CP-Violation in a Composite 2-Higgs Doublet Model

We study CP-Violation (CPV) in a Composite 2-Higgs Doublet Model (C2HDM) based on the global symmetry breaking $SO(6)/[SO(4)\times SO(2)]$, wherein the strong sector is modeled by a two-site moose structure. Non-trivial complex phases in the interactions involving fermions in both the elementary and strong sectors can induce CPV in the Higgs potential as well as the Yukawa coupling parameters. We compute both of the latter and analyse their dependence upon the aforementioned complex phases. Finally, we discuss physics observables which are distinctive of this model. Even in the simplest case with only one complex phase in the strong sector we can get significant CPV effects.


Introduction 2 The SO(6)/[SO(4) × SO(2)] Composite Higgs Model
We consider the spontaneous symmetry breaking of a global symmetry, SO(6) → SO(4)×SO (2), at an energy, f , which we call compositness scale. This dynamics produces 8 pNGBs, which can be identified as two SU (2) weak doublet Higgs fields. Concretely, the pNGB matrix U can be parameterised as where Φ = (φâ 1 , φâ 2 ), with the two real 4-vectors φâ i being rearranged as two SU (2) weak doublet Higgs fields, Φ 1 and Φ 2 , The aforementioned spontaneous symmetry breaking and the strong dynamics emerging in the C2HDM can effectively be described by a two-site moose model [16,17,43,44], which consists of the two sectors, which preserve a global G i = SO(6) × U (1) X (i = 1, 2) symmetry. Hereafter, we call the first and second sectors as the "elementary" and "strong" sectors, respectively. The Lagrangian of the two-site moose model can then be decomposed into two parts, L = L boson + L fermion .
The Lagrangian of the bosonic sector (L boson ) is written as where G 1 and G 2 are gauged, (A A µ , X µ ) and (ρ A µ , ρ X µ ) denote the (SO(6), U (1) X ) gauge fields with (g A , and g X ) with (g ρ , g ρX ) being gauge coupling parameters, respectively. The index A labels the adjoint representation of G. The SM gauge bosons are embedded into the G 1 gauge fields, (A A µ , X µ ). (We regard the other components of the G 1 gauge fields as spurions.) Furthermore, U 1 and Σ 2 are the pNGB matrix fields which are made of the chiral fields, U (recall Eq. (1)): and Σ 0 being an SO(4) × SO(2) invariant vacuum, Σ 0 = 0 4×4 ⊗ iσ 2 . We note that U 1 plays the role of the link field which connects the two sectors while Σ 2 is a linear (adjoint) representation of G 2 . The VEV of U 1 triggers the spontaneous symmetry breaking of G 1 × G 2 to the diagonal component G while the VEV of Σ 2 accounts for the breaking to SO(4) × SO(2) × U (1) X . This breaking pattern provides 24 pNGBs, 16 of which are absorbed in the longitudinal components of the gauge fields, while the remaining 8 can be identified with Higgs fields. The covariant derivatives for the pNGB fields are defined as D µ U 1 = ∂ µ U 1 − iA µ U 1 + iU 1 ρ µ and D µ Σ 2 = ∂ µ Σ 2 − i[ρ µ , Σ 2 ], where A µ = A A µ T A + X µ T X and ρ µ = ρ A µ T A + ρ X µ T X , with T A and T X being the generators of SO (6) and U (1) X , respectively. We thus note that there is no source of CPV in the bosonic sector Lagrangian. We next discuss the fermionic sector Lagrangian (L fermion ). We introduce two fermions, (q 6 L ) t and t 6 R , in the elementary sector, which are a SO(6) 6-plet with U (1) X charge X = 2/3. The third generation of the left-handed quark doublets q L and right-handed top quark t R are embedded into the SO(6) 6-plet fermions as where with θ t being a free parameter valued between −π and π. Hereafter, we consider only the top quark contribution, because the other SM fermions provide only sub-leading corrections to our later analysis, however, they can be included, if necessary, by simply extending the formalism described above. We note that Υ t L does not bring a new source of CPV, because it corresponds to an Hermite operator that projects out a SO(6) generator to a SU (2) L one, while Υ t R does. As we will see below, a non-zero value of sin θ t can induce CPV. We also introduce N SO(6) 6-plet fermions with U (1) X charge X = 2/3, Ψ I (I = 1, 2, · · · , N ), in the strong sector, in order to describe spin-1/2 resonances. The G 1 × G 2 invariant Lagrangian of the fermion sector is given as where the covariant derivatives of the elementary fermions include the interactions with the elementary gauge bosons while the covariant derivative of the spin 1/2 resonance Ψ provides the couplings to the spin-1 resonances introduced above. We do not write down (Σ 2 ) n (n ≥ 3) terms because of (Σ 2 ) 3 = −Σ 2 . It should be noted that the parameters in Eq.
can be taken as complex matrices/vectors and can thus be sources the CPV in our setup. The low-energy effective Lagrangian can be obtained by integrating out the spin-1 (ρ A µ , ρ X µ ) and spin-1/2 (Ψ) resonances. The quadratic terms of the SM fermions and gauge bosons in momentum (p) space are written as where Σ = U Σ 0 U T = U 1 Σ 2 U T 1 . Furthermore,Π's andM 's are form factors which encode the effect of the strong dynamics. The explicit expressions of the form factors for the N = 2 case are summarised in Appendix A. We note that the hermeticity of the Lagrangian implies that the form factorsΠ q 1 andΠ t 1 are purely imaginary, whileΠ qt 0,2 andΠ t 0,2 are real. The form factors M t 0 ,M t 1 andM t 2 can generally be complex. Therefore, CPV effects should appear inΠ q

The Higgs Potential and Top Yukawa Sector
We have introduced the SM gauge bosons and fermions in the elementary sector. Since the embedding of the SM fields explicitly breaks the shift symmetry, the SM field interactions generate a potential for the pNGB (Higgs) fields. Neglecting O(Φ 6 i /f 2 ) (i = 1, 2) terms, the Higgs potential can be parameterised in the form of where the parameters m 2 i and λ i are calculated by the momentum integration of the form factors, Π andM , which are defined in Eq. (9). The explicit relations between the Higgs potential parameters and the form factors are summarised in Appendix B. In contrast to the E2HDM, the Higgs potential parameters in the C2HDM are not independently adjustable parameters. Instead, the structure of the Higgs potential is determined by the strong dynamics. We also note that the C2HDM generally induces non-zero m 2 3 , λ 6 and λ 7 terms, which break (albeit softly) the Z 2 symmetry normally introduced in the E2HDM to control Flavour Changing Neutral Currents (FCNCs) at tree level [45]. Thus, the C2HDM generally introduces FCNCs at tree level. Such FCNCs can, however, be avoided by imposing Yukawa alignment [46], i.e., two Yukawa matrices are assumed to be proportional with each other. Since effects of lighter fermions onto the Higgs potential are negligibly small, we can impose the Yukawa alignment without changing the structure of it.
Let us comment on a technical aspect of the calculation of the Higgs potential. As mentioned in Refs. [16,17], the momentum integration of the form factors generally induces UV divergences. The UV finiteness of the momentum integration requires specific relations among the parameters in the fermion sector, Eq. (8). We summarise the UV finiteness conditions in Appendix B.
Depending on the parameters in the strong sector, the Higgs doublet fields can develop non-trivial VEVs. Without loss of generality, we can parameterise the VEVs of Φ 1 and Φ 2 as where v 1 and v 2 are real and positive while θ v ∈ (−π, π]. Here, v 1 , v 2 and θ v relate to the EW symmetry breaking scale, v EW = ( √ 2G F ) −1/2 246 GeV, with G F being the Fermi constant, as follows where v = v 2 1 + v 2 2 and tan β = v 2 v 1 . It is convenient to define the "Higgs basis" (H 1 , H 2 ) as so that H 1,2 satisfy We can easily identify the physical (pseudo)scalar particles in the Higgs doublet fields by taking the Higgs basis. We parameterise the component fields as where G 0 and G ± are the Nambu-Goldstone bosons which are absorbed by the longitudinal components of the Z and W ± bosons while H ± andφ 0 1,2,3 are the physical charged Higgs boson and neutral Higgs bosons, respectively. If the Higgs potential contains a physical CPV phase, these three neutral Higgs bosonsφ 0 1,2,3 are mixed with each other. Their mass eigenstates and their masses are defined as with M N being the Hessian matrix for the (φ 0 1 ,φ 0 2 ,φ 0 3 ) basis evaluated in vacuum. We identify H 1 as the discovered Higgs boson at the LHC. These masses and the matrix R are calculated via the parameters of the Higgs potential which are in turn determined by the parameters in the strong sector.
The detailed study of the Higgs potential in the C2HDM with CPC was performed in Refs. [16,17]. Here, let us focus on the CPV terms in the Higgs potential. As we discussed in the previous section, the fermion (top) interactions generally include CPV effects. The imaginary part of the Higgs potential parameters are obtained as where .
In our analysis, we neglect terms proportional to ( . These terms are highly suppressed by the factor of (∆ L,R /M * ) 6 and (∆ L,R /M * ) 8 with M * being a typical mass scale of spin 1/2 resonances, respectively, where the ratio ∆ L,R /M * should be smaller than 1 in order to reproduce the top Yukawa coupling, see Eq. (32). With this analysis, we find that Im[λ 5 ] is negligibly small in the C2HDM with CPV. It should also be noted that one of the three complex phases in Eq. (18)- (20) can be eliminated by using the vacuum condition. The remaining physical complex phases may be defined as [47] The other possible quantities carring CPV, such as Im[λ 6 λ * 7 ], can be expressed in terms of J 1 and J 2 . We then observe that a non-zero sin 2θ t , Π q 1 and/or Im[M 2 M * 1 ] onsets CPV in the Higgs sector.
Let us now discuss some phenomenological aspects of CPV in the Higgs sector of our C2HDM, also in relation to the gauge one. There are common and different aspects when comparing it to the E2HDM with CPV. The common one is the relation between the Higgs potential parameters and Higgs mass spectra. Once the Higgs potential parameters are determined by the strong dynamics, one can calculate the properties of the Higgs particles just like in the E2HDM. In particular, we remark that all the three neutral Higgs states are mixed and all have non-zero Higgs-gauge-gauge type interactions if the Higgs potential violates CP symmetry, while only two neutral Higgs fields can couple with gauge-gauge state in the CPC case. Therefore, in case we will observe three neutral scalar particles decaying into W + W − and/or ZZ states at collider experiments, this would be a smoking gun signature of any 2HDM with CPV. Detailed collider studies of such Higgs decays were performed in the context of the E2HDM with CPV in Ref. [48]. Here, we will estimate the Branching Ratios (BRs) of these diboson modes in our scenario in Sect. 4.
In contrast, there are two unique aspects of the C2HDM. Firstly, CPV effects in the Higgs potential correlate with those in the top Yukawa coupling, because the parameters of the former are generated through top quark loop contributions containing the latter. (This is a rather generic aspect in CHMs with CPV.) Secondly, the C2HDM with CPV generally predicts spontaneous CPV, contrarily to the E2HDM case, where both Higgs VEVs can be taken to be real and positive (without loss of generality) by rephasing the Higgs fields and redefining the potential parameters, which is not allowed in our case, so that the relative phase of the two VEVs, θ v , must be retained. In the SO(6) → SO(4) × SO(2) CHM, the complex phase of the Higgs VEVs predicts a non-zeroT -parameter at tree level as follows: where ξ = v 2 EW /f 2 . The bound coming from the EW precision tests is roughly estimated as |T | 10 −3 . Therefore, in our setup, tan β 1 or 1 is required to avoid such a severe constraint onT , while allowing for sizable CPV in the Higgs sector.
In order to see the correlation of the CPV effects between the Yukawa sector and Higgs potential, let us focus on the top Yukawa sector. The Higgs boson couplings with the top quarks can be read from Eq. (9). Up to O(Φ 5 1,2 ), we obtain with the form factors being evaluated in the zero momentum limit. The top mass is then extracted as where Y t | Φ i denotes Y t evaluated by the replacement of Φ i by Φ i . Therefore, it should be noted that, in the C2HDM, the top mass and Yukawa coupling depend on θ t and M 1,2 (0), which also appear in the Higgs potential. This is indeed the correlation between the top and Higgs sectors in the C2HDM that we mentioned in the previous section.

Signatures of the C2HDM with CPV
In this section, we show some numerical results in the C2HDM, particularly focusing on the dependence upon CPV phases of various quantities such as couplings of the SM-like Higgs boson and decays of additional Higgs bosons. For concreteness, we consider the case with two generations of Ψ t fields (N = 2) and assume that We note that the UV finiteness in the momentum integration for the form factors is automatically ensured with these assumptions. (See Appendix B for the details.) In this setup, the input parameters are listed as follows: Without loss of generality, we can take ∆ 1 L , ∆ 2 R , M 11 Ψ , M 22 Ψ , M 12 Ψ andȲ 12 2 ≡ M 12 Ψ − Y 12 2 to be positive and real by rephasing the fields in the strong sector, while Y 12 1 is complex. In the following analysis, we assume Y 12 1 to be real, so that the effect of CPV solely comes from the θ t parameter. In this setup, the top mass is expressed as where In the numerical evaluation, we take into account the O(ξ 3/2 ) contribution to the top mass in order to consistently include the O(ξ) corrections in the modified top Yukawa coupling.
Let us now briefly explain the strategy of our numerical analysis. First, we scan the parameters in the following ranges: withȲ 12 2 ≥ 0 and M 22 Ψ = M 11 Ψ . We note that the case with −π < θ t < 0 shows symmetric results with respect to those with 0 < θ t < π, because the potential parameters and top mass are given as functions of 2θ t . We thus focus on the positive values of θ t . For each point, we evaluate all the form factors given in Appendix A and, by performing the integration on the loop momentum, we obtain numerical values for the parameters of the Higgs potential, i.e., m 2 i (i = 1, . . . , 3) and λ j (j = 1, . . . , 7), from the formulae given in Appendix B. Next, we numerically solve the tadpole conditions where X| 0 denotes the value of X in the limit of zero field values whileφ 0 1,2,3 are the neutral components of the Higgs fields, see Eq. (15). These equations are given in terms of the potential parameters, the VEVs (v 1 , v 2 ) and their relative phase θ v . In the E2DHM, these equations can be solved analytically, so that some of the dimensionful parameters of the potential, e.g., m 2 1 , m 2 2 and Im(m 2 3 ), can be written in terms of the others and the VEVs. In contrast, in the C2HDM, we cannot analytically solve them. Instead, we numerically find a set of values (v 1 , v 2 , θ v ) which satisfy Eq. (35), amongst which we choose the combination giving the deepest potential value if several extrema appear at the same time. Finally, we evaluate various physical quantities (such as masses of the Higgs bosons, their couplings and decay BRs) in a similar way to that in the E2HDM (as intimated). In the last step, we impose the following constraints:  where m H 1 is the mass of the lightest neutral Higgs boson. For the VEV, we impose a much tighter constraint as compared with those on the other two observables, as it is most accurately measured by experiments. In the following analysis, we fix f = 1 TeV as a reference value. First of all, in Fig. 1, we show the behaviour of sin θ v and tan β as functions of θ t . In this figure (and in all the following ones), black points are allowed by the constraints from Eqs. (36) and (37). In addition, we show the case with θ t ≥ 0.01, because that with θ t < 0.01 is almost the same as the CPC scenario. In the left plot, we see that the allowed points mostly appear on the curve of sin 2θ t , which comes from the fact that the imaginary part of the potential parameters is given as a function of sin 2θ t , see Eq. (97). We note that most of the allowed points are numerically found to have θ v ∼ π − 2θ t . There appear a few points above the dashed curve, which typically require tan β 1 due to the constraint from theT parameter, see Eq. (23), as it can be clearly seen by looking at the density of the points between the left and right plots of Fig. 1. In the right plot of this figure, we see the θ t dependence of tan β. When the CPV phase θ t is negligibly small, only the upper limit of tan β ∼ 1 appears from the condition of Eq. (37) 3 . For θ t 0.1, the upper limit of tan β is getting more severe, because of the constraint from theT parameter. For θ t ∼ π/2, points are distributed over a larger tan β region. This is again due to the constraint from theT parameter, but in this region the role of sin β and cos β in the expression of m t given in Eq. (32) is interchanged as compared with the case with θ t 1, so that large tan β values are favoured over small ones.
Next, in Fig. 2, we discuss the behaviour of the elements of the mixing matrix R ij defined in Eq. (16). In particular, the value of 1 − R 11 describes the "alignmentness" of the 125 GeV Higgs boson, i.e., in the limit of 1 − R 11 → 0, the H 1 state coincides with theφ 0 1 state in the Higgs basis. In this case, deviations in the H 1 couplings to ordinary particles from their SM predictions only come from the parameter ξ, which describes the compositeness of the Higgs boson. This should be compared with the E2HDM case, where the Higgs boson couplings become the same as the ones of the SM Higgs boson at tree level in the alignment limit. As seen from the left plot of Fig. 2, our scenario is almost the alignment limit, as the deviations from it are quite small, i.e., O(10 −3 ) or smaller for θ t 0.1. For larger values of θ t , alignmentness becomes stronger, because the masses of the heavier Higgs bosons become larger, see Fig. 4, and thus the decoupling behaviour turns out to be very strong. We also show the different elements of R in the center and right plots. It is seen that the magnitude of R 12 and R 13 can maximally be O(0.01). This small mixing plays an important role to get sizable BRs of H 2 → V V and H 3 → V V (V = W ± , Z), as we will see below.
From these results, we expect that the deviations in the Higgs boson couplings can be well approximated in the alignment limit by analytical formulae. Let us define the usual scaling factors κ X as (X = V and t, where V = W ± , Z) In the alignment limit, κ X can be expressed by using the relation θ v = π − 2θ t as Re where ζ t ≡ Y 12 1 /Y 12 2 . In Fig. 3, we show the predicted κ X values as functions of θ t . We confirm that these values can be well explained by using the approximate formulae given in the above equations. In the CPC limit θ t → 0 with f = 1 TeV, κ V is calculated to be about 0.97 using κ V ∼ 1 − ξ/2. In fact, most of the points appear at κ V 0.97 and θ t 0 with a small fluctuation due to tiny mixing effects, i.e., R = I. It is seen that κ V is slightly enhanced at θ t 0.1, because the phase θ t slightly cancels the negative correction proportional to ξ, see Eq. (39). For Re[κ t ], in the CPC limit, the lowest (largest) value ∼ 0.91 (0.96) can be explained with ζ t tan β ∼ 0 (−4), see also the top-right panel of Fig. 3. When θ t gets larger, the value of Re[κ t ] rapidly shrinks to its lowest value ∼ 0.91, because ζ t tan β approaches zero due to the fact that only small values of tan β are allowed by theT parameter, see Fig. 1. Regarding Im[κ t ], its magnitude gets larger when θ t increases, e.g., |Im[κ t ]| 0.01 at θ t 0.1. The maximal value can be well fit by the dashed curve which is given by the leading term of Eq. (41), i.e., ξθ t ζ t with ζ t = −1.5. In addition, it is important to mention here that the effect of a non-zero θ t on κ V and Re[κ t ] is anti-correlated. Next, we discuss the mass spectrum of the heavy Higgs bosons and their phenomenology. The masses of heavier Higgs states are expressed in the limit of Im[λ 5 ] = 0 and R = I as with In our numerical analysis, the λ i (i = 1, ..., 7) parameters are typically of order 0.1, so that the magnitude of the squared masses is roughly determined by the size of the M 2 parameter. In Fig. 4, we thus show the typical mass scale M of the heavy Higgs bosons (top-left). It is seen that the lowest value of M becomes larger when θ t 0.1, which can be understood from Eq. (46) with tan β 1. We can also see that the masses of the additional Higgs bosons are nearly degenerate, i.e., typical mass differences are at the few GeV level. This means that the additional Higgs bosons almost do not decay into a lighter additional Higgs boson (or, rather, they do so via a very off-shell gauge boson, where possible), while they can decay into the SM-like Higgs boson H 1 . (This is not dissimilar from what found in the C2HDM with CPC.) Finally, we discuss the correlations between CPV effects in the top Yukawa coupling and the Higgs potential. In order to parameterise this effect, we introduce the invariant J 1 defined in Eq. (22). In Fig. 5 (left), we show the correlations between Im[κ t ] and J 1 . We see that, even for small values of |Im[κ t ]|, J 1 can be large. This can be understood by looking at the Im[κ t ] plot in Fig. 3, where the value of Im[κ t ] can be close to zero at θ t = 0 due to cancellations. As a result, there is a mild correlation between these two observables. In contrast, we find a strong correlation between |Im[κ t ]| and the product BR( As expected, such a quantity becomes zero in the CPC limit, because either H 2 or H 3 corresponds to the purely CP-odd state of the C2HDM with CPC, which then does not couple to two gauge bosons (only the CP-even does). Therefore, this product can measure the CPV effects in the Higgs potential, as already proposed in Ref. [48] in the E2HDM with CPV. It is seen that the maximal value of this BR product tends to increase with larger |Im[κ t ]| and can be ∼ 2% for |Im[κ t ]| ∼ 5 × 10 −3 .
In short, at the LHC, our scenario can be probed indirectly by studying the combination of measurements involving κ V , Re[κ t ] and finding two additional neutral Higgs bosons decaying into W + W − and/or ZZ pairs. In fact, Im[κ t ] could also potentially be accessed, e.g., in ttH 1 final states [49].

Summary and Conclusions
In this paper, we have studied a CHM based on the global symmetry breaking SO(6) → SO(4)× SO(2), paying particular attention to CPV generated in the strong sector. The 8 pNGBs that emerge from such a dynamics behave as two SU (2) doublet Higgs fields and their properties are determined by the strong dynamics behind the aforementioned breaking. We have computed the Higgs potential and top Yukawa coupling generated dynamically in this C2HDM by using a two-site moose construction, which effectively describes the strong sector. The Higgs potential and Yukawa coupling are expressed in terms of the momentum integration of several form factors which encode the details of the new strong dynamics, chiefly, of the associated CPV. We have found that complex phases in the partial compositeness parameters can induce CPV in both the Higgs potential and top Yukawa coupling.
We have then discussed typical CPV signatures of such a C2HDM by considering a simple setup, in which there is only one physical complex phase in the strong sector. In the presence of parameter space constraints obtained from reconstructing the SM VEV as well as the top quark and SM-like Higgs boson masses, we have first found that tan β should be much larger or lower than 1 to avoid a largeT parameter while guaranteeing sizable CPV. This is due to the coset structure of SO(6)/[SO(4) × SO(2)], which cannot forbidT contributions at tree level. Next, we have observed that, once we turn on CPV, the mixing effects amongst the neutral Higgs bosons remain negligible and the masses of extra Higgs bosons become large in comparison to that of the SM-like Higgs boson. We have also found that the masses of the additional Higgs bosons are nearly degenerate, which in turn implies that they mainly decay into the SM-like Higgs boson (in pairs or in combination with a gauge boson). Then, we have estimated the couplings of the SM-like Higgs boson to the EW gauge bosons and top quark as a function of the CPV phase θ t . Our results are the following: (i) a non-zero CP phase slightly enhances κ V ; (ii) deviations from the CPC top Yukawa coupling can be of order 10%; (iii) the CPV top Yukawa coupling (Im[κ t ]) can be O(0.01). The upper limit of Im[κ t ] comes from the requirement that v SM 246 GeV, m t 173 GeV and m H 1 125 GeV. We have further provided analytic expressions for the coupling deviation factors, so-called κ's, which clearly justify the above observations. In addition, we have discussed the correlation between CPV effects in the top Yukawa coupling and the Higgs potential. In doing so, we have confirmed that there is a mild positive correlation between Im[κ t ] and the physical complex phase of the Higgs potential. This is one remarkable aspect of the C2HDM with CPV because there should be no such a correlation in the E2HDM with CPV. Finally, we have also estimated the correlation between Im[κ t ] and the product BR(H 2 → V V ) × BR(H 3 → V V ), with V = W ± , Z, as it is sensitive to the mere presence of CPV in the Higgs potential. It was found that the maximal value of this product of BRs depends on the size of Im[κ t ]: e.g., BR( Possible outlooks of our initial investigation will include to assess the testability of our C2HDM with CPV at colliders in great detail, both present and feature ones. Specifically, for what concerns the LHC, it will be worth to investigate physical processes where interactions between Higgs bosons and the top quark occur like, e.g., H 2,3 production and decay into tt and/or ttH 1,2 final states, including exploiting spin and charge asymmetries therein. It would also be interesting to study CPV in other coset models, such as SO(9)/SO (8), as, in this case, we expect to obtain more significant CPV effects in physics observables because this coset structure forbids a largeT parameter. We leave these developments to future publications.
where we ignore cubic terms in the form factors. Notice that Π q and M 1,2 are defined in Eq. (21) while Π t can be calculated as It should also be noted that the momentum integrations for Π q 1,2 and Π t 1,2 have a UV divergence because they behave as ∼ p −4 in the large p 2 regime. We find that, in order to ensure UV finiteness, we need to impose the following conditions: a 0 I 02 + l 00 = a R l 02 + l R0 (for Π q 2 ) , a 0 r 02 + r 00 = a R r 02 + r R0 (for Π t 2 ) , a I l 02 = −l I (for Π q 1 ) , a I r 02 = −r I (for Π t 1 ) , where a, l and r are defined in Eqs. (56)-(75). For the case where the number of spin-1/2 resonances is two, these conditions are equivalent to, respectively, We can easily check that the UV finiteness conditions are automatically satisfied with the choice of Eq. (30).
In the special case where the CPV source is given only by the θ t parameter, the top quark contributions to the Higgs potential parameters are obtained as Re λ where .