Considerations on the suppression of charged particles inhigh energy heavy ion collisions

M.Petrovici,1, 2 A.Lindner,1, 2 and A.Pop1

1National Institute for Physics and Nuclear Engineering - IFIN-HHHadron Physics Department

Bucharest - Romania2Faculty of Physics, University of Bucharest

(Dated: December 14, 2021)

Experimental results related to charged particle suppression obtained at the Relativistic HeavyIon Collider (RHIC) at Brookhaven for Au-Au (Cu-Cu) collisions and at the Large Hadron Collider(LHC) at CERN for Pb-Pb (Xe-Xe) collisions are compiled in terms of RAA, RCP and the ratio ofthe pT spectra for each centrality to the pp minimum bias or to the peripheral one, each of themnormalised with the corresponding charged particle density 〈dNch/dη〉, namely RNAA and RNCP , as afunction of 〈Npart〉 and 〈dNch/dη〉. The studies are focused on a pT range in the region of maximumsuppression evidenced in the experiments. The RAA scaling as a function of 〈Npart〉 and 〈dNch/dη〉 isdiscussed. The core contribution toRAA is presented. The difference inRAA relative to the differencein particle density per unit of rapidity and unit of overlapping area (〈dN/dy〉/S⊥) and the Bjorkenenergy density times the interaction time (εBj · τ) support the model predictions. Considerationson the missing suppression in high charged particle multiplicity events for pp collisions at 7 TeV arepresented. RNCP for the same systems and energies evidence a linear scaling as a function of 〈Npart〉.While (1-RAA)/〈dN/dy〉 shows an exponential decrease with (〈dN/dy〉/S⊥)1/3, (1-RNAA)/〈dN/dy〉shows no dependence on (〈dN/dy〉/S⊥)1/3 for (〈dN/dy〉/S⊥)1/3 ≥2.1 part/fm2/3. The RCP andRNCP , for 4< pT <6 GeV/c, as a function of

√sNN measured at RHIC in Au-Au collisions and at

LHC in Pb-Pb collisions evidence a suppression enhancement from√sNN = 39 GeV up to 200 GeV,

followed by a saturation up to the highest energy of√sNN =5.02 TeV in Pb-Pb collisions. The

√sNN dependences of Rπ

0

AA and (RNAA)π0

in the same pT ranges and for the very central collisions

show the same trends. (1−Rπ0

AA)/(〈dN/dy〉/S⊥) evidences a maximum in the region of√sNN =62.4

GeV, followed by a decrease towards LHC energies.

I. INTRODUCTION

Detailed studies of different observables in heavy ioncollisions at RHIC [1–6] support theoretical predictionspioneered more than 40 years ago [7–10] that at largedensities and temperatures of the fireballs produced atthese energies, the matter is deconfined in its basic con-stituents, quarks and gluons. Obviously, such studiesare rather difficult given that the produced fireballs arehighly non-homogeneous, have a small size and are highlyunstable, their dynamical evolution playing an impor-tant role. One of the powerful tools used to diagnosethe properties of such a deconfined object is the studyof the energy loss of partons traversing the deconfinedmatter [11]. Within the QCD based models, the energyloss of a parton traversing a piece of deconfined matteris due to collisional or radiative processes. Collisionalenergy loss due to elastic parton collisions is expectedto scale linearly with the path length. Radiative energyloss occurs via inelastic processes where a hard partonradiates a gluon. Soft interactions of partons with thedeconfined medium can also induce gluon radiation [12].Radiative energy loss is expected to grow quadraticallywith the path length [13]. There are quite a few theoret-ical approaches for the description of the parton energyloss in expanding deconfined matter [14–23]. However, aproper description of the parton energy loss in the non-equilibrium expanding deconfined matter for the interme-

diate pT range remains a challenging task. The predictedsuppression at LHC energies turned out to be overesti-mated, once the experimental information became avail-able. A comprehensive analysis within CUJET/CIBJETrecently published [24], indicates, similar to the resultsof the JET Collaboration [21], a maximum in q/T 3 asa function of temperature around the critical tempera-ture (Tc) followed by a decrease towards temperaturesreached at LHC energies. A review of the charged parti-cle suppression in terms of the dependence on 〈Npart〉 and〈dNch/dη〉, the core-corona effect and the dependence onparticle density per unit of rapidity and unit of overlap-ping area (〈dN/dy〉/S⊥) or Bjorken energy density timesthe interaction time (εBj · τ) in Cu-Cu and Au-Au at thetop RHIC energy and Xe-Xe and Pb-Pb at LHC energies,is presented in Section II. Section III is dedicated to sim-ilar studies using 〈dNch/dη〉AA/〈dNch/dη〉pp instead of〈Nbin〉 in a model independent estimation of suppression,namely RNAA. In Section IV similar considerations for thecorresponding relative suppression, RCP and RNCP arepresented. (1-RAA)/〈dN/dy〉, (1-RNAA)/〈dN/dy〉 depen-

dences as a function of (〈dN/dy〉/S⊥)1/3 are presented inSection V. The collision energy dependence of RCP , RNCPfor charged particles and RAA, RNAA for π0 is discussedin Section VI. Conclusions are presented in Section VII.

arX

iv:2

008.

0068

6v2

[nu

cl-e

x] 5

Aug

202

0

2

II. RAA (5 < pT < 8 GeV/c) - 〈Npart〉 DEPENDENCE

Usually, the comparisons among different systems anddifferent collision energies in terms of RAA are done asa function of collision centrality. In Figure 1 the aver-age number of participating nucleons (〈Npart〉) [25–29]as a function of centrality obtained within the GlauberMonte Carlo (MC) approach [30–33] is represented. Ascan be seen, the difference in 〈Npart〉 at a given central-ity, for colliding systems with different sizes and incidentenergies, is increasing from peripheral towards centralcollisions.

0 20 40 60 80 100Centrality (%)

0

50

100

150

200

250

300

350

400>pa

rt<N

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 1. The average number of participating nu-cleons 〈Npart〉 as a function of centrality for Cu-Cu,Au-Au collisions at

√sNN = 200 GeV, for Xe-Xe at√

sNN = 5.44 TeV and for Pb-Pb at√sNN = 2.76 and 5.02

TeV.

Therefore, the behaviour of the suppression phenom-ena evidenced in relativistic heavy ion collisions, with thesystem size and collision energy, is better to be studied interms of RAA - 〈Npart〉 dependence. At

√sNN=200 GeV,

the same values of charged particles RAA as a functionof 〈Npart〉 for different bins in pT , for two very differ-ent colliding symmetric systems Au-Au [34] and Cu-Cu,were evidenced [25]. A similar scaling was also evidencedfor a lower collision energy, i.e.

√sNN=62.4 GeV [35].

Such a dependence was studied for pions and protons, for5< pT <8 GeV/c and 5< pT <6 GeV/c respectively, inCu-Cu and Au-Au collisions at

√sNN=200 GeV, by the

STAR Collaboration [36]. A good scaling of Rπ++π−

AA asa function of 〈Npart〉 for the two systems was evidenced.The PHENIX Collaboration has shown that in Au-Aucollisions at

√sNN=62.4 GeV and 200 GeV, the RAA of

π0 for pT > 6 GeV/c has the same value as a functionof 〈Npart〉 [37]. At LHC energies, the CMS Collabora-

0 50 100 150 200 250 300 350 400>part<N

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7>pa

rt>/

<Nsc

<N

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 2. Ratio of the average number of nucleons undergo-ing single collisions to the average number of participatingnucleons (〈Nsc〉/〈Npart〉), as a function of the average num-ber of participating nucleons (〈Npart〉) estimated within theGlauber MC model [30–33].

tion presented a similar scaling for Xe-Xe at√sNN=5.44

TeV and Pb-Pb at√sNN=5.02 TeV [38] with the remark

that RAA for Xe-Xe was obtained using the pT spectrumfrom minimum bias (MB) pp collisions at

√s=5.02 TeV.

Suppression studies at LHC energies up to very large pTvalues [39–41], for charged particles, evidence the maxi-mum suppression in the 5-8 GeV/c pT range. Althoughat RHIC energies the measured pT range is much smallerthan the region where RAA starts to increase, based onthe larger range in pT for π0 [37], one could concludethat the maximum suppression for different centralitiesis in the same range of pT , i.e. 5-8 GeV/c. This isthe main reason to focus the present considerations onsuppression phenomena in this range of transverse mo-menta. Another aspect worth being considered is the socalled core-corona effect [42–53] on the suppression esti-mate. The contribution to the pT spectra in A-A colli-sions from a nucleon suffering a single collision is similarwith the spectra from pp collisions at the same energy. Ifthis is the case, one should first correct the experimentalspectra of A-A collisions with the contribution comingfrom single binary collisions (corona) in order to obtainthe spectra of the core and estimate the correspondingRcoreAA . The percentage of nucleons that suffer more thana single collision (fcore) is reported in [29] for Au-Au col-lisions at

√sNN=200 GeV and for Pb-Pb collisions at√

sNN=2.76 and 5.02 TeV, respectively, while for Xe-Xeand Cu-Cu collisions is presented in Table I. The valuesof the overlapping area of the two nuclei and that corre-sponding to the core contribution, for the two systems,

3

0 50 100 150 200 250 300 350 400>part<N

00.10.20.30.40.50.60.70.80.9

1 <

8 G

eV/c

)T

(5 <

pA

AR

= 200 GeVNNsCu-Cu:

= 200 GeVNNs: PHENIXAu-Au

= 200 GeVNNs: STARAu-Au

0 50 100 150 200 250 300 350 400>core

part<N

00.10.20.30.40.50.60.70.80.9

1

< 8

GeV

/c)

T (5

< p

core

AA

R

= 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

a) b)

FIG. 3. RAA in the 5 < pT < 8 GeV/c region as a function of the average number of nucleons 〈Npart〉; a) experimental values,b) core contribution, see the text.

are also listed. The centrality dependence of the over-lapping area, Svar⊥ , used in the Bjorken energy densityestimate at LHC energies [54], is considered to be given

by ∼√〈σ2x〉〈σ2

y〉 − 〈σ2xy〉 [55], and its estimation and val-

ues for the other systems considered in the present studyare found in [29]. Figure 2 shows the average number ofnucleons undergoing single collisions relative to the aver-age number of participating nucleons (〈Nsc〉/〈Npart〉) asa function of the average number of wounded nucleons.

TABLE I: The percentage of nucleons that suffer more than asingle collision (fcore), the overlapping surface of the collidingnuclei (Svar⊥ ) and the overlapping surface corresponding to thecore contribution ((Svar⊥ )core) for Cu-Cu and Xe-Xe collidingsystems and corresponding collision energies and centralities.

System

√sNN

(GeV)Cen.(%)

fcoreSvar⊥

(fm2)(Svar⊥ )core

(fm2)

Cu-Cu 200

0-10 0.81±0.00 67.9±0.5 51.8±0.410-30 0.69±0.00 53.4±0.4 36.1±0.330-50 0.55±0.00 38.3±0.3 23.3±0.250-70 0.38±0.01 24.7±0.2 13.2±0.1

Xe-Xe 5440

0-5 0.93±0.00 124.1±0.6 105.3±0.55-10 0.89±0.00 114.9±0.6 91.3±0.510-20 0.84±0.00 100.6±0.5 74.9±0.420-30 0.78±0.00 83.7±0.5 57.9±0.330-40 0.72±0.00 69.3±0.4 44.7±0.240-50 0.65±0.00 57.1±0.3 34.2±0.250-60 0.57±0.00 45.9±0.3 25.5±0.160-70 0.47±0.01 35.4±0.2 18.2±0.170-80 0.36±0.01 24.8±0.2 10.9±0.1

As expected, 〈Nsc〉/〈Npart〉 has large values at low〈Npart〉, the system size and collision energy dependence

being rather small. With increasing 〈Npart〉 towardsvery central collisions, although the percentage of nucle-ons undergoing single collisions decreases, the differencebetween the various systems becomes significant. Us-ing the latest results obtained at RHIC for Cu-Cu andAu-Au collisions at

√sNN=200 GeV [25, 26, 34] and

at LHC for Xe-Xe at√sNN=5.44 TeV [27] and Pb-Pb

at√sNN=2.76 and 5.02 TeV [56], we obtained the val-

ues of RAA for 5 < pT < 8 GeV/c presented in Fig-ure 3a. RAA scales as a function of 〈Npart〉 at RHIC(√sNN=200 GeV) and LHC energies, separately, as it

was shown in the above mentioned papers. Within theerror bars, a slight difference, i.e. a bit larger suppres-sion is observed for central Cu-Cu and Xe-Xe collisionsrelative to Au-Au and Pb-Pb respectively, at the corre-sponding 〈Npart〉. The highlighted areas correspondingto the experimental values of RAA represent the system-atic uncertainties and the error bars are the statistical un-certainties, for the cases where both were available (Pb-Pb at

√sNN=2.76 and 5.02 TeV, Xe-Xe at

√sNN=5.44

TeV, Au-Au (PHENIX) and Cu-Cu at√sNN=200 GeV),

while in the case of Au-Au (STAR) the error bars rep-resent the square root of statistical and systematic un-certainties added in quadrature. The suppression due tothe core of the fireball RcoreAA :

RcoreAA =( d2NdηdpT

)cen,core

〈N corebin 〉 · (

d2NdηdpT

)pp,MB(1)

is presented in Figure 3b.The suppression enhances at peripheral collisions by ∼

20-25% and the values for the most central Cu-Cu and

4

0 50 100 150 200 250 300 350 400

>part<N

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 <

8 G

eV/c

)T

(5

< p

AA

R

FIG. 4. The same as Figure 3a with the shape of the overlap-ping area S⊥ at different values of 〈Npart〉 estimated withinthe Glauber MC approach.

Xe-Xe collisions are the same as for Au-Au and Pb-Pbcollisions, respectively, for the same 〈N core

part 〉. While thesuppression for Cu-Cu and Au-Au is the same at the samecollision energy (

√sNN=200 GeV), at LHC the suppres-

sion in Pb-Pb at√sNN=2.76 TeV is the same as at a col-

lision energy 1.82 times higher for Pb-Pb (√sNN=5.02

TeV) and 1.97 times higher for Xe-Xe (√sNN=5.44 TeV).

The small deviation evidenced in Xe-Xe collisions at lowvalues of 〈Npart〉 could be due to the way in which thecorrelation between centrality and 〈Npart〉 is estimatedin the standard Glauber MC approach [57]. As far as,within the error bars, there is a good agreement betweenPHENIX and STAR results, only the results of the STARCollaboration are presented from now on, thus avoidingoverloaded figures.

Considering the 〈Npart〉 dependence of the suppressionhas the advantage that at a given 〈Npart〉 the fireballtransverse area S⊥ is the same for the colliding systemsand collision energies in question [58], small deviationsbeing observed at very central collisions in Cu-Cu andXe-Xe relative to Au-Au and Pb-Pb [29], where the fire-ball shapes are closer to a circular geometry, see Figure4. At LHC energies, with a slight change in the offset(∼ 10fm2) the linear dependence of S⊥ on 〈Npart〉 hasthe same slope as at RHIC energy.

As it is known, all theoretical models predict an en-hancement of the suppression with increasing path lengthand energy density or temperature of the deconfinedmedium traversed by a parton [14–23]. In Figure 5 thesuppression in terms of (1-RAA) in the 5< pT <8 GeV/cregion for the colliding systems and energies under con-sideration, compared with the particle density per unit

0 50 100 150 200 250 300 350 400

>part<N

0.4−

0.2−

0

0.2

0.4

0.6

0.8

1

< 8

GeV

/c)

T (

5 <

pA

A1-

R

0

2

4

6

8

10

12

14

16

18

20 )-2

(f

m<d

N/d

y>/S

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 5. (1-RAA) (full symbols - left scale) and 〈dN/dy〉/S⊥(open symbols - right scale) as a function of 〈Npart〉.

of rapidity and unit of overlapping area (〈dN/dy〉/S⊥)as a function of 〈Npart〉, is represented. The dN/dy val-ues were estimated as in [29, 59]. In the case of Cu-Cuand Au-Au at

√sNN=200 GeV, for the same average

number of participants and 〈dN/dy〉/S⊥, the suppres-sion has the same value, increasing with 〈dN/dy〉/S⊥and size of overlapping area. As far as the suppres-sion in central Cu-Cu collisions is the same as in Au-Aucollisions at the corresponding 〈Npart〉, one could con-clude that the fireball shape plays a negligible role forthe same size of the overlapping area. For 〈Npart〉=200,the differences in 〈dN/dy〉/S⊥ for Pb-Pb at

√sNN=2.76,

5.02 TeV and for Xe-Xe at√sNN=5.44 TeV relative

to Au-Au at√sNN=200 GeV are 5.25±1, 6.77±1 and

7.89±1 (particles/fm2) while the differences in (1-RAA)are 0.10±0.03, 0.11±0.03 and 0.11±0.03. This is a clearevidence of a suppression saturation at LHC energies.For central Au-Au collisions, i.e. 〈Npart〉=350, the differ-ence in 〈dN/dy〉/S⊥ between Pb-Pb at

√sNN=2.76 TeV

and Au-Au at√sNN=200 GeV is 7±1 (particles/fm2)

while the difference in (1-RAA) is 0.082±0.03. With aparton energy loss in the deconfined medium given by[22, 60]:

dE

dx= −k · x · T 3 (2)

where k is the jet-medium coupling, x the path length, Tthe temperature and with the assumption that x2 ∼ S⊥and T 3 ∼ 〈dN/dy〉/S⊥, kLHC ' (0.48 ± 0.03) · kRHICis obtained. Obviously, the hydrodynamic expansionof the deconfined matter traversed by the parton playsa role in the estimated final suppression. Using the√〈dN/dy〉/S⊥ scaling of the average transverse flow ve-

locity, 〈βT 〉 reported in Ref. [29], for the geometricalscaling variable corresponding to the particle densities

5

0 50 100 150 200 250 300 350 400>part<N

0.2−

0

0.2

0.4

0.6

0.8

1A

-A)

AA

- (1-

RPb

Pb: 5

.02

TeV

)A

A(1

-R

2−

0

2

4

6

8

10

)-2

- A

-A (f

mPb

Pb: 5

.02

TeV

<d

N/d

y>/S

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au:

0 50 100 150 200 250 300 350 400>part<N

0.2−

0

0.2

0.4

0.6

0.8

1

A-A

)A

A- (

1-R

PbPb

: 5.0

2 Te

V)

AA

(1-R

2−

0

2

4

6

8

10 c)2 -

A-A

(GeV

/fmPb

Pb: 5

.02

TeV

τBJε

= 2.76 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe: a) b)

FIG. 6. The difference between the suppression in Pb-Pb at√sNN=5.02 TeV and the suppression in Pb-Pb at

√sNN=2.76

TeV, in Xe-Xe at√sNN = 5.44 TeV, in Au-Au and Cu-Cu at

√sNN=200 GeV (full symbols). The corresponding differences in

particle density per unit of rapidity and unit of overlapping area 〈dN/dy〉/S⊥ (Figure 6a - open symbols) and Bjorken energydensity times the interaction time εBj · τ (Figure 6b - open symbols) at the corresponding collision energies can be followedusing the scales on the right sides.

0 500 1000 1500 2000>η/dch<dN

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

< 8

GeV

/c)

T (

5 <

pA

AR

0 500 1000 1500 2000core>η/dch<dN

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

< 8

GeV

/c)

T (

5 <

pco

reA

AR

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 7. RAA as a function of charged particle density per unit of pseudorapidity, 〈dNch/dη〉, for the same systems and collisionenergies as in Figure 4; a) experimental values; b) the core contribution to RAA and 〈dNch/dη〉.

used before for the kLHC/kRHIC estimation, a ratio〈βT 〉LHC/〈βT 〉RHIC '1.09±0.08 is obtained. This couldbe one of the reasons leading to lower values of the jet-medium coupling in Pb-Pb collisions, but not enough toexplain the large difference between RHIC and LHC en-

ergies. In Figure 6 the difference between the suppres-sion in Pb-Pb at

√sNN=5.02 TeV and the suppression

in Au-Au and Cu-Cu collisions at√sNN=200 GeV, Pb-

Pb collisions at√sNN=2.76 TeV and Xe-Xe collisions

at√sNN=5.44 TeV is represented. The corresponding

6

differences in particle density per unit of rapidity andunit of overlapping area, 〈dN/dy〉/S⊥ (Figure 6a) andBjorken energy density times the interaction time, εBj ·τ(Figure 6b) are also represented with the correspondingscales on the right side of the figures.

The Bjorken energy density times the interaction timevalues are estimated based on [61]:

εBj · τ =dETdy· 1

S⊥(3)

where ET is the total transverse energy and S⊥ representsthe overlapping area of the colliding nuclei. The totaltransverse energy per unit of rapidity can be estimatedas follows:

• RHIC√sNN=200 GeV:

dETdy∼ 3

2

(〈mT 〉〈

dN

dy〉)π±

+ 2

(〈mT 〉〈

dN

dy〉)K±,p,p

(4)

• LHC energies:

dETdy∼ 3

2

(〈mT 〉〈

dN

dy〉)π±

+ 2

(〈mT 〉〈

dN

dy〉)K±,p,p,Ξ−,Ξ+

+

(〈mT 〉〈

dN

dy〉)

Λ,Λ,Ω−,Ω+

(5)

The input data used in the estimation of the Bjorkenenergy density times the interaction time are reported in[29, 62–69] and in Table I.

Within the error bars, the suppression in Pb-Pbcollisions at

√sNN=2.76 TeV is the same with the

one corresponding to√sNN=5.02 TeV for all values

of 〈Npart〉, although the difference in 〈dN/dy〉/S⊥ orin εBj · τ increases from 0.88±0.33 particles/fm2 to1.95±0.54 particles/fm2 and from 0.71±0.32 GeV/fm2cto 2.44±0.81 GeV/fm2c, respectively, from the low(〈Npart〉=50) to the highest values of 〈Npart〉. The differ-ence between the suppression in Pb-Pb at

√sNN=5.02

TeV and Au-Au at√sNN=200 GeV decreases from

about 0.27±0.25 to 0.08±0.02 with 〈Npart〉 while thedifferences in 〈dN/dy〉/S⊥ and εBj · τ increase from2.63±0.29 particles/fm2 and 2.13±0.28 GeV/fm2c to8.9±0.43 particles/fm2 and 8.2±0.8 GeV/fm2c, respec-tively.

An alternative representation of RAA could be done asa function of the average charged particle density per unitof pseudorapidity [27]. The 〈dNch/dη〉 experimental datafor heavy ion collisions are taken from [27, 28, 62, 70, 71].The RAA - 〈dNch/dη〉 dependence is presented in Figure7a for the same systems and collision energies as in Fig-ure 4. In such a representation, all systems at all energiesscale as a function of 〈dNch/dη〉. The same represen-tation in terms of RcoreAA and 〈dNch/dη〉core (Figure 7b)shows a larger deviation between RHIC and LHC ener-gies for 〈dNch/dη〉 ≤ 200. Relative to the 〈Npart〉 de-pendence, the difference in the shapes of the overlapping

0 500 1000 1500 2000>η/dch<dN

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

< 8

GeV

/c)

T (

5 <

pA

AR

= 200 GeVNNs = 200 GeVNNs = 2.76 TeVNNs = 5.02 TeVNNs = 5.44 TeVNNs

Cu-Cu: Au-Au: Pb-Pb:Pb-Pb:Xe-Xe:

FIG. 8. The same as Figure 7a, with the shapes of the over-lapping area S⊥ at different values of 〈dNch/dη〉.

areas of different systems for a given 〈dNch/dη〉 is larger,as it can be followed in Figure 8. If we represent εBj ·τ or〈dN/dy〉/S⊥, Figure 9a and Figure 9b, respectively, as afunction of the charged particle density, a difference be-tween the collision energies is evidenced, which increaseswith 〈dNch/dη〉. Therefore, with a few contributionsplaying a role in the observed scaling in 〈dNch/dη〉, itis rather difficult to unravel the importance of each ofthem. The difference between the two representationsis explained by the correlation between 〈dNch/dη〉 and〈Npart〉 presented in Figure 10. While the overlappingarea is very little dependent on the system size and colli-sion energy for a given 〈Npart〉 [29], 〈dNch/dη〉 combinesthe contribution of both, collision energy and system size.

III. WHY RNAA ?

RAA, as a measure of the suppression in heavy ioncollisions, is based on the estimate of the number of bi-nary collisions 〈Nbin〉 within the Glauber MC approachusing straight trajectories as a hypothesis, the depen-dence on the collision energy being introduced by thenucleon-nucleon cross section and the oversimplified as-sumption that every nucleon-nucleon collision takes placeat the same energy,

√s, and consequently the same cross

section, σNN . In Figure 11, the correlation betweenthe number of binary collisions 〈Nbin〉 and 〈Npart〉 es-timated within the standard Glauber MC approach isrepresented. An alternative approach where the energyand σNN change after each collision [72] has shown thatin Pb-Pb collisions at

√sNN=2.76 TeV, the average num-

ber of binary collisions 〈Nbin〉 is significantly lower than

7

0 500 1000 1500 2000>η/dch<dN

2468

10121416

c)2 (G

eV/fm

τ⋅BJε

0 500 1000 1500 2000>η/dch<dN

02468

101214161820)

-2 (

fm<d

N/d

y>/S = 200 GeVNNsCu-Cu:

= 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

a)

b)

FIG. 9. a) The Bjorken energy density times the interactiontime εBj · τ ; b) particle density per unit of rapidity and unitof overlapping area 〈dN/dy〉/S⊥, as a function of the averagecharged particle density per unit of pseudorapidity 〈dNch/dη〉.

the values estimated by the standard Glauber model, thedifference increasing towards central collisions. The dif-ference in 〈Npart〉 is negligible at peripheral and centralcollisions, for mid-central collisions being at the level of∼ 18%.〈Nbin〉/[〈dNch/dη〉A−A/〈dNch/dη〉pp] has to be 1 if

only single collisions take place. A very good cor-relation between 〈Nbin〉 estimated within the stan-dard Glauber model and experimental values of〈dNch/dη〉A−A/〈dNch/dη〉pp is evidenced in Figure 12.However, their ratio as a function of 〈Npart〉 shows an in-crease from close to 1 for the lowest values of 〈Npart〉, upto 〈Npart〉∼150, followed by a tendency towards a satura-tion at ∼3.5 for the largest 〈Npart〉 values, see Figure 13.One should remark that all systems at all investigatedenergies overlap in this representation. In the case of ppcollisions, 〈dNch/dη〉INEL corresponding to the selectionof inelastic collisions, and the parametrisation given in[73] have been used.

Based on these, we will also analyse the model inde-pendent quantity, namely RNAA, obtained as a ratio of thepT spectra in A-A collisions to the one of minimum biaspp collisions at the same energy, each of them normalisedto the corresponding charged particle densities, for all theavailable centralities in A-A collisions, already used in aprevious paper for comparing the behaviour of pT spectrain pp, p-Pb and Pb-Pb collisions as a function of chargedparticle multiplicity and centrality, respectively [74]:

RNAA =( d2NdpT dη

/dNch

dη )cen

( d2NdpT dη

/dNch

dη )pp,MB(6)

In Figure 14, RNAA as a function of 〈Npart〉 for thesystems discussed in the previous section is presented.

0 50 100 150 200 250 300 350 400>part<N

0

200

400

600

800

1000

1200

1400

1600

1800

2000

>η/d

ch<d

N

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 10. The average charged particle density per unit ofpseudorapidity 〈dNch/dη〉 as a function of the average valueof participating nucleons 〈Npart〉.

0 50 100 150 200 250 300 350 400>part<N

0

200

400

600

800

1000

1200

1400

1600

1800

2000>bi

n<N

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 11. Correlation between the average number of binarycollisions 〈Nbin〉 and the average number of participating nu-cleons 〈Npart〉 estimated using the Glauber MC approach.

The scaling as a function of the system size for eachenergy domain, i.e. the highest energy at RHIC andLHC energies remains, the suppression is reduced andthe 〈Npart〉 dependence is close to a linear one. As itis observed in Figure 15, RNAA does not show a similarscaling as RAA as a function of 〈dNch/dη〉 for the twocollision energy domains. However, the scaling at LHC

8

0 100 200 300 400 500pp

>)η/dch

/(<dNA-A>)η/dch

(<dN

0

200

400

600

800

1000

1200

1400

1600

1800

2000>

bin

<N = 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 12. Correlation between the average num-ber of binary collisions 〈Nbin〉 and experimental〈dNch/dη〉AA/〈dNch/dη〉pp.

0 50 100 150 200 250 300 350 400>part<N

0

0.5

1

1.5

2

2.5

3

3.5

4

]pp

>η/d

ch/<

dNA

-A>η

/dch

>/[<

dNbi

n<N

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 13. 〈Nbin〉/[〈dNch/dη〉A−A/〈dNch/dη〉pp] as a functionof 〈Npart〉.

energies remains, a close to linear dependence beingevidenced in this representation as well. The sameconsiderations can be used to estimate the expectedsuppression, (1-RNpp), for pp collisions at

√s=7 TeV and

very high charged particle multiplicity (HM) events.The geometrical scaling [29] shows that for the highestcharged particle multiplicity in pp collisions at

√s=7

TeV, in the case of α=1,√〈dN/dy〉/S⊥=3.3±0.1,

0 50 100 150 200 250 300 350 400>part<N

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

< 8

GeV

/c)

T (

5 <

pN A

AR

FIG. 14. RNAA as a function of 〈Npart〉.

0 500 1000 1500 2000>η/dch<dN

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4 <

8 G

eV/c

)T

(5

< p

N AA

R

FIG. 15. RNAA as a function of 〈dNch/dη〉.

〈βT 〉 in pp and Pb-Pb at√sNN=2.76 TeV is the

same. Therefore, the hydrodynamics should playa negligible role. For this value of

√〈dN/dy〉/S⊥,

Spp⊥ =7.43±0.48 fm2 and SPb−Pb⊥ =70±0.4 fm2. As-

suming the same jet-medium coupling, (1-RN(HM)pp )/(1-

RNAA(〈Npart〉 = 125))∼Spp,HM⊥ /SPb−Pb,〈Npart〉=125⊥ '0.01

±0.01. This could explain why in pp collisions atLHC, in high charged particle multiplicity events, nosuppression was observed, although similarities to Pb-Pb

9

0 50 100 150 200 250 300 350 400>part<N

0

0.2

0.4

0.6

0.8

1

1.2 <

8 G

eV/c

)T

(5

< p

>=30

part

<N AA

/(R

AA

=RC

PR

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au:

= 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb:

= 5.44 TeVNNsXe-Xe:

FIG. 16. RCP for Au-Au and Cu-Cu at√sNN=200 GeV, Pb-

Pb at√sNN=2.76 TeV and 5.02 TeV and Xe-Xe

√sNN=5.44

TeV, as a function of 〈Npart〉.

0 50 100 150 200 250 300 350 400>part<N

0

0.2

0.4

0.6

0.8

1

1.2

1.4

<8 G

eV/c

)T

(5<

pN C

PR

= 200 GeVNNsAu-Au:

= 2.76 TeVNNsPb-Pb:

= 5.02 TeVNNsPb-Pb:

= 5.44 TeVNNsXe-Xe:

FIG. 17. RNCP for Au-Au at√sNN=200 GeV, Xe-Xe at√

sNN=5.44 TeV and Pb-Pb at√sNN=2.76 TeV and 5.02

TeV as a function of 〈Npart〉.

collisions for other observables were evidenced.

IV. RELATIVE SUPPRESSION IN TERMS OFRCP

For energies where the pT spectra in pp collisions werenot measured, the suppression was studied in terms ofRCP , i.e. the ratio of charged particle pT spectra at agiven centrality to the pT spectrum in peripheral colli-sions, each of them divided by the corresponding averagenumber of the binary collisions:

RCP =

(d2NdηdpT

〈Nbin〉

)cen/

(d2NdηdpT

〈Nbin〉

)peripheral(7)

for each centrality in A-A collisions.For a better comparison of the RCP values as a func-

tion of 〈Npart〉, the peripheral centrality of reference waschosen to be the same for all systems and all energies,i.e 〈Npart〉=30. The RCP estimated in this way is rep-resented in Figure 16 for the same systems and energies.The values corresponding to the most central collisionsfor Au-Au and Pb-Pb are, within the error bars, thesame. As for the RAA case, due to the same reasons,using experimental data, we estimated the RNCP :

RNCP =

(d2NdηdpTdNch

dη

)cen/

(d2NdηdpTdNch

dη

)peripheral(8)

The RNCP suppression as a function of 〈Npart〉 (Fig-ure 17) is the same at all values of 〈Npart〉 for all theheavy systems, Au-Au, Xe-Xe and Pb-Pb, although thedifference in the collision energies is ∼ 14-27 times higherenergy at LHC than at RHIC, between the LHC energiesbeing a factor of ∼2. The linear dependence as a functionof 〈Npart〉 follows from the linear dependence observed inRNAA.

V. (1-RAA)/〈dN/dy〉 AND (1-RNAA)/〈dN/dy〉DEPENDENCE ON (〈dN/dy〉/S⊥)1/3

If we assume that the initial entropy is proportional tothe final measured particle density per unit of rapidityand it scales as T 3, based on Eq.2 and taking S⊥ ∼ x2,a qualitative temperature dependence of the jet-mediumcoupling can be obtained. As can be seen in Figure 18, (1-RAA)/〈dN/dy〉 shows an exponential decrease (hatchedline) as a function of (〈dN/dy〉/S⊥)1/3. The line is theresult of the fit with the following expression:

1−RAA〈dN/dy〉

= eα−β·(〈dN/dy〉/S⊥)1/3 (9)

Such a temperature dependence of the jet-medium cou-pling was considered in [22] in order to reproduce thenuclear modification factors at RHIC and LHC energies.

A similar representation for RNAA instead of RAA ispresented in Figure 19. In this case, (1-RNAA)/〈dN/dy〉

10

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3-2/3 (fm)1/3 )(<dN/dy>/S

0

0.0005

0.001

0.0015

0.002

0.0025

0.003<8

GeV

/c)

T (

5<p

<dN

/dy>

AA

1-R

= 200 GeVNNsCu-Cu: = 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 18. (1-RAA)/ 〈dN/dy〉 dependence on

(〈dN/dy〉/S⊥)1/3. The line is the result of the fit withthe expression (9).

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7-2/3 (fm)1/3 )(<dN/dy>/S

0.6−

0.4−

0.2−

0

0.2

0.4

0.63−10×

<8 G

eV/c

)T

(5<

p<d

N/d

y>

N AA

1-R

= 200 GeVNNsAu-Au: = 2.76 TeVNNsPb-Pb: = 5.02 TeVNNsPb-Pb: = 5.44 TeVNNsXe-Xe:

FIG. 19. (1-RNAA)/〈dN/dy〉 dependence on

(〈dN/dy〉/S⊥)1/3.

is constant as a function of (〈dN/dy〉/S⊥)1/3, for(〈dN/dy〉/S⊥)1/3 ≥ 2.1 part/fm2/3, independent on thesize of the heavy colliding systems and collision energy.An impact parameter independence of the jet quenchingparameter was claimed in a series of theoretical estimates[75–77].

VI. THE√sNN DEPENDENCE OF RCP , R

NCP ,

Rπ0

AA, (RNAA)π0

As it is well known, within the Beam Energy Scan(BES) program at RHIC, valuable data were obtained

10 210 310 (GeV)NNs

0.5

1

1.5

2

2.5

3

<6 G

eV/c

)T

(4<p

CP

R

10 210 310 (GeV)NNs

0

1

2

3

4

5

<6 G

eV/c

)T

(4<p

N CP

R

a)

b)

FIG. 20. a) RCP and b) RNCP , for 4< pT <6 GeV/c, as afunction of

√sNN for 0-5% centrality relative to 60-80%.

relative to the behaviour of different observables in Au-Au collisions starting from

√sNN= 7.7 GeV up to 200

GeV. As far as the pT spectra in pp collisions at these en-ergies were not measured, the STAR collaboration stud-ied the

√sNN dependence of RCP [(0-5%)/(60-80%)] for

Au-Au collisions [78]. In order to include as much aspossible the lower energies, where the published data areon a lower pT range, we had to change the pT rangefrom 5< pT <8 GeV/c, used in previous sections, to4< pT <6 GeV/c, for the study of the charged particlesuppression dependence on the collision energy. Theseresults, together with the values obtained in Pb-Pb col-lisions at

√sNN=2.76 and 5.02 TeV, for the most cen-

tral collisions, are presented in Figure 20a. Followingthe arguments from the previous section, in Figure 20bRNCP as a function of the collision energy is presented. Inboth plots is evidenced a decrease of RCP or RNCP from√sNN= 19.6 GeV up to

√sNN= 200 GeV, while the rel-

ative ratios of particle densities per unit of rapidity andunit of overlapping area, within the error bars, are con-stant. Beyond the RHIC energies, RCP and RNCP remainconstant. As far as RAA for charged particles at lowerRHIC energies are not reported, in order to confirm theabove observations, we used RAA of π0 published by thePHENIX collaboration at

√sNN=39, 62.4 and 200 GeV

[37, 79] and by the ALICE Collaboration [80, 81] at LHCenergies.

In order to have an estimate on Rπ0

AA corresponding to0-10% centrality for the collision energies where it wasnot published, we applied the procedure described bel-

11

10 210 310 (GeV)NNs

0

0.5

1

1.5

2

2.5 <

6 G

eV/c

)T

(4 <

pC

PR

a)10 210 310

(GeV)NNs0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.82

< 6

GeV

/c)

T (4

< p

0 π AA

R

Experimental dataInterpolated/extrapolated data

b)

FIG. 21. a) The same as Figure 20a, fitted with the expressiongiven in Eq.10. b) RAA for π0, corresponding to the samerange in pT as a), for experimental values (full symbols) andinterpolated/extrapolated results (open symbols) for 0-10%centrality.

10 210 310 (GeV)NNs

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

20 π AA

R < 6 GeV/cT

4 < p < 8 GeV/c

T5 < p

FIG. 22. π0 RAA for the two pT ranges: 4-6 GeV/c (opensymbols) and 5-8 GeV/c (full symbols) for 0-10% centrality.

low. The RCP -√sNN dependence (Figure 21a) was

fitted with the following expression:

RCP ∝ a+b

sNN+ c ·

√sNN (10)

with a, b, and c as free parameters, the result being pre-sented in Figure 21a. A similar expression was used in

order to fit the measured experimental data of Rπ0

AA -√sNN dependence (Figure 21b), leaving the parameters

free. The result was used for estimating Rπ0

AA at the miss-ing collision energies, i.e. 19.6, 27 and 130 GeV (Figure21b - open symbols). Measured, interpolated and extrap-

10 210 310 (GeV)NNs

00.020.040.060.08

0.10.120.140.160.18)2

) (f

m)/(

<dN

/dy>

/S0 π AA

(1-R 2

468101214161820

0-10

% )

(<dN

/dy>

/S

10 210 310 (GeV)NNs

0.2−

0.15−

0.1−

0.05−

0

0.05

0.1

)2 )

(fm

]/(<d

N/d

y>/S

0 π )N A

A[1

-(R 02468101214161820

0-10

% )

(<dN

/dy>

/S

< 6 GeV/cT

4 < p < 8 GeV/c

T5 < p

a)

b)

FIG. 23. a) (1 − Rπ0

AA)/(〈dN/dy〉/S⊥)0−10% as a function

of collision energy; b) (1 − (RNAA)π0

)/(〈dN/dy〉/S⊥)0−10%

as a function of collision energy (bullets)-left scale and

(〈dN/dy〉/S⊥)0−10% (stars)-right scales for 0-10% centrality.

olated Rπ0

AA values as a function of√sNN are presented in

Figure 22, for both pT ranges used in this paper, namely4-6 GeV/c (open symbols) and 5-8 GeV/c (full symbols).

The Rπ0

AA dependence as a function of√sNN is sim-

ilar with the one evidenced for RCP corresponding tocharged particles presented in Figure 21, i.e. the suppres-sion starts around

√sNN=27 GeV, increases up to the

top RHIC energy and remains constant up to the LHCenergies. The ratios relative to (〈dN/dy〉/S⊥)0−10% as afunction of the collision energy are presented in Figure

23, namely: (1−Rπ0

AA)/(〈dN/dy〉/S⊥)0−10% (Figure 23a)

and [1− (RNAA)π0

]/(〈dN/dy〉/S⊥)0−10% (Figure 23b).These ratios show a maximum around the top RHIC

energies, decreasing towards LHC energies, in qualita-tive agreement with theoretical predictions [21, 24, 82].To what extent such a trend is due to a transition froma magnetic plasma of light monopoles near critical tem-perature region [82] to a deconfined matter dominated byquarks and gluons [24] remains an open question. How-ever, a clear transition in the properties of the deconfinedmatter from RHIC to LHC energies is supported by theexperimental trends.

VII. CONCLUSIONS

Based on the experimental results obtained at RHICfor Au-Au (Cu-Cu) and at LHC for Pb-Pb (Xe-Xe) colli-sions, a detailed analysis of the charged particle suppres-sion in the region of transverse momentum correspondingto the maximum suppression is presented. In order to see

12

to what extent the conclusions based on these studies arenot a consequence of the model estimate of the number ofbinary collisions used in the definition of RAA and RCP ,model independent ratios of the pT spectra for each cen-trality to the pp minimum bias or to the peripheral one,each of them normalised with the corresponding chargedparticle density 〈dNch/dη〉, namely RNAA and RNCP , arepresented. While RAA scales as a function of 〈dNch/dη〉for the top RHIC and all LHC energies, it scales sepa-rately as a function of 〈Npart〉 for RHIC and LHC en-ergies, for all the corresponding measured colliding sys-tems. However, given that 〈dNch/dη〉 depends on thecollision energy and overlapping area of the colliding sys-tems, their relative contribution to suppression is ratherdifficult to be unraveled. This is the main reason whythe considerations on suppression phenomena as a func-tion of collision geometry and collision energy are mainlybased on the 〈Npart〉 dependence.

The influence of the corona contribution on experimen-tal RAA is presented. As expected, the main corona con-tribution is at low values of 〈Npart〉, the core suppressionrelative to the experimental value being larger.

Based on (1-RAA) and 〈dN/dy〉/S⊥ dependences on〈Npart〉, one could conclude that a saturation of sup-pression at LHC energies takes place. At 〈Npart〉=350,corresponding to the most central Au-Au collisions at√sNN=200 GeV, if one considers the parton energy loss

proportional with the squared path length and chargedparticle density per unit of overlapping area, a jet-medium coupling approximately two times lower at LHCthan at RHIC is obtained. The difference in the hy-drodynamic expansion extracted from the 〈βT 〉 scaling

as a function of√〈dN/dy〉/S⊥, of about 10%, cannot

explain the difference observed in the parton-mediumcoupling constant. Such considerations, applied to the

highest charged particle multiplicity measured in pp col-lisions at 7 TeV could explain why no suppression isevidenced in such events while there are similarities toPb-Pb with respect to other observables. RNAA as a func-tion of 〈Npart〉 shows similar separate scaling for RHICand LHC energies, a linear dependence being evidencedat LHC energies. RNCP evidences a very good scalingas a function of 〈Npart〉 for the heavy systems at allcollision energies. The ratio (1-RAA)/〈dN/dy〉 showsan exponential decrease with (〈dN/dy〉/S⊥)1/3 while (1-RNAA)/〈dN/dy〉 is independent on (〈dN/dy〉/S⊥)1/3 for

(〈dN/dy〉/S⊥)1/3 ≥2.1 part/fm2/3, the value being thesame for all the heavy systems and all collision energies.

For the most central collisions, RCP , RNCP and Rπ0

AA,

(RNAA)π0

for 4< pT <6 GeV/c and 5< pT <8 GeV/c,measured at RHIC in Au-Au collisions and at LHC in Pb-Pb collisions, evidence, as a function of the collision en-ergy, a suppression enhancement from

√sNN = 39 GeV

up to 200 GeV, followed by a saturation up to the high-est energy,

√sNN =5.02 TeV for Pb-Pb collision. (1 −

Rπ0

AA)/(〈dN/dy〉/S⊥) and [1 − (RNAA)π0

]/(〈dN/dy〉/S⊥)for 0-10% centrality evidence a maximum around thelargest RHIC energies, in qualitative agreement withmodels predictions. This could be considered as a signa-ture of a transition in the deconfined matter propertiesfrom the top RHIC energy to LHC energies.

ACKNOWLEDGMENTS

This work was carried out under the contractssponsored by the Ministry of Education and Re-search: RONIPALICE-04/10.03.2020 (via IFA Coordi-nating Agency) and PN-19 06 01 03.

[1] I.C. Arsene et al. (BRAHMS Collaboration). Abs.Nucl.Phys.A, 757:1, 2005.

[2] B.B. Back et al. (PHOBOS Collaboration). Nucl.Phys.A,757:28, 2005.

[3] J. Adams et al. (STAR Collaboration). Nucl.Phys.A,757:102, 2005.

[4] K. Adcox et al. (PHENIX Collaboration). Nucl.Phys.A,757:184, 2005.

[5] J.T. Mitchell et al. (PHENIX Collaboration).Nucl.Phys.A, 904-905:903c, 2013.

[6] D. McDonald et al. (STAR Collaboration).Eur.Phys.J.Web of Conferences, 95:01009, 2015.

[7] J.C. Collins and M.J. Perry. Phys.Rev.Lett., 34:1353,1975.

[8] G.F. Chapline and A.K. Kerman. MIT report CTP-695,1978.

[9] E.V. Shuryak. Phys.Lett.B, 78:150, 1978.[10] S.A. Chin. Phys.Lett.B, 78:552, 1978.[11] J.D. Bjorken. FERMILAB-PUB-82-059-THY, 1982.[12] D. d’Enterria. Relativistic Heavy Ion Physics, Landolt-

Bornstein - Group I Elementary Particles, Nuclei and

Atoms, Springer-Verlag Berlin Heidelberg, ISBN 978-3-642-01538-0, 23:471, 2010.

[13] T. Renk. Phys.Rev.C, 76:064905, 2007.[14] R. Baier et al. Nucl.Phys.B, 483:291, 1997.[15] M. Gyulassy, P. Levai, and I. Vitev. Nucl.Phys.B,

571:197, 2000.[16] R. Baier et al. J. High Energ. Phys., 0109:033, 2001.[17] F. Arleo. J. High Energ. Phys., 11:044, 2002.[18] B. Muller and K. Rajagopal. Eur.Phys.J., C43:15, 2005.[19] M. Djordjevic and U.W. Heinz. Phys.Rev.Lett.,

101:022302, 2008.[20] J. Casalderrey-Solana et al. J. High Energ. Phys., 09:175,

2015.[21] K.M. Burke et al. (JET Collaboration). Phys.Rev.C,

90:014909, 2014.[22] B. Betz and M. Gyulassy. J. High Energ. Phys., 08:090,

2014.[23] F. Arleo and G. Falmagne. PoS(HardProbes2018), page

075, 2018.[24] S. Shi, J. Liao, and M. Gyulassy. Chinese Phys. C,

43:044101, 2019.

13

[25] B. Alver et al. (PHOBOS Collaboration). Phys.Rev.Lett.,96:212301, 2006.

[26] J. Adams et al. (STAR Collaboration). Phys.Rev.Lett.,91:172302, 2003.

[27] S. Acharya et al. (ALICE Collaboration). Phys.Lett.B,788:166, 2019.

[28] J. Adam et al. (ALICE Collaboration). Phys.Rev.Lett.,116:222302, 2016.

[29] M. Petrovici et al. Phys.Rev.C, 98:024904, 2018.[30] R.J. Glauber. Phys. Rev., 100:242, 1955.[31] V. Franco and R.J. Glauber. Phys. Rev., 142:119, 1966.[32] M.L. Miller et al. Ann.Rev.Nucl.Part.Sci., 57:205, 2007.[33] M. Rybczynski et al. Comput. Phys. Commun., 185:159,

2014.[34] S.S. Adler et al. (PHENIX Collaboration). Phys.Rev.C,

69:034910, 2004.[35] B.B. Back et al. Phys.Rev.Lett., 94:082304, 2005.[36] B.I. Abelev et al. (STAR Collaboration). Phys.Rev.C,

82:054907, 2010.[37] A.Adare et al. (PHENIX Collaboration). Phys.Rev.Lett.,

109:152301, 2012.[38] A.M. Sirunyan et al. (CMS Collaboration). J. High En-

erg. Phys., 10:138, 2018.[39] S. Chatrchyan et al. (CMS Collaboration). Eur.Phys.J.,

C72:1945, 2012.[40] B. Abelev et al. (ALICE Collaboration). Phys.Lett.B,

720:52, 2013.[41] G. Aad et al. (ATLAS Collaboration). J. High Energ.

Phys., 09:050, 2015.[42] F. Becattini et al. Phys.Rev.C, 69:024905, 2004.[43] P. Bozek. Acta Phys. Pol., B36:3071, 2005.[44] K. Werner. Phys.Rev.Lett., 98:152301, 2007.[45] J. Steinheimer and M. Bleicher. Phys.Rev.C, 84:024905,

2011.[46] F. Becattini and J. Manninen. J.Phys.G, 35:104013,

2008.[47] F. Becattini and J. Manninen. Phys.Lett.B, 673:19, 2009.[48] J. Aichelin and K. Werner. Phys.Rev.C, 79:064907, 2009.[49] J. Aichelin and K. Werner. Phys.Rev.C, 82:034906, 2010.[50] P. Bozek. Phys.Rev.C, 79:054901, 2009.[51] K. Werner C. Schreiber and J. Aichelin. Phys. Atom.

Nucl., 75:640, 2012.[52] M. Gemard and J. Aichelin. Astron. Nachr., 335:660,

2014.[53] M. Petrovici et al. Phys.Rev.C, 96:014908, 2017.[54] J. Adam et al. (ALICE Collaboration). Phys.Rev.C,

94:034903, 2016.

[55] B. Alver et al. Phys.Rev.C, 77:014906, 2008.[56] S. Acharya et al. (ALICE Collaboration). J. High Energ.

Phys., 1811:013, 2018.[57] C. Loizides and A. Morsch. Phys.Lett.B, 773:408, 2017.[58] C. Loizides et al. Phys.Rev.C, 97:054910, 2018.[59] M. Petrovici et al. AIP Conference Proceedings,

2076:040001, 2019.[60] M. Djordjevic et al. Phys.Rev.C, 99:061902, 2019.[61] J.D. Bjorken. Phys.Rev.D, 27:140, 1982.[62] B.I. Abelev et al. (STAR Collaboration). Phys.Rev.C,

79:034909, 2009.[63] B. Abelev et al. (ALICE Collaboration). Phys.Rev.C,

88:044910, 2009.[64] N. Jacazio (ALICE Collaboration). Nucl.Phys.A,

967:421, 2017.[65] J. Adams et al. (STAR Collaboration). Phys.Rev.Lett.,

98:062301, 2007.[66] B. Abelev et al. (ALICE Collaboration). Phys.Rev.Lett.,

111:222301, 2013.[67] D.S. de Albuquerque (ALICE Collaboration). Quark

Matter 2018, 13-19 May 2018.[68] I.C. Arsene et al. (BRAHMS Collaboration).

Phys.Rev.C, 94:014907, 2016.[69] F. Bellini (ALICE Collaboration). Nucl.Phys.A, 982:427,

2019.[70] A. Adare et al. (STAR Collaboration). Phys.Rev.C,

93:024901, 2016.[71] B. Abelev et al. (ALICE Collaboration). Phys.Rev.C,

88:044910, 2013.[72] A. Seryakov and G. Feofilov. AIP Conference Proceed-

ings, 1701:070001, 2016.[73] J. Adam et al. (ALICE Collaboration). Eur.Phys.J.,

C77:33, 2017.[74] M. Petrovici et al. AIP Conference Proceedings,

1852:050003, 2017.[75] C. Andres et al. Eur.Phys.J., C76:475, 2016.[76] C. Andres et al. arXiv:1705.01493[nucl-th], 2017.[77] M. Xie et al. Eur.Phys.J., C79:589, 2019.[78] E. Sangaline et al. (STAR Collaboration). Quark Matter

2012 Conference Proceedings, 2012.[79] A. Adare et al. (PHENIX Collaboration). Phys.Rev.Lett.,

101:232301, 2008.[80] B. Abelev et. (ALICE Collaboration). Eur.Phys.J.,

C74:3108, 2014.[81] D. Sekihata (for the ALICE Collaboration). Quark Mat-

ter 2018, 13-19 May 2018.[82] J. Liao and E. Shuryak. Phys.Rev.Lett., 102:202302,

2009.