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Energy dependence of transverse momentum fluctuations in Au+Au collisions from a
multiphase transport model
Liuyao Zhang ID ,1, 2, 3 Jinhui Chen ID ,2, 3, ∗and Chunjian Zhang ID 2, 3, 4, †
1Institute of Nuclear Science and Technology, Henan Academy of Sciences, Zhengzhou, 450015, China
2Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),
and Institute of Modern Physics, Fudan University, Shanghai 200433, China
3Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China
4Department of Chemistry, Stony Brook University, Stony Brook, NY 11794, USA
Event-by-event mean transverse momentum fluctuations (⟨pT⟩) serve as a sensitive probe of ini-
tial state overlap geometry and energy density fluctuations in relativistic heavy-ion collisions. We
present a systematic investigation of ⟨pT⟩fluctuations in Au+Au collisions at √sNN = 3.0–19.6 GeV,
examining their centrality and energy dependence with the framework of an improved multiphase
transport (AMPT) model. The centrality dependence of the pTcumulants up to fourth order devi-
ates significantly from simple powering-law scaling. Scaled cumulants are performed, with variances
aligning well with the trends observed in the experimental data. Employing a two-subevent method,
short-range correlations are slightly suppressed compared to the standard approach. Furthermore,
baryons exhibit more pronounced ⟨pT⟩fluctuations than mesons, potentially attributable to the ef-
fect of radial flow. These results provide referenced insights into the role of initial state fluctuations
across different energies in heavy-ion collisions.
I. INTRODUCTION
Event-by-event (EbE) fluctuations have been proposed
as a useful probe for exploring initial state and thermal-
ization in heavy-ion collision at relativistic energies [1–
7], offering critical insights into the properties of Quark-
Gluon Plasma (QGP) formation [8]. These fluctuations
are particularly significant for exploring the Quantum
Chromodynamics (QCD) phase diagram, including the
QGP-to-hadron gas transition and the potential exis-
tence of a critical point in strongly interacting mat-
ter [9, 10]. Fluctuations in the positions of participat-
ing nucleons lead to EbE fluctuations in the initial state,
which propagate to final-state observables during system
expansion [11–14]. The temperature fluctuations associ-
ated with phase transition in the QCD phase diagram
can manifest themselves in event-wise mean transverse
momentum (⟨pT⟩) fluctuations of the final-state parti-
cles [15]. ⟨pT⟩is sensitive to the initial energy density
and inversely proportional to the size of the overlap re-
gion [16, 17], while it is also influenced by collective
behavior, fluctuations in the participant nucleons, and
other final-state effects [18].
The dynamical ⟨pT⟩fluctuations estimated by variance
have been extensively studied in the Super Proton Syn-
chrotron (SPS) [1, 19–22], Relativistic Heavy-Ion Col-
lider (RHIC) [2, 3, 23–25], and the Large Hadron Col-
lider (LHC) [4, 26, 27]. These results reveal a univer-
sal multiplicity dependence, where correlations are pro-
gressively diluted with increasing participant numbers,
consistent with dominance by particle pairs originating
from the same nucleon-nucleon collisions. Scaled dy-
∗chenjinhui@fudan.edu.cn
†chunjianzhang@fudan.edu.cn
namical correlations exhibit minimal dependence on the
collision energy, as observed in both small and large
collisions [26, 28]. Recently, higher-order ⟨pT⟩fluctua-
tions have also been investigated in experiments to un-
cover the intricate mechanisms underlying the observed
fluctuations. The ALICE experiment reports a positive
skewness in ⟨pT⟩fluctuations in minimum-bias pp col-
lisions and across all centralities in large collision sys-
tems [4], consistent with hydrodynamic simulation pre-
dictions [18, 29, 30]. Furthermore, the kurtosis of ⟨pT⟩
fluctuations in most central Pb+Pb collisions aligns with
that of Gaussian distribution, suggesting the formation of
a locally thermalized system [4]. The ATLAS collabora-
tion has also observed distinct changes in the mean, vari-
ance, and skewness of ⟨pT⟩distributions in ultra-central
Pb+Pb and Xe+Xe collisions, demonstrating a clear dis-
entanglement of geometrical and intrinsic components
through the analysis of the speed of sound [31]. Neverthe-
less, such EbE ⟨pT⟩fluctuations have also been proposed
as a tool to study nuclear structure effect [17, 32–35].
From a theoretical perspective, ⟨pT⟩fluctuations have
been explored within the color glass condensate (CGC)
framework [36, 37], which has demonstrated strong con-
cordance with data in semicentral and central colli-
sions [38]. The UrQMD model [39, 40], incorporat-
ing only hadronic transport, successfully reproduces the
relative dynamical correlation in Au+Au central colli-
sions at both RHIC [41] and LHC [26] energies. Both
PYTHIA8 [42] and EPOSLHC [43] also have success-
fully modeled relative dynamical correlation at the LHC
energy, and provided insight into the interpretation of
the data [27]. Similarly, the AMPT model, particularly
with the string-melting version demonstrates good agree-
ment with experimental measurements at the LHC en-
ergy [26, 27, 44]. In higher-order ⟨pT⟩fluctuations, HI-
JING simulations exhibit a power-law scaling on charged
particle multiplicity in both skewness and kurtosis, con-
arXiv:2501.08209v1 [nucl-th] 14 Jan 2025
2
sistent with a superposition of independent sources [18].
Moreover, hydrodynamic calculations can explain the
measurements of both skewness and kurtosis quanti-
tively from semicentral to central collisions at LHC en-
ergy [4, 30].
Given that dynamical fluctuations in two-particle pT
correlations have been measured during the RHIC Beam
Energy Scan (BES) program [41], which aims to inves-
tigate the QCD first-order phase transition and identify
the possible critical point, there remains a notable ab-
sence of dynamic studies focused on higher-order ⟨pT⟩
fluctuations stemming from model calculation.
In this paper, we study the collision energy depen-
dence of ⟨pT⟩fluctuations, from mean to kurtosis, us-
ing an improved AMPT model. Variances are calculated
using both the standard and two-subevent methods, re-
vealing underlying decorrelation effects in ⟨pT⟩fluctua-
tions. We further investigate baryon and meson fluctua-
tions across different collision energies and systematically
examine the effects of various configurations and accep-
tances. The paper is organized as follows. Section II in-
troduces the formula for calculating pTcumulants using
standard and subevent methods. Sec. III describes the
improvements to the AMPT model, incorporating system
size and collision energy dependences in the Lund frag-
mentation parameter. The main results are presented in
Sec. IV, followed by a summary in Sec. V.
II. METHODOLOGY
Following the ⟨pT⟩cumulants approach in Refs. [18,
45], the n-particle pTcorrelator in one event is defined
as
cn=P
i1=···=in
wi1···win(pT,i1− ⟨⟨pT⟩⟩)···(pT,in− ⟨⟨pT⟩⟩)
P
i1=···=in
wi1···win
.
(1)
Here, wirepresents the weight of particle i, and the rela-
tion expands algebraically into a simple polynomial form,
pmk =X
i
wk
ipm
i/X
i
wk
i, τk=P
i
wk+1
i
P
i
wik+1 ,
¯p1k≡p1k− ⟨⟨pT⟩⟩,
¯p2k≡p2k−2p1k⟨⟨pT⟩⟩ +⟨⟨pT⟩⟩2,
¯p3k≡p3k−3p2k⟨⟨pT⟩⟩ + 3p1k⟨⟨pT⟩⟩2− ⟨⟨pT⟩⟩3,
¯p4k≡p4k−4p3k⟨⟨pT⟩⟩ + 6p2k⟨⟨pT⟩⟩2−4p1k⟨⟨pT⟩⟩3
+⟨⟨pT⟩⟩4.
(2)
Where ⟨⟨pT⟩⟩ = ¯p11 is the ⟨pT⟩averaged over the event
ensemble, with p≡pT.
Using auxiliary variables, the correlator in Eq. 1 can
be expressed as,
c2=¯p2
11 −τ1¯p22
1−τ1
,
c3=¯p3
11 −3τ1¯p22 ¯p11 + 2τ2¯p33
1−3τ1+ 2τ2
,
c4=¯p4
11 −6τ1¯p22 ¯p2
11 + 3τ2
1¯p2
22 + 8τ2¯p33 ¯p11 −6τ3¯p44
1−6τ1+ 3τ2
1+ 8τ2−6τ3
.
(3)
where particles are selected from pseudorapidity range
|η|<1.0 and 0.5 ≤pT≤3.0 GeV/c, considering all
unique combinations within each event. The advantage
of employing multiparticle pTcorrelator [46, 47] is that
they yield zero values for events with randomly sampled
particles, thereby effectively isolating the non-statistical
fluctuations of interest.
In addition, two-subevent method (2sub) is employed,
where particle combinations are selected from two η-
separated subevents, a (-1.0 < η < η1) and c ( η2< η <
1.0). This ηgap could reduce short-range correlations.
To evaluate the impact of these correlations on high-order
cumulants, the gap is varied to 0.4 and 0.8. The resulting
correlations are given by
c2,sub = (¯p11)a( ¯p11 )c(4)
where the subscripts in Eq. 4 indicate the subevents from
which particles are selected. Since there are two alterna-
tive ways to calculate two-subevent c3, the final value is
taken as the average, c3,2sub = (c3,2sub1 +c3,2sub2)/2.
In Eq. 1, the n-particle pTcorrelator varies with in-
cident energy and collision centrality. To account for
these variations, the measured correlations for variance,
skewness, and kurtosis are scaled by ⟨⟨pT⟩⟩, respectively,
yielding dimensionless scaled cumulants.
k2=p⟨c2⟩
⟨⟨pT⟩⟩,
k3=
3
p⟨c3⟩
⟨⟨pT⟩⟩,
k4=
4
q⟨c4⟩ − 3⟨c2⟩2
⟨⟨pT⟩⟩ .
(5)
For the subevent method, the corresponding scaled cu-
mulants are given by
k2,2sub =s⟨c2,2sub⟩
⟨⟨pT⟩⟩a⟨⟨pT⟩⟩c
.(6)
III. AMPT MODEL
AMPT is an effective Monte Carlo framework, which
is extensively used to study relativistic heavy-ion colli-
sions at SPS, RHIC and the LHC energies [49–62]. The
AMPT model has two versions: the default version (de-
noted as Def-AMPT) and the string-melting (denoted as
3
0.5 1 1.5 2
(GeV/c)
T
p
2−
10
1−
10
1
10
2
10
3
10
-2
(GeV/c)
dy
T
dp
T
pN
2
d
π2
1
STAR 1.0)× 7.7 ( 2.0)×11.5 ( 4.0)×14.5 ( 6.0)×19.6 (
Au+Au (0-5%)
)
+
π(
0.5 1 1.5 2
(GeV/c)
T
p
2−
10
1−
10
1
10
2
10
3
10
-2
(GeV/c)
dy
T
dp
T
pN
2
d
π2
1
Def-AMPT
1.0)× 7.7 ( 2.0)×11.5 ( 4.0)×14.5 ( 6.0)×19.6 (
(p)
0.5 1 1.5 2
(GeV/c)
T
p
0.5
1
1.5
MODEL/STAR
0.5 1 1.5 2
(GeV/c)
T
p
2−
10
1−
10
1
10
2
10
3
10
-2
(GeV/c)
dy
T
dp
T
pN
2
d
π2
1
)
+
(K
0.5 1 1.5 2
(GeV/c)
T
p
0.5
1
1.5
Model/Data
0.5 1 1.5 2
(GeV/c)
T
p
0.5
1
1.5
MODEL/DATA
FIG. 1. (Color online) Transverse momentum spectra at midrapidity (|y|<0.1) for π+(left), K+(middle), and protons (right)
in Au+Au collisions at √sNN = 7.7, 11.5, 14.5, and 19.6 GeV for the 0–5% centrality class. The spectra are scaled for clarity,
with data from Ref. [48]. The lower panels show the model to data ratio.
SM-AMPT) version. Both versions comprise four main
dynamic components: (i) a fluctuating initial condition
from the Heavy Ion Jet INteraction Generator (HIJING)
model [63], (ii) parton cascade simulations using the
Zhang’s Partonc cascade (ZPC) model [64], which in-
cludes only 2 to 2 elastic parton processes, (iii) hadroniza-
tion via the Lund string fragmentation or a quark co-
alescence model, and (iv) hadron scattering described
by A Relativistic Transport (ART) model [65]. This
study employs the latest version, the AMPT-v1.26t9b-
v2.26t9b [66, 67], with the QCD coupling constant αs=
0.33, the screening mass µ=2.265 fm−1corresponding to
a parton scattering cross section of 3.0 mb in the ZPC.
The default AMPT model is utilized primarily in this
study because of its success in describing the transverse
momentum spectra of identified particles in heavy-ion
collisions at SPS and RHIC energies [66, 68]. The ⟨⟨pT⟩⟩
is expected to increase with centrality due to higher ini-
tial temperatures in more central collisions [48]. How-
ever, neither the original default version nor the string-
melting version of AMPT adequately reproduces the
⟨⟨pT⟩⟩ as a function of centrality near midrapidity. [7, 69].
To address the reversed trend predicted by AMPT, the
Lund fragmentation parameters were tuned to introduce
a system-size dependence. According to the Lund string
fragmentation mechanism, the average squared trans-
verse momentum of produced particles is proportional
to the string tension κ, which reflects the energy stored
per unit length of the string and is determined by the
fragmentation parameters aLand bL[50], as shown in
Eq. 7.
κ∝p2
T=1
bL(2 + aL).(7)
Inspired by Ref. [7], the Lund string fragmentation pa-
rameter bLis treated as a local variable dependent on the
nuclear thickness functions of two nuclei, rather than be-
ing a constant parameter. Consequently, bLexhibits an
approximate linear dependence on the impact parameter.
For simplicity, bLis tuned in this study to vary linearly
with the impact parameter, closely reproducing the lin-
ear relationship observed in Ref. [7]. This tuning enables
the AMPT model to reasonably describe the centrality
dependence of ⟨⟨pT⟩⟩. The ⟨⟨pT⟩⟩ also shows a systematic
dependence on incident energy [48], with higher incident
energies yielding larger ⟨⟨pT⟩⟩ due to enhanced collective
effects. To account for this, bLis further adjusted to
increase slightly with collision energy, allowing the im-
proved AMPT to reproduce the energy dependence of
⟨⟨pT⟩⟩, as shown in Fig. 3 (a).
In this study, pTcumulants are calculated for π±,
K±and p(¯p) within |η|<1.0 and 0.5 ≤pT≤
3.0 GeV/c. Centrality classes in Au+Au collisions at
√sNN = 7.7–19.6 GeV are defined by the number of par-
ticipants Npart, directly extracted from AMPT calcu-
lation. Events are divided into the following central-
ity classes: 0–5%, 5–10%, 10–20%, 20–30%, 30–40%,
40–50%, 50–60%, 60-70%, and 70–80%. The mean num-
ber of participating nucleons, ⟨Npart ⟩, corresponds to
these centrality intervals.
IV. RESULTS AND DISCUSSIONS
To evaluate the validity of the enhanced AMPT model
relative to its original version, we first examine the pT
distributions of select particles. Figure 1 presents the pT
spectra for π+(left), K+(middle), and protons (right)
within the midrapidity interval (|y|<0.1) for Au+Au
collisions at √sNN = 7.7, 11.5, 14.5, and 19.6 GeV,
4
specifically in the 0-5% centrality class. The ratios of
the AMPT results to experimental data are computed
to quantify discrepancies. Overall, the improved AMPT
model qualitatively reproduces the experimental observ-
ables for hadrons across a broad pTspectrum, spanning
a variety of incident energies. For clarity, the pTspectra
for different energies have been appropriately scaled.
0.7 0.8 0.9 1 1.1 1.2
(GeV/c)〉
T
p〈
4−
10
3−
10
2−
10
1−
10
1
counts
Def-AMPT
7.7 GeV
9.1 GeV
11.5 GeV
14.5 GeV
17.3 GeV
19.6 GeV
(a) (0-5%)
0.4 0.6 0.8 1 1.2 1.4
(GeV/c)〉
T
p〈
4−
10
3−
10
2−
10
1−
10
1
counts
(60-70%)(b)
FIG. 2. (Color online) Distributions of ⟨pT⟩from AMPT
simulations for Au+Au collisions at √sNN = 7.7, 9.1, 11.5,
14.5, and 19.6 GeV, calculated for π±,K±, and p(¯p) within
0.5 ≤pT≤3.0 GeV/cand (|η|<1.0) for the 0–5% (a), and
60–70% (b) centrality classes, respectively.
The ⟨pT⟩fluctuations can be straightforwardly studied
through its event-wise distributions. Figure 2 shows the
⟨pT⟩distributions for the 0–5% (a) and 50–60% (b) cen-
trality classes in Au+Au collisions at √sNN = 7.7, 9.1,
11.5, 14.5, 17.3, and 19.6 GeV. The analysis reveals ⟨pT⟩
values for both centrality classes steadily increase with
higher collision energies. In peripheral collisions (50–60%
centrality), the ⟨pT⟩distributions exhibit greater vari-
ances, indicating enhanced fluctuations. Additionally,
these distributions show pronounced asymmetry, charac-
terized by a significant rightward tail, suggesting stronger
positive skewness in peripheral collisions compared to
central collisions. This behavior is consistent with exper-
imental observations at RHIC and at LHC [4, 23, 70, 71].
Figure 3 presents the event-ensemble pTcumulants,
including the mean ⟨⟨pT⟩⟩ (a), variance ⟨c2⟩(b), skew-
ness ⟨c3⟩(c), and kurtosis ⟨c4⟩ − 3⟨c2⟩2(d) of pTdis-
tributions in Au+Au collisions over beam energies from
7.7 to 19.6 GeV. These cumulants are shown as a func-
tion of collision centrality, from peripheral 70–80% to
the most central 0–5%. The ⟨⟨pT⟩⟩ values from the en-
hanced AMPT model exhibit a pronounced centrality de-
pendence, with higher values observed in more central
collisions, consistent with previously published AMPT
calculations [7] and the experimental measurements from
STAR [48]. Additionally, a significant beam energy de-
pendence is observed, where lower collision energies yield
smaller ⟨⟨pT⟩⟩ values across all centrality bins. Similar
dependences on centrality or multiplicity, and beam en-
ergies for the ⟨⟨pT⟩⟩ distributions have also been reported
at RHIC [48] and the LHC energies [27, 71]. These trends
can be attributed to increased particle production and
stronger collective flow in more central collisions, where
the overlap region of the colliding nuclei is larger. Higher
beam energies deposit more energy in the collision zone,
leading to higher temperatures and stronger radial flow
in the strongly expanding QGP fireball.
The second-order pTcumulant, ⟨c2⟩, exhibits a non-
zero values across 7.7–19.6 GeV, as shown in panel (b)
of Fig. 3, suggesting significant EbE fluctuations. An
inverse dependence of variance on centrality is consis-
tently observed across all energies, aligning with find-
ings from the STAR and HIJING results [24]. However,
the HIJING calculations show an opposite energy de-
pendence in peripheral versus central collisions, in con-
trast to the consistent energy dependence observed in
published STAR measurements. Notably, our findings
exhibit a systematic collision energy dependence across
various centrality intervals. The dilution of correlations
with increasing centrality may result from a reduction in
particle-pair correlations if they are dominated by parti-
cles originating from the same nucleon-nucleon collisions.
This interpretation is intuitively cross-validated by ⟨pT⟩
distributions across 60–70% and 0–5% centrality classes,
as shown in Fig. 2.
The higher-order cumulants, skewness ⟨c3⟩and kur-
tosis ⟨c4⟩ − 3⟨c2⟩2are presented in panels (c) and (d)
of Fig. 3, respectively. Both correlators exhibit signifi-
cant dependences on centrality and beam energy across
√sNN = 7.7–19.6 GeV. Their magnitudes decrease by
more than one order of magnitude with increasing cen-
trality classes ⟨Npart⟩. Due to the lower statistics, we
admit that the kurtosis calculations at 7.7 GeV are not
shown.
To mitigate the influence of variations in ⟨⟨pT⟩⟩ with
incident energy and/or centrality, scaled quantities in-
cluding variances k2, skewness k3, and kurtosis k4, nor-
malized by ⟨⟨pT⟩⟩, are shown as functions of centrality
for Au+Au collisions across √sNN = 7.7-19.6 GeV in
Fig. 4, respectively. Similarly, the kn(n=2,3,4) exhibits
significant centrality and energy dependence. Notely, the
scaled variance k2qualitatively reproduce the trends ob-
served in the published STAR experiment [41], and also
demonstrates remarkable agreement with the available
STAR measurements at 7.7, 11.5, and 14.5 GeV, vali-
dating the model’s predictive capabilities in these energy
ranges. However, a systematic overestimation merges,
particularly in peripheral to mid-central collisions. Ad-
ditionally, these scaled variances knexhibit an approx-
imate power-law behavior concerning centrality depen-
dence across different collision energies, supporting the
independent source pictures.
In order to minimize the impact of short-range corre-
lations from jets and resonance decays, a two-subevent
method is employed and compared with the standard
method (std) for √sNN = 7.7–19.6 GeV, as shown in
Fig. 5. The ηgap between the two subevents is set to
0.4. The results, calculated for the particles within |η|<
5
10 2
10
〉
part
N〈
0.75
0.8
0.85
0.9
〉〉
T
p〈〈
(a)Au+Au
10 2
10
〉
part
N〈
9−
10
8−
10
7−
10
6−
10
5−
10
4−
10
〉
3
c〈
(c)
10 2
10
〉
part
N〈
4−
10
3−
10
〉
2
c〈
Def-AMPT
7.7 GeV
9.1 GeV
11.5 GeV
14.5 GeV
17.3 GeV
19.6 GeV
(b)
10 2
10
〉
part
N〈
11−
10
10−
10
9−
10
8−
10
7−
10
6−
10
5−
10
4−
10
2
〉
2
c〈 - 3 〉
4
c〈
(d)
FIG. 3. (Color online) The pTcumulants, including event-ensemble transverse momentum ⟨⟨pT⟩⟩ (a), variance ⟨c2⟩(b), skewness
⟨c3⟩(c), and kurtosis ⟨c4⟩ − 3⟨c2⟩2(d), are calculated for selected particles within 0.5 ≤pT≤3.0 GeV/c and |η|<1.0 as a
function of ⟨Npart⟩from peripheral to central collisions across difference collision energies in 7.7, 9.1, 11.5, 14.5, 17.3, and 19.6
GeV.
50 100 150 200 250 300 350
〉
part
N〈
2
4
6
8
10
(%)
n
k
7.7
9.1
11.5
14.5
17.3
19.6
7.7
9.1
11.5
14.5
17.3
19.6
7.7
11.5
14.5
19.6
Def-AMPTSTAR
(a)
n = 2
50 100 150 200250 300 350
〉
part
N〈
2
4
6
8
10
(%)
(b)
n = 3
50 100 150200 250 300 350
〉
part
N〈
2
4
6
8
10
(%)
(c)
n = 4
FIG. 4. (Color online) The scaled variance k2(a), skewness k3(b), and kurtosis k4(c) as a function of centrality in Au+Au
collisions across different collision energies with the AMPT model.
1.0, show that the values obtained using the two-subevent
method are slightly suppressed compared to the standard
method. However, in HIJING simulation for Pb+Pb col-
lisions at LHC energies, the k2is suppressed by a factor of
3 when using the two-subevent method with particles se-
lected from |η|<2.5 [18]. This suppression is attributed
to the decorrelation effect in the dynamic evolution of
the fireball [72].
In high baryon chemical potential regions, baryons
have similar statistics with mesons. To further explore
the radial flow mechanism, we analyzed the scaled vari-
ance, skewness, and kurtosis at √sNN = 19.6 GeV un-
6
50 100 150 200 250 300350 400
〉
part
N〈
1
2
3
4
5
6
(%)
2
k
Def-AMPT: std = 0.4η∆Def-AMPT: 2sub
STAR
(a)
7.7 GeV
50 100 150 200 250 300 350 400
〉
part
N〈
1
2
3
4
5
6
(%)
2
k
(c)
11.5 GeV
50 100 150 200 250 300 350 400
〉
part
N〈
1
2
3
4
5
6
(%)
2
k
(e)
17.3 GeV
50 100 150 200 250 300 350 400
〉
part
N〈
1
2
3
4
5
6
(%)
2
k
(b)
9.1 GeV
50 100 150 200 250 300350 400
〉
part
N〈
1
2
3
4
5
6
(%)
2
k
(d)
14.5 GeV
50 100 150 200 250 300 350 400
〉
part
N〈
1
2
3
4
5
6
(%)
2
k
(f)
19.6 GeV
FIG. 5. (Color online) The scaled variance k2as a function of centrality ⟨Npart⟩in Au+Au collisions at √sNN = 7.7, 9.1, 11.5,
14.5, 17.3, and 19.6 GeV is calculated within AMPT model using both the standard method (std) and two-subevent (2sub)
method, with ∆η= 0.4.
50 100 150 200 250 300 350
〉
part
N〈
1
2
3
4
5
6
7
8
(%)
n
k
hadron
baryon
meson
n = 2(a)
50 100 150 200 250 300 350
〉
part
N〈
1
2
3
4
5
6
7
8
(%)
n = 3(b)
50 100 150 200 250 300 350
〉
part
N〈
1
2
3
4
5
6
7
8
(%)
n = 4(c)
FIG. 6. (Color online) The scaled variance k2(a), skewness k3(b) and kurtosis k4(c) are shown as a function of centrality
⟨Npart⟩for “hadron”, “baryon” and “meson” at a collision energy of 19.6 GeV.
der three scenarios, “hadron”, “baryon”, and “meson”,
as shown in Fig. 6. The results reveal that baryons ex-
hibit more pronounced fluctuations in scaled variance,
skewness, and kurtosis compared to mesons. This be-
havior might be attributed to the effects of radial flow,
which preferentially accelerates heavier particles, push-
ing them from lower to higher pTvalues more effectively
than lighter particles [73].
To dynamically investigate the factors contributing to
EbE ⟨pT⟩fluctuations in the AMPT model, we compared
the scaled variance k2in Au+Au collisions at √sNN =
19.6 GeV using the standard method across three AMPT
configurations: default version, the string-melting ver-
sion without parton scattering (denoted as SM–AMPT:
σp=0 mb), and the string-melting version with a par-
ton cross section of 3 mb (SM–AMPT), as shown in
Fig. 7. The default configuration exhibits more pro-
nounced ⟨pT⟩fluctuations compared to string-melting
version, with and without partonic interactions. The
discrepancy likely arises from differences in hadroniza-
tion mechanisms. Furthermore, the minor variation be-
tween the string-metling AMPT versions with and with-
out partonic interactions suggests that partonic evolu-
tions slightly suppress the EbE fluctuations.
Figure 8 presents the scaled variance k2as a func-
tion of collision energy, ranging from 19.6 to 3.0 GeV
7
50 100 150 200 250 300 350 400
〉
part
N〈
1
2
3
4
5
6
(%)
2
k
Def-AMPT
= 0 mb
p
σSM-AMPT:
SM-AMPT
FIG. 7. (Color online) The scaled variance k2is calcu-
lated for Au+Au collisions as a function of ⟨Npart⟩at 19.6
GeV using three configurations: the standard method in
default (Def-AMPT), normal string-melting (SM-AMPT),
and string-melting without partonic interactions (SM-AMPT:
σp= 0 mb).
0 2 4 6 8 10 12 14 16 18 20
(GeV)
NN
s
0
0.5
1
1.5
2
2.5
(%)
2
k
< 3.0 GeV/c
T
| < 0.5 & 0.5 < pη| < 2.0 GeV/c
T
| < 0.5 & 0.2 < pη| < 3.0 GeV/c
T
| < 1.0 & 0.5 < pη| < 2.0 GeV/c
T
| < 1.0 & 0.2 < pη|
UrQMD
STAR
0-5%
FIG. 8. (Color online) The scaled variance k2within differ-
ent ηrange, calculated using both standard and two-subevent
methods for Au+Au collisions, is shown as a function of col-
lision energy for the 0–5% centrality. Results from AMPT
simulations are compared to STAR experimental data [41]
and UrQMD calculations [41].
in Au+Au collisions for the most central regions (0–5%).
Considering the varying transformation efficiency from
initial geometry to momentum space across different pT
ranges [74], the analysis includes acceptance ranges in-
cluding |η|<1.0 for 0.2 < pT<2.0 GeV/c and 0.5
< pT<3.0 GeV/c, as well as |η|<0.5 for the same
pTintervals. For comparison, data from STAR experi-
ment [41] for |η|<0.5, along with results from UrQMD
simulations are also included. Our findings show a signif-
icant energy dependence of scaled variances, with more
pronounced suppression in the |η|<1.0 range compared
to |η|<0.5. Furthermore, k2is more sensitive to the
transverse momentum pTvariations than to pseudora-
pidity ηacceptance. For 0.5 < pT<3.0 GeV/c, the
scaled variance in both |η|<1.0 and |η|<0.5 ranges are
quantitatively consistent with STAR measurements.
Noted that as the incident energy decreases to √sNN
= 3.0 GeV in |η|<0.5, particularly within 0.2 < pT<
2.0 GeV/c, k2exhibits a significant abnormal increase
compared to the values observed at √sNN = 7.7 GeV.
This indicates an enhancement of dynamical correlations
at lower collision energies and highlights the potentiality
to probe the properties of the medium formed during
collisions.
V. DISCUSSIONS AND CONCLUSION
The default and string-melting versions of the AMPT
model have been improved by implementing a dynamic
tuning mechanism for the Lund string fragmentation pa-
rameter bL, which now varies linearly with the impact pa-
rameter and exhibits increased sensitivity at higher beam
energies. This improvement enables the AMPT model to
accurately reproduce the trends in centrality and beam
energy dependence of the mean transverse momentum
⟨pT⟩fluctuations. With this refined AMPT framework,
we present a comprehensive systematic study of higher-
order dynamical pTcumulants up to fourth order, using
both standard and two-subevent methods as functions of
⟨Npart⟩and collision energies.
Our analysis shows that pTcumulants up to fourth
order, both normalized and unnormalized by ⟨⟨pT⟩⟩, ex-
hibit a strong dependence on centrality across an energy
range of √sNN = 19.6 GeV to 3.0 GeV. Notably, the
scaled variances as a function of centrality within |η|<
0.5 align closely with STAR measurements. In most cen-
tral collisions (0–5%), the scaled variance is more sensi-
tive to transverse momentum pTthan to the acceptance
ηrange. Additionally, for the same pTrange, the scaled
variance demonstrates more significant suppression in the
|η|<1.0 range compared to |η|<0.5. For collisions at
√sNN = 3 GeV, an enhancement in scaled variances is
observed for 0.5 < pT<3.0 GeV/c, particularly within
|η|<0.5. This finding provides valuable references for
the measurements from STAR fixed-target, Beam Energy
Scan program, and possible FAIR-CBM experiments.
We investigate the contributions to ⟨pT⟩fluctuations
from baryons and mesons, separately. The results show
that baryons exhibit considerable ⟨pT⟩fluctuations, likely
due to their heavier masses, which make them more sus-
ceptible to the effect of radial flow compared to mesons.
Additionally, we also evaluated two distinct hadroniza-
tion mechanisms: Lund string fragmentation in the de-
fault version and quark coalescence in the string-melting
version. The findings indicate that the default AMPT
version demonstrates more pronounced ⟨pT⟩fluctuations.
It would be interesting to measure in experiment the
8
high-order transverse momentum fluctuations across col-
lision energies.
VI. ACKNOWLEDGEMENTS
We would like to thank Ziwei Lin, Chao Zhang, and
Liang Zheng for their insightful discussions. We thank
Chen Zhong for the stimulating research environment.
This work was supported in part by the National Key
Research and Development Program of China under Con-
tract Nos. 2022YFA1604900 and 2024YFA1612600, the
National Natural Science Foundation of China (NSFC)
under Contract Nos. 12205051, 12025501, 12147101, the
Natural Science Foundation of Shanghai under Contract
No. 23JC1400200, the Shanghai Pujiang Talents Pro-
gram under Contract No. 24PJA009.
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