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This is to certify that the
dissertation entitled
Using the balance function to search for late
hadronization in Au+Au collisions at a center

of mass energy of 130 GeV per nucleon pair

presented by

Marguerite Belt Tonjes

has been accepted towards fulfillment
of the requirements for

 

PhD degree in Phys ic s

 

 

'

Date ’ o) a?

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

USING THE BALANCE FUNCTION TO SEARCH FOR LATE
HADRONIZATION IN AU+AU COLLISIONS AT A CENTER
OF MASS ENERGY OF 130 GEV PER NUCLECN PAIR

By

Marguerite Belt Tonjes

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2002

ABSTRACT

USING THE BALANCE FUNCTION TO SEARCH FOR LATE HADRONIZATICN IN
Au+Au COLLISIONS AT A CENTER or MASS ENERGY OF 130 GEV PER
NUCLEON PAIR

By

Marguerite Belt Tonjes

Relativistic heavy ion physics is the study of nuclear matter interacting at high
energies and densities. The collisions of gold nuclei at the Relativistic Heavy Ion
Collider (RHIC) provide a source of high density matter for the study and creation
of a novel state of matter, the Quark-Gluon Plasma(QGP). The data set studied in
this dissertation is taken from Au+Au interactions at a center of mass energy of 130
GeV, measured in summer 2000. This is the first such data produced at RHIC. The
analysis presented here focuses upon the measurement of balance functions, which
are new observables in the field of heavy ion physics. The balance function for heavy
ion physics is introduced in Bass, Danielewicz, and Pratt, Phys. Rev. Lett. 85, 2689
(2000). The data were taken with the STAR (Solenoidal Tracker At RHIC) detector,
with analysis performed on charged particles in a pseudorapidity region of In] < 1.3.

The balance function measured for conserving charge/anti-charge pion pairs as a
function of rapidity is predicted to have a width which indicates the time of hadroniza-
tion of the measured particles. Charge/anti-charged particle pairs are created at the
same point in space time, and are correlated in rapidity. Pairs which are created early
have a wide separation in rapidity due to diffusion. However, pairs which are created
late have a narrow separation in rapidity. Balance functions with a broad width show
an early hadronization and are reflective of collisions which can be described as a
superposition of nucleon-nucleon collisions. Balance functions which have a narrow

width suggest late hadronization which iS indicative of the formation of a QGP. In

addition to late hadronization, flow can narrow the balance function width even more
than just the formation of QGP.

The balance function was measured for all charged particle pairs and pion pairs
as a function of pseudorapidity, with respect to four centralities of collisions ranging
from the most central to the peripheral. The balance function was also measured for
pion pairs as a function of rapidity. For these measurements, it is found that the cent-
ral events have a narrow balance function when compared to peripheral events, with
a smooth variation in the intervening centralities. The HIJING simulated nucleon-
nucleon interactions has a width consistent with that of the peripheral data balance
functions, when the simulated events are processed through a STAR detector simu—
lation. A Bjorken thermal model in the Simulated STAR detector gives a pion pair
balance function width which is wider than the central events (although narrower
than peripheral). However the addition of flow narrows the Bjorken model balance
function to that of the central data. The contribution of the acceptance of the det—
ector was studied with various mixed events, pseudorapidity cuts, and with a different
normalization method in the balance function calculation.

These measurements indicate that central events suggest late hadronization, which
is consistent with Bjorken model predictions with the inclusion of additional radial
flow. The balance function width of the peripheral collisions is consistent with model

predictions incorporating a superposition of nucleon-nucleon scattering.

Copyright by
MARGUERITE BELT TONJES
2002

To Dr. Charles Banks Belt, Jr.

ACKNOWLEDGMENTS

Thanks go to a number of people, most of all to Wayne Tonjes, PhD for support
and editing. Hugs to family and friends who put up with thesis—writing and data-
analysis related Silence and moods.

Professionally, I’d like to thank Dr. Michael Cherney for setting me on the path
to research relativistic heavy ion physics. Thanks to Professors Gary Westfall and
Scott Pratt for many enlightening discussions. Many thanks go to the Department of
Energy and the National Science Foundation for providing funding to allow this type
of exciting and innovative research to happen.

Although the work I did on the ElectroMagnetic Calorimeter did not make it into
this dissertation, there is a part of me in those optical fibers. I hope the Calorimeter
brings us interesting data for years to come.

Thanks to Apple computer for OSX, the iPod, and the G4. These were indis-

pensable for fast data analysis, and carrying my thesis text and music to keep me

happy.

King of Swamp Castle “Someday, all this will all be yours.”
Prince Herbert “What, the curtains?”

Monty Python and the Holy Grail

vi

CONT

ENTS

LIST OF TABLES

LIST OF FIGURES

ABBREVIATIONS

1 Introduction
1.1 Quark Gluon Plasma ...........................
1.2 Relativistic Heavy Ion Collider ......................
1.3 STAR ...................................

2 STAR in Year 2000
2.1 Magnet ..................................
2.2 The Time Projection Chamber ......................
2.3 The Trigger ................................
2.4 Centrality .................................
2.5 Event Cuts ................................
2.6 Track Cuts .................................
2.7 Particle Identification ...........................
2.8 Physical Characteristics of a Track ...................

3 The Balance Function

3.1 Definiti

OD .................................

3.2 Predictions ................................

4 Analysis Method

4.1 Software ..................................
4.1.1 STAR to Local Data Conversion .................
4.1.2 Local Data Analysis .......................

4.2 Data Quality ...............................
4.2.1 Logbook Information .......................
4.2.2 Amount of Data ..........................
4.2.3 Graphical Data Quality Check ..................
4.2.4 Programming Quality Assurance ................
4.2.5 Simulation Quality Assurance ..................

vii

xii

xxi

Abel-AH

6
6
8
12
13
14
16
18
21

23
23
27

30
30
32
33

36
37
41
41

5 Data Balance Function Measurements

5.1 Charged Particle Pairs ..........................
5.2 Charged Particle Pairs (No Electrons) ..................
5.3 Pion Pairs .................................
5.4 Other Features of the Balance Function .................
5.4.1 Integral ..............................
5.4.2 Particle Count ...........................
5.5 Kaon Pairs ................................
5.6 Systematic Error .............................
5.7 Balance Function Calculation Checks ..................
5.7.1 Bin Size Variation ........................
5.7.2 No Absolute Value ........................
5.8 Data Summary ..............................
Simulations
6.1 HIJING ..................................
6.1.1 GEANT ..............................
6.1.2 TRS ................................
6.1.3 Charged Particle Pair Balance Function ............
6.1.4 Pion Pairs .............................
6.1.5 Comparison With Data ......................
6.2 Fast Pseudorapidity Simulator ......................
6.2.1 Comparison With HIJING Through GEANT ..........
6.3 Bjorken Thermal Model .........................
6.3.1 Bjorken Model Parameters ....................
6.3.2 Varying Temperature .......................
6.3.3 Varying Time ...........................
6.3.4 Flow ................................
6.4 Simulations Summary ..........................

Understanding the Detector Acceptance in the Measurement
7.1 Mixed Events ...............................

7.1.1 The Traditional Method .....................
7.1.2 Mixing Charges ..........................
7.1.3 Mixing Pseudorapidity ......................
7.2 Pseudorapidity Cuts ...........................
7.2.1 Pseudorapidity Cuts Quantified .................
7.3 New Normalization ............................
7.3.1 Data ................................
7.3.2 Simulations ............................
7.3.3 Mixed Pseudorapidity ......................
7.4 Vertex Asymmetry ............................
7.4.1 Two Randomly Assigned Sub-Events ..............
7.5 Acceptance Summary ...........................

viii

103
106
112
115
118
122

127
129

8 Future of the Balance Function

8.1 Modifications of Pion and Charged Particle Measurements ......
8.2 Event-By-Event Balance Function ....................
8.3 Other Collision Measurements ......................

9 Conclusions

APPENDIX

A Tables: STAR 2000 Data Quality

LIST OF REFERENCES

ix

131
131
134
134

137

139

140

146

LIST OF TABLES

4.1

4.2

5.1

5.2

5.3

5.4

8.1

A.1

A.1

A.1

A2

A2

A2

Particle and absolute value of PID (particle identification) codes used
in balance function analysis. .......................
Amount of events used in analysis for each of the four centralities from
central trigger (top), and minimum bias trigger (bottom) ........

Table of the integral values for each of the balance function measure-
ments shown. ...............................
Table of the acceptance corrected integral values for each of the balance
function measurements shown .......................
Table of the number of positive and negative particles for each particle
type used in balance function measurements. Centralities: c=central,
mc=midcentral, mp=midperipheral, p=peripheral. Particle charged is
all charged particles and charged* refers to all charged particles, no
electrons. Ratios are calculated both for the average number of particles
per event which had two or more of those particles “( part per event)”,
and the average number of particles per event, covering all events used,
“( part per event-all)”. ..........................
Table of the ratio of the amount of particle pairs(charged/pion) used
in balance function analysis for each term (N +-, etc.) ..........

Charged particle balance function weighted average widths, and stat—
istical error bars for Single events from both minimum bias (top) and
central (bottom) triggers ..........................

STAR data runs from 2000 used in this analysis, continues on next two
pages. ...................................
continued. STAR data runs from 2000 used in this analysis, continued
from previous page .............................
continued. STAR data runs from 2000 used in this analysis, continued
from previous two pages ..........................
Suspect STAR data runs from 2000 with logbook notes, continues on
the next two pages .............................
continued. Suspect STAR data runs from 2000 with logbook notes,
continued from previous page. ......................
continued. Suspect STAR data runs from 2000 with logbook notes,
continues from the previous two pages. .................

32

37

58

59

60

135

140

141

142

143

144

145

A.3 HIJING files used in simulation analysis. These have been run through
GSTAR and TRS. ............................

xi

LIST OF FIGURES

1.1

1.2

1.3

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

2.9

2.10

3.1

3.2

Illustration of different phases of nuclear matter, the x—axis is the
Baryon density relative to normal nuclear matter (p/po), and on the

y-axis is temperature (in MeV). ..................... 2
View of Brookhaven lab from the air Showing the RHIC ring (largest
circle) [27] .................................. 4
Schematic of the RHIC ring, showing the Six interaction regions and
the placement of the four experiments [36] ................ 5

Illustration of the STAR detector, pointing out detectors installed for
the year 2000 summer collisions, excluding the RICH. ZDC stands for

Zero Degree Calorimeter. [28]. ...................... 7
Schematic of the STAR Time Projection Chamber [28] ......... 9
Side view of the inner and outer sector of the TPC pad plane showing

the wire geometry, with all measurements in mm [37] .......... 11
Schematic of a TPC sector with inner and outer pads [37]. ...... 12

Minimum bias year 2000 STAR events with four centrality bins labeled. 14
Pseudorapidity histogram for central events in the -62.5 cm vertex bin. 15

Pseudorapidity histogram for central events in the 0 cm vertex bin. . 15
Vertex distribution for year 2000 events used in the analysis, all four
centralities are shown. .......................... 16
dE/dx vs. momentum for negative charged particles identified as pri-

mary in the TPC for July data. The lines drawn through represent the
predicted dE/da: from the Bethe-Bloch equation for different particles
as labeled. The different shades represent density of tracks, with the
highest density being in the center of the pion region, and the lowest

on the edges of the proton region ..................... 19
The No,r values for all negative particles passing track cuts in minimum
bias run 1248015. The Gaussian fit marks the pion Signal ........ 20

A schematic of the collision during hadronization. Arrows to the left
and right represent the spreading of matter. Arrows emerging from
the dots represent particle pairs created together. The jagged lines
represent the interactions of those particles in the medium ....... 25
The Shape of the balance function plotted for a perfect detector. Plusses
are from the statistical measure, whereas squares with circles are from
known balancing pairs ........................... 27

xii

3.3

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

5.1

5.2

5.3

5.4

5.5

5.6

Balance functions for pions (top) and protons (bottom) calculated with
respect to Ay. Bjorken thermal model simulations of a QGP are in
circles and squares. PYTHIA p+p collisions (representing the hadron
gas model) are Shown in triangles [47]. .................

a) Charged particle pair and b) pion pair balance functions measured
for runs 1243059, 1245006, 1245012, and 1246009. Open triangles are
P00hi production library, and closed triangles are P00hm. ......
Multiplicity distribution for both triggers used for the data set analyzed
here. ....................................
All charged particles normalized momentum distribution for a) central
and b) peripheral events ..........................
All charged particles non-normalized vertex distribution for a) central
and b) midcentral events ..........................
All charged particles non-normalized vertex distribution for a) midper—
ipheral and b) peripheral events ......................
All charged particles normalized 77 distribution for a) central and b)
midcentral events ..............................
All charged particles normalized 7] distribution for a) midperipheral
and b) peripheral events ..........................
a) All charged particles normalized pseudorapidity distribution com-
pared to b) pion pseudorapidity distribution ...............
HIJ IN G-GEAN T normalized a) momentum distribution for all charged
particles and b) rapidity distribution for pions ..............
HIJING-GEANT a) non-normalized vertex distribution and b) nor-
malized 77 distribution, both for charged particles. ...........

a) Central (circles) and midperipheral (squares) charged particle bal-
ance function, b) midcentral (diamonds), and peripheral (triangles)
charged particle balance function .....................
a) Central (circles) and midperipheral (squares) charged particle bal-
ance function, b) midcentral (diamonds), and peripheral (triangles)
charged particle balance function. Charged particles do not include
identified electrons. ............................
Summary of widths of charged particle pair balance functions for the
four centrality bins of the data. ......... i ............
Summary of widths of charged particle pair balance functions for the

28

38

39

39

40

40

41

42

43

45

47

four centrality bins of the data, plotted with respect to impact parameter. 49

Charged particle balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events.
The first bin is removed from the Gaussian fit which is Shown by dotted
(central, midcentral), or solid (midperipheral, peripheral) lines. . . . .
Summary of gaussian fit widths of charged particle pair balance func-
tions for the four centrality bins of the data, plotted with respect to
impact parameter. ............................

xiii

50

5.7

5.8

5.9

5.10

5.11

5.12

5.13

5.14

5.15

5.16

5.17

5.18

5.19

Pion pair balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events.
This function is calculated with respect to Ay .............. 51
Pion pair balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events.
This function is calculated with respect to An .............. 51
Pion pair balance function from Jeon & Pratt [63]. The balance func-
tion from the Simple thermal Bjorken model (line) has been parameter—
ized and filtered to roughly provide rough consistency with preliminary
STAR measurements. The inclusion of HBT effects (triangles) gives a.
dip at small Ay, while the extra addition of Coulomb interactions (cir-
cles) modifies the dip ............................ 53
Summary of widths of the pion pair balance functions for the four
centrality bins of the data, plotted with respect to impact parameter.
The first two bins were removed from the data for the width calculation. 54
Summary of widths of the pion pair balance functions (A77) for the four
centrality bins of the data, plotted with respect to impact parameter.
The first two bins were removed from the data for the width calculation. 54
Pion pair balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events.
This function is calculated with respect to Ag, and the first two
bins are removed from the Gaussian fit (shown by dotted (central,
midcentral), or solid (midperipheral, peripheral) lines.) ........ 55
Summary of gaussian fit widths of pion pair balance functions for the
four centrality bins of the data, plotted with respect to impact param-
eter. The first two bins were removed from the data for the fit ..... 55
Plot of the values of D calculated from the balance function acceptance
corrected integral for charged particle pairs. Plot a) is on a scale Show-
ing the QGP and hadronic gas predictions of reference [64], and b) is
on a focused scale. ............................ 58
Kaon pair balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events.
This function is calculated with respect to Ay.) ............ 61
Kaon pair balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events.
This function is calculated with respect to An.) ............ 62
A summary of the balance function widths for charged particle pairs,
comparing both with(errors indicated by ovals) and without(vertical
error bars) a 5% systematic error in the N+_-type terms ........ 64
A summary of the balance function widths for pion pairs, comparing
both with(errors indicated by ovals) and without(vertical error bars) a
5% systematic error in the N+_-type terms. .............. 65
A summary of the balance function widths for charged particle pairs,
comparing both with(errors indicated by ovals) and without(vertical
error bars) a 10% systematic error in the N+_—type terms. ...... 66

xiv

5.20

5.21

5.22

5.23

5.24

5.25

5.26

5.27

5.28

5.29

6.1

6.2

6.3

6.4

A summary of the balance function widths for pion pairs, comparing
both with(errors indicated by ovals) and without(vertica1 error bars) a
10% systematic error in the N+_-type terms. .............
Charged particle pair balance function plotted with an estimated 5%
systematic error on An in addition to the statistical error. ......
Charged particle pair balance function widths plotted with (ovals) and

without (vertical error bars) an additional 5% systematic error on An.

Pion pair balance function widths plotted with (ovals) and without
(vertical error bars) an additional 5% systematic error on An.

Charged particle pair balance function for central (circles), and periph-
eral (triangles) events of the P00hm data set. Measurements are shown
for both a) 20 bins and b) 50 bins in the An region of 0 to 2 ......
Charged particle pair balance function widths for the P00hm data set.
Widths calculated with all bins are shown for both a) 26 bins and b)
65 bins in the A17 region of 0 to 2.6 ....................
Pion pair balance function for central (circles), and peripheral (triang-
les) events of the P00hm data set. Measurements are shown for both
a) 20 bins and b) 50 bins in the Ag region of 0 to 2 ...........
Pion pair balance function widths for the P00hm data set. Widths
calculated with all bins are shown for both a) 26 bins and b) 65 bins
in the Ag region of 0 to 2.6 ........................
Charged particle pair balance function for central (circles), and periph—
eral (triangles) events of the complete data set. Measurements are
shown both a) with all bins and b) omitting two middle bins with
the dashed line showing the central Gaussian fit, and the solid line
showing the peripheral Gaussian fit ....................
Pion pair balance function for central (circles), and peripheral (triang-
les) events of the complete data set. Measurements are shown both
a) with all bins and b) omitting four middle bins with the dashed
line showing the central Gaussian fit, and the solid line Showing the
peripheral Gaussian fit ...........................

Pion pair balance function for HIJING Au+Au (plusses) and p+p
(squares) events. This balance function is calculated for a perfect det-
ector .....................................
a) HIJING-GEAN T central (circles) and midperipheral (squares)
charged particle balance function, b) midcentral (diamonds), and
peripheral (triangles) charged particle balance function .........
a) HIJING-GEANT central (circles) and midperipheral (squares)
charged particle balance function, b) midcentral (diamonds), and
peripheral (triangles) charged particle balance function. Gaussian fits
are shown excluding the first bin from the fit. .............
Summary of widths of charged particle pair balance functions for the
four centrality bins of HIJING-GEANT, plotted with respect to impact
parameter ..................................

XV

67

67

68

70

70

71

71

72

73

78

78

79

79

6.5

6.6

6.7

6.8

6.9

6.10

6.11

6.12

6.13

6.14

6.15

6.16

6.17

6.18

Summary of Gaussian fit widths of charged particle balance functions
for the four centrality bins of HIJING-GEANT, plotted with respect
to impact parameter. Gaussian fits omit the first bin. .........
a) HIJ ING-GEAN T all centralities combined charged particle balance
function. b) The same with a Gaussian fit excluding the first bin. .

a) HIJING-GEAN T central (circles) and midperipheral (squares) pion
pair balance function, b) midcentral (diamonds), and peripheral (tri-
angles) pion pair balance function. ...................
a) BL] IN G-GEANT central (circles) and midperipheral (squares)
charged particle balance function, b) midcentral (diamonds), and
peripheral (triangles) charged particle balance function. Gaussian fits
are Shown excluding the first two bins from the fit. ..........
Summary of widths of pion pair balance functions for the four centrality
bins of HIJING-GEAN T, plotted with respect to impact parameter. .
Summary of Gaussian fit widths of pion pair balance functions for the
four centrality bins of HIJ IN G-GEAN T, plotted with respect to impact
parameter. Gaussian fits omit the first two bins. ............
a) HIJING-GEANT all centralities combined pion pair balance func-
tion. b) The same with a Gaussian fit excluding the first two bins. . .
a) HIJING-GEANT all centralities combined pion pair (A17) balance

function. b) The same with a Gaussian fit excluding the first two bins.

Charged particle pair balance function for a) central(circles) and
peripheral(triangles) data, as well as b) HIJING-GEANT (plusses)
simulated events. .............................
Summary of widths of the charged particle balance functions for the
four centrality bins of data (circles), with HIJ ING-GEAN T represented
by the shaded band. ...........................
Summary of Gaussian fit widths of the charged particle balance
functions for the four centrality bins of data (circles), with HIJING-
GEANT represented by the shaded band. Gaussian fits are done
omitting the first bin ............................
Summary of widths of the pion pair balance functions for the four
centrality bins of data (circles), with HIJING-GEANT represented by
the shaded band. Weighted average widths are calculated excluding
the first two bins ..............................
Summary of Gaussian fit widths of the pion pair balance functions
for the four centrality bins of data (circles), with HIJING-GEANT
represented by the shaded band. Gaussian fits are done omitting the
first two bins. ...............................
Summary of widths of the pion pair (A77) balance functions for the four
centrality bins of data (circles), with HIJING-GEANT represented by
the shaded band. a) Weighted average widths are calculated excluding
the first two bins. b) Gaussian fit widths are Shown, with the first two
bins omitted from the fit ..........................

xvi

80

81

82

82

83

83

84

84

85

86

87

88

88

89

6.19

6.20

6.21

6.22

6.23

6.24

6.25

6.26

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

7.10

Charged particle pair balance function plotted for HIJING-GEANT
(plusses), and HIJING events with the fast n-dependent TPC simula-
tor(squares) ................................. 90
Pseudorapidity histogram of Bjorken thermal model simulations a)
with a perfect detector before and b) after the fast 7} TPC simulator. 92
a) Pion pair balance function (Ag) for Bjorken (x’s) and HIJING
(plusses) Simulations in a perfect detector. b) is the same with Gaussian
fits ...................................... 93
Charged particle pair balance function plotted for Bjorken Simulated
pion pair events with a perfect detector and physical cuts. Open tri-
angles represent 25 particles per event and closed triangles are 100
particles per event. ............................ 93
Pion pair balance function widths plotted for Bjorken events processed
through the fast 17 simulator with varying temperatures. The central
data width (circles) calculated without first bin is shown for comparison. 95
Pion pair balance function widths plotted for Bjorken Simulated for a
perfect detector with varying initial times. The widths are calculated
with all bins of the balance function. .................. 96
Pion pair balance function widths plotted for Bjorken events processed
through the fast 77 Simulator with varying initial times. The central data
width (shaded bar) calculated without first bin is Shown for comparison. 96
Pion pair Bjorken thermal model balance function calculated by Pratt

with (line), and without (box and line) flow [70] ............. 97
Charged particle balance function. Central data are dots, traditional
mixed events are represented by x’s .................... 101
Pion pair balance function. Central data are dots, traditional mixed
events are represented by x’s. ...................... 101

Pion pair mixed event balance function subtracted from the pion pair
balance function. Central data are dots, peripheral events are triangles. 102
Pion pair mixed event balance function subtracted from the pion pair

balance function. Widths of data are plotted with circles ........ 102
Charged particle balance function. Central data are dots, mixed charge
events are represented by squares with slashes .............. 103
Charged particle balance function. Central data are dots, shuffled 77
events are represented by squares with x’s. ............... 104
Charged particle balance function widths. Unmixed data are dots,
shuffled 77 events are represented by squares with x’s. ......... 105
Charged particle balance function. HIJING-GEAN T events are plusses,
shuffled 77 events are represented by squares with +’s .......... 105
Charged particle balance function for HIJING events analyzed with a
perfect detector, with shuffled 17 ...................... 106
Charged particle balance function for central (circles) and midperiph-
eral(squares) data, analyzed with particles that have In] < 0.5 ..... 107

xvii

7.11

7.12

7.13

7.14

7.15

7.16

7.17

7.18

7.19

7.20

7.21

7.22

7.23

7.24

7.25

Pion pair balance function for central (circles) and midperiph-
eral(squares) data, analyzed with pions that have In] < 0.5 .......
Charged particle balance function widths. Data (circles), and HIJING-
GEANT (shaded bar) events are analyzed for particles with a) In] <
0.25, and b) In] < 0.5. ..........................
Charged particle balance function widths. Data (circles), and HIJING—
GEANT (shaded bar) events are analyzed for particles with a) In] <
0.75, and b) In] < 1.0. ..........................
Charged particle balance function widths. Data (circles), and HIJING—

GEANT (Shaded bar) events are analyzed for particles with In] < 1.25.

Pion pair balance function widths. Data (circles), and HIJING-
GEANT (shaded bar) events are analyzed for particles with a)
In] < 0.25, and b) In] < 0.5. .......................
Pion pair balance function widths. Data (circles), and HIJING-
GEANT (Shaded bar) events are analyzed for particles with a)
In] < 0.75, and b) In] < 1.0. .......................
Pion pair balance function widths. Data (circles), and HIJING-

GEANT (Shaded bar) events are analyzed for particles with In] < 1.25.

Charged particle balance function widths with a linear fit over cen-
trality. Data (circles), and HIJING-GEANT (shaded bar) events are
analyzed for particles with a) In] < 0.25, and b) In] < 0.5 ........
Charged particle balance function widths with a linear fit over cen-
trality. Data (circles), and HIJING-GEANT (shaded bar) events are
analyzed for particles with a) In] < 0.75, and b) In] < 1.0 ........
Charged particle balance function widths with a linear fit over cen—
trality. Data (circles), and HIJING—GEANT (Shaded bar) events are
analyzed for particles with In] < 1.25 ...................
Pion pair balance function widths with a linear fit over centrality. Data
(circles), and HIJING-GEANT (Shaded bar) events are analyzed for
particles with a) In] < 0.25, and b) In] < 0.5 ...............
Pion pair balance function widths with a linear fit over centrality. Data
(circles), and HIJING-GEANT (shaded bar) events are analyzed for
particles with a) In] < 0.75, and b) In] < 1.0 ...............
Pion pair function widths with a linear fit over centrality. Data (circles),
and HIJING-GEANT (shaded bar) events are analyzed for particles
with In] < 1.25. ..............................
Slopes from the linear fits of balance function widths for the various
pseudorapidity cuts on the data. Charged particle pairs are closed cir-
cles, and pion pairs are open circles ....................
Charged particle pair balance function for central (circles) and periph—
eral (open triangles) events, analyzed with the new, n-dependent nor—
malization. Gaussian fits are applied omitting the first bin. ......

7.26 Charged particle pair balance function Gaussian fit widths for data

(circles), analyzed with the new, n-dependent normalization. .....

xviii

108

109

109

110

110

111

111

112

112

113

114

114

115

116

118

119

7.27

7.28

7.29

7.30

7.31

7.32

7.33

7.34

7.35

7.36

7.37

7.38

7.39

Pion pair balance function for central (circles) and peripheral (open
triangles) events, analyzed with the new, n-dependent normalization.
Gaussian fits are applied omitting the first two bins ...........
Pion particle pair balance function Gaussian fit widths for data (cir-
cles), analyzed with the new, n-dependent normalization. Gaussian fits
omit the first two bins ...........................
Pion pair balance function (An) for central (circles) and peripheral
(open triangles) events, analyzed with the new, n-dependent normaliz—
ation. Gaussian fits are applied omitting the first two bins. ......
Pion particle pair balance function (An) Gaussian fit widths for data
(circles), analyzed with the new, n-dependent normalization. Gaussian
fits omit the first two bins .........................
Charged particle pair balance function Gaussian fit widths for data
(circles), analyzed with the new, n-dependent normalization. Width
error bars have an additional 5% systematic error ............
a) Charged particle pair balance function for HIJING-GEANT sim-
ulated events, analyzed with the new, n-dependent normalization. b)
Charged particle pair balance function for HIJING p+p events in a
perfect detector, analyzed with the old normalization ..........
Charged particle particle pair balance function Gaussian fit widths for
data (circles), analyzed with the new, n—dependent normalization. The
top shaded bar shows old normalization HIJING p+p events Gaussian
width. The lower shaded bar shows the Gaussian width for HIJING-
GEANT events analyzed with the new normalization ..........
Charged particle pair balance function, analyzed with the new
n-dependent normalization. Central data is represented by closed
circles, and mixed n central data is represented by squares with X’s.

Charged particle balance function for a) central positive (open cir—
cles) and negative (closed circles), and b) midcentral positive (open
diamonds) and negative (closed diamonds) vertex events ........
Charged particle balance function for a) midperipheral positive (open
squares) and negative (closed squares), and b) peripheral positive (open
triangles) and negative (closed triangles) vertex events. ........
Summary of charged particle balance function widths for data with
positive (open circles), and negative (closed circles) vertices in the z
direction. .................................
Summary of pion balance function widths for data with positive (open
circles), and negative (closed circles) vertices in the z direction. . . .
a) Charged particle balance function for simulated HIJING-GEANT
positive (open inverted triangles) and negative (closed inverted triang-
les) vertex events. b) Pion pair balance function for HIJING-GEANT
positive (open inverted triangles) and negative (closed inverted triang-
les) vertex events ..............................

xix

120

120

121

121

122

123

124

125

125

126

126

127

128

7.40

7.41

Charged particle balance function for a) central lst (open circles) and
2nd (closed circles), and b) midcentral lst (open diamonds) and 2nd
(closed diamonds) half of the data set. .................
Charged particle balance function for a) midperipheral lst (open
squares) and 2nd (closed squares), and b) peripheral lst (open
triangles) and 2nd (closed triangles) half of the data set. .......
Summary of charged particle balance function widths for data with the
lst (open circles), and 2nd (closed circles) half of the data set. . . . .
Summary of pion pair balance function widths for data with the lst
(open circles), and 2nd (closed circles) half of the data set .......

Charged particle balance function for one event from a central trigger
data set. Charged particles do not include identified electrons. Note
the scale ...................................

XX

129

130

135

Abbreviations &: Definitions

ADC Analog to Digital Converter

AGS Alternating Gradient Synchrotron

BNL Brookhaven National Laboratory

BRAHMS Broad RAnge Hadron Magnetic Spectrometers experiment at (RHIC)
CERN European Organization for Nuclear Research

charged particle pairs For this dissertation, charged particle pairs refers to mea—

surement without identified electrons unless otherwise stated.
CMOS Complementary Metal Oxide Semiconductor, integrated circuit
CTB Central Trigger Barrel, a detector in STAR
D variable used to measure charge fluctuations
DAQ Data AQuisition
dE/da: Energy loss per length. Usually keV/cm, used for PID in the STAR TPC
DST Data Storage Tape (can be real tape or a data storage file)
ILDST Size and information reduced data storage file

EMC ElectroMagnetic Calorimeter, a detector in STAR

xxi

eta The pseudorapidity, n, defined as n = - ln tan 6/2, where p, = Ip] cos 0. For a

maSSless particle, pseudorapidity % rapidity.

ExB Electric field cross Magnetic field, in this case for corrections of interactions in

the STAR TPC
FTPC Forward TPC, detector in STAR

GEANT A system of detector description and Simulation tools to help physicists in

high energy experiment design and studies
GSTAR framework to run STAR detector Simulations using GEANT
HBT Hanbury-Brown-Twiss, a method of interferometry to determine source size
HIJING Heavy Ion Jet INteraction Generator

HIJING-GEANT in this dissertation, refers to HIJING simulated events which
have been processed through GSTAR and TRS

L3 Level 3 trigger used in STAR (found in logbook notes, Appendix Table A2)
minbias Minimum bias trigger (from logbook notes, Appendix Table A2)

P00hi production library version of STAR 2000 data

P00hm production library version of STAR 2000 data (contains ExB corrections)
P10 Gas of 90% Argon, 10% methane, in the STAR TPC

PHENIX Pioneering High ENergy Ion eXperiment (at RHIC)

PHOBOS Experiment at RHIC

PID Particle IDentification

pQCD Perturbative QCD

xxii

Primvtx % of events with primary vertex (from logbook notes, Appendix Table A2)
PYTHIA Program for the generation of high-energy physics events
QCD Quantum ChromoDynamics, model to describe the strong nuclear interaction

QGP Quark-Gluon Plasma

rapidity Defined as y = -11“ E+pz

.2 E_p , also y = — tanh“l flz

 

RICH Ring Imaging Cherenkov Hodoscope (in STAR)

RHIC Relativistic Heavy Ion Collider

ROOT An Object—Oriented Data Analysis Framework

RQMD Relativistic Quantum Chromo—Dynamics (heavy ion interaction simulator)
SPS Super Proton Synchrotron (at CERN)

STAR Solenoidal Tracker At RHIC, experiment

SVT Silicon Vertex Tracker (in STAR)

TPC Time Projection Chamber (in STAR)

TRS TPC Response Simulator (STAR software)

UNIX Operating system

ZDC Zero Degree Calorimeter (detector common to RHIC experiments)

xxiii

Chapter 1

Introduction

1.1 Quark Gluon Plasma

It is thought that with sufficiently high energy density and temperature, a new phase
of matter could be formed known as the quark-gluon plasma (QGP) [1]. During
formation of this QGP, the quarks and gluons are liberated within this plasma. QGP
is theorized to have existed in the early universe, before a phase transition from QGP
to hadronic matter occurred at ~ 10 us after the Big Bang. The nuclear phase diagram
is shown in Figure 1.1 [2]. The nuclear liquid-gas phase transition has been studied in
nuclear physics, however the transition to QGP is just beginning to be studied and
understood in the field of relativistic heavy ion physics.

The search for QGP is not only important in the formation of new matter, but
also to prove or disprove theories of fundamental particle interaction. The theory of
interactions of particles with the strong force is known as Quantum ChromoDynamics
(QCD) [3]. QCD is a complex and difficult theory which will not be described here,
as more details can be found in reference [3]. QCD predictions are usually made in
the perturbative regime, where the terms are calculable. The ground state of QCD,
the vacuum, is still not understood [4], and by producing and studying QGP, more

can be learned about the vacuum.

Early

 

 

 

 

UnIverse
1+ RHIC
250 t I
Quark-Gluon Plasma
(Ill-1., d3! g, 5.5., 05)
A200
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‘2'“ 150
E
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52
50

 

 

 

Baryon Density

Figure 1.1: Illustration of different phases of nuclear matter, the :r—axis is the Baryon
density relative to normal nuclear matter (p/po), and on the y-axis is temperature (in

MeV).

In 2000, physicists at the CERN(European Organization for Nuclear Research)
SPS(Super Proton Synchrotron) announced that they had observed evidence for the
formation of QGP [5] - [25]. Combining the measurements of several different SPS ex-
periments they claim that there is evidence for a new state of matter in which quarks
and gluons are deconfined. This evidence includes J / \II suppression, strangeness en—
hancement, as well as other observables. CERN studied lead ions colliding on lead at
beam energies of 40 to 160 GeV per nucleon.

In a summary of QGP theory and measurements, S. Bass (reference [26]) states
that “the SPS experiments have created a new state of high energy-density and tem-
perature matter”, and “the concept of a QGP needs to be rethought”. It seems that
both the SPS results and the early results from the RHIC collider (RHIC is discussed
in Section 1.2) have given the theory of QGP a challenge with a set of observations

to explain.

1.2 Relativistic Heavy Ion Collider

The Relativistic Heavy Ion Collider was built to search for the QGP and to study
interactions of heavy nuclei and polarized protons at high energies. RHIC was built
at Brookhaven National Laboratory in Long Island, New York, USA. An aerial photo
of the collider is shown in Figure 1.2.

The two 2.4 mile (3.86 km) circumference beamlines cross at six interaction points.
At four of the interaction regions are experiments which are designed to detect the
particles resulting from the collisions at the interaction points. The two physically
larger experiments are STAR and PHENIX. The two smaller experiments are PHO-
BOS and BRAHMS. Each is located as Shown in Figure 1.3. RHIC is also designed
to provide interactions of polarized protons to study the fundamental nature of the
spin content of protons, as well as to create p+p reference collisions for the heavy ion

program. This dissertation will focus upon Au+Au collisions detected in the STAR

 

Figure 1.2: View of Brookhaven lab from the air showing the RHIC ring (largest
circle) [27].

experiment.

1.3 STAR

STAR is the Solenoidal Tracker At RHIC [28]. It is a large detector with multiple com-
ponents designed to search for signatures of QGP and study the behavior of matter
which interacts at a high energy density. STAR is designed to measure many interest-
ing observables simultaneously, both over many events and on an event—by-event basis.
The design of STAR will allow for measurement of variables which indicate entropy,
temperature, and strangeness chemical potential for collisions. Particle fluctuations
and collective motion of particles and energy (flow) can also be measured. STAR
is designed to measure high transverse momentum processes above 2 GeV/c as well.
The features of STAR are described in more detail in Chapter 2. STAR measurements

performed at RHIC at the time of this writing are published in references [29]— [35].

PHOBOS BBAHMS‘;

 

Figure 1.3: Schematic of the RHIC ring, showing the six interaction regions and the
placement of the four experiments [36].

Chapter 2

STAR in Year 2000

The data used in this analysis was taken in the summer of 2000, during which RHIC
collided Au+Au ions at a center of mass energy of SNN=130 GeV. The collisions
were below the collider’s design energy, but still gave the opportunity to measure
heavy ion collisions at a higher interaction density and energy combination than pre-
viously studied. The main detector used for particle tracking in the summer 2000 data
taking run was the Time Projection Chamber(TPC). The Ring Imaging Cherenkov
Hodoscope (RICH) detector, and a section of the Silicon Vertex Tracker (SVT) were
installed in STAR at the time, but the data analyzed in this dissertation is from the
TPC only. Trigger detectors installed and used were the Zero Degree Calorimeters
(ZDCS), and the Central Trigger Barrel (CTB). Also used was a solenoidal magnet.

An illustration of the STAR setup for year 2000 is shown in Figure 2.1.

2.1 Magnet

The STAR solenoidal magnet is essential for the identification and tracking of charged
particles. The magnet is designed to give a uniform field parallel to the direction of

the beam. The magnet structure also contains the main subdetectors of STAR, the

TPC, CTB, RICH, and eventually the ElectroMagnetic Calorimeter (EMC). The

ElectroMagnetic Calorimeter Silicon Vertex ZDC
Magnet

  
 
 

Electronics
Platforms

ZDC

Figure 2.1: Illustration of the STAR detector, pointing out detectors installed for

the year 2000 summer collisions, excluding the RICH. ZDC stands for Zero Degree
Calorimeter. [28].

EMC is still being installed in stages for use in 2001 and beyond. The magnet can
be operated at a field strength up to 0.5 Tesla, however for the year 2000 run, the
magnet was operated at B3 = 0.25 T. The magnetic field was mapped before the
installation of the TPC, with a precision of 1-2 Gauss for all dimensional components
of the field [37]. The uniform field along the beamline of the magnet eases the tracking

pattern recognition in analysis, as a simple helix model can be used for particle tracks.

2.2 The Time Projection Chamber

The main sub-detector in STAR is the TPC. The TPC is designed to measure many
of the basic physics observables in RHIC’s heavy ion collisions. This detector covers
a total pseudorapidity range of —2.0 < n < 2.0, as well as a full 27r of the azimuthal
direction. The pseudorapidity is defined as n = —ln tan(6/2), where 0 is the angle
radially from the beamline direction, 2. With a magnetic field of 0.25 T, particles
can be measured with pt _>_ 50 MeV/c. The TPC covers a tracking volume with a
length of 4.2 meters, an inner radius of 0.5 meters, and extending to 2 meters outer
radius. The TPC uses time slices of Signals to achieve a three dimensional image of
the charged particle tracks which pass through the detector. An illustration of the
TPC is shown in Figure 2.2 [28].

The TPC contains a gas which is a mixture of 90% Ar and 10% methane (P10).
This gas was chosen due to several properties, including negligible attenuation of
the drifting electrons, a reasonable drift velocity, a high efficiency for dE/das, and
operation at atmospheric pressure. The two endcaps of the TPC each contain 12
sectors, each sector having an inner and outer radius module. Each of these sector
modules has wire planes in front of a cathode plane, creating a pad segmentation. The
pads are arranged in concentric rows to maximize the detection of high transverse
momentum tracks. The central membrane, which can be seen in Figure 2.2, is a

cathode operated at high voltage at the center of the TPC. The inner and outer field

$1,... Sectors
, ‘-""‘/‘

 
  
 
     

Outer Field Cage /
& Support Tub?"’:f,:.~<if.. , Inner
-' ”Field
Cage

 
 
 

I "“51; _\§ ‘ i. W I ‘ ‘ '
I“ ‘ I?» .
\. High Voltage I“
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‘9' ~ , Iv"...- " 4200
SectoL .-""'“i”“ -I_ ., .. mm

Support-Wheeliilfl

I

Figure 2.2: Schematic of the STAR Time Projection Chamber [28].

cages constrain the electric potential as well as providing containment for the P10
gas. The inner radius of the TPC is at 0.5 m, designed with the expectation that in
full luminosity RHIC Au+Au events, the track density would be too high to resolve
individual tracks at smaller radii.

A charged particle which enters the TPC will ionize the P10 gas. The electrons
formed along the particle’s track will then drift towards an endcap due to the electric
potential created between the central membrane and the endcaps. The drift velocity
of the electrons in the TPC was measured to be 5.44 :1: 0.01 cm/ps with the variation
of drift velocity being over a number of days. For the summer 2000 data run, the
position resolution in the z direction was s 500 pm. Figure 2.2 shows the direction
convention used in STAR, that is z is along the beamline, and :2: and y are on the
plane radially outward from the beamline. The field strength near the anode wires in
front of the cathode planes is similar to that of a cylindrical capacitor. Thus, electrons
that have drifted to the pad plane would gain kinetic energy from this electric field,

which ionizes the gas near the wire. This creates an electron avalanche that is detected

on the anode wire. The anode’s pulse is amplified with custom CMOS circuits [37].
Figure 2.3 shows a cross section of the pad wire geometry. There is a third layer of
wires, the gating grid, before the cathode (or shield plane) and anode. The gating
grid wires remove field distortions from the extra ions formed around the anode wires,
yet allows the signal electrons through to be detected. The gating grid may also be
controlled to clean the pad plane area between triggers. To maintain a signal to noise
ratio of 20:1 for the pads including both the small outer sector and large inner sector
pads, the anode voltages were set to a gain of 1100 (3000) in the outer (inner) sector.

The longest drift time in the TPC is 40 us. The electrons reaching the pads are read
in by 512 channels, reading each time slice at 100 ns per channel. The position of the
particle which created the detected electron signal (the hit) is converted from time to
position given the known drift velocity. The position resolution in the radial direction
depends on the size of the pad planes which detected the particle. A schematic of the
pad plane layout is given in Figure 2.4. The outer sector has 3,940 pads, with each
pad having a radial size of 19.5 mm and an azimuthal size of 6.2 mm. The inner sector
contains 1,750 pads, with a pad size of 11.5 mm radially and 2.85 mm azimuthally.
The outer radius of the tracking volume constrains the pseudorapidity coverage of
the TPC to In] < 1, and the inner radius covers In] < 2. The useful tracking volume
extends out to about In] < 1.5, because a track requires at least 10 pad row hits to
be reconstructed as a good track. More about tracking will be covered in Section 2.6.

The TPC is designed to be effective at measuring charged particle tracks at both
high and low multiplicities for RHIC collisions. The TPC has worked well in both
calibration and data taking runs as well as laser calibration tests. For more details

on the STAR TPC, see references [28], [38], [39] and [40].

10

 

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wire geometry, with all measurements in mm [37].

3940 Outer Pads
6.2 mm x 195 mm STAR TPC PAD DIAGRAM

1750 Inner Pads
2.85 mm x 11.5 mm

1" Row Inner Sector
l'_ 60 cm from Center —

   

143 1 6.719581 mm

 

 

8.7 mm Cross Spacing

"-(7 x 48) + (5 x 52) = 596 mm

20 mm RADIAL SPCACING 13! Row Outer Sector
I‘— 3‘ ‘m'mm 1.27 m from Center

 

  

 

 

 

 

Figure 2.4: Schematic of a TPC sector with inner and outer pads [37].
2.3 The Trigger

The two trigger systems used in summer 2000 were the Zero Degree Calorimeters
(ZDCS) [41], and the Central Trigger Barrel (CTB). The ZDCs are hadronic calorime-
ters located :E 18.25 In from the center of STAR along the beamline. All four of the
RHIC experiments have similar systems. The CTB is a cylinder of scintillator slats sur-
rounding the TPC covering a pseudorapidity region of MI < 1.0. The CTB measures
charged particle multiplicity with a 27r azimuthal coverage for every RHIC crossing.
These detectors were designed to operate at a luminosity of 2 x 1026 cm’2 s"1 for
Au+Au collisions at W = 200 GeV, as well as for polarized p+p collisions at
500 GeV with a luminosity of 2 x 10:"2 cm‘2 s“. RHIC is designed to operate with
117 bunches in each of the two rings, with a time between bunch crossings of '2 105
ns [37]. The STAR TPC can record data at a rate of 4 events/second which as the
slowest detector, limits the amount of data taken. To choose which collisions contain

useful physics information, the ZDCs are used to measure the spectator neutrons

12

emerging from a collision. If there are signals in coincidence above threshold in the
ZDCS, then a minimum bias event has occurred. The data set with this trigger is
labeled as minimum bias for any sort of collision event of gold nuclei, whether the
gold ions hit each other peripherally or hit head-on (centrally). Using the CTB, the
efficiency of the coincidence ZDC trigger was found to be above 99% for the whole
range of multiplicities used in this dissertation [37]. A trigger for interesting central
events uses the ZDC signal together with a CTB signal above threshold. The central
trigger events use the highest 15% CTB signals. Central events are those in which
the ions have collided head-0n. In a central collision, a large number of particles are

produced, many of them transverse to the beam direction. More about STAR triggers

can be found in [28] and [42].

2.4 Centrality

There are a number of different types of collisions, from those that are the most
peripheral where the nuclei glance off of each other, to central collisions where the
nuclei hit head-on. To compare these different classes of collisions, an observable is
necessary to define centrality. In heavy ion collisions, the number of particles meas-
ured gives a good estimate of the centrality of the collision, and thus, the impact
parameter. In Figure 2.5, the centrality definitions used in this analysis are shown.
The number of tracks are primary tracks measured in the complete TPC for the min-
imum bias triggered data. Primary tracks are those which reconstruct back towards
the primary collision vertex. The integral of the normalized curve to a specific bin
gives the percentage of centrality. The top 10% of the integrated histogram represents
the most central events, referred to hereafter as the central data set. The next 30%
of centrality are the midcentral events. The next 30% are the midperipheral events,
leaving the last 26% as peripheral events. The most peripheral events, those with

5 tracks or less, were omitted as there are indications that these were non-hadronic

13

 

104 .................
.

.

fl

1

Peripheral 92/33 Mid-Central Central
26 A: eral 30°/o top 10%

 

 

 

 

 

 

 

Figure 2.5: Minimum bias year 2000 STAR events with four centrality bins labeled.

events [43]. The events used had a reconstructed collision vertex, and the tracks used
for centrality were primary tracks which point back to that vertex. This technique
also reduces the number of beam—gas collisions in the peripheral data set. The events
which were used in the balance function measurements also had at least two charged

particle primary tracks within the TPC track cuts (Section 2.6.)

2.5 Event Cuts

During the year 2000 data taking, the RHIC collision region was as large as the TPC
in the z direction. Collisions which had an event vertex within 75 cm of the center of
the TPC in the z direction are chosen in this analysis. A track which would normally
extend over the whole length of the TP C in a vertex(z) = 0 collision might not be fully
detected in a vertex(z) = 100 cm collision. This partial detection of larger vertex(z)
collisions can be seen in Figures 2.6 and 2.7, which show 2 of 10 possible vertex bins.
Figure 2.6 shows the pseudorapidity distribution for the farthest negative vertex bin,

which is centered around 62.5 cm. In contrast, Figure 2.7 shows the pseudorapidity

14

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P 1

 

dN/dn

 

 

104IllIl....1....1...11....1..i.
-l.5 -1 -O.5 O 0.5 1 1.5
n

 

Figure 2.6: Pseudorapidity histogram for central events in the -62.5 cm vertex bin.

(dN/dn) distribution for central trigger events which have a vertex centered around
zero, in the most central vertex bin. The edges of the TPC along with the track cuts
(discussed in Section 2.6) are visible as sharp drops in the pseudorapidity distribution.
Events which had an a: or y position of the vertex more than 1 cm from the center of
the TPC were not analyzed, providing the least amount of variation in 4) given the
TPC resolution.

The distribution of vertices used in events analyzed from the year 2000 data is

 

 

 

 

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Figure 2.7: Pseudorapidity histogram for central events in the 0 cm vertex bin.

15

Central data

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' l’ V V I V

dN/dvertex-z

 

 

 

dN/dvertex-z

 

 

dN/dvertex-z

 

 

O
vertex-z(cm)

Figure 2.8: Vertex distribution
centralities are shown.

shown in Figure 2.8.

2.6 Track Cuts

for year 2000

Midcentral data

 

 

 

 

 

 

 

 

 

 

1000
400 i“ 1
200 ’f {
0 i . . . . 1 ......... n ........ l . U 4
-50 50
vertex-z(crn)
t
500 Peripheral da a
100 :- f
l 1
0 .4 . . 1 ......... l ......... l . . .
~50 50
vertex-2(cm)
events used in the analysis, all four

The energy deposited from charged particles is detected as TPC ADC (Analog to

Digital Converted) signals, giving a 3-dimensional picture of the collision. These sig-

nals are analyzed with a track fitting algorithm that finds the primary vertex of the

collision and fits tracks based upon an expected helical geometry which point back

towards the primary vertex. These tracks are known as primary tracks. Global tracks

16

are all tracks that are found in the detector. In the 2000 data, global tracks are merely
TPC tracks, but in later data runs, they will include other installed and instrumented
detectors such as the EMC and SVT.

For the balance function analysis, the particles that contain the signal are those
which come directly from the primary vertex of the collision. Thus, only primary
tracks are used in the analysis. The initial check of the quality of the track for data
analysis is the flag variable, which is greater than zero for good tracks. All tracks
used in the analysis must be primary tracks which are labeled as good tracks. There
are other measurements which are used to label the tracks which are appropriate for
data analysis use. These measurements, using cuts to remove unusable tracks, are
based upon studies of various track cuts by the STAR collaboration. The cuts used
here were chosen for the analysis in the first STAR publication studying flow [29],
and similar track cuts have been used in many STAR analyses. References [32], [33],
and [37] discuss effects of different cuts.

Tracks are used which have a pseudorapidity measured within I77] < 1.3. Theories
of QGP identification require the measurement of variables over at least two units
of rapidity [44]. The limit of pseudorapidity was chosen to strike a balance between
maximizing statistics and minimizing systematic error from the detector. Smaller
pseudorapidity regions were explored for balance function analysis, as covered in
Section 7.2. Another condition is that tracks must have at least 15 hit points used in
the fit. To eliminate double counting, the ratio of fit points to the possible number
of points must be greater than 0.52. This reduces the possibility of having a single
track labeled as two separate tracks (track splitting), and then being counted twice.
The tracks are limited to a momentum of 0.1 < p < 2 GeV/c. The distance of closest
approach (dca) between the track and the collision vertex can be calculated, and tracks
are used which have a |dca| < 1cm. The dca cut reduces most of the background that

comes from tracks which are the product of secondary decays. These conditions assure

17

that the tracks used in the analysis have a detection efficiency of 85% i 5%. However,
there is an estimated 5% of the particles which are electrons from narrow decays which
happen to point back to the collision vertex. Once a particle track survives all these

tracks cuts, it can be used in balance function analysis.

2.7 Particle Identification

For a balance function measured with all charged particle pairs, some particle identi-
fication is needed to remove the secondary electron contamination from the primary
particle signal. The necessity for removing these electrons is discussed in Section 5.2.
When the balance function is constructed with pion pairs, particle identification is
needed to separate the pions in the events from the other particles. Particle identi-
fication is done using the measured energy loss of the tracks as the particles leave
the TPC. The energy loss values for each hit on a track are used with a truncated
mean method to reduce fluctuations. The dE/dx, mean energy loss per unit length,
for each primary track in a selection of minimum bias events is shown plotted versus
momentum in Figure 2.9. The lines on this logarithmic plot are a calculation of the
expected Bethe-Bloch parameterization for electrons (the line almost parallel to the
:c-axis), pions (the leftmost line), kaons (the next line), and protons (the far right
line). The events shown were not included in the final analysis due to problems with
the trigger, however, the particle types and momenta are consistent with the data
taken in year 2000.

Tracks within events which pass the quality cuts were first compared to the ex-
pected energy loss for pions. The expected dE/dsc for a pion is calculated with the
momentum of the tracks. The number of a the track is from the perfect Bethe-Bloch

particle can be found with the measured mean dE/dzv and momentum as shown in

18

 

 

 

 

 

A _ . .., ,
E I '5»? ., g
3 9E“ ‘ 3"." 1: .
3 E .
x 8.— '1'; .1: ; f _ .
S r '35:- ; ‘ d
'D 7; 33;; g; a . , . é
_ was; 1%.: 1 i
: 1;: ~. -‘ 4
6:- ' ‘ '
5'5-
4; ~
37‘ "
2:. .. e- 'h‘ “1:...
ii i '_
1 f— -,
- l l .L 1 i i 1 I I 1
10'1 1
p GeV/c

Figure 2.9: dE/da: vs. momentum for negative charged particles identified as primary
in the TPC for July data. The lines drawn through represent the predicted dE/da:
from the Bethe-Bloch equation for different particles as labeled. The different shades
represent density of tracks, with the highest density being in the center of the pion
region, and the lowest on the edges of the proton region.

 

10 'VY‘IT'iYIT'rTVYIj‘rTVlIIIVT'YT"

 

 

 

 

 

105 g 1
b“ 104 g E
E 103 _, 1
Z 5 E
"U 102 g g . 1:
10] E. y i 1‘" [a -;
100f“iiiii'n“‘l“‘AlA..-1 A M :
No',t

Figure 2.10: The N0,r values for all negative particles passing track cuts in minimum
bias run 1248015. The Gaussian fit marks the pion signal.

Equation 2.1.

mean dE/dat
n expected dE/da:(7r) ’
Zr
dE/dl‘ resolution.

 

NU,r

 

For the other particles, the 7r is replaced by K, p, or e for each case respectively. In
this method, the number of sigma a track is from any of these four particles can be
used for particle identification. The profile of Na7r for all negative particles in one
minimum bias run (~22,000 events) is shown in Figure 2.10. The shoulder to the left
of the Gaussian curve is probably electrons. The Gaussian shows the pion range, and
to the right are kaons and protons. When the events are analyzed, the track is first
compared with the pion ionization. If the track is within 20‘ of the expected result,
i.e. if [N07,] < 2, then the particle is tagged as a pion.

In Figure 2.9, it can be seen that the dE/dcc signal for pions and kaons begins to
mix at a momentum greater than 0.7 GeV/c. Thus, the particle is kept with the pion

tag only ifits momentum is less than or equal to 0.7 GeV/c, and its ionization is within

20

2a of the calibrated ionization for a pion. Particles which do not fit those qualifications
are then checked against the expected ionization for kaons. The kaon and proton
signals in the dE/da: vs. momentum space begin to mix at 0.8 GeV/c. Thus, particles
that have not been tagged as pions, have p S 0.8 GeV/c, and [NUKI < 0.2 are
tagged as kaons. From the remainder of the tracks, protons are tagged for those
whose momentum is less than or equal to 1 GeV/c to avoid mixing in deuterons
and INUPI < 0.2. The untagged particles which are left are identified as electrons
if they fall within 20 of the electron Bethe-Bloch curve. The remainder particles
are labeled as unidentified particles. All particles are kept for the different types of
analyses. There is an estimated 1% contamination of secondary electrons within the
pion signal, however, the pions compromise the majority of detected particles within
a heavy ion collision. The effect of secondary electrons is strongest at small changes in
rapidity (or pseudorapidity), as discussed in Section 5.1. The resolution of the dE/dx

measurement is estimated to be about 10% [37].

2.8 Physical Characteristics of a Track

Each track is recorded in the STAR TPC as a series of hits in three dimensions. A
track fitting algorithm is used together with the magnetic field information and the
particle hits. In future data taking, STAR will use information for track reconstruction
from all detectors, not just the TPC. However the track data used in this analysis is
from the TPC only. The track fitter used was based upon the expectation that every
particle track would be a section of a helix. The track fitting routine was developed
and tested by STAR collaborators. The track fitting uses the specifics of the track
helix such as the curvature, starting point, azimuthal angle of the track direction at
the starting point, and the fit of a circle in the my plane with the value of the magnetic
field in the z direction to calculate information about the geometry of the particle’s

path. The curvature of the track in the magnetic field gives a measurement of the

21

momentum of the track in three dimensions. The direction of the curvature of the
particle’s track gives the charge of the particle. The pseudorapidity of the track can
be found from the momentum vectors [45] [46]. Once a track is tagged as a primary
track with a good fit and the track passes the track cuts described in Section 2.6, the
necessary information for balance function analysis is written to file. The file creation

and analysis structure is discussed in Chapter 4.

22

Chapter 3

The Balance Function

3. 1 Definition

The balance function is an observable proposed by Bass, Danielewicz, and Pratt [47]
to indicate the hadronization time of particles that emerge from a heavy ion collision.
While there are many standard observables used to probe the interactions of nuclei
at high energies, the balance function is new to the field of heavy ion physics. In p+p
collisions, the charge density balance was measured [48] in a functional form similar to
the balance function. The measurement of the width of the charge density balance for
selected particles in a four jet structure was used to determine that jet fragmentation
is universal. The associated charge density was used in e+e‘ collisions for the study
of strangeness correlations [49] and baryon correlations [50].

In Au+Au collisions, the balance function can be used to clock particle hadroniza-
tion. The time of hadronization could be an indicator of the type of collision that
occurred. It is theorized that at a high enough energy and density, a quark-gluon
plasma (QGP) will be formed [4]. This QGP will exist for a finite amount of time
before it cools off, hadronizing into pions and other particles. The particles detected
in a relativistic heavy ion experiment can be studied to determine features of the

source that created them. There are many observables that can be measured which

23

differentiate a QGP from a hadron gas [51]. The particles which hadronize from a
QGP are theorized to be created at a later time, since the QGP exists for a short
time with quarks and gluons existing without being bound into separate hadrons.
Thus, having an observable which indicates the time of hadronic particle creation
would give a glimpse into the nature of the source that created those particles.

To understand how the balance function measures time, consider the relativis-
tic heavy ion collision just after it occurs, as seen schematically in Figure 3.1. For
a QGP, the long cylinder of material stretched out between the two receding ions
maintains its quark deconfinement for a short time before hadronization [52]. When
hadronization occurs, the particles and their partner anti-particles are produced at
the same space/ time point from quark / anti-quark pairs. At hadronization is when the
fractional charge carried by the quarks is combined into unit charge in particle+/anti—
particle" pairs. These particles do maintain a collective motion, or flow, which can be
measured [29]. However, because hadronization occurs some time after the collision,
the final state interactions of the produced particles have a smaller cross section than
if they are formed earlier. The particles would emerge with a small spread in rapidity
between particle and its partner anti-particle. In contrast, if a hadronic gas is created
instantly, the particle/anti-particle pairs are created very early in the collision, right
after the ions collide. These particles have much opportunity to interact with other
particles in the cylinder, as the system is pulled apart while the beam ions recede.
Thus, the particle pairs which are created from the hadron gas can have a large spread
in rapidity.

The next apparent challenge is to identify the particle/anti-particle pairs which
were created together. When particles are identified in the detector, there is no identi-
fication tag to label which particle pairs with a specific anti—particle. Thus a statistical
method must be used. The balance function is constructed to identify pairs on a stat-

istical basis and provide a measure of the rapidity spread of particle/anti—particle

24

i“ 3% ,
3::er

<—<——>

W/‘Kfl
a/Y/[

Figure 3.1: A schematic of the collision during hadronization. Arrows to the left and
right represent the spreading of matter. Arrows emerging from the dots represent
particle pairs created together. The jagged lines represent the interactions of those
particles in the medium.

pairs. The balance function in its most general form [47] is:

1
3001092) = '2' [9(5) 192]“,1’1) — P(b,P2lb»P1)+ 10013192“): P1) - P(aaP2|aaP1)la

N b,p am)
p(b,p2|a.p1) = (N(a2|pl)1. (3-1)

 

p(b,p2|a,p1) is the conditional probability of observing a particle of type b in bin
p2 given the existence of a particle of type a in bin p1. Thus, if bin p1 refers to a
particle measured in the right half of a detector, then bin p2 could refer to a particle
measured in a smaller subset of the right half of a detector such as one-fourth of that.
These conditional probabilities would then be measured over all possible sub-bins of
the detector. The balance function can be used to consider not only the balance of
charged particle pairs and pion pairs, but also for any quantity which is conserved,
such as strangeness which is measured in the balance function for kaons or baryon
number with proton pairs. However, this thesis will concentrate upon the charged
particle pair and pion pair balance function measurements. The form of the balance

function used in this dissertation is:

25

8mm = 1 [

2 * binsize

 

 

N+_(Ay) — N++(Ay) N_+(Ay) — N__(Ay)

N+ + N- },(3.2)
E +192
E —p,,’

if mass of particle is small, y z 77.

 

l
y = rapidityz—é-ln (3.3)

N+_(Ay) is the histogram of |y(7r+) —y(7r‘)| for all possible pairs within an individual
event. This histogram is summed over all measured collision events. N+(N..) refers
to all the positive( negative) particles used in the pairing. It is possible to construct a
balance function with all possible charged particle pairs instead of pion pairs, except
the argument of B would be A?) instead of Ag. The binsize is included so that the
balance function measured in a perfect detector will normalize to one. Here, the
summations over all possible pairs and then events are not given explicitly, but are
done in the construction of a measurement from events in different Ay (or A77) bins.

The individual terms of the balance function are calculated separately for each
event, which classifies this measurement as an event-by-event observable. The form
of the balance function pulls out the correlation in rapidity (or pseudorapidity) of a
particle/anti-particle pair which has been formed together. The validity of this model
can be demonstrated with a simple simulation. A Bjorken thermal simulation code
was obtained from Pratt [47] which can test that this statistical summing method
works to accurately reconstruct the given input. This simulator first produces a par-
ticle with a random 4-momentum, and then an anti-particle with another random
4-momentum. For a given amount of time, particles are moved forward along straight-
line trajectories and given a random chance weighted by the number of particles at
elastically scattering within a given time. This is followed by more straight-line tra-

jectories and scattering until a final time. Then both particles are spread by the same

26

0.8

 

 

 

 

F5619. 63' i . . O Bjorken - from known pairs]
07 g g; 0 B jorken - measured from 3
E {’3 all possible pairs 1
0.6 : E9 ‘.
0 5 ~_ {:9 8 1t pairs 5
9 ' : [C] perfect detector 3
s 0.4 :- .. .. -
CD .
0.3 :- 99 E, .
: 53 1
0.2 '- J
: u [33 .
: n .
. n -
0.1 E ,
0 ' 1 1 . 1 1 1 1 . . 1 . 1 m 1 1 1 . 1 . 1 . . 1 m
0 0.5 1 1.5 2 2.5

Ay

Figure 3.2: The shape of the balance function plotted for a perfect detector. Plusses
are from the statistical measure, whereas squares with circles are from known balanc-
ing pairs.

random rapidity. The simulated particles are written to a separate file in a format
containing particle identification, momentum, transverse momentum, and pseudorap-
idity (to mimic the real datastream). However, as the particles are written to file, the
difference in rapidity between the co-created particle pairs can also be written to file,
in a histogram which normalizes to one over all possible Ay. The comparison be-
tween these two methods is shown in Figure 3.2. The default settings of the Bjorken
thermal model were used, with a bin width of 0.1, initial temperature of 225 MeV,
final temperature of 120 MeV, with temperature for hadronization of 165 MeV, and
a formation time of 9 fm/c. The rapidity spread is i2 and there are an average of 3

collisions per particle in this simulation.

3.2 Predictions

The comparison of known balancing pairs to a statistical measurement of the bal-
ance function shows visually that the balance function does indeed reconstruct the

rapidity difference between co—created particle/anti-particle pairs. In reference [47],

27

 

1.0 l I I 1 I

 

 

B(Ay)

 

 

 

 

Figure 3.3: Balance functions for pions (top) and protons (bottom) calculated with
respect to Ay. Bjorken thermal model simulations of a QGP are in circles and squares.
PYTHIA p+p collisions (representing the hadron gas model) are shown in triangles
[47].
Bass, Danielewicz, and Pratt not only describe the construction and theory of the
balance function, but make predictions for balance function measurement. Figure
3.3-top shows the balance function measured in a perfect detector for a Bjorken ther-
mal model (circles and squares), and PYTHIA p+p events (triangles). PYTHIA [53]
is a model used to simulates p+p events at \/->—N.\7 2200 GeV, which represents the
hadron gas scenario. PYTHIA models hadronic reactions for p+p collisions as well
as hadron-nucleus and nucleus-nucleus collisions [54]. PYTHIA uses the Lund string
fragmentation model, and its specific predictions have been confirmed by experimen-
tal measurement [53]. The Bjorken thermal model [47], [52] is the same as the one
described above in Section 3.1 without the rapidity spread, which creates an expand-
ing pion gas after a given time at a specified temperature.

It is evident from Figure 3.3 that the Bjorken model has a narrower balance func-

tion than the PYTHIA model. The width of the balance function can be understood as

28

03,, = 2(0‘t2hemal-I-afimusion). The thermal term refers to the thermal rapidity spreading
due to the collective behavior of particles after the collision. The diffusion term repre-
sents the rescattering of particles within the medium. Rescattering of particles would
broaden the balance function by diffusing the charge/anti-charge pairs in space, or
rather rapidity. Considering the discussion in Section 3.1, a QGP model would have a
smaller time to diffuse, so that term in the width could approach zero. This diffusion
term would be affected by whether the created charge moved as a free quark in a
QGP or a as hadron in its early history. Annihilations of particles could also broaden
the balance function, although reference [47] predicts that the balance function for
a QGP would still remain narrower than that of a hadron gas. The predictions in
reference [47] do not cover the issues of resonance decays such as p0 —> 1r+7r‘ and the
effects of experimental acceptance on the balance function. To understand the effects
of having a real detector as opposed to a ‘perfect’ detector, simulations can be used.

To summarize the predictions, the balance function measured from a QGP would
exhibit a narrow shape in comparison to the balance function from a hadron gas
from the superposition of nucleon-nucleon collisions. For small rapidity differences,
the broader hadron gas model is expected to have a lower initial balance function than
that of a QGP. However, experimental and model dependent factors which influence
the shape of the balance function require understanding. The relative time of particle

hadronization may be qualitatively found by measuring the balance function.

29

Chapter 4

Analysis Method

4. 1 Software

Data from the TPC are collected by electronics and analyzed in the first step by the
STAR analysis software. The STAR analysis system is based upon ROOT, a C++
based physics graphing and data analysis package developed at the CERN laboratory.
Collaborators developed software to read this T PC information, with a framework
that allows for many types of data analysis. The TPC information was analyzed and
track reconstruction was performed in production stage. The production was run with
two different libraries, which include the calibrations of the TPC, as well as all the
track-fitting and reconstruction software. Production libraries will be discussed in
Section 4.1.1.

At the time the analysis in this dissertation was performed, the ROOT in-
put/output procedure was time—consuming, as a separate file would need to be
opened for each Data AcQuisition (DAQ) file, or about every 500 events. Thus, the
necessary particle information for the balance function calculation was output to a
binary file. This method created what is referred to as a micro Data Storage Tape
(pDST), a smaller data set created from the total track information on file.

The data set was written out with a header file and a data file. The header file

30

contains the information related to each particular event, the STAR run number, the
STAR event number, and the local analysis event number for that run. Each event
was checked to see if it was within the event quality cuts described in Section 2.5, and
events which were acceptable had the local event number indexed. Two integers were
also written to the header file that contained the number of tracks. The first integer
contains the number of good primary tracks in the TPC. This is the total number of
tracks detected which have the track flag > 0. The track flag is a variable which is
set negative if the track reconstruction routine finds something wrong with the track.
This first number of tracks variable is the one that is used for a centrality cut, as
it covers all the primary tracks detected by the TPC. The next integer contains the
number of accepted tracks which are written to the data file. These are the tracks
which pass the track quality cuts for the balance function calculation.

The data file which is written out contains the passed event number, the passed
track amount, and the needed data for each track. For the P00hi library analysis,
this was the particle ID (PID), the mean dE/da: value in keV/cm of the track, the
magnitude of the momentum (GeV/c), the transverse momentum(GeV/c), and the
pseudorapidity of the track. For the P00hm library analysis, the mean dE/da: variable
was replaced by the number of a the dE/da: was from different particle models. The
different particle models considered were electron, pion, kaon, and proton. In each
case, the particle identification was performed as discussed in Section 2.7, and a PID
was written out, with the sign of the PID integer being the charge of the particle.
The PIDs were assigned as follows in Table 4.1.

The data files were written out with one binary header and one binary data file
per data-taking run. This allowed for simple quality cuts later, when specific runs
were found to contain faulty data, the good and bad runs are tabulated in Appendix
A in Tables A.1 and A2. The binary data format allowed for simple, quick, repetitive

analysis with compiled C++ programs on any computer platform. It also allows

31

 

 

 

 

 

 

particle [PID]
electron 1
pion 2
kaon 3
proton 4
unknown 11

 

 

 

 

Table 4.1: Particle and absolute value of PID (particle identification) codes used in
balance function analysis.
for simple data backup and eliminates dependence on computers at Brookhaven lab

through the network.

4.1.1 STAR to Local Data Conversion

To convert the data from STAR reconstruction files to a local DST, analysis modules
were written. For the P00hi production data, a module was created using the example
of M. Calderon’s ,uDST PionTree maker [37]. This system created a file with ROOT
tree structures that contained the accepted track data. The accepted tracks and event
information were then written to text files which were transferred over the network and
then translated to binary. During each step of the data sort and transfer, histograms of
the variables written were compared to assure that accuracy and information was not
lost. Histograms were created in ROOT directly from the production. Comparative
and analysis histograms were created locally through a C++ program and plotted
in KaleidaGraph. Specific variables were also written out during testing with event
identification to give a specific numeric check.

The same procedure was done for the P00hm production data. However, the data
translation from the STAR ROOT files to binary files was done with a simpler program
than for P00hi which was based upon a template program, StAnalysis Maker [55].
Data from concurrent runs in both P00hi and P00hm were analyzed with the balance
function program to assure that the results did not change with respect to the different

production libraries. The charged particle balance function for the same set of events

32

 

 

0,6---fi,.--- 0.35,--.-,...-..-.-,r-f-‘

 

 

 

 

 

 

. , ' Charg'ed'panicle pairs 2 . . .
0 5 [T f v P00hi (no identified electrons) I 0 3 + % 3151312128531):
~ 1 V P00hm same4runs.‘. ' I + .. . ’1
I T all centralities combined 2 0 25 # + all centralities combined]
Q4? 9? j E vans ]
€- : ? : E. 0-2 r V P00hm *[
3 0.3 f - v
m t y] ; new [i ‘
dzi ‘ 1
. 1 o r
t y? : Ii ] if]
OJ; YY" ,' i 0% §$§ l
. ......... v 1 V
0 -------- - o + AAAAAAA
a) () 05 1 15 2 b) 0 05 I 15 2
An A)’

Figure 4.1: a) Charged particle pair and b) pion pair balance functions measured for
runs 1243059, 1245006, 1245012, and 1246009. Open triangles are P00hi production
library, and closed triangles are P00hm.

analyzed in P00hi and P00hm is shown in Figure 4.1. The shape and structure of
the balance function will be discussed later in Chapter 5. What is important is that
the production both with (P00hm) and without (P00hi) the electric and magnetic

field correction produce the same result within statistical error bars for the balance

function.

4.1.2 Local Data Analysis

The binary files were analyzed with two programs, balance.cp and balanceout.cp. Both
programs were written for use on Macintosh and UNIX platforms. The balance.cp
program analyzed each run separately, calculating the N+_ & N+-type terms. These
terms were then written to a text file, with a separate file for each particle—centrality
combination. The balanceout.cp program would take a specified set of runs and sum
the balance function terms over those runs, calculate the balance function, the error,
the integral, and the weighted average of the balance function. All these variables

calculated would be output to a file in a format readable by KaleidaGraph.

33

4.2 Data Quality

4.2.1 Logbook Information

The quality of the data used was checked with a variety of methods. One of the sim—
plest ones to start with was the fact that the data had been written to file in a STAR
production. This meant that the computer tags on the files used would be labeled
Physics runs, with either central or minimum bias trigger. Logbook information is
also useful for quality, and for STAR this was kept online. During data taking shifts,
the information about the magnetic field, actual trigger parameters used, and any
other important information about the run was entered into the online run log. This
run log can be browsed by a STAR collaborator over the internet, allowing for access
of data logbook information at any time. As the summer 2000 data was the first
physics data that STAR recorded, the information for event quality was scattered
in a number of sources. These sources included the online run log, E—mails to the
Quality Assurance team, and the STAR software team E—mails [55]. Runs such as
1228031, 1240008, and 1246017, were all labeled as Physics data runs, however, the
online run log information showed that the trigger used was a laser trigger. The laser
trigger is used for TPC calibration. A STAR collaborator warned the collaboration
of troublesome runs which were taken on August 15th, 2000. These were runs where
the trigger had lost contact with the RHIC clock, and thus had possibly mistimed
collisions. If the timing is off, it is possible to record an event which includes pieces of
two overlapping events. The TPC track momentum and pseudorapidity would look
normal, but the multiplicity in the TPC would give an inaccurate representation of
the centrality of the collision. Collisions which were recorded with a ultra-peripheral
trigger were also not used in the analysis, as the trigger that was implemented in
summer 2000 for ultra-peripheral collisions was biased as it was optimized for two

track events [56]. Also, when the logbook had runs tagged as being trigger tests, they

34

were not used for similar reasons. Also, the data from the trigger detectors, ZDC and
CTB were checked during the program which converted data from ROOT files to
simple binary files.

Run log information included many details. One type of note found in the log for
particular runs was regarding TPC sector failures. A few times during data taking,
parts of the TPC would fail and require startup. Any run which had this tag on it, as
well as runs close in time would not be used in the balance function analysis. This type
of analysis depends on measuring the conserved charge emerging from the collision
in all directions, given the detector’s known efficiency. With a missing TPC sector,
a piece of the detector is missing which changes the coverage of what is measured.
Events from runs which are missing TPC sectors could be used in other types of data
analyses, but were not used in the balance function analysis. There were a few data
runs where the magnet was off or experienced a failure during or just before that
run. The magnet needed to be operational during the run, as the calibration and
reconstruction depend on having tracks which curve in the magnetic field. Runs were
omitted in which the magnet was off, or had tripped during the run.

In addition to particular run quality, the software used in track reconstruction
was checked for accuracy. Members of the collaboration used GEANT-created tracks
embedded into real data events to check software library reconstruction accuracy as
well as the TPC particle detection efficiency. This work is reported in detail in [37].
There were two software libraries used in this analysis, P00hi and P00hm. P00hi was
the first stable production library used in data analysis. It was shown to give correct
reconstruction and a high reconstruction efficiency for tracks studied. This production
library included all the necessary TPC calibrations and reconstruction software, with
which the end-user would merely need to read the reconstructed event data from a
file and have most variables needed for most analyses. The P00hm software library

was an upgrade of the P00hi library.

35

The change which has the greatest possibility to affect data analysis is the addition
of an ExB field correction. This correction is implemented to add the correction for the
slight misorientation of the electric and magnetic fields to the parallel of the beamline
and to each other. A three dimensional map of the TPC’s magnetic field was made.
The deviations of the magnetic field from parallel together with the electric field can
create an additional force which changes the expected motion of charged particles
away from the primary vertex. As the tracks are reconstructed based upon their
expected behavior in the TPC, the electromagnetic fields would need to be known
precisely. Some runs that were analyzed in the P00hi software library were analyzed
using P00hm, and many runs not analyzed in P00hi were analyzed in P00hm. The
balance function measurement for the two libraries was compared, and there was no
change seen in the P00hm upgrade, as seen in Figure 4.1. Runs used in the final
analysis included those analyzed by both P00hi and P00hm STAR software libraries.

Another piece of information is the runs which were analyzed in production. The
offiine reconstruction (production) of the data would perform simple quality checks.
It was found that there were runs analyzed in the P00hi library that were not re-
constructed in P00hm due to quality. A final check was communication with other
colleagues analyzing the same data. In this way, runs were found which showed trou-
ble with the triggering, which can allow multiple events to overlap. In summary, the
Appendix A Table A.1 shows some logbook information for runs which were used in
the analysis in this dissertation. Appendix A Table A2 shows the logbook informa-
tion for both P00hi and P00hm data runs which were omitted for reasons given in

the last column of that table.

4.2.2 Amount of Data

A total of 244,718 events from the year 2000 passed the event quality cuts, 150,756

minimum bias events, and 102,173 of central events. Table 4.2 shows the amount of

36

 

] centrality [amount] trigger ]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

central 79,844 central
midcentral 25,656 central
midperipheral 897 central
peripheral 581 central
central 15,360 minimum bias
midcentral 46,655 minimum bias
midperipheral 44,267 minimum bias
peripheral 31,458 minimum bias

 

Table 4.2: Amount of events used in analysis for each of the four centralities from
central trigger (top), and minimum bias trigger (bottom).

events for each of the four centralities for the two different triggers.

These are sufficient statistics to perform a charged particle balance function mea-
surement. With the majority of the particles detected in an event being pions, this
also means that a pion balance function may be constructed. However, these statistics
are too low to measure the balance function for kaons and protons. As is discussed in
reference [47], 105 events are necessary to have a clear signal considering the statistical

error bars. The error bars will be discussed in Chapter 5.

4.2.3 Graphical Data Quality Check

The quality of the data was also checked graphically. A multiplicity plot was made
for the combined set of minimum bias and central trigger events, as can be seen in
Figure 4.2. Recalling the shape of the minimum bias multiplicity distribution from
the discussion of centrality in Figure 2.5, it can be seen that there is a bump in the
high multiplicity region. This bump corresponds to the central trigger events. By
studying this multiplicity distribution on a run by run basis, the runs beginning with
1229 recorded on 8/16/2000 were found to contain data which appeared to be of a
minimum bias trigger, although they were apparently recorded with a central trigger.
This study was one aspect that led those runs to be removed from the final analysis.

The distributions for other variables used were also checked for each run. Figure 4.3

37

All data, central and minimum bias trigger

Ta fifi

v r v fiw v

p
1% .-
b
r
>
b
D

4
v' vvv

ch max)

dN/d(n /N
a

 

 

 

 

Figure 4.2: Multiplicity distribution for both triggers used for the data set analyzed

 

 

 

 

 

 

  

 

 

 

 

here.
0 06 Central Data. all particles Peripheral Data. all particles
0.05 : 3 0.06
5 E % 004
A 0.03 , A 003 E
E 0.02 3 § ' g
V : v 0.02;
0'01 Z 0.01 E
a) 0 5 - 1 i . . b) O l
p (GeV/c) p (GeV/c)

Figure 4.3: All charged particles normalized momentum distribution for a) central
and b) peripheral events.
shows the momentum magnitude and pseudorapidity distributions for central events.
The overall shape is typical of primary particles with the specific track cuts that were
used, as detailed in Section 2.6.

At this scale, differences cannot be seen between centralities for the central and
peripheral momentum distributions. Examining the pseudorapidity histograms, there
appears to be a difference between centralities. However, it should be considered that

there are different pseudorapidity distributions for the differing vertex bins. The de-

pendence of pseudorapidity distributions on the vertex is shown in Section 2.5. Thus,

the vertex distribution of the centrality cuts should be examined. Figure 4.4 shows

38

Central Data Midcentral Data

 

 

vvvvvvvvvvvvvvvvvvvvvvvvvv

vertex 2

dN/d

1000

800

600 .4 J
400 ,

200

 

 

 

 

 

 

LA A-.l-.. AAA .4. ‘.Al... AL

‘ 0
a) -80 -60 -40 -20 0 20 40 60 0 b) -80 -60 ~40 -20 0 20 40 6O 80
vertex 2 (cm) vertex 2 (cm)

 

 

Figure 4.4: All charged particles non-normalized vertex distribution for a) central and
b) midcentral events.

Midperipheral Data Peripheral Data

 

 

 

 

 

 

 

 

.1.4AIL4LL..A1.AAI..

"'I.Al“‘l“‘l-AAlAAL11111-..1.A ‘ 0:. ‘ .
a) - -6O -40 -20 0 20 40 60 To 6) -80 -60 -40 -20 0 20 40 6'0‘ 80

 

 

vertex 2 (cm) vertex z (cm)

Figure 4.5: All charged particles non-normalized vertex distribution for a) midper-
ipheral and b) peripheral events.

the non-normalized vertex distributions for central and midcentral events. Figure 4.5
shows the vertex distributions for midperipheral and peripheral events.

The dN /d17 distributions for the four different centrality bins are shown in Figures
4.6 and 4.7. The differences in shape may be partly attributed to the differing vertex
profiles for the different triggers. In general, these histograms may be used as a guide-
line to check validity of the data. The form of the rapidity distribution for the pions
used in the balance function is also useful. Figure 4.8 shows both the pseudorapidity
histogram for all charged particles again, with the rapidity histogram for pions be-
side for comparison. For the pions, the histograms have similar shape to the charged

particle histograms, although the momentum cut does cut the momentum histogram

39

 

 

Central Data, all particles Midcentral Data, all particles
0.025,”. ......... ,----,e---,-- 0.025,”.-vn. .............. ,a-

 

 

(1/N)dN/dn

 

 

 

 

”0.5””1

 

0
11
Figure 4.6: All charged particles normalized 17 distribution for a) central and b) mid-
central events.

 

 

 

 

 

 

 

 

Mid ri heral Data, all particles Peripheral Data, all particles
0,025,“??? vvvvv .fifipfiv.-. 0.025 ......................... ‘
I l 2
0.02 :- 0.02 I- , i
g I :- : .._.-.."‘. ‘,.-.:.-.:,- a: 3 :.,t.‘.=..:"o'-‘~‘-'-,.:* \ 1
'O > n .
20.015 2 0.015 ,-
R . '2
E 0.01 I z 0.01 :
0.005I 0.005 ‘
a) 0 b) 0

   

Figure 4.7: All charged particles normalized 17 distribution for a) midperipheral and
b) peripheral events.

at a p of 0.7 GeV/c.

The plots shown are characteristic of the overall behavior of the data. Individual
runs were examined to be certain that none of the variables deviated markedly from
the data. Overall, the runs examined had reasonable plots. A large number of these
plots were made during analysis, and most will not be reproduced here. The figures
shown above do give a general sense of the typical shape for the data which is useful

in qualitative comparison with simulations.

40

Central Data, all particles Central Data, 1:

 

 

(1/N)dN/dn
.9
o
5‘.
3
(l/N)dN/d'q

 

 

 

 

 

 

 

 

Figure 4.8: a) All charged particles normalized pseudorapidity distribution compared
to b) pion pseudorapidity distribution.

4.2.4 Programming Quality Assurance

During the writing of the programs balance.cp and balanceout.cp, the programs and
subroutines were checked for quality. In this case, quality meant that the calculations
were accurate. One method used was to take a known small set of data, and ensure
that the same balance function measurement was found calculated by the program
and by hand. By hand usually included using a calculator or a simple analysis program

to check the calculation steps.

4.2.5 Simulation Quality Assurance

Simulations of nucleon-nucleon heavy ion collisions were performed with the simulator
HIJING, described in Section 6.1. HIJ IN G stands for Heavy Ion Jet INteraction Gen-
erator [57]. HIJING was processed with the STAR GEANT framework, GSTAR, as
well as the TPC Reconstruction Simulator (TRS), and then analyzed with the STAR
reconstruction chain, treated as if the events were of real data. These simulations
will be referred to as HIJING-GEANT. The GSTAR and TRS code was checked for
quality by STAR collaboration members [58] [59] [60]. Appendix A Table A3 shows
the summary of HIJING-GEANT events which were used in this dissertation, the

impact parameters and number of events.

41

HUING-GEANT, all particles
0,07....,....,,..,,,,.,:
0.06 i

0.05

 

 

.9
E

0.03
0.02
0.01

(1/N)dN/dp

 

 

 

 

 

 

 

l
p (GeV/c)

Figure 4.9: HIJING-GEANT normalized a) momentum distribution for all charged
particles and b) rapidity distribution for pions.

Similar to the data quality check, histograms of variables such as p and 17 can
show the quality of the HIJING—GEANT events. In particular, comparing these his-
tograms to those of the data illustrate the effectiveness of the detector simulation as
well as the underlying physics simulation. For HIJING-GEANT events, there is no
centrality separation, as all impact parameters are expected to have the same balance
function physics. This shall be examined further in Section 6.1.3. Figure 4.9 shows the
momentum and rapidity distribution for HIJING-GEANT events, with the momen-
tum shown for all charged particles, and pseudorapidity for pions. Both plots seem
similar to the data, however the momentum distribution seems narrower at high mo-
menta, and the rapidity distribution has a strange peak. Figure 4.10 shows the vertex
and pseudorapidity distribution for these simulated events. The vertex distribution
is nicely centered, unlike the data. The 77 distribution appears very similar to the
data. Given these quality plots, it appears that there is nothing very wrong about the
HIJING-GEANT simulations. However, the charged particle distributions are more
similar in shape to the data than the pion distributions.

Nothing obviously wrong was found with the data sets or the simulations to be
used in the balance function analysis. Any suspect data was removed, as described in

the previous sections. The quality check proved to be useful, as a large slope change

 

 

HIJlNG-GEANT, all particle, all centralities
0.025--.-Hfl- .......

0.02 {-

VCHCX 1.

0.015

(1/N)dN/dn

0.01 {-

 

 

.6
8
”I

 

. IAALI

a) -80 -60 - -20 0 20 40 60 8 b) 0

 

 

 

vertex 2 (cm)

 

Figure 4.10: HIJING-GEANT a) non-normalized vertex distribution and b) normal-
ized 17 distribution, both for charged particles.

was evident in the midperipheral balance function. Once the runs with faulty trigger
were removed, the large slope change no longer appeared in the measured balance
function. Removing suspect runs does decrease the statistics of data available for
analysis, however it also ensures a cleaner data set with the same detector-dependent

influence on the measurement.

43

Chapter 5

Data Balance Function

Measurements

The balance function measurements shown in this chapter use the same form shown

in Equation 3.2:

 

 

 

B(Ay)= 1 {N+_(Ay)—N++(Ay) + N_+(Ay)— [if—JAM}, (5.1)

2 * binsize N+ N.

where N+_(Ay) is the histogram of |y(7r+) — y(7r‘)| for all possible pairs within an
individual event. N+(N_) refers to all the positive(negative) particles used in the

pairing.

5.1 Charged Particle Pairs

To begin the analysis of the balance function in heavy ion collisions, charged particle
pairs are used. This measurement does not rely on particle identification and includes
the most statistics possible that can indicate an overall behavior of the system. Further
studies will show the behavior of pions as outlined in Section 5.3, kaons, or protons

in particular.

44

 

 

 

 

 

 

 

 

o 7 ........................ 4 , 0.7 ........................ . ,
L Charged particle pairs: 1 Charged particle pairs:
0.6 m 0 Central data 1 0.6 [r o Midcentral data‘:
0.5 I . D Midperipheral data: 05 I1 0 A Peripheral data .1
[$90 {40 3
E 0'4 e 0.4 f A X 0 1
3 gm 3 E 4
°° 0 3 Om m 0 3 ,~ 8 6 1
o I 1
02 W03 1 02* 33 1
O [I] ‘ . A i
0 1 ' 8 E W W 0.1 o a «j
g m I . 8 6 g 1
o ............... 0. b 0 ................. M.
a) 0 0 5 1 1.5 2 2 5 l o o 5 1 1.5 2 2 5
An An

Figure 5.1: a) Central (circles) and midperipheral (squares) charged particle balance
function, b) midcentral (diamonds), and peripheral (triangles) charged particle bal-
ance function.

The balance function measured for charged particle pairs is shown in Figure 5.1.
This measurement consists of events from both central and minimum bias triggered
data which pass the event, track, and quality cuts as discussed in Sections 2.5, 2.6, and
4.2.1. The centrality is defined as shown in Section 2.4, with the central multiplicity
events represented by closed circles, the midcentral by open squares, the midperipheral
by closed diamonds, and the peripheral by open triangles. The error bars shown are

statistical only, calculated as shown in Equation 5.2.

._ 1 \/N+-+N++ 2 \/N-++N-- 2
JB'_2*binsize\/( N+ ) + N. i (5.2)

 

 

 

 

 

N+_ is the number of possible [n(particle+) — n(particle')| in a given A77,- bin. Here,
the implicit summation over events is used as discussed in Chapter 3. The argument
used is pseudorapidity (77), as rapidity cannot be calculated without each particle’s
mass known. The subscript 2 refers to the bins used, in this section the maximum
number of bins is 26, with a binsize of 0.1.

A large peak near Ar] = 0 can be seen in these charged particle pair balance
functions. This peak did not agree with the predictions of the balance function shape

in the simulations of reference [47], which seemed more Gaussian. Looking at the

45

dE/da: vs. p plot used for particle identification in Figure 2.9, one can see that there
are particles which have a dE/da: profile similar to that of electrons in the primary
particle data set. However, there should not be many electrons emerging from the
primary particles of the collision. These electrons are most likely the products of
secondary decays. Therefore, those electrons which can be identified are removed.
Here, the A77 axis is plotted out to the largest possible A7] of 2.6, however there does
not appear to be a significant balance function past A7] of 2.0. Future plots will cut

off at An = 2.0.

5.2 Charged Particle Pairs (No Electrons)

The balance function for charged particle pairs with identified electrons removed
is shown in Figure 5.2. The central multiplicity events are represented by circles,
the midcentral by squares, the midperipheral by diamonds, and the peripheral by
triangles. The peak near A7] = 0 has now been diminished, although there appears to
still be some electron contamination in the particles used. This contamination occurs
when the electron dE/da: band crosses the pion, kaon, and proton bands. Hereafter
all measurements labeled charged particle pair in the text are calculated without
identified electrons, unless otherwise stated. It can be seen in Figure 5.2 that the
balance function for charged particle pairs has a narrower width when measured for
central events than peripheral events. The error bars shown are statistical only.

To make a comparison of the balance function of the different centrality bins, a
method to calculate the width is necessary. One simple procedure is to calculate the

weighted average of the balance function. That is:

(An) = Z—éA—i) (5.3)

46

 

 

 

 

 

 

 

 

0.6 """""""""""""" 0.6 Y '7 V ' I V i r V V ' ' ' v ‘ 'ifi'ifi'fi'
[. Charged particle pairs] I. Charged particle pains]
[m . (no identified electrons): : (no identified electrons”
05 ; 0 Ccnmldm , 0'5 f f o Midcentral data]
: [I] Q W C] Midpenpheraldata: :4 f O A Peripheral data I
0.4 r m « 0.4 f 4. I 0 1
E i Q [I] E I ‘i‘
s 03 r o m s 0.3 i 8
a: . a: t 5
I o : t
0.2 ,r W m m 1 0.2 ,- 3 3
O i I A
. . 6 4
0.1 r .9111 . OJ . e A
: 0 i m m ‘ : e a
. ' a . » o 0 e
a) 0 1 1 Am 1 . - 1 1 1 - - 1:. b) 0 I - - ............. .-$-Q_‘
0 0.5 l 1.5 2 0 0.5 l 1.5 2
An An

Figure 5.2: a) Central (circles) and midperipheral (squares) charged particle balance
function, b) midcentral (diamonds), and peripheral (triangles) charged particle bal-
ance function. Charged particles do not include identified electrons.

where B,- is the balance function for each bin of the histogram. For charged particle
pairs, A77,- is the value of the bin 2' calculated at the middle of the bin. The error on
the balance function for each bin 2' is 68,- as given by Equation. 5.2. The summary
of widths for the charged particle pairs at different centralities can be seen in Figure

5.3. The vertical error bars are calculated from statistics only, as in the equation 5.4.

 

 

2 2.16302
+ Zr Bi

(5.4)

 

 

 

’7 ’ 2,3,. 2.41.8.-

Here, i represents the bins of the histogram, dB,- is from Equation. 5.2, and A77,- is
the middle value of the it“ An bin. Estimates for the systematic error will be covered
in Section 5.6.

The horizontal error bars on the horizontal of Figure 5.3 represent the centrality
width of each of the centrality bins. The widths are shown plotted with respect to
centrality bins, however a more physical quantity can be used. The impact parameter
was calculated with a simple geometrical model. The impact parameter refers to how

central or head-on the different types of collisions are, as discussed in Section 1.2.

47

0.6 WA-
OData

Charged particle pairs I
(no identified electrons) , , :
0.55 f [ Peripheral 1

l .

[ Midperipheral
0.5 3 +
+ Midcentral

045C?"‘-""~-....-......---‘
. 100% 80% 60% 40% 20% 0%
Centrality

 

rfi'T-Vvvvv

<An>

...4Al.ka--

 

 

 

Figure 5.3: Summary of widths of charged particle pair balance functions for the four
centrality bins of the data.

The calculation for impact parameter follows:

18’" 27rbdb

#52

max

2 central fraction. (5.5)

This equation relates the ratio of impact parameter (b) over the maximum impact
parameter (bmax), to the fraction of centrality. The summary of balance function
widths for charged particle pairs plotted with respect to the impact parameter is
shown in Figure 5.4.

Another method to measure the width of the balance function is to use a Gaussian
fit to the data. The balance functions shown in reference [47] appear to be of Gaussian
shape. Exponential and power fits were tried, but the Gaussians appear to give the
best fit to the data. To remove any extraneous electron contamination, the first bin of
the charged particle pair balance function was removed for the Gaussian fit. The fits
of the different events can be seen in Figure 5.5. The width, error of the width, and the
X2 per degree of freedom of the fit are given on the figure for each statistical error bar
weighted Gaussian fit. The summary of these Gaussian widths can be seen in Figure

5.6. For both the Gaussian and the weighted average widths shown in Figures 5.4 and

48

 

0.6 rrrrrr I v v v I v v u {ff
. 0 Data

p

.

Charged particle pairs
(no identified electrons)

0.55 :

fifi I
LLAA+IAAALAAA

<An>

 

 

p
p
p
O45..-1...1...1ALL1.4Lr
.

 

Figure 5.4: Summary of widths of charged particle pair balance functions for the four
centrality bins of the data, plotted with respect to impact parameter.

5.6 respectively, it can be seen that the central events have a narrower balance function
than the peripheral events. As centrality increases from peripheral to midperipheral

towards central, the balance function narrows smoothly.

5.3 Pion Pairs

The pion pair balance function is calculated to obtain a measurement of the rapidity
correlation between pions from the event’s source. Using identified pions, the rapidity
is calculated with mass of the pion = 0.13957018 GeV/c2 [61], and can be used as the
argument of the balance function. As the heavy ion events are expected to produce
mostly pions, this measurement is the next logical measurement to extract the event’s
relative time information with sufficient statistics.

The pion pair balance function measured with respect to rapidity is shown in
Figure 5.7. The central events are represented by circles, midcentral by squares, mid-
peripheral by diamonds, and peripheral by triangles. As a comparison, the pion pair
balance function measured as a function of pseudorapidity is shown in Figure 5.8 with

the symbols following the same convention as in the previous figure.

49

 

 

   
    
  
   

  
    
     
 

 

 

 

 

 

 

Charged particle pairs 1 ’ Charged particle pairs:
> . (no identified electrons): : (no identified electrons) *
0 5 r. 0 5 . o .
l -.,_. -O-Ccntral data ‘ P '°-.. --O--Midcentml data 1
i -B-Midpcn'phcral data 1 i i 9. *pmpmm data 1
0.4 ~ 1 0.4 a. r
A E Central at . A - Midcentral n: l
;_- * 0:0.58420006 j p 0:06.“ $0.007 ;
S 0 3 " . 1% :4.84 1 3 0 3 I’m = 1.74 1
an .. . an .
o Midpcn‘pheral m 1 Peripheral m 1
0.2 0:0.03110010 4, 0.2 . 0:0.72220013 1
led=l.4l , finial.”
0.1 f 0.! .
l ., .
f ..... 1 t .
D I. 1 P I .
0 A A A 1 A A 1 A A A n ) O + ..... A 1 A A A I A A
a) 0 0.5 1 1.5 2 b 0 0.5 l 1.5 2
An All

Figure 5.5: Charged particle balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events. The first bin
is removed from the Gaussian fit which is shown by dotted (central, midcentral), or
solid (midperipheral, peripheral) lines.

 

0.75vn.flfir~n.nvavn
: 0 Data 1
, Charged particle pairs +.
, (no identified electrons) .

.O
\l
"r
4

+

 

 

 

E-
510.65~ .
no I +
b : .
0.6» ,
055
0 0.2 0.4 0.6 0.8 1
b/bmam

Figure 5.6: Summary of gaussian fit widths of charged particle pair balance functions
for the four centrality bins of the data, plotted with respect to impact parameter.

50

 

 

 

 

 

 

 

 

(135 .............. A. - - . . (135 ..c . . . . - . . - . . . .
1! pairs (Ay) . E 3 Pairs (AV);
03, ‘ 03; l
: § f [in #1 0 Central data ‘ : + i f O Midcentraldata I
0.25 i [i] § [l] D Midmphm'dm 1 0.25 $% i.) + 4 A Peripheral data .2
I ¢ I 3
A 0.2 ¢ A 0.2 f 1
51>" Ci] [i3 51>: : i f i
flaw 0 ¢ 1 mom} i
O : ,
01} .¢¢¢ 01} i f4 4
, O ' m E i l
0 05 0 05 - 4‘
i i . i i T Q 0 4
0i AAAAAA We 0 AAAAA g,,,g,4,,tb .
a) 0 05 l 15 2 b) 0 05 l 15 2
Ay A)’

Figure 5.7: Pion pair balance function for a) central (circles), midperipheral (squares),
b) midcentral (diamonds), and peripheral (triangles) events. This function is calcu-
lated with respect to Ay.

 

 

 

 

 

 

 

 

0.3 """"""" ' 1 ' ‘ ' 0.3 if I """""""""
i [i] 1; pairs (A71) i 1! pairs (An):
0.25 F $ § ¢ ¢ . (‘cntral data 0'25 : * ¥ * + O Midcentral d8“?
+$ 4] 1:1 Midpcripheral data . A Peripheral data 1
0.2 E Q ‘ 0.2 f % 1
e . '1' q. . e E 3
£05 $¢¢ i gaw- +$§l 1
0_] 1- . w [i] l 0.1 f % if j
4' 0 r1: l E 9 i 3
0% ¢ m 1 0w: 9 i 1
O . [D . . 4 § i Q 4 .
0 ‘ - . A ‘ .. W .1 b 0 I - ‘ . . 9 LI:
3) 0 05 1A 15 2 l 0 05 l 15 2

n An

Figure 5.8: Pion pair balance function for a) central (circles), midperipheral (squares),
b) midcentral (diamonds), and peripheral (triangles) events. This function is calcu-
lated with respect to An.

51

One feature that stands out is the dip near Ag 2 0 in Figure 5.7. A similar, yet
smaller feature is seen in Figure 5.8 near An = 0. This feature appears stronger in
the balance function for pions measured with respect to rapidity, illustrating the ef-
fect of including the momentum in the observable. To understand this feature, the
Hanbury—Brown-Twiss(HBT) [62] effect must be studied. HBT refers to a method of
two-particle intensity interferometry. This method was originally applied to the mea-
surement of the diameter of stars using photon correlations. In the fields of particle
and nuclear physics, this type of technique has been used to study the size of the
emitting source. With bosons, two-particle interferometry studies show an enhance-
ment at a small momentum difference between identical particles. This HBT effect
would enhance the probability that particles of like charge will have a small Ap. This
enhancement then affects the rapidity difference, making a larger value for like-sign
particle differences in low Ay. This enhancement for HBT creates a dip near small
Ag in the balance function. This feature was modeled in reference [63] to include
HBT and Coulomb corrections for like-sign particle repulsion. The result from [63]
is shown in Figure 5.9. Qualitatively, Figure 5.7 shows the same sort of dip at low
Ag, and with a similar relative magnitude to the amplitude of the function. It is also
possible that other effects are manifest in this dip in the measurement. For example,
electron contamination in the pion signal would give an enhancement at low Ay, thus
the measured dip may not be as strong as a corrected balance function. Another
possibility is that this dip is an effect of two—track merging, or even track splitting,
which are estimated to be on the order of only 1% in a HBT-type correlation analysis,
but may affect the balance function [31]. Figure 5.7 is from data, whereas Figure 5.9
does not include complete acceptance effects, and does not appear to die off at Ag 2
2 as the balance function for data does. Overall, however, the HBT/ Coulomb effect
seems to be the major contributor to the local minimum in the pion rapidity balance

function measured at low Ag.

52

 

0.47nafia1fi7fi ....... vamfih,

—Bjurkcn Thcmul Model

' “a." ---'lhcnnal + HBT

"OI-Themul and HBT and Coulomb

 

 

 

Figure 5.9: Pion pair balance function from Jeon & Pratt [63]. The balance function
from the simple thermal Bjorken model (line) has been parameterized and filtered
to roughly provide rough consistency with preliminary STAR measurements. The
inclusion of HBT effects (triangles) gives a dip at small Ay, while the extra addition
of Coulomb interactions (circles) modifies the dip.

To measure the width of the pion balance function measurement, both the
weighted average as given by Equation. 5.3 and a Gaussian fit are used. In the
case of the pion measurement, the first two bins are removed from both the width
calculation and the Gaussian fit. This removes the HBT and other low Ay effects
from the measured function, leaving a smooth curve to fit. A summary plot of all the
pion widths from the weighted average is shown in Figure 5.10, plotted with respect
to impact parameter. In contrast, Figure 5.11 shows the pion balance function widths
calculated with respect to A17.

The Gaussian fits and their X2 values are shown in Figure 5.12 for all four centrality
bins. The summary of the data’s Gaussian widths is shown in Figure 5.13. In the pion

measurements, the general trend can still be seen, that of the peripheral data having

the widest balance function. The width then narrows smoothly to the central events.

53

 

 

 

 

 

 

0.7. .......

:ODatanpairs
: +3
0.65} l
/\ i i
5 I
0.6: g
I l
0.55 g
0 0.2 0.4 0.6 0.8 l

b/b
max

Figure 5.10: Summary of widths of the pion pair balance functions for the four cen-
trality bins of the data, plotted with respect to impact parameter. The first two bins
were removed from the data for the width calculation.

0.8a...-..
:OData

 

1
A l 4 n A n

0.75

ffi‘rl vv

<An>
O
\J

. .
b 4
b 1
. .
0.65 - 4
> J
> 1
b 1

h 1
O 6 ...................
I

 

 

 

Figure 5.11: Summary of widths of the pion pair balance functions (A17) for the four
centrality bins of the data, plotted with respect to impact parameter. The first two
bins were removed from the data for the width calculation.

54

 

 

 

   
  

   
   

   
  
   
 
  
  
 

 

 

 

 

0.35 M 0.35. -, h--.
1! pairs (Ay) , u pairs (Ay):
0.3 "O-Central am 1 0-3 -. . 7
-a-Midpcriphcmldam ‘*'M'§°¢"""da‘3 ;
0.25 - 0.25 ,- +Penpnmmm 1
Central fit ‘ Midcentral fit i
3 0-2 “057710.010? 9. 0-2 0:0.61610011 1
3 ' x’ld = 0.95 ; S , -, 1% = 1.05 3
£110.15 L 1 m0.15 , -. . 1
’ Midpcn'phcral m . .. Pcnphcral ru .

: “004010.013 : § o:0.686:0.017

0'] ;’ 121d: 1.21 0'1 x’ld=0.94

0.05 r 0.05 ’ I
i , . I
0 f A - L A 1 r n - - 1 ) 0 A - n ..... 1 i
a) o 0.5 1 1.5 2 b o 0.5 1 1.5 2

 

  

 

Figure 5.12: Pion pair balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events. This function
is calculated with respect to Ag, and the first two bins are removed from the Gaussian
fit (shown by dotted (central, midcentral), or solid (midperipheral, peripheral) lines.)

0.7fi..........,...,. J
:ODatanpairs j
A065} 5
3: : + :
S : :
Cos} g
055* MW
0 0.2 0.4 0.6 0.8 l

b/b

max

Figure 5.13: Summary of gaussian fit widths of pion pair balance functions for the
four centrality bins of the data, plotted with respect to impact parameter. The first

 

 

 

 

two bins were removed from the data for the fit.

55

5.4 Other Features of the Balance Function

The main comparison used in the balance function has been the width of the function.
However, there are other features that are useful to understand the physics of the data

set.

5.4.1 Integral

The integral of the balance function measurement may also give useful information.
For a perfect detector, the balance function is constructed so that it normalizes to
unity. However for a physical detector, the integral may show more information.
The possibility of measuring average fluctuations of charge to give a clear signal for
the QGP [64], [65] was discussed in Chapter 3. In Jeon and Pratt [63], a relationship
between the balance function and the fluctuation of charge is derived. This paper
considers the case of a balance function which is measured with respect to rapidity
difference (Ay), over a complete rapidity region Y. This is the same method that is

used in the analysis in this dissertation. For Q = N+ — N_, and NC}, = N+ + N- [63],

 

 

<(Q—<Q>)2)_,_ Y0, , <62)
(m _1 f0 lAyB(Ay)+O(<NCh>), (5.6)

and 0 is the correction, which for electric charge in relativistic heavy ion collisions
is usually less than 5%, as the amount of produced charges is much greater than
the net charge. In the case of this dissertation, Y = 2.6 for the maximum rapidity
difference. Reference [63] emphasizes the convenience that Equation 5.6 modifies the
balance functions into one number that may give more information than just the
width. However, they caution to not analyze charge fluctuations as a function of the
varying rapidity window sizes. It is also possible to see trivially that the integral of
the balance function depends upon the overall rapidity window Y. The changes in

the balance function shape for different rapidity windows are discussed in Chapter 7.

56

 

Balance function integrals ]

 

 

 

 

 

 

 

 

 

 

particle type centrality integral error
pion(Ay) central 0.208 i 0.003
pion(Ay) midcentral 0.233 :1: 0.004
pion(Ay) midperipheral 0.252 :l: 0.005
pion(Ay) peripheral 0.257 :l: 0.006
charged particles(An) central 0.368 :l: 0.003
charged particles(An) midcentral 0.397 :l: 0.004
charged particles(An) midperipheral 0.399 i 0.005
charged particles(An) peripheral 0.384 :l: 0.006

 

 

Table 5.1: Table of the integral values for each of the balance function measurements
shown.

The integral for each of the balance function measurements presented is given in
Table 5.1. For both pions and charged particle pairs, the central events have a smaller
integral than the peripheral events, with a smooth variation in between. Recalling the
dN/dn histograms in Section 4.2.3, the central and peripheral events did not appear
to differ as a function of the acceptance. However, there is a form [63] of Equation.

5.6, which corrects the balance function integral for a detector’s acceptance:

was)” =1——/0YdAyB(Ay)* (1— %) +0 (($52)). (5.7)

 

 

The left hand term of Equation. 5.7 is one—fourth of the D variable for measuring
charge fluctuations which is discussed in Jeon and Koch [64]. Thus, a direct compar-

ison to the predictions can be made. The integral with the acceptance corrections

used to calculate W can be found in Table 5.2. Figure 5.14 shows that for all
charged particle pairs, there does not appear to be a change in D over centrality.
The predictions in reference [64] are that a QGP would have D R: l, and a hadronic
resonance gas would have a D m 3. Measuring no effect with respect to centrality for
the charged particle pairs seems to indicate that there is no QGP formed, however,

that conclusion is dependent upon the theoretical model used in reference [64]. Un-

derstanding the detector acceptance dependency within the measured D is also not

57

[ll l!

 

 

 

 

] Balance function acceptance corrected integrals ]

 

 

 

 

 

particle type centrality integral error
pion(Ay) central 0.170 :l: 0.007
pion(Ay) midcentral 0.188 :1: 0.008
pion(Ay) midperipheral 0.200 i 0.011
pion(Ay) peripheral 0.202 :1: 0.011
charged particles(A77) central 0.301 :l: 0.007
charged particles(A7)) midcentral 0.322 :1: 0.008
charged particles(An) midperipheral 0.316 :1: 0.011
charged particles(An) peripheral 0.300 :l: 0.013

 

 

 

 

 

 

 

Table 5.2: Table of the acceptance corrected integral values for each of the balance
function measurements shown.

 

 

 

 

 

 

 

 

 

 

Charged particle pairs (no electrons) 3 Charged particle pairs (no electrons)
3.5 E- Hadron Gas

3 L J 2.9 f 1
2.5 - '1 2.8 '—+—‘ ' + '.
Q 2 :- 1‘ C3 : :
1.5 E- .3 2.7 ; —+—~ i ' ,
=. GP 5 ’ 3
l : Q : 2.51 -,
0-5 E“ ‘3 : 1

O i 1 1 1 L 1 1 1 1 1 1 1 1 1 m1 1 1 1 1 i 2.5 i 1 1 1 L 1 1 1 1 1 1 1 L 1 1 1 1 1 1 1
a) 0 0.2 0.4 0.6 0.8 1 b) 0 0.2 0.4 0.6 0.8 1

b/b b/b
max max

Figure 5.14: Plot of the values of D calculated from the balance function acceptance
corrected integral for charged particle pairs. Plot a) is on a scale showing the QGP
and hadronic gas predictions of reference [64], and b) is on a focused scale.

trivial. Results are shown here merely for illustration of the relationship between the
balance function and other measurements. Given the large statistical errors in the D
measurement, it appears that the width and shape of the balance function provide

more information about the collision dynamics than the integral alone.

5.4.2 Particle Count

Table 5.3 summarizes the number of positive and negative particles of each type for
the balance function analysis events. The column labeled “( part per event)” gives

the number of particles per events that were used in the particle pairing for that

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

particle cent- total N+ total N - events ( part all ( part per
rality per event) events event-all)
pi c 20,197,232 20,335,228 95,089 426.26 95,204 425.74
pi mc 9,690,679 9,753,709 71,976 270.15 72,311 268.90
pi mp 1,559,441 1,562,546 44,858 69.60 45,164 69.13
pi p 213,323 213,500 31,628 13.50 32,039 13.32
R c 1,433,027 1,341,057 94,825 29.25 95,204 29.14
k mc 669,546 626,425 71,201 18.20 72,311 17.92
k mp 99,101 92,985 37,663 5.10 45,164 4.25
k p 7,510 7,182 6,039 2.43 32,039 0.46
p c 989,831 612,450 94,824 16.90 95,204 16.83
p mc 468,434 297,696 71,003 10.79 72,311 10.59
p mp 73,826 49,694 32,410 3.81 45,164 2.73
p p 5,519 4,026 4,138 2.31 32,039 0.30
charged c 39,422,376 38,271,168 95,204 816.07 95,204 816.07
charged mc 18,202,012 17,708,452 72,311 496.61 72,311 496.61
Charged mp 2,699,237 2,633,753 45,164 118.08 45,164 118.08
charged p 347,772 341,156 32,039 21.50 32,039 21.50
charged* c 38,057,296 36,913,272 95,195 787.55 95,204 787.47
charged* mc 17,589,620 17,096,200 72,287 479.83 72,311 479.68
Charged* mp 2,614,759 2,548,377 45,130 114.41 45,164 114.32
charged* p 337,592 330,531 32,025 20.86 32,039 20.85

 

 

Table 5.3: Table of the number of positive and negative particles for each particle
type used in balance function measurements. Centralities: c=central, mc=midcentral,
mp=midperipheral, p=peripheral. Particle charged is all charged particles and
charged* refers to all charged particles, no electrons. Ratios are calculated both for
the average number of particles per event which had two or more of those particles “(
part per event)”, and the average number of particles per event, covering all events
used, “( part per event-all)”.

59

 

Ratios of (charged) to (pion) contributions to the balance function

[ centrality ] N+~ ] d(N+-) ] N++ ] d(N++) [ N—— ] d(N——)]
central 3.42399 0.00006 3.55447 0.00006 3.29857 0.00006
midcentral 3.21054 0.00010 3.32616 0.00010 3.09944 0.00009
midperipheral 2.76539 0.00040 2.84634 0.00041 2.68847 0.00039
peripheral 2.46190 0.00217 2.52195 0.00225 2.41134 0.00217

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.4: Table of the ratio of the amount of particle pairs(charged/pion) used in
balance function analysis for each term (N+-, etc.).

given particle. For particles such as kaons and protons, it can be seen that in the very
peripheral events, the events used in the analysis had on average 2 of these particles
per event. Considering all events used (“all events”), there is less than 1 proton or
kaon per event on average for the most peripheral events. However, the more central
events do have a larger number of both protons and kaons. These statistics will become
significant for those particular particle’s balance function measurements, addressed
later in Section 5.5.

The contribution of the amount of particle pairs to the balance function measured
is summarized in Table 5.4. Given are the ratios of charged particle pairs to pion pairs
used for each centrality. It can be seen that there are more charged particle pairs
to the pion pairs used in central events compared to peripheral events. This could
be an indication of a changing number of non-pions in the central events, perhaps
more kaons or protons. However, the particle identification cut is fixed based upon
minimum bias events. It is more likely that there are merely more pions in the central
events which are not identified with the fixed number of a cut. With respect to the
balance function width, it is expected that the more massive particles, such as kaons
and protons, should display a strong centrality dependence [47]. The charged particle
pair data set combines the pion, proton, and kaon signals, which could explain the
larger relative difference between central and peripheral balance function widths in

the combined all charged particle data set.

60

 

 

(115.! ...... fifif.rr- ‘ ()_15 -, ..................
. I

l KpIiMAy) ; [ KpuMAy)
: ' 03'de i i o mum
I E] Midpen'pbenldata : r A A W433
0.1 f j 01 1

A ; A

.. , . 1

s 1 s

an 1 + a:

 

 

 

 

0.E-osi[]+[ [31+ 0.[-05[[
i i] 11811111.... E ll i1 1
awor” 3110.5 ‘ .

0.5 1 1.5 2 .
Ay Ay

 

 

 

Figure 5.15: Kaon pair balance function for a) central (circles), midperipheral
(squares), b) midcentral (diamonds), and peripheral (triangles) events. This function
is calculated with respect to Ay.)

5.5 Kaon Pairs

The pre