2022 126 09 June

${\rm f}_{0}(980)$ production in pp collisions

ALICE CollaborationALICE Collaboration

The measurement of the production of ${\rm f}_{0}(980)$ in inelastic pp
collisions at $\sqrt{s}=5.02$ TeV is presented. This is the first reported
measurement of inclusive ${\rm f}_{0}(980)$ yield at LHC energies. The
production is measured at midrapidity, $\left|y\right|<0.5$, in a wide
transverse momentum range, $0<p_{\rm T}<16$ GeV/$c$, by reconstructing the
resonance in the ${\rm f}_{0}(980)$ $\rightarrow\pi^{+}\pi^{-}$ hadronic decay
channel using the ALICE detector. The $p_{\rm T}$-differential yields are
compared to those of pions, protons and $\phi$ mesons as well as to
predictions from the HERWIG 7.2 QCD-inspired Monte Carlo event generator and
calculations from a coalescence model that uses the AMPT model as an input.
The ratio of the $p_{\rm T}$-integrated yield of ${\rm f}_{0}(980)$ relative
to pions is compared to measurements in $\mathrm{e}^{+}\mathrm{e}^{-}$and pp
collisions at lower energies and predictions from statistical hadronisation
models and HERWIG 7.2. A mild collision energy dependence of the ${\rm
f}_{0}(980)$ to pion production is observed in pp collisions from SPS to LHC
energies. All considered models underpredict the $p_{\rm T}$-integrated ${\rm
f}_{0}(980)$/($\pi^{+}$ +$\pi^{-}$) ratio. The prediction from the canonical
statistical hadronisation model assuming a zero total strangeness content of
${\rm f}_{0}(980)$ is consistent with the data within 1.9$\sigma$ and is the
closest to the data. The results provide an essential reference for future
measurements of the particle yield and nuclear modification in p–Pb and Pb–Pb
collisions, which have been proposed to be instrumental to probe the elusive
nature and quark composition of the ${\rm f}_{0}(980)$ scalar meson.

##  1 Introduction

The conventional picture for the classification of hadrons is based on the
constituent quark model introduced in the 1960s [1], in which the observed
mesons and baryons are described as colourless $\mathrm{q\overline{q}}$ and
$\mathrm{qqq}$ bound states, respectively. Most of the known observed states
fit into the quark model picture. At the same time, there are states whose
quantum numbers are known but their mass and width have not been measured, and
observed resonances whose properties suggest an exotic structure [2]. One
remarkable case is that of the light scalar mesons, light-flavoured states
with spin zero, positive parity and charge ($J^{PC}=0^{++}$) and masses below
2 GeV/$c^{2}$, whose identification represents a long-standing puzzle in
particle physics [3, 4, 5, 6, 7, 8]. From a theoretical point of view, the
structure of these states is highly debated [9]: light scalar mesons could be
conventional $\mathrm{q\overline{q}}$ mesons, or compact
(qq)($\mathrm{\overline{q}\overline{q}}$) structures (tetraquarks), or
meson–meson bound states in the form of hadronic molecules, or a superposition
of all these components.

From an experimental point of view, light scalar resonances are typically
reconstructed via their dominant decay channels into pseudoscalar mesons
(e.g., $\pi\pi$, $\eta\pi$, $\eta\eta$…). The states decaying into pions, in
particular, have large characteristic decay widths, of the order of few tens
to few hundreds of MeV/$c^{2}$, due to the large available phase space.
Therefore, the isolation of the particle signals is particularly challenging
as broad signals strongly overlap. In addition, for some of the scalar meson
states, different decay channels can open up within a short mass interval and
distort the line shapes of the nearby resonances.

Among the scalar mesons, the ${\rm f}_{0}(980)$ state is particularly
interesting for two reasons. First, despite a long history of experimental and
theoretical studies, its nature is still controversial as the properties of
the ${\rm f}_{0}(980)$ state are compatible with a conventional
$\mathrm{q\overline{q}}$ meson [10], a tetraquark [11], and a
$\mathrm{K\overline{K}}$ molecular [12] structure. Secondly, the ${\rm
f}_{0}(980)$ represents an interesting probe of the high-density hadronic
final state of heavy-ion collisions and in-medium particle formation
mechanisms [13].

The ${\rm f}_{0}(980)$ couples predominantly to the $\pi\pi$ and
$\mathrm{K\overline{K}}$ channels and the background is mainly given by the
f0(500) and the f0(1270), among the scalar mesons. An indication in favour of
the tetraquark structure of ${\rm f}_{0}(980)$ [14] comes from measurements of
the $\phi$ meson radiative decays by SND [15], CMD2 [16], and KLOE [17, 18]
experiments. This is further supported by a recent analysis [11] of the
$J/\psi$ radiative decay data from BESIII [11, 19]. The ${\rm f}_{0}(980)$ is
also prominently produced in D${}^{+}_{\rm s}$ decays as reported by the E791
collaboration [20], and observed in weak decays of B and Bs mesons measured
with LHCb [21, 22]. There, the appearance of the ${\rm f}_{0}(980)$ in
competition with the $\phi$ meson in these decays could be explained by a
large $\mathrm{s\overline{s}}$ component of this state, combined with the fact
that the $\rm{c}\rightarrow\rm{s}$ coupling is Cabibbo favoured. In this
scenario, the structure of the ${\rm f}_{0}(980)$ would be $|{\rm
f}_{0}(980)\rangle=|(\mbox{$\mathrm{u\overline{u}}$}+\mbox{$\mathrm{d\overline{d}}$})\mbox{$\mathrm{s\overline{s}}$}\rangle/\sqrt{2}$
[2]. An analysis of the measured couplings of the B and Bs mesons to
$J/\psi+$${\rm f}_{0}(980)$ excluded the tetraquark hypothesis [22], a
conclusion that is however challenged by a different analysis of the same data
[23]. Indications that ${\rm f}_{0}(980)$ could be a $\mathrm{K\overline{K}}$
come instead from the study of pion–pion and kaon–kaon scattering via non-
perturbative QCD methods, which use effective meson-exchange models of the
$\pi\pi$ interaction [24, 25] and study the $\mathrm{K\overline{K}}$
interaction for coupled and single channels in chiral effective theory [26,
12].

In addition to measuring the production rates and branching fractions of ${\rm
f}_{0}(980)$ in $\phi$ and heavy-flavour decays, several authors [7, 27, 28,
29] have proposed to investigate its nature by using heavy-ion collisions and
exploiting the unique production (and decay) environment accessible in these
reactions. In high-energy heavy-ion collisions, two extreme states of matter
are reached one after the other. If enough energy is deposited in the
collision region, the state of deconfined strongly interacting matter called
quark–gluon plasma (QGP) is produced and expands as a nearly perfect liquid
until the temperature reaches the pseudo-critical value of $\approx 155$ MeV
[30] and a transition to confined QCD matter takes place. A hot (T $\approx$
100–150 MeV) and dense gas of interacting hadrons is formed in which
resonances decay and particles interact (pseudo)elastically until they
decouple. At the LHC, the system produced in Pb–Pb collisions decouples after
about 10 fm/$c$ [31] and the production of hadronic resonances with lifetimes
of the order of 1 to 10 fm/$c$ is studied to characterise the hadronic stage
of the collision [32, 33, 34]. With its width between 10 and 100 MeV/$c^{2}$
and a corresponding lifetime of $\approx$ 5–10 fm/$c$, the ${\rm f}_{0}(980)$
is a probe for the dense hadron gas formed in the late stage of heavy-ion
collisions [13].

Measurements of the nuclear modification factor [27], the particle yield per
event [28], and the elliptic flow coefficient [29] have been suggested to
provide insights into the internal structure of the ${\rm f}_{0}(980)$. Models
of hadron formation via recombination (coalescence) [35, 36, 37] of quarks in
the quark–gluon plasma that have been successful in describing LHC data,
indicate that the ${\rm f}_{0}(980)$ production in the intermediate transverse
momentum range (2 $<p_{\rm T}<$ 5 GeV/$c$) is sensitive to the number of
constituent quarks. Theory calculations based on a coalescence model [28] show
that the $p_{\rm T}$-integrated production of ${\rm f}_{0}(980)$ in central
heavy-ion collisions at LHC energies is expected to be two orders of magnitude
lower if the state has a tetraquark structure compared to the results for a
non-exotic diquark structure $\mathrm{q\overline{q}}$, or a hadronic molecule
configuration. On the other hand, the production of a tetraquark state would
be enhanced in heavy-ion collisions with respect to pp collisions at the same
energy in the $\approx$ 2–6 GeV/$c$ momentum range [7, 27]. Measurements of
the nuclear modification factor [7, 27] or of the $p_{\rm T}$-dependent yield
ratio of the ${\rm f}_{0}(980)$ to particles with different (but established)
quark content could therefore shed light on the nature of the state. The
authors of [29] also suggest that the azimuthal production asymmetry in the
${\rm f}_{0}(980)$ momentum distributions, quantified by the elliptic flow
coefficient, could be sensitive to the number of constituent quarks in the
kinematic range in which hadron formation occurs predominantly via quark
recombination (coalescence). A measurement of the ${\rm f}_{0}(980)$
production in pp collisions is necessary for the determination of the nuclear
modification factor and constitutes a reference for the study of the particle
production in heavy-ion collisions.

In this letter, the first measurement of the inclusive production of ${\rm
f}_{0}(980)$ in inelastic pp collisions at the LHC is reported. To provide a
baseline for studies in heavy-ion interactions, the data using collisions at
$\sqrt{s}=5.02$ TeV were analysed, corresponding to the centre-of-mass energy
per nucleon of the p–Pb and Pb–Pb data samples collected during the LHC Run 2.
Measurements of ${\rm f}_{0}(980)$ in p–Pb and Pb–Pb collisions at this energy
will be the subject of future publications. The production of ${\rm
f}_{0}(980)$ is measured at midrapidity, $|y|<0.5$, in a broad transverse
momentum range between 0 and 16 GeV/$c$. An overview of the ALICE experimental
setup is given in Sec. 2, followed by a description of the analysis strategy
in Sec. 3. This includes details on the data sample, the ${\rm f}_{0}(980)$
signal reconstruction, the yield extraction and corrections, and the
systematic uncertainty estimation. Results are discussed in comparison to
lower energy data and theoretical models in Sec. 4, while in Sec. 5 the
conclusions are summarised.

##  2 Experimental setup

The experimental setup and details on the performance of the ALICE detector
are described in Refs. [38, 39]. The ALICE detector consists of a central
barrel with a set of detectors devoted to the reconstruction and
identification of the charged particles, a forward muon spectrometer and a set
of backward and forward systems for triggering and event characterisation
purposes. The central barrel detectors are located inside a solenoidal magnet
that provides a magnetic field of 0.5 T. The main detectors employed for the
analysis presented in this work are the V0, the Inner Tracking System (ITS),
the Time Projection Chamber (TPC), and the Time-of-Flight detector (TOF). The
V0 consists of two scintillator arrays placed on both sides of the interaction
point covering the pseudorapidity regions $2.8<\eta<5.1$ (V0A) and
$-3.4<\eta<-1.7$ (V0C), respectively. The V0 provides the minimum bias trigger
of the experiment and is used for suppressing beam-induced background at the
offline analysis level. The position of the collision vertex and the tracks of
charged particles are reconstructed in the central barrel using the ITS and
the TPC. The ITS is a high-resolution tracker that consists of six cylindrical
layers of silicon detectors. The TPC is a large cylindrical drift detector
covering a radial distance of 85 $<r<$ 247 cm from the beam axis and having
longitudinal dimensions of about -250 $<z<$ 250 cm. The TOF is a large area
array of multigap resistive plate chambers, placed at a radius of about
370–399 cm from the beam line. In the central barrel, charged particles can be
identified via measurements of their specific energy loss,
$\textrm{d}E/\textrm{d}x$, provided by the TPC with a resolution of 5$\%$, and
via their time-of-flight measured by the TOF with a resolution of about 80 ps.

##  3 Data analysis

The measurement of ${\rm f}_{0}(980)$ production is performed using a sample
of minimum bias pp collision events at a centre-of-mass energy of
$\sqrt{s}=5.02$ TeV, collected in the years 2015 and 2017. The minimum-bias
trigger requires at least one hit in both V0A and V0C detectors [40]. The
integrated luminosity after trigger selection is $\approx 21.8$ nb-1. Events
are selected for the analysis if the position of the reconstructed collision
vertex along the beam axis is located within 10 cm from the nominal
interaction point. To reduce the pileup caused by multiple interactions in the
same bunch crossing, a criterion based on the offline reconstruction of
multiple primary vertices in the two innermost layers of the ITS, namely the
Silicon Pixel Detector (SPD) is applied [38]. The rejected events account for
less than 1% of the total events. After applying these selection criteria,
$\approx 9.14\times 10^{8}$ collision events have been analysed.

The ${\rm f}_{0}(980)$ resonance signal is reconstructed via its decay into a
pair of oppositely charged pions, ${\rm f}_{0}(980)\rightarrow\pi^{+}\pi^{-}$.
This requires the reconstruction, selection and identification of pion tracks
in the central barrel of ALICE. To ensure a uniform detector acceptance, only
charged tracks with $p_{\rm T}$ $>$ 0.15 GeV/$c$ and pseudorapidity
$|\eta|~{}<$ 0.8 are considered for the analysis. Track selection criteria are
applied to the charged tracks as in previous works [33, 32] to ensure a good
quality of the reconstruction. To this end, each track in the TPC is required
to have crossed at least 70 readout pad rows out of a maximum possible 159. To
reduce the contamination from secondary particles, tracks are accepted if
their distance of closest approach to the collision vertex in the longitudinal
($d_{\rm z}$) and transverse ($d_{\rm{xy}}$) directions satisfy $d_{\rm z}<2$
and $d_{\rm{xy}}<0.0105$ $+0.0350\times p_{\rm T}^{-1.1}$, where $p_{\rm T}$
and distance are in units of GeV/$c$ and cm, respectively.

The identification of pions is performed using the TPC and the TOF detectors
and criteria based on the difference between the measured and expected signals
for a given particle hypothesis, divided by the resolution
($\sigma_{\rm{TPC}}$, $\sigma_{\rm{TOF}}$). In the TPC, charged particles are
identified as $\pi$ if the measured $\textrm{d}E/\textrm{d}x$ is compatible
with the expected pion mean specific energy loss within two standard
deviations (2$\sigma_{\rm{TPC}}$) over the entire momentum range. If a
measurement of the particle time-of-flight by the TOF is available, a TOF-
based 3$\sigma_{\rm{TOF}}$ selection criterion is applied on top of the TPC-
based one, over the measured momentum range.

###  3.1 Raw yield extraction

The ${\rm f}_{0}(980)$ resonance signal is reconstructed via an invariant mass
analysis by combining oppositely-charged pions within the same event into
pairs and imposing the pair to have a rapidity within the range $|y|~{}<$ 0.5.
To remove the combinatorial background, the like-sign method is employed. The
same-charge pion tracks from the same event are combined into $\pi^{+}\pi^{+}$
and $\pi^{-}\pi^{-}$ pairs. The total like-sign invariant mass distribution is
calculated as the geometric mean of the positively-charged and negatively-
charged pair distributions, as $2\sqrt{N^{++}N^{--}}$, where $N^{++}$ and
$N^{--}$ are the number of $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ pairs,
respectively. The $\pi^{+}\pi^{-}$ and like-sign background invariant mass
distributions are extracted for various intervals of the pair $p_{\rm T}$, and
for each of these, the like-sign background is subtracted from the unlike-sign
pair distribution. After the subtraction of the combinatorial background, the
${\rm f}_{0}(980)$ signal peak, sitting on the right-hand tail of the broad
$\mathrm{\rho}(770)$ meson signal, is visible on top of a residual background.
Two examples of the $\pi^{+}$ $\pi^{-}$ invariant mass distributions after
combinatorial background subtraction are shown in Fig. 1 for a low-$p_{\rm T}$
and for a high-$p_{\rm T}$ interval. With increasing $p_{\rm T}$, the
significance of the $\mathrm{f}_{2}(1270)$ resonance signal increases and the
broad $\mathrm{f}_{2}(1270)$ peak becomes visible on the right side of the
${\rm f}_{0}(980)$ signal. The residual background originates from correlated
$\pi^{+}\pi^{-}$ pairs from mini-jets and from misidentified particles. The
main contributions to the correlated background arise from the decay of the
$\mathrm{\rho}(770)$ and the $\mathrm{f}_{2}(1270)$ resonances into
oppositely-charged $\pi$ pairs. In order to extract the ${\rm f}_{0}(980)$
yields in each $p_{\rm T}$ interval, the distributions are fitted in the
invariant mass interval 0.8 $<M_{\pi\pi}<$ 1.6 GeV/$c^{2}$ with a function
that is the sum of three relativistic Breit-Wigner functions (rBW) describing
the $\mathrm{\rho}(770)$, ${\rm f}_{0}(980)$ and $\mathrm{f}_{2}(1270)$
signals [32, 41, 42], and a residual background. Since the resolution on the
invariant mass is negligible with respect to the natural width of the
considered resonances, the resonance shape can be modelled with a rBR with no
need for any additional Gaussian smearing to account for detector resolution
effects. Each of the rBW functions is defined as

${\rm{rBW}}(M_{\pi\pi})=\frac{AM_{\pi\pi}\Gamma(M_{\pi\pi})M_{0}}{(M_{\pi\pi}^{2}-M_{0}^{2})^{2}+M_{0}^{2}\Gamma^{2}(M_{\pi\pi})}$
(1)

where $\Gamma(M_{\pi\pi})$ is given by

$\Gamma(M_{\pi\pi})=\biggl{[}\frac{(M_{\pi\pi}^{2}-4m_{\pi}^{2})}{(M_{0}^{2}-4m_{\pi}^{2})}\biggl{]}^{(2J+1)/2}\times\frac{\Gamma_{0}M_{0}}{M_{\pi\pi}}.$
(2)

Here, $A$ is the normalisation constant, $M_{0}$ and $\Gamma_{0}$ are the rest
mass and width of the resonance, $m_{\pi}$ is the charged pion mass and the
spin is $J=0$ for ${\rm f}_{0}(980)$, $J=1$ for $\mathrm{\rho}(770)$ and $J=2$
for $\mathrm{f}_{2}(1270)$. The shape of the residual background resembles
that of a Maxwell-Boltzmann distribution and therefore it is fitted with a
similar functional form $f_{\rm bg}(M_{\pi\pi})$ [34]

$f_{\rm
bg}(M_{\pi\pi})=\sqrt{(m_{\pi\pi}-m_{\rm{cutoff}})^{n}}~{}C^{3/2}~{}\exp{[-C(m_{\pi\pi}-m_{\rm{cutoff}})^{n}]},$
(3)

where $m_{\rm{cutoff}}$ is the low-mass cutoff expected to be equal to the
rest mass of the $\pi^{+}\pi^{-}$ pair.

For the extraction of the particle yields, the fits are performed with the
following configuration of the fit parameters. The mass and the width of the
$\mathrm{\rho}(770)$, and the width of the $\mathrm{f}_{2}(1270)$ are fixed to
their vacuum values, $m_{\rho}=775.26$ MeV/$c^{2}$, $\Gamma_{\rho}=149.1$
MeV/$c^{2}$, and $\Gamma_{f_{2}}=186.7$ MeV/$c^{2}$ [2]. The width of the
${\rm f}_{0}(980)$ is fixed to the average value of the range reported in Ref.
[2] that corresponds to $\Gamma_{f_{0}}=0.055$ GeV/$c^{2}$. The masses of the
${\rm f}_{0}(980)$ and the $\mathrm{f}_{2}(1270)$, as well as the
$m_{\rm{cutoff}}$, $C$ and $n$ parameters of $f_{\rm bg}$ are left free. The
fit parameter configuration has been varied to take into account possible
imperfections in the description of the background and signal shapes, as
discussed in Section 3.3.

Figure 1: Left (right) plot shows the invariant mass distribution of
$\pi^{+}$$\pi^{-}$ pairs after like-sign background subtraction in low (high)
transverse-momentum interval in pp collisions at $\sqrt{s}=5.02$ TeV in
$|y|<0.5$. Solid blue curves represent fits with the function shown in Eq. 1
and a residual background shown in Eq. 3. Solid red curve represent ${\rm
f}_{0}(980)$ signal while other dashed curves represent the background
contributions from $\mathrm{\rho}(770)$, $\mathrm{f}_{2}(1270)$ and residual
background.

In Fig. 1, the fit result to the invariant mass distribution of
$\pi^{+}$$\pi^{-}$ pairs after like-sign background subtraction is shown for
two $p_{\rm T}$ intervals, namely $0.6<p_{\rm T}<0.8$ GeV/$c$ and $5<p_{\rm
T}<6$ GeV/$c$.

###  3.2 Yield corrections

In order to obtain the ${\rm f}_{0}(980)$ production yield per unit of
rapidity and $p_{\rm T}$ per inelastic event $\big{(}\frac{1}{N_{\rm
INEL}}\frac{{\rm d}^{2}N}{{\rm d}y{\rm d}p_{\rm T}}\big{)}$, several
correction factors are applied to the raw yields obtained from the fit
procedure in each $p_{\rm T}$ interval according to the following formula

$\frac{1}{N_{\rm INEL}}\frac{{\rm d}^{2}N}{{\rm d}p_{\rm T}{\rm
d}y}=\frac{1}{N_{\rm evt}}\frac{N_{{\rm
f}_{0}(980)\rightarrow\pi\pi}}{{\rm\Delta}p_{\rm
T}{\rm\Delta}y}\frac{\epsilon_{\rm{trig}}~{}\epsilon_{\rm{vtx}}}{A\times\epsilon_{\rm{rec}}}\frac{f_{\rm{sig}}}{{\rm
BR}}.$ (4)

Here, $N_{{\rm f}_{0}(980)\rightarrow\pi\pi}$ is the ${\rm f}_{0}(980)$ raw
yield measured in a given rapidity (${\rm\Delta}y$) and transverse momentum
(${\rm\Delta}$$p_{\rm T}$) interval, $N_{\rm evt}$ is the number of collision
events that satisfy the selection criteria. The minimum-bias trigger
efficiency, the vertex reconstruction efficiency and the signal loss
correction factor are represented by $\epsilon_{\rm{trig}}$,
$\epsilon_{\rm{vtx}}$ and $f_{\rm{sig}}$, respectively. The branching ratio
correction amounts to BR = (46 $\pm$ 6)% [43] assuming dominance of $\pi\pi$
and KK channels. The yields of ${\rm f}_{0}(980)$ are normalised to the number
of inelastic pp collisions with a trigger efficiency correction,
$\epsilon_{\rm{trig}}$ = 0.757 $\pm$ 0.019 [44, 45], which takes into account
the efficiency of the V0-based trigger to select inelastic events. The vertex
reconstruction efficiency in pp collisions at $\sqrt{s}=5.02$ TeV is found to
be $\epsilon_{\rm{vtx}}$ = 0.958 [33]. The $A\times\epsilon_{\rm{rec}}$ factor
corrects for the detector acceptance times the ${\rm f}_{0}(980)$
reconstruction efficiency and is evaluated using a detailed Monte Carlo
simulation of the ALICE detector geometry, material, and response. The pp
collision events are simulated using the PYTHIA 8 event generator [46] with
the addition of the ${\rm f}_{0}(980)$ signals. The generated particles in the
simulation are propagated through the detector using GEANT3 [47]. The
$A\times\epsilon_{\rm{rec}}$ is calculated in the rapidity range $|y|<0.5$ as
a function of $p_{\rm T}$ and is defined as the ratio of the number of
reconstructed and generated ${\rm f}_{0}(980)$. The reconstruction of ${\rm
f}_{0}(980)$ in the simulation is performed using the same event and track
selection criteria as employed for the analysis of the data.
The signal loss correction factor, $f_{\rm{sig}}$, accounts for the fraction
of ${\rm f}_{0}(980)$ signal lost due to trigger inefficiencies and can be
determined as a function of $p_{\rm T}$ using Monte Carlo simulations. Because
a simulation with injected ${\rm f}_{0}(980)$ signals may not lead to a
realistic estimate of this correction factor, the correction is taken to be
the same as for the $\phi$ meson at the same collision energy. The earlier
analysis in [33] showed that this correction does not depend significantly on
the particle mass for resonances decaying strongly into two charged particles.
This factor ranges between 1.07 for 0 $<$ $p_{\rm T}$ $<$ 0.2 GeV/$c$ and 1
for $p_{\rm T}$ $>$ 2.5 GeV/$c$.

###  3.3 Systematic uncertainties

Table 1: Contributions to the relative systematic uncertainty of the $p_{\rm T}$-dependent yield of ${\rm f}_{0}(980)$ in pp collisions at $\sqrt{s}=5.02$ TeV. The uncertainties are given for the lowest and the highest $p_{\rm T}$ intervals of the measured spectrum as well as for one intermediate $p_{\rm T}$ interval. The total uncertainty is obtained as the sum in the quadrature of the individual contributions. Values are expressed in percentage (%).  Source of uncertainty |  $p_{\rm T}$ (GeV/$c$)
---|---
| 0–0.2 | 4–4.5 | 12–16
Yield extraction | 7.1% | 8.8% | 15.3%
Track selection | 9.3% | 2.2% | 2.1%
Global tracking efficiency | 2% | 4% | 4%
Particle identification | 6.8% | 1.5% | 6%
Event selection | 7.6% | 2.1% | 3.3%
Material budget | 5.2% | 0% | 0%
Hadronic interaction | 3.4% | 0% | 0%
Total | 16.8% | 10.2% | 17.4%

The sources of systematic uncertainty in the measurement of the ${\rm
f}_{0}(980)$ yields are summarised in Table 1. These include yield extraction,
track and event selection, global tracking efficiency, particle
identification, the knowledge of the ALICE material budget, and that of the
hadron interaction cross section in the detector material. The estimated
values of the uncertainties are reported in Table 1 for low, intermediate and
high-$p_{\rm T}$ intervals. The systematic uncertainty associated with the
yield extraction arises from the fit procedure and is determined by varying
the fitting range as well as the signal and the residual background fit
parameters. In particular, the width of the ${\rm f}_{0}(980)$ was varied by
sampling the range from 10 to 100 MeV/$c^{2}$ given in [2] and the width of
the $\mathrm{f}_{2}(1270)$ was varied within $\pm 7.5$ MeV/$c^{2}$ that
corresponds to a $\pm 3\sigma$ range of the width value reported in [2]. These
variations result in the largest contribution to the uncertainty on the yield
extraction. The uncertainties due to the yield extraction are $p_{\rm T}$
dependent and vary from 7.1% in the lowest $p_{\rm T}$ interval, to 15.3% in
the highest $p_{\rm T}$ interval of this analysis. The systematic uncertainty
due to the track selection is evaluated by varying a single track selection
criterion at a time in both data and simulation, and by repeating all the
steps of the analysis. This contribution ranges from 9.3% to 2.1% from low to
high $p_{\rm T}$. The difference in the efficiency of the matching of TPC
tracks to ITS clusters (global tracking efficiency) between data and
simulations results in a contribution to the systematic uncertainty of 2–4%
depending on $p_{\rm T}$. The systematic uncertainty associated with the
particle identification is due to an imperfection in the description of the
$\textrm{d}E/\textrm{d}x$ in the TPC-based $n\sigma$ selection in the Monte
Carlo simulation as compared to data. The $n\sigma$ selection is varied in
data and simulation simultaneously to a 3$\sigma_{\rm TPC}$ particle
identification criterion and results in a $p_{\rm T}$-dependent relative
systematic uncertainty of 1.5–6.8%. The choice of the event selection criteria
leads to a systematic uncertainty of 2.1–7.6%. The systematic uncertainty
associated with the signal loss correction is estimated by comparing the
correction for $\phi$ mesons, used as a proxy for ${\rm f}_{0}(980)$, with
that of other light-flavour hadrons and is found to be lower than 1%. Finally,
the uncertainty on the knowledge of the ALICE material budget and that of the
hadron interaction cross section in the detector material leads to a
systematic uncertainty lower than 5.3% and 3.4%, respectively [48, 39, 49].
The total relative systematic uncertainty is obtained as the sum in the
quadrature of these contributions.

##  4 Results and Discussion

The $p_{\rm T}$-differential yield of ${\rm f}_{0}(980)$ for $|y|~{}<~{}0.5$
in inelastic pp collisions at $\sqrt{s}=5.02$ TeV is shown in the upper panel
of Fig. 2. The measurement spans a wide $p_{\rm T}$ range from 0 to 16
GeV/$c$.

Figure 2: The $p_{\rm T}$-differential yield of ${\rm f}_{0}(980)$ in pp
collisions at $\sqrt{s}=5.02$ TeV is compared with predictions from the HERWIG
7.2 event generator [50, 51] and with a coalescence calculation [29] based on
the AMPT model [52]. The statistical and systematic uncertainties on data
(full black markers) are shown as bars and boxes, respectively. The middle and
bottom panels show the model to data ratios. The grey boxes at unity represent
the sum in quadrature of the statistical and systematic uncertainty on the
data. The ratios of $\rm{u\bar{u}s\bar{s}}$ tetraquark and $\rm{K^{+}K^{-}}$
molecule configurations from AMPT model predictions to data are multiplied by
100 to improve visibility. In all three panels, the uncertainties associated
with the models are statistical ones.

The normalisation and branching ratio relative uncertainties on the yields are
independent of $p_{\rm T}$ and amount to 2.5% and 13%, respectively [43, 45].

At present, most of the Monte Carlo generators commonly employed to simulate
pp collisions do not implement the generation of ${\rm f}_{0}(980)$ in their
default configurations. One notable exception is the HERWIG 7.2 event
generator [50, 51]. HERWIG 7.2 is a QCD-inspired Monte Carlo event generator
that includes processes like initial and final state QCD radiation, a
description of the underlying event via an eikonal multiple parton–parton
interaction model, and a cluster hadronisation model for the formation of
hadrons from the quarks and gluons produced in the parton shower. To allow for
the comparison, model calculations have been performed in the same $p_{\rm T}$
intervals of the data. HERWIG fails to reproduce the data quantitatively,
underestimating the measured yields by a factor of about (larger than) two at
low and intermediate (high) $p_{\rm T}$ as it can be seen in the HERWIG/Data
ratio shown in the middle panel of Fig. 2. The model reproduces at least
qualitatively the shape of the $p_{\rm T}$ spectrum in the range $1\leq p_{\rm
T}\leq 4$ GeV/$c$. At $p_{\rm T}\geq 4$ GeV/$c$, HERWIG is not able to
reproduce the data neither qualitatively nor quantitatively.

The data are also compared to a recent coalescence calculation [52, 29] that
uses the AMPT multiphase transport model [52], coupled with a coalescence
afterburner with Gaussian Wigner function to generate ${\rm f}_{0}(980)$ in
three configurations, i.e., as a $\rm{s\bar{s}}$ meson, as a
$\rm{u\bar{u}s\bar{s}}$ tetraquark state, and as a $\rm{K^{+}K^{-}}$ molecule.
The AMPT model contains four main components namely initial conditions,
partonic interactions, conversion from partonic to hadronic matter, and
interactions among hadrons based on a relativistic transport (ART) model [53,
54]. The initial conditions are obtained from the HIJING model [55] and the
partonic interactions are determined according to the Zhang’s Parton Cascade
model [56]. In [29], the authors use the phase-space information of quarks
from this stage to implement quark coalescence for the ${\rm f}_{0}(980)$ with
the $\rm{s\bar{s}}$ and tetraquark configurations. In the default version of
AMPT, the conversion of partons to hadrons is then calculated with the Lund
string fragmentation [57, 58, 59], while in the string melting version of the
model [60], a quark coalescence approach is used to combine partons to form
hadrons. The phase-space information of kaons generated at this stage by AMPT
is used as input for the coalescence afterburner for the ${\rm f}_{0}(980)$
molecular state. As shown in Fig. 2, the $\mathrm{s\overline{s}}$, the
molecule, and the tetraquark configuration predictions underestimate the ${\rm
f}_{0}(980)$ $p_{\rm T}$ distribution by a factor of about three, and by two
and three orders of magnitude, respectively. Note that the molecule and the
tetraquark configuration prediction ratios to data are reported in the lowest
panel of Fig. 2 multiplied by a factor of 100 to improve the visibility. In
addition, the shape of the $p_{\rm T}$ spectra for the $\rm{s\bar{s}}$ and the
$\rm{u\bar{u}s\bar{s}}$ tetraquark configurations are found to be
significantly steeper than the measured one. Instead, the ratio between the
model prediction for the $\rm{K^{+}K^{-}}$ molecule configuration and the data
exhibits a milder $p_{\rm T}$ dependence within uncertainties in the
considered $p_{\rm T}$ range (0–3.5 GeV/$c$), indicating that in this
configuration the model can reproduce qualitatively better the measured
spectral shape.

The per-event $p_{\rm T}$-integrated yield, d$N$/d$y$, and average transverse
momentum, $\langle p_{\mathrm{T}}\rangle$, are calculated by integrating the
$p_{\rm T}$-differential yield in the measured transverse momentum range. The
obtained values are the following:

$\displaystyle\frac{{\rm{d}}N}{{\rm{d}}y}=0.0385\pm
0.0001(\rm{stat.})\pm~{}0.0047(\rm{syst.})$ (5) $\displaystyle\langle
p_{\mathrm{T}}\rangle=0.9624\pm
0.0014(\rm{stat.})\pm~{}0.0357(\rm{syst.})\>\rm{Ge}\kern-1.00006pt\rm{V}/c$
(6)

Notably, the yield for $p_{\rm T}$ $>16$ GeV/$c$ has a negligible contribution
to the d$N$/d$y$ and thus no extrapolation was employed.

Figure 3: (Upper panels) Particle yield ratios of ${\rm f}_{0}(980)$ to
$\pi^{+}+\pi^{-}$ [61] (left panel), $\rm{p}+\overline{\rm{p}}$ [61] (middle
panel), and $\phi$ [33] (right panel) measured in inelastic pp collisions at
$\sqrt{s}=$ 5.02 TeV as a function of $p_{\mathrm{T}}$. Data are compared to
HERWIG 7.2 model predictions. The statistical and systematic uncertainties are
shown as bars and boxes, respectively. (Lower panels) Ratio of measured
particle ratios to the HERWIG model calculations (dashed line). The grey boxes
at unity represent the sum in quadrature of the statistical and systematic
uncertainty on the data. In the right panel, the ratio in the region for
$p_{\rm T}<0.8$ GeV/$c$ is off-scale. The relative uncertainty of 13% due to
the branching ratio correction [43] is shown as a green box with an arbitrary
horizontal width for visibility.

The production of ${\rm f}_{0}(980)$ is compared to that of other light-
flavour hadrons in Fig. 3 where the ratios of the ${\rm f}_{0}(980)$ yields to
those of $\pi^{+}$ +$\pi^{-}$ [61], p+$\rm\overline{p}$ [61], and $\phi$ [33]
measured in pp collisions at $\sqrt{s}=5.02$ TeV are reported as a function of
$p_{\rm T}$. The ratio to $\pi^{+}$ +$\pi^{-}$ mesons exhibits an increasing
trend as a function of $p_{\rm T}$ at low $p_{\rm T}$ and for $p_{\rm T}>$ 5
GeV/$c$ it saturates within uncertainties.

The comparison of the production of ${\rm f}_{0}(980)$ to that of protons and
of the $\phi$ meson is particularly interesting as these particles have
similar masses [2] but different quark content. In particular, the $\phi$
meson is a pure $\mathrm{s\overline{s}}$ state, while the ${\rm f}_{0}(980)$
contains a light flavour component ($\mathrm{u\overline{u}}$,
$\mathrm{d\overline{d}}$) as well as a large $\mathrm{s\overline{s}}$
component, as suggested by measurements of ${\rm f}_{0}(980)$ produced in
D${}^{+}_{\rm{s}}$ decays [20]. The ${\rm f}_{0}(980)$ to
$\rm{p}+\overline{\rm{p}}$ ratio shows an increasing monotonic trend as a
function of $p_{\rm T}$, whereas the ${\rm f}_{0}(980)$ to $\phi$ ratio
decreases for $p_{\rm T}<$ 1.5 GeV/$c$, remains flat till $p_{\rm T}\simeq$ 8
GeV/$c$, and increases for $p_{\rm T}>$ 8 GeV/$c$.

The measured $p_{\rm T}$-differential particle yield ratios are compared in
Fig. 3 to the predictions from the HERWIG 7.2 event generator. The shape of
the measured 2${\rm f}_{0}(980)$/($\pi^{+}$ +$\pi^{-}$) ratio is fairly well
reproduced over almost the entire measured $p_{\rm T}$ range, although the
yield is underestimated by about a factor of two by the model. The model
underestimates the 2${\rm f}_{0}(980)$/(p+$\rm\overline{p}$) ratio and fails
to reproduce its $p_{\rm T}$-dependence, with the measured ratio being more
steeply increasing with $p_{\rm T}$ than the predicted one. The trend of ${\rm
f}_{0}(980)$/$\phi$ ratio is flat for $1<p_{\rm T}<10$ GeV/$c$ suggesting that
its $p_{\rm T}$ dependence is qualitatively well reproduced by HERWIG in this
momentum interval. However, the model overestimates the ratio by nearly a
factor of two. For $p_{\rm T}<1$ GeV/$c$, the ${\rm f}_{0}(980)$/$\phi$ ratio
exhibits a steeply decreasing trend, that is qualitatively present also in the
model prediction. At high $p_{\rm T}$, the HERWIG predictions is consistent
with the ${\rm f}_{0}(980)$/$\phi$ data within the uncertainties.

The ratio of the $p_{\rm T}$-integrated ${\rm f}_{0}(980)$ yield relative to
pions in pp collisions at $\sqrt{s}=5.02$ TeV amounts to $2{\rm
f}_{0}/(\pi^{+}+\pi^{-})=(0.0186\pm 0.0026)$, with the uncertainty being the
sum in quadrature of the statistical and systematic uncertainties. The value
is shown in Fig. 4 (red point) in comparison with results from measurements in
pp and $\mathrm{e}^{+}\mathrm{e}^{-}$collisions at lower centre-of-mass
energies as well as with model calculations.

Figure 4: Comparison of the measured $2{\rm f}_{0}/(\pi^{+}+\pi^{-})$ ratio
with measurements in $\mathrm{e}^{+}\mathrm{e}^{-}$collisions at $\sqrt{s}$ =
29 GeV [62], $\sqrt{s}$ = 91 GeV [63] and in pp collisions at $\sqrt{s}$ =
27.5 GeV [64]. The ratios are compared to predictions from statistical
hadronisation model (SHM) calculations for
$\mathrm{e}^{+}\mathrm{e}^{-}$collisions [65] and pp collisions [66], GC-SHM
[67] and HERWIG 7.2 [50, 51]. The hollow boxes represent the total uncertainty
on data. The relative uncertainty of 13% due to the branching ratio correction
[43] applies to all data points and is shown as a yellow box. All error boxes
are drawn with an arbitrary horizontal width for visibility. Figure 5:  The
$2{\rm f}_{0}/(\pi^{+}+\pi^{-})$ ratio measured in pp collisions at
$\sqrt{s}=5.02$ TeV [61] is compared to two distinct predictions for ${\rm
f}_{0}/\pi^{+}$ from a canonical statistical model ($\gamma_{s}-$CSM [68], see
text for details) as a function of
$\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle$. The two calculations
differ by the assumed strangeness content of ${\rm f}_{0}(980)$ and correspond
to zero total strangeness, $|S|=0$ and $|S|=2$. The height of the hollow red
box represents the total uncertainty on the ratio, its width represents the
uncertainty on the $\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle$.
The relative uncertainty of 13% due to the branching ratio correction [43] is
shown as a yellow box with an arbitrary horizontal width for visibility.

The low energy experiment results were originally reported using different
branching ratios, therefore all of them have been updated to take into account
the most recent value of 46% [43] used in this letter. In Fig. 4, the same
uncertainty on the BR is applied to all data points and reported as a shaded
yellow box. The particle ratio value from the fixed-target NA27 experiment at
the CERN SPS, measured in pp collisions at $\sqrt{s}$ = 27.5 GeV [64] is 44.5%
lower than the ratio measured at $\sqrt{s}=5.02$ TeV, suggesting a mild
increase of the ${\rm f}_{0}(980)$ yield relative to pions with increasing
energy of the pp collisions. The particle ratio values from
$\mathrm{e}^{+}\mathrm{e}^{-}$collisions at $\sqrt{s}$ = 29 GeV [62] and
$\sqrt{s}$ = 91 GeV [63] are lower by 61% and higher by a factor of two,
respectively. The particle ratios are compared with predictions based on
statistical hadronisation models [65, 66, 67] and the HERWIG 7.2 event
generator. The statistical hadronisation model predictions by Becattini et al.
for the $\mathrm{e}^{+}\mathrm{e}^{-}$case [65] and for the pp collisions [66]
case underestimate the measurement by about a factor of two, similarly to a
Grand Canonical formulation of the statistical hadronisation model (GC-SHM)
from the GSI-Heidelberg group [67]. The value from HERWIG is also a factor of
about two lower than the measured ratio.

In Fig. 5, the measured $p_{\rm T}$-integrated $2{\rm
f}_{0}/(\pi^{+}+\pi^{-})$ ratio in pp collisions at $\sqrt{s}=5.02$ TeV is
compared to predictions based on the canonical statistical hadronisation model
(CSM) described in [68], as a function of the multiplicity of particles
produced in the collision, expressed in terms of the average pseudorapidity
density of charged particles,
$\langle\rm{d}\it{N}_{ch}/\rm{d}\it{\eta}\rangle$. The prediction spans a
large $\langle\rm{d}\it{N}_{ch}/\rm{d}\it{\eta}\rangle$ interval, reaching the
high multiplicity achieved in central heavy-ion collisions at LHC energies. In
canonical statistical hadronisation models, hadrons are formed from a source
that is assumed to have reached full chemical equilibrium at the chemical
freeze-out temperature $T_{\mathrm{ch}}$, and their yields are determined from
the partition function for a canonical ensemble. The multiplicity dependence
of hadron production is driven by the canonical suppression, namely the exact
conservation of baryon number, electric charge, and strangeness over the
correlation volume. The model considered here, with a temperature
$T_{\mathrm{ch}}=155$ MeV and a correlation volume that spans three units of
rapidity, is able to reproduce the multiplicity dependence of hadron-to-pion
ratios of several species over the charged particle multiplicity range covered
by the ALICE measurements at the LHC, both qualitatively and quantitatively in
most cases (see Fig. 5 of [68]). In addition, to describe the multiplicity
dependence of the $\phi/\pi$ ratio observed at the LHC, the model, henceforth
labelled as $\gamma_{\mathrm{S}}-\mathrm{CSM}$, incorporates the incomplete
equilibration of strangeness by introducing a strangeness saturation factor
$\gamma_{\mathrm{S}}\leq 1$. Notably, in the strangeness nonequilibrium
picture a $\mathrm{s\overline{s}}$ pair like the $\phi$ meson is effectively a
double-strange particle ($|$S$|$ = 2), and ALICE $\phi$ data seem to be best
described with $|$S$|$ = 1–2 [69]. The ${\rm f}_{0}(980)$/$\pi^{+}$ ratio is
calculated for two scenarios, first assuming that the total strangeness
content of ${\rm f}_{0}(980)$ is equal to zero (yellow continuous line) and
second, assuming a total strangeness content equal to two (blue dashed line).
At high multiplicity, where the strangeness content of the system saturates in
presence of a QGP, the calculations for the two scenarios converge and reach
the grand canonical limit.

In the first scenario ($|S|$ = 0), $\gamma_{s}-$CSM predicts higher values for
the ${\rm f}_{0}(980)$ to pion yield ratio as compared to the second scenario
($|S|$ = 2) in the low $\langle\rm{d}\it{N}_{\rm{ch}}/\rm{d}\it{\eta}\rangle$
region. The two predictions match each other for
$\langle\rm{d}\it{N}_{\rm{ch}}/\rm{d}\it{\eta}\rangle$ $\geq$ 100\. The
measured $2{\rm f}_{0}/(\pi^{+}+\pi^{-})$ ratio in pp collisions at
$\sqrt{s}=5.02$ TeV differs by 1.9$\sigma$ from the $\gamma_{s}-$CSM
prediction for the ${\rm f}_{0}(980)$ with net strangeness equal to zero, and
by 4.0$\sigma$ from the $|S|$ = 2 prediction, indicating that the former
scenario is favoured over the latter in this implementation of the model.

##  5 Conclusions

In conclusion, the first measurement of ${\rm f}_{0}(980)$ production in
inelastic pp collisions at $\sqrt{s}=5.02$ TeV at the LHC is presented. The
measurement is performed in a wide $p_{\rm T}$ interval from 0 to 16 GeV/$c$
at midrapidity by reconstructing the resonance in the hadronic decay channel
${\rm f}_{0}(980)$ $\rightarrow\pi^{+}\pi^{-}$. The inclusive ${\rm
f}_{0}(980)$ production is underestimated by HERWIG 7.2 by a factor of two for
$1<p_{\rm T}<4$ GeV/$c$ and by a large factor (up to more than four) in
$4<p_{\rm T}<16$ GeV/$c$. However, this QCD-inspired event generator is able
to describe the $p_{\rm T}$-dependence of the 2${\rm f}_{0}(980)$/($\pi^{+}$
+$\pi^{-}$) and ${\rm f}_{0}(980)$/$\phi$ ratios in a rather broad $p_{\rm T}$
range, while failing to reproduce the 2${\rm
f}_{0}(980)$/(p+$\rm\overline{p}$) ratio over the entire measured $p_{\rm T}$
range. The production of ${\rm f}_{0}(980)$ is also not described by a
AMPT+coalescence model prediction in three configurations ($\rm{s\bar{s}}$
meson, $\rm{u\bar{u}s\bar{s}}$ tetraquark state, and $\rm{K^{+}K^{-}}$
molecule), which use the phase space information of quarks and kaons from the
AMPT model. In order to compare the new measurement of the $p_{\rm
T}$-integrated ${\rm f}_{0}(980)$ to pion ratio with low energy data, the low
energy points were updated with the latest branching ratio. The new result
presented in this letter suggests a mild increase of the production of ${\rm
f}_{0}(980)$ relative to pions in inelastic pp collisions from $\sqrt{s}$ =
27.5 GeV to $\sqrt{s}=5.02$ TeV. For the same ratio, HERWIG 7.2 predicts a
value that is about 43% lower than the measured one whereas different
implementations of the statistical hadronisation model underestimate the data
by up to a factor of about two. Notably, the $\gamma_{s}-$CSM prediction for
the ${\rm f}_{0}(980)$ assuming net strangeness equal to zero is consistent
with the data within 1.9$\sigma$.

In summary, the description of the inclusive ${\rm f}_{0}$ production in pp
collisions provided by the few event generators and theoretical calculations
that attempt its modelling is, at present, unsatisfactory. Future developments
in this direction may help gaining insight over the nature of this particle,
as new data may become available. From the experimental point of view, the
results presented in this letter set the necessary baseline for the future
measurements of the production and the nuclear modification factor of ${\rm
f}_{0}(980)$ in p–Pb and Pb–Pb collisions at the LHC, which have been
suggested as observables that are sensitive to the elusive nature of this
particle.

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graph (‘$\mathcal{P}^{k}\setminus\mathcal{B}_{k}\neq\emptyset$’). We think
that finding a way to instead directly apply the removal lemma is an
interesting problem in its own right. Note that such a direct application of
the removal lemma would likely yield an improvement in the implicit
epsilon–delta quantification of continuity in Theorem 10.7 since it would
allow the use of the bounds famously obtained by Fox [11].

###  10.2. Speed of removal flip processes

Generally, removal flip processes may converge to the destination very slowly.
For example, consider the triangle removal flip process. A well known
construction of Behrend (see [29, Corollary 2.5.2]) implies that for every
$n\in\mathbb{N}$ there exists a graphon $L_{m}$ with
$t(K_{3},L_{m})=O(1/m^{2})$, $\|L_{m}\|_{1}=\exp(-\Theta(\sqrt{\log m}))$, and
whose destination is $\mathrm{dest}(L_{m})=0$. Obviously, for the velocity of
the triangle removal flip process we have, for each $T\geq 0$, that
$\|\mathfrak{V}_{\>}\Phi^{T}L_{m}\|_{1}\leq\|\mathfrak{V}_{\>}L_{m}\|_{1}=6t(K_{3},L_{m})=O(1/m^{2})$.
This example shows that the convergence time777Due to monotonicity,
$\tau_{\mathcal{R}}^{-}=\tau_{\mathcal{R}}^{+}$ for removal processes.
$\tau_{\mathcal{R}}^{-}(\delta)$ cannot be bounded from above by any
polynomial in $1/\delta$. So far the reasoning for the slow speed of
convergence of the triangle removal flip process888which we make formal and
more general in Proposition 10.10 was based on lower bound constructions for
the triangle removal lemma. But a priori we cannot rule out even the case that
$\tau_{\mathcal{R}}^{-}(\delta)=\infty$. This is because the removal lemma
says ‘if the graph is far from all triangle-free graphs then it has many
triangles’ and not ‘if the graph is far from its destination then it has many
triangles’.

Our result shows that the convergence times of each removal rule are finite
and uniformly bounded. Theorem 10.7 plays a key role in the proof.

######  Theorem 10.9.

Let $k\in\mathbb{N}$, $F\in\mathcal{H}_{k}$ and $\mathcal{R}$ be the
$F$-removal rule. Suppose that $e(F)>0$. Then for every $\varepsilon>0$ there
exists $T\geq 0$ so that for every graphon $W$ we have
$\|\Phi^{T}W-\mathrm{dest}(W)\|_{1}<\varepsilon$.

###### Proof.

Suppose that this does not hold. Then there exists $\varepsilon>0$ and a
sequence of graphons $(W_{n})_{n\in\mathbb{N}}$ so that for every
$n\in\mathbb{N}$, we have

(58) $\|\Phi^{n}W_{n}-\mathrm{dest}(W_{n})\|_{1}\geq\varepsilon\;.$

By the Lovász–Szegedy compactness theorem (see [18, Theorem 9.23]) there exist
indices $n_{1}<n_{2}<\cdots$, measure preserving bijections
$\psi_{1},\psi_{2},\ldots$, and a graphon $U$ so that writing
$U_{i}:=W_{n_{i}}^{\psi_{i}}$ we have

(59) $\left\lVert U_{i}-U\right\rVert_{\square}\rightarrow 0\;.$

By Theorem 10.7, we have

(60)
$\left\lVert\mathrm{dest}(U_{i})-\mathrm{dest}(U)\right\rVert_{\square}\rightarrow
0\;.$

Take $T$ such that
$\left\lVert\Phi^{T}U-\mathrm{dest}(U)\right\rVert_{\square}<\varepsilon/4$.
Let $C_{k}$ be as in Theorem 3.8. By (59) and (60) we can choose $\ell\geq T$
such that

(61) $\left\lVert
U_{\ell}-U\right\rVert_{\square}<\varepsilon/(4\exp(C_{k}T))\quad\text{and}\quad\left\lVert\mathrm{dest}(U_{\ell})-\mathrm{dest}(U)\right\rVert_{\square}<\varepsilon/4.$

Since $n_{\ell}\geq\ell\geq T$, we have

$\displaystyle\left\lVert\Phi^{n_{\ell}}U_{\ell}-\mathrm{dest}(U_{\ell})\right\rVert_{1}$
$\displaystyle=\int_{\Omega^{2}}\left(\Phi^{n_{\ell}}U_{\ell}-\mathrm{dest}(U_{\ell})\right)$
$\displaystyle\leq\int_{\Omega^{2}}\left(\Phi^{T}U_{\ell}-\mathrm{dest}(U_{\ell})\right)$
$\displaystyle\leq\left\lVert\Phi^{T}U_{\ell}-\Phi^{T}U\right\rVert_{\square}+\left\lVert\Phi^{T}U-\mathrm{dest}(U)\right\rVert_{\square}+\left\lVert\mathrm{dest}(U_{\ell})-\mathrm{dest}(U)\right\rVert_{\square}$
(61) and Theorem 3.8
$\displaystyle<\frac{\varepsilon}{4}+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}<\varepsilon\;.$

Noting that
$\Phi^{n_{\ell}}U_{\ell}=(\Phi^{n_{\ell}}W_{n_{\ell}})^{\psi_{\ell}}$ and
$\mathrm{dest}(U_{\ell})=(\mathrm{dest}(W_{n_{\ell}}))^{\psi_{\ell}}$ (see
Section LABEL:FIRSTPAPER.ssec:repermuting in [12]), we obtain a contradiction
to (58). ∎

We do not obtain any explicit bounds on the convergence time $T$ in Theorem
10.9. In lieu of the Lovász–Szegedy compactness theorem, one could probably
use the machinery of a regularity lemma (the Frieze-Kannan version seems to be
sufficient) in the proof. Further, to obtain an explicit bound in Theorem
10.9, one would make the $\varepsilon$–$\delta$ quantification of continuity
in Theorem 10.7 explicit as well. Recall that Theorem 10.7 uses Szemerédi’s
regularity lemma (Theorem 2.11) with its infamous tower-type dependency.
Putting them together, we would likely get that
$T\leq\mathsf{tower}((1/\varepsilon)^{O(1)})$; even if we successfully replace
Szemerédi’s regularity lemma with the removal lemma as mentioned in Remark
10.8, we would still get only $T\leq\mathsf{tower}(O(\log(1/\varepsilon)))$.
In our next proposition, however, we prove that a polynomial upper bound
exists if and only if $F$ is bipartite. To this end, some preparatory notation
is needed.

Suppose that $F$ is a bipartite graph. Let $m_{F}$ be the smallest number such
that for each graphon $W$ we have

(62) $t(F,W)\geq\|W\|_{1}^{m_{F}}\;.$

We call $m_{F}$ the _Sidorenko power of $F$_. It is easy to show that $m_{F}$
exists and $m_{F}\in[e(F),(\frac{v(F)}{2})^{2}]$. The Sidorenko conjecture
(see [18]) asserts that $m_{F}=e(F)$ for every bipartite $F$.

######  Proposition 10.10.

Let $k\in\mathbb{N}$, $F\in\mathcal{H}_{k}$ and $\mathcal{R}$ be the
$F$-removal rule. Suppose that $e(F)>0$. Then the following hold.

  1. (i)

If $F$ is bipartite, then the destination of every graphon is the constant-$0$
graphon.

  2. (ii)

Suppose that $F$ is bipartite and its Sidorenko power is $m_{F}$. Then Theorem
10.9 holds with $T=\varepsilon^{-m_{F}+1}$ when $e(F)\geq 2$ and with
$T=\ln(1/\varepsilon)$ when $e(F)=1$.

  3. (iii)

If $F$ is not bipartite then no polynomial bound on $T$ in terms of
$\varepsilon$ exists, that is, for every $M$ there is $\varepsilon_{0}>0$ so
that for all $\varepsilon\in(0,\varepsilon_{0}]$ we have
$\tau_{\mathcal{R}}^{-}(\varepsilon)\geq\varepsilon^{-M}$.

###### Proof.

First suppose that $F$ is bipartite. Let $W$ be a graphon. By (49) and (62) we
have

$\left\lVert\mathfrak{V}_{\>}W\right\rVert_{1}=(k)_{2}\cdot
t(F,W)\geq\left\lVert W\right\rVert_{1}^{m_{F}}\;.$

Then Proposition 3.9 tells us any destination is constant-$0$, proving (i).

We turn to (ii). Let us consider the case $e(F)\geq 2$, which entails
$m_{F}\geq 2$; we shall return to the case $e(F)=1$ later. The required
conclusion is trivial for the constant-$0$ graphon, so consider a graphon $W$
which is not constant-$0$. We claim that the (continuous) function
$f(t):=\left\lVert\Phi^{t}W\right\rVert_{1}$ satisfies

(63) $-f^{\prime}(t)\geq f(t)^{m_{F}}.$

Indeed, Lemma 10.6 and (62) imply

$\displaystyle f(t)-f(t+\varepsilon)$
$\displaystyle=\int_{\Omega^{2}}\left(\Phi^{t}W-\Phi^{t+\varepsilon}W\right)=\sum_{(i,j)\in
E(F)}\int_{\tau=t}^{t+\varepsilon}\int_{(x,y)\in\Omega^{2}}t^{x,y}(F^{(i,j)},\Phi^{\tau}W)$
$\displaystyle\overset{\mbox{\tiny{\eqref{eq:Sido_weak}}}}{\geq}(k)_{2}\int_{t}^{t+\varepsilon}\left\lVert\Phi^{\tau}W\right\rVert_{1}^{m_{F}}\geq(k)_{2}\varepsilon
f(t+\varepsilon)^{m_{F}}\geq\varepsilon f(t+\varepsilon)^{m_{F}}\;.$

Dividing by $\varepsilon$ and taking the limit $\varepsilon\to 0$ we obtain
(63), proving our claim. Now (63) implies that the function
$g(t):=f(t)^{-(m_{F}-1)}$ satisfies

$g^{\prime}(t)=-(m_{F}-1)f(t)^{-m_{F}}f^{\prime}(t)\geq m_{F}-1,$

where we used the fact that $f$ takes positive values. Consequently, we have
$g(t)\geq g(0)+(m_{F}-1)t$, and by substituting $g$ we obtain

$f(t)\leq\left((m_{F}-1)t+f(0)^{1-m_{F}}\right)^{-\frac{1}{m_{F}-1}}\leq
t^{-\frac{1}{m_{F}-1}}\;,$

and the desired conclusion in Theorem 10.9 follows. Similarly, for the case
$e(F)=1$ we have the differential inequality $-f^{\prime}(t)\geq f(t)$ with
auxiliary function $g(t)=-\ln f(t)$. This gives $f(t)\leq f(0)\exp(-t)$, which
then implies the desired conclusion.

To deduce (iii), we recall one of the main results of [2]. Namely, Lemma 3.4
in [2] asserts that if $F$ is not bipartite, then there exists $c>0$ such that
for all sufficiently small $\alpha>0$ and all sufficiently large integers $n$
there exists an $n$-vertex graph $G$ that is $\alpha$-far from being $F$-free
and contains at most $(\frac{\alpha}{c})^{c\log(\frac{c}{\alpha})}n^{v(F)}$
copies of $F$. Here, _$\alpha$ -far_ means that more than $\alpha n^{2}$ edges
need to be removed from $G$ so that it becomes $F$-free. Since the number of
non-injective homomorphisms of $F$ into $G$ is $O(n^{v(F)-1})$, by taking $n$
large enough (depending on $\alpha$), we can assume that the the homomorphism
density $t(F,G)$ is at most $2(\frac{\alpha}{c})^{c\log(\frac{c}{\alpha})}$.
Let $W$ be a graphon representation of $G$. Recalling that $t(F,W)=t(F,G)$, we
have $t(F,W)\leq 2\left(\frac{\alpha}{c}\right)^{c\log(\frac{c}{\alpha})}$.
Also, note that since $t(F,\Phi^{\cdot}W)$ is nonincreasing for the
$F$-removal flip process, by Lemma 10.6 we have

(64) $\left\lVert
W-\Phi^{t}W\right\rVert_{1}=(v(F))_{2}\int_{x=0}^{t}t(F,\Phi^{x}W)\leq
t(v(F))_{2}\cdot t(F,W)\leq t(v(F))_{2}\cdot
2\left(\frac{\alpha}{c}\right)^{c\log(\frac{c}{\alpha})}.$

The notion of being $\alpha$-far translates in a straightforward way into a
graphon representation. In this translation, the $L^{1}$-norm is the graphon
counterpart to measuring the number of edges removed. Let us use this
intuition to show that

(65) $\|W-\mathrm{dest}(W)\|_{1}\geq\alpha\;.$

Indeed, it follows from Proposition LABEL:FIRSTPAPER.prop:twinsstay of [12]
that $\mathrm{dest}(W)$ is a step graphon on steps $(\Omega_{v}:v\in V(G))$
corresponding to the representation of $G$. Let $G^{\prime}\subset G$ be the
subgraph of $G$ containing edges in the support of $\mathrm{dest}(W)$, that
is, $ij\in E(G^{\prime})$ if and only if
$\mathrm{dest}(W){\restriction_{\Omega_{i}\times\Omega_{j}}}>0$. Obviously,
$\frac{e(G)-e(G^{\prime})}{n^{2}}\leq\|W-\mathrm{dest}(W)\|_{1}$. At the same
time, $\mathrm{dest}(W)$ is $F$-free, and therefore so is $G^{\prime}$. Since
$G$ is $\alpha$-far from $F$-freeness, we have
$\frac{e(G)-e(G^{\prime})}{n^{2}}>\alpha$. This allows us to deduce (65).

Thus, for every $t\geq 0$ we have

$\left\lVert\Phi^{t}W-\mathrm{dest}(W)\right\rVert_{1}\geq\left\lVert
W-\mathrm{dest}(W)\right\rVert_{1}-\left\lVert
W-\Phi^{t}W\right\rVert_{1}\overset{\mbox{\tiny{\eqref{eq:destalpha},
\eqref{eq:Lone_diff_upper}}}}{\geq}\alpha-t\cdot
2(v(F))_{2}\left(\frac{\alpha}{c}\right)^{c\log(\frac{c}{\alpha})}\;.$

Given $M>0$, taking $\alpha>0$ small enough, we have
$2(v(F))_{2}(\tfrac{\alpha}{c})^{c\log(\frac{c}{\alpha})}\leq(\alpha/2)^{M+1}$.
Setting $\varepsilon=\alpha/2$ and $t=\varepsilon^{-M}$ we have
$\left\lVert\Phi^{t}W-\mathrm{dest}(W)\right\rVert_{1}\geq\varepsilon$, whence
$\tau_{\mathcal{R}}^{-}(\varepsilon)\geq\varepsilon^{-M}$. This completes the
proof of (iii). ∎

###  10.3. Graphons with the same destination

Fact 10.5 (see also Proposition 10.10) tells us that for non-bipartite graphs
$F$, the $F$-removal rule has a nontrivial set of destinations. An intriguing
problem for any such $F$ is to determine equivalence classes of graphons with
the same destination. A particularly important equivalence class is the class
of graphons whose destination is constant-$0$. We do not have even a guess
what the answer might be.

######  Question 10.11.

Suppose that $F$ is a fixed graph. Determine all graphons whose destination in
the $F$-removal flip process is the constant-0 graphon.

In this section we consider the simplest nontrivial example, $F=K_{3}$, and we
study the destinations of step graphons
$\left\\{W_{\alpha,\beta}:\alpha,\beta\in[0,1]\right\\}$ with two steps
$\Omega_{1}$ and $\Omega_{2}$ of equal size such that
$W_{\alpha,\beta}=\alpha$ on $\Omega_{1}^{2}\cup\Omega_{2}^{2}$ and
$W_{\alpha,\beta}=\beta$ elsewhere. It is easy to see (using [12, Corollary
5.6 and Remark 3.8]) that the trajectory starting at $W_{\alpha,\beta}$ stays
a step graphon with respect to the partition $\Omega_{1}\cup\Omega_{2}$ and
becomes a constant graphon if and only if $\alpha=\beta$. An easy symmetry
argument (using [12, Remark 3.8], say), also implies that the value on
$\Omega_{1}^{2}$ stays equal to the value on $\Omega_{2}^{2}$. To sum up, for
all $\alpha,\beta\in[0,1]$ there are some functions $a,b:[0,\infty)\to[0,1]$
such that we have

(66) $\displaystyle\alpha=\beta\text{ implies }a(t)=b(t)\text{ for every
}t\geq 0\;,$ (67) $\displaystyle\alpha\neq\beta\text{ implies }a(t)\neq
b(t)\text{ for every }t\geq 0\;,$

and for all $t\geq 0$ we have

(68) $\Phi^{t}W_{\alpha,\beta}=W_{a(t),b(t)}\;.$

It is very intuitive that $\mathrm{dest}(W_{\alpha,\beta})=W_{0,\gamma}$, with
$\gamma>0$ whenever $\alpha/\beta$ is small enough and $\gamma=0$ when
$\alpha/\beta$ is large enough. The following result says that the critical
point is $\alpha=\beta$.

######  Proposition 10.12.

Consider the $K_{3}$-removal flip process. For a step graphon
$W_{\alpha,\beta}$ as defined above we have

$\mathrm{dest}(W_{\alpha,\beta})=\begin{cases}W_{0,\beta\exp[-2\alpha/(\beta-\alpha)]}&\text{
if }\alpha<\beta\;,\\\ W_{0,0},&\text{ if }\alpha\geq\beta\;.\end{cases}$

###### Proof.

Since $\mathfrak{V}_{\>}W(x,y)=-6t^{x,y}(K_{3}^{\bullet\bullet},W)$, it is
easy to check that the functions $a$ and $b$ from (68) satisfy the system of
differential equations

$\displaystyle a^{\prime}$ $\displaystyle=-a^{3}/2-ab^{2}/2,$ $\displaystyle
b^{\prime}$ $\displaystyle=-ab^{2}$

with the initial condition $a(0)=\alpha,b(0)=\beta$. Observe that since the
functions $a$ and $b$ take non-negative values, it follows easily from the
system of differential equations that in the case $\alpha=0$ we have $a\equiv
0$ and $b\equiv\beta$. A similar argument tells us that in the case $\beta=0$
we get $b\equiv 0$ and $a(t)=\frac{1}{\sqrt{t+1/\alpha^{2}}}$. Now if
$\alpha=\beta>0$, then by (66) we actually have a single equation
$a^{\prime}=-a^{3}$, which has an explicit solution
$a(t)=b(t)=\frac{1}{\sqrt{2(t+1/(2\alpha^{2}))}}$. Hence, we may assume that
$\alpha\neq\beta$, $\alpha>0,\beta>0$. This implies that $a,b$ take positive
values for every $t\geq 0$ (see [12, Proposition
LABEL:FIRSTPAPER.prop:valueszeroone]) and hence are strictly decreasing
continuous functions. In particular $\lim_{x\to\infty}a(x)$ is nonnegative; it
is actually zero, in view of the differential equation for $a^{\prime}$. Hence
the inverse function $a^{-1}$ is defined on $(0,\alpha]$. Define a function
$f:(0,\alpha]\to(0,1]$ by $f(x):=b(a^{-1}(x))$. Note that

(69) $\mathrm{dest}(W_{\alpha,\beta})=W_{0,\lim_{x\to 0}f(x)}.$

In view of (69), it remains to determine $\lim_{x\to 0}f(x)$. To do so, we
shall seek to understand the function $f$. Since $\alpha\neq\beta$, note that
(67) and continuity implies that

(70) $\text{either}\quad f(x)<x\ \forall x\in(0,\alpha]\quad\text{or}\quad
f(x)>x\ \forall x\in(0,\alpha]$

We have

$\displaystyle f^{\prime}(x)$
$\displaystyle=\frac{b^{\prime}\left(a^{-1}(x)\right)}{a^{\prime}\left(a^{-1}(x)\right)}=-\frac{a\left(a^{-1}(x)\right)b(a^{-1}(x))^{2}}{a\left(a^{-1}(x)\right)^{3}/2+a\left(a^{-1}(x)\right)b\left(a^{-1}(x)\right)^{2}/2}$
$\displaystyle=\frac{2xf(x)^{2}}{x^{3}+xf(x)^{2}}=\frac{2}{(x/f(x))^{2}+1}\;.$

Hence $f$ satisfies a differential equation $f^{\prime}=2/(x^{2}/f^{2}+1)$ for
$x\in(0,\alpha]$ with initial condition $f(\alpha)=\beta$. Denoting
$v(x)=f(x)/x$, we have $f^{\prime}=v^{\prime}x+v$, whence

(71) $\displaystyle xv^{\prime}$
$\displaystyle=f^{\prime}-v=\frac{2}{v^{-2}+1}-v=\frac{-v(v-1)^{2}}{1+v^{2}}\;,$
$\displaystyle\frac{v^{\prime}}{v}$
$\displaystyle=-\frac{v^{2}-2v+1}{x(1+v^{2})}=-\frac{1}{x}+\frac{2v}{x(1+v^{2})}\;.$

Since $(\ln v)^{\prime}=v^{\prime}/v$ and
$\left(2/(v-1)\right)^{\prime}=-2v^{\prime}/(v-1)^{2}$, we get

$\displaystyle\ln v(x)$ $\displaystyle=\ln
v(\alpha)+\int_{\alpha}^{x}\left(\frac{2v}{x(1+v^{2})}-\frac{1}{x}\right)\,\mathrm{d}x\overset{\mbox{\tiny{\eqref{eq:xvprime}}}}{=}\ln\frac{f(\alpha)}{\alpha}+\ln\frac{\alpha}{x}+\int_{\alpha}^{x}\frac{-2v^{\prime}}{(v-1)^{2}}$
$f(\alpha)=\beta$
$\displaystyle=\ln\frac{\beta}{x}+\frac{2}{v(x)-1}-\frac{2}{v(\alpha)-1}.$

Since $v(x)=f(x)/x$ and $v(\alpha)=\beta/\alpha$, it follows that any solution
$f$ satisfies equation

(72)
$\ln\frac{f(x)}{x}-\frac{2}{f(x)/x-1}=\ln\frac{\beta}{x}+\frac{2}{1-\beta/\alpha}.$

So, writing $y=f(x)/x-1$, $F(y):=\ln(1+y)-2/y$ and defining a constant

$c_{0}:=\ln\beta+\frac{2}{1-\beta/\alpha},$

equation (72) reads

$F(y)=c_{0}-\ln x.$

Elementary calculus shows that $F$ is strictly increasing and continuous on
$(-1,0)$ and on $(0,\infty)$ and undefined elsewhere (which is not a problem,
since by (70) we have $y\in(-1,0)\cup(0,\infty)$ for $x\in(0,\alpha]$).
Moreover, $F((-1,0))=F((0,\infty))=\mathbb{R}$. Hence the inverse of $F$
consists of two increasing bijections: $F^{-1}_{-}:\mathbb{R}\to(-1,0)$ and
$F^{-1}_{+}:\mathbb{R}\to(0,\infty)$. By (70), and noting that
$f(\alpha)=\beta$ we get that if $\alpha<\beta$, then
$y(\alpha)=\beta/\alpha-1>0$ and thus $y>0$ for all $x$; similarly, if
$\alpha>\beta$, then $y<0$ for all $x$. Consequently

(73) $f(x)=\begin{cases}x\left(1+F^{-1}_{-}(c_{0}-\ln x)\right),\quad\text{if
}\alpha>\beta,\\\ x\left(1+F^{-1}_{+}(c_{0}-\ln x)\right),\quad\text{if
}\alpha<\beta.\end{cases}$

We shall now determine $\lim_{x\downarrow 0}f(x)$; in view of (69), this will
complete the proof. It clearly equals zero when $\alpha>\beta$, since
$F^{-1}_{-}$ is bounded. In the case $\alpha<\beta$, we have that
$F_{+}^{-1}(x)\to\infty$, as $x\to\infty$, so we estimate $F^{-1}_{+}(x)$ for
large $x$ by choosing an arbitrary constant $\varepsilon>0$ and using
inequalities

$\ln(1+y)-\varepsilon=:L_{\varepsilon}(y)\leq F(y)\leq L_{0}(y)=\ln(1+y)$

that hold for large enough $y$. We note that for large enough $x$

$e^{x}-1=L_{0}^{-1}(x)\leq F^{-1}_{+}(x)\leq
L_{\varepsilon}^{-1}(x)=e^{x+\varepsilon}-1,$

which implies that $xF^{-1}_{+}(c_{0}-\ln
x)\in[e^{c_{0}},e^{c_{0}+\varepsilon}]$. Since $\varepsilon>0$ was arbitrary,
we obtain that

$\lim_{x\downarrow 0}x(1+F^{-1}_{+}(c_{0}-\ln
x))=e^{c_{0}}=\beta\exp\left(-\frac{2\alpha}{\beta-\alpha}\right)\;,$

completing the proof. ∎

###  10.4. Stabilization in a removal flip process

In Section 7.4 we argued that extremist flip processes may take very long to
stabilize. On the other hand, removal flip processes stabilize substantially
faster. Indeed, an $F$-removal flip process started on an $n$-vertex graph
stabilizes no later than the moment that each of its $(n)_{k}$-many distinct
$k$-tuples of vertices is sampled at least once. Then, the solution to the
classical coupon collector’s problem tells us that $\Theta(n^{k}\log n)$ is
asymptotically almost surely an upper bound on the stabilization time. We do
not know whether this bound is optimal.

######  Question 10.13.

For each graph $F$ with the corresponding $F$-removal flip process
$\mathcal{R}_{F}$, we seek a function $b_{F}:\mathbb{N}\rightarrow\mathbb{N}$
growing as slowly as possible so that

$\min_{G\in\mathcal{H}_{n}}\mathbb{P}\left(\kappa_{\mathcal{R}_{F}}(G)\leq
b_{F}(n)\right)\to 1,\quad\text{ as }n\to\infty\;.$

Since the stabilization time is clearly superquadratic, the limitations of
Theorem 1.1 kick in. That means that _a priori_ we cannot connect the concept
of destinations to the structure of a _final graph_ , that is, the graph after
stabilization occurs. Our next theorem shows that, in the case of removal flip
processes, destinations serve as good approximations for the final graphs.

######  Theorem 10.14.

Given a graph $F$, for every $\varepsilon>0$ there exists $n_{0}\in\mathbb{N}$
such that the following holds for all $n\geq n_{0}$. Suppose that $G^{*}$ is
the final graph in the $F$-removal flip process started at an $n$-vertex graph
$G$. Let $W_{G}$ and $W_{G^{*}}$ be the graphon representations of $G$ and
$G^{*}$, respectively. Then with probability at least $1-\varepsilon$ we have
$d_{\square}(\mathrm{dest}(W_{G}),W_{G^{*}})<\varepsilon$.

The proof uses several tools, including Theorem 10.9 and the graph removal
lemma, the latter of which we now recall.

######  Proposition 10.15 (Graph removal lemma).

Given a graph $F$, for every $\alpha>0$ there exists $\beta>0$ with the
following property. Suppose that $G$ is an $n$-vertex graph whose density of
$F$ is less than $\beta$. Then we can remove at most $\alpha n^{2}$ edges from
$G$ and make it $F$-free.

###### Proof of Theorem 10.14.

Let $\varepsilon>0$ and $F$ be a graph. Observe that the statement is trivial
when $e(F)=0$, so we shall assume that $e(F)>0$ and $v(F)\geq 2$. Proposition
10.15 returns $\beta>0$ for input $\alpha:=\frac{\varepsilon}{4e(F)}$. Set
$\gamma:=\min\\{\alpha,\beta/k^{2}\\}$. Theorem 10.9 returns $T\geq 0$ for
error parameter $\gamma$. Lastly, let $n_{0}$ be sufficiently large so that we
have $1-\exp(-\Omega(n_{0}^{2}))\geq 1-\varepsilon$ in Theorem 1.1 with error
parameter $\gamma$ and input $T\geq 0$.

Suppose now $G$ is an $n$-vertex graph for $n\geq n_{0}$. We run the
$F$-removal flip process on $G$ and obtain a sequence of graphs
$G_{0}:=G,G_{1},G_{2},\ldots,G_{\kappa(G)}=:G^{*}$. Let $\mathcal{B}$ be the
event that for all $\ell\in\mathbb{N}\cap[0,Tn^{2}]$ we have
$d_{\square}(W_{G_{\ell}},\Phi^{\ell/n^{2}}W_{G_{0}})<\gamma$. Theorem 1.1
tells us that the probability of this event is at least $1-\varepsilon$.
Assume that $\mathcal{B}$ holds. We shall prove that
$d_{\square}(\mathrm{dest}(W_{G}),W_{G^{*}})<\varepsilon$.

We have $d_{\square}\left(\mathrm{dest}(W_{G}),\Phi^{T}W_{G}\right)\leq\gamma$
and $d_{\square}\left(\Phi^{T}W_{G},W_{G_{Tn^{2}}}\right)\leq\gamma$ by
Theorem 10.9 and Theorem 1.1 respectively. Putting them together, we obtain

(74) $d_{\square}\left(\mathrm{dest}(W_{G}),W_{G_{Tn^{2}}}\right)\leq
d_{\square}\left(\mathrm{dest}(W_{G}),\Phi^{T}W_{G}\right)+d_{\square}\left(\Phi^{T}W_{G},W_{G_{Tn^{2}}}\right)\leq
2\gamma.$

Since $\mathrm{dest}(W_{G})$ is $F$-free by Fact 10.5, it follows from (74)
and Lemma 2.6 that $t(F,W_{G_{Tn^{2}}})<\beta$. Then, Proposition 10.15 tells
us that we can find a set $R\subset E(G_{Tn^{2}})$ of at most
$\frac{\varepsilon n^{2}}{4e(F)}$ edges whose removal would destroy all the
copies of $F$ in $G_{Tn^{2}}$. In particular, every copy of $F$ in
$G_{Tn^{2}}$ has an edge in $R$. Let $X\subseteq[Tn^{2},\ldots,\kappa(G)]$ be
the indices of the non-idle steps. In particular, $X\neq\emptyset$ if and only
if $\kappa(G)>Tn^{2}$. Observe that in each non-idle step indexed in $X$ a
copy of $F$ in $G_{Tn^{2}}$ is removed, so at least one edge from $R$ is
removed. Hence, we have

$e(G_{Tn^{2}})-e(G^{*})=e(F)\cdot|X|\leq e(F)\cdot|R|\leq\frac{\varepsilon
n^{2}}{4}\;,$

which implies
$d_{\square}\left(W_{G_{Tn^{2}}},W_{G^{*}}\right)\leq\varepsilon/4$. Combining
this with (74), we obtain

$d_{\square}\left(\mathrm{dest}(W_{G}),W_{G^{*}}\right)\leq
d_{\square}\left(\mathrm{dest}(W_{G}),W_{G_{Tn^{2}}}\right)+d_{\square}\left(W_{G_{Tn^{2}}},W_{G^{*}}\right)<\varepsilon,$

as was needed. ∎

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