Quantum molecular dynamics and particle production in heavy ion collisions

Pro&.Part.N ~ Phys.,VoL30, pp. 105-114,1993.

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Quantum Molecular Dynamics and Particle Production in Heavy Ion Co!!i~ons 1 S. W. HUANG, A. FAESSLER, G. Q. LI, D. T. KHOA, E. LEHMANN, M. A. MATIN, N. OHTSUKA and R. K. PURl lnstitutJtlr TheoretischePhysik, UniversitdtTiibingen,Auf der Morgenstelle 14, W-7400 Tabingen, Germany

ABSTRACT The production of photons, kaons, antikaons and antiprotons in heavy-ion collisions is calculated in the framework of "quantum" molecular dynamics (QMD). The Skyrme potentials, with parameters chosen to generate the soft and hard nuclear equations of state(EOS), are used in the propagation of nucleons within QMD. The sensitivity of the production of each type of particle to the EOS is discussed. The mechaxdsms of production processes are studied. The theoretical results are compared with the available experimental data.

KEYWOR.DS: QMD, particle production, heavy-ion collisions, EOS

1. I n t r o d u c t i o n One of the most exciting new directions in nuclear physics is the study of the properties of dense nuclear matter and the behavior of hadrons in a dense medium [1-4]. Intermediate heavy-ion (HI) collisions in the energy regime of a few hundred MeV/u up to a few GeV/u are carried out in order to probe the equation of state (EOS) of nuclear matter at high density and/or temperature. In this energy regime the mean field, which is related to the EOS, as well as hLrd collisions still play a important role. Also, the EOS of nuclear matter is of importance for a understanding of astrophysical phenomena like neutron stars or super novae. 1 Supported in part by GSI Darmstadt and BMFT

105

106

S.W. Huang et al.

In central HI collisions at energies of the order of a few GeV per nucleon, a highly compressed and hot nuclear matter is formed. During such a HI collision, much of the kinetic energy of the collision is converted directly into mass in the form of created particles, such as pions and kaons. Besides collective flow and multifrngmentation, the production of very energetic particles, especially at subthreshold energies, seems to offer promising possibilities to probe the nuclear EOS. In order to investigate the particle production theoretically, one can use a transport theory which simulates the time evolution of the HI system and generates stochastic two-body collisions. The semiclassical approaches, such as the one-body theory Boltzmann-Uehling-Uhlenbeck (BUU) equation [5] and the N-body theory "quantum" molecular dynamics (QMD) [6], are at present the most successful microscopic models for the description of time-dependent and highly non-equilibrated systems produced in a HI collisions. Both approaches have been widely used at intermediate bombarding energies. In the Q M D model, each nucleon is represented by a Wigner function in phase space corresponding to a Gaussian wave packet:

fi(r,p,t) = - ~1

_(r_ri(t))2/2L_(p_pi(t))22L/h$

The centroids (ri, pi) of the Gaussian wave packets are propagated by Hamilton's equations of motion. Stochastic two-body collisions are taken into account by Monte-Carlo sampling using a two-body scattering cross section. The Hamiltonian in the equations of motion is constructed from two-body and three-body Skyrme type interactions. Due to the definition of the Hamiltonian for the whole system, the total momentum of the system is conserved. The simulation is done in an event-by-event way. The theory keeps track of the correlations and therefore can describe the entire complex dynamics of heavy ion collision from the initial to the final stage. Hence fragment formation can be described in this model [7]. In addition to its unique ability to describe multifragmentation, the N-body theory Q M D approach has been shown to be successful also in describing particle production in HI collisions[8-11]. In this paper we briefly report some of our results in our systematic calculation of particle production within Q M D approach. In all our calculations of particle production, we assume that the total yield of produced particles in HI collision is an incoherent superposition of the yield from all individual baryon-baryon collisions, and external particles are on the energy shell.

2. P h o t o n production High energy photon production in HI collisions has been well studied at incident energies between 15 and 120 MeV/u. Photons have attracted attention since they are not seriously affected by surrounding

Quantum Molecular Dynamics

107

nuclear medium. In our work, we use a modified Jackson formula, which contains some features of the contribution of exchanged charged pions as an elementary cross section: d2P.r~

dE./dfl~

_

a (2#7 + 3fl~,in20.y) 127r2E~

1

(E.c/m~) 2

×(1 + #~ 1 + (~/m,~)~)' where a is the flue structure constant. #i (/9/) is the velocity of the initial (final) proton. E.~ is the energy of the emitted photon, and 0.~ is its angle with respect to the incident proton, m~ =138 MeV is the mass of exchanged pion. The second term in the second parenthesis in above equation represents the internal contribution. In Fig. 1, the photon production cross section in an SeAr+27A1 collision at 85 MeV/u is compared with the experimental data. The photons are emitted at laboratory angles of 8t~b = 60°(left part) and 900 (right part). The cross section is shown as a function of the photon energy E~ in the laboratory system. The solid and dashed lines represent the theoretical results obtained with the soft and hard EOS, respectively. The solid circles are the experimental data from[12]. We see that the experimental spectrum is nicely reproduced both in ma~_itude and shape. This agreement between theory and experiment allows us to conclude that the basic assumptions concerning the mechanism for particle production is reasonable.

A;+ AI '

•

' 'Ar+ A1 '

' (a)

' (b) '

*'~.,

"4~,0".8~.

3~

v=4

Soft ...... Hard ,

I

40

i

Soft ...... Hard I

80

i

I

120

,

I

160

,

l

4,0

,

I

80

i

I

120

,

I

160

Fig. I. A comparison of theoretical result~ and experimental data for the double differential photon production cro~s section of the 3eAr+27Al collision at 85 MeV/u. (a): s~,~ = 60% (b): S~,~ = 90 °.

It can be seen that the photon cross section is not very sensitive to the nuclear EOS. Our calculation shows also that most photons come from first- and second-chance collisions(figure not shown). Since

S.W. Huang et al.

108

photons can be produced in each proton-neutron collision, so the number of photons produced from the high-temperature and high-density zone is relative small. Also, at this energy, no significant compression is expected to occur.

3. K a o n p r o d u c t i o n It was suggested that the kaons reflect the temperature at an early, hot stage of the reaction process, whereas the pions reflect the temperature at the final, freeze-out stage [2]. Because of the small and mainly elastic cross section of the K + interacting with nucleons(~ 9rob), kaons leave the primary collision zone essentially without reabsorption. Therefore, kaons have been suggested as a sensitive probe for the hot and dense nuclear matter formed in the early stage of HI collisions [10,13-15]. In dealing with K+-production in HI collisions at 1 GeV/u, we assume that kaons are predominantly produced in baryon-baryon ( B B ) collisions in association with another strange particle Y, such as A or E: B1 +B2--* B + Y + K + • The elementary cross section for this process is taken from ref. [13].

'~Au+*~Au_~K++X Elab= 1 GeV/u

~

01ab=44 °

10 ~

e~ ~

10 ~

b i0-,

200

---

HARD

--

SOFT 400

86o

8~o

10~00

1200

Plab(MeV/c) Fig. ~. The differential ~(+-production croaa sections for Au+Au at 1 Ge V/u obtained with 8off and hard mean fielda. The ezperimental data (open square8 with error bars) are taten ~ m reJ.f16].

In this paper, we present results of our calculations for t h e / ( + - p r o d u c t i o n cross section in the

lSTAu+lSTAu collision at 1 GeV/u and make a comparison with the experimental d a t a measured from SIS at GSI [16]. In Fig. 2, theoretical results for the K+-production cross section in this collision are shown as a function of kaon momentum in the laboratory system. The kaons are observed at 0lab = 44 °. Different results obtained with different kinds of mean fields, corresponding to soft and hard EOS, are shown. The experimental d a t a [16] are shown in the figure as open squares. One can see

Quantum Molecular Dynamics

109

that the theoreticalpredictions with soft mean fieldsare in reasonable agreement with experimental data. The results obtained with the soft and hard E O S differby approximately a factor which lies between 2 and 3. In order to see the time scale of kaon production in HI reactions,we decompoee the total differential K+-production cross section into contributions from three differenttime intervalsfrom the start of the collision.The resultsobtained with the soft E O S are shown in Fig. 3. It is found that the ksons produced during the time from 0 to 12 fm/c account for about 45-60% of the total production cross section, while those produced during the time from 12 frn/c to 18 fm/c contribute about 35-50%, The contributions from t > 18fm/c account for 2-4% only. This confirms that the lmons are indeed produced in the early and hot stage of the HI reaction. We also note that the time duration of 18 fm/c is long enough for the multiple BB collisions to take place in the HI system. ~'TAu+~'TAu~K+ + X

10-" Soft

<

Elab= I GeV/u

,o"

#lab=44 °

~"....

,Q

b ~0

.... ~°lS

10" 200

fm/c

460

860

"k

860

,0'0o

200

Plab(MeV/c)

Fig. 8. Decompoaition of the differential kaon.production tion~ from different time intero¢l~.

c r oaa

section into contribu-

To study the subthreshold particle production mechanism in HI collisions, we found that the gh'stchance and second-chance BB collisions both give a negligible contribution to the total croes section, while the later collisions give the main contribution (figure not shown). This is in contrast to the case of photon production, where the first-chance and second-chance collisions give the main contribution to the total cross section, which has been found in our earlier work [8]. This means that, for subthreshold K + production, very few primary nucleons can provide sufficient energy from the Fermi motion to produce a K + in the first-chance and second-chance NN collisions. To see this dearly, we decompose, as shown in Fig. 4, the total cross section into different contributions from di~erent incoming channels.

S. W. Huang et al.

110 107

Au+

197

Au,K

7t-

+X

I0-'

.~

Soft

~

10 -~

"O

10 -~

Elab= 1 GeV/u

91ab=44 °

.... N-A

~to 10-7

..... N - N

200

4~o

o~o

8~o

t0'00

12oo

Plab(MeV/c)

Fig. ~. Decomposition of the differential K +-production cross section into eontributiona from different incoming channels.

We found that the A + N and A + A channels give the main contributions to the total cross section. Since the relative momentum required to produce a K + in A + N and A + A collisions is lower due to the large mass of the A, the majority of the lmons are produced through a two-step process. First, the baryonic resonance, such as A (1232), is produced from a first-chance or a second-chance NN collision. Next, the A, through multiple collisions and the mean field, gains enough energy to produce a K +. This is the reason why the K+-mesons axe mainly produced from A + N and A + A collisions, and these processes occur only after a first-chance collision.

4. Antikaon production Recently, antikaon and antiproton productions have been measured both at Bevalac and SIS. We calculate antikaon and antiproton production in the RQMD approach [11], which is a relativistic generalization of the QMD approach[17]. Antikaon production has a relative high threshold of 2.49 GeV/u in NN collision; however, the K N scattering significantly differs from K + N scattering. Because there are a number of baryons or low-mass baryon resonances with strangeness S = - 1 , the K N cross section is appreciably larger than the K + N cross section, the reabsorption or annihilation of the produced particles should be properly considered. In this work, we consider process B1 +B2--~ N + N + K + K only. For antikaon production we parameterize the total cross section according to the available experimental d a t a of [17]: UK, (V~) = 23.33(V~ --

x/~)pb, x/~ <

3.35GeV

Quantum Molecular Dynamics -- 24.059(x/~ - ~

111

- 0.351pb, V~ > 3.35GeV.

Here, x / ~ -- 2(ran + m/C) is the threshold for antikaon production in free baryon-baryon collision, with mN and m/¢ being the masses of nucleon and kaon, respectively.

We treat reabsorption effects in an empirical way(a more sophisticated treatment can be realized only by non-perturbative calculation). The produced antikaon has to travel a distance D before it leaves the environment and is detected. This distance depends on where the particle is produced and also on the baryon distribution of the system. Within this distance, the particle has the probability T of being absorbed or annihilated. This probability is calculated by the following procedure: for a produced particle at spatial coordinate r and direction of travel ~, we propagate the particle along 15 with steps of 6r = 0.2fro until it reaches the edge of the colliding system where the density is less than pmi,, -- 0.06.fro -s. The probability that the produced antilmon is absorbed or annihilated within this distance 6r = O.2fm is given by:

T(r) = exp[-6r/~(r)], where A is the mean free path of the antikaon. A can be estimated from the relation 1 ~(r) = ~pCr)' where p is the density determined locally along the path by the RQMD simulation of the reaction. m

For the energy regime considered in this work, the total K N isospin averaged cross section is Grin ffi 3¢(I ----1) + ¼¢r(.r _-- 0) ~ 40rob [19]. The inelastic cross-section is about half of the total cross-section. We assume that the elastic K"'N scattering does not influence the antikaon production, so that, in the estimation of the mean free path, only the inelastic part of the total K N cross section is included.

The direct comparisons of antikaon-production cross sections obtained with different nuclear EOS's are presented in Fig. 5 for 2sSi-l-zssi collisions at 1.55 GeV/u. In this figure, we show the Lorentz-invariant double-differential antikaon-production cross section as a function of antikaon kinetic energy. Four different results obtained with the soft and hard EOS's and two different results with and without reabsorption of antikaons axe shown. The experimental data, taken from [20], are shown in the figures as solid circles. We observe a clear difference between the antikaon-production cross sections corresponding to two different EOS's. This difference is almost as large as the difference between the cross sections obtained with and without absorption. The soft EOS predicts a larger antikaon production cross section than the hard one. We also achieve good agreement with experimental data when the soft EOS is used in the theoretical calculation.

S. W. Huang et aL

112 i

•

t

i

Si+Si

0'

i

0~m--0

primordial~oft "\ "'-... ...... with absorption~oft "\'...\.~\'b

--.- primordial,hard

\.

O

\,

.... w i t h a b s o r p t i o n , h a r d ,

I

Ioo

,

I

,

I

2~ ~0 Eem(Me~

,

I

~0

Fig. 5. The Lorentz-invariant double-differential antikaon-production cro~J section in 2sSi-/-2sSi collisions at 1.55 GeV/u. We ~how four se~ of result8 obtained with different EOS'8 and with and without absorption.

5. Antiproton production

The production of antiprotons is the most extreme case of all particle production processes, because it has the highest threshold (5.63 GeV/u) in NN collision. The high threshold energy provides the unique possibility to study particle production simultaneously at high collision energy and far below the threshold. We consider for antiproton production by the process B] +B2--* N + N + p + ~ . The total cross section is taken from ref. [21], which is parametrized from experimental data. The momentum spectrum of ~ is assumed to be proportional to the available phase space of final state. Like antikaons, the produced antiprotons are easily annihilated by the surrounding nuclear medium. We treat the reabsorption of antiproton in the same way as in antikaon case. The ~p inelastic cross section is approximately 60rob in the present energy regime. In Fig. 6 we compare our theoretical results with available experimental data for antiprotonproduction cross sections in nucleus-nucleus collisions. The system under consideration is 12C-I-S3Cu at 3.65 GeV/u. The Lorentz-invariant double-differential antlproton-production cross section is plotted as a function of antiproton momentum in the laboratory system for antiprotons emitted at 01oh = 24°. The experimental data, shown as solid circles in this figure, are taken from [22]. Two sets of results, corresponding to primordial and reabsorption cases, are shown in the figure. The theoretical result which does not contain an annihilation effect is in reasonable agreement with the data. If the absorption factor is considered, which corresponds to about an 80% annihilation of the primordial antiprotons, our theoretical prediction underestimates the experimental data by about a factor of 4.

113

Quantum Molecular Dynamics i

•

i

•

C+(~'

i

'0

I

8~b--24

"~. ..~

"~..4

I••-4 0

......

prtmordml

--.-

.ith absorption I

800

,

|

i

400

I

i

600

I

800

i

l

1000

Plab(MeV/c) Fig. 6. The Lorentz-ineariant double-differenti-I antiproton-produetion cross section in z2C+ss Cu collision at $.65 Ge V/u. We sho~, tu)o sets of theoretical results obtained with and without absorption.

6. S u m m a r y We present, in this paper, the theoretical calculations for the production of several types of particles within the QMD approach. Particle production in HI collisions is expected to be an important probe for hot and compressed nuclear matter formed in HI collisions. Since different particles have different properties, they are expected to be sensitive to the various aspects of HI collisions. Among those properties are: their production threshold, which is determined from a particle rest mass and the elementary production process, and their final state interactions with the surrounding nucleons. The experimental data for photon production in HI collisons can he well understood in the QMD model with assumption of incoherent proton-neutron bremsstralung; however the sensitivity to the EOS in this case is not significant. Our calculations show that kaons are produced from an earlier stage in the HI collision and mainly through a two-step process. At subthreshold energies, the Fermi motion in the nuclear groundstate can not provide enough energy to produce kaons and, hence, resonances dominate kaon production. Kaon cross sections are sensitive to the EOS by a factor between 2 and 3. Theoretical predictions with the soft EOS are in good agreement with the experimental data from SIS at GSI. Certain unambiguous conclusions concerning the properties of the nuclear EOS require, however, further systematic investigations. We parametrized the elementary cross section for antikaon production, the experimental data in HI collisions at Bevalac can be understood rather well with this elementary cross section. The existing

114 S.W. Huanget a~ antiproton production experimental data can be reproduced within the correct order of magnitude. The calculations of antilmon and antiproton production cross sections at SIS energies are in progress.

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