Exceptional regions of the 2HDM parameter space
aa r X i v : . [ h e p - ph ] F e b CFTP/21-001SCIPP-21/01February, 2021
Exceptional regions of the 2HDM parameter space
Howard E. Haber ∗ and Jo˜ao P. Silva † Santa Cruz Institute for Particle Physics,University of California, Santa Cruz, California 95064, USA CFTP, Departamento de F´ısica, Instituto Superior T´ecnico,Universidade de Lisboa, Avenida Rovisco Pais 1, Lisboa 1049, Portugal
Abstract
The exceptional region of the parameter space (ERPS) of the two Higgs doublet model (2HDM)is defined to be the parameter regime where the scalar potential takes on a very special form. In thestandard parameterization of the 2HDM scalar potential with squared mass parameters m , m , m , and dimensionless couplings, λ , λ , . . . , λ , the ERPS corresponds to λ = λ , λ = − λ , m = m and m = 0, corresponding to a scalar potential with an enhanced generalized CPsymmetry called GCP2. Many special features persist if λ = λ and λ = − λ are retained whileallowing for m = m and/or m = 0, corresponding to a scalar potential with a softly-brokenGCP2 symmetry, which we designate as the ERPS4. In this paper, we examine many of the specialfeatures of the ERPS4, as well as even more specialized cases within the ERPS4 framework in whichadditional constraints on the scalar potential parameters are imposed. By surveying the landscapeof the ERPS4, we complete the classification of 2HDM scalar potentials that exhibit an exact Higgsalignment (where the tree-level couplings of one neutral scalar coincide with those of the StandardModel Higgs boson), due to a residual symmetry that is unbroken in the vacuum. One surprisingaspect of the ERPS4 is the possibility that the scalar sector is CP-conserving despite the presenceof a complex parameter of the scalar potential whose complex phase cannot be removed by separaterephasings of the two scalar doublet fields. The significance of the ERPS4 regime for custodialsymmetry is also discussed, and the cases where a custodial symmetric 2HDM scalar potentialpreserves an exact Higgs alignment are elucidated. ∗ E-mail: [email protected] † E-mail: [email protected] . INTRODUCTION After the discovery of the Higgs boson at the LHC [1, 2], the ATLAS and CMS Collabo-rations have ascertained that the observed properties of the Higgs boson are consistent withthe corresponding predictions of the Standard Model (SM). Various production mechanismsand decay channels have been detected, and many of the observed signal strengths are con-sistent with SM expectations given the current precision of the LHC Higgs data, typicallyin the range of 10%–20% depending on the final state observable [3–6].Can it be true that the scalar sector of the SM consists of a single spin-0 boson? In lightof the nonminimal nature of the SM fermions (which consist of three generations of quarksand leptons) as well as the nonminimal nature of the SM gauge group [which is the directproduct of two nonabelian groups and the weak hypercharge U(1) Y ], it would be surprisingif the scalar sector did not possess a nonminimal structure as well. Extended Higgs sectorshave been proposed and explored in the literature since the birth of the Standard Model.Indeed, an important part of the LHC Higgs program is to search for the existence of newscalar states related to the observed Higgs boson, and to study their properties if found.Of course, an arbitrary extended Higgs sector can in many cases be ruled out by currentexperimental data. The observed electroweak ρ parameter [7–10], which is close to 1, andthe absence of tree-level Higgs-mediated flavor changing neutral currents (FCNCs) thatotherwise would lead to observable FCNC effects, in conflict with current experimentalbounds, impose significant constraints on any theory with an extended Higgs sector. The twoHiggs doublet model (2HDM), which is one of the simplest extensions of the SM, possessestwo scalar doublet fields Φ and Φ [11–13], each with the same hypercharge Y = 1 (ina convention where the electric charge is given by Q = T + Y ). Nearly all of the newscalar physics phenomena expected in theories of extended Higgs models can be found inthe 2HDM—charged scalars, CP-odd scalars (in models with a CP-conserving scalar sector)and/or scalars of indefinite CP quantum numbers (in models with a CP-violating scalarsector). Moreover, the 2HDM predicts a tree-level value of ρ = 1 and is also compatiblewith the absence of tree-level Higgs-mediated FCNCs with a suitably chosen Higgs-fermionYukawa interaction [14, 15]. Finally, the 2HDM has often been employed in theories that Radiative corrections to the predicted value of ρ and the size of Higgs-mediated FCNCs impose someinteresting constraints on the 2HDM parameter space (e.g., see Refs. [16–18]). Z discretesymmetry under which one of the Higgs doublet fields is even and the other is odd. An ap-propriate assignment of Z quantum numbers to the fermion fields then provides a symmetryexplanation for the absence of Higgs-mediated tree-level FCNCs [14, 15]. In fact, this resultis robust even in the presence of a soft breaking of the Z symmetry by squared mass param-eters appearing in the scalar potential. The softly-broken Z -symmetric 2HDM is governedby nine independent parameters and is called the complex 2HDM (C2HDM) [30–37].There is some motivation to try to reduce the parameter count even further. For example,imposing CP-invariance on the scalar potential [11] would reduce the number of independentparameters to eight, corresponding to the vacuum expectation value, v ≃
246 GeV, fourscalar masses, two real angles, and one scalar self-coupling. As another example, consider therequirement that one of the scalar states of the 2HDM should resemble the SM Higgs boson.One way of achieving this result is to posit an additional symmetry of the scalar potential,which would further reduce the number of independent scalar sector parameters [38, 39].The exceptional region of the parameter space of the 2HDM, first introduced in Ref. [29],and designated by the acronym ERPS in Ref. [40], corresponds to a special parameterregime in which the coefficients of (Φ † Φ ) and (Φ † Φ ) appearing in the 2HDM scalarpotential are set equal and the coefficient of (Φ † Φ )(Φ † Φ ) is the negative of the coefficientof (Φ † Φ )(Φ † Φ ). In addition, the squared mass coefficients of Φ † Φ and Φ † Φ are set equaland the squared mass coefficient of Φ † Φ + h . c . is set to zero. The number of free parametersof the ERPS is five, consisting of v and the four scalar masses. The ERPS conditions canbe enforced by a global symmetry. Allowing for the conditions on the quadratic terms of3he scalar potential to be relaxed, which would constitute a soft breaking of the globalsymmetry, still yields a rather exceptional region of the 2HDM parameter space, which weshall henceforth denote as the ERPS4 in order to emphasize that the global symmetry ofthe ERPS is still respected by the dimension-four terms of the scalar potential.The ERPS4 is governed by eight parameters in its most general form, and the correspond-ing scalar potential is explicitly CP-violating. If in addition one imposes a CP symmetryon the scalar potential (which may or may not be violated by the vacuum), the numberof parameters is reduced by one. One can identify the seven parameters as v , four scalarmasses, one real angle and one scalar self-coupling. One may also impose additional softly-broken symmetries within the class of the ERPS4 scalar potentials, which yields a subset ofthe ERPS4 with additional exceptional features. All scalar potentials obtained in this wayautomatically possess a CP-conserving scalar potential and vacuum. The 2HDM employedin the MSSM provides one such example.It is worth highlighting a number of the exceptional features of scalar potentials thatreside within the ERPS4. First, in contrast to a generic 2HDM, if the conditions on thescalar potential parameters that define the ERPS4 hold in one scalar field basis, then theyare satisfied in all scalar field bases.Second, in a softly-broken Z -symmetric 2HDM, there are two potentially complex co-efficients of the scalar potential, denoted by m and λ in eq. (2.1), since the other twocomplex coefficients in eq. (2.1) are λ = λ = 0 as a consequence of the Z symmetry.Generically, one finds that the scalar potential is explicitly CP-conserving if and only ifIm( λ ∗ [ m ] ) = 0, since the latter condition implies that one can rephase the scalar fieldsΦ and Φ to remove the complex phases of m and λ . The resulting scalar potentialis then invariant with respect to the CP transformation, Φ → Φ ∗ . Remarkably, the “onlyif” part of this statement is no longer true in the ERPS4. We find that in the specialcase of h Φ i = h Φ i , the scalar potential is explicitly CP-conserving despite the fact thatIm( λ ∗ [ m ] ) = 0. Indeed, we can identify the modified definition of CP that governs the One is always free to change the scalar field basis by redefining the scalar fields, Φ a → U a ¯ b Φ b (summed over b = 1 , U is an arbitrary U(2) matrix. A particular choicefor Φ and Φ is called a choice of scalar field basis. In a generic 2HDM, the squared mass coefficientsand dimensionless quartic coefficients that appear in the scalar potential will be transformed by a changeof scalar field basis. Often, relations among parameters that are valid in one basis cease to be valid in adifferent basis. The ERPS4 conditions are notable in that they hold in all scalar field bases. Z symmetry. It is of interest to classifyall possible symmetries of the scalar potential beyond the Z symmetry of the IDM in whichthe Higgs alignment is exact. All scalar potentials of the ERPS fall within this class. But,one can also maintain exact Higgs alignment in some cases in which the symmetry is softlybroken, corresponding to the ERPS4. Including these cases completes the classification ofall symmetry based explanations for exact Higgs alignment in the 2HDM.Fourth, it is known that custodial symmetry is an accidental symmetry of the SM Higgspotential [46, 47]. In the 2HDM, the custodial symmetry is an accidental symmetry ofthe scalar potential if an additional constraint is imposed [48–51]. A custodial symmetric2HDM scalar potential is automatically CP-conserving. Additional accidental symmetriescan arise in special regions of the parameter space. Of particular interest is the case of acustodial symmetric scalar potential that preserves an exact Higgs alignment. Indeed, withtwo exceptions, the resulting scalar potential is necessarily in the ERPS4 regime.In Section II, we introduce the 2HDM with a softly-broken Z -symmetric scalar potential.The possible enhanced global symmetries of the 2HDM scalar potential beyond the Z symmetry are summarized in Section III, and their connections to the ERPS are exhibited inSection IV. In this section, we provide a set of basis-independent conditions that correspondto the ERPS4 and special subregions of the ERPS4 where additional global symmetries(perhaps softly broken) are imposed.A convenient scalar field basis for the ERPS4 is one where the softly broken Z symmetryand a softly broken permutation symmetry (that interchanges Φ ↔ Φ ) are simultaneouslyimposed. We examine the properties of the resulting scalar sector in Section V and note thatfor generic choices of the parameters, the scalar potential is CP-violating. If the correspond-ing scalar potential is explicitly CP-conserving, then CP may or may not be spontaneously5roken by the vacuum. The CP-conserving ERPS4 is examined in detail and we exhibit thespecial parameter regime where CP is conserved, despite the fact that a simple rephasingof Φ and Φ is not sufficient to produce a scalar potential whose parameters are all real.In Section VI, we extend the softly broken Z symmetry to U(1) and explore the propertiesof this special subregion of the ERPS4. One can show that the corresponding scalar poten-tial respects a generalized CP symmetry (denoted by GCP3) when expressed in a differentscalar field basis. The implications of the scalar potential when expressed in terms of theGCP3 basis of scalar fields are exhibited in Section VII and the relations between the scalarpotential parameters in the two different basis choices is made explicit in Section VIII.As noted above, exact Higgs alignment is realized in the ERPS. If soft-symmetry breakingsquared mass terms are included, the resulting ERPS4 may or may not exhibit exact Higgsalignment. In Section IX, we provide a complete classification of the symmetries (which insome cases is softly broken) that naturally yield a neutral scalar mass eigenstate whose tree-level properties are identical to those of the SM Higgs boson. In Section X, we combine exactHiggs alignment with the constraint of custodial symmetry and exhibit the implications forthe ERPS4 regime. Conclusions and future directions appear in Section XI, followed byfive appendices that provide additional details on the consequences of the ERPS4 for CPsymmetry and other related matters. II. 2HDM WITH A SOFTLY-BROKEN Z -SYMMETRIC SCALAR POTENTIAL Let Φ and Φ denote two complex Y = 1, SU(2) L doublet scalar fields. The most generalgauge invariant renormalizable scalar potential (in the Φ-basis) is given by V = m Φ † Φ + m Φ † Φ − [ m Φ † Φ + h . c . ] + λ (Φ † Φ ) + λ (Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ λ (Φ † Φ )(Φ † Φ ) + n λ (Φ † Φ ) + (cid:2) λ (Φ † Φ ) + λ (Φ † Φ ) (cid:3) Φ † Φ + h . c . o . (2.1)In general, m , λ , λ and λ can be complex. To avoid tree-level Higgs-mediated FCNCs,we shall impose a softly-broken discrete Z symmetry, Φ → +Φ and Φ → − Φ on thequartic terms of eq. (2.1), which implies that λ = λ = 0, whereas m = 0 is allowed. Inthis basis of scalar doublet fields (denoted as the Z basis), the discrete Z symmetry of thequartic terms of eq. (2.1) is manifestly realized. In the Z basis, it is convenient to rephasethe scalar fields such that λ is real. Then, the requirement that V is bounded from below6ields the following conditions [52], λ > , λ > , λ > − ( λ λ ) / , λ + λ ± λ > − ( λ λ ) / . (2.2)The scalar fields will develop non-zero vacuum expectation values (vevs) if the Higgsmass matrix m ij has at least one negative eigenvalue. We assume that the parameters ofthe scalar potential are chosen such that the minimum of the scalar potential respects theU(1) EM gauge symmetry. Then, the scalar field vevs are of the form h Φ i = v √ c β , h Φ i = v √ e iξ s β , (2.3)where c β ≡ cos β = v /v and s β ≡ sin β = v /v with v ≡ ( v + v ) / ≃
246 GeV. Byconvention we take 0 ≤ β ≤ π and 0 ≤ ξ < π .The parameters v , v and ξ are determined by minimizing the scalar potential. Theresulting minimization conditions in the case of λ = λ = 0 and real λ are given by, m v = Re( m e iξ ) v − λ v − λ v v , (2.4) m v = Re( m e iξ ) v − λ v − λ v v , (2.5)Im( m e iξ ) v = λ v v sin 2 ξ , (2.6)Im( m e iξ ) v = λ v v sin 2 ξ , (2.7)where λ ≡ λ + λ + λ cos 2 ξ . (2.8)Note that both eqs. (2.6) and (2.7) are provided in case one of the vevs vanishes. If both v = 0 and v = 0, then the minimization conditions simplify to, m = Re( m e iξ ) tan β − λ v c β − λ v s β , (2.9) m = Re( m e iξ ) cot β − λ v s β − λ v c β , (2.10)Im( m e iξ ) = λ v s β c β sin 2 ξ . (2.11)The value of the potential at the minimum is given by V min = v (cid:2) m c β + m s β − m e iξ ) s β c β + λ v c β + λ v s β + λ v s β c β (cid:3) = − v (cid:2) λ c β + λ s β + 2 λ s β c β (cid:3) , (2.12)7fter making use of eqs. (2.9) and (2.10). In light of eq. (2.2), V min <
0, which means thatthe extremum with v = v = 0 always is less favorable than the asymmetric minimum,assuming that there is a solution to eqs. (2.4)–(2.7) with nonvanishing vevs.If one of the two vevs vanishes, then the minimization conditions are given by m = 0 , m = − λ v , if v = 0 and v = v, (2.13) m = 0 , m = − λ v , if v = 0 and v = v. (2.14)This corresponds to an inert phase in which there exists a Z symmetry that is respectedboth by the scalar potential and the vacuum. This phase exists if and only if m = 0 and m < m <
0] in the case of v = 0 [ v = 0]. These two cases are physically equivalent,as they are related by a basis change where Φ ↔ Φ . The inert phase is stable if all thephysical scalar squared masses are nonnegative. III. ENHANCED GLOBAL SYMMETRIES OF THE SCALAR POTENTIAL
The possible global symmetries of the 2HDM scalar potential have been classified inRefs. [40, 53–55]. Starting from a generic Φ-basis, these symmetries fall into two separatecategories: (i) Higgs family symmetries of the form Φ a → U ab Φ b , and (ii) Generalized CP(GCP) symmetries of the form Φ a → U ab Φ ∗ b , where U resides in a subgroup (either discreteor continuous) of U(2). Although it might appear that the number of possible symmetriesis quite large, it turns out that different choices of U often yield the same constraints on the2HDM scalar potential parameters.The full global U(2) Higgs family symmetry transformation is the largest global symmetrygroup under which the gauge covariant kinetic terms of the scalar fields are invariant. More-over, the scalar potential is invariant under a global hypercharge transformation, U(1) Y ,which is a subgroup of U(2). Thus, any enhanced Higgs family symmetries that are re-spected by the scalar potential would be a subset of the U(2) transformations that do notcontain U(1) Y as a subgroup. We summarize below possible discrete and continuous Higgsfamily symmetries modulo the U(1) Y hypercharge symmetry that can impose constraints onthe 2HDM scalar potential in Tables I and II.Note that the list of symmetries in Table I contains a redundancy. Although it mightappear that the Z and Π discrete symmetries are distinct (as they yield different constraints8 ymmetry transformation law Z Φ → Φ , Φ → − Φ Π (mirror symmetry) Φ ←→ Φ U(1) PQ (Peccei-Quinn symmetry [56]) Φ → e − iθ Φ , Φ → e iθ Φ , for − π < θ ≤ π SO(3) (maximal Higgs flavor symmetry) Φ a → U ab Φ b , U ∈ U(2) / U(1) Y TABLE I: Classification of the Higgs family symmetries of the scalar potential in a generic Φ-basiswhere the symmetries are manifestly realized [40, 53–55]. Note that Z is a subgroup of U(1) PQ .The corresponding constraints on the 2HDM scalar potential parameters are shown in Table III.symmetry transformation lawGCP1 Φ → Φ ∗ , Φ → Φ ∗ GCP2 Φ → Φ ∗ , Φ → − Φ ∗ GCP3 Φ → Φ ∗ cos θ + Φ ∗ sin θ, Φ → − Φ ∗ sin θ + Φ ∗ cos θ , for 0 < θ < π TABLE II: Classification of the generalized CP (GCP) symmetries of the scalar potential in the Φ-basis [40, 53–55]. Note that a GCP3 symmetry transformation with any value of θ that lies between0 and π yields the same constrained 2HDM scalar potential. The corresponding constraints onthe 2HDM scalar potential parameters are shown in Table III. on the 2HDM scalar potential parameters in the Φ-basis), one can show that starting froma Π -symmetric scalar potential, one can find a different basis of scalar fields in which thecorresponding scalar potential manifestly exhibits the Z symmetry, and vice versa [29].In Table III, the constraints of the various possible Higgs family symmetries and GCPsymmetries on the 2HDM scalar potential in a generic Φ-basis are exhibited. In the list ofsymmetries, U(1) corresponds to U(1) PQ (henceforth, we shall suppress the PQ subscript).One can also consider applying two of the symmetries listed above simultaneously inthe same basis. It was shown in Refs. [40, 53] that no new independent models arise inthis way. For example, applying Z and Π in the same basis yields a Z ⊗ Π -symmetricscalar potential that is equivalent to GCP2 when expressed in a different basis. Similarly,applying U(1) and Π in the same basis yields a U(1) ⊗ Π -symmetric scalar potential that isequivalent to a GCP3-symmetric scalar potential when expressed in a different basis. Thisequivalence of GCP3 and U(1) ⊗ Π is explicitly demonstrated in Section VIII.9 ymmetry m m λ λ Re λ Im λ λ λ Z m real λ λ ∗ Z ⊗ Π m λ ⊗ Π m λ m λ λ − λ m λ − λ GCP3 m λ λ − λ − λ , Z ⊗ Π and U(1) ⊗ Π are notindependent from other symmetry conditions, since a change of scalar field basis can be performedin each case to yield a new basis in which the Z , GCP2 and GCP3 symmetries, respectively, aremanifestly realized. There are a number of additional Higgs family symmetries and generalized CP symmetriesthat are closely related to the ones displayed in Tables I and II that will be useful in ourwork. In Tables IV and V, we have listed three additional Higgs family symmetries andtwo additional GCP symmetries that can be used to constrain the parameters of the 2HDMscalar potential. The corresponding constraints are exhibited in Table VI. Given scalarpotentials where Π , U(1) ′ and GCP3 symmetries are manifestly realized, the basis changeΦ → Φ , Φ → i Φ , (3.1)yields m → im , λ → − λ , λ → iλ and λ → iλ , and produces a scalar potentialwhere Π ′ , U(1) ′′ and GCP3 ′ symmetries, respectively, are manifestly realized.The origin of U(1) ′ is slightly more subtle and is derived in Section VIII. It arises in thefollowing way. We have noted above that the U(1) ⊗ Π and GCP3 symmetries are equivalentin the sense that the scalar field bases in which these symmetries are manifestly realized arerelated by a change in basis. Moreover, as shown in Section VIII, by transforming from theU(1) ⊗ Π basis to the GCP3 basis, the U(1) symmetry constraints are mapped onto theU(1) ′ symmetry constraints. 10 ymmetry transformation law related symmetryΠ ′ Φ → Φ , Φ → − Φ Π U(1) ′ Φ → Φ cos θ + Φ sin θ , Φ → − Φ sin θ + Φ cos θ U(1) PQ U(1) ′′ Φ → Φ cos θ + i Φ sin θ , Φ → i Φ sin θ + Φ cos θ U(1) PQ TABLE IV: 2HDM scalar potential Higgs family symmetries in a generic Φ-basis that are relatedby a simple change of basis to the family symmetries of Table I. As in the case of the Peccei-Quinnsymmetry, a scalar potential that respects the U(1) symmetries above must be invariant for anyvalue of − π < θ ≤ π . Note that Π ′ [Π ] is a subgroup of U(1) ′ [U(1) ′′ ], respectively. Thecorresponding constraints on the 2HDM scalar potential parameters are shown in Table VI.symmetry transformation law related symmetryGCP1 ′ Φ → Φ ∗ , Φ → Φ ∗ GCP1GCP3 ′ Φ → Φ ∗ cos θ − i Φ ∗ sin θ, Φ → i Φ ∗ sin θ − Φ ∗ cos θ , for 0 < θ < π GCP3TABLE V: Generalized CP (GCP) symmetries of the scalar potential in the Φ-basis that are relatedby a change of basis to the GCP symmetries of Table II. Note that a GCP3 ′ symmetry with anyvalue of θ that lies between 0 and π yields the same constrained 2HDM scalar potential. Thecorresponding constraints on the 2HDM scalar potential parameters are shown in Table VI. Starting from a GCP1 symmetry transformation in the Φ-basis, consider an arbitrarybasis change, Φ → Φ ′ = U Φ. Then, eqs. (C.18) and (C.28) yield the corresponding GCPtransformation in the Φ ′ -basis, Φ → V Φ ∗ , where V ≡ U U T is a symmetric unitary matrix.The choice of V = (cid:0) (cid:1) corresponds to the definition of GCP1 ′ exhibited in Tables Vand VI. In contrast to the GCP1 symmetry, the GCP1 ′ symmetry transformation is espe-cially noteworthy in that it does not enforce reality conditions on the potentially complexparameters m , λ , λ and λ .Finally, it should be noted that the constraints on the scalar potential in a scalar field basiswhere the GCP3 symmetry is manifestly realized are precisely the same as the constraintsdue to the U(1) ′ ⊗ Z family symmetry when imposed in the same basis of scalar fields . Thisshould be contrasted with the U(1) ⊗ Π -symmetric scalar potential, which is equivalentto the GCP3-symmetric scalar potential when expressed with respect to a different scalarfield basis. Likewise the parameter constraints in a basis where the GCP3 ′ symmetry ismanifestly realized coincide with those that arise from the U(1) ′′ ⊗ Z family symmetry.11 ymmetry m m λ Re λ Im λ λ λ Π ′ m pure imaginary λ − λ ∗ Π ⊗ Π ′ m λ ′ m pure imaginary λ λ − λ − λ ′′ m real λ λ + λ − λ ′ ⊗ Z m λ λ − λ − λ ′′ ⊗ Z m λ λ + λ − λ ′ m λ λ GCP3 ′ m λ λ + λ − λ Z ⊗ Π , GCP3 and GCP3 ′ symmetries coincide with those of the Π ⊗ Π ′ ,U(1) ′ ⊗ Z and U(1) ′′ ⊗ Z symmetries, respectively. IV. AN EXCEPTIONAL REGION OF THE 2HDM PARAMETER SPACE
The exceptional region of the parameter space (ERPS) of the 2HDM corresponds toa regime in which the parameters of the scalar potential satisfy the following conditions: m = m , m = 0, λ = λ and λ = − λ . These conditions can be imposed by a GCP2symmetry, Φ → Φ ∗ , Φ → − Φ ∗ . (4.1)However, in the case of a softly-broken GCP2 symmetry, the conditions on the m ij arerelaxed. In general, one can take m = m and allow for nonzero complex values of m .The resulting parameter regime maintains many of the exceptional characteristics of theERPS and will henceforth be designated as the ERPS4.If the relations, λ = λ and λ = − λ , hold in one scalar field basis, then they holdin all choices of the scalar field basis. Indeed, one can construct a quantity, Z , which isexplicitly given in eq. (B.5), that is manifestly basis invariant under a change of scalar fieldbasis. Evaluating this invariant in a generic Φ-basis, we obtain Z = ( λ − λ ) + | λ + λ | . (4.2)Thus, the invariant condition for the ERPS4 is Z = 0, which yields λ = λ and λ = − λ λ = λ = 0. In such a basis, one can then removeany complex phase associated with the parameter λ by an appropriate rephasing of Φ ,thereby producing a basis where λ = λ = 0 and λ is real.Another possible choice for an enhanced symmetry is to impose simultaneously a Z andΠ symmetry, Z : Φ → Φ , Φ → − Φ , (4.3)Π : Φ ←→ Φ . (4.4)This symmetry adds the constraints, λ = λ = 0 and λ ∈ R to the ERPS4 conditions.Indeed, we have already indicated below eq. (4.2) that in the ERPS4, one can always find achange of basis such that λ = λ = 0 and λ is real in the transformed basis (as first notedin Ref. [29]). That is, a softly-broken GCP2-symmetric scalar potential can be realized as asoftly-broken Z ⊗ Π -symmetric scalar potential in a different scalar field basis.One can impose an additional constraint on the 2HDM parameters by imposing a GCP3symmetry, Φ → Φ ∗ cos θ + Φ ∗ sin θ, Φ → − Φ ∗ sin θ + Φ ∗ cos θ, (4.5)for all 0 < θ < π . This symmetry adds the additional constraint, λ = λ − λ − λ (which implies that λ is real). We will allow for a general soft breaking of the GCP3symmetry so that one can again take m = m and allow for nonzero complex valuesof m . Another possible choice for an enhanced symmetry is to impose simultaneously aU(1) and Π symmetry [40],U(1) : Φ → e − iθ Φ , Φ → e iθ Φ , (4.6)Π : Φ ←→ Φ , (4.7)for any 0 < θ < π . This symmetry adds the constraint λ = 0 to the Z ⊗ Π symmetry. Asnoted at the end of Section III, if the 2HDM scalar potential respects a GCP3 symmetry,then there exists a basis of scalar fields in which the symmetry can be identified as U(1) ⊗ Π .A basis-invariant condition can be found that corresponds to the case in which the quarticterms of the scalar potential respect the GCP3 symmetry in some basis. The invariant was13rst constructed in Ref. [40] and then rederived using a different technique in Appendix B ofRef. [57]. Below, we shall review the method employed in Ref. [57] while providing additionaldetails of the derivation.First, we make use of the notation of eq. (C.4) to assemble the 2HDM scalar potentialcouplings into a rank four tensor denoted by Z ab,cd . It is also convenient to introduce arelated rank four tensor, Z ab,cd ≡ Z ba,cd = Z ab,dc , (4.8)where the two expressions for Z ab,cd given above are equivalent in light of eqs. (C.2) and (C.4).Next, we define a three-vector whose components P B (for B = 1 , ,
3) are given by, P B = ( Z ab,cd + Z ab,cd ) δ ca σ Bdb , (4.9)and a 3 × D AB are given by [57, 58], D AB = ( Z ab,cd + Z ab,cd ) σ Aca σ Bdb − ( Z ab,ab + Z ab,ab ) δ AB , (4.10)where the σ A are the Pauli matrices and there is an implicit sum over repeated indices.Under a change of scalar field basis, Φ → Φ ′ = U Φ, eq. (C.8) yields, P B → P ′ B = R BD P D , D AB → D ′ AB = R AC R BD D CD = ( R D R T ) AB , (4.11)after employing the identity U † σ A U = R AB σ B , where R is a real orthogonal matrix that isexplicitly given by R AB = Tr( U † σ A U σ B ).Using the Kronecker product notation introduced in eqs. (C.9) and (C.14), we can rewriteeq. (4.10) in a convenient form in terms of two 4 × Z and Z , where Z is definedin eq. (C.5) and Z is obtained from Z by interchanging λ ↔ λ . Then, the equivalent formsof eqs. (4.9) and (4.10) are given by, P B = Tr (cid:2) ( × ⊗ σ B )( Z + Z ) (cid:3) , (4.12) The published version of Ref. [40] contains some typographical errors—in eq. (39), det ˜Λ should be replacedby − det ˜Λ and in eq. (44), should be replaced by . All other equations in Section II.E of Ref. [40] arecorrect. Quantities that are invariant with respect to scalar field basis transformations can be constructed out ofobjects such as D AB . Although D AB is not an invariant, related objects such as Tr D , det D and theeigenvalues of D can be used to construct invariant quantities. Ivanov published the first paper thatpresented this strategy in Ref. [58]. × is the 3 × D AB = Tr (cid:2) ( σ A ⊗ σ B )( Z + Z ) (cid:3) − Tr( Z + Z ) δ AB . (4.13)Using Z ab,cd = Z ba,dc [cf. eq. (4.8)], it follows thatTr (cid:2) ( σ A ⊗ σ B )( Z + Z ) (cid:3) = Tr (cid:2) ( σ B ⊗ σ A )( Z + Z ) (cid:3) , (4.14)which shows that D is a symmetric matrix. Moreover, X C Tr (cid:2) ( σ C ⊗ σ C )( Z + Z ) (cid:3) = Tr( Z + Z ) = 2( λ + λ + λ + λ ) , (4.15)which implies that D is a traceless matrix. Indeed, a straightforward calculation yields, P = (cid:16) Re( λ + λ ) − Im( λ + λ ) ( λ − λ ) (cid:17) , (4.16)and D = − ∆ + Re λ − Im λ Re ( λ − λ ) − Im λ − ∆ − Re λ − Im ( λ − λ )Re ( λ − λ ) − Im ( λ − λ ) ∆ , (4.17)where ∆ ≡ ( λ + λ ) − λ − λ . (4.18)In particular, the following condition for the ERPS4, which makes use of the vector P B [cf. eq. (4.16)], reproduces the invariant previously given in eq. (4.2), Z ≡ X B P B P B = ( λ − λ ) + | λ + λ | = 0 . (4.19)Following Ref. [57], we now demonstrate that there exists a basis in which the U(1)symmetry of the quartic terms of the scalar potential is manifestly realized if and only if P B and D AB can be written in the following forms, P B = c q B , D AB = c (cid:0) q A q B − δ AB (cid:1) , (4.20) The matrix D is related to ˜Λ employed in Ref. [40] by D = ˜Λ − (Tr ˜Λ) × . Thus, if λ is an eigenvalueof ˜Λ then λ − Tr ˜Λ is the corresponding eigenvalue of D . Consequently, the condition for degenerateeigenvalues is the same if applied to either D or ˜Λ. There are some advantages to employing eq. (4.17),as the condition Tr D = 0 simplifies the algebraic manipulations. q B is a real three-vector of unit length and c and c are constants. It then followsthat Tr D = 0 , Tr( D ) = c , det D = Tr( D ) = c , (4.21)which yields a characteristic equation for the eigenvalues of D [cf. eqs. (4.24)–(4.27)], x − c x − c = (cid:0) x + c (cid:1) (cid:0) x − c (cid:1) = 0 . (4.22)Hence, the eigenvalues of D are − c , − c , and c . That is, if D = 0 then two of theeigenvalues of D are degenerate. Moreover, in light of the eigenvalue equation, D AB q B = c (cid:0) q A q B − δ AB (cid:1) q B = c q A , (4.23)it follows that q A is the eigenvector of the nondegenerate eigenvalue. Thus, in a scalar fieldbasis in which D as defined by eq. (4.17) is diagonal with two degenerate diagonal elements,it follows that λ = 0 and λ = λ , in which case we can identify c = ∆ and the unit vector q = (0 0 1). Applying this result for q B in eq. (4.20) and comparing with eq. (4.16)then yields c = ( λ − λ ) and λ = − λ . Hence, we conclude that λ = λ = λ = 0in the D -diagonal basis, corresponding to a scalar potential whose quartic terms respect aPeccei-Quinn U(1) symmetry. If we impose in addition the ERPS4 condition then λ = λ ,which implies that P = 0, in which case the quartic terms of the scalar potential respect aU(1) ⊗ Π symmetry.Finally, if D = P = 0 then λ = λ = λ + λ and λ = λ = λ = 0, corresponding to ascalar potential whose quartic terms respect an SO(3) symmetry. Thus, we have successfullyprovided simple basis-invariant conditions for the 2HDM with a softly broken U(1), GCP2[or Z ⊗ Π ], GCP3 [or U(1) ⊗ Π ] and SO(3) symmetry, respectively.Thus, we seek a condition that guarantees that the matrix D given in eq. (4.17) possessestwo degenerate eigenvalues. In general, the characteristic equation of a generic 3 × D is of the form, x + a x + a x + a = 0 , (4.24)where a = − det D = − (Tr D ) + Tr D Tr( D ) − Tr( D ) , (4.25) a = (Tr D ) − Tr( D ) , (4.26) a = − Tr D . (4.27)16he cubic equation given in eq. (4.24) has exactly two degenerate roots if the followingcondition is satisfied [59, 60],
D ≡ a a a − a a + a a − a − a and a = 3 a . (4.28)Since the matrix D given in eq. (4.17) is symmetric and traceless, the condition that D possesses exactly two degenerate eigenvalues simplifies to, D = − a − a = (cid:2) Tr( D )] − (cid:2) Tr( D ) (cid:3) = 0 and D = 0 . (4.29)If the quartic terms of the scalar potential exhibit a U(1) ⊗ Π symmetry, then it followsthat Z = D = 0. Thus, we conclude that the basis-invariant condition, I ≡ Z + D = 0 , (4.30)is satisfied if and only if the quartic terms of the scalar potential exhibit a U(1) ⊗ Π symmetryin some basis (which implies that the quartic terms of the scalar potential exhibits a GCP3symmetry in some other basis). One can determine this condition explicitly by setting λ ≡ λ = λ and λ = − λ when evaluating the characteristic equation of the matrix D ,which yields Tr( D ) = ∆ + 2∆ (cid:0) | λ | − | λ | (cid:1) + 12 Re( λ ∗ λ ) , (4.31)Tr( D ) = ∆ + 2 | λ | + 8 | λ | . (4.32)Inserting these results into eq. (4.29) yields an expression for D . First, we assume that λ = 0, in which case the end result is, D = (cid:2) | λ | − R (∆ + R ) (cid:3) (cid:2) (∆ − R ) + 16 | λ | (cid:3) + CI , (4.33)where ∆ ≡ λ − λ − λ , C ≡ (∆ −| λ | −R ) −R | λ | +2 | λ | (cid:2) ∆ +9(∆+ R ) +3 I +3 | λ | +24 | λ | (cid:3) , (4.34)and R ≡ Re( λ ∗ λ ) | λ | , I ≡ Im( λ ∗ λ ) | λ | . (4.35)Note that R + I = | λ | . 17he product of the first two factors on the right hand side of eq. (4.33) is nonnegativedefinite. Thus, one solution to the equation D = 0 can be obtained by setting | λ | = R (∆ + R ) > , (4.36)which implies that ∆ / R > − R ]. It then follows from eq. (4.33) thateither I = 0 or C = 0. We now demonstrate that the latter possibility is never realized.After inserting eq. (4.36) into the expression for C given in eq. (4.34), we obtain, C = I + (9 R + 6∆ R − ) I + (∆ + R )(∆ + 3 R ) , (4.37)which is a quadratic equation whose discriminant is given by,Disc = (9 R + 6∆ R − ) − R )(∆ + 3 R ) = −R (4∆ + 3 R ) . (4.38)If ∆ / R > − then Disc < C > I . Finally, if − < ∆ / R ≤ − , then eq. (4.37) yields C > I . Thus, wehave shown that for λ = 0, if eq. (4.36) is satisfied then D = 0 if and only if | λ | = R (∆ + R ) > I = 0 . (4.39)One can rewrite the two conditions given in eq. (4.39) as a single complex equation, λ λ ∗ + λ λ ( λ − λ − λ ) − λ = 0 , (4.40)which must hold true for any choice of scalar field basis.If C > λ , then it would immediately follow thateq. (4.39) is the unique solution of the equation D = 0. However one can verify that regionsof the parameter space exist in which C <
0. This seems to leave open the possibility that if λ = 0 then D = 0 can be satisfied with a nonzero value of I due to a cancellation betweenterms in eq. (4.33). Note that R and/or ∆ can be zero. If these quantities are nonvanishing, then their individual signs canbe either positive or negative. If such a solution existed, it would not be continuously connected to the solution given by eq. (4.39),since any small perturbation of the scalar potential parameters from eq. (4.39) would still yield
C >
C <
0, there are no solutions to D = 0 for I = 0 and λ = 0. However, it is disappointingthat we are unable to analytically establish the condition I = 0 directly from D = 0 when λ = 0. λ = 0, the condition D = 0 holds if and only if eq. (4.39) is satisfied. Recall that eq. (4.29) states that the 3 × D (assumed to be nonzero) possesses a doubly degenerateeigenvalue if and only if D = 0. Moreover, any 3 × We can then use the discussion below eq. (4.23) to conclude that in the D -diagonal basis, λ = λ = λ = 0. Performing a basis transformation to an arbitrary basis (e.g., cf. eqs. (A9)and (A10) of Ref. [37]), it follows that Im( λ ∗ λ ) = 0 in any scalar field basis. Thus, we arejustified in setting I = 0 in eq. (4.33), in which case eqs. (4.39) and (4.40) must be validfor any choice of scalar field basis.In the case of λ = 0, one can either evaluate D directly using eq. (4.29) or simply set | λ | = 0 in eq. (4.33) while keeping R and I fixed. Both procedures yield the sameresult, D = | λ | (cid:0) ∆ − | λ | (cid:1) . (4.41)In particular, if λ = 0 then we can rephase Φ such that λ is real, in which case either λ = 0 or λ = ± ( λ − λ − λ ) , (4.42)corresponding to the manifest realization of U(1) ⊗ Π and GCP3/GCP3 ′ , respectively, asindicated by the quartic coupling relations exhibited in Tables III and VI. V. THE Z ⊗ Π SCALAR FIELD BASIS
Since the softly-broken GCP2-symmetric scalar potential is equivalent to a softly-broken Z ⊗ Π -symmetric scalar potential in a different scalar field basis, we henceforth focus onthe Z ⊗ Π basis, where λ ≡ λ = λ , λ = 0 is real and λ = λ = 0. The softly brokenparameters, m , m and m , are arbitrary with m potentially complex. If we demandthat the potential is bounded from below, the following conditions must be satisfied, λ > , λ + λ > , λ + λ + λ − | λ | > . (5.1) Given a 3 × D with eigenvalues − c , − c and 2 c (where c ∈ R ), it followsthat there exists a real orthogonal matrix R such that D = R diag( − c, − c, c ) R T . But we can writediag( − c, − c, c ) AB = c ( q A q B − δ AB ) with q = (0 , ,
1) and c = 3 c . Hence, D = c ( q ′ A q ′ B − δ AB ) withunit vector q ′ A = R AB q B .
19t is convenient to introduce the parameter, R ≡ λ + λ + λ λ . (5.2)It then follows from eq. (2.8) that λ = λR − λ sin ξ . (5.3)We shall first assume that v and v are both nonzero, or equivalently, sin 2 β = 0. Wethen use eqs. (2.9)–(2.11) [with λ ≡ λ = λ ] to fix the values of β and ξ . In particular, c β = m − m m + m + λv , (5.4)cos ξ = 2 Re m s β (cid:2) m + m + λ (1 + R ) v (cid:3) , (5.5)sin ξ = − m s β (cid:2) m + m + (cid:0) λ (1 + R ) − λ (cid:1) v (cid:3) , (5.6)where s β ≡ sin 2 β and c β ≡ cos 2 β . Writing m = | m | e iθ in eqs. (5.5) and (5.6)and imposing cos ξ + sin ξ = 1 yields an equation that determines the phase θ in termsof ξ and the other scalar potential parameters. Thus, the ERPS4 is governed by eight realparameters: λ , λ , λ , λ , | m | , v , β and ξ .It is convenient to introduce the Higgs basis as follows [29, 37, 61–65]. The Higgs basisfields H and H are defined by the linear combinations of Φ and Φ such that hH i = v/ √ hH i = 0. That is, H ≡ c β Φ + s β e − iξ Φ , H = e iη (cid:2) − s β e iξ Φ + c β Φ (cid:3) , (5.7)where we have introduced (following Ref. [37]) the complex phase factor e iη to accountfor the non-uniqueness of the Higgs basis, since one is always free to rephase the Higgsbasis field whose vacuum expectation value vanishes. In particular, e iη is a pseudo-invariantquantity [37] that is rephased under the unitary basis transformation, Φ a → U a ¯ b Φ b , as e iη → (det U ) − e iη , (5.8)where det U is a complex number of unit modulus. In terms of the Higgs basis fields definedin eq. (5.7), the scalar potential is given by, V = Y H † H + Y H † H + [ Y e − iη H † H + h . c . ]+ Z ( H † H ) + Z ( H † H ) + Z ( H † H )( H † H ) + Z ( H † H )( H † H )+ n Z e − iη ( H † H ) + (cid:2) Z e − iη ( H † H ) + Z e − iη H † H ) (cid:3) H † H + h . c . o . (5.9)20he coefficients of the quadratic and quartic terms of the scalar potential in eq. (5.9) areindependent of the initial choice of the Φ-basis. It then follows that Y , Z , Z and Z arealso pseudo-invariant quantities [65] that are rephased under Φ a → U a ¯ b Φ b as[ Y , Z , Z ] → (det U ) − [ Y , Z , Z ] and Z → (det U ) − Z . (5.10)It is straightforward to compute the corresponding Higgs basis parameters. The Y i aregiven by, Y = m c β + m s β − Re( m e iξ ) s β , (5.11) Y = m s β + m c β + Re( m e iξ ) s β , (5.12) Y = (cid:2) ( m − m ) s β − Re( m e iξ ) c β − i Im( m e iξ ) (cid:3) e − iξ . (5.13)Employing eqs. (2.9)–(2.11) [with λ ≡ λ = λ ] to eliminate m , m and Im( m e iξ ), itfollows that, Y = 2 Re( m e iξ ) s β − λv + v (cid:2) λ (1 − R ) + 2 λ sin ξ (cid:3) (1 − s β ) , (5.14) Z = Z = λ − (cid:2) λ (1 − R ) + 2 λ sin ξ (cid:3) s β , (5.15) Z = λ + (cid:2) λ (1 − R ) + 2 λ sin ξ (cid:3) s β , (5.16) Z = λ + (cid:2) λ (1 − R ) + 2 λ sin ξ (cid:3) s β , (5.17) Z = (cid:8) (cid:2) λ (1 − R ) + 2 λ sin ξ (cid:3) s β + λ (cos 2 ξ + ic β sin 2 ξ ) (cid:9) e − iξ , (5.18) Z = − Z = (cid:8) − (cid:2) λ (1 − R ) + 2 λ sin ξ (cid:3) c β + iλ sin 2 ξ ) (cid:9) s β e − iξ . (5.19)One can also check that the scalar potential minimization conditions in the Higgs basis, Y = − Z v , Y = − Z v , (5.20)are satisfied. The eight parameters that specify the ERPS4 can now be identified as v , Y , Z , Z , Z , Re Z , Im Z , and | Z | after using the freedom to rephase the Higgs basis field H to remove the complex phase from Z and Z [cf. eq. (5.10)].The subregion of the ERPS4 where Z = 0 is worthy of special attention. The ERPS4condition, Z = − Z , along with eq. (5.20) yields Y = Z = Z = 0, which signals thepresence of a Z symmetry that is manifestly realized in the Higgs basis and is unbroken bythe vacuum. We recognize this scenario as a special case of the IDM, and hence we shall referto this parameter regime as the inert limit of the softly-broken Z ⊗ Π -symmetric scalarpotential. Moreover, Higgs alignment is exact in the inert limit, as discussed in Section IX.The conditions for achieving the inert limit will be elucidated below eq. (5.59).21hree additional limiting cases are noteworthy. First, if λ = 0, then the quartic terms ofthe scalar potential exhibit a U(1) ⊗ Π symmetry, which will be discussed in more detail inSection VI. Second, if R = 1, then the quartic terms of the scalar potential exhibit a GCP3symmetry, which will be discussed in more detail in Section VII. Both these limits yieldthe same physical scalar sector, since they correspond to the softly-broken GCP3-symmetricscalar potential expressed in two different choices of the scalar field basis. Finally, if λ = 0 and R = 1, then the quartic terms of the scalar potential exhibit an SO(3) symmetry.The charged Higgs mass is given by, m H ± = Y + Z v = 2 Re( m e iξ ) s β − v ( λ + λ cos 2 ξ ) . (5.21)The squared masses of the neutral Higgs bosons are given by the eigenvalues of the neutralscalar squared-mass matrix, M = v Z Re( Z e − iη ) − Im( Z e − iη )Re( Z e − iη ) (cid:2) Z + Re( Z e − iη ) (cid:3) + Y /v − Im( Z e − iη ) − Im( Z e − iη ) − Im( Z e − iη ) (cid:2) Z − Re( Z e − iη ) (cid:3) + Y /v , (5.22)which is expressed with respect to the {√ H − v, √ H , √ H } basis, where Z ≡ Z + Z = λ (cid:2) R + s β (1 − R ) (cid:3) − λ (1 − s β sin ξ ) , (5.23)after making use of eqs. (5.16) and (5.17). The eigenvalues of M are independent of thechoice of η , since these cannot depend on the phase choice used in the definition of the Higgsbasis field H . Hence, in practical calculations, one can choose η to facilitate the analysis.For example, if we choose η = − ξ , then the neutral scalar squared-mass matrix is givenby, M = v λ − Ls β − Ls β c β − λ s β sin 2 ξ − Ls β c β m e iξ ) v s β + Ls β − λ c β sin 2 ξ − λ s β sin 2 ξ − λ c β sin 2 ξ m e iξ ) v s β − λ cos 2 ξ , (5.24)where L ≡ λ (1 − R ) + λ sin ξ . (5.25) The expressions given for m H ± in eq. (5.21) and for M in eqs. (5.22) and (5.23) in terms of the Higgsbasis parameters are valid for the most general 2HDM scalar potential. Z = 0 and/or Z = 0 then the neutral scalar squared-mass matrix has a blockdiagonal form consisting of a 2 × × Z e − iη ) = 0and Re( Z e − iη ) Im( Z e − iη ) = 0. In such cases, the scalar potential and vacuum are CP-conserving, and we shall employ the following convention for the names of the neutral scalarmass eigenstates: the CP-even scalars whose squared masses are the eigenvalues of the 2 × h and H where m h ≤ m H , and the 1 × A .In the case where one of the vevs vanishes (i.e., s β = 0), eqs. (2.13) and (2.14) imply that m = 0. For example, if v = v and v = 0 then Y is a free parameter, Z i = λ i , c β = 1,and ξ is indeterminate. In particular, Y = Z = Z = 0, which signals the presence of a Z symmetry, H → + H , H → −H , that is not broken by the vacuum. This is a specialcase of the IDM and corresponds to the inert limit of the softly-broken Z ⊗ Π -symmetricscalar potential. In particular, Y = m is a free parameter that is generically not equal to Y = m = − λv . To obtain the neutral scalar squared-mass matrix from eq. (5.22), wemust make a choice of η . For reasons discussed below eq. (8.60), we shall choose e − iη = − m A = Y + λv R , (5.26) m H ± = Y + Z v = m A − ( λ + λ ) v , (5.27) m h = λv , (5.28) m H = m A − λ v , (5.29)where we have denoted the mass-eigenstate neutral scalar fields in the inert limit by h ≡ √ H − v , H ≡ √ H , A ≡ √ H . (5.30)This nomenclature (where no mass ordering is implied) will be employed in all subsequentoccurrences of the inert limit in this work, and differs from the convention adopted in theparagraph following eq. (5.25) for the CP-conserving case where Z = 0 and/or Z = 0.If v = 0 and v = v , then one transforms to the Higgs basis via Φ → U Φ with U = ( ).In this case, Y = m is a free parameter, Y = m = − λv , Y = Z = Z = 0, Z i = λ i , c β = −
1, and ξ = 0. The scalar squared masses are again given by eqs. (5.26)–(5.29).In the inert limit where Y = Z = Z = 0, the scalar potential and vacuum are automati-cally CP-conserving. In particular, in the inert limit the neutral scalars consist of a CP-even23eutral scalar h whose properties coincide with those of the SM Higgs boson and two neutralscalars H and A with opposite sign CP quantum numbers. However, one cannot separatelyassign unique CP quantum numbers to H and A , respectively, based on the interactionsof the scalars with the gauge bosons and the scalar self-interactions. CP-conserving in-teractions of the scalars with other sectors of the theory, if present, will often resolve theambiguity and identify A as the neutral CP-odd scalar of the inert scalar doublet.For example, the most general form for CP-conserving neutral Higgs interactions withone generation of fermions in the inert limit is obtained by setting q = 1, q = 1, q = i (with all other q kj = 0), ρ D ∗ = ρ D , and ρ U ∗ = ρ U in eq. (58) of Ref. [37], which yields, − L Y = 1 v (cid:0) m d ¯ dd + m u ¯ uu (cid:1) h + 1 √ (cid:0) ρ D ¯ dd + ρ U ¯ uu (cid:1) H + i √ (cid:0) ρ D ¯ dγ d − ρ U ¯ uγ u (cid:1) A , (5.31)indicating that h behaves like the SM Higgs boson, H is CP-even and A is CP-odd. Notethat ρ D = ρ U = 0 in the IDM, since H is the only Z -odd field of the model, in which casethe individual CP-quantum numbers of H and A are not resolved.Let us examine more closely when a vacuum in which one of the two vevs vanishescan arise. Here, we shall extend the analysis of Ref. [66] to the case of λ = 0. First,we require that R > − v = v and v = 0, then eq. (2.14) yields m = 0 and m = − λv <
0. The value of the scalar potential at the minimum is V min = − ( m ) / (2 λ ). The positivity of m H given in eq. (5.29) yields m + λRv > m > Rm . (5.32)The above inequality is equivalent to(1 + R )( m − m ) < (1 − R )( m + m ) . (5.33)Since 1 + R is always positive, it follows that m − m > − (cid:18) − R R (cid:19) ( m + m ) . (5.34)In the case of v = 0 and v = v , the roles of m and m are interchanged. That is, m − m < (cid:18) − R R (cid:19) ( m + m ) . (5.35) Indeed, the choice of e − iη = 1 would have interchanged the identities of H and A in eqs. (5.26)–(5.29). Introducing Yukawa interactions constitutes a hard breaking of the symmetries responsible for the ERPS.Thus, in this paper we shall not entertain such terms further. m = 0, the converse is notnecessarily true. If m = 0 then two different phases of the 2HDM are possible—an inertphase where one of the two vevs vanishes and a mixed phase where both vevs are nonzero.To analyze the latter possibility in more detail, we again extend the analysis presented inRef. [66] to the case of λ = 0. If m = 0 and v , v = 0, then eqs. (2.4)–(2.7) yield, m = − λ (cid:0) v + Rv (cid:1) , (5.36) m = − λ (cid:0) v + Rv (cid:1) , (5.37)0 = λ sin 2 ξ . (5.38)Since λ = 0 by assumption, it follows that sin 2 ξ = 0 and cos 2 ξ = ±
1. One is always freeto rephase one of the scalar doublet fields so that ξ = 0, since the only possible effect on thescalar potential parameters is a sign change of λ . In the convention where ξ = 0, eq. (5.24)yields m A = − λ v , which implies that λ <
0. Eq. (2.12) then yields V min = − λ ( v + v + 2 Rv v ) . (5.39)It is convenient to eliminate v and v in favor of the scalar potential parameters. Usingeqs. (5.36) and (5.37), one easily obtains, v = 2 λ (cid:18) m R − m − R (cid:19) , v = 2 λ (cid:18) m R − m − R (cid:19) . (5.40)Plugging these values into eq. (5.39) yields, V min = − λ (1 − R ) (cid:2) m + m − Rm m (cid:3) = − λ (cid:20) ( m + m ) R + ( m − m ) − R (cid:21) . (5.41)One can work out a number of inequalities that must be satisfied if the mixed phase isstable. We again require that R > − m = ξ = 0,the trace and determinant of the 2 × m h + m H = λv , m h m H = λ v s β (1 − R ) . (5.42)Hence, the positivity of the scalar squared masses implies that | R | < | R | < m + m = − λv (1 + R ) < , (5.43) m + m + λv = λv (1 − R ) > . (5.44)25sing eq. (5.4) and | c β | ≤
1, it follows that m ≥ − λv and m ≥ − λv . However,it is again more useful to provide inequalities that are independent of the vevs. In light ofeq. (5.40), the requirement that v and v are strictly positive implies that m R > m , m
R > m , (5.45)The above inequalities are equivalent to (cid:18) − R R (cid:19) ( m + m ) < m − m < − (cid:18) − R R (cid:19) ( m + m ) . (5.46)Comparing eq. (5.46) with eqs. (5.34) and (5.35), it follows that the mixed phase and theinert phase do not coexist [53, 66, 67].Returning to the more general case where m = 0, the scalar sector is CP conserving ifand only if Im( Z ∗ Z ) = 0. A straightforward computation yields,Im( Z ∗ Z ) = − λλ (1 − R ) s β c β sin 2 ξ (cid:2) λ (1 − R ) + 2 λ (cid:3) = − λλ ( λ − λ − λ − λ )( λ − λ − λ + λ ) s β c β sin 2 ξ . (5.47)The case of s β = 0 corresponds to the inert limit, which has already been treated above. Inlight of eq. (4.42), the conditions λ = 0 and λ − λ − λ = ± λ correspond to the ERPS4where a U(1) symmetry is manifestly realized in some basis. In particular, λ = λ + λ ± λ correspond to GCP3 and GCP3 ′ , respectively, whereas λ = 0 corresponds to U(1) ⊗ Π ,which is equivalent to GCP3 and GCP3 ′ in different choices of the scalar field basis. Forexample, GCP3 ′ is related to GCP3 via the basis change specified in eq. (3.1). Theseenhanced symmetry cases will be treated separately in Sections VI–VIII.In this section, we shall assume that s β = 0, λ = λ + λ ± λ and λ = 0, in which caseCP is conserved if either (or both) of the following two conditions hold, c β = 0 or sin 2 ξ = 0 . (5.48)In the case of c β = 0, it follows that CP is conserved despite the fact that one cannotseparately rephase Φ and Φ in the Z ⊗ Π basis such that all the parameters of the scalarpotential are real if Im (cid:0) λ ∗ [ m ] ] (cid:1) = 0, as was already noticed in Ref. [37]. To understandthe origin of this result, note that eq. (5.4) implies that m = m when c β = 0. Togetherwith the ERPS4 conditions, it follows that the scalar potential is invariant with respect toa GCP1 ′ transformation [cf. Tables V and VI]. Moreover, the condition of c β = 0 ensuresthat the GCP1 ′ symmetry is preserved by the vacuum.26hen c β = 0, eq. (5.24) is rendered block diagonal, with the 2 × m A = 2 Re( m e iξ ) + λv (1 − R ) + λ v sin ξ , (5.49) m H ± = m A − v (cid:2) λ (1 − R ) + λ + λ (cid:3) , (5.50)where A = ϕ ≡ √ H . The squared-mass matrix of the neutral CP-even scalars is M H = λv (1 + R ) − λ v sin ξ − λ v sin ξ cos ξ − λ v sin ξ cos ξ m A − λv (1 − R ) − λ v cos ξ , (5.51)with respect to the { ϕ , ϕ } basis, where ϕ ≡ √ H − v and ϕ ≡ √ H . Theneutral CP-even scalar mass eigenstates are given by, H = ϕ c β − α − ϕ s β − α , h = ϕ s β − α + ϕ c β − α , (5.52)where 0 ≤ β − α ≤ π , s β − α ≡ sin( β − α ) and c β − α ≡ cos( β − α ), m H,h = (cid:26) m A + ( λR − λ ) v ± q(cid:2) m A − v ( λ + λ cos 2 ξ ) (cid:3) + λ v sin ξ (cid:27) , (5.53)with m h ≤ m H , and c β − α = λ v sin 2 ξ q ( m H − m h ) (cid:2) m H − λv (1 + R ) + λ v sin ξ (cid:3) . (5.54)In the case of sin 2 ξ = 0, eqs. (5.5) and (5.6) imply that Im (cid:2) m (cid:3) = 2 Re m Im m = 0.If sin ξ = 0 then Im m = 0 and all scalar potential parameters are real, whereas if cos ξ = 0then Re m = 0 and a rephasing Φ → i Φ changes the sign of the real parameter λ whileremoving the complex phase of m . Hence a real basis exists, which implies that the scalarpotential and the vacuum are CP conserving. The neutral scalar squared-mass matrix givenin eq. (5.24) is block diagonal when sin 2 ξ = 0, with the 33 element identified as the squaredmass of the CP-odd scalar, A = ϕ ≡ √ H .For sin ξ = 0, it follows from eqs. (5.21) and (5.24) that m A = ± m s β − λ v , m H ± = m A + ( λ − λ ) v , (5.55) In obtaining eq. (5.54), we have employed eqs. (9.1) and (9.5), where the real quantity Z v in theseequations is to be identified with the off-diagonal element of M H given in eq. (5.51). A real basis is defined to be a scalar field basis in which the scalar potential parameters and the vevs aresimultaneously real. ξ = ±
1. The upper 2 × M H = λv (cid:2) − s β (1 − R ) (cid:3) − λv s β c β (1 − R ) − λv s β c β (1 − R ) m A + λ v + λv s β (1 − R ) , (5.56)with respect to the { ϕ , ϕ } basis, where ϕ ≡ √ H − v and ϕ ≡ √ H . Hence, m H,h = (cid:26) m A + ( λ + λ ) v ± q(cid:2) m A + λ v − λv ( c β + Rs β ) (cid:3) + λ s β c β (1 − R ) v (cid:27) , (5.57)with m h ≤ m H , and c β − α = λv s β c β (1 − R )2 q ( m H − m h ) (cid:2) m H − λv (cid:0) − s β (1 − R ) (cid:1)(cid:3) . (5.58)For cos ξ = 0, the results of eqs. (5.55)–(5.58) are modified by the following substitutions, ± Re m → ∓ Im m , λ → − λ , R → R ≡ ( λ + λ − λ ) /λ , (5.59)where the choice of signs in front of Im m corresponds to sin ξ = ±
1. Note that R = 1under the assumption specified above eq. (5.48). If R = 1 then the (softly-broken) Z ⊗ Π symmetry of the scalar potential is promoted to GCP3 ′ , as discussed below eq. (5.47).If c β = sin 2 ξ = 0 then eqs. (5.19) and (5.20) yield Y = Z = Z = 0, correspondingto an inert limit of the softly-broken Z ⊗ Π -symmetric scalar potential. If sin ξ = 0, thenone can obtain the scalar squared masses either by taking the sin ξ = Im m = 0 limitof eqs. (5.49)–(5.51) or by taking the c β = 0 limit of eqs. (5.55)–(5.57). Recall that wehave identified the neutral scalar mass eigenstates in the convention specified in eq. (5.30).Taking into account that M H is exhibited with respect to the { ϕ , ϕ } basis in eq. (5.51)and with respect to the { ϕ , ϕ } basis in eq. (5.56), respectively, it follows that m h = λv (1 + R ) , m A = ± m − λ v ,m H = m A + λ v + λv (1 − R ) , m H ± = m A + ( λ − λ ) v . (5.60)If cos ξ = 0, then eq. (5.60) is modified by applying the substitutions indicated in eq. (5.59).In the inert limit, the vacuum preserves the Π symmetry (whereas the Z symmetry remainssoftly broken since m = 0).Spontaneous CP violation can occur when Im (cid:2) m (cid:3) = 0 (with m = 0) and sin 2 ξ = 0.In addition, as noted in Appendix A below eq. (A.25), one must assume that λ > m = 0 andsin 2 ξ = 0, then eq. (5.6) implies that m + m + λ (1 + R ) v = λ v . Inserting this resultinto eq. (5.5) yields cos ξ = Re m / ( λ v s β c β ); i.e., spontaneous CP violation occurs if [41],0 < | m | < λ v s β c β . (5.61)Likewise, if Re m = 0 and sin 2 ξ = 0, then eq. (5.5) implies that m + m + λ (1+ R ) v = 0.Inserting this result into eq. (5.6) yields sin ξ = Im m / ( λ v s β c β ). Once again, spontaneousCP violation occurs if eq. (5.61) is satisfied.If Im( Z ∗ Z ) = 0 then the scalar potential is explicitly CP-violating. In this case, one mustdiagonalize the 3 × Z ⊗ Π symmetry of the scalar potential is unbroken if m = m and m = 0.First, we suppose that both vevs are nonzero. Then eq. (5.13) implies that Y = 0, whichyields Y = Z = Z = 0 in light of eq. (5.20) and the ERPS condition, corresponding tothe inert limit of the Z ⊗ Π -symmetric scalar potential. Moreover in light of eqs. (2.11)and (5.4), c β = sin 2 ξ = 0 in the Z ⊗ Π symmetry limit, and it follows that the vacuumbreaks Z but conserves Π . The scalar squared masses in the limit of c β = sin ξ = 0 and m = 0 are given by the m = 0 limit of eq. (5.60). Stability of the scalar potential requirespositive squared masses, which yields λ = −| λ | < λ < | λ | and | R | <
1. Likewise, for c β = cos ξ = 0, the scalar squared masses are given by m = 0 limit of eq. (5.60) afterreplacing λ → − λ and R → R [cf. eq. (5.59)], in which case the stability requirementyields λ > λ < λ and | R | < s β = 0) then the Z ⊗ Π symmetry limit of eq. (5.26)corresponds to setting Y = Y = − λv [cf. eqs. (2.13) and (2.14)], which yields m A = λv ( R − , (5.62)which requires that R >
1. The squared masses of H ± , h and H in terms of m A given ineqs. (5.27)–(5.29) remain unchanged. In this case, the vacuum breaks Π but conserves Z .The landscape of scalar potentials in the ERPS4 that respects a softly-broken or exact Z ⊗ Π symmetry (but no larger symmetry) is summarized in Table VII.29 sin 2 ξ m , m m CP-violation? Higgs alignment comment s β = 0 = 0 m = m complex explicit no Im (cid:2) m (cid:3) = 0 s β = 0 = 0 m = m Im (cid:2) m (cid:3) = 0 spontaneous no 0 < | m | < λ v s β s β = 0 = 0 m = m Im (cid:2) m (cid:3) = 0 no no | m | > λ v s β c β = 0 = 0 m = m complex no no m = 0 s β c β = 0 0 m = m Im (cid:2) m (cid:3) = 0 no no s β = 0 m = m c β = 0 0 m = m Im (cid:2) m (cid:3) = 0 no yes m = 0 s β = 0 m = m Z ⊗ Π c β = 0 0 m = m Z ⊗ Π TABLE VII: Landscape of the ERPS4–Part I: Scalar potentials of the 2HDM with either anunbroken or softly broken Z ⊗ Π symmetry that is manifestly realized in the Φ-basis. In all cases, λ ≡ λ = λ = λ + λ ± λ , where λ is real and nonzero, λ = λ = 0, and λ , λ + λ , and λ + λ + λ − | λ | are all positive. Note that if m is pure imaginary, one can rephase Φ → i Φ to obtain a new basis where m is real and λ flips sign. An exact Higgs alignment in the ERPS4is realized in the inert limit where Y = Z = Z = 0. VI. THE U(1) ⊗ Π SCALAR FIELD BASIS
Consider the softly broken U(1) ⊗ Π -symmetric scalar potential where λ ≡ λ = λ and λ = λ = λ = 0. The softly broken parameters, m , m and m , are arbitrary with m potentially complex. The tree-level scalar potential of the 2HDM employed in the MSSMprovides a well-known example of this scenario [12, 22]. If we demand that the potential isbounded from below, the following conditions must be satisfied, λ > , λ + λ > , λ + λ + λ > . (6.1)We shall first assume that v and v are both nonzero, or equivalently sin 2 β = 0. Wethen use eqs. (2.9)–(2.11) [with λ ≡ λ = λ ] to obtain, m = Re( m e iξ ) tan β − λv c β − ( λ + λ ) v s β , (6.2) m = Re( m e iξ ) cot β − λv s β − ( λ + λ ) v c β , (6.3)Im( m e iξ ) = 0 . (6.4)Eqs. (6.2) and (6.3) fix the value of β . In particular, c β = m − m m + m + λv . (6.5)30ince m is the only potentially complex parameter, one can assume without loss ofgenerality that m is real and nonnegative after an appropriate rephasing of one of thetwo Higgs doublet fields. Hence, eq. (6.4) implies that the scalar sector is CP-conserving.Nevertheless, in the analysis presented in this section, we find it convenient to retain allfactors of e iξ for later purposes, which simply means that m e iξ = Re( m e iξ ) ≥
0, in lightof eq. (6.4) and the requirement that m A ≥ λ = 0 ineqs. (5.14)–(5.19), Y = 2 Re( m e iξ ) s β − λv (cid:2) R + s β (1 − R ) (cid:3) , (6.6) Z = Z = λ (cid:2) − s β (1 − R ) (cid:3) , (6.7) Z = λ + λs β (1 − R ) , (6.8) Z = λ + λs β (1 − R ) , (6.9) Z = λs β (1 − R ) e − iξ , (6.10) Z = − Z = − λs β c β (1 − R ) e − iξ , (6.11)where R ≡ λ + λ λ . (6.12)The limit of R = 1 corresponds to the softly-broken SO(3)-symmetric scalar potential, wherethe conditions λ = λ = λ + λ and λ = λ = λ = 0 hold for all choices of the scalarfield basis.The Higgs basis parameters Y and Y are fixed by the potential minimum conditionsgiven in eq. (5.20). Note that Im( Z ∗ Z ) = 0, which implies that a real Higgs basis existsafter an appropriate rephasing of the Higgs basis field H . That is, there is no CP violation(neither explicit nor spontaneous) arising from a scalar potential that exhibits a softly-broken U(1) ⊗ Π symmetry. Using eqs. (6.7)–(6.11), it follows that the following conditionsare satisfied,[Re( Z ∗ Z )] + Re( Z ∗ Z ) | Z | ( Z − Z ) − | Z | = 0 and Im( Z ∗ Z ) = 0 . (6.13)We recognize these conditions as equivalent to eq. (4.39) when applied in the Higgs basis.The squared masses of the neutral Higgs bosons are obtained by computing the eigen-values of eq. (5.22). In light of eqs. (6.10) and (6.11), it is convenient to take η = − ξ in31q. (5.22), since this choice yields Im( Z e − iη ) = Im( Z e − iη ) = 0. One can then immediatelyidentity the squared mass of the CP-odd neutral scalar A = ϕ ≡ √ H , m A = v (cid:2) Z − Re( Z e iξ ) (cid:3) + Y = 2 Re( m e iξ ) s β . (6.14)Combining eqs. (6.2), (6.3) and (6.14) yields an alternative expression, m A = m + m + λv (1 + R ) . (6.15)Likewise, the charged Higgs squared mass is given by m H ± = Y + Z v = m A − λ v , (6.16)after making use of eq. (6.14). Finally, the squared masses of the CP-even neutral scalars,denoted by h and H , are the eigenvalues of the 2 × M H = Z v Re( Z e iξ ) v Re( Z e iξ ) v m A + Re( Z e iξ ) v = λv (cid:2) − s β (1 − R ) (cid:3) − λv s β c β (1 − R ) − λv s β c β (1 − R ) m A + λv s β (1 − R ) , (6.17)with respect to the { ϕ , ϕ } basis, where ϕ ≡ √ H − v and ϕ ≡ √ H . Theneutral CP-even scalar masses are given by, m H,h = (cid:26) m A + λv ± q(cid:2) m A − λv ( c β + Rs β ) (cid:3) + λ s β c β (1 − R ) v (cid:27) , (6.18)with m h ≤ m H , and c β − α = λv s β c β (1 − R )2 q ( m H − m h ) (cid:2) m H − λv (cid:0) − s β (1 − R ) (cid:1)(cid:3) . (6.19)A stable minimum requires that the scalar squared masses should be nonnegative. Thiscondition implies that Re( m e iξ ) ≥ m A ≥ λ v . (6.20)In addition, we demand thatTr M H = m A + λv ≥ , (6.21)1 v det M H = λ v s β (1 − R ) + λm A (cid:2) − s β (1 − R ) (cid:3) ≥ . (6.22) The computation of the squared-mass matrix of the CP-even neutral scalars in the Φ-basis is given inAppendix D. m A ≥ R lies below a critical positive value thatdepends on λ , β and m A /v , − ≤ R ≤ m A λv + s(cid:18) m A λv − (cid:19) + 4 m A λv s β , (6.23)after employing eq. (6.1). It follows that eq. (6.22) is satisfied for all values of β if − ≤ R ≤ m A λv . (6.24)One can fix the parameter space of the softly-broken U(1) ⊗ Π scalar potential by speci-fying the values of six real parameters: λ , λ , R , β , m A and v = 246 GeV. By replacing λ with m h (see eq. (131) of Ref. [66]) and λ with m H ± , the independent parameters of thesoftly-broken U(1) ⊗ Π scalar potential can be taken as m h , m A , m H ± , v , R and β , in whichcase m H = m A − m h + λv [cf. eq. (6.21)] is a derived quantity.The inert limit of the scalar potential, where Y = Z = Z = 0, possesses an exact Z symmetry despite of the presence of squared mass parameters that softly break the U(1) ⊗ Π symmetry. The inert limit arises if either v = 0 or v = 0, but is more general. Indeed,eqs. (2.2) and (6.11) imply that the inert limit arises if any one of the three conditions, R = 1, c β = 0, or s β = 0, is satisfied.The case of R = 1 corresponds to the softly-broken SO(3) scalar potential as noted beloweq. (6.12). In light of eqs. (6.2)–(6.4), it follows that if s β c β = 0 then m = m andRe( m e iξ ) = 0. In this case, the squared masses of the Higgs bosons are given by m h = λv , m H = m A , m H ± = m A − λ v , (6.25)where m A = 2 Re( m e iξ ) /s β . The mass degeneracy of H and A arises due to an unbrokenU(1) symmetry of the scalar potential in the Higgs basis (since Y = Z = Z = Z = 0)that is preserved by the vacuum. The Π symmetry remains softly-broken (since Y = Y ).In the case of c β = 0, eq. (6.5) implies that m = m . Eqs. (6.14)–(6.18) yield, m h = λv (1 + R ) , m H = m A + λv (1 − R ) , m H ± = m A − λ v , (6.26) Apart from the upper bound given in eq. (6.23), one can obtain an independent upper bound by imposingeither tree-level unitarity [68–72] or a perturbativity constraint. One would then expect R/ (4 π ) < ∼ O (1). m A = 2 Re( m e iξ ), in agreement with the λ = 0 limit of eq. (5.60). In this limitingcase, after rephasing one of the two Higgs doublet fields to set ξ = 0, the vacuum preservesthe Π symmetry (whereas the U(1) symmetry remains softly broken since m = 0).The case where one of the vevs vanishes (i.e., s β = 0) should be treated separately andimplies that m = 0 in light of eqs. (2.13) and (2.14). One can check that eqs. (6.7)–(6.11)remain valid after setting s β = 0. In this case, the U(1) symmetry of the scalar potentialis unbroken, whereas the Π symmetry is softly broken if m = m .First, suppose that v = 0 and v = v . Then, eq. (5.26) yields, m A = Y + λv R , (6.27)where Y is a free parameter of the scalar potential that is no longer given by eq. (6.6).Moreover, eq. (6.5) is no longer valid since Y = m is independent of the squared massparameter m ; only the latter is fixed by the scalar potential minimum condition. Thesquared masses of the other scalars are given by eqs. (5.27)–(5.29) by setting λ = 0, m h = λv , m H = m A , m H ± = m A − λ v . (6.28)Note that the U(1) symmetry is preserved by the vacuum, which results in the mass degen-eracy of H and A . Second, if v = 0 and v = v , then it follows that Y = m is a freeparameter and Y = m = − λv . Eqs. (6.7)–(6.11) remain valid after setting β = π .Moreover, the Higgs masses given by eqs. (6.27) and (6.28) also remain valid.Although the vanishing of one of the two vevs requires that m = 0, the converse is notnecessarily true, as previously noted. That is, if m = 0, then both an inert phase and amixed phase of the 2HDM are possible. The inequalities previously obtained that distinguishthe inert and mixed phases in eqs. (5.34), (5.35) and (5.46) still apply (after setting λ = 0),and again imply that the inert and mixed phases do not coexist. In the mixed phase with m = 0, the scalar potential respects the U(1) symmetry, which is spontaneously brokenby the vacuum. Consequently, m A = 0 and the other scalar squared masses are given by, m H,h = λv (cid:2) ± q c β + R s β (cid:3) , m H ± = − λ v , (6.29)with m h ≤ m H . Stability of the vacuum requires that λ < ⊗ Π symmetry of the scalar potential is unbroken if m = m and m = 0.Then, as noted at the end of Section V, the squared mass conditions yield Y = Z = Z = 0,corresponding to the inert limit of the U(1) ⊗ Π -symmetric scalar potential.34 m , m m e iξ R Higgs alignment comment s β c β = 0 m = m > R = 1 no see eq. (6.23) s β c β = 0 m = m | R | < m A = 0 c β = 0 m = m > R = 1 yes − < R < m A / ( λv ) s β = 0 m = m R = 1 yes m H = m A > c β = 0 m = m | R | < s β = 0 m = m R > m H = m A > s β c β = 0 m = m > R = 1 yes m H = m A > c β = 0 m = m > R = 1 yes m H = m A > s β = 0 m = m R = 1 yes m H = m A > m = m R = 1 yes m H = m A = 0TABLE VIII: Landscape of the ERPS4–Part II(a): Scalar potentials of the 2HDM with either anunbroken or softly broken U(1) ⊗ Π symmetry that is manifestly realized in the Φ-basis, where λ ≡ λ = λ , λ = λ = λ = 0, and CP is conserved by the scalar potential and vacuum. Theparameter m e iξ is real and nonnegative [as a consequence of eqs. (6.4) and (6.14)]; if m = 0 and s β = 0 then a massless neutral scalar is present in the neutral scalar spectrum. The parameter R ≡ ( λ + λ ) /λ > −
1; when R = 1 the (softly-broken) U(1) ⊗ Π symmetry is promoted to a(softly-broken) SO(3) symmetry. An exact Higgs alignment in the ERPS4 is realized in the inertlimit where Y = Z = Z = 0. First, we suppose that both vevs are nonzero. Then in the U(1) ⊗ Π symmetry limit,eqs. (6.2)–(6.4) imply that ( R − c β = 0. Hence, the U(1) ⊗ Π symmetry limit arises intwo distinct cases. If m = c β = 0 and R = 1, then eqs. (6.14)–(6.17) yield, m h = λv (1 + R ) , m H = λv (1 − R ) , m A = 0 , m H ± = − λ v . (6.30)Note that a stable minimum exists if λ < | R | <
1. The Π symmetry is preservedby the vacuum, whereas the U(1) symmetry is spontaneously broken by the vacuum andresults in a massless Goldstone boson.If m = m , m = 0 and R = 1, then an SO(3) symmetry is explicitly preserved bythe scalar potential and eqs. (6.14)–(6.17) yield, m h = λv , m H = m A = 0 , m H ± = − λ v . (6.31)The SO(3) symmetry is spontaneously broken by the vacuum, leaving a residual unbrokenU(1) symmetry, which results in two massless Goldstone bosons, H and A .35f one of the vevs vanishes (i.e., s β = 0), then setting λ = 0 in eq. (5.62) and ineqs. (5.27)–(5.29) yields, m h = λv , m H = m A = λv ( R − , m H ± = m A − λ v , (6.32)which corresponds to a stable minimum if R >
1. Note that in this case the Π symmetryis broken by the vacuum, whereas the U(1) symmetry is preserved by the vacuum andresults in the mass degeneracy of H and A . In the limit of R = 1, corresponding to anSO(3)-symmetric scalar potential, the resulting scalar masses are again given by eq. (6.31).Table VIII provides a summary of the landscape of scalar potentials in the subspace ofthe ERPS4 regime where the U(1) ⊗ Π or SO(3) symmetry of the scalar potential is eithersoftly-broken ( m = m and/or m = 0) or unbroken ( m = m and m = 0). VII. THE GCP3 SCALAR FIELD BASIS
Consider a softly broken GCP3 symmetric scalar potential whose parameters (denotedwith prime superscripts) satisfy the following conditions: λ ′ ≡ λ ′ = λ ′ = λ ′ + λ ′ + λ ′ andIm λ ′ = λ ′ = λ ′ = 0. The softly broken parameters m ′ , m ′ and m ′ are arbitrary (with m ′ potentially complex). If we demand that the potential is bounded from below, then λ ′ > , λ ′ + λ ′ > , λ ′ > λ ′ . (7.1)Assuming that v ′ and v ′ are both nonzero, eqs. (2.9)–(2.11) yield, m ′ = Re( m ′ e iξ ′ ) tan β ′ − λ ′ v + λ ′ v s β ′ sin ξ ′ , (7.2) m ′ = Re( m ′ e iξ ′ ) cot β ′ − λ ′ v + λ ′ v c β ′ sin ξ ′ , (7.3)Im( m ′ e iξ ′ ) = λ ′ v s β ′ c β ′ sin 2 ξ ′ . (7.4)Eqs. (7.2)–(7.4) fix the value of β ′ and ξ ′ . In particular, c β ′ = m ′ − m ′ m ′ + m ′ + λ ′ v , (7.5)cos ξ ′ = 2 Re m ′ s β ′ ( m ′ + m ′ + λ ′ v ) , (7.6)sin ξ ′ = − m ′ s β ′ (cid:2) m ′ + m ′ + ( λ ′ − λ ′ ) v (cid:3) . (7.7)36s noted below eq. (5.6), inserting m ′ = | m ′ | e iθ ′ in eqs. (7.6) and (7.7) and imposingcos ξ ′ + sin ξ ′ = 1 yields an equation that determines the phase θ ′ in terms of ξ ′ and theother GCP3 scalar potential parameters.The corresponding parameters of the Higgs basis are obtained by setting R = 1 ineqs. (5.14)–(5.19), Y = 2 Re( m ′ e iξ ′ ) s β ′ − λ ′ v + λ ′ v (cid:0) − s β ′ (cid:1) sin ξ ′ , (7.8) Z = Z = λ ′ − λ ′ s β ′ sin ξ ′ , (7.9) Z = λ ′ + λ ′ s β ′ sin ξ ′ , (7.10) Z = λ ′ + λ ′ s β ′ sin ξ ′ , (7.11) Z = λ ′ e − iξ ′ (cid:0) cos ξ ′ + ic β ′ sin ξ ′ (cid:1) , (7.12) Z = − Z = iλ ′ s β ′ sin ξ ′ e − iξ ′ (cid:0) cos ξ ′ + ic β ′ sin ξ ′ (cid:1) . (7.13)The Higgs basis parameters Y and Y are fixed by the potential minimum conditions givenin eq. (5.20). Note that eq. (6.13) is satisfied, as expected. In addition, in the limit of λ ′ = 0, we recover the softly-broken SO(3)-symmetric scalar potential, where the conditions λ ′ = λ ′ = λ ′ + λ ′ and λ ′ = λ ′ = λ ′ = 0 hold for all choices of the scalar field basis.One can check that CP is conserved in light of the relation, Z = − λ ′ s β ′ sin ξ ′ Z , (7.14)which implies that Im( Z ∗ Z ) = 0. Thus, there exists an appropriate rephasing of the Higgsbasis such that Z , Z and Z are real. This is remarkable in light of the fact that λ is realbut m can be complex, which implies that one cannot perform a simple rephasing of thescalar doublet fields in the GCP3 basis to render all parameters real. In light of the CP-invariance of a softly-broken GCP3-symmetric scalar potential, it must be possible to find aresidual generalized CP transformation under which the scalar potential and the vacuum inthe GCP3 basis is left invariant. In Appendix C, we provide an explicit construction of thisresidual generalized CP transformation. Of course, the existence of such a transformation isa foregone conclusion given that the existence of the residual CP symmetry in the U(1) ⊗ Π basis can be established by inspection.The scalar masses can now be evaluated. First, eq. (5.21) is still valid, m H ± = Y + Z v = 2 Re( m ′ e iξ ′ ) s β ′ − v ( λ ′ + λ ′ cos 2 ξ ′ ) . (7.15)37ext, consider the neutral scalar squared-mass matrix, which is given by eq. (5.22). Notingthat the complex number, cos ξ ′ + ic β ′ sin ξ ′ appears in both eqs. (7.12) and (7.13), it isconvenient to define the complex phase ψ via,cos ξ ′ + ic β ′ sin ξ ′ = (1 − s β ′ sin ξ ′ ) / e iψ . (7.16)In order to make use of eq. (5.22), we must choose a value for η . Following eq. (5.8), weshall transform η = − ξ (which was employed in the U(1) ⊗ Π basis) to the GCP3 basis.The derivation is provided in Section VIII [cf. eqs. (8.58)–(8.60)] and instructs us to choose η = ψ − ξ ′ − π . (7.17)Inserting this result into eq. (5.22), it then follows that Z = − λ ′ (1 − s β ′ sin ξ ′ ) e iη , (7.18) Z = − λ ′ s β ′ sin ξ ′ (1 − s β ′ sin ξ ′ ) / e iη . (7.19)In particular, Im( Z e − iη ) = Im( Z e − iη ) = 0. Thus, we can immediately read off the squaredmass of the CP-odd neutral scale from eq. (5.22), m A = Y + v (cid:2) Z + Z − Re( Z e − iη ) (cid:3) = 2 Re( m ′ e iξ ′ ) s β ′ + λ ′ v sin ξ ′ , (7.20)where A = ϕ ≡ √ H . Combining the results of eqs. (7.2), (7.3) and (7.20) yields, m A = m ′ + m ′ + λ ′ v , (7.21)In addition, eqs. (7.15) and (7.20) yield, m H ± = m A − ( λ ′ + λ ′ ) v . (7.22)The squared masses of the CP-even neutral scalars, h and H are the eigenvalues of the2 × M H = Z v − Im( Z e i ( ξ ′ − ψ ) ) v − Im( Z e i ( ξ ′ − ψ ) ) v m A − Re( Z e i ( ξ ′ − ψ ) ) v = (cid:0) λ ′ − λ ′ s β ′ sin ξ ′ (cid:1) v − λ ′ v s β ′ sin ξ ′ (1 − s β ′ sin ξ ′ ) / − λ ′ v s β ′ sin ξ ′ (1 − s β ′ sin ξ ′ ) / m A − λ ′ v (1 − s β ′ sin ξ ′ ) , (7.23) The computation of the squared-mass matrix of the CP-even neutral scalars starting from the Φ ′ -basis ismuch more difficult. Details are provided in Appendix D. { ϕ , ϕ } basis, where ϕ ≡ √ H − v and ϕ ≡ √ H . Theneutral CP-even scalar masses are given by m H,h = (cid:26) m A +( λ ′ − λ ′ ) v ± q(cid:2) m A − ( λ ′ + λ ′ ) v (cid:3) + 4 λ ′ v ( m A − λ ′ v ) s β ′ sin ξ ′ (cid:27) , (7.24)where m h ≤ m H , and c β − α = λ ′ v s β ′ sin ξ ′ (1 − s β ′ sin ξ ′ ) / q ( m H − m h ) (cid:2) m H − ( λ ′ − λ ′ s β ′ sin ξ ′ (cid:1) v (cid:3) . (7.25)A stable minimum requires that the scalar squared masses should be nonnegative. Thiscondition implies thatRe( m e iξ ′ ) + λ ′ v s β sin ξ ′ ≥ m A ≥ ( λ ′ + λ ′ ) v . (7.26)In addition, we demand thatTr M H = m A + ( λ ′ − λ ′ ) v ≥ , (7.27)1 v det M H = m A ( λ ′ − λ ′ s β ′ sin ξ ′ ) − λ ′ λ ′ v (1 − s β ′ sin ξ ′ ) ≥ . (7.28)Since m A ≥ λ ′ ≤ min ( λ ′ , λ ′ m A λ ′ v + s β ′ sin ξ ′ ( m A − λ ′ v ) ) . (7.29)One can fix the parameter space of the softly-broken GCP3 scalar potential by specifyingthe values of λ ′ , λ ′ , λ ′ , s β ′ sin ξ ′ , m A and v . In particular, once m A is fixed, we see that β ′ and ξ ′ do not appear independently in any 2HDM observable. By replacing λ with m h and λ with m H ± , the independent parameters of the softly-broken GCP3 scalar potentialcan be taken to be m h , m A , m H ± , v , λ ′ and s β ′ sin ξ ′ . That is, just as in the case of thesoftly-broken U(1) ⊗ Π scalar potential, the parameter space is fixed by six real parameters.The inert limit of the softly-broken GCP3-symmetric, corresponding to Y = Z = Z = 0,arises if either v = 0 or v = 0, but is more general. Indeed, eq. (7.13) implies that the inertlimit requires that one of the following three conditions, s β ′ sin ξ ′ = 0, c β ′ = cos ξ ′ = 0, or λ ′ = 0, is satisfied.The case where one of the vevs vanishes (i.e., s β ′ = 0) implies that m ′ = 0 in light ofeqs. (2.13) and (2.14). Then, setting R = 1 in eqs. (5.26)–(5.29) yields, m h = λ ′ v , m H = m A − λ ′ v , m H ± = m A − ( λ ′ + λ ′ ) v , (7.30)39here m A = Y + λ ′ v and Y is a free parameter. Likewise, if sin ξ ′ = 0, then eqs. (7.20)–(7.23) also yield eq. (7.30), where m A = 2 | Re m ′ | /s β ′ . That is, if s β ′ sin ξ ′ = 0 theneq. (7.30) is satisfied where m A = | Re m ′ | s β ′ , if sin ξ ′ = 0 and s β ′ = 0 ,Y + λ ′ v , if s β ′ = 0 , (7.31)Note that sin ξ ′ = 0, s β ′ c β ′ = 0 and m ′ = m ′ , then it follows that Re m ′ = 0 in light ofeqs. (7.2)–(7.4). Using the results of Section VIII, this case corresponds to c β = 0 in theU(1) ⊗ Π basis.Second, if c β ′ = cos ξ ′ = 0, then it follows from eqs. (7.20)–(7.23) that, m h = ( λ ′ − λ ′ ) v , m H = m A = ± m ′ + λ ′ v , m H ± = m A − ( λ ′ + λ ′ ) v . (7.32)Using the results of Section VIII, this case corresponds to s β = 0 in the U(1) ⊗ Π basis.Then the choice of plus [minus] sign in the expression for m H,A in eq. (7.32) correspondsto β = 0 [ β = π ], respectively. Moreover, recall that an unbroken GCP3 symmetry isequivalent to U(1) ′ ⊗ Z [cf. Table VI]. Although the Z symmetry is explicitly broken(due to m ′ = 0), a residual U(1) ′ symmetry survives that is preserved by the vacuum if c β ′ = cos ξ ′ = 0 since, cos θ sin θ − sin θ cos θ ± i = e ± iθ ± i , (7.33)which results in the mass degeneracy of H and A .Third, if λ ′ = 0, then the softly-broken GCP3 symmetry is promoted to a softly brokenSO(3) symmetry. In light of eqs. (7.2)–(7.4), it follows that if s β ′ c β ′ = 0 then m ′ = m ′ and Re( m ′ e iξ ′ ) = 0. We then obtain, m h = λ ′ v , m H = m A = 2 Re( m ′ e iξ ′ ) s β ′ , m H ± = m A − λ ′ v , (7.34)in agreement with eq. (6.25), as expected. Using the results of Section VIII, this casecorresponds to R = 1 in the U(1) ⊗ Π basis. If sin ξ ′ = 0 then Re( m ′ e iξ ′ ) = ± Re m ′ = | Re m ′ | after choosing the sign that yields m A ≥ m ′ = m ′ and m ′ = 0 then the GCP3 symmetry is explicitly preserved bythe scalar potential. In light of eqs. (5.13) and (5.20) and the ERPS condition, it followsthat Y = Z = Z = 0, corresponding to the inert limit of the scalar potential. If bothvevs are nonzero, then it follows from eqs. (7.2)–(7.4) that λ ′ c β ′ sin ξ ′ = λ ′ sin ξ ′ cos ξ ′ = 0.Consequently, the GCP3 symmetry limit arises in the following three distinct cases.First, if λ ′ = 0 and sin ξ ′ = 0, then eqs. (7.30) and (7.31) yield m h = λ ′ v , m H = − λ ′ v , m A = 0 , m H ± = − ( λ ′ + λ ′ ) v , (7.35)which corresponds to a stable minimum if λ ′ < λ ′ < − λ ′ . The GCP3 symmetry isspontaneously broken by the vacuum, resulting in a massless scalar.Second, if λ ′ = 0, c β ′ = 0 and cos ξ ′ = 0, then eq. (7.32) yields, m h = ( λ ′ − λ ′ ) v , m H = m A = λ ′ v , m H ± = ( λ ′ − λ ′ ) v , (7.36)which corresponds to a stable minimum if λ ′ > λ ′ < λ ′ . The mass degeneracy of H and A is again a result of a residual U(1) ′ symmetry that is preserved by the vacuum.Third, if λ ′ = 0, then the GCP3 symmetry of the scalar potential is promoted to anSO(3) symmetry. In this case, m h = λ ′ v , m H = m A = 0 , m H ± = − λ ′ v , (7.37)corresponding to a stable minimum if λ ′ <
0. In particular, the SO(3) symmetry is sponta-neously broken down to U(1), which yields two massless scalars H and A .If only one of the two vevs is nonzero (i.e., s β = 0), then eqs. (2.13) and (2.14) yield m ′ = 0. Setting R = 1 and Y = Y = − λ ′ v in eqs. (5.26)–(5.29), we end up with m h = λ ′ v , m H = − λ ′ v , m A = 0 , m H ± = − ( λ ′ + λ ′ ) v , (7.38)which coincides with the mass spectrum given in eq. (7.35). If in addition we set λ ′ = 0,then we obtain the mass spectrum of eq. (7.37).Tables IX and X provide summaries of the landscape of possible scalar potentials in thesubspace of the ERPS4 regime where the GCP3 or SO(3) symmetry of the scalar potentialis either softly-broken ( m ′ = m ′ and/or m ′ = 0) or unbroken ( m ′ = m ′ and m ′ = 0).Using the results of Section VIII, one can check that each entry of Tables IX and X can bematched up with a corresponding entry of Table VIII (and vice versa).The analysis presented in this section can be repeated for the closely related GCP3 ′ basis,where λ ′ = λ ′ = λ ′ + λ ′ − λ ′ and Im λ ′ = λ ′ = λ ′ = 0. Details are left for the reader.41 ′ ξ ′ m ′ , m ′ m ′ e iξ ′ Higgs alignment comment s β ′ c β ′ = 0 sin 2 ξ ′ = 0 m ′ = m ′ complex no s β ′ c β ′ = 0 cos ξ ′ = 0 m ′ = m ′ real no c β ′ = 0 sin 2 ξ ′ = 0 m ′ = m ′ real ( = 0) no c β ′ = 0 sin ξ ′ = 0 m ′ = m ′ real ( = 0) yes c β ′ = 0 cos ξ ′ = 0 m ′ = m ′ real ( = 0) yes m H = m A > s β ′ c β ′ = 0 sin ξ ′ = 0 m ′ = m ′ real ( = 0) yes s β ′ = 0 m ′ = m ′ s β ′ = 0 sin ξ ′ = 0 m ′ = m ′ c β ′ = 0 cos ξ ′ = 0 m ′ = m ′ m H = m A > s β ′ = 0 m ′ = m ′ λ ≡ λ ′ = λ ′ = λ ′ + λ ′ + λ ′ , (with λ ′ real and nonzero) and λ ′ = λ ′ = 0, and CP is conserved bythe scalar potential and vacuum. An exact Higgs alignment in the ERPS4 is realized in the inertlimit where Y = Z = Z = 0. β ′ m ′ , m ′ m ′ e iξ ′ Higgs alignment comment s β ′ c β ′ = 0 m ′ = m ′ real ( = 0) yes m H = m A > c β ′ = 0 m ′ = m ′ real ( = 0) yes m H = m A > s β ′ = 0 m ′ = m ′ m H = m A > m ′ = m ′ m H = m A = 0TABLE X: Landscape of the ERPS4–Part II(c): Scalar potentials of the 2HDM with either anunbroken or softly broken SO(3) symmetry that is manifestly realized in the Φ-basis. In all cases, λ ≡ λ ′ = λ ′ = λ ′ + λ ′ and λ ′ = λ ′ = λ ′ = 0, and CP is conserved by the scalar potential andvacuum. In all cases of an unbroken or softly broken SO(3) symmetric scalar potential, an exactHiggs alignment is realized as a consequence of Y = Z = Z = 0. VIII. TRANSFORMING FROM THE U(1) ⊗ Π BASIS TO THE GCP3 BASIS
In Ref. [40], it was shown that the U(1) ⊗ Π and GCP3-symmetric scalar potentials arein fact the same scalar potential expressed in different scalar field bases. In this section, weextend this result to the softly-broken U(1) ⊗ Π and GCP3-symmetric scalar potentials byproviding an explicit mapping between the corresponding scalar potential parameters.42onsider the following unitary transformation, U = e iφ √ − i − i , (8.1)where the phase φ is determined in eq. (8.13). Starting from the U(1) ⊗ Π basis defined inSection VI, it then follows that (independently of the choice of φ ), λ ′ = λ ′ = λ ′ = λ (1 + R ) , (8.2) λ ′ = λ + λ (1 − R ) , (8.3) λ ′ = λ + λ (1 − R ) , (8.4) λ ′ = − λ (1 − R ) , (8.5) λ ′ = − λ ′ = 0 , (8.6)where R ≡ ( λ + λ ) /λ . In particular, λ ′ = λ ′ − λ ′ − λ ′ is real and λ ′ = λ ′ = 0,corresponding to the GCP3 basis defined in Section VII. In addition, the correspondingsoft-breaking squared mass parameters are, m ′ = ( m + m ) + Im m , (8.7) m ′ = ( m + m ) − Im m , (8.8) m ′ = Re m + i ( m − m ) . (8.9)Finally, the vevs in the GCP3 basis are given by v ′ = e iφ √ (cid:0) v − iv e iξ (cid:1) , v ′ e iξ ′ = − e iφ i √ (cid:0) v + iv e iξ (cid:1) , (8.10)where v ′ ≡ vc β ′ and v ′ ≡ vs β ′ are real and positive. Hence, c β ′ = 1 √ (cid:0) s β sin ξ (cid:1) / , s β ′ = 1 √ (cid:0) − s β sin ξ (cid:1) / , (8.11)and it immediately follows that s β ′ = 1 − s β sin ξ . (8.12)By convention, 0 ≤ β ′ ≤ π (or equivalently, sin 2 β ′ ≥ φ is determined by the positivity of v ′ . Hence, it follows that e iφ = c β + is β e − iξ (1 + s β sin ξ ) / . (8.13)43hen, eq. (8.10) yields, e iξ ′ s β ′ = − i √ c β + is β cos ξ (cid:0) s β sin ξ (cid:1) / . (8.14)Likewise, the relative phase ξ ′ is given by, e iξ ′ = s β cos ξ − ic β (1 − s β sin ξ ) / . (8.15)That is, sin ξ ′ = − c β (1 − s β sin ξ ) / , cos ξ ′ = s β cos ξ (1 − s β sin ξ ) / . (8.16)Consequently, eqs. (8.12) and (8.16) yield, s β ′ sin ξ ′ = − c β . (8.17)Finally, if β = π and sin ξ = ±
1, then one of the vevs vanishes. It then follows that s β ′ = 0, in which case ξ ′ is indeterminate if s β ′ = 0 and ξ ′ = 0 if c β ′ = 0.Using eqs. (8.9), (8.12) and (8.15), it is instructive to note that2 Re( m ′ e iξ ′ ) s β ′ = 2 Re( m ) s β cos ξ + c β ( m − m )1 − s β sin ξ . (8.18)In light of eq. (6.4), it follows thatRe m = Re( m e iξ ) cos ξ + Im( m e iξ ) sin ξ = Re( m e iξ ) cos ξ . (8.19)Hence, after using eqs. (6.2) and (6.3) for m − m and eq. (8.19) for Re m , it then followsthat eq. (8.18) yields,2 Re( m ′ e iξ ′ ) s β ′ = 2 Re( m e iξ ) s β + λv (1 − R ) c β − s β sin ξ ) . (8.20)In particular, eq. (7.20) yields, m A = 2 Re( m ′ e iξ ′ ) s β ′ + λ ′ v sin ξ ′ = 2 Re( m e iξ ) s β , (8.21)after employing eqs. (8.5) and (8.16). Comparing with eq. (6.14), we see that one obtainsthe same result for m A in the GCP3 basis and the U(1) ⊗ Π basis respectively, as required.Note that the same conclusion can be drawn by plugging the results of eqs. (8.2), (8.7) and(8.8) into eq. (7.21), which reproduces the result of eq. (6.15).For completeness, we check that the scalar mass spectrum derived from the softly brokenU(1) ⊗ Π and GCP3-symmetric scalar potentials coincide, as required. Plugging in the44esults of eqs. (8.4) and (8.5) into eq. (7.22) reproduces the result of eq. (6.16) for m H ± . Tocheck the squared masses of the neutral CP-even scalars, we plug the results of eqs. (8.2)and (8.5) into eq. (7.27), which reproduces the result of eq. (6.21). Finally, we plug in theresults of eqs. (8.2), (8.5) and (8.17) into eq. (7.28), which reproduces the result of eq. (6.22).As a final check of our computations, one can verify that the invariant quantities, Y , Z , . . . , Z , | Z | , | Z | and Z ∗ Z are independent of the choice of basis. For example, startingfrom the GCP3 basis, Z ∗ Z = − λ ′ s β ′ sin ξ ′ | Z | = − λ ′ s β ′ sin ξ ′ (1 − s β ′ sin ξ ′ ) = λ (1 − R ) c β s β , (8.22)in agreement with eqs. (6.10) and (6.11). One can also check that all the other invariantsyield the same values in the GCP3 and U(1) ⊗ Π bases.Given a softly-broken U(1) ⊗ Π -symmetric scalar potential that is displayed in theU(1) ⊗ Π basis where the softly-broken symmetry is manifestly realized, it may turn outthat the scalar potential is invariant under some discrete or continuous subgroup of U(1) ⊗ Π .It is of interest to determine implications of this invariance for the scalar potential parame-ters when expressed in the GCP3 basis. We proceed by assuming that the scalar potential,which is specified in eq. (C.1) in terms of squared mass parameters Y a ¯ b and dimensionlessparameters Z ab,cd [defined in eq. (C.4)], is invariant under the transformation,Φ a → X a ¯ b Φ b , Φ † ¯ a → Φ † ¯ b X † b ¯ a . (8.23)That is [cf. eqs. (C.7) and (C.8)], Y = XY X † , Z = ( X ⊗ X ) Z ( X † ⊗ X † ) . (8.24)Consider a change of scalar field basis specified by U [e.g., U specified in eq. (8.1) trans-forms the U(1) ⊗ Π basis into the GCP3 basis]. In light of eqs. (C.7) and (C.8), Y ′ = U Y U † yields the squared mass parameters in the new basis, and Z ′ = ( U ⊗ U ) Z ( U † ⊗ U † ) yields thedimensionless parameters in the new basis. In the new basis, the symmetry transformationmatrix X will be denoted by X ′ . That is, Y ′ = X ′ Y ′ X ′† , since the parameters in the newbasis are invariant with respect to the transformations induced by X ′ . It then follows that, U Y U † = X ′ ( U Y U † ) X ′† = ⇒ Y = ( U † X ′ U ) Y ( U † X ′ U ) † . (8.25)Comparing this result with eq. (8.24), we can conclude that X ′ = e iζ U XU † , (8.26)45here the complex phase factor e iζ is arbitrary and can be chosen for convenience as itcorresponds to an additional hypercharge U(1) Y transformation, which has no effect on thescalar potential parameters. One can now check that Z ′ = ( X ′ ⊗ X ′ ) Z ′ ( X ′† ⊗ X ′† ) byinserting eq. (8.26) for X ′ and evaluating the products and hermitian conjugates accordingto eqs. (C.10) and (C.11).We shall now employ eq. (8.26) in several examples. First, suppose that the softly-brokenU(1) ⊗ Π -symmetric scalar potential is invariant with respect to Z . Using Table I, it followsthat X = (cid:0) − (cid:1) . Hence, when transformed to the GCP3 basis using U specified in eq. (8.1),and choosing e iζ = − i , it follows that, X ′ = − , (8.27)which corresponds to the Π ′ symmetry defined in Table IV. This is easily checked usingeqs. (8.7)–(8.9). In particular, if Im m = 0 and m = m , then it follows that m ′ = m ′ and Re m ′ = 0 [cf. Table VI].Second, suppose that the softly-broken U(1) ⊗ Π -symmetric scalar potential is invariantwith respect to Π . Using Table I, it follows that X = (cid:0) (cid:1) . Hence, when transformed tothe GCP3 basis using U specified in eq. (8.1), and choosing e iζ = 1, it follows that X ′ = X .That is, the scalar potential in the GCP3 basis also exhibits a Π symmetry. This is easilychecked using eqs. (8.7)–(8.9). Namely, if m = m and Im m = 0 [cf. Table III], thenthe same relations also hold for the primed parameters.Third, suppose that the softly-broken U(1) ⊗ Π -symmetric scalar potential is invariantwith respect to U(1). Using Table I, it follows that X = (cid:16) e − iθ e iθ (cid:17) , where − π < θ ≤ π .Transforming to the GCP3 basis using U given in eq. (8.1), and choosing e iζ = 1, we obtain X ′ = cos θ sin θ − sin θ cos θ , for − π < θ ≤ π, (8.28)which defines the U(1) ′ symmetry transformation. In particular, if λ = λ and m = λ = λ = λ = 0 then eqs. (8.2)–(8.9) yield, m ′ = m ′ , Re m ′ = 0, λ ′ = λ ′ − λ ′ − λ ′ is realand λ ′ = λ ′ = 0, which are the constraints due to U(1) ′ as indicated in Table VI.As our final example, we reconsider the residual unbroken symmetry in the case of asoftly-broken U(1) ⊗ Π -symmetric scalar potential when c β = 0. Below eq. (6.26), wenoted that in this limiting case, after performing a rephasing to set ξ = 0, the residual46ymmetry of the scalar potential and vacuum was Π . However, in this example, we shallkeep ξ arbitrary. In order to accommodate ξ = 0 we define a new discrete symmetry,Π ( α )2 : Φ → e − iα Φ , Φ → e iα Φ , where α is a fixed real parameter . (8.29)Note that when Π ( α )2 is applied twice, one obtains the identity. This means that for any fixedvalue of α , the Π ( α )2 symmetry is equivalent to a Z symmetry that is manifestly realized ina different scalar field basis.Of course, for α = 0, we regain the Π symmetry. Moreover, up to an overall hyperchargeU(1) Y transformation, the Π ′ symmetry corresponds to α = π . If the scalar potential inthe Φ-basis is invariant under Π ( α )2 , then it follows that, m = m , Im( m e iα ) = 0 , λ = λ , Im( λ e iα ) = 0 , λ = λ ∗ e − iα . (8.30)In light of eq. (6.4), it follows that for c β = 0, the residual symmetry of the softly-brokenU(1) ⊗ Π -symmetric scalar potential and vacuum is Π ( ξ )2 (for any fixed value of ξ ).Suppose that the Π ( α )2 symmetry is unbroken by the scalar potential in the U(1) ⊗ Π basis. Then, we can deduce the corresponding symmetry in the GCP3 basis. In this example, X = (cid:16) e − iα e iα (cid:17) , where α is a fixed real parameter. Hence, when transformed to the GCP3basis using U specified in eq. (8.1), and choosing e iζ = 1, it follows that X ′ = sin α cos α cos α − sin α , for a fixed real value of α. (8.31)Without loss of generality, we may take − π < α ≤ π (since α → α + π yields a hyperchargeU(1) Y transformation). We shall denote this symmetry by,Π ( α )2 : Φ → Φ sin α + Φ cos α , Φ → Φ cos α − Φ sin α . (8.32)Note that for α = 0 [ α = π ], the Π ( α )2 symmetry coincides with Π [ Z ]. That is, Π ( α )2 provides an interpolation from the Π to the Z symmetry.Imposing the Π ( α )2 symmetry on the parameters of the scalar potential in the Φ-basis fora fixed value of α = 0, π , it follows that,Im m = Im λ = Im λ = Im λ = 0 , (8.33) m − m = 2 tan α Re m , (8.34) λ − λ = 2 tan α Re( λ + λ ) , (8.35) λ + λ − λ + λ + Re λ ) = 4 cot 2 α Re( λ − λ ) . (8.36)47pplying the above results to the softly-broken GCP3-symmetric scalar potential, we set λ ′ ≡ λ ′ = λ ′ = λ ′ + λ ′ + λ ′ and Im λ ′ = λ ′ = λ ′ = 0. Note that eqs. (8.33)–(8.36) areconsistent with these constraints. In addition, the softly broken GCP3-symmetric scalarpotential preserves the Π ( α )2 symmetry in two cases: (i) if m ′ is real and nonzero then m ′ = m ′ [in which case, α is determined from eq. (8.34)]; or (ii) if m ′ = 0 then m ′ = m ′ (in which case Π ( α )2 , which is a symmetry of the scalar potential for all values of α , is promotedto an unbroken GCP3 symmetry).For example, in the inert limit of the softly-broken GCP3-symmetric scalar potential,where s β ′ c β ′ = 0, sin ξ ′ = 0, m ′ = m ′ and m ′ e iξ ′ is real [See Table IX], we see thatthe conditions for the Π ( α )2 symmetry are satisfied. Moreover, one can show that the Π ( α )2 symmetry is unbroken by the vacuum as follows. Using eqs. (7.2)–(7.4) under the assumptionthat sin ξ ′ = 0, it follows that m ′ e iξ ′ = m ′ cos ξ ′ = ± m ′ is real (where ± corresponds to ξ ′ = 0 or ξ ′ = π , respectively), and m ′ − m ′ = ± m ′ c β ′ s β ′ . (8.37)Hence, eq. (8.34) yields tan α = ± cot 2 β ′ . In the convention where − π < α ≤ π and0 ≤ β ′ ≤ π , it follows that sin α = ± cos 2 β ′ and cos α = sin 2 β ′ , or equivalently α = ± (cid:0) π − β ′ (cid:1) . (8.38)The Π ( α )2 symmetry is unbroken by the vacuum if X ′ v ′ v ′ e iξ ′ = ± v ′ v ′ e iξ ′ , (8.39)where the ± sign reflects the fact that (cid:8) h Φ i , h Φ i (cid:9) is equivalent to (cid:8) −h Φ i , −h Φ i (cid:9) , as thetwo are related by a hypercharge U(1) Y transformation. After using the value of α obtainedin eq. (8.38) to determine X ′ [cf. eq. (8.31)], it follows that the Π ( α )2 symmetry is unbrokenby the vacuum since the following equation is an identity, ± c β ′ s β ′ s β ′ ∓ c β ′ c β ′ ± s β ′ = ± c β ′ ± s β ′ . (8.40)Consequently, one can conclude that the Π ( α )2 symmetry is responsible for the inert limit(and the attendant exact Higgs alignment) in this case.48n some applications, it is useful to invert the relations obtained in eqs. (8.2)–(8.9). Thiscan be achieved by starting from the GCP3 basis and employing the unitary transformation U − = e − iφ √ ii . (8.41)The resulting U(1) ⊗ Π basis parameters are, λ = λ ′ − λ ′ , (8.42) λ = λ ′ + λ ′ , (8.43) λ = λ ′ + λ ′ , (8.44) λR = λ ′ + λ ′ , (8.45) λ = λ = λ = 0 . (8.46)In addition, the corresponding soft-breaking squared mass parameters are: m = ( m ′ + m ′ ) − Im m ′ , (8.47) m = ( m ′ + m ′ ) + Im m ′ , (8.48) m = Re m ′ − i ( m ′ − m ′ ) . (8.49)Finally, the vevs in the U(1) ⊗ Π basis are given by v = e − iφ √ (cid:0) v ′ + iv ′ e iξ ′ (cid:1) , v e iξ = e − iφ i √ (cid:0) v ′ − iv ′ e iξ ′ (cid:1) . (8.50)where v and v are real and positive. Hence, c β = 1 √ (cid:0) − s β ′ sin ξ ′ (cid:1) / , s β = 1 √ (cid:0) s β ′ sin ξ ′ (cid:1) / , (8.51)and it immediately follows that s β = 1 − s β ′ sin ξ ′ . (8.52)By convention, 0 ≤ β ≤ π (or equivalently, sin 2 β ≥ φ is again fixed by the positivity of v , which yields e − iφ = c β ′ − is β ′ e − iξ ′ (1 − s β ′ sin ξ ′ ) / , (8.53)49nd is consistent with eq. (8.13) after employing eqs. (8.51) and (8.55). Then, eq. (8.50)yields, e iξ s β = i √ c β ′ − is β ′ cos ξ ′ (cid:0) − s β ′ sin ξ ′ (cid:1) / . (8.54)Likewise, ξ is given by, e iξ = s β ′ cos ξ ′ + ic β ′ (1 − s β ′ sin ξ ′ ) / . (8.55)That is, sin ξ = c β ′ (1 − s β ′ sin ξ ′ ) / , cos ξ = s β ′ cos ξ ′ (1 − s β ′ sin ξ ′ ) / . (8.56)Hence eqs. (8.52) and (8.56) yield, s β sin ξ = c β ′ . (8.57)Once the U(1) ⊗ Π basis parameters have been derived, one can perform one further rephas-ing to remove the phase ξ (which is unphysical). Finally, if β ′ = π and sin ξ ′ = ±
1, thenone of the vevs vanishes. It then follows that s β = 0, in which case ξ is indeterminate if s β = 0 and ξ = 0 if c β = 0.The scalar masses obtained in the U(1) ⊗ Π basis and the GCP3 basis were derived byapplying eq. (5.22). In employing this equation, a specific value of η was chosen. Theeigenvalues of M are independent of this choice. However, the identification of H and A depend on this choice in the inert limit. For a consistent treatment of the two basis choices,one should also transform η when changing the scalar field basis according to eq. (5.8).In particular, given the choice of η = − ξ that was employed in the U(1) ⊗ Π basis, thecorresponding η ′ in the GCP3 basis is given by e − iη ′ = (det U ) e − iη = e iφ e iξ = ( c β ′ + is β ′ e iξ ′ ) ( s β ′ cos ξ ′ + ic β ′ )(1 − s β ′ sin ξ ′ )(1 − s β ′ sin ξ ′ ) / , (8.58)after employing eq. (8.1) to obtain det U = e iφ and making use of eqs. (8.53) and (8.55).The numerator of eq. (8.58) can be simplified with a little algebra,( c β ′ + is β ′ e iξ ′ ) ( s β ′ cos ξ ′ + ic β ′ ) = e iξ ′ (cid:2) c β ′ cos ξ ′ + i ( s β ′ − sin ξ ′ ) (cid:3) ( s β ′ cos ξ ′ + ic β ′ )= e iξ ′ ( c β ′ sin ξ ′ + i cos ξ ′ )(1 − s β ′ sin ξ ′ ) . = ie i ( ξ ′ − ψ ) (1 − s β ′ sin ξ ′ )(1 − s β ′ sin ξ ′ ) / , (8.59)50here we have used eq. (7.16) in the final step. Inserting this result back into eq. (8.58)yields η ′ = ψ − ξ ′ − π , (8.60)which justifies the choice of η that was employed in eq. (7.17). Note that if s β ′ = 0, thenone of the two vevs in the GCP3 basis vanishes. If c β ′ = 1, then eq. (7.16) yields ψ = ξ ′ andwe conclude that η ′ = − π . If c β ′ = −
1, then ψ = ξ ′ = 0 (since in this case h Φ i = v/ √ η ′ = − π . That is, e − iη ′ = −
1, which is themotivation for the choice of η employed in obtaining eqs. (5.26)–(5.29). For a satisfying check of eq. (8.60), one can compute the value of Z in the GCP3 basisstarting from its value in the U(1) ⊗ Π basis given in eq. (6.10). In performing this com-putation, one must remember to rephase Z as indicated in eq. (5.10). After making use ofeqs. (8.42), (8.45), (8.60) and (7.16), it then follows that Z = λs β (1 − R ) e − iξ e − iφ = − λ ′ (1 − s β ′ sin ξ ′ ) e iη ′ = λ ′ e − iξ ′ (1 − s β ′ sin ξ ′ ) e iψ = λ ′ e − iξ ′ (cos ξ ′ + ic β ′ sin ξ ′ ) , (8.61)in agreement with eq. (7.12).Although cases in which one of the two vevs vanish appear to be isolated from theparameter regimes in which both vevs are nonvanishing, in fact the two parameter regimescan be regarded as being continuously connected. For example, starting from the GCP3basis, the parameter regime in which one of the two vevs vanishes (i.e., s β ′ = 0) impliesthat m ′ = 0 due to eqs. (2.13) and (2.14). In light of eq. (8.9), it follows that in termsof the U(1) ⊗ Π basis parameters, Re m = 0 and m = m , but in general Im m = 0.Moreover, eq. (6.5) then yields β = π . As expected, this parameter regime can be identifiedas the inert limit, independently of the basis choice. Nevertheless, it is clear that β = π is continuously connected to other regions of the parameter space in the U(1) ⊗ Π basis.Similarly, the case of one vanishing vev in the U(1) ⊗ Π basis corresponds to s β ′ sin ξ = ± β ′ = π in the GCP3 basis. Finally, the inert limit in the U(1) ⊗ Π basiswhen R = 1 corresponds to the inert limit in the GCP3 basis when λ ′ = 0. If s β = 0 in the U(1) ⊗ Π basis then Z = 0, in which case m H = m A and the results of eqs. (5.26)–(5.29)do not depend on the choice of η . Note that it follows that the ratio Re( m ′ e iξ ′ ) /s β ′ that appears in eq. (7.20) is indeterminate, in whichcase it can be replaced by m A − λ ′ v sin ξ ′ , with m A being regarded as a free parameter. X. THE HIGGS ALIGNMENT LIMIT
The neutral scalar squared-mass matrix, M , given in eq. (5.22) is expressed with respectto a basis of neutral scalar interaction eigenstates, {√ H − v , Re H , Im H } , where H and H are the neutral components of the Higgs basis fields. The neutral scalar interactioneigenstate, ϕ ≡ √ H − v , possesses tree-level couplings to SM particles that coincideprecisely with those of the SM Higgs boson. Consequently, if the mixing of ϕ with Re H andIm H were to vanish exactly, then ϕ would be a mass-eigenstate with tree-level propertiesthat are indistinguishable from those of the SM Higgs boson. In this case, the direction of ϕ in field space is exactly aligned with the direction of the vacuum expectation value v .Hence, the limit of zero mixing described above is called the Higgs alignment limit [41–45].In light of eq. (5.22), it follows that the Higgs alignment is realized exactly if and onlyif Z = 0, in which case we can identify m ϕ = Z v . In this limit, the masses of thetwo other neutral scalars are not immediately constrained (beyond experimental boundsbased on the absence of any newly discovered scalar states at the LHC). Although theobserved Higgs boson at the LHC is SM-like, the precision of the current data allows for10–20% deviations from SM behavior of the Higgs boson couplings to vector bosons andthird generation quarks and charged leptons [3–6]. Thus, the present Higgs data requiresonly an approximate Higgs alignment, which allows for a small mixing of ϕ with the othertwo neutral scalar interaction eigenstates. This small mixing can be achieved in one of twoways—either | Z | ≪ Y ≫ v . The latter corresponds to the decoupling limit of the2HDM [41, 73].For example, if CP is conserved then the squared-matrix, M , of the neutral scalars,expressed with respect to Higgs basis fields, breaks up into a 2 × × M = Z v Z v Z v m A + Z v
00 0 m A . (9.1)Once again, we see that Z = 0 corresponds to exact Higgs alignment, whereas approximateHiggs alignment is realized when | Z | ≪ m A ≫ v .Diagonalizing the 2 × H and h , Hh = c β − α − s β − α s β − α c β − α √ H − v √ H , (9.2)52here m h ≤ m H , c β − α ≡ cos( β − α ) and s β − α ≡ sin( β − α ) in a convention where0 ≤ β − α ≤ π . In a real Φ-basis with real nonnegative vevs, h Φ a i = v a / √ a = 1 , β = v /v and α is the mixing angle that diagonalizes the CP-even Higgs squared-massmatrix when expressed with respect to the {√ − v , √ − v } basis. Never-theless, the quantity s β − α is independent of the choice of the scalar field basis.After diagonalizing the matrix M H , the neutral CP-even scalar masses are given by, m H,h = (cid:26) m A + ( Z + Z ) v ± q(cid:2) m A − ( Z − Z ) v (cid:3) + | Z | v (cid:27) , (9.3)where m h ≤ m H , and s β − α c β − α = − Z v m H − m h , c β − α − s β − α = m A − ( Z − Z ) v m H − m h . (9.4)We shall henceforth assume that h ≃ √ H − v is SM-like and thus should be identifiedwith the observed Higgs boson with m h ≃
125 GeV. Under this assumption, it follows fromeq. (9.2) that c β − α → c β − α = − Z v p ( m H − m h )( m H − Z v ) . (9.5)in a convention where s β − α ≥
0. Having identified h as SM-like, it follows that m H > Z v ,which confirms that c β − α → One can now ask the following question—is there a symmetry that can be imposed onthe 2HDM scalar potential such that the Higgs alignment limit is exact, corresponding tothe condition that Z = 0. In fact, it is straightforward to identify the complete list of allpossible symmetries that enforce the Z = 0 condition in the 2HDM by noting that thescalar potential minimum condition [cf. eq. (5.20)] would then imply that Y = 0. Thus, inlight of eq. (5.13), the Higgs alignment limit is exact if Y = (cid:2) ( m − m ) s β − Re( m e iξ ) c β − i Im( m e iξ ) (cid:3) e − iξ = 0 . (9.6)A sufficient (but not necessary) condition for satisfying eq. (9.6) can be obtained by setting m = 0, in which case either s β = 0 or m = m . If H were SM-like (with m H ≥ m h ) then m H → Z v in the Higgs alignment limit, in which case eq. (9.5)would not be very useful. Indeed, in this case s β − α → s β − α = − Z v / p ( m H − m h )( Z v − m h ) in an alternativeconvention where c β − α ≥ h is always SM-likein the Higgs alignment limit in light of eq. (5.30), irrespective of the mass ordering of h and H . s β = 0, then the Z symmetry is unbroken by the vacuum. This corresponds to theIDM, where an unbroken Z symmetry is present in the Higgs basis, which implies that Y = Z = Z = 0. In the IDM, the Higgs basis field H is odd under the Z symmetry,whereas H along with all other SM fields are Z -even. If in addition, one imposes thecondition Z = 0, then the IDM scalar potential will exhibit an unbroken U(1) symmetrythat is preserved by the vacuum (resulting in a mass-degenerate pair of inert neutral scalars, m H = m A ). In these two cases, one can identify ϕ √ H − v as the SM Higgs boson attree-level. Deviations of the properties of ϕ from that of the SM Higgs boson can arise dueto the other Higgs fields (beyond ϕ ) contributing to radiative loop corrections to physicalobservables (e.g., the charged Higgs boson loop that contributes to ϕ → γγ decay).If s β = 0 then eq. (9.6) is also satisfied if m = 0 and m = m . Consulting the resultsof Table III, it follows that exact Higgs alignment is automatically implemented if the 2HDMscalar potential respects one of the following symmetries: Z ⊗ Π , U(1) ⊗ Π , SO(3), GCP2,or GCP3. Of course, given that GCP2 is equivalent to Z ⊗ Π in a different scalar fieldbasis and GCP3 is equivalent to U(1) ⊗ Π in a different scalar field basis, it follows thatthere are three inequivalent symmetries of the 2HDM scalar potential beyond the IDM thatyield exact Higgs alignment. What is common to these three inequivalent symmetries isthat they all reside in the ERPS.The conditions that m = m and m = 0 imply that the symmetries identified aboveare preserved by the scalar potential. In particular, consider the expression for Z in theERPS (i.e., where λ ≡ λ = λ and λ = − λ ), Z = e − iξ (cid:26) − s β c β ( λ − λ ) + is β Im( λ e iξ ) + c β Re( λ e iξ ) + ic β Im( λ e iξ ) (cid:27) , (9.7)where λ ≡ λ + λ + Re( λ e iξ ). We have already noted that the inert limit of thescalar potential in the ERPS regime, where exact Higgs alignment is achieved, correspondsto Y = 0, which then implies that Z = 0 via eq. (5.20) and Z = 0 due to the ERPS In Refs. [38, 39], Higgs alignment enforced by a symmetry is called “natural” if Z = 0 is achievedindependently of the value of β . This extra condition eliminates the Z ⊗ Π , GCP2 and U(1) ⊗ Π -symmetric scalar potentials from the list of potentials that exhibit a “natural” Higgs alignment. InRefs. [38, 39], the maximal symmetry groups associated with the GCP3 and SO(3)-symmetric scalarpotentials, which do exhibit a “natural” Higgs alignment, are identified as Z ⊗ O(2) ⊗ O(2) and O(3) ⊗ O(2),respectively. In addition, if the U(1) Y hypercharge gauge coupling g ′ = 0 in the gauge covariant kineticterms of the scalar fields, then an SO(3)-symmetric scalar potential with λ = λ = 0 (cf. Table XII)yields an SO(5)-symmetric scalar Lagrangian that also exhibits a “natural” Higgs alignment. Z ⊗ Π -symmetric scalar potential, applying theconditions exhibited in Table III yields the expression for Z given in eq. (5.19). Because m = m and m = 0, it automatically follows that Y = 0 which implies that Z = 0.The vanishing of Z [although not immediately evident from eq. (9.7)] is a consequence ofthe scalar potential minimum conditions of the ERPS which yield, λ (1 − R ) c β = λ s β sin 2 ξ = 0 or s β = 0 . (9.8)where R ≡ ( λ + λ + λ ) /λ . Since R = 1 and λ = 0 (otherwise, the symmetry group ofthe scalar potential is larger than Z ⊗ Π ), it then follows that either c β = sin 2 ξ = 0 or s β = 0. Inserting these conditions in eq. (9.7) along with the Z ⊗ Π symmetry conditions,Im λ = λ = λ = 0, yields Z = 0 as expected.In the case of an unbroken U(1) ⊗ Π symmetry, we simply add one additional condition, λ = 0 to the scalar potential parameters (while maintaining R = 1). In this case, eq. (9.8)implies that either c β = 0 or s β = 0 (with no restriction on ξ , which is an unphysical phasethat can be rephased away), and again yields Z = 0.It is instructive to consider the GCP3-symmetric scalar potential, which can be obtainedfrom the Z ⊗ Π -symmetric scalar potential by adding one additional constraint, R = 1(while maintaining λ = 0). The case of an unbroken GCP3 symmetry is not physicallydistinct from the previous case since it is equivalent to a U(1) ⊗ Π symmetry in a differentscalar field basis. The minimum conditions of the GCP3-symmetric scalar potential (wherewe now employ primed parameters) yield, λ ′ c β ′ sin ξ ′ = λ ′ s β ′ sin 2 ξ ′ = 0 or s β ′ = 0 . (9.9)These conditions guarantee that Z = 0 [cf eq. (7.13)], independently of the parametersof the GCP3-symmetric scalar potential. In particular, exact Higgs alignment is achievedfor all values of β ′ in cases of an unbroken and some softly-broken GCP3-symmetric scalarpotentials, in contrast to the cases of Z ⊗ Π and U(1) ⊗ Π where exact Higgs alignmentis satisfied only when β = 0, π or π . The reader might wonder how it is possible that exact alignment can be achieved for all values of β ′ butonly special values of β in light of the fact that the U(1) ⊗ Π and GCP3-symmetric scalar potentials canbe transformed into one another by an appropriate change of basis. The answer can be seen by examiningeq. (8.52). In light of eq. (9.9) with λ ′ = 0, it follows that either sin ξ ′ = 0, in which case all values of β ′ are permitted or c β ′ = cos ξ ′ = 0. Using eq. (8.52), it follows that the possible values of β ′ correspond toeither s β = 0 or c β = 0.
55s noted in footnote 21, the authors of Refs. [38, 39] refer to exact Higgs alignment thatis independent of the value of β as “natural” alignment. However, such a designation is notuseful for two reasons. First, the above discussion shows that the GCP3-symmetric scalarpotential exhibits natural alignment in the sense of Refs. [38, 39], whereas the U(1) ⊗ Π -scalar potential does not. This distinction is suspect given that these two scalar potentialsare physically equivalent and can be transformed into each other by an appropriate changeof scalar field basis (as shown explicitly in Section VIII). Second, the concept of naturalness,as introduced by ‘t Hooft in Ref. [77], implies that a small parameter of a theory should beconsidered natural if the symmetry of the Lagrangian is increased by setting the parameterto zero. In the present context, the small parameters are the potentially soft-breakingparameters, m − m and m , of the ERPS4, which could potentially generate departuresfrom exact Higgs alignment. All the symmetry groups discussed in this section provide forexact Higgs alignment naturally in the sense of ‘t Hooft. Indeed, exact Higgs alignmentrealized in this way is stable under renormalization group running, which is further evidencethat the symmetry based approach taken in this section is correct. The case of an unbroken SO(3) symmetry can be examined by taking R = 1 in eq. (9.8) orequivalently taking λ ′ = 0 in eq. (9.9). In both limiting cases, we see Z = 0, independentlyof the scalar potential minimum conditions!The conditions that m = m and m = 0 are sufficient but not necessary for exactHiggs alignment. In particular, exact Higgs alignment arises in any inert limit of the 2HDM.Thus to obtain a complete classification of 2HDM scalar potentials that yield exact Higgsalignment due to a symmetry, it suffices to enumerate the inert limits of the softly-broken Z ⊗ Π , U(1) ⊗ Π or GCP3, and SO(3)-symmetric scalar potentials. These results can befound in Tables VII, VIII, IX and X. We proceed to list all the relevant subcases below.Given a softly-broken Z ⊗ Π -symmetric scalar potential, exact Higgs alignment arisesin two subcases, as shown in Section V: (i) s β = sin 2 ξ = 0, m = m and m = 0,which preserves a Z symmetry that is unbroken in the vacuum, and (ii) c β = sin 2 ξ = 0, m = m and Im (cid:2) m (cid:3) = 2 Re m Im m = 0, which preserves a Π [Π ′ ] symmetry ifIm m = 0 [Re m = 0] that is unbroken in the vacuum. In the absence of soft breaking, In general, renormalization group running does not preserve the scalar field basis. However, the grouptheoretic properties of the symmetries of the scalar potential, whose specific realization may change indifferent choices of the scalar field basis, do not depend on the basis choice. and Π ′ are identical. The Z ,Π or Π ′ residual symmetries are responsible for maintaining the exact Higgs alignment.Given a softly-broken U(1) ⊗ Π -symmetric scalar potential, exact Higgs alignment arisesin two subcases, as shown in Section VI: (i) s β = 0, m = m and m = 0, whichpreserves a U(1) symmetry that is unbroken in the vacuum, and (ii) c β = 0, m = m and m = 0. In light of eq. (6.4), one can rephase Φ → e − iξ Φ to achieve a real basis, inwhich case the scalar potential in subcase (ii) preserves a Π symmetry that is unbroken inthe vacuum. If one does not remove the (unphysical) parameter ξ , then eq. (8.30) can beused to identify the unbroken vacuum symmetry as Π ( ξ )2 , which is Π in the rephased scalarfield basis. In the case of an unbroken U(1) ⊗ Π -symmetric scalar potential, the U(1) ⊗ Π symmetry is spontaneously broken down to U(1) if s β = 0 or to Π if c β = 0.Although a softly-broken GCP3-symmetric scalar potential is equivalent to a softly-broken U(1) ⊗ Π -symmetric scalar potential in a different basis, it is instructive to enu-merate the cases in which a softly-broken GCP3-symmetric scalar potential exhibits exactHiggs alignment. Using the results of Section VII, exact Higgs alignment arises in four sub-cases in terms of the primed GCP3 basis parameters: (i) s β ′ = 0, m ′ = m ′ and m ′ = 0,which preserves a Z symmetry that is unbroken in the vacuum; (ii) c β ′ = cos ξ ′ = 0, m ′ = m ′ and Im m ′ = 0, which preserves a U(1) ′ symmetry that is unbroken in thevacuum; (iii) c β ′ = sin ξ ′ = 0, m ′ = m ′ and Re m ′ = 0, which preserves a Π symmetrythat is unbroken by the vacuum; and (iv) s β ′ c β ′ = 0, sin ξ ′ = 0, m ′ = m ′ and Re m ′ = 0,which preserves a Π ( α )2 vacuum symmetry, where α = (cid:0) π − β ′ (cid:1) cos ξ ′ = ± (cid:0) π − β ′ (cid:1) . Thisresult is derived in Section VIII, where the Π ( α )2 symmetry is introduced in eq. (8.32) andthe relation that yields α in terms of β ′ is obtained in eq. (8.38). Finally, in the case ofan unbroken GCP3-symmetric scalar potential, the GCP3 symmetry, which is equivalentto a U(1) ′ ⊗ Z symmetry, is spontaneously broken down to U(1) ′ [ Z ] if c β = cos ξ ′ = 0[ s β ′ = 0], or to Π ( α )2 with α = ± (cid:0) π − β ′ (cid:1) if s β ′ = 0 and cos ξ ′ = ± Z = 0 independently of the scalar potential minimum con-ditions. This means that all softly-broken SO(3)-symmetric scalar potentials exhibit exactHiggs alignment, since the scalar potential minimum conditions will guarantee that Y = 0even when m = m and/or m = 0.Below eq. (6.25), we noted the presence of mass-degenerate scalars, H and A , which was57ttributed to a Peccei-Quinn U(1) symmetry in the Higgs basis, H → H , H → e iθ H (forany value of 0 ≤ θ < π ), which is unbroken by the vacuum. It is instructive to ascertain theprecise form of the U(1) symmetry in the Φ-basis. To accomplish this, we employ eq. (8.26),where the unitary matrix U = c β − e − i ( ξ + η ) s β e iξ s β e − iη c β , (9.10)transforms the Higgs basis into the Φ-basis. The phase e iη , which appears in eq. (5.9) andreflects the freedom to rephase the Higgs basis field H , cancels exactly when eq. (8.26) isapplied. Starting with X = (cid:0) e iθ (cid:1) , we make use of eq. (8.26) with ζ = − θ to obtain X ′ = cos θ − ic β sin θ − ie − iξ s β sin θ − ie iξ s β sin θ cos θ + ic β sin θ . (9.11)Thus, in the Φ-basis characterized by tan β = |h Φ i / h Φ i| and ξ = arg (cid:2) h Φ i ∗ h Φ i (cid:3) , thePeccei-Quinn symmetry, which we designate by U(1) H (to remind the reader that it hasbeen first applied in the Higgs basis), is given by U (1) H : Φ −→ (cos θ − ic β sin θ )Φ − ie − iξ s β sin θ Φ , Φ −→ − ie iξ s β sin θ Φ + (cos θ + ic β sin θ )Φ . (9.12)Imposing the U(1) H symmetry on the parameters of a general 2HDM scalar potential inthe Φ-basis yields the following constraints,Im( m e iξ ) = λ = λ = λ = 0 , (9.13) λ ≡ λ = λ = λ + λ , (9.14) m − m = 2 cot 2 β Re( m e iξ ) . (9.15)These constraints correspond to a softly-broken SO(3)-symmetric scalar potential and scalarpotential minimum conditions [cf. eqs. (7.2)–(7.4) with λ = λ + λ and λ = 0]. Moreover, X ′ c β s β e iξ = e − iθ c β s β e iξ , (9.16)which confirms that the vacuum is invariant under the U(1) H transformation (up to an overallhypercharge U(1) Y transformation that has no effect on the scalar potential parameters). The version of the Peccei-Quinn symmetry transformation that is used here corresponds to U(1) PQ givenin Table I followed by a hypercharge U(1) Y transformation, which is also a symmetry of the vacuum inthe Higgs basis.
58e conclude that for a generic softly-broken SO(3)-symmetric scalar potential, a U(1) H subgroup remains unbroken and is responsible for the mass-degeneracy of H and A as well asthe exact Higgs alignment. In the case of an unbroken SO(3)-symmetric scalar potential,the SO(3) symmetry is spontaneously broken down to U(1) H , in which case both H and A can be identified as massless Goldstone bosons (of opposite CP quantum numbers).This completes the classification of all unbroken or softly-broken symmetries of the 2HDMscalar potential that yield an exact Higgs alignment. This classification is summarized inTable XI. Many aspects of this table can be easily understood by employing the results ofAppendix E. Applying the ERPS4 conditions ( λ = λ and λ = − λ ) in eqs. (E.1)–(E.6),the parameters of the scalar potential in the ERPS4 regime in the Φ-basis satisfy,Im( m e iξ ) = 0 , (9.17)( m − m ) s β = 2 Re( m e iξ ) c β , (9.18) c β Re( λ e iξ ) = s β c β (cid:2) λ − λ − λ − Re( λ e iξ ) (cid:3) , (9.19) c β Im( λ e iξ ) = − s β Im( λ e iξ ) . (9.20)Eqs. (9.17)–(9.20) provide the necessary and sufficient conditions for the inert limit of thescalar potential in the ERPS4 regime, thereby producing an exact Higgs alignment.One can check that all the entries listed in Table XI (including the first two lines, whichcorrespond to the Z -symmetric and U(1)-symmetric IDMs outside of the ERPS4 regime)satisfy the four conditions specified in eqs. (9.17)–(9.20). For example, starting from thesoftly-broken or unbroken GCP2-symmetric scalar potential transformed to the basis inwhich λ is real and λ = λ = 0, one easily obtains the following correlations of theparameters β and ξ for the symmetry cases listed in Table XI, Z ⊗ Π : s β s ξ = 0 or s β c β = 0 , (9.21)U(1) ⊗ Π : s β c β = 0 , (9.22)GCP3 : s β ′ s ξ ′ = 0 , (9.23)SO(3) : no constraints. (9.24)One can then employ eqs. (9.17) and (9.18) to determine the allowed soft-breaking due to m = m and/or m = 0 that is consistent with exact Higgs alignment. In the case of s β = 0, the Φ-basis coincides with the Higgs basis (up to a possible discrete Π transfor-mation), in which case U(1) H reduces to the standard U(1) PQ symmetry. ymmetry soft-breaking parameter residual unbroken symmetry ofcontraints scalar potential vacuum Z none s β = 0 Z Z U(1) none s β = 0 U(1) U(1) Z ⊗ Π m = m s β = 0 Z Z Z ⊗ Π Re m = 0 c β = sin ξ = 0 Π Π Z ⊗ Π Im m = 0 c β = cos ξ = 0 Π ′ Π ′ Z ⊗ Π none s β = 0 Z ⊗ Π Z Z ⊗ Π none c β = sin 2 ξ = 0 Z ⊗ Π Π U(1) ⊗ Π m = m s β = 0 U(1) U(1)U(1) ⊗ Π Re( m e iξ ) = 0 c β = 0 Π ( ξ )2 Π ( ξ )2 U(1) ⊗ Π none s β = 0 U(1) ⊗ Π U(1)U(1) ⊗ Π none c β = 0 U(1) ⊗ Π Π GCP3 m ′ = m ′ , Re m ′ = 0 s β ′ c β ′ = 0, sin ξ ′ = 0 Π ( α )2 Π ( α )2 GCP3 m ′ = m ′ s β ′ = 0 Z Z GCP3 Re m ′ = 0 c β ′ = 0, sin ξ ′ = 0 Π Π GCP3 Im m ′ = 0 c β ′ = 0, cos ξ ′ = 0 U(1) ′ U(1) ′ GCP3 none s β ′ = 0 U(1) ′ ⊗ Z Z GCP3 none s β ′ = 0, sin ξ ′ = 0 U(1) ′ ⊗ Z Π ( α )2 GCP3 none c β ′ = 0, cos ξ ′ = 0 U(1) ′ ⊗ Z U(1) ′ SO(3) m ′ = m ′ , Re( m ′ e iξ ′ ) = 0 s β ′ c β ′ = 0 U(1) H U(1) H SO(3) Re( m ′ e iξ ′ ) = 0 c β ′ = 0 U(1) H U(1) H SO(3) m ′ = m ′ s β ′ = 0 U(1) U(1)SO(3) none none SO(3) U(1) H TABLE XI: Classification of symmetries of the 2HDM scalar potential that yield exact Higgsalignment, where the tree-level properties of one of the neutral scalar mass-eigenstates coincidewith those of the SM Higgs boson. Note that m = m and Re( m e iξ ) = Im( m e iξ ) = 0(and similarly for the primed parameters) unless otherwise indicated. The unprimed parameterscorrespond to the Z ⊗ Π or U(1) ⊗ Π basis, whereas the primed parameters correspond to theGCP3 basis. All such basis choices are consistent with the ERPS4 with λ = λ = 0 and real λ ;the corresponding parameter constraints for the softly-broken GCP2-symmetric scalar potentialare given in eqs. (9.17)–(9.20). In cases where the residual symmetry is given by Π ( α )2 , the valueof α = (cid:0) π − β ′ (cid:1) cos ξ ′ , where cos ξ ′ = ±
1. Although separate listings are provided for scalarpotentials that exhibit the U(1) ⊗ Π or GCP3 symmetry (either of which may be softly-broken),they represent the same scalar potential expressed in two different choices of the scalar field basis. . IMPLICATIONS OF CUSTODIAL SYMMETRY One of the possible symmetries of the scalar potential that does not appear in Table IIIis custodial symmetry. This symmetry is necessarily violated by the hypercharge U(1) Y gauge interactions. Nevertheless, in the limit of g ′ → ρ -parameter from its custodialsymmetric value of ρ = 1, it is of interest to consider the possibility that the 2HDM scalarpotential respects the custodial symmetry. In more detail, if the 2HDM scalar potentialis symmetric under SU(2) L ⊗ SU(2) R transformations where SU(2) L is identified with theSU(2) part of the electroweak gauge group and SU(2) R is a global symmetry group, thenafter the symmetry breaking of SU(2) L the residual custodial symmetry can be identifiedwith an unbroken diagonal SU(2) L + R global symmetry.Details of the SU(2) L ⊗ SU(2) R transformation laws can be found in Ref. [49–51, 78]. Forexample, it was shown in Ref. [49] that for any custodial symmetric 2HDM scalar potential,there exists a Φ-basis in which, λ = Re λ and m , λ , λ , λ ∈ R . (10.1)Hence, a custodial symmetric potential is explicitly CP-conserving. In the case of an unbro-ken Z -symmetric scalar potential where m = λ = λ = 0, one is always free to rephaseΦ → i Φ , in which case the custodial symmetry condition of eq. (10.1) specializes to λ = ± Re λ and m = Im λ = λ = λ = 0 . (10.2)Additional information is provided by minimizing the scalar potential and determiningthe Higgs basis. Then, as shown in Ref. [50], the scalar potential respects the custodial That is, the custodial symmetry is violated by the gauge covariant kinetic term of the scalar fields that isproportional to g ′ . If λ = λ in the Φ-basis where λ ∈ R , then λ = λ in any real basis, Φ ′ = U Φ, such that U is areal orthogonal matrix. However, if m = λ = λ = 0, then one can perform a basis transformationwhere U is unitary but not real orthogonal that still preserves the reality of the basis. For such basistransformations the relation λ = λ is no longer preserved. Eq. (10.2) provides a trivial example ofthis. A more interesting example is provided by the basis transformation that converts eq. (10.10) intoeq. (10.11). Of course, if U is not real orthogonal, then the basis transformation will not preserve thereality of the vevs. Although this rephasing maintains the reality the scalar potential parameters, it introduces relative phase, ξ = π , in the vevs. Z = Z e − iη ∈ R , Y e − iη = − Z e − iη v ∈ R , Z e − iη ∈ R , (10.3)where the phase η represents the freedom to rephase the Higgs basis field H . It followsthat one can choose η such that the parameters of the scalar potential in the Higgs basisare all real, which implies that GCP1 is a symmetry of the scalar potential and vacuum. Inparticular, in a real Higgs basis, eq. (10.3) yields two possible solutions, Z = Z , (10.4)or Z = ± Z , and Y = Z = Z = 0 . (10.5)Eq. (10.4) is a consequence of choosing η = 0 (mod π ). In the case of Y = Z = Z = 0,the condition Z = − Z is now possible by choosing η = π (mod π ), as indicated ineq. (10.5). Note that if the Yukawa interactions are neglected then the sign of Z in a realHiggs basis is not physical since one can always redefine H → i H while maintaining thereality of the Higgs basis. Thus, eq. (10.5) can be understood to mean that Z = ±| Z | ina real Higgs basis, which correspond to two physically inequivalent conditions.Moreover, employing the results of eq. (10.3) in eqs. (5.21)–(5.23), it follows that if Z = 0then we can identify the squared mass of the CP-odd mass eigenstate as corresponding to the33 element of the squared-mass matrix M given in eq. (5.22), namely m A = m H ± . If Z = 0,then M is diagonal; nevertheless, one can determine the CP-properties of the neutralHiggs mass eigenstates via the three-scalar and four-scalar interaction terms assuming that Z = 0 [65]. One can again confirm that the CP-odd mass eigenstate corresponds to the 33element of M , in which case we also conclude that m A = m H ± .Finally, in the case of a custodially symmetric scalar potential with Y = Z = Z = 0, anexact Higgs alignment is realized and we can identify m h = Z v following the conventionof eq. (5.30), and m H,A = m H ± + ( Z ± Z ) v . Although the CP quantum numbers of H and A are of opposite sign, there are no bosonic interactions that can uniquely identifywhich of the two states H and A is CP-even and which is CP-odd, as previously noted inSection V. Ultimately, the CP quantum numbers of H and A may be fixed by the Higgs-fermion Yukawa couplings (if these interactions are CP-conserving), except in special caseswhere the ambiguity persists [cf. eq. (5.31)]. Assuming that the CP-quantum numbers of62 and A are unambiguously determined by the Yukawa couplings, then the sign of Z ispromoted to a physical parameter in the case of Y = Z = Z = 0. It then follows that [50], m H ± = m A if Z = Z and Z = Z = 0 ,m H if Z = − Z and Z = Z = 0 , (10.6)in a real Higgs basis. In particular, m h < m H if Z = − Z , Z = Z = 0 and m H ± > Z v ,m h > m H if Z = − Z , Z = Z = 0 and m H ± < Z v . (10.7)Indeed, the transformation H → i H changes the sign of Z while also changing the scalarYukawa coupling into a pseudoscalar Yukawa coupling and vice versa.In this section, we propose to classify all 2HDM custodial-symmetric scalar potentials thatexhibit exact Higgs alignment due to an unbroken or softly-broken symmetry. All such scalarpotentials will satisfy the Higgs basis conditions given in eq. (10.5). If the parameters of thecorresponding Higgs potential lie in the ERPS4 regime, then this classification amounts tosupplementing the results of Table XI with the conditions of custodial symmetry.As a first step, we review the classification of custodial symmetric 2HDM scalar potentialsfirst obtained in Refs. [55, 79]. If the scalar potential respects a custodial symmetry, thenthe Higgs Lagrangian can exhibit seven additional global symmetries in the limit of g ′ = 0beyond the symmetries listed in Tables I and II. Three of the seven symmetries correspond toGCP1, the Peccei-Quinn U(1) and Π , which when combined with the custodial symmetryyield maximal symmetry groups of SO(3), SO(4) and Z ⊗ O(3), respectively [79]. The case ofGCP1 corresponds to the minimal implementation of custodial symmetry in the most general2HDM scalar potential. Indeed, the custodial-symmetric scalar potential of the 2HDM mustbe CP-conserving as noted below eq. (10.2). A custodial symmetric, Π -symmetric scalarpotential is equivalent to a custodial symmetric, Z symmetric scalar potential in anotherscalar field basis. To validate this remark, we first combine eq. (10.1) with the constraintsof the Π symmetry shown in Table III to obtain, m = m , λ = λ , λ = Re λ , λ = λ , Im m = Im λ = Im λ = Im λ = 0 , (10.8)in the Φ-basis. We now transform to a new basis by defining Φ ′ = (Φ + Φ ) / √ ′ = (Φ − Φ ) / √
2. In this new basis, the corresponding scalar potential parameters are m ′ = m − Re m , m ′ = m + Re m , m ′ = 0 ,λ ′ = ( λ + λ + 2 λ + 4 λ ) , λ ′ = ( λ + λ + 2 λ − λ ) ,λ ′ = ( λ + λ − λ ) , λ ′ = Re λ ′ = ( λ − λ ) , Im λ ′ = λ ′ = λ ′ = 0 , (10.9)which combines the constraints of eq. (10.1) with the constraints of the Z symmetry.The remaining four symmetry cases of Ref. [79] correspond to Z ⊗ Π , U(1) ⊗ Π , GCP3and SO(3), which when combined with custodial symmetry [cf. eq. (10.2)] yields a maximalsymmetry group of Z ⊗ Z ⊗ SO(3), O(2) ⊗ O(3), Z ⊗ O(4) and SO(5), respectively. Inlight of eq. (9.6), each of these symmetry cases corresponds to the inert limit in the ERPS,and thus satisfy eq. (10.5) in a real Higgs basis. The case of O(2) ⊗ O(3) requires someclarification. In Table 1 of Ref. [79], the constraints on the scalar potential parameterscorresponding to the O(2) ⊗ O(3) symmetry are, m = m , m = 0 , λ = λ = λ , λ = λ = λ = 0 . (10.10)This is a U(1) ⊗ Π -symmetric scalar potential, but it does not satisfy eq. (10.1). However,if we transform to the GCP3 basis then eqs. (8.2)–(8.9) yield, m ′ = m ′ , m ′ = 0 , λ ′ = λ ′ = λ ′ + λ ′ + Re λ ′ , λ ′ = Re λ ′ , Im λ ′ = λ ′ = λ ′ = 0 , (10.11)which corresponds to custodial-symmetric, GCP3-symmetric scalar potential.We are now ready to present the classification of custodial-symmetric 2HDM scalar po-tentials that satisfy exact Higgs alignment due to an unbroken or softly-broken symmetry.Exact Higgs alignment requires Y = Z = 0, and then to achieve Higgs alignment via asymmetry also requires Z = 0. Two immediate examples are the IDM with either an un-broken Z or U(1) symmetry in the Higgs basis. Supplementing these two examples withthe condition that Z = ± Z yields a custodial symmetric scalar potential with exact Higgsalignment. Since a GCP3-symmetric scalar potential is a U(1) ⊗ Π -symmetric scalar potential in a different scalarfield basis, one cannot unambiguously associate O(2) ⊗ O(3) and Z ⊗ O(4) with either ERPS symmetry.A physical criterion for distinguishing these two maximal symmetry groups is provided by the two Higgsbasis conditions specified in eq. (10.19) and exhibited in Table XII and in the discussion that follows.
64n light of the classification of custodial symmetric scalar potentials discussed above, wenow consider cases in which additional unbroken or softly-broken symmetries are present.It is clear that at minimum, a softly-broken Z symmetry that is manifestly realized in theΦ-basis must be present. Consequently, let us consider a softly-broken Z -symmetric scalarpotential in the Φ-basis that is distinct from the Higgs basis which satisfies Y = Z = Z = 0.Since λ = λ = 0 holds in a Φ-basis such that s β = 0, the ERPS4 conditions must besatisfied as we now demonstrate.First, we shall employ eqs. (A.26)–(A.28) of Ref. [37] with s β = 0, λ = λ = 0 and Z = Z = 0 to obtain, s β (cid:2) Z c β − Z s β − Z c β − Re( Z e iξ ) c β − i Im( Z e iξ ) (cid:3) = 0 , (10.12) s β (cid:2) Z s β − Z c β + Z c β + Re( Z e iξ ) c β + i Im( Z e iξ ) (cid:3) = 0 , (10.13)where Z ≡ Z + Z . Adding and subtracting these two equations yields, s β ( Z − Z ) = 0 , (10.14) s β (cid:8) c β (cid:2) Z + Z − Z − Z e iξ ) (cid:3) − i Im( Z e iξ ) (cid:9) = 0 . (10.15)Moreover, we can use eqs. (A20) of Ref. [37] along with eq. (5.20) to obtain, m e iξ = ( Y − Y ) s β . (10.16)It follows that Im( m e iξ ) = 0 . (10.17)Imposing the scalar potential minimum condition given by eq. (2.11), it follows that eithersin 2 ξ = 0 or λ = 0. If λ = 0 then the only remaining complex parameter of the scalarpotential, m , can be arbitrarily rephased. Thus without loss of generality, we may assumethat sin 2 ξ = 0 holds in all cases, or equivalently e iξ = ±
1. Moreover, having assumed that s β = 0, we see that eqs. (10.14) and (10.15) yields Z = Z and Im Z = 0. That is, theHiggs basis parameters satisfy the ERPS4 conditions! In the case of s β c β = 0 and e iξ = ± Z = Z = 0), eqs. (10.14) and (10.15) yield, Z = Z = Z + Z ± Z and Im Z = 0 . (10.18) As expected, if s β = 0 then eqs. (10.14) and (10.15) are automatically satisfied, in which case no enhancedsymmetry beyond Z is present for generic parameters of the scalar potential. ′ symmetryconditions in the Higgs basis. If we now also impose the custodial symmetry condition, Z = ± Z , where the sign choice is uncorrelated with the ± sign appearing in eq. (10.18),then it follows that two relations are possible that are physically distinguishable, Z = Z = Z + 2 Z or Z = Z = Z . (10.19)Moreover, we can use eqs. (A21)–(A28) of Ref. [37] along with eq. (10.18) to obtain thescalar potential parameters in the Φ-basis, λ i = Z i for i = 1 , , . . . , . (10.20)When Y = Z = Z = 0, it is always possible to rephase the Higgs basis field, H → i H ,such that Z = Z = Z + Z + Z . Then, in a GCP3 basis (where parameters in the Φ-basisare indicated with prime superscripts), eqs. (10.19) and (10.20) respectively yield, λ ′ = λ ′ = λ ′ + λ ′ + λ ′ , where λ ′ = λ ′ , (10.21)or λ ′ = λ ′ = λ ′ , where λ ′ = − λ ′ . (10.22)In both cases, the GCP3 conditions are manifestly realized by the quartic terms of thescalar potential in the Φ-basis. In light of eq. (10.11), in the case of unbroken GCP3 andcustodial symmetry, eq. (10.21) yields a maximal symmetry group of O(2) ⊗ O(3) in theclassification of Ref. [79]. To determine the maximal symmetry group of the scalar potentialwhose parameters satisfy eq. (10.22), we transform to the U(1) ⊗ Π basis. Then eqs. (8.42)–(8.46) yield λ = λ = λ and λ = λ = λ = λ = 0, corresponding to a maximal symmetrygroup of Z ⊗ O(4) in the classification of Ref. [79]. Note that the custodial symmetry ispreserved in the presence of soft-breaking of the GCP3 symmetry by allowing for m ′ = 0subject to the condition, m ′ − m ′ = 2 m ′ c β ′ /s β ′ [cf. eqs. (7.2) and (7.3)].If Z = 0, then eq. (10.20) together with the custodial symmetry condition Z = ± Z imply that λ ′ = λ ′ = 0. In light of eqs. (10.21) and (10.22), the SO(3) conditions aremanifestly realized by the quartic terms of the scalar potential. Indeed, in this case thequartic terms are given by V ∋ λ (cid:0) Φ † Φ + Φ † Φ ) , which corresponds to the maximallysymmetric SO(5) limit of the 2HDM (after including the gauge covariant kinetic terms ofthe scalar fields with g ′ = 0) analyzed in Ref. [38]. Soft-breaking of the SO(3) symmetrydue to m ′ = 0 is again allowed subject to the condition, m ′ − m ′ = 2 m ′ c β ′ /s β ′ .66n the case of c β = sin 2 ξ = 0, eqs. (10.14) and (10.15) yield Z = Z and Im Z = 0.Hence, the quartic terms of the scalar potential in the real Higgs basis satisfy the Z ⊗ Π symmetry conditions. Using eqs. (A21)–(A25) of Ref. [37], we obtain λ = λ = Z − ( Z − Z ) , λ i = Z i + ( Z − Z ) , for i = 3 , ,λ = Z ± ( Z − Z ) , ˜ λ = Z + ( Z − Z ) , λ = λ = 0 , (10.23)where Z ≡ Z + Z ± Z and ˜ λ ≡ λ + λ ± λ . Assuming that Z = Z (otherwise,eq. (10.18) is satisfied and we return to the previous case), it follows that λ = ˜ λ . That is,the Z ⊗ Π symmetry of the quartic terms of the scalar potential of the Φ-basis is manifestlyrealized. Soft breaking of the Z ⊗ Π symmetry due to m = 0, where m is real [pureimaginary] if sin ξ = 0 [cos ξ = 0], is allowed subject to the condition m = m .If we now impose the custodial symmetry condition, Z = ± Z , then it follows that twoparameter relations are possible, λ = λ = ˜ λ , where λ = ± λ , (10.24)or λ = λ = λ , where λ = ± λ . (10.25)Eqs. (10.24) and (10.25) are related by a change of scalar field basis. For example, ifsin ξ = 0 then we replace the ± sign with a plus sign in the above expressions. Startingfrom eq. (10.25) and employing eq. (8.1) to transform Φ → Φ = U Φ, the scalar potentialparameters in the new basis satisfy m = m , m = 0 and λ = λ = λ + ( λ − λ ) , λ = λ − ( λ − λ ) , λ = λ = ( λ + λ ) . (10.26)Since λ + λ + λ − λ = 2 λ , eq. (10.24) is satisfied in the Φ-basis, assuming that λ = 0.However, if λ = 0, then λ = λ = λ + λ + λ is satisfied, corresponding to a softly-brokenGCP3-symmetric scalar potential in the Φ-basis. The case of cos ξ = 0 can be similarlytreated by rephasing Φ → i Φ , and yields a softly-broken GCP3 ′ -symmetric scalar potential.Finally, if s β = 0, then one can impose the unbroken or softly-broken symmetries of theERPS4 and the custodial symmetry condition directly in the Higgs basis. In Table XII, weprovide a complete classification of the 2HDM scalar potentials that possess an unbrokencustodial symmetry and exhibit exact Higgs alignment. For convenience, we have includedentries corresponding to both U(1) ⊗ Π and GCP3, which correspond to physically equivalentpoints in the ERPS4 in light of the results of Section VIII.67 iggs basis conditions custodial additional scalar maximal(all cases satisfy symmetry real Φ-basis Lagrangian symmetry Y = Z = Z = 0) conditions constraints symmetry group Z = ± Z = 0 s β = 0 Z Z ⊗ O(3) Z = Z = 0 s β = 0 U(1) SO(4) Z = Z = Z Z = ± Z = 0 c β sin 2 ξ = 0, λ = λ or λ = ± λ Z ⊗ Π Z ⊗ Z ⊗ SO(3) Z = Z = Z Z = ± Z = 0 s β = 0, λ = ± λ Z ⊗ Π Z ⊗ Z ⊗ SO(3) Z = Z = Z + 2 Z Z = ± Z = 0 c β = 0, λ = λ , λ = 0 U(1) ⊗ Π O(2) ⊗ O(3) Z = Z = Z Z = ± Z = 0 c β = 0, λ = λ , λ = 0 U(1) ⊗ Π Z ⊗ O(4) Z = Z = Z Z = Z = 0 s β = 0, λ = λ , λ = 0 U(1) ⊗ Π Z ⊗ O(4) Z = Z = Z + 2 Z Z = Z = 0 s β ′ sin ξ ′ = 0, λ ′ = λ ′ = 0 GCP3 O(2) ⊗ O(3) Z = Z = Z Z = − Z = 0 s β ′ sin ξ ′ = 0, λ ′ = − λ ′ = 0 GCP3 Z ⊗ O(4) Z = Z = Z Z = Z = 0 c β ′ = cos ξ ′ = 0, λ ′ = − λ ′ = 0 GCP3 Z ⊗ O(4) Z = Z = Z Z = Z = 0 λ = λ = 0 SO(3) SO(5)TABLE XII: Classification of 2HDM scalar potentials that possess an unbroken custodial symmetryand satisfy the inert conditions, Y = Z = Z = 0, thereby exhibiting exact Higgs alignment.The Higgs basis field H has been rephased such that Z is real. In the symmetry limit, thescalar Lagrangian symmetry that is manifestly realized in the Φ-basis is shown along with thecorresponding maximal symmetry group (that includes the custodial symmetry in the limit of g ′ →
0) according to the classification provided in Ref. [79]. Excluding the first two lines of thetable, all entries correspond to the ERPS4 regime. The corresponding ERPS symmetry may besoftly-broken if m = m and/or m = 0 as indicated in Table XI. Since GCP3 is equivalent toU(1) ⊗ Π when expressed in a different scalar field basis, there is a one-to-one mapping betweentheir corresponding entries above that is consistent with the results of Section VIII. It is noteworthy that two maximal symmetry groups are associated with both U(1) ⊗ Π and GCP3. These two cases are distinguished by the corresponding Higgs basis conditions.Indeed, one can check that O(2) ⊗ O(3) is physically distinguished from Z ⊗ O(4). Inparticular, in the cases of softly-broken or unbroken U(1) ⊗ Π and GCP3-symmetric scalarpotentials, O(2) ⊗ O(3) is associated with the mass relation, m H ± = m H = m A . In contrast, Z ⊗ O(4) is associated with the mass relation m H ± = m A , which includes the possibility of m H ± = m A = m H if in addition Z = Z = 0. The latter is an example of the more generalresult that any custodial symmetric 2HDM scalar potential with Z = Z = Z = Z = 0exhibits a Peccei-Quinn U(1) symmetry that is unbroken by the scalar potential and vacuumand thus possesses a scalar spectrum where H ± is degenerate in mass with both H and A .68 I. CONCLUSIONS AND FUTURE DIRECTIONS
There is a fascinating region of parameter space of the 2HDM that can be implemented byimposing the generalized CP symmetry, GCP2, on the quartic terms of the scalar potential,which enforces the relations, λ = λ and λ = − λ . We call this region the ERPS4,generalizing the exceptional region of the parameter space (ERPS) of the 2HDM introducedin Refs. [29, 40], where the GCP2 symmetry is also respected by the quadratic terms of thescalar potential and yields the additional constraints, m = m and m = 0. That is, theERPS4 is the parameter space of a softly-broken GCP2-symmetric scalar potential. In thispaper, we have provided a comprehensive study of the many interesting properties of this2HDM parameter regime, including limiting cases of the ERPS4 parameters that extend theunbroken or softly-broken symmetries of the scalar potential beyond GCP2.We have enumerated the basis invariant conditions that characterize the softly-brokenGCP2 symmetries and their extensions and evaluated the scalar squared masses and neutralscalar mixing matrices in each of these cases. We have discussed intricacies that arise whenscalar potentials originating from two different symmetry conditions yield physically equiv-alent results. In such cases, although the parameter constraints imposed by the symmetryconditions may differ, one can show that a unitary transformation relates the scalar fieldbases in which each of the symmetries is manifestly realized. Indeed, a GCP2-symmetricscalar potential is related by a unitary basis transformation to a scalar potential that isinvariant with respect to Z ⊗ Π , and a GCP3-symmetric scalar potential is related by aunitary basis transformation to a scalar potential that is invariant with respect to U(1) ⊗ Π .These considerations persist even if the corresponding symmetries are softly-broken.The proof of the equivalence of GCP2 and Z ⊗ Π is reviewed in Appendix B. However, nosimple analytic formula exists that explicitly relates the parameters of the GCP2 basis andthe Z ⊗ Π basis. Thus, in the present work, our analysis of the ERPS4 always starts fromthe Z ⊗ Π basis, from which all subsequent special cases can be analyzed. In contrast,the translation between the GCP3 basis and the U(1) ⊗ Π basis can be made explicit,and a translation between the parameters defined in each of the two basis choices has beenprovided in Section VIII. The results for the softly-broken SO(3)-symmetric scalar potential,which are different limiting cases of the GCP3 and U(1) ⊗ Π basis choices, ultimately yieldidentical results given that the form of the softly-broken SO(3)-symmetric scalar potential69s invariant with respect to an arbitrary unitary transformation of the scalar field basis.In examining the CP-invariance properties of scalar potentials in the ERPS4, we encoun-tered an interesting feature that runs contrary to a statement usually found in the literature.In a softly-broken Z ⊗ Π -symmetric scalar potential where the magnitudes of the two neu-tral scalar field vevs are equal, we originally noticed in Ref. [37] that the scalar potential andvacuum were both CP-conserving even though the relative phase between the potentiallycomplex parameters m and λ could not be removed by separate rephasings of the scalarfields Φ and Φ . In contrast, outside of the ERPS4 regime, it is straightforward to show thatif λ = λ = 0 and Im (cid:0) λ ∗ [ m ] (cid:1) = 0, then the corresponding scalar potential is explicitlyCP-violating. We were able to identify an alternative definition of CP, denoted by GCP1 ′ inTables V and VI, that provides an explanation for why the softly-broken Z ⊗ Π -symmetricscalar potential with v = v always preserves a CP symmetry.Perhaps even more astonishing was that in a softly-broken GCP3-symmetric scalar po-tential with Im (cid:0) λ ∗ [ m ] (cid:1) = 0, the scalar potential and vacuum are always CP-invariantindependently of the vevs. In this case, the identification of the relevant CP transformationlaw is more obscure (see Appendix C). Of course, this result becomes almost trivial by trans-forming to the U(1) ⊗ Π basis, where a simple rephasing can be performed to remove allpotential complex phases from the corresponding scalar potential parameters. Moreover, inboth CP-conserving examples above where Im (cid:0) λ ∗ [ m ] (cid:1) = 0, a more general unitary trans-formation of the scalar fields exists that can transform directly to a real scalar field basis inwhich the CP-invariance of the scalar potential is manifest. Because the scalar potentials ofthe ERPS4 are quite constrained, such a unitary transformation is still consistent with theparameter constraints imposed by the softly-broken Z ⊗ Π and GCP3 symmetries.A very important subset of the ERPS4 is the so-called inert limit where the Higgs basisparameters satisfy Y = Z = Z = 0. In this parameter regime Higgs alignment is exact,which means that there exists a neutral scalar whose tree-level properties coincide with thoseof the SM Higgs boson. Indeed, the LHC Higgs data have already confirmed at the 10–20%level that the properties of the observed Higgs boson (of mass 125 GeV) are consistentwith the predictions of the Standard Model. Consequently, any phenomenologically viableextended Higgs sector must exhibit at least an approximate Higgs alignment. One canachieve an approximate Higgs alignment automatically in the decoupling limit where themasses of all additional scalars are significantly larger than 125 GeV. However, it is of70nterest to consider the possibility of approximate Higgs alignment without decoupling,as this scenario would provide more options for potential discoveries of new scalars of anextended Higgs sector in future LHC runs. Higgs alignment without decoupling can beachieved without a fine-tuning of scalar sector parameters if a symmetry is present thatcan enforce the Higgs alignment. Thus, in the 2HDM it is especially useful to provide acomplete classification of all such symmetries. The simplest example of a 2HDM with thisproperty is the IDM which possesses an unbroken Z or U(1) symmetry in the Higgs basis.All other 2HDM scenarios that provide a natural explanation for exact Higgs alignment liein the ERPS4 regime. The complete classification has been provided in Table XI.A phenomenologically viable extended Higgs sector must also be consistent with precisionelectroweak constraints. The observation that the electroweak ρ -parameter is approximatelyequal to one strongly suggests that the scalar potential should be invariant under a custodialsymmetry. The ERPS4 enters in these considerations as well, since one of the two ways tosatisfy the requirements of custodial symmetry is provided by the inert limit. Thus, combin-ing the requirements of exact Higgs alignment and custodial symmetry yields a classificationof 2HDM scenarios that is exhibited in Table XII.Finally, there is one aspect of the 2HDM that has been almost completely ignored in ourcomprehensive study of the ERPS4—namely, the Higgs-fermion Yukawa interactions. Thereis a reason for this neglect. In a 2HDM with one generation of quarks and leptons, it isnot possible to construct a Yukawa Lagrangian that respects a GCP2 symmetry or any ofits symmetry extensions. If three generations of quarks and leptons are present, then it ispossible to construct a set of Yukawa interactions that respect a GCP2 or GCP3 symmetryby positing transformation laws that involve fermions of different generations. However, allsuch constructions are inconsistent with the constraints of experimental observations withthe possible exception of one very special implementation of the GCP3 symmetry in Ref. [80].It is not clear whether these remarks also hold if one were to construct a Yukawa Lagrangianthat respects a Z ⊗ Π or U(1) ⊗ Π symmetry. Scalar Lagrangians that are constrained bysymmetries that are physically the same when only the scalar sector is considered might bedifferent once fermions are included. This possibility is presently under study [81].If there is no phenomenologically successful 2HDM Yukawa Lagrangian consistent withthe ERPS4 regime, then there are two possible approaches. In one approach advocated inRef. [38], the ERPS4 conditions are imposed at the Planck scale. The Yukawa interactions71epresent a hard-breaking of the symmetries responsible for the ERPS4 regime. Hence, renor-malization group evolution down to the electroweak scale will generate an effective 2HDMthat deviates from the ERPS4 but might retain some of its best features (e.g., approximateHiggs alignment and approximate custodial symmetry). The second approach follows theproposals of Ref. [66, 82], where vectorlike quark and lepton partners are introduced in anextended Yukawa Lagrangian. In this case, one can easily construct a Yukawa Lagrangianthat is consistent with the ERPS4 regime (even in a one generation model of fermions andthe vectorlike partners). To ensure that the vectorlike fermions are sufficiently heavy toavoid current LHC search limits, one can introduce explicit mass terms for the vectorlikefermions, which then generate the soft-breaking squared mass terms of the ERPS4.In either of these two approaches, one can determine parameter regimes that are consistentwith observed Higgs boson phenomena, while setting useful targets for precision Higgs studiesat the LHC and future Higgs factories now under development. We shall defer such mattersto a future study. Given that the ERPS4 provides a simple framework for an extended Higgssector with a reduced number of free parameters, we would anticipate a number of usefulcorrelations that could emerge, such as relations among various three-scalar couplings, ifdeviations from SM Higgs properties are detected and/or new scalar states are discovered. Acknowledgments
H.E.H. is supported in part by the U.S. Department of Energy Grant No. DE-SC0010107.H.E.H. is grateful for the hospitality and support during his visit to the Instituto SuperiorT´ecnico, Universidade de Lisboa, and he also acknowledges fruitful discussions that tookplace at the University of Warsaw during visits supported by the HARMONIA project ofthe National Science Centre, Poland, under contract UMO-2015/18/M/ST2/00518 (2016–2021). The work of J.P.S. is supported in part by the Portuguese Funda¸c˜ao para a Ciˆencia eTecnologia (FCT) under Contracts No. CERN/FIS-PAR/0008/2019, No. PTDC/FIS-PAR/29436/2017, No. UIDB/00777/2020, and No. UIDP/00777/2020.
Appendix A THE POSSIBILITY OF SPONTANEOUS CP VIOLATION
Consider the case of an explicitly CP-conserving, softly-broken Z -symmetric scalar po-tential written in a real scalar field basis, where λ = λ = 0 and the two potentially complex72calar potential parameters, m and λ , are real and nonzero. In this case, spontaneous CPviolation is possible [64, 83, 84]. It is instructive to examine the minimum and stabilityconditions under the assumption that h Φ i = v / √ h Φ i = v e iξ / √
2, where v and v are real and positive. Following the analysis of Appendix B of Ref. [41], the vacuum valueof the scalar potential is V vac = m v + m v − m v v cos ξ + ( λ v + λ v )+ ( λ + λ − λ ) v v + λ v v cos ξ . (A.1)The scalar potential minimum conditions are0 = ∂V vac ∂v = m v − m v cos ξ + λ v + λ v v , (A.2)0 = ∂V vac ∂v = m v − m v cos ξ + λ v + λ v v , (A.3)0 = 1 v ∂V vac ∂ξ = v v v (cid:0) m − λ v v cos ξ (cid:1) sin ξ , (A.4)where λ is defined in eq. (2.8). The vacuum is CP conserving if sin 2 ξ = 0 [83, 85], whereas the vacuum is potentially CP violating if sin 2 ξ = 0.First consider the case of sin 2 ξ = 0. Having excluded m = 0 from consideration[cf. footnote 32], it follows that sin ξ = 0. Without loss of generality, we may take cos ξ = 1by rephasing Φ → − Φ (which also changes the sign of m but otherwise has no effect onthe other scalar potential parameters). Then, eqs. (A.2) and (A.3) yield, m = m v v − λ v − ( λ + λ + λ ) v , (A.5) m = m v v − λ v − ( λ + λ + λ ) v . (A.6)The stability conditions can be discerned from the Hessian. Computing the relevantsecond derivatives, ∂ V vac ∂v = m + λ v + ( λ + λ + λ ) v = m v v + λ v , (A.7) ∂ V vac ∂v = m + λ v + ( λ + λ + λ ) v = m v v + λ v , (A.8) ∂ V vac ∂v ∂v = − m + ( λ + λ + λ ) v v , (A.9) Spontaneous CP violation is also possible if m is purely imaginary and λ is real. In this case, one canredefine Φ → i Φ , which renders m real while transforming λ → − λ and ξ → ξ + π . If cos ξ = 0 then eq. (A.4) yields m = 0. In this case the Z symmetry of the scalar potential is explicitlypreserved and spontaneous CP violation does not occur [83] (see also Theorem 23.3 of Ref. [64]). H = m v v + λ v − m + ( λ + λ + λ ) v v − m + ( λ + λ + λ ) v v m v v + λ v . (A.10)Stability requires that Tr H >
H >
0. In addition, we demand that the squaredmasses of the neutral Higgs bosons should be positive. Using the results of Ref. [41], thefollowing quantities all must be positive, m A = (cid:18) m v v − λ (cid:19) v , (A.11) m h + m H = m A + λ v + λ v , (A.12) m h m H = m A v (cid:2) λ v + λ v + 2( λ + λ ) v v (cid:3) + (cid:2) λ λ − ( λ + λ ) (cid:3) v v . (A.13)Note that the trace and determinant of the Hessian matrix are related to the squared massesof the neutral scalars,Tr H = m h + m H + λ v , (A.14)det H = m h m H + λ (cid:20)(cid:18) λ + 2 m A v (cid:19) v v + λ v + λ v (cid:21) . (A.15)Next, consider the case of sin 2 ξ = 0. In this case, it is convenient to replace eq. (A.4)with 0 = 1 v ∂V vac ∂ cos ξ = v v v (cid:0) − m + λ v v cos ξ (cid:1) , (A.16)which yields m = λ v v cos ξ . Inserting this result into eqs. (A.2)–(A.3), it follows that m = − λ v − ( λ + λ − λ ) v , (A.17) m = − λ v − ( λ + λ − λ ) v . (A.18)The elements of the 3 × ∂ V vac ∂v = m + λ v + ( λ + λ + λ cos 2 ξ ) v = λ v + λ v cos ξ , (A.19) ∂ V vac ∂v = m + λ v + ( λ + λ + λ cos 2 ξ ) v = λ v + λ v cos ξ , (A.20) ∂ V vac ∂v ∂v = − m cos ξ + ( λ + λ + λ cos 2 ξ ) v v = ( λ + λ − λ sin ξ ) v v , (A.21)1 v ∂ V vac ∂ (cos ξ ) = λ v v v , (A.22)1 v ∂ V vac ∂v ∂ cos ξ = λ v v cos ξv , (A.23)1 v ∂ V vac ∂v ∂ cos ξ = λ v v cos ξv . (A.24)74hus, the Hessian matrix is given by, H = v λ c β + λ s β cos ξ ( λ + λ − λ sin ξ ) s β c β λ v s β c β cos ξ ( λ + λ − λ sin ξ ) s β c β λ s β + λ c β cos ξ λ s β c β cos ξλ s β c β cos ξ λ s β c β cos ξ λ s β c β , (A.25)where s β ≡ v /v and c β ≡ v /v . Stability requires that H is positive definite. By Sylvester’scriterion [86], it follows that the principal minors must all be positive. A necessary (althoughnot sufficient condition) is that all diagonal elements of H must be positive. In light ofeq. (2.2), we conclude that λ > Appendix B AN INVARIANT CHARACTERIZATION OF THE ERPS4 ANDCONSEQUENCES FOR CP SYMMETRY
In eq. (4.19), we provided an invariant characterization of the ERPS4 that is defined by λ = λ and λ = − λ , which if realized in one scalar field basis is then satisfied in all scalarfield bases. Using eq. (4.9) and employing the identify, σ Bab σ Bcd = 2 δ ad δ bc − δ ab δ cd , (B.1)it follows that the ERPS4 invariant can be rewritten in terms of the quartic coefficients ofthe scalar potential, Z = Tr (cid:0) [ Z (1) + Z (2) ] (cid:1) − (cid:0) Tr Z (1) + Tr Z (2) (cid:1) , (B.2)where, following Ref. [29], we have defined Z (1) ad ≡ δ bc Z ab,cd = Z ab,bd = λ + λ λ + λ λ ∗ + λ ∗ λ + λ , (B.3) Z (2) bd ≡ δ ac Z ab,cd = Z ab,ad = λ + λ λ + λ λ ∗ + λ ∗ λ + λ , (B.4)and the Z ab,cd are defined in terms of the quartic couplings of the scalar potential in eq. (C.4).One can simplify eq. (B.2) using the symmetry properties of the Z ab,cd to obtain a slightlymore compact form than was originally obtained in Ref. [29], Z ≡ Tr (cid:8) ( Z (1) ) (cid:9) − (cid:2) Tr( Z (1) (cid:3) = ( λ − λ ) + | λ + λ | . (B.5)75s noted below eq. (4.19), the ERPS4 corresponds to the invariant condition, Z = 0, whichimplies that λ = λ and λ = − λ . This condition is invariant with respect to changes inthe scalar field basis, Φ a → U a ¯ b Φ b for any U ∈ U(2). Thus, if λ = λ and λ = − λ in onebasis, then the same relation holds in any scalar field basis. In particular, it holds in theHiggs basis, which implies that Z = Z and Z = − Z .One can make even a stronger statement which asserts that within the ERPS4, there existsa scalar field basis where λ = λ = 0. Equivalently, a softly-broken GCP2-symmetric 2HDMscalar potential can be rewritten as a softly-broken Z ⊗ Π symmetric scalar potential inanother scalar field basis. This statement was initially derived in Ref. [29] with the assistanceof lemma that was proved in Ref. [87]. A more refined proof is provided in Ref. [37], whichwe summarize here.The first step is to go to the Higgs basis of the ERPS4 where Z = Z and Z = − Z .If Z = 0 then it immediately follows that the quartic terms of the Higgs basis potentialrespect the Z ⊗ Π symmetry. If Z = 0, then consider a U(2) transformation from the Higgsbasis to the Φ-basis with neutral vevs v / √ v e iξ / √ v and v are positive,tan β ≡ v /v , and 0 ≤ ξ < π . It is straightforward to derive a Φ-basis expression for λ = − λ in terms of Higgs basis parameters, λ e iξ = s β c β (cid:2) Z − Z − Re( Z e iξ ) (cid:3) − is β Im( Z e iξ ) + c β Re( Z e iξ ) + ic β Im( Z e iξ ) , (B.6)where Z ≡ Z + Z . If values of β and ξ can be found where λ = 0, then we will haveverified the statement asserting that the quartic terms of a GCP2-symmetric 2HDM scalarpotential always respect a Z ⊗ Π symmetry in some other scalar field basis. Moreover,since one can always interchange the scalar doublet fields, Φ ↔ Φ without modifying the Z ⊗ Π symmetry, it follows that if ( β, ξ ) yield λ = 0 then so does ( π − β, ξ + π ).If Z = 0 then we may write Z = | Z | e iθ . It is then convenient to define¯ ξ ≡ ξ + θ , (B.7)where ¯ ξ is defined modulo π . Consider first the case of Im( Z ∗ Z ) = 0, in which case thesoftly-broken GCP2-symmetric scalar potential respects a residual CP symmetry. Inserting e iξ = e iξ ′ Z ∗ / | Z | and Im( Z ∗ Z ) = 0 into eq. (B.6), we search for values of β and ξ such that, s β Re( Z ∗ Z ) sin 2 ¯ ξ = 2 c β | Z | sin ¯ ξ , (B.8) s β c β (cid:2) | Z | ( Z − Z ) − Re( Z ∗ Z ) cos 2 ¯ ξ (cid:3) = − c β | Z | cos ¯ ξ . (B.9)76e can immediately obtain one solution to eq. (B.8), sin ¯ ξ = 0 or equivalently cos ¯ ξ = ± ξ = ± β ,2 | Z | cot β ± (cid:18) Z − Z − Re( Z ∗ Z ) | Z | (cid:19) cot 2 β − | Z | = 0 , (B.10)which possesses two real roots whose product is equal to −
1. As a result, one ends up withfour choices of ( β, ξ ), where 0 ≤ β ≤ π and cos ¯ ξ = ±
1, for which eqs. (B.8) and (B.9) aresatisfied.Moreover, additional solutions can be found if sin ¯ ξ = 0, in which case one can divideeq. (B.8) by sin ¯ ξ . Solving eq. (B.8) for c β /s β and inserting this result into eq. (B.9) yieldscos ¯ ξ (cid:26) [Re( Z ∗ Z )] + Re( Z ∗ Z ) | Z | ( Z − Z ) − | Z | (cid:27) = 0 . (B.11)One immediate solution to this equation is cos ¯ ξ = 0, which we can then plug back intoeq. (B.8) to obtain cos 2 β = 0. Thus, we learn that ( β = π , ¯ ξ = π ) and ( β = π , ¯ ξ = π )are also solutions to eqs. (B.8) and (B.9).The above results are consistent with the result of Section V. Eq. (5.19) provides a relationbetween Z and the scalar potential parameters of the Z ⊗ Π basis, which is of the form Z = ( x + iy ) e − iξ . Thus, x + iy = | Z | e i ¯ ξ and we can identify,tan ¯ ξ = yx = − λ sin 2 ξ (cid:2) λ (1 − R ) + 2 λ sin ξ (cid:3) c β . (B.12)Taking into account the values of ( β, ¯ ξ ) obtained above that provide solutions to eqs. (B.8)and (B.9), we see that sin ¯ ξ = 0 corresponds to sin 2 ξ = 0 and cos ¯ ξ = 0 correspondsto c β = 0. Since this analysis is based on the assumption that Im( Z ∗ Z ) = 0, we havereproduced the result of eq. (5.48).It is now instructive to compute Im( λ ∗ [ m ] ) in the Z ⊗ Π basis. Using eq. (B11) ofRef. [37], written in a slightly different form under the assumption that c β = 0, we obtain Im (cid:0) λ ∗ [ m ] (cid:1) = − | Z | v s β sin ¯ ξ c β ( c β (cid:18) Y v (cid:19) − Y v (cid:2) s β Z + (1 − c β ) Z (cid:3) − c β R + (cid:18) Z + 2 Y v (cid:19) R + Z ( Z c β − Z s β ) − c β s β | Z | sin ¯ ξ ) , (B.13) If c β = 0, then the expression given in eq. (B11) of Ref. [37] is more useful. R ≡ Re( Z e iξ ) = Re( Z ∗ Z ) cos 2 ¯ ξ + Im( Z ∗ Z ) sin 2 ¯ ξ | Z | . (B.14)Under the assumption that Im( Z ∗ Z ) = 0, the results obtained above imply that eithersin ¯ ξ = 0 or cos ¯ ξ = 0. If sin ¯ ξ = 0 then it immediately follows that Im( λ ∗ [ m ] ) = 0. Thatis, one can rephase the scalar doublet fields in the GCP2 basis to obtain a scalar potentialwhose coefficients are all real in a scalar field basis where the vevs are also real. In contrast,if cos ¯ ξ = 0, which implies that c β = 0 as noted below eq. (B.11), then eq. (B.13) yields,Im (cid:0) λ ∗ [ m ] (cid:1) = ± v | Z | ( | Z | − (cid:18) Z + 2 Y v (cid:19) + (cid:18) Z + 2 Y v (cid:19) (cid:18) Z − Z − Re( Z ∗ Z ) | Z | (cid:19)) , (B.15)which is generically nonzero. Thus, it follows that if β = π and Im m = 0 in the GCP2basis, then one cannot remove all complex phases from the scalar potential with a simplerephasing of the Higgs doublet fields. Nevertheless, the scalar potential and vacuum are CPconserving since Im( Z ∗ Z ) = 0 implies that a real Higgs basis exists (i.e. all Higgs basis scalarpotential parameters are real after an appropriate rephasing of the Higgs basis field H ).In the analysis presented above, we used eq. (B.11) to conclude that cos ¯ ξ = 0. However,there is an alternative solution to eq. (B.11) where the coefficient of cos ¯ ξ vanishes. Indeed,this alternative solution corresponds to the case of the softly-broken GCP3-symmetric 2HDMwhere eq. (6.13) is satisfied. In this case, division of eq. (B.8) by sin ¯ ξ is permitted whensin ¯ ξ = 0, which yields cos ¯ ξ = | Z | cot 2 β Re( Z ∗ Z ) . (B.16)Plugging this result into eq. (B.13) yields Im (cid:0) λ ′ ∗ [ m ′ ] (cid:1) = 0 for generic values of the scalarpotential parameters and β . In particular, in the GCP3 basis (where the scalar potentialparameters are designated with prime superscripts), there exists a residual CP invariancedespite the fact that Im (cid:0) λ ′ ∗ [ m ′ ] (cid:1) = 0 when sin ¯ ξ ′ = 0, independently of the value of β ′ .In contrast, λ = 0 and Z = | Z | e iθ = ±| Z | e − iξ [cf. eq. (6.11)] in the U(1) ⊗ Π basis, inwhich case eq. (B.7) yields sin ¯ ξ = 0.Finally, we consider the case of Im( Z ∗ Z ) = 0, corresponding to a softly-broken GCP2-symmetric 2HDM with no residual CP symmetry. Again, our goal is to demonstrate thata solution ( β, ξ ) to eq. (B.6) exists when λ = 0. The imaginary part of eq. (B.6) yieldsan expression for tan 2 β , which when inserted into the real part of eq. (B.6) produces an78quation for ¯ ξ of the form F ( ¯ ξ ) = 0. One can check that F ( ¯ ξ + π ) = −F ( ¯ ξ ), which impliesthat ¯ ξ is determined modulo π and a solution, F ( ¯ ξ ) = 0, must exist for 0 < ¯ ξ < π . Furtherdetails can be found in Appendix C of Ref. [87] and in Appendix B of Ref. [37].Thus, we have verified that one can always find a scalar field basis for the softly-brokenGCP2-symmetric 2HDM in which λ = λ = 0 and λ real (the latter after an appropriaterephasing of Φ ). This defines the Z ⊗ Π basis that is treated in Section V. Appendix C CP INVARIANCE OF THE SOFTLY-BROKEN GCP3-SYMMETRICSCALAR POTENTIAL
The softly-broken GCP3 scalar potential contains a complex parameter, m ′ , in a basisin which the only other potentially complex parameter, λ ′ , is taken to be real. Moreover,there is a relative phase between the two vevs. Thus, naively one would conjecture thatthe scalar sector of the softly-broken GCP3-symmetric 2HDM is CP violating. However,we have demonstrated that by changing the scalar field basis, this scalar potential can betransformed into a softly-broken U(1) ⊗ Π scalar potential in which λ = 0. Then, one canrephase either Φ or Φ to remove the phase of m , which yields an explicitly CP-conservingscalar potential. Moreover, one can show that the scalar potential minimum condition inthe explicitly CP-conserving basis yields two vevs with no relative complex phase. Hence,it follows that the softly-broken GCP3-symmetric 2HDM is explicitly CP-conserving, andthe vacuum also preserves the CP symmetry. These observations imply that in the originalGCP3 basis, one should be able to identify a residual generalized CP transformation underwhich the GCP3 scalar potential and vacuum are invariant. The purpose of this Appendixis to provide the explicit construction of this generalized CP transformation.We begin by rewriting eq. (2.1) following the notation of Ref. [29], V (Φ) = Y a ¯ b (Φ † ¯ a Φ b ) + Z a ¯ bc ¯ d (Φ † ¯ a Φ b )(Φ † ¯ c Φ d ) , (C.1)where the indices a , ¯ b , c and ¯ d can take one of two values 1, 2 (with an implicit sum overbarred and unbarred index pairs of the same letter), and hermiticity and symmetry underthe interchange of barred and unbarred indices imply that Y a ¯ b = ( Y b ¯ a ) ∗ , Z a ¯ bc ¯ d ≡ Z c ¯ da ¯ b = ( Z b ¯ ad ¯ c ) ∗ . (C.2)79ote that as a matrix, Y = Y Y Y ∗ Y = m − m − ( m ) ∗ m , (C.3)where the minus sign in the definition of m is conventional. It is convenient to assemblethe elements of the tensor Z abcd into a 4 × Z ) as follows.First, we introduce a slightly different notation for the components of Z , Z ac,bd ≡ Z a ¯ bc ¯ d , (C.4)where the first pair of indices of Z ac,bd consists of unbarred indices and the second pairconsists of barred indices. In this notation, it is conventional to omit the bars in the secondpair of indices. With this notation, each element of a row of the matrix Z is denoted by apair of subscripts. These index pairs arranged in the order 11, 12, 21 and 22, and similarlyfor the pair of subscripts denoting each element of a column of Z . The matrix Z is thengiven by, Z = Z , Z , Z , Z , Z , Z , Z , Z , Z , Z , Z , Z , Z , Z , Z , Z , = λ λ λ λ λ ∗ λ λ λ λ ∗ λ λ λ λ ∗ λ ∗ λ ∗ λ . (C.5)Under a basis transformation,Φ a → Φ ′ a = U a ¯ b Φ b , Φ † ¯ a → Φ ′† ¯ a = Φ † ¯ b U † b ¯ a , (C.6)where U ∈ U(2) is a 2 × U † b ¯ a U a ¯ c = δ b ¯ c ). Under this unitary basis transfor-mation, the vevs are transformed as h Φ a i → h Φ ′ a i = U a ¯ b h Φ b i . Moreover, the gauge covariantkinetic terms of the scalar fields are invariant under a unitary basis transformation, whereasthe coefficients Y a ¯ b and Z ab,cd transform covariantly with respect to U(2) transformations as Y a ¯ b → Y ′ a ¯ b = U a ¯ b Y c ¯ d U † d ¯ b = ( U Y U † ) ab , (C.7) Z ab,cd → Z ′ ab,cd = U a ¯ e U b ¯ g Z eg,fh U † f ¯ c U † h ¯ d = (cid:2) ( U ⊗ U ) Z ( U † ⊗ U † ) (cid:3) ab,cd , (C.8) The reader is cautioned that in contrast to eq. (C.4), the symbol Z ab,cd employed in Ref. [40] is equivalentto Z a ¯ bc ¯ d without an interchange of the indices b and c . × × A ⊗ B = A B A BA B A B . (C.9)The Kronecker product of two matrices satisfies the following properties [88]:( A ⊗ B )( C ⊗ D ) = AC ⊗ BD , (C.10)( A ⊗ B ) † = A † ⊗ B † , (C.11)( A ⊗ B ) ⊤ = A ⊤ ⊗ B ⊤ , (C.12)( A ⊗ B ) − = A − ⊗ B − , if A − and B − exist . (C.13)In particular, if A and B are unitary then so is A ⊗ B . The Kronecker product A ⊗ B canbe represented by a rank four tensor whose components are given by,( A ⊗ B ) ab,cd ≡ A ac B bd , (C.14)where a row of the matrix A ⊗ B is denoted by a pair of subscripts that are arranged inthe order 11, 12, 21 and 22 (and similarly for the pair of subscripts denoting the columns of A ⊗ B ). This convention yields the 4 × A ⊗ B given in eq. (C.9).Hence, it follows that, U a ¯ e U b ¯ g Z eg,fh U † f ¯ c U † h ¯ d = ( U ⊗ U ) ab,eg Z eg,fh ( U † ⊗ U † ) fh,cd = (cid:2) ( U ⊗ U ) Z ( U † ⊗ U † ) (cid:3) ab,cd , (C.15)as indicated in eq. (C.8). That is, eqs. (C.7) and (C.8) are equivalent to the matrix equations, Y ′ = U Y U † , Z ′ = ( U ⊗ U ) Z ( U † ⊗ U † ) . (C.16)It is common to consider the standard CP transformation of the scalar fields asΦ a ( t ; ~x ) → Φ CP a ( t ; ~x ) = Φ ∗ a ( t ; − ~x ) , (C.17)where we shall no longer distinguish between barred and unbarred indices, and the referenceto the time ( t ) and space ( ~x ) coordinates will henceforth be suppressed. However, in thepresence of several scalars with the same quantum numbers, U(2) basis transformationscan be included in the definition of the CP transformation. This yields the generalized CPtransformation (GCP) [64, 89], For early work on generalized CP transformations, see Refs. [90–92]. Generalized CP transformations inthe context of the 2HDM have been treated in Refs. [30, 40, 54, 55, 63, 79, 80, 93–97]. GCP a = e iγ X ab Φ ∗ b ≡ e iγ X ab (Φ † b ) ⊤ , (C.18)Φ † GCP a = e − iγ X ∗ ab Φ ⊤ b ≡ e − iγ X ∗ ab (Φ † b ) ∗ , (C.19)where X is an arbitrary unitary matrix of unit determinant and γ ∈ R . We will indicatebelow eq. (C.25) how the complex phase factor e iγ is determined.Note that the transformation Φ a → Φ GCP a , where Φ GCP a is given by eq. (C.18), leavesinvariant the gauge covariant kinetic terms of the scalar fields. The GCP transformation ofa scalar field bilinear yields Φ † GCP a Φ GCP b = X ∗ ac X bd (Φ c Φ † d ) ⊤ , (C.20)which does not depend on the complex phase factor e iγ . Under this GCP transformation,the quadratic terms of the potential may be written as Y ab Φ † GCP a Φ GCP b = Y ab X ∗ ac X bd Φ † d Φ c = X bd Y ∗ ba X ∗ ac Φ † d Φ c = X ca Y ∗ cd X ∗ db Φ † a Φ b = ( X † Y X ) ∗ ab Φ † a Φ b , (C.21)after making use of the hermiticity of Y [cf. eq. (C.2)] and appropriately relabeling theindices. A similar argument can be made for the quartic terms, by employing the propertiesof the Kronecker product given in eqs. (C.10)–(C.13). We conclude that the scalar potentialis invariant under the GCP transformation exhibited in eq. (C.18) if and only if the scalarpotential coefficients obey Y ∗ ab = X ∗ ca Y cd X db = ( X † Y X ) ab , (C.22) Z ∗ ab,cd = X ∗ ea X ∗ gb Z eg,fh X fc X hd , (C.23)or equivalently, Y ∗ = X † Y X , Z ∗ = ( X † ⊗ X † ) Z ( X ⊗ X ) . (C.24)Finally, we must check to see whether the GCP symmetry is preserved by the vacuum,in which case the following condition must be satisfied, h Φ a i = e iγ X ab h Φ † b i . (C.25)The complex phase factor e iγ will be chosen subject to the convention where h Φ i is real andnonnegative. The latter can always be arranged by performing an appropriate hyperchargeU(1) Y transformation on the scalar doublet fields, which has no effect on the coefficients ofthe scalar potential. 82o far, we have assumed that all statements apply in the Φ-basis. If we now perform abasis transformation to the Φ ′ -basis as indicated by eq. (C.6), then we can express the scalarpotential in terms of the Φ ′ -basis scalar potential parameters, V (Φ ′ ) = Y ′ ab (Φ ′† a Φ ′ b ) + Z ′ ac,bd (Φ ′† a Φ ′ b )(Φ ′† c Φ ′ d ) , (C.26)where Y ′ ab and Z ′ ac,bd are given by eqs. (C.7) and (C.8), respectively.Suppose that V (Φ) is invariant under the GCP transformation of Eq. (C.18) with thematrix X . Eq. (C.22) guarantees that Y ∗ = X † Y X . Now, eq. (C.7) relates the coefficientsin the two bases through Y = U † Y ′ U . It then follows that U ⊤ Y ′∗ U ∗ = X † ( U † Y ′ U ) X, (C.27)which implies that Y ′∗ = ( U ∗ X † U † ) Y ′ ( U XU ⊤ ) = X ′† Y ′ X ′ . (C.28)where X ′ = U XU ⊤ . A similar argument can be made for the quartic terms, by employingthe properties of the Kronecker product given in eqs. (C.10)–(C.13). Thus, we conclude that V (Φ ′ ) is invariant under a new GCP transformation with matrix e iγ ′ X ′ = e iγ ′ U XU ⊤ . (C.29)The phase γ ′ is not fixed by this computation and must instead be determined by examiningeq. (C.25) in the Φ ′ -basis in a convention where h Φ ′ i is real and nonnegative.To construct the residual GCP transformation that is a symmetry of the softly-brokenGCP3-symmetric scalar potential, we begin our analysis in the U(1) ⊗ Π basis. The pa-rameters of the quadratic part of the scalar potential are specified in eq. (C.3), where Y ≡ | Y | e iθ is potentially complex. In addition, the parameters of the quartic partof the scalar potential satisfy [cf. eq. (C.5)], λ = Z , = Z , , λ = Z , = Z , , λ = Z , = Z , , (C.30)and all the other Z ab,cd vanish. The softly-broken U(1) ⊗ Π -symmetric scalar potential isinvariant with respect to a GCP transformation with matrix, Here, we are assuming that Y = 0. In the case of Y = 0, one can choose γ = ξ and replace θ with − ξ in the definition of X given in eq. (C.31) to ensure that eqs. (C.32), (C.33) and (C.34) are all satisfied,thereby establishing invariance of the scalar potential and the vacuum under the residual generalized CPtransformation. iγ X = e − iθ . (C.31)The phase γ = − θ has been chosen in anticipation of eq. (C.34) below.To establish the presence of the residual GCP symmetry, we first verify that eq. (C.22)is satisfied, Y | Y | e − iθ | Y | e iθ Y = e iθ Y | Y | e iθ | Y | e − iθ Y e − iθ . (C.32)Next, we verify that eq. (C.23) is satisfied, λ λ λ λ λ
00 0 0 λ = e iθ e iθ
00 0 0 e iθ λ λ λ λ λ
00 0 0 λ e − iθ e − iθ
00 0 0 e − iθ . (C.33)Finally, the GCP symmetry is preserved by the vacuum if eq. (C.25) is satisfied. The vevsare given by v a = ( v , v e iξ ), where v and v are positive and ξ is determined by the scalarpotential minimum condition [cf. eq. (6.4)], Im( Y e iξ ) = 0, which yields sin( θ + ξ ) = 0,under the assumption that Y = 0 [cf. footnote 36]. It then follows that θ + ξ = 0 mod π ,which demonstrates that eq. (C.25) is indeed satisfied. That is, v v e iξ = e − iθ v v e − iξ . (C.34)In the GCP3 basis, λ ′ = λ ′ − λ ′ − λ ′ is real and nonzero. Now, it is not immediatelyobvious that the softly-broken GCP3-symmetric 2HDM preserves a CP symmetry, sinceIm( λ ∗ [ m ] ) = 0, which implies that one cannot rephase the scalar doublet fields to removethe phase of m . Nevertheless, we know that CP is a symmetry of the softly-broken GCP3-symmetric scalar potential and vacuum since it corresponds to a softly-broken symmetricU(1) ⊗ Π scalar potential expressed in a different basis. Thus, it should be possible toexplicitly construct the residual generalized CP transformation that preserves the softly-broken GCP3-symmetric scalar potential and vacuum by employing eq. (C.29), with U givenby eq. (8.1). Indeed, we will construct X ′ below and explicitly verify that eqs. (C.24)and (C.25) are satisfied when expressed in terms of the GCP3 basis parameters.84qs. (8.1) and (C.29) yield, e iγ ′ X ′ = − ie iγ ′ − s c c s , (C.35)where s ≡ sin θ and c ≡ cos θ . Using eq. (8.49), which we can rewrite as Y = Re Y ′ + i ( Y ′ − Y ′ ) , (C.36)it follows that c = Re Y ′ q(cid:0) Re Y ′ (cid:1) + ( Y ′ − Y ′ ) , s = Y ′ − Y ′ q(cid:0) Re Y ′ (cid:1) + ( Y ′ − Y ′ ) . (C.37)One can check that eq. (C.22) is satisfied in the GCP3 basis by rewriting this equation as, Y ′ Y ′∗ Y ′ Y ′ − − s c c s Y ′ Y ′ Y ′∗ Y ′ − s c c s = 0 . (C.38)To verify eq. (C.38), we multiply out the left hand side above to obtain, (cid:2) s Re Y ′ − c ( Y ′ − Y ′ ) (cid:3) c s s − c = 0 , (C.39)which is equal to the zero matrix after making use of eq. (C.37).Next, we check the validity of eq. (C.23) in the GCP3 basis. The explicit form of X ′ ⊗ X ′ is given by X ′ ⊗ X ′ = s − s c − s c c − s c − s c s c − s c c − s s c c s c s c s . (C.40)In the GCP3 basis, Z ′ = λ ′ λ ′ − λ ′ − λ ′ λ ′ λ ′ λ ′ λ ′ λ ′ − λ ′ − λ ′ λ ′ . (C.41) In obtaining eq. (C.35), we have absorbed the phase φ [cf. eq. (8.1)] into the definition of γ ′ .
85t is now straightforward to check that eq. (C.23) is satisfied in the GCP3 basis, indepen-dently of the value of θ .Our final check involves confirming the validity of eq. (C.25) in the GCP3 basis. Thiscomputation will then determine the phase γ ′ . Before performing the computation, we shallrecord an important result that is a consequence of the scalar potential minimum conditionthat is used to fix ξ ′ . In light of eqs. (7.5) and (7.6), it follows that cos ξ ′ = − c c β ′ s s β ′ , (C.42)after employing eq. (C.37).Thus, we must verify c β ′ e iξ ′ s β ′ = − ie iγ ′ − s c c s c β ′ e − iξ ′ s β ′ = − ie iγ ′ (cid:0) − s c β ′ + c s β ′ e − iξ ′ (cid:1) − ie iγ ′ (cid:0) c c β ′ + s s β ′ e − iξ ′ (cid:1) . (C.43)We can immediately determine γ ′ from the equation for c β ′ in eq. (C.43), − ie iγ ′ = − s + c e iξ ′ tan β ′ . (C.44)One can verify that − s + c e iξ ′ tan β ′ is a complex number of unit modulus after makinguse of eq. (C.42), which provides one independent check of the validity of eq. (C.43). Theexplicit form of the residual GCP symmetry in the GCP3 basis has now been fixed.Finally, we must verify the second complex equation for e iξ ′ s β ′ given in eq. (C.43), s β ′ = − ie i ( γ ′ − ξ ′ ) (cid:0) c c β ′ + s s β ′ e − iξ ′ (cid:1) . (C.45)Straightforward algebra shows that eq. (C.45) is an identify after making use of eqs. (C.42)and (C.44) to eliminate cos ξ ′ and e iγ ′ .Thus, we have verified that the scalar potential and vacuum of the softly-broken GCP3-symmetric 2HDM are invariant with respect to a residual GCP transformation with matrix e iγ ′ X ′ = (cid:0) c e iξ ′ tan β ′ − s (cid:1) − s c c s , (C.46)where s and c are given by eq. (C.37) and β ′ and ξ ′ are determined by the GCP3scalar potential parameters as indicated in eqs. (7.5)–(7.7). Note that although the form of Eq. (C.42) can also be deduced from eq. (8.56) after making use of tan ξ = − tan θ , which is a consequenceof the scalar potential minimum condition, sin( θ + ξ ) = 0, obtained above eq. (C.34). iγ ′ X ′ depends on the parameters of the softly-broken GCP3 scalar potential, our calculationdemonstrates that for any choice of the parameters (and in particular for any choice of theparameters that softly breaks the GCP3 symmetry), there exists a residual GCP symmetrycharacterized by the matrix e iγ ′ X ′ .These results are not surprising given that we knew from the beginning that the softly-broken GCP3-symmetric scalar potential is equivalent to a softly broken U(1) ⊗ Π -symmetricscalar potential where the residual CP symmetry transformation law is identified as GCP1,after removing all complex phases from the coefficients of the scalar potential parametersby an appropriate rephasing of the scalar doublet fields. Nevertheless, it is satisfying toexplicitly identify the residual GCP symmetry of the softly-broken GCP3-symmetric scalarpotential independently of the relations between U(1) ⊗ Π basis and the GCP3 basis obtainedin Section VIII. Appendix D SCALAR SQUARED MASS MATRICES IN THE Φ-BASIS
In Sections VI and VII, the neutral scalar squared mass matrices were evaluated in theHiggs basis. Of course, the same scalar squared masses can be obtained by computing theeigenvalues of the neutral scalar squared mass matrices evaluated in the Φ-basis (under theassumption that s β = 0). This computation provides a check of the results obtained inSections VI and VII.For example, for a softly-broken U(1) ⊗ Π -symmetric scalar potential, the calculation ofthe eigenvalues of the neutral scalar squared-mass matrix is most easily done after rephasing m as described below eq. (6.5). In this case, one obtains m A = 2 m /s β and the squaredmasses of h and H (with m H ≤ m H ) correspond to the eigenvalues of the 2 × M H = m A s β + λv c β − s β c β (cid:2) m A − ( λ + λ ) v (cid:3) − s β c β (cid:2) m A − ( λ + λ ) v (cid:3) m A c β + λv s β , (D.1)with respect to the (cid:8) √ − vc β , √ − vs β (cid:9) basis. One can verify that eqs. (6.21)and (6.22) are satisfied, as these equations are independent of the choice of scalar field basis.For a softly-broken GCP3-symmetric scalar potential (where parameters and fields aredenoted with prime superscripts), the computation of the neutral scalar squared-mass matrixin the Φ ′ -basis (where s β ′ = 0) is more challenging. After employing the GCP3 condition,87 ′ + λ ′ = λ ′ − λ ′ , the 4 × M N = s β ′ c β ′ R ′ + λ ′ v c β ′ − cos ξ ′ (cid:2) R ′ − ( λ ′ − λ ′ sin ξ ′ ) v s β ′ c β ′ (cid:3) − cos ξ ′ (cid:2) R ′ − ( λ ′ − λ ′ sin ξ ′ ) v s β ′ c β ′ (cid:3) c β ′ s β ′ R ′ + ( λ ′ s β cos ξ ′ + λ ′ c β sin ξ ′ ) v λ ′ v s β ′ sin 2 ξ ′ sin ξ ′ (cid:2) R ′ + λ ′ v s β ′ c β ′ sin ξ ′ (cid:3) − sin ξ ′ (cid:2) R ′ − ( L ′ − λ ′ cos ξ ′ ) v s β ′ c β ′ (cid:3) λ ′ v s β ′ sin 2 ξ ′ λ ′ v s β ′ sin 2 ξ ′ − sin ξ ′ (cid:2) R ′ − ( L ′ − λ ′ cos ξ ′ ) v s β ′ c β ′ (cid:3) sin ξ ′ (cid:2) R ′ + λ ′ v s β ′ c β ′ sin ξ ′ (cid:3) λ ′ v s β ′ sin 2 ξ ′ s β ′ c β ′ R ′ − λ ′ v s β ′ cos 2 ξ ′ − cos ξ ′ (cid:2) R ′ − λ ′ v s β ′ c β ′ cos ξ ′ (cid:3) − cos ξ ′ (cid:2) R ′ − λ ′ v s β ′ c β ′ cos ξ ′ (cid:3) c β ′ s β ′ R ′ + ( λ ′ s β ′ sin ξ ′ − λ ′ c β ′ cos ξ ′ ) v , (D.2)with respect to the (cid:8) √ ′ − vc β ′ , √ ′ − vs β ′ cos ξ ′ , √ ′ , √ ′ − vs β ′ sin ξ ′ (cid:9) basis, where R ′ ≡ Re( m ′ e iξ ′ ) , L ′ ≡ λ ′ − λ ′ sin ξ ′ . (D.3)The next step is to identify the neutral Goldstone boson, which resides in the Higgs basisfield H , and corresponds to the eigenvector of M N with zero eigenvalue,1 √ G = Im H = − s β ′ sin ξ ′ Re Φ ′ + c β ′ Im Φ ′ + s β ′ cos ξ ′ Im Φ ′ . (D.4)Defining the real orthogonal matrix, R = ξ ′ ξ ′ c β ′ sin ξ ′ s β ′ − c β ′ cos ξ ′ − s β ′ sin ξ ′ c β ′ s β ′ cos ξ ′ , (D.5)one then finds that RM N R T is a matrix with respect to the rotated basis whose fourth rowand column consists entirely of zeros (due to the Goldstone boson). Removing the fourthrow and fourth column yields a 3 × M N = s β ′ c β ′ R ′ + λ ′ v c β ′ − R ′ + L ′ v s β ′ c β ′ λ ′ v s β ′ sin 2 ξ ′ − R ′ + L ′ v s β ′ c β ′ c β ′ s β ′ R ′ + λ ′ v s β λ ′ v c β ′ sin 2 ξ ′ λ ′ v s β ′ sin 2 ξ ′ λ ′ v c β ′ sin 2 ξ ′ R ′ s β ′ c β ′ − λ ′ v cos 2 ξ ′ , (D.6)with respect to the (cid:8) √ ′ − vc β ′ , √ e − iξ ′ Φ ′ ) − vs β ′ , √ (cid:2) s β ′ Im Φ ′ − c β ′ Im( e − iξ ′ Φ ′ ) (cid:3)(cid:9) basis. 88he normalized eigenstate corresponding to A is given by, A = 1 q − s β ′ sin ξ ′ s β ′ cos ξ ′ − c β ′ cos ξ ′ − c β ′ sin ξ ′ , (D.7)with corresponding eigenvalue, m A = Re( m ′ e iξ ′ ) s β ′ c β ′ + λ ′ v sin ξ ′ , (D.8)in agreement with eq. (7.20). In particular, A = (1 − s β ′ sin ξ ′ ) − / √ (cid:8) s β ′ (cid:0) cos ξ ′ Re Φ ′ − c β ′ sin ξ ′ Im Φ ′ (cid:1) + c β (cid:2) c β ′ sin ξ ′ Im( e − iξ ′ Φ ′ ) − cos ξ ′ Re( e − iξ ′ Φ ′ ) (cid:3)(cid:9) = (1 − s β ′ sin ξ ′ ) − / √ (cid:2) i (cos ξ ′ + ic β ′ sin ξ ′ )( s β ′ Φ ′ − c β ′ e − iξ ′ Φ ′ ) (cid:3) = √ (cid:2) ie iψ (cid:0) s β ′ Φ ′ − c β ′ e − iξ ′ Φ ′ ) (cid:3) = √ (cid:2) e iη ′ (cid:0) − s β ′ e iξ ′ Φ ′ + c β ′ Φ ′ (cid:1)(cid:3) = √ H , (D.9)after making use of eqs. (5.7), (7.16) and (8.60).One can then identify two other mutually orthogonal normalized vectors orthogonal to A , √ H − v = c β ′ s β ′ and √ H = 1 q − s β ′ sin ξ ′ − s β ′ c β ′ sin ξ ′ c β ′ c β ′ sin ξ ′ − cos ξ ′ , (D.10)where the identification of the vectors above with the Higgs basis fields follows the sameprocedure that yielded eq. (D.9). Hence, if we construct a 3 × O whose rows are given by the transposes of the column vectors exhibited in eqs. (D.10)and (D.7), respectively, then it is straightforward to verify that OM N O T is a block diagonalmatrix with respect to the (cid:8) √ H − v, √ H , √ H (cid:9) basis. The upper 2 × OM N O T can be identified with the 2 × M H = ( λ ′ − λ ′ s β ′ sin ξ ′ ) v − λ ′ v s β ′ sin ξ ′ (1 − s β ′ sin ξ ′ ) / − λ ′ v s β ′ sin ξ ′ (1 − s β ′ sin ξ ′ ) / m A − λ ′ v (1 − s β ′ sin ξ ′ ) , (D.11)which reproduces the result of eq. (7.23), and the squared mass of the CP-odd scalar, m A [which is given by eq. (D.8)], is the 33 element of OM N O T .89learly this is not the preferred method for computing the squared masses of the neutralscalars that derive from a softly-broken GCP3-symmetric scalar potential, in light of themuch simpler Higgs basis computation given in Section VII. Appendix E THE IDM IN THE Φ-BASIS
The inert doublet model (IDM) can be defined as a 2HDM in which the 2HDM Lagrangianand vacuum are invariant under a Z symmetry, H → H , H → −H , in the Higgs basis.In particular, H is odd under the Z symmetry, whereas all other fields of the 2HDM(i.e., H , the gauge bosons, and the fermions) are even under the Z symmetry. Of course,one is free to transform the scalar field basis from the Higgs basis to an arbitrary Φ-basisby employing the unitary matrix U given in eq. (9.10).Suppose one is given a 2HDM scalar potential in a Φ-basis, where the vevs of the scalarfields yield tan β = |h Φ i / h Φ i| and ξ = arg (cid:2) h Φ i ∗ h Φ i (cid:3) . What are the conditions on thescalar potential parameters that imply that the model under consideration is the IDM? Toanswer this question, we start in the Higgs basis with Y = Z = Z = 0, as mandatedby the Z symmetry. Employing eqs. (A18)–(A28) of Ref. [37], it then follows that in theΦ-basis, the scalar potential parameters must satisfy the following conditions,Im( m e iξ ) = 0 , (E.1)( m − m ) s β = 2 Re( m e iξ ) c β , (E.2) c β Re (cid:2) ( λ − λ ) e iξ (cid:3) = s β c β (cid:2) λ + λ − (cid:0) λ + λ + Re( λ e iξ ) (cid:1)(cid:3) , (E.3) c β Im (cid:2) ( λ − λ ) e iξ (cid:3) = − s β Im( λ e iξ ) , (E.4) c β Re (cid:2) ( λ + λ ) e iξ (cid:3) = s β ( λ − λ ) , (E.5)Im (cid:2) ( λ + λ ) e iξ (cid:3) = 0 . (E.6)When written in the Φ-basis, the form of the Z symmetry that is respected by thescalar potential and vacuum becomes somewhat obscure. Nevertheless, it is straightforwardto check that if eqs. (E.1)–(E.6) are satisfied, then the scalar potential and vacuum areinvariant with respect to the following discrete symmetry of order two in the Φ-basis, Z H : Φ → c β Φ + e − iξ s β Φ , Φ → e iξ s β Φ − c β Φ , (E.7)90hich can be obtained from eq. (9.12) by setting θ = π and applying a hyperchargeU(1) Y transformation to remove an overall factor of − i . Note that the square of the Z H transformation is equal to the identity, as advertised. [1] G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B , 1(2012) [arXiv:1207.7214 [hep-ex]].[2] S. Chatrchyan et al. [CMS Collaboration], “Observation of a New Boson at a Mass of 125GeV with the CMS Experiment at the LHC,” Phys. Lett. B , 30 (2012) [arXiv:1207.7235[hep-ex]].[3] G. Aad et al. [ATLAS Collaboration], “Combined measurements of Higgs boson productionand decay using up to 80 fb − of proton-proton collision data at √ s = 13 TeV collected withthe ATLAS experiment,” Phys. Rev. D , 012002 (2020) [arXiv:1909.02845 [hep-ex]].[4] A.M. Sirunyan et al. [CMS Collaboration], “Combined measurements of Higgs boson couplingsin proton–proton collisions at √ s = 13 TeV,” Eur. Phys. J. C , 421 (2019) [arXiv:1809.10733[hep-ex]].[5] CMS Collaboration, “Combined Higgs boson production and decay measurements with up to137 fb − of proton-proton collision data at √ s = 13 TeV,” CMS-PAS-HIG-19-005 (January,2020) .[6] ATLAS Collaboration, “A combination of measurements of Higgs boson production and decayusing up to 139 fb − of proton–proton collision data at √ s = 13 TeV collected with the ATLASexperiment,” ATLAS-CONF-2020-027 (July, 2020).[7] D.A. Ross and M.J.G. Veltman, “Neutral Currents in Neutrino Experiments,” Nucl. Phys. B , 135(1975).[8] M.J.G. Veltman, “Limit on Mass Differences in the Weinberg Model,” Nucl. Phys. B , 89(1977).[9] M.S. Chanowitz, M.A. Furman and I. Hinchliffe, “Weak Interactions of Ultraheavy Fermions,”Phys. Lett. B , 285 (1978).[10] D. Toussaint, “Renormalization Effects From Superheavy Higgs Particles,” Phys. Rev. D ,1626 (1978).
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