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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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