Physical parameters and basis transformations in the Two-Higgs-Doublet model C. C. Nishi∗ Instituto de Fı́sica “Gleb Wataghin” Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, SP, Brasil and Instituto de Fı́sica Teórica, UNESP – São Paulo State University Rua Pamplona, 145, 01405-900 – São Paulo, Brasil A direct connection between physical parameters of general Two-Higgs-Doublet Model (2HDM) potentials after electroweak symmetry breaking (EWSB) and the parameters that define the potentials before EWSB is established. These physical parameters, such as the mass matrix of the neutral Higgs bosons, have well defined transformation properties under basis transformations transposed to the fields after EWSB. The relations are also explicitly written in a basis covariant form. Violation of these relations may indicate models beyond 2HDMs. In certain cases the whole potential can be defined in terms of the physical parameters. The distinction between basis transformations and reparametrizations is pointed out. Some physical implications are discussed. arXiv:0712.4260v2 [hep-ph] 3 Jan 2008 I. INTRODUCTION The Standard Model (SM) relies on the Higgs mechanism to give masses to all massive gauge bosons and elementary fermions. Such mechanism involves the spontaneous symmetry breaking (SSB) of the electroweak SU (2)L ⊗ U (1)Y gauge group to the electromagnetic U (1)EM as a scalar Higgs doublet acquires a nonzero vacuum expectation value (VEV). Such scheme imposes universality constraints on the couplings between fermions and gauge bosons, establishes relations between the masses of the gauge bosons and fixes the couplings between the physical Higgs and the fermions to be proportional to the masses of the latter [1]. The scalar potential, constituted by only one Higgs doublet, is also very restrictive since, from the knowledge of the electroweak (EW) VEV and the physical Higgs mass, the Higgs trilinear and quartic coupling constants are fixed at tree level. However, the physical Higgs boson, which is the only scalar remnant of the EW Higgs mechanism, was not discovered yet. One then resorts to indirect means to bound the physical Higgs mass, most of them relying on the higher order perturbative behavior of the SM. Such bounds comes from, e.g., the unitarity constraints of the scattering of gauge bosons, the validity of the SM up to the Planck scale and the EW vacuum stability (see Ref.1 and references therein). Studies constraining the Higss mass are very important to its search, in particular, in view of the upcoming LHC experiment. For models extending the electroweak symmetry breaking (EWSB) sector of the SM, the relation of the model parameters before EWSB and the physical parameters identified after EWSB may not be as minimal as in the SM. Specially for N-Higgs-doublet extensions of the SM (NHDMs), the multiplicity of independent parameters may be quite large due to the presence of a horizontal space, i.e., the space of identical gauge multiplets, in ∗ Electronic address: ccnishi@ifi.unicamp.br this case, SU (2)L doublets with quantum numbers identical to the SM Higgs doublet. The simplest Two-HiggsDoublet Model (2HDM) has been extensively studied recently [2, 3, 4, 5, 6, 7, 8, 10, 11] as the effective scalar sector of the MSSM requires two Higgs doublets for anomaly cancelation [1, 12]. Historically, the addition of one or more Higgs doublets were considered to implement the spontaneous CP violation mechanism (SCPV) [13, 14] as an alternative source of CP violation. Technical difficulties that arise when considering NHDMs are twofold: (i) more than one local minimum (orbit), not necessary with the same symmetry breaking pattern, might be present, even at tree level, and (ii) the reparametrization freedom [11] allowed by the presence of the horizontal space formed by the N Higgs-doublets may masquerade the number of relevant independent parameters and symmetry properties such as CP invariance. Difficulty (i) includes the possibility of potentials with no remaining U (1)EM symmetry after EWSB (charge breaking vacuum) [4, 15, 16] and it forces the stability of the vacuum to be a relevant issue at tree level [15]. Item (ii) concerns the reparametrization transformations induced by basis transformations (or horizontal transformations [17]) acting on the identical N Higgsdoublets. Since all the doublets have the same gauge quantum numbers with respect to the SM gauge group, there is no change in the physical content of the theory if one rotates the fields in such space [18, 19, 21]. Such possibility may masquerade the number of relevant independent parameters in the theory. More crucially, transforming real parameters into complex parameters (for complex multiplets), CP invariant theories can be disguised as CP violating theories. This issue can be solved in an objective way by noting that CP invariant theories remain CP invariant through basis transformations but the corresponding CP symmetry transformation also acts differently in different bases. As a practical way of distinguishing the CP property of a theory one can resort to the use of CP-odd basis invariants analogous to the Jarlskog invariant [22] in the quark sector of the SM. In the context of NHDMs, numerous of such invariants can be con- 2 structed [19] but sufficient conditions for CP invariance using a minimum number of invariants could be formulated only for the potentials of 2HDMs [7, 19, 20, 21, 23] and 3HDMs [21]. Recent advances in the study of 2HDMs include the result that at most two local minima can be present whenever there is a discrete set of minima in the orbit space [8, 9]. Such result was obtained by using a Minkowski structure that emerges naturally in the space of the fields through suitable change of variables [10, 21]. Although some controversy remain from numerical examples presenting more than two local minima [24]. Since the counting of the number of local minima can be a very difficult task, the upper bound of the number of minima is an important result. It was also proved for 2HDMs that (a) charge breaking vacua can not coexist with a neutral vacuum [4, 10] and (b) spontaneously CP violating vacua can not coexist with CP invariant vacuum [4, 8, 10]. The result (b) can be extended in a weaker version to NHDMs: a spontaneously CP violating extremum always lies above a CP invariant extremum if the latter exists [16]. The Minkowski structure can be also partially extended to general NHDM potentials [25]. Bearing these results in mind, the present article aims two goals concerning the 2HDM potential: (i) to extract all the physical parameters identifiable only after EWSB and (ii) to study their properties under basis transformations. The first goal involves having a more direct connection between the parameters of the potential and the physical parameters after EWSB. Since basis transformations are allowed before EWSB and they are usually involved to reach the physical basis, it would be desirable to have a basis covariant relation for the physical parameters, which leads to goal (ii). A systematic study of the physical parameters of 2HDMs, including the scalar selfinteractions and the interactions of scalars with fermions and gauge bosons, was carried out in Ref. 6. The basis covariance, however, was not extended to the fields after EWSB. Another aspect of item (i) regards seeking a physical parametrization of the 2HDM potential by rewriting the parameters before EWSB in terms of the physical parameters. As explained in the end of Ref. 25, what prevents the utility of a parametrization depending on physical parameters, such as the masses of the physical charged and neutral Higgs bosons, is the possibility of the potential so defined possess another deeper minimum. The existence of at most two local minima already ameliorate the situation. Such parametrization also excludes by construction the potentials without nontrivial minima. For the cases we know there is only one local minimum and hence it is also the global one, such parametrization is unambiguous. The question that remains is to know if such parametrization can cover all possible 2HDM potential containing only one global minimum. The ouline is as follows: in Sec. II we find the mass matrix for neutral scalars in the basis where the mass matrix for charged Higgs bosons is already diagonal. Re- lations between potential parameters and the mass matrices are found. In Sec. III, covariant relations between the mass matrices and the potential parameters are shown. In Sec. IV we show how to achieve the truly physical basis where all the mass matrices are diagonal, pointing out the distinction between basis transformations before and after EWSB. Finally, the results and physical implications are discussed in Sec. V. Some possibly useful material, including an alternative method to ensure a bounded below potential, is presented in the appendices. II. PHYSICAL PARAMETERS IN THE PCH BASIS A general 2HDM potential can be divided into its quadratic and quartic parts as V = V2 + V4 . (1) The quadratic part is usually written as V2 = Yab Φ†a Φb , a, b = 1, 2, (2) where Y is a hermitian matrix and Φa = (φa1 , φa2 ) are the Higgs doublets for which the notation φa1 = φ(+) a and φa2 = φ(0) a is usually adopted when the vacuum preserves the electromagnetic symmetry. The quartic part can be conveniently written [21] as V4 = 12 Λµν rµ rν , µ, ν = 0, 1, 2, 3, (3) where Λ is a 4 × 4 real symmetric matrix while rµ = 12 (σµ )ab Φ†a Φb , (4) for µ = 0, 1, 2, 3, are real quadratic combinations of the doublets. The matrix σ0 ≡ ✶2 and σi are the Pauli matrices. The quadratic variables rµ = (r0 , r) are functionally free except for the future lightcone constraint [10, 25] r02 − r2 ≥ 0 , r0 ≥ 0 . (5) The convention of summation over repeated indices is adopted with Euclidean metric. For example, rµ rµ = r02 + r2 . The Minkowski metric will not be used to avoid confusion and all indices will be written as lower indices, differently of Refs. [25] and [10]. Using the variables rµ , the quadratic part of the potential in Eq. (2) can be cast into the form V 2 = Mµ rµ , (6) where Mµ has four independent components. The number of free parameters contained in Y and M are the same and they are indeed related by Y = Mµ 12 σµ ↔ Mµ = Tr[σµ Y ] . (7) We want to parametrize the potential in terms of physical quantities that are defined after EWSB. Confining 3 ourselves to neutral vacua, the first choice of physical parameters will be the masses of physical particles, i.e., one charged scalar and three neutral scalars. In addition, more parameters such as the EW vacuum expectation value (VEV), the mixing among the neutral scalars and certain coupling constants will be necessary to completely parametrize the potential that requires 11 essential parameters. To extract the physical masses, we need the quadratic part of the potential after EWSB that is induced by Φa → hΦa i + Φa , (8) where hΦa i is the vacuum expectation value (VEV) of Φa , usually a c-number minimum of the potential in (1). The extremum equations are shown in appendix A. With the shift of Eq. (8), the quadratic part of the potential can be written as [25] V2 SSB = Φ†a hMiab Φb + 21 Λµν sµ sν , (9) where sµ = † 1 2 hΦa i (σµ )ab Φb + h.c., (10) and hMi is the mass squared matrix for the charged Higgs bosons, including the charged Goldstone. (The mass squared matrix will be denoted simply as “mass matrix” from this point on.) Such matrix can be calculated as [25] hMi = Y + 12 σµ Λµν hrν i , (11) hrµ i = 21 (σµ )ab hΦa i† hΦb i . (12) where In the Physical Charged Higgs (PCH) basis [25], for a neutral vacuum, the VEVs are simply     0 0 , hΦ2 i = √v , hΦ1 i = (13) 0 2 while the doublets after the shift (8) can be parametrized as     G+ h+ Φ1 = √1 (t − it ) , Φ2 = √1 (−t + iG0 ) , (14) 2 1 2 2 v2 (1, 0, 0, −1) . 4 (15) In the PCH basis, the mass matrix for the charged scalars can be written hMi = diag(m2 , 0) , V2 (16) SSB = V2 charged + V2 neutral . (17) For the charged fields we have V2 charged = m2 h + h − , (18) while V2 = neutral = 2 1 1 2 2 2 m (t1 + t2 ) + 2 Λµν sµ sν 1 2 ti (MN )ij tj , , (19) (20) where, using the parametrization of Eq. (14), v v s0 = − t3 , si = ti . 2 2 (21) The 3 × 3 matrix MN is the mass matrix for the physical neutral scalars given by v2 MN = m2 diag(1, 1, 0) + Λ̃ 4   0 0 −Λ01 2 v  0 0 −Λ02  , (22) + 4 −Λ −Λ Λ − 2Λ 01 02 00 03 where Λ̃ = {Λij }, i, j = 1, 2, 3. The physical neutral scalars will be orthogonal combinations of ti , defined by the diagonalization of MN in Eq. (22). The mass matrix MN in the PCH basis can be also found in Eqs. (24) and (41) of Ref. 6 in a different notation. From Eqs. (11), (15) and (22) we can find the following relation between Y and MN ,   2Y11 (MN )13 − i(MN )23 1 Y =2 . (MN )13 + i(MN )23 −(MN )33 (23) Except for Y11 , all elements of Y are directly related to MN . Hence, we can completely parametrize the potential in the PCH basis in terms of the set of 12 parameters {v, m, Λ00 , MN , Λ0 } , 3 where v = 246GeV, is the electroweak VEV, ti , i = 1, 2, 3, are normalized neutral scalar fields, h+ is the physical charged Higgs and G+ and G0 are the charged and neutral Goldstone fields respectively. The Goldstone fields G+ and G0 are absorbed by the longitudinal W + and Z 0 gauge bosons by the Higgs mechanism. For the VEV of Eq. (13), we have hrµ i = where the null eigenvalue corresponds to the charged Goldstone. We can divide the quadratic part of the potential of Eq. (9) into (24) where Λ0 = {Λ0i }, i = 1, 2, 3. For fixed values for the set in Eq. (24), we obtain from Eq. (22) the rest of Λµν by 4 [(MN )ij − m2 δij ], i, j = 1, 2 , v2 4 = Λ0i + 2 (MN )i3 , i = 1, 2 , v 4 = −Λ00 + 2Λ03 + 2 (MN )33 . v Λij = (25) Λi3 (26) Λ33 (27) The quadratic parameter Y11 depends on more parameters other than MN as Y11 = m2 − v2 (Λ00 − Λ03 ) + 21 (MN )33 . 4 (28) 4 There are 12 free parameters. Among these, 11 are essential and can not be eliminated by reparametrization [21]. Nevertheless, one parameter can be removed by a remaining U (1) reparametrization freedom due to Φ1 → eiθ Φ1 , Φ2 → Φ2 . (29) h+ → eiθ h+ (30) Since represents the electromagnetic U (1)EM invariance of the potential, the transformation of Eq. (29) amounts effectively to a SO(2) rotation in the t1 , t2 fields. Notice that since hΦ1 i = 0, the transformation of Eq. (29) does not affect the VEVs. Choosing appropriately θ in Eq. (29), one can set one of the following parameters to zero: (MN )12 , (MN )13 , (MN )23 . In particular, choosing (MN )23 = 0, we obtain a real symmetric Y and we constrain Λ23 = Λ03 . For any choice the overall number of free independent parameters should be 11. Of course, different choices, such as Λ01 = 0 or Λ02 = 0, could be alternatively chosen. Obviously, once a choice is made, one can not set more than one parameter to zero. To ensure the vacuum in Eq. (14) to be a local minimum, it is sufficient to pick positive values for m2 (the mass squared of h+ ) and for the three eigenvalues of MN (the masses squared of ti ). Such requirements guarantee that the second derivative of the potential around the extremum is positive semidefinite. There remains the question of boundedness for the potential defined with physical parameters (24). Firstly, we have to choose Λ00 ≥ 0 because taking r0 → ∞ but |r| finite in Eq. (3) would make V4 acquire negative values if Λ00 < 0. Moreover, the following statement can be proved: For a potential V (r) defined as Eq. (1), satisfying Λ00 + λi > 0 for all λi , i = 1, 2, 3, eigenvalues of Λ̃, it is always possible to obtain Λµν rµ rν > 0, for all rµ satisfying Eq. (5), by making the substitution Λ0 → cΛ0 , c > 0 , (31) with appropriately small c. The proof is shown in appendix B The only problem that could make such physical parametrization not viable is the possibility that the potential defined for a given set (24) possess another minimum that lies deeper than the one defined in (13). This possibility is real and numerical examples can be quickly deviced. The problem is not so severe because at most two distinct local minima are possible for bounded below potentials containing two Higgs doublets [8]. Although, in Ref. 24, some numerical examples of 2HDM potentials with more than two minima were apparently devised [24]. On the other hand, this parametrization excludes pontentials without nontrivial minima by construction. III. PHYSICAL PARAMETERS IN AN ARBITRARY BASIS For a general potential (1), the vacuum expectation value (VEV) will not be in the form of Eq. (13). Nonetheless, we can always parametrize a neutral vacuum as     0 0 v v , hΦ2 i = √ . (32) hΦ1 i = √ θ θ iξ 2 cos 2v 2 e sin 2v The VEV in Eq. (15) corresponds to θv = π and ξ = π. In the MSSM, the angle θv corresponds to 2β. To explicit the structure of the horizontal space where basis transformations act, it is more convenient to define [25] u ≡ (φ11 , φ21 )T , w ≡ (φ12 , φ22 )T . (33) We can rewrite Eq. (32) as v hui = (0, 0)T , hwi = √ (cos θ2v , sin θ2v eiξ )T . 2 (34) More generally, we can rewrite v hwi = √ Uv e2 , 2 (35) where Uv is a unitary matrix in SU (2)H and ea , a = 1, 2, are the canonical vectors defined by (ea )b = δab . In terms of rµ we get v2 v2 (1, cos ξ sin θv , sin ξ sin θv , cos θv ) = Rµν nν , 4 4 (36) where nµ = (1, 0, 0, −1) and Rµν can be related to Uv by   1 0 Rµν (Uv ) ≡ 12 Tr[Uv† σµ Uv σν ] = , (37) 0 R̃ hrµ i = with R̃ = {Rij }, i, j = 1, 2, 3, being a rotation matrix in SO(3)H . We can rewrite the quadratic part of the potential in Eq. (9) using u and w of Eq. (33), and their respective VEVs in Eq. (34), V2 SSB = u† hMiu + w† hMiw + 21 Λµν sµ sν , (38) where sµ = 21 hwi† σµ w + h.c. (39) The first term of Eq. (38) corresponds to the mass term of the charged scalars, one physical and one Goldstone, in an arbitrary basis. The respective mass matrix is hMi, which is defined by Eq. (11). The relation between hMi in an arbitrary basis and its diagonal form (16) in the PCH basis is given by Uv† hMiUv = diag(m2 , 0) . (40) 5 We can then reach the PCH basis by the substitutions Φa = (Uv )ab Φ′b , hΦa i = (Uv )ab hΦ′b i , (41) or equivalently w = Uv w′ , u = Uv u′ , (42) with the same substitutions valid for their respective VEVs. Since the basis for which {Y, Λ, hri} is defined is completely arbitrary, the covariance of hMi is valid between any basis and not only with respect to the PCH basis. (A detailed account of the basis covariance of hMi is given in appendix D.) Indeed, we can write [25]   hMi = m2 ✶2 − hŵihŵi† , (43) where hŵi = hwi/|hwi|. We can try to extend the basis covariance for the mass matrix for the physical neutral scalars. We keep the notation MN to denote such mass matrix. Obviously, the second and third term of Eq. (38) is covariant by basis transformations for w, such as the transformation (42) with arbitrary Uv . The question, however, is if we can find appropriate fields ti , with suitable transformation properties, that renders MN covariant by some basis transformation, keeping Eq. (20) form invariant. We obviously want to recover Eq. (22) for MN and Eq. (21) for ti in the PCH basis. The immediate extension of Eq. (14) to define ti in any basis is flawed because a basis transformation over w in Eq. (14) would mix ti with the neutral Goldstone G0 . In other words, with w′ in the PCH basis given by   ′ 1 t1 − it′2 ′ w =√ , (44) ′ 0 2 −t3 + iG we can not define 1 w = √ (iG0 ✶2 + ti σi )hŵi , 2 (49) where hŵi = hwi/|hwi|. One can confirm that Eq. (49) satisfies Eq. (47) using Eq. (10). The covariance can be also checked, 1 w′ = U w = √ (iG0 ✶2 + t′i σi )hŵ′ i , 2 (50) t′i = Rij tj , (51) where and Rij is related to U by a relation similar to Eq.(37). The dependence on G0 is fixed by imposing that Eq. (49) reduces to Eq. (14) in the PCH basis. Notice Eq. (49) differs from Eq. (46) of Ref. 6 as the fields ti transform as vectors under SO(3)H . We can thus rewrite the second term of Eq. (38) in terms of ti using Eqs. (43) and (49) as w† hMiw = m2 2 [t − (hr̂i·t)2 ] . 2 (52) The third term of Eq. (38) can be also easily rewritten in terms of ti by using Eq. (47). The sum of the second and the third terms of Eq. (38) defines the mass matrix for the physical neutral scalars by w† hMiw + 21 Λµν sµ sν = 12 (MN )ij ti tj , (53) giving the basis covariant relation 1 w= √ 2  t1 − it2 −t3 + iG0  (45) because, in general, w 6= Uv w′ , (46) such as for Uv = eiθσ1 /2 . The solution is to promote Eq. (21) to define the real fields ti as ti ≡ where hr̂i is the unit vector in the direction of hri. The relation (48) is proved in appendix C. To write the second and third terms of Eq. (38) in terms of ti it is necessary to find the parametrization of w in terms of ti and G0 . The desired covariant relation is 2 si , i = 1, 2, 3 . v (47) The definition of Eq. (47) ensures that ti would transform as vectors under SO(3)H when SU (2)H transformations are applied to w and hwi in the definition (39) of si . The s0 component depends on ti by the basis invariant relation v s0 = hr̂i· t , (48) 2 h h hrihriT i hrihriT i + |hri| Λ̃ + Λ MN = m2 ✶3 − 00 |hri|2 |hri|2 + hriΛ0 T + Λ0 hriT , (54) where |hri| = v 2 /4. Equation (54) is the generalization of Eq. (22) for an arbitrary basis. Equation (54) illustrates two points: (i) the whole potential after EWSB can be completely defined in terms of the set {m2 , Λ00 , Λ0 , hri, MN } , (55) in an arbitrary basis, since Λ̃ can be written in terms of the set. Moreover, the objects in the set have the same transformation properties under the reparametrization group SU (2)H as the set {M0 , Λ00 , Λ0 , M, Λ̃} that defines the potential before EWSB [21]: two scalars [26], two vectors and one rank-2 tensor. (ii) We can define a Physical Neutral Higgs (PNH) basis, in contrast to the PCH basis, being the basis where MN is diagonal. In 6 general, this basis will coincide neither with the PCH basis nor with the basis with diagonal Λ̃ (the canonical CP basis in Ref. [21]). The relations (23) and (28) can be written in the basis covariant form
Y = 21 [m2 − hr0 iΛ00 − Λ0 ·hri] ✶2 − σi hr̂i i − 21 σi (MN )ij hr̂j i . (56) We made use of the relation ✶2 − hŵihŵi† = 1 2 ✶2 − σi hr̂i i .   (57) One can check Eq. (56) reduces to Eqs. (23) and (28) in the PCH basis. It is also important to remark that Y in Eq. (56) is independent on the particular VEV. For a different minimum of the same potential (or extremum if we do not require positive definite MN and m2 ), MN , m2 and hri differ in such a way that Y is the same. In addition, we can write the (ij) = (33) component of Eq. (23) in the following basis covariant form hŵi† Y hŵi = − 21 hr̂i i(MN )ij hr̂j i . (58) To obtain the interaction terms [25] V3 V4 SSB = Λµν sµ rν , SSB 1 2 Λµν rµ rν = (59) , (60) in terms of the real fields ti , we should calculate rµ in Eq. (4) using Eq. (47). Splitting rµ = xµ + yµ , (61) where xµ = yµ = 1 † 2 u σµ u , 1 † 2 w σµ w , (62) (63) y0 = yi = 2 + t ], 2 − t ]hr̂i i + PHYSICAL BASIS (P-BASIS) The Physical basis (P-basis) should be defined as the basis where all the fields possess definite masses. From Sec. II, we conclude that the mass matrix for physical neutral scalars (MN ) in the PCH basis will be not diagonal in general. From Sec. III, the basis where MN is diagonal (PNH basis) would mix the physical charged Higgs h+ with the charged Goldstone G+ . Thus, neither the PCH basis nor the PNH basis coincide with the Physical basis. To achieve the P-basis, we need independent basis transformations on the upper (u) and lower components (w) of the doublets, i.e., u → Uu u , w → Uw w , MN = diag(m21 , m22 , m23 ) . + 1 0 2 G (t×hr̂i)i (67) The VEV hwi will be in the general form of Eq. (35), different from the PCH basis. The respective hrµ i would be parametrized in the form of Eq. (36). In the PNH basis, the fields ti and G0 , contained in w, already have definite masses. The components of u, however, are combinations of the physical charged fields h+ and G+ . The relation between u and the physical fields is given by (68) where u′ refers to u in the PCH basis, (64) 1 2 (hr̂i·t)ti (66) where Uu , Uw are different transformations in SU (2). The transformations in Eq. (66) are legitimate basis transformations only after EWSB since they still preserve the EM symmetry, for a neutral vacuum, but do not preserve the SU (2)L ⊗ U (1)Y gauge structure of the doublets, except for Uu = Uw . The general group of basis transformations generated by Eq. (66) is SU (2) ⊗ SU (2) instead of SU (2)H valid before EWSB. The P-basis can be achieved either from the PCH basis or from the PNH basis. The latter choice is more convenient. Let us choose the PNH basis for which u = Uv u′ , we obtain the following covariant relations for y, 0 2 1 4 [(G ) 0 2 1 4 [(G ) IV. u′ = (h+ , G+ )T . .(65) Since each component of u = (u1 , u2 )T is a combination of the physical charged Higgs h+ and the charged Goldstone G+ , there is no need to write them in terms of real fields. The expression (62) can be kept as the covariant relation. A last observation about Eq. (65) concerns the transformation properties of G0 under refections, i.e., CP symmetry. To keep the transformation properties of the last term of Eq. (65) to be the same as the preceding terms we conclude that G0 should be a pseudoscalar (scalar under SO(3)H and changing sign under reflection or CP) and consequently CP-odd irrespective of the CP properties of the potential. (See appendix D of Ref. 6.) (69) In a basis invariant form, we know the component of u parallel to hwi is the charged Goldstone, hŵi† u = G+ . (70) The orthogonal direction contains the physical h+ . Obviously, the quadratic part of the potential after EWSB will be V2 SSB = m2 h+ h− + 21 m2i t2i . (71) The remaining task to completely define the potential after EWSB is to write the interaction terms in Eqs. (59) and (60) in terms of {ti , G0 , h+ , G+ }. The sole dependence of those interaction terms on u comes from xµ in 7 Eq. (62). The component x0 is basis independent and can be readily written x0 = 12 (G− G+ + h− h+ ) . (72) The spatial components can be written xi = (R̃v )ij x′j , (73) where (R̃v )ij = Tr[Uv† σi Uv σj ] and x′1 = x′2 x′3 = = − + + − 1 2 (h G + h G ) , − + + − −i 2 (h G − h G ) , − + − + 1 2 (G G − h h ) . (74) (75) (76) The variables sµ can be written in terms of ti using Eqs. (47) and (48) while the variables yµ are defined in Eqs. (64) and (65). V. DISCUSSIONS Equation (58) relates the depth of the potential in the minimum with the mass matrix of the physical neutral scalars. We can obtain bounds on the depth of the potential from the relations V (hri) = 21 V2 (hri) = −V4 (hri) , (77) where hri represents an extremum while V2 and V4 refer respectively to the quadratic and quartic part of the potential before EWSB, defined in Eqs. (2) and (3), evaluated in the extremum. The first equality of Eq. (77) can be written using Eq. (58) as V (hri) = − 12 hr0 ihr̂i iT (MN )ij hr̂j i . (78) From the relation above, we can deduce the following bounds for the depth of a minimum hri, − 12 hr0 im23 ≤ V (hri) ≤ − 12 hr0 im21 , (79) where m23 and m21 are respectively the greatest and the least eigenvalue of MN . We can conclude that a minimum will be deeper if the respective masses for the neutral scalars and the value of v are greater. A different physical bound can be extracted from condition (B11) necessary for bounded below potentials and from the positive definiteness of MN . From Eq. (54) and eT ⊥ (Λ̃ + Λ00 ✶)e⊥ > 0 we arrive at m2 − m23 < v2 Λ00 , 4 (80) where e⊥ is any unit vector orthogonal to hr̂i. The last inequality means the mass of the charged Higgs can not be arbitrarily large compared to the masses of the neutral scalars. In Sec. III we have found the set of Eq. (55) could be chosen as the physical parameters that define the 2HDM potential with a nontrivial vacuum. Among the elements of the set, it is clear that the masses are physical observables. On the other hand, the connection of the coupling constants and mixing matrices appearing in the interaction terms with physical observables is not direct. For example, devising scattering observables to extract the three parameters composing Λ0 , present in V3 and V4 , does not seem a straightfoward task. The form of V3 in terms of the physical fields, given in appendix E, reinforce such difficulty. The explicit form of V3 and V4 in the Physical basis can be also found in Eqs. (57)–(60) of Ref. 6, although the dependence on the mass matrix of the neutral scalars are not explicitly shown. An attempt to extract the observable parameters in the 2HDM, aiming to identify the presence of discrete symmetries through measurements, was made in Ref. 27. Nevertheless, separating the set of Eq. (55) and finding the relation of other parameters with the set is important to establish the number of independent parameters possible. The violation of any relation between parameters would indicate a model with a scalar sector distinct of the 2HDM. These relations should be constrained by experimental data and studies of the bounds on the mass of the physical charged Higgs [28, 29] or of the decay width of the physical Higgs bosons [2, 30] already exist in the literature. Of course, higher order effects, such as the exchange of quarks, would modify these tree level relations. The number of minima may be also modified when higher order contributions are taken into account. The existence of at most two minima, for example, may not be true beyond tree level [31]. Another aspect of the identification of physical parameters concerns the remaining reparametrization freedom such as the one in Eq. (29). That rephasing transformation freedom is particularly important when counting the number of parameters of the mixing matrix R̃v in Eq. (73). Since R̃v appears in the couplings involving the physical charged fields it may seem that it is a physical rotation matrix, needing three angles for its parametrization. However, only two angles are physical. The reason is that the reminiscent reparametrization freedom induced by Eq. (29) can remove one angle. Such reparametrization freedom is equivalent to rotations around hri. An explicit parametrization using two angles is available in Eqs. (E8) and (E9) of appendix E. The case of CP conserving potentials includes the MSSM 2HDM potential (see Ref. 2) and can be easily analyzed by setting Λ2i = 0 for i 6= 2 and M2 = 0 (or real Y ). In addition, if there is no SPCV, we have hr2 i = 0. In this case, from Eq. (54), we see the neutral scalar t2 does not mix with other scalars and corresponds to a CP-odd field with mass m22 = m2 + v2 Λ22 . 4 (81) Relation (81) is equivalent to a known relation encountered in the MSSM [see Eq. (10) of Ref. 12], where the 2 pseudoscalar t2 is usually called A and v4 Λ22 = −m2W . 8 The remaining neutral scalars t1 , t3 are CP-even and their mass matrix can be also read from Eq. (54). It is important to stress that the original basis transformations valid before EWSB forming the SU (2)H group could be explicitly transposed to the fields after EWSB. Although the possibility of transposition could be foreseen, various properties of the transformations after EWSB could not be anticipated. For example, the basis transformations after EWSB mix the physical charged Higgs h+ with the charged Goldstone while the neutral fields ti mix among them [through the same SU (2)H ∼ SO(3)H ] without mixing with the neutral Goldstone that transforms as a scalar of SU (2)H . From the discussions of Sec. IV, we can see there is an important distinction between basis transformation and reparametrization. The transformations of Eq. (66) constitute legitimate basis transformations that preserve the gauge structure after EWSB but they do not configure as reparametrization transformations. On the one hand, only the original SU (2)H basis transformations that preserves the SU (2)L gauge structure configure as reparametrization transformations. On the other hand, the maximal semisimple group of transformations which mix four real fields, ti and G0 , is SO(4). In addition, if we do not impose the kinetic part to be invariant, the reparametrization group SU (2)H can be extended to SL(2, c) [10]. As an terminological issue, the term basis transformations (or horizontal transformations) should be accompanied by the gauge structure that they preserve to be precise. For example, for the 2HDM treated here, it is important to specify if the basis transformations act before [SU (2)L ⊗ U (1)Y ] or after EWSB [U (1)EM ]. In general, the horizontal group after SSB will be larger than the horizontal group before SSB. It should be remarked that usually the physical mixing parameters belong to the additional basis transformations only allowed after SSB. For example, the CKM matrix for quarks comes from the difference between the rotations on the fields {uL , cL , tL } and {dL , sL , bL } necessary to diagonalize the respective mass matrices; applying basis transformations before EWSB, it is only possible to diagonalize one of the up or down quark Yukawa coupling matrices. A similar structure appears in 2HDMs for which the mixing among neutral scalars, the matrix R̃v , appears as the difference between the PCH basis and the PNH basis. For general N -Higgs-doublet models (NHDMs), the covariant relation for the mass matrix of neutral scalars can be easily written by generalizing Eqs. (54) and (49) to N Higgs doublets. The covariant relation for the mass matrix of charged scalars hMi was found in Ref. [25]. The fields ti , however, will transform as a vector of adjSU (N )H , living in a real vector space of N 2 −1 dimensions. Since, in general, a transformation in SO(N 2 − 1), a larger group than adjSU (N )H , will be required to diagonalize MN , the Physical Neutral Higgs (PNH) basis can not be reached by reparametrization but only by general horizontal transformations valid after EWSB. The corresponding basis transformation group will be SO(N 2 − 1) ⊗ SU (N ), the first factor acting on the neutral scalars and the second on the charged scalars independently. The enlargement of the basis transformation group after EWSB compared to the basis transformation group before EWSB is greater in NHDMs, with N > 2, than in the two-Higgs doublet case (N = 2). But the difference is not just quantitative. For the 2HDM potential, the basis transformation group after EWSB, SU (2) ⊗ SU (2), is just the double of the basis transformation group SU (2)H before EWSB, which can be understood as the original basis transformation acting independently on the upper u and lower w components of the doublets, as described in Eq. (66). For N > 2, the factor SO(N 2 − 1) necessary to diagonalize MN and, consequently, necessary to reach the Physical basis, can not be thought as the original reparametrization group SU (N )H acting independently on the lower components w of the doublets. Finally, we can say that a nontrivial horizontal structure in the scalar sector of a theory enriches the latter significantly, opening the possibility of different phenomenology such as different symmetry breaking patterns. At the same time, the theory becomes less predictive as much more free parameters are available. Nevertheless, useful physical information can be extracted from the horizontal structure by classifying the transformation properties of the parameters appearing in the potential. These properties constrain the relations between parameters before and after SSB, relating, for instance, vectors of the horizontal group with vectors. In the 2HDM potential analyzed here, we could relate, for example, the rank-2 tensor Λ̃, appearing before EWSB, with the mass matrix of the neutral scalars MN , only extractable after EWSB. Moreover, these relations were basis invariant. It is important to notice that the transformation properties of the parameters refer to the horizontal group SU (2)H acting on the Higgs doublets before EWSB. Although the horizontal group acting on the fields after EWSB could be larger, the transformation properties of the parameters followed essentially from the original horizontal group valid before EWSB. Obviously, a transformation in the enlarged horizontal group is usually necessary to reach the Physical basis where all fields have definite masses. Acknowledgments The author would like to thank Igor Ivanov for usefull discussions. This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (Fapesp). 9 APPENDIX A: EXTREMUM EQUATIONS Any neutral extremum hrµ i of the potential in Eq. (1) should satisfy the following extremum equations [25] M0 + Λ00 hr0 i + Λ0 ·hri = m2 , 2 Mi + Λi0 hr0 i + Λ̃ij hrj i = −m hr̂i i , (A1) (A2) 2 where i = 1, 2, 3 and m is the mass squared of the physical charged Higgs. The minus sign on the righthand side of Eq. (A2) is the only reminiscent of the Minkowski metric adopted in Ref. 25. The original extremum equation on the doublets reads hMihwi = 0 , APPENDIX B: BOUNDED BELOW CONDITION We seek here the necessary conditions for a bounded below potential using a method distinct to the ones adopted in Refs. 10 and 3. We will restrict ourselves to positive definite V4 . Rewriting V4 for r0 = |r| we obtain (B1) All variables ri will be treated here as c-numbers. We seek the direction r for which the potential increases more slowly. We minimize then V4′ = V4 + 21 λ(r2 − 1) , (B2) constraining r to be in the unit sphere using the Lagrange multiplier method. Differentiating, ∂V4′ = Λ̃ij rj + Λ00 ri + Λ0i |r| + r̂i (r·Λ0 ) + λri ,(B3) ∂ri ∂V4′ = 12 (r2 − 1) . (B4) ∂λ Equation (B3) yields  −1 r̂ = − Λ̃ + (Λ00 + λ + r̂·Λ0 )✶3 Λ0 . V4 extremum = 12 r2 (−λ) (B5) The values of r̂·Λ0 corresponding to an extremum is given by the roots of r̂·Λ0 = f (r̂·Λ0 + λ) , (B6) df (x) = 1, dx (B7) constrained by for x = r̂·Λ0 + λ. The function f (x) is defined by  −1 f (x) ≡ −Λ0 T Λ̃ + (Λ00 + x)✶3 Λ0 . (B8) extremum . (B9) We see all the Lagrange multipliers λ corresponding to an extremum should be negative. In particular, the greatest of them should be negative. In the basis for which Λ̃ + Λ00 ✶3 = diag(a1 , a2 , a3 ), a1 > a2 > a3 , (A3) for hui = 0 and hwi = 6 0. Equation (A3) means hwi is an eigenvector of hMi with null eigenvalue. V4 = 12 rT (Λ̃ + Λ00 ✶3 )r + |r|r·Λ0 . Equation (B6) is found by projecting Eq. (B5) to Λ0 while Eq. (B7) is equivalent to the requirement r̂·r̂ = 1. The components of r̂ perpendicular to Λ0 can be found from Eq. (B5) once r̂·Λ0 is known. For any extremum satisfying Eq. (B5) we find for Eq. (B1) the value f (x) = − 3 X Λ20i . a +x i=1 i (B10) A plot of f (x), with ai > 0, can be seen in Fig. 1 jointly with the solution of greatest λ. We see there are at least two extrema corresponding to the least and greatest λ. The intermediary extrema may not exist depending on the minimum slope of the curves. For example, in Fig. 1, for −a2 ≤ x ≤ −a3 , there is no solution for f ′ (x) = 1. From the schematic view of Fig. 1 we see ai > 0 is necessary to have positive −λ and consequently positive definite V4 , unless Λ0i = 0 for nonpositive ai . In a general basis, it is necessary that Λ00 + eigenvalues(Λ̃) > 0 , (B11) unless Λ0 have null projection in some eigenvector direction. To assure V4 is positive definite is necessary and sufficient to have the greatest Lagrange multiplier max λ < 0 . (B12) From Eq. (B6), the distance between the greatest xmin and the greatest λ is |r̂·Λ0 | = −r̂·Λ0 . Finally, if ai > 0 and one makes |Λ0i | small enough, we can always find λ < 0, proving the assertion preceding Eq. (31). As |Λ0i | get smaller, the curves of f (x) get closer to the x-axis. In special, from f (x) ≥ f (x)|a1 ,a2 →a3 , a3 = min(ai ), we can conclude that 2|Λ0 | < a3 (B13) is a sufficient condition. For |r0 | > |r| we can parametrize r0 = eχ |r|, χ > 0. The analysis of the minimization of V4 for fixed |r| is equivalent to the preceding analysis replacing Λ00 → Λ00 e2χ and Λ0i → Λ0i eχ . If |Λ0 |/Λ00 ≤ 1, condition (B13) is preserved for χ > 0 once it is valid for χ = 0. If |Λ0 |/Λ00 > 1 and λi > 0, a sufficient condition is |Λ0 |2 < min(λi ) , Λ00 where λi are the eigenvalues of Λ̃. (B14) 10 f (x) The equation (40) can thus be written as
Uv† Y + 12 σµ Λµν hrν i Uv = Y ′ + 12 σµ Λ′µν hrν′ i , (D2) where the relation between {Y ′ , Λ′ , hr′ i} in the PCH basis and {Y, Λ, hri} in the original basis is Y ′ = Uv† Y Uv , T hrµ′ i = Rµν hrν i , xmin x λ (D3) (D4) T Λ′µν = Rµα Λαβ Rβν , (D5) and Rµν = Rµν (Uv ) is given by Eq. (37). Hence, the first term of (38) is form invariant, −a1 −a2 −a3 u† hMiu = u′† hM′ iu′ . x=0 ` ´ FIG. 1: Plot of typical f (x). The dot lies at xmin , f (xmin ) ′ where xmin is the greatest value that satisfies f (x) = 1. Equation (B6) defines the value of λ depicted as the intersection of the line x − xmin + f (xmin ) with the x-axis. APPENDIX E: INTERACTION TERMS The interaction terms can be simplified into v V3 2 APPENDIX C: PROOF OF EQ. (48) SSB From the completeness of the σµ matrices [32], 1 2 (σµ )ab (σµ )cd = δad δcb , V4 (C1) we can calculate, for a neutral vacuum hwi = 6 0, hr0 is0 + hri isi = 41 Tr[hwihwi† σµ ]Tr[σµ (whwi† + h.c.)] = 12 Tr[(whwi† + h.c.)hwihwi† ] = 2hr0 is0 , (C2) since hr0 i = can prove p (C3) (C4) diag(m2 , 0) = hM′ i = Y ′ + 21 σµ Λ′µν hrν′ i . = 2 2 1 2 Λ00 (r0 − r ) + T r MN r + + 21 2hr0 i (r0 − hr̂i·r)Λ0 ·r m2
2 r , Λ00 − hr0 i ⊥ |u⊥ |2 = x0 − x·hr̂i , t2⊥ = 2(y0 − y·hr̂i) , (E2) (E3) (E4) (E5) (E6) (E7) and Λ0⊥ is analogous to t⊥ . One can also explicit the matrices Uv and R̃(Uv ) in Eqs. (68) and (73) choosing an explicit parametrization: APPENDIX D: BASIS COVARIANCE FOR hMi It is important to stress that the definition of the charged mass matrix hMi is covariant by basis transformation (41) in the following sense. The definition of the charged mass matrix in Eq. (11) is valid in any basis, in particular, in the PCH basis, SSB u⊥ ≡ u − hŵihŵi† u , tk ≡ t·hr̂i , t⊥ ≡ t − hr̂itk , hri ihri i. With the same reasoning, one s0 y0 = si yi .   = − u† Y u + w† Y w tk + m2 [x0 tk − x·t]   +hr0 iΛ0⊥ ·t⊥ |u⊥ |2 + 21 t2⊥ +(x + y)TMN t , (E1) where where hr0 i = |hwi|2 /2 = v 2 /4. Hence, s0 = hr̂i isi , (D6) Uv = U (θv , ξ)iσ2 , (E8) R̃v = R̃(θv , ξ)R̃(π, 0) = R̃(Uv ) (E9) where 1 1 U (θ, ϕ) ≡ e−i 2 σ3 ϕ e−i 2 σ2 θ . −iJ3 ϕ −iJ2 θ R̃(θ, ϕ) ≡ e e . (E10) (E11) (D1) The generators of rotations are defined by i(Jk )ij = ǫijk . [1] M. Gomez-Bock, M. Mondragon, M. Muhlleitner, M. Spira and P. M. Zerwas, “Concepts of Electroweak Symmetry Breaking and Higgs Physics,” arXiv:0712.2419 [hep-ph]. 11 [2] L. Randall, “Two Higgs Models for Large Tan Beta and Heavy Second Higgs,” arXiv:0711.4360 [hep-ph]. [3] M. Maniatis, A. von Manteuffel, O. Nachtmann and F. Nagel, “Stability and symmetry breaking in the general two-Higgs-doublet model,” Eur. Phys. J. C 48 (2006) 805 [arXiv:hep-ph/0605184]. [4] P. M. Ferreira, R. Santos and A. Barroso, “Stability of the tree-level vacuum in two Higgs doublet models against charge or CP spontaneous violation,” Phys. Lett. B 603 (2004) 219 [Erratum-ibid. B 629 (2005) 114] [arXiv:hep-ph/0406231]; ibidem, “Charge and CP symmetry breaking in two Higgs doublet models,” Phys. Lett. B 632 (2006) 684 [arXiv:hep-ph/0507224]. [5] S. Davidson and H. E. Haber, “Basis-independent methods for the two-Higgs-doublet model,” Phys. Rev. D 72 (2005) 035004 [Erratum-ibid. D 72 (2005) 099902] [arXiv:hep-ph/0504050]. [6] H. E. Haber and D. O’Neil, “Basis-independent methods for the two-Higgs-doublet model. II: The significance of tan(beta),” Phys. Rev. D 74 (2006) 015018 [arXiv:hepph/0602242] [Erratum-ibid. D 74 (2006) 059905]. [7] J. F. Gunion and H. E. Haber, “Conditions for CPviolation in the general two-Higgs-doublet model,” Phys. Rev. D 72, 095002 (2005) [arXiv:hep-ph/0506227]. [8] I. P. Ivanov, “Minkowski space structure of the Higgs potential in 2HDM: II. Minima, symmetries, and topology,” arXiv:0710.3490 [hep-ph]. [9] A set of continuous minima (of different orbits) may exist in certain symmetric potentials [8]. [10] I. P. Ivanov, “Minkowski space structure of the Higgs potential in 2HDM,” Phys. Rev. D 75 (2007) 035001 [Erratum-ibid. D 76 (2007) 039902 (E)] [arXiv:hepph/0609018]. [11] I. F. Ginzburg and M. Krawczyk, “Symmetries of two Higgs doublet model and CP violation,” Phys. Rev. D 72 (2005) 115013 [arXiv:hep-ph/0408011]. [12] M. S. Carena and H. E. Haber, “Higgs boson theory and phenomenology. ((V)),” Prog. Part. Nucl. Phys. 50 (2003) 63 [arXiv:hep-ph/0208209]. [13] T. D. Lee, Phys. Rev. D 8, 1226 (1973); Phys. Rep. 9, 143 (1974). [14] S. Weinberg, Phys. Rev. Lett. 37, 657 (1976). [15] M. Sher, “Electroweak Higgs Potentials And Vacuum Stability,” Phys. Rept. 179 (1989) 273. [16] A. Barroso, P. M. Ferreira, R. Santos and J. P. Silva, “Stability of the normal vacuum in multi-Higgs-doublet models,” Phys. Rev. D 74 (2006) 085016 [arXiv:hepph/0608282]. [17] The terminology “horizontal transformations” and “basis transformations”, as well as “horizontal group” and “basis transformation group”, will be used interchangeably. [18] W. Grimus and M. N. Rebelo, “Automorphisms in gauge theories and the definition of CP and P,” Phys. Rept. 281, 239 (1997) [arXiv:hep-ph/9506272]. [19] F. J. Botella and J. P. Silva, “Jarlskog-like invariants for theories with scalars and fermions,” Phys. Rev. D 51, 3870 (1995) [arXiv:hep-ph/9411288]; L. Lavoura and J. P. Silva, “Fundamental CP violating quantities in a SU(2) x U(1) model with many Higgs doublets,” Phys. Rev. D 50, 4619 (1994) [arXiv:hep-ph/9404276]. [20] G. C. Branco, M. N. Rebelo and J. I. Silva-Marcos, “CPodd invariants in models with several Higgs doublets,” Phys. Lett. B 614, 187 (2005) [arXiv:hep-ph/0502118]. [21] C. C. Nishi, “CP violation conditions in N-Higgs-doublet potentials,” Phys. Rev. D 74 (2006) 036003 [Erratumibid. D 76 (2007) 119901 (E)] [arXiv:hep-ph/0605153]. [22] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985). Also, Phys. Rev. D 35, 1685 (1987); [23] I. P. Ivanov, “Two-Higgs-doublet model from the grouptheoretic perspective,” Phys. Lett. B 632 (2006) 360 [arXiv:hep-ph/0507132]. [24] A. Barroso, P. M. Ferreira and R. Santos, “Neutral minima in two-Higgs doublet models,” Phys. Lett. B 652 (2007) 181 [arXiv:hep-ph/0702098]. [25] C. C. Nishi, “The structure of potentials with N Higgs doublets,” Phys. Rev. D 76 (2007) 055013 [arXiv:0706.2685 [hep-ph]]. [26] The mass squared of the physical charged scalar h+ , m2 , can be regarded as a scalar under SU (2)H by defining m2 = Tr[hMi]. [27] L. Lavoura, “Signatures of discrete symmetries in the scalar sector,” Phys. Rev. D 50 (1994) 7089 [arXiv:hepph/9405307]. [28] G. Barenboim, E. Lunghi, P. Paradisi, W. Porod and O. Vives, “Light charged Higgs at the beginning of the LHC era,” arXiv:0712.3559 [hep-ph]. [29] M. Krawczyk and D. Sokolowska, “The charged Higgs boson mass in the 2HDM: decoupling and CP violation,” arXiv:0711.4900 [hep-ph]. [30] S. Mantry, M. Trott and M. B. Wise, “The Higgs Decay Width in Multi-Scalar Doublet Models,” arXiv:0709.1505 [hep-ph]. [31] This remark is due to Igor Ivanov. [32] C. C. Nishi, “Simple derivation of general Fierz-type identities,” Am. J. Phys. 73, 1160 (2005) [arXiv:hepph/0412245].