Covariant calculation of K+ production in nucleus-nucleus collisions at SIS-energies

Nuclear Physics A541 (1992) 507-524 North-Holland

Covariant calculation of K' production in nucleus-nucleus collisions at -energies Andreas Lang, Wolfgang Cassing, Ulrich Mosel and Klaus Weber Institut fur Theoretische Physik, Universitdt Giessen, ®-6300 Giessen, Germany Received 26 August 1991 (Revised 11 November 1991) Abstract: Within a covariant BUU approach we evaluate perturbatively the production of K + mesons from baryon-baryon collisions and analyse its sensitivity to the nuclear equation of state . At 1 GeV/u we find the K + yield to increase by a factor of three for central collisions of Au+Au when decreasing the nuclear incompressibility from 380 to 210 MeV while keeping the nucleon effective mass in line with transverse momentum distributions as a fur, ..on of rapidity. We furthermore present detailed predictions for the experiments of the KAOS group at SIS .

l. Introduction The first measurements of K+ production in nucleus-nucleus collisions already date back about a decade '.2) . Until recently these experiments have been performed at bombarding energies above 2.1 GeV/u [ref. 3)], which is sizeably larger than the free nucleon-nucleon production threshold (- -1 .56 GeV). Since positive kaons have a relatively long mean free path (-6-8 fm) at normal nuclear matter density due to the conservation of strangeness, they are considered to be favourable probes for the highly compressed, possibly hot stage of the collision. Indeed, the early K+ data have been described rather well within cascade calculations 4-g) or multiple-baryon collision models 9- ' 2) and it was suggested by Aichelin and Ko'3) that K+ yields are sensitive to the nuclear compression modulus. This argument was based on BUU type simulations employing different parametrizations of the nucleon meanfield potential. Further information on the K+ production mechanism has been provided by p + A reactions in the far subthreshold regime from 800 MeV to 1 GeV [ref. '4)], which was found to be dominated by irN-: K+ 11 reaction channels since first chance nucleon-nucleon collisions only gave a minimal contribution 15- ") . For 4°Ca+ 4°Ca collisions below 1 .5 GeV/u, however, the irN channel was shown by Batko et al. 'g) to contribute by only 10-20% ; a result that was later on confirmed by Xiong et al. '9} . Correspondence to: Prof. Dr. W. Cassing, Institut fur Theoretische Physik, Justus-Liebig Universitat Giessen, Heinrich-Buff-Ring 16, D-6300 Giessen, Germany . * Work supported by BMFT and GSI Darmstadt. Part of the dissertation of Andreas Lang. 0375-9474/92/$05 .00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

508

A. Lang et at. / Covariant calculation

the meantime, Aichelin et aL ') had found that momentum-dependent forces have a larger impact on K' yields than variations of the nuclear incompressibility . onsequently, the possible K' signal became questionable . Simultaneously, the r ton-nucleus optical potential had been determined up to I GeV bombarding energy 21-22) providing rather strict limitations for the strength of the momentument forces . Consequently, such forces have to be incorporated properly in traps port-theoreacal approaches to allow for a more consistent interpretation of data. Such momenturn-dependent forces appear quite naturally in relativistic transport theories "') and result from strong scalar and vector pans ofthe nucleon selfenergy in the medium 27-29) . We expect, furthermore, that bombarding energies of IeV/u require a fully covariant treatment of the nucleus-nucleus dynamics . In this paper, we present the first fully covariant calculation of K' production in the SIS energy regime on the basis of incoherent baryon-baryon production processes within the relativistic transport approach 2"2:5-28--'0). We use here a novel and e dent technique for the evaluation of the collision integrals ?") that allows for accurate solutions of the transport equations and for the calculation of K' yields also at low bombarding energy and heavy systems . In sect. 2 we briefly present the relativistic transport model and the parametrizations of the lagrangian and discuss the modelling of the elementary processes where B denotes baryons and Y stands for hyperons, in comparison to experimental data. Sect . 11 contains a detailed study of the sensitivity of K' cross sections to variations in the nucleon effective mass, the nuclear incompressibility as well as different projectile-target combinations and a comparison to the first results of the KAOS group at SIS. Sect. 4, finally, is devoted to a summary and a discussion of open problems . escrioti n of the model 2.1 . THE RELATIVISTIC BUU APPROACH

Since the covariant transport approach 23,25,30,31 ) and its derivation from timedependent Dirac-Brueckner theory has been extensively presented in ref.' s ) we only recall the main steps of the approach. Following Serot and Walecks 32) we parametrize the scalar and vector part of the nucleon selfenergy in terms of scalar and vector fields cr(x) and w,,(x) on the basis of the following lagrangian density, gVW *) ( n, _ gSo,)] V, + 1 19'ffagor _ I M 2,7 2 _ I B0'.3 Y = [ Y' 2 2 S 1 4 1 - Wr - &M,r

2 -t- 2M'V(Ow

,

k I)

with F,,,, = a^. -a,w. . Herein If denotes the nucleon spinor. Since the original alecks model has a quite unrealistic vector coupling constant and a too large incompressibility K, the lagrangian has been extended by a nonlinear self-interaction

509

A. Lang el al. / Covariant calculation

33,34) . e parameter sets of the o-meson as first proposed by oguta and 13odmer used in our actual calculations are given in table 1 . The resulting equation of state in the mean-field limit at temperature T = 0 as well as the Schri5dinger equivalent 28,30) . optical potential have been presented in refs. From the lagrangian (1), one derives the equations of motion for the nucleon spinor and the meson fields :

(2) (YN.0a ,- gvw') - (m - gsa))41= 0,

a A aI'a+ msa+Bo-'+ Ca3 = gsW'P, CIJA F AI'

(3)

+ m vw ° = gV TY ° ' .

(4)

e further use the local-density approximation, which is valid in the energy regime considered here 35 ). The meson fields are then determined by the following equations:

(5)

mSa + BO, 2 + CO,3 = gs( : f'Vf :) -_- gsPS , 2

(6)

Going from the positive-energy nucleon spinor to the expectation value of the Wigner transform f(x, p), , d4 s e - ;"P f(x+Zs)® AP(x - Is) ) f (x, P) = 1 * Tr .f arm (j

(7)

one can derive a Boltzmann-like transport equation that describes the motion of the nucleons in the presence of a scalar o- and a vector w-field, while residual NN interactions are accounted for by the collision integral on the right-hand side 28'36-38,29) : (Ht`a-.+(gvnvF"L°+m*(a`m*)a")) .f(x,17) = Ir.,, _ d3171 d3 1T 4311 j (11 + 171)2 du d1l (2 I I

f

X S(4)(ll

+rj,

- 11,

x (.f (x, IIV(x, Hi)(1-f(x, HMI -f(x, II,)) -.f (x,

17)f (x,111)0 -.f (x,111)(1- .f(x, MI)))

(

TABLE 1 Parametrizations used for the lagrangian of eq. (1)

NLl NL2 NL3

gs

9V

[1/fm]

MS

my [1/fm]

6.91 8.50 9.50

7.54 7.54 10.95

2 .79 2 .79 . .7 ,'

3.97 3.97 3.97

B [1/fm] -40.6 50.37 1 .589

C 384 .4 -6.26 34.23

K [MeV]

m*/m

380 210 380

0.83 0.83 0.70

No

A. Lang et aL / (Ovahwu cwkukabn

where the kinetic momentum is defined as 17, = g, - gvw,,, whereas the effective me* == me -- gs ae In the following we will neglect possible uasiparticle mass is given by in-medium corrections of the NN cross section do,/dfl, as suggested in refs. 28,38,39), and use the parametrization given by Cugnon et al. "), which reproduces the experimental rapidity distributions dN/d Y in simulations of heavy-ion reactions best 41,42) . he major inelastic channel in the energy domain of interest is the A-resonance. herefore we have included the processes N + N - N +J,

N + A -* N+ N ,

N+J --> N+J,

A + A -* A + A ,

in our calculations '8) using again the Cugnon parametrization 4° ). propagated with the same selfenergies as the nucleons.

(9) The A's are

2.2. NUMERICAL SOLUTION OF THE BUU EQUATION

The ususal way to solve eq. (8) is via the test-particle method 43). Here the Lorentz-scalar phase-space distribution function f(x, 17) is approximated by a sum of delta functions, f(r, centered at positions ri and momenta Hi. In the following we will denote each of these phase-space points as a "test particle". One should keep in mind, however, that these "particles" do not represent actual nucleons, but are a simple method for approximating a smooth distribution f(x,17) . These "test particles" move according to classical equations of motion : ar,

at

170

1

ML

of

F

JA

,70-m*(rj)a'jm*(rj)), X

k = 1, 2,3 .

(12)

The first term of eq (12) clearly resembles the form of a Lorentz force proportional to the "particle" velocity Il i /170 as known from electrodynamics . In contrast to our earlier work 23,11-41), we now perform a "full-ensemble" calculation 47), i.e. for the evaluation of the collision integral all particles are allowed to interact w,*+h each other. Therefore the cross section do, /dfl has to be scaled by a factor of , N (N denoting the number of "test particles" per nucleon), leading to a drastic decrease in the effective interaction range R, = vo,/_ 7rN. This is obviously in the spirit of the Boltzmann equation, where the distribution functions in the collision integral should be taken at the same space-time point. In the following computations we use N = 400 for the system Au + Au and N =1000 for ire + Ne.

A. Lang et al. / Covariant calculation

51 1

2.3. PERTURBATIVE TREATMENT OF K + PRODUCTION

Since the kaon multiplicity at subthreshold energies is very low, we use the perturbative approach 48) by summing up incoherently the production probability from all possible baryon-baryon collisions, Lee. E d31V(b) -_ d k

BBEco

f d,f2 47r

, d3p(,~) d k (1-f(r,

In this equation r and t denote the space-time coordinates and %I-s the invariant energy of each baryon-baryon collision (S = (II°+172) 2 - ( , + ,)') involving the quasiparticle mass-shell constraint IIo = 2 + m*`'. The primes denote quantit°tes in the individual BB c.m.s. which have to be transformed to the laboratory frame or the mid-rapidity frame, respectively . ; denotes the kinetic momentum of the final nucleon in the process BB --> K+YN with Y =(A, E) whereas ,f2 is the solid angle of the relative Y, momentum 3 which is not fixed by energy and momentum conservation and has to be integrated out. The factor (1- f(r, H3 ; t)) accounts for Pauli blocking of the final nucleon while a corresponding; blocking for the hyperon Y is neglected. The parameter b denotes the impact parameter of the heavy-ion reactio; which has to be integrated over for the evaluation ofinclusive cross sections . Eq. (13) denotes the primordial differential K+ spectra which should not directly be compared with experimental spectra due to final-state interactions of the kaon with the baryons as shown by Ko and Randrup 49.6) . However, energy integrated K+ yields, in which we are primarily interested, are unaffected by K+ --scattering so that we can neglect the possible final-state interactions throughout this study. The differential production probability d3 p(f )/d3k is finally determined by E

d

3p

~) =Ed3k

(~) d3 k

d3a~

C_ gBly

J)

s

(14)

where d3 QK(v J)/d 3 k denotes the elementary differential K+ production cross section and aBB( -vrs_ ) the total baryon-baryon cross section in a baryon-baryon collision. 2.4. PARAMETRIZATION OF THE ELEMENTARY PROCESS BB , K + YN

Apart from the time-dependent phase-space distribution f(r, ; t) - which can 42) - the be controlled by experimental baryon spectra and rapidity distributions differential K+ multiplicity is entirely determined by the elementary cross section d3 UK(v J)/d 3 k in eq. (14) which is not very well known close to threshold even for pp reactions. Whereas the NN-induced channels car approximately be described within an effective boson-exchange model 5°), this is not clear for A N or even AA channels since there are no experimental data. We thus aim at a parametrization of experimental information within a phase space oriented model that is completely determined by the masses of the particles involved and thus allows for a consistent extrapolation to quasiparticle and nucleon resonances.

A. Lang ei at. / Covarian , catc-Jarian

512

ollowin

wermann ") we write the total it' production cross section as (Pmax( 1TK( 11S) = 0.8

(

rS-) 4 ) C)4 mb

eVy

(15)

with 2 a PM X ::- (S

- ( MB + My + M0 2 )(S - (MB+ MY - MK)2)/(4s),

(16)

where m B , w, and MK denote the masses of the final baryon, hyperon and kwon, respectively . Since the parametrization 0 5) addresses the processes BB -* K + A N and BB - K+ X N simultaneously, we adopt

for v/s- < MI -

My

C -- N1U0.

Ni S

WS - -'~%O

for '/S"' < VrS < %/SC

(17)

for Ws > Oj , with N/7, = mf, + nay- + ate,; = 2.642 GeV and 1sc = 3.2 GeV. In this way, the phase

space is improved at higher energy since the X-channel will dominate above its threshold. In fig. 1 we show the parametrization (15) (solid line) in comparison with the experimental data and the original model of Zwermann ") (dashed line). For the di crential cross section we Lssume the form E

d3 k

3 a, (vrs) E W*2

1 (,_ - k )( k N2 Amax Pmax Pmax)

(18)

250 200

I . -I- . I - I . I . I , I . I - 1 2.0 M U M M 10 12 14 Ebe. [WV] Fig. 1 . Parametrization of the total K' cross section: solid line: this work; dashed line: Zwermann circles: experimental data.

A. Lang et al. / Covariant calculation

51 3

as proposed in refs. 4.') with UK(v J) from eq. (15). This approximation is compare to the differential IC+ cross section in fig. 2 at bombarding energies of 2.5 an 2.88 GeV and found to reproduce the experimental data within 30% . and Since there are no data available for induce channels we assume that they are given by the corresponding cross sections at the same invariant energy, including relative weights due to isospin 4.11), i.e. 3O'N .1--aK++X(*'/S) - 2 trjJ-yK`+X(N'S) - CrNN--"K++X(

) -

(19)

The assumption (19) is based on the argument that only the available energy - an thus the final phase space - will determine the 1{ + yield ; this is not so unreasonable since kaons are not dominantly produced by the decay of a specific resonance as -'s or q's 11 .52). However, we have to keep in mind that there is a considerable uncertainty in the elementary cross sections. In refs. 53,54) it was therefore proposed to study 1{ + production by comparing light and heavy systems at the same bombarding energy so that effects from improper parametrizations cancel out to a large extent; remaining differences between results of calculations that employ different equations of state can then more safely be interpreted. In order to demonstrate this cancellation more quantitatively we show in fig. 3 the ratio of the differential + yield for central collisions of Ne + Ne to Au + Au at 1 GeV/u at a laboratory angle of 41° by employing the parametrizations (15)-(18) and by artificially increasing the N® -, K+ YN cross section by a factor of two. We note that the latter channel is the dominant production mechanism for IC+ at this energy (see below). We find

0.2

0.3

0.4

0.5

0.6

K$ momentum in cms (GeV/c)

Fig. 2. Parametrization ofthe differential K+ cross section : solid line: this work; circles : experimental data.

A. Lang et al. / Covariant cakulation

1; 14

., . N . 1 ;V/u

1045

Kaons in Lab. at 4 1"

0.040 1035 0130 Z 0125

INL2 NL3

0120 0.0111 .0

-

I 11

-

I 12

-

I 13

-

I 0.4

T' [GeV] . ' .Ratio of the differential K' yield NK"'/NK" for the reactions Au+Au and Ne+Ne at 0j,,t,=4l*, Fig b = 0 and I GeV/ u employing different parametrizations for the process NJ - K + YN . Solid line : NLt ; dashed Fie: NL2 ; dotted line : NL3 .

the K " ratio for Ire/ Au to change by less than 41/6 for the parameter sets NLI, NL2 and ND when employing different paramerizations of the elementary cross section, while sizeable differences remain when comparing the results for NLI, NL2 and NL3 . A further question is related to in-medium modifications of the elementary processes 1313- KYIN on the transition matrix elements (apart from the trivial corrections arising from the quasiparticle mass m* and momentum Here it was shown by Wu and Ko 55 ) that the expected changes are only very moderate and well below the uncertainty introduced in the parametrization (15) and (18) . We thus adopt eqs. (15)-(18) in the following and consider explicitly the variation of the K' yield as a function of mass at fixed bombarding energy for different equations of state . cross se ti

s in nucleus-nucleus colki

3.1 . COMPARISON TO FIRST RESULTS OF THE KAOS GROUP

S collaboration 16 ) has recently performed first measurements of K' production in Au+Au reactions at I GeV/u and determined inclusively the ratio of K' to protons in a specific angular range. In order to allow for a proper comparison

A. Lang et al / Covariant calculation

515

with their data (i .e . particle ratios) we first evaluate in our relativistic approach the double differential spectra of protons as a function of impact parameter for the parameter sets LI, L2 and L3 in the angular range from 2®-48® which corresponds to the experimental proton detection efficiency 56). e energyintegrated proton multiplicity is shown in fig. 4 for Au + Au at 1 GeV/ u as a function of impact parameter and shows the proton multiplicity to be identical for the three parameter sets employed within 3% ; the sets NL1 (dashed line) an 2 (solid line), which are of primary interest here, even give the same result within the numerical accuracy. However, the proton multiplicities evaluated from the calculations cannot directly be compared with experimental data, where only single protons are detected . In this respect we adopt the coalescence model developed by 1{och et al. 42) that properly accounts for effective final-state interactions and filter out all events that lead to d, t, ' e, 4 e etc. in the final state. e resulting singleproton multiplicity in the angular range from 12°-48° in the laboratory is shown in fig. 4 as a function of impact parameter b and indicates that about 50% of protons become bound due to final-state interactions for Au + Au at 1 GeV/u. e note that the single-proton multiplicity dNp/dfl in the given angular range directly provides a trigger for the impact parameter b of the collision and thus allows a multiplicity K+ yields. (or impact parameter) selected measurement of the Au + Au, 1 GeV/u protons, A = 30* f 18'

60

50

C

30

10

b [fm] Fig . 4. Proton spectra as a function of the impact parameter b for the reaction Au+A.u at 1 GeV/u for 9;;,,- 12-48° . Solid fine: NLi ; dashed line : NL2; dotted line: NL3 .

516

A. Lang et at / MaAaw cwkutatkn

he resulting ratio of K' mesons to single protons is shown in fig . 5 for the same system as a function of b (or single-proton multiplicity) ; it drops by about one order of magnitude when going from central to peripheral reactions for all three arameter sets PNLI, NL2 and NU . As noted before, this ratio becomes maximal for N1.2 with the softest incompressibility K = 210 MeV. In fig. 6 we show the inclusive single proton spectra and K' spectra at 41" in the laboratory for Au + Au and Ne + Ne at I GeV/ u for the parametersets NL1, N L2 and NU Since the differential K' spectra are rather similar in shape for the three parameter sets, except for the absolute magnitude, a direct comparison of the yields from e + Ne to Au + Au might provide valuable information on the incompressibility e recall that NLI and NU employ the same incompressibility K = 380 MeV, but di er in the effective mass at p = p,, (m */ m = 0.83 for NC.1 ; m */ m = 0 .7 for 1-21), while NLI and NL2 employ the same effective mass (and thus the same strength of the mornenium-diependent forces), but differ in their incompressibility = 210 MeV for N1-2). Fo: the lighter system Ne + Ne we find the K+ yield to be most sensitive to the momentum-dependent forces and almost insensitive to K (NU versus NL2) whereas for Au + Au a sizeable variation with K is seen which is larger than the variation with m* (NLI versus NL3) . The total sensitivity of the inclusive cross section (NL2 versus NU) amounts to a factor of two in line with earlier nonrelativistic calculations 1 1,17,48,55) . In order to understand this sensitivity in more Au + Au; I GeV/U 4.51 0 4 4.0-10"' 15104 'JAP , Ili-4

2 .5-10-4 2.0-10" 1 .5-10-4 1XIM4 5XIW5 0

2

4

28

27

22

6

8

10

12

17

11

6

2

b [gym]

Fig. 5. Ratio of the total K+ to proton yield K+/p as a function of the impact parameter b for 0,,j, = 41'. Solid line: NLI ; dashed line: NL2; dotted line: NU.

A. Lang et at. / Covariant calculation

51 7

inclusive spectrum E, = 1 GeV/u, °0, = 41 Au+Au + p+X

Ne+Ne - p+X

C:

10-s

'b 10-s b 10' b -8

K'+X

i

0

I

500

\ 1

1000

p [Mev]

Fig . 6. Differential inclusive K + and proton spectra for the reactions Au+Au and Ne+Ne at 1 GeV/u for e,,t,=41 °. Solid line: NLI; dashed line: NL2 ; dotted line: NL3 .

detail we will investigate various aspects of the reactions Ne + Ne and Au + Au in the next subsections. When integrating over energy, the resulting inclusive K+,gyp ratio is 1 .7 x 10-4 for NL1, 2.4x 10-4 for NL2 and 1 .4 x 10-4 for NL3. This has to be compared with the experimentally extracted ratio of about 2-3 x 10 "4 for Au + Au [ref. 5')] which seems to favour the softer equation of state NL2 (K = 210 eV). However, the present result still depends on the specific parametrization of the elementary cross section B --> K+ YB and it appears too early to draw any final conclusions about K before appropriate data for lighter systems as Ne + Ne have not been taken at the same bombarding energy. 3.2. PROBING THE HIGH-DENSITY PHASE WITH K +

The reason for the latter sensitivity is demonstrated in fig. 7 where we show the impact-parameter-integrated differential K+ multiplicity d K+/dp as a function of baryon density p/po at the individual production point for Ne+ Ne (dashed line) and Au + Au (solid line) at 1 GeV/u. It is clearly seen that most of the kaons are produced at p > 2po in case of Au + Au whereas for Ne + Ne the distribution is concentrated below 2po. This difference is easily understood in terms of the larg.-r pile-up of density in the heavy system. In case of heavy systems the higher densities,

A . Lang el A / (1vaAwn wkulaton

518

B! cross-section ; I GeV/u; NL2

Fig. 7. Inclusive K' cross section as function ofdensity plp,, for = I GeV/u at 01,,t, = 41' . Parameter set N L2 -, solid line: Au + Au; dashed line: Ne + Ne (x50).

furthermore, are obtained for parameter sets that describe a rather soft EOS 30,58) . Thus ?Q2 leads to the largest density in the reaction zone where the kaon production rate by baryon-baryon collisions (-p2 ) becomes largest. s a consequence the mass dependence of the K' multiplicity for central collisions - that can be extracted experimentally by a trigger on high proton multiplicity scales with different powers (ApAT)' for the different equations of state as demon sawed in hg. 8. We find x == 0.74 for N L 1, x == 0.82 for N L2 and x == 0.72 for N U

Ebeam

8 _ 10-3

NLI NL2 NL3

7- 10-' 6-10-'

5_ 10-3 4-10' 3-10'

K* multiplicity 1 GeV/u, b = 0 fm

s s

2 A 0 -' 110-3 0

0

1 .104

2-104 AT *

Fig. 8. Mass dependence

of

Ap

3-10 4

4-10'

for b = 0 fin, E,,,,,,, = I GeV/ u at 01al = 41*. Solid line : NLI ; dashed line : NL2; dotted lire : NL3 .

the K' multiplicity

A. Lang et at. / Covariant c°alc°ulation

51 9

again expressing the sensitivity to the nuclear incompressibility . This scaling may be used in future experiments to pin-down K more accurately, provided the strength. of the momentum-dependent forces (reflected in m*) is determined from other observables. In this context we note that the transverse momentum per particle r/ versus the longitudinal rapidity y was found to be most sensitive to the value of employed 23,25,28,;") and which can thus be approximately fixed y experimental data. At bombarding energies up to 400 elifu no decisive discrimination between the parameter sets LI, NL2 and NL3 could be found 42) (partly due to efficiency cuts of the plastic ball detector). However, the streamer-chamber data 59) for 40Ar+ KCl and '39 La+' 39La at 800 eV/u for p,/A versus rapidity y appear to exclude strongly momentum-dependent forces as employed in NL3 . On the other hand, NLI and NL2 both provide a satisfactory reproduction of the data 2") with the same effective mass m* but different incompressibilities K. This result was expected since the parameter sets NL1 and NL2 reproduce the experimental proton-nucleus optical potential 2' .22) around I GeV/u whereas NL3 over-estimates it substantially 2x,30) . Consequently, we assume the strength of the momentum-dependent forces to be approximately fixed around 1 GeV/u bombarding energy and are left with variations in K that need to be determined by experimental comparison. 3.3. VARIATION WITH IMPACT PARAMETER

Since central collisions only give a minor cross section in heavy-ion reactions, it is of further interest to study the K+ yield as a function of impact parameter. In this respect we show the calculated K+ multiplicity (multiplied by 27rb) in fig. 9 for Au + Au at 1 GeV/ u and ®,d h = 41° as a function of b (solid line) which can be approximately fitted (dashed line) by d1VK-

db

ni ~~b)

=27rbNK {(b =0)S(b) ,

=2R2

arccos

R =1 .14A' /3 ,

b

() 2R

-b(R`-lb`)''',

(20)

for collisions of symmetric systems where Ap = AT . A similar scaling from exclusive to inclusive yields is found for lighter systems, too. A slightly more significant question is the sensitivity of the K+ yield to variations of the equation of state as a function of impact parameter since the maximum densities differ quite drastically with b [ref. 3(')]. In this respect we show the ratio of the K+ multiplicity for NL2 to NL1 (dashed line) and NL3 to NLI (dotted line)

520

;

A. Lai -

c

at. / , 'ovariant calculation

Au 4 Au; IGeV/u Kaon-s j,- Lab. a t 41*

-r-

28

27

22

17

11

6

2

Fig. 9. Impact parameter dependence of the K' multiplicity for Au + Au at I GeV/u for ®,;,,,=41° and parameter set NI-2; solid line: calculation ; dashed line: analytic formula (cf text).

in fig. 10 for Au + Au at I GeV/ u and Oah == 41 As expected, the sensitivity to the incompressibility K (dashed line) drops with increasing impact parameter whereas for fixed K (dotted line) there is only a small dependence on b. Note, however, that we have already excluded the parameter set NU due to its failure for the pJA versus rapidly distribution (sect 12 ). Thus up to b =t 6 fin a large sensitivity to the incompressibility remains that amounts to roughly 60% of the total K+ cross section (cf. fig . 9). On the other hand, larger impact parameter measurements are most sensitive to the momentum dependence of the EOS which could thus be experiinentally accessible 42) .

3.4. BARYONIC DECOMPOSITION

It is quite well known from cascade studies') that heavy mesons are dominantly produced by multiple baryon-baryon collisions [c.17. the discussion in ref. "fl. One is then it-terested in learning the relative contribution from the various reaction channels in more detail . We thus count the number of collisions N, + N', that two baryons have undergone before producing a K + in a mutual interaction, and

A. Lang et al. / Covariant calculation

2.0

a z

52 1

Au Au; 1 GeV/u Kaons in Lab. at 41

1 .4

e\

1 .2 a

1 .0 0.8 0.6

r a.'. e ' s ] 0 ~ 2 4 6 ~ 8 ~ 10 ~ 12

b [fly] . . 28 ' 27 ' 22 1711 6 . 2



Fig . 10. Ratio ofthe K + multiplicity for Au+ Au at 1 GeV/u, b = 0 fm, O, ; dashed line: NL2/NL1 ; .,,=41' dotted line: NL3/NL1 .

,+ z determine the corresponding production probability as a function of a1)) [ref . for the channels NN, N® and ®® separately . The resulting distributions are shown in fig. 11 for Ne + Ne and for Au + Au at 1 GeV/u. e open histograms denote the channel NN ~ IC+ + X (multiplied by 10), the hatched histograms the channel N® -* I{+ + X and the full histogram the ®® -* K+ + X channel, respectively. It is clearly seen that N processes are of minor importance for both systems and that multiple ® N scattering dominates the cross section at the energy of 1 GeV/u. This is so because the nucleon resonances act as an energy reservoir that can be exploited for subthreshold particle production b°) . . Conclusions

f production In this work we have performed the first covariant calculati ,in of in nucleus-nucleus collisions at subthreshold energses. e leave investigated the sensitivity of the differential 1{+ yield to different pararnetrizations of the underlying lagrangian density that reflects different equations of state for cold nuclear matter. The present work completes a series of studies where the effective baryon-baryon cross section has been approximately "fixed" by dlil/dy distributions in comparison with experiment 12 ) as well as the strength of the momentum-dependent forces (that

522

A. Lang et at. / Cavariant calculation

Ne+Ne at I GeV/u, b=O

14 10 N j+N2

15

AwAu at I GeV/u, b=O

Fig. 11. Baryonic decomposition of the collisions producing a 1C + for Ne + Ne and Au + Au at I GeV/u, h = 0 fin . Open histogram : NN (x 10); hatched histogram : N.1; filled histogram : JA

l naturally arise in a covariant framework) by the proton-nucleus optical potential `' ) 16 .28,49) and experimental transverse momentum distributions ") . As proposed in refs. we have then used the sensitivity of it' yields to the nuclear incompressibility K to extract information on this quantity . Whereas for the light system Ne+ Ne at I GeV/u there is essentially no sensitivity, the K+ yield for central collisions of u + Au at I GeV/ u increases by a factor of three when decreasing K from 380 to 57) 210 MeV. A first comparison to the inclusive -'/p ratios of the KAOS group indeed seems to favour a smaller incompressibility K in contrast to earlier suggestions '2 ) . However, this conclusion might be premawre since it depends on the elementary cross section for the dominant process NI - K' YN employed. Though in-medium modifications of this cross section are not expected to sign~ficantly alter the differential K+ spectra 55 L the absolute values close to thres :ioid are essentially unknown. We have thus exploited the proposal of ref. -4) to use IK.' ratios from light and heavy systems at the same bombarding energy in order to get rid of these uncei .aintiecs (fig. 3 1. For this purpose we have provided detailed spectra for single protons and K' mesons for Ne + Ne and Au + Au at I GeV/u tailored to the detector acceptance of the KAOS collaboration 16), which can be used in the nee round of experiments to obtain closer bounds on the nuclear incompressibility

A. Lang et al. J Covariant calculation

The authors like to thank . Grosse and throughout the course of the present study.

523

. Senger for valuable discussions

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