PHYSICAL REVIEW D 79, 116004 (2009) Generalized CP symmetries and special regions of parameter space in the two-Higgs-doublet model
P.M. Ferreira,1,2 Howard E. Haber,3 and Joa˜o P. Silva1,4 1Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emı´dio Navarro, 1900 Lisboa, Portugal 2Centro de Fı´sica Teo´rica e Computacional, Faculdade de Cieˆncias, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal 3Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, California 95064, USA 4Centro de Fı´sica Teo´rica de Partı´culas, Instituto Superior Te´cnico, P-1049-001 Lisboa, Portugal (Received 27 March 2009; published 15 June 2009) We consider the impact of imposing generalized CP symmetries on the Higgs sector of the two-Higgs- doublet model, and identify three classes of symmetries. Two of these classes constrain the scalar potential parameters to an exceptional region of parameter space, which respects either a Z2 discrete flavor symmetry or a Uð1Þ symmetry. We exhibit a basis-invariant quantity that distinguishes between these two possible symmetries. We also show that the consequences of imposing these two classes of CP symmetry can be achieved by combining Higgs family symmetries, and that this is not possible for the usual CP symmetry. We comment on the vacuum structure and on renormalization in the presence of these symmetries. Finally, we demonstrate that the standard CP symmetry can be used to build all the models we identify, including those based on Higgs family symmetries.
DOI: 10.1103/PhysRevD.79.116004 PACS numbers: 11.30.Er, 11.30.Ly, 12.60.Fr, 14.80.Cp
metry. The need to seek basis-invariant observables in I. INTRODUCTION models with many Higgs was pointed out by Lavoura Despite the fantastic successes of the standard model and Silva [5], and by Botella and Silva [6], stressing (SM) of electroweak interactions, its scalar sector remains applications to CP violation. References [6,7] indicate largely untested [1]. An alternative to the single Higgs how to construct basis-invariant quantities in a systematic doublet of the SM is provided by the two-Higgs-doublet fashion for any model, including multi-Higgs-doublet model (THDM), which can be supplemented by symmetry models. Work on basis invariance in the THDM was requirements on the Higgs fields 1 and 2. Symmetries much expanded upon by Davidson and Haber [8], by leaving the kinetic terms unchanged1 may be of two types. Gunion and Haber [9,10], by Haber and O’Neil [11], and On the one hand, one may relate a with some unitary by other authors [12]. The previous approaches highlight transformation of b. These are known as Higgs family the role played by the Higgs fields. An alternative ap- symmetries, or HF symmetries. On the other hand, one proach, spearheaded by Nishi [13,14], by Ivanov [3,4], may relate a with some unitary transformation of b. and by Maniatis et al. [15], highlights the role played by These are known as generalized CP symmetries, or GCP field bilinears, which is very useful for studies of the symmetries. In this article we consider all such symmetries vacuum structure of the model [16,17]. In this paper, we that are possible in the THDM, according to their impact describe all classes of HF and GCP symmetries in both on the Higgs potential. We identify three classes of GCP languages. symmetries. One problem with two classes of GCP identified here is The study is complicated by the fact that one may that they lead to an exceptional region of parameter space perform a basis transformation on the Higgs fields, thus (ERPS) previously identified as problematic by Gunion hiding what might otherwise be an easily identifiable sym- and Haber [9] and by Davidson and Haber [8]. Indeed, no basis-invariant quantity exists in the literature that dis- 1 It has been argued by Ginsburg [2] and by Ivanov [3,4] that tinguishes between the Z2 and Uð1Þ HF symmetries in the one should also consider the effect of nonunitary global sym- ERPS. metry transformations of the two Higgs fields, as the most If evidence for THDM physics is revealed in future general renormalizable Higgs Lagrangian allows for kinetic experiments, then it will be critical to employ analysis mixing of the two Higgs fields. In this work, we study the possible global symmetries of the effective low-energy Higgs techniques that are free from model-dependent assump- theory that arise after diagonalization of the Higgs kinetic tions. It is for this reason that a basis-independent formal- energy terms. The nonunitary transformations that diagonalize ism for the THDM is so powerful. Nevertheless, current the Higgs kinetic mixing terms also transform the parameters of experimental data already impose significant constraints the Higgs potential, and thus can determine the structure of the remnant Higgs flavor symmetries of effective low-energy Higgs on the most general THDM. In particular, we know that scalar potential. It is the latter that constitutes the main focus of custodial-symmetry breaking effects, flavor changing neu- this work. tral current (FCNC) constraints, and (to a lesser extent)
1550-7998=2009=79(11)=116004(18) 116004-1 Ó 2009 The American Physical Society FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009) CP-violating phenomena impose some significant restric- definition for the electric charge, whereby the upper com- tions on the structure of the THDM (including the Higgs- ponents of the SUð2Þ doublets are charged and the lower fermion interactions). For example, the observed suppres- components neutral. sion of FCNCs implies that either the two heaviest neutral The scalar potential may be written as Higgs bosons of the THDM have masses above 1 TeV, or 2 y 2 y 2 y certain Higgs-fermion Yukawa couplings must be absent VH ¼ m11 1 1 þ m22 2 2 ½m12 1 2 þ H:c: [18]. The latter can be achieved by imposing certain dis- þ 1 ð y Þ2 þ 1 ð y Þ2 þ ð y Þ crete symmetries on the THDM. Likewise, in the most 2 1 1 1 2 2 2 2 3 1 1 y y y 1 y 2 general THDM, mass splittings between charged and neu- ð 2 2Þþ 4ð 1 2Þð 2 1Þþ½2 5ð 1 2Þ tral Higgs bosons can yield custodial-symmetry breaking y y y y effects at one-loop that could be large enough to be in þ 6ð 1 1Þð 1 2Þþ 7ð 2 2Þð 1 2ÞþH:c: ; conflict with the precision electroweak data [19]. Once (2) again, symmetries can be imposed on the THDM to alle- 2 2 where m11, m22, and 1; ; 4 are real parameters. In viate any potential disagreement with data. The implica- 2 tions of such symmetries for THDM phenomenology has general, m12, 5, 6, and 7 are complex. ‘‘H.c.’’ stands for recently been explored by Gerard and collaborators [20] Hermitian conjugation. and by Haber and O’Neil [21]. An alternative notation, useful for the construction of Thus, if THDM physics is discovered, it will be impor- invariants and championed by Botella and Silva [6]is tant to develop experimental methods that can reveal the V ¼ Y ð y Þþ1Z ð y Þð y Þ; (3) presence or absence of underlying symmetries of the most H ab a b 2 ab;cd a b c d general THDM. This requires two essential pieces of input. where Hermiticity implies First, one must identify all possible Higgs symmetries of interest. Second, one must relate these symmetries to basis- Yab ¼ Yba;Zab;cd Zcd;ab ¼ Zba;dc: (4) independent observables that can be probed by experiment. In this paper, we primarily address the first step, although The extremum conditions are we also provide basis-independent characterizations of ½Yab þ Zab;cdvdvc vb ¼ 0 ðfor a ¼ 1; 2Þ: (5) these symmetries. Our analysis focuses the symmetries of the THDM scalar potential. In principle, one can extend Multiplying by va leads to our study of these symmetries to the Higgs-fermion Y ðv v Þ¼ Z ðv v Þðv v Þ: (6) Yukawa interactions, although this lies beyond the scope ab a b ab;cd a b d c of the present work. One should be very careful when comparing Eqs. (2) and This paper is organized as follows. In Sec. II, we in- (3) among different authors, since the same symbol may be troduce our notation and define an invariant that does used for quantities, which differ by signs, factors of two, or distinguish the Z2 and Uð1Þ HF symmetries in the ERPS. complex conjugation. Here, we follow the definitions of In Sec. III, we explain the role played by the vacuum Davidson and Haber [8]. With these definitions expectation values (vevs) in preserving or breaking the Uð1Þ symmetry, and we comment briefly on renormaliza- Y ¼ m2 ;Y¼ m2 ; 11 11 12 12 (7) tion. In Sec. IV, we introduce the GCP transformations and Y ¼ ðm2 Þ Y ¼ m2 ; explain why they are organized into three classes. We 21 12 22 22 summarize our results and set them in the context of the and existing literature in Sec. V, and in Sec. VI, we prove a surprising result: multiple applications of the standard CP Z11;11 ¼ 1;Z22;22 ¼ 2; symmetry can be used to build all the models we identify, Z ¼ Z ¼ ;Z¼ Z ¼ ; including those based on HF symmetries. We draw our 11;22 22;11 3 12;21 21;12 4 (8) conclusions in Sec. VII. Z12;12 ¼ 5;Z21;21 ¼ 5; Z11;12 ¼ Z12;11 ¼ 6;Z11;21 ¼ Z21;11 ¼ ; II. THE SCALAR SECTOR OF THE THDM 6 Z22;12 ¼ Z12;22 ¼ 7;Z22;21 ¼ Z21;22 ¼ 7: A. Three common notations for the scalar potential Let us consider a SUð2Þ Uð1Þ gauge theory with two The previous two notations look at the Higgs fields a Higgs-doublets , with the same hypercharge 1=2, and individually. A third notation is used by Nishi [13,14] and a Ivanov [3,4], who emphasize the presence of field bilinears with vevs y ð a bÞ [17]. Following Nishi [13] we write 0pffiffiffi h ai¼ : (1) V ¼ M r þ r r ; (9) va= 2 H The index a runs from 1 to 2, and we use the standard where ¼ 0,1,2,3and
116004-2 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) 1 who pointed out that r parametrizes the gauge orbits of r ¼ ½ð y Þþð y Þ ; 0 2 1 1 2 2 the Higgs fields, in a space equipped with a Minkowski 1 metric. r ¼ ½ð y Þþð y Þ ¼ Reð y Þ; 1 2 1 2 2 1 1 2 In terms of the parameters of Eq. (2), the 4-vector M (10) i and 4 4 matrix are written, respectively, as r ¼ ½ð y Þ ð y Þ ¼ Imð y Þ; 2 2 1 2 2 1 1 2 2 2 2 2 2 2 1 M ¼ðm þ m ; 2Rem ; 2Imm ;m m Þ; r ¼ ½ð y Þ ð y Þ : 11 22 12 12 11 22 3 2 1 1 2 2 (11) In Eq. (9), summation of repeated indices is adopted with Euclidean metric. This differs from Ivanov’s notation [3,4], and
0 1 ð 1 þ 2Þ=2 þ 3 Reð 6 þ 7Þ Imð 6 þ 7Þð 1 2Þ=2 B C B Reð 6 þ 7Þ 4 þ Re 5 Im 5 Reð 6 7Þ C ¼ @ A: (12) Imð 6 þ 7Þ Im 5 4 Re 5 Imð 6 7Þ ð 1 2Þ=2Reð 6 7Þ Imð 6 7Þð 1 þ 2Þ=2 3
S Equation (9) is related to Eq. (3) through a ! a ¼ Sab b: (19) M ¼ abYba; (13) S is a unitary matrix, so that the gauge-kinetic couplings are also left invariant by this HF symmetry. As a result of 1 ¼ 2Zab;cd ba dc; (14) this symmetry, i 0 S where the matrices are the three Pauli matrices, and Yab ¼ Yab ¼ Sa Y Sb ; (20) is the 2 2 identity matrix. S Zab;cd ¼ Zab;cd ¼ Sa Sc Z ; Sb Sd : (21) B. Basis transformations Notice that this is not the situation considered in Eqs. (15)– 0 We may rewrite the potential in terms of new fields a, (17). There, the coefficients of the Lagrangian do change obtained from the original ones by a simple (global) basis (although the quantities that are physically measurable are transformation invariant with respect to any change of basis). In contrast, 0 Eqs. (19)–(21) imply the existence of a HF symmetry S of a ! a ¼ Uab b; (15) the scalar potential that leaves the coefficients of VH where U 2 Uð2Þ is a 2 2 unitary matrix. Under this unchanged. unitary basis transformation, the gauge-kinetic terms are The Higgs family symmetry group must be a subgroup unchanged, but the coefficients Yab and Zab;cd are trans- of the full Uð2Þ transformation group of 2 2 unitary formed as matrices employed in Eq. (15). Given the most general THDM scalar potential, there is always a Uð1Þ subgroup of Y ! Y0 ¼ U Y U ; (16) ab ab a b Uð2Þ under which the scalar potential is invariant. This is Uð Þ 0 the global hypercharge 1 Y symmetry group Zab;cd ! Z ¼ Ua Uc Z ; U U ; (17) ab;cd b d i i Uð1ÞY: 1 ! e 1; 2 ! e 2; (22) and the vevs are transformed as where is an arbitrary angle (mod 2 ). The invariance ! 0 ¼ va va Uabvb: (18) under the global Uð1ÞY is trivially guaranteed by the in- ð Þ ð Þ Thus, the basis transformations U may be utilized in order variance under the SU 2 U 1 electroweak gauge sym- ð Þ to absorb some of the degrees of freedom of Y and/or Z, metry. Since the global hypercharge U 1 Y is always which implies that not all parameters of Eq. (3)have present, we shall henceforth define the HF symmetries as physical significance. those Higgs family symmetries that are orthogonal to Uð1ÞY. We now turn to the interplay between HF symmetries C. Higgs family symmetries and basis transformations. Let us imagine that, when writ- Let us assume that the scalar potential in Eq. (3) has ten in the basis of fields a, VH has a symmetry S. We then some explicit internal symmetry. That is, we assume that perform a basis transformation from the basis a to the 0 the coefficients of VH stay exactly the same under a trans- basis a, as given by Eq. (15). Clearly, when written in the formation new basis, VH does not remain invariant under S. Rather, it
116004-3 FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009) will be invariant under As previously noted, the Uð2Þ symmetry contains the
0 y global hypercharge Uð1ÞY as a subgroup. Thus, in order to S ¼ USU : (23) identify the corresponding HF symmetry that is orthogonal As we change basis, the form of the potential changes in a to Uð1ÞY, we first observe that way that may obscure the presence of a HF symmetry. In particular, two HF symmetries that naively look distinct Uð2ÞffiSUð2Þ Uð1ÞY=Z2 ffi SOð3Þ Uð1ÞY: (31) will actually yield precisely the same physical predictions if a unitary matrix U exists such that Eq. (23) is satisfied. To prove the above isomorphism, simply note that any HF symmetries in the THDM have a long history. In Uð2Þ matrix can be written as U ¼ ei U^ , where U^ 2 papers by Glashow and Weinberg and by Paschos [18], the SUð2Þ. To cover the full Uð1ÞY group, we must take 0 discrete Z2 symmetry was introduced, <2 . But since both U^ and U^ are elements of SUð2Þ, whereas þ1 and 1 ¼ ei are elements of Uð1Þ , we must Z : ! ; ! ; (24) Y 2 1 1 2 2 identify U^ and U^ as the same group element in order not in order to preclude flavor-changing neutral currents [18]. to double cover the full Uð2Þ group. The identification of U^ This is just the interchange with U^ in SUð2Þ is isomorphic to SOð3Þ, using the well- : $ ; (25) known isomorphism SOð3ÞffiSUð2Þ=Z2. Consequently, 2 1 2 we have identified SOð3Þ as the HF symmetry that con- seen in a different basis, as shown by applying Eq. (23)in strains the scalar potential parameters as indicated in the form Eq. (30). 01 1 11 10 1 11 The impact of these symmetries on the potential parame- ¼ pffiffiffi pffiffiffi : (26) ters in Eq. (2) is shown in Sec. V. As mentioned above, if 10 1 1 0 1 1 1 2 2 one makes a basis change, the potential parameters change Peccei and Quinn [22] introduced the continuous Uð1Þ and so does the explicit form of the symmetry and of its symmetry implications. For example, Eq. (26) shows that the sym- metries Z and are related by a basis change. However, i i 2 2 Uð1Þ: 1 ! e 1; 2 ! e 2; (27) they have a different impact on the parameters in their true for any value of , in connection with the strong CP respective basis. This can be seen explicitly in Table I of problem. Of course, a potential invariant under Uð1Þ is also Sec. V. One can also easily prove that the existence of either the Z , or Peccei-Quinn Uð1Þ symmetry is suffi- invariant under Z2. 2 2 Finally, we examine the largest possible Higgs family cient to guarantee the existence of a basis choice in which symmetry group of the THDM, namely, Uð2Þ. In this case, all scalar potential parameters are real. That is, the corre- a basis transformation would have no effect on the Higgs sponding scalar Higgs sectors are explicitly CP conserving. potential parameters. Since ab is the only Uð2Þ-invariant tensor, it follows that Basis-invariant signs of HF symmetries were discussed extensively in Ref. [8]. Recently, Ferreira and Silva [23] Yab ¼ c1 ab; (28) extended these methods to include Higgs models with more than two Higgs doublets. Zab;cd ¼ c2 ab cd þ c3 ad bc; (29) Consider first the THDM scalar potentials that are in- variant under the so-called simple HF symmetries of where c , c , and c are arbitrary real numbers.2 One can 1 2 3 Ref. [23]. We define a simple HF symmetry to be a easily check from Eqs. (16) and (17) that the unitarity of U symmetry group G with the following property: the re- implies that Y0 ¼ Y and Z0 ¼ Z for any choice of basis, as quirement that the THDM scalar potential is invariant required by the Uð2Þ invariance of the scalar potential. under a particular element g 2 G (where g Þ e and e is Equations (28) and (29) impose the following constraints the identity element) is sufficient to guarantee invariance on the parameters of the THDM scalar potential (indepen- under the entire group G. The discrete cyclic group Z ¼ dently of the choice of basis): n fe; g; g2; ...;gn 1g, where gn ¼ e, is an example of a 2 2 2 possible simple HF symmetry group. If we restrict the m22 ¼ m11;m12 ¼ 0; (30) TDHM scalar potential to include terms of dimension 1 ¼ 2 ¼ 3 þ 4; 5 ¼ 6 ¼ 7 ¼ 0: four or less (e.g., the tree-level scalar potential of the As there are no nonzero potentially complex scalar poten- THDM), then one can show that the Peccei-Quinn Uð1Þ tial parameters, the Uð2Þ-invariant THDM is clearly CP symmetry is also a simple HF symmetry. For example, invariant. consider the matrix ! 2i =3 2 e 0 Note that there is no ac bd term contributing to Zab;cd,as S ¼ 2i =3 : (32) such a term is not invariant under the transformation of Eq. (17). 0 e
116004-4 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) 3 Note that S is an element of the cyclic subgroup Z3 ¼ parameters are real. Having achieved such a basis, fS; S2;S3 ¼ 1g of the Peccei-Quinn Uð1Þ group. As shown Davidson and Haber [8] demonstrate that one may make in Ref. [23], the invariance of the tree-level THDM scalar one additional basis transformation such that potential under a ! Sab b automatically implies the invariance of the scalar potential under the full Peccei- 2 2 2 m22 ¼ m11;m12 ¼ 0; 2 ¼ 1; Quinn Uð1Þ group. In contrast, the maximal HF symmetry, (34) SOð3Þ, introduced above is not a simple HF symmetry, as 7 ¼ 6 ¼ 0; Im 5 ¼ 0: there is no single element of S 2 SOð3Þ such that invari- ance under a ! Sab b guarantees invariance of the tree- These conditions express the ERPS for a specific basis level THDM scalar potential under the full SOð3Þ group of choice. transformations. One might think that since this is such a special region of Typically, the simple HF symmetries take on a simple parameter space that it lacks any relevance. However, the form for a particular choice of basis for the Higgs fields. fact that the conditions in Eq. (33) hold in any basis is a We summarize here a few of the results of Ref. [23]: good indication that a symmetry may lie behind this con- (1) In the THDM, there are only two independent dition. Indeed, as pointed out by Davidson and Haber [8], classes of simple symmetries: a discrete Z2 flavor combining the two symmetries Z2 and 2 in the same symmetry, and a continuous Peccei-Quinn Uð1Þ fla- basis one is lead immediately to the ERPS in the basis of vor symmetry. Eq. (34). Up to now, we considered the impact of imposing (2) Other discrete flavor symmetry groups G that are on the Higgs potential only one symmetry. This was subgroups of Uð1Þ are not considered independent. dubbed a simple symmetry. Now we are considering the That is, if S 2 G (where S Þ e), then invariance possibility that the potential must remain invariant under under the discrete symmetry ! S makes the one symmetry and also under a second symmetry; this scalar potential automatically invariant under the implies further constraints on the parameters of the Higgs full Peccei-Quinn Uð1Þ group; potential. We refer to this possibility as a multiple symme- (3) In most regions of parameter space, one can build try. As seen from Table I of Sec. V, imposing Z2 and 2 in quantities invariant under basis transformations that the same basis leads to the conditions in Eq. (34). detect these symmetries; Incidentally, this example shows that a model that lies in (4) There exists a so-called ERPS characterized by the ERPS, is automatically invariant under Z2. In Sec. IV, we will show that all classes of nontrivial CP m2 ¼ m2 ;m2 ¼ 0; 22 11 12 (33) transformations lead directly to the ERPS, reinforcing the importance of this particular region of parameter space. 2 ¼ 1; 7 ¼ 6:
As shown by Davidson and Haber [8], a theory D. Requirements for Uð1Þ invariance obeying these constraints does have a Z symmetry, 2 In the basis in which the Uð1Þ symmetry takes the form but it may or not have a Uð1Þ symmetry. Within the of Eq. (27), the coefficients of the potential must obey ERPS, the invariants in the literature cannot be used to distinguish the two cases. 02 0 0 0 The last statement above is a result of the following m12 ¼ 0; 5 ¼ 6 ¼ 7 ¼ 0: (35) considerations: In order to distinguish between Z2 and Uð1Þ, Davidson and Haber [8] construct two invariant Imagine that we have a potential of Eq. (2) in the ERPS: 2 2 2 quantities given by Eqs. (46) and (50) of Ref. [8]. m11 ¼ m22, m12 ¼ 0, 2 ¼ 1, and 7 ¼ 6.Wenow Outside the ERPS, these quantities are zero if and only if wish to know whether a transformation U may be chosen Uð1Þ holds. Unfortunately, in the ERPS these quantities such that the potential coefficients in the new basis satisfy vanish automatically independently of whether or not Uð1Þ the Uð1Þ conditions in Eq. (35). Using the transformation holds. Similarly, Ferreira and Silva [23] have constructed rules in Eqs. (A13)–(A23) of Davidson and Haber [8], we invariants detecting HF symmetries. But their use requires find that such a choice of U is possible if and only if the the existence of a matrix, obtained by combining Yab and coefficients in the original basis satisfy Zab;cd, which has two distinct eigenvalues. This does not 3 2 occur when the ERPS is due to a symmetry. Finally, in the 2 6 5 6ð 1 3 4Þ 5 6 ¼ 0; (36) 2 ERPS, Ivanov [3] states that the symmetry might be ‘‘ðZ2Þ or Oð2Þ’’ [our Z or our Uð1Þ] and does not provide a way 2 2 subject to the condition that 5 6 is real. to distinguish the two possible flavor symmetries [24]. Gunion and Haber [9] have shown that the ERPS con- 3Given a scalar potential whose parameters satisfy the ERPS ditions of Eq. (33) are basis independent; if they hold in 2 Þ conditions with Imð 5 6Þ 0, the unitary matrix required to one basis, then they hold in any basis. Moreover, for a transform into a basis in which all the scalar potential parameters model in the ERPS, a basis may be chosen such that all are real can be determined only by numerical means.
116004-5 FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009) E. The D invariant The cubic equation, Eq. (38), has at least two degenerate Having established the importance of the ERPS (as it solutions if [25] can arise from a symmetry), we will now build a basis- 1 1 2 3 1 1 3 2 D ½a1 a þ½ ða1a2 3a0Þ a (44) invariant quantity that can be used to detect the presence of 2 9 2 6 27 2 a Uð1Þ symmetry in this special case. vanishes. The quadratic terms of the Higgs potential are always The expression of D in terms of the parameters in Eq. (2) insensitive to the difference between Z2 and Uð1Þ. is rather complicated, even in the ERPS. But one can show Moreover, the matrix Y is proportional to the unit matrix by direct computation that if the Uð1Þ-symmetry condition 2 in the ERPS. One must thus look at the quartic terms. We of Eq. (36) holds (subject to 5 6 being real), then D ¼ 0. were inspired by the expression of in Eq. (12), which We can simplify the expression for D by changing to a appears in the works of Nishi [13,14] and Ivanov [3,4]. In basis where all parameters are real [9], where we get the ERPS of Eq. (33), breaks into a 1 1 block ( ), 00 D ¼ 1 ½ ð þ Þ 2 2 2 ~ 27 5 1 3 4 5 6 and a 3 3 block ( ¼f ijg; i, j ¼ 1, 2, 3). A basis ½ð Þ2 þ 16 2 : (45) transformation U belonging to SUð2Þ on the a fields 1 3 4 5 6 corresponds to an orthogonal SOð3Þ transformation in the Þ If 6 0, then D ¼ 0 means ri bilinears, given by 2 1 y 2 6 ¼ 5ð 1 3 4 þ 5Þ: (46) Oij ¼ 2 Tr½U iU j : (37) If 6 ¼ 0, then D ¼ 0 corresponds to one of three possible Any matrix O of SOð3Þ can be obtained by considering an conditions: appropriate matrix U of SUð2Þ (unfortunately this property does not generalize for models with more than two Higgs 5 ¼ 0; 5 ¼Æð 1 3 4Þ: (47) doublets). A suitable choice of O can be made that diag- ~ Notice that Eqs. (46) and (47) are equivalent to Eq. (36)in onalizes the 3 3 matrix , thus explaining Eq. (34). In any basis where the coefficients are real. this basis, the difference between the usual choices for Although D can be defined outside the ERPS, the con- Uð1Þ and Z2 corresponds to the possibility that Re 5 might dition D ¼ 0 only guarantees that the model is invariant vanish or not, respectively. under Uð1Þ inside the ERPS of Eq. (33). Outside this region We will now show that, once in the ERPS, the condition one can detect the presence of a Uð1Þ symmetry with the ~ for the existence of Uð1Þ is that has two eigenvalues, invariants proposed by Davidson and Haber [8]. This which are equal. The eigenvalues of a 3 3 matrix are the closes the last breach in the literature concerning basis- solutions to the secular equation invariant signals of discrete symmetries in the THDM. x3 þ a x2 þ a x þ a ¼ 0; (38) Thus, in the ERPS D ¼ 0 is a necessary and sufficient 2 1 0 condition for the presence of a Uð1Þ symmetry. where ~ 1 ~ 3 1 ~ 3 1 ~ ~ 2 III. VACUUM STRUCTURE AND a0 ¼ det ¼ 3 Trð Þ 6ðTr Þ þ 2ðTr Þ Trð Þ RENORMALIZATION 1 3 ð2Þ ð2Þ ¼ Zab;cdðZdc;ghZhg;ba Zdc Zba Þ 3 2 The presence of a Uð1Þ symmetry in the Higgs potential 1 ð1Þ 1 ð2Þ 1 ð1Þ 3 þ 2Zab;cdZdc;ba TrðZ 2Z Þ 6ðTrZ Þ may (or not) imply the existence of a massless scalar, the þ 1ðTrZð1ÞÞ2 TrZð2Þ 1 TrZð1ÞðTrZð2ÞÞ2; (39) axion, depending on whether (or not) the Uð1Þ is broken by 4 2 the vevs. In the previous section we related the basis- invariant condition D ¼ 0 in the ERPS with the presence a ¼ 1ðTr ~ Þ2 1 Trð ~ 2Þ 1 2 2 of a Uð1Þ symmetry. In this section we will show that, 1 ð1Þ 2 ð1Þ ð2Þ ð2Þ 2 whenever the basis-invariant condition D ¼ 0 is satisfied ¼ 2½ðTrZ Þ TrZ TrZ þðTrZ Þ in the ERPS, there is always a stationary point for which a Z Z ; (40) ab;cd dc;ba massless scalar, other than the usual Goldstone bosons, exists. ~ 1 ð2Þ ð1Þ a2 ¼ Tr ¼ 2 TrZ TrZ ; (41) We start by writing the extremum conditions for the THDM in the ERPS. For simplicity, we will be working and in a basis where all the parameters are real [9]. From ð1Þ 1 þ 4 6 þ 7 Eqs. (5) and (8), we obtain Zab Za ; b ¼ ; (42) 6 þ 7 2 þ 4 1 3 2 2 3 0 ¼ Y11v1 þ 2½ 1v1 þ 345v1v2 þ 6ð3v1v2 v2Þ ; ¼ þ 1½ 3 þ 2 þ ð 3 2 Þ ð2Þ 1 þ 3 6 þ 7 0 Y11v2 2 1v2 345v2v1 6 v1 3v2v1 ; Zab Z ;ab ¼ : (43) 6 þ 7 2 þ 3 (48)
116004-6 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) where we have defined 345 3 þ 4 þ 5.Wenow Goldstone boson. Further, this matrix will have an axion if compute the mass matrices. As we will be considering 5 ¼ 0, which is the first condition of Eq. (47). only vacua with real vevs, there will be no mixing between the real and imaginary parts of the doublets. As such, we B. Case 6 ¼ 0, fv1 Þ 0;v2 ¼ 0g can define the mass matrix of the CP-even scalars as given by Returning to Eq. (48), this case gives us 1 2 2 Y11 ¼ 2 1v1; (56) 2 1 @ V ½Mh ij ¼ 0 0 ; (49) 2 @ Reð i Þ@ Reð j Þ which implies Y11 < 0. With this condition, the mass ma- trices become considerably simpler: 0 where i is the neutral (lower) component of the i ! doublet. Thus, we obtain, for the entries of this matrix, v2 0 ½M2 ¼ 1 1 (57) the following expressions: h 1 2 0 2 ð 345 1Þv1 2 1 2 2 ½Mh 11 ¼ Y11 þ 2ð3 1v1 þ 345v2 þ 6 6v1v2Þ and 2 1 2 2 ½Mh 22 ¼ Y11 þ 2ð3 1v2 þ 345v1 þ 6 6v1v2Þ (50) 1 00 ½M2 ¼ : (58) 3 A 2 2 2 2 2 0 ð 3 þ 4 5 1Þv1 ½Mh 12 ¼ 345v1v2 þ 2 6ðv1 v2Þ: Likewise, the pseudoscalar mass matrix is defined as So, we can have an axion in the matrix (57)if
2 345 1 ¼ 0 , 5 ¼ 1 3 4 (59) 2 1 @ V ½MA ij ¼ 0 0 (51) 2 @ Imð i Þ@ Imð j Þ or an axion in matrix (58)if whose entries are given by 5 ¼ 1 þ 3 þ 4: (60)
2 1 2 2 That is, we have an axion if the second or third conditions ½MA 11 ¼ Y11 þ 2½ 1v1 þð 3 þ 4 5Þv2 þ 2 6v1v2 of Eq. (47) are satisfied. The other possible case, fv1 ¼ ½M2 ¼ Y þ 1½ v2 þð þ Þv2 2 v v Þ A 22 11 2 1 2 3 4 5 1 6 1 2 0;v2 0g, produces exactly the same conclusions. 2 1 2 2 ½MA 12 ¼ 5v1v2 þ 2 6ðv1 v2Þ: (52) C. Case 6 Þ 0 The expressions (50) and (52) are valid for all the particular cases we will now consider. This is the hardest case to treat, since we cannot obtain analytical expressions for the vevs. Nevertheless a full analytical treatment is still possible. First, notice that A. Case 6 ¼ 0, fv1;v2g Þ 0 Þ with 6 0 Eqs. (48) imply that both vevs have to be Let us first study the case 6 ¼ 0, wherein we may solve nonzero. At the stationary point of Eqs. (48), the pseudo- the extremum conditions in an analytical manner. It is scalar mass matrix has a Goldstone boson and an eigen- trivial to see that Eqs. (48) have three types of solutions: value given by both vevs different from zero, one vev equal to zero (say, v4 v4 v2) and both vevs zero (trivial noninteresting solution). For ðv2 þ v2Þ 1 2 : (61) Þ 5 1 2 6 a solution with fv1;v2g 0, a necessary condition must be 2v1v2 obeyed so that there is a solution to Eqs. (48): So, an axion exists if we have 2 2 Þ : 1 345 0 (53) v2 v2 2 1 2 ¼ 5 : (62) 2 If we use the extremum conditions to evaluate ½Mh ,we v1v2 6 obtain On the other hand, after some algebraic manipulation, it is 2 simple to obtain from (48) the following condition: 2 1v1 345v1v2 ½Mh ¼ 2 ; (54) 345v1v2 1v2 2 2 v1 v2 4v1v2 1 345 ¼ 6 2 2 : (63) which only has a zero eigenvalue if Eq. (53) is broken. v1v2 v1 v2 Thus, there is no axion in this matrix in this case. As for 2 Substituting Eq. (62) into (63), we obtain ½MA , we get 2 2 5 2 6 2 2 v1 v1v2 1 345 ¼ 6 þ , 2 6 ½MA ¼ 5 2 ; (55) 6 5 v1v2 v2 ¼ 5ð 1 3 4 þ 5Þ: (64) which clearly has a zero eigenvalue corresponding to the Z
116004-7 FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009) Thus, we have shown that all of the conditions stemming where the reference to the time (t) and space (x~) coordi- from the basis-invariant condition D ¼ 0 guarantee the nates will henceforth be suppressed. However, in the pres- existence of some stationary point for which the scalar ence of several scalars with the same quantum numbers, potential yields an axion. Notice that, however, this sta- basis transformations can be included in the definition of tionary point need not coincide with the global minimum the CP transformation. This yields GCP transformations of the potential. GCP y > a ¼ Xa Xa ð Þ ; (70) D. Renormalization group invariance yGCP > y a ¼ Xa Xa ð Þ ; We now briefly examine the renormalization group (RG) behavior of our basis-invariant condition D ¼ 0. It would where X is an arbitrary unitary matrix [27,28].4 GCP GCP be meaningless to say that D ¼ 0 implies a Uð1Þ symmetry Note that the transformation a ! a , where a if that condition were only valid at a given renormalization is given by Eq. (70), leaves the kinetic terms invariant. The scale. That is, it could well be that a numerical accident GCP transformation of a field bilinear yields forces D ¼ 0 at only a given scale. To avoid such a con- clusion, we must verify if D ¼ 0 is a RG-invariant condi- yGCP GCP y > a ¼ Xa Xb ð Þ : (71) tion (in addition to being basis invariant). For a given b renormalization scale , the function of a given parame- Under this GCP transformation, the quadratic terms of the ter x is defined as x ¼ @x=@ . For simplicity, let us rewrite D in Eq. (45)as potential may be written as
1 2 yGCP GCP ¼ y D ¼ 27D1D2; (65) Yab a b YabXa Xb y with ¼ Xb YbaXa y y y 2 ¼ X aY X b a b ¼ðX YXÞab a b: D1 ¼ 5ð 1 3 4 þ 5Þ 2 6 (66) (72) 2 2 D2 ¼ð 1 3 4 5Þ þ 16 6: We have used the Hermiticity condition Y ¼ Y in If we apply the operator @=@ to D, we obtain ab ba going to the second line; and changed the dummy indices $ $ ¼ 1 ð2D D þ D2 Þ: (67) a and b in going to the third line. A similar D 27 1 2 D1 1 D2 argument can be made for the quartic terms. We conclude that the potential is invariant under the GCP transformation If D1 ¼ 0 (which corresponds to three of the conditions presented in Eqs. (46) and (47)) then we immediately have of Eq. (70) if and only if the coefficients obey D ¼ 0. That is, if D ¼ 0 at a given scale, it is zero at all Y ¼ X Y X ¼ðXyYXÞ ; scales. ab a b ab (73) If D ¼ 0 and D Þ 0 we will only have ¼ 0 if 2 1 D Z ¼ X aX cZ ; X bX d: ¼ ab;cd D2 0, or equivalently, ð Þð Þ Introducing 2 1 3 4 5 1 3 4 5 þ ¼ 32 6 6 0: (68) y Yab ¼ Yab X aY X b ¼½Y ðX YXÞ ab; Given that D2 ¼ 0 implies that 6 ¼ 0 and 5 ¼ Zab;cd ¼ Zab;cd X aX cZ ; X bX d; (74) 1 3 4, we once again obtain D ¼ 0. Thus, the condition D ¼ 0 is RG invariant. A direct we may write the conditions for invariance under GCP as verification of the RG invariance of Eqs. (46) and (47), and of the conditions that define the ERPS itself, would Y ¼ 0; (75) require the explicit form of th e functions of the THDM ab involving the 6 coupling. That verification will be made elsewhere [26]. Zab;cd ¼ 0: (76) CP IV. GENERALIZED SYMMETRIES Given Eqs. (4), it is easy to show that It is common to consider the standard CP transformation of the scalar fields as 4Equivalently, one can consider a generalized time-reversal CP transformation proposed in Ref. [29] and considered further in aðt; x~Þ! a ðt; x~Þ¼ aðt; x~Þ; (69) Appendix A of Ref. [9].
116004-8 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) Yab ¼ Yba; Zab;cd Zcd;ab ¼ Zba;dc: (77) the GCP is a standard CP transformation. In contrast, if the unitary matrix X is not symmetric, then no basis exists in
Thus, we need only consider the real coefficients Y11, which Y and Z are real for generic values of the scalar Y22, Z11;11, Z22;22, Z11;22, Z12;21, and the complex potential parameters. Nevertheless, as we shall demon- imposing coefficients Y12, Z11;12, Z22;12, and Z12;12. strate below, by the GCP symmetry on the scalar potential, the parameters of the scalar potential are con- strained in such a way that for an appropriately chosen A. GCP and basis transformations basis change, Y0 ¼ X0yY0X0 ¼ Y0 (with a similar result for We now turn to the interplay between GCP transforma- Z0). tions and basis transformations. Consider the potential of GCP transformations were studied in Refs. [27,28]. In Eq. (3) and call it Vð Þ. Now consider the potential particular, Ecker, Grimus, and Neufeld [28] proved that for obtained from Vð Þ by the basis transformation a ! every matrix X there exists a unitary matrix U such that X0 0 a ¼ Uab b: can be reduced to the form 0 0 0y 0 1 0 0y 0 0y 0 Vð Þ¼Yabð a bÞþ2Zab;cdð a bÞð c dÞ; (78) cos sin X0 ¼ UXU> ¼ ; (82) where the coefficients in the new basis are given by sin cos Eqs. (16) and (17). We will now prove the following theorem: If Vð Þ is invariant under the GCP transforma- tion of Eq. (70) with the matrix X, then Vð 0Þ is invariant where 0 =2. Notice the restricted range for . The under a new GCP transformation with matrix value of can be determined in either of two ways: (i) the eigenvalues of ðX þ X>ÞyðX þ X>Þ=2 are cos , each of X0 ¼ UXU>: (79) which is twice degenerate; or (ii) XX has the eigenvalues eÆ2i . By hypothesis Vð Þ is invariant under the GCP transfor- mation of Eq. (70) with the matrix X. Equation (73) guarantees that Y ¼ XyYX. Now, Eq. (16) relates the B. The three classes of GCP symmetries coefficients in the two basis through Y ¼ UyY0U. Having reached the special form of X0 in Eq. (82), we Substituting gives will now follow the strategy adopted by Ferreira and Silva [23] in connection with HF symmetries. We substitute > 0 y y 0 U Y U ¼ X ðU Y UÞX; (80) Eq. (82) for X in Eq. (73), in order to identify the con- or straints imposed by this reduced form of the GCP trans- formations on the quadratic and quartic couplings. For Y0 ¼ðU XyUyÞY0ðUXU>Þ¼X0yY0X0; (81) each value of , certain constraints will be forced upon the couplings. If two different values of enforce the same as required. A similar argument holds for the quartic terms constraints, we will say that they are in the same class and the proof is complete. > (since no experimental distinction between the two will The fact that the transpose U appears in Eq. (79) rather then be possible). We will start by considering the special than Uy is crucial. In Eq. (23), applicable to HF symme- y cases of ¼ 0 and ¼ =2, and then turn our attention to tries, U appears. Consequently, a basis may be chosen 0 < < =2. where the HF symmetry is represented by a diagonal > matrix S. The presence of U in Eq. (79) implies that, 1. : contrary to popular belief, it is not possible to reduce all CP1 ¼ 0 GCP transformations to the standard CP transformation of When ¼ 0, X0 is the unit matrix, and we obtain the Eq. (69) by a basis transformation. What is possible, as we standard CP transformation shall see below, is to reduce an invariance of the THDM potential under any GCP transformation, to an invariance ! ; ! ; (83) under the standard CP transformation plus some extra 1 1 2 2 constraints. To be more specific, the following result is easily estab- under which Eqs. (73) take the very simple form lished. If the unitary matrix X is symmetric, then it follows 5 that a unitary matrix U exists such that X0 ¼ UXU> ¼ 1, Y ¼ Yab;Z¼ Zab;cd: (84) in which case Y0 ¼ Y0. In this case, a basis exists in which ab ab;cd
We denote this CP transformation by CP1. It forces all 5Here, we make use of a theorem in linear algebra that states that for any unitary symmetric matrix X, a unitary matrix V couplings to be real. Since most couplings are real by the exists such that X ¼ VV>. A proof of this result can be found, Hermiticity of the Higgs potential, the only relevant con- 2 e.g., in Appendix B of Ref. [9]. straints are Imm12 ¼ Im 5 ¼ Im 6 ¼ Im 7 ¼ 0.
116004-9 FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009) 2. CP2: ¼ =2 4 4 1 2 0 ¼ Z11;11 ¼ 1ð1 c Þ 2s 2 345s2 When ¼ =2, 3 3 þ 4Re 6c s þ 4Re 7cs ; 4 4 1 2 01 0 ¼ Z22;22 ¼ 2ð1 c Þ 1s 345s X0 ¼ ; (85) 2 2 10 3 3 4Re 7c s 4Re 6cs ; 1 and we obtain the CP transformation 0 ¼ Z11;22 ¼ 4s2½4Reð 6 7Þc2
þð 1 þ 2 2 345Þs2 ; 1 ! 2; 2 ! 1; (86) 0 ¼ Z12;21 ¼ Z11;220 ¼ Re Z11;12 (90) which we denote by CP2. This was considered by 1 ¼ s½ð 3 1 þ 2 þ 2 345Þc Davidson and Haber [8] in their Eq. (37), who noted that 4 if this symmetry holds in one basis, it holds in all basis ð 1 þ 2 2 345Þc3 choices. Under this transformation, Eq. (75) forces the þ 4Re 6ð2s þ s3Þ 4Re 7s3 ; matrix of quadratic couplings to obey 0 ¼ Re Z ¼ 1s½ð þ 3 2 Þc 22;12 4 1 2 345 m2 m2 2m2 11 22 12 þð 1 þ 2 2 345Þc3 4Re 6s3 0 ¼ Y ¼ 2 2 2 ; (87) 2m12 m22 m11; þ 4Re 7ð2s þ s3Þ ; 2 2 2 leading to m22 ¼ m11 and m12 ¼ 0. Similarly, we may 0 ¼ Re Z12;12 ¼ Z11;22 construct a matrix of matrices containing all coefficients Zab;cd. The uppermost-leftmost matrix corresponds to Z11;cd. The next matrix along the same line corresponds 0 ¼ Im Z11;12 to Z12;cd, and so on. To enforce invariance under CP2,we equate it to zero, 1 ¼ 2½Im 6ð3 þ c2ÞþIm 7ð1 c2Þ Im 5s2 ; 0 1 þ þ 0 0 ¼ Im Z22;12 (91) B 1 2 6 7 6 7 C B þ 0 þ C 1 B 6 7 0 6 7 C ¼ ½Im 6ð1 c2ÞþIm 7ð3 þ c2ÞþIm 5s2 ; 0 ¼ @ A: 2 6 þ 7 0 0 6 þ 7 0 ¼ Im Z12;12 ¼ 2c½Im 5c þ Imð 6 7Þs ; 0 6 þ 7 6 þ 7 2 1 (88) where 345 ¼ 3 þ 4 þ Re 5, c3 ¼ cos3 , and s3 ¼ 2 2 sin3 . We learn that invariance under CP2 forces m22 ¼ m11 and 2 The last three equations may be written as m12 ¼ 0, 2 ¼ 1, and 7 ¼ 6, leading precisely to the ERPS of Eq. (33). Recall that Gunion and Haber [9] found 2 32 3 that, under these conditions we can always find a basis s2 ð3 þ c2Þð1 c2Þ Im 5 46 5746 57 where all parameters are real. As a result, if the potential is 0 ¼ s2 ð1 c2Þð3 þ c2Þ Im 6 : (92) invariant under CP2, there is a basis where CP2 still holds ð1 þ c2Þ s2 s2 Im 7 and in which the potential is also invariant under CP1. The determinant of this homogeneous system of three 3. CP3: 0 < < =2 equations in three unknowns is 32c2, which can never be Þ Finally, we turn to the cases where 0 < < =2. zero since we are assuming that =2. As a result, 5, Imposing Eq. (75) yields 6, and 7 are real, whatever the value of 0 < < =2 2 chosen for the GCP transformation. Since m12 ¼ 0, all 2 2 2 potentially complex parameters must be real. We conclude 0 ¼ Y11 ¼½ðm11 m22Þs 2Rem12c s; that a potential invariant under any GCP with 0 < < =2 0 ¼ Y22 ¼ Y11; is automatically invariant under CP1. Combining this with (89) 0 ¼ Y what we learned from CP2, we conclude the following: if a 12 potential is invariant under some GCP transformation, then 2 2 1 2 2 ¼ Rem12ðc2 1Þ 2i Imm12 þ 2ðm22 m11Þs2; a basis may be found in which it is also invariant under the standard CP transformation, with some added constraints where we have used c ¼ cos , s ¼ sin , c2 ¼ cos2 , and on the parameters. Þ 2 2 s2 ¼ sin2 . Since 0, =2, the conditions m22 ¼ m11 The other set of five independent homogeneous equa- 2 and m12 ¼ 0 are imposed, as in CP2. Similarly, Eq. (76) tions in five unknowns has a determinant equal to zero, yields meaning that not all parameters must vanish. We find that
116004-10 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) ð Þ2 ¼ 1 0 ¼ Z11;11 Z22;22 GCP (or equivalently, requiring the squared of the corresponding generalized time-reversal transformation to ¼ 2s½sð 1 2Þþc2Reð 6 þ 7Þ ; equal the unit matrix) was imposed in Ref. [9] and more (93) recently in Ref. [15]. However, as we have illustrated in 0 ¼ Re Z11;12 Re Z22;12 this section, the invariance under a GCP transformation, in ¼ s½ cð 1 2Þþs2Reð 6 þ 7Þ : which ðGCPÞ2 Þ 1 (corresponding to a unitary matrix X that is not symmetric) is a stronger restriction on the Since s Þ 0, we obtain the homogeneous system parameters of the scalar potential than the invariance under sc a standard CP transformation. 0 ¼ 1 2 ; (94) cs 2Reð 6 þ 7Þ As we see from the results in the previous sections, X is not symmetric for the symmetries CP2 and CP3. In fact, whose determinant is unity. We conclude that ¼ and 2 1 this feature provides a strong distinction among the three ¼ . Thus, GCP invariance with any value of 0 < 7 6 GCP symmetries previously introduced. Let us briefly =2 leads to the ERPS of Eq. (33). Substituting back examine ðGCPÞ2 for the three possible cases CP1, CP2, we obtain Z ¼ Z ¼ Z and 11;11 22;22 11;22 and CP3. Re Z11;12 ¼ Re Z22;12, leaving only two independent equations: 1. ðCP1Þ2 ¼ ¼ 1 ½ð Þ þ 0 Z11;11 2s2 1 345 s2 4 6c2 ; Comparing Eqs. (70) and (83), we come to the immedi- (95) 1 1 ate conclusion that XCP1 ¼ , so that Eq. (98) yields 0 ¼ Re Z22;12 ¼ 2s2½ð 1 345Þc2 4 6s2 ; ðCP1Þ2 ¼ 1: (99) where we have used c þ c3 ¼ 2cc2 and s þ s3 ¼ 2cs2. Þ Since s2 0, the determinant of the system does not This implies that a CP1-invariant scalar potential is invari- vanish, forcing ¼ and ¼ 0. 1 1 345 6 ant under the symmetry group Z2 ¼f ;CP1g. Notice that our results do not depend on which exact value of 0 < < =2 in Eq. (82) we have chosen. If we 2. ðCP2Þ2 require invariance of the potential under GCP with some particular value of 0 < < =2, then the potential is The matrix XCP2 is shown in Eq. (85) so that, by immediately invariant under GCP with any other value of Eq. (98), we obtain 0 < < =2. We name this class of CP invariances CP3. ðCP2Þ2 ¼ 1: (100) Combining everything, we conclude that invariance under CP3 implies Although this result significantly distinguished CP2 from CP1, the authors of Ref. [15] noted (in considering their 2 2 2 m11 ¼ m22;m12 ¼ 0; 2 ¼ 1; ðiÞ CPg symmetries) that the transformation law for a under 2 7 ¼ 6 ¼ 0; Im 5 ¼ 0; Re 5 ¼ 1 3 4: ðCP2Þ can be reduced to the identity by a global hyper- charge transformation. That is, if we start with the sym- (96) 1 1 metry group Z4 ¼f ;CP2; ; CP2g, we can impose an The results of this section are all summarized in Table I of equivalence relation by identifying two elements of Z4 1 Sec. V. related by multiplication by . If we denote ðZ2ÞY ¼ f1; 1g as the two-element discrete subgroup of the global C. The square of the GCP transformation hypercharge Uð1ÞY, then the discrete symmetry group that If we apply a GCP transformation twice to the scalar is orthogonal to Uð1ÞY is given by Z4=ðZ2ÞY ffi Z2. Hence, fields, we will have, from Eq. (70), that the CP2-invariant scalar potential exhibits a Z2 symmetry orthogonal to the Higgs flavor symmetries of the potential. GCP GCP GCP ð a Þ ¼ Xa ð Þ ¼ Xa X b b; (97) 3. 2 so that the square of a GCP transformation is given by ðCP3Þ The matrix X is given in Eq. (82), with 0 < < =2, ðGCPÞ2 ¼ XX : (98) CP3 so that, by Eq. (98), we obtain 2 In particular, for a generic unitary matrix X, ðGCPÞ is a cos2 sin2 Higgs family symmetry transformation. ðCP3Þ2 ¼ ; (101) sin2 cos2 Usually, only GCP transformations with ðGCPÞ2 ¼ 1 (where 1 is the unit matrix) are considered in the literature. which once again is not the unit matrix. However, the y 2 For such a situation, X ¼ X ¼ X , and one can always transformation law for a under ðCP3Þ cannot be reduced find a basis in which X ¼ 1. In this case, a GCP trans- to the identity by a global hypercharge transformation. formation is equivalent to a standard CP transformation in This is the reason why Ref. [15] did not consider CP3. the latter basis choice. For example, the restriction that However, ðCP3Þ2 is a nontrivial HF symmetry of the
116004-11 FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009) 6 CP3-invariant scalar potential. Thus, one can always constructed for those basis choices in which Z2 and Uð1Þ reduce the square of CP3 to the identity by applying a have the specific forms in Eqs. (24) and (27), respectively. suitable HF symmetry transformation. In particular, a If, for example, the basis is changed and Z2 acquires the CP3-invariant scalar potential also exhibits a Z2 symmetry form 2 in Eqs. (25), then the constraints on the coeffi- that is orthogonal to the Higgs flavor symmetries of the cients are altered, as shown explicitly on the fourth line of potential. Table I. However, this does not correspond to a new model. In this paper, we prove that there are three and only three All physical predictions are the same since the specific classes of GCP transformations. Of course, within each forms of Z2 and 2 differ only by the basis change in class, one may change the explicit form of the scalar Eq. (26). The constraints for CP1, CP2, and CP3 shown in potential by a suitable basis transformation; but that will Table I apply to the basis in which the GCP transformation not alter its physical consequences. Similarly, one can set of Eq. (70) is used where X has been transformed into X0 some parameters to zero in some ad hoc fashion, not rooted given by Eq. (82), with ¼ 0, ¼ =2, and 0 < < =2, in a symmetry requirement. But, as we have shown, the respectively. constraints imposed on the scalar potential by a single GCP symmetry can be grouped into three classes: CP1, CP2, B. Multiple symmetries and GCP and CP3. We now wish to consider the possibility of simulta- neously imposing more than one symmetry requirement V. CLASSIFICATION OF THE HF AND GCP on the Higgs potential. For example, one can require that TRANSFORMATION CLASSES IN THE THDM Z2 and 2 be enforced within the same basis. In what follows, we shall indicate that the two symmetries are A. Constraints on scalar potential parameters enforced simultaneously by writing Z2 È 2. Combining Suppose that one is allowed one single symmetry re- the constraints from the appropriate rows of Table I,we quirement for the potential in the THDM. One can choose conclude that, under these two simultaneous requirements an invariance under one particular Higgs family symmetry. 2 2 2 m22 ¼ m11;m12 ¼ 0; 2 ¼ 1; We know that there are only two independent classes of (102) ð Þ such simple symmetries: Z2 and Peccei-Quinn U 1 . One 7 ¼ 6 ¼ 0; Im 5 ¼ 0: can also choose an invariance under a particular GCP symmetry. We have proved that there are three classes of This coincides exactly with the conditions of the ERPS in a GCP symmetries, named CP1, CP2, and CP3. If any of the very special basis, as shown in Eq. (34). Since CP2 leads to above symmetries is imposed on the THDM scalar poten- the ERPS of Eq. (33), we conclude that tial (in a specified basis), then the coefficients of the Z È CP2 in some specific basis: (103) scalar potential are constrained, as summarized in 2 2 Table I. For completeness, we also exhibit the constraints This was noted previously by Davidson and Haber [8]. imposed by SOð3Þ, the largest possible continuous HF Now that we know what all classes of HF and CP symme- symmetry that is orthogonal to the global hypercharge tries can look like, we can ask whether all GCP symmetries Uð1ÞY transformation. can be written as the result of some multiple HF symmetry. Empty entries in Table I correspond to a lack of con- This is clearly not possible for CP1 because of parame- straints on the corresponding parameters. Table I has been ter counting. Table I shows that CP1 reduces the scalar
TABLE I. Impact of the symmetries on the coefficients of the Higgs potential in a specified basis.
2 2 2 Symmetry m11 m22 m12 1 2 3 4 5 6 7
Z2 000 Uð1Þ 0000 2 SOð3Þ m11 0 1 1 3 000 2 2 m11 real 1 real 6 CP1 real real real real 2 CP2 m11 0 1 6 2 CP3 m11 0 1 1 3 4 (real) 0 0
6In Sec. VB, we shall identify ðCP3Þ2 with the Peccei-Quinn Uð1Þ symmetry defined as in Eq. (27) and then transformed to a new basis according to the unitary matrix defined in Eq. (105).
116004-12 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) potential to ten real parameters. We can still perform an invariants are zero, then CP is explicitly conserved, but we orthogonal basis change while keeping all parameters real. cannot tell a priori which GCP applies. Equations (103) This freedom can be used to remove one further parameter; and (106) provide the crucial hint. If we have CP conser- 2 for example, setting m12 ¼ 0 by diagonalizing the Y ma- vation, Z2 È 2 holds, and Uð1Þ does not, then we have trix. No further simplification is allowed. As a result, CP1 CP2. Alternatively, if we have CP conservation, and leaves nine independent parameters. The smallest HF sym- Uð1ÞÈ 2 also holds, then we have CP3. We recall that metry is Z2. Table I shows that Z2 reduces the potential to both CP2 and CP3 lead to the ERPS, and that the general six real and one complex parameter. The resulting eight conditions for the ERPS in Eq. (33) are basis independent. parameters could never account for the nine needed to fully This allows us to distinguish CP2 and CP3 from CP1. But, describe the most general model with the standard CP prior to the present work, no basis-independent quantity invariance CP1.7 had been identified in the literature that could distinguish But one can utilize two HF symmetries in order to obtain Z2 and Uð1Þ in the ERPS. The basis-independent quantity the same constraints obtained by invariance under CP3. D introduced in Sec. II E is precisely the invariant required Þ Let us impose both Uð1Þ and 2 in the same basis. From for this task. That is, in the ERPS D 0 implies CP2, Table I, we conclude that, under these two simultaneous whereas D ¼ 0 implies CP3. requirements One further consequence of the results of Table I can be 2 2 2 seen by simultaneously imposing the Uð1Þ-Peccei-Quinn m22 ¼ m11;m12 ¼ 0; 2 ¼ 1; (104) symmetry and the CP3 symmetry in the same basis. The 7 ¼ 6 ¼ 0; 5 ¼ 0: resulting constraints on the scalar potential parameters are precisely those of the SOð3Þ HF symmetry. Thus, we This does not coincide with the conditions for invariance conclude that under CP3 shown in Eq. (96). However, one can use the transformation rules in Eqs. (A13)–(A23) of Davidson and Uð1ÞÈCP3 SOð3Þ: (107) Haber [8], in order to show that a basis transformation SOð Þ In particular, 3 is not a simple HF symmetry, as the invariance of the scalar potential under a single element of 1ffiffiffi 1 i U ¼ p (105) SOð Þ 2 i 1 3 is not sufficient to guarantee invariance under the full SOð3Þ group of transformations. may be chosen, which takes us from Eqs. (96), where Re 5 ¼ 1 3 4, to Eqs. (104), where 5 ¼ 0 (while C. Maximal symmetry group of the scalar potential maintaining the other relations among the scalar potential orthogonal to Uð1ÞY parameters). We conclude that The standard CP symmetry CP1 is a discrete Z2 sym- Uð1ÞÈ 2 CP3 in some specific basis: (106) metry that transforms the scalar fields into their complex Note that in the basis in which the CP3 relations of Eq. (96) conjugates, and hence is not a subgroup of the Uð2Þ trans- Þ formation group of Eq. (15). We have previously noted that are satisfied with 5 0, the discrete HF symmetry 2 is still respected. However, using Eq. (105), it follows that the THDM scalar potentials that exhibit any nontrivial HF Uð1Þ-Peccei-Quinn symmetry corresponds to the invari- symmetry G is automatically CP conserving. Thus, the ance of the scalar potential under ! O , where actual symmetry group of the scalar potential is in fact the a ab b 8 2 O is an arbitrary SOð2Þ matrix. semidirect product of G and Z2, which we write as G Z2. 2 2 The above results suggest that it should be possible to Noting that Uð1Þ Z2 ffi SOð2Þ Z2 ffi Oð2Þ, and SOð3Þ distinguish CP1, CP2, and CP3 in a basis-invariant fash- Z2 ffi Oð3Þ, we conclude that the maximal symmetry groups of the scalar potential orthogonal to Uð1ÞY for the ion. Botella and Silva [6] have built three so-called J 9 invariants that detect any signal of CP violation (either possible choices of HF symmetries are given in Table II. Finally, we reconsider CP2 and CP3. Equation (103) explicit or spontaneous) after the minimization of the 2 scalar potential. However, in this paper we are concerned implies that the CP2 symmetry is equivalent to a ðZ2Þ HF about the symmetries of the scalar potential independently symmetry. To prove this statement, we note that in the two- of the choice of vacuum. Thus, we shall consider the four dimensional flavor space of Higgs fields, the Z2 and 2 so-called I invariants built by Gunion and Haber [9]in discrete symmetries defined by Eqs. (24) and (25) are given order to detect any signal of explicit CP violation present by (before the vacuum state is determined). If any of these 8 invariants is nonzero, then CP is explicitly violated, and In general, the nontrivial element of Z2 will not commute neither CP1, nor CP2, nor CP3 hold. Conversely, if all I with all elements of G, in which case the relevant mathematical structure is that of a semidirect product. In cases where the nontrivial element of Z2 commutes with all elements of G,we 7 In Ivanov’s language, this is clear since CP1 corresponds to a denote the corresponding direct product as G Z2. 9 2 Z2 transformation of the vector r~, which is the simplest trans- For ease of notation, we denote Z2 Z2 by ðZ2Þ and Z2 3 formation on r~ one could possibly make. See Sec. VD. Z2 Z2 by ðZ2Þ .
116004-13 FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009) TABLE II. Maximal symmetry groups [orthogonal to global 3 potential is ðZ2Þ . Similarly, Eq. (106) implies that the Uð1ÞY hypercharge] of the scalar sector of the THDM. 2 CP3 symmetry is equivalent to a Uð1Þ Z2 HF symmetry. HF symmetry Maximal symmetry This is isomorphic to an Oð2Þ HF symmetry, which is a Designation group group subgroup of the maximally allowed SOð3Þ HF symmetry group. However, the constraints of CP3 on the scalar 2 Z2 Z2 ðZ2Þ potential imply the existence of a basis in which all scalar Peccei-Quinn Uð1Þ Oð2Þ potential parameters are real. Thus, the scalar potential is SOð Þ SOð Þ Oð Þ 3 3 3 explicitly CP conserving. Once again, the Z symmetry CP1 - Z 2 2 associated with this CP transformation is orthogonal to the CP2 ðZ Þ2 ðZ Þ3 2 2 HF symmetry noted above. Thus, the maximal symmetry CP3 Oð2Þ Oð2Þ Z2 group of the CP3-symmetric scalar potential is Oð2Þ Z2. The above results are also summarized in Table II. In all cases, the maximal symmetry group is a direct product of Z2 ¼fS0;S1g; 2 ¼fS0;S2g; (108) the HF symmetry group and the Z2 corresponding to the 1 where S0 is the 2 2 identity matrix and standard CP transformation, whose square is the identity operator. 10 01 One may now ask whether Table II exhausts all possible S ¼ ;S¼ : (109) 1 0 1 2 10 independent symmetry constraints that one may place on the Higgs potential. Perhaps one can choose other combi- If we impose the Z2 and 2 symmetry in the same basis, nations, or maybe one can combine three, four, or more then the scalar potential is invariant under the dihedral symmetries. We know of no way to answer this problem group of eight elements based only on the transformations of the scalar fields a. Fortunately, Ivanov has solved this problem [3] by looking D4 ¼fS0;S1;S2;S3; S0; S1; S2; S3g; (110) at the transformation properties of field bilinears, thus obtaining for the first time the list of symmetries given in where S3 ¼ S1S2 ¼ S2S1. As before, we identify the last column of Table II. ðZ2ÞY fS0; S0g as the two-element discrete subgroup of the global hypercharge Uð1ÞY. However, we have de- D. More on multiple symmetries fined the HF symmetries to be orthogonal to Uð1ÞY. Thus, to determine the HF symmetry group of CP2, we identify We start by looking at the implications of the symme- tries we have studied so far on the vector r~ ¼fr ;r ;r g, as equivalent those elements of D4 that are related by 1 2 3 whose components were introduced in Eq. (10). Notice that multiplication by S0. Grouped theoretically, we identify the HF symmetry group of CP2 as a unitary transformation U on the fields a induces an orthogonal transformation O on the vector of bilinears r~,
D4=ðZ2ÞY ffi Z2 Z2: (111) given by Eq. (37). For every pair of unitary transformations ÆU of SUð2Þ, one can find some corresponding trans- The HF symmetry group of CP2 is not the maximally formation O of SOð3Þ, in a two-to-one correspondence. allowed symmetry group. In particular, the constraints of We then see what these symmetries imply for the coeffi- CP2 on the scalar potential imply the existence of a basis in cients of Eq. (9) [recall the is a symmetric matrix]. which all scalar potential parameters are real. Thus, the Below, we list the transformation of r~ under which the scalar potential is explicitly CP conserving. The Z2 sym- scalar potential is invariant, followed by the corresponding metry associated with this CP transformation is orthogonal constraints on the quadratic and quartic scalar potential to the HF symmetry as previously noted. (This is easily parameters M and . checked explicitly by employing a four-dimensional real Using the results of Table I, we find that Z2 implies representation of the two complex scalar fields.) Thus, the maximal symmetry group of the CP2-symmetric scalar
2 3 2 3 2 3 M 00 r 0 00 03 6 1 7 6 7 6 7 4 5 6 0 7 6 0 11 12 0 7 r~ ! r2 ; 4 5; 4 5; (112) 0 0 12 22 0 r3 M3 03 00 33
Uð1Þ implies
116004-14 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) 2 3 2 3 2 3 M 00 c s 0 6 0 7 6 00 03 7 6 2 2 7 6 0 7 6 0 007 r~ ! 4 s c 0 5r;~ 6 7; 6 11 7; (113) 2 2 4 0 5 4 00 0 5 001 11 M3 03 00 33 and SOð3Þ implies 2 3 2 3 M 000 6 0 7 6 00 7 6 0 7 6 0 11 007 r~ ! Or;~ 46 57; 46 57; (114) 0 00 11 0 0 000 11 where O is an arbitrary 3 3 orthogonal matrix of unit determinant. In the language of bilinears, a basis-invariant ~ ~ condition for the presence of SOð3Þ is that the three eigenvalues of are equal. (Recall that ¼f ijg; i, j ¼ 1,2,3.) As for the GCP symmetries, CP1 implies 2 3 2 3 2 3 M 0 r 0 00 01 03 6 1 7 6 7 6 7 4 5 6 M1 7 6 01 11 0 13 7 r~ ! r2 ; 4 5; 4 5; (115) 0 00 22 0 r3 M3 03 13 0 33 CP2 implies 2 3 2 3 2 3 M 000 r 0 00 6 1 7 6 7 6 7 4 5 6 0 7 6 0 11 12 13 7 r~ ! r2 ; 4 5; 4 5; (116) 0 0 12 22 23 r3 0 0 13 23 33 and CP3 implies 2 3 2 3 2 3 M0 00 000 c2 0 s2 6 7 6 7 6 7 6 0 7 6 0 11 007 r~ ! 4 0 105r;~ 46 57; 46 57: (117) 0 00 22 0 s2 0 c2 0 000 11
Notice that in CP3 two of the eigenvalues of are equal, in requirement that one can place on the Higgs potential via accordance with our observation that D can be used to multiple symmetries. Combining this with our earlier re- distinguish between CP2 and CP3. sults, we conclude that all possible symmetries on the Because each unitary transformation on the fields a scalar sector of the THDM can be reduced to multiple induces an SOð3Þ transformation on the vector of bilinears HF symmetries, with the exception of the standard CP r~, and because the standard CP transformation corresponds transformation (CP1). to an inversion of r2 (a Z2 transformation on the vector r~), Ivanov [3] considers all possible proper and improper VI. BUILDING ALL SYMMETRIES WITH THE transformations of Oð3Þ acting on r~. He identifies the STANDARD CP 2 following six classes of transformations: (i) Z2; (ii) ðZ2Þ ; 3 We have seen that there are only six independent sym- (iii) ðZ2Þ ;(iv)Oð2Þ; (v) Oð2Þ Z2; and (vi) Oð3Þ. Note metry requirements, listed in Table II, that one can impose that these symmetries are all orthogonal to the global on the Higgs potential. We have shown that all possible Uð1ÞY hypercharge symmetry, as the bilinears r0 and r~ symmetries of the scalar sector of the THDM can be are all singlets under a Uð1ÞY transformation. The six reduced to multiple HF symmetries, with the exception classes above identified by Ivanov correspond precisely of the standard CP transformation (CP1). Now we wish to to the six possible maximal symmetry groups identified show a dramatic result: all possible symmetries on the in Table II. No other independent symmetry transforma- scalar sector of the THDM can be reduced to multiple tions are possible. applications of the standard CP symmetry. Our work permits one to identify the abstract transfor- Using Eq. (79), we see that the basis transformation of mation of field bilinears utilized by Ivanov in terms of Eq. (15), changes the standard CP symmetry of Eq. (69) transformations on the scalar fields themselves, as needed into the GCP symmetry of Eq. (70), with for model building. Combining our work with Ivanov’s, we > conclude that there is only one new type of symmetry X ¼ UU : (118)
116004-15 FERREIRA, HABER, AND SILVA PHYSICAL REVIEW D 79, 116004 (2009)
In particular, an orthogonal basis transformation does not where Þ n =2 with n integer. We denote by CP1C the affect the form of the standard CP transformation. Since imposition of the CP symmetry in Eq. (119) with XM ¼ we wish to generate X Þ 1, we will need complex matrices XC (which coincides with the imposition of the standard U. CP symmetry in the basis M ¼ C). Now we wish to consider the following situation: We Finally, we consider have a basis (call it the original basis) and impose the c =2 is =2 c is standard CP symmetry CP1 on that original basis. Next UD ¼ ;XD ¼ ; (123) we consider the same model in a different basis (call it M) is =2 c =2 is c and impose the standard CP symmetry on that basis M.In where Þ n =2 with n integer. We denote by CP1D the general, this procedure of imposing the standard CP sym- imposition of the CP symmetry in Eq. (119) with XM ¼ metry in the original basis and also in the rotated basis M XD (which coincides with the imposition of the standard leads to two independent impositions. The first imposition CP symmetry in the basis M ¼ D). makes all parameters real in the original basis. One way to The impact of the first three symmetries on the coeffi- combine the second imposition with the first is to consider cients of the Higgs potential are summarized in Table III. the basis transformation U taking us from basis M into M Imposing CP1D on the Higgs potential leads to the more the original basis. As we have seen, the standard CP complicated set of equations symmetry in basis M turns, when written in the original 2 2 2 basis, into a symmetry under 2Imðm12Þc þðm22 m11Þs ¼ 0;
CP yCP y 2Imð 6 7Þc2 þ 12345s2 ¼ 0; a ¼ðXMÞa ; a ¼ðXMÞa ð Þ ; (119) (124) ð þ Þc þð Þs ¼ ; > 2Im 6 7 1 2 0 with XM ¼ UMUM. Next we consider several such possibilities. Im 5c þ Reð 6 7Þs ¼ 0; We start with where c =4 is =4 0 i 1 UA ¼ ;XA ¼ : is c 12345 ¼ ð 1 þ 2Þ 3 4 þ Re 5: (125) =4 =4 i 0 2 (120) Combining these results with those in Table I, we have Here and henceforth c (s) with a subindex indicates the shown that cosine (sine) of the angle given in the subindex. We denote CP1 È CP1B ¼ Z2 in some specific basis; by CP1A the imposition of the CP symmetry in Eq. (119) with XM ¼ XA (which coincides with the imposition of the CP1 È CP1C ¼ Uð1Þ; standard CP symmetry in the basis M ¼ A). CP1 È CP1 È CP1 ¼ CP2 in some specific basis; Next we consider A B ! CP1 È CP1A È CP1C ¼ CP3 in some specific basis; e i =4 0 i 0 U ¼ ;X¼ : (121) CP1 È CP1 È CP1 ¼ SOð3Þ: (126) B 0 ei =4 B 0 i C D Let us comment on the ‘‘specific basis choices’’ needed. We denote by CP1B the imposition of the CP symmetry in 2 Imposing CP1 È CP1B leads to m12 ¼ 6 ¼ 7 ¼ 0 and Eq. (119) with XM ¼ XB (which coincides with the impo- 2 Im 5 ¼ 0, while imposing Z2 leads to m ¼ 6 ¼ 7 ¼ 0 sition of the standard CP symmetry in the basis M ¼ B). 12 with no restriction on 5. However, when Z2 holds, one A third possible choice is may rephase by the exponential of i argð Þ=2, thus ! 2 5 making real. In this basis, the restrictions of Z coincide ei =2 0 ei 0 5 2 UC ¼ i =2 ;XC ¼ i ; (122) with the restrictions of CP1 È CP1B. Similarly, imposing 0 e 0 e 2 CP1 È CP1A È CP1C leads to m12 ¼ 5 ¼ 6 ¼ 7 ¼ 0,
TABLE III. Impact of the CP1M symmetries on the coefficients of the Higgs potential. The notation ‘‘imag’’ means that the corresponding entry is purely imaginary. CP1 in the original basis has been included for reference.
2 2 2 Symmetry m11 m22 m12 1 2 3 4 5 6 7 CP1 real real real real 2 CP1A m11 1 6 CP1B imag real imag imag 2 i 2i i i CP1C jm12je j 5je j 6je j 7je
116004-16 GENERALIZED CP SYMMETRIES AND SPECIAL ... PHYSICAL REVIEW D 79, 116004 (2009) 2 2 m22 ¼ m11, and 2 ¼ 1. We see from Table I that CP3 two independent classes (CP2 and CP3), in addition to the has these features, except that 5 need not vanish; it is real standard CP symmetry (CP1). These two classes lead to an and Re 5 ¼ 1 3 4. Starting from the CP3 condi- exceptional region of parameter, which exhibits either a Z2 tions and using the transformation rules in Eqs. (A13)– discrete symmetry or a larger Uð1Þ-Peccei-Quinn symme- (A23) of Davidson and Haber [8], we find that a basis try. We have succeeded in identifying a basis-independent 10 choice is possible such that Re 5 ¼ 0. Perhaps it is easier invariant quantity that can distinguish between the Z2 and to prove the equality Uð1Þ symmetries. In particular, such an invariant is re- quired in order to distinguish between CP2 and CP3, and CP1 È CP1B È CP1D ¼ CP3 in some specific basis: completes the description of all symmetries in the THDM (127) in terms of basis-invariant quantities. Moreover, CP2 and CP3 can be obtained by combining two Higgs family In this case, the only difference between the impositions symmetries and that this is not possible for CP1. from the two sides of the equality come from the sign of We have shown that all symmetries of the THDM pre- Re , which is trivial to flip through the basis change 5 viously identified by Ivanov [3] can be achieved through ! . Finally, imposing CP1 È CP1 È CP1 we 2 2 A B simple symmetries, with the exception of SOð3Þ. However, obtain m2 ¼ Im ¼ ¼ ¼ 0, m2 ¼ m2 , and 12 5 6 7 22 11 the SOð3Þ Higgs family symmetry can be achieved by ¼ . This does not coincide with the conditions of 2 1 imposing a Uð1Þ-Peccei-Quinn symmetry and the CP3 CP2, which lead to the ERPS of Eq. (33). Fortunately, and symmetry in the same basis. Finally, we have demonstrated as we mentioned before, Davidson and Haber [8] proved that all possible symmetries of the scalar sector of the that one may make a further basis transformation such that THDM can be reduced to multiple applications of the Eq. (34) holds, thus coinciding with the conditions im- standard CP symmetry. Our complete description of the posed by CP1 È CP1 È CP1 . A B symmetries on the scalar fields can be combined with Notice that our description of CP2 in terms of several symmetries in the quark and lepton sectors, to aid in model CP1 symmetries is in agreement with the results found by building. the authors of Ref. [15]. These authors also showed a very interesting result, concerning spontaneous symmetry breaking in 2HDM models possessing a CP2 symmetry. ACKNOWLEDGMENTS Namely, they prove (their Theorem 4) that electroweak We would like to thank I. Ivanov and C. Nishi for their symmetry breaking will necessarily spontaneously break helpful comments on the first version of this manuscript. CP2. However, they also show that the vacuum will respect The work of P.M. F. is supported in part by the Portuguese at least one of the CP1 symmetries, which compose CP2. Fundac¸a˜o para a Cieˆncia e a Tecnologia (FCT) under Which is to say, in a model that has a CP2 symmetry, Contract No. PTDC/FIS/70156/2006. The work of spontaneous symmetry breaking necessarily respect the H. E. H. is supported in part by the U.S. Department of CP1 symmetry. Energy, under Grant No. DE-FG02-04ER41268. The work In summary, we have proved that all possible symme- of J. P.S. is supported in part by FCT under Contract tries on the scalar sector of the THDM, including Higgs No. CFTP-Plurianual (U777). H. E. H. is most grateful family symmetries, can be reduced to multiple applications for the kind hospitality and support of the Centro de of the standard CP symmetry. Fı´sica Teo´rica e Computacional at Universidade de Lisboa (sponsored by the Portuguese FCT and Fundac¸a˜o VII. CONCLUSIONS Luso-Americana para o Desenvolvimento) and the Centro de Fı´sica Teo´rica de Partı´culas at Instituto Superior We have studied the application of generalized CP Te´cnico during his visit to Lisbon. This work was initiated symmetries to the THDM, and found that there are only during a conference in honor of A. Barroso.
10 Notice that, in the new basis, 1 differs in general from 3 þ 4; otherwise the larger SOð3Þ Higgs family symmetry would hold.
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