CP violation conditions in N-Higgs-doublet potentials C. C. Nishi∗ arXiv:hep-ph/0605153v3 26 Oct 2007 Instituto de Fı́sica Teórica, UNESP – São Paulo State University Rua Pamplona, 145 01405-900 – São Paulo, Brazil Conditions for CP violation in the scalar potential sector of general N-Higgs-doublet models (NHDMs) are analyzed from a group theoretical perspective. For the simplest two-Higgs-doublet model (2HDM) potential, a minimum set of conditions for explicit and spontaneous CP violation is presented. The conditions can be given a clear geometrical interpretation in terms of quantities in the adjoint representation of the basis transformation group for the two doublets. Such conditions depend on CP-odd pseudoscalar invariants. When the potential is CP invariant, the explicit procedure to reach the real CP-basis and the explicit CP transformation can also be obtained. The procedure to find the real basis and the conditions for CP violation are then extended to general NHDM potentials. The analysis becomes more involved and only a formal procedure to reach the real basis is found. Necessary conditions for CP invariance can still be formulated in terms of group invariants: the CP-odd generalized pseudoscalars. The problem can be completely solved for three Higgs-doublets. PACS numbers: 12.60.Fr, 11.30.Er, 14.80.Cp, 02.20.Sv I. INTRODUCTION It is well known that group automorphisms play an important role in the CP violation phenomenon. In a extensive paper, Grimus and Rebelo [1] have analyzed the CP-type transformations as automorphisms in the gauge symmetry present in the Quantum Field Theory (QFT) models of particle physics. They showed, at the classical level, that general gauge theories with fermions and scalars coupled to gauge bosons through minimal coupling are always CP invariant. In other words, a CP-type transformation that is a symmetry of the theory can always be found. The only terms that can possibly violate the CP symmetry are the Yukawa couplings and scalar potentials. In the Standard Model (SM), the unique source of CP violation comes from the complex phases in the Yukawa couplings that are transferred to the Cabibbo-Kobayashi-Maskawa (CKM) matrix [2] after spontaneous electroweak symmetry breaking (EWSB). Within such context the possibility of (explicit) CP violation is intimately connected with the presence of a horizontal space: the quarks come in three identical families distinguished only by their masses. Another source of CP violation could arise in the scalar potential sector [3]. In such case two patterns can be possible, either the CP symmetry is violated explicitly in the theory before EWSB or the CP violation arises spontaneously jointly with EWSB. Several models with spontaneous CP violation arising from the Higgs sector were constructed after Refs. 4 and 5, aiming to attribute to the violation of CP the same origin of the broken-hidden gauge symmetries. Nonetheless, the available CP violation data seems to be in general accordance with the SM CKM mechanism [6, 7]. Then, concerning the CP violation data, restricted by mixing constraints and strong suppresion of flavor changing neutral currents (FCNC) [8], the challenge is to develop a model that incorporates entirely or partially the CKM mechanism. The scalar potential sector, although phenomenologically rich in CP violating sources (see, e.g., Refs. 5 and 9), has not yet been analyzed for general gauge theories under a group theoretical perspective. One of the reasons for the difficulty for a general treatment is that the scalar potential involves higher order combinations of scalar fields than other sectors of gauge theories, with terms constrained only by the underlying gauge symmetries and, if required, renormalizability. Renormalizability in four dimensions constrains the highest order scalar field combination to be quartic. Another difficulty for analyzing the CP violation properties for general gauge theories is the freedom to change the basis of fields used to describe the theory. The most familiar is the SM’s rephasing freedom for the quark fields: this change of basis transforms the CKM matrix and the complex entries. Such ambiguity can be avoided by using rephasing invariants [10] which depend only on one physically measurable CP phase. More generally, for theories with ∗ Electronic address: ccnishi@ift.unesp.br 2 horizontal spaces, there is a freedom to continuously rotate the basis of such spaces without changing the physical content. For this case, it is also possible to write the observables in a basis independent manner [11, 12, 13, 14, 15, 16], or, in other words, in terms of reparameterization invariants [17]. In any case, it is important to be able to establish general conditions for CP violation to analyze more transparently the possible CP violating patterns for gauge theories with large gauge groups and/or horizontal spaces. Following this spirit of classifying and quantifying CP violation based on basis invariants, it will be treated here the simplest class of extensions of the SM: the multi-Higgs-doublet models [18, 19, 20, 21], which we shall denote by NHDM for N Higgs-doublets. The simplest of them is the two-Higgs-doublet model (2HDM) which has been extensively studied in the literature [9, 22, 23], also employing the basis independent methods [12, 13, 14, 15, 16]. An explicit but not complete study for 3HDM potentials can be found in Ref. 16. The recent interest is based on the fact that the 2HDM can be considered as an effective theory of the minimal supersymmetric extension of the SM (MSSM) [22], which requires two Higgs-doublets from supersymmetry. Concerning the 2HDMs, a throughout analysis of the CP symmetry aspect of the 2HDM potentials was presented recently [14, 15]. The necessary and sufficient conditions for spontaneous and explicit CP violation were presented, expressed in terms of basis independent conditions and invariants. In this respect, in Sec. II, a more compact version of such proofs will be shown. The approach used is much alike the one presented in Ref. 15: from group theoretical analysis, the adjoint representation can be used as the minimum nontrivial representation of the transformation group of change of basis for the two doublets, i.e., the horizontal SU (2) group. Working with the adjoint representation allows for an alternative formulation of the CP invariance conditions which facilitate the analysis and enables one, when the potential is CP invariant, to find the explicit CP transformation and the explicit real basis [12], i.e., the basis for which all the parameters in the potential are real. Such issues were not addressed in previous approaches [14, 15, 16]. The basis independent conditions are formulated in terms of pseudoscalars of the adjoint. In Sec. II A, we also obtain the necessary and sufficient conditions to have spontaneous CP violation. In Sec. III, an extension of the method is attempted to treat general NHDMs. The analysis becomes much more involved than the N = 2 case and further mathematical machinery is necessary. Nevertheless, stringent necessary conditions for CP invariance can be formulated. Generalized pseudoscalars, which should be null for a CP invariant potential, can still be constructed. For N = 3, the conditions found are shown to be sufficient if supplemented by an additional condition. In Sec. III A, a brief account on spontaneous CP violation on NHDMs is presented. At last, in Sec. IV we draw some conclusions and discuss some possible approaches for the complete classification of the CP-symmetry properties for the NHDMs. (Some useful material is also presented in the appendices.) II. N = 2 HIGGS-DOUBLETS For N = 2 Higgs-doublets Φa , a = 1, 2, transforming under SU (2)L ⊗ U (1)Y as (2, 1), the minimal gauge invariant combinations that can be constructed are Aa = Φ†a Φa , a = 1, 2, B = Φ†1 Φ2 and B † . (1) All other invariants can be constructed as combinations of these ones [25]. Thus the most general renormalizable 2HDM potential can be parameterized as [14] λ1 λ2 V (Φ) = m211 A1 + m222 A2 − (m212 B + h.c.) + A21 + A22 + λ3 A1 A2 + λ4 BB † 2 2 λ5 2 + { B + [λ6 A1 + λ7 A2 ]B + h.c.} . 2 (2) From the hermiticity condition, {m211 , m222 , λ1 , λ2 , λ3 , λ4 } are real parameters and {m212 , λ5 , λ6 , λ7 } are potentially complex, summing up to 6 + 2 × 4 = 14 real parameters. The existence of complex parameters per se, though, does not mean the potential in Eq. (2) is CP violating. A U (2)H horizontal transformation can be performed on the two doublets possessing identical quantum numbers, except, perhaps, for their masses, Φ → UΦ where U is a U (2) transformation matrix and Φ denotes the assembly of the two doublets in   Φ1 Φ≡ . Φ2 (3) (4) 3 Actually, the global phase transformation in U amounts for a hypercharge transformation under which the gauge invariants in Eq. (1) do not change. Thus only a U ∈ SU (2)H transformation needs to be considered. This basis transformation freedom suggests, and indeed it can be proved [3, 14], that the necessary and sufficient conditions for V (Φ) to be CP invariant are equivalent to the existence of a basis reached by a transformation U (3) in which all the parameters present in the potential are real. Since these basis transformations can be reformulated as transformations on the parameters, all the analysis resumes in investigating the transformation properties of the parameters of V (Φ) under SU (2)H . Indeed, the parameters can be written as higher order tensors, transforming under the fundamental representation of SU (2)H [12, 13, 14]. Instead of performing the analysis of tensors under the fundamental represention 2 of SU (2)H , as in Ref. 14, we can take advantage of the form of the minimal invariants (1) that transform as 2̄ ⊗ 2 = 3 ⊕ 1, by using as the minimal nontrivial representation the adjoint 3. In fact this property does not depend on the number of doublets N , and the invariants of the type of Eq. (1) always form representations of SU (N )H with decomposition N̄ ⊗ N = adj ⊕ 1 . (5) This property will be exploited in section III to treat general N -Higgs-doublet potentials. For SU (2)H , the decomposition in Eq. (5) can be performed by using instead of {A1 , A2 , B, B † } the real combinations Aµ ≡ 12 Φ† σµ Φ , µ = 0, 1, . . . , 3, (6) where σµ = (1, σ). The Greek index is not a space-time index, which means there is no distinction between covariant or contravariant indices but the convention of summation over repeated indices will be used. The indices running over µ = i = 1, 2, 3 are group indices in the space of the Lie algebra, i.e., in the adjoint representation and the µ = 0 index is the trivial singlet component. The explicit change of basis reads A1 + A2 , 2 A1 − A2 , = 2 † B+B = = ReB , 2 † B−B = ImB , = 2i A0 = A3 A1 A2 (7) which can be readily inverted and inserted in the potential of Eq. (2) to give compactly V (A) = Mµ Aµ + Aµ Λµν Aν , (8) {Mµ } = (m211 + m222 , −2Re m212 , 2Im m212 , m211 − m222 )   λ̄ + λ3 Re(λ6 + λ7 ) −Im(λ6 + λ7 ) ∆λ/2  Re(λ6 + λ7 ) λ4 + Reλ5 −Imλ5 Re(λ6 − λ7 )   , Λ = {Λµν } =   −Im(λ6 + λ7 ) −Imλ5 λ4 − Reλ5 −Im(λ6 − λ7 )  ∆λ/2 Re(λ6 − λ7 ) −Im(λ6 − λ7 ) λ̄ − λ3 (9) where (10) and λ̄ = (λ1 + λ2 )/2, ∆λ = λ1 − λ2 . Notice that all parameters in this basis are real and the criterion for CP violation have to be different of the reality condition. Furthermore, Λ is real and symmetric. The coefficients of M can be more conveniently written as Mµ ≡ Tr[σµ Y ] , (11) where Y = m211 −m212 2 −m2∗ 12 m22 ! = Mµ 12 σµ , (12) is the mass matrix for V (Φ) Φ2 = Φ† Y Φ . (13) 4 These relations can be easily extended to general N doublets by replacing the {σi } matrices by the proper generators of SU (N )H , {λi }, and the corresponding identity matrix. The relation (11) follows from the completeness of the basis {σµ } in the space of complex 2 × 2 matrices [26]. Expanding Eq. (8) in terms of the irreducible pieces of 3 ⊕ 1, V (A) = M0 A0 + Λ00 (A0 )2 + Mi Ai + 2Λ0i A0 Ai + Ai Λ̃ij Aj , (14) we identify two vectors M ≡ {Mi }, Λ0 ≡ {Λ0i } and one rank two tensor Λ̃ = {Λij } with respect to 3. Further mention to the representation will be suppressed and it will be assumed that the representation in question is the adjoint if otherwise not stated. (For example, “vectors” and “tensors” transform under the adjoint.) The tensor Λ̃ can be further reduced into irreducible pieces following 3 ⊗ 3 = 5 ⊕ 1 as Λ̃ = Λ̃(0) + Λ̃(5) . (15) The singlet component is just Λ̃(0) = 31 Tr[Λ̃] 13 , and the remaining of Λ̃ is the 5-component. This last decomposition of Λ̃, though, will not be necessary for the analysis because of the particular fact that the adjoint of SU (2) ∼ SO(3) and all analysis can be done considering the rotation group in three dimensions, which is very much known. The SU (2) → SO(3) two-to-one mapping is given by the transformation induced by Eq. (3) over the invariants Aµ , A0 → A0 Ai → Oij (U )Aj , (16) where Oij (U ) ≡ 12 Tr[U † σi U σj ] , ∈ SO(3) . (17) If U = exp(iσ · θ/2), Oij (θ) = [exp(iθk Jk )]ij , where (iJk )ij = εkij are the generators of SU (2) [SO(3)] in the adjoint representation. At this point we have to introduce the transformation properties of the scalar doublets under CP. One possible choice is CP Φa (x) −→ Φ∗a (x̂) , (18) where x̂ = (x0 , −x) for x = (x0 , x). The transformation of Eq. (18) induces in the invariants Aµ the transformation CP A0 (x) −→ A0 (x̂) CP Ai (x) −→ (I2 )ij Aj (x̂) , (19) where I2 ≡ diag(1, −1, 1) represents the reflection in the 2-axis. We shall denote the transformations (18) and (19) as canonical CP-transformations and, in particular, the second equation of Eq. (19) as the canonical CP-reflection. Since horizontal transformations are also allowed, the most general CP transformation is given by the composition of Eqs. (18) and (19) with SU (2)H transformations; any additional phase can be absorbed in those transformations. Thus the CP transformation over Ai involves a reflection and it does not belong to the proper rotations SO(3) induced by horizontal transformations. The question of CP invariance, then, resumes in the existence of horizontal transformations composite with a reflection that leaves the potential invariant. Since the reflection along the 2-axis can be transformed into the reflection along any axis through the composition with rotations, the natural choice of ⊤ basis is the basis for which Λ̃ is diagonal: OCP Λ̃OCP = diag({λ̃i }). It is always possible to find OCP ∈ SO(3) because Λ̃ is real and symmetric. Furthermore, OCP is unique up to reordering of the diagonal values {λ̃i }, or up to CP rotations in the subspace of degenerate eigenvalues in case {λ̃i } are not all different. In such basis, Ai → A′i = Oij Aj , the potential in Eq. (14) becomes V (A) = M0 A0 + Λ00 (A0 )2 + Mi′ A′i + 2Λ′0i A0 A′i + λ̃i (A′i )2 , (20) M′ = OCP M and Λ′0 = OCP Λ0 . (21) where The last term of Eq. (20) is reflection invariant along any of the principal axes of Λ̃, e′1 , e′2 , e′3 (if Λ̃ does not have degenerate eigenvalues, the three principal axes are the only directions leaving the tensor invariant by reflection; with degeneracies, a continuous set of directions exist in the degenerate subspace). The only terms that must be considered 5 are the third and the fourth ones. They depend on two vectors M′ and Λ′0 : for the potential in Eq. (20) to be invariant by reflection, and consequently by CP, it is necessary that the vectors M′ and Λ′0 be null for the same component. In such case, it is always possible by a suitable π/2 rotation to choose that direction to be the 2-axis. This is the canonical CP-basis (the real basis in Ref. 14) which have all the parameters in the potential V (Φ) real, since there is no A′2 components, which are the only possible source of complex entries in the change of basis of Eq. (7). The CP transformation in terms of the original variables is recovered with the inverse transformation as CP † ∗ Φa (x) −→ (U CP U CP )ab Φ∗b (x̂) , ⊤ CP Ai (x) −→ (OCP I2 OCP )ij Aj (x̂) , (22) where OCP = O(U CP ). The conditions we have found rely on a systematic procedure to find the canonical CP-basis. The basis may not exist and the theory is CP violating. In this two doublet case, the change of basis can be easily achieved by a diagonalization. However, sometimes it is more useful to have a direct criterion to check if the CP invariance holds before going to the procedure of finding the CP-basis. For N > 2 doublets the procedure of finding the CP-basis is not straightforward and direct criteria are much more helpful. The criteria for N = 2 can be formulated with the pseudoscalar invariants I(v1 , v2 , v3 ) = εijk v1i v2j v3k = (v1 × v2 ) · v3 . (23) It is common knowledge that the pseudoscalars defined by Eq. (23) are invariant by rotations but changes sign under a reflection or a space-inversion. Consequently, if the potential V (A) is reflection invariant (CP-invariant), then all pseudoscalar invariants of the theory are null. The lowest order non-trivial pseudoscalars that can be constructed with two vectors {M, Λ0 } and one rank-2 tensor Λ̃ are IM = I(M, Λ̃M, Λ̃2 M) , 2 IΛ0 = I(Λ0 , Λ̃Λ0 , Λ̃ Λ0 ) , (24) (25) I1 = I(M, Λ0 , Λ̃M) , (26) I2 = I(M, Λ0 , Λ̃Λ0 ) , (27) with dimensions M 3 Λ3 , Λ6 , M 2 Λ2 and M Λ3 respectively. The following statements will be proved: (A) If M × Λ0 6= 0, I1 = 0 and I2 = 0 are the necessary and sufficient conditions to V (A) be CP invariant. The CP reflection direction is M × Λ0 and it is also an eigenvector of Λ̃. (B) If M k Λ0 , IM = 0 (or IΛ0 = 0) is the necessary and sufficient condition to V (A) be CP invariant. The CP reflection direction is either M × Λ̃M (6= 0) or an eigenvector of Λ̃ perpendicular to M (if Λ̃M k M) and the CP reflection direction is an eigenvector of Λ̃. (C) All higher order pseudoscalar invariants are null if (A) or (B) is true. The statements (A) and (B) are proved by noting that I(v1 , v2 , v3 ) = 0 implies that v1 , v2 , v3 lie in the same plane. For (A), if I1 = I2 = 0, we can write Λ̃M = αM + βΛ0 and Λ̃Λ0 = α′ M + β ′ Λ0 , which means the application of Λ̃ on M or Λ0 lie on the plane perperpendicular to M × Λ0 ; IM = 0 and IΛ0 = 0 are automatic. Then IM = 0 implies Λ̃2 M = α′′ M + β ′′ Λ̃M, which means Λ̃n M remains in the plane defined by {M, Λ0 }. The same reasoning apply to Λ̃m Λ0 from IΛ0 = 0. Then, the set {M, Λ0 } defines a principal plane of Λ̃, i.e., a plane perpendicular to a principal axis of Λ̃, the vector M × Λ0 , which is then an eigenvector of Λ̃. The latter can be seen from I1 = (M × Λ0 ).(Λ̃M) = (Λ̃(M × Λ0 )).M = 0, and Λ̃(M × Λ0 ) is perpendicular to M; analogously I2 = 0 implies Λ̃(M × Λ0 ) is also perpendicular to Λ0 , therefore Λ̃(M × Λ0 ) ∝ (M × Λ0 ). At last, choose (M × Λ0 ) as the reflection direction (e′2 -axis), then M and Λ0 have null projection with respect to that direction and the CP-basis is found. This proves that I1 = I2 = 0 is a sufficient condition. That it is also necessary, can be seen through the search of the CP-basis: a CP-basis requires both {M, Λ0 } to be in the same principal plane, then Λ̃n M or Λ̃m Λ0 remain in that plane and I1 = I2 = 0. For the disjoint case (B), I1 = I2 = 0 is automatic. There is only one independent direction and a rank-2 tensor. IM = 0 (or IΛ0 = 0) implies that either Λ̃M k M and M is an eigenvector, or Λ̃n M = αM + β Λ̃M and {Λ̃M, M} defines a principal plane. Then, use either M × Λ̃M (6= 0) or an eigenvector of Λ̃ perpendicular to M (Λ̃M k M) as 6 the CP-reflection direction and the CP-basis is achieved. The converse is also true, if a CP-basis can be found, the invariants are null. A subtlety arises when Λ̃ have degeneracies. When only two eigenvalues are equal, still one principal direction and a perpendicular principal plane is defined; every vector in the latter plane is an eigenvector and any plane containing the non-degenerate eigenvector is also a principal plane. With these extended definitions the proofs above are still valid. For the trivial case when the three eigenvalues are degenerate, Λ̃ is proportional to the identity and a CP-basis can always be found by using M × Λ0 as the CP-reflection direction. It is also important to remark that only for Λ̃ nondegenerate, the CP-basis is unique up to a discrete subgroup of SO(3)H ; for the remaining cases there is a continuous infinite of possible CP-basis, when one exists. As for the CP-reflection direction, only when Λ̃ non-degenerate and M × Λ0 6= 0 or M × Λ̃M 6= 0 (M k Λ0 ) the direction is unique; for M k Λ0 and M k Λ̃M (Λ̃ non-degenerate), there are two possible directions. At last, all higher order pseudoscalar invariants are either combinations of lower order scalars or pseudoscalars, or is of the form Eq. (23) and involves vectors with further applications of Λ̃, for example, Λ̃n M; if the conditions (A) or (B) are valid, they all remain in the principal plane defined by {M, Λ0 } or {M, Λ̃M}, which implies all pseudoscalars of the form Eq. (23) are also null. This completes (C). Conditions (A) and (B) solve the problem of finding the minimum set of reparameterization invariant conditions to test the CP-invariance of a 2HDM potential, a problem that was not completely solved in previous approaches [14, 15, 16]. For completeness, we compare the invariants of Eq. (24) with that of Ref. 14 and arrive at the equalities IY 3Z I2Y 2Z I6Z I3Y 3Z = I2 , = − 12 I1 , = −2IΛ0 , = 41 [IM + (M·Λ0 )I1 − M02 I2 ] . (28) (29) (30) (31) For Eqs. (28)–(30), the proportionality is assured by dimensional counting, since these are the lowest order invariants by SU (2)H but not invariant by the corresponding CP-type transformation. The proportionality constant can be found by restricting to particular values, for example, λ6 = −λ7 [14]. For Eq. (31), the full calculation is necessary. Also, the statement in Ref. 14 that it is always possible to find a basis when λ6 = −λ7 can be seen here as the possibility of rotating Λ0 in Eq. (10) to the 3-direction. Another example is the special point λ1 = λ2 and λ7 = −λ6 , which corresponds to Λ0 = 0 and the condition for CP invariance only imposes conditions on M and Λ̃. However, from the perspective of the development of this section, we see M = 0 is as special a point as Λ0 = 0 is. Only if the theory is CP-violating and one wants to classify the violation in soft or hard violation [17], the two cases are different. A. Conditions for spontaneous CP violation This issue has already been investigated in Ref. 14 using as the minimal representation the fundamental representation of SU (2)H . We will work out, instead, the conditions for spontaneous CP violation in the 2HDM using the adjoint representation. We already explored the conditions to have explicit CP violation in V (Φ), Eq. (2). In such case after EWSB, the CP violating property will remain in the potential. On the other hand, if the potential is CP conserving, after EWSB, the theory could become CP violating if the vacuum is not invariant by the CP-type transformation of the original potential. We will concentrate on this spontaneously broken CP case. Given the potential V (A) in Eq. (8), the spontaneously broken potential is given by shifting the fields Φ → Φ + hΦi Aµ → Aµ + hAµ i + Bµ , (32) (33) where hAµ i ≡ Bµ ≡ † 1 2 hΦi σµ hΦi † 1 1 † 2 hΦi σµ Φ + 2 Φ σµ hΦi (34) . (35) 7 The vacuum expectation values (VEVs), invariant by the U (1)EM , can be parametrized by   0 !  v1   v  hΦ1 i v  , hΦi = =√   hΦ2 i 2 0   v2 iξ e v p where v = v12 + v22 = 246 GeV. The parameters of Eq. (36) have to obey the minimization constraints ∂ ∗ V (Φ) ∂φ(0) a Φ=hΦi = ∂Aµ ∂ V (A) (0)∗ ∂Aµ ∂φa Φ=hΦi = (Mµ + 2Λµν hAν i) 12 (σµ )ab hφ(0) b i= 0 . (36) (37) (0) For the charged component the condition is trivial hφ(+) 6 0) solution for Eq. (37) is a i = 0. The nontrivial (hφa i = conditioned by the existence of solutions hAµ i = 6 0 of det[(Mµ + 2Λµν hAν i)σµ ] = (M0 + 2Λ0ν hAν i)2 − (Mi + 2Λiν hAν i)2 = 0 , (38) provided that V (hΦi) < V (0) and hΦi corresponds to an absolute minimum. When Eq. (36) is used, the parameterization of hAµ i is hAµ i = v2 v2 (1, v) = (1, sin θv cos ξ, sin θv sin ξ, cos θv ) , 2 2 (39) where tan θ2v = vv12 ; Eq. (39) is just the projective map of the complex number v2 eiξ /v1 to the unit sphere. Notice that the connection of the parameter θv used here with the more usual parameter β used in the MSSM description [13, 22] is given by tan θ2v = tan β. In case the potential in Eq. (2) has a CP symmetry, it can be written in the CP-basis (or the real basis [14]) in the form of Eq. (20). The CP transformations are just Eqs. (18) and (19). The potential after EWSB can be written as V (Φ + hΦi) = V (A) + V (hAi) + Λµν Bµ Bν + 2Λµν Aµ (hAν i + Bν ) , (40) which, in the CP-basis, have M2 = 0, Λ02 = 0 and Λ̃ = diag({λ̃i }). The condition (Mµ + 2Λµν hAν i)Bµ = 0, derived from Eq. (37) was used. By construction, if hΦi also transformed under CP as Φ, the potential in Eq. (40) would be CP invariant. However the invariance of the vacuum under any symmetry implies the VEVs have to be invariant under the CP transformation. Looking into the details, if we apply the transformations of Eqs. (18) and (19) into Eq. (40), since V (Φ∗ + hΦi) = V (Φ + hΦi∗ ) for an initial CP invariant potential, the potential remains CP invariant after EWSB if, and only if, (I2 hAi)i = hAi i Bi (hΦi∗ ) = Bi (hΦi) . (41) (42) Equation (41) implies hA2 i = 0 and from the parameterization of Eq. (39) it implies ξ = 0, π. Then Eq. (42) is automatically satisfied with hΦi∗ = hΦi. Actually, any solution of the form hΦi∗ = eiα hΦi satisfies Eq. (41) but not Eq. (42). The parameterization of Eq. (36), however, automatically takes into account Eq. (42) when Eq. (41) is satisfied. Thus, using such parameterization the analysis can be carried out exclusively in the adjoint representation. In a general basis, the conditions on the CP-basis investigated so far can be translated to the following condition: if V (Φ) is CP invariant and it has a nontrivial minimum hΦi = 6 0, V (Φ + hΦi) is CP invariant if, and only if, hAi i is in the principal plane defined by {M, Λ0 , Λ̃M}. The more specific conditions for {hAi i} to be in the latter principal plane are: a) If M × Λ0 6= 0, I({hAi i}, M, Λ0 ) = 0. b) If M k Λ0 and Λ̃M × M 6= 0, I({hAi i}, M, Λ̃M) = 0. c) If M k Λ0 and Λ̃M k M, I({hAi i}, M, Λ̃{hAi i}) = 0. The CP-reflection directions for (a) and (b) are the same as in (A) and (B) of sec. II. For (c), if {hAi i} k M the CP-reflection direction is an eigenvector of Λ̃ perpendicular to M; otherwise {hAi i} × M is an eigenvector of Λ̃ and it is the CP-reflection direction. In the CP-basis, hΦi is real. 8 III. N ≥ 2 HIGGS-DOUBLETS For N ≥ 2 Higgs-doublets Φa , a = 1, . . . , N , transforming as (2,1) under SU (2)L ⊗ U (1)Y , the general gauge invariant potential can be written as [24] V (Φ) = Yab Φ†a Φb + Z(ab)(cd) (Φ†a Φb )∗ (Φ†c Φd ) , (43)  Φ1    Φ2   Φ=  ..  .  .  (44) Aab ≡ Φ†a Φb , (45) where  ΦN We define then the minimal SM gauge invariants and define a column vector (ab) = (11), (12), (13), . . . , (1N ), (21), . . . (N N ), of length  A11 A12 .. . N2 by the ordering              † A ≡ Φ ⊗ Φ =  A1N  .    A21     ..   .  AN N (46) Additionally we denote the pair of indices as (ab) ≡ σ, running as Eq. (46), and define the operation of change of labelling σ b = (ba), if σ = (ab), in such a way that if Aσ = Aab , then A∗σ = Aba = Aσb . With this notation the quartic part of V (Φ) can be written V (Φ) Φ4 = A∗σ Zσσ′ Aσ′ ≡ A† ZA . (47) This parameterization constrains Z to be hermitian Z † = Z, (Z(ab)(cd) )∗ = Z(dc)(ba) or Zσ∗1 σ2 = Zσ2 σ1 . (48) At the same time, because of A∗σ1 Aσ2 = A∗σc2 Aσc1 , Z has the property Zσ1 σ2 = Zσc2 σc1 . (49) Thus, Z is a N 2 × N 2 hermitian matrix with the additional property of Eq. (49). To count the number of independent variables of Z we have to divide its (complex) elements into four sets: (d1) N diagonal (σ1 = σ2 ≡ σ and σ = σ̂) and (d2) N (N − 1) diagonal (σ1 = σ2 ≡ σ and σ 6= σ̂) real elements because of the Hermiticity condition (48); (o1) N (N − 1) off-diagonal (σ1 6= σ2 but σ1 = σˆ2 ) and (o2) N 2 (N 2 − 1) − N (N − 1) off-diagonal (σ1 6= σ2 and σ1 6= σˆ2 ) complex but not all independent elements. The total is N 4 elements as it should be. The number of independent real parameters is then N (d1) + N (N − 1)/2 (d2) in the diagonal real elements [Eq. (49) only imposes conditions on the elements in (d2)] and N (N − 1) (o1) + 21 [N 2 (N 2 − 1) − N (N − 1)] (o2) in the off-diagonal complex elements [Eq. (49) only imposes conditions on the elements in (o2)] summing up to N 2 (N 2 + 1)/2. For example, for N = 2 there were 4(4 + 1)/2 = 10 real parameters corresponding to the real and complex parameters, {λ1 , λ2 , λ3 , λ4 } and {λ5 , λ6 , λ7 }, respectively. The horizontal transformation group is now G = SU (N )H ; a global phase can be absorbed by U (1)Y symmetry as in the N = 2 case. We can define new variables, equivalent to Eq. (6), as Aµ ≡ 21 Φ† λµ Φ , µ = 0, 1, . . . , d, (50) 9 q where λ0 = N2 1 and {λi } are the d = dim SU (N ) = N 2 − 1 hermitian generators of SU (N )H in the fundamental representation obeying the normalization Tr[λi λj ] = 2δij , such that Tr[λµ λν ] = 2δµν . The new second order variables Aµ transform as N̄ ⊗ N = d ⊕ 1, where d denotes the adjoint representation. The index µ = 0 corresponds to the singlet component while the indices µ = i = 1, ..., d correspond to the adjoint, transforming under SU (N )H as Ai → Rij Aj . (51) The matrix Rij can be obtained from the fundamental represention U , acting on Φ as Eq. (3), from the relation Rij (U ) = 12 Tr[U † λi U λj ] . (52) Rij (θ) = exp[iθk Tk ]ij , (53) If U = exp(iθ · λ/2), where (iTk )ij = fkij . The coefficients fijk , are the structure constants of SU (N )H defined by [Ti , Tj ] = ifijk Tk , (54) for any {Ti } spanning the compact SU (N )H algebra G. In particular, Eq. (54) is valid for {λi /2}, the fundamental representation generators. Since the structure constants of Eq. (54) are real, the adjoint representation of Eqs. (52) and (53) is real and thus it represents a subgroup of SO(d). It is only for N = 2 the adjoint representation is the orthogonal group itself. The transformation matrix from Aσ (46) to Aµ (50) can be obtained from the completeness relation of {λµ } [26]: 1 2 (λµ )ab (λµ )cd = δad δcb . (55) In the notation where σ1 = (ab) and σ2 = (cd), we can write Eq. (55) in the form 2Cµσ1 Cµσ2 = δσ1 σc2 , (56) Cµσ1 ≡ 12 (λµ )σ1 . (57) where −1 Equation (56) implies Cσµ ≡ 2Cµbσ , since the inverse is unique. The definition of Eq. (57) enable us to write Eq. (50) in the form Aµ = Cµσ Aσ . (58) By using the inverse of Eq. (58) we can write the potential of Eq. (43) in the same form of Eq. (8), V (A) = Mµ Aµ + Λµν Aµ Aν , = M 0 A0 + Λ00 A20 (59) + Mi Ai + 2Λ0i A0 Ai + Λ̃ij Ai Aj , (60) where Mµ ≡ Tr[Y λµ ] , Λµν ≡ Cσ−1∗ Zσ1 σ2 Cσ−1 2ν 1µ (61) , Λ̃ = {Λij } , i, j = 1, 2 . . . , N. (62) (63) Using the properties of Eqs. (48) and (49) of Z, we can see Λ is a N 2 × N 2 real and symmetric matrix, hence with N 2 (N 2 + 1)/2 real parameters, the same number of parameters of Z. The rank-2 tensor Λ̃ transforms under G as (d ⊗ d)S and it forms a reducible representation. (See appendix D.) The procedure to find the CP-basis can be sought in some analogy with the N = 2 case. The difficulty for N > 2, however, is that the existence of a horizontal transformation on the vector Ai , defined by Eq. (51), capable of diagonalizing Λ̃, is not always guaranteed and it depends on the form of Λ̃ itself. 10 Nevertheless, the diagonalization of Λ̃ is not strictly necessary. To see this, we have to analyze the CP properties of Φ and Ai . Firstly, any CP-type transformation can be written as a combination of a horizontal transformation and the canonical CP-transformations Φ(x) → Φ∗ (x̂) , CP (64) CP A0 (x) → A0 (x̂) , (65) CP Ai (x) → −ηij Aj (x̂) . (66) Equation (66) represents the canonical CP-reflection defined by the CP-reflection matrix η given by ηij = − 12 Tr[λ⊤i λj ] , (67) ψ which means λ⊤i = −ηij λj . The mapping λi → −λ⊤i = ηij λj is the contragradient automorphism in the Lie algebra [1], which I will denote by ψ. Such automorphism maps the fundamental representation to the antifundamental representation, D(g) → D∗ (g) (these two representations are not equivalent for N > 2). All the irreducible representations (irreps) we are treating here [d and all components in (d ⊗ d)S ] are self-conjugate [1, 27] and, indeed, they are real representations. To set a convention, we will use the following ordering for the basis of the Lie algebra G of G: {Ti } = {hi , Sα , Aα }. (68) The set {hi } spans the Cartan subalgebra (CSA) tr and the set {Aα }, denoted by tq , are the generators of the real H = SO(N ) subgroup of G = SU (N ). The remaining subspace spanned by {Sα } will be denoted by t̃q and the sum tr ⊕ t̃q ≡ tp represents the generators of the coset G/H. Notice that tp and tq are invariant by the action of the subgroup H and hence they form representation spaces for H (see appendix B). We will use the symbols {hi , Sα , Aα } to denote either the abstract algebra in the Weyl-Cartan basis or the fundamental representation of them. The dimensions of these subspaces of G are respectively, r = rankG = N − 1, q = (d − r)/2 = N (N − 1)/2 and p = (d + r)/2 = N (N + 1)/2 − 1; q denotes the number of positive roots in the algebra and α are the positive roots that label the generators Eα + E−α , 2 Eα − E−α = . 2i Sα = (69) Aα (70) The Eα are the “ladder” generators in the Cartan-Weyl basis. For example, for the fundamental representation of SU (3), we have in terms of the Gell-Mann matrices [28], {hi } = {λ3 /2, λ8 /2}, {Sα } = {λ1 /2, λ6 /2, λ4 /2} and {Aα } = {λ2 /2, λ7 /2, λ5 /2}. The two last subspaces are ordered according to α1 , α2 and α3 = α1 + α2 . Notice that in such representation Sα are symmetric matrices and Aα are antisymmetric matrices. With such ordering the CP-reflection matrix is ! − 1p 0 . (71) η= 0 1q Thus, we see that the application of the automorphism ψ separates G into an odd part tp and an even part tq , which constitutes a subalgebra [29]. The condition for CP-invariance of the term containing Λ̃ in Eq. (60) is then the existence of a group element g such that cond. 1: ηR(g)Λ̃R(g −1 )η = R(g)Λ̃R(g −1 ) , (72) where R(g) ≡ R(U ) is an element in the adjoint representation of SU (N ) (52). Equation (72) is equivalent, in this ⊤ representation, to the statement: exists a R(U ) in the adjoint representation of SU(N), such that RΛ̃R is block diagonal p × p superior and q × q inferior. Thus full diagonalization is not necessary. Now suppose a g satisfying cond. 1 exists [30]. We can write Eq. (60) in the basis defined by one representative g, V (A) = M0 A0 + Λ00 A20 + Mi′ A′i + 2Λ′0i A0 A′i + Λ̃′ij A′i A′j , (73) where Mi′ = R(g)ij Mj and Λ′0i = R(g)ij Λ0j . The necessary conditions for V (A) to be CP invariant are cond. 2a: Mi′ = −ηij Mj′ cond. 2b: Λ′0i = −ηij Λ′0j (74) . (75) 11 Of course, there can be more than one distinct coset satisfying cond. 1, and, then, conditions (74) and (75) have to be checked for all these cosets. If for every coset satisfying cond. 1, there is no coset satisfying cond. 2a and cond. 2b, then the potential is CP violating. Otherwise, g satisfying conds. 1, 2a and 2b defines a CP-basis and the potential is CP invariant. Let us analyze further the conds. 1, 2a and 2b. To do that, we denote by V = Rd ∼ G the adjoint representation space, isomorphic to the algebra vector space. The automorphism ψ separates the space V into two subspaces V = Vp ⊕ Vq , one odd (Vp ) and one even (Vq ) under the automorphism: ηv = −v, if v ∈ Vp , ηv = v, if v ∈ Vq . (76) They correspond respectively to tp and tq , subspaces of G = tp ⊕ tq . The correspondence between G and V is given by Eq. (A26). With this notation, considering the matrix Λ̃ is a linear transformation over V , conds. 1, 2.a and 2.b imply that there should be two subspaces Vp′ and Vq′ of V = Vp′ ⊕ Vq′ invariant by Λ̃ and both M and Λ0 should be in Vp′ . Moreover, the two subspaces should be connected to Vp and Vq by a group transformation, i.e., Vp′ = R(g)Vp and Vq′ = R(g)Vq for some g in G. The explicit search for the matrices satisfying conds. 1, 2.a and 2.b is a difficult task. We can seek, instead, invariant conditions based on group invariants, analogously with what was done to the N = 2 case. For that purpose, it will be shown in the following that generalized pseudoscalar invariants [31], analogous to the true pseudoscalars (23), can still be constructed with respect do SU (N ) and any such quantity should be zero for a CP-invariant potential. The generalized pseudoscalar is defined as a trilinear totally antisymmetric function of vectors in the adjoint, defined by I(v1 , v2 , v3 ) ≡ fijk v1i v2i v3i . (77) We keep the same notation as in Eq. (23), noticing that Eq. (77) corresponds to a more general case. We can also define the analogous of the vector product in three dimensions, as (v1 ∧ v2 )i ≡ fijk v1j v2k . (78) fijk ηia ηjb ηkc = fabc , (79) From Eq. (A3) and λ⊤i = −ηij λj we see that which means ψ indeed represents an automorphism in the algebra. However, the CP-reflection of Eq. (66) acts with the opposite sign compared to the automorphism ψ. Therefore, the quantity in Eq. (77) is invariant by SU (N )H transformations but changes sign under a CP transformation, i.e., a CP-reflection. Thus we see the trilinear function (77) behaves as a pseudoscalar under a CP-reflection. Such property means that any pseudoscalar of the form Eq. (77), constructed with the parameters of a CP-invariant potential V (A) should be zero. The pseudoscalar invariants of lowest order, constructed with {M, Λ0 , Λ̃}, are of the same form as in Eqs. (24). We will see, however, that the vanishing of these quantities may not guarantee the CP-invariance of V (A). Let us exploit further the properties of the CP-reflection (66). For that end, it is known that an additional trilinear scalar, which is totally symmetric, can be defined for N > 2 as J(v1 , v2 , v3 ) ≡ dijk v1i v2j v3k , (80) (v1 ∨ v2 )i ≡ dijk v1j v2k . (81) as well as a “symmetric” vector product The coefficient dijk is the totally symmetric 3-rank tensor of SU (N ) defined by Eq. (A5). The behavior of the scalar J (80) is opposite to the scalar I (77), since it changes sign under the contragradient automorphism, as can be seen by Eq. (A6), and remain invariant under a CP-type reflection. Using the two trilinear invariants I and J, the following relations can be obtained for any Vp′ = R(g)Vp and Vq′ = R(g)Vq : I(Vp′ , Vp′ , Vp′ ) I(Vq′ , Vq′ , Vp′ ) J(Vq′ , Vq′ , Vq′ ) J(Vp′ , Vp′ , Vq′ ) = = = = 0 0 0 0 , , , . (82) 12 These relations can be proved by noting that they are invariant under transformations in G and it can be evaluated with the corresponding vectors in the original subspaces Vp and Vq . Using Eqs. (79) (for dijk the opposite sign is valid) and (76), the invariants in Eq. (82) are equal to their opposites, which imply they are null. For example, fijk v1i v2j v3k = fijk ηia ηjb ηkc v1a v2b v3c = −fijk v1i v2j v3k = 0 for v1 , v2 , v3 in Vp . (The relations (82) can also be proved by using the explicit representations for fijk and dijk , appendix A 2.) Moreover, the relations in Eq. (82) imply Vp′ ∧ Vp′ ⊂ Vq′ , Vp′ ∧ Vq′ ⊂ Vp′ , Vq′ ∧ Vq′ ⊂ Vq′ , (83) and Vp′ ∨ Vp′ ⊂ Vp′ , Vp′ ∨ Vq′ ⊂ Vq′ , Vq′ ∨ Vq′ ⊂ Vp′ , (84) since the choice of vectors in each subspace is arbitrary and the two subspaces are disjoint and covers the whole vector space V . The first relation in Eq. (82) confirms that if conds. 1, 2.a and 2.b are satisfied, all I invariants are indeed null. Let us now analyze cond. 1. If cond. 1 is true, Λ̃ should have two invariant subspaces Vp′ and Vq′ connected to Vp and Vq by the same group element. Since Λ̃ is real and symmetric, it can be diagonalized by SO(d) transformations with real eigenvalues. The d orthonormal eigenvectors of Λ̃, denoted by e′i , i = 1, 2, . . . , d, form a basis for V (if Λ̃ is degenerate, find orthogonal vectors in the degenerate subspace). Any set of eigenvectors spans an invariant subspace of Λ̃. q of them should span a subspace connected to Vq while the remaining p eigenvectors should span the orthogonal complementary subspace connected to Vp . There is a criterion to check if a given vector v is in some Vq′ . For q vectors, additional criteria exist to check if they form a vector space and if they are closed under the algebra G. These criteria follow from the fact that Vq is isomorphic to tq which forms a subspace of the (i times) N × N real antisymmetric matrices. Any antisymmetric matrix M has Tr[M 2k+1 ] = 0. The converse is true in the sense that for M hermitian, Tr[M 2k+1 ] = 0 for all 2k + 1 ≤ N imply M can be conjugated by SU (N ) to an antisymmetric matrix, i.e., in tq . (See appendix C for the proof.) Therefore, v belongs to a Vq′ if, and only if, 1 2 Tr[(v · λ)2k+1 ] = J2k+1 (v, v, . . . , v) = 0 , for all 2k + 1 ≤ N. (85) We have introduced the n-linear symmetric function v v , . . . , vn in , Jn (v1 , v2 , . . . , vn ) ≡ Γi(n) 1 i2 ···in 1 i1 2 i2 (86) which depends on the rank-n totally symmetric tensor 1 X1 ≡ Γi(n) Tr[λσ(i1 ) λσ(i2 ) . . . λσ(in ) ] , n ≥ 2 , 1 i2 ···in n! σ 2 (87) where σ denotes permutations among n elements and the sum runs over all possible permutations. The tensor in Eq. (87) are the tensors used to construct the r Casimir invariants of any representation of SU (N ) [32]. In particular, Γ(3) ijk = dijk and J3 = J. For two vectors v1 and v2 , each one satisfying Eq. (85), the linear combination c1 v1 + c2 v2 is also in some Vq′ if, and only if, 1 2 Tr[(c1 v1 · λ + c2 v2 · λ)2k+1 ] = 0 , for all 2k + 1 ≤ N . (88) In general, 1 2 Tr[(c1 v1 · λ + c2 v2 · λ)n ] = n   X m m n−m c c Jn (v1 , v1 , . . . , v1 , v2 , . . . , v2 ) . | {z } | {z } n 1 2 m=0 m n−m (89) Since the coefficients c1 and c2 are arbitrary, Eq. (88), requires J2k+1 (v1 , . . . , v1 , v2 , . . . , v2 ) = 0 (90) 13 for 2k + 1 ≤ N and all combinations of v1 and v2 . The generalization for a set of q normalized eigenvectors of Λ̃, labelled as e′p+i , i = 1, . . . , q, is straightforward. They form a q-dimensional vector space Vq′ if, and only if, each vector satisfy Eq. (85) and any combination of m ≤ q vectors satisfy Eq. (90). To guarantee that they are closed under the algebra G, compute the I invariants using any two vectors in Vq′ and one vector in Vp′ , as in the second relation of Eq. (82): they should all be null. This conditions attest that the vector space Vq′ is q-dimensional and forms a subalgebra of G. That the subalgebra isomorphic to Vq′ is semisimple and compact can be checked by Cartan’s criterion: the Cartan metric, as in Eq. (A4), have to be positive definite [29]. It remains to be checked if Vq′ is indeed connected to Vq by a group element. At this point, we can see an example for which the vanishing of the I-invariants (24) [generalized to N > 2 using Eq. (77)] does not guarantee the CP-invariance of the potential: If M and Λ0 are orthogonal eigenvectors of Λ̃, all I-invariant are null, but nothing can be said about Λ̃ satisfying cond. 1. Even if a Vq′ can be found using the procedure above, if it contains at least one of M or Λ0 , the potential is CP violating. For N = 3, the problem of finding the necessary and sufficient conditions for CP-invariance can be completely solved. In this case, the only nontrivial symmetric function is the J invariant in Eq. (80). The numerology is d = 8, r = 2, q = 3 and p = 5 and thus, Vq′ is three dimensional. It can be proved [33] that any three dimensional subalgebra is either conjugated to the SU (2) subalgebra spanned by {λ1 /2, λ2 /2, λ3 /2} or to the real SO(3) subalgebra spanned by {λ2 /5, λ7 /2, λ5 /2}, in Gell-Mann’s notation. If three eigenvectors e′6 , e′7 , e′8 of Λ̃ satisfy conditions (85) and (90), an additional condition to distinguish between the two equidimensional subalgebras is to use [33] |I(e′6 , e′7 , e′8 )| = 1 , 2 (91) which is satisfied only if the subalgebra is conjugate to SO(3). For the subalgebra conjugate to SU (2) the value for Eq. (91) is unity. This fact can be understood by observing that {λ2 /5, λ7 /2, λ5 /2} are half of the usual generators of SO(3) in the defining representation, giving for the structure constants restricted to the subalgebra, I(e′p+i , e′p+j , e′p+k ) = 12 εijk , i, j, k = 1, 2, 3, after choosing appropriately the ordering for those three vectors. In addition, if cond. 1 is true, an appropriate basis for the V (A) can be chosen to be the one with q × q inferior block of Λ̃ diagonal. Such choice is possible because when Λ̃ is block diagonal p × p and q × q, a transformation in SO(3) ⊂ SU (3) can still make the inferior block diagonal. For N > 3 that procedure is no longer guaranteed since the q × q blocks is transformed by the adjoint representation of SO(N ), which differs from the defining representation (see appendix B). Once the vector space Vq′ is found, when such space is unique up to multiplication by the subgroup SO(3), the vectors M and Λ0 should be in the orthogonal subspace Vp′ , i.e., M · e′p+i = 0 and Λ0 · e′p+i = 0 for all i = 1, 2, 3. Otherwise the potential V (A) is CP-violating. A. Conditions for spontaneous CP violation Let us briefly analyze the conditions for spontaneous CP-violation for a potential V (Φ) (59) that is CP-invariant before EWSB. The analysis can be performed in complete analogy with Sec. II A. The minimization equation in this case are identical to Eqs. (37), after replacing the matrices σµ by the corresponding λµ in Eq. (50). The same replacement applies to the first member of Eq. (38). The conditions (41) and (42) in the CP-basis are replaced by hΦi∗ = hΦi , hAi i = −ηij hAj i . (92) (93) A suitable generalization of parameterization (36) for N > 2 can be defined as v hΦi = √ (Uv eN ) ⊗ e2 , 2 (94) where eN = (0, 0, . . . , 0, 1)⊤ ∈ CN , e2 = (0, 1)⊤ is the SU (2)L breaking direction and Uv is a SU (N )H transformation. The parameterization (94) is justified because any vector z = (z1 , z2 , . . . , zN )⊤ in CN can be transformed by a SU (N ) transformation into z ′ = (0, 0, . . . , |z|)⊤ [32]. If Eq. (92) is true Uv is real and belongs to the real subgroup SO(N ). In a general basis, it is necessary that {hAi i} ∈ Vp′ , i.e., the vector corresponding to the VEV have to be in the same subspace as M and Λ0 , which is true if {hAj i} · e′p+i = 0, i = 1, . . . , q, for {e′p+i } spanning the subspace Vq′ invariant by Λ̃. For N = 3, such conditions are sufficient to guarantee that V (Φ + hΦi) is also CP-invariant. 14 IV. CONCLUSIONS AND DISCUSSIONS The NHDMs are simple extensions of the SM for which the presence of a horizontal space allows the possibility of “rotating” the basis in such space without modifying the physical content of the theory, e.g., CP symmetry or asymmetry. For N similar SM Higgs-doublets, which are complex, the relevant reparameterization transformations form a SU (N ) group. Restricted to the scalar potential sector, due to the rather restricted bilinear form of the minimal gauge invariants, the NHDM potential can be written in terms of the adjoint representation of SU (N )H . The CP-type transformations act as “reflections”, the CP-reflections, on the parameters written as vectors and tensors of the adjoint. Therefore, the scalar potential of the NHDMs are CP-invariant if, and only if, one can find a CPreflection that leaves the potential invariant. In addition, the analysis in the adjoint representation was shown to be much easier to carry out than the tensor analysis based on the fundamental and antifundamental representations. Of course if other representations that can not be written in terms of the adjoint are present, the analysis invariably would require the fundamental representations. For example, to extend this analysis to the Yukawa sector of the NHDMs, the fundamental representation is necessary there. For N = 2, with the fortunate coincidence of the adjoint of SU (2) being the rotation group in three dimensions, the full analysis is facilitated by the possible geometrical description. All the necessary and sufficient conditions for CP violation can be formulated for the 2HDM scalar potential sector. Those conditions can be formulated in terms of basis invariants which coincided with previously found ones [14], except for proportionality constants. (A comparison between the invariants in Refs. 14 and 16 is given in Ref. 15.) For CP-invariant potentials, this method also enabled us to find the explicit CP transformation in any basis and the procedure to reach the real basis. For CP-violating potentials, the canonical form of Eq. (20) still defines a standard form, besides the physical Higgs-basis [12, 20], to compare among the various 2HDMs: two 2HDM potentials are physically equivalent if they have the same form in the canonical CP-basis. (For convention, use the basis for which the eigenvalues of Λ̃ is in decreasing order.) This CP-basis also makes the soft/hard classification of CP-violation [17] easier to perform: From Eq. (20), we see the potential V (A) violates the CP-symmetry hardly only if the fourth term is CP violating, i.e., if IΛ0 (25) is not null; otherwise, the potential has soft CP violation through the third term or it is CP symmetric. From Eq. (40), we see the spontaneous CP violation only occurs softly. For N = 3, the necessary and sufficient conditions for CP-violation can still be formulated in a systematic way. However, these conditions may possibly be reduced to fewer and more strict conditions. Such reduction requires a more detailed study of the relation between the invariants (24)–(27) and the described procedure to check the CP symmetry or asymmetry. In case the potential is CP-invariant, the explicit procedure to reach a real basis (among infinitely many) is also lacking in this context and for N > 3 as well. For N > 3, necessary conditions for CP-invariance in the NHDM potential can be found but whether those conditions are sufficient or can be supplemented to be sufficient is an open question. The answer lies in the classification and perhaps parameterization of the orbital structure of the adjoint representation of the SU (N ) group. In any case, if a result similar to N = 3 can be found, i.e., if any SO(N ) subalgebra of the SU (N ) algebra is conjugated to the real SO(N ) subalgebra, the problem is practically solved. Another possible approach would be the study of the automorphism properties of the irreducible representation of SU (N ) contained in Λ̃ that are larger than the adjoint. For example, Λ̃ for N ≥ 3 contains a component transforming under the adjoint representation (see table I in appendix D and appendix E). For this component it always exists a transformation capable of transforming it to satisfy Eq. (72). For higher dimensional irreps a detailed study is not known to the author. To conclude, the method presented here illustrates that using the adjoint representation as the minimal nontrivial representation can have substantial advantage over the fundamental representation treatmens to handle the freedom of change of basis within a large horizontal space. Inherent to that was the notion of CP-type transformations as automorphisms in the group of horizontal transformations. Such notion was useful to distinguish the CP invariance/violation (explicit/spontaneous) properties of the theory and to construct the CP-odd basis invariants. APPENDIX A: NOTATION AND CONVENTIONS We use for the fundamental representation of SU (N ) the N × N traceless hermitian matrices {Fa } ≡ { 12 λa } normalized as Tr[Fi Fj ] = 21 δij . (A1) The number of generators is d = dimSU (N ) = N 2 − 1. The matrices λa are generalizations of the Gell-Mann matrices for SU (3) [28]. 15 The compact semisimple Lie algebra is defined by [Fi , Fj ] = ifijk Fk , (A2) which is satisfied for any represention D(Fa ). By using the convention of Eq. (A1), we have the relation fijk =   2  1  Tr [Fi , Fj ]Fk = Tr [λi , λj ]λk . i 4i (A3) The Cartan metric in the adjoint representation reads d X fajk fbjk = N δab . (A4) j,k=1 In the enveloping algebra implicit in the fundamental representation, we have also {Fi , Fj } = 1 2N δij 1 + dijk Fk . (A5) The coefficients dijk are totally symmetric under exchange of indices and they are familiar for SU(3) [28]. These coefficients can be obtained from the fundamental representation dijk = 2Tr[{Fi , Fj }Fk ] = 41 Tr[{λi , λj }λk ] , (A6) and obey the property d X dajk dbjk = j,k=1 N2 − 4 δab . N (A7) Taking the trace of Eq. (A5) we obtain the value of the second order Casimir invariant d X (Fi )2 = C2 (F )1 , (A8) i=1 d . The second order Casimir invariant for the adjoint representation is already given by Eq. (A4) where C2 (F ) = 2N which imply C2 (ad) = N . The fundamental representation for SU (N ) 1. We show here a explicit choice of matrices for the fundamental representation of SU (N ) in the Cartan-Weyl basis. With certain choice of phases and cocycles implicit, such choice coincides with the Gell-Mann type matrices (except for a factor one-half). The SU (N ) algebra G is the algebra of the hermitian and traceless N × N matrices. This is the defining and a fundamental (and minimal) representation. An orthogonal basis for this algebra can be chosen to be the d matrices 1 diag(1k , −k, 0, . . . , 0) , k = 1, . . . , r, 2k(k + 1) hk = p Sij = Aij = 1 2 (eij + eji ) , 1 2i (eij − eji ) , (A9) i < j = 1, . . . , N, (A10) i < j = 1, . . . , N, (A11) where r = N − 1, and eij denotes the canonical basis defined by (eij )kl = δik δjl . Each type of matrices spans the algebra subspaces {hk } ∼ tr , {Sij } ∼ t̃q and {Aij } ∼ tq in Eq. (68) and the normalization satisfies Eq. (A1). If we associate (i, j) = (i, i + 1) ↔ αi [i = 1, . . . , r], (i, j) = (i, i + 2) ↔ αr+i [i = 1, . . . , r − 1], . . ., (1, N ) ↔ αq , we obtain the correspondence Sij ↔ Sα and Aij ↔ Aα ; q = (d − r)/2 = N (N − 1)/2 is the number of positive roots of the algebra denoted by α, used for labeling Sα and Aα . The first r roots are the simple roots. All positive roots can be written as combinations of the simple roots. Since SU (N ) is a simply laced algebra [27], the positive roots are given, in terms of the simple roots, h=1 α1 , α2 , . . . , αr , 16 h=2 h=3 .. . α1 + α2 , α2 + α3 , . . . , αr−1 + αr , α1 + α2 + α3 , α2 + α3 + α4 , . . . , αr−2 + αr−1 + αr , .. . h=r α1 + α2 + . . . + αr . (A12) The height h is the sum of the expansion coefficients of the positive roots in terms of the simple roots. Only the sum of neighbor simple roots are also roots. The roots live in an Euclidean r-dimensional space and explicit coordinates can be obtained from the matrices (A9), (A10) and (A11), and the relation [hk , Eα ] = (α)k Eα . (A13) Using Eα = ei,i+1 , the simple roots αi , which are normalized as (αi , αi ) = 1, have coordinates (αi )k = (hk )ii − (hk )i+1,i+1 . (A14) The weight system of the fundamental representation have highest weight λ1 , which is just the first primitive weight defined by 2(λi , αj )/(αj , αj ) = δij , i, j = 1, . . . , r. The r + 1 weights of this representation can be obtained by subtracting positive roots from the highest weight: µ0 = λ1 µ1 = λ1 − α1 µ2 = λ1 − α1 − α2 .. .. . . µr = λ1 − α1 − · · · − αr ∼ (10 . . . 0) ∼ (−110 . . . 0) ∼ (0−110 . . . 0) .. . (A15) ∼ (0 . . . 0−1) . The last column corresponds to the weights P in Dynkin basis [27] µa ∼ (n1 n2 . . . nr ), which are the expansion coefficients in terms of the primitive weights µa = ri=1 ni λi . These r + 1 = N weights can label all the states, which are not degenerate in this case. The matrices (A9) represents the Cartan subalgebra, and in the Cartan-Weyl basis they are diagonal. The diagonal elements are just the components of the weights in Eq. (A15), i.e., hk ≡ hµ|hk |µ′ i = diag(µ0 , µ1 , . . . , µr )k . 2. (A16) fijk and dijk tensors By using Eqs. (A3), (A6) and the properties of the fundamental representation described in the preceding subsection, we can deduce some general features of the rank-3 tensors fijk and dijk with the ordering defined by Eq. (68). Firstly, we define f (Fi , Fj , Fk ) ≡ fijk , d(Fi , Fj , Fk ) ≡ dijk . (A17) (A18) Then, the following properties can be proved, f (hi , hj , Fk ) f (hi , Sα , Sβ ) f (hi , Aα , Aβ ) f (hi , Aα , Sβ ) f (hi , Sα , Aβ ) = = = = = 0 0 0 −(α)i δαβ (α)i δαβ . (A19) The zeros above can be obtained from the general relations [tp , tp ] ⊂ tq [tq , tq ] ⊂ tq [tq , tp ] ⊂ tp , (A20) 17 and [tr , tr ] = 0 [tr , tq ] ⊂ t̃q [tr , t̃q ] ⊂ tq , (A21) and the fact that the trace of the product of two elements of distinct subspaces is null (orthogonality). The properties (A20) are easily seen in the fundamental representation by the symmetric character of the elements of tp , the antisymmetric character of the elements of tq , and the properties of a commutator [A, B]⊤ = −[A⊤ , B⊤ ]. For illustration, we will show how to obtain the non-null elements of Eq. (A19), knowing the properties of the fundamental representation in the Cartan-Weyl basis. The procedure is as follows, X f (hi , Aα , Sβ ) = 2i hµ|[hi , Aα ]Sβ |µi µ = −2(α)i = − 21 (α)i X µ X µ −(α)i δαβ 21 = hµ|Sα Sβ |µi hµ|Eα E−β + E−α Eβ |µi X [δ(µ − α) + δ(µ + α)] , (A22) µ where δ(µ) = ( 1 , µ is a weight, 0 , µ is not a weight. (A23) In addition, we have used Eqs. (69), (70), (A13) and hµ|E−α Eα |µi = δ(µ + α). We can see from Eq. (A15) that the last sum of Eq. (A22) gives 2 for any positive root α since there are always one positive root connecting two weights. For dijk , the fundamental representation is essential since we can not use the Lie algebra properties. Some properties are d(hi , hj , Sα ) = d(hi , hj , Aα ) = 0 d(hi , Sα , Aβ ) = 0 X d(hi , Sα , Sβ ) = δαβ (µ)i [δ(µ − α) + δ(µ + α)] µ d(hi , Aα , Aβ ) = d(h i , Sα , Sβ ) X d(hi , hj , hk ) = 4 (µ)i (µ)j (µ)k . (A24) µ Basis independent properties can be extracted by defining a symmetric algebra [32, 33] in the space of the N × N tracesless hermitian matrices. Such space will be denoted by Mh (N, C), and it is isomorphic to a Rd vector space. Given x, y ∈ Mh (N, C), the symmetric algebra is defined as ˜ ˜ x ∨ y ≡ 12 {x, y } − ˜ ˜ ˜ ˜ 1 N Tr[xy ] ˜˜ . (A25) Obviously x ∨ y ∈ Mh (N, C). The tilde in x means ˜ ˜ ˜ x ≡ x · λ = xi λi , ˜ (A26) where x lives in Rd , in the adjoint representation space. In terms of the vectors x and y in the adjoint, the symmetric algebra (A25) can be written x ∨ y ≡ (x ∨ y) · λ , ˜ ˜ (A27) where the ∨ in the righthand side of Eq. (A27) is the product defined on the adjoint vectors, Eq. (81). We use the same symbol for both of the products. 18 From the symmetric and antisymmetric nature of the elements of tp and tq , respectively, we can conclude that tp ∨ tp ⊂ tp tq ∨ tq ⊂ tp tq ∨ tp ⊂ tq . (A28) Then, the equality dijk = 2Tr[{Fi , Fj }Fk ] = 4Tr[Fi ∨ Fj Fk ] yields d(tp , tp , tq ) = 0 d(tq , tq , tq ) = 0 . (A29) The results above are invariant if the positions of any pair are interchanged. For particular elements of G, we also have d(Fi , Fi , Fi ) = 0, for Fi = Sα and Fi = h1 . However, any element F in t̃q does not satisfy d(F, F, F ) = 0, differently of tq . Thus tq forms a subspace (and subalgebra) of G but not t̃q . APPENDIX B: BRANCHING OF ADJSU (N ) WITH RESPECT TO REAL SO(N ) The separation of the SU (N ) algebra in tp and tq , as in Eq. (68), naturally induces two representations of the SO(N ) subalgebra generated by tq = {Aα }. One of them is just the adjoint of SO(N ) carried by the real antisymmetric N × N matrices spanned by {iAα }, for which the subgroup action is eiθα Aα (aβ iAβ )e−iθα Aα = iAα D1 (eiθ·A )αβ aβ . (B1) The representation D1 is just the lower q × q block of the adjoint representation of exp(θα D(iAα )) of SU (N ) with the ordering (68). This is an irrep of dimension q = N (N − 1)/2. The other representation is carried by the real N × N symmetric traceless matrices spanned by tp = {hi , Sα }. The subgroup action is given by ! ai iθ·A −iθα Aα iθα Aα . (B2) = (hi , Sα )D2 (e ) (ai hi + bβ Sβ )e e bβ The representation D2 is just the upper p × p block matrix in the adjoint representation exp(θα D(iAα )) of SU (N ) whose dimension is p = N (N + 1)/2 − 1 and it is irreducible. APPENDIX C: PROPERTIES OF MATRICES SIMILAR TO ANTISYMMETRIC MATRICES The following proposition will be proved: For any complex or real n × n diagonalizable [34] matrix X, Tr[X 2m+1 ] = 0 for all 2m + 1 ≤ n (C1) imply X is similar to an antisymmetric matrix A = U XU −1 . The converse is trivial since the trace of a matrix is equal to the trace of the transpose. For the proof, we need the characteristic equation [33] det(X − λ1) = (−1)n [λn − n X γk (X)λn−k ] , (C2) k=1 where γ1 (X) = Tr[X] , γ2 (X) = γ3 (X) = 1 Tr[X 2 2 1 Tr[X 3 3 .. . γn (X) = (C3) − γ1 (X)X] , (C4) − γ1 (X)X − γ2 (X)X] , (C5) 2 (C6) 1 Tr[X n n − n−1 X k=1 γk (X)X n−k ] . (C7) 19 The same coefficients enter in the matricial equation Xn − n X γk (X)X n−k = 0 , (C8) k=1 for which X 0 = 1n is implicit. If Eq. (C1) is satisfied, all odd coefficients γ2k+1 (X) = 0 and the characteristic equation reads det(X − λ1) = (−1)n [λn − γ2 (X)λn−2 − γ4 (X)λn−4 − . . . − γn (X)] . (C9) If n even we rewrite n = 2m and Eq. (C9) yields det(X − λ1) = (λ2 )m − m X γ2k (X)(λ2 )m−k = f (λ2 ) . (C10) γ2k (X)(λ2 )m−k ] = −λf (λ2 ) . (C11) k=1 If n odd we rewrite n = 2m + 1 and Eq. (C9) yields det(X − λ1) = −λ[(λ2 )m − m X k=1 For both Eqs. (C10) and (C11), f (λ2 ) is a polynomial in λ2 of order m and it has, including degeneracies, m (complex) roots λ2i , i = 1, 2, . . . , m. Then f (λ2 ) = m m Y Y (λ − λi )(λ + λi ) , (λ2 − λ2i ) = (C12) i=1 i=1 which implies that for each eigenvalue λi of X an opposite eigenvalue −λi exists (both might be zero), except for a unique additional zero eigenvalue when n is odd, as can be seen from Eq. (C11). The existence of opposite eigenvalues guarantees the existence of a similarity transformation U1 that leads X to the diagonal form ( diag(λ1 σ3 , λ2 σ3 , . . . , λm σ3 ) for n = 2m, −1 U1 XU1 = (C13) diag(λ1 σ3 , λ2 σ3 , . . . , λm σ3 , 0) for n = 2m + 1. Then, one can use the matrix U2 = ( 1m ⊗ e−iσ π/4 diag(1m ⊗ e−iσ π/4 , 0) 1 1 for n = 2m, for n = 2m + 1, to transform Eq. (C13) into antisymmetric form ( diag(λ1 σ2 , λ2 σ2 , . . . , λm σ2 ) −1 −1 U2 U1 XU1 U2 = diag(λ1 σ2 , λ2 σ2 , . . . , λm σ2 , 0) for n = 2m, for n = 2m + 1. (C14) (C15) When X is hermitian, U1 can be unitary and the eigenvalues λi are real. For X in the SU (N ) algebra, condition (C1) is necessary and sufficient for X to be in the orbit of an element in tq . APPENDIX D: THE DECOMPOSITION OF (d ⊗ d)S OF SU (N ) In the last term of Eq. (60), Λ̃ transforms under the (d ⊗ d)S representation of SU (N )H . Such representation is reducible. Though, unlike the N = 2 case, it has more than two components, as shown in the table I for N = 2, . . . , 6. (It can be proved that the number of components are at most four.) Table I shows the branchings of the direct product representation d ⊗ d for the symmetric part denoted by the subscript S [27]; the last column shows the dimension of the representation space of (d ⊗ d)S , which is just the space of the real symmetric d × d matrices. 20 TABLE I: SU (N ) decompositions N 2 3 4 5 6 d = N2 − 1 (d ⊗ d)S 3 5⊕1 8 27 ⊕ 8 ⊕ 1 15 84 ⊕ 20 ⊕ 15 ⊕ 1 24 200 ⊕ 75 ⊕ 24 ⊕ 1 35 405 ⊕ 189 ⊕ 35 ⊕ 1 d(d+1) 2 6 36 120 300 630 APPENDIX E: REAL SYMMETRIC ADJOINT REPRESENTATION IN (d ⊗ d)S We know the adjoint representation for a Lie group can be obtained from the vector space spanned by the algebra itself in any representation. In particular, any compact semisimple Lie algebra can be represented by the d × d real antisymmetric matrices given by the structure constants i(Ti )jk = fijk . In contrast, there is also a d × d real symmetric representation spanned by the real symmetric matrices {di } given by (di )jk = dijk , (E1) which is the rank-3 totally symmetric tensor from Eq. (A6). That {di } represents the SU (N ) in the adjoint representation can be seen by [Ti , dj ] = ifijk dk . (E2) Equation (E2) can be proved by using Eqs. (A3), (A6) and the completeness relation of Eq. (55). Moreover dijk Tj Tk = − N di . 2 (E3) Thus, for N ≥ 3, the component in the adjoint of the tensor Λ̃ can be extracted as Λ̃ ad = Λ̃(ad) i di , (E4) where, from Eq. (A7), Λ̃(ad) = i N N 2 −4 Tr[Λ̃di ] = N N 2 −4 dijk Λ̃jk . (E5) This is a practical way of extracting the symmetric adjoint representation of (d ⊗ d)S . ACKNOWLEDGMENTS This work was supported by Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico (CNPq). The author would like to thank Prof. J. C. Montero and Prof. V. Pleitez for pointing up Refs. 13 and 14 which motivated this work, and Prof. L. A. 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