Single production of vector-like bottom quark at the LHeC

Existences of vector-like quarks (VLQs) are predicted in many new physics scenarios beyond the Standard Model (SM). We study the possibility of detecting the vector-like bottom quark (VLQ-$B$) being the $SU(2)$ singlet with electric charge $-1/3$ at the Large Hadron Electron Collider (LHeC) in a model independent framework. The decay properties and single production of VLQ-$B$ at the LHeC are explored. Three types of signatures are investigated. By carrying out a fast simulation for the signals and the corresponding backgrounds, the signal significances are obtained. Our numerical results show that detecting of VLQ-$B$ via the semileptonic channel is better than via the fully hadronic or leptonic channel.


I. Introduction
With the discovery of a 125 GeV Higgs boson in July 2012 by ATLAS and CMS collaborations at the CERN Large Hadron Collider (LHC) [1,2], the Standard Model (SM) has acquired remarkably success at explaining most of the available experimental phenomena with great accuracy. As yet, there are still unresolved theoretical issues in the SM, such as the nature of the electroweak symmetry breaking and the hierarchy between the electroweak and the Planck scales. One solution is by introducing new heavy particles called vector-like quarks (VLQs) which regulate the Higgs boson mass-squared divergence [3,4]. Since VLQs can obtain the gauge invariant mass terms of the form mψψ directly, they are not subject to the constraints from Higgs production. Therefore, VLQs as a class of interesting particles have not been excluded by precision measurements.
The left-and right-handed components of VLQs have the same transformation properties under the SM electroweak symmetry group [15,16]. VLQs can be embedded in singlet or multiplets representations of SU (2) and have four possible charge assignments: Q T = +2/3, Q B = −1/3, Q X = +5/3, and Q Y = −4/3. A common feature of these new fermions is that they are assumed to decay to a SM quark with a SM gauge boson, or a Higgs boson. In this paper, we focus on the SU (2) singlet vector-like bottom quark (VLQ-B) in a model independent way.
A lot of phenomenological studies for VLQs have been presented in vast literatures [17]. For example, Ref. [18] has considered single production of VLQ-B which decays into Hb at the LHC in context of the composite Higgs model, Ref. [19] introduced an effective Lagrangian to study the possibility of detecting VLQ-B via the decay channel B → W t at the LHC. Ref. [20] performed global fits of the constraints of VLQ-B by using the CKM unitarity violation, excess in Higgs signal strength, and bottom quark forward-backward asymmetry. In our previous work [21], we have considered the capability of detecting VLQ-B at the LHC via single production channel, which is a more potential process than pair production since its less phase-space suppression when the mass of VLQ-B is more heavier.
By now, the direct searches for VLQ-B have been performed by ATLAS and CMS collaborations at the LHC with center-of-mass (c.m.) energy of √ s = 13 TeV and an integrated luminosity of 35-36 fb −1 [22][23][24]. Although there are not any signatures be detected, the constraints on VLQ-B have been obtained. The most stringent bounds on the VLQ-B mass are in the range of 700-1800 GeV depending on the production modes, the considered final states and the assumed branching ratios.
In fact, the collider has become and will remain an important tool to test wide classes of new physics models. Thus, it is highly motivated to investigate all sensitive search strategies within the possibly available accelerator and detector designs.
Here, we intend to study the possibility of detecting VLQ-B in the proposed powerful high energy ep collider, the Large Hadron Electron Collider (LHeC) [25] with a 60-140 GeV electron beam and a 7 TeV proton beam from the LHC. It is supposed to run synchronously with the Compared to the previous ep ollider, HERA, the LHeC extends one order of magnitude in the c.m. energy and 1000 times in the integrated luminosity. Refs. [26][27][28] have studied the discovery potential of the vector-like top quarks through various channels at the LHeC, where the vector-like top partner is the SU (2) singlet with charge 2/3. To the best of our knowledge, so far, no work has been done to search single production of the SU (2) singlet VLQ-B at the LHeC. Hence, we mainly study the observability of the single VLQ-B production at the LHeC combine with the B → W t decay channel in our work. Considering the final state have two W -boson (one of those come from top quark decaying), we analyze three types of signatures, which come from the fully hadronic decay channel, the fully leptonic decay channel, and the semileptonic decay channel, respectively.
We expect that such work may become a complementary to other production processes in searches for the heavy VLQ-B at the LHC.
This paper is organized as follows: in section II, we brief review the couplings of VLQ-B with the SM particles, and discuss its possible decay modes. Section III devotes to a detailed analysis of the relevant signals and backgrounds. Finally, we summarize our results in section IV.

II. Effective Lagrangian and decay modes of the vector-like bottom quark
VLQs can interact with the SM quarks and the Higgs boson through Yukawa couplings. After the Higgs developing a nonzero vacuum expectation value (VEV), VLQs are allowed to mix with the SM quarks. The total mass matrices of quarks are determined by the chosen SU (2) representations of VLQs. By diagonalizing the mass matrices, one can obtain the couplings between physical states which can be found in Ref. [29]. Ref. [30] proposed a more compact parameterization for the vector-like top quark couplings. Similarly, we consider the same parameterization in the case of VLQ-B and assume that it is the SU (2) singlet, only couple to the third generation SM quarks.
The generic parametrization of an effective Lagrangian of VLQ-B is given by (showing only the couplings relevant for our analysis): +h.c., where g is the SU (2) L coupling constant, υ 246 GeV is the electroweak symmetry breaking scale. We have abbreviated cos θ W as c W , where θ W is the Weinberg angle. There are only three parameters that fully describe the relevant interactions we consider. Besides the mass parameter M B , there are two coupling parameters appearing in Eq.(1): • κ B , the coupling strength to SM quarks in units of standard couplings, which is only relevant to single production; • R L , the generation mixing coupling, which describes the rate of decays to first two generation quarks with respect to the third generation, where the subscript L represents the chirality of the fermions. For the singlet VLQ-B, we neglect the mixing of right-handed quarks since it is suppressed [4].  According to above discussions, VLQ-B has three decay modes: W t, Zb, and Hb. The corresponding partial widths are given by where and the function λ(a, b, c) is given by In the limit of M B m t , the partial widths can be approximate written as From above equations, we can see that, for heavy VLQ-B, is a good approximation as expected by the Goldstone boson equivalence theorem [31]. From the Lagrangian given in Eq.1, one may expect the branching ratios of Hb channel is the largest one since the coupling coefficient of BHb is proportional to M B . Actually, the partial width Γ (B → Hb) is proportional to M 3 B , which is similarly with that for other two decay channels as shown in Eqs. (7 -9). The branching ratios of these decay channels are plotted as functions of the mass parameter

III. Signal analysis and discovery potentiality
For the single production of VLQ-B at the LHeC, the dominant way is mediated by the exchange of a Z boson in the t-channel. The Feynman diagram for the single production and decaying into W t is presented in Fig. 2. For the chosen decay channel of VLQ-B, the final state contains two W -bosons (one of those coming from top quark decay). There are three types of signatures, which come from the fully hadronic, the fully leptonic and the semileptonic decay channel, respectively.
To proceed signal analysis, we need to know the values of some parameters. The SM input parameters which relevant to our calculations are taken from Ref. [32] as follow: Considering the current constraints on the VLQ-B mass [22][23][24], we choose three benchmark points:  M B = 800, 900, 1000 GeV, which are referred to as B 800 , B 900 , B 1000 , respectively. The stringent bounds on the coupling parameter κ B come from the experimental data about the Zbb couplings, which give the upper limit as κ B < 0.23 [30]. In our numerical estimation, we will take κ B = 0.2.
For taking the e − and p beam energy as 140 GeV and 7 TeV respectively, the c.m. energy of the LHeC is √ s = 1.98 TeV. In Fig. 3, we show the cross sections of the process e − p → e − B with different κ B and e − beam polarization at the LHeC. Obviously, the cross sections are insensitive to the polarization of e − beam, we only discuss the unpolarized case.
The model file [33] which realize the Lagrangian given by Eq.1 can be found in the dedicated FeynRules package [34]. Signal and background events are simulated at the leading order using MadGraph5-aMC@NLO [35] with the CTEQ6L parton distribution function (PDF) [36]. Showering, fragmentation and hadronization are performed with customized Pythia [37]. The PGS is applied for detector simulation, and the relevant parameters are taken for LHeC Detector Design [25,38]. The anti-κ t algorithm [39] with parameter ∆R = 0.4 is used to reconstruct the jets.
Finally, MadAnalysis5 [40] is applied for data analysis and plotting.

A. The fully hadronic channel
In this subsection, we analyze the signal and background events and explore the sensitivity of the singlet VLQ-B at the LHeC ( √ s = 1.98 TeV) through the fully hadronic decay channel: For this channel, the typical signal is exactly one charged electron, one b-jet and four jets. The main SM backgrounds come from the following five processes the one light jet mentioned above can be faked as b-jet.
The signal and background processes are simulated at the LHeC with an integrated luminosity of 1000 fb −1 . Firstly, we apply the basic cuts to the signal and background events, which are used to simulate the geometrical acceptance and detection threshold of the detector. These basic cuts are selected as follows in our simulation p l T > 10 GeV, |η l | < 2.5; p j T > 20 GeV, |η j | < 5; ∆R(x, y) > 0.4, x, y = l, j, where the particle separation ∆R xy is defined as (∆η xy ) 2 + (∆φ xy ) 2 with ∆η xy and ∆φ xy being the rapidity and azimuthal angle gaps between the two particles in a pair. We use the characteristics of the signal as a handle to reduce the backgrounds. Hence, we dipict the normalized distributions of H T , p j T , θ e − j and θ jj for signals and backgrounds in Fig. 4. According to the information of these kinematic distributions, we impose the following cuts to get a high statistical significance. All cuts applied are given in the following list.
• Cut 1: The first kinematical selection involves the total hadronic energy H T , which is shown in • Cut 2: The distributions of the transverse momentum p T of the jet for signals and backgrounds are shown in Fig 4(b). Based on these normalized distributions, we require the second cut selection is p j T > 250 GeV.
• Cut 3 : Fig 4(c) and Fig 4(d) show the normalized distributions of θ ej and θ jj , which denote the angles between electron momentum and jet momentum, and two jet momentum. Here, the jet represents each of the four jets. From these distributions, we can efficiently suppress the backgrounds by impose the cuts: θ e − j > 2.5 and θ jj > 3.
where S and B denote the numbers of the signal and background events, respectively. We define SS = 5 and 3 as the discovery significance and the possible evidence, respectively. Note that we do not consider the theoretical and systematic uncertainties. In Table 1

B. The fully leptonic channel
Similar with the previous section, we analyze the observation potential and explore the sensitivity of the singlet VLQ-B at the LHeC through the leptonic decay channel: For this channel, the typical signal is exactly three charged leptons, one b jet, and missing energy.
The main SM backgrounds come from the following processes: where the light jet can be faked as b-jet.
In order to get some hints of further cuts for reducing the backgrounds, we analysis the normalized distributions of / H T , p l T and η l for the signals and backgrounds as shown in Fig. 6. Then, to get high statistical significance, a set of further cuts are given as followings.
• Cut 1: The first cut involves the missing hadronic transverse energy / H T , which is plotted in Fig. 6(a) for signals and backgrounds. Only events with / H T > 120 GeV is selected. This cut is specially useful for reducing background events.
• Cut 2: The normalized distributions of transverse momenta of leptons for signals and backgrounds are shown in Fig. 6(b), we can see that the transverse momenta of signal events are distributed mostly at large p l T values, which are different from the distributions of background events. So, we require p l T > 60 GeV to enhance the signal significance.
• Cut 3: We plot the normalized distributions of the pseudo-rapidity of the lepton in Fig. 6(c) for the signals and backgrounds. From these distributions, we can efficiently reduce the backgrounds by requiring the lepton to have the cut: η l < −1.0.  with the integrated luminosity L = 1000 fb −1 in Table 2. And the contour plots of integrated luminosity in the SS-M B plane are shown in Fig. 7. We can see that, for M B =800, 900 and 1000 For this channel, the typical signal is exactly two charged leptons, one b-jet, two jets (which coming from the top quark decay) and missing energy.
The dominant SM backgrounds come from the following processes where one light jet might be faked as b-jet.
We apply the following basic cuts on the signal and background events in our simulation: p l T > 10 GeV; |η l | < 2.5; p j T > 20 GeV; |η j | < 5; ∆R(x, y) > 0.4, x, y = l, j Further, we apply some general preselections as following.
To carry out the cut-based analysis, we discuss the normalized distributions of p l T , p j T , / E T and H T for signals and backgrounds at the LHeC with an integrated luminosity of 1000 fb −1 shown in Fig. 8. All cuts are applied one after the other in the order given in the following list.
• Cut 1 : In Fig.8(a), we show the normalized distributions of the transverse momenta p l T for signals and backgrounds. Based on the normalized distributions, we impose the first cut to get a high significance: p l T > 100 GeV.
• Cut 2 : The normalized distribution of the transverse momenta p j T is given in Fig.8(b). We require the transverse momentum p T of the jet is larger than 100 GeV, which can efficiently reduce the backgrounds and enhance the signal significance.  (d) Figure 8: Normalized distributions of p l T , p j T , / E T and H T for signals and backgrounds at the LHeC with an integrated luminosity of 1000 fb −1 .
• Cut 3 : From Fig. 8(c), we can see that the missing transverse energy / E T should be larger than 100 GeV.
• Cut 4 : From Fig. 8(d), we can see that the events with H T > 300 GeV are kept. This cut can further reduce the backgrounds.   In Table 3 From above discussions, we can see that, for single production of VLQ-B at the LHeC, it is possible to detect its signal via the fully hadronic, the fully leptonic and the semileptonic final states. However, the VLQ-B, which is the SU (2) singlet with electric charge −1/3 can be more easy detected via the semileptonic decay channel at the LHeC.

V. Conclusions
In this paper, we study the discovery potential of the single production of VLQ-B at the LHeC, through three types of the characteristic signals, which come from the fully hadronic, the fully leptonic and the semileptonic decay channels. We focus our attention on the SU (2)