Searches for the Anomalous FCNC Top-Higgs Couplings with Polarized Electron Beam at the LHeC

In this paper, we study the single top and Higgs associated production $\rm e^- p\rightarrow \nu_e \bar{t} \rightarrow \nu_e h \bar{q}(h\rightarrow b\bar{b})$ in the top-Higgs FCNC couplings at the LHeC with the electron beam energy of $E_{e}$ = 60 GeV and $E_{e}$ = 120 GeV, combination of a 7 TeV and 50 TeV proton beam. With the possibility of e-beam polarization ($p_{e}$ = 0, $\pm0.6$), we distinct the Cut-based method and the Multivariate Analysis (MVA) based method, and compare with the current experimental and theoretical limits. It is shown that the branching ratio Br $\rm(t\to uh)$ can be probed to 0.113 (0.093) $\%$, 0.071 (0.057) $\%$, 0.030 (0.022) $\%$ and 0.024 (0.019) $\%$ with the Cut-based (MVA-based) analysis at ($E_{p}$, $E_{e}$) = (7 TeV, 60 GeV), ($E_{p}$, $E_{e}$) = (7 TeV, 120 GeV), ($E_{p}$, $E_{e}$) = (50 TeV, 60 GeV) and ($E_{p}$, $E_{e}$) = (50 TeV, 120 GeV) beam energy and 1$\sigma$ level. With the possibility of e-beam polarization, the expected limits can be probed down to 0.090 (0.073) $\%$, 0.056 (0.045) $\%$, 0.024 (0.018) $\%$ and 0.019 (0.015) $\%$, respectively.


I. INTRODUCTION
The Large Hadron Electron Collider (LHeC) is the second electron-hadron collider following HERA [1]. With remarkable higher energy and luminosity, the LHeC is a major step towards understanding the Higgs physics and QCD. For the LHeC colliding energy, the 7 TeV proton beam at the LHC as well as the 50 TeV proton beam at the future FCC-he [2] and a new 60 GeV electron beam [1] are envisaged. To probe new physics, the anomalous flavor changing neutral current (FCNC) Yukawa interactions, between the top-Higgs and either an up or charm quark, would provide a clear signal. The SM Lagrangian can be extended by the following terms, where the real parameters κ tuh and κ tch denote the FCNC couplings of the Higgs to up-type quarks. The total decay width of the top-quark Γ t is where the decay width Γ SM t→W − b and Γ t→u(c)h can be found in [3] and [4], respectively. Thus, the branching ratio for t → u(c)h can be approximately given by where G F is the Fermi constant and τ W = m W mt . The W boson and top quark masses are chosen to be m W = 79.82 GeV and m t = 173.2 GeV, respectively.
Up to now, the investigation of t → qh anomalous couplings have been experimented by many groups, which gives the stronge limits on the top-Higgs FCNC couplings. For instance, according to the ATLAS and CMS collaborations, the upper limits of Br (t → qh) < 0.79 % [5,6] and Br (t → qh) < 0.45 % [7] have been set at 95 % confidence level (C.L.). Except for the direct collider measurements, the low energy observable, by bounding the tqH vertex from the observed D 0 −D 0 mixing [8], the upper limit of Br (t → qh) < 5 × 10 −3 may be produced. Furthermore, through Z → cc decay and electroweak observables, the upper limit of Br (t → qh) < 0.21 % [9] can be obtained.
On the other hand, based on the experimental data, many phenomenological studies are performed from different channels. For instance, [10] found that the branching ratios Br (t → qh) can be probed to 0.24 % at 3σ level at 14 TeV LHC with an integrated luminosity of 3000 fb −1 through the process Wt → Whq → νbγγq. [11] explored the top-Higgs FCNC couplings through tt → Wbqh → νbγγq and found the branching ratios Br (t → uh) can be probed to 0.23 % at 3σ sensitivity at 14 TeV LHC with L = 3000 fb −1 . And [12] obtained the Br (t → qh) to be 0.112 % based on the process of tt → tqh → νbbbq. The process of th → νbτ + τ − has been studied in [13] and they estimated the upper limits of Br (t → uh) < 0.15 % at 100 fb −1 of 13 TeV data for multilepton searches. The results from different experiments and theoretical channels are summarized in Table I. In this study, we examined the e − p → ν et → ν e hq at the LHeC where the Higgs boson decays to bb, at a 7 (50) TeV with a 60 (120) GeV electron beam and 1000 fb −1 integrated luminosity. The possibility of e-beam polarization is also considered. The Feynman diagram is plotted in Fig. 1. The main backgrounds which yield the same or similar final states to the signal are listed as below:

II. TOOLS AND METHOD
During the simulation, we first extract the Feynman Rules by using the FeynRules package [14] and generate the event with MadGraph@NLO [15]. PYTHIA6.4 [16] was set to solve the initial and final state parton shower, hadronization, heavy hadron decays, etc. We use CTEQ6L [17] as the parton distribution function and set the renormalization and fac-torization scale to be µ r = µ f . We take the input heavy particle masses as m h = 125.7 GeV, m t = 173.2 GeV, m Z = 91.1876 GeV and m W = 79.82 GeV, respectively. We employ the following basic pre-selections cuts to select the events: where ∆R = ∆Φ 2 + ∆η 2 is the separation with ∆η and ∆Φ in the rapidity-azimuth

A. Cut-based method
In order to distinguish between signal-related events and background-related events as much as possible, we set a series of cuts. We list all the Cut-based selections here: • cut1: the basic pre-selection cuts.
• cut5: the reconstructed W boson mass window m W < 50 GeV or m W > 90 GeV.
• cut6: the reconstructed Z boson mass window m Z < 55 GeV or m Z > 95 GeV.

B. MVA-based method
We implemented the MVA method using the Root Toolkit for Multivariate Analysis (TMVA) [18]. After cut1, cut2 and cut3, we especially select several input variables to discriminate the signal and background events, thus resulting better signal significance.
Specifically, we define a set of totally 44 kinematic variables and choose the most effective ones for Boosted Decision Trees (BDT) training, which are: the b-jet number (N bjet ), the separation in the Φ − η plane between jets (∆R B 1 B 2 , ∆R B 1 J 1 ), the difference in azimuthal angle between jets (∆Φ B 1 B 2 , ∆Φ B 1 J 1 ), the transverse momentum of the jet (p J 1 T ), the difference in |η| between Higgs jet system (∆η hJ 1 ). It is worth noting that e-beam polarization is considered in both Cut-based method and MVA-based method.

III. RESULTS
In Fig. 2 (60) GeV and Fig. 3 (120) GeV, we show the dependence of the cross section σ on the top-Higgs FCNC couplings κ tqh at E e = 60 (120) GeV with p e = ±0.6 electron beam polarization combination of a 7 (50) GeV proton beam for three different cases. (I) κ tqh = κ tuh , κ tch = 0, (II) κ tqh = κ tch , κ tuh = 0 and (III) κ tqh = κ tuh = κ tch . Obviously, the cross section of κ tqh = 0.1 can be 100 times larger than that of κ tqh = 0.01, and the cross section of 50 TeV can be 9.1 (6.6) times larger than that of 7 TeV with a 60 (120) GeV electron beam. We also find that the cross section between polarized and unpolarized electron beam cases are related as: FIG. 3: The same as Fig. 2 but for E e = 120 GeV.
The cross section of the signal and backgrounds (in units of fb) are summarized in Table II (Cut-based method) and In order to estimate the sensitivity to the anomalous tqH couplings, we used chi-square (χ 2 ) function [19,20]: where σ tot is the total cross section and δ is the statistical error. In Fig. 4 (Cut-based Analysis) and    Table II but  Finally, we give a precise integrated luminosity (L) corresponding to the critical limits obtained by the experimental results (Table IV) and other phenomenological studies (Table   V) . With the e-beam polarization p 2 = -0.6, the L needed to get the upper bounds on the Br (t → qh) is reduced significantly. A detailed comparison between the LHeC collider(s) and the LHC or linear colliders are given.