Poisson distribution and multiperipheralism

Nuclear Physics B35 (1971) 406-412. North-Holland Publishing Company

POISSON DISTRIBUTION

AND MULTIPERIPHERALISM

Luca CANESCHI Institute of Theoretical Physics, Department of Physics, Stanford University, Stanford, California 94305* Received 29 March 1971

Abstract: The asymptotic behavior of the production cross sections and the particle distributions are related to the J-plane structure of the forward elastic amplitude in a general multiperipheral framework. The validity of the Poisson distribution is investigated, and it is suggested that certain sums over production cross sections can be of particular physical relevance.

The difficulty involved in the analysis of many-particle reactions has focused both experimental and theoretical interest on highly integrated observable quantities, which can still provide a clue to the mechanisms responsible for many-particle production. The study of the total production cross sections O(na, n b .... s) (where a, b, etc. are labels corresponding to the various kinds of elementary particles and s is the square of the total c.m. energy) is particularly appealing, because no kinematical variable (but the incoming momentum) needs to be measured. Several papers have been recently devoted to finding a suitable modification to an assumed original distribution o(n, s) (that should hold in a fictitious world of identical bosons with no quantum numbers) in order to make it consistent with the conservation laws (like charge [1,2], or better s-channel isospin [3] ) that yield nontrivial constraints on the amplitudes for producing the physical particles. The Poisson distribution for o(n, s) seems everybody's favorite starting point: it is simple, in reasonable agreement with the known data, and it is suggested by models of statistical independent emission, of classical coherent state production, and, to some extent, by multiperipheral models [4]. The purpose of this paper is to investigate more in detail the validity of the Poisson distribution in the multiperipheral scheme. We will therefore work in a world of identical bosons, and, without yet committing ourselves to a particular model, define our multiperipheral scheme through the following requirements: * Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, US Air Force, under AFOSR Contract No. F44620-68-C-0075.

L. Caneschi, Poisson distribution

407

(a) It is possible to define a coupling constant g such that o(n, s) = gnf(n, s) when f(n, s) is independent of g*. (b) The high-energy behavior of the total cross section can be expressed in the form

Otot(S)-- ~ o(,, s)= f s~ u03) d~ n

C

1 -gu03)

'

(1)

where the contour C leaves all the singularities of the integral to the left. Through the optical theorem the same integral also determines the high-energy behavior of the imaginary part of the forward elastic amplitude. (c) We also assume that the function u03) is real and analytic for/3 >/3 c and that

lim [U03c + e ) ] - I = O. e--*O

Our goal is to relate properties of a(n, s) to the structure of u03)**. Let us start with the study of the behavior of o(n, s) at flexed n, s --> oo. The validity of the Poisson distribution o(n, s) = ((n(s))n/n !)e-~(s) together with the logarithmic behavior of ~ common to all multiperipheral models would imply

a(n, s) ~ (lns) n s - c otot(s ) , but we know from the analysis of Finkelstein and Kajantie [6] that in the multiRegge model (that satisfies all our requirements) the asymptotic behavior is given by a(n, s) = (In In s) n s - c o~S)ot. The property of u03) that determines the asymptotic behavior of a(n, s) is the type of singularity at/3 =/~c (a singularity is required by assumption (c)). Let us start from the simplest case in which (i) the singularity of u03) at/3 =/3c is a isolated pole; (ii) this is the only singularity of u03). Then u03) --- 1/03 -/3c) and from (1) ~

g~ f(n, s) = s ~c÷g .

The left-hand side is analytic in g at g = O, and equating the terms in a power series expansion in gn, we obtain _ ( l n s ) n sac f(n, s ) - ~ . *'The definition is not of obvious physical significance, since it requires the possibility of changing the value ofg in order to be meaningful. ** In the case in which the singularities of u(/3) are simple poles, a similar approach has been studied in ref. [5].

L. Caneschi, Poisson distribution

408

This simple situation occurs in the extreme approximations to the multi-Regge model studied by Chew and Pignotti [4]. If we relax assumption (ii), the total cross section has the form Otot(S) = s ~(g) (1 + O(s-A)) ,

(2)

with A -- A(g) uniformly positive. The function a(g), defined in an implicit form by the most positive root of the equation 1 =gu(a),

is analytic at g = 0, because a -+ 3c when g --> 0, and u(fl)- 1 is analytic around fl = 3c. Therefore we can neglect the non-leading term in (2) and expand in power series of g, to obtain

s 3c d n sa(g): (Ctl lns) nsoc' ( n ( n - 1 ) .fin, s ) = ~ - ,

dgn

"

n!

1+

2Ins

ct2+ ... ) . a2

(3)

Here a n = (dn~/dgn)g=O, and these derivatives can be determined in terms of the derivatives (dn/d3) (1/u(fl)) computed at 3 = 3c. This situation occurs for instance in the ladder ¢3 theory where u(3) has poles at the negative integer values of 3. Note that even if the asymptotic behavior o f o(n, s) is not changed by the non-zero value of an(n > 1), to keep only the first (or the first few) terms in the expression (3) is dangerous for n ~ ~(s), because ~ increases with In s. Let us now assume that u(fl) has a branch cut at 3 = 3c. We will consider a powerlike and a logarithmic branch cuts. In both cases the function [u(fl)] - 1 is not analytic at 3 = 3c, and consequently afg) is not analytic at g = 0. Furthermore the total cross section behaves like Otot(S) = s a(g) + cut contribution, but as g ~ 0 the pole terms is no longer leading over the cut term, and therefore both must be taken into account. To find the asymptotic behavior off(n, s) in this case we must actually " u n d o " the solution of the integral equation that presumably led to (1), and expand the integrand in (1) in powers o f g to obtain

.fin, s) = f s~uf~)n+ l d~ . c Assuming now u0~) = ~ -

t3c)-P,

L. Caneschi, Poisson distribution

409

we obtain f(n, s) = s oc (In s)np- l+p [F(p(n + 1))]- 1

(4)

In the multi-Regge model the presence of shrinkage in the subenergies associated with the produced particles is responsible for the existence of a logarithmic branch cut in u(fl) at the usual position/3 c = 2t~in - 2 where t~in is the intercept of the Regge-pole that is repeatedly exchanged along the multiperipheral chain. The leading Regge pole in Otot depends on g in an exponential form close to g = 0: or(g) ~ e - 1/g and again the point g = 0 is not an analytic point for a~g). The asymptotic behavior o f f ( n , s) is determined by: f(n, s) ~

f s ~ ( - In (fl -/3c)) n + 1 d$ . C

The result of Finkelstein and Kajantie [6] can be easily rederived using the identity: (lnx)n =

dn

xele=0.

de n We obtain s#C dn+l (lns)e _ - sac In s (In In s) n + non-leading terms in In In s . f(n's)=lns den+l r(e) e=0 -

From the preceding analysis the following facts emerge: (i) The asymptotic behavior of o(n, s) is usually not the one predicted by the Poisson distribution: this happens only if u(fl) has an isolated pole at/3 =/3c. Therefore we expect substantial deviations from the Poisson distributions for n ~ n. From another point of view, we also expect substantial deviations from this distribution also for n >> ~, where threshold effects should be very important. Nevertheless, the 15oisson distribution can be approximately valid [6] in the most important region n,~n.

(ii) The study of o(n, s) is equivalent to an expansion around g = 0. This is a difficult operation to perform, because ct(g) is in general non-analytic at g = 0, and consequently the natural expansion in powers of In s breaks down. Furthermore, even if the total cross section is dominated by a leading pole at the physical value of g = ~, the cut contributions are essential to determine a(n, s)*. * The necessity of keeping into account both the leading pole and the cut contributions in Otot(S) can be ~en in simple examples. If for instance u(#) = (B - #c)-~, the leading pole has the form s#c+g . At any finite value ofg the pole leads the cut by a power of s but to disregard the cut contribution yields the absurd result o(n, s) = 0 for n odd.

L.Caneschi,Poissondistribution

410

In the physical world we know from the energy dependence of ~, that the value o f ~ is of the order of one, and this value is rather large*. Therefore we expect c~(g) to be a smooth function around g = ~, and one wonders whether it would not be better to consider an expansion around this point instead of an expansion around g = 0. To do this let us define a set of functions~n, s) as follows

°t°t(s)=?0-- 0~-~)~/~n,s).

(5)

The relations k f( - k 's)(--g)k-n f(n,s)= ~k (n) and its inverse

are easy to derive. The definition of f depends on ~: if we choose this to be the physical value of the coupling constant, then

where a(n, s) are the physical production cross sections (in particular j~0, s) = Otot; j~l, s) = n Otot and so on). From the definition (5) 1 dk j~k, s) = k.l ig--/'Otot/=~" dgr

(6)

Because the expansion (6) is performed around g = ~ :~ 0, only the contribution from the leading pole survives asymptotically. Furthermore a(g) is analytic around g = ~, therefore in analogy to the case of an isolated leading singularity (eq. (3)): "t~k' s ) - (~1 klns)k ~

sa (1 +k(k-2In s l ) ~, 2 ~+ 2...)

* The scale is set in the multi-Regge model by t~'0°2>.See ref. [7].

(7)

411

L. Caneschi, Poisson distribution

where ~ = or(g-) and Nn = (dn/dgn) t~(g)lg=~. The exact Poisson distribution again corresponds to Nn = 0 for n > 1. It is clear that the analysis in terms of f is more convenient than the conventional analysis in terms of f if the leading singularity of u(fl) is a branch cut. Let us therefore concentrate our attention on the multi-Regge model, in which this is the case. In the model considered in ref. [7], u(fl) is given by a

e - x dx/q3-

u ~ ) = 3' if(l, 1 , 7 ~ -/3c) ) =

+

=

o Here a parametrizes the vertex residue function as (a eat)k and t~' is the slope of the multiply exchanged Regge trajectory. In the self-consistent solution of ref. [7] 3, ~ 2, and the Chew-Pignotti model [4] is recovered in the limit 3, ~ ~. From the relation 1 = gu(a) we obtain - u2(~)

> 0,

~1 = d

The ratio (~2/~ 2) vanishes as 5 - 1 for 8 = ~'(~ -/3c) ~ ~. In general ~n is of order 81 - n , and exact Poisson distribution holds in the limit 6 ~ oo. The deviations from Poisson distribution measure therefore the ratio between the width of the cut in J and the distance between the leading pole and/3 c. If this ratio is small, the quantities fdeviate little from the one predicted by the Poisson distribution. The significant contribution to.~n, s) with n ~ ~ comes essentially from the large cross sections o(k, s) with k ~ ~, which therefore also approximately follow the Poisson distribution, even if the tails for k '~ ~ or k >> n can substantially deviate. In conclusion we suggest that a systematic study of the sums Y.(n/k) e(n, s) can provide a simple and more direct interpretation than the study of the o(n, s) themselves. The deviations of the behavior of these sums from the predictions of the Poisson distribution measures the importance of the cut, and can be used to extimate the distance of the relevant cut in J for Otot(S) from the leading pole. A nice conversation with R.Blanckenbecler is acknowledged with gratitude. NOTE ADDED IN PROOF Muller [8] remarks that the ~ k , s) of eq. (5) can be obtained by integrating the k-particle inclusive distribution over the k-particle phase space. The approximation of independent emission trivially yields fl(k, s) = [/7(1, s)] k, i.e. a Poisson distribution, and eq. (7) establishes a link between the cuts in angular momentum and the correlations in many-particle inclusive reactions.

412

L. Caneschi, Poisson distribution

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]

C.P.Wang, Phys. Rev. 180 (1969) 1463; Phys. Letters 30B (1969) 115; 32B (1970) 125. D.Horn and R.Silver, Phys. Rev. D2 (1970) 2032; Ann. of Phys., to be published. L.Caneschi and A.Schwimmer, Phys. Letters 33B (1970) 557; Phys. Rev. D3 (1971) 1588. G.F.Chew and A.Pignotti, Phys. Rev. 176 (1968) 2112. J.S.BaU and G.Maxchesini, Phys. Rev. D2 (1970) 2665. J.Finkelstein and K.Kajantie, Nuovo Cimento 56A (1968) 659. L.Caneschi and AoPignotti, Phys. Rev. 180 (1969) 1525. A.H.Muller, Phys. Rev. D4 (1971) 150.