Study of Single Particle Spectra and
Two Particle Correlations
in Au+Au Collisions at 4 - 11 A GeV

Tatsuya CHUJO

A dissertation submitted to the Doctoral Programin Physics, the University of Tsukuba
in partial ful llment of the requirements for the
degree of Doctor of Philosophy (Science)
February, 2000

Abstract
We present the results of single particle spectra in Au+Au collisions at the energy of 4 and 11 A
GeV, and the results of high statistic + + two-particle correlations in Yano-Koonin Podgoretskii(YKP) parameterization in Au+Au collisions at 11 A GeV, measured by the E866 experiment using
the Alternating Gradient Synchrotron (AGS) at Brookhaven National Laboratory (BNL).
For the systematic studies of hadron productions in relativistic heavy-ion collisions near midrapidity region, we have constructed a rotatable magnetic spectrometer, called the Forward Spectrometer. It is designed for the good particle identi cation capability under the condition of highparticle density and for the wide range of kinematic coverage in Au+Au collisions at AGS. Using thedata taken by the Forward Spectrometer, single particle spectra and HBT correlations are analyzed.
For single particle analysis, we have measured transverse mass spectra for  , K +, protons anddeuterons at 4 and 11 A GeV as a function of centrality. For the transverse mass spectra, we haveobserved the mass dependence of mean transverse mass < mt > ;m0 ; for the particle with heavier
mass, the larger < mt > ;m0 is seen. In addition, the shapes of the mt spectrum for protons havea shoulder-arm shape in central collisions at low mt region. On the other hand, the pion spectra at
11 A GeV have a concave shape at low mt region, which can be attributed, in part, to the e ectsof a transverse velocity eld of a created system. Observed shoulder-arm shapes of protons and themass dependence of mean transverse mass are consistent with an expanding source.
In + + HBT correlation analysis in YKP parameterization, it is found that the extracted
source size parameter in the transverse direction, Rt , decreases from 5.21  0.17 fm to 3.73  0.26
fm with increasing the transverse pair momentum KT from 0.1 GeV/c to 0.45 GeV/c. This gradualdecease of RT with increasing KT is also consistent with the expanding source.
Motivated by these observations that both single particle spectra and HBT correlations are
qualitatively consistent with the picture of an expanding source, the simultaneous analysis of bothdata using the nite expanding source model is carried out for the rst time at AGS energy. From
this analysis, the freeze-out temperature T and the transverse expansion velocity t in Au+Au
central collisions at 11 A GeV are extracted uniquely; T = 95  5 MeV and < t >= 0:53  0:03,
which can be considered as more reliable values than those from the conventional analysis of single
particle spectra alone.
Compared with T = 120  12 MeV and < t >= 0:37  0:08, reported from the former analysis
at CERN-SPS 158 A GeV Pb+Pb collisions, T is larger at SPS than at AGS, while < t > is larger
at AGS than at SPS at the con dence level of 95 %. The softening of equation of state by the quark
gluon plasma (QGP) formation is one of the qualitative interpretation for the observed small < t >at SPS energy than that at AGS. Thus, the reduction of < t > at SPS dose not contradict the
picture of the QGP formation in Pb+Pb collisions at 158 A GeV, which is indicated by the recent
observation of anomalous J= suppression and low-mass dilepton enhancement.

Contents
1 Introduction

1

2 Relativistic Heavy Ion Collisions

11

1.1 Physics of High Energy Heavy-Ion Collisions . . . . . . . . . . . . . . 1
1.1.1 Signatures of the quark gluon plasma . . . . . . . . . . . . . . 2
1.2 Features Observed in Heavy Ion Collisions at the Energy of 158 A GeV 5
1.2.1 J/ suppression . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Low-mass dilepton enhancement . . . . . . . . . . . . . . . . . 5
1.2.3 Collective behavior . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Contributions of Author . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Participant-Spectator Picture . . . . . . . . . . . . . . . . . . . . . .
2.2 Single Particle Spectra at AGS Energies . . . . . . . . . . . . . . . .
2.2.1 mt spectra in p+A collisions . . . . . . . . . . . . . . . . . . .
2.2.2 mt spectra in A+A collisions . . . . . . . . . . . . . . . . . . .
2.2.3 Interpretation with radial ow model . . . . . . . . . . . . . .
2.3 Two Particle Correlation . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Hanbury-Brown and Twiss (HBT) e ect . . . . . . . . . . . .
2.3.2 KT dependence of source radius and its interpretation with radial ow model . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Simultaneous Interpretation of Single Particle Spectra and HBT with
Radial Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 The E866 Experimental Setup

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 AGS Accelerator Facility . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Global Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i

11
12
12
14
17
19
19
23
23

26

26
27
29

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3.3.1 Beam counter system (BTOT and HOLE) . . . . . . . . . . .
3.3.2 Bull's eye counter (BE) . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Zero-degree calorimeter (ZCAL) . . . . . . . . . . . . . . . . .
3.3.4 New multiplicity array (NMA) . . . . . . . . . . . . . . . . . .
3.4 Forward Spectrometer (FS) . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Forward spectrometer coordinate . . . . . . . . . . . . . . . .
3.4.2 Dipole magnets (FM1 and FM2) . . . . . . . . . . . . . . . .
3.4.3 Tracking detectors (FT and TPC) . . . . . . . . . . . . . . . .
3.4.4 Time-of- ight wall (FTOF) . . . . . . . . . . . . . . . . . . .
3.4.5 Heavy metal shields . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Henry Higgins Spectrometer (HH) . . . . . . . . . . . . . . . . . . . .
3.6 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . .

29
32
32
33
35
36
36
37
40
42
42
43

4 Run Conditions

44

5 Data Reductions

48

4.1
4.2
4.3
4.4

Beam Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trigger Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analyzed Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Geometry calibration . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 FT calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 TPC calibration . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 FTOF calibration . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Track Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Track nding by TPC . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Track con rmation by FT . . . . . . . . . . . . . . . . . . . .
5.2.3 Track matching at analyzing magnet . . . . . . . . . . . . . .
5.2.4 Target cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.5 Fiducial cut . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6 Association with FTOF . . . . . . . . . . . . . . . . . . . . .
5.3 Momentum Determination . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Particle Identi cation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Tracking Eciency Correction . . . . . . . . . . . . . . . . . . . . . .

44
44
44
46
48
48
49
49
50
51
52
52
53
54
54
56
56
58
62

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5.5.1 TPC hardware eciency . . . . . . . . . . . . . . . . . . . . .
5.5.2 Track Reconstruction Software Eciency . . . . . . . . . . . .
5.6 FTOF Detection Eciency . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Decay Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Experimental Results I { Single Particle Spectra
6.1
6.2
6.3
6.4

6.5
6.6
6.7
6.8

Interaction Trigger Cross Section . . . . . . . . . . . . . . . . . . . .
Invariant Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . .
Centrality Cut Criteria . . . . . . . . . . . . . . . . . . . . . . . . . .
Transverse Mass Spectra . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Centrality dependence at Ebeam = 4 A GeV . . . . . . . . . .
6.4.2 Centrality dependence at Ebeam = 11 A GeV . . . . . . . . . .
Rapidity Density Distributions . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Centrality dependence at Ebeam = 4 A GeV . . . . . . . . . .
6.5.2 Centrality dependence at Ebeam = 11 A GeV . . . . . . . . . .
Centrality Dependence of Mean Transverse Mass . . . . . . . . . . . .
Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of the Results for Single Particle Spectra . . . . . . . . . .

62
64
65
67

68

68
69
70
75
75
75
76
76
83
90
90
91

7 Experimental Results II { Two-Particle Correlations

93

8 Discussions

110

7.1 Coulomb Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Cut Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.2.1 Two-track separation . . . . . . . . . . . . . . . . . . . . . . . 96
7.2.2 Other cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Statistics and Event Mixing . . . . . . . . . . . . . . . . . . . . . . . 99
7.4 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.4.1 YKP parameterization . . . . . . . . . . . . . . . . . . . . . . 100
7.4.2 Standard side-out-long parameterization . . . . . . . . . . . . 100
7.5 Results in YKP Parameterization (KT dependence of RT ) . . . . . . 102
7.6 Results in the Standard Side-Out-Long Parameterization . . . . . . . 106
7.7 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.8 Summary of the Results for Two Particle Correlations . . . . . . . . . 109
8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

iv
8.2 Expanding Source Model . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2.1 Emission function . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2.2 Static Gaussian source . . . . . . . . . . . . . . . . . . . . . . 112
8.2.3 Boost-invariant source . . . . . . . . . . . . . . . . . . . . . . 113
8.2.4 Finite expanding source . . . . . . . . . . . . . . . . . . . . . 113
8.2.5 mt spectrum in nite expanding source . . . . . . . . . . . . . 115
8.3 Fitting Results by the Finite Expanding Source Model . . . . . . . . 116
8.3.1 Fitting results of single particle spectra . . . . . . . . . . . . . 116
8.3.2 Fitting results of two-particle correlations . . . . . . . . . . . 122
8.3.3 Simultaneous analysis of single particle spectra and HBT . . . 123
8.4 Excitation Function of < t > and T . . . . . . . . . . . . . . . . . . 125
8.4.1 Excitation functions from SIS to AGS and SPS . . . . . . . . 125
8.4.2 Comparison of t and T between AGS and SPS energies . . . 125

9 Conclusions

130

A Kinematic Variables

132

B The BNL-AGS E802/866 Collaboration List

133

Acknowledgments
I would like to express my great appreciation to Prof. Y. Miake for his introducing
me to the exciting eld of physics. His appropriate and valuable advice show me theway to promote the present work, and I could learn a lot from him as an experimental
physicist. I express my thanks to Prof. K. Yagi for his encouragement to complete
this work.
I wish to express my thanks to Dr. K. Kurita for leading me to the E866 experiment. His appropriate and kind advice helped me a lot to accomplish this work. I
appreciate Dr. Y. Akiba for his useful advice and discussions in the data analysis
and for his hospitality. I express my special thanks to Dr. K. Shigaki for his advice
on this work and for his friendship at BNL. I would like to express my appreciation
to Prof. S. Nagamiya for his continuous encouragement.
I wish to express great thanks to Prof. C. Chasman for his gladly accepting me
as a member of the E866 collaborator and for his guidance and hospitality as thegroup leader of Heavy Ion Research Group at BNL and the co-spokes person of theE866 experiment. I appreciate Dr. H. Hamagaki, the co-spokes person of the E866,
for his stimulate advice. I express great thanks to all the collaborators of the E866
experiment. I owe Dr. J. H. Lee for his appropriate advice in HBT analysis and data
reductions, Dr. K. Ashktorab for his help and advice. I wish to express thanks to
Ms. E. Kim for her extensive calibration of the drift chambers and data reductions.
Here I acknowledge all the members of the high energy nuclear physics group at
University of Tsukuba. I would like to thank Dr. A. Kumagai for his help in thedata analysis, especially the FTOF calibration part. I would like to appreciate to
Dr. H. Sako, who gave me useful advice on many aspects of the analysis. I express
my thanks to Dr. S. Nishimura for useful discussion of HBT analysis and valuableadvice. I express my thanks Dr. M. Nishimura, Mr. S. Sato, Mr. S. Kato and Dr.
K. Enosawa for their friendship and discussions on the present work. I appreciate
Mr. Y. Okuma, Mr. Y. Yokota, Ms. Y. Miyamoto and Ms. R. Higuchi for theirv

vi
encouragement and friendship. I appreciate Mr. A. Kiyomichi, Mr. T. Hirano, Mr.
T. Yoshikawa, Ms. M. Suzuki, Mr. K. Koseki, Ms. M. Aizawa, Mr. D. Hakozaki, Mr.
Y. Akutsu, Mr. M. Ono, Mr. N. Maeda and Mr. H. Tsuruoka for their friendshipand kindness.
This work is supported in part by the U.S. Department of Energy under contracts with ANL, BNL, the University of California at Riverside, Columbia University,
LLNL, and MIT, in part by NASA under a contract with the University of California,
by the U.S.-Japan High Energy Physics Collaboration Treaty, Grant-in-Aid for Scienti c Research (Specially Promoted Research and International Scienti c Research)of the Ministry of Education, the University of Tsukuba Special Research Projects,
and by the Research Fellowships of the Japan Society for the Promotion of Sciencefor Young Scientists.
Finally, I would like to express my great appreciation to my parents for theirsupport and encouragement to accomplish this work.

Chapter 1
Introduction
In this chapter, the physics of high energy heavy-ion collisions are introduced. After reviewing new results observed at CERN-SPS, we present
the thesis motivation.

1.1 Physics of High Energy Heavy-Ion Collisions
Heavy-ion collisions at high energies (E=m  1) o er an unique opportunity to probehighly excited dense nuclear matter in the laboratory. The driving force for suchstudies both on experimental and theoretical side, is the expectation that a entirely
new form of nuclear matter may be created from relativistic heavy-ion reactions.
That form of matter is called the quark-gluon plasma (QGP), which is quantum
chromodynamics (QCD) analogue of the plasma phase of ordinary atomic matter.
According to the lattice QCD calculations, it is predicted that the phase transition
from hadronic phase to QGP phase occurs at energy densities   2 GeV/fm3, corresponding to a critical temperature Tc = 140200 MeV in baryon-free matter [1, 2].
Fig. 1.1 shows the schematic phase diagram of strongly interacting matter. As the diagram shows, it is expected that a QGP phase transition occurs at a high temperature
and/or a high baryon density.
In order to understand the properties of the nuclear matter under the extreme
conditions and hopefully to capture the signal of QGP formation, various experiments have been taken place both at the Alternating Gradient Synchrotron (AGS) at
Brookhaven National Laboratory (BNL), and the Super Proton Synchrotron (SPS)at CERN. In the near future, the Relativistic Heavy Ion Collider (RHIC) at BNL and1

2

Figure 1.1: The phase diagram of strongly interacting matter, showing the hadronic
phase at low temperature and baryon density, the mixed phase (transition region) andthe QGP phase. The curves and lines illustrate possible trajectories in supernovae
explosions, Big Bang evolution, and in heavy-ion reactions at present (AGS and SPS)
and future (RHIC and LHC) accelerators. This picture was produced by LHC-ALICE
Collaboration.
the Large Hadron Collider (LHC) at CERN will be operated. Table 1.1 shows thelist of current and proposed facilities dedicated to accelerations of heavy-ion beams.

1.1.1 Signatures of the quark gluon plasma
Unlike such ordinary plasmas, however, the decon ned quarks and gluons of a QGP
can not be observed directly because of the fundamental con ning property of thephysical QCD vacuum. What we can observe is hadronic and leptonic remnantsof the transient QGP state or hadronic gas state. In order to catch the signals of
QGP formation, a variety of probes are proposed. As the distinct signature of theexistence of QGP is still under the study, the best way to prove the presence of a QGP
phase is to measure several signatures simultaneously. Recent reviews of the possible
QGP signatures and experimental status can be found in the literatures [3, 4]. The
signatures can be categorized in terms of decon nement, chiral symmetry restorationand collectivity.

1. Introduction3pAccelerator Location Ion beam Momentums y Commissioning date[A  GeV/c] [GeV]AGS
BNL 16 O, 28 Si
14.6
5.4
Oct. 1986
197 Au11.4
4.8
Apr. 1992
SPS
CERN 16O, 32S
200
19.4
Sep. 1986
208 Pb
158
17.4
Nov. 1994
197
Au
RHIC
BNL
100
200
Spring 2000
208 Pb
LHC
CERN
3150
6300
2005 (project)
Table 1.1: List of current and proposed facilities, the ion beams, the corresponding
beam momentum and the center of mass energy. (y per nucleon-nucleon pair for asymmetric colliding system.)(1) Decon nement

In QGP phase, quarks and gluons are not con ned to hadrons but move freely insigni cantly large volume. In such decon ned medium, long range term of quark
potential is expected to be suppressed due to the screening by surrounding quarks.
Such e ect in electromagnetism is known as Debye screening and the reduced rangeof potential is characterized by Debye radius. Application of Debye screening e ect
to the QCD was proposed as a signature of QGP formation by Matsui and Satz [5, 6]
in 1986. If radius of a meson is larger than the Debye radius, which is determinedby the plasma temperature and density, the meson cannot survive in the plasma.
It is proposed that J/ meson made of cc quarks is suitable for the detection of
Debye screening e ect because of the following reasons; 1) as J/ is leptonic decay
object, decay products do not interact strongly with other hadrons, thus it is expected
penetrative probe for the early stage of the collisions, 2) J/ 's are produced in thevery early stage of the collision, 3) hadronic interaction of J/ is expected to be not
too high ( N  6 mb), thus it has the information of initial state condition of thecollisions.
As discussed in 1.2.1, J/ suppression e ect is indeed observed in Pb+Pb collisions at the energy of 158 GeV per nucleon [7]. It is highly controversial whether it
can be interpreted as \true" signal of QGP formation.

(2) Chiral symmetry restoration

4
At low temperature, the QCD vacuum is characterized by non vanishing density ofvirtual quark - anti quark pairs and broken chiral symmetry. At high temperature,
not only the decon nement phase transition takes place, but the chiral symmetry is
expected to be restored.
Particle mass is determined by its interaction in the medium. The mass of mesonis expected to be modi ed in high density and high temperature in the medium. Thisphenomenon is closely related to the restoration of chiral symmetry of the QCD.
Mass of vector mesons have been proposed as a probe of the chiral symmetry
restoration. One of the important proposal is detection of the change of  meson
mass, which is expected to dominate at the low-mass region of dilepton spectrum. As
discussed in 1.2.2, strong enhancement in the low-mass region of di-electron spectra
are reported in Pb+Au collisions at CERN-SPS [9, 10]. But the same enhancement
has been also observed in S+Au collisions, this is the one of the hot subject to interpretthe results.

(3) Collision dynamics and equation of state

Study of collective motion of produced hadrons in nal state is expected to provide
information on the dynamics of heavy ion collisions. With a hydrodynamical view ofcollisions, collective motion is governed by a pressure gradient of compressed nuclear
matter at the early stage of collision. In the case of a phase transition from theordinal nuclear to the quark gluon plasma, it is expected that the equation of stateshould exhibit a corresponding softening due to the increased number of degrees of
freedom [11]. Thus, the observation of collective motion is crucially important tovalidate the hydrodynamical description of dynamics.
If the phase transition is the rst order, the equation of state will be \softest"at the critical temperature Tc. Such softening is expected to a ect the dynamic
evolution of the system because the internal pressure drops at Tc. Thus an observationof excitation function of the transverse collective ow can be a probe for the QGP
formation; drop in the excitation function of collective ow indicates threshold energyof the QGP formation.

1. Introduction

5

1.2 Features Observed in Heavy Ion Collisions at
the Energy of 158 A GeVBased on the motivation to create QGP in the laboratory, lead ion beams at theenergy of 158 GeV per nucleon are accelerated at CERN-SPS. Several interesting
features of the observation have been reported, among which are the suppression ofJ/ production in Pb+Pb collisions, the enhancement of low mass lepton pairs andthe collective behavior of hadron production. These features have been interpreted as
signatures of QGP formation under the key words of decon nement, chiral restorationand collectivity, although clear conclusion has not been made yet.

1.2.1 J/ suppression
As we mentioned in 1.1.1, the formation of the cc bound states would be suppressed in
the QGP due to the screening of the attractive color force which binds the two quarks
together. Charmonia are produced in the very early stage of the collision. Strongly
bound states, like the J/ , are expected to interact weakly with other hadrons. As aresult, and more than other signatures, J/ suppression have its original information
through the di erent stages of the reacting medium.
Recently, NA50 collaboration have been reported the anomalous J/ suppressionin central Pb+Pb collisions at 158 A GeV [7]. Fig. 1.2 shows the ratio of the J/ to the
Drell-Yan cross-sections divided by the exponentially decreasing function accounting
for normal nuclear absorption. The horizontal axis is the mean nuclear path length
L. In this gure, the J/ cross-section measured for peripheral Pb+Pb collisions
agrees with the general pattern determined from proton, oxygen and sulphur-induced
reactions which can be accounted for by nuclear absorption. On the other hand,in central collisions a clear departure from the pattern of the nuclear absorption isobserved and is claimed as an anomalous J/ suppression. This experimental nding
can be considered as a strong indication of the production of a QGP phase in central
Pb+Pb collisions [7].

1.2.2 Low-mass dilepton enhancement
Since leptons interact only electromagnetically and their mean free path is considerably larger than the size of the collision volume, the production of lepton pairs is

6

Figure 1.2: The ratio of the J/ to the Drell-Yan cross-sections (J= /DY ) divided
by the exponentially decreasing function accounting for normal nuclear absorption [7].
The horizontal axis is the mean nuclear path length L.
commonly accepted as a promising probe to study the dynamical evolution of nuclear
collisions [8]. They are created during the entire space-time evolution of the system,
beginning at the early hot stage up to the point where the hadrons cease to interact. Because of this penetrative properties, the production of dileptons re ects theinitial temperature of the system and the matter created in the reactions, i.e., a QGP
through quark annihilation (qq ! l+ l;) or a high density hadron gas through pionannihilation (+; ! l+l;).
The CERES/NA45 Collaboration have been carried out the systematic study of+;e e pair productions in low-mass region at CERN energies with proton, S and Pb
beams. Fig. 1.3 shows the inclusive e+e; mass spectrum in Pb+Au at 158 A GeV [9].
The data compared to the sum of the expected contributions from hadron decay (solid
curve). It is observed the enhancement at low-mass region in the spectrum comparedto the contributions from hadron decay.
This dielectron spectrum at the SPS energy seems explainable in hydrodynamic
approach assuming the creation of a thermalized QGP [12]. On the other hand, thesame data can be reproduced in the framework of microscopic hadronic transportmodels incorporating mass shifts of vector mesons [13, 14].

1. Introduction

7

Figure 1.3: Inclusive e+ e; mass spectrum in Pb+Au collisions at 158 A GeV normalized to the observed charged-particle density [9]. The data compared to the sum of
the expected contributions from hadron decays (solid curve).

1.2.3 Collective behavior
Transverse collective expansion is expected to be more prominent in the heavy ion
collision than in that of lighter projectiles, owing to the smaller surface-to-volume
ratio, the larger \ reball" volume and the longer duration of expansion. Such indication can be visible in the single particle transverse mass spectra. The measured
distributions can be reproduced by invoking a radial velocity eld in transverse direction with a temperature at freeze-out stage. But the single particle spectra alone
can not distinguish between an elevated temperature and a boost due to transverse
collective expansion, because of the strong anti-correlation between transverse ow
velocity t and temperature T in the model. To avoid its uncertainty, the combination of single particle spectra with two-particle correlation results can resolve thisessential ambiguity and determine t and T uniquely.
Recently the simultaneous analysis of single particle spectra and two particle correlation function have been reported by the NA49 experiment in central Pb+Pb
collisions at 158 A GeV [17]. In the analysis, The correlation functions for negative
hadrons have been tted with the Yano-Koonin-Podgoretskii (YKP) parameterization [15] as formulated by Heinz [16] and the extracted source parameters have been

8

Figure 1.4: Allowed regions of freeze-out temperature T vs. transverse velocity t
for central Pb+Pb collisions near mid-rapidity [17]. The boundaries are derived
from negative hadron two-particle correlation functions and from negative hadronand deuteron transverse mass spectra within the framework of the expanding source
model. Contours are drawn at  around tted values.
studied as a function of rapidity and transverse momentum of the pair.
In the framework of an expanding hadronic source model, the analysis of correlation functions yields information about the source size and the duration of thefreeze-out process, as well as the ratio t2=T , where t is the transverse velocity and
T is the freeze-out temperature. The ambiguity between temperature and expansion velocity implicit in the ratio can then be lifted by considering single particle
spectra. Fig. 1.4 shows the allowed regions of freeze-out temperature vs. transverse
velocity for central Pb+Pb collisions near mid-rapidity. The boundaries are derived
from two-particle correlation functions for negative hadrons and from negative hadronand deuteron transverse mass spectra within the framework of the expanding source
model. Contours are drawn at  around tted values. In the paper [17], they
conclude that both features of single particle spectra and the average pair transverse momentum KT dependence of two-particle correlations can be described by
T = 120  12 MeV and t = 0:55  0:12 at CERN-SPS energy.

1. Introduction

9

1.3 Thesis Motivation
As reviewed in the previous section, it is claimed that the signals of the QGP formation emerge at 158 GeV per nucleon with lead ion beams. To understand the validityof QGP formation in Pb+Pb collisions at CERN-SPS, however, it is important toinvestigate whether not only a single but also several signals could be explained simultaneously by a consistent model. Furthermore, since transverse expansion velocityt and temperature T at freeze-out are the most fundamental values to characterize
the system created in the reactions, the observation of these parameters at lower
beam energy, for example at AGS, and comparisons of extracted ( t ; T ) at the two
di erent energies can provide the important input about the question of QGP formation at SPS energy. From this point of view, the simultaneous analysis of single
particle spectra and two-particle correlation at AGS energy can be considered as asuitable experimental observable both for the consistency of the picture of expansion
and the evolution of the system from AGS to SPS energy.
In this thesis, we present the results of single particle spectra in Au+Au collisions at the energy of 4 and 11 A GeV, and the rst results of high statistic ++
two-particle correlations in Yano-Koonin Podgoretskii (YKP) parameterization in
Au+Au collisions at 11 GeV per nucleon, measured by the E866 experiment using
the Alternating Gradient Synchrotron (AGS) at Brookhaven National Laboratory(BNL). Within the framework of the expanding source model, t and T are extracted
from the two di erent analysis, then these results are compared to that of other beamenergies.
The thesis is organized as follows: Chapter 2 describe the features of relativistic
heavy ion collisions, particular emphasis on the single particle spectra and two particle correlation at AGS energy. Chapter 3 describes the experimental setup, Chapter 4gives the run condition, Chapter 5 gives calibrations of detectors, particle identi cation and corrections. In Chapter 6, the experimental results of single particle spectraand in Chapter 7, that of two particle correlations are presented. In Chapter 8, it
is discussed the transverse velocity eld and freeze-out temperature using the nite
expanding source model. Finally, conclusions are made in Chapter 9.

10

1.4 Contributions of Author
The author has contributed to AGS-E866 experiment as following.

 Calibrated the Time Projection Chambers and the Time-of-Fight counter.
 Analyzed the single particle spectra at 4 A GeV beam energy and publishedthe results [18].

 Analyzed single particle spectra and two-particle correlations at 11 A GeV beamenergy.

Chapter 2
Relativistic Heavy Ion Collisions
In this chapter, participant-spectator picture are introduced for theunderstanding of global features of relativistic heavy-ion collisions. Next,
we review the results of single particle spectra and its interpretations
at AGS energies. For two-particle correlation, the formalism of HBT,
existing experimental results and its understanding are shown. Finally,
simultaneous interpretation of single particle spectra and HBT with radialow model are presented.

2.1 Participant-Spectator Picture
Nuclei are extended objects, and therefore their geometry plays an important role inheavy ion collisions. Fig. 2.1 shows a sketch of reaction at a high energy between symmetric Lorentz contracted nuclei in the center of mass frame. The impact parameterb separates the nucleons into participants with primary nucleon-nucleon collisions,and spectators, which proceed with little perturbation along the original direction.
The use of this clear cut geometry between spectators and participants is justi ed by
the fact that at very high energies the size of nucleons is larger than their Compton
wavelength and the nuclear radius is larger than the interaction length of  1.8 fm.
Cross sections, number of participants and related kinematic quantities are usuallycalculated from simple geometrical considerations [19].
Based on Participant-Spectator picture, the centrality of the Au+Au collision are
de ned by the measurements of the energy deposition of the calorimeter located atzero-degree in the beam direction (see Section 6.3).
11

12

Figure 2.1: Participant-Spectator picture of a high energy heavy-ion collision with
impact parameter b. On the left (a), the two incoming nuclei in the center of massframe are shown. On the right (b), after the collision, the nucleons are separated toparticipants, projectile spectators and target spectators.

2.2 Single Particle Spectra at AGS Energies
Single particle spectra, i.e., transverse momentum pt distributions are often used
to study the properties of hadrons at the freeze-out stage. The term of freeze-out
refers here to the nal decoupling of hadrons from the strong interaction. In thissection, previous experimental results of single particle spectra in proton-nucleus (pA)
collisions and nucleus-nucleus (AA) collisions at AGS energies are reviewed.

2.2.1 mt spectra in p+A collisions
Transverse momentum spectra are usually presented in terms of the invariant di erential cross section,3
1 d2 = 1 d2 ;E ddp3 = 2p
dp dy 2m dm dy

t

t

t

t

(2.1)

q

where pt is the transverse momentum, y is the rapidity, mt = p2t + m20 is the transverse mass and m0 is the rest mass of the particle. The use of the transverse mass
is suggested by the experimental fact that the cross section of given particle species
in proton-proton collisions is better described by an exponential in mt rather thanpt [20]. The spectra shape and even the absolute cross sections of di erent mass

2. Relativistic Heavy Ion Collisions

13

particles are similar when plotted as a function of mt [4]. This observation leads tothe notion of \mt scaling". For parameterizations of the invariant cross section, theinverse slope parameter T0 is used as,3
 t
E ddp3 / exp ; m
T :

(2.2)

0

A number of theoretical explanations have been proposed for mt scaling [21, 22, 23],
often assuming a local thermal equilibrium and the inverse slope parameter T0 as
temperature.
In a model of purely thermalized system, the transverse mass spectrum is obtainedby integrating over rapidity using the modi ed Bessel function K1 , which behaves
asymptotically like a decreasing exponential [22]:1 dN  m K  mt 
t 1
mt dmtT0
1 dN  pm exp; mt t
mt dmt
T0

(2.3)
for

mt  1:T0(2.4)

At the AGS energy, transverse mass spectra for , K  and protons have beenmeasured by E802 experiment in p+Be and p+Au reactions at 14.6 GeV/c close
to mid-rapidity (y = 1.2-1.4). Within the pt range covered by the experiment, thespectra have a single exponential shape as a function of mt in both p+Be and p+Au
reactions as shown in Fig 2.2 [24]. And all these spectra are almost in parallel, i.e.,roughly the same inverse slope parameter. In Fig. 2.3, the inverse slope parameters,
extracted by the tting in the single exponential function for the spectra in p+Be,
p+Au and central Si+Au system are summarized. The extracted slopes are 140 
160 MeV for , K  and protons in proton induced reactions. Thus, so called \mtscaling" is seen at p+p and p+A reactions at the AGS energy.

14

Figure 2.2: Invariant cross sections as a function of transverse kinetic energy mt ; m0for , K  and protons in (a) p+Be and (b) p+Au collisions at the beam momentumof 14.6 A GeV/c [24]. The rapidity range is 1.2  y  1.4.

2.2.2 mt spectra in A+A collisions
In contrast to pA reactions, mt spectra in nucleus-nucleus central collisions at midrapidity do not scale in the same inverse slope parameter for all particle species.
Fig. 2.4 shows the particle mass dependence of inverse slope parameters in p+Au,
Si+Al (central) and Au+Au (central) collisions at AGS energies. A distinct di erenceof the inverse slope parameters between pA and AA system can be seen. While theinverse slope parameter in p+Au system remain almost constant T0 ' 150 MeV as afunction of particle mass, the parameters in AA system increase as a function of themass. Furthermore, for a given mass, the heavier the colliding system, the higher the
inverse slope parameter.
Fig. 2.5 shows the invariant cross section for pions and proton at mid-rapidity
in Au+Au central collisions at 11.6 A GeV/c [25]. The proton spectrum has visibly
convex shape at low mt region (mt ; m0  0.3 GeV/c2). Unlike the measured spectrain p+A and Si+A at similar incident energy presented above, the proton spectrumin Au+Au reactions cannot be described by a single exponential function. On theother hand,  spectra show strong rise at low mt region, and therefore these spectra

2. Relativistic Heavy Ion Collisions

15

Figure 2.3: Comparison of inverse slope parameters T0 in p+Be, p+Au and central
Si+Au collisions [24]. Inverse slope parameters are obtained by exponential ts with
the invariant cross section.
also cannot describe in the single exponential function. The rise of  spectra at
low mt might be attributed to, in part, by the decay of resonances, e.g.,  which
tend to produce relatively low momentum pions, and in part by the collective ow
phenomenon. The shape of K  spectra in these reactions have the single exponential
shape [26].
In summary, three important qualitative features in AA collisions can be listed asfollowing.
1. For a given collision system, the inverse slope parameter T0 is a function of theparticle mass, i.e., the heavier the particle, the higher T0 .
2. The values of T0 is larger for heavier collision systems.
3. The deviation from the single exponential shape in the spectra for protons andpions.
At the next subsection, it will be shown that these features in AA collision system
are qualitatively explained by introducing a transverse velocity eld in the createdsystem.

16

Figure 2.4: Inverse slope parameters as a function of particle mass in p+Au, Si+Al(central) at 14.6 A GeV/c and Au+Au (central) at 11.6 A GeV/c (E802 data).

Figure 2.5: Invariant cross section for pions and protons at the mid-rapidity as afunction of transverse kinetic energy mt ; m0 in Au+Au central collisions at 11.6 A
GeV/c [25]. The data for protons (solid points) is scaled by a factor of 0.5. The open
circles are the data for ; and the solid squares are that for +. The inset shows theratio of ;=+ as a function of mt ; m0 .

2. Relativistic Heavy Ion Collisions

17

2.2.3 Interpretation with radial ow model
Most successful description of the di erent slope parameters and the change of shapeobserved in mt spectra in AA collisions is given by the model including the transverse expanding velocity eld together with a moderate temperature of a thermalized
system. With the presence of the transverse velocity eld, the mass dependence of
inverse slope parameters can be understood qualitatively as follows.
In case of a purely thermal motion, all particles (irrespective of their mass) would
move with the same average kinetic energy determined by the temperature, i.e.,< Ekine > Tthermal:(2.5)
On the other hand, in case of a purely collective motion, all particles would movewith the same velocity c and consequently, the average kinetic energy would increase
proportional to their mass m0, since2< Ecollective > m02 c :(2.6)
Under the assumption of complete decoupling between the thermal and collective
motion of the particle, superpositions of both types of motion will give the massdependence of;< Ekine > = < Ethermal > + < Ecollective >2= Tthermal + m0 <2 t > ;(2.7)

where < t > is the averaged collective velocity for all particle species. The inverse
slope parameter T0 is proportional to the average transverse kinetic energy and isgiven as,T0 / Tthermal + m0 < t >2 :(2.8)
Furthermore, because of this velocity dependence, for heavier collision system,
which presumably stronger collective transverse ow, the value of T0 is expected to belarger. The above observations are thus qualitatively consistent with the hypothesisof transverse hydrodynamic ow produced in the heavy-ion collisions. Quantitatively,
the phenomenological hydrodynamical model proposed by Schnedermann et. al. [27]18

Figure 2.6: Deuteron, proton, K + and + spectra in Si+Au at 14.6 A GeV/c (E802
data [28]) at y = 1.3 compared to calculated spectra including transverse expansionof the source at temperature T [29]. Dashed lines are the spectra of a source at T =120 MeV expanding transversely < t > = 0.39 (left) and a source at T = 140 MeV
and < t > = 0.33 (right). The arrows indicate the lower limit of the t region.
can be applied to the single particle spectra for the extraction of transverse velocity
and temperature at freeze-out.
In this model, collective ow e ects are incorporated into transverse mass spectra
as following:dN / Z R rdrm I ( pt sinh() )K ( mt cosh() );
(2.9)
t 0
1
m dm
T
T
t

t

0

where modi ed Bessel function I0 and K1 , a parameter  = tanh;1( t ) and a transverse velocity pro le of t (r) = tmax(r=R) . In the above equation, the parameter T
is interpreted as a temperature at freeze-out stage in the model.
Fig. 2.6 shows the tting results by the hydrodynamical ow model in central
Si+Au data at 14.6 A GeV/c near mid-rapidity. Dashed lines are the spectra of asource at T = 120 MeV and < t > = 0.39 (left) and a source at T = 140 MeV
and < t > = 0.33 (right). As show in Fig. 2.6, all the particle spectra from pions
to deuterons are shown to be reproduced very well with two parameters, T and t .
Problem with this tting is, however, no unique parameter set for ( t , T ); there isstrong correlation between t and T and it is dicult to determine the unique set ofparameter without ambiguity from single particle spectra alone.

2. Relativistic Heavy Ion Collisions

19

2.3 Two Particle Correlation
In addition to single particle spectra of hadrons, two particle correlation of identicalparticle provides information on the space-time evolution of the system created inheavy-ion collisions at freeze-out 1 .
In a pioneering experiment in 1956, Hanbury-Brown and Twiss measured the angular diameter of a star using the correlation between two photons [30]. Their ideas
have been applied to subatomic physics to measure source sizes using di erent typesof particles. The phenomenon of the space-time correlation or the energy-momentum
correlation of detected identical particles emitted from an extended source is known
as the Hanbury-Brown and Twiss (HBT) e ect. HBT e ect originates from thesymmetrization of the wave function of bosons (Bose-Einstein correlation), or antisymmetrization for fermions (Fermi-Dirac correlation). In the following subsections,the formalism of two particle correlation function for identical bosons [31], the transverse momentum dependence of source size extracted by HBT measurements and thee ect of the expanding source to the extracted source size are described.

2.3.1 Hanbury-Brown and Twiss (HBT) e ect
We consider a source of identical bosons (see Fig. 2.7) and describe in plane waves.
To calculate the two particle momentum distribution P2(p1 ; p2), we need the source
density (x) in space-time, the random phase (x) of a boson emitted at x and theprobability amplitude A(p; x) for production of a boson of momentum p at x. The
probability amplitude of detecting a boson of four-momentum p1 at x01 and anotherboson of four-momentum p2 at x02 which are emitted at x1 and x2 , is given after
symmetrization by:
12= p1 fA(p1 ; x1)ei(x1 ) A(p2; x2 )ei(x2 )eip1 (x1;x1 )eip2(x2 ;x2)
2
+ A(p1; x2 )ei(x2 )A(p2 ; x1)ei(x1 ) eip1(x2 ;x1) eip2(x1 ;x2) g
 ei(x1 )+i(x2 ) (p1; p2 ; x1; x2 ; x01; x02 );
0

0

0

0

(2.10)

where the particles are on-shell, eip1 (x1;x1 ) and the other similar terms describe thepropagation of a particle. As we can not distinguish which points of the source0

1 Cease to interact strongly among hadrons in a reaction zone. The particle are emitted freely

without nal state interaction after freeze-out.

20

x1
p1

x1’

p2

x2

x2’

Emitting Sourceof Particles

Figure 2.7: A boson of four-momentum p1 is detected at the space-time point x01 andanother identical boson with four-momentum p2 is detected at x02 . They are emitted
from the points x1 and x2 of an extended source. The solid lines joining x1 to x01 andx2 to x02 and the dashed lines joining x1 to x02 and x2 to x01 are possible trajectories
for the bosons.
the detected bosons are coming from, we have to integrate the above probability
amplitude over all pairs of points of the source. Then we have to take the modulus
square to obtain the two-particle momentum distribution:Z
1
P2(p1; p2) = 2! d4x1 d4x2 d4y1d4y2(x1 )(x2 )(y1)(y2)j 12 j2:(2.11)
Due to the randomness of the phase , above integral is non-zero contribution
only when x1 = y1 and x2 = y2 or x1 = y2 and x2 = y1. By using the symmetry ofunder exchange of x1 and x2 (Bose-Einstein statistics):Z

P2(p1 ; p2) = d4x1 d4x2(x1 )(x2 )j (p1; p2 ; x1; x2 ; x01; x02 )j2 :Using the de nition of 12 in Eq. 2.10, we nd:(2.12)Z
1
P2 (p1; p2) = 2 d4x1 d4x2 (x1 )(x2 )fA2(p1 ; x1)A2 (p2; x2 ) + A2(p1 ; x2)A2 (p2; x1 )+ A(p1 ; x1)A(p2 ; x1)e;i(p1 ;p2)x1 A(p1; x2 )A(p2; x2 )ei(p1;p2)x2

2. Relativistic Heavy Ion Collisions

21+ A(p1 ; x1)A(p2 ; x1)ei(p1 ;p2)x1 A(p1; x2 )A(p2; x2 )e;i(p1;p2)x2 g: (2.13)
After same renaming of the integration variables, we nally get: Z2P2(p1; p2) = P1 (p1)P1(p2 ) 1 + d4 xeiqxe (x; p1 ; p2) ;(2.14)

where

q = Zp1 ; p2;(2.15)P1 (p) = d4xf (p; x);(2.16)
f (p; x) = A2 (p; x)(x)(2.17)
andqq
f (p1; x)f (p2; x)f (p1; x)f (p2; x)qe (x; p1; p2) = qR 4
(2.18)=
R
P1(p1)P2 (p1)d x1 f (p1; x1 ) d4 x2f (p2; x2 )P1 (p) is the single-particle momentum distribution and f (p; x) is the phase space
distribution function.
The two-particle correlation function C2 (p1; p2) is de ned by the ratio of the probability of observing p1 and p2 in coincidence over to the probability of observing themseparately:where

C2(p1; p2)  PP(2p(p)1P; p(2p) ) = 1 + j~e (q; p1; p2)j2;
1 1(2.19)1 2Z

~e (q; p1; p2) = d4xeiqxe (x; p1; p2)(2.20)
is the Fourier transform of e . Thus, the two particle correlation function C2(p1 ; p2) isdirectly related to the Fourier transform of the e ective density, as given by Eq. 2.18.
In many applications, one parameterizes the e ective density in the form of a Gaussian
distribution,
2
2
2
2 

e (x; p1; p2) = 42R NR R  exp ; 2xR2 ; 2yR2 ; 2zR2 ; 2t2 ;x y z t

x

y

z

t

(2.21)

where the normalization constant N and the standard deviations Rx, Ry , Rz and tdepend on p1 and p2 . The constant coecients in Eq. 2.21 are chosen in such a waythat the e ective density is normalized to

22
C2
2

1.75

1.5

1/R

1.25

1

0.75

0.5

0.25

0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Qinv (GeV/c)

Figure 2.8: Two-particle correlation function of a source with Rinv = 6 fm and  =1 as a function of qinv .

Z

N (p1 ; p2) = d4xe (x; p1 ; p2):
(2.22)We choose the collision axis as the longitudinal z-axis and we place the source
center-of-mass at the spatial origin. The parameter Rz measures the longitudinal'radius' of the source along the beam direction. Rx measures the transverse radius of
the source along the direction towards the detectors and Ry measures the transverse
radius of the source perpendicular to the line joining the source and the detector.
The Fourier transform of the e ective source density Eq. 2.21 is2 2
2 2 2 2 R2 q 2
~e (q; p1; p2) = N exp ; Rx2qx ; y2 y ; Rz2qz ; t2qt :(2.23)
Finally, we get the correlation function for identical bosons in Gaussian parameterization by:

C2(q) = 1 + Ne;R2x qx2 ;R2y qy2 ;R2z qz2;t2 qt2 :(2.24)
In one-dimensional case, correlation function is expressed as a function of qinv :
where

2
;
C2(qinv ) = 1 + e;R2inv qinv

(2.25)

q
qinv = qx2 + qy2 + qz2 ; q02;

(2.26)

2. Relativistic Heavy Ion Collisions

23

where q0 is the energy di erence between two particles, and qx, qy and qz are the momentum di erence for each direction. Fig. 2.8 shows a typical two-particle correlation
function of a source with Rinv = 6 fm and  = 1 as a function of qinv .

2.3.2 KT dependence of source radius and its interpretation
with radial ow modelTwo particle correlations have been rst used to study the properties of a hadron
emitting system created in hadronic collisions under the assumption of a static source.
The expansion of a source that is not static but exhibits a hydrodynamical scaling
evolution has been discussed by Shuryak [32] and Bjorken [33]. Within the frameworkof such models, it has been further shown that the dependence of the correlation
function on kinematic variables of the pion pair, such as average transverse momentumof pair KT , is an important diagnostic tool for the study of an expanding system [34][40].
The E877 Collaboration has measured positively charged pion correlation as afunction of the transverse mass of the pair as shown in Fig. 2.9 [41]. The transverseradii (RT ) are found to systematically decrease with increasing transverse mass ofthe pair and increasing rapidity of the pair. This behavior is consistent with theassumption of a collectively expanding system.
q 2 2Fig. 2.10 shows the transverse mass of the pair MT (= KT + m0 ) dependence of
RT parameter at mid-rapidity in the expanding source model [42]. Solid lines in thegure is for pion pairs and dashed lines is for kaon pairs in di erent transverse ow
in terms of rapidity f . As shown in this gure, the decrease of RT parameter withincreasing KT appear only if the t (or f ) is non-zero. Thus the KT dependence of
RT parameter can be a diagnostic probe to prove the existence of the transverse owof the system.

2.4 Simultaneous Interpretation of Single Particle
Spectra and HBT with Radial Flow ModelAs shown in Section 2.2.3 and 2.3.2, existing results of both single particle spectra
and two-particle correlations in Au+Au collisions at AGS energy are qualitatively
consistent with the picture of expanding source.

24

Figure 2.9: Transverse radius RT from 2-dimensional ++ correlation as a function of
the transverse mass of the pair, MT for di erent rapidites from AGS-E877 preliminary
data [41]. The data point from K +K + correlations (star) is also shown.

Figure 2.10: RT as a function of the transverse mass of pair MT at mid-rapidity ofpair in the expanding source model [42]. Solid lines is for pions and dashed lines isfor kaons in di erent transverse ow rapidity f .

2. Relativistic Heavy Ion Collisions

25

Motivated by the such experimental observations, simultaneous analysis of single particle spectra and two-particle correlations using the nite expanding source isconsidered as one of the best way to access to the dynamical parameters, such astransverse velocity t and freeze-out temperature T of the system created in relativistic heavy-ion reactions. This simultaneous analysis can resolve the ambiguities
between t and T of the source, that are unavoidable when single and two-particle
spectra are analyzed separately.
At CERN-SPS energy, as we mentioned in Section 1.2.3, NA49 collaboration has
measured both single particle spectra and two-particle correlations, and extracted theparameter set of t and T by the simultaneous analysis of both type of spectra using
the nite expanding source model. The comparison of the ( t ; T ) in the same analysis
method at AGS with SPS energy is expected to shed light on the understanding ofevolution of the collision dynamics with constancy of the model.

Chapter 3
The E866 Experimental Setup
The E866 experimental setup are described in this chapter. AGS accelerator facility, the global detectors, and spectrometer system are explainedin each section.

3.1 Overview
In this chapter, the experimental setup of BNL-AGS-E866 experiment is described.
Fig. 3.1 shows the plan view of the E866 experiment. We have adopted two-arm
spectrometer system to cover the wide kinematic acceptance. The upper part withrespect to the beam line in Fig. 3.1 is called the Henry Higgins spectrometer (HH),
the lower part is the Forward Spectrometer (FS). In addition to these spectrometersystem, the global detectors are equipped to determine the event geometry of thecollision; the beam counter system (BTOT and HOLE), the zero-degree C erenkov
counter (Bull's Eye), the zero-degree calorimeter (ZCAL), and the multiplicity counter(NMA). By using the apparatus, E866 experiment focus on the studies of systematic
measurements of hadron production in Au+Au collisions at  11 A GeV as described
below [43].
1. To measure particle spectra as a function of rapidity and transverse momentum.
2. To study the dependence of these spectra on centrality of the collisions byutilizing the event characterization detectors.
3. To measure two particle correlations.
26

3. The E866 Experimental Setup

27

Figure 3.1: The plan view of the E866 spectrometer. The upper part with respect tothe beam line is called the Henry Higgins spectrometer (HH), the lower part is the
Forward Spectrometer (FS).
4. To look for systematic variations of cross sections with di erent beams andtargets for understanding of the reaction dynamics.
5. To search an evidence for production of highly dense nuclear matter, or ultimately, the QGP.
In the following sections, features and speci cations of above detectors are described. In section 3.2, AGS accelerator facility is explained. Section 3.3 is on the
global detectors, and section 3.4 is on the Forward Spectrometer, which is used forthe analysis. In section 3.5, only the features of the Henry Higgins spectrometer(HH) are concisely described. Finally, in section 3.6, the data acquisition system are
mentioned.

3.2 AGS Accelerator Facility
Alternating Gradient Synchrotron (AGS) at Brookhaven National Laboratory is oneof the accelerator to be able to accelerate heavy ions like Au up to 10  15 A GeV

28

Figure 3.2: A schematic diagram of the AGS accelerator facility. The future collider
ring, RHIC, is also shown in the upper part of the gure.

3. The E866 Experimental Setup

29

beam energy. The schematic diagram of the AGS accelerator facility is shown in
Fig. 3.2. The AGS complex consists of Tandem Van de Graa , the AGS Booster andthe AGS main ring.
In order to extract Au beams at 11 A GeV/c, the Au ions are accelerated up to
1 A MeV at the Tandem Van de Graa with a charge state of 13+, rst. Then the
Au beams are transfered to the Booster and accelerated to 192 A MeV with a charge
state 33+. Before the injection to the AGS main ring, Au+33 ions are stripped theirelectrons up to Au+77 , then are transported to the AGS main ring and accelerated up
to the nal momentum, i.e., 11 A GeV/c [45]. At the experimental area, Au beams
are transported with a charge 79 with 37  104 per spill at the target position. The
repetition cycle is 4 sec, and at top is 1 sec/spill. In 1995/1996 run, in addition
to 11 A GeV/c beam energy, two di erent lower energy of Au beams are available forthe rst time. The exact numbers of these energy in the 1995/1996 run are 2.003 and4.04 GeV per nucleon in terms of a kinematic energy and 11.69 GeV/c per nucleon
in terms of a momentum [46].

3.3 Global Detectors
In this section, the beam counter system (BTOT and HOLE), Bull's Eye counter(BE), the zero-degree calorimeter (ZCAL) and the New multiplicity array (NMA) aredescribed. BTOT and HOLE are used to de ne the BEAM trigger (see Section 4.3).
BE is set to determine the interaction trigger (INT ) by means of a measurement of
the total number of projectile spectator's charges. ZCAL is placed to measure thetotal kinetic energy of the charged and neutral particles and give an information for a
event characterization. NMA is placed around the target to measure the multiplicity
of pions and it also give a measure of the centrality event by event.

3.3.1 Beam counter system (BTOT and HOLE)
The beam counter system consists of BTOT and HOLE counters. BTOT is aC erenkov counter with a quartz radiator to detect the incoming Au beam signals.
The quartz has 51 mm height, 76 mm width and 200 m thickness and positioned
at 171 cm upstream of the target. Two 2-inch photo-multiplier tubes, Hamamatsu
R2083, are equipped on its both sides for readout the signals. The resolution (q =q)
is estimated to be  1.9 %. BTOT provides a time-zero signal for the time-of- ight

30

2pt(GeV/c)pt(GeV/c)

Acceptance for Single Particle (FS)
+

π

1.5

π-

1.5

1

0.5
0

2

1

0.5

0

1

2

3

0

4

0

1

2

3

2

K

1.5

+

0.5
0

Proton

1

0.5

0

1

2

3

4

y
pt(GeV/c)

2

1.5

1

4

y
pt(GeV/c)pt(GeV/c)

y

0

0

1

2

3

4

y

2

Deuteron

1.5
1

0.5
0

0

1

2

3

4

y

Figure 3.3: The coverage in the rapidity y and transverse-momentum pt space of the
Forward Spectrometer; for + (top left), ; (top right), K + (middle left), protons(middle right) and deuterons (bottom left). In each panel, mid-rapidity of 4 A GeV(solid arrow) and 11 A GeV (broken arrow) are also shown.

3. The E866 Experimental Setup

+

31

+

π π Acceptance in Yππ - KT space

KT (GeV/c)

0.8
0.7

0.6
0.5

0.4
0.3
0.2

0.1
0

1

1.2

1.4

1.6

ycm

1.8

2

2.2

2.4

2.6

2.8

3

Yππ

Figure 3.4: The acceptance of ++ pairs. The horizontal axis is the rapidity of pair,and the vertical axis is the average transverse momentum of pair. The arrow indicatesthe mid-rapidity ycm at the energy of 11 A GeV.

32
counter. HOLE is a scintillation counter with a circular 10 mm diameter hole at thecenter, positioned at 145cm upstream from the target. To reject beam halo, only particles passing through its 10 mm hole are regarded as valid Au beams. Speci cationsof the beam counter system is listed in Table 3.1.
Counter
BTOT
HOLE
BE
Material
Quartz Plastic Scintillator
QuartzThickness [mm]
0.2
3.2
0.3
Width [mm]
76.2
76.2
20 cm diameter
Height [mm]
50.8
50.8
;
Hole Diameter [mm];10.
;Distance from the Target [cm] ;171
;145
950
Table 3.1: Speci cations of the beam counter system (BTOT and HOLE) and Bull's
Eye counter (BE). A negative value in the row of \distance from the target" denotes
upstream of the target.

3.3.2 Bull's eye counter (BE)
Bull's Eye counter, located at 9.5 m downstream from the target, is equipped to de ne
an interaction trigger of beam particles with target nucleus by means of measurementsof the total number of projectile spectator's charges. BE is a C erenkov counter, using
a quartz (20 cm diameter and 300 m thickness) as a radiator, with a light di user
on the surface. As shown in Fig. 3.5, the C erenkov light is read out by eight 2-inch
photo-multiplier tubes, Hamamatsu H1161, arranged radially around the radiator.
The measured charge smaller than that of the beam particle provides the minimum
bias interaction trigger. The resolution, q =Z is  6.7 % [47]. Speci cations of Bull's
Eye counter is also listed in Table 3.1.

3.3.3 Zero-degree calorimeter (ZCAL)
To determine a geometry of a Au+Au collisions, Zero Degree Calorimeter (ZCAL) is
placed at the zero degree in the laboratory frame, 11.7 m downstream from the target.
ZCAL is the Fe-Scint. sandwiched-type hadronic calorimeter. ZCAL measured theenergy of projectile fragments after collisions and it provides an estimation of the3. The E866 Experimental Setup

33

Figure 3.5: A beam view of Bull's Eye counter [47].
impact parameter of the interaction. ZCAL has 1.8 m length in the beam direction,
which is divided into two components, called H1 and H2. The basic stacking unit iscomposed of a 10 mm iron plate followed by a 3 mm scintillator sheet, sandwiched
between two TiO2 coated Mylar sheets in order to di use the light and to protect thesurfaces [48]. There are 32 stacking units in H1, 106 in H2. Eight light signals fromeach section are transmitted via wave length shifter plates to photo-multipliers at therear of ZCAL. The sizes of the ZCAL have 60 cm height, 60 cm width and 193 cm
length corresponding to 8.9 interaction lengths. The energy resolution of the ZCALp
was E = E = 2:88  0:02 [GeV1=2 ] in 1993 run [49].

3.3.4 New multiplicity array (NMA)
In addition to the Zero-degree colorimeter (ZCAL), an additional global detector, theNew Multiplicity Array (NMA) has been built for E866 [50, 51, 52]. It is designedwith the aim of providing a measure of the total number of particles and pseudorapidity distributions in a large solid angle. A second goal is to obtain a measure of
the reaction plane for intermediate impact parameter reaction.
The NMA is an array of 346 C erenkov light detecting modules. A typical module

34

Figure 3.6: New Multiplicity Array (NMA) module [51].
shown in Fig. 3.6. Each module is composed of a clear plastic mated directly to a
photo-multiplier tube. Since the index of refraction of the plastic is about 1.5, theC erenkov light begins to turn on at = v=c = 0:67. This feature allows the detector
to be sensitive mostly to fast charged particles, bulk of which are the produced charged
pions. Furthermore, we need the detector to be sensitive to the produced neutral pions
as well. We achieved this by placing a sheet of lead in front of the clear plastic. The
photons from the decayed 0s would then convert to e+ e; pairs in the lead. The
thickness of lead is chosen so that the conversion probability is 25 %. This yields,on the average, one particle response for each 0 , thus putting 0 roughly on equal
footing with the charged pion. The lead sheet does not a ect the charged hadrons
much, since the nuclear interaction length in lead is much longer. The added bene tof having a lead sheet in front of the module is that essentially all of the low-energy
;electrons from the target are blocked out.
The modules are arranged in close-packed fashion to cover the polar angle  from7o to 112:3o. Instead of using the variable , we usually use the variable pseudorapidity  = ;ln(tan(=2)). For relativistic particles, (  1),   y, the rapidity
variable. In our case, particle identi cation is not possible, we resort to using . The
array is segmented in such a way that the ;bite  is some convenient number,most of them begin 0.20. The array covers the azimuth angle , with gaps left open
for free passage of particles from the target to the spectrometers. The array is also
segmented in , with most of the modules having a -bite  of 10o. Fig. 3.7 showsa schematic view of New Multiplicity Array (NMA). The response of each module issuch that it can distinguish a single hit, double hits, etc. One then adds up hits fromall the modules to yield a raw multiplicity. The segmentations in  and  allow us toobtain the dN=d and dN=d (reaction plane) on an event-by-event basis.

3. The E866 Experimental Setup

35

Figure 3.7: A schematic view of New Multiplicity Array (NMA) [50].

3.4 Forward Spectrometer (FS)
To cover the mid-rapidity region in Au+Au collisions at 11 A GeV/c, the Forward
Spectrometer (FS) is newly equipped in 1993. FS is the rotatable spectrometer from
6 degrees to 27 degrees. In order to handle high multiplicity events and to achievea reasonable tracking and particle identi cation, the Forward Spectrometer has been
designed by considering the following points.

 Small solid angle (5 msr).
 Sweeping magnet (FM1) and heavy metal shields.
 Combination of time projection chambers and drift chambers.
 Finely segmented Time-of- ight wall detector.
The small solid angle reduces the number of tracks per event in the tracking
chambers. The sweeping magnet and heavy metal shields reject background particles. Time projection chambers reconstruct tracks eciently under high multiple
tracks. The nely segmented Time-of- ight wall reduces ineciencies caused by pileup events. The performance of the spectrometer is summarized in Table 3.2 [44].
The Forward Spectrometer consists of heavy metal shields, dipole magnets, two
tracking stations and nely segmented high resolution Time-of- ight wall. Each tracking station has a TPC with two drift chambers on both sides. In the following sections,each detector of FS is described.

36
FTOF

TPC2

Z

TPC1
Y
Snout

X

O TargetFigure 3.8: The Forward spectrometer coordinate.

3.4.1 Forward spectrometer coordinate
A Forward Spectrometer (FS) coordinate is de ned as follows: The origin is at thetarget position; the z-axis called the 6-degree line is a line perpendicular to the surfacesof the magnets and the tracking chambers, the y-axis is upward direction; the x-axis istaken so that the coordinate system to be the right handed system. The spectrometer
angle is de ned as the angle between the incident beam line and the z-axis of the FS
coordinates.

3.4.2 Dipole magnets (FM1 and FM2)
The Forward Spectrometer has two dipole magnets, named FM1 and FM2, respectively. FM1 is placed in front of the rst tracking chamber station to remove background particles with low momentum. FM2 is placed between the tracking chamber
stations as an analyzing magnet. FM2 has an 94 cm e ective length and about 0.9
Tm bending power at its full strength. Field maps of FM1 and FM2 are measured
with Hall probes. The speci cations are summarized in Table 3.3.

3. The E866 Experimental Setup

37
Performance
5 [msr]6 ; 27 [degree]
1:5 ; 5 [%]Solid angle
Setting angle
Momentum resolution (p=p)(proton, 0 < p  5[GeV/c])
Time of ight resolution, TOF ()76 [psec]
 ; K PID separation (2:5)
0 < p  2:5 [GeV/c]
K ; p PID separation (2:5)
0 < p  3:5 [GeV/c]
Table 3.2: Performance of the Forward Spectrometer [44].
MagnetFM1
FM2
Distance from the target [mm]
1754.
3505.
Field Gap Size (x  y  z) [mm3]229:  159:  508:
597:  283:  762:
Yoke Size (x  y  z) [mm3]546:  616:  800:
1346:  1442:  1156:
E ective Field Length [cm]
623.2
943.4
Maximum B  dL [Tm]
0.249 (at 4 kG setting) 0.572 (at 6 kG setting)
Table 3.3: Speci cations of the dipole magnets.

3.4.3 Tracking detectors (FT and TPC)
The spectrometer has two tracking chamber stations, named FTR1 and FTR2 infront and behind the analyzing magnet, FM2. Each tracking station has one TPC
with two drift chambers on both sides of a TPC. The time projection chambers areused as the primary track- nding devices, using three-dimensional track information.
Drift chambers improve spatial resolution of the tracks.

Time projection chamber (TPC)
A time projection chamber (TPC) is able to get three-dimensional hit positions ofa charged particle's trajectory by simultaneous measurements a drift time and a hit
position on an end cap Multi Wire Proportional Chambers (MWPC). TPC has an
excellent pattern recognition capability even under the multi-track events, comparedwith a two dimensional chamber, e.g. a drift chamber.
TPC consists of a eld cage, an end-cap wire chamber, a gating grid plane and agas cage, as shown in Fig. 3.9 [53] with the coordinate system. There are two TPCs

38

Y

X

Z

Figure 3.9: A schematic structure of the time projection chamber [54].
at each tracking station, called TPC1 and TPC2, placed upstream and downstreamof FM2, respectively. We use Ar-isobutane (75:25) gas mixture for TPC1 and P-20
for TPC2 [54].
After the passage of charged particles in a TPC cage, electron-positive ion pairs
are created along the path, and electrons drift in the eld cage along the drift eld tothe gating grid plane. When the gating grid is opened by a trigger signal, the electrons
pass through to the Multi Wire Proportional Chamber (MWPC). Strong eld aroundthe anode wires multiplies electrons and form an avalanche of secondary electrons.
These electrons are observed as the anode current output. A eld distortion due tothe positive ions from the anode wires are suppressed by the gating grid systems. The
performance of TPC1 and TPC2 is summarized in Table 3.4.

3. The E866 Experimental Setup

39

Chamber
TPC1
TPC2
Frame Size (x  y  z) [mm3]406:  450:  352: 965:  660:  450:
Field Cage Size (x  y  z) [mm3] 300:  248:  280: 800:  409:  330:
Window Size (x  y) [mm2]300:  220:  280:
800:  380:
E ective Field Length [cm]
623.2
943.4
E ective Volume (x  y  z) [mm3] 290  200  280 770  350  330
Number of Rows
6
6
Row-to-Row Pitch [mm]46.
54.
Drift Field [V/cm]
800.
640.
Maximum Drift Length [cm]26.
42.
Anode-to-Anode Pitch [mm]3.
4.
Anode-to-Cathode Gap [mm]3.
4.
E ective Anode Wire Length [mm]7.
7.
Anode Wire Diameter [m]15.
15.
Operation Voltage [-kV]
1.90
2:15 ; 2:20
Number of Anode Wires [/row]
96
192
Total Number of Readout
576
1152
Gating Grid Wire Pitch [mm]1.
1.
Grid-to-Cathode Distance [mm]8.
8.
Gating Grid Operation Voltage [V]
50:
80:
Table 3.4: Speci cations of the time projection chambers.

Drift chambers (FT)Four drift chambers, FT1, FT2, FT3 and FT4, are equipped for determination of
precise hit positions of tracks using its good spatial track de ning capability. The
typical resolution is 250 m  370m. The speci cations of drift chambers are listed
in Table 3.5 and the plane con guration of the drift chambers are summarized in Table 3.6. FT1 and FT2 have 10 planes with the wire con guration of (xx0 uu0yy0vv0xx0 ),and FT3 and FT4 have 8 planes with (xx0 uu0vv0xx0 ). Each pair planes are staggeredby a half cell to cover inecient region near wires and to solve the left-right ambiguity.
The gas mixture are Ar-isobutane (50:50) for each FT. FT2 and FT3 are also used
to de ne the Forward Spectrometer trigger (FSPEC ) (see Section 4.3).

40
Chamber
Frame Size

FT1
x [mm] 338.
y [mm] 394.
z [mm] 60.
Window Size x [mm] 238.
y [mm] 174.
Number of Planes
10
Anode Plane Pitch [mm] 3.2

FT2
503.
394.
60.
353.
233.
10
3.2

FT3
844.
520.
81.
644.
320.
8
4.8

FT4
1003.
520.
81.
803.
360.
8
4.8

Table 3.5: Speci cations of the drift chambers.

Figure 3.10: The schematic view of the Time-of- ight wall, FTOF.

3.4.4 Time-of- ight wall (FTOF)
The Time-of- ight wall, FTOF, is used to measure the time-of- ight of particles fromthe target to the FTOF wall. By combining the time-of- ight with the momentum
and the path length of the particles, the particle identi cation is carried out. Fig. 3.10
shows the schematic view of FTOF. FTOF is positioned at 6 m from the target, mostdownstream of the Forward Spectrometer. FTOF consists of 100 slats of plastic
scintillator, BC404 of Bicron. The dimensions of the slat is 1.24 cm  42 cm  1.25
cm (width  hight  depth). The speci cations of FTOF are summarized in Table
3.7.
Each slat is read out with two 3/4 inch photo-multiplier tubes (PMT), Hamamatsu-

3. The E866 Experimental Setup

41

Chamber Plane Order Wire Orientation # Wires Drift Length
FT1 xx0 uu0yy0vv0xx0 x
0. deg.
56
2.0 mmy
90. deg.
40
2.0 mmu;30. deg.
64
2.0 mmv
30. deg.
64
2.0 mm
FT2 xx0 uu0yy0vv0xx0 x
0. deg.
56
3.0 mmy
90. deg.
36
3.0 mmu;30. deg.
64
3.0 mmv
30. deg.
64
3.0 mmFT3
xx0 uu0vv0xx0x
0. deg.
78
4.0 mmu;30. deg.
80
4.0 mmv
30. deg.
80
4.0 mmFT4
xx0 uu0vv0xx0x
0. deg.
78
5.0 mmu;30. deg.
80
5.0 mmv
30. deg.
80
5.0 mmTable 3.6: Plane con guration of the drift chambers.
R3478S, through light guides on its both ends. Each PMT is mounted on a board
which have a bleeder circuit for PMT and a discriminator, called on-tube discriminator [55]. The on-tube discriminator have two outputs: passively internally split analog
signal and discriminated output. The discriminated signals are re-discriminated in
the counting house to reshape the signals before sending them to TDC, and then
are recorded with Phillips CAMAC TDC model 7186 with 25 psec/bin. The analog
signals are transfered to LeCroy FASTBUS ADC 1882N in the counting house. ADC
values are used in o -line analysis for a slewing correction as well as for a charge cut.
Distance from the target [m]6.0
E ective Area (x  y) [mm2]1260  420
Number of Slats
100
Slat Size (x  y  z) [mm3] 12:4  419:  12:5
Table 3.7: Speci cations of FTOF.

42

Figure 3.11: A schematic view the Henry Higgins spectrometer system [56].

3.4.5 Heavy metal shields
To shield particles from chamber frames of the rst tracking station (FTR1) and thereturn yoke of FM1 magnet, the collimator made of Tungsten is put between thetarget and FM1. The heavy metal shield is also equipped on the beam side wall of
FM1 to reject the particles from the beam pipes to the rst tracking station.

3.5 Henry Higgins Spectrometer (HH)
The Henry Higgins Spectrometer is placed on the opposite side of the Forward Spectrometer with respect to the beam line. It has been used for the E802/E859 experiments as an main spectrometer, and for the E866 experiment, and it is used to cover
the backward region from  20 degrees with respect to the beam line. By combining
these two spectrometers, it is possible to cover a wide y ; pt acceptance. The apertureis 25 msr. In the analysis, the data taken by HH is not used. Details of the Henry
Higgins Spectrometer are described in [56].

3. The E866 Experimental Setup

43

3.6 Data Acquisition System
The data acquisition system consists of 11 CPU's in a single VME create, and gathereddata from 8 CAMAC crates and 6 FASTBUS crates [57]. The DAQ system is managedby two Tadpole TP40V 68040-based CPU's which function together as event builder,data reformatter, network interface, and tape logger. Data from CAMAC modules areread by crate controllers based on SBE VPU25 modules with 68020 CPU's. FASTBUS
data are collected by a LeCroy 1821 Segment Manager in each FASTBUS crate, andtransferred via ber-optic link to a custom-built interface to the VME bus. Datatransferred from CAMAC and FASTBUS crates are assembled and nally recorded on
8 mm tape using an Exabyte Model 8510 drive with write speed up to 500 kByte/sec.

Chapter 4
Run Conditions
In this chapter, run conditions used for 4 and 11 A GeV data analysis
are summarized. The beam energy, targets, trigger de nition, the spectrometer setting and data statistics are described in the following sections.

4.1 Beam Energy
In 1995/1996 run, two di erent lower energy of Au beams have been available forthe rst time in addition to 11 A GeV/c beam energy. The exact beam energy is
2.003 and 4.04 GeV per nucleon in terms of a kinematic energy and 11.69 GeV/c per
nucleon in terms of a momentum [46]. In the analysis, the data taken from 4 and 11
A GeV beam energy run are used.

4.2 Targets
For 4 A GeV run, 197 Au target, 1.961 g/cm2 thickness, corresponding to the 3.0%
interaction rate is used. For 11 A GeV run, targets of 1.5%, 3.0%, 4.0% and 5.0%
interaction rate are used. Data without a target, called empty target is also taken forthe background subtraction.

4.3 Trigger Conditions
We have collected data with four di erent triggers de ned as follows.
44

4. Run Conditions

45

1. BEAM trigger
The rst one is the BEAM trigger for a good beam particle. It is de ned withthe beam counters as;

BEAM  PRE \ BTOT \ HOLE;(4.1)
where PRE shows that there is no other particle arrived within 500 nsec [46] (pile-up
rejection).
2. INT trigger
The second is the interaction trigger, INT , de ned with Bull's Eye counter foran event in which a beam particle interact with a target particle. The de nition is asfollows;

INT  BEAM \ BE;(4.2)
where BE indicates that the measured charge of particle with Bull's Eye counter islower than a threshold.
3. FSPEC trigger
The third is FSPEC for an event in which at least one particle enter the Forward
Spectrometer. The de nition is as follows;

FSPEC  INT \ FT 2 \ FT 3;(4.3)
where FT 2 (FT 3) is required to have at least one of all wires on the rst and second
planes of the FT2 (FT3).
4. ZCAL trigger
This de nes central events, which means a beam particle and target particle collidewith a small impact parameter.

ZCAL  INT \ HZCAL;(4.4)
where HZCAL means an event with the hardware sum of ZCAL signals below thethreshold, which is set that the ratio of trigger counts, ZCAL=INT is 8  11%.

46
Angle Polarity No. of FSPEC triggers
6
2A/4A
0.2 M
2B/4B
3.2 M
9
2A/4A
0.2 M
2B/4B
3.5 M
12 2A/4B
2.6 M
15 2A/4B
2.6 M
18 2A/4B
1.5 M
21 2A/4B
1.5 M
24 2A/4B
1.5 M
27 2A/4B
1.5 M
Total
;
18.3 M
Table 4.1: The statistics of FSPEC trigger at 4 A GeV beam energy. These data
set are used for the single particle spectra analysis.

4.4 Analyzed Data Set
We have taken data of FSPEC trigger with 8 spectrometer angle settings from 6 to
27 degrees in 3 degrees step for 4 A GeV run, 10 angle setting from 6 to 24 degrees
in 2 degrees step for 11 A GeV run. For ZCAL trigger, we have taken data only 10
degrees angle setting at the energy of 11 A GeV. We have also taken data with 0k
G/0 kG, called zero- eld run, for geometry calibration of tracking chambers and forbackground subtraction. Table 4.1 and 4.2 summarize the data statistics of FSPEC
trigger at 4 and 11 A GeV beam energy. In Table 4.3, statistics of ZCAL trigger
at 11 A GeV beam energy are summarized. In this thesis, the data with FSPEC
trigger are used for single particle analysis in order to evaluate in various centrality,
and the high statistic data with ZCAL are used for HBT analysis.
The polarity which bent positive particles toward the beam is called as A polarity,
which is the negative-favor setting for the Forward Spectrometer; the other polarity
is called as B polarity. The magnetic eld of FM1 is set at 0.2 [T] except for zero- eld
runs, and that of FM2 is at 0.4 [T] (0.249 [Tm]) and 0.6 [T] (0.572 [Tm]).

4. Run Conditions

47

No. of FSPEC triggers
A-polarity
B-polarity
2A/4A 2A/6A A Total
2B/4B
6
3.2M 0.8M
4.0M
2.7M
8
3.4M 1.1M
4.5M
2.6M
10
3.3M 0.9M
4.2M
3.2M
12
2.7M 0.7M
3.4M
2.4M
14
4.3M 0.5M
4.8M
2.4M
16
4.4M
4.4M
2.7M
18
4.6M
4.6M
1.6M
20
4.3M
4.3M
0.1M
22
1.3M
1.3M
0.3M
24
1.7M
1.7M
0.2M
Total 33.2M 4.0M 37.2M
18.2M

Angle

Table 4.2: The statistics of FSPEC trigger at 11 A GeV beam energy. These data
set are used for the single particle spectra analysis.

Angle Polarity No. of ZCAL triggers
10 2A/4B
23.6 M
Total;
23.6 M
Table 4.3: The statistics of ZCAL trigger at 11 A GeV beam energy. These data areused for the two-particle correlation analysis.

Chapter 5
Data Reductions
The outline of this chapter is as follows. In section 5.1, the geometry
calibration and chambers and the time-of- ight counter calibrations are
explained. In section 5.2 to 5.4, the methods of track reconstruction,
momentum determination and particle identi cation are described. In
section 5.5, Tracking eciency correction, in section 5.6, the time-of- ight
counter detection eciency, in section 5.7 decay corrections are described.

5.1 Calibration
A position alignment, called geometry calibration, is carried out at the rst stage ofthe data reduction in order to reconstruct tracks and momenta. After the geometry
calibration, drift chambers (FT), time projection chambers (TPC) and time-of- ight
counter (FTOF) are calibrated.

5.1.1 Geometry calibration
Before the run started, the positions of each detector have been surveyed in x-, yand z- directions and placed at the designed positions. We assume that the z- positions of each detector is correct, since there is no way to calibrate in this direction.
In x- and y- directions, the geometry of FT and TPC are calibrated so as to reconstruct the straight tracks using the data of zero- eld run for each angle setting.
The calibration procedure is as follows. First, hit positions on four drift chambers
are tted to a straight line, and adjust its x and y position so as to have minimum
2 values for reconstructed tracks. At this stage, TPC information is not used in48

5. Data Reduction

49

order to avoid a ection by non-calibrated TPC tracks to FT's geometry calibration.
Next, TPC's geometry in x direction is calibrated by using the calibrated geometry
parameters of FT. All six rows at each TPC are aligned so as to reconstruct straightline. This calibration is carried out at each row and at each TPC for all zero- eld
run, independently.

5.1.2 FT calibration
A drift time t, a drift velocity v and a distance from an anode wire l are calibrated
for each drift chamber, and v and l are parameterized as following;t = TDC ; t0 ;v(t) = ap0 + a1 + 2a2t;
2 ptl(t) = a0 t + a1 t + a2 t2 ;

(5.1)
(5.2)
(5.3)

where TDC is the TDC value, t0 is the time-o set, and ai are tting parameters.
The calibration constants and t0 are calibrated on plane-by-plane every angle settingby using straight tracks of zero- eld run.

5.1.3 TPC calibration
After the FT's geometry calibrations, each TPC's geometry is calibrated at every
angle setting using zero- eld run. At this stage, every rows on each TPC are aligned
so as to be consistent with FT. For calibration of y-position of a track in TPCs,
time-zero t0 and drift velocity of electrons v are adjusted at each row. y-position ofa track in TPC is given asy(t) = v(t ; t0);
(5.4)
as a function of drift time t. In addition to these calibration, TPC2 needed morecalibration parameters, because of shift of a hit position due to the drift eld distortionand the magnet fringing eld. Here, we summarize the TPC calibration procedure.
The procedure can be divided into four categories as follows.
1. X-o set, x0, calibration at each raw for the geometry calibration.
2. Time-zero, t0, and drift velocity, v0 calibration at each row.
3. Field distortion correction in x- direction, Xcor (x; y), as a function of the x- andy-coordinates depending on the FM2 eld at each row of TPC2.

50
4. Field distortion correction in y- direction, Ycor (y), as a function of the ycoordinate at each row of TPC2.
The dependence of these calibration parameters are summarized in Table 5.1. In theanalysis, TPC is calibrated as the same way that was carried out in '94 run data
analysis. See [58] for detail description.
parameter applied TPC name dependencex0
TPC1, TPC2
angle
t0
TPC1, TPC2
angle, timev
TPC1, TPC2
time
Xcor
TPC2
FM2 eld
Ycor
TPC2;
Table 5.1: The calibration parameters of TPC and its dependence on angle setting,time and magnetic eld setting of FM2.

5.1.4 FTOF calibration
FTOF calibration is performed after the track reconstruction and the momentum
determination (see Section 5.2 and 5.3). Time of ight, TOF , and y position of acharged particle, yTOF, are calculated as follows,

TOF = tStop +2 tSbot ; BTOT ; t0;
yTOF = (TDCbot ;2 TDCtop )  v + y0;
SitSi = TDCi  Ci ; p;
ADCi ; PEDi
BTOT = TDCL  CL +2 TDCR  CR ;
where

(5.5)
(5.6)
(5.7)
(5.8)

5. Data Reduction

51

TDCi : TDC value
ADCi : ADC value
PEDi : ADC pedestal [ADC bin]t0 : time o set [nsec]
y0 : y-position o set [mm]Ci : TDC time conversion factor [nsec/TDC bin]v : propagation velocity of light in plastic scintillator [mm/TDC bin]Si : slewing correction factor [nsec/(ADC bin)1=2]
and suces i = \top" or \bot" denotes a PMT mounded on the top or bottom of theslat of FTOF wall, suces "L" or \R" denotes a PMT mounted on the left or rightside of BTOT counter. The y0 is calibrated with the FS tracks (see Section 5.2) for
each slat. The slewing correction factors are calibrated with pions, roughly identi ed
with squared mass of ;0:2 < m2 < 0:15, for each PMT. After the slewing correction,
the t0 are re-calibrated with . The calibration of the t0 or y0 is performed every
10  15 runs.

5.2 Track Reconstruction
A track reconstruction code for the Forward Spectrometer, called FAReAST 1 has
been developed for the E866 collaboration. The basic algorithm of the track reconstruction code is follows. First, the tracks are formed independently in each TPC
complex, called TPC tracks. Second, TPC tracks are extended to the both upstream
and downstream drift chambers, and are associated with the hits of drift chambers
and can be reconstructed tracks at each tracking stations, called FTR1 and FTR2
tracks, respectively. Third, if distances between a FTR1 and a FTR2 are close within
3 of their distributions, FTR1 and FTR2 tracks are connected at the mid-plane of
FM2 and formed FS tracks. Then their momenta are calculated using a transfer matrix. Finally, the track is traced back to the target z position through FM1 magnet
using the calculated momentum. And they are also associated to the FTOF position.
FAReAST searches for FTOF hits which have within a 3 of the track positions.

1 Forward Arm Reconstruction Algorithm Starting from the TPC's.

52

Figure 5.1: A typical event at the 14 degree, 2A/4A magnetic setting with tworeconstructed tracks. A side view and a top view of the Forward Spectrometer are
shown.

5.2.1 Track nding by TPC
TPC is used as an initial track nder, taking advantage of its high three dimensional
pattern recognition performance. As the drift electrons di use in both x- and ydirections while drifting in the chambers, a few wires on a row could be red due totheir nite cluster size in x-direction. And because of the feature of the ampli ers,a few pulses could be made in the y-direction. We perform a clustering from several
close hits and calculate the incident particle position as the center of the clusters.
After the clustering, TPC track is formed by searching hit rows which have aligned
clusters and combining them. These series of clusters are tted by a straight line tocalculate a track position and a track vector. For the reconstruction of TPC tracks,
at least 3 rows hits are required.

5.2.2 Track con rmation by FT
TPC tracks are extrapolated onto FTs on both sides of accompanying TPC in orderto get more precise position information of tracks. If the number of planes without
close FT hits (missing planes) is  4, the track is combined with the closest FT hits,
and regarded as FTR tracks at each tracking station. Each FTR track is re- tted witha straight line using the position information of both TPC and FT. The left/rightambiguity on each plane of FTs is solved at the same time, in an iterative way to5. Data Reduction

53
Top View of FSm(perpendicular
P1(Intersection
bisector of AB)
of m and FTR1)
P2(Intersectionof m and FTR2)

A
FTR1

B
X

FTR2

M2
Z
Y(upward)

Effective Edges

Figure 5.2: A illustration of the de nition of track matching at FM2. Points A and
B are intersections of FTR1 and FTR2 with entrance and exit edges of FM2, respectively. Plane \m" is the perpendicular bisector of AB . P1 and P2 are intersectionsof FTR1 and FTR2 with m. Coordinates of P1 and P2 are P1(x1 , y1), P2(x2 , y2),where xi and yi are de ned in Eq. 5.9 and 5.10, respectively.
minimize the 2 of the tting.

5.2.3 Track matching at analyzing magnet
After reconstruction of FTR tracks, FTR1 and FTR2 are extended to FM2 magnet
and connected on the plane m in the FM2. Fig. 5.2 shows the de nition of the planem. The track position matchings in the x- and y-directions (x and y) and thevector matching in y-direction (y ) are de ned as follows.
x  x2 ; x1 ;y  y2 ; y1;
2 dy1
y  dydz ; dz ;
2

1

(5.9)
(5.10)(5.11)

where xi and yi are the x- and y-position of FTRi track projected onto the plane m,and dyi=dxi is the slope of the track in the y-z plane. The cut criteria of distance
between FTR1 and FTR2 track at the matching plane should depend on momentum due to multiple Coulomb scattering. Since the momenta of the tracks are not54
determined yet at this stage, the cut are set loose as,x < 5x;
y < 5y ;y < 5 dydz ;(5.12)(5.13)(5.14)

where  is the width in r.m.s of their distributions. The pairs of FTR1 and FTR2
tracks which ful lled the loose criteria, are regarded as FS tracks. The widths of theirdistributions are determined by their intrinsic detector resolutions and the multiple
Coulomb scattering. The widths are tted with a function as follows,s(5.15)
i( p) = a0 + ( ap1)2
where i is x or y or dy=dz, a0 and a1 are tting parameters. After momentum
determination and TOF association, Eq. 5.12- 5.14 are replaced to re ned 3 cuts.

5.2.4 Target cut
To reject the wrong combination between FTR1 and FTR2, the target cut is applied.
After momentum determination, each FS track is traced back to the target position
through FM1. The width of the target image in x- and y-directions also depended on
momentum. The widths are also tted with Eq. 5.15, and the target cut is applied
using the widths as a function of p. Target position residuals in x- and y-directions,xt and yt are de ned as follows;xt  xt ; xt ;
yt  yt ; yt;(5.16)(5.17)

where xt (yt ) is the target position in x (y)-direction, and xt (yt) represents theirmean values. The target position is de ned as the track position at z = 0, whichcalculated by tracing back the FTR1 track M1, using a ray-tracing method.

5.2.5 Fiducial cut
As the acceptance of the spectrometer is determined by the collimator, the emission angles of the accepted particles should have a clear cuto both in the x- andy- directions. But in practice, the spectrometer has the nite angular resolution,

6

55

Track Position Matching in x
(mm)

(mm)

5. Data Reduction

4
2
0

6

Track Position Matching in y

4
2

1

2

3

0

4

βp (GeV/c)

1

2

3

4

βp (GeV/c)(rad)Track Angle Matching in y
0.008
0.006
0.004
0.002
0

1

2

3

4

βp (GeV/c)
Target Position in y
(mm)

(mm)

Target Position in x
10
5
0

10
5

1

2

3

0

4

βp (GeV/c)

15
10
5
0

2

3

4

βp (GeV/c)

FTOF-FTR2 Association in y
(mm)

(mm)

FTOF-FTR2 Association in x

1

15
10
5

1

2

3

4

βp (GeV/c)

0

1

2

3

4

βp (GeV/c)

Figure 5.3: Widths of seven cut parameters in r.m.s as a function of p [GeV/c]
for protons (solid line), + ( lled circle). These plots show from the top left gure;widths of the track matching [mm] in the x-direction (the rst row left), in the ydirection (the rst row right), the track angle matching [rad] in the y-direction (thesecond row left), target positions [mm] in the x-direction (the third row left), in they-direction (the third row right), and the FTOF hit residual [mm] with respect to thetrack position in the x-direction (bottom row left), and in the y-direction (bottom
row right).

56
the measured distribution of emission angle has blunt edges. In order to avoid the
distortion of the particle distribution near the edges of the geometrical acceptance, aducial cut is applied on the emission angle at the target.

5.2.6 Association with FTOF
Each FS track is extrapolated onto FTOF wall for association with FTOF hits. FTOF
position residuals with respect to FTR2 track positions in the x- and y- directions,xTOF and yTOF are de ned as;
xTOF  xTOF ; xFTR2 ;yTOF  yTOF ; yFTR2:(5.18)(5.19)

The cut for the association between FTOF hit and FTR track is applied as the same
way as the track matching cut at FM2 and the target cut using Eq. 5.18 and 5.19 inboth x- and y- directions.

5.3 Momentum Determination
Using the eld map of FM2 which is measured with the Hall probe, the momentum
calculation of the FS tracks are performed.
In general, the relation between the momentum kick of a track passing throughmagnetic eld is as follows,Ip = q B  ds;(5.20)
where q is the charge of a particle, B the magnetic eld, and s the path of the particle
in the magnetic eld. If we take units of them as p[GeV/c], B[T], s[m], and q[e], theabove equation becomes as follows;Ip = 0:29979  q B  ds [GeV/c];(5.21)

where c is the velocity of light. For the x-component, the following equation is derived;IIpx = 0:29979  q (By dz ; Bz dy)H' 0:29979  q By dz [GeV/c]:(5.22)

We ignore Bz dy for the following reasons. The main component of B both in M1
and M2 is By . Typically Bx and Bz are less than 100 G, which are only a few percent

5. Data Reduction

57

M2
Zin
ZcloseZinner

ZouterZout

Effective Edge

H

Figure 5.4: A demonstration of By dz sampling points for the momentum calculation.
of By (2 kG for M1, 4 or 6 kG for M2). Furthermore, the main component of s inH
the magnetic eld is z. The path length dz is 508 mm for M1, and 762 mm for M2,
H
H
H
while x and y are less than 100 mm. As a consequence, Bz dy is 3 order smallerH
H
than By dz. We make the table of Bz dy for every (xi ; yi) mesh point, along z-axis
Hfrom entrance e ective edge to exit e ective edge. In order to calculate Bz dy witha reasonable accuracy, we take two points, xinner and xouter as shown in Fig. 5.4. The
two points are de ned as,
xinner = xin +2xclose ;(5.23)(5.24)
xouter = xclose 2+ xout ;where xin = (xin; yin; zin) is the FTR1 track position at zin (entrance e ective edge),
xout = (xout ; yout; zout )is the FTR2 track position at zout (exit e ective edge), andxclose is the closest approach point between a pair of FTR1 and FTR2 tracks. The
momentum kick in the x-direction is calculated with the average of the two eld
Hintegration By dz, as follows.
H
H
0:29979q( Bydz(inner) + Bydz(outer));(5.25)
px =2
Hwhere Bydz(inner) is calculated along z-axis through the point (xinner, yinner), so as

58

H Bydz(outer) . Finally, the momentum of the particle is calculated as

p = u2;pxu1x(5.26)

xwhere uix denotes dx=dz of FTRi track (i=1,2).
As the check of the absolute value of calculated momentum, we examine a lambda
mass peak formed from p ; ; pairs. Fig. 5.5 shows the invariant mass distributionof p ; ; pairs observed in 4 A GeV Au+Au collisions. The peak corresponds to ,
and it is tted by a Gaussian function. We obtained the mean (m ) and the width(m ) as,m = 1116:0  0:1 [MeV=c2];
m = 3:25  0:19 [MeV=c2];(5.27)(5.28)The error of the peak position relative to the rest mass (1115.7 MeV=c2 ) is 0:3 
0:1 MeV=c2. By the kinematics of decay ( ! p + ;), the momentum of decayed pand ; in the rest frame of , pcm is expressed as,qM 1 ; ( mpM+m )2
pcm == 0:106 [GeV=c];(5.29)
2
where M , mp and m are mass of , p and ;, respectively. The relation between
the error of momentum pcm and the error of the  mass peak, M , is derived by
Eq. 5.29 with pcm as,pcm =  M2 ; 1 M :(5.30)
p
2p2
M
cm

cm



Using Eq. 5.30, the corresponding momentum error is estimated as 1:5  0:5 %.

5.4 Particle Identi cation
The particle species are identi ed by its squared mass, m2 and momentum. The
squared mass is calculated from momentum p, time-of- ight TOF and path length L
as, TOF 2
2
2
; 1g:
(5.31)
m =p f L
Fig. 5.6 shows a scatter plot of squared mass versus signed momentum, positively
charged particles (+ sign); negatively charged particles (; sign), in a 9 degrees angle
setting at 4 A GeV beam energy.

5. Data Reduction59-

π - p Invariant Mass
12.37
Constant
Mean
Sigma

80

/ 7
74.68
1.116
0.3245E-02

4.785
0.9823E-04
0.1916E-03

70

60

50

40

30

20

10

0

1.1

1.105

1.11

1.115

1.12

1.125

1.13

1.135

1.14

1.145

1.15

2

Invariant Mass (GeV/c )Figure 5.5: The invariant mass distribution of p ; ; pairs observed in 4 A GeV
Au+Au collisions. The peak corresponds to  particle. The peak is tted to Gaussianfunction.
The resolution of m2 is determined by two factors; the momentum resolutionof the tracking system and the timing resolution of the time-of- ight system. The
momentum resolution p is calculated as,!
!
p 2 = (C p)2 + Cmult 2 ;angpCang = p
x
s 

x
x
0
:
0136
Z
1 + 0:038ln X ;Cmult = p
X
x
0
0
I
px = 0:29979 Bydz;(5.32)(5.33)(5.34)(5.35)

where p, , and Z are the momentum [GeV/c], velocity, and the charge of the particle,
 is the intrinsic angular resolution of the tracking system, x=X0 is the thickness ofmaterial in radiation lengths, and Cang and Cmult are the contributions of the angularresolution and Multiple Coulomb scattering, respectively. Then the resolution ofsquared mass m2 is written as,

m2 2

=

4m4

!
p 2 + 4p4  TOF 22p
L(5.36)

60

Squared Mass vs Momentum

Figure 5.6: A scatter plot of measured squared mass versus signed momentum, positively charged particles (+ sign); negatively charged particles (; sign), in a 9 degrees
angle setting at 4 A GeV beam energy. Solid lines represents the applied particle
identi cation cuts.

5. Data Reduction

61

2

4

σm2 (GeV /c )Squared-Mass Resolution
0.5

deuteron
proton
kaon
pion

0.4

0.3

0.2

0.1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Momentum p (GeV/c)

Figure 5.7: The resolution of squared mass for 10 degree run at 11 A GeV beamenergy, as a function of momentum for deuterons, protons, kaons and pions. See text
for details.
=

2 + 4 m4
4m4p2  Cang

Ctof = TOF
L ;

2!
m
2 + 4p2 (m2 + p2 )  C 2
1 + p2  Cmult
tof (5.37)(5.38)

where m is the mass of the particle, TOF is the time-of- ight resolution, and L is
the ight-path length. Fig. 5.7 shows the width of squared mass as a function of
momentum for deuterons, protons, K + and + using the real data.
The basic boundary of the particle identi cation is determined using the squared
mass resolution as a function of momentum as follows;

jm2measured ; m2realj < 2:5m2 ;(5.39)

where m2measured is the squared mass calculated from the measured momentum andthe measured time-of- ight, mreal is the real mass of the particle.
In addition to the boundaries determined with mass square widths, the following
boundaries are applied to suppress the contamination of other particle species as in
Table 5.2.

62Particle Squared mass boundary [GeV2 /c4] Momentum boundary [GeV/c]< 0.1250.45 ; 4.0
K0.18 ; 0.5
0.45 ; 3.0p0.5 ; 2.0
0.45 ; 5.0d2.0 ; 6.0
0.45 ; 5.0
Table 5.2: Particle identi cation boundary of squared mass and momentum for , K ,p and d.

5.5 Tracking Eciency Correction
Track reconstruction eciency, track is divided into three parts as,

track = FT  TPC  soft(5.40)

whereFT : FT hardware eciency
TPC : TPC hardware eciency
soft : track reconstruction software eciency.
FT hardware eciency is regarded as 100 % because the eciency per plane of FT's
is higher than 95 %. FT1 and FT2 are required to have hits at least 6 out of 10
planes, and for FT3 and FT4, they are at least 4 out of 8 planes. Therefore FT isevaluated as 100 %.

5.5.1 TPC hardware eciency
TPC hardware eciency depends on the position of the hits and the momentum of
the particle. The position dependence in x-direction of the TPC hardware eciency
is due to eld distortion. The y-position dependence of the eciency is regarded as
due to electron di usion and absorption during drift. The momentum dependence ofthe eciency is due to the change of energy deposit of the particle in TPC, dE=dx.
Thus the momentum dependence of the eciency is estimated as a function of ,pwhere = v=c, and = 1= 1 ; 2. The TPC hardware eciency per row is writtenas;TPCr;x;y ( ) = 1 ; Ar;x;y  Ine ( );(5.41)

5. Data Reduction

63
TPC2 Inefficiency
Inefficiency

Inefficiency

TPC1 Inefficiency
0.12
0.1
0.08
0.06

0.3
0.25
0.2
0.15

0.04

0.1

0.02

0.05

0

0

10

20

30

0

0

10

20

βγ

Figure 5.8: Ineciency of TPC1 and TPC2 as a function ofand pions. Ineciency is averaged in all raws of TPCs.

30

βγ

for deuterons, protons

where suces r, x and y denote the row number of the TPC, hit position in x-direction
and y-direction at each row, respectively, A is the normalization factor indicating thedegree of ineciency, and Ine ( ) is TPC hardware ineciency averaged over row
and hit position. The TPC ineciency of row i, Ine i is calculated as follows;
Ine i = 1 ; TPCi ;of tracks with hits on all row i
TPCi = No. ofNo.
tracks with hits on all rows except row i :(5.42)(5.43)

Fig. 5.8 shows the averaged ineciency as a function of of TPC1 and TPC2,
respectively. The TPC hardware eciency is calculated track by track with TPCr;x;yof the hit position as,

TPC = TPC 0 + TPC 1 + TPC 2 + TPC 3 ;(5.44)

TPC 0 = 1  2  3  4  5  6
TPC 1 = (1 ; 1 )  2  3  4  5  6
+ 
+ 1  2  3  4  5  (1 ; 6)
TPC 2 = (1 ; 1 )  (1 ; 2 )  3  4  5  6
+ 
+ 1  2  3  4  (1 ; 5 )  (1 ; 6 )(5.45)

where(5.46)(5.47)

64

TPC 3 = (1 ; 1 )  (1 ; 2 )  (1 ; 3 )  4  5  6
+ 
+ 1  2  3  (1 ; 4 )  (1 ; 5 )  (1 ; 6 )(5.48)

where n of TPC n denotes the number of missing rows, i (1 = 1; 2; 3; 4; 5; 6), denotes
the eciency at row i.

5.5.2 Track Reconstruction Software Eciency
Track reconstruction software eciency due to the multiple Coulomb scattering isevaluated using a Monte Carlo single track of each particle species. A thousand eventswith single track with a xed p are generated with simulation code with multiple
scattering switched on or o , respectively. The energy loss is always considered.
Furthermore the output of simulation is converted to same data format with realdata. Then the track reconstruction is performed with the same tracking code. The
software eciency due to the multiple scattering for a xed p is evaluated as;
No. of reconstructed tracks with multi. scattering on
soft = No.
of reconstructed tracks with multi. scattering o(5.49)In the simulation, Forward Spectrometer is set at 6 degree and the magnetic
strength is set to 2 K Gauss and 4 K Gauss for FM1 and FM2, respectively. The
magnetic eld is set to bent positive particles away from the beam line. Then the
single tracks of deuteron, proton or positive pion are generated with a xed emission
angle with  of 6:5o and  of 180o with each p from 0.2 to 5 GeV/c. The eciency
is parameterized as,soft = a0 + a1 log( p);(5.50)
where a0 and a1 are tting parameters. The tting parameters a0 and a1 are 0.967 and
0.0306, respectively. It is found that the eciency goes down as the p decreasing,from 99 % to 94 %.
In former analysis by Sako [44], the track multiplicity dependence of the trackreconstruction software eciency has been studied. Fig. 5.9 shows the track reconstruction eciency as a function of p for multiplicity 1 (solid), 5 (dashed) and 9(dotted). The eciency decreases as the multiplicity increases at momentum higher
than 2 GeV/c, while it is stable at momentum less than 2 GeV/c. The number of5. Data Reduction

65

efficiency

Track reconstruction software efficiency

β p [GeV/c]

Figure 5.9: Track reconstruction eciency of anti-protons as a function of p [GeV/c]
for reconstructed track multiplicity of 1 (solid), 5 (dashed), and 9 (dotted) [44]. Theeciency at high p decreases as the multiplicity increases.
tracks in the inclusive events is less than 2. Thus, the track multiplicity dependenceof the software eciency is small.

5.6 FTOF Detection Eciency
Since FTOF TDC's do not have a multiple hit capability, if two particles hit thesame FTOF slat, only the earlier hit is recorded. We evaluated the ineciency due
to the FTOF occupancy. The ineciency depends not only on the angle and magnet
settings but also on the trigger types.
The eciency is written as follows;

R t dN (t) dt
TOF
i (t) = 1 ; 0N dt ;event

(5.51)

where i is a slat number (0  i  99), t is a TDC value [nsec], dNdt(t) is the particle
rate [nsec;1]. One can understand this equation by resolving the ineciency into two
factors as follows;
R t dN (t) dt
TOF(5.52)
1 ; i (t) = 0 dt  Ni ;NiNeventwhere Ni represents the number of tracks on slat i. The rst factor is the probability
that a hit is observed on a slat i at a time earlier than t, namely, the probability that

66
6deg, Minimum Bias

24deg, Minimum Bias

slat

slat
tof [ns]

tof [ns]

6deg, 6% Central

24deg, 6% Central

slat

slat
tof [ns]

tof [ns]

Figure 5.10: FTOF detection eciency as a function of the slat number (the leftside axis) and a time-of- ight in nsec (the right-side axis). Four plots correspond to
FSPEC events at 6 degrees (top left), at 24 degrees (top right), 6 % central events
at 6 degrees (bottom left), and at 24 degrees (bottom right) [44].
suppose the rst particle hits on slat i at t and the second particle hits the slat, thesecond particle hits earlier than the rst particle. The second factor is the probability
that the second particle hits the same slat, i.
We calculate TOF
i (t) for all the angle and magnet settings and for minimum bias
and several centrality events. Fig. 5.10 shows FTOF detection eciency as a functionof the slat number and the time-of- ight. The eciency decreases as the TDC value
increases, and as the slat number decreases, namely, the slat is closer to the beam.
Also the eciency is lower for central events.

5. Data Reduction

67

Figure 5.11: Survival rates of + and K + after track reconstruction and PID as afunction of L=p [mmc/GeV]. Solid lines show tted curves, and dashed lines show
expected curves from the decay time constants [44].

5.7 Decay Correction
For pions and kaons, we estimate the e ect of decay on eciency. We use Monte
Carlo single tracks with and without decay, and calculated the survival rate of themafter track reconstruction. The eciency is de ned as follows;
No. of PID'ed tracks with decay :(5.53)
decay  No.
of PID'ed tracks without decay
We calculate decay as a function of L=p, where L is the path length, p is the momentum. Fig. 5.11 shows decay as a function of L=p for + and K +.

Chapter 6
Experimental Results I { Single
Particle Spectra
In this chapter, the procedure to calculate the cross sections and theexperimental results of single particle spectra of deuterons, protons, K +and  in Au+Au collisions at 4 and 11 A GeV are presented. In Section 6.1, the interaction trigger cross section is de ned. In Section 6.2,
procedures to evaluate the invariant cross sections is given. The centrality
cut criteria is given in Section 6.3. The centrality dependence of transverse mass spectra for d, p, K + and  at 4 and 11 A GeV beam energy
are presented in Section 6.4. The centrality dependence of rapidity distribution dN=dy and mean transverse mass < mt > ;m0 are presented
in Section 6.5 and 6.6, respectively. Finally, the results of single particle
spectra are summarized in Section 6.8.

6.1 Interaction Trigger Cross Section
The trigger cross section trig of the interaction trigger (INT ) is de ned as follows.

0
bg N bg 1 24
S
S
N
trig
trig trig A 10 A [barn];trig  @ trig
;
bg
Nbeam
NAD
Nbeam

where

68

(6.1)

6. Experimental Results I { Single Particle Spectra
Nbeam
bg
Nbeam
Ntrig
bg
Ntrig
Strig
bg
Strig
A
NA
D

:
:
:
:
:
:
:
:
:

69

number of beam particles
number of beam particles with empty target
number of triggers
number of triggers with empty target
scale-down factor
scale-down factor with empty target
mass number of the target nuclei [amu]Avogadro's number (6:02  1023 [mol;1])
the target thickness [g/cm2].

The calculated the interaction trigger cross section trig is 5.3  0.2 barn, which issmaller than the total geometrical cross section ( 6.8 barn). This will be due tosome of peripheral events are dropped by the BE threshold.

6.2 Invariant Cross Sections
The Lorentz invariant cross section of particle production is described as,33
d3
E ddp3 = p dpd dyd= m dm(6.2)
t t
t t dydqwith transverse momentum pt or transverse mass mt = p2t + m20 , where m0 , p, E ,y, , is the rest mass, momentum, total energy, rapidity, and azimuthal angle of theparticle, respectively. The invariant cross section is evaluated at each rapidity bin
and transverse momentum bin, as
24 A 13
1
[barn  GeV;2 c3 ]; (6.3)
E ddp3 = StrigNNi (y; pt)  10
NAD  ptypt i(y; pt)
beam
where Sbeam , Nbeam , NA, and D are as de ned in Section 6.1, and Ni(y; pt) is numberof particles (i = ; p; ),  is total analysis eciency, y is bin size in rapidity, pt
is bin size in transverse momentum and i(y; pt) is azimuthal acceptance.
For geometrical acceptance calculation, we make acceptance maps for each y ; pt
bin by Monte Carlo method. Around the spectrometer aperture, 2 M particles ofboth charges are generated and tracked using the GEANT [59] without the physicalinteraction. If the particles are in heavy materials such as the collimator or theframes of the tracking chambers, these particles are forced to stop there. By counting

70
the number of particles which passed all detectors of the Forward Spectrometer, thegeometrical acceptance =2 is calculated as a function of momentum and polar
emission angle with respect to the incident beam. Then =2 is converted to afunction of rapidity and transverse momentum for each particle species. The ducial
cut and the momentum cuto s are included. The analyzed momentum ranges forvarious particle species are listed in Table 6.1. Left gures in Fig. 6.1 show samplesof the acceptance map, for pions and protons at the 12 degrees angle setting. Rightgures in Fig. 6.1 show the statistical errors associated with the calculation of themaps. If the statistical errors are larger than 5% at the edges of the maps, thecorresponding bins are rejected.
Particle Momentum Cuto range [GeV/c]
0.45 ; 4.0K0.45 ; 3.0p0.45 ; 5.0d0.45 ; 5.0
Table 6.1: The analyzed momentum ranges for each particle species.

6.3 Centrality Cut Criteria
We de ne the centrality of events with the total kinetic energy of beam spectators
deposited in ZCAL. The procedure to de ne the centrality is as follows. We rst
subtract the ZCAL spectrum normalized with BEAM trigger counts in an empty
target run from that in a Au target run. We calculate the integral of the ZCAL
energy spectrum (software sum), over the ZCAL energy from 0 to zc. We de ne thecentrality with the integrals as follows;
R zc dINT dz(6.4)
Centrality  R z0max ddzINT :0dz dz
Fig. 6.2 shows ZCAL energy spectra for INT events, with a 6-degree, 2A/4A
run. Table 6.2 shows the relation between the centrality and its corresponding ZCAL
energy at the beam energy of 11 A GeV. We set 4 centrality windows; 0 ; 10%,
10 ; 30%, 30 ; 50%, and 50 ; 100% of interaction trigger cross section. Table 6.3
shows the ZCAL energy ranges and mean ZCAL energies in the 4 centrality windows.

6. Experimental Results I { Single Particle Spectra

Acceptance map for protons and pions at 12 degrees

protons

protons, error %pions(+)

pions(+), error %Figure 6.1: Samples of acceptance maps for protons and pions at 12 degrees

71

72

Figure 6.2: ZCAL spectra [GeV] for INT events (top gure, solid line) and for run
without the target (top gure, dashed line), normalized with BEAM counts at 11 A
GeV beam energy. The net ZCAL spectrum obtain by subtracting the two spectra(bottom gure).

6. Experimental Results I { Single Particle Spectra

73

Centrality (0=most central) ZCAL energy [GeV]
10 %
461
20 %
821
30 %
1114
40 %
1417
50 %
1679
60 %
1845
70 %
1968
80 %
2057
90 %
2147
100 %
3000
Table 6.2: Centrality and ZCAL cut energy.
Centrality range
ZCAL energy range [GeV] Mean ZCAL energy [GeV]
0 ; 10 %0 ; 461307
10 ; 30 %461 ; 1114
803
30 ; 50 %1114 ; 1679
1412
50 ; 100 %1679 ; 3000
2044
0 ; 100 % (Minimum Bias)
0 ; 3000
1517
Table 6.3: Centrality ranges and corresponding ZCAL energy ranges and mean ZCAL
energies.
At lower beam energy, i.e., Ebeam = 4 A GeV, ZCAL energy is not a good measurefor the centrality, because of the aggravation of ZCAL resolution at lower energy.
Fig. 6.3 (top) shows the correlation between NMA multiplicity and ZCAL spectra[GeV] at the beam energy of 4 A GeV. The non-linearity can be seen at lower energyof ZCAL spectra. For this reason, NMA multiplicity is used for the centrality cut at
4 A GeV beam energy. Fig. 6.3 (bottom) is the NMA multiplicity distribution after
subtraction of the distribution in empty target in INT events. Table 6.4 shows the
NMA multiplicity range in the 4 centrality windows.

NMA multiplicity

74

250

45

225
40

200
35

175

30

150

25

125
100

20

75

15

50

10

25

5

0

0

0

200

400

600

800

1000

1200

ZCAL software sum GeV

10

-1

-2

10

-3

10

10

-4

25

50

75

100

125

150

175

200

225

250

NMA(INT)/BEAM target IN - OUT

Figure 6.3: The correlation between NMA multiplicity and ZCAL spectra [GeV] atthe beam energy of 4 A GeV (top gure). The centrality is de ned by the NMA
multiplicity distribution after subtraction of NMA distribution in empty target in
INT events (bottom gure).

6. Experimental Results I { Single Particle Spectra

75

6.4 Transverse Mass Spectra
6.4.1 Centrality dependence at Ebeam = 4 A GeV
Fig. 6.4 shows the invariant di erential cross sections of protons and deuteron in four
centrality bins as a function of transverse kinetic energy near mid-rapidity in Au+Au
collisions at 4 A GeV. The invariant di erential cross section of  and K + are shownin Fig. 6.5 and Fig. 6.6, respectively. Each successive spectrum is divided by 10.

6.4.2 Centrality dependence at Ebeam = 11 A GeV
Fig. 6.7 shows the invariant di erential cross sections of protons and deuteron in four
centrality bins as a function of transverse kinetic energy near mid-rapidity in Au+Au
collisions at 11 A GeV. The invariant di erential cross section of  and K + are
shown in Fig. 6.8 and Fig. 6.9, respectively. Each successive spectrum is divided by
10.

Centrality range
NMA multiplicity
0 ; 10 %134 ; 225
10 ; 30 %76 ; 134
30 ; 50 %35 ; 76
50 ; 100 %0 ; 35
0 ; 100 % (Minimum Bias)
0 ; 225
Table 6.4: Centrality ranges and corresponding NMA multiplicity at 4 A GeV beamenergy.

76

6.5 Rapidity Density Distributions
It is empirically known that the di erential cross sections of particles produced from
p+p, p+A, and A+A collisions generally show a shape of a single exponential or
double exponentials of mt ; m0, where m0 is mass of each particle species.
When a transverse mass spectrum shows a single exponential form, as is in thecase of kaons, we parameterize the spectrum as follows;d2 = N exp (; mt ; m0 );(6.5)
02mt dmt dy
T0
where T0 [GeV/c2 ] is called the inverse slope parameter, which is a measure of thetemperature of each particle species. The N0 is obtained by integrating Eq. 6.5 overmt ;d = Z 1 d2  2m dm = 2N T (T + m ):
(6.6)
t t
0 0 0
0
dy
2m dm dy
m0tt

By substituting N0 to the Eq. 6.5, we obtain a parameterization of the transverse
mass spectrum;
1  d2
1dN  exp (; mt ; m0 );=trig 2mtdmt dy 2T0(T0 + m0 ) dy
T0

(6.7)

1 d
where dNdy  trig dy is called the rapidity density. We obtain dN=dy and T0 as t
parameters.
When a transverse-mass spectrum is clearly deviated from a single exponential
form, as is in the case of protons and pions, we parameterize the spectrum with adouble exponential form as follows;

1  d21
1  dN  exp (; mt ; m0 )=trig 2mtdmt dy2T1(T1 + m0) 1 + w dy
T1
mt ; m0 ): (6.8)+ 2T (T1 + m )  1 +w w  dN

exp
(
;
dy
T
2 2

0

2

We obtain dN=dy, w, T1, and T2 as t parameters.

6.5.1 Centrality dependence at Ebeam = 4 A GeV
Fig. 6.10 shows the rapidity density distributions of protons and deuteron in four
centrality bins. The rapidity density distributions of  and K + are shown in Fig. 6.11
and Fig. 6.12, respectively.

6. Experimental Results I { Single Particle Spectra

77

Figure 6.4: Transverse mass spectra of protons (left) and deuterons (right) in four
centrality ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % near mid-rapidity at
4 A GeV beam energy. The units of the horizontal and the vertical axes are GeV/c2
and GeV;2c4, respectively. Each successive spectrum is divided by 10.

78

Figure 6.5: Transverse mass spectra of + (left) and ; (right) in four centrality
ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % near mid-rapidity at 4 A
GeV beam energy. The units of the horizontal and the vertical axes are GeV/c2 andGeV;2c4 , respectively. Each successive spectrum is divided by 10.

6. Experimental Results I { Single Particle Spectra

79

Figure 6.6: Transverse mass spectra of K + in four centrality ranges, 0 - 10 %, 10 -30%, 30 - 50 % and 50 - 100 % near mid-rapidity at 4 A GeV beam energy. The unitsof the horizontal and the vertical axes are GeV/c2 and GeV;2 c4, respectively. Eachsuccessive spectrum is divided by 10.

80

Figure 6.7: Transverse mass spectra of protons (left) and deuterons (right) in four
centrality ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % near mid-rapidity at
11 A GeV beam energy. The units of the horizontal and the vertical axes are GeV/c2
and GeV;2c4, respectively. Each successive spectrum is divided by 10.

6. Experimental Results I { Single Particle Spectra

81

Figure 6.8: Transverse mass spectra of + (left) and ; (right) in four centrality
ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % near mid-rapidity at 11 A
GeV beam energy. The units of the horizontal and the vertical axes are GeV/c2 andGeV;2c4 , respectively. Each successive spectrum is divided by 10.

82

Figure 6.9: Transverse mass spectra of K + in four centrality ranges, 0 - 10 %, 10 -30%, 30 - 50 % and 50 - 100 % near mid-rapidity at 11 A GeV beam energy. The unitsof the horizontal and the vertical axes are GeV/c2 and GeV;2 c4, respectively. Eachsuccessive spectrum is divided by 10.

6. Experimental Results I { Single Particle Spectra

83

6.5.2 Centrality dependence at Ebeam = 11 A GeV
Fig. 6.13 shows the rapidity density distributions of protons and deuteron in four
centrality bins. The rapidity density distributions of  and K + are shown in Fig. 6.14
and Fig. 6.15, respectively.

84

Figure 6.10: Rapidity density distributions of protons(left) and deuterons (right) in
four centrality ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % at 4 A GeV beamenergy.

6. Experimental Results I { Single Particle Spectra

85

Figure 6.11: Rapidity density distributions of + (left) and ; (right) in four centrality ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % at 4 A GeV beamenergy.

86

Figure 6.12: Rapidity density distributions of K + in four centrality ranges, 0 - 10 %,10 -30 %, 30 - 50 % and 50 - 100 % at 4 A GeV beam energy.

6. Experimental Results I { Single Particle Spectra

87

Figure 6.13: Rapidity density distributions of protons (left) and deuterons (right) in
four centrality ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % at 11 A GeV
beam energy.

88

Figure 6.14: Rapidity density distributions of + (left) and ; (right) in four centrality ranges, 0 - 10 %, 10 -30 %, 30 - 50 % and 50 - 100 % at 11 A GeV beamenergy.

6. Experimental Results I { Single Particle Spectra

89

Figure 6.15: Rapidity density distributions of K + in four centrality ranges, 0 - 10 %,10 -30 %, 30 - 50 % and 50 - 100 % at 11 A GeV beam energy.

90

Figure 6.16: < mt > ;m0 [GeV/c2] of protons, K + and  at 4 A GeV beam energy(left) and that at 11 A GeV(right) as a function of centrality given in percentage ofinteraction trigger cross sections.

6.6 Centrality Dependence of Mean Transverse Mass
In this section, mean transverse mass < mt > ;m0 as a function of centrality. We
obtained < mt > ;m0 with the t parameter of transverse mass spectra of either
Eq. 6.7 or Eq. 6.8 for each particle species. Fig. 6.16 shows < mt > ;m0 [GeV/c2] of
deuterons, protons, K + and  at 4 A GeV beam energy (left) and protons, K + and at 11 A GeV(right) as a function of centrality near mid-rapidity.

6.7 Systematic Errors
The systematics of the single particle analysis is checked using the kinematic symmetry of the collision system with respect to the center of mass rapidity ycm ' 1:17.
Fig 6.17 shows the ratios of invariant di erential cross sections between kinematic

6. Experimental Results I { Single Particle Spectra

91

Figure 6.17: Ratio of inclusive proton di erential yield at 4 A GeV beam energy
between 1:1 < y < 1:2 and 1:2 < y < 1:3 as a function of transverse kinetic energy
mt ; m0 .
re ected rapidity regions as a function of transverse kinetic energy for protons. The
error in the gure are the statistical errors of the ratios. The proton di erential crosssection in the range 1:1 < y < 1:2 is consistent to that in 1:2 < y < 1:3 within  10%.
The applied corrections and the systematic errors in the analysis are summarized inTable. 6.5. The overall systematic error of the invariant cross sections is estimated to
be 8%.

6.8 Summary of the Results for Single Particle
SpectraFor single particle analysis, we have measured transverse mass spectra and rapidity
distributions dN=dy for , K +, protons and deuterons in Au+Au collisions at 4
and 11 A GeV as a function of centrality. For the transverse mass spectra, we haveobserved the mass dependence of mean transverse mass < mt > ;m0 ; for the particle
with heavier mass, the larger < mt > ;m0 is seen. For the shapes of the spectra, mt
spectra of protons have shoulder-arm shape in central collisions at low mt region, On
the other hand, the pion spectra have an enhancement at low mt , in other words, theconcave shape at low mt region at 11 A GeV beam energy. Though the enhancementof pions could be explained in part by the contribution from baryon resonances such

92
Correction Range Typical Correction Sys. Error
Inclusive
TPC Hardware E .
0;7 %
2%
1%
Track Recon. Soft. E .
1;7 %
2%
5%
FTOF Detection E .
0 ; 30 %
7%
2%
PID
1%
1%
1%
Sub total
1 ; 40 %12 %6%
Acceptance{{5%
Centrality Cut{{
2%
INT Cross Section
{
{
3%
Total
1 ; 40 %12 %8%Table 6.5: The table of correction amplitude and systematic errors. The typicalcorrections are at 14 degrees, with momentum of 2.5 GeV/c.
as , the concave shape can be also reproduced by invoking a transverse velocity
eld of the system with a freeze-out temperature. This expanding source picture canalso explain observed shoulder-arm shapes in mt spectra for protons and deuterons,
and the mass dependence of the mean transverse mass. For the proton's dN=dy, thedistributions have a maximum at mid-rapidity at both 4 and 11 A GeV beam energy
in central events, which indicates a large baryon stopping in Au+Au collisions at
AGS energies.

Chapter 7
Experimental Results II {Two-Particle Correlations
In this chapter, experimental results of two-particle correlations for++ pairs in Au+Au central collisions at 11 A GeV beam energy arepresented. After the descriptions of Coulomb correction and cut criteria
applied in the analysis, the experimental results of two-particle correlations are shown in two di erent frame.

7.1 Coulomb Correction
Experimentally, the normalized two-particle correlation function is obtained from theratio between correlated and uncorrelated particle distribution as,

dN=d3p1 d3p2
C2(Q  p1 ; p2 ) = dN=d3 p1  dN=d3 p2= A(Q)=B (Q):

(7.1)
(7.2)

where A(Q) is the measured two particle probability distribution as a function ofrelative four-momentum Q = p1 ; p2 , and B (Q) is the uncorrelated (background)
distribution, calculated in the same way but using two particles taken from di erent
events.
First, we consider the two particle as p1 = (E1; px1; py1; pz1) = (E1; p~1) and p2 =~ Q0 and Qinv as,(E2 ; px2; py2 ; pz2) = (E2; ~p2). We de ne the variable Q;

Q~ = ~p1 ; ~p2 = (Qx ; Qy ; Qz );
93

(7.3)

94
C2

x 10
1.5
10000

1.4
1.3

8000

1.2
1.1

6000

1
0.9

4000

0.8
0.7

λ : 0.428586 ± 0.0106416

0.6

RINV : 5.39073 ± 0.0812275 [fm]

0.5

0.1

0.2

2000

0.3

0

0

0.1

QINV (GeV/c)

0.2

0.3

QINV (GeV/c)

Figure 7.1: Left panel: Correlation functions for ++ pairs as a function of Qinv .
Open circles correspond to the data before Coulomb correction, lled circles correspond to that of after standard Gamov Coulomb correction. Right panel: Qinv
distribution of real data with standard Gamov Coulomb correction (open square)and normalized mixed background ( lled square).

Q0 = E1 ; E2 ;q
Qinv = Q2x + Q2y + Q2z ; Q20

(7.4)
(7.5)

The right panel of Fig. 7.1 shows Qinv distribution of ++ pair of real data with standard Gamov Coulomb correction (open square) and normalized mixed background( lled square).The ratio of the two distributions give the normalized correlation function C2 as a function of Qinv as shown in the left panel of Fig. 7.1.
To calculate the two-particle correlation function C2 for identical boson, we assume that particles su er no nal state interaction after emission. However for theexperimental analysis of HBT, the Coulomb interaction between these charged particles in the nal state is not negligible. The Coulomb repulsion e ect suppresses thepair of lower relative momentum. To extract the \true" information of the sourcefrom the correlation function, the Coulomb correction must be incorporated. In thisanalysis, the standard Gamov correction is applied on a pair-by-pair bias in real data
used to correct the Coulomb e ect in the correlation function.
The method of the standard Gamov correction is as follows. The correction factor
depends only the relative momentum di erence, Q = p1 ; p2 . The standard Gamov
correction factor is expressed as,G() = e22; 1(7.6)

7. Experimental Results II { Two-Particle Correlations

95

where  = m =Q, with a particle mass m and the ne structure constant . The
Gamov factor G is the square of the relative Coulomb wave-function of a particle
pair at zero separation in con guration space. Using Eq. 7.6, the Gamov factor isweighted in real event in pair-by-pair bias as a function of Qinv . For three dimensional
decomposition of Q as described in Section 7.4, the Gamov factor is weighted in eachQi bin calculated by the corresponding Qinv as,

C2(Qi) = C2raw (Qi )  G;1(Qinv ):(7.7)
The left panel in Fig. 7.1 shows the correlation function as a function of Qinv withand without the standard Gamov Coulomb correction.
While the assumptions of a non-relativistic and point-like source made in theformulation of the standard Gamov factor are perhaps appropriate for the sources
investigated in e+ e; or proton-proton collisions, they are certainly not valid for thesources created in the relativistic heavy-ion experiments at AGS and SPS.
NA49 collaboration has been extracted the e ects of the Coulomb interaction fromcorrelations between measured \+;" charged pairs in the experiment [17]. Because
opposite sign pions do not exhibit a Bose-Einstein correlation, any signal obtained bythis method may be interpreted as due to the Coulomb e ect.
The Coulomb correction applied in NA49 is based on the correlation function of
the correlation of \+;" pairs in the variable Qinv with the same cuts used in thecalculations of a conventional correlation function. The Coulomb correction is thenparameterized in an analytic form, and the factor is weighted for each pair in termsof their Qinv . The phenomenological Coulomb correction is taken as an exponentiallydamped Gamov factor as;(G() ; 1) exp;Qinv =Qeff +1;(7.8)
where G() is the Gamov factor as de ned in Eq. 7.6, and Qeff is a parameter thatis t to the data. We call this type of correction factor as \modi ed Gamov factor".
In the limit of Qeff ! 1, the correction approaches the Gamov factor (a point likesource), while in the limit of Qeff ! 0, the function goes to 1 (an in nite source).
Fig. 7.2 shows the Coulomb correction factor as a function of Qinv in standard Gamov(solid line) and modi ed Gamov (broken line) Coulomb correction. In general thefactor in standard Gamov correction is bigger than that in the modi ed Gamov
correction, and the e ect is not vanished in high Qinv region in standard Gamov
correction.

Coulomb corection factor

96
2

Gamov

1.8

modified Gamov
1.6

1.4

1.2

1

0.8

0.6

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

QINV (GeV/c)

Figure 7.2: Coulomb correction factor as a function of Qinv in standard Gamov (solidline) and modi ed Gamov (broken line) method.
Fig. 7.3 shows the correlation function C2 as a function of Qinv with standard
Gamov correction, modi ed Gamov correction and without Coulomb correction. For
C2 with modi ed Gamov correction, the value of Qeff is set as 46 MeV, which is usedin the NA49 two-particle correlation analysis in Pb+Pb collisions [60]. The extracted
Rinv parameters in modi ed Gamov Coulomb correction is 10 % larger than thatin standard Gamov method. This uncertainty due to the di erence of two types of
Coulomb corrections is included as the systematic error in this analysis.

7.2 Cut Criteria
7.2.1 Two-track separation
Due to the nite resolution of two-track separation in the Forward Spectrometer, wehave to take into account the cut of the close tracks of pair, for the the correlation
function. Fig. 7.4 and Fig. 7.5 show the distributions of ++ pair as a function of
the position di erence of the two-particle in x and y direction at the mid-plane ofeach TPC in 1-dimensional and 2-dimensional mesh.
The position di erence of a pair at TPCs in x and y directions, i.e.,X = jx1 ; x2 j;

(7.9)

C2

7. Experimental Results II { Two-Particle Correlations

97

1.5

Gamov

1.4

modified Gamov
without Coulomb

1.3

1.2

1.1

1

0.9

0.02

0.04

0.06

0.08

0.1

0.12

QINV (GeV/c)

Figure 7.3: The correlation function C2 as a function of Qinv with standard Gamov
correction (solid circles), modi ed Gamov correction (open triangles) and without
Coulomb correction (open circles). The tting results by the function, C2 (Qinv ) =2 Q2 ) to data are also shown. For C with modi ed Gamov correction,
1+  exp(;Rinv
2
inv
the value of Qeff in Eq. 7.8 is set as 46 MeV, which is used in the NA49 two-particle
correlation analysis in Pb+Pb collisions [60].

98
Two-track Separation at TPCs
Accepted tracks

7000

14000

6000

12000

5000

10000

4000

8000

3000

6000

2000

4000

1000

2000

0

0

20

40

60

80

100

0

0

20

40

TPC1 ∆X (mm)

60

80

100

TPC1 ∆Y (mm)
7000

4000
3500

6000

3000

5000

2500

4000

2000
3000
1500
2000

1000

1000

500
0

0

20

40

60

80

100

0

0

20

TPC2 ∆X (mm)

40

60

80

100

TPC2 ∆Y (mm)Figure 7.4: The distributions of ++ pairs as a function of the position di erenceof the two-particle in x and y direction at the mid-plane of each TPC. The hatchedarea are accepted as a track for the calculation of correlation function, if the otherdirections and TPC planes are also within the cut criteria. The unit in the gures is[mm].
Y = jy1 ; y2j;

(7.10)

are used for the cut. As shown in Fig. 7.5, the resolution of two track separation isestimated  1 [cm], we set the cut-o parameter as 1 [cm] in both x and y directions
at each TPC.

7.2.2 Other cuts
In addition with the cut of two-track separation, it is applied that the rapidity cut
due to the limited acceptance and centrality cut for the selection of 10 % central by
ZCAL as, Rapidity cut : 1.6 < Y < 2.3

7. Experimental Results II { Two-Particle Correlations
TPC2

30

∆Y (mm)∆Y (mm)

TPC1
25

30
25

20

20

15

15

10

10

5

5

0

0

10

99

20

30

0

0

10

∆X (mm)

20

30∆X (mm)Figure 7.5: The distributions of ++ pairs as a function of the position di erenceof the two-particle in 2-dimensional (x ; y) mesh at TPC1 (left) and TPC2 (right).
The lines in the gures represent the boundary for the two-track separation cut.

 Centrality cut : 10 % most central event selection by ZCAL
By these cuts, the event of the most central and near mid-rapidity can be selected.

7.3 Statistics and Event Mixing
As described in Chapter 4, we used the data set 23.6 M in ZCAL trigger event forhigh statistic ++ two-particle correlations analysis. After the several cuts and the
Coulomb correction, the extracted ++ pairs in Au+Au 10% most central collisions
at 11 A GeV are about 2.5 M.
In order to calculate the correlation function, the uncorrelated distribution of pion
pairs is needed for the normalization. A convenient way is to make the uncorrelated
distribution from di erent events with the same cut criteria, called the event mixing
method.
In this analysis, the event mixing has been done by sampling in each successive veevent, and make the pair from di erent event in the same way as making combination
for the real events. The mixed event distribution is normalized by the factor, which
is obtained by the ratio of entries between the real and mixed events.

100

7.4 Parameterization
In this section, we describe the two di erent parameterization, Yano-Koonin Podgoretskii (YKP) parameterization and the standard side-out-long parameterization.

7.4.1 YKP parameterization
For extraction of the source parameters from the correlation functions, we employ
the Gaussian approximation in a representation given by Heinz [16] that is reveal
longitudinal collective motion. This parameterization is known as \Yano-Koonin
Podgoretskii parameterization" [15], and we abbreviate it as \YKP" hereafter.
The YKP parameterization procedure uses a Gaussian t for the two-particle
correlation function in boost invariant form. The relative momentum of a pair, Q
is decomposed into QT , QL and Q0 , where the momentum di erence of transverse
direction, that of longitudinal direction and the energy di erence of the two particles,
respectively. The frame is set in the local center of mass system of the pair. The
tting function in YKP parameterization can be expressed as,h
C2 (QT ; QL; Q0; KT ) = 1 +  exp ;Q2T RT2
; Y2 K QL ; Y K Q0 2 RL2 i; Y2 K Q0 ; Y K QL 2 R02 :(7.11)

This expression is the explicitof the energy di erence Q0 , transverse moq function
mentum di erence of QT = Q2x + Q2y , longitudinal momentum di erence QL = Qz .
The tting parameters in this parameterization are correlation intensity parameter ,source size parameters; RT , RL , R0 and Yano-Koonin velocity Y K , which describes
the source's longitudinal collective motion.

7.4.2 Standard side-out-long parameterization
Another parameterization, which is often used in 3-dimensional HBT analysis is thestandard side-out-long parameterization1 .
In this parameterization, the frame is chosen the Longitudinal Center of Mass
System (LCMS), which de ned as p1z + p2z = 0. As illustrated in Fig. 7.6, the relative
1 This parameterization is also call as Bertsch-Pratt parameterization.

7. Experimental Results II { Two-Particle Correlations

101

Figure 7.6: The illustration of the standard side-out-long parameterization. See text
for detail descriptions.
momentum can be decomposed into Q~ L and Q~ T , where Q~ L is parallel to the beamaxis and Q~ T is perpendicular to the beam axis. The same decomposition can be donewith the average momentum K~ = 21 (p~1 + p~2 ), the corresponding notation is K~ L and
K~ T . The transverse component Q~ T can be decomposed further into Q~ side and Q~ out ,where Q~ out is parallel to K~ T and Q~ side is perpendicular to K~ T . For the longitudinal
direction, the sux is replaced from L to long, QL ! Qlong . The tting function of
the standard side-out-long parameterization in the Gaussian approximation can be
expressed as,
2 ; Q2 R2 ; Q2 R2 ): (7.12)
C2(Qside; Qout ; Qlong ; KT ) = 1 +  exp(;Q2sideRsideout outlong long

The tting parameters are , Rside, Rout and Rlong .

102

7.5 Results in YKP Parameterization (KT dependence of RT )In this section, the results of averaged momentum KT dependence of RT parameter in
YKP parameterization are presented. First, we de ne the class of KT range. Fig. 7.7
shows the KT distribution of ++ pairs. The distribution is divided in three di erent
class of KT range, as listed in the Table 7.1.
Fig. 7.8 shows correlation functions in YKP parameterization as a function of QT
in three di erent KT range; (I) the low KT region (top left), (II) the mid KT region(top right) and (III) the high KT region (bottom left). For the other two variables,
QL, Q0 are projected on QT axis, if jQLj; jQ0j are less than 50 MeV/c.
For the tting function, number of parameters in Eq. 7.11 can be reduced bylimiting Q0 and QL variables. We use the tting function to extract the RT parameteras,

C2(QT ; KT ) = 1 +  exp(;Q2T RT2 ):(7.13)In Fig. 7.8, the tting results are also shown in each panel. The extracted RT
parameters are plotted as a function of KT in Fig. 7.9. It is observed that the gradualdecrease of RT from 5.21  0.17 fm to 3.73  0.26 fm with increasing KT from 0.1
GeV/c to 0.45 GeV/c.
As shown in section 7.1, Coulomb correction is important for the extraction ofsource parameters from two-particle correlations. In order to check the e ect ofCoulomb correction, RT parameters in each KT range with modi ed Gamov Coulomb
correction are extracted. Fig. 7.10 shows the two-particle correlation functions in
YKP parameterization in two di erent Coulomb corrections; standard Gamov (solid
circles) and modi ed Gamov (open circles) correction. Fig. 7.11 shows the KT dependence of RT parameters in standard Gamov correction and modi ed Gamov correcClass KT range [GeV/c]
Class I
0.1 { 0.25
Class II
0.25 { 0.35
Class III0.35 { 0.45
Table 7.1: The class of KT range.

7. Experimental Results II { Two-Particle Correlations

103

x 10 2
2000
1750
1500
1250
1000
750
500

I II III

250
0

0

0.2

0.4

0.6

0.8

KT (GeV/c)

Figure 7.7: The averaged transverse momentum KT distribution of ++ pairs. The
distribution is divided in three di erent class of KT range, as listed in the Table 7.1.
tion. For modi ed Gamov correction, Qeff is set as 46 MeV, which is used in NA49
Pb+Pb collisions HBT analysis. The extracted source size parameters are listed inTable 7.2. The extracted RT in modi ed Gamov correction are larger than that instandard Gamov correction and depend on the range of KT . In KT range I and II,
it is about 2.5 % and in range III, it is estimated about 1.0 %. These uncertainty
caused by coulomb correction are included as an systematic error for the model tting
in Chapter 8.
Standard Gamov
Modi ed Gamov
KT Class RT [fm]RT [fm]I
5.21  0.17 0.39  0.01 5.34  0.19 0.35  0.01
II
4.64  0.27 0.30  0.02 4.76  0.31 0.26  0.02
III
3.73  0.26 0.46  0.04 3.75  0.28 0.41  0.03
Table 7.2: Extracted RT and  parameters in two di erent Coulomb correction,
standard Gamov and modi ed Gamov correction.

104

C2

C2

Correlation function in YKP as a function of QT
(I)

1.4
1.2

1.2

1

1

λ : 0.385811 ± 0.0114791
RT : 5.2108 ± 0.170826 [fm]

0.8
0.6
0

0.05

(II)

1.4

λ : 0.300394 ± 0.0181208
RT : 4.64371 ± 0.272907 [fm]

0.8
0.6

0.1

0.15

0

0.05

C2

QT (GeV/c)

0.1

0.15

QT (GeV/c)

(III)

1.4
1.2

|QL|, |Q0| < 50 MeV/cprojection in QT1

λ : 0.455184 ± 0.0352542
RT : 3.72684 ± 0.257681 [fm]

0.8
0.6
0

0.05

0.1

0.15

QT (GeV/c)

RT (fm)Figure 7.8: ++ correlation functions in Au+Au central collisions at 11 A GeV
beam energy. The correlation functions are presented in YKP parameterization as afunction of QT in three di erent KT range; the low KT region (I; top left), the mid
KT region (II; top right) and the high KT region (III; bottom left). For the othertwo variables, QL, Q0 are projected on QT , if jQLj; jQ0j < 50 MeV/c. Solid curvesin the panels represent the tting results by Eq. 7.13 and the numbers correspond to
the extracted tting parameters,  and RT [fm] are also shown.
5
4
3
2

I

II

III

0.2

0.3

0.4

1
0

KT (GeV/c)

Figure 7.9: KT dependence of RT parameters extracted in YKP parameterization.
The error bars indicate the statistical errors only.

C2

C2

7. Experimental Results II { Two-Particle Correlations
(I)

1.4

1.2

1

1

0

0.05

0.1

0.15

(II)

1.4

1.2

0.8

0.8

0

0.05

QT (GeV/c)
C2

105

0.1

0.15

QT (GeV/c)

(III)

1.4
1.2

|QL|, |Q0| < 50 MeV/c

1
0.8

0

0.05

0.1

projection in QT

0.15

QT (GeV/c)

RT (fm)Figure 7.10: Two-particle correlation functions (YKP parameterization) in two di erent Coulomb corrections, standard Gamov (solid circle) and modi ed Gamov (opencircle) correction in three KT ranges. The tting results by Eq. 7.13 to the data arealso shown.

5
4
3
2

I

II

III

0.3

0.4

1
0

0.2

KT (GeV/c)

Figure 7.11: KT dependence of RT parameters extracted in two di erent Coulomb
corrections, standard Gamov correction (solid circles) and modi ed Gamov correction(open circles) with Qeff = 46 MeV in Eq. 7.8, respectively. The error bars indicate
the statistical errors only.

106

7.6 Results in the Standard Side-Out-Long Parameterization
To check the consistency, the correlation functions in the standard side-out-long parameterization are studied using the same data set and the same cut criteria, which
is used in the HBT analysis in YKP parameterization. Fig 7.12 shows the correlation
functions in the standard side-out-long parameterization. From left to right in thegure, the correlation function as a function of Qside, Qout and Qlong are shown. From
top to bottom, C2 functions at each KT range, classi ed in Table 7.1 are arranged.
These correlation functions in Qi is obtain by the limited in jQj j; jQk j < 50 MeV/c
where i, j and k indicate \side", \out" and \long".
The extracted source size parameters, Rside, Rout and Rlong are plotted as a function of KT in Fig. 7.12 and these values are listed in Table 7.3. It is observed thatthe gradual decrease of all source size parameters with increase of the average transverse momentum of pair, and it is consistent with the results obtained in the YKPparameterization.

KT range
Rside [fm] Rout [fm] Rlong [fm]I (0.1 < KT < 0.25 GeV/c) 4.71  0.50 4.27  0.12 3.71  0.20
II (0.25 < KT < 0.35 GeV/c) 4.24  0.57 3.33  0.13 3.20  0.20
III (0.35 < KT < 0.45 GeV/c) 3.17  0.35 3.47  0.22 2.66  0.20
Table 7.3: Extracted source size parameters in the standard side-out-long parameterization at each KT range.

C2

7. Experimental Results II { Two-Particle Correlations

107

1.4
1.2

I

1
0.8
0.6

C2

0

0.1

0.2 0

0.1

0.2 0

0.1

0.2

Qside (GeV/c)

Qout (GeV/c)

Qlong (GeV/c)

0.1

0.1

0.1

1.4
1.2

II

1
0.8
0.6

C2

0

0.2 0

0.2 0

0.2

Qside (GeV/c)

Qout (GeV/c)

Qlong (GeV/c)

0.1

0.1

0.1

1.4
1.2

III

1
0.8
0.6
0

0.2 0

Qside (GeV/c)

0.2 0

Qout (GeV/c)

0.2

Qlong (GeV/c)

Figure 7.12: The correlation functions of ++ pairs in the standard side-out-longparameterization. From left to right, the correlation functions as a function of Qside,
Qout and Qlong are shown. From top to bottom, C2 functions at each KT , classi ed
in Table 7.1 are arranged. The solid curves in each panel correspond to the tting
results with Eq. 7.12.

108

Figure 7.13: The average transverse momentum dependence of Rside (solid circle),
Rout (open square) and Rlong (open diamond) parameters in the standard side-outlong parameterization. The error bars indicate the statistical errors only.

7.7 Systematic Errors
Systematic error on the extracted HBT radii are itemized as follows.
1. Coulomb correction
We used the standard Gamov factor for the Coulomb correction in the analysis. The Gamov factor is \point-like", in that it describes the emission of an
isolated pair of charged particles from a small and otherwise neutral spatial
region (r < 1fm). Because of the nite size of realistic sources [61] and themulti-particle Coulomb screening which occurs when a large number of charged
pions are simultaneously emitted, the Gamov coulomb corrections is expected
to cause systematic errors in the case of Au+Au collisions at AGS energies. The
more realistic method for Coulomb correction has been developed by Pratt [37],which includes the Coulomb and the strong interaction corrections in the nitesource size. Another realistic way of Coulomb correction is the extraction of thefactor from measurements of correlation function of unlike-sign pairs as shownin Section 7.1 and 7.5. By the comparison between the extracted RT parameters with standard Gamov correction and that with modi ed Gamov correction,
it is estimated 2.5 % in KT range I and II, and 1.0 % in KT range III.

7. Experimental Results II { Two-Particle Correlations

109

2. Contamination in particle identi cation
In the analysis, the correlation functions are extracted identi ed +. The contamination from the other particles is estimated less than 1%, the error due tothe contamination is expected to be negligible.

7.8 Summary of the Results for Two Particle Correlations
For two-particle correlation analysis, 2.5 millions of ++ pairs near mid-rapidity at
11 A GeV beam energy are used to extract correlation functions. In the frame of
Yano-Koonin Podgoretskii (YKP) parameterization, it is found that the extracted
source size parameter in the transverse direction, RT , decrease from 5.21  0.17 fm
to 3.73  0.26 fm with increasing averaged transverse pair momentum KT from 0.1
GeV/c to 0.45 GeV/c. This gradual decease of RT parameters with increasing of KT
is consistent with the picture with a expanding source.
As a consistency check, the correlation functions in the standard side-out-longparameterization are studied using the same data set and the same cut criteria, which
is used in the HBT analysis in YKP parameterization. In that frame, it is also
observed that the gradual decrease of all source size parameters with increase theaverage transverse momentum of pair, and it is consistent with the results obtainedin the YKP parameterization.

Chapter 8
Discussions
In this Chapter, using the experimental results of single particle spectra in Chapter 6 and two-particle correlation in Chapter 7, we discuss
the transverse velocity t and the freeze-out temperature T of the systemcreated in Au+Au collisions at AGS energies within the framework of the
expanding source model. Then the extracted t and T are compared withthe results in Pb+Pb collisions at SPS, and we discuss validity of thepicture of QGP formation in Pb+Pb central collisions at SPS from theviewpoint of t and T .

8.1 Overview
As shown in Chapter 6, we have observed the di erent mean transverse mass and thechange of shapes in mt spectra for di erent particle species in Au+Au collisions at
AGS energies. In Chapter 7, the gradual decrease of RT parameters with increasing
averaged transverse momentum KT of pair has been observed in ++ correlation
function measurements at the energy of 11 A GeV central Au+Au collisions. As
mentioned in Chapter 2, above features of both single particle and HBT's can bequalitatively explained by a expanding source picture.
On the other hand, in Pb+Pb central collisions at CERN-SPS energy, the sametendency has been reported by several experiments [65, 66, 67]. The transverse expansion velocity t and the freeze-out temperature T have been extracted by NA49 collaboration from the simultaneous analysis of single particle spectra and two-particle
correlations using the framework of the nite expanding source scenario. Further110

8. Discussions

111

more, as we introduced in Chapter 1, interesting experimental results which indicate
the QGP formation have been reported in Pb+Pb at SPS energy in leptonic probes.
Thus the comparison of t and T at AGS energies with that of at SPS should provide
an important input to investigate the QGP formation.
In this chapter, the transverse velocity and the freeze-out temperature are extracted by tting the data in Au+Au central collisions at AGS energies with the
expanding source model. After the brief description of the nite expanding source
model in Section 8.2, tting results of the measured data in Au+Au central collisions
with the model are shown in Section 8.3. Finally, the comparison with other beamenergies are made, and the validity of picture of QGP formation at SPS energy are
examined qualitatively in Section 8.4.

8.2 Expanding Source Model
We start from review of the general approach and de nition of the notation with regard to the expanding source model [39, 40]. First, we mention the emission function,
which de nes the single particle momentum spectrum and two-particle correlation
function. Next, the expanding source model is introduced in various sources, thestatic Gaussian source, the boost-invariant source and the nite expanding source.
At the last part of this section, the functions of mt spectrum and RT including t
and T are given.

8.2.1 Emission function
The emission function S (x; p) describes the probability that a particle with on-shell
momentum p (p2 = m2 ) is emitted from the space-time point x. S (x; p) determines
the single particle momentum spectrum P1(p) as well as the two-particle correlation
function C2(K; q) as,Z
dN
P1 (p) = E d3p = d4xS (x; p);(8.1)
2
C2(K; q) = < N<(NN ;>1) > P P(2p(p)1; ;Pp2())(8.2)
1 1 1 p2 )R d4xS (x; K )eiqxj2
j= 1 + R d4xS (x; p ) R d4xS (x; p ) ;
(8.3)
1

1

where K = (1=2)(p1 + p2) = (KT ; KL) is the average momentum of the two particles,
q = p1 ; p2 is their relative momentum and P1(p), P2(p1; p2) are 1- and 2-particle

112
distributions. In the numerator of Eq. 8.3, we have introduced o -shell momentum
four-vectors K and q for the total and relative momentum of the particles in the pairpby de ning K0 = (1=2)(E1 + E2 ) and q0 = E1 ; E2 where Ei = m2 + pi2. Bypputting K on shell, K0 ' EK = m2 + K2 , and setting p1 = p2 = K in the 1-particle
distribution in the denominator, the correlation function is rewritten as,

R d4xS (x; K )eiqxj2
jC2(K; q) ' 1 + j R d4 xS (x; K )j2 :(8.4)

Therefore, if the emission function S (x; K ) is known, the single particle spectrum andtwo particle correlation can be calculated by Eq. 8.1 and Eq. 8.4, respectively.

8.2.2 Static Gaussian sourceFirst, we consider a static emission function which is de ned by a space-time Gaussian
at a position (x; y; z) and time t in its center of mass frame as; 2 y2 z2 (t ; t0 )2 S (x; K ) = h(K ) exp ; x 2+
R2 ; 2L2 ; 2(t)2 ;(8.5)

where the z axis is along the beam (longitudinal) direction, L and R are the sizeof the source in the beam direction and in the perpendicular (transverse) direction
with respect to the beam axis, respectively. The average and the root mean squareof freeze-out time of the source is shown as t0 and t, respectively. We de ne therapidity corresponding to K as; K0 + Kz :
(8.6)
Y = 21 ln K
0 ; KzThe transverse mass MT and the energy K0 corresponding to K are de ned by;q
MT = m20 + KT2 ;K0 = MT cosh(Y );

(8.7)
(8.8)

where m0 is the rest mass of the particle. Assuming a thermalized system, the factor h(K ) in Eq. 8.5 can be expressed by the Lorentz invariant Boltzmann factor,
K0 exp(;K0 =T ), where T is the temperature at freeze-out. In this case, the source
function S (x; K ) can be rewritten as;
2 y2z2 ; (t ; t0 )2 ;;S (x; K ) = NMT cosh(Y ) exp ; KT0 ; x 2+
R2
2L2 2(t)2(8.9)

8. Discussions

113

where N is a normalization factor. In the Yano-Koonin-Podgoretskii parameterization, abbreviation as YKP hereafter, tting parameters are given as;

RT = R;
RL = L;R0 = t:(8.10)(8.11)(8.12)

8.2.3 Boost-invariant source
Next we consider a longitudinally expanding source which invariant under a Lorentz
boost along the beam axis. The emission function can be expressed as,
2 y 2 ( ;  )2 0S (x; K ) = NMT cosh(Y ; ) exp ; KT u ; x 2+;;
(8.13)
2R
2( )2p

where  = t2 ; z2 is the longitudinal proper time, 0 and  are the averageand the root mean square of the freeze-out proper time of the source, respectively.
The longitudinal expansion of the source is described by the ow four-velocity u(x)relative to some xed frame as,u(x) = (cosh ; 0; 0; sinh ):(8.14)In the expression, we assume longitudinal boost invariance by setting vL = z=t, i.e.,
identifying the longitudinal ow rapidity  ow = (1=2) ln[(1+ vL)=(1 ; vL )] with spacetime rapidity  = (1=2) ln[(t + z)=(t ; z)].
The YKP tting procedure gives:

RT = R s

RL = 0 MT ;T
R0 = :(8.15)(8.16)(8.17)

8.2.4 Finite expanding source
Finally, we consider the nite expanding source model as an extension of the above
models. For the assumption used in the model, it can be summarized as below,
1. Local thermal equilibration of the system.

114

Figure 8.1: Flow velocity pro le of a medium, u, which is a time-like unit vector
parallel to the world-line of particle. Reproduced from [68].
2. Transverse and longitudinal motion can be decoupled.
3. Longitudinal boost invariant.
4. Azimuthally symmetric Gaussian source.
5. Sudden freeze-out at a temperature T and a time 0 for all particle species.
6. No resonance contributions.
Considering these assumptions, the emission function in the case of the nite expanding source can be expressed as,
2 y 2 ( ;  )2
2 : (8.18)
0
S (x; K ) = NMT cosh(Y ; ) exp ; KT u ; x 2+;;
R2
2( )2 2()2
The cuto in space-time rapidity  is introduced due to the terms in cosh(Y ; )in the above expression. In the approximation that MT =T  1, the YKP tting
procedure gives as,

RT = R;1  12 ;1
RL = 0 MTT cosh(Y ; ) ;+cosh2 (Y ; ) ()2R0 = ;(8.19)(8.20)(8.21)

where the parameter v is the velocity di erence between the measurement frame andthe local rest frame of the source. The parameter  is solution of :tanh(Y ; ) ; MTT sinh(Y ; ) + ()2 = 0:(8.22)

8. Discussions

115

In the case of a boost-invariant source, the above equation is simply  = Y . In thelimit  ! 1, the nite expanding source reduces to the boost-invariant source.
Next, We consider now a source function with a transverse velocity t in the ow
four-velocity as,u(x) = (cosh  cosh t ; sinh t cos ; sinh t sin ; sinh  cosh t);(8.23)The transverse ow t is taken azimuthally () symmetric. It is parameterized as theuniform expansion in r direction as: (t r) = t r ;(8.24)
Rp
where t is a constant and r = x2 + y2. In the YKP tting procedure, the measured
RT is given as,
1
RT ' 1 + MTT f2 2
q 2 2

K + m 2  21
(8.25)= 1+ Tf
T
As seen in Eq. 8.25, RT decreases with increasing the expansion velocity and withincreasing average transverse momentum KT of pair. We use the function in Eq. 8.25
to t the measured data of KT dependence of RT parameter. It should be noted that
only if t is incorporated in the model, the KT dependence on RT appear.

8.2.5 mt spectrum in nite expanding source
In this section, we will discuss the transverse mass spectrum in the framework of
the nite expanding source model given by Chapman et al. [69]. The single particle
distribution P1 in the case can be approximated by the expression as a function of
the transverse mass mt and the rapidity y as,
"
#
2
R
m
t

2
2
4
P1(mt ; y) / mtR  1 + 2R2 ( ) ; 8T ( )G"#2
2 ; m2
2
m
(
m
0
:
5
y
t
t
t
0
exp ; T + 2T (T + m 2 ; ()2 + T=m ;(8.26)t tt
where
1 = 1 1 + 2 mt  ;(8.27)t T
R2
RG2
1 = 1 + mt :(8.28)
2()2 T

116
In Eq. 8.28,  is the width of the space-time rapidity  = (1=2) ln(t + z)=(t ; z).
This is related to the width of the rapidity distribution y as,(y)2 = ()2 + T :(8.29)
mt
In Fig. 8.2, it is demonstrated the change of the shapes and slopes of mt spectra
for deuterons, protons, kaons and pions at the xed temperature T = 150 MeV andvarious transverse velocity from t = 0.0 to 0.6 in 0.2 step. It is clearly shown thatthe shoulder arm shape arises in deuteron and proton spectra by increasing t , incontrast, in pions spectra, t changes the shapes from exponential scaling to concave
shape particularly at low mt region. Spectra shapes of kaons are almost the same, but
increasing its inverse slope parameters with increasing t . These features are clearlyseen in 4 and 11 A GeV Au+Au collisions as discussed in Chapter 6.

8.3 Fitting Results by the Finite Expanding Source
Model
8.3.1 Fitting results of single particle spectra
Using Eq. 8.26 as the tting function to the data, t and T are extracted in Au+Au
central collisions at 4 and 11 A GeV, near mid-rapidity region.
Fig. 8.3 and Fig. 8.4 show the results of the tting by the nite expanding source
model for deuterons, protons, kaons and pions in Au+Au collisions at 4 and 11 A
GeV beam energy, respectively. For the pion spectra at both 4 and 11 A GeV, averageof + and ; spectra is taken for the tting, and low mt range is excluded for thetting to avoid the contribution from resonance decay.
As shown in the Fig. 8.3 and Fig. 8.4, all the spectra at 4 and 11 A GeV are well
reproduced by the model. Fig. 8.5 and Fig. 8.6 show allowed regions of freeze-out
temperature T vs. transverse velocity t , derived from the t of Eq. 8.26 to single
particle spectra of deuterons, protons, kaons and pions in Au+Au collisions near midrapidity at 4 and 11 A GeV, respectively. Contours in each gure are drown at two
standard deviations from the best t values. The overlapped region of contours in
the gure can be consider to the allowed region in t and T of the system. For the
results at 4 A GeV, the spectra have the overlapped region in T = 80  90 MeV andt = 0:6  0:7, except for the deuteron. For the results at 11 A GeV, there are the8. Discussions

117

10

10

10

10

10

-5

deuteron-6βt = 0.6(a.u.)(a.u.)

Single Paritcle distribution βt=0.0-0.6, T=150 MeV-2

proton

10
-3

10

-7

-4

10
-8

-5

10

βt = 0.0
-9

-6

10
0

0.5

1

1.5

0

0.5

2

10

10

10

-1

K

π

1
-1

-2

10

-3

10

-2

-3

-4

10

-5

10

1.5

mt-m0 (GeV/c )(a.u.)(a.u.)

101

2

mt-m0 (GeV/c )

10
0

0.5

1

1.5

mt-m0 (GeV/c2)

-4

0

0.5

1

1.5

mt-m0 (GeV/c2)

Figure 8.2: mt spectra for deuterons, protons, kaons and pions calculated by the nite
expanding source model (Eq. 8.26) at T = 150MeV and various transverse velocitiest.

118
allowed region T = 90  100 MeV and t = 0:65  0:85. It is found the extracted
value of both t and T are larger at 11 A GeV beam energy than that at 4 A GeV
in central Au+Au collisions. The stronger transverse expansion velocity with higher
temperature at 11 A GeV are created than that of 4 A GeV beam energy.
As discussed in Section 1.2.3, overlapped region is rather broad, so that the unique
set of parameter ( t; T ) can not be determined by the single particle spectra alone,
because of the strong anti-correlation between t and T . In order to resolve thisuncertainty, KT dependence of RT parameter from two-particle correlations provide
the another boundary in t vs. T plane, which is expected in perpendicular to thecontours from single particle spectra.

8. Discussions

119+(a.u.)(a.u.)

Au+Au at 4 A GeV Central-(π +π )/2

K+

1

1

10

10

10

-1

-1

-2

10
0.25

0.5

0.75

1

-2

1.25

0.2

0.4

2

proton0.8

mT-m0 (GeV/c )(a.u.)(a.u.)

100.6

2

mT-m0 (GeV/c )deuteron

1

10
10

-1

-1

0.25

0.5

0.75

1

mT-m0 (GeV/c2)

0.2

0.4

0.6

0.8

mT-m0 (GeV/c2)

Figure 8.3: Fitting results by nite expansion source model for deuterons, protons,
kaons and pions in Au+Au collisions at 4 A GeV beam energy. The centrality ofdata is 10% most central. For pions, cross-section are deduced by taking average(+ + ;)=2.

120

10

2

+

(a.u.)(a.u.)

Au+Au at 11 A GeV Central-(π +π )/2

K

10

+

10

1

1

-1

10

0.5

1

1.5

0.2

proton0.4

0.6

mT-m0 (GeV/c2)(a.u.)(a.u.)

mT-m0 (GeV/c2)

deuteron

10
10

-1

1

10

-1

0.5

1

0.2
2

mT-m0 (GeV/c )

0.4

0.6
2

mT-m0 (GeV/c )Figure 8.4: Fitting results by nite expansion source model for deuterons, protons,
kaons and pions in Au+Au collisions at 11 A GeV beam energy. The centrality ofdata is 10% most central. For pions, cross-section are deduced by taking average(+ + ;)=2.

8. Discussions

T (MeV)

121

200

Au+Au at 4 A GeV Central

180
160

K+

140

deuteron

120
+

100

π

80

proton

60

0.2

0.4

0.6

0.8

1
βt

T (MeV)

Figure 8.5: Allowed regions of freeze-out temperature T vs. transverse velocity t ,derived from the t of Eq. 8.26 to single particle spectra of deuterons, protons, kaonsand pions in central Au+Au collisions at 4 A GeV, near mid-rapidity. Contours areshown for two standard deviations from the best t.

200
180

Au+Au at 11 A GeV Central
K

deuteron

+

160
140

proton

120
+

100

π

80
60

0.2

0.4

0.6

0.8

1
βt

Figure 8.6: Allowed regions of freeze-out temperature T vs. transverse velocity t ,derived from the t of Eq. 8.26 to single particle spectra of deuterons, protons, kaonsand pions in central Au+Au collisions at 11 A GeV, near mid-rapidity. Contours areshown for two standard deviations from the best t.

RT (fm)

122

6
5
4
3
2
1
0

0.2

0.3

0.4

KT (GeV/c)

Figure 8.7: Fitting result of KT dependence of RT parameter in YKP parameterization by the nite expanding source model in Eq. 8.25 (solid line). The solid circles
correspond to the experimental data obtained in Section 7.4.

8.3.2 Fitting results of two-particle correlations
Next, we consider KT dependence of RT by the same expanding source model, which isused in the tting to the single particle spectra. The tting function is the expressionin Eq. 8.25, and tting parameter is the ratio of t2=T . Fig. 8.7 shows the tting
result, which is included systematic error from Coulomb corrections in the error bars.
The extracted parameter by the tting is t2=T = 8:00  0:70[GeV;1]. From the ratio,
the allowed region of T ; t can be determined. Fig. 8.8 shows the the allowed regionof t vs. T , derived from the t of Eq. 8.25 to the data of KT dependence of RT
parameter in YKP parameterization. Contours are drown at two standard deviations
from the best t values.
The contours obtained from the analysis of two-particle correlation provide thefurther restriction to determine t and T uniquely, because of the di erent domain int ; T plane between single particle spectra and HBT.

8. Discussions
T (MeV)

123
2

120

β t/T = 8.00 ± 0.70

100

(GeV-1)

80
60
40
20
0

0

0.2

0.4

0.6

0.8

1

βt

Figure 8.8: The allowed region of freeze-out temperature T vs. transverse velocityt , derived from the t of Eq. 8.25 to two-particle correlations of  +  + pairs incentral Au+Au collisions at 11 A GeV, near mid-rapidity. Contours are drown at two
standard deviations from the best t values.

8.3.3 Simultaneous analysis of single particle spectra andHBT
In single particle analysis, strong anti-correlation between t and T is seen, while in
HBT analysis, t and T is found to be proportional. Therefore, t and T cannot
be determined uniquely neither from the analysis of single particle nor HBT alone.
But by combining these two regions, it can be resolved this essential ambiguity anddetermine t and T uniquely.
Fig. 8.9 shows the overlay of the tting results (Fig. 8.8) obtained in HBT on
that from single particle spectra contours (Fig. 8.6) in Au+Au central collisions near
mid-rapidity. From Fig. 8.9, we can obtain the overlap region between the contours
from single particle spectra and that from HBT analysis, at T = 95  5 MeV andt = 0:80  0:05, which characterize the system created in Au+Au central collisions
at the energy of 11 A GeV.

T (MeV)

124

200
180

deuteron

K+

160

+ +

π π HBT

140
120
100

π+

proton

80
60

0.2

0.4

0.6

0.8

1

βt
Figure 8.9: Allowed regions of T vs t for central Au+Au collisions near mid-rapidity,
derived from the t of Eq. 8.26 to transverse mass spectra of deuterons, protons, kaonsand pions, and from the t of Eq. 8.25 to to two-particle correlations of ++ pairs
in central Au+Au collisions at 11 A GeV, near mid-rapidity. Contours are shownfor two standard deviations from the best t for transverse spectra and two-particle
correlations.

8. Discussions

125

8.4 Excitation Function of < t > and T
As introduced in Chapter 1, it is predicted that the QGP formation causes the softening of the equation of state because the internal pressure relative to the internalenergy drops at a critical temperature Tc [11, 62, 63, 64]. Thus the excitation functionof transverse expansion velocity is considered as an important clue for the detectionof the QGP formation; a sudden drop of t may indicate the QGP formation. In thissection, we discuss the excitation functions of transverse velocity t and the freeze-out
temperate T from the viewpoint of the QGP formation.

8.4.1 Excitation functions from SIS to AGS and SPS
The mean transverse expansion velocity < t > and freeze-out temperature T havebeen studied in various collision system in wide range of the beam energies from GSISIS energy regime, 100 A MeV to CERN-SPS energy regime, 200 A GeV by the tting
the single particle spectra with the radial ow model [27]. Fig. 8.11 and Fig. 8.12
show the excitation function of < t > and T from various experiments, respectively.
The extracted values from the present analysis at AGS and from NA49 at SPS arealso plotted in the gures, extracted by the simultaneous analysis of single particle
spectra and HBT with the expanding source model. It should be noted that meantransverse velocity in this analysis and that of NA49 collaboration are deduced by
taking average of the velocity pro le of the expanding source model, t (r) = t Rr
as,
R R dr (r) 2< t >= 0 R R t = 3 t ;(8.30)
0 dr rwhere R is the radius of the system and r is the distance from the center of thesystem. As seen in Fig. 8.11, the values of < t > rise monotonically with Ebeam up
to the AGS energy region ( 10 A GeV), but it seems to saturate at the SPS energy.
From Fig. 8.12, on the other hand, freeze-out temperatures rise continuously withincreasing of Ebeam .

8.4.2 Comparison of t and T between AGS and SPS energies
At CERN-SPS energy (158 A GeV) in Pb+Pb collisions, the similar simultaneous
analysis of single particle spectra with HBT have been carried out. The extracted
values in Pb+Pb central collisions by NA49 experiment is T = 120  12 MeV and126
t = 0:55  0:12 [17].

In comparison of these at AGS and at SPS, the stronger
transverse velocity t is obtained in central Au+Au collisions at 11 A GeV than thatof SPS, while the freeze-out temperature is larger at SPS than at AGS energy at thecon dence level of 95 %. How can we interpret the tendencies of t and T ?
First, we consider the freeze-out temperature. The extracted T from the simultaneous analysis of single particle spectra with HBT is not the value for the simple
parameterization like a inverse slope parameter, but is the more reliable value as thetemperature of the system at freeze-out stage, because of 1) less ambiguity betweent and T (as shown in Section 8.3.1) and 2) consistency; the same model are used fortting with both data.
According to the lattice QCD calculations, the phase transition from hadronic
phase to QGP phase occurs at the critical temperature Tc = 140  200 MeV inbaryon-free matter [1]. We cannot compare directly the results of the lattice QCD
with extracted freeze-out temperature. But as regarding the reduction of extracted
t at SPS energy and extracted T is not so far from Tc , it is not so hard to assume
that the temperature of the system created in Pb+Pb at SPS is above Tc, then thesystem expands and cools down to the freeze-out temperature T = 120 MeV, andnally, the particles are emitted from the reaction zone freely. On the other hand, in
Au+Au central collisions at 11 A GeV, extracted T = 95  5 MeV seems far from thecritical temperature. Therefore, it is likely that beam energy corresponding to the
QGP phase transition is in between AGS and SPS.
Several interesting features of the observation have been reported in Pb+Pb collisions at SPS, among which are the suppression of J= production [7] by NA50
experiment and the enhancement of low-mass dilepton [9] by CERES experiment.
NA50 collaboration have claimed in the paper [7] that \an observed anomalous J=
suppression can be considered as a strong indication of the production of a QGP
phase in central Pb+Pb collisions". Ordinal hadronic scenario has been shown to failin qualitative explanation of J/ data.
The observed e ect of the smaller t at SPS compared to that at AGS energy
seems to be explained by the softening of the equation of state (EoS) due to a QGP
formation at SPS energy in Pb+Pb collisions. Fig. 8.10 shows the equation of state,
derived by simple parameterization of the results of lattice QCD calculations [11]. The
ideal hadron gas (dashed lines), a rst order QGP phase transition (solid lines) and asmooth QGP transition (dotted lines) are shown in the gure. Each panel in Fig. 8.10

8. Discussions

127

Figure 8.10: The equation of state: (a) entropy density divided by T 3, (b) energy
density divided by T 4 as function of temperature, (c) pressure, and (d) the squared
velocity of sound as functions of energy density. Solid lines are for a rst ordertransition (T =0), dotted lines for a smooth transition with T = 0:1Tc, anddashed lines are for an ideal gas with dH degrees of freedom. The ratio of degrees of
freedom is dQ=dH = 37=3 [11].
is; (a) the entropy density, (b) the energy density as functions of temperature, and (c)the pressure and (d) the squared velocity of sound c2S  dp=d as functions of energy
density. T is the width of temperature at Tc. As can be seen in (c) in Fig. 8.10, forT = 0 the pressure stays constant in the mixed phase H    Q, where H is theenergy density at the phase boundary between mixed phase and hadronic phase, andQ is that between mixed phase and QGP. As hydrodynamical expansion is driven bypressure gradients, the square of the velocity of sound c2S = dp=d is the most relevantmeasure of the system's tendency to expand as shown in Fig 8.10 (d). For T = 0,
the velocity of sound vanishes in the mixed phase, i.e., mixed phase matter does notexhibit expansion at all. For nite T , pressure gradients are nite, but still smaller
than for an ideal hadronic gas, and therefore the expansion velocity is also reduced(Fig. 8.10 (d)).
The reduction of c2S in the transition region is commonly referred to as \softening"of the EoS. For matter passing through that region during the expansion phase, thevelocity of expansion will temporarily slow down. Based on the above picture, theobserved reduction of t at SPS may be attributed to these softening of EoS by theformation of QGP.

128

Figure 8.11: Excitation function of mean transverse velocity < t > extracted by thetting single particle spectra with the expanding source model [27]. Extracted values
in this analysis and NA49 collaboration by the simultaneous analysis of single particle
spectra and HBT are also shown in terms of < t >. See also [70] for the compilationof < t > extracted from single particle spectra alone in various experiments.

8. Discussions

129

Figure 8.12: Excitation function of freeze-out temperature T extracted by the tting
single particle spectra with the expanding source model [27]. Extracted values inthis analysis and NA49 collaboration by the simultaneous analysis of single particle
spectra and HBT are also shown. See also [70] for the compilation of T extracted
from single particle spectra alone in various experiments.

Chapter 9
Conclusions
We have presented the results of single particle spectra in Au+Au collisions at theenergy of 4 and 11 A GeV, and the results of high statistic ++ two-particle correlations in Yano-Koonin Podgoretskii (YKP) parameterization in Au+Au collisions
at 11 A GeV, measured by the E866 experiment using the Alternating Gradient Synchrotron (AGS) at Brookhaven National Laboratory (BNL).
For the systematic studies of hadron productions in relativistic heavy-ion collisions
near mid-rapidity region, we have constructed a rotatable magnetic spectrometer,
called the Forward Spectrometer. It is designed for the good particle identi cation
capability under the condition of high particle density and for the wide range ofkinematic coverage in Au+Au collisions at AGS. Using the data taken by the Forward
Spectrometer, single particle spectra and HBT correlations are analyzed.
For single particle analysis, we have measured transverse mass spectra for ,
K +, protons and deuterons at 4 and 11 A GeV as a function of centrality. For thetransverse mass spectra, we have observed the mass dependence of mean transverse
mass < mt > ;m0 ; for the particle with heavier mass, the larger < mt > ;m0 is
seen. In addition, the shapes of the mt spectrum for protons have a shoulder-arm
shape in central collisions at low mt region. On the other hand, the pion spectra at
11 A GeV have a concave shape at low mt region, which can be attributed, in part, tothe e ects of a transverse velocity eld of a created system. Observed shoulder-arm
shapes of protons and the mass dependence of mean transverse mass are consistent
with an expanding source.
For the proton's dN=dy, the distributions have a maximum at mid-rapidity atboth 4 and 11 A GeV beam energy in central events, which indicates a large baryon
1309. Conclusions

131

stopping in Au+Au collisions at AGS energies.
In ++ HBT correlation analysis in YKP parameterization, it is found that theextracted source size parameter in the transverse direction, Rt , decreases from 5.21
 0.17 fm to 3.73  0.26 fm with increasing the transverse pair momentum KT from
0.1 GeV/c to 0.45 GeV/c. This gradual decease of RT with increasing KT is also
consistent with the expanding source.
As a consistency check, the correlation functions in the standard side-out-longparameterization are studied using the same data set and the same cut criteria, which
is used in the HBT analysis in YKP parameterization. In that frame, it is also
observed that the gradual decrease of all source size parameters with increase theaverage transverse momentum of pair, and it is consistent with the results obtainedin the YKP parameterization.
Motivated by these observations that both single particle spectra and HBT correlations are qualitatively consistent with the picture of an expanding source, thesimultaneous analysis of both data using the nite expanding source model is carriedout for the rst time at AGS energy. From this analysis, the freeze-out temperature T
and the transverse expansion velocity t in Au+Au central collisions at 11 A GeV are
extracted uniquely; T = 95  5 MeV and < t >= 0:53  0:03, which can be considered
as more reliable values than those from the conventional analysis of single particle
spectra alone. Compared with T = 120  12 MeV and < t >= 0:37  0:08, reported
from the former analysis at CERN-SPS 158 A GeV Pb+Pb collisions, T is larger at
SPS than at AGS, while < t > is larger at AGS than at SPS at the con dence levelof 95 %. The softening of equation of state (EoS) by the QGP formation is one of
the qualitative interpretation for the observed small < t > at SPS energy than thatat AGS. Thus, the reduction of < t > at SPS dose not contradict the picture of the
QGP formation in Pb+Pb collisions at 158 A GeV, which is indicated by the recent
observation of anomalous J= suppression and low-mass dilepton enhancement.

Appendix A
Kinematic Variables
Here, we introduce kinematic variables used in the thesis. It is useful to describe
them with Lorentz invariant variables or variables which have simple Lorentz transformation properties, because we deal with relativistic particles and system .
We take a beam line to be z-axis of a frame. For a particle which has momentum
p = (px; py ; pz )qand mass m, we use a longitudinal momentum pz , a transverse mop
mentum pt  p2x + p2y , and total energy E = p2 + m2 . Transverse mass mt andrapidity y are de ned as

mt 

q

p2t + m2
(A.1)!
E + pz :(A.2)
y  12 ln E
; pz
The total energy and longitudinal momentum of a particle can be easily related to
its transverse mass and rapidity as
E = mt cosh(y)pz = mt sinh(y):

132

(A.3)
(A.4)

Appendix B
The BNL-AGS E802/866
Collaboration List
BNL-UC Berkeley-UC Riverside-Columbia-INS(Tokyo)-Kyoto-LLNL-Maryland-MITTokyo-Tsukuba-Yonsei
L.Ahleh, Y.Akibae, K.Ashktoraba , M.D.Bakerh, D.Beavisa, H.C.Brittg , J.Changc,
C.Chasmana , Z.Chena, C.-Y.Chid , Y.Y.Chua , T.Chujok , V.Cianciolog , B.A.Coled ,H.J.Crawfordb , J.B.Cumminga , R.Debbea, J.C.Dunloph , W.Eldredgec, J.Engelageb,
S.-Y.Fungc, J.Gaardhojei , E.Garciam, M.Gonina , S.Gushuea, H.Hamagakie ,A.Hanseni, L.Hanseni, R.S.Hayanoj , G.Heintzelmanh, S.Hommae , E.Juddb ,H.Kanekof , J.Kangl , E.-J.Kiml , A.Kumagaik , K.Kuritak , J.-H.Leea, M.J.Levinea ,J.Lukeg , Y.Miakek , A.Mignereym, B.Moskowitza , M.Moulsond , C.Muentza ,S.Nagamiyad , M.N.Namboodirig , C.Ogilvieh, J.Olnessa, L.P.Remsberga, H.Sakok ,T.C.Sangsterg , R.Setoc, J.Sheam, K.Shigakij , R.Soltzg , S.G.Steadmanh,
G.S.F.Stephansh, M.J.Tannenbauma , J.H.Thomasg , S.Ueno-Hayashik , F.Videbka ,F.Wangd , Y.Wangd , Y.Wud , H.Xiangc, G.H.Xuc, K.Yagik , H.Yaoh , W.A.Zajcd ,Q.Zhuc, F.Zhua
a Brookhaven National Laboratory, Upton, NY 11973

b University of California, Space Sciences Laboratory, Berkeley, CA 94720
c University of California, Riverside, CA 92507

d Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY

133

134
10533
e Institute for Nuclear Study, University of Tokyo, Tokyo 188, Japanf Kyoto University, Sakyo-ku, Kyoto 606, Japang Lawrence Livermore National Laboratory, Livermore, CA 94550
h Massachusetts Institute of Technology, Cambridge, MA 02139
i Niels Bohr Institute for Astronomy, Physics, and Geophysics, University of
Copenhagen, Denmark
j Department of Physics, University of Tokyo, Tokyo 113, Japank University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japanl Yonsei University, Seoul 120-749, Korea
m University of Maryland,College Park, MD 20742

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