Conditions for the early onset of KNO scaling

Volume 47B, number 1

PHYSICS LETTERS

15 October 1973

CONDITIONS FOR THE EARLY ONSET OF KNO SCALING* D.W. SCHLITT* Institute for Theoretical Physics, University of Utrecht, Utrecht, The Netherlands Received 2 July 1973 A precise def'mition of "early onset of KNO scaling" is given and the conditions for its experimental observation are examined. It is argued that the observed early onset of KNO scaling in the charged particle distribution in p-p scattering is due to a fortuitous maximum in the normalized moments C~ = qand not due to any significant underlying physical mechanism.

In recent months there has been considerable discussion [1-5] of an early onset of KNO scaling [6]. It is the purpose of this paper to define the meaning of "early onset of KNO scaling" and to discuss the conditions which seem to be responsible for the experimental observations. The meaning of the phrase in question is usually vague and left for the reader to deduce from the context. The easiest interpretation is that the scaling behavior reported in refs. [2, 3], occurs at an energy lower than expected by the writer. There seem to be two criteria that justify the use of the term. They are related to the assumptions made in ref. [6] which lead to the scaling hypothesis. Early KNO scaling can be said to occur in some process if (n)o n/Oinel is well represented by a function ff (n/(n)) and 1) (n> is not yet well represented by A + B Ins or more generally, when 2) the inclusive cross-sections have not yet developed a well def'med rapidity plateau. Under either of these criteria the use of the term "early" seems to be well justified for proton-proton collisions in the 50-300 GeV/c momentum range [7]. What is responsible for the observed scaling? Since the arguments on which the hypothesis of KNO scaling are based are the existence of a well defined rapidity plateau and the consequent Ins behavior of(n>, the above conditions for early scaling are good reasons for expecting scaling to be absent. There is an additional reason why the scaling that has been observed in unSupported in part by the University of Nebraska Research Council. * Permanent address: Department of Physics, University of Nebraska 68508, USA.

expected. The scaling is seen in the distribution of charged particles. The discussion of rapidity plateaus etc. is suited to the case where one is considering a single type of particle, e.g. pions. Even if one neglects the production of anti-nucleons and strange particles, the charged particle distribution includes protons in addition to pions. The rapidity distribution of the protons is quite different from that Qf the pions because of the large mass difference and because the protons will predominently be leading particles. If the scaling in this energy range is even remotely connected with the arguments used in ref. [6], then one would expect to see the scaling in the negative particles, which are predominently 7 , rather than in the charged particles. In fig. 1 we plot (n_)o n/Oinel as a function ofn_[(n_), and in fig. 2 the charged particle distribution, (N c )ONc/Oinel,as a function of

A

50

•

=e

•

•O"

o 69 • 102

•

•

0.4

• 205 =

o

303

o

Q2~

ool ao

'e •

12o

2:o

n_l

te~ t

"'. 0 3

-

~.o..---J- 04

4.0

-

Fig. 1. The experimental values of (n_)on/Oinel v~sus

n_/
47

Volume 47B, n u m b e r 1

PHYSICS L E T T E R S

$#

15 October 1973 2.0

(a)

•

%,

1.5

*

.'

50

v=


o 69

,'

,102 ,205 =303

oo

°G

,c~•

. t , col

1.0

** ,

*a~

==

e,

QO 0.5

I

I

I

I

(20

o

•

Q~

&e a

m

3.O

% 0.0 0.0

I

,

1.0

2.0

"

•

xc; ,, c~

• , ,~,

"us

x

#~

e~l~--

2.0

3.0

~ m

°°

Nc/

1.o

Fig. 2. The experimental values of ONc/Oinelversus Nc/
=/(n_)q,

O~Z <~.

(1)

(3)

Through charge conservation all of these quantities are interrelated, since Ne = 2n_ + 2.

(4)

The existence of a scaling function ~ follows from the Cq's being independent of Z and not too rapidly increasing as a function ofq. The differences between fig. 1 and fig. 2 can be summarized by the statement that the Cff are essentially independent of Z while the Cq are not. The existence of Feynman scaling in the inclusive cross section and the rapidity plateau does not give direct information about the Cq. Instead one finds 48

a

xl

0.0 0.0

G q = ( n _ ( n _ - 1) .... (n_ - q +l))/
Z=(,Nc)-I;

In

n

Nc/(Nc>. It is obvious without the aid of fits to hypothetical functions and ) 2 values that scaling is best satisfied for the charged particles. In order to understand why this happens let us turn our attention to the moments of the distributions. We will be interested in four sets of normalized moments Cq

o

a=

t

n

I

Z

&

Fig. 3. a) The normalized m o m e n t s for q = 2, b) The normalized m o m e n t s for q ---3. The values are c o m p u t e d from the data in ref. [71.

that it is the Gq- which should become constant for small Z (the first few correction terms are powers of Z as can be seen from eq. (2.9) of ref. [6]). To go from Gq- to Cq- involves additional arguments which establish that the difference between Gq and Cq is a polynomial in Z for small Z and vanishes at Z=0. * Normal onset of KNO scaling would occur by Gq becoming nearly constant due to the development of the rapidity plateau, and simultaneously or later the Cq would become constant as the polynomial differemce becomes insignificant. In this picture the behavior of C~ and Go and would follow along in some complicated way due~o eq. (4). The same assumptions lead to C~ and Gff being polynomials and consequently scaling for the charged particles. Thus all four quantities approach constant values; it is simple to show that for a given q they approach a common value which we will denote by aq. In fig. 3 we show the experimental values of these quantities for q=2 and 3. All of them except C~ and * Even if one bases early KNO scaling on a c o m p o u n d Poisson distribution as suggested in ref. [5] it is the Gq which are expected to b e c o m e constant.

Volume 47B, number 1

PHYSICS LETTERS

C~ are far from being constant. Also included is GeV/c. According to Krzywicki [8]this quantity should also approach ot2 if, within the framework of Mueller's optical theorem, the forward scattering amplitude is dominated by isosinglet exchange. There are two possibilities for explaining the exceptional position of the Cff. Either there is a fundamental physical reason why they are constant or it is an accident. We will argue that the second explanation is correct and that the accident occurs in a neutral way as the rapidity plateau begins to develop. We propose that the apparent constancy of Cq is due to the fact that in the range 0.1 ~ Z ~ 0.2 the Cff have maxima. This happens because of the qualitative behavior of the Gq. From general physical principles and the limited experimental information available we can sketch the qualitative behavior of~Gq as a function of Z. At Z=0 it has the limiting value 0tq and slope Oq. Since the experimental evidence indicates that scaling is approached from below in both the central and fragmentation regions and s i n c e / 3 q is primarily a correction term due to the fragmentation region, we expect it to be negative. As Z increases we expect Gq to continue to fall as a smooth low order polynomial until ~ 0.1. Somewhere near this value of Z the rapidity plateau will have disappeared and Gq will begin to fall much more rapidly. Since the multiplicity distribution of negative particles is nearly Poisson in the interval Plab = 3 5 - 5 0 GeV/c [9], the Gq must be near 1 at the corresponding values of Z ( ~ 0.2). As Z increases further they continue to fall to zero at thresholds determined by conservation of energy; Gq is identically zero until it is possible to produce q negative pions (or equivalently 2 q charged pions). It is obvious that this discussion is limited to sufficiently small values of q,~say q ~ 5. This is evidently sufficient to determine the main features of the probability distribution as found, for example, in ref. [9]. The importance of higher moments in determining the probability distribution and the function ~k is an interesting question which has been raised elsewhere [ 10] and we will not consider it here. Next we investigate the implications of the behavior of the G~ on Cff. It is easy to show that

15 October 1973

Cq = 2qz q + (2 q -

1) 2q-lzq-l(1-2Z)

(5)

(non_)/(n o) (n_) for 205

+

~ (q) S (l)'2q-lzq-t(1 - 2Z)IGt

k=2=

and

~C~/OZ

= 2 q-1 (q _ 2q +1)zq-2 +q2q (2q-1-1)zq-2(1-2Z) q k + k=2 ~ 1~=2 (q)S(~)2q-tZq-t-l(1-2Z)t-l= (6) X [(q-l-2qZ)Gi + Z ( 1 - 2 Z ) aGflaZ]. where (if) is a binomial coefficient and .q(z./)is a Sterling number of the second kind [11 ]. From these two equations, the qualatative behavior of the Gq, and some experimental information about the Cqc we can deduce some general properties of the C qc. 1 Let us start at Z - ~ and work toward smaller valu1 CI es of Z. At Z = ~ Cq- 1 and as Z decreases the value of C~ rises above 1. If all the Gq were to remain zero, then C~e would have a maximum in the interval " ~
c~=% +[q(q +1)%_1 -

2q% +2t3qlz+ 0 (z2). (7) 49

Volume 47B, number 1

PHYSICS LETTERS

where a o = a 1 = 1. If C~ goes through a maximum then the coefficient o f Z should be positive. This requires that/~q is not t o o negative and that the Otq do not increase too fast with q. These are just the conditions for the existence of normal KNO scaling. Harari and Rabinovici [12] have made a Ins fit to the data for Plab in the range 5 0 - 3 0 0 GeV/c. Their fit can be written as C~ --- 1.20 + 1.42 Z - 6.24 Z 2. When written in this form it gives a reasonable fit to the data and has a maximum at Z = 0.11. Finally, if one examines the experimental values o f Cff given in ref. [2] one notes that except f o r q = 9 and 10 the value is lower for 300 GeV/c than it is for 200 GeV/c. In all cases the data is consistent with the idea o f a maximum. In conclusion we see that there is substantial evidence that the scaling behavior that has been observed is the result o f the general qualitative features o f the moments of the negative pion distribution and is not due to any deep and significant underlying physical mechanism which requires the moments o f the charged particle distribution to be constant for energies where the term early scaling would apply. An additional conclusion is that the tendancy o f (n)On/Oin d to fall roughly on a smooth curve as seen in figs. 1 and 2 is a consequence o f this combination being rather insensitive to the slow changes in the multiplicity distribution [4]. The moments are more revealing o f the physics than the KNO scaling function. The author expresses his appreciation to the Insti-

50

15 October 1973

tute for Theoretical Physics for their hospitality during his leave from the University o f Nebraska.

References [1] [2] [3] [4]

P. Slattery, Phys. Rev. Lett. 29 (1972) 1627. P. Slattery, Phys. Rev. D7 (1973) 2073. P. Oleson, Phys. Lett. 41B (1972) 602. K. Fialkowski and H.I.Miettinen, Phys. Lett. 43B (1973) 493. [5] M. Le BeUac, J.L. Meunier and G. Plaut, Compound Poisson distributions KNO scaling, University of Nice preprint (March 1973). [6] Z. Koba, H.B. Nielsen and P. Olesen, Nuc. Phys. B40 (1972) 317. [ 7 ] The experimental data of interest are contained in the following: W. Richter, Diplomarbeit, Mfinehen (1972), unpublished. H. Boggild et al., Nucl. Phys. B21 (1971) 285. French-Soviet Union Collaboration, Phys. Lett. 42B (1972) 519. Michigan-Rochester Collaboration, Phys. Rev. Lett. 29 (1972) 1686. G. Charlton et al., Phys. Rev. Lett. 29 (1972) 515. NAL-UCLA Collaboration, Phys. Rev. Lett. 29 (1972) 1627. G. Charlton et al., Phys. Rev. Lett. 29 (1972) 1759. [8] A. Krzywicki, comments on the systematics of multiplicity distributions, Cern preprint (March 1973) [9] J. Lach and E. Malamud, Phys. Lett. 44B (1973) 474. [10] A. Chodos, M.H. Rubin and R.L. Sugar, Theoretical physics with KNO scaling, University of Pennsylvania Preprint (January 1973). [ 11 ] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington (1964), p. 824. [12] H. Harari and E. Rabinovici, Phys. Lett. 43B (1973) 49.