Coalescence of deuterons in relativistic heavy ion collisions
J. L. Nagle and B. S. Kumar
A.W. Wright Nuclear Structure Laboratory, Yale University, New Haven, Connecticut 06520
D. Kusnezov
Physics Department, Yale University, New Haven, Connecticut 06520
H. Sorge and R. Mattiello
Institut fu ¨r Theoretische Physik, D-60054 Frankfurt am Main 11, Germany
Received 18 July 1995
We explore the process of coalescence as a way to create deuterons, antideuterons, and other composite
particles in proton+nucleus and nucleus+nucleus collisions. We discuss several approaches to coalescence
calculations, and describe in detail some work using an extension to the transport theoretical approach RQMD.
We compare our calculations to measured yields of composite particles produced in proton+nucleus and
nucleus+nucleus collisions.
PACS numbers: 25.75.Dw, 02.70.Ns, 24.10.Jv
I. INTRODUCTION
The creation of novel states of nuclear and possibly quark
matter using relativistic heavy ion collisions has interested
many scientists over the past several years 1. In order to
understand the properties of the collision region, it is impera-
tive that experiments be able to give us information on its
lifetime and thermodynamic attributes such as temperature,
volume, density, and entropy. In particular, a transition of
nuclear matter to quark matter is expected to result in a
strongly interacting region that lives for a long time, and thus
expands to a large volume with large concomitant entropy
production.
We discuss below some ways by which one can calculate
the abundances of deuterons produced in heavy ion colli-
sions via the mechanism of coalescence. Our initial interest
in such calculations was to investigate the dimensions of the
collision volume at freeze-out 2. Our success in describing
the yields of deuterons prompted us to investigate how well
our techniques applied to studies of the antideuteron 3. Our
work has resulted in an improved understanding of some of
the subtleties in doing coalescence calculations. A study of
the production of light nuclei should enable us to obtain
information that constrains the temperature, baryon density,
entropy, and lifetime of the collision volume at ‘‘freeze out.’’
We attempt to elucidate some of our ideas in this paper.
We discuss the shortcomings of the simple coalescence
and thermal models, and how they can be remedied through
use of a coalescence extension to a transport model. We will
investigate the effects of source expansion and hydrody-
namic flow on the yields of composite particles. We compare
the predictions of phenomenological and wave function ap-
proaches to coalescence calculations and compare our results
with data for light and heavy colliding systems.
II. SIMPLE COALESCENCE MODEL
In 1963, Butler and Pearson developed a model for deu-
teron formation in proton-nucleus collisions 4. According
to them, ‘‘the proton-neutron pair interacts with the static
nuclear optical potential which, together with the usual p -n
strong force, allows them to bind together to form a deu-
teron.’’ Their calculation used second-order perturbation
theory to obtain a relation between the density of deuterons
in momentum space and the density of protons and neutrons
in momentum space. The key result is that, on account of
simple momentum phase space considerations, the deuteron
density in momentum space, d
3
N
d
/ dK
3
, is proportional to
the proton density in momentum space, d
3
N
p
/ dk
3
, times the
neutron density in momentum space, d
3
N
n
/ dk
3
, at equal
momentum per nucleon ( K =2 k ), and can be expressed as
d
3
N
d
dK
3
=B
2
d
3
N
p
dk
3
d
3
N
n
dk
3
. 1
Since many experiments measure protons but not neuterons,
it is useful to rewrite this equation assuming the neutron and
proton densities to be identical:
d
3
N
d
dK
3
=B
2
d
3
N
p
dk
3
2
, 2
where
B
2
=| V
0
|
2
1 +
m
2
k
2
J
R . 3
Here m is the nucleon mass,
2
/ m =2.225 MeV is the bind-
ing energy of the deuteron, | V
0
| is the depth of the optical
potential, and J ( R ) is a dimensionless function depending
on the optical potential of the target nucleus. Schwarzchild
and Zupanc ˇic extended this phase space relation to describe
the production of various light nuclei in nucleus-nucleus col-
lisions 5. However, the constant coefficient B
A
was no
longer thought to represent an admixture of the binding en-
ergy of the deuteron and the nuclear optical potential of the
target nucleus. For more violent nucleus-nucleus collisions
PHYSICAL REVIEW C JANUARY 1996 VOLUME 53, NUMBER 1
53 0556-2813/96/531/36710/$06.00 367 © 1996 The American Physical Society