Basis-independent treatment of the complex 2HDM

The complex 2HDM (C2HDM) is the most general CP-violating two Higgs-doublet-model that possesses a softly broken Z2 symmetry. However, the physical consequences of the model cannot depend on the basis of scalar fields used to define it. Thus, to get a better sense of the significance of the C2HDM parameters, we have analyzed this model by employing a basis-independent formalism. This formalism involves transforming to the Higgs basis (which is defined up to an arbitrary complex phase) and identifying quantities that are invariant with respect to this phase degree of freedom. Using this method, we have obtained the constraints that enforce the softly broken Z2 symmetry. One can then relate the C2HDM parameters to basis-independent quantities up to a twofold ambiguity. We then show how this remaining ambiguity is resolved. We also examine the possibility of spontaneous CP violation when the scalar potential of the C2HDM is explicitly CP conserving. Basis-independent constraints are presented that govern the presence of spontaneous CP violation.


I. INTRODUCTION
The two-Higgs-doublet model (2HDM) is one of the most well-studied extensions of the Standard Model (SM). Various motivations for adding a second hypercharge-one complex Higgs doublet to the Standard Model have been advocated in the literature [1][2][3][4][5][6][7][8][9][10][11][12]. In most cases, the structure of the 2HDM scalar potential is constrained in some way. For example, many papers assume a CPconserving scalar potential and vacuum in order to simplify the resulting Higgs phenomenology. In such models, the three neutral Higgs bosons are states of definite CP, consisting of two CP-even scalars and one CP-odd scalar.
The assumption of CP conservation in the bosonic sector of the 2HDM may not be tenable in light of the CPviolating effects that necessarily exist in the Higgs-fermion Yukawa couplings [which are the source of the phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix that governs flavor physics]. However, the most general 2HDM scalar potential and Yukawa couplings generically yield Higgs-mediated flavor-changing neutral currents (FCNCs) at tree level in conflict with experimental observations (which imply that FCNCs are significantly suppressed). The simplest way to avoid tree-level Higgs-mediated FCNCs is to impose a discrete Z 2 symmetry on the Higgs Lagrangian [13][14][15]. Remarkably, such a symmetry, if exact, removes tree-level Higgs-mediated FCNCs in the Yukawa sector while eliminating all CP-violating phases in the bosonic sector of the theory. However, the imposition of an exact Z 2 symmetry is too restrictive. For example, no decoupling limit exists in the Z 2 -symmetric 2HDM [16]. Since the LHC Higgs data imply that the observed Higgs boson is SM-like in its properties, one can only achieve approximate Higgs alignment without decoupling by a finetuning of the Higgs scalar potential parameters [16][17][18][19][20][21][22][23].
It is possible to satisfy the phenomenological constraint of suppressed Higgs-mediated FCNCs by introducing a soft breaking of the Z 2 symmetry. Having introduced such a symmetry breaking term in the Higgs Lagrangian, it is now possible that unremovable complex phases in the scalar potential exist, in which case Higgs-mediated CP-violating effects will be present. The 2HDM with a softly broken Z 2 symmetry and unremovable complex phases in the scalar potential is called the complex 2HDM (often denoted as the C2HDM) [24][25][26][27][28][29][30][31][32].
The C2HDM is typically exhibited in a scalar field basis in which the Z 2 symmetry of the dimension-four terms of the Higgs Lagrangian is manifest. Nevertheless, the physical consequences of the C2HDM are independent of the choice of basis. It is often convenient to employ a basis-independent formalism [33], in which the relevant parameters of the model are manifestly independent of the basis choice. Indeed, basis-independent couplings (in principle) can always be directly related to physical observables. Thus, it is useful to express the parameters of the C2HDM, defined in the basis in which the Z 2 symmetry is manifestly realized, in terms of basis-independent quantities.
To see utility of the basis-independent approach, consider the well-known quantity, given by the ratio of the absolute values of the two neutral Higgs field vacuum expectation values defined in some basis of the scalar fields. In the most general 2HDM, this quantity is basis dependent and thus no physical observable can depend on it. In the C2HDM, tan β is defined via the Higgs-fermion Yukawa couplings in a basis where the Z 2 symmetry of the dimension-four terms of the Higgs Lagrangian is manifestly realized. However, even given such a definition, some residual basis dependence remains. Moreover, no coupling in the bosonic sector of the C2HDM depends on tan β [34].
In this paper, we follow the basis-independent formalism of Refs. [33,34], which was inspired by an elegant formulation of the 2HDM in Ref. [8] that was subsequently described in more detail in Ref. [9]. An alternative approach to basis-independent methods in the 2HDM based on employing a set of independent physical couplings is given in Refs. [31,35]. The translation between these two approaches can be found in Appendix D of Ref. [23]. The bilinear formalism of the 2HDM employed in Refs. [36][37][38][39][40][41] also provides a powerful framework for establishing basis-independent results that can be applied in numerous applications.
In order to make this paper self-contained, we recapitulate in Sec. II the ingredients of the basis-independent treatment of the 2HDM developed in Refs. [33,34] in full detail. In particular, we emphasize the singular importance of the Higgs basis (defined to be a basis in which one of the two neutral scalar fields has zero vacuum expectation value), which possesses some important invariant features. In this regard, we tweak the formalism of Ref. [34] to emphasize the significance of the complex phase degree of freedom associated with the definition of the Higgs basis. This allows us to define invariant Higgs basis scalar fields, which simplifies the subsequent analysis.
In Sec. III, we obtain expressions for the charged and neutral Higgs mass-eigenstate fields in terms of the invariant Higgs basis fields, which can then be expressed in terms of the scalar fields of the original basis. The neutral Higgs mass eigenstates arise after the diagonalization of a 3 × 3 squared-mass matrix, which yields three invariant mixing angles. Although we have slightly modified the formalism of Ref. [34], we can explicitly show that one invariant mixing angle combines additively with a parameter that represents the phase dependence implicit in the definition of the Higgs basis. Hence, only two of the three invariant mixing angles can be related to physical observables.
In Sec. IV, we introduce a basis-invariant description of the Higgs-fermion Yukawa interactions. We again tweak the formalism of Ref. [34] in order to construct matrix invariant Yukawa couplings. We then introduce the Type-I and Type-II Yukawa Higgs-quark couplings [42][43][44] by imposing a (softly broken) Z 2 symmetry that defines the parameter tan β and guarantees the absence of tree-level Higgs-mediated FCNCs. Although the physics literature treats tan β as a physical parameter of the 2HDM, 1 we emphasize that a residual basis dependence is still present and associated with the freedom to interchange the two Higgs fields in a basis where the softly broken Z 2 symmetry is manifestly realized.
In Sec. V, a basis-independent treatment of the softly broken Z 2 symmetry (which is needed in the construction of the Type-I and Type-II Yukawa interactions) is presented. Formal basis-independent expressions were originally given in Ref. [33], and explicit results in the case of the CP-conserving 2HDM were presented in Ref. [45]. In this paper, we provide the corresponding results that are applicable if CP violation is present in the 2HDM, with a careful analysis of all possible special cases. We subsequently noticed that some equivalent results can also be found in a paper by Lavoura [46], although the basisindependent nature of Lavoura's results was not initially appreciated.
In Sec. VI, we are finally ready to carry out the basisindependent treatment of the C2HDM. In the literature, the parameters of the C2HDM are typically defined in the basis where the softly broken Z 2 symmetry is manifest and where the two scalar field vacuum expectation values are real and positive. Our goal was to provide a translation between these parameters and the corresponding parameters of the basis-independent formalism. In doing so, one gains insight into the nature of the original C2HDM parameters and their relations to physical quantities. We again emphasize the significance of the residual basis dependence associated with the interchange of the two scalar fields.
In Sec. VII, we return to the paper of Lavoura [46]. We provide the necessary detail to derive Lavoura's results and indicate where his results fall short (i.e., special cases 1 The definition of the term "physical parameter" requires some care. In this paper, we identify a Lagrangian parameter as a physical parameter if it can be uniquely related to quantities that can be obtained (in principle) from direct experimental measurements. Note that parameters that cannot be defined in terms of quantities that are invariant with respect to field redefinitions are not physical parameters. in which Lavoura's results do not apply). Lavoura attempted to find two invariant conditions for identifying the presence of spontaneous CP violation in the 2HDM. He was able to find one of the conditions but unable to find the second one. We complete his search and discuss various special cases in which only one invariant condition is required.
We briefly summarize our conclusions in Sec. VIII. Additional details are relegated to five appendices. Appendix A provides the necessary formulae for transforming between two scalar field bases. In particular, we exhibit how the parameters of the original basis of the 2HDM are expressed in terms of the parameters of the Higgs basis. Appendix B treats the so-called exceptional region of the 2HDM parameter space (the nomenclature was introduced in Ref. [47]). Indeed, in this parameter regime, special attention is mandated as some of our derivations of basis-independent conditions provided in the main text are not applicable in this case. Appendix C demonstrates that the formal basis-independent conditions for a (softly broken) Z 2 symmetry given in Ref. [33] are equivalent to the results of the explicit derivation given in Sec. V. Appendix D provides a simple proof for the existence of a particular basis of scalar field in which the CP-odd invariants employed in Sec. VII take on especially convenient forms. Finally, Appendix E examines the mixing of the three neutral physical scalars of the 2HDM in a generic basis of the two scalar fields.

II. BASIS-INDEPENDENT FORMALISM OF THE 2HDM
The fields of the two-Higgs-doublet model (2HDM) consist of two identical complex hypercharge one, SU (2) doublet scalar fields Φ a ðxÞ ≡ ðΦ þ a ðxÞ; Φ 0 a ðxÞÞ, where the "Higgs flavor" index a ¼ 1; 2 labels the two-Higgs-doublet fields. The most general renormalizable SUð2Þ L × Uð1Þ Y invariant scalar potential is given by where m 2 11 , m 2 22 , and λ 1 ; …; λ 4 are real parameters and m 2 12 , λ 5 , λ 6 and λ 7 are potentially complex parameters. We assume that the parameters of the scalar potential are chosen such that the minimum of the scalar potential respects the Uð1Þ EM gauge symmetry. Then, the scalar field vacuum expectations values (vevs) are of the form where v 1 and v 2 are real and non-negative, 0 ≤ ξ < 2π, and v is determined by the Fermi constant, In Eq. (5), the covariant derivative of the electroweak gauge group acting on the scalar fields yields where s W ≡ sin θ W and c W ≡ cos θ W . Since the scalar doublets Φ 1 and Φ 2 have identical SUð2Þ × Uð1Þ quantum numbers, one is free to express the scalar potential in terms of two orthonormal linear combinations of the original scalar fields. The parameters appearing in Eq. (2) depend on a particular basis choice of the two scalar fields (denoted henceforth as the Φ basis). The most general redefinition of the scalar fields that leaves L KE invariant corresponds to a global U(2) trans- In our convention of employing unbarred and barred indices, there is an implicit sum over unbarred-barred index pairs such as a andā. 2 Following Refs. [8,9,33], the scalar potential can be written in U(2)-covariant form, 2 Note that replacing an unbarred index with a barred index is equivalent to complex conjugation. An alternative but equivalent convention makes use of lower and upper Higgs flavor indices in place of barred and unbarred indices, in which case there is an implicit sum over a repeated upper-lower index pair.
where the quartic couplings satisfy Z abcd ¼ Z cdab . The hermiticity of the scalar potential implies that Y ab ¼ ðY bā Þ Ã and Z abcd ¼ ðZ bādc Þ Ã . Under a flavor-U(2) transformation, the tensors Y ab and Z abcd transform covariantly: Y ab → U ac Y cd U † db and Z abcd → U aē U † fb U cḡ U † hd Z efgh . The coefficients of the scalar potential depend on the choice of basis. The transformation of these coefficients under a U(2) basis change, exhibited explicitly in Eqs. (A2)-(A11), are precisely the transformation laws of Y and Z given above.
For the convenience of the reader, we recapitulate the ingredients of the basis-independent approach employed in Ref. [34], in order to make this paper self-contained. In an arbitrary scalar basis, the vevs of the two-Higgs-doublet fields [cf. Eq. (3)] can be written compactly as Since the tensors Y ab and Z abcd exhibit tensorial properties with respect to global U(2) transformations in the Higgs flavor space, one can easily construct invariants with respect to the U(2) by forming U(2)-scalar quantities. It is convenient to define two Hermitian projection operators, The matrices V and W can be used to define the following manifestly basis-invariant real quantities that depend on the scalar potential parameters [cf. Eq. (7)]: In addition, we shall define the following pseudoinvariant (potentially complex) quantities: In particular, Eq. (11) implies that under a basis trans- Note that Z Ã 5 Z 2 6 , Z Ã 5 Z 2 7 , and Z Ã 6 Z 7 are basis-invariant quantities that can be obtained from the pseudoinvariants Z 5 , Z 6 , and Z 7 .
Once the scalar potential minimum is determined, which definesv a , one can introduce new Higgs-doublet fields that define the Higgs basis, The definitions of H 1 and H 2 imply that where we have used Eq. (8) and the fact thatv andŵ are complex orthogonal unit vectors. Note that the definition of the scalar field H 1 is basis independent, whereas the scalar field H 2 is a pseudoinvariant field due to the transformation properties ofŵ given in Eq. (11). That is, In contrast, Y 1 , Y 2 , and Z 1;2;3;4 are invariant when transforming between two Higgs bases.
Finally, we note that the 2HDM scalar potential and vacuum are CP invariant if one can find a choice of η such that all the coefficients of the scalar potential in Eq. (28) are real after imposing the scalar potential minimum conditions given in Eq. (24). This condition is satisfied if and only if [48] (see also Refs. [33,34])

III. THE CHARGED AND NEUTRAL HIGGS MASS EIGENSTATES
To determine the Higgs mass eigenstates, one must examine the terms of the scalar potential that are quadratic in the scalar fields (after imposing the scalar potential minimum conditions and defining shifted fields with zero vevs). We have slightly tweaked the procedure that was carried out in Ref. [34], and we summarize the results here.
We parameterize the invariant Higgs basis fields H 1 and H 2 as follows: where G þ (and its Hermitian conjugate) are the charged Goldstone bosons and G 0 is the neutral Goldstone boson. The three remaining neutral fields mix, and the resulting neutral Higgs squared-mass matrix in the φ 0 1 -φ 0 2 -a 0 basis is where The squared-mass matrix M 2 is real symmetric; hence, it can be diagonalized by a special real orthogonal transformation, where R is a real matrix such that RR T ¼ I, det R ¼ 1 and the m 2 i are the eigenvalues of M 2 . A convenient form for R is where c ij ≡ cos θ ij and s ij ≡ sin θ ij . We have writtenc 23 ≡ cosθ 23 ands 23 ≡ sinθ 23 to distinguish between the angle θ 23 defined in Ref. [34] and the angleθ 23 defined above. Indeed, the angles θ 12 , θ 13 , andθ 23 defined above are all invariant quantities since they are obtained by diagonalizing M 2 whose matrix elements are manifestly basis invariant. The neutral physical Higgs mass eigenstates are denoted by h 1 , h 2 , and h 3 , where the q kl are listed in Table I. Employing Eqs. (21) and (35), it follows that for k ¼ 1; 2; 3, where the shifted neutral fields are defined byΦ 0 a ≡ Φ 0 a − vv a = ffiffi ffi 2 p . It is straightforward to verify that Eq. (37) also applies to the neutral Goldstone boson if we denote h 0 ≡ G 0 and define q 01 ¼ i and q 02 ¼ 0 as indicated in Table I.
We have also introduced the quantity, 3 Note that e −iθ 23 is a pseudoinvariant quantity. In particular, in light of Eq. (26), it follows that under a U(2) basis transformation, Φ a → U ab Φ b . This transformation law is consistent with Eq. (11) and the fact that the neutral Higgs mass-eigenstates h k are invariant fields. 4 3 Note that θ 23 corresponds precisely to the angle of the same name employed in Ref. [34]. 4 The remaining freedom to define the overall sign of h k is associated with the convention adopted for the domains of the mixing angles θ ij , as discussed in Ref. [34], and is independent of scalar field basis transformations. 5 Here we differ slightly from Ref. [34] where a noninvariant charged Higgs field, H þ ¼ŵ Ã a Φ þ a , is employed.
For completeness, we note that Eqs. (21) and (31) yield expressions for the massless charged Goldstone field, G þ ¼v Ã a Φ þ a , and the charged Higgs field, H þ ¼ e iηŵÃ a Φ þ a , with corresponding squared mass, Nevertheless, one is always free to rephase the charged Higgs field without affecting any observable of the model. It is convenient to rephase, H þ → e −iθ 23 H þ , which yields Note that this rephasing is conventional and does not alter the fact that H þ is an invariant field with respect to scalar field basis transformations. Finally, one can invert Eq. (37) and include the charged scalars to obtain 5 Althoughθ 23 is an invariant parameter, it has no physical significance, since it only appears in Eq. (42) in the combination defined in Eq. (38). Indeed, if we now insert Eq. (42) into the expression for the scalar potential given in Eq. (7) to derive the bosonic couplings of the 2HDM, one sees thatθ 23 never appears explicitly in any observable. Consequently, one can simply setθ 23 ¼ 0 without loss of generality, which would identify η ¼ θ 23 as the pseudoinvariant phase angle that specifies the choice of Higgs basis.
It is useful to rewrite the neutral Higgs mass diagonalization equation [Eq. (33)] as follows. With R ≡ R 12 R 13R23 given by Eq. (34), we define where A 2 is the auxiliary quantity, Note that we have employed Eq. (38), which results in the appearance of e −iθ 23 in the appropriate places given that the matrix elements of f M 2 are invariant quantities (but with no separate dependence on the invariant angleθ 23 ). The diagonal neutral Higgs squared-mass matrix is then given bỹ where the diagonalizing matrixR ≡ R 12 R 13 depends only on the invariant angles θ 12 and θ 13 , after making use of Eq. (40) in the evaluation of Eq. (48), and The conditions for a CP-invariant scalar potential and vacuum were given in Eq. (30). These conditions are satisfied in the following two cases: In both cases, the neutral scalar squared-mass matrix given in Eq. (43) assumes a block diagonal form consisting of a 2 × 2 mass matrix that yields the squared masses of two neutral CP-even Higgs bosons and a 1 × 1 mass matrix corresponding to the squared mass of a neutral CP-odd Higgs boson. In this paper, our primary focus is the 2HDM with a scalar sector that exhibits either explicit or spontaneous CP violation, in which case neither Eq. (30) nor Eqs. (51) and (52) are satisfied.

IV. HIGGS-FERMION YUKAWA INTERACTIONS
The Higgs boson couplings to the fermions arise from the Yukawa Lagrangian. We shall slightly tweak the results that were initially presented in Ref. [34] (with some corrections subsequently noted in Ref. [49]). In terms of the quark mass-eigenstate fields, the Yukawa Lagrangian in the Φ basis is given by where Q R;L ≡ P R;L Q, with P R;L ≡ 1 2 ð1 AE γ 5 Þ [for Q ¼ U; D], K is the CKM mixing matrix, and the h U;D are 3 × 3 Yukawa coupling matrices. We can construct invariant matrix Yukawa couplings κ Q and ρ Q by defining 6 Inverting these equations yields Inserting the above result into Eq. (53) and employing Eqs. (21), (25), and (38), we end up with the Yukawa Lagrangian in terms of the invariant Higgs basis fields, In light of Eq. (22), κ U and κ D are proportional to the (real non-negative) diagonal quark mass matrices M U and M D , respectively. In particular, In contrast, the matrices ρ U and ρ D are independent complex 3 × 3 matrices. 6 We have modified the definition of ρ Q as compared to the one employed in Refs. [33,34,49] by including a factor of e iθ 23 . This new definition has been adopted as a matter of convenience since ρ Q defined as in Eq. (54) is invariant with respect to basis transformations of the scalar fields.
One can now reexpress the Higgs basis fields in terms of mass-eigenstate charged and neutral Higgs fields by inverting Eq. (35) and employing Eq. (41) to obtain the Yukawa couplings of the quarks to the physical scalars and to the Goldstone bosons. Of course, the same result can be obtained directly by inserting Eq. (42) into Eq. (53). The end result is where there is an implicit sum over k ¼ 0, 1, 2, 3 (and h 0 ≡ G 0 ). As expected, the Higgs-quark Yukawa couplings depend only on invariant quantities, namely, M Q and ρ Q (for Q ¼ U, D) and the invariant angles θ 12 , θ 13 , while all dependence onθ 23 has canceled. Since ρ Q is in general a complex matrix, Eq. (58) exhibits CP-violating neutral-Higgs-fermion interactions. Moreover, Higgs-mediated FCNCs are present at tree level in cases where the ρ Q are not flavor diagonal.
To avoid tree-level Higgs-mediated FCNCs, we shall impose a Z 2 symmetry on the Higgs Lagrangian specified by Eqs. (2), (5), and (53). If the scalar potential respects the discrete symmetry Φ 1 → Φ 1 and Φ 2 → −Φ 2 , then it follows that m 2 12 ¼ λ 6 ¼ λ 7 ¼ 0. However, phenomenological considerations allow for the presence of a soft Z 2 -breaking term, m 2 12 ≠ 0. Consequently, we shall henceforth apply the Z 2 symmetry exclusively to the dimension-four terms of the Higgs Lagrangian. Note that the action of the Z 2 symmetry on the scalar fields is basis dependent. In Sec. V, we shall recast this action in a basisindependent form.
One must also impose the Z 2 symmetry on the Yukawa Lagrangian, which defines the so-called Z 2 basis. Four possible Z 2 charge assignments are exhibited in Table II, Of course, the above conditions are basis dependent. Types Ia and Ib (collectively denoted by Type I) and Types IIa and IIb (collectively denoted by Type II) are essentially equivalent, respectively, differing only in which scalar is denoted by Φ 1 and which is denoted by Φ 2 . In Ref. [34], the following basis-independent conditions were given: which are clearly satisfied in the Z 2 basis. Employing Eq. (55) yields the invariant conditions, where we have used the fact that κ Q is a real matrix [cf. Eq. (57)].
In the Z 2 basis, Eq. (3) yieldsv ¼ ðcos β; e iξ sin βÞ and w ¼ ð−e −iξ sin β; cos βÞ, where tan β ≡ jv 2 j=jv 1 j. Hence, using Eqs. (54) and (57), one obtains TABLE II. Four possible Z 2 charge assignments that forbid tree-level Higgs-mediated FCNC effects in the 2HDM Higgs-quark Yukawa interactions and the corresponding invariant Yukawa coupling matrix parameters. The Type Ia and Ib cases (collectively referred to as Type I) and the Type IIa and IIb cases (collectively referred to as Type II) differ, respectively, by the interchange of Φ 1 → Φ 2 or equivalently by the interchange of cot β → tan β. The presence of the Z 2 symmetry fixes ρ U and ρ D to be diagonal matrices as exhibited below.
which we have also recorded in Table II. Indeed, ρ U and ρ D are proportional to the diagonal quark matrices M U and M D , respectively, indicating that the tree-level Higgs-quark couplings are flavor diagonal. Since the ρ Q are basis invariants, the quantity, e iðξþθ 23 Þ tan β, is a physical parameter in the 2HDM with Type-I or Type-II Yukawa couplings.
In particular, note that one still has the freedom to make a transformation that interchanges Φ 1 ↔ Φ 2 in the Z 2 basis. In performing such a basis transformation, one must also interchange tan β ↔ cot β while changing the sign of the quantity e iðξþθ 23 Þ [as we shall demonstrate in Eq. (75)]. These two parameter transformations simply result in the interchange of the a and b versions of the Type-I and Type-II Yukawa couplings. Once a specific discrete symmetry is chosen (among the four specified in Table II), tan β is promoted to a physical parameter of the model. It then follows that e iðξþθ 23 Þ is also physical. However, the parameters ξ and θ 23 separately retain their basis-dependent nature.
In contrast, the parameter tan β does not appear in the bosonic couplings of the 2HDM. This statement is easily checked by inserting Eq. (42) into Eqs. (2) and (5), which yields the Higgs self-couplings and the Higgs couplings to vector bosons [34]. The couplings of the Higgs bosons to the gauge bosons depend only on the gauge couplings and the invariant mixing angles θ 12 and θ 13 by virtue of Eqs. (5) and (42). 7 The Higgs self-couplings will additionally depend on invariant combinations of the Z i and e −iθ 23 . If there exists a scalar field basis in which λ 6 ¼ λ 7 ¼ 0, then this basis is related to the Higgs basis by a rotation by the angle β. The existence of such a basis will yield an invariant relation among the Z i that will be derived in the next section. It is only through this relation [cf. Eqs. (82) and (83)] that tan β can be indirectly probed via the Higgs selfcouplings.
The Z 2 symmetry of the 2HDM scalar potential is manifestly realized in a scalar field basis where Of course, such a description is basis dependent. In this section, we explore a basisindependent characterization of the Z 2 symmetry, where the symmetry is either exact or softly broken. We obtain conditions in terms of Higgs basis parameters that are independent of the initial choice of scalar field basis. Our analysis generalizes results previously obtained in Refs. [25,45,46]. The connection of the results obtained in this section with the basis-independent conditions that are independent of the vacuum, derived in Ref. [33], is discussed in Appendix C. An alternative basis-independent treatment of the Z 2 symmetry based on the bilinear formalism of the 2HDM scalar potential can be found in Refs. [36,39,40].

A. The inert doublet model
A very special case of the 2HDM is the so-called inert doublet model (IDM). In this model, the Higgs basis exhibits an exact Z 2 symmetry, H 1 → H 1 and H 2 → −H 2 . Imposing this symmetry on the scalar potential given in Eq. (28) yields The conditions given in Eq. (69) are basis independent given that Y 3 , Z 6 , and Z 7 are pseudo-invariant quantities. Note that it is sufficient to impose the Z 2 symmetry on the dimension-four terms of Eq. (28), since if Z 6 ¼ 0 then Y 3 ¼ 0 due to the scalar potential minimum conditions of Eq. (24). Thus, in this case, it is not possible to softly break the Z 2 symmetry.
To complete the definition of the IDM, the Higgsfermion Yukawa couplings are fixed by imposing the condition that all fermion fields are even under the Z 2 symmetry. This corresponds to Type-Ib Yukawa couplings as specified in Table II with tan β ¼ 0. In this case, ρ U ¼ ρ D ¼ 0, which implies that the doublet H 2 does not couple to the fermions. Consequently, H 2 is called an inert doublet. Due to the fact that Z 6 ¼ 0, the tree-level couplings of the neutral scalar that resides in the doublet H 1 are precisely those of the SM Higgs boson. Moreover, in the bosonic sector of the theory, the scalar fields that reside in the doublet H 2 can only couple in pairs to the gauge bosons and to the SM Higgs boson.
In light of Eq. (69), Z 5 is the only potentially complex parameter of the IDM scalar potential. This means that one is free to rephase the pseudoinvariant Higgs basis field H 2 such that all Higgs basis scalar potential parameters are real. Hence, the IDM scalar potential and vacuum are CP conserving. Since the main interest of this paper is the 2HDM with a softly broken Z 2 symmetry and CP violation, we shall henceforth assume that the Z 2 symmetry of the dimension-four terms of the scalar potential is manifestly realized in a basis that is not the Higgs basis. That is, Z 6 and Z 7 are not both simultaneously equal to zero. This assumption will allow for the possibility of a 2HDM scalar sector that exhibits either explicit or spontaneous CP violation.

B. A softly broken Z 2 symmetry
Suppose that the Z 2 symmetry of the dimension-four terms of the scalar potential is manifestly realized in some scalar field basis (henceforth denoted as the Z 2 basis), which implies that λ 6 ¼ λ 7 ¼ 0 in this basis. In light of Eqs. (A29) and (A30), it follows that the Z 2 basis exists if and only if values of β and ξ can be found such that The real and imaginary parts of Eqs. (70) and (71) yield four independent real equations. The Z 2 basis is not unique. Suppose, we choose a Φ basis in which λ 6 ¼ λ 7 ¼ 0. To maintain the conditions, In particular, by noting that s β it immediately follows that β 0 ¼ 1 2 π − β and ξ 0 ¼ ζ. Moreover, after employing Eq. (20) which shows that the phase factor, e iðξþθ 23 Þ , appearing in the expressions for ρ Q exhibited in Eqs. (65)-(68), changes sign when transforming from the Φ basis to the Φ 0 basis. Consequently, the effect of this scalar field transformation is to interchange the a and b versions of the Type-I and Type-II Yukawa couplings as asserted below Eq. (68). Returning to Eqs. (70) and (71), we first take the imaginary part of Eq. (70) to obtain ImðZ 67 e iξ Þ ¼ 0: Assuming that Z 67 ≠ 0 (we will return to the case of Z 67 ¼ 0 later), we shall denote Then, Eq. (76) implies that ξ þ θ 67 ¼ nπ, for some integer n, or equivalently The two possible sign choices in Eq. (78) correspond to the Φ and Φ 0 basis choices identified above in which (70) and (71) yields Assuming Z 1 ≠ Z 2 (we will return to the case of Since 0 ≤ β ≤ 1 2 π, it follows that In particular, which demonstrates that tan β in the Φ basis corresponds to cot β in the Φ 0 basis. Moreover, It then follows that Taking the real and imaginary parts of Eq. (86) and massaging the real part yield It is convenient to multiply Eq. (88) by −i and add the result to Eq. (87). This yields a single complex equation. Finally, since Z 67 ≠ 0 by assumption, one can divide this complex equation by Z Ã 67 and take the complex conjugate of the result to obtain The cases where Z 1 ¼ Z 2 and/or Z 67 ¼ 0 are easily treated. First, if Z 1 ¼ Z 2 and Z 67 ≠ 0, then Eqs. (79) and (80) imply that s 2β ¼ 1 and c 2β ¼ 0, and it follows that the Z 2 symmetry is manifest in the Higgs basis, as noted in Sec. VA. In this latter case, one must employ the Type-Ib Yukawa interactions, which yield ρ U ¼ ρ D ¼ 0. This corresponds to the case of tan β ¼ 0 in Eq. (66). 8 Likewise, in the case of Type-II couplings, M U ¼ ρ D ¼ 0 and ρ U is a arbitrary complex matrix. In the IDM (corresponding to a Type-Ib Yukawa sector with , the fermions couple only to the Z 2 -even scalar doublet H 1 , whose tree-level interactions exactly coincide with those of the SM Higgs doublet. Finally, the case of Z 1 ¼ Z 2 and Z 67 ¼ 0 requires special treatment; this case has been dubbed the "exceptional region" of the 2HDM parameter space in Ref. [47]. The analysis of Appendix B shows that in this exceptional case, there always exists a scalar field basis in which the softly broken Z 2 symmetry is manifestly realized. Furthermore, Eqs. (88) and (89) are trivially satisfied in the exceptional region of the 2HDM parameter space.
In conclusion, Eq. (89) is a necessary condition for the presence of a softly broken Z 2 symmetry. It is also a sufficient condition in all cases with one exception. Namely, if Z 1 ¼ Z 2 , Z 5 ≠ 0, and Z 67 ≠ 0, then Eq. (89) must be supplemented with the additional constraint of ImðZ Ã 5 Z 2 67 Þ ¼ 0.
In the case of the CP-conserving 2HDM, it is possible to rephase the pseudoinvariant Higgs basis field H 2 such that all of the Z i are real. In this real basis, Eq. (89) reduces to a result previously given in eq. (54) of Ref. [45]. The scalar basis in which λ 6 ¼ λ 7 ¼ 0 is obtained from the Higgs basis by a rotation by an angle β, which is determined by Eq. (81), in a convention where v 1 and v 2 are non-negative [in which case ξ ¼ 0 so that sgnZ 67 ¼ AE1 in light of Eq. (78)]. Once again, the exceptional region of parameter space where Z 1 ¼ Z 2 and Z 67 ¼ 0 must be treated separately. Using Eqs. (B2) and (B3) with ξ ¼ 0 and real Z i , it follows that cot 2β is a solution of Eq. (B7), where Z 5 and Z 6 are real and AE is identified with sgnZ 6 (or equivalently, replace jZ 6 j with Z 6 and replace AE with a plus sign).
C. Softly broken Z 2 symmetry and spontaneously broken CP symmetry Suppose that the conditions for a softly broken Z 2symmetric scalar potential obtained in Sec. V B are satisfied. Then a Z 2 basis exists (which is not unique) in which λ 6 ¼ λ 7 ¼ 0. If in addition, then one can rephase one of the scalar fields such that m 2 12 and λ 5 are simultaneously real. In this case, the scalar potential is explicitly CP invariant. In addition, if in this socalled real Z 2 basis there is an unremovable complex phase in the vevs, that is, then the CP symmetry of the scalar potential is spontaneously broken. Using Eqs. (A20) and (A25), where Z 345 ≡ Z 34 þ ReðZ 5 e 2iξ Þ. Next, we employ the potential minimum conditions [Eq.
and we make use of Eq. (82) for s 2β and c 2β . To make further progress, we first assume that Z 1 ≠ Z 2 and Z 67 ≠ 0. In this case, we can use Eqs. (77) and (78) to write e iξ ¼ AEZ Ã 67 =jZ 67 j. It is convenient to introduce the following notation: It then follows that ImðZ 6 e iξ Þ ¼ AE Finally, we employ Eqs. (87) and (88) to obtain Plugging the above results into Eq. (94), we end up with where the function F is given by 9 Thus, Imðλ Ã 5 ½m 2 12 2 Þ ¼ 0 if one of two conditions are satisfied: This implies that one can rephase the Higgs basis field H 2 such that Z 5 , Z 6 , and Z 7 are simultaneously real [which also implies that Y 3 is real by Eq. (24)]. That is, all the coefficients of the scalar potential in the Higgs basis and the corresponding vevs are real, implying that the scalar potential and the vacuum are CP conserving. In contrast, if f 3 ≠ 0 and F ¼ 0, then the scalar potential is explicitly CP conserving as noted below Eq. (92). However, in the Z 2 basis in which all scalar potential parameters are real, the vevs exhibit a complex phase that cannot be removed by a basis transformation while maintaining real coefficients in the scalar potential. In particular, ImðZ 6 Z Ã 7 Þ ≠ 0 implies that no real Higgs basis exists, which is a signal of CP violation. 10 Thus, f 3 ≠ 0 and F ¼ 0 is a basisindependent signal of spontaneous CP violation. 11 If F ¼ 0, then Eq. (103) provides a quadratic equation for Y 2 that yields Y 2 ∼ OðZ i Þ. In contrast, the decoupling limit of the 2HDM corresponds to Y 2 ≫ v [34]. Since jZ i j=4π ≲ Oð1Þ as a consequence of tree-level unitarity [49][50][51][52][53][54][55], it follows that the 2HDM with a softly broken Z 2 symmetry and spontaneous CP violation possesses no decoupling limit [56].
To complete the analysis of this subsection, we must address the special cases in which either Z 1 ¼ Z 2 and/or Z 67 ¼ 0. As noted below Eq. (89), if Z 1 ¼ Z 2 and Z 67 ≠ 0, then Eqs. (79) and (80) imply that s 2β ¼ 1 and c 2β ¼ 0, and it follows that ImðZ Ã 5 Z 2 67 Þ ¼ 0 and jZ 6 j ¼ jZ 7 j in light of Eqs. (87) and (88). Then, Eq. (94) yields Note that in contrast to Eq. (100), ReðZ Ã 5 Z 2 67 Þ is not determined in terms of the Z i , f 1 , and f 2 , since in the case of Z 1 ¼ Z 2 , this quantity is not constrained by Eqs. (87) and (88). Indeed, another way to derive Eq. (104) is to use Eq. (100) to solve for f 2 in terms of ReðZ Ã 5 Z 2 6 Þ and substitute this result back into Eq. (103). In this way, the factor of Z 1 − Z 2 in the denominator of Eq. (102) is canceled. The resulting expression is significantly more complicated than the one given in Eq. (103). Nevertheless, by setting Z 1 ¼ Z 2 in this latter expression, we have checked that one recovers the result of Eq. (104). Thus, we again conclude that spontaneous CP violation occurs if f 3 ≠ 0 and the following basis-independent condition is satisfied: Next, as noted below Eq. (89), if Z 67 ¼ 0 and Z 1 ≠ Z 2 , then Eqs. (70) and (71) imply that Z 6 ¼ Z 7 ¼ 0. Thus, an unbroken Z 2 symmetry is manifestly realized in the Higgs basis. That is, in this case, one identifies m 2 12 ¼ 0 and thus Imðλ Ã 5 ½m 2 12 Þ ¼ 0 is trivially satisfied. Moreover, one can rephase the Higgs basis field H 2 such that Z 5 is real. Hence, 9 An expression for F was first derived by Lavoura in Ref. [46], although his eq. (22) contains a misprint in which the factor of f 2 in the coefficient of was inadvertently dropped. 10 We define a real Higgs basis to be the basis in which the potentially complex parameters Z 5 , Z 6 , and Z 7 are simultaneously real. In this case, Y 3 is also real in light of Eq. (24). Note that a real Higgs basis exists if and only if ImðZ in which case one can rephase the Higgs basis field H 2 appropriately to achieve the real Higgs basis. In the 2HDM, the existence of a real Higgs basis is a necessary and sufficient condition for a CP-conserving scalar potential and vacuum. 11 Basis-independent conditions for spontaneous CP violation have also been obtained in the bilinear formalism of the 2HDM in Refs. [37,38]. a real Higgs basis exists which implies that both the scalar potential and the vacuum are CP conserving.
So far, in all cases considered above, the conditions λ 6 ¼ λ 7 ¼ 0 and Imðλ Ã 5 ½m 2 12 2 Þ ¼ 0 in the Φ basis were necessary and sufficient for an explicitly CP-conserving scalar potential. One encounters a surprising result when considering the final case of the exceptional region of parameter space, where Z 1 ¼ Z 2 and Z 7 ¼ −Z 6 ≠ 0, where the only potentially CP-violating invariant is ImðZ Ã 5 Z 2 6 Þ. Suppose that the Higgs basis parameters satisfy ImðZ Ã 5 Z 2 6 Þ ¼ 0, Z 1 ¼ Z 2 and Z 7 ¼ −Z 6 ≠ 0. Then, there exists a Φ basis that satisfies λ 6 ¼ λ 7 ¼ 0, β ¼ 1 4 π, and cosðξ þ θ 6 Þ ¼ 0, where θ 6 ¼ arg Z 6 . It follows that where the sign choice in Eq. (107) is correlated with sinðξ þ θ 6 Þ ¼ AE1. In light of Eqs. (A26) and (A27), it follows that λ 6 ¼ λ 7 ¼ 0. If we now insert the above results into Eqs. (A20) and (A25) and employ the scalar potential minimum conditions [Eq. (24)], then Hence, for generic choices of the remaining scalar potential parameters, one can conclude that a parameter regime within the exceptional region of the parameter space exits where in which the scalar potential is explicitly CP conserving, and moreover CP is not spontaneously broken! In this case, CP is conserved despite the fact that no Z 2 basis exists in which all the scalar potential parameters are real (for further details, see Ref. [57]).
D. Imposing the convention of non-negative real vevs in the Z 2 basis In some applications, it is convenient to adopt a convention in which ξ ¼ 0 in the basis where λ 6 ¼ λ 7 ¼ 0. If this condition is not satisfied initially, it is straightforward to impose this condition by an appropriate rephasing of the Higgs-doublet field Φ 2 . In this convention, the real and imaginary parts of Eqs. (70) and (71) yield Equations (110)-(112) are equivalent to eq. (3.16) of Ref. [58]. Because we have fixed ξ ¼ 0 in the Φ basis, we must choose ξ ¼ ζ ¼ 0 in Eq. (72) in defining the Φ 0 basis in order to maintain our convention in which the vevs v 1 and v 2 are real and non-negative. That is, it follows that pseudoinvariant quantities will change sign between the Φ and Φ 0 bases. Indeed, the effect of transforming from the Φ basis to the Φ 0 basis is to modify the Φ-basis parameters such that whereas λ 3 , λ 4 , and λ 6 ¼ λ 7 ¼ 0 are unchanged. In light of Eq. (20), the Higgs basis parameters obtained starting from the Φ 0 basis differ from those obtained starting from the Φ basis by the following sign changes: In particular, the Higgs basis parameter Z 5 is unchanged since ðdet UÞ 2 ¼ 1.
As previously noted, tan β is not yet a physical parameter, since the effect of transforming from the Φ basis to the Φ 0 basis is to modify β → 1 2 π − β. In light of these remarks, one can check that Eqs. (110)-(113) are invariant with respect to the transformation Φ 0 a ¼ U ab Φ b , and thus define the invariant conditions for the existence of a scalar field basis with λ 6 ¼ λ 7 ¼ 0 and non-negative real scalar vevs (i.e., ξ ¼ 0).
Consider first the case of Z 67 ≠ 0. By virtue of Eq. (111), it follows that the pseudo-invariant quantity Z 67 is real. This condition fixes the Higgs basis up to a twofold ambiguity that depends on the sign of Z 67 . This ambiguity is simply a consequence of the freedom to change from the Φ basis to the Φ 0 basis, while maintaining the Z 2 -basis conditions, λ 6 ¼ λ 7 ¼ 0, as discussed above. Likewise, the pseudoinvariant quantity e iθ 23 is determined up to a twofold ambiguity, as its sign can be flipped by transforming from the Φ basis to the Φ 0 basis.
One can obtain an explicit expression for e iθ 23 in terms of pseudoinvariant quantities by setting ξ ¼ 0 in Eq. (84), Under Φ 1 ↔ Φ 2 , c 2β changes sign, and we conclude that θ 23 is determined modulo π. However, a more practical expression can be obtained as follows. Writing Z 6 ≡ jZ 6 je iθ 6 and Z 7 ≡ jZ 7 je iθ 7 , Eq. (111) is equivalent to the equation, jZ 6 j sin θ 6 þ jZ 7 j sin θ 7 ¼ 0. One can eliminate θ 7 and solve for θ 6 to obtain which implies that θ 6 is determined modulo π. Under the assumption that Z 6 ≠ 0, one can obtain an explicit formula for e iθ 23 , where the numerator and denominator on the right-hand side of Eq. (118) are evaluated by employing Eqs. (117) and (50), respectively. As expected, θ 23 is thus determined modulo π. If Z 6 ¼ 0, then Eq. (111) yields sin θ 7 ¼ 0, which implies that Z 2 7 ¼ jZ 7 j 2 . In this case, assuming Z 5 ≡ jZ 5 je iθ 5 ≠ 0, it follows that in the case of Z 6 ¼ 0: Hence, where the numerator and denominator on the right-hand side of Eq. (120) are evaluated by employing Eqs. (119) and (49), respectively. Taking the square root of Eq. (120) determines θ 23 modulo π. If Z 5 ¼ Z 6 ¼ 0, then the squared-mass matrix of the neutral Higgs scalars is diagonal. In this case, the mass basis and the Higgs basis (with Z 7 real) coincide and the scalar potential and vacuum are CP conserving.
The case of Z 67 ¼ 0 must be separately considered. If Z 67 ¼ 0 and Z 1 ≠ Z 2 , then as discussed below Eq. (89), it follows that Z 6 ¼ Z 7 ¼ 0 corresponding to the IDM. The exceptional region of parameter space corresponding to Z 67 ¼ 0, Z 6 ≠ 0, and Z 1 ¼ Z 2 is treated in Appendix B. In this case, Eq. (78) is replaced by where Z 6 ≡ jZ 6 je iθ 6 and ξ 0 ≡ ξ þ θ 6 is a pseudoinvariant quantity that is determined modulo π in Appendix B. Once again, we see that in a convention where ξ ¼ 0, the Z 2 basis is uniquely defined up to a twofold ambiguity corresponding to the fact that ξ 0 , and hence θ 6 and θ 23 , have been determined modulo π.
Finally, in light of the remarks at the end of Sec. IV, we can conclude that in a convention in which ξ ¼ 0, once a specific discrete symmetry is chosen (among the four specified in Table II), both tan β and θ 23 are promoted to physical parameters of the model.

E. An exact Z 2 symmetry
In Sec. V B, we defined the Z 2 basis to be the scalar basis in which λ 6 ¼ λ 7 ¼ 0. If in addition m 2 12 ¼ 0 in the same basis, then the scalar potential possesses an exact Z 2 symmetry; i.e., it is invariant under Φ 1 → Φ 1 and Φ 2 → −Φ 2 . In this case, the condition m 2 12 ¼ 0 yields additional constraints. In light of Eq. (A20), where ξ and β have been determined previously by Eqs. (78) and (81), respectively, under the assumption that Z 67 ≠ 0. Hence, employing e iξ ¼ AEe −iθ 67 ¼ AEZ Ã 67 =jZ 67 j in Eq. (122), it follows that The analysis above relied on the assumption that Z 67 ≠ 0. Thus, we now examine the relevant conditions for an exactly Z 2 -symmetric scalar potential when Z 67 ¼ 0.
If Z 67 ¼ 0 and Z 6 ¼ 0, then we also have Z 7 ¼ Y 3 ¼ 0 [the latter of Eq. (24)], in which case the exact Z 2 symmetry is manifest in the Higgs basis. Consequently, in what follows, we shall assume that Z 67 ¼ 0 and Z 6 ≠ 0.
In this paper, we are primarily interested in the case where either the scalar potential or the vacuum is CP violating. However, it is easy to see that if the Z 2 symmetry is exact, then both the scalar potential and vacuum are CP conserving. In the Z 2 basis, since m 2 12 ¼ λ 6 ¼ λ 7 ¼ 0, the only potentially complex scalar potential parameter is λ 5 , whose phase can be removed by an appropriate rephasing of the Higgs fields. Moreover, if hΦ † 1 Φ 2 i ¼ 1 2 v 1 v 2 e iξ , then the ξ-dependent term of the scalar potential is of the form V ∋ 1 4 λ 5 v 2 1 v 2 2 cos 2ξ, which is minimized when ξ ¼ 0, 1 2 π or π (depending on the sign of λ). If ξ ¼ 1 2 π, then one can rephase Φ 2 → iΦ 2 , which simply changes the sign λ 5 while rendering the two vevs relatively real. Hence, the vacuum is CP conserving. Having achieved a scalar potential with only real parameters and real vevs, it immediately follows that a real Higgs basis exists. That is, a Higgs basis exists such that Z 5 , Z 6 , and Z 7 (and Y 3 ¼ − 1 2 Z 6 v 2 via the scalar potential minimum condition) are simultaneously real.
Nevertheless, it is instructive to show directly that the existence of a real Higgs basis can be deduced solely from the relations satisfied by the Higgs basis parameters when an exact Z 2 symmetry is present. First, consider the case where the exact Z 2 symmetry is manifest in the Higgs basis, i.e., Y 3 ¼ Z 6 ¼ Z 7 ¼ 0. In this case, the only potentially complex parameter in the Higgs basis is Z 5 . The phase of Z 5 can be removed by a rephasing of the Higgs basis field H 2 . Hence, if the Z 2 symmetry is manifest in the Higgs basis, then a real Higgs basis exists and the scalar potential and the vacuum are CP conserving.
Next, suppose that Z 67 ≠ 0. Then, if we combine Eqs. (88) and (124) and employ the scalar potential minimum condition, it follows that if the Z 2 symmetry is exact, then Given that Z 67 ≠ 0, the two conditions exhibited in Eq. (132) are sufficient to guarantee the existence of a real Higgs basis in which Z 5 , Z 6 , and Z 7 are simultaneously real. If Z 67 ¼ 0 and Z 1 ≠ Z 2 , then Eq. (87) implies that Z 6 ¼ Z 7 ¼ 0 in which case the Z 2 symmetry is manifest in the Higgs basis and the previous considerations apply. Finally, if Z 67 ¼ 0, Z 6 ≠ 0, and Z 1 ¼ Z 2 , then Eq. (129) implies the existence of a real Higgs basis. Thus, in all possible cases, if an exact Z 2 symmetry is present in some scalar field basis, then a real Higgs basis exists and the scalar potential and vacuum in any scalar basis are CP conserving.
If the Z 2 symmetry is exact, then a real Higgs basis exists, and the Higgs basis parameters in Eq. (123) can be taken to be real. Employing Eq. (24) then yields Equations (90) and (133) are equivalent to eqs. (18) and (19) of Ref. [46]. Note that Eq. (133) is trivially satisfied if Z 67 ¼ 0 and Z 1 ¼ Z 2 . In this latter case, one must also impose Eq. (131) to guarantee the presence of an exact Z 2 symmetry. This last observation was missed in Ref. [46].

VI. THE C2HDM IN THE Z 2 BASIS
The C2HDM is a two-Higgs-doublet model in which either the scalar potential or the vacuum is CP violating. To avoid tree-level Higgs-mediated FCNCs, one imposes a Z 2 symmetry on the dimension-four terms of the Higgs Lagrangian. The symmetry is manifest in the Φ basis by setting λ 6 ¼ λ 7 ¼ 0 in Eq. (2). The Z 2 symmetry is assumed to be softly broken by taking m 2 12 ≠ 0. If the CP violation in the scalar potential is explicit, then Imðλ Ã 5 ½m 2 12 2 Þ ≠ 0. Imposing the Z 2 symmetry on Eq. (53) implies that the Higgs-quark Yukawa couplings are either of Type I or Type II as discussed in Sec. IV.
In Sec. V D, we noted that one is always free to rephase the Higgs-doublet fields such that the vevs are real. (The corresponding results prior to rephasing the vevs are given in Appendix E.) Henceforth, we define the C2HDM in the in the notation of Eqs. (3) and (4), where c β ≡ cos β and s β ≡ sin β, with 0 ≤ β ≤ 1 2 π. In this convention, one may parametrize the scalar doublets in the Φ basis as Setting λ 6 ¼ λ 7 ¼ ξ ¼ 0 in Eq. (E3) yields the C2HDM scalar potential minimum conditions, Im m 2 12 ¼ After eliminating m 2 12 , m 2 22 , and Im m 2 12 , we are left with nine real parameters that govern the C2HDM: v, tan β, Re m 2 12 , λ 1 , λ 2 , λ 3 , λ 4 , Re λ 5 , and Im λ 5 . By adopting the convention where both vevs are real and positive, it follows that if s 2β ≠ 0 and Im λ 5 ≠ 0 [which implies that Im m 2 12 ≠ 0 via Eq. (138)], then CP is violated in the scalar sector.
If CP is violated in the scalar sector, then the violation is either explicit or spontaneous. A scalar potential of the 2HDM is explicitly CP conserving if and only if a real basis exists [59] (i.e., a basis of scalar fields exists in which all the scalar potential parameters are real). However, in transforming to a real basis, the vevs (which were real in the original basis by convention) may acquire a relative complex phase that is unremovable by any further basis change that maintains the reality of the scalar field basis. This latter scenario corresponds to the case of spontaneous CP violation. Consequently, both spontaneous and explicit CP violation are treated simultaneously in the convention adopted in Eq. (134).
It is instructive to perform the counting of parameters using the invariants quantities discussed in previous sections. After employing Eq. (24), one is left initially with six real parameters, v, Y 2 , Z 1 , Z 2 , Z 3 , and Z 4 , and three complex parameters, Z 5 , Z 6 , and Z 7 , for a total of 12 parameters. Since one can rephase the pseudoinvariant Higgs basis field H 2 , this freedom removes one phase from the three complex parameters. Finally, since a softly broken Z 2 symmetry is present, one obtains one complex constraint equation (derived in Sec. V) that removes two additional parameters. This leaves nine independent real parameters in agreement with our previous counting.
If s 2β ¼ 0, then the model corresponds to the IDM which is CP conserving. Consequently, in our considerations of the C2HDM, we shall henceforth assume that s 2β ≠ 0, which is a necessary ingredient for the presence of CP violation, as noted below Eq. (138). Since λ 6 ¼ λ 7 ¼ 0 (in the Z 2 basis), it then follows from Eq. (D1) that if λ 1 ≠ λ 2 , then λ 6 þ λ 7 is nonzero when evaluated in any other scalar field basis. In particular, λ 1 ≠ λ 2 implies that Z 67 ≠ 0. In contrast, if λ 1 ¼ λ 2 in the Z 2 basis, then it follows that Z 1 ¼ Z 2 and Z 67 ¼ 0, which corresponds to the exceptional region of the parameter space (see Appendix B).
In light of Eqs. (21), (25), and (31), one can identify the massless would-be neutral Goldstone boson with G 0 ¼ c β χ 1 þ s β χ 2 . Thus, the neutral scalar state orthogonal to G 0 is After diagonalizing the squared-mass matrix of the neutral scalar fields, η 1 , η 2 , and η 3 , the three neutral mass-eigenstate scalar fields, h 1 , h 2 , and h 3 , can be identified as 0 In the C2HDM literature, the 3 × 3 orthogonal mixing matrix R is parametrized as [60] R It is now straightforward to relate the angles α 1 , α 2 , and α 3 of the C2HDM literature to basis-independent quantities introduced in Sec. II. In Appendix E, we have examined the mixing of the neutral scalars in the Z 2 basis. Setting ξ ¼ 0 in Eqs. (E9)-(E11) yields One can relate the mixing angles α 1 , α 2 , and α 3 to invariant (or pseudoinvariant) quantities by setting ξ ¼ 0 in Eqs. (E12) and (E13). It is convenient to defineᾱ 1 ≡ α 1 − β. We then obtain the results exhibited in Table III.
In the presence of a softly broken Z 2 symmetry, Eq. (75) implies that the quantity e iðξþθ 23 Þ is determined up to a twofold ambiguity associated with a residual basis dependence corresponding to the interchange of the two scalar doublets while maintaining λ 6 ¼ λ 7 ¼ 0. Having adopted the C2HDM convention where ξ ¼ 0, it therefore follows that e iθ 23 is determined up to a twofold ambiguity. In particular, one no longer has the freedom to rephase the Higgs basis field H 2 , which would result in an additive shift of the parameter θ 23 [cf. Eq. (39)]. In light of Eqs. (72)-(75), it follows that under the basis transformation that simply interchanges Φ 1 and Φ 2 (with no rephasing), s β ↔ c β and e iθ 23 → −e iθ 23 . Moreover, which yields R k1 ↔ R k2 and R k3 → −R k3 . These results are consistent with Eqs. (142)-(144) since the q k1 and q k2 are basis-invariant quantities. Finally, we note that the free parameter Rem 2 12 can also be related to basis-invariant quantities by employing Eq. (A20) with ξ ¼ 0 and Eq. (24), and making use of the results of Sec. V D. If λ 1 ≠ λ 2 , then Z 67 ≠ 0, in which case Eqs. (110) and (111) yield where s 2β is given by Eq. (82). The case of λ 1 ¼ λ 2 in the Z 2 basis corresponds to the exceptional region of parameter space, where Z 1 ¼ Z 2 and Z 67 ¼ 0, as previously noted. In this case, Eq. (146) does not apply and one must employ the results of Appendix B. The resulting expression for Rem 2 12 is unwieldy and we do not present it here. It is instructive to identify the nine real parameters of the C2HDM in terms of the scalar masses and mixing angles. In order to perform the correct counting, we note the following sum rule: which is derived at the end of Appendix E. This sum rule imposes one relation among the ten real quantities, v, tan β, α 1 , α 2 , α 3 , m 1 , m 2 , m 3 , Re m 2 12 , and m H AE , resulting in nine independent parameters. One can repeat the counting of parameters using basis-invariant quantities. In light of Eq. (40) and Eqs. (47)-(50), one can eliminate Z 1 , Z 3 , Z 4 , Z 5 e −2iθ 23 , and Z 6 e −iθ 23 in terms of scalar masses and the invariant mixing angles θ 12 and θ 13 . This leaves three TABLE III. The relation between the neutral Higgs mixing angles α i of the C2HDM defined in the Z 2 basis and (pseudo)invariant combinations of mixing angles defined in the Higgs basis. In the notation used below,c 1 ≡ cosᾱ 1 ands 1 ≡ sinᾱ 1 , invariant parameters, Z 2 , ReðZ 7 e −iθ 23 Þ and ImðZ 7 e −iθ 23 Þ, of which two are determined from the one complex constraint equation arising from the condition of a softly broken Z 2 symmetry. For example, if we eliminate the complex parameter Z 7 using Eq. (89), we are left with the following nine real parameters: v, Y 2 , Z 2 , θ 12 , θ 13 , m 1 , m 2 , m 3 , and m H AE . The complete set of Feynman rules for the C2HDM in terms of the Z 2 -basis parameters can be found in Refs. [32,61]. One can check that all the Higgs couplings obtained this way (after using Eq. (41) to define an invariant charged Higgs field) are invariant with respect to basis transformations. As previously noted, all the bosonic couplings of the most general 2HDM (without any imposed discrete symmetries) can be found in Ref. [34] expressed directly in terms of invariant quantities q k1 , q k2 , and the Higgs basis scalar potential coefficients (including appropriate factors of e −iθ 23 to ensure basis-independent combinations). The bosonic couplings of the most general 2HDM also apply to the C2HDM, since as emphasized in Sec. V, tan β does not appear explicitly in any of these couplings. It is a straightforward to verify that the cubic and quartic Higgs self-couplings, which appear in Ref. [34], match precisely the corresponding C2HDM couplings given in Ref. [61].
Finally, the Type-Ia and Type-IIa Higgs-quark couplings are obtained from Eq.
where there is an implicit sum over the three neutral Higgs mass-eigenstates h k . Using the results of Table III, one can reproduce the results of Ref. [32]. Indeed, as previously noted, tan β and e −iθ 23 now appear explicitly in the Yukawa couplings. However, these quantities are not quite physical parameters, since under the basis change Φ 1 ↔ Φ 2 , it follows that cot β ↔ tan β and e −iθ 23 change sign. This has the effect of interchanging the a and b versions of the Type-I and Type-II Yukawa couplings (cf. footnote 12). In order to promote tan β and e iθ 23 to physical parameters, one must remove the remaining freedom to interchange Φ 1 ↔ Φ 2 in the C2HDM. This corresponds to making a specific choice of the discrete symmetry among the four specified in Table II. In practice, this can be achieved by declaring, e.g., that tan β < 1 corresponds to an enhanced coupling of the neutral Higgs bosons to uptime quarks. Given this additional proviso, it follows that the signs of c 2β and e iθ 23 are then fixed and can now be considered as physical parameters of the model. Indeed, c 2β can be expressed in terms of basis-invariant parameters as specified in Eq. (82), where the sign ambiguity is fixed by the sign of λ 1 − λ 2 [cf. Eq. (A16)], under the assumption that λ 1 ≠ λ 2 . Likewise, e iθ 23 is uniquely determined by the formal basis-independent expression given by Eq. (116) [after employing Eq. (82) for s 2β =c 2β with the sign ambiguity fixed as indicated above]. Finally, the exceptional region of the parameter space where λ 1 ¼ λ 2 in the Z 2 basis is treated in Appendix B.

VII. DETECTING DISCRETE SYMMETRIES
In Ref. [46], Lavoura described ways to detect the presence of discrete symmetries exhibited by the scalar potential of the 2HDM. Four cases of discrete symmetries were examined: (i) exact Z 2 symmetry; (ii) explicit CP breaking by a complex soft Z 2 -breaking squared-mass term (which defines the C2HDM); (iii) softly broken Z 2 and spontaneously broken CP symmetries [62]; and (iv) the Lee model of spontaneous CP violation [1], where no (unbroken or softly broken) Z 2 symmetry is present. For the reader's convenience, we provide a translation between Lavoura's notation and the notation of this paper, In case (i), Lavoura asserts that Eqs. (18) and (19) of Ref. [46] are the conditions for an exact Z 2 -symmetric scalar potential. We have confirmed that these conditions are both necessary and sufficient in Sec. V E, as indicated below Eq. (133).
In case (ii), Lavoura asserts that Eqs. (20) and (21) of Ref. [46] are the conditions for explicit CP breaking by a complex soft Z 2 -breaking term. We have confirmed that these results are a consequence of Eqs. (87) and (88) Indeed, Eq. (88) is equivalent to eq. (20) of Ref. [46]. In addition, by multiplying Eq. (89) by Z 6 − Z 7 and then taking the imaginary part of the resulting expression, one reproduces eq. (21) of Ref. [46], 12 As discussed in Sec. IV, the Yukawa couplings for Type Ib and IIb can be obtained from Eqs. (148) and (149), respectively, by replacing cot β ↔ tan β and changing the sign of e −iθ 23 .
In case (iii), Lavoura asserts that Eqs. (20)-(22) of Ref. [46] are the conditions for a softly broken Z 2symmetric scalar potential and spontaneously broken CP symmetry. We have confirmed Lavoura's results in Sec. V C, while noting a typographical error in eq. (22) of Ref. [46] (see footnote 9). The corresponding corrected equation (with a different overall normalization) was given in Eq. (103). Moreover, Lavoura's results are not applicable in cases of Z 1 ¼ Z 2 and/or Z 67 ¼ 0. The correct expressions that replace Eq. (103) in these special cases have been obtained in Sec. V C and Appendix B. Note that if Z 6 ≠ AEZ 7 , then only two of the three equations among Eqs. (87), (88), and (151) are independent. 13 In case (iv), Lavoura attempts to discover the conditions on the 2HDM Higgs basis parameters that govern the Lee model of spontaneous CP violation [1]. In this model, the Z 2 symmetry is absent, i.e., there is no basis of scalar fields in which λ 6 ¼ λ 7 ¼ 0. A scalar field basis exists in the Lee model in which all the scalar potential parameters are simultaneously real, implying that the scalar potential is explicitly CP conserving. However, there is an unremovable relative complex phase between the two vevs hΦ 0 1 i and hΦ 0 2 i. Moreover, no real Higgs basis exists. In terms of the Higgs basis parameters, the nonexistence of a real Higgs basis implies that at least one of the following three quantities, ImðZ 2 6 Z Ã 5 Þ, ImðZ 2 7 Z Ã 5 Þ, and ImðZ 6 Z Ã 7 Þ must be nonvanishing [cf. Eq. (30)]. Hence, the vacuum is CP violating; that is, the Lee model exhibits spontaneous CP violation.
When considering the Lee model, Lavoura noted in Ref. [46] that there should be two relations among the parameters of the Lee model, corresponding to the two independent CP-odd invariants. Lavoura found one relation, that appears in eq. (27) of Ref. [46]. But he was unable to identify the second invariant condition. We now proceed to confirm Lavoura's invariant quantity and to complete his mission by finding the second invariant quantity that was missed. Moreover, we shall demonstrate that in certain regions of the parameter space of the Lee model, Lavoura's invariant vanishes, in which case two additional invariant quantities must be introduced in order to cover all possible special cases.
Consider the scalar potential of the general 2HDM given in Eq. (2), with no constraints initially imposed on the scalar potential parameters. To check for the presence of explicit CP violation in all possible regions of the 2HDM parameter space, it is necessary and sufficient to consider four CP-odd basis-invariant quantities, identified in Ref. [59], as follows 14 : If all four of these CP-odd invariants vanish, then there exists a real Φ basis, in which case the scalar potential is explicitly CP conserving. Aside from special regions in parameter space, at most two of these invariants are independent, as we will demonstrate below. Explicit forms for the above four CP-odd invariants can be found in Ref. [59]. We proceed to evaluate them in the Higgs basis. After employing Eq. (24), it follows that where the f i are defined in Eq. (95). One can check that −I Y3Z =v 2 corresponds precisely to the left-hand side of eq. (27) of Ref. [46]. Thus, I 2Y2Z is the second invariant quantity that governs the Lee model, which is the one that Lavoura was unable to find. Apart from special regions of the Lee model parameter space, I Y3Z ¼ I 2Y2Z ¼ 0 provide nontrivial relations among the parameters that must hold for a spontaneously CPviolating scalar sector. However, there exist special regions 13 Note that Im½ðZ 6 þ Z 7 ÞE ¼ 0 yields Eq. (88) and Im½ðZ 6 − Z 7 ÞE ¼ 0 yields Eq. (151), where E denotes the left-hand side of Eq. (89). It then follows that Re½ðZ 6 þ Z 7 ÞE ¼ 0, which is equivalent to Eq. (87). 14 Three CP-odd invariants that are equivalent to Eqs. (152)-(154) were also identified in Ref. [63]. Subsequently, a grouptheoretic formulation of the 2HDM scalar potential was developed in Refs. [36,37] that provided an elegant form for the basis-independent conditions governing explicit CP conservation in the 2HDM. The bilinear formalism exploited in the latter two references has also been employed in the study of the CP properties of the 2HDM scalar potential in Refs. [38][39][40][41]. of the Lee model parameter space where one or both of the invariants exhibited in Eqs. (156) and (157) automatically vanish. One such example arises in the case of a softly broken Z 2 symmetry, corresponding to λ 6 ¼ λ 7 ¼ 0 in the Φ basis in which the Lee model is initially defined. This case was studied in detail in Sec. V C, where it was shown that I Y3Z automatically vanishes and thus provides no constraint. Lavoura was well aware of this in Ref. [46].
Indeed, he noted that I Y3Z is a linear combination of the left-hand sides of Eqs. (88) and (151). Since both of these quantities vanish if λ 6 ¼ λ 7 ¼ 0 in some scalar field basis, it follows that I Y3Z ¼ 0 is automatic in a model with a softly broken Z 2 symmetry. One can check this explicitly as follows. First, if Z 67 ¼ 0, then f 3 ¼ 0, and Eq. (156) immediately yields I Y3Z ¼ 0. Next, if Z 67 ≠ 0, then Employing Eqs. (100) and (101) in Eqs. (156) and (158), one can easily verify that I Y3Z ¼ 0.
In Eqs. (102) and (103), an invariant condition was identified that guarantees that the scalar sector of the 2HDM with a softly broken Z 2 exhibits spontaneous CP violation. We now demonstrate that this invariant condition is equivalent to the requirement that f 3 ≠ 0 and I 2Y2Z ¼ 0. Assuming that Z 67 ≠ 0, we shall make use of the following formulae: ð159Þ which are derived in the same manner as Eq. (158). One can now evaluate I 2Y2Z given in Eq. (157) with the help of Eqs. (158)-(160). Imposing the conditions of a softly broken Z 2 symmetry by employing Eqs. (100) and (101), the end result of this computation is where F is given explicitly in Eq. (103). This result confirms that f 3 ≠ 0 and I 2Y2Z ¼ 0 are the invariant conditions for spontaneous CP violation in the softly broken Z 2 -symmetric 2HDM. As discussed in Sec. V C, Eq. (161) can be used in the case of Z 1 ¼ Z 2 by employing Eq. (100) to eliminate f 2 in favor of ReðZ Ã 5 Z 2 67 Þ. This procedure will remove the potential singularity due to the factor of Z 1 − Z 2 in the denominator of Eq. (161).
Because λ 6 ¼ λ 7 ¼ 0 in the Φ basis, the only potentially nontrivial phase is the relative phase between m 2 12 and λ 5 . Thus, only one invariant condition is needed to determine whether or not the model exhibits spontaneous CP violation. In the special case of Z 67 ¼ 0 and Z 1 ≠ Z 2 , the conditions for a softly broken Z 2 symmetry given in Eqs. (70) and (71) yield Y 3 ¼ Z 6 ¼ Z 7 ¼ 0 [after using Eq. (24)], corresponding to the (CP-conserving) IDM treated in Sec. VA. In the exceptional region of parameter space defined by Z 67 ¼ 0 and Z 1 ¼ Z 2 , it follows that I 2Y2Z ¼ 0, and one must discover another invariant condition to determine whether the model exhibits spontaneous CP violation.
In order to exhibit cases where Eqs. (156) and (157) are not sufficient to determine whether or not the scalar potential is explicitly CP conserving, we shall make use of the observation of Ref. [59] that it is always possible to perform a basis transformation such that in the transformed basis of scalar fields, λ 7 ¼ −λ 6 (a simple proof of this result is presented in Appendix D). Since basis-invariant quantities can be evaluated in any basis without changing their values, we shall evaluate the four CP-odd invariants listed in Eqs. (152)-(155) in a basis where λ 7 ¼ −λ 6 , where these invariants take on the following simpler forms: where λ 34 ≡ λ 3 þ λ 4 . If I Y3Z ¼ 0, then additional CP-odd invariants may need to be considered. In a Φ basis of scalar fields where λ 6 ¼ −λ 7 , the invariant I Y3Z ¼ 0 if any one of the following four conditions hold: We now examine each of these four cases in turn. Subsequently, we shall examine two additional special cases of interest in which I Y3Z does not vanish.
Case 1.-λ 6 ¼ 0 and λ 1 ≠ λ 2 . This case corresponds to a scalar potential with a softly broken Z 2 symmetry, since λ 6 ¼ λ 7 ¼ 0 in the Φ basis. Equations (162)-(165) yield I Y3Z ¼ I 6Z ¼ 0 and The above results imply that in this case only one invariant quantity, I 2Y2Z , is needed to determine whether the scalar potential is explicitly CP conserving. Indeed, Eq. (166) immediately shows that Eq. (102) is proportional to I 2Y2Z , a result that was obtained above by a rather tedious computation that yielded Eq. (161). Moreover, Eq. (166) provides a very simple method for computing I 2Y2Z in terms of Higgs basis parameters. Using Eqs. (A21) and (A22), it follows that , it follows that if λ 1 ¼ λ 2 and λ 6 ¼ −λ 7 , then these relations hold in any basis of scalar fields. Hence, it follows that Z 1 ¼ Z 2 and Z 6 ¼ −Z 7 . This is the exceptional region of the 2HDM parameter space, which is treated in more detail in Appendix B. In this case, Eqs. (162)-(165) yield I Y3Z ¼ I 2Y2Z ¼ I 6Z ¼ 0 and after evaluating I 3Y3Z in the Higgs basis and employing Eq. (24). If ImðZ Ã 5 Z 6 Þ ¼ 0, then a real Higgs basis exists and both the scalar potential and vacuum are CP conserving. If ImðZ Ã 5 Z 6 Þ ≠ 0 and I 3Y3Z ¼ 0, then the model exhibits spontaneous CP violation. This result provides the previously missing invariant condition for spontaneous CP violation in the exceptional region of the 2HDM parameter space.
To be complete, we examine two further cases in which I Y3Z ≠ 0, where only one CP-odd invariant is needed to determine whether the scalar potential is explicitly CP conserving.
That is, only one invariant quantity, I Y3Z , is needed to determine whether the scalar potential is explicitly CP conserving.
In summary, in generic regions of the 2HDM parameter space, it is sufficient to examine two CP-odd invariant quantities, I Y3Z and I 2Y2Z given in Eqs. (156) and (157) in order to determine whether or not the scalar potential explicitly breaks the CP symmetry. In special regions of parameter space examined in the six cases above, one CPodd invariant quantity is sufficient, although in some cases a third CP-odd invariant, I 6Z , or a fourth CP-odd invariant, I 3Y3Z , is needed to determine the CP property of the scalar potential. In the Lee model of spontaneous CP violation, all four CP-odd invariants vanish, and the scalar potential is explicitly CP conserving, but at least one of the invariants, ImðZ 2 6 Z Ã 5 Þ, ImðZ 2 7 Z Ã 5 Þ, and ImðZ 6 Z Ã 7 Þ is nonvanishing, signaling that in the absence of explicit CP violation, the source of the CP violation must be attributed to the properties of the vacuum.

VIII. CONCLUSIONS
The C2HDM is the most general two-Higgs-doublet model that possesses a softly broken Z 2 symmetry (the latter is imposed to eliminate tree-level Higgs-mediated FCNCs). In the so-called Z 2 basis where the Z 2 symmetry of the quartic terms in the scalar potential is manifestly realized, one can rephase the scalar fields such that the vevs v 1 and v 2 are real and non-negative. After minimizing the scalar potential and fixing v ¼ ðv 2 1 þ v 2 2 Þ 1=2 ¼ 246 GeV, the C2HDM is governed by nine additional real parameters: four scalar masses, one additional squared-mass parameter, Re m 2 12 , tan β ¼ v 2 =v 1 , and three mixing angles arising from the diagonalization of the neutral scalar squared-mass matrix. One sum rule [cf. Eq. (147)] reduces the total number of independent degrees of freedom (including v) to nine.
In this paper, we have provided a basis-invariant treatment of the C2HDM. This involves a number of steps. First, we transformed to the Higgs basis, which is defined up to an arbitrary rephasing of the Higgs basis field H 2 (which by definition possesses no vacuum expectation value). Consequently, the real parameters of the Higgs basis scalar potential are invariant quantities, whereas the complex parameters are pseudoinvariant quantities that are rephased under H 2 → e iχ H 2 . This allows us to easily identify basis-independent quantities, which are related to physical observables of the model. The softly broken Z 2 symmetry constrains the Higgs basis parameters and yields one complex invariant constraint equation. Our results are consistent with the more formal results of Ref. [33] and a recent computation of Ref. [58] that was carried out in a convention of real vevs in the Z 2 basis. For completeness, we have also provided the corresponding constraints if the Z 2 symmetry is extended to incorporate the dimension-two squared-mass terms of the scalar potential.
Having obtained the constraints due to the presence of a softly broken Z 2 symmetry, one can check that the C2HM is governed by nine basis-independent parameters in agreement with our previous counting above. Moreover, one can now identify the behavior of the parameters of the C2HDM under basis transformations. Our analysis revealed that some combinations of the mixing angles α 1 , α 2 , and α 3 and the parameter tan β possess a residual basis dependence due to the freedom to interchange the two complex scalar doublet fields of the C2HDM. In practice, this residual basis dependence is removed by declaring that tan β < 1 corresponds to an enhanced coupling of some of the neutral Higgs bosons to up-type quarks. Having adopted this convention (which is implicitly assumed in the literature but never stated explicitly), the angle parameters of the C2HDM and the parameter tan β are promoted to basisindependent quantities that can be directly related to physical observables.
Our work also resolves an apparent conflict between the number of physical phases in the matrices that diagonalize the squared-mass matrix of the neutral Higgs fields that arise in the two approaches. Indeed, the basis-invariant calculation exhibited in Sec. III involves two basis-invariant angles (θ 12 and θ 13 ), and one unphysical angle (θ 23 ), whereas the calculations in the C2HDM resulting in Eq. (141) yields three physical angles α 1;2;3 . The resolution of this conundrum is associated with the observation that the C2HDM is initially defined in a Z 2 basis where both vevs are real. The constraint imposed by the reality of the two vevs ultimately allows one to ascribe physical significance to the pseudoinvariant quantity, θ 23 . This can be seen directly in Eq. (77) which relates the relative phase of the two vevs to the phase of the pseudoinvariant quantity Z 67 . Thus, by fixing the phase of the two vevs to be zero, one fixes the quantity Z 67 to be real. This leaves a sign ambiguity that is resolved once a twofold ambiguity in the definition of tan β is fixed as indicated above. We have also examined special cases in which Z 67 ¼ 0, where the phase of Z 6 is similarly fixed in the convention of real vevs. 15 The so-called exceptional region of the 2HDM parameter space where Z 1 ¼ Z 2 and Z 67 ¼ 0 requires special attention and is treated in Appendix B.
Finally, we have reanalyzed the techniques for detecting the presence of discrete symmetries originally presented by Lavoura in Ref. [46]. We have obtained results that are in agreement with the corresponding results in Lavoura's paper (after correcting one typographical error in Ref. [46]). In addition, we have extended Lavoura's results in two directions. First, we noted that the invariant constraints obtained by Lavoura do not apply in all parameter regimes of the C2HDM. Some special cases require additional analysis, and we have provided the appropriate modifications in cases that cannot be obtained directly from considerations of the generic regions of the parameter space. Second, Lavoura was only able to obtain one of two relations that must be satisfied in the 2HDM with an explicitly CP-conserving scalar potential but with no (unbroken or broken) Z 2 symmetry that exhibits spontaneous CP violation (i.e., the Lee model [1]). We have provided the second relation that was missed by Lavoura (using the results obtained in Ref. [59]), and we have clarified a number of special cases in which only one relation is sufficient (although that relation is typically not the one found by Lavoura). It is also instructive to apply this analysis in the presence of a softly broken Z 2 symmetry. In doing so, we noted a surprising aspect of a subset of the exceptional region of the parameter space where no Z 2 basis exists where all the scalar potential parameters are real, and yet the corresponding 2HDM is CP conserving.
In conclusion, the basis-independent formalism possesses many advantages. For example, just like covariance in relativistic theories where an equation can be checked by ensuring that both sides of the equation behave similarly under Lorentz transformations in the same way, the basisindependent formalism affords similar benefits. Indeed, errors in numerous equations in this paper were avoided by such considerations. In addition, due to the close connection of basis-independent quantities to physical observables, one obtains confidence in appreciating the significance of the relations among the various 2HDM parametrizations. 15 If Z 6 ¼ Z 7 ¼ 0, then the model reduces to the IDM discussed in Sec. VA. This model is necessarily CP conserving and thus is not of further interest to us in this work.
We hope that the application of basis-independent methods in the analysis of the C2HDM presented in this paper has contributed to a better understanding of this model and will be useful in future phenomenological studies of CP-violating Higgs phenomena. Since the scalar doublets Φ 1 and Φ 2 have identical SUð2Þ × Uð1Þ quantum numbers, one is free to define two orthonormal linear combinations of the original scalar fields. The parameters appearing in Eq. (2) depend on a particular basis choice of the two scalar fields. Relative to an initial (generic) basis choice, the scalar fields in the new basis are given by

ACKNOWLEDGMENTS
up to an overall complex phase factor e iψ that has no effect on the scalar potential parameters, since this corresponds to a global hypercharge transformation. With respect to the new Φ 0 basis, the scalar potential takes on the same form given in Eq. (2) where s β ≡ sin β, c β ≡ cos β, etc., and We shall make use of Eqs. (A2)-(A11) to write out the explicit relations between the scalar potential parameters of a generic basis and the Higgs basis. We can employ the unitary matrix given by Eq. (A1), where and v 1 and v 2 are the magnitudes of the vevs of the neutral components of the Higgs fields in the generic basis, defined in Eq. (3). In particular, are non-negative quantities, which implies that we may assume that 0 ≤ β ≤ 1 2 π. It follows that the invariant Higgs basis fields defined in Eq. (25) are given by Consequently, we can identify the primed scalar potential parameters with the scalar potential coefficients of the Higgs basis, fH 1 ; H 2 g, as specified in Eq. (28).
In the exceptional case of Z 1 ¼ Z 2 and Z 7 ¼ −Z 6 , it follows from Eqs. (A21)-(A27) that λ 1 ¼ λ 2 and λ 7 ¼ −λ 6 in all scalar field bases. 17 In this appendix, we show that in this exceptional case, there exists a Φ basis in which λ 6 ¼ λ 7 ¼ 0. That is, there exists a scalar field basis where the Z 2 symmetry of the quartic terms of the scalar potential is manifest.
If Z 6 ¼ 0, then the scalar potential in the Higgs basis manifestly exhibits the Z 2 symmetry, so we shall henceforth assume that Z 6 ≠ 0, in which case we may write Z 6 ≡ jZ 6 je iθ 6 . It is convenient to introduce Under the basis transformation Φ a → U ab Φ b , where U is given by Eq. (72), it follows that e iξ 0 → −e iξ 0 , in light of Eq. (74). That is, ξ 0 is only determined modulo π, corresponding to the twofold ambiguity anticipated above. Inserting e iξ ¼ e iξ 0 Z Ã 6 =jZ 6 j into Eqs. (B2) and (B3) yields We now consider two cases. First, if we assume that ImðZ Ã 5 Z 2 6 Þ ¼ 0 then sin ξ 0 ¼ 0 is a solution to Eq. (B5), which implies that cos ξ 0 ¼ AE1; the twofold ambiguity was anticipated in light of the comment following Eq. (B4). 17 We note in passing that the exceptional region of parameter space where λ 1 ¼ λ 2 and λ 7 ¼ −λ 6 was identified in Ref. [47] as the conditions for a softly broken CP2-symmetric scalar potential, where CP2 is the generalized CP transformation, Φ 1 → Φ Ã 2 and Φ 2 → −Φ Ã 1 . In general, dimension-two soft CP2-breaking squared-mass terms are present and violate the CP2-symmetric conditions, m 2 11 ¼ m 2 22 and m 2 12 ¼ 0. However, the CP2 symmetry is also violated by the dimension-four Yukawa interactions, which constitute a hard breaking of the symmetry [64]. Consequently, the exceptional region of the parameter space is unnatural and must be regarded as finely tuned. Inserting cos ξ 0 ¼ AE1 into Eq. (B6) then yields a quadratic equation for cot 2β ¼ c 2β =s 2β , As expected from Eq. (74), changing the sign of cos ξ 0 from þ1 to −1 simply changes the sign of cot 2β. Moreover, Eq. (B7) possesses two real roots whose product is equal to −1. This observation implies that if β is one solution of Eq. (B7), then the second solution is β AE 1 4 π (where the sign is chosen such that the second solution lies between 0 and Since the coefficient of cos ξ 0 is generically nonzero, it follows that cos ξ 0 ¼ 0. Plugging this result back into Eq. (B5) yields cos 2β ¼ 0. Hence, ðβ ¼ 1 4 π; ξ 0 ¼ 1 2 πÞ and ðβ ¼ 1 4 π; ξ 0 ¼ 3 2 πÞ are also solutions to Eqs. (B5) and (B6) when ImðZ Ã 5 Z 2 6 Þ ¼ 0. These two solutions are again related by the basis transformation Φ a → U ab Φ b , where U is given by Eq. (72).
We end this appendix with a discussion of spontaneous CP violation. Starting from Eq. (94), we can eliminate ReðZ 5 e 2iξ Þ and ImðZ 5 e 2iξ Þ by employing Eqs. (B2) and (B3). If we denote R ≡ ReðZ 6 e iξ Þ ¼ jZ 6 j cos ξ 0 and I ≡ ImðZ 6 e iξ Þ ¼ jZ 6 j sin ξ 0 , the end result is where λ 5 and m 2 12 are parameters of the scalar potential in the Z 2 basis, and β and ξ are solutions to Eqs. (B2) and (B3).

APPENDIX C: BASIS-INVARIANT CONDITIONS FOR THE Z 2 SYMMETRY REVISITED
In Sec. V, conditions for the presence of a Z 2 symmetry in the scalar potential (which may or may not be softly broken) were derived. These conditions were expressed in terms of the Higgs basis scalar potential parameters and were invariant with respect to an arbitrary rephasing of the Higgs basis field H 2 that defines the set of all possible Higgs bases. In Ref. [33], a set of manifestly basis-invariant expressions were presented which were sensitive to the presence of a Z 2 symmetry in the 2HDM scalar potential. 18 In this appendix, we demonstrate that if these expressions are evaluated in the Higgs basis, then the results of Sec. V are recovered.
Hence, it follows that the condition for the existence of a softly broken Z 2 symmetry that is manifest in some scalar field basis is given by [33] ½Z ð1Þ ; Z ð11Þ ¼ 0: where Z 34 ≡ Z 3 þ Z 4 and Z 67 ≡ Z 6 þ Z 7 . 18 The group theoretic analysis of the 2HDM scalar potential developed in Ref. [36] and the geometric picture of Ref. [40] provide alternative approaches for obtaining a basis-independent condition for the presence of a softly broken Z 2 symmetry. Thus, we arrive at two conditions for the Higgs basis scalar potential parameters that imply the existence of a softly broken Z 2 symmetry, ðZ 1 − Z 2 Þ½Z 34 Z 67 − Z 2 Z 6 − Z 1 Z 7 þ Z 5 Z Ã 67 − 2Z 67 ðjZ 6 j 2 − jZ 7 j 2 Þ ¼ 0; ðC9Þ which reproduce the results of Eqs. (89) and (88), respectively. If the Z 2 symmetry is exact, then in addition to Eq. (C5), one must impose a second condition [33], in the Φ 0 basis that are related to the corresponding coefficients of the Φ basis by the U(2) transformation given by Eq. (A1). It then follows that We assume that λ 7 ≠ −λ 6 . The goal of this appendix is to show that there exists a choice of β and ξ such that λ 0 7 ¼ −λ 0 6 . Consider the diagonalization of the matrix Z ð1Þ ab ≡ δ cd Z acdb , which is explicitly given by Under a basis transformation, Φ a → Φ 0 a ¼ U ab Φ b , it follows that Z ð1Þ ab → U ac Z ð1Þ cd U † db , where the unitary matrix U is given by Eq. (A1). It is possible to choose η, β, and ξ such that where the λ AE are the eigenvalues of Z ð1Þ , In determining the diagonalization matrix U, one is free to take η ¼ 0 without loss of generality. 21 By convention, we shall also take 0 ≤ β ≤ 1 2 π and 0 ≤ ξ < 2π. It is convenient to introduce the notation, λ 67 ≡ λ 6 þ λ 7 ≡ jλ 67 je iθ 67 : It is then straightforward to check that the diagonalization of Z ð1Þ is achieved if U is given by Eq. (A1) with η ¼ 0, ξ ¼ −θ 67 , and Indeed, by inserting ξ ¼ −θ 67 into Eq. (D1) and using Eq. (D6), one readily verifies that after making use of Eq. (D6) in the final step. Hence, we conclude that λ 0 6 þ λ 0 7 ¼ 0. That is, it is always possible to find a basis change such that λ 0 6 ¼ −λ 0 7 . For the record, we verify the diagonalization of Z ð1Þ by computing Note that this result is consistent with Eq. (D6).

APPENDIX E: MIXING OF THE NEUTRAL HIGGS SCALARS IN THE Φ BASIS
In Sec. III, the mixing of the neutral Higgs scalars was obtained in the Higgs basis. In this appendix, we examine the mixing in the Φ basis, where the scalar potential is given by Eq. (2). In the Φ basis, the two scalar doublet fields can be parametrized by 21 In light of Eq. (D3), it follows that the columns of U −1 ¼ U † are the normalized eigenvectors of Z ð1Þ , which are only defined up to an overall complex phase. Hence, one is free to rephase the second row of Eq. (A1) in order to set η ¼ 0.