Some regularities of charged-particle multiplicity distributions and branching processes

Volume 223, number 1

PHYSICS LETTERS B

1 June 1989

SOME REGULARITIES OF CHARGED-PARTICLE MULTIPLICITY DISTRIBUTIONS PROCESSES

AND B R A N C H I N G

P.V. C H L I A P N I K O V and O.G. T C H I K I L E V Institutefor HighEnergyPhysics,Serpukhov,SU-142284Protvino(MoscowRegion), USSR Received 27 March 1989

The multiplicity distributions of charged particles produced in hadronic collisions are considered from the point of view of stationary branching processes. It is shown that the linear dependence of the dispersion on the average multiplicityD =A + B ( n ) - the empirical Wroblewski relation - is an intrinsic property of a wide variety of branching processes. The partially coherent laser distribution is also deduced from one particular case of the branching process.

In our previous p a p e r [ 1 ] we established a relation between the p a r a m e t e r s o f the negative b i n o m i a l distribution ( N B D ) , describing the multiplicity distribution o f the negatively charged particles, by considering the multiparticle p r o d u c t i o n in h a d r o n i c pp ( p p ) collisions as a stationary branching process with the generating function satisfying the K o l m o g o r o v reverse differential equation. In the present p a p e r we make the next step in the same direction, assuming that multiparticle production in h a d r o n i c collisions is a two-step process. On the first stage o f a collision, several sources (fireballs or superclusters) are p r o d u c e d a n d initiate, independently o f each other, the evolution process according to a stationary branching process with a single birth. As a result a n u m b e r o f clusters are f o r m e d which, at the second stage, decay into the observed particles with the generating function q~( x ) . The generating function

the initial state at t=to into a state with i clusters at the t-value o f the evolution parameter. F o r a branching process with a single birth, when one cluster can d i s a p p e a r with p r o b a b i l i t y / t ( t ) A t or can create one ( a n d only o n e ) new cluster with probability 2 ( t ) A t for an infinitely small increment o f the evolution parameter from t to t + At, the f u n c t i o n f ( x , t) can be written in the form

f(x, t)=/z(t)(1--x)+X(t)(xZ--x). In this case the K o l m o g o r o v equation can be solved analytically:

F=l-[i

dz2(z) exp(- i to

[2(~)-/t(~) ] d~)

to t

-1

to

(3) F(x, to,

t)= ~ Pi(to, t)x ~

(1)

i=0

o f the branching process, for the case o f one type o f particles only and for a continuous p a r a m e t e r t, satisfies the K o l m o g o r o v reverse differential equation

[2]

dF /dto= -f(F(x, to, t),

F o r the stationary processes with 2 a n d / t independent oft, the expression ( 3 ) transforms to the following simple form:

Fl = [a-btp(x) ] / [c-d~o(X) ] ,

(4)

where to) •

(2)

Pi(to, t) in ( 1 ) is the p r o b a b i l i t y o f transition from

a=/z(T-1),

b=#T-2,

d=2(T-

T=exp[O,-#)(t-to)]

1),

c=2T-It, (5a) 119

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PHYSICS LETTERS B

for 2 ~/~ (for the so-called noncritical processes) and a=T,

b=T-1,

c=T+l,

d=T,

T=2(t-to) (5b)

for 2 = / t (for the so-called critical processes). If, in addition, the so-called immigration (with coefficient t,) from an outside source is present in the evolution process (as in the case when charged particles are formed from the evolution of neutral clusters or superclusters), then the generating function takes the form Fz=(c-d)"[a-b~o(x)]/[c-d~o(x)]

~+' ,

(6)

where c~= v/2, and the parameters a, b, c and d are the same as in (5), if the process has started from one cluster. However, if the immigration has started from zero, (6) transforms into ~ F3=(c-d)"/[c-d~o(x)

] '~ .

(7)

All formulae presented above are well known in the theory of branching processes (for an acquaintance with it, see, for example, refs. [ 3,4 ] ). Attempts to apply the branching processes for the analysis of multiparticle production have also been made previously [ 5 - 1 0 ] . However, the majority of authors have usually based their calculations on the direct (also called forward) but not the reverse (also called backward) Kolmogorov equation, as we do. As one can see from ( 5 ), the parameters a, b, c and d depend linearly on T. Writing the first ( ( n ) ) and second (f2 = ( n ( n - 1 ) ) ) factorial moments as the first and second derivative, respectively, o f the generating function F at x = 1 ( n ) = (dFl.z.3/dx)x=l , f2 = (dZFi,2,3/dxZ)x=l , one can easily show that ( n ) is a linear and f2 a quadratic function of T for all three expressions (4), ( 6 ) and (7) for F. But then the variance D 2 = (n 2) ( n ) 2 =f2 + ( n ) - ( n ) 2 of the multiplicity distribution for each of the processes with the generating functions (4), (6) or (7) is a quadratic function o f T o r ( n ) , i.e. -

DZ=Ao +A1 ( n ) +A2 ( n ) 2

(8)

#' One can easily see that F2=F,F3. Notice also that the generating function (7) is close to the one for the NBD, and the value of a, as in the experiment, can be noninteger. 120

1 June 1989

Obviously it is not necessary that the sources created at the first stage of collision be the same. One can envisage that the mixture of different sources with the generating functions (4), (6) and ( 7 ) is created. In such a case the generating function can be written, for example, in the following quite general form: N

M= Z Pif(F~)gi(Fz)hi(F3),

(9)

i=1

where 7£Pi= 1 and f , gi and hi are some functions of the generating functions (4), (6) and (7) in the ith component o f the mixture. But again (n) = (dM/dX)x=,= ~ Pi(d(fgihi)/dx):,=t, A = (d2M/dx2)x=, = X P i ( d 2 ( f g i h i ) / d x 2 ) ~ = , are linear and quadratic functions of T respectively and the expression (8) is again satisfied ~2 Thus, starting from a sufficiently general assumption about the two-step mechanism in which the evolution of the created sources into the clusters proceeds according to the stationary branching process with single birth, we obtain expression (8) explaining the empirical Wroblewski relation [ 11,12 ] D=A+B(n)

.

(10)

Attempts to explain the Wroblewski relation have already been made earlier, for example, in a twocomponent model of Van Hove [ 13 ], with the first component being responsible for the diffraction and the second one for the pionization (for a more recent attempt to describe the charged-particle multiplicity distribution using the two-component model, see ref. [ 14 ] ). However, it is difficult to reconcile such approaches with the observation of the dependence (8) in the processes of e + e - annihilation into hadrons [ 1 5 - 2 0 ] . In our case, the quadratic dependence o f D e on ( n ) in (8) is a property inherent to each component in (9). For negatively charged particles produced in pp interactions the Wroblewski formula (10) can be written in a good approximation as D=(1/,~)(

( n ) +0.5) ,

(11)

~2 The parameters 2 and ,u as well as the function ~0(x), can be different for the functions F,, F2and F3 and for different terms in (9). This does not influence the conclusion about the linear and quadratic dependence of (n) and D 2 on T.

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because the fit ( 1O) to the experimental data yields [211 A=0.245 ±0.006,

1 June 1989

P~(N, S, k) =

B = 0 . 5 8 4 ± 0.003. XL~k_,)(

But it is known [22] that (11 ) is in contradiction with the relation

D2/ (n)2=

l/(n) +

1/k

(with 1/ k increasing with energy ) following from the NBD. In our approach this contradiction is solved in favour of the Wroblewski formula, if one considers the case when three identical sources are produced in the collision and then develop according to the stationary branching process with single birth with the generating function (4), i.e., when formula (9) can be written in the form

M(x, T)=(lt(T-1)-(PT-})f°(x)'~ 3 \ (2T-#)-2(T-

(12)

1 )~0(x)J '

and ~0(x) obeys the Poisson distribution ~3

~o(x)=exp[O(x-1)] . One can easily show that in this case

(1

(N/k)" f -S ) +N/k) "+k exPel ~N-/k

-kS/U~ l+N/k]'

(14)

recently applied widely for the analysis of charged multiplicities [24-28]. In (14) L~')(x) is the Laguerre polynomial of the order r; k, N and S are three parameters, with k the integer number of sources, ( n ) = S+N, with S and N the coherent and incoherent parts. Let us show that the distribution (14) can be obtained as a special case of formula (19) with only one term and the following functions:

f(x)=exp[O(x-1)],

g(x)=l,

h(x)=x. (15)

The possible interpretation is that the Poisson distribution of sources without immigration and one source with immigration (7) are created at the first stage of interaction. For simplicity, we consider the branching process without absorption (Ft=0) and with ~0(x) =x. Then the generating function of process ( 15 ) can be written as

D21(n) =A+B(n) ,

=( c - d ~

F [a-bx

where

B=l(l+fl)/(1-fl),

,O=ul2.

(13)

Besides, for fl= 0 the coefficient in (13) completely coincides with the value 1/v/3 in (11 ). We notice also that the last experimental data of the UA5 Collaboration at x ~ = 9 0 0 GeV [23] for the full kinematical region are badly described by the NBD and indicate that 1/k ceases to grow with increasing energy, again in agreement with the constant value of the asymptotic limit D2/(n) 2=B( = } ) in ( 11 ) and (13) ~4. Let us consider now, from the point of view of branching processes, the partially coherent laser distribution (PCLD)

X exp[0(2T-2~T-1 )x-1)1"

(16)

Using the generating function for the Laguerre polynomials [ 29 ] ( l - -1z ) kexp (--~ZXz) =

~mz"L~-l)(x) '

one can obtain the following distribution Pn from (t6): P, = e x p ( - 0 )

X[(T-1)"/T~'+"]L~'-')(-O/(T-1)). ~3 In this case the generating function (12) corresponds to the compound Poisson distribution. ~4 The same limit in the two-componentmodel [ 14] equals ½.

(17)

One can easily notice that the distributions (14) and (17) coincide if k=a,

N=a(T-1),

S=OT. 121

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T h i s p r o v e s that the P C L D is a special case o f the b r a n c h i n g process ( 9 ) ~5. M o r e o v e r the c o h e r e n t and i n c o h e r e n t c o n t r i b u t i o n s into ( n ) in o u r case are correlated:

N /o~-S/O+ 1 =0, a n d the ratio N / S is a f u n c t i o n o f ( n ) :

N / S = (o~/0) ( ( n ) - 0 ) / ( ( n ) +or) .

(18)

In ( 1 4 ) the p a r a m e t e r s N and S are i n d e p e n d e n t . H o w e v e r , the p h e n o m e n o l o g i c a l analysis o f the inelastic pp d a t a in the C M energy range f r o m 13 G e V to 63 G e V using the P C L D gives [27]

N/S=a((n)-3), w h i c h is similar to ( 18 ). T h e p r e s e n t e d e x a m p l e s w i t h the g e n e r a t i n g functions ( 12 ) a n d ( 1 7 ) s h o w the great possibilities o f the m a t h e m a t i c s d e v e l o p e d for the b r a n c h i n g processes. E x p e r i m e n t only, h o w e v e r , can d e t e r m i n e w h i c h b r a n c h i n g process is really t a k i n g place. We close w i t h one final remark. A l t h o u g h all results p r e s e n t e d h a v e b e e n o b t a i n e d for an a r b i t r a r y e v o l u t i o n p a r a m e t e r t, it is o f interest to relate it w i t h the p a r a m e t e r s c h a r a c t e r i z i n g an i n t e r a c t i o n . Q C D suggests the f o r m t ~ l n x/s. T h e n , for n o n c r i t i c a l b r a n c h i n g processes, o n e has ( n ) ~ T ~ (,~/j)~-u, in a g r e e m e n t with the energy d e p e n d e n c e o f ( n ) at high energies.

~5 In contradistinction to (14), the order of the Laguerre polynomials in ( 17 ) can be noninteger.

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