Classification of finite reparametrization symmetry groups in the three-Higgs-doublet model

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 which symmetry groups can be implemented via the scalar sector. In N-Higgs-doublet models, this classification problem was solved only for N=2 doublets. Very recently, we suggested a method to classify all realizable finite symmetry groups of Higgs-family transformations in the three-Higgs-doublet model (3HDM). Here, we present this classification in all detail together with an introduction to the theory of solvable groups, which play the key role in our derivation. We also consider generalized-CP symmetries, and discuss the interplay between Higgs-family symmetries and CP-conservation. In particular, we prove that presence of the ^sub 4^ symmetry guarantees the explicit CP-conservation of the potential. This work completes classification of finite reparametrization symmetry groups in 3HDM.

Eur. Phys. J. C (2013) 73:2309DOI 10.1140/epjc/s10052-013-2309-x

Regular Article - Theoretical Physics

Classication of nite reparametrization symmetry groups in the three-Higgs-doublet model

Igor P. Ivanov1,2,a, E. Vdovin2

1IFPA, Universit de Lige, Alle du 6 Aot 17, btiment B5a, 4000 Lige, Belgium

2Sobolev Institute of Mathematics, Koptyug avenue 4, 630090 Novosibirsk, Russia Received: 1 November 2012 / Revised: 11 January 2013 / Published online: 9 February 2013 Springer-Verlag Berlin Heidelberg and Societ Italiana di Fisica 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 which symmetry groups can be implemented via the scalar sector. In N-Higgs-doublet models, this classication problem was solved only for N = 2 doublets. Very recently,

we suggested a method to classify all realizable nite symmetry groups of Higgs-family transformations in the three-Higgs-doublet model (3HDM). Here, we present this classication in all detail together with an introduction to the theory of solvable groups, which play the key role in our derivation. We also consider generalized-CP symmetries, and discuss the interplay between Higgs-family symmetries and CP-conservation. In particular, we prove that presence of the Z4 symmetry guarantees the explicit CP-conservation of the potential. This work completes classication of nite reparametrization symmetry groups in 3HDM.

1 Introduction

The nature of the electroweak symmetry breaking is one of the main puzzles in high-energy physics. Very recently, the CMS and ATLAS collaborations at the LHC announced the discovery of the Higgs-like resonance at 126 GeV [1, 2], and their rst measurements indicate intriguing deviations from the Standard Model (SM) expectations. Whether these data signal that a non-minimal Higgs mechanism is indeed at work and if so what it is, are among the hottest questions in particle physics these days.

In the past decades, many non-minimal Higgs sectors have been considered [3]. One conceptually simple and phenomenologically attractive class of models involves several Higgs doublets with identical quantum numbers (N-Higgs-doublet models, NHDM). Its simplest version with only two

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doublets, 2HDM, was proposed decades ago [4], but it is still actively studied, see [5] for a recent review, and it has now become a standard reference model of the physics beyond the Standard Model (bSM). Constructions with more than two doublets are also extensively investigated [621].

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 the model. These are not related with the gauge symmetries of the SM but rather reect extra symmetry structures in the horizontal space of the model. One of the main phenomenological motivations in working with several doublets is the ease with which one can introduce various symmetry groups. Indeed, Higgs elds with identical quantum numbers can mix, and it is possible that some of these Higgs-family mixing transformations leave the scalar sector invariant. Even in 2HDM, presence of such a symmetry in the Lagrangian and its possible spontaneous violation can lead to a number of remarkable phenomena such as various forms of CP-violation [4, 2224], non-standard thermal phase transitions which may be relevant for the early Universe [2527], natural scalar dark matter candidates [2830]. For models with three or more doublets, an extra motivation is the possibility to incorporate into the Higgs sector non-abelian nite symmetry groups, which can then lead to interesting patterns in the fermionic mass matrices (for a general introduction into discrete symmetry groups relevant for particle physics, see [31]). In this respect, the very popular symmetry group has been A4 [1115], the smallest nite group with a three-dimensional irreducible representation, but larger symmetry groups also received some attention [6, 16, 17].

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 years ago [3236], see also [5]

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for a review. Focusing on discrete symmetries, the only realizable Higgs-family symmetry groups are Z2 and (Z2)2. The

Z2 group can be generated, for example, by the sign ip of one of the doublets (and it does not matter which, because once we focus on the scalar sector only, the simultaneous sign ip of both doublets does not change the Lagrangian), while the (Z2)2 group is generated by sign ips and the exchange 1 2. If generalized-CP transformations are also

included, then (Z2)3 becomes realizable as well, the additional generator being simply the CP conjugation.

With more than two doublets, the problem remains open.

Although several attempts have been made in past to classify at least some symmetries in NHDM, [18, 19, 21], they led only to 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 how to prove that the given potential does not have other symmetries. An even more difcult problem is to prove that no other symmetry group can be implemented for a given N.

In the recent paper [37] we found such a criterion for abelian symmetry groups in NHDM for arbitrary N. Since abelian subgroups are the basic building blocks of any group, classication of realizable abelian symmetry groups in NHDM was an important milestone. We stress that this task is different from just classifying all abelian subgroups of SU(3), because invariance of the Higgs potential places strong and non-trivial restrictions on possible symmetry groups.

In this paper, we solve the classication problem for all nite symmetry groups in 3HDM, including non-abelian groups. We do this by using the abelian groups in 3HDM found in [37] and by applying certain results and methods from the theory of solvable groups. Some of these results were already briey described in [38]. Here, we present a detailed derivation of this classication together with an introduction to the relevant methods from nite group theory. In addition, we extend the analysis to symmetry groups which include both Higgs-family and generalized-CP transformations. This work, therefore, solves the problem of classication of nite reparametrization symmetry groups in 3HDM.

We would like to stress one important feature in which our method differs from more traditional approaches to symmetry classication problem, at least within the bSM physics. Usually, one starts by imposing invariance under certain transformations, and then one tries to recognize the symmetry group of the resulting potential. In this way it is very difcult to see whether all possible symmetries are exhausted. We approach the problem the other way around.We rst restrict the list of nite groups which can appear as symmetry groups of 3HDM, and then we check one by one whether these groups can indeed be implemented.

The structure of this paper is the following. In Sect. 2 we describe different types of symmetry in the scalar sector of

NHDM and discuss the important concept of realizable symmetry groups. Section 3 contains an elementary introduction into the theory of (nite) solvable groups. Although it contains pure mathematics, we put it in the main text because it is a key part of the group-theoretic step of our classication, which is presented in Sect. 4. Then, in Sect. 5 we describe the methods which we will use to prove the absence of continuous symmetries. Sections 6 and 7 contain the main results of the paper: explicit constructions of the realizable symmetry groups and of the potentials symmetric under each group. Finally, in Sect. 8 we summarize and discuss our results. For the readers convenience, we list in the Appendix potentials for each of the realizable non-abelian symmetry groups.

2 Symmetries of the scalar sector of multi-Higgs-doublet models

2.1 Reparametrization transformations

In NHDM we introduce N complex Higgs doublets with the electroweak isospin Y = 1/2, which interact with the

gauge bosons and matter elds in the standard way, and also self-interact via a Higgs potential. The generic renormalizable Higgs potential can contain only quadratic and quartic gauge-invariant terms, and it can be compactly written as [3941]:

V = Yab ab + Zabcd ab cd , (1)

where all indices run from 1 to N. Coefcients of the potential are grouped into components of tensors Yab and Zabcd;

there are N2 independent components in Y and N2(N2 +

1)/2 independent components in Z.

In this work we focus only on the scalar sector of the NHDM. Therefore, once coefcients Yab and Zabcd are given, the model is completely dened, and one should be able to express all its properties (the number and the positions of extrema, the spectrum and interactions of the physical Higgs bosons) via components of Y s and Zs. This explicit expression, however, cannot be written via elementary functions, and it remains unknown in the general case for any N > 2.

A very important feature of the most general potential is that any non-degenerate linear transformation in the space of Higgs doublets belonging to the group GL(2, C) keeps the generic form of the potential, changing only the coefcients of Y and Z. We call such a transformation a Higgs-basis change. In addition, the CP transformation, which maps doublets to their hermitian conjugates a a, also keeps the generic form of the potential, up

to coefcient modication. Its combination with a Higgs-basis change represents a transformation which is usually

Eur. Phys. J. C (2013) 73:2309 Page 3 of 25

called a generalized-CP transformation [4246]. The Higgs basis changes and generalized-CP transformations can be called together reparametrization transformations because they preserve the generic structure of the potential and lead only to its reparametrization.

A reparametrization transformation changes the basis in the space of Higgs doublets but does not modify the structural features of the model such as the number and the properties of minima, the symmetries of the potential and their spontaneous breaking at the minimum point. These properties must be the same for all the potentials linked by reparametrization transformations. Therefore, these properties must be expressible in terms of reparametrization-invariant combinations of Y s and Zs [39, 40, 47, 48].

If a reparametrization transformation maps a certain potential exactly to itself, that is, if it leaves certain Y s and Zs invariant, we say that the potential has a reparametrization symmetry. 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 fermion sectors. Therefore, understanding which groups can appear as reparametrization symmetry groups in NHDM with given N is of much importance for phenomenology of the model.

2.2 The group of kinetic-term-preserving reparametrization transformations

Often, one restricts the group of reparametrization transformations only to those transformations which keep the Higgs kinetic term invariant. In this case, a generic basis change becomes a unitary transformation a Uabb

with U U(N). A kinetic-term-preserving generalized-CP

transformation is an anti-unitary map a Uabb, which

can be written as UCP = 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 included in the gauge group U(1)Y .

Since we want to study structural symmetries of the NHDM potentials, we should disregard transformations which leave all the potentials invariant by construction. This leads to the group U(N)/U(1) PSU(N). Note that SU(N), which is

often considered in these circumstances, still contains transformations which only amount to the overall phase shift of all doublets. They form the center of SU(N), Z(SU(N))

ZN , and act trivially on all NHDM potentials. Being invariant under them does not represent any structural property of the Higgs potential, therefore, we are led again to the factor group SU(N)/Z(SU(N)) = PSU(N). This allows us to

write the group of kinetic-term-preserving reparametrization transformations as a semidirect product of the Higgs basis change group and the Z2 group generated by J (for a more detailed discussion, see [37]):

Grep = PSU(N)

Z2. (2)

Here the asterisk indicates that the generator of the corresponding group is an anti-unitary transformation; we will use this notation throughout the paper.

Below, when discussing symmetry groups of the 3HDM potential, we will be either looking for subgroups of PSU(3) (if only unitary transformations are allowed) or subgroups of this Grep (when anti-unitary reparametrization transformations are also included). This should always be kept in mind when comparing our results with the groups which are discussed as symmetry groups in the 3HDM scalar sector. For example, in [16, 17] a 3HDM potential symmetric 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 Higgs potentials. Therefore, the structural properties of that model are dened by the factor groups (27)/Z(SU(3))

Z3

Z3

and (54)/Z(SU(3)) (

Z3

Z3) Z2, which belong to

PSU(3).

2.3 Realizable symmetry groups

There is an important technical point which should be kept in mind when we classify symmetry groups of NHDM. When we impose a reparametrization symmetry group G on the potential, we restrict its coefcients in a certain way. It might happen then that the resulting potential becomes symmetric under a larger symmetry group

G properly contain-

ing G.

One drawback of this situation is that we do not have control over the true symmetry properties of the potential: if we construct a G-symmetric potential, we do not know a priori what is its full symmetry group

G. This might be especially dangerous if G is nite while

G turns out to be continuous, as it might lead to unwanted Goldstone bosons. Another undesirable feature is related with symmetry breaking. Suppose that we impose invariance of the potential under group G but we do not check what is the true symmetry group

G. After electroweak symmetry breaking, the symmetry group of the vacuum is Gv

G, and it can happen that Gv is not a subgroup of G. This is not what we normally expect when we construct a G-symmetric model, and it is an indication of a higher symmetry.

Examples of these situations were encountered in literature before. For instance, the authors of [19] explicitly show that trying to impose a Zp, p > 2, group of rephasing transformations in 2HDM unavoidably leads to a potential with continuous PecceiQuinn symmetry. For 3HDM they nd an even worse example, when a cyclic group immediately leads to a U(1) U(1)-symmetric potential. Another

well-known example is the A4-symmetric 3HDM potential, which at certain values of parameters admits vacua with the S3 symmetry, although S3 is not a subgroup of A4, see an explicit study in [15]. The explanation is that the potential

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at these values of parameters becomes symmetric under S4 which contains both A4 and S3.

In order to avoid such situations altogether, we must always check for each G whether the G-symmetric potentials are invariant under any larger group. We are interested only in those groups G, for which there exists a G-invariant potential with the property that no other reparametrization transformation leaves it invariant (either within PSU(3) or within Grep, depending on whether we include anti-unitary transformations). Following [21, 37], we call such groups realizable.

Using the terminology just introduced we can precisely formulate the two main questions which we address in this paper:

1. considering only non-trivial kinetic-term-preserving Higgs-basis transformations (i.e. group PSU(3)), what are the realizable nite symmetry groups in 3HDM?

2. more generally, considering non-trivial kinetic-term-preserving reparametrization transformations, which can now include generalized-CP transformations (i.e. group Grep), what are the realizable nite symmetry groups in 3HDM?

For abelian groups, these questions were answered in [37] for general N. Here we focus on non-abelian nite realizable groups for N = 3.

3 Solvable groups: an elementary introduction

Our classication of realizable groups of Higgs-family symmetries in 3HDM contains two essential parts: the group-theoretic and the calculational ones. The group-theoretic part will make use of some methods of pure nite group theory, which are not very familiar to the physics community (although they are quite elementary for a mathematician with expertise in group theory). To equip the reader with all the methods needed to understand the group-theoretic part 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 problem of this paper. For a deeper introduction to solvable groups and nite group theory in general, see e.g. [49].

3.1 Basics

We assume that the reader is familiar with the basic denitions from group theory. We only stress here that we will work with nite groups, therefore the order of the group G (the number of elements in G) denoted as |G| is always

nite, and so is the order of any element g (the smallest positive integer n such that gn = e, the identity element of the

group).

A group G is called abelian if all its elements commute. An alternative way to formulate it is to say that all commutators in the group are trivial: [x, y] = xyx1y1 = 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, g2 = e, then the group is abelian. Indeed, for

any x, y G we have

[x, y] = xyx1y1 = xyxy = (xy)2 = e, (3) which means that x and y commute.

A group G can have proper subgroups H < G (whenever we do not require that the subgroup H is proper, we write H G), whose order must, by Lagranges theorem, divide

the order 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 con

sider its powers: if order of the element g is n, we will get

the cyclic group Zn < G.

The inverse of Lagranges theorem is not, generally speaking, true: namely, if p is a divisor of |G|, the group G

does not necessarily have a subgroup of order p. However, if p is a prime which enters the prime decomposition of |G|,

then according to Cauchys theorem such a subgroup must exist (this group is Zp because there are no other groups of prime order). It immediately follows that if we have the list of all abelian subgroups of a given nite group G, then the prime decomposition of |G| can only contain primes which

are present in the orders of these abelian subgroups.

In fact, there is an existence criterion stronger than Cauchys theorem. Namely, if pa is the highest power of the prime p that enters the prime decomposition of |G|,

then G contains a 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, see Chap. 1 in [49].

There are several ways to present a nite group. One possibility is to list all its elements and write down the |G||G|

multiplication table. Clearly, this presentation becomes impractical for a sufciently large group. A more compact and powerful way is known as presentation by generators and relations. We call a subset M = {g1, g2, . . . } of the elements

of G a generating set (and its elements are called generators) if every g G can be written as a product of elements

of M or their inverses. The fact that G is generated by the set M is denoted as G = M . Finding a minimal generat

ing set for a given group and listing equalities which these generators satisfy is precisely presentation of the group by generators and relations. For example, the symmetry group of the regular n-sided polygon has the following presentation by generators and relations:

D2n = a, b | a2 = b2 = (ab)n = e . (4)

Eur. Phys. J. C (2013) 73:2309 Page 5 of 25

This group is known as the dihedral group and has order

|D2n| = 2n (note that there exists an alternative convention

for denoting dihedral groups: Dn; the one which we use has its order in the subscript).

3.2 Normal subgroups and extensions

Consider two groups G and H . Suppose we have a map f from G to H , f : G H , which sends every g G into its

image f (g) H . If this map preserves the group operation,

f (g1)f (g2) = f (g1g2), then it is called a homomorphism. If

this map is surjective (i.e. it covers the entire H ) and injective (distinct elements from G have distinct images in H ), then f is invertible and is called an isomorphism.

In the case when H = G, we deal with an isomorphism of

the group onto itself, which is called an automorphism. One can note that composition of two automorphisms is also an automorphism, and dene the group structure on the set of all automorphisms of G. This automorphism group is denoted as Aut(G). The trivial automorphism which xes every element 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 dene f : x g1xg for every x G.

It can be immediately checked that f is an automorphism, and that it sends a subgroup of G into a (possibly another) subgroup of G. It can, however, happen that certain subgroups will be mapped onto themselves: g1Hg = H . Sub

groups which satisfy this invariance criterion for every possible g G are called normal, or invariant subgroups. The

fact that H is a normal subgroup of G is denoted as H [triangleleft] G.

Even when a subgroup H is not normal in G, one can pick up some elements g G such that g1Hg = H . The

set of elements of G with the property g1Hg = H forms a

group, which is called the normalizer of H in G and denoted as NG(H). We then have H [triangleleft] NG(H) G. Working with

normalizers is a useful intermediate step in situations when it is not known whether the subgroup H is normal in G.

Having a normal subgroup H [triangleleft] G gives some information about the structure of G. One can dene 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 simplies its study. Given a normal subgroup H [triangleleft] G, one can dene the canonical homomorphism : G G/H which sends every element

g G into its coset gH . Its kernel (all elements g which

are mapped by into the identity element of G/H ) is precisely H . Thus, every normal subgroup is the kernel of the corresponding canonical homomorphism. The reverse statement is also true: kernels of homomorphisms are always normal subgroups.

The group-constructing procedure inverse to factoring is called extension. Given two groups, N and H , a group G is

called an extension of H by N (denoted as N . H ), if there exists N0 [triangleleft] G such that N0 N and G/N0 H . In the case

when, in addition, H is also isomorphic to a subgroup of G and G = NH , we deal with a split extension. The criterion

for G to be a split extension can also be written as existence of N [triangleleft] G and H G such that NH = G and N H = 1,

so that G/N = H . The group G is then called a semidirect

product G = N

H .

Even if two groups N and H are xed, they can support several extensions and split extensions. Therefore one faces the problem of classifying of all extensions of two given groups.

For the most elementary example, consider extensions of H =

Z2 (generated by a) by N =

Z2 (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 choice is G =

Z2

Z2, which can be presented as a, b | a2 = b2 =

(ab)2 = e . For a non-split extension, we require that only N

is isomorphic to a subgroup of G. Thus, we still have b2 = e,

while a2 must not be the unit element. Then we have to set a2 = b producing the group

Z4. So, Z4 does not split over

Z2, while Z2

Z2 does.

3.3 Characteristic subgroups

In what concerns embedding of groups, normality is a relatively weak property. Namely, if K [triangleleft] H and H [triangleleft] G, then K is not necessarily normal in G (it is instead called subnormal in G). Indeed, recall that a normal subgroup K [triangleleft]H stays invariant under all inner automorphisms on H . Here inner

is meant with respect to the group H , namely, h1Kh = K

for all h H . However, since H [triangleleft] G, one can x g G

but g / H and consider an automorphism on H dened by

H g1Hg. This is indeed an automorphism on H be

cause it induces a permutation of elements of H preserving its group property, but it is not inner, because g does not belong to H . Therefore K does not have to be invariant under it: g1Kg = 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 characteristic in 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 some subgroups are characteristic gives even more information than their normality.

There is one simple rule which guarantees that certain subgroups are characteristic. If we have a rule dened in terms of the group G which identies its subgroup H uniquely, then H is characteristic in G. Two important examples are:

Page 6 of 25 Eur. Phys. J. C (2013) 73:2309

the center of the group G denoted as Z(G), which is the

set of all elements z G such that 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 de

noted as G and dened as the subgroup generated by all commutators:

G = [x, y] , x, y G. (6)

Note that the word generated is needed because the set of commutators is generally speaking not closed under the group multiplication. Clearly, the commutator subgroup of an abelian group is trivial, therefore the size of G can be used to qualitatively characterize how far G is from being abelian.

3.4 Consequences of existenceof a normal maximal abelian subgroup

Let us now prove a rather simple group-theoretic result, which, however, will be important for our classication of symmetries in 3HDM. This result, loosely speaking, is the observation that a mere existence of a subgroup of G with some special properties can strongly restrict the structure of the group G.

First, an abelian subgroup A < G is called a maximal abelian subgroup if there is no other abelian subgroup B with property A < B G. Note that the word maximal

refers not to the size but to containment. This denition does not specify a unique subgroup; in fact a group can have several maximal abelian subgroups. They correspond to terminal points in the partially ordered tree of abelian subgroups of G.

Suppose that A is an abelian subgroup of a nite group G.

Elements of A, of course, commute among themselves. But it can also happen that there exist other elements g G,

g / A, which also commute with all elements of A. The set

of all such elements is called the centralizer of A in G:

CG(A) = g G | [g, a] = e a A . (7)

It is easy to check that CG(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 CG(A).

Clearly, A CG(A). If A is a proper subgroup of CG(A),

then it means that A is not a maximal abelian subgroup. Indeed, we take an element g CG(A), g / A, and consider

another subgroup B = A, g . This subgroup is abelian and

is strictly larger than A: A < B G. On the other hand,

an element x G which commutes with all elements of B

will certainly commute with all elements of A, while the converse is not necessarily true. Therefore, we get the following chain: A < B CG(B) CG(A). Next, we check

whether B is a proper subgroup of CG(B). If so, we can enlarge it again in the same way by considering C = B, g ,

where g CG(B), g /

B. We can continue this procedure

until it terminates with an abelian subgroup K which is self-centralizing:

A < B < < K = CG(K) CG(B) CG(A). (8) Since there exists no other element in G which would commute with all elements of K, we conclude that K is a maximal abelian subgroup in G.

Let us now see what changes if the abelian subgroup A is normal. Any element g G acting on A by conjugation

induces an automorphism of A. Thus, we have a map from G to the group of automorphisms of A, f : G Aut(A).

The kernel of f consists of such gs which induce the trivial automorphism of A, that is, which leave every a A un

changed: g1ag = a a A. But this coincides with the

denition of centralizer. Therefore we conclude that ker f =

CG(A).

The fact that CG(A) is the kernel of the homomorphism f implies that CG(A) is a normal subgroup of G. Note that it is essential that the abelian subgroup in question, A, is normal; if it were not, CG(A) would not have to be normal.

Now, if A is a normal maximal abelian subgroup of G, then ker f = CG(A) = A. In other words, the kernel of

G/A Aut(A) is trivial, and therefore, G/A is isomorphic

to a subgroup of 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 A . K, where K Aut(A). (9) This is a powerful structural implication for the group G of existence of a normal maximal abelian subgroup.

3.5 Automorphism groups

For future reference, we give some details on the automorphism groups Aut(A) of certain abelian groups A. In this subsection we will use the additive notation for the group operation.

Suppose A =

Zn is the cyclic group of order n with generator e: ne = e + + e

n times

= 0. An automorphism act

ing on A is a group-structure-preserving permutation of elements of A. Since A is generated by e, this automorphism is completely and uniquely dened once we assign the value of (e) = k and make sure that m(e) = 0 for all

0 < m < n. This holds when k and n are coprime (k = 1 is

coprime to any n). The number of integers less than n and

Eur. Phys. J. C (2013) 73:2309 Page 7 of 25

coprime to n is called the Euler function (n). Thus, we have |Aut(

Zn)| = (n). For a prime p, the Euler function

is obviously (p) = p 1. In general, if pk11 pkss is the

prime decomposition for n, then

pk1

1 pkss = pk11 pkss

= pk11 pk111 pkss pks1s .

Suppose now that p is prime and

A =

Zp

Zp

n times

= (

Zp)n.

Then G can be considered as an n-dimensional vector space over a nite eld Fp of order p. Vectors in this space can be written as

x = k1e1 + + knen,

where numbers ki

Fp and basis vectors ei are certain non-zero elements of the ith group Zp. The group of all automorphisms on (Zp)n is then the general linear group in this space GLn(p).

Again, in order to dene an automorphism acting on A, it is sufcient to assign where the basis vectors ei are sent by and to make sure that they stay linearly independent: that is, if m1(e1) + + mn(en) = 0, with mi

Fp, then

all mi = 0. In order to calculate |GLn(p)|, we just need to

nd to how many different bases the initial basis {e1, . . . , en}

can be mapped to. The rst vector, e1, can be sent to pn 1

vectors, the second vector, e2, can then be sent to pn p

vectors linearly independent with (e1), and so forth. The result is

GLn(p) = pn 1 pn p pn pn1

= p

n(n1)

2 (p 1) p2 1 pn 1 . (10)

In particular, |Aut(

Zp

Zp)| = |GL2(p)| = p(p 1)(p2

1), and the p-subgroup of Aut(Zp

Zp) can only be Zp.

3.6 Nilpotent groups

In group theory, a powerful tool to investigate structure and properties of groups is to establish existence of subgroup series with certain properties. For example, a nite collection of normal subgroups Ni [triangleleft] G is called a normal series for G if

1 = N0 N1 N2 Nr = G. (11)

Restricting the properties of the factor groups Ni/Ni1 for

all i, one can infer non-trivial consequences for the group G.

If all the factor groups in the normal series lie in the centers, Ni/Ni1 Z(G/Ni1) for 1 i r, then (11) be

comes a central series, and the group G is then called nilpo-tent. The smallest 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). A non-abelian group G whose

factor group by its center G/Z(G) gives an abelian group is a nilpotent group of class 2, etc. So, nilpotent groups are often regarded as close relatives of abelian groups in the class of non-abelian ones. One important class of nilpotent groups is p-groups, i.e. nite groups whose order is a power of a prime p.

Nilpotent groups bear several remarkable features. We mention here only two of them which we will use below. First, a nilpotent group has a normal self-centralizing, and therefore maximal, abelian subgroup (Lemma 4.16 in [49]), whose implications were discussed above. Second, if H is a proper subgroup of a nilpotent group G, then H is also a proper subgroup of NG(H) (Theorem 1.22 in [49]). In other words, the only subgroup of a nilpotent group G which happens to be self-normalizing is the group G itself.

3.7 Solvable groups

A group G is called solvable if it has a normal series (11) in which all factor groups Ni/Ni1 are abelian. This is a

broader denition than the one of nilpotent groups. Therefore we can expect that both criteria and properties of solvable groups will be weaker than for nilpotent groups.

One particular example is that unlike nilpotent groups, a solvable group does not have to possess 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 rst introduce another series of nested subgroups, called the derived series. We rst nd G , the derived subgroup of G, then we nd its derived subgroup,

G = (G ) , then the third derived subgroup, G(3) = (G ) ,

and so on. The derived series is simply

G(3) G G G. (12)

The relation of the derived series with solvability is the following: G is solvable if and only if its derived series terminates, i.e. G(m) = 1 for some integer m 0 (Lemma 3.9

in [49]). The basic idea behind the proof of this statement is the observations that G is the unique smallest normal subgroup of G with an abelian factor group. Indeed, if N [triangleleft] G and : G G/N is the canonical homomorphism, then

(G ) = (G/N) (commutators are mapped into commu

tators). If we want G/N to be abelian, then (G/N) = 1,

and G ker = N. Therefore, whatever Nr1 we choose

in (11), it will contain G . This argument can be continued

Page 8 of 25 Eur. Phys. J. C (2013) 73:2309

through the series, and since the normal series terminates, so does the derived series.

Now, since G(m) = 1 for some nite m, we can consider

G(m1). It is an abelian group because its derived subgroup is trivial. Being a characteristic subgroup of G(m2), it is denitely normal 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 it to a maximal abelian subgroup, but then it is not guaranteed to be normal. Thus, in order to use the result (9), we need to prove the existence of an abelian subgroup which combines both properties. This situation is not generic: a solvable groups does not have to possess a normal maximal abelian subgroup. However, it can possess it in certain cases, and we will show below that in what concerns nite symmetry groups in 3HDM, they do contain such a subgroup.

4 Structure of the nite symmetry groups in 3HDM

4.1 Abelian subgroups and Burnsides theorem

Our goal is to understand which nite groups G can be realized as Higgs-family symmetry groups in the scalar sector of 3HDM. We stress that we look for realizable groups only, see discussion in Sect. 2.3.

Since nite groups have abelian subgroups, it is natural rst to ask which abelian subgroups G can have. This can be immediately inferred from our paper [37] devoted to abelian symmetry groups in NHDM. In the particular case of 3HDM, only the following groups can appear as abelian subgroups of a nite realizable symmetry group G:

Z2, Z3, Z4, Z2

Z2, Z3

Z3. (13)

The rst four are the only realizable nite subgroups of maximal tori in PSU(3). The last group, Z3

Z3, is on its own a maximal abelian subgroup of PSU(3), but it is not realizable because a Z3

Z3-symmetric potential is automatically symmetric under (Z3

Z3) Z2, see explicit expressions below. However, since it appears as an abelian subgroup of a nite realizable group, it must be included into consideration. Trying to impose any other abelian Higgs-family symmetry group on the 3HDM potential unavoidably makes it symmetric under a continuous group.

Let us rst see what order the nite (non-abelian) group G can have. We note that the orders of all abelian groups in (13) have only two prime divisors: 2 and 3. Thus, by Cauchys theorem, the order of the group G can also have only these two prime divisors: |G| = 2a3b. Then according

to Burnsides paqb-theorem the group G is solvable (Theorem 7.8 in [49]), and this means that G contains a normal abelian subgroup, which belongs, of course, to the list (13).

In order to proceed further, we need to prove that one can in fact nd a normal maximal (that is, self-centralizing) abelian subgroup of G, a property which is not generic to solvable groups but which holds in our case.

4.2 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 < CG(A), then there exists another abelian subgroup B > A, which is normal and self-centralizing in G.

Suppose that A < CG(A). Then for every b CG(A)\A,

the group Ab = A, b is an abelian subgroup of G, which

properly contains A. Figure 1 should help visualize embedding of various abelian subgroups of this kind in CG(A).

Note that CG(A) can be non-abelian. There are two possibilities compatible with the list (13):

(i) A =

Z2, and then Ab can be either Z2

Z2 or Z4,

(ii) A =

Z3, and then Ab =

Z3

Z3.

Thus CG(A) is either a 2-group or a 3-group. Below we assume that p = 2 if CG(A) is a 2-group, and p = 3 if CG(A)

is a 3-group.

Since CG(A) is a p-group, it is nilpotent, and according to discussion in Sect. 3.6, it possesses a normal maximal abelian subgroup B (which of course can be represented as Ab for some b), while B properly includes A =

Zp:

A < B CG(A). In particular, B is self-centralizing in

CG(A), so according to our discussion in Sect. 3.4, the factor group CG(A)/B is a subgroup of Aut(B). If B = CG(A),

then CG(A) is abelian and, being a centralizer of a normal subgroup, it is normal in G. Clearly B CG(B)

CG(A) = B, therefore CG(A) is the desired normal abelian

self-centralizing subgroup of G.

Assume now that B = CG(A):

A < B = CCG(A)(B)

=CG(B)

< CG(A) < G. (14)

Fig. 1 Illustration of CG(A) and some of its subgroups

Eur. Phys. J. C (2013) 73:2309 Page 9 of 25

The illustration in Fig. 1 refers to this case. Since B is an abelian subgroup of G, it must be in list (13). So, either B =

Zp

Z4, so H is the desired normal self-centralizing subgroup of G. If G = Q8 is quaternion then, as we describe

in Sect. 6.3.3, trying to impose a Q8 symmetry group on the 3HDM potential will result in a potential symmetric under a continuous group. Thus, this situation cannot happen if we search for nite realizable groups G. Note that this feature is purely calculational and does not rely on the existence of a normal maximal abelian subgroup which we prove here.

In the case p = 3, we nd that CG(A) is a non-abelian

group of order p3 = 27 and exponent 3, i.e. for every

g CG(A) we have g3 = 1. It is non-abelian and cannot

contain elements of order 9 because (13) does not contain abelian groups of orders 9 or 27.

In this case we do not yet know whether B is normal in G, but it is denitely normal in its own normalizer B [triangleleft] NG(B) G. Moreover CG(A) NG(B), since B

is normal in CG(A). These relations are visualized by the following relations:

B [triangleleft] CG(A) NG(B) G < PSU(3). (15) We can then consider the factor group NG(B)/B. We know that B =

Z3

Summarizing the group-theoretic part of our derivation, we proved that any nite group G which can be realized as a Higgs-family symmetry group in 3HDM is solvable, and in addition it 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 all the ve candidates for A, whose explicit realization were already given in [37], and by means of direct calculations see which extension can work in 3HDM.

5 Detecting continuous symmetries

Before we embark on analyzing each particular abelian group and its extensions, let us discuss an important issue. In this paper, we focus on discrete symmetries of the scalar sector in 3HDM. The symmetry groups we study must be realizable, that is, we need to prove that a potential symmetric under a nite 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 any continuous symmetry.

In principle, it would be desirable to derive a basis-invariant criterion for existence or absence of a continuous symmetry. Such condition is known for 2HDM [3236], while for the more than two doublets a necessary and sufcient condition is still missing. However, in certain special but important cases it is possible to derive a sufcient condition for absence of any continuous symmetry. Since this method relies on the properties of the orbit space in 3HDM, we start by briey describing it.

5.1 Orbit space in 3HDM

The formalism of representing the space of electroweak-gauge orbits of Higgs elds via bilinears was rst developed for 2HDM [3236, 5052], and then generalized to N doublets in [20]. Below we focus on the 3HDM case.

The Higgs potential depends on the Higgs doublets via their gauge-invariant bilinear combinations ab, a, b =

1, 2, 3. These bilinears can be organized into the following real scalar r0 and real vector ri, i = 1, . . . , 8:

r0 =

(11) + (22) + (33)

3 ,

r3 =

Zp or B =

Zp2 (the last case occurs only if p = 2), and in any of these cases we obtain |B| = p2.

Now, recall that CG(A) is a p-group, and so is CG(A)/B. If B =

Zp

Zp, then CG(A)/B is a p-subgroup of GL2(p), in particular, |CG(A)/B| = p. If B =

Zp2 , then CG(A)/B is a p-subgroup of Aut(Zp2 ). Since (p2) = p(p 1), it follows

that |CG(A)/B| = p. So in any case we have |CG(A)| = p3.

Now the arguments depend on p.

In the case p = 2, we see that CG(A) is a non-abelian

group of order 8. Thus CG(A) is either dihedral group D8 or the quaternion group Q8. If CG(A) is dihedral, then it possesses the unique (and hence characteristic) subgroup H =

Z3 is a maximal abelian group in PSU(3) [37]; therefore it is self-centralizing in PSU(3) and, consequently, in G and in its subgroup NG(B).

Then, in particular, we see that NG(B)/B is a subgroup of Aut(B) = GL2(3). Moreover, the analysis which will

be exposed in detail in Sect. 7 allows us to state that NPSU(3)(B)/B = SL2(3), so NG(B)/B is a subgroup of

SL2(3). We show in Sect. 7 that one cannot use elements of order 3 from SL2(3) because the potential will then become invariant under a continuous symmetry group. Therefore, NG(B)/B cannot have elements of order 3, which implies that B is a Sylow 3-subgroup of NG(B). The same statement holds for every group that lies between NG(B) and B, in particular, to CG(A). This contradicts the fact that |CG(A) : B| = 3 and CG(A)

NG(B). So this case is impossible.

(11) (22)

2 ,

r8 =

(11) + (22) 2(33)

23 ,

Page 10 of 25 Eur. Phys. J. C (2013) 73:2309

r1 = Re 12 , r2 = Im 12 , r4 = Re 31 , r5 = Im 31 , r6 = Re 23 , r7 = Im 23 .

(16)

The last six components can be grouped into three complex coordinates:

r12 = 12 = r1 + ir2,

r45 = 31 = r4 + ir5,

r67 = 23 = r6 + ir7.

(17)

It is also convenient to dene the normalized coordinates ni = ri/r0. The orbit space of the 3HDM is then represented

by an algebraic manifold lying in the 1 + 8-dimensional Eu

clidean space of r0 and ri and is dened by the following (in)equalities [20]:

r0 0,

n2 1, 3dijkninjnk =

3

n2 1

2 , (18)

where dijk is the fully symmetric SU(3) tensor. It can also be derived that | n| is bounded from below:

n2 = ,

14 1. (19)

The value of parametrizes SU(3)-orbits inside the orbit space. In particular, we will use this relation below when substituting r23 + r28 by r20 |r12|2 |r45|2 |r67|2.

Any U(3) transformation in the space of doublets 1, 2, 3 leaves r0 invariant and induces an SO(8) rotation of the vector ri. Note that this map is not surjective, namely not every SO(8) rotation of ri can be induced by a U(3) transformation in the space of doublets. Therefore, unlike in 2HDM, we do not expect the orbit space of 3HDM to be SO(8)-symmetric, and the last condition in (18) stresses that.

Let us take a closer look at the (n3, n8)-subspace. It follows from (18) that the orbit space intersects this plane along the equilateral triangle shown in Fig. 2. Its vertices P , P ,

P lie on the neutral manifold, which satisfy the condition

n2 = 1 and which would correspond to the neutral vacuum

if the minimum of the potential were located there, while the line segments joining them correspond to the charge-breaking vacuum, see details in [20]. The orbit space in this plane clearly lacks the rotational symmetry and has only the symmetries of the equilateral triangle.

5.2 Absence of continuous symmetries

The convenience of the formalism of bilinears is that the most general Higgs potential becomes a quadratic form in this space:

V = M0r0 Miri +

1

200r20 + 0ir0ri +

1

2ijrirj . (20)

Fig. 2 The orbit space of 3HDM in the (n3, n8)-subspace (all other ni = 0). The outer and inner circles correspond to | n| = 1 and | n| = 1/2, respectively

The real symmetric matrix ij has eight real eigenvalues (counted with multiplicity). In order for the potential to be symmetric under a continuous group of transformations, ij must have eigenvalues of multiplicities > 1. Note that any statement about eigenvalues of ij is basis-invariant and therefore it can be checked in any basis. Furthermore, if we nd a basis in which ij has a block-diagonal form, and if eigenvalues from different blocks are distinct, then a continuous symmetry requires that each block is either invariant under this symmetry, or contains eigenvalues with multiplicity >1.

Let us consider an important special case of this situation. Suppose that the potential has no terms of type (aa)(bc), where a, b, c are all distinct. This implies the absence of terms r0,3,8r1,2,4,5,6,7, and the block-diagonal form of ij , in which two blocks correspond to the (r3, r8) subspace and to its orthogonal complement. Suppose also that the eigenvalues of ij in the (r3, r8) subspace are distinct from those in the orthogonal complement. It follows then that any possible continuous symmetry must act trivially in the (r3, r8) subspace, because the orbit space here lacks the rotational invariance. However, if r0, r3, and r8 are xed, then 11, 22, and 33 are also xed. So, the doublets do not mix, and the possible continuous symmetry group can only be a subgroup of the group of pure phase rotations, which were studied in [37].

If in addition it is known that a given potential is not symmetric under continuous phase rotations, then we conclude that it does not have any continuous symmetry from PSU(3). It turns out that all the cases of various nite symmetry groups we consider below, except the last one, are of this type. Since the arguments of this section provide a sufcient condition for absence of continuous symmetries, they guarantee that the corresponding potentials can have only nite symmetry groups. Absence of a continuous symmetry in the very last case will be proved separately.

Eur. Phys. J. C (2013) 73:2309 Page 11 of 25

6 Possible extensions: the torus chain

We now check all the candidates for A from the list (13) and see which extension can work in 3HDM. In this section we will deal with the rst four groups from the list, which arise as subgroups of the maximal torus; the last group will be considered later. For each group A, we use its explicit realization given in [37] as a group of rephasing transformations, and then we search for additional transformations from PSU(3) with the desired multiplication properties.

6.1 Representing elements of PSU(3)

Before we start analysis of each case, let us make a general remark on how we describe the elements of PSU(3).Using the bar notation for the canonical homomorphism SU(3) PSU(3), we denote H < PSU(3) if its full preim

age in SU(3) is H . Denoting the center of SU(3) as Z =

Z(SU(3))

Z3, we have Z = {1, z, z2}, wherez = diag(, , ), = e2i/3. (21)

The elements of the group H (a, b, . . . H ) will be writ

ten as 3 3 matrices from SU(3). The elements of H

(, b, . . . H ) are the corresponding cosets of Z in H . Ex

plicit manipulation with these cosets is inconvenient, therefore in our calculation we represent an element PSU(3)

by any of the three representing elements from SU(3): a, az, or az2. We will usually choose a and then prove that this representation is faithful (does not depend on the choice of representing element).

6.2 Extending Z2 and Z3

The smallest group from the list is A =

Z2, whose automorphism group is Aut(Z2) = {1}, so that G =

Z2. This case

the center, and therefore all of them correspond to the same generator from PSU(3). It is straightforward to check that

selecting a to represent is a faithful representation.

The explicit solution of the matrix equation ab = ba2

shows that b SU(3) must be of the form

b =

0 ei 0

ei 0 0

0 0 1

, (23)

with an arbitrary . The choice of the mixing pair of doublets (1 and 2 in this case) is xed by the choice of invariant doublet in a.

The fact that b is not uniquely dened means that there exists not a single D6 group but a whole family of D6 groups parametrized by the value of . Below, when checking whether a potential is D6 symmetric, we will need to check its invariance under all possible D6s from this family.

The generic Z3-symmetric potential contains the part invariant under any phase rotation

V0 =

1i3

m2i

i i +

1ij3

ij

i i jj

+

1i<j3

ij

i j ji ,

and the following additional terms:

VZ3 = 1 21 31 + 2 12 32

+3 13 23 + h.c. (24)

with complex 1, 2, 3. At least two of them must be nonzero, otherwise the potential will be symmetric under a continuous group of Higgs-family transformations [37]. Let us denote their phases by 1, 2, and 3, respectively. If the parameters of V0 satisfy

m211 = m222, 11 = 22, 13 = 23, 13 = 23,

(25)

was already considered in [37].

The next possibility is A =

Z3, whose Aut(Z3) =

Z2. The

only non-trivial case to be considered is G/A =

Z2, which

implies that G can be either Z6 or D6 S3, the symmetry

group of the equilateral triangle. The former can be disregarded because it does not appear in the list (13), thus we focus only on the D6 case.

6.2.1 Constructing D6

The group D6 is generated by two elements a, b with the following relations: a3 = 1, b2 = 1, ab = ba2. Following

[37], we represent the Z3 group by phase rotations:

a = diag , 2, 1 . (22)

There are in fact three such groups which differ only by the choice of the doublet which is xed. However, their generators, a, az, and az2, differ only by a transformation from

and if, in addition, |1| = |2|, then the whole potential be

comes symmetric under one particular D6 group constructed with b in (23) with the value of = (2 1 + )/3 +

2k/3. The extra freedom given by 2k/3 corresponds to three order-two elements of D6: b, ab, a2b. We opt to dene b by setting k = 0. Alternatively, we can be compactly

write the condition as

3 = 1 + 2. (26)

To summarize, the criterion of the D6 symmetry of the potential is that, after a possible doublet relabeling, conditions (25) and (26) are satised.

Page 12 of 25 Eur. Phys. J. C (2013) 73:2309

Let us also note that when constructing the group D6 we could have searched for b satisfying not ab = ba2 but

ab = ba2 zp, with p = 1, 2. Solutions of this equation ex

ist, but they do not lead to any new possibilities. Indeed, let us introduce a = azp. Then, we get a b = ba 2. Thus, we

get the same equation for b as before, up to a cyclic permutation of doublets, the possibility which we already took into account.

6.2.2 Proving that D6 is realizable

This construction allows us to write down an example of the D6-symmetric potential: it is V0 restricted by conditions (25) plus VZ3 in (24) subject to |1| = |2|. In order to

show that D6 is realizable, we need to demonstrate that this potential is not symmetric under any larger Higgs-family transformation group.

This proof is short and contains two steps. First, we note that the conditions described in Sect. 5 are fullled: the (r3, r8)-subspace does not couple to its orthogonal complement via ij , and that the eigenvalues in these two subspaces are dened by different sets of free parameters. The extra terms (24) guarantee that there is only nite group of phase rotations, the group Z3. Therefore, the sufcient conditions described in Sect. 5 are satised, and the generic D6-symmetric potential has no continuous symmetry.

Second, we need to show that the generic D6-symmetric potential has no higher discrete symmetries. This is proved by the simple observation that all other nite groups to be discussed below which could possibly contain D6 lead to stronger restrictions on the potential than (25) and

|1| = |2|. Therefore, not satisfying those stronger restric

tions will yield a potential symmetric only under D6.

6.2.3 Including antiunitary transformations

Any generalized-CP (antiunitary) transformation acting on three doublets is of the form

J = c J, c PSU(3). (27)

Here J is the operation of hermitian conjugation of the doublets. If G is the symmetry group of unitary transformations, then it is normal in G, J , and J induces automorphisms

in G. So, when we search for J , we require that

J 2 G, J 1aJ G, (28) where a generically denotes the generators of G. If such a transformation is found, the group is extended from G to G Z2, where the asterisk on the group indicates that its generator is antiunitary.

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 Z2, with

G > G. But at the end of this paper we will have a complete list of all nite realizable symmetry groups of unitary transformations, and this list will contain G anyway. So, this pos

sibility is not overlooked but will be studied in its due time after construction of G.

Now, turning to extension of D6 by an antiunitary symmetry, we rst note that the resulting group D6 Z2 is a non-abelian group of order 12 containing a normal subgroup D6.

Among the three non-abelian groups of order 12, there exists only one group, namely D6

Z2, with a subgroup D6 (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 order 12. Note that it contains, among other, the subgroup Z6; its presence does not contradict the list (13) because that list refers only to the groups of unitary transformations.

Next, let us denote the generator of Z2 by J = cJ . Since

J centralizes the entire D6, it follows that (J )1aJ = a,

(J )1bJ = b, and (J )2 = cJ cJ = cc = 1. The matrix c

satisfying these conditions must be of the form

c =

0 ei 0

ei 0 0

0 0 e2i

, (29)

with arbitrary . Requiring the potential to stay invariant under J , we obtain the following conditions on : 6 =

2(1 + 2) = 23. Therefore, if the following extra con

dition is fullled:

2(1 + 2 + 3) = 0 (30)

the D6-invariant potential becomes symmetric under the group D6

Z2. If this condition is not satised, the symmetry group remains D6 even in the case when antiunitary transformations are allowed. We conclude that both D6 and

D6

Z2 are realizable in 3HDM.

It is interesting to note that if we set 3 = 0, then the po

tential would still be invariant under D6. However, in this case it becomes symmetric under J with 6 = 2(1 +

2), without any extra condition on 1 and 2, and the potential becomes automatically invariant under D6

Z2. So,

we conclude that the fact that D6 is still realizable even if antiunitary transformations are included is due to the special feature of the Z3-symmetry: we have three, not two terms in the Z3-symmetric potential, and it is the third term that prevents an automatic antiunitary symmetry.

6.3 Extending Z4

Let us now take A =

Z4 generated by a. Then Aut(Z4) =

Z2,

so that G =

Z4 . Z2 generated by a and some b /

Z4. The

Eur. Phys. J. C (2013) 73:2309 Page 13 of 25

two non-abelian possibilities for G are the dihedral group D8 representing symmetries of the square, and the quaternion group Q8. In both cases b1ab = a3, with the only

difference that b2 = 1 for D8 while b2 = a2 for Q8. Note

that extension leading to the dihedral group is split, D8 =

Z4 Z2, while Q8 is not.

6.3.1 Constructing D8

Representing a by phase rotations a = diag(i, i, 1), we

nd that b satisfying these conditions is again of the form (23) with arbitrary . However, now we do not have the freedom to choose the pair of doublets which are mixed by b: this pair is xed by a. Also, unlike the Z3 case, the matrix equation ab = ba3 z does not have solutions for b SU(3).

The Z4-symmetric potential (for this choice of a) is V0 +

VZ4 , where

VZ4 = 1 31 32 + 2 12 2 + h.c. (31)

The phases of 1 and 2 are, as usual, denoted as 1 and 2, respectively. Upon b, the rst term here remains invariant, while the second term transforms as

12 2 e4i 21 2. (32)

This means that the potential (31) is always symmetric under (23) provided that we choose

= 2/2. (33)

Therefore, in order to get a D8-symmetric potential we only require that V0 satises conditions (25). The proof that D8 is realizable (as long as only unitary transformations are concerned) follows along the same lines as in Sect. 6.2.2.

6.3.2 Including antiunitary transformations

In [37] we found that exactly the same conditions, namely (25) and (33), must be satised for existence of an antiunitary transformation commuting with the elements of Z4.

This transformation is again J = cJ , where c is given by

(29) with 6 = 21, and it commutes with all elements

of D8. Therefore, if we include antiunitary transformations, we automatically get the group D8

By checking how VZ4 in (31) transforms under it, we nd that the rst term simply changes its sign. The only way to make the potential symmetric under Q8 is to set 1 = 0. But

then we know from [37] that the potential becomes invariant under a continuous group of phase rotations. Therefore, Q8 is not realizable.

6.4 Extending Z2

Z2

If A =

Z2

Z2, then Aut(Z2

Z2) = GL2(2) = S3. The

group Z2

Z2 can be realized as the group of independent sign ips of the three doublets with generators a1 =

diag(1, 1, 1) (equivalent to the sign ip of the rst dou

blet) and a2 = diag(1, 1, 1) (equivalent to the sign ip

of the second doublet), so that a1a2 is equivalent to the sign ip of the third doublet. The potential symmetric under this group contains V0 and additional terms

VZ2

Z2

=

12

12 2 +

23

23 2 +

31

31 2 + h.c. (35)

with at least two among coefcients

ij being non-zero. The coefcients can be complex; as usual we denote their phases as ij . This model is also known as Weinbergs 3HDM [6].

The non-abelian nite group G can be constructed as extension of A by Z2, by Z3, or by S3.

6.4.1 Extension (Z2

Z2) . Z2

Consider rst the extension (Z2

Z2) . Z2. The only extension leading to a non-abelian group is (Z2

Z2) . Z2 = D8,

and we already proved that this group is realizable. Nevertheless, we prefer to explicitly work it out to see the reduction of free parameters.

The element b which we search for must act on {a1, a2,

a1a2} as a transposition of any pair. In addition, b2

Z2

Z2. It does not matter which pair of generators is transposed, as this choice can be changes by renumbering the doublets. So, we take b such that b1a1b = a2 and

b1a2b = a1. Then, b2 can be either 1 or a1a2, because

choices b2 = a1 or a2 lead to inconsistent relations. Indeed,

if we assume b2 = a1, then

a2 = b1a1b = b1b2b = b2 = a1,

which is a contradiction. In both cases (b2 = 1 and b2 =

a1a2) we get the group D8. Even more, we get the same D8 group: if b2 = a1a2, then b = ba1 satises b 2 = 1, while its

action on a1 and a2 remains the same. So, it is sufcient to focus on the b2 = 1 case only.

Again, explicitly solving the matrix equations, we get b of the form (23) with arbitrary . Then, we check how the potential (35) changes upon b and nd that we need to set

Z2, while D8 becomes non-realizable. Note that the resulting group does not contain Z8. Indeed, we showed in [37] that imposing Z8 symmetry group leads to a potential with continuous symmetry.

6.3.3 Attempting at Q8

Solving matrix equations ab = ba3 and b2 = a2, we get the

following form of b:

b(Q8) =

0 ei 0

ei 0 0 0 0 1

. (34)

Page 14 of 25 Eur. Phys. J. C (2013) 73:2309

4 = 212, 2 = (23 + 31), |

23| = |

31|.(36)

Equations on the phase can be satised if

2(12 + 23 + 31) = 0 Im(

12

23

31) = 0. (37)

So, if: (1) this condition is satised, (2) two among |

ij| are

equal, (3) condition on V0 (25) is satised, then the potential is D8-symmetric. Note also that if

12 = 0 (which we are

allowed to consider because (35) contains three rather than two terms), then condition on the phases is not needed.

It might seem that these conditions on the potential to make it D8-symmetric are more restrictive than in the Z4 extension we studied above. However, note that the Z2

Z2-

V0 = m2 11 + 22 + 33

+ 11 + 22 + 33 2

+ 11 22 + 22 33

+ 33 11

+ |12|2 + |23|2 + |31|2 . (41)

6.4.3 Constructing (Z2

Z2) S3 = O

The last extension, (Z2

Z2) . S3, is also split, otherwise we would obtain Z6. It leads to the group O S4, the symme

try group of the octahedron and the cube. As it includes T as a subgroup, the most general O-symmetric potential is V0 from (41) plus VT from (40) with the additional condition that

symmetric potential (35) has six free parameters, and we placed two conditions to reduce the number of free parameters in the D8 potential to four (apart from V0). On the other hand, (31) had only four from the beginning, and without any restriction this number survives. Therefore we have the same number of degrees of freedom when constructing D8 in either way.

6.4.2 Constructing (Z2

Z2) Z3 = T

The extension by Z3 is necessarily split, (Z2

is real (the extra symmetry with respect to the T -symmetric case is a transposition of any two doublets).

6.4.4 Including antiunitary transformations

The case of D8 has been already considered in Sect. 6.3.2.

The tetrahedral potential VT + V0 from (40) and (41) is

symmetric under the following antiunitary transformation:

J =

Z2) Z3,

leading to the group T A4, the symmetry group of the

tetrahedron. To construct it, we need b such that b3 = 1 with

the property that b acts on {a1, a2, a1a2} by cyclic permuta

tions. Fixing the order of permutations by b1a1b = a2, we

nd that b must be of the form

b =

0 1 0 1 0 0 0 0 1

0 ei1 0

0 0 ei2

ei(1+2) 0 0

, (38)

with arbitrary 1, 2. It then follows that if the coefcients in (35) satisfy

|

12| = |

23| = |

31|, (39)

then VZ2Z2 is symmetric under one particular b with

1 =

212 31 236 , 2 =

J, (42)

which generates a Z2 group. Therefore the symmetry group of this potential is the full achiral tetrahedral group Td

T Z2, which is isomorphic to S4.

The octahedral potential is a particular case of the tetrahedral one, therefore it is also invariant under an antiunitary transformation. The extra Z2 subgroup is generated by the complex 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 symmetry group Oh O

Z2.

223 31 12 6 .

Then, by a rephasing transformation one also make the phases of all

ij equal and bring (35) to the following form:

VT =

12 2 + 23 2 + 31 2 + h.c. (40) with a complex

. In this form, the parameters 1 = 2 = 0,

and the matrix b is just the cyclic permutation of the doublets. In addition, the symmetry under b places stronger conditions on the parameters of V0, so that the most general V0 satisfying them is

6.5 Extensions of abelian groupsby an antiunitary transformation

The last type of extension we need to consider is of the type A . Z2, where A is one of the four abelian groups of

Higgs-family transformations lying in a maximal torus, that is, the rst four groups in the list (13), while the Z2 is as usual generated by an antiunitary transformation J = cJ .

This problem was partly solved in [37], where such extensions leading to abelian groups were analyzed. It was established that only the following four abelian groups of this type are realizable: Z2, Z4, Z2

Z2, and Z2

Z2

Z2.

Here, we consider non-abelian extensions of this type.

Eur. Phys. J. C (2013) 73:2309 Page 15 of 25

6.5.1 Anti-unitary extension of Z3

The smallest non-abelian group we can have is Z3

Z2 D6. We stress that this D6 group we search for

is different from what we analyzed in Sect. 6.2, because there the D6 group contained only unitary transformations, see a discussion in Sect. 8.3. Using the same notation for the generator a of the Z3 group, we nd that the transformation c in the denition of J must be diagonal: c = diag(ei1, ei2, ei(1+2)). Then, studying how the

Z3-

Second, its automorphism group Aut(Z3

Z3) is sufciently large and requires an accurate description.

Let us rst remind the reader how this group is constructed. We rst consider the subgroup of SU(3) generated by

a =

1 0 0 0 0 0 0 2

, b =

0 1 0 0 0 1 1 0 0

symmetric potential V0 + VZ3 changes under J = cJ , we

nd that the only condition to be satised is (30).

If this condition is satised, then the potential is invariant under Z3 Z2 D6, if not, then the symmetry group

remains Z3. This proves that both groups are realizable in 3HDM. Note that in contrast with the D6

. (43)

This group known as (27) is non-abelian because a and b do not commute, but their commutator lies in the center of SU(3):

[a, b] = aba1b1 = z2 Z SU(3) . (44)

Therefore, its image under the canonical homomorphism SU(3) PSU(3) becomes the desired abelian group

(27)/Z3 =

Z3

Z3. The true generators of Z3

Z3 are

cosets = aZ(SU(3)) and b = bZ(SU(3)) from PSU(3),

and they obviously commute: [, b] = 1. Note that since

Z3

Z3 is a maximal abelian subgroup in PSU(3), there is no other element in PSU(3) commuting with all elements of this group, so CPSU(3)(Z3

Z3) =

Z3

Z3.

Z2 case, we do not place any extra condition such as (25).

6.5.2 Anti-unitary extension of Z4

A priori, the two non-abelian extensions here are again D8 and Q8. With the usual convention for a, the generator of

Z4, we again nd that c must be of the same diagonal form. This immediately excludes the Q8 case because we have (J )2 = cc = 1.

The case of Z4 Z2 D8 is possible. Even more, it turns

out that the Z4-symmetric potential V0 +VZ4 is always sym

metric under some J of this type. It means, therefore, that if anti-unitary transformations are included, Z4 is not realizable anymore: the true symmetry group of the potential is

Z4 Z2 D8. In more physical terms, we conclude that

the presence of a Z4 group of Higgs-family transformations makes the potential explicitly CP-conserving.

6.5.3 Anti-unitary extension of Z2

Z2

If the normal self-centralizing abelian subgroup of G, whose existence was proved in Sect. 4.2, is A =

Z3

Z3,

then G can be constructed as an extension of A by a subgroup of Aut(Z3

Z3) = GL2(3), the general linear group

of transformations of two-dimensional vector space over the nite eld F3. The order of this group is |GL2(3)| = 48, and

it will prove useful if we now digress and describe the structure of this group in some detail.

7.1.1 Z3

Z2) . Z2

can produce only D8, which was already considered. We only remark here that c turns out to be of the type (29), which places extra constraints on V0. Not satisfying these constraints will keep the symmetry group Z2

Z2, which

The only non-abelian extension of the type (Z2

Z3 as a vector space over F3

The nite eld F3 is dened as the additive group of integers mod 3, in which the multiplication is also introduced. It is convenient to denote the elements of this eld as 0, 1, 1

with obvious addition and multiplication laws. Unlike the integers themselves, F3 is closed under division by a nonzero number, the property that makes F3 a eld.

A vector space over a nite eld is dened just as over any usual eld. The group Z3

means that it is realizable.

7 The Z3

Z3 can be thought of as a 2D vector space over F3; its elements are (with the additive notation for the group operation) x = qa + qb b,

where qa, qb

F3, and, b are, as before, the generators of

the group Z3

Z3. In the multiplicative notation, we write

Z3 chain

7.1 The group and its extensions

The last abelian group from the list (13), Z3

Z3, requires a special treatment due to a number of reasons. First, it does not belong to any maximal torus of PSU(3) but is a maximal abelian subgroup of PSU(3) on its own [37], and its full preimage in SU(3) is the non-abelian group (27) [53].

x = qa bqb.

It is possible to dene an antisymmetric scalar product in this space. For any x

Z3

Z3, take any element of its preimage, x (27). Then, for any two elements x, y

Z3

Z3, construct the number ( x, y) as [x, y]

F3.

This map is faithful: although we can select different x for a given x, all of them give the same [x, y].

Page 16 of 25 Eur. Phys. J. C (2013) 73:2309

Clearly, ( x, y) = ( y, x), in the additive notation. Be

sides, the so dened product is linear in both arguments:

( x1 + x2, y) = ( x1, y) + ( x2, y), ( x, y1 + y2) = ( x, y1) + ( x, y2).

= g1[x, y]g = g1zrg

zr = ( x, y), if g is unitary,

= (z)r = (z1)r = ( x, y), if g is anti-unitary.

(45)

Indeed, for any three elements of any group the following relation holds:

[xy, z] = xyzy1x1z1 = xyzy1 z1x1xz x1z1

= x[y, z]x1[x, z]. (46) If in addition all commutators take values in the center of the group SU(N), then x and x1 can be canceled, and we get [xy, z] = [y, z][x, z]. In our case we represent x1 =

aqa1 bqb1zr1 and similarly for x2 and y, and noting that all

zri are inessential, we recover the above linearity in the rst argument. Thus, Z3

Z3 becomes a vector space over F3 equipped with an antisymmetric scalar product.

Note that all antisymmetric products in Z3

Z3 are proportional to (, b). Indeed, if two elements x and x are de

ned by their vectors

q = (qa, qb) and

q = (q a, q b), then

(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 1xJ = x. So, unitary transformations preserve the anti

symmetric product, while anti-unitary ones ip its sign.

Generically, the subgroup of a general linear group which conserves an antisymmetric bilinear product in a vector space is called symplectic. Here we have the group Sp2(3) <

GL2(3). It turns out that Sp2(3) = SL2(3). Indeed, sup

pose g GL2(3) acts in the 2D space over

due to bilinearity we get

x, x = qaq b qbq a (, b) = ijqiq j (, b), (47)

where ij is the standard antisymmetric tensor with 12 = 21 = 1, 11 = 22 = 0.

7.1.2 The automorphism group of Z3

Z3

F3 by mapping qi g(q) = gii qi . Then, the product transforms as( x, y) g( x), g( y) = ijgii gjj q(x)i q(y)j (, b)

= det g ( x, y). (50)

Since det g = 1, we get two kinds of transformations:

those which conserve all products (det g = 1, so that g

SL2(3)) and those which ip their signs (det g = 1), hence

the identication of Sp2(3) and SL2(3) follows.

We conclude that the nite symmetry group G of unitary transformations with the normal self-centralizing abelian subgroup Z3

Z3 can be constructed as extension (Z3

Z3) . K, where K SL2(3).7.1.3 Explicit description of SL2(3)

The structure of the group SL2(3) is well-known, but it will prove useful to have the explicit expressions for some of its elements.

The order of the group is |SL2(3)| = 24. It contains ele

ments of order 2, 3, 4, and 6, generating the corresponding cyclic subgroups. The subgroup Z2 is generated by the center of the group

c =

1 0

0 1

The automorphism group of Z3

Z3 can then be viewed as the group of non-degenerate matrices with elements from F3 acting in this 2D space, which explains why Aut(Z3

Z3) =

GL2(3). Each matrix q can be dened by its action on the generators, b: qaa + qab b, b qba + qbb b, and

can therefore be written as

q =

qaa qab qba qbb

, det q = 0. (48)

The group operation in GL2(3) is just the matrix product.Recall now that the elements of both the Z3

, (51)

which in the multiplicative notation means 2, b b2.

There are four distinct Z3 subgroups generated by

f1 =

1 1 0 1

, f2 =

1 0

1 1

,

Z3 group

and its automorphism group are represented in our case as unitary or antiunitary transformations of the three doublets (that is, we work not with the abstract groups but with their three-dimensional complex representations). Since Z3

Z3

, f4 =

0 1

1 1

,

(52)

f3 =

0 1

1 1

is assumed to be normal in G, the elements g Aut(

Z3

Z3)

act on the elements of Z3

Z3 by conjugation: x g1 xg,

which we denoted by g( x). Then the antisymmetric prod

uct dened above changes upon this action in the following way:

g( x), g( y)

three Z4 subgroups generated by

d1 =

0 1

1 0

, d2 =

1 1

1 1

,

(53)

,

d3 =

1 1

1 1

Eur. Phys. J. C (2013) 73:2309 Page 17 of 25

and four Z6 subgroups, which we do not write explicitly because they are absent in the list (13).

Every element of SL2(3) can be represented by a unique (up to center) SU(3) matrix, which can be found by explicitly solving the corresponding matrix equations dening the action of this element. For example, the transformation c is dened by

c(a) = c1ac = a2, c(b) = c1bc = b2. (54)

Rewriting these equations as 3 3 matrix equations ac =

ca2, bc = cb2 and solving them explicitly, we nd the ma

trix c:

c =

The rst eigenvalue corresponds to the subspace (r3, r8), while the rest are three 2D subspaces within its orthogonal complement (r1, r2, r4, r5, r6, r7). For generic values of the coefcients, they do not coincide. Then, according to our discussion in Sect. 5, a continuous symmetry group, if present, must consist only of phase rotations of the doublets. But the 3 term selects only the Z3 group of phase rotations, which proves that no continuous symmetry leaves this potential invariant.

7.3 Extension (Z3

Z3) Z2

It turns out that Z3

Z3 is not realizable because the potential (56) is symmetric under a larger group (Z3

Z3) Z2 =

(54)/Z3, which is generated by, b, c with the following

relations:

a3 = b3 = 1, c2 = 1, [, b] = 1,

c c = 2, c b c = b2.

In terms of explicit transformation laws, c is the coset

cZ(SU(3)), with c being the exchange of any two doublets, for example (55). Note that , c = S3 is the group

of arbitrary permutations of the three doublets. Thus, if G = (

Z3

Z3) . K, then a G-symmetric potential must be a restriction of (56), and K must contain a Z2 subgroup.

There are three kinds of subgroups of SL2(3) containing Z2 but not containing Z6: Z2, Z4, and Q8. In each case it would give a split extension, so G must contain a subgroup isomorphic to one of these groups. Since, as we argued above, the quaternion group Q8 is not realizable in 3HDM, K can only be Z2 or Z4. Therefore, the only additional case to consider is (Z3

Z3) Z4, the group also

known as (36) [53].

7.4 Extension (Z3

1 0 0

0 0 1

0 1 0

. (55)

7.2 Generic potential

A generic potential symmetric under Z3

Z3 is

V = m2 11 + 22 + 33

+0 11 + 22 + 33 2

+

13 11 2 + 22 2 + 33 2

11 22 22 33 33 11

+2 12 2 + 23 2 + 31 2

+ 3 12 13 + 23 21

+ 31 32 + h.c. (56) with real m2, 0, 1, 2 and complex 3. All values here are generic. This potential can be found by taking the potential symmetric under the Z3 group of phase rotations described above 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 = 3m2r0 + 30r20 + 31 r23 + r28

+2 |r12|2 + |r45|2 + |r67|2

+3 r12r45 + r67r12 + r45r67

+3 r12r45 + r67r12 + r45r67

= 3m2r0 + 30r20 + ijrirj . (57)

It is important to prove that this potential has no continuous symmetry. Using the approach described in Sect. 5, we calculate the eigenvalues of ij and nd that it has four distinct eigenvalues of multiplicity two:

31, 2 + 3 + 3, 2 + 3 + 23, 2 + 23 + 3.

(58)

Z3) Z4

There are three distinct Z4 subgroups in SL2(3) generated by d1, d2, and d3, listed in (53). In principle, all of them are conjugate inside SL2(3), but for our purposes all of them need to be checked. Explicit solutions of the matrix equations give the following transformations:

d1 =

i 3

1 1 1 1 2

1 2

,

1 1 1 1 2

,

d2 =

i 3

(59)

1 1 2 1

1

.

d3 =

i 3

Page 18 of 25 Eur. Phys. J. C (2013) 73:2309

Note that the prefactor i/3 can also be written as 1/(2 ).

Let us mention here that when searching for explicit SU(3) realizations of the transformations d1, we solve equations d11ad1 = b, d11bd1 = a2. However, we could also

use other representative matrices, a and b , which differ from a and b by transformations from the center. For example, we can also ask for solutions of

d 11ad 1 = zn1b, d 11bd 1 = zn2a2. (60)

However, the solution of this equation can be written as

d 1 = d1an1bn2. (61)

Therefore the resulting group d 1,, b coincides with d1,, b . The similar results hold for d2 and d3.

7.4.1 Conditions for the (Z3

Z3) Z4 symmetry

We should now check how the potential (56) changes under these transformations and when it remains invariant. The calculation is simplied if we introduce the following combinations of bilinears (here ij stands for ij ):

A0 = 11 + 22 + 33, A1 = 11 + 22 + 233, A2 = A1, B0 = 12 + 23 + 31,

B1 = 12 + 23 + 231, B2 = 12 + 223 + 31,

B0 = 21 + 32 + 13, B1 = 21 + 232 + 13, B2 = 21 + 32 + 213.

Next, introducing

X =

13 11 2 + 22 2 + 33 2 11 22

22 33 33 11 ,

=

0 3 3 3 3 2 1 1

3 1 1 2

3 1 2 1

,

0 3 23 3 3 2 2

3 1 22

23 2 2 1

,

T (d2) =

1 3

1 3|A1|2,

Y = 12 2 + 23 2 + 31 2 (62)

= |

B0|2 + |B1|2 + |B2|2

3 ,

Z = 12 13 + 23 21 + 31 32

= |

and T (d3) = [T (d2)]. It can be also noted that T (d2)

acts in the space of (X, Y, 2Z, Z) by the matrix T (d1). So, T (d1), T (d2) and T (d3) represent the same type of transformations acting in the spaces (X, Y, Z, Z), (X, Y, 2Z, Z), or (X, Y, Z, 2Z), respectively. That is, if (x, y, z, z) is an eigenvector of T (d1), then (x, y, z, 2z)

is an eigenvector of T (d2) and (x, y, 2z, z) is an eigenvector of T (d3). This observation restores the expected symmetry among the three types of Z4 subgroup inside SL2(3).

Since these matrices are hermitian and unitary, they act by pure reections, which implies that each of them is diagonalizable and has eigenvalues 1. If we want the potential to

be symmetric under one of these di, it must induce the same transformations in the space of i = (1, 2, 3, 3). There

fore, in order to nd conditions that the potential is invariant under di, we need to nd eigenvectors of T (di) corresponding to the eigenvalue 1 and require that is projection on

these eigenvectors is zero.

Consider rst T (d1). It has two eigenvectors corresponding to the eigenvalue 1: (3, 1, 1, 1) and (0, 0, 1, 1).

B0|2 + 2|B1|2 + |B2|2 3 ,

we write the potential (56) as

V = 3m2r0 + 30r20 + iXi,where iXi = 1X + 2Y + 3Z + 3Z (63)

is the scalar product of the vector of coefcients and the vector of coordinates. Now, it follows from explicit calculations that the action of di can be compactly represented by the following transformations:

d1: A1 B0, B0 A1,

B1 2B2, B2 B1, d2: A1 B1, B1 A1,

B0 B2, B2 B0, d3: A1 B2, B2 A1,

B0 B1, B1 B0,

or even more compactly

d1: |A1|2 |B0|2, |B1|2 |B2|2,d2: |A1|2 |B1|2, |B0|2 |B2|2, (64)

d3: |A1|2 |B2|2, |B0|2 |B1|2.

Therefore, their action in the space of (X, Y, Z, Z) is given by the following hermitian and unitary matrices:

T (d1) =

1 3

Eur. Phys. J. C (2013) 73:2309 Page 19 of 25

Therefore, we obtain the following condition for the potential to be symmetric under d1:

3 is real and 3 =

31 2

There exist only two elements in the algebra su(3) with this property:

t1 =

0 i i i 0 i i i 0

and t2 =

0 1 1 1 0 1 1 1 0

2 . (65)

Similarly, for d2 we have

3 is real and 3 =

31 2

2 . (66)

For d3 we have the complex conjugate condition. Therefore, the potential (56) is symmetric under (Z3

Z3) Z4 if

. (69)

t2 generates pure phase rotations. It is explicitly S3-invariant, therefore the corresponding U(1) group is also invariant. t1 induces SO(3) rotations of the doublets around the axis (1, 1, 1). It is Z3-invariant, while reections from S3 ip the sign of t1. However, the U(1) group is still invariant. Since t1 and t2 realize different representations of S3, one cannot take their linear combinations. So, the list of possibilities is restricted only to t1 and t2 themselves.

The eigenvalues and eigenvectors of t1 are

= 0:

23

3 = 1, (67)

which encompasses all these cases. Let us also mention that when these conditions are taken into account, the spectrum of the matrix ij given in (58) becomes even more degenerate: it contains two eigenvalues of multiplicity four (we refer to this spectrum as 4 + 4).

7.4.2 Absence of a continuous symmetry

In order for the group (Z3

31 2

1 1 1

, = 3:

1 2

,

(70)

= 3:

1 2

.

The presence of the eigenvalue = 0 implies that the com

bination 1 +2 +3 is invariant under the U(1) group gen

erated by t1. Bilinear invariants are

|1 + 2 + 3|2, 1 + 22 + 3 2,

1 + 2 + 23 2,

Z3) Z4 to be realizable, we need to show that the potential (56) with parameters satisfying (67) is not symmetric under any continuous group.

We rst note that even if such a continuous symmetry group existed, it could only be U(1). Indeed, the spectrum of ij in our case is 4+4, while for U(1)U(1) and SU(2)

it must be 6 + 2, and for SO(3) it must be 5 + 3.

Let us now consider, for example, the d1-symmetric potential. Using 8

i=1 r2i = r20, where 1/4 1 parametr

izes SU(3)-orbits in the orbit space, we can rewrite it as

V = 3m2r0 + (30 + 31)r20

31 22 |r12 r45|2 + |r45 r67|2

+ |r67 r12|2 . (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 invariant under the S3 group of arbitrary permutations of the doublets, then the same potential must be also invariant under other U(1) subgroups which are generated by various tg, which are obtained by acting on t by g S3. If t = tg (or to be more

accurate, if their corresponding U(1) groups are different), then the continuous symmetry group immediately becomes larger than U(1), which is impossible. Therefore, tg must be equal (up to sign) to t for all g S3. In other words, S3 must

stabilize the U(1) symmetry group.

(71)

which simply means that r1 + r4 + r6 and r2 + r5 + r7 are,

separately, invariant. So, if the potential depends only on r0 and these two combinations, then it is symmetric under the U(1) generated by t1. The point is that our potential (68)

cannot be written via these combinations only, therefore it is not invariant under this group.

Consider now t2. Its eigensystem is

= 2:

.

(72)

There is no zero eigenvalue, therefore no linear combination of s is invariant. The independent bilinear combinations are

|1 + 2 + 3|2, |2 3|2,

|21 2 3|2,

2 3 (21 2 3).

(73)

1 1 1

, = 1:

0

1

1

and

2

1

1

Page 20 of 25 Eur. Phys. J. C (2013) 73:2309

In addition, there exists a triple product of s which is also invariant but it is irrelevant for our analysis because our potential contains only two s and two s. These invariants can also be rewritten as the following linearly independent invariants (here i = ii):

1 + 2r6, 2 + 2r4, 3 + 2r1, r2 + r5 + r7.(74)

Despite the fact that we now have more invariants than in the previous case, it is still impossible to express (68) via these combinations. This means that (68) is not symmetric under t2.

This completes the proof that the potential (56) subject to conditions (67) is not invariant under any continuous group.

7.4.3 Absence of a larger nite symmetry group

Although the group-theoretic arguments guarantee that no other extension can be used, it is still instructive to check what happens if we try to impose invariance under other subgroups of SL2(3).

Let us rst note that if we try to impose simultaneous invariance under two among di (trying to get Q8), we must set 3 = 0. But then the potential has an obvious continuous

symmetry, and our attempt fails.

Next, let us assume that the potential is invariant under (Z3

The spectrum of ij becomes of the type 6 + 2. This high

symmetry hints at existence of a possible continuous symmetry of the potential, and it is indeed the case. For example, the following SO(2) rotations among three doublets, a Rab()b, leave r12 + r45 + r67 invariant:

R() =

1 3

1 + 2 cos 1 + 2 cos 1 + 2 cos

1 + 2 cos 1 + 2 cos 1 + 2 cos

1 + 2 cos 1 + 2 cos 1 + 2 cos

,

(79)

with [0, 2) and = + 2/3 and = + 4/3.

Note that at = 0, 2/3 and 4/3 we recover the

Z3

group b .

We conclude therefore that imposing invariance under Z3 < SL2(3) makes the potential symmetric under a continuous group. In this way, we completely exhausted possibilities offered by SL2(3).

7.5 Anti-unitary transformations

We showed in Sect. 7.1 that antiunitary transformations correspond to elements of GL2(3) not lying in SL2(3) as they have negative determinant and ip the sign of the antisymmetric scalar product in A =

Z3

Z3. The complex conjugation operator, J , acts in A by sending a to a2 and leaving b invariant. Therefore, the corresponding matrix is

J =

1 0

0 1

Z3) Z3, where the last Z3 is generated by one of the generators f in (52), for example f = f1. Its representative

matrix in SU(3) is

f =

i 3

1 2 1

1 1 2 2 1 1

, f 3 = 1. (75)

An analysis similar to what was described above allows us to nd the corresponding transformation matrix in the space of X, Y, Z, Z:

T (f1) =

1 3

. (80)

Since any antiunitary transformation can be written as J =

qJ , where q is unitary, it follows that q must belong to SL2(3).

Next, we need to nd which qs can be used. Clearly, (J )2 = qJ qJ = qq SL2(3). If we are looking for an an

tiunitary symmetry of a (Z3

0 3 32 3 3 2 2

32 2 2

3 2 2

Z3) Z2-symmetric potential, then qq must be either 1 or c, which generates the center of SL2(3).

Let us rst consider the second possibility.

If q =

x y z t

, then q =

x y z t

. (76)

It leads to the following conditions for the potential to be symmetric under (Z3

Z3) Z3:

3 . (77)

In the space of bilinears, the potential can then be compactly written as

V = 3m2r0 + 30 + 31 r20

+3|r12 + r45 + r67|2. (78)

3 = 3 and 1 =

2 3

. (81)

Using this to solve qq = c, we get six possible solutions,

but all of them have det q = 1, that is, they do not belong

to SL2(3). Therefore, the only possibility is qq = 1.

But then we can apply the results of our search for antiunitary transformations for the D6 case. Our group (Z3

Z3) Z2 contains the D6 subgroup with = . Therefore,

we arrive at the conclusion: in order for our potential to be symmetric under an antiunitary transformation, we must require

6 arg 3 = 0. (82)

Eur. Phys. J. C (2013) 73:2309 Page 21 of 25

If this criterion is satised, the symmetry group becomes (Z3

Z3) (Z2

Z2); otherwise the group remains (Z3

Z3) Z2. Therefore, both groups are realizable in 3HDM.

Now, consider the case of the extended symmetry group, (Z3

Z3) Z4 (36). In this case (82) is satised auto

matically due to (67). We then conclude that in this case the realizable symmetry is (36) Z2.

8 Summary and discussion

8.1 List of realizable nite symmetry groups in 3HDM

Bringing together the results of the search for abelian symmetry groups [37] and of the present work, we can nally give the list of nite groups which can appear as the symmetry groups of the scalar sector in 3HDM. If only Higgs-family transformations are concerned, the realizable nite groups are

Z2, Z3, Z4, Z2

Z2,

Z3 Z2 D6,

Z4 Z2 D8, D6, D6

Z2, D8

Z2, (84)

A4 Z2 Td, S4

Z2 Oh,

(Z3

Z3) Z2, (Z3

Z3) Z2

Z2 ,

(36) Z2.

As usual, an asterisk here indicates that the generator of the corresponding group is an anti-unitary transformation. Note that Higgs-family transformation groups Z4, D8, A4,

S4, and (36) become non-realizable in this case, because potentials symmetric under them are automatically symmetric under an additional anti-unitary transformation. In all cases apart from A4 this is a consequence of our nding in Sect. 6.5 that presence of the Z4 group of Higgs-family transformations always leads to an additional anti-unitary symmetry.

These lists complete the classication of realizable nite symmetry groups of the scalar sector of 3HDM. Conditions for the existence and examples of the potentials symmetric under each of these groups have been given in [37] and in the present work. For the readers convenience, we collect examples with non-abelian groups in the appendix.

8.2 Interplay between Higgs-family symmetries and explicit CP-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 2HDM potential with any Higgs-family symmetry. In this sense, generalized-CP symmetries can be viewed 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 Z4 group of

Higgs-family transformations guarantees that the potential is explicitly CP-conserving. This is, of course, a sufcient but not necessary condition for explicit CP-violation. Put in other words, explicit CP-violation is incompatible with the Higgs-family symmetry group Z4.

8.3 Two different D6 groups

It is interesting to note that the list (84) contains two different D6 groups. One is Z3 Z2, generated by a Higgs-family transformation of order 3 and a generalized-CP transformation. The other D6 is a group of Higgs-family transformations only, and a potential invariant under it does not have any generalized-CP symmetry. Clearly, they lead to different phenomenological consequences, as the rst case

D6, D8, T A4, O S4, (Z3

Z3) Z2 (54)/

Z3,

(83)

This list is complete: trying to impose any other nite symmetry group of Higgs-family transformations leads to the potential with a continuous symmetry.

Figure 3 should help visualize relations among different groups from this list. Going up along a 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 under the upper group.

If both unitary (Higgs-family) and antiunitary (generalized-CP) transformations are allowed, the list becomes

Z2, Z3, Z2

Z2, Z2, Z4,

Z2

(Z3

Z3) Z4 (36).

Z2, Z2

Z2

Z2,

Fig. 3 Tree of nite realizable groups of Higgs-family transformations in 3HDM

Page 22 of 25 Eur. Phys. J. C (2013) 73:2309

is explicitly CP-conserving, while the latter is explicitly CP-violating.

Such a situation was absent in the two-Higgs-doublet model, where xing the symmetry group uniquely dened 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 the Higgs-family D6 symmetry group without any generalized-CP symmetry. 2HDM does not offer this kind of freedom: any non-trivial Higgs-family symmetry group automatically leads to a generalized-CP symmetry.

8.4 Further directions of research

Certainly, our results do not provide answers to all symmetry-related questions which can be posed in 3HDM. Our paper should rather be regarded as the rst step towards systematic exploration of all the possibilities offered by three Higgs doublets. Here are some further questions which deserve closer study:

Continuous symmetry groups should also be included in

the list. There exist only few Lie groups inside PSU(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 nite groups and with anti-unitary transformations (the case of abelian groups was analyzed in [37]).

It is well-known that the vacuum state does not have to

respect all the symmetries of the Lagrangian, so the nite symmetry groups described here can be broken upon electroweak symmetry breaking. What are the symmetry breaking patterns for each of these groups? Clearly, if the symmetry group is very small, then the vacuum state can either conserve it or break it, either completely or partially. But when the nite group becomes sufciently large, there are two important changes. First, some of the groups can never be conserved upon EWSB; the origin of this feature and some 3HDM examples were discussed in [54]. Second, a sufciently large symmetry group cannot break down completely, as it would create too many degenerate vacua, which is not possible from the algebraic-geometric point of view. Indeed, in the geometric reformulation of the Higgs potential minimization problem [21], the points of the global minima in the (r0, ri)-space are precisely the contact points of two nine-dimensional algebraic manifolds: the orbit space and a certain quadric. Intersection of two algebraic manifolds of known degrees is also an algebraic manifold of a certain degree (the planar analog of this statement is Bezouts theorem). In the degenerate case when this manifold is reduced to a set 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 beyond any doubt.

What are possible symmetries of the potential beyond

the unitary and antiunitary transformations? For example, the full reparametrization group of the 2HDM potential is GL(2, ) Z2 rather than SU(2) Z2, [3436]. It means that a potential can be left invariant by transformations which are neither unitary nor anti-unitary. Although these transformations played important role in the geometric constructions in the 2HDM orbit space, they did not produce new symmetry groups beyond what was already found from the unitary transformations. It would be interesting to check the situation in 3HDM. Unfortunately, the geometric method which worked well for 2HDM becomes much more intricate with more than two doublets [20, 21].

It would also be interesting to see if the potential can have

symmetries beyond reparametrization transformations. In the case of 2HDM, this problem was analyzed in [55, 56]. Although these additional symmetries cannot be extended to kinetic term, they could still provide useful information on the structure of the Higgs potential and properties of the physical Higgs bosons.

In summary, we found all nite groups which can be realized as symmetry groups of Higgs-family or generalized-CP transformations in the three-Higgs-doublet model. Our list (84) is complete: trying to impose any other discrete symmetry group on the 3HDM Higgs potential will make it symmetric under a continuous group.

Acknowledgements This work was supported by the Belgian Fund F.R.S.-FNRS, and in part by grants RFBR 11-02-00242-a, RFBR 12-01-33102, RF President grant for scientic schools NSc-3802.2012.2, and the Program of Department of Physics SC RAS and SB RAS Studies of Higgs boson and exotic particles at LHC.

Appendix: 3HDM potentials with non-abelian Higgs-family symmetry group

Here, for the readers convenience, we list once again Higgs potentials with a given symmetry group. We focus here on cases with non-abelian groups from the list (84) because abelian ones were already discussed in detail in [37]. In each case we start from the most general potential compatible with the given realizable group presented in the main text and use the residual reparametrization freedom to simplify the coefcients of the potential (usually, it amounts to rephasing of doublets which makes some of the coefcients real). For each group G, the potential written below faithfully represents all possible Higgs potentials with realizable symmetry group G. In this sense, the symmetry group uniquely denes the phenomenology of the scalar sector of

Eur. Phys. J. C (2013) 73:2309 Page 23 of 25

3HDM, the only exception being D6 with its two distinct realizations.

Group D6

Z3 Z2 Consider the most general phase-independent part of the Higgs potential

V0 =

1i3

m2i

i i +

1ij3

ij

i i jj

+

1i<j3

ij

i j ji ,

0 1 0 1 0 0 0 0 1

and the additional terms

VZ3 = 1 21 31 + 2 12 32

+3 13 23 + h.c. (A.1)

For generic i, these terms are symmetric only under the group Z3 generated by

a3 =

. (A.7)

There are no other Higgs-family or generalized-CP transformations which leave this potential invariant. Any explicitly CP-violating D6-symmetric 3HDM potential can always be brought into this form.

Group D6

Z2 If in the previous case we set sin 3 =

0 in (A.6), then the potential becomes symmetric under D6

. (A.2)

If it happens that the product 123 is purely real, then by rephasing of doublets one can make all coefcients in (A.1) real. The resulting potential, V0 + VZ3, is symmetric under

D6

Z3 Z2 generated by a3 and the CP-transformation.

Group D8

0 0

0 2 0

0 0 1

, = exp

2i

3

Z4 Z2 Consider now terms

VZ4 = 1 31 32 + 2 12 2 + h.c., (A.3)

which are symmetric under the group Z4 generated by

a4 =

Z2 generated by a3, b, and the generalized CP-transformation b CP.

Group D8

Z2 The potential V1 +VZ4 is symmetric under

the group D8

Z2 generated by a4, b, and b CP.

Group A4 Z2 A potential symmetric under A4 Z2 can be brought into the following form:

VA4 Z2

= m2 11 + 22 + 33

+ 11 + 22 + 33 2

+ 11 22 + 22 33

+ 33 11

+ 12 2 + 23 2 + 31 2

+ 12 2 + 23 2 + 31 2 + h.c. (A.8)

with complex

i 0 0 0 i 0

0 0 1

. (A.4)

It is always possible to compensate the phases of 1 and 2 by an appropriate rephasing of the doublets. Therefore, the potential V0 + VZ4 is symmetric under the group D8

Z4 Z2 generated by a4 and the CP-transformation.

Group D6 of unitary transformations Let us restrict the coefcients of V0 in the way that guarantees the symmetry under 1 2. Then, V0 turns into

V1 = m211 11 + 22 m233 33

+11 11 2 + 22 2 + 33 33 2

+13 11 + 22 33 + 12 11 22

+ 13 13 2 + 23 2 + 12 12 2, (A.5)

where all coefcients are real and generic. Imposing the same requirement on VZ3 and performing rephasing, we obtain

VD6 = 1 21 31 12 32

+ |3|ei3 13 23 + h.c., (A.6)

where 1 is real and sin 3 = 0. The resulting potential,

V1 + VD6, is symmetric under D6 generated by a3 and

b =

. Its symmetry group is generated by independent sign ips of the individual doublets, by cyclic permutations of 1, 2, 3, and by the exchange of any pair of doublet together with the CP-transformation. An alternative form of this potential is

VA4 Z2

= m2 11 + 22 + 33

+ 11 + 22 + 33 2

+ 11 22 + 22 33

Page 24 of 25 Eur. Phys. J. C (2013) 73:2309

+ 33 11

+ Re Re 12 2 + Re 23 2 + Re 31 2

+ Im Im 12 2 + Im 23 2 + Im 31 2

+ ReIm Re 12 Im 12 + Re 23 Im 23

+ Re 31 Im 31 . (A.9)

Group S4

Z2 If the parameter

in (A.8) is real or, equivalently, ReIm = 0 in (A.9), the potential becomes symmetric

under S4

Z2 generated by sign ips, all permutation of the three doublets, and the CP-transformation.

Group (Z3

Z3) Z2 (54)/

Z3 Consider the follow-

ing potential:

Here, (54) is generated by the same a3 and b as before and, in addition, by the cyclic permutation

c =

0 1 0 0 0 1 1 0 0

, (A.11)

while the subgroup Z3 is the center of SU(3).

Group (Z3

Z3) (Z2

Z2) The potential (A.10) becomes symmetric under a generalized-CP transformation if 3 = k /3 with any integer k. In this case, one can make

3 real by a rephasing transformation. The extra generator then is the CP-transformation.

Group (36) Z2 The same potential (A.10) becomes symmetric under the group (36) Z2 if, upon rephasing, 3 = (31 2)/2. The potential can then be rewritten as

V(36) Z2

= m2I0 + 0I20 + 31I1

+

2 31

V(54)/Z3

= m2 11 + 22 + 33

+ 0 11 + 22 + 33 2

+ 1 11 2 + 22 2 + 33 2 11 22

22 33 33 11

+ 2 12 2 + 23 2 + 31 2

+ 3 12 13 + 23 21

+ 31 32 + h.c. (A.10)

with generic real m2, 0, 1, 2 and complex 3. The symmetry group of this potential is (Z3

Z3) Z2 = (54)/

Z3.

2 12 23 2 + 23 31 2

+ 31 12 2 . (A.12)

Here I0 and I1 are the SU(3)-invariants

I0 =

r03 = 11 + 22 + 33,

I1 =

i

r2i

(A.13)

=

(11)2 + (22)2 + (33)2 (11)(22) (22)(33) (33)(11)

3 +

12 2 +

23 2 +

31 2.

It is remarkable that this potential has only one structural free parameter, and the term containing it reduces the full SU(3) symmetry group to a nite subgroup (36).

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