One-loop corrections to the $Zb\bar{b}$ vertex in models with scalar doublets and singlets

We study the one-loop corrections to the $Zb\bar{b}$ vertex in extensions of the Standard Model with arbitrary numbers of scalar doublets, neutral scalar singlets, and charged scalar singlets. Starting with a general parameterization of theories with neutral and singly-charged scalar particles, we derive the constraints that must be obeyed by the couplings in order for the divergent contributions to cancel. Then, we show that those constraints are obeyed by the models that we are interested in, and we write down the full finite expression for the vertex in those models. We apply our results to some particular cases, highlighting the importance of diagrams with neutral scalars that are often neglected in the literature.


Introduction
The discovery of a scalar particle at the LHC [1,2] urges the questions of whether there are more neutral scalars and whether there are charged scalars. Multi-scalar models have long been studied-for reviews see, for example, Refs. [3,4,5]. Here, we concentrate on models with n d scalar doublets, n c charged-scalar singlets, and n n neutral-scalar singlets. The scalar-particle content is, thus, 2n ≡ 2 (n d + n c ) charged scalars H ± a (a = 1, . . . , n) and m ≡ 2n d + n n neutral scalars S 0 l (l = 1, . . . , m). In our notation, H ± 1 = G ± and S 0 1 = G 0 are, respectively, the charged and neutral would-be Goldstone bosons. Light extra scalars may be detected directly through their production, while heavy scalars may be detected indirectly through their impact on the radiative corrections. We focus on the coupling Zbb: 1 where P L,R are the projectors of chirality and, at the tree level, in models without extra gauge fields. As usual, s W and c W are the sine and the cosine, respectively, of the Weinberg angle θ W . Haber and Logan [7] have considered the one-loop corrections to the vertex Zbb in models with extra scalars in any representation of the gauge group SU (2) L . We extend their analysis by considering CP-violating scalar sectors and by exploring the constraints arising from the cancellation of divergences. We write down the final results in models with singlets and doublets in a simple and usable form. This is possible due to a convenient parameterization that was introduced in Refs. [8,9,10], following earlier work [11]. We explicitly calculate the contributions due to diagrams with neutral scalars, which, although nominally suppressed by the bottom-quark mass m b , may be enhanced by ZZS 0 l couplings proportional to ratios of vacuum expectation values (VEVs).
We present the Lagrangian and the relevant calculations in Section 2, ending with the constraints imposed by the cancellation of divergences. In Section 3 we introduce the parameterization relevant for doublets and singlets, showing that the constraints are indeed satisfied, and we simplify the final expressions. The connection with experiment is reviewed in Section 4, and then applied in Section 5 to some simple cases, looking in particular at the importance of diagrams with neutral scalars. We draw our conclusions in Section 6. An appendix summarizes the definitions of the Passarino-Veltman functions used in this paper.

The one-loop calculation
We use the approximation where the CKM matrix element V tb = 1, requiring us to consider only the quarks bottom with mass m b and top with mass m t . We will neglect m b in the propagators and loop functions, but we keep generic couplings. Indeed, enhancements due to ratios of VEVs may arise in some couplings, and this must be properly taken into account. One example is the tan β enhancement that appears in some two-Higgs-doublet models (2HDMs), which allows some diagrams proportional to m b tan β to compete, in the region of large tan β, with diagrams proportional to m t . This also extends the results in ref. [7].

Couplings
In addition to the couplings in Eqs. (1) and (2), we need In Eq. (3), at the tree level From Eqs. (2) and (5), The charged scalars H ± a and the neutral scalars S 0 l interact with the quarks through and with the Z gauge boson through where M Z is the mass of the Z. In general, the coefficients c a , d a , and r l in Eqs. (7) and (8) are complex, while the y l in Eq. (11) are real. The n × n matrix X in Eq. (9) is Hermitian. The m × m matrix Y in Eq. (10) is real and antisymmetric. We let m a denote the mass of H ± a and m l denote the mass of S 0 l .

One-loop diagrams
At one-loop level, the diagrams contributing to the Zbb vertex are shown in Figs. 1 and 2, for charged and neutral scalars, respectively. This classification of the diagrams was proposed in Ref. [7], wherein the diagrams in Fig. 3 were also mentioned, but then neglected. The diagrams in Fig. 3 involving the charged scalars do not give new contributions beyond the Standard Model (SM) in models with only scalar singlets and doublets, because in these models there are no ZW ± H ∓ a couplings other than the ZW ± G ∓ already present in the SM. The diagrams in Fig. 3 involving neutral scalars are proportional to m b . This is because, if one considers (for example) two external left-handed bottom quarks, then one of them connects to a chirality-conserving Z while the other one connects to a chiralityflipping scalar; this necessitates a mass insertion in the bottom-quark propagator. Since the diagrams in Fig. 3 are convergent, one may neglect them by taking m b = 0, and this is what was done in Ref. [7]. Nevertheless, because m b could appear multiplied by a large coefficient (such as tan β = v 2 /v 1 in the Z 2 -symmetric 2HDM) we will also present their The diagrams in Figs. 1 and 2 are divergent and must be renormalized. We follow the on-shell renormalization scheme of Hollik [12,13]. Applying multiplicative renormalization, the renormalized vertex acquires some terms leading to a correction to the Z propagator; these are part of the oblique parameters and were shown to be very small in Ref. [7]. Here we are looking for the terms that change the tree-level couplings, which after renormalization may be written as Figure 3: Diagrams referred as to "type d)" in Ref. [7].
where ∆g ℵ (ℵ = L, R) represent all the one-loop corrections after renormalization, including the ones involving G ± , G 0 , and the already-observed neutral scalar with mass 125 GeV (more on this in Section 4). To perform the renormalization one needs to evaluate the renormalization constants that are obtained from the self-energies. We therefore need to evaluate the contributions of both the charged and neutral scalars to the self-energies, shown in Fig. 4. The self-energy iΣ (p) receives contributions proportional to pP L , pP R , m b P L , and m b P R . In our approximation of neglecting m b , we write Following Hollik's renormalization scheme [12,13], the self-energy produces contributions to ∆g Lb and ∆g Rb given by Note that Ref. [7] follows an equivalent procedure, ignoring renormalization and calculating simply the reducible diagrams with self-energy corrections in the external bottom quarks, which they dub "type c) diagrams". Although we do perform the renormalization, we will name the contributions arising from it as "type c)", allowing for an easy comparison with Ref. [7]. Our calculations of the various diagrams have been performed analytically and then confirmed by automatic computation. In order to automatically perform the calculation we use common techniques that include FeynRules [14], QGRAF [15], and FeynCalc [16,17].
Recently, two of us (DF and JCR) have developed the new software FeynMaster [18] that handles, in an automated way, all these steps. The results involve Passarino-Veltman loop functions [19]; our conventions for them coincide with those in FeynCalc and LoopTools [20,21], and are summarized in Appendix A.
We next turn to the computation of each diagram.

Calculating the diagrams involving charged scalars
The diagrams in Fig. 1a) lead to where C 00 is a Passarino-Veltman function defined through Eq. (106). We have set m b = 0 inside all the Passarino-Veltman functions; however, when evaluating them numerically it is sometimes better to keep m b = 0 in order to avoid numerical instabilities. We should note that the sums in Eqs. (15) start at a = 1, i.e. they include the charged Golsdtone bosons G ± . However, one may show that X 1a = X a1 = 0, and therefore the sum in Eq. (15a) may start at a, a = 2, while the term with a = a = 1 is separately included in the SM contribution. The diagrams in Fig. 1b) lead, after taking into account that (d − 2) C 00 (. . . ) = 2 C 00 (. The Passarino-Veltman function C 0 is defined in Eq. (104), while C 12 is defined through Eq. (106). As for the type c) contributions, arising through renormalization from diagram a) in Fig. 4, we find The Passarino-Veltman function B 1 is defined in Eq. (103). In the CP-conserving limit, Eqs. (15)-(17) agree with Eqs. (4.1) of Ref. [7], and also with Ref. [22].
The functions B 1 and C 00 are divergent; all the other Passarino-Veltman functions appearing in this paper are finite. In dimensional regularization, defining the divergent quantity one has Therefore, the divergent terms in Eqs. (15)- (17) are (20b) We thus conclude that in any sensible theory one must have where we have used Eq. (6). Equations (21) are presented here for the first time. Of course, the cancellation of divergences must hold separately for ∆g Lb and ∆g Rb .

Calculating the diagrams involving neutral scalars
The diagrams in Fig. 2a The diagrams in Fig. 2b) lead to As for the type c) contributions, arising through renormalization from the second Fig. 4, we find In the CP-conserving limit, Eqs. (22)-(24) agree with Eqs. (5.1) of Ref. [7]. Collecting all the divergent terms in Eqs. (22a), (23a), and (24a) we find This is a new constraint, which can also be obtained by collecting all the divergent terms in Eqs. (22b), (23b), and (24b). Diagrams c) and d) in Fig. 3 involve neutral scalars. They are not divergent and they are proportional to m b . However, we keep them because they might be enhanced when the coupling of neutral scalars to the bottom quark gets enhanced, as in the type-II 2HDM. From them we get The function C 1 is defined through Eq. (105). At this juncture we want to make a clarification. The one-loop results for ∆g Lb and ∆g Rb have imaginary parts. If there are no scalars with mass below M Z /2, then the imaginary parts only appear through cuts of the internal bottom-quark lines of Fig. 2b), thus affecting only the contributions with neutral scalars. Although those imaginary parts may be of the same order of magnitude as the real parts, they are unimportant because the observables will depend on, for example, where the last line follows from the fact that g 0 Lb is real. As a result, the impact of an imaginary ∆g Lb on the observables (see the next section) effectively appears only at higher order.

Models with doublet and singlet scalars
We now focus on extensions of the SM with n d scalar doublets, n c singly-charged scalar SU (2) L singlets, and n n real scalar gauge-invariant fields. The particle content is then 2n ≡ 2 (n d + n c ) charged scalars H ± a and m ≡ 2n d + n n neutral scalars S 0 l ; this counting includes the Goldstone bosons H ± 1 = G ± and S 0 1 = G 0 . Without loss of generality, one may assume that the scalar with mass 125 GeV found at the LHC is S 0 2 ; generality is lost if one makes the further assumption that the masses are ordered, since there might be massive scalar(s) below 125 GeV.
The scalar doublets are The fields ϕ 0 where the v k may be complex. Obviously, the charged and neutral SU (2) L singlets have no Yukawa couplings. The Yukawa Lagrangian is where the e k and the f k (k = 1, . . . , n d ) are the Yukawa coupling constants.

Formalism
We use the formalism in Refs. [8,9,10]. We write ϕ + k and ϕ 0 k as superpositions of the physical (= eigenstate of mass) fields as The matrix U is n d × n and the matrix V is n d × m.
Since H ± 1 and S 0 1 are Goldstone bosons, the first columns of U and V are fixed and given by where v 2 ≡ n d k=1 |v k | 2 (v is real and positive by definition). There is an n × n matrixŨ that is unitary, implying that The matrix T in Eq. (34) only exists when the number n c of charged scalar SU (2) L singlets is nonzero. There is an m × m matrix that is real and orthogonal. Therefore, The matrix R in Eq. (36) only exists in models with n n = 0. One can show [9] that in this class of models Moreover, leading to y l=1 = 0, because V † V is Hermitian and therefore Im V † V 11 = 0. Thus, the sum in Eq. (11) really starts at l = 2, viz. there is no vertex ZZG 0 , just as there is no vertex ZZZ.

Cancellation of the divergences
It follows from Eqs. (7), (8), and (30)-(32) that where we have defined the n d × 1 vectors From Eqs. (33) and (42)-(44), We further define the m × 1 column vector It then follows from Eq. (44) that m l=1 We now use Eqs. (37) to obtain m l=1 |r l | 2 = 1 2 From Eqs. (42) and (35), Notice that the two sums in Eq. (50) run over different spaces (up to n and n d , respectively). Similarly, where the last equality follows from Eq. (50). This proves that this class of models obeys the consistency Eq. (21a). Similarly, one can show that Eq. (21b) is also obeyed, confirming within this class of models the cancellation of the divergences of the contributions from charged scalars. Next we compute m l,l =1 We use once again Eqs. (37) to obtain m l,l =1 where in the last step we have used Eq. (49). Taking into account Eq. (40), we conclude that in this class of models the consistency Eq. (26) also holds.

Simplification of the charged-scalars contribution
In this class of models, from Eqs. (39) and (6), Therefore, one may write the charged-scalars contribution as The first column of the matrix T is zero, because k |U k1 | 2 = k |v k | 2 v 2 = 1. Thus, T T T * 1a = T T T * a1 = 0 and the charged Goldstone boson does not contribute to the sum in the last line of Eq. (56). On the other hand, the Goldstone boson does contribute to the sum over a in the first four lines, but |c 1 | has the same value as in the SM, cf. Eq. (46a); therefore, the contribution of the charged Goldstone boson is always the same and should be subtracted out. The simplified expression for the charged-scalar contributions to ∆g Rb is obtained from Eq. (56) through the changes c a → d * a and L ↔ R.
Suppose a model with no charged SU (2) L singlets. Then the matrix T does not exist. If one furthermore makes the approximation M Z = 0, then the contribution of the charged scalars in Eq. (56) becomes and similarly for ∆g Rb , with c a → d a and L ↔ R. One easily finds that and that The dependence on θ W disappeared! This must indeed happen because, in the limit M Z = 0, the Z gauge boson is indistinguishable from the photon-since they are both massless-, and therefore the Weinberg angle loses its meaning and must disappear from any physically meaningful quantity. The function has been given in Eq. (4.5) of Ref. [7] and has been used in all the subsequent analyses, by many authors, of models with extra doublets (and possibly neutral singlets). In our more general result (56), though, we keep CP violation, we allow for charged singlets, and do not make M Z = 0. As a consequence of Eqs. (60), in a 2HDM, where there is only one physical charged scalar, ∆g Lb In general, as long as there are no charged singlets and the approximation M Z ≈ 0 is good, ∆g Lb and ∆g Rb have opposite signs when the contribution of the neutral scalars is not taken into account.

Connection with experiment
The couplings g Lb and g Rb in Eq. (1) may be determined experimentally from: 2 1. The rate 2. Several asymmetries, including (a) the Z-pole forward-backward asymmetry measured at LEP1 where b F (b B ) stands for final-state bottom quarks moving in the forward (backward) direction with respect to the direction of the initial-state electron; (b) the left-right forward-backward asymmetry measured by the SLD Collaboration Introducing the vector-and axial-vector bottom-quark couplings one has [7, 24] In Eq. (67), η QCD = 0.9953 and η QED = 0.99975 are QCD and QED corrections, respectively. Moreover, Neglecting µ b ≈ 10 −3 and setting the QCD and QED corrections to unity, one gets Equation (73)  There are, thus, two possible approaches. The first one consists in taking as good the values (74) obtained from the SM fit and using R fit b and A fit b as constraints on New Physics (NP). The second one is seeking NP that might explain an R b just slightly above the SM, together with an A b that undershoots the SM by 2.8σ.
It is convenient to switch from the parameterization in Eq. (12), which splits the couplings g ℵb as g 0 ℵb + ∆g ℵb , where g 0 ℵb is the tree-level piece and ∆g ℵb is the one-loop piece, to the alternative parameterization which splits them into the SM piece g SM ℵb (which includes the SM loop correction) and the NP piece δg ℵb . A simple rule of thumb can be obtained by expanding to first order in the deviations; one finds [7] This shows that, assuming (rather arbitrarily) δg Rb ≈ −δg Lb , δR b is pulled down (up) and δA b is pulled up (down) by a positive (negative) δg Lb . Inverting Eqs. (76) [7], If one wishes to follow the second approach above, viz. using NP to keep R b close to its SM value while reducing A b significantly, then one needs to get a small δg Lb together with a significant positive δg Rb .

The inert doublet model
This is a 2HDM characterized by Thus, all the VEV is in the first doublet and the matrices We have while, from the matrix V in Eq. (79), (82)

Charged-scalar contribution
The contribution of the charged Goldstone boson can be separated and included in the SM. The genuine new contribution is If we plot δg Lb /|c 2 | 2 and δg Rb /|d 2 | 2 , we get general results for any 2HDM. We have used LoopTools [20] to perform the numerical integrations contained in the Passarino-Veltman functions. The results are shown in Fig. 5. One sees that 0 < δg Lb 0.002 |c 2 | 2 and that Eq. (62) holds to an excellent approximation; this indicates that the approximation M Z = 0 is in fact very good. This is vindicated by Fig. 6, which displays the asymmetries R g L,R between the values of δg Lb |c 2 | 2 and δg Rb |d 2 | 2 computed with M Z = 0 and with M Z = 0 One observes that both asymmetries are at most of order 1%.
To have an idea of the impact of these contributions on the experimental constraints on R b and A b we display the latter in Fig. 7. One sees that A b does not impose a relevant constraint, but R b does. The result is dominated by the variation in |c 2 |, although there is also dependence on the charged Higgs mass as can be seen in Fig. 5. In order to have agreement with the data |c 2 | tends to be small, approaching the SM point, although larger values are possible if the charged Higgs mass is also large.

Neutral-scalar contribution
We have evaluated these contributions by using LoopTools [20]. 3 We have checked in the numerical simulation that the divergences indeed cancel, by verifying that the results are independent of the ∆ parameter of LoopTools. Without loss of generality, we have required that M 4 > M 3 . The results are shown in Fig. 8. It is seen that δg Lb > 0 but δg Rb < 0 (recall that a negative δg Rb goes in the wrong direction if one wishes to explain  Comparing Figs. 5 and 8, one sees that, unless the masses of the two NP neutral scalars are close to each other, there is in general no rationale for neglecting the neutral-scalar contribution as compared to the charged-scalar one, as most authors do in their analyses.

The complex 2HDM
The complex 2HDM (C2HDM) is a two-Higgs-doublet model with a softly broken Z 2 symmetry. The scalar potential is where all the parameters, except m 2 12 and λ 5 , are real. In general, Im (m 2 12 ) 2 λ * 5 is allowed to be nonzero. By rephasing Φ 1 and Φ 2 , we go to a basis where the VEVs are real and positive: 0 |ϕ 0 where v = 246 GeV. Thenceforth, c θ , s θ , and t θ represent the cosine, sine, and tangent, respectively, of whatever angle θ is in the subindex. We write the scalar doublets as We transform the fields into the so-called Higgs basis through [25] Then H 2 does not have a VEV: G + and G 0 are the Goldstone bosons. There is a charged pair H ± with mass m H ± . In a standard C2HDM notation, η 3 := I 2 and the neutral mass eigenstates are obtained from the three neutral components as The orthogonal matrix R diagonalizes the neutral mass matrix through where 4 m 2 = 125 GeV ≤ m 3 ≤ m 4 are the masses of the neutral scalars (m 1 is the unphysical mass of the Goldstone boson S 0 1 = G 0 ). We parameterize the orthogonal matrix R as [26] Without loss of generality, the angles may be restricted to [26] − π/2 < α 1 ≤ π/2, −π/2 < α 2 ≤ π/2, 0 ≤ α 3 ≤ π/2.
Taking the limit α 2 , α 3 → 0 one recovers a 2HDM with softly broken Z 2 symmetry and no CP violation; this is the 'real 2HDM', in which S 0 4 = A is the massive CP-odd scalar. Comparing with Eqs. (31) and (32), we find (99) Equation (81) still holds and Assuming the Yukawa couplings to follow the type-II 2HDM pattern, viz. e 2 = f 1 = 0 and we have Note that, contrary to the assumptions in the previous subsection, here |c 2 | and |d 2 | usually are of vastly different orders of magnitude; in particular, |d 2 | |c 2 | for large values of tan β.
This model was studied in detail in Ref. [27], which introduced the code C2HDM HDECAY implementing the C2HDM in HDECAY [28,29]. For illustrative purposes, we take points from that fit, where, invoking constraints from Flavour Physics, tan β was taken below 20. We denote this by the "low tan β set". We also show results for a new dedicated run with very large tan β > 20, which we dub the "large tan β set". Such high values of tan β may be in contradiction with certain Flavour Physics observables, notably (as we will now show) Z → bb. 5 Nevertheless, we will consider such extreme values since we wish to stress that the details of such a bound may require both charged and neutral contributions. As shown in Fig. 8 of Ref. [30], very large tan β is only consistent with current measurements at LHC if α 1 lies in a very restricted range, which we impose in this run ab initio.
As in the inert doublet model discussed previously, the contribution due to the charged Goldstone bosons decouples, it is included in the SM and subtracted out, and the result is still given by Eqs. (83) and Fig. 5. Note that δg Lb is positive while δg Rb is negative. Recall that δg Lb tends to make R b smaller and from there comes a bound in the m H ± -tan β plane. The correction of δg Rb is too small to have an impact on R b . On the other hand, δg Rb has a substantial impact on A b , but it goes in the wrong direction when compared with the direct experimental measurements.
We are particularly interested in the corrections arising from neutral scalars, because they are usually discarded. Firstly, we would like to know under which circumstances can contributions from the neutral scalars be large. Secondly, we are interested in studying the specific neutral scalar contributions of "type d)", in Fig. 3. Recall that the latter are proportional to m b and they are always neglected, despite the fact that they could possibly be enhanced by tan β.
Let us denote by a superscript c (n) the new physics contributions coming from the charged (neutral) scalars. We separated the data from our scans in three different sets, • Small tan β ∈ [0, 10], blue in the plots • Intermediate tan β ∈ [10,30], green in the plots • Large tan β > 30, red in the plots In the left panel of Fig. 9 we show δg n Lb versus δg c Lb for all three sets. In the right panel of Fig. 9 we show −δg n Rb versus −δg c Rb (because both δg n Rb and δg c Rb are negative). We see that, contrary to popular belief, |δg n Rb | is of order but usually larger than |δg c Rb |. They approach each other for moderate values of tan β and eventually both become quite large for very large values of tan β. In contrast, for low tan β ∼ 1, δg c Lb is of order 10 −1 and much larger than δg n Lb . For intermediate values of tan β, though, δg c Lb ∼ δg n Lb ∼ 10 −5 and, for larger tan β, the result (although small) is dominated by the contributions from the neutral scalars. The sums are shown as functions of tan β in Fig. 10. We see that a significant impact on A b and R b can only occur, either for very low or very high values of tan β.  The impact on A b and R b is shown in Fig. 11 for the low tan β set. The points in Fig. 11 only stray from the 2σ R b bounds for tan β < 0.8. This is why only points above tan β = 0.8 were taken in Ref. [27]. In the low tan β region, the new contributions to δg b are dominated by the diagrams with charged scalars, as can seen from Fig. 9.
We are now interested in studying what would happen in the large tan β region, especially in what concerns the contributions from neutral scalars. In Fig. 12a) we see that, for δg L , the neutral scalar contributions (in dark blue) are typically larger than the charged scalar contributions (in green) by orders of magnitude, when tan β > 30. This Figure 11: A b versus R b in the C2HDM for low tan β (tan β < 10). In this regime the charged scalars contribution is dominant.
is an important conclusion that we draw, as we had already seen in Fig. 9. One cannot neglect the effect from neutral scalars to δg Lb . We expect this to be even more important in models with more than two Higgs doublets and/or extra singlets. It is interesting to inquire about the importance of the type d) neutral scalar contributions (in red). They are nominally of "order m b ", and were discarded in Ref. [7]. We see in Fig. 12a) they dominate up to tan β ∼ 160, when the other neutral scalar diagrams (in purple) take over. But, because they have the opposite sign, we see a pronounced dip in δg n Lb around tan β ∼ 160. Fig. 12b) shows the neutral scalar contributions to δg Rb from type d) diagrams in red. We see that these diagrams are basically irrelevant for δg Rb . However, the overall contribution from neutral scalars (in green) is a relevant percentage of the contribution from charged scalars (in blue). The curve in purple shows the sum of all contributions.
The effect of the total contribution to A b and R b in the C2HDM, for all values of tan β, is shown in Fig. 13. The contribution of the charged scalars is shown in red for tan β < 0.8, in orange for 0.8 < tan β < 250, and in brown for tan β > 250. For low values of tan β (red curve) the contribution from the charged scalars tends to make R b small and no effect on A b (see Fig. 11). However, for the large tan β regime, the charged scalars (brown curve) would tend to make R b small and A b large. The contribution from neutral scalars is shown in cyan for tan β < 350 and in blue for tan β > 350. We see that, in contrast with the charged scalars, the neutral scalars contributions, for large values of tan β, tend to make R b large and A b small. And these are precisely the dominant contributions for very large tan β, as we see from the total (green) curve.
We conclude that, when studying the impact of Z → bb on multi scalar models at very large couplings (very large tan β in our example of the C2HDM) the neutral scalar contributions must be taken into account. Of course, we will need to include in any model all theoretical and experimental constraints, which may curtail a large part of such extreme couplings. This must be evaluated in a case by case basis.

Conclusions
We have studied the one-loop contributions to Z → bb in models with extra scalars. We have started by deriving the conditions on generic couplings that must hold for the divergences to cancel. We have then concentrated on models with extra SU (2) L doublets and singlets, either neutral, as in Ref. [7], or charged. The final expressions are greatly simplified, due to the parametrization in Refs. [8,9,10,11]. We also extend the analysis in Ref. [7] to models with CP violation in the scalar sector. We highlight the importance of the neutral-scalar contributions. In particular, Fig. 13 shows that, for extreme values of tan β, the bounds coming from A b and R b are mostly due to the neutral scalars and not the charged scalars. This highlights the need to consider both contributions in generic models with extra singlet and doublet scalars.