Classification of finite reparametrization symmetry groups in the three-Higgs-doublet model
CClassification of finite reparametrization symmetrygroups in the three-Higgs-doublet model
Igor P. Ivanov , , Evgeny Vdovin IFPA, Universit´e de Li`ege, All´ee du 6 Aoˆut 17, bˆatiment B5a, 4000 Li`ege, Belgium Sobolev Institute of Mathematics, Koptyug avenue 4, 630090, Novosibirsk, Russia
February 18, 2013
Abstract
Symmetries play a crucial role in electroweak symmetry breaking models with non-minimal Higgs content. Within each class of these models, it is desirable to know whichsymmetry groups can be implemented via the scalar sector. In N -Higgs-doublet mod-els, this classification problem was solved only for N = 2 doublets. Very recently, wesuggested a method to classify all realizable finite symmetry groups of Higgs-family trans-formations in the three-Higgs-doublet model (3HDM). Here, we present this classificationin all detail together with an introduction to the theory of solvable groups, which playthe key role in our derivation. We also consider generalized- CP symmetries, and dis-cuss the interplay between Higgs-family symmetries and CP -conservation. In particular,we prove that presence of the Z symmetry guarantees the explicit CP -conservation ofthe potential. This work completes classification of finite reparametrization symmetrygroups in 3HDM. The nature of the electroweak symmetry breaking is one of the main puzzles in high-energyphysics. Very recently, the CMS and ATLAS collaborations at the LHC announced the dis-covery of the Higgs-like resonance at 126 GeV, [1], and their first measurements indicateintriguing deviations from the Standard Model (SM) expectations. Whether these data signalthat a non-minimal Higgs mechanism is indeed at work and if so what it is, are among thehottest questions in particle physics these days.In the past decades, many non-minimal Higgs sectors have been considered, [2]. Oneconceptually simple and phenomenologically attractive class of models involves several Higgsdoublets with identical quantum numbers ( N -Higgs-doublet models, NHDM). Its simplestversion with only two doublets, 2HDM, was proposed decades ago, [3], but it is still activelystudied, see [4] for a recent review, and it has now become a standard reference model of thebeyond the Standard Model (bSM) physics. Constructions with more than two doublets arealso extensively investigated, [5–14].Many bSM models aim at providing a natural explanation for the numerical values of(some of) the SM parameters. Often, it is done by invoking additional symmetries in the1 a r X i v : . [ h e p - ph ] F e b odel. These are not related with the gauge symmetries of the SM but rather reflect extrasymmetry structures in the “horizontal space” of the model. One of the main phenomenologicalmotivations in working with several doublets is the ease with which one can introduce varioussymmetry groups. Indeed, Higgs fields with identical quantum numbers can mix, and itis possible that some of these Higgs-family mixing transformations leave the scalar sectorinvariant. Even in 2HDM, presence of such a symmetry in the lagrangian and its possiblespontaneous violation can lead to a number of remarkable phenomena such as various formsof CP -violation, [3, 15], non-standard thermal phase transitions which may be relevant for theearly Universe, [16], natural scalar dark matter candidates, [17]. For models with three ormore doublets, an extra motivation is the possibility to incorporate into the Higgs sector non-abelian finite symmetry groups, which can then lead to interesting patterns in the fermionicmass matrices (for a general introduction into discrete symmetry groups relevant for particlephysics, see [18]). In this respect, the very popular symmetry group has been A , [7, 8], thesmallest finite group with a three-dimensional irreducible representation, but larger symmetrygroups also received some attention, [5, 9, 10].Given the importance of symmetries for the NHDM phenomenology, it is natural to ask: which symmetry groups can be implemented in the scalar sector of NHDM for a given N ? In the two-Higgs-doublet model (2HDM), this question has been answered several yearsago, [19, 20], see also [4] for a review. Focusing on discrete symmetries, the only realizableHiggs-family symmetry groups are Z and ( Z ) . The Z group can be generated, for example,by the sign flip of one of the doublets (and it does not matter which, because once we focuson the scalar sector only, the simultaneous sign flip of both doublets does not change thelagrangian), while the ( Z ) group is generated by sign flips and the exchange φ ↔ φ . Ifgeneralized- CP transformations are also included, then ( Z ) becomes realizable as well, theadditional generator being simply the CP conjugation.With more than two doublets, the problem remains open. Although several attempts havebeen made in past to classify at least some symmetries in NHDM, [11, 12, 14], they led onlyto very partial results. The main obstacle here was the lack of the completeness criterion.Although many obvious symmetry groups could be immediately guessed, it was not clear howto prove that the given potential does not have other symmetries. An even more difficultproblem is to prove that no other symmetry group can be implemented for a given N .In the recent paper [21] we found such a criterion for abelian symmetry groups in NHDM forarbitrary N . Since abelian subgroups are the basic building blocks of any group, classificationof realizable abelian symmetry groups in NHDM was an important milestone. We stress thatthis task is different from just classifying all abelian subgroups of SU (3), because invarianceof the Higgs potential places strong and non-trivial restrictions on possible symmetry groups.In this paper, we solve the classification problem for all finite symmetry groups in 3HDM,including non-abelian groups. We do this by using the abelian groups in 3HDM found in [21]and by applying certain results and methods from the theory of solvable groups. Some of theseresults were already briefly described in [22]. Here, we present a detailed derivation of thisclassification together with an introduction to the relevant methods from finite group theory.In addition, we extend the analysis to symmetry groups which include both Higgs-family andgeneralized- CP transformations. This work, therefore, solves the problem of classification offinite reparametrization symmetry groups in 3HDM.We would like to stress one important feature in which our method differs from moretraditional approaches to symmetry classification problem, at least within the bSM physics.2sually, one starts by imposing invariance under certain transformations, and then one triesto recognize the symmetry group of the resulting potential. In this way it is very difficult tosee whether all possible symmetries are exhausted. We approach the problem the other wayaround. We first restrict the list of finite groups which can appear as symmetry groups of3HDM, and then we check one by one whether these groups can indeed be implemented.The structure of this paper is the following. In Section 2 we describe different types ofsymmetries in the scalar sector of NHDM and discuss the important concept of realizablesymmetry groups. Section 3 contains an elementary introduction into the theory of (finite)solvable groups. Although it contains pure mathematics, we put it in the main text because itis a key part of the group-theoretic step of our classification, which is presented in Section 4.Then, in Section 5 we describe the methods which we will use to prove the absence of continu-ous symmetries. Sections 6 and 7 contain the main results of the paper: explicit constructionsof the realizable symmetry groups and of the potentials symmetric under each group. Finally,in Section 8 we summarize and discuss our results. For the reader’s convenience, we list in theAppendix potentials for each of the realizable non-abelian symmetry groups. In NHDM we introduce N complex Higgs doublets with the electroweak isospin Y = 1 / V = Y ab ( φ † a φ b ) + Z abcd ( φ † a φ b )( φ † c φ d ) , (1)where all indices run from 1 to N . Coefficients of the potential are grouped into componentsof tensors Y ab and Z abcd ; there are N independent components in Y and N ( N + 1) / Z .In this work we focus only on the scalar sector of the NHDM. Therefore, once coefficients Y ab and Z abcd are given, the model is completely defined, and one should be able to expressall its properties (the number and the positions of extrema, the spectrum and interactions ofthe physical Higgs bosons) via components of Y ’s and Z ’s. This explicit expression, however,cannot be written via elementary functions, and it remains unknown in the general case forany N > GL (2 , C ) keeps thegeneric form of the potential, changing only the coefficients of Y and Z . We call such a trans-formation a Higgs-basis change . In addition, the CP transformation, which maps doubletsto their hermitean conjugates φ a → φ † a , also keeps the generic form of the potential, up tocoefficient modification. Its combination with a Higgs-basis change represents a transforma-tion which is usually called a generalized- CP transformation , [25]. The Higgs basis changesand generalized- CP transformations can be called together reparametrization transformations reparametrization-invariant combinations of Y ’s and Z ’s, [23, 26].If a reparametrization transformation maps a certain potential exactly to itself, that is,if it leaves certain Y ’s and Z ’s invariant, we say that the potential has a reparametrizationsymmetry . Usually, there is a close relation between the reparametrization symmetry group G of the potential and its phenomenological properties, both within the scalar and the fermionsectors. Therefore, understanding which groups can appear as reparametrization symmetrygroups in NHDM with given N is of much importance for phenomenology of the model. Often, one restricts the group of reparametrization transformations only to those transfor-mations which keep the Higgs kinetic term invariant. In this case, a generic basis changebecomes a unitary transformation φ a (cid:55)→ U ab φ b with U ∈ U ( N ). A kinetic-term-preservinggeneralized- CP transformation is an anti-unitary map φ a (cid:55)→ U ab φ † b , which can be written as U CP = U · J , with a unitary U and with J being the symbol for the CP -transformation.The group U ( N ) contains the group of overall phase rotations, which are already includedin the gauge group U (1) Y . Since we want to study structural symmetries of the NHDMpotentials, we should disregard transformations which leave all the potentials invariant byconstruction. This leads us to the group U ( N ) /U (1) (cid:39) P SU ( N ). Note that SU ( N ), whichis often considered in these circumstances, still contains transformations which only amountto the overall phase shift of all doublets. They form the center of SU ( N ), Z ( SU ( N )) (cid:39) Z N ,and act trivially on all NHDM potentials. Being invariant under them does not representany structural property of the Higgs potential, therefore, we are led again to the factor group SU ( N ) /Z ( SU ( N )) = P SU ( N ). This allows us to write the group of kinetic-term-preservingreparametrization transformations as a semidirect product of the Higgs basis change groupand the Z group generated by J (for a more detailed discussion, see [21]): G rep = P SU ( N ) (cid:111) Z ∗ . (2)Here the asterisk indicates that the generator of the corresponding group is an anti-unitarytransformation; we will use this notation throughout the paper.Below, when discussing symmetry groups of the 3HDM potential, we will be either lookingfor subgroups of P SU (3) (if only unitary transformations are allowed) or subgroups of this G rep (when anti-unitary reparametrization transformations are also included). This shouldalways be kept in mind when comparing our results with the groups which are discussedas symmetry groups in the 3HDM scalar sector. For example, in [9, 10] a 3HDM potentialsymmetric under ∆(27) or ∆(54) was considered, both groups being subgroups of SU (3).However, they contains the center of SU (3), which, we repeat, acts trivially on all Higgs4otentials. Therefore, the structural properties of that model are defined by the factor groups∆(27) /Z ( SU (3)) (cid:39) Z × Z and ∆(54) /Z ( SU (3)) (cid:39) ( Z × Z ) (cid:111) Z , which belong to P SU (3).
There is an important technical point which should be kept in mind when we classify symmetrygroups of NHDM. When we impose a reparametrization symmetry group G on the potential,we restrict its coefficients in a certain way. It might happen then that the resulting potentialbecomes symmetric under a larger symmetry group (cid:101) G properly containing G .One drawback of this situation is that we do not have control over the true symmetryproperties of the potential: if we construct a G -symmetric potential, we do not know a prioriwhat is its full symmetry group (cid:101) G . This might be especially dangerous if G is finite while (cid:101) G turns out to be continuous, as it might lead to unwanted goldstone bosons. Anotherundesirable feature is related with symmetry breaking. Suppose that we impose invarianceof the potential under group G but we do not check what is the true symmetry group (cid:101) G .After electroweak symmetry breaking, the symmetry group of the vacuum is G v ≤ (cid:101) G , and itcan happen that G v is not a subgroup of G . This is not what we normally expect when weconstruct a G -symmetric model, and it is an indication of a higher symmetry.Examples of these situations were encountered in literature before. For instance, theauthors of [12] explicitly show that trying to impose a Z p , p >
2, group of rephasing transfor-mations in 2HDM unavoidably leads to a potential with continuous Peccei-Quinn symmetry.For 3HDM they find an even worse example, when a cyclic group immediately leads to a U (1) × U (1)-symmetric potential. Another well-known example is the A -symmetric 3HDMpotential, which at certain values of parameters admits vacua with the S symmetry, although S is not a subgroup of A , see an explicit study in [8]. The explanation is that the potentialat these values of parameters becomes symmetric under S which contains both A and S .In order to avoid such situations altogether, we must always check for each G whetherthe G -symmetric potentials are invariant under any larger group. We are interested only inthose groups G , for which there exists a G -invariant potential with the property that no otherreparametrization transformation leaves it invariant (either within P SU (3) or within G rep ,depending on whether we include anti-unitary transformations). Following [14, 21], we callsuch groups realizable .Using the terminology just introduced we can precisely formulate the two main questionswhich we address in this paper:1. considering only non-trivial kinetic-term-preserving Higgs-basis transformations (i.e. group P SU (3)), what are the realizable finite symmetry groups in 3HDM?2. more generally, considering non-trivial kinetic-term-preserving reparametrization trans-formations, which can now include generalized- CP transformations (i.e. group G rep ),what are the realizable finite symmetry groups in 3HDM?For abelian groups, these questions were answered in [21] for general N . Here we focus onnon-abelian finite realizable groups for N = 3.5 Solvable groups: an elementary introduction
Our classification of realizable groups of Higgs-family symmetries in 3HDM contains two essen-tial parts: the group-theoretic and the calculational ones. The group-theoretic part will makeuse of some methods of pure finite group theory, which are not very familiar to the physicscommunity (although they are quite elementary for a mathematician with expertise in grouptheory). To equip the reader with all the methods needed to understand the group-theoreticpart of our analysis, we begin by giving a concise introduction to the theory of solvable groups.In doing so, we mention only methods and results which are relevant for the particular problemof this paper. For a deeper introduction to solvable groups and finite group theory in general,see e.g. [27].
We assume that the reader is familiar with the basic definitions from group theory. We onlystress here that we will work with finite groups, therefore the order of the group G (the numberof elements in G ) denoted as | G | is always finite, and so is the order of any element g (thesmallest positive integer n such that g n = e , the identity element of the group).A group G is called abelian if all its elements commute. An alternative way to formulate itis to say that all commutators in the group are trivial: [ x, y ] = xyx − y − = e for all x, y ∈ G .Working with commutators is sometimes easier than checking the commutativity explicitly.For example, it is easy to prove that if every non-trivial element of the group has order two, g = e , then the group is abelian. Indeed, for any x, y ∈ G we have[ x, y ] = xyx − y − = xyxy = ( xy ) = e , (3)which means that x and y commute.A group G can have proper subgroups H < G (whenever we do not require that thesubgroup H is proper, we write H ≤ G ), whose order must, by Lagrange’s theorem, divide theorder of the group: | H | divides | G | . If proper subgroups exist, some of them must be abelian.A simple way to obtain an abelian subgroup is to pick up an element g ∈ G and consider itspowers: if order of the element g is n , we will get the cyclic group Z n < G .The inverse of Largrange’s theorem is not, generally speaking, true: namely, if p is a divisorof | G | , the group G does not necessarily have a subgroup of order p . However, if p is a primewhich enters the prime decomposition of | G | , then according to Cauchy’s theorem such asubgroup must exist (this group is Z p because there are no other groups of prime order). Itimmediately follows that if we have the list of all abelian subgroups of a given finite group G ,then the prime decomposition of | G | can only contain primes which are present in the ordersof these abelian subgroups.In fact, there is an existence criterion stronger than Cauchy’s theorem. Namely, if p a isthe highest power of the prime p that enters the prime decomposition of | G | , then G containsa subgroup of this order, which is called the Sylow p -subgroup of the group G . This theorem(known as the Sylow-E theorem) is the starting point of the theory of Sylow subgroups, seeChapter 1 in [27].There are several ways to present a finite group. One possibility is to list all its elements andwrite down the | G | × | G | multiplication table. Clearly, this presentation becomes impracticalfor a sufficiently large group. A more compact and powerful way is known as presentation by enerators and relations . We call a subset M = { g , g , . . . } of the elements of G a generatingset (and its elements are called generators ) if every g ∈ G can be written as a product ofelements of M or their inverses. The fact that G is generated by the set M is denoted as G = (cid:104) M (cid:105) . Finding a minimal generating set for a given group and listing equalities whichthese generators satisfy is precisely presentation of the group by generators and relations. Forexample, the symmetry group of the regular n -sided polygon has the following presentationby generators and relations: D n = (cid:104) a, b | a = b = ( ab ) n = e (cid:105) . (4)This group is known as the dihedral group and has order | D n | = 2 n (note that there exists analternative convention for denoting dihedral groups: D n ; the one which we use has its orderin the subscript). Consider two groups G and H . Suppose we have a map f from G to H , f : G → H , whichsends every g ∈ G into its image f ( g ) ∈ H . If this map preserves the group operation, f ( g ) f ( g ) = f ( g g ), then it is called a homomorphism . If this map is surjective (i.e. it coversthe entire H ) and injective (distinct elements from G have distinct images in H ), then f isinvertible and is called an isomorphism.In the case when H = G , we deal with an isomorphism of the group onto itself, whichis called an automorphism . One can note that composition of two automorphisms is alsoan automorphism, and define the group structure on the set of all automorphisms of G .This automorphism group is denoted as Aut ( G ). The trivial automorphism which fixes everyelement of G is the identity element of Aut ( G ).Let us now consider a special class of automorphisms called inner automorphisms, or conjugations . Fix an element g ∈ G and define f : x (cid:55)→ g − xg for every x ∈ G . It canbe immediately checked that f is an automorphism, and that it sends a subgroup of G intoa (possibly another) subgroup of G . It can however happen that certain subgroups will bemapped onto themselves: g − Hg = H . Subgroups which satisfy this invariance criterion forevery possible g ∈ G are called normal , or invariant subgroups. The fact that H is a normalsubgroup of G is denoted as H (cid:67) G .Even when a subgroup H is not normal in G , one can pick up some elements g ∈ G suchthat g − Hg = H . The set of elements of G with the property g − Hg = H forms a group, whichis called the normalizer of H in G and denoted as N G ( H ). We then have H (cid:67) N G ( H ) ≤ G .Working with normalizers is a useful intermediate step in situations when it is not knownwhether the subgroup H is normal in G .Having a normal subgroup H (cid:67) G gives some information about the structure of G . One candefine the group structure on the set of (left) cosets of H , which is now called the factor group G/H . Thus, one breaks the group into two smaller groups, which often simplifies its study.Given a normal subgroup H (cid:67) G , one can define the canonical homomorphism φ : G → G/H which sends every element g ∈ G into its coset gH . Its kernel (all elements g which aremapped by φ into the identity element of G/H ) is precisely H . Thus, every normal subgroupis the kernel of the corresponding canonical homomorphism. The reverse statement is alsotrue: kernels of homomorphisms are always normal subgroups.7he group-constructing procedure inverse to factoring is called extension . Given twogroups, N and H , a group G is called an extension of H by N (denoted as N . H ), if thereexists N (cid:67) G such that N (cid:39) N and G/N (cid:39) H . In the case when, in addition, H is alsoisomorphic to a subgroup of G and G = N H , we deal with a split extension . The criterionfor G to be a split extension can also be written as existence of N (cid:67) G and H ≤ G such that N H = G and N ∩ H = 1, so that G/N = H . The group G is then called a semidirect product G = N (cid:111) H .Even if two groups N and H are fixed, they can support several extensions and splitextensions. Therefore one faces the problem of classifying of all extensions of two given groups.For the most elementary example, consider extensions of H = Z (generated by a ) by N = Z (generated by b ), which should produce a group of order 4. Then, for a split extension,we need a group G which has two distinct subgroups isomorphic to N and H . The only choiceis G = Z × Z , which can be presented as (cid:104) a, b | a = b = ( ab ) = e (cid:105) . For a non-splitextension, we require that only N is isomorphic to a subgroup of G . Thus, we still have b = e , while a must not be the unit element. Then we have to set a = b producing thegroup Z . So, Z does not split over Z , while Z × Z does. In what concerns embedding of groups, normality is a relatively weak property. Namely, if K (cid:67) H and H (cid:67) G , then K is not necessarily normal in G (it is instead called subnormal in G ).Indeed, recall that a normal subgroup K (cid:67) H stays invariant under all inner automorphismson H . Here “inner” is meant with respect to the group H , namely, h − Kh = K for all h ∈ H . However since H (cid:67) G , one can fix g ∈ G but g (cid:54)∈ H and consider an automorphismon H defined by H → g − Hg . This is indeed an automorphism on H because it induces apermutation of elements of H preserving its group property, but it is not inner, because g doesnot belong to H . Therefore K does not have to be invariant under it: g − Kg (cid:54) = K .However there is a stronger property which guarantees normality for embedded groups.Let us call a subgroup K characteristic in H if it is invariant under all (not only inner)automorphisms of H . Then, repeating the above arguments, we see that if K is characteristicin H , and H is normal in G , then K is also normal in G . Also, if K is characteristic in H and H is characteristic in G , then K is also characteristic in G . Thus, knowing that somesubgroups are characteristic gives even more information than their normality.There is one simple rule which guarantees that certain subgroups are characteristic. If wehave a rule defined in terms of the group G which identifies its subgroup H uniquely, then H is characteristic in G . Two important examples are: • the center of the group G denoted as Z ( G ), which is the set of all elements z ∈ G suchthat they commute with all elements of G : Z ( G ) = { z ∈ G | [ z, g ] = e ∀ g ∈ G } . (5)The center of an abelian group coincides with the group itself. • the commutator subgroup (or derived subgroup ) of G denoted as G (cid:48) and defined as thesubgroup generated by all commutators: G (cid:48) = (cid:104) [ x, y ] (cid:105) , x, y ∈ G . (6)8ote that the word “generated” is needed because the set of commutators is generallyspeaking not closed under the group multiplication. Clearly, the commutator subgroupof an abelian group is trivial, therefore the size of G (cid:48) can be used to qualitatively char-acterize how far G is from being abelian. Let us now prove a rather simple group-theoretic result, which however will be important forour classification of symmetries in 3HDM. This result, loosely speaking, is the observationthat a mere existence of a subgroup of G with some special properties can strongly restrictthe structure of the group G .First, an abelian subgroup A < G is called a maximal abelian subgroup if there is no otherabelian subgroup B with property A < B ≤ G . Note that the word “maximal” refers notto the size but to containment. This definition does not specify a unique subgroup; in facta group can have several maximal abelian subgroups. They correspond to terminal points inthe partially-ordered tree of abelian subgroups of G .Suppose that A is an abelian subgroup of a finite group G . Elements of A , of course,commute among themselves. But it can also happen that there exist other elements g ∈ G , g (cid:54)∈ A , which also commute with all elements of A . The set of all such elements is called the centralizer of A in G : C G ( A ) = { g ∈ G | [ g, a ] = e ∀ a ∈ A } . (7)It is easy to check that C G ( A ) is a subgroup of G , and it can be non-abelian. The name“centralizer” refers to the fact that although A is not the center in G , it is the center in C G ( A ).Clearly, A ≤ C G ( A ). If A is a proper subgroup of C G ( A ), then it means that A is nota maximal abelian subgroup. Indeed, we take an element g ∈ C G ( A ), g (cid:54)∈ A , and consideranother subgroup B = (cid:104) A, g (cid:105) . This subgroup is abelian and is strictly larger than A : A
Aut ( A ). Summarizing our discussion, if A is a normal maximal abelian subgroup of G , then G can be constructed as an extension of A by a subgroup of Aut ( A ): G (cid:39) A . K , where K ≤ Aut ( A ) . (9)This is a powerful structural implication for the group G of existence of a normal maximalabelian subgroup. For future reference, we give some details on the automorphism groups
Aut ( A ) of certainabelian groups A . In this subsection we will use the additive notation for the group operation.Suppose A = Z n is the cyclic group of order n with generator e : ne = e + · · · + e (cid:124) (cid:123)(cid:122) (cid:125) n times = 0.An automorphism σ acting on A is a group-structure-preserving permutation of elements of A . Since A is generated by e , this automorphism is completely and uniquely defined once weassign the value of σ ( e ) = k and make sure that mσ ( e ) (cid:54) = 0 for all 0 < m < n . This holdswhen k and n are coprime ( k = 1 is coprime to any n ). The number of integers less than n and coprime to n is called the Euler function ϕ ( n ). Thus, we have | Aut ( Z n ) | = ϕ ( n ). For aprime p , the Euler function is obviously ϕ ( p ) = p −
1. In general, if p k · · · p k s s is the primedecomposition for n , then ϕ ( p k · · · p k s s ) = ϕ ( p k ) · · · ϕ ( p k s s ) = ( p k − p k − ) · · · ( p k s s − p k s − s ) . Suppose now that p is prime and A = Z p × · · · × Z p (cid:124) (cid:123)(cid:122) (cid:125) n times = ( Z p ) n . Then G can be considered as an n -dimensional vector space over a finite field F p of order p .Vectors in this space can be written as x = k e + · · · + k n e n , where numbers k i ∈ F p and “basis vectors” e i are certain non-zero elements of the i -th group Z p . The group of all automorphisms on ( Z p ) n is then the general linear group in this space GL n ( p ).Again, in order to define an automorphism σ acting on A , it is sufficient to assign wherethe basis vectors e i are sent by σ and to make sure that they stay linearly independent: thatis, if m σ ( e ) + · · · + m n σ ( e n ) = 0, with m i ∈ F p , then all m i = 0. In order to calculate | GL n ( p ) | , we just need to find to how many different bases the initial basis { e , . . . , e n } canbe mapped to. The first vector, e , can be sent to p n − e , canbe then sent to p n − p vectors linearly independent with σ ( e ), and so forth. The result is | GL n ( p ) | = ( p n − p n − p ) · · · ( p n − p n − ) = p n ( n − ( p − p − · · · ( p n − . (10)10n particular, | Aut ( Z p × Z p ) | = | GL ( p ) | = p ( p − p − p -subgroup of Aut ( Z p × Z p )can only be Z p . In group theory, a powerful tool to investigate structure and properties of groups is to establishexistence of subgroup series with certain properties. For example, a finite collection of normalsubgroups N i (cid:67) G is called a normal series for G if1 = N ≤ N ≤ N ≤ · · · ≤ N r = G . (11)Restricting the properties of the factor groups N i /N i − for all i , one can infer non-trivialconsequences for the group G .If all the factor groups in the normal series lie in the centers, N i /N i − ≤ Z ( G/N i − ) for1 ≤ i ≤ r , then (11) becomes a central series , and the group G is then called nilpotent . Thesmallest number r for which the central series exists is called the nilpotency class of G .Clearly, abelian groups are nilpotent groups of class 1 because for them G ≤ Z ( G ). Anon-abelian group G whose factor group by its center G/Z ( G ) gives an abelian group is anilpotent group of class 2, etc. So, nilpotent groups are often regarded as “close relatives” ofabelian groups in the class of non-abelian ones. One important class of nilpotent groups is p -groups, i.e. finite groups whose order is a power of a prime p .Nilpotent groups bear several remarkable features. We mention here only two of themwhich we will use below. First, a nilpotent group has a normal self-centralizing, and thereforemaximal, abelian subgroup (Lemma 4.16 in [27]), whose implications were discussed above.Second, if H is a proper subgroup of a nilpotent group G , then H is also a proper subgroupof N G ( H ) (Theorem 1.22 in [27]). In other words, the only subgroup of a nilpotent group G which happens to be self-normalizing is the group G itself. A group G is called solvable if it has a normal series (11) in which all factor groups N i /N i − are abelian. This is a broader definition than the one of nilpotent groups. Therefore we canexpect that both criteria and properties of solvable groups will be weaker than for nilpotentgroups.One particular example is that unlike nilpotent groups, a solvable group does not have topossess a normal self-centralizing abelian subgroup. However what it does possess is just a normal abelian subgroup . In order to prove this statement, let us first introduce another seriesof nested subgroups, called the derived series . We first find G (cid:48) , the derived subgroup of G ,then we find its derived subgroup, G (cid:48)(cid:48) = ( G (cid:48) ) (cid:48) , then the third derived subgroup, G (3) = ( G (cid:48)(cid:48) ) (cid:48) ,and so on. The derived series is simply · · · ≤ G (3) ≤ G (cid:48)(cid:48) ≤ G (cid:48) ≤ G . (12)The relation of the derived series with solvability is the following: G is solvable if and onlyif its derived series terminates, i.e. G ( m ) = 1 for some integer m ≥ G (cid:48) is the unique smallestnormal subgroup of G with an abelian factor group. Indeed, if N (cid:67) G and φ : G → G/N is the11anonical homomorphism, then φ ( G (cid:48) ) = ( G/N ) (cid:48) (commutators are mapped into commutators).If we want G/N to be abelian, then (
G/N ) (cid:48) = 1, and G (cid:48) ≤ ker φ = N . Therefore, whatever N r − we choose in (11), it will contain G (cid:48) . This argument can be continued through the series,and since the normal series terminates, so does the derived series.Now, since G ( m ) = 1 for some finite m , we can consider G ( m − . It is an abelian group be-cause its derived subgroup is trivial. Being a characteristic subgroup of G ( m − , it is definitelynormal in G . Thus, we obtain the desired normal abelian subgroup.A normal abelian subgroup is not guaranteed to be maximal. One can, of course, extend itto a maximal abelian subgroup, but then it is not guaranteed to be normal. Thus, in order touse the result (9), we need to prove the existence of an abelian subgroup which combines bothproperties. This situation is not generic: a solvable groups does not have to possess a normalmaximal abelian subgroup. However it can possess it in certain cases, and we will show belowthat in what concerns finite symmetry groups in 3HDM, they do contain such a subgroup. Our goal is to understand which finite groups G can be realized as Higgs-family symmetrygroups in the scalar sector of 3HDM. We stress that we look for realizable groups only, seediscussion in section 2.3.Since finite groups have abelian subgroups, it is natural first to ask which abelian subgroups G can have. This can be immediately inferred from our paper [21] devoted to abelian symmetrygroups in NHDM. In the particular case of 3HDM, only the following groups can appear asabelian subgroups of a finite realizable symmetry group G : Z , Z , Z , Z × Z , Z × Z . (13)The first four are the only realizable finite subgroups of maximal tori in P SU (3). The lastgroup, Z × Z , is on its own a maximal abelian subgroup of P SU (3), but it is not realizablebecause a Z × Z -symmetric potential is automatically symmetric under ( Z × Z ) (cid:111) Z ,see explicit expressions below. However, since it appears as an abelian subgroup of a finiterealizable group, it must be included into consideration. Trying to impose any other abelianHiggs-family symmetry group on the 3HDM potential unavoidably makes it symmetric undera continuous group.Let us first see what order the finite (non-abelian) group G can have. We note that theorders of all abelian groups in (13) have only two prime divisors: 2 and 3. Thus, by Cauchy’stheorem, the order of the group G can also have only these two prime divisors: | G | = 2 a b .Then according to the Burnside’s p a q b -theorem the group G is solvable (Theorem 7.8 in [27]),and this means that G contains a normal abelian subgroup, which belongs, of course, to thelist (13).In order to proceed further, we need to prove that one can in fact find a normal maximal(that is, self-centralizing) abelian subgroup of G , a property which is not generic to solvablegroups but which holds in our case. 12 .2 Existence of a normal abelian self-centralizing subgroup Suppose
A < G is a normal abelian subgroup, whose existence follows from the solvability of G . In this subsection we prove that even if it is not self-centralizing, i.e. A < C G ( A ), thenthere exists another abelian subgroup B > A , which is normal and self-centralizing in G . A A bb b' C (A) G A b' Figure 1: Illustration of C G ( A ) and some of its subgroups.Suppose that A < C G ( A ). Then for every b ∈ C G ( A ) \ A , the group A b = (cid:104) A, b (cid:105) is anabelian subgroup of G , which properly contains A . Fig. 1 should help visualize embedding ofvarious abelian subgroups of this kind in C G ( A ). Note that C G ( A ) can be non-abelian. Thereare two possibilities compatible with the list (13):(i) A = Z , and then A b can be either Z × Z or Z ,(ii) A = Z , and then A b = Z × Z .Thus C G ( A ) is either a 2-group or a 3-group. Below we assume that p = 2 if C G ( A ) is a2-group, and p = 3 if C G ( A ) is a 3-group.Since C G ( A ) is a p -group, it is nilpotent, and according to discussion in section 3.6, itpossesses a normal maximal abelian subgroup B (which of course can be represented as A b for some b ), while B properly includes A = Z p : A < B ≤ C G ( A ). In particular, B is self-centralizing in C G ( A ), so according to our discussion in section 3.4, the factor group C G ( A ) /B is a subgroup of Aut ( B ). If B = C G ( A ), then C G ( A ) is abelian and, being a centralizer of anormal subgroup, it is normal in G . Clearly B ≤ C G ( B ) ≤ C G ( A ) = B , therefore C G ( A ) isthe desired normal abelian self-centralizing subgroup of G .Assume now that B (cid:54) = C G ( A ): A < B = C C G ( A ) ( B ) (cid:124) (cid:123)(cid:122) (cid:125) = C G ( B ) < C G ( A ) < G . (14)The illustration in Fig. 1 refers to this case. Since B is an abelian subgroup of G , it must bein list (13). So, either B = Z p × Z p or B = Z p (the last case occurs only if p = 2), and in anyof these cases we obtain | B | = p . Now, recall that C G ( A ) is a p -group, and so is C G ( A ) /B .If B = Z p × Z p , then C G ( A ) /B is a p -subgroup of GL ( p ), in particular, | C G ( A ) /B | = p . If B = Z p , then C G ( A ) /B is a p -subgroup of Aut ( Z p ). Since ϕ ( p ) = p ( p − | C G ( A ) /B | = p . So in any case we have | C G ( A ) | = p .Now the arguments depend on p . 13 In the case p = 2, we have that C G ( A ) is a nonabelian group of order 8. Thus C G ( A )is either dihedral group D or the quaternion group Q . If C G ( A ) is dihedral, then itpossesses the unique (and hence characteristic) subgroup H = Z , so H is the desirednormal self-centralizing subgroup of G . If G = Q is quaternion then, as we describe inSection 6.3.3, trying to impose a Q symmetry group on the 3HDM potential will resultin a potential symmetric under a continuous group. Thus, this situation cannot happenif we search for finite realizable groups G . Note that this feature is purely calculationaland does not rely on the existence of a normal maximal abelian subgroup which we provehere. • In the case p = 3, we have that C G ( A ) is a nonabelian group of order p = 27 andexponent 3, i.e. for every g ∈ C G ( A ) we have g = 1. It is nonabelian and cannotcontain elements of order 9 because (13) does not contain abelian groups of orders 9 or27.In this case we do not yet know whether B is normal in G , but it is definitely normal inits own normalizer B (cid:67) N G ( B ) ≤ G . Moreover C G ( A ) ≤ N G ( B ), since B is normal in C G ( A ). These relations are visualized by the following relations: B (cid:67) C G ( A ) ≤ N G ( B ) ≤ G < P SU (3) . (15)We can then consider the factor group N G ( B ) /B . We know that B = Z × Z isa maximal abelian group in P SU (3), [21]; therefore it is self-centralizing in
P SU (3)and, consequently, in G and in its subgroup N G ( B ). Then, in particular, we havethat N G ( B ) /B is a subgroup of Aut ( B ) = GL (3). Moreover, the analysis which willbe exposed in detail in Section 7 allows us to state that N P SU (3) ( B ) /B = SL (3), so N G ( B ) /B is a subgroup of SL (3). We show in Section 7 that one cannot use elements oforder 3 from SL (3) because the potential will then become invariant under a continuoussymmetry group. Therefore, N G ( B ) /B cannot have elements of order 3, which impliesthat B is a Sylow 3-subgroup of N G ( B ). The same statement holds for every group thatlies “between” N G ( B ) and B , in particular, to C G ( A ). This contradicts the fact that | C G ( A ) : B | = 3 and C G ( A ) ≤ N G ( B ). So this case is impossible.Summarizing the group-theoretic part of our derivation, we proved that any finite group G which can be realized as a Higgs-family symmetry group in 3HDM is solvable, and in additionit contains a normal self-centralizing abelian subgroup A belonging to the list (13). Then,according to (9) the group G can be constructed as an extension of A by a subgroup of Aut ( A ).This marks the end of the group-theoretic part of our analysis. We now need to check allthe five candidates for A , whose explicit realization were already given in [21], and by meansof direct calculations see which extension can work in 3HDM. Before we embark on analyzing each particular abelian group and its extensions, let us discussan important issue. In this paper, we focus on discrete symmetries of the scalar sector in3HDM. The symmetry groups we study must be realizable, that is, we need to prove that a14otential symmetric under a finite group G is not symmetric under any larger group containing G . In particular, we must prove that a given G -symmetric potential does not have anycontinuous symmetry.In principle, it would be desirable to derive a basis-invariant criterion for existence orabsence of a continuous symmetry. Such condition is known for 2HDM, [19, 20], while forthe more than two doublets a necessary and sufficient condition is still missing. However, incertain special but important cases it is possible to derive a sufficient condition for absenceof any continuous symmetry. Since this method relies on the properties of the orbit space in3HDM, we start by briefly describing it. The formalism of representing the space of electroweak-gauge orbits of Higgs fields via bilinearswas first developed for 2HDM, [19, 20, 28], and then generalized to N doublets in [13]. Belowwe focus on the 3HDM case.The Higgs potential depends on the Higgs doublets via their gauge-invariant bilinear com-binations φ † a φ b , a, b = 1 , ,
3. These bilinears can be organized into the following real scalar r and real vector r i , i = 1 , . . . , r = ( φ † φ ) + ( φ † φ ) + ( φ † φ ) √ , r = ( φ † φ ) − ( φ † φ )2 , r = ( φ † φ ) + ( φ † φ ) − φ † φ )2 √ ,r = Re( φ † φ ) , r = Im( φ † φ ) , r = Re( φ † φ ) ,r = Im( φ † φ ) , r = Re( φ † φ ) , r = Im( φ † φ ) . (16)The last six components can be grouped into three “complex coordinates”: r = ( φ † φ ) = r + ir , r = ( φ † φ ) = r + ir , r = ( φ † φ ) = r + ir . (17)It is also convenient to define the normalized coordinates n i = r i /r . The orbit space of the3HDM is then represented by an algebraic manifold lying in the 1 + 8-dimensional euclideanspace of r and r i and is defined by the following (in)equalities, [13]: r ≥ , (cid:126)n ≤ , √ d ijk n i n j n k = 3 (cid:126)n − , (18)where d ijk is the fully symmetric SU (3) tensor. It can also be derived that | (cid:126)n | is boundedfrom below: (cid:126)n = α , ≤ α ≤ . (19)The value of α parametrizes SU (3)-orbits inside the orbit space. In particular, we will usethis relation below when substituting r + r by αr − | r | − | r | − | r | .Any U (3) transformation in the space of doublets φ , φ , φ leaves r invariant and inducesan SO (8) rotation of the vector r i . Note that this map is not surjective, namely not every SO (8) rotation of r i can be induced by a U (3) transformation in the space of doublets. There-fore, unlike in 2HDM, we do not expect the orbit space of 3HDM to be SO (8)-symmetric, andthe last condition in (18) stresses that. 15 n PP n P Figure 2: The orbit space of 3HDM in the ( n , n )-subspace (all other n i = 0). The outer andinner circles correspond to | (cid:126)n | = 1 and | (cid:126)n | = 1 /
2, respectively.Let us take a closer look at the ( n , n )-subspace. It follows from (18) that the orbit spaceintersects this plane along the equilateral triangle shown in Fig. 2. Its vertices P , P (cid:48) , P (cid:48)(cid:48) lie onthe “neutral” manifold, which satisfy the condition (cid:126)n = 1 and which would correspond to theneutral vacuum if the minimum of the potential were located there, while the line segmentsjoining them correspond to the charge-breaking vacuum, see details in [13]. The orbit space inthis plane clearly lacks the rotational symmetry and has only the symmetries of the equilateraltriangle. The convenience of the formalism of bilinears is that the most general Higgs potential becomesa quadratic form in this space: V = − M r − M i r i + 12 Λ r + Λ i r r i + 12 Λ ij r i r j . (20)The real symmetric matrix Λ ij has eight real eigenvalues (counted with multiplicity). In orderfor the potential to be symmetric under a continuous group of transformations, Λ ij must haveeigenvalues of multiplicities >
1. Note that any statement about eigenvalues of Λ ij is basis-invariant and therefore it can be checked in any basis. Furthermore, if we find a basis inwhich Λ ij has a block-diagonal form, and if eigenvalues from different blocks are distinct, thena continuous symmetry requires that each block is either invariant under this symmetry, orcontains eigenvalues with multiplicity > φ † a φ a )( φ † b φ c ), where a, b, c are all distinct. This implies the absence ofterms r , , r , , , , , , and the block-diagonal form of Λ ij , in which two blocks correspond tothe ( r , r ) subspace and to its orthogonal complement. Suppose also that the eigenvalues ofΛ ij in the ( r , r ) subspace are distinct from those in the orthogonal complement. It followsthen that any possible continuous symmetry must act trivially in the ( r , r ) subspace, becausethe orbit space here lacks the rotational invariance. However, if r , r , and r are fixed, then φ † φ , φ † φ , and φ † φ are also fixed. So, the doublets do not mix, and the possible continuous16ymmetry group can only be a subgroup of the group of pure phase rotations, which werestudied in [21].If in addition it is known that a given potential is not symmetric under continuous phaserotations, then we conclude that it does not have any continuous symmetry from P SU (3). Itturns out that all the cases of various finite symmetry groups we consider below, except thelast one, are of this type. Since the arguments of this section provide a sufficient condition forabsence of continuous symmetries, they guarantee that the corresponding potentials can haveonly finite symmetry groups. Absence of a continuous symmetry in the very last case will beproved separately.
We now check all the candidates for A from the list (13) and see which extension can workin 3HDM. In this section we will deal with the first four groups from the list, which arise assubgroups of the maximal torus; the last group will be considered later. For each group A , weuse its explicit realization given in [21] as a group of rephasing transformations, and then wesearch for additional transformations from P SU (3) with the desired multiplication properties.
P SU (3)
Before we start analysis of each case, let us make a general remark on how we describe theelements of
P SU (3). Using the bar notation for the canonical homomorphism SU (3) → P SU (3), we denote ¯
H < P SU (3) if its full preimage in SU (3) is H . Denoting the center of SU (3) as Z = Z ( SU (3)) (cid:39) Z , we have Z = { , z, z } , where z = diag( ω, ω, ω ) , ω = e πi/ . (21)The elements of the group H ( a, b, . . . ∈ H ) will be written as 3 × SU (3). Theelements of ¯ H (¯ a, ¯ b, · · · ∈ ¯ H ) are the corresponding cosets of Z in H . Explicit manipulationwith these cosets is inconvenient, therefore in our calculation we represent an element ¯ a ∈ P SU (3) by any of the three representing elements from SU (3): a , az , or az . We will usuallychoose a and then prove that this representation is faithful (does not depend on the choice ofrepresenting element). Z and Z The smallest group from the list is A = Z , whose automorphism group is Aut ( Z ) = { } , sothat G = Z . This case was already considered in [21].The next possibility is A = Z , whose Aut ( Z ) = Z . The only non-trivial case to beconsidered is G/A = Z , which implies that G can be either Z or D (cid:39) S , the symmetrygroup of the equilateral triangle. The former can be disregarded because it does not appearin the list (13), thus we focus only on the D case.17 .2.1 Constructing D The group D is generated by two elements a, b with the following relations: a = 1, b = 1, ab = ba . Following [21], we represent the Z group by phase rotations: a = diag( ω, ω , . (22)There are in fact three such groups which differ only by the choice of the doublet which is fixed.However their generators, a , az , and az , differ only by a transformation from the center, andtherefore all of them correspond to the same generator ¯ a from P SU (3). It is straightforwardto check that selecting a to represent ¯ a is a faithful representation.The explicit solution of the matrix equation ab = ba shows that b ∈ SU (3) must be of theform b = e iδ e − iδ − , (23)with an arbitrary δ . The choice of the mixing pair of doublets ( φ and φ in this case) is fixedby the choice of invariant doublet in a .The fact that b is not uniquely defined means that there exists not a single D group buta whole family of D groups parametrized by the value of δ . Below, when checking whethera potential is D symmetric, we will need to check its invariance under all possible D ’s fromthis family.The generic Z -symmetric potential contains the part invariant under any phase rotation V = − (cid:88) ≤ i ≤ m i ( φ † i φ i ) + (cid:88) ≤ i ≤ j ≤ λ ij ( φ † i φ i )( φ † j φ j ) + (cid:88) ≤ i 3. Theextra freedom given by 2 πk/ D : b , ab , a b . Weopt to define b by setting k = 0. Alternatively, we can be compactly write the condition as3 δ = π − ψ + ψ . (26)To summarize, the criterion of the D symmetry of the potential is that, after a possibledoublet relabeling, conditions (25) and (26) are satisfied.Let us also note that when constructing the group D we could have searched for b satisfyingnot ab = ba but ab = ba · z p , with p = 1 , 2. Solutions of this equation exist, but they donot lead to any new possibilities. Indeed, let us introduce a (cid:48) = az p . Then, we get a (cid:48) b = ba (cid:48) .Thus, we get the same equation for b as before, up to a cyclic permutation of doublets, thepossibility which we already took into account.18 .2.2 Proving that D is realizable This construction allows us to write down an example of the D -symmetric potential: it is V restricted by conditions (25) plus V Z in (24) subject to | λ | = | λ | . In order to show that D is realizable, we need to demonstrate that this potential is not symmetric under any largerHiggs-family transformation group.This proof is short and contains two steps. First, we note that the conditions describedin section 5 are fulfilled: the ( r , r )-subspace does not couple to its orthogonal complementvia Λ ij , and that the eigenvalues in these two subspaces are defined by different sets of freeparameters. The extra terms (24) guarantee that there is only finite group of phase rotations,the group Z . Therefore, the sufficient conditions described in section 5 are satisfied, and thegeneric D -symmetric potential has no continuous symmetry.Second, we need to show that the generic D -symmetric potential has no higher discretesymmetries. This is proved by the simple observation that all other finite groups to be dis-cussed below which could possibly contain D lead to stronger restrictions on the potentialthan (25) and | λ | = | λ | . Therefore, not satisfying those stronger restrictions will yield apotential symmetric only under D . Any generalized- CP (antiunitary) transformation acting on three doublets is of the form J (cid:48) = c · J , c ∈ P SU (3) . (27)Here J is the operation of hermitean conjugation of the doublets. If G is the symmetry groupof unitary transformations, then it is normal in (cid:104) G, J (cid:48) (cid:105) , and J (cid:48) induces automorphisms in G .So, when we search for J (cid:48) , we require that( J (cid:48) ) ∈ G , ( J (cid:48) ) − aJ (cid:48) ∈ G , (28)where a generically denotes the generators of G . If such a transformation is found, the groupis extended from G to G (cid:111) Z ∗ , where asterisk on the group indicates that its generator isantiunitary.Note the crucial point of our method: when extending G by an antiunitary transformation,we require that the unitary transformation symmetry group remains G . The logic is simple.If we start with a realizable group G of unitary transformations but do not impose condition(28), we will end up with a potential being symmetric under ˜ G (cid:111) Z ∗ , with ˜ G > G . But at theend of this paper we will have a complete list of all finite realizable symmetry groups of unitarytransformations, and this list will contain ˜ G anyway. So, this possibility is not overlooked butwill be studied in its due time after construction of ˜ G .Now, turning to extension of D by an antiunitary symmetry, we first note that the resultinggroup D (cid:111) Z ∗ is a non-abelian group of order 12 containing a normal subgroup D . Amongthe three non-abelian groups of order 12, there exists only one group, namely D × Z ∗ , witha subgroup D (which is automatically normal because all subgroups of index 2 are normal).This fact can also be proved in a more general way without knowing the list of groups of order12. Note that it contains, among other, the subgroup Z ∗ ; its presence does not contradict thelist (13) because that list refers only to the groups of unitary transformations.19ext, let us denote the generator of Z ∗ by J (cid:48) = cJ . Since J (cid:48) centralizes the entire D , itfollows that ( J (cid:48) ) − aJ (cid:48) = a , ( J (cid:48) ) − bJ (cid:48) = b , and ( J (cid:48) ) = cJ cJ = cc ∗ = 1. The matrix c satisfyingthese conditions must be of the form c = e iγ e iγ − e − iγ , (29)with arbitrary γ . Requiring the potential to stay invariant under J (cid:48) , we obtain the followingconditions on γ : 6 γ = − ψ + ψ ) = 2 ψ . Therefore, if the following extra condition isfulfilled, 2( ψ + ψ + ψ ) = 0 . (30)the D -invariant potential becomes symmetric under the group D × Z ∗ . If this condition is notsatisfied, the symmetry group remains D even in the case when antiunitary transformationsare allowed. We conclude that both D and D × Z ∗ are realizable in 3HDM.It is interesting to note that if we set λ = 0, then the potential would still be invariantunder D . However in this case it becomes symmetric under J (cid:48) with 6 γ = − ψ + ψ ), withoutany extra condition on ψ and ψ , and the potential becomes automatically invariant under D × Z ∗ . So, we conclude that the fact that D is still realizable even if anti-unitary trans-formations are included is due to the special feature of the Z -symmetry: we have three, nottwo terms in the Z -symmetric potential, and it is the third term that prevents an automaticanti-unitary symmetry. Z Let us now take A = Z generated by a . Then Aut ( Z ) = Z , so that G = Z . Z generatedby a and some b (cid:54)∈ Z . The two non-abelian possibilities for G are the dihedral group D representing symmetries of the square, and the quaternion group Q . In both cases b − ab = a ,with the only difference that b = 1 for D while b = a for Q . Note that extension leadingto the dihedral group is split, D = Z (cid:111) Z , while Q is not. D Representing a by phase rotations a = diag( i, − i, b satisfying these conditionsis again of the form (23) with arbitrary δ . However now we do not have the freedom to choosethe pair of doublets which are mixed by b : this pair is fixed by a . Also, unlike the Z case,the matrix equation ab = ba · z does not have solutions for b ∈ SU (3).The Z -symmetric potential (for this choice of a ) is V + V Z , where V Z = λ ( φ † φ )( φ † φ ) + λ ( φ † φ ) + h.c. (31)The phases of λ and λ are, as usual, denoted as ψ and ψ , respectively. Upon b , the firstterm here remains invariant, while the second term transforms as( φ † φ ) (cid:55)→ e − iδ ( φ † φ ) . (32)This means that the potential (31) is always symmetric under (23) provided that we choose δ = ψ / , (33)20herefore, in order to get a D -symmetric potential we only require that V satisfies conditions(25). The proof that D is realizable (as long as only unitary transformations are concerned)follows along the same lines as in section 6.2.2. In [21] we found that exactly the same conditions, namely (25) and (33), must be satisfiedfor existence of an antiunitary transformation commuting with the elements of Z . Thistransformation is again J (cid:48) = cJ , where c is given by (29) with 6 γ = 2 ψ , and it commuteswith all elements of D . Therefore, if we include antiunitary transformations, we automaticallyget the group D × Z ∗ , while D becomes non-realizable. Note that the resulting group does notcontain Z ∗ . Indeed, we showed in [21] that imposing Z ∗ symmetry group leads to a potentialwith continuous symmetry. Q Solving matrix equations ab = ba and b = a , we get the following form of b : b ( Q ) = e iδ − e − iδ . (34)By checking how V Z in (31) transforms under it, we find that the first term simply changesits sign. The only way to make the potential symmetric under Q is to set λ = 0. But thenwe know from [21] that the potential becomes invariant under a continuous group of phaserotations. Therefore, Q is not realizable. Z × Z If A = Z × Z , then Aut ( Z × Z ) = GL (2) = S . The group Z × Z can be realized asthe group of independent sign flips of the three doublets with generators a = diag(1 , − , − a = diag( − , , − 1) (equivalent to thesign flip of the second doublet), so that a a is equivalent to the sign flip of the third doublet.The potential symmetric under this group contains V and additional terms V Z × Z = ˜ λ ( φ † φ ) + ˜ λ ( φ † φ ) + ˜ λ ( φ † φ ) + h.c. (35)with at least two among coefficients ˜ λ ij being non-zero. The coefficients can be complex; asusual we denote their phases as ψ ij . This model is also known as the Weinberg’s 3HDM, [5].The non-abelian finite group G can be constructed as extension of A by Z , by Z , orby S . ( Z × Z ) . Z Consider first the extension ( Z × Z ) . Z . The only extension leading to a non-abelian groupis ( Z × Z ) . Z = D , and we already proved that this group is realizable. Nevertheless, weprefer to explicitly work it out to see the reduction of free parameters.21he element b which we search for must act on { a , a , a a } as a transposition of anypair. In addition, b ∈ Z × Z . It does not matter which pair of generators is transposed, asthis choice can be changes by renumbering the doublets. So, we take b such that b − a b = a and b − a b = a . Then, b can be either 1 or a a , because choices b = a or a lead toinconsistent relations. Indeed, if we assume b = a , then a = b − a b = b − b b = b = a , which is a contradiction. In both cases ( b = 1 and b = a a ) we get the group D . Evenmore, we get the same D group: if b = a a , then b (cid:48) = ba satisfies b (cid:48) = 1, while its actionon a and a remains the same. So, it is sufficient to focus on the b = 1 case only.Again, explicitly solving the matrix equations, we get b of the form (23) with arbitrary δ .Then, we check how the potential (35) changes upon b and find that we need to set4 δ = 2 ψ , δ = − ( ψ + ψ ) , | ˜ λ | = | ˜ λ | . (36)Equations on the phase δ can be satisfied if2( ψ + ψ + ψ ) = 0 ⇔ Im(˜ λ ˜ λ ˜ λ ) = 0 . (37)So, if: (1) this condition is satisfied, (2) two among | ˜ λ ij | are equal, (3) condition on V (25) issatisfied, then the potential is D -symmetric. Note also that if ˜ λ = 0 (which we are allowedto consider because (35) contains three rather than two terms), then condition on the phasesis not needed.It might seem that these conditions on the potential to make it D -symmetric are morerestrictive than in the Z extension we studied above. However note that the Z × Z -symmetricpotential (35) has six free parameters, and we placed two conditions to reduce the number offree parameters in the D potential to four (apart from V ). On the other hand, (31) had onlyfour from the beginning, and without any restriction this number survives. Therefore we havethe same number of degrees of freedom when constructing D in either way. ( Z × Z ) (cid:111) Z = T The extension by Z is necessarily split, ( Z × Z ) (cid:111) Z , leading to the group T (cid:39) A , thesymmetry group of the tetrahedron. To construct it, we need b such that b = 1 with theproperty that b acts on { a , a , a a } by cyclic permutations. Fixing the order of permutationsby b − a b = a , we find that b must be of the form b = e iδ 00 0 e iδ e − i ( δ + δ ) , (38)with arbitrary δ , δ . It then follows that if coefficients in (35) satisfy | ˜ λ | = | ˜ λ | = | ˜ λ | , (39)then V Z × Z is symmetric under one particular b with δ = 2 ψ − ψ − ψ , δ = 2 ψ − ψ − ψ . λ ij equal and bring (35)to the following form V T = ˜ λ (cid:104) ( φ † φ ) + ( φ † φ ) + ( φ † φ ) (cid:105) + h.c. (40)with a complex ˜ λ . In this form, the parameters δ = δ = 0, and the matrix b is just the cyclicpermutation of the doublets. In addition, the symmetry under b places stronger conditions onthe parameters of V , so that the most general V satisfying them is V = − m (cid:104) ( φ † φ ) + ( φ † φ ) + ( φ † φ ) (cid:105) + λ (cid:104) ( φ † φ ) + ( φ † φ ) + ( φ † φ ) (cid:105) (41)+ λ (cid:48) (cid:104) ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) (cid:105) + λ (cid:48)(cid:48) (cid:16) | φ † φ | + | φ † φ | + | φ † φ | (cid:17) . ( Z × Z ) (cid:111) S = O The last extension, ( Z × Z ) . S , is also split, otherwise we would obtain Z . It leads to thegroup O (cid:39) S , the symmetry group of the octahedron and the cube. As it includes T as asubgroup, the most general O -symmetric potential is V from (41) plus V T from (40) with theadditional condition that ˜ λ is real (the extra symmetry with respect to the T -symmetric caseis a transposition of any two doublets). The case of D has been already considered in section 6.3.2.The tetrahedral potential V T + V from (40) and (41) is symmetric under the followingantiunitary transformation: J (cid:48) = − · J , (42)which generates a Z ∗ group. Therefore the symmetry group of this potential is the full achiraltetrahedral group T d (cid:39) T (cid:111) Z ∗ , which is isomorphic to S .The octahedral potential is a particular case of the tetrahedral one, therefore it is alsoinvariant under an antiunitary transformation. The extra Z ∗ subgroup is generated by thecomplex conjugation, J , and this transformation commutes with the entire Higgs-family group O . Therefore, the symmetry group of the potential is the full achiral octahedral symmetrygroup O h (cid:39) O × Z ∗ . The last type of extension we need to consider is of the type A . Z ∗ , where A is one of the fourabelian groups of Higgs-family transformations lying in a maximal torus, that is, the first fourgroups in the list (13), while the Z ∗ is as usual generated by an antiunitary transformation J (cid:48) = cJ . This problem was partly solved in [21], where such extensions leading to abelian groups were analyzed. It was established that only the following four abelian groups of thistype are realizable: Z ∗ , Z ∗ , Z × Z ∗ , and Z × Z × Z ∗ . Here, we consider non-abelian extensionsof this type. 23 .5.1 Anti-unitary extension of Z The smallest non-abelian group we can have is Z (cid:111) Z ∗ (cid:39) D . We stress that this D groupwe search for is different from what we analyzed in section 6.2, because there the D groupcontained only unitary transformations, see a discussion in section 8.3. Using the same nota-tion for the generator a of the Z group, we find that the transformation c in the definitionof J (cid:48) must be diagonal: c = diag( e iξ , e iξ , e − i ( ξ + ξ ) ). Then, studying how the Z -symmetricpotential V + V Z changes under J (cid:48) = cJ , we obtain that the only condition to be satisfied is(30).If this condition is satisfied, then the potential is invariant under Z (cid:111) Z ∗ (cid:39) D , if not,then the symmetry group remains Z . This proves that both groups are realizable in 3HDM.Note that in contrast with the D × Z ∗ case, we do not place any extra condition such as (25). Z A priori, the two non-abelian extensions here are again D and Q . With the usual conventionfor a , the generator of Z , we again obtain that c must be of the same diagonal form. Thisimmediately excludes the Q case because we have ( J (cid:48) ) = c ∗ c = 1.The case of Z (cid:111) Z ∗ (cid:39) D is possible. Even more, it turns out that the Z -symmetricpotential V + V Z is always symmetric under some J (cid:48) of this type. It means, therefore, thatif anti-unitary transformations are included, Z is not realizable anymore: the true symmetrygroup of the potential is Z (cid:111) Z ∗ (cid:39) D . In more physical terms, we conclude that presence ofa Z group of Higgs-family transformations makes the potential explicitly CP -conserving . Z × Z The only non-abelian extension of the type ( Z × Z ) . Z ∗ can produce only D , which wasalready considered. We only remark here that c turns out to be of the type (29), which placesextra constraints on V . Not satisfying these constraints will keep the symmetry group Z × Z ,which means that it is realizable. Z × Z chain The last abelian group from the list (13), Z × Z , requires a special treatment due to a numberof reasons. First, it does not belong to any maximal torus of P SU (3) but is a maximal abeliansubgroup of P SU (3) on its own, [21], and its full preimage in SU (3) is the non-abelian group∆(27), [29]. Second, its automorphism group Aut ( Z × Z ) is sufficiently large and requiresan accurate description.Let us first remind how this group is constructed. We first consider the subgroup of SU (3)generated by a = ω 00 0 ω , b = . (43)24his group known as ∆(27) is non-abelian because a and b do not commute, but their com-mutator lies in the center of SU (3):[ a, b ] = aba − b − = z ∈ Z ( SU (3)) . (44)Therefore, its image under the canonical homomorphism SU (3) → P SU (3) becomes thedesired abelian group ∆(27) / Z = Z × Z . The true generators of Z × Z are cosets ¯ a = aZ ( SU (3)) and ¯ b = bZ ( SU (3)) from P SU (3), and they obviously commute: [¯ a, ¯ b ] = 1. Notethat since Z × Z is a maximal abelian subgroup in P SU (3), there is no other element in P SU (3) commuting with all elements of this group, so C P SU (3) ( Z × Z ) = Z × Z .If the normal self-centralizing abelian subgroup of G , whose existence was proved in sec-tion 4.2, is A = Z × Z , then G can be constructed as an extension of A by a subgroup of Aut ( Z × Z ) = GL (3), the general linear group of transformations of two-dimensional vectorspace over the finite field F . The order of this group is | GL (3) | = 48, and it will prove usefulif we now digress and describe the structure of this group in some detail. Z × Z as a vector space over F The finite field F is defined as the additive group of integers mod 3, in which the multiplicationis also introduced. It is convenient to denote the elements of this field as 0 , , − F is closed under divisionby a non-zero number, the property that makes F a field.A vector space over a finite field is defined just as over any “usual” field. The group Z × Z can be thought of as a 2D vector space over F ; its elements are (with the additive notationfor the group operation) ¯ x = q a ¯ a + q b ¯ b , where q a , q b ∈ F , and ¯ a, ¯ b are, as before, the generatorsof the group Z × Z . In the multiplicative notation, we write ¯ x = ¯ a q a ¯ b q b .It is possible to define an antisymmetric scalar product in this space. For any ¯ x ∈ Z × Z ,take any element of its preimage, x ∈ ∆(27). Then, for any two elements ¯ x, ¯ y ∈ Z × Z ,construct the number (¯ x, ¯ y ) as [ x, y ] ∈ F . This map is faithful: although we can selectdifferent x for a given ¯ x , all of them give the same [ x, y ].Clearly, (¯ x, ¯ y ) = − (¯ y, ¯ x ), in the additive notation. Besides, the so defined product is linearin both arguments:(¯ x + ¯ x , ¯ y ) = (¯ x , ¯ y ) + (¯ x , ¯ y ) , (¯ x, ¯ y + ¯ y ) = (¯ x, ¯ y ) + (¯ x, ¯ y ) . (45)Indeed, for any three elements of any group the following relation holds:[ xy, z ] = xyzy − x − z − = xyzy − · z − x − xz · x − z − = x [ y, z ] x − [ x, z ] . (46)If in addition all commutators take values in the center of the group SU ( N ), then x and x − can be cancelled, and we get [ xy, z ] = [ y, z ][ x, z ]. In our case we represent x = ¯ a q a ¯ b q b z r andsimilarly for x and y , and noting that all z r i are inessential, we recover the above linearityin the first argument. Thus, Z × Z becomes a vector space over F equipped with anantisymmetric scalar product.Note that all antisymmetric products in Z × Z are proportional to (¯ a, ¯ b ). Indeed, if twoelements ¯ x and ¯ x (cid:48) are defined by their vectors (cid:126)q = ( q a , q b ) and (cid:126)q (cid:48) = ( q (cid:48) a , q (cid:48) b ), then due tobilinearity we get (¯ x, ¯ x (cid:48) ) = ( q a q (cid:48) b − q b q (cid:48) a )(¯ a, ¯ b ) = (cid:15) ij q i q (cid:48) j (¯ a, ¯ b ) , (47)where (cid:15) ij is the standard antisymmetric tensor with (cid:15) = − (cid:15) = 1, (cid:15) = (cid:15) = 0.25 .1.2 The automorphism group of Z × Z The automorphism group of Z × Z can then be viewed as the group of non-degeneratematrices with elements from F acting in this 2D space, which explains why Aut ( Z × Z ) = GL (3). Each matrix q can be defined by its action on the generators ¯ a , ¯ b : ¯ a (cid:55)→ q aa ¯ a + q ab ¯ b ,¯ b (cid:55)→ q ba ¯ a + q bb ¯ b , and can therefore be written as q = (cid:18) q aa q ab q ba q bb (cid:19) , det q (cid:54) = 0 . (48)The group operation in GL (3) is just the matrix product.Recall now that the elements of both the Z × Z group and of its automorphism group arerepresented in our case as unitary or antiunitary transformations of the three doublets (that is,we work not with the abstract groups but with their 3-dimensional complex representations).Since Z × Z is assumed to be normal in G , the elements g ∈ Aut ( Z × Z ) act on the elementsof Z × Z by conjugation: ¯ x (cid:55)→ g − ¯ xg , which we denoted by g (¯ x ). Then the antisymmetricproduct defined above changes upon this action in the following way:( g (¯ x ) , g (¯ y )) = g − [ x, y ] g = g − z r g = (cid:26) z r = (¯ x, ¯ y ) , if g is unitary , ( z ∗ ) r = ( z − ) r = − (¯ x, ¯ y ) , if g is anti-unitary . (49)Here we used the fact the commutator of any two elements of ∆(27) lies in the center Z ( SU (3)),and that the CP conjugation operator J acts on any x ∈ SU (3) by J − xJ = x ∗ . So, unitarytransformations preserve the antisymmetric product, while anti-unitary ones flip its sign.Generically, the subgroup of a general linear group which conserves an antisymmetricbilinear product in a vector space is called symplectic. Here we have the group Sp (3) 1, we get two kinds of transformations: those which conserve all products(det g = 1, so that g ∈ SL (3)) and those which flip their signs (det g = − Sp (3) and SL (3) follows.We conclude that the finite symmetry group G of unitary transformations with the normalself-centralizing abelian subgroup Z × Z can be constructed as extension ( Z × Z ) . K , where K ≤ SL (3). SL (3)The structure of the group SL (3) is well-known, but it will prove useful to have the explicitexpressions for some of its elements.The order of the group is | SL (3) | = 24. It contains elements of order 2, 3, 4, and 6,generating the corresponding cyclic subgroups. The subgroup Z is generated by the centerof the group c = (cid:18) − − (cid:19) , (51)26hich in the multiplicative notation means ¯ a (cid:55)→ ¯ a , ¯ b (cid:55)→ ¯ b . There are four distinct Z subgroups generated by f = (cid:18) (cid:19) , f = (cid:18) (cid:19) , f = (cid:18) − − (cid:19) , f = (cid:18) − − (cid:19) , (52)three Z subgroups generated by d = (cid:18) − (cid:19) , d = (cid:18) − (cid:19) , d = (cid:18) − (cid:19) , (53)and four Z subgroups, which we do not write explicitly because they are absent in the list(13).Every element of SL (3) can be represented by a unique (up to center) SU (3) matrix, whichcan be found by explicitly solving the corresponding matrix equations defining the action ofthis element. For example, the transformation c is defined by c ( a ) = c − ac = a , c ( b ) = c − bc = b . (54)Rewriting these equations as 3 × ac = ca , bc = cb and solving themexplicitly, we find the matrix c : c = − − − , (55) A generic potential symmetric under Z × Z is V = − m (cid:104) φ † φ + φ † φ + φ † φ (cid:105) + λ (cid:104) φ † φ + φ † φ + φ † φ (cid:105) + λ √ (cid:104) ( φ † φ ) + ( φ † φ ) + ( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) (cid:105) + λ (cid:16) | φ † φ | + | φ † φ | + | φ † φ | (cid:17) + (cid:16) λ (cid:104) ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) (cid:105) + h.c. (cid:17) (56)with real m , λ , λ , λ and complex λ . All values here are generic. This potential canbe found by taking the potential symmetric under the Z group of phase rotations describedabove and then requiring that it be invariant under the cyclic permutations on the doublets.Written in the space of bilinears, the potential has the form V = −√ m r + 3 λ r + √ λ ( r + r ) + λ ( | r | + | r | + | r | )+ λ ( r r ∗ + r r ∗ + r r ∗ ) + λ ∗ ( r ∗ r + r ∗ r + r ∗ r )= −√ m r + 3 λ r + Λ ij r i r j . (57)It is important to prove that this potential has no continuous symmetry. Using the approachdescribed in section 5, we calculate the eigenvalues of Λ ij and find that it has four distincteigenvalues of multiplicity two: √ λ , λ + λ + λ ∗ , λ + ωλ + ω λ ∗ , λ + ω λ + ωλ ∗ . (58)27he first eigenvalue corresponds to the subspace ( r , r ), while the rest are three 2D subspaceswithin its orthogonal complement ( r , r , r , r , r , r ). For generic values of the coefficients,they do not coincide. Then, according to our discussion in section 5, a continuous symmetrygroup, if present, must consist only of phase rotations of the doublets. But the λ term selectsonly the Z group of phase rotations, which proves that no continuous symmetry leaves thispotential invariant. ( Z × Z ) (cid:111) Z It turns out that Z × Z is not realizable because the potential (56) is symmetric under alarger group ( Z × Z ) (cid:111) Z = ∆(54) / Z , which is generated by ¯ a, ¯ b, ¯ c with the followingrelations ¯ a = ¯ b = 1 , ¯ c = 1 , [¯ a, ¯ b ] = 1 , ¯ c ¯ a ¯ c = ¯ a , ¯ c ¯ b ¯ c = ¯ b . In terms of explicit transformation laws, ¯ c is the coset cZ ( SU (3)), with c being the exchangeof any two doublets, for example (55). Note that (cid:104) ¯ a, ¯ c (cid:105) = S is the group of arbitrary permu-tations of the three doublets. Thus, if G = ( Z × Z ) . K , then a G -symmetric potential mustbe a restriction of (56), and K must contain a Z subgroup.There are three kinds of subgroups of SL (3) containing Z but not containing Z : Z , Z , and Q . In each case it would give a split extension, so G must contain a subgroupisomorphic to one of these groups. Since, as we argued above, the quaternion group Q is notrealizable in 3HDM, K can only be Z or Z . Therefore, the only additional case to consideris ( Z × Z ) (cid:111) Z , the group also known as Σ(36), [29]. ( Z × Z ) (cid:111) Z There are three distinct Z subgroups in SL (3) generated by d , d , and d , listed in (53). Inprinciple, all of them are conjugate inside SL (3), but for our purposes all of them need to bechecked. Explicit solutions of the matrix equations give the following transformations: d = i √ ω ω ω ω , d = i √ ωω ω ω ω , d = i √ ω ω ω ω ω . (59)Note that the prefactor i/ √ / ( ω − ω ).Let us mention here that when searching for explicit SU (3) realizations of the transfor-mations d , we solve equations d − ad = b , d − bd = a . However, we could also use otherrepresentative matrices, a (cid:48) and b (cid:48) , which differ from a and b by transformations from the center.For example, we can also ask for solutions of d (cid:48)− ad (cid:48) = z n b , d (cid:48)− bd (cid:48) = z n a . (60)However, the solution of this equation can be written as d (cid:48) = d a n b n . (61)Therefore the resulting group (cid:104) ¯ d (cid:48) , ¯ a, ¯ b (cid:105) coincides with (cid:104) ¯ d , ¯ a, ¯ b (cid:105) . The similar results hold for d and d . 28 .4.1 Conditions for the ( Z × Z ) (cid:111) Z symmetry We should now check how the potential (56) changes under these transformations and whenit remains invariant. The calculation is simplified if we introduce the following combinationsof bilinears (here i ∗ j stands for φ † i φ j ): A = 1 ∗ ∗ ∗ , A = 1 ∗ ω ∗ ω ∗ , A = A ∗ B = 1 ∗ ∗ ∗ , B = 1 ∗ ω ∗ ω ∗ , B = 1 ∗ ω ∗ ω ∗ ,B ∗ = 2 ∗ ∗ ∗ , B ∗ = 2 ∗ ω ∗ ω ∗ , B ∗ = 2 ∗ ω ∗ ω ∗ . Next, introducing X = 1 √ (cid:104) ( φ † φ ) + ( φ † φ ) + ( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) (cid:105) , = 1 √ | A | ,Y = | φ † φ | + | φ † φ | + | φ † φ | = | B | + | B | + | B | ,Z ∗ = ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) = | B | + ω | B | + ω | B | , (62)we write the potential (56) as V = −√ m r + 3 λ r + λ ∗ i X i , where λ ∗ i X i = λ X + λ Y + λ Z ∗ + λ ∗ Z (63)is the scalar product of the vector of coefficients and the vector of coordinates. Now, it followsfrom explicit calculations that the action of d i can be compactly represented by the followingtransformations: d : A → B , B → A ∗ , B → ω B , B → B ∗ ,d : A → B , B → ωA ∗ , B → ωB ∗ , B → B ,d : A → B , B → ωA ∗ , B → B , B → ωB ∗ , or even more compactly d : | A | ↔ | B | , | B | ↔ | B | ,d : | A | ↔ | B | , | B | ↔ | B | ,d : | A | ↔ | B | , | B | ↔ | B | . (64)Therefore, their action in the space of ( X, Y, Z, Z ∗ ) is given by the following hermitean andunitary matrices T ( d ) = 13 √ √ √ √ − − √ − − √ − − , T ( d ) = 13 √ ω √ ω √ √ − ω − ωω √ − ω − ω ω √ − ω ω − , and T ( d ) = [ T ( d )] ∗ . It can be also noted that T ( d ) acts in the space of ( X, Y, ω Z, ωZ ∗ )by the matrix T ( d ). So, T ( d ), T ( d ) and T ( d ) represent the same type of transformations29cting in the spaces ( X, Y, Z, Z ∗ ), ( X, Y, ω Z, ωZ ∗ ), or ( X, Y, ωZ, ω Z ∗ ), respectively. Thatis, if ( x, y, z, z ∗ ) is an eigenvector of T ( d ), then ( x, y, ωz, ω z ∗ ) is an eigenvector of T ( d ) and( x, y, ω z, ωz ∗ ) is an eigenvector of T ( d ). This observation restores the expected symmetryamong the three types of Z subgroups inside SL (3).Since these matrices are hermitean and unitary, they act by pure reflections, which impliesthat each of them is diagonalizable and has eigenvalues ± 1. If we want the potential to besymmetric under one of these d i , it must induce the same transformations in the space of λ i = ( λ , λ , λ ∗ , λ ). Therefore, in order to find conditions that the potential is invariantunder d i , we need to find eigenvectors of T ( d i ) corresponding to the eigenvalue − λ i ’s projection on these eigenvectors is zero.Consider first T ( d ). It has two eigenvectors corresponding to the eigenvalue − 1: ( −√ , , , , , , − d : λ is real and λ = √ λ − λ . (65)Similarly, for d we have ωλ is real and ωλ = √ λ − λ . (66)For d we have the complex conjugate condition. Therefore, the potential (56) is symmetricunder ( Z × Z ) (cid:111) Z if (cid:18) λ √ λ − λ (cid:19) = 1 , (67)which encompasses all these cases. Let us also mention that when these conditions are takeninto account, the spectrum of the matrix Λ ij given in (58) becomes even more degenerate: itcontains two eigenvalues of multiplicity four (we refer to this spectrum as 4 + 4). In order for the group ( Z × Z ) (cid:111) Z to be realizable, we need to show that the potential (56)with parameters satisfying (67) is not symmetric under any continuous group.We first note that even if such a continuous symmetry group existed, it could only be U (1).Indeed, the spectrum of Λ ij in our case is 4 + 4, while for U (1) × U (1) and SU (2) it must be6 + 2, and for SO (3) it must be 5 + 3.Let us now consider, for example, the d -symmetric potential. Using (cid:80) i =1 r i = αr , where1 / ≤ α ≤ SU (3)-orbits in the orbit space, we can rewrite it as V = −√ m r +(3 λ + α √ λ ) r − √ λ − λ (cid:0) | r − r | + | r − r | + | r − r | (cid:1) . (68)Suppose the potential (68) is invariant under a U (1) group of transformations of doublets,generated by the generator t from the algebra su (3). Since the potential (68) is invariantunder the S group of arbitrary permutations of the doublets, then the same potential mustbe also invariant under other U (1) subgroups which are generated by various t g , which areobtained by acting on t by g ∈ S . If t (cid:54) = t g (or to be more accurate, if their corresponding U (1) groups are different), then the continuous symmetry group immediately becomes larger30han U (1), which is impossible. Therefore, t g must be equal (up to sign) to t for all g ∈ S .In other words, S must stabilize the U (1) symmetry group.There exist only two elements in the algebra su (3) with this property: t = i − i − i ii − i and t = . (69) t generates pure phase rotations. It is explicitly S -invariant, therefore the corresponding U (1) group is also invariant. t induces SO (3) rotations of the doublets around the axis(1 , , Z -invariant, while reflections from S flip the sign of t . However the U (1)group is still invariant. Since t and t realize different representations of S , one cannot taketheir linear combinations. So, the list of possibilities is restricted only to t and t themselves.The eigenvalues and eigenvectors of t are ζ = 0 : , ζ = √ ω ω , ζ = −√ ωω . (70)The presence of the eigenvalue ζ = 0 implies that the combination φ + φ + φ is invariantunder the U (1) group generated by t . Bilinear invariants are | φ + φ + φ | , | φ + ω φ + ωφ | , | φ + ωφ + ω φ | , (71)which simply means that r + r + r and r + r + r are, separately, invariant. So, if thepotential depends only on r and these two combinations, then it is symmetric under the U (1)generated by t . The point is that our potential (68) cannot be written via these combinationsonly, therefore it is not invariant under this group.Consider now t . Its eigensystem is ζ = 2 : , ζ = − − and − − . (72)There is no zero eigenvalue, therefore no linear combination of φ ’s is invariant. The indepen-dent bilinear combinations are | φ + φ + φ | , | φ − φ | , | φ − φ − φ | , ( φ † − φ † )(2 φ − φ − φ ) . (73)In addition, there exists a triple product of φ ’s which is also invariant but it is irrelevant forour analysis because our potential contains only two φ ’s and two φ † ’s. These invariants canalso be rewritten as the following linearly independent invariants (here ρ i = φ † i φ i ): ρ + 2 r , ρ + 2 r , ρ + 2 r , r + r + r . (74)Despite the fact that we now have more invariants than in the previous case, it is still impossibleto express (68) via these combinations. This means that (68) is not symmetric under t .This completes the proof that the potential (56) subject to conditions (67) is not invariantunder any continuous group. 31 .4.3 Absence of a larger finite symmetry group Although the group-theoretic arguments guarantee that no other extension can be used, it isstill instructive to check what happens if we try to impose invariance under other subgroupsof SL (3).Let us first note that if we try to impose simultaneous invariance under two among d i (trying to get Q ), we must set λ = 0. But then the potential has an obvious continuoussymmetry, and our attempt fails.Next, let us assume that the potential is invariant under ( Z × Z ) (cid:111) Z , where the last Z is generated by one of the generators f in (52), for example f = f . Its representative matrixin SU (3) is f = − i √ ω 11 1 ω ω , f = 1 . (75)An analysis similar to what was described above allows us to find the corresponding transfor-mation matrix in the space of X, Y, Z, Z ∗ : T ( f ) = 13 √ √ ω √ ω √ − ω − ω √ ω − ω − ω √ ω − ω − ω . (76)It leads to the following conditions for the potential to be symmetric under ( Z × Z ) (cid:111) Z : λ = λ ∗ and λ = λ − λ √ . (77)In the space of bilinears, the potential can then be compactly written as V = −√ m r + (3 λ + √ λ α ) r + λ | r + r + r | . (78)The spectrum of Λ ij becomes of the type 6 + 2. This high symmetry hints at existence of apossible continuous symmetry of the potential, and it is indeed the case. For example, thefollowing SO (2) rotations among three doublets, φ a (cid:55)→ R ab ( α ) φ b , leave r + r + r invariant: R ( α ) = 13 α α (cid:48)(cid:48) α (cid:48) α (cid:48) α α (cid:48)(cid:48) α (cid:48)(cid:48) α (cid:48) α , (79)with α ∈ [0 , π ) and α (cid:48) = α + 2 π/ α (cid:48)(cid:48) = α + 4 π/ 3. Note that at α = 0, 2 π/ π/ Z group (cid:104) b (cid:105) .We conclude therefore that imposing invariance under Z < SL (3) makes the potentialsymmetric under a continuous group. In this way, we completely exhausted possibilities offeredby SL (3). We showed in section 7.1 that antiunitary transformations correspond to elements of GL (3)not lying in SL (3) as they have negative determinant and flip the sign of the antisymmetric32calar product in A = Z × Z . The complex conjugation operator, J , acts in A by sending a to a and leaving b invariant. Therefore, the corresponding matrix is J = (cid:18) − (cid:19) . (80)Since any antiunitary transformation can be written as J (cid:48) = qJ , where q is unitary, it followsthat q must belong to SL (3).Next, we need to find which q ’s can be used. Clearly, ( J (cid:48) ) = qJ qJ = qq ∗ ∈ SL (3). If weare looking for an antiunitary symmetry of a ( Z × Z ) (cid:111) Z -symmetric potential, then qq ∗ must be either 1 or c , which generates the center of SL (3).Let us first consider the second possibility.If q = (cid:18) x yz t (cid:19) , then q ∗ = (cid:18) x − y − z t (cid:19) . (81)Using this to solve qq ∗ = c , we get six possible solutions, but all of them have det q = − SL (3). Therefore, the only possibility is qq ∗ = 1.But then we can apply the results of our search for antiunitary transformations for the D case. Our group ( Z × Z ) (cid:111) Z contains the D subgroup with δ = π . Therefore, wearrive at the conclusion: in order for our potential to be symmetric under an antiunitarytransformation, we must require 6 arg λ = 0 . (82)If this criterion is satisfied, the symmetry group becomes ( Z × Z ) (cid:111) ( Z × Z ∗ ); otherwise thegroup remains ( Z × Z ) (cid:111) Z . Therefore, both groups are realizable in 3HDM.Now, consider the case of the extended symmetry group, ( Z × Z ) (cid:111) Z (cid:39) Σ(36). Inthis case (82) is satisfied automatically due to (67). We then conclude that in this case therealizable symmetry is Σ(36) (cid:111) Z ∗ . Bringing together the results of the search for abelian symmetry groups [21] and of the presentwork, we can finally give the list of finite groups which can appear as the symmetry groups ofthe scalar sector in 3HDM. If only Higgs-family transformations are concerned, the realizablefinite groups are Z , Z , Z , Z × Z , D , D , T (cid:39) A , O (cid:39) S , ( Z × Z ) (cid:111) Z (cid:39) ∆(54) / Z , ( Z × Z ) (cid:111) Z (cid:39) Σ(36) . (83)This list is complete: trying to impose any other finite symmetry group of Higgs-family trans-formations leads to the potential with a continuous symmetry.Fig. 3 should help visualize relations among different groups from this list. Going up alonga branch of this tree means that, starting with a potential symmetric under the lower group,one can restrict its free parameters in such a way that the potential becomes symmetric underthe upper group. 33 e } Z Z × Z Z Z D A S D ∆(54) / Z Σ(36)Figure 3: Tree of finite realizable groups of Higgs-family transformations in 3HDMIf both unitary (Higgs-family) and antiunitary (generalized- CP ) transformations are al-lowed, the list becomes Z , Z , Z × Z , Z ∗ , Z ∗ , Z × Z ∗ , Z × Z × Z ∗ , Z (cid:111) Z ∗ (cid:39) D , Z (cid:111) Z ∗ (cid:39) D ,D , D × Z ∗ , D × Z ∗ , A (cid:111) Z ∗ (cid:39) T d , S × Z ∗ (cid:39) O h , ( Z × Z ) (cid:111) Z , ( Z × Z ) (cid:111) ( Z × Z ∗ ) , Σ(36) (cid:111) Z ∗ . (84)As usual, an asterisk here indicates that the generator of the corresponding group is an anti-unitary transformation. Note that Higgs-family transformation groups Z , D , A , S , andΣ(36) become non-realizable in this case, because potentials symmetric under them are au-tomatically symmetric under an additional anti-unitary transformation. In all cases apartfrom A this is a consequence of our finding in section 6.5 that presence of the Z group ofHiggs-family transformations always leads to an additional anti-unitary symmetry.These lists complete the classification of realizable finite symmetry groups of the scalarsector of 3HDM. Conditions for the existence and examples of the potentials symmetric un-der each of these groups have been given in [21] and in the present work. For the reader’sconvenience, we collect examples with non-abelian groups in the Appendix. C P -violation In 2HDM, presence of any Higgs-family symmetry immediately leads to a generalized- CP symmetry. In other words, it is impossible to write down an explicitly CP -violating 2HDMpotential with any Higgs-family symmetry. In this sense, generalized- CP symmetries can beviewed as the smallest building blocks of any symmetry group in 2HDM.By comparing lists (83) and (84), we see that this conclusion is no longer true for 3HDM,namely there are some Higgs-family symmetry groups which are compatible with explicit CP -violation. However we found another, quite remarkable feature in 3HDM: the presence of a Z roup of Higgs-family transformations guarantees that the potential is explicitly CP -conserving .This is, of course, a sufficient but not necessary condition for explicit CP -violation. Put inother words, explicit CP -violation is incompatible with the Higgs-family symmetry group Z . D groups It is interesting to note that the list (84) contains two different D groups. One is Z (cid:111) Z ∗ ,generated by a Higgs-family transformation of order 3 and a generalized- CP transformation.The other D is a group of Higgs-family transformations only, and a potential invariant under itdoes not have any generalized- CP symmetry. Clearly, they lead to different phenomenologicalconsequences, as the first case is explicitly CP -conserving, while the latter is explicitly CP -violating.Such a situation was absent in the two-Higgs-doublet model, where fixing the symmetrygroup uniquely defined the (tree-level) phenomenological consequences in the scalar sector.What makes it possible in 3HDM is a looser relation between Higgs-family and generalized- CP symmetries just discussed. In particular, it is possible to have a potential with theHiggs-family D symmetry group without any generalized- CP symmetry. 2HDM does notoffer this kind of freedom: any non-trivial Higgs-family symmetry group automatically leadsto a generalized- CP symmetry. Certainly, our results do not provide answers to all symmetry-related questions which can beposed in 3HDM. Our paper should rather be regarded as the first step towards systematicexploration of all the possibilities offered by three Higgs doublets. Here are some furtherquestions which deserve a closer study: • Continuous symmetry groups should also be included in the list. There exist only fewLie groups inside P SU (3): U (1), U (1) × U (1), SU (2), SU (2) × U (1), SO (3). The non-trivial question is which of these groups can be merged with some of the finite groupsand with anti-unitary transformations (the case of abelian groups was analyzed in [21]). • It is well-known that the vacuum state does not have to respect all the symmetriesof the Lagrangian, so the finite symmetry groups described here can be broken uponelectroweak symmetry breaking. What are the symmetry breaking patterns for each ofthese groups? Clearly, if the symmetry group is very small, then the vacuum state caneither conserve it or break it, either completely or partially. But when the finite groupbecomes sufficiently large, there are two important changes. First, some of the groupscan never be conserved upon EWSB; the origin of this feature and some 3HDM exampleswere discussed in [30]. Second, a sufficiently large symmetry group cannot break downcompletely, as it would create too many degenerate vacua, which is not possible from thealgebraic-geometric point of view. Indeed, in the geometric reformulation of the Higgspotential minimization problem [14], the points of the global minima in the ( r , r i )-spaceare precisely the contact points of two 9-dimensional algebraic manifolds: the orbitspace and a certain quadric. Intersection of two algebraic manifolds of known degreesis also an algebraic manifold of a certain degree (the planar analogue of this statementis the Bezout’s theorem). In the degenerate case when this manifold is reduced to a35et of isolated points, there must exist an upper limit for the number of these points.Unfortunately, we have not yet found this number for 3HDM, but its existence is beyondany doubt. • What are possible symmetries of the potential beyond the unitary and antiunitary trans-formations? For example, the full reparametrization group of the 2HDM potential is GL (2 , C ) (cid:111) Z ∗ rather than SU (2) (cid:111) Z ∗ , [20]. It means that a potential can be left in-variant by transformations which are neither unitary nor anti-unitary. Although thesetransformations played important role in the geometric constructions in the 2HDM orbitspace, they did not produce new symmetry groups beyond what was already found fromthe unitary transformations. It would be interesting to check the situation in 3HDM.Unfortunately, the geometric method which worked well for 2HDM becomes much moreintricate with more than two doublets, [13, 14]. • It would also be interesting to see if the potential can have symmetries beyond repara-metrization transformations. In the case of 2HDM, this problem was analyzed in [31].Although these additional symmetries cannot be extended to kinetic term, they couldstill provide useful information on the structure of the Higgs potential and properties ofthe physical Higgs bosons.In summary, we found all finite groups which can be realized as symmetry groups of Higgs-family or generalized- CP transformations in the three-Higgs-doublet model. Our list (84) iscomplete: trying to impose any other discrete symmetry group on the 3HDM Higgs potentialwill make it symmetric under a continuous group.This work was supported by the Belgian Fund F.R.S.-FNRS, and in part by grants RFBR11-02-00242-a, RFBR 12-01-33102, RF President grant for scientific schools NSc-3802.2012.2,and the Program of Department of Physics SC RAS and SB RAS ”Studies of Higgs bosonand exotic particles at LHC.” A 3HDM potentials with non-abelian Higgs-family sym-metry group Here, for the reader’s convenience, we list once again Higgs potentials with a given symmetrygroup. We focus here on cases with non-abelian groups from the list (84) because abelianones were already discussed in detail in [21]. In each case we start from the most generalpotential compatible with the given realizable group presented in the main text and use theresidual reparametrization freedom to simplify the coefficients of the potential (usually, itamounts to rephasing of doublets which makes some of the coefficients real). For each group G , the potential written below faithfully represents all possible Higgs potentials with realizablesymmetry group G . In this sense, the symmetry group uniquely defines the phenomenologyof the scalar sector of 3HDM, the only exception being D with its two distinct realizations. Group D (cid:39) Z (cid:111) Z ∗ . Consider the most general phase-independent part of the Higgspotential V = − (cid:88) ≤ i ≤ m i ( φ † i φ i ) + (cid:88) ≤ i ≤ j ≤ λ ij ( φ † i φ i )( φ † j φ j ) + (cid:88) ≤ i 00 0 1 , ω = exp (cid:18) πi (cid:19) . (86)If it happens that the product λ λ λ is purely real, then by rephasing of doublets one can makeall coefficients in (85) real. The resulting potential, V + V Z , is symmetric under D (cid:39) Z (cid:111) Z ∗ generated by a and the CP -transformation. Group D (cid:39) Z (cid:111) Z ∗ . Consider now terms V Z = λ ( φ † φ )( φ † φ ) + λ ( φ † φ ) + h.c., (87)which are symmetric under the group Z generated by a = i − i 00 0 1 . (88)It is always possible to compensate the phases of λ and λ by an appropriate rephasing ofthe doublets. Therefore, the potential V + V Z is symmetric under the group D (cid:39) Z (cid:111) Z ∗ generated by a and the CP -transformation. Group D of unitary transformations. Let us restrict the coefficients of V in the waythat guarantees the symmetry under φ ↔ φ . Then, V turns into V = − m (cid:104) ( φ † φ ) + ( φ † φ ) (cid:105) − m ( φ † φ ) + λ (cid:104) ( φ † φ ) + ( φ † φ ) (cid:105) + λ ( φ † φ ) (89)+ λ (cid:104) ( φ † φ ) + ( φ † φ ) (cid:105) ( φ † φ ) + λ ( φ † φ )( φ † φ ) + λ (cid:48) (cid:104) | φ † φ | + | φ † φ | (cid:105) + λ (cid:48) | φ † φ | , where all coefficients are real and generic. Imposing the same requirement on V Z and per-forming rephasing, we obtain V D = λ (cid:104) ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) (cid:105) + | λ | e iψ ( φ † φ )( φ † φ ) + h.c. (90)where λ is real and sin ψ (cid:54) = 0. The resulting potential, V + V D , is symmetric under D generated by a and b = − . (91)There are no other Higgs-family or generalized- CP transformations which leave this potentialinvariant. Any explicitly CP -violating D -symmetric 3HDM potential can always be broughtinto this form. Group D × Z ∗ . If in the previous case we set sin ψ = 0 in (90), then the potentialbecomes symmetric under D × Z ∗ generated by a , b , and the generalized CP -transformation b · CP . 37 roup D × Z ∗ . The potential V + V Z is symmetric under the group D × Z ∗ generatedby a , b , and b · CP . Group A (cid:111) Z ∗ . A potential symmetric under A (cid:111) Z ∗ can be brought into the followingform V A (cid:111)Z ∗ = − m (cid:104) φ † φ + φ † φ + φ † φ (cid:105) + λ (cid:104) φ † φ + φ † φ + φ † φ (cid:105) (92)+ λ (cid:48) (cid:104) ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) (cid:105) + λ (cid:48)(cid:48) (cid:16) | φ † φ | + | φ † φ | + | φ † φ | (cid:17) + (cid:16) ˜ λ (cid:104) ( φ † φ ) + ( φ † φ ) + ( φ † φ ) (cid:105) + h.c. (cid:17) with complex ˜ λ . Its symmetry group is generated by independent sign flips of the individualdoublets, by cyclic permutations of φ , φ , φ , and by the exchange of any pair of doublettogether with the CP -transformation. An alternative form of this potential is V A (cid:111)Z ∗ = − m (cid:104) φ † φ + φ † φ + φ † φ (cid:105) + λ (cid:104) φ † φ + φ † φ + φ † φ (cid:105) (93)+ λ (cid:48) (cid:104) ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) (cid:105) + λ Re (cid:104) (Re φ † φ ) + (Re φ † φ ) + (Re φ † φ ) (cid:105) + λ Im (cid:104) (Im φ † φ ) + (Im φ † φ ) + (Im φ † φ ) (cid:105) + λ ReIm (cid:104) Re φ † φ Im φ † φ + Re φ † φ Im φ † φ + Re φ † φ Im φ † φ (cid:105) . Group S × Z ∗ . If the parameter ˜ λ in (92) is real or, equivalently, λ ReIm = 0 in (93),the potential becomes symmetric under S × Z ∗ generated by sign flips, all permutation of thethree doublets, and the CP -transformation. Group ( Z × Z ) (cid:111) Z (cid:39) ∆(54) / Z . Consider the following potential V ∆(54) / Z = − m (cid:104) φ † φ + φ † φ + φ † φ (cid:105) + λ (cid:104) φ † φ + φ † φ + φ † φ (cid:105) + λ (cid:104) ( φ † φ ) + ( φ † φ ) + ( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) (cid:105) + λ (cid:104) | φ † φ | + | φ † φ | + | φ † φ | (cid:105) + λ (cid:104) ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) + ( φ † φ )( φ † φ ) (cid:105) + h.c. (94)with generic real m , λ , λ , λ and complex λ . The symmetry group of this potential is( Z × Z ) (cid:111) Z = ∆(54) / Z . Here, ∆(54) is generated by the same a and b as before and, inaddition, by the cyclic permutation c = , (95)while the subgroup Z is the center of SU (3). Group ( Z × Z ) (cid:111) ( Z × Z ∗ ). The potential (94) becomes symmetric under a generalized- CP transformation if λ = k · π/ k . In this case, one can make λ real by arephasing transformation. The extra generator then is the CP -transformation.38 roup Σ(36) (cid:111) Z ∗ . The same potential (94) becomes symmetric under the group Σ(36) (cid:111)Z ∗ if, upon rephasing, λ = (3 λ − λ ) / 2. The potential can then be rewritten as V Σ(36) (cid:111)Z ∗ = − m I + λ I + 3 λ I + λ − λ (cid:16) | φ † φ − φ † φ | + | φ † φ − φ † φ | + | φ † φ − φ † φ | (cid:17) . (96)Here I and I are the SU (3)-invariants I = r √ φ † φ + φ † φ + φ † φ ,I = (cid:88) i r i = ( φ † φ ) + ( φ † φ ) + ( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ ) − ( φ † φ )( φ † φ )3+ | φ † φ | + | φ † φ | + | φ † φ | . (97)It is remarkable that this potential has only one “structural” free parameter, and the termcontaining it reduces the full SU (3) symmetry group to a finite subgroup Σ(36). References [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B , 1 (2012); S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 30 (2012).[2] E. Accomando et al. , “Workshop on CP studies and non-standard Higgs physics,”arXiv:hep-ph/0608079.[3] T. D. Lee, Phys. Rev. D , 1226 (1973).[4] G. C. Branco et al, Phys. Rept. , 1 (2012).[5] S. Weinberg, Phys. Rev. Lett. , 657 (1976);[6] R. Erdem, Phys. Lett. B , 222 (1995); S. L. Adler, Phys. Rev. D , 015012 (1999)[Erratum-ibid. D , 099902 (1999)]; A. Barroso, P. M. Ferreira, R. Santos and J. P. Silva,Phys. Rev. D , 085016 (2006); T. Fukuyama, H. Sugiyama, K. Tsumura, Phys. Rev.D , 036004 (2010).[7] E. Ma and G. Rajasekaran, Phys. Rev. D , 113012 (2001); L. Lavoura and H. Kuhbock,Eur. Phys. J. C , 303 (2008); S. Morisi and E. Peinado, Phys. Rev. D , 113011 (2009);A. C. B. Machado, J. C. Montero, and V. Pleitez, Phys. Lett. B , 318 (2011).[8] R. de Adelhart Toorop, F. Bazzocchi, L. Merlo and A. Paris, JHEP , 035 (2011).[9] G. C. Branco, J. M. Gerard, and W. Grimus, Phys. Lett. B , 383 (1984).[10] I. de Medeiros Varzielas and D. Emmanuel-Costa, Phys. Rev. D , 117901 (2011).[11] C. C. Nishi, Phys. Rev. D , 055013 (2007).[12] P. M. Ferreira and J. P. Silva, Phys. Rev. D , 116007 (2008).3913] I. P. Ivanov and C. C. Nishi, Phys. Rev. D , 015014 (2010).[14] I. P. Ivanov, JHEP , 020 (2010).[15] M. Maniatis, A. von Manteuffel and O. Nachtmann, Eur. Phys. J. C , 739 (2008);M. Maniatis and O. Nachtmann, JHEP , 028 (2009); P. M. Ferreira and J. P. Silva,Eur. Phys. J. C , 45 (2010).[16] I. P. Ivanov, Acta Phys. Polon. B , 2789 (2009); I. F. Ginzburg, I. P. Ivanov andK. A. Kanishev, Phys. Rev. D , 085031 (2010); I. F. Ginzburg, K. A. Kanishev,M. Krawczyk and D. Sokolowska, Phys. Rev. D , 123533 (2010).[17] N. G. Deshpande and E. Ma, Phys. Rev. D , 2574 (1978); R. Barbieri, L. J. Hall andV. S. Rychkov, Phys. Rev. D , 015007 (2006); L. Lopez Honorez, E. Nezri, J. F. Oliverand M. H. G. Tytgat, JCAP , 028 (2007).[18] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Prog.Theor. Phys. Suppl. , 1 (2010).[19] C. C. Nishi, Phys. Rev. D , 036003 (2006) [Erratum-ibid.D , 119901 (2007)];P. M. Ferreira, H. E. Haber, M. Maniatis, O. Nachtmann, J. P. Silva, Int. J. Mod. Phys.A , 769-808 (2011).[20] I. P. Ivanov, Phys. Lett. B , 360 (2006); Phys. Rev. D , 035001 (2007) [Erratum-ibid. D , 039902 (2007)]; Phys. Rev. D , 015017 (2008).[21] I. P. Ivanov, V. Keus and E. Vdovin, J. Phys. A , 215201 (2012) [arXiv:1112.1660[math-ph]].[22] I. P. Ivanov and E. Vdovin, Phys. Rev. D , 095030 (2012).[23] L. Lavoura and J. P. Silva, Phys. Rev. D , 4619 (1994); F. J. Botella and J. P. Silva,Phys. Rev. D , 3870 (1995).[24] G. C. Branco, L. Lavoura, and J. P. Silva, CP Violation (Oxford University Press, Oxford,1999).[25] G. Ecker, W. Grimus and W. Konetschny, Nucl. Phys. B , 465 (1981); G. Ecker,W. Grimus and H. Neufeld, Nucl. Phys. B , 70 (1984); G. Ecker, W. Grimus andH. Neufeld, J. Phys. A , L807 (1987); H. Neufeld, W. Grimus and G. Ecker, Int. J.Mod. Phys. A , 603 (1988); P. M. Ferreira, H. E. Haber and J. P. Silva, Phys. Rev. D , 116004 (2009).[26] S. Davidson and H. E. Haber, Phys. Rev. D , 035004 (2005) [Erratum-ibid. D ,099902 (2005)]; H. E. Haber and D. O’Neil, Phys. Rev. D , 015018 (2006) [Erratum-ibid. D , 805 (2006).[29] W. M. Fairbairn, T. Fulton, and W. H. Klink, J. Math. Phys. , 1038 (1982).[30] I. P. Ivanov, V. Keus, Phys. Lett. B695 , 459 (2011).[31] R. A. Battye, G. D. Brawn and A. Pilaftsis, JHEP , 020 (2011); A. Pilaftsis, Phys.Lett. B706