[1]Marek Tasevsky

[1]Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2,
Prague, 18221, Czech Republic

2]Department of Physics, University of Durham, Durham, DH1 3LE, United Kingdom

3]Petersburg Nuclear Physics Institute, NRC “Kurchatov Institute”, Gatchina,
St. Petersburg, 188300, Russia

# Searches for QCD instantons with forward proton tagging

<EMAIL_ADDRESS>Valery Khoze<EMAIL_ADDRESS>Daniel Milne
<EMAIL_ADDRESS>Michail Ryskin<EMAIL_ADDRESS>*  [  [

###### Abstract

We study the possibility to observe heavy ($M_{\rm inst}>60$ GeV) QCD
instantons at the LHC in events with one or two tagged leading protons
including fast simulation of detector and pile-up effects. We show that the
expected instanton signal in a single-tagged configuration is strongly
affected by central detector and pile-up effects. For double-tagged approach,
where larger integrated luminosities and hence larger pile-up contaminations
need to be considered, the combinatorial background overwhelms the expected
signal. We suggest that additional time information about tracks at central
and forward rapidities would be crucial for potential improvements.

###### keywords:

Quantum Chromodynamics, Instanton, Diffraction, Large Rapidity Gap, Forward
Proton Tagging

##  1 Introduction

Instantons are non-perturbative classical solutions of Euclidean equations of
motion in non-abelian gauge theories Belavin:1975fg . In the semi-classical
limit, instantons describe quantum tunneling between different vacuum sectors
of the theory tHooft:1976snw ; Callan:1976je ; Jackiw:1976pf . They are either
directly responsible for generating or at least contributed to many key
aspects of non-perturbative low-energy dynamics of strong interactions
tHooft:1986ooh ; Callan:1977gz ; Novikov:1981xi ; Shuryak:1982dp ;
Diakonov:1984vw ; Schafer:1996wv . These include the role of instantons in the
breaking of the $U(1)_{A}$ symmetry and the spontaneous breakdown of the
chiral symmetry, the formation of quark and gluon condensates,
$<0\lvert\bar{q}q\rvert 0>$ and $<0\lvert G^{a}_{\mu\nu}G^{a}_{\mu\nu}\rvert
0>$ and so on.

Unfortunately, up to now, the QCD instanton has never been observed
experimentally. 111The QCD instanton production in inclusive events at the LHC
was considered in Refs. Khoze:2019jta ; Khoze:2020tpp ; Amoroso:2020zrz . The
problem is that the large-size instanton is very challenging to distinguish
from various possible soft QCD contributions, while the cross-section of the
heavy (small-size) instanton production is exponentially suppressed by the
$e^{-2S_{I}}$ factor, where the corresponding to instanton action
$S_{I}=2\pi/\alpha_{s}$. Nevertheless, it would be very appealing to observe
the small-size instanton signal since in such a case, the uncertainties due to
the soft QCD and other non-perturbative effects are better controllable.

The characteristic signature of small-size instanton is the production of a
large number of isotropically distributed (mini)jets. That is, we are
searching for the high multiplicity events with large (close to 1) sphericity
$S$.

It was shown in Ref. Sas:2021pup  that at high multiplicities the role of the
multiple parton interactions (MPI) strongly increases and the resulting
sphericity of these MPI events becomes also close to one. Therefore, first of
all, we have to suppress the MPI contributions. This can be done by selecting
the events with large rapidity gaps (LRGs). Indeed, it was demonstrated in
Ref. Khoze:2021jkd  that in this case there is a good chance to observe the
instanton signal at the LHC. Recall that the kinematics considered in Ref.
Khoze:2021jkd  corresponds to the LRG events, selected by observing the
leading proton which carries away a very close to 1 ($x_{L}\sim 0.997$)
fraction of its initial momentum in the ALFA or TOTEM detectors
AbdelKhalek:2016tiv ; TOTEM:2008lue . Since the remaining energy is quite
small, the mass of the instanton is not too large: $M_{\rm inst}\sim 20-40$
GeV.

In the present paper we discuss the possibility to observe the higher mass
instantons ($M_{\rm inst}>60$ GeV) by tagging the leading protons with the
dedicated forward proton detectors (FPDs): AFP AFP ; Tasevsky:2015xya  on the
ATLAS side or CT-PPS CT-PPS ; CMS:2021ncv  on the CMS side, when the remaining
fraction of beam energy, $\xi=1-x_{L}\sim 0.03$. They were both installed in
Run 2 and first experience shows that they cover a $0.02<\xi<0.15$ region.
They are also equipped by time-of-flight (ToF) detectors with a time
resolution of 10 ps expected to be achieved in Run 4. The goal for the Run 3
data taking is 20 ps.

Due to the strong, $e^{-2\pi/\alpha_{s}}$, suppression of a heavy instanton
amplitude the expected cross-section becomes rather small. Thus, we have to
consider the possibility to work at a large luminosity and account for the
pile-up background. We study the ’one LRG’ kinematics, where only one leading
proton is detected, and the central instanton production, when both leading
protons are observed.

We will follow here and below the results of the previous Khoze:2021jkd ;
Khoze:2021pwd  papers. In section 2 we recall the definition and basic
formulae for the instanton-induced processes. In section 3 the main details of
the generation and selection of instanton events with one or two LRGs are
discussed. Searching strategy and the optimal cuts are described ed in section
4, while the results are presented in section 5. We conclude in section 6.

##  2 The QCD instanton

In QCD, the instanton configuration consists of the gauge field,

$\displaystyle A_{\mu}^{a\,{\rm inst}}(x)$ $\displaystyle=$
$\displaystyle\frac{2\rho^{2}}{g}\frac{\bar{\eta}^{a}_{\mu\nu}(x-x_{0})_{\nu}}{(x-x_{0})^{2}((x-x_{0})^{2}+\rho^{2})}\,,$
(1)

along with the fermion components for light ($m_{f}<1/\rho$) fermions,

$\bar{q}_{Lf}=\psi^{(0)}(x)\,,\quad q_{Rf}=\psi^{(0)}(x)\,.$ (2)

The gauge field $A_{\mu}^{a\,{\rm inst}}$ is the Belavin-Polyakov-Schwartz-
Tyupkin (BPST) instanton solution Belavin:1975fg  of the self-duality
equations in the singular gauge. Here $\rho$ is the instanton size and $x_{0}$
is the instanton position. Constant group-theoretic coefficients
$\bar{\eta}^{a}_{\mu\nu}$ are the ’t Hooft eta symbols defined in Ref.
tHooft:1976snw . The fermionic components $\psi^{(0)}$ are the corresponding
normalised solutions of the Dirac equation
$\gamma^{\mu}D_{\mu}[A_{\mu}^{a\,{\rm inst}}]\psi^{(0)}\,=0$. These are the
fermion zero modes of the instanton. The instanton configuration is a local
minimum of the Euclidean action, and the action on the instanton is given by
$S_{I}=\frac{8\pi^{2}}{g^{2}}=\frac{2\pi}{\alpha_{s}}$.

The instanton configuration in Eq. (1) has a topological charge equal to one
and thus, due to the chiral anomaly, the instanton processes violate
chirality. If the instanton is produced by a two-gluon initial state, the
final state of this instanton-mediated process will have $N_{f}$ pairs of
quarks and anti-quarks with the same chirality,

$g+g\to n_{g}\times g+\sum^{N_{f}}_{f=1}(q_{Rf}+\bar{q}_{Lf})\ ,$ (3)

where $N_{f}$ is the number of light flavours relative to the inverse
instanton size, $m_{f}<1/\rho$. The instanton contribution to the amplitude
for this process comes from expanding the corresponding path integral in the
instanton field background. At leading order in the instanton perturbation
theory, the amplitude takes the form of an integral over the instanton
collective coordinates (see e.g. Refs. Khoze:2019jta ; Khoze:2020tpp  for more
detail)

${\cal A}^{\rm\,L.O.}_{\,2\to\,n_{g}+2N_{f}}=\int
d^{4}x_{0}\int_{0}^{\infty}d\rho\,D(\rho)\,e^{-S_{I}}\,\prod_{i=1}^{n_{g}+2}A_{{\rm
LSZ}}^{\rm inst}(p_{i}\mathchar
24635\relax\;\rho)\,\prod_{j=1}^{2N_{f}}\psi^{(0)}_{{\rm LSZ}}(p_{j}\mathchar
24635\relax\;\rho).$ (4)

The factors of $A_{{\rm LSZ}}^{\rm inst}(p_{i}\mathchar 24635\relax\;\rho)$
and $\psi^{(0)}_{{\rm LSZ}}(p_{j}\mathchar 24635\relax\;\rho)$ are the
standard insertions of the LSZ-reduced instanton fields in the momentum
representation and $D(\rho)$ is given by the known expression for the
instanton density tHooft:1976snw .

From the point of view of Feynman graphs the leading order instanton amplitude
(Eq. (4)) reveals itself as a family of multi-particle vertices (with
different numbers of emitted gluons), integrated over the instanton position
and size. It describes the emission of a large number of gluons, $n_{g}\propto
E^{2}/\alpha_{s}$, together with a fixed number of quarks and anti-quarks, one
pair for each light flavour in accordance with Eq. (3). The semi-classical
suppression factor, $\exp(-S_{I})=\exp(-2\pi/\alpha_{s})$, will be partially
compensated by the growth with jet energy, $E$, of the high multiplicity
cross-section for the process in Eq. (3). The fully factorised structure of
the field insertions on the right-hand side of Eq. (4) implies that at leading
order in instanton perturbation theory there are no correlations between the
momenta of the external legs in the instanton amplitude. The momenta of
individual particles in the final state are mutually independent, apart from
overall momentum conservation.

Thus, to discover the QCD instanton we have to observe in the final state a
multi-particle cluster or a fireball which contains in general a large number
of isotropically distributed gluon (mini)jets accompanied by $N_{f}$ pairs of
light quark jets generated by a subprocess such as in Eq. (3).

It is quite challenging, however, to identify the instanton on top of the
underlying event. Recall, that the instanton is not a particle and there will
be no peak in the invariant mass, $M_{\rm inst}$, distribution 222What we mean
by the instanton mass is the partonic energy $\sqrt{\hat{s}}$ of the initial
2-gluon state in the process in Eq. (3). As we integrate over the Bjorken x
variables when computing hadronic cross-section, we sum over a broad range of
instanton masses. . The mean value of $M_{\rm inst}$ can at least in principle
be “measured" or reconstructed as the mass of the minijet system created by
the instanton fireball in each given event. Talking about the instanton we
actually mean a family of objects of different sizes, $\rho$ and different
orientations in the colour and the Lorentz spaces. The mean value of $M_{\rm
inst}$ depends on $\rho$, increasing when $\rho$ decreases. Since
experimentally it is impossible to measure the instanton size, $\rho$, below
we use the mass $M_{\rm inst}$ to characterise the properties of the instanton
production event.

##  3 Generation of events

As discussed above, we expect a large ‘underlying event’ background. However
the background caused by multi-parton interactions can be effectively
suppressed by selecting events with Large Rapidity Gaps (LRGs) or the leading
proton. Indeed, each additional ’parton-parton$\to$ dijet’ scattering needs
additional energy and is accompanied by the colour flow created by the parton
cascade needed to form the incoming partons.This colour flow produces
secondaries that fill the LRG. The LRG survival probability, $S^{2}$, (i.e.
the probability not to destroy the LRG) is rather small, $S^{2}\leq 0.1$, see
e.g. Ref. Khoze:2017sdd 333The value $S^{2}=0.1$ is consistent with the
experimental results CMS:2020dns ; ATLAS:2015yqo  for diffractive high $E_{T}$
dijet production. Strictly speaking, the gap survival factor $S^{2}$ depends
on the particular process. However, in our case both the instanton and the
high $E_{T}$ dijet are mainly produced in the collision of two virtual
incoming gluons. Moreover, the virtuality of the gluons producing the QCD
instanton is expected to be smaller than that for the case of the dijet
production, corresponding to a somewhat larger impact parameter and relatively
larger $S^{2}$. Therefore, our approach should be considered rather as a
conservative estimate.. Thus, the probability to observe $n$ additional
branches of parton-parton interactions in LRG events is suppressed by the
factor $(S^{2})^{n}$.

For the case of one leading proton, the instanton events are generated as
being produced in the proton-Pomeron collision (see Fig. 1 for illustration).
When the two leading protons are detected, the Pomeron-Pomeron inelastic
collision is considered (see Ref. Khoze:2021pwd  for more details). The
incoming Pomeron parton distribution is taken from the HERA data. In
particular, we use the fit B of H1 collaboration H1:2006zyl . The obtained
cross-section is multiplied by the gap survival probability $S^{2}=0.1(0.05)$
for the one (two) leading proton kinematics. To diminish the possible scale
uncertainty we use the $k_{T}$-factorization approach as it was described in
Khoze:2021pwd . The unintegrated PDFs are calculated based on the LO KMR
prescription Kimber:2001sc . We account for the Mueller form factor
Mueller:1990qa , describing quantum corrections to the gluon-gluon collision,
and for the incoming gluon virtuality, $Q^{2}$, via the corresponding
instanton form factor

$J(\rho Q)~{}=~{}\rho QK_{1}(\rho Q)\ ,$ (5)

where $\rho$ is the instanton radius and $K_{1}$ is the Bessel function.

In this study we work at $\sqrt{s}=14$ TeV and we scrutinize two mass regions,
namely $M_{\rm inst}>60$ GeV and $M_{\rm inst}>100$ GeV, both in the FPD
acceptance $0.02<\xi<0.05$ and for the single-tagged (ST) configuration, the
latter also for the double-tagged (DT) configuration. In total, we then work
with three instanton signal event record samples. All are generated using the
RAMBO algorithm Kleiss:1985gy  at $\xi=0.03$, with proton-Pomeron collision
type for the ST approach ($2.5\cdot 10^{6}$ events for $M_{\rm inst}>60$ GeV
and $5\cdot 10^{5}$ events for $M_{\rm inst}>100$ GeV) and with Pomeron-
Pomeron collision type for the DT approach ($10^{5}$ events). The respective
instanton production cross-sections integrated over the $0.02<\xi<0.05$ region
are 1004.6 pb and 39.6 pb for the proton-Pomeron sample and for $M_{\rm
inst}>60$ GeV and 100 GeV, respectively, and 500 fb for the Pomeron-Pomeron
sample and $M_{\rm inst}>100$ GeV. All samples are showered and hadronised
using PYTHIA 8.2 Sjostrand:2014zea  with initial-state radiation (ISR), final-
state radiation (FSR) and the MPI switched on. For simplicity, in all cases
the instanton signal is generated with a forward leading proton only in one
hemisphere with $p_{z}<0$. For this reason, in the following all studies are
done and all cuts are tailored for one hemisphere. Assuming a full symmetry in
$z$ coordinate, final numbers are then obtained by doubling those from the
studied hemisphere.

For both, ST and DT approaches, we consider two backgrounds, dijet production
in Single-diffractive (SD) and Non-diffractive (ND) interactions, which are by
far most dominant due to resemblance of their final states to the final state
of the signal and due to their huge production cross-sections with respect to
that of the signal. In both, again ISR, FSR and MPI are switched on. For the
SD dijet background, we use the dynamical gap survival approach with MPI
between Pomeron and proton switched on Rasmussen:2015qgr . Production cross-
sections for $\hat{p}_{\rm T,min}>10$ GeV are 80 $\mu$b for SD and 8.64 mb for
ND. At truth level, we have generated roughly $5\cdot 10^{11}$ SD events and
$8.5\cdot 10^{11}$ ND events.

Since the relatively heavy instanton produces a rather large number of jets
with the energy $E_{T}\sim 1/\rho$ we are looking for high multiplicity events
which:

  * •

do not contain very high-$E_{T}$ jets, and

  * •

still have a large density of the transverse energy,
$\sum_{i}dE_{Ti}/d\eta\sim M_{\rm inst}/3$ (the sum is over all secondary
particles in the given $\eta$ interval).

Moreover, since each jet from the instanton cluster contains a leading hadron
we can select events with a large multiplicity (say, $N_{\rm ch}>20$) of
charged particles with $p_{T}>0.5$ GeV in a limited rapidity interval.

Figure 1: Instanton production in a diffractive process with an LRG. The
Pomeron exchange is shown by the thick doubled line. The red bar shows the
range of $\eta$ considered in this paper. $Y$ indicates the incoming proton
position in rapidity. As shown in the diagram secondaries will be produced
also outside this range but they will not be used when calculating $E_{T}$ or
$N_{ch}$.

Since the LHC detectors never cover the whole ($4\pi$) rapidity interval,
there is no chance to adequately measure the value of $M_{\rm inst}$. To
select the events with appropriate $M_{\rm inst}$ we introduce the cut on the
total transverse energy measured within the given rapidity interval
$\sum_{i}E_{Ti}>M_{0}$.

Here we consider the instanton production in the proton-Pomeron collision (in
terms of Regge theory the Pomeron exchange is responsible for the presence of
the LRG) selecting events with a large multiplicity and relatively large
transverse energy 444The idea to search for the instanton in events with very
large multiplicity but not too large transverse energy and the first
evaluation of the instanton cross-section at collider energies were discussed
long ago in Ref. Balitsky:1993sg .. We expect that these events will be more
or less spherically symmetric, that is, in such events there should be a large
probability to observe the sphericity $S$ close to 1. Since we can not observe
the particles in the whole $4\pi$ sphere we consider the “transverse
sphericity" (or cylindricity) defined as
$S_{T}=2\lambda_{2}/(\lambda_{1}+\lambda_{2})$ where $\lambda_{i}$ are the
eigenvalues of the matrix

$S^{\alpha\beta}=\frac{\sum_{i}p^{\alpha}_{i}p^{\beta}_{i}}{\sum_{i}\lvert\vec{p}^{2}_{i}\rvert}\
,$ (6)

and $\lambda_{2}<\lambda_{1}$. Here $p_{i}^{\alpha}$ is the two-dimensional
transverse component of the momentum of the $i$-the particle and we sum over
all particles observed in the event within a given rapidity interval.

##  4 Search strategy

The search for instanton signal in the harsh environment of various
backgrounds consists of two steps. In the first step, we work at generator
level, examine several cut scenarios and select the one giving the best signal
to background (S/B) ratio, which we then call a “golden scenario”. In the
second step, a fast simulation of the detector and pile-up effects is added,
and their impact on the S/B ratio for this golden scenario is investigated.

As reported in publications by ATLAS and CMS where forward proton detectors
AFP and CT-PPS, respectively, were used, the lowest $\xi$ value reachable in
Run 2 was 0.02 and a similar reach is expected for Run 3. Regions of higher
$\xi$ may be contaminated by Reggeon contributions and ND events that survive
the $\xi>0.02$ cut thanks to fluctuations in the hadronization process and the
large cross-section. Therefore, to suppress both these contributions, we
decided to work in a rather narrow $0.02<\xi<0.05$ range which is used as a
baseline acceptance region in all considerations below. We study two
approaches, using single-tagged and double-tagged protons. While in the
former, the selection efficiency is higher but background contamination is
usually higher and one cannot use the ToF detector to suppress pile-up
background, in the latter, all backgrounds are usually better suppressed
including the pile-up by utilizing the ToF detector but we pay for that by
lower selection efficiencies. Nevertheless in the end it depends on production
cross-sections of signal and backgrounds and on the impact of various cut
scenarios for a chosen FPD acceptance and a given luminosity scenario. Since
pile-up effects can easily diminish advantages of the final state of the
signal (including the presence of LRG), we concentrate on low pile-up
scenarios in the case of single-tagged approach and on medium pile-up amounts
in the double-tagged approach.

###  4.1 Combinatorial background

With the increasing average number of pile-up events per bunch crossing,
$\langle\mu\rangle$, the probability to detect a pile-up proton in the FPD
acceptance increases. When overlaid with a hard-scale event (triggered by
various L1 triggers) it forms a combinatorial background which is usually
dangerous due to resemblance of its final state to that of the signal and due
to the fact that both, the pile-up proton seen in FPD and the hard-scale event
can occur much more frequently than the signal. Detailed discussions and
dependences on $\langle\mu\rangle$ of this probability to fake either single-
tagged (ST) or double-tagged (DT) signal in FPDs, and how this background can
be tamed by using time-of-flight (ToF) detectors installed in FPDs at LHC can
be followed in Refs. Cerny:2020rvp ; Tasevsky:2014cpa ; Harland-Lang:2018hmi .
Frequency of both, the fake ST and DT signal in FPD, depends on the
probability to see a proton from minimum bias events in the acceptance of FPD
on one side, $P_{\rm ST}$. For the acceptance considered here, namely
$0.02<\xi<0.05$, $P_{\rm ST}=0.0048$ as predicted by Pythia 8.2 at
$\sqrt{s}=14$ TeV for minimum bias events with ISR, FSR and MPI switched on.
For a given amount of pile-up, $\langle\mu\rangle$, the probability per bunch
crossing to have a pile-up proton in the acceptance of FPD on one side is

$P_{\rm comb}=1-(1-P_{\rm ST})^{\langle\mu\rangle}.$ (7)

This combinatorial factor serves to estimate the total combinatorial
background for both, the ST and DT strategies. The most dangerous situation
for ST occurs when one hard-scale event (for example SD or ND dijets with low
jet $p_{T}$) is overlayed with another soft SD event producing a forward going
proton. For the DT strategy, the danger lies in an overlay of three events:
one hard-scale SD or ND dijet event and two soft SD events each giving a
forward going proton on opposite sides from the IP. The total combinatorial
background is then estimated as a product of the fiducial cross section for
the SD or ND dijet process (i.e. after applying all cuts except the $\xi$
acceptance) times the combinatorial factor $P_{\rm comb}$ in the ST case or
$P_{\rm comb}^{2}$ in the DT case. We remind that the $\xi$ acceptance cut is
applied already on the one soft SD event for ST (two soft SD events for DT),
so it cannot be applied anymore on the hard-scale event.

The values of the combinatorial factor are then 0.48%, 0.96% and 2.38% for
$\langle\mu\rangle$ of 1, 2 and 5, considered for ST, and 0.84% and 4.58% for
$\langle\mu\rangle$ of 20 and 50, considered for DT. The last two can be
further reduced by making use of ToF detectors (we need to detect a proton on
each side in one event). If we assume a resolution of 10 ps, the ToF
suppression of the combinatorial background by a factor of about 18 (16) for
$\langle\mu\rangle$ of 20 and 50, respectively, can be achieved. We also note
that production cross section of the ST (DT) instanton signal is roughly 100
(1000) times smaller than the corresponding instanton signal cross-section in
inclusive events. The reasoning is based on the presence of $S^{2}$ in the
case of diffractively produced instanton and on the ratio of diffractive to
inclusive PDFs being about 10.

###  4.2 Generator level

In addition to the FPD acceptance, the following variables based on properties
of particles measured in the central detector were examined (note that we are
limiting ourselves to positive pseudorapidities because the leading protons in
signal samples are generated only with negative rapidities):

  * •

$N_{\rm ch05}$ = number of charged particles with $p_{T}>0.5$ GeV and detected
in a part of the central tracker $0.0<\eta<2.0$

  * •

$N_{\rm ch20(25,30)}$ = number of charged particles with $0<\eta<2.0$ and
$p_{T}>2.0,2.5$ or 3.0 GeV

  * •

$\sum E_{T}$ = sum of $E_{T}$ of charged particles with $p_{T}>0.5$ GeV and
$0.0<\eta<2.0$

  * •

$N_{\rm ch05fw}$ = number of charged particles with $p_{T}>0.5$ GeV detected
in one half of the forward calorimeter $2.5<\eta<4.9$

  * •

$\sum E_{T}^{\rm fw}$ = sum of $E_{T}$ of charged particles with $p_{T}>0.5$
GeV and $2.5<\eta<4.9$

From each variable we constructed at least one cut as follows: $N_{\rm
ch05}>30(40)$, $N_{\rm ch20(25,30)}=0$, $\sum E_{T}>30(40)$ GeV, $N_{\rm
ch05fw}<6$ and $\sum E_{T}^{\rm fw}<4$ GeV. While vetoing particles with a
relatively high transverse momentum aims at rejecting events with high-$E_{T}$
jets, the cuts on the activity in the forward calorimeter are introduced in
order to suppress a fake signal caused by the MPI. In total we have examined
42 cut scenarios on the proton-Pomeron sample with $M_{\rm inst}>100$ GeV.
From comparing fiducial cross-sections of signal and background for each cut
scenario using the signal sample and a very large SD sample, we conclude that
the best S/B ratio of 2.1 is obtained for two cut scenarios, namely one with

$N_{\rm ch05}>40{\rm\ \ and\ \ }N_{\rm ch25}=0{\rm\ \ and\ \ }\sum E_{T}^{\rm
fw}<4~{}{\rm GeV}$ (8)

and one with the same cuts as in Eq.(8) but with $\sum E_{T}>30$ GeV in
addition. Since the ratio S/B after adding the cut $\sum E_{T}>30$ GeV remains
the same, we drop this second one and call the cuts in Eq.(8) the “golden cut
scenario”. For this scenario the corresponding S/B for the signal sample with
$M_{\rm inst}>60$ GeV is 2.3. When applying the same set of cuts on a large
sample of ND dijet events (over 8.5$\cdot 10^{11}$ events were generated),
none survives. This translates to an estimated fiducial cross-section for ND
background to be smaller than 10.2 fb. The differential cross-section as a
function of transverse sphericity, $S_{T}$, for the signal and SD dijet
background and for the golden cut scenario is shown in Fig. 2. For both
proton-Pomeron signal samples, $M_{\rm inst}>60$ GeV and $M_{\rm inst}>100$
GeV, the signal is clearly seen to be well-separated from the SD background
and peaking at higher $S_{T}$ values as expected. By comparing the left with
the right plot, we can say that the cuts from the golden scenario reduce to a
large extent the contribution from $M_{\rm inst}<100$ GeV.

Figure 2: Differential cross-section as a function of transverse sphericity at
generator level separately for the instanton signal from proton-Pomeron
collisions ($M_{\rm inst}>60$ GeV on the left and $M_{\rm inst}>100$ GeV on
the right) generated by RAMBO and SD dijet background generated by PYTHIA 8.2
for the golden cut scenario (Eq. (8)). Only statistical uncertainties are
plotted.

###  4.3 Detector level

The detector and pile-up effects for events surviving the golden cut scenario
are then studied using Delphes 3.5 fast simulation package Forthomme:2018ecc
with an ATLAS input card. On both, the signal and background samples of events
at generator level, we run Delphes with various amounts of pile-up, defined by
an average number of pile-up events per bunch crossing, $\langle\mu\rangle$.
For each pile-up amount, we are also considering an adequate integrated
luminosity, which leads us to the following working ($\langle\mu\rangle$,$\cal
L$ [fb-1]) points: (0,0.1), (1,0.1), (2,1) and (5,10) for the ST approach and
(20,60) and (50,300) for the DT approach. Given the assumed difficulties to
distinguish the signal from all relevant backgrounds including pile-up, we
rather concentrate on low (in ST approach) and medium (in DT approach) amounts
of pile-up. At the same time, we are convinced that these are conceivable
luminosity scenarios for Run 2 and Run 3.

At detector level, we work with tracks in the central tracker and with
calorimeter clusters in the forward calorimeter. On the sample of signal
events, we first need to tune track selection criteria and cuts that are based
on tracks. Following the track selection procedure used in the analysis of
charged tracks at 13 TeV by ATLAS ATLAS:2016zba , we reduce tracks from
overlaid pile-up events to an acceptable minimum. Although we observe more
than twice as many tracks with $p_{T}>0.5$ GeV for $\langle\mu\rangle=5$ than
for no pile-up, after requiring $(z_{\rm vtx}-z_{\rm trk})\sin\theta<1.5$ mm
and $d_{0}^{\rm trk}<1.5$ mm in addition, the number of tracks with
$p_{T}>0.5$ GeV drops considerably and exceeds the one at zero pile-up by only
2.5%. Here $z_{\rm vtx}$ is the z-coordinate of the primary vertex, $z_{\rm
trk},\theta$ and $d_{0}^{\rm trk}$ are z-coordinate, polar angle and
transverse impact parameter of a given track, respectively. Track
reconstruction efficiencies and resolutions as functions of track $p_{T}$ and
$\eta$ are accounted for via Delphes input card. A nominal track efficiency is
roughly 80% on average for $p_{T}>0.5$ GeV and not close to the tracker edges
(see e.g. Ref. ATLAS:2016vxz , Fig. 13) and this one is applied in all
studies. As a systematic cross-check, we also examine an optimized track
efficiency (see Ref. Schillaci:2021zcr ) which reaches almost 90% on average
for $p_{T}>0.5$ GeV. Our aim is to illustrate the situation after data taking,
thus to estimate real numbers of collected events, therefore we do not correct
for these track reconstruction inefficiencies but rather adapt the first cut
in Eq.(8) to $N_{\rm tr05}>X$ (where $N_{\rm tr05}$ is number of tracks with
$p_{T}>0.5$ GeV and $0<\eta<2.0$ and $X<40$). The energy flow measurement from
Run 1 by ATLAS ATLAS:2012vre  indicates that the third cut in Eq.(8) needs to
be adapted as well. As Fig. 2 a in Ref. ATLAS:2012vre  shows, Pythia 6
predicts a steeper distribution of $\sum E_{T}$ in the forward region
$4.0<\lvert\eta\rvert<4.8$ than data, both at detector level. While for $\sum
E_{T}<$ 3–4 GeV, it overestimates the data, for $\sum E_{T}>$ 3–4 GeV it
underestimates them. To account for this Pythia 6 discrepancy, one would need
to re-weight it by a ratio data to MC at detector level as is done in Ref.
ATLAS:2012vre . But taking into account two small caveats, namely that in our
analysis we work with Pythia 8.2 and in the larger region, $2.5<\eta<4.9$, we
decide to take rather a simplistic approach and adapt the third cut in Eq.(8)
to $\sum E_{T}^{\rm fwcalo}<5$ (or 6) GeV (where the sum runs over clusters in
the forward calorimeter). At the same time, this new threshold should not be
too far from the original 4 GeV threshold used at generator level. The reason
is that clusters in the forward calorimeter lie outside the tracker coverage,
so they cannot be linked with tracks and consequently we do not have
information about which part of cluster energy comes from pile-up. Therefore,
if departing too far from the 4 GeV threshold, we would be picking up an
uncontrollable amount of pile-up. Furthermore, we should also take into
account that detector and pile-up effects may cause migrations of events from
regions below to regions above cut thresholds (for example at generator level,
$N_{\rm ch05}<40$ but pile-up events may cause $N_{\rm tr05}>40$). To this
end, out of many generated ND and SD background events ($2\cdot 10^{11}$ for
SD, $6\cdot 10^{11}$ for ND) we retain subsamples of events surviving cuts
that are more relaxed than those in Eq.(8), namely we apply cuts $N_{\rm
ch05}>30$ and $N_{\rm ch30}=0$ and $\sum E_{T}^{\rm fw}<6$ GeV. Those samples
(about 100 (1500) thousand of SD (ND) events) are then processed with Delphes
and subsequently four samples with different amounts of pile-up events are
created for each instanton mass region and analyzed in detail. For the sake of
completeness, we note that to save computational power, the proton $\xi$ cut
has not been applied even for the zero pile-up scenario and its effect has
been accounted for by scaling the fiducial cross section for SD (ND) dijets by
$P_{\rm ST}(\mu=0)$ = 4.3% (0.015%). These $P_{\rm ST}$ values correspond to
SD (ND) dijet backgrounds with $\hat{p_{\rm T}}>$ 10 GeV. On another large-
statistics SD dijet sample we have applied the proton $\xi$ acceptance cut and
got very similar results as using the SD dijet sample without this
preselection. This confirms the validity of this factorization of the FPD
$\xi$ acceptance cut.

As explained above, to avoid double-counting, the generator-level FPD
$\xi$-acceptance cut obtained from the proton information cannot be applied
for the hard-scale event since it is already applied on overlayed soft pile-up
events. But we can make use of information from the central detector and try
to restrict ourselves to a $\xi$-region similar to that of the applied FPD
acceptance $0.02<\xi<0.05$ at generator level - note that a strict matching is
not possible using tracker information only due to its too narrow
$\eta$-acceptance but the $\xi$ quantity evaluated using the calorimeter
(whose coverage is $\mid\eta\mid<4.9$), $\xi^{\rm calo}$, can bring us much
closer to the generator-level $\xi$-range. The additional cut on $\xi^{\rm
calo}<0.025$ turns out to be reasonably efficient. Following the discussion
above, we adapt the golden cut scenario in Eq.(8) to these two sets of cuts at
detector level:

$N_{\rm tr05}>25{\rm\ \ and\ \ }N_{\rm tr20}=0{\rm\ \ and\ \ }\sum E_{T}^{\rm
fwcalo}<5~{}{\rm GeV\ \ and\ \ }\xi^{\rm calo}<0.025$ (9)

$N_{\rm tr05}>30{\rm\ \ and\ \ }N_{\rm tr25}=0{\rm\ \ and\ \ }\sum E_{T}^{\rm
fwcalo}<5~{}{\rm GeV\ \ and\ \ }\xi^{\rm calo}<0.025$ (10)

where $N_{\rm tr05}$, $N_{\rm tr25}$ and $N_{\rm tr20}$ are numbers of tracks
in the region $0.0<\eta<2.0$ and for $p_{T}>0.5$ GeV, $p_{T}>2.5$ GeV and
$p_{T}>2.0$ GeV, respectively, and $E_{T}^{\rm fwcalo}$ is a sum of $E_{T}$ of
clusters in the forward calorimeter with $p_{T}>0.5$ GeV and $2.5<\eta<4.9$.
The $\xi^{\rm calo}$ quantity is calculated as a sum of $E_{\rm T}e^{-\eta}$
over calorimeter clusters with $E_{\rm T}>0.2$ GeV. The cuts in Eq. (9) (Eq.
(10)) are used as nominal for $M_{\rm inst}>60$ GeV ($M_{\rm inst}>100$ GeV).

##  5 Results

###  5.1 Single tag

To illustrate the situation after data taking at detector level, the expected
event yields for signal generated with $M_{\rm inst}>60$ GeV and $M_{\rm
inst}>100$ GeV together with both backgrounds, the SD dijets and ND dijets,
are shown in Figs. 3 and 4 as functions of $S_{T}$ for four luminosity
scenarios, defined above after applying detector-level cuts defined in Eq.(9)
and Eq.(10), respectively.

Figure 3: Distributions of expected event yields as functions of transverse
sphericity at detector level for instanton signal from proton-Pomeron
collisions generated by RAMBO for $M_{\rm inst}>60$ GeV and backgrounds from
ND dijets and SD dijets generated by PYTHIA 8.2 after applying detector-level
cuts in Eq. (9) for four luminosity scenarios. Only statistical uncertainties
are shown, estimated using expected event numbers from Table 1.

Figure 4: Distributions of expected event yields as functions of transverse
sphericity at detector level for instanton signal from proton-Pomeron
collisions generated by RAMBO for $M_{\rm inst}>100$ GeV and backgrounds from
ND dijets and SD dijets generated by PYTHIA 8.2 after applying detector-level
cuts in Eq. (10) for four luminosity scenarios. Only statistical uncertainties
are shown, estimated using expected event numbers from Table 1.

To estimate the influence of detector effects, we remove the $\xi^{\rm
calo}<0.025$ cut and compare the S/B ratio at zero pile-up with the generator-
level S/B ratio. While the latter amounts to 2.3 (2.1) for $M_{\rm inst}>60$
(100) GeV, the former drops to about 0.6 (0.3) which suggests that the
detector effects are noticeable. To study their origin in more detail, we
investigate event samples without the $\xi^{\rm calo}<0.025$ cut in the
following. For the $M_{\rm inst}>60$ GeV signal, the mean values of $\sum
E_{T}$ in the forward region for these two cases (9.2 GeV vs. 10.1 GeV) do not
differ much hence the track reconstruction inefficiencies and resolutions seem
to be responsible for the observed S/B deterioration. By adding more and more
pile-up, the mean values of $\sum E_{T}^{\rm fwcalo}$ in signal increase to
17, 24 and 37 GeV for $\langle\mu\rangle$ of 1, 2 and 5, respectively, giving
rise to fewer and fewer events surviving the $\sum E_{T}^{\rm fwcalo}$ cut
with increasing pile-up. But the same holds for the SD background, where
corresponding values are 3.5 and 4.1 GeV for generator level and detector
level without pile-up, which then rise with increasing pile-up to 11.5, 18.7
and 41 GeV. The lower mean values for the SD background are due to the pre-
selection procedure applied to the SD sample. We remind that we pre-select SD
events at generator level by cuts defined in Section 4.3 where one of cuts is
$\sum E_{T}^{\rm fw}<6$ GeV. It is also evident that the pile-up effect is
slightly more pronounced for SD background than for the signal.

As demonstrated, however, on the zero pile-up case in Fig. 3 (top left), the
effect of the detector-level $\xi^{\rm calo}<0.025$ cut is remarkable – it
brings the S/B ratio back above 1.0. Thanks to the much bigger resemblance of
the SD background to the signal, it is the SD process which dominates the
total contamination. The situation dramatically changes when combinatorial
effects enter the game: here the ND background dominates thanks to roughly two
orders of magnitude higher cross section than the SD background and to the
fact that the size of the combinatorial factor $P_{\rm comb}$ stays the same
for both ND and SD backgrounds. The mean value of the $\xi^{\rm calo}$
distribution rises with increasing $\langle\mu\rangle$, but more for signal
than for both backgrounds. Convoluted with the increasing combinatorial factor
$P_{\rm comb}$ this leads to a decreasing S/B ratio as the level of pile-up
increases.

The final event yields for signal, SD and ND dijet backgrounds are shown in
Table 1. As explained above, they correspond to doubling those obtained from
analyzing all generated event samples since the signal has been generated with
the forward proton scattered in one hemisphere only and consequently the
selection cuts have been tailored to one hemisphere as well.

For the specific cut scenarios in Eq. (9) and Eq. (10) considered for the two
mass intervals, we observe that the S/B ratio safely exceeds unity when pile-
up is not considered, thanks to the efficient $\xi^{\rm calo}$ cut. If we
require $\xi^{\rm calo}<0.025$, the S/B stays above unity for
$\langle\mu\rangle=1$ and $M_{\rm inst}>60$ GeV, all other luminosity
scenarios give S/B below 1.0. Its value further decreases with increasing
$\langle\mu\rangle$ and reaches minima of 0.3 (0.2) at $\langle\mu\rangle=5$
for $M_{\rm inst}>60$ (100) GeV. It is therefore a clear preference to collect
data at rather low amounts of pile-up, and we believe that a special run with
$\langle\mu\rangle\sim$ 1 and $\cal L\sim$ 0.1 fb-1 is realistic to consider.

($\langle\mu\rangle,\cal L$[fb-1]) |  $M_{\rm inst}>60$ GeV |  $M_{\rm inst}>100$ GeV
---|---|---
(0, 0.1) | 19.0/(0.4+3.5) | 5.8/(0.2+3.5)
(1.0, 0.1) | 8.7/(6.5+0.2) | 3.2/(4.7+0.2)
(2.0, 1.0) | 52.2/(58.1+2.5) | 15.4/(55.3+2.2)
(5.0, 10.0) | 56.2/(205.6+13.3) | 23.8/(137.1+7.6)
Table 1: Summary of event yields after applying cuts in Eq. (9) and Eq. (10)
for the single-tag search approach for $M_{\rm inst}>60$ GeV and $M_{\rm
inst}>100$ GeV, respectively, and for four luminosity scenarios
($\langle\mu\rangle$, $\cal L$). For each scenario, a ratio of number of
signal to background events, $N_{\rm S}/(N_{\rm ND}+N_{\rm SD})$, is shown.

###  5.2 Systematic studies or searching for an optimum working point

In the effort to improve S/B ratios, we investigate variations of the
selection cuts or tracking efficiency. We observe very similar numbers for a
loosened cut $\sum E_{T}^{\rm fwcalo}<6$ GeV. As anticipated, we also study
the effect of a more efficient tracking. This leads to an increase of the
average efficiency from 80% to 90% and hence of the track multiplicity. This
allows us to apply a more strict cut on the track multiplicity for $M_{\rm
inst}>60$ GeV, namely $N_{\rm tr05}>28$, but the S/B ratios appear to be
unchanged. Tightening the track multiplicity cut to $N_{\rm tr05}>26$ and
$N_{\rm tr05}>27$ for $M_{\rm inst}>60$ GeV and the original tracking or to
$N_{\rm tr05}>30$ for the optimal tracking does not change the S/B ratio
significantly either. For $M_{\rm inst}>100$ GeV tightening the track
multiplicity to $N_{\rm tr05}>32$ or $N_{\rm tr05}>35$ increases the S/B ratio
with respect to that achieved for the scenario in Eq. (10), however, since
very few signal events survive, statistical significances would be rather
small thus do not make this option viable.

The numbers in Table 1 correspond to the whole $S_{T}$ spectrum shown in Fig.
3. Restricting ourselves to the region where the instanton signal is expected,
namely $S_{T}>0.5$, would in general lead to an increase of S/B by roughly
30%, only for the highest pile-up point the increase is above 100%. The
improvement of significances is rather modest.

###  5.3 Double tag

As discussed in Khoze:2021pwd , the central instanton production processes
have some promising advantages. Thus, in such a case, the Pomeron-Pomeron
colliding energy is relatively low, which strongly reduces the multiplicity of
the background underlying events. Moreover, in this energy range, the central
detector becomes almost hermetic (close to $4\pi$) for the Pomeron-Pomeron
secondaries, and only a small part of the finally produced hadrons will avoid
detection.555The idea to observe instantons in a 2-Pomeron collision was first
put forward in Ref Shuryak:2003xz  in the context of Pomeron collisions at
very low invariant mass, $2<M_{\rm inst}<5$ GeV.. Note also that detecting two
outgoing protons would allow one to place an upper limit on the instanton
mass.

For the double-tag approach, we use the signal sample generated using Pomeron-
Pomeron collisions. We studied a limited set of cut scenarios which are close
to the golden one (Eq.(8)) and observe selection efficiencies that are in the
ballpark with those obtained on the proton-Pomeron signal sample. Given about
80 times smaller production cross-section compared to the proton-Pomeron case,
that would give fiducial cross-sections of the order of fractions of
femtobarns. We have to consider higher values of $\langle\mu\rangle$ with
correspondingly larger integrated luminosities, for example
($\langle\mu\rangle$, $\cal L$ [fb-1]) = (20, 60) and (50, 300). This would
lead to increasing the event yields for all, signal and backgrounds, by a
factor of 60 or 300 with respect to e.g. the (2,1) scenario considered in the
ST approach.

As explained in Section 4.1, the combinatorial factor $P_{\rm comb}^{2}$ from
the fake protons seen in the acceptance of FPDs due to additional pile-up
collisions is not large, due, mainly, to the narrow $\xi$ range chosen. It can
then be further reduced by utilizing ToF detectors. The ToF detectors with an
assumed 20 ps resolution (which is a goal of AFP ToF in Run 3 and slightly
better than achieved by AFP ToF in Run 2 ToFPUBNote ) would be able to
suppress the combinatorial background by a factor of 9 and 8 for
$\langle\mu\rangle$ of 20 and 50, respectively. Taking into account the values
of $P_{\rm comb}^{2}$ evaluated above (0.84% and 4.58% for $\langle\mu\rangle$
of 20 and 50, respectively), we can make a very rough extrapolation from the
ST results discussed in the previous section to much higher pile-up amounts
considered for the DT analysis. If we for example compare the luminosity
scenarios (1,0.1) for ST and (20,60) for DT, the fiducial cross section of ND
and SD backgrounds would be scaled by a 5 times lower factor (0.84%/9 vs.
0.48%) for DT but the ratio of signal cross sections is 1/80. This leads to a
reduction of the S/B ratio obtained for the (1,0.1) scenario for ST by a
factor of about 16. For the (50,300) scenario in DT, the S/B ratio would even
drop by two orders of magnitude. It should be noted that outlooks for
significances may be more favourable since in both luminosity scenarios for DT
we collect much more statistics than for ST but it has to be added that the
expected effect of such a high level of pile-up on both, signal and
backgrounds, is a considerable drop of selected events. A central instanton
production either in the DT or ST approach deserves a detailed analysis
including the simulation of detector and pile-up effects, and is in our plans
for a future paper.

###  5.4 Potential improvements

Triggers were not discussed in this note and to our knowledge, there are no
dedicated instanton triggers being used by the LHC experiments, so instanton
signals would have to be searched in data collected by other triggers, e.g.
very-low $E_{T}$ jets or minimum bias triggers used to study properties of
charged particles. It would, however, be extremely useful to propose and build
triggers with conditions tailored to collect instanton-enhanced data samples,
e.g. those in Eqs. (8, 9, 10). So far tracking information is not available at
L1 and so are not vertices since vertex reconstruction is a time-consuming
procedure but triggering an instanton-like signal already at L1 would make the
instanton search in collected data more efficient. At HLT, a full-scan
tracking (collecting information about track multiplicities, $\eta$, $p_{T}$
or $\sum E_{T}$) can be obtained in special low pile-up runs. In standard runs
other approaches should be taken, for example to run the full-scan online
tracking in coordination with very low-$E_{T}$ jet or other-soft scale
triggers, getting possibly also the full information about primary vertices.
Tracking in a limited region-of-interest (similar to that in Eq. (9) or Eq.
(10)) should be possible at HLT too.

As we have discussed in Section 4.3, pile-up increases enormously $\sum E_{T}$
in the forward region ($2.5<\lvert\eta\rvert<4.9$), according to Delphes with
the ATLAS input card, from 9 GeV at zero pile-up up to 37 GeV at
$\langle\mu\rangle$ of 5 for signal. This is because the tracking information
is missing there so we do not have control over what fraction of energy comes
from pile-up. One can of course require just one primary vertex which would
greatly suppress pile-up effects but would drastically reduce available
statistics which is not affordable. Alternatively, adding time information of
individual calorimeter cells, if possible already in Run 2 and Run 3, should
help to identify those coming from pile-up vertices distant from the primary
one. This should be much improved in Run 4 where both, trackers in ATLAS as
well as CMS will be upgraded to cover a region $\lvert\eta\rvert<4.9$ and time
information of tracks should be available as well CMSMIP ; ATLASHGTD . This
time information about tracks in the central detector, whether alone or
together with the ToF information about the leading forward proton on one side
from the interaction point will not only help in searches for the proton-
Pomeron signal but could also potentially allow us to use single-tagged events
to separate the Pomeron-Pomeron-induced instanton signal from all backgrounds
(in this case the undetected proton will often have $\xi<0.02$, below the
current FPD acceptance). Indeed this combined time information and the
increased selection efficiency expected for single-tagged events promises to
significantly increase the S/B ratio compared to double-tagged approach for
the search for instantons produced in the Pomeron-Pomeron collisions.

Next, we can exploit the fact that the density of secondary particles ${\rm
d}N_{\rm ch}/{\rm d}\eta$ reaches its maximum at the rapidity $\eta=\eta_{\rm
inst}$ equal to the instanton rapidity. Indeed, in terms of $\eta$ the
spherical distribution caused by the instanton decay reads

${\rm d}N_{\rm ch}/{\rm d}\eta\propto 1/\cosh^{2}(\eta-\eta_{\rm inst}).$

In the present study we chose the interval $0<\eta<2$ which, on average,
corresponds to this maximum, however, the statistical significance of the
result may be improved if the $N_{\rm ch}$ and $\sum E_{T}$ cuts are imposed
in the central detector within the $\eta_{0}<\eta<\eta_{0}+2$ interval, where
the particle density is maximal in each particular event. We are planning to
implement this idea in future studies.

Finally we remind that the decay of the instanton produces one additional pair
of each flavour of light ($m_{f}<1/\rho$) quark Khoze:2019jta ;
Amoroso:2020zrz ; Khoze:2021jkd . So in the case of the signal, we expect to
observe a larger number of strange and charm particles than in background
events. While this fact has not been examined in this study, we believe it has
potential to improve the S/B ratio.

##  6 Conclusions

In Ref. Khoze:2021jkd ; Khoze:2021pwd  it was proposed to search for QCD
instantons at the LHC in events with Large Rapidity Gaps. Here we investigate
the search strategy for the case of heavy instanton by tagging leading protons
with dedicated AFP or CT-PPS forward proton detectors at $\sqrt{s}=14$ TeV.
Since the expected cross-sections are quite small we consider scenarios with
relatively large integrated luminosities and, thus, we have to account for
effects of pile-up. In addition, we include fast simulation of central
detector response. For instanton signal, we use a dedicated MC event generator
RAMBO, while the dominant backgrounds, Single-diffractive and Non-diffractive
dijets are estimated using PYTHIA 8.2, all including full underlying event
simulation. The small size (heavy) instanton produces a large number of mini-
jets and large transverse energy in a limited rapidity interval, therefore we
are selecting events with a large multiplicity. To suppress the main
background caused by multiple parton interactions and pile-up events, we
introduce an additional cut, transverse energy in the forward ($2.5<\eta<4.9$)
calorimeter should be less than 5–6 GeV. The combinatorial background, caused
by fake protons from pile-up interactions seen in the acceptance of FPD, is
greatly reduced if we limit the acceptance interval to $0.02<\xi<0.05$. We
investigate two proton tagging strategies, the single-tag and double-tag with
appropriate initial collisions, namely proton-Pomeron and Pomeron-Pomeron,
respectively, both simulated by RAMBO. By concentrating on instanton masses
larger than 60 GeV, we show that by applying appropriate cuts and requiring
one leading proton, we could expect the signal-to-background ratio S/B $>2.3$
at generator level. To keep the pile-up effects in both, the central and
forward parts of the main detector at a tolerable level, we choose to work at
a relatively low pile-up rate, with $\langle\mu\rangle<5$. We show that the
detector and pile-up effects are kept under control and consequently the S/B
ratio kept well above unity if $\langle\mu\rangle\lesssim 1$ and integrated
luminosity is around 0.1 fb-1.

The kinematics with two tagged leading protons have some advantages. In this
case, the central detector becomes almost hermetic for the Pomeron-Pomeron
collision. Unfortunately, the expected cross-section becomes about 80 times
smaller compared to proton-Pomeron collisions. That is, we have to consider a
larger $\langle\mu\rangle=$ 20–50. In such a case, even with a good forward
proton timing resolution the combinatorial background caused by the non-
diffractive events, accompanied by leading protons from two other soft events
turns out to be too large to reach advantageous S/B ratios, although a
detailed analysis including detector and pile-up effects is necessary to make
firm conclusions.

For any instanton searches in both, the proton-Pomeron and Pomeron-Pomeron
collisions, the additional time information about tracks in the central and
forward rapidities seems to be beneficial.

## Acknowledgments

Computational resources were provided by the Computing Center Adam:2020qor
and by CEICO (Central European Institute of Cosmology) both at the Institute
of Physics of the Czech Academy of Sciences. The authors are also grateful to
Valya Khoze for valuable discussions and encouragement. DLM is supported by an
STFC studentship. MT is supported by the Ministry of education, youth and
sport of the Czech Republic within the project LTT17018.

## References

  *   * (1) Belavin, A.A., Polyakov, A.M., Schwartz, A.S., Tyupkin, Y.S.: Pseudoparticle Solutions of the Yang-Mills Equations. Phys. Lett. B 59, 85–87 (1975). https://doi.org/10.1016/0370-2693(75)90163-X
  * (2) ’t Hooft, G.: Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle. Phys. Rev. D 14, 3432–3450 (1976). https://doi.org/10.1103/PhysRevD.14.3432. [Erratum: Phys.Rev.D 18, 2199 (1978)]
  * (3) Callan, C.G. Jr., Dashen, R.F., Gross, D.J.: The Structure of the Gauge Theory Vacuum. Phys. Lett. B 63, 334–340 (1976). https://doi.org/10.1016/0370-2693(76)90277-X
  * (4) Jackiw, R., Rebbi, C.: Vacuum Periodicity in a Yang-Mills Quantum Theory. Phys. Rev. Lett. 37, 172–175 (1976). https://doi.org/10.1103/PhysRevLett.37.172
  * (5) ’t Hooft, G.: How Instantons Solve the U(1) Problem. Phys. Rept. 142, 357–387 (1986). https://doi.org/10.1016/0370-1573(86)90117-1
  * (6) Callan, C.G. Jr., Dashen, R.F., Gross, D.J.: Toward a Theory of the Strong Interactions. Phys. Rev. D 17, 2717 (1978). https://doi.org/10.1103/PhysRevD.17.2717
  * (7) Novikov, V.A., Shifman, M.A., Vainshtein, A.I., Zakharov, V.I.: Are All Hadrons Alike? . Nucl. Phys. B 191, 301–369 (1981). https://doi.org/10.1016/0550-3213(81)90303-5
  * (8) Shuryak, E.V.: The Role of Instantons in Quantum Chromodynamics. 2. Hadronic Structure. Nucl. Phys. B 203, 116–139 (1982). https://doi.org/10.1016/0550-3213(82)90479-5
  * (9) Diakonov, D., Petrov, V.Y.: CHIRAL CONDENSATE IN THE INSTANTON VACUUM. Phys. Lett. B 147, 351–356 (1984). https://doi.org/10.1016/0370-2693(84)90132-1
  * (10) Schäfer, T., Shuryak, E.V.: Instantons in QCD. Rev. Mod. Phys. 70, 323–426 (1998) arXiv:hep-ph/9610451. https://doi.org/10.1103/RevModPhys.70.323
  * (11) Khoze, V.V., Krauss, F., Schott, M.: Large Effects from Small QCD Instantons: Making Soft Bombs at Hadron Colliders. JHEP 04, 201 (2020) arXiv:1911.09726 [hep-ph]. https://doi.org/10.1007/JHEP04(2020)201
  * (12) Khoze, V.V., Milne, D.L., Spannowsky, M.: Searching for QCD Instantons at Hadron Colliders. Phys. Rev. D 103(1), 014017 (2021) arXiv:2010.02287 [hep-ph]. https://doi.org/10.1103/PhysRevD.103.014017
  * (13) Amoroso, S., Kar, D., Schott, M.: How to discover QCD Instantons at the LHC. Eur. Phys. J. C 81(7), 624 (2021) arXiv:2012.09120 [hep-ph]. https://doi.org/10.1140/epjc/s10052-021-09412-1
  * (14) Sas, M., Schoppink, J.: Event shapes and jets in $e^{+}e^{-}$ and pp collisions. Nucl. Phys. A 1011, 122195 (2021) arXiv:2101.12367 [hep-ph]. https://doi.org/10.1016/j.nuclphysa.2021.122195
  * (15) Khoze, V.A., Khoze, V.V., Milne, D.L., Ryskin, M.G.: Hunting for QCD instantons at the LHC in events with large rapidity gaps. Phys. Rev. D 104(5), 054013 (2021) arXiv:2104.01861 [hep-ph]. https://doi.org/10.1103/PhysRevD.104.054013
  * (16) Abdel Khalek, S., et al.: The ALFA Roman Pot Detectors of ATLAS. JINST 11(11), 11013 (2016) arXiv:1609.00249 [physics.ins-det]. https://doi.org/10.1088/1748-0221/11/11/P11013
  * (17) Anelli, G., et al.: The TOTEM experiment at the CERN Large Hadron Collider. JINST 3, 08007 (2008). https://doi.org/10.1088/1748-0221/3/08/S08007
  * (18) Adamczyk, L., et al.: Technical Design Report for the ATLAS Forward Proton Detector CERN-LHCC-2015-009; ATLAS-TDR-024
  * (19) Tasevsky, M.: Status of the AFP project in the ATLAS experiment. AIP Conf. Proc. 1654, 090001 (2015). https://doi.org/10.1063/1.4916008
  * (20) Albrow, M., et al.: CMS-TOTEM Precision Proton Spectrometer. Technical Report CERN-LHCC-2014-021. TOTEM-TDR-003. CMS-TDR-13, CERN, Geneva (Sep 2014). https://cds.cern.ch/record/1753795
  * (21) Albrow, M., et al.: The CMS Precision Proton Spectrometer at the HL-LHC – Expression of Interest (2021) arXiv:2103.02752 [physics.ins-det]
  * (22) Khoze, V.A., Khoze, V.V., Milne, D.L., Ryskin, M.G.: Central instanton production. Phys. Rev. D 105(3), 036008 (2022) arXiv:2111.02159 [hep-ph]. https://doi.org/10.1103/PhysRevD.105.036008
  * (23) Khoze, V.A., Martin, A.D., Ryskin, M.G.: Multiple interactions and rapidity gap survival. J. Phys. G 45(5), 053002 (2018) arXiv:1710.11505 [hep-ph]. https://doi.org/10.1088/1361-6471/aab1bf
  * (24) Sirunyan, A.M., et al.: Measurement of single-diffractive dijet production in proton-proton collisions at $\sqrt{s}=$ 8 TeV with the CMS and TOTEM experiments. Eur. Phys. J. C 80(12), 1164 (2020) arXiv:2002.12146 [hep-ex]. https://doi.org/10.1140/epjc/s10052-020-08562-y. [Erratum: Eur.Phys.J.C 81, 383 (2021)]
  * (25) Aad, G., et al.: Dijet production in $\sqrt{s}=$ 7 TeV $pp$ collisions with large rapidity gaps at the ATLAS experiment. Phys. Lett. B 754, 214–234 (2016) arXiv:1511.00502 [hep-ex]. https://doi.org/10.1016/j.physletb.2016.01.028
  * (26) Aktas, A., et al.: Measurement and QCD analysis of the diffractive deep-inelastic scattering cross-section at HERA. Eur. Phys. J. C 48, 715–748 (2006) arXiv:hep-ex/0606004. https://doi.org/10.1140/epjc/s10052-006-0035-3
  * (27) Kimber, M.A., Martin, A.D., Ryskin, M.G.: Unintegrated parton distributions. Phys. Rev. D 63, 114027 (2001) arXiv:hep-ph/0101348. https://doi.org/%****␣inst.bbl␣Line␣450␣****10.1103/PhysRevD.63.114027
  * (28) Mueller, A.H.: First Quantum Corrections to Gluon-gluon Collisions in the One Instanton Sector. Nucl. Phys. B 348, 310–326 (1991). https://doi.org/10.1016/0550-3213(91)90521-X
  * (29) Kleiss, R., Stirling, W.J., Ellis, S.D.: A New Monte Carlo Treatment of Multiparticle Phase Space at High-energies. Comput. Phys. Commun. 40, 359 (1986). https://doi.org/10.1016/0010-4655(86)90119-0
  * (30) Sjöstrand, T., Ask, S., Christiansen, J.R., Corke, R., Desai, N., Ilten, P., Mrenna, S., Prestel, S., Rasmussen, C.O., Skands, P.Z.: An introduction to PYTHIA 8.2. Comput. Phys. Commun. 191, 159–177 (2015) arXiv:1410.3012 [hep-ph]. https://doi.org/10.1016/j.cpc.2015.01.024
  * (31) Rasmussen, C.O., Sjöstrand, T.: Hard Diffraction with Dynamic Gap Survival. JHEP 02, 142 (2016) arXiv:1512.05525 [hep-ph]. https://doi.org/10.1007/JHEP02(2016)142
  * (32) Balitsky, I.I., Ryskin, M.G.: The Possibility of experimental observation of the QCD instanton. Phys. Atom. Nucl. 56, 1106–1109 (1993)
  * (33) Černý, K., Sýkora, T., Taševský, M., Žlebčík, R.: Performance studies of Time-of-Flight detectors at LHC. JINST 16(01), 01030 (2021) arXiv:2010.00237 [hep-ph]. https://doi.org/10.1088/1748-0221/16/01/P01030
  * (34) Tasevsky, M.: Review of Central Exclusive Production of the Higgs Boson Beyond the Standard Model. Int. J. Mod. Phys. A 29, 1446012 (2014) arXiv:1407.8332 [hep-ph]. https://doi.org/10.1142/S0217751X14460129
  * (35) Harland-Lang, L.A., Khoze, V.A., Ryskin, M.G., Tasevsky, M.: LHC Searches for Dark Matter in Compressed Mass Scenarios: Challenges in the Forward Proton Mode. JHEP 04, 010 (2019) arXiv:1812.04886 [hep-ph]. https://doi.org/10.1007/JHEP04(2019)010
  * (36) Forthomme, L.: CepGen – A generic central exclusive processes event generator for hadron-hadron collisions. Comput. Phys. Commun. 271, 108225 (2022) arXiv:1808.06059 [hep-ph]. https://doi.org/10.1016/j.cpc.2021.108225
  * (37) Aaboud, M., et al.: Charged-particle distributions at low transverse momentum in $\sqrt{s}=13$ TeV $pp$ interactions measured with the ATLAS detector at the LHC. Eur. Phys. J. C 76(9), 502 (2016) arXiv:1606.01133 [hep-ex]. https://doi.org/10.1140/epjc/s10052-016-4335-y
  * (38) Aad, G., et al.: Measurement of the charged-particle multiplicity inside jets from $\sqrt{s}=8$ TeV $pp$ collisions with the ATLAS detector. Eur. Phys. J. C 76(6), 322 (2016) arXiv:1602.00988 [hep-ex]. https://doi.org/10.1140/epjc/s10052-016-4126-5
  * (39) Schillaci, Z.M.: Improvements to ATLAS Inner Detector Track reconstruction for LHC Run-3. EPJ Web Conf. 251, 03048 (2021). https://doi.org/10.1051/epjconf/202125103048
  * (40) Aad, G., et al.: Measurements of the pseudorapidity dependence of the total transverse energy in proton-proton collisions at $\sqrt{s}=7$ TeV with ATLAS. JHEP 11, 033 (2012) arXiv:1208.6256 [hep-ex]. https://doi.org/10.1007/JHEP11(2012)033
  * (41) Shuryak, E., Zahed, I.: Semiclassical double pomeron production of glueballs and eta-prime. Phys. Rev. D 68, 034001 (2003) arXiv:hep-ph/0302231. https://doi.org/10.1103/PhysRevD.68.034001
  * (42) Adamczyk, L., et al.: Performance of the ATLAS Forward Proton Time-of-Flight Detector in 2017. Technical report, CERN, Geneva (2021). https://cds.cern.ch/record/2749821
  * (43) Apresyan, A., et al.: Technical proposal for a MIP timing detector in the CMS experiment Phase 2 upgrade. Technical report, CERN, Geneva (2017). https://doi.org/10.17181/CERN.2RSJ.UE8W. https://cds.cern.ch/record/2296612
  * (44) Allaire, C., et al.: A High-Granularity Timing Detector (HGTD) in ATLAS : Performance at the HL-LHC. Technical report, CERN, Geneva (2018). https://cds.cern.ch/record/2302827
  * (45) Adam, M., et al.: Distributed resources of Czech WLCG Tier-2 center. EPJ Web Conf. 245, 03034 (2020). https://doi.org/10.1051/epjconf/202024503034