Fitting the $Z b \bar b$ vertex in the two-Higgs-doublet model and in the three-Higgs-doublet model

We investigate the new contributions to the parameters $g_L$ and $g_R$ of the $Z b \bar b$ vertex in a multi-Higgs-doublet model (MHDM). We emphasize that those contributions generally worsen the fit of those parameters to the experimental data. We propose a solution to this problem, wherein $g_R$ has the opposite sign from the one predicted by the Standard Model; this solution, though, necessitates light scalars and large Yukawa couplings in the MHDM.


Introduction
In this paper we focus on the Zbb coupling where c w is the cosine of the weak mixing angle and P L and P R are the projection operators of chirality. At tree level, where s w is the sine of the weak mixing angle. With s 2 w = 0.22337 [1], one obtains g tree L = −0.42554 and g tree R = 0.07446. The Standard Model (SM) prediction is [2] g SM L = −0.420875, g SM R = 0.077362.
In the presence of New Physics, we write Experimentally, we get at g L and g R by measuring two quantities called A b and R b ; their precise experimental definitions may be found in refs. [2][3][4] and in appendix A. One has where r b = (g L + g R )/(g L − g R ) and µ b = [m b (m 2 Z )] 2 m 2 Z . We use the numerical values m b (m 2 Z ) = 3 GeV and m Z = 91.1876 GeV [1]. Equation (5) may be inverted to yield Notice the existence of two solutions for . The other measured quantity is where c QCD = 0.9953 and c QED = 0.99975 are QCD and QED corrections, respectively, and s c + s u + s s + s d = 1.3184. The solution to equations (7) and (8b) is Notice the two possible signs of g R in equation (9).
An overall fit of many electroweak observables gives [4] R fit b = 0.21629 ± 0.00066, (10a) A fit b = 0.923 ± 0.020. (10b) On the other hand, A b has been directly measured at LEP1 and at SLAC in two different ways, see appendix A. The averaged result of those measurements is While the A b value of equation (10b) deviates from the Standard-Model A b value 0.9347 by just 0.6σ, the A b value of equation (11) displays a much larger disagreement of 2.6σ. In this work we consider both the set of values (10), which we denote through the superscript "fit," and the set formed by the values (10a) and (11), which we denote through the superscript "average." Plugging the central values of those two sets into equations (6) and (9), we obtain solutions 1, 2, 3, and 4 for g L and g R in table 1. We also display in that table the corresponding values of δg L = g L + 0.420875 and δg R = g R − 0.077362. We see that  Table 1: The results of equations (6) and (9) for g L and g R and the corresponding values of δg L and δg R extracted from equations (3) and (4). The superscript "fit" corresponds to the input values (10), while the superscript "average" corresponds to the input values (10a) and (11).
solutions 3 and 4 have a much too large δg L ; we outright discard those solutions. 1 Solution 1 seems to be preferred over solution 2 because it has much smaller |δg R |. 2 Still, in this work we shall also consider solution 2.
In this paper we seek to reproduce solutions 1 and 2 by invoking New Physics, specifically either the two-Higgs-doublet model (2HDM) or the three-Higgs-doublet model (3HDM). The 2HDM is one of the simplest possible extensions of the SM. One of the many motivations for 1 Solutions 3 and 4 are good when one only measures R b and A b at the Z 0 peak; when one gets away from that peak, the diagram with an intermediate photon becomes significant and one easily finds that solutions 3 and 4 are not really experimentally valid [5]. So, there are both theoretical and experimental reasons for discarding them.
2 A recent preprint [6] claims that there are already a couple LHC points that favour solution 1 over solution 2 and that in the future the two solutions could be decisively discriminated through the highluminosity-LHC data. On the other hand, the older ref. [5] claims that the PETRA (35 GeV) data actually favour solution 2 over solution 1. the 2HDM is supersymmetry: the Minimal Supersymmetric Standard Model has two Higgs doublets. Also, the 2HDM may generate a Baryon Asymmetry of the Universe sufficiently large, due to the flexibility of its scalar mass spectrum. We recommend the review [7] on the 2HDM in general, and refs. [8,9] on the aligned 2HDM. In recent years the 3HDM has received increased attention, see e.g. refs. [10][11][12]. The aligned 3HDM is discussed in refs. [9,13].
The plan of this work is as follows. In section 2 we present the general formulas of δg L and δg R in the n-Higgs-doublet model. In section 3 we consider the particular case of an aligned 2HDM and we specify the constraints on the scalar masses that we have used in that case. We do the same job for an aligned 3HDM in section 4. We then present numerical results in section 5, followed by our conclusions in section 6. Appendix A deals on the definition of R b and A b and on the experimental data for them. Appendix B works out the derivation of the neutral-scalar contributions to δg L and δg R .
2 The Zbb vertex in the aligned nHDM

Mixing formalism
In a general n-Higgs-doublet model (nHDM) and utilizing, without loss of generality, the 'charged Higgs basis' [14], the scalar doublets Φ 1 , . . . Φ n are written where S + 1 is a charged Goldstone boson, S 0 1 is the neutral Goldstone boson, v ≈ 246 GeV is the (real and positive) vacuum expectation value (VEV), and S + 2 , . . . , S + n are physical charged scalars with masses m C2 , . . . , m Cn , respectively. Without loss of generality, we order the doublets Φ k through m C2 ≤ m C3 ≤ · · · ≤ m Cn . We are free to rephase each of the Φ k , thereby mixing R k and I k through a 2 × 2 orthogonal matrix.
The real fields H, R k , and I k (k = 2, . . . , n) are not eigenstates of mass, rather where V is an n × 2n matrix with (1, 1) matrix element V 11 = i. The physical neutral-scalar fields S 0 2 , . . . , S 0 2n are real and have masses m 2 , . . . , m 2n , respectively. An important property of V is that For the sake of simplicity, we assume alignment. This means that H ≡ S 0 2 is a physical neutral scalar that does not mix with the R k and I k . Hence, V 12 = 1 and V 1j = 0, ∀j = 3, . . . , 2n; also, V k1 = V k2 = 0, ∀k = 2, . . . , n. The scalar H is assumed to be the particle with mass m 2 ≈ 125 GeV that has been observed at the LHC. In this paper, alignment is just a simplifying assumption that we do not pretend to justify through any symmetry imposed on the nHDM. We order the S 0 j through m 3 ≤ m 4 ≤ . . . ≤ m 2n . Notice that, in principle, one or more of these masses may be lower than m 2 .
We define the real antisymmetric matrix To compute the one-loop corrections to the Zbb vertex in the nHDM, we make the simplifying assumption that only the top and bottom quarks exist and the (t, b) Cabibbo-Kobayashi-Maskawa matrix element is 1. The relevant part of the Yukawa Lagrangian is [15] (16) where the e k and f k are Yukawa coupling constants.

The charged-scalar contribution
In the nHDM at the one-loop level, both δg L and δg R are the sum of a contribution, which we denote through a superscript c, from diagrams having charged scalars and top quarks in the internal lines of the loop, and another contribution, which we denote through a superscript n, from diagrams with neutral scalars and bottom quarks in the internal lines: The charged-scalar contribution has been computed long time ago [3]. It corresponds to the computation of the diagrams in figure 1. It is Figure 3 1 Figure 1: Feynman diagrams that produce equations (21) and (22).
where the functions f L and f R are defined through In equations (22) m t is the top-quark mass and m Z is the mass of the gauge boson Z 0 . In the approximation m Z = 0, the functions f L and f R do not depend of s w 3 and are symmetric of each other:

The neutral-scalar contribution
The neutral-scalar contribution to δg L and δg R has been recently emphasized in ref. [15], following the original computation in ref. [3]; it corresponds to the computation of the diagrams in figure 3 and it is recapitulated in appendix B. 5 Assuming alignment and discarding 3 When m Z = 0 the Z 0 is indistinguishable from the photon and therefore the weak mixing angle is arbitrary and unphysical. 4 We have performed the numerical computation of Passarino-Veltman functions by using the Fortran library Collier [16] through interface CollierLink [17]. 5 The diagrams in figure 4 do not contribute to δg L and δg R in our case. This is so because diagrams (a) and (b) only exist, if there are only scalar doublets, when H + a is the charged Goldstone boson, and because diagrams (c) and (d) are proportional to the bottom-quark mass.    (24) and (25).  Figure 4: Feynman diagrams that do not contribute to δg L and δg R in the specific case of this paper.
the Standard-Model contributions that involve S 0 1 and S 0 2 , one has where F is an n × 1 vector with k th component F k = f k for k = 2, . . . , n, and The functions h L and h R are independent of s w when m Z = 0; however, that approximation is not a good one for those functions. We depict their real parts in figure 5. 6 One sees that,

The aligned 2HDM
In a two-Higgs-doublet model with alignment [15], the doublet Φ 2 may be rephased so that R 2 ≡ S 0 3 and I 2 ≡ S 0 4 are the new physical neutral scalars. Then, There are five New-Physics parameters on which δg L and δg R depend: the neutral-scalar masses m 3 and m 4 , the charged-scalar mass m C2 , and the Yukawa couplings e 2 and f 2 . One has [15] We now consider the scalar potential of the 2HDM [18], In the Higgs basis, µ 1 = − λ 1 v 2 / 2 and µ 3 = − λ 6 v 2 / 2. Because of alignment, λ 6 (and µ 3 ) are zero and From the masses of the scalars we compute where Λ 5 := |λ 5 |.
The masses m C2 , m 3 , and m 4 are not completely free, because they must comply with unitarity (UNI) and bounded-from-below (BFB) requirements [18]. For the sake of simplicity, in our analysis we assume λ 2 = λ 7 = 0. We enforce the UNI conditions on the quantities (30). Additionally, there are • UNI conditions [18] • the condition to avoid the situation of 'panic vacuum', namely [19] After computing λ 4 and Λ 5 through equations (30) and after checking inequalities (31), we verify whether there is any value of λ 3 that satisfies the inequalities (32)-(34); if there is, then the inputed masses m C2 , m 3 , and m 4 are valid; else, they are not.
We also compute the contribution of the new scalars to the oblique parameter where m W = 80.4 GeV is the mass of the gauge bosons W ± and Additionally, we apply constraints on the oblique parameter [20] This parameter has been computed in ref. [21] to be Here, We either enforce the phenomenological constraint [1] or we allow for other New Physics beyond the 2HDM and apply milder requirements by allowing values of T and S within its 3σ and 2σ bounds, respectively.

Parameterization of the neutral-scalar mixing
In the three-Higgs-doublet model with alignment, where T is a 4 × 4 real orthogonal matrix. We parameterize where O pq (θ) represents a rotation through an angle θ in the (p, q) plane. Now, O 13 (θ 5 ) is a rotation that mixes R 2 and I 2 , and O 24 (θ 6 ) is a rotation mixing R 3 and I 3 , viz. they represent rephasings of the doublets Φ 2 and Φ 3 , respectively. Since such rephasings are unphysical, one may without loss of generality drop those two rotations from the parameterization (43), obtaining where c p = cos θ p and s p = sin θ p for p = 1, 2, 3, 4. Then, with and The contribution of the new scalars to the oblique parameter T , given in equation (23) of ref. [22], is For the oblique parameter S one has

The scalar potential
The parameters The scalar potential of the 3HDM has lots of couplings and it is impractical to work with it. So we concentrate on a truncated version of the potential, viz. we discard from the quartic part of the potential all the terms that either do not contain Φ 1 or are linear in Φ 1 . 7 The remaining potential is where µ 1,2,3 and λ 1,4,5,7,8 are real and the remaining parameters are in general complex. In order that the VEV of Φ 1 is v √ 2 and the VEVs of Φ 2 and Φ 3 are zero, one must have In order that the charged-scalar mass matrix is diag (m 2 C2 , m 2 C3 ), one must have Equations (51) and (52) are the conditions for the charged Higgs basis. Next we write down the conditions for alignment, i.e. for H ≡ S 0 2 to have mass m 2 and not to have mass terms together with either R 2 , R 3 , I 2 , or I 3 : hence µ 4 and µ 5 are zero too. The mass terms of R 2 , R 3 , I 2 , and I 3 are given by where N is a 4 × 4 real symmetric matrix. Using equations (42) and (44), one finds that Equations (55) allow one to compute λ 7 , λ 8 , λ 10 , λ 11 , λ 22 , and λ 25 by using as input the masses of the charged scalars and the masses and mixings of the neutral scalars. On the other hand, λ 4 , λ 5 , and λ 19 constitute extra parameters that we input by hand-just as we did with λ 3 in section 3.
UNI constraints These constraints state that the moduli of the eigenvalues of some matrices must be smaller than 8π. The method for the derivation of those matrices in a general nHDM was explained in ref. [14]. In our specific case, the UNI constraints are (using Λ i ≡ |λ i | for i = 10, 11,19,22,25), (57) must be smaller than 8π. In the inequalities (56), all the square roots are taken positive and λ 1 = m 2 2 /v 2 is positive too.

Numerical results
In this section we display various scatter plots obtained by using the formulas in sections 3 and 4. In all the plots, we have restricted the Yukawa couplings e 2 , e 3 , f 2 , and f 3 to have moduli smaller than 4π. The charged-scalar masses m C2 and m C3 were assumed to be between 150 GeV and 2 TeV. The neutral-scalar masses m 3 , . . . , m 6 were supposed to be lower than 2 TeV, but they have sometimes been allowed to be as low as 50 GeV. In practice, the upper bound on the scalar masses is mostly irrelevant, since the contributions of the new scalars to δg L and δg R tend to zero when the scalars become very heavy. Constraints on the masses of the new scalars of the 2HDM and 3HDM may be derived from collider experiments on the production and subsequent decay of on-shell Higgs bosons. The sensitivity is limited by the kinematic reach of the experiments; moreover, the constraints usually depend on the assumed Yukawa couplings of the scalars and the fermions, which we do not want to specify in our work.
Constraints from the process b → sγ are most stringent. For a 2HDM of type II, a lower bound m C2 > 480 GeV at 95% CL has been derived in ref. [26]. However, in the case of the 3HDM, it has been shown in refs. [10,27] that, due to the increased number of parameters, m C2 and m C3 may actually be lighter than the mass of top quark while complying with the constraints from b → sγ.
In the 2HDM, a bound m C2 150 GeV on the mass of the charged Higgs boson has been derived from searches at the LHC [28,29]. Recent global fits [30,31] give bounds on the scalar masses for various types of Yukawa couplings in the 2HDM. In ref. [30] it is claimed that the mass of the heavy CP-even Higgs boson m H > {450, 700} GeV, the mass of the CP-odd Higgs boson m A > {500, 750} GeV, and the mass of the charged Higgs boson m C2 > {460, 740} GeV; the first values in the curled brackets correspond to the "lepton specific" type of 2HDM while the second values correspond to the type II and the "flipped" 2HDM. In the fit [31] of the aligned 2HDM one finds a lower bound of the new-scalar masses m C2 , m 3 , and m 4 around 500 GeV or around 750 GeV, depending on the fitted mass range.
We depict in figure 6 the confrontation between experiment and the values of δg L and δg R attainable in the aligned 2HDM. One sees that, if one forces the 2HDM to comply with  (10a); similarly, the violet lines correspond to the value (10b) and the light-blue lines to the value (11). Red points agree with the 1σ intervals (10); green points agree with the 2σ, but not with the 1σ, intervals (10); and blue points agree either with the 1σ or the 2σ intervals (10). Both the red and the green points satisfy the 1σ limits in equation (41) the S and T -oblique parameter constraints (41), then the 2HDM cannot achieve a better agreement with solution 1 for g L and g R than the SM; in particular, when one uses the A b value (11), the 2HDM cannot even reach the 2σ interval. Only when one allows both for a laxer S-and T -oblique parameters constraints and for a very low neutral-scalar mass m 3 60 GeV are the central values of both solutions 1 fit and 1 average attainable. In the right panel of figure 6 one sees that, if both new neutral scalars of the 2HDM have masses larger than 150 GeV, then the fit to solution 1 is never better than in the SM case, even if one does not take into account the S and T -parameter constraints. 8 In figure 7 we display the same points as in figure 6, now distinguishing the neutral-scalar contribution to δg L from the charged-scalar contribution to the same quantity. The same exercise is performed in figure 8 for the contributions to δg R . In the left panels of figures 7  and 8 one can see that the agreement of some blue points with solution 1 fit is obtained not just by using very light neutral scalars and laxer oblique parameters S and T , but also through a fine-tuning where large neutral-scalar and charged-scalar contributions almost cancel each other. In the right panels of those figures one sees that, when both neutral scalars have masses above 150 GeV, the signs of the neutral-scalar and charged-scalar contributions are the same-this explains the agreement worse than in the SM observed in figure 6. One also sees in figures 7 and 8 that the neutral-scalar contributions δg n L and δg n R are often comparable in size to, or even larger than, the charged-scalar contributions δg c L and δg c R , respectively. Thus, the usual practice of taking into account just the charged-scalar contribution may lead to erroneous results.
One might hope the situation of disagreement with experiment to be milder in the 3HDM relative to the 2HDM, but one sees in figure 9 that this hardly happens. The agreement of the 3HDM with experiment may be better than the one of the 2HDM, but only in the case where very light neutral scalars exist. We have checked that, just as in the 2HDM, the better agreement occurs through an extensive finetuning where δg n L ≈ −δg c L and δg n R ≈ −δg c R . In figures 6-9 we have tried, and failed, to make the fits of solution 1 in the 2HDM and in the 3HDM better than in the SM. Things are different with solution 2, which the nHDM models can easily reproduce-with some caveats. We remind the reader that in solution 2 the parameter g L is about the same as predicted by the SM, but the parameter g R has sign opposite to the one in the SM, viz. g R ≈ −0.08 in solution 2 while g R ≈ +0.08 in the SM. In the left panel of figure 10 and in figure 11 we see how the fit of solution 2 works out in the case of the 2HDM. One sees that one can attain the 1σ intervals and the best-fit points both of solution 2 fit and of solution 2 average , but this requires (1) the new scalars of the 2HDM to be lighter than 440 GeV, (2) the Yukawa coupling f 2 to be quite large, and (3) the Yukawa coupling e 2 to be relatively small, possibly even zero. In practice, the upper bound on the masses of the scalars originates in the upper bound that unitarity imposes on f 2 , as seen in the middle panel of figure 11; we have taken (rather arbitrarily) that upper bound to be |f 2 | < 4π ≈ 12.5. In the left panel of figure 11 one sees that |f 2 | must be larger than 9 anyway. It is also clear from figure 10 that, the lighter the new scalars are allowed to be, the easier it is to reproduce solution 2; moreover, it is easier to reproduce solution 2 average , viz. with the value (11) for A b , than solution 2 fit , viz. with the value (10b) for A b , because solution 2 average does not necessitate m 3 to be as low as solution 2 fit .
In the right panel of figure 10 and in figures 12 and 13 we illustrate the fitting of solution 2 in the 3HDM. Comparing the left and right panels of figure 10, we see that the 2HDM and the 3HDM give similar results, but in the 3HDM it is possible to reach solution 2 average with larger masses of the new scalars. Indeed, in the 3HDM the lightest neutral scalar m 3 may be as heavy as 620 GeV, while in the 2HDM m 3 < 420 GeV. Like in figure 11, in figures 12 and 13 we have used points that satisfy the S-and T -parameters 1σ bounds (41), that fall into the 1σ intervals of δg L and δg R for solution 2 average , 9 and that have m 3 > 100 GeV. In figure 12 we display the charged-and neutral-scalar contributions to δg L and δg R . One sees that solution 2 may be considered a finetuning, with |δg c R | |δg n R , δg c L , δg n L |. We stress once again that the neutral-scalar contributions are as instrumental as the chargedscalar ones in obtaining decent fits. In figure 13 we illustrate the moduli of the f Yukawa couplings and their relationship to the masses of the scalars. One sees that there is a bound |f 2 | 2 + |f 3 | 2 9, but each one of the Yukawa couplings f 2 and f 3 may separately vanish. One also observes that there is a simple straight-line correlation between the maximum possible value for the mass of the lightest charged scalar, m C2 , and the minimum possible value for the largest of the Yukawa couplings f 2 and f 3 . It is worth pointing out that in the 3HDM, just as in the 2HDM, the masses m 3 , m 4 , and m C2 must be low (because of the unitarity upper bound on the f Yukawa couplings), but in the 3HDM the masses m 5 , m 6 , and m C3 do not need to be low-they may be of order TeV.

Conclusions
The Standard Model (SM) has a slight problem in fitting the Zbb vertex, since it produces a g R smaller than what is needed to reproduce the fit (10); this discrepancy becomes larger when one uses for A b the value (11). In this paper we have found that this small problem can only worsen when one extends the SM through a nHDM. This is because the contributions of the new scalars usually produce a negative δg R , i.e. they go in the wrong direction to alleviate the problem, aggravating it instead.
There is one possible escape from this conclusion if the extra neutral scalars of the nHDM are very light, i.e. lighter than the Fermi scale, because the contribution of the neutral scalars to δg R may in that case be positive and partially compensate for the inevitably negative contribution of the charged scalars. This is a contrived effort, though, both because it is experimentally difficult to accomodate very light neutral scalars and because, from the theoretical side, light neutral scalars together with heavy charged scalars easily lead to a much-too-large oblique parameter T .
In this paper we have considered the possibility that, in nHDM models, we might look instead at an alternative fit of the Zbb vertex, wherein g R has the opposite sign from the one predicted by the SM. This is what we have called "solution 2" in table 1. That solution necessitates a very large negative δg R (together with a small δg L ), that may seem like a finetuning, but is easy to obtain in a nHDM. This solution, though, also works only if the new scalars are relatively light and if at least one of the Yukawa couplings denoted f k in equation (16) is quite large, viz. larger than 9 or so.

A Definition and measurements of R b and A b
The experimental quantities R b and A b are defined for e + e − collisions at the Z 0 peak, i.e. with √ s ≈ m Z . Let the quark q (q = u, d, s, c, b) couple to the Z 0 as in e + e − collisions at the Z 0 peak; thus, R b is the fraction of the produced hadrons that contain a bb pair. Clearly, in the absence of QCD, QED, and mass corrections, When one includes QCD, QED, and mass corrections equation (A7a) gets substituted by equation (7) and R b decreases from the value in equation (A7b) to the SM prediction [4] 0.21581. Similarly, s c + s u + s s + s d becomes 1.3184 instead of 1.32827 as in equation (A5).
If the mass of the bottom quark was zero, equation (5) would read at the tree level. The quantity A b was accessed at LEP 1 through the forward-backward asymmetry of the produced bb quark pairs, where A e , that is when the polarization P e of the electron beam is zero. The SLD Collaboration has used polarized beams (P e = 1) and therefore it could directly access where the subscripts L and R refer to the electron's polarization and the subscripts F and B refer to the forward or backward direction of travel of the final-state bottom quarks. The value of A b obtained from the SLD measurement is A fit b = 0.923 ± 0.020 and is 0.6σ below the SM value [1]. However, this good agreement only applies to the overall fit of many observables. Extracting A b from A 0,b FB when A e = 0.1501±0.0016 leads to A b = 0.885±0.0017, which is 2.9σ below the SM prediction. The combined value A b = 0.901 ± 0.013 deviates from the SM value by 2.6σ. These discrepancies in A b could be an evidence of New Physics, but they could also be due to a statistical fluctuation or another experimental effect in one of asymmetries; more precise experiments are needed.
A direct measurement of the Zbb couplings at the LHC is challenging because of the large backgrounds in the detection of a Z 0 decaying into a bottom quark-antiquark pair. A recent study [6] has proposed a novel method to probe the anomalous Zbb couplings through the measurement of the cross section of the associated production gg → Zh at the High Luminosity LHC.
Lepton colliders of the next generation, vg. the CEPC, ILC, or FCC-ee offer great opportunities for further studies of the Zbb vertex, because they could collect a large amount of data around the Z 0 pole. In the analysis [25] there is a list of the observables that are most important for improving the constraints on the Zbb coupling, and of the expected precision reach of those three proposed future e + e − colliders. These estimates, for the observables directly related to the Zbb coupling, are summarized in table 2. We see that, with an increase of precision of more than one order of magnitude, a future collider has the potential to solve the A 0,b FB discrepancy found at LEP. If its results are SM-like, a future lepton collider can provide strong constraints on models beyond the SM; if the A 0,b FB discrepancy found at LEP does come from New Physics, then any of the three next-generation e + e − colliders will be able to rule out the SM with more than 5σ significance [25].

B General formula for the neutral-scalar contribution
According to ref. [15], the contributions to δg L and δg R of loops with internal lines of neutral scalars and bottom quarks are the sums of three types of Feynman diagrams. Thus, δg n L = δg n L (a) + δg n L (b) + δg n L (c), δg n R = δg n R (a) + δg n R (b) + δg n R (c). A ll V T F l V † F * l C 00 0, m 2 Z , 0, 0, m 2 l , m 2 l , # of Z 0 s ∼ 2 × 10 7 ∼ 2 × 10 9 ∼ 10 9 ∼ 10 12 Table 2: The estimated precision reach for Zbb observables at future e + e − colliders according to ref. [25]. The present result for each observable is shown in the second column. The third column shows the σ of the present measurements at LEP and SLC, respectively, while the other columns show the estimates of the precision reach for the future colliders. In each entry, the number in the top line shows the total uncertainty and the number (in parenthesis) in the bottom line shows the systematic uncertainty. The last row shows the expected number of Z 0 events that will be collected.
where A is the matrix defined in equation (15), and