, 
       K^(3)_5 = i φ^(3)_3^†φ^(3)_1 - i φ^(3)_1^†φ^(3)_3, 

    K^(3)_6 = i φ^(3)_3^†φ^(3)_2 - i φ^(3)_2^†φ^(3)_3, 
       K^(3)_7 = φ^(3)_1^†φ^(3)_1 - φ^(3)_2^†φ^(3)_2, 

   K^(3)_8 = 1/√(3) (
 φ^(3)_1^†φ^(3)_1 + 
 φ^(3)_2^†φ^(3)_2 - 
 2 φ^(3)_3^†φ^(3)_3 ) .
  
The inversion of these relations allow us to replace all the scalar products of the type φ^(m)_i^†φ^(m)_j by the bilinears,

    φ^(2)_1^†φ^(2)_1 = 1/2(
    K_0^(2) + K_3^(2)),
       φ^(2)_2^†φ^(2)_1 = 1/2(
    K_1^(2) - i K_2^(2)),
           φ^(2)_2^†φ^(2)_2 = 1/2(
    K_0^(2) - K_3^(2)) ,
    
    and   φ^(3)_1^†φ^(3)_1 = 
    K_0^(3)/√(6) + K_7^(3)/2 +K_8^(3)/2√(3), 
       φ^(3)_2^†φ^(3)_1 = 1/2(
    K_1^(3) - i K_4^(3)),
           φ^(3)_3^†φ^(3)_1 = 1/2(
    K_2^(3) - i K_5^(3)),
       φ^(3)_2^†φ^(3)_2 = 
    K_0^(3)/√(6) - K_7^(3)/2 +K_8^(3)/2√(3),
       φ^(3)_3^†φ^(3)_2 = 1/2(
    K_3^(3) - i K_5^(3)), 
       φ^(3)_3^†φ^(3)_3 = 
    K_0^(3)/√(6) - K_8^(3)/√(3) .

Note that we get the remaining expressions of the scalar products by recognizing that 
(φ_i^(m)^†φ_j^(m))^* = φ^(m)_j^†φ^(m)_i and recalling that the bilinears K_μ^(m) are real.

Now we express the potential (<ref>) in terms of bilinears by using (<ref>), 
(<ref>), giving,

    V_example =
    m_2^2 K^(2)_0 + √(3/2) m_3^2 K^(3)_0 
    + λ_2 (K^(2)_0)^2 
    + 3/2λ_3 (K^(3)_0)^2 
    + λ_23 K^(2)_1 K^(3)_1 .


















We study the potential with respect to standard CP symmetries. From (<ref>) 
we find the explicit transformation matrices for the two doublets and the three triplets,

    C^(2) = (1, -1, 1),     
    C^(3) = (1,1, -1,-1,-1,1, 1) .

We can now check the conditions of the parameters, (<ref>), to see whether or not the potential is explicitly CP conserving. In particular we find the only non-trivial condition
λ_23 C^(2)_11 C^(3)_11 = λ_23. With view at (<ref>) this condition is fulfilled, therefore, the potential is explicitly symmetric under CP transformations.


 

§ CONCLUDSIONS

 
 
There is substantial motivation to explore models with Higgs multiplets beyond singlets and doublets. 
Despite the stringent constraints imposed by the electroweak ρ parameter, viable scenarios still exist where Higgs bosons with higher multiplicities can play a role. 

Building on our previous work, we have demonstrated that gauge redundancies in the description of Higgs multiplets generally obscure the physical interpretation of these models. For an arbitrary number of Higgs-boson multiplets, we have employed gauge-invariant methods to determine the domain, utilizing bilinear representations similarly to the approach used for multiple Higgs-boson doublets.

However, the potential may include terms that, while gauge-invariant, are not bilinear in all the present multiplets. Using methods akin to those discussed in <cit.>, we treated the multiplets within the irreducible symmetric representation, with  indices. Just as Lorentz invariance is reflected by contracted indices, contracted  indices signify  invariance. This approach allows for the construction of all gauge-invariant potential terms for an arbitrary assignment of hypercharge to the multiplets.

Finally, we have investigated general symmetries in terms of bilinears and symmetric representations. Specifically, we illustrated how standard CP transformations manifest in multi-Higgs potentials and provided an explicit example to demonstrate this.



 We are very thankful to Otto Nachtmann, Lohan Sartore, and Kristjan Kannike for valuable comments and suggestions.
 This work is supported by Chile ANID FONDECYT project 1200641. 
 
 
 

§ ONE-TO-ONE MAP OF BILINEARS TO HIGGS MULTIPLETS

 
Here we want to show the one-to-one correspondence of the Higgs bilinears to the Higgs multiplets - except for unphysical gauge degrees of freedom. The proof of this is a generalization of the proof given in theorem 5 of <cit.>.
We consider the general case of n_m copies of Higgs multiplets φ^(m)_1(x), …,  φ^(m)_n_m(x). As outlined above these n_m multiplets of the same type are written together in one matrix,

    ψ^(m) = 
    [   φ_1^(m)^;          ⋮; φ_n_m^(m)^ ] 
    =
    [   ϕ^1_1       ⋯   ϕ^m_1;       ⋮       ⋮; ϕ^1_n_m       ⋯ ϕ^m_n_m ]    ∈ (n_m × m) .


We shall proof now that for any hermitean, positive semidefinite matrix K^(m) of dimension n_m × n_m of rank less than or equal to m, there are Higgs multiplets φ^(m)_1(x), …,  φ^(m)_n_m(x) such that 
K^(m) = ψ^(m)ψ^(m)^†. For a given matrix K^(m) of dimension n_m × n_m the Higgs multiplets are determined uniquely - except for a electroweak gauge transformation .


Suppose we have a hermitean, positive semidefinite matrix K^(m) of dimension n_m × n_m of rank less than or equal to m. We can diagonalize this matrix such that

    K^(m) = W(x)
    ( κ_1(x), …, κ_m(x), 0, …, 0) W^†(x)

The matrices W(x) are unitary matrices of dimension (n_m × n_m) and the eigenvalues fulfill

    κ_i(x) ≥ 0,      i ∈{1, …, m} .

Having diagonalized K^(m) we set 

    ψ(x) = W(x) 
    (
    [ √(κ_1(x))                   0;         0         ⋱         0;         0           √(κ_m(x));                   0;           ]).

We see that ψ(x) has the form (<ref>) and that K^(m)(x) = ψ(x) ψ^†(x).

It remains to show that the ψ(x) fulfilling K^(m)(x) = ψ(x) ψ^†(x) are unique up to gauge transformations. Suppose, we have two field configurations ψ(x) and ψ'(x) of the multiplets giving the same matrix

    K^(m)(x) = ψ(x) ψ^†(x) = ψ'(x) ψ'^†(x)

Diagonalization of K^(m) as in (<ref>) then reads

    ( [ [ κ_1(x)             0;             ⋱;      0        κ_m(x) ]                                                             0;                                                             0                                                             0 ])
      =
    ( W^†ψ) ( W^†ψ)^† =
    ( W^†ψ' ) ( W^†ψ' )^† .

From this follows that we can write 

    ( W^†ψ) = 
    (
    [   χ^(m)_1(x)^;             ⋮; χ^(m)_n_m(x)^;             0 ]),
        ( W^†ψ' ) = 
    ([   χ'^(m)_1(x)^;              ⋮; χ'^(m)_n_m(x)^;              0 ]),

where

    χ^(m)_i^†(x) χ^(m)_j(x) =
    χ'^(m)_i^†(x) χ'^(m)_j(x) = κ_i(x) δ_ij .

From this we conclude that there is a unitary matrix U_G ∈ U(m × m) such that
 
    χ'^(m)_i = U_G χ^(m)_i,      i ∈{1, … m} .

 With this we find for the multiplets ψ' and ψ
 
    W^†(x) ψ'(x) = W^†(x) ψ(x) U_G^ (x),

 and noting that W(x) is unitary we find eventually
 
    ψ'(x) = ψ(x) U_G^ (x),

 that is, as stated, ψ'(x) and ψ(x) are related by a gauge transformation.
 

JHEP

