The ratios of the MC predictions to  the experimental results are shown.

Bands represent the uncertainties of the experimental results.

Taken from Ref. <cit.>.

Top panel:

The average transverse momentum \(\langle p_{\mathrm{T}}\rangle \) as
a function of the  multiplicity, at the centre-of-mass energy  \(\sqrt{s}= 7\)
<cit.>	   with
(a) \(n_{\mathrm{ch}} \ge 2\), \(p_{\mathrm{T}} >100\)
(b) \(n_{\mathrm{ch}} \ge 1\), \(p_{\mathrm{T}} >500\) .

The data represented by dots are compared to various particle-level MC predictions,
which are shown by curves.

The shaded areas around the data points represent the total statistical and systematic uncertainties added in quadrature.

Bottom panel:

The ratios of the MC predictions to  the experimental results are shown.

Bands represent the uncertainties of the experimental results.

Taken from Ref. <cit.>.

Top panel:

The average transverse momentum \(\langle p_{\mathrm{T}}\rangle \) as
a function of the  multiplicity at the centre-of-mass energy  \(\sqrt{s}= 0.9\)
<cit.>	   with
(a) \(n_{\mathrm{ch}} \ge 2\), \(p_{\mathrm{T}} >100\)
(b) \(n_{\mathrm{ch}} \ge 1\), \(p_{\mathrm{T}} >500\) .

The data represented by dots are compared to various particle-level MC predictions,
which are shown by curves.

The shaded areas around the data points represent the total statistical and systematic uncertainties added in quadrature.

Bottom panel:

The ratios of the MC predictions to  the experimental results are shown.

Bands represent the uncertainties of the experimental results.

Taken from Ref. <cit.>.

The charged-particle average  transverse momentum distributions
are shown in Figs. <ref> – <ref>
at  the CM energies  \(\sqrt{s}= 0.9,\ 2.36,\ 7,\ 8,\) and  \(13\) .

The average  transverse momentum versus the primary charged-particle multiplicity is shown in
Fig. <ref> at  \(\sqrt{s}= 13\)   for
\(n_{\mathrm{ch}} \ge 2\), \(p_{\mathrm{T}} >100\)
<cit.> and \(n_{\mathrm{ch}} \ge 1\),   \(p_{\mathrm{T}} >500\)
<cit.>,  respectively.

For \(p_{\mathrm{T}} >100\)  in Fig. <ref>(a)
it increases towards higher \(n_{\mathrm{ch}}\),
as modelled by a colour reconnection mechanism in  Pythia 8
and by the hydrodynamical evolution model in  EPOS.

QGSJET-II  generator, which has no model for colour coherence effects,  describes the data poorly.

For low \(n_{\mathrm{ch}}\),
Pythia 8 A2 and EPOS underestimate the data,
where Pythia 8 Monash agrees within the uncertainties.

For higher \(n_{\mathrm{ch}}\) all generators overestimate the data, but for \(n_{\mathrm{ch}} > 40\),
there is a constant offset for both Pythia 8 tunes, which describe the data to within  \(10\)%.

EPOS describes the data reasonably well and to within \(2\)%.

Figure <ref>(b) for
\(n_{\mathrm{ch}} \ge 1\),  \(p_{\mathrm{T}} >500\)
shows the mean transverse momentum versus the charged-particle multiplicity.

The  \(\langle p_{\mathrm{T}}\rangle\) rises with \(n_{\mathrm{ch}}\),  from \(0.8\)  to \(1.2\) .

This increase is expected due to colour coherence effects  being important in dense parton environments and
is modelled by the  colour reconnection mechanism in  Pythia 8  or by
the hydrodynamical evolution model used in  EPOS.

If the high-\(n_{\mathrm{ch}}\) region is assumed  to be dominated by events with numerous MPI,
without colour coherence effects the   \(\langle p_{\mathrm{T}}\rangle\)
is approximately independent of  \(n_{\mathrm{ch}}\).

Inclusion of
colour coherence effects leads to fewer  additional charged particles produced with every
additional MPI, with an equally large \(p_{\mathrm{T}}\)  to be shared among the produced hadrons

EPOS predicts a slightly lower  \(\langle p_{\mathrm{T}}\rangle\)

but describes the dependence on \(n_{\mathrm{ch}}\) very well.

The Pythia 8  tunes predict a steeper rise of
\(\langle p_{\mathrm{T}}\rangle\)
with \(n_{\mathrm{ch}}\) than the data, predicting lower values in the low-\(n_{\mathrm{ch}}\)
region and higher values in the high-\(n_{\mathrm{ch}}\) region.

predicts a \(\langle p_{\mathrm{T}}\rangle\) of  \( \sim  1\) ,
with very little dependence on \(n_{\mathrm{ch}}\); this is expected
as it contains no model for colour coherence effects.

Similar plots as for \(13\)  are also shown for \(8\)  in Fig. <ref>
for transverse momentum thresholds of \(100\)  and \(500\) , respectively.

The average \(p_{\mathrm{T}}\) rises with multiplicity although the
rise becomes progressively less steep as the multiplicity increases.

This is expected due to colour coherence effects in dense parton environments, which are modelled
by a colour reconnection mechanism in Pythia 8 or by the hydrodynamical evolution model
used in EPOS.

It is assumed that numerous MPI
dominate the high-multiplicity events, and that colour coherence effects thereby lead to fewer additional
charged particles produced with every additional MPI, which share a higher average \(p_{\mathrm{T}}\).

The EPOS and Pythia 8 models provide a fair description of the data.

The QGSJET-II
model fails to predict the mean transverse momentum over the entire multiplicity range,
as it does not simulate colour coherence effects and therefore shows very little dependence on the multiplicity.

Figures <ref> and  <ref> show  the results for events  at the
CM energies  \(\sqrt{s}= 7\)  and  \(\sqrt{s}= 0.9\) for
\(n_{\mathrm{ch}} \ge 2\), \(p_{\mathrm{T}} >100\)
\(n_{\mathrm{ch}} \ge 1\),  \(p_{\mathrm{T}} >500\) ,

Globally one can say that
at \(\sqrt{s}= 0.9\)  the slope versus \(n_{\mathrm{ch}}\)
for high values of \(n_{\mathrm{ch}}\) seems to be well described by most
models,  but the absolute value is best modelled by
Pythia 6 DW.

At the highest  CM energy (\(8\) and \(13\) ) above  multiplicity of \(20\)
the models vary widely both in slope and in absolute value;
at low values of \(n_{\mathrm{ch}}\)  none of the models describe the data very well.

In the more inclusive  PS region,
Figs.<ref>(a) and  <ref>(a),
the models vary widely, especially at \(\sqrt{s}= 7\) .

The measurement of \(\langle p_{\mathrm{T}}\rangle \) as a function of
the charged multiplicity at  \(\sqrt{s}=  2.36\)
is not shown because different track reconstruction methods are used for determining
\(p_{\mathrm{T}}\) and multiplicity distributions.

Top panel:

The Pythia 8 A3, A2 and Monash  tune predictions  compared with  the
ATLAS charged-particle  average transverse momentum,  \(\langle p_{\mathrm{T}}\rangle \),
distributions for events with
\(n_{\mathrm{ch}} \ge 1\) with \(p_{\mathrm{T}} >500\)
at  the center-of-mass energies (a) \(13\) , (b) \(8\) , (c) \(7\) ,
and (d) \(0.9\) .

The yellow shaded areas represent the measurement uncertainty.

Bottom panel:

The ratios of the MC predictions to  the experimental results  are shown.

Bands represent the uncertainties of the experimental results.

Taken from Ref. <cit.>.

In Fig. <ref>,  which shows the mean transverse momentum,
\(\langle p_{\mathrm{T}}\rangle \), against  the  charged particle multiplicity correlation
the choice of lower colour reconnection strength led to slight improvement over
Pythia 8 A2.

Although  \(\sqrt{s} = 2.36\)   <cit.>
and  \(\sqrt{s} = 8\)     charged particle distributions
were not used in tuning, comparisons are  made with those distributions for completeness.

In  Figs. <ref>,  <ref>,  <ref>  and  <ref>
distributions at   \(\sqrt{s} = 7\)  and \(\sqrt{s} = 13\)
predicted by  Pythia 8 A3,  in compared to
Pythia 8 A2,  show a broadly comparable,  or better, level of agreement.

Pythia 8 A2   demonstrates  that an acceptable description of data
can be achieved by using the

DL model for diffraction  and can be viewed
as a possible starting point  for further systematic studies of soft-QCD tunes.

The results of  Pythia 8 A3  provide good reasons to believe that an improved and more
reliable simulation of pile-up overlay can be obtained.

Top panel:

The average transverse momentum,  \(\langle p_{\mathrm{T}}\rangle \),
as a function of the scaled multiplicity  \(z\)  defined  by  Eq. (<ref>) for events with
(a)  \(n_{\mathrm{ch}} \ge 2\),   \(p_{\mathrm{T}} >100\) MeV  and
(b)  \(n_{\mathrm{ch}} \ge 1\),   \(p_{\mathrm{T}} >500\) MeV
for  \(\mid\eta\mid  < 2.5\)  measurement   at the  centre-of-mass energies
\(0.9\), \(7\), \(8\) and  \(13\)   by the ATLAS

(a) The gray curve and  the  band of the uncertainties are the result of the interpolation
of the charged-particle multiplicity distribution at  \(13\) .

The gray curve and band of the uncertainties are the result of the interpolation
of the charged-particle multiplicity distribution at  \(13\) .

The error bars and boxes represent the statistical and systematic contributions, respectively.

Bottom panel:

The ratios of the average transverse momentum distributions to the interpolated distribution
at \( \sqrt{s}  = 13\)  are shown.

Bands represent the uncertainties for the ratios
as results of statistical and systematic uncertainties added in quadrature for both distributions.

Taken from Ref. <cit.>.

The correct comparison  of the primary charged-particle average transverse momentum,
\(\langle p_{\mathrm{T}}\rangle \),  as a function of the  scaled multiplicity
\( z\) for events with
\(n_{\mathrm{ch}} \ge 2\) and \(p_{\mathrm{T}} >100\) ;
\(n_{\mathrm{ch}} \ge 1\) and \(p_{\mathrm{T}} >500\)
measure for  \(\mid\eta\mid  < 2.5\)  at the   CM energies
from \(0.9\)  to  \(13\)   by the ATLAS
are presented in  Fig. <ref> <cit.>.

The \(\langle p_{\mathrm{T}}\rangle \) distribution  as a function of \( z \) acquires
a higher value at higher collision energies.

The values of \(\langle p_{\mathrm{T}}\rangle \)  distributions increases by \( 18\)%  and  \(13\)%
for \(z > 1\)   with energy increase from \(0.9\) to \(13\)    for
\(p_{\mathrm{T}} >100\)  and \(p_{\mathrm{T}} >500\) ,  respectively.

The results at \(7\) and \(8\)  are in agreement within error bars.

The values of \(\langle p_{\mathrm{T}}\rangle \) distributions  increases by
\(\approx 3\)% for \(p_{\mathrm{T}} >100\)
and  by \(\approx 2.5\)% for \(p_{\mathrm{T}} >500\) with increase in energy from \(8\) to \(13\)   for  \(z > 0.5\).

The ratio of  \(\langle p_{\mathrm{T}}\rangle \) distributions
for  \(8\) to \(13\)   are  \(\approx 6\) times  smaller than the ratio for \(0.9\)  to  \(13\) .

For \( p_{\mathrm{T}} > 100\)  and  \( p_{\mathrm{T}} > 500\)   at
the highest energies distributions  increase towards higher \( n_{\mathrm{ch}}\),  as modelled by
CR mechanism in  Pythia 8 and by the hydrodynamical evolution model in

The QGSJET-II  generator describes the data poorly.

For low \( n_{\mathrm{ch}}\),  Pythia 8  A2, EPOS
underestimate the data and for higher \( n_{\mathrm{ch}}\) all generators overestimate the data.

EPOS describes the data reasonably well and to within \(2\)%.

As discussed in  Ref. <cit.>,
the  \(\langle p_{\mathrm{T}} (n) \rangle\)  of distributions of primary charged particles  was
produced via jet fragmentation,   slowly  increases with collision energy,  as shown in
Fig. <ref>.

This is caused by the stronger absorption  (at larger \(\sqrt{s}\))  of the gluons with a smaller
\( k_{\mathrm{T}} \)  (\(\sigma^{abs} \propto  1 / k_{\mathrm{T}}^2\)).

The growth of  \( \langle p_{\mathrm{T}}  \rangle\)  with multiplicity  can be explained by the
fact that events with larger  \( n_{\mathrm{ch}} \)  correspond to a smaller impact parameter, \(b\),
where the absorption of  the low  \( k_{\mathrm{T}} \)   component is stronger,  and
larger multiplicity can be originated  by the events with jets/minijets with higher  \( p_{\mathrm{T}} \).

Since  \( \langle p_{\mathrm{T}}  \rangle\)  of primary charged particles grows with \(\sqrt{s}\),
the increase with \(\sqrt{s}\) of transverse energy flow   is a bit faster than that of  the particle density.

§.§.§ Average transverse momentum distributions of the LHC experiments

Top panel:

A comparison of CMS  results  <cit.> for  the average transverse momentum,
\(\langle p_{\mathrm{T}} \rangle\),   as a function of of  the charge-particle multiplicity, \(n_{\mathrm{ch}}\),
for inclusive pseudorapidity  region  \( \mid\eta\mid < 2.4\) with  the prediction of
the Pythia  D6T tune,
the Pythia 8
and PHOJET models  at
\(\sqrt{s} = 0.9,\ 2.36\) and \(7\) .

For clarity,  the results for different energies are shifted by the  values  shown in the plots.

Fits to the high-multiplicity part  (\(n_{\mathrm{ch}} > 15 \))  with a linear form in
\(\sqrt{n_{\mathrm{ch}}}\)  are superimposed.

Bottom panel:

The ratios of the higher-energy data to the fit at \(\sqrt{s} = 0.9\)
indicate the approximate energy independence of
\(\langle p_{\mathrm{T}} \rangle\)  at fixed  \(n_{\mathrm{ch}}\).

Taken from Ref. <cit.>.

Figure <ref> (top)  show  a CMS comparison of   the
average transverse momentum,  \(\langle p_{\mathrm{T}} \rangle\),
as a function  of  the charge-particle multiplicity, \(n_{\mathrm{ch}}\),
for  the inclusive pseudorapidity  region  \( \mid\eta\mid < 2.4\)
with prediction of  the Pythia  D6T tune,
the Pythia 8  and PHOJET models  at  \(\sqrt{s} = 0.9,\ 2.36\) and \(7\)

In Fig. <ref> (bottom)  the ratios of the higher-energy data to the fit at
\(\sqrt{s} = 0.9\)   indicate the approximate energy independence of
\(\langle p_{\mathrm{T}} \rangle\)  at fixed  \(n_{\mathrm{ch}}\).

These results are in disagreement with  the ATLAS results presented in
Fig. <ref>,  where a ratio  depends on the multiplicity.

The ATLAS  ratio of  \( \langle p_{\mathrm{T}} \rangle \) distributions for
\( 7 \)   to  \( 0.9 \)  is  \(\approx 1.18\)
for \(z \gtrsim 2\) as shown in Fig. <ref>(a).

According to the  CMS,  the same ratio shown in Fig. <ref>
is  \(\approx 1.05\)  for  \( n_{\mathrm{ch}} \gtrsim 30\)  or  \(z \gtrsim 1\),
because   \(\langle n_{\mathrm{ch}} \rangle = 30.4 \) at \(7\) in Table <ref>.

That is  \(\approx 3.5\) times smaller than for ATLAS.

Among the three classes of models,  Pythia 8
gives the best overall description of the multiplicity  distribution and the dependence of the
average transverse momentum on  \(n_{\mathrm{ch}}\).

Inspired by  <cit.> the fit  by the first-degree polynom in  \(\sqrt{ n_{\mathrm{ch}} }\)
to the multiplicity dependence of  \(\langle p_{\mathrm{T}} (n_{\mathrm{ch}} ) \rangle\) for
\(n_{\mathrm{ch}} > 1.5\) at each energy, yielding a good description which is valid at all three energies.

The ratios of the data obtained at  \(\sqrt{s} = 7\)    and  \(\sqrt{s} = 2.36\)
with respect to the data at  \(\sqrt{s} = 0.9\)    show that the rise of the average
transverse momentum with the multiplicity  weakly depends on energy.

average charged-particle transverse momentum,  \( \langle p_{\mathrm{T}} \rangle \),  and
normalized on  \( \langle p_{\mathrm{T}} \rangle_{\mathrm{incl}} \),
\( \langle p_{\mathrm{T}} \rangle / \langle p_{\mathrm{T}}\rangle_{\mathrm{incl}} \),
distributions as a function of the scaled multiplicity  \(z\)
or \( N_{\mathrm{ch}} /\langle N_{\mathrm{ch}} \rangle \) for
\(pp\), \(p\)-\(Pb\),  \(Xe\)-\(Xe\) and \(Pb\)-\(Pb\) collisions
at the different centre-of-mass energies
\(\sqrt{s} = 2.36,\ 5.02,\ 7,\ 8\) and  \(13\)  for \(pp\),
\(\sqrt{s} = 5.02\) and  \(8.16\)  for \(p\)-\(Pb\),
\(\sqrt{s} = 5.44\)  for \(Xe\)-\(Xe\)
and \(\sqrt{s} = 2.76\) and  \(5.02\)  for \(Pp\)-\(Pb\)
for events in the kinematic range \( N_{\mathrm{ch}} > 0\),  \( \mid\eta\mid < 0.8 \)
and \( 0.15 < p_{\mathrm{T}} < 10 \) .

Statistical and systematic uncertainties are shown as bars and semi-transparent bands, respectively.

The ratio of  Pythia 8 <cit.> and EPOS LHC  <cit.>
model predictions to data for \(pp\) collisions at various energies are shown  for
\( \langle p_{\mathrm{T}} \rangle \)  distributions.

The semi-transparent bands indicate the relative systematic uncertainties of the data.

Taken from Ref. <cit.>.

The  average  charged-particle transverse momenta
for \( pp\)  collisions at the different centre-of-mass energies
\(\sqrt{s} = 2.36,\ 5.02,\ 7,\ 8\) and  \(13\)
for the kinematic region
\( \mid\eta\mid < 0.8 \)  and  \( 0.15 < p_{\mathrm{T}} < 10 \) were obtained by the ALICE experiment
and are presented

in Fig. <ref>.

Fig. <ref>(a)

the average charged-particle transverse momentum
\( \langle p_{\mathrm{T}} \rangle \)  spectra
in Fig. <ref>(b)
\( \langle p_{\mathrm{T}} \rangle \) spectra

divided by their respective multiplicity-integrated values,

\( \langle p_{\mathrm{T}} \rangle_{\mathrm{incl}} \),
as a function of relative multiplicity

\( N_{\mathrm{ch}} /\langle N_{\mathrm{ch}} \rangle \),
same as the scale variable \(z\),
are  shown.

The value of
\( \langle p_{\mathrm{T}} \rangle_{\mathrm{incl}} \)
for \(pp\)  collisions
increase from
\( 6.05 \pm 0.17 \) at \(\sqrt{s} = 2.76\)
\( 9.48 \pm 0.07 \) at \(\sqrt{s} =13\) (see in  Table 2 <cit.>).

The values for each collision system align almost perfectly for the
\( \langle p_{\mathrm{T}} \rangle / \langle p_{\mathrm{T}} \rangle_{\mathrm{incl}} \).

In \(pp\) collisions, the overall shapes of the  \( \langle p_{\mathrm{T}} \rangle \)
distributions are shown in Fig. <ref>(c)
in comparison with predictions from  Pythia 8  <cit.> (solid lines)
and  EPOS LHC   <cit.> (dashed lines).

Pythia 8  underpredicts the experimental data on
\( \langle p_{\mathrm{T}} \rangle \)
at the lowest values of
\(N_{\mathrm{ch}} \)
by up to \(4\)%.

The \(N_{\mathrm{ch}} \)  dependent
\( \langle p_{\mathrm{T}} \rangle \)
values produced by
Pythia 8
increase faster than the measurements with an almost linear dependence up to
\(N_{\mathrm{ch}} \approx 20 \),

after which the ratio shows a flat multiplicity dependence with an offset from unity varying from
\(0.5\)% at\(\sqrt{s} = 5.02\)    up to \(4\)% at the highest CM energy.

EPOS LHC is further off at low multiplicities by up to \(5\)%
and increases slower than the measurements, underestimating themby up to \(6\)% around
\(N_{\mathrm{ch}} \approx 9 \).

At higher multiplicities, the increase is faster with a linearly rising ratio up to
\(N_{\mathrm{ch}} \approx 20 - 30 \),
reaching a plateau which describes the measurements within \(\pm 2\)%.

§ KNO SCALING

§.§ Study of the KNO scaling using the ATLAS results

Deviation from  the KNO scaling was already observed long ago at  the ISR energies
in \(pp\) collisions at \(\sqrt{s}\) from  \(0.0304\) to \(0.0622\) ,
in the full  PS, for inelastic events

For hadron-hadron collisions,   the approximate KNO scaling holds up to  the ISR energies

On the other hand, for NSD collisions, scaling was still found to be present
<cit.>, suggesting that diffractive processes might also play a role in KNO scaling violations.

In \(e^+ e^-\) collisions, at  \(\sqrt{s}\) from  \(0.005\) to \(0.034\) ,
the KNO scaling was found to  hold within \(\pm 20\)%  <cit.>.

Clear scaling violations become manifested  above   \( \sqrt{s} \approx  0.2\)
both for the multiplicity distributions in full  PS and  in central pseudorapidity ranges

In \(p\bar{p}\)) collisions at the CERN collider at  \(\sqrt{s} = 0.2\), \(0.546\) and \(0.9\) ,
the KNO scaling was found to be violated for NSD collisions in full  PS

Nevertheless, for NSD collisions, in limited central pseudorapidity intervals,  the
KNO scaling was still found to hold up to \(0.9\) ,   and at  (\(\sqrt{s} = 0.546\) ,  the
KNO scaling was found to hold in the pseudorapidity interval  \(\mid\eta\mid < 3.5\)

In  \(  p \bar{p}\) collisions,  and for large rapidity ranges,
the UA5 experiment was the first to observe a larger-than-expected  high-multiplicity tail and a change of slope
which was interpreted as evidence for  a multi-component structure  of the final states

In NSD  \(pp\) collisions at the LHC, at  \(\sqrt{s} = 2.36\)   and \(7\)
and in \(\mid\eta\mid < 0.5\),  ALICE
and  CMS  <cit.> observed no significant deviation from  the KNO scaling.

On the other hand the CMS  observation of strong KNO scaling violations at \(\sqrt{s} = 7\) ,
as well as a change of slop in \( P_{\mathrm{n}} \),  confirm the earlier measurements.

Top panel:

KNO-scaled primary charged-particle  multiplicity distributions
as a function of the scaled multiplicity  \(z\)
for events with  \(n_{\mathrm{ch}} \ge 2\), \(p_{\mathrm{T}} >100\)
and \(\mid\eta\mid  < 2.5\)  measured   at the centre-of-mass energies
\(0.9\),  \(7\),  \(8\) and \(13\)   by the ATLAS
(a) the complete multiplicity region and
(b) the zoom multiplicity region with \(z \le 3\)
at  the  \(\sqrt{s} = 7\),  \(8\) and \(13\) .

The gray curve and  the band of the uncertainties are the result of the interpolation
of the charged-particle multiplicity distribution at  \(13\) .

The uncertainties represent the sum in quadrature of the statistical and systematic contributions.

Bottom panel:

The ratios of the KNO-scaled primary charged-particle distributions
to the interpolated distribution at \( \sqrt{s}  = 13\)  are shown.

Bands represent the uncertainties for the ratios
as results of statistical and systematic uncertainties added in quadrature for both distributions.

Taken from Ref. <cit.>.

Top panel:

KNO-scaled primary charged-particle multiplicity distributions  as a function of the scaled multiplicity
\(z\) for events with  \(n_{\mathrm{ch}} \ge 1\), \(p_{\mathrm{T}} >500\)
and \(\mid\eta\mid  < 2.5\)  measurement  at the centre-of-mass energies
\(0.9\),  \(2.36\), \(7\),  \(8\) and \(13\)   by the ATLAS
(a) the complete multiplicity region and
(b) the zoom multiplicity region with \(z \le 3\)  at  the  \(\sqrt{s} = 7\),  \(8\) and \(13\) .

The gray curve and band of the uncertainties are the result of the interpolation
of the charged-particle multiplicity distribution at  \(13\) .

The uncertainties represent the sum in quadrature of the statistical and systematic contributions.

Bottom panel:

The ratios of the KNO-scaled primary charged-particle
distributions to the interpolated distribution at \( \sqrt{s}  = 13\)  are shown.

Bands represent the uncertainties for the ratios
as results of statistical and systematic uncertainties added in quadrature for both distributions.

Taken from Ref. <cit.>.

The KNO variable \( z \) provides another way to study the evolution of the shape of
multiplicity distributions with varying CM energies  and pseudorapidity intervals.

For the verification of  the KNO scaling hypothesis  the following
equation with  dependence  on the CM energy and a kinematic region,
\( p_{\mathrm{T}}^{\mathrm{min}} \),
was used in Ref. <cit.>:

\begin{equation}
\label{eq_Psi_nch}
\Psi ( z , \sqrt{s})
= { \langle n_{\mathrm{ch}} (\sqrt{s}, p_{\mathrm{T}}^{\mathrm{min}}) \rangle }
\cdot
P (n_{\mathrm{ch}}, \sqrt{s}, p_{\mathrm{T}}^{\mathrm{min}})
= \frac{ \langle n_{\mathrm{ch}} (\sqrt{s}, p_{\mathrm{T}}^{\mathrm{min}}) \rangle }{ N_{\mathrm{ev}} (\sqrt{s},  p_{\mathrm{T}}^{\mathrm{min}} ) } \cdot \frac{d N_{\mathrm{ev}} (\sqrt{s}, p_{\mathrm{T}}^{\mathrm{min}} )}{ d n_{\mathrm{ch}}},
\end{equation}

\(n_{\mathrm{ch}}\)  is the number of primary charged particles within the kinematic acceptance in an event,
\( P (n_{\mathrm{ch}}, \sqrt{s}) \)  is   the  probability distributions  of producing
\( n_{\mathrm{ch}} \)  particles,
\(N_{\mathrm{ev}} \)  is the number of events with  primary charged particles in
the kinematic acceptance,  \( \langle n (\sqrt{s}) \rangle \)   is the average multiplicity of primary particles at
the  CM energy,  and  \( \Psi ( z ) \) is the  particle distribution as a function of the scaled multiplicity.

The KNO scale variable  \(z \) provides  a way to study evolution of shapes of the KNO charged-particle
multiplicity distributions (see Eq. (<ref>)) with varying  CM energy and kinematic region,
for example   \(p_{\mathrm{T}}^{\mathrm{min}}\) threshold.

The  KNO distributions and their ratios, studied using ATLAS results,
are presented in Fig. <ref>
for charged particles with \(p_{\mathrm{T}} >100\)  and in
Fig. <ref> for those with \(p_{\mathrm{T}} >500\) .

These figures are similar to  Fig. <ref> and Fig. <ref>
but  the vertical axis is stretched by the factor
\( \langle n_{\mathrm{ch}} (\sqrt{s}, p_{\mathrm{T}}^{\mathrm{min}})\rangle \).

The quantities of interest are derived from the original set of

KNO distributions and the  ratios  of  these distributions to the one at \(13\) .

The high-multiplicity tail of  the  distributions is pushed up and the maximum of the distribution
is shifted towards small values of \(z\) with  increasing collision energy.

Ratios of the KNO distributions between the smallest  CM energy  \(0.9\)  to  \(13\)
reach the maximum  value at  \( z \approx 0.8 \) and the

minimum value for the highest multiplicity  at \( z \approx 5.5 \)
for \(p_{\mathrm{T}} >100\) , as can be seen in
Fig. <ref>(a), and
\( z \approx 6.5 \) for \(p_{\mathrm{T}} >500\) , in
Fig. <ref>(a).

There is an intersection point for all distributions at  \( z \approx 2\).

A test of the KNO scaling distributions between  \(\sqrt{s} = 0.9\) and \(13\)
confirms that KNO scaling violation increases with decreasing collision energy.

Ratios of the KNO distributions  between the highest energies  \(8\)  and \(13\)
exceed   the maximum value of  \(+8\)% at  \( z \approx 0.5 \)  and
the minimum value of   \(-15\)% at \( z \approx 0.1 \)
for \(p_{\mathrm{T}} >100\) ,  as can be seen in
Fig. <ref>(b), and
the maximum value of  \(+5\)% at  \( z \approx 0.5 \)  and  \( -13 \)% at \( z \approx 0.1 \)
for \(p_{\mathrm{T}} >500\) ,  in  Fig. <ref>(b).

For the high multiplicity tail, these ratios are in agreement within error bars with the KNO distribution at
\(13\) .

Single- and double-diffractive processes  make an important  contribution only for the
low-multiplicity  region,  \(z \lesssim 0.3\).

The typologies of diffractive and non-diffractive events are different and their  KNO  behaviour
may also be  different.

The negative spread, \( \lesssim -8 \)%, for the low multiplicity may  be the  result
of  the contribution from  diffractive processes.

The KNO scaling   tends  to be  valid in the energy region from
\(\sqrt{s} = 7\) to \(\sqrt{s} =13\)  within
\( \approx \mbox{}^{\mbox{~}+8}_{-15} \)% for \( z \lesssim 2 \)
and  within error bars for \( z \gtrsim 2  \)   for events with  the
charged-particle transverse momentum  \(p_{\mathrm{T}} >100\)
(Fig. <ref>(b)),
and  within   \( \mbox{}^{\mbox{~}+5}_{-13} \)% for \( z \lesssim 3 \)
and  within error bars for \( z \gtrsim 3  \)   for events with  the
charged-particle transverse momentum  \(p_{\mathrm{T}} >500\)
(Fig. <ref>(b)).

The tendency of the KNO scaling to  hold  for the highest collision energies is observed.

The MC QGSM predictions are made for the KNO
non-diffractive charged-particle multiplicity distributions
for  \(pp\) collisions including at  the highest LHC  CM energy  \(\sqrt{s}= 14\)
for \(\mid\eta\mid <2.4\)  in Fig. 12 in Ref. <cit.>.

These distributions have the same qualitative behaviour as those presented in
Fig. <ref>(a).

The MC QGSM described the KNO  distributions as the contribution of
the cylinder diagram and diagrams with multi-Pomeron scattering.

The pronounced peak in the low \(z\)  arises solely due to a single Pomeron exchange,
and the maxima of  the distributions for multi-Pomeron processes are moved in the direction of
high \(z\)  thus  pushing  up the tail <cit.>.

The  normalised order-\(q\) moments   \(C_{\mathrm{q}}(\sqrt{s})\)
of the primary charged-particle multiplicity distributions   measured by  the ATLAS  experiment
for events  collected at \( \sqrt{s} = 0.9,\ 2.36,\ 7, 8\) and \(13\)  for
(a) the  pseudorapidity region  \( \mid\eta\mid < 2.5\).

The results of  CMS   <cit.> and lower-energy experiments
NA22   <cit.>,
UA1   <cit.>,   and
UA5  <cit.>  are included.

The ATLAS results for
\( \mid\eta\mid < 2.5\), \(n_{\mathrm{ch}} \ge 1\),  \(p_{\mathrm{T}} >500\) .

The ATLAS results for
\( \mid\eta\mid < 2.5\), \(n_{\mathrm{ch}} \ge 2\), \(p_{\mathrm{T}} >100\) MeV
with additional scaled multiplicity thresholds:  \( z  > 0.5 \) and  \( z > 1.0 \).

\(C_{\mathrm{2}}\),
\(C_{\mathrm{3}}\)
\(C_{\mathrm{4}}\),
\(C_{\mathrm{5}}\)
results are shown in the  bottom and top panels,  respectively.

The vertical bars  are the statistical uncertainties and  the  squares  are the systematic uncertainties.

The coloured symbols  are the data.

Fits of the  \( \log{\sqrt{s}}\)   dependence of the  \( C_{\mathrm{q}} (\sqrt{s}) \)
of the multiplicity distribution   (assuming linear dependence)   are shown.

In (a) for \(\sqrt{s} = 0.9\) ,
data from experiments other than ATLAS were drawn shifted to lower  \(\sqrt{s}\) for clarity.

The  lines show the results of the fits for  \(C_{\mathrm{q}}  (\sqrt{s}) \)  with statistical and systematic uncertainties added in quadrature.

Taken from Ref. <cit.>.

The energy independence of the moments of  the probability distributions  defined as
\( P (n_{\mathrm{ch}}, \sqrt{s}) \)

\begin{equation}
\label{eq_Cq}
C_{\mathrm{q}} (\sqrt{s}) =
\frac{ \sum_{n=1}^{n_{max}} n_{\mathrm{ch}}^\mathrm{q} (\sqrt{s}) P (n_{\mathrm{ch}}, \sqrt{s})
\left( \sum_{n=1}^{n_{max}} n_{\mathrm{ch}}  (\sqrt{s}) P (n_{\mathrm{ch}}, \sqrt{s}) \right)^\mathrm{q}
\end{equation}

in the energy asymptotic was the precise finding of the KNO scaling   <cit.>.

The analysis results for the validity of KNO scaling  is shown quantitatively in
Fig. <ref>  by the  \( C_{\mathrm{q}} (\sqrt{s}) \)
of the multiplicity distributions measured by the ATLAS and complemented with
the CMS  measurements at  \(\sqrt{s} =0.9,\ 2.36\) and \(7\)
<cit.> and   results of the  lower-energy experiments by
NA22  <cit.>, UA1  <cit.>,  and  UA5  <cit.>.

The \( C_{\mathrm{q}} (\sqrt{s}) \)  calculations based on  the ATLAS results  for  the kinematic region
\( \mid\eta\mid < 2.5\), \(n_{\mathrm{ch}} \ge 2\) and  \(p_{\mathrm{T}} >100\) are shown in  Fig. <ref>(a).

The  ATLAS and CMS  results are  agree within the errors.

The values of  \( C_{\mathrm{q}} (\sqrt{s}) \)

for all  experiments   linearly increase with \( \log{\sqrt{s}}\) as illustrated by the fits in
Fig. <ref>(a).

Since, as mentioned above,  the KNO scaling requires  that \( C_{\mathrm{q}} (\sqrt{s}) \)
be independent of  energy,  one can state that the KNO scaling is violated at least for
the full region of scaled multiplicity.

Figure <ref>(b)  shows for the first time the values of  \( C_{\mathrm{q}} (\sqrt{s}) \)
calculated using multiplicity distributions  measured by ATLAS for the kinematic region
\( \mid\eta\mid < 2.5\), \(n_{\mathrm{ch}} \ge 1\) and  \(p_{\mathrm{T}} >500\) .

Similarly as in Fig. <ref>(a)  the values of  \( C_{\mathrm{q}} (\sqrt{s}) \)
linearly  increase  with \( \log{\sqrt{s}}\).

The \( C_{\mathrm{q}} \) values at \(\sqrt{s} = 2.36\) TeV  in Fig. <ref>(b)
are much smaller  than those for other energies.

This is because the  region of primary charged-particle multiplicity distributions at  \(2.36\)
is  smaller (up to \(z \approx 3.5\))  than that for higher  CM  energies (up to \(z \approx 9\))

Therefore,  the \( C_{\mathrm{q}} \) values at \(\sqrt{s} = 2.36\)   were note used in the fits.

The  \( C_{\mathrm{q}} (\sqrt{s}) \)    for  \(p_{\mathrm{T}} >500\)
have  higher   bias  (\( \alpha \))  and  slope (\( \beta \))   of the fits than those for
minimum \(p_{\mathrm{T}}\) threshold,  the bias   increasing
from \(1.1\) at  \(q=2\) up to  \(2.1\) at  \(q=5\),
and  the slope   increasing from \(1.4\) at  \(q=2\) up to  \(2.6\) at  \(q=5\).

This is the result of  stronger interactions with  a higher \(p_{\mathrm{T}}\) threshold.

Figure <ref>(c)  shows moments  \(C_{\mathrm{q}}\)   for events with
\(n_{\mathrm{ch}} \ge 2\), \(p_{\mathrm{T}} >100\)   and for  \( z > 0.5 \)
without  the fraction of single and double diffraction events, which was accepted  by
the ATLAS minimum-bias trigger

In this case,  the values of  \( C_{\mathrm{q}} (\sqrt{s}) \)  are systematically higher than those for
full distributions  with  \( z > 0\) and   show a similar linear increase with \( \log{\sqrt{s}}\)
as is  illustrated in   Fig. <ref>(c).

For multiplicity distributions for \( z > 1.0 \)  the  values of  \( C_{\mathrm{q}} (\sqrt{s}) \)
at the highest energies  \( \sqrt{s} =7,\ 8\) and \(13\)  are  in agreement within error uncertainties,
as can be seen  in Fig. <ref>(c).

Therefore,  the energy independence of  the moments  of various orders can be considered
as a confirmation of  the KNO scaling.

§.§ Study of the KNO scaling at the LHC experiments

The CMS KNO-scaled primary charged-particle  multiplicity distributions    <cit.>
as a function of the scaled multiplicity  \(z\)  at  the centre-of-mass energies \(\sqrt{s} = 0.9\) and \(7\)
in two pseudorapidity intervals:
(a) \( \mid\eta\mid < 2.4\)
(b) \( \mid\eta\mid < 0.5\).

Taken from Ref. <cit.>.

Top panel:

Comparison of multiplicity distributions in KNO variables measured by  UA5
in \(  p \bar{p}\)  collisions at   \( \sqrt{s} = 0.2 \)  and  \(0.9\)     and
by ALICE   <cit.>
at   \(\sqrt{s} = 0.9 \)  and  \(2.36\) ,    for NSD events in  \( \mid\eta\mid < 0.5 \).

Bottom panel:

The ratio between  the ALICE measurements at \( \sqrt{s} = 0.9 \)  and \( 2.36 \)   is shown.

The error bars represent the combined statistical and systematic uncertainties.

Taken from Ref. <cit.>.

The KNO scaling violation   was studied   for  different pseudorapidity ranges  in LHC experiments by the CMS
<cit.>  and  the ALICE  <cit.>
at  the CM energies  from \( \sqrt{s} = 0.9 \) to \(8\) .

The multiplicity distributions obtained by the  CMS detector   are shown in  the KNO form
<cit.>  for  the pseudorapidity interval of   \(\mid\eta\mid < 2.4\)
in  Fig. <ref>(a),  which is close to  the similar ATLAS results with \(\mid\eta\mid < 2.5\),
and for a more  central pseudorapidity interval  \(\mid\eta\mid < 0.5\)   in  Fig. <ref>(b).

The variation of the ratio for  the central region  of \(0.9\) to \(7\)  with \(\mid\eta\mid < 0.5\) is
about \(\pm 15\)%  and agree with  \(1\) within error bars;   therefore  the KNO scaling  holds.

The variation of the ratio for  the full region with \(\mid\eta\mid < 2.4\) is  twice wider  \(\approx \pm 30\)%
and does not agree with \(1\) in error bars,   therefore  the KNO scaling is violated similar to  the ATLAS  data
in Fig. <ref>(a).

Scaling is a characteristic property of the multiplicity distribution in  cascade processes of a single jet
with self-similar branching  and   a fixed coupling constant

A similar conclusion  about the shape evolution of  the multiplicity distributions
like from Fig. <ref>(b) can be  extracted
from  Fig. <ref>(c),
where are compared  the ALICE measurements   plotted in terms of KNO variables
at the two energies and UA5 \(p \bar{p} \)  data at  \(\sqrt{s} = 0.2\) and \(0.9\) ,
for NSD collisions and pseudorapidity interval  \( \mid\eta\mid < 0.5\).

While the KNO scaling gives a reasonable description of the data from  \(\sqrt{s} = 0.2\) and \(2.36\) ,
the ratio between the  \(\sqrt{s} = 0.9\) and \(2.36\)
data shows a slight departure from unity above  \(z = 4\),
but  it is in agreement with unit within error bars.

Left side panel:

The ALICE  KNO scaled primary charged-particle  multiplicity distributions
as a function of the scaled multiplicity  \(z\) at  the centre-of-mass energies
\(\sqrt{s} = 0.9,\ 2.36,\ 7\) and \(8\)    in pseudorapidity intervals:
(top) 		\( \mid\eta\mid < 0.5\),
(middle) 	\( \mid\eta\mid < 1.0\)
(bottom) 	\( \mid\eta\mid < 1.5\)

The uncertainties represent the sum in quadrature of the statistical and systematic contributions.

Right side panel:

The ratios of the KNO scaled primary charged-particle distributions to the interpolated distribution
at  \( \sqrt{s}  = 0.9 \)   are shown.

Bands represent the uncertainties for the ratios
as results of statistical and systematic uncertainties added in quadrature for both distributions.

Taken from Ref. <cit.>.

The KNO test on the ALICE  results in the range   of \(0.9\) to \(8\)
<cit.> is presented in Fig. <ref>.

The KNO-scaled distributions and their ratios were obtained for each of the available combinations of
corrections with the same procedure used for multiplicity distribution measurements.

Bin-to-bin correlations were ignored when comparing KNO distributions and \(q\)-moments
at various CM energies.

Consequently, the relative errors obtained on the ratios are somewhat overestimated.

The ratios between   two highest energies and  \(0.9\)   exceed the value of \(2\) at
\(z > 5.5\), \(5\) and \(4.5\),
for \( \mid\eta\mid < 0.5\), \( \mid\eta\mid < 1.0\)  and \( \mid\eta\mid < 1.5\),  respectively,
Fig. <ref>.

This confirms that KNO scaling violation increases with the size of  increasing pseudorapidity interval.

The shape of the KNO scaling violation reflects the fact that the high-multiplicity tail  of the distribution increases
with energy and with size of pseudorapidity interval  faster  than that for low-multiplicity tail
(\( n_{\mathrm{ch}} \le 20\)).

A test of  the KNO scaling between  \(\sqrt{s} = 0.9\) to \(8\) confirms that KNO scaling violation increases with increasing \(\sqrt{s}\) and,  at a given  CM energy,
with increasing width of   pseudorapidity intervals.

This is similar to  the ATLAS result in Fig. <ref>(a).

The ALICE KNO scaled primary charged-particle  multiplicity distributions
as a function of the scaled multiplicity  \(z\) for \( pp\)  collisions at the different centre-of-mass energies
\(\sqrt{s} = 2.36,\ 5.02,\ 7,\ 8\) and  \(13\)  for events in the kinematic range
\( N_{\mathrm{ch}} > 0\),  \( \mid\eta\mid < 0.8 \)  and
\( 0.15 < p_{\mathrm{T}} < 10 \) .

Statistical and systematic uncertainties are shown as bars and semi-transparent bands, respectively.

The ratios of the KNO scaled primary charged-particle distributions to the interpolated distribution
at   \( \sqrt{s}  = 13 \)    are shown.

Statistical and systematic uncertainties are shown as bars and semi-transparent bands, respectively.

The ratio of  Pythia 8 <cit.> and EPOS LHC  <cit.>
model predictions to data for \(pp\) collisions at various energies for
the KNO scaling of primary charged-particle  multiplicity distributions

The semi-transparent bands indicate the relative systematic uncertainties of the data.

Taken from Ref. <cit.>.

The KNO test on the ALICE results  for \( pp\)  collisions at the different centre-of-mass energies
\(\sqrt{s} = 2.36,\ 5.02,\ 7,\ 8\) and  \(13\)    for the kinematic region
\( \mid\eta\mid < 0.8 \)  and  \( 0.15 < p_{\mathrm{T}} < 10 \) is presented in Fig. <ref>(a).

Figure <ref>(b)  shows the corresponding ratios of the KNO scaled multiplicity distributions
at various CM energies  relative to \(\sqrt{s} = 13\) .

The KNO scaling apparently holds  within \(\approx 30\)% for
CM energies from \(2.36\) to \(8\)  in relative to  \(\sqrt{s} = 13\) .

Figure <ref>(c)  compare measured results
for the respective KNO scaled multiplicity distributions with predictions from
Pythia 8  <cit.> (solid lines)
and  EPOS LHC   <cit.> (dashed lines).

Like for multiplicity distributions in Fig. <ref>(b),
the overall shapes of the  KNO-scaled distribution shown in Fig. <ref>(c)
are better described by  EPOS LHC,  while  Pythia 8  falls sharply off above
\( N_{\mathrm{ch}}/ \langle N_{\mathrm{ch}}\rangle \approx 4\)
and these models within \(25\)% agree with the experimental distributions
with larger deviations at highest multiplicities.

The ALICE KNO scaled primary charged-particle  multiplicity distributions
as a function of the scaled multiplicity  \(z\) for \( pp\)  collisions at the different centre-of-mass energies
\(\sqrt{s} = 2.36,\ 5.02,\ 7\) and  \(13\)  for events in the kinematic range
\( \mid\eta\mid < 0.8 \),  \( p_{\mathrm{T}} > 0.5\)   and
\( 5 \leq p_{\mathrm{T}}^{\mathrm{trig}} \leq 40\)   in
(a) the tran-max  and
trans-min regions for \(pp\) collisions at
\( \sqrt{s}=2.76,\ 5.02,\  7\) and \(13\) .

the KNO multiplicity distributions are normalized to that for \(pp\) collisions at
\( \sqrt{s} = 7\) .

The boxes and the error bars represent the systematic and statistical uncertainties, respectively.

Taken from Ref. <cit.>.

Figure <ref>  shows the  ALICE results  for the trans-max and trans-min UE regions
for charged-particle multiplicity distributions in KNO variables  for \(pp\) collisions at
\( \sqrt{s}=2.76,\ 5.02,\  7\) and \(13\)   <cit.>.

The trans-max and trans-min regions of UE refer to the sub-transverse regions  with
the largest and smallest charged-particle multiplicity  which have an enhanced sensitivity to
ISR-FSR and UE, respectively  <cit.>.

In the trans-max region, within 20%, the KNO-like scaling is observed in a wider range of multiplicity
(\(0<z<4\))  relative to the results reported in <cit.>,  while for higher  \(z\)  values (\( z > 4 \))
the scaling is broken.

It is worth noticing that for trans-max both contributions are considered:  UE and ISR-FSR.

If the effect of ISR-FSR is suppressed, i.e., exploiting the features of trans-min region,
the KNO-like scaling also holds for
\( 0 < z < 4\), and then for \( z > 4 \) the KNO-like scaling is still broken but a higher
\( z \) reach is achieved, especially for \( z>6\),  a larger violation is observed.

Events with high-multiplicity jets can contribute to the large violation of the scaling properties.

It was observed that for \(z > 3 \),  the number of uncorrelated seeds (or MPI) deviate from
the linear trend suggesting  the presence of high-multiplicity jets <cit.>.

The normalized moments \(C_{\mathrm{q}}\) of the primary charged-particle multiplicity distributions
measurement by the  CMS for events with CM energies  at \(\sqrt{s} = 0.9,\ 2.36\) and \(7\) for pseudorapidity regions
\( \mid\eta\mid < 0.5\).

\(C_{\mathrm{2}}\),
\(C_{\mathrm{3}}\)
\(C_{\mathrm{4}}\),
\(C_{\mathrm{5}}\)
are shown in  the  bottom and  top panels,  respectively.

The results of  lower-energy experiments
NA22  <cit.> and
UA5 <cit.>
are included.

Fits of the  \( \log{\sqrt{s}}\)   dependence of the normalized moments  \( C_{\mathrm{q}} \)
of the multiplicity distribution for   \( \mid\eta\mid < 0.5\) (assuming no dependence) are shown.

The uncertainties represent the sum in quadrature of the statistical and systematic contributions.

Taken from Ref. <cit.>.

The ALICE  CM energy dependence of the  moments   \(C_{\mathrm{q}}\)
(\(q = 2\) to \(4\), left-hand  scale, and  \(q = 5\), right-hand  scale)
of the multiplicity distributions for NSD events in three different pseudorapidity intervals:
(top) 		\( \mid\eta\mid < 1.5\),
(middle) 	\( \mid\eta\mid < 1.0\)
(bottom) 	\( \mid\eta\mid < 0.5\)

ALICE   data (black) are compared to
UA5  <cit.> (red) for \( \mid\eta\mid < 0.5\) and \( \mid\eta\mid < 1.0\)
at \(\sqrt{s} = 0.9\)
and CMS  <cit.> (blue) at \(\sqrt{s} = 0.9\) and \(7\)   for \( \mid\eta\mid < 0.5\).

The error bars represent the combined statistical and systematic uncertainties.

The data at \(\sqrt{s} = 0.9\) and \(7\)   are slightly displaced horizontally for visibility.

Taken from Ref. <cit.>.

Multiplicity distributions may be characterized by their normalized  \(C_{\mathrm{q}}\)-moments
where  \(q\)  is a positive integer studied here for  the values  \(2,\ 3,\ 4\) and \(5\), for NSD events.

The results obtained by different experiments  for  the \(C_{\mathrm{q}}\)-moment  dependence
on \(\sqrt{s}\)  are shown in Fig. <ref>.

For  three pseudorapidity intervals
\( \mid\eta\mid < 0.5 \),
\( \mid\eta\mid < 1.0 \)  and
\( \mid\eta\mid < 1.5\),
\(C_{\mathrm{2}}\) remains constant over the energy range,
\(C_{\mathrm{3}}\) shows a small increase with increasing energy for
two largest \(\eta\) intervals,
\(C_{\mathrm{4}}\) and
\(C_{\mathrm{5}}\) show an increase with increasing energy,
which becomes stronger for larger \( \eta \)  intervals.

These ALICE data are in agreement with   UA5 <cit.>  and
CMS <cit.>.

The results of KNO scaling research according to the data of   the ALICE,  CMS and  ATLAS experiments
have been  analysed.

The shape evolution of the multiplicity distributions with a collision energy at ATLAS
is studied in terms of KNO scaling variables  at \(\sqrt{s}\) from \(0.9\) to \(13\)
with   the inclusive \( \mid\eta\mid < 2.5 \) one.

The KNO scaling and \(C_{\mathrm{q}}\)-moments were studied by  the CMS
at \(\sqrt{s}\) from \(0.9\) to \(7\)   in central pseudorapidity  \( \mid\eta\mid < 0.5 \) region
and more inclusive  \( \mid\eta\mid < 2.4 \) regions,  and the ALICE  at \(\sqrt{s}\) from \(0.9\)
to \(8\)  in three pseudorapidity regions:  \( \mid\eta\mid < 0.5 \),  \( \mid\eta\mid < 1.0 \)
and  \( \mid\eta\mid < 1.5\).

The charged-particle multiplicity distributions on the KNO scale for all experiments
have the similar shape and   decrease with increasing collision energy.

For all experiments the KNO scaling  is  violated for energies from  \(0.9\) to \(7\) if taking into account more inclusive pseoudorapidity regions.

The ATLAS data  demonstrate the tendency for  the KNO scaling  to be independent of energy
for the highest energies.

The CMS results  show that the KNO scaling holds  for central pseudorapidity region, \( \mid\eta\mid < 0.5 \),
and is  dependent  of  the energy from \(\sqrt{s} = 0.9\) to \(7\) ,
because \(C_{\mathrm{q}}\)-moments demonstrate  independence of energy
and the shape of  the KNO function is  similar.

Another situation  is  for  the inclusive region \( \mid\eta\mid < 2.4 \) where
\(C_{\mathrm{q}}\)-moments  demonstrate the linear increase with energy.

The ALICE results have the KNO scaling violation for all pseudorapidity regions
in depending on the  energy from \(\sqrt{s} = 0.9\) to \(8\) ,
because \(C_{\mathrm{q}}\)-moments  linear increase with  \( \log{\sqrt{s}}\).

Ratios of the KNO distributions between the smallest   \( \sqrt{s} = 0.9\)   and  \(8\)
exceed the maximum positive value at  \( z \approx 0.5 \)  and the maximum positive  value for the
multiplicity  at  \( z \approx 4.5 \),  \( z \approx 5.5 \) and  \( z \approx 6.0 \)
for  the pseudorapidity intervals
\( \mid\eta\mid < 1.5 \),  \( \mid\eta\mid < 1.0 \)  and  \( \mid\eta\mid < 0.5\),  respectively.

There is an intersection point for all distributions at \( z \approx 2\).

The shapes at \( \sqrt{s} =7 \) and \( 8 \)   are similar and agree within error bars.

The  ALICE results show the tendency for  the KNO scaling   to be independent of energy for the highest energies.

Therefore, an investigation of the KNO scaling at energies   higher than \(13\)  is  important.

The validity of KNO scaling is shown more quantitatively in Fig. <ref>(a) for wider pseudorapidity region  and  for smaller pseudorapisity region,  \( \mid\eta\mid < 0.5\),
in Fig. <ref>(a)   by the normalized order-\(q\) moments \( C_{\mathrm{q}} \)
of the multiplicity distribution,  complemented with measurements at lower energies  experiments
NA22  <cit.> and  UA5 <cit.>.

For  \( \mid\eta\mid < 0.5\)   the values of  \( C_{\mathrm{q}} \)  remain constant
over the full  CM energy range,  as illustrated by the fits in  Fig. <ref>(a).

The KNO-scaling  study by ALICE  is carried out for the NSD event class only  so that
SD events,  which may have a different behaviour,  are not included in the data samples.

The ALICE data are consistent with  the UA5  \( p \bar{p} \) measurements at  \(0.9\) <cit.>.

The energy dependence of the reduced moments  \(C_{\mathrm{q}}\)   shown in
Fig. <ref>(b)
indicates a slight increase, which is not significant given the size of our systematic uncertainties.

Systematic uncertainties are assumed to be uncorrelated between energies.

§ CONCLUSIONS

The ATLAS  studied MB events in \(pp\)  interactions  at
the CM energies    \(\sqrt{s} = 0.9\), \( 2.36\), \(7\), \(8\) and \(13\)
for the absolute pseudorapidity region less than \(2.5\)  in five separate
PS regions
\(n_{\mathrm{ch}} \ge 2\),  \(p_{\mathrm{T}} > 100\) and
\(n_{\mathrm{ch}} \ge 1,\ 6,\ 20,\ 50\),  \(p_{\mathrm{T}} > 500\) recorded in 2010 – 2015.

The data were taken in the special  configuration of the
LHC with low beam currents and reduced beam focusing,
producing a low mean number of interactions per
bunch-crossing in the range \(0.003\) – \(0.007\).

The charged-particle multiplicity dependences on pseudorapidity,  charged-particle multiplicity  and
transverse momentum, as  well as  the dependence of the mean transverse momentum on multiplicity
were presented for
study the soft-QCD phenomena.

The measured distributions are presented as inclusive-inelastic distributions within a

region with minimal model-dependent corrections
to facilitate the comparison with models.

There variables are tuned in event generators using these MB measurements,
because there is a variability in modelling since non-perturbative QCD is used.

The results are compared to the predictions of more than ten
MC models tuned to a wide range of measurements.

Then variables  in  the MC event generators were tuned using  the MB measurements of
the LHC and Tevatron experiments,
because there was a variability in modelling since non-perturbative QCD was used.

This review  reported that  the multiplicity distribution is not described perfectly
by any of the models, there are large discrepancies especially at large multiplicities.

Having observed similar discrepancies at all measured energies, we conclude that
for every collision energy, model parameters usually need to be re-tuned in every
MC generator.

Reasonable agreement of  the tunes used in   the MC models  with the data were presented.

The  models
EPOS LHC,

Pythia 6
Pythia 8
show big troubles in describing the whole
spectrum in
but the best agreement is achieved with

A new ATLAS Pythia 8  A3 tune was presented
for result predictions at Run 3 of the LHC.

The comparisons of the charged-particle multiplicity   and   the average transverse momentum
distributions  on  the basis of  the scaled multiplicity

using the LHC experiments results

were presented.

The charged-particle multiplicity distributions on the KNO scale
have the  similar shape  and decrease with increasing energy.

The KNO scaling  was studied  using

the LHC  experiments  results.

A test of the KNO scaling between \(0.9\)  and \(13\)
confirms that the KNO scaling violation increases with decreasing collision energy.

The KNO distributions tend to be independent of energy for the highest energies.

The mean transverse momentum on the KNO scale has the same shape and increases with increasing energy.

§ ACKNOWLEDGEMENTS

Our thank the ATLAS collaboration for the excellent experimental results which were used in this review.

Special  thanks are to Edward K. Sarkisyan-Grinbaum  and   Stanislav Tokar  for very productive discussions.

Grateful to Pavel Tsiareshka for the technical support.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

titleReview of Particle Physics

titlePhenomenology of Soft QCD: The Role of Minimum-Bias
journaltitleAdv. Ser. Direct. High Energy Phys.

titleLHC Machine

family=ATLAS Collaboration,

titleThe ATLAS Experiment at the CERN Large Hadron Collider

family=ATLAS Collaboration,

titleCharged-particle multiplicities in \(pp\) interactions at
\(\sqrt{s} = 900\,\textrm{GeV}\) measured with the ATLAS detector at the
journaltitlePhys. Lett. B

family=ATLAS Collaboration,

titleCharged-particle multiplicities in \(pp\) interactions
measured with the ATLAS detector at the LHC
journaltitleNew J. Phys.

family=ATLAS Collaboration,

family=ATLAS Collaboration,

titleCharged-particle distributions in \(\sqrt{s} =
13\,\textrm{TeV}\) \(pp\) interactions measured with the ATLAS detector at
the LHC
journaltitlePhys. Lett. B

family=ATLAS Collaboration,

titleCharged-particle distributions at low transverse momentum in
\(\sqrt{s} = 13\,\textrm{TeV}\) \(pp\) interactions measured with the ATLAS
detector at the LHC
journaltitleEur. Phys. J. C

titleMeasurement of charged particle spectra in \(pp\) collisions
and nuclear modification factor $R_\mathrm{pPb}$ at
$\sqrt{s_{\mathrm{NN}}}=5.02$  with the ATLAS detector at the LHC

titleCharged hadron production in $p+Pb$ collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$  measured at high transverse momentum by
the ATLAS experiment

family=ALICE Collaboration,

titleThe ALICE experiment at the LHC
journaltitlePhys. Part. Nucl.

titleFirst proton-proton collisions at the LHC as observed with
the ALICE detector: Measurement of the charged particle pseudorapidity
density at s**(1/2) = 900-
journaltitleEur. Phys. J. C

family=ALICE Collaboration,

titleCharged-particle multiplicity measurement in proton–proton
collisions at $\sqrt{s} = 0.9$ and $2.36$  with ALICE at LHC
journaltitleEur. Phys. J. C

titleTransverse momentum spectra of charged particles in
proton-proton collisions at $\sqrt{s} = 900$  with ALICE at the LHC
journaltitlePhys. Lett. B

family=ALICE Collaboration,

titleCharged-particle multiplicity measurement in proton–proton
collisions at $\sqrt{s}=7$  with ALICE at LHC
journaltitleEur. Phys. J. C

titleMultiplicity dependence of the average transverse momentum
in $pp$, $p-Pb$, and $Pb-Pb$ collisions at the LHC
journaltitlePhys. Lett. B

titleCharged-particle multiplicity measurement with Reconstructed
Tracks in $pp$ Collisions at $\sqrt{s} = 0.9$ and $7$  with ALICE at the

family=ALICE Collaboration,

titleEnergy Dependence of the Transverse Momentum Distributions
of Charged Particles in \(pp\) Collisions Measured by ALICE
journaltitleEur. Phys. J. C

family=ALICE Collaboration,

titleCharged-particle multiplicities in proton–proton
collisions at $\sqrt{s} = 0.9$ to $8$
journaltitleEur. Phys. J. C

family=ALICE Collaboration,

titlePseudorapidity and transverse-momentum distributions of
charged particles in proton–proton collisions at $\sqrt{s} =
journaltitlePhys. Lett. B

family=ALICE Collaboration,

titleCharged-particle multiplicity distributions over a wide
pseudorapidity range in proton–proton collisions at $\sqrt{s} = 0.9$, $7$,
and $8$
journaltitleEur. Phys. J. C

family=ALICE Collaboration,

titleCharged-particle production as a function of multiplicity
and transverse spherocity in \(pp\) collisions at $\sqrt{s} =5.02$ and
journaltitleEur. Phys. J. C

titleMultiplicity dependence of charged-particle production in
\(pp\), \(p\)-\(Pb\), \(Xe\)-\(Xe\) and \(Pb\)-\(Pb\) collisions at the LHC

family=CMS Collaboration,

titleThe CMS Experiment at the CERN LHC

family=CMS Collaboration,

titleTransverse Momentum and Pseudorapidity Distributions of
Charged Hadrons in $pp$ Collisions at $\sqrt{s} = 0.9$ and $2.36$

family=CMS Collaboration,

journaltitlePhys. Rev. Lett.

family=CMS Collaboration,

titleCharged Particle Multiplicities in $pp$ Interactions at
$\sqrt{s}=0.9$, $2.36$, and $7$

titlePseudorapidity distributions of charged particles in $pp$
collisions at $\sqrt{s}= 7$  with at least one central charged

family=CMS Collaboration,

titleCharged particle transverse momentum spectra in $pp$
collisions at $\sqrt{s} = 0.9$ and $7$

family=CMS Collaboration,

note[Erratum: JHEP 02, 055 (2012)]
titleMeasurement of energy flow at large pseudorapidities in $pp$
collisions at $\sqrt{s} = 0.9$ and $7$

family=CMS Collaboration,

titlePseudorapidity distribution of charged hadrons in
proton–proton collisions at $\sqrt{s} = 13$
journaltitlePhys. Lett. B

titleMeasurement of pseudorapidity distributions of charged
particles in proton-proton collisions at $\sqrt{s} = 13$  by the CMS

family=CMS Collaboration,

titleMeasurement of charged particle spectra in minimum-bias
events from proton–proton collisions at $\sqrt{s} = 13$
journaltitleEur. Phys. J. C

family=TOTEM Collaboration,

titleThe TOTEM experiment at the CERN Large Hadron Collider

family=CMS, TOTEM Collaborations,

titleMeasurement of pseudorapidity distributions of charged
particles in proton–proton collisions at $\sqrt{s} = 8$  by the CMS and
TOTEM experiments
journaltitleEur. Phys. J. C

family=LHCb Collaboration,

titleThe LHCb Detector at the LHC

family=LHCb Collaboration,

titleMeasurement of charged particle multiplicities in $pp$
collisions at ${\sqrt{s} = 7}$  in the forward region
journaltitleEur. Phys. J. C

family=LHCb Collaboration,

family=LHCf Collaboration,

titleThe LHCf detector at the CERN Large Hadron Collider

family=TOTEM Collaboration,

titleMeasurement of the forward charged particle pseudorapidity
density in $pp$ collisions at $\sqrt{s} = 8$  using a displaced
interaction point
journaltitleEur. Phys. J. C

titleCharged Particle Multiplicity Distributions in \(p p\)
Collisions at ISR Energies
journaltitleNucl. Phys. B

titleCharged Multiplicity Distribution in p p Interactions at ISR
journaltitlePhys. Rev. D

titlePhase Space Dependence of the Multiplicity Distribution in
$\pi^+ p$ and $p p$ Collisions at 250-GeV/$c$
journaltitleZ. Phys. C

titleTransverse Momentum Spectra for Charged Particles at the
CERN Proton anti-Proton Collider
journaltitlePhys. Lett. B

family=UA1 Collaboration,

titleA Study of the General Characteristics of $p\bar{p}$
Collisions at $\sqrt{s} = 0.2$- to $0.9$-
journaltitleNucl. Phys. B

family=UA4 Collaboration,

titlePseudorapidity Distribution of Charged Particles in
Diffraction Dissociation Events at the CERN SPS Collider
journaltitlePhys. Lett. B

family=UA5 Collaboration,

titleCharged Particle Multiplicities at the CERN SPS Collider
journaltitlePhys. Lett. B

family=UA5 Collaboration,

titleComparison of $p \bar{p}$ and $p p$ Interactions at
journaltitlePhys. Lett. B

family=UA5 Collaboration,

titleScaling Violation Favoring High Multiplicity Events at
$540$- CMS Energy
journaltitlePhys. Lett. B

family=UA5 Collaboration,

titleAn Investigation of Multiplicity Distributions in Different
Pseudorapidity Intervals in anti-p p Reactions at a CMS Energy of
journaltitlePhys. Lett. B

family=UA5 Collaboration,

titleScaling Violations in Multiplicity Distributions at
$200$- and $900$-
journaltitlePhys. Lett. B

family=UA5 Collaboration,

titleScaling of Pseudorapidity Distributions at c.m. Energies Up
to $0.9$-
journaltitleZ. Phys. C

titleDiffraction Dissociation at the CERN Pulsed Collider at
CM Energies of $900$- and $200$-
journaltitleZ. Phys. C

titleUA5: A general study of proton-antiproton physics at
$\sqrt{s} = 546$-
journaltitlePhys. Rept.

titleCharged Particle Correlations in $\bar{P} P$ Collisions at
c. m. Energies of $200$-, $546$- and $900$-
journaltitleZ. Phys. C

family=UA5 Collaboration,

titleCharged Particle Multiplicity Distributions at $200$-  and $900$- Center-Of-Mass Energy
journaltitleZ. Phys. C

titleThe CDF Detector: An Overview
journaltitleNucl. Instrum. Meth. A

family=CDF Collaboration,

numberFERMILAB-DESIGN-1996-01, FERMILAB-PUB-96-390-E
titleThe CDF-II detector: Technical design report,
FERMILAB-DESIGN-1996-01, FERMILAB-PUB-96-390-E

family=CDF Collaboration,

titleTransverse Momentum Distributions of Charged Particles
Produced in $\bar{p}p$ Interactions at $\sqrt{s} = 630$  and
journaltitlePhys. Rev. Lett.

family=CDF Collaboration,

family=Tran ThanhVan,

booktitle24th Rencontres de Moriond: New Results in Hadronic
Interactions, 12–18 March 1989. Les Arcs, France (C89-03-12)
titlePseudorapidity distributions of charged particles produced
in $\bar{p}p$ interactions at $\sqrt{s} = 630$  and $1800$
journaltitlePhys. Rev. D

titleSoft and Hard Interactions in $p\bar{p}$ Collisions at
$\sqrt{s}= 1800$- and $630$-
journaltitlePhys. Rev. D

family=CDF Collaboration,

note[Erratum: Phys.Rev.D 82, 119903 (2010)]
titleMeasurement of Particle Production and Inclusive
Differential Cross Sections in $p \bar{p}$ Collisions at $\sqrt{s} =
journaltitlePhys. Rev. D

titleStudy of Soft QCD at the Tevatron
journaltitleNuovo Cim. C

titleMultiplicity dependence of transverse momentum spectra of
centrally produced hadrons in anti-p p collisions at $0.3$-, $0.54$-,
$0.9$-, and $1.8$- center-of-mass energy
journaltitlePhys. Lett. B

titleA Similarity hypothesis in the strong interactions. 1.
Multiple hadron production in \( e^+ e^- \) annihilation
journaltitleSov. Phys. JETP

titleScaling of multiplicity distributions in high-energy hadron
journaltitleNucl. Phys. B

seriesCERN Yellow Reports: School Proceedings
titleMulti-body phenomena in strong interactions – description
of hadronic multi-body final states (p. 171) in Proceedings CERN-JINR School
of Physics, Ebeltoft, Denmark, 17-13 Jun 1973

titleVery high-energy collisions of hadrons
journaltitlePhys. Rev. Lett.

titleStudy of KNO scaling in \(pp\) collisions at $\sqrt{s}$ from
\(0.9\) to \(13\)  using results of the ATLAS at the LHC
journaltitleEur. Phys. J. C

titleThe Quark-Gluon Structure of the Pomeron and the Rise of
Inclusive Spectra at High-Energies
journaltitlePhys. Lett. B

titlePomeron as Quark-Gluon Strings and Multiple Hadron
Production at SPS Collider Energies
journaltitlePhys. Lett. B

titleA REGGEON DIAGRAM TECHNIQUE
journaltitleSov. Phys. JETP

titleSemihard Processes in QCD
journaltitlePhys. Rept.

titleHow many of the scaling trends in \(pp\) collisions will be
violated at \(\sqrt{s_{\mathrm{NN}}} = 14\) ? – Predictions from Monte
Carlo quark-gluon string model
journaltitlePhys. Rev. D

titleScaling trends in proton–proton collisions from SPS to LHC
in quark-gluon string model
journaltitleJ. Phys. Conf. Ser.

titleSoft multihadron dynamics. Book. Publisher: Singapore,
Hackensack, NJ: World Scientific, 652 pages

titleHadron multiplicities
journaltitlePhys. Rept.

titleCharged-Particle Multiplicity in proton–proton Collisions
journaltitleJ. Phys. G

titleKNO scaling 30 years later
journaltitleNucl. Phys. B Proc. Suppl.

family=ATLAS Collaboration,

titleMeasurement of underlying event characteristics using
charged particles in \(pp\) collisions at $\sqrt{s} = 900$  and
$7$  with the ATLAS detector
journaltitlePhys. Rev. D

family=ATLAS Collaboration,

titleMeasurements of underlying-event properties using neutral
and charged particles in $pp$ collisions at $900$  and $7$  with
the ATLAS detector at the LHC
journaltitleEur. Phys. J. C

family=ATLAS Collaboration,

titleMeasurement of charged-particle distributions sensitive to
the underlying event in $ \sqrt{s}=13$  proton–proton collisions with
the ATLAS detector at the LHC

family=ATLAS Collaboration,

titleTwo-particle Bose–Einstein correlations in
\(pp\) collisions at $\sqrt{s} = 0.9$ and $7$  measured with the ATLAS
journaltitleEur. Phys. J. C

family=ATLAS Collaboration,

titleTwo-particle Bose–Einstein correlations in
\(pp\) collisions at $\sqrt{s} = 13$  measured with the ATLAS detector
at the LHC
journaltitleEur. Phys. J. C

family=ATLAS Collaboration,

titleMeasurement of the Inelastic proton–proton Cross-Section at
$\sqrt{s}=7$  with the ATLAS Detector
journaltitleNature Commun.

family=ATLAS Collaboration,

titleMeasurement of the Inelastic proton–proton Cross Section at
$\sqrt{s} = 13$  with the ATLAS Detector at the LHC
journaltitlePhys. Rev. Lett.

booktitleProceedings. 28th International Symposium on Lepton
Photon Interactions at High Energies, Guangzhou (Guangdong), China, August
7-12, 2017, 241-259
titleSoft QCD measurements at LHC

titleSoft QCD at ATLAS and CMS, Proceedings. 26th Cracow
Epiphany Conference on LHC Physics: Standard Model and Beyond (Epiphany),
Krakow, Poland, January 7-10, 2020
journaltitleActa Phys. Polon. B

titleSoft QCD in ATLAS: Bridging non-perturbative and
perturbative aspects of multiparticle production
journaltitleNucl. Phys. B Proc. Suppl.

family=CMS Collaboration,

titleMeasurement of diffraction dissociation cross sections in
$pp$ collisions at $\sqrt{s} = 7$
journaltitlePhys. Rev. D

titleReview of Particle Physics, Chapter 20. High Energy Soft
QCD and Diffraction, pp. 385–408

titleHigh-energy strong interactions: from `hard' to `soft'
journaltitleEur. Phys. J. C

booktitle100 Years of Subatomic Physics

titleParton ladder splitting and the rapidity dependence of
transverse momentum spectra in deuteron-gold collisions at RHIC
journaltitlePhys. Rev. C

titlepQCD based approach to parton production and equilibration
in high-energy nuclear collisions
journaltitlePhys. Rept.

titleTaming the Energy Rise of the Total proton–proton

titleMonte Carlo treatment of hadronic interactions in enhanced
Pomeron scheme: I. QGSJET-II model
journaltitlePhys. Rev. D

titleConsistent treatment of soft and hard processes in hadronic
journaltitleJ. Phys. G

titleA Model for Strong Interactions at High Energy Based on
CGC/Saturation Approach and the BFKL Pomeron
journaltitleActa Phys. Polon. Supp.

titleOn the Pomeranchuk Singularity in Asymptotically Free
journaltitlePhys. Lett. B

titleMultiRegge Processes in the Yang-Mills Theory
journaltitleSov. Phys. JETP

titleThe Pomeranchuk Singularity in Nonabelian Gauge Theories
journaltitleSov. Phys. JETP

titleThe Pomeranchuk Singularity in Quantum Chromodynamics
journaltitleSov. J. Nucl. Phys.

Cambridge Univ. Press

isbn978-1-107-42475-3, 978-0-521-63148-8, 978-0-511-71744-4
titleQuantum chromodynamics: Perturbative and nonperturbative

Cambridge University Press

isbn978-0-521-11257-4, 978-1-139-55768-9
titleQuantum chromodynamics at high energy

titleSoft and hard interactions in $p \bar{p}$ collisions at
$\sqrt{s}$ = 1800-GeV and 630-GeV
journaltitleNucl. Phys. B Proc. Suppl.

booktitle30th International Symposium on Multiparticle Dynamics
howpublishedIn: Proceedings of the “30th International Symposium
on Multiparticle Dynamics”, Tihany, Lake Balaton, Hungary, 9-15 October
2000, FERMILAB-CONF-01-042-E
titleSoft and Hard Interactions in $p\bar{p}$ Collisions at
$\sqrt{s} = 1800$  and $630$

titleSoft and hard interactions in $p \bar{p}$ collisions at
$\sqrt{s}$ = 1800-GeV and 630-GeV
journaltitlePhys. Atom. Nucl.

titlePerformance of the ATLAS Trigger System in 2010
journaltitleEur. Phys. J. C

titlePerformance of the ATLAS Trigger System in 2015
journaltitleEur. Phys. J. C

howpublishedCERN-LHCC-2010-013, ATLAS-TDR-19
titleATLAS Insertable B-Layer Technical Design Report

howpublishedCERN-LHCC-2012-009, ATLAS-TDR-19-ADD-1
titleATLAS Insertable B-Layer Technical Design Report Addendum

howpublishedATL-PHYS-PUB-2009-033, ATL-COM-PHYS-2009-104
titleA Study of Minimum Bias Events

titleThe ATLAS Collaboration Software and Firmware

* Invalid format of field 'month'

titleThe Pythia 8 A3 tune description of ATLAS minimum bias and
inelastic measurements incorporating the Donnachie–Landshoff diffractive

titleA Brief Introduction to PYTHIA 8.1
journaltitleComput. Phys. Commun.

titleParton distributions with LHC data
journaltitleNucl. Phys. B

: the Monash 2013 Tune
journaltitleEur. Phys. J. C

titleFurther ATLAS tunes of PYTHIA6 and Pythia8

titleParton distributions for the LHC
journaltitleEur. Phys. J. C

titleEvidence for hydrodynamic evolution in proton–proton
scattering at \(900\)
journaltitlePhys. Rev. C

given= Iu.,

titleEPOS LHC: Test of collective hadronization with data
measured at the CERN Large Hadron Collider
journaltitlePhys. Rev. C

titlePYTHIA 6.4 Physics and Manual

titleATLAS tunes of PYTHIA 6 and Pythia 8 for MC11

titleNew generation of parton distributions with uncertainties
from global QCD analysis

titleATLAS Monte Carlo tunes for MC09

titleTevatron-for-LHC Report of the QCD Working Group

titlePhotoproduction within the two component dual parton model.
1. Amplitudes and cross-sections
journaltitleZ. Phys. C

booktitle1st International Workshop on Multiple Partonic
Interactions at the LHC
titleThe Perugia Tunes, FERMILAB-CONF-09-113-T

titleHigh-energy physics event generation with PYTHIA 6.1
journaltitleComput. Phys. Commun.

titleAn introduction to PYTHIA 8.2
journaltitleComput. Phys. Commun.

numberLU-TP 22-16, MCNET-22-04, FERMILAB-PUB-22-227-SCD
titleA comprehensive guide to the physics and usage of PYTHIA

titleInterleaved parton showers and tuning prospects

booktitle3rd Les Houches Workshop on Physics at  Colliders
titleNew showers with transverse-momentum-ordering

Oxford University Press

titleThe Black Book of Quantum Chromodynamics: A Primer for the
LHC Era

titleGeneral-purpose event generators for LHC physics
journaltitlePhys. Rept.

titleEffect of color reconnection on forward-backward
multiplicity and mean transverse momentum correlation

family=Tran ThanhVan,

titleJets in Small p(T) Hadronic Collisions, Universality of
Quark Fragmentation, and Rising Rapidity Plateaus
journaltitlePhys. Lett. B

family=Tran ThanhVan,

titleDual parton model
journaltitlePhys. Rept.

titleGeant4 – a simulation toolkit
journaltitleNucl. Instrum. Meth. A

titleHadronic diffractive cross-sections and the rise of the
total cross-section
journaltitlePhys. Rev. D

titleTotal cross-sections
journaltitlePhys. Lett. B

titleHerwig\(+\)\(+\) physics and manual
journaltitleEur. Phys. J. C

titleATLAS inner tracking detectors: Run 1 performance and
developments for Run 2
journaltitleNucl. Part. Phys. Proc.

titleQCD motivated approach to soft interactions at high
energies: nucleus-nucleus and hadron-nucleus collisions
journaltitleNucl. Phys. A

titleLight hadron production in inclusive \(pp\)-scattering at
journaltitlePhys. Rev. D

titlePredictions of Quark-Gluon String Model for \(pp\) at LHC
journaltitleEur. Phys. J. C

titleGluon saturation and inclusive hadron production at LHC
journaltitlePhys. Rev. D

titleCharged Particle Multiplicity Distributions in Proton
Anti-proton Collisions at 540-GeV Center-of-mass Energy
journaltitlePhys. Lett. B

booktitle23rd International Symposium on Ultra-High Energy
Multiparticle Phenomena, Aspen, Colorado, September 12–17
titleMultiplicity distributions in $\bar{p} p$ interactions at
$\sqrt{s} = 1800$-, FERMILAB-CONF-93-359-E CDF

titleCharged particle multiplicities in deep inelastic scattering
at HERA
journaltitleZ. Phys. C

booktitle38th International Symposium on Multiparticle Dynamics
titleMulti-parton interactions and underlying events from
Tevatron to LHC
journaltitleIn proceedings: “38th International Symposium on
Multiparticle Dynamics” (ISMD 2008), Hamburg, Germany, September 15-20

titlePrediction for minimum bias and the underlying event at LHC
journaltitleEur. Phys. J. C

titleSystematic event generator tuning for the LHC
journaltitleEur. Phys. J. C

titleTuning Monte Carlo Generators: The Perugia Tunes
journaltitlePhys. Rev. D

titleHadronic photon-photon interactions at high-energies
journaltitlePhys. Rev. D

titleA Unitary model for structure functions and diffractive
production at small x
journaltitleNucl. Phys. B

titleConstraints from the first LHC data on hadronic event
generators for ultra-high energy cosmic-ray physics
journaltitleAstropart. Phys.

titleSystematic Measurements of Identified Particle Spectra in $p
p$, $d+ Au$ and $Au+Au$ Collisions from STAR
journaltitlePhys. Rev. C

family=PHOBOS Collaboration,

titlePseudorapidity distributions of charged particles in \(d +
Au\) and \(p + p\) collisions at \(\sqrt{s_{\mathrm{NN}}} = 200\)-
journaltitleJ. Phys. G

titleTotal and elastic cross-sections and global event
characteristics in $p \bar{p}$ and $p p$ collisions at $\sqrt{s}=53$
journaltitleAIP Conf. Proc.

titlePhobos results on charged particle multiplicity and
pseudorapidity distributions in $Au+Au$, $Cu+Cu$, $d+Au$, and $p+p$
collisions at ultra-relativistic energies
journaltitlePhys. Rev. C

family=ALICE Collaboration,

titleCharged-particle multiplicity density at mid-rapidity in
central $Pb-Pb$ collisions at $\sqrt{s_{\mathrm{NN}}} = 2.76$
journaltitlePhys. Rev. Lett.

titleInclusive Hadron Production in the CERN-LHC Era
journaltitlePhys. Rev. D

family=ALICE Collaboration,

titleNeutral pion and $\eta$ meson production in proton–proton
collisions at $\sqrt{s}=0.9$  and $\sqrt{s}=7$
journaltitlePhys. Lett. B

titleROOT: A C\(+\)\(+\) framework for petabyte data storage,
statistical analysis and visualization
journaltitleComput. Phys. Commun.

titleThe role of double parton collisions in soft hadron
journaltitlePhys. Lett. B

titleSoft and semihard components structure in multiparticle
production in high-energy collisions
journaltitleNucl. Phys. B Proc. Suppl.

booktitleLU-TP-13-37, MCNET-13-16
titleColour reconnection and its effects on precise measurements
at the LHC

titleLHAPDF6: parton density access in the LHC precision era
journaltitleEur. Phys. J. C

titleSaturation and Scaling of Multiplicity, Mean $p_T$ and $p_T$
Distributions from $200$  $\leq \sqrt{s} \leq 7$
journaltitleActa Phys. Polon. B

titleJet Production and Fragmentation in \(e^+ e^-\) Annihilation
at $12$- to $43$-
journaltitleZ. Phys. C

family=Tran ThanhVan,

titleRapidity Dependence of Negative and All Charged
Multiplicities in Nondiffractive $\pi^+ p$ and $p p$ Collisions at
journaltitlePhys. Lett. B

titleRemarks on Multiplicity Fluctuations and KNO Scaling in
$\bar{p} p$ Collider Experiments
journaltitlePhys. Lett. B

titleON APPROXIMATE KNO SCALING AT ANTI-P P COLLIDER
journaltitleZ. Phys. C

titleMultiplicity distribution and mechanisms of the high-energy
hadron collisions
journaltitlePhys. Rev. D

booktitleConference on Multiparticle Dynamics, CERN-EP-89-135
titleUA5 MULTIPLICITY DISTRIBUTIONS AND FITS OF VARIOUS
FUNCTIONS, Conference on Multiparticle Dynamics, La Thuile, Italy, March

titleMoments of multiplicity distributions for KNO scaling study
using the ATLAS results

titleSimilarity hypothesis in strong interactions. 2. Cascade
formation of hadrons and their energy distribution in \(e^ + e^-\)
journaltitleSov. Phys. JETP

titleOn the Connection Between Branching Processes and Scale
Invariant Field Theory for \(e^ + e^-\) Annihilation
journaltitlePhys. Rev. D

titleJet Model With Exclusive Parton Hadron Duality
journaltitleZ. Phys. C

titleImproved QCD treatment of the KNO phenomenon
journaltitlePhys. Lett. B

titleThe Log normal distribution for cascade multiplicities in
hadron collisions
journaltitlePhys. Lett. B

titleScaling Predictions for Multiplicity Distributions at
journaltitleZ. Phys. C

titleScaling of multiplicity distributions and collision dynamics
in \(e^+ e^-\) and \(p p\) interactions
journaltitleMod. Phys. Lett. A

booktitle20th International Conference on Strangeness in Quark
Matter 2022, Busan, Republic of Korea
titleParticle production as a function of underlying-event
activity and very forward energy with ALICE

titleMultiplicity dependence of two-particle azimuthal
correlations in \(pp\) collisions at the LHC

titleExtraction of the multiplicity dependence of multiparton
interactions from LHC \(pp\) data using machine learning techniques
journaltitleJ. Phys. G