# Single production of an exotic vector-like $Y$ quark
at future high energy $pp$ colliders

Liangliang Shang1,2111Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Yuxiao Yan1222Email<EMAIL_ADDRESS>Stefano Moretti2,3333Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Bingfang Yang1444E-mail:
<EMAIL_ADDRESS>1School of Physics, Henan Normal University, Xinxiang
453007, PR China
2Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20
Uppsala, Sweden
3School of Physics and Astronomy, University of Southampton, Highfield,
Southampton SO17 1BJ, UK

###### Abstract

Vector-like quarks have been predicted in various new physics scenarios beyond
the Standard Model (SM). In a simplified modelling of a $(B,Y)$ doublet
including a vector-like quark $Y$, with charge $-\frac{4}{3}$e, there are only
two free parameters: the $Y$ coupling $\kappa_{Y}$ and mass $m_{Y}$. In the
five flavor scheme, we investigate the single production of the $Y$ state
decaying into $Wb$ at the Large Hadron Collider (LHC) Run-III and High-
Luminosity LHC (HL-LHC) operating at $\sqrt{s}$ = 14 TeV, the possible High-
Energy LHC (HE-LHC) with $\sqrt{s}$ = 27 TeV as well as the Future Circular
Collider in hadron-hadron mode (FCC-hh) with $\sqrt{s}$ = 100 TeV. Through
detailed signal-to-background analyses and detector simulations, we assess the
exclusion capabilities of the $Y$ state at the different colliders. We find
that this can be improved significantly with increasing collision energy,
especially at the HE-LHC and FCC-hh, both demonstrating an obvious advantage
with respect to the HL-LHC case in the case of high $m_{Y}$. Assuming a 10%
systematic uncertainty on the background event rate, the exclusion
capabilities are summarized as follows: (1) the LHC Run-III can exclude the
correlated regions of $\kappa_{Y}\in[0.044,0.5]$ and $m_{Y}\in[1000\text{
GeV},3099\text{ GeV}]$ with integrated luminosity $L=300\text{ fb}^{-1}$; (2)
the HL-LHC can exclude the correlated regions of $\kappa_{Y}\in[0.027,0.5]$
and $m_{Y}\in[1000\text{ GeV},3653\text{ GeV}]$ with $L=3$ ab-1; (3) the HE-
LHC can exclude the correlated regions of $\kappa_{Y}\in[0.030,0.5]$ and
$m_{Y}\in[1000\text{ GeV},4936\text{ GeV}]$ with $L=3$ ab-1; (4) the FCC-hh
can exclude the correlated regions of $\kappa_{Y}\in[0.051,0.5]$ and
$m_{Y}\in[1000\text{ GeV},6610\text{ GeV}]$ with $L=3$ ab-1.

##  I introduction

In 2012, the ATLAS and CMS experiments at the Large Hadron Collider (LHC) made
a significant discovery by confirming the existence of the Higgs boson,
thereby providing further validation for the Standard Model (SM) ATLAS:2012yve
; CMS:2012qbp . However, the SM has certain limits in addressing several
prominent issues, such as neutrino masses, gauge hierarchy, dark matter and
dark energy. In various new physics scenarios like little Higgs models Arkani-
Hamed:2002ikv ; Han:2003wu ; Chang:2003vs ; Cao:2007pv , extra dimensions 4 ,
composite Higgs models Agashe:2004rs ; Bellazzini:2014yua ; Low:2015nqa ;
Bian:2015ota ; He:2001fz ; He:1999vp  and other extended models 6 ; 7 ; 8 ,
Vector-Like Quarks (VLQs) are predicted to play a role in resolving the gauge
hierarchy problem by mitigating the quadratic divergences of the Higgs field.
Such VLQs are fermions with spin $\frac{1}{2}$ and possess the unique
characteristic of undergoing both left- and right-handed component
transformations under the Electro-Weak (EW) symmetry group of the SM 9 .
Unlike chiral quarks, VLQs do not acquire masses through Yukawa couplings to
the Higgs field and therefore have the potential to counterbalance loop
corrections to the Higgs boson mass stemming from the top quark of the SM.
Furthermore, VLQs can generate characteristic signatures at colliders and have
been widely studied (see, for example, Banerjee:2023upj ; Benbrik:2023xlo ;
Zeng:2023ljl ; Canbay:2023vmj ; Belyaev:2023yym ; Shang:2023ebe ; Yang:2023wnv
; Bhardwaj:2022nko ; Bhardwaj:2022wfz ; Bardhan:2022sif ; Shang:2022tkr ;
Freitas:2022cno ; Benbrik:2022kpo ; Corcella:2021mdl ; VLX2021 ;
Belyaev:2021zgq ; Deandrea:2021vje ; Dasgupta:2021fzw ; King:2020mau ;
Liu:2019jgp ; Benbrik:2019zdp ; Xie:2019gya ; Bizot:2018tds ;
Cacciapaglia:2018qep ; Cacciapaglia:2018lld ; Carvalho:2018jkq ; CMS:2018kcw ;
Barducci:2017xtw ; CMS:2017voh ; Chen:2016yfv ; Arhrib:2016rlj ;
Cacciapaglia:2015ixa ; Angelescu:2015kga ; Panizzi:2014dwa ; Panizzi:2014tya ;
Cacciapaglia:2012dd ; Okada:2012gy ; Cacciapaglia:2011fx ; delAguila:1989rq ).

A VLQ model typically introduces four new states: $T$, $B$, $X$ and $Y$, their
electric charges being $+\frac{2}{3}$, $-\frac{1}{3}$, $+\frac{5}{3}$ and
$-\frac{4}{3}$, respectively. In such kind of model, VLQs can be categorized
into three types: singlets $(T)$, $(B)$, doublets $(X,T)$, $(T,B)$, $(B,Y)$
and triplets $(X,T,B)$, $(T,B,Y)$. Notably, the $Y$ quark cannot exist as a
singlet. Further, it is expected to decay with a 100% Branching Ratio (BR)
into a $b$ quark and $W$ boson when $Y$ is lighter than the other VLQs,
whether in a doublet or triplet.

In this study, we will focus on the observability of single $Y$ production at
the Large Hadron Collider (LHC) Run-III, the High-Luminosity LHC (HL-LHC)
Gianotti:2002xx ; Apollinari:2017lan , the High-Energy LHC (HE-LHC)
FCC:2018bvk  and the Future Circular Collider operating in hadron-hadron mode
(FCC-hh) FCC:2018vvp , specifically, within the $(B,Y)$ doublet realisation.

The ATLAS Collaboration conducted a search for a VLQ $Y$ at 13 TeV with an
integrated luminosity of 36.1 fb-1 ATLAS:2018dyh . They found that the upper
limits on the mixing angle are as small as $\left|\sin{\theta_{R}}\right|$ =
0.17 for a $Y$ quark with a mass of 800 GeV in the $(B,Y)$ doublet model, and
$\left|\sin{\theta_{L}}\right|$ = 0.16 for a $Y$ quark with a mass of 800 GeV
in the $(T,B,Y)$ triplet model. The CMS Collaboration also conducted a search
for $Y$ states in the $Wb$ channel at 13 TeV using 2.3 fb-1 of data
CMS:2017fpk . They searched for final states involving one electron or muon,
at least one $b$-tagged jet with large transverse momentum, at least one jet
in the forward region of the detector plus (sizeable) missing transverse
momentum. Their findings indicate that the observed (expected) lower mass
limits are 1.40 (1.0) TeV for a VLQ $Y$ with a coupling value of 0.5 and a
BR($Y\to W^{-}b$) = 1. The ATLAS Collaboration recently presented a search for
the pair-production of VLQ $T$ in the lepton+jets final state using 140 fb-1
at 13 TeV ATLAS:2023shy . They pointed out that the most stringent limits are
set for the scenario BR($T\to W^{+}b$)$=1$, for which $T$ masses below 1700
GeV (1570 GeV) are observed (expected) to be excluded at 95% Confidence Level
(CL). And the limits can also apply to a VLQ $Y$ with BR($Y\to W^{-}b$)$=1$.
All such limits stem from VLQ pair production, induced by Quantum Chromo-
Dynamics (QCD).

Furthermore, there are comparable exclusion limits on the mixing parameter
$\sin{\theta_{R}}$ from EW Precision Observables (EWPOs), for example within
the $(B,Y)$ doublet model, Ref. 9  found that the upper limits on
$\sin{\theta_{R}}$ are approximately 0.21 and 0.15 at $m_{Y}=1000\text{ GeV}$
and 2000 GeV respectively at 95% CL from the oblique parameters $S$ and $T$.
Ref. Cao:2022mif  highlighted that, considering the $W$ boson mass measurement
by the CDF collaboration CDF:2022hxs , the $2\sigma$ bounds on
$\sin{\theta_{R}}$ from the oblique parameters $S,T$ and $U$ are approximately
$[0.15,0.23]$ and $[0.09,0.13]$ at $m_{Y}=1000\text{ GeV}$ and 3000 GeV in a
conservative average scenario, respectively. They also pointed out that the
constraints from the $Zb\bar{b}$ coupling are weaker than those from the EWPOs
for about $m_{Y}>1600\text{ GeV}$.

The single production of a VLQ is instead model dependent, as the couplings
involved are EW ones, yet they may make a significant contribution to the
total VLQ production cross section, compared to the pair production, due to
less phase space suppression, in the region of high VLQ masses. In this work,
we will in particular focus on the process $pp\to Y(\to W^{-}b)\bar{b}j\to
l^{-}\bar{\nu}_{l}b\bar{b}j$ (with $l^{-}$ standing for electron or muon and
$j$ standing for first two-generation quark jets), combined with its charged
conjugated process $pp\to\bar{Y}bj$. We expect that the forthcoming results
will provide complementary information to the one provided by VLQ pair
production in the quest to detect a doublet $Y$ quark at the aforementioned
future colliders.

The paper is structured as follows. In Section II, we introduce the simplified
VLQ model used in our simulations. In Section III, we analyze the properties
of the signal process and SM backgrounds. Subsequently, we conduct simulations
and calculate the $Y$ state exclusion and discovery capabilities at the HL-
LHC, HE-LHC and FCC-hh. Finally, in Section IV, we provide a summary. (We also
have an Appendix where we map the $Y$ state of our simplified model onto the
$(B,Y)$ doublet representation.)

##  II Doublet $Y$ VLQ in a simplified model

As mentioned, in a generic VLQ model, one can include four types of states
called $T$, $B$, $X$ and $Y$, with electric charges $+\frac{2}{3}$,
$-\frac{1}{3}$, $+\frac{5}{3}$ and $-\frac{4}{3}$, respectively. Under the SM
gauge group, $SU(3)$C $\times$ $SU(2)$L $\times$ $U(1)$Y, there are seven
possible representations of VLQs as shown in Table 1.

| $T$ | $B$ | $(T,B)$ | $(B,Y)$ | $(X,T)$ | $(T,B,Y)$ | $(X,T,B)$
---|---|---|---|---|---|---|---
$SU(3)_{C}$ |  3 | 3 | 3 | 3 | 3 | 3 | 3
$SU(2)_{L}$ |  1 | 1 | 2 | 2 | 2 | 3 | 3
$U(1)_{Y}$ |  $\frac{2}{3}$ | $-\frac{1}{3}$ | $\frac{1}{6}$ | $-\frac{5}{6}$ | $\frac{7}{6}$ | $-\frac{1}{3}$ | $\frac{2}{3}$
Table 1: Representations of VLQs and their quantum numbers under the SM gauge
group.

These representations allow for couplings between VLQs and SM gauge bosons and
quarks. The kinetic and mass terms of the VLQs are described as Cao:2022mif ,

$\mathcal{L}=\sum_{F}\bar{F}(i\not{D}-M_{F})F$ (1)

where $F=\left\\{U,D,Q_{1},Q_{5},Q_{7},T_{1},T_{2}\right\\}$,
$D_{\mu}=\partial_{\mu}+ig_{1}Y_{F}B_{\mu}+ig_{2}S^{I}W_{\mu}^{I}+ig_{s}T^{A}G^{A}_{\mu}$,
$\lambda^{A}$($A=1,2,\cdots,8$) and $\tau^{I}$($I=1,2,3$), related to the
Gell-Mann and Pauli matrices via $T^{A}=\frac{1}{2}\lambda^{A}$ and
$S^{I}=\frac{1}{2}\tau^{I}$, respectively. In our simplified model, we use an
effective Lagrangian framework for the interactions of a VLQ $Y$ with the SM
quarks through $W$ boson exchange, including as $Y$ free parameters
$\kappa^{i,L/R}_{Y}$ (couplings) and $m_{Y}$ (mass) Buchkremer:2013bha :

$\mathcal{L}=\left\\{\kappa_{Y}^{i,L/R}\sqrt{\frac{\zeta_{i}}{\Gamma_{W}^{0}}}\frac{g}{\sqrt{2}}\left[\bar{Y}_{L/R}W_{\mu}^{-}\gamma^{\mu}d^{i}_{L/R}\right]+\text{H.c.}\right\\}+m_{Y}\bar{Y}Y,$
(2)

where $d^{i}_{L/R}$($i=1,2,3$) represent the three types of quarks in the SM
while $L$ and $R$ stand for the left-handed and right-handed chiralities,
respectively. We assume that the $Y$ only couples to the third generation
quarks of the SM, that is, $Y$ decays 100% into $Wb$ and therefore
$\zeta_{1}=\zeta_{2}=0,\zeta_{3}=1$. Considering that the $Y$ mass is much
greater than any SM quark mass ($m_{q}$), that is, $m_{Y}\gg m_{q}$, the
kinematic function can be approximated as $\Gamma^{0}_{W}=1$
Buchkremer:2013bha , so that the above Lagrangian can be simplified as

$\mathcal{L}=\left\\{\frac{g\kappa_{Y}^{3,L/R}}{\sqrt{2}}\left[\bar{Y}_{L/R}W_{\mu}^{-}\gamma^{\mu}b_{L/R}\right]+\text{H.c.}\right\\}+m_{Y}\bar{Y}Y,$
(3)

where $g$ is the EW coupling constant. Comparing the Lagrangian for the
$(B,Y)$ doublet and $(T,B,Y)$ triplet, we observe that the relationship
between the coupling $\kappa_{Y}^{3,L/R}$ and mixing angle $\theta^{L/R}$ is
$\sin\theta^{L/R}=\kappa_{Y}^{3,L/R}$ for the doublet and
$\sin\theta^{L/R}=\sqrt{2}\kappa_{Y}^{3,L/R}$ for the triplet, with details to
be found in Appendix A. Taking into account the relationship
$\tan\theta^{L}=\frac{m_{b}}{m_{B}}\tan\theta^{R}$ and
$\tan\theta^{R}=\frac{m_{b}}{m_{B}}\tan\theta^{L}$ as well as the condition
$m_{B}\gg m_{b}$, we can assume $\kappa_{Y}^{3,L}=0$ for the doublet and
$\kappa_{Y}^{3,R}=0$ for the triplet. (In the subsequent content, we will use
$\kappa_{Y}$ to denote $\kappa_{Y}^{3,R}$ for the sake of simplicity.) The
decay width of the VLQ $Y$ can be expressed as Cetinkaya:2020yjf ,

$\Gamma(Y\to
Wq)=\frac{\alpha_{e}\kappa^{2}_{Y}}{16\sin^{2}\theta_{W}}\frac{(m^{2}_{W}-m^{2}_{Y})^{2}(2m^{2}_{W}+m^{2}_{Y})}{m^{2}_{W}m^{3}_{Y}},$
(4)

where $\alpha_{\rm EM}=\frac{{g^{\prime}}^{2}}{4\pi}$, $g^{\prime}$ is the
Electro-Magnetic (EM) coupling constant and $\theta_{W}$ the EW mixing angle.
In this paper, we solely focus on the Narrow Width Approximation (NWA), which
we use for the purpose of simplifying scattering amplitude calculations.
However, it is worth noting that several studies Carvalho:2018jkq ;
Berdine:2007uv ; Moretti:2016gkr ; Deandrea:2021vje  have highlighted the
limitations of the NWA in scenarios involving new physics with VLQs.
Specifically, it becomes imperative to consider a finite width when this
becomes larger than $\alpha_{\rm EM}\approx 1\%$, given the substantial
interference effects emerging between VLQ production and decay channels,
coupled with their interactions with the corresponding irreducible
backgrounds. To address the limitations of our approach then, we will also
present the ratio $\Gamma_{Y}/m_{Y}$ in our subsequent results and we
emphasise since now that, crucially, for the region where
$\Gamma_{Y}/m_{Y}>1\%$, our sensitivities may be under- or over-estimated, as
such interferences could be positive or negative, respectively. Also, before
starting with our numerical analysis, we remind the reader that one can apply
the results of our forthcoming simulations to a specific VLQ representation,
such as, e.g., $(B,Y)$ or $(T,B,Y)$, by utilizing the aforementioned
relationships.

Figure 1: Representative Feynman diagram of single $Y$ (in red) production
followed by its subsequent decay $Y\to W^{-}(\to l^{-}\bar{\nu}_{l})b$. Here,
$q$ in the initial state represents one of the first two-generation quarks and
bottom quark, $j$ in the final state represents one of the first two-
generation jets, $b$ in the intermediate (final) state represents a $b$-quark
(jet) while $l^{-}$ represents either an electron or muon. Notice that, since
we use the five flavor scheme, the $g\to b\bar{b}$ splitting in the diagram is
actually accounted for through the PDF evolution.

In Figure 1, we show a representative Feynman diagram of the signal production
$pp\to Y\bar{b}j$ and decay chain $Y\to W^{-}(\to l^{-}\bar{\nu}_{l})b$. We
expect the $W$ boson and the high-momentum $b$-jet to exhibit a back-to-back
alignment in the transverse plane, originating from the decay of the massive
$Y$ quark. The topology also encompasses an outgoing light quark, often
resulting in a forward jet within the detector. Furthermore, the second
$b$-jet arising from the splitting of a gluon into a pair of $b$-quarks can be
observed in either the forward or central region. According to these features
of signal events, the primary SM backgrounds include $pp\to t\bar{b}j$, $pp\to
W^{+}W^{-}b$, $pp\to Zbj$, $pp\to W^{+}bj$, and their charge conjugated
processes. Among them, $pp\to t\bar{b}j$ and $pp\to W^{+}W^{-}b$ are
irreducible backgrounds, while the others are reducible backgrounds. We have
also assessed additional backgrounds, such as $pp\to t\bar{t}$, and found that
their contribution can be ignored based on the selection criteria that will be
discussed later.

The signal production cross section is determined not only by the mass $m_{Y}$
but also by the coupling strength $\kappa_{Y}$. The cross section is directly
proportional to $\kappa_{Y}^{2}$ for a fixed $m_{Y}$ as long as the NWA is met
Moretti:2016gkr . In Figure 2, we show the tree-level cross sections for
single $Y$ production as a function of the mass $m_{Y}$. We can see that, as
$m_{Y}$ increases, the cross section gradually decreases due to a smaller
phase space.

Figure 2: The tree-level cross sections for single $Y$ production as a
function of the mass $m_{Y}$ for various values of the coupling $\kappa_{Y}$.
The charge conjugated process has also been taken into account.

In Figure 3, we show the tree-level cross sections for the signal benchmarks
$m_{Y}=1000\text{ GeV}$ (labeled as $Y_{1000}$) and $m_{Y}=1500\text{ GeV}$
(labeled as $Y_{1500}$) with $\kappa_{Y}=0.1$ and $\kappa_{Y}=0.5$ as well as
the tree-level cross sections for the background processes. It is evident that
the rates for the latter are significantly larger than those for the former.
Consequently, we should design efficient selection criteria (in terms of
kinematic cuts) to reduce the number of background events while preserving the
signal events. Furthermore, the cross sections for both signal and backgrounds
increase with increasing collider energy.

Figure 3: The tree-level cross sections as a function of the center-of-mass
energy $\sqrt{s}$ for the signal benchmarks and backgrounds. Solid lines
represent the signal processes and dashed lines represent the background
processes. The cross sections also include the corresponding charge conjugated
processes.

The Next-to-Leading Order (NLO) (or even higher order) QCD corrections for the
SM background cross sections at the LHC have been extensively explored in
Refs. Czakon:2012zr ; Campbell:2005zv ; Campbell:2006cu ; Kidonakis:2018ncr ;
Boos:2012vm . The $K$ factors associated with the background cross sections
adopted in our calculations are summarized in Table 2. (Note that, despite
they change somewhat with energy, we neglect here changes of $K$ factors
values at different colliders, like in Ref. Yang:2022wfa .)

Processes | $Zbj$ | $W^{+}bj$ | $W^{+}W^{-}b$ | $t\bar{b}j$
---|---|---|---|---
$K$ factor | 1.3 Campbell:2005zv  | 1.9 Campbell:2006cu  | 2.1 Campbell:2006cu  | 1.4 Kidonakis:2018ncr ; Boos:2012vm
Table 2: $K$ factors representing the QCD corrections for the background
processes.

There are stringent limits from the oblique parameters $S$, $T$ and $U$ in
EWPOs Hollik:1988ii ; Peskin:1990zt ; Grinstein:1991cd ; Peskin:1991sw ;
Lavoura:1992np ; Burgess:1993mg ; Maksymyk:1993zm ; Cynolter:2008ea ; 9 ;
Chen:2017hak ; Cao:2022mif ; He:2022zjz ; Arsenault:2022xty . These oblique
parameters relate to the weak isospin current $J^{\mu}_{1,2,3}$ and the
electromagnetic current $J^{\mu}_{Q}=J^{\mu}_{3}+J^{\mu}_{Y}$, involving their
vacuum-polarization amplitudes as defined in references Peskin:1990zt ;
Peskin:1991sw :

$\displaystyle S$ $\displaystyle\equiv$
$\displaystyle-\frac{16\pi}{m_{Z}^{2}}\left\\{\Sigma_{33}(m_{Z}^{2})-\Sigma_{33}(0)-\Sigma_{3Q}(m_{Z}^{2})\right\\}$
(5) $\displaystyle=$
$\displaystyle\frac{16\pi}{m_{Z}^{2}}\left\\{\Sigma_{3Y}(m_{Z}^{2})-\Sigma_{3Y}(0)\right\\},$
$\displaystyle T$ $\displaystyle\equiv$
$\displaystyle\frac{4\pi}{\sin^{2}\theta_{W}\cos^{2}\theta_{W}m_{Z}^{2}}\left\\{\Sigma_{33}(0)-\Sigma_{11}(0)\right\\},$
(6) $\displaystyle U$ $\displaystyle\equiv$
$\displaystyle\frac{16\pi}{m_{Z}^{2}}\left\\{\Sigma_{33}(m_{Z}^{2})-\Sigma_{33}(0)\right\\}-\frac{16\pi}{m_{W}^{2}}\left\\{\Sigma_{11}(m_{Z}^{2})-\Sigma_{11}(0)\right\\},$
(7)

where $m_{W}$ and $m_{Z}$ denote the mass for $W$ and $Z$ boson, respectively.
The $Z$-boson current, represented by
$e(J^{\mu}_{3}-s_{W}^{2}J^{\mu}_{Q})/(\sin\theta_{W}\cos\theta_{W})$, involves
$e$ linked to the fine-structure constant $\alpha$ through $e^{2}\equiv
4\pi\alpha$. Consequently, the oblique parameters can be reformulated using
the vacuum polarizations of the SM gauge bosons as:

$\displaystyle\alpha T$ $\displaystyle=$ $\displaystyle\frac{\Sigma_{ZZ}^{\rm
new}\left(0\right)}{m_{Z}^{2}}-\frac{\Sigma_{WW}^{\rm
new}\left(0\right)}{m_{W}^{2}}$ (8)
$\displaystyle\frac{\alpha}{\sin^{2}2\theta_{W}}\,S$ $\displaystyle=$
$\displaystyle-\frac{\Sigma_{ZZ}^{\rm
new}\left(m_{Z}^{2}\right)-\Sigma_{ZZ}^{\rm
new}\left(0\right)}{m_{Z}^{2}}+\left.\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}+\frac{\cos
2\theta_{W}}{\cos\theta_{W}\sin\theta_{W}}\left.\frac{\partial\Sigma_{\gamma
Z}^{\rm new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}$ (9)
$\displaystyle\simeq$ $\displaystyle\left.-\frac{\partial\Sigma_{ZZ}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}\right|_{p^{2}=0}+\left.\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}+\frac{\cos
2\theta_{W}}{\cos\theta_{W}\sin\theta_{W}}\left.\frac{\partial\Sigma_{\gamma
Z}^{\rm new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}$
$\displaystyle\frac{\alpha}{4\sin^{2}\theta_{W}}\,U$ $\displaystyle=$
$\displaystyle-\frac{\Sigma_{WW}^{\rm
new}\left(m_{W}^{2}\right)-\Sigma_{WW}^{\rm
new}\left(0\right)}{m_{W}^{2}}+\cos^{2}\theta_{W}\,\frac{\Sigma_{ZZ}^{\rm
new}\left(m_{Z}^{2}\right)-\Sigma_{ZZ}^{\rm new}\left(0\right)}{m_{Z}^{2}}$
(10)
$\displaystyle+\sin^{2}\theta_{W}\left.\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}+\sin
2\theta_{W}\left.\frac{\partial\Sigma_{\gamma Z}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}\right|_{p^{2}=0}$
$\displaystyle\simeq$ $\displaystyle-\frac{\partial\Sigma_{WW}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}|_{p^{2}=0}+\cos^{2}\theta_{W}\frac{\partial\Sigma_{ZZ}^{\rm
new}\left(p^{2}\right)}{\partial
p^{2}}|_{p^{2}=0}+\sin^{2}\theta_{W}\frac{\partial\Sigma_{\gamma\gamma}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}|_{p^{2}=0}$ $\displaystyle+\sin
2\theta_{W}\frac{\partial\Sigma_{\gamma Z}^{\rm
new}\left(p^{2}\right)}{\partial p^{2}}|_{p^{2}=0}.$

The contributions in the doublet $(B,Y)$ model to these oblique parameters can
be approximated as follows Cao:2022mif :

$S\simeq\frac{1}{2\pi}\left\\{-\frac{2}{3}\kappa_{Y}^{2}\ln\frac{\mathcal{M}^{2}}{m_{b}^{2}}+\frac{11}{3}\kappa_{Y}^{2}\right\\},\,U\simeq-\frac{\kappa_{Y}^{2}}{2\pi},\,T\simeq\frac{3m_{t}^{2}}{8\pi\sin^{2}\theta_{W}m_{W}^{2}}\kappa_{Y}^{4}\frac{2\mathcal{M}^{2}}{3m_{t}^{2}}$
(11)

Here, $\mathcal{M}^{2}=(m_{Y}^{2}-m_{b}^{2}\kappa_{Y}^{2})/(1-\kappa_{Y}^{2})$
and $m_{W}=m_{Z}\cos\theta_{W}$. For the numerical calculation, the $\chi^{2}$
function for the oblique parameter fit should be less than 8.02 for three
degrees of freedom to compute the $2\sigma$ limits, respectively.
${S}=-0.02\pm 0.1$, ${T}=0.03\pm 0.12$, ${U}=0.01\pm 0.11$; there exists a
strong correlation of 92% between the $S$ and $T$ parameters, while the U
parameter exhibits an anti-correlation of -80% (-93%) with $S$ ($T$)
ParticleDataGroup:2022pth . Specific numerical values of the input parameters
are detailed in Eq. 12.

##  III Signal to background analysis

The signal model file is sourced from FeynRules feynruls  and parton-level
events are generated using MadGraph5_aMC$@$NLO Alwall:2014hca  with the
NNPDF23LO1 NNPDF  Parton Distribution Function (PDFs). Dynamic factorization
and renormalization scales, set as default in MadEvent website_factor , are
utilized. Subsequently, fast detector simulations are conducted using Delphes
3.4.2 deFavereau:2013fsa  with the built-in detector configurations of the LHC
Run-III, HL-LHC, HE-LHC website_hllhc  and FCC-hh website_fcchh . Jets are
clustered by FastJet Cacciari:2011ma  employing the anti-$kt$ algorithm
Cacciari:2005hq  with a distance parameter of $\Delta R=0.4$. Furthermore,
MadAnalysis 5 Conte:2012fm  is used to analyze both signal and background
events. Finally, the EasyScan_HEP package Shang:2023gfy  is utilized to
connect these programs and scan the VLQ parameter space.

The numerical values of the input SM parameters are taken as follows
ParticleDataGroup:2022pth :

$\displaystyle m_{b}=4.18{\rm~{}GeV},\quad m_{t}=172.69{\rm~{}GeV},\quad
m_{Z}=91.1876{\rm~{}GeV},$
$\displaystyle\sin^{2}\theta_{W}=0.22339,\quad\alpha(m_{Z})=\frac{1}{127.951},\quad\alpha_{s}(m_{Z})=0.1179.$
(12)

Considering the general detection capabilities of detectors, the following
basic cuts are chosen:

$\displaystyle\Delta R(x,y)>0.4$ $\displaystyle(x,y=l,j,b),$ $\displaystyle
p^{l}_{T}>25\ {\rm GeV},$ $\displaystyle\ \ |\eta_{l}|<2.5,$ $\displaystyle
p^{j}_{T}>20\ {\rm GeV},$ $\displaystyle\ \ |\eta_{j}|<5.0,$ $\displaystyle
p^{b}_{T}>25\ {\rm GeV},$ $\displaystyle\ \ |\eta_{b}|<2.5,$

where $\Delta R=\sqrt{\Delta\Phi^{2}+\Delta\eta^{2}}$ denotes the separation
in the rapidity($\eta$)–azimuth($\phi$) plane.

To handle the relatively small event number of signal ($s$) and background
($b$) events, we will use the median signiﬁcance $\mathcal{Z}$ to estimate the
expected discovery and exclusion reaches Cowan:2010js ; Kumar:2015tna ,

$\displaystyle\mathcal{Z}_{excl}=\sqrt{2\left[s-b\ln\left(\frac{b+s+x}{2b}\right)-\frac{1}{\delta^{2}}\ln\left(\frac{b-s+x}{2b}\right)\right]-(b+s-x)\left(1+\frac{1}{\delta^{2}b}\right)},$
(13)

$\displaystyle\mathcal{Z}_{disc}=\sqrt{2\left[(s+b)\ln\left(\frac{(s+b)(1+\delta^{2}b}{b+(s+b)\delta^{2}b}\right)-\frac{1}{\delta^{2}}\ln\left(1+\frac{\delta^{2}s}{1+\delta^{2}b}\right)\right]},$
(14) $x=\sqrt{(s+b)^{2}-\frac{4\delta^{2}sb^{2}}{1+\delta^{2}b}},$ (15)

where $\delta$ is the uncertainty that inevitably appears in the measurement
of the background. In the completely ideal case, that is $\delta$=0, Eq. (13)
and (14) can be simplified as follows, respectively:

$\mathcal{Z}_{excl}=\sqrt{2\left[s-b\ln\left(1+\frac{s}{b}\right)\right]},$
(16)

and

$\mathcal{Z}_{disc}=\sqrt{2\left[(s+b)\ln\left(1+\frac{s}{b}\right)-s\right]}.$
(17)

###  III.1 LHC Run-III and HL-LHC

Firstly, we establish a trigger that emulates the LHC Run-III and HL-LHC
detector response based on the count of final state particles detected in each
event. Given the limited efficiency of the detector in identifying jets, we
adopt a lenient approach towards the number of jets. Consequently, the final
trigger criteria are defined as follows: $N_{l}=1$, $N_{j}\geq 2$, $N_{j}\leq
4$ and $N_{b}\geq 2$.

Considering that the mass of $Y$ is notably greater than that of its decay
products, the latter exhibit distinct spatial characteristics in
pseudorapidity $\eta$ and spatial separation $\Delta R$ compared to
backgrounds. These differences inform our selection criteria.

Cuts |  $Y_{1500}$ (fb) |  $Y_{1800}$ (fb) |  $t\bar{b}j$ (fb) |  $W^{+}bj$ (fb) |  $W^{+}W^{-}b$ (fb) |  $Zbj$ (fb)
---|---|---|---|---|---|---
Basic Cuts | 1.99 | 0.97 | 13855.00 | 15016.00 | 18967.00 | 13897.00
Trigger | 0.29 | 0.13 | 2227.40 | 775.10 | 1251.50 | 312.80
Cut 1 | 0.25 | 0.12 | 40.09 | 11.95 | 39.12 | 2.63
Cut 2 | 0.23 | 0.11 | 7.46 | 4.07 | 8.07 | 0.63
Cut 3 | 0.16 | 0.08 | 4.51 | 3.02 | 4.93 | 0.39
Cut 4 | 0.08 | 0.05 | 0.08 | 1.35 | 1.18 | 0.15
Cut 5 | 0.08 | 0.04 | 0.08 | 1.00 | 0.89 | 0.13
Cut 6 | 0.04 | 0.03 | 0.01 | 0.02 | 0.05 | 0.00
Cut 7 | 0.03 | 0.02 | 0.01 | 0.00 | 0.03 | 0.00
Table 3:  Cut flows of the signal with $\kappa_{Y}=0.1$ and backgrounds at the
14 TeV HL-LHC, where the conjugate processes $pp\to\bar{t}bj$,
$W^{-}\bar{b}j$, $W^{+}W^{-}\bar{b}$, $Z\bar{b}j$ have been included.

Furthermore, since the mass range of $Y$ is much heavier than the particles
originating from background processes, we anticipate that the transverse
momentum (referred to as $\vec{p}_{T}$ and its magnitude denoted as $p_{T}$)
of decay products of the $Y$ state will be substantially larger than those of
the same particles from background processes. Besides, we will also consider
variables such as $\not{E}_{T}$, $\not{H}_{T}$ and $M_{T}$ to distinguish the
signal from the background. Here, $\not{E}_{T}$ represents the magnitude of
the sum of the transverse momenta of all visible final state particles,
$\not{H}_{T}$ is analogous to $\not{E}_{T}$ but only considers all visible
hadronic momenta while the transverse mass $M_{T}$ is defined as follows:

$\displaystyle M_{T}^{2}$
$\displaystyle\equiv[E_{T}(1)+E_{T}(2)]^{2}-[\vec{p}_{T}(1)+\vec{p}_{T}(2)]^{2}$
$\displaystyle=m_{1}^{2}+m_{2}^{2}+2[E_{T}(1)E_{T}(2)-\vec{p}_{T}(1)\cdot\vec{p}_{T}(2)],$

where $E_{T}(i)=\sqrt{p^{2}_{T}(i)+m^{2}_{i}}$ and $m^{2}_{i}=p_{i}^{2}$ with
$p_{i}$ representing a 4-vector.

In Figure 4, we present the normalized distributions of $p_{T}^{j_{1}}$,
$M_{b_{1}l_{1}}$, $M_{j_{1}j_{2}}$, $M_{T}^{b_{2}l_{1}}$,
$M_{T}^{b_{1}l_{1}}$, $\Delta R_{j_{1},b_{1}}$, $\not{H}_{T}$ and
$\not{E}_{T}$ for both $m_{Y}=1500\text{ GeV}$ and $m_{Y}=1800\text{ GeV}$
with $\kappa_{Y}=0.1$ as well as for the background processes. Based on these
distributions, we have devised the following selection criteria to distinguish
the signal from the various backgrounds555The subscript on the particle symbol
is arranged according to the magnitude of the particle transverse momentum:
e.g., in the case of $b$-jets, $p_{T}^{b_{1}}$ is greater than
$p_{T}^{b_{2}}$.:

  * •

Trigger: $N_{l}=1$, $N_{j}\geq 2$, $N_{j}\leq 4$, and $N_{b}\geq 2$;

  * •

Cut-1: $p_{T}^{j_{1}}>300\text{ GeV}$;

  * •

Cut-2: $M_{b_{1}l_{1}}>500\text{ GeV}$;

  * •

Cut-3: $M_{j_{1}j_{2}}>500\text{ GeV}$;

  * •

Cut-4: $M_{T}^{b_{1}l_{1}}>200\text{ GeV}$ and $M_{T}^{b_{2}l_{1}}>200\text{
GeV}$;

  * •

Cut-5: $\Delta R_{j_{1},b_{1}}<1.0$;

  * •

Cut-6: $\not{H}_{T}>600\text{ GeV}$;

  * •

Cut-7: $\not{E}_{T}>200\text{ GeV}$.

By applying these cuts, we can see that the signal efficiencies for
$m_{Y}=1500\text{ GeV}$ and $m_{Y}=1800\text{ GeV}$ are 1.35% and 2.41%,
respectively. The higher efficiency for the latter can be attributed to the
larger transverse boost of the final state originating from an heavier $Y$.
Meanwhile, the background processes are significantly suppressed. For
reference, we provide the cut flows in Table 3.

We present the exclusion capability ($\mathcal{Z}_{\text{excl}}=2$) and
discovery potential ($\mathcal{Z}_{\text{disc}}=5$) for $Y$ with two different
integrated luminosities, 1000 fb-1 and 3000 fb-1, at the HL-LHC, as shown in
the top line of Figure 7. This analysis considers both the ideal scenario
without systematic uncertainties and the case with a 10% systematic
uncertainty. In the presence of 10% systematic uncertainty, the $Y$ can be
excluded in the correlated parameter space of $\kappa_{Y}\in[0.044,0.5]$ and
$m_{Y}\in[1000\text{ GeV},3099\text{ GeV}]$ with an integrated luminosity of
$L=300\text{ fb}^{-1}$, which corresponds to the maximum achievable integrated
luminosity during LHC Run-III. If the integrated luminosity is raised to 3000
fb-1, aligning with the maximum achievable at the HL-LHC, the excluded
parameter zones extend to $\kappa_{Y}\in[0.027,0.5]$ and $m_{Y}\in[1000\text{
GeV},3653\text{ GeV}]$. Furthermore, the discovery regions are
$\kappa_{Y}\in[0.072,0.5]$ ([0.047, 0.5]) and $m_{Y}\in[1000\text{
GeV},2621\text{ GeV}]$ ($[1000\text{ GeV},3047\text{ GeV}]$) with $L=300\text{
fb}^{-1}$ (3000 fb-1).

Figure 4:  Normalized distributions for the signals of $m_{Y}=$ 1500 GeV and
1800 GeV and SM backgrounds at the HL-LHC. The conjugated processes have been
included.

###  III.2 27 TeV HE-LHC

Cuts |  $Y_{1500}$ (fb) |  $Y_{1800}$ (fb) |  $t\bar{b}j$ (fb) |  $W^{+}bj$ (fb) |  $W^{+}W^{-}b$ (fb) |  $Zbj$ (fb)
---|---|---|---|---|---|---
Basic Cuts | 16.86 | 10.01 | 41398.00 | 38670.00 | 69303.00 | 69658.00
Trigger | 1.78 | 0.10 | 6224.50 | 2149.10 | 4445.40 | 1700.70
Cut 1 | 1.50 | 0.91 | 86.07 | 29.51 | 133.60 | 10.73
Cut 2 | 1.36 | 0.85 | 18.30 | 11.14 | 29.52 | 2.37
Cut 3 | 0.95 | 0.62 | 12.83 | 9.05 | 19.27 | 1.53
Cut 4 | 0.35 | 0.27 | 0.17 | 2.94 | 3.10 | 0.35
Cut 5 | 0.33 | 0.25 | 0.17 | 2.01 | 2.36 | 0.28
Cut 6 | 0.12 | 0.16 | 0.00 | 0.04 | 0.37 | 0.00
Cut 7 | 0.09 | 0.12 | 0.00 | 0.04 | 0.14 | 0.00
Table 4:  Cut flows of the signal with $\kappa_{Y}=0.1$ and backgrounds at the
27 TeV HE-LHC.

Figure 5:  Normalized distributions for the signals with $m_{Y}=$ 1500 GeV and
1800 GeV and backgrounds at the HE-LHC.

This section delves into the prospective signal of $Y$ at the future 27 TeV
HE-LHC. In Figure 5, we exhibit the normalized distributions for both signal
and background processes, forming the basis for our distinctive selection
criteria:

  * •

Trigger: $N_{l}=1$, $N_{j}\geq 2$, $N_{j}\leq 4$, and $N_{b}\geq 2$;

  * •

Cut-1: $p_{T}^{j_{1}}>350\text{ GeV}$;

  * •

Cut-2: $M_{b_{1}l_{1}}>550\text{ GeV}$;

  * •

Cut-3: $M_{j_{1}j_{2}}>550\text{ GeV}$;

  * •

Cut-4: $M_{T}^{b_{2}l_{1}}>250\text{ GeV}$ and $M_{T}^{b_{1}l_{1}}>250\text{
GeV}$;

  * •

Cut-5: $\Delta R_{j_{1},b_{1}}<0.5$;

  * •

Cut-6: $\not{H}_{T}>650\text{ GeV}$;

  * •

Cut-7: $\not{E}_{T}>200\text{ GeV}$.

The kinematic variables remain consistent with those of the 14 TeV case, but
the cut threshold values for transverse momentum-based variables, such as
$\not{H}_{T}>650\text{ GeV}$, are higher than those in the 14 TeV case. This
adjustment accounts for the increased center-of-mass energy. Detailed cut
flows are outlined in Table 4 and the exclusion capability and discovery
potential are shown in the second row of Figure 7. The $Y$ quark can be
excluded within the correlated parameter space of $\kappa_{Y}\in[0.033,0.5]$
and $m_{Y}\in[1000\text{ GeV},4783\text{ GeV}]$ with 10% systematic
uncertainty for $L=1000\text{ fb}^{-1}$. If the integrated luminosity is
raised to the highest designed value 10 ab-1, the excluded parameter regions
can be extended to $\kappa_{Y}\in[0.029,0.5]$ and $m_{Y}\in[1000\text{
GeV},4987\text{ GeV}]$. For $L=3000\text{ fb}^{-1}$, the discovery regions are
$\kappa_{Y}\in[0.053,0.5]$ and $m_{Y}\in[1000\text{ GeV},3885\text{ GeV}]$. If
the integrated luminosity is raised to the highest designed value 10 ab-1, the
discovery parameter regions can be extended to $\kappa_{Y}\in[0.051,0.5]$ and
$m_{Y}\in[1000\text{ GeV},3943\text{ GeV}]$.

###  III.3 100 TeV FCC-hh

Cuts |  $Y_{1500}$ (fb) |  $Y_{1800}$ (fb) |  $t\bar{b}j$ (fb) |  $W^{+}bj$ (fb) |  $W^{+}W^{-}b$ (fb) |  $Zbj$ (fb)
---|---|---|---|---|---|---
Basic Cuts | 261.26 | 183.18 | 237538.00 | 206093.00 | 573258.00 | 291603.00
Trigger | 13.44 | 8.42 | 33633.00 | 17209.00 | 40939.00 | 6363.90
Cut 1 | 6.37 | 4.20 | 209.30 | 112.70 | 605.90 | 18.95
Cut 2 | 5.63 | 3.89 | 54.16 | 48.43 | 163.30 | 7.58
Cut 3 | 3.30 | 2.51 | 3.33 | 23.91 | 53.74 | 3.21
Cut 4 | 3.14 | 2.43 | 3.33 | 17.72 | 45.12 | 3.21
Cut 5 | 1.40 | 1.70 | 0.48 | 1.65 | 6.15 | 0.00
Cut 6 | 0.81 | 1.16 | 0.24 | 0.21 | 2.87 | 0.00
Table 5:  Cut flows of the signal with $\kappa_{Y}=0.1$ and backgrounds at the
100 TeV FCC-hh.

Figure 6:  Normalized distributions for the signals with $m_{Y}=$ 1500 GeV and
1800 GeV, and backgrounds at the FCC-hh.

Here, we explore the anticipated signal of ${Y}$ in the context of the future
100 TeV FCC-hh. The figures in Figure 6 portray normalized distributions for
both signal and background processes, laying the groundwork for our
distinctive selection criteria:

  * •

Trigger: $N_{l}=1$, $N_{j}\geq 2$, $N_{j}\leq 4$, and $N_{b}\geq 2$;

  * •

Cut-1: $p_{T}^{j_{1}}>350\text{ GeV}$, $|\eta_{j_{1}}|<1$;

  * •

Cut-2: $M_{b_{1},l_{1}}>550\text{ GeV}$;

  * •

Cut-3: $M_{T}^{b_{2}l_{1}}>150\text{ GeV}$ and $M_{T}^{b_{1}l_{1}}>250\text{
GeV}$;

  * •

Cut-4: $\Delta R_{j_{1},b_{1}}<0.5\text{ GeV}$;

  * •

Cut-5: $\not{H}_{T}>650\text{ GeV}$;

  * •

Cut-6: $\not{E}_{T}>300\text{ GeV}$.

Compared to previous cases, an additional variable, $\eta_{j_{1}}$, is
introduced here. Upon analyzing the distributions of $\eta_{j_{1}}$, it is
apparent that the signal tends to be more central than the backgrounds. Thus,
we require $|\eta_{j_{1}}|<1$. The signal efficiencies for $m_{Y}=1500\text{
GeV}$ and $m_{Y}=1800\text{ GeV}$ are 0.20% and 0.45%, respectively. Notably,
there is a significant suppression in the background processes. Comprehensive
cut flows are provided in Table 5. The exclusion capability and discovery
potential are illustrated in the final row of Figure 7. It is evident that
systematic uncertainty has a considerable impact on the results. Even with a
10% systematic uncertainty, the parameter space region will significantly
shrink. Accounting for the 10% systematic uncertainty, the $Y$ quark can be
excluded within the correlated parameter space of $\kappa_{Y}\in[0.051,0.5]$
and $m_{Y}\in[1000\text{ GeV},6610\text{ GeV}]$ at the highest design value of
luminosity, $L=30\text{ ab}^{-1}$. Additionally, the $Y$ state can be
discovered within $\kappa_{Y}\in[0.088,0.5]$ and $m_{Y}\in[1000\text{
GeV},4624\text{ GeV}]$ at $L=30\text{ ab}^{-1}$.

Figure 7: The exclusion capability ($\mathcal{Z}_{\text{excl}}=2$) and
discovery potential ($\mathcal{Z}_{\text{disc}}=5$) for the $Y$ state at the
LHC Run-III and HL-LHC, $\sqrt{s}=27\text{ TeV}$ HE-LHC and
$\sqrt{s}=100\text{ TeV}$ FCC-hh. Solid lines represent the ideal scenario
without systematic uncertainty, the dotted lines represent the scenario with a
10% systematic uncertainty. Dashed lines denote the contours of
$\Gamma_{Y}/m_{Y}$. The blue (grey) shaded area indicates the exclusion region
of the current LHC at $\sqrt{s}=$ 13 TeV with $L=$36.1 fb-1 (140 fb-1), as
reported in Ref. ATLAS:2018dyh  (Ref. ATLAS:2023shy ). Meanwhile, the yellow
shaded area denotes the allowed region for the oblique parameters $S,T$ and
$U$, considering the current measurements in Ref. ParticleDataGroup:2022pth .

##  IV Summary

Colliders | $L/\text{fb}^{-1}$ | Uncertainty | Exclusion | Discovery
---|---|---|---|---
|  |  | $\kappa_{Y}$ |  $m_{Y}$(GeV) | $\kappa_{Y}$ |  $m_{Y}$(GeV)
LHC Run-III | 300 | 0 | [0.043,0.5] | [1000,3111] | [0.069,0.5] | [1000,2665]
300 | 10% | [0.044,0.5] | [1000,3099] | [0.072,0.5] | [1000,2621]
14 TeV HL-LHC | 1000 | 0 | [0.031,0.5] | [1000,3486] | [0.049,0.5] | [1000,2988]
3000 | 0 | [0.023,0.5] | [1000,3820] | [0.037,0.5] | [1000,3267]
1000 | 10% | [0.033,0.5] | [1000,3398] | [0.055,0.5] | [1000,2880]
3000 | 10% | [0.027,0.5] | [1000,3653] | [0.047,0.5] | [1000,3047]
| 1000 | 0 | [0.026,0.5] | [1000,5213] | [0.042,0.5] | [1000,4359]
27 TeV HE-LHC | 3000 | 0 | [0.020,0.5] | [1000,5811] | [0.031,0.5] | [1000,4863]
10000 | 0 | [0.015,0.5] | [1000,6476] | [0.024,0.5] | [1000,5513]
1000 | 10% | [0.033,0.5] | [1000,4783] | [0.057,0.5] | [1000,3783]
3000 | 10% | [0.030,0.5] | [1000,4936] | [0.053,0.5] | [1000,3885]
| 10000 | 10% | [0.029,0.5] | [1000,4987] | [0.051,0.5] | [1000,3943]
| 1000 | 0 | [0.022,0.5] | [1000,9953] | [0.035,0.5] | [1000,7933]
| 3000 | 0 | [0.016,0.5] | [1000,11259] | [0.026,0.5] | [1000,9000]
100 TeV FCC-hh | 10000 | 0 | [0.014,0.5] | [1000,12254] | [0.021,0.5] | [1000,10425]
30000 | 0 | [0.010,0.5] | [1000,13771] | [0.015,0.5] | [1000,11649]
1000 | 10% | [0.051,0.5] | [1000,6610] | [0.088,0.5] | [1000,4624]
3000 | 10% | [0.051,0.5] | [1000,6610] | [0.088,0.5] | [1000,4624]
| 10000 | 10% | [0.051,0.5] | [1000,6610] | [0.088,0.5] | [1000,4624]
| 30000 | 10% | [0.051,0.5] | [1000,6610] | [0.088,0.5] | [1000,4624]
Table 6: Summary for 2$\sigma$ exclusion limits and 5$\sigma$ signal
discoveries at the LHC Run-III and HL-LHC, $\sqrt{s}=27\text{ TeV}$ HE-LHC and
$\sqrt{s}=100\text{ TeV}$ FCC-hh.

In a simplified model, we have investigated the single production of a doublet
VLQ denoted by $Y$ in the $Wb$ decay channel at the the $\sqrt{s}=14$ TeV HL-
LHC, $\sqrt{s}=27$ TeV HE-LHC and $\sqrt{s}=100$ TeV FCC-hh, following its
production via $pp\to Ybj$, with the $W$ decaying leptonically (into electrons
and muons plus their respective neutrinos). We have performed a detector level
simulation for the signal and relevant SM backgrounds. Considering a
systematic uncertainty of 10% with an integrated luminosity of 3000 fb-1, the
exclusion and discovery capabilities, as displayed in Table VI, can be
described as follows: (1) The HL-LHC can exclude (discover) the correlated
regions of $\kappa_{Y}\in[0.027,0.5]$ $([0.047,0.5])$ and $m_{Y}\in[1000\text{
GeV},3653\text{ GeV}]$ $([1000\text{ GeV},3047\text{ GeV}])$; (2) The HE-LHC
can exclude (discover) the correlated regions of $\kappa_{Y}\in[0.030,0.5]$
$([0.053,0.5])$ and $m_{Y}\in[1000\text{ GeV},4936\text{ GeV}]$ $([1000\text{
GeV},3885\text{ GeV}])$; (3) The FCC-hh can exclude (discover) the correlated
regions of $\kappa_{Y}\in[0.051,0.5]$ $([0.088,0.5])$ and $m_{Y}\in[1000\text{
GeV},6610\text{ GeV}]$ $([1000\text{ GeV},4624\text{ GeV}])$.

Furthermore, we highlight that the stringent constraint on the VLQ $Y$,
derived from the $Y$ pair production search with BR($Y\to W^{-}b$) $=1$,
imposes $m_{Y}>1700\text{ GeV}$. In this context, we reassess the potential of
LHC Run-III to explore the VLQ $Y$, revealing that the associated parameter
regions of $\kappa_{Y}\in[0.044,0.5]$ $([0.072,0.5])$ and $m_{Y}\in[1000\text{
GeV},3099\text{ GeV}]$ $([1000\text{ GeV},2621\text{ GeV}])$ can be excluded
(discovered) based on LHC Run-III luminosity. We foresee that our
investigation will spur complementary explorations for a potential $Y$ quark
at forthcoming $pp$ colliders.

## Acknowledgements

This work of LS, YY and BY is supported by the Natural Science Foundation of
Henan Province under Grant No. 232300421217, the National Research Project
Cultivation Foundation of Henan Normal University under Grant No. 2021PL10,
the China Scholarship Council under Grant No. 202208410277 and also powered by
the High Performance Computing Center of Henan Normal University. The work of
SM is supported in part through the NExT Institute, the Knut and Alice
Wallenberg Foundation under the Grant No. KAW 2017.0100 (SHIFT) and the STFC
Consolidated Grant No. ST/ L000296/1.

##  Appendix A Appendix: Relationship between Eq. (1) and Eq. (3)

In the appendix, we provide the relationship between the $(B,Y)$ doublet
representation and the simplified model used in the simulation. However, we do
not present the relationship between the $(X,B,Y)$ triplet representation and
the simplified model here because it can be easily derived from the remainder
of this Appendix.)

The Lagrangian for the $Y$ coupling with the SM gauge fields and the $Y$ mass
term is

$\displaystyle\mathcal{L}=\bar{Q}_{5}(i\not{D}-M_{F})Q_{5}$ (A.1)

where one has

$\displaystyle Q_{5}=\begin{pmatrix}B_{0}\\\
Y_{0}\end{pmatrix},\bar{Q}_{5}=(\bar{B}_{0},\bar{Y}_{0}),\not{D}=\gamma^{\mu}D_{\mu},D_{\mu}=\partial_{\mu}+i{g}^{\prime}Y_{F}B_{\mu}+\frac{i}{2}g\tau^{I}W_{\mu}^{I}$
(A.2)

and the weak isospin $g$ and weak hypercharge $g^{\prime}$ are the SU(2)L and
U(1)Y couplings, respectively. We use a subscript $0$ to represent the
interaction eigenstates. The unphysical fields $B_{\mu}$ and $W^{I}_{\mu}$
($I=1,2,3$) can be transformed into the physical fields of the photon
$A_{\mu}$, the neutral $Z$ boson $Z_{\mu}$ and charged $W$ bosons
$W^{\pm}_{\mu}$ via the following equations:

$\displaystyle B_{\mu}$
$\displaystyle=\cos\theta_{W}A_{\mu}-\sin\theta_{W}Z_{\mu},W^{3}_{\mu}=\sin\theta_{W}A_{\mu}+\cos\theta_{W}Z_{\mu},$
$\displaystyle W^{1}_{\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}(W^{+}_{\mu}+W^{-}_{\mu}),W^{2}_{\mu}=\frac{i}{\sqrt{2}}(W^{+}_{\mu}-W^{-}_{\mu})$
(A.3)

where $\theta_{W}$ is the Weinberg angle, which can be expressed via
$\sin\theta_{W}=\frac{e}{g}$ and $\cos\theta_{W}=\frac{e}{g^{\prime}}$. Here,
$M_{F}$ is a free mass parameter. Considering the charge of $Y$, the
Lagrangian for the $Y$ coupling with the SM gauge fields is

$\displaystyle\mathcal{L}_{Q_{5}Q_{5}V}$
$\displaystyle=\bar{Q}_{5}\left(\frac{5}{6}{g}^{\prime}B_{\mu}-\frac{g}{2}\begin{bmatrix}W_{\mu}^{3}&W_{\mu}^{1}-iW_{\mu}^{2}\\\
W_{\mu}^{1}+iW_{\mu}^{2}&-W_{\mu}^{3}\end{bmatrix}\right)\gamma^{\mu}Q_{5}$
$\displaystyle=\bar{Q}_{5}\begin{bmatrix}\frac{1}{3}eA_{\mu}-\frac{g}{2\cos\theta}\left(1+\frac{2}{3}\sin^{2}\theta\right)Z_{\mu}&-\frac{g}{\sqrt{2}}W_{\mu}^{+}\\\
-\frac{g}{\sqrt{2}}W_{\mu}^{-}&\frac{4}{3}eA_{\mu}+\frac{g}{2\cos\theta}\left(1-\frac{8}{3}\sin^{2}\theta\right)Z_{\mu}\end{bmatrix}\gamma^{\mu}Q_{5}$
$\displaystyle=\frac{1}{3}e\bar{B}_{0}A_{\mu}\gamma^{\mu}B_{0}-\frac{g}{2\cos\theta}\left(1+\frac{2}{3}\sin^{2}\theta\right)\bar{B}_{0}Z_{\mu}\gamma^{\mu}B_{0}$
$\displaystyle\,\,\,\,\,\,+\frac{4}{3}e\bar{Y}_{0}A_{\mu}\gamma^{\mu}Y_{0}+\frac{g}{2\cos\theta}\left(1-\frac{8}{3}\sin^{2}\theta\right)\bar{Y}_{0}Z_{\mu}\gamma^{\mu}Y_{0}$
$\displaystyle\,\,\,\,\,\,-\frac{g}{\sqrt{2}}\bar{Y}_{0}W^{-}_{\mu}\gamma^{\mu}B_{0}-\frac{g}{\sqrt{2}}\bar{B}_{0}W^{+}_{\mu}\gamma^{\mu}Y_{0}.$
(A.4)

In our study, $(B,Y)$ states exclusively couple with the third-generation
quarks of the SM. Therefore, the Lagrangian for the mass term of the bottom
quark mass eigenstate $b$ and its partner mass eigenstate $B$ can be written
as

$\displaystyle\mathcal{L}_{\text{mass}}=-\begin{pmatrix}\bar{b}_{0}^{L}&\bar{B}_{0}^{L}\end{pmatrix}\begin{pmatrix}y_{33}^{d}\frac{v}{\sqrt{2}}&y_{34}^{d}\frac{v}{\sqrt{2}}\\\
y_{43}^{d}\frac{v}{\sqrt{2}}&M^{0}\end{pmatrix}\begin{pmatrix}b_{0}^{R}\\\
B_{0}^{R}\end{pmatrix}+\text{H.c.}$ (A.5)

where $v=246\text{ GeV}$ is the Vacuum Expectation Value (VEV) of the Higgs
field, $M^{0}$ is a bare mass term, $y_{33}^{d}$ and $y_{43}^{d}$ are Yukawa
coupling coefficients while $y_{34}^{d}=0$ for doublets. The mass matrix can
be diagonalized by the two mixing matrices $V^{L}$ and $V^{R}$, as follows:

$\displaystyle\begin{pmatrix}b_{0}^{L,R}\\\
B_{0}^{L,R}\end{pmatrix}=V^{L,R}\begin{pmatrix}b^{L,R}\\\
B^{L,R}\end{pmatrix}$ (A.6)

where $L$ and $R$ stands for the left-hand and right-hand chiralities,
respectively. There exists the following relationship too:
$B_{0}=B_{0}^{L}+B_{0}^{R}$ and $Y_{0}=Y_{0}^{L}+Y_{0}^{R}$. The $2\times 2$
unitary matrices $V^{L}$ and $V^{R}$ can be parameterized by the mixing angles
$\theta^{L}$ and $\theta^{R}$, respectively, as

$\displaystyle V^{L,R}=\begin{pmatrix}\cos\theta^{L,R}&\sin\theta^{L,R}\\\
-\sin\theta^{L,R}&\cos\theta^{L,R}\end{pmatrix}$ (A.7)

We can then determine the expressions
$B_{0}^{L,R}=-\sin\theta^{L,R}b^{L,R}+\cos\theta^{L,R}B^{L,R}$. For $Y$, it is
as simple as $Y_{0}^{L,R}=Y^{L,R}$, where $Y$ represents the mass eigenstate.
This is because there are no $\pm 4/3$ particles in the SM. Therefore, we can
derive the interactions between the $Y$, $W$ and $b$ states as follows:

$\displaystyle\mathcal{L}_{YW^{\pm}b}$
$\displaystyle=-\frac{g}{\sqrt{2}}\left(\bar{Y}^{L}+\bar{Y}^{R}\right)W^{-}_{\mu}\gamma^{\mu}\left(-\sin\theta^{L}b^{L}-\sin\theta^{R}b^{R}\right)+\text{H.c.}$
$\displaystyle=\frac{g}{\sqrt{2}}\sin\theta^{L}\bar{Y}^{L}W^{-}_{\mu}\gamma^{\mu}b^{L}+\frac{g}{\sqrt{2}}\sin\theta^{R}\bar{Y}^{R}W^{-}_{\mu}\gamma^{\mu}b^{R}+\text{H.c.}$
(A.8)

Using unitary matrices, we can finally obtain

$\displaystyle
V^{L}\begin{pmatrix}y_{33}^{d}\frac{v}{\sqrt{2}}&y_{34}^{d}\frac{v}{\sqrt{2}}\\\
y_{43}^{d}\frac{v}{\sqrt{2}}&M^{0}\end{pmatrix}(V^{R})^{\dagger}=\begin{pmatrix}m_{b}&0\\\
0&m_{B}\end{pmatrix}$ (A.9)

After performing calculations involving trigonometric function identities, we
can obtain666For the $(T,B,Y)$ triplet, $y_{43}^{d}=0$, we can deduce instead
that $\tan\theta^{R}=\frac{m_{b}}{m_{B}}\tan\theta^{L}$.:

$\displaystyle\tan\theta^{L}=\frac{m_{b}}{m_{B}}\tan\theta^{R}$ (A.10)

Since $m_{B}\gg m_{b}$, we can conclude that $\sin\theta^{L}\gg\sin\theta^{R}$
in the ($B,Y$) doublet. Therefore, our study primarily concentrates on the
right-handed coupling part of the interactions involving the $Y$, $W$ and $b$
states.

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