# A fresh look at the LHC limits on scalar leptoquarks

Arvind Bhaskar<EMAIL_ADDRESS>Center for Computational Natural
Sciences and Bioinformatics, International Institute of Information
Technology, Hyderabad 500 032, India  Institute of Physics, Sachivalaya Marg,
Bhubaneswar 751005, India  Arijit Das<EMAIL_ADDRESS>Indian
Institute of Science Education and Research Thiruvananthapuram, Vithura,
Kerala, 695 551, India  Tanumoy Mandal<EMAIL_ADDRESS>Indian
Institute of Science Education and Research Thiruvananthapuram, Vithura,
Kerala, 695 551, India  Subhadip Mitra<EMAIL_ADDRESS>Center for
Computational Natural Sciences and Bioinformatics, International Institute of
Information Technology, Hyderabad 500 032, India  Rachit Sharma
<EMAIL_ADDRESS>Indian Institute of Science Education and Research
Thiruvananthapuram, Vithura, Kerala, 695 551, India

###### Abstract

The scalar Leptoquark (sLQ) parameter space is well-explored experimentally.
The direct pair production searches at the LHC have excluded light sLQs almost
model agnostically, and the high-$p_{\rm T}$ dilepton tail data have put
strong bounds on the leptoquark-quark-lepton Yukawa couplings for a wide range
of sLQ masses. However, these do not show the complete picture. Previously,
Ref. Mandal:2015vfa  [JHEP, 07, 028 (2015)] showed how the dilepton-dijet data
from the pair production searches could give strong limits on these couplings.
This was possible by including the single-production contribution to the
dilepton-dijet signal. In this paper, we take a fresh look at the LHC limits
on all sLQs by following the same principle and combine all significant
contributions—from pair and single productions, $t$-channel sLQ exchange and
its interference with the Standard Model background—to the $\mu\mu jj$ final
state and recast the limits. We notice that the LQ exchange and its
interference with the background processes play significant roles in the
limits. The $\mu\mu jj$-recast limits are comparable to or, in some cases,
significantly better than the currently known limits (from high-$p_{\rm T}$
dilepton data and direct searches), i.e., the LHC data rules out more
parameter space than what is considered in the current literature. For the
first time, we also show how including the QED processes can noticeably
improve the LQ mass exclusion limits from the QCD-only limits.

##  I Introduction

Hypothetical bosons called Leptoquarks (LQs) have found significant importance
in theoretical and experimental particle physics. These scalar/vector colour-
triplet particles carry fractional electric charges and connect the Standard
Model (SM) quark and lepton sectors. Many beyond the SM (BSM) theories predict
their existence at the TeV scale Pati:1974yy ; Georgi:1974sy ; Fritzsch:1974nn
; Farhi:1980xs ; Schrempp:1984nj ; Wudka:1985ef ; Barbier:2004ez . Their
current popularity in the literature mainly stems from their role in
simultaneously resolving the flavour and other persistent anomalies (for
example muon $g-2$ Muong-2:2021vma ; Muong-2:2021ojo ; Muong-2:2023cdq ). This
has motivated phenomenologists to speculate on new top-down/bottom-up models
Goncalves:2023qpz ; DaRold:2023hmx  and propose novel collider signatures and
search strategies for LQs Dorsner:2014axa ; Diaz:2017lit ;
Bandyopadhyay:2018syt ; Schmaltz:2018nls ; Bhaskar:2020kdr ; Greljo:2020tgv ;
Dorsner:2021chv ; Bandyopadhyay:2021pld ; Bhaskar:2022ygp ; Parashar:2022wrd ;
Ghosh:2023ocz ; Florez:2023jdb . The strong theoretical motivation for the
low-mass LQ states within the reach of the Large Hadron Collider (LHC) makes
their search an active and popular pursuit among experimental particle
physicists. In recent years, research in these directions has become
progressively comprehensive. The ATLAS and CMS collaborations have conducted
extensive searches for the pair and single productions of LQs (see the ATLAS
and CMS summary plots for LQ searches and the references therein). In the
absence of any discovery, these searches have put strong limits on LQ
parameters.

Most commonly, these searches focus on LQ pair production (PP), where the LQs
decay into lepton-jet pairs forming dilepton-dijet ($\ell\ell jj$, where
$\ell=e,\mu$) final states. From these, model-independent limits on the LQ
mass are drawn assuming the LQ-quark-lepton Yukawa couplings ($y$) responsible
for LQ decays are small and the PP process is essentially strong-interaction
mediated in this limit. The current upper limit on the PP cross section in the
$\ell\ell jj$ channel is about $0.1$ fb for scalar LQ (sLQ) masses around
$1.5$ TeV ATLAS:2020dsk . The cross sections of single production (SP)
processes depend on the new coupling(s), and hence direct SP searches in the
$\ell\ell j$ final states provide LQ mass-dependent upper limits on $y$ (which
can also be interpreted as $y$-dependent lower bounds on LQ masses).

Apart from the direct searches, there are also some less obvious sources of
LHC bounds. For example, the high-$p_{\rm T}$ tail of the dilepton or the
lepton + $\not{E}_{\rm T}$ resonance (like $Z^{\prime}$ or $W^{\prime}$)
search data can limit the LQ parameter space Raj:2016aky ; Greljo:2017vvb ;
Bansal:2018eha ; Alvarez:2018jfb ; Alves:2018krf ; Fuentes-Martin:2020lea ;
Buonocore:2020erb ; Crivellin:2021egp ; Haisch:2022lkt ; Allwicher:2022gkm . A
$t$-channel LQ exchange can produce two high-$p_{\rm T}$ leptons (or a lepton
and a neutrino) from the quark fusion processes which will affect the tails of
the distributions. The $t$-channel LQ exchange [we refer to it as the indirect
production (IP)] is highly sensitive to the new couplings as the amplitude is
proportional to the second power in the Yukawa couplings. Hence, one can use
the high-$p_{\rm T}$ dilepton or lepton + $\not{E}_{\rm T}$ data to constrain
$y$ for a range of masses of the LQ involved (wider than the mass range
accessible to the direct searches). Interestingly, the IP amplitude interferes
with the gauge bosons-mediated dilepton or monolepton + $\not{E}_{\rm T}$
productions; the interference terms play major roles in determining the limits
on the couplings Mandal:2018kau .

In principle, we can combine all the latest ATLAS and CMS bounds to obtain the
allowed regions in the LQ parameter space. That, however, will only show a
partial picture for a couple of reasons. Just as we recast the dilepton data
to draw limits on the LQ parameters, we can recast the direct-search data by
including the contribution of all processes in the signal to draw stronger
limits. For example, let us suppose some of the new Yukawa couplings are
$\mathcal{O}(1)$ (as needed by the anomalies). In that case, the $\ell\ell jj$
signal can receive significant contributions from the SP and IP (and its
interference) processes and lead to stronger (albeit model-dependent) bounds
than the model-independent PP bounds. This is not entirely a new
observation—we illustrated how systematic inclusion of SP events in the PP-
search ($\ell\ell jj$) signal and PP events in the SP-search ($\ell\ell j$)
signal lead to stronger limits in Ref. Mandal:2015vfa  (also see
Aydemir:2019ynb ; Chandak:2019iwj ; Bhaskar:2021pml ; Bhaskar:2021gsy ;
Bhaskar:2022vgk ; Aydemir:2022lrq ). However, this point—not specific to LQs
but applicable to many other BSM searches as well Mandal:2012rx ;
Mandal:2016csb —has largely been overlooked in the literature. Second, these
limits ignore the contributions of a class of diagrams. Since LQs have both
electric and colour charges, they can couple to both photons and gluons.
Interestingly, the gauge symmetries allow for a gluon-photon-sLQ-sLQ
interaction Crivellin:2021ejk . So far, the contributions from the photon-
initiated diagrams (or those involving this particular interaction term) are
not accounted for in the collider analyses in general.

Given the important roles the TeV-scale LQs play in a wide range of BSM
scenarios addressing various open problems, it is crucial to scrutinise the
LHC limits on the LQs in detail. The LHC limits are independent of (and
competitive to) other low-energy bounds (like those from various flavour
observables); they exclude various LQ scenarios which are allowed by the low-
energy limits (see, e.g., Bhaskar:2021pml ). In this paper, we re-examine the
LHC limits on the scalar LQs from the latest ($\sim 140$ fb-1) data in light
of the above points. Since, at the LHC, muons are the easiest to identify
among the leptons, we consider the $\mu\mu jj$ ATLAS:2020dsk  and $\mu\mu$
CMS:2021ctt  data for our purpose (there is no $\mu\mu j$ data at this
luminosity). In addition, we also consider the latest $\mu b\mu b$
ATLAS:2020dsk  and $\mu t\mu t$ CMS:2022nty  data in recasting the exclusion
limits.

The plan of the paper is as follows. We review the interactions and different
production mechanisms for the sLQs that can couple with the muon in Sec. II;
explain the recast methodologies in Sec. III; present the recast limits in
Sec. IV; and finally, we conclude in Sec. V.

Table 1: The Yukawa interactions in up and down-aligned scalar-LQ scenarios (in the notation of Ref. Dorsner:2016wpm , except for the $R_{2}$ couplings where the roles of $i$ and $j$ are interchanged). We ignore the diquark interactions as those are not relevant to our analysis. Here, $V\equiv V_{\rm CKM}$ denotes the Cabibbo-Kobayashi-Maskawa matrix. In the rest of the paper, we use a slightly different but more explicit notation for the Yukawa couplings—see Appendix A for details.  Model | Down-aligned Yukawa Interactions | Up-aligned Yukawa Interactions
---|---|---
$S_{1}$ | $-y^{LL}_{1ij}\ \overline{d^{C}_{L}}^{i}\nu^{j}_{L}S_{1}+(V^{*}y_{1}^{LL})_{ij}\ \overline{u^{C}_{L}}^{i}e^{j}_{L}S_{1}+y^{RR}_{1ij}\ \overline{u^{C}_{R}}^{i}e^{j}_{R}S_{1}$ | $-(V^{T}y^{LL}_{1})_{ij}\ \overline{d^{C}_{L}}^{i}\nu^{j}_{L}S_{1}+y^{LL}_{1ij}\ \overline{u^{C}_{L}}^{i}e^{j}_{L}S_{1}+y^{RR}_{1ij}\ \overline{u^{C}_{R}}^{i}e^{j}_{R}S_{1}$
$\widetilde{S}_{1}$ | $\widetilde{y}^{RR}_{1ij}\ \overline{d^{C}_{R}}^{i}e^{j}_{R}\widetilde{S}_{1}$
$R_{2}$ | $-y^{RL}_{2ij}\ (\overline{u}_{R}^{i}e^{j}_{L}R^{5/3}_{2}-\overline{u}_{R}^{i}\nu^{j}_{L}R^{2/3}_{2})$ | $-y^{RL}_{2ij}\ (\overline{u}_{R}^{i}e^{j}_{L}R^{5/3}_{2}-\overline{u}_{R}^{i}\nu^{j}_{L}R^{2/3}_{2})$
$+(Vy_{2}^{LR})_{ij}\ \overline{u}_{L}^{i}e^{j}_{R}R^{5/3}_{2}+y^{{LR}}_{2ij}\ \overline{d}_{L}^{i}e^{j}_{R}R^{2/3}_{2}$ | $+y^{LR}_{2ij}\ \overline{u}_{L}^{i}e^{j}_{R}R^{5/3}_{2}+(V^{\dagger}y_{2}^{LR})_{ij}\ \overline{d}_{L}^{i}e^{j}_{R}R^{2/3}_{2}$
$\widetilde{R}_{2}$ | $-\widetilde{y}^{RL}_{2ij}\ (\overline{d}^{i}_{R}e^{j}_{L}\widetilde{R}^{2/3}_{2}-\overline{d}^{i}_{R}\nu^{j}_{L}\widetilde{R}^{-1/3}_{2})$
$S_{3}$ | $-y^{LL}_{3ij}\ \overline{d^{C}_{L}}^{i}\nu^{j}_{L}S^{1/3}_{3}-(V^{*}y_{3}^{LL})_{ij}\ \overline{u^{C}_{L}}^{i}e^{j}_{L}S^{1/3}_{3}$ | $-(V^{T}y_{3}^{LL})_{ij}\ \overline{d^{C}_{L}}^{i}\nu^{j}_{L}S^{1/3}_{3}-y^{LL}_{3ij}\ \overline{u^{C}_{L}}^{i}e^{j}_{L}S^{1/3}_{3}$
$-\sqrt{2}y^{LL}_{3ij}\ \overline{d^{C}_{L}}^{i}e^{j}_{L}S^{4/3}_{3}+\sqrt{2}(V^{*}y^{LL}_{3})_{ij}\ \overline{u^{C}_{L}}^{i}\nu^{j}_{L}S^{-2/3}_{3}$ | $-\sqrt{2}(V^{T}y_{3}^{LL})_{ij}\ \overline{d^{C}_{L}}^{i}e^{j}_{L}S^{4/3}_{3}+\sqrt{2}y^{LL}_{3ij}\ \overline{u^{C}_{L}}^{i}\nu^{j}_{L}S^{-2/3}_{3}$

((a)) (a)

((b)) (b)

((c)) (c)

((d)) (d)

((e)) (e)

((f)) (f)

((g)) (g)

((h)) (h)

((i)) (i)

((j)) (j)

((k)) (k)

((l)) (l)

Figure 1: Representative Feynman diagrams for pair, single and indirect
productions of sLQs. A generic sLQ is denoted by $\Phi$.

((a))  (a) ((b))  (b)

((c))  (c) ((d))  (d)

Figure 2: The PP, SP, IP, and II contributions for $S_{1}$ (subjected to a
$M_{\mu^{+}\mu^{-}}\geq 120$ GeV cut) as functions of $M_{S_{1}}$ for
$y^{LL}_{10,12}=1$ at different orders of $\alpha_{s}$, $\alpha_{e}$, and
$\alpha_{\lambda}$. The constructive/destructive interference contributions
are indicated with $(\pm)$ signs in front.

##  II Producing scalar leptoquarks at the LHC

In the notation of Ref. Dorsner:2016wpm , all sLQ species except
$\overline{S}_{1}$ (which exclusively couples to right-handed neutrinos),
namely, $S_{1}$, $\widetilde{S}_{1}$, $R_{2}$, $\widetilde{R}_{2}$, and
$S_{3}$ can directly produce $\ell\ell jj$ final states at the LHC. We display
the possible Yukawa interactions of these SLQs with quarks and leptons in
Table 1. Depending on whether the mass eigenstates of sLQs are aligned with
the mass eigenstates of the up-type quarks or the down-type ones, we consider
two scenarios that couple with left-handed quarks—up-aligned and down-aligned.
For simplicity, we do not consider the more general setup where these sLQs are
neither up or down-aligned. We also ignore the effect of neutrino mixing and
take $U_{\rm PMNS}=1$ as all neutrinos produce missing energy, and the
neutrino oscillation length is irrelevant to our analysis.

LQs can be produced at the LHC in two ways—resonantly or nonresonantly. In
resonant modes, the pair and single production (PP and SP) processes produce
on-shell LQs, whereas in nonresonant production (indirect production—IP) a LQ
is exchanged in the $t$-channel. The IP process can interfere (constructively
or destructively, depending on the sLQ species) with SM processes. Since all
these production topologies can lead to $\ell\ell jj$ final states, we briefly
review them.

Pair production: Two sLQs (of the same or different species) can be produced
resonantly. These decay to produce the $\ell\ell jj$ final states. The leading
tree-level contributions to the PP cross section are:

  1. 1.

$\mathcal{O}(\alpha_{s}^{2}\alpha_{e}^{0}\alpha_{y}^{0})$:111We use
$\alpha_{y}=y^{2}/4\pi$ where $y$ denotes a sLQ-quark-lepton Yukawa coupling.
The diagrams contributing to this order are purely QCD-mediated; see, e.g.,
Fig. 1(a). Hence, the contribution is model-independent.

  2. 2.

$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{2}\alpha_{y}^{0})$: There are $qq$ and
$\gamma\gamma$-initiated processes that depend only on the electric charge of
the sLQ [Figs. 1(b) and 1(c)].

  3. 3.

$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{0}\alpha_{y}^{2})$: A lepton exchange
in the $t$-channel can lead to sLQ pair production—see Fig. 1(d). This purely
new physics (NP) contribution is highly sensitive to the LQ Yukawa couplings
and the parton distribution functions (PDFs) of the initial quarks.

  4. 4.

$\mathcal{O}(\alpha_{s}^{1}\alpha_{e}^{0}\alpha_{y}^{1})$: The contribution to
the total PP cross section at this order comes from the interference between
the diagrams that separately produce
$\mathcal{O}(\alpha_{s}^{2}\alpha_{e}^{0}\alpha_{y}^{0})$ and
$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{0}\alpha_{y}^{2})$ contributions to the
cross section. This interference is destructive in nature for all sLQ species.

  5. 5.

$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{1}\alpha_{y}^{1})$: This contribution
comes from the interference of the diagrams that separately produce
$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{2}\alpha_{y}^{0})$ and
$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{0}\alpha_{y}^{2})$ contributions to the
cross section. Generally, this interference is a minor contribution to the PP
cross section.

  6. 6.

$\mathcal{O}(\alpha_{s}^{1}\alpha_{e}^{1}\alpha_{y}^{0})$: Finally, there is
also a QCD-QED mixed order contribution to the PP [see Fig. 1(e)], which,
unlike the above two cases, does not come from any interference term (colour
conservation stops the $qq\to g\to\ell_{q}\ell_{q}$ and
$qq\to\gamma/Z\to\ell_{q}\ell_{q}$ processes from interfering) but the gluon-
photon-sLQ-sLQ term in the kinetic Lagrangian Crivellin:2021ejk .

Putting these together, we can make the parameter dependence of the PP cross
section explicit as

$\displaystyle\sigma_{\rm PP}(M_{\ell_{q}},y)=$ $\displaystyle\ \sigma_{\rm
PP}^{200}(M_{\ell_{q}})+\sigma_{\rm PP}^{110}(M_{\ell_{q}})+\sigma_{\rm
PP}^{020}(M_{\ell_{q}})$ $\displaystyle+y^{2}\,\overline{\sigma_{\rm
PP}^{101}(M_{\ell_{q}})}+y^{2}\,\overline{\sigma_{\rm
PP}^{011}(M_{\ell_{q}})}+y^{4}\,\overline{\sigma_{\rm
PP}^{002}(M_{\ell_{q}})},$ (1)

where $\sigma_{\rm P}^{ijk}$ denotes the contribution of order
$\alpha_{s}^{i}\alpha_{e}^{j}\alpha_{y}^{k}$ to the process “${\rm P}$” and
the overlines indicate the functions under are evaluated at $y=1$, i.e.,
$\overline{\sigma^{ijk}_{P}(M_{\ell_{q}})}=\sigma^{ijk}_{P}(M_{\ell_{q}},y=1)$.

Single production: The new Yukawa couplings allow only one sLQ to be produced
resonantly in association with a lepton and jets ($pp\to\ell_{q}\ell$ or
$pp\to\ell_{q}\ell j$). The decay of the sLQ can give the desired final
states. There are two-body single productions (2BSPs) and three-body single
productions (3BSPs). At the LHC, the $\ell\ell jj$ signal will essentially
come from the 3BSP processes, as we discuss below.

2BSP: There are two leading-order (LO) contributions to this process:

  1. 1.

$\mathcal{O}(\alpha_{s}^{1}\alpha_{e}^{0}\alpha_{y}^{1})$: There are
$qg$-initiated contributions to $pp\to\ell_{q}\ell$ [shown in Fig. 1(f)].

  2. 2.

$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{1}\alpha_{y}^{1})$: Similarly, a
$q\gamma$-initiated diagram is shown in Fig. 1(g).

3BSP: There are two types of diagrams that contribute to the 3BSP process.
First, a (hard) jet can be radiated off a 2BSP process [i.e., an initial/final
state radiation (ISR/FSR) jet]:

  1. 1.

$\mathcal{O}(\alpha_{s}^{1+(1)}\alpha_{e}^{0}\alpha_{y}^{1})$:222The
parentheses in the exponents count the number of hard radiations. A (hard) jet
is emitted from the $\mathcal{O}(\alpha_{s}^{1}\alpha_{e}^{0}\alpha_{y}^{1})$
contribution to the 2BSP process; see, e.g., Fig. 1(h) where the jet is
emitted from the LQ; the jet can also come off the initial partons.

  2. 2.

$\mathcal{O}(\alpha_{s}^{(1)}\alpha_{e}^{1}\alpha_{y}^{1})$: An (hard) ISR or
FSR from the $\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{1}\alpha_{y}^{1})$
contribution to the 2SBP can contribute to this order. See, e.g., Fig. 1(i).

  3. 3.

$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{1+(1)}\alpha_{y}^{1})$: Similarly, when
an initial state quark splits into a hard quark (producing a hard ISR) and an
electroweak boson, which further interacts with another quark to produce a LQ
and a lepton [as in Fig. 1(g)], we count it as a separate 3BSP diagram; see
Fig. 1(j).

Second, as shown in Ref. Mandal:2015vfa  (also see Mandal:2012rx ),
$pp\to\ell_{q}\ell j$ contain some genuine NP contributions where the jet is
neither an ISR nor a FSR off the $pp\to\ell_{q}\ell$ processes:

  1. 4.

$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{0}\alpha_{y}^{3})$: This come from
another set of $qq$-initiated diagrams that depend only on the LQ Yukawa
coupling(s)—a representative Feynman diagram is shown in Fig. 1(k).

  2. 5.

There is another possibility where the lepton and the jet come from an off-
shell sLQ, i.e., $pp\to\ell_{q}\ell_{q}^{*}\to\ell_{q}\ell j$. Hence, the
process contributes at one higher power of $\alpha_{y}$ than the PP
subprocess—$\mathcal{O}(\alpha_{s}^{2}\alpha_{e}^{0}\alpha_{y}^{1})$,
$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{2}\alpha_{y}^{1})$,
$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{0}\alpha_{y}^{3})$,
$\mathcal{O}(\alpha_{s}^{1}\alpha_{e}^{0}\alpha_{y}^{2})$,
$\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{1}\alpha_{y}^{2})$, and
$\mathcal{O}(\alpha_{s}^{1}\alpha_{e}^{1}\alpha_{y}^{1})$. If the sLQ is far
off-shell, it will have very different kinematics than that of the PP
processes.

Figure 3: Indirect interference contributions for different sLQs.

Estimating the SP contribution to $\ell\ell jj$ states systematically requires
some care Mandal:2015vfa . Even though the final state of the 2BSP processes
is $\ell\ell j$ ($pp\to\ell_{q}\ell\to\ell q\ell$) at the Born level, not
$\ell\ell jj$, these processes can emit an extra jet while showering and
contribute to the $\ell\ell jj$ signal. However, if the radiation is
soft/collinear, it will be hard to distinguish it experimentally from the
Born-level process. Hence, to avoid confusion, we club such processes where
the extra jet is hard with 3BSP. This also agrees with the experimental
schemes where jets are identified with reasonably hard $p_{T}$ cuts. Following
Ref. Mandal:2015vfa , we can mathematically express the SP contribution to the
$\ell\ell jj$ final state with the following equation.

$\displaystyle\sigma_{\rm SP}=$ $\displaystyle\ \underbrace{\big{(}\sigma^{\rm
LO}_{2BB(\ell_{q}\ell)}+\overbrace{\sigma^{\rm
virtual}_{(\ell_{q}\ell)}+\sigma_{{2BR}(\ell_{q}\ell j)}^{\rm
soft+collinear}}^{\rm Divergent~{}terms}+\sigma_{{3BNS}(\ell_{q}\ell j)}^{\rm
soft}\big{)}}_{\rm Negligible~{}contribution}$ $\displaystyle\
+\underbrace{\big{(}\sigma_{{2BR}(\ell_{q}\ell j)}^{\rm
hard}+\sigma_{{3BNS}(\ell_{q}\ell j)}^{\rm hard}\big{)}}_{\rm
Main~{}contribution\ (\approx\ 3BSP)}\ +\ \cdots,$ (2)

where the suffix $2BB$ indicates two-body Born-level processes (2BSP), $2BR$
denotes 2BSP$+$jet processes and $3BNS$ indicates the NP contribution in 3BSP.
In other words, the main contribution to the $\ell\ell jj$ signal from the
single productions comes from the 3BSP processes. Hence, the parametric
dependence of the SP can be made explicit as,

$\displaystyle\sigma_{\rm SP}(M_{\ell_{q}},y)=$ $\displaystyle\
y^{2}\left\\{\overline{\sigma_{SP}^{021}(M_{\ell_{q}})}+\overline{\sigma_{SP}^{201}(M_{\ell_{q}})}+\overline{\sigma_{SP}^{111}(M_{\ell_{q}})}\right\\}$
$\displaystyle\ +y^{6}\overline{\sigma_{SP}^{003}(M_{\ell_{q}})}.$ (3)

There is a chance of double counting in generating SP Monte Carlo (MC) events
separately as $\ell_{q}\ell j$ events can also come from PP diagrams as
$pp\to\ell_{q}\ell_{q}\to\ell_{q}(\ell j)$. One needs to make sure the lepton
and jet in the final state are not coming from an on-shell LQ Mandal:2015vfa .

Indirect production: The main contribution from this topology to $\ell\ell jj$
final states comes from a single order:

  1. 1.

$\mathcal{O}(\alpha_{s}^{(2)}\alpha_{e}^{0}\alpha_{y}^{2})$: We show an
illustrative diagram of the $qq^{(\prime)}\to\ell\ell$ process via a
$t$-channel sLQ exchange in Fig. 1(l). This process can contribute to the
$\ell\ell jj$ signal in the presence of two additional hard radiation jets.

Indirect interference (II): The IP process ($qq\to\ell\ell$) interferes with
the SM $s$-channel $Z/\gamma*$-mediated processes with the same initial and
final states. We count this process separately from the IP as the kinematics
of these two contributions are vastly different.

  1. 1.

$\mathcal{O}(\alpha_{s}^{(2)}\alpha_{e}^{1}\alpha_{y}^{1})$: Because of the
large cross sections of the SM processes, the interference contribution
usually plays a major (sometimes, the determining) role in the exclusion
limits in the high mass and large coupling(s) region. Its sign depends on the
sLQ involved in the diagram; e.g., it is negative for the $S_{1}$, it is
overall positive for the $R_{2}$, it is negative for the $S_{3}$ if the
initial quarks are up-type but positive if they are down-type, etc.
Raj:2016aky ; Bansal:2018eha .

The total indirect contribution can be expressed as:

$\sigma_{\rm ind}(M_{\ell_{q}},y)=y^{2}\overline{\sigma_{\rm
II}^{011}(M_{\ell_{q}})}+y^{4}\overline{\sigma_{\rm IP}^{002}(M_{\ell_{q}})}.$
(4)

For an illustration, we plot the cross sections of different orders and
topologies for a down-aligned $S_{1}$ with the coupling $y^{LL}_{10,12}=1$
(see Appendix A for the notation) with respect to its mass at the $13$ TeV LHC
in Fig. 2. There, we see that as $M_{S_{1}}$ increases, the photon-initiated
contribution to the 3BSP process becomes more important than the gluon-
initiated ones due to the lighter $t$-channel exchange (see the corresponding
Feynman diagrams in Fig. 1). The II contributions for various sLQs are shown
in Fig. 3.

Table 2: The cross sections ($\sigma$), the numbers of events ($\mathcal{N}$) surviving the $\mu\mu jj$-selection cuts ATLAS:2020dsk  for $\mathcal{L}_{exp}=139$ fb-1 of integrated luminosity, and the cut efficiencies [$\varepsilon$, defined in Eq. (6)] for three benchmark masses of $S_{1}$. As in Fig. 2, we have set $y_{10,12}^{LL}=1$ to obtain the numbers. We see that the importance of the indirect modes increases with the mass of the LQ. Mass ($S_{1}$) | Pair Production | Single Production | Indirect Production | Indirect Interference
---|---|---|---|---
(TeV) |  $\sigma_{\rm PP}$ (fb) | $\epsilon_{\rm PP}$ | $\mathcal{N}_{\rm PP}$ |  $\sigma_{\rm SP}$ (fb) | $\epsilon_{\rm SP}$ | $\mathcal{N}_{\rm SP}$ |  $\sigma_{\rm IP}$ (fb) | $\epsilon_{\rm IP}$ | $\mathcal{N}_{\rm IP}$ |  $\sigma_{\rm II}$ (fb) | $\epsilon_{\rm II}$ | $\mathcal{N}_{\rm II}$
$1.5$ | $1.4\times 10^{-1}$ | $0.42$ | $8.2$ | $8.4$ | $0.17$ | $198.5$ | $32.0$ | $0.018$ | $111.2$ | $-275.0$ | $0.007$ | $-267.6$
$2.5$ | $6.9\times 10^{-4}$ | $0.32$ | $3.1\times 10^{-2}$ | $5.4\times 10^{-1}$ | $0.13$ | $9.8$ | $5.4$ | $0.018$ | $13.4$ | $-102.0$ | $0.007$ | $-99.2$
$4.0$ | $4.6\times 10^{-7}$ | $0.30$ | $2.0\times 10^{-5}$ | $3.4\times 10^{-2}$ | $0.20$ | $0.7$ | $1.0$ | $0.028$ | $3.7$ | $-40.2$ | $0.010$ | $-55.9$

##  III Recasting LHC searches

###  III.1 Direct-search data

We recast the current ATLAS limits on sLQ pair production from the $\ell\ell
jj$ channel ATLAS:2020dsk  by recalculating the signal. We make the signal
inclusive by including all the contributions to the $\ell\ell jj$ final states
discussed in the previous section. To recast, we rely on the following
relation between the number of observed events at a given $M_{\ell_{q}}$ and
the signal cross sections:

$\displaystyle\frac{\mathcal{N}_{obs}(M_{\ell_{q}})}{\mathcal{L}_{exp}}=$
$\displaystyle\ \sigma_{obs}(M_{\ell_{q}})\times\epsilon_{exp}(M_{\ell_{q}})$
$\displaystyle=$ $\displaystyle\ \sum_{i~{}\in~{}{\rm
topologies}}\sigma_{i}(M_{\ell_{q}},\vec{y})\times\beta_{i}(\vec{y})\times\epsilon_{i}(M_{\ell_{q}}),$
(5)

where $\epsilon$’s denote the final selection efficiencies (including the
acceptance), $\beta$ the BRs and $\mathcal{L}_{exp}$ the integerated
luminosity. In the second line, the sum goes over different topologies—PP, SP,
IP, and II. Here, we have assumed the efficiencies to be largely independent
of the couplings ($\vec{y}$), which is a reasonable assumption since the
selection cuts are kinematic in nature.

To estimate the new efficiencies, we implement all the sLQ Lagrangians in
FeynRules Alloul:2013bka  to get the UFO model files Degrande:2011ua ;
Darme:2023jdn  for MadGraph5 Alwall:2014hca . We use the NNPDF PDFs
NNPDF:2021uiq  with dynamical renormalisation and factorisation scales to
generate LO events; which are passed through Pythia8 Bierlich:2022pfr  for
showering and hadronisation, and Delphes deFavereau:2013fsa  for detector
effects. Jets are formed using anti-$k_{T}$ Cacciari:2008gp  clustering
algorithm in FastJet Cacciari:2011ma . In our analysis, we use LO cross
sections for all signal processes except the PP, for which the NLO corrections
are known Kramer:2004df ; Mandal:2015lca . We include the higher-order effects
with a constant $K$ factor of $1.6$ for this process.

We pass the simulated events through the ATLAS selection criteria to estimate
the efficiencies for different topologies as

$\epsilon_{i}=\frac{\mathcal{N}_{i}}{\mathcal{N}_{i}^{gen}}\mbox{~{}with~{}}i=\\{\mbox{PP,
SP, IP, II}\\},$ (6)

where $\mathcal{N}_{i}$ is the number of events surviving the cuts and
$\mathcal{N}_{i}^{gen}$ is the total number of generated Monte Carlo events
without any cuts for the $i^{\rm th}$ topology. For validation, we reproduced
the PP limits in Refs Dorsner:2022ibm  and ATLAS:2020dsk  on $\lambda-M_{\rm
LQ}$ and $\beta-M_{\rm LQ}$ planes for all LQs.We have validated the model-
independent limits on all sLQ masses from Ref. Dorsner:2022ibm  within a few
GeVs. For an illustration, we list the cross sections, the numbers of events
surviving the $\ell\ell jj$-channel cuts, and the cut efficiencies for three
benchmark masses of $S_{1}$ in Table 2.

Before we move on, we note that in Ref. ATLAS:2020dsk , the observed limits go
up to $M_{\ell_{q}}=2$ TeV. For our purpose, we have extrapolated the limits
beyond $2$ TeV assuming the experimental selection cuts remain the same in
this range, i.e., $\mathcal{N}_{obs}(M_{\ell_{q}}>2~{}{\rm
TeV})=\mathcal{N}_{obs}(M_{\ell_{q}}=2~{}{\rm TeV})$.

###  III.2 Dilepton-search data

We also consider the latest dimuon search data from CMS CMS:2021ctt  to put
bounds on the new Yukawa couplings as functions of sLQ masses. Since the
leptons produced from TeV-scale LQ decays or in association with LQs will have
high $p_{T}$, they will affect the high-$p_{T}$ tail of the dilepton (in our
case, dimuon) distribution in general. Following the method illustrated in
Refs. Mandal:2018kau ; Bhaskar:2021pml  (also see Aydemir:2019ynb ;
Bhaskar:2022vgk ), we perform $\chi^{2}$ estimations to obtain the $2\sigma$
limits on $y$’s by fitting the dimuon-$p_{T}$ distribution for any fixed
$M_{\ell_{q}}$ hypothesis. We compute the binwise signal efficiencies by
passing the events from all the production processes mentioned above through
the cuts used in the dilepton search analysis and then combining them. We note
that since there is no restriction on the number of hard jets in this case, a
priori, more processes will contribute to the dilepton signal than the
$\ell\ell jj$ one. For example, the 2BSP processes or the $qq\to\mu\mu$
process via $t$-channel LQ exchange (without the hard radiations), which
essentially did not contribute earlier, will contribute to the dilepton
signal. We include all contributions systematically.

Table 3: Pure QCD [$\mathcal{O}(\alpha_{s}^{2}\alpha_{e}^{0}\alpha_{y}^{0}$] vs pure QCD+QED [$\mathcal{O}(\alpha_{s}^{2}\alpha_{e}^{0}\alpha_{y}^{0})$ $+$ $\mathcal{O}(\alpha_{s}^{0}\alpha_{e}^{2}\alpha_{y}^{0})$ $+$ $\mathcal{O}(\alpha_{s}^{1}\alpha_{e}^{1}\alpha_{y}^{0})$] mass exclusion limits: Model-independent limits on various sLQ masses (in GeV) with and without the QED contributions. We assume sLQ decays through only one small coupling (shown in parentheses). The same limits are also applicable when second-generation quark is involved.  Model | QCD | QCD+QED
---|---|---
$S_{1}(y^{LL}_{10,12})$ | $1418$ | $1423$
$S_{1}(y^{RR}_{10,12})$ | $1733$ | $1741$
$\tilde{S}_{1}(y^{RR}_{11,12})$ | $1733$ | $1854$
$R_{2}(y^{LR}_{20,12})$ | $1852$ | $2012$
$R_{2}(y^{RL}_{20,12})$ | $1733$ | $1917$
$\tilde{R}_{2}(y^{RL}_{21,12})$ | $1733$ | $1767$
$S_{3}(y^{LL}_{30,12})$ | $1772$ | $1882$

((a))  (a)

((b))  (b)

((c))  (c)

((d))  (d)

((e))  (e)

((f))  (f)

((g))  (g)

((h))  (h)

((i))  (i)

((j))  (j)

((k))  (k)

((l))  (l)

((m))  (m)

((n))  (n)

Figure 4: Exclusion limits on all sLQs when only one Yukawa coupling is
nonzero (see Appendix A for the notation). The contributions from the PP,
PP+SP, and PP+SP+IP+II processes are separately marked; UA and DA indicate up
and down-aligned scenarios, respectively (see Table 1).

((a))  (a)

((b))  (b)

((c))  (c)

((d))  (d)

((e))  (e)

((f))  (f)

((g))  (g)

((h))  (h)

((i))  (i)

((j))  (j)

((k))  (k)

((l))  (l)

((m))  (m)

((n))  (n)

((o))  (o)

Figure 5: Same as Fig. 4, except here, two Yukawa couplings are nonzero.

((a))  (a)

((b))  (b)

((c))  (c)

((d))  (d)

((e))  (e)

((f))  (f)

((g))  (g)

((h))  (h)

((i))  (i)

Figure 6: Single-coupling exclusion limits when the quark involved is a third-
generation quark.

##  IV Exclusion limits

We summarise the coupling-independent exclusion limits on the sLQs obtained
from the $\mu\mu jj$ data in Table 3. The differences between the limits come
from the differences in the BRs.

###  IV.1 The singlets ($S_{1}$ and $\widetilde{S_{1}}$)

The $S_{1}$ LQ has four Yukawa couplings through which it can decay to $\mu j$
final states: $y^{LL}_{10,12}$, $y^{RR}_{10,12}$, $y^{LL}_{10,22}$, and
$y^{RR}_{10,22}$. We show the recast ($95\%$ CL) exclusion limits on these
couplings separately in Figs. 4(a)–4(d). In these plots, we show how the
limits change if we consider only the PP or PP+SP or PP+SP+IP+II processes. We
also show the $2\sigma$ limits from the high-$p_{T}$ dimuon tail. We see that
for $S_{1}$, the dimuon limits are the hardest. There are some essential
points to note here. First, the indirect interference is destructive in the
case of the $S_{1}$, making the PP+SP+IP+II limits weaker than the PP+SP ones
but, at the same time, making the dimuon limits stronger. Second, bigger
regions are excluded when $S_{1}$ mainly couples to the first-generation
quarks as compared to the second-generation quarks because of the difference
between the PDFs of valance and sea quarks. Third, for the left-handed
couplings, BR$(S_{1}\to\mu j)=50$% but it is 100% for the right-handed
couplings. As a result, for the same quark generation, the resonant (PP and
PP+SP) limits are stronger for the right-handed couplings than for the left-
handed couplings. Finally, as indicated earlier, the direct limits on the
left-handed couplings vary depending on whether $S_{1}$ is up-aligned or down-
aligned. However, the difference is visible only in the case of the
$y^{LL}_{10,22}$ coupling [see Fig. 4(c)], i.e., when $S_{1}$ couples mainly
to the second-generation quarks. Because of the smaller second-generation
quark PDFs, the relative difference is only marginal for $y^{LL}_{10,12}$.

Limits on these couplings taken two at a time are shown in Figs. 5(a)–5(e) for
different $M_{S_{1}}$ values. Like the one-coupling plots, regions excluded by
the LHC data (only $\mu\mu jj$) are shown with colours.

For $\widetilde{S}_{1}$, we plot the exclusion limits on $y^{RR}_{11,12}$ and
$y^{RR}_{11,22}$ on the mass-coupling planes in Figs. 4(e) and 4(f),
respectively. The interference in this case is positive. Hence, unlike the
case of $S_{1}$, the combined direct limits (PP+SP+IP+II) are the strongest,
and the dimuon limits are relatively weaker. The simultaneous limits on these
two couplings are shown in Fig. 5(g).

Figure 7: The LHC data excludes parts of the parameter space (blue) where the sLQs can explain the muon $g-2$ anomaly.  Table 4: Extrapolated maximum mass exclusion limit (TeV) for perturbative $y\approx\sqrt{4\pi}$ for different sLQs. To obtain these, we have assumed only one coupling is nonzero at a time. sLQ | $y$ | Limit | $y$ | Limit | $y$ | Limit | Limit
---|---|---|---|---|---|---|---
($\mu b\mu b$) |  ($\mu t\mu t$)
$S_{1}$ | $y^{LL}_{10,12}$ | $3.5$ | $y^{LL}_{10,22}$ | $2.5$ | $y^{LL}_{10,32}$ | — | $1.6$
$y^{RR}_{10,12}$ | $4.4$ | $y^{RR}_{10,22}$ | $2.8$ | $y^{RR}_{10,32}$ | — | $1.9$
$\tilde{S_{1}}$ | $y^{RR}_{11,12}$ | $27.0$ | $y^{RR}_{11,22}$ | $12.0$ | $y^{RR}_{11,32}$ | $2.7$ | —
$R_{2}$ | $y^{LR}_{20,12}$ | $28.0$ | $y^{LR}_{20,22}$ | $12.0$ | $y^{LR}_{20,32}$ | $2.6$ | $2.3$
$y^{RL}_{20,12}$ | $27.0$ | $y^{RL}_{20,22}$ | $9.0$ | $y^{RL}_{20,32}$ | — | $2.1$
$\tilde{R_{2}}$ | $y^{RL}_{21,12}$ | $8.2$ | $y^{RL}_{21,22}$ | $5.6$ | $y^{RL}_{21,32}$ | $2.6$ | —
$S_{3}$ | $y^{LL}_{30,12}$ | $33.0$ | $y^{LL}_{30,22}$ | $18.0$ | $y^{LL}_{30,32}$ | $3.1$ | $1.6$

###  IV.2 The doublets ($R_{2}$ and $\widetilde{R}_{2}$)

For $R_{2}$, we show the separate limits on $y^{LR}_{20,12}$,
$y^{LR}_{20,22}$, $y^{RL}_{20,12}$, and $y^{RL}_{20,22}$ on the mass-coupling
planes in Figs. 4(g)–4(j), respectively. The interference is positive in the
case of $R_{2}$ and, with the $y^{LR}_{20,ij}$ couplings, both its components
($R_{2}^{5/3}$ and $R_{2}^{2/3}$) can significantly contribute to the
$\mu\mu(jj)$ data, making the recast limits stronger than the singlet cases in
general. The two-coupling bounds on $R_{2}$ are shown in Figs. 5(h)–5(m).

For $\widetilde{R}_{2}$, the limits on $y^{RL}_{21,12}$ and $y^{RL}_{21,22}$
are shown in Figs. 4(k) and 4(l), respectively. The simultaneous limits on
these two couplings for different $M_{\widetilde{R_{2}}}$ are shown in Fig.
5(n).

One can also make similar observations for the doublets as we did in the case
of $S_{1}$ (e.g., the difference coming from first and second-generation PDFs,
etc.). However, there is a difference—if only one (muon) coupling is nonzero,
BR$(R_{2}^{n}/\widetilde{R}_{2}^{5/3}\to\mu j)=100\%$, irrespective of the
chiralities of the quark and the lepton.

###  IV.3 The triplet $S_{3}$

There are only two couplings for $S_{3}$ to contribute to $\mu\mu(jj)$ final
states: $y^{LL}_{30,12}$ and $y^{LL}_{30,22}$. We show the limits on the
couplings separately in Figs. 4(m) and 4(n), and simultaneously in Fig. 5(o).

###  IV.4 Heavy jets: the third-generation quarks

So far, we have implicitly assumed that the jets in the direct searches are
light. However, the ATLAS $\mu\mu jj$ search allows the jets to be $b$ jets
and puts bound on the $\mu b\mu b$ final state. We recast the $\mu b\mu b$
mass limits to draw bounds on the sLQ couplings with the bottom quark and the
muon ($y_{32}$). For completeness, we also recast the $\mu t\mu t$ bounds
obtained by CMS CMS:2022nty . We show the limits on these couplings for
various sLQs in Fig. 6. In these cases, we do not show the relatively minor
indirect or interference contributions to the limits. The reason these are
small is because the third-generation quarks have small/vanishing PDFs and
they are unlikely to appear as radiation.

###  IV.5 Illustration: the muon $g-2$ anomaly and the LHC limits

As mentioned in the introduction, one of the major reasons for the recent
popularity of sLQs in the literature is their ability to explain experimental
anomalies like the one seen in muon $g-2$. In Fig. 7, we draw the LHC bounds
on two sLQs ($S_{1}$ and $R_{2}$) known to address the muon $g-2$ anomaly to
illustrate how the current LHC $\mu\mu jj$ data impinge on the relevant
parameter space. We have used the expressions available in Refs.
Dorsner:2016wpm ; De:2023acg  to obtain the $g-2$ parameter regions in these
plots. One can draw similar comparisons with sLQ-parameter spaces relevant to
other anomalies as well.

###  IV.6 Extrapolating the limits

From the one-coupling plots in Figs. 4 and 6, we see that the mass exclusion
bound strengthens as a coupling increases. Naturally, one can ask about the
maximum mass bound for a perturbative new coupling. Table. 4 shows the maximum
(extrapolated) limits for all possible perturbative couplings, assuming only
one nonzero coupling. Of course, the limits weaken if more than one coupling
is nonzero and BR$($LQ $\to\ell j)$ is reduced.

##  V Summary and Discussions

Since LQs play essential roles in a wide range of BSM scenarios, it is crucial
to interpret the experimental limits on them closely. In this paper, we
revisited the exclusion limits on sLQs from the latest LHC $\mu\mu jj$ data.
The pair production searches at the LHC usually obtain largely model-agnostic
exclusion bounds on sLQ masses. On the other hand, the high-$p_{\rm T}$-tails
of the dilepton (or monolepton $+~{}\not{E}_{\rm T}$) resonance search data
provide bounds on the sLQ-quark-lepton Yukawa couplings. Our study showed
that, in some cases, the LHC data exclude more parameter space than what we
get by putting the above limits together.

Our study essentially expanded the basic idea of Ref. Mandal:2015vfa , which
argued for a systematic combination of the events from the signal processes
that could produce the same (or experimentally indistinguishable) final
states. There, it was shown how one could include single-production events in
the pair production signal (and vice versa) to obtain bounds on the Yukawa
couplings from the pair production (dilepton-dijet) data. In our analysis, we
extended the scope of the signal and included events from pair and single
productions, $t$-channel sLQ exchange and its interference with the SM
background processes. Our study is comprehensive as we considered all sLQs
that can couple with a muon and a quark. Our results point to some interesting
observations. First, the $\mu\mu jj$-recast limits are very restrictive and,
in some cases, are the strongest. Second, when calculating the model-
independent limits (i.e., assuming the $y\to 0$ limit), one ignores the QED-
mediated contributions. However, Table 3 showed that the QED effects were not
negligible in all cases, especially when sLQs carry higher electric charges.
Moreover, the model-independent limits on the doublet and triplet sLQs are
stronger than that on the singlet $S_{1}$ that decays to the $\mu j$ final
state with $100\%$ BR. This happens because there are multiple components of
the doublet and triplet sLQs which contribute to the same final state and
hence contribute to the limits.

## Acknowledgements

We thank Anirudhan for the discussions on muon $g-2$. We also thank Gokul and
Rithika for helping with FeynRules model files. RS would like to acknowledge
the Prime Minister’s Research Fellowship (PMRF ID: 0802000).

##  Appendix A Notation for the Yukawa couplings

We use an intuitive notation for the LQ Yukawa couplings in our paper and the
FeynRules model files, publicly available at GitHub. In this notation, the
sLQ-quark-lepton Yukawa couplings are generically denoted as
$y_{pq,\,ij}^{AB}$, where the subscript $p$ indicates the $SU(2)$
representation of the sLQ; $q$ is $1$ or $2$ depending on whether there is a
tilde ($\sim$) or bar ($-$) over the LQ, it is zero otherwise; $i$ and $j$
denote the quark and the lepton generations, respectively. The superscripts
$A$ and $B$ show the quark and lepton chiralities, respectively. For example,
$y^{RR}_{11,\,22}$ indicates the coupling of $\widetilde{S}_{1}$ with a right-
handed second-generation quark and a right-handed second-generation lepton.
Similarly, $y^{RL}_{21,\,12}$ is the coupling of the $SU(2)$-doublet $R_{2}$
with a right-handed first-generation quark and left-handed second-generation
lepton.

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