CP violation in 2HDM and EFT: the ZZZ vertex

We study the CP violating ZZZ vertex in the two-Higgs doublet model, which is a probe of a Jarlskog-type invariant in the extended Higgs sector. The form factor $f_4^Z$ is evaluated at one loop in a general $R_\xi$ gauge and its magnitude is estimated in the realistic parameter space. Then we turn to the decoupling limit of the two-Higgs doublet model, where the extra scalars are heavy and the physics can be described by the Standard Model supplemented by higher-dimensional operators. The leading operator contributing to $f_4^Z$ at one loop is identified. The CP violating ZZZ vertex is not generated in the effective theory by dimension-8 operators, but instead arises only at the dimension-12 level, which implies an additional suppression by powers of the heavy Higgs mass scale.


Introduction
The LHC experiments announced in 2012 the discovery of a neutral scalar (h) of mass 125 GeV [1,2], consistent with the 1964 prediction of a Higgs Boson as a by-product of the spontaneous symmetry breaking of a gauge symmetry [3][4][5][6]. This opened a very exciting program addressing two fundamental questions: i) how many scalars are there?, and ii) do the couplings of the 125 GeV state conform to the prediction in the Standard Model (SM) of electroweak interactions [7,8]? Thus far, there is no definite sign of an inconsistency with the SM.
A model independent way to interpret the data is provided by the SM effective field theory (SMEFT), where one allows for all operators constructed from the SM fields, organized in an expansion A complementary approach consists in investigating the constraints that the data place on well motivated theories. One simple example consists in adding one more scalar doublet to the SM. Two Higgs doublet models (2HDMs) are interesting because they contain many properties which one may find in more complicated theories, such as the presence of extra neutral scalars, charged scalars, CP-odd or admixtures of CP-even and CP-odd scalars, the possibilities for spontaneous CP violation or flavour changing neutral scalar interactions, among others [9,10]. Here we focus on the so-called "complex 2HDM" (C2HDM) [11][12][13][14][15][16][17][18].
2HDMs can be approximated by the SMEFT at energies below the mass scale of the new scalars, where the indirect effects of the new scalars are represented by a tower of the higher-dimensional operators in the Lagrangian. There has been some interest in the matching between the 2HDM parameters and the SMEFT Wilson coefficients [19][20][21][22][23][24][25]. This exercise allows one to get some intuition about the pattern of operators expected from realistic extensions of the SM, and to identify the leading new physics effects in a model-independent language. In this paper we discuss how CP violation of the C2HDM is manifested in the SMEFT. More precisely, we concentrate on the CP violating ZZZ vertex, which appears at one loop in the C2HDM. This is an especially interesting observable because it measures directly a Jarlskog-type invariant in the Higgs-gauge sector [26], first introduced in [27,28]. At the technical level, the C2HDM computation involves loops with both heavy and light particles, which require special care when matching to the low-energy effective theory [29][30][31][32][33][34]. In fact, we will show that the effective description of the CP violating ZZZ vertex is quite non-trivial in this case. Generally, the lowest order in the SMEFT expansion where the ZZZ vertex may appear is dimension-8 (O(Λ −4 )) [35,36]. However, in turns out that in the SMEFT matched to the C2HDM at one loop the CPviolating ZZZ vertex is only generated at (O(Λ −8 )), that is by a dimension-12 operator.
Our paper is organized as follows. In section 2 we review the observable form factors associated with the ZZZ vertex. Our C2HDM notation and conventions are summarized in section 3. Section 4 contains the calculation of the CP-violating contribution to the ZZZ vertex in a general R ξ gauge, which we compare with that in the previous literature [26]. The corresponding calculation and the CP violating operator in the SMEFT is discussed in section 5. Finally, section 6 presents our conclusions. Some technical details concerning approximation of the loop integrals using the method of regions are given in Appendix A, while derivation of the CP-violating dimension-12 SMEFT operators using gauge invariant functional methods is given in Appendix B.

The ZZZ vertex
We start by reviewing the formalism to describe the effective Z 3 vertex [37,38]. Consider the diagram in figure 1 with two on-shell Z bosons characterized by outgoing 4-momenta p 1 , p 2 and polarization vectors (p 1 ), (p 2 ), and an off-shell Z boson with the incoming The blob in figure 1 may represent a contact interaction or particles running in loops. The Lorentz and Bose symmetries constrain the Z 3 vertex function Γ µαβ to the following form: In the first line we have pulled out a function of q 2 from the form factors so as to match the standard notation of Ref. [38]. 1 The form factor f Z 4 (q 2 ) corresponds to a C-odd and P -even (thus CP -odd) interaction. One way to see this is to note that an effective Lagrangian with the Z 3 interaction, 2) leads to the tree-level vertex in eq. (2.1) with f Z 4 (q 2 ) =κ ZZZ . The CP properties are then easily obtained given C acting as Z µ → −Z µ and P acting as By similar argument one shows that f Z 5 (q 2 ) corresponds to a C-odd and P -odd (thus CP -even) interaction.
It is important to stress that the full vertex Γ µαβ is not an observable. Nevertheless, the form factors f Z 4 and f Z 5 can be related to observable quantities in the following sense. Consider the amplitude to produce a pair of Z bosons. This process will receive a contribution from the diagram with an intermediate off-shell Z boson in the s-channel: where j µ is the f current to which the Z boson couples in the Lagrangian. We assume that j µ is conserved, q µ j µ = 0, which is the case in the relevant situation of qq and e + e − collisions in the limit where the fermions are treated as massless. Moreover, p α 1 α (p 1 ) = p β 2 β (p 2 ) = 0. Then the s-channel part of the ZZ production amplitude reduces to 3) As long as it is possible to isolate the s-channel production, the form factors f Z 4 and f Z 5 are measurable. In particular, f Z 4 can be related to experimentally observable CP asymmetries in ZZ production in colliders [26,39]. On the other hand, the remaining form factorsf i (q 2 ) in eq. (2.1) are not observable; in fact, they may be gauge-dependent in specific calculations.

Complex Two-Higgs Doublet Model
In this section we summarize the salient features of the C2HDM, for a review see e.g. [9,10]. The most general renormalizable scalar potential is where Φ 1 and Φ 2 are complex scalar SU (2) L doublets, with vacuum expectation values (VEVs) v 1 / √ 2 and v 2 / √ 2. The parameters m 2 11 , m 2 22 , and λ 1 . . . λ 4 are real parameters, while m 2 12 and λ 5 . . . λ 7 can be complex. In general, both Φ 1 and Φ 2 can have Yukawa couplings to all the SM fermions. However, this leads to flavour changing neutral scalar interactions (FCNSI), which are tightly constrained by experiment. As a result, it is usually assumed that there is a Z 2 symmetry [40,41], acting on the scalars as with appropriate transformations on the fermions, guaranteeing that fermions of a given charge couple exclusively to one of the two scalar fields. Of course, one can perform a basis change on the scalar fields. The couplings in the scalar potential and in the Yukawa interactions, as well as the specific implementation of the Z 2 symmetry, change from one basis to the next; but any physical observable cannot depend on such a choice. We denote by the "Z 2 basis", the basis in which the transformation has the specific form in eq. (3.2). For an exact Z 2 symmetry, m 2 12 , λ 6 , and λ 7 vanish. Since the absence of m 2 12 precludes a decoupling limit [42], one usually breaks it softly through m 2 12 = 0. If arg(λ 5 ) = 2 arg(m 2 12 ), then we may take both couplings real, and the potential preserves CP. When v 1 and v 2 are also real, there is no CP violation (explicit or spontaneous) and the model is known as the "real 2HDM". In contrast, if arg(λ 5 ) = 2 arg(m 2 12 ), then the potential violates CP explicitly; this is known as the "complex 2HDM" (C2HDM) [11][12][13][14][15][16][17][18]. Here, we will choose a basis where v 1 and v 2 are real, without loss of generality.
It is convenient to introduce the "Higgs basis" [27,28], defined as the basis where only the first scalar has a VEV. This is obtained through the unitary transformation The doublets in the Higgs basis may be parametrized as where G ± and G 0 are the Goldstone bosons which, in the unitary gauge, are absorbed as the longitudinal components of W ± and Z, while H ± are the charged scalars. The scalar potential in the Higgs basis has the form where we follow the notation of [43]. The parameters Y 1,2 and Z 1,2,3,4 are all real; the others are, in general, complex. Note that, in the Higgs basis, H 1 and H 2 are not eigenstates of the Z 2 symmetry and, therefore, the cross terms proportional to Z 6 and Z 7 are in general present. The stationarity conditions in the Higgs basis read The last equation means that only Z 5 , Z 6 , and Z 7 are independently complex. Thus, all sources of CP violation in the Higgs potential must be related to the invariant quantities Im(Z 7 Z * 6 ), Im(Z 2 7 Z * 5 ), and Im(Z 2 6 Z * 5 ) [27]. 2 The dictionary between the Z 2 basis and the Higgs basis for the quadratic terms Y i is The generated cross-term can be present even if m 2 12 = 0, unless m 2 11 = m 2 22 (the masses of Φ 1 and Φ 2 being equal). Similarly, for the quartic terms Z i : 14) where λ 345 = λ 3 + λ 4 + Re(λ 5 ). What is relevant is that not all Z i are independent, as they satisfy the relations: , and the first two equations are those relevant for the real 2HDM discussed in [43]. Using these relations we can eliminate for example Z 6 and Z 7 , and express our results in terms of the remaining Z i . Thus, in the C2HDM, all CP violation invariants in the Higgs potential are proportional to a single phase, which comes from Im(m 2 12 λ * 5 ) in the original basis. One goes from the neutral scalars {h, R, I} written in the Higgs basis into the neutral scalar mass basis where and s i = sin α i and c i = cos α i (i = 2, 3). Similarly,s 1 = sinα 1 andc 1 = cosα 1 , wherẽ We have defined T to agree with the definition in ref. [10]. In the C2HDM one usually defines a matrix R such that The angles α 1 , α 2 , and α 3 were introduced in [13], and, without loss of generality, may be restricted to The real 2HDM may be obtained by setting s 2 = s 3 = 0, and the usual α = α 1 − π/2. In the Higgs basis h is the only scalar field that has a coupling to two gauge bosons, and it coincides with the SM one g sm hV V . Thus, In the C2HDM, 22) and the coupling of the lightest Higgs to gauge bosons is given by and the coupling of each scalar mass eigenstate with two vector bosons must be smaller than the corresponding coupling in the SM. This property generalizes to any multi Higgs doublet model, so that a value well above the SM would exclude the SM and also all such models. Conversely, since the measurements are consistent with a coupling of the 125 GeV scalar with two gauge bosons very close to the SM value, then the mixing angles in T must be such that this scalar almost coincides with the h in the Higgs basis. This translates into the so-called alignment limit of and in the C2HDM and real 2HDM, respectively.

CP-violating ZZZ vertex in C2HDM
We turn to the calculation of one-loop contributions to the ZZZ vertex in the C2HDM. The goal is to determine the CP-violating form factor f Z 4 (q 2 ) defined by eq. (2.1) (the other physical form factor f Z 5 (q 2 ) vanishes at one loop). To that end, we can neglect all the Lorentz structures that are not of the form η µα p β 1 or η µβ p α 2 . A good consistency check is to verify that these two Lorentz structures have the same coefficient, as the result should be invariant for the exchange (p 1 , α) ↔ (p 2 , β). We perform the calculation in a general R ξ gauge, and verify gauge invariance at the end of the calculation. We express the results in terms of the Passarino-Veltman (PV) functions [44], following the LoopTools conventions [45]. To evaluate the loop integrals we use the Mathematica packages FeynCalc [46], and we cross-checked the result with Package-X [47]. Our final result disagrees with the previous literature [26], therefore we will present in some detail the intermediate steps of our calculation.

Couplings and Propagators
For our calculation, we need the following vertices [48], where c W = cos θ W , all momenta are incoming, and the i of the Feynman rules is already included. 3 The coefficient x i above is related to the C2HDM parameters as That is, x i coincide with the T 1i in eq. (3.22). As we are doing the calculation in a general R ξ gauge, we also need the propagators for the Goldstone G 0 and the Z in this gauge [49], (4.6)

Diagrams with
We start with the diagrams containing only Higgs bosons in the internal lines, as shown in figure 2. Because of the coupling structure in eq. (4.1), all the three scalars have to be different. We get the same result as in Ref. [26], We consider now the diagram with one Goldstone boson in one of the internal lines (there are no diagrams with either two or three Goldstone bosons), as shown in figure 3. There are two more diagrams with the G 0 in the other internal lines. Each of them will have all the possible combinations of h i , h j . As before, due to the coupling structure in eq. (4.1) and eq. (4.2), we must have i = j in all possible combinations. In the R ξ gauge, we get the result, which agrees with Ref. [26] in the Feynman gauge limit ξ = 1.

Diagrams with
(4.9) In the limit ξ = 1 this result differs in the overall sign from that in Ref. [26].

Final Result
Summing the different contributions, , we find that the ξ dependent parts of Eqs. (4.8) and (4.9) cancel out, ensuring gauge invariance of the final result. Also the antisymmetry of each term in eqs. (4.7), (4.8) and (4.9) implies that the divergences originating from the PV function C 001 cancel and the final result is finite. All in all, the CP violating Z 3 form factor expressed by the PV functions takes the form The dependence of the form factor on q 2 is illustrated in figure 5 for several choices of the heavy scalar spectrum. The order of magnitude of |f Z 4 | that can be achieved in the realistic parameter space of the C2HDM is shown in figure 6, reaching the values of order 10 −5 . For comparison, the recent ATLAS [50] and CMS [51] analyses of ZZ production at the LHC set upper bounds on |f Z 4 | (assumed real) on the order of 10 −3 . When considering a generic framework beyond the SM, one must check whether effects other than f Z 4 may contribute to the actual experimental observable being measured (and from which f Z 4 is inferred). For example, one can see from Fig. 1 in Ref. [51] that there is a contribution from h 125 → ZZ to the four lepton events from which f Z 4 is extracted. For the SM Higgs this is not a problem, since this is merely an order 5% contribution to the cross section and, moreover, the measurement of f Z 4 is made by requiring in addition that each Z in the final state has a mass in the range 60-120 GeV. But it could be a concern if a heavier Higgs were to decay into ZZ, competing with the signal from the ZZZ vertex. This problem is  Figure 6. Scatter plots showing the absolute value of the CP violating form factor f Z 4 (q 2 ) for two values of q 2 for points in the parameter space of the type-1 C2HDM satisfying theoretical (unitarity, bounded from below) and experimental (LHC Higgs, electric dipole moments, and electroweak precision measurements) constraints. mitigated in the C2HDM because of a combination of two facts. First, we know from the h 125 → ZZ measurements that the corresponding coupling in the C2HDM lies very close to the SM value (the so-called alignment limit). Second, the sum rule in eq. (3.24) guarantees that any heavier scalar will have a very small coupling to ZZ. Nevertheless, once statistics improve at LHC, a precise constraint on f Z 4 can best be achieved by a detailed simulation of the C2HDM within the experimental analysis of the collaborations, which is beyond the scope of this work. Our results for the maximum of |f Z 4 | are slightly below those reported in Ref. [26]. This is mainly due to the effect of including in our scan the bound on the electron EDM [52]. The sign difference that we have found does not affect much the absolute value, because the diagram where it occurs is typically the dominant one (in the gauge ξ = 1) [26].
For future reference, we also give the final form of the Z 3 vertex before evaluating the loop integrals: where ILS stands for (in general divergent and gauge dependent) irrelevant Lorentz structures that do not contribute to the observable form factors. In this form the vertex is manifestly symmetric under interchanging p 1 ↔ p 2 , α ↔ β. Performing the momentum integral and extracting from the coefficient of the tensor structure η µα p 1,β + η µβ p 2,α , cf.eq. (2.1), one obtains the result in eq. (4.10).

CP-violating ZZZ vertex in SMEFT
In this section we discuss how the CP-violating ZZZ vertex arises in the low-energy EFT where the heavy non-SM scalars of the C2HDM are integrated out. We denote m 1 = m h = 125 GeV, m 2 = m H , m 3 = m 2 H + δ 2 with δ ∼ v, and we are interested in the decoupling limit m H m h . In such a case, only the SM degrees of freedom are available at the energies E ∼ v m H . In this regime the dynamics is described by the SMEFT, with the SM Lagrangian augmented by higher-dimensional operators. At the level of dimension-6 operators the matching of the SMEFT Lagrangian to the 2HDM UV completion was discussed e.g. in Refs. [19-21, 23, 25, 53, 54]. However, within the EFT framework studied in these references, there is no source of CP violation contributing to the ZZZ vertex. Below we will identify the higher-dimensional CP-violating operator and discuss how the ZZZ vertex is generated.
The first step toward this goal is to expand the form factor f Z 4 (q 2 ) in powers of 1/m H . In principle, one could expand the result in eq. (4.9) using the known expressions for the PV functions. In practice, this path is difficult due to a complicated form and nonanalytic behavior of the PV functions involved. Instead, we find it easier to apply the method of regions [55]. A loop integral containing two disparate mass scales m light m heavy can be calculated by A) expanding the integrand for k ∼ m light and performing the integral, B) expanding the integrand for k ∼ m heavy and performing the integral, and then adding these two contributions together. An important point here is that both A) and B) have a clear counterpart on the EFT side where the scales m heavy are integrated out. Namely, A) corresponds to 1-loop Feynman diagrams with the light particles in the loop and an insertion of tree-level-generated effective operators, while B) corresponds to tree-level diagrams with an insertion of operators whose Wilson coefficients are suppressed by a loop factor. Applying the method of regions to the integrals in eq. (4.11) we find that the leading contributions to the sum of the integrals are O(m −4 H ) and come from the diagrams with one heavy scalar and two SM particles in the loop (h, Z, or the corresponding Goldstone boson). Moreover, we find that it is the soft region A), k m H , which dominates. Other diagrams and integration regions contribute only at O(m −6 H ) or higher. This immediately tells us that, in the EFT for the 2HDM, the CP-violating ZZZ vertex is generated at one loop via diagrams with h, Z in the loop and an insertion of a tree-level-generated effective operator.
In Appendix A we give the details of the method of regions applied to eq. (4.11), albeit for the sake of brevity we work there in the simplified limit m h → m Z . Here we write down the leading contribution to the CP-violating form factor f Z 4 for a general m h , valid for m h m H and q 2 m 2 H : where we introduced the DiscB function as defined in [47]: We have checked numerically that eq. (5.1) correctly reproduces the f Z 4 form factor in the C2HDM in the decoupling limit. This is illustrated in figure 7 where, as long as q 2 m 2 H , both the real and imaginary parts of the two results converge as we increase m H .
We find that the CP violating Z 3 form factor is strongly suppressed in the decoupling limit. First, the momentum integration brings the suppression factor 1/m 4 H in eq. (5.1), which is stronger than the naive estimate from dimensional analysis due to cancellations between diagrams with h 2 and h 3 . Moreover, in the decoupling limit the mixing angles  Red: the normalized form factorf Z 4 (q 2 ) as defined in eq. (4.10). We show the dependence on the second neutral Higgs mass m H for two values of q 2 . For q 2 = 1 TeV we also show separately the real and imaginary parts (for q 2 = 200 GeV the form factor is purely real). The heaviest neutral Higgs mass is assumed to be m 2 H + v 2 . Blue: the same observable calculated in the matched SMEFT (cf.eq. (5.1)).
between the Higgs scalars are also suppressed: All in all, we find that f Z 4 ∼ 1 (16π 2 )m 8 H in the decoupling limit. This tells us that, in the SMEFT matched to C2HDM at one loop, the CP violating ZZZ vertex arises from a dimension-12 operator! Note that f Z 4 (q 2 ) has an imaginary part for q 2 > (m Z + m h ) 2 . Indeed, the DiscB function has an imaginary part and a branch cut for p 2 > (m 1 + m 2 ) 2 , while it is real for p 2 < (m 1 + m 2 ) 2 . This confirms the argument above eq. (5.1) that the ZZZ vertex should arise from loop diagrams in the EFT where Z and h can simultaneously go on-shell and that the CP-violating dimension-12 operator should be present in the EFT matched at tree level to the C2HDM. Figure 8. A one-loop diagram contributing to the Z 3 vertex in the EFT. There are two other diagrams corresponding to permutations of the external legs.
In Appendix B, using the functional integral methods [30,33], we sketch how to systematically derive the tree-level EFT Lagrangian for the C2HDM in the manifestly SU (3) × SU (2) × U (1) gauge invariant language up to an arbitrary order in 1/m H expansion. We follow that procedure and find that the leading CP-violating operator in the bosonic sector indeed occurs at O(m −8 H ). The operator in question is identified as where X 0 ≡ H † H − v 2 /2. Expanding the Higgs doublet around its VEV, the operator in eq. (5.4) yields (among others) the Z 3 h interaction term: which is P-even and C-odd (thus CP-odd). Hence, the dimension-12 operator in eq. (5.4) leads to CP violation when Im (Z * 5 Z 2 6 ) = 0. An equivalent way to derive the effective interaction in eq. (5.5) from the C2HDM is to consider a tree-level exchange of the heavy Higgs scalars between the ZZ and Zh vertices.
In the presence of the CP-violating interaction in eq. (5.5), one indeed finds 1-loop contributions to the Z 3 vertex, see the diagrams in figure 8. Working in the unitary gauge we find Evaluating the integral in Package-X [47] and extracting f Z 4 , we exactly recover the result in eq. (5.1). This confirms that the dimension-12 operator in eq. (5.4) fully accounts for the leading 1/m H behavior of the ZZZ vertex in the C2HDM at one loop in the decoupling limit.
It may be surprising that the Z 3 vertex in the EFT arises only at the dimension-12 level. After all, there are lower-dimensional CP-violating operators that lead to Z 3 interactions. It is well known that the Z 3 vertex cannot be generated by dimension-6 operators, however it does arise from the dimension-8 operator and other similar operators with B µν → W i µν [36]. These operators lead to the contact Z 3 interaction in eq. (2.2), and thus directly contribute to the f Z 4 form factor without going through a loop diagram. However, one can prove that the dimension-8 operators like the one in eq. (5.7) cannot be generated from the C2HDM at one loop. The underlying reason is that in the C2HDM all new CP-violating effects are proportional to the Jarlskog-type invariant [27]: In the decoupling limit, eq. (5.3) shows that this invariant is proportional to the Higgs potential couplings Z i in the 3rd power. The rest follows from power counting using the Planck constant as a proxy [56][57][58][59][60]. Reinstating the Planck constant in the path integral, Dφe i d 4 xL/ , the Lagrangian should carry the dimension [L] = 1 . One can assign the power 1/2 to each propagating field, the power 1−n/2 to the coupling multiplying the term with n fields in the Lagrangian, and the power 1 for each loop factor 1 16π 2 . In this scheme, the electroweak couplings carry the power [g] = −1/2 , while for the quartic Higgs couplings [Z i ] = −1 . It follows that the CP violating invariant is proportional to J CP ∼ −3 . On the other hand, the Wilson coefficient of the dimension-8 operator in eq. (5.7) should have [c 8 ] = −1 . If that operator arises at l loops in the C2HDM then c 8 ∼ g 2 J CP (16π 2 ) l and the power counting fixes l = 3: the dimension-8 operator in eq. (5.7) cannot appear before the 3 loop level in the matching of the SMEFT to the C2HDM. The same power counting shows that the dimension-12 operator in eq. (5.7) is allowed at tree level.

Conclusions
In this paper we have studied the CP violating triple-Z vertex in the C2HDM and in its effective description within the SMEFT framework. From the point of view of the highenergy theory, the ZZZ vertex probes one of the two independent Jarlskog-type invariants in the extended Higgs sector. The leading contributions arise from triangle one-loop diagrams with both SM particles and the new Higgs scalars. We obtained the CP violating form factor f Z 4 in a general R ξ gauge, thus demonstrating the gauge invariance of the result and reassessing previous calculations in the literature. Starting from the (complicated) full result, we extracted an analytic approximation valid in the decoupling limit when the mass scale m H of the heavy scalars is much larger than m Z and the momentum flowing through the vertex. Given that approximation, we were able to identify the operators and diagrams responsible for the generation of the ZZZ vertex in the low-energy effective theory where the heavy scalars are integrated out. Even though the ZZZ vertex can in principle be generated by dimension-8 operators in the SMEFT, such contributions are absent in the effective theory matched to the C2HDM at one loop. This fact may be surprising at first, but it follows from simple power counting, given the dependence of the Jarlskog invariants on the masses and couplings of the C2HDM. Instead, we found that the CP violating ZZZ vertex appears in the effective theory only at the level of dimension-12 operators. In practice, this means the CP violating effects in diboson production will be extremely suppressed (by a loop factor multiplied by (v/m H ) 8 ) if the mass scale of the heavy Higgs partners is well above the weak scale.

Acknowledgments
We are grateful to Celine Degrande and to Alexander Savin for discussions. This work was funded in part by a bilateral agreement, sponsored in Portugal by FCT -

A Effective ZZZ vertex via method of regions
In this appendix we discuss how to isolate the leading contribution to the CP-violating ZZZ form factor f Z 4 (q 2 ) in the C2HDM in the limit where the extra scalars are much heavier than the Higgs boson. To this end we will utilize the method of regions [55]. A loop integral with two disparate mass scales m light m heavy can be calculated by A) expanding the integrand for soft momenta k ∼ m light and performing the integral, B) expanding the integrand for hard momenta k ∼ m heavy and performing the integral, and adding the contributions A) and B). Note that the separate soft and hard contributions may be UV or IR divergent. If that is the case, the integrals have to be regulated, with the regulator dependence canceling out when the full result is finite. In the following we will implicitly use the dimensional regularization which is convenient because the EFT expansion is not complicated by the presence of massive regulators.
We apply this method to the ZZZ vertex in the C2HDM, whose integral representation is given in eq. (4.11). For presentation purposes, in this appendix we work in the unphysical limit m 1 = m Z . The reason is that in this limit the ZZZ vertex simplifies considerably, as the first four lines in eq. (4.11) cancel against each other: Taking that simplified limit allows us to illustrate the gist of the argument. The discussion for the general case m 1 = m h is completely analogous, but much more tedious and paperconsuming. We can rewrite eq. (A.1) as sums the contributions with one heavy scalar in the loop: while I HH β sums the contributions with two heavy scalars in the loop: . Solving the equation of motion and deriving L EFT can be readily performed perturbatively in the 1/Y 2 expansion. As explained in section 5, power counting arguments show that the relevant CP-violating operators arise only at the level at O(Y −4 2 ), corresponding to dimension-12 operators. Deriving the complete effective Lagrangian up to dimension 12 would be quite a task. For the present purpose, we focus only on its small fragment containing purely bosonic CP-violating interactions. We know that these have to be proportional to the Jarlskog-like invariant Im (Z * 5 Z 2 6 ). Therefore we will only trace the terms in L EFT that contain Z 5 or Z 6 , and ignore everything else. We thus consider the C2HDM Lagrangian where the dots stand for other terms in the Higgs potential, gauge kinetic terms, and all fermionic terms, which are not relevant for the present discussion. We work in the Higgs basis where Y 3 = −Z 6 v 2 /2, H 1 = v/ √ 2, H 2 = 0. Then Y 3 + Z 6 |H 1 | 2 = Z 6 X 0 , where we defined X 0 ≡ |H 1 | 2 − v 2 /2. The equation of motion for H 2 takes the form Plugging that solution back into eq. (B.1) one gets the EFT Lagrangian in 1/Y 2 expansion: L EFT = L SM + ∞ n=1 Y −n 2 L (2n+4) , where each term contains local operators composed of H 1 and its (covariant) derivatives. In the low-energy theory H 2 is integrated out and H 1 remains as the only doublet scalar, so in the following we relabel H 1 → H. It is now a trivial if tedious exercise to determine the EFT operators L (2n+4) at each given order. For example, this procedure yields L (6) ⊃ |Z 6 | 2 X 2 0 |H| 2 which shifts the triple Higgs boson coupling away from the SM prediction, or L (8) ⊃ |Z 6 D µ (X 0 H)| 2 which renormalizes the Higgs boson kinetic terms, thus uniformly shifting all the Higgs boson couplings. At n = 8 we also encounter a term proportional to Z * 5 Z 2 6 , L (8) ⊃ − Z * 5 Z 2 6 X 2 0 |H| 4 2 + h.c., but it is CP-conserving and only yields interactions proportional to Re Z * 5 Z 2 6 . At n = 10 we find L (10) ⊃ − To derive the second equality we used the recursion in eq. (B.3). Plugging in the solution in eq. (B.4) and integrating by parts we find that L (12) contains the following terms proportional to Z * 5 Z 2 6 : Ignoring again interactions proportional to ∂ µ Z µ , only the second term in the bracket leads to non-trivial CP-violating interactions: In the last step we used the classical equation of motion for the Higgs boson field. These are the leading CP-violating interactions in the bosonic sector of the low-energy effective theory of C2HDM. At one loop in the EFT, the interaction term ∼ ∂ ν hZ ν Z µ Z µ generates the CP-violating ZZZ vertex via the Feynman diagram in figure 8.