PROPERTIES OF CERTAIN COLLECTIVE VARIABLES USED IN HIGH-ENERGY PHYSICS

G. G. Arushanov,* R. K. Zayanov, and A. I. Ergashev UDC 539.101

The properties of the collective variables ~m (m = i, 2), T, and P, which are used extensively in high-energy physics, are investigated. It is shown that the variables Sm are intimately related to the dipole (m = i) and quadrupo]e (m = 2) moments. A general equation is given for calculating the variance of Sm for azimuth angles having any probability density function; well-known special cases are deduced from this equation. Two collective variables, the "oblateness" T and the "prolateness" P, have the property of stabiIity when the momentum distribution of particles in the ground state is symmetric about some axis. This property should be useful in searches for events of the "jet" and "vortex" type.

The difficulties of representing experimental data on high-energy multiparticle pro- duction stimulate interest in the study of kinematic variables characterizing many-particle reactions as a whole. These variables are constructed from single-variable particles and are determined for each event in succession. Here we investigate the properties of certain collective variables widely used in high-energy physics.

Abduzhamilov et al. [i] (see also [2]) have used the following variables for the in- vestigation of azimuthal correlations:

n n

/ : 1 / = 1

( m = 1, 2), (1)

where ~i is the azimuth angle of the i-th particle, and n is the multiplicity of secondary charged particies. We first of all show that these variables are intimately related to the dipole (m = i) and quadrupole (m = 2) moments.

We consider the symmetric tensor of rank 2

n

K:~ --- ' ~ (2K~. t~ - ' ~ ) (,,, ~ = x, y), ( 2 ) l : l

where K~ =rl/Irll and r I is the radius vector of the i-th charge situated in the xy plane. If all the charges are equal, K~ has the physical significance of a two-dimensional quadru- pole moment tensor and can be reduced to the principal (x' and y') axes. We verify by di- rect computation that

I = I i = I (3)

The relationship of ~a to the quadrupole moment K is evident from Eqs. (i) and (3). It can be shown analogously that 6z is related to the dipole moment.

We now consider the statistical properties of Sm- Let n angles ~ be mutually indepen- dent and identicalIy distributed in the interval 0 ~ ~i < 2~. We calculate the expected

*Deceased.

A. R. Beruni State Technical University, Tashkent. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 79-83, March, 1993. Original article submitted April 8, 1992; revision submitted July 13, 1992.

1064-8887/93/3603-0253512.50 © 1993 Plenum Publishing Corporation 253

value V(~m) and the variance oa($m). For azimuth angles having an arbitrary probability density function f(~i) we have

2~ 2~ r/

o 0 I

~ (.~) ---- ~ ( ~ ) --~,' (~,,). (6)

S u b s t i t u t i n g Eq. (1) in ( 4 ) - ( 6 ) , we o b t a i n a s imple e x p r e s s i o n f o r t h e e x p e c t e d v a l u e [1, 2] :

v (~,) = ] /n (n-- 1) [v~ (cosm~) q -~ (sinm~) ], ( 7 )

in which ~ is interpreted as one of the angles ~i, say ~ . For the variance o~'(~m) we have derived the rather complicated, hitherto unpublished expression

o~ ( ~ ) = 2q +4c~ (n--2) --2c~ (2n--3), ( 8 )

where

ca =- ~ (cos 2 m~) + ,-~ (sin z m~) + 2 -I v 2 (sin 2m~),

c~ = ~ (cos ~ m,~) ,2 (cos m~) q- ~ (sin ~ m?) ~2 (sin m~) q- ~ (sin 2m?) , (cos m~) ~ (sin m?),

c3 = [~2 (cos m,~) q- ~ (sin ,m~)] 2 "

(9)

(lO) (2_1)

Similar results have been obtained in [i, 2], but only for two special cases of the distribu- tion function f(~):

i) in the case of an isotropic angular distribution

f(~)---- (2n)-!; (2.2)

2) on the assumption of an anisotropic distribution of the type

f(~)---- (2~)7 t ( l+acos2~) . (1.3)

The s p e c i a l r e s u ] . t s in [1, 2] a r e e a s i l y deduced from our g e n e r a l e q u a t i o n s ( 8 ) - ( 1 1 ) , which shou ld be u s e f u l f o r t he a n a l y s i s o f h i g h - e n e r g y h a d r o n - h a d r o n c o l l i s i o n s [3 ] .

Next we consider a tensor of the form

Q~ qiq~ (~, 8 : x., y, z), (]4) i : I

where ql =pl/Ipi[ is a unit vector, and Pl is the momentum of the i,th secondary particle in the center-of-mass system (CMS). Let lj (j = i, 2, 3) denote the eigenvalues of the tensor Q~B' where

and, as a result of normalization,

Upper and lower bounds for Iz, 12, and 1 s follow from (].5) and (16):

{

(15)

(16)

(17)

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'2_ < a I a < 6 3,3 < a .a a < 4 , 2 3 ,3 < a t a < 3 , 6

s,o

2,5

0

O< z12w<2 I I I ¢ dd

N ~P 7,5

f dN N ~T

7,5

2,5

s% 0,5 1

I

ii [ t 1 : ,

o,5 ~ 0,5

Fig. 1

0,6< a~1< 1,2 I I

r i t. ' l

" I ,, I 0,5

O~g < a,1, <f,2

t i i ;

I

q i

[ ...I

[ i

! i i T

Fig. 2

The following collective variables can be constructed from the eigenvalues of the tensor

O~ [4]:

p = ! l x ' - h ~,~-h / 2 + e •

(18)

( lg)

The variables T and P characterize the degree of deviation of the angular distribution of the particles from spherical symmetry and indicate the degree to which it is prolate (I?) or oblate (P). Their range of variation is 0-1. If the angular distribution of the particles in an individual event has the configuration of two ideally thin jets, we have T = i. If, on the other hand, this distribution is in the shape of an ideally thin vortex, we have P = i. Finally, if it is spherically symmetric, then T = P = 0. The collective variables T ("prolateness") and P ("oblateness") have been used [4-7] to investigate the form of events in ~-p interactions at 200 GeV/sec and in pp interactions at 200 GeV/sec and 400 GeV/sec also, a sudden change in the average values <T> and <P> has been observed [8] in the same reac- tions as the multiplicity is increased; this event can be interpreted as a second-order phase transition.

Let the variables ~l and ~2 assume equal values for a system of n particles in the ground state:

~'? = ~ ' (20)

255

and let us suppose that slight variations of the densities of particles in momentum space induce fluctuations

A~l,A~2=/=O, A~3-----O. (21)

We i n t e n d t o show t h a t t h e f l u c t u a t i o n s o f t h e v a r i a b l e T a r e e q u a l t o z e r o i n s u c h a s y s - tem:

AT=0

Indeed, it follows from Eqs. (21) and (16) that

(22)

so that

%1 =n- -A~- -%2 , ( 2 3 )

Ak~=--Ak2. (24)

We now confirm the validity of Eq. (22) by direct calculation of the increment AT. The sub- stitutions T + P and X 3 + X I render the assertion (20)-(22) true as well.

Consequently, although the collective variables T and P fluctuate together with the particle density in momentum space in the general case, fluctuations of T and P do not occur in the special case where the momentum distribution of the particles is symmetric about some axis.

We note that the stability of T and P against slight momentum fluctuations of a definite species of particles is attributable to the canceling effect created by the particular struc- ture of these variables [the second term in Eqs. (18) and (19)]. To graphically demonstrate this property of T and P, we have created i0,000 random stars with multiplicity n = 6. Only the angles of emission of secondary particles in the common CMS were sampled in each star, and they were done so isotropically. This sampling mode made it easy to choose stars with the properties that we required (e.g., events of the jet or vortex type with distinct sym- metry axes in momentum space) by means of X 3 and kz.

Figures 1 and 2 show the distributions of the random stars with respect to T and P and how they change with the intervals of variation of the respective variables X 3 and kz. Clearly, the smaller the widths Ak 3 and AX I of the intervals, the narrower are the corre- sponding distributions with respect to T and P, consistent with the assertion (20)-(22). The modeling results indicate that the collective variables T and P, by virtue of their stability against slight particle-density fluctuations in momentum space, can be used in ex- perimental searches for events of the jet or vortex type predicted by certain multiple-pro- duction theories [9].

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