Correlation between mean transverse momentum and anisotropic flow in heavy-ion collisions

Abstract

The correlation between the mean transverse momentum of outgoing particles, ⟨pt⟩\langle p_t \rangle, and the magnitude of anisotropic flow, vnv_n, has recently been measured in Pb+Pb collisions at the CERN Large Hadron Collider, as a function of the collision centrality. We confirm the previous observation that event-by-event hydrodynamics predicts a correlation between vnv_n and ⟨pt⟩\langle p_t \rangle that is similar to that measured in data. We show that the magnitude of this correlation can be directly predicted from the initial condition of the hydrodynamic calculation, for n=2,3, if one replaces vnv_n by the corresponding initial-state anisotropy, εn\varepsilon_n, and ⟨pt⟩\langle p_t\rangle by the total energy per unit rapidity of the fluid at the beginning of the hydrodynamic expansion.

Full text

Correlation between mean transverse momentum and anisotropic flow
in heavy-ion collisions
Giuliano Giacalone,1Fernando G. Gardim,2Jacquelyn Noronha-Hostler,3and Jean-Yves Ollitrault1
1Universit´e Paris Saclay, CNRS, CEA, Institut de physique th´eorique, 91191 Gif-sur-Yvette, France
2Instituto de Ciˆencia e Tecnologia, Universidade Federal de Alfenas, 37715-400 Po¸cos de Caldas, MG, Brazil
3Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
(Dated: April 7, 2020)
The correlation between the mean transverse momentum of outgoing particles, hpti, and the
magnitude of anisotropic flow, vn, has recently been measured in Pb+Pb collisions at the CERN
Large Hadron Collider, as a function of the collision centrality. We confirm the previous observation
that event-by-event hydrodynamics predicts a correlation between vnand hptithat is similar to
that measured in data. We show that the magnitude of this correlation can be directly predicted
from the initial condition of the hydrodynamic calculation, for n= 2,3, if one replaces vnby the
corresponding initial-state anisotropy, εn, and hptiby the total energy per unit rapidity of the fluid
at the beginning of the hydrodynamic expansion.
I. INTRODUCTION
Anisotropic flow has been much studied in ultrarela-
tivistic heavy-ion collisions, as it probes the properties
of the little quark-gluon plasma formed in these colli-
sions [1]. The event-by-event fluctuations of vn, the nth
Fourier harmonic of the azimuthal distribution of the
emitted hadrons, have been precisely characterized [2–
4], as well as the mutual correlations between different
flow harmonics [5–8]. Recently, following a suggestion
by Bo˙zek [9], the ATLAS Collaboration has measured
a correlation of a new type, namely, the correlation be-
tween the mean transverse momentum, hpti, and v2
n[10],
a quantity dubbed ρn. Although event-by-event hydro-
dynamic results on ρnare in fair agreement with exper-
imental data [11], a clear picture of the physical mecha-
nism that produces this correlation is still missing.
In this paper, we explain the origin of the correlation
between hptiand vnin hydrodynamic calculations. We
first confirm, in Sec. II, that state-of-the-art hydrody-
namic calculations yield results on ρnthat are in agree-
ment with recent Pb+Pb data. We use results from hy-
drodynamic calculations [12] obtained prior to the AT-
LAS analysis, so that they can be considered as predic-
tions. We then unravel, in Sec. III, the physical mech-
anism behind ρn. While it is well established that v2
and v3originate on an event-by-event basis from the
anisotropies of the initial density profile, εn[13–18], the
new crucial feature that we shall elucidate here is the
origin of hptifluctuations in hydrodynamics. These fluc-
tuations are thought to be driven by fluctuations in the
fireball size, R[19, 20], a phenomenon referred to as size-
flow transmutation [21]. We show in Sec. III A that, in
fact, a much better predictor for hptiis provided by the
initial total energy per rapidity of the fluid [22]. On this
basis, in Sec. III B we evaluate ρnusing a standard initial-
state model for Pb+Pb collisions, and we obtain results
that are in good agreement with ATLAS data. We fur-
ther show that agreement with data is instead lost if one
uses Ras an event-by-event predictor of hpti.
II. RESULTS FROM EVENT-BY-EVENT
HYDRODYNAMICS
The ATLAS Collaboration measured the Pearson cor-
relation coefficient between the mean transverse momen-
tum and the anisotropic flow of the event [10]. Exper-
imentally, this is obtained from a three-particle correla-
tion introduced by Bo˙zek [9]. Since self correlations are
subtracted in the measure of the correlation coefficient
(i.e., one does not correlate a particle with itself ), this
observable is insensitive to trivial statistical fluctuations
and probes genuine dynamical fluctuations [23], due to
correlations. In hydrodynamics, ρncan be evaluated as:
ρn≡hptiv2
n− hhptiiv2
n
σptσv2
n
,(1)
where, following the notation of Ref.[20], hptidenotes an
average over the single-particle momentum distribution,
f(p), at freeze-out in a given event1, and the outer an-
gular brackets denote an average over events in a given
multiplicity (centrality) window. σptand σvndenote, re-
spectively, the standard deviation of hptiand of v2
n:
σpt≡qhhpti2i − hhptii2,
σv2
n≡qhv4
ni−hv2
ni2.(2)
We evaluate ρnusing hydrodynamic simulations. The
setup of our calculation is the same as in Ref. [12]. We
evolve 50000 minimum bias Pb+Pb collisions at √sNN =
5.02 TeV through the 2+1 viscous relativistic hydrody-
namical code V-USPHYDRO [24–26]. The initial condition
of the evolution is the profile of entropy density gener-
ated using the TRENTo model [27], tuned as in Ref. [28].2
1Using this notation, one can also write vn≡
heinϕi
, where ϕis
the azimuthal angle of the particle momentum.
2That is, we implement a geometric average of nuclear thickness
arXiv:2004.01765v1 [nucl-th] 3 Apr 2020

2
Npart
−0.2
0.0
0.2
ρ2
(a)
ρ2ATLAS
ρ2v-USPHydro
Npart
−0.2
0.0
0.2
ρ3
(b)
Pb+Pb, √sNN = 5.02 TeV
ρ3ATLAS
ρ3v-USPHydro
200 400
Npart
−0.2
0.0
0.2
ρ4
(c)
ρ4ATLAS
ρ4v-USPHydro
FIG. 1. (Color online) Value of ρnfor n= 2 (a), n= 3 (b),
n= 4 (c), as a function of the number of participant nucleons
in Pb+Pb collisions at √sNN = 5.02 TeV. Empty symbols
are experimental results from the ATLAS Collaboration [10],
while full symbols are hydrodynamic results [12].
We neglect the expansion of the system during the pre-
equilibrium phase [29–31], and start hydrodynamics at
time τ0= 0.6 fm/c after the collision [32]. We use the lat-
tice QCD equation of state [33], and we implement a con-
stant shear viscosity over entropy ratio η/s = 0.047 [34].
functions (parameter p= 0), where the thickness function of a
nucleus is given by a linear superimposition of the thicknesses
of the corresponding participant nucleons, modeled as Gaussian
density profiles of width w= 0.51 fm. The thickness of each par-
ticipant nucleon is further allowed to fluctuate in normalization
according to a gamma distribution of unit mean and standard
deviation 1/√k, with k= 1.6.
Fluid cells are transformed into hadrons [35] when the
local temperature drops below 150 MeV. All hadronic
resonances can be formed during this freeze-out process,
and we implement subsequent strong decays into stable
hadrons. To mimic the centrality selection performed
in experiment, we sort events into centrality classes ac-
cording to their initial entropy (5% classes are used).
We evaluate hadron observables by integrating over the
transverse momentum range 0.2< pt<3 GeV/c, and
over |η|<0.8.3
In Fig. 1 we show our results along with ATLAS
data. We choose data integrated over the 0.5< pt<
2 GeV/c, which is close to our setup. We conclude that
event-by-event relativistic hydrodynamics captures semi-
quantitatively the magnitude and the centrality depen-
dence of ρ2,ρ3and ρ4.
III. PHYSICAL ORIGIN OF ρ2AND ρ3
A. Initial energy as a predictor for hpti
The full hydrodynamic calculation allows us to repro-
duce the experimental data, but it does not give much in-
sight into the physics underlying the observed ρn. In the
same way as v2and v3, and their higher-order moments,
are driven by the initial spatial eccentricity, ε2, and tri-
angularity, ε3, on an event-by-event basis, it would be
highly insightful to trace the origin of ρnback to the ini-
tial state of the hydrodynamic calculation. For this pur-
pose, one must identify the property of the initial state
which drives the event-by-event fluctuations of the mean
transverse momentum.
It has recently been shown [22] that if one fixes the
total entropy (which in an experiment amounts to fixing
the centrality of the collision), then hptiis essentially
determined by the energy of the fluid per unit rapidity
at the initial time τ0, which we denote by Ei:
Ei≡τ0Z(τ0, x, y)dxdy, (3)
where is the energy density, and the integral runs over
the transverse plane. This is at variance with the earlier
claims [19] that hptiis determined by the initial trans-
verse size of the fireball, R, defined as [20, 21]:
R2≡2R(x2+y2)s(τ0, x, y)dxdy
Rs(τ0, x, y)dxdy (4)
where sis the entropy density.4
To illustrate the difference between these two predic-
tors, we have evaluated Ei,Rand hptiin event-by-event
3Note that the ATLAS detector has a wider acceptance in η, but
this difference is unlikely to have any sizable effects on ρn.
4The factor 2 ensures that for a uniform entropy density s(τ0,x, y )
in a circle of radius R, the right-hand side gives R2.

3
5.5 6.0 6.5
R(fm)
700
750
800
hpti(MeV)
(a)
55.0 57.5 60.0 62.5
Ei(105MeV)
Pb+Pb
√sNN = 5.02 TeV
(b)
FIG. 2. (Color online) Results from ideal hydrodynamic simulations of Pb+Pb collisions at √sNN = 5.02 TeV at impact
parameter b= 2.5 fm. 850 events have been simulated, where each event has the same total entropy, but a different entropy
density profile. Each symbol represents a different event. (a) Scatter plot of the mean transverse momentum of charged
particles, hpti, versus the initial size, R, defined by Eq. (4). (b) Scatter plot of hptiversus the initial energy per unit rapidity,
Ei, defined by Eq. (3).
hydrodynamics at fixed initial entropy. Note that the
minimum bias calculation performed in Sec. II is not well-
suited for this purpose, as, even if we narrowed down the
width of our centrality bins by a factor 2, there would
still be significant entropy fluctuations in our sample.
For this reason, we resort to the calculation shown in
Ref. [22]. Here the events are evaluated at fixed impact
parameter b= 2.5 fm, and at fixed total entropy cor-
responding to the mean entropy of Pb+Pb collisions at
√sNN = 5.02 TeV in the 0-5% centrality window. Also,
we perform an ideal hydrodynamic expansion, which en-
sures conservation of entropy. The initial condition of the
calculation, the equation of state, and the initialization
time, τ0, are the same used in the calculation of Sec. II.5
This calculation is evolved through the MUSIC hydrody-
namic code [37–39].
Figure 2(a) displays the scatter plot of Rvs. hptiob-
tained in this calculation. These two quantities are nega-
tively correlated, as already shown by other authors [21].
The explanation is that for a fixed total entropy, a smaller
size generally implies a larger entropy density, hence
a larger temperature, which in turn implies a larger
hpti[40]. There is, however, a significant spread of the
values of hptifor a fixed R. By contrast, there is an al-
most one-to-one correspondence between hptiand the ini-
tial energy, Ei, as displayed in Fig. 2 (b). We shall show
now that, in order to understand the measured correla-
tion between hptiand vn, it is indeed crucial to employ
Eias a predictor of the average transverse momentum.
5The sole differences are that we implement slightly smaller en-
tropy fluctuations, using k= 2 in TRENTo, which provide a
better fit of LHC data [12], as well as a slightly higher freeze-out
temperature Tf= 156.5 MeV, which has now become a more
standard choice [36].
B. Results from models of initial conditions
We first explain how ρn, defined in Eq. (1), can be eval-
uated from the initial conditions of the hydrodynamic
calculations. First, one uses the approximate propor-
tionality between vnand the initial anisotropy εn[17]
for n= 2,3:
vn=κnεn,(5)
where κnis a response coefficient which is the same for
all events at a given centrality.6Next, one uses the ob-
servation, made in Sec. III A, that hpti=f(Ei), where
f(Ei) is some smooth function of the initial energy, Ei.
Linearizing in the fluctuations of Eiand hptiaround their
mean values, hEiiand hhptii, one obtains
hpti − hhptii =f0(hEii) (Ei− hEii).(6)
Inserting Eqs. (5) and (6) into Eq. (1), one obtains:
ρn=Eiε2
n− hEiiε2
n
σEiσε2
n
f0(hEii)
|f0(hEii)|,(7)
where σEiand σε2
ndenote the standard deviations, ob-
tained by replacing hptiand vnwith Eiand εnin Eq. (2).
Remarkably enough, the dependence on the unknown
function f(Ei) cancels, except for the sign of f0(hEii).
An important advantage of Eq. (7) is that it allows us
to evaluate ρnin millions of simulated initial conditions
with little computational effort, and thus to overcome the
6This linear response works for v2and v3, but not for v4[16], so
that we do not discuss ρ4in this section.

4
100 200 300 400
Npart
−0.1
0.0
0.1
0.2
0.3
ρ2
(a)
100 200 300 400
Npart
−0.1
0.0
0.1
0.2
0.3
ρ3
(b)
ATLAS
ρn(Ei, ε2
n)
ρn(R, ε2
n)
FIG. 3. (Color online) Variation of ρ2(a) and ρ3(b) with the number of participants in Pb+Pb collisions at √sNN = 5.02 TeV.
As in Fig. 1, symbols are experimental results from the ATLAS Collaboration [10]. The shaded band is our result using Eq. (7).
The width of the band is the statistical error evaluated through jackknife resampling. The dashed line is obtained by replacing
Eiwith Rin Eq. (7).
issue of large entropy fluctuations within a finite central-
ity bin [11]. This allows us, hence, to evaluate ρnin the
strict limit of fixed initial entropy, and to reproduce the
situation of Fig. 2 in a minimum bias calculation.
To this aim we have generated 20 million minimum bias
Pb+Pb events using the same TRENTo parametrization
as in Fig. 2. We sort the events into narrow 0.25% cen-
trality bins, and in each bin we evaluate ρnaccording to
Eq. (7). To evaluate Eiin each event, we assume that
the entropy profile returned by TRENTo, s, is related to
the energy density, , of the event through ∝s4/3. This
is typically a very good approximation at the high tem-
peratures achieved in the initial state of nucleus-nucles
collisions. Our result is displayed in Fig. 3 as a shaded
band. Note that we recombine 0.25% bins into 1% bins
for sake of visualization. Our TRENTo calculation is in
good agreement with ATLAS data (open symbols) for
both ρ2and ρ3, and is consistent with the full hydro-
dynamic calculation shown in Fig. 1, in the sense that
both evaluations slightly underestimates ρ2while they
overestimates ρ3. Note that ρ2and ρ3measured by the
ATLAS Collaboration have a slight dependence on the
ptcut used in the analysis [10]. The difference between
our results and experimental data is of the same order, or
smaller, than the dependence of experimental results on
the ptcuts. This feature is not captured by our predic-
tion, which is independent of these cuts by construction.
It would be therefore interesting to have new measure-
ments of ρnwith a lower ptcut, of order 0.2 or 0.3 GeV,
which is where the bulk of the produced particles sits.
This may improve agreement between our evaluations
and data.
While the quantitative results shown in Fig. 3 de-
pend on the parametrization of the TRENTo model, we
show in Appendix A that the main qualitative features,
for instance the fact that ρnis positive in central colli-
sions, are robust and model-independent. It is interesting
though that the choice of parameters made here, namely,
p= 0, preferred from previous comparisons [18, 28], and
k= 2 [12], also optimizes agreement with ρndata. We
have also checked that the stringent condition of having a
fixed total entropy, S, can be relaxed by replacing Eiwith
Ei/S in Eq. (7). With this choice, results are essentially
unchanged if one uses 2% centrality bins. A moderate
variation starts to be visible if one uses 5% bins, as with
other observables [41].
Finally, we show how the results are changed if one uses
the initial size, R, as a predictor of hpti. If one replaces
Eiwith Rin Eq. (7), the resulting value of ρnis com-
pletely different, as shown by the dashed lines in Fig. 3.7
These results show that, at fixed centrality, the correla-
tion between hptiand v2
nis not driven by the event-by-
event fluctuations of the fireball size. Note however that a
better geometric predictor can be constructed following
Schenke, Shen and Teaney [42], whose paper appeared
while we were finalizing this manuscript. They show that
if one replaces the rms size, R, with the area of the el-
liptical region of nuclear overlap, R2p1−ε2
2, then the
quality of the predictor for ρ2improves greatly.
7Note that the correlation between Rand hptiis negative, as
shown in Fig. 2 (a). This implies f0(hRi)<0 in the right-hand
side of Eq. (7), resulting in an overall negative sign.

5
100 200 300 400
Npart
−0.1
0.0
0.1
0.2
0.3
ρ2
(a) ATLAS
100 200 300 400
Npart
−0.1
0.0
0.1
0.2
0.3
ρ3
(b)
p= 1
p= 0
p=−1
FIG. 4. (Color online) Dependence of ρ2(a) and ρ3(b) on the parameter pin the TRENTo model. Full lines: p= 0 (entropy
density s∝√TATB), as in Fig. 3. Dotted lines: p= 1 (s∝TA+TB). Dashed lines: p=−1 (s∝TATB/(TA+TB)). We use
broader centrality bins for this calculation than for Fig. 3, which explains the small differences between the p= 0 results of the
two figures. As in Figs. 1 and 3, symbols are ATLAS data [10].
IV. CONCLUSIONS
We have shown that ATLAS results on ρnin Pb+Pb
collisions can be explained by hydrodynamics. The mech-
anism driving the correlation between the mean trans-
verse momentum and anisotropic flow in Pb+Pb colli-
sions can be traced back to the initial density profile,
i.e., to the early stages of the collision. This implies in
turn that this observable has limited sensitivity to the
details of the hydrodynamic expansion in general, and
to the transport coefficients of the fluid in particular, as
nicely confirmed by the hydrodynamic results (Fig. 9) of
Ref. [42]. We have found that hptifluctuations are driven
by fluctuations of the initial energy over entropy ratio
Ei/S, and not by the fluctuations of the fireball size as
previously thought. By use of Eq. (7), models of initial
conditions that fit anisotropic flow data and multiplic-
ity fluctuations also naturally reproduce the centrality
dependence of ρ2and ρ3measured by the ATLAS Col-
laboration without any further adjustment. Note that
experimental data are also available for p+Pb collisions,
the study of which we leave for future work.
ACKNOWLEDGMENTS
FGG was supported by CNPq (Conselho Nacional
de Desenvolvimento Cientifico) grant 312932/2018-9, by
INCT-FNA grant 464898/2014-5 and FAPESP grant
2018/24720-6. G.G., M.L. and J.-Y.O. were supported by
USP-COFECUB (grant Uc Ph 160-16, 2015/13). J.N.H.
acknowledges the support of the Alfred P. Sloan Founda-
tion, support from the US-DOE Nuclear Science Grant
No. de-sc0019175. J.-Y. O. thanks Piotr Bo˙zek for
discussions. G.G. acknowledges useful discussions with
Derek Teaney and Bj¨orn Schenke.
Appendix A: Varying the parametrization of the
initial profile
We check the sensitivity of ρn, as defined by Eq. (7),
to the parametrization of initial conditions. Figure 4 dis-
plays the variation of ρnfor three different values of p
in the TRENTo model. Several qualitative trends are ro-
bust: ρ2and ρ3are both positive for central collisions; As
the number of participants decreases from its maximum
value, ρ2steeply increases and then decreases, eventually
becoming negative, while the centrality dependence of ρ3
is milder. But significant differences appear at the quan-
titative level, and the value p= 0, which is the preferred
value also for other observables [18, 28], agrees best with
the recent ρndata. We have also studied the dependence
on the parameter kgoverning the magnitude of fluctu-
ations in TRENTo. Results in Fig. 3 are obtained with
k= 2, but we have also carried out calculations with
k= 1, corresponding to larger fluctuations. We have
found (not shown) that the results for ρ2are essentially
unchanged except for a minor increase in central colli-
sions, while the variation of ρ3becomes flatter, similar
to the p=−1 results in Fig. 4.
[1] U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci.
63, 123 (2013) doi:10.1146/annurev-nucl-102212-170540
[arXiv:1301.2826 [nucl-th]].

6
[2] G. Aad et al. [ATLAS Collaboration], JHEP 1311, 183
(2013) doi:10.1007/JHEP11(2013)183 [arXiv:1305.2942
[hep-ex]].
[3] A. M. Sirunyan et al. [CMS Collaboration], Phys. Lett.
B789, 643 (2019) doi:10.1016/j.physletb.2018.11.063
[arXiv:1711.05594 [nucl-ex]].
[4] S. Acharya et al. [ALICE Collaboration],
Phys. Rev. Lett. 123, no. 14, 142301 (2019)
doi:10.1103/PhysRevLett.123.142301 [arXiv:1903.01790
[nucl-ex]].
[5] G. Aad et al. [ATLAS Collaboration], Phys. Rev. C 90,
no. 2, 024905 (2014) doi:10.1103/PhysRevC.90.024905
[arXiv:1403.0489 [hep-ex]].
[6] J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett.
117, 182301 (2016) doi:10.1103/PhysRevLett.117.182301
[arXiv:1604.07663 [nucl-ex]].
[7] A. M. Sirunyan et al. [CMS Collaboration],
arXiv:1910.08789 [hep-ex].
[8] S. Acharya et al. [ALICE Collaboration],
arXiv:2002.00633 [nucl-ex].
[9] P. Bozek, Phys. Rev. C 93, no. 4, 044908 (2016)
doi:10.1103/PhysRevC.93.044908 [arXiv:1601.04513
[nucl-th]].
[10] G. Aad et al. [ATLAS Collaboration], Eur. Phys. J. C
79, no. 12, 985 (2019) doi:10.1140/epjc/s10052-019-7489-
6 [arXiv:1907.05176 [nucl-ex]].
[11] P. Bozek and H. Mehrabpour, arXiv:2002.08832 [nucl-th].
[12] G. Giacalone, J. Noronha-Hostler, M. Luzum and
J. Y. Ollitrault, Phys. Rev. C 97, no. 3, 034904 (2018)
doi:10.1103/PhysRevC.97.034904 [arXiv:1711.08499
[nucl-th]].
[13] B. Alver et al. [PHOBOS Collabora-
tion], Phys. Rev. Lett. 98, 242302 (2007)
doi:10.1103/PhysRevLett.98.242302 [nucl-ex/0610037].
[14] B. Alver and G. Roland, Phys. Rev. C 81,
054905 (2010) Erratum: [Phys. Rev. C 82, 039903
(2010)] doi:10.1103/PhysRevC.82.039903, 10.1103/Phys-
RevC.81.054905 [arXiv:1003.0194 [nucl-th]].
[15] D. Teaney and L. Yan, Phys. Rev. C 83, 064904 (2011)
doi:10.1103/PhysRevC.83.064904 [arXiv:1010.1876
[nucl-th]].
[16] F. G. Gardim, F. Grassi, M. Luzum and
J. Y. Ollitrault, Phys. Rev. C 85, 024908 (2012)
doi:10.1103/PhysRevC.85.024908 [arXiv:1111.6538
[nucl-th]].
[17] H. Niemi, G. S. Denicol, H. Holopainen and P. Huovi-
nen, Phys. Rev. C 87, no. 5, 054901 (2013)
doi:10.1103/PhysRevC.87.054901 [arXiv:1212.1008
[nucl-th]].
[18] G. Giacalone, J. Noronha-Hostler and J. Y. Ol-
litrault, Phys. Rev. C 95, no. 5, 054910 (2017)
doi:10.1103/PhysRevC.95.054910 [arXiv:1702.01730
[nucl-th]].
[19] W. Broniowski, M. Chojnacki and L. Obara, Phys. Rev.
C80, 051902 (2009) doi:10.1103/PhysRevC.80.051902
[arXiv:0907.3216 [nucl-th]].
[20] P. Bozek and W. Broniowski, Phys. Rev. C 85,
044910 (2012) doi:10.1103/PhysRevC.85.044910
[arXiv:1203.1810 [nucl-th]].
[21] P. Boek and W. Broniowski, Phys. Rev. C 96,
no. 1, 014904 (2017) doi:10.1103/PhysRevC.96.014904
[arXiv:1701.09105 [nucl-th]].
[22] F. G. Gardim, G. Giacalone, M. Luzum and J. Y. Olli-
trault, arXiv:2002.07008 [nucl-th].
[23] J. Adams et al. [STAR Collaboration], Phys. Rev. C 72,
044902 (2005) doi:10.1103/PhysRevC.72.044902 [nucl-
ex/0504031].
[24] J. Noronha-Hostler, G. S. Denicol, J. Noronha,
R. P. G. Andrade and F. Grassi, Phys. Rev. C 88,
no. 4, 044916 (2013) doi:10.1103/PhysRevC.88.044916
[arXiv:1305.1981 [nucl-th]].
[25] J. Noronha-Hostler, J. Noronha and F. Grassi,
Phys. Rev. C 90, no. 3, 034907 (2014)
doi:10.1103/PhysRevC.90.034907 [arXiv:1406.3333
[nucl-th]].
[26] J. Noronha-Hostler, J. Noronha and M. Gyu-
lassy, Phys. Rev. C 93, no. 2, 024909 (2016)
doi:10.1103/PhysRevC.93.024909 [arXiv:1508.02455
[nucl-th]].
[27] J. S. Moreland, J. E. Bernhard and S. A. Bass,
Phys. Rev. C 92, no. 1, 011901 (2015)
doi:10.1103/PhysRevC.92.011901 [arXiv:1412.4708
[nucl-th]].
[28] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu
and U. Heinz, Phys. Rev. C 94, no. 2, 024907 (2016)
doi:10.1103/PhysRevC.94.024907 [arXiv:1605.03954
[nucl-th]].
[29] J. Vredevoogd and S. Pratt, Phys. Rev. C 79,
044915 (2009) doi:10.1103/PhysRevC.79.044915
[arXiv:0810.4325 [nucl-th]].
[30] W. van der Schee, P. Romatschke and S. Pratt,
Phys. Rev. Lett. 111, no. 22, 222302 (2013)
doi:10.1103/PhysRevLett.111.222302 [arXiv:1307.2539
[nucl-th]].
[31] A. Kurkela, A. Mazeliauskas, J. F. Paquet, S. Schlicht-
ing and D. Teaney, Phys. Rev. Lett. 122, no.
12, 122302 (2019) doi:10.1103/PhysRevLett.122.122302
[arXiv:1805.01604 [hep-ph]].
[32] P. F. Kolb, P. Huovinen, U. W. Heinz and H. Heisel-
berg, Phys. Lett. B 500, 232 (2001) doi:10.1016/S0370-
2693(01)00079-X [hep-ph/0012137].
[33] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz,
S. Krieg and K. K. Szabo, Phys. Lett. B 730, 99 (2014)
doi:10.1016/j.physletb.2014.01.007 [arXiv:1309.5258
[hep-lat]].
[34] P. Alba, V. Mantovani Sarti, J. Noronha, J. Noronha-
Hostler, P. Parotto, I. Portillo Vazquez and
C. Ratti, Phys. Rev. C 98, no. 3, 034909 (2018)
doi:10.1103/PhysRevC.98.034909 [arXiv:1711.05207
[nucl-th]].
[35] D. Teaney, Phys. Rev. C 68, 034913 (2003)
doi:10.1103/PhysRevC.68.034913 [nucl-th/0301099].
[36] A. Bazavov et al. [HotQCD Collaboration], Phys. Lett.
B795, 15 (2019) doi:10.1016/j.physletb.2019.05.013
[arXiv:1812.08235 [hep-lat]].
[37] B. Schenke, S. Jeon and C. Gale, Phys. Rev.
C82, 014903 (2010) doi:10.1103/PhysRevC.82.014903
[arXiv:1004.1408 [hep-ph]].
[38] B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett.
106, 042301 (2011) doi:10.1103/PhysRevLett.106.042301
[arXiv:1009.3244 [hep-ph]].
[39] J. F. Paquet, C. Shen, G. S. Denicol, M. Luzum,
B. Schenke, S. Jeon and C. Gale, Phys. Rev. C 93,
no. 4, 044906 (2016) doi:10.1103/PhysRevC.93.044906
[arXiv:1509.06738 [hep-ph]].
[40] F. G. Gardim, G. Giacalone, M. Luzum and J. Y. Olli-
trault, arXiv:1908.09728 [nucl-th].

7
[41] F. G. Gardim, F. Grassi, M. Luzum and J. Noronha-
Hostler, Phys. Rev. C 95, no. 3, 034901 (2017)
doi:10.1103/PhysRevC.95.034901 [arXiv:1608.02982
[nucl-th]].
[42] B. Schenke, C. Shen and D. Teaney, [arXiv:2004.00690
[nucl-th]].