# Improved Monte Carlo Glauber predictions at present and future nuclear
colliders

Constantin Loizides  ORNL, Oak Ridge, USA  Jason Kamin  UIC, Chicago, USA
David d’Enterria  CERN, Geneva, Switzerland

###### Abstract

We present the results of an improved Monte Carlo Glauber (MCG) model of
relevance for collisions involving nuclei at center-of-mass energies of BNL
RHIC ($\sqrt{s_{\rm NN}}=0.2$ TeV), CERN LHC ($\sqrt{s_{\rm NN}}=2.76$–$8.8$
TeV), and proposed future hadron colliders ($\sqrt{s_{\rm NN}}\approx 10$–$63$
TeV). The inelastic pp cross sections as a function of $\sqrt{s_{\rm NN}}$ are
obtained from a precise data-driven parametrization that exploits the many
available measurements at LHC collision energies. We describe the nuclear
density of a lead nucleus with two separated 2-parameter Fermi distributions
for protons and neutrons to account for their different densities close to the
nuclear periphery. Furthermore, we model the nucleon degrees of freedom inside
the nucleus through a lattice with a minimum nodal separation, combined with a
“recentering and reweighting” procedure, that overcomes some limitations of
previous MCG approaches. The nuclear overlap function, number of participant
nucleons and binary nucleon–nucleon collisions, participant eccentricity and
triangularity, overlap area and average path length are presented in intervals
of percentile centrality for lead–lead (PbPb) and proton–lead (pPb) collisions
at all collision energies. We demonstrate for collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV that the central values of the Glauber quantities change by up
to 7% in a few bins of reaction centrality, due to the improvements
implemented, though typically remain within the previously assigned systematic
uncertainties, while their new associated uncertainties are generally smaller
(mostly below 5%) at all centralities than for earlier calculations. Tables
for all quantities versus centrality at present and foreseen collision
energies involving Pb nuclei, as well as for collisions of XeXe at
$\sqrt{s_{\rm NN}}=5.44$ TeV, and AuAu and CuCu at $\sqrt{s_{\rm NN}}=0.2$
TeV, are provided. The source code for the improved Monte Carlo Glauber model
is made publicly available.

##  I Introduction

The interpretation of many results measured in high-energy heavy ion
collisions relies on the use of a model of the initial matter distribution
resulting from the overlap of the two colliding nuclei at a given impact
parameter $b$. Indeed, quantities such as (i) the centrality dependence,
expressed by the number of participating nucleons in the collision $N_{\rm
part}(b)$, of any observable, (ii) the nuclear overlap function $T_{\rm
AA}(b)$ or the number of binary nucleon-nucleon collisions $N_{\rm coll}(b)$
used to derive the nuclear modification factor ($R_{\rm AA}$) from the ratio
of AA over pp spectra, (iii) the elliptic and triangular flow parameters
$v_{2}$ and $v_{3}$ normalized by the eccentricity $\varepsilon_{2}(b)$ and
triangularity $\varepsilon_{3}(b)$ of the overlap region, the average (iv)
surface area $A_{\rm T}(b)$ and (v) path-length $L(b)$ of the interaction
region, all depend on a realistic model of the collision geometry
Miller:2007ri .

The standard method employed in high-energy heavy-ion collisions describes the
initial transverse shape of the nuclei in terms of 2-parameter Fermi (2pF)
distributions (also often called Wood-Saxon distributions) with half-density
radius $R$ and diffusivity $a$ parameters obtained from fits to elastic
lepton-nucleus data DeJager:1974liz ; DeJager:1987qc , and determines the
underlying multi-nucleon interactions in the overlap area between the nuclei
through a Glauber eikonal approach Glauber:1970jm . In the Monte Carlo Glauber
(MCG) models (e.g. Wang:1991hta ; Alver:2008aq ; Loizides:2014vua ;
Broniowski:2007nz ; Rybczynski:2013yba ; Loizides:2016djv ), individual
nucleons are sampled event-by-event from the underlying 2pF distributions and
the collision properties are calculated by averaging over multiple events.
However, neutron-rich nuclei, such as 208Pb may have differing proton and
neutron density distributions at the nuclear periphery. Indeed, measurements
have recently been able to extract the neutron profile of several nuclei that
show differences with respect to their proton distribution Klos:2007is ;
Tarbert:2013jze , and various works have already studied its impact on
different isospin–dependent observables in nuclear collisions
Paukkunen:2015bwa ; De:2016ggl ; Helenius:2016dsk .

In this article, we present the results of improved Glauber Monte Carlo model
calculations for $N_{\rm coll}(b)$, $N_{\rm part}(b)$, $T_{\rm AA}(b)$,
$\varepsilon_{2}(b)$, $\varepsilon_{3}(b)$, $A_{\rm T}(b)$ and $L(b)$ in PbPb
and pPb collisions at LHC ($\sqrt{s_{\rm NN}}=2.76$, $5.02$, $5.5$, $8.16$ and
$8.8$ TeV), High-Energy LHC ($\sqrt{s_{\rm NN}}=10.6$, $17$ TeV), and Future
Circular Collider FCC ($\sqrt{s_{\rm NN}}=39$ and $63$ TeV) Dainese:2016gch
energies, by considering for the first time separated transverse profiles for
protons and neutrons in the lead nucleus. The corresponding values for the
inelastic pp cross section are obtained from a data-driven parametrization
with reduced uncertainties thanks to the many available measurements at LHC
collision energies. The nucleon degrees of freedom inside a nucleus are
modeled using a lattice with a minimum nodal separation, that mimics hard-core
repulsion between nucleons without distorting the nuclear density. Residual
small distortions in the generated nuclear densities resulting from adjusting
the nucleon center-of-mass (c.m.) with that of the nucleus are overcome by
reweighting the original nuclear density. We exemplify for collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV that the central values of $N_{\rm coll}(b)$,
$N_{\rm part}(b)$, $T_{\rm AA}(b)$, and $\varepsilon_{2}(b)$ change due to the
inclusion of the separated proton and neutron transverse distributions, but
typically remain within the previously assigned systematic uncertainties.
Their new associated uncertainties are generally smaller than for earlier
calculations except for mid-peripheral events where they are slightly larger
in some cases. Tables for all quantities versus centrality at present and
foreseen collision energies involving Pb nuclei are provided. Results for
other studied systems, such as AuAu and CuCu collisions at $\sqrt{s_{\rm
NN}}=0.2$ TeV and XeXe collisions at $\sqrt{s_{\rm NN}}=5.44$ TeV, are
provided also for completeness. As for previous versions of the model, the
source code for “TGlauberMC” (v3.1) has been made publicly available at
HepForge glaucode .

The paper is organized as follows: Section II describes the basic quantities
of interest computed in the article. Section III presents a parametrization of
the c.m. energy dependence of the nucleon inelastic cross section
($\sigma_{\rm NN}$) based on existing proton–proton (pp) and proton–antiproton
($\rm p\bar{p}$) data. Section IV introduces the basic details of the MCG
calculation. Section V discusses the improvements of the MCG modeling, namely
using a more realistic nuclear matter density with separated protons and
neutrons profiles (Sec. V.1), incorporating a minimum inter-nucleon separation
without distorting the nuclear profile (Sec. V.2), reweighting the nuclear
density to compensate residual distortions introduced by the nucleon center-
of-mass recentering (Sec. V.3), and using a more precise parameterization of
the $\sigma_{\rm NN}$ (Sec. V.4). Section VI presents the results of the
improved MCG calculation and Section VII summarizes our main conclusions.
Appendix A illustrates the difference between an optical and a Monte Carlo
Glauber calculation. Appendix B briefly discusses the inclusion of
subnucleonic degrees of freedom in the MCG calculation. Appendix C provides an
updated user’s guide for running the publicly available MCG code. Appendix D
provides tables with calculated quantities for all relevant collision energies
involving Pb nuclei, including XeXe collisions at $\sqrt{s_{\rm NN}}=5.44$ TeV
as well as AuAu and CuCu collisions at $\sqrt{s_{\rm NN}}=0.2$ TeV.

##  II Glauber formalism

The standard procedure to determine the transverse overlap area, and other
derived quantities in a generic proton–nucleus (pA) or nucleus–nucleus
collision (AB) at impact parameter $b$, is based on a simple Glauber multi-
scattering eikonal model that assumes straight-line trajectories of the
nucleons from the two colliding nuclei Glauber:1970jm . A review that
describes the basic formalism can be found in Miller:2007ri , of which we
briefly summarize the main concepts here.

To simplify the mathematical description, the reaction plane of the two
colliding nuclei, i.e. the plane defined by the impact parameter and the beam
direction, is given by the $x$\- and $z$-axes, while the transverse plane is
given by the $x$\- and $y$-axes. The collision impact parameter $b$ is
distributed assuming ${\rm d}N/{\rm d}b\propto b$, and the centers of the
nuclei are shifted to $(-\frac{b}{2},0,0)$ and $(\frac{b}{2},0,0)$.

In “optical” Glauber calculations a smooth nuclear matter density, $\rho$, for
each nucleus is used and properties of the reaction zone and all derived
quantities are analytically calculated. In Monte Carlo based approaches
individual nucleons are distributed for each nucleus according to $\rho$ in an
event-by-event basis and collision properties as well as derived quantities
are calculated by averaging over multiple events. In both cases, following the
eikonal ansatz, the nucleons are assumed to move in straight trajectories
along the beam axis. The nuclear reaction is modeled by successive independent
interactions between two nucleons from different nuclei, where the interaction
strength between two nucleons is typically modeled using the nucleon–nucleon
inelastic cross section ($\sigma_{\rm NN}$) in the transverse plane. The
transverse positions of nucleons are assumed to be constant during the short
passage time of the two high-energy nuclei, while their longitudinal
coordinate does not play a role in the calculation.

The optical calculations are based on the thickness function of a nucleus
which quantifies the transverse nucleon density as
$T(x,y)=\int\,\rho(x,y,z)\,{\rm d}z$, which is usually normalized to the
number of nucleons in the nucleus A. The nuclear overlap function of nuclei
$A$ and $B$ colliding at impact parameter $b$, $T_{{}_{\it{AB}}}(b)$, can then
be expressed as the convolution of the corresponding thickness functions of
$A$ and $B$

$\displaystyle T_{{}_{\it{AB}}}(b)$ $\displaystyle=$
$\displaystyle\int\rho_{\rm coll}(x,y,b)\,{\rm d}x{\rm d}y$ (1)
$\displaystyle=$ $\displaystyle\int T_{\rm
A}\left(x-\frac{b}{2},y\right)T_{\rm B}\left(x+\frac{b}{2},y\right)\,{\rm
d}x{\rm d}y$

usually normalized so that $\int T_{{}_{\it{AB}}}(b)\,b\,{\rm d}b=AB$.

The number of nucleons in the target and projectile nuclei that interacted at
least once in a collision at impact parameter $b$ is called the number of
participants (or “wounded nucleons”), and calculated as Bialas:1976ed ;
Kharzeev:1996yx

$\displaystyle N_{\rm part}(b)$ $\displaystyle=$ $\displaystyle\int{\rho_{\rm
part}(x,y,b)\,{\rm d}x{\rm d}y}$ (2) $\displaystyle=$ $\displaystyle
A\int{T_{\rm A}^{-}\left({1-\left[{1-\sigma_{\rm NN}T_{\rm
B}^{+}}\right]^{B}}\right){\rm d}x{\rm d}y}$ $\displaystyle+$ $\displaystyle
B\int{T_{\rm B}^{+}\left({1-\left[{1-\sigma_{\rm NN}T_{\rm
A}^{-}}\right]^{A}}\right){\rm d}x{\rm d}y}$

with $T_{\rm X}^{\pm}\equiv T_{\rm X}(x\pm\frac{b}{2},y)$. Similarly, the
total number of binary nucleon–nucleon collisions at impact parameter $b$ is
calculated as

$\displaystyle N_{\rm coll}(b)$ $\displaystyle=$ $\displaystyle\sigma_{\rm
NN}\,\int\rho_{\rm coll}(x,y,b)\,{\rm d}x{\rm d}y$ (3) $\displaystyle=$
$\displaystyle\sigma_{\rm NN}T_{{}_{\it{AB}}}(b)\,.$

Hence, the nuclear overlap function, $T_{{}_{\it{AB}}}(b)=N_{\rm
coll}(b)/\sigma_{\rm NN}$, can be thought of as the nucleon-nucleon luminosity
(reaction rate per unit cross section) in an AB collision at a given impact
parameter $b$.

MCG calculations obtain the quantities (2) and (3) by simply counting either
the number of nucleons that interacted at least once ($N_{\rm part}$), or the
total number of individual nucleon–nucleon collisions ($N_{\rm coll}$), where
the collisions between the nucleons of the two incoming nuclei are determined
by a $\sigma_{\rm NN}$-dependent interaction probability in the transverse
plane.

The second moment, also called eccentricity Alver:2006wh , the third moment,
also called triangularity Alver:2010gr , and higher moments Teaney:2010vd  of
the collision region at impact parameter $b$, which are used to characterize
the initial geometrical shape, are given by

$\varepsilon_{n}(b)=\frac{\left<r^{n}\cos(n\phi-n\psi)\right>}{\left<r^{n}\right>}$
(4)

where $n$ denotes the moment ($n=2$ for eccentricity, $n=3$ for
triangularity), $r=\sqrt{x^{2}+y^{2}}$ and $\psi=\tan^{-1}\frac{y}{x}$. The
averages are performed by considering the central positions of either,
participant nucleons or binary nucleon–nucleon collisions, or of an admixture
of the two.

The effective transverse overlap area between the two nuclei is often taken to
be proportional to the widths of the participant distributions

$A_{\perp}(b)\propto\sqrt{\left<x^{2}\right>\left<y^{2}\right>}$ (5)

where the averages are taken over participant nucleons. There is no commonly
accepted definition of the absolute normalization of the overlap area.
Historically, either $\pi$ Alver:2008zza  or $4\pi$ Drescher:2007cd  have been
used, where the latter essentially coincides with the geometrical overlap area
of two uniform disks. Recently, it was proposed to directly calculate the area
in the MCG by evaluating the area of wounded nucleons with a fine-grained grid
Loizides:2016djv .

The average path-length through a static medium with a density parameterized
with $\rho_{\rm part}$ can be calculated using

$L(b)=\frac{\int l\,\rho_{\rm
part}(x_{0}+l\cos\phi_{0},y_{0}+l\sin\phi_{0},b)\,{\rm d}l\,{\rm
d}P_{0}}{0.5\int\rho_{\rm
part}(x_{0}+l\cos\phi_{0},y_{0}+l\sin\phi_{0},b)\,{\rm d}l\,{\rm d}P_{0}}$ (6)

where the initial point $(x_{0},y_{0})$ is usually distributed according to
$\rho_{\rm coll}$, and the azimuthal direction $\phi_{0}$ uniformly
Drees:2003zh ; Dainese:2004te .

The total inelastic cross sections for pA or AB collisions are

$\sigma_{\rm pA}=2\pi\,\int b\,\left[1-e^{-\sigma_{\rm NN}\,T_{\rm
A}(b)}\right]\,{\rm d}b\;,$ (7)

and

$\sigma_{\rm AB}=2\pi\,\int b\,\left[1-e^{-\sigma_{\rm
NN}\,T_{{}_{\it{AB}}}(b)}\right]\,{\rm d}b\,.$ (8)

MCG calculations obtain the cross sections by simply multiplying the fraction
of accepted events with $\pi b^{2}_{\rm max}$, where $b_{\rm max}$ is the
maximum generated impact parameter (usually $20$ fm).

Observables are often studied in intervals of cross sections, called
“centrality percentiles”, whose experimental ranges are typically obtained by
ordering the events according to their particle multiplicity or transverse
energy and, in the case of AA collisions, can be translated into equivalent
ranges of impact parameter (see e.g. Abelev:2013qoq ). Instead of reporting
results as a function of centrality percentiles, often the mean number of
participants in the centrality interval is used, which, like all quantities in
a Glauber calculation, can be obtained by performing the calculation over
their respective impact parameter range.

Figure 1: Inelastic pp cross section as a function of c.m. energy in the range $\sqrt{s}=0.2$–$100$ TeV. Experimental data points at various colliders and cosmic-ray energies from UA5 Alner:1986iy , E710 Amos:1990jh ; Amos:1991bp , CDF Abe:1993xy ; Abe:1993xx , ALICE Abelev:2012sea , ATLAS Aad:2011eu ; Aad:2014dca ; Aaboud:2016ijx ; Aaboud:2016mmw , CMS Chatrchyan:2012nj ; VanHaevermaet:2016gnh , LHCb Aaij:2014vfa , TOTEM Antchev:2011vs ; Antchev:2013iaa ; Antchev:2013paa  and AUGER Collaboration:2012wt . Fits of $\ln s$, $\ln^{2}s$ and $\ln^{n}s$ to the data are shown (for details see text). Type | $A$ | $B$ | $n$ | $\chi^{2}/N_{\rm dof}$
---|---|---|---|---
$\ln s$ | $-3.33\pm 1.58$ | $4.195\pm 0.103$ |  $1$ (fixed) | $1.52$
$\ln^{2}s$ | $25.0\pm 0.9$ | $0.146\pm 0.004$ |  $2$ (fixed) | $0.97$
$\ln^{n}s$ | $29.8\pm 4.7$ | $0.038\pm 0.060$ | $2.43\pm 0.50$ | $0.98$
Table 1: Fit values and $\chi^{2}/N_{\rm dof}$ for the collision-energy
dependence of $\sigma_{\rm NN}$ parameterized by Eq. 9 and displayed in Fig.
1.

##  III Parameterization of the inelastic nucleon–nucleon cross section

A fundamental ingredient of any Glauber calculation is the inelastic nucleon-
nucleon cross section, $\sigma_{\rm NN}$, at the same c.m. energy
$\sqrt{s_{\rm NN}}$ of the nuclear collision under consideration. The value of
$\sigma_{\rm NN}$ includes particle production contributions from both
(semi)hard parton-parton scatterings, computable above a given $p_{\rm
T}\approx$ 2 GeV cutoff by perturbative QCD approaches, as well as from softer
“peripheral” scatterings of diffractive nature, with a scale not very far from
$\Lambda_{\rm QCD}\approx 0.2$ GeV. Today, $\sigma_{\rm NN}$ cannot be
computed from first-principle QCD calculations (although future developments
in lattice QCD computations could improve this situation) and one resorts to
phenomenological approaches to fit the experimental data and predict their
evolution as a function of $\sqrt{s_{\rm NN}}$ dEnterria:2016oxo . At high
c.m. energies, above a few tens of GeV, pp and $\rm p\bar{p}$ (as well as nn
and np) collisions all feature the same inelastic cross sections. Any
potential differences due to their different valence-quark structure are
increasingly irrelevant, and all existing experimental measurements can be
combined to extract $\sigma_{\rm NN}$. The $\sqrt{s}$ dependence of the
inelastic cross section $\sigma_{\rm NN}$ is shown in Fig. 1 for all the
available data from $\rm p\bar{p}$ and pp colliders, and the AUGER result at
$\sqrt{s}=57$ TeV derived from cosmic-ray data Collaboration:2012wt . We
include $\rm p\bar{p}$ measurements from UA5 Alner:1986iy  at $\sqrt{s}=200$
and $900$ GeV, E710 Amos:1990jh ; Amos:1991bp  and CDF Abe:1993xy ; Abe:1993xx
at $\sqrt{s}=1.8$ TeV, as well as pp results from ALICE at 7 TeV
Abelev:2012sea , ATLAS at $7$, $8$ and $13$ TeV Aad:2011eu ; Aad:2014dca ;
Aaboud:2016ijx ; Aaboud:2016mmw , CMS at $7$ and $13$ TeV Chatrchyan:2012nj ;
VanHaevermaet:2016gnh , LHCb at $7$ TeV Aaij:2014vfa , and TOTEM at $7$ and
$8$ TeV Antchev:2011vs ; Antchev:2013iaa ; Antchev:2013paa . The experimental
$\sigma_{\rm NN}$ values plotted are either obtained (i) from the subtraction
$\sigma_{\rm inel}=\sigma_{\rm tot}-\sigma_{\rm el}$, where $\sigma_{\rm tot}$
and $\sigma_{\rm el}$ have been accurately measured in dedicated forward Roman
pot detectors (TOTEM Antchev:2011vs ; Antchev:2013iaa ; Antchev:2013paa  and
ALFA Aad:2014dca ; Aaboud:2016ijx ; Aaboud:2016mmw ), or (ii) from
measurements of inelastic particle production data in the central detectors
collected with “minimum bias” triggers. The latter measurements are less
accurate than the former, as they require an extrapolation, dominated by
diffractive contributions, to forward regions of phase space not covered by
detectors, and therefore have larger uncertainties.

$\sqrt{s}$ (TeV) |  $\sigma_{\rm NN}$ (mb)
---|---
0.2 | $41.6\pm 0.6$
0.9 | $52.2\pm 1.0$
2.76 | $61.8\pm 0.9$
5.02 | $67.6\pm 0.6$
5.44 | $68.4\pm 0.5$
5.5 | $68.5\pm 0.5$
7 | $70.9\pm 0.4$
8 | $72.3\pm 0.5$
8.16 | $72.5\pm 0.5$
8.8 | $73.3\pm 0.6$
10.6 | $75.3\pm 0.7$
13 | $77.6\pm 1.0$
14 | $78.4\pm 1.1$
17 | $80.6\pm 1.5$
27 | $86.0\pm 2.4$
39 | $90.5\pm 3.3$
63 | $96.5\pm 4.6$
100 | $102.6\pm 6.0$
Table 2:  Values of the nucleon-nucleon inelastic cross section $\sigma_{\rm
NN}$ extracted from the $\ln^{2}$ fit, with the uncertainties estimated from
the difference of the $\ln s$ and $\ln^{n}s$ ($n=2.43$) fits at collision-
energies $\sqrt{s}$ relevant for RHIC, LHC, and FCC.

The collision-energy dependence of $\sigma_{\rm NN}$ has been fit to the
parameterization

$\sigma_{\rm NN}(s)=A+B\,\ln^{n}(s)\,,$ (9)

where $n$ was fixed to either $n=1$ or $n=2$, or otherwise left free in the
fit. The values and $\chi^{2}/N_{\rm dof}$ for the three cases are given in
Table 1. The $n=2$ case, which represents the asymptotic $\sqrt{s}$-dependence
expected to saturate the Froissart bound Froissart:1961ux , is used as central
value for the interpolation (and extrapolation) of $\sigma_{\rm NN}$ versus
$\sqrt{s}$, listed in Table 2 for relevant LHC and FCC energies. The
difference (normalized by 2.4 to account for the full width at half maximum)
of the so-derived $\sigma_{\rm NN}$ values to those obtained for $n=1,2.43$ is
assigned as systematic uncertainty (shown as a band in Fig. 1). The resulting
cross section at 100 TeV of $\sigma_{\rm NN}=102.6\pm 6.0$ mb is consistent
with the value $105.1\pm 2.0$ mb, obtained from the average of various model
calculations dEnterria:2016oxo . The value extracted for the top RHIC energy
of $\sqrt{s_{\rm NN}}=0.2$ TeV is $41.6\pm 0.6$ mb and is consistent with the
typically used value of $42\pm 3$ mb Alver:2008zza .

Other more complicated functional forms were also tried, motivated by the
Ansatz used by the COMPETE collaboration Cudell:2002xe , such as $\sigma_{\rm
NN}(\sqrt{s})=A+B\,\ln^{2}(s)+C\,s^{-\eta}$, and $\sigma_{\rm
NN}(\sqrt{s})=A+B\,\ln^{2}(s)+D\ln(s)$. The corresponding fits resulted in
$A=24.4\pm 1.4$, $B=0.1008\pm 0.1537$, $C=1.454\pm 1.768$, $\eta=0.131\pm
0.0180$ with $\chi^{2}/N_{\rm dof}=1.09$, and $A=39.7\pm 1.4$, $B=0.2212\pm
0.0708$, $D=-2.154\pm 2.035$ with $\chi^{2}/N_{\rm dof}=0.96$, respectively.
Both parameterizations turned out to be numerically close to Eq. 9 for
$n\approx 2.43$ as determined by the simpler $\ln^{n}s$-fit.

From the obtained values of $\sigma_{\rm NN}$, one can then easily derive the
corresponding proton-nucleus and nucleus-nucleus inelastic collisions making
use of Eq. 7 and Eq. 8. The computed $\sigma_{\rm pA}$ and $\sigma_{\rm AB}$
results for all relevant collision systems in this work are listed in Table 5.
The Glauber calculation gives $\sigma_{\rm PbPb}^{\rm MC}=7.57\pm 0.03$ b and
$\sigma_{\rm pPb}^{\rm MC}=2.08\pm 0.01$ b for the hadronic PbPb and pPb cross
sections, in good agreement with the measured values of $\sigma_{\rm
PbPb}=7.7\pm 0.6$ b at $\sqrt{s_{\rm NN}}=2.76$ TeV ALICE:2012aa  and
$\sigma_{pPb}=2.06\pm 0.08$ b Khachatryan:2015zaa  as well as $\sigma=2.10\pm
0.07$ Abelev:2014epa  at $\sqrt{s_{\rm NN}}=5.02$ TeV, respectively.

##  IV Details of the MCG calculation

The implementation of the MCG calculation is described in detail in
Alver:2008aq ; Loizides:2014vua . It consists of two steps: first,
constructing the nuclei and, second, colliding the nuclei.

To construct a nucleus, the position of each nucleon is determined according
to a probability density function usually taken from measurements of the
charge density distribution of the nucleus DeJager:1974liz ; DeJager:1987qc .
For spherical nuclei, the nucleon positions can be determined in polar
coordinates with a uniform distribution for the azimuthal and polar angles,
coupled with a 2pF distribution in the radial direction

$\rho(r)=\frac{\rho_{0}}{1+\exp{[(r-R)/a]}}\,,$ (10)

where $\rho_{0}$ is a normalization constant so that $\int d^{3}r\,\rho(r)=1$.
The half-density or central radius $R$ describes the mean location of the
nucleus area (i.e. $R$ is indicative of the extension of the bulk part of the
density distribution). The diffusivity parameter $a$ describes the tail of the
density profile. Values for Pb nuclei are listed in Table 4, while a complete
list of parameters for other nuclei can be found in LABEL:Loizides:2014vua. To
mimic a hard-core repulsion potential between nucleons, a minimum inter-
nucleon separation ($d_{\rm min}$) of usually $0.4$ fm between their centers
is enforced when sampling the positions of the nucleons inside a nucleus. In
order to ensure that the center-of-mass of each constructed nucleus is at
$(0,0,0)$, the nucleons are individually “recentered” through a procedure
discussed in more detail later.

To simulate the collision, the centers of the nuclei are then shifted to
$(-b/2,0,0)$ and $(b/2,0,0)$. The collision of two nuclei is then modeled by
assuming that the nucleons of each nucleus travel in a straight line along the
beam axis in the transverse plane (eikonal approximation), ignoring their
longitudinal coordinates in the calculation. The impact parameter of the
collision is chosen randomly from ${\rm d}N/{\rm d}b\propto b$ up to some
large maximum $b_{\rm max}\simeq 20\,$fm, chosen to be significantly greater
than twice the nuclear radius. Two nucleons from different nuclei are usually
assumed to collide if their relative transverse distance is less than a
diameter given by

$D=\sqrt{\sigma_{\rm NN}/\pi}$ (11)

which geometrically parameterizes the interaction strength of two nucleons for
a given value of $\sigma_{\rm NN}$. If no nucleon–nucleon collision is
registered for any pair of nucleons, then no nucleus–nucleus collision
occurred. Counters for determination of the total (geometric) cross section
are updated accordingly. The inelastic nucleon-nucleon cross section
$\sigma_{\rm NN}$ is either directly taken from measurements in pp collisions,
or extracted from interpolations of the available data, as explained in Sec.
III.

Constructing the nucleus is a principal ingredient of the MCG model and the
dominant source of systematic uncertainties in the Glauber quantities, in
particular after reducing the uncertainties of the interpolated $\sigma_{\rm
NN}$ values. In the following, we will discuss improvements of the MCG model
aiming at achieving a more accurate baseline description with reduced
systematic uncertainties. The new results will be labeled improved MCG, and
discussed in detail in the next Section. In order to compare with previous
baseline results, we compare the results of our new calculations with those
from a set of traditional parameters of the MCG model, typically used in
previous studies Abelev:2013qoq ; Adam:2014qja ; Loizides:2016djv , given by
$\sigma_{\rm NN}=64$ mb for $\sqrt{s_{\rm NN}}=2.76$ TeV and $\sigma_{\rm
NN}=70$ mb for $\sqrt{s_{\rm NN}}=5.02$ TeV with an uncertainty of $\pm 5$ mb,
with charge radius and diffusivity of the nuclear density profile varied
within their measured $1\sigma$ uncertainties, and minimum inter-nucleon
separation distance varied by 100%, i.e. between $0$ and $0.8$ fm. The
algorithmic definitions, as well as central values and uncertainties of the
parameters, for the traditional and improved MCG setups are summarized in
Table 3.

MCG model | Traditional | Improved
---|---|---
Density for Pb | Charge, 2pF (“Pb”) | Point, D2pF (“Pbpnrw”)
NN separation (fm) | $d_{\rm min}=0.4\pm 0.4$ | $d_{\rm node}=0.4\pm 0.4$
$\sigma_{\rm NN}$ (mb) | $70\pm 5$ | $67.6\pm 0.5$
Recentering | Shift |  $d_{\rm max}=0.1$ fm + reweight
TGlauberMC |  $\equiv$v2.x |  $\equiv$v3.x
Table 3: Parameters with corresponding uncertainties for the traditional and
improved MCG model used to compute Glauber quantities for nuclear collisions
at a reference c.m. energy of $\sqrt{s_{\rm NN}}=5.02$ TeV.

##  V Improvements of the MCG modeling

###  V.1 Nuclear matter density

The nuclear density parameters used for the 2pF distributions are typically
taken from the Atomic Data and Nuclear Data Tables (ADND) DeJager:1987qc ;
DeJager:1974liz ; Fricke:1995zz ; Angeli:2004kvy ; Angeli:2013epw . They are
extracted via Coulomb scattering in electron–nucleus and muon–nucleus
measurements and therefore dominantly probe the charge density of the nucleus.
Since 208Pb is a “doubly-magic” nucleus (both the number of protons, 82, and
number of neutrons, 126, are arranged in fully closed energy shells), it is
rather immune to shape deformations, and hence its charge density is well
described by a 2pF distribution, with $R$ and $a$ determined to within $1$ and
$2$%, respectively. Traditionally, the values for 207Pb from DeJager:1987qc
have been used instead of those for 208Pb from DeJager:1974liz  when modeling
208Pb in MCG calculations. 111There is no clear reason for that, and we
speculate that it may simply be an oversight because the 208Pb parameters were
only collected in the earliest, but not in the later ADND publications. In any
case, the two sets of parameters are essentially the same, as can be seen in
Table 4.

Figure 2: Nuclear density of 208Pb for the charge distribution (2pF) as well as the sum of the proton and neutron point density distributions (D2pF), corresponding to the parameters listed in Table 4. The grey band indicates the $1\sigma$ uncertainty for the D2pF distribution. Charge density | Name |  $R$ (fm) |  $a$ (fm)
---|---|---|---
207Pb DeJager:1987qc  | Pb | $6.620\pm 0.060$ | $0.546\pm 0.010$
208Pb DeJager:1974liz  | Pb* | $6.624\pm 0.035$ | $0.549\pm 0.008$
Point density |  |  |
208Pb | Pbpn |  |
proton Klos:2007is  |  | $6.68\pm 0.02$ | $0.447\pm 0.01$
neutron Tarbert:2013jze ; Klos:2007is  |  | $6.69\pm 0.03$ | $0.560\pm 0.03$
Table 4:  Nuclear density parameters of Pb for charge and point density
distributions. For the neutron point density, the values are averaged as
explained in the text. The name of the corresponding profile in the TGlauberMC
implementation glaucode  is also listed.

However, since the MCG uses the charge density to place the central locations
of each nucleon, a preferred representation is the point density distribution,
which parameterizes the 2pF function for the centers of the nucleons.
Transforming from the charge to point distribution involves parameter
unfolding which is performed using the proton root-mean-square (rms) charge
radius $\sqrt{\langle r^{2}\rangle}=0.875$ fm Mohr:2015ccw  via the
prescription given in Klos:2007is ; Warda:2010qa . The point density 2pF
parameter values are slightly smaller than the charge density ones due to the
proton’s finite spatial extension. The corresponding uncertainties on the
proton radius density $R_{\rm p}$ have become smaller over the years and are
now below $0.5$% Fricke:1995zz ; Angeli:2004kvy ; Angeli:2013epw . However,
the diffusivity parameter for protons, $a_{\rm p}$, is no longer quoted in the
more recent ADND tables. Moreover, it has been shown that at very large radii
(distances greater than $\approx R+3a$) the 2pF parameterization begins to
fail as the measured density falls off faster than a Woods-Saxon distribution
Wycech:2007jb . This observation can be modeled by letting the diffusivity
parameter shrink with increasing $r$, and while the authors provide $a(r)$ for
40Ca and 48Ca, they do not provide it for 208Pb. Thus, we maintain the
constant-$a$ 2pF form and, in turn, sustain the traditional relatively large
uncertainty on the diffusivity parameter of about $2$% ($\pm 0.01$ fm).

Using the nucleon point density distribution leads to a more realistic
placement of the nucleons. However, there is evidence that the proton and
neutron distributions may not be exactly the same at the surface of heavy
stable nuclei barrett1977nuclear . This effect is particularly important in
neutron-rich nuclei, such as 208Pb with a neutron excess of $N/Z\approx 1.5$.
Protons near the center of the nucleus feel electrostatic repulsion from all
directions resulting in an electrostatic equilibrium and a constant charge
density. However, at $r\\!\gtrsim\\!6$ fm, where the nucleon density begins to
drop, the outermost protons need additional “skin” or “halo” neutrons in the
periphery to counteract the outward Coulomb repulsion and maintain a
sufficient nuclear surface tension thereby increasing the overall binding
energy.

To extract the 2pF parameters for neutrons, the Crystal Ball collaboration has
performed a measurement via coherent pion photoproduction Tarbert:2013jze
while the Low Energy Proton Ring (LEAR) at CERN has investigated
antiproton–nucleus interactions coupled with radiochemistry techniques
Klos:2007is . The former extracts neutron point density parameters of $R_{\rm
n}=6.70\pm 0.03$ (stat.) fm and $a_{\rm n}=0.55\pm 0.01$
(stat.)${}^{+0.02}_{-0.03}$ (syst.) fm, while the latter reports comparable
values of $R_{\rm n}=6.684\pm 0.020$ (stat.) fm and $a_{\rm n}=0.571$ fm.
These data favor the peripheral neutron distribution in the form of a neutron
“halo” rather than a neutron “skin”, i.e. the neutron distribution is slightly
broader than the proton distribution because of its larger diffusivity
($a_{\rm n}-a_{\rm p}\approx 0.1$ fm), but has the same half-radius as the
proton distribution ($R_{\rm p}\approx R_{\rm n}\approx 6.7$ fm). For the LEAR
measurement no uncertainty was explicitly reported for $a_{\rm n}$ though the
central value is consistent with Tarbert:2013jze  and we assume $\pm 0.03$ fm;
both use the same proton charge density parameters taken from the ADND. The
neutron parameters are then averaged and listed in Table 4 together with the
proton point 2pF parameters. The combined point density distribution for
proton and neutrons is then the weighted sum of the individual 2pF
distributions, which we simulate in the MCG by drawing 82 protons from the
proton point 2pF and 128 neutrons from the neutron point 2pF. The D2pF
distribution is displayed in Fig. 2 with its corresponding $1\sigma$
uncertainty and compared to the traditionally-used charge density
distribution.

Figure 3: Relative change in $N_{\rm coll}$ in PbPb and pPb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV due to the updated nuclear density profile. The
baseline uses the standard 2pF charge density, while the new results are
obtained with the D2pF density.

The relative change in $N_{\rm coll}$ due to switching from the 2pF charge
density to the D2pF point density representation (while everything else is
computed in the traditional approach) is illustrated in Fig. 3. In mid-
peripheral PbPb collisions, the change results in a maximum $\sim 4$% increase
in $N_{\rm coll}$ and approximately $2$% for pPb collisions. This is largely
driven by the increase of the central radius in the D2pF compared to the 2pF
parameterization.

###  V.2 Minimum nucleon separation

Prior to this work, varying the inter-nucleon separation from the default
value ($0.4$ fm) to its assumed upper limit ($0.8$ fm) led to uncertainties of
about $2$% in the derived Glauber quantities. Such a result is somewhat
surprising given that, if uniform spherical packing is naïvely assumed for
nucleons near the center of the nucleus, the typical distance between any two
nucleons should be $1.5$–$2$ fm, significantly larger than $d_{\rm min}$, and
hence the results should not be dramatically affected when varying the latter.
Traditional MCG implementations place nucleons by first sampling the 2pF
distribution and then checking the $d_{\rm min}$ requirement with respect to
the already placed nucleons. When the $d_{\rm min}$ requirement is not
satisfied, the algorithm discards that nucleon and resamples the 2pF
probability distribution. This approach results in an overall bias in the
constructed radial distribution that propagates to all computed quantities.
Figure 4 shows the resulting deformation in the radial profile due to this
bias, which increases with increasing $d_{\rm min}$. Nucleons are
preferentially pushed to larger radii where there is more physical phase-space
to fill.

Figure 4: Nuclear radial density distributions for various values of the
inter-nucleon distance $d_{\rm min}$ using the “traditional” MCG
implementation (top panel) and their ratio to the 2pF profile (bottom panel).
The deviations from 1 at large $r$ increase with increasing $d_{\rm min}$.

One approach to overcome this effect is to rescale the input profile
parameters until the bias brings the resulting density back to the desired 2pF
distribution Hirano:2009ah ; Shou:2014eya . This iterative procedure is
cumbersome, unphysical, and not universal for all collisions systems. Instead,
to remove this bias, we introduce a uniform three-dimensional lattice with a
minimum nodal separation ($d_{\rm node}$) equivalent to $d_{\rm min}$. The
full physical phase space is sampled by pre-calculating all lattice nodes
within a cubic space of $40\times 40\times 40$ fm3. These nodes are sampled
uniformly in Cartesian space and subsequently populated with a nucleon
according to the 2pF probability distribution. Once a node has been populated,
it is removed from the sampling. By apriori restricting the allowable phase-
space to exclude overlapping nucleons, the 2pF probability distribution can be
sampled without introducing artificial distortions. To ensure that
regularities in the lattice are avoided, the lattice is randomized event-by-
event in azimuthal and polar orientation in addition to being randomly
translated laterally in Cartesian space. After the implementation of the
lattice framework, the density profile remains largely intact and subsequently
the centrality variables become stable with respect to $d_{\rm min}$
variations. This is demonstrated in Fig. 5, which shows the resulting density
profiles when varying $d_{\rm min}$ by 100% ($0.4\pm 0.4$ fm). The results are
insensitive to the specific lattice basis used hcp : Hexagonal Close Packing
(HCP), Face Centered Cubic, Body Centered Cubic, and Simple Cubic. Generally,
lattices with packing fractions above about $50$% are indistinguishable for
$d_{\rm min}<1.2$ fm. The HCP lattice was used as the default configuration as
it has the most optimal packing fraction of $74$%. The insensitivity to the
lattice structure is intuitive when considering that less than 0.5% of nodes
inside a radius of about $6.7$ fm are populated when $d_{\rm min}=0.4$ fm. As
either $d_{\rm min}$ is increased to larger than $1.2$ fm or the packing
fraction drops significantly below 50%, the fraction of nodes available will
be greatly reduced and distortions start to impact the density distribution.
It should be noted that, from a technical standpoint, the same result can be
achieved with the traditional MCG implementation by discarding the entire
nucleus in the event of two nucleons overlapping (rather than only the
offending nucleon). This, however, is computationally prohibitive and
therefore impractical.

Figure 5: Nuclear radial density distributions for various values of the
inter-nucleon lattice distance $d_{\rm node}$ ($\equiv d_{\rm min}$) using the
lattice MCG implementation (top panel) and their ratio to the 2pF profile
(bottom panel). Figure 6: Relative change in $N_{\rm coll}$ for PbPb and pPb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV after introducing the nucleon
lattice placement algorithm with $d_{\rm node}=0.4$ fm. The baseline uses the
traditional MCG implementation with $d_{\rm min}=0.4$ fm.

Figure 6 quantifies the relative change in $N_{\rm coll}$ with respect to the
traditional MCG implementation for $d_{\rm min}=0.4$ fm for PbPb and pPb
collisions. The mean value of $N_{\rm coll}$ as a function of centrality
changes by less than 0.2%. Since the radial profile is not affected by
variations of $d_{\rm min}$, introducing the lattice to construct the nuclei
effectively removes the uncertainty due to the minimum distance between
nucleons (see Sec. VI).

Figure 7: Ratio of radial density distributions constructed with the standard
MCG for $d_{\rm min}=0$ fm over that from the 2pF profile for different
methods to recenter the nucleons. The green distribution is obtained with the
standard approach including recentering, while the grey distribution is
without recentering. The other two cases impose $d_{\rm max}=0.1$ fm (as
explained in the text). The red distribution is obtained by dividing
(reweighting) the Pb nucleus radial profile with the 2nd-order polynomial
(blue line).

###  V.3 Recentering

Inspecting the bottom panel of Fig. 5 closely, reveals that there are still
residual differences of up to a few percent in the radial profile, even when
the lattice is used. Indeed, even for $d_{\rm min}=0$ fm, i.e. without a
requirement on the nucleon–nucleon separation, a non-monotonic structure
emerges, as can be seen in the “zoomed-in” ratio relative to the 2pF profile
displayed in Fig. 7. It originates from the recentering algorithm that is
usually applied in MCG calculations, since without recentering the ratio
relative to the 2pF is exactly one. The traditional MCG approaches
Alver:2008aq ; Loizides:2014vua ; Broniowski:2007nz ; Rybczynski:2013yba ;
Loizides:2016djv , except the HIJING model Wang:1991hta , recenter the
nucleons by the average of the displacement after having distributed them
individually according to the nuclear density profile. This is also the case
for the advanced MC calculation of LABEL:Alvioli:2011sk, which includes
realistic nucleon–nucleon correlations Alvioli:2009ab . The recentering step
is applied to ensure that the center-of-mass of the constructed nucleus
coincides with that of the nuclear density from which the nucleon positions
were stochastically determined. Shifting the nucleons by the average
displacement, however, introduces a distortion of the radial profile, which
increases with decreasing degrees of freedom. This effect has been recently
discussed in Glauber approaches accounting for subnucleonic degrees of
freedom, where the distortion is particularly large when only three partons
(quarks) are distributed inside a proton Mitchell:2016jio . For a 208Pb
nucleus, the width of the center-of-mass shift is about $0.2$ fm in each
direction, and the effect of the associated distortions have so far been
ignored. In order to ensure that the center-of-mass of the constructed nucleus
is at $(0,0,0)$, one can only accept constructed nuclei where the average
displacement in each direction is small, e.g. smaller than $d_{\rm max}=0.1$
fm. This requirement leads to more (less) dense radial distributions than the
2pF profile for small (large) radii, as can be seen in the corresponding ratio
(blue curve) in Fig. 7. The corresponding ratio can be empirically described
by a second-order polynomial as $f(r)=1.00863-0.00045r-0.00021r^{2}$.
Reweighting the original radial profile with $f$, i.e. using $\rho/f$ to
distribute nucleons in the radial direction, allows to correct for the
residual bias. The ratio of the resulting radial distribution relative to the
2pF deviates by less than $0.2$% from unity.

The effect on $N_{\rm coll}$ of the reweighted radial profile relative to the
standard 2pF profile is quantified in Fig. 8, and leads to variations below
1%. In particular, for pPb collisions the residual change from recentering to
reweighting is much smaller than the modification introduced by recentering
alone, which for peripheral collisions is larger than 10% (see Fig. 20 in the
Appendix).

Figure 8: Relative change in $N_{\rm coll}$ in PbPb and pPb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV using the reweighted profile for $d_{\rm
min}=0.1$ fm and $d_{\rm min}=0.4$. The baseline uses the traditional MCG
implementation with $d_{\rm min}=0.4$ fm. Figure 9: Relative change in $N_{\rm
coll}$ in PbPb and pPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV due to the
updated nucleon-nucleon cross section. The baseline uses $\sigma_{\rm NN}=70$
mb, while the updated value is $67.6$ mb.

###  V.4 Nucleon–nucleon collision modeling

Given that the nucleon-nucleon interaction probability depends on the
condition given by Eq. 11, the improved $\sqrt{s}$-parameterization and
uncertainty of $\sigma_{\rm NN}$ discussed in Section III leads to Glauber
quantities that are both more accurate and more precise than before.

Figure 10: Ratio of $N_{\rm coll}$, $N_{\rm part}$, $T_{\rm AA}$ and
$\varepsilon_{2}$ as a function of centrality obtained using the improved
approach (including $\sqrt{s_{\rm NN}}$, lattice, and D2pF changes) over the
same quantities obtained with the traditional approach, for PbPb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV. The grey band illustrates the traditionally
reported uncertainties. Figure 11: Ratio of $N_{\rm coll}$, $N_{\rm part}$,
$T_{\rm AA}$ and $\varepsilon_{2}$ as a function of centrality obtained using
the improved approach (including $\sqrt{s_{\rm NN}}$, lattice, and D2pF
changes) over the same quantities obtained with the traditional approach, for
pPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV. The grey band illustrates the
traditionally reported uncertainties. Figure 12: Relative uncertainties in
$N_{\rm coll}$, $N_{\rm part}$, $T_{\rm AA}$ and $\varepsilon_{2}$ arising
from varying $\sigma_{\rm NN}$, as well as inter-nucleon separation and
density parameters for the improved and traditional approaches in PbPb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV. Figure 13: Relative uncertainties
in $N_{\rm coll}$, $N_{\rm part}$, and $T_{\rm AA}$ and $\varepsilon_{2}$
arising from varying $\sigma_{\rm NN}$ as well as inter-nucleon separation and
density parameters for the improved and traditional approaches in pPb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV.

To demonstrate the effect of this change for $\sqrt{s_{\rm NN}}=5.02$ TeV, the
relative change of $N_{\rm coll}$ with respect to the previously used
$\sigma_{\rm NN}$ value is shown in Fig. 9. The baseline uses the value of
$\sigma_{\rm NN}=70$ mb, commonly used at the LHC, while the updated value is
$67.6$ mb. As expected, the change is largest for central collisions, namely
equal to the ratio of $67.6/70\approx 0.97$, while for the most peripheral
collisions there is no observable numerical change.

##  VI Results

The improvements considered here, including the $\sqrt{s}$-parameterization of
$\sigma_{\rm NN}$, the use of the D2pF profile, plus lattice regularization as
well as the recentering and reweighting approach, comprise the improved MCG
approach glaucode , whose parameters are summarized in Table 3. To illustrate
the differences with the traditional approach, we compare the values of
$N_{\rm coll}$, $N_{\rm part}$, $T_{\rm AA}$ and $\varepsilon_{2}$ computed
with both approaches for PbPb and pPb collisions at $\sqrt{s_{\rm NN}}=5.02$
TeV in Figs. 10 and 11. The uncertainties due to the 2pF and D2pF parameters,
given in Table 4, were calculated by running 100k MCG events for 100 parameter
set variations. Each variation allowed each parameter to take a random value
within a Gaussian distribution with a width of the $1\sigma$ uncertainty on
each parameter. The spread of the resulting values quantified by their
standard deviation was used as the reported resulting $1\sigma$ uncertainty
due to the 2pF and D2pF parameters. The uncertainties due to $\sigma_{\rm NN}$
as well as due to the minimum inter-nuclear separation ($d_{\rm min}$ and
$d_{\rm node}$), given in Table 3, were obtained by running with nominal
settings varying each one of the parameters by $\pm\sigma$ at a time, and
assigning half of the difference as the corresponding $1\sigma$ uncertainty.
To obtain the total uncertainties the individual uncertainties due to density
profile, $\sigma_{\rm NN}$, and inter-nucleon separation were added in
quadrature.

Figures 10 and 11 quantify the changes as the ratio of $N_{\rm coll}$, $N_{\rm
part}$, $T_{\rm AA}$ and $\varepsilon_{2}$ in $5$%-wide centrality intervals
using the improved and traditional approach for PbPb and pPb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV, respectively. The ratios are compared to the
total uncertainties of the traditional approach to illustrate that the central
values of the improved results are generally within the previously assigned
uncertainties, which were typically dominated by the large uncertainty on
$\sigma_{\rm NN}$, except in the case of $T_{\rm AA}$ for peripheral PbPb
collisions. Since in $T_{\rm AA}$ the quite large uncertainties on
$\sigma_{\rm NN}$ cancel out, this quantity is especially sensitive to other
small changes introduced by the improvements. Our results indicate that, for
the relevant centrality classes, previous experimental results on $R_{\rm AA}$
would have to be scaled up by up to $3$–$5$%, however ratios of results taking
at $\sqrt{s_{\rm NN}}=5.02$ and $\sqrt{s_{\rm NN}}=2.76$ would not be affected
because $T_{\rm AA}$ would change similarly in both cases. We checked that the
lattice and traditional approaches lead to identical results for identical
settings. Hence, since it is less computationally intensive, one can also use
$d_{\rm min}=0.4$ fm in the traditional way (i.e. without the lattice) but
ignoring the uncertainties introduced from variation to $d_{\rm min}=0$ and
$0.8$ fm.

Figure 14: Average fraction of pp, pn, and nn collisions for PbPb and pPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV obtained with our improved MCG model. $\sqrt{s_{\rm NN}}$ (TeV) |  $\sigma_{\rm NN}$ (mb) | System |  $\sigma$ (b) | Table
---|---|---|---|---
2.76 | $61.8\pm 0.9$ | PbPb | $7.57\pm 0.03$ | Table 9
5.02 | $67.6\pm 0.6$ | PbPb | $7.66\pm 0.03$ | Table 10
5.5 | $68.5\pm 0.5$ | PbPb | $7.67\pm 0.03$ | Table 11
10.6 | $75.3\pm 0.7$ | PbPb | $7.77\pm 0.03$ | Table 12
39 | $90.5\pm 3.3$ | PbPb | $7.90\pm 0.03$ | Table 13
5.02 | $67.6\pm 0.6$ | pPb | $2.08\pm 0.01$ | Table 14
8.16 | $72.5\pm 0.5$ | pPb | $2.12\pm 0.01$ | Table 15
8.8 | $73.3\pm 0.6$ | pPb | $2.13\pm 0.01$ | Table 16
17 | $80.6\pm 1.5$ | pPb | $2.18\pm 0.01$ | Table 17
63 | $96.5\pm 4.6$ | pPb | $2.28\pm 0.01$ | Table 18
5.44 | $68.4\pm 0.5$ | XeXe | $5.67\pm 0.02$ | Table 19
0.2 | $41.6\pm 0.6$ | AuAu | $6.80\pm 0.03$ | Table 20
0.2 | $41.6\pm 0.6$ | CuCu | $3.43\pm 0.03$ | Table 21
Table 5:  Values for total PbPb and pPb cross sections (with statistical
uncertainties) at collision energies relevant for the LHC and FCC. For
completeness, results for XeXe at $\sqrt{s_{\rm NN}}=5.44$ TeV, as well as as
AuAu and CuCu collisions at $\sqrt{s_{\rm NN}}=0.2$ TeV are also included. The
values for $\sigma_{\rm NN}$ are from Table 2. For every collision system the
corresponding centrality dependent Glauber quantities can be found in the
specified table provided in Appendix D. Figure 15: Number of binary collisions
($N_{\rm coll}$), number of participants ($N_{\rm part}$), and overlap
function ($T_{\rm AA}$) as a function of centrality for PbPb collisions at
$\sqrt{s_{\rm NN}}=2.76$, $5.02$ and $39$ TeV (left columns) and for pPb
collisions at $\sqrt{s_{\rm NN}}=5.02$, $8.16$ and $63$ TeV (right columns)
using the improved MCG. Figure 16: Overlap area ($A_{\rm T}$), average path
length ($L$), participant eccentricity ($\varepsilon_{2}$) and triangularity
($\varepsilon_{3}$) as a function of centrality for PbPb collisions at
$\sqrt{s_{\rm NN}}=2.76$, $5.02$ and $39$ TeV (left columns) and for pPb
collisions at $\sqrt{s_{\rm NN}}=5.02$, $8.16$ and $63$ TeV (right columns)
using the improved MCG.

Figures 12 and 13 show the respective individual and total uncertainties as
dotted lines for the traditional and full lines for the improved model. The
2pF uncertainties and the minimum inter-nucleon separation ($d_{\rm min}$)
reach up to about $2$%, while the (previously) large uncertainty on
$\sigma_{\rm NN}$ propagated into up to $7$% on $N_{\rm coll}$ for central
collisions and $T_{\rm AA}$ for peripheral collisions, and typically dominated
the final uncertainty . In contrast, the uncertainties due to the minimum
separation enforced between nucleons by the lattice as well as due to the more
precise parameterization of $\sigma_{\rm NN}$($\sqrt{s_{\rm NN}}$) are quite
small, and, in particular, for the inter-nucleon separation nearly negligible
in the improved approach. This is particularly apparent in the case of
$\varepsilon_{2}$, where the uncertainty related to $d_{\rm min}$ was
substantial in central collisions for the traditional approach. The
uncertainty due to the D2pF parameters, however, have grown. Since the
uncertainty on the neutron diffusivity has actually increased to about $5$%
and there are about $50$% more neutrons than protons in 208Pb, the inclusion
of the D2pF nuclear density description results in a more accurate, though
less precise, determination of Glauber quantities. For this reason, coupled
with the fact that traditional 2pF forms represented the charge density rather
than the point density, previously quoted uncertainties based on the 2pF
parameters were slightly underestimated. For pPb collisions, other
experimental uncertainties become dominant, such as those resulting from the
event activity class used to determine the centrality Adam:2014qja .
Furthermore, it is important to realize that the spread of the computed
quantities in each centrality class is rather large, in particular for
peripheral collisions, where the ratios of the standard deviation over the
mean of each Glauber quantity can reach up to 80% (as can be seen in the
Tables of Appendix D).

Figure 14 shows the average fraction of pp, pn, and nn collisions for PbPb and
pPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV from the D2pF calculation. In
peripheral collisions, the pn and nn interactions become more probable due to
the extended neutron “halo” or “skin”, and therefore are particularly relevant
for precision measurements involving isospin– (or electric charge–) dependent
observables, such as electroweak boson production, in nuclear collisions
Paukkunen:2015bwa ; De:2016ggl ; Helenius:2016dsk .

Finally, we present the number of binary collisions ($N_{\rm coll}$), number
of participants ($N_{\rm part}$), and overlap ($T_{\rm AA}$) in Fig. 15, as
well as the overlap area ($A_{\rm T}$), average path length ($L$), participant
eccentricity ($\varepsilon_{2}$), and triangularity ($\varepsilon_{3}$) in
Fig. 16 as a function of centrality for PbPb collisions at $\sqrt{s_{\rm
NN}}=2.76$, $5.02$ and $39$ TeV (left plots), and in pPb collisions at
$\sqrt{s_{\rm NN}}=5.02$, $8.16$ and $63$ TeV (right plots) using the improved
MCG. The inelastic cross sections for all collision systems computed with the
improved approach are given in Table 5. Appendix D provides detailed tables
for the corresponding quantities in $5$%-wide centrality classes.

##  VII Summary

We have presented the results of an improved Monte Carlo Glauber model for the
calculation of quantities of relevance for collisions involving nuclei at
center-of-mass energies of BNL RHIC ($\sqrt{s_{\rm NN}}=0.2$ TeV), CERN LHC
($\sqrt{s_{\rm NN}}=2.76$–$8.8$ TeV), and proposed future hadron colliders
($\sqrt{s_{\rm NN}}\approx 10$–$63$ TeV). The corresponding values for the
inelastic pp cross section are obtained from a data-driven parametrization
resulting in a tenfold reduction of the uncertainties due to the many
available measurements at LHC collision energies (Fig. 1). We describe the
nuclear transverse density with two independent 2-parameter Fermi
distributions for protons and neutrons to account for their different
densities close to the nuclear periphery. Furthermore, we model the nucleon
degrees of freedom inside a nucleus using a lattice with a minimum nodal
separation to enforce the exclusion of overlapping nucleons without distorting
the nuclear density. Residual small distortions in the generated nuclear
densities, resulting from adjusting the nucleon center-of-mass with that of
the nucleus, are overcome by appropriately reweighting the original nuclear
density. We demonstrate for collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV that
the central values of the first four quantities change due to the inclusion of
the separated proton and neutron transverse distributions, though they remain
typically within the previously assigned systematic uncertainties, while their
new associated uncertainties are generally smaller than for earlier
calculations (Figs. 10–13). The number of participant nucleons, binary
nucleon–nucleon collisions, nuclear overlap function, participant eccentricity
and triangularity, overlap area and average path length are presented in
intervals of percentile centrality for lead–lead (PbPb) and proton–lead (pPb)
collisions at all collisions energies (Figs. 15 and 16). Tables for all
quantities versus centrality at present and foreseen collision energies
involving Pb-nuclei, but also for XeXe at $\sqrt{s_{\rm NN}}=5.44$ TeV, and
for AuAu and CuCu collisions at $\sqrt{s_{\rm NN}}=0.2$ TeV, are provided (see
Table 5). The source code for the Monte Carlo Glauber program is made publicly
available in LABEL:glaucode. The authors welcome comments on the code and
suggestions on how to make it more useful to both experimentalists and
theorists.

## Acknowledgments

We are grateful to Klaus Reygers for common work on a first study of the
impact of separated neutron and proton densities on the Glauber quantities. We
thank Dan Watts for fruitful discussions regarding LABEL:Tarbert:2013jze. J.K.
and C.L. are supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under contract numbers DE-FG02-94ER40865 and DE-
AC05-00OR22725, respectively.

## References

  * (1) M. L. Miller, K. Reygers, S. J. Sanders, and P. Steinberg, “Glauber modeling in high energy nuclear collisions,” Ann. Rev. Nucl. Part. Sci. 57 (2007) 205–243, arXiv:nucl-ex/0701025 [nucl-ex].
  * (2) C. W. De Jager, H. De Vries, and C. De Vries, “Nuclear charge and magnetization density distribution parameters from elastic electron scattering,” Atom. Data Nucl. Data Tabl. 14 (1974) 479–508.
  * (3) H. De Vries, C. W. De Jager, and C. De Vries, “Nuclear charge and magnetization density distribution parameters from elastic electron scattering,” Atom. Data Nucl. Data Tabl. 36 (1987) 495–536.
  * (4) R. J. Glauber and G. Matthiae, “High-energy scattering of protons by nuclei,” Nucl. Phys. B21 (1970) 135–157.
  * (5) X.-N. Wang and M. Gyulassy, “HIJING: A Monte Carlo model for multiple jet production in pp, pA and Å collisions,” Phys. Rev. D44 (1991) 3501–3516.
  * (6) B. Alver, M. Baker, C. Loizides, and P. Steinberg, “The PHOBOS Glauber Monte Carlo,” arXiv:0805.4411 [nucl-ex].
  * (7) C. Loizides, J. Nagle, and P. Steinberg, “Improved version of the PHOBOS Glauber Monte Carlo,” SoftwareX 1-2 (2015) 13–18, arXiv:1408.2549 [nucl-ex].
  * (8) W. Broniowski, M. Rybczynski, and P. Bozek, “GLISSANDO: Glauber initial-state simulation and more…,” Comput. Phys. Commun. 180 (2009) 69–83, arXiv:0710.5731 [nucl-th].
  * (9) M. Rybczynski, G. Stefanek, W. Broniowski, and P. Bozek, “GLISSANDO 2: GLauber Initial-State Simulation AND mOre…, ver. 2,” Comput. Phys. Commun. 185 (2014) 1759–1772, arXiv:1310.5475 [nucl-th].
  * (10) C. Loizides, “Glauber modeling of high-energy nuclear collisions at the subnucleon level,” Phys. Rev. C94 no. 2, (2016) 024914, arXiv:1603.07375 [nucl-ex].
  * (11) B. Klos et al., “Neutron density distributions from antiprotonic 208Pb and 209Bi atoms,” Phys. Rev. C76 (2007) 014311, arXiv:nucl-ex/0702016 [NUCL-EX].
  * (12) C. M. Tarbert et al., “Neutron skin of 208Pb from coherent pion photoproduction,” Phys. Rev. Lett. 112 no. 24, (2014) 242502, arXiv:1311.0168 [nucl-ex].
  * (13) H. Paukkunen, “Neutron skin and centrality classification in high-energy heavy-ion collisions at the LHC,” Phys. Lett. B745 (2015) 73–78, arXiv:1503.02448 [hep-ph].
  * (14) S. De, “The effect of neutron skin on inclusive prompt photon production in PbPb collisions at Large Hadron Collider energies,” J. Phys. G44 no. 4, (2017) 045104, arXiv:1609.09608 [nucl-th].
  * (15) I. Helenius, H. Paukkunen, and K. J. Eskola, “Neutron-skin effect in direct-photon and charged hadron-production in PbPb collisions at the LHC,” Eur. Phys. J. C77 no. 3, (2017) 148, arXiv:1606.06910 [hep-ph].
  * (16) A. Dainese et al., “Heavy ions at the Future Circular Collider,” CERN Yellow Report no. 3, (2017) 635–692, arXiv:1605.01389 [hep-ph].
  * (17) “TGlauberMC on HepForge.” http://tglaubermc.hepforge.org/.  latest version v3.1.
  * (18) A. Bialas, M. Bleszynski, and W. Czyz, “Multiplicity distributions in nucleus-nucleus collisions at high energies,” Nucl. Phys. B111 (1976) 461–476.
  * (19) D. Kharzeev, C. Lourenco, M. Nardi, and H. Satz, “A Quantitative analysis of charmonium suppression in nuclear collisions,” Z. Phys. C74 (1997) 307–318, arXiv:hep-ph/9612217 [hep-ph].
  * (20) PHOBOS Collaboration, B. Alver et al., “System size, energy, pseudorapidity, and centrality dependence of elliptic flow,” Phys. Rev. Lett. 98 (2007) 242302, arXiv:nucl-ex/0610037 [nucl-ex].
  * (21) B. Alver and G. Roland, “Collision geometry fluctuations and triangular flow in heavy-ion collisions,” Phys. Rev. C81 (2010) 054905, arXiv:1003.0194 [nucl-th]. [Erratum: Phys. Rev.C82,039903(2010)].
  * (22) D. Teaney and L. Yan, “Triangularity and dipole asymmetry in heavy ion collisions,” Phys. Rev. C83 (2011) 064904, arXiv:1010.1876 [nucl-th].
  * (23) B. Alver et al., “Importance of correlations and fluctuations on the initial source eccentricity in high-energy nucleus-nucleus collisions,” Phys. Rev. C77 (2008) 014906, arXiv:0711.3724 [nucl-ex].
  * (24) H.-J. Drescher, A. Dumitru, C. Gombeaud, and J.-Y. Ollitrault, “The Centrality dependence of elliptic flow, the hydrodynamic limit, and the viscosity of hot QCD,” Phys. Rev. C76 (2007) 024905, arXiv:0704.3553 [nucl-th].
  * (25) A. Drees, H. Feng, and J. Jia, “Medium induced jet absorption at RHIC,” Phys. Rev. C71 (2005) 034909, arXiv:nucl-th/0310044 [nucl-th].
  * (26) A. Dainese, C. Loizides, and G. Paic, “Leading-particle suppression in high energy nucleus-nucleus collisions,” Eur. Phys. J. C38 (2005) 461–474, arXiv:hep-ph/0406201 [hep-ph].
  * (27) ALICE Collaboration, B. Abelev et al., “Centrality determination of PbPb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV with ALICE,” Phys. Rev. C88 no. 4, (2013) 044909, arXiv:1301.4361 [nucl-ex].
  * (28) UA5 Collaboration, G. J. Alner et al., “Antiproton-proton cross sections at 200 and 900 GeV c.m. energy,” Z. Phys. C32 (1986) 153–161.
  * (29) E710 Collaboration, N. A. Amos et al., “A luminosity independent measurement of the $\bar{\rm p}$p total cross section at $\sqrt{s}=1.8$ Tev,” Phys. Lett. B243 (1990) 158–164.
  * (30) E710 Collaboration, N. A. Amos et al., “Measurement of $\rho$, the ratio of the real to imaginary part of the $\bar{Rev.~{}Mod.~{}Phys.~{}}$p forward elastic scattering amplitude, at $\sqrt{s}=1.8$ TeV,” Phys. Rev. Lett. 68 (1992) 2433–2436.
  * (31) CDF Collaboration, F. Abe et al., “Measurement of the $\bar{p}p$ total cross-section at $\sqrt{s}=546$ GeV and 1800 GeV,” Phys. Rev. D50 (1994) 5550–5561.
  * (32) CDF Collaboration, F. Abe et al., “Measurement of small angle $\bar{p}p$ elastic scattering at $\sqrt{s}=546$ GeV and 1800 GeV,” Phys. Rev. D50 (1994) 5518–5534.
  * (33) ALICE Collaboration, B. Abelev et al., “Measurement of inelastic, single- and double-diffraction cross sections in proton–proton collisions at the LHC with ALICE,” Eur. Phys. J. C73 no. 6, (2013) 2456, arXiv:1208.4968 [hep-ex].
  * (34) ATLAS Collaboration, G. Aad et al., “Measurement of the inelastic proton–proton cross section at $\sqrt{s}=7$ TeV with the ATLAS Detector,” Nature Commun. 2 (2011) 463, arXiv:1104.0326 [hep-ex].
  * (35) ATLAS Collaboration, G. Aad et al., “Measurement of the total cross section from elastic scattering in pp collisions at $\sqrt{s}=7$ TeV with the ATLAS detector,” Nucl. Phys. B889 (2014) 486–548, arXiv:1408.5778 [hep-ex].
  * (36) ATLAS Collaboration, M. Aaboud et al., “Measurement of the total cross section from elastic scattering in pp collisions at $\sqrt{s}=8$ TeV with the ATLAS detector,” Phys. Lett. B761 (2016) 158–178, arXiv:1607.06605 [hep-ex].
  * (37) ATLAS Collaboration, M. Aaboud et al., “Measurement of the inelastic proton-proton cross section at $\sqrt{s}=13$ TeV with the ATLAS detector at the LHC,” Phys. Rev. Lett. 117 no. 18, (2016) 182002, arXiv:1606.02625 [hep-ex].
  * (38) CMS Collaboration, S. Chatrchyan et al., “Measurement of the inelastic proton-proton cross section at $\sqrt{s}=7$ TeV,” Phys. Lett. B722 (2013) 5–27, arXiv:1210.6718 [hep-ex].
  * (39) CMS Collaboration, H. Van Haevermaet, “Measurement of the inelastic proton–proton cross section at $\sqrt{s}$ = 13 TeV,” in Proceedings, 24th International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS 2016): Hamburg, Germany, April 11-25, 2016.  2016\.  arXiv:1607.02033 [hep-ex].
  * (40) LHCb Collaboration, R. Aaij et al., “Measurement of the inelastic pp cross-section at a centre-of-mass energy of $\sqrt{s}=7$ TeV,” JHEP 02 (2015) 129, arXiv:1412.2500 [hep-ex].
  * (41) TOTEM Collaboration, G. Antchev et al., “First measurement of the total proton-proton cross section at the LHC energy of $\sqrt{s}=7$ TeV,” Europhys. Lett. 96 (2011) 21002, arXiv:1110.1395 [hep-ex].
  * (42) TOTEM Collaboration, G. Antchev et al., “Luminosity-independent measurements of total, elastic and inelastic cross-sections at $\sqrt{s}=7$ TeV,” Europhys. Lett. 101 (2013) 21004.
  * (43) TOTEM Collaboration, G. Antchev et al., “Luminosity-independent measurement of the proton-proton total cross section at $\sqrt{s}=8$ TeV,” Phys. Rev. Lett. 111 no. 1, (2013) 012001.
  * (44) Pierre Auger Collaboration, P. Abreu et al., “Measurement of the proton-air cross-section at $\sqrt{s}=57$ TeV with the Pierre Auger Observatory,” Phys. Rev. Lett. 109 (2012) 062002, arXiv:1208.1520 [hep-ex].
  * (45) D. d’Enterria and T. Pierog, “Global properties of proton-proton collisions at $\sqrt{s}$ = 100 TeV,” JHEP 08 (2016) 170, arXiv:1604.08536 [hep-ph].
  * (46) M. Froissart, “Asymptotic behavior and subtractions in the Mandelstam representation,” Phys. Rev. 123 (1961) 1053–1057.
  * (47) COMPETE Collaboration, J. R. Cudell, V. V. Ezhela, P. Gauron, K. Kang, Yu. V. Kuyanov, S. B. Lugovsky, E. Martynov, B. Nicolescu, E. A. Razuvaev, and N. P. Tkachenko, “Benchmarks for the forward observables at RHIC, the Tevatron Run II and the LHC,” Phys. Rev. Lett. 89 (2002) 201801, arXiv:hep-ph/0206172 [hep-ph].
  * (48) ALICE Collaboration, B. Abelev et al., “Measurement of the cross section for electromagnetic dissociation with neutron emission in PbPb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV,” Phys. Rev. Lett. 109 (2012) 252302, arXiv:1203.2436 [nucl-ex].
  * (49) CMS Collaboration, V. Khachatryan et al., “Measurement of the inelastic cross section in proton–lead collisions at $\sqrt{s_{NN}}=$ 5.02 TeV,” Phys. Lett. B759 (2016) 641–662, arXiv:1509.03893 [hep-ex].
  * (50) ALICE Collaboration, B. B. Abelev et al., “Measurement of visible cross sections in proton-lead collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV in van der Meer scans with the ALICE detector,” JINST 9 no. 11, (2014) P11003, arXiv:1405.1849 [nucl-ex].
  * (51) ALICE Collaboration, J. Adam et al., “Centrality dependence of particle production in pPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV,” Phys. Rev. C91 no. 6, (2015) 064905, arXiv:1412.6828 [nucl-ex].
  * (52) G. Fricke, C. Bernhardt, K. Heilig, L. A. Schaller, L. Schellenberg, E. B. Shera, and C. W. de Jager, “Nuclear Ground State Charge Radii from Electromagnetic Interactions,” Atom. Data Nucl. Data Tabl. 60 (1995) 177.
  * (53) I. Angeli, “A consistent set of nuclear rms charge radii: properties of the radius surface R(N,Z),” Atom. Data Nucl. Data Tabl. 87 no. 2, (2004) 185–206.
  * (54) I. Angeli and K. P. Marinova, “Table of experimental nuclear ground state charge radii: An update,” Atom. Data Nucl. Data Tabl. 99 no. 1, (2013) 69–95.
  * (55) P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA Recommended values of the fundamental physical constants: 2014,” Rev. Mod. Phys. 88 no. 3, (2016) 035009, arXiv:1507.07956 [physics.atom-ph].
  * (56) M. Warda, X. Vinas, X. Roca-Maza, and M. Centelles, “Analysis of bulk and surface contributions in the neutron skin of nuclei,” Phys. Rev. C81 (2010) 054309, arXiv:1003.5225 [nucl-th].
  * (57) S. Wycech, F. J. Hartmann, J. Jastrzebski, B. Klos, A. Trzcinska, and T. von Egidy, “Nuclear surface studies with antiprotonic atom X-rays,” Phys. Rev. C76 (2007) 034316, arXiv:nucl-th/0702029 [NUCL-TH].
  * (58) R. C. Barrett and D. F. Jackson, Nuclear sizes and structure.  Oxford University Press, 1977.
  * (59) T. Hirano and Y. Nara, “Eccentricity fluctuation effects on elliptic flow in relativistic heavy ion collisions,” Phys. Rev. C79 (2009) 064904, arXiv:0904.4080 [nucl-th].
  * (60) Q. Y. Shou, Y. G. Ma, P. Sorensen, A. H. Tang, F. Videbæk, and H. Wang, “Parameterization of deformed nuclei for Glauber modeling in Relativistic Heavy Ion Collisions,” Phys. Lett. B749 (2015) 215–220, arXiv:1409.8375 [nucl-th].
  * (61) P. Krishna and D. Pandey, Close-packed structures.  University College Cardiff Press, 1981.  Download here.
  * (62) M. Alvioli, H. Holopainen, K. J. Eskola, and M. Strikman, “Initial state anisotropies and their uncertainties in ultrarelativistic heavy-ion collisions from the Monte Carlo Glauber model,” Phys. Rev. C85 (2012) 034902, arXiv:1112.5306 [hep-ph].
  * (63) M. Alvioli, H. J. Drescher, and M. Strikman, “A Monte Carlo generator of nucleon configurations in complex nuclei including nucleon–nucleon correlations,” Phys. Lett. B680 (2009) 225–230, arXiv:0905.2670 [nucl-th].
  * (64) J. T. Mitchell, D. V. Perepelitsa, M. J. Tannenbaum, and P. W. Stankus, “Tests of constituent-quark generation methods which maintain both the nucleon center of mass and the desired radial distribution in Monte Carlo Glauber models,” Phys. Rev. C93 no. 5, (2016) 054910, arXiv:1603.08836 [nucl-ex].
  * (65) Dariusz Miskowiec, “Nuclear Overlap Calculation.” http://web-docs.gsi.de/~misko/overlap/.  Code adapted to match the MC implementation as closely as possible.
  * (66) R. Hofstadter, “Electron scattering and nuclear structure,” Rev. Mod. Phys. 28 (1956) 214–254.
  * (67) M. Alvioli and M. Strikman, “Color fluctuation effects in proton-nucleus collisions,” Phys. Lett. B722 (2013) 347–354, arXiv:1301.0728 [hep-ph].
  * (68) R. Corke and T. Sjostrand, “Multiparton Interactions with an x-dependent Proton Size,” JHEP 05 (2011) 009, arXiv:1101.5953 [hep-ph].
  * (69) M. Rybczyński and Z. Wlodarczyk, “The nucleon–nucleon collision profile and cross section fluctuations,” J. Phys. G41 (2013) 015106, arXiv:1307.0636 [nucl-th].
  * (70) J. Jia, “Influence of the nucleon-nucleon collision geometry on the determination of the nuclear modification factor for nucleon-nucleus and nucleus-nucleus collisions,” Phys. Lett. B681 (2009) 320–325, arXiv:0907.4175 [nucl-th].
  * (71) C. Loizides and A. Morsch, “Absence of jet quenching in peripheral nucleus–nucleus collisions,” Phys. Lett. B773 (2017) 408–411, arXiv:1705.08856 [nucl-ex].
  * (72) R. Brun and F. Rademakers, “ROOT: An object oriented data analysis framework,” Nucl. Instrum. Meth. A389 (1997) 81–86.
  * (73) K. Tsukada et al., “First elastic electron scattering from 132Xe at the SCRIT facility,” Phys. Rev. Lett. 118 no. 26, (2017) 262501, arXiv:1703.04278 [nucl-ex].

##  Appendix A Comparison with optical Glauber

As described in Sec. II, the underlying Glauber formalism is the same for
optical and MC calculations. Nevertheless, as discussed in
LABEL:Miller:2007ri, there are differences in their results, in particular in
peripheral collisions. This is demonstrated in Fig. 17, where $N_{\rm coll}$
in peripheral collisions deviates strongly between an optical and the Monte
Carlo Glauber calculation. The optical Glauber calculation overlap  was
performed with the same parameters for the 2pF distribution of Pb. Similarly,
the proton was described in the same way, namely with an exponential
distribution $\exp\left(-r/R\right)$ with $R=0.234$ fm based on the measured
form factor of the proton Hofstadter:1956qs . For peripheral PbPb collisions
beyond 60% centrality the two calculations differ by more than 20%, and in the
case of pPb collisions the ratio is even non-monotonous. Optical calculations,
which assume a smooth matter distribution, and by construction cannot capture
event-by-event fluctuations, should not be trusted in this regime.

Figure 17: Relative change in $N_{\rm coll}$ in PbPb and pPb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV due to the use of optical Glauber model. The
baseline uses the traditional MCG approach with $\sigma_{\rm NN}=70$ mb.

##  Appendix B Subnucleonic degrees of freedom

Potential improvements can be added to the MCG model to take into account
subnucleonic dynamics in the nuclear collision, by adding parton degrees of
freedom Loizides:2016djv , or fluctuations in the nucleon shape (also known as
Glauber-Gribov fluctuations) Alvioli:2013vk . The TGlauberMC includes the
possibility that pp collisions themselves have an impact parameter dependence,
as e.g. regularly taken into account in the PYTHIA event generator
Corke:2011yy . A convenient way to include the $b_{\rm NN}$ dependence in MCG
models is to replace the hard-sphere collision condition, $P(b_{\rm
NN})=\Theta(D-b_{\rm NN})$ from Eq. 11, with

$P(b_{\rm NN})=\Gamma\left(1/\omega,\frac{b_{\rm
NN}^{2}}{D^{2}\omega}\right)/\Gamma\left(1/\omega\right)$ (12)

where $b_{\rm NN}$ is the difference between two nucleon centers in the
transverse plane, $\Gamma$ is the Gamma function, and $\omega$ a parameter
which covers from the hard-sphere ($\omega=0$) to the Gaussian ($\omega=1$)
limits.

As can be seen in Fig. 18, the resulting probability distribution approaches
the hard-sphere step function for $\omega\rightarrow 0$ and a Gaussian for
$\omega\rightarrow 1$. The proposed value, $\omega=0.4$, for the collisions at
the LHC energies reproduces the measured values of both the total and elastic
pp cross sections Rybczynski:2013mla ; Rybczynski:2013yba . Using $\omega=0.4$
leads to an effective reduction of the number of collisions relative to the
hard-sphere condition, by about $5$% and $10$% in peripheral pPb and PbPb
collisions, respectively, as shown in Fig. 19. Since the MCG calculation uses
$P(b_{\rm NN})$ to determine whether there is a NN collision, using $\omega>0$
in such calculations will impact the set of generated nucleus–nucleus
collisions, and hence all Glauber quantities will change with respect to the
typically applied hard-sphere ($\omega=0$) condition, not only $N_{\rm coll}$.
The resulting change in $N_{\rm coll}$ is qualitatively similar to earlier
studies Jia:2009mq  on the influence of the nucleon–nucleon collision geometry
on the determination of the $R_{\rm AA}$. However, a realistic modeling of the
number of hard collisions per NN collision, and in general of the correlation
between soft and hard particle production, is needed to be able to compare
experimental data with calculations Morsch:2017brb .

Figure 18: Nucleon–nucleon collision impact parameter dependence $P(b_{\rm
NN})$ from Eq. 12 for various values of $\omega$, at 5.02 TeV. The vertical
dashed line ($\omega=0$) corresponds to the hard-sphere limit. Figure 19:
Relative change in $N_{\rm coll}$ in PbPb and pPb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV due to the use of $\omega=0.4$ in Eq. 12 instead of the hard-
sphere condition.

##  Appendix C User’s guide

The source code, which relies on the ROOT Brun:1997pa  framework (version
4.00/08 or higher), can be obtained at the TGlauberMC page on HepForge
glaucode . All functionality is implemented in the macro runglauber_vX.Y.C,
where version “X.Y==3.0” described here. For generating events with 3H or 3He,
the additional text files called “h3_plaintext.dat” or “he3_plaintext.dat” are
needed. Compiling the code is done as in earlier versions, namely by executing

    root [0] gSystem->Load("libMathMore")
    root [1] .L runglauber_3.0.C+

Three classes, TGlauNucleon, TGlauNucleus and TGlauberMC and three functions
(macros) runAndSaveNtuple(), runAndSaveNucleons(), and runAndSmearNtuple() are
defined in the provided macro. In the following, we only describe the improved
functionality, see LABEL:Loizides:2014vua for the complete guide.

Executing the program can be steered by the provided runAndSaveNtuple() macro
that takes the following arguments:

     Int_t n,          = number of events
     char *sysA        = name of nucleus A
     char *sysB        = name of nuclear B
     Double_t signn    = inelastic pp cross section
     Double_t sigwidth = width of signn
     Double_t mind     = minimum distance
     Double_t omega    = parameter for NN collision
     Double_t noded    = node distance
     char *fname       = file name

The macro will generate $n$ many Monte Carlo events and store event-by-event
computed quantities in a ROOT tree, further described below, saved on disk for
a given file name. If no argument for the file name is given, the code will
provide it based on values given for the other arguments. The names for
various nuclear profiles are listed in Table 4 and Table 6, and for the
corresponding reweighted profiles in Table 7, and Table 8. A complete list for
other nuclei can be found in LABEL:Loizides:2014vua. All implemented cases can
also be found in the TGlauNucleus::Lookup function in the code. The value for
$\sigma_{\rm NN}$ is given in mb, and a variety of values for high energy
collisions can be found in Table 2. In case a positive value for the width of
$\sigma_{\rm NN}$ is given, then Glauber-Gribov fluctuations (useful for pA
collisions studies) will be simulated. As a default, a minimum separation
distance of $d_{\rm min}=0.4$ fm is recommended. If a positive value for the
node distance is given, then the nucleons will be placed on a lattice (HCP, if
not otherwise specified). For values below $1$ fm the results do not depend on
the node distance, but is is recommended to use $0.4$ fm for consistency with
$d_{\rm min}$. By default no lattice will be used, and the calculation will be
identical to version 2 of the code. If a positive value of $\omega$ is given,
as per Eq. 12, the determination of the number of NN collisions will use an
NN-dependent impact parameter distribution as shown in Fig. 18. Otherwise, by
default, the hard-sphere condition is used.

Nucleus | Name |  $R$ (fm) |  $a$ (fm)
---|---|---|---
63Cu | Cu | $4.20\pm 0.02$ | $0.596\pm 0.008$
129Xe | Xe | $5.36\pm 0.10$ | $0.590\pm 0.070$
197Au | Au | $6.38\pm 0.06$ | $0.535\pm 0.027$
Table 6:  Nuclear density parameters for charge density distributions of Cu, Xe and Au (see LABEL:Loizides:2014vua). The name of the corresponding profile in the TGlauberMC implementation is also listed. See Table 4 for Pb. Separate proton/neutron point densities are not known. The values for xenon are obtained from $R=5.4\pm 0.1$ fm and $a=0.61^{+0.07}_{-0.09}$ fm for 132Xe from LABEL:Tsukada:2017llu, where the radius was scaled down by $0.99=(129/132)^{1/3}$ and $a$ was reduced by $0.02$ fm to symmetrize the uncertainty and to approximate the smaller neutron skin of 129Xe. Nucleus | Name | $p_{0}$ | $p_{1}\cdot 10^{4}$ | $p_{2}\cdot 10^{4}$
---|---|---|---|---
63Cu | Curw | $1.0090$ | $-7.9040$ | $-3.8990$
129Xe | Xerw | $1.0091$ | $-7.2230$ | $-2.6630$
197Au | Aurw | $1.0090$ | $-5.9091$ | $-2.1050$
207Pb | Pbrw | $1.0086$ | $-4.4808$ | $-2.0587$
208Pb | Pbpnrw | $1.0087$ | $-4.6148$ | $-2.0357$
Table 7: Values for the coefficients of the 2nd-order polynomial used to correct the radial nuclear density distribution to cancel the effects of the recentering. The name of the corresponding profile in the TGlauberMC implementation is also listed. In case one of the reweighted parameterizations is chosen, the code will by default only generate events with $d_{\rm max}<0.1$ fm. Name | $\beta_{2}$ | $\beta_{4}$ | $p_{0}$ | $p_{1}\cdot 10^{3}$ | $p_{2}\cdot 10^{5}$
---|---|---|---|---|---
Cu2rw | $0.162$ | $-0.006$ | $1.0127$ | $-2.9808$ | $-9.9722$
Xe2arw | $0.18$ | $0$ | $1.0125$ | $-2.4851$ | $-5.7246$
Au2rw | $-0.131$ | $-0.031$ | $1.0126$ | $-2.2552$ | $-3.7151$
Table 8: Same as Table 7 for deformed nuclear profiles corresponding to
density parameters given in Table 6. Figure 20: Relative change in $N_{\rm
coll}$ in PbPb and pPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV for $d_{\rm
min}=0.4$ fm without recentering compared to recentering, using the
traditional MCG implementation with $d_{\rm min}=0.4$ fm.

In addition to quantities described in LABEL:Loizides:2014vua, the following
quantities are stored in the ROOT tree:

  * •

Nhard: Number of hard collisions (based on fHardFrac)

  * •

Ncollpp: Number of pp collisions

  * •

Ncollpn: Number of pn collisions

  * •

Ncollnn: Number of nn collisions

  * •

AreaW: Area defined by width of participants

  * •

AreaO: Area by ”or” of participants in grid

  * •

AreaA: Area by ”and” of participants in grid

  * •

X0: Production point in x

  * •

Y0: Production point in y

  * •

Phi0: Direction in $\phi$

  * •

Length: Length in $\phi_{0}$

The following set of functions controls additional behavior of the TGlauberMC
class: SetHardFrac(Double_t) sets the fraction of cross section used for the
calculation of hard collisions (by default $0.65$). SetCalcArea(Bool_t) and
SetCalcLength(Bool_t) enable the calculation of the overlap area using a fine
grid and the length (starting from a randomly chosen binary collision with
($x_{0},y_{0}$) in a random direction $\phi_{0}$ of the transverse plane).
They are by default not computed since the calculation is rather slow.
SetRecenter(Int_t) specifies if and how to recenter nucleons in a nucleus,
where 0 means no recentering, 1 (default) means recentering by shifting all
nucleons by the average displacement, 2 means recentering by shifting only one
nucleon, and 3 recentering by shifting only along the $z$-direction after
rotating the nucleus to align the $x$ and $y$ coordinates of its center with
0. Figure 20 demonstrates the relative change of $N_{\rm coll}$ in PbPb and
pPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV for $d_{\rm min}=0.4$ fm
without recentering compared to recentering using the traditional MCG
implementation with $d_{\rm min}=0.4$ fm. SetShiftMax(Double_t) specifies the
maximum displacement ($d_{\rm max}$) of the nucleon center-of-mass in every
direction from zero (by default any shift is accepted). SetLattice(Int_t)
specifies the lattice type to use (HCP by default), and SetSmearing(Double_t)
specifies the width of a Gaussian by which the nucleon position will be
smeared around the lattice node position (by default not done).
SetBMin(Double_t) and SetBMax(Double_t) can be used to restrict the impact
parameter (by default between $0$ and $20$ fm). SetDetail(Int_t) allows one to
restrict the number of variables written to the ROOT tree (by default
everything is written). SetMinDistance(Double_t) defines the minimum
separation distance (by default $0.4$ fm). SetNodeDistance(Double_t) sets the
node separation in the lattice mode. This value should be as large as $d_{\rm
min}$. By default it is negative, i.e. the lattice mode is not enabled. Using
SetNNProf(TF1 *) one can set another profile than that defined by Eq. 12. See
the code how it is done for getNNProf.

##  Appendix D Tables for all computed MCG quantities
in $5$%-wide centrality classes

In the following, we present the results for $N_{\rm coll}$, $N_{\rm part}$,
$T_{\rm AA}$, $\varepsilon_{2}$, $\varepsilon_{3}$, $A_{\rm T}$, and $L$ in
$5$%-wide centrality classes for all systems, summarized in Table 5. The
centrality classes are defined by slicing the impact parameter ($b$)
distribution. For all systems at least 5M events were computed. For each
quantity, the average and the standard deviation (labeled as rms) of the
quantity in the centrality class are reported. The settings for the improved
MCG model are given in Table 3.

PbPb at $\sqrt{s_{\rm NN}}=2.76$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 3.47 1615 $\pm$ 135 382 $\pm$ 17.1 26.1 $\pm$
2.2 0.0737 $\pm$ 0.0401 0.0853 $\pm$ 0.0455 5.15 $\pm$ 3 96.3 $\pm$ 7.44 5–10%
3.47 4.91 1268 $\pm$ 104 328.1 $\pm$ 17.6 20.5 $\pm$ 1.7 0.116 $\pm$ 0.059
0.113 $\pm$ 0.059 4.85 $\pm$ 2.85 78 $\pm$ 5.54 10–15% 4.91 6.01 1002 $\pm$ 86
280 $\pm$ 16.7 16.2 $\pm$ 1.4 0.163 $\pm$ 0.0745 0.135 $\pm$ 0.0699 4.54 $\pm$
2.71 64.8 $\pm$ 4.76 15–20% 6.01 6.94 790.1 $\pm$ 74.4 238.2 $\pm$ 15.8 12.8
$\pm$ 1.2 0.208 $\pm$ 0.087 0.154 $\pm$ 0.0802 4.26 $\pm$ 2.57 54.2 $\pm$ 4.29
20–25% 6.94 7.76 618.1 $\pm$ 66 201.5 $\pm$ 15.1 10 $\pm$ 1.1 0.25 $\pm$
0.0986 0.173 $\pm$ 0.0899 4 $\pm$ 2.45 45.3 $\pm$ 3.99 25–30% 7.76 8.5 478.9
$\pm$ 59.1 169.4 $\pm$ 14.4 7.75 $\pm$ 0.96 0.29 $\pm$ 0.11 0.193 $\pm$ 0.0993
3.74 $\pm$ 2.33 37.8 $\pm$ 3.74 30–35% 8.5 9.18 366.3 $\pm$ 52.7 141.4 $\pm$
13.9 5.93 $\pm$ 0.85 0.325 $\pm$ 0.121 0.213 $\pm$ 0.11 3.5 $\pm$ 2.2 31.2
$\pm$ 3.55 35–40% 9.18 9.81 275.5 $\pm$ 46.7 116.8 $\pm$ 13.4 4.46 $\pm$ 0.76
0.36 $\pm$ 0.132 0.235 $\pm$ 0.12 3.27 $\pm$ 2.1 25.6 $\pm$ 3.36 40–45% 9.81
10.4 203.1 $\pm$ 40.5 95.12 $\pm$ 12.9 3.29 $\pm$ 0.65 0.393 $\pm$ 0.143 0.258
$\pm$ 0.13 3.03 $\pm$ 1.99 20.6 $\pm$ 3.18 45–50% 10.4 11 146.6 $\pm$ 34.3
76.37 $\pm$ 12.2 2.37 $\pm$ 0.56 0.426 $\pm$ 0.155 0.283 $\pm$ 0.14 2.81 $\pm$
1.89 16.4 $\pm$ 2.99 50–55% 11 11.5 103.4 $\pm$ 28.6 60.3 $\pm$ 11.5 1.67
$\pm$ 0.46 0.458 $\pm$ 0.168 0.31 $\pm$ 0.151 2.58 $\pm$ 1.78 12.7 $\pm$ 2.79
55–60% 11.5 12 70.98 $\pm$ 22.9 46.59 $\pm$ 10.7 1.15 $\pm$ 0.37 0.492 $\pm$
0.179 0.339 $\pm$ 0.162 2.35 $\pm$ 1.68 9.63 $\pm$ 2.55 60–65% 12 12.5 47.54
$\pm$ 18.2 35.15 $\pm$ 9.8 0.769 $\pm$ 0.29 0.529 $\pm$ 0.191 0.366 $\pm$
0.171 2.12 $\pm$ 1.56 7.1 $\pm$ 2.3 65–70% 12.5 13 31.04 $\pm$ 14.1 25.84
$\pm$ 8.79 0.502 $\pm$ 0.23 0.567 $\pm$ 0.201 0.392 $\pm$ 0.181 1.88 $\pm$
1.43 5.08 $\pm$ 2.02 70–75% 13 13.4 19.78 $\pm$ 10.6 18.46 $\pm$ 7.65 0.32
$\pm$ 0.17 0.606 $\pm$ 0.21 0.408 $\pm$ 0.192 1.64 $\pm$ 1.29 3.51 $\pm$ 1.72
75–80% 13.4 13.9 12.43 $\pm$ 7.85 12.89 $\pm$ 6.43 0.201 $\pm$ 0.13 0.645
$\pm$ 0.219 0.414 $\pm$ 0.211 1.42 $\pm$ 1.12 2.37 $\pm$ 1.42 80–85% 13.9 14.4
7.794 $\pm$ 5.62 8.928 $\pm$ 5.12 0.126 $\pm$ 0.091 0.683 $\pm$ 0.231 0.409
$\pm$ 0.241 1.22 $\pm$ 0.935 1.59 $\pm$ 1.13 85–90% 14.4 14.9 4.958 $\pm$ 3.94
6.224 $\pm$ 3.9 0.0802 $\pm$ 0.064 0.724 $\pm$ 0.243 0.388 $\pm$ 0.278 1.06
$\pm$ 0.756 1.07 $\pm$ 0.869 90–95% 14.9 15.6 3.206 $\pm$ 2.68 4.405 $\pm$ 2.8
0.0519 $\pm$ 0.043 0.775 $\pm$ 0.25 0.346 $\pm$ 0.312 0.94 $\pm$ 0.579 0.729
$\pm$ 0.658 95–100% 15.6 20 2.007 $\pm$ 1.67 3.078 $\pm$ 1.76 0.0325 $\pm$
0.027 0.849 $\pm$ 0.238 0.252 $\pm$ 0.326 0.844 $\pm$ 0.408 0.486 $\pm$ 0.48
0–100% 0 20 353.3 $\pm$ 464 112.5 $\pm$ 115 5.72 $\pm$ 7.5 0.447 $\pm$ 0.278
0.273 $\pm$ 0.206 2.76 $\pm$ 2.35 25.7 $\pm$ 28

Table 9:  Various quantities for PbPb collisions at $\sqrt{s_{\rm NN}}=2.76$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 12.

PbPb at $\sqrt{s_{\rm NN}}=5.02$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 3.49 1762 $\pm$ 147 384.3 $\pm$ 16.6 26.1
$\pm$ 2.2 0.072 $\pm$ 0.0391 0.0843 $\pm$ 0.0451 5.19 $\pm$ 3 101 $\pm$ 7.87
5–10% 3.49 4.93 1380 $\pm$ 113 331.2 $\pm$ 17.7 20.4 $\pm$ 1.7 0.112 $\pm$
0.0575 0.112 $\pm$ 0.0585 4.89 $\pm$ 2.85 82.1 $\pm$ 5.86 10–15% 4.93 6.04
1088 $\pm$ 93.4 283 $\pm$ 16.8 16.1 $\pm$ 1.4 0.158 $\pm$ 0.073 0.134 $\pm$
0.0697 4.6 $\pm$ 2.71 68.2 $\pm$ 5.03 15–20% 6.04 6.98 855.3 $\pm$ 80.8 240.9
$\pm$ 16 12.7 $\pm$ 1.2 0.202 $\pm$ 0.086 0.154 $\pm$ 0.0797 4.32 $\pm$ 2.57
57.1 $\pm$ 4.55 20–25% 6.98 7.8 667.6 $\pm$ 71.6 204 $\pm$ 15.3 9.88 $\pm$ 1.1
0.243 $\pm$ 0.0977 0.173 $\pm$ 0.0895 4.06 $\pm$ 2.45 47.9 $\pm$ 4.22 25–30%
7.8 8.55 515.7 $\pm$ 63.9 171.6 $\pm$ 14.7 7.63 $\pm$ 0.95 0.282 $\pm$ 0.109
0.192 $\pm$ 0.0992 3.81 $\pm$ 2.33 40 $\pm$ 3.97 30–35% 8.55 9.23 392.9 $\pm$
57 143.2 $\pm$ 14.1 5.81 $\pm$ 0.84 0.318 $\pm$ 0.12 0.212 $\pm$ 0.109 3.57
$\pm$ 2.21 33.1 $\pm$ 3.76 35–40% 9.23 9.87 294.5 $\pm$ 50 118.3 $\pm$ 13.6
4.36 $\pm$ 0.74 0.352 $\pm$ 0.131 0.234 $\pm$ 0.119 3.33 $\pm$ 2.1 27.2 $\pm$
3.56 40–45% 9.87 10.5 216.4 $\pm$ 43.3 96.49 $\pm$ 13 3.2 $\pm$ 0.64 0.385
$\pm$ 0.142 0.257 $\pm$ 0.129 3.1 $\pm$ 1.99 22 $\pm$ 3.38 45–50% 10.5 11
155.5 $\pm$ 36.6 77.48 $\pm$ 12.4 2.3 $\pm$ 0.54 0.418 $\pm$ 0.154 0.281 $\pm$
0.139 2.88 $\pm$ 1.89 17.5 $\pm$ 3.18 50–55% 11 11.6 109.2 $\pm$ 30.2 61.19
$\pm$ 11.7 1.62 $\pm$ 0.45 0.45 $\pm$ 0.166 0.307 $\pm$ 0.15 2.65 $\pm$ 1.79
13.6 $\pm$ 2.97 55–60% 11.6 12.1 74.73 $\pm$ 24.3 47.31 $\pm$ 10.9 1.11 $\pm$
0.36 0.484 $\pm$ 0.177 0.335 $\pm$ 0.159 2.42 $\pm$ 1.68 10.3 $\pm$ 2.73
60–65% 12.1 12.6 49.88 $\pm$ 19.1 35.74 $\pm$ 9.96 0.738 $\pm$ 0.28 0.52 $\pm$
0.189 0.362 $\pm$ 0.169 2.2 $\pm$ 1.58 7.64 $\pm$ 2.46 65–70% 12.6 13.1 32.38
$\pm$ 14.7 26.26 $\pm$ 8.95 0.479 $\pm$ 0.22 0.557 $\pm$ 0.2 0.385 $\pm$ 0.179
1.96 $\pm$ 1.45 5.47 $\pm$ 2.18 70–75% 13.1 13.5 20.54 $\pm$ 11.1 18.75 $\pm$
7.79 0.304 $\pm$ 0.16 0.596 $\pm$ 0.209 0.402 $\pm$ 0.19 1.72 $\pm$ 1.31 3.79
$\pm$ 1.86 75–80% 13.5 14 12.85 $\pm$ 8.16 13.09 $\pm$ 6.55 0.19 $\pm$ 0.12
0.634 $\pm$ 0.219 0.409 $\pm$ 0.209 1.49 $\pm$ 1.14 2.56 $\pm$ 1.54 80–85% 14
14.4 8.006 $\pm$ 5.82 9.038 $\pm$ 5.22 0.118 $\pm$ 0.086 0.672 $\pm$ 0.232
0.407 $\pm$ 0.239 1.28 $\pm$ 0.963 1.71 $\pm$ 1.22 85–90% 14.4 14.9 5.084
$\pm$ 4.08 6.304 $\pm$ 3.98 0.0752 $\pm$ 0.06 0.715 $\pm$ 0.245 0.39 $\pm$
0.277 1.12 $\pm$ 0.784 1.15 $\pm$ 0.944 90–95% 14.9 15.6 3.27 $\pm$ 2.77 4.452
$\pm$ 2.86 0.0484 $\pm$ 0.041 0.766 $\pm$ 0.254 0.349 $\pm$ 0.313 0.993 $\pm$
0.608 0.786 $\pm$ 0.715 95–100% 15.6 20 2.035 $\pm$ 1.72 3.103 $\pm$ 1.8
0.0301 $\pm$ 0.025 0.844 $\pm$ 0.241 0.255 $\pm$ 0.328 0.889 $\pm$ 0.428 0.522
$\pm$ 0.521 0–100% 0 20 382.2 $\pm$ 506 113.8 $\pm$ 116 5.65 $\pm$ 7.5 0.439
$\pm$ 0.277 0.272 $\pm$ 0.205 2.82 $\pm$ 2.35 27.2 $\pm$ 29.4

Table 10:  Various quantities for PbPb collisions at $\sqrt{s_{\rm NN}}=5.02$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 12.

PbPb at $\sqrt{s_{\rm NN}}=5.5$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 3.5 1782 $\pm$ 150 384.3 $\pm$ 16.7 26 $\pm$
2.2 0.0719 $\pm$ 0.0391 0.0843 $\pm$ 0.0451 5.19 $\pm$ 3 102 $\pm$ 7.97 5–10%
3.5 4.95 1395 $\pm$ 114 331.2 $\pm$ 17.6 20.4 $\pm$ 1.7 0.112 $\pm$ 0.0574
0.112 $\pm$ 0.0583 4.9 $\pm$ 2.85 82.6 $\pm$ 5.87 10–15% 4.95 6.05 1100 $\pm$
94.5 283.3 $\pm$ 16.9 16.1 $\pm$ 1.4 0.157 $\pm$ 0.0732 0.134 $\pm$ 0.0697
4.61 $\pm$ 2.7 68.7 $\pm$ 5.06 15–20% 6.05 6.99 864.6 $\pm$ 81.8 241.1 $\pm$
16.1 12.6 $\pm$ 1.2 0.201 $\pm$ 0.0859 0.154 $\pm$ 0.0798 4.31 $\pm$ 2.57 57.5
$\pm$ 4.6 20–25% 6.99 7.81 674.2 $\pm$ 72.4 204.1 $\pm$ 15.3 9.84 $\pm$ 1.1
0.243 $\pm$ 0.0974 0.173 $\pm$ 0.0895 4.06 $\pm$ 2.44 48.2 $\pm$ 4.26 25–30%
7.81 8.56 520.4 $\pm$ 64.6 171.7 $\pm$ 14.7 7.6 $\pm$ 0.94 0.281 $\pm$ 0.109
0.192 $\pm$ 0.0993 3.82 $\pm$ 2.32 40.2 $\pm$ 4.01 30–35% 8.56 9.25 396 $\pm$
57.6 143.2 $\pm$ 14.2 5.78 $\pm$ 0.84 0.317 $\pm$ 0.12 0.212 $\pm$ 0.109 3.58
$\pm$ 2.21 33.3 $\pm$ 3.8 35–40% 9.25 9.89 296.1 $\pm$ 50.5 118.2 $\pm$ 13.6
4.32 $\pm$ 0.74 0.351 $\pm$ 0.131 0.233 $\pm$ 0.119 3.34 $\pm$ 2.1 27.3 $\pm$
3.59 40–45% 9.89 10.5 217.9 $\pm$ 43.6 96.51 $\pm$ 13.1 3.18 $\pm$ 0.64 0.385
$\pm$ 0.142 0.256 $\pm$ 0.129 3.11 $\pm$ 2 22.1 $\pm$ 3.4 45–50% 10.5 11 156.7
$\pm$ 36.8 77.59 $\pm$ 12.4 2.29 $\pm$ 0.54 0.416 $\pm$ 0.154 0.281 $\pm$
0.139 2.88 $\pm$ 1.89 17.6 $\pm$ 3.2 50–55% 11 11.6 109.9 $\pm$ 30.4 61.26
$\pm$ 11.7 1.6 $\pm$ 0.44 0.449 $\pm$ 0.166 0.307 $\pm$ 0.149 2.66 $\pm$ 1.79
13.7 $\pm$ 2.99 55–60% 11.6 12.1 75.17 $\pm$ 24.5 47.38 $\pm$ 10.9 1.1 $\pm$
0.36 0.482 $\pm$ 0.178 0.335 $\pm$ 0.159 2.44 $\pm$ 1.69 10.4 $\pm$ 2.76
60–65% 12.1 12.6 50.13 $\pm$ 19.3 35.75 $\pm$ 9.99 0.732 $\pm$ 0.28 0.518
$\pm$ 0.189 0.362 $\pm$ 0.169 2.21 $\pm$ 1.57 7.71 $\pm$ 2.49 65–70% 12.6 13.1
32.49 $\pm$ 14.8 26.26 $\pm$ 8.95 0.474 $\pm$ 0.22 0.557 $\pm$ 0.2 0.385 $\pm$
0.178 1.97 $\pm$ 1.46 5.51 $\pm$ 2.19 70–75% 13.1 13.5 20.62 $\pm$ 11.2 18.75
$\pm$ 7.81 0.301 $\pm$ 0.16 0.595 $\pm$ 0.209 0.401 $\pm$ 0.19 1.73 $\pm$ 1.31
3.82 $\pm$ 1.88 75–80% 13.5 14 12.85 $\pm$ 8.18 13.08 $\pm$ 6.56 0.188 $\pm$
0.12 0.633 $\pm$ 0.219 0.409 $\pm$ 0.209 1.5 $\pm$ 1.15 2.58 $\pm$ 1.55 80–85%
14 14.4 8.045 $\pm$ 5.86 9.056 $\pm$ 5.23 0.117 $\pm$ 0.086 0.67 $\pm$ 0.232
0.406 $\pm$ 0.239 1.3 $\pm$ 0.976 1.73 $\pm$ 1.23 85–90% 14.4 15 5.114 $\pm$
4.11 6.325 $\pm$ 3.99 0.0747 $\pm$ 0.06 0.713 $\pm$ 0.246 0.39 $\pm$ 0.276
1.13 $\pm$ 0.786 1.17 $\pm$ 0.954 90–95% 15 15.6 3.284 $\pm$ 2.78 4.466 $\pm$
2.87 0.0479 $\pm$ 0.041 0.765 $\pm$ 0.254 0.35 $\pm$ 0.313 1 $\pm$ 0.615 0.796
$\pm$ 0.723 95–100% 15.6 20 2.037 $\pm$ 1.72 3.104 $\pm$ 1.81 0.0297 $\pm$
0.025 0.845 $\pm$ 0.242 0.255 $\pm$ 0.327 0.896 $\pm$ 0.431 0.528 $\pm$ 0.529
0–100% 0 20 386.1 $\pm$ 512 113.8 $\pm$ 116 5.64 $\pm$ 7.5 0.438 $\pm$ 0.277
0.272 $\pm$ 0.205 2.83 $\pm$ 2.35 27.4 $\pm$ 29.6

Table 11:  Various quantities for PbPb collisions at $\sqrt{s_{\rm NN}}=5.5$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 12.

PbPb at $\sqrt{s_{\rm NN}}=10.6$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 3.51 1954 $\pm$ 163 386.7 $\pm$ 16 26 $\pm$
2.2 0.0703 $\pm$ 0.0381 0.0832 $\pm$ 0.0445 5.24 $\pm$ 2.99 107 $\pm$ 8.36
5–10% 3.51 4.97 1528 $\pm$ 125 334.8 $\pm$ 17.5 20.3 $\pm$ 1.7 0.107 $\pm$
0.0556 0.111 $\pm$ 0.0581 4.96 $\pm$ 2.84 86.8 $\pm$ 6.2 10–15% 4.97 6.08 1202
$\pm$ 103 286.8 $\pm$ 17 16 $\pm$ 1.4 0.151 $\pm$ 0.0712 0.133 $\pm$ 0.0692
4.67 $\pm$ 2.7 72.2 $\pm$ 5.35 15–20% 6.08 7.02 943.1 $\pm$ 89 244.5 $\pm$
16.2 12.5 $\pm$ 1.2 0.194 $\pm$ 0.0843 0.153 $\pm$ 0.0795 4.4 $\pm$ 2.57 60.7
$\pm$ 4.86 20–25% 7.02 7.85 734.5 $\pm$ 78.8 207.4 $\pm$ 15.6 9.75 $\pm$ 1
0.234 $\pm$ 0.0962 0.172 $\pm$ 0.0889 4.13 $\pm$ 2.45 50.9 $\pm$ 4.51 25–30%
7.85 8.6 565.5 $\pm$ 70.4 174.7 $\pm$ 14.9 7.51 $\pm$ 0.93 0.272 $\pm$ 0.107
0.191 $\pm$ 0.0986 3.88 $\pm$ 2.32 42.6 $\pm$ 4.24 30–35% 8.6 9.29 429 $\pm$
62.6 145.8 $\pm$ 14.4 5.7 $\pm$ 0.83 0.308 $\pm$ 0.118 0.211 $\pm$ 0.108 3.65
$\pm$ 2.22 35.4 $\pm$ 4.03 35–40% 9.29 9.93 320.2 $\pm$ 54.8 120.5 $\pm$ 13.9
4.25 $\pm$ 0.73 0.342 $\pm$ 0.129 0.232 $\pm$ 0.118 3.41 $\pm$ 2.11 29.1 $\pm$
3.81 40–45% 9.93 10.5 234.2 $\pm$ 47.1 98.36 $\pm$ 13.3 3.11 $\pm$ 0.63 0.375
$\pm$ 0.14 0.254 $\pm$ 0.128 3.19 $\pm$ 2 23.6 $\pm$ 3.62 45–50% 10.5 11.1
167.6 $\pm$ 39.6 79.08 $\pm$ 12.7 2.23 $\pm$ 0.53 0.407 $\pm$ 0.152 0.278
$\pm$ 0.138 2.96 $\pm$ 1.9 18.9 $\pm$ 3.42 50–55% 11.1 11.6 117 $\pm$ 32.5
62.42 $\pm$ 12 1.55 $\pm$ 0.43 0.439 $\pm$ 0.164 0.303 $\pm$ 0.147 2.74 $\pm$
1.8 14.7 $\pm$ 3.2 55–60% 11.6 12.2 79.64 $\pm$ 26 48.3 $\pm$ 11.1 1.06 $\pm$
0.35 0.473 $\pm$ 0.176 0.329 $\pm$ 0.157 2.52 $\pm$ 1.69 11.2 $\pm$ 2.95
60–65% 12.2 12.7 52.8 $\pm$ 20.4 36.49 $\pm$ 10.2 0.701 $\pm$ 0.27 0.508 $\pm$
0.188 0.355 $\pm$ 0.166 2.28 $\pm$ 1.58 8.32 $\pm$ 2.67 65–70% 12.7 13.1 34.12
$\pm$ 15.6 26.81 $\pm$ 9.15 0.453 $\pm$ 0.21 0.544 $\pm$ 0.199 0.378 $\pm$
0.175 2.05 $\pm$ 1.47 5.97 $\pm$ 2.37 70–75% 13.1 13.6 21.54 $\pm$ 11.7 19.15
$\pm$ 7.98 0.286 $\pm$ 0.16 0.584 $\pm$ 0.208 0.395 $\pm$ 0.188 1.81 $\pm$
1.34 4.14 $\pm$ 2.04 75–80% 13.6 14.1 13.34 $\pm$ 8.53 13.32 $\pm$ 6.7 0.177
$\pm$ 0.11 0.62 $\pm$ 0.219 0.405 $\pm$ 0.207 1.58 $\pm$ 1.18 2.79 $\pm$ 1.68
80–85% 14.1 14.5 8.321 $\pm$ 6.12 9.216 $\pm$ 5.36 0.111 $\pm$ 0.081 0.659
$\pm$ 0.233 0.404 $\pm$ 0.238 1.36 $\pm$ 0.999 1.87 $\pm$ 1.34 85–90% 14.5 15
5.26 $\pm$ 4.28 6.423 $\pm$ 4.09 0.0698 $\pm$ 0.057 0.701 $\pm$ 0.248 0.392
$\pm$ 0.276 1.2 $\pm$ 0.823 1.26 $\pm$ 1.04 90–95% 15 15.7 3.364 $\pm$ 2.89
4.526 $\pm$ 2.94 0.0447 $\pm$ 0.038 0.756 $\pm$ 0.258 0.353 $\pm$ 0.313 1.06
$\pm$ 0.641 0.859 $\pm$ 0.787 95–100% 15.7 20 2.071 $\pm$ 1.78 3.135 $\pm$
1.85 0.0275 $\pm$ 0.024 0.838 $\pm$ 0.246 0.26 $\pm$ 0.329 0.947 $\pm$ 0.449
0.568 $\pm$ 0.575 0–100% 0 20 420.7 $\pm$ 561 115.4 $\pm$ 117 5.59 $\pm$ 7.4
0.429 $\pm$ 0.275 0.269 $\pm$ 0.203 2.9 $\pm$ 2.35 29 $\pm$ 31.1

Table 12:  Various quantities for PbPb collisions at $\sqrt{s_{\rm NN}}=10.6$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 12.

PbPb at $\sqrt{s_{\rm NN}}=39$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm max}$
(fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm part}\rangle\,\pm$
rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1) $\varepsilon_{2}\,\pm$ rms
$\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm) $A_{\perp}\pm$ rms (fm2) 0–5% 0
3.55 2334 $\pm$ 196 390.9 $\pm$ 14.8 25.8 $\pm$ 2.2 0.0675 $\pm$ 0.0362 0.0808
$\pm$ 0.0432 5.34 $\pm$ 2.97 116 $\pm$ 9.21 5–10% 3.55 5.02 1817 $\pm$ 150 341
$\pm$ 17.4 20.1 $\pm$ 1.7 0.0991 $\pm$ 0.0522 0.109 $\pm$ 0.057 5.06 $\pm$
2.84 94.2 $\pm$ 6.8 10–15% 5.02 6.15 1422 $\pm$ 123 293.1 $\pm$ 17.3 15.7
$\pm$ 1.4 0.139 $\pm$ 0.0679 0.131 $\pm$ 0.0683 4.8 $\pm$ 2.7 78.6 $\pm$ 5.9
15–20% 6.15 7.11 1109 $\pm$ 106 250.3 $\pm$ 16.6 12.2 $\pm$ 1.2 0.18 $\pm$
0.0811 0.151 $\pm$ 0.0784 4.52 $\pm$ 2.57 66.1 $\pm$ 5.36 20–25% 7.11 7.94
857.8 $\pm$ 93 212.6 $\pm$ 16 9.48 $\pm$ 1 0.219 $\pm$ 0.093 0.17 $\pm$ 0.0883
4.28 $\pm$ 2.45 55.7 $\pm$ 4.99 25–30% 7.94 8.7 655.9 $\pm$ 82.5 179.3 $\pm$
15.4 7.25 $\pm$ 0.91 0.255 $\pm$ 0.104 0.189 $\pm$ 0.0975 4.03 $\pm$ 2.33 46.8
$\pm$ 4.69 30–35% 8.7 9.4 494.6 $\pm$ 72.8 149.8 $\pm$ 14.8 5.47 $\pm$ 0.8
0.29 $\pm$ 0.115 0.207 $\pm$ 0.106 3.79 $\pm$ 2.22 39.1 $\pm$ 4.46 35–40% 9.4
10 366.4 $\pm$ 63.3 124 $\pm$ 14.3 4.05 $\pm$ 0.7 0.323 $\pm$ 0.126 0.227
$\pm$ 0.116 3.56 $\pm$ 2.1 32.3 $\pm$ 4.24 40–45% 10 10.7 265.8 $\pm$ 54 101.3
$\pm$ 13.8 2.94 $\pm$ 0.6 0.355 $\pm$ 0.137 0.249 $\pm$ 0.125 3.33 $\pm$ 2
26.3 $\pm$ 4.03 45–50% 10.7 11.2 189 $\pm$ 45 81.63 $\pm$ 13.1 2.09 $\pm$ 0.5
0.386 $\pm$ 0.148 0.271 $\pm$ 0.134 3.11 $\pm$ 1.9 21.2 $\pm$ 3.82 50–55% 11.2
11.8 131 $\pm$ 36.7 64.58 $\pm$ 12.4 1.45 $\pm$ 0.41 0.418 $\pm$ 0.16 0.295
$\pm$ 0.143 2.89 $\pm$ 1.8 16.6 $\pm$ 3.59 55–60% 11.8 12.3 88.42 $\pm$ 29.2
49.99 $\pm$ 11.5 0.977 $\pm$ 0.32 0.451 $\pm$ 0.172 0.319 $\pm$ 0.152 2.68
$\pm$ 1.71 12.8 $\pm$ 3.33 60–65% 12.3 12.8 58.06 $\pm$ 22.6 37.76 $\pm$ 10.6
0.642 $\pm$ 0.25 0.485 $\pm$ 0.184 0.344 $\pm$ 0.161 2.45 $\pm$ 1.6 9.51 $\pm$
3.04 65–70% 12.8 13.3 37.12 $\pm$ 17.1 27.71 $\pm$ 9.47 0.41 $\pm$ 0.19 0.522
$\pm$ 0.196 0.366 $\pm$ 0.171 2.21 $\pm$ 1.5 6.84 $\pm$ 2.71 70–75% 13.3 13.8
23.23 $\pm$ 12.7 19.78 $\pm$ 8.27 0.257 $\pm$ 0.14 0.559 $\pm$ 0.206 0.383
$\pm$ 0.182 1.97 $\pm$ 1.38 4.76 $\pm$ 2.34 75–80% 13.8 14.2 14.37 $\pm$ 9.28
13.82 $\pm$ 6.97 0.159 $\pm$ 0.1 0.595 $\pm$ 0.218 0.395 $\pm$ 0.202 1.74
$\pm$ 1.23 3.24 $\pm$ 1.95 80–85% 14.2 14.7 8.862 $\pm$ 6.59 9.532 $\pm$ 5.59
0.0979 $\pm$ 0.073 0.635 $\pm$ 0.233 0.401 $\pm$ 0.234 1.52 $\pm$ 1.06 2.17
$\pm$ 1.57 85–90% 14.7 15.2 5.539 $\pm$ 4.6 6.606 $\pm$ 4.28 0.0612 $\pm$
0.051 0.681 $\pm$ 0.251 0.393 $\pm$ 0.273 1.33 $\pm$ 0.877 1.46 $\pm$ 1.22
90–95% 15.2 15.9 3.517 $\pm$ 3.1 4.641 $\pm$ 3.09 0.0389 $\pm$ 0.034 0.74
$\pm$ 0.263 0.359 $\pm$ 0.313 1.18 $\pm$ 0.696 0.99 $\pm$ 0.929 95–100% 15.9
20 2.15 $\pm$ 1.93 3.204 $\pm$ 1.99 0.0238 $\pm$ 0.021 0.828 $\pm$ 0.253 0.267
$\pm$ 0.332 1.05 $\pm$ 0.488 0.655 $\pm$ 0.682 0–100% 0 20 494.1 $\pm$ 668
118.1 $\pm$ 119 5.46 $\pm$ 7.4 0.412 $\pm$ 0.272 0.265 $\pm$ 0.201 3.04 $\pm$
2.36 31.8 $\pm$ 33.7

Table 13:  Various quantities for PbPb collisions at $\sqrt{s_{\rm NN}}=39$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 12.

pPb at $\sqrt{s_{\rm NN}}=5.02$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 1.82 13.68 $\pm$ 3.51 14.68 $\pm$ 3.51 0.202
$\pm$ 0.052 0.299 $\pm$ 0.157 0.328 $\pm$ 0.168 1.23 $\pm$ 0.434 1.64 $\pm$
0.123 5–10% 1.82 2.58 13.11 $\pm$ 3.4 14.11 $\pm$ 3.4 0.194 $\pm$ 0.05 0.307
$\pm$ 0.161 0.337 $\pm$ 0.171 1.22 $\pm$ 0.436 1.63 $\pm$ 0.135 10–15% 2.58
3.16 12.5 $\pm$ 3.3 13.5 $\pm$ 3.3 0.185 $\pm$ 0.049 0.315 $\pm$ 0.166 0.345
$\pm$ 0.175 1.21 $\pm$ 0.437 1.62 $\pm$ 0.146 15–20% 3.16 3.65 11.83 $\pm$
3.18 12.83 $\pm$ 3.18 0.175 $\pm$ 0.047 0.326 $\pm$ 0.17 0.356 $\pm$ 0.179 1.2
$\pm$ 0.439 1.61 $\pm$ 0.161 20–25% 3.65 4.08 11.13 $\pm$ 3.07 12.13 $\pm$
3.07 0.165 $\pm$ 0.045 0.338 $\pm$ 0.176 0.369 $\pm$ 0.184 1.19 $\pm$ 0.442
1.6 $\pm$ 0.179 25–30% 4.08 4.47 10.36 $\pm$ 2.96 11.36 $\pm$ 2.96 0.153 $\pm$
0.044 0.354 $\pm$ 0.183 0.384 $\pm$ 0.19 1.17 $\pm$ 0.443 1.58 $\pm$ 0.201
30–35% 4.47 4.83 9.529 $\pm$ 2.83 10.53 $\pm$ 2.83 0.141 $\pm$ 0.042 0.373
$\pm$ 0.191 0.402 $\pm$ 0.196 1.15 $\pm$ 0.445 1.55 $\pm$ 0.227 35–40% 4.83
5.16 8.646 $\pm$ 2.7 9.646 $\pm$ 2.7 0.128 $\pm$ 0.04 0.396 $\pm$ 0.201 0.422
$\pm$ 0.203 1.12 $\pm$ 0.445 1.51 $\pm$ 0.26 40–45% 5.16 5.47 7.721 $\pm$ 2.57
8.721 $\pm$ 2.57 0.114 $\pm$ 0.038 0.424 $\pm$ 0.212 0.445 $\pm$ 0.21 1.1
$\pm$ 0.444 1.46 $\pm$ 0.299 45–50% 5.47 5.77 6.766 $\pm$ 2.41 7.766 $\pm$
2.41 0.1 $\pm$ 0.036 0.459 $\pm$ 0.224 0.467 $\pm$ 0.218 1.06 $\pm$ 0.439 1.39
$\pm$ 0.344 50–55% 5.77 6.05 5.836 $\pm$ 2.25 6.836 $\pm$ 2.25 0.0863 $\pm$
0.033 0.498 $\pm$ 0.236 0.488 $\pm$ 0.227 1.03 $\pm$ 0.433 1.3 $\pm$ 0.39
55–60% 6.05 6.32 4.949 $\pm$ 2.07 5.949 $\pm$ 2.07 0.0732 $\pm$ 0.031 0.539
$\pm$ 0.248 0.504 $\pm$ 0.24 0.989 $\pm$ 0.425 1.19 $\pm$ 0.432 60–65% 6.32
6.58 4.132 $\pm$ 1.87 5.132 $\pm$ 1.87 0.0611 $\pm$ 0.028 0.586 $\pm$ 0.261
0.508 $\pm$ 0.259 0.952 $\pm$ 0.412 1.07 $\pm$ 0.464 65–70% 6.58 6.84 3.415
$\pm$ 1.66 4.415 $\pm$ 1.66 0.0505 $\pm$ 0.025 0.636 $\pm$ 0.271 0.497 $\pm$
0.284 0.919 $\pm$ 0.396 0.932 $\pm$ 0.48 70–75% 6.84 7.1 2.802 $\pm$ 1.45
3.802 $\pm$ 1.45 0.0415 $\pm$ 0.021 0.689 $\pm$ 0.276 0.47 $\pm$ 0.314 0.89
$\pm$ 0.379 0.798 $\pm$ 0.48 75–80% 7.1 7.36 2.294 $\pm$ 1.23 3.294 $\pm$ 1.23
0.0339 $\pm$ 0.018 0.745 $\pm$ 0.275 0.42 $\pm$ 0.342 0.866 $\pm$ 0.361 0.673
$\pm$ 0.467 80–85% 7.36 7.65 1.877 $\pm$ 1 2.877 $\pm$ 1 0.0278 $\pm$ 0.015
0.805 $\pm$ 0.262 0.349 $\pm$ 0.357 0.845 $\pm$ 0.341 0.561 $\pm$ 0.441 85–90%
7.65 7.99 1.55 $\pm$ 0.784 2.55 $\pm$ 0.784 0.0229 $\pm$ 0.012 0.865 $\pm$
0.234 0.26 $\pm$ 0.348 0.828 $\pm$ 0.324 0.467 $\pm$ 0.412 90–95% 7.99 8.49
1.287 $\pm$ 0.556 2.287 $\pm$ 0.556 0.019 $\pm$ 0.0082 0.923 $\pm$ 0.188 0.159
$\pm$ 0.303 0.816 $\pm$ 0.307 0.389 $\pm$ 0.379 95–100% 8.49 14.7 1.082 $\pm$
0.295 2.082 $\pm$ 0.295 0.016 $\pm$ 0.0044 0.976 $\pm$ 0.109 0.0516 $\pm$
0.189 0.805 $\pm$ 0.291 0.327 $\pm$ 0.348 0–100% 0 14.7 6.723 $\pm$ 4.87 7.723
$\pm$ 4.87 0.0994 $\pm$ 0.072 0.543 $\pm$ 0.306 0.378 $\pm$ 0.271 1.03 $\pm$
0.433 1.17 $\pm$ 0.574

Table 14:  Various quantities for pPb collisions at $\sqrt{s_{\rm NN}}=5.02$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 13.

pPb at $\sqrt{s_{\rm NN}}=8.16$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 1.84 14.65 $\pm$ 3.59 15.65 $\pm$ 3.59 0.202
$\pm$ 0.05 0.286 $\pm$ 0.151 0.315 $\pm$ 0.162 1.29 $\pm$ 0.445 1.77 $\pm$
0.11 5–10% 1.84 2.6 14.01 $\pm$ 3.51 15.01 $\pm$ 3.51 0.193 $\pm$ 0.048 0.295
$\pm$ 0.155 0.325 $\pm$ 0.166 1.28 $\pm$ 0.447 1.76 $\pm$ 0.122 10–15% 2.6
3.19 13.34 $\pm$ 3.4 14.34 $\pm$ 3.4 0.184 $\pm$ 0.047 0.304 $\pm$ 0.159 0.333
$\pm$ 0.169 1.27 $\pm$ 0.45 1.76 $\pm$ 0.136 15–20% 3.19 3.68 12.62 $\pm$ 3.27
13.62 $\pm$ 3.27 0.174 $\pm$ 0.045 0.314 $\pm$ 0.164 0.344 $\pm$ 0.174 1.26
$\pm$ 0.451 1.74 $\pm$ 0.152 20–25% 3.68 4.11 11.87 $\pm$ 3.16 12.87 $\pm$
3.16 0.164 $\pm$ 0.044 0.326 $\pm$ 0.17 0.357 $\pm$ 0.179 1.24 $\pm$ 0.453
1.73 $\pm$ 0.169 25–30% 4.11 4.51 11.01 $\pm$ 3.03 12.01 $\pm$ 3.03 0.152
$\pm$ 0.042 0.341 $\pm$ 0.177 0.372 $\pm$ 0.185 1.22 $\pm$ 0.456 1.71 $\pm$
0.194 30–35% 4.51 4.87 10.07 $\pm$ 2.91 11.07 $\pm$ 2.91 0.139 $\pm$ 0.04
0.361 $\pm$ 0.186 0.391 $\pm$ 0.192 1.2 $\pm$ 0.457 1.68 $\pm$ 0.224 35–40%
4.87 5.2 9.116 $\pm$ 2.78 10.12 $\pm$ 2.78 0.126 $\pm$ 0.038 0.384 $\pm$ 0.196
0.413 $\pm$ 0.199 1.18 $\pm$ 0.459 1.64 $\pm$ 0.261 40–45% 5.2 5.52 8.091
$\pm$ 2.62 9.091 $\pm$ 2.62 0.112 $\pm$ 0.036 0.413 $\pm$ 0.207 0.437 $\pm$
0.207 1.14 $\pm$ 0.456 1.59 $\pm$ 0.303 45–50% 5.52 5.82 7.06 $\pm$ 2.46 8.06
$\pm$ 2.46 0.0974 $\pm$ 0.034 0.449 $\pm$ 0.219 0.461 $\pm$ 0.215 1.11 $\pm$
0.454 1.51 $\pm$ 0.355 50–55% 5.82 6.1 6.064 $\pm$ 2.29 7.064 $\pm$ 2.29
0.0836 $\pm$ 0.032 0.486 $\pm$ 0.232 0.485 $\pm$ 0.224 1.07 $\pm$ 0.448 1.42
$\pm$ 0.406 55–60% 6.1 6.38 5.108 $\pm$ 2.1 6.108 $\pm$ 2.1 0.0705 $\pm$ 0.029
0.529 $\pm$ 0.246 0.503 $\pm$ 0.237 1.03 $\pm$ 0.438 1.29 $\pm$ 0.456 60–65%
6.38 6.64 4.241 $\pm$ 1.9 5.241 $\pm$ 1.9 0.0585 $\pm$ 0.026 0.575 $\pm$ 0.258
0.51 $\pm$ 0.256 0.992 $\pm$ 0.425 1.15 $\pm$ 0.493 65–70% 6.64 6.9 3.49 $\pm$
1.69 4.49 $\pm$ 1.69 0.0481 $\pm$ 0.023 0.627 $\pm$ 0.27 0.501 $\pm$ 0.281
0.958 $\pm$ 0.41 1 $\pm$ 0.512 70–75% 6.9 7.15 2.856 $\pm$ 1.47 3.856 $\pm$
1.47 0.0394 $\pm$ 0.02 0.681 $\pm$ 0.277 0.474 $\pm$ 0.312 0.929 $\pm$ 0.391
0.854 $\pm$ 0.513 75–80% 7.15 7.42 2.328 $\pm$ 1.24 3.328 $\pm$ 1.24 0.0321
$\pm$ 0.017 0.738 $\pm$ 0.277 0.426 $\pm$ 0.341 0.903 $\pm$ 0.373 0.717 $\pm$
0.498 80–85% 7.42 7.71 1.905 $\pm$ 1.02 2.905 $\pm$ 1.02 0.0263 $\pm$ 0.014
0.798 $\pm$ 0.265 0.357 $\pm$ 0.358 0.881 $\pm$ 0.355 0.597 $\pm$ 0.47 85–90%
7.71 8.05 1.556 $\pm$ 0.787 2.556 $\pm$ 0.787 0.0215 $\pm$ 0.011 0.862 $\pm$
0.237 0.265 $\pm$ 0.351 0.861 $\pm$ 0.335 0.496 $\pm$ 0.438 90–95% 8.05 8.55
1.291 $\pm$ 0.56 2.291 $\pm$ 0.56 0.0178 $\pm$ 0.0077 0.921 $\pm$ 0.191 0.162
$\pm$ 0.306 0.849 $\pm$ 0.316 0.41 $\pm$ 0.402 95–100% 8.55 14.8 1.083 $\pm$
0.296 2.083 $\pm$ 0.296 0.0149 $\pm$ 0.0041 0.976 $\pm$ 0.11 0.0517 $\pm$
0.189 0.838 $\pm$ 0.302 0.343 $\pm$ 0.369 0–100% 0 14.8 7.085 $\pm$ 5.19 8.085
$\pm$ 5.19 0.0977 $\pm$ 0.072 0.534 $\pm$ 0.306 0.374 $\pm$ 0.269 1.08 $\pm$
0.448 1.26 $\pm$ 0.618

Table 15:  Various quantities for pPb collisions at $\sqrt{s_{\rm NN}}=8.16$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 13.

pPb at $\sqrt{s_{\rm NN}}=8.8$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm max}$
(fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm part}\rangle\,\pm$
rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1) $\varepsilon_{2}\,\pm$ rms
$\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm) $A_{\perp}\pm$ rms (fm2) 0–5% 0
1.84 14.83 $\pm$ 3.63 15.83 $\pm$ 3.63 0.202 $\pm$ 0.05 0.285 $\pm$ 0.15 0.314
$\pm$ 0.161 1.3 $\pm$ 0.447 1.79 $\pm$ 0.109 5–10% 1.84 2.6 14.18 $\pm$ 3.52
15.18 $\pm$ 3.52 0.193 $\pm$ 0.048 0.293 $\pm$ 0.154 0.322 $\pm$ 0.165 1.29
$\pm$ 0.447 1.79 $\pm$ 0.12 10–15% 2.6 3.19 13.49 $\pm$ 3.41 14.49 $\pm$ 3.41
0.184 $\pm$ 0.047 0.301 $\pm$ 0.158 0.331 $\pm$ 0.168 1.28 $\pm$ 0.45 1.78
$\pm$ 0.133 15–20% 3.19 3.68 12.75 $\pm$ 3.29 13.75 $\pm$ 3.29 0.174 $\pm$
0.045 0.312 $\pm$ 0.164 0.342 $\pm$ 0.173 1.26 $\pm$ 0.454 1.77 $\pm$ 0.149
20–25% 3.68 4.12 11.97 $\pm$ 3.16 12.97 $\pm$ 3.16 0.163 $\pm$ 0.043 0.324
$\pm$ 0.169 0.355 $\pm$ 0.179 1.25 $\pm$ 0.457 1.75 $\pm$ 0.168 25–30% 4.12
4.51 11.11 $\pm$ 3.04 12.11 $\pm$ 3.04 0.152 $\pm$ 0.041 0.34 $\pm$ 0.177
0.369 $\pm$ 0.184 1.23 $\pm$ 0.457 1.73 $\pm$ 0.193 30–35% 4.51 4.87 10.18
$\pm$ 2.93 11.18 $\pm$ 2.93 0.139 $\pm$ 0.04 0.36 $\pm$ 0.184 0.389 $\pm$
0.191 1.21 $\pm$ 0.46 1.7 $\pm$ 0.223 35–40% 4.87 5.21 9.179 $\pm$ 2.78 10.18
$\pm$ 2.78 0.125 $\pm$ 0.038 0.383 $\pm$ 0.195 0.411 $\pm$ 0.199 1.18 $\pm$
0.461 1.67 $\pm$ 0.259 40–45% 5.21 5.53 8.151 $\pm$ 2.64 9.151 $\pm$ 2.64
0.111 $\pm$ 0.036 0.411 $\pm$ 0.206 0.435 $\pm$ 0.207 1.15 $\pm$ 0.459 1.61
$\pm$ 0.305 45–50% 5.53 5.83 7.113 $\pm$ 2.47 8.113 $\pm$ 2.47 0.097 $\pm$
0.034 0.446 $\pm$ 0.218 0.461 $\pm$ 0.215 1.11 $\pm$ 0.455 1.54 $\pm$ 0.355
50–55% 5.83 6.11 6.082 $\pm$ 2.29 7.082 $\pm$ 2.29 0.083 $\pm$ 0.031 0.487
$\pm$ 0.232 0.481 $\pm$ 0.224 1.07 $\pm$ 0.45 1.43 $\pm$ 0.41 55–60% 6.11 6.38
5.148 $\pm$ 2.11 6.148 $\pm$ 2.11 0.0702 $\pm$ 0.029 0.527 $\pm$ 0.245 0.501
$\pm$ 0.237 1.03 $\pm$ 0.44 1.31 $\pm$ 0.459 60–65% 6.38 6.65 4.266 $\pm$ 1.91
5.266 $\pm$ 1.91 0.0582 $\pm$ 0.026 0.574 $\pm$ 0.258 0.509 $\pm$ 0.256 0.999
$\pm$ 0.427 1.16 $\pm$ 0.499 65–70% 6.65 6.91 3.496 $\pm$ 1.69 4.496 $\pm$
1.69 0.0477 $\pm$ 0.023 0.625 $\pm$ 0.27 0.501 $\pm$ 0.281 0.964 $\pm$ 0.413
1.01 $\pm$ 0.516 70–75% 6.91 7.16 2.858 $\pm$ 1.47 3.858 $\pm$ 1.47 0.039
$\pm$ 0.02 0.679 $\pm$ 0.278 0.475 $\pm$ 0.312 0.934 $\pm$ 0.395 0.863 $\pm$
0.518 75–80% 7.16 7.43 2.328 $\pm$ 1.24 3.328 $\pm$ 1.24 0.0318 $\pm$ 0.017
0.737 $\pm$ 0.277 0.428 $\pm$ 0.341 0.909 $\pm$ 0.375 0.721 $\pm$ 0.502 80–85%
7.43 7.72 1.898 $\pm$ 1.01 2.898 $\pm$ 1.01 0.0259 $\pm$ 0.014 0.8 $\pm$ 0.264
0.354 $\pm$ 0.357 0.888 $\pm$ 0.357 0.601 $\pm$ 0.476 85–90% 7.72 8.06 1.557
$\pm$ 0.79 2.557 $\pm$ 0.79 0.0212 $\pm$ 0.011 0.863 $\pm$ 0.236 0.263 $\pm$
0.349 0.869 $\pm$ 0.338 0.498 $\pm$ 0.443 90–95% 8.06 8.56 1.294 $\pm$ 0.564
2.294 $\pm$ 0.564 0.0177 $\pm$ 0.0077 0.921 $\pm$ 0.191 0.163 $\pm$ 0.306
0.854 $\pm$ 0.319 0.415 $\pm$ 0.407 95–100% 8.56 14.4 1.08 $\pm$ 0.292 2.08
$\pm$ 0.292 0.0147 $\pm$ 0.004 0.977 $\pm$ 0.109 0.0508 $\pm$ 0.188 0.843
$\pm$ 0.302 0.345 $\pm$ 0.371 0–100% 0 14.4 7.146 $\pm$ 5.25 8.146 $\pm$ 5.25
0.0975 $\pm$ 0.072 0.532 $\pm$ 0.307 0.373 $\pm$ 0.269 1.08 $\pm$ 0.45 1.27
$\pm$ 0.626

Table 16:  Various quantities for pPb collisions at $\sqrt{s_{\rm NN}}=8.8$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Typical uncertainties,
due to the MCG model parameters for each quantity at any centrality bin, can
be read off from Fig. 13.

pPb at $\sqrt{s_{\rm NN}}=17$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm max}$
(fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm part}\rangle\,\pm$
rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1) $\varepsilon_{2}\,\pm$ rms
$\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm) $A_{\perp}\pm$ rms (fm2) 0–5% 0
1.86 16.26 $\pm$ 3.78 17.26 $\pm$ 3.78 0.202 $\pm$ 0.047 0.27 $\pm$ 0.143
0.297 $\pm$ 0.153 1.38 $\pm$ 0.461 1.99 $\pm$ 0.0941 5–10% 1.86 2.63 15.54
$\pm$ 3.65 16.54 $\pm$ 3.65 0.193 $\pm$ 0.045 0.278 $\pm$ 0.146 0.305 $\pm$
0.157 1.37 $\pm$ 0.464 1.98 $\pm$ 0.103 10–15% 2.63 3.23 14.75 $\pm$ 3.54
15.75 $\pm$ 3.54 0.183 $\pm$ 0.044 0.285 $\pm$ 0.15 0.315 $\pm$ 0.161 1.36
$\pm$ 0.467 1.97 $\pm$ 0.119 15–20% 3.23 3.73 13.92 $\pm$ 3.42 14.92 $\pm$
3.42 0.173 $\pm$ 0.042 0.296 $\pm$ 0.155 0.327 $\pm$ 0.166 1.34 $\pm$ 0.47
1.96 $\pm$ 0.135 20–25% 3.73 4.17 13.01 $\pm$ 3.27 14.01 $\pm$ 3.27 0.161
$\pm$ 0.041 0.307 $\pm$ 0.16 0.34 $\pm$ 0.172 1.33 $\pm$ 0.472 1.95 $\pm$
0.156 25–30% 4.17 4.56 12.02 $\pm$ 3.14 13.02 $\pm$ 3.14 0.149 $\pm$ 0.039
0.324 $\pm$ 0.169 0.355 $\pm$ 0.178 1.31 $\pm$ 0.473 1.93 $\pm$ 0.181 30–35%
4.56 4.93 10.98 $\pm$ 3.01 11.98 $\pm$ 3.01 0.136 $\pm$ 0.037 0.343 $\pm$
0.177 0.375 $\pm$ 0.186 1.28 $\pm$ 0.477 1.9 $\pm$ 0.212 35–40% 4.93 5.27
9.849 $\pm$ 2.87 10.85 $\pm$ 2.87 0.122 $\pm$ 0.036 0.368 $\pm$ 0.188 0.398
$\pm$ 0.194 1.25 $\pm$ 0.478 1.86 $\pm$ 0.256 40–45% 5.27 5.59 8.701 $\pm$ 2.7
9.701 $\pm$ 2.7 0.108 $\pm$ 0.034 0.398 $\pm$ 0.2 0.424 $\pm$ 0.202 1.22 $\pm$
0.478 1.8 $\pm$ 0.31 45–50% 5.59 5.89 7.539 $\pm$ 2.53 8.539 $\pm$ 2.53 0.0935
$\pm$ 0.031 0.433 $\pm$ 0.212 0.451 $\pm$ 0.212 1.18 $\pm$ 0.473 1.72 $\pm$
0.369 50–55% 5.89 6.18 6.412 $\pm$ 2.36 7.412 $\pm$ 2.36 0.0796 $\pm$ 0.029
0.473 $\pm$ 0.226 0.477 $\pm$ 0.221 1.14 $\pm$ 0.468 1.6 $\pm$ 0.437 55–60%
6.18 6.46 5.376 $\pm$ 2.17 6.376 $\pm$ 2.17 0.0667 $\pm$ 0.027 0.517 $\pm$
0.24 0.497 $\pm$ 0.233 1.09 $\pm$ 0.459 1.46 $\pm$ 0.496 60–65% 6.46 6.73
4.425 $\pm$ 1.95 5.425 $\pm$ 1.95 0.0549 $\pm$ 0.024 0.562 $\pm$ 0.255 0.508
$\pm$ 0.251 1.05 $\pm$ 0.447 1.29 $\pm$ 0.539 65–70% 6.73 6.98 3.626 $\pm$
1.73 4.626 $\pm$ 1.73 0.045 $\pm$ 0.021 0.609 $\pm$ 0.268 0.507 $\pm$ 0.276
1.02 $\pm$ 0.429 1.12 $\pm$ 0.564 70–75% 6.98 7.24 2.938 $\pm$ 1.49 3.938
$\pm$ 1.49 0.0365 $\pm$ 0.019 0.665 $\pm$ 0.279 0.483 $\pm$ 0.308 0.988 $\pm$
0.413 0.945 $\pm$ 0.566 75–80% 7.24 7.51 2.37 $\pm$ 1.27 3.37 $\pm$ 1.27
0.0294 $\pm$ 0.016 0.729 $\pm$ 0.279 0.434 $\pm$ 0.34 0.958 $\pm$ 0.392 0.788
$\pm$ 0.55 80–85% 7.51 7.8 1.925 $\pm$ 1.03 2.925 $\pm$ 1.03 0.0239 $\pm$
0.013 0.793 $\pm$ 0.268 0.363 $\pm$ 0.358 0.934 $\pm$ 0.371 0.653 $\pm$ 0.518
85–90% 7.8 8.14 1.575 $\pm$ 0.801 2.575 $\pm$ 0.801 0.0195 $\pm$ 0.0099 0.857
$\pm$ 0.24 0.272 $\pm$ 0.353 0.917 $\pm$ 0.354 0.54 $\pm$ 0.483 90–95% 8.14
8.64 1.295 $\pm$ 0.561 2.295 $\pm$ 0.561 0.0161 $\pm$ 0.007 0.919 $\pm$ 0.193
0.166 $\pm$ 0.309 0.902 $\pm$ 0.335 0.442 $\pm$ 0.439 95–100% 8.64 14.9 1.084
$\pm$ 0.298 2.084 $\pm$ 0.298 0.0134 $\pm$ 0.0037 0.976 $\pm$ 0.112 0.0529
$\pm$ 0.191 0.891 $\pm$ 0.319 0.371 $\pm$ 0.403 0–100% 0 14.9 7.677 $\pm$ 5.72
8.677 $\pm$ 5.72 0.0952 $\pm$ 0.071 0.52 $\pm$ 0.308 0.367 $\pm$ 0.266 1.14
$\pm$ 0.47 1.41 $\pm$ 0.693

Table 17:  Various quantities for pPb collisions at $\sqrt{s_{\rm NN}}=17$ TeV
obtained with the improved MCG model for centrality classes defined by slicing
the impact parameter ($b$) distribution. The mean and standard deviation of
each quantity (denoted as rms) are given. Typical uncertainties, due to the
MCG model parameters for each quantity at any centrality bin, can be read off
from Fig. 13.

pPb at $\sqrt{s_{\rm NN}}=63$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm max}$
(fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm part}\rangle\,\pm$
rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1) $\varepsilon_{2}\,\pm$ rms
$\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm) $A_{\perp}\pm$ rms (fm2) 0–5% 0
1.91 19.37 $\pm$ 4.11 20.37 $\pm$ 4.11 0.201 $\pm$ 0.043 0.244 $\pm$ 0.129
0.27 $\pm$ 0.139 1.54 $\pm$ 0.493 2.4 $\pm$ 0.0711 5–10% 1.91 2.7 18.45 $\pm$
3.96 19.45 $\pm$ 3.96 0.191 $\pm$ 0.041 0.251 $\pm$ 0.133 0.277 $\pm$ 0.144
1.53 $\pm$ 0.494 2.39 $\pm$ 0.0801 10–15% 2.7 3.31 17.47 $\pm$ 3.8 18.47 $\pm$
3.8 0.181 $\pm$ 0.039 0.259 $\pm$ 0.136 0.286 $\pm$ 0.148 1.52 $\pm$ 0.496
2.39 $\pm$ 0.0904 15–20% 3.31 3.81 16.39 $\pm$ 3.64 17.39 $\pm$ 3.64 0.17
$\pm$ 0.038 0.27 $\pm$ 0.141 0.297 $\pm$ 0.153 1.5 $\pm$ 0.497 2.38 $\pm$
0.105 20–25% 3.81 4.26 15.24 $\pm$ 3.5 16.24 $\pm$ 3.5 0.158 $\pm$ 0.036 0.281
$\pm$ 0.147 0.312 $\pm$ 0.159 1.49 $\pm$ 0.503 2.37 $\pm$ 0.128 25–30% 4.26
4.67 13.97 $\pm$ 3.34 14.97 $\pm$ 3.34 0.145 $\pm$ 0.035 0.298 $\pm$ 0.155
0.328 $\pm$ 0.166 1.46 $\pm$ 0.505 2.35 $\pm$ 0.156 30–35% 4.67 5.04 12.61
$\pm$ 3.18 13.61 $\pm$ 3.18 0.131 $\pm$ 0.033 0.318 $\pm$ 0.164 0.35 $\pm$
0.174 1.43 $\pm$ 0.509 2.32 $\pm$ 0.196 35–40% 5.04 5.39 11.19 $\pm$ 3.02
12.19 $\pm$ 3.02 0.116 $\pm$ 0.031 0.346 $\pm$ 0.175 0.374 $\pm$ 0.184 1.39
$\pm$ 0.512 2.28 $\pm$ 0.251 40–45% 5.39 5.72 9.76 $\pm$ 2.83 10.76 $\pm$ 2.83
0.101 $\pm$ 0.029 0.377 $\pm$ 0.188 0.403 $\pm$ 0.193 1.35 $\pm$ 0.513 2.21
$\pm$ 0.318 45–50% 5.72 6.03 8.359 $\pm$ 2.66 9.359 $\pm$ 2.66 0.0866 $\pm$
0.028 0.413 $\pm$ 0.201 0.433 $\pm$ 0.204 1.3 $\pm$ 0.508 2.1 $\pm$ 0.403
50–55% 6.03 6.32 7.046 $\pm$ 2.45 8.046 $\pm$ 2.45 0.073 $\pm$ 0.025 0.451
$\pm$ 0.214 0.462 $\pm$ 0.215 1.26 $\pm$ 0.504 1.96 $\pm$ 0.49 55–60% 6.32 6.6
5.837 $\pm$ 2.26 6.837 $\pm$ 2.26 0.0605 $\pm$ 0.023 0.492 $\pm$ 0.23 0.488
$\pm$ 0.227 1.21 $\pm$ 0.494 1.78 $\pm$ 0.571 60–65% 6.6 6.88 4.771 $\pm$ 2.03
5.771 $\pm$ 2.03 0.0494 $\pm$ 0.021 0.534 $\pm$ 0.246 0.508 $\pm$ 0.243 1.17
$\pm$ 0.484 1.56 $\pm$ 0.63 65–70% 6.88 7.14 3.856 $\pm$ 1.79 4.856 $\pm$ 1.79
0.04 $\pm$ 0.019 0.583 $\pm$ 0.265 0.513 $\pm$ 0.269 1.13 $\pm$ 0.467 1.34
$\pm$ 0.662 70–75% 7.14 7.4 3.088 $\pm$ 1.56 4.088 $\pm$ 1.56 0.032 $\pm$
0.016 0.642 $\pm$ 0.28 0.494 $\pm$ 0.303 1.09 $\pm$ 0.446 1.12 $\pm$ 0.667
75–80% 7.4 7.67 2.467 $\pm$ 1.31 3.467 $\pm$ 1.31 0.0256 $\pm$ 0.014 0.709
$\pm$ 0.284 0.45 $\pm$ 0.337 1.06 $\pm$ 0.426 0.92 $\pm$ 0.643 80–85% 7.67
7.96 1.976 $\pm$ 1.06 2.976 $\pm$ 1.06 0.0205 $\pm$ 0.011 0.779 $\pm$ 0.274
0.378 $\pm$ 0.36 1.04 $\pm$ 0.408 0.755 $\pm$ 0.606 85–90% 7.96 8.31 1.599
$\pm$ 0.819 2.599 $\pm$ 0.819 0.0166 $\pm$ 0.0085 0.849 $\pm$ 0.246 0.283
$\pm$ 0.357 1.02 $\pm$ 0.385 0.617 $\pm$ 0.558 90–95% 8.31 8.8 1.309 $\pm$
0.578 2.309 $\pm$ 0.578 0.0136 $\pm$ 0.006 0.915 $\pm$ 0.198 0.173 $\pm$ 0.314
1 $\pm$ 0.366 0.504 $\pm$ 0.51 95–100% 8.8 14.9 1.085 $\pm$ 0.301 2.085 $\pm$
0.301 0.0112 $\pm$ 0.0031 0.975 $\pm$ 0.114 0.0546 $\pm$ 0.195 0.988 $\pm$
0.35 0.419 $\pm$ 0.466 0–100% 0 14.9 8.789 $\pm$ 6.74 9.789 $\pm$ 6.74 0.0911
$\pm$ 0.07 0.5 $\pm$ 0.309 0.357 $\pm$ 0.263 1.27 $\pm$ 0.509 1.71 $\pm$ 0.839

Table 18:  Various quantities for pPb collisions at $\sqrt{s_{\rm NN}}=63$ TeV
obtained with the improved MCG model for centrality classes defined by slicing
the impact parameter ($b$) distribution. The mean and standard deviation of
each quantity (denoted as rms) are given. Typical uncertainties, due to the
MCG model parameters for each quantity at any centrality bin, can be read off
from Fig. 13.

XeXe at $\sqrt{s_{\rm NN}}=5.44$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 3.01 942.5 $\pm$ 92.1 236.5 $\pm$ 10 13.8
$\pm$ 1.3 0.116 $\pm$ 0.06 0.112 $\pm$ 0.059 4.33 $\pm$ 2.52 66.2 $\pm$ 5.39
5–10% 3.01 4.26 734.1 $\pm$ 72.8 206.1 $\pm$ 11.7 10.7 $\pm$ 1.1 0.141 $\pm$
0.0734 0.145 $\pm$ 0.0748 4.11 $\pm$ 2.42 53.9 $\pm$ 4.56 10–15% 4.26 5.22
571.9 $\pm$ 62 177.1 $\pm$ 12.2 8.36 $\pm$ 0.91 0.173 $\pm$ 0.0887 0.173 $\pm$
0.0887 3.89 $\pm$ 2.32 44.6 $\pm$ 4.16 15–20% 5.22 6.02 443.9 $\pm$ 55.5 151.1
$\pm$ 12.4 6.49 $\pm$ 0.81 0.207 $\pm$ 0.103 0.199 $\pm$ 0.101 3.67 $\pm$ 2.23
37.2 $\pm$ 3.96 20–25% 6.02 6.73 341.7 $\pm$ 50.8 127.9 $\pm$ 12.6 5 $\pm$
0.74 0.243 $\pm$ 0.117 0.223 $\pm$ 0.113 3.45 $\pm$ 2.13 30.9 $\pm$ 3.82
25–30% 6.73 7.38 260.5 $\pm$ 46.2 107.4 $\pm$ 12.6 3.81 $\pm$ 0.68 0.278 $\pm$
0.13 0.247 $\pm$ 0.124 3.24 $\pm$ 2.03 25.6 $\pm$ 3.68 30–35% 7.38 7.97 196.1
$\pm$ 41.7 89.36 $\pm$ 12.6 2.87 $\pm$ 0.61 0.314 $\pm$ 0.143 0.271 $\pm$
0.134 3.05 $\pm$ 1.95 21 $\pm$ 3.56 35–40% 7.97 8.52 145.5 $\pm$ 36.8 73.53
$\pm$ 12.4 2.13 $\pm$ 0.54 0.349 $\pm$ 0.156 0.295 $\pm$ 0.145 2.85 $\pm$ 1.86
17.1 $\pm$ 3.41 40–45% 8.52 9.04 106.5 $\pm$ 31.7 59.75 $\pm$ 12.1 1.56 $\pm$
0.46 0.386 $\pm$ 0.168 0.32 $\pm$ 0.154 2.64 $\pm$ 1.77 13.7 $\pm$ 3.25 45–50%
9.04 9.53 76.83 $\pm$ 26.8 47.94 $\pm$ 11.6 1.12 $\pm$ 0.39 0.424 $\pm$ 0.18
0.343 $\pm$ 0.162 2.45 $\pm$ 1.69 10.8 $\pm$ 3.05 50–55% 9.53 9.99 54.64 $\pm$
22.1 37.9 $\pm$ 10.9 0.799 $\pm$ 0.32 0.463 $\pm$ 0.191 0.366 $\pm$ 0.17 2.26
$\pm$ 1.6 8.39 $\pm$ 2.81 55–60% 9.99 10.4 38.28 $\pm$ 18 29.43 $\pm$ 10.1
0.56 $\pm$ 0.26 0.504 $\pm$ 0.202 0.386 $\pm$ 0.178 2.05 $\pm$ 1.5 6.4 $\pm$
2.55 60–65% 10.4 10.9 26.61 $\pm$ 14.4 22.56 $\pm$ 9.17 0.389 $\pm$ 0.21 0.545
$\pm$ 0.211 0.401 $\pm$ 0.186 1.85 $\pm$ 1.39 4.8 $\pm$ 2.27 65–70% 10.9 11.3
18.25 $\pm$ 11.3 16.98 $\pm$ 8.06 0.267 $\pm$ 0.16 0.588 $\pm$ 0.218 0.41
$\pm$ 0.198 1.65 $\pm$ 1.27 3.54 $\pm$ 1.96 70–75% 11.3 11.7 12.49 $\pm$ 8.7
12.68 $\pm$ 6.89 0.183 $\pm$ 0.13 0.626 $\pm$ 0.226 0.414 $\pm$ 0.216 1.47
$\pm$ 1.14 2.58 $\pm$ 1.66 75–80% 11.7 12.1 8.627 $\pm$ 6.62 9.503 $\pm$ 5.74
0.126 $\pm$ 0.097 0.663 $\pm$ 0.235 0.41 $\pm$ 0.24 1.31 $\pm$ 0.995 1.89
$\pm$ 1.37 80–85% 12.1 12.5 6.011 $\pm$ 4.93 7.152 $\pm$ 4.61 0.0879 $\pm$
0.072 0.7 $\pm$ 0.244 0.399 $\pm$ 0.267 1.17 $\pm$ 0.848 1.39 $\pm$ 1.11
85–90% 12.5 13.1 4.232 $\pm$ 3.64 5.422 $\pm$ 3.6 0.0619 $\pm$ 0.053 0.74
$\pm$ 0.25 0.375 $\pm$ 0.295 1.06 $\pm$ 0.706 1.03 $\pm$ 0.891 90–95% 13.1
13.8 2.967 $\pm$ 2.58 4.116 $\pm$ 2.67 0.0434 $\pm$ 0.038 0.786 $\pm$ 0.252
0.333 $\pm$ 0.319 0.965 $\pm$ 0.568 0.759 $\pm$ 0.697 95–100% 13.8 20 1.95
$\pm$ 1.64 3.007 $\pm$ 1.72 0.0285 $\pm$ 0.024 0.857 $\pm$ 0.235 0.241 $\pm$
0.324 0.88 $\pm$ 0.414 0.533 $\pm$ 0.524 0–100% 0 20 199.6 $\pm$ 270 71.26
$\pm$ 72 2.92 $\pm$ 3.9 0.455 $\pm$ 0.288 0.303 $\pm$ 0.215 2.42 $\pm$ 2.01
17.6 $\pm$ 19.3

Table 19: Various quantities for XeXe collisions at $\sqrt{s_{\rm NN}}=5.44$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Deformed profile for Xe
was used.

AuAu at $\sqrt{s_{\rm NN}}=0.2$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 3.31 1053 $\pm$ 92.2 351 $\pm$ 17.8 25.3 $\pm$
2.2 0.0989 $\pm$ 0.0518 0.0941 $\pm$ 0.05 4.76 $\pm$ 2.91 67.4 $\pm$ 5.19
5–10% 3.31 4.68 831.4 $\pm$ 72.1 298.1 $\pm$ 17 20 $\pm$ 1.7 0.14 $\pm$ 0.0704
0.122 $\pm$ 0.0633 4.43 $\pm$ 2.76 54.6 $\pm$ 4.05 10–15% 4.68 5.73 660.1
$\pm$ 61 252.7 $\pm$ 16 15.9 $\pm$ 1.5 0.186 $\pm$ 0.0861 0.144 $\pm$ 0.0741
4.12 $\pm$ 2.61 45.1 $\pm$ 3.55 15–20% 5.73 6.61 523 $\pm$ 54.4 213.8 $\pm$
15.4 12.6 $\pm$ 1.3 0.231 $\pm$ 0.0985 0.164 $\pm$ 0.0846 3.86 $\pm$ 2.48 37.5
$\pm$ 3.29 20–25% 6.61 7.39 412 $\pm$ 49.5 180.1 $\pm$ 14.9 9.9 $\pm$ 1.2
0.273 $\pm$ 0.11 0.184 $\pm$ 0.0949 3.6 $\pm$ 2.36 31.1 $\pm$ 3.11 25–30% 7.39
8.1 321.1 $\pm$ 45.3 150.8 $\pm$ 14.6 7.72 $\pm$ 1.1 0.313 $\pm$ 0.121 0.205
$\pm$ 0.105 3.35 $\pm$ 2.24 25.6 $\pm$ 2.96 30–35% 8.1 8.75 247.2 $\pm$ 41.3
125.1 $\pm$ 14.3 5.94 $\pm$ 0.99 0.349 $\pm$ 0.132 0.227 $\pm$ 0.116 3.1 $\pm$
2.13 21 $\pm$ 2.84 35–40% 8.75 9.35 187.8 $\pm$ 37 102.8 $\pm$ 13.9 4.51 $\pm$
0.89 0.384 $\pm$ 0.143 0.251 $\pm$ 0.127 2.87 $\pm$ 2.03 17 $\pm$ 2.71 40–45%
9.35 9.92 139.9 $\pm$ 32.5 83.36 $\pm$ 13.4 3.36 $\pm$ 0.78 0.418 $\pm$ 0.154
0.277 $\pm$ 0.139 2.64 $\pm$ 1.91 13.5 $\pm$ 2.54 45–50% 9.92 10.5 102.4 $\pm$
27.8 66.65 $\pm$ 12.7 2.46 $\pm$ 0.67 0.452 $\pm$ 0.166 0.305 $\pm$ 0.151 2.41
$\pm$ 1.8 10.6 $\pm$ 2.37 50–55% 10.5 11 73.35 $\pm$ 23.4 52.37 $\pm$ 11.9
1.76 $\pm$ 0.56 0.485 $\pm$ 0.178 0.337 $\pm$ 0.163 2.18 $\pm$ 1.69 8.17 $\pm$
2.17 55–60% 11 11.5 51.45 $\pm$ 19.2 40.39 $\pm$ 11 1.24 $\pm$ 0.46 0.522
$\pm$ 0.19 0.367 $\pm$ 0.174 1.95 $\pm$ 1.57 6.15 $\pm$ 1.96 60–65% 11.5 11.9
35.33 $\pm$ 15.4 30.5 $\pm$ 9.95 0.849 $\pm$ 0.37 0.56 $\pm$ 0.202 0.397 $\pm$
0.184 1.72 $\pm$ 1.43 4.53 $\pm$ 1.73 65–70% 11.9 12.4 23.74 $\pm$ 12 22.5
$\pm$ 8.79 0.571 $\pm$ 0.29 0.603 $\pm$ 0.211 0.42 $\pm$ 0.194 1.49 $\pm$ 1.28
3.24 $\pm$ 1.48 70–75% 12.4 12.8 15.64 $\pm$ 9.17 16.23 $\pm$ 7.5 0.376 $\pm$
0.22 0.646 $\pm$ 0.218 0.431 $\pm$ 0.207 1.28 $\pm$ 1.11 2.27 $\pm$ 1.24
75–80% 12.8 13.2 10.22 $\pm$ 6.83 11.55 $\pm$ 6.17 0.246 $\pm$ 0.16 0.688
$\pm$ 0.223 0.427 $\pm$ 0.226 1.1 $\pm$ 0.944 1.56 $\pm$ 1 80–85% 13.2 13.7
6.699 $\pm$ 4.96 8.193 $\pm$ 4.86 0.161 $\pm$ 0.12 0.728 $\pm$ 0.229 0.41
$\pm$ 0.253 0.943 $\pm$ 0.777 1.08 $\pm$ 0.788 85–90% 13.7 14.2 4.426 $\pm$
3.49 5.852 $\pm$ 3.67 0.106 $\pm$ 0.084 0.767 $\pm$ 0.233 0.378 $\pm$ 0.283
0.828 $\pm$ 0.624 0.743 $\pm$ 0.604 90–95% 14.2 14.9 2.949 $\pm$ 2.38 4.216
$\pm$ 2.6 0.0709 $\pm$ 0.057 0.81 $\pm$ 0.235 0.329 $\pm$ 0.31 0.739 $\pm$
0.48 0.517 $\pm$ 0.454 95–100% 14.9 20 1.867 $\pm$ 1.43 2.957 $\pm$ 1.57
0.0449 $\pm$ 0.034 0.874 $\pm$ 0.218 0.231 $\pm$ 0.316 0.665 $\pm$ 0.335 0.345
$\pm$ 0.327 0–100% 0 20 235.1 $\pm$ 304 100.9 $\pm$ 105 5.65 $\pm$ 7.3 0.477
$\pm$ 0.284 0.285 $\pm$ 0.213 2.4 $\pm$ 2.24 17.6 $\pm$ 19.6

Table 20: Various quantities for AuAu collisions at $\sqrt{s_{\rm NN}}=0.2$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Deformed profile for Au
was used.

CuCu at $\sqrt{s_{\rm NN}}=0.2$ TeV Centrality $b_{\rm min}$ (fm) $b_{\rm
max}$ (fm) $\langle N_{\rm coll}\rangle\,\pm$ rms $\langle N_{\rm
part}\rangle\,\pm$ rms $\langle T_{\rm AA}\rangle\,\pm$ rms (mb-1)
$\varepsilon_{2}\,\pm$ rms $\varepsilon_{3}\,\pm$ rms $L\,\pm$ rms (fm)
$A_{\perp}\pm$ rms (fm2) 0–5% 0 2.34 203.6 $\pm$ 24.9 106.5 $\pm$ 6.21 4.9
$\pm$ 0.6 0.164 $\pm$ 0.0849 0.181 $\pm$ 0.0933 3.12 $\pm$ 2.08 21.2 $\pm$ 2.1
5–10% 2.34 3.31 162.9 $\pm$ 20.6 91.68 $\pm$ 6.41 3.91 $\pm$ 0.5 0.199 $\pm$
0.102 0.218 $\pm$ 0.11 2.92 $\pm$ 2 17.4 $\pm$ 1.84 10–15% 3.31 4.06 130.1
$\pm$ 18 78.42 $\pm$ 6.52 3.13 $\pm$ 0.43 0.236 $\pm$ 0.119 0.251 $\pm$ 0.124
2.73 $\pm$ 1.91 14.4 $\pm$ 1.71 15–20% 4.06 4.68 103.7 $\pm$ 16.3 66.83 $\pm$
6.65 2.49 $\pm$ 0.39 0.274 $\pm$ 0.135 0.282 $\pm$ 0.138 2.54 $\pm$ 1.83 11.9
$\pm$ 1.64 20–25% 4.68 5.24 82.13 $\pm$ 15 56.58 $\pm$ 6.78 1.97 $\pm$ 0.36
0.313 $\pm$ 0.149 0.312 $\pm$ 0.151 2.36 $\pm$ 1.74 9.86 $\pm$ 1.59 25–30%
5.24 5.73 64.7 $\pm$ 13.8 47.63 $\pm$ 6.86 1.56 $\pm$ 0.33 0.353 $\pm$ 0.164
0.34 $\pm$ 0.162 2.19 $\pm$ 1.66 8.1 $\pm$ 1.54 30–35% 5.73 6.19 50.63 $\pm$
12.5 39.83 $\pm$ 6.86 1.22 $\pm$ 0.3 0.393 $\pm$ 0.177 0.367 $\pm$ 0.171 2.02
$\pm$ 1.57 6.62 $\pm$ 1.48 35–40% 6.19 6.62 39.28 $\pm$ 11.3 33.03 $\pm$ 6.8
0.944 $\pm$ 0.27 0.434 $\pm$ 0.19 0.394 $\pm$ 0.179 1.85 $\pm$ 1.48 5.36 $\pm$
1.41 40–45% 6.62 7.02 30.23 $\pm$ 10.2 27.14 $\pm$ 6.66 0.727 $\pm$ 0.24 0.477
$\pm$ 0.202 0.416 $\pm$ 0.187 1.69 $\pm$ 1.39 4.3 $\pm$ 1.34 45–50% 7.02 7.4
23.11 $\pm$ 8.95 22.11 $\pm$ 6.43 0.556 $\pm$ 0.22 0.522 $\pm$ 0.213 0.433
$\pm$ 0.193 1.53 $\pm$ 1.29 3.42 $\pm$ 1.25 50–55% 7.4 7.77 17.54 $\pm$ 7.79
17.84 $\pm$ 6.08 0.422 $\pm$ 0.19 0.567 $\pm$ 0.22 0.444 $\pm$ 0.199 1.38