#  Is there (no) collective flow in pp collisions?

Gábor Bíró  0000-0003-2849-0120<EMAIL_ADDRESS>Leonid Serkin
0000-0003-4749-5250<EMAIL_ADDRESS>Guy Paić
0000-0003-2513-2459<EMAIL_ADDRESS>Gergely Gábor Barnaföldi
0000-0001-9223-6480<EMAIL_ADDRESS>HUN-REN Wigner
Research Center for Physics, 29–33 Konkoly–Thege Miklós Str., H-1121 Budapest,
Hungary.  ELTE Eötvös Loránd University, Institute of Physics, 1A Pázmány
Péter Sétány, H-1117 Budapest, Hungary.  Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México, Apartado Postal 70-543, Ciudad de
México 04510, México.

###### Abstract

The transverse momentum spectra and their multiplicity dependence serve as key
tools for extracting parameters that can be compared with theoretical models.
This comparison aims to establish the behaviour and nature of the system
created in the collision. Over the past decade, the scientific community has
extensively studied the possibility of a system analogous to quark-gluon
plasma, predicted in heavy nuclei collisions, also existing in collisions
involving light nuclei and proton-proton collisions. We have reanalysed the
experimental data published by the ALICE Collaboration at the LHC, exploring a
seemingly universal feature of transverse momentum spectra of charged
particles. We have identified a specific range where the contribution of the
hard part is nonexistent for the studied multiplicities. We present the
dependence of the mean transverse momenta obtained in the soft and soft+hard
(mixes) parts and discuss the results in the context of the ongoing
controversy regarding the existence of collectivity in small systems. Finally,
we also discuss possible refinements of the analyses concerning the use of
statistical parameters of higher order, aimed at better distinguishing the
agreement of theoretical and Monte Carlo models with the data.

###### keywords:

collectivity flow transverse momentum charged hadron spectra mean

sort&compress

##  1 Introduction

The exploration of collective effects in proton-proton (pp) collisions has
been a highly active area of research over the past decade Adam et al. (2017);
Khachatryan et al. (2017); Aaboud et al. (2019); Acharya et al. (2022); Nagle
and Zajc (2018); Strickland (2019); Mishra et al. (2021); Bierlich et al.
(2022). Numerous studies have documented the presence of collective flow in pp
collision spectra, see e.g. Jiang et al. (2015); Ghosh et al. (2014); Gu et
al. (2022); Ortiz et al. (2017); Blok and Wiedemann (2019); Ji et al. (2023)
and the references therein. Recent measurements of high-multiplicity events in
both pp and proton-nucleus collisions have raised new questions regarding the
potential production of hot QCD matter in such small systems. This is
particularly intriguing given that in high-multiplicity pp collision events,
the energy density may reach levels comparable to those observed in nucleus-
nucleus collisions. Hence, it is essential to investigate both traditional and
novel observables across various event multiplicity classes Ortiz and Paic
(2022); Ortiz et al. (2023); Bíró et al. (2020); Mishra et al. (2023, 2019);
Horvath et al. (2023).

A compelling argument in favour of the hydrodynamical description of the
results has been provided by the so-called “blast-wave model” Schnedermann et
al. (1993). The standard approach in terms of hydrodynamical expansion
involves fitting a blast-wave function across the entire range of charged-
particle transverse momentum, typically including the part with $p_{\rm T}$
below 1 GeV/c. In Mazeliauskas and Vislavicius (2020), the authors have
observed considerable flow values with a fit within the $p_{\rm T}$ range of
$0.5-3$ GeV/c, where a significant contribution from low-$p_{\rm T}$ minijets
can be expected Wang (1993). However, the foundation of blast-wave models for
extracting flow patterns in collisions of nucleons or light nuclei remains a
subject of discussion and contention Ortiz Velásquez (2015); Lim and Nagle
(2021); Giacalone et al. (2021). Studies claiming to detect collective flow
often achieve their results with somewhat arbitrary choice of fitting ranges
within the blast-wave models, potentially skewing the extracted temperatures
and flow parameters Mazeliauskas and Vislavicius (2020); Bíró et al. (2020);
Abelev et al. (2013, 2014); Acharya et al. (2020). Our particular interest has
been raised by the observation that models such as Pythia 8 Sjöstrand et al.
(2015), which do not incorporate hydrodynamical expansion like Epos LHC Pierog
et al. (2015), still manage to reasonably reproduce the data. In this paper,
we endeavour to clarify the controversy regarding the existence of
collectivity in small systems, which holds significance within the broader
context of nuclear collisions.

Inspired by the results obtained by the ALICE Collaboration Aamodt et al.
(2010a, b); Abelev et al. (2013); Adam et al. (2016); Acharya et al. (2019,
2018, 2019, 2021, 2020), we notice that, traditionally, collision data
observables are presented as mean values of more differential distributions,
such as mean charged-particle multiplicity $\left<N_{\mathrm{ch}}\right>$,
mean of transverse momentum $\left<p_{\rm T}\right>$, mean values of
anisotropic flow, etc. However, an accurate interpretation of theoretical
model parameters from measured data may encounter inconsistencies due to the
QCD theory’s dual, “Janus-faced” nature, embracing both perturbative and
nonperturbative, or soft and hard, aspects. While comparing these mean values
across different theoretical models aims to confirm or refute theoretical
premises, the typical outcome is that models A or B demonstrate “fair
agreement with the data”. However, agreeing on the means of two models does
not always lead to a genuine understanding of their nature because they may
yield the same mean with significantly different underlying assumptions. In
this paper, we present a case that partially illustrates the above criticism:
the $p_{\rm T}$ evolution of charged particles as a function of their
multiplicity, extensively studied as the main source of collective effects in
pp collisions Gu et al. (2022); Ortiz et al. (2017); Liu et al. (2023);
Bierlich et al. (2018); Zhao et al. (2018).

Figure 1:  $N_{\mathrm{ch}}$-integrated transverse momentum distributions of
charged particles for three different multiplicity classes as measured by the
ALICE Collaboration Aamodt et al. (2010a, b); Abelev et al. (2013); Adam et
al. (2016); Acharya et al. (2019, 2018, 2019, 2021, 2020). Results for pp
collisions at (a) $\sqrt{s}=0.9$ TeV, (b) $\sqrt{s}=5.02$ TeV, (c)
$\sqrt{s}=7$ TeV and (d) $\sqrt{s}=13$ TeV are presented. Statistical
uncertainties are represented by error bars, while total systematic
uncertainties are depicted as boxes around the data points. The top panels
display ratios of the $p_{\rm T}$ spectra to the inclusive spectrum for the
lowest-multiplicity class. Similarly, the middle panels show ratios of the
$p_{\rm T}$ spectra to the inclusive spectrum for the closest multiplicity
class to the inclusive class. In the bottom panels, ratios of the $p_{\rm T}$
spectra to the inclusive spectrum are displayed on a logarithmic scale for the
highest-multiplicity class. The solid line in each figure represents the
Pythia 8 simulation at the corresponding $\sqrt{s}$. (colour online)

##  2 Experimental $p_{\text{T}}$ spectra and Monte Carlo modelling

In recent years, the ALICE Collaboration has investigated the charged-particle
transverse momentum spectra as a function of charged-particle multiplicity at
various centre-of-mass energies, $\sqrt{s}$. Our particular interest lies in
exploring the soft, low-$p_{\rm T}$ region of the spectrum within the context
of the ongoing debate surrounding the existence of collectivity in small
systems.

The top panels of Figure 1 present the $N_{\mathrm{ch}}$-integrated $p_{\rm
T}$ distributions of unidentified charged particles at mid-pseudorapidity
($|\eta|<0.5$ or $|\eta|<0.8$), measured by the ALICE detector in inelastic
(INEL) pp collisions at $\sqrt{s}$ = 0.9, 5.02, 7 and 13 TeV Aamodt et al.
(2010a, b); Abelev et al. (2013); Adam et al. (2016); Acharya et al. (2019,
2018, 2019, 2021, 2020). The charged-particle pseudorapidity densities,
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$, are based on
tracklets in the Silicon Pixel detector and multiplicities recorded in the
forward V0 detectors (V0M amplitude). The values of the
$\left<{\mathrm{d}}N_{\mathrm{ch,Incl}}/{\mathrm{d}}\eta\right>$ in the
inclusive ($\mathrm{INEL}>0$) multiplicity class are: $4.20\pm 0.03$ at
$\sqrt{s}=0.9$ TeV and $|\eta|<0.5$ Aamodt et al. (2010b), $5.91\pm 0.45$ at
$\sqrt{s}=5.02$ TeV and $|\eta|<0.8$ Acharya et al. (2019), $5.96\pm 0.23$ at
$\sqrt{s}=7$ TeV and $|\eta|<0.5$ Acharya et al. (2019), and $7.60\pm 0.50$ at
$\sqrt{s}=13$ TeV and $|\eta|<0.8$ Acharya et al. (2019).

For each collision energy, several multiplicity classes are examined, and then
categorised into three ranges for simplicity. The lowest-multiplicity class
includes events with
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ much smaller than
$\left<{\mathrm{d}}N_{\mathrm{ch,Incl}}/{\mathrm{d}}\eta\right>$, the middle
class contains encompasses events where
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ is of the order of
$\left<{\mathrm{d}}N_{\mathrm{ch,Incl}}/{\mathrm{d}}\eta\right>$, and within
the highest-multiplicity class,
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ is greater than
$\left<{\mathrm{d}}N_{\mathrm{ch,Incl}}/{\mathrm{d}}\eta\right>$.

We compare the experimental results with Pythia 8 (version 8.309) Monte Carlo
(MC) simulation Sjöstrand et al. (2015), widely used for studying LHC physics.
We generated $\sim$50 million inelastic events at each $\sqrt{s}$, including
non-diffractive and diffractive components, with the latter comprising 31% and
27% of the total cross-section at $\sqrt{s}$ = 0.9 and 13 TeV, respectively.
We employed the Monash 2013 tune, calibrated using early LHC measurements
Skands et al. (2014), and processed the MC simulation with Rivet v3 Bierlich
et al. (2020). Simulations were also carried out employing a core-corona
approach based on new microcanonical hadronisation procedures in Epos 4
(version 4.0.0) framework Werner (2023); Werner and Guiot (2023) without any
LHC-dedicated tune. Epos 4 incorporates collective (flow-like) effects and
integrates knowledge from $S$-matrix theory, perturbative QCD and saturation.
Following the definition provided in Acharya et al. (2019), the results are
presented for primary charged particles in the kinematic range $0.15$ GeV/c
$\leq p_{\rm T}\leq 20$ GeV/c.

As can be seen in Figure 1, Pythia 8 predictions successfully describe the
qualitative features of the evolution of the $p_{\rm T}$ spectra, in
particular at low values. The bottom panels of Figure 1 present the ratios of
the normalised $p_{\rm T}$ spectra to the inclusive spectrum for the three
multiplicity classes. The features of the $p_{\rm T}$ spectra, namely the
change in spectral shape from low- to high-multiplicity values, are
qualitatively consistent across all energies. The only significant difference
is the higher multiplicity reach at $\sqrt{s}$ =13 TeV compared to that at 0.9
TeV. Surprisingly, the behaviour of the ratios to the inclusive spectrum for
the three multiplicity classes resulted a singular pattern, exhibiting a
crossing at the unity value at $\sim$0.6 GeV/c for all studied $\sqrt{s}$
energies. Above this value, the ratios diverge significantly.

Since the description of the so-called underlying event is known to modify the
$p_{\rm T}$ spectral shape, different configurations within the Pythia 8 MC
model were tested. Events were simulated with the default parametrisation of
the so-called multi-parton interactions-based model of colour reconnection
(CR) Sjöstrand and van Zijl (1987); Sjostrand and Skands (2004), as well as
with modified CR based on the gluon-move model Argyropoulos and Sjöstrand
(2014), without any colour reconnection, and without initial-state (ISR) or
final-state radiation (FSR). Additionally, given the sensitivity of the
low-$p_{\rm T}$ region to diffractive contributions Sirunyan et al. (2020),
events produced only via diffractive processes were also tested.

Figure 2 presents the ratios of the normalised $p_{\rm T}$ spectra to the
inclusive spectrum for the three multiplicity classes at $\sqrt{s}=13$ TeV
using different Pythia 8 settings, along with the prediction from Epos 4. The
ratios present the same behaviour as it has been observed earlier: there is a
significant hardening of the $p_{\rm T}$ distributions towards larger
multiplicities, resulting in a change of the spectral shape compared to the
inclusive spectrum and attributed to the presence of energetic jets Aamodt et
al. (2010a); Abelev et al. (2013); Adam et al. (2016). However, the crossing
point at the unity value is practically independent of the different Pythia 8
settings. A deviation in terms of the crossing point at the unity value is
observed only for the Epos 4 case.

Figure 2:  Ratios of the normalised $p_{\rm T}$ spectra to the inclusive
spectrum for the three multiplicity classes at $\sqrt{s}=13$ TeV. Predictions
from Pythia 8 simulation with different settings are shown: (a) default Pythia
8 model, (b) Pythia 8 with modified CR, (c) Pythia 8 with CR turned off, (d)
Pythia 8 without ISR, (e) Pythia 8 without FSR, (f) Pythia 8 events produced
only via diffractive processes. The prediction from Epos 4 is shown in (g).
The red line denotes the observed crossing value at $\sim$0.6 GeV/c. (colour
online)

It is well known that transverse momentum distributions for different
multiplicities result from distinct contributions: the low-$p_{\rm T}$ part of
the spectrum is primarily represented by soft contributions, whereas the
high-$p_{\rm T}$ part is attributed to a mixture of jet and minijet
contributions Acharya et al. (2023a, b); Ortiz et al. (2024). The boundary
between soft and hard processes is not precisely defined but is typically
assumed to fall within the range of $1-2$ GeV/c Prasad et al. (2015); Wang and
Wang (2001); Aaboud et al. (2016).

Our remarkable observation of a “crossing-point” drives us to the same
conclusion: the $p_{\rm T}$ spectra in a given multiplicity class stems from
at least two different processes. Hence, it seems legitimate to separate each
individual $p_{\rm T}$ spectrum into two parts at the observed crossing value
at $\sim$0.6 GeV/c:

  1. -

Soft region (low-$p_{\rm T}$): $0.15$ GeV/c $\leq p_{\rm T}\leq 0.6$ GeV/c,

  2. -

Mixed region (higher-$p_{\rm T}$): $0.6$ GeV/c $<p_{\rm T}\leq 4$ GeV/c.

Within the limited event multiplicity range provided by ALICE data at the four
$\sqrt{s}$ energies investigated, the maximum $p_{\rm T}$ range studied is
capped at $4.0$ GeV/c. This limit is set by the highest $p_{\rm T}$ bin
available at the collision energy of $\sqrt{s}=0.9$ TeV Aamodt et al. (2010a).

##  3 Statistical moments of the $p_{\text{T}}$ spectra in different regions

Figure 3:  Mean transverse momentum as a function of the charged-particle
pseudorapidity density as measured by the ALICE Collaboration in inelastic pp
collisions at $\sqrt{s}$ = 0.9, 5.02, 7 and 13 TeV. Values of $\left<p_{\rm
T}\right>$ are presented for the high-$p_{\rm T}$ region with $0.6$ GeV/c
$<p_{\rm T}\leq 4$ GeV/c (top panel), full-$p_{\rm T}$ region with $0.15$
GeV/c $\leq p_{\rm T}\leq 4.0$ GeV/c (middle panel) and low-$p_{\rm T}$ region
of $0.15$ GeV/c $\leq p_{\rm T}\leq 0.6$ GeV/c (bottom panel). Statistical
uncertainties on data are represented by error bars. The solid line in each
panel represents the Pythia 8 simulation at the corresponding $\sqrt{s}$.
(colour online)  Figure 4:  Variance of transverse momentum as a function of
the charged-particle pseudorapidity density as measured by the ALICE
Collaboration in inelastic pp collisions at $\sqrt{s}$ = 0.9, 5.02, 7 and 13
TeV. Variance values are presented for the high-$p_{\rm T}$ region with $0.6$
GeV/c $<p_{\rm T}\leq 4$ GeV/c (top panel), full-$p_{\rm T}$ region with
$0.15$ GeV/c $\leq p_{\rm T}\leq 4.0$ GeV/c (middle panel) and low-$p_{\rm T}$
region of $0.15$ GeV/c $\leq p_{\rm T}\leq 0.6$ GeV/c (bottom panel).
Statistical uncertainties on data are represented by error bars. The solid
line in each panel represents the Pythia 8 simulation at the corresponding
$\sqrt{s}$. (color online)

Based on our observation that the ratios of various experimental spectra
across a wide range of collision energies exhibit a consistent crossing-point
at approximately the same $p_{\rm T}\sim 0.6$ GeV/c for all investigated event
multiplicities, a pattern also observed in both Pythia 8 and Epos 4
predictions, it is imperative to conduct a comprehensive investigation into
the statistical properties of these two distinct $p_{\rm T}$ regions: the
region below the crossing-point and the region above it.

In the following, we explore the first two statistical moments of the $p_{\rm
T}$ spectra, its mean and variance, as a function of the charged-particle
pseudorapidity density measured by the ALICE Collaboration. The mean
transverse momentum, $\left<p_{\rm T}\right>$ , a widely used metric in
scientific literature, is defined by Eq. (1) and calculated using a weighted
averaging approach, which involves computing the weighted average of the
values present in the $p_{\rm T}$ histogram. Variance, $\sigma^{2}$, defined
in Eq. (2) as the second central moment of the $p_{\rm T}$ distribution,
quantifies the variability within the data and offers insights into the
distribution’s shape and deviation from the mean:

$\displaystyle\left<p_{T}\right>$ $\displaystyle=$ $\displaystyle\sum
wp\left/\sum w\right.,$ (1) $\displaystyle\sigma^{2}$ $\displaystyle=$
$\displaystyle\frac{\sum(wp^{2})\sum(w)-\sum(wp)^{2}}{\sum(w)^{2}-\sum(w^{2})},$
(2)

where $p$ and $w$ represent the bin centres and weights of the $p_{\rm T}$
histogram, respectively. Both the mean and variance of the $p_{\rm T}$ spectra
are evaluated within the three regions: one below the crossing-point, one
above it, and within the full-$p_{\rm T}$ region studied ($0.15$ GeV/c $\leq
p_{\rm T}\leq 4.0$ GeV/c).

Figure 3 presents the mean transverse momentum as a function of the charged-
particle pseudorapidity density, as measured by the ALICE Collaboration in pp
collisions at $\sqrt{s}$ = 0.9, 5.02, 7 and 13 TeV. The measured values of
$\left<p_{\rm T}\right>$ are shown for the three regions (mixed/higher-$p_{\rm
T}$, full-$p_{\rm T}$ and soft/low-$p_{\rm T}$) and are compared to Pythia 8
simulations. In a similar manner, Figure 4 presents the variance of the
$p_{\rm T}$ spectra measured by the ALICE Collaboration at four collision
energies and its comparison with MC simulation.

We observe a striking feature in Figure 3 and 4: both $\left<p_{\rm T}\right>$
and variance of the $p_{\rm T}$ spectra remain constant with respect to
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ within the
low-$p_{\rm T}$ region. This observation is corroborated by Pythia 8
simulations. Furthermore, within this $p_{\rm T}$ range, both $\left<p_{\rm
T}\right>$ and $\sigma^{2}$ show no dependence on the centre-of-mass energy,
spanning from 0.9 to 13 TeV. To further explore this phenomenon, we compare
the predictions of Pythia 8 to those of Epos 4 for $\left<p_{\rm T}\right>$
and $\sigma^{2}$ in the three $p_{\rm T}$ regions at $\sqrt{s}=13$ TeV, as
shown in Figure 5.

Figure 5:  Ratio between Pythia 8 and Epos 4 predictions with default settings
for (a) $\left<p_{\rm T}\right>$ and (b) $\sigma^{2}$ as a function of
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ in inelastic pp
collisions at $\sqrt{s}$ = 13 TeV. Ratios are presented for the high-$p_{\rm
T}$ region with $0.6$ GeV/c $<p_{\rm T}\leq 4$ GeV/c (top panels),
full-$p_{\rm T}$ region with $0.15$ GeV/c $\leq p_{\rm T}\leq 4.0$ GeV/c
(middle panels) and low-$p_{\rm T}$ region of $0.15$ GeV/c $\leq p_{\rm T}\leq
0.6$ GeV/c (bottom panels). Statistical uncertainties on the MC predictions
are indicated by error bars. (color online)

##  4 Discussion

While reanalysing the experimental charged particles $p_{\rm T}$ spectra
published by the ALICE Collaboration, as presented in Figure 1, a consistent
crossing-point at around $\sim$0.6 GeV/c was observed when comparing the ratio
of the spectra to the inclusive distribution. This pattern, observed across
different $\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ classes
and over a wide range of collision energies, is also present in the
predictions from both Pythia 8 and Epos 4 models, as can be seen in Figure 2.

The separation of the charged particle $p_{\rm T}$ spectra into two regions
defined by the observed crossing-point resulted in two markedly different
behaviours of $\left<p_{\rm T}\right>$ in each region. The results shown in
Figure 3 and 4 indicate the absence of any variation with
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ and/or collision
energy in the values of $\left<p_{\rm T}\right>$ and $\sigma^{2}$ within the
soft, low-$p_{\rm T}$ region. Our results are consistent with the hypothesis
of centre-of-mass energy invariance within the low-$p_{\rm T}$ part of the
spectra, a phenomenon observed by the CDF collaboration in pp collisions at
$\sqrt{s}$ = 0.63 and 1.8 TeV Acosta et al. (2002).

We do not observe the predicted feature by blast-wave models of a flow
contribution increasing with multiplicity. Hence, our results suggest the
absence of collectivity effects in pp collisions within the sloft, low-$p_{\rm
T}$ region (below 0.6 GeV/c). We attribute the reported flow values in the
literature to the choice of fitting ranges within the blast-wave formalism.

While based on significantly different underlying assumptions, where Pythia 8
is rooted in perturbative QCD and string fragmentation and Epos 4 incorporates
collective effects and hydrodynamic considerations, the results in Figure 5
demonstrate that both MC models exhibit the same independence of $\left<p_{\rm
T}\right>$ and $\sigma^{2}$ on
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ within the
low-$p_{\rm T}$ range.

Our reanalysis of the ALICE Collaboration data indicates that the region below
0.6 GeV/c is predominantly associated with soft processes, while the region
beyond comprises a mixture of soft and hard ones. Therefore, this low-$p_{\rm
T}$ region serves as an excellent laboratory for studying soft physics in pp
collisions.

We advocate for the use of statistical parameters of higher order, such as
variance and/or skewness Giacalone et al. (2021) to better to better discern
the agreement between theoretical models, MC simulations, and experimental
data. This is particularly important since the weight of soft and hard
processes may differ, yet yield the same ratio.

##  5 Summary

We have conducted a detailed analysis of the multiplicity dependence of
normalised transverse momentum distributions of charged particles, using the
extensive dataset of publicly available pp collision data at $\sqrt{s}$ = 0.9,
5.02, 7 and 13 TeV from the ALICE Collaboration. We observe that the mean
transverse momentum $\left<p_{\rm T}\right>$ is highly sensitive to the
selected $p_{\rm T}$ range. Our investigation reveals a remarkable and
previously unidentified phenomenon: a universal crossing point of the ratios
to the inclusive spectrum at $p_{\rm T}\approx 0.6$ GeV/c. Below 0.6 GeV/c,
$\left<p_{\rm T}\right>$ exhibits an intriguing independence from the charged-
particle pseudorapidity densities and collision energy. This result is
supported by Pythia 8 with different underlying event and generator settings,
as well as by Epos 4 that incorporates collective flow effects. We examine a
higher-order statistical parameter, such as variance, to gain further insights
into how well the MC models agree among themselves and with experimental data.

The absence of any significant $\left<p_{\rm T}\right>$ and $\sigma^{2}$
variation in the low-$p_{\rm T}$ region challenges the conventional
interpretations and applications of the blast-wave models, suggesting that the
selection of $p_{\rm T}$ ranges in such analyses demands greater scrutiny and
justification to avoid potentially misleading conclusions. Our study thus
contributes significantly to the ongoing controversy regarding the presence of
collective effects in small systems. Finally, we would like to emphasise that
the current conclusions apply to the existing
$\left<{\mathrm{d}}N_{\mathrm{ch}}/{\mathrm{d}}\eta\right>$ range measured
with sufficient statistics by the ALICE Collaboration. Future LHC runs should
enable us to extend our investigation beyond the current multiplicity reach.

## Acknowledgments

The research was supported by the Hungarian National Research, Development and
Innovation Office OTKA K135515, 2019-2.1.11-TÉT-2019-00078,
2022-4.1.2-NEMZ_KI-2022-00031, 2022-4.1.2-NEMZ_KI-2022-00008 and
2022-4.1.2-NEMZ_KI-2022-00009 grants, and by the Wigner Scientific Computing
Laboratory and the HUN-REN Wigner Cloud. Support for this work has been
received from the Mexican National Council of Humanities, Sciences and
Technologies CONAHCYT under Grants No. CF-2042 and No. A1-S-22917. L.S.
acknowledges the support received from CONAHCYT for the postdoctoral
fellowship.

## References

  * Adam et al. (2017) J. Adam, et al. (ALICE Collaboration),  Enhanced production of multi-strange hadrons in high-multiplicity proton-proton collisions,  Nature Phys. 13 (2017) 535–539. doi:10.1038/nphys4111. arXiv:1606.07424.
  * Khachatryan et al. (2017) V. Khachatryan, et al. (CMS Collaboration),  Evidence for collectivity in pp collisions at the LHC,  Phys. Lett. B 765 (2017) 193–220. doi:10.1016/j.physletb.2016.12.009. arXiv:1606.06198.
  * Aaboud et al. (2019) M. Aaboud, et al. (ATLAS Collaboration),  Correlated long-range mixed-harmonic fluctuations measured in $pp$, $p$+Pb and low-multiplicity Pb+Pb collisions with the ATLAS detector,  Phys. Lett. B 789 (2019) 444–471. doi:10.1016/j.physletb.2018.11.065. arXiv:1807.02012.
  * Acharya et al. (2022) S. Acharya, et al. (ALICE Collaboration),  The ALICE experiment – A journey through QCD (2022). arXiv:2211.04384.
  * Nagle and Zajc (2018) J. L. Nagle, W. A. Zajc,  Small System Collectivity in Relativistic Hadronic and Nuclear Collisions,  Ann. Rev. Nucl. Part. Sci. 68 (2018) 211–235. doi:10.1146/annurev-nucl-101916-123209. arXiv:1801.03477.
  * Strickland (2019) M. Strickland,  Small system studies: A theory overview,  Nucl. Phys. A 982 (2019) 92–98. doi:https://doi.org/10.1016/j.nuclphysa.2018.09.071.
  * Mishra et al. (2021) A. N. Mishra, G. Paić, C. Pajares, R. P. Scharenberg, B. K. Srivastava,  Deconfinement and degrees of freedom in pp and $A-A$ collisions at LHC energies,  Eur. Phys. J. A 57 (2021) 245. doi:10.1140/epja/s10050-021-00552-2. arXiv:2006.10169.
  * Bierlich et al. (2022) C. Bierlich, S. Chakraborty, G. Gustafson, L. Lönnblad,  Strangeness enhancement across collision systems without a plasma,  Phys. Lett. B 835 (2022) 137571. doi:10.1016/j.physletb.2022.137571. arXiv:2205.11170.
  * Jiang et al. (2015) K. Jiang, Y. Zhu, W. Liu, H. Chen, C. Li, L. Ruan, Z. Tang, Z. Xu,  Onset of radial flow in p+p collisions,  Phys. Rev. C 91 (2015). URL: https://www.osti.gov/biblio/1188210. doi:10.1103/PhysRevC.91.024910.
  * Ghosh et al. (2014) P. Ghosh, S. Muhuri, J. K. Nayak, R. Varma,  Indication of transverse radial flow in high-multiplicity proton–proton collisions at the Large Hadron Collider,  J. Phys. G 41 (2014) 035106. doi:10.1088/0954-3899/41/3/035106. arXiv:1402.6813.
  * Gu et al. (2022) J. Gu, C. Li, Q. Wang, W. Zhang, H. Zheng,  Collective expansion in pp collisions using the Tsallis statistics,  J. Phys. G 49 (2022) 115101. doi:10.1088/1361-6471/ac9074. arXiv:2201.02091.
  * Ortiz et al. (2017) A. Ortiz, G. Bencedi, H. Bello,  Revealing the source of the radial flow patterns in proton–proton collisions using hard probes,  J. Phys. G 44 (2017) 065001. doi:10.1088/1361-6471/aa6594. arXiv:1608.04784.
  * Blok and Wiedemann (2019) B. Blok, U. A. Wiedemann,  Collectivity in pp from resummed interference effects?,  Phys. Lett. B 795 (2019) 259–265. doi:https://doi.org/10.1016/j.physletb.2019.05.038.
  * Ji et al. (2023) S. Ji, M. Virta, T. Kallio, S. H. Lim, D. J. Kim,  Toward unbiased flow measurements in pp collisions at the CERN Large Hadron Collider,  Phys. Rev. C 108 (2023) 034909. doi:10.1103/PhysRevC.108.034909. arXiv:2303.05806.
  * Ortiz and Paic (2022) A. Ortiz, G. Paic,  A look into the “hedgehog” events in pp collisions,  Rev. Mex. Fis. Suppl. 3 (2022) 040911. doi:10.31349/SuplRevMexFis.3.040911. arXiv:2204.13733.
  * Ortiz et al. (2023) A. Ortiz, A. Khuntia, O. Vázquez-Rueda, S. Tripathy, G. Bencedi, S. Prasad, F. Fan,  Unveiling the effects of multiple soft partonic interactions in pp collisions at s=13.6 TeV using a new event classifier,  Phys. Rev. D 107 (2023) 076012. doi:10.1103/PhysRevD.107.076012. arXiv:2211.06093.
  * Bíró et al. (2020) G. Bíró, G. G. Barnaföldi, T. S. Biró,  Tsallis-thermometer: a QGP indicator for large and small collisional systems,  J. Phys. G 47 (2020) 105002. doi:10.1088/1361-6471/ab8dcb. arXiv:2003.03278.
  * Mishra et al. (2023) A. N. Mishra, G. G. Barnaföldi, G. Paić,  Quantifying the underlying event: investigating angular dependence of multiplicity classes and transverse-momentum spectra in high-energy pp collisions at LHC energies,  J. Phys. G 50 (2023) 095004. doi:10.1088/1361-6471/ace924. arXiv:2108.13938.
  * Mishra et al. (2019) A. N. Mishra, A. Ortiz, G. Paic,  Intriguing similarities of high-$p_{T}$ particle production between $pp$ and $A-A$ collisions,  Phys. Rev. C 99 (2019) 034911. doi:10.1103/PhysRevC.99.034911. arXiv:1805.04572.
  * Horvath et al. (2023) A. Horvath, E. Frajna, R. Vertesi,  Event-Shape-Dependent Analysis of Charm–Anticharm Azimuthal Correlations in Simulations,  Universe 9 (2023) 308. doi:10.3390/universe9070308. arXiv:2306.05910.
  * Schnedermann et al. (1993) E. Schnedermann, J. Sollfrank, U. W. Heinz,  Thermal phenomenology of hadrons from 200-A/GeV S+S collisions,  Phys. Rev. C 48 (1993) 2462–2475. doi:10.1103/PhysRevC.48.2462. arXiv:nucl-th/9307020.
  * Mazeliauskas and Vislavicius (2020) A. Mazeliauskas, V. Vislavicius,  Temperature and fluid velocity on the freeze-out surface from $\pi$, $K$, $p$ spectra in pp, p-Pb and Pb-Pb collisions,  Phys. Rev. C 101 (2020) 014910. doi:10.1103/PhysRevC.101.014910. arXiv:1907.11059.
  * Wang (1993) X.-N. Wang,  Studying minijets via the ${\mathit{p}}_{\mathit{T}}$ dependence of two-particle correlation in azimuthal angle $\varphi$,  Phys. Rev. D 47 (1993) 2754–2760. URL: https://link.aps.org/doi/10.1103/PhysRevD.47.2754. doi:10.1103/PhysRevD.47.2754.
  * Ortiz Velásquez (2015) A. Ortiz Velásquez,  Mean $p_{T}$ scaling with $m/n_{q}$ at the LHC: Absence of (hydro) flow in small systems?,  Nucl. Phys. A 943 (2015) 9–17. doi:10.1016/j.nuclphysa.2015.08.003. arXiv:1506.00584.
  * Lim and Nagle (2021) S. H. Lim, J. L. Nagle,  Exploring origins for correlations between flow harmonics and transverse momentum in small collision systems,  Phys. Rev. C 103 (2021) 064906. doi:10.1103/PhysRevC.103.064906. arXiv:2103.01348.
  * Giacalone et al. (2021) G. Giacalone, F. G. Gardim, J. Noronha-Hostler, J.-Y. Ollitrault,  Correlation between mean transverse momentum and anisotropic flow in heavy-ion collisions,  Phys. Rev. C 103 (2021) 024909. doi:10.1103/PhysRevC.103.024909. arXiv:2004.01765.
  * Abelev et al. (2013) B. Abelev, et al. (ALICE Collaboration),  Centrality dependence of $\pi$, K, p production in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV,  Phys. Rev. C 88 (2013) 044910. doi:10.1103/PhysRevC.88.044910. arXiv:1303.0737.
  * Abelev et al. (2014) B. B. Abelev, et al. (ALICE Collaboration),  Multiplicity Dependence of Pion, Kaon, Proton and Lambda Production in p-Pb Collisions at $\sqrt{s_{NN}}$ = 5.02 TeV,  Phys. Lett. B 728 (2014) 25–38. doi:10.1016/j.physletb.2013.11.020. arXiv:1307.6796.
  * Acharya et al. (2020) S. Acharya, et al. (ALICE Collaboration),  Production of charged pions, kaons, and (anti-)protons in Pb-Pb and inelastic $pp$ collisions at $\sqrt{s_{NN}}$ = 5.02 TeV,  Phys. Rev. C 101 (2020) 044907. doi:10.1103/PhysRevC.101.044907. arXiv:1910.07678.
  * Sjöstrand et al. (2015) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, P. Z. Skands,  An introduction to PYTHIA 8.2,  Comput. Phys. Commun. 191 (2015) 159–177. doi:10.1016/j.cpc.2015.01.024. arXiv:1410.3012.
  * Pierog et al. (2015) T. Pierog, I. Karpenko, J. M. Katzy, E. Yatsenko, K. Werner,  EPOS LHC: Test of collective hadronization with data measured at the CERN Large Hadron Collider,  Phys. Rev. C 92 (2015) 034906. doi:10.1103/PhysRevC.92.034906. arXiv:1306.0121.
  * Aamodt et al. (2010a) K. Aamodt, et al. (ALICE Collaboration),  Transverse momentum spectra of charged particles in proton-proton collisions at $\sqrt{s}=900$~GeV with ALICE at the LHC,  Phys. Lett. B 693 (2010a) 53–68. doi:10.1016/j.physletb.2010.08.026. arXiv:1007.0719.
  * Aamodt et al. (2010b) K. Aamodt, et al. (ALICE Collaboration),  Charged-particle multiplicity measurement in proton-proton collisions at $\sqrt{s}=0.9$ and 2.36 TeV with ALICE at LHC,  Eur. Phys. J. C 68 (2010b) 89–108. doi:10.1140/epjc/s10052-010-1339-x. arXiv:1004.3034.
  * Abelev et al. (2013) B. B. Abelev, et al. (ALICE Collaboration),  Energy Dependence of the Transverse Momentum Distributions of Charged Particles in pp Collisions Measured by ALICE,  Eur. Phys. J. C 73 (2013) 2662. doi:10.1140/epjc/s10052-013-2662-9. arXiv:1307.1093.
  * Adam et al. (2016) J. Adam, et al. (ALICE Collaboration),  Pseudorapidity and transverse-momentum distributions of charged particles in proton–proton collisions at $\sqrt{s}=$ 13 TeV,  Phys. Lett. B 753 (2016) 319–329. doi:10.1016/j.physletb.2015.12.030. arXiv:1509.08734.
  * Acharya et al. (2019) S. Acharya, et al. (ALICE Collaboration),  Multiplicity dependence of light-flavor hadron production in pp collisions at $\sqrt{s}$ = 7 TeV,  Phys. Rev. C 99 (2019) 024906. doi:10.1103/PhysRevC.99.024906. arXiv:1807.11321.
  * Acharya et al. (2018) S. Acharya, et al. (ALICE Collaboration),  Transverse momentum spectra and nuclear modification factors of charged particles in pp, p-Pb and Pb-Pb collisions at the LHC,  JHEP 11 (2018). doi:10.1007/JHEP11(2018)013. arXiv:1802.09145.
  * Acharya et al. (2019) S. Acharya, et al. (ALICE Collaboration),  Charged-particle production as a function of multiplicity and transverse spherocity in pp collisions at $\sqrt{s}=5.02$ and 13 TeV,  Eur. Phys. J. C 79 (2019) 857. doi:10.1140/epjc/s10052-019-7350-y. arXiv:1905.07208.
  * Acharya et al. (2021) S. Acharya, et al. (ALICE Collaboration),  Production of light-flavor hadrons in pp collisions at $\sqrt{s}~{}=~{}7\text{ and }\sqrt{s}=13\,\text{ TeV}$,  Eur. Phys. J. C 81 (2021) 256. doi:10.1140/epjc/s10052-020-08690-5. arXiv:2005.11120.
  * Acharya et al. (2020) S. Acharya, et al. (ALICE Collaboration),  Multiplicity dependence of $\pi$, K, and p production in pp collisions at $\sqrt{s}=13$ TeV,  Eur. Phys. J. C 80 (2020) 693. doi:10.1140/epjc/s10052-020-8125-1. arXiv:2003.02394.
  * Liu et al. (2023) L. Liu, Z.-B. Yin, L. Zheng,  Universal scaling of kinetic freeze-out parameters across different collision systems at LHC energies*,  Chin. Phys. C 47 (2023) 024103. doi:10.1088/1674-1137/aca38d. arXiv:2211.08103.
  * Bierlich et al. (2018) C. Bierlich, G. Gustafson, L. Lönnblad,  Collectivity without plasma in hadronic collisions,  Phys. Lett. B 779 (2018) 58–63. doi:10.1016/j.physletb.2018.01.069. arXiv:1710.09725.
  * Zhao et al. (2018) W. Zhao, Y. Zhou, H. Xu, W. Deng, H. Song,  Hydrodynamic collectivity in proton–proton collisions at 13 TeV,  Phys. Lett. B 780 (2018) 495–500. doi:10.1016/j.physletb.2018.03.022. arXiv:1801.00271.
  * Skands et al. (2014) P. Skands, S. Carrazza, J. Rojo,  Tuning PYTHIA 8.1: the Monash 2013 Tune,  Eur. Phys. J. C 74 (2014) 3024. doi:10.1140/epjc/s10052-014-3024-y. arXiv:1404.5630.
  * Bierlich et al. (2020) C. Bierlich, et al.,  Robust Independent Validation of Experiment and Theory: Rivet version 3,  SciPost Phys. 8 (2020) 026. doi:10.21468/SciPostPhys.8.2.026. arXiv:1912.05451.
  * Werner (2023) K. Werner,  Revealing a deep connection between factorization and saturation: New insight into modeling high-energy proton-proton and nucleus-nucleus scattering in the EPOS4 framework,  Phys. Rev. C 108 (2023) 064903. doi:10.1103/PhysRevC.108.064903. arXiv:2301.12517.
  * Werner and Guiot (2023) K. Werner, B. Guiot,  Perturbative QCD concerning light and heavy flavor in the EPOS4 framework,  Phys. Rev. C 108 (2023) 034904. doi:10.1103/PhysRevC.108.034904. arXiv:2306.02396.
  * Sjöstrand and van Zijl (1987) T. Sjöstrand, M. van Zijl,  A multiple-interaction model for the event structure in hadron collisions,  Phys. Rev. D 36 (1987) 2019–2041. URL: https://link.aps.org/doi/10.1103/PhysRevD.36.2019. doi:10.1103/PhysRevD.36.2019.
  * Sjostrand and Skands (2004) T. Sjostrand, P. Z. Skands,  Multiple interactions and the structure of beam remnants,  JHEP 03 (2004) 053\. doi:10.1088/1126-6708/2004/03/053. arXiv:hep-ph/0402078.
  * Argyropoulos and Sjöstrand (2014) S. Argyropoulos, T. Sjöstrand,  Effects of color reconnection on $t\bar{t}$ final states at the LHC,  JHEP 11 (2014) 043\. doi:10.1007/JHEP11(2014)043. arXiv:1407.6653.
  * Sirunyan et al. (2020) A. M. Sirunyan, et al. (CMS and TOTEM Collaborations),  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 (2020) 1164. doi:10.1140/epjc/s10052-020-08562-y. arXiv:2002.12146, [Erratum: Eur.Phys.J.C 81, 383 (2021)].
  * Abelev et al. (2013) B. B. Abelev, et al. (ALICE Collaboration),  Multiplicity dependence of the average transverse momentum in pp, p-Pb, and Pb-Pb collisions at the LHC,  Phys. Lett. B 727 (2013) 371–380. doi:10.1016/j.physletb.2013.10.054. arXiv:1307.1094.
  * Acharya et al. (2023a) S. Acharya, et al. (ALICE Collaboration),  Multiplicity dependence of charged-particle production in pp, p-Pb, Xe-Xe and Pb-Pb collisions at the LHC,  Phys. Lett. B 845 (2023a) 138110. doi:10.1016/j.physletb.2023.138110. arXiv:2211.15326.
  * Acharya et al. (2023b) S. Acharya, et al. (ALICE Collaboration),  Search for jet quenching effects in high-multiplicity pp collisions at $\sqrt{s}$ = 13 TeV via di-jet acoplanarity (2023b). arXiv:2309.03788.
  * Ortiz et al. (2024) A. Ortiz, L. Valencia Palomo, V. M. M. Neriz,  Study of Transverse-Spherocity Biased pp Collisions at the LHC Energies Using the PYTHIA 8 Event Generator,  Universe 10 (2024) 30. doi:10.3390/universe10010030. arXiv:2310.20406.
  * Prasad et al. (2015) S. K. Prasad, S. Das, S. K. Ghosh, P. Ghosh, S. Muhuri, T. K. Nayak, R. Ray,  ”Soft” and ”hard” interactions in proton-proton collisions at LHC energies,  Proc. Indian Natl. Sci. Acad. 81 (2015) 213–216. doi:10.16943/ptinsa/2015/v81i1/48071.
  * Wang and Wang (2001) E. Wang, X.-N. Wang,  Interplay of soft and hard processes and hadron ${p}_{T}$ spectra in $\mathrm{pA}$ and $\mathrm{AA}$ collisions,  Phys. Rev. C 64 (2001) 034901. URL: https://link.aps.org/doi/10.1103/PhysRevC.64.034901. doi:10.1103/PhysRevC.64.034901.
  * Aaboud et al. (2016) M. Aaboud, et al. (ATLAS Collaboration),  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 (2016) 502. doi:10.1140/epjc/s10052-016-4335-y. arXiv:1606.01133.
  * Acosta et al. (2002) D. Acosta, et al. (CDF Collaboration),  Soft and Hard Interactions in $p\bar{p}$ Collisions at $\sqrt{s}=$ 1800-GeV and 630-GeV,  Phys. Rev. D 65 (2002) 072005. doi:10.1103/PhysRevD.65.072005.
  * Giacalone et al. (2021) G. Giacalone, F. G. Gardim, J. Noronha-Hostler, J.-Y. Ollitrault,  Skewness of mean transverse momentum fluctuations in heavy-ion collisions,  Phys. Rev. C 103 (2021) 024910. doi:10.1103/PhysRevC.103.024910. arXiv:2004.09799.