Top and Beauty synergies in SMEFT-fits at present and future colliders

We perform global fits within Standard Model Effective Field Theory (SMEFT) combining top-quark pair production processes and decay with $b\rightarrow s$ flavor changing neutral current transitions and $Z \to b \bar b$ in three stages: using existing data from the LHC and $B$-factories, using projections for the HL-LHC and Belle II, and studying the additional new physics impact from a future lepton collider. The latter is ideally suited to directly probe $\ell^+\ell^-\rightarrow t\bar t$ transitions. We observe powerful synergies in combining both top and beauty observables as flat directions are removed and more operators can be probed. We find that a future lepton collider significantly enhances this interplay and qualitatively improves global SMEFT fits.


I. INTRODUCTION
Physics beyond the Standard Model (BSM) has and is being intensively searched for at the Large Hadron Collider (LHC) and predecessor machines. However, despite the large amount of data analyzed, no direct detection of BSM particles has been reported to date. Thus, BSM physics could be feebly interacting only, has signatures not covered by the standard searches, or is simply sufficiently separated from the electroweak scale. The latter scenario opens up a complementary approach to hunt for BSM physics at high energy colliders, in a similar spirit as the high luminosity flavor physics programs in pursuit of the precision frontiers with indirect searches. In this regard, the Standard Model Effective Field Theory (SMEFT) offers both a systematic and model-independent way to parametrize BSM contributions in terms of higher-dimensional operators constructed out of Standard Model (SM) fields and consistent with SM symmetries. At energies below the scale of BSM physics, Λ, this framework allows to perform global fits which could give hints for signatures of BSM physics in different observables and sectors simultaneously.
In this work, we extend previous works and analyze sensitivities to semileptonic four-fermion operators. The reason for doing so goes way beyond of making the fit more model-independent: Firstly, semileptonic four-fermion operators are presently of high interest as they are the agents of the flavor anomalies, hints of a breakdown of the SM in semileptonic b → s decay data [24].
This work is organized as follows: In Sec. II we introduce the dimension-six SMEFT operators considered in this work and the low-energy effective field theories (EFTs) employed to compute SM and BSM contributions to B observables. We also present the matching between SMEFT and weak effective theory (WET) and highlight how SU (2) L invariance of the SMEFT Lagrangian allows to relate top-quark physics and b → s flavor-changing neutral currents (FCNCs). In Sec. III we discuss the sensitivity of different observables to the various effective operators considered. Fits to present top-quark, Zbb, and B data are presented in Sec. IV. We analyze how the complementary sensitivity of the observables from top-quark, Zbb, and B sectors improves constraints on the SMEFT coefficients. In Sec. V we consider different future scenarios, and detail on the question how measurements at a future lepton collider can provide additional information on SMEFT coefficients.
In Sec. VI we conclude. Additional information is provided in several appendices.

II. EFFECTIVE THEORY SETUP
In this section we give the requisite EFT setup to describe BSM contributions to top-quark and beauty observables. We introduce the SMEFT Lagrangian in Sec. II A, and identify the effective operators contributing to interactions of third-generation quarks. Consequences for FCNCs that arise from flavor mixing are worked out in Sec. II B, where we also highlight the complementarity between contributions from up-type and down-type quarks. The matching conditions for B observables in the low energy effective Lagrangian in terms of SMEFT coefficients are detailed in Sec. II C.

A. SMEFT dimension-six operators
At energies sufficiently below the scale of new physics, Λ, the effects of new interactions and BSM particles can be described by a series of higher-dimensional effective operators with mass dimension d > 4 [44,45].
Contributions from odd-dimensional operators lead to lepton-and baryon-number violation [46,47] and are neglected in this work. In the following, we employ the Warsaw basis [48] of dimension-six operators, and consider operators with gauge bosons and semileptonic four-fermion operators lq = l L γ µ τ I l L q L γ µ τ I q L , O qe = (q L γ µ q L ) (ē R γ µ e R ) , Here, q L , l L are the quark and lepton SU (2) L doublets, and u R , e R the up-type quark and charged lepton SU (2) L singlets, respectively. Flavor indices that exist for each SM fermion field are suppressed here for brevity but will be discussed in Sec. II B. With B µν , W I µν and G A µν we denote the gauge field strength tensors of U (1) Y , SU (2) L and SU (3) C , respectively. T A = λ A /2 and τ I /2 are the generators of SU (3) C and SU (2) L in the fundamental representation with A = 1, . . . , 8 and I = 1, 2, 3, and λ A and τ I are the Gell-Mann and Pauli matrices, respectively. The SM Higgs doublet is denoted by ϕ with its conjugate given asφ Further dimension-six operators exist that contribute at subleading order to top-quark observables such as dipole operators O dX with X = B, W, G and right-handed b quarks, with contributions suppressed by m b /m t . We neglect those as well as all other SMEFT operators involving right-handed down-type quarks. Scalar and tensor operators O (1/3) lequ are not included in our analysis since these operators do not give any relevant contributions at O(Λ −2 ) for the interactions considered in this work [14,40]. Contributions from four-quark operators to ttγ, ttZ and tt production are neglected as tt production at the LHC is dominated by the gg channel [8] 1 . In addition we also neglect leptonic dipole operators, i.e., vertex corrections to lepton currents because they are severely constrained by Z-precision measurements [50].
Note that dipole operators are in general non-hermitian which allows for complex-valued Wilson coefficients. However, the dominant interference terms are proportional only to the real part of the coefficients. For the sake of simplicity, we thus assume all coefficients to be real-valued.

B. Flavor and mass basis
The dimension-six operators (2), (3) are given in the flavor basis. In general, quark mass and flavor bases are related by unitary transformations S k L/R , k = u, d, where u and d denote up-and down-type quarks in the mass basis, respectively, and i, j = 1, 2, 3 are flavor indices. The CKM matrix V is then given as The rotation matrices of right handed quarks S u/d R can simply be absorbed in the flavor-basis Wilson coefficient C i , giving rise to coefficients in the mass basis, denoted byĈ i [51]. In contrast, the flavor rotations S u/d L of quark doublets relate different physical processes by SU (2) L -symmetry. Consider a contribution involving a doublet quark current with SU (2) L -singlet structure, i.e., the C (1) O (1) terms with quark flavor indices restored. For instance, Since we are interested in top-quark physics, in the last line we have chosen to work in the upmass basis, the basis in which up-quark flavor and mass bases are identical and flavor mixing is entirely in the down-sector. Irrespective of this choice for the mass basis, C As a result, up-type and down-type quarks probe different combinations of C (1) and C (3) , a feature recently also exploited in probing lepton flavor universality and conservation with processes involving neutrinos [52]. Further details on SMEFT coefficients and operators in the up-mass basis are given in App. B and App. C, respectively.
In this analysis, we only consider contributions from (flavor basis) Wilson coefficients with third generation quarks,Ĉ 33 i . Such hierarchies may arise in BSM scenarios with enhanced couplings to third-generation quarks, similar to the top-philic scenario discussed in Ref. [11]. As can be seen in Eqs. (6), (7), flavor mixing induces contributions to d i defined in the up-mass basis.
Lepton universality does not have to be assumed for fits to present data since the bulk of the existing b-physics precision distributions is with muons. In the future, Belle II is expected to deliver both b → se + e − and b → sµ + µ − distributions, and to shed light on the present hints that electrons and muons may be more different than thought [53]. In the far future, the b → se + e − results can be combined with tt-production data from an e + e − -collider; the muon ones could be combined with data from a muon collider, to improve the prospects for lepton flavor-specific fits. We also note that lepton flavor violating operators could also be included in the future. On the other hand, once data on dineutrino modes are included in the fit, assumptions on lepton flavor are in order, since the branching ratios are measured in a flavor-inclusive way: (11), in top-quarks with charged leptons (upper row), and beauty with charged leptons and neutrinos (lower row). The black circles denote SMEFT operators, wavy lines are electroweak gauge bosons.
Universality dictates that the total dineutrino branching ratio is given by three times a flavor- Here, i is fixed, but could be any of the three flavors.
We do assume universality when we include dineutrino modes in the fits to future data.
As is customary, in the following we use rescaled coefficients and drop the superscript for brevitỹ where v = 246 GeV is the Higgs vacuum expectation value. To highlight SU (2) L complementary between top and beauty, we introducẽ The sensitivities are illustrated in Fig. 1.

C. Matching and Running: SMEFT and WET
To constrain the Wilson coefficients of the SMEFT operators in Eqs. (2) and (3) using B physics measurements, the SMEFT Lagrangian has to be matched onto the WET Lagrangian, see App. A for details. The procedure to compute BSM contributions at the scale µ b in terms of coefficients given at a higher scale µ is described in detail in Ref. [18] and adapted here. Throughout this work, we consider values for Wilson coefficients at the scale µ = 1 TeV.

SMEFT RGE
The values of the Wilson coefficients depend on the energy scale µ of the process considered.
The renormalization group equations (RGEs) allow to combine measurements at different scales in one analysis. The RGEs for Eqs. (2) and (3) have been computed in Refs. [54][55][56][57]. We include these effects at one-loop level by applying the wilson [58] package.

Matching SMEFT onto WET
Flavor rotations allow for contributions fromĈ 33 i coefficients to b → s transitions whenever two SU (2) L quark doublets are present in the operator. We obtain finite tree level contributions from lq and O qe to the WET coefficients of the semileptonic four-fermion operators O 9,10,L , defined in App. A, as [51,59]: where sin 2 θ w = 0.223 denotes the weak mixing angle. We used for ∆C tree 9 in the second step the well-know suppression of Z-penguins due to the numerical smallness of the Z's vector coupling to charged leptons [60].
Note that there is sensitivity, although only at the one-loop level, to the semileptonic operators with up-type singlet quarks, O eu and O lu . The numerical values of the matching conditions at µ W = m W are computed with wilson following Ref. [62] and are provided in App. E. In the actual analysis, RGE effects in SMEFT and WET are taken into account as well.

WET RGE
We employ flavio [65] and wilson to compute the values of the SM and BSM contributions at the scale µ b .

III. OBSERVABLES
In this section we give details on how theory predictions and distributions for top-observables (Sec. III A), Z → bb transitions (Sec. III B), and Bphysics (Sec. III C) are obtained, and discuss the sensitivities of the observables to SMEFT coefficients (Sec. III D).

A. Top-quark observables
We employ the Monte Carlo (MC) generator MadGraph5_aMC@NLO [66] to compute the tt, ttγ and ttZ production cross sections at the LHC and the tt production cross section and the forward-backward symmetry at CLIC in LO QCD. The cross sections can be parametrized in terms of the Wilson coefficients as where σ int.
i and σ BSM ij denote interference terms between SM and dimension-six operators and purely BSM terms, respectively. The forward-backward asymmetry is defined as where θ denotes the angle between the three-momenta of the top quark and the positron in the center-of-mass frame. BSM contributions in both numerator and denominator are parametrized according to Eq. (19).
To obtain σ int.
i and σ BSM ij we utilize the dim6top_LO UFO model [11]. For the computation of the fiducial cross sections of ttγ production we generate samples as a 2 → 7 process including BSM contributions in the top-quark decay. The fiducal acceptances are obtained by showering the events with PYTHIA8 [67] and performing an event selection at particle level with MadAnalysis [68][69][70].
For the jet clustering we apply the anti-k t algorithm [71] with radius parameter R = 0.4 using FastJet [72]. The computation is discussed in detail in Ref. [18].
We compute the helicity fractions according to Ref. [73] with the difference that we also include quadratic contributions. In our analysis, we consider only O uW as only this operator gives contributions O(Λ −2 ) that are not suppressed by a factor m b /m t . The top-quark decay width is computed following Ref. [74] including quadratic contributions.

B. Zbb observables
To compute Z → bb observables we employ MadGraph5_aMC@NLO together with the dim6top_LO UFO model for both the forward-backward asymmetry A 0,b FB and the ratio of partial widths for Z → ff BSM contributions to A 0,b FB are computed using Eq. (20), and for R b we include BSM contributions in both numerator and denominator. where and C L (µ b ) SM = Xs sin 2 θw with X s = 1.469 ± 0.017, and lepton flavor universality is assumed. We also consider the B s −B s mass difference ∆M s , which can be computed as [75] ∆M s = ∆M SM an asterisk receive additional contribution at one-loop level, which are suppressed by at least one order of magnitude, see Eqs. (25), (26) and Appendix E for details.
Total cross sections of the top-quark production channels, the top-quark decay width, and the helicity fractions measured at the LHC allow to test six coefficients of the operators in Eq. (2),

Process
Observable Two-fermion operators Four-fermion operators  combinationC − ϕq (see Eq. (11)), the total decay width is sensitive toC Observables of Z → bb decay are sensitive toC + ϕq , and the other operators considered here do not contribute to this process.
Including b → s observables allows to put new and stronger constraints on SMEFT coefficients.
The interplay of b → sγ transitions with ttγ has been worked out in [18]. BSM contributions to the former are induced at one-loop level byC uB ,C uW ,C uG , andC (3) ϕq .
For b → s + − transitions, tree level contributions to ∆C 9,10 arise fromC + ϕq ,C + lq , defined in Eq. (11), andC qe . The latter cancels, however, in the left-chiral combination ∆C 9 − ∆C 10 , which is the one that gives the dominant interference term in semileptonic B decays with the SM. We therefore expect only little sensitivity toC qe from these modes. On the other hand, this highlights the importance of B s → µµ, which is sensitive to C 10 only. At one-loop level, all eleven SMEFT operators considered here contribute to ∆C 9,10 (C uG only via mixing). In the case ofC lq ,C lu ,C qe ,C eu , and partiallyC (3) ϕq , these contributions can simply be absorbed by redefining the fit degrees of freedom Numerically, these loop-level corrections are typically below percent-level compared to tree-level contributions. For the remaining contributions fromC ϕq ,C uB ,C uW (andC uG ) to ∆C 9,10 such a redefinition is not possible and additional degrees of freedom arise. However, these remaining contributions to ∆C 9,10 are at least one order of magnitude smaller than the tree-level ones.
lq andC lu can be absorbed intoC Dineutrino observables depend only on the sum of these coefficients. Meson mixing is sensitive at one-loop level toC uW ,C ϕq , andC ϕq while contributions fromC uG arise only through SMEFT O(α s ) RGE evolution. Electroweak RGE effects in B physics [76] as well as in top-quark physics are included in the numeric fits but are not shown here for clarity.
In summary, while all SMEFT coefficients contribute to the B physics observables considered, these effects are mostly induced at one-loop level and thus naturally suppressed. Notable exceptions are tree-level contributions fromC + ϕq ,C + lq ,C qe , andC − lq +C + ϕq . In addition,C uB is important as it contributes with sizable coefficient to ∆C 7 [18]. Thus, we expect that B physics data constrains these SMEFT-coefficients rather strongly, and the others much less.
Measurements of top-quark pair production cross sections and the forward-backward asymmetry at a lepton collider are sensitive to four linear combinations of two-fermion operatorsC uB ,C uW , ϕq , andC ϕu . The sensitivity toC uG is smaller because contributions arise only through RGE evolution. While these coefficients affect the ttZ and ttγ vertex, four-fermion operators can also contribute in following linear combinations:C − lq ,C qe ,C eu , andC lu . Thus, combining + − → tt observables with top-quark ones at LHC and B physics observables allows to test the complete 11dimensional parameter space. In particular, coefficientsC eu andC lu remain only poorly constrained by Belle II and the HL-LHC. A summary of the dominant contributions to the different observables is illustrated in Fig. 2.

IV. FITS TO PRESENT DATA
We employ EFTfitter [77], which is based on the Bayesian Analysis Toolkit -BAT.jl [78], to constrain the Wilson coefficients in a Bayesian interpretation. We include systematic and statistical experimental and SM theory uncertainties. All uncertainties on the measured observables are assumed to be Gaussian distributed. The procedure of our fit is detailed in our previous analyses in Refs. [18,49], and is based on Ref. [77].
BSM contributions are parametrized as in (19), which includes quadratic dimension-six terms.
While these purely BSM contributions are formally of higher order in the EFT expansion, it has been shown [12,49] that the inclusion of such quadratic terms has only a negligible effect on constraints of coefficients for which the linear term in the EFT expansion gives the dominant contribution, as expected in regions where the EFT is valid.
We include measurements of observables from both top-quark pair production processes and top-quark decay at the LHC, Z → bb transitions, and b → s transitions from different experiments.
Measurements of the same observable from different experiments can in principle be correlated [79].
Correlations are included as long as they are provided, comprising mainly bin-to-bin correlations and correlations between the W boson helicity fractions. Unknown correlations can affect the result  of the fit significantly [49]. Therefore, we follow a strategy similar to the ones of Refs. [14,16] and include only the most precise measurement of an observable in the fit. Especially, if no complete correlation matrices for differential distributions are given by the experiments, we do not include these measurements in the analysis. For B physics observables, a variety of measurements have been combined by the Heavy Flavour Averaging Group (HFLAV) [80]. Wherever possible, we include their averaged experimental values in our analysis. For all remaining unknown correlations between different observables, we make the simplifying assumptions that the measurements included in the fit are uncorrelated.
We work out current constraints from top-quark measurements in Sec. IV A, from Z → bb data in Sec. IV B, from B-physics in Sec. IV C, and perform a global analysis in Sec. IV D.
A. Current constraints from top-quark measurements at the LHC In Tab. II we summarize the measurements and the corresponding SM predictions of the topquark observables at the LHC included in the fit. This dataset comprises measurements of fiducal cross sections σ fid (ttγ, 1 ) (σ fid (ttγ, 2 )) of ttγ production in the single lepton (dilepton) channel, inclusive cross sections σ inc (tt) and σ inc (ttZ) of tt and ttZ production, respectively, measurements of the W boson helicity fractions F 0,L , and a measurement of the total top-quark decay width Γ t . The SM predictions for ttγ cross sections include NLO QCD corrections Refs. [81,82], while predictions for ttZ cross sections are computed at NLO QCD including electroweak corrections [84][85][86]. For tt production, the SM prediction at NNLO QCD is taken from Ref. [87], and has been computed following Ref. [88]. Predictions for helicity fractions and the total decay width include NNLO QCD corrections [90,92].
In Fig. 3 we give constraints on SMEFT Wilson coefficients detailed in Tab. I obtained in a fit of six coefficients to the data in Tab. II. The strongest constraints are found forC uG andC uW , Top LHC 5 × 10 2 10 1 5 × 10 1 10 0 total width of smallest 90% interval

B. Constraints from Zbb measurements
Precision measurements of Z pole observables have been performed at LEP 1 and SLC, and the results are collected in Ref. [50]. In our analysis, we focus on those that are sensitive to BSM contributions which affect the Zbb vertex. The measurements included are those of the forwardbackward asymmetry and the ratio of partial widths for Z → ff [93] The corresponding SM values are given as [50,93] A These observables are sensitive to BSM contributions fromC + ϕq , which alter the Zbb vertex, and allow to derive strong constraints on this coefficient. The results of a fit of one (C + ϕq ) and two (C ϕq plane we find, as expected, strong correlations, and only a very small slice of the two-dimensional parameter space is allowed by present data.

C. Current constraints from B physics measurements
In Tab. III we give the B physics observables and the corresponding references of the measurements and SM predictions considered in our fit. This dataset includes both inclusive and exclusive branching ratios of b → sγ transitions, total and differential branching ratios of various b → sµ + µ − processes, inclusive branching ratios and asymmetries of b → s + − transitions, and angular dis- In the case of B s → µ + µ − , we consider the latest results presented by the LHCb collaboration [98]. We compute SM predictions and uncertainties with flavio [65]. In addition, we also include the mass difference ∆M s measured in B s −B s mixing, with SM prediction from Ref. [75]. Note that we do not take into account measurements of the B → K ( * ) νν branching ratios as only upper limits are presently available by Belle [103] and BaBar [104], which can not be considered in EFTfitter.
In Fig. 5     Constraints on the remaining coefficients are around one (C uB ,C ϕq ,C ϕq ) to two (C uG ,C uW ,C ϕu ) orders of magnitude weaker. Note that deviations from the SM, which are present in the onedimensional projection of the posterior distribution of ∆C 9 , can not be seen in the one-dimensional results in the SMEFT basis. This is due to the strong correlations among the SMEFT coefficients induced by the matching conditions. 1 10 0 total width of smallest 90% interval

D. Combined fit to current data
Combining top-quark, Zbb, and B observables allows to constrain a larger number of SMEFT coefficients compared to fits using only the individual datasets. Specifically, the coefficients constrained by data in Tabs. II and III and Zbb data arẽ 1.0 0.5 0.0 0.5 For the prior we assume a uniform distribution over the interval −1 ≤C i ≤ 1.
From the fit of these eight coefficients to the combined dataset we obtain the results shown in Fig. 6. The strongest constraints are onC qe andC + lq , whose width of the smallest interval is around (5 − 6) × 10 −3 . This is expected, since both coefficients give sizable contributions to ∆C 9 and ∆C 10 at tree level (12). ForC uB ,C uG ,C uW ,C (1) ϕq andC (3) ϕq the constraints are about one order of magnitude weaker, with a width of around (5 − 7) × 10 −2 . While constraints onC uG and C uW coincide with those derived from fits to top-quark data, the combination of the three datasets significantly tightens constraints on the other three coefficients. ForC uB this enhancement stems from different sensitivities of top-quark and B data, as already discovered in Ref. [18]. The effect of the different datasets is shown in detail in Fig. 7 (left), where we give the two-dimensional projection of the posterior distributions obtained in fits to different datasets in theC uB -C uW plane. Here, the effects are even more pronounced compared to Ref. [18], since a larger set of B observables is considered here. Constraints onC (1) ϕq andC (3) ϕq (Fig. 6) are tightened by the inclusion of Zbb data, which strongly constraintsC + ϕq , as well as the strong constraints onC (3) ϕq , which arise from the combination of top-quark and B physics data (see Fig. 7). As can be seen, in the combined fit the SM is included in the smallest intervals containing 90 % of the posterior distribution ofC ϕq , which is shown in detail in Fig. 13. The weakest constraints are found forC ϕu , since contributions to B physics data are strongly suppressed, and ttZ production offers only a limited sensitivity, as we can already see in Fig. 3.
Interestingly, we find two solutions for several coefficients; one of which is SM like, while the other one deviates from the SM:C uB ,C ϕq ,C ϕq , and the four-fermion coefficientsC qe andC + lq . As can be seen in Fig. 7, the second solutions stem from the correlations between the coefficients introduced by matching the SMEFT basis onto the WET basis. Since the number of degrees of freedom is smaller in WET, correlations among the coefficients arise. Inclusion of top-quark data reduces these correlations, however, for the five coefficients the sensitivity of top-quark observables does not suffice to exclude the non-SM branches completely given present data and theory predictions.
Without further input this ambiguity cannot be resolved.
We compare our results to those reported in a recent study on b → s + − transitions [106].
In contrast to our analysis, operators are defined in a basis of diagonal down-type quark Yukawa couplings, which leads to an additional factor of 1/(V tb V * ts ). Taking this factor into account, the results from [106] correspond toC + lq ,C qe ∼ 10 −3 , consistent with Fig. 6. Repeating our fit withC qe andC + lq only, we find agreement with Ref. [106]. We also comment on Drell-Yan production at the LHC. Amongst the couplings with top-quark focus considered in this works, (8), this concernsC + ϕq ,C qe andC + lq , just like b → s + − and Z → bb. Drell-Yan limits from existing data and a 3000 fb −1 future projection for the semileptonic four-fermion operators with b-quarks are at the level of O(10 −2 ) [107,108], and weaker than in the combined fit, Fig. 6. Note, with the flavor of the initial quarks in pp-collisions undetermined an actual measurement of a quark flavor-specific coefficient is not possible. A detailed study of the implications of Drell-Yan processes for a global fit is beyond the scope of this work.

V. IMPACT OF FUTURE COLLIDERS
Both the HL-LHC operating at 14 TeV with an integrated luminosity of 3000 fb −1 [42] and Belle II at 50 ab −1 [43] are going to test the SM at the next level of precision. In Sec. V A, we work out the impact of future measurements at these facilities on the SMEFT Wilson coefficients.
A first study of top-quark physics at the proposed lepton collider CLIC has been provided in Ref. [27]. CLIC is intended to operate at three different center-of-mass energies: 380 GeV, We combine existing data with HL-LHC, Belle II and CLIC projections in Sec. V C.

A. Expected constraints from HL-LHC and Belle II
For the expected experimental uncertainties at the HL-LHC and Belle II we adopt estimates of the expected precision by ATLAS, CMS and Belle II collaborations [42,43,[109][110][111]. If no value for the systematic uncertainties is given, we assume that these uncertainties shrink by a factor of two compared to the current best measurement, which is the case for the tt and ttZ cross sections, the W boson helicity fractions, and the top-quark decay width. In addition, we make the assumption that theory uncertainties shrink by a factor of two compared to the current SM uncertainties due to improved MC predictions and higher-order calculations. We summarize the observables and references for the expected experimental and theory precision at HL-LHC and Belle II in Tab. IV. For the purpose of the fit, we consider present central values of measurements for the future projections. If no measurement is available, we consider the SM for central values.
For fiducial cross sections of ttγ production, an analysis with the expected uncertainties is provided in Refs. [42,109]. For both the dilepton and single-lepton cross section we consider the precision of the channel with the largest experimental uncertainty as our estimate. For ttZ production we follow the analysis in Refs. [42,110] and scale statistical uncertainties according to the luminosity. For systematic uncertainties we assume for simplicity a reduction by a factor of 2. For estimating the expected precision of the total tt production cross section, we base our assumptions on the study of differential tt cross sections in Ref. [42,111]. For the uncertainties we apply the same assumptions as for ttZ. As the W boson helicity fractions and the top-quark    decay width are not discussed in Ref. [42], we treat them in the same way as the tt cross section for simplicity.
For measurements of b → s transitions we take the estimates in Ref. [43] into account. For the b → sγ inclusive branching ratio we take the precision for the BR(B → X s γ) Eγ >1.9 GeV measurement and assume that the same uncertainties apply for E γ > 1.6 GeV. In case of B (+) → K (+) * γ, we directly include the estimated precision in Ref. [43]. Similarly, for the inclusive decayB → X s + − we use the expected precision for the 3.5 GeV 2 ≤ q 2 ≤ 6 GeV 2 bin. We also considered other bins for this observable and found very comparable sensitivity. Finally, for B → K * µ + µ − we include the angular distribution observable P ( ) i in different q 2 bins, and study the implications of the anomalies found in present data of b → sµ + µ − angular distributions.
Combining top-quark and B observables at HL-LHC and Belle II allows to test a total of nine SMEFT coefficients, see Fig. 8. In order to derive these constraints with EFTfitter, we have chosen a smaller prior |C i | ≤ 0.1 for the four-fermion coefficients because the posterior distribution lies only in a very small region, and a larger prior would lead to convergence issues. At this point, we neglect subleading contributions fromC eu andC lu , which are considered in Sec. V C. As can be Combined current Combined current + near 10 2 5 × 10 2 10 1 5 × 10 1 10 0 total width of smallest 90% interval Combined current Combined current + near  independently due to the orthogonal sensitivity compared to b → s + − observables, as indicated in Fig. 9. We observe that the interval obtained in the combined fit is significantly smaller than   [27]. SM predictions are taken from [40].
expected from the simple overlay of constraints from b → sνν and b → s + − observables. The reason is, that the posterior distribution is constrained in the multi-dimensional hyperspace, and the combination significantly reduces correlations among different coefficients. In addition, we find that two solutions forC lq are allowed: one is close to the SM, while the other is around C lq ∼ 10 −3 , and deviates strongly from the SM. Without further input, this ambiguity can not be resolved. Constraints on the remaining coefficientsC uB ,C uG ,C uW ,C

B. CLIC projections
In Tab. V we list the top-quark observables for the CLIC future projections considered in this work. This set comprises total cross sections of tt production and forward-backward asymmetries A FB as observables for different energy stages and beam polarizations [27]. We use the current SM predictions as nominal values, which include NLO QCD corrections [40].
In Fig. 10 we give the results for a fit to the CLIC projections in Tab. V. A smaller prior |C i | ≤ 0.1 is employed for the four-fermion coefficients due to the small size of the posterior distribution. We explicitly checked that we do not remove any solutions. Constraints onC uG , which contributes via mixing only, are at the level of 4 × 10 −1 , and weaker compared to the ones on the remaining Wilson coefficients. ForC − ϕq andC ϕu the width of the smallest 90 % interval is at the level of 10 −1 . In comparison, constraints onC uB andC uW are found to be stronger by one order of magnitude.
Even tighter constraints are obtained for four-fermion interactions, where the width of the smallest interval is at the level of (2 − 6) × 10 −4 . Interestingly, while Fig. 10 shows results of a fit treating CLIC only 5 × 10 4 0 5 × 10 4 10 3 10 2 10 1 total width of smallest 90% interval Combined current + near CLIC only Combined all 10 4 10 3 10 2 10 1 10 0 10 1 total width of smallest 90% interval coefficients. In the case ofC lq . While b → s + − and b → sνν observables allow to test both coefficients simultaneously, the inclusion of CLIC observables is mandatory to remove the second solution, see Fig. 12 (right). Correlations, which are induced by CLIC observables, between both coefficients are still present, and sizable deviations from the SM can be found, which is shown in more detail in Fig. 14 in App. F. These deviations stem from the assumption that Belle II confirms present LHCb data. Interestingly, even though CLIC observables strongly constrainC − lq (assuming that the SM value is measured), the exact position of the smallest 90 % interval on theC subspace is determined by Belle II results (Fig. 14). A scenario, in which we assume SM values for Belle II observables, is shown in Fig. 15 in App. F, and we find agreement with the SM in this case.
While indeed constraints from CLIC projections and top-quark and B data and projections in the near-future scenario have a different sensitivity, the 90 % region forC and Fig. 12 only shows two-dimensional projections.
As anticipated in Sec. II B the full, global fit results including CLIC projections are obtained assuming lepton-flavor universality. While in BSM scenarios where lepton generations couple differently the results cannot be applied directly, the findings on the orthogonality of the constraints and synergies between top and beauty continue to hold.

VI. CONCLUSIONS
We performed fits within SMEFT to top-quark pair production, decay, Z → bb transitions, and b → s transitions. We highlight how each of the individual datasets constrains different sets of Wilson coefficients of dimension-six operators affecting top-quark physics at present and future colliders. Extending previous works [18], we put an emphasis on semileptonic four-fermion operators, which are of high interest as they may be anomalous according to current flavor data and moreover are essentially unconstrained for top quarks. SU (2) L invariance leads to relations between up-type and down-type quark observables, a well-known feature with recent applications in semileptonic processes within SMEFT [52]. Here, we exploit this symmetry link between top and beauty observabes at the LHC and a future lepton collider.
Using existing data in Tabs. II and III  TeV LHC with 41.5 fb −1 [112]. The constraints on four-fermion coefficientsC qe andC − lq are more than one order of magnitude weaker compared to ours using current data, Fig. 6. However, the CMS-analysis is sensitive toC eu ,C lu , otherwise unconstrained by present data. A study of the future physics potential of this type of analysis would be desirable, however, requires detector-level simulations and is beyond the scope of this work. by the doctoral scholarship program of the Studienstiftung des deutschen Volkes.

Appendix A: Weak effective theory
At energies below the scale µ W ∼ m W , physical processes are described by the Weak Effective Theory (WET). All BSM particles which are heavier than m W as well as the top quark and the W , Z and Higgs bosons are integrated out. Both b → s + − and b → sγ transitions are described by the following Lagrangian: Here, G F is the Fermi-constant, C i are Wilson coefficients and Q i are the corresponding effective operators which are defined as follows: with chiral left (right) projectors L (R) and the field strength tensor of the photon F µν . We denote charged leptons with and neglect contributions proportional to the subleading CKMmatrix element V ub and to the strange-quark mass.
The effective Lagrangian for b → sνν transitions can be written as with effective operators Assuming flavor universality, only diagonal terms i = j contribute, and all three flavors couple with identical strength. The B s −B s mass difference ∆M s can be described as with the effective operator

Appendix B: SMEFT coefficients in the mass basis
In the up-mass basis we absorb the unitary rotations S u L,R between the flavor and mass basis into the Wilson coefficients. The ones of the operators (2) are then given aŝ Similarly, we obtain for the coefficients of the four-fermion operators (3) Appendix C: SMEFT operators in the mass basis In the up-mass eigenbasis, with coefficients defined according to Eq. (B1) we find for the effective operators in Eq. (2) Similarly, we find for the four-fermion operators in Eq. (3) with coefficients defined in Eq. (B2) These results are in agreement with Ref. [51].
The remaining functions relevant for contributions from dipole operators read [51] The following functions relevant for the matching of up-type dipole operators on C 9 and C 10 are taken from Ref. [51] and read Z uB (x t ) = − x 2 t + 3x t − 2 4(x t − 1) 2 + 3x t − 2 2(x t − 1) 3 ln (x t ) .
The functions H i relevant for the matching ofC Finally, functions relevant for the matching of SMEFT coefficients onto C L at one-loop level are taken from [62]. Here, we give results with all evanescent coefficients set to 1: