aainstitutetext: University of Split, Faculty of Electrical Engineering,
Mechanical Engineering and Naval Architecture in Split (FESB), R. Boškovića
32, HR-21000 Split, Croatiabbinstitutetext: CERN, Theoretical Physics
Department, CH-1211 Geneva 23, Switzerlandccinstitutetext: J. Stefan
Institute, Jamova 39, P. O. Box 3000, SI-1001 Ljubljana,
Sloveniaddinstitutetext: Department of Physics, University of Ljubljana,
Jadranska 19, SI-1000 Ljubljana, Sloveniaeeinstitutetext: Department of
Physics, Faculty of Science, University of Zagreb, Bijenička cesta 32,
HR-10000 Zagreb, Croatia

# Novel Leptoquark Pair Production at LHC

Ilja Doršner c,d Svjetlana Fajfer e and Ajla Lejlić<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We introduce a novel mechanism for the leptoquark pair production at LHC that
is of a $t$-channel topology and is quark-quark initiated. This mechanism
operates under fairly general conditions. One of them is that the two
leptoquarks in question couple to the same lepton and the other one is that
the fermion numbers of these two leptoquarks differ by two. The strength of
the proposed mechanism provides an alternative way to the conventional
processes to efficiently constrain the parameter space of the two leptoquark
scenarios at LHC whenever the aforementioned conditions are met. We
accordingly present one case study to outline the physics potential of this
novel production mechanism.

##  1 Introduction

Scalar leptoquarks exhibit very reach phenomenology and are frequently used
either singly or in pairs to address some of the disagreements between the
Standard Model predictions and experimental observations. For example, scalar
leptoquarks can generate neutrino masses Chua:1999si ; Mahanta:1999xd ;
Dorsner:2017wwn ; Cai:2017wry , produce observable shifts in anomalous
magnetic moments of charged leptons Queiroz:2014zfa ; Biggio:2014ela ;
ColuccioLeskow:2016dox ; Crivellin:2018qmi ; Kowalska:2018ulj ; Mandal:2019gff
; Dorsner:2019itg ; Bigaran:2020jil ; Dorsner:2020aaz , provide a source of
gauge coupling unification Murayama:1991ah ; Dorsner:2005fq ; Dorsner:2005ii ;
Dorsner:2013tla ; Cox:2016epl ; Becirevic:2018afm , and/or accommodate the
$B$-physics anomalies Kosnik:2012dj ; Sakaki:2013bfa ; Hiller:2014yaa ;
Sahoo:2015pzk ; Pas:2015hca ; Becirevic:2015asa ; Bauer:2015knc ; Cox:2016epl
; Das:2016vkr ; Popov:2016fzr ; Becirevic:2016yqi ; Dorsner:2017ufx ;
Chen:2017hir ; Buttazzo:2017ixm ; Angelescu:2018tyl ; Azatov:2018kzb ;
Marzocca:2018wcf ; Kumar:2018kmr ; Becirevic:2018afm ; Crivellin:2019dwb ;
Popov:2019tyc ; Saad:2020ihm ; Saad:2020ucl . There is thus a strong physics
case to search for the scalar leptoquark states in order to probe the relevant
parameter space of these hypothetical particles.

The CMS and ATLAS analyses of the leptoquark pair production signatures at LHC
are currently some of the most robust sources of experimental constraints on
the leptoquark parameter space. (See, for example, Refs. CMS:2020gru ;
Aad:2020iuy ; Aad:2020jmj ; Sirunyan:2020zbk ; Aad:2021rrh  for a sample of
the latest bounds.) The relevant cross sections for the leptoquark pair
production in proton-proton collisions are already available at the next-to-
leading order Kramer:1997hh ; Kramer:2004df ; Mandal:2015lca ; Dorsner:2018ynv
; Borschensky:2020hot  as well as the next-to-next-to-leading order
Beenakker:2016lwe ; Beenakker:1997ut ; Beenakker:2010nq ; Beenakker:2016gmf
in strong coupling constant and the associated tools to produce these cross
sections are readily accessible for the signal simulation if and when needed.
Apart from the QCD driven contribution towards the pair production cross
section at hadron colliders there is only one diagram that exhibits any
dependence on the leptoquark Yukawa coupling(s) to the quark-lepton pairs. It
is of the $t$-channel type, as shown in panel $(a)$ of Fig. 1, where the
initial state is always made out of a quark-antiquark combination.

Figure 1:  Panel $(a)$ shows a $t$-channel Feynman diagram relevant for a pair
production of leptoquarks at LHC. Panels $(b)$ and $(c)$ feature contributions
towards a single leptoquark production. Here, $y$ represents Yukawa coupling
of a lepton ($l$) and a quark ($q$) with a leptoquark (LQ).

As it turns out, the $t$-channel contribution of Fig. 1 towards the leptoquark
pair production at LHC becomes relevant only when the leptoquark interacts
very strongly, i.e., with the couplings of order one, to the quarks of the
first generation Dorsner:2014axa . However, in this particular regime other
processes such as a single leptoquark production Dorsner:2014axa ;
Alves:2002tj ; Hammett:2015sea ; Mandal:2015vfa , induced by the diagrams in
panels $(b)$ and $(c)$ of Fig. 1, the Drell-Yan di-lepton production
Faroughy:2016osc ; Raj:2016aky ; Greljo:2017vvb ; Bansal:2018eha ;
Schmaltz:2018nls ; Fuentes-Martin:2020lea , and/or a resonant leptoquark
production Ohnemus:1994xf ; Eboli:1997fb ; Buonocore:2020erb ; Greljo:2020tgv
might become equally capable of or even more important in constraining the
leptoquark parameter space. Moreover, one also needs to take into account the
experimental constraints from flavor physics, such as the Atomic Parity
Violation ones, for a given $l$-$q$-LQ scenario to ensure its viability,
especially within a large Yukawa coupling regime Dorsner:2016wpm . (For an
example of a combined analysis of the relevant experimental constraints with
regard to the potential viability of the leptoquark parameter space see Ref.
Crivellin:2021egp .) It is thus often stated that the pair production of
leptoquarks at LHC, within the otherwise viable parameter space, is purely QCD
driven. This is assumed to hold even when a given leptoquark multiplet has
more than one electric charge component Diaz:2017lit  and/or when there is
more than one leptoquark multiplet present within a given New Physics scenario
Becirevic:2018afm . It is our intention to show that one needs to be careful
with this assumption, especially in the latter case. We will demonstrate this
within one explicit flavor construction just to prove our point. It is
straightforward to adopt our analysis to address other scenarios with the same
underlying features that will be clearly outlined in our study.

The main premise behind our work is a possibility that one might have a New
Physics scenario that allows for a production of two scalar leptoquarks via
the $t$-channel type of contribution of Fig. 1 but with two quarks instead of
the quark-antiquark pair in the initial state, where the two quarks do not
have to necessarily be of the same flavor. The main benefits of that
possibility, for the leptoquark pair production at LHC, are $(i)$ the
potential absence of the parton distribution function (PDF) suppression of the
associated cross section and $(ii)$ the introduction of additional production
channels. The quark-quark initiated contribution can thus start to dominate
over the QCD production for much smaller values of Yukawa couplings when
compared to the quark-antiquark initiated $t$-channel case. Moreover, it can
even start to dominate over the single leptoquark production due to a more
pronounced scaling, i.e., quartic vs. quadratic, with respect to the Yukawa
couplings. The quark-quark initiated $t$-channel contribution thus has a
potential to be even more important than the single leptoquark production, the
Drell-Yan di-lepton production, and/or the resonant leptoquark production in
constraining certain parts of the parameter space spanned by the relevant
Yukawa couplings and leptoquark masses.

Before we present our case study we outline the general features of the
processes we aim to investigate. The relevant diagrams are shown in Fig. 2.
The $u$ and $d$ in Fig. 2 are the up quark and the down quark, respectively,
and we explicitly denote the electric charges of the two final state
leptoquarks with superscripts. We will concentrate our attention, in Sec. 2,
on one particular scenario that induces both the $uu$ and $ud$ diagrams as
shown in panels $(a)$ and $(b)$ of Fig. 2, respectively. Of course, one might
also need to look at the contributions from other initial state quark flavor
combinations within a given leptoquark scenario.

It is clear from Fig. 2 that, due to a conservation of the fermion number
$F=3B+L$, where $B$ and $L$ are the baryon and lepton numbers, respectively,
the final state should comprise one leptoquark with $F=0$ and the other one
with $|F|=2$. Concretely, $\mathrm{LQ}_{1}$ and $\mathrm{LQ}_{2}$ of Fig. 2
are $F=0$ and $|F|=2$ scalar fields, respectively. In the leptoquark parlance,
the production processes we intend to analyse in this work can occur whenever
the New Physics scenario combines $S_{3}$, $\tilde{S}_{1}$ or $S_{1}$ with
either $R_{2}$ or $\tilde{R}_{2}$. We stress that the leptoquark pairs in
question do not need to mix via the Higgs boson for this production mechanism
to work.

Figure 2:  The leading order Feynman diagrams for the novel leptoquark pair
production at LHC, where the final state leptoquarks $\mathrm{LQ}_{1}$ and
$\mathrm{LQ}_{2}$ have fermion numbers that differ by two. The electric
charges of the two leptoquarks, in units of an absolute value of the electron
charge, are denoted with superscripts.

With these preliminaries out of the way, we turn our attention to the case
study.

##  2 Case Study

We study in detail one particular flavor realisation of the $S_{1}$-$R_{2}$
leptoquark pair scenario. The relevant interaction terms in the Lagrangian are

$\mathcal{L}\supset+y_{1\,ij}\bar{e}_{R}^{i}R_{2}^{a\,*}Q_{L}^{j,a}+y_{2\,ij}\bar{u}_{R}^{C\,i}S_{1}e_{R}^{j}+\textrm{h.c.},$
(1)

where $y_{1}$ and $y_{2}$ are Yukawa coupling matrices in the flavor space and
$a$ is the $SU(2)$ index. We take for simplicity that both electric charge
eigenstates in $R_{2}$ are mass degenerate. We furthermore assume that the
fields comprising $R_{2}$ have the same mass as $S_{1}$ leptoquark, if
simultaneously present, and we denote this common mass parameter with
$m_{\mathrm{LQ}}$ in our study.

If we expand Eq. (1) in terms of the leptoquark and fermion mass eigenstates
we obtain

$\mathcal{L}\supset+(y_{1}V^{\dagger})_{ij}\bar{e}^{i}_{R}u^{j}_{L}R_{2}^{+5/3\,*}+y_{1\,ij}\bar{e}^{i}_{R}d^{j}_{L}R_{2}^{+2/3\,*}+y_{2\,ij}\bar{u}_{R}^{C\,i}S^{+1/3}_{1}e_{R}^{j}+\textrm{h.c.},$
(2)

where $V$ represents a Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix. The
leptoquark superscript in Eq. (2) denotes its electric charge in units of an
absolute value of the electron charge. To simplify things even further we
assume that the only entry in $y_{1}$ matrix that is different from zero is
the 21 element while in $y_{2}$ we take the only non-zero entry to be the 12
element. If both of these couplings are present, we take them to be equal to
each other and denote that parameter with $y$, i.e., $y_{1\,21}=y_{2\,12}=y$,
where $y$ is always assumed to be real. Finally, we set the CKM matrix to be
equal to the identity matrix in our numerical simulations. The departure from
the CKM assumption will only make a tiny quantitative difference without any
change of the qualitative picture we are interested in. With all these
assumptions we have a flavor scenario where all three leptoquarks, i.e.,
$S^{+1/3}_{1}$, $R_{2}^{+5/3}$, and $R_{2}^{+2/3}$, couple exclusively to
muons. The only difference is that $S^{+1/3}_{1}$ and $R_{2}^{+5/3\,*}$ couple
to an up quark whereas $R_{2}^{+2/3\,*}$ couples to a down quark. Since
$R_{2}$ and $S_{1}$ are $F=0$ and $|F|=2$ leptoquarks, respectively, and since
they both couple to the same lepton, we have all the necessary ingredients to
generate the pair production processes we are interested in. In particular,
this scenario has an ability to simultaneously generate the diagrams given in
panels $(a)$ and $(b)$ of Fig. 2, where both leptoquarks in the final state,
i.e., $\mathrm{LQ}_{1}$ and $\mathrm{LQ}_{2}$, will exclusively decay into
muons and jets, where the jets are initiated by either $u$ or $d$ quarks and
are hence indistinguishable from the experimental point of view. Again, the
final state for the leptoquark pair production, within our flavor ansatz, will
always comprise a muon-antimuon pair and two jets.

We are finally ready to make some quantitative comparisons. In order to do
that we implement the aforementioned $S_{1}$-$R_{2}$ scenario in the universal
FeynRules output format Alloul:2013bka  and generate relevant cross sections
within the MadGraph5_aMC@NLO framework Alwall:2014hca  using the nn23lo1 PDF
set Ball:2014uwa . All our simulations are of the leading order nature to
allow for fair and simple comparisons. The center-of-mass energy for proton-
proton collisions is set at 13 TeV and we always use central values for
relevant production cross sections using fixed factorisation ($\mu_{F}$) and
renormalisation scales ($\mu_{R}$) set at $\mu_{F}=\mu_{R}=m_{\mathrm{LQ}}/2$.

Our analysis follows closely the approach advocated in Ref. Dorsner:2014axa .
Namely, we use the fact that the functional dependence for the conventional
leptoquark pair production at LHC, within our flavor construction, can be
written as

$\sigma_{\textrm{P}}(y,m_{\mathrm{LQ}})=a_{0}(m_{\mathrm{LQ}})+a_{2}(m_{\mathrm{LQ}})y^{2}+a_{4}(m_{\mathrm{LQ}})y^{4}.$
(3)

The first term in Eq. (3) represents the QCD pair production contribution
towards cross section. It only depends on the mass of the leptoquark for a
given center-of-mass energy and the associated PDF set. The third term in Eq.
(3) is the $t$-channel production contribution whereas the second term is the
interference between the $t$-channel and the QCD amplitudes. It is important
to note that both $a_{2}(m_{\mathrm{LQ}})$ and $a_{4}(m_{\mathrm{LQ}})$ depend
on the flavor of the quark that the leptoquark couples to. Clearly, it is
sufficient to numerically evaluate the pair production cross section for three
different values of Yukawa coupling $y$ for a fixed value of $m_{\mathrm{LQ}}$
in order to extract information on $a_{0}(m_{\mathrm{LQ}})$,
$a_{2}(m_{\mathrm{LQ}})$, and $a_{4}(m_{\mathrm{LQ}})$ and thus provide an
input for subsequent interpolation. To accommodate the latest bounds on the
leptoquark mass we start this process for $m_{\mathrm{LQ}}=1.6$ TeV and
proceed with our numerical simulation up to $m_{\mathrm{LQ}}=2.4$ TeV in the
200 GeV increments. (A leptoquark that couples exclusively to the first
generation quarks and a muon cannot have a mass below 1.7 TeV Aad:2020iuy .)
Note that $S^{+1/3}_{1}$ and $R_{2}^{+5/3}$ will have exactly the same
functional dependence for $\sigma_{\textrm{P}}(y,m_{\mathrm{LQ}})$ since they
both couple purely to $u$ quark. We accordingly denote this cross section with
$\sigma_{\textrm{P}u}$. The cross section
$\sigma_{\textrm{P}}(y,m_{\mathrm{LQ}})$ to produce a
$R_{2}^{+2/3}$-$R_{2}^{+2/3\,*}$ pair is analogously denoted with
$\sigma_{\textrm{P}d}$. Note, again, that the QCD contribution, given by
$a_{0}(m_{\mathrm{LQ}})$ of Eq. (3), will be the same regardless whether one
looks at $pp\rightarrow S_{1}^{+1/3}S^{+1/3\,*}_{1}$, $pp\rightarrow
R_{2}^{+5/3}R_{2}^{+5/3\,*}$, or $pp\rightarrow R_{2}^{+2/3}R_{2}^{+2/3\,*}$
processes.

The cross section for the single leptoquark production, on the other hand,
scales quadratically with respect to the Yukawa coupling as

$\sigma_{\textrm{S}}(y,m_{\mathrm{LQ}})=a(m_{\mathrm{LQ}})y^{2},$ (4)

where $a(m_{\mathrm{LQ}})$ encapsulates its mass and flavor dependence. We
accordingly denote with $\sigma_{\textrm{S}u}$ the single leptoquark
production cross section for $pp\rightarrow S^{+1/3}_{1}\mu^{-}$ together with
$pp\rightarrow S^{+1/3\,*}_{1}\mu^{+}$. The same cross section is applicable
for $pp\rightarrow R_{2}^{+5/3}\mu^{-}$ together with $pp\rightarrow
R_{2}^{+5/3\,*}\mu^{+}$. For the single leptoquark production of
$R_{2}^{+2/3}$, on the other hand, we introduce $\sigma_{\textrm{S}d}$. To
numerically extract information on the leptoquark mass and flavor dependence
of $a(m_{\mathrm{LQ}})$ we evaluate $\sigma_{\textrm{S}}(y,m_{\mathrm{LQ}})$
for aforementioned leptoquark mass range at the fixed value of $y$ and perform
an interpolation.

Finally, when we consider the $uu$ contribution of panel $(a)$ from Fig. 2
towards the pair production cross section that corresponds to the $t$-channel
production of a $R_{2}^{+5/3}$-$S^{+1/3\,*}_{1}$ pair we denote it with
$\sigma_{uu}$ whereas the cross section for the production of a
$R_{2}^{+2/3}$-$S^{+1/3\,*}_{1}$ pair that corresponds to the process of panel
$(b)$ from Fig. 2 is denoted with $\sigma_{ud}$. Both the $uu$ and $ud$
contributions individually scale as

$\sigma_{uu\,(ud)}=a_{uu\,(ud)}(m_{\mathrm{LQ}})y^{4},$ (5)

where the functional dependence of $a_{uu}(m_{\mathrm{LQ}})$ and
$a_{ud}(m_{\mathrm{LQ}})$ is obtained by evaluating the cross sections
$\sigma_{uu}$ and $\sigma_{ud}$ for $pp\rightarrow
R_{2}^{+5/3}S^{+1/3\,*}_{1}$ and $pp\rightarrow R_{2}^{+2/3}S^{+1/3\,*}_{1}$,
respectively, for aforementioned leptoquark mass range at the fixed value of
$y$.

Our initial intent is just to show the strength of $\sigma_{uu\,(ud)}$
contributions when compared to the usual pair and single leptoquark production
cross sections. To that end we show in Fig. 3 the contours of constant value
for the aforementioned cross sections as a function of $m_{\mathrm{LQ}}$ and
Yukawa coupling(s) $y$. The left panel of Fig. 3 features the production
mechanisms that are associated with the states that couple to the up quarks,
i.e., $S^{+1/3}_{1}$ and $R_{2}^{+5/3}$, while the right panel concerns the
production mechanisms for the state that couples to the down quarks, i.e.,
$R_{2}^{+2/3}$. The only intricacy here is that in order to have the
$\sigma_{ud}$ contribution one needs the presence of both $S^{+1/3}_{1}$ and
$R_{2}^{+2/3}$, as shown in panel $(b)$ of Fig 2. The solid lines in Fig. 3
represent the contours of constant value for cross section of the conventional
pair production, the dashed contours correspond to the constant values of the
single leptoquark production cross sections whereas the dot-dashed contours
correspond to the constant cross sections for the novel pair production via
quark-quark initial state. We also specify the values of the actual cross
sections in femtobarn units in Fig. 3.

It is clear from the left panel of Fig. 3 that, within the leptoquark mass
range we consider, there are four distinct regions that correspond to the
following situations, going from the bottom to the top of that panel:
$\sigma_{\textrm{P}u}>\sigma_{\textrm{S}u}>\sigma_{uu}$,
$\sigma_{\textrm{S}u}>\sigma_{\textrm{P}u}>\sigma_{uu}$,
$\sigma_{\textrm{S}u}>\sigma_{uu}>\sigma_{\textrm{P}u}$, and
$\sigma_{uu}>\sigma_{\textrm{S}u}>\sigma_{\textrm{P}u}$. It is the single
leptoquark production that starts to dominate first over the pair production.
But, it is soon followed by the $\sigma_{uu}$ contribution. In fact, as
advocated at the beginning of this manuscript, our simulation shows that there
is even a region where the novel contribution encapsulated solely by
$\sigma_{uu}$ starts to dominate over all the conventional production
mechanisms. The exact same qualitative picture holds for the scenario that
concerns the leptoquarks that couple to the down quarks as can be seen in the
right panel of Fig. 3. What is important to notice is that $\sigma_{uu}$ and
$\sigma_{ud}$ represent individual contributions towards the pair production
cross section which simply means that the overall enhancement effect can and,
in our case study, will be much more pronounced, as we show next.

Figure 3:  The contours of constant value for the novel contributions
($\sigma_{uu\,(ud)}$) and the usual pair
($\sigma_{\textrm{P}u\,(\textrm{P}d)}$) and single leptoquark production
($\sigma_{\textrm{S}u\,(\textrm{S}d)}$) cross sections as a function of
$m_{\mathrm{LQ}}$ and coupling $y$. Colored regions correspond to the
particular orderings of the cross sectional strengths.

What remains to be seen is how the pair production cross sections compare for
different scenarios. Namely, we want to see what the difference is between the
case when only $S_{1}$ leptoquark is present, when only $R_{2}$ is present,
and, finally, when both $S_{1}$ and $R_{2}$ are simultaneously present. Note
that in the $R_{2}$ case one needs to take into consideration an additional
process that is depicted in Fig. 4 for our particular flavor ansatz. There are
thus four contributions towards the leptoquark pair production in the $R_{2}$
scenario, i.e., $pp\rightarrow R_{2}^{+5/3}R_{2}^{+5/3\,*}$, $pp\rightarrow
R_{2}^{+2/3}R_{2}^{+2/3\,*}$, $pp\rightarrow R_{2}^{+5/3}R_{2}^{+2/3\,*}$, and
$pp\rightarrow R_{2}^{+5/3\,*}R_{2}^{+2/3}$, that one needs to consider.

Figure 4:  A self-induced $t$-channel contribution towards a pair production
of the $R_{2}$ leptoquark multiplet components at LHC within our flavor
ansatz.

In the $S_{1}$-$R_{2}$ scenario one has to account for the additional
production modes such as $R_{2}^{+5/3}$-$S^{+1/3\,*}_{1}$ and
$R_{2}^{+2/3}$-$S^{+1/3\,*}_{1}$ as well as the associated charge conjugate
terms. Note, once again, that upon decay one would, in all three scenarios,
have two opposite sign muons and two jets in the final state. So, this
comparison will yield an accurate estimate of the overall effect of the
proposed mechanism within this flavor ansatz.

Before we present the results of our numerical analysis, let us briefly
discuss the naive expectation for these pair production cross sections that we
denote with $\sigma_{\textrm{P}\,S_{1}}$, $\sigma_{\textrm{P}\,{R_{2}}}$, and
$\sigma_{\textrm{P}\,{S_{1}\textnormal{-}R_{2}}}$. To that end we introduce a
dimensionless parameter $\mu_{\mathrm{LQ}}$ that is defined as the ratio of an
actual pair production cross section for a given leptoquark scenario and the
QCD driven pair production for a single leptoquark scenario, as encapsulated
by $a_{0}(m_{\mathrm{LQ}})$ of Eq (3), for a fixed leptoquark mass. If the
cross section does not depend on Yukawa coupling, or Yukawa coupling is small,
one would expect to get $\mu_{S_{1}}=1$. Of course, following the same
reasoning, one would expect that $\mu_{R_{2}}=2$ since $R_{2}$ is made out of
two physical mass-degenerate leptoquark states. Finally, one would naively
expect to find $\mu_{S_{1}\textnormal{-}R_{2}}=3$ for the $S_{1}$-$R_{2}$
scenario if $S_{1}$ and the two states in $R_{2}$ are degenerate in mass.

The outcome of our numerical analysis for $\mu_{S_{1}}$, $\mu_{R_{2}}$, and
$\mu_{S_{1}\textnormal{-}R_{2}}$ is shown in Fig. 5. Indeed, for small $y$ one
does observe the naive scaling of $\mu_{\mathrm{LQ}}$. However, this behaviour
breaks down for relatively small values of Yukawa couplings, as can be seen
from the left panel of Fig. 5. In fact, the breaking of the naive picture is
especially strong in the case of the $S_{1}$-$R_{2}$ scenario, where the novel
contributions towards pair production that correspond to the Feynman diagrams
shown in panels $(a)$ and $(b)$ of Fig. 2 start to kick in at about $y\approx
0.1$. Since the novel contributions scale with the fourth power of Yukawa
coupling, the effect is very strong at large $y$ and we accordingly present
its full range in the right panel of Fig. 5. The enhancement of
$\mu_{\mathrm{LQ}}$ parameter is more pronounced for larger values of
$m_{\mathrm{LQ}}$ since the phase space suppression for the conventional pair
production is more relevant in that regime. We accordingly show in Fig. 5
behaviour of $\mu_{S_{1}}$, $\mu_{R_{2}}$, and
$\mu_{S_{1}\textnormal{-}R_{2}}$ for two different leptoquark masses, i.e.,
$m_{\mathrm{LQ}}=1.6$ TeV and $m_{\mathrm{LQ}}=2.4$ TeV. For example, in the
$S_{1}$ case, at $y=1.0$, one would see a signal that is approximately 2 times
larger than the naive expectation. For the $R_{2}$ case, at $y=1.0$, a signal
would exceed naive expectation by a factor of 4 for $m_{\mathrm{LQ}}=2.4$ TeV.
However, in the $S_{1}$-$R_{2}$ case, at $y=1.0$, one would see a signal that
is approximately 30 times larger than the expected value for
$m_{\mathrm{LQ}}=1.6$ TeV and approximately 80 times larger for
$m_{\mathrm{LQ}}=2.4$ TeV. And, for $y=1.2$, all the aforementioned
enhancements would double with respect to the $y=1.0$ case due to the quartic
nature of the effect. As advocated, one needs to be rather careful if more
than one leptoquark multiplet is present to make sure that all the relevant
contributions towards the scalar leptoquark pair production are taken into
account.

Figure 5:  Ratios of the factual cross sections for the pair production of
leptoquarks at LHC for three scenarios — $S_{1}$, $R_{2}$, and $S_{1}$-$R_{2}$
— and the QCD cross section for the pair production of one leptoquark as a
function of the associated Yukawa coupling(s). In all three instances the
leptoquarks of mass $m_{\mathrm{LQ}}$, as indicated in the plots, couple to
the first generation quarks and a right-chiral muon with Yukawa coupling
strength $y$ while the center-of-mass energy of the $pp$ collisions is 13 TeV.

We specify, for the benefit of the reader, the cross sections associated with
Fig. 5 in Table 1.

$m_{\textrm{LQ}}$ (TeV) | $y$ |  $\sigma_{\textrm{P}\,S_{1}}$ (fb) |  $\sigma_{\textrm{P}\,R_{2}}$ (fb) |  $\sigma_{\textrm{P}\,S_{1}\textnormal{-}R_{2}}$ (fb)
---|---|---|---|---
| $0.1$ | $0.141$ | $0.283$ | $0.425$
1.6 | $0.5$ | $0.132$ | $0.293$ | $1.19$
| $1.0$ | $0.287$ | $0.869$ | $13.3$
| $0.1$ | $0.0143$ | $0.0286$ | $0.0432$
2.0 | $0.5$ | $0.0131$ | $0.0299$ | $0.177$
| $1.0$ | $0.0330$ | $0.100$ | $2.27$
| $0.1$ | $0.00155$ | $0.00311$ | $0.00469$
2.4 | $0.5$ | $0.00139$ | $0.00330$ | $0.0284$
| $1.0$ | $0.00413$ | $0.0126$ | $0.396$
Table 1: The leading order cross sections for the leptoquark pair production
within the $S_{1}$, $R_{2}$, and $S_{1}$-$R_{2}$ scenarios through the proton-
proton collisions at 13 TeV center-of-mass energy, where in all three
instances leptoquarks of mass $m_{\textrm{LQ}}$ couple exclusively to a right-
chiral muon and the first generation quarks, as allowed by the Standard Model
gauge group, with the coupling strength $y$.

Before we conclude some final remarks are in order. The main enhancement of
the pair production in our case study originates from the diagrams in panels
$(a)$ and $(b)$ of Fig. 2. However, there could be additional contributions
from other diagrams in Fig. 2 within a given two leptoquark scenario. The
point is that the effect we advocate could be even larger than what we have
presented in Fig. 5. Also, it could be possible to construct a flavor scenario
where the enhancement is through, for example, $uu$ mode but the two final
state leptoquarks decay preferentially into quarks of heavier flavor(s) and/or
into different lepton(s) than the one that is exchanged in the $t$-channel.
Moreover, the two final state leptoquarks do not have to be mass degenerate
which is often one of the underlying assumptions to reconstruct the source of
the final state particles in the data analysis of the pair production at LHC.
This would open up other kinematically distinct signatures for the final state
leptons and quarks. This mechanism can lead to enhancement even when the
initial state is not associated with the valence quarks simply because it
allows for additional production channels. Again, there are multiple effects
of different nature that are associated with the proposed pair production
mechanism that certainly deserve additional attention within a well-defined
leptoquark scenario. Finally, we have noticed that the additional self-induced
contribution towards the leptoquark pair production, as given by the Feynman
diagram of Fig. 4 for the $R_{2}$ scenario, has not been considered in the
literature at all. We accordingly plan to purse the associated phenomenology
in future publications DFLS .

##  3 Conclusions

We introduce and discuss a novel production mechanism to generate scalar
leptoquarks in pairs at LHC. The proposed mechanism is based on the
$t$-channel Feynman diagram topology and it has quark-quark pairs in the
initial state which makes it perfectly suited for the LHC studies. The
mechanism in question requires a presence of two scalar leptoquarks that
couple to the same lepton and whose fermion numbers differ by two. So, the
mechanism can be operational whenever the New Physics scenario combines
$S_{3}$, $\tilde{S}_{1}$ or $S_{1}$ with either $R_{2}$ or $\tilde{R}_{2}$.

To demonstrate the strength of this mechanism we study one particular flavor
realisation based on the $S_{1}$-$R_{2}$ scenario, where both $S_{1}$ and
$R_{2}$ couple to a right-chiral muon and the quarks of the first generation,
as governed by the Standard Model gauge group. We explicitly show that the
pair production signal at LHC can be easily enhanced by a factor of ten to a
hundred even for the moderate values of the Yukawa couplings when compared
with the naive expectation as given by the QCD driven production mechanisms.
It can thus serve as an alternative way to the conventional processes to
efficiently constrain the parameter space of the two leptoquark scenarios at
LHC within otherwise allowed parameter space spanned by the Yukawa couplings
and leptoquark masses.

###### Acknowledgements.

I.D. would like to thank the CERN Theory Department and Institute Jožef Stefan
on hospitality. The work of S.F. was in part financially supported by the
Slovenian Research Agency (research core funding No. P1-0035).

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