The Zbb Couplings at Future e+e- Colliders

Many new physics models predict sizable modifications to the SM Zbb couplings, while the corresponding measurements at LEP and SLC exhibit some discrepancy with the SM predictions. After updating the current results on the Zbb coupling constraints from global fits, we list the observables that are most important for improving the Zbb coupling constraints and estimate the expected precision reach of three proposed future e+e- colliders, CEPC, ILC and FCC-ee. We consider both the case that the results are SM-like and the one that the Zbb couplings deviate significantly from the SM predictions. We show that, if we assume the value of the Zbb couplings to be within 68% CL of the current measurements, any one of the three colliders will be able to rule out the SM with more than 99.9999% CL (5 sigma). We study the implications of the improved Zbb coupling constraints on new physics models, and point out their complementarity with the constraints from the direct search of new physics particles at the LHC, as well as with Higgs precision measurements. Our results provide a further motivation for the construction of future e+e- colliders.


Introduction
The LHC has just started to run at an unprecedented center-of-mass energy, 13 TeV, and will be able to probe new physics (NP) at higher energies. At the same time, precision measurements of electroweak physics at future e + e − machines will also offer powerful probes of Beyond the Standard Model (BSM) physics.
The next lepton collider may not be too distant in the future from us. Several compelling plans exist, including the International Linear Collider (ILC) [1], FCC-ee, formerly known as TLEP [2], and the Circular Electron-Position Collider (CEPC) [3]. With the discovery of the Higgs boson, the primary goal of such future e + e − colliders will be to produce a large sample of Higgs boson events at around ∼ 250 GeV to precisely measure the Higgs boson's properties, acting as a "Higgs factory". On the other hand, such e + e − colliders could also collect large amount of data around the Z-pole, producing several orders of magnitude more Z-bosons than what was produced at LEP. The large amount of Z-pole data would greatly improve the measurement of the several electroweak precision observables (EWPOs), which could provide strong constraints on NP.
Several studies on the measurement of EWPOs at future e + e − colliders have been performed [4][5][6][7], mainly focusing on the oblique corrections parameterized by the Peskin-Takeuchi parameters S and T [8]. NP could also have sizable non-universal corrections.
The corrections to the Zbb vertex is particularly interesting, and quite generic in NP models. For example, being the left-handed top and bottom quarks in the same electroweak (EW) doublet, new physics that couples to the top quark usually also affects the Zbb couplings [9]. Composite Higgs models with light top partners usually predict a large correction to the Zb LbL coupling, unless it is protected by some symmetry analogous to the custodial symmetry that protects the weak isospin [10]. Additionally, heavy Higgs bosons typically couple more strongly to the third generation quarks and modify the Zbb coupling through loops [11].
The story is even more interesting on the experimental side. At LEP, the left and right handed Zbb couplings are mainly determined by two measurements at the Z-pole: R 0 b , the ratio of the Z → bb partial width to the inclusive hadronic width, and A 0,b FB , the forward-backward asymmetry of the bottom quark. The measured value of R 0 b agrees with the most recent two-loop calculation of its SM prediction within 1 σ [12][13][14]. A 0,b FB , instead, exhibits a long-standing discrepancy with the SM prediction with a significance at around 2.5 σ [12,14]. In addition, SLD directly measured the bottom quark asymmetry 0.923 ± 0.020 0.93463 ± 0.00004 Table 1: The measured values and SM predictions of R 0 b , A 0,b FB and A b according to the most recent result from the Gfitter group [14].
with longitudinal beam polarizations, A b , which is consistent with the SM prediction within 1 σ but slightly prefers a shift in the same direction as the LEP A 0,b FB measurement does. The measured values and SM predictions of R 0 b , A 0,b FB and A b are summarized in Table 1 4 . To obtain the desired modification for A 0,b FB without violating the experimental constraint on R 0 b and A b , a simultaneous modification of both the Zb LbL and Zb RbR couplings is required. To obtain the best estimation for the preferred values of the Zb LbL and Zb RbR couplings, a global fit to all precision data has to be performed (see e.g. [16,17], for earlier studies).
A future e + e − collider offers great opportunities for further studies on the Zbb couplings. With the huge improvement on statistics at Z-pole, it will surely have the potential to resolve the A 0,b FB discrepancy at LEP. If the result agrees with the SM predictions, a e + e − collider can provide very strong constraint on NP models; if the LEP A 0,b FB discrepancy does come from NP, a e + e − collider will have the potential to rule out the SM with enough significance, therefore providing strong indirect evidence for physics beyond the SM. In either case, the results would greatly improve our understanding of fundamental particle physics. In this paper, we perform a study of the constraints on non-universal modifications of the Zbb couplings from prospective precisions at future e + e − colliders, which, to our best knowledge, is the first study of such kind.
The rest of this paper is organized as follows. In Section 2, we review the current constraints on the Zbb couplings and discuss the importance of including the strong coupling constant in the global fit. In Section 3, we compare the precision reaches of the three proposed e + e − colliders and outline the most important measurements needed for improving the Zbb coupling constraints. We then perform a model independent analysis to constrain the effective Lagrangian responsible of the modifications of the Zbb vertex, in both the case that the results are SM-like and the one that NP causes a significant 4 Ref. [15] quotes a value of 0.00015 as the theoretical uncertainty for the SM prediction for R 0 b . Either way, the current theoretical uncertainty of R 0 b is too small to have an impact on the current precision data. deviation in the bottom asymmetries. In Section 4, we interpret these constraints on specific NP scenarios including two Higgs doublet models (2HDMs), composite Higgs models and the Beautiful Mirror Model. We will compare these constraints to the direct reach of the NP particles at the LHC, as well as to the constraints from oblique parameters and Higgs precision measurements. In Section 5, we present our conclusions. Finally, in Appendix A, we discuss our treatment of theory uncertainties.

Current constraints on the Zbb couplings
In this section we present the constraints on the Zbb couplings from the global fit of the current precision electroweak data. We follow closely the fit procedure of the Gfitter group [14,18] 5 , with the Z-pole data from LEP, SLD [12] along with the measurements of the W , top and Higgs masses [24][25][26][27] and the hadronic contribution to the running fine structure constant [28] 6 . In particular, our procedure uses the input parameters and the electroweak and QCD corrections to the electroweak observables of the latest GFitter analysis [14], even though it does not make use of the GFitter public code.
There are, in fact, a few important differences between our procedure and the one of the Gfitter group. First, we include the world average of the strong coupling constant α S (M 2 Z ) [24,30] (denoted as α S (M 2 Z ) avg. ) as a constraint in the fit 7 . Later in this section, we will show that the inclusion of α S (M 2 Z ) avg. has a moderate, but non-negligible, impact on the constraints of the Zbb couplings 8 . Second, the observable sin 2 θ l eff (Q FB ) is not included in our fit. sin 2 θ l eff (Q FB ) is a direct measurement of the leptonic effective weak mixing angle at LEP, using the charge forward backward asymmetry. Its measurement has a strong model dependence and the LEP result explicitly assumes the SM, therefore it is difficult to interpret this measurement in the presence of vertex corrections. In practice, this observable is not precisely measured and has a small impact in the global fit. To summarize, the SM free parameters considered in our fit are M H , M Z , m top , ∆α (5) had and α S (M 2 Z ). The additional observables included in our fit are 5 For a global fit in the context of the Standard Model effective field theory, see e.g. [19][20][21][22][23]. 6 It should be noted that there also exist non-trivial bounds on the Zbb couplings from hadron colliders, as pointed out in e.g. Ref. [29], although the precision can not compete with the one obtained from lepton colliders. 7 The value we use is ∆α (5) had = (2757 ± 10) × 10 −5 . 8 Ref. [17], [31] and [24] included α S (M 2 Z ) avg. but did not comment on its impact on the Zbb couplings.
The coupling of the Z to left and right handed bottoms, denoted by g Lb and g Rb , are given in the following interaction term, where c W ≡ cos θ W , with θ W the weak mixing angle, and g is the SU (2) gauge coupling.
Through out this paper we shall use δg Lb and δg Rb to parameterize the modification of the Zbb couplings, defined as where g SM Lb and g SM Rb are the SM predictions for g Lb and g Rb , which at the tree level are given by At the tree level, R 0 b , A b and A 0,b FB at Z-pole can be written as , where q denotes a sum over all quarks except the top quark, and (2.5) These expressions will be modified once loop corrections are included.
It should be pointed out that the measurements at Z-pole alone could not determine the signs of g Lb and g Rb . The off-peak measurements can resolve the sign ambiguities due to interference of the Z diagram with the s-channel photon diagram. However, as pointed out in Ref. [32], the LEP data at scales different from m Z are limited in statistics and could definitely resolve the sign of g Lb , but not the one of g Rb . Nevertheless, NP theories able to flip the sign of g Rb are typically in tension with other EW precision data such as the constraints on S and T parameters [16,32]. Future lepton colliders will collect a large amount of data at higher scales, which should completely resolve this ambiguity 9 . 9 As an example, the value of A 0,b FB at around 240 GeV is changed by ∼ 0.2 if the sign of g Rb is flipped with respect to the SM prediction. On the other hand, the proposed CEPC run at ∼ 240 GeV would collect 5 ab −1 of data over ten years with two detectors [3], resulting in a statistical uncertainty of ∼ 0.0003 for A 0,b FB , which is sufficient for resolving the sign of g Rb as long as enough events are left after the event selection and the systematics are under control.
In this paper we do not consider the possibility that g Rb has the opposite sign of its SM prediction.
The potential NP that modifies the Zbb couplings can also change other EW observables. To capture the most relevant corrections without relying too much on the model assumptions, we will consider NP scenarios that contribute to the oblique parameters S and T along with the modified Zbb couplings. Therefore, our minimal model assumption is SM together with S, T , δg Lb and δg Rb treated as free parameters. For later convenience we will denote it as (SM+S, T, δg Lb , δg Rb ).
With the model assumptions and fit procedure described above, we obtain the constraints on δg Lb and δg Rb , shown in Fig. 1. The blue (orange) region corresponds to a confidence level (CL) smaller than 68% (95%), while the green dot is the SM prediction (δg Lb = δg Rb = 0). In addition, in the left plot we show the individual constraints from As shown in Table 1, the current experimental (1σ) uncertainties are 0.00066 for R 0 b and 0.0016 (0.020) for A 0,b FB (A b ). These numbers together with Eq. (2.6) provide a good analytical understanding of the Zbb coupling constraints. First, R 0 b is more constraining than A 0,b FB (A b ) and leads to a large positive correlation between δg Lb and δg Rb . Second, R 0 b is numerically more sensitive to δg Lb while A 0,b FB (A b ) is more sensitive to δg Rb . As mentioned previously, in our fit we have included α S (M 2 Z ) avg. (world average) as a constraint in the global fit. α S (M 2 Z ) avg. includes several different measurements, but is dominated by the lattice calculation [24,30]. More explicitly, to avoid double counting, we use the PDG world average excluding electroweak precision test (EWPT) results [24], which is The Gfitter group [14,18] does not include this constraint in the fit, since the global fit for the SM and also for the oblique parameters S and T is not sensitive to α S (M 2 Z ) avg. . One can simply extract the value of α S (M 2 Z ) from the EW global fit, which is in good agreement with α S (M 2 Z ) avg. assuming the SM (+S and T ). This can be seen by our first two results in Table 2. However, the extraction of α S (M 2 Z ) from EW global fit has some model dependence and, with our model assumption, (SM+S, T, δg Lb , δg Rb ), the agreement with α S (M 2 Z ) avg. becomes a bit worse (but still < 1σ). This result is shown in the last row of Table 2. This suggests that including α S (M 2 Z ) avg. in the fit can have some impact on the Zbb coupling constraints. Indeed, as shown in the right plot of Fig. 1, the inclusion of α S (M 2 Z ) avg. has some small but non-negligible effect on δg Lb and on its correlation with δg Rb . This is because α S (M 2 Z ) avg. has a stronger effect on R 0 b than on A 0,b FB and prefers smaller δR 0 b s, leading to an increase in δg Lb and a smaller decrease in δg Rb (see Eq. (2.6)). In the future, while both the precision of the Z-pole data and α S (M 2 Z ) avg. will be improved, α S (M 2 Z ) avg. will at least provide an important consistency check and it will be interesting to include it in the global fit for the Zbb coupling constraints.
To summarize this Section, in Table 3  0.1153 ± 0.0035 Table 2: Prediction for α S (M 2 Z ) from EW global fit with different model assumptions and without the world average α S (M 2 Z ) avg. = 0.1185 ± 0.0005 as a constraint.  Table 3, the correlation between the two groups, (S, T ) and (δg Lb , δg Rb ), are not very strong, and we expect a similar behavior at future colliders, given that the relative improvements are not extremely different for different observables. For simplicity, in the next Section we shall focus on the constraint on δg Lb and δg Rb and marginalize over S and T . We refer the reader to other literature, e.g. Ref. [5], for prospective constraints on S and T at future e + e − colliders.
3 Zbb coupling constraints from future e + e − colliders Future e + e − colliders will be able to significantly improve the precision of the measurements at the Z-pole thanks to a much larger statistics. The reaches have been estimated in the Technical Design Report (TDR) for ILC [1], the TLEP whitepaper for FCC-ee [2] and the preliminary Conceptual Design Report (preCDR) for CEPC [3]. However, these estimations usually either contain only a subset of EW observables, or have combined several observables into one (e.g. the effective leptonic mixing angle sin θ l eff ), and are therefore not straight forward to apply in our study. In addition, some of the estimations are rather preliminary, having strong dependence on the assumptions for systematic uncertainties and whether or not beam polarization will be implemented. In Section 3.1, we outline the key observables that are needed for improving the Zbb coupling constraints and try to estimate their precision reach at the three future colliders. Using these estimations, we proceed to study the constraints on the Zbb couplings by the method of global fit, and the results are shown in Section 3.2 and 3.3.
In our study we consider the following benchmark scenarios for the three colliders: • CEPC with a relative conservative estimation for the systematic uncertainties and with a statistics of only 2 × 10 9 Z events. While beam polarization could be a potential option for the run at the Z-pole, here we assume that it is not implemented.
• CEPC+, which is CEPC with a more aggressive estimation for the systematic uncertainties, and assuming 10 10 Z events.
• ILC, with a lower statistics (10 9 Z events), but with beam polarization.
• FCC-ee with 10 12 Z events and beam polarization.

Precision of the EWPOs at future e + e − colliders
The observables directly related to the Zbb couplings are R 0 b , A 0,b FB (measured without beam polarization) and A b (measured with beam polarization). However, the three observables also have explicit dependence on the effective weak mixing angles, sin 2 θ l eff (for leptons) and sin 2 θ b eff (for bottom) 10 . In particular, A 0,b FB is quite sensitive to sin 2 θ l eff as it is proportional to A e . In fact, at present, the LEP measurement of A 0,b FB provides one of the best determination of sin 2 θ l eff assuming SM (the other one being A LR from SLD). Therefore, to extract the Zbb couplings from A 0,b FB , it is important to obtain an independent determination of sin 2 θ l eff , while the most precise such determination is provided by the leptonic asymmetry observables. On the other hand, R 0 b and A b are numerically not very sensitive to sin 2 θ b eff . Without beam polarization, the forward-backward leptonic asymmetry A 0,l FB (= 3 4 A 2 l ) can be measured. In addition, a measurement of A l (A e and A τ ) can be obtained using the average final-state longitudinal τ polarization and its forward-backward asymmetry [12], which we denote as A l (P τ ). With beam polarization, the left-right asymmetry A LR (= A e ) can be directly measured, but it is rather irrelevant in terms of the Zbb coupling constraints as A b can be directly measured as well, and we have checked that the impact of the improvement of A LR on the Zbb coupling is rather negligible. However, A LR can still be helpful for constraining the Zbb couplings in a global fit, for example, in the case that two (or more) colliders are built and only one of them have beam polarization, similar to the situation of LEP and SLC.
It is worth noting that, apart from R 0 b , a number of additional observables are also sensitive to the coupling combination g 2 Lb + g 2 Rb through the dependence on the total hadronic decay width, among which R 0 l , which is the ratio of the total hadronic Z decay width and the Z decay width to one lepton species, is relatively well measured and provides the best sensitivity. We find that a significant improvement of the precision of R 0 l can have a significant impact on the Zbb coupling constraints. In particular, with the estimated precision reach at FCC-ee shown later, the measurement of R 0 l turns out to be more constraining than the one of R 0 b to the Zbb couplings. However, the constraint from R 0 l depends strongly on the assumption that the coupling of Z to other fermions are SM-like, and one should be cautious when applying the Zbb coupling constraints to a model for which this assumption is not true. In the end of Section 3.2 we will also show the results for FCC-ee with a more conservative estimation for the precision of R 0 l . To obtain the estimation of the precision reach of the several observables, the following procedure is performed. For each observable, we use the estimation in the corresponding literature, if it is provided. In particular, if a range of values is provided, we choose the more conservative one. If the estimation is not provided in the literature, we estimate the precision with the following strategies: we assume that the systematic uncertainties at CEPC is a factor of 1/3 the ones at LEP and the systematic uncertainties for the scenario "CEPC+" is reduced by an additional factor of 1/2 from CEPC 11 . For CEPC and CEPC+, we assume the statistical uncertainty simply scales with 1/ √ N , where N is the total number of Z events expected to be collected. Additionally, for ILC, Ref. [1] does not provide an estimation for the uncertainty of R 0 l (∆R 0 l ), for which we adopt the estimation in Ref. [18] by Gfitter, ∆R 0 l = 0.004. Finally, for the FCC-ee, Ref. [2] does not provide an estimation for the uncertainty of A b (∆A b ). We naïvely scale it from the estimation for the ILC,  Table 4: The estimated precision reach for the observables (which current experimental measurement is shown in the first column) most relevant to constrain the Zbb coupling at future colliders. The second column shows the uncertainty of the present measurements from LEP and SLC, while the other columns show the estimations of the precision reach for different future colliders and scenarios. The numbers highlighted in blue are our own estimation. In each entry, the number at the top shows the total uncertainty while the number at the bottom (in parenthesis) shows the corresponding systematic uncertainty. A blank entry denotes an observable that is either not measured or not important for our global fit. The last row shows the expected number of Z events that will be collected.
The estimations for the observables mentioned above are summarized in Table 4. A similar method is used to estimate the precision reach for the additional EW observables not listed in Table 4, even if we have checked that they have a much smaller impact on the Zbb coupling constraints. In Table 4 ILC and FCC-ee can measure A 0,b FB , A 0,l FB and A l (P τ ), but the corresponding observable with beam polarization, A b and A LR , can be measured significantly more precisely, so, for simplicity, we do not include the former observables.
A potential issue for the interpretation of future measurements is the effects of the theoretical uncertainties, which could become important if they are much larger than the experimental uncertainties. In our study, we assume that the electroweak three-loop corrections will be computed in the future. In that case, the effects from the theoretical uncertainties on the Zbb coupling constraints are very small and can be safely neglected even for FCC-ee. This is either because the theoretical uncertainty is numerically small (such as δ th R 0 b which is estimated to be a few times 10 −5 [6]) or the observable itself has little effect on the Zbb coupling constraints, such as the top quark mass. The theoretical uncertainty of sin 2 θ b eff also has little impact, since A b is not very sensitive to it. More details on the treatment of the theoretical uncertainties can be found in Appendix A.

SM-like measurements and constraints on NP
In this Section, we assume the future experimental results agree perfectly with the SM predictions and the estimated precision of future measurements as described in the previous Section. The preferred regions in the (δg Lb , δg Rb ) plane obtained by our global fit are shown in Fig. 2 We report the 1σ uncertainties of δg Lb and δg Rb as well as their correlation (ρ) in Table 5. Due to the strong correlation between δg Lb and δg Rb (in particular at CEPC), one need to be careful when using these results to constrain NP models, since in some models only one of δg Lb and δg Rb can receive a sizable contribution while the other one is close to zero. Therefore, in Table 5 we also show the 1σ uncertainties for δg Lb (δg Rb ), while δg Rb (δg Lb ) is fixed to zero. For CEPC, the estimation for ∆R 0 l in the preCDR [3] seems to be very conservative, suggesting little improvement of its systematic uncertainty from LEP to CEPC (see Table   4). A scaling of a factor of 1/3 on the systematic uncertainty would give a value of ∆R 0 l = 0.003 for the total uncertainty. The results for this scenario are shown in Table 6, which exhibits a slight improvement. In Table 6 we also show the results for FCC-ee with a more conservative estimation of ∆R 0 l , also using ∆R 0 l = 0.003 (instead of ∆R 0 l = 0.001).

Discovering NP through
A more interesting possibility is that the long standing A 0,b FB discrepancy does come from NP, in which case the precision reach at any of the three future e + e − colliders should be able to rule out the SM with very high significance and therefore provide strong indirect evidence for physics beyond SM. To illustrate this point, we consider the following two scenarios. Scenario I: we assume that the true values for δg Lb and δg Rb (denoted by δg 0 Lb and δg 0 Rb ) are given by the best fit values of the current data, δg 0 Lb = 0.0030 and δg 0 Rb = 0.0176 (see Table 3). Scenario II: we assume that the the true values of δg 0 Lb and δg 0 Rb are closer to zero, while still being consistent with the current measurements within   Table 6: Same as Table 5, but for CEPC and FCC-ee both with ∆R 0 l = 0.003, which serves as a reasonably optimistic estimation for CEPC and a conservative one for FCC-ee.
68%CL. As a benchmark point, we choose δg 0 Lb = 0.0009 and δg 0 Rb = 0.0075. In principle, one would expect the NP to have non-zero contributions to the S and T parameters, as well. We find that changing the central values of S and T of the hypothetical measurement within the current constraints has very small impact on the Zbb couplings. For simplicity we assume that the hypothetical data agrees with SM other than the modification to g Lb and g Rb .
The preferred regions in the (δg Lb , δg Rb ) plane are shown in Fig. 3. The two plots correspond to the two scenarios described above, and each shows the 99.9999% CL (corresponding to 5σ for a one-dimensional Gaussian distribution) constraints from different colliders. From the figure, it is clear that the SM prediction at zero (denoted by a green dot) can be ruled out at 99.9999% CL by all the e + e − colliders we discuss, even if we assume that the future measurements will point towards smaller values of δg Lb and δg Rb within 68% CL of the current measurements.

Implication on NP models
In this Section, we analyze the implications of the future measurements of the Zbb couplings on specific NP models. We start with a brief discussion of the constraints on  Figure 3: The preferred regions in the (δg Lb , δg Rb ) plane, given by the global fit of the future measurements at CEPC (in cyan), CEPC+ (in blue), ILC (in red) and FCC-ee (in black). The solid and dotted purple contours correspond to the 68% and 95% CL constraints from the current measurements. The two panels correspond to Scenario I and Scenario II presented in the text, and each plot shows the 99.9999% CL constraints from different colliders with dashed contours. The green dot is the SM prediction (δg Lb = δg Rb = 0). effective Lagrangians. At dimension 6, the only operators that modifies directly the Zbb couplings are (see e.g. [17,21]) After electroweak symmetry breaking, these operators lead to a shift in the Zbb couplings: where a Hb , a s HQ , a t HQ are the coefficients of the O Hb , O s HQ , O t HQ operators, respectively and v is the vacuum expectation value of the Higgs (v = 246 GeV). In Table 7, we present the constraints on these operators at the several future e + e − machines, assuming that a s HQ = a t HQ = a Hb = 1/Λ 2 . Scales as large as (20 − 30) TeV can be probed by the future measurement of the Zbb couplings.
Next, we pass to the analysis of specific NP frameworks that can generate some of the operators forementioned, including two Higgs doublet models, composite Higgs models current CEPC CEPC+ ILC FCC-ee Λ(TeV) 6.8 13 20 15 27

Two Higgs doublet models
As shown in e.g. [11], models with an extended Higgs sector can predict sizable NP contributions to the Zbb vertex. In particular, focusing on two Higgs doublet models (2HDMs), based on discrete symmetries to avoid flavor changing neutral currents (FCNCs) at the tree level, the most important contribution generically comes at the one loop level, from the charged Higgs exchange. The sign of the charged Higgs NP contribution to δg Lb (δg Rb ) is fixed and is always positive (negative). In a Type II 2HDM, the contribution to δg Lb (δg Rb ) increases at small (large) values of tan β, since the coupling H ±b L t R (H ±t L b R ) leading to a non-zero δg Lb (δg Rb ) is proportional to m t / tan β (m b tan β). In a Type I 2HDM, instead, both δg Lb and δg Rb increase at large values of tan β 13 , leading always to a NP contribution δg Lb δg Rb . In Fig. 4, we show the constraints on the m H ± − tan β plane, using the present measurement of the Zbb coupling (in purple) as well as the expected more precise measurement at CEPC (in cyan), CEPC+ (in blue), ILC (in red) and FCC-ee (in black). For the figure, we have assumed that the future measurements perfectly agree with the SM predictions and we have marginalized over the values of the S and T parameters.
In Type II models, interesting constraints arise at low values of tan β for which δg Lb δg Rb 14 . Type I models, instead, are only allowed in the region with small tan β unless m H ± is very large. If we specify the full spectrum of a 2HDM, including the masses of the 13 Here we use the tan β convention such that the two charged Higgs couplings are proportional to m t tan β and m b tan β.  In purple is the constraint we obtain using the present uncertainties on the EWPOs; in cyan, blue, red and black are the constraints expected with the future measurements at CEPC, CEPC+, ILC and FCC-ee, respectively. In the Type II 2HDM, the region below the curves is excluded. In the Type I 2HDM, the exclusion is above the curves. We assume that the future measurements perfectly agree with the SM predictions and marginalized over the values of the S and T parameters.
neutral scalar H and pseudoscalar A, as well as the mixing angle α − β between the two doublets, S and T are not free parameters. In general, the constraints will be stronger. Presently, LHC charged Higgs searches almost totally exclude charged Higgs bosons with a mass below the top mass in Type II 2HDMs [33]. There are no LHC searches at around the top mass, for 160 GeV < m H ± < 180 GeV up to date. Above 180 GeV, constraints are rather weak and cover only models with large values of tan β (tan β ≥ 40), for which the production cross section of the charged Higgs in association with a top quark is in the O(1) pb range. In this regime, the two most important bounds come from the searches for H ± → τ ν [33] and for H ± → tb [34]. At the 14 TeV LHC, also charged Higgs boson with mass above the top mass will be relatively well probed. In particular, searches for H ± → tb will have the potential to probe tan β 3 and tan β 15 for m H ± ∼ 500 GeV at the High Luminosity (HL)-LHC [35]. Comparing to our results of Fig. 4 (left panel), we see that constraints from future measurements of EWPOs can be complementary to direct searches for Type II 2HDMs, being able to probe low values of tan β even for m H ± m t , as well as the challenging region 160 GeV < m H ± < 180 GeV, presently not covered by direct searches.
Finally, one can interpret the searches for charged Higgs bosons in terms of Type I 2HDMs. Below the top mass, only a small region with tan β < 1 has not been yet probed by the H ± → τ ν search. Above the top mass, the exclusion is very week and is not covering any part of the plane shown in Fig. 4 (right panel). At the HL-LHC, this region will be very well probed by a H ± → tb search, with potential exclusions for the entire range of tan β presented in the figure, up to m H ± ∼ 500 GeV.

Composite Higgs models
Composite Higgs models usually predicts a large correction to the Zb LbL coupling, since a sizable mixing between the third generation quarks and the strong dynamics is needed to generate the large top mass. The correction to the Zb RbR coupling is usually much smaller, unless one specifically extend the fermion sector to generate a large correction (e.g., as in Ref. [36]). It was pointed out in Ref. [10] that an O(4) symmetry, which is the SU (2) L ⊗ SU (2) R symmetry, analogous to the custodial symmetry protecting the weak isospin, with the addition of a left-right parity P LR , could be used to protect the Zb LbL coupling, such that a natural composite Higgs model can be consistent with EW precision constraints. Nevertheless, in realistic models there still exist corrections to the Zb LbL coupling because 1) the P LR symmetry can only protect the Zb LbL coupling at zero momentum and 2) there are also several contributions that explicitly break P LR . These corrections could become relevant if the constraint on the Zb LbL coupling is significantly improved at future e + e − colliders. Ref. [37] estimates the size of different contributions to the Zb LbL coupling in minimal composite Higgs models with custodial protection.
(Also see Ref. [38] for a recent review.) While the magnitudes and signs of different contributions are rather model dependent, the leading correction usually comes from P LR breaking effects of fermion loops and is . Each contour represents the 95% CL constraint and the region in the top-left side of the contour is excluded. The grey horizontal line corresponds to g ρ /g ψ = 3/0.7 . In purple is the constraint we obtain using the present uncertainties on the EWPOs; in cyan, blue, red and black are the constraints expected with the future measurements at CEPC, CEPC+, ILC and FCC-ee, respectively.
as an equality, the results in Table 5 (assuming δg Rb = 0) can be interpreted in terms of constraints in the (g ρ /g ψ , f ) plane where g ρ ≡ m ρ /f and g ψ ≡ m 4 /f . This is shown in Fig. 5, where each contour represents the 95% CL constraint and the region in the top-left side of the contour is excluded. The grey horizontal line has g ρ /g ψ = 3/0.7 which corresponds to the benchmark point m ρ = 3 TeV and m 4 = 700 GeV of Ref. [37].
Since g ρ /g ψ is typically bounded to be a few times one, future e + e − colliders can constrain f to be at least a few TeVs thanks to the measurement of the Zbb couplings.
This is comparable to the constraints from the direct searches of top partners at the next run of the LHC, given that the mass of the top partner can not be much larger than f in order to obtain the correct Higgs mass [39]. The constraints from the Zbb couplings is significantly stronger than the ones from oblique parameters but weaker than the ones from Higgs precision measurement, and in particular from the HZZ vertex, quoted in Ref. [5]. The latter can, in fact, constrain f at the level of ∼ 2.8 TeV (CEPC) and ∼ 3.9 TeV (FCC-ee) at 95%CL. Other studies of future constraints on composite Higgs models can be found in Ref. [40,41]. To conclude, while the constraints from Zbb couplings has a stronger model dependence, it is complementary to the constraints from oblique corrections, Higgs precision measurements and direct searches and can be very helpful for the discrimination of different composite Higgs models.

The Beautiful Mirror Model
In the Beautiful Mirror Model proposed in Ref. [32], the modifications to the Zbb couplings are caused by the mixing of the bottom quark and new heavy vector-like quark(s) 15  singlet. If B has t 3 = − 1 2 , δg Rb would be negative and one would need a large shift in the coupling to flip its sign, g Rb ≈ −g SM Rb , in order to resolve the A 0,b FB discrepancy. Such a large shift requires a very light B quark and a large custodial symmetry breaking and has been almost completely probed by LHC direct searches for vector-like quarks [16,43,44].
A more appealing choice is t 3 = 1 2 , that can lead to a good fit of the EWPOs, without a too light B quark, thanks to a positive contribution to g Rb . This scenario was denoted as the "Top-less Mirror" in the original paper [32], since there is no top-partner in the new doublet quark. The new doublet quark alone can not simultaneously generate a sizable enough δg Lb , but this can be easily achieved by introducing an additional singlet. Therefore, we discuss an extension of the SM with the following vector-like quarks, where the numbers in the bracket denotes representations under SU (3) c , SU (2) L , and the U (1) Y hypercharge. The relevant terms in the Lagrangian are given by which leads to the following 3 × 3 mass matrix M B for the bottom-like quarks while the mass of the charge −4/3 quark X is simply given by M 1 , The shifts in the Zbb couplings are thus given by (4.10) The new quarks contributes also to the T parameter through fermion loops. Ignoring the small bottom mass and the higher order terms, the T parameter is given by Therefore, M 1 can not be much larger than a few TeV without the Yukawa coupling, violating perturbativity bounds.
The LHC is directly searching for the mirror quarks. As pointed out in Ref. [44], the charge −4/3 exotic quark X decays to b and W with the same sign of electric charges, which is extraordinary in theory but hard to capture in experiments, since it is very hard (if not impossible) to measure the charge of b-jets. As such, the strongest bounds on X come from searches of top partners decaying to b W . The recent CMS analysis [45] sets a lower limit of 912 GeV at the 95% CL for a pair produced top vector-like quark with 100%BR to b W , using 8 TeV data. Currently, this is the most stringent constraint on M 1 . There also exist bounds from bottom partner searches (e.g. Ref. [46]) which are slightly weaker.
The current constraint from the LHC is consistent with the one from EW precision data shown in the right plot of Fig. 6. The situation may get much more interesting in the future: the bounds on the mirror quark masses are expected to reach (2-2.5) TeV at the 14 TeV LHC [44,47] using the single production channel. The HL-LHC with 3000 fb −1 data could probably push the bound further to above 3 TeV, if the data is consistent with SM [48]. Such a bound would be in tension with the current EW precision data. If no mirror quark is found during the LHC run, it could be an indication that either 1) the LEP A 0,b FB discrepancy is not due to NP or it comes from some NP other than the Beautiful Mirror Model 16 or 2) the underlying NP is some extension or modification of the "minimal" Beautiful Mirror Model which evades the constraints from direct searches, still producing a good fit to EW precision data.

Conclusion
Precision measurements of SM couplings are the central goal of future lepton colliders.
Such measurements are complementary to direct searches at high energy proton colliders.
In this paper, we have extracted the constraints on possible modifications of the Zbb couplings from the SM predictions by performing global fits of both the current precision data and the prospective data from future e + e − colliders. We pointed out that the world average value of the strong coupling constant contains non-trivial information and should be included in the global fit of models with non-zero NP contributions to the Zbb vertex, which has not been pointed out elsewhere. For the future colliders, we summarized the set of observables most important for improving the Zbb coupling constraints and compared the precision reaches at CEPC, ILC and FCC-ee. We studied both the case that the results are SM-like and the one that the Zbb couplings deviate significantly from the SM prediction as suggested by the LEP A 0,b FB discrepancy. For the latter case, we showed that even if we assume that the future measurements will point towards smaller values of δg Lb and δg Rb within 68% CL of the current measurements, any one of the proposed e + e − colliders will be able to rule out the SM with more than 99.9999% CL, equivalent to 5 standard deviations for a one-dimensional Gaussian distribution.
Finally, we studied the implication on NP models from the improvements of the Zbb coupling constraints. We first considered generic 2HDMs, in which the limits from pre- We have also shown that the particular models considered in this paper generically predict deviations in the Higgs couplings, which can also be measured very precisely at the Higgs factory stage of the lepton colliders. The interplay between Higgs and Z precision measurements will be very valuable at future e + e − machines, in extracting maximal information about new physics.
x is predicted to be some certain value, x th , up to some uncertainty, δ. This uncertainty could come from the omission of higher order terms in a fixed order calculation, or the subtlety in the definition of certain quantities (e.g. the top mass). As such, it is strictly speaking not a statistical quantity and there is no good reason to assume it follows a Gaussian distribution. We assume x takes a probability density function h(x; θ) that is flat within x th ± δ and zero elsewhere, On the other hand, we assume the measured value, x mea , takes a Gaussian distribution g(x mea |x) centered around the true value x with standard deviation σ.
The distribution of the measure value x mea given a set of model parameters θ is thus obtained by convolution: f (x mea ; θ) = dx g(x mea |x)h(x; θ) which reduces to the Gaussian distribution in the limit δ → 0. Eq. (A.2) can be implemented in a global fit with N observables with a modified χ 2 function (assuming no correlation) where for each observable j, M j is the measured value, O j is the predicted value, σ j is the experimental uncertainty and δ j is the theoretical uncertainty. R 0 b and A 0,b FB (A b ) are directly related to the Zbb couplings, and their theoretical uncertainties are most important to our study. The theoretical uncertainty of R 0 b (δ th R 0 b ) is estimated to be 1.5 × 10 −4 from two-loop diagrams without closed fermion loops and higher-order contributions [15] and could be reduced to a few times 10 −5 assuming the 2-loop and 3-loop computations will be completed in the future [6]. Naively, one would expect it to have an impact, especially for FCC-ee which will be able to measure R 0 b to a precision of about 6 × 10 −5 . However, even with a conservative estimation, δ th R 0 b = 5 × 10 −5 , and for FCC-ee, we find the change of the total uncertainty from the inclusion of the theoretical uncertainty rather small, as the 68% CL bound changes from 6 × 10 −5 to 6.65 × 10 −5 . The corresponding probability density functions are shown in Fig. 7, for which we have set the central value to zero and scaled up the uncertainties by 10 5 for convenience. Given that the estimations of the future experimental uncertainties are still very preliminary, we ignore this small effect due to the theoretical uncertainty of R 0 b in our study. . The light vertical lines shows the corresponding 68% and 95% CL bounds. Without the theoretical uncertainty, the 68% and 95% CL bounds are ±6 and ±12; with the theoretical uncertainty, the 68% and 95% CL bounds are ±6.65 and ±13.0.
With (longitudinal) beam polarization, A b can be directly measured. Without beam polarization, A 0,b FB can be measured, which is related to A b by A 0,b FB = 3 4 A e A b . The value of A e can be extracted experimentally by the measurement of A 0,l FB . Therefore, the theoretical uncertainty of A 0,b FB also only comes from A b , which can be parameterized by the theoretical uncertainty of sin 2 θ b eff (The overall form factors of g Lb and g Rb cancel in the ratio.) A b is numerically not very sensitive to sin 2 θ b eff . At the leading order in the SM, one has where δ th denotes the theoretical uncertainty of the corresponding quantity. Even with the current theoretical uncertainty of sin 2 θ b eff , which is about 5 × 10 −5 [51], δ th A b and δ th A 0,b FB are much smaller than the future experimental precisions and can be safely ignored.
The effects of the theoretical uncertainties of other quantities, such as the top mass and W mass, are important in general (e.g. for the S and T parameters, as pointed out in Ref. [5]) but do not directly affect the Zbb couplings. We implemented these theoretical uncertainties and found that the changes of the Zbb coupling constraints are negligible.