Subthreshold pion production in nucleus-nucleus collisions within the quantum molecular dynamics approach

Nuclear Physics A534 (1991) 697-719 North-Holland

NUCLEAR PHYSICS A

SUBTHRESHOLD PION PRC)UCTION IN NUCLEUS-NUCLEUS COLLISIONS WITHIN THE QUANTUM MOLECULAR DYNAMICS APPROACH* Guoqiang LI', DAO TIEN KHOA?, Tomoyuki MARUYAMA, S.W. HUANG, N. OHTSUKM and Amand FAESSLER Institut jûr 7heoretische Physik-, Universitâi Tâbingen, 7400 Tübingen, Germany J. AICHELIN

Institut J-ur Theoretische Physik, Universitdi Heidelberg, 69oo Heidelberg, Germany Received 18 January 1991 Abstract: The subthreshold pion production from 12C + 12C, 2O Ne + 2O Ne, 4OCa+ 4OCa, 2ONe+ 4Cu and 93Nb +93 Nb collisions is studied witt in the quantum molecular dynamics (QMD) model. For the mean field part of the theory, we use both the soft and the hard equations of state (EOS) with and without the momentum dependence in the interaction, while for the in-medium NN cross section, the Brueckner G-matrix is utilized. The pion yield from a nucleus-nucleus collision is assumed to be the incoherent sum of the corresponding yields from the individual nucleon-nucleon collisions. For the latter process, parametrized cross sections are adopted. Tlhe dependence of the pion cross section on the different nuclear EOS, on the momentum dependence in the interaction and on the pion reabsorption factor is studied. The time sequence of the pion production is also analyzed. The theoretical results f0Ir 2O Ne + 2ONe (Elab/ A = 183 MeV) and 2O Ne + 64Cu (E blA =250 MeV) collisions are compared with available experimental data.

1. Introduction One ofthe ultimate goals of heavy-ion nuclear physics is to determine the equation of state (EOS) of nuclear matter over a wide range of densities and temperatures 1-4 ) . Probing the nuclear EOS in the regions of high density and/or temperature is of fundamental importance not only in nuclear physics (nuclear viscosity, heat conductivity, possible pbase transitions, such as liquid-gas, pion condensation, A-isomers, etc.) and field theory (QCD phase transition to a quark-gluon plasma), but is also a basic prerequisite for an understanding of many astrophysical problems, such as the dynamics of the early universe as well as the dynamics of supernova explosions and the stability of neutron stars. It is expected that during intermediate- and high-energy heavy-ion collisions, a small piece of nuclear matter with high density and/or temperature might be created * Supported by the GSI Darmstadt under the contract 96621 and the Alexander von Humboldt Foundation . ' Permanent address: Department of Physics, Hangzhou University, Hangzhou, P.R. China. 2 Alexander von Humboldt Research Fellow . 3 Present address : Onomichi Junior College, Onomichi, Japan. 0375-9474/91/$ 03.50 @ 1991 - Elsevier Science Publishers B.V. All rights reserved

698

Cr. Li et aL / Subthreshold pion production

as a transient stage of the collision . One knows the initial setups of the collision, so one can also detect and analyse its final results, but one cannot detect directly the properties of the possible transient compressional phase in which one is interested. A suitable theoretical approach is indispensable in this respect in order to follow the time evolution of the colliding sysem. and to infer the properties of the interm ate stage from the known initial conditions and the final outcome of a Collis n. The Boltzmann-Uehling-Uhlenbeck (BUU/VUU) equation -5-") and the uantu molecular dynamics (QM D) model ' 2`5) play central roles forthis purpose. many final observables of a given collision, one should try to select those which are sensitive to the nuclear EOS in order to determine it unambiguously. 1.4,16.17 A Collective flows (transv-erse momenta in and out ofthe reaction plane) [refs . frequently " '8) have been the collision , ,d particles that are produced during ered as suitable probes of the nuclear EOS. In the present paper, we study uction of subthreshold pions in nucleus-nucleus collisions, first investigated ichelin "), within the QMD approach and analyse its sensitivity to the ients of the calculation. It is dear from previous studies '9,20) that the momentum dependence in the interaction plays a significant role in the collective behavior and in particle production in nucleus-nucleus collisions. No unambiguous conclusion can be drawn on the incompressibility of nuclear matter until we know the momentum dependence 21,22), we sufficiently well. Continuing our previous works on photon production will in the present work analyse the influence of the momentum dependence in the in-medium interaction on the pion production as well as the sensitivity of the pion production cross section to the nuclear EOS. esides the mutual two- and three-body interactions (or the mean field in the UU approach), another important ingredient that determines the dynamic evolution of the colliding system is the in-medium nucleon-nucleon (NN) cross section which differs from the free NN cross section because of the medium effects. In the early works, either a constant isotropic cross section of 40 mb or the free cross section was utilized. The Cugnon parametrization 23), which is claimed to have taken into account medium effects in an empirical way, has also been used by several groups. In order to remove as much as possible the ambiguities, one should use in principle an in-medium NN cross section that is based on a solid microscopic foundation . This has been realized by the works of our group, with the use of an NN cross section derived from a G-matrix, which is a solution of the Bethe24-26) . This G-matrix cross section will Goldstone equation in the nuclear medium be used in the present work. There are two possibilities to create pions from subthreshold nucleus-nucleus collisions. (The threshold for pion production in a free NN collision is around 300 MeV.) The first suggestion depends on the coupling of the projectile beam velocity with the Fermi momenta of the nucleons which gives enough energy to a few nucleons so that they can undergo intranuclear collisions with sufficient energy

G. Li el at. / Subthreshold pion production

699

to produce pions. Another possibility involves the cooperative effect which coherently pools the kinetic energy of several nucleons. In this work we adopt the first picture within which the total pion yield from a nucleus-nucleus collision is the incoherent sum of the corresponding yields from individual NN collisions. One thus needs, in addition, an elementary cross section for particle (photon or mesons) production in these NN processes. For the subthreshold pions considered in this paper, we use the parametrization of VerWest and Arndt 27), which has been 11 .28) . exploited by some groups We give in sect . 2 a short description of the QMD approach and the formula for the evaluation of pion cross sections in a nucleus-nudeus collision from the elementary NN --3- NNir cross sections. Our results are presented and compared with available experimental data in sect. 3. The final section contains conclusions and the outlook. 2. Outline of the theoretical approach 2.1. QMD MODEL

In the QMD model the wavefunction of the ith nucleon is expressed as a --, centered around ri.(t) and pio(t) : wavepacket with a width -,IL rio( t))214L] exp, [-ipio - r] . (1) vi(rio(t)' pidt)) - (2 7rL)314 exp [-(r The corresponding Wigner representation is 1 exp fi(r, P, 0 = -P

[_( r _ rio)2/2L] exp [( P _Pio(,»22L] .

(2)

The dynamical evolution of the phase space consists of the propagation governed by the Hamilton equations of motion, which represents the mean field part of the NN interaction, and the stochastic binary collisions between the nucleons which account for the residual part of the interaction. The mean field part of the interaction includes a Skyrme-type B-force (two-body and three-body), a Yukawa-type finite-range force and an effective Coulomb interaction : Vtot = V(2) + V(3) , (3) where

V(2) = t,&(ri - r2) + VyUk+ ivcOui , V(3) = 6 t2'6(rl - r2),b(r, - rj) -"lt2'6(rl

= , e-1r1_r211M 3 I rl _ I/M') r2 Vc0u, _ (ZIAe) 2 ji,,Yuk

Irl -

r2l

r2) p'y[!(r, + r2)], 2

(4a) (4b) (4c) (4d)

G. Li et aL / Subihreshold pion production

e use m = 1.5 fin and 1~; = -6 .66 MeV in this work. The parameters t,, t, and y are connected with the nuclear equation of state, or ultimately with the saturation properties of the nuclear matter: (5) U(P) = a(P/PO)+P(P/PO)" -

e momentum dependence in the interaction is introduced through the following form with its parameters (t4 = 1.57 MeV, tS = 5 X 10-4 MeV-2) fitted to the protonnucleus optical potential: UMD1 =

t4ln'[t .,(

)2 + 118 (ri - J%') -

en the momentum dependence is introduced, the parameters of the nuclear S should be readjusted in order to have the right saturation density and binding . In this work we use both the hard (K = 380 MeV) and soft (K = 200 MeV) S with and without the momentum dependence in the interaction . All the meters are listed in table 1 . e propagation of the nucleons under the influence of these interactions is tre classically using the Hamilton equations of motion: drio M dpio aH di aPiO " di ario e e single- article hamiltonian Hi given by , U~2. ) +.! 1: U(3) Hi = Ti +151 2 3

U

jk

ijk

where the two-body and three-body potentials respectively are U~2)

f

UQ)

fi(ri, Pi, t )fj( r

ijk

i Pi

t)fj (rj, pj, t) V42)( ri - rj) dri drj dpi dpj , j

( Pj t)fkI rk

Pk

t) V(3)( ri

rj, rk)

dri drj drk dpi dpj dpk .

(9)

Two nucleons are allowed to collide when they approach each other close enough . This is done in exactly the same way as in BUU calculations : one checks whether two nucleons will collide or not with the criterion that the closest distance dnin TABLE I The parameters of four equations of state used in the present work

S H SMD HMD

a (MeV)

P (MeV)

-356 -124 -390 -130

303 70.5 320 59

y

K (MeV)

2 1.14 2.09

200 380 200 380

2

6

G. Li el at. / Subthreshold pion production

701

between two nucleons within a given time interval should be less than -..Ia -_ NN/ ir. One also checks whether the collision is Pauli blocked or not. If a collision does occur, one determines the magnitudes of the final momenta from the conservation of energy and momentum, whereas the directions of these momenta are fixed by a Monte Carlo procedure from a given angular distribution. In-medium corrections for the NN cross section enter by solving the Bethe-Goldstone equation for the momentum and spatial density distribution of the nucleons at the collision point IRrio - rio) G = V+ VQF : L.I- G, e

(10)

where V is the bare NN interaction and QF is the Pauli operator for two colliding nuclear matters with the momentum distribution F. The detailed description of this procedure is given in refs. 24-26) . 2.2. CONNECTION WITH ELEMENTARY NN-+NNv PROCESS

With the assumption of particle production by incoherent on-shell NN collisions, the differential particle multiplicity emerging from a nucleus-nucleus collision at a given impact parameter b is obtained by the sum over all possible NN collisions with Pauli blocking for the final nucleon states taken into account "): I d 3 N,(b) lcx dE,, dflx

NN

f

I d3p-, ('fs) -

k'X dE'X dfl'X

[I -f(r, p3, t)][ I -f(r, p4 , t)] dil,

(11)

where x stands for the particle under consideration (they are 7r', iro and 7- in this work) . The primed and the unprimed quantities are in the center-of-mass system of two nucleons and the center-of-mass system of the two colliding nuclei, respectively. 0 denotes the solid angle of the relative momentum p3 -p4 which is not fixed by the energy and momentum conservation and has to be averaged out. The invariant differential production probability d 3px (vrs)IdE'X d1l'X for a given invariant energy vrs- is ususally expressed as the ratio between the invariant differential NN cross section for the production of particle x and the total NN cross section: 1 d3pX ( V1--S) d 3 o-x (-I-s-) 1 /ff,.t J/ NN (VIS) [k x dEx df2x kx dEx dûx

(12)

Here we need again the in-medium total NN cross section cr" NN . In order to be consistent, we will use for this quantity the cross section obtained from the Brueckner G-matrix [see eq. (10)], whereas in early works, this consistency was not explicitly exploited. For the comparison of the theoretical results with experimenta! data, one needs the double-differential particle production cross section which is obtained by

G. Li et aL / Subthreshold pion production

702

inte rating over the impact parameter b : 1 d3Gr, = 1 d3 Nx(b) (13) 2r bdb k., dF, dû., dE,, dû.', In the present case, for the production of subthreshold pions, we adopt the metrizations ofVerWest and Arndt 27) which have been fitted to the experimental e cross section of pion production in NN collisions can be expressed in s of four elementary ones with different isospin combinations of the initial and two-nucleon systems. These cross sections are denoted by cr, I- where I and I' e total isospins of two nucleons before and after the collision. In this way we istinguish among three kinds of pions (7r', 7ro, ir - ). The channels for the uction of different pions together with the corresponding relations with arlt. are in table 2. e etrizations of ref. :!7) are based on the assumption that the pions are produced through baryonic resonances. Since in the isospin-0 channel ((70,) the A cannot contribute, the lowest lying I=-!:2 resonance, N*(1430), is also taken into account. The cross section is parametrized as .W(,hc )2

2p2

[ 111

er

M01 2,2 (q l qO )3

]f

PO

(S

*_

2)2+M2.r2» MO 0

where s* = ( P

2= I

is - mi~,

2= PO ;VMO+

)2_M2

MN N ) _ (M»2][S + (M»2]14s, Pr S (MN - (MN ~S*) M~jr )2][ S * _ M~,)2]14s*, q 2(,~~ =[S*-(MN(MN + -9

2 = [ M2_ M ).][M2_ Mj2 qo ]l4mo2 0 (MN - ', 0 (MN + TABLE 2

The different channels and the corresponding relations with the isospin cross sections for three kinds of pions Pions

Channels

Tr

p+p-> n+p+ 1r + n+p-> n+n+ 1r +

1701 + 0~1 I 2( 47*11 + aOl

P+P-P+P+Ir o n+n-> n+n+ 7r o n+p--> n+p+ ir o

0"I I 4711 2( 0'10+ tT01)

n+n-3, n+p+ n+p-3- p+p+

0,10+'711 1 2( 0'11 + 0701

V0

1r -

Cross sections

ir 7r -

(14)

G. Li et al, / Subthreshold pion production

703

and (M (%Fs» = Mo + -41,ro(arctan Z, - arctan Z-)- ' In Z+ = (21FO)(%IS - MN

MO)

Z_ = (2/Fo)(M N + Mir

MO)

(1

+z2

Here both A and N* are assumed to have a mass distribution with a peak at Mo and a width ro . (M) is their mean mass (invariant energy) obtained by averaging the Breit-Wigner distribution. p, is the maximum momentum of A or N* in the NA or NN* center-of-mass system for a given NN invariant energy Vs , whereas q is the maximum momentum of the pion in the A or N* rest frame originating from the decay of a A or N* with invariant energy v s*. q0 is the corresponding maximum momentum of the pion when the invariant energy of the A or N* is Mo . m., r, il and f are four free parameters adjusted to fit the experimental data. All the necessary parameters are presented in table 3. Since the parametrizations given above are energy integrated, and because we need the spectrum of the produced pions for a correct determination ofthe final-state phase space, we adopt here the same parametrization for this invariant differential cross section as used by many groups "-29), which originates from an idea of Randrup and Ko 30): I 'd3IT. fSO = 12 )2 (1 - Ik.,Ilk ax)(1k7r1/kmax (15) ()'-(V S) 1rk2 k, dE r dD ir 4 ir kmax -

f

(-,

with

s=4( P 2 + M 2N), where a is expressed in terms of a as shown in table 2, p denotes the nucleon momentum in the center-of-mass system of two colliding nucleons . k2max

=[s - (m.,, +

MN

2

)2][S

-(2MN-

,,

M

)2]14s,

W

,,,,

2.3 . PION REABSORPTION

Unlike photons and dileptons, the pions interact with suffounding nucleons very strongly. They might be rescattered or reabsorbed once they are created. No definite TABLE 3

The parameters used in the parametrizations of the isospin cross sections

1

9

mo (GeV) Mo (GeV)

0', 1

0, 10

0~01

3.772 1 .262 1 .118 0.09902

15 .28 0 1 .245 0.1374

146.3 0 1.472 0.02249

704

G. Li et A / Subthreshold pion production

conclusion c-, n be drawn on the in-medium pion production unless this reabsorption is taken into -iccount seriously. There have already been some microscopic calculaAt present we do tions on the pion absorption in the dense nuclear matter not refer to these complicated microscopic approaches but rather adopt a simple empirical consideration. We assume that the pion is created in the center of two colliding nue!eons . It has to travel through a distance D before it leaves the nuclear environment and is detected . In our approach, this distance depends not only on the geometry of the colliding nuclei as in ref. "), but also on the nucleon-density distribution of the system. Since different equations of state result in different nucleon-densit.-y distributions and evolutions, we can in this way obtain some correlation be- ween pion re-absorption and the nu~:-lear EOS which will be discussed in secL 3. Within the distance the pion has a probability T to be absorbed by the nucleons, which is expressed as 31 .32

).

T = exp (-D/Aj .

(16)

The probability of pion reabsorption depends mainly on two facts. First, it depends on where the pion is created. When the pion is created near the center of the dinuclear system, the reabsorption will be very strong, since the pion will have to travel a longer distance through the nuclear surrounding before it is detected . Secondly, it depends on the mean free path A,, of pions . This quantity, which is certainly a function of pion energy and momentum, has already been studied microscopically based on the pion-nucleon interaction. However, in this work, we will use a constant mean free path for pions with different energies and momenta. This is a reasonable approximation for pions produced in subthreshold nucleusnucleus collisions. Several values are assigned to this mean free path in order to analyze the sensibility of the results to the choice of A.,. 3. Results and discussions We consider the dynamic evolution of the colliding system over time interval of 60 frn/ c which is divided into 200 steps with a constant time step 0.3 frn/c. For all the systems considered here, 6 impact parameters with constant ',-teps are used for the integration appearing in eq . (12). The particle production is treated perturbatively, which is -justified for subthreshold processes. During the evolution of the system, we record all the attempted NN collisions generated by the QMD code, be it Pauli blocked or not, for the later calculation of particle production . The evolution of the system and the production of particles are treated separately . The production of particles does not influence the evolution of the system. After the total time interval (60 frn/ c) has been covered and all the in6ependent simulations have been run, we switch on the code for pion production which evaluates pions from all those attempted nucleon-nucleon colE_

G. Li et aL / Subthreshold pion production

705

lisions. Here we should check if the available invariant energy of two colliding nucleons is larger than the threshold for pion production, and if according to energy and momentuan conservation the final states oftwo nucleons are Pauli blocked after the pion is emitted. The probabilities of Pauli blocking might be different with and without the emission of the pion, since the pion will take away part. ofthe available energy and momentum, resulting in a different final-state phase space distribution. The present results are divided into two parts. The first one concentrates on the influence of different ingredients of the model, such as the equation of state, the momentum dependence in the interaction, the reabsorption of produced pions and the different time intervals, on the double-differential cross section of pions. For this purpose, we consider four systems 12C + 12C, 2ONe+2ONe, 40Ca+ 40Ca and 93 Nb +93Nb, with bombarding energy El.b/A= 200 MeV. The pions, are calculated at angle Oc.,,,. = 90* in the center-of-mass system of two colliding nuclei. It should be noted that, when calculating the pion production in the nucleus-nucleus collisit,n, there are three different reference system- There is the laboratory system in which the target is at rest and the projectile bombards it with an energy Elab/A = 200 Me V. The initialization and the time evolution of the system are accomplished in Che center-of-mass system of the two colliding nuclei which ususally has a velocity of v/ c - 0.2-0.3 with respect to the laboratory system for the bombarding energies considered here. The production ofpions in the individual nucleon-nucleon collision is calculated, on the other hand, in the center-of-mass system of these two nucleons . There are frequently transforms among these three reference systems which are always done with the relativistic kinematics. Caution should be taken when comparing theoretical results with the experimental data, since the former are usually calculated in the center-of-mass system of the colliding nuclei, while the latter are frequently measured with respect to the laboratory system . The initialization plays a significant role in the success of a simulation approach . There are at least two criteria to be satisfied for a meaningful initialization . First, the initialization should reproduce as accurately as possible the grourld-state properties of the nucleus. Second, the initialized nucleus should keep its stability as long as possible when it is not boosted. In the present work, we notice that the five initialized nuclei 12C, 2ONe, 4OCa , 64Cu and 93 Nb have the right root mean squat~ (RMS) radii and binding energies, for both the soft and hard equations of state with and without the momentum dependence . The particle production is closely related with the time evolution in phase space, especially the number of NN collisions and the density of the nuclear environment . We show in figs . I and 2 respectively the time evolution of the average number of NN collisions and maximum central density for 93 Nb +93Nb at Elab/ A= 200 MeV. The impact parameter used in this example is 3 fin. The maximum central density reaches its peak at a time of about 20 fm/ c, whereas the number of NN collisions gets to the maximum a little later which means more collisions occur at the expansion stage. With the momentum independent interaction one obtains a higher density

G. Li et al. / Subthreshold pion production

Nb+Nb 200 MeV/A

I

Nb+Nb 200 MeV/A

Soft f

- SMD

-------- HMD

-------- Hard

24

71me (fm,/c)

48

0

24

48

71me (fWQ

Fig. 1. The tinie evolution of the average number of NN collisions for '13 Nb +93 Nb at and b = 3 fin .

C)

E

I .

X tu CQ ~Ci

I I

Fig. 2. Same as fig. I but for the maximum central demsity.

Elab/

A = 200 MeV

G. Li et A/ Subthreshold pion production

707

and a larger number of NN collisions for the soft EOS. The inclusion of the momentum dependence into the interaction leads to a lower density and a smaller number of collisions. We arrange the results for pion production cross sections in several different ways, in order to clarify the influence of different ingredients of our model on these observables. We present, in fig. 3, the pion double-differential cross section da/dE dD for different equations of state, as indicated in the figure (for the definition of these abbreviations and their parameters, cf. table 1). I'he absorption of pions in the surrounding nuclear matter is included by eq. (16) where the distance D is calculated for each pion from the distribution of nucleons up to a minimum density p = 0.06 fIn-3. But the mean free path A,, is assumed to be a constant as given below. In fig. 3 we stress the importance of pion reabsorption and its correlation with the nuclear EOS so that we compare the results obtained with different pion mean free paths for the same EOS. The solid, dashed and chain-dotted lines in fig. 3 correspond to the cases with pion mean free paths A..= 00, 10 fm and 5 fin, respectively. The system under consideration is 20Ne + 20Ne at Elab/ A = 200 MeV. In fig. 4, the pion cross section with four different equations of state are directly compared for a light ( 20Ne + 2ONe) and a heavy (93Nb+93Nb) system. Figs. 5-7 demonstrate also the difference in the pion cross sections with different equations of state, for three systems 12C + 12C, 'Ca+Ca and 93M +931,qb, respectively. In these figures, however, we want to stress the importance of different time intervals (early-time versus later-time contributions) to the pion cross sections and its correlation with the mass number of the colliding system under consideration.Therefore, we present in each figure the total cross section (summed over the whole time interval of 60 fm/ c) as well as the contributions from time intervals (0, 15 fm/ c), (15, 21 fm/ c) and (21, 60 fm/ c) with solid, dashed, chain-dotted and dash-dotted lines, respectively . All the resukul depicted in figs. 4-7 are obtained with A, = 5 fin. We want to draw the following obset-vations from these figures: (i) The pion reabsorption has a very strong effect on the pion production in nuclear medium. The difference between the cross section with pion reabsorption (A = 5 fin) and without the reabsorption (A = infinity) amounts to about 50% for all the system considered here. (ii) The influeiice of pion reabsorption on the pion cross section depends in principle on the nuclear EOS used in the calculation through the dependence of pion mean free path on the nuclear density. This effect is more apparent for a soft EOS than for a hard EOS. When a soft EOS is used, the nuclear matter is easier to be compressed, and a higher density can be reached, thus the reabsorption of pions is more significant. But since we are using here a density-independent mean free path A, = 5 fin, this effect is not included and cannot be seen in the present results. (iii) The difference between the cross sections with different incompressibilities (soft or hard EOS) is not very significant, and depends very much on the mass

G. Li et aL / Subthreshold pion production

709 Né+

'eV/A (S)

Ne+Ne 200 MeV/A (H)

Ne+Ne 2W

eV/A (SMD)

Ne+Ne 200 MeV/A (HWJ)

5__~'b %U -I :x

-4

0

40 E7T

80

(MéV)

1200

40

80

E7T

(MéV)

120

Fig-31 . Double-differential cross section for neutral pions emitted at 0,.,,, . = 90' from 20Ne + 20 Ne collision at Elab/A = 200 MeV. The different equations of state used in the calculation are indicated by (a) S and H and (b) SMF) and HMD. We stress in this figure the importance of pion reabsorption .

G. Li el al. / Subthreshold pon production

Ne+Ne 200 MeV/A

Eir (MéV)

709

-Nb+Nb 200 'eV/A

Eir (MéV)

Fig. 4. A direct comparison of ion production cross section with four equations of state as indicated for two colliding systems: 20Ne+ 20Ne and 93Nb+ 3Nb.

number of the colliding system . For a heavy system, the difference is more apparent. This difference is a direct result of the different collision numbers obtained with different EOS. With a hard EOS, we have stronger repulsion and fewer nucleonnucleon collisions, leading to a slightly smaller cross section . For a heavy system, the reduction of collision numbers with a hard EOS is obviously more drastic . But we should note again that the difference in the pion cross section induced by different incompressibilities is strongly correlated with the pion reabsorption factor, or the pion mean free path. It is clear that from this difference no definite conclusion can be drawn on the incompressibility of nuclear matter, unless the pion reabsorption factor is seriously and soundly determined. (iv) On the other hand, we observe a very large difference between the pion production cross sections with and without the momentum dependence. The pion production cross sections with a momentum-dependent interaction are much smalfler than those with a momentum-independent interaction . In most cases, !~he aiffe.rence amounts to a factor of three for a small incompressibility, whilst the difference. in cross sections because of different incompressibilities is only about 30% . The momentum dependence of the interaction affects the cross section also through the reduction of the collision numbers. When the momentum dependence is introduced, we get additional repulsion among the nucleons . The repulsion is stronger for two nucleons with larger relative momentum [cf. eq. (6)] . This fact has such a direct and important consequence that the number of effective NN collisions which

G. Li et at. / Subthreshold pion production

710

C+C 200

~.'eV/A (S)

C+C 200 MeV/A (H)

C+C 14,00

eV/A (S

C+C 200 MeV/A (HMD)

TOW O
Fig. 5. Double-differentiai cross section for neutral pions emitted at 0C.~. = goo from 12C + 12C collision at Elab/A = 200 MeV . The different equations of state used in the calculation are indicated by (a) and S and H, and (b) SMD and HMD. We stress in this figure the time evolution of the pion cross section.

G. Li et at. / Subthreshold pion production

Ca+Ca 200

Ca+Ca 200 MeV/A (S)

'eV/A (H)

lb

M 'bM-O 'b V-4 zz tD Ici

el 10 ,..-4 0

40

80

Eir (MéV)

120

Ca+Ca 203 MeV/A (SMD)

Fig. 6. Same

as fig. 3 but

1600

40

so

Eir (MéV)

120

Ca+Ca 200 MeV/A UM)

for a 4OCa +40Ca collision at

Elab/ A = 200

MeV.

160

G. Li

712

at. / Subthreshold pion production

ef

Nb+Nb 200 MeV/A (H)

Nb+Nb 2W .;'eV/A (S)

%

101ïm 1-4

V-4

0

TOW ------ O
"

1.

11

b

N

1.

1%

1..

IT

a

40

160

E7T

0

TOW ------ O
80 E-ff

Nb+Nb 200 .eV/A (SMD)

11, 1

a

lm Offl)

1 ()

Nb+Nb 200 MeV/A (HMD)

"0 -4

4)

C:

-4

tD Iq

ce ,CD

-4

------- O
40 7.

Same

,

so 120 E7T (MêV) as

fig .

5

but

for

160 0 a

93

40

80 120 ElT (MêV)

Nb +93 Nb collision at Elab/ A= 200 MeV .

160

G. Li et al. / Subthreshold pion production

713

otherwise would lead to the production of pions is drastically reduced . This is probably the reason why the momentum dependence is much more effective than the incompressibility in reducing the pion cross section . For the kaon "') and the photon 22) productions, one observes a similar behavior. But the reduction of the photon production cross section with momentum dependence is not so drasstic, because there we have no threshold . We stress here again that, because of the szrong; pion reabsorption and because ofthe significant effect ofthe momentum dependence in the interaction on the pion production cross section, pions are not very promising probes for the nuclear EOS, especially not for its incompressibility. (v) When looking ito figs. 5-7, we notice some interesting phenomena on the time evolution of the pion cross section, which we have also observed in the case of photon production. The contributions to the pion production cross section from different time intervals depend strongly on the mass number of the colliding system. For a small system, such as 12C+ 12C, the contribution from the first 15 fmlc (dashed line in fig. 5) is much more important than that from the second time interval (chain-dotted line in fig. 5). In the case of momentum dependence, the former nearly exhausts the total cross section (solid line in fig. 5). For this small system, there are essentially no pions from a time later than 21 fm/c, since after that time NN collisions are very rare. With the increase of the mass number of the colliding system, the contributions from the second (15,21 fm/c) and third time intervals (21, 60 fm/ c) become increasingly important. For a medium heavy system, such as 4OCa+ 4OCa, the contributior s from the first two time intervals are of the same order of magnitude [the most substantial contribution is from the time interval (15,21 fm/c) within which both maximum central density and NN collision rate reach their peaks], whereas for a heavy system such as 93 Nb +93 Nb, the contribution from the last time interval (dash-dotted line in fig. 7) is more significant than those from the first two time interval (dashed and chain-dotted lines in fig. 7). We have checked that in this case the contribution to the pion production cross section is essentially from the time interval (15, 30 m/ c). This behavior follows closely the time evolution of the number of NN collisions and the maximum central density. In the second part of this section, we present some comparisons between our theoretical results with the available experimental data. We want to point out that, although there have been many theoretical calculations on the pion production in nucleus-nucleus collisions, most of them are concerned either with very low energy (below Elab/A = 100 MeV) [refs . 11,28,33-35)] or with very high energy (around 36,37 )] . As far as we know, there have been few theoretical Elab/A = I GeWu) [refs. attempts to describe the pion production in an energy region between -E la b/A= 200 MeV and the pion threshold, although there exist experimental data in this region. In this view we think that our present comparison is of interest, beyond the fact that we use for the first time in both the simulation and the elementary pion production probability a medium-dependent NN cross section obtained from the Bethe-Goldstone equation .

6. Li et al. / Subihreshold pion

714

production

ID

4

C:

M ,CD

400

300

200

PL,b (M, eV/c)

OLI,b=96*

'12 0 -4

0

F

0

a

a 0 X 0

40

0

0 F

. -4

0 0

A 0

M ,CD

lo

.0

- (d)

i

0

f. 1

100

1

i

400 200 300 PL,b (MeV/C) Fig. 8. The Lorentz-invariant double-differential production cross section for negative pions from a 20Ne + 2' Ne collision at Ejb/ A = 183 MeV as a function of the pion momentum, emitted at Olab = (a) 20', (b) 40*, (c) 70' and (d) 90*. The results obtained with different equations of state are compared with experimental data from ref. 38 ). The open symbols indicate the four EOS used in the calculation: triangle: S, plus: H; cross: SMD, diamond: HMD. The full circles give the experimental data.

G. Li et A / Subthreshold pion production

715

-4

OLab -:~

M J3

9

N4.,, I-N -4

0

4 go

,CD -f m

,CD 4

200

100 PL,b

(MeV/C)

Fig. 8-continued

300

400

eshold pion production

716

ntial cross section for negative pions from a The Lorentz-inv "Ne+ 'Ne collîsio eV is depicted in fig. 8 as a function of pion vhere (a), (b), (c) and (d) correspond to pions momentum in the la ctively. The theoretical results have been emitted at Oi,,b = 20', in the figure . The experimental data are obtairied with differe taken from ref. ") for a roximately equal in mass to the projectile) collision ai the same bom In figs. 9 and 10, the Lorentz-invariant double-differential cross sM-i sitive (fig. 9) and negative (fig. 10) pions emitted at Olab = 00 in a "O Ne+ collision at El.b/ A = 250 Me are presented as a function of the pion kinetic e the laboratory system, together with the 39) . experimental data from ref erally speaking, our theoretical results are in agreement with the experiment data. In most cases, the discrepancy between theoret', cal and experimental results is within a factor of two, depending on the nuclear EOS used . Here we observe again that, for light ( 29 Ne + 20Ne) and heavier ( 20 Ne + 64Cu) systems, the influence of the EOS on the pion production cross section is not the same. From a small system, soft EOS leads to a smaller cross section, especially for low energy pions, whereas for a heavier system the cross section with

to

Fig. 9. The Lorentz-invariant do-able-differential production cross section for positive pions emitted at from 20Ne + 64CU collision at E1 /A = 250 Olab ": 00 MeV . The results obtained with different equations ,b of state are compafed with experimental data from ref. '9). For the meaning of the symbols in the figure, see the caption of fig. 8.

G. Li et al. / Subthreshold pion production

717

OLab~::~

*0

1

>0

A

'b

X O

1.4

'bI

0

X 0

+ A

0

V,

*2 ,CD

_f

to 0

0

100

Ej.b (MOV)

200

Fig. 10. Same as fig. 9 but for negative pions.

a soft EOS is larger than that with a hard EOS. Looking at these figures more carefully, we find that the theoretical results for high-energy (momentum) pions are always smaller than the corresponding experimental data. There is also an indication that we have more low-energy (momentum) pions than we should have .

4. Conclusions and perspectives

The present work is concerned with the subthreshold pion production in the + 12C nucleus-nucleus collisions around Ej,b/ A= 200 MeV. Five., systems, from to 93 Nb+ 93 Nb, have been studied . The reason we choose this energy rt,.,gion is threefold. First, there have been few theoretical calculations for pion prodli- iction in nucleus-nucleus collisions in this region . Second, this is the energy range to which the nonrelativistic QMD approach is applicable . Third, one expects that the niedium dependence on the NN cross section is in this energy region very large so that the use of in-medium NN cross sections is necessary. We already noticed that the theoretical cross sections for high-energy (momentum) pions are always smaller than the corresponding experimental data. This is in fact an intrinsic deficiency of QMD which cannot reproduce accurately the Fermi momenta of the initialized !~

It

71 8

G. Li et al. / Subthreshold pion production

nuclei and at the same time keep their stability for long time. In addition, QMD is a classical approach and does not include the high momentum tail above the Fermi momentum described quaritum mechanically . These high momenta around and above the Fermi momentum play a very important role in the subthreshold pion production. For nucleus-nucleus collisions at energies lower than the pion threshold, one should, on the one hand, improve the initialization in the QMD approach in order to have the right Fermi momentum and include the high momentum tail. On the other hand, this is also the energy region where a possible coherent production of subthreshold pions should show up. For nucleus-nucleus collisions at around Elab/ A= I GeV, where one has also many experimental data, relativistic effects might be important. In the present QMD approach, the kinematics in solving the Hamilton equations of motion is already relativistic. But the dynamics is essentially nonrelativistic and violates Lorentz covariance. The relativistic QMD (RQMD) approach has already been established based on the constraint Hamilton dynamics 40-42) . In the next step, we are going to calculate the pion production from nucleus-nucleus collisions around Elab/ A = I GeV, within both the relativistic and nonrelativisic QMD approaches, in order to see the importance of a fully relativistic treatment of high-energy nucleus-nucleus collisions. We have also made a detailed analysis of the dependence of the pion production cross section on the different ingredients of our calculation in the present paper. The important conclusions are: (i) The difference in the pion production cross sections with different incompressibilities ofnuclear matter is not significant enough to render subthreshold pions as suitable probes for this quantity . (ii) The situation is even worse because of uncertainties in the pion reabsorption and the momentum dependence of the interaction, which influence strongly the pion production cross sections. Thus we cannot draw definite conclusions on the relation between the stiffness of the nuclear EOS and the pion production cross section. (iii) The contributions to the pion production cross section from different time intervals are strongly correlated with the mass number of the colliding systems. With the increase of the mass number of the colliding system, the contribution from later-time NN collisions becomes increasingly significant. We find that not all subthreshold pions are from the first-chance NN collisions . For subthreshold 'q particles, ref. ") gives a similar conclusion . We would like to point out that the elementary cross section for pion production in NN collisions is not very small. Thus the perturbative approach as used in the present work will not be very sound if used in high-energy nucleus-nucleus collisions around Elab/ A = I GeV. (Below the subthreshold, about one out of 500 NN collisions is effective for pion production so that the perturbative approximation is reliable.) One possibility to avoid this unreliable assumption is to take into account explicitly the production, propagation and reabsorption of pions in the simulation code (QMD/RQMD calculation), such that the pions are no longer treated perturbatively but on equal footing with nucleons .

G. Li et at. / Subthreshold pion production

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The authors thank Professor M. Ismail for help with the modifications in QMD code. One of us (D.T.K.) acknowledges a research grant from the von Humboldt Foundation. Referewn I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42)

H. St6cker and W. Greiner, Phys. Reports 137 (1986) 277 N.K. Glendening, Phys. Rev. C37 (1988) 2733 G.W. Welke, M. Prakash, T.T.S. Kuo and S. Das Gupta, Phys. Rev. C38 (I H .H . Gubrod, K.H. Kampert, B. Kolb, A.M. Poskanzer, H.G. Ritter, P. Schi Phys. Rev. C42 (1990) 640 G.F. Bertsch, H. Kruse and S. Das Gupta, Phys. Rev. C29 (1984) 673 J. Aichelin and G.F. Bertsch, Phys. Rev. C31 (1985) 1730 J.J. Molitoris and H. St6cker, Phys. Rev. C36 (1987) 220 W. Bauer, G.F. Bertsch, W. Cassing and U. Mosel, Phys. Rev. C34 (I )217 G.F. Bertsch and S. Das Gupta, Phys. Reports 160 (1988) 189 C. Gregoire, B. Remaud, F. Sebiller and L. Vinet, Nud. Phys. A468 (1987) 321 W. Cassing, V. Metag, U. Mosel and K. Nitta, Phys. Reports 18B (1990) 363 J. Aichelin and H. St6cker, Phys. Lett . B176 (1986) 14 J. Aichelin, G. Peilert, A. Bohnet, A. Rosenhauser, H. St6&er and W. Greiner~ (1988)2451 G. Peilert, A. Rosenhauser, J. Aichelin, H. St6cker and W. Greiner, Phys. Rev. C39 (I A. Bohnet, N. Ohtsuka, J. Aichelin, R- Linden and A. Faessler, Nucl. Phys. A494 (I P. Danielwicz and G. Odyniec, Phys. Lett. B157 (1985) 146 H. Stdcker, J. Maruhn and W. Greiner, Phys. Rev. Lett. 44 (1980) 725 J. Aichelin, Phys. Lett. B164 (1985) 261 7)1926 J. Aichelin, A. Rosenhauser, G. Peilert, H. St6cker and W. Greiner, Phys. Rev . Lett. C. Gale, G.F. Bertsch and S. Das Gupta, Phys. Rev. C35 (1987) 1666 N. Ohtsuka, M. Shabshiry, M. Ismail, A. Faessler and J. Aichelin, J. of Phys. G16 (1990) L155 Dao Tien Khoa, N. Ohtsr-ka, S.W. Huang, M. Ismail, A- Faessler and J. Aichelin, AS" (1991) 363 J. Cugnon, T. Mizutani and J. Vandermeulen, Nucl. Phys. AM2 (1981) 505 T. Izumoto, S. Krewald and A. Faessler, Nud . Phys. A341 (1980) 319 M. Treft, A. Faessler and W.H. Dickhoff, Nucl. Phys. A443 (1989) 499 A. Faessler, Nucl. Phys. A495 (1989) 103c B.J. VerWest and R.A. Arndt, Phys. Rev. C25 (1982) 1979 C. Guet, Nucl. Phys. A428 (1984) 119c W. Zwermann and B. Schiirmann, Nucl. Phys. A423 (1984) 525 J. Randrup and C.M. Ko, Nucl. Phys. A343 (1980) 519; A411 (1983) 537 B.T. Haar and R. Malfliet, Phys. Rev. C36 (1987) 1611 B. Schfirmann, W. Zwermann and R.Malfielt, Phys. Reports 147 (1987) 71 B. Blann, Phys. Rev. C32 (1985) 1231 M. Tohyama, R. Kaps, D. Masak and U. Mosel, Nucl. Phys. A437 (1985) 739 M. Tohyma and U. Mosel, Nucl. Phys. A459 (1986) 739 Y. Yariv and Z. Fraenkel, Phys. Rev. C20 (1979) 2227 J. Cugnon, Phys. Rev. C22 (1980) 1885 S. Nagamiya, et aL, Phys. Rev. Lett. 48 (1982) 1780 W. Benenson et al., Phys. Rev. Lett. 43 (1979) 683 P.M. Dirac, Rev. Mod. Phys. 21 (1949) 392 H. Sorge, H. St6cker and W. Greiner, Ann. of Phys . 192 (1989) 266 T. Maruyama, S.W. Huang, N. Ohtsuka, G.Li and A. Faessler, Nucl. Phys. A534 (1991) 720