Generalised P and CP transformations in the 3-Higgs-doublet model

We study generalised P and CP transformations in the three-Higgs-doublet model (3HDM) with Higgs and gauge fields only. We find that there are two equivalence classes, with respect to flavour transformations, of generalised P transformations and there is only one class of CP transformations. We discuss the conditions the potential has to satisfy in order to be invariant under these transformations. We apply the method of bilinears which we briefly review. We discuss the relation to the conventional basis, where the potential is written in terms of scalar products of the doublet fields. In particular we reproduce the known result that a potential is invariant under CP transformations if and only if there is a conventional basis where all parameters are real. Eventually we study standard P and CP transformations in the $n$-Higgs-doublet model (nHDM). We show that for the bilinears of the nHDM the standard CP transformation corresponds to a diagonal linear transformation with only $\pm 1$ as diagonal elements. We give this matrix explicitly for arbitrary $n$.


INTRODUCTION
One motivation, decades ago, to study models with an extension of the number of Higgs-boson doublets was to investigate possible sources of CP violation. In [1] it was shown that in a model with more than one Higgs field one can have spontaneous CP violation. In [2] not only the famous Cabibbo-Kobayashi-Maskawa (CKM) matrix was introduced, governing CP violation in the standard model (SM) of particle physics, but also the possibility of having CP violation from the scalar sector was explored. In [3] the CP properties of a model with four quarks and three Higgs bosons were investigated. In the two-Higgs-doublet model (THDM) much effort has been spent to study CP transformations; see for instance [4][5][6][7]. The introduction of bilinears has led to an enormous simplification of the description of any n-Higgs-boson-doublet model (nHDM) [8][9][10][11][12][13][14][15]. In particular, a study of generalised CP transformations (CP g ) in the THDM was presented in [5] using the method of bilinears. It turned out that these CP g transformations have a simple geometric interpretation. In the space of the bilinears they correspond to reflections on planes or to a point reflection. A THDM has been studied in detail which is symmetric under this CP g point reflection [16][17][18][19]. This model has been shown to have interesting consequences: a viable model of this kind has to have at least two fermion families with a large mass hierarchy. In this way, a CP g symmetry gives a theoretical argument for family replication. Recent studies of 3HDM's can, for instance, be found in [20][21][22][23][24][25][26][27][28]. Symmetries of the 3HDM have been studied in [29,30].
Here we want to study generalised parity (P g ) and charge conjugation times parity (CP g ) transformations for the three-Higgs-doublet model, 3HDM. Our paper is organised as follows.
First we review briefly the bilinear approach for the case of three Higgs-boson doublets. This is done in section 2, whereas basis transformations are briefly discussed in section 3. Followed by these preparations we study in section 4 the standard P s and CP s transformations and generalised P g and CP g transformations. In section 5 we consider flavour transformations of the three Higgs-boson doublets in order to bring the generalised P and CP transformations to a standard form. Eventually in section 6 we classify all generalised P g and CP g transformations by suitable choices of bases. Details of the calculation can be found in the appendices A, B, and C. In appendix D we discuss our results in the context to the conventional basis of the potential, written in terms of scalar products of the Higgs-boson doublets. In appendix E we briefly discuss the standard P and CP transformations for the case of an arbitrary number of n Higgs-boson doublets, that is, for the nHDM. The standard CP s transformations correspond to reflections in the space of bilinears for the THDM as well as the 3HDM. We show that this does not hold in the nHDM for certain values of n.

BILINEARS IN THE 3HDM
We will consider models with three Higgs-boson doublets which all carry the same hypercharge y = +1/2 and denote the complex doublet fields by , i = 1, 2, 3. (2.1) We shall consider Yang-Mills-Higgs Lagrangians of the form where L YM (x) is the standard Yang-Mills Lagrangian for the gauge bosons W j λ (x) (j =1,2,3) of SU (2) L and B λ (x) of U (1) Y ; see for instance [31]. Furthermore, D µ is the SU (2) L × U (1) Y -covariant derivative and V (ϕ i ) is the gaugeinvariant potential term. A detailed study of this type of models with respect to stability and symmetry breaking was presented in [14]. In this article we discussed in detail the bilinears for the 3HDM which will also play an essential role in our present article. In order to make our present paper self contained we repeat here the main points of the bilinear method for the 3HDM.
The most general SU (2) L × U (1) Y gauge-invariant Higgs potential can only be a function of products of the Higgsboson doublets in the form We now introduce the 3 × 2 matrix of the Higgs-boson fields (see section 2 of [14]) All possible SU (2) L × U (1) Y invariant scalar products (2.3) may be arranged into the hermitian 3×3 matrix is the conveniently scaled unit matrix and λ a , a = 1, . . . , 8 are the Gell-Mann matrices. Here and in the following we will assume that greek indices (α, β, . . .) run from 0 to 8 and latin indices (a, b, . . .) from 1 to 8. We have The matrix K (2.5) can be decomposed as With the matrix K(x), as defined in terms of the doublet fields in (2.5), as well as the decomposition (2.9), (2.10), we may immediately express the scalar products in terms of the bilinears; see appendix A. The matrix K(x) (2.5) is positive semidefinite which follows directly from its definition K(x) = φ(x)φ † (x). The nine coefficients K α (x) of its decomposition (2.9) are completely fixed given the Higgs-boson fields. The 3 × 2 matrix φ(x) has trivially rank less than or equal to two, from which it follows that this holds also for the matrix K. As has been shown in detail in [9], (see the theorem 5 there), to any hermitian 3 × 3 matrix K(x) with rank less than or equal to two there correspond Higgs-boson fields ϕ i (x), i = 1, 2, 3, which are determined uniquely, up to gauge transformations. The bilinears parametrise the gauge orbits of the three Higgs fields (2.1). The space of the bilinears is the subset of the nine-dimensional space of real vectors (K 0 , . . . , K 8 ) satisfying where the constants G αβγ are given in (A3) of appendix A (see (2.16), (A.31), and (A.32) of [14]). Any 3HDM potential leading to a renormalisable theory can, in terms of bilinears, be written in the form with real parameters: ξ 0 , η 00 , ξ a , η a and the symmetric parameter matrix E ab = E ba with a, b = 1, . . . , 8 [14]. Note that a constant term in the potential can always be dropped. Defining the eight-component vectors K, ξ, η, and 8 × 8 matrix E by we can write the general 3HDM potential in the form

CHANGE OF BASIS
Let us now study an arbitrary unitary mixing of the Higgs-boson doublets of the form (see section 3 of [14]) . This change of basis corresponds to the following transformations of the 3 × 2 matrix φ(x) and of the 3 × 3 matrix K(x) defined in (2.4) and (2.5), respectively, The bilinears K a (x) transform under a change of basis as where the matrix R(U ) = (R ab (U )) is given by that is, R(U ) ∈ SO (8). Let us note that the R(U ) form only a subset of SO (8).
Under the replacement (3.4), the Higgs potential (2.15) remains unchanged if we simultaneously transform the parameters as follows The standard parity transformation, P s , reads where i ∈ {1, 2, 3} and For the bilinears we find from (4.1) The Lagrangian (2.2) is invariant under P s . Of course, once we include fermions in the usual way, parity invariance is lost. But in the present article we shall consider only the Lagrangian (2.2) and its possible symmetries. Next we consider the standard CP transformation, CP s , Here x and x are again given by (4.2). For φ(x) (2.4) and K(x) (2.5) we get from (4.4) and for the bilinears (2.10) where we define the 8 × 8 matrixĈ s = Ĉ s ab by Explicitly we get from the Gell-Mann matriceŝ Obviously, this matrix has the propertieŝ The CP s transformation gives, applied twice, again the trivial transformation in terms of the doublet fields: (4.10) In appendix C we discuss the standard P and CP transformations for the case of n Higgs-boson doublets, that is, for the nHDM.
We shall now define generalised parity (P g ) and CP transformations (CP g ) for the 3HDM. We do this at the level of the bilinears with the following requirements.
(1) Both, P g and CP g transformations are required to be linear in the K α of the form We require furthermore, that the length of the vector (K a ) is left invariant Note that this is the case for the P s and CP s transformations; see (4.3), (4.6), and (4.8), respectively.
(2) The allowed space of the bilinears K α must not be left. This requires that the K α (x) must fulfil (2.12) if the original K α (x) do so.
(3) Application of a P g or CP g transformation twice should give back the original bilinears K α (x). That is, we requireĈĈ (4.14) From (4.14) we see that we have We shall call transformations where det(Ĉ) = +1 generalised P (P g ) and where det(Ĉ) = −1 generalised CP (CP g ) transformations.
We have seen in section 3 that we can make flavour U (3) rotations of the Higgs fields. If we make a corresponding transformation of the parameters of the potential we get the same theory but written in a different basis.
We now want to study how the matrixĈ of (4.12) looks like in a new basis. We have under a change of basis U from (3.4) and (4.11) (4.16) Therefore, the matrixĈ of a generalised P or CP transformation in a new basis readŝ Generalised transformations where the corresponding matricesĈ are related by a flavour transformation (4.17) will be called equivalent. The main purpose of our present article is to determine all equivalence classes of generalised parity and generalised CP transformations, P g and CP g , respectively.

STANDARD FORMS OF GENERALISED P AND CP TRANSFORMATIONS
The problem is now to find standard forms for the matricesĈ which satisfy our conditions (1)-(3) to which general matricesĈ can be brought using only the flavour transformations (4.17).

(5.2)
But since the K a (x ) have to fulfil the condition (2.12), equation (5.2) does not follow immediately. We present the proof of (5.2) in appendix A. The technique which we use there is to consider (5.1) for a suitable number of special cases where the model with three Higgs fields reduces to one with only two Higgs fields. Next we consider the last equation of (2.12) which must be fulfilled both for K α (x) and K α (x) from (4.11); see condition (2) above: With the explicit form of the constants G αβγ from (A3) we get from (5.3) Using now (4.11) and (4.13) we get Again, (5.6) does not follow immediately from (5.5) since the K a (x ) are not independent. They have to fulfil (2.12). The proof of (5.6) is presented in appendix A considering (5.5) for a suitable number of special cases.
To summarise: the equations which determine the matricesĈ (4.12) of a P g or a CP g transformation are (4.14), (5.2), and (5.6).

Flavour transformations of the matricesĈ
In this section we shall use the flavour transformations (4.17) to bring the matricesĈ (4.12) to a standard form. From (4.14) and (5.2) we see thatĈ is a symmetric matrix Therefore,Ĉ can be diagonalised by an SO(8) matrix. Due to (4.14) the eigenvalues ofĈ can only be ±1. Note that, a priori, we do not know if such an SO(8) matrix diagonalisingĈ can be written as a flavour transformation R(U ) as in (4.17). In any case,Ĉ has eight eigenvectors which we can, without loss of generality, assume to be real. Suppose c (8) is one of these eigenvectors, which we assume to be normalised, Under a basis transformation (4.17) this eigenvector transforms as We use this in order to bring c (8) to a standard form. For this we consider the matrix Under a basis transformation (5.9) we get We have furthermore Through a basis transformation U we may diagonalise Λ (8) . Taking the explicit form of the Gell-Mann matrices into account we get Therefore, taking into account (5.8), we can, by a basis change, achieve the form Since an overall sign of c (8) is irrelevant we can restrict the parameter χ to −π/2 < χ ≤ π/2, corresponding to cos(χ) ≥ 0. But we may further restrict χ in the following way. Let us consider the matrix Λ (8) (χ): Since we can, by SU (3) basis transformations, exchange the eigenvalues, we can require that the eigenvalues of Λ (8) (χ) are in decreasing order, that is, From these requirements we get 0 ≤ χ ≤ π/2 and χ ≤ π/3, that is 0 ≤ χ ≤ π/3. We consider now the range π/6 < χ ≤ π/3 and set From (5.15) we get then with 0 ≤ χ < π/6 Since the overall sign of Λ (8) and the order of the eigenvalues do not matter we see that (5.18) is equivalent to (5.15) with χ replaced by χ . Taking everything together we see that by flavour transformations we can bring Λ (8) and correspondingly c (8) to the forms (5.15) and (5.14), respectively, with For the standard form ofĈ we choose now in addition to c (8) (5.14) with χ from (5.19) seven orthonormal vectors c (1) to c (7) , which are also orthogonal to c (8) : Explicitly we use The matrixĈ has then the formĈ From (4.14) and (5.2) we must havẽ We can further simplify (C ij ). For χ = 0 we have from (5.15) This matrix is invariant under the following U (3) flavour transformations . For the case χ = 0 we haveC ij =Ĉ ij and we can, using the flavour transformations R(U ) (4.17) with U from (5.25) achieve that For the general case, 0 < χ ≤ π/6, all three eigenvalues of Λ (8) (χ) (5.15) are different. Therefore, we can only make the following U (3) transformations leaving Λ (8) With the corresponding flavour transformations R(U ) from (4.17) we can achievẽ see appendix B.

THE SOLUTIONS FORĈ
In this section we give the solutions for the matricesĈ (4.12). The equations to be solved are the following: we have from (4.14) and (5.2)ĈĈ =Ĉ TĈ = 1 8 .
Using this we can write (5.6) in the form With the help of the flavour transformations, as explained in section 5, we can, without loss of generality, assume that one eigenvector c (8) ofĈ has the form (5.14) with 0 ≤ χ ≤ π/6; see (5.19). We shall first treat the case χ = 0, where we can transformĈ such that (5.26) holds. We have then C a8 = 0, for a = 1, . . . , 7, We shall now consider special values of a, r, s in (6.2), take into account (6.3), and determine from this all elementŝ C ab . For a = r = s = 8 we get from (6.2) and (6.3) Since d 888 = 0, see table II in appendix A, we getĈ 88 = 1. (6.5) Next we set a = 8, r = s = 1. From (6.2), (6.3), and (6.5) we get then Since all eigenvalues ofĈ are ±1 we must have This shows that the r.h.s. and l.h.s. of (6.6) are ≥ 0 and ≤ 0, respectively. Therefore, both have to be zero, which impliesĈ In a similar way we show, setting in (6.2) a = 8, r = s = 2, and a = 8, r = s = 3, that we must havê Now we consider the cases These give the relations with the solution whereĈ 33 = ±1; see (6.10). At this point we have to distinguish two cases. We start with the caseĈ 33 = +1. From (6.13) we get then Choosing now in (6.2) a = 4, r = 1, s = 6 and a = 5, r = 1, s = 7 we get With (6.8), (6.14) and the orthogonality ofĈ this implieŝ Taking everything together we see that we have already shown thatĈ must be diagonal withĈ 33 =Ĉ 88 = 1 and C aa = ±1 for a = 1, 2, 4, 5, 6, 7. From (5.6) we get now that we must havê The solutions of (6.17) are now easily obtained using the d abc values from table II in appendix A. We label the solutions by (S1), . . . , (S8); see table I. There we also list the values of det(Ĉ) and of N ± , where N + (N − ) = the number of eigenvalues ofĈ equal to + 1 (−1) .
We have given here the detailed derivation of the solution matricesĈ of (6.1) and (6.2) for the case χ = 0,Ĉ 33 = 1 in (6.13). In appendix C we show that for χ = 0,Ĉ 33 = −1 in (6.13) there is no solution. Furthermore we discuss in appendix C the cases with 0 < χ ≤ π/6. It turns out that also there only the solutions of table I exist. Thus, in table I we have listed indeed all solutions of (6.1) and (6.2), of course, apart from flavour transformations of them.
We shall now discuss the meaning of the solutions (S1), . . . , (S8) from table I. That is, we will discuss the transformations (4.11) (6.19) for theĈ aa from table I. In the following the solution matrix for (Si) will be labeledĈ (i) , i = 1, . . . , 8.
(S2) Here det(Ĉ (2) ) = +1. InsertingĈ (2) in (6.19) we obtain a generalised parity transformation P g which, at the field level, reads This is the standard parity transformation followed by a flavour transformation, see (B4), Since the sets of eigenvaluesĈ aa for (S1) and (S2) are different, P s and the P g above are inequivalent.
(S4) This case corresponds to a standard CP transformation followed by a flavour transformation (6.21) ButĈ (4) is equivalent to the standard CP transformationĈ (3) . We have with (B4), (B5), and (4.17) (S5) This corresponds to the standard CP transformation followed by a flavour transformation U (−π/2, π, π/2) from (B4). We get Also this generalised CP transformation is equivalent to the standard one since we havê Here we have a standard CP transformation followed by a flavour transformation U (−π/2, π, −π/2); see (B4). We get AlsoĈ (6) is equivalent to the standard CP transformation since we have, see (B5), (S7) This case corresponds to the standard parity transformation followed by a flavour transformation U (−π/2, π, π/2); see (B4). We get This is equivalent to the generalised parity transformation P g from (S2) since we havê (S8) Here we have a standard parity transformation followed by a flavour transformation U (π/2, π, −π/2) from (B4) Also here we find equivalence to the generalised parity transformation from (S2) since we havê (6.33) To summarise: we have found that for the 3HDM there are two equivalence classes of generalised parity transformations. Convenient representatives of these classes are the standard parity transformation with the matrixĈ (1) from (S1) and the generalised parity transformation with the matrixĈ (8) from (S8); see table I. All generalised CP transformations form only one equivalence class with the standard CP transformation as representative; seeĈ (3) from (S3) in table I. In this way we have obtained a complete answer to the question of generalised P and CP transformations in the 3HDM.

INVARIANT POTENTIALS
We consider now the potential of the 3HDM in the form (2.13) respectively (2.15). Note that all parameters (2.14) of the potential, written in this form, must be real. We consider now a generalised parity (P g ) or CP transformation (CP g ) satisfying the conditions (1), (2), (3) of section 4; see (4.11)-(4.14). The potential V is invariant under this transformation if Writing this out we get This must hold for all allowed K α (x ). From this it follows, using (A27) and (A28), that the potential V of (2.13), In appendix D we discuss the relation of these conditions to statements on CP violation using the conventional form of the basis. Finally we consider the generalised P transformations P g of the class with representative (S8). A potential V allows such a generalised P g invariance if and only if there is a basis where we have, insertingĈ (8)  In this way we have obtained a complete overview of the invariance conditions of the potential for generalised P and CP transformations.

CONCLUSIONS
In this paper we have considered the three-Higgs-doublet model (3HDM) with Higgs and gauge fields only. We have investigated generalised P and CP transformations in this model. We have shown that there are two equivalence classes  (with respect to flavour transformations) of generalised P transformations and only one class of CP transformations. Convenient representatives for these classes are given in table I: the standard P transformation (S1), the generalised P transformation (S8), and the standard CP transformation (S3). We have discussed the conditions which a potential has to fulfil in order to be invariant under any of these transformations. In all our work we made use of the method of bilinears. The relation to the conventional basis for the potential is discussed in detail in appendix D.
To summarise, we have investigated the general 3HDM in view of generalised P and CP transformations which applied twice give back the unit transformation up to possible gauge transformations. Our work gives a complete overview of such transformations and of their consequences for the potential.
A very interesting type of "CP" transformations was considered in [32]. There one has to apply the transformation four times in order to get back the unit transformation. Of course, also such transformations can be analysed with the methods developed in the present paper. This will be dealt with in a separate work.
Finally we have made some remarks on the standard P and CP transformations in the nHDM in appendix E. We have shown that in the space of bilinears the standard CP transformation CP s is again given by a linear transformation of the corresponding bilinears with an (n 2 − 1) × (n 2 − 1) matrixĈ which is given explicitly.Ĉ is diagonal with diagonal elements ±1. The number of −1 elements is n(n − 1)/2. Here we recall some formulae for the bilinears of the 3HDM and the THDM. Then we prove equations (5.2) and (5.6).
The bilinears of the 3HDM as defined in (2.10) are given explicitly by The K α satisfy (2.12) where G αβγ is given by (see (A.31) of [14]) which is completely symmetric in α, β, γ. Explicitly we get The d abc are the usual symmetric constants of SU (3). We list the non-zero elements of the d abc in table II; see for instance [31].
In the following we also need the bilinears for the THDM given in [8,9]. Therefore, we reproduce some results of these references here. Let be two Higgs-doublet fields with hypercharge y = 1/2. We define the matrix L by and the bilinears of the THDM by L = 1 2 where σ i are the Pauli matrices. Explicitly we get and from this see section 3 of [9]. To distinguish in our present article 3HDM and THDM quantities we use ψ and L for the fields and bilinears of the THDM, respectively, instead of ϕ and K in [9]. The bilinears L α satisfy see (36) of [9]. Now we are in the position to prove (5.2). We have for all K α satisfying (2.12) from (5.1) where We want to show that G bc = δ bc . The technique for this is to use special cases, corresponding to THDM fields in (A10). First we set (a) This gives, using (A7), (A8), and (A1), and from (A10) Since the L 0 , L i only have to satisfy (A9) the polynomials on the left-and right-hand sides of (A14) must be equal. This implies We consider then six more special cases Proceeding in each case as shown explicitly for case (a) we obtain that we must have which was to be proven. Here, all special cases (a)-(e) can easily be treated by hand, as we have shown by (A14), (A15). But we also have written a computer program to deal with these cases.
To prove (5.6) we proceed in the same way. We write (5.5) as where Clearly, D abc is completely symmetric. We insert now again special cases, corresponding to THDMs, in (A21) and compare the polynomials in L 0 , L 1 , L 2 , L 3 which we obtain on the right-and left-hand sides of (A21). The special cases which we use here are again (a) to (e) from (A12), (A16)-(A19). In addition we need here the choices The comparison of the two sides of (A21) for all these special cases is then done with the help of a computer program and gives This proves (5.6).

Appendix C: Details of the calculation for the solutionsĈ
Here we complete the discussion of the solutions of (6.1) and (6.2) for the matricesĈ. Here we consider the case χ = 0 in (5.19). We have treated this case generally up to (6.13) where we had to distinguish two cases,Ĉ 33 = ±1. The caseĈ 33 = +1 is discussed in section 6. Here we discuss the casê C 33 = −1. From (6.3) and (6.12) we get then, using here and in the following the d abc values from table II, Now we set in (6.2) a = 4, r = 1, s = 6 and a = 5, r = 1, s = 7. This gives, taking into account (6.3) to (6.10) and, therefore, with (C1)Ĉ 66 =Ĉ 77 = 0. (C3) Next we choose a = 2, r = 5, s = 6 in (6.2). This gives with (C1) (C4) Using (5.26) and (C1) this givesĈ But sinceĈ is an orthogonal matrix we have and we get from (C5)Ĉ Using again the orthogonality property ofĈ we find Finally we choose a = 1, r = 4, s = 6 in (6.2) and find This is a contradiction and shows that there is no solution forĈ for the case χ = 0,Ĉ 33 = −1.
We emphasise that S will in general not be a flavour transformation R(U ) as in (3.5), (3.6). From (C10) we have Now we transformĈ with S and setĤ = SĈS T . (C13) a b cd abc a b cd abc general χ χ = π/6 general χ χ = π/6 The non-zero elements ofd abc of (C16) as function of χ and for the special value χ = π/6. Here s = sin(χ), c = cos(χ).

(C14)
Form (6.2) we get the following condition forĤ whered In table III we list the non-zero elements ofd abc .
By construction of S (C10) and from (5.28) we havê H a8 = 0 for a = 1, . . . , 7 ,Ĥ 12 =Ĥ 45 = 0, Now we choose special values for a, b, c in (C15) in order to determine the possible solutions of this equation. We start with a = b = c = 8. This gives with (C17) For 0 < χ < π/6 we haved 888 = 0; see table III. Therefore we find Next we choose in (C15) a = 8, b = c = 1. This gives For 0 < χ < π/6 we have Furthermore we have, sinceĤ is an orthogonal matrix (see (C14)), Therefore, the r.h.s. of (C21) is greater than or equal to zero, the l.h.s. less than or equal to zero. Thus, both sides must be zero and we getĤ Taking into account (C24) andĤ 81 =Ĥ 21 = 0 from (C17) we get from the orthogonality relation forĤ From (C25) and (C26) we get with (C22)Ĥ Next we choose a = 8, b = c = 2 in (C15). Here the argumentation is as for a = 8, b = c = 1 above and we find With (C22) we conclude from (C29) Next we consider a = 8, b = c = 4 and a = 3, b = c = 4 in (C15). We get then From (C31) we find According to (C30) we haveĤ 33 = ±1. ButĤ 33 = −1 leads to a contradiction in (C32). Thus we must havê Looking back at (C31) this impliesĤ Now we choose a = 8, b = c = 5 and a = 3, b = c = 5 in (C15). This gives As above we conclude from (C35) From a = 4, b = 1, c = 6 in (C15) we getĤ Finally, from a = 5, b = 1, c = 7 in (C15) we get Collecting now everything together we have shown thatĤ has diagonal form witĥ But now we can transform back toĈ using (C13) and we find From there on we can follow the analysis as in section 6 from (6.16) onwards. We find then also here exactly the same solutions (S1) to (S8) listed in table I.
Appendix D: The potential in the conventional basis In our paper we have always worked with the potential expressed as a polynomial in the bilinears K α (2.10); see (2.13) and (2.15). In this case all the parameters (2.14) of the potential are necessarily real. But frequently the potential is written as a polynomial in the field products ϕ † 1 ϕ 1 , ϕ † 2 ϕ 2 , etc.; see for instance [4]. Then the parameters of this polynomial for the potential need not all be real. In the following we shall discuss the connection of these two ways of writing the potential. We shall also discuss how the conditions of CP invariance look like in such a basis.
We start by writing the transformation (A1) from the products of the Higgs fields to the K α in matrix form. For this we introduce the 9 dimensional vector We have then from (A1)K with the 9 × 9 matrixÃ given byÃ The reverse transformation readsP We introduce now from (2.13) to (2.15)ξ With this we can write the potential V as follows Note thatζ andF will have imaginary parts. Indeed, we splitÃ (D3) into its real (Ã R ) and imaginary (Ã I ) parts, and similarly forFF We find then In tables IV and V we list the values forF R andF I , respectively. Suppose now that the potential V allows CP invariance. This holds if and only if there is a basis where (7.5) is true for the parameters ξ, η, E. From (D12) and table V we see that in the conventional basis (D1) this requires all imaginary parts of the parameters to vanish and vice versa. In this way we recover the statement, first shown for the THDM in [4], that a potential allows CP invariance if and only if there is a conventional basis where all parameters are real. α βF Rαβ α βF Rαβ 0 0 1 3 3E33 + 2 In this appendix we investigate the standard CP transformation in the general case of n ≥ 2 Higgs boson doublets which all carry the same hypercharge y = +1/2. We denote the complex doublet fields by We now introduce the n × 2 matrix of the Higgs-boson fields and define the hermitian matrix A basis for the n × n matrices is given by the matrices λ α , α = 0, 1, . . . , n 2 − 1 , α βF Iαβ α βF Iαβ  The matrix K (E3) can be written in the basis of the scaled unit matrix and the generalised Gell-Mann matrices as where the real coefficients K α are given by The standard parity transformation, P s , reads with x and x given in (4.2). For the bilinears we get from (E3) and (E11) Next we consider the standard CP transformation where we define the (n 2 − 1) × (n 2 − 1) matrixĈ s by λ T a =Ĉ s ab λ b ,Ĉ s = Ĉ s ab , a, b ∈ {1, . . . , n 2 − 1}.
Note that this form of the matrixĈ s ensures that two subsequent standard CP s transformations give back the original bilinears: Of course, we see this also immediately at the level of the fields from (E14) Now let us count the number of antisymmetric generalised Gell-Mann matrices. In the representation as given in Fig. 1 the antisymmetric matrices are those of (E7). For a given n the second row of table VI gives the indices a of the antisymmetric matrices λ a . That is, for given n all matrices with indices listed in the second row up to the entry n are the antisymmetric ones. The third row of table VI shows the total number of antisymmetric matrices for a given n.
Let us look at the simplest cases. For n = 2, that is, for the THDM, there is only one antisymmetric matrix λ 2 (a = 2), which is the second Pauli matrix. The total number of antisymmetric matrices is one. Therefore we get from (E17) in this case for the CP s transformation matrix, as shown in section 3 of This is again a reflection in K space, that is, det(Ĉ s ) = −1. Proceeding with the 4HDM, following table VI, we see that the corresponding CP s -transformation matrix is 4HDM:Ĉ s = diag(1, −1, 1, 1, −1, 1, −1, 1, 1, −1, 1, −1, 1, −1, 1).
Here we obviously have det(Ĉ s ) = +1 unlike the cases of the THDM and the 3HDM where the determinant is −1.
For the general case of n let I n as be the set of the indices of the antisymmetric generalised Gell-Mann matrices λ a as given in table VI, I n as = {k 2 − 2k + 2, k 2 − 2k + 4, . . . , k 2 − 2 | k = 2, . . . , n}, that is,