c

Physics Letters B 321 (1994) 431-434 North-Holland

PHYSICS LETTERS B

Factorial cumulant moments and correlations in pp collisions

at 400 GeV/c W a n g S h a o s h u n , Z h a n g Jie, Y e Y u n x i u , X i a o C h e n g u o a n d Z h o n g Y u

Department of Modern Physics, Universityof Scienceand Technologyof Chain, Hefei 230027, China Received 18 November 1993 Editor:. L. Montanct

The pseudorapidity distribution of charged particles produced in pp collisions at 400 GeV/c was measured by using LEBC films offered by the CERN NA27 Collaboration. The scaled factorial cumulant moments have been calculated. The results show that the second-order cumulants have positive values, while the cumulants of higher order are consistent with zero except for the situation of not>I4 events, where the third-order cumulants have positive values beyond the statistical uncertainties. It means that the observed increase of the higher-order factorial moments Fq is almost due to the short range dynamical two-particle correlations in pp collisions at 400 GeV/c.

Intermittent behavior o f multiparticle production is now observed in high energy leptonic, hadronic and nuclear collisions by computing the scaled factorial m o m e n t s suggested by Bialas and Peschanski [ 1 ]. Several models have been proposed to explain the intermittent phenomenon, such as self-similar r a n d o m cascade models, jet models with a self-similar branching structure, the second-order phase transition from the quark-giuon plasma to normal hadronic matter and short-range correlations. Carruthers and Sarcevic [2] have shown that in hadronic collisions the observed increase o f the scaled factorial moments is a consequence o f the short-range correlations, and so it m a y not be necessary to invoke power-law behavior. However, from scaled factorial moments only, we cannot investigate the correlations o f various orders, because any factorial m o m e n t o f a given order includes contributions from all lower orders o f multiparticle correlations, especially from twoparticle correlations, and therefore we need a m e t h o d to remove all lower order correlations from any given order. The factorial cumulant m o m e n t s are designed to do this job, and the factorial m o m e n t s Fq can be expressed in terms o f the factorial cumulant moments as follows [ 3 ]: F2 =K2 + 1 , Elsevier Science B.V.

SSDI 0370-2693 (93) E 1557-E

F3=K3 + 3K2 + 1 , F4=K4 +4K3 + 3(--~2) +6K2+ l , F5 = K s + 5K4 + 10K3 + 10K3K2 + 1 5 ( K 2 ~ + 10K2+ 1 ,

(1 c o n t ' d )

where the vertically averaged scaled factorial moments Fq ( q = 2 , 3, 4, 5, ...) are defined as

1 ~

(km(km-1)...(k~-q+l))

Fq(Jr/) = ~ . ~__, 1

~ [ " !--,,

(km)q Pq(ql,..., r/q)

(2)

and 1 Jve~

( k , , , ) = N~---~ ,=, ~' k,,,,=~m&l.

(3)

The pseudorapidity window At/is divided into M b i n s o f equal width J q = AF1/M, km is the number o f particles in a bin m, ( ) denotes the average over m a n y events, andpq (ql, ..-, r/q) is the q-particle density correlation function. The qth order scaled factorial cumulant m o m e n t is given by

(1) 431

Volume 321, number 4

1

1

PHYSICS LETTERS B

g

Cq(rtl,..., rtq)

f m) (4)

- .~,1~2,=1 ( k m ) q ,

where C2(rtl, rt2) =P2(rtl, ?72) --Pl(rtl)Pl (rt2) ,

C3(rtl, rt2,/'/3) =P3(rtl,

t'/2, ?73)--PI (rtl)P2(rt2, 1/3)

--Pl (rt2)P2(rt3,711) --Pl (rt3)P2(rtl, ?72)

+2p, (rt,)p~ (rt2)P, (rt3), :

(5)

are the two-, three-,...-particle correlation functions. In eq. ( 1 ) the overbar average is defined as 1

g

(6)

KiKj = - ~ ~--1Ki( m )Kj( m ) .

Here f qt,n) Ka(m) = (kin) a

(7)

is the qth order factorial cumulant moment for bin m and, f~'~)=(k,~(km-l))-(k,,)

2,

f~'~)=(km(k,~-l)(k,~-2))

3 February 1994

tions, while in hadronic collisions K3 and K4 are nonzero. In the present investigation the pseudorapidity distributions for charged particles produced in pp collisions at 400 G e V / c have been measured by using the LEBC films offered by the CERN NA27 Collaboration. The space geometrical acceptance was 47r for the high resolution LEBC bubble chamber. The diameter of the bubble was 17 ~tm. The density of the bubbles was 80 cm -~, without magnetic field, the tracks were linear and the pictures were dear, this helped to measure the angular distributions of the reaction products accurately. The details about the measurement are described elsewhere [ 7 ]. A total of 3436 events (rich/> 4) have been measured. The pseudorapidity distribution in the CM frame is shown in fig. 1. The accuracy of the pseudorapidity in the region of interest ( - 2 < rt < 2) is of the order of 0.1 pseudorapidity units. The scaled factorial cumulant moments have been calculated for all events and for 2026 events with charged multiplicity rich>_,10 in the pseudorapidity window Art = 4 ( - 2 < rt < 2). The calculated results of the cumulant moments of order q = 2, 3, 4 and 5 for all events and for higher multiplicity events (nCh~> 10) are plotted against 1/~rt in fig. 2 and fig. 3 respectively. It is observed that the values of the second order cumulants K2 for different average multiplicity events are significantly different from zero. In each case, the value of K2 increases initially with decreasing bin size, then approaches a saturation value.

- 3 (kin(kin - 1 ) ~ (kin) +2(kin )3 :

(8)

are just the integrals of the corresponding cumulant correlation functions C o (rtl, ..., rta). The relations between the vertically averaged factorial moments of different order and the corresponding cumulant moments, as given by ( 1 ), are exact. So we can calculate recursively the cumulant moments K a from the measured data according to eq. ( 1 ). If there are no true dynamical correlations, the cumulant moments thus obtained will be vanishing. Recently, some experimental results on cumulant moments have been reported [ 4-6 ]. It is found that the cumulants of order q = 2 (q = 3 for some samples) are nonzero, and other higher order moments vanish within their statistical errors for heavy-ion interac432

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--6.0

--~.6

0.0

~.6

5.0

7? Fig. 1. The pseudorapidity distribution in the center-of-mass frame for n~>4 events.

Volume 321, n u m b e r 4

PHYSICS LETTERS B

,

I

3 February 1994

'

'

(a) q = 2

0.4

|

(,.) q=Z

0.4

V'

I

b~ 02,

0.2

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.~.I.,,.T~I11'I"TTT,'11"

0.0 0.4

(b)

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(~) q=4

....... ,,.,

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i

b~ o.o

t

,, I

10.0

-

(d) 5.0

¢r

O'

b~

o~o

O.0

6.0

f

x 2 / d o f = 0.27,

K2 =27( 2 (&//O

7=0.126__0.005,

1 + e x p ( - &//()

(9)

7=0.277_+0.005,

_,

A,,..I..t ~.,

I

i

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t

'

.,~,..*..,..,1"~,~

......

" .... "'T~P'"T ' I 8.0

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(=1.14_+0.10,

for n~ >/10:

x2/dof=

The best values of the fitted parameters are given as follows:

,

--....,,~,..,.,~tlfttlt

Fig. 3. Variation of cumulant m o m e n t s K,, q = 2 , 3, 4 and 5 with 1/J~/for rich>t 10 events. The solid curve represents the best fitted values of K2 obtained from ( 9 ) with parameters r = 0.126, ~ ffi0.53.

The value of K2 for nch >t 4 events is larger than the one for higher multiplicity events (n~ >t 10). As suggested in ref. [4], the values of/(2 versus &/can be fitted with the following formula:

&12

|

.......

-5.0 t 0.0

12.0

Fig. 2. Variation o f eumulant m o m e n t s Kq, q = 2 , 3, 4 and 5 with 1 / b / f o r nca >/4 events. The solid curve represents the best fitted values of K2 obtained from (9) with parameters ~,= 0.277, ~-- 1.14.

-

i

I

i

(d) q=5 5.0

0.0

I

|

. . . . . .

-2.0 10.0

i T

-5.0

i

(c) q=4

o.o

for rich >t 4 :

i

-0.4

-0.4

ce

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,

(b) q=a

0.0

'1

I

~ '11

(=0.53_+0.07,

0.20.

Here the values of 7 are decided on by the saturation values of K2 and the values of(dependent on the initial increase with decreasing bin size, and so a differ433

Volume 321, number 4

PHYSICS LETTERS B

ent fitting region m a y cause different values o f (. In the statistical model for multiparticle production based on the analogy with a F e y n m a n - W i l s o n "gas", the strength o f the correlation length ( indicates how far the physical system is from the critical point [ 8 ]. The values o f ( obtained from this experiment coincide with the results in ref. [ 8 ], but they are significantly smaller than those obtained from heavy-ion interactions [ 5 ]. For q = 3, 4, and 5, the values o f the cumulant moments are compatible with zero within their statistical uncertainties except for the case with n~h>~4 events, where the third-order cumulant moments are non-negligible. The results are similar to those of hadron-nucleus and nucleus-nucleus collisions at approximately the same EM energy per particle. In summary, we conclude that scaled factorial cumulant moments can be used to investigate the correlations o f various orders, because the contributions from lower-order correlations have been subtracted out o f the higher order correlation functions. It is found that the second-order cumulants have positive values, while the cumulants o f higher order are consistent with zero except for the case with nch>~4 events, where the third-order cumulants have positive values beyond the statistical uncertainties. This

434

3 February 1994

means that there are no statistically significant correlations o f order higher than q = 3 for pp collisions at 400 GeV/c. The observed increase o f the higherorder factorial moments Fqis almost due to dynamical two-particle correlations. We are grateful to the C E R N NA27 Collaboration for offering the LEBC films, we are also grateful to the National Natural Science Foundation o f China for financial support.

References [ 1] A. Bialas and R. Peschanski, Nuel. Phys. B 273 (1986) 703; B 308 (1988) 857. [2] P. Carruthers and I. Sarcevic, Phys. Rev. Lett. 63 (1989) 1562. [3 ] P. Carruthers, H.C. Eggersand I. Sareevic, Phys. Lett. B 254 (1991) 258. [4] P. Carruthers, H.C. Eggersand I. Sareevic, Phys. Rev. C 44 (1991) 1629. [5] P.L. Jain, A. Mukhopadhyay and G. Singh, Z. Phys. C 58 (1993) 1. [6] M.I. Adamovich et al., Phys. Rex,. D 47 (1993) 3726. [ 7 ] Wang Shaoshun et al., High En. Phys. Nu¢l. Phys. 15 ( 1991 ) 331. [8] P. Carruthers and I. Sarcevic, Phys. Lett. B 189 (1987) 442.