-
Entropy production in pp and Pb-Pb collisions at energies
available at the CERNLarge Hadron Collider
Patrick Hanus and Klaus ReygersPhysikalisches Institut,
Universität Heidelberg, Im Neuenheimer Feld 226, D-69120
Heidelberg, Germany
Aleksas MazeliauskasTheoretical Physics Department, CERN,
CH-1211 Genève 23, Switzerland and
Institut für Theoretische Physik, Universität
Heidelberg,Philosophenweg 16, D-69120 Heidelberg, Germany
(Dated: December 6, 2019)
We use experimentally measured identified particle spectra and
Hanbury Brown-Twiss radii todetermine the entropy per unit rapidity
dS/dy produced in
√s = 7 TeV pp and
√sNN = 2.76 TeV
Pb–Pb collisions. We find that dS/dy = 11 335 ± 1188 in 0–10%
Pb–Pb, dS/dy = 135.7 ± 17.9in high-multiplicity pp, and dS/dy =
37.8 ± 3.7 in minimum bias pp collisions and compare
thecorresponding entropy per charged particle (dS/dy)/(dNch/dy) to
predictions of statistical models.Finally, we use the QCD kinetic
theory pre-equilibrium and viscous hydrodynamics to model
entropyproduction in the collision and reconstruct the average
temperature profile at τ0 ≈ 1 fm/c for highmultiplicity pp and
Pb–Pb collisions.
I. INTRODUCTION
Ultrarelativistic collisions of nuclei as studied at RHICand the
LHC are typically modeled assuming rapid ther-malization within a
time scale of 1–2 fm/c [1]. The sub-sequent longitudinal and
transverse expansion of the cre-ated quark-gluon plasma (QGP) is
then described by vis-cous relativistic hydrodynamics [2]. In this
picture thebulk of the entropy is created during the
thermalizationprocess and the later stages of the evolution add
rela-tively little [3]. By correctly accounting for the
entropyproduction in different stages of the collisions, one
cantherefore relate the measurable final-state particle
multi-plicities to the properties of system, e.g. initial
tempera-ture, at the earlier stages of the collisions.
Two different methods are frequently used to estimatethe total
produced entropy in nuclear collisions. In thefirst method,
pioneered by Pal and Pratt, one calculatesthe entropy based on
transverse momentum spectra ofdifferent particle species and their
source radii as deter-mined from Hanbury Brown-Twiss correlations
[4]. Theoriginal paper analyzed data from
√sNN = 130 GeV Au–
Au collisions and is still the basis of many entropy
esti-mations at other energies [3]. The second method usesthe
entropy per hadron as calculated in a hadron reso-nance gas model
to translate the final-state multiplicitydN/dy per unit of rapidity
to an entropy dS/dy [5, 6].Even though the estimate of the entropy
from the mea-sured multiplicity dNch/dη is relatively
straightforwardone finds quite different values for the conversion
fac-tor between the measured charged-particle multiplicitydNch/dη
and the entropy dS/dy in the literature [6–9].
This paper provides an up-to-date calculation of en-tropy
production in pp and Pb–Pb collisions at the LHCenergies and uses
state-of-the-art modeling of the QGP toreconstruct the initial
conditions at the earliest momentsin the collision. In Sec. II we
recap the method of Ref. [4],
which we use in Sec. III A and Sec. III B to calculate thetotal
produced entropy per rapidity, and the entropy perfinal-state
charged hadron S/Nch ≡ (dS/dy)/(dNch/dy)from the identified
particle spectra and femtoscopy datafor√s = 7 TeV pp and
√sNN = 2.76 TeV Pb–Pb col-
lisions at LHC [10–16]. In Sec. IV the result for theentropy per
particle is then compared to different esti-mates of the entropy
per hadron calculated in hadronresonance gas models at the chemical
freeze-out temper-ature of Tch ≈ 156 MeV [17]. Finally in Sec. V A
we usedifferent models of the QGP evolution to track
entropyproduction in different stages of the collisions and to
de-termine the initial temperature profile at τ = 1 fm/c.
II. ENTROPY FROM TRANSVERSEMOMENTUM SPECTRA AND HBT RADII
In this section we recap the entropy calculation fromphase-space
densities obtained from particle spectra andfemtoscopy [4].
Foundations for this method were laid inRefs. [18, 19]. The entropy
S for a given hadron speciesat the time of kinetic freeze-out is
calculated from thephase space density f(~p, ~r) according to
S = (2J + 1)
∫d3rd3p
(2π)3 [−f ln f ± (1± f) ln (1± f)]
(1)
where + is for bosons and the − for fermions. The factor2J + 1
is the spin degeneracy. The total entropy in thecollision is then
given by the sum of the entropies of theproduced hadrons species.
The integral in Eq. (1) can beevaluated using the series
expansion
± (1± f) ln (1± f) = f ± f2
2− f
3
6± f
4
12+ . . . . (2)
arX
iv:1
908.
0279
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[he
p-ph
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201
9
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2
Three-dimensional source radii measured through Han-bury
Brown-Twiss two-particle correlations [20] are usu-ally determined
in the longitudinally co-moving system(LCMS) in which the component
of the pair momentumalong the beam direction vanishes. The density
profileof the source in the LCMS is parametrized by a
three-dimensional Gaussian so that the phase space density canbe
written as
f(~p, ~r) = F(~p) exp(− x
2out
2R2out− x
2side
2R2side−
x2long2R2long
)(3)
with
F(~p) = (2π)3/2
2J + 1
d3N
d3p
1
RoutRsideRlong. (4)
The radii in Eqs. (3) and (4) are functions of the momen-tum
~p.
In many cases only the one-dimensional source radiusRinv, which
is determined in the pair rest frame (PRF),can be determined
experimentally owing to limited statis-tics. In Ref. [4] the
relation between Rinv in the PRF andthe three-dimensional radii in
the LCMS was assumed tobe
R3inv ≈ γRoutRsideRlong, (5)
where γ = mT/m ≡√m2 + p2T/m. This is also our stan-
dard assumption. In [21] and [22] the ALICE collabora-tion
reported values for both Rinv and Rout, Rside, Rlongobtained from
two-pion correlations in Pb–Pb collisionsat√sNN = 2.76 TeV and pp
collisions at
√s = 7 TeV,
respectively. From these results one can determine amore general
version of Eq. (5) of the form R3inv ≈h(γ)RoutRsideRlong with h(γ)
= αγ
β . Results for theentropy dS/dy obtained under this assumption
are givenin Appendix A.
Using Eq. (5) one arrives at
dS
dy=
∫dpT 2πpT E
d3N
d3p
(5
2− lnF ± F
25/2
− F2
2 · 35/2 ±F3
3 · 45/2)
(6)
with
F = 1m
(2π)3/2
2J + 1
1
R3inv(mT)E
d3N
d3p(7)
where m is the particle mass and + is for bosons and −for
fermions. Note that Eq. (6) includes the terms up tof4i /12 of the
Taylor expansion in Eq. (2).
Pions have the highest phase space density of the con-sidered
hadrons and the approximation made in Eq. (6)is better than 1% for
pions in central Pb–Pb collisionsat√sNN = 2.76 TeV. In pp
collisions at
√s = 7 TeV the
maximum pion phase space density F(pT) exceeds unity
at low pT rendering the series expansion in Eq. (2) unreli-able.
For pions in pp collisions we therefore approximatethe (1 + f) ln(1
+ f) term of Eq. (1) by a polynomial oforder 8. This gives an
approximate expression with nu-merical coefficients ai which is
also valid for values of Fobtained for pions in high-multiplicity
pp collisions:
dS
dy=
∫dpT 2πpT E
d3N
d3p
(5
2− lnF +
7∑i=0
aiF i). (8)
III. RESULTS
A. Entropy in Pb–Pb collisions at√sNN = 2.76TeV
We determine the entropy in Pb–Pb collisions at√sNN = 2.76 TeV
for the 10% most central collisions
considering as final-state hadrons the particles given inTable
I. The calculation uses transverse momentum spec-tra of π, K, p
[10], Λ [11], and Ξ, Ω [12] from the ALICEcollaboration as
experimental input. We also use HBTradii measured by ALICE
[23].
For the entropy determination the measured transversemomentum
spectra need to be extrapolated to pT = 0.To this end we fit
different functional forms to the pTspectra (Tsallis,
Bose-Einstein, exponential in transverse
mass mT =√p2T +m
2, Boltzmann, as defined in [10]).In the entropy calculation we
only use the extrapolationsin pT regions where data are not
available, otherwise weused the measured spectra. Differences of
the entropyestimate for different functional form are taken as a
con-tribution to the systematic uncertainty. We have checkedthat
the pT-integrated π, K, p multiplicities (dn/dy)y=0agree with the
values published in [10].
The one-dimensional invariant HBT radii Rinv are onlyavailable
for π, K, and p. When plotted as a function oftransverse mass mT
=
√m2 + p2T the Rinv values for
these particles do not fall on a common curve. How-ever, in [26]
it was shown that the HBT radii Rinv di-vided by ((
√γ + 2)/3)1/2 where γ = mT/m are approx-
imately a function of mT only. This empirical scalingfactor for
Rinv is related to the fact that for a three-dimensional Gaussian
parametrization of the source theone-dimensional source
distribution in general cannot bedescribed by a Gaussian (see
Appendix of [26]). Weuse this mT scaling of the scaled HBT radii to
obtainRinv(mT) for all considered particles. The bottom panelof
Fig. 1 shows parametrizations of the scaled HBT radiiwith a power
law function and with an exponential func-tion which provide
different extrapolation towards thepion mass. We propagate the
systematic uncertainties ofthe measured HBT radii as well as the
uncertainty relatedto the two different parametrizations to the
uncertaintyof the extracted entropy.
For the entropy calculation the particle species in Ta-ble I are
considered stable. The entropy carried by neu-trons, neutral kaons,
η, η′, and Σ baryons is estimated
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3
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4pT (GeV/c)
0
200
400
600
800
1000
12001 N e
vtd2
Ndp
Tdy
[(GeV
/c)
1 ]pions
0.25 0.50 0.75 1.00 1.25 1.50 1.75mT (GeV)
0
2
4
6
8
10
12
R inv
/((+
2)/3
)1/2 (
fm) power law
exppionskaonsprotons
FIG. 1. Transverse momentum spectrum of positive pions(top) and
scaled HBT radii Rinv (bottom) in 0–10% Pb–Pbcollisions at
√sNN = 2.76 TeV. A Tsallis function [24, 25]
is fitted to the spectrum to extrapolate to pT = 0.
Theone-dimensional HBT radii divided by ((
√γ + 2)/3)1/2 [26]
where γ = mT/m as a function of the transverse mass mTare
parametrized by a power law function αmβT and by anexponential
function a exp(−mT/b) + c.
based on measured species assuming that the entropy perparticle
is similar for particles with similar masses. Theentropy carried by
neutrons is assumed to be the sameas the entropy carried by
protons. The entropy associ-ated with neutral kaons and η mesons is
determined fromcharged kaons, the entropy of η′ from protons, and
theentropy of Σ baryons from Λ.
The yields of particles in Table I contain contributionsfrom
strong decays. To take into account mass differencesand to estimate
the contributions from strong decays tothe different particle
species we simulate particle decayswith the aid of Pythia 8.2 [27,
28]. To this end we gen-erate primary particles available in Pythia
8.2 with ratesproportional to equilibrium particle densities in a
nonin-teracting hadron gas [29, 30]:
n =
∞∑k=1
Tg
2π2(±)k+1k
m2K2
(km
T
)ekµ/T . (9)
Here g = 2J + 1 is the spin degeneracy factor and K2the modified
Bessel function of the second kind. The +
is for bosons and the − for fermions. For the chemicalpotential
we use µ = 0. For the temperature we takeT = 156 MeV as obtained
from statistical model fits toparticle yields measured at the LHC
[17]. We then sim-ulate strong and electromagnetic decays of the
primaryparticles. Particle ratios after decays are used to
esti-mate the entropy of unmeasured particles. In case of theη
meson we find that after decays the η/K+ ratio is 0.69while the
primary ratio is ηprim/K
+prim = 0.79. For the η
′
we find η′prim/pprim = 0.45 and η′/p = 0.25 after decays.
The primary Σ−prim/Λprim ratio is about 0.66. The en-tropy
carried by the Σ baryons is derived from the ratiosΣ−/Λ ≈ 0.26 and
Σ0/Λ ≈ 0.27 after decays.
The η, η′ mesons and Σ0 baryons decay electromagnet-ically.
Decay products from these decays (η, η′ → pions,and Σ0 → Λγ) are
not subtracted from the experimen-tally determined particle
spectra. As η, η′, and Σ areconsidered stable in the entropy
calculation (see Table I)we correct for this feeddown
contributions. In the par-ticle decay simulation described above we
determine thefeeddown fraction
Rfd(X → Y ) =number of Y from X
total number of Y(10)
and find Rfd(η → π+) = 3.6%, Rfd(η′ → π+) = 1.2%,Rfd(η
′ → η) = 5.9%, and Rfd(Σ0 → Λ) = 27.0%.The entropies for the
particle species considered sta-
ble are summarized in Table I. These values representthe average
of the entropies obtained for the power lawand the exponential
parametrization of the scaled invari-ant HBT radii. In both cases
the Tsallis function wasused to extrapolate the measured transverse
momentumspectra to pT = 0. We considered the uncertainties ofthe
measured transverse momentum spectra, the choiceof the
parametrization of the pT spectra, the uncertain-ties of the
measured HBT radii, and the choice of theparametrization of the HBT
radii as a function of mTas sources of systematic uncertainties.
The estimated to-tal entropy in 0–10% most central Pb–Pb collisions
at√sNN = 2.76 TeV is 11 335 ± 1188. The uncertainty of
the estimated entropy is the quadratic sum of the un-certainties
related to the transverse momentum spectra(σspectra = 629) and
invariant HBT radii (σRinv = 1007).
It is interesting to calculate the entropy per chargedhadron in
the final state from the total entropy. From[31] we obtain for
0–10% most central Pb–Pb collisions at√sNN = 2.76 TeV a
charged-particle multiplicity at mid-
rapidity of dNch/dη = 1448±54. From our parametriza-tions of the
pion, kaon, and proton spectra we find aJacobian for the change of
variables from pseudorapid-ity to rapidity of (dNch/dy)/(dNch/dη) =
1.162± 0.008.This yields an entropy per charged hadron in the
finalstate of S/Nch = 6.7± 0.8.
In the paper by Pratt and Pal the entropy was de-termined for
the 11% most central Au–Au collisions ata center-of-mass energy
of
√sNN = 130 GeV. The to-
tal entropy per unit of rapidity around midrapidity was
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4
TABLE I. Estimate of the entropy (dS/dy)y=0 for 0–10%most
central Pb–Pb collisions at
√sNN = 2.76 TeV. The ta-
ble shows the hadrons considered as stable final-state
particlesand their contribution to the total entropy.
particle (dS/dy)one statey=0 factor (dS/dy)totaly=0
π 2182 3 6546
K 605 4 2420
η 399 1 399
η′ 66 1 66
p 266 2 532
n 266 2 532
Λ 160 2 320
Σ 58 6 348
Ξ 39 4 156
Ω 8 2 16
total 11 335
found to be dS/dy = 4451 with an estimated uncer-tainty of 10%.
Using dNch/dy = 536 ± 21 from [32]and (dNch/dy)/(dNch/dη) ≈ 1.15 we
find an entropyper charged particle of S/Nch ≡ (dS/dy)/(dNch/dy)
=7.2± 0.8. This value for Au–Au collisions at a center-of-mass
energy of
√sNN = 130 GeV agrees with the value of
S/Nch = 6.7 ± 0.8 we obtain for the LHC energy in thispaper.
B. Entropy in pp collisions at√s = 7TeV
Not only in high-energy nucleus-nucleus collisions butalso in
proton-proton and proton-nucleus collisions trans-verse momentum
spectra and azimuthal distributions ofproduced particles can be
modeled assuming a hydrody-namic evolution of the created matter
[33–35]. This pro-vides a motivation to determine the entropy dS/dy
withthe Pal-Pratt method also in pp collisions. Moreover,
theexperimentally determination of the entropy is of interestin the
context of models which are based on entropy pro-ductions
mechanisms not related to particle scatterings(see, e.g., [36,
37]). Here we focus on minimum bias andhigh-multiplicity pp
collisions at
√s = 7 TeV.
Transverse momentum spectra for both minimum biascollisions (π,
K, p [13], Λ [11], and Ξ, Ω [14]) and high-multiplicity pp
collisions (π, K, p [15], Λ, Ξ, Ω [16]) aretaken from the ALICE
experiment. The high-multiplicitysample (class I in [16] and [15])
roughly corresponds tothe 0-1% percentile of the multiplicity
distribution mea-sured at forward and backward pseudorapidities.
HBTradii are taken from [22]. In minimum bias pp collisionsthere is
little dependence of Rinv on transverse mass anda constant value
Rinv = 1.1± 0.1 fm is assumed. For thehigh-multiplicity sample mT
scaling of Rinv is assumedand the same power law and exponential
functional formsas in the Pb–Pb analysis are fit to the data from
[22](Nch = 42–51 class in [22]). Taking into account the un-
certainty of associating the multiplicity class in [15, 16]with
the one in [22] we assume an uncertainty of Rinv forthe
high-multiplicity sample of about 10%.
With the same assumptions for the contribution ofunobserved
particles and feeddown as in Pb–Pb colli-sions we obtain dS/dy|MB =
37.8 ± 3.7 in minimumbias (MB) collisions and dS/dy|HM = 135.7 ±
17.9 forthe high-multiplicity (HM) sample. The contribution ofthe
different particles species to the total entropy aregiven in Tables
II and III. With dNch/dη = 6.0 ± 0.1[38] and (dNch/dy)/(dNch/dη) =
1.21 ± 0.01 for mini-mum bias pp collisions we obtain S/Nch|MB =
5.2 ± 0.5for the entropy per final-state charged particle. Forthe
high-multiplicity sample with dNch/dη = 21.3 ± 0.6[15] and
(dNch/dy)/(dNch/dη) = 1.19 ± 0.01 we findS/Nch|HM = 5.4± 0.7.
TABLE II. Estimate of the entropy (dS/dy)y=0 in minimumbias pp
collisions at
√s = 7 TeV
particle (dS/dy)one statey=0 factor (dS/dy)totaly=0
π 6.7 3 20.1
K 2.1 4 8.4
η 1.4 1 1.4
η′ 0.3 1 0.3
p 1.2 2 2.4
n 1.2 2 2.4
Λ 0.6 2 1.2
Σ 0.2 6 1.2
Ξ 0.1 4 0.4
Ω 0.01 2 0.02
total 37.8
TABLE III. Estimate of the entropy (dS/dy)y=0 in
high-multiplicity pp collisions (class I in [16] and [15])) at
√s =
7 TeV
particle (dS/dy)one statey=0 factor (dS/dy)totaly=0
π 23.8 3 71.4
K 7.5 4 30.0
η 4.9 1 4.9
η′ 1.0 1 1.0
p 4.2 2 8.4
n 4.2 2 8.4
Λ 2.3 2 4.6
Σ 0.8 6 4.8
Ξ 0.5 4 2.0
Ω 0.1 2 0.2
total 135.7
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5
IV. COMPARISONS TO STATISTICALMODELS
In order to compare the S/Nch value determined fromthe measured
final state particle spectra to calculationsin which particles
originate from a hadron resonance gasone needs to know the ratio
N/Nch of the total number ofprimary hadrons N(≡ Nprim) to the total
number of mea-sured charged hadrons in the final state Nch(≡
Nfinalch ).The latter contains feed-down contributions from
strongand electromagnetic hadron decays. If only pions wereproduced
one would get N/Nch = 3/2. With the afore-mentioned Pythia 8.2
simulation and the list of stablehadrons implemented in Pythia
(again with hadron yieldsgiven by Eq. (9) for T = 156 MeV and µb =
0) we obtaina value of (N/Nch)Pythia = 1.14. In this calculation
par-ticles with a lifetime τ above 1 mm/c were considered sta-ble.
Using the implementation of the hadron resonancegas of ref. [39] we
find (N/Nch)TF = 1.09. In followingwe use N/Nch = 1.115 ± 0.03,
i.e., we take the averageof the two results as central value and
the difference as ameasure of the uncertainty.
In the simplest form of the description of a hadronresonance gas
the system is treated as a non-interactinggas of point-like hadrons
where hadronic resonances havezero width. The entropy density for a
primary hadronwith mass m at thermal equilibrium with temperatureT
and vanishing chemical potential µ = 0 is then givenby [30]
s =4gT 3
2π2
∞∑k=1
(±)k+1k4
[(km
T
)2K2
(km
T
)+
1
4
(km
T
)3K1
(km
T
)](11)
where + is for bosons and − for fermions. K1 and K2are modified
Bessel functions of the second kind. UsingEqs. (9) and (11) the
entropy per primary hadron in thethermal hadron resonance gas can
be calculated as
S/N =
∑i si∑i ni
(12)
where the index i denotes the different particles species.For
illustration, the entropy per hadron is shown in Fig-ure 2 as a
function of the upper limit on the mass for allparticles listed in
the particle data book [40].
More sophisticated implementations of the hadron res-onance gas
take the volume of the hadrons and the finitewidth of hadronic
resonances into account [17, 39, 41–48]. Some of these models
implement chemical non-equilibrium factors which we do not consider
here. Mod-els can also differ in the set of considered hadron
states.In the following we concentrate on the models by
Braun-Munzinger et al. [17] (“model 1”) and Vovchenko/Stöcker[39]
(“model 2”). The corresponding values for the en-tropy per primary
hadron S/N and the entropy per final
0.5 1.0 1.5 2.0 2.5 3.0m (GeV/c2)
3.54.04.55.05.56.06.57.07.5
S/N
FIG. 2. Entropy per primary hadron S/N for a non-interacting
thermal hadron resonance gas at a temperatureof T = 156 MeV as
given by Eq. (12) as a function of theupper mass limit for
particles listed in the particle data book[40]. The entropy per
hadron saturates for high upper masslimits at a value of S/N =
6.9.
TABLE IV. Entropy per primary hadron S/N at a tem-perature of T
= 156 MeV for different hadron resonance gasmodels. The entropy per
final state charged hadron is cal-culated from S/N by multiplying
with the factor N/Nch =1.115 ± 0.03. The volume correction of model
2 is thebased on the Quantum van der Waals model. Within 1–2σ the
S/Nch values of these models agree with the valueof S/Nch = 6.7±
0.8 obtained from data.Model S/N S/Nch
Simple HRG (Eq. (12)) 6.9 7.7± 0.2Model 1 (Braun-Munzinger et
al. [17, 49])
without volume correction 7.3 8.1± 0.2with volume correction 7.6
8.5± 0.2Model 2 (Vovchenko, Stöcker [39])
ideal 6.9 7.7± 0.2with volume correction, zero width 7.2 8.1±
0.2with volume correction, finite width 7.1 7.9± 0.2
state charged hadron S/Nch are given in Tab. IV. TheS/Nch values
for these models are somewhat larger thanthe measured value of
S/Nch = 6.7± 0.8, but the devia-tions are not larger than 1–2σ. We
note here that the twoapproaches calculate slightly different
quantities. Our es-timate is based on the non-equilibrium
distributions of afew final state hadrons, while Eq. (12) sums the
entropycontributions of all primary hadrons in a thermal
statebefore the decays. Although on general grounds we ex-pect the
total entropy to increase during the decays andre-scatterings in
the hadronic phase, there are some de-cay products, e.g. photons,
which are not included in ourcurrent entropy count. Accounting for
such differencesbetween the two approaches might bring the
estimatescloser together.
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6
V. INITIAL CONDITIONS AND ENTROPYPRODUCTION
A. Pb–Pb collisions
The entropy in nuclear collisions, which we calcu-lated in
previous sections, is not created instantaneously,but rather the
entropy production takes place in sev-eral stages in nuclear
collisions [3]. In this section wewill use different models to
describe the boost invariantexpansion and, in particular, to
determine the averageinitial conditions in 0–10% most central Pb–Pb
collisionsat√sNN = 2.76 TeV at time τ0 = 1 fm.
First, we can make an estimate of the initial tempera-ture T
(τ0) by simplifying the early time evolution of theQGP. As it was
done in the original work by Bjorken [50],we will consider a
one-dimensional boost-invariant ex-pansion of homogeneous plasma
with the transverse ex-tent determined by the geometry of the
nuclei, i.e. thetransverse area A. The time evolution of the energy
den-sity is then solely a function of proper time τ and theassumed
constituent equation for the longitudinal pres-sure PL = τ
2T ηη. For collisionless gas with PL = 0,the energy density
falls as e ∝ τ−1, i.e. energy per ra-pidity dE/dy = eτA is constant
[50]. For an ideal fluidwith equation of state p = c2se, where cs
is the (con-stant) speed of sound, the energy density decreases
faster,
e ∝ τ−1−c2s , because of the work done against the lon-gitudinal
pressure [50–52]. However, total entropy perrapidity dS/dy = sτA
stays constant irrespective of theequation of state as long as
viscous dissipation can beneglected [50–53]. Below we compare the
initial tem-perature estimates at time τ0 = 1 fm obtained from
thefinal entropy dS/dy and energy dE/dy using, respec-tively,
isentropic ideal fluid and free-streaming evolutionsof QGP 1.
Assuming that the subsequent near ideal hydrody-namic evolution
does not change the total entropy perrapidity dS/dy (which is also
true for free-streaming ex-pansion) the initial entropy density is
equal to
s(τ0) =1
Aτ0
dS
dy
∣∣∣∣y=0
. (13)
Using for the transverse area A = πR2Pb with RPb =6.62 fm [55,
56] gives an initial entropy density for the 0–10% most central
Pb–Pb collisions at
√sNN = 2.76 TeV
s(τ0) = 82.3 fm−3. (14)
According to the lattice QCD equation of state [57, 58],this
corresponds to a temperature
T (τ0) ≈ 340 MeV. (15)
1 For initial energy estimates at even earlier times, see a
recentpublication [54], where a generalised constitutive equation
of ahydrodynamic attractor was considered.
The transverse energy at midrapidity for the 10% mostPb–Pb
collisions at
√sNN = 2.76 TeV was measured to
be dET/dy ≈ 1910 GeV [59, 60]. Using again A = πR2Pbwith RPb =
6.62 fm as an approximation for the trans-verse overlap area the
initial energy density can be cal-culated according to the Bjorken
formula [50]
e(τ0) =1
Aτ0
dETdy
∣∣∣∣y=0
, (16)
which gives an energy density e(τ0) ≈ 13.9 GeV/fm3.This would
correspond to much lower initial tempera-ture T (τ0) ≈ 305 MeV.
This is because Eq. (16) is de-rived under the assumption of a
constant energy per ra-pidity [50]. This holds for a free-streaming
(or pressure-less) expansion, but in hydrodynamics the system
coolsdown faster due to work done against the longitudinalpressure.
Taking τf = RPb as a rough estimate for thelifetime of the
fireball, ideal hydrodynamics predicts an
(τf/τ0)13 ≈ 1.9 times larger initial energy density, which
would revise the initial temperature estimate upwards toT (τ0) ≈
355 MeV and closer to the value we obtainedfrom the entropy
method.
Instead of assuming a constant entropy density in a col-lision,
it is more realistic to use an entropy density profiles(τ, ~r),
where ~r is a two-dimensional vector in the trans-verse plane (we
still assume boost-invariance in the lon-gitudinal direction). We
will employ the two-componentoptical Glauber model to construct the
transverse profileof initial entropy density [61]. In this model
the initialentropy is proportional to the participant nucleon
num-ber and the number of binary collisions. For a collision
at impact parameter ~b, the entropy profile is then
s(τi, ~r;~b) =κsτi
(1− α
2
dNpart(~r,~b)
d2r+ α
dN coll(~r,~b)
d2r
),
(17)where κs(1 − α)/2 is entropy per rapidity per partici-pant
and κsα is entropy per rapidity per binary collision.The number
densities are calculated using the nucleon-nucleon thickness
functions (see Appendix B for details),and the value α = 0.128
reproduces centrality depen-dence of multiplicity [56]. We average
over the impact
parameter |~b| ≤ 4.94 fm to produce entropy profile
corre-sponding to 0-10% centrality bin of Pb–Pb collisions at√sNN =
2.76 TeV [56]. The overall normalization fac-
tor κs is adjusted to reproduce the final state entropyestimated
in Sec. III A, which depends on the expansionmodel.
To simulate the evolution and entropy productionin
nucleus-nucleus collisions we employ two recentlydeveloped models:
kinetic pre-equilibrium propagatorKøMPøST [62, 63], and viscous
relativistic hydrodynam-ics code FluiduM [64]2. For simplicity we
employ a con-
2 We neglect the entropy production in the hadronic phase
andmatch the entropy on the freeze-out surface.
-
7
−10 −5 0 5 10x (fm)
0
2
4
6
8
10
12τ(fm)
0
0.1
0.2
0.3
0.4
0.5T (GeV)
1
2
30-10% PbPb
FIG. 3. Temperature in hydrodynamic evolution of an av-eraged
0–10% Pb–Pb event at
√sNN = 2.76 TeV. Lines 1
and 3 are the freeze-out Tfo = 156 MeV contour, whereasline 2
indicates a constant time contour in the QGP phase,i.e. T (τ, ~r)
> Tfo. Initial conditions for hydrodynamics atτhydro = 0.6 fm
were provided by KøMPøST pre-equilibriumevolution from the starting
time of τEKT = 0.1 fm.
stant value of specific shear viscosity η/s and vanishingbulk
viscosity ζ/s throughout the evolution.
KøMPøST uses linear response functions obtainedfrom QCD kinetic
theory3 to propagate and equilibratethe highly anisotropic initial
energy momentum tensor,which can be specified at an early starting
time τEKT =0.1 fm. We specify the initial energy-momentum
tensorprofile to be4
Tµν(τEKT, ~r) = e(τEKT, ~r) diag(1, 12 ,
12 , 0). (18)
At the end of KøMPøST evolution all components of theenergy
momentum tensor, the energy density, transverseflow and the
shear-stress components, are passed to thehydrodynamic model at
fixed time τhydro = 0.6 fm.
The FluiduM package solves the linearized Israel-Stewart type
hydrodynamic equations around an az-imuthally symmetric background
profile. In this work wepropagate the radial background profile
until the freeze-out condition is met, which we define by a
constantfreeze-out temperature Tfo = 156 MeV. Above this
tem-perature the equation of state is that of lattice QCD
[58].Unless otherwise stated, we use a constant specific
shearviscosity η/s = 0.08 and vanishing bulk viscosity ζ/s = 0.
3 The current implementation of KøMPøST uses results of pureglue
simulations, but recent calculations with full QCD de-grees of
freedom indicate that the evolution of the total energy-momentum
tensor will not be significantly altered by chemicalequilibration
[65, 66].
4 As a purely practical tool we use a lattice equation of state
toconvert entropy density profile obtained from the Glauber
modelEq. (17) to the energy density needed to initialize
KøMPøST,even though the system at τEKT = 0.1 fm is not in
thermody-namic equilibrium.
We start by showing the temperature evolution in thehydrodynamic
phase in Fig. 3. The combined solid anddotted white lines represent
the freeze-out line at Tfo =156 MeV. The dashed horizontal line
indicates the spatialslice of the fireball at some fixed time τ and
above thefreeze-out temperature. We now can define entropy as
anintegral of the entropy current suµ over a hypersurface Σiwhere
Σi is one or more of the contours shown in Fig. 3.We define the
total entropy in the QGP state at time τas the integral over the
contour 2:
S(τ)|QGP ≡∫
Σ2(τ)
dσµsuµ. (19)
To include the entropy outflow from the QGP because offreeze-out
we also define entropy on the contours Σ1(τ
′ <τ) + Σ2(τ):
S(τ)|QGP+freeze-out ≡∫
Σ1(τ ′ τ). In Fig. 4(a)we show the time dependence of entropy
per rapidityin the QGP phase (yellow line) and including freeze-out
outflow (green line) in hydrodynamically expandingplasma. The solid
lines are for the simulation with withη/s = 0.08 and dashed lines
correspond to η/s = 0.16.In both cases the initial entropy profile,
Eq. (17), is ad-justed so that after the pre-equilibrium (KøMPøST)
andhydrodynamic (FluiduM) evolution the final entropy onthe
freeze-out surface is equal to dS/dy = 11 335 esti-mated in Sec.
III A. We see that at early times entropyis produced rapidly, but
there is little entropy outflowthrough the freeze-out surface. At τ
≈ 2 fm the entropyin the hot QGP phase starts to drop because
matter iscrossing the freeze-out surface and at τ ≈ 10 fm there
isno hot QGP phase left.
Here we note that the early time viscous entropy pro-duction in
the hydrodynamic phase depends strongly onthe initialization of the
shear-stress tensor. In this workwe use the pre-equilibrium
propagator KøMPøST, whichprovides all components of energy-momentum
tensor athydrodynamic starting time and the shear-stress
tensorapproximately satisfies the Navier-Strokes
constitutiveequations [62, 63]. We determine that for evolution
withη/s = 0.08 the entropy per rapidity at time τ0 = 1.0 fmis ≈ 95%
of the final entropy on the freeze-out. For twicelarger shear
viscosity the entropy production doubles andto produce the same
final entropy we need only ≈ 90%at τ0 = 1.0 fm. Such entropy
production is neglected inthe naive estimate of Eq. (13).
Analogously to entropy, we use the same contours todefine energy
in the collision, that is, as integrals of theenergy current euµ.
In Fig. 4(b) we show the energyper rapidity in different phases of
the collision. We con-firm that the energy per rapidity decreases
rapidly in the
-
8
0
2000
4000
6000
8000
10000
12000
0 1 2 3 4 5 6 7 8 9 10
0-10% PbPb
dS/dy
τ (fm)
η/s = 0.08QGP T > 156MeV
QGP+freeze-outη/s = 0.16
(a)
0
1000
2000
3000
4000
5000
0 1 2 3 4 5 6 7 8 9 10
0-10% PbPb
dE/dy(G
eV)
τ (fm)
η/s = 0.08QGP T > 156MeV
QGP+freeze-outKøMPøST
(b)
FIG. 4. (a) Entropy per rapidity in viscous hydrodynamic
expansion with specific shear viscosity η/s = 0.08 for central√sNN
= 2.76 TeV Pb–Pb collisions (centrality class 0-10%) as a function
of time of contour 2 in Fig. 3. The yellow line
corresponds to entropy in the QGP phase (T (τ, r) > Tfo)
(contour 2 in Fig. 3), whereas the green line shows the total
cumulativeentropy (contour 1 + 2 in Fig. 3). Dashed red lines show
the corresponding result for a simulation with η/s = 0.16. In
bothcases the initial conditions, i.e. parameter κs in Eq. (17),
were tuned to reproduce the final freeze-out entropy dS/dy = 11
335after the pre-equilibrium and hydrodynamic evolutions. (b)
Analogous plot for energy per rapidity in hydrodynamic
expansionwith η/s = 0.08. The additional points show energy per
rapidity in the pre-equilibrium stage.
0
50
100
150
200
−10 −5 0 5 10
0-10% PbPb
τs(fm
−2)
x (fm)
τ = 1.0 fmτ = 3.0 fmτ = 6.0 fm
(a)
0
0.1
0.2
0.3
0.4
0.5
−10 −5 0 5 10
0-10% PbPb
T(G
eV)
x (fm)
τ = 1.0 fmτ = 3.0 fmτ = 6.0 fm
(b)
FIG. 5. (a) Entropy density profile (multiplied by τ) in viscous
hydrodynamic simulation with η/s = 0.08 at times τ = 1, 3, 6 fm.The
black dotted square indicates the initial entropy density estimate
τ0s(τ0) = 82.3 fm
−2, see Eq. (13). (b) Temperature profileat τ = 1, 3, 6 fm. The
black dotted line corresponds to T = 340 MeV.
hydrodynamic phase and at τ0 = 1.0 fm is nearly twicelarger than
on the freeze-out surface and therefore inval-idating the naive
initial energy density estimates usingBjorken formula Eq. (16).
However we do note that themagnitude of the final energy per
rapidity in our event isbelow the measured value. In addition we
show points forthe energy per rapidity in the pre-hydro phase
simulatedby KøMPøST. Despite the large anisotropy in the
initialenergy-momentum tensor (T zz ≈ 0 initially), the energyper
rapidity is rapidly decreasing in this phase. We notethat at the
same time there is a significant entropy pro-duction in the kinetic
pre-equilibrium evolution [63].
Next in Fig. 5(a) we look at the transverse entropy den-sity
profile τs(τ, ~r) at different times τ = 1.0, 3.0, 6.0 fmin the
hydrodynamic evolution with η/s = 0.08. We seethat the profile
changes only little between 1 fm and 3 fm,which is because of an
approximate one dimensional ex-pansion and viscous entropy
production. At later timesthe profile expands radially and drops in
magnitude. Theblack-dotted line indicates the naive estimate of
entropydensity τ0s(τ0) = 82.3 fm
−2 for a disk-like profile withradius RPb = 6.62 fm, see Eq.
(14). Despite an over-estimation of the net entropy at τ0 = 1 fm,
the actualdensity at the center of entropy profile is twice
larger
-
9
−2 −1 0 1 2r (fm)
0
0.5
1
1.5
2
2.5τ(fm)
0
0.1
0.2
0.3
0.4
0.5T (GeV)
1
2
30-1% pp
FIG. 6. Temperature in hydrodynamic evolution of an aver-aged
0–1% pp event at
√s = 7 TeV. Lines 1 and 3 are the
freeze-out Tfo = 156 MeV contour, whereas line 2 indicates
aconstant time contour in the QGP phase, i.e. T (τ, ~r) >
Tfo.Initial conditions for hydrodynamics at τhydro = 0.4 fm
wereprovided by KøMPøST pre-equilibrium evolution from thestarting
time of τEKT = 0.1 fm.
than the naive estimate. Correspondingly, the
transversetemperature profile at τ0 = 1 fm, shown in Fig. 5(b),
islarger than the simple estimate and can reach 400 MeVin the
center of the fireball.
B. Central pp collisions
In this section we present a similar analysis of en-tropy
production in ultra-central pp collisions. Becauseof much smaller
initial size, the QGP fireball (if created),has a much shorter
life-time than the central Pb–Pb col-lisions. This should enhance
the relative role of the pre-equilibrium physics of QGP
formation.
To model the initial entropy density in pp collision, weuse a
Gaussian parametrization of the transverse entropydistribution
s(τ0, ~r) =κs
τ02πσ2e−
r2
2σ2 (21)
with a width σ = 0.6 fm, as used in other parametriza-tions
[67]. We use a fixed value of η/s = 0.08 and, inview of the range
of applicability of the linearized pre-equilibrium propagator, we
use KøMPøST for a shortertime from τEKT = 0.1 fm to τhydro = 0.4
fm.
First we show the temperature evolution in Fig. 6 andindicate
the freeze-out contour (lines 1 and 3). We notethat because of the
compact initial size, the transverseexpansion is so explosive that
the center of the fireballactually freezes-out before the edges
(similar results werefound in Refs. [68, 69]). Next in Fig. 7(a) we
show the en-tropy evolution in the QGP phase and together with
theoutflow from through the freeze-out surface. In a smallersystem,
the radial flow builds up faster and the QGP
and the combined QGP+freeze-out surface contributionsstarts to
deviate early. This does not capture the entropywhich already left
T > Tfo region in the KøMPøST phase,but for the early hydro
starting time τhydro, this fractionis small. We see that as a
fireball of QGP ultra centralpp collisions have a lifetime just
above τ = 2 fm. There-fore the τ0 = 1 fm reference time is no
longer adequatetime to discuss the “initial conditions” in such
collisions.Next, in Fig. 7(b) we show the energy per rapidity in
thehydrodynamic and pre-equilibrium stages. Here againwe see that
energy per rapidity decreases more rapidly incomparison of entropy
production.
For the transversely resolved picture of entropy andtemperature
profiles, we supply figures Figs. 8(a) and8(b) correspondingly. At
τ0 = 1 fm the maximum en-tropy density is much smaller than in
0-10% centralityPb–Pb collisions and only at τ = 0.5 fm the
temperatureat the center reaches above T = 300 MeV.
VI. SUMMARY AND CONCLUSIONS
We provide independent determination of the final-state entropy
dS/dy in
√s = 7 TeV pp and
√sNN =
2.76 TeV Pb–Pb collisions from the final phase space den-sity
calculated from the experimental data of identifiedparticle spectra
and HBT radii. In addition, we havecalculated the entropy per
final-state charged hadron(dS/dy)/(dNch/dy) in different collision
systems. Wefind the following values for pp and Pb–Pb
collisions:
system dS/dy (dS/dy)/(dNch/dy)
Pb–Pb, 0–10% 11 335± 1188 6.7± 0.8pp minimum bias 37.8± 3.7 5.2±
0.5
pp high mult. 135.7± 17.9 5.4± 0.7
We compare our results for (dS/dy)/(dNch/dy) ratiobased on
experimental data, to the values obtained fromthe statistical
hadron resonance gas model at the chem-ical freeze-out temperature
of Tch = 156 MeV. For the0–10% most central Pb–Pb collisions
statistical modelvalues are systematically higher than our
estimate, butin agreement at the 1–2σ level. However the
measured(dS/dy)/(dNch/dy) values in minimum bias and
high-multiplicity pp collisions at
√s = 7 TeV are below the
theory predictions for a chemically equilibrated reso-nance gas
at Tch = 156 MeV, perhaps indicating thatfull chemical equilibrium
is not reached in these col-lisions. Here we note that,
interestingly, in pp colli-sions the estimated soft pion
phase-space density ex-ceeds unity. Finally, we have checked the
dependence ofour results on the relation between one-dimensional
andthree-dimensional HBT radii, Eq. (5), in Appendix A.We found no
significant change for Pb-Pb results, butpp entropy increased by
10%, which corresponds to 1σdeviation from the results above using
Eq. (5).
-
10
0
20
40
60
80
100
120
140
0 0.5 1 1.5 2 2.5
0-1% pp
dS/dy
τ (fm)
η/s = 0.08QGP T > 156MeV
QGP+freeze-out
(a)
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5
0-1% pp
dE/dy(G
eV)
τ (fm)
QGP T > 156MeVQGP+freeze-out
KøMPøST
(b)
FIG. 7. (a) Entropy per rapidity in viscous hydrodynamic
expansion with specific shear viscosity η/s = 0.08 for√s = 7 TeV
pp
collisions (0–1% collisions with the highest multiplicity) as a
function of time of contour 2 in Fig. 6. The yellow line
correspondsto entropy in the QGP phase (T (τ, r) > Tfo) (contour
2 in Fig. 6), whereas the green line shows the total cumulative
entropy(contour 1 + 2 in Fig. 6). The initial conditions, i.e.
parameter κs in Eq. (21), was tuned to reproduce the final
freeze-outentropy dS/dy = 135.7 after the pre-equilibrium and
hydrodynamic evolutions. (b) Analogous plot for energy per rapidity
inhydrodynamic expansion with η/s = 0.08. Additional points show
energy per rapidity in the pre-equilibrium stage.
0
10
20
30
40
50
60
−2 −1 0 1 2
0-1% pp
τs(fm
−2)
x (fm)
τ = 0.5 fmτ = 1.0 fmτ = 1.5 fm
(a)
0
0.1
0.2
0.3
0.4
0.5
−2 −1 0 1 2
0-1% pp
T(G
eV)
x (fm)
τ = 0.5 fmτ = 1.0 fmτ = 1.5 fm
(b)
FIG. 8. (a) Entropy density profile (multiplied by τ) in viscous
hydrodynamic simulation with η/s = 0.08 at times τ =0.5, 1.0, 1.5
fm. (b) Corresponding temperature profiles at τ = 0.5, 1.0, 1.5
fm.
The precise knowledge of the total produced entropyin heavy ion
collisions and the entropy per final-statecharged hadron is
important for constraining the bulkproperties of the initial-state
from the final state ob-servables [54, 62, 66]. To determine the
initial mediumproperties for high multiplicity pp and Pb–Pb
collisions,we performed simulations of averaged initial
conditionsstarting at τ0 = 0.1 fm/c with kinetic
pre-equilibriummodel KøMPøST [62, 63, 70] and viscous relativistic
hy-drodynamics code FluiduM [64]. Importantly, these cal-culations
take into account the produced entropy andwork done in both the
pre-equilibrium and hydrodynamicphases of the expansion [50–54]. We
find that for simu-lations with the specific shear viscosity value
η/s = 0.08
the initial pre-equilibrium energy per unity rapidity isabout
three times larger than at the final state in 0–10%most central
Pb–Pb collisions at
√sNN = 2.76 TeV, and
approximately twice larger in high multiplicity pp col-lisions
at
√s = 7 TeV. At the time τ = 1 fm/c, the
temperature in the center of the approximately equili-brated QGP
fireball is about T ≈ 400 MeV for Pb–Pband T ≈ 250 MeV for
high-multiplicity pp collision sys-tems. Finally, we note that in
our simulations of Pb–Pbcollisions with η/s = 0.08 only about 5% of
the total fi-nal entropy is produced after τ = 1 fm/c, meaning
thatmost of entropy production occurs in the
pre-equilibriumphase.
-
11
ACKNOWLEDGMENTS
The authors thank Stefan Floerchinger, EduardoGrossi, and Jorrit
Lion for sharing the FluiduM pack-age and useful discussions. A.M.
thanks Giuliano Gi-acalone, Oscar Garcia-Montero, Sören
Schlichting andDerek Teaney for helpful discussions. Moreover,
P.H.and K.R. thank Dariusz Miskowiec and Johanna Stachelfor
valuable discussions. This work is part of and sup-ported by the
DFG Collaborative Research Centre “SFB1225 (ISOQUANT)”.
Appendix A: Relation between the 1D HBT radiusRinv and the 3D
HBT radii Rout, Rside, Rlong
A transformation of the three-dimensional GaussianHBT radii from
the longitudinally co-moving system(LCMS) to the pair rest frame
(PRF) only affects the out-wards direction according to RPRFout =
γR
LCMSout where γ =
mT/m ≡√m2 + p2T/m. For a three-dimensional Gaus-
sian parametrization of the source, the
one-dimensionaldistribution in radial distance from the origin is
not ingeneral a one-dimensional Gaussian. Therefore, thereis no
exact formula relating the radius Rinv of the one-dimensional
Gaussian parametrization of the source andRout, Rside, Rlong
[26].
The maximum phase space density F for a more gen-eral version of
Eq. (5) of the form
R3inv ≈ h(γ)RoutRsideRlong (A1)is given by
F = h(γ)mT
(2π)3/2
2J + 1
1
R3invE
d3N
d3p. (A2)
In this section we use assume h(γ) = αγβ and use datafrom ALICE
[21, 22] to determine the values of α and βwhich best describe the
relation between the measuredone-dimensional radii Rinv and the
three-dimensionalradii Rout, Rside, Rlong. The results are shown in
Fig. 9.The best fit values for α and β turn out be
significantlydifferent between Pb–Pb and pp collisions. We then
usethese values to calculate maximum phase space densitiesaccording
to Eq. (A2). The corresponding results for theentropy and the
entropy per final-state charged particleare:
system dS/dy (dS/dy)/(dNch/dy)
Pb–Pb, 0–10% 11 534± 1188 6.9± 0.8pp minimum bias 41.7± 4.1 5.7±
0.6
pp high mult. 159.0± 19.8 6.3± 0.8
We note that for Pb–Pb collisions the entropy esti-mates
increases only very slightly. For pp collisions, thevalues for the
entropy are higher than our standard re-sults obtain using α = β =
1, but they agree at the 1σlevel.
0.3 0.4 0.5 0.6 0.7 0.8 0.9kT (GeV/c)
0
2
4
6
8
10
12
R inv
(fm
)
( Rout Rside Rlong)1/3
( Rout Rside Rlong)1/3, = 1.52, = 0.51data, Pb Pb, 0 5%
0.2 0.3 0.4 0.5 0.6kT (GeV/c)
0.000.250.500.751.001.251.501.752.00
R inv
(fm
)
( Rout Rside Rlong)1/3
( Rout Rside Rlong)1/3, = 0.48, = 1.18data, pp, min. bias (Nch =
12 16 in Aamodt 2011)
0.2 0.3 0.4 0.5 0.6kT (GeV/c)
0.000.250.500.751.001.251.501.752.00
R inv
(fm
)
( Rout Rside Rlong)1/3
( Rout Rside Rlong)1/3, = 0.54, = 0.93data, pp, high mult. (Nch
= 42 51 in Aamodt 2011)
FIG. 9. Comparison of measured one-dimensional HBTradii Rinv as
a function of the transverse pair momentumkT from the ALICE
collaboration and Rinv values calcu-lated from measured
three-dimensional HBT radii Rout, Rside,Rlong according to R
3inv ≈ αγβRoutRsideRlong. The bands
reflect the uncertainties of the measured Rinv values.
Thecalculated Rinv values are shown for α = β = 1 and forthe values
of α and β which fit the measured Rinv radiibest. Results are shown
for Pb–Pb collisions (centrality 0–5%) at
√sNN = 2.76 TeV [21] (upper panel), “minimum
bias” pp collisions (Nch = 12–16 in (Aamodt 2011: [22])) at√s =
7 TeV [22] (middle panel), and high-multiplicity pp col-
lisions (Nch = 42–51 in (Aamodt 2011: [22])) at√s = 7 TeV
[22] (lower panel).
-
12
Appendix B: Two-component Glauber model
In this section we recap the details of the two-component
Glauber model used to generate initial condi-tions for KøMPøST
evolution. The nuclear charge den-sity distribution of lead nuclei
is parametrized by Wood-Saxon distribution [61]
ρ(~r) = ρ01
1 + exp(|~r|−Ra
) , (B1)where for our purposes we will choose ρ0 such that the
to-tal volume integral of ρ is equal to the number of nucleonsNA =
208. Then ρ0 = 0.160391 fm
−3, R = 6.62 fm, anda = 0.546 fm. For Lorentz contracted nuclei,
the lon-gitudinal direction can be integrated out to obtain
thedensity per unit transverse area
T (~r⊥) =∫ ∞−∞
ρ (~r⊥, z) dz. (B2)
Then the collision probability of two nuclei with NA andNB
nucleons is given by
dN coll(~r,~b)
d2~r= TA(~r)TB(~r −~b)σNNinel , (B3)
where the radius is implicitly assumed to be in the trans-verse
plane and σNNinel = 6.4 fm
2 is the inelastic nucleon-nucleon cross-section. The number of
participant nucle-ons per transverse area is given by
dNpart(~r,~b)
d2~r= TA(~r)
[1−
(1− TB(~r −~b)σNNinel /NB
)NB]+ TB(~r −~b)
[1−
(1− TA(~r)σNNinel /NA
)NA].
(B4)
These probabilities are combined in the two-componentGlauber
model [56, 61] where α is an adjustable param-eter
(sτ)0 = κs
(1− α
2
dNpart(~r,~b)
d2r+ α
dN coll(~r,~b)
d2r
)(B5)
We use α = 0.128, which is the same value as in AL-ICE
publication [56], but with different parametrization
of Eq. (B5), namely α = 1−f1+f .
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Entropy production in pp and Pb-Pb collisions at energies
available at the CERN Large Hadron ColliderAbstractI IntroductionII
Entropy from transverse momentum spectra and HBT radiiIII ResultsA
Entropy in Pb–Pb collisions at sNN = 2.76TeVB Entropy in pp
collisions at s = 7TeV
IV Comparisons to statistical modelsV Initial conditions and
entropy productionA Pb–Pb collisionsB Central pp collisions
VI Summary and Conclusions AcknowledgmentsA Relation between the
1D HBT radius Rinv and the 3D HBT radii Rout, Rside, RlongB
Two-component Glauber model References