Subthreshold kaon production and the nuclear equation of state

Abstract

We reexamine in the relativistic transport model the dependence of kaon yield on the nuclear equation of state in heavy ion collisions at energies that are below the threshold for kaon production from the nucleon-nucleon interaction in free space. For Au+Au collisions at GeV/nucleon, we find that the kaon yield measured by the Kaos collaboration at GSI can be accounted for if a soft nuclear equation of state is used. We also confirm the results obtained in non-relativistic transport models that the dependence of kaon yield on the nuclear equation of state is more appreciable in heavy ion collisions at lower incident energies. We further clarify the difference between the predictions from the relativistic transport model and the non-relativistic transport model with a momentum-dependent potential.

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arXiv:nucl-th/9411024v1 22 Nov 1994
Subthreshold kaon production and the nuclear equation of state
G. Q. Li and C. M. Ko
Cyclotron Institute and Physics Department
Texas A&M University, College Station, Texas 77843
Abstract
We reexamine in the relativistic transport model the dependence of kaon yield
on the nuclear equation of state in heavy ion collisions at energies that are
below the threshold for kaon production from the nucleon-nucleon interaction
in free space. For Au+Au collisions at 1 GeV/nucleon, we find that the kaon
yield measured by the Kaos collaboration at GSI can be accounted for if a
soft nuclear equation of state is used. We also confirm the results obtained in
the non-relativistic transport model that the dependence of kaon yield on the
nuclear equation of state is more appreciable in heavy ion collisions at lower
incident energies. We further clarify the difference between the predictions
from the relativistic transport model and the non-relativistic transport model
with a momentum-dependent potential.
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About ten years ago, Aichelin and Ko [1] pointed out that in heavy ion collisions at
incident energy per nucleon that was below the kaon production threshold in nucleon-nucleon
interaction in free space (which is about 1.58 GeV) kaon production was sensitive to the
nuclear equation of state (EOS) at high densities. In the non-relativistic Boltzmann-Uehling-
Uhlenbeck (BUU) model, they found that in central heavy ion collisions at an incident energy
of 0.7 GeV/nucleon the kaon yield obtained with a soft EOS (compressibility K=200 MeV)
was about 2-3 times larger than that obtained with a stiff EOS (K=380 MeV). This finding
was later confirmed in calculations based on the quantum molecular dynamics (QMD) [2,3].
The determination of the nuclear EOS at high densities has been one of the main mo-
tivations for recent experimental measurements of kaons in heavy ion collisions around 1
GeV/nucleon by the Kaos collaboration at GSI [4,5]. Since the experimental data have
become available, there has been a resurgence of theoretical studies on kaon production
in heavy ion collisions, based on both non-relativistic [6–8] and relativistic [9,10] transport
models. Huang et al., [6] have carried out the first comparison of theoretical results, obtained
with the QMD model, with the experimental data from Au+Au collisions at 1 GeV/nucleon.
Good agreements with the experimental data have been obtained when a soft EOS is used in
the model. With a stiff EOS, their results are about a factor of two below the experimental
data. These findings have recently been confirmed by Hartnack et al., [7], using also the
QMD model. A similar calculation has been carried out by Li [8] in the non-relativistic
BUU approach. Again, a soft EOS has been found to give reasonable agreements with the
experimental data.
In both BUU and QMD calculations, the nuclear EOS is modeled by a simple Skyrme
parameterization. In this model, the energy density of the nuclear matter at density ρ is
given by
E =α
2
ρ2
ρ0
+
β
γ + 1
ργ+1
ργ
0
+
4
(2π)3
?kF
0
d3k(m2+ k2)1/2,(1)
where kF is the Fermi momentum and m is the nucleon mass. The parameters α, β and
γ are determined by requiring that in normal nuclear matter one has a saturation density
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of ρ0 = 0.17 fm−3, a binding energy of 15.6 MeV, and a compressibility of K=200 MeV
(soft EOS with α = −356 MeV, β = 303 MeV, and γ = 7/6) or 380 MeV (stiff EOS with
α = −124 MeV, β = 70.5 MeV, and γ = 2). These two sets of EOS (i.e., the energy per
nucleon as defined by E/A=E/ρ−m) are shown in Fig. 1 by the dashed curves. At 3ρ0, the
two differ by about 55 MeV. We note that in the Skyrme parameterization, the compressional
energy at high densities depends mainly on the second term in Eq. (1), or more specifically,
on the magnitude of γ, which is directly proportional to the nuclear compressibility K at
the saturation density.
On the other hand, the calculations of Fang et al. [9] and Maruyama et al. [10] have been
carried out in the relativistic transport model where the nuclear EOS is modeled using the
non-linear σ-ω model [11]. The energy density of the nuclear matter in this model is given
by
E =
g2
2m2
ω
ω
ρ2+m2
σ
2g2
σ
(m − m∗)2+
b
3g3
σ
(m − m∗)3+
c
4g4
σ
(m − m∗)4
+
4
(2π)3
?kF
0
d3k(m∗2+ k2)1/2,(2)
where m∗is the nucleon effective mass. The parameters gσ, gω, b, and c are determined by not
only the normal nuclear matter properties, such as the saturation density, the binding energy,
and the compressibility, but also the nucleon effective mass. In Ref. [9], we have used two
sets of parameters which correspond to the same nucleon effective mass, m∗= 0.83 m, but
different values of nuclear compressibility, i.e., K=380 MeV (with Cσ= (gσ/mσ)m = 11.27,
Cω= (gω/mω)m = 8.498, B = b/(g3
σm) = −2.83×10−2, and C = c/g4
σ= 0.186) and K=200
MeV (with Cσ= 13.95, Cω= 8.498, B = 1.99 × 10−2, and C = −2.96 × 10−3). It has been
found in Ref. [9] that the kaon yield at 1 GeV/nucleon obtained with K=200 MeV is only
about 15% larger than that with K=380 MeV. This result is inconsistent with the findings
from the non-relativistic transport models [6–8], where the difference between the results
obtained with the two Skyrme EOS’s is about a factor of two.
It has recently been realized that the so-call “stiff” EOS used in Ref. [9] is not as stiff
as that in the Skyrme parameterization. As can be seen from Eq. (2), in the relativistic
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approach the EOS at high densities largely depends on the vector repulsion from the omega
meson. From the Hugenholtz-Van Hove theorem, which requires that the Fermi energy is
equal to the average single particle energy at saturation, we have the following relation [12]
?gω
mω
?2=m − E∗
F(ρ0) −¯B
ρ0
,(3)
where ρ0 and¯B are the saturation density and binding energy, respectively, and E∗
F=
(m∗2+ k2
F)1/2. Since the two EOS’s in Ref. [9] have the same nucleon effective mass, the
vector coupling constants are thus the same. At 3ρ0, the energy per nucleon with the “stiff”
EOS is only about 15 MeV larger than that of the soft EOS. This is much smaller than the
difference between the stiff and the soft EOS in the Skyrme parameterization (cf. dashed
curves in Fig.1). As a result, the kaon yield is not very sensitive to the relativistic nuclear
EOS’s used in Ref. [9].
Up to 4ρ0(see Fig. 1), the soft EOS used in Ref. [9] is very close to the soft EOS given
by the Skyrme parameterization (Eq. (1)). To compare results between relativistic and
non-relativistic approaches, we need to use a stiff EOS in the relativistic approach which
is also similar to the stiff EOS used in the non-relativistic approach. Such an EOS can
be obtained by using a smaller nucleon effective mass m∗= 0.68 m but the same nuclear
compressibility (K=380 MeV) at saturation density. The parameters for this EOS are
Cσ= 15.94, Cω= 12.92, B = 8.0 × 10−4, C = 2.26 × 10−3.
The two relativistic EOS’s are shown in Fig. 1 by the solid curves. At 3ρ0, the two differ
by about 56 MeV as in the Skyrme parameterization.
To see the sensitivity of subthreshold kaon production to the two relativistic nuclear
EOS’s, we have carried out a perturbative calculation of kaon production in Au+Au collisions
using the relativistic transport model developed in Ref. [13]. Kaons are mainly produced
from baryon-baryon interactions, and the production cross sections are taken from the linear
parameterization of Randrup and Ko [14]. Contributions from meson-baryon interactions
[15], higher resonances [16], and multi-baryon interactions [17] have been neglected as they
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are unimportant for kaon production at energies around 1 GeV/nucleon. The rescattering of
produced kaons with nucleons is treated by the perturbative test particle method introduced
in [18]. Details of the calculations can be found in Ref. [9].
For a head-on Au+Au collision at 1 GeV/nucleon, we show in the left panel of Fig. 2
the total number of baryon-baryon collisions that have energies above the kaon production
threshold. With the soft EOS, this number is about 95 but is reduced to about 54 when the
stiff EOS is used. The reduction is partly due to the fact that the maximum central density
reached with the soft EOS (about 2.9ρ0) is higher than that with the stiff EOS (about 2.4ρ0).
As a result, the average density at which kaons are produced is also higher for the soft EOS
(about 2.5ρ0) than for the stiff one (about 2.1ρ0). Furthermore, the energy per nucleon at
these densities is about 5 MeV for the former and 15 MeV for the latter. Thus, more kinetic
energy is converted into the compressional energy in the case of the stiff EOS. This effect
can be seen from the the right panel in Fig. 2, where we show the distribution of pmaxin
the collision, with pmaxbeing the maximum momentum of the produced kaon in a given
baryon-baryon collision. The average value of pmaxis about 0.272 GeV/c in the case of the
soft EOS and is reduced to about 0.245 GeV/c for the stiff EOS. Overall, the kaon yield
with the soft EOS is about a factor of two larger than that with the stiff one, consistent
with the findings of non-relativistic transport models [6–8].
Our results thus demonstrate that the kaon yield from heavy ion collisions is similar
in both relativistic and non-relativistic transport models if similar nuclear equations of
state are used. In Ref. [2], the non-relativistic transport model was generalized to include
a momentum-dependent potential, and it was shown that this would reduce significantly
the kaon yield due to the lower number of collisions and deceleration as a result of the
momentum-dependent potential. This result is in contrary to ours based on the relativistic
transport model, which includes the momentum-dependent potential via the nucleon effec-
tive mass. We believe that the kaon yield calculated in Ref. [2] is incorrect as it has not
taken into account the difference in the initial and final potential energies in the reaction
BB → NY K, where B and Y denote a baryon (nucleon or delta) and a hyperon (lambda
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