Single top quark production as a probe of anomalous $tq\gamma$ and $tqZ$ couplings at the FCC-ee

In this paper, a detailed study to probe the top quark Flavour-Changing Neutral Currents (FCNC) $tq\gamma$ and $tqZ$ at the future $e^{-}e^{+}$ collider FCC-ee in three different center-of-mass energies of 240, 350 and 500 GeV is presented. A set of useful variables are proposed and used in a multivariate technique to separate signal $e^- e^+ \rightarrow Z/\gamma \rightarrow t \bar{q} ~ ( \bar{t} q )$ from standard model background processes. The study includes a fast detector simulation based on the {\sc delphes} package to consider the detector effects. The $3 \sigma$ discovery regions and the upper limits on the FCNC branching ratios at 95\% confidence level (CL) in terms of the integrated luminosity are presented. It is shown that with 300 fb$^{-1}$ of integrated luminosity of data, FCC-ee would be able to exclude the effective coupling strengths above ${\cal O} (10^{-4}-10^{-5})$ which is corresponding to branching fraction of ${\cal O}(0.01-0.001)$\%. We show that moving to a high-luminosity regime leads to a significant improvement on the upper bounds on the top quark FCNC couplings to a photon or a $Z$ boson.


Introduction
Top quark with its large mass and very short life time is one of the most interesting discovered particles in the Standard Model (SM). Studying top quark enables us to investigate the electroweak symmetry breaking mechanism (EWSB) as well as searching for extensions of the SM. In the framework of the SM, top-quark Flavour-Changing Neutral Currents (FCNC) only arise at loop level and highly suppressed because of the GIM (Glashow-Iliopoulos-Maiani) mechanism [1]. For example, the SM predictions for the branching fractions of FCNC processes like t → γu(c) and t → Zu(c) are of the order of 10 −14 and 10 −16 , respectively. The ability of the present experiments is far from measuring such tiny branching ratios. On the other hand, several extensions of the SM such as Technicolor, SUSY models, Higgs doublet models predict much higher branching ratios up to 10 8 − 10 10 order of magnitude larger than SM values [2][3][4][5][6][7][8]. Consequently, any observation of these rare FCNC transitions would be a clear signal of new physics beyond the SM.
Recently, both CMS and ATLAS experiments have looked for FCNC transitions of the top quark to γ, gluon, Z and Higgs bosons through different channels [9][10][11][12][13][14][15][16][17][18][19][20]. The most stringent observed upper limits at 95% confidence level (CL) have been found to be [13,20]: BR(t → Zq) < 0.05% , BR(t → uγ) < 0.0161% , BR(t → cγ) < 0.182%. (1) The limit on the t → qZ has been set using the combination of LHC data at 7 and 8 TeV while the bounds on t → u(c)γ are based on only 8 TeV data. It is notable to mention here that even at the future upgrades of the LHC, these bounds would not be improved considerably. For example, the future upper bounds on BR(t → qZ) have been predicted to be 0.01% at 95% CL at 14 TeV centerof-mass energy with 3000 fb −1 of integrated luminosity of data [21,22]. Therefore, an important task is to look at the future colliders potential to search for the anomalous FCNC couplings, in particular the e − e + colliders such as ILC, CLIC and the Future Circular Collider (FCC) [23][24][25][26][27][28][29] In [30], an analysis has been performed to probe the sensitivity of a future e − e + collider to top FCNC to the photon and a Z boson in the e − e + → Z/γ → tq (tq) channel. This analysis has been done at the center-of-mass energies of 500 GeV and 800 GeV with the integrated luminosity of up to 1000 fb −1 without including parton showering, hadronization, and decay of unstable particles. However, the analysis considers cases with no beam polarization, with polarization of electron and with the case of polarization of electron and positron. Then the sensitivity to tqγ and tqZ FCNC couplings have been estimated. The future large scale circular electron-positron collider (FCC-ee) would be one of the highprecision and high-luminosity machines which will be able to perform precise measurements on the Higgs boson, top-quark, Z and W bosons [31,32]. Due to the expected large amount of data and large production rates, FCC-ee can provide an excellent opportunity for precise studies, in particular in the top quark sector. FCC-ee is designed to be working at the center-of-mass energy up to the tt threshold mass, i.e. √ s = 350 GeV which is upgradeable to 500 GeV. The goal is to reach to a luminosity of L = 1.3 × 10 34 cm −2 s −1 [31,32]. In this paper, our aim is to study the anomalous FCNC of tqγ and tqZ via single top quark production in the FCC-ee at three different center-of-mass energies of 240 GeV,350 GeV and 500 GeV. The final state consists of a top quark in association with a light-quark. We consider the leptonic decay of the W boson in top quark decay, (t → W b → ℓν ℓ b). In the analysis, we perform parton shower, hadronization and decays of unstable particles as well as a raw detector simulation. We present the 3σ discovery ranges and upper limits on the branching ratios at 95% C.L in terms of the integrated luminosity. Finally, the results are compared with the present and future results from the LHC experiments The paper is organized as follows. In Section 2 we present the theoretical framework which describes the top quark FCNC couplings to a photon and Z boson. The Monte Carlo event generation, detector simulation and signal separation from backgrounds are described in Section 3. In Section 4, the results of the sensitivity estimation are presented. Finally, Section 5 concludes the paper.

Theoretical formalism
The anomalous FCNC couplings of a top quark with a photon and Z boson can be written in a model independent way using an effective Lagrangian approach. The lowest order terms describing tqγ and tqZ couplings has the following form [17,[33][34][35][36]: where the λ tq , κ tq and X tq are dimensionless real parameters that denote the strength of the anomalous FCNC couplings. In the effective Lagrangian, the complex chirality parameters are normalized to |λ a | 2 + |λ v | 2 = |x L | 2 + |x R | 2 = |κ v | 2 + |κ a | 2 = 1 and P L,R are the left-and righthanded projection operators, P L,R = 1 2 (1 ∓ γ 5 ). It is notable that the term which contains γ µ is dimension four and the terms with σ µν are dimension five. The anomalous FCNC interactions tqγ and tqZ lead to production of a top quark in association with a light quark in electronpositron collisions. The Feynman diagram for this process is shown in Figure 1 including the subsequent leptonic decay of the W boson in top quark decay. In Table 1, the cross sections of e − + e + → tū + tc +tu +tc including the branching ratio of top quark decays into a W boson and a b-quark with W boson decays into a charged lepton (muon and electron) and neutrino are presented. The cross sections are shown at three different center-of-mass energies of 240, 350 and 500 GeV. It should be pointed out that the cross sections due to photon and Z boson exchange are different and depends on the type of coupling. The contribution of photon exchange and Z boson with σ µν coupling increases with the energy of the center-of-mass. This is because of the presence of an additional momentum factor q ν in the effective Lagrangian.
According to the three independent terms of the Lagrangian, there are three separate ways to produce single top quark plus a light quark. In this analysis, all three terms of the Lagrangian are investigated independently with the following sets of the parameters: λ v = 1, λ a = 0 for tqγ, for vector like coupling of tqZ: x L = x R = 1 while for tensor FCNC coupling of tqZ: κ v = 1, κ a = 0. In case of observing an excess indicating FCNC signal, the angular distribution of the outgoing particles can be used to determine the chirality of the FCNC couplings. In Figure  2, the distributions of the cosine of the angle between the outgoing charged lepton with respect to the z−axis (beam axis) are depicted for the tqγ signal scenario with three independent types of couplings: (λ v = 1, λ a = 0), (λ v = 1, λ a = 1) and (λ v = 1, λ a = −1). As it can be seen, for the type of coupling with no γ 5 , the angular distribution is quite flat while for the type of coupling with projection operator 1 ± γ 5 the distribution has a behaviour like a parabola with opposite shapes depending on the sign of γ 5 .  Table 1: Cross-sections (in fb) of σ(e − + e + → tū + tc +tu +tc) × Br(t → W b → lνb) with l = e, µ for three signal scenarios, tqγ, tqZ (vector-tensor) before applying cuts. The cross section of the main background process W ± jj including the branching ratio of the leptonic decay of the W boson is also presented.

Analysis strategy
As we have mentioned before, this study is dedicated to probe the tqγ and tqZ FCNC couplings via single top quark production at FCC-ee. The results will be presented at different center-ofmass energies of the colliding electron-positron. In this section, the details of the event generation and Monte Carlo simulation for signal and background, event selection, and multivariate analysis to separate signal from SM background will be presented.

Event generation and simulation
Now, we present the signal and the corresponding backgrounds generation in the e − e + collisions. The signal is defined as e − e + → Z/γ → tq (tq), where q is an up or a charm quark. The top quark decays through SM, Therefore, the final state consists of a charged lepton, missing energy, a b-jet and a light jet.
In order to simulate and generate the signal events, the effective Lagrangian describing the FCNC couplings is implemented with the FeynRules package [37][38][39][40][41], then the model has been imported to a UFO module [42] and inserted in MadGraph 5 [43,44].
Based on the expected signature of the signal events, the main background contribution is originating from W W production when one of the W bosons decays hadronicly and another one decays leptonically, i.e. e + e − → W + W − → ℓ + ν ℓ jj(ℓ − ν ℓ jj). Again, we use MadGraph 5 to generate the background events. The signal and background events are generated in the centerof-mass energies of √ s=240, 350 and 500 GeV.
We employ Pythia 8.1 package [45][46][47][48] for parton showering, hadronization and decay of unstable particles. To reconstruct jet the FastJet package [49][50][51] with an anti-k t algorithm [52,53] with a cone size of R = 0.4 is used. Where R = (∆η) 2 + (∆φ) 2 , with η = − ln(tan(θ/2)). The parameters φ and η are the azimuthal and polar angles with respect to the z-axis. We present the results with 70% for the efficiency of b-tagging and a 5% mistagging rates is also considered. In our analysis, b-tagging plays an important role to reject the contribution of W W background. More background rejection affects the final upper limits on the branching ratios.
To account for the detector resolution, the final state particles (leptons and jets) are smeared according to a Gaussian distribution with the following parameterization [54,55]: where ℓ stands for lepton which can be an electron or a muon and j stands for final state jets. The symbol ⊕ represents a quadrature sum. In reality, the electron and muon energy resolutions show different dependencies on the electromagnetic calorimetry and tracking of charged particle. Nevertheless, in our analysis the uniform values for energy resolution are used for the final state lepton. It is a more conservative assumption for the energies under consideration in this analysis than the capabilities of tracking. Now, we apply the following detector acceptance cuts on the final state objects: where p jet,ℓ,b−jet T are the transverse momenta of jets, leptons and b-quark jet, respectively. / E miss T is the missing transverse energy. In addition to these cuts, to have well separated objects, we require the distances in the (η, φ) space between each two objects to be greater than 0.4. The presence of a charged lepton (electron or muon) with p ℓ T ≥ 10 GeV and |η ℓ | < 2.5, which is tagged as originating from W boson decay, is required for the leptonic decays channel of top quark.
In order to reconstruct the top quark, all components of the neutrino momentum is needed. The missing transverse energy ( / E miss T ) is taken as the transverse component of neutrino momentum. We obtain the z-component of the neutrino momentum by using the W boson mass constraint: M 2 W = (p ℓ + p ν ) 2 . In most cases, two solutions are obtained for the z-component of the neutrino. Consequently, the combination of the charged lepton and two neutrinos gives two W bosons, which is combined with the highest-p T b-tagged jet. The combination which leads to closest mass to the top quark mass is then chosen. Among all reconstructed untagged jets, the one with highest-p T is chosen to be jet originating from the light quark in the signal final state. Finally, the mass distribution of the reconstructed top-quark is illustrated in Figure 3 for tqγ signal and for W ± jj SM background. These distributions are at the center-of-mass energy of 350 GeV. It is worth mentioning that as expected the signal distribution has a peak near the top mass while the background events have an almost flat distribution with no peak. Moving to higher center-of-mass energies causes to more separation of top mass distributions of signal from background.    Table 2: Cross-sections (in fb) for three signal scenarios, tqγ, tqZ (vector and tensor) after including the branching ratios and after applying the preliminary kinematic cuts.

Separation of signal from background
In order to reduce the main SM background W ± jj which have a different topology than the signal events, a multivariate technique [57][58][59][60][61] is used. After the pre-selection cuts described in the previous section which consists of detector acceptance cuts, b-tagging efficiency and misidentification rate around 48-50% of the signal events and about 35-40% of background events are survived. The cross sections of signal in all scenarios and backgrounds at three center-of-mass energies after the pre-selection cuts are presented in Table 2. These pre-selection cuts are generally loose selections on single variables, which remove a large fraction of the background events while barely reducing also the signal events. Then in order to obtain a better separation of signal from background events, a multivariate technique is used. The choice of proper set of variables is very efficient in surviving the signal, while reducing the large fraction of SM background events. We select those variables which have the best possible discrimination power between signal and background. The following variables are used in the analysis: For all signal scenarios tqγ, tqZ(γ µ ) and tqZ(σ µν ) the same variables are used while multivariate analyses are different because of different behaviours in shapes of these variables. The analyses are performed separately at the center-of-mass energies of 240,350 and 500 GeV. It is remarkable that in this analysis, the events with only one b-tagged jet are kept and the total number of jets is free to be two or three. These requirements with the preliminary requirement of the presence of only one isolated charged lepton in the final state help suppress the contribution of background events from top pair production.
The cross sections of the signals tqγ and tqZ(γ µ , σ µν ) and the main background W ± jj after performing the multivariate analysis are presented in Table 3. As it be seen from the Table 3, the background rejection rate varies at different center-of-mass energies. For all signal scenarios, the background rejection rates are ∼ 10 −1 , ∼ 10 −2 and ∼ 10 −3 at the center-of-mass energies of 240, 350, 500 GeV, respectively. The discriminating power of the input variables are increasing with the center-of-mass energies of the collision. When we go to higher energies the overlapping between the signal and background distributions reduces. In particular, this happens for the top mass, lepton energy and top quark transverse momentum distributions. Larger background suppression is achieved for the tqZ signal with σ µν coupling. Since the signal-to-background ratio for all signal scenarios increases with the increment of the center-of-mass energy, more sensitivity is expected at larger energies.   Table 3: Cross-sections (in fb) for three signal scenarios, tqγ, tqZ (vector and tensor) after performing the multivariate analysis to separate signal from background events.

Sensitivity estimation
For estimation of the FCNC sensitivity, the expected 3σ significance and the upper limits on the anomalous couplings and the branching ratios at 95% C.L are presented. The 3σ discovery ranges are obtained using the significance S/ √ B where S and B are the number of signal and background events after all selections, respectively. The number of signal and background events can be obtained by: S = σ signal ×L and B = σ bkg ×L. The cross sections of signal and backgrounds are taken from Table 3 which are the obtained after all selection cuts. Without including any systematic effects, the 3σ discovery regions of the branching ratios are presented in Table 4 for three signal scenarios at three center-of-mass energies of the electron-positron collision. The 3σ discovery regions in terms of the integrated luminosity are also depicted in Figure 5. We observe that at 3σ significance level branching ratios at the order of 10 −3 is achievable at the center-of-mass energy of 240 GeV while going to larger energies of 350 and 500 GeV leads to an improvement of one order of magnitude for tqγ and tqZ(σ µν ) with an integrated luminosity of 100 fb −1 . The FCNC transition of t → qZ with γ µ -type couplings would not be measured better than the order of 10 −4 . This is because of the lack of a momentum factor q ν in the effective Lagrangian with respect to the other signal scenarios that causes to lower signal cross section. According to Figure  5, going to high luminosity regime at the center-of-mass energies of 240 and 350 leads to a reach sensitivity at the order of 10 −5 . To achieve a better sensitivity of the order of 10 −6 , we need to increase the center-of-mass energy of 500 GeV.
√ s (GeV) 240 350 500 Table 4: The sensitivity for a significance level of 3σ at the center-of-mass energies of 240, 350, 500 GeV and with 100 fb −1 of integrated luminosity of data.
In order to set 95% C.L upper limits on the anomalous FCNC couplings and consequently on the branching ratios, the CLs procedure is used [62]. The CLs technique is currently used by the big experiments such as ATLAS and CMS experiments to provide upper limits on the new physics cross sections and constraining the theory parameters. For the limits calculations the RooStats [63] package has been used. The 95% C.L upper limits on the branching ratios of t → qγ and t → qZ at the center-of-mass energies of 240, 350 and 500 GeV are shown in Table  5 based on an integrated luminosity of 100 fb −1 . As we expected, at each center-of-mass energy, Br(t → qZ) (σ µν ), and (c) for Br(t → qZ) (γ µ ).
the loosest limits belong to the FCNC transition of t → qZ with γ µ −type coupling (10 −3 − 10 −4 ). We note that the larger center-of-mass energy leads to even the level of one order of magnitude tighter bounds. In Figure 6, we present the current observed upper limits on the Br(t → qZ) versus Br(t → qγ) at 95% C.L from the recent analyses of the ATLAS and CMS experiments. The expected sensitivity from the ATLAS experiment with 300 fb −1 is in proton-proton collisions at the centerof-mass energy of 14 TeV is also shown by the dashed lines. The sensitivity of the FCC-ee with 100 fb −1 at the center-of-mass energy of 350 GeV is compared with the CMS and ATLAS results. With an integrated luminosity of 300 fb −1 , ATLAS is expected to to reach to an upper limit of 7.8 × 10 −5 on the branching ratio of t → qγ and 2.3 × 10 −4 on the branching ratio of t → qZ (σ µν −type coupling). The FCC-ee potential upper limits are expected to be significantly smaller than the attainable limits by the future LHC program.
√ s (GeV) 240 350 500 It is worth mentioning that the FCNC transitions can also be probed in tt production when a top quark decays anomalously into q +γ or q +Z. However, it has been found that the limits would be looser than the ones obtained in single top productions [64]. It is remarkable that although the limits obtained in this study are expected to be more stringent than the LHC future however in case of observing the signal, in contrary to FCC-ee, LHC is able to discriminate between anomalous tqγ and tqZ. In case of signal observation, LHC would also be able to discriminate between anomalous tuV and tcV using for example the charge ratio technique [65]. This also would be possible at the FCC-ee in case of having detectors with good efficiency of charm tagging.

Summary and conclusions
Top quark flavor-changing neutral current interactions are extremely forbidden in the SM framework because of the GIM (Glashow-Iliopoulos-Maiani) mechanisms. The SM predictions for branching ratios of the top quark decay into a photon or a Z boson and an up-type quark are at the order of 10 −14 . However, several extensions of the SM can enhance the branching ratios by a factor of 10 8−9 depending on the model. Therefore, precise measurement of these branching ratios provide an excellent possibility to probe the extensions of SM in the top quark sector. While it is impossible to measure the branching ratios with the precisions of order of 10 −14 to test the SM, observation of sizeable branching ratios would indicate new physics beyind the SM.
FCC-ee/TLEP with a clean environment and large luminosity would provide a unique opportunity to measure the properties of particles and their interactions. In this work, we investigate the sensitivity and discovery prospects of FCC-ee/TLEP to the top quark FCNC transitions. We look for the FCNC tqγ and tqZ couplings in single top-quark production in the process of e − + e + → tq +tq. We perform the analysis in a model independent way using the effective Lagrangian approach at the center-of-mass energies of √ s = 240, 350 and 500 GeV. In the analysis,  we only consider the leptonic (electron and muon) decay of the W boson in the top quark decay.
In the analysis, parton shower, hadronization and decay of unstable particles are performed using Pythia and jets are reconstructed using the anti−k t algorithm available in the FastJet package. The main background arises from the W W events, with one W boson decays leptonically. A set of kinematic variables has been proposed as the input variables to a multivariate analysis for discrimination between signal from background events. We find the 3σ discovery ranges and the 95% C.L sensitivity for three signal scenarios versus the integrated luminosity at the center-of-mass energies of 240, 350 and 500 GeV. We find that with increasing the center-of-mass energy stronger bounds would be reachable. With an integrated luminosity of 100 fb −1 at the center-of-mass energy of 350 GeV, upper limits of 2.3 × 10 −5 , 6.7 × 10 −5 would be obtained on Br(t → qγ) and Br(t → qZ) (σ µν −type), respectively. A looser upper limit of 1.9 × 10 −4 on Br(t → qZ) with γ µ −type interaction is obtained. It is found that a sensitivity of the order of 10 −6 at large integrated luminosities would be attainable. The results are compared with the expected LHC results at the center-of-mass energy of 14 TeV with 300 fb −1 . FCC-ee will be able to reach a considerable sensitivity to the anomalous FCNC couplings of tqγ and tqZ. Finally, it is remarkable that in all signal scenarios and even with large amount of data ∼ 3 ab −1 , the branching ratios would not be measured better than 10 −6 .