Prospects for vectorlike leptons at future proton-proton colliders

TL;DR

This paper evaluates the potential of future proton-proton colliders to discover or exclude vectorlike leptons, focusing on their mass reach and detection prospects through multi-lepton signatures.

Contribution

It provides detailed estimates of discovery and exclusion mass limits for vectorlike leptons at various future collider energies and luminosities, highlighting the challenges for isosinglet types.

Findings

27 TeV collider could exclude leptons up to 2300 GeV

100 TeV collider could exclude leptons up to 5750 GeV

Discovery is more feasible for isodoublet leptons than isosinglet ones

Abstract

Vectorlike leptons are an intriguing possibility for physics beyond the Standard Model. We study the reach for discovering (at 5$\sigma$ significance) or excluding (at 95 per cent confidence) models of charged vectorlike leptons that mix predominantly with the tau, using multi-lepton signatures at various future proton-proton collider options: a high-luminosity LHC with $\sqrt{s} =$ 14 TeV, a high-energy LHC with $\sqrt{s} =$ 27 TeV, and possible new longer-tunnel colliders with $\sqrt{s} =$ 70 or 100 TeV. For weak isodoublet vectorlike leptons, we estimate that a 27 TeV high-energy LHC with 15 ab$^{-1}$ could exclude masses up to about 2300 GeV, or discover them if the mass is less than about 1700 GeV, while a 100 TeV collider with 30 ab$^{-1}$ could exclude masses up to about 5750 GeV, or make a discovery if the mass is less than about 4000 GeV. However, we find that weak isosinglet vectorlike leptons present a much more difficult challenge, with some reach for exclusion, but not for discovery at any of the collider options considered.

Tables

Table 1. Table 7.1: Summary of mass reaches for the 95% exclusion and 5 ​ σ 5 𝜎 5\sigma discovery of Doublet VLL at future p ​ p 𝑝 𝑝 pp colliders. The fractional uncertainty in the background is assumed to be Δ b / b = 0.1 subscript Δ 𝑏 𝑏 0.1 \Delta_{b}/b=0.1 .

Collider	Exclusion reach	Discovery reach
14 TeV HL-LHC, 3 $a{b}^{-1}$	1250 GeV	900 GeV
27 TeV HE-LHC, 15 $a{b}^{-1}$	2300 GeV	1700 GeV
70 TeV $pp$ collider, 30 $a{b}^{-1}$	4700 GeV	3400 GeV
100 TeV $pp$ collider, 30 $a{b}^{-1}$	5750 GeV	4000 GeV

Table 2. Table 7.2: Summary of mass reaches for the exclusion of Singlet VLL in the non-minimal models at future p ​ p 𝑝 𝑝 pp colliders. The fractional uncertainty in the background is assumed to be Δ b / b = 0.1 subscript Δ 𝑏 𝑏 0.1 \Delta_{b}/b=0.1 .

Collider	$Z$-philic	$W$-phobic	Higgs-philic
14 TeV HL-LHC, 3 $a{b}^{-1}$	600 GeV	360 GeV	300 GeV
27 TeV HE-LHC, 15 $a{b}^{-1}$	1200 GeV	670 GeV	$-$
70 TeV $pp$ collider, 30 $a{b}^{-1}$	1700 GeV	850 GeV	$-$
100 TeV $pp$ collider, 30 $a{b}^{-1}$	2850 GeV	1200 GeV	$-$

Equations

$$Z=\sqrt{2}{\mathrm{erfc}}^{-1}(2p).$$

$$Z_{\mathrm{disc}}$$

$$\displaystyle Z_{\rm exc}=\biggl{\{}2\biggl{[}s-b\ln\biggl{(}\frac{b+s+x}{2b}\biggr{)}-\frac{b^{2}}{\Delta_{b}^{2}}\ln\biggl{(}\frac{b-s+x}{2b}\biggr{)}\biggr{]}-(b+s-x)(1+b/\Delta_{b}^{2})\biggr{\}}^{1/2},\phantom{xx}$$

$$x=\sqrt{(s+b{)}^2-4s{\Delta }_b^2/(1+{\Delta }_b^2/b)}.$$

$$Z_{\mathrm{disc}}$$

$$Z_{\mathrm{exc}}$$

$${\tau }^{\prime }+{\overline{\tau }}^{\prime }$$

$$L+\overline{L}=\left(\begin{array}{c}{\nu }^{\prime }\\ {\tau }^{\prime }\end{array}\right)+\left(\begin{array}{c}{\overline{\tau }}^{\prime }\\ {\overline{\nu }}^{\prime }\end{array}\right)$$

$$\mathcal{M}$$

$$M_{\tau }$$

$$M_{{\tau }^{\prime }}$$

$$pp$$

$$\mathrm{BR}({\nu }^{\prime }\to W^{+}{\tau }^{-})=\mathrm{BR}({\overline{\nu }}^{\prime }\to W^{-}{\tau }^{+})=1.$$

$$\Gamma ({\tau }^{\prime }\to h\tau )$$

$$\Gamma ({\tau }^{\prime }\to Z\tau )$$

$$\Gamma ({\tau }^{\prime }\to W\nu )$$

$$\left[\text{mbox}BR({\tau }^{\prime }\to h\tau ),\text{mbox}BR({\tau }^{\prime }\to Z\tau ),\text{mbox}BR({\tau }^{\prime }\to W\nu )\right]$$

$$ZZ{\tau }^{-}{\tau }^{+},Zh{\tau }^{-}{\tau }^{+},hh{\tau }^{-}{\tau }^{+}$$

$$W^{+}{W}^{-}{\tau }^{-}{\tau }^{+}$$

$$Z{W}^{\pm }{\tau }^{-}{\tau }^{+},h{W}^{\pm }{\tau }^{-}{\tau }^{+}$$

$$Z{W}^{\pm }{\tau }^{\mp }+E_T^{\mathrm{miss}},h{W}^{\pm }{\tau }^{\mp }+E_T^{\mathrm{miss}},W^{+}{W}^{-}+E_T^{\mathrm{miss}},$$

$$\mathrm{BR}({\tau }^{\prime }\to h\tau )$$

$$\mathrm{BR}({\tau }^{\prime }\to Z\tau )$$

$$\mathrm{BR}({\tau }^{\prime }\to h\tau )$$

$$|\eta |$$

$$\Delta R_{l,l^{\prime }}$$

$$\Delta R_{l,j}$$

$$N(e,\mu ,\tau )$$

$$N(e,\mu )$$

$$p_T^{e_1}\text{mbox}or{p}_T^{{\mu }_1}$$

$$\displaystyle N(\mbox{$b$-jets})$$

$$e/\mu +\ge 1\tau$$

$$\ge 3e/\mu +1\tau$$

Full text

1. I Introduction

2. II Minimal models for production and decay of vectorlike leptons

3. III Results for the HL-LHC collider

4. III.1 Doublet VLL model

5. III.2 Singlet VLL models

6. IV Results for the HE-LHC collider

7. IV.1 Doublet VLL model

8. IV.2 Singlet VLL models

9. V Results for a $pp$ collider with $\sqrt{s}=70$ TeV

10. V.1 Doublet VLL model

11. V.2 Singlet VLL models

12. VI Results for a $pp$ collider with $\sqrt{s}=100$ TeV

13. VI.1 Doublet VLL model

14. VI.2 Singlet VLL models

15. VII Outlook

I Introduction

The Large Hadron Collider (LHC) has conducted searches for new physics beyond the Standard Model, at proton-proton center of mass energies up to $\sqrt{s}=13$ TeV. Following the discovery LHCdiscovery of the Higgs scalar boson with mass near $M_{h}=125$ GeV, the present evidence from the LHC is consistent with the minimal version of the Standard Model, with direct and indirect limits on new particles extending well into the TeV range. However, there are many candidate new physics models that obey decoupling as the mass scale of new physics is raised. These include supersymmetric theories, where agreement with the Standard Model gets better as the scale of supersymmetry breaking is raised. An example of a non-decoupling theory is a new chiral 4th family of fermions. Because of the necessity of generating the large chiral fermion masses through Yukawa couplings and the Higgs mechanism, decoupling is violated, which causes significant contributions to precision electroweak observables as well as Higgs boson production and decay rates, in conflict Lenz:2013iha with the observations.

New fermions are still allowed, because they will obey decoupling, if they obtain their mass entirely or mostly as bare electroweak-singlet terms in the Lagrangian, rather than from the Yukawa couplings to the Higgs field. This implies that the new fermions are vectorlike (the antonym of chiral). It is a notable common feature of many new physics theories that are motivated for other reasons (such as the need for a cosmological and astrophysical dark matter candidate, or to address the hierarchy problem) that they often contain vectorlike fermions. For example, in supersymmetry Primer , the Higgsinos have the same gauge quantum numbers as would a vectorlike pair of weak isodoublet leptons. It is plausible that additional vectorlike fermion masses are at the TeV scale, because whatever mechanism is responsible for the Higgsino masses can also be applied to the vectorlike fermion masses, which have the same structure. Vectorlike fermions can therefore be added on to the minimal version of supersymmetry, with one possible benefit that they can contribute to the lightest Higgs boson mass Moroi:1991mg ; Moroi:1992zk ; Babu:2004xg ; Babu:2008ge ; Martin:2009bg ; Graham:2009gy ; Martin:2010dc ; Endo:2011mc ; Evans:2011uq ; Li:2011ab ; Moroi:2011aa ; Endo:2011xq ; Endo:2012rd ; Nakayama:2012zc ; Martin:2012dg , which allows a lower supersymmetry breaking scale consistent with $M_{h}=125$ GeV. For example, adding vectorlike quarks to the simplest models of gauge-mediated supersymmetry breaking allows Endo:2011xq ; Endo:2012rd ; Nakayama:2012zc ; Martin:2012dg them to be still discoverable at the LHC. However, even ignoring such motivations, it is also worthwhile to consider vectorlike fermions merely on the merits of being a simple and consistent candidate extension of the Standard Model.

For vectorlike quarks, the LHC pair production cross-section is determined from QCD, and is therefore large and essentially model-independent.†††Single production of vectorlike quarks is also subject to LHC limits, but in a much more model-dependent way. The searches for vectorlike quarks are consequently quite advanced, with current LHC limits (depending on the specific decay modes) on pair-produced vectorlike quarks that can be found in refs. Aaboud:2018pii ; Sirunyan:2018omb .

In contrast, vectorlike leptons (VLLs) Dimopoulos:1990kc ; Sher:1995tc ; Thomas:1998wy ; Frampton:1999xi ; delAguila:2008pw ; delAguila:2010vg ; Ishiwata:2011hr ; Dermisek:2012as ; Joglekar:2012vc ; Kearney:2012zi ; Arina:2012aj ; Batell:2012zw ; Dermisek:2012ke ; Feng:2013mea ; Joglekar:2013zya ; Schwaller:2013hqa ; Dermisek:2013gta ; Ishiwata:2013gma ; Fairbairn:2013xaa ; Altmannshofer:2013zba ; Fichet:2013gsa ; Falkowski:2013jya ; Halverson:2014nwa ; Dobrescu:2014fca ; Ellis:2014dza ; Falkowski:2014ffa ; Holdom:2014rsa ; Dermisek:2014cia ; Dobrescu:2014esa ; Dermisek:2014qca ; Holdom:2014boa ; Aad:2015dha ; Ishiwata:2015cga ; Bizot:2015zaa ; Dermisek:2015oja ; Bhattacharya:2015qpa ; Kumar:2015tna ; Dermisek:2015hue ; Abdullah:2016avr ; Dermisek:2016via ; Feng:2016ysn ; Angelescu:2016mhl ; Bahrami:2016has ; Choudhury:2017fuu ; Poh:2017tfo ; Lu:2017uur ; Araz:2018uyi ; Bell:2019mbn ; Zheng:2019kqu ; CMS:2018cgi ; Sirunyan:2019ofn are pair-produced by $s$-channel electroweak vector boson diagrams, leading to cross-sections that are much smaller and dependent on the choices of weak isospin and weak hypercharge of the new states.‡‡‡Here we concentrate on models with new charged vectorlike leptons. Signatures from models with neutrino masses from electroweak singlet states at the TeV scale are reviewed in Cai:2017jrq ; Pascoli:2018heg , for example. For this reason, until recently the LHC detector collaborations had not put limits on VLLs beyond those that could be inferred from the CERN LEP $e^{-}e^{+}$ collider, which excluded Achard:2001qw masses up to $101.2$ GeV. There had been several phenomenological studies that looked into the LHC discovery and exclusion capabilities. In ref. Dermisek:2014qca , search strategies and capabilities were discussed for the optimistic case that the VLLs decayed mostly to muons, and an ATLAS search was conducted for that case in Aad:2015dha . In ref. Kumar:2015tna , the more pessimistic case of decays to tau-rich final states were considered, and it was argued that even with the (then current, but now old) data at $\sqrt{s}=8$ TeV it should be possible to exclude such $SU(2)_{L}$-doublet VLLs up to about $M_{\tau^{\prime}}=275$ GeV, but not $SU(2)_{L}$-singlet charged VLLs, which have a much smaller production cross-section and unfavorable branching ratios. Since then, the CMS collaboration has published CMS:2018cgi ; Sirunyan:2019ofn the results of dedicated searches for doublet VLLs, based on 41.4 fb*-1* and 77.4 fb*-1* data samples at $\sqrt{s}=13$ TeV. Aided in part by a deficit of events in the signal regions compared to the background expectation, CMS was able to obtain an exclusion for 120 GeV $<M_{\tau^{\prime}}<$ 790 GeV for a pure $SU(2)_{L}$ doublet VLL pair that mixes with and decays to tau leptons.

In this paper, we will consider the prospects for exclusion or discovery of VLLs at planned future proton-proton colliders. While there are a wide variety of possible VLL models, we will consider as benchmark models two simple minimal cases as defined in ref. Kumar:2015tna , a “Singlet VLL” and a “Doublet VLL” model, with the mixing to the Standard Model leptons assumed to be entirely with the tau, and small. One advantage of these models as benchmarks is that the pair production cross-sections are uniquely determined as a function of the mass, as discussed in ref. Kumar:2015tna and reviewed in the next section. As was the case in ref. Kumar:2015tna , our analysis below shows that the minimal Singlet VLL model is very difficult to probe at proton-proton colliders. We will therefore also consider a class of non-minimal models which are similar to the minimal Singlet VLL model and have the same production cross-section, but have different branching ratios for the lightest charged new lepton.

We will consider the following proton-proton collider options:

• •

a High-Luminosity LHC (HL-LHC), defined as 3 ab*-1* of proton-proton collisions at $\sqrt{s}=14$ TeV,

• •

a High-Energy LHC (HE-LHC), defined as 15 ab*-1* at $\sqrt{s}=27$ TeV,

• •

a very high energy $pp$ collider, defined as up to 30 ab*-1* at either $\sqrt{s}=70$ or 100 TeV, which could be realized as e.g. a CERN hadron-hadron Future Circular Collider (FCC-hh) FCChh or a Super proton proton Collider (SppC) SppC in China.

The physics potential for HL-LHC and HE-LHC has been studied for various other beyond-the-Standard-Model scenarios in ref. CidVidal:2018eel and references therein.

For reach estimates, we will use a simple cut-based counting experiment criteria, where the significance

[TABLE]

of an experimental result is related to the probability $p$ of observing, in an ensemble of many repeated experiments, a result of equal or greater incompatibility with the null hypothesis. Let the number of background events be Poisson distributed with mean $b$, with the variance in $b$ (corresponding intuitively to an uncertainty in the expected number of background events) given by $\Delta_{b}$, and suppose the number of signal events is Poisson distributed with predicted mean $s$. Then the median expected significance for discovery (where the null hypothesis is background only, and the signal is assumed to be present in the data) is approximately LiMa ; Cousins:2008zz ; Cowan:2010js ; Cowan :

[TABLE]

The median expected significance for exclusion (where the null hypothesis is now the background + signal model, but the signal is assumed to be actually absent in the data) can be approximated by Kumar:2015tna :

[TABLE]

where

[TABLE]

In the idealized limit of a perfectly known background prediction, $\Delta_{b}=0$, these would reduce to

[TABLE]

In the further limit of large $b$, these both approach $Z_{\rm disc}=Z_{\rm exc}=s/\sqrt{b}$. In the following, we will use as a criteria for expected 5-sigma discovery that $s$ is at least 10 and eq. (1.2) should exceed $Z_{\rm disc}>5$, and for an expected 95% exclusion that eq (1.3) should exceed $Z_{\rm exc}>1.645$ corresponding to $p=0.05$. We will use the somewhat arbitrary (since the capabilities of the detectors are unknown at present) choices $\Delta_{b}/b=$ 0.1, 0.2, and 0.5, corresponding to a 10%, 20%, and 50% uncertainty in the background. We have also assumed that $b$ is always at least 1 event, to be conservative.

II Minimal models for production and decay of vectorlike leptons

Following the terminology and definitions in ref. Kumar:2015tna , the “Singlet VLL Model” contains the Standard Model fields and interactions plus an $SU(2)_{L}$-singlet charged VLL $\tau^{\prime-}$ and its antiparticle $\tau^{\prime+}$, which consist of 2-component left-handed fermions transforming under $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ as

[TABLE]

The “Doublet VLL Model” contains a new charged lepton and its neutral heavy Dirac neutrino partner, realized as 2-component left-handed fermions transforming as

[TABLE]

In both models, a single weak-isosinglet bare fermion mass parameter $M$ is mostly responsible for the new fermion masses, with a small Yukawa couplings $\epsilon$ to the Higgs field providing the mixing mass with the Standard Model $\tau$ lepton, which also has its own Yukawa coupling $y_{\tau}$ to the Higgs field. The charged fermion mass matrix for the $\tau,\tau^{\prime}$ system in each case then can be written in the form

[TABLE]

where $v\approx 174$ GeV is the Standard Model Higgs vacuum expectation value. The tree-level mass eigenvalues, obtained from the square roots of the eigenvalues of ${\cal M}^{\dagger}{\cal M}$ after expanding for $y_{\tau}v,\epsilon v\ll M$, are:

[TABLE]

where the ellipses in both cases represent terms suppressed by $\epsilon^{4}v^{4}/M^{4}$ or $\epsilon^{2}y_{\tau}^{2}v^{4}/M^{4}$, while the tree-level neutral VLL mass in the Doublet model is simply $M_{\nu^{\prime}}=M$. There is also a 1-loop radiative correction to the physical mass splitting $M_{\nu^{\prime}}-M_{\tau^{\prime}}$, but it is also small, only of order a few hundred MeV Thomas:1998wy .

In the special case of no mixing with the tau lepton, $\epsilon=0$, the lightest VLL would be absolutely stable due to a conserved global symmetry. Here, we assume instead that $\epsilon$ is small enough to be treated as a tiny perturbation in the mass matrix, but that it exceeds about $2\times 10^{-7}$, which is large enough to allow Kumar:2015tna the VLLs to decay promptly†††If the $\tau^{\prime}$ in the Singlet VLL model is quasi-stable, then search strategies based on time-of-flight and ionization rate should apply, as in the ATLAS search in ref. Aaboud:2019trc . Assuming that the $\tau^{\prime}$ interacts similarly to a chargino, we infer from the cross-section limit in Figure 10 of Aaboud:2019trc that its mass should exceed about 750 GeV. This does not apply to the Doublet VLL model, where the $\tau^{\prime}$ can decay to $\nu^{\prime}$ inside the detector. on collider detector length scales to Standard Model states, with widths that dominate over the competing mode $\tau^{-\prime}\rightarrow\pi^{-}\nu^{\prime}$ Thomas:1998wy in the Doublet VLL case. The fermion mass eigenstates then consist of, besides the usual $\tau^{+},\tau^{-},\nu_{\tau}$ and the rest of the Standard Model states, a charged Dirac pair $\tau^{\prime-},\tau^{\prime+}$ in both models, and a neutral Dirac pair $\nu^{\prime},\overline{\nu}^{\prime}$ in the Doublet VLL model only. Due to the small size of $\epsilon$ and the 1-loop radiative splitting, we can take $M_{\nu^{\prime}}\approx M_{\tau^{\prime}}$ for the purposes of collider simulations.

The collider pair production modes are

[TABLE]

involving couplings to the $\gamma$, $Z$, and $W^{\pm}$ vector bosons, which are listed in ref. Kumar:2015tna . Of course, only the first of these processes occurs for the Singlet VLL model.

In Figure 2.1, we show the total pair production cross-sections at a proton-proton collider as a function of $M_{\tau^{\prime}}=M_{\nu^{\prime}}$, for the choices $\sqrt{s}=8,13,14,27,70,100$ TeV.

Note that the Doublet VLL model has a considerably larger total cross-section, which is partly responsible for the much better search prospects as found in ref. Kumar:2015tna and below in the present paper. For example, at $M_{\tau^{\prime}}=1000$ GeV, the total cross-section is about 12.5 times larger in the Doublet VLL model than for the Singlet VLL model, nearly independent of the proton-proton collision energy over the range from 14 to 100 TeV. In the Doublet VLL model, the total contribution is dominated by the $W$-boson-mediated $\tau^{\prime-}\overline{\nu}^{\prime}$ and $\tau^{\prime+}\nu^{\prime}$ final states. However, even if one restricts attention to only the $\tau^{\prime-}\tau^{\prime+}$ final state common to both models, the Doublet VLL model has a significantly larger cross-section. This is illustrated in Figure 2.2, which compares the individual cross-sections for each of the final states at $\sqrt{s}=27$ TeV. (Results are similar at other values of $\sqrt{s}$.)

The total pair-production cross-sections for these models are also shown in Figure 2.3 as a function of $\sqrt{s}$, for various choices of $M_{\tau^{\prime}}=M_{\nu^{\prime}}$.

Clearly, the cross-sections are getting larger at higher collider energies. In the Doublet VLL model, one obtains roughly the same cross-section for $M_{\tau^{\prime}}=4000$ GeV at $\sqrt{s}=100$ TeV as one would have for $M_{\tau^{\prime}}=1870$ GeV at $\sqrt{s}=27$ TeV, or for $M_{\tau^{\prime}}=1220$ GeV at $\sqrt{s}=14$ TeV.

As for the decays, the lepton-flavor conserving charged current process $\tau^{\prime}\rightarrow\nu^{\prime}\pi^{-}$ is kinematically allowed in the Doublet VLL model, but has a width smaller than the direct lepton-flavor-violating decays to Standard Model fermions as long as \epsilon\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\sim}\hss}\raise 1.0pt\hbox{>}}2\times 10^{-7} Kumar:2015tna , and is therefore neglected. In the Doublet VLL model, the neutral VLLs decay 100% of the time to a $W$ boson and an ordinary tau lepton:

[TABLE]

This reflects our assumptions that there is no $\nu^{\prime}$ mixing with the left-handed neutrinos of the Standard Model and that the mixing Yukawa coupling $\epsilon$ involves the tau lepton, rather than muon or the electron; otherwise the discovery and exclusion strategies would be much easier, due to the higher detection efficiency and lower fake rates for $e,\mu$ compared to $\tau$.

In both models, the charged VLLs can decay into the final states $Z\tau$, $h\tau$, and $W\nu_{\tau}$. The decay widths (neglecting the tau lepton mass) are Kumar:2015tna :

[TABLE]

in both the Singlet and Doublet VLL models, while

[TABLE]

where $r_{X}\equiv M_{X}^{2}/M_{\tau^{\prime}}^{2}$ for each of $X=h,Z,W$. In the decays to $Z$ and $W$, the factors $(1+2r_{Z})$ and $(1+2r_{W})$ can be understood as coming from the longitudinal (1) and transverse $(2r_{X})$ components of the weak vector bosons. The longitudinal components can in turn be understood as essentially the Goldstone modes that are eaten by the vector bosons to obtain their masses. This illustrates the usual Goldstone equivalence principle, which implies that for the limit of large $M_{\tau^{\prime}}$ the branching ratios should approach:

[TABLE]

In Figure 2.4, we plot the branching ratios for the $\tau^{\prime}$ decays in the two models, as a function of the mass $M_{\tau^{\prime}}$, showing the asymptotic approach to the Goldstone equivalence limit.

The Doublet VLL model therefore has the following final states:

[TABLE]

from $\tau^{\prime}$ pair production eq. (2.6),

[TABLE]

from $\nu^{\prime}$ pair production eq. (2.7), and

[TABLE]

from $\tau^{\prime}\nu^{\prime}$ production eq. (2.8) and eq. (2.9). Besides the three signals in eq. (2.20), the Singlet VLL model also has final states

[TABLE]

with the last being the largest in the minimal Singlet VLL model.

With the above branching ratios for the minimal Singlet VLL model, the analysis below does not find any reach for any of the signal regions at any of the collider options considered below, due to the low cross-section and the unfortunate large branching ratio for $\tau^{\prime}\rightarrow W\nu$. Therefore, we instead consider a class of non-minimal models which have the feature that the lightest new fermion is still a charged, mostly isosinglet $\tau^{\prime}$, but with other new fermions much heavier and therefore not contributing to the production. Thus, the new particle content first accessible to colliders is the same as in the minimal Singlet VLL model, but the mixing with the much heavier vectorlike fermions allows the branching ratios of $\tau^{\prime}$ to be essentially arbitrary among the three final states listed above. We therefore present results for modified Singlet VLL models, with the same $\tau^{\prime+}\tau^{\prime-}$ production cross-section as the minimal Singlet VLL model, but with branching ratios set to the following three sub-cases:

[TABLE]

Realizations of these simplified models, from appropriate limits of mixing a vectorlike isosinglet fermion with a much heavier vectorlike isodoublet fermion, are discussed in an Appendix.

For signal simulation, we have input a Lagrangian which governs the production and decay of the Doublet VLL and the Singlet VLL models (as discussed above) into FeynRules v2.3 Alloul:2013bka , a Mathematica package, to obtain the Feynman rules as a Universal FeynRules Output (UFO) file, which is then imported into MadGraph5_aMC@NLO v2.6.5 Alwall:2011uj ‡‡‡Electronic input files relevant for VLL models are available from the authors on request. We considered Standard Model production of the final states $WZ$, $ZZ$, $t\overline{t}Z$, $t\overline{t}W$, $hW$, $hZ$, $t\overline{t}h$, $WWW$, $WWZ$, $ZZW$, and $ZZZ$ as the main backgrounds contributing to multi-lepton final states. We did not include reducible backgrounds such as $W+jets$, $Z+jets$, and $WZ+jets$, where one or more jets fake a tau. While they could be important, we do not have a way of reliably estimating them for unknown future detectors. Therefore it must be kept in mind that if they are large, our projections may be too optimistic.

Both signal and background events were generated using MadGraph at leading order (LO). The decay couplings in the (VLL) Lagrangian are left as free parameters in the model file of FeynRules, so that one has the flexibility to adjust them during the run time of MadGraph. The numerical values of the decay couplings that are used are actually not important (as long as they are not too small or too large), because in each separate Madgraph run the decays of VLL were forced into one of the individual channels mentioned above, and later the event samples were normalized using the branching ratios shown in Figure 2.4. We calculated the cross-sections for signal at LO and normalized the background cross-sections as calculated at next-to-leading order (NLO), using MadGraph. While using MadGraph, we used their default set of parton distribution functions (PDFs) based on NNPDF2.3 set nnpdf23qed_lo . We used Pythia 8.2 Sjostrand:2006za ; Sjostrand:2015za for showering and hadronization. Then, Delphes 3.4 deFavereau:2013fsa was used for detector simulation. We have used the default Delphes tagging efficiencies and misidentification rates for taus and b-jets, where the tau tagging efficiency is 0.6, and the tau misidentification rate for QCD jets is 0.01. To increase the yield for background events to pass cuts described below, we forced every particle, except for the SM Higgs boson ($h$), to decay into leptons (including taus) and/or jets, such that they contribute to multi-lepton final states with at least 3 leptons. For generation of both signal and background events, we only considered the decay of $h$ into $W^{+}W^{-}$, $ZZ$, and $\tau^{+}\tau^{-}$, ignoring all other decays, and have normalized using the branching ratios of $h$ from HDECAY Djouadi:1997yw . Thus each signal/background containing one or more $h$ is split into 3 signal/background components per $h$. We generated at least $10^{5}$ events for each signal component at each mass point, and at least $10^{6}$ events for each background component, at each of $\sqrt{s}=(14,27,70,100)$ TeV. These numbers of generated events are usually sufficient, with a few exceptions noted below. In the real world, better background determinations may come from data.

In our analysis, we first impose object cuts on leptons $\ell=e,\mu,\tau$, where, from now on, $\tau$ refers to a tau lepton that undergoes a hadronic decay. These include minimum $p_{T}$ cuts that depend on the collider option, and are therefore listed in the subsections below. In all cases, the lepton candidates are required to pass the following further cuts on pseudo-rapidity $\eta$ and isolation from other lepton candidates or jets:

[TABLE]

Here, $\Delta R=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}$ as is usual.

Events are then selected with at least 3 three leptons, of which at least 2 must be $e$ or $\mu$. The leading $e$ or $\mu$ lepton is also required to satisfy a minimum $p_{T}$ trigger requirement that depends on the collider option, and is therefore listed separately in each of the subsections below. We also veto $b$-jets to reduce large backgrounds from $t\overline{t}$ production processes. Thus our event pre-selection common to all signal regions is:

[TABLE]

Events with no $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pair with invariant mass within 15 GeV of $M_{Z}$ are referred to below as “no-$Z$”, and events with exactly two $e/\mu$ that have same-sign charges will be labeled as “SS”. We then considered the following 6 distinct types of signal regions:

[TABLE]

Finally, in each of these signal regions, we imposed a minimum lower bound cut on $L_{T}$, which is defined to be the sum of the transverse momentum of all leptons:

[TABLE]

We varied the choice of this cut to obtain good exclusion and discovery reach simultaneously for each signal region and collider option and assumed fractional background uncertainty $\Delta_{b}/b=(0.1,\>0.2,\>0.5)$. However, we have only done a very rough optimization for the $L_{T}$ cut, in part because the optimization is different for exclusion and for discovery, and also because the Monte Carlo simulations are only an approximation to the actual experimental capabilities, which will rely on detector designs yet to be determined. In general, the results found below reflect that the choice of cut on $L_{T}$ increases with the mass $M_{\tau^{\prime}}$ at the edge of the reach. Also, for convenience, we always chose the same $L_{T}$ cut for both exclusion and discovery, even though an optimized cut would likely be somewhat different for the two cases.

Note that the signal regions considered in eqs. (2.34)-(2.39) are far from exclusive of each other. Therefore, to be conservative we have not attempted to combine them, although doing so could lead to some extension of the reach prospects.

III Results for the HL-LHC collider

In this section, we discuss the possibility of exclusion or discovery of both Doublet and Singlet VLL (including the minimal and non-minimal versions) at $\sqrt{s}=14$ TeV with 3 ab*-1* of $pp$ collisions. In addition to the pseudo-rapidity, isolation and other requirements of eqs. (2.27)-(2.33), we require all leptons including hadronic tau candidates to satisfy

[TABLE]

Additionally, the leading $e$ or $\mu$ in each event is required to satisfy a trigger requirement:

[TABLE]

We then considered the six signal regions mentioned in eqs. (2.34)-(2.39).

III.1 Doublet VLL model

In Figure 3.1, we show the $L_{T}$ distributions for the best four of these signal regions, for five different choices of $M_{\tau^{\prime}}$ as labeled, and for the total of all backgrounds shown as the shaded histogram. For $pp$ collisions with $\sqrt{s}=14$ TeV with 3 ab*-1*, we found that the expected reach is approximately maximized if we then choose a cut $L_{T}>800$ GeV.

Figure 3.2 shows the $L_{T}$ distributions for all background components, for the four best signal regions as labeled. The $L_{T}$ cut is shown in the figure as a vertical dashed line. After imposing the $L_{T}$ cut, the dominant SM backgrounds are $WZ$, $t\bar{t}V$ and $VVV$ (where $V=W,Z$) in the two signal regions with 2 SS $e/\mu\>+\!\geq 1\tau$, while the dominant SM backgrounds are $t\bar{t}V$ and $ZZ$ in the signal region with $\geq 2e/\mu+2\tau$, and $t\bar{t}V$ and $t\bar{t}h$ in the signal region with $\geq 2e/\mu+2\tau$ (no-$Z$).

Figure 3.3 shows the resulting median expected significances for exclusion (left panels) and discovery (right panels), for $\Delta_{b}/b=0.1$ (top row), $0.2$ (middle row), and $0.5$ (bottom row), with the cut requirement $L_{T}>800$ GeV imposed. In all cases, the best signal regions for both exclusion and discovery scenarios are the ones with 2 SS $e/\mu\>+\!\geq 1\tau$, with the additional requirement $E_{T}^{\rm miss}>150$ GeV providing slightly more reach. The two signal regions with $\geq 2e/\mu+2\tau$ give slightly less reach at higher masses, but could actually provide comparable exclusion significances for lower masses.

From Figure 3.3 we conclude that a 14 TeV high-luminosity LHC with 3 ab*-1* should be able to exclude Doublet VLLs with $M_{\tau^{\prime}}$ up to about 1250 GeV if they are indeed absent, or discover them if the mass is less than about 900 GeV, assuming that the future background determination from data is subject to uncertainties of order 10% or less. The figure also shows that prospects for exclusion are much less sensitive to uncertainty in the background than the prospects for discovery. For $\Delta_{b}/b=0.5$, one can still expect to exclude Doublet VLLs up to about $M_{\tau^{\prime}}=1190$ GeV, or discover them if the mass is less than about 690 GeV, but the latter has already been excluded at 95% confidence level by CMS CMS:2018cgi ; Sirunyan:2019ofn .

In the case that there are enough events for a clear discovery, one can also hope to measure the mass of the $\tau^{\prime}$. In figure 3.4, we show the event distributions for the 3-body invariant mass of $\tau^{\pm}e^{+}e^{-}$ or $\tau^{\pm}\mu^{+}\mu^{-}$, for the signal region with $\geq 2e/\mu+2\tau$, for two different choices of input $M_{\tau^{\prime}}$, and for the total of all backgrounds shown as the shaded histogram. (Combinatorial backgrounds from wrong associations of the lepton pair and tau in the signal sample are of course also present and included.) Here, we require the 2-body invariant mass of the $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pair to be within 10 GeV of $M_{Z}$. Additionally, we also impose the cut $L_{T}>800$ GeV. From Figure 3.4, we observe that the distributions for Doublet VLLs are peaked just below their respective masses, which gives a possibility to measure the masses of Doublet VLLs, if they are indeed discovered. At $\sqrt{s}=14$ TeV, the range of masses that one can hope to measure in this way is limited by the present exclusion up to $M_{\tau^{\prime}}=790$ GeV by CMS Sirunyan:2019ofn and by the fact that the cross-section decreases rapidly for higher masses.

III.2 Singlet VLL models

Singlet VLLs are much more challenging than Doublet VLL, due to their much smaller production cross sections. For the minimal Singlet model, we find no possible exclusion or discovery at $\sqrt{s}=14$ TeV with integrated luminosity of 3 $ab^{-1}$. And, for the non-minimal Singlet models, we find some exclusion possibility, but with no prospects for discovery. In view of the difficulties involved, below we consider only the case that the uncertainty in the background is $\Delta_{b}/b=0.1$.

To maximize the exclusion reach in the non-minimal cases, we chose a cut $L_{T}>700$ GeV for the $Z$-philic Singlet VLL, $L_{T}>400$ GeV for the $W$-phobic Singlet VLL and $L_{T}>200$ GeV for the Higgs-philic Singlet VLL. Figure 3.5 shows the resulting median expected significances for exclusion, for $\Delta_{b}/b=0.1$, for the best signal region for each of the Singlet VLL models, with the cuts on $L_{T}$ imposed. In all the models, the best signal region for exclusion is the one which requires $\geq 2e/\mu+2\tau$. However, for the Higgs-philic Singlet VLL model, the additional requirement of no-$Z$ proved to be beneficial.

From Figure 3.5, we conclude that assuming $\Delta_{b}/b=0.1$, a 14 TeV $pp$ collider with 3 ab*-1* should be able to exclude Singlet VLLs with masses up to about 600 GeV if they are $Z$-philic, or exclude masses up to about 360 GeV if they are $W$-phobic, or exclude masses up to about 300 GeV if they are Higgs-philic. We find that there is no possible exclusion of Singlet VLLs in the minimal model. Furthermore, there are no discovery prospects in minimal or non-minimal Singlet VLL models.

IV Results for the HE-LHC collider

In this section, we discuss the prospects for exclusion and discovery of VLLs at $\sqrt{s}=27$ TeV with 15 ab*-1* of $pp$ collisions in the six signal regions mentioned in eqs. (2.34)-(2.39). All leptons including hadronic tau candidates are required to satisfy:

[TABLE]

along with the same pseudo-rapidity, isolation and other requirements of eqs. (2.27)-(2.33), with a trigger requirement of:

[TABLE]

IV.1 Doublet VLL model

In Figure 4.1, we show the $L_{T}$ distributions for the best four of these signal regions, for four different choices of $M_{\tau^{\prime}}$ as labeled, and for the total of all backgrounds shown as the shaded histogram. The cut chosen to enhance the reach for exclusion and discovery in this case was $L_{T}>1500$ GeV.

Figure 4.2 shows the $L_{T}$ distributions for all background components, for the four best signal regions as labeled. The $L_{T}$ cut is shown in the figure as a vertical dashed line. After imposing the $L_{T}$ cut, the most important SM backgrounds are $t\bar{t}V$ and $VVV$ in all four of these signal regions.

Figure 4.3 shows the $Z_{\rm excl}$ (left panels) and $Z_{\rm disc}$ (right panels) as a function of $M_{\tau^{\prime}}$, for $\Delta_{b}/b=0.1$ (top row), $0.2$ (middle row), and $0.5$ (bottom row), with the cut $L_{T}>1500$ GeV imposed. The signal region with 2 SS $e/\mu\>+\!\geq 1\tau$ with $E_{T}^{\rm miss}>150$ GeV has the farthest mass reach. However, at lower masses the two signal regions with $\geq 2e/\mu+2\tau$ give slightly better exclusion and discovery significances.

By looking at Figure 4.3, we conclude that a 27 TeV high-energy LHC with 15 ab*-1* could exclude Doublet VLLs with masses up to about 2300 GeV or discover them if the mass is less than about 1700 GeV, assuming the fractional uncertainty in the background to be 0.1. On the other hand, if we assume $\Delta_{b}/b=0.5$, we would still be able to exclude Doublet VLLs up to $M_{\tau^{\prime}}=2050$ GeV, or discover them if the mass is less than about 1250 GeV. Just as in the case of the HL-LHC, from Figure 4.3, a larger uncertainty in the background has a moderate effect on the prospects for exclusion, but a much larger impact on the prospects for discovery.

We again consider the prospects for observing a mass peak in the case that there are enough events for a clear discovery. In figure 4.4, we show the event distributions for the 3-body invariant mass of $\tau^{\pm}e^{+}e^{-}$ or $\tau^{\pm}\mu^{+}\mu^{-}$, for the signal region with $\geq 2e/\mu+2\tau$, for three different choices of $M_{\tau^{\prime}}$. The total of all backgrounds is shown as the shaded histogram. We require the 2-body invariant mass of the $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pair to be within 10 GeV of $M_{Z}$. Additionally, we also impose the cut $L_{T}>1500$ GeV. From Figure 4.4, we observe that the distributions for Doublet VLLs are peaked just below their respective masses, which gives a possibility to measure the masses of Doublet VLLs, if they are indeed discovered. For higher masses, statistical limitations and the combinatorial background mean that the mass determinations will be problematic.

IV.2 Singlet VLL models

At $\sqrt{s}=27$ TeV with integrated luminosity of 15 $ab^{-1}$, we find that there is no exclusion or discovery reach for the minimal and the Higgs-philic Singlet VLL models. For the other two non-minimal Singlet models, we find some exclusion possibility, but again with no prospects for discovery.

We chose a cut $L_{T}>1400$ GeV for the $Z$-philic Singlet VLL, $L_{T}>800$ GeV for the $W$-phobic Singlet VLL, to maximize the exclusion reach. A cut $L_{T}>600$ GeV for the Higgs-philic Singlet VLL and $L_{T}>800$ GeV for the minimal Singlet VLL model were chosen for the best possible exclusion significance. Figure 4.5 shows the resulting median expected significances for exclusion, for $\Delta_{b}/b=0.1$, for the best signal region for each of the Singlet VLL models, with the cuts on $L_{T}$ imposed, as mentioned above. The best signal region for exclusion, for all the Singlet VLL models is the one which requires $\geq 2e/\mu+2\tau$, except that an additional requirement of no-$Z$ gave better results for the Higgs-philic model.

From Figure 4.5, we can conclude that a 27 TeV $pp$ collider with 15 ab*-1* could possibly exclude Singlet VLLs with masses up to about 1200 GeV in the $Z$-philic model, or exclude masses up to about 670 GeV in the $W$-phobic model, but with no prospects for discovery. These results assume a fractional uncertainty in the background of $\Delta_{b}/b=0.1$. In both minimal and the Higgs-philic Singlet VLL models, there is no possibility for exclusion or discovery. From the results for Singlet VLL analyses for 14 and 27 TeV colliders, we can note that the exclusion reach scaled approximately linearly with $\sqrt{s}$, for both $Z$-philic and the $W$-phobic Singlet VLL models.

V Results for a $pp$ collider with

$\sqrt{s}=70$ TeV

In this section, we turn our attention to prospects for exclusion and discovery of VLLs at possible future $pp$ collider at $\sqrt{s}=$ 70 TeV, with integrated luminosity of 30 ab*-1* in the six signal regions mentioned in eqs. (2.34)-(2.39). All the leptons including hadronic tau candidates are required to satisfy:

[TABLE]

along with the same pseudo-rapidity, isolation and other requirements of eqs. (2.27)-(2.33), with at least one $e$ or $\mu$ satisfying a trigger requirement:

[TABLE]

V.1 Doublet VLL model

In Figure 5.1, we show the $L_{T}$ distributions for the best four of the signal regions, for four different choices of $M_{\tau^{\prime}}$ as labeled, and for the total of all backgrounds shown as the shaded histogram. We found that choosing the cut $L_{T}>2800$ GeV provides a good reach for both exclusion and discovery in this case.

Figure 5.2 shows the $L_{T}$ distributions for all background components, for the four best signal regions as labeled. The $L_{T}$ cut is shown in the figure as a vertical dashed line. After imposing the $L_{T}$ cut, the most important SM backgrounds are $t\bar{t}V$ and $VVV$ in the two signal regions with 2 SS $e/\mu\>+\!\geq 1\tau$, while the most important SM backgrounds are $WZ$ and $t\bar{t}V$ in the signal region with $\geq 2e/\mu+2\tau$, and $t\bar{t}h$ and $t\bar{t}V$ in the signal region with $\geq 2e/\mu+2\tau$ (no-$Z$). We note that the fluctuation of about 66 normalized events in the bin from 4000 to 4500 GeV in the lower left panel (i.e. signal region with $\geq 2e/\mu+2\tau$) of Figure 5.1 is due to a single simulated event of $WZ$ background. This is because of the large cross-section but extremely low yield for this background component in this signal region even after forcing decays to leptons. This is an unavoidable source of uncertainty for our analysis; given that our sample size was already $5.5\times 10^{6}$ simulated events for this component, no practically feasible increase in sample size would yield significantly better statistics. However, in the real world the background can perhaps be determined more accurately from data.

Figure 5.3 shows the median expected significances for exclusion $Z_{\rm excl}$ (left panels) and discovery $Z_{\rm disc}$ (right panels) as a function of $M_{\tau^{\prime}}$, for $\Delta_{b}/b=0.1$ (top row), $0.2$ (middle row), and $0.5$ (bottom row), with the cut $L_{T}>2800$ GeV imposed. The two signal regions with 2 SS $e/\mu\>+\!\geq 1\tau$ and the signal region with $\geq 2e/\mu+2\tau$ (no-$Z$) have comparable exclusion significances and reaches. However, the latter has higher discovery significances at lower masses, as well as at higher fractional uncertainties in the background, e.g. $\Delta_{b}/b=0.5$ .

Figure 5.3 shows a possibility of excluding Doublet VLLs of masses up to about 4700 GeV or discovering them if mass is less than about 3400 GeV with a 70 TeV $pp$ collider with 30 ab*-1*, assuming the fractional uncertainty in the background to be $\Delta_{b}/b=0.1$. If $\Delta_{b}/b=0.5$, we can still expect to exclude Doublet VLLs up to $M_{\tau^{\prime}}=4150$ GeV, or discover them if the mass is less than about 2400 GeV. Again, a larger uncertainty in the background has a moderate effect on the prospects for exclusion, but a much larger impact on the prospects for discovery.

V.2 Singlet VLL models

For $pp$ collisions with $\sqrt{s}=70$ TeV with 30 $ab^{-1}$, we again find that there is no possible exclusion or discovery reach for the minimal and the Higgs-philic Singlet VLL models. While we find some exclusion possibility for the $Z$-philic and the $W$-phobic Singlet VLL models, there are again no prospects for discovery. Figure 5.4 shows the resulting median expected significances for exclusion, for $\Delta_{b}/b=0.1$, for the best signal region for each of the Singlet VLL models, with appropriate cuts on $L_{T}$ imposed. To approximately maximize the exclusion reaches for the $Z$-philic and $W$-phobic Singlet VLL models, we chose a cut $L_{T}>2100$ GeV for the former and $L_{T}>950$ GeV for the latter. The best signal region for exclusion for the $Z$-philic Singlet VLL model is the one which requires $\geq 3e/\mu+1\tau$, while it was $\geq 2e/\mu+2\tau$, no-$Z$ for all other Singlet VLL models.

From Figure 5.4, we conclude that a 70 TeV $pp$ collider with 30 ab*-1* could possibly exclude Singlet VLLs with masses up to about 1700 GeV in the $Z$-philic model, or exclude masses up to 850 GeV in the $W$-phobic model, but with no prospects for $5\sigma$ discovery. In both minimal and the Higgs-philic Singlet VLL models, there is unfortunately no possibility for exclusion or discovery, at least with the signal regions we considered.

VI Results for a $pp$ collider with

$\sqrt{s}=100$ TeV

Finally, we consider the possibility of excluding or discovering VLLs at a future very high energy $pp$ collider at $\sqrt{s}=100$ TeV, with integrated luminosity of 30 $ab^{-1}$, in the six signal regions mentioned in eqs. (2.34)-(2.39). We require all leptons including hadronic tau candidates to satisfy:

[TABLE]

along with the same pseudo-rapidity, isolation and other requirements of eqs. (2.27)-(2.33). We then require the leading $e$ or $\mu$ in each event to satisfy a trigger requirement:

[TABLE]

VI.1 Doublet VLL model

In Figure 6.1, we show the $L_{T}$ distributions for the best four of these signal regions, for four different choices of $M_{\tau^{\prime}}$ as labeled, and for the total of all backgrounds shown as the shaded histogram. To obtain enhanced expected reaches for both exclusion and discovery, we then chose a cut $L_{T}>3500$ GeV.

Figure 6.2 shows the $L_{T}$ distributions for all background components, for the four best signal regions as labeled. The $L_{T}$ cut is shown in the figure as a vertical dashed line. After imposing the $L_{T}$ cut, the most important SM backgrounds are $t\bar{t}V$ and $VVV$ in the two signal regions with 2 SS $e/\mu\>+\!\geq 1\tau$ and in the signal region with $\geq 2e/\mu+2\tau$, while the most important SM background in the signal region with $\geq 2e/\mu+2\tau$ (no-$Z$) is $VVV$.

Figure 6.3 shows the median expected significances for exclusion $Z_{\rm excl}$ (left panels) and discovery $Z_{\rm disc}$ (right panels) as a function of $M_{\tau^{\prime}}$, for $\Delta_{b}/b=0.1$ (top row), $0.2$ (middle row), and $0.5$ (bottom row), with the cut $L_{T}>3500$ GeV imposed. The signal regions which require 2 SS $e/\mu\>+\!\geq 1\tau$ usually provide the best exclusion and discovery reaches. An exception is that with the fractional uncertainty in the background $\Delta_{b}/b=0.5$, the signal regions with $\geq 3e/\mu+1\tau$ have the farthest discovery reach.

From Figure 6.3, we conclude that a 100 TeV $pp$ collider with 30 ab*-1* could exclude Doublet VLLs with $M_{\tau^{\prime}}$ up to about 5750 GeV or discover them if the mass is less than about 4000 GeV, assuming the fractional uncertainty in the background to be 0.1. For $\Delta_{b}/b=0.5$, one can still expect to exclude Doublet VLLs if the mass is up to about 5100 GeV, or discover them if the mass is less than about 3100 GeV. Again, as a recurring theme at all the collider options considered, a larger uncertainty in the background produces a moderate reduction of the exclusion reach, but has a much greater impact on the prospects for discovery.

We again consider the possibility of observing a mass peak for the $\tau^{\prime}$ when a clear discovery can be made. Figure 6.4 shows the event distributions for 3-body invariant mass of $\tau^{\pm}e^{+}e^{-}$ or $\tau^{\pm}\mu^{+}\mu^{-}$, for the signal region with $\geq 2e/\mu+2\tau$, for three different choices of $M_{\tau^{\prime}}$, and for the total of all backgrounds shown as the shaded histogram. We require the 2-body invariant mass of $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pair to be within 10 GeV of $M_{Z}$. We also impose the cut $L_{T}>3500$ GeV. From Figure 6.4, we note that there are peaks in the distributions for Doublet VLLs corresponding to, and slightly below, their respective masses, giving a possibility to measure the masses of Doublet VLLs, if they are indeed discovered.

VI.2 Singlet VLL models

We find no possible exclusion of Singlet VLLs in the minimal and the Higgs-philic Singlet VLL models, for $pp$ collisions with $\sqrt{s}=100$ TeV with integrated luminosity of 30 $ab^{-1}$. We find some exclusion prospects for the $Z$-philic and $W$-phobic Singlet VLL models, assuming that the fractional uncertainty in the background is $\Delta_{b}/b=0.1$. Again, there are no discovery prospects in any of the Singlet VLL models.

We chose cuts of $L_{T}>3000$ GeV for the $Z$-philic Singlet VLL, and $L_{T}>1200$ GeV for the $W$-phobic Singlet VLL, to approximately maximize the exclusion reach in each case. Figure 6.5 shows the resulting median expected significances for exclusion, for $\Delta_{b}/b=0.1$, for the best signal region for each of the Singlet VLL models, with the cuts on $L_{T}$ imposed, as mentioned above. The best signal region for exclusion for the $Z$-philic Singlet VLL model is the one which requires $\geq 3e/\mu+1\tau$, while it was $\geq 2e/\mu+2\tau$, no-$Z$ for all other Singlet VLL models.

From Figure 6.5, assuming $\Delta_{b}/b=0.1$, we conclude that a 100 TeV $pp$ collider with 30 ab*-1* should be able to exclude Singlet VLLs with masses up to about 2850 GeV, if they are $Z$-philic, or exclude masses up to 1200 GeV, if they are $W$-phobic. We find that there are no prospects for exclusion of Singlet VLLs in both minimal and the Higgs-philic Singlet VLL models.

VII Outlook

Vectorlike leptons are a common feature of motivated theories of physics beyond the Standard Model. In this paper, we have studied the prospects for discovering simple VLL models at future $pp$ colliders, assuming that the new particles decay promptly by mixing with taus, using multi-lepton signatures at high transverse momentum. The results of this paper should be considered as first estimates, which certainly should be reassessed as the plans and parameters for future colliders and their detectors become clearer. We have not attempted to make use of $h\rightarrow b\overline{b}$ decays, due in part to a relative lack of confidence in what should be assumed about $b$-jet tagging capabilities compared to $e,\mu$ identification. We have also based our analysis entirely on a cut-and-count strategy, while a more sophisticated method such as a likelihood analysis would certainly be able to do better.

Summaries of our results are shown in Table 7.1 for the Doublet VLL model, and in Table 7.2 for the non-minimal Singlet VLL models. In the weak isodoublet case, we have found excellent reach prospects for both 95% exclusion (if the VLLs are absent) or $5\sigma$ discovery (if the VLLs are present). In the latter case, we also showed that there is a possibility to observe an invariant mass peak, although we did not pursue this in detail and the mass peak observation is often quite limited by statistics. In the weak isosinglet case, the situation is much more difficult. We found no discovery or exclusion reach at all,†††The only limits on such a particle at present are $M_{\tau^{\prime}}>101.2$ GeV from LEP Achard:2001qw , and our estimate above in a footnote in section II, inferred from the ATLAS long-lived chargino search of ref. Aaboud:2019trc , of $M_{\tau^{\prime}}>750$ GeV if it is quasi-stable on detector scales. at any $pp$ collider, for the simple and well-motivated (e.g. Martin:2009bg ) case of a promptly decaying pure isosinglet $\tau^{\prime}$ that mixes with the Standard Model tau. This should stand as a challenge to future work.

Appendix A: Mixed singlet/doublet VLL models

Consider a generalized model framework that contains, besides the Standard Model chiral third-family leptons $\ell_{3}$ and $\overline{e}_{3}$, a vectorlike weak isosinglet pair $E,\overline{E}$ and a weak isodoublet pair $L,\overline{L}$. (Here we use 2-component fermion notation Dreiner:2008tw ; Martin:2012us , so that barred fields have electric charge $+1$ and unbarred fields have electric charge $-1$. The bar is part of the name of the 2-component fermion field, and does not denote any kind of conjugation.) After electroweak symmetry breaking, the Lagrangian mass terms for the charged leptons can be written in the form

[TABLE]

where the charged lepton mixing mass matrix is

[TABLE]

where $v=174$ GeV is the Standard Model Higgs expectation value, $y_{\tau}$, $\epsilon_{1}$, $\epsilon_{2}$, $x_{1}$, and $x_{2}$ are Yukawa couplings, and $M_{1}$ and $M_{2}$ are electroweak-singlet bare mass terms for the isosinglet and isodoublet vectorlike lepton pairs, respectively. This can be diagonalized to obtain mass eigenstates according to:

[TABLE]

where $R$ and $L$ are unitary $3\times 3$ matrices and, by convention, $M_{\tau}<M_{\tau^{\prime}}<M_{\tau^{\prime\prime}}$, where $\tau$ is the usual tau lepton with $M_{\tau}=1.777$ GeV, and in the special cases considered here $\tau^{\prime\prime}$ will be taken to be heavy enough to decouple from direct experimental observation. The neutral VLL $\nu^{\prime}$ has mass $M_{2}$.

For the lighter new charged VLL $\tau^{\prime}$, we have partial decay widths:

[TABLE]

where $r_{X}=M_{X}^{2}/M_{\tau^{\prime}}^{2}$ for each of $X=h,Z,W$, and

[TABLE]

for $(\tau_{1},\tau_{2},\tau_{3})$ corresponding to mass eigenstates $(\tau,\tau^{\prime},\tau^{\prime\prime})$ respectively.

In general, the decay widths to Standard Model states are quadratic in $\epsilon_{1},\epsilon_{2}$ for small values of those parameters. For example, expanding to obtain the terms proportional to $\epsilon_{1}^{2}$, $\epsilon_{1}\epsilon_{2}$, and $\epsilon_{2}^{2}$, and then keeping only the leading order in a further expansion in $y_{\tau}$, $x_{1}$, and $x_{2}$ in each of these terms, we obtain for $M_{2}^{2}\ll M_{1}^{2}$:

[TABLE]

and for $M_{1}^{2}\ll M_{2}^{2}$:

[TABLE]

Some special limits of interest follow:

• •

If $M_{2}^{2}\ll M_{1}^{2}$ and $\epsilon_{1}=0$, then the isosinglet heavier fermion mass eigenstate $\tau^{\prime\prime}$ decouples from experiment, and the lighter states $\tau^{\prime}$ and $\nu^{\prime}$ form the minimal Doublet VLL model as discussed above in section II with $\epsilon=\epsilon_{2}$ and $M_{\tau^{\prime}}=M_{\nu^{\prime}}=M_{2}$, and branching ratios that asymptotically approach ${\rm BR}(\tau^{\prime}\rightarrow h\tau)={\rm BR}(\tau^{\prime}\rightarrow Z\tau)=0.5$ and ${\rm BR}(\nu^{\prime}\rightarrow W\tau)=1$.

• •

If $M_{1}^{2}\ll M_{2}^{2}$ and $\epsilon_{2}=0$, then the heavy isodoublet fermions $\tau^{\prime\prime}$ and $\nu^{\prime}$ decouple, and the result is the minimal Singlet VLL model as discussed above in section II with $\epsilon=\epsilon_{1}$ and $M_{\tau^{\prime}}=M_{1}$, and branching ratios that asymptotically approach ${\rm BR}(\tau^{\prime}\rightarrow h\tau)={\rm BR}(\tau^{\prime}\rightarrow Z\tau)=0.25$ and ${\rm BR}(\tau^{\prime}\rightarrow W\nu_{\tau})=0.5$.

• •

If $M_{1}^{2}\ll M_{2}^{2}$ and $\epsilon_{1}=0$ and $\epsilon_{2}\not=0$, then the lightest new fermion will again be a mostly isosinglet $\tau^{\prime}$ vectorlike lepton with mass approximately $M_{\tau^{\prime}}=M_{1}$. Since the $\tau^{\prime}$ is mostly isosinglet, its production cross-section is nearly the same as in the minimal Singlet VLL model. However, due to its mixing with the heavier isodoublets rather than direct mixing with the Standard Model tau lepton, the decay $\tau^{\prime}\rightarrow W\nu_{\tau}$ is highly suppressed. The possibilities include the sub-cases:

• $\star$

$W$-phobic Singlet VLL: if $x_{1}=0$, then BR$(\tau^{\prime}\rightarrow h\tau)\approx{\rm BR}(\tau^{\prime}\rightarrow Z\tau)\approx 0.5$.

• $\star$

Higgs-philic Singlet VLL: if $x_{2}=0$, then BR$(\tau^{\prime}\rightarrow h\tau)\approx 1$.

• $\star$

$Z$-philic Singlet VLL: if $x_{1}\approx-x_{2}M_{1}/2M_{2}$, then ${\rm BR}(\tau^{\prime}\rightarrow Z\tau)\approx 1$.

These of course do not exhaust the possibilities, and in a more general search it would be sensible to simply take $\Gamma(\tau^{\prime}\rightarrow Z\tau)$ and $\Gamma(\tau^{\prime}\rightarrow h\tau)$ and $\Gamma(\tau^{\prime}\rightarrow W\nu_{\tau})$ to be free parameters.

Acknowledgments: P.N.B. thanks Olivier Mattelaer for all the helpful suggestions regarding the usage of Madgraph. P.N.B. also thanks Sergey A. Uzunyan for his kind assistance in using the NICADD compute cluster at Northern Illinois University, and Ramanpreet Singh and Elliot Parrish for technical advice. P.N.B. would also like to express special thanks to Nicholas T. Karonis, John Winans and the entire support team of Gaea cluster at Northern Illinois University for their help and support in using their high performance computing facility. This work was supported in part by the National Science Foundation grant number PHY-1719273.