Nuclear Physics A515 (1990) 747-760

North-Holland

THE RELATIVISTIC BUU APPROACH:

Analysis of retardation effects and meson-field radiation*

Klaus WEBER, Bernhard BLATTEL, Volker KOCH, Andreas LANG,

Wolfgang CASSING and Ulrich MOSEL

Institut fiir Theoretische Physik, Universitiit Giessen, D-6300 Giessen, Fed. Rep. Germany

Received 12 February 1990

(Revised 1 May 1990)

Abstract: We simulate relativistic heavy-ion collisions within the relativistic BUU model employing a

full solution of the meson-field equations. The results are compared to those obtained in the local density approximation used in previous calculations. Both with and without a collision term, we

find good agreement up to projectile energies of 5 GeV/nucleon. Retardation and radiation effects

are visible; however, their influence on the reaction dynamics and observables is negligible. The

amount of energy radiated off the system is found to be less than 3 MeV/nucleon.

1. Introduction

In the last decade the study of relativistic heavy-ion collisions with projectile

energies ranging from several hundred MeV/nucleon to a few GeV/nucleon has

become an important section of experimental nuclear physics. If offers the unique

possibility to investigate hot nuclear matter at high densities or nuclei far from

equilibrium.

A variety of nonrelativistic models has been developed to describe low- and

intermediate-energy heavy-ion collisions. Especially transport theories like the BUU

or VUU approach have proven to be successful approximations to the respective

many-body problem. However, once the kinetic energy per nucleon reaches its rest

mass, one expects genuine relativistic effects to become important and therefore

should use a fully relativistic theory, which does not only employ relativistic

kinematics but is based on a covariant dynamical field theory.

An effective field theory - which has had considerable success in recent years -

is the Walecka model (QHD-I) ‘). Its elementary degrees of freedom are nucleons

interacting with a Lorentz-scalar meson field u as well as a Lorentz-vector meson

field w. Based on this theory a relativistic BUU approach has been formulated 2-6)

which in the semiclassical limit h + 0 yields a transport equation for the nucleon

phase-space distribution function similar to the nonrelativistic BUU-equation ‘3’).

Residual nucleon-nucleon interactions in this model are described by a collision

term modeling two-particle correlations. A more rigorous derivation of covariant

transport theories can be found in refs. 93’o).

l Supported by BMFT and GSI Darmstadt.

0375-9474/90/$03.50 @ Elsevier Science Publishers B.V. (North-Holland)

748 K. Weber et al. / Relativistic 5UiJ approach

The big advantage of transport equations lies in the fact that they can be solved fairly easily within the test-particle method **“). The meson fields, on the other hand, obey inhomogeneous Klein-Gordon equations of motion. Since the solution of these equations involves some numerical effort, previous calculations have made use of the local-density approximation (LDA), in which all derivatives with respect to space and time in the Klein-Gordon equations are neglected. In doing so, one destroys the dynamical independence of the meson fields, relating them directly to the sources.

Whereas neglecting the finite range of the interactions should be justifiable due to the large mass of the mesons and consequently short range of the forces, the question arises whether at relativistic energies the neglect of the time derivative remains justified. The full solution of the equations of motion might exhibit retarda- tion effects as proposed by Cusson et ai. ‘*), free radiation of the meson fields, as well as dynamical instabilities 13*14). All these effects cannot be reproduced by the LDA.

In this work we will, therefore, compare simulations of heavy-ion collisions employing the LDA with those of the full solution (FS). To distinguish finite-range effects from those mentioned above, we present the quasistatic approximation (QSA), where finite-range effects as well as some trivia1 time dependence are included.

2. The model

The relativistic BUU approach is based on a transport equation for the nucleon phase-space distribution function f(x, II) [ref. “>I

(n& + (gJI”F&“” + m*(a:m*))a:)f(x, IT)

1 d311, d317’ d317: --‘- -- 2 s

x (f(& ml-(x, Z)(l -sw3f(~, W)(l --at2~)‘_f(x, 111))

-ffx, J7).f(x, fll)(l -%w3m, W)(l 4(W3fk Cl)) (2.1)

where all momenta on the r.h.s. are on-mass-shell, e.g,

(2.2)

Here 17” = pee - g,wP is the kinetic momentum, m” = M - g,a the effective mass of the nucleons and F *“y = Y”w” --a”~” the field strength tensor for the w-field.

Neglecting the r.h.s. of the equation we are left with a Vlasov-type equation describing nucleons moving on smooth trajectories in the mean field generated by the meson fields. The term 17,,Fp” in eq. (2.1) shows that this relativistic approach automatically leads to a momentum dependence of the mean-held potential. This momentum dependence has been investigated experimentally via nucleon-nucleus scattering 15); one observes a potential which is attractive for bombarding energies

K. Weber et al. / Relativistic BUlJ approach 149

below 200 MeV and turns repulsive above. Whereas this potential increases almost

linearly up to 300 MeV, it saturates at higher energies. In the Walecka model the

optical potential increases linearly with energy ‘). The slope of this increase depends

on the inverse of the Fermi-liquid effective mass, which is close to the relativistic

effective mass m* in nuclear matter ‘6-‘8). Thus, especially for a low effective mass

(m* = 0.56M in case of QHD-I) the repulsive force will be grossly overestimated

at high energies 16). One thus has to keep in mind that we might overestimate flow

effects and stopping for bombarding energies larger than about 600 MeV/nucleon.

The r.h.s. of eq. (2.1) is the collision term which to some extent takes into account

two-particle correlations: It is given by the probability to find two nucleons at the

same space-time point, multiplied by a scattering rate involving the in-medium cross

section du/dfl and by the available final-state phase space. Pauli blocking factors

are included in this expression. This collision term can be derived from Dirac-

Brueckner theory in the local density approximation assuming on-shell collision

processes 9,1o). I n our model, the real part of the nucleon self-energy .Z is described

by a scalar field u and a vector field We [refs. 5*‘o)]. The imaginary part of .Z defines

the in-medium cross section which at high energies can be well approximated by

free transition rates lo). In present calculations we thus use a parametrization of the

free nucleon-nucleon cross section adopted from Cugnon 19) without taking care

of possible further medium corrections 9,10,20).

Eq. (2.1) describes the dynamics of the nucleons only. It depends on the meson

fields which in this relativistic model have their own dynamics governed by

inhomogeneous Klein-Gordon equations

a2 ~~--V2u+m:o=gSp,,

d2

SW p -V2wp + mzwcL = gvjw ,

with the nucleon current

jp(x) = d’nF/(x, n) 0

and the scalar density

ps(x) = d’L$$(x, n) . 0

(2.3)

(2.4)

(2.5)

(2.6)

The closed set of equations of motion (2.1) to (2.4) has to be solved simultaneously.

We note that within the derivation of the transport equation (2.1) we have assumed

that the products hd,ua,J(x, n) and hd,w@‘d=j(x, n) are smaller than unity. In

solving eqs. (2.3) and (2.4) one should therefore not take into account very high

frequencies and short wavelengths. In practice, this turns out to be no problem

750 K. Weber et al. / Relativistic BUU approach

since the phase-space distribution is found to be a rather smooth function of the

momentum (especially when including residual nucleon-nucleon scattering)

whereas oscillations of the meson fields show a minimum wavelength of about 2 fm

and corresponding frequencies.

All calculations presented in this work were performed using the original para-

meter set QHD-I [ref. ‘)] (cf. table 1). It leads to a very stiff equation of state (with

a compressibility of K = 540 MeV and an effective mass of m* = 0.56M) as well as

an unrealistic saturation density of p0 = 0.19 fme3. However, these deficiencies are

not important for our arguments.

The meson-field equations may be treated in several ways: - In the full solution (FS) we solve these equations exactly. Thus, the FS contains

all retardation and radiation effects. - In the quasistatic approximation (QSA), one approximates the time derivative by

the value for an undisturbed field moving with constant velocity u in the z-direction.

For symmetric systems in the center-of-mass frame this yields

(2.7)

2

vZLo”-V2w”+m$0’=g,jP. dZ2

(2.8)

Thus the QSA accounts for the finite range of the meson fields as well as for some

“trivial” time dependence and should be reasonable as long as the nuclei are not

significantly stopped during the collision and retardation and radiation effects are

small. - In the local-density approximation (LDA) the equations of motion are reduced to

m5a = g,p,, (2.9)

m$oP = gvjp. (2.10)

Due to the large meson masses this should be valid for fields varying smoothly in

space and time, e.g. large nuclei colliding at low bombarding energy. The LDA has

already been successfully applied in previous calculations 2’-23).

Another possible approximation, which is related to the QSA and will therefore

not be discussed in detail here, is the small acceleration approximation proposed

by Feldmeier and collaborators 24).

Here we will present results for the full solution of the meson field equations and

compare to those obtained in the LDA as well as QSA.

TABLE 1

Parameters used in our calculations

m,(fm-‘) m, (fm-‘) g\ g”

2.79 3.91 9.51 11.67

K. Weber et al. / ~e~atiuisti~ BUU a~p~ouc~

3. Numerical realization

751

To solve eq. (2.1) numerically, we employ the test-particle method as introduced

by Wong into the nuclear physics context ‘I), i.e. we represent the distribution

function f(x, II) by a sum of a-functions

f(x, II)+ y s3(x-xj(t))63(rI-rIj(f)) * I I

(3.1)

Here IV is the number of test particles per nucleon, whereas X, and & are the

coordinates in phase space of the ith test particle. This ansatz leads to the following

equations of motion for a single test particle

a&i = IIi,/ITP, (3.2)

a,n: = g, 2 Fk’“(Xi) ++ m*(x,)ak,( m*(x,)) , I I

(3.3)

for k = 1,2,3 which are solved by a standard predictor-corrector scheme. In our

calculations we use 100 to 1000 test particles per nucleon to minimize fluctuations

in the density; the collisions are treated in the usual way by employing the parallel

ensemble method “).

In contrast to the test particle dist~bution the densities and fields are evaluated

on a fixed grid. The integrals in (2.5) and (2.6) then are replaced by a sum over all

test particles in one grid cell. Most of our calculations are performed on a grid of

dimensions 32 x 32 x 64 with mesh sizes of 0.5 to 1.0 fm. To determine the radiation

energy produced in the case of the FS we employed a 65 x 65 x 85 grid with mesh

size 0.7 fm.

At the initial time step the test particles are distributed such that the resulting

densities and meson fields correspond to the self-consistent solution of the static

Walecka model in the Thomas-Fermi approximation 25X26).

In the LDA the meson fields can be calculated directly from the source terms;

the selfconsistency relation connnecting c and ps in eq. (2.6) and eq. (2.9) (with

m” = M - g,a( m*)) is solved by iteration at each grid point.

In the case of the QSA a Fourier transformation of eq. (2.7) leads to

(3.4)

with y = m and the corresponding equations for the four components of the

vector field 6~~. These equations are solved using fast Fourier transformation tech-

niques. Here, too, the selfconsistency relation between scalar field and scalar density

has to be accounted for by iteration.

‘To obtain the full solution we employ a predictor-corrector scheme of variable

order for direct integration of second-order differential equations. The necessary

starting vaiues for the integration scheme are obtained using the QSA.

752 EC. Weber et al. / Relativistic BUU upproach

To check our numerical methods we have investigated the stability of finite nuclei in time as well as the energy ~onse~ation during a collision. This is much more critical than in corresponding nonrelativistic treatments. Since the mean field in the Walecka model is given as the difference of two quite large potentials (several hundred MeV each), relatively small errors in scalar and vector field can lead to large errors in the mean field and thus in the time propagation of the system. In our calculations we find a decrease in the total energy (nucleon mass subtracted) of about 5% during a collision. This is accurate enough for our purposes.

The r.m.s. radius of static nuclei exhibits oscillations with an amplitude of about 5% of the mean value. However, since the nucleus remains stable for at least 80 fm and the period of these oscillations is of the order 30 fm, they are irrelevant for simulations of high”energy collisions, where the whole reaction takes place in about 20 fm or less (see for example fig. 1 below).

4. Validity of the LDA and QSA

We first investigate the range of validity for the approximate solutions of the meson field equations. For this purpose we first concentrate on pure Vlasov calcula- tions. We have performed Vlasov calculations for the systems 160+ 160, 4oCa + 40Ca and 19’Au + 197Au at projectile energies ranging from 0.6 to 14 GeV/nucleon. At the highest energy the physical content of our mode1 is probably not sufficient to describe the baryon-meson dynamics correctly, but we expect our more general conefusions on relativistic effects to be quite reliabIe.

Fig. 1 shows the time evolution of the baryon density for 40Ca+40Ca at 1 GeV/nucieon employing the FS for a central collision. We find that the nuclei pass through each other without any significant stopping. This is the expected result of a pure mean-field calculation. Another feature is the strong inflation of the nuclei in transverse direction, leading to a rapid decrease of the central density.

The momentum-space density ~(17, t) for the same times also exhibits this sideward expansion (fig. 2): during the maximum overlap the momentum distributions acquire a typical triangle shape. While there is only little filling of p(fl, t) at II = 0, transverse momentum is created, especially at high longitudinal momenta. The reason for this behaviour can be found in eq. (3.3). The first term on the r.h.s. is analogous to the well-known Lorentz force of relativistic electrodynamics. This force leads to a transverse momentum of the test particles proportional to their longitudinal velocity which then yields the observed shape.

We find that the changes in momentum space are completed long before the nuclei separate again. The further evolution of the system consists basically of uniform motion.

Fig. 3 shows the final projectile rapidity distribution for the same collision at t = 22.2 fm. The initial distribution at t = 0 is included to demonstrate the stopping and widening which has taken place. Obviously the approximate solutions reproduce

K. Weber et al. / Relativistic 3fJU approach 753

Fig. 1. Contour plot of the baryon density in the plane y = 0 for the collision 4”Ca +?Za at 1 GeV/nucleon

and b = 0 fm. Six consecutive timesteps from first overlap to almost complete separation are shown.

-I - -a- _t , , , ( ( a? b I b a ’ * ‘ /--I * s t= 14.7 fIw 6- t= 17.8 iwn t= 22.2 ha’

I-

- 7 t-

2 “y ‘: _s_ - f1.00~10-2 a - -.-‘. +1.00~10-’

-4 - : ...... +2*()().10-’ -I - : ---- +3.00~10-’

-(I-, , , , , , -8 -6 -1 -1 0 2 1 6 LI -8 -6 -1 -2 0 2 1 I * -II -8 -, -1 0 z 1 I 8

p,(fm-‘) p&m-‘) pAfm_‘)

Fig. 2. Contour plot of the integrated momentum-space density in the p._ p,-plane for the collision

40Ca+40Ca at 1 GeV/nucleon and b = 0 fm. The time steps are the same as in fig. 1.

754 K. Weber et al. / Relativistic BUU approach

Y

Fig. 3. Rapidity distribution of the target after the collision for 40Ca+40Ca at 1 GeV/nucleon and

b = 0 fm. Solid line: FS; dashed line: QSA; dash-dotted line: LDA; dotted line: initial configuration.

the FS almost perfectly. This statement holds up to at least 5 GeV/nucleon for all systems. Whereas the stopping is almost zero for light systems as 160+ %, we find a considerable increase in stopping with mass number. However, even for ‘97Au+ 19’Au target and projectile distribution remain well separated in momentum space, in contrast to calculations including the collision term (cf. fig. 6).

At 14 GeV/nucleon noticeable differences between LDA on one and QSA and FS on the other hand can be found. In fig. 4 we show the rapidity distribution for 40Ca+40Ca at 14 GeV/nucleon and zero impact parameter. Since only the LDA shows deviations from the FS, we conclude that they are caused by the neglect of finite range in this approximation. This in general leads to larger gradients in the equations of motion (3.3). Because of Lorentz contraction this effect becomes more important at higher energies and causes an enhanced stopping.

Fig. 4. Rapidity distribution of the target after the collision for 4oCa+40Ca at 14GeV/nucleon and

b = 0 fm. Solid line: FS; dashed line: QSA; dash-dotted line: LDA; dotted line: initial configuration.

K. Weber et ai. / Relativisric BUU approach 755

I ‘ I I I ( I I I <I -4 -2 0 2 4

p,(fm-‘l

Fig. 5. Transverse momentum per particle after the collision for 40Ca+4”Ca at 1 GeV/nucleon and b = 2 fm. Solid line: FS; dashed line: QSA; dash-dotted line: LDA.

Let us next turn to the transverse momentum distribution. An important observable in collisions with finite impact parameter is the transverse flow. Fig. 5 depicts the transverse momentum per particle for the collision 40Ca+40Ca at 1 GeV/nucleon and b = 2 fm impact parameter. Again, LDA, QSA and FS yield exactly the same results within the numerical accuracy. This is true for all systems and energies up to 5 GeV/nucleon.

An interesting side-result comes from the comparison of the transverse flow for different mass numbers, provided the collision geometry is the same for all systems. This is achieved by setting the impact parameter equal to some fraction of the r.m.s. radius, in our case roughly b = 0.6R. One finds that, in contrast to the system I60 + IhO, the results for 40Ca + 40Ca and 19’Au + 19’Au are very similar. We conclude that the transverse momentum per particle produced by the mean field is independent of the mass once surface effects become unimportant.

Fig. 6. Rapidity distribution after the collision for 40Ca+40Ca at 1 GeV/nucIeon and b = 0 fm. Solid line: FS; dashed line: LDA. The collision term is included in this calculation.

756 K. Weber et al. / Relativistic l3Ul.i approach

In conclusion, we can state that in pure Vlasov calculations both LDA and QSA reproduce the dynamical evolution very well. This could be explained by the fact that in these calculations the nuclei pass through each other without any significant stopping.

One would therefore expect larger discrepancies once the collision term is included since it leads to almost total stopping and hence to a much more drastic difference between initial and final state of the system. Surprisingly, calculations which include the collision term yield the same results. Fig. 6 demonstrates this for the final rapidity distribution of the system 40Ca+40Ca at 1 GeV/nucleon.

5. Retardation effects

In the previous section we have shown that the baryonic observables are hardly affected by genuine relativistic effects. In this section we will now look for such effects directly in the meson fields.

One of the phenomena that cannot be reproduced by the local density approxima- tion is retardation, since it is caused by the finite range as well as the finite group velocity of the meson fields. The quasistatic approximation yields the correct field for sources moving at constant velocity, but fails to reproduce nontrivial retardation effects due to some acceleration of the system or dynamical changes in the source densities. Therefore, discrepancies between the QSA solution and the full solution should become observable in the fields when two criteria are met: First, there must be a genuine change in time in the source densities which is not due to some trivial time dependence like e.g. uniform motion. Second, the rate of change must be large enough to lead to observable effects. To illustrate the first point, imagine two nuclei passing through each other without any interaction. Although the baryon density changes rapidly, the w-field calculated in the QSA will be identical to the FS since - due to the additivity of the baryon current - it is always simply the sum of two properly boosted fields. This is not so in the case of the scalar field whose source term depends - via the effective mass - on the field itself.

This already gives us a first hint where to look for retardation effects: Since, in contrast to the baryon current, the scalar densities of target and projectile do not simply add up to the total scalar density in the overlap zone, the scalar field may experience a nontrivial change at the beginning and the end of the overlap phase. To detect these effects, we compare the field amplitudes calculated employing the QSA and the FS.

Indeed, we find almost no differences for the w-field, whereas clear evidence of retardation can be seen in the a-amplitude at a projectile energy of 14 GeV/nucleon. In fig. 7 we compare the a-amplitude calculated using QSA and FS for the collision 40Ca+40Ca at this energy. Some timesteps during and after the collision are shown. It can be seen very clearly that during the first three time steps the amplitude of the FS “follows” the one of the instantaneous QSA with some delay. For later times

Fig. 7. Amplitude of the a-fietd during and after the collision for *°Ca+40Ca at 14 GeV/nucfenn and b = 0 fm. Solid iine: FS; dotted fine: QSA.

we observe oscilIations in the baryon-free space between the nuclei even after they are well separated again. These die out quite slowly in time. At lower energies this effect vanishes; it cannot be found below 5 GeV/nucleon. At the beginning of the overlap phase we observe some discrepancies, too, but these die out again.

Although retardation leads to substantial differences in the field amplitude, it obviously has no detectable influence on the reaction dynamics, as we have already demonstrated in the previous section (cf. fig. 4). Especially the amount of energy deposited in the free oscillations is not large enough to lead to a noticeable loss of kinetic energy of the system.

Note that the effect shown in fig. 7 is particular to VIasov calculations; in calculations including the collision term it does not occur due to the sizeable stopping of the nuclei and the corresponding scalar source. In this case, however, one would expect a much larger effect in the initial phase of the reaction; this is discussed in the next section.

6. Radiation

Another feature of the FS is its capability to describe the occurrence of radiation in the meson fields produced by dynamical changes in the source densities. Since VIasov calculations exhibit aImost no stopping, whereas the inclusion of the collision

758 K. We&r et al. / Relativistic BlJU approach

term leads to a sizeable stopping, one would expect the amount of energy produced to be much higher when the collision term is included.

Indeed we find free oscillations in the meson field amplitude, but, especially for light systems, their amplitude is very small. Fig. 8 shows as an example the u- amplitude for 40Ca + 4oCa at 1 GeV/nucleon and zero impact parameter. The collision term is included in this calculation.

To get an estimate of the amount of energy contained in this radiation and to distinguish near-zone fields from radiation, it would be necessary to calculate the flow of field energy through a surface which is sufficiently far away from the remnants of the collision. Integration of this flow over a sufficiently long time interval would then yield the total amount of radiation energy lost by the system.

Because of numerous numerical problems, especially the need for a very large grid (to take care of the rapidly expanding remnants of the collision) and long simulation time (until most of the radiation has passed through the surface), this is not possible. Instead we calculate the total energy content of all grid cells with zero baryon density which are not adjacent to cells with baryons. Employing this rough method guarantees that we do not include surface fields, but we are obviously unable to dinstinguish near-zone fields from radiation fields and therefore can only find an upper limit for the amount of radiation energy produced.

Table 2 shows the maximum radiation energy reached for collisions of “Ca+ 40Ca and rg7Au + ‘97A~ at 1,.5 and 14 GeV/nucleon. Because of the large numerical errors involved, these should be understood only as estimates of the upper limit for the radiation energy produced. Note that most of this energy is contained in the a-field.

+6.30.10-’ +3.73.10-’ +7.56~10-’ +1s4.10- t 252.10- +5.04.10-’ +7.56.10-’

Fig. 8. Contour plot of the tr-amplitude in the plane y = 0 for the collision @Ca +@Ca at 1 GeV/nucleon and b = 0 fm. The collision term is included in this calculation.

X. Weber et al. / Rel~tivisti& BUU approach 759

TABLE 2

Maximum field energy in the density free region for the systems “Ca+‘%a and “‘AU + 19’Au

at different projectile energies with coilision term (CT) and without collision term (Vlasov)

Projectile energy: 1 GeV/nucleon 5 GeV/nucIeon 14 GeV/nucleon

@Ca, Vlasov 0.4 0.1 0.15

?Za, CT 0.5 0.1 0.15

19’Au, Vlasov 1.1 0.3 0.4

19?Au, CT 2.1 0.3 0.4

In the case of Masov calculations we excluded the energy content of the retardation oscillations discussed in the last section since these are clearly near-zone fields.

The inclusion of the collision term leads to a significantly enhanced production of radiation energy only for the heavy system ‘97Au+ 19’Au at 1 GeV/nucleon. However, even in this case the total amount of energy radiated is small. It seems that the characteristic stopping time of roughly 5-12 fm, depending on system and energy, is too large to lead to a sizeable production of radiation. The characteristic frequency of changes in the system has to be larger than the meson mass to produce free radiation (in contrast to near zone fields). Inserting the masses of 2.79 fm-’ for the a-meson and 3.97 fm-’ for the w-meson yields an upper limit for the characteristic time of 2.3 fm and 1.6 fm, respectively.

The main source of the radiation, therefore, seems to be the sudden change in the scalar density at the beginning overlap which is rather independent of the collision term. The decrease of the radiated energy when going from I GeV/nucleo~ to higher energies may be due to the fact that at these energies the scalar density does not have sufficient time to reach its selfconsistent value and therefore more or less behaves like the vector density, i.e. the scalar density in the overlap region is just the sum of the scalar densities of target and projectile 27).

In general we can state that the observed maximum radiation energy of about 2 MeV per nucleon is very small compared to the kinetic energy per nucleon and therefore does not lead to any dramatic energy loss of the system.

7. Conclusions

In conefusion we find that for all systems and projectile energies below a few GeV per nucleon, the local density approximation as well as the quasistatic approxi- mation reproduce the results obtained by full solution of the meson field equations of motion very well. This is especially true for calculations employing the collision term which are essential to achieve realistic dynamics and to be able to compare to experiment.

Retardation and radiation effects are visible but their influence on the nucleon reaction dynamics is obviously negligible. Since the TDHF dynamics is usually quite

760 K. Weber et al. / Relativistic BUU approach

well simulated by transport theories we feel that there is a genuine disagreement

between this result and the one obtained in ref. 12).

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