The nuclear equation of state in quantum hadrodynamics and pion yields in heavy-ion collisions

Nuclear Physics A495 (1989) 347c-354~ North-Holland, Amsterdam

34%

THE NUCLEAR EQUATION OF STATE IN QUANTUM PION YIELDS IN HEAVY-ION COLLISIONS

M.CU.BERO, MSCHeNHOFEN, W.NORENBERG

M.GERING,

Gesellschaft fur Sch~rionenf~schung D-6100 Darmstadt, West-Germany

HADRODYNAMICS

MSAMBATARO,

(GSI),

Postfach

AND

H.FELDMEIER

and

110552

Delta and pion abundances in hot dense nuclear matter produced in heavv-ion collisions are calculated selfconsistently within the framework of quantum hadrodynamics for different eauations of state IEOS). The densitv of deltas turns out to be much more sensitive to theeffective masses ‘of th; baryons than to the compression modulus at normal nuclear matter. The behaviour of the compressional energy per baryon as function of temperature is studied, as well as the influence of vacuum polarization and deviations from thermal momentum distributions of the nucleons on the pion production. 1.

INTRODUCTION We have studied

sensitivity

the

pion and delta

on the physical

mean-field

theory

Lorentz-scalar

properties

3, where

field

abundances

of different

the nucleons

interact

per,.

We extend

this Walecka

scalar meson field *, so that compression

modulus

In a relativistic

I{;

model

theory

nucleons to the repulsive vector to be counterbalanced a smaller effective function

nucleon

meson field.

>.

and their relativistic

meson fields,

i.e.

The two free parameters at normal

cubic and quartic

two parameters

the stiffness

by a large coupling

< 0

isoscalar

en of nuclear matter

by including

the additional

and the effective

mean-field

field

matter

‘*’ . We use an effective

via classical

< (p > and a Lorentz-vector

this model are chosen to fit the energy density

in hot dense nuclear

models

a of

nuclear density

selfcoupling

terms of the

allow us to vary the values of the

mass rn&

at normal

of the EOS is governed

In order to fit pno and err, a strong

to the attractive

5, 6 .

nuclear density

by the coupling

of the

repulsion

has

scalar meson field and this implies

mass of the baryons at puO and a steeper decline of the effective

mass as a

of the baryon density.

In order to deal with additional

spin-312

hot compressed

baryon

which

nuclear matter

we include

has the same coupling

to the classical

nucleons

7, and pions which

coupling

between the pions and the baryons alters considerably

in nuclear matter

‘.

However,

are assumed to be in thermal

since the total

the sum of delta and pion abundances of the delta density,

the influence

pion yield is neglected. results are compared

the delta

equilibrium

meson fields

with

the dispersion

pion yield in heavy-ion

of the ( modified)

in medium

fluctuations

relation

collisions

dispersion

as an as the

the baryons.

(cf. sect 2), and since the pion density

Effects from the vacuum

resonance

The

of the pion

is determined

by

is less than 10%

relation

on the total

are discussed in section 4 and the

to those of sections 2 and 3 which are calculated

disregarding

the negative

energy states in the Dirac-sea (no-sea approximation). In order to allow for a simple lisions,

we employ

the fireball

reference model

to experimental

within

0375-9474l89~SO3,50 Elsevier Science Pub~she~ B.V. (North-Holland Physics Publishing Division)

pion yields from

the one-dimensional

shock

nucleus-nucleus approximation

colwhere

M. Cuber0 et al. / The nuclear equation of state

348c

the conservation

of baryon

mines the baryon the bombarding

density energy

number,

energy and momentum

~8, the energy by the relativistic

we assume thermal

and chemical

equilibrium

momentum

distributions

FIREBALL

MODEL

The number barding

AND

lo.

in the initial

9.

of deltas

nucleon.

If one assumes that

expansion

of the fireball,

and in section 5 we study the influ-

in intermediate-energy

of nucleons which

nucleons.

in the fireball,

the total

number

as was indicated

collisions

participate

with the picture

by the participant

and pions produced

heavy-ion

at a fixed bom-

in nucleon-nucleon

of a fireball

which is formed

The pions finally

each delta

decaying

of pions plus deltas

into

observed,

is conserved

during

relates the pion yield to the EOS. In this model the multiplicity

AIrr of pions per

is given by

and ~rs = /IN + ,,A denote the densities of deltas, pions and baryons (i.e. However,

shock fronts,

the transverse

there are several uncertainties flow which develops

the degree of equilibration

in the fireball

(cf.

5,15. These effects render large uncertainties results so that

Depending

on the bombarding

for two different

EOS which

and compression

modulus

with the bombarding

correspond

ICC1are shown in fig.1.

energy per nucleon,

density the

in

= JIN + PA.

nucleon

becomes energetically

part

favourable

gain of the Fermi-energy

comparison

2 400AleT’

in the collisions

the delta

abundance.

and 10% depending

saturates

for

to transform

is essentially

values of the density ~)n and

nucleon

the fireball

density

E ,,,,,3/,4,

density

2 400AdeT’.

of a nucleon

up to Er,,,rj/A,

= 2GcT’,

energy, pn is

grows with The

With

into a delta

to the

increasing

reason

increasing

is that density

compensates

it the

for the

energy for

which

and temperatures

the pion abundance

in fig.la

in addition

In this way the additional

For the densities

increases

is illustrated

of Er,nrr/A,

baryon

Results

mass m&o in

and, at the same bombarding

nucleons into deltas.

in fig.la.

We rather

models.

used for an increase of the delta abundance

z 1.2GeV

At El,n,j/rl,,

pion yields.

of different

equations.

the total

“t,

of deltas and pions

values of the effective

is shown as a function

of the

of the fireball

with the experimental

of the delta degree of freedom

Whereas

nucleons

thickness

the Rankine-Hugoniot

= rn.a - mN more and more.

Er,nn/A,

produced

properties

As expected,

E,,,rr3/A,,

in the transformation

m> - rnh

30% around

from

to difFerent

where the nuclear part PN in the fireball

E~anlA,

5) and the reabsorption

to a quantitative

are obtained

larger the softer the EOS. The importance

mass difference

sect.

the finite

the formation

energy EL,,,3 and the EOS, different

T in the fireball

energy,

already during

of the pion yields on the physical

the temperature

baryon

in the model:

we do not try to deduce an EOS from the experimental

explore the sensitivity

about

the

‘I, one obtains a model

in some cascade calculations

plus deltas).

beam

are

a pion and a

where PA, in

total

of

2 and 3

In sections

of the nucleons on the delta and pion abundances.

result is consistent

stage of the collision

I2813 which

in the fireball

to the number

This experimental

remnants

participant

equations

PION YIELDS

of pions, observed

energy, is proportional

collisions

deter-

and the pressure I’ as a function

Rankine-Hugoniot

ence of non-isotropic

2.

across a narrow shock front

E/A

per baryon

reaches

which

are

is small compared

to

= 2GcI’- the ratio Prr/PA reaches only values between 5%

on the EOS used for the nuclear matter

in the fireball.

349c

M. Cuber0 et al. / The nuclear equation of state

5

1

(a)

’

’

I

.....

,, ./” ,....... ,,_/~~~_.;~~~~.:.

\“3-rr ‘g

8

’ ...‘.

/._...

_.,_....

A”..-

qa

-

tjo

’

i

1. ....’

........‘. __.__,.

_

-

,,“’ ,:;.<~’

k-

t-

I

0 0

*

i

500

I

t 1500

1000

I

f

0 0

2000

1

1

1

0.3 0.2 0.1 -

,.*-; .,.. ., /.. j, d /S& * , 500

0.0 0

I.”

’

I

I

200

/

t

I 1 1500 2000

1000

EuB/A,

,

,

8

I

1500

1000

. 2000

ELAB/Ap (MeV)

,/ ,/ _,... / ,...’ / ,...‘. ,.“ .,... -’ ,,.’ / -’ .’,/ ,...I’ f : ,/’ ,,.,,/’,/ 5 ,.<’,,..” ” 5% ,i ,,.C .’ ,.,.*I...‘; z lzi

0.4 - w

I

500

ELAB/Ap IMeW

r=

I

..A

~ 3

QN

-

.A ___...---~ ,(.*...“_,_.A ,... _./’ ,..,..‘“x/ .,/,...’ ,,,... ;::.., ,...;:~/’

100 -(b)

_~.+_._ _.e.-.-.-.-.-.-...- -.--- -.e._.___._ _.___________._,_, ;:

2-y-J’

1

.“’

_...... .‘.’

_ _._._._..7 _,_~____._.----~ “.~~...,.~+““r ~~... (,,_

,,).,....” ”

B4-

I

QB

id)

0

I

1

(MeV)

I

,

i .i’

2

3

4

5

QB~QBO

FIGURE 1 Results for two different EOS with the same compression modulus Ko = 400MeV but different effective nucleon masses m;to = 0.85rn~ (dotted curve) and rnhO = 0.65m~ (dash-dotted curve): (a)Total baryon density ~8, nucleon density PN, (b) temperature T, (c) pion multiplicity iV.frr as functions of the bombarding energy per particle El,nn /_.4, and (d) total energy per baryon as function

Fig.lc EOS (cf.

of in

&OS).

Experimental

shows the pion multiplicity fig Id)

two different

which

correspond

to the standard

the stiffer

EOS with

We understand the Fermi-energy effective production

we find,

nucleon

mass rnko

argumentation,

as depicted

Ko = 406MeVY

the strong

influence

much faster.

modulus

= 6.85mN

energy for two different K0 = 400MeV,

and nz;Ve =

mass

in fig.

of the effective

’.

For equal values of the effective

2a, the well-known

behaviour

mass on the pion multiplicity

of a nucleon

of the nucleons

but to

0.65m~.

In

the pion yields are now larger for the stiffer

yields less pions than the softer EOS with

in the transformation

mass the Fermi-energy

of the bombarding

to the same compression

EOS which has the smaller value for the effective mass m&o = 0.85m~

I”.

Mrr as function

values of the effective

contradistinction

data from

into a delta

approaches

(cf.

figla).

the threshold

nucleon

‘“~12*‘6

that

K0 = 2lOMeV. by the gain of For the smaller

ma - mN for delta

35oc

M. Cuber0 et al. / The nuclear equation of state

I 0

500

,

I

EL&A,

, 2000

1500

1000

0

1

2

(MeV)

3

4

5

QB/QBO

FIGURE

2

Pion multiplicity i’tln (a) as function of the beam energy for two different EOS (b) with the same * mNO = 0.85m~ but different values of the compression modulus I&, = 210kfel’ (solid curve) and If0 = 400ilIeV (dotted curve).

3.

RELATION

BETWEEN

COMPRESSIONAL

In order to get a deeper understanding baryon

into thermal,

collective

and compressional

sum of terms which correspond

Note that

to nucleons,

the terms 2o~ and ?a& contain

via the effective

ENERGY

c

EOS

we decompose

parts.

deltas,

The total

the total

energy

energy

per baryon

per

UI is the

pions and mean fields,

parts from

masses. We can rearrange w =

AND

of the fireball

the interaction

with

the mean scalar field

the sum in the form

~&,tm*,+ u~ccol,t,

+

~~)coAlt~n

(3)

i=N,A,lI

by introducing

the thermal

W&BtlM

parts

=

(mA

-

~~N)pA/Ptl 4,,tm

of

the

nuclons,

which is defined

=

7”AlF

perature

+

7flN(T

by subtracting

we can test the conjecture

wcobrr,n

7u

=

0, PN)

all thermal

I3 that

The results are depicted with

=

7oA(T,

w,(T,

PA)

+

7UA(T

WA@

=

0, PA)

-

energy

(m,A

-

energy

(7)

mN)PA/m

7~.

Within

resembles the total

Up to T Y 4OAfeV

of the mean field part 11:nrr,‘.

(5)

0, PA)

part

at T = 0. However for larger T the differences

dependence

=

(6)

energies from the total

this compressional

in fig.3.

-

Prr)

deltas and pions, and the compressional

WCOMPR

T = 0.

+

our model energy

w at

it is indeed possible to identify become large due to the tem-

M. Cuber0 et al. / The nuclear equation of state

I 1

I

I

I

2

3

L

351c

I

QdQBO FIGURE 3 Compressional energy as function of the baryon density at different temperatures T = 7OAlel/ (dashed curve) and T = 1OOIileY (dash-dotted curve) compared to the EOS at T = 0 (solid curve). 4.

EFFECTS

OF VACUUM

The calculations, the occupied

FLUCTUATlONS

presented so far, have been performed

negative-energy

avoids the infinite

contributions

energy-momentum

tensor.

obtained

which

set~nteraction

the effects of vacuum

In particular,

the Dirac sea of the baryons (nucleons in agreement

with

which

in the

l7 with those

we want to find out to what extent

on the equation

and deltas)

the cubic

in the no-sea approximation

of state.

yield a softening

the results in the no-sea approximation,

the EOS in the renormalized

theory

in the Waiecka model.

Fig.4a shows the effects of renormalization

Thus,

from

In this way one

states for example

the results of the renormalized

terms in the scalar field # can simulate

fluctuations

the contribution

approximation).

arise from the negative-energy

Here we compare

in the no-sea approximation.

and quartic

by neglecting

states in the Dirac sea (no-sea

model is correlated

The corrections

due to

of the EOS (dotted

we find that

with an increase of the effective

line).

the softening nucleon

of

mass,

reduces the pion yield (cf. fig 4b).

_._

0

500

1000

EM/A,,

1500

(MeV)

2000

0

t

2

3

6

5

QB/QBO

FIGURE 4 Pion multiplicities (a as function of the incident energy for three EOS (b) corresponding to the renormafized model 2dotted curve), the model with selfinteractions in the no-sea approximation with the parameters fitted to the same values for m;Yo and KQ (solid curve) and the Walecka model in the no-sea approximation (dashed curve).

3.52~

M. Cubero et al. / The nuclear equation

As we have seen in the previous obtained

by introducing

sections,

appropriate

while still keeping the no-sea approximation. J&

for the model without

quartic

selfinteraction

effective

These

identical

Fig.4 also shows the EOS and the pion multiplicity but with cubic and to the values for the

vacuum fluctuation

modulus

of the renormalized

as well as the corresponding show that

the scalar field together

5.

of state can also be

terms of the scalar field

effects (no-sea approximation)

results therefore

negative

of the equation

selfinteraction

terms of the scalar field r$ which have been adjusted

mass and the compression

are almost

a softening

cubic and quartic

ofstate

with

Walecka

pion yields as calculated

a lagrangian

which

includes

model.

appropriate

the use of the no-sea approximation

The two EOS

in the fireball non-linear

can simulate

model. terms

in

the effect of the

energy states in the vacuum.

INCOMPLETE

THERMALIZATION

The observed

momentum

the beam direction

indicating

13*18.

In this section

During

the formation

non-equilibrium

that

we study

the effects

thermal

nucleons

equilibrium

momentum

elongated

in

is not attained distr;butions

In.

that there are nucleons which suffer only one or

thermalizing

we parametrize

are slightly

in the fireball

of such non-isotropic

we expect

and escape without

component

FIREBALL

of the participant

a complete

of the fireball

two elastic collisions

IN THE

distributions

in the fireball.

the nucleonic

In order to account

momentum

distribution

for this

as ’

1 %4P)

= (1 -

.vpf/f)(Jp’

which

results from

tional

to o in the argument

parallel

to the beam.

of an operater

momentum

The parameter

distributions

increasing

components

and are produced be in thermal

for the temperature, the bombardjng

n of the nucleonic

inelastic

(temperature

the baryon density,

energy.

of the distribution

plays value

distribution.

Fig.5 illustrates

r3,tR the anisotropy

of the

by the ratio

and parallel to the beam axis. The isotropic

to 6 < (Y < 0.6). decreasing

masses of the colliding

nuclei.

No anisotropy

with of the

for deltas and pions is assumed because they are not present primarily

only in strongly

equilibrium

propor-

component

and fixes the expectation

of the momentum

perpendicular

energy and decreasing

distributions

the term

the momentum

defines the anisotropy thermodynamics

values for R range from 1 to x 0.6 (corresponding

bombarding

by introducing

to the value R = 1. For central collisions of equal nuclei the

with cy = 0 corresponds

momentum

function

Here pii denotes

for o = 6.6 and o = 6. Experimentally

has been measured

of the mean momentum

experimental

rw which

in generalized

measures the anisotropy

distributions

distribution

distribution

of the exponential.

parameter

which

the momentum

T

the usual Fermi-Dirac

the role of a Lagrange

(6)

+ m;;2 - rnk) - (v - mk)

1-h exp

nucleon-nucleon

the nucleon

The qualitative

momentum

collisions,

T) with other fireball

distribution

behaviour

and hence are expected

particles.

density

and the pion yields as functions

of the curves with

is understood

to

Figure 6 shows the results

as follows.

increasing

of

deformations

Values (Y > 6 imply

that

353c

M. Cuber0 et al. / The nuclear equation of state

there exists some collective kinetic energy which does not contribute to the thermal energy but increases the value of Tz,, (the z-component pressure”) in the Rankine-Hugoniot

of the energy-momentum

FIGURE 5 (a) Isotropic (a = 0, R = 1) and (b) non-isotropic (CY of the nucieons at T = 14OAfeV.

5

tensor or “longitudinal

equations.

0.6, R M 0.6) momentum distributions

=

5.

(a)

I

,

,

,

I

,

I

03)

ZP ?

Pi ?

....,,,,, ,,,(

~,,(,,...._.,....................

g3-

63

-..‘-‘_“.

---------------------____

2 100

2 E l-

50

0 0

500

E,,,/A,

1000

1500

(MeV)

2000

0

500

1000

ELmlAp

1500

2000

(MeV)

FIGURE 6 Baryon (a} and nucleon (b) densities, temperature (c) as well as pion yields (df as functions of the bombarding ener y for three different values of the deformation of the nucleonic momentum distribution: (Y = 0 f solid curve), cr = 0.3 (dashed curve) and cv = 0.6 (dotted curve). (The value5 Ka = ZlOMcV, rn>a = 0.85m~ have been used in these calculations).

M. Cuber0 et al. / The nuclear equation of state

354c

Thus,

both the temperature

fixed bombarding

energy.

T and the baryon density

Furthermore,

prr decrease with

increasing

since T decreases and m,; stays nearly constant,

N at the the delta

density as well as the pion density decreases while the nucleon density increases with increasing cy.

REFERENCES

1)

M. Cubero,

M. Schonhofen,

H. Feldmeier

and W. Norenberg,

Phys.

Lett.

2018 (1988)

11.

2) M.

Schijnhofen, M. Cubero, H. Feldmeier and W. Nijrenberg, in: Proceedings of the International Workshop on Gross Properties of Nuclei and Nuclear Excitations XVI (Hirschegg, Austria, 1988) pp. 72-78.

3) 8. Serot and J.D. Walecka, Phys. Vol.16 (1986). 4) J. Boguta

and A. Bodmer,

“The

Nucl.

51 N.K. Glendenning,

LBL-preprints

6) B.M.

J.A. Maruhn,

Waldhauser,

Relativistic

Nuclear

Phys. A292 (1977) 23081

(April

H. Stocker

Many-Body

Problem”,

Adv.

Nucl.

Rev. C38 (1988)

1003.

413.

1987). 23912 (August

and W. Greiner,

Phys.

1987).

7) D. Lurie, “Particles and Fields”, John Wiley and Sons (1968) pp. 44. S.I.A. Garpman, N.K. Glendenning and Y.J. Karant, Nucl. Phys. A322 (1979)

382.

8) B. Friedman,

483.

9) A.H. Taub,

V.R. Pandharipande Phys.

Rev. 74 (1948)

and Q.N.

Usmani,

Nucl.

Phys. A372 (1981)

328.

10) J.W. Harris, G. Odyniec, H.G. Pugh, L.S. Schroeder, M.L. Tincknell, W. Rauch, R. Stock, R. Bock, R. Brockmann, A. Sandoval, H. Strijbele, R.E. Renfordt, D. Schall, D. Bangert, J.P. Suilivan, K.L. Wolf, A. Dacal, C. Guerra and M.E. Ortiz, Phys.Rev.Lett. 58 (1987) 463; J.W. Harris, R. Bock, R. Brockmann, A. Sandoval, R. Stock, H. Strobele, G. Odyniec, H.G. Pugh, L.S. Schroeder, R.E. Renfordt, D. Schall, D. Banger& W. Rauch and K.L. Wolf, Phys. tett. 153B (1985) 377. 11)

J. Cugnon, J. Cugnon,

121 H. Stocker, 131 H. Stock,

T. Mitzutani and J. Vandermeulen, Nucl. Phys. A352 (1981) 505; D. Kinet and J. Vandermeulen, Nucl. Phys. A379 (1982) 553. W. Greiner and W. Scheid, Z. Physik A286 (1978) Phys.

Rep. 135 (1986)

121.

259.

and J. Kapusta,

Phys. Rev. C33 (1986) 1288.

14)

R.B. Clare, D. Strottman

15)

H.W. Barz, B. Lukacs and J. Zimanyi, Phys. Rev. C27 (1983) 2695; Y. Kitazoe, M. Sano, H. Toki and S. Nagamiya, Phys. Rev. Lett. 58 (1987)

16)

M. Sano, M. Gyulassy, M. Wakai and Y. Kitatoe,

17)

S.A. Chin, Phys. Lett. 62B (1976 263.; R.A. Freedman, Phys. Lett. 718 11977) 369.

18)

H.A. Gustafsson, H.H. Gutbrod, Renner, H. Riedesel, H.G. Ritter, 141.

Phys.

Lett.

1568 (1985)

1508. 27.

B. Kolb. H. Liihner, B. Ludewigt, A.M. Poskanzer, T. A. Warwick and H. Wieman, Phys. Lett. 142B (1984)

191 I. Lovas, G. Wolf and N.L. Balazs, Phys.

Rev. C3.5 (1987)

141.