aainstitutetext: University of Split, Faculty of Electrical Engineering,
Mechanical Engineering and Naval Architecture in Split, Ruđera Boškovića 32,
HR-21000 Split, Croatiabbinstitutetext: J. Stefan Institute, Jamova 39, P. O.
Box 3000, SI-1001 Ljubljana, Sloveniaccinstitutetext: Department of Physics,
Faculty of Science, University of Zagreb, Bijenička cesta 32, HR-10000 Zagreb,
Croatiaddinstitutetext: Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland

# Asymmetric Leptoquark Pair Production at LHC

Ilja Doršner c Ajla Lejlić d Shaikh Saad<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate asymmetric leptoquark pair production mechanism at the Large
Hadron Collider to advocate its potential relevance to establish reliable
constraints on the leptoquark parameter space and its ability to aid in
correct identification of these attractive sources of new physics. The main
feature of asymmetric pair production that genuinely distinguishes it from the
usual leptoquark pair production is given by the fact that the two leptoquarks
that are produced in proton-proton collisions through a $t$-channel lepton
exchange are not charge conjugates of each other. Hence the proposed name of
asymmetric leptoquark pair production for this type of process. We spell out
prerequisite conditions for the asymmetric leptoquark pair production
mechanism to be operational and enumerate all possible combinations of
leptoquark multiplets that can potentially generate it. We finally reinterpret
existing leptoquark pair production search results within several simple
scalar leptoquark extensions of the Standard Model, assuming that the
leptoquarks exclusively couple to either electrons or muons and the first
generation quarks, to demonstrate proper inclusion of asymmetric pair
production. We consequently present accurate parameter space constraints for
the $S_{1}$, $S_{3}$, $R_{2}$, $S_{1}$+$S_{3}$, and $S_{1}$+$R_{2}$ leptoquark
scenarios.

##  1 Introduction

Leptoquark pair production is the only available process to efficiently look
for these hypothetical particles at hadron colliders when the coupling
strength between the relevant quark-lepton pairs and a leptoquark is small. It
is thus clear that the search for leptoquarks via pair production is always
going to be an integral part of the Large Hadron Collider (LHC) experimental
agenda in years, if not decades, to come. (For a sample of the leptoquark pair
production search results, see Refs. ATLAS:2020dsk ; ATLAS:2020xov ;
CMS:2020wzx ; ATLAS:2021oiz ; CMS:2022zks .) The leptoquark pair production
cross sections applicable to LHC are accordingly available at the next-to-
leading order Kramer:1997hh ; Kramer:2004df ; Mandal:2015lca ; Dorsner:2018ynv
as well as the next-to-next-to-leading order Beenakker:2016lwe ;
Beenakker:1997ut ; Beenakker:2010nq ; Beenakker:2016gmf  in strong coupling
constant, and, more recently, at the next-to-leading order in both the strong
coupling constant and the leptoquark Yukawa coupling(s) Borschensky:2020hot .

As the strength of interaction between the quark-lepton pairs and a leptoquark
is gradually increased, the collider searches for signals from several other
processes start to be relevant in constraining the leptoquark parameter space.
These processes, at the LHC, are a single leptoquark production Alves:2002tj ;
Dorsner:2014axa ; Hammett:2015sea ; Mandal:2015vfa ; Dorsner:2018ynv ;
Schmaltz:2018nls , a non-resonant production of the Drell-Yan type
Faroughy:2016osc ; Raj:2016aky ; Greljo:2017vvb ; Bansal:2018eha ;
Schmaltz:2018nls ; Fuentes-Martin:2020lea ; Allwicher:2022gkm , and a resonant
leptoquark production Ohnemus:1994xf ; Eboli:1997fb ; Buonocore:2020erb ;
Greljo:2020tgv ; Buonocore:2022msy .

Note, however, that even the leptoquark pair production exhibits dependence on
the Yukawa coupling strength Dorsner:2014axa . This is especially true in the
case of a novel mechanism of leptoquark pair production that has been recently
introduced in Ref. Dorsner:2021chv . The main feature of this novel mechanism
that distinguishes it from the usual leptoquark pair production at the LHC is
the fact that the two leptoquarks that are produced in proton-proton
collisions through a $t$-channel exchange of a lepton do not comprise a charge
conjugate pair. This is a primary reason why we refer to it as an asymmetric
leptoquark pair production in this study. The novel production mechanism,
though, can yield the same final state as the conventional pair production. In
fact, the final state kinematics should be exactly the same if the leptoquarks
in question are degenerate in mass. This work aims to address the correct
interpretation of existing and future experimental search results for those
final states that are due to the leptoquark pair production processes and
subsequent leptoquark decays if one appropriately incorporates the
aforementioned asymmetric mechanism contributions. It dovetails the initial
analysis of Ref. Dorsner:2021chv  and extends the scope of the
phenomenological discussion of asymmetric pair production presented therein.
It also nicely complements recent work on the inclusion of the asymmetric pair
production mechanism in the next-to-leading order in QCD cross section
determinations for the leptoquark pair production Borschensky:2022xsa . We
stress that the asymmetric contributions to the leptoquark pair production
have not been included in any of publicly available experimental search
analyses thus far.

We will, for definiteness, focus our attention solely on the scalar leptoquark
extensions of the Standard Model (SM). We accordingly present in Table 1 a
list of pertinent scalar leptoquarks and associated transformation properties
under the SM gauge group $SU(3)\times SU(2)\times U(1)$. Since the chirality
of the leptons that the scalar leptoquark couples to is very important for our
discussion, we indicate relevant chiralities of both quarks and leptons using
$R$ and $L$ for right- and left-chiral fields, respectively, in the third
column of Table 1. Our convention is such that the first (second) letter, in
that column, denotes chirality of quarks (leptons). For example, the fact that
$S_{1}$ leptoquark can directly couple to the $SU(2)$ doublets of quarks and
leptons or/and the $SU(2)$ singlets of quarks and leptons is indicated by
simultaneous presence of $LL$ and $RR$ designations in the third column of
Table 1.

$(SU(3),SU(2),U(1))$ | LQ SYMBOL | CHIRALITY TYPE (LQ-$q$-$l$) | $F$
---|---|---|---
$(\overline{\mathbf{3}},\mathbf{3},1/3)$ | $S_{3}$ | $LL$ | $-2$
$(\mathbf{3},\mathbf{2},7/6)$ | $R_{2}$ |  $RL$, $LR$ | $0$
$(\mathbf{3},\mathbf{2},1/6)$ | $\tilde{R}_{2}$ | $RL$ | $0$
$(\overline{\mathbf{3}},\mathbf{1},4/3)$ | $\tilde{S}_{1}$ | $RR$ | $-2$
$(\overline{\mathbf{3}},\mathbf{1},1/3)$ | $S_{1}$ |  $LL$, $RR$ | $-2$
Table 1: Scalar leptoquark multiplets, chiralities of the leptoquark
interactions with the SM quark-lepton pairs, and associated leptoquark fermion
numbers.

We also specify fermion number $F$ of scalar leptoquark multiplets in Table 1,
where $F$ is defined as the sum of the lepton number and three times the
baryon number of leptons and quarks that a given leptoquark couples to.
Leptoquarks with $F=-2$ exclusively couple/decay to quarks and leptons whereas
$F=0$ leptoquarks couple/decay to quark-antilepton or antiquark-lepton pairs.

Since our hyper-charge normalization is $Q=I_{3}+Y$, where $Q$ corresponds to
electric charge in units of the positron charge, $I_{3}$ stands for the
diagonal generator of $SU(2)$, and $Y$ represents $U(1)$ hyper-charge
operator, the electric charge eigenvalues of scalar leptoquarks in Table 1 are
$S_{3}^{+4/3}$, $S_{3}^{+1/3}$, $S_{3}^{-2/3}$, $R_{2}^{+5/3}$,
$R_{2}^{+2/3}$, $\tilde{R}_{2}^{+2/3}$, $\tilde{R}_{2}^{-1/3}$,
$\tilde{S}_{1}^{+4/3}$, and $S_{1}^{+1/3}$. We will always denote leptoquarks
using this notation and furthermore write, for simplicity, that
$(\mathrm{LQ}^{+Q})^{*}=\mathrm{LQ}^{-Q}$ and
$(\mathrm{LQ}^{-Q})^{*}=\mathrm{LQ}^{+Q}$. Note that leptoquarks of the same
electric charge can, in principle, mix with each other upon the breaking of
the SM symmetry down to $SU(3)\times U(1)_{\mathrm{em}}$ even if they have
different fermion numbers. This type of mixing can lead to interesting
physical phenomena that are somewhat orthogonal to our study. This is the main
reason why we neglect all such possible mixings terms.

There are several prerequisite conditions for the asymmetric leptoquark pair
production mechanism under consideration to be operational Dorsner:2021chv .
First, it requires non-negligible Yukawa coupling(s) between leptoquarks and
the SM quarks and leptons. Second, this mechanism is relevant whenever there
exist at least two leptoquark states originating from the same or two
different leptoquark multiplets that couple to a lepton of the same chirality
and flavor. This, then, leads to a simple schematic representation shown in
Fig. 1 of all possible minimal leptoquark combinations that can potentially
generate asymmetric pair production at hadron colliders and, consequentially,
LHC.

Figure 1: Schematic classification of potential sources of asymmetric
leptoquark pair production. See text for details.

The double-headed arrows in Fig. 1 connect those leptoquark multiplets that
can simultaneously couple to a lepton of the same flavor and chirality. For
example, $R_{2}$ can couple to the SM leptons of both chiralities
Dorsner:2016wpm  as indicated with the $RL$ and $LR$ designations in Fig. 1
and Table 1. Again, it is the second letter that denotes the lepton chirality.
If $R_{2}$ couples to the left-chiral leptons, it can, in principle,
participate in the asymmetric pair production with all those multiplets that
can also couple to the left-chiral leptons such as $S_{3}$, $S_{1}$, and
$\tilde{R}_{2}$. If $R_{2}$ couples to the right-chiral leptons, it can
potentially contribute to asymmetric pair production on its own, as indicated
in Fig. 1, and/or in conjunction with $S_{1}$ and $\tilde{S}_{1}$.

The double-headed arrows in Fig. 1 are color-coded either blue or red to
distinguish between two different initial state configurations behind the
relevant asymmetric pair production processes even though the leptoquark pairs
in question are always generated in proton-proton collisions via a $t$-channel
lepton exchange. If the two leptoquarks $\mathrm{LQ}_{1}$ and
$\mathrm{LQ}_{2}$ have different fermion numbers, i.e., $\Delta
F=|F(\mathrm{LQ}_{1})|-|F(\mathrm{LQ}_{2})|=\pm 2$, the initial states are of
the $qq^{\prime}$ and $\overline{q}\overline{q^{\prime}}$ nature, where $q$
and $q^{\prime}$ denote the quark fields and can, in principle, be equal to
$u$, $d$, $s$, $c$, and $b$. These scenarios are indicated with blue double-
headed arrows in Fig. 1. If, on the other hand, leptoquarks have the same
fermion number, i.e., $\Delta F=0$, the initial states are of the
$q\overline{q^{\prime}}$ and $\overline{q}q^{\prime}$ nature, where, again,
$q,q^{\prime}=u,d,s,c,b$. The $\Delta F=0$ scenarios are depicted with red
double-headed arrows in Fig. 1. In view of all these requirements, we note
that it is entirely possible to have a new physics scenario with only one
scalar leptoquark multiplet and only one non-zero Yukawa coupling and still be
able to asymmetrically produce leptoquark pairs at the LHC Dorsner:2021chv .
There are two such scenarios, as indicated in Fig. 1. One is generated if a
single non-zero Yukawa coupling exists between $R_{2}$ and any right-chiral
charged lepton. The other one requires presence of a single non-zero Yukawa
coupling for $S_{3}$.

It is possible to succinctly depict all the relevant diagrams that result in
asymmetric pair production at hadron colliders. There are, all in all, six
such $t$-channel diagrams that can potentially generate asymmetric pair
production. We present these diagrams in Fig. 2 and then summarize in Tables 2
and 3 associated scenarios that require presence of, at most, two scalar
leptoquark multiplets when $\Delta F=0$ and $\Delta F=\pm 2$, respectively.

Figure 2: Types of diagrams for asymmetric production. $q$, $q^{\prime}$, $l$, $\mathrm{LQ}_{1}$, and $\mathrm{LQ}_{2}$ for $\Delta F=|F(\mathrm{LQ}_{1})-|F(\mathrm{LQ}_{2})|=\pm 2$ and $\Delta F=|F(\mathrm{LQ}_{1})|-|F(\mathrm{LQ}_{2})|=0$ are specified in Tables 2 and 3, respectively. Here $l$ refers to a charged lepton or a neutrino. Diagram Type | $q$ | $\overline{q^{\prime}}$ | $l$ | LQ1 | LQ2 | LQ scenario
---|---|---|---|---|---|---
A | $u_{L}$ | $\overline{d}_{L}$ | $\ell_{R}$ | $R_{2}^{+5/3}$ | $R_{2}^{+2/3}$ |  $R_{2}$
A | $d_{L}$ | $\overline{u}_{L}$ | $\ell_{R}$ | $R_{2}^{-2/3}$ | $R_{2}^{-5/3}$
A′ | $d_{L}$ | $\overline{u}_{L}$ | $\nu_{L}$ | $S_{3}^{-1/3}$ | $S_{3}^{-2/3}$ |  $S_{3}$
A′ | $u_{L}$ | $\overline{d}_{L}$ | $\nu_{L}$ | $S_{3}^{+2/3}$ | $S_{3}^{+1/3}$
A′ | $d_{L}$ | $\overline{u}_{L}$ | $\ell_{L}$ | $S_{3}^{-4/3}$ | $S_{3}^{+1/3}$
A′ | $u_{L}$ | $\overline{d}_{L}$ | $\ell_{L}$ | $S_{3}^{-1/3}$ | $S_{3}^{+4/3}$
A | $d_{R}$ | $\overline{u}_{R}$ | $\ell_{L}$ | $\widetilde{R}_{2}^{+2/3}$ | $R_{2}^{-5/3}$ |  $\widetilde{R}_{2}$+$R_{2}$
A | $u_{R}$ | $\overline{d}_{R}$ | $\ell_{L}$ | $R_{2}^{+5/3}$ | $\widetilde{R}_{2}^{-2/3}$
A | $d_{R}$ | $\overline{u}_{R}$ | $\nu_{L}$ | $\widetilde{R}_{2}^{-1/3}$ | $R_{2}^{-2/3}$
A | $u_{R}$ | $\overline{d}_{R}$ | $\nu_{L}$ | $R_{2}^{+2/3}$ | $\widetilde{R}_{2}^{+1/3}$
A′ | $d_{R}$ | $\overline{u}_{R}$ | $\ell_{R}$ | $\widetilde{S}_{1}^{-4/3}$ | $S_{1}^{+1/3}$ |  $\widetilde{S}_{1}$+$S_{1}$
A′ | $u_{R}$ | $\overline{d}_{R}$ | $\ell_{R}$ | $S_{1}^{-1/3}$ | $\widetilde{S}_{1}^{+4/3}$
A′ | $d_{L}$ | $\overline{u}_{L}$ | $\nu_{L}$ | $S_{1}^{-1/3}$ | $S_{3}^{-2/3}$ |  $S_{1}$+$S_{3}$
A′ | $u_{L}$ | $\overline{d}_{L}$ | $\nu_{L}$ | $S_{3}^{+2/3}$ | $S_{1}^{+1/3}$
A′ | $u_{L}$ | $\overline{d}_{L}$ | $\ell_{L}$ | $S_{1}^{-1/3}$ | $S_{3}^{+4/3}$
A′ | $d_{L}$ | $\overline{u}_{L}$ | $\ell_{L}$ | $S_{3}^{-4/3}$ | $S_{1}^{+1/3}$
Table 2: Asymmetric production with $q\overline{q^{\prime}}$ and $\overline{q}q^{\prime}$ initial states. See Fig. 2 for the diagram type. Diagram Type | $q/\overline{q}$ | $q^{\prime}/\overline{q^{\prime}}$ | $l$ | LQ1 | LQ2 | LQ scenario
---|---|---|---|---|---|---
B′ | $d_{R}$ | $u_{L}$ | $\ell_{R}$ | $\widetilde{S}_{1}^{-4/3}$ | $R_{2}^{+5/3}$ |  $\widetilde{S}_{1}$+$R_{2}$
B′ | $d_{R}$ | $\overline{d}_{L}$ | $\ell_{R}$ | $\widetilde{S}_{1}^{-4/3}$ | $R_{2}^{+2/3}$
C′ | $\overline{u}_{L}$ | $\overline{d}_{R}$ | $\ell_{R}$ | $R_{2}^{-5/3}$ | $\widetilde{S}_{1}^{+4/3}$
C′ | $\overline{d}_{L}$ | $\overline{d}_{R}$ | $\ell_{R}$ | $R_{2}^{-2/3}$ | $\widetilde{S}_{1}^{+4/3}$
B | $u_{R}$ | $d_{L}$ | $\nu_{L}$ | $R_{2}^{+2/3}$ | $S_{1}^{-1/3}$ |  $S_{1}$+$R_{2}$
B | $u_{R}$ | $u_{L}$ | $\ell_{L}$ | $R_{2}^{+5/3}$ | $S_{1}^{-1/3}$
B | $u_{L}$ | $u_{R}$ | $\ell_{R}$ | $R_{2}^{+5/3}$ | $S_{1}^{-1/3}$
B | $d_{L}$ | $u_{R}$ | $\ell_{R}$ | $R_{2}^{+2/3}$ | $S_{1}^{-1/3}$
C | $\overline{d}_{L}$ | $\overline{u}_{R}$ | $\nu_{L}$ | $S_{1}^{+1/3}$ | $R_{2}^{-2/3}$
C | $\overline{u}_{L}$ | $\overline{u}_{R}$ | $\ell_{L}$ | $S_{1}^{+1/3}$ | $R_{2}^{-5/3}$
C | $\overline{u}_{R}$ | $\overline{u}_{L}$ | $\ell_{R}$ | $S_{1}^{+1/3}$ | $R_{2}^{-5/3}$
C | $\overline{u}_{R}$ | $\overline{d}_{L}$ | $\ell_{R}$ | $S_{1}^{+1/3}$ | $R_{2}^{-2/3}$
B | $d_{R}$ | $u_{L}$ | $\ell_{L}$ | $\widetilde{R}_{2}^{+2/3}$ | $S_{1}^{-1/3}$ |  $S_{1}$+$\widetilde{R}_{2}$
C | $\overline{u}_{L}$ | $\overline{d}_{R}$ | $\ell_{L}$ | $S_{1}^{+1/3}$ | $\widetilde{R}_{2}^{-2/3}$
B | $u_{R}$ | $d_{L}$ | $\ell_{L}$ | $R_{2}^{+5/3}$ | $S_{3}^{-4/3}$ |  $S_{3}$+$R_{2}$
B | $u_{R}$ | $u_{L}$ | $\ell_{L}$ | $R_{2}^{+5/3}$ | $S_{3}^{-1/3}$
B | $u_{R}$ | $d_{L}$ | $\nu_{L}$ | $R_{2}^{+2/3}$ | $S_{3}^{-1/3}$
C | $\overline{d}_{L}$ | $\overline{u}_{R}$ | $\ell_{L}$ | $S_{3}^{+4/3}$ | $R_{2}^{-5/3}$
C | $\overline{u}_{L}$ | $\overline{u}_{R}$ | $\ell_{L}$ | $S_{3}^{+1/3}$ | $R_{2}^{-5/3}$
C | $\overline{d}_{L}$ | $\overline{u}_{R}$ | $\nu_{L}$ | $S_{3}^{+1/3}$ | $R_{2}^{-2/3}$
B | $d_{R}$ | $d_{L}$ | $\ell_{L}$ | $\widetilde{R}_{2}^{+2/3}$ | $S_{3}^{-4/3}$ |  $S_{3}$+$\widetilde{R}_{2}$
B | $d_{R}$ | $u_{L}$ | $\ell_{L}$ | $\widetilde{R}_{2}^{+2/3}$ | $S_{3}^{-1/3}$
B | $d_{R}$ | $u_{L}$ | $\nu_{L}$ | $\widetilde{R}_{2}^{-1/3}$ | $S_{3}^{+2/3}$
C | $\overline{d}_{L}$ | $\overline{d}_{R}$ | $\ell_{L}$ | $S_{3}^{+4/3}$ | $\widetilde{R}_{2}^{-2/3}$
C | $\overline{u}_{L}$ | $\overline{d}_{R}$ | $\ell_{L}$ | $S_{3}^{+1/3}$ | $\widetilde{R}_{2}^{-2/3}$
C | $\overline{u}_{L}$ | $\overline{d}_{R}$ | $\nu_{L}$ | $S_{3}^{-2/3}$ | $\widetilde{R}_{2}^{+1/3}$
Table 3: Asymmetric production with $qq^{\prime}$ and
$\overline{q}\overline{q^{\prime}}$ initial states. See Fig. 2 for the diagram
type.

The schematics in Fig. 1, diagrams of Fig. 2, and Tables 2 and 3 give a
complete classification of the asymmetric pair production processes at hadron
colliders. With this exhaustive classification completed we turn our attention
towards more quantitative discussion of aforementioned mechanism.

The rest of the manuscript is organised as follows. In Sec. 2 we address
subtleties associated with both the asymmetric and conventional leptoquark
pair productions and present several specific instances of inclusion of
asymmetric pair production into the usual search strategy for leptoquarks,
assuming that the leptoquarks in question exclusively couple to either
electrons or muons and the first generation quarks. We consequently present
accurate parameter space constraints for the $S_{1}$, $S_{3}$, $R_{2}$,
$S_{1}$+$S_{3}$, and $S_{1}$+$R_{2}$ leptoquark scenarios, where, for the
electron coupling case, we generate in Sec. 3 the latest limits from the
atomic parity violation (APV) searches. We briefly conclude in Sec. 4.

##  2 Asymmetric pair production

Asymmetric pair production mechanism we want to investigate produces two
leptoquarks $\mathrm{LQ}_{1}$ and $\mathrm{LQ}_{2}$ that are not charge
conjugates of each other through one or more of the $t$-channel diagrams of
Fig. 2. There is thus no interference between the asymmetric and conventional
leptoquark pair productions at the amplitude level even though the final state
signatures of both processes, upon the $\mathrm{LQ}_{1}$ and $\mathrm{LQ}_{2}$
subsequent decays, can be exactly the same. We can accordingly focus our
attention solely on the asymmetric pair production cross sections that can be
simply added, if and when appropriate, to the conventional pair production
cross sections. We work, for simplicity, at the leading order in QCD and
denote the cross sections of interest with

$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}_{1}},m_{\mathrm{LQ}_{2}})=a_{q_{1}q_{2}}(m_{\mathrm{LQ}_{1}},m_{\mathrm{LQ}_{2}})|y_{q_{1}}y_{q_{2}}|^{2},$
(1)

where
$q_{1},q_{2}=u,\overline{u},d,\overline{d},s,\overline{s},c,\overline{c},b,\overline{b}$.
Here, leptoquark $\mathrm{LQ}_{i}$ of mass $m_{\mathrm{LQ}_{i}}$ couples to a
quark $q_{i}$ and a lepton $l$ of a given chirality and flavor with strength
$y_{q_{i}}$, where $i=1,2$.

Note that the cross sections of Eq. (1) do not depend on whether
$\mathrm{LQ}_{1}$ couples to a quark $q_{1}$ while $\mathrm{LQ}_{2}$ couples
to a quark $q_{2}$ or vice versa. This is only relevant for subsequent
leptoquark decays. The cross sections of Eq. (1) also do not depend on the
type of lepton that leptoquarks $\mathrm{LQ}_{1}$ and $\mathrm{LQ}_{2}$
simultaneously couple to. They are proportional to a square of the product
$|y_{q_{1}}y_{q_{2}}|$ and can thus be trivially rescaled as a function of
Yukawa couplings once they are determined for one particular value of
$|y_{q_{1}}y_{q_{2}}|$ product.

We will make an assumption that $\mathrm{LQ}_{1}$ and $\mathrm{LQ}_{2}$ are
mass-degenerate, i.e., $m_{\mathrm{LQ}_{1}}=m_{\mathrm{LQ}_{2}}\equiv
m_{\mathrm{LQ}}$, and furthermore take all Yukawa couplings to be real. These
two assumptions allow us to introduce cross section
$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}})$
that is symmetric in flavor, i.e.,
$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}\equiv\sigma^{\mathrm{pair}}_{q_{2}q_{1}}$,
where

$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}})=\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}_{1}}=m_{\mathrm{LQ}_{2}}\equiv
m_{\mathrm{LQ}},m_{\mathrm{LQ}_{2}}).$ (2)

Note that the cross sections of Eq. (2) allow us to extract limits on the
leptoquark parameter space from existing experimental searches in a
straightforward fashion since the current analyses rely on an explicit
assumption of mass degeneracy for hypothetical leptoquark pairs being
produced.

There are fifteen cross sections
$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}})$ of
interest, at the LHC, when the initial states are quark-quark pairs and twenty
five when the initial states are quark-antiquark pairs. We are not interested
in the cross sections that are antiquark-antiquark initiated as these are
highly suppressed at the LHC although we include them for completeness in the
numerical simulation once we reinterpret current leptoquark search analyses
results.

The quark-quark initiated cross sections are given in Fig. 3 under the
assumption that $|y_{q_{1}}y_{q_{2}}|=1$, where $q_{1}=u,d,s,c$ and
$q_{2}=u,d,s,c,b$, while the quark-antiquark initiated cross sections are
given in Fig. 4 under the same assumption that $|y_{q_{1}}y_{q_{2}}|=1$, but,
this time around, with $q_{1}=u,d,s,c$ and
$q_{2}=\overline{u},\overline{d},\overline{s},\overline{c},\overline{b}$. We
also present in Figs. 3 and 4, for comparison purposes, conventional scalar
leptoquark pair production cross section at the LHC that is evaluated under
the assumption that the leptoquark Yukawa couplings are negligible but still
large enough to ensure prompt leptoquark decay. This particular cross section
is simply denoted with
$\sigma^{\mathrm{pair}}_{\mathrm{QCD}}(m_{\mathrm{LQ}})$ to stress that it is
purely QCD induced and it is represented by a thick dashed black curve in both
Figs. 3 and 4.

The asymmetric leptoquark pair production cross sections of Figs. 3 and 4 are
extracted from the new physics scenarios of Fig. 1, where all of them are
generated by the $t$-channel processes of Fig. 2. These scenarios are
implemented using FeynRules Alloul:2013bka  and subsequently imported in
MadGraph5_aMC@NLO framework Alwall:2014hca  to produce numerical results for
$m_{\mathrm{LQ}}$ values between 1.6 TeV and 2.6 TeV. We exclusively use the
nn23lo1 PDF set NNPDF:2014otw  to generate leading order cross sections for
the center-of-mass energy of proton-proton collisions set at 13 TeV, where the
factorisation ($\mu_{F}$) and renormalization ($\mu_{R}$) scales are taken to
be $\mu_{F}=\mu_{R}=m_{\mathrm{LQ}}/2$. Note that we only quote central values
for all cross sections as we are solely interested in relative strengths of
various potential contributions.

Figure 3: Asymmetric leptoquark pair production cross sections
$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}})$ for
quark-quark initial states, where $q_{1}=u,d,s,c$ and $q_{2}=u,d,s,c,b$.
Figure 4: Asymmetric leptoquark pair production cross sections
$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}})$ for
quark-antiquark initial states, where $q_{1}=u,d,s,c$ and
$q_{2}=\overline{u},\overline{d},\overline{s},\overline{c},\overline{b}$.

One can observe from Fig. 3 that the quark-quark initiated asymmetric pair
production cross sections of mass-degenerate scalar leptoquarks
$\mathrm{LQ}_{1}$ and $\mathrm{LQ}_{2}$, i.e., when $\Delta
F=|F(\mathrm{LQ}_{1})|-|F(\mathrm{LQ}_{2})|=\pm 2$ and
$m_{\mathrm{LQ}_{1}}=m_{\mathrm{LQ}_{2}}=m_{\mathrm{LQ}}$, can be comparable
to or be even substantially larger than the QCD driven leptoquark pair
production cross section at the LHC if at least one of the leptoquarks couples
to a valence quark and the product of relevant Yukawa cuplings is of order
one. For example,
$\sigma^{\mathrm{pair}}_{uu}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}})$ for
$|y_{q_{1}}y_{q_{2}}|=1$ is approximately two orders of magnitude larger than
$\sigma^{\mathrm{pair}}_{\mathrm{QCD}}(m_{\mathrm{LQ}})$. In all other
instances, i.e., for $q_{1},q_{2}=s,c,b$, the cross sections for asymmetric
pair production are, at best, a tiny correction of the QCD driven one, again,
for order one Yukawa coupling strengths. In the case of the quark-antiquark
initiated asymmetric pair production cross sections the only truly relevant
scenarios, once again, are those where the initial quark is a valence quark.
This is nicely illustrated in Fig. 4 with the direct comparison between the
QCD cross section rendered with a black thick dashed curve and the quark-
antiquark induced cross sections
$\sigma^{\mathrm{pair}}_{q_{1}q_{2}}(y_{q_{1}},y_{q_{2}},m_{\mathrm{LQ}})$,
where $q_{1}=u,d,s,c$ and
$q_{2}=\overline{u},\overline{d},\overline{s},\overline{c},\overline{b}$ and
$|y_{q_{1}}y_{q_{2}}|=1$. The reason why we opted to plot combinations
$\sigma^{\mathrm{pair}}_{u\overline{u}}+\sigma^{\mathrm{pair}}_{d\overline{d}}$,
$\sigma^{\mathrm{pair}}_{u\overline{c}}+\sigma^{\mathrm{pair}}_{d\overline{s}}$,
$\sigma^{\mathrm{pair}}_{c\overline{u}}+\sigma^{\mathrm{pair}}_{s\overline{d}}$
and
$\sigma^{\mathrm{pair}}_{c\overline{c}}+\sigma^{\mathrm{pair}}_{s\overline{s}}$
in Fig. 4, instead of individual cross sections, will be elaborated on in Sec.
2.1.5.

Figs. 3 and 4 demonstrate that it is entirely possible to have substantial
cross sections for the asymmetric leptoquark pair production even when one of
the leptoquarks couples weakly to the first generation of quarks whereas the
other leptoquark couples strongly to the second or third generation of quarks
as long as they both couple to a lepton of the same flavor and chirality. Of
course, quark-quark initiated processes of asymmetric leptoquark pair
production, i.e., when $\Delta F=|F(\mathrm{LQ}_{1})|-|F(\mathrm{LQ}_{2})|=\pm
2$, are potentially much more relevant at the LHC whereas quark-antiquark
initiated processes are naturally enhanced at the Tevatron like machines. Note
that the QCD cross section drops faster than the asymmetric cross sections as
the mass of leptoquarks is increased. This is due to the fact that the gluon-
gluon initiated processes start to be subdominant with respect to the
processes initiated by quarks once one goes towards the large leptoquark mass
limit.

Before we give an explicit example of the potential importance of the
asymmetric scalar leptoquark pair production mechanism we want to address one
subtlety associated with the conventional leptoquark pair production that has
not been discussed in the literature before.

Conventional leptoquark pair production amplitude, when the leptoquarks
comprising a pair are charge conjugates of each other, has two distinct
contributions at the leading order. The first one is of purely QCD nature
whereas the second one exhibits quadratic dependence on the leptoquark Yukawa
coupling $y_{q}$. For a single scalar leptoquark $\mathrm{LQ}$ that couples to
a quark $q$ and any lepton $l$ with Yukawa coupling $y_{q}$ the conventional
leptoquark pair production cross section can thus be written as

$\sigma^{\mathrm{pair}}_{q\overline{q}}(y_{q},m_{\mathrm{LQ}})=\sigma^{\mathrm{pair}}_{\mathrm{QCD}}(m_{\mathrm{LQ}})+a^{\mathrm{interference}}_{q\overline{q}}(m_{\mathrm{LQ}})y_{q}^{2}+\sigma^{\mathrm{pair}}_{q\overline{q}}(y_{q},y_{q},m_{\mathrm{LQ}}),$
(3)

where $\sigma^{\mathrm{pair}}_{\mathrm{QCD}}(m_{\mathrm{LQ}})$ and
$\sigma^{\mathrm{pair}}_{q\overline{q}}(y_{q},y_{q},m_{\mathrm{LQ}})$ have
been featured before and we assume, for consistency, that $y_{q}$ is real. If
$y_{q}$ is small, the cross section depends solely on the leptoquark mass
$m_{\mathrm{LQ}}$ and the particularities associated with the hadron machine
itself and it is given by
$\sigma^{\mathrm{pair}}_{\mathrm{QCD}}(m_{\mathrm{LQ}})$. In fact,
$\sigma^{\mathrm{pair}}_{\mathrm{QCD}}(m_{\mathrm{LQ}})$ has been known
analytically at the next-to-leading order in QCD for a long time Kramer:2004df
.

The last term in Eq. (3) corresponds to a $t$-channel exchange of a lepton $l$
with the $q\overline{q}$ pair in the initial state. Again, it does not depend
on the type of lepton that the leptoquark couples to and the relevant cross
sections are already introduced in Fig. 4 for $y_{q}=1$. Finally, there is the
interference term
$a^{\mathrm{interference}}_{q\overline{q}}(m_{\mathrm{LQ}})y_{q}^{2}$ that
turns out to always be negative. There is thus a dip in the pair production
cross section below the
$\sigma^{\mathrm{pair}}_{\mathrm{QCD}}(m_{\mathrm{LQ}})$ value as Yukawa
coupling is increased before $y_{q}$ becomes sufficiently large to make the
third term in Eq. (3) that is of quartic nature in terms of $y_{q}$ to start
to dominate over the interference term that is of quadratic nature in $y_{q}$.
We note, that the automated inclusion of the $t$-channel term at the next-to-
leading order in QCD has been recently introduced in the literature
Borschensky:2021hbo .

The things, though, can change with regard to interference effect if a scalar
leptoquark $\mathrm{LQ}$ couples to a quark $q$ and any lepton $l$ with Yukawa
coupling $y_{q}$ and another quark $q^{\prime}$ and the same lepton with
Yukawa coupling $y_{q^{\prime}}$. There will then exist four interference
terms $a^{\mathrm{interference}}_{q\overline{q}}(m_{\mathrm{LQ}})y_{q}^{2}$,
$a^{\mathrm{interference}}_{q^{\prime}\overline{q^{\prime}}}(m_{\mathrm{LQ}})y_{q^{\prime}}^{2}$,
$a^{\mathrm{interference}}_{q\overline{q^{\prime}}}(m_{\mathrm{LQ}})y_{q}y_{q^{\prime}}$,
and
$a^{\mathrm{interference}}_{q^{\prime}\overline{q}}(m_{\mathrm{LQ}})y_{q}y_{q^{\prime}}$,
where the last two can obviously exhibit constructive interference if $y_{q}$
and $y_{q^{\prime}}$ differ in sign. In fact, it might be even possible for
both $a^{\mathrm{interference}}_{q\overline{q}}(m_{\mathrm{LQ}})y_{q}^{2}$ and
$a^{\mathrm{interference}}_{q^{\prime}\overline{q^{\prime}}}(m_{\mathrm{LQ}})y_{q^{\prime}}^{2}$
to be less relevant than either
$a^{\mathrm{interference}}_{q\overline{q^{\prime}}}(m_{\mathrm{LQ}})y_{q}y_{q^{\prime}}$
or
$a^{\mathrm{interference}}_{q^{\prime}\overline{q}}(m_{\mathrm{LQ}})y_{q}y_{q^{\prime}}$.
The point we want to make here is that the conventional pair production of
leptoquarks might be sensitive not only to Yukawa coupling strengths but also
to the relative sign between relevant Yukawa couplings even when these
couplings are taken to be real.

With these preliminary considerations out of the way we now turn towards
quantitative analysis of the asymmetric pair production mechanism within
several concrete scenarios of new physics.

###  2.1 Case studies

Our primary aim is to advocate importance of inclusion of the asymmetric pair
production mechanism in a quantitative determination of the viable leptoquark
parameter space if and when appropriate. To that end, we discuss five
different leptoquark extensions of the SM and derive, for several particular
realisations of these extensions, accurate limits using two specific
experimental searches. More specifically, we recast the ATLAS Collaboration
analysis ATLAS:2020dsk  of the leptoquark pair production searches via
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ and
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$
processes, where $j$ is taken to generically represents a light jet, i.e.,
$j=u,\overline{u},d,\overline{d},s,\overline{s}$, while it is implicitly
understood that both $ee$ and $\mu\mu$ stand for oppositely charged lepton
pairs. All five scenarios provide a setting for pedagogical illustration of
various phenomenological intricacies associated with the leptoquark pair
production signatures.

First of these five scenarios involves a presence of a single scalar
leptoquark $S_{1}$. The second scenario of new physics is an $R_{2}$ extension
of the SM, where $R_{2}$ multiplet comprises two states, i.e., $R_{2}^{+5/3}$
and $R_{2}^{-2/3}$. Third scenario extends the SM particle content with both
$S_{1}$ and $R_{2}$ while forth scenario concerns addition of an $S_{3}$
leptoquark multiplet to the SM particle content, where $S_{3}$ contains
scalars $S_{3}^{+4/3}$, $S_{3}^{+1/3}$, and $S_{3}^{-2/3}$. Fifth scenario
regards simultaneous extension of the SM with both $S_{1}$ and $S_{3}$. We
will assume that all these leptoquarks exclusively couple to either electrons
or muons and the first generation quarks, to simplify discussion, where, for
the electron coupling case, we also produce in Sec. 3 the latest APV search
limits on the leptoquark parameter spaces.

Relevant parts of the $S_{1}$ lagrangian, for our study, are

$\displaystyle\mathcal{L}_{S_{1}}=$
$\displaystyle+y^{LL}_{1\,ij}\bar{Q}_{L}^{C\,i,a}S_{1}\epsilon^{ab}L_{L}^{j,b}+y^{RR}_{1\,ij}\bar{u}_{R}^{C\,i}S_{1}e_{R}^{j}+\textrm{h.c.}$
$\displaystyle=$
$\displaystyle-(y^{LL}_{1}U)_{ij}\bar{d}_{L}^{C\,i}\nu_{L}^{j}S_{1}^{+1/3}+(V^{*}y^{LL}_{1})_{ij}\bar{u}_{L}^{C\,i}e_{L}^{j}S_{1}^{+1/3}+y^{RR}_{1\,ij}\bar{u}_{R}^{C\,i}e_{R}^{j}S_{1}^{+1/3}+\textrm{h.c.},$
(4)

where $a,b(=1,2)$ are $SU(2)$ indices, $V$ is a Cabibbo–Kobayashi–Maskawa
(CKM) mixing matrix, and $U$ represents a Pontecorvo–Maki–Nakagawa–Sakata
(PMNS) unitary mixing matrix. We set the CKM matrix to be an identity matrix
whereas the exact form of the PMNS matrix is irrelevant for our considerations
as long as it resides entirely in the neutrino sector. Note that the CKM
matrix, in our convention, is in the up-type quark sector. We will address
validity of our assumption that the off-diagonal CKM matrix elements can be
neglected and whether the exact placement of the CKM matrix is of any
importance.

Pertinent parts of the $R_{2}$ lagrangian are

$\displaystyle\mathcal{L}_{R_{2}}=$
$\displaystyle-y^{RL}_{2\,ij}\bar{u}_{R}^{i}R_{2}^{a}\epsilon^{ab}L_{L}^{j,b}+y^{LR}_{2\,ij}\bar{e}_{R}^{i}R_{2}^{a\,*}Q_{L}^{j,a}+\textrm{h.c.}$
$\displaystyle=$
$\displaystyle-y^{RL}_{2\,ij}\bar{u}^{i}_{R}e^{j}_{L}R_{2}^{+5/3}+(y^{RL}_{2}U)_{ij}\bar{u}^{i}_{R}\nu^{j}_{L}R_{2}^{+2/3}+$
$\displaystyle+(y^{LR}_{2}V^{\dagger})_{ij}\bar{e}^{i}_{R}u^{j}_{L}R_{2}^{-5/3}+y^{LR}_{2\,ij}\bar{e}^{i}_{R}d^{j}_{L}R_{2}^{-2/3}+\textrm{h.c.}.$
(5)

One can note that all unitary transformations of the right-chiral fermions can
be completely absorbed, for both the $S_{1}$ and $R_{2}$ scenarios, into
associated Yukawa coupling matrices. We accordingly take all unitary
transformations of right-chiral quarks and charged leptons to be unphysical in
our study.

The $S_{3}$ Lagrangian, in our notation, is

$\displaystyle\mathcal{L}_{S_{3}}$
$\displaystyle=y^{LL}_{3\,ij}\bar{Q}_{L}^{C\,i,a}\epsilon^{ab}(\tau^{k}S^{k}_{3})^{bc}L_{L}^{j,c}+\mathrm{h.c.}\,$
$\displaystyle=-(y^{LL}_{3}U)_{ij}\bar{d}_{L}^{C\,i}\nu_{L}^{j}S^{+1/3}_{3}-(V^{*}y^{LL}_{3})_{ij}\bar{u}_{L}^{C\,i}e_{L}^{j}S^{+1/3}_{3}+$
$\displaystyle+\sqrt{2}(V^{*}y^{LL}_{3}U)_{ij}\bar{u}_{L}^{C\,i}\nu_{L}^{j}S^{-2/3}_{3}-\sqrt{2}y^{LL}_{3\,ij}\bar{d}_{L}^{C\,i}e_{L}^{j}S^{+4/3}_{3}+\text{h.c.},$
(6)

where $\tau^{k}$, $k=1,2,3$, are Pauli matrices and we define
$S^{+4/3}_{3}=(S^{1}_{3}-iS^{2}_{3})/\sqrt{2}$, $S^{+1/3}_{3}=S^{3}_{3}$, and
$S^{-2/3}_{3}=(S^{1}_{3}+iS^{2}_{3})/\sqrt{2}$ to be electric charge
eigenstates.

Again, in all of these scenarios we will always assume a presence of a single
non-zero Yukawa coupling to either electron or muon and the first generation
quarks for each of these leptoquark multiplets, if and when they are featured,
in order to simplify discussion.

####  2.1.1 Case study: $S_{1}(\overline{\mathbf{3}},\mathbf{1},1/3)$

Let us first address the $S_{1}$ scenario.

  * •

If we assume that $y^{LL}_{1\,11}\equiv y$ is the only non-zero Yukawa
coupling present in Eq. (4), we have that the branching ratios for the $S_{1}$
decays are $B(S_{1}^{\pm 1/3}\rightarrow j\nu)=1/2$ and $B(S_{1}^{\pm
1/3}\rightarrow je)=1/2$. A recast of the ATLAS Collaboration analysis
ATLAS:2020dsk  of the leptoquark pair production search via
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process at 13
TeV center-of-mass energy of proton-proton collisions, using an integrated
luminosity of 139 fb-1, then yields a limit on the mass of $S_{1}$ leptoquark,
as a function of $y\equiv y^{LL}_{1\,11}$, which is rendered with a thick
dashed black curve in Fig. 5. The exclusion region is to the left of that
curve and it is based on the ATLAS Collaboration observed 95% C.L. limit. The
Yukawa dependent limit we present in Fig. 5, for small values of
$y^{LL}_{1\,11}\equiv y$, needs to agree with the outcome of the ATLAS
Collaboration analysis when $B(S_{1}^{\pm 1/3}\rightarrow je)=1/2$, i.e.
$m_{\mathrm{LQ}}\geq 1380$ GeV, which is based on the next-to-leading order
cross section in QCD calculation ATLAS:2020dsk . We accordingly rescale our
leading order simulation when presenting the limits in Fig. 5 and note that
the cross section obtained in that way indeed corresponds to the next-to-
leading order cross section in QCD as given in Ref. Dorsner:2018ynv . We also
plot in Fig. 5 the leptoquark parameter constraint with a vertical thin dashed
black line if one would use
$\sigma_{\mathrm{QCD}}^{\mathrm{pair}}(m_{\mathrm{LQ}})$ instead of the Yukawa
dependant cross section, for this particular branching fraction scenario. That
vertical line is additionally marked with “w/o $t$-channels” to stress
exclusion of the $t$-channel lepton exchange diagrams during evaluation of
$\sigma_{\mathrm{QCD}}^{\mathrm{pair}}(m_{\mathrm{LQ}})$.

We note two subtleties with regard to the $y^{LL}_{1\,11}\equiv y\neq 0$ case.
First, there are two $t$-channel contributions towards the $S_{1}$ pair
production that need to be included in this analysis. One contribution is due
to the first term in the second line of Eq. (4) and it is $d\overline{d}$
initiated. The other contribution is $u\overline{u}$ initiated and it is due
to the second term in the second line of Eq. (4). Another subtlety concerns
the CKM mixing matrix placement. Namely, if the CKM matrix is taken to be in
the up-type quark sector it would induce coupling between $S_{1}$, a charm
quark, and an electron through the second term in the second line of Eq. (4).
This would primarily impact the branching ratio $B(S_{1}^{\pm 1/3}\rightarrow
je)$ by reducing it to 80% of its initial value and would also introduce
$B(S_{1}^{\pm 1/3}\rightarrow ce)$ at the level of 10%. These changes in
branching fractions would consequentially impact interpretation of the ATLAS
Collaboration analysis ATLAS:2020dsk  that can distinguish between light jets
and, for example, a $c$-quark induced jet. The bounds on the $S_{1}$ parameter
space would accordingly shift to the left in Fig. 5. The placement of the CKM
mixing matrix in the down-type quark sector, on the other hand, would not
produce any such shift.

  * •

If we take $y^{LL}_{1\,12}\equiv y\neq 0$ in Eq. (4), the branching ratios for
the $S_{1}$ decays read $B(S_{1}^{\pm 1/3}\rightarrow j\nu)=1/2$ and
$B(S_{1}^{\pm 1/3}\rightarrow j\mu)=1/2$. A recast of the ATLAS Collaboration
analysis ATLAS:2020dsk  of the leptoquark pair production search via
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$ process
then yields a limit on the mass of $S_{1}$ leptoquark, as a function of
$y\equiv y^{LL}_{1\,12}$, which is rendered with a thick dashed black curve in
Fig. 6. The limit we present in Fig. 6, for small values of
$y^{LL}_{1\,12}\equiv y$, corresponds to the outcome of the ATLAS
Collaboration analysis when $B(S_{1}^{\pm 1/3}\rightarrow j\mu)=1/2$, i.e.
$m_{\mathrm{LQ}}\geq 1420$ GeV, that is shown as a vertical thin dashed black
line in Fig. 6.

  * •

If we take that $y^{RR}_{1\,11}\equiv y$ is the only non-zero Yukawa coupling
in Eq. (4), we get that $B(S_{1}^{\pm 1/3}\rightarrow je)=1$ and the correct
interpretation of the ATLAS Collaboration results ATLAS:2020dsk  would
correspond to a bound rendered with a thick dashed blue curve in Fig. 5. This
bound, for small values of $y^{RR}_{1\,11}\equiv y$, yields
$m_{\mathrm{LQ}}\geq 1790$ GeV ATLAS:2020dsk  and thus coincides with the
constraint presented with a vertical thin dashed blue line that is generated
if one were to use $\sigma_{\mathrm{QCD}}^{\mathrm{pair}}(m_{\mathrm{LQ}})$
instead of the more appropriate
$\sigma_{u\overline{u}}^{\mathrm{pair}}(y,m_{\mathrm{LQ}})$ to interpret the
ATLAS Collaboration analysis.

  * •

If we take that $y^{RR}_{1\,12}\equiv y\neq 0$, a recast of the ATLAS
Collaboration results ATLAS:2020dsk  on the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$ process
search yields a bound rendered with a thick dashed blue curve in Fig. 6. This
bound, for small values of $y^{RR}_{1\,12}\equiv y$, reads
$m_{\mathrm{LQ}}\geq 1730$ GeV ATLAS:2020dsk  and is given with a vertical
thin dashed blue line in Fig. 6.

Note that the exclusion regions, in all four cases, feature negative
interference effects, as discussed in connection to Eq. (3), for intermediate
values of Yukawa couplings $y^{LL}_{1\,11}\equiv y$, $y^{LL}_{1\,12}\equiv y$,
$y^{RR}_{1\,11}\equiv y$, and $y^{RR}_{1\,12}\equiv y$.

Figure 5: The leptoquark parameter space limits for the $S_{1}$, $R_{2}$, and
$S_{1}$+$R_{2}$ scenarios extracted from the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
search ATLAS:2020dsk  performed at 13 TeV center-of-mass energy of proton-
proton collisions at the LHC, using an integrated luminosity of 139 fb-1. See
text for more details. Figure 6: The leptoquark parameter space limits for the
$S_{1}$, $R_{2}$, and $S_{1}$+$R_{2}$ scenarios extracted from the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$ process
search ATLAS:2020dsk  performed at 13 TeV center-of-mass energy of proton-
proton collisions at the LHC, using an integrated luminosity of 139 fb-1. See
text for more details.

####  2.1.2 Case study: $R_{2}(\mathbf{3},\mathbf{2},7/6)$

We consider, in what follows, scenarios when we switch on, individually,
$y^{RL}_{2\,11}$, $y^{RL}_{2\,12}$, $y^{LR}_{2\,11}$, and $y^{LR}_{2\,12}$ of
Eq. (5) while all other Yukawa matrix elements are taken to be negligible.

  * •

If we turn on Yukawa coupling $y^{RL}_{2\,11}\equiv y$ in Eq. (5), we have
that $B(R_{2}^{\pm 5/3}\rightarrow je)=1$ and $B(R_{2}^{\pm 2/3}\rightarrow
j\nu)=1$. Since the members of the $R_{2}$ multiplet need to be mass-
degenerate for all practical purposes, the limit on the $R_{2}^{\pm 5/3}$
parameter space, as extracted from the ATLAS Collaboration pair production
analysis ATLAS:2020dsk , should also be applicable to $R_{2}^{\pm 2/3}$ and
vice versa. If we furthermore take into account the fact that the experimental
limit on $pp\rightarrow R_{2}^{+5/3}R_{2}^{-5/3}\rightarrow jjee$ is certainly
more relevant than the limit that could be extracted from $pp\rightarrow
R_{2}^{+2/3}R_{2}^{-2/3}\rightarrow jj\nu\nu$, the constraint on the viable
$R_{2}^{\pm 5/3}$ and $R_{2}^{\pm 2/3}$ parameter spaces is given with a thick
dashed blue curve in Fig. 5. Note that this particular limit on the
$R_{2}^{\pm 5/3}$ and $R_{2}^{\pm 2/3}$ parameter spaces, when
$y^{RL}_{2\,11}\equiv y\neq 0$, is the same as for $S_{1}$ leptoquark when
$y^{RR}_{1\,11}\equiv y\neq 0$.

  * •

If we assume that $y^{RL}_{2\,12}\equiv y\neq 0$ in Eq. (5), we have that
$B(R_{2}^{\pm 5/3}\rightarrow j\mu)=1$ and $B(R_{2}^{\pm 2/3}\rightarrow
j\nu)=1$. In analogy to the previous case the reinterpretation of the ATLAS
Collaboration pair production analysis ATLAS:2020dsk  of the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$ process
search yields a constraint on the viable $R_{2}^{\pm 5/3}$ and $R_{2}^{\pm
2/3}$ parameter spaces that is given with a thick dashed blue curve in Fig. 6.
A vertical thin dashed blue line in Fig. 6 represents a limit that is based
solely on the QCD cross section and corresponds to $m_{\mathrm{LQ}}\geq 1730$
GeV ATLAS:2020dsk . The limits on the $R_{2}^{\pm 5/3}$ and $R_{2}^{\pm 2/3}$
parameter spaces, when $y^{RL}_{2\,12}\equiv y\neq 0$, are the same as for
$S_{1}$ leptoquark when $y^{RR}_{1\,12}\equiv y\neq 0$.

  * •

If we set $y^{LR}_{2\,11}\equiv y\neq 0$, we have that $B(R_{2}^{\pm
5/3}\rightarrow je)=1$ and $B(R_{2}^{\pm 2/3}\rightarrow je)=1$. Since the
pair productions of both components of $R_{2}$ produce the same final state,
i.e., $pp\rightarrow R_{2}^{+5/3}R_{2}^{-5/3}\rightarrow jjee$ and
$pp\rightarrow R_{2}^{+2/3}R_{2}^{-2/3}\rightarrow jjee$, we need to take that
into account. Naive combination of these two processes, i.e., based purely on
the $\sigma_{\mathrm{QCD}}^{\mathrm{pair}}(m_{\mathrm{LQ}})$ value, results in
a bound given by a vertical thin dashed red line in Fig. 5 and yields
$m_{\mathrm{LQ}}\geq 1920$ GeV. If we furthermore include the Yukawa
dependence of the cross sections to pair produce both components of $R_{2}$
multiplet, we obtain a limit rendered in a thick dashed red curve in Fig. 5.
It should be noted that the generation of the thick dashed red curve denoted
with “w/o asymm. contributions” calls for separate evaluation of cross
sections for both $pp\rightarrow R_{2}^{+5/3}R_{2}^{-5/3}(\rightarrow jjee)$
and $pp\rightarrow R_{2}^{+2/3}R_{2}^{-2/3}(\rightarrow jjee)$ and their
subsequent addition. Since $R_{2}^{-5/3}$ couples to the up quark while
$R_{2}^{-2/3}$ couples to the down quark, these two cross sections, as
functions of $y^{LR}_{2\,11}\equiv y$, are clearly not identical.

Note, however, that simple addition of cross sections to produce
$R_{2}^{+5/3}R_{2}^{-5/3}$ and $R_{2}^{+2/3}R_{2}^{-2/3}$ pairs does not
account for the asymmetric pair production mechanism effects that we want to
advocate. To take into account asymmetric pair production we also need to
include cross sections for $pp\rightarrow R_{2}^{+5/3}R_{2}^{-2/3}(\rightarrow
jjee)$ and $pp\rightarrow R_{2}^{-5/3}R_{2}^{+2/3}(\rightarrow jjee)$.
Relevant diagrams for these two processes are presented in Fig. 7. The
diagrams of Fig. 7 explicitly show that the two leptoquarks that are produced
do not comprise a charge conjugate pair. These processes thus do not
interfere, at the amplitude level, with the conventional pair production
mechanisms even though they yield the exact same $jjee$ final state.

If we combine both the conventional and asymmetric pair production cross
sections, and apply the constraints obtained by the ATLAS Collaboration on the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
ATLAS:2020dsk , we obtain a proper bound rendered with a thick dot-dashed red
curve in Fig. 5. The relevance of the asymmetric contribution is, in our view,
self-evident.

We finally present, for completeness, the leading order cross sections for
$pp\rightarrow R_{2}^{+5/3}R_{2}^{-5/3}$, $pp\rightarrow
R_{2}^{+2/3}R_{2}^{-2/3}$, $pp\rightarrow R_{2}^{+5/3}R_{2}^{-2/3}$, and
$pp\rightarrow R_{2}^{-5/3}R_{2}^{+2/3}$ in Table 4 as functions of $y\equiv
y^{LR}_{2\,11}$ and $m_{\mathrm{LQ}}$. Note, again, that the cross sections
for $pp\rightarrow R_{2}^{+5/3}R_{2}^{-5/3}$ and $pp\rightarrow
R_{2}^{+2/3}R_{2}^{-2/3}$ behave differently with respect to change of
$y^{LR}_{2\,11}\equiv y$.

  * •

If we set $y^{LR}_{2\,12}\equiv y\neq 0$, we have that $B(R_{2}^{\pm
5/3}\rightarrow j\mu)=1$ and $B(R_{2}^{\pm 2/3}\rightarrow j\mu)=1$. The
limits presented in Fig. 6 in red with a vertical thin dashed line, a thick
dashed curve, and a thick dot-dashed curve correspond to a QCD only limit, a
conventional Yukawa coupling dependent limit, and a proper limit that includes
asymmetric production effects, respectively. These limits converge to the same
bound of $m_{\mathrm{LQ}}\geq 1850$ GeV, as required by the observed 95% C.L.
limit of the ATLAS Collaboration search for the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$ process
ATLAS:2020dsk .

Figure 7: Asymmetric pair production for the case of $R_{2}$ leptoquark. $m_{\textrm{LQ}}$ | $y$ |  $\sigma_{R_{2}^{+2/3}R_{2}^{-2/3}}$ (fb) |  $\sigma_{R_{2}^{+5/3}R_{2}^{-5/3}}$ (fb) |  $\sigma_{R_{2}^{+5/3}R_{2}^{-2/3}}$ (fb) |  $\sigma_{R_{2}^{-5/3}R_{2}^{+2/3}}$ (fb)
---|---|---|---|---|---
| $0.1$ | $0.141$ | $0.141$ | $2.98\times 10^{-5}$ | $8.39\times 10^{-6}$
1.6 TeV | $0.5$ | $0.138$ | $0.132$ | $0.0187$ | $0.00523$
| $1.0$ | $0.201$ | $0.287$ | $0.298$ | $0.0837$
| $0.1$ | $0.0143$ | $0.0143$ | $3.67\times 10^{-6}$ | $9.93\times 10^{-7}$
2.0 TeV | $0.5$ | $0.0139$ | $0.0131$ | $0.00292$ | $0.000622$
| $1.0$ | $0.0210$ | $0.0331$ | $0.0367$ | $0.00996$
| $0.1$ | $0.00157$ | $0.00156$ | $4.80\times 10^{-7}$ | $1.28\times 10^{-7}$
2.4 TeV | $0.5$ | $0.00151$ | $0.00139$ | $2.99\times 10^{-4}$ | $0.796\times 10^{-4}$
| $1.0$ | $0.00239$ | $0.00414$ | $0.00482$ | $0.00128$
Table 4: The leading order cross sections for the leptoquark pair production
for the $R_{2}$ scenario in the proton-proton collisions at 13 TeV center-of-
mass energy when the $R_{2}$ components of mass $m_{\textrm{LQ}}$ couple
exclusively to a right-chiral leptons and the first generation quarks, as
allowed by the SM gauge group, with the coupling strength $y\equiv
y^{LR}_{2\,11}$.

####  2.1.3 Case study:
$S_{1}(\overline{\mathbf{3}},\mathbf{1},1/3)$+$R_{2}(\mathbf{3},\mathbf{2},7/6)$

Since both $S_{1}$ and $R_{2}$ multiplets can couple to the SM leptons of both
chiralities, there are four different scenarios to consider even if only one
Yukawa coupling for each of these two multiplets is turned on at a given time.
To avoid overburdening the reader with too many details and to drive our point
of potential importance of the asymmetric pair production inclusion, we will
investigate only one of these four possibilities for both the $jjee$ and
$jj\mu\mu$ final state scenarios.

  * •

We first take that $y^{RR}_{1\,11}=y^{LR}_{2\,11}\equiv y\neq 0$ so that all
three leptoquarks decay into the same final state, i.e., $B(S_{1}^{\pm
1/3}\rightarrow je)=B(R_{2}^{\pm 5/3}\rightarrow je)=B(R_{2}^{\pm
2/3}\rightarrow je)=1$. If we furthermore take that the masses of $S_{1}$ and
of the two charged components in $R_{2}$ are the same, we obtain that a pure
QCD cross section generates bound given with a vertical thin dashed green line
whereas a simplistic addition of cross sections for processes $pp\rightarrow
S_{1}^{+1/3}S_{1}^{-1/3}(\rightarrow jjee)$, $pp\rightarrow
R_{2}^{+5/3}R_{2}^{-5/3}(\rightarrow jjee)$, and $pp\rightarrow
R_{2}^{+2/3}R_{2}^{-2/3}(\rightarrow jjee)$ yields a bound given by a thick
dashed green curve in Fig. 5. These bounds are based on the observed 95% C.L.
limits, as given by the ATLAS Collaboration results on the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
search ATLAS:2020dsk , and yield $m_{\mathrm{LQ}}\geq 2000$ GeV in the small
Yukawa coupling limit. Since the ATLAS Collaboration analysis ATLAS:2020dsk
provides results for the leptoquark masses up to 2 TeV only, we conservatively
assume that the observed limits above 2 TeV would have the ATLAS Collaboration
2 TeV level values.

If we finally include all six asymmetric contributions, i.e., $pp\rightarrow
S_{1}^{\pm 1/3}R_{2}^{\pm 5/3}(\rightarrow jjee)$, $pp\rightarrow S_{1}^{\pm
1/3}R_{2}^{\pm 2/3}(\rightarrow jjee)$, and $pp\rightarrow R_{2}^{\pm
5/3}R_{2}^{\mp 2/3}(\rightarrow jjee)$, we obtain a bound given by a thick
dot-dashed green curve in Fig. 5. Once again, the importance of inclusion of
the asymmetric contribution is, in our view, self-evident.

  * •

If we take that $y^{RR}_{1\,12}=y^{LR}_{2\,12}\equiv y\neq 0$ so that all
three leptoquarks decay into muons and light jets, i.e., $B(S_{1}^{\pm
1/3}\rightarrow j\mu)=B(R_{2}^{\pm 5/3}\rightarrow j\mu)=B(R_{2}^{\pm
2/3}\rightarrow j\mu)=1$, we obtain the limits rendered in Fig. 6 in green.
These limits are obtained under the assumption that all three leptoquarks are
mass-degenerate and represent a recast of the ATLAS Collaboration search for
the $pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$
process ATLAS:2020dsk . A vertical thin dashed line is generated by a pure QCD
cross section, a thick dashed curve is produced by a simple inclusion of the
Yukawa dependent terms in the relevant cross sections while a thick dot-dashed
curve corresponds to the correct inclusion of both the conventional and
asymmetric contributions towards total cross section that yields the
$jj\mu\mu$ final state. These three limits, rendered in green in Fig. 6,
converge to $m_{\mathrm{LQ}}\geq 1930$ GeV for small values of
$y^{RR}_{1\,12}=y^{LR}_{2\,12}\equiv y$.

####  2.1.4 Case study: $S_{3}(\overline{\mathbf{3}},\mathbf{3},1/3)$

We consider two particular scenarios for the $S_{3}$ case. One scenario is
when $y^{LL}_{3\,11}\neq 0$ and the other one is when $y^{LL}_{3\,12}\neq 0$.

  * •

We assume that $y^{LL}_{3\,11}\equiv y$ of Eq. (6) is the only non-zero Yukawa
coupling and take all three leptoquarks within $S_{3}$ multiplet to be
degenerate in mass that we denote by $m_{\mathrm{LQ}}$. The branching
fractions for the $S_{3}$ components, when $y^{LL}_{3\,11}\neq 0$, are
$B(S_{3}^{\pm 4/3}\rightarrow je)=1$, $B(S_{3}^{\pm 2/3}\rightarrow j\nu)=1$,
$B(S_{3}^{\pm 1/3}\rightarrow je)=1/2$, and $B(S_{3}^{\pm 1/3}\rightarrow
j\nu)=1/2$.

If we are to use the ATLAS Collaboration results on the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
ATLAS:2020dsk  to generate accurate constraints on the $S_{3}$ parameter
space, we need to take into account several factors. Namely, in the regime of
the QCD dominated leptoquark pair production, i.e., for small
$y^{LL}_{3\,11}$, there are two different processes that yield the $jjee$
final state. These are $pp\rightarrow S_{3}^{+4/3}S_{3}^{-4/3}\rightarrow
jjee$ and $pp\rightarrow S_{3}^{+1/3}S_{3}^{-1/3}\rightarrow jjee$, where the
$S_{3}^{+4/3}S_{3}^{-4/3}$ pair goes exclusively into $jjee$ whereas the
$S_{3}^{+1/3}S_{3}^{-1/3}$ pair decays into $jjee$ only 25% of the time. If
$y^{LL}_{3\,11}$ is not small, we need to include conventional $t$-channel
contributions that were discussed in the context of Eq. (3). These
contributions, however, are not the same for $pp\rightarrow
S_{3}^{+4/3}S_{3}^{-4/3}$ and $pp\rightarrow S_{3}^{+1/3}S_{3}^{-1/3}$ since
the former process is $d\overline{d}$ initiated whereas the latter one is both
$u\overline{u}$ and $d\overline{d}$ initiated. Moreover, the $S_{3}^{\pm 4/3}$
couplings to the quark-lepton pairs are always a factor of $\sqrt{2}$ larger
than that of $S_{3}^{\pm 1/3}$ due to the $SU(2)$ symmetry of the SM. If we
account for all these intricacies, we obtain the Yukawa dependent limit given
in Fig. 9 with a thick dashed red curve. A vertical thin dashed red line in
Fig. 9, on the other hand, denotes a naive limit if we were to use purely QCD
dominated leptoquark pair production cross sections and yields
$m_{\mathrm{LQ}}\geq 1830$ GeV. We opted not to present the $S_{3}$ results
with all other scenarios discussed previously in Fig. 5 in order to provide
ease of readability. Note, however, that we provide in Fig. 9 limits on the
$S_{1}$ scenarios that were already discussed in Sec. 2.1.1 and also presented
in Fig. 5 for comparison purposes.

Our considerations, up to this point, did not incorporate potential asymmetric
production contributions towards the $jjee$ final state. There are, in
general, four asymmetric pair production contributions in any $S_{3}$ scenario
and we present associated diagrams in Fig. 8. Two diagrams in the second row
of Fig. 8 can give the $jjee$ final state via $pp\rightarrow
S_{3}^{-4/3}S_{3}^{+1/3}\rightarrow jjee$ and $pp\rightarrow
S_{3}^{+4/3}S_{3}^{-1/3}\rightarrow jjee$ with 50% probability, each, where
$pp\rightarrow S_{3}^{-4/3}S_{3}^{+1/3}$ and $pp\rightarrow
S_{3}^{+4/3}S_{3}^{-1/3}$ are $u\overline{d}$ and $d\overline{u}$ initiated,
respectively. If we account for these effects, we obtain a limit given in Fig.
9 with a thick dot-dashed red curve. It is this limit that represents correct
interpretation of the ATLAS Collaboration results on the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
ATLAS:2020dsk  when $y^{LL}_{3\,11}\equiv y$ of Eq. (6) is the only non-zero
Yukawa coupling. Again, the importance of the asymmetric production inclusion
is self-evident.

Figure 8: Asymmetric pair production for the case of $S_{3}$ leptoquark.

  * •

We assume that $y^{LL}_{3\,12}\equiv y$ of Eq. (6) is the only non-zero Yukawa
coupling and take all three leptoquarks within $S_{3}$ multiplet to be
degenerate. The branching fractions for the $S_{3}$ components, when
$y^{LL}_{3\,12}\neq 0$, are $B(S_{3}^{\pm 4/3}\rightarrow j\mu)=1$,
$B(S_{3}^{\pm 2/3}\rightarrow j\nu)=1$, $B(S_{3}^{\pm 1/3}\rightarrow
j\mu)=1/2$, and $B(S_{3}^{\pm 1/3}\rightarrow j\nu)=1/2$. Following the
procedure already outlined for the recast of the $y^{LL}_{3\,11}\equiv y\neq
0$ case and applying it on the results of the ATLAS Collaboration search for
the $pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$
process ATLAS:2020dsk , we obtain limits rendered in red in Fig. 10. These
converge at $m_{\mathrm{LQ}}\geq 1770$ GeV for small values of
$y^{LL}_{3\,12}\equiv y$.

Figure 9: The leptoquark parameter space limits for the $S_{1}$, $S_{3}$, and
$S_{1}$+$S_{3}$ scenarios extracted from the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
search ATLAS:2020dsk  performed at 13 TeV center-of-mass energy of proton-
proton collisions at the LHC, using an integrated luminosity of 139 fb-1. See
text for more details. Figure 10: The leptoquark parameter space limits for
the $S_{1}$, $S_{3}$, and $S_{1}$+$S_{3}$ scenarios extracted from the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$ process
search ATLAS:2020dsk  performed at 13 TeV center-of-mass energy of proton-
proton collisions at the LHC, using an integrated luminosity of 139 fb-1. See
text for more details.

####  2.1.5 Case study:
$S_{1}(\overline{\mathbf{3}},\mathbf{1},1/3)$+$S_{3}(\overline{\mathbf{3}},\mathbf{3},1/3)$

Since $S_{3}$ couples exclusively to the left-chiral leptons we will assume
that the only non-zero Yukawa couplings in this scenario, comprising $S_{1}$
and $S_{3}$ leptoquarks, are either $y^{LL}_{1\,11}$ and $y^{LL}_{3\,11}$ or
$y^{LL}_{1\,12}$ and $y^{LL}_{3\,12}$. This ansatz will allow us to present an
analysis of the asymmetric pair production effects within the $\Delta F=0$
system.

  * •

We first consider scenario with $y^{LL}_{1\,11}\neq 0$ and $y^{LL}_{3\,11}\neq
0$. The branching fractions of leptoquarks are $B(S_{3}^{\pm 4/3}\rightarrow
je)=1$, $B(S_{3}^{\pm 2/3}\rightarrow j\nu)=1$, $B(S_{3}^{\pm 1/3}\rightarrow
je)=1/2$, $B(S_{3}^{\pm 1/3}\rightarrow j\nu)=1/2$, $B(S_{1}^{\pm
1/3}\rightarrow je)=1/2$, and $B(S_{1}^{\pm 1/3}\rightarrow j\nu)=1/2$. We
furthermore assume that $S_{1}$ and the components of $S_{3}$ are degenerate
in mass and also take that $y^{LL}_{1\,11}=y^{LL}_{3\,11}\equiv y$ to simplify
discussion.

A naive QCD limit on the parameter space of this scenario, set by the ATLAS
Collaboration data on the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
ATLAS:2020dsk , is presented in Fig. 9 with a vertical thin dashed green line
and corresponds to $m_{\mathrm{LQ}}\geq 1860$ GeV. If we also include the
usual $t$-channel contributions for both $S_{1}$ and $S_{3}$, as discussed
previously in Secs. 2.1.1 and 2.1.4, respectively, we obtain the limit given
with a thick dashed green curve in Fig. 9.

In order to numerically evaluate the asymmetric pair production contributions
we need to account for $pp\rightarrow S_{3}^{-4/3}S_{3}^{+1/3}\rightarrow
jjee$ (50%), $pp\rightarrow S_{3}^{+4/3}S_{3}^{-1/3}\rightarrow jjee$ (50%),
$pp\rightarrow S_{3}^{-4/3}S_{1}^{+1/3}\rightarrow jjee$ (50%), $pp\rightarrow
S_{3}^{+4/3}S_{1}^{-1/3}\rightarrow jjee$ (50%), $pp\rightarrow
S_{3}^{-1/3}S_{1}^{+1/3}\rightarrow jjee$ (25%) and $pp\rightarrow
S_{3}^{+1/3}S_{1}^{-1/3}\rightarrow jjee$ (25%), where we specify in
parentheses the associated decay rate into the $jjee$ final state for each of
these processes. Note that the last two processes are both $u\overline{u}$ and
$d\overline{d}$ initiated. In fact, $S_{1}$+$S_{3}$ scenario is the only
$\Delta F=0$ scenario that features asymmetric production initiated with the
$q\overline{q^{\prime}}$ combination, where both $q$ and $q^{\prime}$ are of
the same type of flavor. Moreover, these same-flavor contributions always come
in pairs as they are simultaneously generated by the up-type and down-type
quarks. This is the reason why we opted to present combinations
$\sigma^{\mathrm{pair}}_{u\overline{u}}+\sigma^{\mathrm{pair}}_{d\overline{d}}$,
$\sigma^{\mathrm{pair}}_{u\overline{c}}+\sigma^{\mathrm{pair}}_{d\overline{s}}$,
$\sigma^{\mathrm{pair}}_{c\overline{u}}+\sigma^{\mathrm{pair}}_{s\overline{d}}$
and
$\sigma^{\mathrm{pair}}_{c\overline{c}}+\sigma^{\mathrm{pair}}_{s\overline{s}}$
in Fig. 4 instead of individual $q\overline{q^{\prime}}$ contributions.

If we properly include all the relevant processes that yield the $jjee$ final
state, we obtain a limit on the $S_{1}$+$S_{3}$ scenario parameter space that
is given by a thick dot-dashed green curve in Fig. 9. The parameter space to
the left of that curve is excluded by the ATLAS Collaboration search for the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jjee$ process
ATLAS:2020dsk .

  * •

If we assume that $y^{LL}_{1\,12}\neq 0$ and $y^{LL}_{3\,12}\neq 0$, the
branching fractions of leptoquarks read $B(S_{3}^{\pm 4/3}\rightarrow
j\mu)=1$, $B(S_{3}^{\pm 2/3}\rightarrow j\nu)=1$, $B(S_{3}^{\pm
1/3}\rightarrow j\mu)=1/2$, $B(S_{3}^{\pm 1/3}\rightarrow j\nu)=1/2$,
$B(S_{1}^{\pm 1/3}\rightarrow j\mu)=1/2$, and $B(S_{1}^{\pm 1/3}\rightarrow
j\nu)=1/2$. If we also assume that $S_{1}$ and the components of $S_{3}$ are
degenerate in mass and take that $y^{LL}_{1\,12}=y^{LL}_{3\,12}\equiv y$, we
obtain the set of limits rendered in green in Fig. 10 that converge to
$m_{\mathrm{LQ}}\geq 1800$ GeV for small values of
$y^{LL}_{1\,12}=y^{LL}_{3\,12}\equiv y$. These limits use the ATLAS
Collaboration search results for the
$pp\rightarrow\mathrm{LQ}\overline{\mathrm{LQ}}\rightarrow jj\mu\mu$ process
ATLAS:2020dsk . A vertical thin dashed line is the bound based on the QCD
cross section. A thick dashed curve is generated if one accounts for the usual
$t$-channel contributions whereas a thick dot-dashed curve is the accurate
limit that incorporates both the conventional and asymmetric leptoquark pair
production mechanism effects.

###  2.2 Final remarks

Before we conclude this section, several remarks are in order.

  * •

We have explicitly assumed in our analysis that the asymmetrically produced
leptoquarks are mass degenerate. If that is not the case the asymmetric pair
production mechanism would allow for an unambiguous and unique test of
existence of multiple leptoquarks if the leptoquarks in question couple to the
lepton(s) of the same chirality. The asymmetric pair production search would
thus be complementary to other detection methods, either direct or indirect,
to ascertain the existence of these hypothetical particles at hadron
colliders.

  * •

We have not included the CKM matrix effects in our numerical study. We have,
however, commented in Sec. 2.1.1 on the fact that the CKM matrix effects can
reduce the branching ratio of specific channels we considered, thereby
affecting the associated bounds on the leptoquark parameter space. We have
also not performed a full next-to-leading order simulation of the leptoquark
pair production cross sections. These effects can be accounted for with
available tools but would only affect quantitative aspect of our study without
compromising our main message.

  * •

It is important to note that all leptoquark scenarios that we presented
require individual attention if one is to extract accurate parameter space
constraints. In fact, even in the small Yukawa coupling limit, different
scenarios would usually yield different lower bounds on the mass of relevant
leptoquark(s). Our study should thus be seen as a blueprint for inclusion of
the asymmetric leptoquark pair production effects and proper interpretation of
available experimental data.

##  3 Atomic parity violation

In the proceeding section we produce leptoquark pair production search limits
for various scalar leptoquark scenarios when the leptoquarks in question
exclusively couple to either electrons or muons and the first generation
quarks. In the former case it is also important to address the impact of the
APV search constraints on otherwise viable leptoquark parameter space.

The effective APV interactions can be parametrized as Langacker:1990jf

$\displaystyle\mathcal{L}_{\mathrm{PV}}=\frac{G_{F}}{\sqrt{2}}\left(\overline{e}\gamma^{\mu}\gamma^{5}e\right)\left(\sum_{q=u,d}\hat{C}_{1q}\overline{q}\gamma_{\mu}q\right),$
(7)

where coefficients $\hat{C}_{1q}=C^{\mathrm{SM}}_{1q}+C^{\mathrm{NP}}_{1q}$
capture both the the SM and the New Physics (NP) contributions. In particular,
$C^{\mathrm{SM}}_{1u}=-0.1887$ and $C^{\mathrm{SM}}_{1d}=0.3419$ Qweak:2018tjf
, whereas the NP contributions $C^{\mathrm{NP}}_{1q}$ for our scenarios are
given in Table 5 Barger:2000gv ; Crivellin:2021egp .

Non-zero Yukawas | $C_{1u}^{\mathrm{NP}}$ | $C_{1d}^{\mathrm{NP}}$ | LQ scenario
---|---|---|---
$y_{1\;11}^{\mathrm{LL}}\equiv y$ | $-\frac{v^{2}}{4m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ | $0$ |  $S_{1}$
$y_{1\;11}^{\mathrm{RR}}\equiv y$ | $\frac{v^{2}}{4m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ | $0$
$y_{2\;11}^{\mathrm{LR}}\equiv y$ | $-\frac{v^{2}}{4m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ | $-\frac{v^{2}}{4m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ |  $R_{2}$
$y_{2\;11}^{\mathrm{RL}}\equiv y$ | $\frac{v^{2}}{4m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ | $0$
$y_{1\;11}^{\mathrm{RR}}=y_{2\;11}^{\mathrm{LR}}\equiv y$ | $0$ | $-\frac{v^{2}}{4m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ |  $S_{1}$+$R_{2}$
$y_{3\;11}^{\mathrm{LL}}\equiv y$ | $-\frac{v^{2}}{4m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ | $-\frac{v^{2}}{2m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ | $S_{3}$
$y_{1\;11}^{\mathrm{LL}}=y_{3\;11}^{\mathrm{LL}}\equiv y$ | $-\frac{v^{2}}{2m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ | $-\frac{v^{2}}{2m^{2}_{\mathrm{LQ}}}\left|y\right|^{2}$ |  $S_{1}$+$S_{3}$
Table 5: $C^{\mathrm{NP}}_{1q}$ coefficients Barger:2000gv ; Crivellin:2021egp
for the scalar leptoquark scenarios under consideration. The mass and Yukawa
coupling degeneracies of leptoquarks are understood while $v=246$ GeV.

Figure 11: APV limits for the proton (left) and Cs (right) measurements
juxtaposed with the leptoquark pair production search limits for the
corresponding leptoquark scenarios, as indicated. Shaded regions are ruled out
by the APV measurements.

The content of Table 5 clearly shows that each leptoquark scenario, except for
the $S_{1}$ scenario with $y_{1\;11}^{\mathrm{RR}}\equiv y$ and $R_{2}$ with
$y_{2\;11}^{\mathrm{RL}}\equiv y$ that are identical, is to be treated
differently when it comes to APV constraints. This situation exactly mirrors
our findings with regard to constraints originating from the leptoquark pair
production search. We accordingly have six distinct cases to consider, all in
all.

Figure 12: APV limits for the proton (left) and Cs (right) measurements
juxtaposed with the leptoquark pair production search limits for the
corresponding leptoquark scenarios, as indicated. Shaded regions are ruled out
by the APV measurements.

To proceed one defines a nuclear weak charge

$\displaystyle
Q_{W}\left(Z,N\right)=-2\left(2Z+N\right)2\hat{C}_{1u}-2\left(Z+2N\right)\hat{C}_{1d},$
(8)

where $Z$ is a nuclear charge number and $N$ represents a number of neutrons.
The experimental measurements of the nuclear weak charge of the proton
($Q_{\mathrm{W}}(p)$) and ${}^{133}\mathrm{Cs}$
($Q_{\mathrm{W}}({}^{133}\mathrm{Cs})$) are ParticleDataGroup:2020ssz ;
Crivellin:2021egp

$\displaystyle
Q_{\mathrm{W}}(p)=-2\left(2\hat{C}_{1u}+\hat{C}_{1d}\right)=0.0719\pm 0.0045,$
(9)

and

$\displaystyle
Q_{\mathrm{W}}({}^{133}\mathrm{Cs})=-2\left(188\hat{C}_{1u}+211\hat{C}_{1d}\right)=-72.82\pm
0.42,$ (10)

respectively. It is important to note that the measurement of
$Q_{\mathrm{W}}(p)$ is in agreement with the SM prediction whereas the
measured value of $Q_{\mathrm{W}}({}^{133}\mathrm{Cs})$ is not. In fact, the
value of $Q_{\mathrm{W}}({}^{133}\mathrm{Cs})$ prefers finite negative NP
contributions, as, for example, given with negative entries in Table 5. We
accordingly opt to show separately constraints generated by measurements of
$Q_{\mathrm{W}}(p)$ and $Q_{\mathrm{W}}({}^{133}\mathrm{Cs})$ in Figs. 11, 12,
and 13, juxtaposing them with the leptoquark pair production search limits,
where we also group aforementioned six different leptoquark scenarios
pairwise. The first column in Figs. 11, 12, and 13 is reserved for the
$Q_{\mathrm{W}}(p)$ generated constraint, whereas the second column reflect
the impact of the $Q_{\mathrm{W}}({}^{133}\mathrm{Cs})$ measurement on the
leptoquark parameter space. The shaded regions in Figs. 11, 12, and 13 are
ruled out by the APV measurements at the $1\sigma$ level, where different
leptoquark scenarios are shown in separate rows for clarity.

Fig. 13 clearly shows that the pair production constraint for the
$S_{1}$+$R_{2}$ scenario is superior to the existing APV constraints. It is
also clear that the the $S_{1}$ scenario with $y_{1\;11}^{\mathrm{RR}}\neq 0$
and $R_{2}$ with $y_{2\;11}^{\mathrm{RL}}\neq 0$ are completely ruled out by
the $Q_{\mathrm{W}}({}^{133}\mathrm{Cs})$ measurement. Of course, the APV
constraints are irrelevant for the scenarios when leptoquarks couple to muons.
We also note that it is possible to arrange for cancellation between
individual leptoquark contributions towards APV interactions of Eq. (7). This
possibility of having the NP coefficients $C^{\mathrm{NP}}_{1q}$ vanish can
be, for example, trivially realised within the $S_{1}$ scenario with
$y_{1\;11}^{\mathrm{LL}}=y_{1\;11}^{\mathrm{RR}}\equiv y$ or within the
$S_{1}$+$R_{2}$ scenario with
$y_{1\;11}^{\mathrm{LL}}=y_{2\;11}^{\mathrm{RL}}\equiv y$.

Figure 13: APV limits for the proton (left) and Cs (right) measurements
juxtaposed with the leptoquark pair production search limits for the
corresponding leptoquark scenarios, as indicated. Shaded regions are ruled out
by the APV measurements.

##  4 Conclusions

This work investigates the asymmetric leptoquark pair production mechanism at
the LHC. A sharp difference between the conventional leptoquark pair
production and the asymmetric one is that for the latter, which is produced
via $t$-channel lepton exchange, the pairs of produced leptoquarks are not
conjugate states of each other. We spell out necessary conditions for an
operational asymmetric leptoquark pair production mechanism and catalog all
possible combinations of leptoquark multiplets that can potentially generate
it. We, furthermore, demonstrate how to properly combine asymmetric and
conventional pair production mechanism effects by considering several
scenarios where the SM is extended with either one or two scalar leptoquark
multiplets. Finally, based on our analysis, we advocate that contributions
from asymmetric pair production should be included when deriving reliable
constraints on leptoquark parameter space as well as be used when attempting
to perform correct identification of these promising new physics sources.

###### Acknowledgements.

I.D. would like to thank Svjetlana Fajfer for numerous fruitful discussions
with regard to this project.

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