aainstitutetext: Albert Einstein Center for Fundamental Physics, Institut für
Theoretische Physik, Universität Bern,
Sidlerstrasse 5, CH-3012 Bern, Switzerland.bbinstitutetext: CERN, Theoretical
Physics Department, CH-1211 Geneva 23, Switzerland.ccinstitutetext: Physik-
Institut, Universität Zürich, CH-8057 Zürich, Switzerland.

# Lepton-Quark Fusion at Hadron Colliders, precisely

Admir Greljo c and Nudžeim Selimović<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

When a TeV-scale leptoquark has a sizeable Yukawa coupling, its dominant
production mechanism at hadron colliders is the partonic-level lepton-quark
fusion. Even though the parton distribution functions for leptons inside the
proton are minuscule, they get compensated by the resonant enhancement. We
present the first computation of higher order radiative corrections to the
resonant leptoquark production cross section at the Large Hadron Collider
(LHC). Next-to-leading (NLO) QCD and QED corrections are similar in size but
come with the opposite sign. We compute NLO $K$-factors for a wide range of
scalar leptoquark masses, as well as, all possible combinations of quark and
lepton flavors and leptoquark charges. Theoretical uncertainties due to the
renormalisation and factorisation scale variations and the limited knowledge
of parton distribution functions are quantified. We finally discuss how to
disentangle the flavor structure of leptoquark interactions by exploiting the
interplay between different production channels.

††preprint: CERN-TH-2020-204, ZU-TH 49/20††arxiv: 2012.02092

##  1 Introduction

Leptoquarks (LQs) are hypothetical new bosons that convert quarks into leptons
and vice versa. The discovery of a leptoquark would represent a major
breakthrough in our understanding of particle interactions, pointing towards
an underlying quark-lepton unification at short distances. The phenomenology
of TeV-scale leptoquarks is a very rich and mature subject, for a recent
review see Ref. Dorsner:2016wpm . Leptoquarks at the TeV-scale are consistent
with the non-observation of proton decay and can be found in wildly different
settings beyond the Standard Model (SM). For example, they are in the spectrum
of low-scale quark-lepton unification models à la Pati-Salam (see e.g.
Pati:1974yy ; DiLuzio:2017vat ; Bordone:2017bld ; Greljo:2018tuh ;
Fornal:2018dqn ; Heeck:2018ntp ; Cornella:2019hct ; Blanke:2018sro ;
Balaji:2019kwe ). TeV-scale leptoquarks also appear as pseudo-Nambu–Goldstone
bosons of a new strongly interacting dynamics possibly related to the origin
of the electroweak symmetry breaking (see e.g. Gripaios:2009dq ; Fuentes-
Martin:2020bnh ; Barbieri:2017tuq ; Sannino:2017utc ; Marzocca:2018wcf ), or
as a consequence of R-parity violation in supersymmetry (see e.g.
Giudice:1997wb ; Csaki:2011ge ; Altmannshofer:2020axr ). On the one hand, they
lead to distinct indirect modifications of low-energy flavor transitions,
neutrino properties, top quark, electroweak precision, and Higgs physics. On
the other hand, the direct production of a leptoquark at the LHC leaves a
remarkable signature in the detector. Namely, a leptoquark would appear as a
resonance in the invariant mass of a lepton and a quark jet.

Leptoquarks are colored just like quarks. Therefore, they are copiously
produced in pairs in proton-proton collisions at the LHC by strong force
Blumlein:1996qp ; Kramer:1997hh ; Kramer:2004df ; Diaz:2017lit ;
Dorsner:2018ynv ; Borschensky:2020hot ; Allanach:2019zfr . A representative
Feynman diagram is shown in Fig. 1 (a). In the limit of a small leptoquark
coupling to quark and lepton ($y_{q\ell}$), the scalar leptoquark production
at hadron colliders is determined entirely by the strong coupling $\alpha_{s}$
and the leptoquark mass $m_{\rm LQ}$. The phenomenology becomes more
interesting once $y_{q\ell}$ is increased. This is particularly relevant when
establishing a connection with the low-energy flavor physics. The present
indirect constraints on a TeV-scale leptoquark suggest that $y_{q\ell}$ flavor
matrix has a peculiar structure with some entries left unconstrained, and
therefore possibly large. Taking a different perspective on the current data,
in order to explain the existing experimental anomalies in $B$-meson decays
Lees:2013uzd ; Hirose:2016wfn ; Aaij:2015yra ; Aaij:2014ora ; Aaij:2017vbb ;
Aaij:2013qta ; Aaij:2015oid ; Aaij:2019wad  or muon $g-2$ Bennett:2006fi ,
some leptoquark couplings are required to be large.

If leptoquarks are indeed behind the origin of these discrepancies, there will
be other production mechanisms beyond the QCD-induced pair production. To
begin with, for a sizeable $y_{q\ell}$, there is an additional contribution
with $t$-channel lepton exchange in $q\bar{q}$ fusion Bessaa:2014jya ;
Dorsner:2014axa . However, the production of two leptoquarks becomes quickly
phase-space suppressed with increasing leptoquark mass. Therefore, often
discussed in the literature is the single leptoquark plus lepton production
from quark-gluon scattering Alves:2002tj ; Hammett:2015sea ; Mandal:2015vfa ;
Dorsner:2018ynv . A representative Feynman diagram is shown in Fig. 1 (b). The
production cross section for this process is proportional to
$|y_{q\ell}|^{2}$, but suffers less phase-space suppression. For a heavier
leptoquark and a larger coupling, this production mechanism starts to dominate
over the pair production. In this work, we are interested in a sizeable (yet
perturbative) coupling range (i.e. $0.1\lesssim y_{lq}\lesssim\sqrt{4\pi}$
depending on the quark flavor), for which the production of a single
leptoquark plus lepton becomes comparable or even favorable.

For example, Fig. 2 shows the relative comparison of different channels in the
mass versus coupling plane when the leptoquark couples to down quark (left
panel) and bottom quark (right panel). The upper edge of the vertical axis is
chosen such that the $t$-channel induced pair production is suppressed
compared to the pure QCD contribution. Nonetheless, the single leptoquark
production plus the charge-conjugated (c.c.) process, dominates over the pair
production in the large portion of the parameter space shown in Fig. 2.
Relevant information on these parameters can also be extracted from indirect
leptoquark effects at high-$p_{T}$, such as Drell-Yan tails Faroughy:2016osc ;
Schmaltz:2018nls ; Greljo:2017vvb ; Fuentes-Martin:2020lea ; Greljo:2018tzh ;
Marzocca:2020ueu ; Baker:2019sli . These probe complementary parameter space
compared to both single and pair production (see Section 4 in Ref.
Dorsner:2018ynv ).

Figure 1:  Representative Feynman diagrams for three different leptoquark
production mechanisms at hadron colliders: ($a$) pair $pp\to S_{Q_{\rm
LQ}}^{\dagger}S_{Q_{\rm LQ}}$ , ($b$) single $pp\to S_{Q_{\rm LQ}}\ell$ and
($c$) resonant $pp\to S_{Q_{\rm LQ}}$ .

The collider phenomenology of TeV-scale leptoquarks had a new twist recently.
The precise extraction of lepton parton distribution functions (PDFs)
Buonocore:2020nai  based on the LUX method Manohar:2016nzj ; Manohar:2017eqh
(see also Bertone:2015lqa ; Bertone:2017bme ) facilitated another leptoquark
production mechanism, the resonant leptoquark production Ohnemus:1994xf ;
Eboli:1997fb ; Buonocore:2020erb . The tree-level Feynman diagram is shown in
Fig. 1 (c). The production cross section for the direct lepton-quark fusion is
also proportional to $|y_{q\ell}|^{2}$, but suffers even less phase-space
suppression than the single leptoquark plus lepton channel. The difference
between the two is the absence (presence) of a high-$p_{T}$ lepton. Therefore,
the resonant channel cross section is always larger as shown in Fig. 2.
Interestingly, this applies to all combinations of quarks and leptons
involved. The ATLAS and CMS collaborations have extensively searched for
leptoquarks in pair production and a single leptoquark plus lepton channel
Sirunyan:2017yrk ; Sirunyan:2018jdk ; Sirunyan:2018ryt ; Aaboud:2019bye ;
Aad:2020jmj ; ATLAS:2020sxq ; Aad:2020iuy ; Aad:2020sgw ; Sirunyan:2018btu ;
Sirunyan:2018vhk ; Sirunyan:2018xtm ; Aaboud:2019jcc ; CMS:2020gru , however,
the resonant production was not considered so far. Nonetheless, the
phenomenological collider simulation in Ref. Buonocore:2020erb  shows that the
resonant channel has a potential to probe the uncharted territory of interest
in the mass versus coupling plane. It is therefore of utmost importance for
leptoquark hunters at the LHC to place the resonant production mechanism at
the top of their to-do list.

In this paper we fill in the gap on the theory side. Leptoquark toolbox for
precision collider studies Dorsner:2018ynv  includes leptoquark pair and
single production at NLO in QCD. The scope of this work is to precisely
calculate the resonant leptoquark production cross at the LHC including for
the first time higher order radiative corrections and quantify the
uncertainties from the missing orders and limited knowledge of parton
distribution functions. The main result of this paper are the resonant
leptoquark production cross sections at the LHC at NLO QCD plus QED with the
corresponding uncertainties. These are reported in Tables 1 and 2, together
with the complete set of NLO $K$-factors reported in Figs. 8, 9 and 10.
Interestingly, we find that NLO QED corrections are as important as QCD
corrections. Along the way, we discuss the interplay between different
production mechanism and propose methods to determine the quark flavor inside
the proton from which the leptoquark was created. The present study is limited
to scalar leptoquarks and will be extended to include vectors in the future.
Radiative corrections in the scalar leptoquark models are not sensitive to the
details of the ultraviolet completion, in contrast to the vector case Fuentes-
Martin:2019ign ; Fuentes-Martin:2020luw ; Fuentes-Martin:2020hvc .

The paper is organised as follows. In Section 2 we set up the framework and
present compact analytic expressions for the relevant partonic cross sections
stemming from loop calculations detailed in Appendices A and B. In Section 3
we perform a numerical calculation of the hadronic cross section for the
resonant leptoquark production at the LHC using the most recent lepton parton
distribution functions. Supplemental numerical results are left for Appendix
C. We finally conclude in Section 4.

Figure 2:  Comparison of cross sections for three leptoquark production
mechanisms at the LHC ($\sqrt{s}=13$ TeV). Shaded regions show the parameter
space in the leptoquark mass versus coupling plane for which the corresponding
cross sections are $>0.1$ fb. The pair production $pp\to S_{Q_{\rm
LQ}}^{\dagger}S_{Q_{\rm LQ}}$ cross section is shown in black, while the
single $(pp\to S_{Q_{\rm LQ}}\,\ell)+$ c.c. and the resonant $(pp\to S_{Q_{\rm
LQ}})+$ c.c. cross sections are shown in blue and red, respectively. In the
left (right) panel, the leptoquark interacts primarily with the down (bottom)
quark. The lepton flavors in the resonant production are shown with solid
($\tau$), dashed ($\mu$) and dotted ($e$) lines. The electric charge of
$S_{Q_{\rm LQ}}$ is set to $Q_{\rm LQ}=2/3$, however, the difference is
negligible for $Q=4/3$. For consistency, all cross sections are computed at
NLO QCD (plus NLO QED for the resonant process) with the same central PDF set
LUXlep-NNPDF31_nlo_as_0118_luxqed (v2) Buonocore:2020nai . The first two
processes are computed using the leptoquark toolbox Dorsner:2018ynv , while
the resonant production is taken from Section 3. The pair production from the
$t$-channel leptoquark exchange is negligible in this coupling range.

##  2 Scalar leptoquark resonant production

The inevitable condition for a field coupling quarks and leptons at the tree
level is to transform in the (anti)fundamental representation of the $SU(3)$
part of the SM gauge group. The interaction between leptoquark and gluons is
then completely specified and forms the basis for NLO QCD calculations. In
contrast, the electroweak part of the SM allows for leptoquark representations
involving different $SU(2)_{L}\times U(1)_{Y}$ multiplets with the
corresponding hypercharges $Y$. As we are interested in evaluating the NLO QED
corrections, the only relevant information is that after electroweak symmetry
breaking, the possible absolute electric charges for any component of the
$SU(2)_{L}$ multiplet are $|Q_{\rm LQ}|=\\{1/3,2/3,4/3,5/3\\}$, in the units
of the positron charge. Therefore, to assess the NLO QCD plus QED corrections
to the resonant leptoquark production, we can treat one component inside the
multiplet at a time, with the production of other components corresponding to
separate processes.

The fermion content is the SM one, and the quark-lepton interaction with the
scalar leptoquark $S_{Q_{\rm LQ}}$ of charge $Q_{\rm LQ}$ is given by

$\mathcal{L}\supset-y^{L}_{q\ell}~{}\bar{q}P_{L}\ell~{}S_{Q_{\rm
LQ}}-y^{R}_{q\ell}~{}\bar{q}P_{R}\ell~{}S_{Q_{\rm LQ}}~{}+~{}{\rm h.c.}\,,$
(1)

where $y^{L,R}_{q\ell}$ are $3\times 3$ matrices in flavor space, encoding the
most general form of Yukawa couplings. The chiral fermionic fields $q_{L,R}$
and $\ell_{L,R}$ (note the left- and right-handed chiral projectors $P_{L,R}$)
correspond to charge and mass eigenstates after the electroweak symmetry
breaking. (Fermion mixings when going from the interaction to the mass basis,
both in the quark and lepton sectors, are already absorbed in Yukawa matrices
$y^{L,R}_{q\ell}$.) Depending on $Q_{\rm LQ}$, some fermionic fields in Eq.
(1) are charge-conjugated from the usual SM definitions, for example
$\mathcal{L}\supset-y^{R}_{ue}~{}\overline{u_{L}}\
e_{L}^{C}~{}S^{\dagger}_{1/3}$. We can use the proton composition to
precipitate lepton-quark fusion involving quark flavors $u,d,s,c,b$ and
charged leptons $e,\mu,\tau$. When calculating partonic cross sections, we
will work in the limit of disregarding all fermion masses, which is an
excellent approximation given the energy of the collisions. Neutrinos are not
created in photon splitting and cannot be generated inside the proton at the
order we are interested in. In the absence of fermion masses, possible
interference terms involving left- and right-handed Yukawa couplings vanish.
This allows us to independently treat processes in which leptoquark is
resonantly produced by the same flavor combination of quarks and leptons, but
of the opposite chirality. Additionally, the resonant leptoquark production is
specified by one entry in the chiral Yukawa matrix irrespective of all other
entries. When several flavor couplings contribute to the production of the
same leptoquark, the individual contributions to production cross section
factorise, and we add them separately.

In full generality, we summarize that scalar leptoquarks (SLQs) are $SU(3)$
triplets, with four possible values of the electric charge, and their resonant
production cross section is determined by the entries in the Yukawa matrices
without interference. This exhausts all possibilities for SLQs and we conclude
that our computation could be easily matched to any model containing these
particles. Moreover, we note that neglecting fermion masses causes all one-
loop corrections proportional to Yukawa couplings to vanish. Accordingly, for
the case of SLQ, the dominant NLO effects originate from QED and QCD.111The
situation is different in the case of vector leptoquarks (VLQs). The
calculation of NLO corrections for these particles necessarily involves
details depending on the UV completion that embeds them. For instance, in many
popular extensions of the SM, VLQs are accompanied by the massive color octet
affecting NLO QCD contributions to processes involving VLQ in a nontrivial way
Fuentes-Martin:2019ign ; Fuentes-Martin:2020luw . For the moment, we focus on
the resonant production of the SLQs, while postponing a detailed analysis of
spin-1 leptoquarks for future work.

The relevant NLO QED (QCD) corrections to partonic cross section are
calculated in Appendix A (B). The hadronic cross section is obtained after
convoluting the relevant partonic cross sections, $\hat{\sigma}$, with the
parton distribution functions, $f_{i}$ and $f_{j}$, in the following way,

$\sigma(s)=2\
\sum_{ij}\int_{\xi}^{1}dy\,f_{i}(y)\int_{\xi/y}^{1}dz\,\frac{\xi}{yz^{2}}\
f_{j}\left(\frac{\xi}{yz}\right)\ \hat{\sigma}_{ij}(z)~{},$ (2)

where $\xi=m^{2}_{{\rm LQ}}/s$, $\sqrt{s}$ is the collider center of mass
energy, $y$ is the fraction of proton momentum carried by the parton labeled
by $i$, and $z=m_{\rm LQ}^{2}/\hat{s}$, with $\hat{s}$ being the partonic-
level center of mass energy. The sum goes over $ij=\\{q\ell$, $g\ell$,
$q\gamma$}, with the individual cases corresponding to Eqs. (3), (4), and (6),
respectively.

###  2.1 Next-to-leading order QCD corrections

The hadronic cross section for resonant leptoquark production is set by the
size of the colliding parton densities, and the size of the parton level cross
section. The Yukawa couplings are $\mathcal{O}(1)$, and at the leading order
(LO), the partonic cross section scales as
$\hat{\sigma}_{0}\propto|y_{ql}|^{2}$. The parton density for gluons and
quarks can be viewed as a sum of terms $\sum_{n}(\alpha_{s}L)^{n}$, where
$\alpha_{s}$ is the QCD coupling, $L=\log(\mu_{F}^{2}/\Lambda^{2})$, with
$\mu_{F}$ representing the factorisation scale, and $\Lambda$ is the typical
hadronic scale. The QCD coupling is evaluated at the factorisation scale and
its size is set by $\alpha_{s}\approx 1/L$. We conclude that gluon and quark
PDFs are non-perturbative objects of $\mathcal{O}(1)$. In contrast, the photon
density is the first order QED effect and its size is determined by $\alpha
L\sum_{n}(\alpha_{s}L)^{n}$, where $\alpha$ is the QED coupling. Further, as a
result of photon splitting, lepton PDFs are generated at the next order in QED
and their size is given by $\alpha^{2}L^{2}\sum_{n}(\alpha_{s}L)^{n}$. We
apply the same QED to QCD coupling comparison already employed in
Buonocore:2020nai ; Manohar:2016nzj ; Manohar:2017eqh  and use that
$\alpha\approx\alpha_{s}^{2}$. Accordingly, in terms of $\alpha_{s}$, the size
of the photon density is $\mathcal{O}(\alpha_{s})$, while the lepton densities
are $\mathcal{O}(\alpha_{s}^{2})$. The size of the LO hadronic cross section
for resonant leptoquark production is then $\int(f_{q}\otimes f_{\ell})\
\hat{\sigma}_{0}\sim\mathcal{O}(\alpha_{s}^{2})$. Therefore, the typical QCD
correction coming from $\mathcal{O}(\alpha_{s})$ diagrams represents the
contribution to hadronic cross section which is $\mathcal{O}(\alpha_{s}^{3})$.

The virtual corrections from gluon loops (Fig. 5) are summed with the diagrams
involving the real gluon emission (Fig. 6) to obtain the IR safe partonic
cross section

$\displaystyle\hat{\sigma}_{q\ell}(z)=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\Bigg{\\{}\left[1+\frac{\alpha_{s}}{2\pi}C_{F}\left(\frac{3}{2}\log\left(\frac{\mu_{\rm
R}^{2}}{\mu_{\rm
F}^{2}}\right)-\frac{\pi^{2}}{3}\right)\right]\delta(1-z)-\frac{2z}{(1-z)_{+}}$
(3)
$\displaystyle+2(1+z^{2})\left(\frac{\log(1-z)}{(1-z)}\right)_{+}-\frac{1+z^{2}}{(1-z)_{+}}\log\left(\frac{z\mu_{\rm
F}^{2}}{m_{\rm LQ}^{2}}\right)\Bigg{\\}}\,,$

where $C_{F}=4/3$ and $\mu_{\rm F}$, $\mu_{\rm R}$ are the factorisation and
renormalisation scales, respectively. The remaining $\mathcal{O}(\alpha_{s})$
diagrams that contribute to the resonant leptoquark production involve gluons
in the initial state (Fig. 7). The partonic cross section in this case reads

$\hat{\sigma}_{g\ell}(z)=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\frac{\alpha_{s}}{2\pi}T_{R}\left[-\log\left(\frac{z\mu^{2}_{\rm
F}}{(1-z)^{2}m_{\rm LQ}^{2}}\right)(z^{2}+(1-z)^{2})+2z(1-z)(2+\log
z)\right],$ (4)

where $T_{R}=1/2$. The $\overline{\rm MS}$ scheme was utilized both for
factorisation and renormalisation. The NLO QCD corrections are universal for
all leptoquark types. More details about the partonic cross section
calculation can be found in Appendices A and B.

###  2.2 Next-to-leading order QED corrections

The NLO QED corrections are provided by processes where the initial lepton is
replaced by a photon splitting into lepton pairs (Fig. 4). We estimate the
size of these corrections by $\alpha_{s}$ power counting for the leptoquark
production via $\gamma+q\to\ell+$ LQ. When convoluted with the corresponding
PDFs, the size of the resonant cross section is

$\int\left(f_{q}\otimes f_{\gamma}\right)\hat{\sigma}_{\gamma q}\approx
1\times\alpha L\times\alpha=\alpha_{s}^{3}~{}.$ (5)

Interestingly enough, the QED corrections are of the same order as the typical
QCD corrections and their inclusion is essential in assessing the NLO effects
in resonant leptoquark production. Employing the $\overline{\rm MS}$
factorisation scheme, the partonic cross section reads

$\hat{\sigma}_{q\gamma}=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\frac{\alpha}{2\pi}\left(-\log\left(\frac{z\ \mu_{\rm
F}^{2}}{(1-z)^{2}m_{\rm LQ}^{2}}\right)(z^{2}+(1-z)^{2})+X_{Q_{\rm
LQ}}(z)\right).$ (6)

The logarithmic part is universal for all leptoquark types since it originates
from photon splitting to charged lepton pair, while the charge dependence is
encoded in the functions $X_{Q_{\rm LQ}}(z)$,

$\displaystyle X_{1/3}(z)$
$\displaystyle=-\frac{2}{9}(1-z)(5-13z)+\frac{2}{9}(1-5z)z\log z,$ (7)
$\displaystyle X_{2/3}(z)$
$\displaystyle=-\frac{11}{18}(1-z)(1-5z)+\frac{8}{9}(1-2z)z\log z,$ (8)
$\displaystyle X_{4/3}(z)$
$\displaystyle=\frac{1}{18}(1-z)(13+103z)+\frac{16}{9}(2-z)z\log z,$ (9)
$\displaystyle X_{5/3}(z)$
$\displaystyle=\frac{2}{9}(1-z)(7+37z)+\frac{10}{9}(5-z)z\log z,$ (10)

where subscripts $\\{1/3,\ 2/3,\ 4/3,\ 5/3\\}$ correspond to electric charge
of the leptoquark. Since the CP is conserved, the same formulas hold for the
charge-conjugated processes. Note that loop diagrams involving photons are
higher order in QCD coupling. Namely, the size of these diagrams, for the
1-loop corrections to partonic-level cross section involving photons,
$\hat{\sigma}_{q\ell}^{(1)}$, is given by

$\int\left(f_{q}\otimes f_{\ell}\right)\hat{\sigma}_{q\ell}^{(1)}\approx
1\times\alpha^{2}L^{2}\times\alpha=\alpha_{s}^{4}~{},$ (11)

and we neglect them. The detailed derivation of the NLO QED corrections is
presented in Appendix A.

##  3 Numerical results and discussion

We carry out a numerical calculation of the hadronic cross section for the
resonant leptoquark production in $pp$ collisions. We consider the most
general flavor structure of the leptoquark coupling $y_{q\ell}$ to a quark
$q\equiv d_{L},d_{R}$, $u_{L},u_{R}$, $s_{L},s_{R}$, $c_{L},c_{R},b_{L}$ or
$b_{R}$, and a lepton $\ell\equiv e_{L},e_{R}$, $\mu_{L},\mu_{R},\tau_{L}$, or
$\tau_{R}$. All options for the leptoquark electric charge are considered,
$|Q_{{\rm LQ}}|=1/3$, $2/3$, $4/3$ and $5/3$. Cross sections are calculated
for every $q\ell$ combination separately assuming $y_{q\ell}=1$. As a
reminder, the total cross section is simply the sum over different channels,
$\sigma=\sum_{q,\ell}|y_{q\ell}|^{2}\sigma_{q\ell}$. We compute the process
and its charge conjugate at leading and next-to-leading order in QCD and QED.
We scan over the large leptoquark mass window $m_{{\rm LQ}}=[500-5000]$ GeV
relevant for the future studies at the LHC. As a benchmark, the collider
center of mass energy is set to $\sqrt{s}=13$ TeV.

$m_{\mathrm{LQ}}$ [TeV] | Partons |  $\sigma_{\mathrm{S^{1/3}}}$ [pb] |  $\sigma_{\mathrm{S^{5/3}}}$ [pb]
---|---|---|---
0.9 | $u\ +\ e$ | $(1.45\times 10^{-1})^{+3.1\%}_{-3.7\%}\pm 1.8\%$ | $(1.58\times 10^{-1})^{+2.9\%}_{-3.4\%}\pm 1.8\%$
$u\ +\ \mu$ | $(1.39\times 10^{-1})^{+3.1\%}_{-3.8\%}\pm 1.9\%$ | $(1.52\times 10^{-1})^{+2.9\%}_{-3.5\%}\pm 1.8\%$
$u\ +\ \tau$ | $(1.11\times 10^{-1})^{+3.6\%}_{-4.0\%}\pm 2.0\%$ | $(1.23\times 10^{-1})^{+3.4\%}_{-3.6\%}\pm 2.0\%$
$c\ +\ e$ | $(1.32\times 10^{-2})^{+4.2\%}_{-5.1\%}\pm 12.1\%$ | $(1.44\times 10^{-2})^{+3.9\%}_{-4.7\%}\pm 12.2\%$
$c\ +\ {\mu}$ | $(1.29\times 10^{-2})^{+4.3\%}_{-5.2\%}\pm 12.0\%$ | $(1.40\times 10^{-2})^{+3.9\%}_{-4.8\%}\pm 12.0\%$
$c\ +\ {\tau}$ | $(1.01\times 10^{-2})^{+4.6\%}_{-5.5\%}\pm 12.2\%$ | $(1.12\times 10^{-2})^{+4.1\%}_{-5.0\%}\pm 12.2\%$
1.6 | $u\ +\ {e}$ | $(1.40\times 10^{-2})^{+2.8\%}_{-3.3\%}\pm 2.0\%$ | $(1.49\times 10^{-2})^{+2.7\%}_{-3.1\%}\pm 2.0\%$
$u\ +\ {\mu}$ | $(1.36\times 10^{-2})^{+2.9\%}_{-3.4\%}\pm 2.0\%$ | $(1.46\times 10^{-2})^{+2.7\%}_{-3.1\%}\pm 2.0\%$
$u\ +\ {\tau}$ | $(1.11\times 10^{-2})^{+3.3\%}_{-3.5\%}\pm 2.2\%$ | $(1.20\times 10^{-2})^{+2.9\%}_{-3.2\%}\pm 2.2\%$
$c\ +\ {e}$ | $(7.31\times 10^{-4})^{+3.8\%}_{-4.4\%}\pm 24.2\%$ | $(7.80\times 10^{-4})^{+3.6\%}_{-4.1\%}\pm 24.3\%$
$c\ +\ {\mu}$ | $(7.16\times 10^{-4})^{+3.8\%}_{-4.4\%}\pm 24.0\%$ | $(7.65\times 10^{-4})^{+3.6\%}_{-4.1\%}\pm 24.1\%$
$c\ +\ {\tau}$ | $(5.78\times 10^{-4})^{+4.1\%}_{-4.7\%}\pm 24.2\%$ | $(6.28\times 10^{-4})^{+3.8\%}_{-4.3\%}\pm 24.3\%$
2.5 | $u\ +\ {e}$ | $(1.53\times 10^{-3})^{+2.6\%}_{-3.0\%}\pm 2.4\%$ | $(1.61\times 10^{-3})^{+2.5\%}_{-2.8\%}\pm 2.4\%$
$u\ +\ {\mu}$ | $(1.50\times 10^{-3})^{+2.6\%}_{-3.0\%}\pm 2.4\%$ | $(1.59\times 10^{-3})^{+2.5\%}_{-2.9\%}\pm 2.4\%$
$u\ +\ {\tau}$ | $(1.25\times 10^{-3})^{+2.9\%}_{-3.2\%}\pm 2.5\%$ | $(1.33\times 10^{-3})^{+2.6\%}_{-3.0\%}\pm 2.5\%$
$c\ +\ {e}$ | $(5.52\times 10^{-5})^{+3.3\%}_{-3.8\%}\pm 41.9\%$ | $(5.83\times 10^{-5})^{+3.1\%}_{-3.6\%}\pm 42.1\%$
$c\ +\ {\mu}$ | $(5.43\times 10^{-5})^{+3.3\%}_{-3.8\%}\pm 41.8\%$ | $(5.74\times 10^{-5})^{+3.2\%}_{-3.6\%}\pm 41.9\%$
$c\ +\ {\tau}$ | $(4.48\times 10^{-5})^{+3.5\%}_{-4.0\%}\pm 42.0\%$ | $(4.79\times 10^{-5})^{+3.3\%}_{-3.8\%}\pm 42.1\%$
4.0 | $u\ +\ {e}$ | $(7.21\times 10^{-5})^{+2.3\%}_{-2.6\%}\pm 3.1\%$ | $(7.49\times 10^{-5})^{+2.2\%}_{-2.5\%}\pm 3.0\%$
$u\ +\ {\mu}$ | $(7.14\times 10^{-5})^{+2.3\%}_{-2.6\%}\pm 3.1\%$ | $(7.42\times 10^{-5})^{+2.2\%}_{-2.5\%}\pm 3.1\%$
$u\ +\ {\tau}$ | $(6.02\times 10^{-5})^{+2.4\%}_{-2.7\%}\pm 3.2\%$ | $(6.30\times 10^{-5})^{+2.3\%}_{-2.6\%}\pm 3.2\%$
$c\ +\ {e}$ | $(2.35\times 10^{-6})^{+2.6\%}_{-3.0\%}\pm 63.0\%$ | $(2.45\times 10^{-6})^{+2.6\%}_{-2.9\%}\pm 63.1\%$
$c\ +\ {\mu}$ | $(2.33\times 10^{-6})^{+2.7\%}_{-3.0\%}\pm 62.9\%$ | $(2.42\times 10^{-6})^{+2.6\%}_{-2.9\%}\pm 63.1\%$
$c\ +\ {\tau}$ | $(1.96\times 10^{-6})^{+2.8\%}_{-3.1\%}\pm 63.1\%$ | $(2.05\times 10^{-6})^{+2.7\%}_{-3.0\%}\pm 63.2\%$
Table 1:  Inclusive cross sections in pb for the resonant leptoquark production from up-type quarks, $pp\to$ LQ + charge-conjugated process, as a function of the leptoquark mass $m_{{\rm LQ}}$ at $\sqrt{s}=13$ TeV. The cross section $\sigma_{\mathrm{S^{1/3}}}\;(\sigma_{\mathrm{S^{5/3}}})$ corresponds to the resonant production of scalar LQ with absolute electric charge $1/3\ (5/3)$ when the associated Yukawa coupling strength is set to one, $y_{q\ell}=1$. The second column denotes which quark-lepton pair couples to the corresponding leptoquark. First (second) uncertainty is due to the renormalisation and factorisation scale variations (PDF replicas), and is given in per cent units. See Section 3 for details. $m_{\mathrm{LQ}}$ [TeV] | Partons |  $\sigma_{\mathrm{S^{2/3}}}$ [pb] |  $\sigma_{\mathrm{S^{4/3}}}$ [pb]
---|---|---|---
0.9 | $d\ +\ {e}$ | $(8.85\times 10^{-2})^{+3.3\%}_{-3.7\%}\pm 2.0\%$ | $(9.21\times 10^{-2})^{+3.2\%}_{-3.5\%}\pm 2.0\%$
$d\ +\ {\mu}$ | $(8.54\times 10^{-2})^{+3.4\%}_{-3.7\%}\pm 2.0\%$ | $(8.90\times 10^{-2})^{+3.3\%}_{-3.6\%}\pm 2.0\%$
$d\ +\ {\tau}$ | $(6.80\times 10^{-2})^{+4.0\%}_{-3.9\%}\pm 2.1\%$ | $(7.15\times 10^{-2})^{+3.9\%}_{-3.7\%}\pm 2.1\%$
$s\ +\ {e}$ | $(2.41\times 10^{-2})^{+3.8\%}_{-4.2\%}\pm 5.4\%$ | $(2.51\times 10^{-2})^{+3.7\%}_{-4.1\%}\pm 5.4\%$
$s\ +\ {\mu}$ | $(2.34\times 10^{-2})^{+3.9\%}_{-4.3\%}\pm 5.4\%$ | $(2.44\times 10^{-2})^{+3.8\%}_{-4.1\%}\pm 5.4\%$
$s\ +\ {\tau}$ | $(1.85\times 10^{-2})^{+4.3\%}_{-4.5\%}\pm 5.5\%$ | $(1.95\times 10^{-2})^{+4.2\%}_{-4.3\%}\pm 5.5\%$
$b\ +\ {e}$ | $(9.01\times 10^{-3})^{+4.9\%}_{-5.8\%}\pm 1.8\%$ | $(9.39\times 10^{-3})^{+4.7\%}_{-5.6\%}\pm 1.8\%$
$b\ +\ {\mu}$ | $(8.76\times 10^{-3})^{+4.9\%}_{-5.8\%}\pm 1.8\%$ | $(9.14\times 10^{-3})^{+4.7\%}_{-5.6\%}\pm 1.8\%$
$b\ +\ {\tau}$ | $(6.87\times 10^{-3})^{+5.3\%}_{-6.3\%}\pm 2.1\%$ | $(7.25\times 10^{-3})^{+5.0\%}_{-5.9\%}\pm 2.0\%$
1.6 | $d\ +\ {e}$ | $(7.23\times 10^{-3})^{+3.0\%}_{-3.2\%}\pm 2.3\%$ | $(7.47\times 10^{-3})^{+2.8\%}_{-3.1\%}\pm 2.3\%$
$d\ +\ {\mu}$ | $(7.07\times 10^{-3})^{+3.0\%}_{-3.3\%}\pm 2.3\%$ | $(7.30\times 10^{-3})^{+2.9\%}_{-3.1\%}\pm 2.3\%$
$d\ +\ {\tau}$ | $(5.76\times 10^{-3})^{+3.7\%}_{-3.4\%}\pm 2.4\%$ | $(6.00\times 10^{-3})^{+3.5\%}_{-3.3\%}\pm 2.4\%$
$s\ +\ {e}$ | $(1.40\times 10^{-3})^{+3.5\%}_{-3.7\%}\pm 8.8\%$ | $(1.44\times 10^{-3})^{+3.4\%}_{-3.5\%}\pm 8.8\%$
$s\ +\ {\mu}$ | $(1.37\times 10^{-3})^{+3.5\%}_{-3.7\%}\pm 8.7\%$ | $(1.42\times 10^{-3})^{+3.4\%}_{-3.6\%}\pm 8.7\%$
$s\ +\ {\tau}$ | $(1.12\times 10^{-3})^{+4.1\%}_{-3.9\%}\pm 8.8\%$ | $(1.16\times 10^{-3})^{+4.0\%}_{-3.7\%}\pm 8.8\%$
$b\ +\ {e}$ | $(4.40\times 10^{-4})^{+4.5\%}_{-5.1\%}\pm 2.4\%$ | $(4.55\times 10^{-4})^{+4.3\%}_{-5.0\%}\pm 2.4\%$
$b\ +\ {\mu}$ | $(4.32\times 10^{-4})^{+4.5\%}_{-5.1\%}\pm 2.4\%$ | $(4.47\times 10^{-4})^{+4.4\%}_{-5.0\%}\pm 2.4\%$
$b\ +\ {\tau}$ | $(3.49\times 10^{-4})^{+4.8\%}_{-5.5\%}\pm 2.6\%$ | $(3.63\times 10^{-4})^{+4.6\%}_{-5.3\%}\pm 2.6\%$
2.5 | $d\ +\ {e}$ | $(6.63\times 10^{-4})^{+2.6\%}_{-2.9\%}\pm 2.9\%$ | $(6.80\times 10^{-4})^{+2.4\%}_{-2.8\%}\pm 2.9\%$
$d\ +\ {\mu}$ | $(6.54\times 10^{-4})^{+2.6\%}_{-2.9\%}\pm 2.9\%$ | $(6.71\times 10^{-4})^{+2.5\%}_{-2.8\%}\pm 2.9\%$
$d\ +\ {\tau}$ | $(5.43\times 10^{-4})^{+3.3\%}_{-3.0\%}\pm 3.0\%$ | $(5.60\times 10^{-4})^{+3.1\%}_{-2.9\%}\pm 3.0\%$
$s\ +\ {e}$ | $(9.66\times 10^{-5})^{+2.9\%}_{-3.3\%}\pm 16.0\%$ | $(9.90\times 10^{-5})^{+2.8\%}_{-3.2\%}\pm 16.0\%$
$s\ +\ {\mu}$ | $(9.53\times 10^{-5})^{+3.0\%}_{-3.2\%}\pm 15.9\%$ | $(9.77\times 10^{-5})^{+2.9\%}_{-3.2\%}\pm 15.9\%$
$s\ +\ {\tau}$ | $(7.89\times 10^{-5})^{+3.6\%}_{-3.4\%}\pm 16.0\%$ | $(8.14\times 10^{-5})^{+3.4\%}_{-3.3\%}\pm 16.0\%$
$b\ +\ {e}$ | $(2.42\times 10^{-5})^{+4.2\%}_{-4.7\%}\pm 3.8\%$ | $(2.48\times 10^{-5})^{+4.2\%}_{-4.5\%}\pm 3.8\%$
$b\ +\ {\mu}$ | $(2.39\times 10^{-5})^{+4.3\%}_{-4.7\%}\pm 3.8\%$ | $(2.45\times 10^{-5})^{+4.2\%}_{-4.6\%}\pm 3.8\%$
$b\ +\ {\tau}$ | $(1.97\times 10^{-5})^{+4.6\%}_{-5.0\%}\pm 3.9\%$ | $(2.03\times 10^{-5})^{+4.5\%}_{-4.8\%}\pm 3.9\%$
4.0 | $d\ +\ {e}$ | $(2.41\times 10^{-5})^{+2.1\%}_{-2.5\%}\pm 4.7\%$ | $(2.45\times 10^{-5})^{+2.1\%}_{-2.4\%}\pm 4.7\%$
$d\ +\ {\mu}$ | $(2.39\times 10^{-5})^{+2.1\%}_{-2.5\%}\pm 4.7\%$ | $(2.43\times 10^{-5})^{+2.1\%}_{-2.4\%}\pm 4.7\%$
$d\ +\ {\tau}$ | $(2.02\times 10^{-5})^{+2.6\%}_{-2.6\%}\pm 4.7\%$ | $(2.06\times 10^{-5})^{+2.4\%}_{-2.5\%}\pm 4.7\%$
$s\ +\ {e}$ | $(2.84\times 10^{-6})^{+2.4\%}_{-2.8\%}\pm 37.5\%$ | $(2.89\times 10^{-6})^{+2.4\%}_{-2.8\%}\pm 37.7\%$
$s\ +\ {\mu}$ | $(2.81\times 10^{-6})^{+2.4\%}_{-2.8\%}\pm 37.5\%$ | $(2.87\times 10^{-6})^{+2.4\%}_{-2.8\%}\pm 37.6\%$
$s\ +\ {\tau}$ | $(2.37\times 10^{-6})^{+2.5\%}_{-3.0\%}\pm 37.5\%$ | $(2.42\times 10^{-6})^{+2.5\%}_{-2.9\%}\pm 37.7\%$
$b\ +\ {e}$ | $(4.32\times 10^{-7})^{+3.9\%}_{-4.1\%}\pm 10.2\%$ | $(4.40\times 10^{-7})^{+3.8\%}_{-4.1\%}\pm 10.2\%$
$b\ +\ {\mu}$ | $(4.29\times 10^{-7})^{+3.9\%}_{-4.2\%}\pm 10.2\%$ | $(4.37\times 10^{-7})^{+3.8\%}_{-4.1\%}\pm 10.2\%$
$b\ +\ {\tau}$ | $(3.59\times 10^{-7})^{+4.5\%}_{-4.4\%}\pm 10.3\%$ | $(3.67\times 10^{-7})^{+4.3\%}_{-4.3\%}\pm 10.3\%$
Table 2:  Inclusive cross sections in pb for the resonant leptoquark
production from down-type quarks, $pp\to$ LQ + charge-conjugated process, as a
function of the leptoquark mass $m_{{\rm LQ}}$ at $\sqrt{s}=13$ TeV. The cross
section $\sigma_{\mathrm{S^{2/3}}}\;(\sigma_{\mathrm{S^{2/3}}})$ corresponds
to the resonant production of scalar LQ with absolute electric charge $2/3\
(4/3)$ when the associated Yukawa coupling strength is set to one,
$y_{q\ell}=1$. The second column denotes which quark-lepton pair couples to
the corresponding leptoquark. First (second) uncertainty is due to the
renormalisation and factorisation scale variations (PDF replicas), and is
given in per cent units. See Section 3 for details.

Partonic cross sections are convoluted with LUXlep-NNPDF31_nlo_as_0118_luxqed
(v2) parton distribution functions derived in Ref. Buonocore:2020nai . To this
purpose, we employ the Mathematica package ManeParse Clark:2016jgm  for
manipulating the LHAPDF grids Buckley:2014ana . The PDF extrapolation in
$Q^{2}$ is checked by solving the corresponding DGLAP equations using Hoppet
Salam:2008qg  in accordance with the prescription from Buonocore:2020nai .
Also, the running of the gauge couplings with the renormalisation scale is
appropriately included. The central renormalisation and factorisation scales
are set to $\mu_{\rm R}=\mu_{\rm F}=m_{{\rm LQ}}$. We estimate the uncertainty
from the missing higher order corrections by varying the scales in the range
$\\{\mu_{\rm R},\mu_{\rm F}\\}\in[0.5-2]\,m_{{\rm LQ}}$ , while respecting
$0.5\leq\mu_{\rm R}/\mu_{\rm F}\leq 2$. Independently of $\mu_{\rm R}$ and
$\mu_{\rm F}$ scale variations, the renormalisation group running of the
leptoquark coupling $y_{q\ell}(\mu)$ in the range $\mu\in[0.5-2]~{}m_{{\rm
LQ}}$ leads to the cross section prediction uncertainty of about $4\%$ across
the entire $m_{{\rm LQ}}$ window. The uncertainties due to the parton
distribution functions are calculated by the method of replicas Ball:2017nwa ;
Bertone:2017bme . In particular, we report the standard deviation of the
result calculated over one hundred replicas as the PDF error.

The lepton and antilepton PDFs are numerically the same. However, this is not
the case for the light quarks, implying that e.g. $ue^{-}$ induced cross
section is different from $\bar{u}e^{+}$. We therefore report the cross
sections for $pp\to$ LQ + the charge-conjugated process in Table 1 (up-type
quarks) and Table 2 (down-type quarks), at NLO QCD + QED accuracy. Thanks to
the inclusion of radiative corrections computed in Section 2, the
uncertainties due to the $\\{\mu_{\rm R},\mu_{\rm F}\\}$ scale variations are
at the level of few per cent for all leptoquark charges, as well as, quark and
lepton flavors and benchmark masses. The uncertainties due to the parton
distribution functions strongly depend on the quark flavor and the leptoquark
mass. In particular, the total uncertainty becomes dominated by the limited
knowledge of the heavy quark PDFs when $m_{{\rm LQ}}$ is several TeV.

Figure 3:  NLO $K$-factor ($\sigma_{{\rm NLO}}/\sigma_{{\rm LO}}$) for the
resonant scalar leptoquark production at 13 TeV LHC. Shown with orange (red)
solid lines are the NLO QCD (NLO QCD + QED) predictions normalised to the LO
when setting the central scales to $\mu_{\rm R}=\mu_{\rm F}=m_{\mathrm{LQ}}$.
The colored bands are obtained by varying factorisation and renormalisation
scales in the NLO calculations within $\\{\mu_{\rm F},\mu_{\rm
R}\\}\in[0.5,2]~{}m_{\mathrm{LQ}}$ while respecting $1/2\leq\mu_{\rm
R}/\mu_{\rm F}\leq 2$. We use the central PDF set from LUXlep-
NNPDF31_nlo_as_0118_luxqed (v2) Buonocore:2020nai . The benchmark example in
the plot is a leptoquark with $|Q_{{\rm LQ}}|=4/3$ coupled to the down quark
and electron. All other cases are shown in Figs. 8, 9, and 10 (see Appendix
C).

Next-to-leading order $K$-factors, defined as the ratio of NLO to LO results,
are shown in Appendix C in Figs. 8 (electron), 9 (muon), and 10 (tau) for all
possible quark and lepton flavors and leptoquark charges. One notable example
is shown in Fig. 3 in the main text. These are calculated using the central
PDF set and the central scales $\mu_{\rm R}=\mu_{\rm F}=m_{{\rm LQ}}$ for the
LO cross section, while at NLO, we consider $\\{\mu_{\rm R},\mu_{\rm F}\\}$
scale variation with the central PDF set. The red (orange) bands are with
(without) NLO QED corrections.222Our calculation is also an important test of
the lepton PDFs derived in Ref. Buonocore:2020nai . In all cases considered,
the error band dramatically shrinks, illustrating the importance of the NLO
QED corrections. Interestingly, both QCD and QED corrections are large,
however, they partially cancel in the total cross section. Inspecting Figs. 8,
9, and 10 we conclude that $K$-factors typically exhibit only a slight
dependence on the leptoquark mass and electric charge, as well as, lepton
flavors. In this calculation, we sum up cross sections for the process $pp\to$
LQ and the charge-conjugated process before taking the ratio. We checked that
the individual $K$-factors for the two are very close to each other, thus we
report only the $K$-factors for the sum.

We also study the dependence of the NLO $K$-factors on the PDF uncertainties.
In particular, for every PDF replica we compute $\sigma_{{\rm
NLO}}/\sigma_{{\rm LO}}$. We then derive the $68\%$ confidence level range
around the central PDF prediction. Interestingly, this band does not exceed
the NLO QCD + QED scale variation band, except for a very heavy leptoquark
close to the edge of the considered mass range, where the PDF errors are
$\mathcal{O}(1)$ for some flavors. In other words, PDF uncertainties cancel in
the ratio to a good approximation. We therefore conclude that the $K$-factors
reported in Appendix C are robust, and will not change significantly by more
precise PDFs in the future.

On the practical side, the existing leading order generators are missing the
leptonic shower crucial to properly simulate the resonant leptoquark events.
However, this shortcoming will soon be resolved, see the third footnote in
Ref. Buonocore:2020erb . Once this is in place, the $K$-factors derived in
this paper can be directly applied to the future LHC resonant leptoquark
searches to correct the overall signal yield. The main experimental difference
between the single leptoquark plus lepton production and the resonant
leptoquark production is the $p_{T}$ spectrum of the accompanied lepton. In
particular, the lepton is hard (soft) in the former (latter) case. Therefore,
measuring the lepton (or the leptoquark) $p_{T}$ distribution will enable
efficient discrimination between different leptoquark production mechanisms at
the LHC. To this purpose, it is crucial to have a good theoretical control
over the $p_{T}$ spectrum. Our Appendix could serve as a starting point for
this calculation. Note, that the leptoquark searches so far required the
presence of two charged leptons which effectively vetos the resonant
mechanism.

The leptoquark signature is quite unique; it will show up as a resonance in
the jet-lepton invariant mass distribution. To study the flavor structure of
the underlying interactions, one can make use of the flavor tagging of the
decay products. Unfortunately, on the quark side there is a big degeneracy
among light quarks $u,d,s$ and $c$ which are somewhat distinguished from the
$b$ quark. The task is even more difficult on the production side. The ratio
of the rates for the single leptoquark plus lepton production and the resonant
leptoquark production does not depend on the value of the leptoquark Yukawa
coupling, however, it is sensitive to the initial quark flavor. This can be
used to determine the flavor structure of the dominant leptoquark coupling in
production. We have checked that the ratio drops quickly with the leptoquark
mass and the discrepancy is more pronounced for sea quarks than for valence
quarks.

Another observable relevant for the leptoquark flavor physics in high-$p_{T}$
collisions is the ratio of the resonant rate $pp\to$ LQ to its charge-
conjugated process. For heavy $c$ and $b$ quarks the two rates are the same,
while for the valence quarks the two rates can differ by a factor of
$\mathcal{O}(10)$. We have checked that this observable indeed has a
discriminating power, however, a dedicated analysis is needed to make a
quantitative statement. We also noticed a large PDF uncertainties in the
prediction of this ratio attributed to the poor knowledge of sea quarks at
large $x$. Therefore, the success of this method depends on the improvements
in measuring sea quark parton distribution functions.

##  4 Conclusions

A discovery of a leptoquark at the Large Hadron Collider would fundamentally
change our understanding of particle physics, pointing towards a microscopic
theory where quarks and leptons unify. Viable extensions of the Standard Model
with TeV-scale leptoquarks exist, and are safe on proton decay and dangerous
flavor changing neutral currents. Moreover, these models have recently
received a large attention within the community. Namely, leptoquarks in the
TeV mass range provide an elegant explanation of the long-standing hints on
the lepton flavor universality violation in $B$-meson decays, as well as, the
anomalous magnetic moment of the muon.

Leptoquark collider searches so far were mainly focused on the pair production
mechanism driven by QCD interactions, while the role of the defining
leptoquark interaction to a quark and a lepton was invoked in decays. However,
interesting flavor effects occur when the leptoquark coupling is large(ish)
Buttazzo:2017ixm ; Angelescu:2018tyl ; Aebischer:2019mlg ; Dorsner:2019itg ;
Gherardi:2020qhc ; Aaboud:2017efa ; Mandal:2019gff ; Bordone:2020lnb ;
Fuentes-Martin:2019mun ; Crivellin:2020mjs ; Altmannshofer:2020ywf ;
Bause:2020xzj ; Brdar:2020quo ; Bause:2019vpr , consequently predicting richer
collider phenomenology on the production side. Building on Refs.
Buonocore:2020erb ; Buonocore:2020nai , in this paper we study the resonant
leptoquark production mechanism. Namely, the quantum fluctuations allow for a
small presence of a lepton inside the proton which fuses with a quark from the
other proton, to produce a leptoquark. The smallness of the lepton
distribution is overcome by the resonant enhancement, providing this mechanism
with the largest cross sections of all when $m_{{\rm LQ}}\gtrsim 1$ TeV and
$y_{q\ell}\sim\mathcal{O}(1)$, see Fig 2.

We calculate for the first time next-to-leading order QCD and QED corrections
to the resonant leptoquark production at hadron colliders. The present study
is limited to scalar leptoquarks while the vector leptoquark case is left for
the future work. The total cross section is given in closed form in Eqs. (2),
(3), (4), and (6), and the detailed derivation is carried out in Appendices A
and B. This formula is numerically integrated with the most recent lepton PDFs
Buonocore:2020nai  to obtain the hadronic cross sections at the LHC. The main
numerical results are reported in Tables 1 and 2, and in Figs. 8, 9 and 10.
The calculation is performed for a set of benchmark points in the mass range
relevant for the future searches, as well as, for all possible lepton and
quark flavors and leptoquark charges. Importantly, our results are applicable
for a general scalar leptoquark model with arbitrary flavor couplings.

We find that both QCD and QED corrections are large and are of similar size.
However, they come with the opposite sign and cancel out in the final cross
section, leading to somewhat smaller corrections of the tree-level result than
initially expected. However, the advantage of our calculation is that we are
now in position to reliably estimate the theoretical uncertainties. On this
note, we observed a dramatic reduction of the renormalisation and
factorisation scale variation uncertainties after the inclusion of QED
corrections on top of the QCD ones. This is nicely illustrated in Fig. 3 with
the red band. The leading source of theoretical error at this point is the
limited knowledge of the parton distribution functions, in particular, the sea
quark PDFs at large $x$. The breakdown of different uncertainties is
summarised in the predictions for the total cross sections in Tables 1 and 2.
The complete set of NLO $K$-factors is reported in Appendix C and can be
straightforwardly applied in the future experimental searches at the LHC for
the most general leptoquark model.

Finally, should a leptoquark be discovered at the LHC, precision measurements
of the resonant process and its charge-conjugate, as well as, the single
leptoquark plus lepton production, would help to deduce the flavor of the
leptoquark interactions. Hopefully, synchronised deviations would show up in
the low-energy flavor transitions to confirm this picture.

###### Acknowledgements.

We thank Gino Isidori, Javier Fuentes-Martín, Matthias König and Pier
Francesco Monni for useful discussions. The work of AG has received funding
from the Swiss National Science Foundation (SNF) through the Eccellenza
Professorial Fellowship “Flavor Physics at the High Energy Frontier” project
number 186866. The work of AG and NS is supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme, grant agreement 833280 (FLAY).

##  Appendix A NLO QED corrections to resonant production

The QED corrections to the resonant leptoquark production correspond to
processes involving a photon in the initial state splitting into a lepton
pair. As explained in Section 2.2 , the inclusion of these corrections is
necessary for calculating the resonant production at
$\mathcal{O}(\alpha_{s}^{3})$, which is a typical size of the NLO QCD
corrections. Due to different electric charges, the QED corrections to
production cross section will differ for various leptoquark types. We can
resonantly produce all types of scalar leptoquarks using the suitable partons
inside the colliding protons. The possible combinations are

$\displaystyle\bar{u}+e^{+}$ $\displaystyle\longrightarrow S_{1/3}\,,$ (12)
$\displaystyle d+e^{+}$ $\displaystyle\longrightarrow S_{2/3}\,,$
$\displaystyle\bar{d}+e^{+}$ $\displaystyle\longrightarrow S_{4/3}\,,$
$\displaystyle u+e^{+}$ $\displaystyle\longrightarrow S_{5/3}\,,$

along with the corresponding charge conjugated processes, where the fraction
in the subscript denotes the leptoquark electric charge. The QED correction
for each of the listed process is given by three diagrams shown in Fig. 4. The
initial state quark and photon create the complementary scalar leptoquark
together with a soft charged lepton in the final state.

Figure 4: Diagrams for the process $\gamma+q\to l^{-}+LQ$ contributing to the
resonant leptoquark production at $\mathcal{O}(\alpha)$.

Generically, the amplitude for the process $\gamma(p_{1})+q(p_{2})\to
l^{-}(k)+LQ^{+}(q)$ obtained by interfering diagrams reads

$i\mathcal{M}=-iy_{q\ell}\ e\
\bar{u}(k)\text{P}_{L,R}\left[\gamma^{\mu}\frac{\left(\not{p_{1}}-\not{k}\right)}{\left(p_{1}-k\right)^{2}}+Q_{\rm
LQ}\frac{\left(2q-p_{1}\right)^{\mu}}{\left(q-p_{1}\right)^{2}-m^{2}}+Q_{\rm
q}\frac{\left(\not{p_{1}}+\not{p_{2}}\right)}{\left(p_{1}+p_{2}\right)^{2}}\gamma^{\mu}\right]\zeta(p_{2})\epsilon_{\mu}(p_{1})\,,$
(13)

where $p_{1}\ (p_{2})$ is the four-momentum of the photon ((anti-)quark) in
the initial state, while $q\ (k)$ is the four-momentum of the leptoquark (soft
charged lepton) in the final state. The fermionic wave-function $\zeta(p_{2})$
could either stand for the particle or anti-particle, depending on the
produced leptoquark type. The partonic cross section calculation presented is
the same for all types of leptoquarks, with the only difference provided by
different particle charges, $Q_{\rm q}$ denoting the (anti-)quark and $Q_{\rm
LQ}$ the leptoquark charge. To express the kinematics, it is convenient to use
the center of mass frame in which

$p_{1}^{\mu}=\frac{\sqrt{\hat{s}}}{2}(1,0,0,1),\quad
p_{2}^{\mu}=\frac{\sqrt{\hat{s}}}{2}(1,0,0,-1),\quad
k=\frac{\sqrt{\hat{s}}}{2}(1-z)(1,0,\sin{\theta},\cos{\theta})\,,$ (14)

with $z=m^{2}_{{\rm LQ}}/\hat{s}$. Additionally, the relation between partonic
Mandelstam variables $\hat{s}=(p_{1}+p_{2})^{2}$ and $\hat{t}=(p_{1}-k)^{2}$
becomes $\hat{t}=-\hat{s}w(1-z)$, where $w=(1-\cos{\theta})/2$. The collinear
divergences that appear when soft lepton is emitted parallel to the photon are
regulated using dimensional regularisation with $d=4-2\epsilon$. Averaging
over the initial, and summing over the final polarisations and colors, the
averaged squared matrix element can be written as

$|\overline{\mathcal{M}}|^{2}=\frac{|y_{q\ell}|^{2}e^{2}}{d-2}\left(\mathcal{M}^{2}_{\text{div}}+\mathcal{M}^{2}_{\text{fin}}\right)\,,$
(15)

where $d-2$ in the denominator counts the polarisations of the massless gauge
bosons in $d$-dimensions, and $\mathcal{M}^{2}_{\text{div}}$
($\mathcal{M}^{2}_{\text{fin}}$) denotes the part of the averaged squared
matrix element that will produce the IR-divergent (IR-finite) contributions to
the cross section after integration over the phase-space. In terms of $w$ and
$z$, they can be written as

$\displaystyle\mathcal{M}^{2}_{\text{div}}$ $\displaystyle=$ $\displaystyle\
\frac{1}{w}\left[\frac{d-2}{2(1-z)}+2Q_{\rm q}z+Q_{\rm
LQ}\left(1-\frac{1+2z}{1-w(1-z)}\right)\right],$ (16)
$\displaystyle\mathcal{M}^{2}_{\text{fin}}$ $\displaystyle=$ $\displaystyle\
\frac{(1-w)(1-z)}{1-w(1-z)}\left[Q_{\rm
LQ}^{2}\left(1-\frac{2z}{1-w(1-z)}\right)-Q_{\rm q}Q_{\rm
LQ}\left(1-2z\right)\right]$ (17) $\displaystyle\ +Q_{\rm
q}\left(d-2(1+z)\right)-Q_{\rm
LQ}\left(1-\frac{1+2z}{1-w(1-z)}\right)+\frac{d-2}{2}Q_{\rm q}^{2}w(1-z).$

Moreover, the integration over the 2-body phase-space in $d$-dimensions with
the corresponding flux factor, expressed in terms of $w$ and $z$, can be
performed as

$\frac{1}{16\pi\hat{s}}\left(\frac{4\pi\mu^{2}}{\hat{s}}\right)^{\frac{4-d}{2}}\int_{0}^{1}|\overline{\mathcal{M}}|^{2}\frac{\left[w(1-w)\right]^{\frac{d-4}{2}}(1-z)^{d-3}}{\Gamma(\frac{d-2}{2})}\
dw.$ (18)

The integral over $\mathcal{M}^{2}_{\text{fin}}$ is IR safe for $d=4$. The
result for the finite contribution to the partonic cross section is

$\displaystyle\hat{\sigma}_{\text{fin}}$ $\displaystyle=\
\frac{\pi|y_{q\ell}|^{2}}{4\hat{s}}\frac{\alpha}{2\pi}\bigg{[}$ $\displaystyle
Q_{\rm q}Q_{\rm LQ}(1-2z)(z-z\log z-1)+Q_{\rm LQ}^{2}(1+z-2z^{2}+3z\log z)$
(19) $\displaystyle\ +2Q_{\rm q}\left(1+\frac{Q_{\rm
q}}{4}\right)(1-z)^{2}+Q_{\rm LQ}(z-(1+2z)\log z-1)\bigg{]}.$

On the other hand, the phase-space integration over
$\mathcal{M}^{2}_{\text{div}}$ induces IR-poles in the partonic cross section.
Regulating the integral we obtain

$\displaystyle\hat{\sigma}_{\text{div}}=\frac{\pi|y_{q\ell}|^{2}}{4\hat{s}}\frac{\alpha}{2\pi}\left(\frac{4\pi\mu^{2}}{\hat{s}}\right)^{\epsilon}\frac{1}{\Gamma(1-\epsilon)}\int
w^{-1-\epsilon}(1-w)^{-\epsilon}(1-z)^{1-2\epsilon}\mathcal{F}(w,z)dw\,,$ (20)
$\displaystyle\mathcal{F}(w,z)=\frac{1}{1-z}+(1+\epsilon)\left[2Q_{\rm
q}z+Q_{\rm LQ}\left(1-\frac{1+2z}{1-w(1-z)}\right)\right].$ (21)

The IR-pole in $\hat{\sigma}_{\text{div}}$ becomes explicit after
$w^{-1-\epsilon}$ is expanded around $\epsilon=0$ to give a distribution

$w^{-1-\epsilon}=-\frac{1}{\epsilon}\delta(w)+\frac{1}{w_{+}}+\mathcal{O}(\epsilon),$
(22)

with the plus distribution defined such that

$\int_{0}^{1}\frac{f(1-w)}{w_{+}}\ dw=\int_{0}^{1}\frac{f(1-w)-f(1)}{w}\ dw.$
(23)

The contribution to the partonic cross section containing a collinear
divergence is then

$\displaystyle\hat{\sigma}_{\text{div}}$
$\displaystyle=\frac{\pi|y_{q\ell}|^{2}}{4\hat{s}}\frac{\alpha}{2\pi}\left(\frac{4\pi\mu^{2}}{\hat{s}}\right)^{\epsilon}\frac{1}{\Gamma(1-\epsilon)}\bigg{[}-\frac{1}{\epsilon}\bigg{(}1+2(Q_{\rm
q}-Q_{\rm LQ})z(1-z)\bigg{)}+2\log(1-z)$ $\displaystyle\quad+Q_{\rm
LQ}(1-z)(1+2z)\log z-2(Q_{\rm q}-Q_{\rm LQ})z(1-z)(1-2\log(1-z))\bigg{]}.$
(24)

Note that the divergence is universal for all combinations listed in (12) as
$(Q_{\rm q}-Q_{\rm LQ})=-1$, for all of them, and the corresponding
coefficient is identified with the leading-order photon-to-charged lepton
splitting function

$P_{\ell\leftarrow\gamma}(z)=z^{2}+(1-z)^{2}.$ (25)

In order to calculate the measured hadronic cross section for these processes,
we need to convolute the partonic cross section with the corresponding quark
and photon PDFs. In the procedure, the collinear singularity can be absorbed
into the bare PDFs at a factorisation scale $\mu_{\rm F}$. Here, we utilize
the $\overline{\rm MS}$ factorisation scheme by adding the counter term

$\hat{\sigma}^{CT}=\frac{\pi|y_{q\ell}|^{2}}{4\hat{s}}\frac{\alpha}{2\pi}(4\pi)^{\epsilon}\frac{1}{\epsilon\Gamma(1-\epsilon)}P_{\ell\leftarrow\gamma}(z),$
(26)

to the partonic cross section, consistent with the $\overline{\rm MS}$
prescription used in extracting parton density functions presented in
Buonocore:2020nai . Combining relations (A), (24) and (26), we find that the
result is finite, factorisation scale dependent, and can be written in the
following form

$\hat{\sigma}_{q\gamma}(z)=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\frac{\alpha}{2\pi}\left(-\log\left(\frac{\mu_{\rm
F}^{2}}{\hat{s}(1-z)^{2}}\right)P_{\ell\leftarrow\gamma}(z)+X_{Q_{LQ}}(z)\right).$
(27)

The non-universal pieces for different scalar leptoquarks, after the electric
charges are replaced, read

$\displaystyle X_{1/3}(z)$
$\displaystyle=-\frac{2}{9}(1-z)(5-13z)+\frac{2}{9}(1-5z)z\log z,$ (28)
$\displaystyle X_{2/3}(z)$
$\displaystyle=-\frac{11}{18}(1-z)(1-5z)+\frac{8}{9}(1-2z)z\log z,$ (29)
$\displaystyle X_{4/3}(z)$
$\displaystyle=\frac{1}{18}(1-z)(13+103z)+\frac{16}{9}(2-z)z\log z,$ (30)
$\displaystyle X_{5/3}(z)$
$\displaystyle=\frac{2}{9}(1-z)(7+37z)+\frac{10}{9}(5-z)z\log z,$ (31)

where $\\{1/3,\ 2/3,\ 4/3,\ 5/3\\}$ correspond to electric charge of the
leptoquark in the final state.

##  Appendix B NLO QCD corrections to resonant production

###  B.1 Virtual QCD corrections

The calculation of the $\mathcal{O}(\alpha_{s})$ virtual corrections to the
resonant leptoquark production proceeds as shown in Fig. 5: (a) the quark
wave-function correction, (b) the leptoquark wave-function correction, and (c)
the vertex correction. The UV finiteness of the results is achieved by adding
the $\overline{\text{MS}}$ counter-terms.

Figure 5: The virtual corrections to the resonant leptoquark production at
$\mathcal{O}(\alpha_{s})$.

Adding the virtual contributions to the tree-level amplitude, the NLO
amplitude may be written as

$\mathcal{A}_{\text{NLO}}=\mathcal{A}_{\text{tree}}\left[1+\frac{\alpha_{s}}{4\pi}\left(\frac{1}{2}\delta
Z_{\rm q}(0)+\frac{1}{2}\delta Z_{\rm LQ}(m_{\rm LQ}^{2})+\delta V_{\rm
LQ}(m_{\rm LQ}^{2})\right)\right]\ .$ (32)

The chiral Yukawa couplings result in the vanishing leptoquark contribution to
the massless fermion wave-function, with gluon providing the only contribution

$\delta Z_{\rm q}(0)=C_{F}\left(\frac{1}{\epsilon_{\rm IR}}+L_{\mu}^{\rm
IR}-L_{\mu}^{\rm UV}\right)\,,$ (33)

where $L_{\mu}^{\rm IR(UV)}=\log(\mu_{\rm F(R)}^{2}/m_{\rm LQ}^{2})$, with
$\mu_{\rm F}$ and $\mu_{\rm R}$ denoting the factorisation and renormalisation
scales, respectively, and $C_{F}=4/3$. Similarly, the leptoquark two-point
function $\Sigma_{LQ}(q^{2})$, with $q$ being the leptoquark four-momentum,
receives no contribution from fermions, and the only effect is caused by the
leptoquark coupling to gluons. We renormalise the leptoquark mass on-shell,
with the wave-function correction defined as

$\displaystyle\delta Z_{\rm LQ}(q^{2})=\frac{\Sigma_{\rm
LQ}(q^{2})-\Sigma_{\rm LQ}(m_{\rm LQ}^{2})}{q^{2}-m_{\rm LQ}^{2}}\ .$ (34)

Taking the on-shell limit for the resonant leptoquark production $q^{2}=m_{\rm
LQ}^{2}$, the correction becomes

$\lim_{q^{2}\to m_{\rm LQ}^{2}}\delta Z_{\rm
LQ}(q^{2})=2C_{F}\left(L_{\mu}^{\rm UV}-\frac{1}{\epsilon_{\rm
IR}}-L_{\mu}^{\rm IR}\right)\,,$ (35)

while the vertex correction, evaluated on-shell reads

$\delta V_{\rm LQ}(m_{\rm LQ}^{2})=C_{F}\left[L_{\mu}^{\rm
UV}-2\left(\frac{1}{\epsilon_{\rm IR}}+L_{\mu}^{\rm
IR}\right)-\frac{1}{\epsilon_{\rm IR}^{2}}-\frac{1}{\epsilon_{\rm
IR}}L_{\mu}^{\rm IR}-\frac{1}{2}\left(L_{\mu}^{\rm
IR}\right)^{2}-\frac{\pi^{2}}{12}-2\right]\ .$ (36)

Combining the individual contributions listed above and integrating the
averaged matrix element $|\overline{\mathcal{A}_{\text{NLO}}}|^{2}$ over the
leptoquark phase-space, we obtain the virtual correction to the partonic cross
section for the leptoquark resonant production

$\displaystyle\hat{\sigma}^{V}(z)=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\Bigg{\\{}1+\frac{\alpha_{s}}{2\pi}C_{F}$
$\displaystyle\Bigg{[}\frac{3}{2}L_{\mu}^{\rm
UV}-\frac{5}{2}\left(\frac{1}{\epsilon_{\rm IR}}+L_{\mu}^{\rm
IR}\right)-\frac{1}{\epsilon_{\rm IR}^{2}}-\frac{1}{\epsilon_{\rm
IR}}L_{\mu}^{\rm IR}$ (37) $\displaystyle-\frac{1}{2}\left(L_{\mu}^{\rm
IR}\right)^{2}-\frac{\pi^{2}}{12}-2\Bigg{]}\Bigg{\\}}\ \delta(1-z).$

###  B.2 Real QCD corrections

The calculation of the real QCD corrections closely follows the steps
described in Appendix A for the QED case. We note that the results of this
calculation already exist in the literature Kunszt:1997at ; Plehn:1997az  (see
also Djouadi:1989md ), which we have checked and found complete agreement.
With that in mind, here we present the corresponding results, and commit to
these references for more details. The first process to consider is the one
with the soft gluon in the final state, which can happen either by emission
from the quark or the leptoquark as shown in diagrams in Fig. 6.

Figure 6: Diagrams for the process $q+l\to g+LQ$ contributing to the resonant
leptoquark production at $\mathcal{O}(\alpha_{s})$.

The partonic cross section for this process is given by

$\displaystyle\hat{\sigma}^{R}(z)=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\frac{\alpha_{s}}{2\pi}C_{F}$
$\displaystyle\Bigg{\\{}\delta(1-z)\Bigg{[}-\frac{3}{2}L_{\mu}^{\rm
IR}+\frac{5}{2}\left(\frac{1}{\epsilon_{\rm IR}}+L_{\mu}^{\rm
IR}\right)+\frac{1}{\epsilon_{\rm IR}^{2}}+\frac{1}{\epsilon_{\rm
IR}}L_{\mu}^{\rm IR}+\frac{1}{2}\left(L_{\mu}^{\rm
IR}\right)^{2}-\frac{\pi^{2}}{4}+2\Bigg{]}$
$\displaystyle+2(1+z^{2})\left(\frac{\log(1-z)}{(1-z)}\right)_{+}-\frac{2z}{(1-z)_{+}}-\frac{1+z^{2}}{(1-z)_{+}}\log\left(\frac{z\mu_{\rm
F}^{2}}{m_{\rm LQ}^{2}}\right)\Bigg{\\}}.$ (38)

As expected, the inclusion of the gluon radiation provides the IR divergences
that exactly cancel the ones present in the virtual contribution (37). The
combined result reads

$\displaystyle\hat{\sigma}_{q\ell}(z)=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\Bigg{\\{}\left[1+\frac{\alpha_{s}}{2\pi}C_{F}\left(\frac{3}{2}\log\left(\frac{\mu_{\rm
R}^{2}}{\mu_{\rm
F}^{2}}\right)-\frac{\pi^{2}}{3}\right)\right]\delta(1-z)-\frac{2z}{(1-z)_{+}}$
(39)
$\displaystyle+2(1+z^{2})\left(\frac{\log(1-z)}{(1-z)}\right)_{+}-\frac{2z}{(1-z)_{+}}-\frac{1+z^{2}}{(1-z)_{+}}\log\left(\frac{z\mu_{\rm
F}^{2}}{m_{\rm LQ}^{2}}\right)\Bigg{\\}}\,.$

The second process which we need to take into account is the one with the soft
quark in the final state, corresponding to the diagrams in Fig. 7.

Figure 7: Diagrams for the process $g+l\to q+LQ$ contributing to the resonant
leptoquark production at $\mathcal{O}(\alpha_{s})$.

The partonic cross section for this process is given by

$\hat{\sigma}_{g\ell}(z)=\frac{\pi z|y_{q\ell}|^{2}}{4m_{\rm
LQ}^{2}}\frac{\alpha_{s}}{2\pi}T_{R}\left[-\log\left(\frac{z\mu^{2}_{\rm
F}}{(1-z)^{2}m_{\rm LQ}^{2}}\right)(z^{2}+(1-z)^{2})+2z(1-z)(2+\log
z)\right]\,,$ (40)

where $T_{R}=1/2$ is the appropriate $SU(3)$ color factor. Due to the massless
quark which can be collinear to the gluon in diagram (a) of Fig. 7, this
process needs the inclusion of the $\overline{\text{MS}}$ counter-term. This
is exactly the same situation we already studied in the process $\gamma+q\to
l^{-}+LQ$, and the universality of the $\log(\mu_{\rm F})$ terms for the two
processes in (40) and (27), up to color factors, becomes evident.

##  Appendix C Supplemental numerical results

Figs. 8, 9, and 10 show the NLO K-factors for three lepton flavors: electron,
muon and tau, respectively. Each figure contains ten plots for different quark
flavors and leptoquark charge. Down-type quarks couple the $|Q_{{\rm
LQ}}|=2/3,4/3$, while the up-type quarks couple the $|Q_{{\rm LQ}}|=1/3,5/3$.
For more details see Section 3.

Figure 8:  (electron) NLO $K$-factors ($\sigma_{{\rm NLO}}/\sigma_{{\rm LO}}$)
for resonant scalar leptoquark production at 13 TeV LHC. Shown with orange
(red) are the NLO QCD (NLO QCD + QED) predictions normalised to the LO when
setting the central scales to $\mu_{R}=\mu_{F}=m_{\mathrm{LQ}}$. The colored
bands are obtained by varying factorisation and renormalisation scales in the
NLO calculations within $\\{\mu_{F},\mu_{R}\\}\in[0.5,2]~{}m_{\mathrm{LQ}}$
while respecting $1/2\leq\mu_{R}/\mu_{F}\leq 2$. We use the central PDF set
from LUXlep-NNPDF31_nlo_as_0118_luxqed (v2) Buonocore:2020nai .

Figure 9:  (muon) NLO $K$-factors ($\sigma_{{\rm NLO}}/\sigma_{{\rm LO}}$) for
resonant scalar leptoquark production at 13 TeV LHC. Shown with orange (red)
are the NLO QCD (NLO QCD + QED) predictions normalised to the LO when setting
the central scales to $\mu_{R}=\mu_{F}=m_{\mathrm{LQ}}$. The colored bands are
obtained by varying factorisation and renormalisation scales in the NLO
calculations within $\\{\mu_{F},\mu_{R}\\}\in[0.5,2]~{}m_{\mathrm{LQ}}$ while
respecting $1/2\leq\mu_{R}/\mu_{F}\leq 2$. We use the central PDF set from
LUXlep-NNPDF31_nlo_as_0118_luxqed (v2) Buonocore:2020nai .

Figure 10:  (tau) NLO $K$-factors ($\sigma_{{\rm NLO}}/\sigma_{{\rm LO}}$) for
resonant scalar leptoquark production at 13 TeV LHC. Shown with orange (red)
are the NLO QCD (NLO QCD + QED) predictions normalised to the LO when setting
the central scales to $\mu_{R}=\mu_{F}=m_{\mathrm{LQ}}$. The colored bands are
obtained by varying factorisation and renormalisation scales in the NLO
calculations within $\\{\mu_{F},\mu_{R}\\}\in[0.5,2]~{}m_{\mathrm{LQ}}$ while
respecting $1/2\leq\mu_{R}/\mu_{F}\leq 2$. We use the central PDF set from
LUXlep-NNPDF31_nlo_as_0118_luxqed (v2) Buonocore:2020nai .

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