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# Searching for singlet vector-like leptons via pair production at ILC

Chong-Xing Yue1,2 Yue-Qi Wang1,2 Han Wang1,2 Yi-Hang Wang1,2 Si Li1,2
1Department of Physics, Liaoning Normal University, Dalian 116029, China
2Center for Theoretical and Experimental High Energy Physics, Liaoning Normal
University, China

###### Abstract

Vector-like leptons (VLLs) as one kind of the most intriguing particles, have
been widely concerned in several extensions of the Standard Model (SM). In
this work, we explore the discovery potential of VLLs via pair production in
the context of models that satisfy asymptotic safety at the International
Linear Collider (ILC). The expected sensitivities of the ILC with the center
of mass energy $\sqrt{s}=$ 1 TeV and the integrated luminosity $\mathcal{L}$ =
1 ab-1 to the parameter space of this kind of VLL models are derived. The
results we obtained show that, for the VLL with mass $M_{F}$ in the region of
$M_{F}\in$ [101GeV, 463GeV], the Yukawa coupling $\kappa$ can be as low as
$\kappa\in$ [0.032, 0.098].

##  I Introduction

Although the discovery of the Higgs boson Aad et al. (2012); Chatrchyan et al.
(2012) indicates the completion of the Standard Model (SM) of elementary
particles, there still remain some unresolved issues that require to make
clear. For example, it can not deal with the existence of gauge hierarchy Feng
(2013) and neutrino masses Gonzalez-Garcia and Maltoni (2008). Moreover, mass
and mixing patterns of the SM leptons Workman et al. (2022) cannot be stated
clearly. So, large number of new physics models beyond the SM have been
proposed and these models predict existence of new particles. The vector-like
fermions including vector-like quarks (VLQs) and vector-like leptons (VLLs) as
one kind of the most intriguing particles have recently attracted widespread
attention both in theory and experiment. It is well known that vector-like
quarks have been studied extensively in literatures, for example see the
classical papers He and Xianyu (2014); Wang et al. (2013); He et al. (2002,
2000) and recent review article Alves et al. (2024). However, it is important
to note that, from the point of view of particle phenomenology, vector-like
leptons have the same status as vector-like quarks.

So far, there are many extensions of the SM that predict the existence of
VLLs, such as composite models Panico and Wulzer (2016); Cacciapaglia et al.
(2020), left-right symmetric models Pati and Salam (1974); Mohapatra and Pati
(1975); Senjanovic and Mohapatra (1975); Mohapatra et al. (1978),
supersymmetric models Martin (2010); Endo et al. (2011); Araz et al. (2018),
and grand unified theories Morais et al. (2021, 2020). The extensively
searches of the VLLs with masses from a few GeV to TeV consist of an important
component of the experimental and theoretical expectations. The parameter
space and the variety of mass ranges of VLLs have been limited by different
high energy collider experiments. An early study for VLLs at the Large
Electron-positron (LEP) excluded the heavy leptons with mass up to 101.2 GeV
Achard et al. (2001). At the Large Hadron Collider (LHC), CMS excluded VLLs
transforming as singlets under $SU(2)_{L}$ in the range from 125 to 150 GeV at
95% CL Tumasyan et al. (2022). For doublet vector-like $\tau$ lepton extension
of the SM, a recent ATLAS study based on 139 fb-1 at the 13 TeV LHC excluded
VLLs in the mass range 130-900 GeV at 95% CL Aad et al. (2023). Using the CMS
search based on 77.4 fb-1 at the 13 TeV LHC, the authors of Ref. Bißmann et
al. (2021) have found that flavorful $SU(2)_{L}$ singlet VLLs are excluded for
mass below around 300 GeV and the doublet VLLs are excluded below around 800
GeV. Additionally, other noteworthy discussions regarding VLLs in the context
of different new physics scenarios have been made in Refs. Halverson et al.
(2014); Schwaller et al. (2013); Ishiwata and Wise (2013); Dermisek et al.
(2014, 2016); Chen and Nomura (2016); Xu et al. (2018); Zheng (2020); Freitas
et al. (2021); Dermisek et al. (2021a, b); Kawamura and Raby (2021); Raju et
al. (2023); Kawamura and Shin (2023); Cao et al. (2023); Liu and Yang (2010).

At present, most of the research is based on different VLL models or in model-
independent way. In this work, we explore the discovery potential of VLLs at
the International Linear Collider (ILC) in the context of the models that
satisfy asymptotic safety Litim and Sannino (2014); Bond and Litim (2019);
Bond et al. (2020); Bond and Litim (2018, 2017a, 2017b); Bednyakov and
Mukhaeva (2023); Litim et al. (2023) and introduce new matter fields and
Yukawa couplings Bißmann et al. (2021); Hiller et al. (2020a, 2019, b). The
new models called VLS models in this paper, contain VLLs and new scalars,
which tame the UV behavior Gross and Wilczek (1973) of the Standard Model
towards the Planck scale or beyond. After spontaneous symmetry breaking, the
Z, W and Higgs boson couplings will be modified due to the mixing of VLLs, new
scalars with the SM leptons, Higgs boson, respectively. In this paper, we
focus on the couplings of VLLs to neutrino and W boson and investigate the
possibility of searching for the VLLs at the 1 TeV ILC for an integrated
luminosity of $\mathcal{L}=1$ ab-1 Aihara et al. (2019); Bambade et al.
(2019).

The paper is structured as follows. The interaction lagrangian of VLLs with
the SM particles is introduced in section II. Detailed analysis for the
possibility of probing VLLs via pair production at the ILC is provided in
section III. In section IV, we summarize the main results about the projected
sensitivity of the ILC to the couplings of VLLs with W bosons for different
values of the VLL mass.

##  II Effective interactions of VLLs

The VLS models Hiller et al. (2020a, 2019, b) as one kind of the new VLL
models predict the existence of the three generations of VLLs $\psi_{L,R}$,
which may be $SU(2)_{L}$ singlets with hypercharge $Y=-1$ and $SU(2)_{L}$
doublets with $Y=-\frac{1}{2}$ corresponding the singlet and doublet VLS
models, respectively. Two models also involve complex scalars $S_{ij}$ as
singlets under the SM gauge interactions, where $i,j=1,2,3$ are two flavor
indices. The new Yukawa lagrangians of the VLS models read

$\displaystyle\begin{split}\mathcal{L}_{\text{Y}}^{\text{singlet}}=&-\kappa\bar{L}H\psi_{R}-\kappa^{\prime}\bar{E}S^{\dagger}\psi_{L}-y\bar{\psi}_{L}S\psi_{R}+h.c.,\end{split}$
(1)
$\displaystyle\begin{split}\mathcal{L}_{\text{Y}}^{\text{doublet}}=&-\kappa\bar{E}H^{\dagger}\psi_{L}-\kappa^{\prime}\bar{L}S\psi_{R}-y\bar{\psi}_{L}S\psi_{R}+h.c..\end{split}$
(2)

Where E, L and H denote the SM singlet, doublet leptons and Higgs boson,
respectively. On account of $SU(3)$-flavor symmetries of VLS models, each
lepton fulfills flavor-conservation and the new Yukawa couplings
$y,\kappa,\kappa^{\prime}$ become single couplings, instead of being tensors
Hiller et al. (2020a).

After spontaneous symmetry breaking, the effective interaction among VLLs with
the new scalar particles and SM particles in the singlet VLS model is given by
Ref. Bißmann et al. (2021).

$\displaystyle\begin{split}\mathcal{L}_{\text{int}}^{\text{singlet}}=&-e\bar{\psi}\gamma^{\mu}\psi
A_{\mu}+\frac{g}{\cos\theta_{w}}\bar{\psi}\gamma^{\mu}\psi
Z_{\mu}+(-\frac{\kappa}{\sqrt{2}}\bar{\ell}_{L}\psi_{R}h-\kappa^{\prime}\bar{\ell}_{R}S^{\dagger}\psi_{L}+g_{S}\bar{\ell}_{R}S^{\dagger}\ell_{L}\\\
&+g_{Z}\bar{\ell}_{L}\gamma^{\mu}\psi_{L}Z_{\mu}+g_{W}\bar{\nu}\gamma^{\mu}\psi_{L}W_{\mu}^{+}+h.c.),\end{split}$
(3)

where

$\displaystyle\begin{split}&g_{S}=\frac{\kappa^{\prime}\kappa}{\sqrt{2}}\frac{\upsilon_{h}}{M_{F}},\\\
&g_{Z}=-\frac{\kappa g}{2\sqrt{2}\cos\theta_{w}}\frac{\upsilon_{h}}{M_{F}},\\\
&g_{W}=\frac{\kappa g}{2}\frac{\upsilon_{h}}{M_{F}}.\end{split}$ (4)

The values of the electromagnetic coupling $e$, the weak mixing angle
$\theta_{w}$ and the $SU(2)_{L}$ coupling $g$ are taken from Particle Data
Group (PDG) Workman et al. (2022). We denote the VLL mass as $M_{F}$, which
assumed to be equal for three generations of VLLs.

It is well known that measurements of the muon anomalous magnetic moment
(AMM), $a_{\mu}=(g-2)_{\mu}/2$ Jegerlehner and Nyffeler (2009); Lindner et al.
(2018); Athron et al. (2021), indicate the discrepancy from the SM prediction.
The E989 experiment at Fermilab recently released an update regarding the
measurement of $a_{\mu}$ from Run-2 and Run-3 Grange et al. (2015). The
analysis of the Muon g-2 collaboration using the new combined value has led to
$\Delta{a_{\mu}}\equiv a_{\mu}^{\text{exp}}$ $-$
$a_{\mu}^{\text{SM}}=(116592059\pm 22)\times 10^{-11}-(116591810\pm 43)\times
10^{-11}=(249\pm 48)\times 10^{-11}$, a discrepancy of 5.1$\sigma$ CL
Aguillard et al. (2023); Aoyama et al. (2020). The contributions of the VLS
models to the muon AMM have been carefully studied in Refs.Hiller et al.
(2020a, 2019), which only appear at one loop level, are dependent on the free
parameter $\kappa^{\prime}$ and the masses of new particles including $M_{F}$
and $M_{S}$, and scale quadratically with the muon mass,

$\displaystyle\begin{split}&\Delta{a_{\mu}=\frac{\kappa^{\prime}}{32\pi^{2}}\frac{m_{\mu}^{2}}{M_{F}^{2}}f(\frac{M_{S}^{2}}{M_{F}^{2}})},\end{split}$
(5)

with $f(t)=(2t^{3}+3t^{2}-6t^{2}\ln{t}-6t+1)/(t-1)^{4}$ positive for any $t$,
and $f(0)=1$. If we demand the VLS models to solve the muon AMM anomaly, then
the coupling $\kappa^{\prime}$ can be expressed in terms of the mass
parameters $M_{F}$ and $M_{S}$. The contributions of the VLS models to
$(g-2)_{e}$ Fan et al. (2023) come from one-loop level as well and mainly
depend on the coupling $\kappa$, the mass parameters $M_{F}$ and $M_{S}$, see
Hiller et al. (2019, 2020b) for details. Furthermore, the
$Z\rightarrow\ell\ell$ data Workman et al. (2022) constrains the mixing angles
$\theta$ as $\theta\simeq\kappa v_{h}/\sqrt{2}M_{F}<\mathcal{O}(10^{-2})$.
According to both AMMs and $Z\rightarrow\ell\ell$ data, we simply fix
$\kappa/\kappa^{\prime}=10^{-2}$ with $\kappa^{\prime}$ computed according to
Eq.(5) for $M_{S}=$ 500 GeV in our numerical calculation, as done in Ref.
Bißmann et al. (2021).

In the singlet VLS model, the charged VLLs can decay into the final states
$W\nu$, $Z\ell$, $S\ell$, $h\ell$ and the partial decay widths are

$\displaystyle\begin{split}&\Gamma(\Psi\rightarrow
W\nu)=g_{W}^{2}\frac{M_{F}}{32\pi}(1-r_{W}^{2})^{2}(2+1/r_{W}^{2}),\\\
&\Gamma(\Psi\rightarrow
Z\ell)=g_{Z}^{2}\frac{M_{F}}{32\pi}(1-r_{Z}^{2})^{2}(2+1/r_{Z}^{2}),\\\
&\Gamma(\Psi\rightarrow S^{\ast}\ell)=\kappa^{\prime
2}\frac{M_{F}}{32\pi}(1-r_{S}^{2})^{2},\\\ &\Gamma(\Psi\rightarrow
h\ell)=\kappa^{2}\frac{M_{F}}{64\pi}(1-r_{h}^{2})^{2},\\\ \end{split}$ (6)

where $r_{X}=M_{X}/M_{F}$ for $X=W,Z,S,h$. The branching ratios as function of
$M_{F}$ are given in Ref. Bißmann et al. (2021), which show that the branching
ratio $Br(\psi\rightarrow W\nu)$ is the highest one for $M_{F}\leq$ 500 GeV.
So in this work, we focus on the channel of VLLs decaying into $W\nu$. To
facilitate following calculations, we plot the VLL total decay width in FIG. 1
as function of the VLL mass $M_{F}$ for $M_{S}=$ 500GeV and different values
of the coupling $\kappa$.

Figure 1: The total decay width as function of the VLL mass $M_{F}$ for
different values of $\kappa$.

The VLS models satisfying asymptotic safety, possible up to the Planck scale,
allow to explain the discrepancies between the SM prediction and current data
of AMMs and other observables. The new particles $\psi$ and $S$ predicted by
VLS models can be as light as a few hundred GeV and thus can be probed at high
energy colliders. In this work, we mainly consider the possibility of
detecting the singlet VLS model through pair production of VLLs at the ILC
with $\sqrt{s}=$ 1 TeV and $\mathcal{L}=$ 1 ab-1 and only discuss the first
generation of VLLs as an example.

##  III EVENT GENERATION AND NUMERICAL RESULTS

The Feynman diagrams for the pair production of VLLs via process
$e^{+}e^{-}\rightarrow\psi\overline{\psi}$ at the ILC are shown in FIG. 2. We
implement the singlet VLS model into `FeynRules` package Alloul et al. (2014)
and export the model files to the `UFO` format Degrande et al. (2012). The
numerical results are given by using Monte-Carlo (MC) simulation with the
`MadGraph5_aMC@NLO` toolkit Alwall et al. (2014); Frederix et al. (2018).
Since the polarized $e^{+}$ beams and $e^{-}$ beams can enhance the cross
section effectively, we set the polarization options as $P_{e^{+}}=-$ 0.3,
$P_{e^{-}}=$ 0.8 and assume the VLL mass in the range of 100 - 500 GeV within
the detection capability of the 1 TeV ILC Abramowicz et al. (2013);
Chakrabarty and Chakraborty (2023), the Yukawa coupling $\kappa$ in the range
of 0.01 - 0.1 which is consistent with the relevant experimental data Bißmann
et al. (2021). Considering the most large branching ratio $Br(\psi\rightarrow
W\nu)$ and assuming the electroweak boson W decays into $\ell\nu$, the signal
of pair production of VLLs at the ILC is a pair of charged leptons plus
missing energy. The production cross sections of the signal process
$e^{+}e^{-}\rightarrow\psi\overline{\psi}\rightarrow
W^{+}\nu_{e}W^{-}\nu_{e}\rightarrow\ell^{+}\ell^{-}\nu_{e}\bar{\nu_{e}}\nu_{\ell}\bar{\nu_{\ell}}$
at the 1 TeV ILC are shown in FIG. 3 as function of the VLL mass $M_{F}$ for
different values of $\kappa$. The main SM background we concerned comes from
the processes
$e^{+}e^{-}\rightarrow\ell^{+}\ell^{-}\nu_{\ell}\bar{\nu_{\ell}}$ and
$e^{+}e^{-}\rightarrow\ell^{+}\ell^{-}\nu_{\ell}\nu_{\ell}\bar{\nu_{\ell}}\bar{\nu_{\ell}}$
with $\ell=e,\mu$. It is obvious that the signal cross section is smaller than
that of the corresponding SM background (0.08931pb). The parton level events
of the signal and background are required to pass through the basic cuts, so
our numerical results have employed the basic cuts as follows

$\begin{array}[]{l}\textit{p}_{T}^{\ell}>10\mathrm{GeV},~{}~{}\quad\Delta
R_{\ell\ell}>0.4,~{}~{}\quad\lvert\eta_{\ell}\rvert<2.5,\\\ \end{array}$

where $\textit{p}_{T}$ denotes the transverse momentum, $\Delta
R(x,y)=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}$ is the separation in the
rapidity-azimuth plane. We transmit the parton-level events to `PYTHIA8`
Sjöstrand et al. (2015) for showering and hadronization. Then we make a fast
detector simulations by `DELPHES` de Favereau et al. (2014) using the ILD
detector card. Finally, we perform the event analysis by `MADANALYSIS5` Conte
et al. (2013, 2014); Conte and Fuks (2018).

Figure 2: The Feynman diagrams for the process
$e^{+}e^{-}\rightarrow\psi\overline{\psi}$. Figure 3: The cross sections of
the process $e^{+}e^{-}\rightarrow\psi\overline{\psi}\rightarrow
W^{+}\nu_{e}W^{-}\nu_{e}\rightarrow\ell^{+}\ell^{-}\nu_{e}\bar{\nu_{e}}\nu_{\ell}\bar{\nu_{\ell}}$
as function of the VLL mass $M_{F}$ for different values of the coupling
$\kappa$.

To improve the signal and suppress the background, we serve the z-component of
the momentum and the pseudorapidity of $\ell^{+},\ell^{-}$ in the final states
as effective cuts, severally. The normalized distributions of these
observables are depicted in FIG. 4 with the solid lines for signal events and
dashed lines for background events. The benchmark points are $M_{F}=$ 100 GeV,
200 GeV, 300 GeV, 400 GeV and 500 GeV selected with the assumption of
$\kappa=$ 0.06, $\mathcal{L}=$ 1 ab-1.

Figure 4: The normalized distributions of the observables
$\textit{p}_{\textrm{z}}^{\ell^{+}}$ (a), $\textit{p}_{\textrm{z}}^{\ell^{-}}$
(b), $\eta_{\ell^{+}}$ (c) and $\eta_{\ell^{-}}$ (d) in signal and background
events for various VLL masses at the 1 TeV ILC with $\mathcal{L}=$ 1 ab-1.

From FIG. 4$(a)$, we can see that more signal events are distributed in the
region with smaller $\textit{p}_{\textrm{z}}^{\ell^{+}}$ compared to the
background. The reason is that the process
$e^{+}e^{-}\rightarrow\ell^{+}\ell^{-}\nu_{\ell}\bar{\nu_{\ell}}$ dominates
the background. The missing particles of background carried less energy lead
to the z-component of the momentum of visible particles $\ell^{+}$ to be
distributed at large momentum, whereas the momentum of the signal is mainly
distributed at small masses. The z-component of the momentum of $\ell^{-}$ is
similar to $\textit{p}_{\textrm{z}}^{\ell^{+}}$ as shown in FIG. 4(b).
Therefore, we take the cut $\textit{p}_{\textrm{z}}^{\ell^{+}}<$ 180 GeV and
$\textit{p}_{\textrm{z}}^{\ell^{-}}>-$ 180 GeV, respectively. The
$\eta_{\ell^{+}}$ distribution given by FIG. 4(c) shows that a large
proportion of the beam axis, which is significantly distinct from the
background. The same goes for $\eta_{\ell^{-}}$ distribution, so we
implemented the cut of $\eta_{\ell^{+}}<$ 0.8 and $\eta_{\ell^{-}}>-$ 0.8 to
filter the background. Based on the characteristics of the kinematics
distributions, the selected cuts are listed in table 1 to further reduce the
background.

Table 1: All cuts on the signal and background. Basic cuts | $~{}~{}~{}~{}~{}~{}~{}\textit{p}_{T}^{\ell}>10\mathrm{GeV},~{}~{}\quad\Delta R_{\ell\ell}>0.4,~{}~{}\quad\lvert\eta_{\ell}\rvert<2.5~{}~{}~{}~{}~{}~{}~{}$
---|---
Cut 1 | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\textit{p}_{\textrm{z}}^{\ell^{+}}<180~{}GeV~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
Cut 2 | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\textit{p}_{\textrm{z}}^{\ell^{-}}>-180~{}GeV~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
Cut 3 | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\eta_{\ell^{+}}<0.8~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
Cut 4 | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\eta_{\ell^{-}}>-0.8~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
Table 2: The cross sections of the signal and the background processes after the improved cuts applied for $\kappa=0.06$ at the 1TeV ILC with benchmark points.  Cuts | cross sections for signal (background) [pb]
---|---
|  $M_{F}=100$ GeV |  $M_{F}=200$ GeV |  $M_{F}=300$ GeV |  $M_{F}=400$ GeV |  $M_{F}=500$ GeV
Basic Cuts |  $1.8351\times 10^{-3}$ $(0.08931)$ |  $8.9060\times 10^{-4}$ $(0.08931)$ |  $4.3086\times 10^{-4}$ $(0.08931)$ |  $1.8733\times 10^{-4}$ $(0.08931)$ |  $2.2202\times 10^{-6}$ $(0.08931)$
Cut 1 |  $1.4419\times 10^{-3}$ $(0.04564)$ |  $7.2116\times 10^{-4}$ $(0.04564)$ |  $3.5429\times 10^{-4}$ $(0.04564)$ |  $1.5573\times 10^{-4}$ $(0.04564)$ |  $1.8926\times 10^{-6}$ $(0.04564)$
Cut 2 |  $1.1614\times 10^{-3}$ $(0.01443)$ |  $5.9084\times 10^{-4}$ $(0.01443)$ |  $2.9374\times 10^{-4}$ $(0.01443)$ |  $1.3048\times 10^{-4}$ $(0.01443)$ |  $1.6176\times 10^{-6}$ $(0.01443)$
Cut 3 |  $1.0406\times 10^{-3}$ $(0.01135)$ |  $5.2073\times 10^{-4}$ $(0.01135)$ |  $2.5001\times 10^{-4}$ $(0.01135)$ |  $1.0487\times 10^{-4}$ $(0.01135)$ |  $1.2301\times 10^{-6}$ $(0.01135)$
Cut 4 |  $9.9005\times 10^{-4}$ $(7.0357\times 10^{-3})$ |  $4.7868\times 10^{-4}$ $(7.0357\times 10^{-3})$ |  $2.1930\times 10^{-4}$ $(7.0357\times 10^{-3})$ |  $8.6010\times 10^{-5}$ $(7.0357\times 10^{-3})$ |  $9.3007\times 10^{-5}$ $(7.0357\times 10^{-3})$
$SS$ |  $11.04$ |  $5.52$ |  $2.57$ |  $1.019$ |  $0.011$

After these improved cuts are applied, the SM background is substantially
suppressed. The cross sections of the signal and background after applying the
above selection cuts for the five benchmark points picked for 100 GeV
$\leq{M_{F}}\leq$ 500 GeV are displayed in table 2 at 1 TeV ILC. Then we
further calculate the statistical significance $SS=S/\sqrt{S+B}$, where $S$
and $B$ are the number of events for the signal and background, respectively.
Considering the integrated luminosity of $\mathcal{L}=$ 1 ab-1, a statistical
significance of 5.521$\sigma$ (2.574$\sigma$) can be reached when $M_{F}$ is
taken as 200 GeV (300 GeV). The significance can be greater than 5$\sigma$ for
$M_{F}$ less than 200 GeV and will be small as $M_{F}$ approaches 500 GeV.

Figure 5: The 2$\sigma$, 3$\sigma$ and 5$\sigma$ curves for the process
$e^{+}e^{-}\rightarrow\psi\overline{\psi}$ at the ILC with $\sqrt{s}=$ 1 TeV
and $\mathcal{L}=$ 1 ab-1 in the $M_{F}-\kappa$ plane.

In FIG. 5, the 2$\sigma$, 3$\sigma$ and 5$\sigma$ curves for the process
$e^{+}e^{-}\rightarrow\psi\overline{\psi}\rightarrow
W^{+}\nu_{e}W^{-}\nu_{e}\rightarrow\ell^{+}\ell^{-}\nu_{e}\bar{\nu_{e}}\nu_{\ell}\bar{\nu_{\ell}}$
at the ILC with $\sqrt{s}=$ 1 TeV and $\mathcal{L}=$ 1 ab-1 are plotted in the
plane of $M_{F}$-$\kappa$. As shown in FIG. 5, the Yukawa coupling $\kappa$
increases with $M_{F}$ increasing and breaks when $M_{F}$ approaches to 463
(437, 386) GeV. That is mainly because the cross section of the signal sharply
decreases with increasing $M_{F}$ as shown in FIG. 3. For the mass $M_{F}$ in
the range of 100 - 500 GeV, the values of the statistical significance $SS$
can reach 2$\sigma$, 3$\sigma$ and 5$\sigma$ as long as the coupling
coefficient $\kappa$ greater than 0.032, 0.036 and 0.044, respectively. Thus,
the ILC has the potential to discover the VLLs in the considered mass range.

##  IV conclusions

Asymptotically safe extensions of the SM are one kind of interesting new
physics scenarios, which predict the existence of new singlet scalars and
vector-like fermions producing significant contributions to some low-energy
observables and rich collider phenomenology. The singlet and doublet VLS
models are two specific models featuring three generations of either
$SU(2)_{L}$ singlet or doublet VLLs. In this paper, we investigate the
probability of probing the VLLs at the ILC with $P(e^{+},e^{-})=(-0.3,0.8)$
beam polarization via the signal process
$e^{+}e^{-}\rightarrow\psi\overline{\psi}\rightarrow
W^{+}\nu_{e}W^{-}\nu_{e}\rightarrow\ell^{+}\ell^{-}\nu_{e}\bar{\nu_{e}}\nu_{\ell}\bar{\nu_{\ell}}$
in the framework of the singlet VLS model. The expected sensitivities of the
ILC with $\sqrt{s}=$ 1 TeV and $\mathcal{L}$ = 1 ab-1 to the parameter space
of this kind of VLL models are derived. The results show that the parameter
space $M_{F}\in$ [101, 463] GeV and $\kappa\in$ [0.032, 0.098] can be covered
by the proposed ILC. Our conclusions are obtained for the first generation of
VLLs, which are also apply to the second generation of VLLs. Certainly, due to
the rapid decay of $\tau$, making the identification very challenging,
searching for the third generation VLLs at ILC needs to be further
investigated, which is one of our future tasks.

## ACKNOWLEDGMENT

This work was partially supported by the National Natural Science Foundation
of China under Grant No. 11875157, No. 12147214 and No. 11905093. We would
like to thank Ji-Chong Yang for useful discussions.

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