ENERGY DEPENDENCE OF FLUCTUATION AND CORRELATION OBSERVABLES
OF TRANSVERSE MOMENTUM IN HEAVY-ION COLLISIONS
By
John F. Novak

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
Physics - Doctor of Philosophy
2013

ABSTRACT
ENERGY DEPENDENCE OF FLUCTUATION AND CORRELATION
OBSERVABLES OF TRANSVERSE MOMENTUM IN HEAVY-ION
COLLISIONS
By
John F. Novak
In collisions of heavy ions of su cient energy, cold nuclear matter can be forced into a strongly
interacting state of quark-gloun plasma (QGP). To study the properties of QGP and the
phase transition to hadronic matter, Au+Au collisions were performed at the Relativistic
Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and studied using the
Solendoidal Tracker at RHIC (STAR) detector. These Au+Au collision were taken during
2010 and 2011 as part of the RHIC Beam Energy Scan (BES) at energies

p

sNN = 7.7, 11.5,

19.6, 27, 39, 62.4, and 200 GeV. The primary goal of the BES was to search for the critical
point of the phase transition between the QGP phase and the hadronic matter phase of
nuclear matter.
In this dissertation two analyses on these data are presented which focus on ﬂuctuations of the
average transverse momentum (hpt i) of the particles produced in heavy-ion collisions. hpt i
is related to the temperature of the systems produced in the collisions [35], and ﬂuctuations
of hpt i should be sensitive to ﬂuctuations of the temperature [40]. The moments of the hpt i
distributions has also been proposed to be sensitive to the correlation length of the QGP
medium [41, 42], which will diverge at the critical point.
Fluctuations of hpt i will depend upon both dynamic ﬂuctuations of the produced systems,
and statistical ﬂuctuations due to limited statistics. The ﬁrst analysis presented in this dis-

sertation is of the two particle relative momentum correlator

⌦

↵
pt,i , pt,j which is a direct

2
measure of the dynamic ﬂuctuations of the variance of the hpt i distribution, hp i,dynamic .
t

The second analysis presented in this dissertation is of the higher moments of the hpt i distribution. The dynamic higher moments are inferred by comparison of the measured data
with mixed events and statically sampled events which reproduce the statistical ﬂuctuations
while having no dynamic ﬂuctuations.
No consistent non-monotonic behavior, which would be a conclusive indication of the QGP
critical point, is observed. Some anomalous behavior of the higher moments is noted which
will require further analysis. Dynamic ﬂuctuations of the hpt i distribution, as measured by
⌦
↵
the two particle correlator
pt,i , pt,j and the higher moments of the hpt i distribution,
are observed to increase with energy. There is a strong energy dependence below

p

19.6 GeV, and the dynamic ﬂuctuations of hpt i are consistent with zero at 7.7 GeV.

sNN =

Copyright by
JOHN F. NOVAK
2013

Dedicated to my wife, Christine. Without you this would not have happened.

v

ACKNOWLEDGMENTS

There are many people for whom I am thankful, and without whom this dissertation would
not exist. First and foremost, I would like to thank my wife, Christine, for her patience and
support. There were many long days and late nights, and she was always willing to let me
disappear into my o ce to work. I also have to thank her parents, Belinda and Joe Venner,
the best family one could ask for. They were always there to provide support and love.
I am also thankful for my thesis advisor, Dr. Gary Westfall. Gary taught me, guided me,
and was patient with me when, in my enthusiasm, I would blow past “subtleties” which
usually turned out to be rather important. I will always remember his expression each time
I would appear in his o ce to tell him that I rewrote all of the analysis code, again. I always
got there in the end, and his faith in me was worth more than he can know.
I also want to thank the army of people that made my results possible: the sta↵ at
Brookhaven National Lab, my collaborators in STAR, the many people at the National
Superconducing Cyclotron Laboratory who made it possible for me to work, and all of the
people at Michigan State University who worked behind the scenes so that I could be a
graduate student. I need to thank the people who are Michigan State University and the
Department of Physics and Astronomy for the opportunity to be a graduate student. The
National Science Foundation also has my sincere thanks for funding my research.

vi

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . .
1.1 The Standard Model and Quantum ChromoDynamics
1.1.1 Color Conﬁnement . . . . . . . . . . . . . . .
1.1.2 Asymptotic Freedom . . . . . . . . . . . . . .
1.2 Quark Gluon Plasma . . . . . . . . . . . . . . . . . .
1.2.1 The QCD Phase Diagram . . . . . . . . . . .
1.2.2 The QCD Critical Point . . . . . . . . . . . .
1.2.2.1 Critical Opalescence . . . . . . . . .
1.3 Heavy-Ion Collisions . . . . . . . . . . . . . . . . . .
1.3.1 System Evolution . . . . . . . . . . . . . . . .
1.3.2 Collision Centrality . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

1
2
3
4
5
5
8
9
10
10
12

Chapter 2 Experimental Setup . . . . . . . . .
2.1 The Relativistic Heavy-Ion Collider (RHIC)
2.2 STAR . . . . . . . . . . . . . . . . . . . . .
2.3 Time Projection Chamber . . . . . . . . . .
2.4 Time of Flight Detector . . . . . . . . . . .
2.5 Other STAR Subsystems . . . . . . . . . . .
2.5.1 Vertex Position Detectors . . . . . .
2.5.2 Electromagnetic Calorimeters . . . .
2.5.3 Beam Beam Counters . . . . . . . .
2.5.4 Zero Degree Calorimeters . . . . . .

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

13
13
16
19
22
27
27
29
29
31

Chapter 3 Data from the Beam
3.1 RHIC Beam Energy Scan .
3.2 Good Run Determination .
3.3 Calibrations . . . . . . . . .
3.3.1 TPC Calibrations . .
3.3.2 ToF Calibrations . .
3.4 Track Cuts . . . . . . . . . .
3.5 Multiplicity Corrections . .
3.6 Centrality Determination . .
3.7 Trigger Deﬁnition . . . . . .

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

32
32
34
35
35
36
37
38
40
44

Energy
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .

vii

Scan
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .

3.8

The UrQMD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Chapter 4 Motivation and Construction of Analyses
4.1 Event-by-event Observables . . . . . . . . . . . . . .
4.2 Fluctuations versus Correlations . . . . . . . . . . . .
4.3 hpt i Fluctuations . . . . . . . . . . . . . . . . . . . .
4.4 Mixed and Statistically Sampled Events . . . . . . .
4.5 Analyses in this Dissertation . . . . . . . . . . . . . .
4.6 Sources of Fluctuations . . . . . . . . . . . . . . . . .
4.6.1 Jets . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Flow . . . . . . . . . . . . . . . . . . . . . . .
4.6.3 Resonance Decays . . . . . . . . . . . . . . . .
4.6.4 Changing Chemistry . . . . . . . . . . . . . .
4.6.5 Temperature Fluctuations . . . . . . . . . . .
4.6.6 Correlation Length . .⌦ . . . . . . .↵ . . . . . .
4.7 Two-Particle pt Correlations,
pt,i , pt,j
. . . . .
4.7.1 Mathematical Construction . . . . . . . . . .
4.7.2 Scalings . . . . . . . . . . . . . . . . . . . . .
4.7.2.1 By hhpt ii 1 . . . . . . . . . . . . . .
4.7.2.2 By Multiplicity . . . . . . . . . . . .
4.7.2.3 By hhpt ii 1 and Multiplicity . . . .
4.8 Higher Moments of hpt i . . . . . . . . . . . . . . . .
4.8.1 Mathematical Construction . . . . . . . . . .
4.8.2 Baselines . . . . . . . . . . . . . . . . . . . . .
4.8.2.1 Gamma Distributions . . . . . . . .
4.8.2.2 Statistical Baseline . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

46
46
47
48
50
53
53
54
56
57
59
60
61
64
65
68
69
69
70
70
71
72
73
74

Chapter 5 Results of the pt Correlation Analysis
5.1 Behavior of hpt i . . . . . . . . . . . . . . . . . .
5.2 Unscaled Correlations . . . . . . . . . . . . . . .
5.3 Correlations Scaled with hhpt ii 1 . . . . . . . .
5.4 Correlations Scaled with Multiplicity . . . . . .
5.5 Correlations Scaled with Multiplicity and hhpt ii
5.6 Comparison with Published Results . . . . . . .
Chapter 6 pt Correlation Analysis Checks
6.1 ⌘ Cut Dependence . . . . . . . . . . . .
6.2 Detector E ciency Dependence . . . . .
6.3 Bin Width Study . . . . . . . . . . . . .
6.4 Auto-correlations Study . . . . . . . . .
6.5 Short Range Correlations . . . . . . . . .
6.6 Errors Calculations . . . . . . . . . . . .

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

78
78
80
83
85
87
89

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

92
92
95
101
101
104
106

.
.
.
.

1

Chapter 7 Results of the Higher Moments Analysis . . . . . . . . . . . . . . 108
7.1 Higher Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
viii

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

108
109
109
112
117
117
119
120
123
123

Chapter 8 Checks of the Higher Moments Analysis
8.1 ⌘ Cut Dependence . . . . . . . . . . . . . . . . . .
8.2 pt Cut Dependence . . . . . . . . . . . . . . . . . .
8.3 Detector E ciency Dependence . . . . . . . . . . .
8.4 Error Calculation . . . . . . . . . . . . . . . . . . .

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

126
126
136
136
142

Chapter 9 Conclusion . . . . . . . .
9.1 Summary of Correlations Analysis
9.2 Summary of Moments Analysis .
9.3 Looking Forward . . . . . . . . .
9.4 In Summary . . . . . . . . . . . .

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

144
144
146
149
150

7.2

7.1.1 Comparisons with Baselines .
7.1.2 µ1 (1 ) . . . . . . . . . . . . .
2
7.1.3 µ2 ( hp i , 2 ) . . . . . . . . .
t
7.1.4 µ3 (3 ) . . . . . . . . . . . . .
7.1.5 µ4 . . . . . . . . . . . . . . .
7.1.6 4 . . . . . . . . . . . . . . .
7.1.7 Comparisons of Moments with
Cumulant Ratios . . . . . . . . . . .
7.2.1 S . . . . . . . . . . . . . . .
7.2.2 K 2 . . . . . . . . . . . . . .

APPENDIX

.
.
.
.
.

.
.
.
.
.

. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
UrQMD
. . . . .
. . . . .
. . . . .

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

. . . . . . . . . . . . . . .

BIBLIOGRAPHY

. . . . . . . . . . . . .

ix

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

. . . . . . . . . . . . . . . . . . . 152
. . . . . . . . . . . . . . . . . 158

LIST OF TABLES

Table 3.1

Data sets used in this dissertation. . . . . . . . . . . . . . . . . . . .

33

Table 3.2

Analysis cuts used in this dissertation . . . . . . . . . . . . . . . . .

38

Table 3.3

Centrality cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Table 3.4

The number of events per centrality bin. . . . . . . . . . . . . . . .

43

Table 5.1

Parameters of the gamma distributions derived from the hpt i spectra. 80

Table 8.1

Systematic error of the moments analysis. . . . . . . . . . . . . . . . 143

x

LIST OF FIGURES

Figure 1.1

The force between two quarks. . . . . . . . . . . . . . . . . . . . . .

4

Figure 1.2

QCD renormalization coupling constant . . . . . . . . . . . . . . . .

6

Figure 1.3

The QCD phase diagram . . . . . . . . . . . . . . . . . . . . . . . .

7

Figure 1.4

System evolution in heavy-ion collisions . . . . . . . . . . . . . . . .

11

Figure 2.1

Aerial view of RHIC. . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Figure 2.2

STAR layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Figure 2.3

STAR detector subsystems. . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.4

STAR TPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Figure 2.5

Location of one STAR ToF tray. . . . . . . . . . . . . . . . . . . . .

24

Figure 2.6

MRPC Layout.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Figure 2.7

MRPC Layout.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Figure 2.8

ToF matching e ciency.

. . . . . . . . . . . . . . . . . . . . . . . .

28

Figure 2.9

BBC layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Figure 3.1

Laser calibration event in the TPC. . . . . . . . . . . . . . . . . . .

36

Figure 3.2

Illustration of collision centrality. . . . . . . . . . . . . . . . . . . . .

42

Figure 4.1

hpt i spectra for 200 GeV. . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 4.2

Momentum dependence of the nuclear modiﬁcation factor. . . . . . .

55

Figure 4.3

hhpt ii (refMult) for 7.7 GeV. . . . . . . . . . . . . . . . . . . . . . .

66

Figure 4.4

hhpt ii (refMult) for 200 GeV. . . . . . . . . . . . . . . . . . . . . . .

67

xi

Figure 4.5

Toy model demonstrating the e↵ect of non-uniform pt acceptance on
the hpt i distribution. . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Figure 4.6

E ciency as a function of pt for 62.4 GeV. . . . . . . . . . . . . . .

77

Figure 5.1

hpt i distributions with gamma distributions. . . . . . . . . . . . . .

79

Figure 5.2

hhpt ii vs

Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8

⌦

p

sNN

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

↵
p
pt,i , pt,j vs sNN . . . . . . . . . . . . . . . . . . . . . . . . .

q⌦

↵
p
pt,i , pt,j / hhpt ii vs sNN . . . . . . . . . . . . . . . . . . . .

↵
p
pt,i , pt,j vs sNN . . . . . . . . . . . . . . . . . . . . . . .
⌦
↵
hN i pt,i , pt,j vs Npart . . . . . . . . . . . . . . . . . . . . . . .
hN i

⌦

q
⌦
↵
hN i pt,i , pt,j / hhpt ii vs N part . . . . . . . . . . . . . . . . . .
q
⌦
↵
p
hN i pt,i , pt,j / hhpt ii vs sNN . . . . . . . . . . . . . . . . .

Figure 5.9

q⌦

Figure 6.1

Symmetric ⌘-cut study of

Figure 6.2

Symmetric ⌘-cut study of

↵
p
pt,i , pt,j / hhpt ii vs sNN for the 0-5% centrality bin. . . . .
q⌦
q⌦

q⌦

↵
pt,i , pt,j / hhpt ii for 19.6 GeV . . .
↵
pt,i , pt,j / hhpt ii for 200 GeV . . .

↵
pt,i , pt,j / hhpt ii for 11.5 GeV . .

Figure 6.3

Asymmetric ⌘-cut study of

Figure 6.4

Asymmetric ⌘-cut study of

Figure 6.5

hhpt ii from UrQMD with and without momentum dependent ine ciency simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 6.6

Figure 6.7

q⌦

⌦

↵
pt,i , pt,j / hhpt ii for 39 GeV . . .

↵
pt,i , pt,j from UrQMD with and without momentum dependent
ine ciency simulation . . . . . . . . . . . . . . . . . . . . . . . . . .
q⌦

81
82
84
85
86
88
89
90
93
94
95
96

98

99

↵
pt,i , pt,j / hhpt ii from UrQMD with and without momentum
dependent ine ciency simulation . . . . . . . . . . . . . . . . . . . . 100
xii

Figure 6.8
Figure 6.9
Figure 6.10

⌦

↵
pt,i , pt,j as calculated using refMult and refMult2 for 19.6 GeV. 102
⌦
↵
pt,i , pt,j as calculated using refMult and refMult2 for 62.4 GeV. 103
⌦

↵ q⌦

↵
sign
pt,i , pt,j
pt,i , pt,j / hhpt ii as a versus energy with
and without the short range correlation correction. . . . . . . . . . . 105

Figure 7.1

The ﬁrst moment, µ1,hp i , of the hpt i distribution. . . . . . . . . . . 110
t

Figure 7.2

The second moment, µ2,hp i , of the hpt i distribution. . . . . . . . . . 111
t

Figure 7.3

The average multiplicity hN i. . . . . . . . . . . . . . . . . . . . . . . 113

Figure 7.4

The average variance of the inclusive pt distribution,

Figure 7.5

The third moment, µ3,hp i , of the hpt i distribution. . . . . . . . . . . 115
t

Figure 7.6

The fourth moment, µ4,hp i , of the hpt i distribution. . . . . . . . . . 118
t

Figure 7.7

The fourth cumulant, 4,hp i , of the hpt i distribution. . . . . . . . . 119
t

Figure 7.8

The ﬁrst moment, µ1 , from UrQMD with and without pt dependent
e ciency simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 7.9

The second moment, µ2 , from UrQMD with and without pt dependent e ciency simulation. . . . . . . . . . . . . . . . . . . . . . . . . 122

Figure 7.10

S for the hpt i distribution. . . . . . . . . . . . . . . . . . . . . . . 124

Figure 7.11

K 2 for the hpt i distribution. . . . . . . . . . . . . . . . . . . . . . . 125

Figure 8.1

Symmetric ⌘-cut study of µ1 . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 8.2

Symmetric ⌘-cut study of µ1 . . . . . . . . . . . . . . . . . . . . . . . 129

Figure 8.3

Symmetric ⌘-cut study of µ2 . . . . . . . . . . . . . . . . . . . . . . . 130

Figure 8.4

Symmetric ⌘-cut study of µ2 . . . . . . . . . . . . . . . . . . . . . . . 131

Figure 8.5

Asymmetric ⌘-cut study of µ1 . . . . . . . . . . . . . . . . . . . . . . 132

Figure 8.6

Asymmetric ⌘-cut study of µ1 . . . . . . . . . . . . . . . . . . . . . . 133

xiii

D

2
pt

E

. . . . . . 114

Figure 8.7

Asymmetric ⌘-cut study of µ2 . . . . . . . . . . . . . . . . . . . . . . 134

Figure 8.8

Asymmetric ⌘-cut study of µ2 . . . . . . . . . . . . . . . . . . . . . . 135

Figure 8.9

pt cut study of µ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 8.10

pt cut study of µ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 8.11

pt cut study of µ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 8.12

pt cut study of µ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 8.13

S for UrQMD with and without pt dependent e ciency simulation. 141

Figure 8.14

K 2 for UrQMD with and without pt dependent e ciency simulation.142

xiv

Chapter 1
Introduction
In the ﬁrst moments of time, only a few microseconds after the big bang, the universe existed
in a state far di↵erent from that which is seen almost anywhere in the universe today. All
matter was exceedingly dense and hot (a few trillion degrees), so much so that not only
atoms but also their nuclei and the nucleons which form them could not exist. It was too
dense and too hot for baryons and mesons to form from quarks and gluons and all partonic
matter existed as a plasma of free quarks and gluons.
Within a few microseconds the expansion and cooling of the universe caused the plasma
to hadronize into color neutral particles. Today the only place where those primordial
conditions are accessibly reproduced is at the center of heavy-ion collisions like those studied
in this dissertation.
The study of heavy-ion physics is clearly applicable to the cosmological investigation of the
universe’s earliest moments, but it is also a powerful tool in the study of nuclear matter. The
available phase space for nuclear matter is enormous when compared to the region of it which
can be sampled experimentally. Large regions of the nuclear phase diagram are uncertain
and theoretical predictions there are poorly constrained. Heavy-ion collisions sample speciﬁc
regions of the nuclear phase space and can constrain theoretical predictions and the phase
diagram.
1

1.1

The Standard Model and Quantum ChromoDynamics

The theory which currently o↵ers the best description of the fundamental particles and
the interactions of those particles is the Standard Model. The Standard Model includes
6 leptons (the electron, muon, tau, and their respective neutrinos), 6 quarks (up, down,
charm, strange, top, and bottom), four gauge bosons as force carriers (the photon, gluon, Z
boson, and the two W bosons), and the Higgs boson (also known as the Englert-Brout-HiggsGuralnik-Hagen-Kibble boson). Forces between the fundamental particles are manifested as
exchanges of gauge bosons. The interactions between the gauge bosons and the quarks and
leptons are described by the theories of Quantum ElectroDynamics (QED), Electro-Weak
theory (EWT), and Quantum ChromoDynamics (QCD).
The electromagnetic force carrier is the photon, and the interaction between photons and
charged particles is described by QED. The weak force carriers are the Z bosons and the
two W bosons, and is described by EWT. The fact that the Z and W bosons are massive
particles limits the e↵ective range of the weak force. The interactions of gluons with quarks is
described as an exchange of ‘color charge’ (in analogy with electric charge) and it governed
by QCD. While QCD is analogous to QED in some ways, its behavior is fundamentally
di↵erent. Two of the most deﬁning behaviors of QCD are ‘color conﬁnement’ and ‘asymptotic
freedom’.

2

1.1.1

Color Conﬁnement

Gluons, the QCD force carrier, mediate the forces between objects with color charge just
as photons mediate the force between objects with electric charge. However, gluons have
color charge themselves, which gives rise to the phenomenon know as “color conﬁnement”.
The photon, having no charge, does not interact with itself, or other photons. This means
that as two charged particles are separated, the force between them decreases with distance.
This is because photons radiate in all directions, so the intensity falls o↵ with distance r
like, 1/ (surface area) / 1/r2 . In the analogous QCD example, as two quarks are separated
in space, the gluons between them interact with themselves and each other and produce
a ‘color ﬂux tube’. The e↵ect of this tube is that the force between the quarks does not
decrease with distance, but remains constant. In principle, it would require inﬁnite force to
separate two quarks. In practice however, once there is enough energy in the ﬂux tube, the
tube will ‘break’ by producing a quark-antiquark pair (Figure 1.1).
The spatial potential between two interacting quarks is given by,

Vs (r) =

4 ↵s
+ kr,
3 r

(1.1)

were r is the distance between the quarks, ↵s is the strong coupling constant, and k describes
the long range interactions. An equivalent statement is that the force between the same two
quarks is given by1

Fs (r) =
1F

(r) =

4 ↵s
+ k,
3 r2

dV (r)
dr .

3

(1.2)

Figure 1.1: The force between two quarks as a function of the distance between the quarks. A
color ﬂux tube forms between the quarks, which causes force to stay constant with distance.
Adapted from Ref. [1]. For interpretation of the references to color in this and all other
Figures, the reader is referred to the electronic version of this dissertation.
Which is to say that for large r the force is constant.

1.1.2

Asymptotic Freedom

Similar to QED, in QCD the renormalization coupling constant, ↵s (µ), depends on the
renormalization scale. In QED, due to charge screening from vacuum-ﬂuctuation virtualparticles, the coupling, ↵, decreases with distance. This is known as the running coupling
constant in QED. In QCD something similar happens, but gluon self interaction changes
2

g (µ)
the behavior of ↵s (µ). The coupling constant ↵s (µ) can be written as ↵s (µ) = s4⇡ ⇡
⇣4⇡
⌘,
ln µ2 /⇤2
0

where ⇤ is the QCD scale, µ is the momentum transfer scale, and 0 is the ﬁrst

order beta-function which encodes the energy independent part of the coupling parameter.
When 0 > 0 the coupling decreases logarithmically with energy, which is called asymptotic
freedom.
To say this is simpler terms: the quarks become asymptotically less bound as energy in-

4

creases, and quarks can only be completely ‘free’ in the limit of inﬁnite energy. Figure
1.2 shows the QCD renormalization coupling constant, ↵s (µ), as a function of energy (or
momentum) transfer showing the QCD prediction versus experimental measurements.
The practical consequence of asymptotic freedom is that the only way to study free quarks
is in systems at very high energies like those produced in heavy-ion collisions.

1.2

Quark Gluon Plasma

Due to the conﬁnement of quarks, as discussed in Section 1.1, at normal temperatures and
densities quarks are bound inside of hadrons.

1.2.1

The QCD Phase Diagram

A cartoon representation of the phase diagram for quark matter can be seen in Figure 1.3.
This representation of the phase diagram shows estimates of the phases of quark matter as
a function of temperature and baryon chemical potential (µB ), which can be thought of as
a proxy for density. At high enough temperature the phase becomes independent of µB and
the only possible phase is QGP. At temperatures below ⇠170 MeV (the critical temperature)
the phase depends upon both the temperature and µB .
At µB = 0, the density will also be zero, so the lower left point of Figure 1.3 corresponds to
vacuum. For low, but non-zero µB , below the phase transition to QGP, quarks and gluons
will coalesce into hadrons and take the phase of a hadron gas. Increasing µB is equivalent
to increasing density, and a point can be noted at temperature ⇠ 0 MeV and µB = nucleon

5

Figure 1.2: The QCD renormalization coupling constant, ↵s (µ), as a function of energy
(or momentum) transfer. The QCD calculation is shown with experimental measurements.
Taken from Ref. [2].

6

Figure 1.3: The QCD phase diagram. Adapted from Ref. [3].

7

mass (⇠ 939 MeV) which corresponds to to the transition to nuclear matter.
For µB greater than ⇠ 900 MeV, the density is greater than that of nuclear matter. Exotic
phases are theorized to exist at very large µB . Such phases may possibly exist at the cores
of neutron stars and could have unique properties such as color super conductivity.
In heavy-ion collisions, if there is su cient center of mass energy, the system will be forced
non-adiabaticly to a point in the QGP phase. The temperature of this initial state increases
with collision energy, while µB will decrease with collision energy.
In heavy-ion collisions after the initial QGP state is formed the system will cool and expand.
dµ

B
The expansion is isentropic and follows a trajectory so that dS = const, where S is the

entropy. Both the temperature and µB decreases until the system undergoes a phase transition and hadronizes as it passes into the hadron gas phase. If the system passes through
a ﬁrst order phase transition, it will move along the phase transition line toward lower µB
due to latent heat. After passing through the phase transition the system will continue to
cool isentropically.

1.2.2

The QCD Critical Point

For zero baryon chemical potential, µB = 0, lattice calculations indicate the there should
be a smooth crossover between the QGP phase and the hadron gas phase with a critical
temperature in the region of 160 to 170 MeV [4, 5, 6, 7, 8, 9]. The systems produced
in heavy-ion collisions at high incident energies are very near to vanishing µB [10]. For
temperature T = 0, lattice calculations, nuclear theory, and analogies with other physical
systems suggest that there should be a ﬁrst order phase transition most likely to a color

8

superconducting phase [11, 12, 8, 9]. If both results are true, then in the region of ﬁnite
temperature and ﬁnite µB there must exist a critical point. Lattice calculations in the region
of both ﬁnite temperature and ﬁnite µB are computationally expensive, but they also indicate
that there should be a critical point [7, 8, 9]. The only way to locate the critical point, if it
exists, is to explore the phase diagram experimentally with heavy-ion collisions.

1.2.2.1

Critical Opalescence

Systems which pass near a critical point between a smooth crossover and a ﬁrst or second
order phase transition will undergo a phenomena known as critical opalescence. Critical
opalescence was ﬁrst observed in 1869 in liquid CO2 . As the temperature was increased
through 304.25 K and was held at 72.9 atm the medium became cloudy [13, 14]. This
cloudiness indicated density ﬂuctuations occurring in subdomains which were large compared
to the wavelength of light. These ﬂuctuations scatter the light passing through the medium,
resulting in the cloudy appearance. This phenomena was explained by Albert Einstein forty
years later [15], and is related to the divergence of the correlation length, ⇠. The divergence
of the correlation length is common to all critical points between smooth crossovers and ﬁrst
or second order phase transitions.
In an idealized case the correlation length would diverge to inﬁnity; in practice however,
correlations are restricted to propagating at ﬁnite speeds and the systems are ﬁnite in volume
and time. This is important in heavy-ion collisions, because the system lifetime is very short
(a few fm/c), the volume is ﬁnite, and the volume is rapidly expanding. This means that
if observables are constructed which are supposed to be sensitive to the correlation length,
the magnitude of their deviation at the critical point will be strongly related to the size,

9

lifetime, and inherent properties of the system being studied. The observables analyzed for
this dissertation and their relationship to the correlation length are addressed in depth in
Chapter 4.

1.3
1.3.1

Heavy-Ion Collisions
System Evolution

The ﬁrst few fm/c seconds of a heavy-ion collision are dominated by hard processes such
as fragmentation, quark pair production, and jet production. Through the process of hard
interactions, the system evolves to local thermal equilibrium, and a strongly interacting
QGP phase is formed. The QGP phase has very high temperature and density, so it rapidly
expands and cools. Once the system reaches the phase transition it undergoes chemical freeze
out, at which point the QGP hadronizes into a hadron gas and the chemical composition is
approximately ﬁxed. For most of the energies studied the transition from QGP to hadron
gas is a smooth crossover so it should be noted that di↵erent parts of the medium undergo
the phase transitions at di↵erent points in time. This means that there is a period in the
evolution of the system where there is a core of QGP matter surrounded by a hadron gas.
After chemical freeze out, the system continues to interact kinetically and the produced
hadrons will scatter o↵ each other. Some of the produced hadrons will be unstable and will
decay during this phase of the system. The scattering continues until the system reaches
kinetic freeze out, at which point the volume is di↵use enough that the particles no longer
interact at all. After kinetic freeze out the thermal and kinetic information of the system is
ﬁxed.
10

Figure 1.4: A representation of the system evolution in heavy-ion collision showing the phases
and transitions. The ordinate corresponds to time, and the abscissa corresponds to distance
along the beam axis. The origin is the center of the collision at the moment of impact. Taken
from Ref. [16].

11

After the ﬁnal freeze out, the produced particles stream out into the detectors where their
masses, charges, energies, and other properties are measured. The only information available
to a physicist is the data read out from each detector, so any information about the earlier
states of the system has to be inferred from the particles.

1.3.2

Collision Centrality

An important parameter in heavy-ion collisions is the event centrality. The collision centrality
is analogous to the impact parameter of the colliding nuclei. If the two nuclei collide exactly
head-on, with an impact parameter b = 0 fm, we would say the collision was central. As
the impact parameter is made larger, we say the collisions become more peripheral, until we
reach the most peripheral possible collisions at b ⇡ 14 fm in the case of gold (14 fm = 2 ⇥ 7
fm, where 7 fm is approximately the radius of a gold nucleus).
Experimentally the collision centrality for an event is determined using the multiplicity of
the event assuming that more central collisions have higher multiplicity. Multiplicity can be
deﬁned experimentally in many ways, but they are all attempts to quantify the number of
particles emitted from an event. In practice only charged particles are observed, and the
geometrical acceptance is limited by the detector. Some analyses (those which are studying
properties of the multiplicity distributions) need careful multiplicity deﬁnitions in order to
avoid biasing their results. In all cases the objective is to use a measure which is a proxy for
the absolute number of particles produced in a collision. To relate experimentally measured
multiplicities to inferred centralities, Glauber Monte-Carlo simulations are used and ﬁt to
the data. This is addressed in more depth in Section 3.6.

12

Chapter 2
Experimental Setup

2.1

The Relativistic Heavy-Ion Collider (RHIC)

All data presented in this dissertation were taken at the Relativistic Heavy-Ion Collider
(RHIC) which is an experimental facility located at Brookhaven National Laboratory (BNL)
in Upton, NY. RHIC consists of a collection of accelerators and storage rings capable of
colliding both high intensity polarized protons as well as heavy-ions. Figure 2.1 shows an
aerial view of RHIC. RHIC’s accelerating chain has four steps: the Tandem Van de Graa↵
linear accelerator, the Booster synchrotron, the Alternating Gradient Synchrotron (AGS),
and the main rings. The main rings consist of two super-cooled concentric storage rings,
through which two beams can be circulated in opposite directions. The main rings run
through a tunnel 3.8 km in circumference, and there are six interaction areas where the
beams can be collided. Only four of RHIC’s six interaction regions are used for experiments.
The experiments at RHIC are BRHAMS (Broad RAnge Hadron Magnetic Spectrometers) at
2 o’clock, PHENIX (Pioneering High Energy Nuclear Interactions experiment) at 8 o’clock,
PHOBOS1 at 10 o’clock, and STAR (Solenoidal Tracker at RHIC) at 6 o’clock. Of these
1 PHOBOS

is not an acronym. According to an interview with Wit Busza, spokesman
for PHOBOS, “we ﬁrst proposed a slightly more expensive experiment called the Modular
Array for RHIC Spectra, or MARS. That was considered too expensive, so we came up with
a reduced version and one of my colleagues at MIT said that, since Mars was too expensive,
13

Figure 2.1: Aerial view of RHIC. From Ref. [18].
four, only STAR and PHENIX are still collecting data. Both BRHAMS and PHOBOS have
completed their experimental programs and were decommissioned once their physics goals
had been reached.
The production of heavy-ions begins at the ion source which feeds the Tandem Van de Graa↵
accelerator. All data used in this thesis were taken for Au + Au collisions. Negatively charged
gold ions are produce by a cesium sputter source operated in pulsed beam mode [19]. The
cesium sputter source in pulsed mode can deliver 500 µs pulses with a peak intensity of 290
µA without damaging the accelerator. In 2010 the ion source was upgraded to the Electron
Beam Ion Source (EBIS), which can deliver much higher intensities on the order of mA
why not build the moon of Mars, which is Phobos. And if that was still considered too
expensive, we ﬁgured we’d come up with Deimos, a still smaller moon of Mars. So that’s
the origin of the name.” [17]
14

[20].
From the ion source the ions are injected into the Tandem Van de Graa↵. They are partially
stripped to a positive charge state at the terminal (the ﬁltering after the Tandem selects
those ions which were stripped to a +12 state) with a thin carbon foil (2 µg/cm2 ) [21], and
they are accelerated to an energy of 1 AMeV by the time they exit the Tandem. At the exit
of the Tandem, the ions are further stripped, and then ﬁltered with dipole bending magnets
to select only ions with a charge state of +32 (which had been accelerated to 1 AMeV by
the Tandem) [22].
These are then sent into the Booster synchrotron, where they are bunched and accelerated
to 95 AMeV. After the Booster, the beam is again stripped and ﬁltered so that all ions have
a charge state of +77 (helium-like). The ions are then sent into the AGS which accelerates
them to the RHIC injection energy of 10.8 AGeV, and fully strips the ions to a charge state
of +79. After being stripped and ﬁltered, each bunch contains 109 ions and they are injected
into one of the two counter-rotating main RHIC rings via a switching magnet at the end of
the AGS-to-RHIC Beam Transfer Line. The injection of bunches into the main RHIC rings
continues until both beam lines are full which takes on the order of a minute [21].
Once in the main ring, the ions are accelerated up to the desired energy. The main ring can
collide energies from the injection energy of 10.8 AGeV up to a top operating energy of 100
p
p
AGeV ( sNN = 200 GeV, where sNN is the energy per nucleon pair) for heavy ions and
p
250 GeV ( sNN = 500 GeV) for protons. Au + Au collisions have been run at RHIC at and
below the injection energy of

p

sNN = 19.6 GeV by lowering the energy of the ions from the

AGS and not accelerating the ions in the main ring (just using the rings for storage and not
ramping the beam). Energies down to

p

sNN = 7.7 GeV have been produced for Au+Au
15

collisions.
The RHIC main ring has two independent beam lines separated 90 cm horizontally which
circulate in opposite directions [22]. As viewed from above, the clockwise beam is referred
to as blue, while counter-clockwise is yellow. The independent beam lines allow RHIC to
collide not just symmetric systems (p + p, Cu + Cu, and Au + Au), but also asymmetric
systems (d + Au). The main ring is composed of six 356 m ring arc segments and six 277
m long insertion segments. Each arc segment contains 11 sections which contain two dipole
superconducting magnets which bend the beams, and two quadrupole and two sextupole
superconducting magnets which focus the beams.
All of the data used in the analyses presented in this dissertation are from Au + Au collisions
collected during Runs 10 and 11 (taken in 2010 and 2011 respectively). Speciﬁcally, the data
for energies
p

p

sNN = 7.7, 11.5, 39, and 62.4 GeV were collected in Run 10. The energies

sNN = 19.6, 27, and 200 GeV were collected during Run 11. During Run 10,

p

sNN =

200 GeV was also run, but in Run 11 the data collection triggers were improved to prevent
pile-up in the detector, so only the Run 11 data are presented (see Section 3.7).

2.2

STAR

The data used in this dissertation were collected with the STAR detector. STAR is a
large acceptance detector with full azimuthal coverage (2⇡), good track resolution, and good
particle identiﬁcation. The STAR detector is a combination of many detector subsystems
which are azimuthally symmetric about the beam pipe which run through the center of
STAR. All together, STAR is three stories tall and weighs 1,200 tons. Most of STAR’s

16

Figure 2.2: Artist rendering of STAR with detector subsystems labeled. Figure produced by
Maria and Alexander Schmah.
weight comes from the room-temperature solenoid magnet which surrounds most of the
detector subsystems and weighs 1,100 tons. The STAR magnet can be run at either it’s full
ﬁeld strength (0.5 T) or at half-full strength (0.25 T). The magnetic ﬁeld is parallel to the
beam axis, and is uniform within 0.0040 T [23].
Many of the detector subsystems in STAR have changed since STAR’s commissioning in 2000.
The heart of STAR, the STAR Time Projection Chamber (TPC) has remained unchanged.
Section 2.3 is devoted to it. The Time of Flight detector (ToF) was added before Run 10 in
2010, and is discussed in Section 2.4. Other detector subsystems such as the Electromagnetic

17

Figure 2.3: STAR detector subsystems. Adapted from Ref. [24].
Calorimeter (EMC), the upgraded pseudo-Vertex Position Detectors (upVPD), and the Zero
Degree Calorimeter (ZDC) are addressed in Section 2.5.
Figure 2.2 shows the STAR detector with the subsystems labeled. In this ﬁgure, the end
cap is pulled back so that the internals of STAR can be viewed. Coordinates in STAR are
often given in terms of the cartesian directions x, y, and z. These are deﬁned such that
the x-axis points south, the y-axis points up, and the z-axis points west along the beam
line. The origin of the coordinate system (x = y = z = 0) is located at the geometric
center of STAR. Sometimes STAR coordinates are given in angular coordinates such that ✓
is measured from the positive z-axis, and

is measured from the positive y-axis. In practice,

✓-angles are rarely used and angular coverage with respect to the z-axis is usually given in
terms of pseudorapidity, ⌘, which is deﬁned as:

18

⌘=

✓

✓
ln tan
2

◆

.

(2.1)

Starting from the beam pipe, the ﬁrst detector that the particles will interact with is the
TPC. Near the beam pipe, outside of the TPC, are the upVPDs. Continuing radially aways
from the beam pipe, outside of the TPC is the ToF detector. Radially after the ToF is one
part of the EMC, the Barrel EMC (BEMC). The other part of the EMC, the End-cap EMC
(EEMC), is located in the positive z direction, outside of the upVPD. Outside of all of this
is the STAR magnet. Along the beam pipe, just outside the STAR magnet are the Beam
Beam Counters (BBC). The ZDCs are located 18 m down the beam line at the ﬁrst bends
of the beam pipe.

2.3

Time Projection Chamber

The Time Projection Chamber is the heart of the STAR detector. The TPC is a gas detector
that measures particle’s tracks and can determine path length ( s [cm]), momentum (p
[GeV/c]), and ionization energy loss (dE/dx [MeV/cm]) in the TPC gas. Figure 2.4 shows
a drawing of the STAR TPC. The STAR TPC was the largest TPC in the world for some
time until it was surpassed in 2004 by the TPC built for ALICE at the Large Hadron
Collider. The TPC is 4 m in diameter and 4.2 m in length. The TPC has full azimuthal
coverage (0 <

< 2⇡), and covers a pseudorapidity range of

1 < ⌘ < 1 (equivalent to

45 < ✓ < 135 ).
The TPC can measure charged particle momenta for particles in the range of 100 MeV/c
up to 30 GeV/c [25]. The TPC is ﬁlled with a mixture of 10% methane and 90% argon gas,
19

which is kept at 2 mb above atmospheric pressure. The positive pressure is maintained in
order to insure purity of the TPC gas; if there are any leaks, the TPC gas will leak out,
rather than air leaking in.
When a charged particle passes through the TPC, it ionizes the TPC gas. The released
electrons drift toward the ends of the TPC pushed there via an 135 V/cm electric ﬁeld
maintained between the central membrane (located at z = 0) and the ends of the TPC via
an inner and outer ﬁeld cage and the end caps. When the electrons reach the ends of the
TPC they trigger electron avalanches which amplify the track signals 1000-3000 times. The
electron drift velocity (5.45 cm/µs) and the time it takes the electrons to drift to the end
caps give a measure of the particle position in the z direction (parallel to the beam axis).
The position of the particle in the plane perpendicular to the beam axis is given by the
Multi-Wire Proportional Chambers (MWPC). The MWPC measures the temporary image
charge induced by the electron avalanche. The MWPC is highly segmented, with 12 sectors
each containing 5692 pads. The optimal e ciency of the MWPC is 96%, with the ine ciency
due to sector boundaries.

20

Figure 2.4: Diagram of the STAR TPC. From Ref. [25].
21

Particle identiﬁcation (PID) is possible with the TPC, because dE/dx is mass and charge
dependent. Pions and kaons can be reliably distinguished in the momentum range of 0.1 <
p < 0.7 GeV/c, and protons can be identiﬁed up to ⇠1.0 GeV/c.

2.4

Time of Flight Detector

While the TPC can perform PID, it is limited to low momentum particles. The STAR Time
of Flight detector (ToF), in conjunction with the TPC, provides PID capabilities at much
larger momentum than the TPC alone. The ToF measures particle ﬂight times ( t [s]),
which when paired with track information from the TPC, gives the particle velocities ( ),
and with it, mass (m [GeV/c2 ]).
Flight time determination requires two time measurements: a start time, t0 , and a stop time,
tstop . Only the “stop” times are measured by the ToF, and the “start” times are provided by
the upVPDs. The upVPDs will be addressed in more detail in Section 2.5.1. There are two
upVPDs, located near the beam pipe to the east and west of the collision area. The upVPDs
measure photons from the collisions, so neither measures t0 , but rather they measure teast
and twest , which can be written as:
teast = t0 +

L + Vz
c

(2.2)

twest = t0 +

L

Vz
c

(2.3)

Where L is the distance from the upVPDs to the center of STAR, and Vz is the z-position
of the collision in STAR.2 From these, it can be seen that

t0 =
2 This

teast + twest
2

L
.
c

simpliﬁed formulation assumes that Vx = Vy = 0.
22

(2.4)

From the start time t0 provided by the upVPDs, and the stop time tstop provided by the
ToF itself, the time of ﬂight,

t, can be calculated,

t = tstop

Using the path length

t0 .

(2.5)

s from the TPC, it is possible to calculate the inverse velocity 1/

via3 ,

1

t
.
s

=

(2.6)

Additionally, using the momentum p from the TPC, we can calculate mass:

m=

p

=

p

p

1

2

=p

s
✓ ◆2
1

1

(2.7)

The relationship between 1/ and p is more sensitive to mass at larger values of p, so using
the ToF, pions and kaons can be identiﬁed up to ⇠1.8 GeV/c and protons up to ⇠3 GeV/c.
This extends the momentum region over which PID is possible by almost a factor of ⇠3
compared to the TPC alone.
The ToF detector consists of 120 trays of Multigap Resitive Plate Chambers (MRPC). Each
ToF tray contains 32 MRPCs, for a total of 3840. The ToF trays are arranged in two rings
of 60 trays each surrounding the TPC inside the STAR magnet. A diagram showing the
placement of the ToF trays with respect to the TPC can be seen in Figure 2.5. The ToF covers
the full azimuthal range (0 <
3 In

< 2⇡) and a pseudorapidity range of

all of the equations presented here, we set c = 1
23

0.94 < ⌘ < 0.94.

Figure 2.5: Location of one tray of the STAR ToF. From the ToF proposal and documentation Ref. [26].
Each MRPC consists of a stack of seven glass plates with 220-µm wide gaps between each
plate as can bee seen in Figures 2.6 and 2.7 [26]. On either side of this stack are graphite
electrodes which generate a large potential di↵erence across the stack (of order 15.5-17 kV).
Outside of the peripheral graphite electrode is a PCB with six copper readout pads. When
a charge particle passes through the MRPC, it will ionize the gas between the plates. The
large potential di↵erence causes an electron avalanche, which is detected by the readout
pads.

24

Figure 2.6: Layout of each MRPC from the STAR ToF. From the ToF proposal and documentation, Ref. [26]. Continued in
Figure 2.7.

25

26
Figure 2.7: Layout of each MRPC from the STAR ToF. From the ToF proposal and documentation, Ref. [26]. Continued from
Figure 2.6.

26

For each event, all ToF hits are mapped and compared to a map of the reconstructed tracks
from the TPC. For ToF cells that mapped to only one reconstructed track, the ToF information is associated with the TPC track. The ToF-TPC matching introduces an additional
ine ciency, which can be seen plotted in Figure 2.8. The e ciency is both mass and pt
dependent for pt < 1 GeV/c, and above pt < 1 GeV/c the e ciency is constant at about
70%.

2.5

Other STAR Subsystems

The TPC and ToF are the primary detectors which measure individual track data, but
there are many other detector subsystems which are used for triggering, vertex position
determination, background suppression, and analysis speciﬁc data.

2.5.1

Vertex Position Detectors

There are two upgraded pseudo-vertex position detectors (upVPDS), which are located near
the beam line, inside of STAR, outside of the TPC. Their location can be seen variously in
Figures 2.2, 2.3, and 2.5. The upVPDs were installed in 2006 as an upgrade to replace the
previous pseudo-vertex detectors (pVPDs).
Each upVPD consists of 19 assemblies, where each assembly contains two cylindrical disks
of lead (total thickness of 1/4 inch) which act as a converter, backed by a cylindrical disk
of plastic scintillator (thickness 1 cm), all mounted on a photomultiplier tube (PMT). All
parts of the assembly (lead, scintillator, and PMT) have a diameter of 1.5 inches [27].

27

Figure 2.8: ToF matching e ciency for identiﬁed particle species. From Ref. [1].

28

The upVPDs are not only used to determine the start-time for each event, as described in
Section 2.4, their secondary purpose is to determine the location of the event vertex along
the beam axis. From Equations 2.2 and 2.3 we can see that,

Vz =

2.5.2

(teast

twest ) c
.
2

(2.8)

Electromagnetic Calorimeters

The Electromagnetic Calorimeter (EMC), consists of two independent detectors: the Barrel
EMC (BEMC), and End cap EMC (EEMC). The BEMC is installed outside the ToF detector, inside of the STAR magnet, has full azimuthal acceptance (0 <
pseudorapidity

< 2⇡), and covers

1 < ⌘ < 1. The EEMC is installed in the west end cap, and covers the

forward pseudorapidity region 1.09 < ⌘ < 2, also with full azimuthal coverage. Both the
BEMC and EEMC are lead- scintillator sampling calorimeters which can measure particle
energy and be used for triggering.

2.5.3

Beam Beam Counters

The Beam Beam Counters (BBC) are scintillator annuli installed around the beam pipe just
outside the pole tips of the STAR magnet. They were built to be used as local polarimeters
for use with polarized protons beams, although they are also used as triggers. Each of the
BBC detectors consists of 36 scintillator tiles divided into two groups of 18. The two groups
of tiles are di↵erent sizes, and arranged into two rings, as can be seen in Figure 2.9. Each of
the detectors is placed 3.75 meters aways from the center of STAR.

29

Figure 2.9: The layout of the BBC detectors. Ref. [28].

30

2.5.4

Zero Degree Calorimeters

The Zero Degree Calorimeters (ZDCs) are small hadronic calorimeters located approximately
18 meters from the center of STAR on the beam axis after the ﬁrst bends in the beam line.
The acceptance area is very small, extending only approximately 5 cm away from the beam
axis. The ZDCs consist of three modules, each a layer of lead and a layer of plastic scintillator.
The ZDCs mostly detects spectator neutrons and is used for triggering as well as monitoring
the beam luminosity.

31

Chapter 3
Data from the Beam Energy Scan

3.1

RHIC Beam Energy Scan

All of the data used in this dissertation were collected during Runs 10 and 11, taken during
years 2010 and 2011 respectively. The data collected during Runs 10 and 11 were from the
RHIC Beam Energy Scan (BES). The purpose of the BES was to systematically explore a
large energy range, which corresponds to a large part of µB -T space of the nuclear phase
diagram (see Figure 1.3). All of the BES energies were done with Au+Au collisions. The
energies run during the BES, and also the energies analyzed in this thesis, are

p

sNN = 7.7,

11.5, 19.6, 27, 39, 62.4, and 200 GeV. The data sets used in this analysis are listed in Table
3.1 along with calculated µB values corresponding to the collision energy, the year the data
were taken, and the number of events used for each energy.
There are various units of measurement for the data collected at STAR. Each year’s total data
is designated by “Run” number. The ﬁrst year that data were taken was 2001, corresponding
to Run 1, resulting in a convenient correlation between year in the decade and Run number.
Over the course of one year’s Run, many energies and species may be collided. While running
each day there are several “ﬁlls”, where the RHIC ring is ﬁlled with nuclei, the nuclei are
ramped up to the desired energy, collisions are performed and physics is taken. When the
32

p

sNN (GeV) hµB i (MeV)
7.7
421
11.5
316
19.6
206
27
156
39
112
62.4
73
200
24

Year
2010
2010
2011
2011
2010
2010
2011

Species Events (M)
Au+Au
1.6
Au+Au
2.4
Au+Au
16.3
Au+Au
44.4
Au+Au
10.7
Au+Au
34.5
Au+Au
23

Table 3.1: Table of run data for data sets used in this dissertation. The baryon chemical
potential (hµB i) values are from Ref. [10].
beam luminosity becomes too low, the ﬁll is dumped. Over the course of a single ﬁll many
“runs” are taken. It is an unfortunate that the term “run” is used twice, but it is convention.
A run may last a few minutes or a few hours depending on the energies and species being
collided, the data triggers being used, and the physics goals. Usually, the primary constraint
on the length of a run is the size of the resulting data ﬁles. Most people prefer multiple
manageably-sized data ﬁles to a few unwieldy ones. An additional advantage of taking many
smaller runs is that if the detector changes over the course of a ﬁll it can be identiﬁed.
For each energy, only a subset of the full data taken was used in the analysis. The full data
set was processed, and periods with a minimum of detector problems and strong consistency
of quality assurance observables were selected as good runs for use in the analysis. The good
run determination is discussed in more detail in Section 3.2. Many of the detectors used
in the analysis require calibrations before and during each run; some of these calibrations
are discussed in Section 3.3. Some of the important measures, notably the multiplicity of
each event, depend upon other observables such as beam luminosity and vertex position.
Those observables are corrected for these e↵ects, and the procedure and the corrections are
discussed in Section 3.5. In addition to all of these methods of quality assurance, analysis
cuts are placed on the tracks of each event to diminish backgrounds, suppress secondary

33

interactions, and to ensure uniform detector performance. These track cuts are discussed in
Section 3.4. As was discussed in Section 1.3.2, event centrality strongly a↵ects many physics
observables. The method of centrality determination is discussed in Section 3.6.

3.2

Good Run Determination

During the course of an experiment many things can go wrong and many things can change;
the magnet may trip, detector power supplies may fail, TPC read out boards may stop
working, ToF trays may stop working, TPC anodes may short out, the beam may get
dumped, and the beam may even accidentally get dumped in the detector. The reason why
things go wrong is often not clear, and often the ﬁx is to power cycle the system which is
having problems. If power cycling the system doesn’t work, often the next step is to power
cycle the power supply crates for that system as soon as an experimental access is possible.
In some cases, when systems aren’t ﬁxed by simple solutions, they are masked out of the
data acquisition system until an expert can address them or an extended access is possible.
These variations in the state and quality of the detectors means that often a substantial part
of the data taken is unusable for particular analyses.
These variations are addressed by “good run selection”, where by the entire data set is looked
over, and sections of the data are selected where quality assurance observables remained
constant, and there were no major problems with the detector.

34

3.3

Calibrations

The data from every detector have to be digitized and processed before they are useful for
analysis. Here we will address calibrations that needed to be performed on the data from
two of the primary detectors: the TPC and the ToF. Calibrating detectors is a large undertaking which is shared by all members of the STAR collaboration. The types of calibrations
which are performed and the general approaches are explained here, but for more detailed
information the reader should consult the referenced material.

3.3.1

TPC Calibrations

The STAR TPC is calibrated by use of two Nd:YAG (neodymium-doped yttrium aluminum
garnet; Nd:Y3 Al5 O12 ) lasers [29]. This type of laser was selected because it could deliver
a beam of su cient energy density(2-10 µJ/mm2 ) to produce ionization of the TPC gas
equivalent to minimum-ionizing particles without the addition of organic gas additives which
would accelerate aging. The beams from the two lasers are expanded to 30 mm in diameter
then directed at splitters on the TPC end caps that split each beam into six beams of equal
intensity, all of which are directed parallel to the beam axis. Each of these twelve beams
is incident upon a six separate bundles of seven mirrors, placed at intervals along the TPC
length. This arrangement produces a total of 504 laser beams that cover the entire TPC
volume. The laser mirrors are mounted on dielectrics, and the laser beams can be positioned
to an accuracy of 50 µ [29]. Figure 3.1 shows an example of a reconstructed laser calibration
run in the TPC.
Laser calibrations are taken at regular intervals during data collection to ensure that any

35

Figure 3.1: An event reconstruction of the lasers used in the TPC calibration. From Ref.
[30].
changes in the performance of the TPC is noted. The TPC calibration procedure consists
of reconstructing the signals produced by the TPC so that the reconstructed laser tracks
coincide with their known spatial locations.

3.3.2

ToF Calibrations

There are many calibrations done to the data from the ToF detector before it is used in any
analysis. These calibrations fall into two categories: calibrations done to address hardware
properties which remain unchanged with time, and run-by-run calibrations. There are two
hardware calibrations of signiﬁcance: correcting for the integrated non-linearity (INL) of the

36

timing chips, and correcting for signal delays due to di↵erences in cabling length. There are
two run-by-run calibrations of note: correcting for “slewing”, a detector e↵ect where particles
which deposit more energy into the detector get read as arriving earlier, and correcting for
the “Zhit ” e↵ect.
All of the hardware calibration measurements were carried out at Rice University during the
assembly and testing of the ToF detector. The INL e↵ect is a consequence of the timing chips
used in the ToF trays, and arrises from the fact that when signals come in they are binned
according to arrival time, but the widths of these bins are not identical. Each timing chip
has di↵erent INL behavior, and these were all measured extensively before being installed in
STAR. The other hardware calibration is to address cable-length delays. These are delays in
signals from ToF trays which are further from the data acquisition system in terms of cable
length.

3.4

Track Cuts

Even after good run and good event selection, there can still be bad events in good runs.
Also, in good events, not all tracks are usable for analysis. Events can be rejected if the
vertex is located too far from the center of the STAR detector. The vertex position with is
given given as a point (Vx , Vy , Vz ) with the origin at the center of STAR. For all energies the
q
2
vertex is required to be located within 2 cm of the center of the beam axis ( Vx + Vy2 < 2
cm). This is to ensure that the vertex is not to close to the beam pipe, also the further the
reconstructed vertex is from the center of the beam axis, the more likely it is that it was
mis-reconstructed and is not correct. Additionally, the vertex is required to be within 30 cm

37

p

sNN

Year

pt (GeV/c)

⌘

Vz (cm)

DCA (cm)

7.7 GeV

2010 0.15 < pt < 2.0 |⌘| < 1.0 |Vz | < 30.0 DCA < 1.0

11 GeV

2010 0.15 < pt < 2.0 |⌘| < 1.0 |Vz | < 30.0 DCA < 1.0

19.6 GeV

2011 0.15 < pt < 2.0 |⌘| < 1.0 |Vz | < 30.0 DCA < 1.0

27 GeV

2011 0.15 < pt < 2.0 |⌘| < 1.0 |Vz | < 30.0 DCA < 1.0

39 GeV

2010 0.15 < pt < 2.0 |⌘| < 1.0 |Vz | < 30.0 DCA < 1.0

62.4 GeV

2010 0.15 < pt < 2.0 |⌘| < 1.0 |Vz | < 30.0 DCA < 1.0

200 GeV

2011 0.15 < pt < 2.0 |⌘| < 1.0 |Vz | < 30.0 DCA < 1.0

q
2
Vx + Vy2 (cm)
q
V 2 + Vy2 < 2.0
q x
V 2 + Vy2 < 2.0
q x
V 2 + Vy2 < 2.0
q x
V 2 + Vy2 < 2.0
q x
V 2 + Vy2 < 2.0
q x
V 2 + Vy2 < 2.0
q x
2
Vx + Vy2 < 2.0

Table 3.2: The data sets and analysis cuts used in this analysis. The pt and ⌘ cuts were
selected to agree with previous analyses.
of STAR along the beam axis for all energies (Vz < 30 cm). Especially at lower energies,

where the beam is less focused, it is possible for collisions to happen far from the center of
STAR, but for these events the STAR acceptance is not symmetric and the probability of
having missing tracks grows. Tracks are also cut on the distance of closest approach (DCA)
between the track and the event vertex.

3.5

Multiplicity Corrections

As was discussed in Section 1.3.2, the centrality of a collision is determined from the multiplicity the event deposits in the detector. The method of centrality determination will be
discussed in depth in Section 3.6. Here we will introduce two multiplicity measures which
are used in STAR and discuss how they are corrected for dependence on other observables.
Multiplicity corrections are a large undertaking which is shared by all members of the STAR
collaboration.
The primary multiplicity value used in most analyses, and used throughout most of this
38

dissertation is refMult. RefMult is deﬁned as the number of charged particles detected
within the rapidity range

0.5 < ⌘ < 0.5. This value, however, is observed to depend on

several other measures: beam luminosity (as measured by the ZDC, see Section 2.5.4), and
vertex position along the beam axis (Vz ). The dependence upon beam luminosity can be
explained logically by the fact that at higher beam luminosities there are large backgrounds,
so naturally the number of tracks in the detector should be higher. The dependence upon
Vz happens because events with larger Vz values occur closer to the ends of the detector,
and more of the tracks are lost, resulting in a lower observed multiplicity.
The other multiplicity deﬁnition used in this dissertation is refMult2. There are observables
that are based on the distribution of event-by-event multiplicties. Centrality determination is
still necessary in these analyses, but using the same multiplicity observable for centrality cuts
as is being studied introduces bias which is known as “auto-correlations”. Several alternative
multiplicity deﬁnitions have been proposed, one of which is refMult2. The analyses presented
in this dissertation are not directly related to multiplicity, but event multiplicity and hpt i are
known to be correlated, so checks were performed to ensure there where no auto-correlations
(see Section 6.4).
refMult2 is deﬁned as the number of charged particles seen in the detector in the rapidity range 0.5 < |⌘| < 1.0. Observables using refMult2 for the centrality determination are
restricted to the tracks within |⌘| < 0.5, so that none of the tracks used in the centrality
determination are included in the analysis. RefMult2 has similar dependence upon beam
luminosity and Vz as refMult, and the same corrections done for refMult were done independently for refMult2.

39

3.6

Centrality Determination

After the multiplicity has been corrected, centrality determination is done using wounded
nucleon Glauber Monte Carlo simulations. Centrality determination assumes that there is
a monotonic relationship between impact parameter and multiplicity, dN/db, where N is
the multiplicity and b is the impact parameter. There are many ways to determine the
relationship between impact parameter and multiplicity, we use a wounded nucleon Monte
Carlo simulation. Wounded nucleon Glauber Monte Carlo is a type of heavy-ion collision
simulation. In it each nucleus is modeled as a spherical bundle of nucleons. An impact
parameter is chosen randomly from an appropriate distribution, and then the nuclei are
collided. All nucleons are assumed to travel in straight lines, and nucleon collisions are determined using a nucleon-nucleon cross-section (taken from proton-proton collisions). From
just this information, the number of participants (Npart ), and number of binary collisions
(Ncoll ) can be extracted. The Glauber Ncoll distribution can then be extrapolated to a
multiplicity distribution by assuming that each collision will produce a random number of
particles given by a negative binomial distribution (NDB). The Glauber simulation and NBD
are ﬁt to the experimental multiplicity distribution by tuning the parameters of the model
and NBD.
Using the wounded nucleon model allows an approximate mapping between the impact
parameter of the colliding nuclei and the observed multiplicity. The model also allows the
multiplicity distribution to be extended all the way to multiplicties of one. Experimentally,
the multiplicity distribution is truncated for small multiplicity values. The e ciency of
detecting events begins to drop for events with multiplicity less than 20 (Ref. [31]) due to
trigger ine ciencies (Section 3.7 for trigger deﬁnition).
40

Having the full multiplicity distribution is important because centrality bins are deﬁned as
fractions of the whole distribution. Centrality bins are given as percentages of the whole, and
are deﬁned such that the most central events are those in the ﬁrst few percent (0-5%), and
the most peripheral events are those in the last few percent (90-100%). These centrality bins
are calculated by integrating the entire Glauber simulated multiplicity distribution to get the
total multiplicity, then integrating the distribution from the highest observed multiplicity
value down towards zero until the appropriate fraction of the total has been accounted
for.
Figure 3.2 illustrates the relationship between impact parameter, Npart , the total number of
observed particles (Ntotal ), and centrality for Au+Au. The ﬁgure is not data, and is only
meant to illustrate the relationship between centrality and multiplicity. In the ﬁgure, Ntotal
corresponds to approximately twice refMult because refMult is equivalent to Ntotal measured
in the region |⌘| < 0.5.
The centrality bin cuts for the refMult multiplicity observable used in this dissertation are
listed in Table 3.3. Centrality bins were calculated in steps of 10% of the total refMult
distribution, except for the two most central bins which were done in steps of 5%. Only the
ﬁrst eight centrality bins were used.
Table 3.4 lists the total number of events in each centrality bin for each energy used in the
analyses of this dissertation. Note that the number of events in each 10% bin is approximately
equal, and is twice that as in the 5% bins, as should be expected.

41

Figure 3.2: An illustration of the relationship between impact parameter, Npart , the total
number of observed particles (Ntotal ), and centrality for Au+Au collisions. This ﬁgure is
not experimental data, and is only meant for illustrative purposes. The values of Ntotal will
change with collision energy. From Ref. [32].

42

p

sNN
7.7 GeV
11.5 GeV
19.6 GeV
27 GeV
39 GeV
62.4 GeV
200 GeV

0-5% 5-10% 10-20% 20-30% 30-40% 40-50% 50-60% 60-70% 70-80%
185
154
106
72
46
28
16
8
4
221
184
127
86
56
34
19.6
10
5
263
220
152
102
66
40
23
12
6
288
241
168
114
74
45
26
13
6
316
265
185
125
81
50
28
15
7
339
285
196
135
88
54
30
16
7
466
396
281
193
125
76
43
22
10

Table 3.3: Lower reference multiplicity cuts (refMult) used per centrality bin. The upper multiplicity cut for the most central
bin was set arbitrarily high, and the lower cut for the most peripheral bin was set to 0.
Energy
7.7 GeV
11.5 GeV
19.6 GeV
27 GeV
39 GeV
62.4 GeV
200 GeV

0-5% 5-10% 10-20% 20-30% 30-40% 40-50% 50-60% 60-70% 70-80%
Total
80k
83k
173k
173k
183k
180k
178k
189k
159k
1,607k
127k
128k
262k
264k
263k
271k
276k
265k
236k
2,373k
921k
882k
1,836k 1,878k 1,853k 1,879k 1,808k 1,831k 1,603k 16,319.6k
2,361k 2,407k 4,879k 4,954k 4,985k 5,051k 4,849k 5,249k 4,928k
44,385k
587k
584k
1,179k 1,206k 1,196k 1,169k 1,210k 1,129k 1,176k
10,707k
1,858k 1,886k 3,827k 3,822k 3,776k 3,776k 3,908k 3,605k 3,995k
34,479k
1,480k 1,507k 2,884k 2,764k 2,763k 2,691k 2,599k 2,507k 2,124k
23,030k

Table 3.4: The number of events in each centrality bin which passed the analysis cuts. Note that for each energy the number
events in each 10% bin is approximately equal and twice that in the two 5% bins, as should be expected.

43

3.7

Trigger Deﬁnition

The Star Data Acquisition system (DAQ) records events when the appropriate trigger conditions are satisﬁed. For this dissertation most of the data were collected using the minimumbias trigger, which was deﬁned as a coincidence of the signals from the ZDC, the upVPD
and/or BBC. The 200 GeV data were collected using a pile-up rejection trigger. The pile-up
rejection trigger was deﬁned the same as the minimum-bias trigger, but events would not be
recorded if there had been a event ⇡ 40 µs before or after the triggered event. The timing is
set by the size of the TPC and the electron drift velocity. It is set so that there is su cient
time for all of the electrons to clear the TPC volume. Pile-up occurs when charges from
tracks associated with previous or following events are within the TPC volume when the
current event is recorded. These “piled up” tracks can possibly be assigned to the wrong
event, and can bias analyses. Pile-up becomes more of a problem as beam luminosity and
beam energy increase, which is why the pile up rejection trigger was used for 200 GeV.

3.8

The UrQMD Model

The Ultrarelativistic Quantum Molecular Dynamics model (UrQMD) is a microscopic model
used to simulate (ultra)relativistic heavy ion collisons [33, 34]. UrQMD is a microscopic
transport model. Nuclei are modeled in three-dimensions as clusters of hadrons. Two nuclei are collided, and hadron interactions are determined by experimentally measured cross
sections. For energies

p

sNN < 5 GeV the phenomenology of hadronic interactions is de-

scribed in terms of interactions between known hadrons and their resonances. For energies
p

sNN > 5 GeV particle production is dominated by the excitation of color strings and their
44

subsequent fragmentation into hadrons.
Throughout this dissertation, results are compared with those given by UrQMD. In addition
to the data from the BES, a set of model data was produced by UrQMD. We use version 3.3
and all parameters are set to default settings. Only events for the most central centrality
bin (0-5%) were generated, using randomly selected impact parameters in the range of 0.0 3.12 fm. The analyses performed on the UrQMD data sets where done in the same manner
as done on the BES data sets, with the exception that hpt i was treated as invariant with
multiplicity.

45

Chapter 4
Motivation and Construction of
Analyses

4.1

Event-by-event Observables

Prior to the 1960’s, the number of statistics taken in heavy-ion experiments, both the number
of collisions detected and the number of tracks from an event observed, were so small that
all measurements had to be constructed such that they were aggregates over many events.
For example, particle multiplicities and track spectra were both studied by averaging or
summing over many events. Modern experiments produce both su cient statistics and have
su cient geometric coverage that event-by-event observables can be constructed. All of the
analyses presented in this dissertation are event-by-event analyses.
The advantage of event-by-event analyses over inclusive analyses is that they are more sensitive to state changes. This is well illustrated by a clever analogy: on a rainy day hold a piece
of paper outside. If you have the paper in the rain long enough, it will become completely
wet, this corresponds to taking an average. If however, you only leave the paper outside
for a few seconds, you will observe only wet spots from the droplets of rain. The uniformly
wet paper would suggest that the rain is a uniform mist, however the second quick measure
46

demonstrates the droplet nature of rain1 . One can imagine how varying the orientation of
the paper may reveal the speed and direction of the rain, or how a careful study of the spots
may reveal information about rain droplet size or surface tension, or perhaps if su cient
statistics were taken rarer forms of precipitation may be observed such as snow or hail.
In central collisions at 200 GeV, on the order of 103 particles are deposited into the detector. At each energies presented in this dissertation there are millions of events. Using
event-by-event observables opens the possibility of observing phase transitions and gaining
qualitatively di↵erent information than using event-averaged observables.

4.2

Fluctuations versus Correlations

Throughout this dissertation, the terms ‘correlations’ and ‘ﬂuctuations’ are often used interchangeably. The reason for this is that their meaning is very similar and in some cases
synonymous. Strictly speaking they mean slightly di↵erent things, but their meaning varies
by usage and individual. A correlation is typically a function of a multi-particle property (for
example the invariant momentum of two particles) which measures how many multi-particle
groups are observed with a given value of that property (how the number of observed pairs
of particles changes with invariant momentum). Fluctuation measurements on the other
hand are generally a integral or summation of a correlation function. In this way, correlation
measurements generally have ﬁner ‘resolution’ than ﬂuctuation measurements. In this dissertation I present a two-particle correlation, but it is actually an average over a two particle
correlation, so it could be more accurately called a ﬂuctuation observable.
1 This

analogy is originally from Prof. A.D. Jackson

47

Experimental ﬂuctuation and correlation measurements have generally followed two approaches. The ﬁrst approach has been to characterize the distribution of the observation
parameter under consideration by variances, covariances, or higher moments. The second
approach has been to characterize the ﬂuctuations with correlation functions such as balance
functions or multi-particle correlators. These two approaches are equivalent, which will be
demonstrated in the following sections.

4.3

hpti Fluctuations

Event-by-event ﬂuctuation analyses can be done for several observables such as multiplicity,
transverse energy, and momentum. Transverse momentum, and momentum more generally,
are relevant quantities to be studied because they are proxies of the state variable temperature. To a large degree, the pt distributions are thermal [35] so their shape is determined
principally by the masses of the particles and the temperature of the body from which they
were emitted.
Referring to the ‘temperature’ of the pt distribution makes a number of assumptions about
the underlying system, namely that the system can be treated thermodynamically and that
it was (at least locally) thermally equilibrated. These assumptions are still uncertain, and
the second in particular can be strongly questioned just from the basis of the short system
lifetime, but the particle spectra do suggest thermal behavior. In the case of nucleon-nucleon
collisions, a thermal treatment seems less well founded, and heavy-ion collisions tend toward
nucleon-nucleon collisions as they become less central, so at some point the thermal treatment
of heavy-ion collisions becomes invalid. At what point this occurs has not been thoroughly

48

established. Some of the analyses in this dissertation may be illuminating in trying to
quantify this transition.
Before beginning, it is interesting to consider the general nature of hpt i ﬂuctuations. As
previously stated, the ﬂuctuations of hpt i are of interest because is hpt i related to the temperature of the system. In an ideal situation, the ﬂuctuations of hpt i would correspond
directly to ﬂuctuations of the temperature of the system. Our systems are not ideal: the
number of particles per event is ﬁnite, the statistics are limited, and there are physical
processes which could cause hpt i to ﬂuctuate.
In an idealized scenario, we can imagine that our events are observations of a thermal system
of ﬁxed chemical composition, the temperate of every event is identical, and the multiplicity
N is uncorrelated to any other properties of the system and can be arbitrarily large (and
we will say it is the same in every event for simplicity). This system is clearly non-physical,
but it serves to illustrate a point. In this case, each event is a sampling from an identical
2
thermal pt distribution with mean hpt i and variance pt . From the central limit theorem we

can calculate the mean and variance of the hpt i distribution,

hhpt ii = hpt i
2
hpt i

=

2
pt
N .

(4.1)

The point of note is that the observed ﬂuctuations of hpt i are purely statistical. In the limit
2
of N ! 1 we see that hp i ! 0.
t

We can imagine now a similar scenario, but where the temperature of the underlying distribution ﬂuctuates. Because the distribution is still purely thermal, ﬂuctuations of the
2
2
temperature ( T ) will result in ﬂuctuations of both hpt i and pt event-by-event. For clarity

49

of notation, we will say that the event-by-event ﬂuctuations of temperature results in dy2
namic ﬂuctuations of hpt i, hp i,dynamic . In this case, we will still ﬁnd that hhpt ii = hpt i, but
t

the ﬂuctuations of the hpt i distribution are no longer purely statistical. The ﬂuctuations of
the hpt i distribution will be,

2
hpt i

2
= hp i,dynamic +
t

* 2 +
pt

N

.

(4.2)

2
2
2
Now, in the limit of N ! 1 we see that hp i ! hp i,dynamic . In this ideal case hp i,dynamic
t
t
t

is entirely a consequence of temperature ﬂuctuations. In real events, there are many physical
processes which could give rise to event-by-event, non-statistical ﬂuctuations in hpt i. These
2
non-thermal contributions to hp i,dynamic are addressed in length in Section 4.6.
t

In addition to the second moment of hpt i shown in Eq. 4.2, similar arguments can be made
for higher moments of hpt i. Each higher moment will have a dynamic component and a
purely statistical component resulting from ﬁnite multiplicity. Throughout this dissertation
the dynamic component of the higher moments are called the higher dynamic moments to
distinguish them from the measured higher moments.

4.4

Mixed and Statistically Sampled Events

Throughout the analyses presented in this dissertation, ‘mixed’ events and ‘statistically
sampled’ events are used. Both are di↵erent approaches to creating ‘data’ which have all
of the same detector e↵ects, analysis e↵ects, and physics as the real data, but without any
correlations.

50

Mixed events are more complicated to generate that statistically sampled events, but they
more faithfully fulﬁll the intention of reproducing the data while removing correlations and
have utility outside of pt analyses. Mixed events are events generated from the data by
combining tracks selected from di↵erent events. The tracks of all events are “mixed” in that
a particular mixed event will have tracks from many di↵erent real events. Because each track
came from a di↵erent event, they cannot be correlated except statistically. Mixed events have
been used previously in experimental analyses [1, 36]. The procedure for generating mixed
events is the same as in [1].
Statistically sampled events contain much less information than real or mixed events. Unlike
mixed events they do not have individual tracks, but are rather just an hpt i and a multiplicity.
From the real data for each energy and centrality bin two spectra were generated: a spectrum
of the event multiplicity, and a spectrum of each track’s pt from all events. Statistically
sampled events were generated from these two spectra by randomly sampling the multiplicity
distribution for a number N , then sampling N times from the pt distribution. These N
samplings were averaged, giving an hpt i. This was repeated many times (several million,
depending on energy, centrality bin, and the particular analysis), which generated a new,
statistical hpt i distribution. An example of one of these statistical distributions, compared
with an experimental distribution, is shown in Figure 4.1.
The intention of both of these approaches is to generate events where the higher dynamic
moments are zero by construction. The measured higher moments of mixed and statistically
sampled events are identical to the statistical component of the measured higher moments
of the data. Their utility is that they allow us to extract the higher dynamic moments by
subtracting the statistical component from the measured higher moments.

51

Data
Sampled

Figure 4.1: Spectra of hpt i for 200 GeV from data and from the statistically sampled method.
The real data are wider (has a larger variance) which indicates that there are dynamic
correlations in the real data which are not reproduced by the statistically sampled baseline.

52

4.5

Analyses in this Dissertation

There are two analyses presented in this dissertation, one measuring the two particle cor⌦
↵
relator
pt,i , pt,j , and another measuring the moments of the hpt i distributions. These
two analyses take di↵erent approaches but are closely related. The ﬁrst analysis corresponds
to directly measuring

2
,
hpt i,dynamic

while the second corresponds to measuring the total

ﬂuctuations, both dynamic and statistical, of the higher moments.
The theoretical motivation for these analyses is presented along with competing physical
e↵ect in the Section 4.6. Each of these analyses, their respective mathematical constructions,
subtleties, and necessary consideration will be introduced in Sections 4.7 and 4.8, then they
each have several chapters devoted to results, analysis details, and comparisons.

4.6

Sources of Fluctuations

There are many physical processes which may give rise to ﬂuctuations. The most exciting
of these processes are the critical point and a change in the order of the QCD phase transition which may give rise to dynamic ﬂuctuations. There are however many other possible
sources of ﬂuctuations. The myriad of possible sources for statistical and dynamic hpt i ﬂuctuations may limit our capacity to draw concrete physical conclusions from the results of
these analyses. For many of these extra sources of ﬂuctuations the degree to which they
a↵ect dynamic hpt i ﬂuctuations has not been quantiﬁed experimentally or theoretically. As
with many heavy-ion analyses, these results on their own may be inadequate for making
deﬁnitive conclusions. The results of the analyses presented in this dissertation will have to
be weighed hand-in-hand with results from other forthcoming analyses in order to make con53

crete statements about the physical behavior of the systems we have produced. Establishing
the existence of the QGP was similar in that there were several analyses which suggested
the formation of a QGP, but each on their own was not deﬁnitive.
Below is presented a list of possible sources of statistical and dynamic ﬂuctuations. The
list is not exhaustive, but e↵ort has been made to include those sources which should most
largely a↵ect the present analyses.

4.6.1

Jets

During the initial collision, sometimes a parton picks up a large amount of energy from hard
interactions and is ejected from the medium. The parton will hadronize as it exits the volume, and will produce a spatially localized collection of strongly-correlated high momentum
particles, referred to as a ‘jet’. Jets will a↵ect the shape of the pt distribution, increasing
the variance pt . Jets increase with collision energy, as does the initial temperature. Both
2
of these e↵ects increase pt and may be hard to disentangle. We minimize the e↵ects of jets

by analyzing particles in the range of 0.15 < pt < 2.0 GeV/c.
A related e↵ect is high pt particle suppression and jet suppression. High pt particle suppression was one of the key signals of QGP formation [37]. This e↵ect has been observed
to decrease with energy, disappearing at energies below

p

sNN = 19.6 GeV (see Figure

4.2).
An important question to ask is how these e↵ects will a↵ect ﬂuctuations of hpt i. The sta2
tistical component of hpt i depends upon pt , so it will increase in the presence of jets and

decrease when there is high pt particle suppression. Dynamic ﬂuctuations in jets or high

54

Figure 4.2: The momentum and energy dependence of the nuclear modiﬁcation factor RCP .
RCP is the ratio of charged particles observed in the data with the number of binary collisions in a Wounded Nucleon Glauber Monte Carlo simulation ﬁt to the data. For energies
p
sNN 19.6 Ge, the decrease in RCP above pt ⇡ 2.5 GeV/c indicates high pt particle
suppression. From Ref. [38]

55

pt particle suppression will result in dynamic ﬂuctuations of hpt i, the magnitude of which
will depend upon the magnitude of the ﬂuctuations and the magnitude of the e↵ects on
hpt i.

4.6.2

Flow

Flow is an e↵ect where non-symmetric spatial anisotropy results in non-symmetric momentum anisotropy. When two nuclei collide in a mid-peripheral collision, the overlap region
will be almond or football shaped (see Figure 3.2). The nuclei are taken to be in a plane
that passes through the beam axis and the center of both nuclei, called the event plane.
Elliptic ﬂow arises because the larger spatial gradient along the short axis of the collision
region will result in a larger pressure gradient. Particles emitted along this axis (in the event
plane) will have on average larger momenta than those emitted perpendicular to the event
plane. Elliptic ﬂow corresponds to variations in pt of the form cos (2 ), where

is the angle

with respect to the event plane. There are other types of ﬂow corresponding to higher order
harmonics, cos (n ).
Elliptic ﬂow will a↵ect the shape of the pt spectra, broadening the pt spectra (and increasing
the statistical ﬂuctuations). Because ﬂow is a consequence of geometry, for events of the
same energy and centrality it should largely be constant event-by-event and so should not
change hpt i. In general ﬂow should not e↵ect dynamic hpt i correlations. Elliptic ﬂow will
only produce dynamic correlations if it itself ﬂuctuates dynamically, or if the experimental
acceptance is not axially symmetric (for example, if there was a dead sector in the TPC).
Non-uniform acceptance will produce dynamic correlations because events in which the event
plane points into the dead sector will have smaller measured hpt i (because some of the event’s
56

high pt tracks will be lost), while events in which the event plane points perpendicular to
the dead sector will have larger measured hpt i (because some of the event’s low pt tracks will
be lost). It is important to note that while the true values of hpt i may not be changing, the
measured values of hpt i will ﬂuctuate which will result in dynamic correlations.
The magnitude of this e↵ect will depend on many e↵ects such as centrality, energy, the
species of the nuclei, as well as how much of the detector is a↵ected and to what magnitude.
This e↵ect has been studied experimentally, and the results are presented in Ref [39]. In
the data presented in this thesis, great care has been taken to minimize detector acceptance
e↵ects to minimize any contributions to the observed pt ﬂuctuations.

4.6.3

Resonance Decays

Many of the particles produced in heavy-ion collisions are unstable. Instead of the original
unstable particle (the mother particle) it is the decay products (the daughter particles) which
are measured by STAR. Each of the daughter particles will carry some fraction of the mother
particle’s momentum. This will a↵ect the pt distribution, decreasing pt because some tracks
will be replaced by several lower momentum tracks. This will also a↵ect the hpt i distribution
because the average momentum of the daughter particles will be substantially smaller than
the momentum of the mother particle. To estimate the scale of this e↵ect we can imagine a
simpliﬁed case where an event has N particles, average transverse momentum hpt i, and one
particle with transverse momentum pt,i , decays in Nd particles, each with identical transverse
PNd
1
momentum pt,i ✏ where ✏ = N 2 . Note i=1 pt,i ✏ = pt,i . The average transverse momentum
d

2 This

assumes that all of the daughter particles continue in the exact same direction of
the mother particle and that the mass di↵erence in the decay is negligible.

57

p

t,i
of the daughter particles is then N . The new measured total transverse momentum of the
d

event will be,

hpt imeasured =

N hpt i
N

p

t,i
pt,i + Nd N
d

1 + Nd

.

(4.3)

We will assume that N >> 1 and simplify to,

hpt imeasured =

N hpt i
.
N + Nd

(4.4)

To summarize, for each decay the measured average transverse momentum will decrease by
a factor f =

1 .
Nd
1+ N

If N >> Nd , then f ⇡ 1. If there are many resonance decays, or if the

multiplicity is very low, this may have a substantial e↵ect.
It is important to note that while the presence of resonance decays will shift the value of
hpt i, it is the ﬂuctuations of hpt i which are being studied here. What is of actual interest
is how ﬂuctuations in resonance decays will results in ﬂuctuations of hpt i. To quantify this
e↵ect, the number of resonance decays and the scale of their ﬂuctuations would have to be
determined. Both of these properties will vary with energy and centrality. To estimate the
e↵ect, we can say hpt i = hpt io F , then the ﬂuctuations of hpt i will be,

hpt i = hpt i

s✓

◆
✓ ◆2
hpt io 2
F
+
,
hpt io
F

(4.5)

where hpt io is the average transverse momentum of the system before the decay, F is the
product of f for each resonance decay, and x is the ﬂuctuation of quantity x.

58

To study the e↵ect of resonance decays on the analyses presented in this dissertation, one
could identify the prominent resonances, then suppress the daughter particles by cutting out
all pairs of particles of the correct species with qinv 3 corresponding the mass of the mother
particle. Correlations would be induced by the cut which would need to be corrected for
with mixed events. This was not done in this analysis because no particle identiﬁcation was
performed.

4.6.4

Changing Chemistry

Variations in the chemistry of events (the ratios of various particle species) could have an
e↵ect on the observed non-statistical ﬂuctuations of hpt i. In the case of a perfectly thermal
QGP system, di↵erent particle species will have di↵erent momenta because of their masses.
Particle ratios are determined by the collision energy and temperature of the system, but will
have statistical variation. Even in the case of constant temperature, variations in particle
ratios will result in additional ﬂuctuations of hpt i.
This e↵ect could be investigated by performing the analyses using identiﬁed particles. Performing the analysis on only one particle species at a time should mitigate the e↵ects of
changing chemistry. Doing pt analyses on identiﬁed particles has the additional complication that particle identiﬁcation is done by cutting on pt and p. Truncating the underlying
pt distributions will a↵ect the ﬂuctuations of hpt i, this e↵ect would be non-trivial and have
to be investigated.
3 Center

of mass invariant momentum di↵erence.

59

4.6.5

Temperature Fluctuations

As event-by-event analyses became practical, one of the original motivations for studying hpt i
ﬂuctuations was to measure temperature ﬂuctuations and the speciﬁc heat of the produced
systems. The speciﬁc heat, CV , would provide insight into the equation of state of the
system and could be sensitive to the order of the phase transition. In the case of a ﬁrst
order phase transition, the speciﬁc heat will be much larger than in a smooth crossover. In
a smooth cross-over ﬂuctuations in the energy of the system will result in a ﬂuctuation in
the temperature, but in a ﬁrst order phase transition ﬂuctuations in energy may go into the
latent heat of the phase transition and the temperature may ﬂuctuate much less.
The motivation for using hpt i ﬂuctuations to measure speciﬁc heat is as follows. In a thermal
system the hpt i of a system should be related to the temperature T by some function hpt i =
F (hT i). Likewise hhpt ii = F (hT i). Assuming the temperature ﬂuctuations are Gaussian,
arguments from the theory of error propagation give [40],

hpt i,dynamic

= F 0 (hT i)

T,dynamic ,

(4.6)

where F 0 (hT i) is the derivative of F (hT i). We can then write,

hpt i,dynamic

hhpt ii

=

F 0 (hT i)
F (hT i)

T,dynamic

hT i

.

(4.7)

Treating the system as an ideal gas, there are two limiting cases for the pt spectra, a nonrelp
ativistic ideal gas where hhpt ii / hT i and an ultrarelativistic ideal gas where hhpt ii / hT i.
In reality the system will be somewhere between these limiting cases.
60

It then follows

that,

hpt i,dynamic

hhpt ii

T,dynamic

=✏

hT i

where ✏ is some value between 0.5 and 1. Noting that CV =

✓

hpt i,dynamic

hhpt ii

◆2

=

,

✓

(4.8)

hT i
T,dynamic

◆2

, we can say,

✏2
.
CV

(4.9)

It is interesting to note that there will be statistical ﬂuctuations of the temperature. Fluctuations in the temperature will go like T,stat ⇡ p 1
hT i

hN i

. Where hN i is the mean multiplicity of

the events. A ballpark value of hN i ⇡ 500, which gives T,stat = 0.045. The statistical ﬂuchT i
tuations of temperature come from the limited number of particles, which is also the source
of the statistical ﬂuctuations of hpt i, so it is unnecessary to make an additional correction
explicitly for statistical temperature ﬂuctuations.

4.6.6

Correlation Length

One of the most exciting possible sources of dynamic ﬂuctuations of hpt i is the QCD critical
point (see Section 1.2.2).
In Ref. [41], M. Stephanov works out a simpliﬁed theory of pions coupling to a ﬁeld

of

mass m, with interactions given by the Lagrangian
˜

L ⇡⇡ = 2G ⇡ + ⇡

61

(4.10)

with coupling G. He calls this model the Linear Sigma Model. He deﬁnes a universal
D
E
ﬂuctuation correlator
n↵ , nk which is a two particle correlation between occupation
p
numbers. The subscripts and superscripts denote that n↵ is the occupation number of a
p
quantum state deﬁned by property ↵ with value p. The value of n↵ will vary event to event,
p
⌦ ↵
⌦ ↵↵
and can be averaged over many events to n↵ . Then n↵ = n↵
np . He demonstrates
p
p
p

that the G2 order correction to the universal ﬂuctuation correlator can be shown, to ﬁrst
order near the critical point to be,

D

E
1
n↵ , nk 2 / 2 .
p
G
m
˜

(4.11)

This is notable because m is related to the correlation length, ⇠ = m 1 (see also Ref. [42]).
˜
˜
Near the critical point, when ⇠ diverges, the universal ﬂuctuation correlator should also
diverge.
In Ref. [41] it is shown that the variance of hpt i is related to

⌦

ni , nj

↵

(the absence

of superscripts is because the occupation numbers are implied to be of momentum states)
by,

2
hpt i

=

1

XX⌦

hN i2 i

ni , nj

j

↵

pt,i

hpt i

pt,j

hpt i .

(4.12)

However, the theoretical derivation assumes that there are no statistical ﬂuctuations,
⌧ 2
pt
2
2
2
=
= 0 and so hp i = hp i,dynamic (from Eq. 4.2). The physical
N
hpt i,statistical
t
t
2
2
quantity of interest is then hp i,dynamic and not hp i .
t
t

In another paper on the Linear Sigma Model, Stephanov also relates the third moment of

62

the hpt i spectra to a generalized three particle correlator [42]

D

3

( hpt i)

E

=

1

XXX⌦

hN i3 i

j

ni , nj , nk

k

↵

pt,i

⌦

hpt i

↵
ni , nj , nk by,

hpt i

pt,j

pt,k

hpt i .
(4.13)

The three particle correlator is related to the correlation length by

⌦

ni , nj , nk

↵

/

⇠ 6 . As with the two particle correlator, the physical quantity of interest will actually be
D
E
D
E
( hpt i)3
and not ( hpt i)3 because of non-physical statistical ﬂuctuations.
dynamic

These general multi-particle correlators are related to the moments of the zero-momentum
R
mode of sigma ﬁeld, 0 ⌘ d4 (x) /V . In Stephanov’s Linear Sigma Model, the relationship
between the moments of this sigma ﬁeld and the correlation length is given by [42],

⌦ 2↵ T 2
0 = V⇠
⌦ 3↵ 2 T
3
= 0 = V ⇠6
⌦ 4↵
⌦ 4 ↵ ⌦ 2 ↵2 6T h
= 0 c= 0
= V 2 ( 3 ⇠)2
0

2 =
3
4

(4.14)
4

i

⇠8

Like in Eqs. 4.12 and 4.13, we can relate the moments of the sigma ﬁeld to dynamic moments
of the hpt i distribution. However, we cannot expect the dynamic moments of the hpt i spectra
to necessarily follow the ﬂuctuations of the sigma ﬁeld because there are many e↵ects which
can modify these ﬂuctuations, and there are physical processes which will manifest in the
dynamic correlations, but are not related to thermodynamics of the system. Also, the system
has continued to interact for some time after passing near the critical point.
The critical contribution to the higher moments of the hpt i spectra have not been calculated,
and will depend upon momentum distributions, quantum momentum states, momentum
63

e ciencies, and potentially other factors. There are other analyses, outside of the scope of
this dissertation which attempt to detect critical opalescence through the moments of the
multiplicity distributions. For these analyses the relationship between the correlation length
and the moments of the multiplicity distributions has been estimated to be [43, 44, 45],

2 / ⇠ 2 ,
3 / ⇠ 9/2 ,
4 / ⇠ 7 ,

(4.15)

S / ⇠ 5/2 ,
K 2 / ⇠5,
where S is the ‘skewness’4 and K is the kurtosis5 . A similar relationship is expected for the
higher moments of hpt i.

4.7

Two-Particle pt Correlations, h pt,i, pt,j i

The correlation function which is used in this dissertation is the two-particle relative trans⌦
↵
verse momentum correlator,
pt,i , pt,j . This correlator is equivalent the dynamic com-

2
ponent of the variance of hpt i, hp i,dynamic . By construction it will be 0 for mixed events
t

and statistically sampled events. The mathematical deﬁnition of this correlator is given in
Section 4.7.1. It is also equivalent to the di↵erence of the variance of the hpt i distribution
and the average variance of the event-by-event pt distribution scaled by average multiplicity.
4S
5K

µ
3
= 3 = 3/2
3

µ
= 4
4

2


3= 4
2
2

64

This can be inferred from Eq. 4.2, but is also rigorously demonstrated in Appendix . The
experimental analysis is presented in Chapters 5 and 6.

4.7.1

Mathematical Construction

The two-particle transverse momentum correlation observable,

⌦

↵
pt,i , pt,j , is constructed

as follows. For each event an average pt is calculated by averaging the pt values of each track
within the analysis acceptance:

hpt ik =

PNk

i=1 pt,i

Nk

(4.16)

The values of hpt ik are then averaged over multiple events. This event averaged pt is done
as a function of multiplicity (refMult) because the event average pt is multiplicity dependent
as has already been observed in other analyses [46].

hhpt ii =

PNevent

hpt ik
k=1
Nevent

(4.17)

At high multiplicities the number of events available decreases to zero, and the calculated
value of hhpt ii ﬂuctuates due to limited statistics. To compensate for this statistical variation,
hhpt ii as a function of multiplicity is ﬁt with a linear form at large multiplicites and the ﬁtted
values are used in further steps of the calculation. Examples of this ﬁtting can be seen in
Figures 4.3 and 4.4.
For each event the correlator is deﬁned as a double sum, which is also equivalent to a single
sum over all pairs of particles. The sum is taken over the product of the deviation of pt for
65

Raw values
Fit for high pt

Figure 4.3: hhpt ii (refMult) for 7.7 GeV. The raw distribution is plotted in blue points, and
the smoothed hhpt ii values in green. The error bands are statistical errors. The errors above
refMult ⇡ 250 grow very large due to low statistics.

66

Raw values
Fit for high pt

Figure 4.4: hhpt ii (refMult) for 200 GeV. The raw distribution is plotted in blue points, and
the smoothed hhpt ii values in green. The error bands are statistical errors. The errors above
refMult ⇡ 600 grow very large due to low statistics.

67

all pairs of particles in each event.

N

Ck =

N

k
k
X X

hhpt ii

pt,i

i=1 j=1,j6=i

pt,j

hhpt ii

(4.18)

The correlator is divided by the total number of pairs of particles in the event, and then
averaged over all events.

⌦

↵

1

pt,i , pt,j =
Nevents

Nevents

X

k=1

Ck
Nk (Nk

1)

(4.19)

Just as with hhpt ii, this average is done as a function of reference multiplicity. The values
⌦
↵
of
pt,i , pt,j are averaged over each centrality bin (weighted by the number of events in
each multiplicity bin).

4.7.2

Scalings

The purpose of the two-particle correlator is to isolate critical behavior. However, the twoparticle correlator is also sensitive to physical e↵ects not related to critical behavior. In order
to remove these e↵ects, and isolate critical behavior, various scalings are used. Scalings can
also be used to attempt to equate results from di↵erent systems, for example: p+p, Cu+Cu,
and Au+Au. If correlations between di↵erent systems could be shown to be equivalent using
some scaling related to system parameters, it may be indicative of some underlying behavior
which is constant between the di↵ering systems.

68

4.7.2.1

By hhpt ii 1

The correlation observable may have an dependence arising from the energy dependence
of hhpt ii, so scaling with hhpt ii should in principle remove it. The correlation observable
⌦
↵
pt,i , pt,j has units of momentum squared ((GeV/c)2 ), while hhpt ii has units of momentum (GeV/c), so when scaling with hhpt ii we must either square hhpt ii or take the

square root of the correlation observable. In addition to addressing possible energy depenq⌦
↵
dence, this scaling provides two other beneﬁts, ﬁrst when scaled as
pt,i , pt,j / hhpt ii
our observable is equivalent to the CERES observable ⌃pt [36]. Second, the e ciency deq⌦
↵
pendence of
pt,i , pt,j is the same as that of hhpt ii (see Section 6.2) so the scaled
observable is independent of e ciency.

The strongest motivation for so scaling with hhpt ii comes from Section 4.6.5. Noting that
⌦
↵
2
pt,i , pt,j = hp i,dynamic , from Equations 4.8 and 4.9 we can write,
t

q⌦

↵
pt,i , pt,j
✏
T,dynamic
=✏
=p
,
hhpt ii
hT i
CV

(4.20)

where, as in Equations 4.8 and 4.9, ✏ is some number between 0.5 and 1 coming from the
relationship between hhpt ii and hT i, and CV is the speciﬁc heat of the system.

4.7.2.2

By Multiplicity

There are two independent motivations for scaling with variants of multiplicity. The ﬁrst
motivation is to explore the possibility that heavy-ion collisions behave like a large number
of nucleon-nucleon collisions. If that were the case, then one would expect the correlations to
scale with the number of individual nucleon-nucleon collisions (Ncoll ). The value of Ncoll is
69

not experimentally accessible, but could be estimated using Glauber simulations; it should,
in principle, be related to to the collision energy and the detected multiplicity. The Ncoll
scaling is not presented in this dissertation.
The other motivation for scaling with multiplicity comes from the fact that

⌦

↵
pt,i , pt,j is

calculated as an average over all pairs. As the number of particles in an event (N ) increases,
the number of pairs increases like N 2 , so the summation ‘dilutes’ the correlations. This
dilution goes as 1/N , so it can be addressed by scaling with the number of particles used in
the double summation.

4.7.2.3

By hhpt ii 1 and Multiplicity

The arguments presented for each scaling are both valid, so in addition to presenting them
individually, results will be presented of the correlation observable scaled by both hhpt ii 1
and N .

4.8

Higher Moments of hpti

The moments of distributions corresponding to ﬂuctuation observables are potential measures of critical behavior because they relate to the correlation length of the medium. The
mathematical description of the higher moments is presented in Section 4.8.1. Baseline
expectations of statistical behavior are presented in Section 4.8.2. The corresponding experimental analysis is presented in Chapters 7 and 8.

70

4.8.1

Mathematical Construction

All of the moments of the hpt i distributions have the same form. The moment of order n is
deﬁned as,

µn =

PNevents
i=1

(hpt ii hhpt ii)n
.
Nevents

(4.21)

The ﬁrst moment is more commonly known as the mean, and the second moment as the variance. The moments of the distribution are also related to the cumulants of the distribution.
The ﬁrst six cumulants are related to the moments by,

1 = µ1 ,
2 = µ2 ,
3 = µ3 ,

(4.22)

4 = µ4

3µ2 ,
2

5 = µ5

10µ2 µ3 ,

6 = µ6

15µ2 µ4

10µ2 + 30µ3 .
3
2

In addition to being expressed as cumulants, the higher moments are sometimes expressed
as classical named properties of distributions: variance ( 2 ), skewness (S), and kurtosis (K),
which are related to the moments and cumulants by,

71

2

= µ2 = 2 ,
µ



3
S = 3 = 3/2 ,
3

2

µ

(4.23)


3 = 4.
2

K= 4
4

2

These are also expressed as moments products, which can be written in terms of ratios of
the cumulants,



S = 3 ,
2
K

2

=

4
2 .

(4.24)

Each moment and cumulant is calculated individually for each multiplicity bin. This is
done so that the variation in the value of hhpt ii with multiplicity does not skew the other
moments. After all of the calculations are done for each multiplicity, the centrality bin values
are calculated by taking an average over all of the multiplicity bins in each centrality bin,
weighting each multiplicity bin by the number of events in that bin.
Unlike with the two-particle correlator analysis, the value of hhpt ii is not smoothed at large
multiplicity values with a linear ﬁt.

4.8.2

Baselines

The hpt i distributions are largely statistical, and the e↵ects of early-state ﬂuctuations are
expected to be very small (a few % change in the variance, for example). Additionally, the
STAR detector subsystems have non-trivial e ciencies which depend on pt , among other
things. These e ciencies do a↵ect the hpt i distributions. As such, it is instructive to have a
72

statistical baseline to compare the experimental results with. Also, the moments products
which are related to the correlation length are expected to diverge near the critical point, but
the scale of their divergence is uncertain and moments products themselves may vary due
to analysis methods and experimental e↵ects. By comparing the moments with an expected
baseline, critical behavior can be highlighted. Lastly, as discussed in Section 4.3, there are
both dynamic and statistical contributions to hpt i ﬂuctuations. Having a purely statistical
baseline allows us to extract the dynamic component of the hpt i ﬂuctuations. Here I present
two baselines which have been considered in the higher moments analysis presented in this
dissertation.

4.8.2.1

Gamma Distributions

The hpt i distributions has been observed in many experiments to be well described by a
gamma distributions [47, 36, 48]. The gamma distribution has the form,

f (y) = f (y, ↵, ) =

with parameters ↵ and , and

(↵)

( y)↵ 1 e

y,

(4.25)

(x) is the gamma function. The parameters of the gamma

distribution are related to the mean and variance of the distribution,

↵=
=

hhpt ii2
,
2
hpt i
2
hpt i
.
hhpt ii

The ﬁrst four moments of the gamma distribution are given in terms of ↵ and

73

(4.26)

by,

µ1 = ↵ = hhpt ii ,
2
µ2 = ↵ 2 = hp i ,
t

µ3 = 2↵

3

=

(4.27)

4
hpt i
,
hhpt ii

µ4 = 3↵ (2 + ↵) 4 .

The gamma distribution comes from the underlying pt distributions being approximately
exponential. If the underlying distribution is perfectly exponential, e

y,

then physically

the ↵ parameter is the number of samples taken from the underlying distribution. When
↵ = 1 the underlying distribution is reproduced. When ↵ becomes large, the distribution
approaches a normal distribution, as is expected from the central limit theorem. The gamma
behavior of the hpt i distribution is a consequence of statistics and the thermal character of
the pt distribution.

4.8.2.2

Statistical Baseline

While to ﬁrst order the hpt i distributions are gamma distributions, they may deviate due to
the pt dependent track detection e ciency of the detector (see Figure 4.6). Additionally, the
underlying pt distributions are truncated, and are not perfectly exponential. To demonstrate
the e↵ect of pt dependent track detection e ciency a toy model was constructed. Figure 4.5
shows the result of this toy model. In this toy model, every event has a multiplicity taken
from a gaussian distribution with µ = 100 and
distribution with µ = 1 and

= 10, and hpt i sampled from a Gaussian

= 0.25. The track pt distribution was also assumed to be

gaussian with it’s mean given by the sampling from the hpt i distribution and

= 1. As each

track was drawn, it’s value was compared to a simulated e ciency step at pt = 1.25. If the

74

Real Data
Data with Eff Cut

Figure 4.5: This is a simulated result showing the e↵ects of pt track cuts on the hpt i distribution. The blue histogram is the true hpt i distribution, while the green histogram the
result of running each track from each event through a simulated e ciency.
value of the track was less than 1.25, the track was always kept (100% e ciency), if the track
had a value of greater than 1.25, the track was kept only 75% of the time (75% e ciency).
After this e ciency simulation, the hpt i of the event was recalculated, giving the e ciency
simulated hpt i distribution. In reality the e ciency usually varies smoothly with pt . The
e↵ect of this e ciency step is that events with a mean at or above the step have more tracks
in the lower e ciency region. Events far below the step are shifted only marginally, but
events at or above the step are shifted lower with a higher probability, resulting in a skewing
of the hpt i distribution.
75

This e↵ect would be pronounced if the moments analysis were done with identiﬁed particle
species, particularly if particle identiﬁcation was done using a combination of TPC and
TOF. Because the TPC and TOF have di↵erent momentum ranges, the PID e ciency as a
function of pt changes and is very sensitive to the PID method used. The moment analysis
for identiﬁed particles is not presented in this dissertation, but pt dependent track detection
e ciency may still skew the hpt i distributions.
Experimentally, the pt dependent track detection e ciency of the detector is determined by
using ‘embedding’. Embedding consists of taking real physics events, adding artiﬁcial tracks
(embedding them into the event), running them through a full simulation of the detector,
performing track reconstruction, and then calculating the fraction of embedded events which
have survived through reconstruction.
To address concerns about pt dependent track detection e ciency, statistically sampled
events were used as a baseline (see Section 4.4). The pt dependent track detection e ciency will change the underlying pt distribution, so the e↵ect should be reproduced in the
statistically sampled events.

76

Figure 4.6: E ciency as a function of pt for 62.4 GeV in the 0-5% centrality bin as determined
by embedding. The e ciency was done independently for each particle species.

77

Chapter 5
Results of the pt Correlation
Analysis

5.1

Behavior of hpti

For all seven energies, the hpt i spectra for each centrality bin was observed to be well described by gamma distributions, particularly in the central bins, as has been found in previous
analyses [36]. The hpt i spectra with gamma distributions for the 0-5% centrality bin can
be seen in Figure 5.1, and the corresponding parameter values for the gamma distributions
are listed in table 5.1. Note that the distributions are not ﬁts, they are simply distributions
with the same mean and variance as the data, and both the data and the distributions have
been normalized to 1.
The mean value of the hpt i gamma distributions are equivalent to hhpt ii for that centrality
bin. The values of hhpt ii for all centralities, plotted versus
Above

p

p

sNN , is shown in Figure 5.2.

sNN = 19.6 GeV, hhpt ii are observed to increase smoothly with energy for all

centrality bins. For all but the two most peripheral centrality bins, the values of hhpt ii is
observed to increase with decreasing energy below

p

sNN = 19.6 GeV. This is due to changes

in the particle ratios. Below 19.6 GeV the pion to proton ratio decreases with energy. For
78

11.5 GeV

19.6 GeV

27 GeV

39 GeV

Counts (normalized)

7.7 GeV

62.4 GeV

200 GeV
Data
Gamma

Figure 5.1: hpt i distributions with gamma distributions. These spectra have been normalized
so that they integrate to one. No ﬁtting was performed, the gamma distributions simply
have the same mean and variance as the data.

79

Energy
7.7 GeV
11.5 GeV
19.6 GeV
27 GeV
39 GeV
62.4 GeV
200 GeV

↵
789.5 ± 0.31
930.9 ± 0.26
1089.1 ± 0.09
1166.3 ± 0.05
1241.8 ± 0.12
1271.5 ± 0.06
1465.3 ± 0.08

0.00070137
0.00058381
0.00049720
0.00046754
0.00044513
0.00044472
0.00040798

±
±
±
±
±
±
±

2.3e-7
1.3e-7
2.8e-8
1.4e-8
3.4e-8
1.6e-8
1.6e-8

µ
0.55371 ±
0.54346 ±
0.54148 ±
0.54528 ±
0.55278 ±
0.56546 ±
0.59781 ±

7.0e-5
5.0e-5
1.7e-5
1.0e-5
2.0e-5
1.2e-5
1.3e-5

0.0197067
0.0178123
0.0164082
0.0159670
0.0156863
0.0158578
0.0156173

±
±
±
±
±
±
±

2.9e-6
1.9e-6
3.9e-7
1.9e-7
5.2e-7
2.2e-7
2.6e-7

Table 5.1: The parameters of the gamma distributions ﬁt to the hpt i spectra for the 0-5%
centrality bin. The gamma distribution is fully deﬁned by two parameters, either ↵ and
or µ and ; both pairs of values are listed here for convenience.
individual particle species hhpt ii increases smoothly and continuously with

p

sNN . We expect

hhpt ii to increase with energy because it is related to the temperature of the system. Di↵erent
particle species have di↵erent values of hhpt ii due to their di↵erent masses. Protons have
hhpt ii values approximately 3 times that for pions. The decrease in the number of pions
versus the number of protons at the lower energies is why we observe hhpt ii to decrease with
energy for

p

sNN < 19.6 GeV. This trend is reproduced by UrQMD (see Figure 6.5 and Sec

3.8).

5.2

Unscaled Correlations

The correlation observable

⌦

↵
p
pt,i , pt,j versus sNN for all incident energies and centrali-

ties is shown in Figure 5.3. It increases smoothly with both energy and decreasing centrality.
The increase with centrality can be explained, in part, by the decrease in multiplicity: in
central collisions where the observed multiplicity N increases, the number of pairs of particles goes like N 2 . This will result in a ‘dilution’ of the correlations. Also, hpt i varies with
⌦
↵
centrality, which will e↵ect that scale of
pt,i , pt,j .
80

p
Figure 5.2: hhpt ii vs sNN for all seven energies and eight centrality bins. The error bars
are statistical. If the particle ratios are constant, hhpt ii will vary with the temperature of
p
the system and should increase with energy. The decrease with energy up to sNN = 19.6
GeV is due to the decrease in the number of pions relative to the number of protons.

81

⌦
↵ p
Figure 5.3:
pt,i , pt,j vs sNN for all seven energies and eight centrality bins.↵ The error
⌦
bars are statistical errors, and the error bands are systematic errors.
pt,i , pt,j is related
to the dynamic correlations of hpt i.
There could be any number of reasons why

⌦

↵
pt,i , pt,j is observed to increase with collision

energy. The increase could be due to the increase in the number of jets, increase in the

number of resonance decays, or other phenomena. As with the centrally dependence, there
will be some e↵ect due to variations in multiplicity and hpt i. The dramatic decrease below
p

sNN = 19.6 GeV could be due to a change in the phase transition.

82

5.3

Correlations Scaled with hhptii

1

Figure 5.4 shows the correlation observable scaled with average transverse momentum,
q⌦
↵
pt,i , pt,j / hhpt ii as a function of energy for all eight centrality bins. The scaled
q⌦
↵
observable,
pt,i , pt,j / hhpt ii, still increases with both centrality and energy, but be-

cause hhpt ii also increases with energy above 19.6 GeV, the energy dependence weakens
above 19.6 GeV.
The centrality dependence is largely a consequence of the ‘dilution’ e↵ect discussed above,
⌦
↵
and can be addressed by scaling with the multiplicity used in the calculation of
pt,i , pt,j .

As discussed above, source of the energy dependence is ambiguous, but may be due to an
increase in jets or resonance decays. There may also be a ‘dilution’ e↵ect due to changing
multiplicity with collision energy.
This version of the scaled observable is equivalent to the correlation observable ⌃pt which was
studied by CERES [49]. Comparisons with their observations are shown below in Section
5.6. This scaling is also of interest because it may be related to the speciﬁc heat of the
system, CV , as discussed in Section 4.6.5. Rewriting Eq 4.9 for CV ,

CV / q⌦

hhpt ii
pt,i , pt,j

↵,

(5.1)

we ﬁnd that CV is proportional to the inverse of the current scaling. The statistical error
bars at 7.7 GeV are large, but CV appears to decrease with energy up to 19.6 GeV then
q⌦
↵
remains approximately constant. This increase in hhpt ii /
pt,i , pt,j with decreasing

energy at 11.5 and 7.7 GeV could indicate a change to a ﬁrst order phase transition although

83

q⌦

↵
p
Figure 5.4:
pt,i , pt,j / hhpt ii vs sNN for all seven energies and eight centrality bins.
The error bars are statistical errors, and the error bands are systematic errors.

84

⌦
↵
p
Figure 5.5: hN i pt,i , pt,j vs sNN for all seven energies and eight centrality bins. The
error bars are statistical errors. The solid lines are shown to guide the eye.
the errors are large at the lower energies.

5.4

Correlations Scaled with Multiplicity

Figure 5.5 is the correlation observable scaled with the multiplicity of tracks used in the
⌦
↵
calculation, hN i pt,i , pt,j , as a function of energy for eight centrality bins. Figure 5.6 is
⌦
↵
the same scaled observable, hN i pt,i , pt,j , as a function of Npart for eight centrality bins.
⌦
↵
⌦
↵
When comparing with
pt,i , pt,j (Figure 5.3) it can be noted that hN i pt,i , pt,j has
very little centrality dependence.

85

7.7 GeV
11.5 GeV
19.6 GeV

27 GeV
39 GeV
62.4 GeV

200 GeV

⌦
↵
Figure 5.6: hN i pt,i , pt,j vs Npart for all seven energies and eight centrality bins. The
error bars are statistical errors. The solid lines are shown to guide the eye.

86

5.5

Correlations Scaled with Multiplicity and hhptii

1

Figures 5.7 and 5.8 show the correlation observable scaled with both multiplicity and average
q
⌦
↵
transverse momentum, hN i pt,i , pt,j / hhpt ii. Figure 5.7 shows
q
⌦
↵
hN i pt,i , pt,j / hhpt ii as a function of Npart for the seven energies. Figure 5.8 shows
q
⌦
↵
hN i pt,i , pt,j / hhpt ii as a function of energy for the eight centrality bins.
Figure 5.7 shows that this scaling has, to a large degree, removed the centrality dependence of the correlation observable. The increase in the correlation observable with Npart
for very peripheral bins (bins with small Npart values) has been proposed as a sign of thermalization [50]. This behavior may also arise from changes in the underlying system; very
peripheral collisions are better approximated as N+N (nucleon-nucleon) collisions than A+A
(nuclei-nuclei) collisions. This scaling seems to plateau and exhibit only a weak centrality dependence in the central and mid-peripheral bins, decreasing with increasing centrality. The
weak centrality dependence in the central and mid-peripheral bins may be due to centrality
dependent e ciency. The e ciency in central bins is slightly lower than the centrality in
peripheral bins due to the larger number of tracks in the detector.
q
⌦
↵
Figure 5.8 shows the same scaling, hN i pt,i , pt,j / hhpt ii, as a function of energy for

the eight centrality bins. This scaling appears to increase monotonically with energy, and the
centrality dependence is signiﬁcantly weaker than the energy dependence. The dependence
q
⌦
↵
on energy but not on centrality suggests that hN i pt,i , pt,j / hhpt ii is not dependent
upon thermodynamics, but on the hard interactions of the initial system before thermal-

ization and QGP formation. In mid-peripheral collisions, hhpt ii is smaller, indicating that
the temperature is lower. The absence of centrality dependence suggests that either the dy-

87

7.7 GeV
11.5 GeV
19.6 GeV

27 GeV
39 GeV
62.4 GeV

200 GeV

q

⌦
↵
Figure 5.7:
hN i pt,i , pt,j / hhpt ii vs Npart for all seven energies and eight centrality
bins. The error bars are statistical. The solid lines are shown to guide the eye.

88

q
⌦
↵
p
Figure 5.8: hN i pt,i , pt,j / hhpt ii vs sNN for all seven energies and eight centrality
bins. The error bars are statistical.
namic correlations do not depend on temperature (and only depend upon collision energy),
or this scaling has suppressed the temperature dependence while magnifying the energy
dependence.

5.6

Comparison with Published Results

Figure 5.9 shows a comparison of the results of this analysis for the scaled observable
q⌦
↵
pt,i , pt,j / hhpt ii in the 0-5% centrality bin with other analyses. The green points
89

Current Analysis
CERES, Nucl. Phys. A727, 97 (2003) [49]
ALICE, J. Phys .G 38, 124095 (2011) [51]
STAR, Phys. Rev. C 72, 044902 (2005) [36]
UrQMD
Recalculated Run 4

q⌦
↵
p
Figure 5.9:
pt,i , pt,j / hhpt ii vs sNN for the 0-5% centrality bin for the present
analysis and several other analyses. The error bars are statistical errors. The error bands
are the systematic error of the STAR analyses.
are from the CERES experiment, which was a ﬁxed target Pb beam onto Au foil at 8.7,
12.3, and 17.3 GeV [49]. The red triangle is a preliminary point from the ALICE collaboration from Pb+Pb collisions at 2760 [51]. The cyan stars are the 2005 STAR results from
Run 4 [36]. The magenta line represents UrQMD model calculations (Section 3.8). Lastly,
the yellow diamonds correspond to values obtained from the present analysis applied to the
data from the 2005 STAR analyses.
The two highest energy points from the CERES study are in good agreement with the present
analysis, however the lowest energy point, done at
90

p

sNN = 8.7 GeV, appears to be in strong

disagreement with the present point at 7.7 GeV. This disagreement may be partially due
to the ﬁducial cuts used in the CERES analysis. During the run for that energy, they had
problems with the detector electronics, and were forced to apply a cut of 17⇡/24 <

< 2⇡.

In the discussion of the systematic errors, they state that the e↵ect of this cut was found to
be small [49], but no values are given. However, a STAR analysis done with Cu+Cu, which
investigated the e↵ect of ﬁducial cuts, suggests that a cut of that size could increase the
q⌦
↵
value of
pt,i , pt,j / hhpt ii by approximately 10% [39]. A shift of 10% would not put
the analyses in agreement, but they would disagree less.
The comparison with ALICE is suggests that

q⌦

↵
pt,i , pt,j / hhpt ii may plateau, but with-

out additional data points the trend is inconclusive.

The results at 62.4 GeV and 200 GeV deviate slightly from the previous STAR results. In
order to test if this was an error in the previous analysis, the full analysis was rerun on two
subsets of the Run 4 data. The analysis code, and some aspects of the analysis procedure
have been improved since 2005, but the produced results are in agreement with the published
results. While this analysis suggests a stronger energy dependence than the 2005 analysis,
the error bars of the two analyses overlap when including estimates of systematic errors and
the results are in excellent agreement.
Taken as a whole, and noting that the CERES 8.7 GeV point had detector problems,
q⌦
↵
pt,i , pt,j / hhpt ii is observed to increase strongly with energy up to 19.6 GeV, and

then have much weaker energy dependence above 19.6 GeV. The ALICE point suggests that
q⌦
↵
pt,i , pt,j / hhpt ii may plateau. The strong decrease below 19.6 GeV may indicate a
change to a ﬁrst order phase transition.

91

Chapter 6
pt Correlation Analysis Checks
Many analysis checks have been performed to insure that the results are robust against small
changes in analysis cuts and detector e ciency. Studies have also been performed to check
for bin width e↵ects, auto-correlations from the method of centrality deﬁnition, and to insure
no dependence on detector pile-up.

6.1

⌘ Cut Dependence

The STAR detector has wide uniform acceptance about mid-rapidity, but previous experiments to which we desire to make experimental comparisons have di↵erent, potentially
non-symmetric acceptances in rapidity [49]. The CERES experiment used a ﬁxed target geometry, so both the width of the rapidity acceptance and the center of the rapidity window
shift with incident energy. A thorough study was performed to insure that the correlation
observable was robust to both small changes in the width of the rapidity window and it’s
symmetry about mid rapidity.
The full analysis was run for every center of mass energy for several symmetric (|⌘| < 1.0,
|⌘| < 0.5, |⌘| < 0.25, |⌘| < 0.1) and asymmetric ( 1.0 < ⌘ < 0.0, 0.0 < ⌘ < 1.0) analysis
q⌦
↵
cuts. The results for
pt,i , pt,j / hhpt ii for two energies can be seen in Figures 6.1
92

q⌦
↵
Figure 6.1: Comparisons of
pt,i , pt,j / hhpt ii calculated using di↵erent ⌘ cuts for 19.6
GeV. In all cases the ⌘ cut was symmetric about ⌘ = 0. Three centrality bins are shown:
0-5%, 20-30%, and 50-60%. The error bars are statistical.
and 6.2. Within errors, the value for all centrality bins is constant. In the case of tighter
⌘ cuts, the values for each centrality bin are seen to shift up slightly, due to the decreased
multiplicity.
The e↵ect of non-symmetric ⌘ cuts was studied for all energies with two di↵erent ⌘ cuts:
q⌦
↵
1.0 < |⌘| < 0, and 0 < |⌘| < 1.0. The results for
pt,i , pt,j / hhpt ii for two energies

are shown in Figures 6.3 and 6.4 in addition to the analysis done for |⌘| < 0.5. In Figures
6.3 and 6.4 the absolute width of the ⌘ window is 1 in all cases, and it can be seen that the
values for each centrality bin are constant within errors for all three ⌘ cuts. In all cases the
93

q⌦
↵
Figure 6.2: Comparisons of
pt,i , pt,j / hhpt ii calculated using di↵erent ⌘ cuts for 200
GeV. In all cases the ⌘ cut was symmetric about ⌘ = 0. Three centrality bins are shown:
0-5%, 20-30%, and 50-60%. The error bars are statistical.

94

q⌦
↵
Figure 6.3: Comparisons of
pt,i , pt,j / hhpt ii calculated using di↵erent ⌘ cuts for 11.5
GeV. In all cases the ⌘ cut had an absolute width of 1. Three centrality bins are shown:
0-5%, 20-30%, and 50-60%. The error bars are statistical.
values of hhpt ii are identical. It has been concluded that the symmetry of the ⌘ cut has no
e↵ect on the results of this analysis.

6.2

Detector E ciency Dependence

To study the e↵ect of detector e ciency on the correlation observable, two e ciency studies
were performed. The ﬁrst investigated the e↵ect of a uniform decrease in e ciency, and the
second a decrease of pt dependent e ciency.
95

q⌦
↵
Figure 6.4: Comparisons of
pt,i , pt,j / hhpt ii calculated using di↵erent ⌘ cuts for 39
GeV. In all cases the ⌘ cut had an absolute width of 1. Three centrality bins are shown:
0-5%, 20-30%, and 50-60%. The error bars are statistical.

96

The case of uniform ine ciency was simulated by randomly dropping tracks. The e ciency,
E, was deﬁned as a percentage value 0.0 < E < 1.0, , and for each track a random number
was selected from the range ↵ 2 (0, 1]. If the random number was less that the chosen
e ciency, ↵ < E, the track was accepted. Decreasing the e ciency uniformly had no
substantial e↵ect on the results of the analysis, as expected. With decreasing e ciency, the
statistical errors grew due to limited statistics, and the correlation values ﬂuctuated, but
they remained constant within errors.
Simulated events produced with UrQMD were used to study the e↵ect of pt dependent
e ciency (Section 3.8). No detector reconstructions were used, but the same acceptance
cuts (⌘, pt , Vz , etc) were applied to the generated events. The pt dependent e ciency was
ﬁt to embedding data and done separately for each particle species. The analysis was done
twice on the UrQMD events, once using all tracks within the acceptance window, and once
with the simulated pt dependent e ciency applied. A plot of the e ciency as a function of
pt was previously shown in Figure 4.6.
When simulating the e ciency, hhpt ii increased 1-2% because the e ciency is lower for lower
momentum tracks, which shifts the means of the distributions up. The magnitude of the
shift in hhpt ii was correlated with

p

sNN because it is related to the mean and width of

the pt spectra. In Figure 6.5 are shown hhpt ii as a function of

p

sNN with and without the

simulated pt dependent ine ciency.
⌦
↵
With the simulated e ciency, the value of
pt,i , pt,j shifted up 7-8%, again dependent
⌦
↵
on energy. The increase in
pt,i , pt,j is due to the decrease in multiplicity, which results
⌦
↵
p
in less dilution of the correlations. Figure 6.6 shows
pt,i , pt,j as a function of sNN
with and without the simulated pt dependent ine ciency.
97

UrQMD

All Data
pt Inefficiency Simulated

Figure 6.5: The average momentum hhpt ii from UrQMD with (green squares) and without
(blue circles) momentum dependent ine ciency simulation for the most central centrality
bins plotted versus center of mass energy. The error bands in this plot are statistical.
When using the simulated e ciency and calculating the scaled observable

q⌦

↵
pt,i , pt,j / hhpt ii,

the values ﬂuctuated on the scale of 2-4%. The e ciency and multiplicity dependence of
q⌦
↵
⌦
↵
pt,i , pt,j / hhpt ii is less than that of hhpt ii or
pt,i , pt,j because they have the
same e ciency dependence, that cancels when scaling.

98

All Data
pt Inefficiency Simulated

UrQMD

⌦
↵
Figure 6.6: The correlation observable
pt,i , pt,j from UrQMD with (green squares)
and without (blue circles) momentum dependent ine ciency simulation for the most central centrality bins plotted versus center of mass energy. The error bands in this plot are
statistical.

99

UrQMD

All Data
pt Inefficiency Simulated

q⌦
↵
Figure 6.7: The scaled correlation observable
pt,i , pt,j / hhpt ii from UrQMD with
(green squares) and without (blue circles) momentum dependent ine ciency simulation for
the most central centrality bins plotted versus center of mass energy. The error bands in
this plot are statistical.

100

6.3

Bin Width Study

Some analyses have been shown to be sensitive to the width of the centrality bins used, this is
⌦
↵
commonly referred to as a bin-width e↵ect. Having deﬁned both hhpt ii and
pt,i , pt,j as

functions of multiplicity, it should be apparent that this analysis is not sensitive to bin-width
e↵ects. The correlation observable, given as a function of multiplicity, is equivalent to the
ﬁnest possible binning. In order to demonstrate this insensitivity to binning, the analysis
was performed also with a uniform centrality binning of 2.5% wide bins in addition to the
standard centrality bins. The results showed no indications of bin-width e↵ects.

6.4

Auto-correlations Study

Artiﬁcial correlations can be induced by using the same multiplicity value to deﬁne collision
centrality and in the calculation of the observables. For these cases alternative centrality
deﬁnitions have been proposed to be used in deﬁning the system centrality, so as to not
induce auto-correlations.
This analysis was checked for auto-correlations by performing the full analysis for every
energy with the standard reference multiplicity (refMult) and with an alternative reference
multiplicity (refMult2). RefMult2 is deﬁned as the number of tracks seen in the detector in
the region 0.5 < |⌘| < 1.0, and then the analysis is performed in the region |⌘| < 0.5. This is
done so that the tracks used to deﬁne the reference multiplicity are not the same track used
in the analysis of pt ﬂuctuations. The comparisons shown between refMult and refMult2 are
done with the analysis cut of |⌘| < 0.5.

101

⌦
↵
Figure 6.8:
pt,i , pt,j as calculated using refMult and refMult2 for 19.6 GeV for the
central 8 centrality bins plotted by Npart . The error bars are statistical.
Both refMult and refMult2 have standardized STAR centrality cuts and multiplicity corrections to address minor dependences on Vz and coincidence rate as discussed in Section
3.5. The o cial STAR refMult and refMult2 centrality cuts were used when checking for
auto-correlations.
The results for refMult and refMult2 were in exact agreement for all energies with the
exception of peripheral bins. Two energies can be seen in Figures 6.8 and 6.9. Throughout
this analysis refMult has been used.

102

⌦
↵
Figure 6.9:
pt,i , pt,j as calculated using refMult and refMult2 for 62.4 GeV for the
central 8 centrality bins plotted by Npart . The error bars are statistical.

103

6.5

Short Range Correlations

An additional e↵ect which was investigated was the e↵ect of short range correlations on
the correlation observable. Short range correlations are correlations which arise from detector e↵ects (two track resolution) and quantum e↵ects (HBT, Bose-Einstein statistics, and
Coulomb e↵ect), and are di↵erent from the bulk correlation e↵ect which we are trying to
study. These have been investigated in previous experimental analyses [49, 36]. An estimate
of the e↵ect of these short range correlations is presented here, but all other results in the
dissertation are not corrected for short range correlations.
An attempt to suppress short range correlations was made using a two step procedure.
⌦
↵
First the correlation observable,
pt,i , pt,j , was calculated for all pairs of particles with
qinv > 100 MeV/c, where qinv is the two particle invariant momentum di↵erence. This
cuto↵ was used because it represents the range in qinv where HBT and Coulomb e↵ect have
been shown to be negligible [49]. When this qinv pair cut is applied, it artiﬁcially induces
large anti-correlations. This artiﬁcial anti-correlation is addressed in the second step of the
⌦
↵
procedure:
pt,i , pt,j is calculated for mixed events with the same qinv cut, and the
value from the mixed events is subtracted from the real events.

Applying the qinv cut to the data does two things: it removes the short range correlations,
and it induces a purely statistical anti-correlation. Applying the qinv cut to mixed events
induces the same, purely statistical anti-correlation, and because there are no correlations (by
construction) this statistical anti-correlation is the only thing induced. So, by subtracting
the mixed events results from the real data, only the short range correlations should be
suppressed.

104

Uncorrected Analysis
With SRC Removed

⌦
↵ q⌦
↵
p
Figure 6.10: sign
pt,i , pt,j
pt,i , pt,j / hhpt ii as a function of sNN with and
without the short range correlation correction for the q
0-5% centrality bin. The error bars
⌦
↵
are statistical errors. This observable is equivalent to
pt,i , pt,j / hhpt ii, but the sign
⌦
↵
of
pt,i , pt,j is moved outside of the square root.
Figure 6.10 shows sign

⌦

pt,i , pt,j

↵ q⌦

↵
pt,i , pt,j / hhpt ii for 0-5% centrality bin as a

function of energy with and without short range correlations suppressed. After the correction,
the results are systematically higher for all energies above 11.5 GeV. This observable is
q⌦
↵
consistent with
pt,i , pt,j / hhpt ii, but in the case of 7.7 GeV with SRC suppressed
⌦
↵
pt,i , pt,j was negative, so the negative sign was moved outside of the square root. The
results corrected for SRC show less incident energy dependence that the uncorrected results,
but still increase with incident energy.

105

6.6

Errors Calculations

Excluding systematic sources of error, the errors in this analysis come from two primary
source: statistical ﬂuctuations, and variation of observables over the width of a centrality
bin.
The statistical error was estimated with a sub-event method. For each energy, at each step
of the calculation, the data set was divided into ﬁve sub-sets. The analysis was performed on
each sub-set of the data. The mean of the result from the ﬁve samples is then the calculated
p
result using the full data set, and the standard deviation of the ﬁve samples divided by 5
was taken as a measure of the statistical variation.
Observing hhpt ii and

⌦

↵
pt,i , pt,j as functions of multiplicity, one can see that they are

not uniform over the width of a centrality bin. Except for in central bins, this variation is

a larger contribution to the error than the statistical ﬂuctuations. When averaging over a
centrality bin, both the mean and the variance are calculated weighted with the number of
events in each multiplicity bin taken from the multiplicity spectra. The variance from this
averaging is added in quadrature with the statistical variance of each bin weighed with the
number of events in that bin.
We estimate the systematic errors of hhpt ii by using studies of pt dependent e ciency (2.4%),
sensitivity to the ⌘ acceptance window (0.5%), and sensitivity to the lower value of the track
pt cut (1%). We estimate the total systematic error of hhpt ii to be 3%.
We estimate the systematic errors of the correlation observable

⌦

↵
pt,i , pt.j by using studies

of pt dependent e ciency (7.5%), sensitivity to the ⌘ acceptance window (9.2%). We estimate
⌦
↵
the total systematic error of
pt,i , pt.j to be 16.5%.
106

We estimate the systematic errors of the scaled observable

q⌦

↵
pt,i , pt.j / hhpt ii by using

studies of pt dependent e ciency (1.3%), sensitivity to the ⌘ acceptance window (⇠4%),
and sensitivity to the lower value of the track pt cut (1%). We estimate the total systematic
q⌦
q⌦
↵
↵
error of
pt,i , pt.j / hhpt ii to be 7%. The systematic error of
pt,i , pt.j / hhpt ii is
⌦
↵
smaller than that of
pt,i , pt.j because e ciency e↵ects cancel.

107

Chapter 7
Results of the Higher Moments
Analysis

7.1

Higher Moments

Using Eq. 4.21, the moment of an arbitrary order of the hpt i spectra can be calculated. In
practice, we are limited to the ﬁrst few moments by statistics. The ﬁrst moment is determined
by the ‘center’ of the distribution, the second by it’s ‘width’, and higher moments are sensitive
to the behavior in the tails of the distribution. Higher moments are increasingly sensitive to
behavior further out in the tails, which are also the parts of the distribution with the fewest
statistics. The behavior of the cumulants is analogous to that of the moments since the nth
cumulant is some combination of the moments up to the nth .

7.1.1

Comparisons with Baselines

Here are presented the behavior of the ﬁrst four moments for the data and the two baselines:
the gamma distribution baseline and the statistically sampled baseline.

108

7.1.2

µ1 (1 )

Figure 7.1 shows ﬁrst moment of the hpt i spectra (the mean) for the 0-5% centrality bin as
a function of energy. In addition to the data, the two baselines are also plotted: the gamma
baseline, and the statistically sampled baseline. This ﬁgure is equivalent to Figure 5.2 except
for the addition of the baselines. We see the same behavior as before: hpt i increases with
energy above 19.6 GeV, but decreases with energy up to 19.6 GeV. This is a consequence of
the changing particle ratios, speciﬁcally the changing pion to proton ratio as pion production
decreases. For identiﬁed particles, hpt i increases with energy across the entire incident energy
range.
For the ﬁrst moment, both baselines exactly reproduce the data. This is to be expected for
the gamma baseline, because the ﬁrst and second moments are equivalent to the parameters
used to deﬁne the gamma distribution. It is promising that the sampled baseline reproduces
the ﬁrst moment. This need not be the case for several reasons: the sampling method
does not take into account particle species, and the sampling method assumes that within a
centrality bin the multiplicity and the pt spectra are uncorrelated.

7.1.3

µ2 (

2
hpt i ,

2 )

Figure 7.2 shows the second moment of the hpt i spectra (the variance) for the 0-5% centrality
bin as a function of energy. As is Figure 7.1 the sampled baseline and gamma baseline are
also plotted. To ﬁrst order the variance decreases monotonically with energy. The second
moment (and all higher moments as well) are non-trivially sensitive to pt cuts, ⌘ cuts,
centrality cuts, and detector e ciencies, so it is unclear if the kink near 39 and 62.4 GeV is

109

Data
Sampled Baseline
Gamma Baseline

Figure 7.1: The ﬁrst moment, µ1 , for the 0-5% centrality bin plotted versus energy. Two
baselines are also plotted: the gamma baseline, and the statistically sampled baseline. The
lines are error bands which represent statistical error.

110

Data
Sampled Baseline
Gamma Baseline

Figure 7.2: The ﬁrst moment, µ2 , for the 0-5% centrality bin plotted versus energy. Two
baselines are also plotted: the gamma baseline, and the statistically sampled baseline. The
lines are error bands which represent statistical error.
physical, or an experimental artifact. We have extensively carried out quality assurance for
the data, but it is possible that some unknown e↵ects still remain.
As with the ﬁrst moment we see, as anticipated, that the gamma baseline reproduces the
data. Unlike with the ﬁrst moment, the sampled baseline is smaller than the data. This
observation is equivalent to the statement that the hpt i spectra is narrower for the sampled
baseline than for the data. This has been observed previously when comparing data with
mixed events [36]. This indicates that the data are not purely statistical, and that there
are correlations in the data. The di↵erence between the data and the statistically sampled

111

baseline is in agreement with the values of

⌦

↵
pt,i , pt,j presented in Chapter 5. This is no

2
surprise because they are equivalent approaches for measuring hp i,dynamic as discussed in
t

Section 4.3.
2
The increase in hp i,dynamic could indicate many di↵erent physical phenomena: increase in
t

jets (and jet ﬂuctuations), increase in resonance decays (and increase in resonance ﬂuctuations), or an increase in temperature ﬂuctuations. The decrease in µ2 with energy is largely
due to the increase in multiplicity with energy (See Figure 7.3). The increase in multiplicity
decreases the statistical ﬂuctuations. The statistical ﬂuctuations are also related to the avD E
2
2
erage variance of the underlying pt distribution, pt (see Fig 7.4). The magnitude of pt
is dominated by the temperature of the distribution, but is modiﬁed by other e↵ects. The
decrease with energy up to

p

sNN = 19.6 GeV is due to the change in particle ratios, just

as was observed with hhpt ii (Section 5.1). Jets, ﬂow, high-pt particle suppression and other
D E
2
e↵ects may all modify pt .

7.1.4

µ3 (3 )

Figure 7.5 shows the third moment of the hpt i spectra for the 0-5% centrality bin as a
function of energy for the data and the two baselines. As with the second moment, we see
that the third moment decreases with energy, except for some non-monotonic behavior in
the region from 27 to 62.4 GeV.
This moment shows interesting deviation between the data and the baselines. Unlike with
the previous moments, here the gamma baseline does not exactly reproduce the data. The
trend of the data are reproduced but the gamma baseline underestimates the third moment
for all energies except 7.7 GeV, and does not reproduce the ‘kink’, which may be indicative
112

Figure 7.3: The average multiplicity hN i as a function of energy for the 0-5% centrality bin.

113

Figure 7.4: The average variance of the underlying pt distribution,
energy for the 0-5% centrality bin.

114

D

2
pt

E

, as a function of

Data
Sampled Baseline
Gamma Baseline

Figure 7.5: The third moment, µ3 , for the 0-5% centrality bin plotted versus energy. Two
baselines are also plotted: the gamma baseline, and the statistical sampled baseline. Solid
lines are drawn to guide the eye. Error bands represent statistical error.

115

of interesting physics.
The sampled baseline also does not reproduce the data. At high energies (62.4 GeV and 200
GeV) the sampled baseline is lower than the data, while at 7.7 GeV the sampled baseline
is in agreement with the data within errors. This, like in the second moment, indicates
a disappearance of correlations in the data as the energy decreases. It is interesting that
the sampled baseline does not reproduce the ‘kink’ in the region from 27 to 62.4 GeV. The
hypothesis with the sampled baseline was that experimental e↵ects which change the hpt i
distribution do so by their e↵ect on the track pt distribution. So, by statistically sampling the
experimental track pt distribution, we should create a statistical baseline which perseveres
the purely analysis and experimentally dependent e↵ects. The disagreement between the
data and the sampled baseline means either the sampled baseline hypothesis was incorrect
and that the sampled baseline does not preserve the experimental and analysis e↵ects (or
those e↵ects those e↵ects become too dilute), or we are seeing the ﬁrst tantalizing indications
of interesting incident energy dependent physics.
The di↵erence between the data and the sampled baseline is the dynamic third moment,
µ3,dynamic , and is a shortcut to the three particle relative transverse-momentum correlator
⌦
↵
⌦
↵
pt,i , pt,j , pt,k , which would be deﬁned analogously to
pt,i , pt,j . The absence of

dynamic ﬂuctuations of the hpt i distribution at 7.7 and 11 GeV is an interesting physics
observation which has already been made several times. This could indicate a change to a
ﬁrst order phase transition, or something more mundane. There could be physical e↵ects
which may decrease the correlation of the system, increasing thermalization and washing out
dynamic correlations, the systems at 7.7 and 11 GeV may not be forming a QGP, or there
may simply be too few high pt particles to accurately sample the temperature and hpt i of
116

the system.

7.1.5

µ4

Figure 7.6 shows the fourth moment of the hpt i spectra for the 0-5% centrality bin as a
function of energy for the data and the two baselines. The behavior of the fourth moment
of the data is similar to that of the second and third moments, it decreases with energy
except for some small non-monotonic behavior near 62.4 GeV. The similar behavior of the
di↵erent higher moments is not unexpected, because they are all powers of the symmetry or
asymmetry of the distribution.
The data and baselines look very similar to the second moment, µ2 . The gamma baseline
reproduces the data very well. The di↵erence between the data and the sampled baseline is
the dynamic fourth moment, µ4,dynamic . As has been previously observed serval times, there
clearly are dynamic ﬂuctuations of hpt i but they decrease as the energy decreases, becoming
consistent with zero at 7.7 GeV.

7.1.6

4

The ﬁrst three moments are equivalent to the ﬁrst three cumulants, but the fourth moment
and fourth cumulant di↵er (Eq. 4.14). Figure 7.7 shows the fourth cumlant, 4 , for the
0-5% centrality bin as a function of energy for the data and the two baselines. The fourth
cumulant is related to the moments by, 4 = µ4

3µ2 . The two terms µ4 and µ2 are close
2
2

in magnitude, so the di↵erence is a small number with a large uncertainty. Not much can be
said about the fourth moment except that it does not invalidate the conclusion that dynamic

117

Data
Sampled Baseline
Gamma Baseline

Figure 7.6: The fourth moment, µ4 , for the 0-5% centrality bin plotted versus energy. Two
baselines are also plotted: the gamma baseline, and the statistical sampled baseline. The
lines are error bands which represent statistical error.

118

Data
Sampled Baseline
Gamma Baseline

Figure 7.7: The fourth cumulant, 4 , for the 0-5% centrality bin plotted versus energy. Two
baselines are also plotted: the gamma baseline, and the statistical sampled baseline. Solid
lines are drawn to guide the eye. Error bands represent statistical error.
correlations are disappearing from the data at 11.5 and 7.7 GeV.

7.1.7

Comparisons of Moments with UrQMD

The full higher moments analysis was also performed on results generated by UrQMD (Section 3.8). All of the UrQMD results presented here were calculated with simulated pt dependent ine ciency. The e↵ect of this e ciency simulation on the higher moments analysis
is addressed in Section 8.3. The UrQMD analysis was done with the same analysis and
geometric cuts as the data. Only the ﬁrst two moments are presented here because all mo119

ments greater than the second moment exhibit the same trend and behavior as the second
moment.
Figure 7.8 shows the ﬁrst moment of the hpt i spectra as a function of energy for the 0-5%
centrality bin for the data and UrQMD. The behavior of the ﬁrst moment for UrQMD was
already shown in Figure 6.5. The magnitude and trend of the data are only approximately
reproduced by UrQMD. The trend of decreasing with energy for lower energies, then increasing with energy at higher energies is present, but the inﬂection energy is signiﬁcantly higher
(39 GeV as opposed to 19.6 GeV). As a result, the value of the ﬁrst moment is signiﬁcantly
lower for UrQMD at high energies than in the data.
Figure 7.9 shows the second moment of the hpt i spectra as a function of energy for the 0-5%
centrality bin for the data and UrQMD. As with the ﬁrst moment, UrQMD only generally
reproduces the trend and magnitude of the data. UrQMD exhibits a much stronger energy
dependence than the data, over estimating the second moment at 7.7 and 11.5 GeV, and
under estimating it for all energies 19.6 GeV and above.

7.2

Cumulant Ratios

As was discussed in 4.6.6, ratios of the cumulants have been proposed as sensitive probes of
the correlation length. Taking a ratio should cause e ciencies to cancel. Several ratios have
been proposed, and here we will examine two: S , K 2 and compare them with the gamma
baseline and the statistically sampled baseline.

120

Data
UrQMD

Figure 7.8: The ﬁrst moment, µ1 , for three centrality bins plotted versus energy. In addition
to the experimental result, the result from UrQMD with pt dependent e ciency is plotted.
Statistical errors are plotted as error bands, but are smaller than the point markers.

121

Data
UrQMD

Figure 7.9: The second moment, µ2 , for three centrality bins plotted versus energy. In
addition to the experimental result, the result from UrQMD with pt dependent e ciency is
plotted. Statistical errors are plotted as error bands, but are smaller than the point markers.

122

7.2.1

S


Figure 7.10 shows S (equivalent to 3 ) for the 0-5% centrality as a function of energy for
2
data, the gamma baseline, and the statistically sampled baseline.
A number of things are apparent: neither baseline reproduces the data for all energies. Both
baselines reproduce the data at 7.7 GeV, and the sampled baseline also reproduces the data
at 11.5 GeV, but for all other energies the baselines underestimate the data. Also notable is
the pronounced ‘kink’ in the data centered around 27 to 39 GeV. This kink was previously
observed in the moments of the distribution. Neither baseline shows this behavior, and it
may be an indication of interesting physics.

7.2.2

K

2


Figure 7.11 shows K 2 for the 0-5% (equivalent to 4 ) centrality as a function of energy
2

for data, the gamma baseline, and the statistically sampled baseline. The sensitivity of
this measurement to statistics is apparent. Event the sampled baseline, which was done
with 5 million events per energy (substantially more that the data, see Table 3.4), exhibits
statistical ﬂuctuations. The behavior of this ratio is dominated by the fourth cumulant. As
with the fourth cumulant, this does not invalidate the conclusion that correlations disappear
at 11.5 and 7.7 GeV, but it o↵ers little additional insight.
The trend of the data appears erratic, and more statistics and more detailed analysis is
necessary before any strong conclusions can be drawn.

123

Data
Sampled Baseline
Gamma Baseline

Figure 7.10: The cumulant product S for two centrality bins plotted versus energy. Two
baselines are plotted along with the data. Solid lines are drawn to guide the eye. Error
bands represent statistical error.

124

Data
Sampled Baseline
Gamma Baseline

Figure 7.11: The cumulant product K 2 for two centrality bins plotted versus energy. Two
baselines are plotted along with the data. Solid lines are drawn to guide the eye. Error
bands represent statistical error.

125

Chapter 8
Checks of the Higher Moments
Analysis
This is the ﬁrst time a systematic analysis of the moments of the hpt i spectra has been
attempted. A detailed study of the e↵ects of the analysis cuts and detector e ciency on the
moments and cumulant ratios was undertaken. In Section 8.1 the e↵ect of both the width
and symmetry of the ⌘ cut is studied, in Section 8.2 the e↵ect of the upper pt cut is studied,
and in Section 8.3 the e↵ect of pt dependent e ciency is studied. Additionally, in Section 8.4
the method used for calculating the errors of the moments and cumulant ratios is presented,
and the systematic error of the higher moments analyses is estimated.

8.1

⌘ Cut Dependence

The e↵ect of the ⌘ cut on the analysis was studied by investigating two di↵erent sets of
cases: one set of cases where ⌘ cut was kept centered at ⌘ = 0 but it’s width was varied, and
another set where the width of the ⌘ cut was kept constant but where it was centered in the
detector was varied. These are the same cases investigated while checking the ⌘dependence
⌦
↵
of
pt,i , pt,j .
126

Figures 8.1 and 8.2 shows the ﬁrst moment of the hpt i spectra for four energies and three
centralities as calculated using two di↵erent symmetric ⌘ cuts. Figures 8.3 and 8.4 is the
same Figure for the second moment.
Both the ﬁrst and second moments are sensitive to the width of the ⌘ cut. The sensitivity of
the second moment to the width of the ⌘ cut is largely independent of the energy, while for the
ﬁrst moment di↵erent energies di↵er in sensitivity with 7.7 GeV being notably sensitive.
As the ⌘ cut gets narrower, both the ﬁrst and second moments increase. The ﬁrst moments
sensitivity to the width of the ⌘ cut implies that the transverse momentum deposited in the
detector is larger at mid-rapidity than away from mid-rapidity.
The increase in the second moment with the narrowing of the ⌘ cut seems contrary to the
increase in the ﬁrst moment. The decrease in the ﬁrst moment implies that there are more
low pt tracks at large rapidity than small rapidity, and as these tracks are removed by the
⌘ cut, the hpt i distribution would be expected to become narrower in addition to the mean
shifting to higher pt . Instead the second moment increases, which is a consequence of the
poorer sampling of the underlying track pt distribution because less tracks are within the
acceptance.
Figures 8.5, 8.6, 8.7, and 8.8 show the ﬁrst and second moments for two energies and the
ﬁrst three centrality bins using three ⌘ cuts of di↵ering symmetry. All three ⌘ cuts have the
same width (⌘ = 1), only where they are centered in the detector is shifted.
There is no consistent trend for either the ﬁrst or second moment as the symmetry of the ⌘ cut
is varied. While the STAR detector has a symmetric acceptance about ⌘ = 0, experimental
di culties do not arise symmetrically. The variations between these asymmetric ⌘ cuts arises

127

0-5%
5-10%

10-20%

39 GeV

Figure 8.1: The ﬁrst moment, µ1 , for three centrality bins for 39 GeV plotted with symmetric
⌘ cuts of various widths. The lines are to guide the eye. There are error bars which represent
statistical error, but are smaller than the point markers.

128

0-5%
5-10%

200 GeV
10-20%

Figure 8.2: The ﬁrst moment, µ1 , for three centrality bins for 200 GeV plotted with symmetric ⌘ cuts of various widths. The lines are to guide the eye. There are error bars which
represent statistical error, but are smaller than the point markers.

129

0-5%
5-10%
10-20%

39 GeV

Figure 8.3: The second moment, µ2 , for three centrality bins for 39 GeV plotted with
symmetric ⌘ cuts of various widths. The lines are to guide the eye. There are error bars
which represent statistical error, but are smaller than the point markers.

130

0-5%
5-10%
10-20%

200 GeV

Figure 8.4: The second moment, µ2 , for three centrality bins for 200 GeV plotted with
symmetric ⌘ cuts of various widths. The lines are to guide the eye. There are error bars
which represent statistical error, but are smaller than the point markers.

131

7.7 GeV

0-5%
5-10%
10-20%

Figure 8.5: The ﬁrst moment, µ1 , for three centrality bins for 7.7 GeV plotted with three
asymmetric ⌘ cuts of width 1. The lines are to guide the eye. There are error bars which
represent statistical error, but are smaller than the point markers.
from detector asymmetries. If we were performing an analysis of ⌘ correlations this would be
a concern. Since we are interested in the bulk properties of the hpt i distribution we simply
selected a wide symmetric ⌘ acceptance. Also note that the changes in the ﬁrst and second
moments are small.

132

0-5%

200 GeV

5-10%

10-20%

Figure 8.6: The ﬁrst moment, µ1 , for three centrality bins 200 GeV plotted with three
asymmetric ⌘ cuts of width 1. The lines are to guide the eye. There are error bars which
represent statistical error, but are smaller than the point markers.

133

10-20%

7.7 GeV

5-10%
0-5%

Figure 8.7: The second moment, µ2 , for three centrality bins for 7.7 GeV plotted with three
asymmetric ⌘ cuts of width 1. The lines are to guide the eye. There are error bars which
represent statistical error, but are smaller than the point markers.

134

10-20%

200 GeV

5-10%
0-5%

Figure 8.8: The second moment, µ2 , for three centrality bins and for 200 GeV plotted with
three asymmetric ⌘ cuts of width 1. The lines are to guide the eye. There are error bars
which represent statistical error, but are smaller than the point markers.

135

8.2

pt Cut Dependence

Figures 8.9, 8.10, 8.11, and 8.12 show the ﬁrst and second moment of the hpt i distributions
for two energies and three centralities done with ﬁve di↵erent pt analysis cuts. All ﬁve
analysis cuts have the same lower pt cut (0.15 GeV/c), and their upper pt cut is varied from
0.5 to 3.0 GeV/c.
Both the ﬁrst and second moments decrease as the pt cut is made narrower. This behavior
is anticipated because the underlying pt distribution is being truncated, which will truncate
the hpt i distribution.
The pt cut 0.15 GeV/c < pt < 2.0 GeV/c was selected for the results shown in this dissertation so that they could be compared with previous analysis.

8.3

Detector E ciency Dependence

The e↵ect of pt dependent e ciency was studied with UrQMD (Section 3.8). The higher
moments analysis was run twice, once using the full set of UrQMD simulated events, and
again where the tracks in each event were used or not used based on pt e ciency as determined by embedding. The pt dependent e ciency was done independently for individual
particle species.
The behavior of the ﬁrst moment has already been shown in Figure 6.5, and it was observed
to increase with ine ciency because the e ciency is lower for low pt tracks. The second
moment is not shown in this dissertation, but it is also observed to increase with ine ciency.
This is because there are fewer tracks per event.

136

0-5%
5-10%
10-20%

19.6 GeV

Figure 8.9: The ﬁrst moment, µ1 , for three centrality bins for 19.6 GeV plotted with pt cuts
which vary in their upper bound. The lines are to guide the eye. There are error bars which
represent statistical error, but are smaller than the point markers.

137

0-5%
5-10%
10-20%

39 GeV

Figure 8.10: The ﬁrst moment, µ1 , for three centrality bins for 39 GeV plotted with pt cuts
which vary in their upper bound. The lines are to guide the eye. There are error bars which
represent statistical error, but are smaller than the point markers.

138

0-5%
5-10%
10-20%

19.6 GeV

Figure 8.11: The second moment, µ2 , for three centrality bins for 19.6 GeV plotted with pt
cuts which vary in their upper bound. The lines are to guide the eye. There are error bars
which represent statistical error, but are smaller than the point markers.

139

0-5%
5-10%
10-20%

39 GeV

Figure 8.12: The second moment, µ2 , for three centrality bins for 39 GeV plotted with pt
cuts which vary in their upper bound. The lines are to guide the eye. There are error bars
which represent statistical error, but are smaller than the point markers.

140

pt Inefficiency Simulated
All Data

UrQMD

Figure 8.13: The cumulant ratio S for UrQMD with and without pt dependent e ciency
simulation. The error bands represent statistical error.
Figure 8.13 shows the cumulant ration S

as a function of energy for UrQMD with and

without the simulated pt e ciency. Two things can be noted: there is statistical ﬂuctuation
and this simulation would be improved with additional statistics, and the result with simulated pt dependent e ciency is higher for energies up to 39 GeV. Above 39 GeV the results
with and without the simulated e ciency are within statistical agreement.
Figure 8.14 shows the cumulant ratio K 2 as a function of energy for UrQMD with and
without the simulated pt e ciency. Due to statistical error, not much can be said. The
result seems suggestive that pt ine ciency increases K 2 at the lower energies, but the two

141

pt Inefficiency Simulated
All Data

UrQMD

Figure 8.14: The cumulant ratio K 2 for UrQMD with and without pt dependent e ciency
simulation. The error bands represent statistical error.
results are within statistical agreement at 7.7, 19.6, and 27 GeV.

8.4

Error Calculation

The statistical errors of higher moments have been derived using the Delta theorem in Ref.
[52]. The statistical error of a moment are related non-trivially to higher moments of the
distribution. Because the derivation of the statistical errors is not related to the underlying
physics, but it rather an exercise in mathematics, the derivation will not be reproduced here.
Interested readers are referred to Ref. [52].
142

Energy
7.7 GeV
11.5 GeV
19.6 GeV
27 GeV
39 GeV
62.4 GeV
200 GeV

µ1
0.751%
0.504%
0.551%
0.135%
0.493%
0.265%
0.450%

µ2
µ3
µ4
4.426% 11.107% 8.522%
4.114% 5.88% 7.578%
3.655% 10.336% 7.574%
1.97% 2.684% 3.914%
4.777% 8.899% 9.656%
0.386% 0.506% 1.061%
3.222% 6.570% 6.384%

Table 8.1: The systematic error for the 0-5% centrality bin of the ﬁrst four moments as
calculated from the variation noted when changing the symmetry of the ⌘ cut.
The statistical error of the higher moments has been estimated from the investigation of
the symmetry of the ⌘ cut. The width of the ⌘ cut and the upper bound of the pt cut
were not used in this estimation because the variation of the moments with these two cuts
is systematic and understood. The systematic error in the 0-5% centrality bin of the ﬁrst
moment is ⇠ 0.5%, for the second moment is ⇠ 3%, for the third moment is ⇠ 7%, and for
the fourth moment is ⇠ 7%. The estimated systematic errors of the ﬁrst four moments for
each energy is given in Table 8.1.

143

Chapter 9
Conclusion
Two analyses have been presented in this dissertation, the study of the two-particle transverse
⌦
↵
momentum correlator
pt,i , pt,j , and the study of the higher moments of the hpt i spectra.
Both of these studies, but in particular the study of the moments, were intended to be
⌦
↵
searches for the QCD critical point. The study of
pt,i , pt,j only has utility as a critical
point observable due to it’s relationship with the second moment of the hpt i spectra, but it
has additional use as a measure of system equilibration.

9.1

Summary of Correlations Analysis

As mentioned above,

⌦

↵
pt,i , pt,j can be used as a critical point observable because of

it’s relationship to the second moment of the hpt i spectra (see App. ). Additionally,
⌦
↵
pt,i , pt,j can be used to explore the thermal behavior of the system and look for behavior such as equilibration and “thermalization”.
The ﬁrst part of the

⌦

↵
pt,i , pt,j analysis was the determination of hhpt ii. hhpt ii is given as

a function of energy for all centrality bins in Figure 5.2. In central and mid-peripheral bins

that for energies up to 19.6 GeV hhpt ii decreases with energy, and above 19.6 GeV hhpt ii
increases with energy for all centrality bins. This behavior is a consequence of changing
144

particle ratios, speciﬁcally the pion-proton ratio. For thermal distributions of the same
temperature, protons will have larger a hhpt ii than pions because they are more massive.
This behavior is qualitatively reproduced by UrQMD.
⌦
↵
The unscaled correlator
pt,i , pt,j is shown in Figure 5.3 as a function of energy for
⌦
↵
all centrality bins.
pt,i , pt,j increases with both energy and as collisions become more

peripheral. The variation with centrality is to variation in hhpt ii and the changing number
of particles in each event. Peripheral events, corresponding to lower multiplicties, have fewer
tracks. As collisions become more central, and the number of tracks increases like N , the
number of pairs increases like N 2 . This results in a “dilution” of the correlations because
⌦
↵
pt,i , pt,j is an average over all pairs of tracks.

↵
pt,i , pt,j , also presented were several scaled variants. The ﬁrst scaled
q⌦
↵
correlation presented was
pt,i , pt,j / hhpt ii which is shown in Figure 5.4 as a function
In addition to

⌦

of energy for all centrality bins. The advantage of scaling with hhpt ii is that the result
q⌦
↵
is unitless and e ciency independent. The behavior of
pt,i , pt,j / hhpt ii is similar
⌦
↵
to that of
pt,i , pt,j , increasing with both energy and in peripheral collisions, but the
energy dependence is weakened.

↵
⌦
↵
pt,i , pt,j . Figure 5.5 is hN i pt,i , pt,j as a func⌦
↵
tion of energy for all centrality bins. The motivation for examining hN i pt,i , pt,j is to
⌦
↵
check if the centrality dependence of
pt,i , pt,j is due only to the dilution e↵ect discussed
⌦
↵
above. hN i pt,i , pt,j increases with collision energy.
Another scaling presented was hN i

The ﬁnal scaling presented was

⌦

q
q
⌦
↵
⌦
↵
hN i pt,i , pt,j / hhpt ii.
hN i pt,i , pt,j / hhpt ii is

shown in two Figures: Figure 5.7 all seven energies are plotted as functions of Npart and
q
⌦
↵
in Figure 5.8 hN i pt,i , pt,j / hhpt ii is given as a function of energy for each centrality
145

bin. Figure 5.7 shows that this scaling has largely removed the centrality dependence of
⌦
↵
pt,i , pt,j . This makes sense because we have addressed both the dilution e↵ect and
the variation of hhpt ii with centrality. The remaining centrality dependence is indicative of
new physics. It is possible that we are observing the threshold where the produced systems
are too small and short lived to reach thermal equilibration and produce QGP. Another
q
⌦
↵
interesting property is observed in 5.8, hN i pt,i , pt,j / hhpt ii is observed to increase
⌦
↵
smoothly and continuously with collision energy. This is the dependence of
pt,i , pt,j ,
q
⌦
↵
or more precisely hN i pt,i , pt,j / hhpt ii, on the temperature of the system.
Other experimental analyses have used

q⌦

↵
pt,i , pt,j / hhpt ii [36, 51], or equivalent observ-

ables [49], and comparisons with those results is shown in Figure 5.9. The results presented

in this dissertation are in excellent agreement with previous results from the STAR collaboration [36], and the agreement with the results from CERES [49] is reasonable, and the
disagreements are believed to be understood (see Section 5.6). The addition of a prelimq⌦
↵
inary point from ALICE [51] suggests that
pt,i , pt,j / hhpt ii may plateau at higher
energies.

9.2

Summary of Moments Analysis

The higher moments analysis of the hpt i spectra is a wholly original study, and the results
are of interest because they are strongly related to the correlation length of the system and
are therefore a excellent candidate as a critical point observable. In addition to the ﬁrst four
moments and cumulants, the two cumulant ratios S

and K 2 were presented. The hpt i

spectra, with the gamma distributions used for the baselines are in Figure 5.1.

146

The ﬁrst moment of the hpt i spectra, µ1 , is shown in Figure 7.1 as a function of energy for the
0-5% centrality bin for the data and the two baselines. The ﬁrst moment is identical to hhpt ii
presented in the two-particle correlation analysis and the results are in exact agreement.
The second moment, µ2 , is given in Figure 7.2 as a function of energy for the 0-5% centrality
bin for the data and the two baselines. The gamma baseline is ﬁt to the ﬁrst and second
moments of the data, so it’s agreement with the data are expected. The statistical baseline is
lower for the data for all energies except 7.7 GeV. This means that the statistically sampled
distribution is narrower than the data, and that there are correlations in the data which
are not reproduced in the statistical baseline. The hpt i spectra for 200 GeV in the 0-5%
centrality bin for both data and the statistically sampled baseline is shown in Figure 4.1. It
is notable that the deviation of the sampled baseline from the data are largest at the highest
energies, and that the deviation becomes smaller as the energy decreases until at 7.7 GeV the
data and sampled baseline are indistinguishable. This indicates that the correlations in the
data are decreasing with energy, which is the same conclusion measured by the two-particle
correlator.
Figure 7.5 is the third moment, µ3 , as a function of energy for the 0-5% bin for the data
and the two baselines. This is the only indication that the data are note exactly gamma
distributions. We see that the third moment decreases with collision energy, except for an
increase in the region of 39 to 62.4 GeV. Both baselines are lower than the data above
11.5 GeV, with the exception that the gamma baseline is within errors at 200 GeV. Neither
baseline reproduces the ‘kink’ in the 39 to 62.4 GeV region. The deviation between the data
and the statistical baseline could again be indicative of correlations in the data which are
not present in the statistical baseline.

147

The fourth moment, µ4 , as a function of energy for the 0-5% centrality bin for the data and
two baselines is Figure 7.6. The fourth moment looks very similar to the second moment, as
could be anticipated because they are both measures of the symmetry of the distributions.
We see that the gamma baseline reproduces the data very well, and that the statistical
baseline is lower than the data for all energies except 7.7 GeV indicating that there are
non-statistical correlations in the data.
The fourth cumulant, 4 , as a function of energy for the 0-5% centrality bin for the data and
two baselines is Figure 7.7. The fourth cumulant a di↵erence between two values, resulting a
small number with very large error bars. It is clear that more statistics are needed in order
to draw any conclusions from the fourth cumulant. While the fourth cumulant is of little
use, it does not invalidate the conclusion that dynamic correlations disappear from the data
at 11.5 and 7.7 GeV.
The cumulant ratio S

is given as a function of energy for the 0-5% centrality bin for the

data and the two baselines in Figure 7.10. We see exciting behavior similar to that which was
observed in the third moment: S

decreases with energy except for a pronounced increase

at 39 to 62.4 GeV, for all energies except 7.7 GeV the baselines are lower than the data (the
statistical baseline also agrees at 11.5 GeV), and neither baseline reproduces the kink at 39
to 62.4 GeV. This is one of the most exciting results, but additional analysis will be required
to insure that this result is physical and not just an artifact.
The other cumulant ratio presented was K 2 , given in Figure 7.11 as a function of energy for
the 0-5% centrality bin for the data and the two baselines. The need for additional statistics
is apparent, and it’s behavior is driven by the behavior of 4 .

148

9.3

Looking Forward

While there are many results presented in this dissertation, there are ultimately two results
of note: ﬁrst we observe that dynamic correlations of hpt i decrease below 19.6 GeV becoming
negligible at 7.7 GeV, and second no dramatic non-monotonic behavior is observed in any
pt ﬂuctuations as collision energy or centrality is varied.
The decrease of dynamic correlations with decreasing energy could be an indication of the
onset of the deconﬁned phase, change in the type of phase transition, or it could be caused
by many less novel physical e↵ects: changing chemistry, ﬂow, particle decays, charge correlations, jets, or other e↵ects. The Hanbury Brown and Twiss (HBT) e↵ect is one that can
already be discounted by the short range correlations check (cutting on pairs by qinv . See
Section 6.5). The e↵ect of changing chemistry could be investigated by performing the analyses independently for identiﬁed particle species. Performing these analyses for identiﬁed
particles presents additional challenges. The e↵ect of ﬂow can be studied using momentum
correlations in

✓ for various angles with respect to the event plane. Identiﬁed particle

decays, such a as the

meson to two kaons, can be excluded by using a similar method used

in the short range correlation analysis. Pairs of particles of the correct species and with qinv
corresponding to the mass of the mother particle can be cut out and mixed events can be
used to correct the induced auto-correlation.
The absence of dramatic non-monotonic behavior is an important null result which suggests
that we have not observed the critical point. A more quantitative statement will require
both more sophisticated analysis and more detailed theoretical work. The dependence of pt
ﬂuctuations on the correlation length is a function of chemistry, experimental pt dependent

149

e ciencies, and analysis cuts. There may be many competing physical e↵ects which are all
manifest in hpt i ﬂuctuations and cannot be distinguished without other analyses. The many
uncertainties are why most critical point searches have approached the problem empirically.
The results presented in this dissertation will have to be taken hand-in-hand with other
forthcoming analyses in order to make truly conclusive statements about the critical point
of QCD.

9.4

In Summary

⌦
↵
The two-particle transverse momentum correlator
pt,i , pt,j and the scaled variant
q⌦
↵
pt,i , pt,j / hhpt ii both suggest that there are non-statistical correlations in the data
which increase with energy and are small, if not in agreement with zero, at 7.7 GeV. A
similar result is also seen in the analysis of the higher moments of the hpt i spectra where
moments of the data are larger than those of the statistically sampled baseline at the higher
energies. The deviation between the data and the sampled baseline is energy dependent and
disappears at 7.7 GeV.
Both analyses,

⌦

↵
pt,i , pt,j and moments of the hpt i spectra, show no indications of non-

monotonic behavior with changing collision energy with one exception. The examination of
the odd higher moments of the hpt i spectra shows anomalous behavior and increases in the
higher moments in the region around 39 GeV to 62.4 GeV.

150

APPENDIX

151

APPENDIX
Derivation of
⌦
↵
2
pt,i, pt,j = hp i
t

⌧

2
pt

N

Here we will break down the two-particle transverse momentum (pt ) correlation observable,
⌦
↵
pt,i , pt,j , which has been used in several STAR analyses [36, 39]. We will demonstrate
⌦
↵
that
pt,i , pt,j can be written as a sum of two parts, one corresponding to the average
covariance of the average pt spectrum, and another corresponding to the second moment of
the hpt i spectra and the average number of tracks used in the calculation.

A.1

Additional Notation

The correlation observable

⌦

↵
pt,i , pt,j is given in Section 4.7.1. Before deriving our result

we will need to deﬁne a few more terms. First, without too much concern as to the physical
signiﬁcance, we deﬁne a transverse momentum sample self-covariance for an event k,

2
k,k

=

PNk PNk
i =1

j=1

pt,i

hpt ik

2
Nk

pt,j

hpt ik

.

(A.1)

2
The self-covariance k,k is always zero. The only assumption necessary to demonstrate this

152

is that the underlying distribution is bound. In the experimental case, the underlying pt
⇥
⇤
distributions are bound to a range pt,min , pt,max by analysis cuts, but even in the ideal
case all that is needed is that the the underlying distribution is positive and integrable.
First we write Eq. A.1 expanding the hpt i terms,

2
k,k

PN PN
i=1

=

!
PN
pt,k
k
N

pt,i

j=1

!
PN
pt,k
k
N

pt,j

N2

.

(A.2)

We can then multiply and regroup,

PN PN
i=1

2
k,k

=

j=1

2

4pt,i pt,j

PN
k pt,k
N

pt,i + pt,j +

N2

3

!2
PN
pt,k
k
5
N

.

(A.3)

Performing the summation on the last two terms in the numerator and simplifying,

2
k,k

=

PN PN
i=1

j=1 pt,i pt,j

N2

The ﬁnal step is to note that

⇣P

N
k

pt,k

⌘2

=

PN PN
i=1

j=1 pt,i pt,j

PN PN
i=1

j=1 pt,i pt,j
N2

= N hpt i

PN

j=1 pt,i

N 2 hpt i2

.

(A.4)

= N 2 hpt i2 . This may

not be immediately obvious, but it arrises from the fact that a one dimensional distribution
cannot be correlated with itself, and the mean of the product of two uncorrolated distributions is the product of the means. Stated mathematically, hxyi = hxi ⇥ hyi. Therefore,
2
k,k

= 0.

We also deﬁne an “idealized” variant of Ck (Eq. 4.18),

153

N

{Ck } =

N

k
k
X X

hpt ik

pt,i

i=1 j=1,j6=i

pt,j

hpt ik .

(A.5)

{Ck } is “idealized” in that {Ck } = Ck if all events have the same hpt i.
Note that Eq. A.1 and Eq. A.5 can be related by,

N

{Ck } = Nk (Nk

Noting that

2
k,k

1) k,k

k
X

pt,i

i=1

= 0 and that the sample variance

2
pt ,k

hpt ik

2

.

is deﬁned

(A.6)

2
x,k

=

PNk
i=1 (xi hxik )
Nk

we can simplify further to,

{Ck } =
Lastly, before returning to

⌦

2
Nk p ,k .
t

(A.7)

↵
pt,i , pt,j , we deﬁne for each event k,
hpt ik = hpt ik

hhpt ii ,

(A.8)

which is the deviation of the mean transverse momentum with respect to the mean over
many events.

154

A.2

Reconstruction of h pt,i, pt,j i

With this notational machinery in place, we can begin by expanding Eq. 4.19 with Eq.
4.18,

⌦

↵

Nevents

1

pt,i , pt,j =
Nevents

X

k=1

PNk PNk
i=1

pt,i

j=1,j6=i

Nk (Nk

hhpt ii

hhpt ii

pt,j

1)

.

(A.9)

Using Eq. A.8, we can replace the hhpt ii terms,

⌦

↵

pt,i , pt,j =

Nevents

1
Nevents

X

k=1

2 PN PN
k
k
pt,i
4 i=1 j=1,j6=i
Nk (Nk
·

hpt ik +

hpt ik

1)

pt,j

hpt ik +
...

pt,i

hpt ik

hpt ik

#

(A.10)
.

Multiplying and regrouping gives us,

⌦

↵
1
pt,i , pt,j =
Nevents

Nevents

X

k=1

2 PN PN
k
k
i=1
j=1,j6=i
4

...

pt,j

hpt ik

+ hpt i2
k ·· ·
···
Nk (Nk 1)
+ hpt ik pt,i + pt,j
...

(A.11)
2 hpt ik

#

.

This can be further simpliﬁed by noting three things: the ﬁrst term of the interior of the

155

double summation is the same as in Eq. A.5, the second term is unchanged by the summation
except for picking up a pre-factor of N (N

1), and the last term will go to zero after the

summation.
We can then simplify Eq. A.11 to,

⌦

↵

1

pt,i , pt,j =
Nevents

Using Eq. A.7, we can replace

⌦

↵

Nevents ✓

X

k=1

◆
{Ck }
2
+ ( hpt ik ) .
Nk (Nk 1)

(A.12)

{Ck }
,
Nk (Nk 1)

1

Nevents

X

pt,i , pt,j =
Nevents

k=1

2
pt ,k

Nk

1

+ ( hpt ik )2

!

.

(A.13)

Performing the average we get,

⌦

↵

D

pt,i , pt,j = ( hpt i)2

E

*

2
pt

N

1

+

.

(A.14)

D
E
hp i
Note that the ﬁrst term, ( hpt i)2 , is 2 t , the second moment of the hpt i distribution, or
2
equivalently hp i . We can also replace (N
t

1) with N because they will be almost identical

for large values.

⌦

↵

2
pt,i , pt,j = hp i
t

So the two particle correlation observable

⌦

* 2 +
pt

N

.

(A.15)

↵
pt,i , pt,j is equivalent to the second moment

of the hpt i distribution minus the variance in the underlying event pt distributions scaled by
156

the number of tracks.

157

BIBLIOGRAPHY

158

BIBLIOGRAPHY
[1] H. Wang, Study of Particle Ratio Fluctuations and Charge Balance Functions at RHIC.
PhD thesis, Michigan State University, East Lansing, Michigan, 2012.
[2] S. Bethke, “Experimental Tests of Asymptotic Freedom,” Progress in Particle and
Nuclear Physics, vol. 58, pp. 351–386, 2007.
[3] T. Ullrich, “Figures for the 2007 NSAC/LRP,” http://rhig.physics.yale.edu/
~ullrich/lrpfigs/.
[4] A. Bazavovi, T. Bhattacharya, M. Cheng et al., “Chiral and Deconﬁnement Aspects of
the QCD Transition,” Physical Review D, vol. 85, p. 054503, Mar 2012.
[5] C. Bernard, T. Burch, C. DeTar et al., “QCD Thermodynamics with Three Flavors of
Improved Staggered Quarks,” Physical Review D, vol. 71, p. 034504, Feb 2005.
[6] Y. Aoki, G. Endr˝di, Z. Fodor et al., “The Order of the Quantum Chromodynamics
o
Transition Predicted by the Standard Model of Particle Physics,” Nature, vol. 443,
pp. 675–678, Oct. 2006.
[7] M. Cheng, P. Hegde, C. Jung et al., “Baryon Number, Strangeness, and Electric Charge
Fluctuations in QCD at High Temperature,” Physical Review D, vol. 79, p. 074505, Apr
2009.
[8] A. M. Halasz, A. Jackson, R. Shrock et al., “On the Phase Diagram of QCD,” Physical
Review D, vol. D58, p. 096007, 1998.
[9] J. Berges and K. Rajagopal, “Color Superconductivity and Chiral Symmetry Restoration at Nonzero Baryon Density and Temperature,” Nuclear Physics B, vol. B538,
pp. 215–232, 1999.
[10] J. Cleymans, H. Oeschler, K. Redlich et al., “Comparison of Chemical Freeze-Out Criteria in Heavy-Ion Collisions,” Physical Review C, vol. 73, p. 034905, Mar 2006.

159

[11] F. R. Brown, F. P. Butler, H. Chen et al., “On the Existence of a Phase Transition for
QCD with Three Light Quarks,” Physical Review Letters, vol. 65, pp. 2491–2494, Nov
1990.
[12] A. M´csy, F. Sannino, and K. Tuominen, “Conﬁnement versus Chiral Symmetry,”
o
Physical Review Letters, vol. 92, p. 182302, May 2004.
[13] T. Andrews, “The Bakerian Lecture: On the Continuity of the Gaseous and Liquid
States of Matter,” Philosophical Transactions of the Royal Society of London, vol. A159,
Part II, pp. 575–590, 1869.
[14] M. R. Moldover, “Visual Observation of the Critical Temperature and Denisity: CO2
and C2 H4 ,” Institute for Basic Standards, National Bureau of Standards, May 1974.
[15] A. Einstein, “Theorie der Opaleszenz von Homogenen Fl¨ssigkeiten und
u
Fl¨ssigkeitsgemischen in der N¨he des Kritischen Zustandes,” Annalen der Physik,
u
a
vol. 338, no. 16, pp. 1275–1298, 1910.
[16] S. Shi, Event Anisotropy ⌫2 at STAR.
Wuhan, China, March 2010.

PhD thesis, Huazhong Normal University,

[17] G. Taubes, “Wit Busza Interview- Special Topic of Hadron Colliders,” Science Watch,
April 2010.
[18] “Arial
View
of
RHIC,”
GeneralMisc/arial.jpg.

http://www.star.bnl.gov/public/tpc/photos/

[19] D. B. Steski, J. Alessi, J. Benjamin et al., “Operation of the Relativistic Heavy Ion
Collider Au Ion Source,” Review of Scientiﬁc Instruments, vol. 73, no. 2, pp. 797–799,
2002.
[20] J. G. Alessi, D. Barton, E. Beebe et al., “The Brookhaven National Laboratory Electron
Beam Ion Source for RHIC,” Review of Scientiﬁc Instruments, vol. 81, no. 2, p. 02A509,
2010.
[21] J. Benjamin, C. Carlson, I. Feigenbaum et al., “Injecting RHIC from the Brookhaven
Tandem Van de Graa↵,” in Particle Accelerator Conference, 1999. Proceedings of the
1999, vol. 4, pp. 2277–2279 vol.4, 1999.

160

[22] H. Hahn, E. Forsyth, H. Foelsche et al., “The RHIC Design Overview,” Nuclear
Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers,
Detectors and Associated Equipment, vol. 499, no. 2-3, pp. 245 – 263, 2003. The Relativistic Heavy Ion Collider Project: RHIC and its Detectors.
[23] R. L. Brown, A. Etkin, K. J. Foley et al., “The STAR Detector Magnet Subsystem,” in
Particle Accelerator Conference, 1997. Proceedings of the 1997, vol. 3, pp. 3230–3232
vol.3, 1997.
[24] “STAR Detector Conﬁguration,” http://www.star.bnl.gov/public/tpc/photos/
GeneralMisc/STARdetector.jpg.
[25] M. Anderson, J. Berkovitz, W. Betts et al., “The STAR Time Projection Chamber: a
Unique Tool for Studying High Multiplicity Events at RHIC,” Nuclear Instruments and
Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment, vol. 499, no. 2-3, pp. 659 – 678, 2003. The Relativistic Heavy
Ion Collider Project: RHIC and its Detectors.
[26] STAR TOF Collaboration, “Proposal for a Large Area Time of Flight System for
STAR,” tech. rep., 2004.
[27] J. Zhou, “Construction of a New Detector, and Calibration Strategies, for Start Timing
in the STAR Experiment at RHIC,” Master’s thesis, Rice University, Houston, Texas,
October 2006.
[28] “BBC Layout,”
hexagon.pdf.

http://www.star.bnl.gov/public/trg/BBC/Schematics/bbc_

[29] A. Lebedev, “A Laser Calibration System for the STAR TPC,” Nuclear Instruments
and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors
and Associated Equipment, vol. 478, no. 1-2, pp. 163 – 165, 2002. Proceedings of the
ninth Int.Conf. on Instrumentation.
[30] J. Abele, J. Berkovitz, J. Boehm et al., “The Laser System for the STAR Time Projection Chamber,” Nuclear Instruments and Methods in Physics Research A, vol. 499,
pp. 692–702, Mar. 2003.
[31] H. Masui. Private communication.

161

[32] M. L. Miller, K. Reygers, S. J. Sanders et al., “Glauber Modeling in High-Energy Nuclear
Collisions,” Annual Review of Nuclear and Particle Science, vol. 57, no. 1, pp. 205–243,
2007.
[33] S. A. Bass, M. Belkacem, M. Bleicher et al., “Microscopic Models for Ultrarelativistic
Heavy Ion Collisions,” Progress in Particle and Nuclear Physics, vol. 41, pp. 255–369,
1998.
[34] M. Bleicher, E. Zabrodin, C. Spieles et al., “Relativistic Hadron-Hadron Collisions in
the Ultra-Relativistic Quantum Molecular Dynamics Model,” Journal of Physics G:
Nuclear and Particle Physics, vol. 25, no. 9, p. 1859, 1999.
[35] N. Davidson, H. Miller, D. Oertzen et al., “Thermal Gas Models and Particle Production
in Heavy Ion Collisions,” Zeitschrift fur Physik C Particles and Fields, vol. 56, no. 2,
pp. 319–324, 1992.
[36] J. Adams et al., “Incident Energy Dependence of pt Correlations at Relativistic Energies,” Physical Review C, vol. 72, p. 044902, Oct 2005.
[37] B. Mohanty et al., “STAR Experiment Results from the Beam Energy Scan Program at
RHIC,” Journal of Physics G: Nuclear and Particle Physics, vol. 38, no. 12, p. 124023,
2011.
[38] S. P. Horvat et al., “Measurement of Beam Energy Dependent Nuclear Modiﬁcation
Factors at STAR,” Journal of Physics: Conference Series, vol. 446, no. 1, p. 012017,
2013.
[39] L. Adamczyk et al., “System Size Dependence of Transverse Momentum Correlations
at RHIC,” Physical Review C, vol. C87, p. 064902, 2013.
[40] S. A. Voloshin, V. Koch, and H. G. Ritter, “Event-by-Event Fluctuations in Collective
Quantities,” Physical Review C, vol. 60, p. 024901, Jul 1999.
[41] M. Stephanov, “Thermal Fluctuations in the Interacting Pion Gas,” Physical Review
D, vol. 65, p. 096008, May 2002.
[42] M. A. Stephanov, “Non-Gaussian Fluctuations near the QCD Critical Point,” Physical
Review Letters, vol. 102, p. 032301, Jan 2009.

162

[43] M. A. Stephanov, “Sign of Kurtosis near the QCD Critical Point,” Physical Review
Letters, vol. 107, p. 052301, Jul 2011.
[44] M. Stephanov, K. Rajagopal, and E. Shuryak, “Signatures of the Tricritical Point in
QCD,” Physical Review Letters, vol. 81, pp. 4816–4819, Nov 1998.
[45] C. Athanasiou, K. Rajagopal, and M. Stephanov, “Using Higher Moments of Fluctuations and their Ratios in the Search for the QCD Critical Point,” Physical Review D,
vol. 82, p. 074008, Oct 2010.
[46] H. Heiselberg, “Event-by-Event Physics in Relativistic Heavy Ion Collisions,” Physics
Reports, vol. 351, pp. 161–194, 2001.
[47] Moriond, Event by Event Averages in Heavy Ion Collisions, 2002.
[48] M. Tannenbaum, “The Distribution Function of the Event-by-Event Average pt for
Statistically Independent Emission,” Physics Letters B, vol. 498, no. 1-2, pp. 29 – 34,
2001.
[49] D. Adamova et al., “Event by Event Fluctuations of the Mean Transverse Momentum
in 40, 80 and 158 AGeV/c Pb-Au Collisions,” Nuclear Physics A, vol. A727, pp. 97–119,
2003.
[50] S. Gavin, “Traces of Thermalization from pt Fluctuations in Nuclear Collisions,”
Physical Review Letters, vol. 92, p. 162301, Apr 2004.
[51] S. Heckel and the ALICE Collaboration, “Event-by-Event Mean pt Fluctuations in pp
and Pb-Pb Collisions Measured by the ALICE Experiment at the LHC,” Journal of
Physics G: Nuclear and Particle Physics, vol. 38, no. 12, p. 124095, 2011.
[52] X. Luo, “Error Estimation for Moment Analysis in Heavy-Ion Collision Experiment,”
Journal of Physics G: Nuclear and Particle Physics, vol. 39, no. 2, p. 025008, 2012.

163