# Quantifying the Underlying Event:
Investigating Angular Dependence of Multiplicity Classes and Transverse-
momentum Spectra in High-energy pp Collisions at LHC Energies

Aditya Nath Mishra1, Gergely Gábor Barnaföldi1 and Guy Paić2 1Wigner Research
Centre for Physics, 29-33 Konkoly-Thege Miklós str., 1121 Budapest, Hungary
2Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Apartado Postal 70-543, México City 04510, México

###### Abstract

We present a study of the transverse momentum spectra and their evolution in
function of the position of the azimuthal of the particles associated to the
leading particle. Additionally, the behavior of the spherocity distribution in
the same azimuthal bins is reported. The studies were made using proton-proton
collisions at $\sqrt{s}$ = 13 TeV using PYTHIA8 Monte Carlo event generator.
The Multiplicity and midrapidity transverse momentum spectra of charged
hadrons have been analyzed in the non-extensive statistical framework. The
results on the findings corresponding to the Underlying Event are reported.

##  I Introduction

Numerous experimental measurements over a wide range of energies and collision
systems have been devoted to studying the properties of the QCD matter formed
in ultra-relativistic nuclear collisions. A number of measurements have
provided strong indications that the matter formed in the heavy-ion collisions
behaves as a strongly interacting fluid that exhibits collectivity Arsene:2005
; BBBack:2005 ; Adams:2005 ; Adcox:2005 ; Chatrchyan:2012 ; Abelev:2013 ;
Aad:2013 . Initially, small collision systems, like proton-proton and proton-
nucleus, have been used to study initial and final state effects in “cold”
nuclear matter, treating them as a baseline for the interpretation of heavy-
ion results. However, recently, collective, fluid-like features strikingly
similar to those observed in heavy-ion collisions, have been also observed in
small collisions systems especially in high-multiplicity events Cole:2007 .
This suggests, that in compressed, dense matter, the correlation length is
getting shorter, thus the number of degrees of freedom are increased.

Strongly correlated statistical systems can be investigated by correlation
methods. However, while angular correlations with triggers refer on the
momentum space, the modern analytical tools and the high statistics allows us
to study more in detail the distributions in phase space and the associated
entropy production. The use of event structure tools like the spherocity can
also be applied. Finite and strongly correlated systems have been investigated
recently in the non-extensive statistical framework. Applying the Tsallis-
entropy Biro:2020kve  and its connection to the high-energy nuclear
collisions, opened new windows by using the so called Tsallis – Pareto
distribution Biro:2020kve . The Tsallis parameters of the transverse momentum
($p_{T}$) distributions of the produced hadron states provide a better
description of the full spectra than other functional forms. The distribution,
which can be derived from the Tsallis-entropy formula Biro:2020kve , fits well
both at low- and high $p_{T}$, better than either thermal or perturbative QCD
models. Indeed, the spectra in the intermediate transverse momentum range (2
GeV/$c$ $\lesssim p_{T}\lesssim 6$ GeV/$c$), is well described, providing a
unified description between the bulk, hydro-like, and the perturbative ranges.
Another aspect of the non-extensive statistical framework is that the spectrum
parameters: the temperature, $T$ and non-extensivity, $q$, carry physical
information to interpret the system created in the high-energy collision. In a
recent work, the parameter-evolution of Tsallis – Pareto transverse momentum
distributions has been identified with high predictive power at various event
multiplicities. This presents a new view of the equation of state of the high
temperature and extremely dense nuclear matter, which can be investigated in
geometrical and event shape aspects as well as by angular-correlation
triggers.

In this work, the aim is to qualitatively and quantitatively classify high-
energy events in comparison to the well known event shape observable. The
method here is to generate hadron spectra in ultra-relativistic proton-proton
collisions, extract the Tsallis-parameters in given multiplicity classes,
which provide an entropy-based event classification. This led us to understand
the strongly correlated, non-perturbative regime and to determine the absolute
and quantitative properties of the Underlying Event (UE).

The structure of this work is as follows: we present the event generator
settings and the simulated data. We explain our method and we define global
quantities and the two cases for our correlation analysis. Global and event
shape variables are given, by fitting the hadron spectra in the angular bins,
where Tsallis – Pareto parameters, multiplicity values and transverse
spherocities are determined. Introducing different angular-correlation slices
as cases, allow us to compare model parameters with each other and with
various theoretical approaches. We also provide the Tsallis-thermometer for
angular regions. In a final step, the correlation of these parameters with
multiplicity and event-shape variables are carried out and cross checked for
consistency.

##  II The Simulated Data

PYTHIA pythia6 ; pythia8.1 ; pythia8.2  is a general-purpose Monte Carlo event
generator, widely used for the generation of events in high-energy collisions
of leptons, protons and nuclei. It has undergone decades of development and
tuning to collider and other data. The event generation in PYTHIA consists of
several steps starting typically from a hard scattering process, followed by
initial and final state parton showering, multiparton interactions, and the
final hadronization process.

The results reported in this paper are obtained from 1 billion non-diffractive
events for pp collisions at $\sqrt{s}$ = 13 TeV simulated using PYTHIA version
8.240 with the default Monash 2013 tune pythiaMonash .

The events and particles are selected for this analysis according to the
following criteria. The events having at least three primary charged particle
with transverse momentum $p_{T}>$ 0.15 GeV/$c$ within the pseudorapidity
$|\eta|<$ 0.8 are analyzed, which selection is required by the later-defined
event-shape variable calculations. The primary charged particles are defined
as all final state particles including decay products except coming form weak
decays of strange particles, this definition is similar to the one used by the
ALICE experiment alice:2017 , thus might be considered for experimental
investigations in the future.

##  III Analysis Method and Definitions

In this section we present the definitions, tools, and observables used
further in the paper.

###  III.1 Leading Particles

The particle with maximum transverse momentum in a particular event is
considered the leading particle. The azimuthtal angle reflects the angle of
all associated charged particles with respect to the leading particle.

Figure 1: The usual definition of regions in the azimuthal angle with respect
to the leading (highest-$p_{T}$) charged particle, where leading charged
particle is highlighted by a red arrow.

###  III.2 Underlying Event

The Underlying Event (UE) consists of particles that accompany a hard
scattering such as the beam-beam remnants and particles produced through
Multiple-Parton Interactions (MPI). Initial-State Radiation (ISR) and Final-
State Radiation (FSR) contribute to the UE. As illustrated in Fig 1, the
azimuthal angular difference between leading charged particle and associated
charged particles of the event,
$\Delta\phi_{0}=(\phi^{leading}-\phi^{assoc.})$, is used to define three
distinct topological regions following the CDF Collaboration CDFPaper1 ;
CDFPaper2 :

  * •

Near Side (NS): $|\Delta\phi_{0}|<\pi/3$, ($|\Delta\phi_{0}|<60^{\circ}$)

  * •

Transverse Side (TS): $\pi/3<|\Delta\phi_{0}|<2\pi/3$,
($60^{\circ}<|\Delta\phi_{0}|<120^{\circ}$)

  * •

Away Side (AS): $|\Delta\phi_{0}|>2\pi/3$, ($|\Delta\phi_{0}|<120^{\circ}$)

The NS and the AS regions are dominated by string fragments originating from
the hardest partonic process of the event. On the other hand, the TS is
dominated largely by the contributions of the UE.

To understand the particle production mechanism, by exploring the
contributions of UE activity, in the azimuthal space we consider two Cases,
shown in Fig 2, to divide azimuthal space, $\Delta\phi$, in 18 different
sections, where $\Delta\phi$ is the angle between the leading charged particle
and associated charged particles of the event.

Figure 2: Definitions of two Cases for $\Delta\phi$ bin selections. Case I
(left): Open $\Delta\phi$ angle by 20∘, named “opening angle”. In this Case
the binning starts from –10 - 10∘ and the last bin covers full azimuthal space
i.e. Minimum Bias (MB). Case II (right): make slices of the $\Delta\phi$ of
size 20∘, named “sliding angle”. In Case II the first bin 0 - 20∘ is treated
in both in with and without leading particle scenarios.

Case I

: We open $\Delta\phi$ angle in steps of 20∘, named “opening angle”. The
binning starts from –10 - 10∘ and the last bin covers full azimuthal space
i.e. –180 - 180∘. It is easy to recognize, that the last bin is the Minimum
Bias (MB) and, therefore the ratio of the largest $\Delta\phi$ bin to the MB
found exactly one. Case I is useful to investigate the evolution of the
thermodynamical observables of the system.

Case II

: We make slices of the $\Delta\phi$ of size 20∘, named “sliding angle”. In
this case, the results for the first bin 0 - 20∘ are reported in two ways:
including and excluding the leading particle in the result. Case II is a tool
for exploring the geometrical structure of the Underlying Event.

Note, Case I and Case II can be cross-checked within the 1st bin with and
without including the leading particle. This first bin is important, since
this is the perturbative region, therefore well-described by theoretical
models. Moreover one has to take into account, that angle opening for Case I
and Case II is treated differently, which will present as different structures
in the $\Delta\phi$ plots. Case I definition includes both sides of the
leading particles, while Case II has mirror symmetry for $\Delta\phi=\pi$.

###  III.3 Charged Hadron Multiplicity

The multiplicity of produced charged particles is one of the basic
characteristics of high-energy hadron collisions and has been the subject of
longstanding experimental and theoretical studies, which have shaped the
understanding of the particle production mechanism and entropy balance. We
note, the number of charged particles is calculated by the selection defined
in Section II. Since we select the events having at least three primary
charged particles in one of the $\Delta\phi$ bins in case second, we loose
some of the events. Therefore, minimum bias reported in this paper is
different than earlier studies using all events Mishra:PRC2019 ; Pranjal:2021
.

###  III.4 Transverse Momentum Spectra

The transverse momentum spectra of charged particles produced in high energy
proton-proton collisions at the Large Hadron Collider (LHC) energies offer
unique information about soft and hard interactions. The $p_{T}$-spectra
consists of a low $p_{T}$-region where soft processes dominate particle
production, whereas the high $p_{T}$-region is mostly dominated by hard
processes. In this paper, we study $p_{T}$-spectra of charged particles using
PYTHIA8 Monte-Carlo generator in pp collisions at $\sqrt{s}$ = 13 TeV to
characterise the events by analysing UE contributions in the azimuthal space.

###  III.5 Event shape variable spherocity

The mid-rapidity charged hadron transverse spherocity ($S_{0}$), hereinafter
referred to as spherocity, has been used as a valuable tool to characterise
the events through the geometrical distribution of the $p_{T}$’s of the
charged hadrons, which is by definition collinear and infrared safe
Salam:2009jx . By restricting it to the transverse plane, transverse
spherocity avoids the bias from the boost along the beam axis Banfi:2010xy .
Transverse spherocity is defined for a unit transverse vector $\hat{n}$ which
minimizes the ratio Cuautle:2014yda ; Cuautle:2015kra :

Figure 3: Schematic picture of event shape variable (spherocity) showing jetty
and isotropic events in the transverse plane.

$\displaystyle
S_{0}=\frac{\pi^{2}}{4}\bigg{(}\frac{\Sigma_{i}~{}|\vec{p}_{T_{i}}\times\hat{n}|}{\Sigma_{i}~{}p_{T_{i}}}\bigg{)}^{2}.$
(1)

The extreme limits of $S_{0}$ are related to specific configurations of events
in the transverse plane. The value of $S_{0}$ ranges from 0 to 1, which is
ensured by multiplying the normalisation constant $\pi{{}^{2}}/4$ in Eq. (1).
The lower limit of spherocity ($S_{0}\rightarrow 0$) corresponds to event
topologies where all transverse momentum vectors are (anti)parallel or the sum
of the $p_{T}$ is dominated by a single track, called pencil like or jetty
events. The upper limit of spherocity ($S_{0}\rightarrow 1$) corresponds to
event topologies where transverse momentum vectors are “isotropically”
distributed, called isotropic events. The jetty events are usually the hard
events while the isotropic ones are the result of soft processes. The
schematic picture of the event shape variable (spherocity) showing jetty and
isotropic events in the transverse plane is shown in Fig. 3. Note, for
obtaining spherocity values one needs at least three primary charged particles
in the event, which was set in Section II accordingly.

##  IV Global & Event-shape Properties

To explore the structure and properties of events, the geometrical and
momentum quantities need to be handled in parallel. Our aim is to connect the
geometrical structure of multiplicity distributions and the transverse
momentum spectra. These two aspects can be interconnected by the spherocity.

Although this task is well defined for a jet-like event structure, where both
the momentum and the spatial dimensions are limited, this method is not
unequivocal for the isotropic Underlying Event. We investigate here the
angular dependence of these global observables in parallel with the event
shape properties to explore the geometrical connection between them in both
Case I and Case II were defined in Section III.

###  IV.1 Multiplicity Variations with $\Delta\phi$

(a) Case I (b) Case II

Figure 4: Multiplicity in $\Delta\phi$ bin is scaled with the multiplicity
from MB in the same $\Delta\phi$ bin are shown for Case I (left) and Case II
(right) for both Monash tune (open symbols) and CR off (blue solid line).
First bin of the Case II is plotted by full symbols for both Monash tune
(black point) and CR off (blue rectangle).

To understand the multiplicity variation with $\Delta\phi$, we plot charged
particle multiplicity scaled it with minimum bias multiplicity ($\frac{\rm
N_{\rm ch(\Delta\phi)}/\Delta\phi}{\rm N_{\rm ch(MB)}/2\pi}$) in each
$\Delta\phi$ bin for both cases. Figure 4(a) shows that relative multiplicity
decreases with opening angle, $\Delta\phi$ and get saturated to 1 at high
$\Delta\phi$ in Case I. Here, as widening the opening angle, the hadronic
degrees of freedom are increasing, which suppress the effect around the
leading particle and weaken the dominance of the pencil-like (jetty) structure
at the Near Side. In Fig. 4(b), Case II, relative multiplicity attain value
more than one in the first Near Side bin. The relative multiplicity shows a
decrease from Near Side to Transverse Side and then a small rise (close to
one) from Transverse Side to Away Side. In Case II the differential properties
present rapidly-changing effects than in Case I as we will see later in
Section V.

(a) Case I (b) Case II

Figure 5: Average spherocity ($\bar{S^{\prime}}_{0}$) in $\Delta\phi$ bins are
shown for Case I (left) and Case II (right). The horizontal dashed line
corresponds to the minimum bias (MB) average spherocity for both cases.

###  IV.2 Event shape variations with $\Delta\phi$

Geometrical structure of the UE can be investigated by calculating the
spherocity distribution as a function of $\Delta\phi$ for both Case I and Case
II. We obtained spherocity distributions for each $\Delta\phi$ bin with
respect to the leading particle of the event to study the geometrical
structure of the angular distributed produced charged particle of the event.
Then we plot the average spherocity ($\bar{S^{\prime}}_{0}$) as a function of
$\Delta\phi$ in Fig. 5. The average spherocity ($\bar{S}_{0}$) of the MB
events is also shown by a black dashed line. In Case I, we observe that the
$\bar{S^{\prime}}_{0}$ increases with $\Delta\phi$ and for the last
$\Delta\phi$ bin it gets the same value as MB. Fig. 5(a) does not give any
fruitful information about the geometrical construction of the UE, apart from
that by opening the angle and entering to the Away Side regime,
$\bar{S^{\prime}}_{0}$ presents more isotrope situation relative to the MB.
The distribution is more clear, in Case II, where we can get an interesting
observation with $\Delta\phi$. One can see that $\bar{S^{\prime}}_{0}$
increases with $\Delta\phi$ and attain higher values (shaded area) than the MB
in the TS region which suggest that TS region is more isotropic than the MB
(displayed by a horizontal dashed line). One can make an observation here that
the charged particle multiplicity in the Transverse Side always lies lower
than the MB. We observed that the range of $\Delta\phi$ for TS region is more
wider than the traditional definition (see double-shaded area). This study
gives the range of the TS from 40∘ to 140∘ ($2\pi/9$ to $7\pi/9$) in the one
half of the $\Delta\phi$ plane. This suggests that the NS and AS regions are
narrower than the traditional definition, thus the UE is in reality wider than
usually considered by about $66\%$ as suggested in Ref. Agocs . This
observation is in agreement with 4(a), where the normalized multiplicity
($\frac{\rm N_{\rm ch(\Delta\phi)}/\Delta\phi}{\rm N_{\rm ch(MB)}/2\pi}$)
crossing 1 (dashed line), suggesting the border between the jet-like,
perturbative regime and the bulk, soft UE. This however is around
$\Delta\phi\approx\pm$ 0.6 around the Near Side and similarly, the effect is
present with less magnitude at the Away Side.

Note, both spherocity and multiplicity show similar structure. One can
associate this with the recently-discovered feature of the jet-structure fix
point, $R_{fix}$ and the observed jet density scaling, while considering
fixed-sized jets in various event multiplicity classes Varga:2019 ; Gemes:2020
; Vertesi:2020 .

(a) Case I (b) Case II

Figure 6: Charged particle transverse momentum spectra (upper panel) and their
ratios to minimum bias spectra (lower panel) are shown for both Case I (left)
and Case II (right). First bin of the Case II is plotted for two different
scenario: (1) with leading particle (black points) and (2) without leading
particle (black circles).

###  IV.3 Transverse Momentum Spectra with Non-extensive Tsallis Statistics

Non-extensive thermodynamics can be applied to high energy physics. In high-
energy heavy-ion, and especially in proton-proton collisions we are far from a
canonical thermal state, described by the Boltzmann – Gibbs statistic. More
than three decades ago, the Tsallis distribution tsallis  was first proposed
as a generalisation of the Boltzmann – Gibbs distribution which is
characterised by an additional parameter $q$. The parameter $q$ measures the
deviation from a standard Boltzmann – Gibbs distribution and known as the
measure of ‘non-extensivity’. Over the past few years, Tsallis distribution
has been successful in describing the transverse momentum distributions to
high-energy heavy-ion, and especially in proton-proton collisions bialas ;
grigoryan ; wong ; ristea ; biro ; parvan1 ; parvan2 ; tripathy ; zheng1 ;
marques ; azmi ; sorin ; sena ; Azmi:2014dwa ; Biro:2004qg ; Khuntia:2017ite ;
zheng ; de .

In the present study, we use one particular form of Tsallis distribution,
named Tsallis – Pareto distributions tsbiro:pa2015 ; tsb:epja2013 , which
satisfy the thermodynamic consistency relations and given by:

$f(m_{T})=A\cdot\left[1+\frac{q-1}{T_{s}}(m_{T}-m)\right]^{-\frac{1}{q-1}}\ \
\ ,$ (2)

where $A$ is scale parameter, $q$ is the non-extensive parameter, $T_{s}$ is a
temperature-like parameter, called Tsallis temperature and
$m_{T}=\sqrt{p_{T}^{2}+m^{2}}$ is the transverse mass of the given
(identified) hadron species. Note, that we neglect radial flow in this
approach, and we use the pion mass for the unidentified charged hadron fits.
Parameter, $q$ is manifest as the power of the Tsallis – Pareto distribution’s
tail, which can be associated with the power of the transverse momentum
spectra in the perturbative (high-$p_{T}$) regime, $n=\frac{1}{1-q}$.

Experimental observations show that the charged multiplicity in high-energy
collisions is dependent on the collision energy alice:prl2010  by the
following relation:

$\frac{\left<{\rm d}N_{ch}/{\rm
d}\eta\right>}{\left<N_{part}\right>/2}\propto\left\\{\begin{array}[]{ll}s_{NN}^{0.15}&\textrm{for
AA,}\\\ s_{NN}^{0.11}&\textrm{for pp.}\end{array}\right.$ (3)

Using the Eq. (3), one can use a given parametric form of identified hadron
multiplicity per unit rapidity $\left(\left.\frac{{\rm d}\rm N_{\rm ch}}{{\rm
d}y}\right|_{y=0}\right)$ as a function of Tsallis – Pareto fitting parameters
$A$, $T_{s}$ and $q$ as

$\displaystyle\left.\frac{\rm dN_{ch}}{\rm dy}\right|_{y=0}=2\pi
AT_{s}\left[\frac{(2-q)m^{2}+2mT_{s}+2T_{s}^{2}}{(2-q)(3-2q)}\right]$
$\displaystyle\hskip
59.75095pt\times\left[1+\frac{q-1}{T_{s}}m\right]^{-\frac{1}{q-1}}.$ (4)

Since our aim here is to characterise the events by analysing different the UE
contributions in the azimuthal space therefore, we fit $p_{T}$-spectra of
charged particle with Tsallis – Pareto fitting function and explored the
evolution of the Tsallis – Pareto parameters $A$, $T_{s}$ and $q$ in different
$\Delta\phi$ bins for both cases.

The charged particle transverse momentum spectra and their ratio to the MB for
proton-proton collisions at $\sqrt{s}$ = 13 TeV using event generator PYTHIA
(Monash tune) are shown in Fig. 6. For the Case I, we observe an increase in
yield with $\Delta\phi$ at low-$p_{T}$ ($p_{T}<3$ Gev/$c$) while at
high-$p_{T}$ spectra remain nearly unchanged, (see Fig. 6(a)) which indicates
that most of the high-$p_{T}$ particle are produced in the first $\Delta\phi$
bin (–10 - 10∘). ALICE has recently reported that charged particle density
shows a sharp rise with the leading charged particle $p_{T}$
($p_{T}^{leading}$) at low-$p_{T}$ ($p_{T}<3$ Gev/$c$) and then gets saturated
for Transverse Side but Near and Away Side show a slow rise with
$p_{T}^{leading}$ ALICE:vz . The rise in the Away Side is smaller than the
Near Side. As we open more and more the $\Delta\phi$ angle we include
Transverse and Away Side regions which have no sufficient high-$p_{T}$
contributions to change the slope of the spectra at high-$p_{T}$ ($p_{T}>3$
GeV/$c$). This difference at low-$p_{T}$ can be seen clearly in the lower
panel of the Fig. 6(a). In Case II, as we go for higher sliding $\Delta\phi$
bins, we move from Near Side to Transverse Side and finally to Away Side
region. Here, we observe that at high-$p_{T}$ ($p_{T}>3$ Gev/$c$) the spectra
get softer as we move from Near Side to Transverse Side and again start
getting harder as we move to Away Side but never reach to the values observed
in the Near Side. The shape of the spectra again confirm that Near Side region
contains most of the high-$p_{\rm T}$ particles while Transverse Side which is
sensitive to the UE has low-$p_{\rm T}$ particles contributions. At
low-$p_{T}$, the spectra in all three region are almost same which again
matches with the ALICE results ALICE:vz  where they show that all three region
spectra show similar sharp rise with $p_{T}^{leading}$. This difference at
high-$p_{T}$ can be seen clearly in the lower panel of the Fig. 6(b). In
general, the Case I more sensitive to the low transverse momentum regime,
while Case II presents differences for the high and intermediate $\rm p_{\rm
T}$ apart from the first bin with leading charged particle.

###  IV.4 The $\Delta\phi$-scaling of the spectrum-fit parameters

The $\Delta\phi$ dependent charged particle transverse momentum spectra are
then fitted with the Tsallis – Pareto function given in Eq. (2). The fitting
parameters $A$, $T_{\rm s}$ and q are shown in Fig. 7 and Fig. 8 for Case I
and Case II, respectively. The fitting parameters for both cases are given in
Appendix A. One can make following observations from the fitted Tsallis –
Pareto parameters:

Figure 7: Tsallis – Pareto scale parameter $A$, Tsallis – Pareto temperature
$T_{s}$ and non-extensive parameter $q$ for pp collisions at $\sqrt{s}$ = 13
TeV using PYTHIA8 Monash tune (open markers) and without color reconnection
(solid blue line) are shown in Fig. 7(a), (b) and (c) respectively, for Case
I. (d) Multiplicity as a function of $\Delta\phi$ for PYTHIA8 Monash tune
(open markers) and without color reconnection (solid blue line) are shown. The
recalculated multiplicity values using Eq. (4) are also shown by solid red
line. Figure 8: Tsallis – Pareto scale parameter $A$, Tsallis – Pareto
temperature $T_{s}$ and non-extensive parameter $q$ for pp collisions at
$\sqrt{s}$ = 13 TeV using PYTHIA8 Monash tune (open markers) and without color
reconnection (solid blue line) are shown in Fig. 8(a), (b) and (c)
respectively, for Case II. (d) Multiplicity as a function of $\Delta\phi$ for
PYTHIA8 Monash tune (open markers) and without color reconnection (solid blue
line) are shown. The recalculated multiplicity values using Eq. (4) are also
shown by solid red line.

The scale parameter $A$

shows the expected rise with $\Delta\phi$ in Case I because of addition of
more charged particles at low-$p_{T}$ while in Case II it achieves its highest
value in Away Side region and have points independent of $\Delta\phi$ in
Transverse Side region. We find similar results with the events generated
using PYTHIA without Color Reconnection (CR). PYTHIA8 without CR (solid curve)
produces more particles than the Monash tune, we get higher values for scale
parameter $A$.

Tsallis – Pareto temperature, $T_{s}$

in Case I, decreases with $\Delta\phi$ (Fig. 7(b)). The fall is faster with
$\Delta\phi$ for Near Side region but as we add Transverse and Away Side
particles this fall gets smaller and become nearly independent of
$\Delta\phi$. We also observe similar behaviour for Case II shown in Fig.
8(b). Near Side has the highest $T_{s}$ values and Transverse and Away Side
show nearly independent behaviour. Considering the first bin of $\Delta\phi$
without $p_{T}^{leading}$ then $T_{s}$ values lie within 15% for the full
range of $\Delta\phi$. The PYTHIA8 without CR (solid curve) also gives the
similar trend with lower values.

Non-extensivity parameter $q$

is sensitive to the high-$p_{T}$ part of the spectra. Fig. 7(c) shows $q$ as a
function of $\Delta\phi$ for Case I. Since spectra at high particle momenta,
$p_{T}>3$ GeV/$c$, are nearly unchanged with $\Delta\phi$ in Case I, $q$ show
a small rise in Near Side region and then get saturated with $\Delta\phi$. In
Case II, we observe a sharp fall in $q$ from Near Side to Transverse Side then
nearly constant in Transverse Side and then a small rise in Away Side with
$\Delta\phi$ shown in Fig. 8(c). The change in $q$ with $\Delta\phi$ is bigger
in Case II than the Case I. The PYTHIA8 without CR (solid curve) also result
in similar trend but with higher values.

Multiplicity dependence

on the opening/sliding angles is presented on Fig. 7(d)and Fig. 8(d),
respectively. We use Eq. (4) to calculate multiplicity for the different
$\Delta\phi$ bins using the $A$, $T_{s}$ and $q$ parameter obtained from the
Tsallis – Pareto fitting Biro:2020kve . This calculation is shown by red color
line in Fig. 7(d) and Fig. 8(d). Once can see that Eq. (4) works very well for
both cases. Further observation is that calculations without CR (solid blue
lines) generates larger multiplicity, especially getting away from the Leading
Side.

Confirming the validity of Eqs. (2)-(4), provides solid basis for the general
idea presented in Ref Biro:2020kve . It seems the model is not only valid in a
certain multiplicity bin and at a given c.m. energy, but fits in all
geometrical bins as well. This suggests that Tsallis – Pareto distribution
fits carry information parallel on both the soft and the hard regimes.
Analysing the two cases makes clear, that Tsallis-parameters represent the
properties of the event well, thus we can apply this in our approach to
identify the UE.

##  V Identifying the Underlying Event

Motivated by the observations in the spatial/geometrical structure and
momentum space from Figs. 5(b) and 6, respectively, we try to identify the
Underlying Event quantitatively by the Tsallis – Pareto parameters and by the
variation of the investigated global- and event-shape observables.

Figure 9: The Tsallis-thermometer, which presents the relation between Tsallis
temperature $T_{s}$ and non-extensivity parameter ($q$) for both Case I and
Case II. Red box represents the region obtained for new UE definition for Case
II as defined in Fig. 5(b).

The Tsallis-thermometer, is dedicated to visualize the interconnection between
Tsallis temperature ($T_{s}$) and non-extensivity parameter ($q$). This
relation presents not only the specific values, but also provides quantitative
information on the (non-)extensivity of the system. Since $T_{s}$ and $q$ are
interconnected and depends well on the thermodynamics of the system,
investigation of the two parameters is required in parallel. The relation
between the $\Delta\phi$-scaling Tsallis temperature and non-extensivity
parameter is shown in Fig. 9. In Case I, with the increase of $\Delta\phi$,
$T_{s}$ decreases in contrary to a small change in parameter, $q$. The highest
value of $T_{s}$ corresponds to the first $\Delta\phi$ bin ($-10^{\circ}$ to
$10^{\circ}$) while the lowest $T_{s}$ value were found in the last (full)
$\Delta\phi$ bin ($-180^{\circ}$ to $180^{\circ}$). As approaching to the
minimum bias, the non-extensitivity inherited from the Near Side is preserved
with $q\approx 1.15$, while $T_{s}$ drops below to its half value due to the
increasing multiplicity. This supports the idea, that the evolution in Case I
is mainly driven by the increase of the degrees of freedom. (Black points are
for Monash tune, black line is with no CR, purple horizontal band is for the
parameter region in $q$).

In Case II, in one hand, once we take into account the leading particle, the
first bin is different form the others carrying the highest values for
$T_{s}=0.3$ GeV and $q=1.14$. On the other hand we still observed big
variation in $q$ while the change is small in $T_{s}$ after excluding the
first bin ($0^{\circ}$ \- 20∘) with the leading charged particle. (Blue points
are for Monash tune, blue line is with no CR, green horizontal band is for the
full $T_{s}$ parameter variation region.) Since in Case II the systems (bins)
have the same cone structure, this investigation represent more qualitatively
the correlation between the degrees of freedom. Taking the quantitative values
of $q$, the lowest values appears in the non-perturbative Transverse Side.
Away Side bins has the lowest $T_{s}$ values, but in the Transverse Side
values are a factor of 2 lower, than in the first bin nearby the Leading
Particle.

Plotting the corresponding $T_{s}$ and $q$ values one can identify a compact
locations at $T_{s}=160\pm 10$ MeV and at $q=1.085\pm 0.05$ marked as red box
in Fig. 9. Interestingly this region is the lowest in non-extensivity, thus as
the most isotrope Transverse Side bins seem to be the closest to the special
case of the Tsallis – Pareto distribution, the $q=1$ Boltzmann – Gibbs
description. Following the obtained geometrical structure from Fig. 5, the
Underlying Event can be associated with the $[\pm 40^{\circ},\pm 140^{\circ}]$
($[\pm 2\pi/9,\pm 7\pi/9]$), region in Case II, indeed the more strict CDF-
definition in $[\pm 60^{\circ},\pm 120^{\circ}]$ ($[\pm\pi/3,\pm 2\pi/3]$).

The Tsallis thermometer presents stable parameter values within the extended
UE region. We assume that the variation of the parameters with $\Delta\phi$
carries further information. To understand the geometrical variation of the
physical quantities, we investigated the derivatives of the Tsallis – Pareto
scale parameters as a function of $\Delta\phi$ ($\frac{\delta
X}{\delta(\Delta\phi)}$, here ’$X$’ can be $A$, $T_{s}$, $q$ and $N_{ch}$) can
give us the information about the particle production mechanism in NS, AS and
TS. We show the absolute value of the first derivatives of the $\Delta\phi$
dependence of Tsallis – Pareto scale parameter $A$, Tsallis – Pareto
temperature $T_{s}$, non-extensive parameter $q$ and multiplicity for pp
collisions at $\sqrt{s}$ = 13 TeV using PYTHIA8 Monash tune (red) and PYTHIA8
without color reconnection (blue) for Case I and Case II in Figs. 10 and 11,
respectively.

Figure 10: First derivatives (absolute values) of the $\Delta\phi$ dependence
of Tsallis – Pareto scale parameter $A$, Tsallis – Pareto temperature $T_{s}$,
non-extensive parameter $q$ and multiplicity for pp collisions at $\sqrt{s}$ =
13 TeV using PYTHIA8 Monash tune (red) and without color reconnection (blue)
for Case I. Figure 11: First derivatives (absolute values) of the $\Delta\phi$
dependence of Tsallis – Pareto scale parameter A, Tsallis – Pareto temperature
$T_{s}$, non-extensive parameter q and multiplicity for pp collisions at
$\sqrt{s}$ = 13 TeV using PYTHIA8 Monash tune (solid red/dashed magenta lines
are with/without leading particle) and without CR (blue) are shown for Case
II.

For Case I, the first derivative of Tsallis – Pareto parameters $A$ show
change only in first few bins which correspond to the Near Side and then get
almost constant with $\Delta\phi$. The $T_{s}$ and $q$ have nearly zero values
of first derivatives except in Near Side. This suggests that adding more and
more particles, and increasing the phase-space with opening $\Delta\phi$, from
Transverse and Away Side does not change the thermodynamics of the system. As
constant multiplicity derivatives present, the system size increase is linear
beyond the first bins. On the other hand for Case II, first derivatives of
Tsallis – Pareto parameters are zero in Transverse Side while first
derivatives are non-zero for parameters $A$, $T_{s}$ and $q$ in Near Side and
Away Side. The amplitudes of the first derivatives are smaller in Away Side
than the Near Side because of the high-$p_{T}$ contribution in Near Side. In
Case I, one can conclude from the above discussions that $\Delta\phi$
dependence of the Tsallis – Pareto parameters $T_{s}$ and $q$ remain nearly
unchanged beyond NS region, which is confirmed by their derivatives having
values close to zero, since summing up more-and-more degrees of freedom from
the TS and AS washes out the strong effect in the NS, therefore,

$\displaystyle\frac{\delta T_{s}}{\delta(\Delta\phi)}\rightarrow
0~{}~{}~{}\&~{}~{}~{}\frac{\delta q}{\delta(\Delta\phi)}\rightarrow
0~{}~{}~{}~{}{\rm(for~{}~{}\Delta\phi>\pi/3)}\ .$ (5)

Interesting aspect is that, although multiplicity has different slopes for
Monash tune and calculations without CR on Fig. 7, the derivatives, thus the
variation of the $T_{s}$ and $q$ are the same on Fig. 10. This support, that
the thermodynamics of the system is not changing in Case I apart from the
model-dependent ”system size”, reflected by the variation in the
normalization, $A$ and in the multiplicity.

In Case II, we observed changes in the parameter values within the NS and AS
region, while in the TS region almost constant figures were found. One can see
the zero values for the Tsallis-parameter derivatives in TS regions while non-
zero in NS and AS regions as plotted in Fig. 11,

$\displaystyle\frac{\delta T_{s}}{\delta(\Delta\phi)}\neq
0~{}~{}~{}\&~{}~{}\frac{\delta q}{\delta(\Delta\phi)}\neq
0~{}~{}~{}~{}{\rm(for~{}NS~{}\&~{}AS)}$ (6) $\displaystyle\frac{\delta
T_{s}}{\delta(\Delta\phi)}\approx 0~{}~{}~{}\&~{}~{}\frac{\delta
q}{\delta(\Delta\phi)}\approx
0~{}~{}~{}~{}{\rm(for~{}TS)~{}~{}~{}~{}~{}~{}~{}~{}}$ (7)

In Case II derivatives of the multiplicity and the Tsallis-temperature present
zero value outside of the NS on Fig. 11, which means that the total energy
(the product of multiplicity and $T_{s}$) of a given $\Delta\phi$ bin, should
be constant everywhere within the TS and AS regions. If both total energy and
phase-space are equally distributed outside the NS, then transverse momentum
spectra is the one, which carries information on the thermodynamics.
Furthermore, the tail of the $p_{T}$ distribution is connected to the $q$
value, which relation can provide a bin-by-bin measure to describe
quantitatively the hadron production of the Underlying Event.

The conclusion from the above discussions is that charged particle production
in the UE region is fully interconnected with the Transverse Side, and since
it does not change much in the neighbouring $\Delta\phi$ bins, the extension
of the UE presented in Fig. 5 is advocated. In these widened regions due to
the stability of the multiplicity, the Tsallis parameters reflect the UE’s
limits well. The constancy of the Tsallis parameters $A$, $T_{s}$, $q$, and
multiplicity in the TS, can be interpreted in the two cases: (i) the linear
increase of the system size in Case I by washing out the strong effect at the
NS, and (ii) selecting constant total energy bins as it is given in Case II.

##  VI Corss-check with Spherocity

Spherocity values carry compound information on the angular structure and the
(absolute value) distribution of the transverse momenta vector. It is easy to
imagine, that the $p_{T}$ distribution and the Tsallis-parameters of the
$\Delta\phi$ correlations will be different for ”jetty” or ”isotrope” event
selections. Therefore the obtained Tsallis-parameter values will vary in
different spherocity classes. A possible cross-check of our presented results
is to compare the quantified underlying-event properties by applying event-
shape-variable classified events. To investigate this we divide $S_{0}$ in
four regions (0$-$0.25, 0.25$-$0.50, 0.50$-$0.75 and 0.75$-$1.0) to select
different kind of events for further study. We fit Tsallis – Pareto function
to the particle $p_{T}$ spectra for all four $S_{0}$ regions for Case I and
Case II. The obtained Tsallis parameters for different $\Delta\phi$ are
plotted for Case I and Case II in Fig. 12 and Fig. 13, respectively. Note, the
averages of these detailed results are equivalent to the proper minimum bias
curves in (Figs. 7 and 8), where no $S_{0}$ selection were applied.

Figure 12: $\Delta\phi$ dependence of Tsallis – Pareto parameters ($A$,
$T_{S}$ and $q$) and multiplicity for different spherocity classes in pp
collisions at $\sqrt{s}$ = 13 TeV using PYTHIA Monash tune are shown for Case
I. Figure 13: $\Delta\phi$ dependence of Tsallis – Pareto parameters ($A$,
$T_{S}$ and $q$) and multiplicity for different spherocity classes in pp
collisions at $\sqrt{s}$ = 13 TeV using PYTHIA Monash tune are shown for Case
II.

In Case I, $T_{s}$ and $q$ show a similar trend with $\Delta\phi$ as observed
in the minimum bias. Following the expectations $T_{s}$ gets the highest
values for jetty events and the lowest for isotropic events while $q$ gets its
highest values for isotropic events and lowest for jetty events ($\delta
q=0.15$). The multiplicity always increases with $\Delta\phi$ and its slope is
higher for isotropic events than the jetty one. One can see that multiplicity
has the same values in the first few $\Delta\phi$ bins for all four sections
of the $S_{0}$ and then multiplicity values rises faster in isotropic events
than the jetty ones. Since jetty events have more contribution from the NS,
and less from the TS, the multiplicity values are flat for this class. This
behaviour of multiplicity is shown in Fig. 12(d). The derivatives of the
Tsallis parameters with $\Delta\phi$ are the largest for the isotrope event
classes and parameters present minimal variation for jetty events.

In Case II one can observe that isotrope events achieve the highest $T_{s}$ in
the first bin of $\Delta\phi$ and then start decreasing when it enters in TS
and then rise again a bit in AS. Interestingly the change in $T_{s}$ become
almost zero for jetty events, since there is no relevant contribution from the
UE relative to the NS. This is in agreement with our earlier observation
presented in the Tsallis-thermometer, Fig. 9. The $\frac{\delta
T_{s}}{\delta(\Delta\phi)}\rightarrow 0$ confirms this behaviour for jetty
events more

The change in parameter $q$ shows also $\delta q\approx 0.15$ decrease while
going from jetty to isotropic events. Further observation on multiplicity: as
we move from jetty to isotropic events, multiplicity is rising for isotrope
event ($0.75<S_{0}<1.0$) and decreasing for jetty ones ($0.0<S_{0}<0.25$). Is
is interesting to see, that the variation of this effect is the strongest in
the TS.

In summary we can confirm, that quantified Underlying Event properties by the
Tsallis – Pareto fits from jetty to isotrope events, correlates with our
expectation of the geometrical distribution within an event as we have
obtained in Section V.

##  VII Summary

We investigated the geometrical structure of charged particle production in
proton-proton collisions at $\sqrt{s}$ = 13 TeV using PYTHIA8 Monte Carlo
event generator. The properties of the Underlying Event have been quantified
in angular bins, which led us to explore the event structure. In parallel the
multiplicity and midrapidity transverse momentum spectra of charged hadrons
have been analyzed. We used the non-extensive statistical approach for our
study. We quantified the properties of the event slices in two different
angular selection cases to identify the Underlying Event. Case I provided a
general trend summing up the hadronic degrees of freedoms as opening the
phase-space from the leading particle, while Case II described the geometrical
structure within equal, $20^{\circ}$ angular bin slices.

We found that the standard definition of the Underlying Event can be extended
with about 66% in geometry within the range, $[\pm 40^{\circ},\pm
140^{\circ}]$ in comparison to the widely-used CDF-definition, $[\pm
60^{\circ},\pm 120^{\circ}]$. Indeed we found an upper transverse-momentum
threshold, $p_{T}\lesssim$ 3-4 GeV/$c$ for hadrons with Underlying Event
origin. Location of the Underlying Event on the Tsallis-thermometer has been
also presented at $T_{s}=160\pm 10$ MeV and at $q=1.085\pm 0.05$. Non-
extensivity value was found to be the closest to the Boltzmann – Gibbs limit
($q=1.0$) of the Tsallis – Pareto distribution’s in isotropic events.

These findings were cross-checked with the parameter derivatives and
spherocity-classified events. The obtained, nearly-zero angular variation of
the spectral parameters in the Transverse Side region can support the
identification and better localization of the Underlying Event by the Tsallis
– Pareto parameters. Finally our quantified values correlated well with the
spherocity-classified parameter trends.

## Acknowledgements

Authors would like to thank the Hungarian National Research, Development and
Innovation Office (NKFIH) under the contract numbers OTKA K135515, K123815 and
NKFIH 2019-2.1.11-TET-2019-00078, 2019-2.1.11-TET-2019-00050, Wigner GPU
Laboratory, Mexican Conacyt project CF2042 and the support of the Coordinacion
de la Investigacion Cientifica, Mexico. Discussion on the fit procedure with
G. Bíró and T.S. Biró is highly appreciated.

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##  Appendix A Tsallis – Pareto fitting parameters

Table 1: Tsallis – Pareto fitting parameters for the fitting of PYTHIA (Monash tune) spectra at $\sqrt{s}=13$ TeV in pp collisions for Case I $\Delta\phi$ bin | A |  ${\rm\bf T}_{s}$ (GeV) | q | $\bf\chi^{2}/ndf$
---|---|---|---|---
-10∘ – 10∘ | 1.759${\displaystyle\pm}$0.133 | 0.298${\displaystyle\pm}$0.009 | 1.133${\displaystyle\pm}$0.004 | 2.511
-20∘ – 20∘ | 5.540${\displaystyle\pm}$0.479 | 0.232${\displaystyle\pm}$0.007 | 1.150${\displaystyle\pm}$0.004 | 1.975
-30∘ – 30∘ | 10.573${\displaystyle\pm}$0.903 | 0.202${\displaystyle\pm}$0.006 | 1.157${\displaystyle\pm}$0.003 | 1.808
-40∘ – 40∘ | 16.547${\displaystyle\pm}$1.386 | 0.185${\displaystyle\pm}$0.005 | 1.159${\displaystyle\pm}$0.003 | 1.646
-50∘ – 50∘ | 23.091${\displaystyle\pm}$1.904 | 0.174${\displaystyle\pm}$0.005 | 1.161${\displaystyle\pm}$0.003 | 1.582
-60∘ – 60∘ | 30.027${\displaystyle\pm}$2.443 | 0.166${\displaystyle\pm}$0.005 | 1.162${\displaystyle\pm}$0.003 | 1.431
-70∘ – 70∘ | 37.195${\displaystyle\pm}$2.991 | 0.159${\displaystyle\pm}$0.004 | 1.162${\displaystyle\pm}$0.003 | 1.449
-80∘ – 80∘ | 44.544${\displaystyle\pm}$3.544 | 0.156${\displaystyle\pm}$0.004 | 1.162${\displaystyle\pm}$0.003 | 1.209
-90∘ – 90∘ | 52.005${\displaystyle\pm}$4.093 | 0.152${\displaystyle\pm}$0.004 | 1.162${\displaystyle\pm}$0.002 | 1.099
-100∘ – 100∘ | 59.586${\displaystyle\pm}$4.642 | 0.149${\displaystyle\pm}$0.004 | 1.161${\displaystyle\pm}$0.002 | 1.081
-110∘ – 110∘ | 67.257${\displaystyle\pm}$5.186 | 0.147${\displaystyle\pm}$0.004 | 1.160${\displaystyle\pm}$0.002 | 1.065
-120∘ – 120∘ | 74.980${\displaystyle\pm}$5.725 | 0.146${\displaystyle\pm}$0.004 | 1.159${\displaystyle\pm}$0.002 | 1.097
-130∘ – 130∘ | 82.733${\displaystyle\pm}$6.256 | 0.144${\displaystyle\pm}$0.003 | 1.159${\displaystyle\pm}$0.002 | 1.053
-140∘ – 140∘ | 90.482${\displaystyle\pm}$6.778 | 0.143${\displaystyle\pm}$0.003 | 1.158${\displaystyle\pm}$0.002 | 1.047
-150∘ – 150∘ | 98.206${\displaystyle\pm}$7.292 | 0.142${\displaystyle\pm}$0.003 | 1.157${\displaystyle\pm}$0.002 | 1.045
-160∘ – 160∘ | 105.856${\displaystyle\pm}$7.795 | 0.142${\displaystyle\pm}$0.003 | 1.156${\displaystyle\pm}$0.002 | 1.030
-170∘ – 170∘ | 113.428${\displaystyle\pm}$8.289 | 0.141${\displaystyle\pm}$0.003 | 1.155${\displaystyle\pm}$0.002 | 0.987
-180∘ – 180∘ | 121.037${\displaystyle\pm}$8.784 | 0.141${\displaystyle\pm}$0.003 | 1.155${\displaystyle\pm}$0.002 | 0.924
Table 2: Tsallis – Pareto fitting parameters for the fitting of PYTHIA (Monash tune) spectra at $\sqrt{s}=13$ TeV in pp collisions for Case II  $\Delta\phi$ bin | A |  ${\rm\bf T}_{s}$ (GeV) | q | $\bf\chi^{2}/ndf$
---|---|---|---|---
0∘ – 20∘ (with $p_{\mathrm{T}}^{\mathrm{leading}}$) | 1.912${\displaystyle\pm}$0.152 | 0.296${\displaystyle\pm}$0.009 | 1.134${\displaystyle\pm}$0.004 | 2.471
0∘ – 20∘ (without $p_{\mathrm{T}}^{\mathrm{leading}}$) | 6.814${\displaystyle\pm}$0.456 | 0.158${\displaystyle\pm}$0.003 | 1.133${\displaystyle\pm}$0.002 | 0.968
20∘ – 40∘ | 4.663${\displaystyle\pm}$0.300 | 0.178${\displaystyle\pm}$0.003 | 1.078${\displaystyle\pm}$0.002 | 2.044
40∘ – 60∘ | 5.105${\displaystyle\pm}$0.336 | 0.164${\displaystyle\pm}$0.003 | 1.082${\displaystyle\pm}$0.002 | 1.792
60∘ – 80∘ | 4.852${\displaystyle\pm}$0.329 | 0.163${\displaystyle\pm}$0.003 | 1.082${\displaystyle\pm}$0.002 | 2.111
80∘ – 100∘ | 4.825${\displaystyle\pm}$0.334 | 0.162${\displaystyle\pm}$0.003 | 1.084${\displaystyle\pm}$0.002 | 2.152
100∘ – 120∘ | 4.959${\displaystyle\pm}$0.345 | 0.161${\displaystyle\pm}$0.003 | 1.087${\displaystyle\pm}$0.002 | 2.230
120∘ – 140∘ | 5.313${\displaystyle\pm}$0.365 | 0.157${\displaystyle\pm}$0.003 | 1.093${\displaystyle\pm}$0.002 | 1.928
140∘ – 160∘ | 5.947${\displaystyle\pm}$0.405 | 0.152${\displaystyle\pm}$0.003 | 1.103${\displaystyle\pm}$0.002 | 1.779
160∘ – 180∘ | 6.565${\displaystyle\pm}$0.440 | 0.146${\displaystyle\pm}$0.003 | 1.114${\displaystyle\pm}$0.002 | 1.254
180∘ – 200∘ | 6.678${\displaystyle\pm}$0.449 | 0.145${\displaystyle\pm}$0.003 | 1.116${\displaystyle\pm}$0.002 | 1.355
200∘ – 220∘ | 5.935${\displaystyle\pm}$0.403 | 0.152${\displaystyle\pm}$0.003 | 1.103${\displaystyle\pm}$0.002 | 1.726
220∘ – 240∘ | 5.249${\displaystyle\pm}$0.357 | 0.159${\displaystyle\pm}$0.003 | 1.092${\displaystyle\pm}$0.002 | 1.852
240∘ – 260∘ | 5.014${\displaystyle\pm}$0.348 | 0.160${\displaystyle\pm}$0.003 | 1.087${\displaystyle\pm}$0.002 | 2.145
260∘ – 280∘ | 4.693${\displaystyle\pm}$0.329 | 0.164${\displaystyle\pm}$0.003 | 1.083${\displaystyle\pm}$0.002 | 2.375
280∘ – 300∘ | 4.841${\displaystyle\pm}$0.333 | 0.163${\displaystyle\pm}$0.003 | 1.083${\displaystyle\pm}$0.002 | 2.182
300∘ – 320∘ | 5.106${\displaystyle\pm}$0.340 | 0.164${\displaystyle\pm}$0.003 | 1.082${\displaystyle\pm}$0.002 | 1.891
320∘ – 340∘ | 4.698${\displaystyle\pm}$0.302 | 0.177${\displaystyle\pm}$0.003 | 1.079${\displaystyle\pm}$0.002 | 1.960
340∘ – 360∘ | 6.093${\displaystyle\pm}$0.405 | 0.162${\displaystyle\pm}$0.003 | 1.131${\displaystyle\pm}$0.002 | 1.041