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

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 [1,2] 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 [3] and neutrino masses [4].Moreover, mass and mixing patterns of the SM leptons [5] 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 vectorlike 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 [6][7][8][9] and recent review article [10].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 [11,12], left-right symmetric models [13][14][15][16], supersymmetric models [17][18][19], and grand unified theories [20,21].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 Electronpositron (LEP) excluded the heavy leptons with mass up to 101.2 GeV [22].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 [23].For doublet vector-like τ 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 [24].Using the CMS search based on 77.4 fb −1 at the 13 TeV LHC, the authors of Ref. [25] 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.[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].
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 [42][43][44][45][46][47][48][49] and introduce new matter fields and Yukawa couplings [25,[50][51][52].The new models called VLS models in this paper, contain VLLs and new scalars, which tame the UV behavior [53] 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 L = 1 ab −1 [54,55].
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 [50][51][52] as one kind of the new VLL models predict the existence of the three generations of VLLs ψ L,R , which may be SU (2) L singlets with hypercharge Y = −1 and SU (2) L doublets with Y = − 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 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, κ, κ ′ become single couplings, instead of being tensors [50].
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. [25]. where The values of the electromagnetic coupling e, the weak mixing angle θ w and the SU (2) L coupling g are taken from Particle Data Group (PDG) [5].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 µ = (g − 2) µ /2 [56][57][58], indicate the discrepancy from the SM prediction.The E989 experiment at Fermilab recently released an update regarding the measurement of a µ from Run-2 and Run-3 [59].The analysis of the Muon g-2 collaboration using the new combined value has led to [60,61].The contributions of the VLS models to the muon AMM have been carefully studied in Refs.[50,51], which only appear at one loop level, are dependent on the free parameter κ ′ and the masses of new particles including M F and M S , and scale quadratically with the muon mass, 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 κ ′ can be expressed in terms of the mass parameters M F and M S .The contributions of the VLS models to (g−2) e [62] come from one-loop level as well and mainly depend on the coupling κ, the mass parameters M F and M S , see [51,52] for details.Furthermore, the Z → ℓℓ data [5] constrains the mixing angles θ as θ ≃ κv h / √ 2M F < O(10 −2 ).According to both AMMs and Z → ℓℓ data, we simply fix κ/κ ′ = 10 −2 with κ ′ computed according to Eq.( 5) for M S = 500 GeV in our numerical calculation, as done in Ref. [25].
In the singlet VLS model, the charged VLLs can decay into the final states W ν, Zℓ, Sℓ, hℓ and the partial decay widths are where

III. EVENT GENERATION AND NUMERICAL RESULTS
The Feynman diagrams for the pair production of VLLs via process e + e − → ψψ at the ILC are shown in FIG. 2. We implement the singlet VLS model into FeynRules package [63] and export the model files to the UFO format [64].The numerical results are given by using Monte-Carlo (MC) simulation with the MadGraph5_aMC@NLO toolkit [65,66].Since the polarized e + beams and e − beams can enhance the cross section effectively, we set the polarization options as P e + = − 0. From FIG. 4(a), we can see that more signal events are distributed in the region with smaller p ℓ + z compared to the background.The reason is that the process e + e − → ℓ + ℓ − ν ℓ νℓ dominates the background.The missing particles of background carried less energy lead to the z-component of the momentum of visible particles ℓ + 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 ℓ − is similar to p ℓ + z as shown in FIG.4(b).Therefore, we take the cut p ℓ + z < 180 GeV and p ℓ − z > − 180 GeV, respectively.The η ℓ + 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 η ℓ − distribution, so we implemented the cut of η ℓ + < 0.8 and η ℓ − > − 0.8 to filter the background.Based on the characteristics of the kinematics distributions, the selected cuts are listed in table I to further reduce the background.the first generation of VLLs, which are also apply to the second generation of VLLs.Certainly, due to the rapid decay of τ , 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.

FIG. 1 :
FIG. 1: The total decay width as function of the VLL mass M F for different values of κ.
FIG. 5: The 2σ, 3σ and 5σ curves for the process e + e − → ψψ at the ILC with √ s = 1 TeV and L = 1 ab −1 in the M F − κ plane.

TABLE I :
All cuts on the signal and background.

TABLE II :
The cross sections of the signal and the background processes after the improved cuts applied for κ = 0.06 at the 1TeV ILC with benchmark points.