††institutetext: School of Physics, Nanjing University, Jiangsu Province, CN,
210000

# Search for Low-Mass Leptoquarks using the $llqq$ final states in Pb-Pb
Ultra-Peripheral Collisions

Li-Gang Xia<EMAIL_ADDRESS>
###### Abstract

After a review over past experiments and theoretical requirements from low-
energy flavour physics, we argue that the possibility of low-mass leptoquarks
(LQ) cannot be fully excluded due to the assumptions made in the measurements.
Therefore we propose to search for the pair production of low-mass leptoquarks
in Pb-Pb ultra-peripheral collisions with the least model dependence (assuming
a small coupling constant $\lesssim 0.1$). There are a couple of advantages:
1) high photon flux provides high production rate for low mass LQs; 2) the
background contamination is much lower than that in $p$-$p$ collisions. The
analysis strategy permits a leptoquark to decay to all possible lepton-plus-
quark modes. Taking the scalar LQ, $S_{3}$, with an electric charge
$|q|=\frac{4}{3}e$, as example, the mass point of 100 GeV can be excluded at
the 95 % confidence level using a dataset of 4 pb-1 Pb-Pb ultra-peripheral
collisions at $\sqrt{s}=5.02$ TeV and the performance of the ATLAS detector in
Run 2. The proposed method also applies to searching for high-mass LQs in
$p$-$p$ collisions as long as the LQ pair production mechanism dominates.

††arxiv: 2012.15618

##  1 Introduction

Leptoquarks (LQ) are bosons to connect the quark and lepton sectors. They
carry fractional electric charge and color. LQs are predicted in many
extensions of the Standard Model (SM), such as grand unified theories
PatiSalam1 ; PatiSalam2 ; GeorgiGlashow ; Unify1 , technicolor models
Buchmuller ; DStechnicolor ; Dtechnicolor ; FStechnicolor ; LRtechnicolor ,
composite LQ models lightLQ ; CompositeLQ1 ; CompositeLQ2  and other models
ELdynabreaking ; Ellis . They are able to explain the evidences of the lepton
universality violation (LUV) measured in the $B$-meson decays Hiller ; BLQ1 ;
BLQ2 ; BLQ3 ; BLQ4 ; BLQ5 ; BLQ6 ; BLQ7 ; BLQ8 . Most recently, the LHCb
experiment reported another evidence for LUV in the decays $B^{+}\to
K^{+}l^{+}l^{-}$ with a significance of 3.1 $\sigma$ LHCbnew . Introducing one
or more LQs seems a natural explanation LHCbwater1 ; LHCbwater2 ; LHCbwater3 ;
LHCbwater4 ; LHCbwater5 ; LHCbwater6 ; LHCbwater7 .

Various LQs have been searched for in the LEP, Tevatron, LHC and other
experiments. Ref. LQreview  is a good review on the LQ physics and
experimental searches up to 2016. Table 1 summarizes a list of experimental
searches from different colliders. From the table, it seems that LQs are
excluded up to 1 TeV or more. However, all searching results are interpreted
with assumptions. One common assumption is that a LQ couples only to the same
generation of fermions (this assumption is denoted by “SG” in Table 1). For
example, a so-called “third-generation” LQ with a charge $+\frac{2}{3}e$ only
couples to $b\tau^{+}$ or $t\bar{\nu}_{\tau}$. This assumption is not
appropriate because of more and more strong evidences for LUV. Taking the
decay $B^{+}\to K^{+}l^{+}l^{-}$ as in the LHCb measurement LHCbnew  as
example, a LQ must be able to couple to $\mu/e$ and $b/s$ quark if LQ is the
right solution. A much looser assumption is that a LQ couples to a single
lepton plus all possible quarks (this assumption is denoted by “SL” in Table
1). This assumption was used in the OPAL experiment OPAL2 ; OPAL3 . Inspired
by their strategy, a model-independent method is proposed in Sec. 3. For the
measurements using single LQ production processes, we have to interpret the
results as a function of the LQ mass and the coupling constant between a LQ
and the fermions (denoted by $\lambda$ or $\lambda_{lq}$ for the lepton $l$
and the quark $q$). One common way is to set constraints on the LQ mass
assuming a coupling strength which is equal to that in the electromagnetic
interaction, namely, $\lambda=\sqrt{4\pi\alpha_{em}}$ in Table 1, where
$\alpha_{em}\approx\frac{1}{137}$ is the fine structure constant.

Table 1:  Overview of the experimental searches for the leptoquarks. “SG” represents the assumption that a LQ couples to the same generation of fermions. “SL” represents the assumption that a LQ couples to single lepton and all possible quarks. It should be noted that the presented exclusion region may be just part of the experiment results.  Experiment | Exclusion region (condition) | Assumption
---|---|---
AMYAMY  | 4.3-26.3 GeV ($|q|=\frac{2}{3}e,B(\text{LQ}\to eq)=1$) | SG, LQ$\to eq/\mu q/\nu q$ ($q=d,s$)
CELLOCELLO  | 7-20.5 GeV | LQ$\to\mu q/\nu q$
ALEPHALEPH  | 4-44 GeV ($|q|=\frac{1}{3}e$) | SG, LQ$\to eu/\nu_{e}d$
| 6-44 GeV ($|q|=\frac{1}{3}e$) | SG, LQ$\to\mu c/\nu_{\mu}s$
DELPHIDELPHI  | 134 GeV ($B(\text{LQ}\to eq)=1$,$\lambda=\sqrt{4\pi\alpha_{em}}$) | SG, LQ$\to eq/\nu_{e}q$ ($q=u,d$)
OPALOPAL1  |  $\lambda<1$ for 200 GeV, ($|q|=\frac{4}{3}$, B(LQ$\to eq$)=1) | LQ$\to eq/\nu_{e}q$, ($q=u,d,s,c,b$)
OPALOPAL2  | 46.3 GeV ($|q|=\frac{4}{3},B(\text{LQ}\to eq/\mu q)=1$) | LQ$\to eq/\mu q$, ($q=u,d,s,c,b$)
| 45.5 GeV ($|q|=\frac{4}{3},B(\text{LQ}\to\tau q)=1$) | SG, LQ$\to\tau q$ , ($q=u,d,s,c,b$)
OPALOPAL3  | 50-100 GeV ($|q|=\frac{4}{3},B(\text{LQ}\to eq)=1$) | SL, LQ $\to eq$, ($q=u,d,s,c,b$)
| 50-101 GeV ($|q|=\frac{4}{3},B(\text{LQ}\to\mu q)=1$) | SL, LQ $\to\mu q$, ($q=u,d,s,c,b$)
| 50-99 GeV ($|q|=\frac{4}{3},B(\text{LQ}\to\tau q)=1$) | SL, LQ $\to\tau q$, ($q=u,d,s,c,b$)
H1H11  | 257 GeV ($B(\text{LQ}\to eq)=1$,$\lambda=\sqrt{4\pi\alpha_{em}}$) | LQ$\to eq/\nu_{e}q$ ($q=u,d$)
H1H12  | 712 GeV ($\lambda=\sqrt{4\pi\alpha_{em}}$) | LQ$\to eq/\mu q$, ($q=u,d,s,c,b$)
H1H13  | 800 GeV ($B(\text{LQ}\to eq)=1,\lambda=\sqrt{4\pi\alpha_{em}}$) | LQ$\to eq/\nu_{e}q$
CDFCDF0  | 99 GeV (B(LQ$\to\tau b$)=1) | SG, LQ$\to\tau b$
CDFCDF1  | 236 GeV (B(LQ$\to eq$)=1) | SG, LQ$\to eq/\nu_{e}q$ ($q=u,d$)
CDFCDF2  | 224 GeV (B(LQ$\to\mu q$)=1) | SG, LQ$\to\mu q/\nu_{\mu}q$ ($q=c,s$)
CDFCDF3  | 317 GeV (B(Vector LQ$\to\tau b$)=1) | SG, Vector LQ$\to\tau b$
D0D03  | 229 GeV (B(LQ$\to\nu_{\tau}b$)=1) | SG, LQ$\to\nu_{\tau}b$
D0D05  | 210 GeV (B(LQ$\to\tau b$)=1) | SG, LQ$\to\tau b$
D0D02  | 316 GeV (B(LQ$\to\mu q$)=1) | SG, LQ$\to\mu q/\nu_{\mu}q$ ($q=c,s$)
D0D01  | 295 GeV (B(LQ$\to eq$)=1) | SG, LQ$\to eq/\nu_{e}q$ ($q=u,d$)
D0D04  | 247 GeV (B(LQ$\to\nu_{\tau}b$)=1) | SG, LQ$\to\nu_{\tau}b$
ATLASATLAS1  | 200-534 GeV (B(LQ$\to\tau b$)=1) | SG, LQ$\to\tau b$
ATLASATLAS2  | 200-800 GeV | SG, LQ$\to\nu_{\tau}t/\tau b/\nu_{\tau}b/\tau t$
ATLASATLAS3  | 0.4-1.8 TeV (B(LQ$\to eq$)=1) | SL, LQ$\to eq$ ($q=u,d,s,c,b$)
| 0.4-1.7 TeV (B(LQ$\to\mu q$)=1) | SL, LQ$\to eq$ ($q=u,d,s,c,b$)
ATLASATLAS4  | 0.9-1.48 TeV (B(LQ$\to et$)=1) | SL, LQ $\to et$
| 0.9-1.47 TeV (B(LQ$\to\mu t$)=1) | SL, LQ $\to\mu t$
ATLASATLAS5  | 0.5-1.43 TeV ($|q|=\frac{1}{3}$,B(LQ$\to\tau t$)=1) | SG, LQ $\to\tau t/\nu_{\tau}b$
CMSCMS1  | 200-740 GeV (B(LQ$\to\tau b$)=1) | SG, LQ$\to\tau b/\nu_{\tau}t$
CMSCMS2  | 200-1730 GeV (B(LQ$\to e^{-}u$)=1, $\lambda=1$) | SG, LQ$\to e^{-}u$
| 300-660 GeV (B(LQ$\to\mu^{-}c$)=1, $\lambda=1$) | SG, LQ$\to\mu^{-}c$
CMSCMS3  | 200-850 GeV (B(LQ$\to\tau b$)=1) | SG, LQ$\to\tau b$
CMSCMS4  | 200-740 GeV (B(LQ$\to\tau b$)=1, $\lambda=1$) | SG, LQ$\to\tau^{+}b$
CMSCMS5  | 900 GeV ($|q|=\frac{1}{3},B(\text{LQ}\to t\mu/t\tau/b\nu)=1$) | LQ$\to t\mu/t\tau/b\nu)=1$
CMSCMS6  | 250-1020 GeV (B(LQ$\to\tau b$)=1) | SG, LQ$\to\tau b$
CMSCMS7  | 950 GeV (B(LQ$\to\tau^{-}t$)=B(LQ$\to\nu_{\tau}b$)=0.5) | SG, LQ$\to\tau^{-}t/\nu_{\tau}b$

A combined fit to all measurements is desirable to avoid the assumptions, but
not available for the moment. In Sec. 2, we will show that 80 GeV is a robust
lower limit for the LQ with an electric charge $|q|=\frac{4}{3}e$ based on the
OPAL measurements OPAL2 ; OPAL3 . Considering a LQ with an electric charge
$|q|=\frac{4}{3}e$, it has 9 decay modes ($\text{LQ}\to l+q$ with
$l=e,\mu,\tau$ and $q=d,s,b$). Except the OPAL experiment, there is no search
for the final state with a tau lepton plus light quarks. Therefore, it seems
that we cannot deny the possibility of a low-mass LQ based on previous
searches without a global fit. On the other hand, we will show that low-energy
flavour physics theory_bound1 ; theory_bound2 ; theory_bound3 ; theory_bound4
can impose strong constraint on combinations of $\lambda_{lq}$ and
$m_{\text{LQ}}$ in Sec. 2. We can only set upper limits on $\lambda_{lq}$ for
a fixed LQ mass or vice versa. Given these difficulties, model-independent
methods to search for LQs are highly necessary.

For the low mass LQs (less than 200 GeV), we find that the nuclear ultra-
peripheral collision (UPC) is an ideal place to perform such searches. In
UPCs, relativistic nuclei collide at impact parameters ($b$) that are so large
that there are no hadronic interactions. Thus it provides a unique opportunity
to study photon-photon interactions. Ref. UPCreview1 ; UPCreview2  are two
good reviews about the physics of UPCs. Given high photon flux at low energies
(proportional to the 4-th power of the nucleus charge) and much cleaner
collision environment, we propose to search for the LQ pair production in Pb-
Pb UPC events. Most of previous searches use the property that LQs carry color
charge. This proposal uses the property that LQs have electric charge, and is
much less model-dependent compared to the existing searching strategies.
Taking the scalar LQ with an electric charge $|q|=\frac{4}{3}e$, denoted as
$S_{3}$ as in the review LQreview , as example, a feasibility study using all
possible $llqq$ ($l=e,\mu,\tau$, $q=d,s,b$ and charge conjugation is always
assumed) final states is conducted based on the performance of the ATLAS
detector in Run 2 and Pb-Pb UPCs at the center-of-mass (c.m.) energy
$\sqrt{s}=5.02$ TeV.

In Sec. 2, the constraints on the LQ mass and coupling constant from the
experiments and the low-energy flavour physics are reviewed. In Sec. 3, a much
less model-dependent method is proposed to search for LQ pair production,
which process dominates if the coupling strength of the LQ to the fermions is
not big ($\lesssim 0.1$ as shown in Fig. 1). It is worth emphasizing that the
proposed method itself is also applicable to search for high-mass LQs in
$p$-$p$ collisions as long as the pair production mechanism dominates. In Sec.
4, we describe the cross sections in UPCs and the preparation of MC samples.
In Sec. 5, we elaborate the selection criteria and the analysis strategy. In
Sec. 6, we will analyze the dominant systematic uncertainties and propose some
solutions. In Sec. 7, we will describe the statistical interpretation scheme
and present the sensitivity results. In Sec. 8, we apply the method to the
available $p$-$p$ dataset from ATLAS ATLAS2  to investigate the possibility of
excluding a 100 GeV LQ unambiguously. Sec. 9 is a short summary.

##  2  The bounds from the experimental searches and low-energy flavour
physics

###  2.1 A robust lower mass bound from the OPAL experiments

For a LQ, both its mass and coupling constant are unknown. The partial width
of $\text{LQ}\to l+q$ is

$\Gamma_{lq}=\frac{\lambda_{lq}^{2}m}{C\pi}\>,$ (1)

where $C=16$ for a scalar LQ and $24$ for a vector LQ. The LQ would decay
outside a detector for a mass of $100$ GeV and a coupling constant of less
than $10^{-8}$. Since a LQ carries both electric and color charge, a long-
lived one would interact with the detector matter and may appear like a
$b$-tagged jet. There is no dedicated search for such objects. If the lifetime
is not too long, some direct searches are summarized in Table 1. Among these
results, two of the OPAL measurements OPAL2 ; OPAL3  seem to use the weakest
assumption, which forbids a LQ to couple to different generations of leptons,
but allow to couple to all possible quarks. In addition, it used the pair
production in $e^{+}e^{-}$ collisions. The pair production dominates if
$\lambda\lesssim 10^{-2}$ and its cross section does not depend upon
$\lambda$. Collecting the branching fractions, we can naively interpret their
results in a different way (taking the LQ with the electric charge
$|q|=\frac{4}{3}e$ and the measurement OPAL3  as example).

$\displaystyle\sigma(m)B_{ex}^{2}$ $\displaystyle<$
$\displaystyle\sigma(100\>\text{GeV})\>,$ (2) $\displaystyle\sigma(m)B_{\mu
x}^{2}$ $\displaystyle<$ $\displaystyle\sigma(101\>\text{GeV})\>,$ (3)
$\displaystyle\sigma(m)B_{\tau x}^{2}$ $\displaystyle<$
$\displaystyle\sigma(99\>\text{GeV})\>,$ (4)

where $\sigma(m)$ is the production cross section of a pair of LQs with a mass
$m$ in the process $e^{+}e^{-}\to\text{LQ}\overline{\text{LQ}}$; $B_{lx}$ is
the branching fraction of $\text{LQ}\to lx$ with $x=d,s,b$ for a fixed lepton
$l$. Using $B_{ex}+B_{\mu x}+B_{\tau x}=1$ and the Cauchy-Schwarz inequality,
we have

$\displaystyle\sigma(m)(B_{ex}+B_{\mu x}+B_{\tau x})^{2}\leq$
$\displaystyle\sigma(m)3(B_{ex}^{2}+B_{\mu x}^{2}+B_{\tau x}^{2})$ (5)
$\displaystyle<$ $\displaystyle
3(\sigma(100\>\text{GeV})+\sigma(101\>\text{GeV})+\sigma(99\>\text{GeV}))$ (6)
$\displaystyle<$ $\displaystyle 9\sigma(99\>\text{GeV})\>.$ (7)

Using the cross section formula in Ref. OPAL3  and the highest collision
energy, a conservative bound is obtained.

$m>80\>\text{GeV}$ (8)

This interpretation neither considers the shape information of the invariant
mass reconstructed from a lepton candidate and a jet, nor considers the fact
that there are multiple collision energy points with different luminosities.
It does not take into account the contribution from $\text{LQ}\to\tau+q$ with
a subsequent leptonical $\tau$ decay in the $eeqq$ or $\mu\mu qq$ signal
region, either. A full consideration shall give a better bound (but would not
exceed $\sim$100 GeV anyway). Thus we claim this is a robust bound with little
model dependence.

A recent ATLAS measurement looked at the final states $eq$ and $\mu q$ with
$q$ being all possible quarks except the top quark and searched for the LQ
mass above 400 GeV ATLAS3 . Exclusion regions as a function of the LQ mass and
the branching fraction is reported (more details see ATLAS3 ). Unfortunately,
there seems no search for the final state with a tau lepton and light quarks
except the OPAL experiment. It is difficult to exclude LQs with higher mass
without assumptions.

Other experiments as summarized in Table 1 either focused on only specific
final state or made very strong assumptions. No further bounds can be obtained
without a combined fit.

###  2.2 The bounds from low-energy flavour physics

Low-energy flavour physics like rare meson decays and lepton decays can
provide stringent bounds on combinations of the coupling constant and mass
theory_bound1 ; theory_bound2 ; theory_bound3 ; theory_bound4 . In most cases,
a LQ propagator would inevitably couple to a lepton and a quark on both ends.
The extra amplitude induced by a LQ would be proportional to a factor of
$\lambda_{lq}\lambda_{l^{\prime}q^{\prime}}/m_{\text{LQ}}^{2}$ if the involved
energy scale is much smaller than the LQ mass. This factor is invariant under
the scaling transformation $\lambda\to k\lambda$ and $m_{\text{LQ}}\to
km_{\text{LQ}}$. Here we quote a few bounds from the literature. For $K_{L}\to
e^{\pm}\mu^{\mp}$, Ref. theory_bound2  obtained

$|\lambda|^{2}<6\times 10^{-7}(\frac{m_{\text{LQ}}}{100\>\text{GeV}})^{2}\>.$
(9)

For the $\mu^{-}\to e^{-}$ conversion on the Titanium mu2e_exp , Ref.
theory_bound3  obtained

$|\lambda_{\mu q}\lambda_{eq}|<1.7\times
10^{-7}(\frac{m_{\text{LQ}}}{100\>\text{GeV}})^{2}\>.$ (10)

Interestingly, the neutral meson ($K^{0},D^{0},B^{0}$) mixing is found to be
able to constrain $\lambda^{2}/m_{\text{LQ}}$. A recent calculation
theory_bound4  gives roughly (the exact bound is model-dependent)

$|\lambda|^{2}<2.7\times 10^{-3}\frac{m_{\text{LQ}}}{100\>\text{GeV}}\>.$ (11)

Due to the scaling invariance above, one cannot provide definite bounds on a
LQ’s mass due to the mixing with the coupling constant. The existing anomalies
in $B$-meson decays need significant contribution from new physics. The bounds
based on those anomalies are highly model-dependent. We shall not expand the
discussion here. In order to effectively bound a LQ’s mass without touching
its coupling to the fermions, we shall consider the physics processes where
only LQs and gauge bosons are involved. For example, we can measure the
differential cross section of $\gamma\gamma\to\gamma\gamma$, $\gamma\gamma\to
gg$ or $gg\to\gamma\gamma$. Both the SM particles and LQs contribute at loop
level. The first process is the cleanest, but the first and the second ones
shall only be sensitive to low-mass LQs in heavy-ion collisions. The third one
might be sensitive to high-mass LQs in $p$-$p$ collisions, though it suffers
from the irreducible background $q\bar{q}\to\gamma\gamma$. The resonance
effect near the threshold of a pair of LQs shall enhance the sensitivity.
Anyway, the method presented in next section is more available and
straightforward though more experimental efforts are needed.

##  3  A model-independent searching method using the LQ pair production

The LQ, $S_{3}$, considered in this feasibility study has a charge
$|q|=\frac{4}{3}e$ and is allowed to decay to all possible $lq$ modes
($l=e,\mu,\tau$ and $q=d,s,b$). The pair production in $p$-$p$ collisions and
Pb-Pb UPCs dominate if the coupling constant is not too big (e.g.
$\lambda\lesssim 0.1$ as shown in Fig. 1). This is consistent with the bounds
from the low-energy physics in last section. The cross section of pair
production does not depend on the coupling constant, but on the branching
fractions of the decay modes. Thus we consider all possible final states,
namely, $eejj$, $e\mu jj$, $e\tau_{\text{had}}jj$, $\mu\mu jj$,
$\mu\tau_{\text{had}}jj$, and $\tau_{\text{had}}\tau_{\text{had}}jj$, where
$\tau_{\text{had}}$ denotes a hadronic tau decay candidate and $j$ denotes a
jet candidate. We do not attempt to identify the jet flavour for simplicity.
For each final state, all LQ’s contributions are considered. For example, all
LQ$\to eq/\mu q/\tau q$ decays contribute in the $e\mu jj$ final state.

Three floating parameters, $\kappa,B_{ex}$ and $B_{\mu x}$, are introduced in
the statistical interpretation. The branching fraction $B_{\tau x}$ is
constrained to be $1-B_{ex}-B_{\mu x}$. $\kappa$ is the ratio of the measured
cross section to the model prediction. We further impose the conditions $0\leq
B_{ex},B_{\mu x}\leq 1$ and $B_{ex}+B_{\mu x}\leq 1$. For a given LQ mass, the
measurement can determine the upper limit at 95 % confidence level (C.L.) of
$\kappa$. If the upper limit is less than 1, then a LQ with the mass is
excluded.

The method is model-independent in the sense that it allows a LQ to couple to
all possible fermions.

##  4 Cross sections and MC simulation

###  4.1 Cross sections in UPC

For a nucleus $A$, the cross section for the process $A+A\to A+A+X$ is

$\sigma_{X}=\int
dk_{1}dk_{2}\frac{d^{2}L_{\gamma\gamma}}{dk_{1}dk_{1}}\sigma^{\gamma\gamma\to
X}(k_{1},k_{2})\>,$ (12)

where $\sigma^{\gamma\gamma\to X}(k_{1},k_{2})$ is the production cross
section of the final state $X$ in two-photon interactions with energy $k_{1}$
and $k_{2}$; $d^{2}L_{\gamma\gamma}/dk_{1}dk_{2}$ is the corresponding two-
photon luminosity.

$~{}\frac{d^{2}L_{\gamma\gamma}}{dk_{1}dk_{2}}=\int_{b_{1}>R_{A}}\int_{b_{2}>R_{A}}d^{2}b_{1}d^{2}b_{2}\frac{d^{3}N_{\gamma}}{dk_{1}d^{2}b_{1}}\frac{d^{3}N_{\gamma}}{dk_{2}d^{2}b_{2}}\>,$
(13)

where $R_{A}$ is the nucleus radius; and $d^{3}N_{\gamma}/dkd^{2}b$ is the
photon flux from a nucleus at a distance $b$. For a charge $Z$ nucleus with
velocity $\beta$ and Lorentz boost factor $\gamma$, the photon flux is

$\frac{d^{3}N_{\gamma}}{dkd^{2}b}=\frac{Z^{2}\alpha
x^{2}}{\pi^{2}kb^{2}}[K_{1}^{2}(x)+\frac{K_{0}^{2}(x)}{\gamma}]\>,$ (14)

where $x=\frac{kb}{\beta\gamma}$ and $K_{0}(x)$ and $K_{1}(x)$ are modified
Bessel functions. The photon flux decreases exponentially above the cut off
$k_{\text{max}}\approx\frac{\gamma}{R_{A}}$.

In Eq. 13, we have assumed that the probability of not having a hadronic
interaction, denoted by $P_{0\text{had}}(b)$, is 0 if $b<R_{A}$ and 1 if
$b>R_{A}$. In general, we can replace the integration $\int_{b>R_{A}}\cdots
d^{2}b$ in Eq. 13 by $\int_{b=0}^{\infty}P_{0\text{had}}(b)\cdots d^{2}b$.
Using a more complicated description of $P_{0\text{had}}(b)$ leads to a
$10-15$% difference in the flux UPCreview1 .

###  4.2 MC sample preparation

The MadGraph 5 MG5  is used to compute cross sections by convolving the
Weizäcker-Williams equivalent photon flux (Eq. 14) of the nucleus with the
elementary two-photon cross section, and to produce the signal and background
samples. The parton showers are simulated with Pythia 8 pythia  and the
detector response is simulated with Delphes 3 delphes  based on the Run 2
performance of the ATLAS detector.

The signal model is obtained from LQtoolboxurl  and documented in Ref.
LQtoolbox . The LQ of interest in this work is the scalar boson $S_{3}$, which
is allowed to couple to all charged leptons and down-type quarks. Six signal
samples corresponding to the processes $\gamma\gamma\to S_{3}\bar{S}_{3}\to
eeqq/e\mu qq/e\tau qq/\mu\mu qq/\mu\tau qq/\tau\tau qq$ are produced with a
mass 100 GeV and all coupling constant fixed at $10^{-2}$. The possible
interference effect with the SM process is not considered. The fiducial region
at generator level is defined as $p_{\text{T}}(l)>13$ GeV,
$p_{\text{T}}(j)>15$ GeV, $|\eta(l)|<2.5$, $|\eta(j)|<5$, $\Delta R(j,l)>0.3$
and $\Delta R(j,j)>0.3$, where $p_{\text{T}}$, $\eta$ and $\Delta R$ represent
the transverse momentum, rapidity and the angular distance111The angular
distance between two objects is defined as $\Delta
R(a,b)\equiv\sqrt{(\eta(a)-\eta(b))^{2}+(\phi(a)-\phi(b))^{2}}$, where $\eta$
and $\phi$ are the rapidity and azimuthal angle, respectively., respectively.

Figure 1 shows the fiducial cross section of
Pb$+$Pb$\to$Pb$+$Pb$+e^{+}e^{-}d\bar{d}$ as a function of the coupling
constant $\lambda_{ed}$. Little dependence is seen for $\lambda_{ed}\lesssim
0.1$. The cross section used in the normalization does not consider the
branching fractions, which will be treated as the parameters of interest in
the statistical interpretation, and is scaled by a factor of $10^{-2}$ in the
plots in Sec. 5 just for better illustration.

Figure 1:  Fiducial cross section of Pb$+$Pb$\to$Pb$+$Pb$+e^{+}e^{-}d\bar{d}$
driven by the leptoquark $S_{3}$ in ultra-peripheral collisions at
$\sqrt{s}=5.03$ TeV as a function of the coupling constant $\lambda_{ed}$ with
a mass of 100 GeV (blue curve) or 200 GeV (red curve, scaled by a factor of
$10^{4}$).

The dominant background is from the SM process $\gamma\gamma\to
l^{+}l^{-}q\bar{q}$. It is irreducible. The background events with multiple
leptons or multiple jets are also considered and their contribution is minor.
In addition, the background from Pb-Pb hadronic interactions is inevitable. It
is very difficult to simulate this background in reality. We usually design
tight cuts to suppress it and estimate it in a data-driven way. We assume this
background is negligible after the event selection in next section.

##  5 Event selection and Observable reconstruction

Full reconstruction of all 4 objects is not desirable because of low
efficiency. We require two lepton candidates and at least one jet candidates
to be reconstructed. A charged lepton candidate ($e/\mu$) is required to have
transverse momentum $p_{\text{T}}>15$ GeV and rapidity $|\eta|<2.5$. A
hadronic tau candidate $\tau_{\text{had}}$ is required to have
$p_{\text{T}}>20$ GeV and $|\eta|<2.5$. The associated number of charged
tracks is required to be $1$ or $3$ with unit sum of charges. A jet candidate
denoted by $j$ is required to have $p_{\text{T}}>20$ GeV and $|\eta|<5.0$.

An overlapping removal is performed according to the priority order $\mu\to
e\to\tau_{\text{had}}\to j$. We first select muon candidate and then select
electron candidate. The electron candidate is required not to overlap with the
muon candidate within the angular distance $\Delta R=0.4$. $\tau_{\text{had}}$
candidate is required not to overlap with electron and muon candidates within
the same angular distance. Similarly the $j$ candidate is required not to
overlap with $\mu$, $e$ and $\tau_{\text{had}}$ candidates. 12 signal regions
(SR) are designed below ($x=1,2$).

  * •

“ee$x$j” signal region: 2 electron candidates with opposite charge sign and
$x$ jets.

  * •

“emu$x$j” signal region: 1 electron candidate and 1 muon candidate with
opposite charge sign and $x$ jets.

  * •

“eta$x$j” signal region: 1 electron candidate and 1 $\tau_{\text{had}}$
candidates with opposite charge sign and $x$ jets.

  * •

“mumu$x$j” signal region: 2 muon candidates with opposite charge sign and $x$
jets.

  * •

“muta$x$j” signal region: 1 muon candidate and 1 $\tau_{\text{had}}$ candidate
with opposite charge sign and $x$ jets.

  * •

“tata$x$j” signal region: 2 $\tau_{\text{had}}$ candidate and $x$ jets. For
the signal region “tata2j”, the missing transverse energy 222The missing
transverse energy is defined as negative vectorial sum of all visible objects’
transverse momenta. $E_{\text{T}}^{\text{miss}}$ is required to be greater
than 10 GeV to suppress the multi-jet background.

If we have real data samples, we can apply further requirements to suppress
the background due to the hadronic Pb-Pb interactions: 1) no additional
charged tracks in the inner detector and no additional energy deposit from
neutral particles in the calorimeters as used in Ref. exclWW ; yy2WW ; 2) no
significant energy deposit in the Zero Degree Calorimeters ZDC  for the ATLAS
detector ( because the Pb nuclei are intact in UPCs) as used in Ref. yy2mumu ;
3) adoption of multi-variate-analysis (MVA) methods TMVA ; weaklearner  to
separate signal events from background events. In Appendix A, it is
demonstrated that the signal purity in Pb-Pb collisions is much better than
that in $p$-$p$ collisions.

The signal is identified as a peak structure in the invariant mass spectrum of
a lepton and a jet candidates. For the events with only one jet, the jet is
used twice to reconstruct the LQ mass. For the events with two jets, the best
alignment is determined by minimizing their relative mass difference, namely,

$\min\frac{|m(l_{1}j_{1})-m(l_{2}j_{2})|}{m(l_{1}j_{1})+m(l_{2}j_{2})}\>.$
(15)

The subscript associated with the lepton candidate represents the charge sign,
namely, $l_{1,2}=l^{\pm}$, in the signal regions with the same flavour leptons
(ee$x$j, mumu$x$j and tata$x$j). Otherwise the subscript represents the lepton
flavour. For the eta$x$j and muta$x$j signal regions, we assume that the
missing transverse energy is due to the neutrinos from the tau decays and thus
added to the $\tau_{\text{had}}$ candidate, namely,

$\displaystyle p_{x}(\tau)$ $\displaystyle=$ $\displaystyle
p_{x}(\tau_{\text{had}})+E_{\text{T}}^{\text{miss}}\cos\phi^{\text{miss}}\>,$
(16) $\displaystyle p_{y}(\tau)$ $\displaystyle=$ $\displaystyle
p_{y}(\tau_{\text{had}})+E_{\text{T}}^{\text{miss}}\sin\phi^{\text{miss}}\>,$
(17) $\displaystyle p_{z}(\tau)$ $\displaystyle=$ $\displaystyle
p_{z}(\tau_{\text{had}})+E_{\text{T}}^{\text{miss}}\sinh\eta(\tau_{\text{had}})\>,$
(18)

where $\phi^{\text{miss}}$ is the azimuthal angle of the missing transverse
energy; and the collinear approximation, which assumes that the flight
direction of the invisible neutrino(s) in a tau decay is the same as that of
the visible tau decay products mmc ; mxlg , is used in Eq. 18.

Assuming a luminosity of 1 fb-1, Fig. 2-7 show the distribution of the
invariant mass of a lepton and a jet candidates in all signal regions. We can
see that the $S_{3}$ resonance is well reconstructed, which also confirms that
the resonant production of a $S_{3}$ pair dominates. Other features in these
distributions are also expected: 1) the mass resolution of $m(ej)$ and $m(\mu
j)$ is better than that of $m(\tau j)$; 2) The resolution of
$m(\tau_{\text{had}}j)$ is even worse because it is only a partial
reconstruction of the $S_{3}$ resonance; 3) the background level in the emu2j
signal region is quite low because lepton flavour is conserved in the SM.

In order to improve the sensitivity to the mode $\text{LQ}\to\tau q$, two mass
reconstruction methods are tried in the tata2j signal region. One method is
using the collinear approximation (technical details can be found in Ref. mmc
; mxlg ), the other method is from Ref. mxlg  which reconstructs the mass by
sampling the neutrino(s)’ momentum distribution in the center-of-mass frame of
the tau lepton. The two mass variables, $m_{\text{coll}}$ and
$m_{\text{xlg}}$, are shown in Fig. 8. The latter mass $m_{\text{xlg}}$ has a
better resolution and accurately reconstructs the $S_{3}$ mass. Thus it is
used as the final observable for the tata2j signal region.

Figure 2:  The distribution of the invariant mass of a lepton candidate and a
jet candidate in the signal region “ee1j” (Top) and “ee2j” (Bottom). The left
one is $m(e^{+}j)$ and the right one is $m(e^{-}j)$. The black points in the
upper pads show the asimov data which is just the sum of all background events
with uncertainties completely suppressed for better illustration. The ratio of
the asimov data and the total background is shown in the lower pads, where the
error bars on the black points represent the expected data uncertainty while
the green bands represent the total MC statistical uncertainty.

Figure 3:  The distribution of the invariant mass of a lepton candidate and a
jet candidate in the signal region “emu1j” (Top) and “emu2j” (Bottom). The
left one is $m(ej)$ and the right one is $m(\mu j)$. The black points in the
upper pads show the asimov data which is just the sum of all background events
with uncertainties completely suppressed for better illustration. The ratio of
the asimov data and the total background is shown in the lower pads, where the
error bars on the black points represent the expected data uncertainty while
the green bands represent the total MC statistical uncertainty.

Figure 4:  The distribution of the invariant mass of a lepton candidate and a
jet candidate in the signal region “eta1j”(Top) and “eta2j”(Bottom). The left
one is $m(ej)$ and the right one is $m(\tau j)$. The black points in the upper
pads show the asimov data which is just the sum of all background events with
uncertainties completely suppressed for better illustration. The ratio of the
asimov data and the total background is shown in the lower pads, where the
error bars on the black points represent the expected data uncertainty while
the green bands represent the total MC statistical uncertainty.

Figure 5:  The distribution of the invariant mass of a lepton candidate and a
jet candidate in the signal region “mumu1j”(Top) and “mumu2j”(Bottom). The
left one is $m(\mu^{+}j)$ and the right one is $m(\mu^{-}j)$. The black points
in the upper pads show the asimov data which is just the sum of all background
events with uncertainties completely suppressed for better illustration. The
ratio of the asimov data and the total background is shown in the lower pads,
where the error bars on the black points represent the expected data
uncertainty while the green bands represent the total MC statistical
uncertainty.

Figure 6:  The distribution of the invariant mass of a lepton candidate and a
jet candidate in the signal region “muta1j”(Top) and “muta2j”(Bottom). The
left one is $m(\mu j)$ and the right one is $m(\tau j)$. The black points in
the upper pads show the asimov data which is just the sum of all background
events with uncertainties completely suppressed for better illustration. The
ratio of the asimov data and the total background is shown in the lower pads,
where the error bars on the black points represent the expected data
uncertainty while the green bands represent the total MC statistical
uncertainty.

Figure 7:  The distribution of the invariant mass of a lepton candidate and a
jet candidate in the signal region “tata1j”(Top) and “tata2j”(Bottom). The
left one is $m(\tau_{\text{had}}^{+}j)$ and the right one is
$m(\tau_{\text{had}}^{-}j)$. The black points in the upper pads show the
asimov data which is just the sum of all background events with uncertainties
completely suppressed for better illustration. The ratio of the asimov data
and the total background is shown in the lower pads, where the error bars on
the black points represent the expected data uncertainty while the green bands
represent the total MC statistical uncertainty.

Figure 8:  The distribution of the reconstructed mass of a lepton candidate
and a jet candidate in the signal region “tata2j” based on the collinear
approximation method (Top) and the method in Ref. mxlg  (Bottom). The left one
is $m(\tau^{+}j)$ and the right one is $m(\tau^{-}j)$. The black points in the
upper pads show the asimov data which is just the sum of all background events
with uncertainties completely suppressed for better illustration. The ratio of
the asimov data and the total background is shown in the lower pads, where the
error bars on the black points represent the expected data uncertainty while
the green bands represent the total MC statistical uncertainty.

##  6 Systematical uncertainties

The systematical uncertainties can be categorized into experimental
uncertainties and theoretical uncertainties. Here is a summary of experimental
uncertainties based on the ATLAS detector in Run 2 and main theoretical
uncertainties.

  * •

Luminosity uncertainty: 4%, which is quoted from Ref. ATLAS_LbyL  following
the method in Ref. ATLAS_lumi_meas .

  * •

Object triggering, reconstruction and identification efficiencies: the
performance of final object triggering, reconstruction and identification can
found in Ref. ATLAS_electron_perf , Ref. ATLAS_muon_perf , Ref. ATLAS_tau_perf
, Ref. ATLAS_met_perf , and Ref. ATLAS_jet_perf ; ATLAS_jet_perf2  for
electron, muon, hadronic tau, missing transverse energy and jet, respectively.
The efficiency uncertainties are usually expressed as a function of the
transverse momentum. Table 2 shows the combined efficiency uncertainties
(reconstruction and identification) for the transverse momentum range involved
in this analysis. For jets, the dominant uncertainty is due to jet energy
scale and energy resolution. In the transverse momentum range involved in this
study, the systematical uncertainty is 5 % on the energy scale and 6 % on the
energy resolution ATLAS_jet_perf2 .

Table 2:  The combined efficiency uncertainty (reconstruction and identification) of the final objects.  Object | Combined efficiency uncertainty
---|---
Electron | about 7%-3% for $p_{\text{T}}=15-40$ GeV ATLAS_electron_perf
Muon | about 1% for $p_{\text{T}}=15-40$ GeV ATLAS_muon_perf
Hadronic tau | about 5% for $p_{\text{T}}=20-60$ GeV ATLAS_tau_perf
  * •

Photon flux: there are a number of theoretic uncertainties in the photon flux
calculations. Some of them need precise experimental measurements. One
important uncertainty is about the probability of not having a hadronic
interaction, $P_{0\text{had}}(b)$. As emphasized in the end of Sec. 4.1, the
different choices of $P_{0\text{had}}(b)$ show a photon flux uncertainty of
10-15% UPCreview1 . More discussions about these theoretic factors can be
found in Ref. UPCreview1 ; UPCreview2 . Photon flux uncertainty is directly
related to the uncertainty of the signal production cross section. Here we
present a data-driven estimation of this uncertainty. Because we have found a
robust mass bound $m_{\text{LQ}}>80$ GeV in Sec. 2.1 and the irreducible
background with $m(lj)<80$ GeV share the same final objects as the signal,
thus the region with $m(lj)<80$ GeV can be used as a control region to
constrain the photon flux uncertainty.

  * •

Uncertainty of the background due to Pb-Pb hadronic interactions: the MC
simulation of this background is unavailable and we have assumed it is
negligible in the signal regions. One of the key features of photon-photon
interactions in UPCs is that the net transverse momentum of the final state is
very small. This feature can be used to estimate the Pb-Pb hadronic background
in the signal regions “ee2j”, “emu2j”, and “mumu2j” where no neutrinos are in
the signal events. In other signal regions, we may use empirical functions to
model it based on the fact that the background shape is expected to be smooth
compared to the signal resonance structure.

  * •

Jet flavour: no attempt is made to identify the jet flavour. The jet
reconstruction efficiency may differ for different flavours. This can be
estimated by MC and the systematical uncertainty can be controlled by studying
the process of $\gamma\gamma\to q\bar{q}$. A more detailed analysis by
categorizing the events into signal regions with light jets (from $u,d,s$
quarks), $c$-tagged jets and $b$-tagged jets will surely improve the
sensitivity. But the methodology is the same.

##  7 Statistical interpretation and sensitivity results

A simultaneous binned likelihood fit is performed to the distributions shown
in Figs. 2-8. The likelihood function is

$\mathcal{L}(\kappa,B_{ex},B_{\mu
x})=\Pi_{i=1}^{12}\Pi_{j=1}^{N_{\text{bins}}^{i}}P(n_{j}^{i}|b_{j}^{i}+\kappa\sum_{k=1}^{N_{\text{sig}}^{i}}s_{j,k}^{i}(B_{ex},B_{\mu
x}))\>,$ (19)

where the index $i$ represents the $i$-th signal region; the index $j$
represent the $j$-th bin; the index $k$ represent the $k$-th signal mode;
$N_{\text{bins}}^{i}$ represents the number of bin in the $i$-th signal
region; $N_{\text{sig}}^{i}$ represent the number of signal modes in the
$i$-th signal region; and $s_{j,k}^{i}(B_{ex},B_{\mu x})$ represents the yield
contribution of the $k$-th signal mode to the $j$-th bin in the $i$-th signal
region. The yield of each signal mode should be associated to the branching
fractions appropriately. For example, we need to multiply the signal yield of
$\text{LQ}\overline{\text{LQ}}\to eq\tau q$ by a factor of
$2B_{ex}(1-B_{ex}-B_{\mu x})$. Furthermore, we should impose the constraint
$0\leq B_{ex}+B_{\mu x}\leq 1$ as stated in Sec. 3.

Allowing $B_{ex}$ and $B_{\mu x}$ to float, the left plot of Fig. 9 shows the
$-2\Delta\ln\mathcal{L}(\kappa)\equiv-2\ln\frac{\mathcal{L}(\kappa)}{\mathcal{L}_{\text{max}}}$
as a function of $\kappa$ assuming the absence of new physics. We can see that
$\kappa>1.1\times 10^{-2}$ can be excluded at 95 % confidence level for a
dataset with the luminosity $L=1$ fb-1. To exclude $\kappa>1$, a dataset of
$L\approx 4.0$ pb-1 is needed as shown in right plot of Fig. 9. The
sensitivity results are summarized in Table 3.

Figure 9:  Left: the $-2\Delta\ln\mathcal{L}$ as a function of $\kappa$ for a luminosity of 1 fb-1. The red lines indicate the interval at 95 % confidence level. Right: the upper limit of $\kappa$ at 95 % confidence level as a function of the luminosity. The red lines indicate the luminosity corresponding to the upper limit of $\kappa$ being 1.  Table 3:  Expected sensitivity to search for $S_{3}$ with a mass of 100 GeV using 1 fb-1 of Pb-Pb UPCs at $\sqrt{s}=5.02$ TeV.  $S_{3}$ mass | Upper limit of $\kappa$ (95 %C.L.) for 1 fb-1 | Luminosity to exclude $\kappa=1$
---|---|---
100 GeV | $1.1\times 10^{-2}$ | 4.0 pb-1

##  8 Can we exclude a 100 GeV LQ unambiguously?

In this section, we allow a LQ to decay to all possible fermions and
investigate the possibility of a 100 GeV LQ with an electric charge
$\frac{4}{3}e$ considering all available experimental constraints. The results
from the H1 experiment H11 ; H12 ; H13 , CDF experiment CDF0 ; CDF1 ; CDF2 ,
and ATLAS experiment ATLAS2  are used. We do not include other experiments
which did not look at this mass region although they may potentially provide
stronger constraints. Table I summarizes the available constraints with the
orignal interpretation with the corresponding assumption. We need to
reinterpret them to remove the assumption. If an experiment does not
explicitly identify the flavour of a jet, we will always include either all
light down-type quarks (denoted by $q=d,s$) or all down-type quarks (denoted
by $Q=d,s,b$) in the reinterpretation.

Table 4:  Reinterpretation of the experimental constraints on a 100 GeV LQ with an electric charge $\frac{4}{3}e$. In the table, $q$ denotes the light down-type quarks $d$ and $s$; $Q$ denotes all down-type quarks $d$, $s$ and $b$.  Exp. | Source | Original constraint (assumption) | Re-interpretation
---|---|---|---
H1 H11  | Fig. 13 in H11  |  $\lambda_{ed}\lesssim 0.01$ ($B_{ed}=1$) | $\lambda_{ed}^{2}B_{eq}<0.01^{2}$
H1 H12  | Fig. 3 in H12  |  $\lambda_{ed/\mu d}\lesssim 0.001$ ($B_{ed/\mu d}=\frac{1}{2}$) | $\lambda_{ed}^{2}\frac{B_{\mu q}}{\frac{1}{2}}<(0.001)^{2}$
H1 H13  | Fig. 3 in H13  |  $\lambda_{ed}\lesssim 0.006$ ($B_{ed}=1$) | $\lambda_{ed}^{2}B_{eq}\lesssim(0.006)^{2}$
CDF CDF0  | Table I in CDF0  |  $\sigma<10$ pb ($B_{\tau Q}=1$) |  $\sigma_{\text{th}}B_{\tau Q}^{2}<10$ pb, $\sigma_{\text{th}}=9.8$ pb
CDF CDF1  | Table I in CDF1  |  $\sigma<1.1$ pb ($B_{eq}=1$) |  $\sigma_{\text{th}}B_{eq}^{2}<1.1$ pb, $\sigma_{\text{th}}=11.13$ pb
CDF CDF2  | Table II in CDF2  |  $\sigma<1.35$ pb ($B_{\mu q}=1$) |  $\sigma_{\text{th}}B_{\mu q}^{2}<1.35$ pb, $\sigma_{\text{th}}=11.13$ pb
ATLAS ATLAS2  | Fig. 4 in ATLAS2  | Exclude 200-800 GeV (SG) | see text

In the H1 experiment, a LQ is produced via the single production mode
$e+p\to\text{LQ}+X$. For $\text{LQ}\to lq$, the final yield is proportional to
$\lambda_{ed}^{2}B_{lq}$ (we ignore the possible interference with the SM
process). The constraints are then re-expressed as

$\displaystyle\lambda_{ed}^{2}B_{eq}\lesssim(0.01)^{2}$ (20)
$\displaystyle\lambda_{ed}^{2}B_{\mu q}<2\times(0.001)^{2}$ (21)
$\displaystyle\lambda_{ed}^{2}B_{eq}\lesssim(0.006)^{2}\>,$ (22)

which are from Ref. H11 ; H12 ; H13 , respectively.

In the CDF experiment, a pair of LQs is produced in the reaction of
$p\bar{p}\to\text{LQ}\overline{\text{LQ}}+X$. The final yield is proportional
to the product of the pair-production cross section and the branching
fractions. Using the cross section from the theory, we can obtain the
constraints on the branching fractions, namely,

$\displaystyle B_{\tau Q}<1.01$ (23) $\displaystyle B_{eq}<0.31$ (24)
$\displaystyle B_{\mu q}<0.35\>,$ (25)

which are from Ref. CDF0 ; CDF1 ; CDF2 , respectively.

Based on the constraints above, it is impossible to exclude a 100 GeV LQ
definitely because all couplings are not covered. The ATLAS and CMS
experiments have also performed comprehensive searches for various LQs via
either single- or pair-production mechanism. However, we find only Ref. ATLAS2
from ATLAS has a public dataset which can be used to probe the existence of a
100 GeV LQ with an electric charge $\frac{4}{3}e$. The original motivation of
Ref. ATLAS2  is to search for a LQ decaying to $t\tau/b\nu$ or $b\tau/t\nu$.
We will use their data with the final state
$\tau_{\text{lep}}\tau_{\text{had}}$ plus one or more $b$-tagged jets, namely,
Fig. 4 (middle left) from Ref. ATLAS2  and Fig. 6b, Fig. 6c and Fig. 7b in its
auxiliary material ATLAS2aux . The data with the
$\tau_{\text{had}}\tau_{\text{had}}$ final state is not used due to the
complicated trigger requirements which depend upon the real-time data-taking
status.

Figure 10 shows the distribution of the pair mass of a tau-lepton candidate
and its matched jet candidate in the signal region with one $b$-tagged jets
(denoted by $\tau_{\text{lep}}\tau_{\text{had}}1j_{b}$) and signal region with
two $b$-tagged jets (denoted by $\tau_{\text{lep}}\tau_{\text{had}}2j_{b}$).
The matching criterion is to minimize the mass difference of the LQ pair
candidate (see Ref. ATLAS2 ).

We allow all possible decay modes, and do not differentiate $e$ and $\mu$ or
$d$ and $s$. We thus have four decay modes, $\text{LQ}\to b\tau,bl,q\tau,ql$
where $l=e,\mu$ and $q=d,s$ and 10 individual signal topologies from a LQ
pair. They are also shown in Fig. 10. In preparing signal samples, we have
tried our best to apply the same selection criteria as described in Ref.
ATLAS2 . Given that no significant deviation from the SM prediction is seen
and following the methodology presented in this work, the upper limit at 95 %
C.L. of the pair-production cross section modifier $\kappa$ is determined to
be 0.83 as shown in Fig. 11, for which the CDF constraints in Eq. 24\- 25 have
been considered. It means that a LQ with a mass 100 GeV is excluded at 95 %
C.L. under the general condition that a LQ can decay to all possible fermions.

It is worthwhile to make two comments: 1) the sensitivity is dominated by the
modes $q\tau q\tau$ and $q\tau ql$ (marked as squares and open circles in Fig.
10) for which we should design a signal region without any $b$-tagged jet in
reality; 2) Ref. ATLAS2  claims to exclude a LQ up to 1030 GeV assuming
$B_{b\tau}=1$. It is likely that the exclusion region would shrink much
dropping the assumption. Given the observation in this section, this work is
better seen as a proof of concept in order to search for LQs in a least model-
dependent way.

Figure 10:  The distribution of the pair mass of a tau-lepton candidate and
its mass-matched jet candidate in the signal region with one $b$-tagged jet
(Top) and two $b$-tagged jets (Bottom). The left one is $m(lj)$ and the right
one is $m(\tau j)$. The black points represent the data from Ref. ATLAS2 ;
ATLAS2aux . The open circles, black squares and histograms represent different
signal topologies for a 100 GeV LQ.  Figure 11:  The $-2\Delta\ln\mathcal{L}$
as a function of $\kappa$ for a 100 GeV LQ. The red lines indicate the
interval at 95 % confidence level.

##  9 Summary

The possibility of low-mass leptoquarks cannot be excluded due to the
assumptions made in previous experimental measurements. In this work, we
propose a least model-dependent method (assuming $\lambda\lesssim 0.1$) to
search for them in the Pb-Pb UPCs thanks to the high photon flux and
potentially low background contamination. It allows a LQ to couple to all
possible lepton-quark pairs. Taking the scalar LQ $S_{3}$ as example, the
feasibility is investigated using all $llqq$ final states based on the Pb-Pb
UPCs at $\sqrt{s}=5.02$ TeV and the performance of the ATLAS detector in Run
2. The LQ with a mass of 100 GeV can be excluded at 95 % confidence level
using a dataset of 4.0 pb-1. This proposal uses the property that LQs carry
electric charge and is complementary to the searches in $p$-$p$ collisions.
The method also works for searching for high-mass LQs in $p$-$p$ collisions as
long as the pair production mechanism dominates.

###### Acknowledgements.

I am greatly indebted to David d’Enterria for support with MadGraph and the
theoretical discussions about UPC. I also wish to express my appreciation to
Prof. Shan Jin for helping to resume my research career. Especially I would
like to thank Fang Dai for her encouragement and financial support.

##  Appendix A Signal purity comparison between in Pb-Pb collisions and in
$p$-$p$ collisions

In this appendix, we compare the signal purity between in Pb-Pb collisions at
5.02 TeV and in $p$-$p$ collisions at 13 TeV using the $bb\tau\tau$ final
state (an experimental search see Ref. ATLAS_LQ3 ). The final objects are
reconstructed in the same way as described in Sec. 5. The $b$-jet is
reconstructed at a working point with an efficiency of 80 %. The events with
one charged lepton candidate ($e$ or $\mu$), one hadronic tau decay candidate
and one $b$-jet candidate are selected. The luminosity is 1 fb-1 for Pb-Pb
collisions and 36 fb-1 for $p$-$p$ collisions.

Figure 12 shows the distribution of the invariant mass of a hadronic tau
candidate and a $b$-jet candidate. Both the production cross section in Pb-Pb
collisions and in $p$-$p$ collisions are scaled by a factor $10^{-2}$ for
better illustration (not including the scaling factor as shown in the legend
in Fig. 12). We can see that the signal-to-background ratio in Pb-Pb
collisions is around 50 times higher than that in $p$-$p$ collisions. The
reason is that the $t\bar{t}$ background dominates in $p$-$p$ collisions while
it is negligible in Pb-Pb collisions (the cross section is about 1 % that of
$bb\tau\tau$ production in the SM). This is one of the advantages of searching
for low-mass LQs in Pb-Pb UPCs.

Figure 12:  The distribution of the invariant mass of a hadronic tau candidate
and a $b$-jet candidate in the Pb-Pb collisions (L) and the $p$-$p$ collisions
(R). The signal histograms are scaled by a factor of $10^{-2}$ besides the
scaling factor in the legends. The black points in the upper pads show the
asimov data which is just the sum of all background events with uncertainties
completely suppressed for better illustration. The ratio of the asimov data
and the total background is shown in the lower pads, where the error bars on
the black points represent the expected data uncertainty while the green bands
represent the total MC statistical uncertainty.

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