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This is to certify that the
dissertation entitled
A Hadronic Transport Model for
Relativistic Heavy Ion Collisions

presented by
Bao-An Li

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in PhYSiCS

 

 

 

 

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MSU Is An Affirmative Action/Equal Opportunity Institution
chS-m

 

 

 

 

 

 

 

 

 

A HADRONIC TRANSPORT MODEL FOR
RELATIVISTIC HEAVY ION COLLISIONS

By

Bao-An Li
A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1991

ABSTRACT

A HADRONIC TRANSPORT MODEL FOR RELATIVISTIC HEAVY ION
COLLISIONS

By

Bao-An Li

A hadronic transport model for relativistic heavy ion collisions is developed by de-
riving and solving numerically a coupled set of transport equations for the phase
space distribution functions Of nucleons, Delta resonances and pions. Starting from
an effective hadronic Lagrangian density with minimal couplings between baryons
and mesons, we first derive coupled equations of motion for the density matrices of
nucleons, Delta resonances, and pi mesons as well as for the pion-baryon interaction
vertex function. By truncating at the level of two-body correlations a closed set
of equations of motion for the one body density matrix is obtained. A subsequent
Wigner transformation then leads to a tractable set of relativistic transport equa-
tions for interacting nucleons, Delta resonances and pions. The transport equations
are then solved numerically with the test particle method to study relativistic heavy

ion collisions.

The validity of the model can be seen from its ability of reproducing available
experimental data, explaining experimental and theoretical puzzles, as well as its
predicting power for new phenomena. The experimentally observed concave shape of
the pion spectra in relativistic heavy ion collisions is well reproduced. The mecha-

nism that causes the concave shape of the pion spectra is found to be the different

contributions of the delta resonance produced during the early and the late stages
of the heavy ion collision and due to the energy dependence of the pion and delta
absorption cross sections. The dependence of the shape of the pion spectra on the
beam energy, the target and projectile mass, and the impact parameter is also stud-
ied. An approximate scaling function for the shape parameter of the pion spectra is
predicted. Another new phenomenon that the model is able to explain is the prefer-
ential emission of pions in asymmetric nucleus-nucleus collisions. It is found that the
preferential emission of pions away from the interaction zone towards the projectile
side in the transverse direction and longitudinal direction is due to the stronger pion

absorption by the heavier target spectator.

This thesis is dedicatedwith love to my wife Wei Sun

iv

ACKNOWLEDGEMENTS

First and foremost, I am deeply grateful to my thesis advisor, Dr. Wolfgang
Bauer, for his invaluable tutelage and constant support. The freedom given to me by
Wolfgang to study various topics as well as his insight, encouragement, patience and
enthusiasm in trying to understand these different research topics have been critical
for my being able to write this thesis and to broaden my knowledge in the vast field

of nuclear physics.

I would like to thank Dr. George F. Bertsch for all of his help and enlightening
advice during my graduate study at MSU. From him I learned details of the BUU
transport model for heavy ion collisions. Collaborations with him in the study of pion
collectivity and pion flow have resulted in two published papers and one part of this
thesis. Dr. Pawel Danielewicz deserves a great deal of my appreciation for his many
pertinent comments and suggestions during my thesis research and the friendship of
his family. I would like to thank Dr. Walt Benenson for answering my questions
regarding the experimental aspects of pion physics and his encouragement by telling
me details of his planned experiments. I would like to express my gratitude to Shari
Conroy for her invaluable assistance and friendship through out the course of this
research. I am very grateful to Dr. Dan Stump and Dr. Philip Duxbury for serving
on my thesis guidance committee. I would also like to acknowledge all other faculty
and staff members of the Department of Physics and the National Superconducting
Cyclotron Laboratory for their quality education and their support in many ways. I

sincerely thank all of my fellow graduate students for their friendship.

I would like to take this space to express my gratitude to the nuclear theory group
at Oak Ridge National Laboratory and the Department of Physics and Astronomy

at the University Of Tennessee for the hospitality and the intellectual stimulation I

V

received when I was visiting there from August, 1986 to July, 1987. Particularly, I am
greatly indebted to Dr. Cheuk-Yin Wong for his friendship and guidance in research.
From him I learned the Glauber-type multiple collision model for relativistic heavy ion
collisions and much basic knowledge which has been very essential in doing research.
His continuous encouragement and valuable advice haveibeen very helpful during my

graduate study.

A heartfelt “thank you” goes to Dr. Philip J. Siemens now at Oregon State
University. His invaluable assistance in many ways enabled me to enter graduate
school in the U.S. and be successful. More importantly, I benefited from his guidance
and teaching when we were at Oak Ridge. From him I learned the basic idea. of liquid-
gas phase transition and nuclear multifragmentation. I have profited from knowing

him as a teacher and take pleasure from knowing him as a friend.

I would like to express my gratitude to Dr. Jakob Bondorf at the Neils Bohr
Institute and Dr. Christerfer J. Pethick at NORDITA for their warm hospitality

during my visit of Copenhagen in the summer of 1987.

I greatly appreciate the essential contributions from all of my collaborators. Dr.
Shun-Jin Wang at University of Lanzhou and Dr. J¢rgen Randrup at Lawrence
Berkeley Laboratory contributed a great deal to this thesis through the collaboration
in the study of the relativistic transport theory for hadronic matter. I benefited
from the collaboration with Dr. Scott Pratt now at the University of Wisconsin,
Madison in the study of nuclear liquid-gas phase transition, the collaboration with
Dr. Volker Koch at the State University of New York at Stony Brook in the study
of pion collectivity in relativistic heavy ion collisions, and the collaboration with Dr.
Mahir S. Hussein at the University of 35.0 Paulo in the study of pion production with
radioactive nuclei. I give my special thanks to Dr. Jing-Ye Zhang at the Institute of

Modern Physics, Chinese Academy of Science, who taught me the cranking model for

vi

studying nuclear structure at high spin states. With him I published my first paper
entitled “The shape coexistence of Krypton isotopes” when I was an undergraduate

student. His influence has been essential to me in studying nuclear physics.

I owe a great deal to my family, my parents and my sisters for their encouragement
and moral support throughout the years. Finally, I would like to deeply thank my
wife, Wei Sun, for her love, understanding and encouragement and so I dedicate this

thesis to her with love.

vii

Contents

LIST OF TABLES x

LIST OF FIGURES xi

1 Introduction 1

2 Relativistic Transport Equations for Hadronic Matter 7
2.1 Model for hadronic matter ........................
2.1.1 Model Lagrangian .........................

2.1.2 Equations of motion for the hadron fields ............ 10

2.1.3 Hamiltonian for hadronic matter ................ 11

2.2 Density matrix treatment ........................ 14

2.2.1 Density operators ......................... 15

2.2.2 Equation of motion for baryons ......... ~ ........ 16

2.2.3 Equation of motion for pions ................... 19

2.3 Transport equations for hadronic matter ................ 21

2.3.1 The Wigner transformations ................... 22

2.3.2 The Vlasov terms ......................... 23

2.3.3 The baryon-baryon collision terms ................ 25

2.3.4 The baryonopion collision terms ................. 26

2.3.5 Transport equations ....................... 30

3 Numerical Realization of the Model 34

3.1 Equations of motion for test particles .................. 34

3.2 Thenuclearequationofstate ...................... 36

3.3 Elementary nucleon-nucleon cross sections ............... 39

3.3.1 Elastic scattering channels .................... 39

3.3.2 Inelastic scattering channels ................... 40

3.3.3 Direct pion production channels ................. 47

viii

3.3.4 The pion-nucleon resonance and decay .............

3.3.5 Pauli blocking for fermions and enhancement factors for bosons

4 Pion production dynamics

5 The concave shape of pion spectra

5.1 Mechanisms for the concave shape of pion spectra .......... , .

5.1.1 Model calculation for the pion spectrum ............

5.1.2 Concave pion spectra .......................

5.1.3 Comparison with experimental data ...............

5.2 Systematics of pion spectra

5.2.1 Energy dependence

5.2.2 Mass dependence .........................

5.2.3 Impact parameter dependence ..................

5.3 Approximate analytic scaling function

6 Preferential emission of pions

6.1 The mechanism for the preferential emission of pions .........

6.2 Comparison to the experimental data ..................

7 Summary and Outlook
A Derivation of If, and 1;;

LIST OF REFERENCES

ix

48
52

54

66
68
69
72
80
85
86
88
88
91

96
99
105

108

111

120

List of Tables

3.1 Isospin cross section parameters ..................... 42

List of Figures

2.1

2.2

3.1

3.2
3.3
3.4

3.5

3.6

4.1

4.2

4.3

4.4

Diagrammatic representation of the gain and loss terms responsible
for changing the baryon phase-space distribution fb(x, p) as a result of
baryon-pion collisions. The terms on the left pertain to A resonances,
b = A, while those on the right are for nucleons, b = N .........

Diagrammatic representation of the gain and loss terms responsible
for changing the pion phase-space distribution f,,(a:,k) as a result of
baryon-pion collisions. ..........................

The nuclear equation of state, the solid line is for the stiff equation of
state and the dotted line is for the soft one. ..............

Energy dependence of the isospin cross sections ............

27

29

44

Energy dependence of the A production cross section in p + p collisions 45

Energy dependence of the A and N " production cross section in n +
p collisions .................................

Pion kinetic energy distribution in the direct process. The solid line
is from Fermi statistical model and the histogram is from the Monte
Carlo simulation .............................

Pion spectrum in nucleon-nucleon collisions. The solid histogram is
the spectrum from the direct process. The dashed histogram is the
spectrum from the N" decay and the dotted histogram is that from
the A decay .................................

Upper figure: Accumulation of the total number of baryon-baryon col-
lisions in the reaction of Ca+Ca at E/ A = 1.8 GeV and impact pa-
rameter b = 0 fm. Lower figure: Time evolution of the reaction rate
for the specified processes in the same reaction. ............

Time evolution of the population of free pions, A’s and N"s in central

collisions of La + La at E/ A = 1350 MeV ................

Beam energy dependence of the time evolution of the pion multiplicity
in central collisions of La + La ......................

Excitation function of the pion multiplicity in central collisions of La
+ La. The squares are the experimental data of Ref. [Harr87], and the
round plot symbols on the solid line are the model calculations.

xi

46

49

51

55

57

59

60

4.5

4.6
4.7

4.8

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

Mass dependence of the time evolution of the pion multiplicity in cen-
tral collisions ...............................

Mass dependence of the pion multiplicity ................

The impact parameter dependence of the pion production in Ar + Kcl
reaction at the beam energy of 1800 MeV per nucleon .........

The impact parameter dependence of the pion multiplicity in Ar + Kcl
reaction at the beam energy of 1800 MeV per nucleon .........

The concave shape of the pion spectrum in Ar + Kcl reaction at a beam
energy of 1.8 GeV / nucleon. The experimental data shown by the plot
symbols are from ref. [Broc84], the solid curve is the one-temperature
fit to the experimental data with a temperature of T=63 MeV. . . . .

Calculated contribution to the pion spectrum from pion already free
(solid histogram) and still bound in baryonic resonances (dashed his-
togram) as well as their sum (circles) at t = 20 fm/c. The straight line
is a thermal distribution with a temperature of 78 MeV. .......

Calculated contribution to the pion spectrum from pion already free
(solid histogram) and still bound in baryonic resonances (dashed his-
togram) as well as their sum (circles) at t = 40 fm/c. .........

Upper part: Rate of A production during the La + La reaction. Lower
part: Probability distribution for the C .M.S. energy of nucleon-nucleon
collisions. The solid histogram is for t2 l2 fm/c and the dashed his-
togram is for t3 6 fm/c. ........ ‘ .................

The time dependence of the average center of mass energy of nucleon-
nucleon collisions in which baryon resonance can be produced during
the reaction of La + La at a beam energy of 1.35 GeV/ nucleon and an
impact parameter of 1.0 fm. ............... ‘ ........

Schematic illustration of the rapidity distribution of nucleons. The
upper window shows the rapidity distribution of nucleons in the ini-
tial state. The lower window shows the rapidity distribution during
the reaction process. The dotted curve shows nucleons having been
scattered into mid~rapidity range. ....................

Comparison between the spectra of primordial pions (solid histogram)
and final pions(dotted histogram) .....................

Comparison between calculations (histogram) and the experimental
data of ref. [Odyn88] (plot symbols). The dashed line is the one-
temperature fit to the experimental data with temperature T = 49

MeV .....................................

Local slope T, as a function of the pion kinetic energy for the central
collision of La + La at E/ A = 1350 MeV. Circles are the experimental
data and the histogram is the model calculation. The solid line repre-

sents T; as extracted from the two-temperature fit to the experimental
data .....................................

xii

61
62

63

64

67

70

71

73

74

76

77

81

83

5.10

5.11

5.12

5.13

5.14

6.1

6.2

6.3

6.4

Comparison between calculations (histogram) and the experimental

data of ref. [Broc84] (plot symbols). the dashed line is the one-temperature

fit to the experimental data with temperature T=63 MeV. ......

Upper figure: Calculated energy dependence of the pion spectra at
90 :1: 30 degrees in the center of mass frame for La+La reactions at
impact parameter b = 1 fm. Lower figure: Energy dependence of the
shape parameter R of the pion spectra shown in the upper figure. The
solid line represents the analytic scaling function of equation 5.6. . . .

Upper figure: Calculated mass dependence of the pion spectra at
90 :l: 30 degrees in the center of mass frame for a beam energy of
1.5 GeV/ nucleon and impact parameter b = 1 fm. Lower figure: Mass
dependence of the shape parameter R of the pion spectra shown in the
upper figure. The solid line represents the analytic scaling function of
equation 5.6. ...............................

Upper figure: Calculated impact parameter dependence of the pion
spectra at 90 :l: 30 degrees in the center of mass frame for Ar+KCl
reactions at beam energy of 1.8 GeV/nucleon. Lower figure: Impact
parameter dependence of the shape parameter R of the pion spectra
shown in the upper figure. The solid line represents the analytic scaling
function of equation 5.6. .........................

The systematics of the average number of nucleon-nucleon collisions
suffered by each participant nucleon ...................

Transverse momentum analysis for La + La collisions at a beam energy

of 800 MeV per nucleon. The data are from ref.[Dani88] .......

Upper figure: 1r+ rapidity distribution calculated with ( solid histogram
) and without ( dashed histogram ) the pion reabsorption channels for
the reaction of Ne + Pb. Lower figure: Calculated 1r+ transverse mo-
mentum distributions in the true reaction plane with ( solid histogram
) and without ( dashed histogram ) the pion reabsorption channels.

Rapidity distribution and transverse momentum distribution calcu-
lated for La + La reaction at E/ A = 800 MeV and the impact pa-
rameter of 1 fm ...............................

Upper figure: calculated 1r+ rapidity distribution after using the de-
tector filter cut for the Ne + Pb reaction at E/ A = 800 MeV. Lower
figure: Comparison between the experimental pion transverse momen-
tum distribution ( round plot symbols )and the model calculation (
histogram ) for the same reaction .....................

xiii

84

87

89

90

93

98

102

104

106

Chapter 1

Introduction

In relativistic heavy ion collisions at beam energies from a few hundred MeV up to
about 2 GeV per nucleon, nuclear matter at high density and high temperature can
be formed transiently. From a theoretical standpoint, this energy region provides
interesting challenges. Since the beam energy is comparable to the mass of the nu-
cleon, nonrelativistic approximations to the nuclear dynamics problem are no longer
suitable. In addition, the energies are high enough to create baryonic excitations, and
mesonic degrees of freedom become important as well. However, the achieved energy
density is well below what is required to dissolve the hadrons into deconfined quarks

and gluons.

Relativistic heavy ion collisions offer us a unique opportunity to study the be-
haviour of nuclear matter at extreme conditions as well as the reaction dynamics of
finite-size hadronic systems. Aspects of particular interest are both of microscopic
nature, such as the in-medium hadron-hadron interactions and the dispersion relation
of mesons in hot and dense matter, and macroscopic, such as the nuclear equation of
state, the transport properties (6. g. viscosity and heat conductivity), and the collec-

tive motion during the compression and decompression phase of the reaction.

A thorough understanding of these aspects has consequences which reach far be-

yond the scope of nuclear physics. The explosion mechanism of supernovae, the inte-
rior structure of neutron stars and the formation of matter during the early universe
strongly depends on the properties of hadronic matter over a wide range of densi-
ties and temperatures. For example, in the gravitational collapse of massive stars
the matter density can reach 2-4 times the normal nuclear matter density[Brow89a],
and the stiffness of the nuclear equation of state then determines whether a prompt

explosion of the supernova can take place.

The beam energy range considered here is below what is required to produce a
quark-gluon plasma[Qmat87, Qmat90], a state of matter in which the building blocks
of nucleons and mesons become deconfined and move in an extended volume. How-
ever, an understanding of the physics mentioned above is mandatory for any reliable
analysis of the ongoing experimental search for this new state of matter. This is be-
cause the created quark-gluon plasma would be surrounded by and finally hadronize
to the “colder” highly compressed hadronic matter. The properties of the hot and
dense hadronic matter must therefore be well understood so that particles emitted
from it will not be confused with those from the quark-gluon plasma[Kapu91] and

therefore clear signals for the formation of the quark-gluon plasma can be detected.

As for the reaction dynamics, relativistic heavy ion collisions provide us the pos-
sibility to study the evolution of hadronic matter towards the equilibrium, the details
of the reaction mechanisms, and how the nuclear force acts in the hot and dense
hadronic environment. In particular, particles produced in relativistic heavy ion colli-
sions are good probes of the reaction dynamics and the elementary particle production

mechanism[Ca5390a, Cass90b, Mose91].

However, the extraction of the properties of hot and dense nuclear matter from
experimental data is complicated by the reaction dynamics. When we use the term

hot and dense nuclear matter, about which we would like to gain information, we

refer to nuclear matter in equilibrium. However, the initial state of the reaction is far
from equilibrium. In momentum space two Fermi spheres are separated by the beam
momentum. The hot and dense nuclear matter is formed transiently, its properties
can only be inferred from observables in the final state. Equilibrium situations can
only be formed in very low energy and possibly very high energy nuclear reactions.
At lower energies we find compound nuclear reactions where nucleon-nucleon colli-
sions are severly suppressed due to the Pauli blocking, and the mean field keeps the
nucleons together long enough to equilibrate. At high beam energies, on the con-
trary, frequent nucleon-nucleon collisions cause the thermalization whereas effects of
the mean field are small. However, in the energy range considered here, the Pauli
blocking of the collisions is neither strong enough to avoid particle emission during
the mean field equilibration time, nor is it weak enough to allow a sufficient number
of nucleon-nucleon collisions to happen before the system desintegrates. Therefore,
the extraction of the properties of nuclear matter at high density and temperature
strongly depends on our understanding of the reaction dynamics. It is the complexity
involved in the reaction dynamics which makes the properties of nuclear matter under

extreme conditions extracted up to now far from being accurate.

Considerable amounts of experimental data have been accumulated during the
past decade by observing the products resulting from relativistic heavy-ion collisions,
such as nucleons, light and heavy nuclear fragments, pions, dileptons, photons and
kaons (see, for example, refs. [Harw87, Rand90]). Yet, many of the quantitative
interpretations of these data remain rather uncertain, as the properties of the hot and
dense matter extracted by comparing the experimental observations with theoretical
calculations vary considerably with the specific model employed. Some properties
extracted with different methods are even mutually conflicting. One example for this

is the numerical value of the nuclear compressibility which varies greatly depending

on the model assumptions employed [Glen88].

The most successful models, in terms of reproducing a variety of the experimen-
tal observables in intermediate-energy nuclear collisions, are the Boltzmann-Uhling-
Uehlingbeck model (BUU) [Bert88a, St6c86] and its relativistic extensions[Koli87,
Blat88], quantum molecular dynamics (QMD) [Aich91], and quantum-correlation dy-
namics [Ca5590a, Cass90b]. These models are developed by numerically solving the
BUU transport equation either in its quantum version or in its relativistic exten-
sions. However, the BUU transport equation describes the time evolution of the
phase space distribution function for only one kind of fermions. The collision integral
in this equation was derived by assuming particles can make elastic collisions only.
The dynamics in these models developed for heavy ion collisions at intermediate ener-
gies is therefore restricted to the baryonic level. Mesonic degrees of freedom enter via
nuclear potentials only. In relativistic heavy ion collisions at beam energies around
1 GeV/ nucleon, however, about one half of the nucleon-nucleon cross section is in-
elastic, mainly through pion production. Therefore, the energy range to which these
models can be successfully applied is limited to beam energies below a few hundred
MeV / nucleon, although meson production has been treated in a perturbative manner
in the subthreshold regions [Ca3590a, Mose91, Baue89]. Also within these models, one
has studied pion production by assuming that Delta resonances have lifetimes longer
than the nuclear reaction time; this is the so-called Frozen Delta Approximation.
The number of Deltas at the end of the calculation then is equated to the total pion

multiplicity[Bert84].

Another approach which has been successfully applied to high-energy nucleus-
nucleus collisions is to treat all nucleons as essentially free particles interacting with
each other with their free nucleon-nucleon cross sections. Intranuclear cascade models

[Cugn81, Cug1182, Rand79, Cugn88] are based on this approach. Here, pion produc-

U‘

tion and reabsorption is included into the dynamical process through the formation
and decay of A resonances. These models were able to calculate properly the overall
features of nuclear equilibration [Rand79] and pion production [Cugn82, Kita86] in
heavy-ion collisions with beam energies around 1 GeV per nucleon. Although they
have some shortcomings when quantitatively compared to experimental data, the

intranuclear cascade models have been remarkably successful.

However, in this energy range the long range nucleon-nucleon interactions are still
sufficiently significant that the particles are not free but are moving in a varying
mean field. Recent computer simulations of relativistic heavy-ion collisions[Wolf90,
Xion90a, Liba91a] have extended the original BU U model to contain pion production
and reabsorption in the dynamical process. They indicate that it is important to
include the mesonic degrees of freedom explicitly, while keeping the mean field, in
order to explain the dilepton production data [Roch88] and quantitative aspects of

pion spectra such as the two-temperature shape observed at the BEVALAC [Odyn88,
Cha390].

Nevertheless, a complete set of transport equations which govern the dynamical
process in the hadronic matter was not available until recently. With this situation in
mind, several groups have set out to provide a derivation of such transport equations
[Siem89, Sc11689, Davi91, Bote90]. These attempts, however, are still at an early

stage, and a complete numerical realization is not available as of yet.

It is the purpose of this thesis to present a complete hadronic transport model
for relativistic heavy ion collisions[Libana, Baue91a, Liba91b, Wlbr91a, Liba91c,
Baue91b]. The framework for describing nuclear reactions is extended from the baryon
dynamics level to the hadron dynamics level. The model provides a framework for
the theoretical understanding of the nuclear physics phenomena observed, expected

and unexpected, in the energy range from a few hundred MeV/nucleon to a few

GeV / nucleon.
The thesis is organized as follows.

By starting from a hadronic Lagrangian density we derive in Chapter 2 a cou-
pled set of transport equations for the phase space distribution functions of nucleons,
deltas, and pions, which are the main constituents of hadronic matter formed in rel-
ativistic nuclear collisions. These equations reflect the physics of relativistic nuclear
collisions in an instructive manner. Moreover, an approximate solution of the equa-
tions is possible with present computers (Chapter 3). Our derivation is rather similar
to the approach taken in ref. [Wang89, CassQOc], but we go beyond that work by in-
cluding both A resonances and dynamical pions, which are expected to be significant

at relativistic energies.

Chapter 3 is devoted to the numerical realization of the hadronic transport model.
We discuss in detail how the transport equations are solved numerically and present

inputs of the model.
In Chapter 4 we apply our model to study the dynamics of pion production.

In Chapter 5 we study the concave shape of the pion spectra in relativistic heavy
ion collisions. The mechanism that causes the concave shape of the pion spectra is
found to be the different contributions of the delta resonance produced during the
early and the late stage of the heavy ion collisions and the energy dependence of the

pion and delta reabsorption cross sections.

Chapter 6 is devoted to the study of the preferential emission of pions in asym-
metric nucleus-nucleus collisions. The experimentally observed preferential emission
of pions away from the interaction zone towards the projectile side in the transverse
direction is found to be due to the stronger pion reabsorption by the heavier target

spectator.

We will summarize in Chapter 7.

Chapter 2

Relativistic Transport Equations
for Hadronic Matter

In this chapter, we derive a coupled set of transport equations for the phase space
distribution functions of nucleons, baryon resonances and pions, and establish no—
tations for succeeding chapters. The transport equations furnish a computationally
manageable scheme for treating the dynamics of the interacting baryons and pions,

reducing the many-body problem to a set of coupled one-body problems.

First, in section 1, we construct the Hamiltonian for the hadronic matter, starting
from the effective Lagrangian density containing free fields of nucleons, deltas,
0-, w- and w-mesons, as well as the minimum coupling between them. In section
2 we derive equations of motion for the one-body density matrix of nucleons, delta
resonances and pions. Subsequently, in section 3, we make Wigner transformations
of these equations, in order to obtain a set of transport equations for the phase-space
distribution functions of baryons and pions. These equations contain a Vlasov term of
the usual form and several collision terms, in analogy with the standard Boltzmann-

Uehling-Uhlenbeck equation[Bert88a].

2.1 Model for hadronic matter

This section introduces the model description of hadronic matter in terms of inter-
acting baryonic and mesonic fields. Throughout the developments, we employ units

in which 71 and c are unity.
2.1.1 Model Lagrangian

For nuclear collisions at beam energies of up to around one GeV per nucleon, the
main baryonic excitation is the A(1236) resonance. Higher resonances have negli-
gible excitation functions. Therefore, a first step towards a complete description of
hadronic matter should include nucleons and A resonances, in addition to 1r, 0 and
w mesons. Accordingly, we adopt a model hadronic Lagrangian density involving the
baryon fields N(.r) and A”(:r), the meson fields 1r(:1:), 0(3), and w“(:r), and their
interactions in the minimal coupling scheme commonly used for relativistic hadronic

systems [Wang89, CassQOc, Wlbr9la],
£(zr) = £0(.1:)+ Lima) , ’ (2.1)

where £0 and Elm are the free—field and interaction Lagrangian densities, respec-
tively. We have used :c to denote the Minkowski four-vector (t, r). Moreover, the free
Lagrangian density is
50(3) = N(I)(i7“3u - mAIM/(10+ Ku($)(i7”3u - MA)A"(2:)
+ glam) . we) - mire) - «(sun (22)
+ gamma) - mzo’en
imam» + gmiwxwm ,

and the interaction Lagrangian density is

We) = —z‘g.~~W(x)75rN(x) - «(1) + ga~~N(x>N(x)o(x)

10

- ngNN($)7"N(I)wu(I)
+ gfi~A[Z#(:c)TN(;r) - 6“1r(.r) + i—V-(I)TTA“($) - 8u1r(;r)]
- ignAAKu(Il7sTA"(1l ' “(53) + gaAAEu($lA“($)0($l

— gwAAEu(I)7VAqu(I) v , (23)

with F“, = ',,w,, —- 8...)“. The nucleon field N(;z:) is an isospinor, the A field is
described by the Rarita-Schwinger formalism[Rari41] as a four-vector with each com-
ponent as an isospinor. The pion field «(1) is an isovector and a Minkowski pseudo-
scalar. Furthermore, the sigma field 0(3) is a scalar in both Minkowski and isospin

space, whereas the omega field 7.5,,(1) is a Minkowski vector and an isoscalar.

It is convenient to employ the isospin generators r, T and t which act on the

isospinor N(.r), the isospinor A(.r), and the isovector 1r(a:), respectively. They satisfy

1

T = t 69 31" (2.4)
and
_ 1 for p 9
T3 — { —1 for n ’ (2.5)
1 for 1r+
t3 = 0 for 1r° , (2.6)
—l for 1r'

It is also convenient to employ the isospin transition operator T = (73., ’13, 7.), as in
refs. [Brow75, Oset82]. The matrix representation of 7;, p=1, 0, -1, can be obtained

from the following equation,

(mm. = Z (ngllkémWr. (2.7)

kz—l

11

where t*1 = (1, :l:i,0)/\/2 and t0 = (0,0,1), and only k = MT — m, contributes. The

above two expressions imply

7‘; 0 0 0 0 0
0 1 O 0 3 0
7+ _ 75 , 7. = 1 , To = 3 . (2.8)
0 0 0 2 '
0 0 If ‘ 0 3
7; 0 0

2.1.2 Equations of motion for the hadron fields

The equations of motion for the hadron fields can be obtained from the Lagrangian

density given in equations (2.1-2.3) by means of the Euler-Lagrange equations. The

result is
(W791. - mN)N(I) = ingN77($) ' 75TN($) - gaNN0(I)-N(I) (2-9)
+ gwwwwt(I)7“x’V($) - gnNATlA“($) ° 3M”) ,
(2'7“8,‘ - MA)A“(.r) = igMA‘flr) . 75TA“(.1:) - gaAAa(;r)A“(x) (2.10)
+ gwaawu(x)7"A“(I) - gnma‘WU) - TN(~'€) ,
(3“3“ + mild—T) = gaNN-N-($)N($) ‘l' gaAAZufxlAu($) a (2'11)
(5715“ + milflxl = -i9«NNN($)‘YsTN(~Tl - i9wAAZu(I)7sTA“($)

- gn~A[5“(3u($)TN(x)) + 3u('1—V-($)TIA“($))] , (2-12)

auFffl’ + miwfix) = ngNW(x)7:/N($) "l" gwAAXu(x)7uAu(x) ° (2°13)

12

The above set of equations form a complete closed set of equations for the evo-
lution of the various hadronic fields considered. In principle, the solution of these
equations can be used to describe nuclear reactions. However, due to computer lim-
itations, their solution is presently not within reach. An alternative way is to solve
transport equations for the hadron fields. For the purpose of deriving the transport
equations within the framework of the density matrix formalism we construct the

model Hamiltonian for hadronic matter in the next subsection.

2.1.3 Hamiltonian for hadronic matter

The main role of the meson fields in the Lagrangian (2.1) is to mediate the strong
interaction between the baryons. This is strictly true for the fields 0(x) and (12,,(1)
which have no manifestations in terms of real physical particles. Therefore, these
fields can be regarded as representing virtual mesons, and they can be eliminated
in exchange for effective potentials acting among the baryons. However, the pion
field plays a dual role; not only can the pion mediate interactions (and in this role
it acts as a virtual meson, similar to a and w) but it can alSo be manifested as
a real physical particle that can be observed. Thus, the pion field is somewhat
akin to the electromagnetic field; the transverse component of the electromagnetic
field represents real photons while the longitudinal component mediates the Coulomb

interaction between charged particles.

In order to construct an effective Hamiltonian for hadronic matter, we eliminate all
virtual meson fields, leaving only the real pion, in addition to the baryons, since real
particles are amenable to numerical simulation. Of course, such a separation can not
be made in an exact manner, because the pion has a finite mass, in contradistinction
to the photon in QED. Nevertheless, a useful approximate treatment can be made,

as we shall now describe.

13

The elimination of the virtual meson fields can be accomplished by means of the
Green’s function technique. However, the equation of motion for the baryon fields
and pion field obtained in this way will be non-local in space and time. This is the
price paid for the elimination of virtual meson fields. Since we wish to formulate a
transport theory within the density-matrix framework”, non-locality in time is incon-
venient. Fortunately, meson retardation effects are not significant in the energy range
considered here. Consequently, the instantaneous meson exchange approximation can
be made to establish time locality and thus make the equations of motion amenable
to the density-matrix treatment [Wang85, CassQOb, Wang89, CassQOc, Webe90]. For
the a' and w mesons this approximation seems to be well justified. The pion, however,
has a smaller mass, our approximation has to be used with caution. In terms of the
meson propogator the instantaneous meson exchange approximation can be expressed

as
C(x — :r’) z C(r — r', t)6(t — t'). (2.14)

We have to keep in mind, however, that by using the instantaneous meson exchange
approximation, we give up relativistic covariance. With this approximation, an effec-

tive Hamiltonian for hadronic matter can be constructed as[Wlbr91a]

H”

e

a = H 3 + H”+ Vb" . (2.15)

The effective baryon Hamiltonian He“, the Hamiltonian for the real pion H ”, and the

baryon-pion interaction Vb" are

[1121(I)E(;r)r/) (:13)dr (2.16)
‘lré/W(()$1)(¢l(12lv($1 — $2)¢(x2)¢(x1)dr1dr2 ,

--%/[1'r()o :I:);2:+V«( ) V«(;r )+m,2r «() «() )]dr, (2.17)

14

where
__ 111N075) _. “(33)
W) - ( wits) ) " i A”) l
and
N($2)N($1)
¢($2)¢($1)= 2331283
A(2:2)A($1)

EN(:I:) = Cid—mi) + 7omN ,

1326:) = (14—16") + 70MA ,

(2.18)

(2.19)

(2.20)

(2.21)

(2.22)

(2.23)

and a; = 70'), and do = 7070 = 1. Furthermore, U" and V(xl - 2;) are defined as

follows,
171(2) = U. . «(2),
with

U =(QNN a...)
R UAN UAA ’

(2.24)

(2.25)

15

UNN = ignNN‘rWsT 9 UNA = "grrNA70Tlau , (2.26)

UAN = —g«NA7OTaV , UAA = igma‘Vo‘YsTaw . (2.27)
and

17(21 — :2) = a,“ , 1,331,: = N, A. (2223)

Keeping in mind that U, is a matrix of isovectors, we can omit the boldface in the

following derivations without causing any confusion.

The equations of motion for the baryon fields and pion fields can be derived
from the effective hadronic Hamiltonian under the instantaneous meson exchange

approximation by virtue of the following Heisenberg equations,

 

.BN .

1‘15,— = [Hfm Nl , (229)
,BA“ ~

. a, = [Haw] . (2.30)
.8« .

238- = [11:3, «] . . (2.31)

The salient features of the hadronic Hamiltanion (eq. (2.15)) and the above Heisenberg
equations are 1) they have structure similar to those of non-relativistic quantum
many-body theory, and, as we shall see below, 2) they can be cast into density-
matrix form, which, when augmented with correlation dynamics, is suitable for a

non-perturbative treatment of quantum many-body problems.

2.2 Density matrix treatment

In this section we shall reformulate the hadron dynamics in terms of the density
matrix. Within the two-body correlation approximation, we obtain a set of coupled

equations of motion for the one-body density matrix of baryons and pions.

16

2.2.1 Density operators

The n-body baryon density Operator 6,, is defined as

fin(1,2,---n;1',2',---n') = 111*(1’)72*(2’) . --1l)l(n')1/J(n) . . . 211(1), ' (2.32)
with
n = mu 2 (t,rn,mn) , n’ = 1;: (t,r;,m:,) , (2.33)

where mn is the spin-isospin quantum number and we have used that t' = t due to
the instantaneous approximation (2.14). The n-body density matrix is defined as the

expectation value of fin, namely

p.<1.2,---n;1',2'.---n') = <2.(1,2,---n;1’,2’---n’)> . (2.34)
The baryon number operator is defined as

N = Trlrbl(1)1/)(1) = TrlNl(1)N(1) + TrlAL(1)A“(1) . (2.35)

It is straightforward to show that[Wlbr91a]

A

[N.¢(1)]=—w(1). [N,¢*(1)1=u>*(1), N*=N. (2.36)

From eqs. (232,235,236) we obtain the following reduction relations

1 1
in = . Tr n in = Tr n in . . 2.37
P N—n (+1)P +1 (+1)P +1 N—n ( )

 

 

To obtain the correlation dynamics the key step is to separate out many-body

correlations from the reduced density matrices. This can be realized by a non-linear

17

transformation [Wang85],

pn(1,2,~-,n;1',2',-~,n') (2.38)

= ASnEpmpUU-wan - P;1’,2’.~-.(n - Pl')
:1

pr(:z -p+1,---,n;(n — p+1)',---,n')+ Cn(1,2,---,n;1',2',-~,n') ,
where the operation ASn should be understood as follows. The operator A denotes
the antisymmetrization operation among those of the variables (1,2, - - - , n) that refer
to identical particles. (Thus, labels referring to nucleons are antisymmetrized sepa-
rately from those referring to deltas.) Furthermore, the subsequent operation by S
symmetrizes among variable pairs (1, 1’), - - - , (n,n’) of identical particles. The com-
bined operation AS” then acts among 71 particles, and the repeated terms should

be omitted. Thus, for the oneparticle density we have p 5 p1 = C1, while the

two-particle density p2 is given by
p2(1.‘2;1’.‘2’) = A52p(1;1’)p(‘2;‘2’) + (720.213?) (239)
= p(1;1')p(2;2') - p(l; 2')p(2; 1') + C2(1,2;1',2').
The most common approximations in quantum many-body theory can be obtained by
making the lowest order truncation. The simplest truncation approximation assumes
that all the many-body correlations vanish and leads to the mean-field approximation.
The next order truncation, namely, 02 75 0 and C02 = 0 leads to the two-body

correlation dynamics. In the following we restrict ourself to the two-body correlation

dynamics.
2.2.2 Equation of motion for baryons

By using of the basic anticommutation relation among 1b and 112* we obtain the

equation of motion for the n-body density operator,

as,

. a, = [Wm/3.1+ Tf(n+1)lV(n + 1). a...) . (2.40)

18

Here H"(n) can be written as,
Hh(n) = $801+ 01(1)] + 213(1)) = H(n) + 2 01(1). (2.41)
1:] i<j 121
where H(n) is the single particle Hamiltonian
= $2120) +iz‘1(i,j), ’ (2.42)
i=1 1<J
and the potential U"(:r) is given by eqs. (2.24-2.27). The operator V(n+ 1) appearing

in eq. (2.40) is defined as
(n+1) =Zv(i,n+l). (2.43)

The equation of motion for pn can then be obtained by taking the expectation

value of the operator fin,

a n “ 7r(
. a", =(H(np..1+Tr1..11(Vn+11p..11+(1ZU(111.11 (2.441

 

Since (7"(x) contains the pion fields, the last term in the above equation depends on

the quantity

(fin($1,$2,-H$n;$’1,$’2,'--:E:,)7|’(y)) 1 (245)

which contain the irreducible vertices associated with the pion-baryon interactions.
At present, we are not able to treat these in general. Nevertheless, we do find a way

to include the lowest-order vertex, which suffices for our present purpose.

Truncating eq. (2.44) at the second order, Le. assuming Cn>2 = 0, we obtain

2.2—At) 2 [E110] + Tr2[t3(1,2),p2l + [07" P] ’ (246)
gait? = [E(I)+E(2l+5(112lipzl+Tr3[fi(113)+”(2’3)’p3l

+ (10"(11 +U"(2 111.11 (2.471

19

where the baryon-pion vertex function F (1:, y, r' ) is an isovector and it is defined as
Miami“) = (WWWWWUH - (2-48)

Noticing that in the two-body correlation approximation C3 :- 0 and therefore

P2 = A52(PP)+C'21 (249)

P3 = A53(PPP + P02) 1 (2-50)

the equations of motion, eqs. (2.46, 2.47), become a coupled set of equations of motion

for p and C2. Explicitly, C2 satisfies

.802

:3,— = (133(1) + 13(21 + (11,21,021 + (210.221.115.11111

+ Tr3l13(1, 3) + 13(213). A53(PPP + PC‘Z)lL . (2.51)

where []L means linked terms in which any multi-variable functions can not be fac-

torized according to particle variables[Wang85].

To further simplify the equation of motion for the one-body baryon density matrix
eq. (2.46) we need to solve eq. (2.51) for C2, this can be done by closely following
the time-dependent G-matrix theory of ref.[Cas588, Wang89]. As we have shown in
ref.[Wlbr91a] this leads to

P2 = l1 + @12(E)612@(EllA32(P(1)P(2))ll + Gl(E)912§i2(E)l- (252)
Where

5,2(E) = [E — 11(1) — 11(2) + 151-1 , (2.53)

11(1) = 12(1) + (“1(1) , (2.54)

(“1(1) = T13=3113(2'3)(1 — P.3)p(33') , (2.55)

012 = l - TI‘3_—_3I(P13 + p23)p(33’) . (2.56)

20

Here the operator 15,5 interchanges the coordinates of the identical particles i and j;

for non-identical particles it is zero. The C-matrix obeys the equation
C(E) = 5 + 0912(E)912@(E). (2.57)

From the above relations we can express the second term on the right hand side of

eq. (2.46) as
Tram/121 = 101.1(01,pl+ 1:. . (2.581

where the mean field UHF relates to the real part of the G-matrix and the baryon-

baryon collision term [:6 relates to the imaginary part of the G-matrix

UHF/1 = 112114611120), (2.59)
and
1:11 = —iTr2[iClélgéIm(§12)p20 - ionlmfgizléléizG
+ szoéléizfiiz — @12é12GP2oéil - (2-60)

The equation of motion for the one-body baryon density matrix then reduces to

.6 ~
26—: = (UHF(G1,pl + 1:.+ 1:. . (2.611

where the baryon-pion collision term If, is given by
1:, = [1]., r] . (2.62)
2.2.3 Equation of motion for pions

In order to close the equations of motion for baryons, it is necessary to determine the
equations of motion for the pion density matrix and for the pion—baryon interaction

vertex function F.

21

We start from the equation of motion for the pion field which can be written as

(8%,. + m§)1r(11) = —11,3(m) , (2.63)
where

a=(:i:::: 2214.063. ‘33:) i 1......
with

fiNN = ingN‘YO'YST 1 aNA = gnNa‘YoTTau 1 (2-65)

flAN = 911111117078” 1 i‘AA = i91aa‘7075T5m) - (255)

Similarly to U, 11 is a matrix of isovectors, and we omit the boldface in the following.
Eq. (2.63) is the Klein-Gordon equation for the pion field, which contains the second-
order time derivative. Therefore this equation is not a convenient starting point for
the derivation of the equation of motion for the pion density matrix. In order to
linearize the equation, we find the following identity useful,

82

6.6“ + m3, = fl — v2 + 1113, (2.61)

8 8
_ __ '_ 1/—V2 2 '__ ./_v2 2
— (2m + + 111,) (1 at + 111,) .

If this relation is combined with the free-particle approximation to the Klein—Gordon

operator, namely

1% 3 ,/_V2 + m3, = E. , (2.68)

the Klein-Gordon equation for the pion can be approximated by the following two

equations,

 

111(1) = E.«(1)+ . fifi(:c:r), (2.69)

ii" = —E,2,«(:1:)— u,6(;1::1:) . (2.70)

Ix?
Iv

The pion density operator is defined as
p,(:r; x') = 1r(2:') ~1r(:z:) . (2.71)

The Wigner function for the pions, f,(r, p, t), can be obtained from the density matrix

p,(2:; x’) by means of a Fourier transformation (see eq. (2.76) in sect. 2.3.1), "

8 8 _' . f«('»P,t)
_. _ _ 'P 3d = —, .
[Tr p,(r+2,r 2)e s E,(p) (2 72)

The associated equation of motion for )6, is readily obtained from eqs. (2.69270)

and so is the expectation value of p‘,,

 

.apfl' A W 6
1 at = [E,.,p,,] + lb,r (2.73)
where
1 1
I" = — —.-‘ F . .
bx 2[E,ru, l (2 74)

The equation of motion for the pion—baryon interaction vertex operator F can be
obtained by using of the Heisenberg equation. It is shown that an analytical solution
for the baryompion interaction vertex can be obtained in the two-body approximation
as[Wlbr91a]

1
Edy)

1

f‘(a:,y,x') = —i7r6(iz(x) + 13330!) " (1617,)” E (y)

 

 

+ gm.) - 0w»

1

Saw) — we»?

 

1 ) ];3(x; I')fi(y)fi(y; y) . (2.75)

E, y 3
2.3 Transport equations for hadronic matter

It should in principle be possible to solve equations (2.61 and 2.73) numerically by, for

example, expansion on a basis of TDHF wave functions. This was done by Tohyama

23

et al.[Gong90] for the nucleonic systems. This approach, however, is severely limited
by the available computing resources and has only lent itself to very few exploratory

studies and comparisons with experimental observables.

A more tractable procedure is to introduce the Wigner transformation of the
density matrix. Within a semiclassical approximation, numerical solutions of the
equations of motion for the Wigner transform can be obtained by utilizing the test-
particle method. For a purely baryonic system this approach was introduced on
a mean field level by Wong[Wong82] and later utilized to study nuclear transport

phenomena by Bertsch ct al.[Bert84, Bert88a].

We follow this latter approach and perform the Wigner transformation of our

equations of motion in this section.

2.3.1 The Wigner transformations

The Wigner transformations for the baryon and pion density matrices are

fb(r,p,t) = /p(r + %,nz;r — 3,77%) e"p’sds , ’ (2.76)

G

f.(r,k,t) = /p,.(r + %,m; r — 1;,m') e-‘P'sds , (2.77)

where the caret is used to remind of the fact that these quantities are still matrices

with respect to the spin-isospin labels m and m’. They can be expanded as

 

 

. . 1/2

f1(r,p,t) = Zfaalxp) ( 53.12:???» u;.(n)ua(n) , (2.78)

Hr k t) -Ef (xk)( 1 )mvT (kw (k) (279)
1r 3 a —3gt 33 Eg(k)Ept(k) fi’ )0 a .

where the summation is taken over all possible single particle states. Here v3 is the

isospinor of the pion, ua(l'l) is the spin-isospinor of the baryon which has the effective

24

mass M; and the momentum II in accordance with the fact that the baryons are
moving in scalar and vector fields. Furthermore, a = (b, m,,m¢) is used to specify
quantum numbers of the baryon, where b = N or A, and m,/m¢ is the spin/isospin

of the baryon. These spinors satisfy the following orthonormality conditions

<u.(n)lu..,(r1)> = 566,5;(Hl/M; » (2-80)

and
(”8(kllvfi'(kll = 653’ - (2.81)

In the above equations we have used p = (E, P) for the four-momentum of baryons
and k = (E,, k) for that of pions. The effective momentum and mass of the baryon

are related to the vector and scalar fields through

r1), = pp - U» , (2.82)

M; = M0, + U, , _ (2.83)
and the energy of the baryon in the nuclear medium is given by

E; = (IL-II" + Mfr/2 . (2.84)
2.3.2 The Vlasov terms

In order to bring out the physics of the kinetic equations for the hadron density
matrices more clearly, it is instructive to compare their form with the standard BU U

equation. For this purpose, it is useful to recast our equations of motion on the form

i— — [E + 02m] = 1:. + 1:. . (2.85)

— [12”, p.) = 1;; . (2.86)

 

25

The left-hand sides of these two equations are the Vlasov terms, corresponding to the
collisionless one-body propagation. These terms will be rewritten in the usual form
below. The mean field UHF in these equations has been assumed to only contain

scalar and vector components. It can therefore be decomposed as
UHF(G) = —a,.U“(x) + 70Us($) , (2.87)

with a), = (707,-,7070) = (0,31). Considering only the diagonal elements of p and p,r
in isospin space, as is usually done, we may now perform a Wigner transformation
of eqs. (2.85, 2.86) and subsequently take the trace in spin space. Employing the
semi-classical limit for the mean-field terms, the following equations of motion for
the baryon and pion phase space distributions are then obtained[Wlbr91a], with the

Vlasov terms in an explicit form

 

 

 

afb($p) Hi V; :1: — H“ V-x V" a: M; V‘.‘ i
at + 137(1)) :fb( 20) E509) .U..(a:) pfb( p)+ 55(1)) ,U,V,f.,(xp)
= 156(xp)+l£.(xp) (2.88)

for the particular state b of the baryon. For any charge state of the pion we have

 

af.(xk) + k . v3

(91! EWUC) f,(xk) = 1:"(13’6) , (2.89)

where the collision terms are given by

1:5(zp) = —i/Tr[fb(:c, 1;:r’, 1') e"p‘rdr , (2.90)
1:,(xp) = —i/TrIg,(x,1;x',1')e‘ip'rdr , (2.91)
and

1:,(xk) = —i/Tr1;',,(:r, 1;:13',1') e-‘k’dr . (2.92)

26

2.3.3 The baryon-baryon collision terms

The collision term 13,, represent the rate of change of the baryon phase-space dis-
tribution function as a result of baryon-baryon collisions. To calculate the collision
term, we use the spin-isospinor uo, to represent a baryon; they satisfy the orthonor-
mality relation eq. (2.80). In this notation, the different N and A charge states can
be considered as identical particles with different intrinsic quantum numbers a. The
two-body density matrix p20=ASp(1)p(2) can then be antisymmetrized even between
N and A. The interactions used here does not contain the exchange term between N
and A, and so the exchange term of the matrix elements of the interaction between N
and A automatically vanishes, due to the above orthonormality relations. Therefore,
the spurious terms in p20 due to the antisymmetrization between N and A have no
contribution to the collision terms [86. With this in mind, the calculation of If), is

straightforward, although lengthy. The final result is[Wlbr91a]

, = 7r / / MgMglMgLMga
[66(Ip) (2“,)9 Z . [dpldPdeCi 1353;117:2135,

axazasnm’,’

 

5(5509) + E;.(P1) - E34172) - E;3(P3))5(P + P1 - P2 - P3)
((Pabpiallélpzazpaaall (2-93)
[<(P‘102P3a3lélpabplalll - <(P202P303l6lplalpablll

[f02($m)faa($p3)7al (zp1)76(xp) - To... ($1097... (mafia. (mp1 )fb(xp)l ,

where

((PabPI 011 lélpzazpsasll = (2.94)

/<ua<p)u.. (P1)léluaz(P2)um(pa))e““p‘pl"‘1’2‘P3)]"/2dr.

The collision term I 3,, respects the Pauli exclusion principle as shown in the appearance

of 70(3):) = 1 - fa($p) and 76(xp) == 1 - fb(2:p). The effective interaction 6, and

27

hence 6', contains the p H n and N H A transition operators, and therefore all the
possible collision processes are included in this collision integral. The above baryon
collision term is of the same form as the N N collision term appearing in the standard

BUU equations, but generalized to accommodate the four A states of the baryon.

2.3.4 The baryon-pion collision terms

In this section we calculate the rate of change of the hadron phase-space distribution
function due to baryon-pion interactions. Let us first consider If”. It is given by

131(1, 1’) = (f£,(11’)), where the collision operator is
i:,(11') = U,(;z:)F(2:, 2:, x') — I‘(x,:r',x')U,r(:r'). (2.95)

Taking the average of the collision operator is tedious and the details of this process
can be found in Appendix A. As in the BU U equation, the collision terms can be
separated into gain terms and loss terms. (The same is true for 13,.) The phys-
ical processes represented by the gain and loss terms of If, are shown in Fig.2.l.

Explicitly,

Ib1r(r pvt)- - Igain($ P) — Ilboss(xp) ' (2'96)
The gain term is given by

1" - :(zp) = (2.97)

ZN/j Mb )(Ua'p'lfi(P’)fl(k)a(P)luap) ' ("aplfi(k)lua'p’)
E;(p )E‘a(p’

8(2—1r-_1r)6 E710“)

comb

[7,(1k)fa'($P’)71($P)5(33(1)) + E..(k) - E;I(P')l5(P' - k - P)

+f«(xk)fal($P')76(-’€P)5(E500) - Ex(k) - ESv(P’))5(P’ + k - 10)] dp'dk ,

while the loss term is

150.3(8).) = (2.98)

The collision term If,

fb($P) Mir?) f«($kl
/
/
gain: +
\
\
fa’(xpl) flak) fa’(xp’)
fa'(xp') fume) fa (1319')
/
/
loss: +
\
\
NIP) MM?) fwd“)

Figure 2.1: Diagrammatic representation of the gain and loss terms responsible for
changing the baryon phase-space distribution fb(:c, p) as a result of baryon-pion colli-
sions. The terms on the left pertain to A resonances, b = A, while those on the right
are for nucleons, b = N.

Z(// 0130,01'p'lu()E(klufplluap)'(uaplffi(k)lua’p’)
E;(p E:(k)

Gra'm 2

[f«( 351: )72(1P ’<5)fb(IP) (55(1)) + 132(k) - E;'(P'))5(P' - k - Pl

+72(Ik)7.22(rp')fb(rp)5(55(1)) - 522(k) - E;'(P’))5(P' + k - Pl] dp'dk -

Here 7,.(xlc) = 1 + f,r(.z:k), and the subscript 77' has been used to specify the isospin
of the pion. These are the general expressions, and the matrix elements in these
equations assure that only physical processes can happen. For example, only when
6 specifies a nucleon do the first terms in these two equations contribute, while only
when I) specifies a A will the second terms contribute. Each of the matrix elements is
a vector in spin-isospin space, as dictated by the underlined quantities, and the dot

signifies a contraction with respect to these labels.

We now turn to the collision term 1);. It is given by 15', = (1.2;), where the collision

operator is

 

 

. 2 2 1
ilf,(;r;.r’) = %{E 1(x)11(.r)F(.r,;r',x)- F(:c',x,x')fi(.r')E ”0} . (2.99)

It can also be separated into gain terms and loss terms,
132(r2k2tl— " 1:22.2(xkl- 117,24“) - (‘2-100)

The physical processes represented by the gain and loss term have been depicted in

Fig.2.2. Explicitly,
M)‘ M‘(p’
212”“ = 22.2 -—2§//2

("0’7”lil—(kli‘luJ + Pllzluap) ’ (“aplLi-(kllua’p’l
523(k)

5( 32(17'1- 152(k) - Ea(P)) 5(P' - P - 1:)

72(Ik)f2'(rp')7..(rp) dpdp’ . (2-101)

 

30

The collision term 1;;

mp) Lek)

gain:

fa’(xp’)

loss:

\
\

f2(xp) Mask)

Figure 2.2: Diagrammatic representation of the gain and loss terms responsible for
changing the pion phase-space distribution f,(.r, k) as a result of baryon-pion collie
sions.

3 1
and

.11; M"

1122203“ = 16(27r)5 Gig/[5' 20’) ”73

(ua'p'lflkWP + Pllzluapl ’ ("aplfiiklluq'p’l

 

 

 

532(k)
5(E;2(p') - 5,.(12) — Ea(p)) 6(p’ - p - k)
f2(rk)f2(rp)7..r(rp’) dpdp'- (2.102)

It should be noted that the fermion suppression factors To and the boson enhancement
factors 7,, are included in these collision terms automatically and follow from our

derivation.

2.3.5 Transport equations

In the previous section, we have given the semiclassical equations of motion, eqs. (2.88,
2.89), for the phase-space distributions fb(zp) and f,(zp) for baryons (nucleons and
deltas) and 7r mesons, respectively. The collision terms appearing in these equations
of motion are presented in eq. (2.93) for baryon-baryon collisions, and in eqs. (2.97,
2.98) and (2.101, 2.102) for pion-baryon collisions. Together, these equations form
a complete set of coupled transport equations for nucleons, A resonances, and 7r
mesons, including all many-particle correlations up to and including the two-body
level. Three—body and higher correlations are neglected. This limits the applicability
of our theory to heavy-ion collisions up to only a few GeV per nucleon. For higher

energies, three- and more-particle effects are expected to play a more significant role
[Dani90].
ln the following chapters, we will not study the full equation with both scalar and

vector mean field potentials. Rather we wish to focus on the dynamics induced by

the coupled collision terms for pions, Deltas and nucleons. For this purpose, we limit

32

ourselves to the case of the presence of only a zeroth component of the vector mean

field potential to be able to compare with previous approaches.

With this assumption, eq. (2.88), the transport equation for the particular state

b of the baryon (nucleon or Delta) leads to

91:51” + g . v.7.(2p) — V2U(I) - vpmzp) = 15.6212) + 1:.(212). (2.103)

For the pion we still have

a 1? k k 1?
—f:—'d({_l + E: . Vrf”(1-k) : [b1r('rk) . (2.104)

The collision terms can be written in more compact forms by using of four-dimensional

delta functions and introducing the square of the transition matrix element

152(2)) = (2.105)

M MGIMO A40:
11’ 2 [f] ngOIEazzEaaaWbbb(plahpzaz,p303,pab).

 

b
010303.113,

[fa2(rp2)f22(xpa)72,(WHY-dz?) - 7.,(rp2)72,(xpa)fm(Ip1)fb(rp)l -
1
5(“(13 + P: - P2 - Palmdpidpzdpa-
Where Wfb(plal,pgag,p303,pab) is the square of the transition matrix element in

baryon-baryon collisions, which determines the transition rate. Explicitly, it is given

by

W:b(P1012P2022P3032Pab) = ((Pabplailélpzazpsasll (2-105)
[<(P202P303lélpabplalll — ((Pzazmaalélplalpablll-
Since we are not pursuing a first principle theory at the present stage, the effect of the

transition matrix element will be simulated by using of the free space cross sections

as we will discuss in detail in the next chapter. In this respect, we take the same path

33

as all other presently available relativistic dynamical models. The collision terms due

to baryon-pion interactions can be rewritten as

 

I:r(xp) =

7r MbAlaI b i I

‘ w ,
8 2§ngf/ BMW/32.412) 62(ap,7rk,ap)

{[(1+ f2(rk))f2'(xp')72(rp) - f2(£197.22(IP’)fb(1p)l<5“’(p’ - k - 12) (‘2-107)

+[f2(rk)f22(rp’)72(rp) - (1 + f2(rk))722(rp’)f2(rp)l6“’(p’ + k - p1} -
l
(2216

 

dp’dk .
and
1);,(212) =
-§./ / 2.;‘13'21721’221
[(1 + f2(rk))f2'(rp')72(rp) - fn($k)fa(IP)7a'($P’)l'
6W(p' — p _ Hzfidpdp’. (2.108)

W'bb,(a'p’, Irk, up) and Wb’;(ap, a’p’, 7rk) are the square of the transition matrix element

 

for the corresponding processes, again their effect will be simulated by using of the

free space cross sections and the width of the resonances. Explicitly,

(“0'17’lfiipllfliklaU’Huap) ' (“Grill-(kllua’p’)

 

 

b I I _
Wb,(ap,7rk,ap) — E:(Ic) , (2.109)
and
A A I 2 - . I I
(Ii/62(0)), alpl, Irk) = (“a :1 ly_(k)u(p 'l’ P ) luapl (“apllfkllua p) (2.110)

133(k) '

In summary of this chapter, we have started from a relativistic field theoretical
Lagrangian of the Walecka type, including 0', w and w mesons. By integrating over

the degrees of freedom of the virtual mesons, we were able to obtain mean-field terms

34

for the baryon interactions. However. the pion is treated as a real particle. In this

way, we are able to incorporate the formation and decay of A resonances.

Even though our theoretical framework fully utilizes relativistic kinematics, and
although we start from a fully covariant Lagrangian, our final results do not include

the true relativistic effects of retardation.

This is because we made use of the instantaneous meson exchange approximation,
eq. (2.14), replacing the Green’s function C(a: — :r’) by G(r — I", t)6(t - t’). This was
done to remove the non-locality in time introduced by the elimination of the virtual

meson fields.

The advantage of the transport equations as derived here is that one can represent
the phase-space distribution functions for nucleons, deltas and pions by test parti-
cle distributions for the different species. With this, one should be able to extend
the powerful simulation techniques developed for the non-relativistic case of the dy-
namical simulation of the phase-space distribution function of nucleons in heavy-ion
collisions to the relativistic coupled problem for nucleons, deltas and pions. Although
the transport equations presented here are derived for a system of pions, nucleons
and Delta resonances, they can be easily extended to contain higher resonances, such
as the N‘. In the following we will treat N "s in a manner similar to that for Delta

reson ances .

Chapter 3

Numerical Realization of the

Model

3.1 Equations of motion for test particles

In this chapter, we present details of the numerical realization procedure of the

hadronic transport model.

The transport equations 2.103 and 2.104 for hadronic matter, are highly coupled
through the collision integrals. However, their solutions can be obtained numerically
within the test particle method which was first introduced to nuclear physics by
Wong[Wong82]. The details of the application of the test particle method to solve

the standard BUU equation can be found in Ref. [Bert88a].

In the test particle method one discretizes the continuous distribution function

with a finite number of test particles representing individual phase space cells, i.e.

f( (r ."’p.t)= %;6(— 6(-p p.) (3.1)

where r; and p, are coordinates of the individual test particles. N. is the number
of test particles per nucleon. To obtain the equations of motion for the test parti-

cles corresponding to the solution of the transport eqs. (2.103 and 2.104) we take

35

derivatives of eq. (3.1):

v! = %,t:6'r(

fo = —26()6’(r—r,
0f _

‘37 - 1(1);ng

36

)6__r.-
6(_p p! at

+6(r -r.-)6(p— p.13—

(3.4)

Substituting the above derivatives into the transport equations (2.103 and 2.104) the

equations of motion for the coordinates of test particles can be obtained. For the

baryon test particles

(3.5)

(3.2)

(3.7)

(3.8)

Here D “(12), D: (p ) are the changing rate of the baryon momentum distribution due

to baryonobaryon collisions and baryon-pion collisions, respectively, in accordance

with the collision integrals If, and 13,. DLUC) is the corresponding change in the pion

momentum distribution due to baryon-pion collisions corresponding to the collision

integral I 1;- They are calculated in the same manner as in the cascade models[Rand79,

Cugn81, Cugn82, Cugn88], namely by discretizing the reaction time into small time

steps and solving the collision integrals within each time step via a Monte Carlo

simulation method. It is seen that the motion of test particles in the phase Space

37

is governed by the mean field U and the vertex of nucleon-nucleon collisions. In the

following we describe these two ingredients of the model in detail.

3.2 The nuclear equation of state

The nuclear equation of state describes the response of the nuclear matter to different
temperatures and densities. Current knowledge comprehends a narrow region around

the nuclear ground state.

In the present work we use a Skyrme-type density dependent mean field potential

for nucleons,

U = 1 bid. .
(p) apo+ (po) (39)

The potential energy density is then given by

 

(” )“p. (3.10)

2 b
W(p) = /U(p)dp = Ep- + 53

2pc) 1+0

The properties of nuclear matter are still not very well known. Only its saturation
point at equilibrium, namely, density p0 = 0.17 f m"3 and binding energy per nucleon
E/ A =—15.75 MeV, is well determined while even the compressibility coefficient at
equilibrium is only known to lie between 210 MeV[Blai280] and 310 MeV[Shar88]. For
high density and high temperature nuclear matter no reliable information available.

Here the compressibility coefficient K of nuclear matter is defined as
K = 9p(6P/3p),, (3.11)

where the derivative is taken adiabatically and P is the pressure. For the parameter-

ization of the mean field given in eq. (3.9), K is given by

p2
K = 9(—’+a+ba). (3.12)

3m

:38

 

A L 1 A

A A A l L L LL 1

 

 

 

 

P/Po

Figure 3.1: The nuclear equation of state, the solid line is for the stiff equation of
state and the dotted line is for the soft one.

3‘)

Here p; = 1.36fm"1 is the Fermi momentum. The energy per nucleon for nuclear

matter at zero temperature is given by

E WM) 3 p a
_= _._+:E _ 3, 3.13
A p 0 ((p0) ( )

where E f=37.26 MeV is the Fermi energy. The parameters a. b and a are then deter-
mined by the binding energy per nucleon at p0, a specified compressibility coefficient
K and the equilibrium condition of pressure P = 0. For the so called stiff equation

of state K = 377 MeV, a = —123.6 MeV, b = 70.4 MeV and a' = 2. For the soft
equation of state K: ‘201 MeV, a = —358.1 MeV, b = 304.8 MeV and a' = %. To
see the difference between the two equation of states and their density dependence,

the two equations of state have been drawn in Fig. 3.1 as a function of the nucleon

density.

It has been shown, however, that the stiff momentum independent equation of
state produces about the same amount of transverse momentum and similar flow
angle distribution in heavy ion collisions as the available momentum dependent pa-
rameterizations, and that the soft momentum independent equation of state produces
less transverse momentum and smaller flow angles[Gale87a]. We will study the sen-
sitivity of the observables to the variation of the momentum-independent equation
of state. The mean field potentials of the A and N ‘ are still very uncertain[Gino78].
However, relativistic heavy ion collisions are expected to provide information about
these potentials[Siem88, Siem89]. [n the following we assumed that the potential

energies of the A and N ' are the same as that of nucleons.

40
3.3 Elementary nucleon-nucleon cross sections

3.3.1 Elastic scattering channels

We have taken into account the following elastic scattering channels

N+N—HV+N, (3H)
N+A~N+A, (mm
N+NN+N+M, (um
A+A~A+A, QM)
M+NH»W+NI (um
iw+A—ww+a. am)

Where the nucleon N, the A resonance and the N ' resonance can be in any charge
state allowed by the charge conservation law. Here we adopt the well known Cugnon’s
parameterization for nucleon-nucleon elastic cross sections, both for the angular dis-
tribution and for the total cross section. Since no information about the collision
between a nucleon and a baryon resonance or the collision between two baryon reso-
nances is available, we assumed that all the elastic scattering channels have the same

cross section. The total cross section is parameterized as

ael( fi)

35
= 2 . . .
0.1M?) l+100(\/§-1.8993)+ 0. (\/§>18993)

55, («3 < 1.8993), (3.20)

 

In this parameterization, \/3 is the center of mass energy of nucleon-nucleon collisions

and measured in GeV and a' is in mb.

The differential cross section is given by

_ = . .21
dt cc (3 )

41

Where t is the negative of the square of the momentum transfer in the C.M.S. and b

is parameterized as

6[3.65(\/§ - 1.866116
1+ (3.65(,/3 -1.866)]6'

 

b(\/3) =

3.3.2 Inelastic scattering channels

The following inelastic scattering channels have been included in the model

p+pHrz+A++,
P+PHP+A+.
n+pHp+AR
n+pHn+Afl
n+an+A',
n+an+A°,
P+pHp+N‘+.
n+pHp+N'°.
n+p4—+n+N'+,

n+n¢—>n+N'°.

A
S»)
IQ
[Ca

V

(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)

(3.32)

A and N' production cross sections for each charge state in all of the above inelastic

channels can be estimated by using of VerWest and Arndt’s isospin decomposition

formula[Verw82] for pion production in nucleon-nucleon collisions.

According to VerWest and Arndt the reaction cross section for single pion pro-

duction in nucleon-nucleon collisions can be reduced to four independent isospin cross

sections 0,, with i, f = 1 or 0, here i and f are the initial and final isospin of the two

nucleon system. Explicitly,

UlP+P—'P+P+7rol = 011) (3.33)
U(p+p->p+n+vr+) = clown). (3.34)
0 1 .

a(n+p—>p+n+7r) = 5(0104'001), _ (3.35)
l

0(n+p—vn+n+1r+) = gland-001), (3.36)
l

“(n+P—’P+P+"-) = 307113-001)- (3.37)

Assuming that pions are produced through the intermediate state of A and N‘ res-
onances, the following parameterization for the isospin cross section 0,; results in a

good fit to the experimental single pion production data of nucleon-nucleon collisions.

 

 

rr()"1c)'2 p. 3 "1.3132(9/‘Iol3
O’,f(\/§) = 2p2 CAI—)3) (8" _ 771(2))2 + "231.29 (3.38)

where 01,6, mo and I‘ are parameters, it has been listed in table 3.1. s is the square

of the center of mass energy. Other quantities in the parameterization are defined as

P2 = 3- M13). , (3.39)

50 = (MN + m0)2. (3.40)

pg = :—° -111}, (3.41)
p313) = 4—13-14 - (MN - (4402113 — (MN + (mm, (3.42)
q2(s') = i- s’ - (MN — 1w,)2][s' — (MN + M,)2], (3.43)
90 = q(mg), (3.44)

3- = (My, (3.45)

_ Po 1 + Z2
(.M(s)) = Mo + (arctan(Z+) — arctan(Z-)) l(2") ln(1—_—Z—;-). (3.46)
Where
Z. = NS - MN - 2110);: (3.47)
0

43

Table 3.1: Isospin cross section parameters

 

 

 

parameter an 0'10 (701
a 3.772 15.28 146.3
3 1.262 O 0

mo(MeV) 1188 1245 1472

r(MeV) 99.02 137.4 26.49

 

 

2
Z- = (1”,); + M. — Mo)F—, (3.48)
0

for A resonance Mo=l220 MeV, I‘o=l20 MeV and for N' resonance Mo=l430 MeV,

Fo=200 MeV.

To gain some familarity with the energy dependence of the isospin cross section,
the isospin cross section have been calculated and plotted in Fig. 3.2 for a nucleon

with kinetic energy E). to collide with a nucleon at rest in the laboratory frame.

As shown in ref. [Wolf90], in terms of the isospin cross sections the inelastic cross

sections can be estimated as

0(P+P_’"+A++) =010+ £011, (3.49)
0(P+P'*P+A+) = 3011. (3.50)
”(n+P-’P+AO)=%0’11+:11'0'10, (3.51)
a(n 'l' P -’ n + A+) = $011 '1' 21-010, (3.52)
a(p+p—*p+N‘+) =0, (3.53)
0(n + p -* p + N‘°) = :E'O'Ol, (3.54)

a(n+p—>n+N'+) = 001. (3.55)

Ale...)

44

These cross sections have been shown in Fig. 3.3 and Fig. 3.4. Cross sections for the n
+ 11 channels can be obtained from that of p + p channels by isospin symmetry. The
differential cross sections for the inelastic collisions have been assumed to be isotropic

in the nucleon-nucleon center of mass frame.

The reabsorption cross section for baryon resonances have been obtained from the

detailed balance. Explicitly, for N‘ resonance

2

0(n + N‘+ —-> n + p) = :—£0(p +11 —+ n + N”), (3.56)
p2

a(p+N‘°—)n+p)=#0(p+n-*p+N'°). (3.57)

1 p'2
01A++p~p+p)=I-Eé-0(p+p-'A++p). (3858)

1 p}
0(A+++n—>p+p)=Z-F-a(p+p—1A+++n), (3.59)

p2
0(A++n—*p+n)=7-;§~0(p+n—+A++n), . (3.60)

0 1 P3 0

0(A +p-+p+n)=-2---p—_2oa(p+n-+A +p). (3.61)

By using isospin symmetry and averaging over the isospin degree of freedom the above

relations reduce to

2
a(N+A—+N+N)=%-%.0(N+N—+N+A). (3.62)

Here p f is the momentum in the final N N channel in the C.M. of the colliding particles.
This assumes that the resonance is narrow; in general eq.(3.62) underestimates the

absorption rate of Iow~energy A’s[Dani91].

 

15.0%..3T-...T....,....,....,

ALlAJAl

12.5 " 010

10.0 L —‘

A _ .
.0 t .
E l.

V 7.5 T j
3:! ) .
b :

5.0 L— 011 "'
)- /’ —————————————— 1
: I, 4
2.5 r / a j
. , 01 .
P / -(

)-

0.0 -rfrh.-.1.......1-1-1‘... .1.. .. 1.
500 750 100012501500

Ek (MeV)

 

 

 

Figure 3.2: Energy dependence of the isospin cross sections

46

 

 

 

~ p+p4n+A++
)- 4
A ’ 7
.0 . 1
E 10 - a
v r 1
b
/ p+p-*p+
7' /
L / .
/
l 4
_ /
/
O ’{l A 1 L L l n #4 L l A L L L1 1 14

 

500 750 10001250 1500
Ek (MeV)

Figure 3.3: Energy dependence of the A production cross section in p + p collisions

47

 

r I I T V Y Y 1 l' r V T I ‘l’ V V V T ‘l’ V f l T I
6

n+p4n+A*,p+A° 1

A
“1‘23
v
b . .
2 L , 3
/
* n+p+n+N°+, p+N°°/ -
b /
/
/
//
)- 1’ 4

 

 

”
O AQL._.__—L-rfrml A L l ;l A A 1 l

500 750 100012501500
Ek (MeV)

 

Figure 3.4: Energy dependence of the A and N‘ production cross section in n + p
collisions

48

3.3.3 Direct pion production channels

Direct processes for pion production of the form
1V1 + 1V2 -* (V3 '1' N4 'i' W (3.03)

account about 20 percent of the total pion production cross section[Kita86]. However.
the energy dependence of the direct pion production cross section is unknown. We
will set the percentage of the direct process in the total inelastic cross section as a
free parameter. meanwhile, reduce the A and A" production cross section estimated
in the previous section by this percentage, so that by turning on or off and adjusting
this parameter we can study the effect of the direct processes on the experimental
observables. The kinematics of the three-body final state is not easy to treat and
one usually assumes that the pion takes all of the available kinetic energy in the
nucleon-nucleon center of mass frame of the initial state[Kita86, Aich85]. Another
way of determining the momentum distribution of the three-body final state is to use
the Fermi statistical model for pion production[Ferm54]. The essential assumption in
this model is that the production of pions in nucleon-nucleon collisions is governed by
statistics rather than by dynamics, with cross sections determined by the available
phase space subject to conservation laws. In the C.M.S. of the two initial nucleons

the momentum distribution for one of the three particles in the direct process is given

by [Bloc56]

dP 2W2

_ 2(m3 + m3) (m3 - m5)“
dPi

‘Tfim’lE-nv-s «E-av-firW3

 

 

(3.64)

(m3 - m3)”

«E-av-fifl

 

-{3(E- E,)2[1—

2("13+ "13) ("13- m3)”

—-2 .—
P111 (E-E1)2-Pi ((E-EiV-prn'

 

 

49
The maximum of p1 is given by

( _ {[E2 - (m1 ‘1' 7712 + T77,3)2][EI2 — (m1 .. n12 _ m3)‘2]}1/2
pl)max — 2E ,

 

(3.65)

where E = \/E is the total energy of the system. The momentum distribution of
the other two nucleons can be obtained from the energy-momentum conservation.
We have developed a Monte Carlo procedure to simulate the three-body momentum
distribution. In Fig. 3.5 we compare the pion kinetic energy distribution obtained
from the above equation and that from the Monte Carlo procedure for nucleon-nucleon
collisions at the center of mass energy of 3 GeV. It is seen that the momentum

distribution of the three-body final state can be simulated satisfactorily.

Taking the cross section for the direct pion production as 20 percent of the total
inelastic cross section, independent of the center of mass energy of the nucleon-nucleon
collisions, we find that the calculated pion spectrum for a nucleus—nucleus at E /A 2
lGeV is rather unchanged from that assuming all pions are produced through the
intermediate state of A and N ‘ resonances[Liba91a]. To avoid the ambiguity caused
by the unknown energy dependence of the direct process and to compare with other

model calculations we will neglect the- direct process in the following calculations.

3.3.4 The pion-nucleon resonance and decay

In the model we allow the decay of baryon resonances and the pion-nucleon resonance

of the form
AH7r+N,N‘o—>1r+N. (3.66)

Where A, N‘, 1r and nucleon can be in any charge state allowed by charge conserva-

tion. The branching ratio of the possible final state is determined by the appropriate

 

 

 

0,0025....,....,....,...‘
$00020:— .3
>. .

Q)

5 0.0015 _— _
= : I

% 0.0010 — _:

\,= \/s=3000 MeV 3

95 0.0005 _— J

 

 

0 200 400 600 800
T,(MeV)

Figure 3.5: Pion kinetic energy distribution in the direct process. The solid line is
from Fermi statistical model and the histogram is from the Monte Carlo simulation

.31

Clebsch-Gordan coefficients. The mass of the baryon resonance fromed in the pion-
nucleon resonance is uniquely determined by the reaction kinematics. The width for

the A resonance is parameterized following Kitazoe et al.[Kita86] as

0.47q3 .
F = . . 7

 

where q is the momentum of the pion in the A rest frame. For N" resonance a

constant width of 200 MeV has been used[Verw82].

During each time step of duration dt, the decay probability of the A’s and N "5
present in the system is determined by an exponential law using the proper time

obtained from their widths,

—dtr
Pdecayzl-e laT=

"Mb"

(3.68)

Since one of our purposes is to study properties of the pion spectrum in heavy ion
collisions, it is necessary to discuss how the pion spectrum in a single nucleon-nucleon
collision is calculated. The A or N' resonance is assumed to be produced isotropically
in the nucleon—nucleon center of mass frame and we also assume that the decay of
the resonance has an isotropic angular distribution in the rest frame of the resonance.
The decay of the resonance is then calculated using the Monte Carlo integration
technique. This leads to a pion spectrum in the rest frame of the resonance which is
finally Lorentz transformed into the desired frame. As an illustration, we show in Fig.
3.6 the contribution to the pion spectrum from the A resonance (dotted histogram),
N‘ resonances (dashed histogram) and the direct process (solid histogram) in nucleon-
nucleon collisions at the center of mass energy of 3 GeV. The spectra from the three

processes are normalized to 1, respectively.

The cross section for the pion-nucleon resonance is also parameterized using

the Breit-Wigner formula with the maximum cross section from the experimental

us
(0

 

0.0030 b T V Tr V r I V I T ‘I T I I Y I Y Y I

0 0025 L 57.53.. vs = 3000 MeV .1

T g. = r j
%, 0.0020 5 -J-—«-.. g.
E :5 P." "‘1 3
= 0.0015 3% g 1- _:
Ed ,._' : 1
Q :2 1 -1 j
96 I. (_ 1_: 1
0.0005 P:_: ‘7 _‘

7" |_I 4

., .

L'l‘)

 

 

 

 

0.0000011LLLllL*l4llnlah'1'~L

Figure 3.6: Pion spectrum in nucleon-nucleon collisions. The solid histogram is the
spectrum from the direct process. The dashed histogram is the spectrum from the
N‘ decay and the dotted histogram is that from the A decay.

data[Part88],

O’max(7r+ + p -+ A++) = 01115111717 -+- n -v A‘) = 200 mb. (3.69)
am..(rr° + p —. 3+) = 0.....(rro + n —. A0) =135 mb, (3.70)
0.....(7‘ + p —-> 48°) = an...(vr+ + n —+ 3+) = 70 mb, (3.71)
0......(4‘ + p -1 N'o) = 0......(00 + n —* N'O) = 50 mb. (3.72)

47,,,,,‘(7r+ + n —» N”) = amax(7r° + p —* N“) = 50 mb. (3.73)

3.3.5 Pauli blocking for fermions and enhancement factors
for bosons

The phase-space occupation factors for the final state of the fermions, 1 - fb(:rp), are
treated via a Monte-Carlo rejection method. Since the computation of all possible
final-state phase-space occupation factors is a very time-intensive task, we have de-
veloped a technique to store the six-dimensional phase—space occupation probability
at every time step on a lattice[Baue90]. In this way we are able to use a large number
of test particles (> 100) to represent a real particle in the reaction of two heavy nuclei
while using a reasonable amount of CPU time. BUU-type of calculations for heavy
system have been hindered using the'old way of evaluating Pauli, blocking factors.
For reactions involving intermediate mass nuclei, as many as 400 test particles per

real particle have been used.

The final state phase-space occupation probability factors for bosons, l + f,(.rp),
cannot be treated by conventional rejection methods, because the possible range of
values of this function is not between 0 and 1. However, it is possible to introduce a

cutoff, F, such
F > max(l + f,.(:rp))

for all coordinate values (1:,p) during the course of the nuclear collision. By

54

multiplying the interaction matrix element by F and dividing (1 + f,(.rp))/F one

can use the conventional rejection technique on this scaled occupation probability

factor[Wlbr91a, Welk91].

In the present calculation the mean boson phase space occupation probabilities
are on the order of (f,) z 5 - 10", because we have, for‘example, z 50 pions Of three
different isospin substates distributed over a total phase space volume of a: 400 h3
for central La + La collisions at 1.35 GeV/ nucleon beam energy. Thus the effect
of stimulated emission of pions due to the effect of the boson enhancement factor is
negligible in the case studied here. In nucleus-nucleus collisions at CERN-energies,
however, this may not be the case. Depending on the assumptions for pion freeze
out and expansion of the hadronic system, pion phase space occupation probabilities
may become comparable to 1, and one may introduce a non-zero chemical potential
for pions. Kataya and Ruuskanen[Kata90] have shown that then one can also obtain

a concave p, spectrum for negative pions due to this effect.

Chapter 4

Pion production dynamics

In this chapter we study the dynamics of pion production. In relativistic heavy ion
collisions of beam energy around 1 GeV / nucleon about one half of the nucleon-nucleon
collision cross sections are inelastic, mainly through pion production. Pion production

may reveal interesting properties of the reaction dynamics.

To gain some insight into the dynamical properties of relativistic heavy ion colli-
sions and to get some familiarity with the dynamical characteristics of our model, we
show in Fig. 4.1 the time evolution of the accumulated total number of baryon-baryon
collisions and the reaction rates for several relevant collision processes. In the model
explicit isospin degrees of freedom have been used, the quantities shown in the figure

are the sums over all possible isospin channels.

The particular choice of the system Ca+Ca at a beam energy of E/A=l.8 GeV
and impact parameter b=0 is made in order to compare with Cugnon’s cascade
calculation[Cugn81, Cugn82, Cugn88]. The overall time dependence of the total col-
lision number and reaction rates is similar to that of the cascade calculation. The
accumulated baryon-baryon collision numbers saturates at around 10-15 fm/c. Af-
ter this time, mainly A decays and pion reabsorptions are present. We see that the

A destruction processes, NA -> N N and A —+ N 1r, set in slightly later than the

55

56

 

200 ..I....I....,....l....l..

(

 

   
  

 

 

 

 

 

 

 

 

 

 
      

 

 

 

: -(

C +

150.— '1

a: » .

m : Ca+Ca :

z 100; E/A=1.B GeV ‘3

1- 4

C 1

0" .1....L.-..L....1....Ln..41

1g_...-l...-Tr...,-fi..l....,...-:

10;- ‘3

a;— 1:

63- 3.

.. L .3

I 4E 3

’8 2:- 1‘

313 10:' :

- -(

.3 8:- r
‘5 :
02'. 6.-
4;-

: 2’ .

2:- ..-N1r—-°A ,.

0: .~'.1...r1-...1....1.”.1

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Time (fm/c)

Figure 4.1: Upper figure: Accumulation of the total number of baryon-baryon colli-
sions in the reaction of Ca+Ca at E/ A = 1.8 GeV and impact parameter b = 0 fm.
Lower figure: Time evolution of the reaction rate for the specified processes in the
same reaction.

OI
~l

N N -+ NA process, because the former processes need an appreciable accumulation
of A’s, and also because the A has a finite lifetime. The A decay rate is always higher

than that of the formation of this resonance.

It is of interest to note the quantitative differencesbetween our model calculations
and the cascade model calculations. The saturated number of total baryon-baryon
collisions calculated in the present model is about 15 percent smaller than that of the
cascade model. This is mainly due to the better treatment of the Pauli blocking factor
and the inclusion of the mean field in our model. The time integrated cross section for
the A reabsorption process (NA —+ NN) in the present model is about twice of that
in the cascade model. Consequently, the total number of pions observed in the final
state is about 30 percent less than that in the cascade model. A later modified cascade
calculation[Cugn88] shows that the discrepancies between the experimental data and
the cascade calculation on pion production in proton, pion and heavy ion induced
reactions can be removed if one artifically multiplies the cross section 0(N A —-v N N )
by a factor of three without changing the cross section 0(N N —+ NA). This has been
explained as an indication of the enhancement of the pion reabsorption in nuclear
medium and the underestimation of the pion reabsorption in their model. However,
in our calculation the medium effect explanation is not necessary, and the total pion
production cross sections are in agreement with data (see below).

Since we are interested in the properties of pions, it is crucial to know the time
evolution of the source of the pions. In Fig. 4.2 the population of free pions and pions
still bound inside excited baryons (unborn pions) is displayed for the system La+La
at a beam energy of 1350 MeV per nucleon and b=1fm. It is seen that the total
number of pions, A’s and N"s freezes out at around t=20 fm/c at a value which is
in good agreement with experimental data[Harr87]. The overall time dependence of

our bound and unbound pion multiplicities is similar to the one obtained from the

(,1!
Ct;

 

 

 

 

 

60_..e.l.e..,..-.r...4
: a j
).
407 1
h n :
: .-'«
z 30:- 1
20;- \ T
.- \\\\‘j
: ‘H
10? N‘+A 1
0'...-'..L...-1....1...‘

 

0 10 20 30 40
Time (fm/c)

Figure 4.2: Time evolution of the population of free pions, A’s and N"s in central
collisions of La + La at E/ A = 1350 MeV.

cascade calculation.

In Fig. 4.3 we study the beam energy dependence of the time evolution of the
pion multiplicity. First, we note that the reaction time scale is getting shorter as the
beam energy is increased. Secondly, the final pion multiplicity is prOportional to the

beam energy.

In Fig. 4.4, We compare the-final pion multiplicity with that of the experimental
data for central collisions of La+La. The round plot symbols on the solid line are
our calculated results, and the square symbols with error bars are the experimental
data[Harr87]. A good agreement can be seen in the whole energy range. Similarly
good results have been obtained with a VUU-model[Moli87]. In Ref. [Moli87] a. com-
parison of the data to existing cascade model calculations is shown as well, and a

clear overprediction is observed for the cascade model.

In Fig. 4.5 we display the mass dependence of the time evolution of the pion
multiplicity in central collisions at a beam energy of 1.5 GeV/ nucleon. It is seen that
the reaction time scale is rather insensitive to the size of the colliding system .The
final pion multiplicity as a function of 2A, which approximates the mass number of the
participant region in central collisions, has been shown in Fig. 4.6. It is seen that the
total pion multiplicity is almost linear with respect to 2A. This is in accordance with
the experimental findings[Harr87]. The linear dependence of the pion multiplicity on
the mass number of the participant region reflects the fact that pion production is a

bulk nuclear matter probe rather a surface probe.

The impact parameter dependences of the time evolution of the pion population
and the final pion multiplicity have been shown in Fig. 4.7 and Fig. 4.8 for the reaction
of Ar + Kcl at a beam energy of 1800 MeV per nucleon. It is seen that for central

collisions at an impact parameter b S 1 fm, the time scale for pion production and

6O

 

 

 

 

 

80 ...,....l...fir....l....
i La+La
E A=1700 V
>‘ 60 __ boun/ Me _
.2
4., 4
'3 1300MeV
2 4o — a
g - noouev
0!!
OI! J
900MeV
20 " 700MeV "
)- 500Mev
O ..l....l....l....l.

 

 

 

0 10 20 30 40 50
t (fm/c)

Figure 4.3: Beam energy dependence of the time evolution of the pion multiplicity in
central collisions of La + La

61

 

0.25 T-...,....[HmlfifiTmnh

0.20 T 4

< - .
A 0.15 f- -j
5 (
C‘. . 4
V L 4
0.10 L j

0 Data - j

0.05 f ° Model j

 

 

f.l...Ll....lr...l...4L....l

0.50 0.75 1.00 1.25 1.50 1.75
Blah/A (GBV)

 

Figure 4.4: Excitation function of the pion multiplicity in central collisions of La
+ La. The squares are the experimental data of Ref. [Harr87], and the round plot
symbols on the solid line are the model calculations.

 

 

 

 

 

 

 

80 ....I....r....l...-l....l..
’ Ebem/A=1500MeV
+ .
>. 60 - 159Tb+159Tb -4
+9 .. ‘
g 139 139
“a La+ La
E i 1198n+1198n *
:3 * .
2 40 L— 1°°Ag+1°°Ag-
.3 * <
0'. _ 80Br+3°Br
MCu+°4Cu
20 - _
’ 4c’Ca+"’°Ca
i 2°Ne+z°Ne
0 ..1....La.L.L-.-.i..-.1..
0 10 20 30 40 50
t (fm/c)

Figure 4.5: Mass dependence of the time evolution of the pion multiplicity in central
collisions

63

 

50...]....,..fi1..-.,...

 
 

Tl Y

50 E/A =1500 MeV

 

40—
30-
20-

10:-

bjAJ_Al_LJ_Lllj.lll[Lilllllljllj

50 100 150 200 250
2A

    

 

 

 

Figure 4.6: Mass dependence of the pion multiplicity

64

 

 

 

 

 

 

 

 

 

 

 

25 ..l.-..,....,-.fil.fi..r...-
: Ar+Kc1 i
~ Em/A=1800Mev
20+ 1
: k b=0 ‘
>5 —=—
r: - b=1fm .
o . ,
z
a... 15 _ _.
33‘ . b=2frn «
3 1 «
2 ' b=3frn
C: F .
.2 10 - a
an ’ (
b=4frn <
f d
5: b=5frn
Op .1LL.LLLLTLI....I....LTM.
0 5 10 15 20 25 30
t (fm/c)

Figure 4.7: The impact parameter dependence of the pion production in Ar + Kcl
reaction at the beam energy of 1800 MeV per nucleon

 

   
  

Ar+Kcl —:

17.5
' E/A=1.8 GeV 3

15.0:—
" 12.5_
10.0}

7.5 1-

 

 

 

50 L...l....
C

b (fm)

Figure 4.8: The impact parameter dependence of the pion multiplicity in Ar + Kcl
reaction at the beam energy of 1800 MeV per nucleon

66

the final pion multiplicity are insensitive to the impact parameter because of the
complete overlap of the projectile and the target. For larger impact parameters the
pion production takes longer time, and the final pion multiplicity decrease linearly
with the increasing impact parameter. This is in consistence with the mass and the

energy dependence.

Due to the energy degradation in the reaction process, inelastic nucleon-nucleon
collisions mainly occur in the early stage of the reaction. In the later expansion phase
of the reaction only the resonance decay and the pion-nucleon resonance persist.
The relative time dependence of these processes has been shown in Fig. 4.1. This
property of the pion production dynamics cause the saturation of pion multiplicity
in the expansion phase of the reaction. The saturation of the pion multiplicity can
be seen clearly in Fig. 4.3 and Fig. 4.5. This property was first observed in Cugnon’s
cascade model calculation. Since then it has stimulated a lot of studies of properties
of the compressional phase of heavy ion collisions using of the pion multiplicity. One
typical example is the effort of extracting the nuclear equation of state from the
excitation function of the pion multiplicity by assuming the discrepancy between the
experimental data and the cascade model prediction comes completely from the lack
of compression energy in the model[Stoc82, Harr87]. More elaborated models like
the Boltzmann-Uehling-Uhlenbeck transport model (BUU)[Bert84, Bert88a, Kru585]
and the Quantum Molecular Dynamics model (QMD)[Aich91], which include the
mean field in addition to the two-body collisions, found that the sensitivity of the
pion multiplicity is not so obvious, especially when momentum dependent forces are
taken into account[Gale87c]. One therefore has been investigating experimentally
other global properties of pions like pion spectra and pion flow. In the following two

chapters we will concentrate on the study of pion spectra and pion flow, respectively.

Chapter 5

The concave shape of pion spectra

Some features of the pion kinetic energy spectrum in relativistic heavy ion collisions
are expected to provide information about the space-time dynamics of the reaction.
One interesting feature observed is that pion spectra in central heavy ion collisions
show a concave shape on a logrithmic scale, which can be well fitted by a superpo-
sition of two Boltzmann distributions with widely different slope parameters. This
phenomenon has been observed in Ar+KCl collisions at a beam energy of 1.8 GeV
per nucleon[Broc84] and La+La at E/ A = 1.35 GeV [Odyn88]. Preliminary results
from Au+Au at E/ A = 1.15 GeV[Cha590] also show this feature. A typical example
of the concave shape of the pion. spectrum. has been given in Fig. 5.1 for the reaction
of Ar + Kcl at a beam energy of 1.8 GeV/ nucleon. Plot symbols with error bars are
the experimental data, the solid curve is the Maxwell-Boltzmann distribution with a
temperature of 63 MeV. It is seen that the experimental spectrum deviates from the

Maxwell-Boltzmann thermal distribution for the total energies of pions larger than

0.5 GeV.

In ultrarelativistic heavy ion collisions and proton induced reactions, pion trans-
verse momentum spectra also show a concave shape[Harr89]. This has generated a
lot of interest, and the origin of this phenomenon has been vigorously debated in the

literature[Shur88, Leeh89, Kusn89, Kata90, Brow91]. Therefore, the understanding of
67

68

 

Y I Y V V U l I I V V l V T r T l T I V

Ar+Kcl -.
E/A = 1.8 GeV 1

H
010
OO
00

u 'l

f
9?

L11

H
010
00
1'1

9
45’
4'
l

H
O
I

/

dza/dEdfl (mb/GeV sr)
()1
l

 

 

 

1 .rrilr...lnr
0.2 0.4 0.6

E (GeV)

 

Figure 5.1: The concave shape of the pion spectrum in Ar + Kcl reaction at a beam
energy of 1.8 GeV/nucleon. The experimental data shown by the plot symbols are
from ref. [Broc84], the solid curve is the one-temperature fit to the experimental data
with a temperature of T=63 MeV.

69

the mechanism that causes the concave shape of the pion kinetic energy spectra in rel-
ativistic heavy ion collisions may shed some light on the origin of the concave shape of

the pion transverse momentum spectra in ultrarelativistic nuclear collisions[Libana].

In relativistic heavy ion collisions of beam energies around 1 GeV / nucleon, several
hypotheses have been made by the groups who discovered this effect in order to
explain their experimental results. These include the superposition of thermal pions
and the pions from the final state A decays, higher resonances[Broc84] and the effect
of baryon flow on the pions[Cha590]. Based on a simplified hydrodynamical model
calculation[Hahn88], it was also conjectured that the concave shape of the pion spectra
may come from an isotropic hydrodynamical expansion of the hot compressed nuclear

matter.

The cascade model predicts purely thermal pion spectra[Broc84], although it has
been very successful in predicting many other experimental observables in relativistic
heavy ion collisions. The original BUU model uses the frozen Delta approximation

and also fails to explain the origin of the concave shape of the pion spectra[Cha390].

We propose that the concave shape of the pion spectra is a result of different
contributions of A resonances produced early and late during the course of the heavy
ion collision. In the following we explore this idea in detail and compare our model

calculations with the available experimental data.

5.1 Mechanisms for the concave shape of pion
spectra

In this section, we apply our model to study the mechanism that causes the concave
shape of pion spectra in central heavy ion collisions at beam energies around 1 GeV

per nucleon.

70

5.1.1 Model calculation for the pion spectrum

In Fig. 5.2, we show the number of pions per energy interval for the La+La reaction.
-P'—EdN/dE, as a function of the pion kinetic energy. where P is the momentum and

E is the total energy of the pions.

The time chosen for the figure, t=20 fm/c after the start of the calculation; by
this time most of the baryon-baryon collisions have ceased, but a large fraction of the
excited baryons produced have not decayed yet. The real pions which are not bound
in resonances are represented by the solid histogram. For a thermally equilibrated

dilute pion gas at a temperature T, we can use the Boltzmann distribution function

1 (UV
FEB-ET - C'eXP(-Ekin/T)- (5'1)

As we can observe from Fig. 5.2. the free pions at 20 fm/c can be well described with

a Boltzmann distribution of temperature 78 MeV (straight line fit).

By assuming sudden decay of all A’s and N"s present at 20 fm/c, the contribution
to the pion spectrum from bound pions can be obtained. These are shown by the
dashed histogram. It is clear, that these pions do not show the same temperature as

the pions which are already free at this time, but indicate a lower temperature.

If we superimpose the two contributions to the pion spectrum, we obtain the result
which is represented by the round plot symbols. The error bars are of statistical
nature since we solve the transport equations (eqs. 2.103 and 2.104) with a Monte
Carlo integration procedure. The concave shape obtained in this way clearly hints at
a pion spectrum with a two-temperature appearance. The low temperature is about
50 MeV for pions with Emu < 0.2 GeV and the higher one is about 78 MeV for pions

with Elan 2 0.2 GeV.

Different contributions to the pion spectrum at t = 40 fm/c is shown in Fig. 5.3.

 

'0
1 4 LL ALLA

 

 

(PE)"‘dN/dE
if;
f.

l AJAL‘]

 

100 ...1..1L1...-1..-114..
0.1 0.2 0.3 0.4 0.5

Ek-m(GeV)

 

 

 

Figure 5.2: Calculated contribution to the pion spectrum from pion already free (solid
histogram) and still bound in baryonic resonances (dashed histogram) as well as their
sum (circles) at t = 20 fm/c. The straight line is a thermal distribution with a
temperature of 78 MeV.

A}
[V

 

 

 

""

(PE)"l dN/dE

I 1
P-_.-‘
L

 

 

100 ...1....-1-....1..-.1..6.
0.1 0.2 0.3 0.4 0.5
Em (GeV)

 

Figure 5.3: Calculated contribution to the pion spectrum from pion already free (solid
histogram) and still bound in baryonic resonances (dashed histogram) as well as their

sum (circles) at t = 40 fm/c.

73

We note that the total pion spectrum obtained at t = 20 fm/c (Fig. 5.2) is almost
the same as that obtained at t = 40 fm/c when we stop our calculation. The reason
for this is that between these two time instances the A’s and N"s are almost moving

freely during this expansion phase before they decay.

5.1.2 Concave pion spectra

What is the reason for the pions that are still bound at t=20 fm/c or 40 fm/c to show
a lower temperature? We attempt to answer this question in Fig. 5.4 and Fig. 5.5.

The upper part of Fig. 5.4 shows the rate of processes
N + N —v N + A (5.2)

during the La + La reaction. The lower part of the figure displays the probability
distribution of baryon-baryon center of mass energies, \/§, for two different time
intervals during the course of the heavy ion reaction, as extracted from the computer
calculation. The dashed histogram corresponds to all baryon-baryon collisions of the
type Eq. 5.2 during the initial compressional phase of the reaction (dashed hatched
area in the upper part of the figure), t S. 6 fin/c. The solid histogram corresponds
in the same way to all collisions for t Z 12 fm/c (solid hatched area). We see that
the center of mass energy distribution in the early stage of the reaction peaked at a
higher energy than that in the later stage of the reaction. To be more quantitative
on the energy degradation in the reaction process, we show in Fig. 5.5 the average

center of mass energy of nucleon-nucleon collisions of the type Eq. 5.2.

We can clearly see that the early baryon~baryon collisions are on average more
energetic than the later ones. This is because the central rapidity region is initially
free of baryons, but is increasingly more populated as the reaction proceeds. To ease

the understanding of the dynamical effect we illustrate schematically the rapidity

 

 

 

   

 

 

 

 

 

 

 

 

I r I I r I ,
.. 15:- €
l . .
E E 3
a.) 5 f' '3
+3 . .
CU . I
Q: 0 i mmlnn . . . l . _
0 5 10 15 20
Time (fm/c)
. ' . I v I . I 1
0.15 E— '5
g”: 0.10} _,..I'_“' E -{
CL. .. "‘1' 1
0.05 _— .-J-' l—, -j
t .1" '-~_
000 .fiJ . l L . . . l r . . . ‘ -0 ,.
2000 2200 2400 2600 2800
\/s (MeV)

Figure 5.4: Upper part: Rate of A production during the La + La reaction. Lower
part: Probability distribution for the C.M.S. energy of nucleon-nucleon collisions.
The solid histogram is for t2 12 fm/c and the dashed histogram is for t3 6 fm/c.

*l

0‘

 

 

 

 

 

 

 

92.5_....,...-,....,...
8 I La+La .
x 2.5:- E/A=1.350ev —:
g E b=1.0fm :
/I\ 2.4:- 1
z' 2.3}- -j
+ ; *
E . I
A 2.27 T
N ' .4
> I ‘
In L _‘
v 2.1: III“ 1
0 10 20 30 40

t (fm/c)

Figure 5.5: The time dependence of the average center of mass energy of nucleon-
nucleon collisions in which baryon resonance can be produced during the reaction of
La + La at a beam energy of 1.35 GeV/ nucleon and an impact parameter of 1.0 fm.

76

distribution of nucleons in the reaction process in Fig. 5.6. A subsequent interaction
of a nucleon at central rapidity with a nucleon at target or projectile rapidity thus
becomes more and more probable towards the later time in the reaction. Since it is
less energetic than a reaction of a nucleon at projectile rapidity and one at target
rapidity (the only kind possible in the initial stage of the reaction), the A’s produced
later are less energetic than the ones produced earlier, and the different contributions

to the kinetic energy spectrum of the pions can be understood.

The pion spectra shown in Fig. 5.2 are the results of the full dynamical evolution of
the system, and this also includes the reabsorption of pionic excitations. We therefore
have to ask what the effect of pion reabsorption and rescattering on the pion spectrum
is.

In the present model pions can be reabsorbed through the two-step mechanism,
namely, N1r —> A(N‘) and N A(N ‘) —-> N N . To illustrate the effect of pion reabsorp-
tion and rescattering on the pion spectrum, we compare in Fig. 5.7 the primordial 7r"
spectrum with the final one obtained at t=40 fm/c from the full dynamical evolution
for the La+La reaction. The primordial pions are obtained by recording the momen-
tum, mass and isospin of all A’s and. N"s when they. arefirst producedduring the.
course of the reaction, and by calculating the pion spectrum which would be obtained
if all of these baryonic resonances were to decay right after they are formed, and the
resulting pions would propagate without further interaction from then on. The pri-
mordial pions are represented by the round plot symbols on the solid histogram and

the final pions are represented by the plot symbols on the dotted histogram.

We first notice the large difference in the normalizations of the two histograms:
The number of finally escaping pions is only on the order of 10% to 20% of the total

number of pionic excitations generated during the course of the heavy ion reaction.

~l

~l

 

1 v I r v v v T t Y r I
I I I .
q
-(

Initial State i

I UVIYYIIIU'Y‘
l

I
ll

 

:>. c 1
”o .— j
\ I 4
- . L . L L 1 . r L l 4
% h r l . . . r I . l 4
: , 1
:- Intermedlate State -:
I 1
I- J
1- ..... ‘

 

 

 

   

Figure 5.6: Schematic illustration of the rapidity distribution of nucleons. The upper
window shows the rapidity distribution of nucleons in the initial state. The lower
window shows the rapidity distribution during the reaction process. The dotted
curve shows nucleons having been scattered into mid-rapidity range.

 

 

 

 

100 ..........II ..
0.1 0.2 0.3
Ekin (GeV)

 

0.4 0.5

Figure 5.7: Comparison between the spectra of primordial pions (solid histogram)
and final pions(dotted histogram).

79

Second, we see that the primordial pion spectrum does not show the concave shape
observed in the final distribution. Due to the effect described above, the primordial
pion spectrum is, however, not quite of the Boltzmann type as predicted by the
thermal equilibrium model calculations[Hahn88, Pirn79, Bar281]. The reason for
this additional shape change lies in the energy dependence of the elementary pion
absorption cross sections obtained from detailed balance. High energy pions with
kinetic energy larger than 0.2 GeV are rescattered or reabsorbed at a higher rate

than low energy ones.

In addition, pion absorption is a two-step process and thus dependent on the
square of the nucleon-density. This also favors the reabsorption of the higher energy
pions produced early in the reaction (when the baryon density was high) over the
reabsorption of lower energy pions produced later in the heavy ion reaction (when
the baryon density was lower). Clearly, a dynamical calculation of pion reabsorption
is thus essential for the correct explanation of pion energy spectra, and a calculation

utilizing an energy independent pion mean free path in nuclear matter is insufficient.

From the above arguments based on Figs. 5.4 and 5.7, it is clear that we do not
have only two contributions of different temperature to the pion spectrum, but rather
a continuous change from the initial high temperature contribution to the final low
temperature. The formation of the concave shape of the pion spectra is due to the
gradual change of mean energies of the formed baryon resonances and the gradual

change of reabsorption conditions during the course of the reaction.

In the study of pion spectra in relativistic heavy ion collisions an important ques—
tion is to what extent the slope of pion spectra reflects the true temperature of the
system in the early high-compression phase of the reaction. From the comparison of
the two spectra shown in Fig. 5.2 and Fig. 5.7, we see that the higher temperature

extracted from the pion spectra in the final state more accurately reflects properties

80

of the system in its early phase. However, one should use caution in applying the term
“temperature”, because what we observe is not the consequence of an equilibrated
system, but rather of a non-equilibrium transport process of a system on its path

towards kinetic equilibration.

Within the present transport model, one can also test the other hypotheses made
to explain the concave shape of the pion spectra. By studying the dependence of the
shape of the pion spectra on the nuclear equation of state, we can study the effect of
the baryon collective flow on the pion spectra. It is found that nuclear collective flow
are very sensitive to the nuclear equation of state. With a stiff nuclear equation of
state one predicts a higher average transverse momentum in the reaction plane and
a larger flow angle for nucleons[Bert88b]. Within the present model calculations, it
is found that pion spectra calculated with a stiff nuclear equation of state and a soft
nuclear equation of state are not distinguishable within the statistical error bars. It
indicates that one should at most expect a small effect on the pion spectra from the

baryon collective flow.

The effect of higher resonances on the pion spectrum, such as that of N '(1440)’s
included in the present model, has been studied by turning off the reaction channels
involving N "s. It is found that the presence of N "s has only a very small effect due
to their small production cross sections in the energy range of interest here. This
finding is in agreement with that of Randrup from the study of the hadronic matter
equilibration process[Rand79]. It was found that as long as the beam energy remains
on the order of 1 GeV / nucleon it suffices to include only the A resonance. However, at
higher energies, Ebeam 2 2 GeV/ nucleon, higher baryon resonances as well as direct
multiple pion production grow increasingly important in the reaction process, and

their effects on the pion spectra remains to be explored.

81

5.1.3 Comparison with experimental data

In Fig. 5.8, we compare our model calculation with experimental data for central (b
< 2.8 fm) reactions of La + La at a beam energy of 1350 MeV per nucleon. The
experimental 1r" number distribution, (PE)"dN/dE at 90:1:30 degrees in the center
of mass of the target and projectile, is shown as a function of pion kinetic energy and
represented by the round plot symbols. The data can be well fit by a two-temperature

fit of the form
(PE)‘1dN/dE = A1 exp(-Ek;n/T1) + A2 exp(—Ekj,,/Tg), (5.3)

with a x2 per degree of freedom of 0.9 (T1 =45 MeV and T2 =101 MeV), whereas
the minimum x2 per degree of freedom is 3.4 for a one-temperature fit with T = 58
MeV[Odyn88]. For comparison, the one-temperature fit (T = 49 MeV) to the data
is shown with the dashed line. Our calculations are represented by the histogram.
Calculation and data both show a clear deviation from the one temperature fit. The
data are in reasonable agreement with our calculation. A slight tendency of under-
predicting the higher energy pions of energy Bun 2 0.4 GeV is noticed. One of the
reasons is that a semiclassicalmomentum distribution has been used to initialize the
nucleons; therefore the calculation lacks the quantum high momentum components.

This is a problem our model shares with all the other semiclassical dynamical models.

In the lowest energy bin, we overpredict the data by a factor of 2-3, which is
due to the fact that the detailed balance of eq. 3.56-eq. 3.61 was derived assuming
that the baryon resonance has a zero energy width. As we will discuss in Chapter
7, the use of this detailed balance underpredicts the reabsorption of low energy A’s.
Using a new detailed balance which takes into account the finite width of the baryon
resonance, the low wnergy part of the pion spectrum can well reproduce that of the

experimental data.

 

(PE)"ldN/dE

 

 

 

10-1 .-.1...-I .\. 11%

0.2 0.4 0.6
Em (GeV)

 

Figure 5.8: Comparison between calculations (histogram) and the experimental data

of ref. [Odyn88] (plot symbols). The dashed line is the one-temperature fit to the
experimental data with temperature T = 49 MeV.

83

To show the change in apparent temperature as a function of the kinetic energy

of the pions, we introduce a local slope[Shur88, Baue91b]

n: ”(imp—155%) (5.4)

and plot it as a function of EM...

In Fig. 5.9 we perform such an analysis and compare our calculations (histgrams)
with the experimental data (circles). The errors bars were in both cases obtained
by taking forward and backward difference formulas to compute T; and using the
difference in the results as an indication for the errors. It is clear that data and
calculations are in good agreement within the error bars, and that they both show
a change in local slope not compatible with a one-temperature picture. In this fig-
ure, a one-temperature spectrum would show up as a straight horizontal line. For
comparison, we also show the local slope extracted from the best two.temperature fit

according to eq. 5.3 to the data (T1 =45 MeV, T2 = 101 MeV, and (41/242 = 5.0).

In Fig. 5.10, we perform another comparison between our calculation and the
experimental data for central collisions of Ar+KCl at a beam energy of E/ A = 1.8
GeV. This is the experimental data set in which the two-temperature‘structure of the
pion spectrum was first found [Broc84]. Again, the experimental data are displayed
by the round plot symbols, and. the solid histogram is our calculation. The dashed
curve is a one-temperature fit to the spectrum with T = 63 MeV.

As can be seen from the figure, the pion energy spectrum shows a deviation from
the simple exponential law for pion energies greater than 0.5 GeV. A fit with two
slope parameters (T1 = 58 MeV and T2 = 110 MeV) reproduces the whole spectrum.
Similar to the situation in La + La reactions shown in Fig. 5.8, experimental data

and model calculation agree quite well.

 

1:30 TYrTlI’TYWIYI'TIYUVUIVUYT

 

 

 

100

 

T1 (MeV)

 

 

 

I;
'

 

 

50

 

 

 

 

 

 

 

1
0 LL LlJLJI'lLILlAJlllIIAJlAL

O 100 200 300 400 500

 

Em (MeV)

Figure 5.9: Local slope T, as a function of the pion kinetic energy for the central
collision of La + La at E/ A = 1350 MeV. Circles are the experimental data and the
histogram is the model calculation. The solid line represents T; as extracted from the
two-temperature fit to the experimental data.

 

7 I ' r V r I Y I I’ r I v r t I 1' T t Y

131000 at. Ar+KCl -
'0 500 . E/A = 1.8 GeV -j
> - V. 1
m b ..

Q n i
g 50:- a... —
g '- \\:" A ‘
g 10 f— \\ j ‘1
E 5i- " l
cu - \ ll

“U

\

b . I ”I. 1'
1 L l A L I. J l L A A l l l l L J | I

0.2 0.4 0.6 0.8 1.0
E (GeV)

 

 

 

Figure 5.10: Comparison between calculations (histogram) and the experimental data
of ref. [Broc84] (plot symbols). the dashed line is the one-temperature fit to the
experimental data with temperature T=63 MeV.

86
It is well known that the cascade models with A resonances and their decays
predict a single exponential spectrum. In principle. the main differences between our

hadronic transport model and the cascade models are the inclusion of the mean field

for baryons and a more proper treatment of the Pauli blocking in the present model.

The agreement between our model calculations and the available experimental
data indicates that to correctly describe the experimental observables in heavy ion
collisions of 1-2 GeV/ nucleon beam energies, it is necessary to include the mesonic
degrees of freedom explicitly, while still keeping the baryon mean field. This is because
in this energy domain the long range nucleon-nucleon interactions are still sufficiently
significant that the particles are not free but moving in a varying mean field both in

space and time.

5.2 Systematics of pion spectra

From our previous discussions, we see that what is important for reproducing the
experimentally observed concave shape of the pion spectra is the correct description
of the effects of the reaction dynamics on pion production and absorption. Pions in
the high temperature component are mainly produced inthe. early. high-compression.
phase of the reaction, probably during the first one or two nucleon-nucleon colli-
sions per nucleon. On the other hand, low energy pions are mainly produced in the
late expansion phase of the reaction. Pion spectra therefore indeed carry interest-
ing information about the space-time dynamics of heavy ion collisions. More of this
information can be obtained by studying the dependence of the shape of the pion

spectra on the beam energy, mass and impact parameter.

Since the systematic experimental study of the concave shape of the pion spectra is

underway on SIS / GSI by the KaoS collaboration,[Oesc89] it is therefore interesting to

87

study these systematics based on our model calculations also. Moreover. a comparison
between the two systematic studies will further determine the mechanism that causes

the concave shape of the pion spectra and further test our model ingredients.

5.2.1 Energy dependence

We first study the energy dependence of the shape of the pion spectra in central
collisions of La+La. In experiments one usually measures the pion spectrum at around
90 degrees in the C.M. system in order to avoid the ‘corona effect’,[Broc84] so that
reliable information about the dynamics and properties of the hot and dense matter
in the participant region can be extracted. All the spectra presented in the following
are then calculated in the C.M. system at 90:1:30 degrees in accordance with the

experimental situation of the KaoS collaboration.[Oesc89]

Pion spectra in central collisions have been calculated for beam energies from 0.5
GeV/ nucleon to 2.1 GeV/ nucleon; typical ones are displayed in the upper part of Fig.
5.11. Pion spectra at beam energies below 0.7 GeV / nucleon can be well described by
a one-temperature Boltzmann distribution. This is because in this energy range the
nucleon-nucleoninelastic cross section issmall, only the first collision of. a nucleonwith
target rapidity and a nucleon with projectile rapidity can effectively produce pions.
For beam energies greater than 0.7 GeV/ nucleon, if one fits the lower energy part of
the spectrum with a single exponential distribution it is seen that the deviation of
the spectrum from the single exponential'distribution increases as the beam energy
increases up to around 1.5 GeV/ nucleon. The general feature observed here is in
agreement with the experimental findings.[Odyn88] However, the tendency is not so
obvious at beam energies above 1.5 GeV/ nucleon. As can be seen the slope of high
energy pions also increases as the beam energy increases. As we have discussed in the

last section, pions in this high “temperature” component are produced in the early

88

 

 

 

 

 

 

1010
108
2
\ 6
% 10
L 104
a 102
100
..l....l....l.4.nl-. .l....
0.1 0.2 0.3 0.4 0.5 0.6
Em (GeV)
02'.

 

, b=1fm .
05’-..l.4¥nl....l ..l.

 

 

 

0.5 1.0 1.5 2.0
Ebeam/A (GeV)

Figure 5.11: Upper figure: Calculated energy dependence of the pion spectra at 90i30
degrees in the center of mass frame for La+La reactions at impact parameter b = 1
fm. Lower figure: Energy dependence of the shape parameter R of the pion spectra
shown in the upper figure. The solid line represents the analytic scaling function of
equation 5.6.

89

high-compression phase of the reaction. The slope of the high-energy pions reflects
the amount of energy deposited in the participant region via baryon excitations, which

is monotonically increasing with beam energy.
5.2.2 Mass dependence

In Fig. 5.12, the upper part shows our predictions on the mass dependence of the pion
spectrum. All calculations are performed for symmetric systems at a beam energy of
1.5 GeV per nucleon and an impact parameter of 1 fm. For light systems, such as
C +C and Ne+ N e (not shown), the spectra can be well described by a one-temperature
Boltzmann distribution. The reason for this is that, for light systems the size of the
participant region is small which is compatible with the mean free path of the nucleon,
and therefore on average particles only suffer one collision during the course of the
reaction. For heavier systems from Ca+Ca to Tb+Tb, the situation is different. If
one fits the lower energy part of the spectrum with a single exponential distribution
one sees that the deviation of the spectrum from the single exponential distribution
increases as the mass of the system increases. This feature of the mass dependence
also agrees with the experimental results[Chas90]. Looking at the experimental pion
spectra shown in Figs. 5.8 and 5.10, the deviation of the spectrum from the single
exponential law in La+La is much larger than that in Ar+KCl, given the fact that
the beam energy per nucleon in the La+La reaction is 450 MeV smaller than in the
reaction of Ar+KCl. We also observed that the slope of high- energy pions is almost

independent of the mass of the colliding nuclei.

5.2.3 Impact parameter dependence

Little knowledge of the impact parameter dependence of the pion spectra is presently

available from experimental data. The KaoS collaboration has thus planned to study

90

 

(PE)‘1dN/dE

 

 

 

...l....l....l..4LL.. 11...,
0.1 0.2 0.3 0.4 0.5 0.6

 

l-OJ‘T'IIIr'I'Ir'Irr;
0.93 -5
m 0.8
0.7
0.6
0.5
0.4’

 

7' I

-9-

E/A==1.5 GeV -;
b=1fm -2

l L I I l A l I l 1 l L l l l T

100 200 300
AP +A,.

Figure 5.12: Upper figure: Calculated mass dependence of the pion spectra at 90 :1: 30
degrees in the center of mass frame for a beam energy of 1.5 GeV / nucleon and impact
parameter b = 1 fm. Lower figure: Mass dependence of the shape parameter R of the
pion spectra shown in the upper figure. The solid line represents the analytic scaling
function of equation 5.6.

VIVIUTV‘W'VIVVVVI

 

 

)-

 

91

 

 

 

 

 

 

 

 

Li]
”C
\
z
LU
La
m
n.
V
..l....l....l-...l.. .l'-.—i.-
0.1 0.2 0.3 0.4 0.5 0.6
Em (GeV)
05 ...-,-...,....,...-I....1..--j
0.7“— I -Z
0:: " 3

  
  

0.6
Ar+KCl

0-5 E/A=1.8 GeV (
0.4 ....l....l....l....l.... ....

0 1 2 3 4 5 6
b (fm)

 

Figure 5.13: Upper figure: Calculated impact parameter dependence of the pion
spectra at 90 :l: 30 degrees in the center of mass frame for Ar+KCl reactions at beam
energy of 1.8 GeV/ nucleon. Lower figure: Impact parameter dependence of the shape
parameter R of the pion spectra shown in the upper figure. The solid line represents
the analytic scaling function of equation 5.6.

92

this dependence. In particular, a change in the slope of high-energy pions with cen-
trality might indicate a thermal origin for these pions, while a similar slope would
favour a dynamical decay process. Our predictions of the impact parameter depen-
dence of the pion spectrum are shown in the upper part of Fig. 5.13. The reaction
of Ar+KCl at a beam energy of 1.8 GeV per nucleon has been chosen. It'can be
seen that the spectra are almost parallel to each other and show a concave shape
for impact parameters smaller than 3fm. For impact parameters larger than 5fm
the spectra show only one temperature component. This is, because at large impact
parameters too few particles are inside the collision zone, and pions are produced in

first collisions only.

In experiments the centrality of the reaction are usually measured in terms of the
charged particle multiplicity, in this case charged pions. In our model calculatiOns
there is the impact parameter and the multiplicity of charged pions in the final state.
It will therefore be possible to make a direct comparison between our calculations

and the coming experimental data from the KaoS collaboration.

5.3 Approximate analytic scaling function

To describe the shape of the pion spectrum quantitatively and further study its de-
pendence on beam energy, mass and impact parameter we perform a least square fit
to the spectrum with a two-temperature distribution function of eq. 5.3 and define
the shape parameter as

AITI

R = .
AITI + .4ng

 

(5.5)

With T1 we refer to the lower temperature in the two temperature fit. It should be

noted that this temperature is not the same as obtained by fitting the lower energy

93

part of the spectrum with a single exponential function. R represents approximately

the fraction of the pion yield from the first exponential[Odyn88].

The shape parameters as extracted from our computer calculations are plotted in
Figs. 5.11, 5.12, and 5.13 as functions of beam energy. mass and impact parameter,
respectively. The energy dependence of the shape parameter is rather flat. As a
function of the total mass of the system, R increases from about 0.5 to 0.8 when the
total masses of the system grows from 24 to 160. For heavier systems, R saturates
at around 0.8. For Ar+KC1 collisions at a beam energy of 1.8 GeV per nucleon,
the shape parameter is about 0.7 for impact parameters smaller than 2 fm; it then

decreases to about 0.4 at b = 5 fm.

In order to understand the scaling behaviour of the shape parameter R we may
attempt to formulate approximate scaling laws, based on our knowledge of the mech-
anism that causes the concave shape of the pion spectra. As we have discussed
previously, pions in the higher temperature component are mainly produced in the
early high-compression phase during the course of the reaction. To obtain an approx-
imate scaling function we assume that pions in the higher temperature component are
completely from first collisions in the early phase of the reaction. This approximation.
is valid for beam energies which are not too high. To contribute to the lower energy
component, at least one of the colliding nucleons has to have had at least one previous
collision to transport it into the ‘mid-rapidity source’. We use Poisson statistics for
the probability distribution of the number of nucleon-nucleon collisions. Under these

simplifying assumptions, the scaling function for the shape parameter is given by

R: 1‘6- “"e- . (5.6)

l-e“a

 

Here, the numerator represents the probablity to have had at least two collisions,

whereas the denominator is the probability for nucleons to have had at least one col-

94

 

 

La+La -I
b=1fm.

[ILLJLIALLAIJAALLAIIII

0.5 1.0 1.5 2.0 2.5 3.0

Em/A(Gev)
-..,--..]..-fifi..T...fi,:

§~ s/A-iscov —;

 

Allin]

 

OHNCJIP

 

 

LA 11

 

 

 

A 1" LaLa _.
5 5...-1-.+..1...44-..-1.-..)i
2.5 5.0 7.5 10.0 12.5

b(fm)

 

Y FT r r‘T rt V r TiT
1

EL 4
I E/A= 1.5601! -‘

E....I....1....I.-
100 200 300
Apt-AP

Figure 5.14: The systematics of the average number of nucleon-nucleon collisions
suffered by each participant nucleon

4
L1 AA

A

 

 

 

onwe oonwe
v 'o It:

95
lision. Here ‘5' is the average number of nucleon-nucleon collisions per nucleon[Baue88].

APAdUnJE) fo dxdy (3°00 dZIdZ2P:($2 + y2 + zf)l/2p,,,(2:2 + y2 + 2.3)"2

H = . . . . . 9
§fo dxdy [3°00 d:(p,,(.rz + y2 + 201/2 + pdl‘z + y2 + 22W?)

(5.7)

Here 0...,(E) is the energy dependent total nucleon-nucleon cross section. The inte-
grations over x and y are performed over the geometrical overlap area O. 2,, is a
correction factor resulting from the fact that the final-state phase space for the scat-
tering nucleons is partially Pauli forbidden. With use of geometrical considerations.

it can be approximated by

 

23—.1h23 —h '2
Ap=(1— PF 2 (PF )) 9 (58)

(PF + P513
with h = (pp — pb)0(pp — pb). Ad is another correction factor used to include properly
the effect of energy degradation and pion reabsorption. A constant of 0.6 has been
used in our calculations. The systematics of the collision number calculated from eq.

5.7 have been displayed in Fig. 5.14.

Even though we expect the above scaling function for the shape parameter to be
only an approximation, we can still compare it with the numerical values extracted
from the calculated spectra. The results obtained from Eq. (5.6) are displayed by
the solid lines in the lower parts in Figs. 9, 10 and 11. As can be seen, the qualitative
features of the numerical calculations are reproduced, and the gross features of the
shape parameter and therefore the systematics of pion spectra in relativistic heavy
ion collisions at beam energies around 1 GeV/ nucleon can be understood in terms of

the simple scaling arguments presented here.

In summary of this chapter, an application of the hadronic model to the study of
the pion spectrum shows that the shape of the pion spectrum reflects the effect of the
reaction dynamics on pions, the mechanism for the concave shape of the pion spectra

in central heavy ion collisions is found to be due to the different contributions of

96

the Delta resonances produced early and late during the course of the reaction. The
available experimental pion spectra have been reproduced. The systematic study of
the shape of the pion spectra indicates that the concavity of the pion spectra increases
with both the beam energy and the mass of the colliding nuclei, and decrease with

impact parameter.

An approximate scaling function for the shape parameter of the pion spectra has
been derived from a Glauber-type multiple collision model to understand the system-
atics of the pion spectra. Upcoming experimental results from the KaoS collaboration
on the systematics of the concave shape of the pion spectra are expected to further

test our model predictions.

We conclude from our study in this chapter that in heavy ion collisions of beam
energies around 1 GeV / nucleon it is necessary to treat mesonic degrees of freedom
explicitly, without neglecting the nuclear mean field. This is important to the under-
standing of the observables sensitive to the dynamical degrees of freedom in relativistic
heavy ion collisions and to reliably infer properties of hot and dense matter produced

in these collisions.

Chapter 6

Preferential emission of pions

The existence of a collective flow signature among the final state baryons of rela-
tivistic heavy ion collisions at beam energies around 1 GeV/ nucleon has been firmly
established by the sphericity analysis[Gyul82, Dani83] and the in-plane transverse
momentum analysis[Dani85] of the rich harvest of data from the Berkeley plastic ball
detector. These methods have revealed the collective motion following the decompres-
sion of the hot and dense nuclear matter in both the reaction plane and perpendicular
to the reaction plane. In the reaction plane a sideward deflectionof spectator particles,
the so called “bounce-off”, and an azimuthally asymmetric emission of participant
particles, the so called “side-splash” have been observed[St5c80, Buch83, Gust84}. A
collective flow of nucleons perpendicular to the reaction plane, the so called “squeeze-

out”, has also been found[Gutb89].

For ease of the following discussions, we discuss the basic idea of the transverse
momentum analysis here in more detailed. The idea is to investigate whether a
single particle of the event knows something about the whole event, i.e. the question
whether the particles are collectively correlated. Danielewicz and Odyniec proposed
to construct the reaction plane form the beam direction and the vector

Q; = Ewa’fili (6-11

i¢j
97

98

determined from the detected particles in each event of mass symmetric nucleus-
nucleus collisions. pl,- is the momentum component of the particle i perpendicular to
the beam direction. The weight w, is taken as 1 for y,- > ycm and —1 for y.- < ycm,
where y...m is the center of mass rapidity of the nucleus-nucleus collision and y,- is the

rapidity of the particle i defined as

 

E P.
' = ln Eipzi (6.2)
The transverse momentum of the particle j in each event is defined as
p... = {0. $1.000}. (6.3)

The average transverse momentum (p,) is obtained by averaging over all events.
The average transverse momentum analysis, namely plotting the average transverse
momentum (p...) vs. the rapidity y, can now be performed. A positive (negative)
average transverse momentum for positive (negative) rapidity indicates that particles
flow collectively. Fig. 6.1 shows as an example the average transverse momentum in
the reaction plane, (p,), as a function of the rapidity in the laboratory frame for La
+ La collisions at a beam energy of 800 MeV per nucleon. The characteristic S-shape

of the resulting curves, is a clear signature of collective nuclear matter flow.

Due to the small mass of pions compared to that of baryons, it has been pointed
out that the pions might serve as a good probe of any hydrodynamical flow[Goss89].
Moreover, as pions are mainly coming from the decay of A resonances in the relativis-
tic heavy ion collisions, the remnant of the collective flow carried by A resonances

might be seen in the final state pions.

Looking for flow signatures among the final state pions, several groups[Kean86,
Dani88, Goss89] have studied the transverse momentum distribution in the reaction

plane for pions. One of the most striking results from the DIOGEN E collaboration[GossS9]

99

 

 

 

 

 

400»-..,-...l....,.-..,...eT...
a 3 3
\ - <
% 200~ , _
* ¢
2 ~ ¢
v ,_
r
A o *
>~ .
v L 4
of . t .
v L ,
—200- j
t +
—4OOfL l L1.l..n.l. Ll.n.l...‘

0.00 0.25 0.50 0.75 1.00

Y/YB

Figure 6.1: Transverse momentum analysis for La + La collisions at a beam energy
of 800 MeV per nucleon. The data are from ref.[Dani88]

100

is that the in-plane transverse momentum of pions is always positive even for back-

ward rapidities, for the asymmetric (Ne or Ar) + ( Nb or Pb) systems.

However, the Intra—Nuclear-Cascade model predicts values compatible with zero
over the whole range of rapidity[(305589]. The Quantum-Molecular-Dynamics model
calculation of the pion transverse momentum distribution[Hart88] indicates that the
introduction of the mean field describes some of the experimental effect. but the
model predicts less asymmetry than observed experimentally. Therefore, the ques-
tion whether the experimentally observed preferential emission of pions away from
the interaction zone towards the projectile side in the asymmetric nucleus-nucleus
collisions is due to the collective flow of pions or due to the shadowing effect of the

heavier target spectator has not been resolved.

In this chapter, we report on the results of a study about the pion transverse
momentum distribution in the reaction plane by using of our hadronic transport
model. The preferential emission of pions towards the projectile side in the transverse
direction in the reaction Ne + Pb at a beam energy of 800 MeV/ nucleon is found
to be due to the stronger absorption of pions by the heavier target spectator. The
calculated transverse momentum distributionof pions in the reaction plane agrees

with that of the experimental data.

6.1 The mechanism for the preferential emission
of pions

In our model calculation, the reaction plane is known a priori and we refer this
plane as the true reaction plane in the following discussions. The reaction plane
estimated from the observed charged particles will be referred to as the estimated

reaction plane. To study the mechanism for the preferential emission of pions in the

101

transverse direction, we first study the pion transverse momentum distribution in the

true reaction plane without using the experimental detector filter.

In Fig. 6.2, the rapidity distribution and the transverse momentum distribution
(scaled with the mass of the pion) in the true reaction plane for 1r+ from the reaction
Ne + Pb at a beam energy of 800 MeV/nucleon are shown with the solid histograms.
The calculation was done at an impact parameter of 3 fm which coincides with the
condition of the experimental data[G05589]. It is seen that the rapidity distribution
peaks near the center of mass rapidity of 0.1 unit and is asymmetric about the center of
mass rapidity. The transverse momentum of pions in the true reaction plane is positive
even for negative rapidities, which reflects the fact that the pions are preferentially

emitted towards one side of the participant region in the transverse direction.

As discussed earlier, in the baryon transverse momentum analysis, the S shaped
distribution in the reaction plane with the average in-plane transverse momentum (p,)
positive (negative) for positive (negative) rapidities in the C.M. system for repulsive
interactions has been taken as a signature of the collective baryon flow[Dani85], and
S-shaped distributions of opposite sign are also found in the beam energy region
below. 100 MeV/ nucleon. where attractive interactions dominate[Krof89]. Both the
flow parameter and the average in-plane transverse momentum have been found to

be sensitive to the nuclear equation of state[Moli85, Gale87a, Bert88b].

Is the nonzero in-plane transverse momentum of pions a remnant of the baryon
collective flow carried by A resonances ? To answer this question we have studied the
dependence of the pion transverse momentum distribution on the nuclear equation
of state. Within statistical fluctuations results from the calculations done with a stiff
equation of state corresponding to a nuclear matter compressibility of K = 380 MeV
and with a soft equation of state corresponding to K = 210 MeV are the same. This

indicates that the effect of baryon collective flow on the pion transverse momentum

102

distribution is negligible and the origin of the positive in-plane transverse momentum

of pions is not predominantly the remnant of the A flow.

It has been speculated that the mechanism that causes the positive pion transverse
momentum in the reaction plane might be due to the target shadowing effect[Kean86,
GossS9], and this idea has been demonstrated in a phenomenological model assuming

that pions have a constant mean free path in nuclear matter[Goss89].

In the present dynamical model calculation. pions are reabsorbed through a two-

step mechanism, namely,

1r+.’V->A.

N+A~N+N. we

The cross section for these processes in each isospin channel have been discussed in

detail in Chapter 3.

To study the effect of the pion reabsorption and rescattering and therefore check
the shadowing effect in forming the positive in-plane transverse momentum of pions,
we calculated the pion transverse momentum distribution and the rapidity distribu-
tion by turning off thepion reabsorptionchannels (6.4)and the A rescattering channel-
N + A —v N + A. Results of this calculation are shown with the dashed histograms
in Fig. 6.2. (For ease of comparison, we have normalized the total production cross
section of these primordial pions to the one for the pious produced including the
reabsorption and rescattering channels.) In this case the in-plane transverse momen-
tum is zero within statistical error bars and the rapidity distribution is symmetric
about half-beam rapidity of 0.6 unit, which reflects the fact that the pions are emitted

isotropically in the center of mass frame of two colliding nucleons.

Comparing the rapidity distributions obtained with and without the reabsorp-

tion and rescattering channels ( solid histogram and dashed histogram ), we first

103

 

1.59-
1.05—
055—

0,5-.,....,....,....

 

 

 

 

 

dN,,+/dY

 

 

 

 

 

 

 

 

 

 

 

 

\ 0.2 I- -;
N '_’ .
0H 0.0 #Jq—i-_'_f-L’-A+JP+JP‘:=#*‘1

P'l'TIf
.l

v
—0.2

 

 

 

Ylab.

Figure 6.2: Upper figure: 1r+ rapidity distribution calculated with ( solid histogram
) and without ( dashed histogram ) the pion reabsorption channels for the reaction
of Ne + Pb. Lower figure: Calculated 1r+ transverse momentum distributions in the
true reaction plane with ( solid histogram ) and without ( dashed histogram ) the

pion reabsorption channels.

104

notice that pions with positive rapidities emitted towards the target side are more
reabsorbed compared to pions with negative rapidities emitted towards the projectile
side. This reflects the preferential emission of pions in the longitudinal direction. as
one would expect for the highly mass-asymmetric system. Second, the reabsorption
and reemission of pions as well as the rescattering of A’s help to thermalize the sys-
tem. This effect appears as the change of the peak of the rapidity distribution from
the mid-rapidity to the center of mass rapidity as the reabsorption, reemission and

the rescattering channels are turned on.

From the results of these calculations, it is clear that the positive in-plane trans-
verse momentum of pions in the asymmetric nucleus-nucleus collisions is due to the
stronger reabsorption of pions by the heavier target and therefore the speculation

about the shadowing effect of the target is confirmed.

In symmetric nucleus-nucleus collisions the spectators are the same on both sides
of the interaction zone. It is interesting to study the pion transverse momentum
distribution and the rapidity distribution in symmetric systems to test the sensitivity
of the model to the geometry of the pion absorbing matter. In Fig. 6.3, the rapidity
distribution and the transverse momentum distribution are shown for. pions from
central collisions of La + La at a beam energy of 800 MeV / nucleon. It is seen that both
the rapidity and the in-plane transverse momentum distributions are symmetric about
the center of mass rapidity of 0.6 unit and the transverse momentum distribution has

a typical S shape.

6.2 Comparison to the experimental data

In order to compare the model predictions and the experimental data of the pion

transverse momentum distribution. we have made a full simulation of the central

105

 

IrrlvTrv rrrv rvrrlrrvtlrrr

+125

'V'V'T
1
11111

 

114A]

 

-()-

 

 

 

 

 

 

 

 

'U'VIYVW‘I'V"

 

dN,,+/dY
O H N (.0 4> U'I
,.
l

 

 

 

 

 

 

 

 

 

 

0,2:....,....,....,....,..-.l.,.-:
(a) 0.1:— .f
E, 0.0: I‘m
_0.2L...1-...1.-.-i..-..1-...1... i

-1.0—0.5 0.0 0.5 1.0 1.5 2.0
Ylab.

Figure 6.3: Rapidity distribution and transverse momentum distribution calculated
for La + La reaction at E/ A = 800 MeV and the impact parameter of 1 fm.

106

Pictorial Drift Chamber (PDC) acceptance of the DIOGENE collaboration[Goss89].
The PDC detector covers the polar angle 0 from 20 to 132 degrees. Due to the
low energy limit of the detector, only particles satisfying the following relations are

accounted. For pions

pj/m > 0.66 + 0.77y, y < 0; (6,5)

pl/m > 0.66 — 0.63y , y > 0. (6.6)
For baryons

pl/m > 0.36 + 0.72y , y < 0; (6,7)

pi/m > 0.36 — 0.8y , y > 0. (6.8)

In the same way as in the experimental data analysis[GossSQ], we estimate the reaction
plane for each event from the beam direction and the vector

Q=Zwm we

#1'

determined from the detected protons. Here the weight are w,- = y,- — 37, and g is the
average rapidity of the detected protons. This weight is different from the one that
was originally proposed for symmetric systems since the center of mass rapidity of the
participant system is not known a priori in each event for asymmetric nucleus-nucleus

collisions.

In Fig. 6.4, we perform a comparison between the experimental data and the model
calculations for the Ne + Pb reaction. The experimental data are represented by the
round plot symbols. The solid histograms are the model calculations, the error bars in

the model calculations are statistical in nature, since we solve the coupled transport

107

 

OB:-...,....Tfi.e.,..-.j
0.6:- * d
0.4

0.2
00- ...1.l_._..1....|.-.-—-.1.1

0.6 l- I I I I I I I I I I I I I I I I I r
)-
b
" . q
'— —(
0.4 . .
' d
h 1

 

 

 

 

 

 

 

 

cm". /dY

 

 

 

 

 

 

 

 

 

0.0 . ' t

" 4

h 'd

" 1

—02— I -
o I-

1 l 1 A I 1 L n n 1 A n 1 1

< Px/m >

O

N

l
'6'

'6-

O

-6-

.1

 

 

 

Figure 6.4: Upper figure: calculated 1r”r rapidity distribution after using the detector
filter cut for the Ne + Pb reaction at E/ A = 800 MeV. Lower figure: Comparison be-
tween the experimental pion transverse momentum distribution ( round plot symbols
)and the model calculation ( histogram ) for the same reaction.

108

equations for the hadronic matter with a Monte Carlo integration technique. The
experimental data are in reasonable agreement with our model predictions. To show
the effect of the detector filter cut, the rapidity distribution of the detected 1r"”s in
the model calculation has been shown in the upper part of Fig. 6.4. Since the cascade
model did not reproduce the preferential emission of pions, it has been conjectured
that in-medium effects and pion production channels involving more than two nu-
cleons could be important in the energy range studied here[Goss89]. However, our
calculations indicates that it is not necessary to introduce additional medium effects
and many-particle processes beyond the nuclear mean field and the Pauli exclusion
principle for final state nucleons to understand the phenomenon of the preferential

emission of pions.

In summary of this chapter, we performed hadronic transport model calculations
of the pion rapidity distribution and the in-plane transverse momentum distribution.
We discussed the effects of the target shadowing and the A flow in forming the positive
in-plane transverse momentum of pions in asymmetric nucleus-nucleus collisions. We
found that the mechanism for the preferential emission of pions from the interaction
zone towards the projectile side in the transverse direction is due to the stronger
reabsorption of pions by the heavier target. The model prediction of the pion in-

plane transverse momentum distribution agrees with the experimental data.

Chapter 7

Summary and Outlook

In this work we have developed a new hadronic transport model for relativistic heavy
ion collisions at beam energies around 1 GeV/ nucleon by deriving and solving numer-
ically a coupled set of transport equations for the phase space distribution functions

of nucleons, Delta resonances and pions.

Starting from an effective hadronic Lagrangian density with minimal couplings
between baryons and mesons, we first derived coupled equations of motion for the
density matrices of nucleons, Delta resonances, and pion mesons as well as for the
pion-baryon interaction vertex function. By truncating at the level of two-body corre-
lations a closed set of equations of motion‘for‘the one'body density matrix is obtained.
A subsequent Wigner transformation then leads to a tractable set of relativistic trans-
port equations for interacting nucleons, Delta resonances and pions. The transport

equations are then solved numerically with the test particle method.

The model aims at formulating a framework for the theoretical understanding of
the nuclear physics phenomena in relativistic heavy ion collisions. We first applied
our model to study the dynamics of pion production and the pion multiplicity. The
experimental excitation function of pion multiplicity was reproduced. The application

of the model to the study of pion spectra reveals that the mechanism that causes the

109

110

concave shape of the pion spectra is due to the different contributions of the delta
resonance produced during the early and the late stages of the heavy ion collision
and due to the energy dependence of the pion and delta absorption cross sections.
The dependence of the shape of the pion spectra on the beam energy, the target and
projectile mass, and the impact parameter has also been studied. An apprOximate
scaling function for the shape parameter of the pion spectra is predicted. Another
new phenomenon that the model is able to explain is the preferential emission of
pions in asymmetric nucleus-nucleus collisions. We have found that the preferential
emission of pions away from the interaction zone towards the projectile side in the
transverse direction and longitudinal direction is due to the stronger pion absorption

by the heavier target spectator.

The success of the model in reproducing different experimental data sets for tOtal
pion excitation functions, pion kinetic energy spectra, the two-temperature appear-
ance of pion spectra, and preferential emission of pions in asymmetric nucleus-nucleus
collisions indicates that our model is able to describe most features of pion produc-
tion physics in relativistic heavy ion collisions. This supports the conclusion that the

approximations entering our model should be approximately valid.

Our transport equations for baryons contain a vector field and a scalar field, which
are momentum dependent. As a first step of the model, only the zeroth component
of the vector field has been taken into account, which has been parameterized by a
density dependent functional. Further improvement of the model should incorporate

the momentum dependent vector and scalar potentials.

Our transport equations include in principle the possibility for a changed disper-
sion relation for pions in nuclear matter. It was first pointed out by .I. Kapusta and
C. Gale that the pion dispersion in hot and dense nuclear matter can be studied

by looking at the dilepton spectra in relativistic heavy ion collisions[Gale87b], since

111

W+

Ir’ annihilation is the main source for the production of dileptons with large invari-
ant masses. Interesting phenomena have been found in the dilepton spectra[Roch88]
in BEVALAC heavy ion collisions. Conventional transport models[Xion90a, Wolf90]
with the free space dispersion relation for pions seems unable to completely under-
stand the observed features of the dilepton spectra. Extension of our hadronic trans-

port model to include the in-medium dispersion relation for pions would be useful for

the study of dilepton physics.

Further extensions to the study of two-pion correlations and kaon production are

presently in the planning stages.

Appendix A

o o b
Derivatlon of I,” and If):r

To calculate If, and If”, we need to use the explicit expressions for 0"(1), 0,,(3),

and 0(1), i.e. eqs. (2.24), (2.25) and (2.64). The explicit expressions for p‘(:c:r’) is

 

p(m') = tb'(x’)w(r) . (A 1)
where

(Mr) - .2. (53%))1/2 auger-z , (A 2)
and

The explicit expression for p,(x:r’) is

p,(:rz’) = 1r(a:’)- «(2) (A.4)
where
_ 1 1/2 than-1541:): t —ik-z+iE.(k)t
1r(2)- 2L, ) [b.te + b,,,e ] , (A.5)
«k "E'(k)

where the creation and annihilation operators bf”, and b“. are isovectors. In the above

expansions, only the positive energy components are included.

112

113

Before proceeding to calculate the collision terms due to pion-baryon collisions.

we list the approximations to be used in the following

(“L'p'aapai'lprlamml ~ for”) )fai(pl) 601(31’6pp’601c215p1p'l (A0)

+ fa’ip’)“ " faipllfsaajfsppifsa’mép’p) 9

<03...) z imam-t . (11.7)
(0.1.0:...) z (1 + f«(k))5n'5kk' , (A.8)
(brkbr’k’l = (birkbir’k’) = (brkl = (bit) = 0 - (At-9)

Now we are ready to calculate the collision terms, first for 13,,(2p). The collision

operator can be written as
i:,,(.n') = 1‘3; +1}, +1}, — i5}, - if; — 1‘53, , (A.10)

where I L and I R are Hermitian conjugate of each other. The zero-order terms in

1r(.r) can be expressed as

ism)

‘ fl

= —«b(zz')6(§ (z) + E.(x)- 3 (500.1%) s ‘(I)a(§—xm(n')

= 11' Z : [uLp’flpj- p)'uap] [ua(p;l(p’l- Pluaipil

a’p’.” 0119', .am

 

l MgMg'Mgi M5;
5.0»: - p1)[5;(p)E;.(¢>E;.1p.)E;, (Pi)
615:0») + 3.0»; — p.) - 8:10))

l i ip’z'-ipz+ip' z-ip) 1:
aa'P'aaPaa'lp'laalPle l ° (A-lll

 

11/2

114

The expectation value of this quantity is

 

If; 11"
15“,,(21') = (1%,,(11) —7r— ZE . (11;. .g(p —p')ua )
cup ap EO( ME 0(pl) p p

 

l
(u “apylP- P'H)uap)m
6(E;(p) + E«(p - p’) - 33.4103)
fa'(p')(1 - fa(p))e"’"""’ (A12)
In phase space it has the form
L -7r My”; unit} —'u -u 11 —' 1.
10.4171?) — g E 01(1) )E" a(p,)( cpl—(P Pll op) ( cpl—(P Plluap)
l . I
m5(EM@)+E(P)-501(P))
5(P -P")fa'(P)(1- fa(P)) - (A-13)

To go further, we notice that the interaction matrix 11 (eq.2.64) contains differen-
tial operator 8,. in it’s off-diagonal elements, namely the N H A transition matrix
elements. Correspondingly, it has terms linear in p and terms independent of p in
momentum space. For terms linear in p, 31(p — p’) and therefore 16;, must be van-
ish in accordance with 6(p - p’). Since E;(p) + E,(0) — E;.(p) ¢ 0 with a = N
orA, terms independent of p, namely (uA:,:|fi(l)|uA,) and (qu,:|fl(1)|u~p) also van-
ish in accordance with 6(E;(p) + E,(0) — E;.(p)). We then have 16;,(xp) = 0. and
1&(zp) = 0. Because of (b,).) and (03”,) are zero, the linear terms in w-field vanish

identically, If,(:rp) = [3,(zp) = 0. Therefore, only bilinear terms in «(:r) contribute

to 13,. Formally
if. =13". — 1'5. . (A-l4)
where

11.1”)

= —%[U"(1)—U”()II)]2((I51')+E(17) 32(6))

dug-9pm) (Emu-3 :(fimm)
)+

= —%[U"(1)—U"(1’)2]6(iz

'5 . _ . 5 .
my») pm' ) (5422)) 3 -2(5;)p(rr)

,, M; M;,M;,1 M;1 1’ 2
_ gt: 2:, Z (4Er(k)Er'(kilE;(P)E;’(p’)E;,(PllE;;(#1))

k r’k’ ap. a ”p mp; 4:11,}?l

 

6(E;<p) + £24k) — E;.(p'))a;'piaapa;;,;am

‘- I l_l' I I _ A ’1
1

[(bflkeikx-iEdk)! + bik€-ikx+igw(k)t)

“we“ "”5"“ " + bltie"""*‘5~"""‘) - (21;.ka _ k')i2(P)2uap)
+ (ui'p'“(pl)21£(0)uap)(barkeikxl—igdk)‘+ bike-ikr’+i5'(k)t)
(bf’k’eik’zi-iEudk’)‘ + b:'kle-ik'r'+ig*,(kl)t)

+ 2(bwkeik1-iEdk)! + bike-ikxi-iEflkh)
(b,.,.e‘*’=’-‘E~“")‘ + b;,k.e-“=’=’-‘E«<*’)‘)(u;,,,a(p')g(k)a(p)ua,)] , (A.15)
and analogously for 12’}... The expectation value of [2L1r is then

125(13') = (21135)) (A-16)

 

MGM; 1 . ,
= é‘z Z .280: )E;(p)E;:(p’)E3[(p’-p)(u;’g(p ‘P)"°"')

«I: ap.a"p

5(ES(P) + E,(Ic) — E;,(p’))fa,(p')(1 _ fa(p))eip'(z'-r)
(Wk) + (1 + f«(k))][u2',:(fi(p’)’a(0) + fi(0)fi(p)2)ua,]

+ ‘2lf«(k)e“"""" + (1 + f«(k))€"‘“”"l(“Lipvfi(P’)fl(k)fi(P)uap)) -

116

The baryon component of the Wigner transformation of 1.21;,(11’) is therefore
1 I --1' -
1214,0131?) = W/Tr(b)lf,(11)e prdr (.A.l7)

_ : 1115.111; 1 H .. ,, — I 1
— 8 2 ;” QE,(k)E;(p”)E;.(p’) Eflp” __ I") (“an ll“) P)lua p)

rkapp

([f«(k) + (1 + f«(k))lfa'(P')(1 - fb(P”))

-5(EJ(P") + E«(k) - E;1(P'))5(Pl- P) - (ua'p'|fi(P')’fi.(0) + i(0)fi(p)’luap)
+2lf1r(k)fa’(p’)(l - f1(P”))<5(P' + k - P)5(E§(P") - 511(k) - E;I(P'))
+(1+ fN(k))fa’(p’)(1 - f1(p”))6(p’ - k - P)5(E§(P") + 131(k) - E;'(P'))l

'(ua’p’lfi(P,)l(kla(P)luap)) -

 

This expression for the collision terms can be simplified, as we shall show. As discussed
previously 1n the calculation of [5“,(1p), the terms containing (11.. ., IMO) )Iuap) vanish.
Moreover, 31(11) consists of terms linear in k and terms independent of 1:. For the first
kind, Mk = O) = 0. For the second kind, 324(k) = 11(1). We also have (uNp:|&(p’)2_11(l)+
fi(1)fi(p)2|uAp) = 0. Therefore only the “diagonal” terms survive, (uap:|fi(p’)2fi(l) +
fi(1)1‘1(p)2|uap) 74 0, with a=N or A. With the on-shell approximation, we also have
p” = p, and 6(E;(p) + E,(Ic) — ;.(p’))6(p’ — p) = 0. Therefore the terms containing
11(0) vanish. With the above conditions and approximations, the collision terms for
baryons due to pion-baryon interactions can be simplified and, furthermore, they can

be separated into gain terms and loss terms,

 

[5.311(11’) (A.18)
(“a'p'1&(P’)fi(k)fi(P)luap) ' (“apllukllua'p'l
“8:41;...511M15; G11!) 5:11)

[fx(k)fa’(p,)(l - f1(P))5(EJ (P) - 151(k) - ESa(P’))5(P' + k - P)

+ (1+ f«(k))fa1(P’)(1 - f1(P))5(EE(P) + E«(k) - E;'(P'))5(P' - k - 10)]

117

 

and
157131?) (A.19)
__ .1: MgAlg’ (“0"13'l‘u'0J )E(k) “(p)luap> ' (uapll‘uknua’p’)
3 210%... E;(p)E;:(p’) 13;:(1)

[Mk )(1 — fa(p ’))f1(p)6(E§(p) + 1511(1) - E;'(p’))6(p’ - k - p)
+ (1+ f»(k))(1 -fa(p ’11))1f1()6(E 'p( )- 131(k) - E;r(p'))5(p’+k -p)] -

Finally, let us calculate 1,3,. Its operator form is

 

 

 

 

i:«(‘rx)_ _ igai11(” I’)— il:u(xx’) 7 (A°20)
where
« _ 1 - ‘ , I
[5814“) -' 2E,(I)y-(I)F(I,x ,l‘) 1 (A21)
and
‘r I _ ‘ I I . I l
[1055(II) - r(1 1311: )y-(‘r )2E.',r(l") ' (A.22)
In more detail
1r 1 - ‘ “ I ‘
1.....1n 1 = {6943)1111111111. 5.151— 515.11 111.23)
10*1x-1— 0*1x.112p1x-x.1m(1,). 11x'111u'1 .

where 1* = 1:1:e(e -> 0) means that f1(1-), fi(1+), U'(1-) and U'(1+) should operate
on fi(1-1+). After having operated, 1.. should assume the value 1. Since U'(1)
contains the pion field «(1), we encounter the difficulty of calculating the expectation
value («(1) - «(1)) for the gain term and («(1’) - «(1’)) for the loss term. Since we

know that

118
—» /—1 f(k)d3k
E,(Ic) "
1
-+ —-—b1'bl'b1 bl
$25410“ ,4 1+ mm 14 1+ 1:111

_, 22E“) $111M )+bL( 1b,,(x )+bk(¢’)bk(r1+b£(z’1bl(2=1>
k

«(
Z-i—jfik) , (A24)
I: 134k)

as 1’ -> 1, there is an uncertainty for the order of operators bk(1) and bl(1). Since

the gain term for the pion due to pion-baryon interactions is related to the pion

production process and the loss term is related to the pion reabsorption process, to

eliminate the above uncertainty one should use

 

((bk(x1bz(x1+ bk(x1bz(x1+ 12421121421 + bumble»

 

1
_ 2:3
and

(“(I') - 3(5))

2— 2E ( k) <b*(x1bk( 1+ 61(x1bk(x1+ 11211421 + human»

2

k E,(k

 

(A.26)

Apart from the above exceptions, the calculation of 1:5,, is similar to that of IL.

The expectation value of the gain term Igunls

[gain(xx’) = (181”)
1 M51";
$2 2 E E (k1E;(p1E;.(r1

ark apa "p alma’p

 

119

 

eI’II(z-y)+i(p’ -p)y I I
-d/ I: dy 211,135 6(801121— Ego) -p11— Ea(p11

eiq'(r’-y’)+i(p1-pn )y'

(u ;Hp&(q)fl(pl +P)::OP)/dqidy, (27,.)3E3( qI)
(“LIP 113((1 )Uaim)(1 +ftlk l) fa’P( )(1_f0(p))60'°1600169’P16PP’1

1k M“ , ,
__. 11.62 2 E(E;(p’)OE;(p)6(Ea'(p)-E1r(p—p)“130(1)”

Irk apa’ p’

 

 

1 ,
mm + fx(k))faI(p)(1- fa(P))

(uLpImp — mm + p’12uap1 - (amp - #121111. (A271

ei(p’-P)(r-r’)

The gain term in phase space is then 4

1* (1k) = [Tr1;m(u)e-‘P"dr15,.(k)

 

gain
1r MJM;
_ 160,21, 551p)E;I(p’)

(ua'p'lg(k)fi(p + p,)2luap) ' (“aplfi(k)lua’p’)

 

Ewe)
5(E;I(P') - E,(k) - Ea(P))5(P’ - P - ’3)
(1+ f«(k))faI(P')(1- fa(P)) - (A28)

The loss term can be found analogously to be

[lirmlxkl

= [Tr1h(11')c"p‘rdrE,(k)
1r MéMg.
“ 16 Zi.1=«‘,:.(1a113,:1(zr1

crap

 

(ua’P'lg-(k)a(p + p’)2luap)' (“aplylk )‘lua ”p ’>
531(k)

5(E;I(P') - EIUC) - Ea(P))5(P’ - P - 5)

f1r(k)fa(P)(l - faI(P')) . (A29)

120
In the continuous limit, changing the summation over momentum into integra-

tions, we obtain the expressions for gain and Il’fm as in eqs. (2.101) and (2.102).

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