Single production of singlet vector-like leptons at the ILC

With the discovery of the Higgs boson by the a toroidal large hadron collider apparatus (ATLAS) and compact muon solenoid (CMS) collaborations at the Large Hadron Collider (LHC) in 2012 Aad et al. (2012); Chatrchyan et al. (2012), the standard model (SM) of particle physics has was further validated. So far, the SM is a remarkably successful theory capable of explaining a large number of observations. However, it is clear that the SM lacks some of the vital ingredients to help us understand nature, such as the candidates of dark matter Bertone et al. (2005); Feng (2010), the excess of matter over antimatter Ade et al. (2014), gauge hierarchy Feng (2013), neutrino masses Gonzalez-Garcia and Maltoni (2008), as well as the fact that the masses and mixing patterns of the SM leptons cannot yet be clearly determined Morais et al. (2023). Besides, there are also deviations between the experimental measurements and the SM predictions for certain physical observations. For instance, the measurement of the muon anomalous magnetic moment (AMM) $a_{\mu}$ indicates that the significance of the discrepancy between the experimental average and the SM prediction approaches to 4.2$\sigma$ Abi et al. (2021). Therefore, we need new physical scenarios to provide a more natural solution to the above problems and the existed anomalies.

Considering the flavor democracy (FD) hypothesis that may enlighten mass pattern of the SM fermions, the existence of extra fermions beyond the three generations of the SM is necessary. The Higgs data from LHC imposes strong constraints on the additional chiral fermions, while the non-chiral fermions, generally referred to as vector-like fermions, are much less constrained Kuflik et al. (2013). Due to the vector-like nature of non-chiral fermions, they can evade constraints from fourth-generation searches Eberhardt et al. (2012). Vector-like fermions with masses generated independently of the Higgs mechanism become increasingly attractive and appear in numerous new physics models beyond the SM, such as little Higgs models Arkani-Hamed et al. (2002); Perelstein et al. (2004); Carena et al. (2007); Han et al. (2003), composite Higgs models Anastasiou et al. (2009); Kong et al. (2012); Gillioz et al. (2012); Keren-Zur et al. (2013); Falkowski et al. (2014), models of warped or universal extra-dimension Agashe et al. (2007); Contino et al. (2007); Li et al. (2012); Huang et al. (2012); Redi (2013), grand unified theories with non-minimal supersymmetric extensions Kang et al. (2008); Babu et al. (2008); Martin (2010); Graham et al. (2010). Very recently, Ref. Adhikary et al. (2024) has reexamined theoretical constraints on the new physics models with vector-like fermions.

Vector-like fermions include vector-like quarks (VLQs) and vector-like leptons (VLLs), where VLLs are defined as colourless, spin $-1/2$ fermions. The left- and right-handed components of VLLs transform in the same way under the electroweak symmetry group. Like the SM leptons, VLLs have three generations and the same lepton numbers with their corresponding SM leptons. VLLs might explain the ”Cabibbo angle anomaly” Crivellin et al. (2020), the muon AMM anomaly Megias et al. (2017) and help obtain the realistic Higgs mass within the framework of the composite Higgs model Carmona and Goertz (2015, 2016, 2018), while also being used to study the phenomenology of dark matter Guedes and Santiago (2022).

The current research status of VLLs can be discovered in Ref. Baspehlivan et al. (2022). Although VLLs have the same status as VLQs from particle phenomenology, there are not many direct searches for VLLs. Prior to direct searches at the LHC, a lower bound on the VLLs mass $M_{VLL}>101.2$ GeV is placed by L3 collaboration at the large electron-positron collider (LEP) Achard et al. (2001). Using the LHC data about multilepton events, collected in 2016-2018 corresponding to an integrated luminosity of 138 fb^-1, the CMS collaboration has given the constraints on the VLL masses, which excludes the doublet VLLs with mass below 1045 GeV at 95% confidence level (CL), while the singlet VLLs with mass in the range from 125 GeV to 150 GeV are excluded Tumasyan et al. (2022). The analytical results of the ATLAS collaboration show that the doublet VLLs with mass in the range from 130 GeV to 900 GeV are excluded at 95% CL Aad et al. (2023). We believe that future runs of the LHC (LHC Run-3 and High Luminosity-LHC) will give more interesting results about searching for VLLs.

The vector-like lepton with scalar (VLS) models Hiller et al. (2020a, b, 2019) satisfy asymptotically safety that tames the UV behavior towards the Planck scale Litim and Sannino (2014); Bond and Litim (2019, 2018); Bednyakov and Mukhaeva (2023); Litim et al. (2023) and are allowed to explain the discrepancies between the measured values of the electron and muon AMMs and the SM predicted values. The VLS models, which can be divided into the singlet and doublet VLS models, predict the existence of VLLs and new scalars. The couplings of these new particles with the SM particles through renormalizable Yukawa interactions. The masses of VLLs can be as light as a few hundred GeV, so they can be probed at different colliders Achard et al. (2001); Bißmann et al. (2021); Yue et al. (2024); Das et al. (2020); Das and Mandal (2021); Bhattacharya et al. (2022).

It is well known that lepton colliders can offer a cleaner environment compared to hadron colliders and the polarized incident beams can improve the discovery potential of new particles. Thus, in this work, we will explore the possibility of detecting VLLs at the International Linear Collider (ILC), taking into account the appropriate polarization for our analysis. Considering the lower limits on the masses of the singlet and doublet VLLs, we concentrate on the searches for VLL within the framework of the singlet VLS model through its single production with subsequent decays to the pure leptonic final state and the fully hadronic final state at the ILC with the center of mass energy $\sqrt{s}=$ 1 TeV and the integrated luminosity $\mathcal{L}$ = 1 ab^-1 Aihara et al. (2019); Bambade et al. (2019).

This paper is organized as follows. In section II, we briefly present the main content of the singlet VLS model, which is related our numerical calculation. In section III, we offer a detailed analysis of the possible of detecting singlet VLL through its single production processes based on the ILC detector simulation. For the analysis of signal, we consider two kinds of decay modes for the W boson, i.e. pure leptonic and fully hadronic channels. Our conclusions are given in section IV.

The VLS models  Hiller et al. (2020a, b, 2019) predict three generations of VLLs, denoted as $\psi_{L,R}$ and each generation of VLLs is colorless with hypercharge $Y=-1$ and $-1/2$, which can be regarded as $SU(2)_{L}$ singlets and doublets, respectively. Considering the constraints on the masses of the doublet and singlet VLLs from the LHC Tumasyan et al. (2022); Aad et al. (2023), we will focus our attention on the possibility signal of the singlet VLL via its single production at the 1 TeV ILC.

The new Yukawa couplings in the framework of the singlet VLS model read Hiller et al. (2020a)

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

(1)

where E, L and H denote the SM singlet, doublet leptons and Higgs boson, respectively. S represents new scalar introduced to the singlet VLS model that is singlet under the SM gauge groups. Due to the conservation of each lepton flavor meeting with $SU(3)$-flavor symmetry, the new Yukawa couplings $y,\kappa,\kappa^{\prime}$ become single couplings instead of tensors.

After electroweak symmertry breaking, the couplings of the singlet VLL with the electroweak gauge bosons and new scalars are given by Bißmann et al. (2021); Hiller et al. (2020a)

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

(2)

Here, $A_{\mu}$, $Z_{\mu}$ and $W_{\mu}$ denote photon, Z boson and W boson, respectively. $e$, $\theta_{w}$, $g$ correspond to the electromagnetic coupling, the weak mixing angle and the $SU(2)_{L}$ coupling. The coupling constants $g_{S}$, $g_{Z}$ and $g_{W}$ are represented by the following formulas

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

(3)

Where $M_{F}$ is regarded as the common mass of the VLLs of all flavors, and $\upsilon_{w}\simeq$ 246 GeV is the electroweak symmetry breaking scale.

Then the partial decay widths of the singlet VLL to SM final states can be written as

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

(4)

where $r_{X}=M_{X}/M_{F}$ for $X=W,Z,h$. Certainly, if the singlet VLL mass is larger than the mass of the new scalar $S$, it is possible that the singlet VLL decays into $S$ and a SM charged lepton. However, in this paper, we assume that the new scalar $S$ is generally heavier than the singlet VLL, and thus do not consider this decay channel.

From Eq.(1) we can see that VLL can be single produced at the ILC via the process $e^{+}e^{-}\rightarrow\psi\ell$ mediated by the gauge boson $Z$ or the scalars $S$ and $H$. The relevant Feynman diagrams are shown in Fig. 1. However, the main contributions come from exchange of the gauge boson $Z$. Furthermore, the t-channel contributions are smaller than those of the s-channel and only contribute to the final state $\psi e$ in the case of flavor conservation. All the same, we will consider all of these contributions in our numerical estimations.

If the lepton mass of the final state is disregarded, then the production cross sections of the processes $e^{+}e^{-}\rightarrow$ $\psi e$, $\psi\mu$ and $\psi\tau$ are approximately equal to each other. Nevertheless, for $\tau$ lepton, it decays more rapidly and the background is more intricate. For simplicity, we take the first generation of leptons as an example to conduct numerical calculation and the analysis of the signal and SM background in this paper. From above discussions we can see that the production cross sections are contingent upon the free parameters ${M_{F}},{M_{S}},\kappa^{\prime}$ and $\kappa$. For better clarity, we give the expression form of the production cross section of the s-channel process $e^{+}e^{-}\rightarrow$ $\psi e$ only induced by exchange of the gauge boson $Z$

$\displaystyle\begin{split}\sigma(e^{+}e^{-}\rightarrow\psi~{}e)=\frac{e^{2}\kappa^{2}\upsilon_{w}^{2}(8\sin^{4}\theta_{w}-4\sin^{2}\theta_{w}+1)(s-M_{F}^{2})^{2}(M_{F}+2s)}{3072\pi\sin^{4}\theta_{w}\cos^{4}\theta_{w}s^{2}M_{F}(s-M_{Z}^{2})}.\end{split}$

(5)

The constraints on the free parameters of the VLS models have been extensively studied in Refs. Hiller et al. (2020a, b); Bißmann et al. (2021); Yue et al. (2024), which mainly come from measurements of the muon and electron AMMs and the relevant electroweak data. The contributions of the VLS models to the muon AMM, $a_{\mu}$, are given by Hiller et al. (2020a, b)

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

(6)

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 $M_{S}$. Similar to $a_{\mu}$, the contributions of the VLS models to the electron AMM, $a_{e}$, is related to the coupling $\kappa$, the mass parameters $M_{F}$ and $M_{S}$. According to both AMMs and electroweak data, we simply fix $\kappa/\kappa^{\prime}=10^{-2}$ with $\kappa^{\prime}$ computed from Eq.(6) for $M_{S}=$ 500 GeV and $\kappa\in$ (0.01, 0.1) in our following numerical calculation.

As polarized $e^{+}$ and $e^{-}$ beams can change the production cross sections, in Fig. 2, we plot the production cross section of the process $e^{+}e^{-}\rightarrow\psi e^{+}$ with different polarization options as a function of the mass parameter $M_{F}$ at the 1 TeV ILC for the coupling parameter $\kappa=$ 0.03. It can be observed from Fig. 2 that the values of the production cross section attains maximum when the polarization is $-$30% for positron beam and 80% for electron beam, which are in the range of 0.023 - 0.883 pb for 100 GeV$<M_{F}<$ 700 GeV. In order to produce a heavier VLL in a larger parameter space of the singlet VLS model, we choose polarized beams with $P(e^{+},e^{-})=(-30\%,~{}80\%)$ in the following analysis of the signal and SM background.

According to the discussions given in Ref. Bißmann et al. (2021) about the branching ratios for various possible decay channels of singlet VLL and the decay formulas given above, we find that the value of the branching ratio $Br(\psi\rightarrow W\nu)$ is the largest. So, we prefer to choose the most outstanding decay channel for our analysis. Furthermore, in the following analysis of the signal and SM background, we assume that W boson decays through two kinds of mode, that are pure leptonic and fully hadronic decay channels. The production and decay chain are presented as follows:
(1) pure leptonic decay channel:         $e^{+}e^{-}\rightarrow\psi e^{+}\rightarrow W^{-}\nu_{e}e^{+}\rightarrow\ell^{-}\bar{\nu_{\ell}}\nu_{e}e^{+}$;
(2) fully hadronic decay channel:       $e^{+}e^{-}\rightarrow\psi e^{+}\rightarrow W^{-}\nu_{e}e^{+}\rightarrow jj\nu_{e}e^{+}$.

For further calculation, we use FeynRules package Alloul et al. (2014) to generate the model file and export it to the UFO format Degrande et al. (2012). The simulation of the signal and SM background at leading order (LO) is performed with MadGraph5_aMC@NLO Alwall et al. (2014); Frederix et al. (2018). The basic cuts we use first are given by

$$\begin{array}[]{l}~{}~{}~{}~{}~{}~{}\textit{p}_{T}^{\ell}>10\mathrm{GeV},~{}~{}\quad\lvert\eta_{\ell}\rvert<2.5,\\ ~{}~{}~{}~{}~{}~{}\textit{p}_{T}^{j}>20\mathrm{GeV},~{}~{}\quad\lvert\eta_{j}\rvert<5.0,\\ ~{}\quad\Delta R(x,~{}y)>0.4,~{}with~{}x,y=\ell,~{}j(u,d,c,s).\\ \end{array}$$

$\textit{p}_{T}^{\ell}$ and $\textit{p}_{T}^{j}$ are the transverse momentum of leptons and jets. $\lvert\eta_{\ell}\rvert$ and $\lvert\eta_{j}\rvert$ are the pseudo-rapidity of leptons and jets. $\Delta R(x,~{}y)$ is defined as the separation in the rapidity-azimuth plane, $\Delta R(x,y)=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}$ with $\phi$ being azimuthal angle. Then we use PYTHIA8 Sjöstrand et al. (2015) to simulate initial state radiation, final state radiation and hadronization. The fast detector simulation is made by DELPHES 3 de Favereau et al. (2014) using the International Linear Detector (ILD) detector card. To realize the kinematic and cut-based analysis, the package MADANALYSIS 5 Conte et al. (2013, 2014); Conte and Fuks (2018) is adopted.

So far, a large number of new physics models have been proposed to solve the SM problems and the existed anomalies, which predict the existence of new particles. Searching for the possible physics signals of various new particles are the main goals of current and future collider experiments. VLLs as one kind of interesting new particles can produce rich phenomenology at low- and high-energy experiments, which should be extensively studied.

In this paper, we investigate the possibility of detecting singlet VLL through its single production at the 1 TeV ILC with $\mathcal{L}$ = 1 ab^-1 and the polarization option $P(e^{+},e^{-})=(-30\%,80\%)$. For the signal and SM background analysis, we have considered two kinds of decay channels for the W boson, i.e. pure leptonic and fully hadronic decay channels, which produce two kinds of the signals: one is two charged leptons plus missing energy, the other is two jets, one charged lepton plus missing energy. Our analytic results for the first and second generations of singlet VLLs show that the parameter space $M_{F}\in$ [300, 675] GeV and $\kappa\in$ [0.0294, 0.1] might be detected by the proposed ILC for pure leptonic decay channel. For fully hadronic decay channel, larger detection regions of $M_{F}\in$ [300, 700] GeV and $\kappa\in$ [0.0264, 0.0941] are derived. Comparing the results of the two cases, we can see that singlet VLL might be more easily detected at the ILC via its single production in fully hadronic decay than that in pure leptonic decay channel.

This work was partially supported by the National Natural Science Foundation of China under Grant No. 11875157, No. 12147214 and No. 11905093.