CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

OLVASÓIÉIT

^{LLiJiiü 1}

T(L >СГлГ£бЗ>

KFKI-1981-12

H.W. B A R Z B. L U K Á C S J. Z I M Á N Y I G. FÁI

В, J A K O B S S O N

ON THE ROLE OF THE DELTA RESONANCES IN HIGH ENERGY HEAVY ION REACTIONS

6Hungarian Academy o f Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

J W

ON THE ROLE OF THE DELTA RESONANCES IN HIGH ENERGY HEAVY ION REACTIONS

H.W. Barz+ , B. Lukács and J. Zimányi Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary

G. Fái

Eötvös University, Budapest, Hungary H-1088 Budapest, Puskin u. 5.

B. Jakobsson

Dept, of Physics, University of Lund, Sölvegatan 14, S-223 62 Lund, Sweden

HU ISSN 0368 5330 ISBN 963 371 789 2

On leave from the Central Research Institute

,

ABSTRACT

Proton and pion spectra from relativistic heavy ion collisions are calculated in the framework of a hadrochemical mo d e l . An explanation of the striking absence of the delta decay pion peak in the observed pion spectra is suggested.

АННОТАЦИЯ

В релятивистском столкновении тяжелых ионов рассчитаны спектры протонов и пионов с помощью модели, содержащей превращение частиц. В экспериментальном спектре отсутствует вклад пионов из распада дельта-частиц. Дается объяснение этого факта.

KI VONAT

Relativisztikus nehézion-ütközésben keletkező protonok és pionok spek­

trumát számoljuk egy hadrokémiai modellben. Javaslunk egy magyarázatot arra, hogy a megfigyelt pion-spektrumban miért nem lép fel a delta-rezonanciák bomlásából várható csúcs.

R e c e n t l y it has been a c h a l l e n g e to e x p l a i n the pion i n c l u s ­ ive s p e c t r a o b t a i n e d in high e n e r g y h e a v y ion c o l l i s i o n s /Е^ е а т / /А 0.5 GeV/ wi t h pa r a m e t e r s fixed by the p r o t o n inclusive

spectra [1-3]. In m o s t models [4,5] pions are p r o d u c e d through the c r e a t i o n and d e c a y of d e l t a resonances. The e x p e r i m e n t a l pion spectrum, h o w e v e r , do e s not show the peak c h a r a c t e r i s t i c to d e l t a decay. T h e r e are several p r o c e s s e s w h i c h m a y c o n t r i b u t e to the smearing of this peak. The a i m of the p r e s e n t wor k is to i n v e s ­ tigate t h e s e p r o c esses in the frame w o r k of a d e t a i l e d h e avy ion reac t i o n m o d e l [6]. We intend to show th a t some m o d i f i c a t i o n of the a s s u m p t i o n on the c r e a t i o n of pions seems to be necessary.

We c o n s i d e r a central or nea r central symmetric heavy ion c o l l i s i o n as the i n t e r p e n e t r a t i o n of two spheres o r i g i n a l l y f i l l ­ ed wi t h c o l d nuc l e o n gas. In the o v e r l a p v o l u m e a p i e c e of h o t and d e n s e hadr o n i c matter, the firecloud, is created. We shall use the h a d r o - c h e m i c a l e q u a t i o n s of [5] to d e s c r i b e the d e v e l o p ­ m e n t of the firecloud until the b r e a k - u p of the system. The h a d r o - c h e m i c a l p r o c e s s e s l e a d i n g to the forma t i o n of Д r e s o n ­ ance, тг- a n d p -me s o n s are t a k e n into account.

A f t e r the c o m p l e t e o v e r l a p of the two spheres the r e a c t i o n is m o d e l e d by a s p h e r i c a l l y s y m m e t r i c expansion. In Ref. [5]

this e x p a n s i o n was governed b y the a n a l y t i c solution v(x,t) of the h y d r o d y n a m i c a l equations. In the p r e s e n t work, however, w e use a r e l a t i v i s t i c a p p r o x i m a t i o n to d e s c r i b e the h y d r o d y ­ namical flow. This is done by i d e n t i f y i n g the r a dial f l o w - v e l o ­ ci t y u s e d in [5-7] to the radial c o m p o n e n t of the f o u r - v e l o c i t y u(x,t) as follows:

v(x,t) / l - v ( x , t )2 1c 2

= X V

’/ л ?

u(x,t) (1)

w h e r e v(x,t) is the radial velocity, x = r/R(t) w i t h R(t) b e i n g the radius of the e x p a n d i n g sphere as a function of time, w h i l e v and t are c o n s t a n t s d e t e r m i n e d by the initial v a l u e of the

о о

radius and energy.

We a s s u m e u n i f o r m density, t e m p e r a t u r e and c h e m i c a l c o m ­ p o s i t i o n in the e x p a n d i n g sphere. To in c l u d e the e f f e c t of the s p a c e - d e p e n d e n t h y d r o d y n a m i c a l flow we subdivide the total v o l ­ u m e to small cells /515 in the p r e s e n t n u m e rical calculation/.

T h e flow v e l o c i t y and the thermal d i s t r i b u t i o n are c a l c u l a t e d s e p a r a t e l y for each cell. In a tot a l l y h o m o g e n e o u s case the n u m b e r of c e l l s into w h i c h the sphere is divided, is irrelevant, t h e final p a r t i c l e sp e c t r a c a n n o t d epend o n this number. Even t h e fluctuations do d e p e n d on the total p a r t i c l e number. This s u b d i v i s i o n is a m a t h e m a t i c a l o n e only. The a p p l i c a b i l i t y of t h e h y d r o d y n a m i c s is a d i f f e r e n t question, however. There a

"physical cell size", n amely t h a t d e t e r m i n e d by the ave r a g e i n ­ t e r p a r t i c l e distance, is to be c o m p a r e d w i t h the c h a r a c t e r i s t i c lengths of the h y d r o d y n a m i c a l fields. The pr e s e n t an a l y t i c a l h y d r o d y n a m i c model, however, y i e l d s s m o o t h l y v a r y i n g field qu a n t i ties thus e n s u r i n g its a p p l i c a b i l i t y to the e x p a n s i o n of the f i r e c l o u d .

The b r e a k - u p ti m e is c a l c u l a t e d b y m e a n s of the r e q u i r e m e n t th a t the c h a n g e of t e m p e r a t u r e d u r i n g the average c o l l i s i o n time T , cannot be larger tha n the t e m p e r a t u r e itself. Thus the b r e a k - u p time, t^, is d e t e r m i n e d by the c o n d i t i o n

W " “W

w h e r e

TT (t) /dT(t)

k dt /T(t) ) -1

(3)

is the time c h a r a c t e r i s t i c for the c o o l i n g of the s y s t e m w i t h T (t) being the t e m p e r a t u r e and the average c o l l i s i o n time ^{t c}= X /^v w i t h X the m e a n free path, v the average thermal velocity. In

c a l c u l a t i n g X=l/ap w e use a=60 m b as a thermal a v e r a g e of the n u c l e o n - n u c l e o n cross section in the f i r e cloud and for p the

3

number d e n s i t y of the firecloud. Sim i l a r c o n d i t i o n s are u s e d in a s t r o p h y s i c a l c a l c u l a t i o n s for n e u t r i n o d e c o u p l i n g and t h ere the c o n s t a n t a is 0.5 [10]. If a=2.0, the k i n e m a t i c s of the e x ­ p ansion prevents t h e c o l l i s i o n s b e t w e e n particle pairs of d i s ­ tance X a f ter t^. We regard X to be an a d j u s t a b l e p a r a m e t e r to a c e r t a i n extent.

To o b t a i n the spectra first the thermal d i s t r i b u t i o n s of the d i f f e r e n t p a r t i c l e s are to be c a l c u l a t e d for each individual cell in its rest frame. We u s e relati v i s t i c B o l t z m a n n d i s t r i b u ­ tions f o r the n u c l e o n s and d e l t a s and a r e l a t i v i s t i c Bose d i s ­ t r i b u t i o n for the pions /the z e r o - e n e r g y pion c o n t r i b u t i o n is added e x p l i c i t l y [8]/. S i n c e the d e l t a p a r t i c l e s decay w e l l b efore r e a c h i n g the detector, we a s s u m e that all the s u r v i v i n g deltas d e c a y at the b r e a k - u p time c o n t r i b u t i n g to the sp e c t r a of pions a n d nucleons. Thus the spectra are k n own in the cell frames.

C h o o s i n g any individual cell m o v i n g w i t h v e l o c i t y v^ one c a n take a m o m e n t u m p in the firecloud center of mass system, t r a n s f o r m it bac k to the cell frame b y a Lorentz t r a n s f o r m a t i o n r e s u l t i n g in = q ^ ( p , v ^ ) . In the cel l the n u m b e r of p a r t i c l e s of m o m e n ­ tum q^ c a n be c a l c u l a t e d f r o m the k n o w n d i s t r i b u t i o n functions.

E s p e c i a l l y for the d e l t a - d e c a y pions this p r o c e d u r e yields the L o r e n t z - i n v a r i a n t cross section:

Ncell

V 16

(4)

2 z .

l- 2 ) (z+l)e

zi+

where

(5)

4

and

. 2 2 , 2 (ш. ~m.,+m Д N it

2 2 4 -ш с

TT

(б)

are the energy and m o m e n t u m of the p i o n in t h e Д rest frame, w h i l e is the chem i c a l potential in cell i. E x p r e s s i o n (4) is s i m i l a r to th a t given in Ref. 9. The n u m e r i c a l calcul a t i o n s hav e b e e n c a r r i e d out for the r e a c t i o n U + U a t the b o m b a r d i n g e n e r g y Еъеат1A=8o° M e V '

The short lifetime o f the Д p a r t i c l e s a n d the low t e m p e r a ­ ture of the firecloud c l o s e to the break-up, however, raise some d o u b t about the t r e a t m e n t of the Д p a r t i c l e s as an i n d ependent c o m ponent. Due to the s h o r t lifetime of the deltas, t h e i r number is a function of the n u m b e r of n u c l e o n s and p i o n s and the t e m ­ p e r a t u r e only. We have a h o t i n t e r a c t i n g gas o f two com p o n e n t s w i t h the only h a d r o c h e m i c a l reaction

N + N + N + fr (7)

In this p i c t u r e the simplest w a y to i n c l u d e the i n t e r a c ­ tion is to a p p r o x i m a t e its effect b y a c o l l i s i o n scheme. On

i d e n t i f y i n g the Nit cross section w i t h the c r o s s section of delta p r o d u c t i o n and the d u r a t i o n of the c o l l i s i o n b y h /Гд w e arrive at a simple r e i n t e r p r e t a t i o n of the earlier h a d r o c h e m i c a l c a l c u l a ­ tions namely tha t the n e w d e n s ities v' and v' should be cal- c u l a t e d a c c o r d i n g to the p r escription:

v ' = v + v .

it TT Д

VN = VN + V A

(8)

w h e r e v N , and are the old densities. T h e q u a n t i t y Vд is n o w inte r p r e t e d as the d ensity of pairs just being in the p r o c e s s of collision. The e nergy of these p a i r s is the sum of the ener g i e s of their c o n s t i tuents. At low t e m p e r a t u r e s close to the bre a k u p the s p e c t r u m of pions e m e r g i n g from these pairs

5

remains a p p r o x i m a t e l y thermal. /By low t e m p e r a t u r e w e mean th a t the a v e r a g e thermal kinetic e n e r g y of the Ntt c o l l i s i o n is e s s e n ­ tially s m a l l e r than the e n e r g y of the r e s o nance in the N^tt c r oss section./ W i t h this m o d i f i c a t i o n the n u m e rical r e s u l t s of the previous m o d e l yield a good a p p r o x i m a t i o n to the c h e m i s t r y of the f i r e c l o u d in spite of the d oubts r a i s e d above. W e shall see, however, t h a t the two d i f f e r e n t i n t e r pretations of the Ntt i n t e r ­ actions l e a d to d i f f e r e n t i n c l u s i v e pi o n spectra. In the first case the s p e c t r u m has a d e lta peak or shoulder at » 130 MeV, while in the other case one c a n n o t e x p e c t serious de v i a t i o n s

from the thermal distribution. In the following w e p resent the results of the numerical c a l c u l a t i o n s s h o w i n g the ef f e c t s of the d i f f e r e n t p r o c e s s e s d i s c u s s e d above.

On Fig. 1 the effect of the h y d r o d y n a m i c a l f l o w is shown.

We d i s p l a y proton and pion s p e c t r a o b t a i n e d i) as d e s c r i b e d a b o v e and ii) a r t i f i c i a l l y p utting the flow v e l o c i t y v ^ = 0 for each cell.

Coulomb e f f e c t s are not included. P roton and pion s pectra in the c o m o v i n g c e l l - f r a m e s are c o n s i d e r e d to be thermal. It is c l e a r from Fig. 1 that the flow has a much sma l l e r e f f e c t on the p i o n s than on t h e protons. This has also bee n pointed o u t in [3] and can be u n d e r s t o o d t a king into ac c o u n t the mass d i f f e r e n c e b e ­ tween p r o t o n s and pions. In Fig. 2 we sho w pion s p e c t r a o b t a i n e d a c c o r d i n g to the d i f f e r e n t p h i l o s o p h i e s d e s c r i b e d above. The dashed c u r v e represents the pu r e thermal /Bose/ spectrum. The dotted a n d the continuous c u r v e s include the c o n t r i b u t i o n of the delta d e c a y pions cal c u l a t e d w i t h sharp and finite w i d t h d e l t a mass, r e spectively. The v a lue of the b r e a k - u p p a r a m e t e r of eq.

/2/ is f i x e d to a=0.35. The f inite w idth delta m a s s w a s s i m u ­ lated by u s i n g six d i f f e r e n t m a s s d e lta p a r t i c l e s w i t h i n the

Гд = 120 M e V width and w e i g h t i n g them c o r r e s p o n d i n g to the B r e i t - W igner s h a p e of the delta resonance.

It c a n be seen from Fig. 2 that the i n c l usion of the d e l t a decay p i o n s m a n i fests itself in a p r o n o u n c e d d i f f e r e n c e from the thermal d i s t r i b u t i o n even in the case of the f i nite w i dth d e l t a mass. To sho w the d e p e n d e n c e of the sp e c t r a on t h e b r e a k - u p time, on Fig. 3 pion spectra o b t a i n e d wi t h d i f f e r e n t b r e a k - u p c o n d i ­

6

tions are presented. The d e l t a particles are treated as real p a r ­ ticles a c c o r d i n g to the first i n t e r p retation. It is seen that d i f f e r e n t b r e a k - u p t imes r e s u l t in q u a l i t a t i v e l y d i f f e r e n t s p e c ­ tral shapes. /They are i m p l e m e n t e d by c h o o s i n g the p a r a m e t e r a in eq. (2) a = 0.5, 1,2 respectively./ T h i s is a c o n s e q u e n c e of the fact that the n o n m o n o t o n i c c o m p o n e n t o f the p i o n spec t r u m a r i s i n g fro m delta d e c a y /see Fig. 2/ is rapidly d e c r e a s i n g w i t h i n c r e a s i n g b r e a k - u p time. It is seen t h a t none of these b r e a k - u p c o n d i t i o n s produces a spectral shape r e s e m b l i n g the e x p e r i m e n t a l pi o n spectra. On Fig. 4 the spe c t r a o b t a i n e d on the basis of the second i n t e r p r e t a t i o n /i.e. t h a t the d e l t a p a r t icles do not for m an inde p e n d e n t t h e r m o d y n a m i c a l component/ are c o m p a r e d to the e x p e r i m e n t a l data [1]. The a g r e e m e n t is q u i t e reasonable.

We conc l u d e t h a t the h y d r o d y n a m i c a l flow is v e r y i m p o rtant to exp l a i n s i m u l t a n e o u s l y the e x p e r i m e n t a l proton and pion s p e c ­ tra. N e i t h e r the h y d r o d y n a m i c a l flow n o r a finite w i d t h mass d i s ­ t r i b u t i o n of the d e l t a p a r t i c l e s is enough, however, to p r o d u c e a g r e e m e n t w i t h e x p e r i m e n t a l spe c t r a if t h e deltas are treated as real particles in the e x p a n d i n g firecloud. T h e r e f o r e in the e x p a n s i o n stage of the h e avy ion r e a c t i o n the Ntt i n t e r a c t i o n should be a p p r o x i m a t e d by some other m e t h o d . In the present w o r k d e s c r i b e d such a p o s s i b i l i t y in the form of a c o l l i s i o n sheme.

The authors a c k n o w l e d g e fruitful d i s c u s s i o n s w i t h J.P.

B ondorf and P.J. S i e m e n s and the war m h o s p i t a l i t y of the N i els Bohr I n s t itute e x t e n d e d to them.

7

R E F E R E N C E S

[1] S. Nagamiya, P r o c e e d i n g s of the 41" th Hi g h E n e r g y Heavy Ion Summer Study, Berkeley, Ju l y 1978. LBL-776, UC-34C,

C o n f .-780766

[2] S. Nagamiya, talk p r e s e n t e d at the C o n f e r e n c e "Nuclear Physics in H e a v y Ion Col l i s i o n s b e t w e e n 10 and 300 MeV/

nucleon", Copenhagen, S e p t e m b e r 1979

[3] P.J. Siemens and J.O. Rasmussen, Phys. Rev. Lett. 42:

/1979/ 880

[4] G.D. Westfall, J.G. Gosset, P.J. Johansen, A.M. Poskanzer, W.G. Meyer, H.H. Gutbrod, A. S a n d o v a l and R. Stock, Phys.

Rev. Lett. 31_ /1976/ 1202

A. Mekjian, Phys. Rev. Lett. 2§. /1977/ 640

[5] I. Montvay a n d J. Zimányi, N u c l . Phys. A316 /1979/ 490 [6] J. Zimányi, P r o c e e d i n g s of the EPS C o n f e r e n c e "Large

A m p l itude C o l l e c t i v e N u c l e a r M o t i o n s " Keszthely, June 1979., p. 824

[7] J.P. Bondorf, S.I.A. G a r p m a n and J. Zimányi, Nucl. Phys.

A29 6 /1978/ 320

[8] J. Zimányi, G. Fái and B. Jakobsson, Phys. Rev. Lett. £3 /1979/ 1705

[9] J.I. Kapusta, Phys. Rev. 160 /1977/ 1493 [10] T. de Graaf, Lett. N u o v o Cim. 4, 638 /1970/

8

F I G U R E C A P T I O N S

Fig. 1 The effect of a hydro dynamic at flow on the particle spectra. Continuous curves are obtained by taking

into account both thermal and hydrodynamical velocities.

Dashed curves show spectra calculated by artificially putting the hydrodynamical velocities equal to zero

in computing the momentum distributions. Proton and pion spectra in the comoving cell-frames are considered to be thermal. /The calculations are made for central U + U collisions at the bombarding energy E ^ ^ = 8 0 0 MeV/A

d^ о

Invariant cross sections E — - are plotted with arbit- dpó

rary normalization against center of mass kinetic energy E*/. On the figure the temperature of the firecloud kT as well as the apparent temperatures for protons and pions defined by the slope factors /kT and kT appj тг respectively/ are qiven.r a' a

Fig. 2 Invariant pion cross sections /arbitrary normalization/

in the U + U reaction at ^ - 800 MeV /nucleon. The dashed line refers to pure thermal distribution /Bose statistics with hydrodynamical flow/, the other two curves both contain the contribution from delta decay, the dotted line corresponding to the sharp delta mass mдС - 1236 MeV while the continuous one corresponding

to a delta mass distribution around m.c^ - 1236 MeV

A

with a width of Гд - 120 MeV.

Fig. 3 Invariant pion cross section/arbitrary normalization/

in the U + U reaction at = 800 MeV/ nucleon. All three curves contain both the thermal and delta decay contribution /with sharp delta mass/. The curves differ only in break-up times and are labelled accordingly /see text/.

Fig. 4 Invariant proton and pion cross sections /arbitrary normalization/ in the U + U reaction at - 800 MeV/

nucleon. The curve for protons is obtained using Boltzmann statistics/including the effect of hydrody­

namical flow/. For pions Bose statistics with the con­

tribution of the condensate [8] is used and hydrodynami­

cal effects are included. Dots refer to the 800 MeV/A Ar+KCl, QCM=90°, high multiplicity experimental data of Ref. 1.

(mb / sr /(MeV ) с

9

A

(MeV)

Fig. 1

(mb/sr/(MeV)

I O

(mb/sr/(MeV)

12

Fig. 4

jp.

.

*

0

.

.

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Szegő Károly

Szakmai lektor: Révai János Nyelvi lektor: Perjés Zoltán Gépelte: Beron Péterné

Példányszám: Törzsszám: 81-96 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1981. február hó