The shape of the KNO scaling function in QCD

Volume 121B, number 6

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17 February 1983

THE SHAPE OF THE KNO SCALING FUNCTION IN QCD Fernand HAYOT 1 and George STERMAN Institute for Theoretical Physics, State University o f New York at Stony Brook, N Y 11794, USA

Received 8 November 1982

We present a simple, heuristic derivation of the shape of the KNO scaling function in a leading log model for QCD.

A remarkable feature of the charged particle multiplicity distribution at the p - ~ collider [1] 01/2 = 540 GeV) is that it satisfies KNO scaling [2]. The data at lower energies (ISR, Fermilab and Petra) already exhibit such scaling [3,4], and its general applicability seems well established in the light of the recent results, where the average charged multiplicity is thirty, and events with more than a hundred particles have been observed. In ref. [5], KNO scaling is derived from QCD in the leading log approximation (LLA). The arguments are rather technical. In this paper, we try to make the QCD description more accessible by showing that it is in fact easy to understand the overall shape of the KNO distribution. We share with ref. [5] the picture of gluon bremsstrahlung and branching. Other sources of particle multiplicity, such as cut reggeons [6], will not be treated. The only technical result we will need is that the average multiplicity of a jet o f m a s s M increases like exp (In M2) 1/2 [5,7]. The intuitive picture of hadronization familiar from "preconfinement" [5,8] arguments is that highly offshell lines branch to give a large number of color singlet clusters whose masses fall in a narrow distribution about some fixed, energy-independent, mass Qo- We shall present a quite different but complementary picture in which a small number of clusters, whose masses grow with energy, give rise to observed hadrons. Such a schemewas put forward some time ago by L6vy [9], and, as we shall see, it gives a remarkably accurate prediction for the KNO scaling function. Our goal is to

derive the qualitative features of this model from QCD. First, let us review L6vy's work. We begin by supposing that in inelastic collisions a number c of clusters is produced, each one decaying into a number p of particles. The number n of charged particles is then simply (1)

n=cp .

If we take p as a fixed number at a given energy (as an idealization of a narrow distr~ution), the charged particle probability distribution P (n)will directly reflect that of the clusters, P(c). The average charged multiplicity is then simply (2)

(n } = ( c ) p .

This corresponds to the QCD picture of ref. [5] ifc is identified with the number of color singlet clusters with masses near Qo, each of which should produce p hadrons, p is then energy independent; KNO scaling holds for the hadrons because it holds for the clusters. In L6vy's picture just the opposite is true. (c) is taken as a fixed, energy independent number, while p grows with energy. The distribution of particles is now indirectly related to the distribution of clusters L~vy picked for the cluster probability a Poisson distribution (3)

P(c) = (c )C e-/c ! .

(c) is fixed in the following way. The experimental ratio H = (n 2 ) / ( ( n 2) _ (n 2

1 Permanent address: DPh-T, CEN Saclay, 91191 Gif-surYvette Cedex, (France). 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

)2)

(4)

is constant and nearly equal to four. This is true all the way from Fermilab to ~ - p collider energies [2]. For 419

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the Poisson distribution (3), (c 2) - (c) 2 = (c). Then (2) gives

H = .

(5)

Thus, the experimental quantity H is just the average number of clusters. What is remarkable is that a Poisson distribution for the clusters, with only one ~arameter (c) = H, not only gives KNO scaling for P ( n ) , but gives an explicit form for P(n),

P'(n) = (2/,o) P(n/p) = (2 /) e((n/). ).

k~ ~ p - ~ k i ,

(8)

i=1

(6)

With (c) = 4, and replacing c I by V(c + 1) in order to obtain a continuous function, it is (n)'fi(n) = ~b( n / ( n ) ) ,

Fig. 1. Diagramsin the LLA.

where p is the quark's final momentum. Then we may use the eikonal approximation for each quark denominator:

(7) (p +~ki)2 ~ 2p.~k

i .

(9)

¢ ( x ) = 8 e - 4 4 4 x / r ' ( 4 x + 1).

As shown by L6vy, this function is in astonishingly good agreement with the data: the first nine moments c a = (n'~)/(n) a (tx = 2, 3 .... 10) are in close agreement with those determined experimentally from the data obtained over the Fermilab range (50-300 GeV) [3]. Other forms ofP(c), with (c) constant, would lead to KNO scaling for P (n), but it seems significant that the data appears to choose a Poisson distribution. It might seem difficult to derive in QCD the picture of a fixed number of heavy clusters following a Poisson distribution. For instance, on-shell gluons themselves are not emitted in a Poisson distribution. Nevertheless, we shall see the picture emerge in the LLA. As a field theoretic model we turn, as in ref. [5], to the decay of an off-shell quark of mass Q2. The relevant LLA graphs in the axial gauge are shown in fig. 1. These graphs have been studied extensively in the jet calculus and related investigations. We assume they give a qualitative picture of the cross section. We observe that the graphs already arrange themselves into a number of clusters, each associated with the emission of a single, usually off-shell, gluon, which branches into real gluons without interacting with the products of any other of the "original" gluons. Such interactions are lower order in the LLA. In the process of branching, the quark loses at most a small fraction of its energy at each step, and it remains energetic throughout the process, although it eventually reaches the mass shell as its invariant mass is radiated away. We consider that region in momentum space where, ifn gluons of momentum ki, i = 1 ... n, are emitted, then 420

The momenta k i must then be strongly ordered in both energy and angle (to p), and by standard manipulations [10] we find that, in the LLA, the effect of gluons on the cross section may be exponentiated,

R(k 2)

d4k' ~ (k2 - k'2) disc ~r"v(k ~, =g2p~PV df (27r)4 ( p ° k ' ) 2

(lO) where e is a small fLxed number and o0 is the "Born" amplitude, for the lowest-order process with no gluons. nUV(k2) is the full gluon two-point vertex in the axial gauge. From (10)we see that the production of clusters, if not gluons, does follow a Poisson distribution. This gives the first confirmation of L6vy's picture. I f P c is the probability of emitting c clusters, then expanding o/o 0 gives e-(CT )(CT)c Pc = c! '

(11)

where

f

(c T) = ~2Q2 dk ~ P(k 2)

(12)

is the average total number of clusters emitted. Since

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I

(o)

(b)

Fig. 2. Real and virtual diagrams whose mass dependence cancels.

the gluon is massless, (c T ) is an IR divergent quantity unless a cutoff is introduced. If we assume this is done, then (c T ) is finite, but energy dependent. It will be dominated by the emission of nearly on-shell clusters in a nearly forward direction. The branching of these clusters into gluons is, by definition, included in P in eq. (12). It does not change the number of clusters. The number of clusters is therefore much smaller than the number of gluons appearing in the final state. We would like to estimate their number. First we estimate the total number of clusters, and then go on to show that only a small subset of these contribute most of the multiplicity. To estimate the total number of clusters we consider fig. 2. Fig. 2a represents eq. (12). Fig. 2b is the corresponding diagram with no cut gluons. It is known to behave as In Q2/m2 In(In Q2/m2) in the axial guage taking asymptotic freedom into account [11]. m is a fixed finite mass. It gives the leading behavior of the Sudakov form factor. A variation [12] of the Kinoshita -Lee-Nauenberg theorem [ 13] requires that fig. 2a cancel the m dependence of fig. 2b. But fig. 2a is proportional to (c T ), the average total number of clusters. Thus we conclude (c T) ~ In Q2 -In In Q2 .

(13)

The exact dependence (13) is not crucial, only the fact that (c T ) increases as some power of a log, and hence much more slowly than does (n), asymptotically. In fact we can do better and put an even more slowly varying upper bound on the number of clusters. To do this we use the following observation. If ti(Q) represents the average multiplicity for a jet of mass Q, then

h(e20 2) ~ exp [c(ln 0 2 ) 1/2 ] exp (ln(e)/21n 0 2) ~ h(Q 2) exp(c ln(e)/21n 0 2 ) ,

(14)

where e is a small but a fixed number. That is, although h increases faster than any log, it shares with the logarithms the property that scaling its argument makes no

17 February 1983

difference in the scale of the function for large values of the argument. Thus, asymptotically, any cluster whose mass is even a small fraction of Q2 has an average multiplicity of the same order as the whole jet. It is the fluctuations in the number of such clusters that dominate the fluctuations in the total multiplicity. "Lighter" clusters, whose masses grow much less rapidly than Q2, are not numerous enough to build up large multiplicities. Define 0 2 by (c T) exp[c(ln Q2m)1/2 ] ~ 6 exp[c(ln Q2)1/2] , (15) with 6 ,< 1. This gives Q2m ~ Q2exp [ - ( 2 / c ) (In Q2)1/21n(1 n Q2 In ln(Q2)/6)] (16) The effect of all clusters with masses less than Q2m must be small. Their average number, (about equal to (eT)) times exp [cOn Q2m)], which is the maximum average number of particles into which one of them can decay, is a small fraction, 6, of the average total multiplicity. This suggests that we define an average multiplicity of "hard" clusters by e2Q2 dk 2

(CH)= f

k 2 p(k2)

(l/c) (In 02) 1/2- [In(In 0 2 ) 2 + O(ln In Q2. In In In Q2)] .

(17)

Here we have again used the analysis leading to (13). In terms of CH, the cross section may be written as o = OOrsoftexp((cH )), I-

Q2m dk2

rsoft -- e x p L f

-p-g(k2)J.-I

(18)

Once again, exp(--(CH)) times the nth term in the expansion of exp ((C H)) may be interpreted as the probability for the emission of the n hard clusters, which therefore also follow a Poisson distribution. In summary, we see that the number of clusters which determine the gross features of the multiplicity distribution is a slowly varying function of energy. This, along with the Poisson distribution for hard clusters, completes the heuristic justification for L~vy's form of the KNO 421

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scaling function. Now we go on to make several explanatory points. (i) We must emphasize that the clusters themselves have no direct physical interpretation. No experiment can distinguish which cluster a particle in the final state came from. They are not to be identified with separate jets, either. All clusters are emitted preferentially in the forward direction, although hard ones will tend to spread out more than soft. What is important is simply that the cross section can be written in the form (10), and that, as we shall see below, we can analyze the relation of multiplicity fluctuations to the structure of the exponent. (ii) Levy's clusters each decay into a fLxed number of particles. Ours do not, since they describe the decay of gluon jets. To study this problem we introduce a Fourier transform for n-particle cross sections, O n , by f ~ exp(-int~)G(ot),

o n ~ o Orsoft

02

k "~ (19)

Q(k 2, #) is the probability for a gluon of mass k 2 to decay into # particles. In (19), we have neglected the contribution of r s to o n . Moments of o n with respects to n give, for example

(n) = (CH)(#H), (n 2) = (CH) (#2) + (c H)2(#H)2 ,

<.n> = f

k2 g d. #a(k2,").

(20)

Q~m The first moment does not depend on the distribution of#H, but of course higher moments do. For the distribution used by L~vy, for which #n is a f'txed number, (n2)L = (n)2(1 + 1/(CH)).

(21)

Then the second moment for a general distribution ~k is given by

(n2 )q, (n2)L 422

-

2 --(#H)~o) 2 ~ (/'tH), . I + (CH)\ (/.tH)~ /" 1

1 + .

.

.

(22)

17 February 1983

To get a rough estimate of the change in the second moment, we observe that #H is itself the multiplicity for the decay of a hard jet, in this case one of the clusters. If for self-consistency, we assume KNO scaling in the decay of jets, ratios like the one in (22) should not depend on energy. We assume as well they do not depend crucially on particle type. If we ignore other sources of large multiplicity, the ratio of the variance of #H to (#H)2 ought to be about the same for each cluster as for the total multiplicity. Taking the experimental value of 1/4 for this ratio, and L~vy's value of c H = 4, we estimate (n2)~ ~ (n2)L X 1.05.

(23)

Thus the second moment need not change much if the clusters' distribution in multiplicity is sharp, as would be expected from experiment. These considerations can be extended to higher moments. They are only meant to test the consistency of our picture, and are not meant to be taken seriously beyond that. (iii) Can this picture survive an extension beyond ladder diagrams and that the LLA? We believe that it is possible that it does. As we pointed out above, the concept of the cluster is just a convenient construct. The real information is that the cross sections exponentiate as in eqs. (10) and (19). That the complete cross sections might have such a form is possible. In fact, leading log exponentiation, associated with ladder diagrams, has been found to generalize beyond the LLA in, for instance the Sudakov form factor [10,11] and jet cross sections [14]. (iv) The observation that only the "fat" gluons contribute to the multiplicity asymptotically has conse. quences for the universality of the KNO function. Quite generally, we expect the overall shape of the function not to depend on the details of process since it is in a sense determined by short distance behavior. (v) Our pictures and that of ref. [5] are not in contradiction. They are based on different organizations o f the basic diagrams. We simply point out that the f'Lxedcluster interpretation offers a simpler way of understanding the overall shape of KNO scaling function. We wish to thank C.N. Yang for useful comments and discussions. One of us (FH) thanks the Institute for Theoretical Physics at Stony Brook for its hospitality. This work was supported in part by the National Science Foundation Grant No. PHY 81-09110.

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