Dynamical and Bose-Einstein correlations of centrally produced pion pairs in hadron-hadron collisions

Nuclear Physics B136 (1978) 349-364

© North-Holland Publishing Company

D Y N A M I C A L A N D BOSE-EINSTEIN C O R R E L A T I O N S O F C E N T R A L L Y PRODUCED PION P A I R S IN H A D R O N - H A D R O N COLLISIONS * J. ENGELS

Faculty o f Pkysics, Bielefeld University, Germany K. S C H I L L I N G **

CERN, Geneva, Switzerland Received 18 October 1977

We calculate dynamical and Bose-Einstein inclusive correlations of pion pairs (as functions of the invariant pair mass MTrn) from the decay of independently produced clusters, whose finite size is explicitly taken into account. We show that recent pp bubble chamber data from FNAL on the correlation functions C2(M;rrr) for centrally produced 7r+Tr - and ~r-Tr- pairs can be understood within this picture. Some implica- tions for Hanbury-Brown-Twiss type analyses are discussed.

1. I n t r o d u c t i o n

E x p e r i m e n t s have revealed strong short-range correlations in rapidity a m o n g cen- trally p r o d u c e d particle pairs at energies Elab ~> 100 G e V [1]. These rapidity corre- lations can be u n d e r s t o o d a s being due to the i n t e r m e d i a t e p r o d u c t i o n o f clusters which s u b s e q u e n t l y decay i n t o the observed final-state h a d r o n s [2]. In rapidity space the most sensitive observable q u a n t i t y from which to d e t e r m i n e cluster prop- erties from e x p e r i m e n t is the strength o f the short-range correlations, which is related to the ratio (n(n - 1))c/(n) c o f the first t w o m o m e n t s o f the cluster decay m u l t i p l i c i t y d i s t r i b u t i o n (averaged over the p r o d u c e d cluster spectrum). A s s u m i n g i n d e p e n d e n t emission o f clusters, one arrives at an average cluster mass 3~t c ~-- 1.3 GeV and an average charge m u l t i p l i c i t y (nch)c ~- 2 ***. These n u m b e r s are consis- t e n t with other final-state densities observed, such as gap distributions, zone charac.

teristics, semi-inclusive correlations as well as charge-transfer correlations measured in rapidity space [3].

* Work supported by the Deutsche Forschungsgemeinschaft, Grant: SCHI 123/3.

** On leave of absence until 15 October 1977, from the University of Wuppertal.

The most recent accounts of the present experimental status of cluster models can be found in ref. [3a]. A dissident view with regard to cluster properties is maintained in ref. 13b].

349

350 J. Engels, K. Schilling /Dynamical and Bose-Einstein correlations

These small numbers and the very fact that high-statistics bubble chamber exper- iments at Ela b ~< 20 GeV have revealed the importance of inclusive resonance produc tion [4] suggest the use of the invariant mass of the pair,M~. = (Pl +P2) 2 , rather than its relative rapidity, Ay = Yl - Y2, as the variable against which to plot the cor- relation C2 ; while the shape of C2(2xy) is only determined by the (assumed) iso- tropy of cluster decay [5], the shape of C2 (M~rTr) would be expected to exhibit addi- tional information such as, e.g., resonance production [6,7].

Besides these d y n a m i c a l effects due to intermediate cluster production, one expects identical pions emerging from hadron collisions to show B o s e - E i n s t e i n (BE) correlations *. In fact it has been repeatedly suggested that this analogue of the Han- bury-Brown-Twiss effect be utilized to determine the size of the pion-emitting source in hadron collisions experimentally [9]. Here again, the variable MnTr is much more suitable than Ay, because there is a one-to-one correspondence between the BE symmetry point Pl = P2 and the threshold mass M~Tr = 2m (m being the pion mass). Although the BE effect is clearly established experimentally in azimuthal correlations [10] and in the Kopylov variables q = Pl - P2 [11] as well as in q2 = 4m 2 _ M 2 [12], the procedures to actually extract the interaction volume from data are plagued by the problem of how to disentangle background, (i.e., dynamical) from BE short-range correlations [ 1 0 - 1 2 ] . Moreover, the situation is confused by the expectation that abundant production of resonances of width F ~< 100 MeV would result in a tremendously narrow BE correlation width, AMTrTr -~ 10 MeV [13,14], which is below present experimental resolution.

In this paper, therefore, we set out to investigate the relative importance of Bose-Einstein and dynamical correlations from cluster decay using the invariant pair mass M~,~ as the appropriate variable. As far as dynamics is concerned we need, of course, some theoretical prejudice. We shall adopt the attitude to work in a frame- work which is as simple as possible. Therefore we return to the old independent emission model of clusters with mean properties borrowed from "rapidity physics".

We differ however from previous authors in the treatment of cluster decay. In a preceding paper we have shown that neither a factorized form of the two-particle density nor the thermodynamical manner of including Bose statistics is appropriate when evaluating the decay of a cluster of small mass such as 1.3 GeV [15]. There- fore in this paper we will carefully take account of the lightness of clusters by imposing energy-momentum conservation, isospin invariance and Bose statistics in cluster decay.

We should mention that this paper was motivated by the recent publication [6]

of first high-energy two-particle correlations as functions of MTr~r and subsequent interpretations [6,16] of these data. The conclusion reached in ref. [16] was that the cluster model fails in explaining the large correlations observed in the region M ~ , <~ m o . Confirming our earlier preliminary results [17], we shall demonstrate in

* The implications o f BE effects in the uncorrelated jet model have been investigated in ref.

[8].

J. Engels, K. Schilling / Dynamical and Bose-Einstein correlations 351

this paper that this conclusion is false and that the cluster picture can, indeed, accom- modate the inclusive correlation data from the 205 GeV/c pp bubble chamber experiment * as presented in ref. [6].

2. The model

Intuitively, the most attractive way to view clusters is to describe them as fire- balls fi la Hagedorn [18], since his fireballs imply both resonances and their inter- actions in an average sense. Interactions of resonances are expected to occur due to the observed density of secondaries in rapidity space [19]. To put it differently, it is unlikely that established resonances are produced independently; they rather cluster into larger structures, i.e., fireballs.

It is important to note that the thermodynamic limit cannot be taken when treat- ing fireballs of mass Mc = 1.3 GeV [15]. Therefore one is led to the microcanonical description o f fireballs as given by Frautschi and Hagedorn [20]. Unfortunately, their statistical bootstrap scheme has not yet been solved with Bose statistics and isospin included. Being faced with fairly light clusters, however, one may well accept a phase-space description (fi la Fermi) as a good approximation to fireball decay (fi la Hagedorn).

So we actually calculate our distributions from the density of states r(Q 2) for a finite ideal Bose gas o f mass Mc = x/Q 2. For isospin-zero pions, the formula for r(Q 2) has been given by Chaichian et al. [21] in terms of a "cluster expansion":

with

a o

Br(Q 2) = ~ ~ k ( n i i Bt~2(t)(Q 2, m,..., m ... krn,._ss2km ) , (1) k=-I n 1 ... nk=O

n I times n k times

k k y _ 3 n r

l= ~ nr , h(n,) = I-I

r = l r = l n r ]

Here Q stands for the cluster four-momentum, and ~2 (t) is the Lorentz-invariant momentum-space integral for l particles of the indicated masses. B is a constant o f dimension [L 2 ] that one would expect to be of the order

B ~ ~ 7r = 4.1 GeV-2 hLm ,

i f m = 140 MeV.

The terms in eq. (1) with k > 1 can be thought of as being composed out of one

* U n f o r t u n a t e l y , the results of the ISR e x p e r i m e n t p u b l i s h e d in ref. [7] are n o t useful for an i n d e p e n d e n t analysis since t h e y have n o t been corrected for acceptance biases.

352 J. Engels, K. Schilling/Dynamical and Bose-Einstein correlations

or more B E clusters which are defined as "condensate droplets" consisting of i pions of identical momenta. Thus each contribution to r(Q 2) is characterized by the num- ber of times, ni, a BE cluster appears in it, where 2 <~ i ~< k. The partial sum corre- sponding to k = 1 is nothing but the Boltzmann-statistics expression for 7. The total pion number from the term described by the set {ni) = (hi .... , nk ~, is given by z k r= 1 m r . Therefore, if one wants to include all contributions up to a given pion number N, one is faced with a partition problem. For N = 16, e.g., one has 914 such partitions.

The above defined BE clusters should not be confused with the usual notion o f clusters as intermediate structures in h a d r o n - h a d r o n collisions. To ensure clarity, we shall henceforth call the latter clusters strong interaction (SI) clusters.

The generalization of eq. (1) to the case of pions with isospin 1 has been derived by Kripfganz [22]. The result is an expansion in BE clusters that are still degenerate in their charge composition: instead of being characterized by one number i, the BE cluster is now characterized by a set (i} = (i+, i_, i0), the numbers o f n +, n - , n °, respectively, which it contains. The resulting number of partitions grows so rapidly with N that the problem exceeds the capacity o f a big computer. We observe, however, that the coefficients h (ni) given in ref. [22] are small and even negative whenever they involve BE clusters that consist of pions o f different charges. There- fore, we approximate the problem by retaining only the manifestly positive "dia- gonal" terms, whose BE clusters contain only equally charged pions. In this case we obtain from eqs. (22) and (23) of ref. [22] the following natural generalization of eq. (1) for the level density of an SI cluster with isospin (L I3):

B~3(Q2 ) = ~ ~ h l 3 (n(i, q)} B t a (l) (Q2, ~ ... k m ... k m ) ,

k=l {n(i,q) l n' 1 times n~ time'---"~

with

k

n'= G n(r,q), l : n;.

q=+,--,O r=l

(2)

n(r, q) denotes the number of times a BE cluster composed of r pions of charge q appears. The coefficients h depend on the whole set (n(/, q)} and read

k rt3n(r, t)

-I'I3 H H - -

hi3 {n(i, q)} = l"rN.~ .

n (r, t)!

L--t) t=+,--,0 rt=l

(3)

p / , / 3 denotes the standard Cerulus coefficients for an isospin (I, 13) object decay-

"L~vt ;

ing into the set {Nt } = (N+, N _ , N o ) o f N+, N _ , N o pions of charges +, - , 0, res- pectively [23]. Here N t = ~,kr= 1 rn(r, t) and of course 13 = N+ - N _ . For N~< 16

J. Engels, K. Schilling/Dynamical and Bose-Einstein correlations 353 and I = 1,13 = 0 this leads to 9742 terms in the expansion, eq. (2)!

It is straightforward to obtain the inclusive two-particle density F2 (P l, P2) of an SI cluster o f isospin (L 13) into two pions o f charges t, t':

f~2't,lt3(pl,P2)=TJ3(Q2) -1 G G hI3(n(i,q)}B l

k=l {n(i,q)}

k

X [j,~'j= 1 j3j'3n(j't)(n(J"

t')

~jj'~tt')a(l--2)(O 2 ... fm ... fdm .... )

k (4)

+ 2PlO(5(3)(p 1 --p2)6tt' G /3(j._ 1)nQ.,t)~2(t-1)((Q _]./71)2 .... ,/am, ...)1 , j=l

with

9 = a - J P l - I ' P z .

Note that the integral over F2 is normalized to the second moment over the SI cluster decay multiplicity distribution

h ^d3pi F2t t'(Pl, 1'13" P 2 ) " = (rlt(nt' - 8tt') }1'I3 .

i= 1 2Pio '

(s)

The &function term in eq. (4) is the most prominent consequence of Bose-Einstein statistics: it is due to two equally charged pions emanating out of one BE cluster.

This contribution will in the following be called the "condensation" term.

So far we have been only concerned with the SI cluster properties themselves. To make contact with experiment, we must formulate the independent emission hypoth- esis [24]. According to this hypothesis, the normalized inclusive one- and two-particle densities from a hadron-hadron collision are given in terms of an SI cluster yield function ~,(Q2) that is independent on the SI cluster momentum (we suppress iso- spin for the moment)

Pl (Pl) = f d4Q~v(Q z)

FIQ(P 1) ,

2 d3pi 8 ( ~ ) 2 -MTra)

p=(M..) =f H

^2Pio

(6)

×(fd4Q ~(Q2)FO(p,,

P z ) + P, (P,) P, (P2)} • (7) F Q, F o are inclusive one- and two-particle decay densities of an SI cluster at four-

momentum Q. The structure o f f Q for Q = 0 has been given in eq. (4). The indepen- dent emission hypothesis is reflected in the structure o f the last term in eq. (7).

354 J. Engels, K. Schill&g /Dynamical and Bose-Einstein correlations Using the convolution

2

p l ® P l ( M n n ) = f l - I d3pi ~ ( ~ ) 2 _ M r r T r ) / o l ( P l ) P l ( P 2 ) ' (8)

2Pio

one defines the inclusive two-particle correlation

C2(MnTr)=p2(MnTr)

/91 ® Pl (MTrn) - (9)

Within the independent emission hypothesis, eq. (7), C2 is simply given by

C2 (MnTr) = f d 4 Q ~(Q2) 72 (.Qz, M~rTr), (10)

with

72(QLM=)=fFI

2 ^d3p~ i 6(X/-(pT~P-~p2) 2

MTr~r)F2Q(pl,p2).

(11)

i= 1 2pio

Note that the Q dependence in F 2 (Q2, MT.r) has dropped out, since F? is an invari- ant function. The d4Q integration in eq. (10) is essentially over longitudinal phase space. We thus arrive at the simple form for C2 :

C2(M..)

= f d M c w(Mc)

7 2 ( M 2 , M . . ) ,

(12) where

w(Mc) = f d 4 Q ~(x/Q 2 - Mc)~(Q2). ⁽¹³⁾

It is now a trivial matter to include SI cluster isospin and pion charges

LI3 LI3 2

C~"(M.O=fdMc(~ w (Mc) 72t, t,(Mc,Mn~r)} ,

(14) I,I 3

where

t, t'

denote the observed pion charges. To leave matters simple, we choose the shape

ofwI'I3(Mc)

to be independent o f ( L 13):

W I' I3 (Mc) = o~I, I 3 W ( M c ) . This leads us to

--t'I3 (M2,M~r~r)

(15)

c~t'(M~D: f dMc w(Mc) ~-J

Otl, I3~-l:2t, t, I,I 3

Here the function w(Mc) contains the cluster production dynamics. In the following we shall choose a simple parametrization for it:

w(Mc) = (M c - M0) ~ exp(-TMc) . (16)

J. Engels, K. Schilling /Dynamical and Bose-Einstein correlations 355 We adjust the model to the empirical findings from rapidity space distributions, i.e., we adjust/3, 7 , M 0 to reproduce Mc ~ 1.3 GeV. A reasonable choice consists in putting M 0 = 0.5 GeV,/3 = 4, 3' = 6 GeV - l . Note that this value of 7 is also suggested by inclusive PT spectra. The parameter B in eq. (4) was taken to be 4 GeV - 2 , which yields (rich) c = 2.1, and therefore the reasonable average pion energy in the SI cluster rest system is 413 MeV. The normalizations

ai,i3

control the actual numbers of produced SI clusters; they must evidently depend on the primary energy and will be discussed in sect. 3.

3. Numerical results

We first admitted both isoscalar and isovector SI clusters but the correlations turned out to depend little on isospin I. Therefore we shall only present results for isovector SI clusters in the following. The calculations were done on a computer, using a program of Kajantie and Karim/iki to evaluate the momentum-space inte- grals [25]. We allowed for SI cluster decay multiplicities up to 16.

We first display in fig. 1 the correlation functions C ~ - ( M ~ r ) and C2--(M~r~) due to single SI clusters of various charges and discrete representative mass Mc = 1.32 GeV. The quantum number dependence of the shapes of these distribu- tions demonstrates our previous claim that the thermodynamic limit is too rough an approximation in this context. Note that C~-- has been multiplied by factors of 4 and 16 in figs. la, b, respectively.

C~-- is clearly narrower than C ~ - for neutral and positive clusters. The suppres- sion of C ~ - can be understood as being mainly due to charge conservation: it needs at least a four-particle state to accommodate two negatives in an 13 = 0 configura- tion and a five-particle state to get two negatives out o f a I3 = +1 configuration.

Thus the suppression factors 4 and 16 are consequences of the decay multiplicity distribution of the SI clusters. For negatively charged SI clusters, it is equally prob- able to find pion pairs o f the same and opposite charges. Therefore, the correlations C~-- and C~-- are fairly much the same. Note that the "condensate" terms have not been added in fig. 1. Their inclusion would in fact render the integrated correlations from negatively charged SI clusters exactly equal.

Next we compare the relative strength (nt(n t, - ~

tt,))/(nt)

per cluster of definite mass Mc as a function o f M c (fig. 2). The relative ordering of the different curves is again expected from charge conservation and charge symmetry. Since all these mean values soon show a straight-line behaviour, the relative magnitudes of C~- - and C~- - become sensitively dependent on SI cluster mass for both neutral and positively charged SI clusters. More interesting information is contained in fig. 3, where we have plotted the integrals of the F 2 densities, again per SI cluster of given M c versus Mc, fig. 3a, and the "condensate" terms contained in these integrals (in per cent) in the case of identical pions (see fig. 3b). Typically, this contribution is of the order of 2 0 - 3 0 % in the region of the observed average SI cluster mass, and it becomes

356 J. Engels, K. Schilling /Dynamical and Bose-Einstein correlations

3 . 0 J

2.5 2,0 1.5 1.0 0.5 0

] :1, 13 = 0 (Q)

. . . . C~-

I m

'7 2.5 2.O 1.5 1.0 0.5 0

[ : 1 , 13=+1

\ \ ' x \

i i i i ~ i \1

Cb)

1 6 . C~'-

. . . . C~-

2.5 2.0 1.5 1.0 0.5

(c)

_ _ C2--

. . . . C2"- I =1, 13 = -1

I [ I 1 I I

0.4 0.8 1.2

M . n [ G e V ]

i I I . ~

0 1.6

• . . ÷ - - - - - - ^.

Fig. 1. Correlation functlonsC~ (MTrlr) (dotted curves) and C 2 (M~rrr) (full curves) from single isovector SI clusters of mass/I,/c = 1.32 GeV and (a) zero charge, (b) positive charge, (c) negative charge. The correlations C 2 - have been scaled up by factors 4 and 16 in cases (a) and (b), respec tively.

the d o m i n a n t t e r m in the region o f low SI cluster masses. The d e p e n d e n c e on the SI cluster charge is appreciable.

A f t e r this prelude on cluster p r o p e r t i e s , let us smear o u t the SI cluster mass as described in sect. 2, eqs. ( 1 5 ) and (16). We thus have three p a r a m e t e r s , a t i 3 , to adjust the m o d e l to the m e a n central p i o n m u l t i p l i c i t i e s ~ + = 3.8 and g - = 2.7 m e a s u r e d in the Ela b = 205 G e V pp b u b b l e c h a m b e r e x p e r i m e n t [6] ( " c e n t r a l " was d e f i n e d in ref. [6] by a c u t in F e y n m a n x , ix[ ~< 0.6). These n u m b e r s suffice to fix

3. Engels, K. Schilling / D y n a m i c a l and Bose-Einstein correlations 357

J

1.8 1.6 1.4 1.2 ..= 1.o

t ;-- 0.8

¢

c- 0.6

y

OA, 02

f

1 / / f

/ j

J ' ~ " t "

/ j "

/ . j "

/ / I " .

/ . / " .,..'

/ / . J .--' ,.'"

j . j . . , , . " "

. j , " . . . * " *

I A . . , ~ : . q ~ . . . ' l ' " ' I I I I ) I ,.-

0 2 0.6 1.0 1.4 1.8 2.2

M, [Gev]

Fig. 2. The correlation strength <nt(n t' - 8 tt'))/(nt) from a single SI cluster as a function of its massM c. Full curve: t = +, t' = - , neutral clusters. Dashed curve: t = +, t' = - , non-neutral clus- ters, and t = - , t' = - , negatively charged clusters. Dashed-dotted curve: t = - , t' = - , neutral clusters. Dotted curve: t = - , t' = - , positively charged clusters.

two o f our parameters, or to p u t it differently, we can still play ( w i t h i n certain limits) with the n u m b e r of, say, negative SI clusters. Setting their p r o d u c t i o n rate equal to zero clearly m i n i m i z e s C ~ - - . I n this case, we o b t a i n fro = 1.97 a n d ~-+ = 1.1 for the m e a n n u m b e r o f n e u t r a l a n d positive SI clusters, respectively. The overall cluster d i s t r i b u t i o n f u n c t i o n aw(Mc), a = E z 3 a l l 3, is p l o t t e d in fig. 4. It corresponds to an average charge m u l t i p l i c i t y per average SI cluster o f (nch)c = 2.12 a n d )~c = 1.28 GeV.

The c o m p a r i s o n with the e x p e r i m e n t a l data [6] is c o n t a i n e d in fig. 5. The full curves s h o w n there are the m o d e l predictions w i t h o u t the " c o n d e n s a t i o n " t e r m in eq. (4). We find that the observed e x p e r i m e n t a l ratio

f

⁵

is correctly b o r n e o u t b y the model. F u r t h e r m o r e , the strong peaking o f ~r+Tr - corre- lations at M . . < m n is described b y the m o d e l as well. The ~ r - n - correlation, h o w - ever, is evidently m u c h larger in the l o w - M . , region t h a n the full curve predicts. I n fact, the w i d t h o f this curve is o n l y slightly narrower ( b y a b o u t 40 MeV) t h a n for 7r+Tr - . So we do n e e d the " c o n d e n s a t i o n " terms, to be discussed below. As far as the large M.~r is c o n c e r n e d , o n the o t h e r h a n d , we have to r e m e m b e r t h a t our ideal gas a p p r o x i m a t i o n to fireball decay becomes p o o r at large Me. I n fact, the large- mass taft o f C 2 ( M . . ) is s o m e w h a t sensitive to the size o f 7 in eq. (16) w h i c h is n o t the case for the proper fireball m o d e l [17]. Therefore one should n o t a t t a c h t o o m u c h weight to the l a r g e - M . , behaviour o f o u r a p p r o x i m a t i o n . In a n y case, one can

358 J. Engels, K. Schilling/Dynamical and Bose-Einstein correlations

J 3.0

"~ 2S 2.0 a.-,2 1.5 10

% 0.5

0 0.2

( a )

1 / / .~.., , ~ , . . ~ " " . . . . . . . .

0.6 10 1.4 1.8 2.2

Mo [Gev]

100

~ 9 0

o

70

~ 6 0 ' E

c 5 0 8

~ 4 0

(/1 o

"' 30

"6 .~- 20

a

lO o

0.2 Fig. 3. (a) The integral

2

fi~__l --d3pi F 2 ( P l , P 2 )

_ 2pi 0

Cb)

i -

\ \ \

,,,

. ".

\. ...

\ \ ...

\ _\ \ _. -. _•

\ \ . ' ...

~ . " ' . . .

" - ~ "~-... ~ ...

L I t I I t I i [ I --

0 . 6 1 0 1 4 1 B 2 2

M¢ [ G e V ]

from single-cluster decay plotted versus SI cluster mass M c. The meaning of the various distribu- tions is the same as in fig. 2. (b) The percentage of the "condensation" term (see text after eq.

(5)) contained in the curves of fig. 3a in the case of rr-Tr- distributions. (dashed curve) negatively charged; (dotted curve) positively charged; (dash-and-dot curve) neutral SI clusters.

easily u n d e r s t a n d t h e b a c k g r o u n d u n d e r t h e p signal in C~-(M~r~r) as c o m i n g f r o m t h r e e - a n d m o r e - p i o n d e c a y o f SI clusters.

T h e " c o n d e n s a t i o n " t e r m t h a t s h o u l d be a d d e d to t h e full curve d r a w n in fig. 5b f ' ~ C ~ - - a m o u n t s t o 0 . 1 7 6 ( M , Tr - 2 m ) . In o r d e r to visualize this t e r m , w e r e m e m b e r

J. Engels, K. Schilling /Dynamical and Bose-Einstein correlations 3 5 9

4 0 ~ 3.5 3O

v

~ 2.0

1.5 1.0 0.5

0.2 Q6 1.0 1.~ 1.8 2.2

Mc [0ev]

Fig. 4. SI cluster distribution assumed to be produced during collision (see eq. (16)). It corre- sponds to ~ = 4, 7 = 6 GeV -1 , M 0 = 500 MeV. Average cluster mass is ~'~c = 1.28 GeV. Mean SI cluster numbers: ~-_ = 0, ~-+ = 1.1, ~-0 = 1.97.

that its 5-function character stems from the usual limit o f continuous counting o f states in the derivation o f the level density r [21 ] . With a proper choice o f a finite quantization volume, the 6 function becomes smeared out to a function C ~ (M'Tr~r) with a width o related to the spatial extension o f an SI cluster. Hence we make a properly normalized Gauss ansatz for it:

C ~ - (M~Tr) = 0 . 1 7 ~ 2 o x p ( - ( M ~ , - 2m) ~ / 2 o 2 ) . (17)

Naturally one would expect 02 = m 2 = 0.02 GeV 2. The corresponding prediction for the n n correlations is shown as a dashed curve in fig. 5b. F o r the purpose o f illus- tration, we have also included in this figure the cases 02 = 0.01 GeV 2 and 02 - 0.03 GeV 2. We see that the " c o n d e n s a t i o n " terms add appreciably but not sufficiently to the low M ~ peaking o f C~--. Roughly speaking the SI cluster decay accounts for 80% o f the observed low-mass enhancement. One might be t e m p t e d to increase the number o f negative clusters in order to better reproduce the peak in C ~ - - . In fig.

6, we display the situation for the (arbitrary) choice ~-0 = Y-. As expected, the C~-- correlation does show a stronger enhancement for small M~,~ than previously found;

but the general behaviour o f the n n correlation is now significantly above the data.

The data indicate a dip structure at M ~ = 700 MeV, which, if it persisted, would be a serious problem for such simple SI cluster emission models as ours.

To give an impression o f the importance o f energy-momentum conservation in SI cluster decay, we have included in fig. 6a the results o f the most simple-minded

360 J. Engels, K. Schilling / Dynamical and Bose-Einstein correlations

8

"T

6

~ s

:E

' 4

t j

3

_ [

- ( a )

0.4 0.8 1.2 1.6

M~=[GeV]

i I

2.0

,~0!-

3.5 - (b)

~I.S -

3.0- t J 2.5 -

~ 2 . 0 -

- 0 . 5 I ~ I I [ _ I I [ I I I =.

0 0. z, 0.B 1.2 1.6 2.0

M.~[GeV]

Fig. 5. Inclusive correlations C~-(Mn~) and C2-(Mrr n) calculated from ttle SI cluster distribu- tion shown in fig. 4 (assuming no production of negative SI clusters). The full curve in fig. 5b does not contain the " c o n d e n s a t i o n " term. The latter has been added with various widths (see eq. (17)): (dashed curve) a 2 = 0.01 GeV 2, (dash-and-dot curve) a 2 = 0.02 GeV 2, (dotted curve)

~r 2 = 0.03 GeV 2. Data are taken from ref. {6].

t h e r m o d y n a m i c a l a n s a t z

F 2 ( P l , P 2 ) ~ ( e x p ( E 1 / k T ) - 1 } - I { e x p ( E 2 / k T ) _ 1 ) - ! , ( 1 8 ) a r b i t r a r i l y n o r m a l i z e d , a n d f o r t h e t w o t e m p e r a t u r e s T = 120 M e V a n d T = 180 M e V

J. Engels, K. Schilling /Dynamical and Bose-Einstein correlations 361

8

7

.~. 5 4

i 3

, (a)

\ '...

2 I\ "....

",,

1 \

\~. ""...

"'"

I J I m

0 O.t, 0.8 1.2 1.6 2.0

M=n [GeV]

/*.0 ^3.5 t ^{( b )}

3.0

~ 2.0

"

. . . . 2.0

M~ [ GeV'I

Fig. 6. (a) Inclusive correlation C~2-(Mnn). Full curve: prediction from SI cluster decay, with equal amounts of neutral and negative clusters, v 0 = v = 0.6, v-+ = 1.7. Dashed (dotted) curve:

calculated shapes from simple thermodynamical ansatz eq. (18) for T = 120 (180) MeV. (b) Inclu- sive correlation C~-(MTr~r) calculated from S1 cluster decay as in fig. 6a. The full curve does not contain the "condensation" term. The dashed-dotted curve included the "condensation" term with o 2 = 0.02 GeV 2.

4. Discussion and s u m m a r y

We have d e m o n s t r a t e d that the m a i n features o f the present inclusive c o r r e l a t i o n data for C~-(M~,) and C~-(M~n) can be naturally u n d e r s t o o d and e c o n o m i c a l l y described in a c o n v e n t i o n a l i n d e p e n d e n t emission m o d e l for fireballs o f isospin o n e and average p r o p e r t i e s k n o w n f r o m rapidity space investigations. The d y n a m i c a l correlations for zr+Tr - and n - n - pairs are p r e d i c t e d to be v e r y similar in shape, w i t h

362 J. Engels, K. Schilling/Dynamical and Bose-Einstein correlations 7r ~r suppressed by about a factor o f 5.5. This result suggests that we use the experimental C ~ - ( M . ~ ) , normalized to match the experimental C { - ( M , ~ ) for M 2 700 MeV as dynamical background when analyzing data to extract the Hanbury- B r o w n - T w i s s effect.

The " c o n d e n s a t i o n " terms from SI cluster decay amount to about 30% o f the observed 7r 7r correlations for pair masses below 450 MeV. They seemingly need some augmentation by BE correlations between equal pions out of different clusters.

Such correlations have been evaluated in resonance approximations by Thomas [ 13]

and Grassberger [ 14]. Inserting decay widths and presently known p r o d u c t i o n rates o f established resonances they arrive at effects which are concentrated within the mass region 280 MeV <~ M~,r < 300 MeV. Taking clusters instead, we could expect a broadening o f this region. However, the evaluation o f these effects within our pres- ent statistical framework would need a n u m b e r o f additional assumptions on details of the cluster emission mechanism (such as long-range correlations and overall energy.

m o m e n t u m conservation) and should only be pursued in connection with semi-inclu- sive data *. Therefore, as far as the Bose-Einstein effect is concerned, our present t r e a t m e n t is c o m p l e m e n t a r y to refs. [13,14] **

To make further progress in the understanding o f the short-range correlation phenomena, one evidently needs:

(i) semi-inclusive correlations as functions o f the invariant masses o f two and more particles (including neutrals) to observe higher resonances directly and test p r o d u c t i o n mechanisms more severely;

(ii) higher-statistics data o f good mass resolution to see details o f resonance inter- ference and to settle the above-mentioned dip problem in C 2 - ( M ~ r , ) ;

(iii) relevant ISR data to see possible energy dependences of correlation functions C2 (M.,r ).

We thank Dr. R. Hagedorn for m a n y useful discussions and a critical reading o f the manuscript. KS appreciates his support in the use o f the CERN c o m p u t e r sys- tem SIGMA, without which this work would have been much less fun.

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** Recently, Giovannini and Veneziano [27] suggested the elimination of correlations from within clusters (named jets in their paper) by forming certain combinations of differently charged pion-pair distributions. Unfortunately, though, they assume their jets to decay independently of quantum numbers (like charge), which certainly does not hold for short- range correlation phenomena.

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