Search for single production of vector-like B quark decaying to bZ at future linear colliders

New vector-like quarks are predicted in many new physics scenarios beyond the Standard Model (SM). Based on a model-independent framework, we investigate the prospect of discovering the SU(2) singlet vector-like bottom quark (VLQ-B) in e+e-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{+}e^{-}$$\end{document} collisions at 3 TeV Compact Linear Collider. We study the single VLQ-B production process e-e+→Bb¯→Zbb¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-}e^{+}\rightarrow B{\bar{b}}\rightarrow Zb{\bar{b}}$$\end{document} with two types of decay channel: Z→ℓ+ℓ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z\rightarrow \ell ^{+}\ell ^{-}$$\end{document}, and Z→νν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z\rightarrow \nu {\bar{\nu }}$$\end{document}. By carrying out a full simulation for the signals and the relevant SM backgrounds, the 2σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\sigma $$\end{document} exclusion limit and 5σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\sigma $$\end{document} discovery prospects are, respectively, obtained on the B quark mass and the coupling strength g∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^{*}$$\end{document} with the integrated luminosity of 5 ab-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {ab}^{-1}$$\end{document}. Our numerical results show that the possible signals of the singlet VLQ-B might be detected at the future high-energy e+e-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{+}e^{-}$$\end{document} linear colliders.


Introduction
The discovery of the Higgs boson by the ATLAS and CMS Collaborations at the Large Hadron Collider (LHC) [1,2] opened a new field for explorations in the realm of particle physics. To solve the gauge hierarchy problem, the vectorlike quarks (VLQs) are predicted to regulate the Higgs boson mass-squared divergence [3] in many new physics (NP) scenarios beyond the Standard Model (SM), such as little Higgs models [4], extra dimensions [5], composite Higgs models [6], and other extended models [7][8][9][10]. These new particles are color-triplet spin-1/2 fermions and the left-and right-handed components transform with the same properties under the SM electroweak symmetry group [11].
The electric charges of VLQs could be +2/3e (T quark), −1/3e (B quark), +5/3e (X quark or −4/3e (Y quark), and they could be appear in electroweak singlet [ A common feature of these new fermions is that they could decay into a SM quark and a SM gauge boson, or a Higgs boson, which might generate characteristic signatures at the current and future high-energy colliders (for example see ). Here we focus on the singlet VLQ-B quark, which couples exclusively to third-generation SM quarks, B → W t, Hb, Zb.
Up to now, the direct searches for VLQ-B have been performed by the ATLAS and CMS Collaborations in Run 2 [43][44][45][46][47][48][49][50][51]. Although there is no experimental evidence of such VLQ-B, the constraints on VLQ-B have been obtained at 95% condifence level (CL). For instance, an analysis from CMS including single-lepton, dilepton, and multilepton final states probed all decay modes of the VLQ-B, and excluded their masses in the range 910-1240 GeV [49], depending on the combination of the VLQ branching fractions. Very recently, the CMS Collaboration presented a search for VLQ-B pair production in the fully hadronic final state using Run 2 data with a total integrated luminosity 137 fb −1 [50], and excluded their masses up to 1570, 1390, and 1450 GeV for 100% B → bh, 100% B → t Z, and BY doublet cases, respectively. In addition, a combination of searches utilizing various final states were performed by the ATLAS Collaboration [51], and excluded values of the VLQ-B mass up to 1220, 1370, and 1140 GeV for the singlet, the (T, B) doublet and (B, Y ) doublet cases, respectively.
So far, most of the work about the VLQ-B focuses on phenomenological analysis at the hadron colliders [52][53][54]. Compared with the hadron colliders, a future linear e + e − collider has a particularly clear background environment [55][56][57], i.e., the final stage of Compact Linear Collider (CLIC) operating at an energy of 3 TeV is expected to directly examine the pair production of new heavy top partner of mass up to 1.5 TeV [58]. The high-energy linear collider is thus a precision machine with which the properties of new VLQs can be measured precisely [59][60][61][62][63][64][65]. To the best of our knowledge, no work has been done to search single production of the SU (2) singlet VLQ-B in e + e − collisions. Therefore, we focus on the observability of the single VLQ-B production at the CLIC via the process e + e − → Bb combined with the B → bZ and the subsequent decay channels Z → + − , and Z → νν, respectively. The benefit is that it is a more highly potential process than pair production owing to less phase-space suppression. We expect that such work may become a complementary option to other production processes in searches for the heavy VLQ-B at the future high-energy linear colliders. This paper is organized as follows: in Sect. 2, we brief review the couplings of VLQ-B with the SM particles, and discuss its single production at future 3 TeV CLIC and the possible decay modes. Section 3 is devoted to a detailed analysis of the relevant signals and backgrounds at the CLIC. Finally, we summarize our results in Sect. 4.

Singlet vector-like bottom quark in the simplified model
Following the notation of Ref. [14], a simple Lagrangian that parametrizes the VLQ-B couplings to quarks and electroweak boson can be expressed as (showing only the couplings relevant for our analysis) where g is the SU (2) L gauge coupling constant and θ W is the Weinberg angle. Thus there are only two model parameters: the VLQ-B quark mass m B and the coupling strength to SM quarks in units of standard couplings, g * .
In order to make a prediction for the signal, we calculate the cross section for the process e + e − → Bb + bB at leading order (LO) by using MadGraph5-aMC@NLO [66]. Note that the model file of the singlet VLQ-B quark could be downloaded from the Feynrules Model Database [67] and could also be implemented via the FeynRules package [68].
The numerical values of the input parameters are taken from [69].
In Fig. 1a, we show the dependence of the cross sections for the process e + e − → Bb + bB on the VLQ-B quark mass m B at a 3 TeV CLIC for three typical values of g * . As the VLQ-B quark mass grows, the cross section of single production decreases slowly due to a larger phase space. For g * = 0.2 and m B = 1.5 (2) TeV, the cross section can reach 0.41 (0.24) fb. Obviously, the cross section of single B−quark production is proportional to the square of the coupling strength g * for a given VLQ-B quark mass. In Fig. 1b, we also show the branching ratios (BRs) of the three decay channels B → t W , bZ, and bh by varying the B quark mass at fixed g * = 0.1. One can see that is a good approximation, as expected from the Goldstone boson equivalence theorem [70][71][72][73][74].

Collider simulation and analysis
From the above discussions, we know that the VLQ-B can be singly produced through s-channel Z boson exchange by e + e − collisions. The relevant Feynman diagrams for single production and decaying into bZ are depicted in Fig. 2.
Considering the subsequent leptonic decay channel Z → + − , and invisible decay channel Z → νν, there are two typical final states, 2b2 , and 2b + / E T , respectively. Monte Carlo event simulations for signal and SM background are interfaced to Pythia 8.20 [75] for fragmentation and showering. All event samples are fed into the Delphes 3.4.2 [76] with the CLIC detector card designed for 3 TeV [77]. In our analysis, jets are clustered with the Valencia Linear Collider (VLC) algorithm [78,79] in exclusive mode with a fixed number of jet (N = 2 where N corresponds to the number of partons expected in the tell level final state) and fixed one size parameter R = 1.0. The b-tagging efficiency and misidentification rates are taken as the medium working points (WP) (70% b-tagging efficiency) and the misidentification rates are given as a function of energy and pseudorapidity, i.e., in a bit where E > 500 GeV and 1.53 < |η| ≤ 2.09, misidentification rates are 9 × 10 −3 for the medium WP. Finally, event analysis is performed by using MadAnalysis5 [80].

Analysis of decay channel Z → + −
In this subsection, we analyze the signal and background events at 3 TeV CLIC through the Z → + − ( = e, μ) decay channel.
For this channel, the typical signal is two b-jet and two Opposite-Sign and Same-Flavor (OSSF) leptons. The main SM backgrounds come from the following processes: To identify objects, we choose the basic cuts at parton level for the signals and SM backgrounds as follows: where R = 2 + η 2 denotes the separation in the rapidity-azimuth plane, p ,b, j T are the transverse momentum of leptons, b-jets, and light jets.
For the signal, the leptons 1 and 2 are two OSSF leptons that are assumed to be the product of the Z -boson decay, and at least two b-tagged jet are present. In Fig. 3, we plot some differential distributions for signals and SM backgrounds at the CLIC, such as the transverse momentum distributions of the leading and sub-leading b-jets ( p b 1 T , p b 2 T ), the separations R b 1 ,b 2 , the transverse momentum distributions of the leading and sub-leading leptons ( p 1 2 T ), the separations R 1 , 2 , and the invariant mass distribution for the B quark M b 1 2 . Based on these kinematical distributions, we can impose the following set of cuts: We present the cross sections of three typical signal (m B = 1200, 1500, 2000 GeV) and the relevant backgrounds after imposing the cuts in Table 1. One can see that all the SM backgrounds are suppressed very efficiently, while the signals still have a relatively good efficiency at the end of the cut flow. The large background comes from the e + e − → bb + − process, with a total cross section of 1.92 ab.

Analysis of invisible decay channels Z → νν
In this subsection, we analyze the signal and background events through the invisible decays Z → νν decay channel.
For this channel, the main SM backgrounds come from the following processes: • e + e − → ννbb, • e + e − → νν j j.
In our simulations, we apply the following basic cuts on the signal and background events: In order to get some hints of further cuts for reducing the SM backgrounds, we analyzed the normalized distributions of the missing transverse energy / E T , p b 1 T , p b 2 T , R b 1 ,b 2 , and the transverse mass distribution for the B quark M T (b 1 ) for Note that the definition of the transverse mass of the system can be seen from Ref. [81].
Based on these kinematical distributions, a set of further cuts are given as follows: • Cut-1: The transverse missing energy is required / E T > 300GeV. We summarize the cross sections of three typical signal (m B = 1200, 1500, 2000 GeV) and the relevant backgrounds after imposing the cuts in Table 2. One can see that the large background comes from the e + e − → ννbb process, with a total cross section of 17.4 ab.

Discovery and exclusion significance
To handle the relatively small event number, we will use the median significance Z to estimate the expected discovery and exclusion significance [82]: Here, s and b stand for the numbers of signal and SM background events after cuts in which the integrated luminosity are taken as 5 ab −1 at 3 TeV CLIC. In Fig. 5, we plot the 2σ and 5σ sensitivity reaches for the coupling strength g * as a function of m B at 3 TeV CLIC with integral luminosity 5 ab −1 . One finds that, for the Z → + − decay channel, the VLQ-B quark can be excluded in the region of g * ∈ [0.25, 0.5] and m B ∈ [1200 GeV, 1850 GeV] at the 3 TeV CLIC with the integrated luminosity of 5 ab −1 ,  Then, we combine its sensitivity with the above two types of decay channels by using Z comb = Z 2 ¯ + Z 2 νν . For comparison, we further present in Fig. 6 the combined sensitivity reaches for the coupling strength g * as a function of the VLQ-B quark mass m B . One can see that the B quark can be excluded in the region of g * ∈ [0.16, 0.5] and m B ∈ [1200 GeV, 2400 GeV] at the 3 TeV CLIC with the integrated luminosity of 5 ab −1 , while the discovery region can reach g * ∈ [0.26, 0.5] and m B ∈ [1200 GeV, 2150 GeV]. From above discussions, we can see that, for single production of VLQ-B with the subsequent B → bZ decay channel at 3 TeV CLIC, it is possible to detect its signal via 2b2 and 2b + / E T final states, respectively.

Conclusion
In this work, we analyzed single production of the SU(2) singlet VLQ-B at the future 3 TeV CLIC via the process e + e − → Bb → Zbb in a simplified model, in which only two free parameters are included, the VLQ-B quark mass m B and the EW coupling constant g * . We performed a full simulation for the signals and the relevant SM backgrounds based on two types of decay channels Z → + − ( = e, μ) and Z → νν. The 5σ discovery prospects and 2σ exclusion limits in the parameter plane of the two variables m B and g * were obtained at 3 TeV CLIC with the integral luminosity of 5 ab −1 , including the combined reaches of both leptonic and invisible channels. Our numerical results show that the VLQ-B can be excluded in the region of g * ∈ [0.16, 0.5] and m B ∈ [1200 GeV, 2400 GeV] and the discovery region can reach g * ∈ [0.26, 0.5] and m B ∈ [1200 GeV, 2150 GeV]. We therefore expect that the signatures studied here will provide complementary information for detecting such singlet VLQ-B quark in the future 3 TeV CLIC.
Acknowledgements We thank Yao-Bei Liu for helpful discussion. The work is supported by the Foundation of the Xinxiang Medical University (Grant no. 505035).

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: All data has been included in the table and figures.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .