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

2v2

[he

p-ph

] 5

Dec

201

9

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

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

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

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.

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