UC Davis
UC Davis Previously Published Works
Title
Nonmonotonic Energy Dependence of Net-Proton Number Fluctuations.
Permalink
https://escholarship.org/uc/item/19m4w50f
Journal
Physical review letters, 126(9)
ISSN
0031-9007
Authors
Adam, J
Adamczyk, L
Adams, JR
et al.
Publication Date
2021-03-01
DOI
10.1103/physrevlett.126.092301
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California
Non-monotonic energy dependence of net-proton number fluctuations
J. Adam
6
, L. Adamczyk
2
, J. R. Adams
39
, J. K. Adkins
30
, G. Agakishiev
28
, M. M. Aggarwal
41
, Z. Ahammed
61
, I. Alekseev
3,35
,
D. M. Anderson
55
, A. Aparin
28
, E. C. Aschenauer
6
, M. U. Ashraf
11
, F. G. Atetalla
29
, A. Attri
41
, G. S. Averichev
28
,
V. Bairathi
53
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10
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52
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20
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27
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14
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6
,
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3
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6
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35
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45
, H. Caines
64
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´
on de la Barca S
´
anchez
8
,
D. Cebra
8
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29,6
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14
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9
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37
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6
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28
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49
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1
Abilene Christian University, Abilene, Texas 79699
2
AGH University of Science and Technology, FPACS, Cracow 30-059, Poland
3
Alikhanov Institute for Theoretical and Experimental Physics NRC ”Kurchatov Institute”, Moscow 117218, Russia
4
Argonne National Laboratory, Argonne, Illinois 60439
5
American University of Cairo, New Cairo 11835, New Cairo, Egypt
6
Brookhaven National Laboratory, Upton, New York 11973
7
University of California, Berkeley, California 94720
8
University of California, Davis, California 95616
9
University of California, Los Angeles, California 90095
10
University of California, Riverside, California 92521
11
Central China Normal University, Wuhan, Hubei 430079
2
12
University of Illinois at Chicago, Chicago, Illinois 60607
13
Creighton University, Omaha, Nebraska 68178
14
Czech Technical University in Prague, FNSPE, Prague 115 19, Czech Republic
15
Technische Universit
¨
at Darmstadt, Darmstadt 64289, Germany
16
ELTE E
¨
otv
¨
os Lor
´
and University, Budapest, Hungary H-1117
17
Frankfurt Institute for Advanced Studies FIAS, Frankfurt 60438, Germany
18
Fudan University, Shanghai, 200433
19
University of Heidelberg, Heidelberg 69120, Germany
20
University of Houston, Houston, Texas 77204
21
Huzhou University, Huzhou, Zhejiang 313000
22
Indian Institute of Science Education and Research (IISER), Berhampur 760010 , India
23
Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India
24
Indian Institute Technology, Patna, Bihar 801106, India
25
Indiana University, Bloomington, Indiana 47408
26
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000
27
University of Jammu, Jammu 180001, India
28
Joint Institute for Nuclear Research, Dubna 141 980, Russia
29
Kent State University, Kent, Ohio 44242
30
University of Kentucky, Lexington, Kentucky 40506-0055
31
Lawrence Berkeley National Laboratory, Berkeley, California 94720
32
Lehigh University, Bethlehem, Pennsylvania 18015
33
Max-Planck-Institut f
¨
ur Physik, Munich 80805, Germany
34
Michigan State University, East Lansing, Michigan 48824
35
National Research Nuclear University MEPhI, Moscow 115409, Russia
36
National Institute of Science Education and Research, HBNI, Jatni 752050, India
37
National Cheng Kung University, Tainan 70101
38
Nuclear Physics Institute of the CAS, Rez 250 68, Czech Republic
39
Ohio State University, Columbus, Ohio 43210
40
Institute of Nuclear Physics PAN, Cracow 31-342, Poland
41
Panjab University, Chandigarh 160014, India
42
Pennsylvania State University, University Park, Pennsylvania 16802
43
NRC ”Kurchatov Institute”, Institute of High Energy Physics, Protvino 142281, Russia
44
Purdue University, West Lafayette, Indiana 47907
45
Rice University, Houston, Texas 77251
46
Rutgers University, Piscataway, New Jersey 08854
47
Universidade de S
˜
ao Paulo, S
˜
ao Paulo, Brazil 05314-970
48
University of Science and Technology of China, Hefei, Anhui 230026
49
Shandong University, Qingdao, Shandong 266237
50
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800
51
Southern Connecticut State University, New Haven, Connecticut 06515
52
State University of New York, Stony Brook, New York 11794
53
Instituto de Alta Investigaci
´
on, Universidad de Tarapac
´
a, Arica 1000000, Chile
54
Temple University, Philadelphia, Pennsylvania 19122
55
Texas A&M University, College Station, Texas 77843
56
University of Texas, Austin, Texas 78712
57
Tsinghua University, Beijing 100084
58
University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
59
United States Naval Academy, Annapolis, Maryland 21402
60
Valparaiso University, Valparaiso, Indiana 46383
61
Variable Energy Cyclotron Centre, Kolkata 700064, India
62
Warsaw University of Technology, Warsaw 00-661, Poland
63
Wayne State University, Detroit, Michigan 48201 and
64
Yale University, New Haven, Connecticut 06520
(STAR Collaboration)
Non-monotonic variation with collision energy (
√
s
NN
) of the moments of the net-baryon number distribu-
tion in heavy-ion collisions, related to the correlation length and the susceptibilities of the system, is suggested
as a signature for the Quantum Chromodynamics (QCD) critical point. We report the first evidence of a non-
monotonic variation in kurtosis times variance of the net-proton number (proxy for net-baryon number) dis-
tribution as a function of
√
s
NN
with 3.1σ significance, for head-on (central) gold-on-gold (Au+Au) collisions
measured using the STAR detector at RHIC. Data in non-central Au+Au collisions and models of heavy-ion
collisions without a critical point show a monotonic variation as a function of
√
s
NN
.
3
One of the fundamental goals in physics is to understand
the properties of matter when subjected to variations in tem-
perature and pressure. Currently, the study of the phases of
strongly interacting nuclear matter is the focus of many re-
search activities worldwide, both theoretically and experimen-
tally [1, 2]. The theory that governs the strong interactions
is Quantum Chromodynamics (QCD), and the correspond-
ing phase diagram is called the QCD phase diagram. From
different examples of condensed-matter systems, experimen-
tal progress in mapping out phase diagrams is achieved by
changing the material doping, adding more holes than elec-
trons. Similarly it is suggested for the QCD phase diagram,
that adding more quarks than antiquarks (the energy required
is defined by the baryonic chemical potential, µ
B
), through
changing the heavy-ion collision energy, enables a search for
new emergent properties and a possible critical point in the
phase diagram. The phase diagram of QCD has at least two
distinct phases: a Quark Gluon Plasma (QGP) at higher tem-
peratures, and a state of confined quarks and gluons at lower
temperatures called the hadronic phase [3–5]. It is inferred
from lattice QCD calculations [6] that the transition is consis-
tent with being a cross over at small µ
B
, and that the transi-
tion temperature is about 155 MeV [7–9]. An important pre-
dicted feature of the QCD phase structure is a critical point
[10, 11], followed at higher µ
B
by a first order phase transi-
tion. Attempts are being made to locate the predicted critical
point both experimentally and theoretically. Current theoreti-
cal calculations are highly uncertain about the location of the
critical point. Lattice QCD calculations at finite µ
B
face nu-
merical challenges in computing [12, 13]. Within these lim-
itations, the current best estimate from lattice QCD is that if
there is a critical point, its location is likely above µ
B
∼ 300
MeV [12, 13]. The goal of this work is to search for possible
signatures of the critical point by varying the collision energy
in heavy ion collisions to cover a wide range in effective tem-
perature (T ) and µ
B
in the QCD phase diagram [14].
Another key aspect of investigating the QCD phase diagram
is to determine whether the system has attained thermal equi-
librium. Several theoretical interpretations of experimental
data have the underlying assumption that the system produced
in the collisions should have come to local thermal equilib-
rium during its evolution. Experimental tests of thermaliza-
tion for these femto-scale expanding systems are non-trivial.
However, the yields of produced hadrons and fluctuations of
multiplicity distributions related to conserved quantities have
been studied and shown to have characteristics of thermody-
namic equilibrium for higher collision energies [12, 15–20].
Upon approaching a critical point, the correlation length di-
verges and thus renders, to a large extent, microscopic details
irrelevant. Hence observables like the moments of the con-
served net-baryon number distribution, which are sensitive to
the correlation length, are of interest when searching for a crit-
ical point. A non-monotonic variation of these moments as a
function of
√
s
NN
has been proposed as an experimental sig-
nature of a critical point [10, 14]. However, considering the
complexity of the system formed in heavy-ion collisions, sig-
natures of a critical point are detectable only if they can sur-
vive the evolution of the system, including the effects of finite
size and time [21]. Hence, it was proposed to study higher
moments of distributions of conserved quantities (N) due to
their stronger dependence on the correlation length [11]. The
promising higher moments are the skewness, S =
(δN)
3
/σ
3
,
and kurtosis, κ = [
(δN)
4
/σ
4
] – 3, where δN = N – M, M
is the mean and σ is the standard deviation. The magnitude
and the sign of the moments, which quantify the shape of
the multiplicity distributions, are important for understanding
the critical point [14, 22]. An additional crucial experimental
challenge is to measure, on an event-by-event basis, all of the
baryons produced within the acceptance of a detector [23–25].
However, theoretical calculations have shown that the proton-
number fluctuations can also reflect the baryon-number fluc-
tuations at the critical point [23, 26].
The measurements reported here are from Au+Au colli-
sions recorded by the STAR detector [27] at RHIC from
the years 2010 to 2017. The data is presented for
√
s
NN
=
7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV as
part of phase-I of the Beam Energy Scan (BES) program at
RHIC [15]. These
√
s
NN
values correspond to µ
B
values rang-
ing from 420 MeV to 20 MeV at chemical freeze-out [15].
All valid Au+Au collisions occurring within 60 cm (80 cm
for
√
s
NN
= 7.7 GeV) of the nominal interaction point along
the beam axis are selected. For the results presented here, the
number of minimum bias Au+Au collisions ranges between
3 million for
√
s
NN
= 7.7 GeV and 585 million at
√
s
NN
=
54.4 GeV. These statistics are found to be adequate to make
the measurements of the moments of the net-proton distribu-
tions up to the fourth order [28]. The collisions are further
divided into centrality classes characterised by their impact
parameter, which is the closest distance between the centroid
of two nuclei passing by. In practice, the impact parame-
ter is determined indirectly from the measured multiplicity of
charged particles other than protons (p) and anti-protons ( ¯p) in
the pseudo-rapidity range |η| < 1, where η = −ln[tan(θ/2)],
with θ being the angle between the momentum of the parti-
cle and the positive direction of the beam axis. We exclude
p and ¯p while classifying events based on impact parame-
ter specifically to avoid self-correlation effects [29]. The ef-
fect of self-correlation potentially arising due to the decay of
heavier hadrons into p( ¯p) and other charged particles has been
checked to be negligible from a study using standard heavy-
ion collision event generators, HIJING [30] and UrQMD [31].
The effect of resonance decays and the pseudo-rapidity range
for centrality determination have been understood and opti-
mized using model calculations [32, 33]. The results pre-
sented here correspond to two event classes: central collisions
(impact parameters ∼ 0-3 fm, obtained from the top 5% of
the above-mentioned multiplicity distribution) and peripheral
collisions (impact parameters ∼ 12-13 fm, obtained from the
70-80% region of the multiplicity distribution).
The protons and anti-protons are identified, along with their
momenta, by reconstructing their tracks in the Time Projec-
tion Chamber (TPC) placed within a solenoidal magnetic field
of 0.5 Tesla, and by measuring their ionization energy loss
(dE/dx) in the sensitive gas-filled volume of the chamber.
The selected kinematic region for protons covers all azimuthal
angles for the rapidity range |y| < 0.5, where rapidity y is the
4
10− 0 10 20 30 40
0
0.02
0.04
0.06
0.08
0.1
7.7
11.5
14.5
19.6
27
39
54.4
62.4
200
(GeV)
NN
s
Au+Au Collisions
0-5% Central
< 2.0 (GeV/c), |y| < 0.5
T
0.4 < p
Normalized Number of Events
Net-proton (∆N
p
= N
p
- N
p
)
FIG. 1. Event-by-event net-proton number distributions for head-on
(0-5% central) Au+Au collisions for nine
√
s
NN
values measured by
STAR. The distributions are normalized to the total number of events
at each
√
s
NN
. The statistical uncertainties are smaller than the sym-
bol sizes and the lines are shown to guide the eye. The distributions
in this figure are not corrected for proton and anti-proton detection
efficiency. The deviation of the distribution for
√
s
NN
= 54.4 GeV
from the general energy dependence trend is understood to be due to
the reconstruction efficiency of protons and anti-protons being dif-
ferent compared to other energies.
inverse hyperbolic tangent of the component of speed parallel
to the beam direction in units of the speed of light. The pre-
cise measurement of dE/dx with a resolution of 7% in Au+Au
collisions allows for a clear identification of protons up to 800
MeV/c in transverse momentum (p
T
). The identification for
larger p
T
(up to 2 GeV/c, with purity above 97%) is made by
a Time Of Flight detector (TOF) [34] having a timing resolu-
tion of better than 100 ps. A minimum p
T
threshold of 400
MeV/c and a maximum distance of closest approach to the
collision vertex of 1 cm for each p( ¯p) candidate track is used
to suppress contamination from secondaries and other back-
grounds [15, 35]. This p
T
acceptance accounts for approx-
imately 80% of the total p + ¯p multiplicity at mid-rapidity.
This is a significant improvement from the results previously
reported [35] which only had the p + ¯p measured using the
TPC. The observation of non-monotonic variation of the kur-
tosis times variance (κσ
2
) with energy is much more signif-
icant with the increased acceptance. For the rapidity depen-
dence of the observable see Supplemental Material [34].
Figure 1 shows the event-by-event net-proton (N
p
−N
¯p
=
∆N
p
) distributions obtained by measuring the number of pro-
tons (N
p
) and anti-protons (N
¯p
) at mid-rapidity (|y| < 0.5) in
the transverse momentum range 0.4 < p
T
(GeV/c)< 2.0 for
Au+Au collisions at various
√
s
NN
. To study the shape of
the event-by-event net-proton distribution in detail, cumulants
(C
n
) of various orders are calculated, where C
1
= M, C
2
= σ
2
,
C
3
= Sσ
3
and C
4
= κσ
4
.
Figure 2 shows the net-proton cumulants (C
n
) as a function
of
√
s
NN
for central and peripheral (see Supplemental Mate-
rial [34] for a zoomed version). Au+Au collisions. The cumu-
lants are corrected for the multiplicity variations arising due
to finite impact parameter range for the measurements [32].
These corrections suppress the volume fluctuations consid-
erably [32, 36]. A different volume fluctuation correction
method [37] has been applied to the 0-5% central Au+Au col-
lision data and the results were found to be consistent with
those shown in Fig 2 . The cumulants are also corrected for
finite track reconstruction efficiencies of the TPC and TOF
0
10
20
30
40
(1) C
1
0 - 5%
70 - 80%
5 10 20 50 100 200
0
10
20
30
40
(3) C
3
0
10
20
30
40
(2) C
2
0
20
40
60
80
100
5 10 20 50 100 200
(4) C
4
Stat. uncertainty
Syst. uncertainty
Net-proton Cumulants
(GeV)
NN
sCollision Energy
Au+Au CollisionsAu+Au Collisions
Net-proton
< 2.0 (GeV/c)
T
|y| < 0.5, 0.4 < p
FIG. 2. Cumulants (C
n
) of the net-proton distributions for central
(0-5%) and peripheral (70-80%) Au+Au collisions as a function of
collision energy. The transverse momentum (p
T
) range for the mea-
surements is from 0.4 to 2 GeV/c and the rapidity (y) range is -0.5 <
y < 0.5.
detectors. This is done by assuming a binomial response of
the two detectors [35, 38]. A cross-check using a different
method based on unfolding [34] of the distributions for central
Au+Au collisions at
√
s
NN
= 200 GeV has been found to give
values consistent with the cumulants shown in Fig. 2. Further,
the efficiency correction method used has been verified in a
Monte Carlo calculation. Typical values for the efficiencies
in the TPC (TOF-matching) for the momentum range stud-
ied in 0-5% central Au+Au collisions at
√
s
NN
= 7.7 GeV are
83%(72%) and 81%(70%) for the protons and anti-protons,
respectively. The corresponding efficiencies for
√
s
NN
= 200
GeV collisions are 62%(69%) and 60%(68%) for the protons
and anti-protons, respectively. The statistical uncertainties
are obtained using both a bootstrap approach [28, 38] and
the Delta theorem [28, 38, 39] method. The systematic un-
certainties are estimated by varying the experimental require-
ments to reconstruct p ( ¯p) in the TPC and TOF. These require-
ments include the distance of the proton and anti-proton tracks
from the primary vertex position, track quality reflected by the
number of TPC space points used in the track reconstruction,
the particle identification criteria passing certain selection cri-
teria, and the uncertainties in estimating the reconstruction ef-
ficiencies. The systematic uncertainties at different collision
energies are uncorrelated.
The large values of C
3
and C
4
for central Au+Au collisions
show that the distributions have non-Gaussian shapes, a possi-
ble indication of enhanced fluctuations arising from a possible
critical point [11, 22]. The corresponding values for periph-
eral collisions are small and close to zero. For central colli-
sions, the C
1
and C
3
monotonically decrease with increasing
√
s
NN
.
We employ ratios of cumulants in order to cancel volume
variations to first order. Further, these ratios of cumulants
are related to the ratio of baryon-number susceptibilities. The