Probing Top Quark FCNC tqgamma and tqZ Couplings at Future Electron-Proton Colliders

The top quark flavor changing neutral current (FCNC) processes are extremely suppressed within the Standard Model (SM) of particle physics. However, they could be enhanced in a new physics model Beyond the Standard Model (BSM). The top quark FCNC interactions would be a good test of new physics at present and future colliders. Within the framework of the BSM models, these interactions can be described by an effective Lagrangian. In this work, we study tqgamma and tqZ effective FCNC interaction vertices through the process e-p->e-Wq+X at future electron proton colliders, projected as Large Hadron electron Collider (LHeC) and Future Circular Collider-hadron electron (FCC-he). The cross sections for the signal have been calculated for different values of parameters lambda_q for tqgamma vertices and kappa_q for $tqZ$ vertices. Taking into account the relevant background we estimate the attainable range of signal parameters as a function of the integrated luminosity and present contour plots of couplings for different significance levels including detector simulation.

Phenomenologically, the sensitivities to the top quark FCNC interactions have been estimated on the branching ratio BR(t → uZ/uγ) ≃ 10 −5 for the HL-LHC with √ s = 14 TeV and L int = 3 ab −1 , and the branching ratio BR(t → uZ/uγ) ≃ 10 −6 for Future Circular Collider-hadron hadron (FCC-hh) with √ s = 100 TeV and L int = 10 ab −1 in Ref. [10], while the bounds have been estimated an order of magnitude larger for BR(t → cZ/cγ).
The future hadron electron collider projects currently under consideration are the Large he) [12]. The LHeC comprises a 60 GeV electron beam that will collide with the 7 TeV proton beam of LHC, having an integrated luminosity of L int = 100 fb −1 per year, and planning to reach 1 ab −1 over the years. On the other hand, the FCC-he mode is considered to be realized by accelerating electrons up to 60 GeV and colliding them with the proton beam at the energy of 50 TeV. A number of recent work exploring the new physics capability and potential of the projected ep colliders have been reported in Refs. [11][12][13][14][15].
In this work, we study the process e − p → e − W q + X including tqγ and tqZ effective FCNC interaction vertices at future hadron electron colliders, namely LHeC and FCC-he.
The effective Lagrangian is introduced and used in Section II to calculate the top quark FCNC decay widths Γ(t → qγ) and Γ(t → qZ) and the branching ratios. The cross sections for the signal have been calculated for different values of parameters λ q for tqγ vertices and κ q for tqZ vertices. We estimate the attainable range of top quark FCNC parameters depending on the integrated luminosity of the future ep colliders in section III. The signal and background analysis including realistic detector effects have been performed, and the contour plots of couplings κ q and λ q at different significance levels have been presented.
Finally, we summarize our results and conclude on the better limits for the top FCNC branchings.

II. TOP QUARK FCNC tqγ AND tqZ INTERACTIONS
At the electron-proton collision environment, top quark anomalous FCNC interactions in the tqγ and tqZ vertices can be described in a model independent effective Lagrangian where g e (g W ) is the electromagnetic (weak) coupling constant; c W is the cosine of weak mixing angle; λ L(R) q and κ L(R) q are the strengths of anomalous top FCNC tqγ and tqZ couplings (where q = u,c), which vanish at the leading order in the SM; P L(R) denotes the left (right) handed projection operators. The photon field strength tensor is A µν and Z boson field strenght tensor is Z µν , and the anti-symmetric tensor is σ µν = i 2 [γ µ , γ ν ]. The effective Lagrangian is used to calculate both decay widths (for the channels t → qγ and t → qZ) and production cross sections.
In addition to the usual decay channel t → W + b, the top quark can also decay into uptype quarks (u or c) associated with a vector boson via FCNC as given in Eq. 1. Considering only the SM decay width and the FCNC interactions with electroweak neutral gauge bosons, the top quark decay width (Γ t ) can be written as The dominant SM decay mode of top quark is t → W + b, the decay width for this mode is given as at the leading order (LO), and it is improved to the next to leading order (NLO) expression as given in Ref. [16]. The ratios of the SM decay widths are calculated as Γ(t → W + s)/Γ(t → 6.318 × 10 −5 [17]. The top quark FCNC partial decay widths are for the t → qγ channel, while the other partial decay widths are for the t → qZ channel, where q = u, c. The branching ratios for t → qγ and t → qZ decay channels depending on the FCNC tqγ and tqZ couplings are shown in Fig. 1.

III. SENSITIVITIES AT FUTURE EP COLLIDERS
The production subprocess (e − q → e − W + b, where q = u,c) including signal diagrams with tqγ and tqZ interaction vertices is presented in Fig. 2. The similar diagrams for the subprocess (e −q → e − W −b ) have also been included in the calculation. The cross sections for the process e − p → e − W ± q + X at different values of couplings κ q and λ q in the range of (0.00 − 0.05) at LHeC and FCC-he are given in Table I. The cross section increases when the coupling parameters κ q and λ q grow in the interested range. We plot the contours using Table I to estimate the sensitivity to FCNC coupling parameters. The contour lines correspond to different values of the signal cross sections (where ∆σ denotes the signal cross section (in pb) when the interfering background cross section is subtracted from the total cross section) as shown in Fig. 3 for LHeC and FCC-he. For a cross section value of the signal the sensitivity to coupling parameter λ q is higher than the coupling parameter κ q .
The process e − p → e − W ± q + X includes both the signal and the background interfering with the signal. We calculate the cross sections for this process to normalize the distributions from the signal and background events. We take into account the main background (B1: e − W ± q) and include other background (B2: e − Zq) which contain at least three jets and one electron in the final state. Here, QCD multijet backgrounds are not included in the analysis of top quark FCNC tqγ and tqZ interactions.
In our calculations, we produce signal and background events by using MadGraph 5 aMC@NLO [18], with an effective Lagrangian implementation through FeynRules [19] for the signal. Afterwards the parton showering and detector fast simulations are carried out with Pythia 6 [20] and Delphes 3.4 [21], respectively.  The kinematical distributions for signal and interfering background are given in Fig. 4 for LHeC and FCC-he. The transverse momentum (p T ) (on the left) and rapidty (η) (on the right) distributions of the leading jet, second leading jet and third leading jet are presented in these figures. These distributions are obtained after preselection of the events. For the analysis of signal and background events, we also apply analysis cuts after the generator level pre-selection. In order to select signal events we require having one electron and three jets ordered according to the highest transverse momentum p T . Since there is an energy asymmetry in the electron-proton collisions, the jets from the process mainly peaks in the In the analysis, we require at least three jets and one electron in the events, one of the jets should be b-tagged with leading jet p T (j) > 40 GeV and other jets having p T (j) > 30 GeV and | η(j) |< 2.5, the electron with p T (e) > 20 GeV and | η(e) |< 2.5 as the cut flow given in Table II. Further steps in the cut flow table include invariant mass intervals for selecting events for the analysis.
The cut efficiencies have been calculated after pre-selection for signal and background as shown in Fig. 6 for LHeC and FCC-he. We have larger cut efficiencies for higher values of the FCNC couplings. Fig. 6 shows that the cut efficiency for the background changes from 6% to 1% for Cut-1 to Cut-5, whereas the cut efficiencies for the signal decrease from 11% to 3.2% for couplings κ q = λ q = 0.05.
After Cut-5, the number of events for background and signal (different values of couplings κ q and λ q ) are given in Table III for LHeC and for FCC-he with an integrated luminosity of 100 fb −1 . For the coupling parameters κ q = λ q = 0.05 we obtain the number of events   Cut-1 Cut-2 Cut-3 Cut-4 Cut-5 Ef ciency1(%) Cut-1 Cut-2 Cut-3 Cut-4 Cut-5 Ef ciency1(%) factor is 0.17 for κ q = λ q = 0.01. For each cut step the number of events can be obtained from Table III with the relative cut efficiency factors from Fig. 6.
We plot the invariant mass distribution of top quark reconstructed from three jets (one of them is b−tagged) for different coupling scenarios (at first row) λ q = 0.0, κ q = 0.05, (second row) λ q = 0.05, κ q = 0.0 and (third row) λ q = 0.05, κ q = 0.05 as shown in Fig. 7 for LHeC and FCC-he. The ratio of the S + B and B is more enhanced at top mass for equal coupling scenario (c) when it is compared with the other scenarios (a) and (b) as seen from Fig. 7.
In order to quantify statistical significance (SS), we calculate signal (S) and background (B) events after final cut. Here the SS is defined by  The SS values depending on the integrated luminosity ranging from 1 fb −1 to 1 ab −1 at the LHeC and FCC-he are presented in Fig. 8 for the coupling scenarios (at first row) λ q = 0.0, κ q = 0.05, (second row) λ q = 0.05, κ q = 0.0 and (third row) λ q = 0.05, κ q = 0.05.
The significance corresponding to 2σ, 3σ and 5σ lines (dotted) are also shown in these figures. In Fig. 8, the SS values depending on the integrated luminosity ranging from 1 fb −1 to 1 ab −1 at the FCC-he are presented for these coupling scenarios with the 2σ, 3σ and 5σ significances.
Using the corresponding statistical significances, we fit the significance as a function of two parameters κ q and λ q at the integrated luminosity of 100 fb −1 and 1 ab −1 . We obtain contour lines from the fit procedure. In Fig. 9, we estimate the reach for couplings κ q and λ q corresponding to 2σ, 3σ and 5σ significance for integrated luminosity at the LHeC and FCC-he, respectively. We obtain the 2σ significance for the couplings κ q = 0.014, λ q = 0.012 and κ q = 0.008, λ q = 0.007 at LHeC with the integrated luminosities 100 fb −1 and 1 ab −1 , respectively. The sensitivities to the couplings are enhanced at FCC-he as the obtained values κ q = 0.008, λ q = 0.006 and κ q = 0.0037, λ q = 0.0025 for L int = 100 fb −1 and 1 ab −1 , respectively.
The limits on couplings can be translated into the branching ratio via Fig. 1. We find the upper limits on branching ratio BR(t → qZ)≤ 4.0 × 10 −5 and BR(t → qZ)≤ 1.0 × 10 −5 at 2σ significance level for L int = 1 ab −1 at LHeC and FCC-he, respectively. The HL-LHC will produce a large number of top quarks, which also provide opportunity to search for FCNC processes to improve existing constraints on the branching ratios BR(t → qZ)< 10 −5  with the upgraded LHC experiments. We find better limits when compared to the current experimental limits and estimations for HL-LHC. In our previous studies given in Refs. [13] and [14], we have obtained the limits on the top quark FCNC tqγ couplings depending on the integrated luminosity of future ep colliders. As a complementary to these studies, here we have analyzed both tqγ and tqZ couplings in three different scenarios and obtained sensitivities to the couplings κ q and λ q .
Finally, extending the analysis for higher luminosities, we present the expected sensitivities on BR(t → qγ) and BR(t → qZ) as a function of the integrated luminosity (in the range between 100 fb −1 and 3000 fb −1 ) at the LHeC and FCC-he in Fig. 10. For the integrated luminosities of 1 ab −1 , 2 ab −1 and 3 ab −1 , the sensitivities on BR(t → qγ) and BR(t → qZ) are given in Table IV at the LHeC and FCC-he.

IV. CONCLUSION
The top quark FCNC interactions are important probes for new physics beyond the SM.
It is also worth to mention that the analysis include the signal and background interference TeV will allow us to significantly improve the sensitivity to the top quark FCNC.