Corrections to KNO scaling

Nuclear Physics B82 (1974) 221-236. North-Holland Publishing Company

CORRECTIONS

TO KNO SCALING

Steven P. A U E R B A C H *

Department of Physics, University of California, Berkeley, Ca. 94720 Received 28 September 1973 (Revised 3 July 1974)

Abstract: Recent data from NAL, together with a reanalysis of older data, indicate that there are substantial corrections to KNO Scaling. With this in mind, we use a simple argument, involving the generating function'of the power moments (nq), to deduce the form of corrections to KNO scaling at finite energies. Mueller's gas analog model with long range order is worked out explicity; the corrections are -30% for n = (n) at 400 GeV, and -20% at l 0 TeV, if experimental values of (n> and f2 are used. A test of the experimental data for these correction terms is presented. Correction terms of the order of 30% are not ruled out, although no positive evidence for their existence is found. A discussion of the area under the KNO scaling curve is given. This, and related questions, turn out to be somewhat subtle. A surprising result is that the area under the KNO scaling curve is not guaranteed to equal unity.

1. Introduction R e c e n t data f r o m N A L , together with a reanalysis o f older data [l ], indicate that there are substantial corrections to K N O scaling [2]. With this in mind, it seems w o r t h w h i l e to ask what sort o f corrections to KNO scaling might be e x p e c t e d at finite energies. In this paper we use a fairly simple a r g u m e n t to deduce corrections to K N O scaling at finite energy. Thse corrections have the interesting f o r m (n)Pn=~2

~)

+in-5 q~ ( n )

+ ....

(1)

A calculation o f the c o r r e c t i o n term is w o r k e d out in Mueller's gas m o d e l [3], where t h e y produce a fractional change ( - f l / 2 f 2 ) w h e n n = (n). Using actual data, this w o u l d be about - 3 0 % at 400 G e V [1] and a b o u t - 2 0 % at at cosmic ray energies [4] (10 TeV). In sect. 4 we present a test o f the experimental data far the existence o f the non-leading t e r m in (1). Due to the rather limited range o f values o f (n) in the present data, and also to the necessity to interpolate to values o f ( n ) for * Research supported by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Contract number F44620-70-C-0028.

S.P. Auerbach, KNO scaling

222

which measurements do not exist, the test is rather insensitive, and in fact correction terms as large as 30% are not ruled out by present data. One rather amusing point which arises in the course of this work is the question of the normalization of the KNO scaling function g2(x). It seems to be an almost trivial consequence of (1) that the area under fZ(x) should be unity. However, in reality, due to the asymptotic nature of the series in (1), this is rather subtle issue, and in fact the area under ~2(x) is not necessarily unity. This, and related questions, are discussed in sect. 5. In the course of this work we systematically use a generating function for the power moments (nq), and show the relation of this function to Muellers cluster expansion. We also give a proof of Chodos, Rubin and Sugar's result [5] that, in a certain sense, exact KNO scaling is impossible. The proof is very similar in spirit to theirs, but we proceed by deriving various relations among the power moments. We use these relations to show that exact KNO scaling is impossible, and also to give certain restrictions on the approach to KNO scaling. It should be pointed out that the form of the correction term in (1) was also derived by KNO [2]. The contribution of this paper is an explicit expression for ~2 and q5 in terms of the multiplicity moments. The formalism in this paper is similar in some respects to work by Weisberger [6], by Tomozawa [7] and by Weingarten [81.

2. Power moment

generating function

Let Pn be the probability of producing n particles. Mueller's generating function

[3]

is

53

(2)

n=O

and Mueller's cluster moments are defined by ~(z)=exp

~ (z-l) n n =1 n! fn

(3)

Let us further define the factorial moments (4)

gk = ( n ( n - - 1 ) . . . ( n - k + 1))

-

~

n=k

n(n-

1) . . .

(n-k +

1)

Pn,

S.P. Auerbach, KNO scaling

223

and the power moments (nq) = ~

(5)

nqP n.

n=O

The power moments are generated by G(x)= ~

enXp

n

n=O

x q

= q~o ~! <~q>

(6)

In analogy to (3) we introduce Xn

G(x) = exp n__~l ~ . D n

(7)

In order to deduce relations between gk, (nq), fk, and D r, we need to know some properties of Stirlin~'s numbers of the first and second kind [9]. Stirling's numbers of the first kind, S (q) may be defined by [log(l +z)] q = q!

~

s 2(n) zn ~..

(8)

rt=q

Stirling's numbers of the second kind, $n(q), are defined by

n=m

The Stifling numbers obey many identities, among which are H

X(X--I)...(X-rt+ | ) =

~ m=

x q=

q ~

S (m) x m, 0

rt

x ( x - 1 ) . . . ( x - m + l ) • • . ,~(m).

(10)

m=O

We first derive closed relations between the factorial and power moments. From (3) and (6) we have ¢,(1 +z) =

q=0

(log (1 +z)) q (n q) q!

= ~ (1 +z)nG n=O

_._~z n n-- 0 ~. gn"

(11)

S.P. Auerbach, KNO scaling

224

Using (8), and expanding in powers of z, we find m

(n(n-1)...(n-m+

1) = ~ (nq)S(qn). q=0

(12)

The inverse relation is q (nq)= ~ $(m)(n(n-1)...(n-m+l)).

(13)

"q

m =0

We next derive closed relations between the power moments,fk, and D k. In (2) put z = ex, and use (9). The result is p Dp = k~= l fk $(pk).

(14)

The inverse relation is n

Differentiating (7) q times and setting z -- 0 we have, using Faa di Bruno's formula [91, S

(n q ) -

t

(D1)

a1

a

. . . (Dq) q

a1,a2 . . . . aq

q!

,

(16)

(1 !)ala 1 ! . . . (q!)aqaq!

where N' indicates that the sum is over all integers al, a2, ... aq such that a 1 + 2a 2 + . . . + qaq = q. The inversion relation is

fn

=

~'

-

(G1;1

...

(Gn)an n!(al + ' " + a n - l ) !

an

a1

,

(17)

(1 !)alal! . . . (n!)anan !

where m q=l

3. Impossibility of Exact KNO scaling Chodos, Rubin and Sugar [5] (hereafter called CRS) have shown that, in a certain technical sense, exact KNO scaling is impossible. To be precise, they have shown that the ratios Ck, defined by

S.P. Auerbach, KNO scaling

(n~)

225

(19)

cannot all be exactly constant. Since KNO's proof of their scaling property relied on the constancy of the Ck, CRS's result makes the theoretical underpinnings of KNO scaling a bit shaky. As our first application of all this formalism we give a version of the CRS proof that exact KNO scaling is impossible. This proof is very similar in spirit to that of CRS, but in addition we derive constraints on Ck. From (2) and (6) we have

(l°gz)q(nq). q!

6(z) = ~ q=0

(20)

Now, if(z) is analytic at z = 0, and so, circling z about z = 0 exactly once, we have

(z) = ~,(e2'~iz) = ~ q=0

(logz + 2r~i)q (nq)" q!

(21)

Equating powers of log z we find that Cr = 1 ~ 0 Cr+l (n)/

(2~i) l I!

(22)

This establishes relations between the various Cr, and (n). If the Cr are exactly independent of energy, and hence of (n), there are just two independent relations, the others being derivable by differentiation. Letting 11 = 27r (n), these relations are 113 115 '7 c l - Y. c3 +7. cs " = o,

112

114

116

2! C2 - - - C4 +--6~-. ~ C6 " " = 0. 4!

(23)

The left-hand side in (23) are analytic functions of 11 which vanish identically, and thus their derivatives at 11 = 0 vanish. Thus, C r --- 0 for all r, which shows that nontrivial exact KNO scaling is impossible. There are other restrictions which follow from (23). For example, it is simple to deduce that the behaviour

Ci = ~i + (n~Y

(24)

for all i, where c~i and/3 i are exactly independent of (n), is also not compatible with (23). Further, any behaviour similar to (24), but with a finite, fixed, number of terms, is impossible.

S.P. Auerbach, KNO scaling

226

It should be remarked, however, that this argument is, to a certain extent, irrelevant. Nothing in the argument prevents the construction of a probability distribution for which KNO scaling is very close to exact, and yet whose moments exactly obey eq. (22). For example, define a probability distribution by

enwhere ~ is a smooth function which satisfies

j( 0

-

?

dx a ( x ) = 1,

d x x a ( x ) = 1,

(26)

0

N(07)) being a normalization factor

and (n~) a number which is, for the moment, arbitrary. If ~2 is sufficiently smooth, then N((~) ~ 1 and (n} ~ ( ~ , and so KNO scaling will be almost exact. However, as is easy to verify, eq. (22) is exactly satisfied for the moments defined by this probability distribution.

4. Corrections to KNO scaling Using the formalism developed in sect. 2 we can derive, in a simple manner, corrections to KNO scaling. Our technique involves a systematic use of the generating function exp D(x) . In eq. (8) let us replace x by ( - x ) , to obtain e~nXpn = exp

D(-x)

(28)

n=O

If the Pn are a smooth function of n, we can replace the sum by an integral. Then (28) looks like a Laplace transform. Inverting, we find that c+i~o

e

1

- 2~ri

f

dx e nx exp

D(-x)

,

(29)

c-ioo

Strickly speaking, approximating the sum in (28) by an integral is a somewhat suspect procedure. Note, in fact, that if we carry out the inverse Laplace transform indicated in (29) we get

S.P. Auerbach, KNO scaling

227

c+i~

1 f 27ri

e nx exp O ( - x )

=

c-i~

m : 0

Pm 6 ( n - m ) ,

(30)

rather than getting Pn, as claimed in (29). We could base our subsequent work on the exact result (30). However, it is more convenient to adopt the following method. Let P(n) be a smooth interpolating function for Pn, with the properties

? p(n) dn = 1.

P ( n ) = Pn; n--O, 1,2 . . . . .

(31)

0 We then redefine the various moments in terms o f P (n)' e.g.

(n) -+ (~) = ~-f n p(n) dn, 0

f 0

dn e -nx F ( n ) = exp D ( - x ) .

(32)

Inverting, we now have an exact result c+i,,~

?(n):

~ {1- f

dx nx exp D ( - x )

.

(33)

c--i~

If the Pn are a smooth function of n, then )7k will be very~close to fk. In sect. 5 a more quantitative discussion of the connection between fk and fk will be given, but it seems reasonably clear that, i f f k has theenergy dependence fk = a~ (n)k + /3k (n) k-1 + . . . , then it will also be true that fk = ffk (n)k + ~k (n)k-1 + ".'- - • In order to derive KNO scaling we shall suppose that

fk = t~k (n)k +~k (n)k-1 + . . . .

(34)

As pointed out by KNO, this is the sort of energy dependence which arises when the inclusive distributions obey Feynman scaling. As mentioned above, the same energy dependence will hold for the smoothed quantities jTk. From now on we drop the " ~ " for notational convenience. To leading order in (n)

D p = C~p (n)P + (n)P- l (ap_ l + ½p ( p - 1 ) ~p) =- ~p (n) p + "gp (n) p-1 ,

(35)

since $ ; p - 1 ) = ½p(p _ 1).

(36)

228

S.P. Auerbach, K N O scaling

Therefore,

_ 1 Pn 2zri f

c+i~

c-ioo

dxe nx exp

+1 ~ (-1) p (-1)Pe~p(x(n))P p! P = 1 -~" (n) p = 1

( ~

7p (x (n)) p }.

(37)

/

For large values of (n), we then expand the exponential in (39) in powers of 1/(n), change variables to ~"= (n)x, and, distorting the contour appropriately, Pn

(n)

~

+-(n) 2

~

(38)

,

where

/

c+io~

a(x)=

1 c-ioo c+io~

q~(x) = 1 c-i~

deX'expI

} (39)

The first term in (38) gives KNO scaling, while the second term gives corrections to scaling. The first term precisely reproduces Weisberger's result [6], of course. The second term shows that corrections to KNO scaling are of the form ((n))- 1 times a function of the scaled variable n/(n). As may be seen from (35), corrections to KNO scaling arise from two distinct sources: (i) non-leading terms in fk, and (ii) terms in Dp proportional to fp_ 1. It should be pointed out again that KNO also obtained a formula like (38). However, they did not obtain a result for the correction term in terms of t h e f k. In fig. 1 we present a test for the non-leading term in the formula = l__a

Pn(n)

n__'~+ 1 ~ n ( (n) ] ((n))2 ( ~ n ) )

(40)

Defining c~ = n/(n),

n2Pn = na~2(¢~) + a2q~(c0 ,

(41)

and so, plotting n2Pn versus n for fixed ~, we should get a straight line with slope c~gZ(c0 and intercept c~2q5(c0. As a practical matter, this test has the disadvantage that, fixing c~ and varying n, (n) takes on values for which measurmentS do not exist. In fig. 1 we present a plot ofn2Pn versus n for c~= 1.5, 1.88 and 2.25, using linear interpolation (i.e. for fixed n, we interpolated linearly in (n)) to guess values ofP n where measurements do not exist. The stridght lines in fig. 1 are drawn solely to guide the eye. There is no clear evidence for a non-vanishing q~. However, this is

S.P. Auerbach, KNO scaling

251

' I l l '

I

'

I'

I'

229

,T,I

I

I

I

I

I

2O

15

aY ed

C~ = 1 . 8 8

I0

1

¢--

c~ :

2.25--

5

0

J

-5

~ 0

I 2

J

I,

I,

1,

1,

I,

I,

I

4

6

8

I0

12

14

16

I

I 18

I

1 20

n Fig. l. A test for the existence of the non-leading term.

not a terribly sensitive test. Consider, for example, a = 1.88, for which one might estimate ~2 = 0.3 -+ 0.06 and q5 = 0 -+ 0.6. Then, if we allow q5 = 0.6 as being compatible with the data, the ratio of non-leading to leading terms in Pn is aq~(a)/(nS2(cO) 3.8/n, which, for 8 ~ n ~< 16, is a sizeable percentage. The reader might well wonder about the correctness of the conclusion that present data do not rule out the existence of substantial correction terms (say, 30%), when the "traditional" KNO plot ((n) Pn versus n/(n)) shows points from different energies clustering on the same curve to within 5-10%. However, over the energy range 50 to 400 GeV, where KNO scaling has been tested by Slattery [10], (n) varies only by a factor of about 2 (from 4 to 9). Consider what happens to (n) Pn as we vary (n), with n/(n) fixed. Say that the correction term ((n))-1 ~(n/(n))is 30% of the leading term ~2(n/(n)) for (n) = 4. Then for (n) = 8, the correction term will be 15% of the leading term, and so (n)P n will vary by a factor of(1 .+ 0.3)/(1 + 0.15) = 1.13, i.e., (n)P n will vary only by 13%. Moreover, if the scaling curve is drawn half way between the values of (n) Pn for
S.P. Auerbach, KNO scaling

230 $(1)= 1,

$p(2)= 2P-1 _ 1,

(42)

we find (43)

Dp = ( f l - f 2 ) + 2p-1 f2'

If we wished to exactly evaluate the Pn, we could use the exact result (30), together with

D(-x) = (~f2 - f l ) + (fl - f 2 ) e-x + if2 e-2X,

(44)

However, to evaluate corrections to KNO scaling, we shall instead use the scaling argument. We assume t h a t f 2 ~ f l 2 - ( n ) 2 and scale x by x - ~"

(45)

I1'

then f2

2,,

D3

3

+],

where we have collected inside the square bracket terms which go like ((n}) -1 . Then,

Pn-2rrl/fl

c-ioo f

dfen~/flexp-~+

f2

1+~'2 _6f-~2fl

'"]i47)

Defining

n-f 1 r?- 2 x ~ 2 ,

(48)

the final result is

-g-2

+2 3

(49)

This form asymptotically exhibits KNO scaling, since (f2)~ ~ (n). At finite energies, scaling in terms of (f2)~ rather than (n} is presumably more accurate, at least for the gas model. We could, of course, write f 2 ~ a(n) 2 + b (n} + c, and explicitly derive scaling in terms of (n). The result of such a calculation would be numerically identical to (49), but the correction terms would be larger and more complicated. The alert reader will notice that the normalization of the KNO scaling piece is not quite correct, i.e., its integral is not unity. This is discussed further in sect. 5.

S.P. Auerbach, KNO scaling

231

The correction at n = (n) -~fl is

-

(5o)

26'

If we just take the experimental values for the charged moments [4, 10], we would find a correction of about - 3 0 % at 400 GeV and about - 2 0 % at cosmic ray energies (10 TeV). Of course, the higher moments f 3 . . . . do not vanish, so this estimate is not terribly reliable. As mentioned before, however, correction terms of this size are not ruled out by present data. At first sight it might seem that the Poisson distribution model, and some simple two component models, might be thought of as counterexamples which show that (38) is incorrect. The reason for this is that, for the Poisson distribution,
(51)

as (n) -~ o~. Thus, if (38) applied to the Poisson distribution, one would apparently see the delta function at all values of (n), plus a term which decreases as (n) ~ ~. It seems to us to be more reasonable to think of these as cases which do not exhibit KNO scaling at all. To understand this point, let us s e t f 2 = 0 in (46). We find, again scaling as in (43), that ~2 D(-g') : --~" +

2f 1

¢3 +... 3!f~

(52)

and so

c+i~ _

Pn

1

27rifl

d~ e n~/I'

ioo

-¢

exp

+

+ ....

(53)

In this.case, it is not legitimate to expand the exponential as a power series in 1/f 1 , since the resulting integrals do not converge. Because of this Pn does not depend on (nffl) alone, to leading order, but also o n f 1 . Thus, this case should not be said to exhibit KNO scaling. If one insisted on expanding the exponent, one would obtain a power series in (1/fl) times delta functions and derivatives of delta functions, whose first term is just (5 I). However, since derivatives of delta functions are more singular than delta functions, the series does not make much sense, and it seems preferable to agree that the Poisson case does not exhibit KNO scaling. Similar remarks apply to those simple two component models which are just a sum of a Poisson distribution plus a finite number of constant terms.

5. Normalization of the KNO scaling function In this section we discuss the area under the KNO scaling curve. At first sight this appears to be an utterly trivial question. After all, one might argue, we know that

S.P. Auerbach, KNO sealing

232

fi(n) dn= 1 ,

(54)

0 and we have shown, given (34), that (n)

~

+ -~ (n) 2

an

NY

= d,

~

+

....

(55)

Now,


(56)

(n) o where c and d are constants independent of n. Thus,

1 = c + (-~ + . . . .

(57)

and it seems perfectly obvious that c = 1 and d = 0. This line of argument is in fact incorrect, for rather subtle reasons. As evidence for this, note that, in our calculation with Mueller's gas model with long range correlations, the integral of the KNO function is =

1 + erf

- -

(2%)' 0

Since, by assumption, f2 ~ f 2 , this is indeed a constant, but it is not equal to unity For charged particles at 400 GeV in proton-proton collisions, f l ~ 9,f2 ~ 13.5, and the integral is about 0.992. This is a tiny effect, but as a matter of principle it is amusing. How could it be that the simple argument leading to c = 1, d = 0 is incorrect? The series in (38) is asymptotic; to be precise, for a given value of nfln), it is an asymptotic series in powers of 1/
S.P. Auerbach, KNO scaling

233

only to the non-leading terms in (38) and do not affect ~ at all. In order to do so, it is necessary to introduce a precise interpolation method. For our present purposes it is sufficient to use linear interpolation. To be specific, we let fi(n) = (Pm+t -

Pro) (n-m) +P,n'

m ~
(59)

We then have +1 e nx P(n) dn

eD(X) =

m=O m = eD(X)

ex + e - x - 2

x2

PO

x - 1 + e-x

(60)

x2

The discussion at this point is most simple if we assume P0 = 0. This will be the case, for example, if we are dealing with proton-proton collisions, and n refers to the number of charged particles in the final state. By charge conservation, P0 = 0, since at least two charged particles must occur in the final state. I f P 0 4= 0, the simplest procedure would be to use the mathematical trick of allowing n = - 1 to occur, and s e t t i n g P 1 = 0. The sum in (60) would then run from m = - 1 to % and the P0 term in (60) would be replaced by the analogous term with P-1 instead o f P 0. This new term would of course vanish, since P-1 = 0. The net result of these machinations is (whether or not P0 = 0) D(x) = D ( x ) + log ex + e - x - 2 x2

,

(61)

from which we deduce that the first few moments are changed to D1 = D I '

L)2 =D2

+ gl,

/~ 3 = D 3 ,

1~4 = D 4

g~. 1

(62)

The change in these moments depends only on the interpolation method, and not on the probability distribution itself. Using (15) we find

71 =fl'

d~=f2 + -~'

f3 =f3 -7,'

~=f4+g-O-.'°9

(63)

It is clear that these changes due to interpolation do not affect the KNO scaling function, since the changes are independent of (n). Another legitimate question the reader might have is whether, ifP(n) is computed exactly, using (33), the resulting function P(n) would necessarily have the correct moments. Although the reader would probably be willing to accept this, we shall present a brief proof of this fact, which has the virtue that it also illustrates how the area (and other moment integrals) of the KNO function could be "incorrect".

234

S.P. Auerbach, KNO scaling

We wish to show that, iffi(n) is computed using (33), the resulting function has the correct moments. This is equivalent to proving that c+i~

e~(-y) = f e -ny 0

1 f 27ri c-i~

dxenXeTJ(_x)

(64)

The first step is to show that interchanging the order of integration is allowed. By Fubini's theorem, this is legitimate if the integrand is absolutely integrable in one order of integration. Assuming P0 = 0, and using (6), (7) and (60), as Re x ~ % e/~(-x)

PI -- , x2

(65)

and so the integrand is absolutely integrable if we do the n integral first. Thus, interchange of orders of integration is allowed, and we next must show that c+i~

1

eJ~(-Y) = 27ri f

dx

e~ ( - x )

1

-y--.x

(66)

It is simple to show, using the definition of exp D(x), that exp D ( - x ) is analytic for x in the right half plane. Using again the asymptotic behaviour of exp D ( - x ) , we can close the contour of integration in (66) to the right, picking up the pole a t y = x, and we get the desired answer. The asymptotic behaviour of exp D ( - x ) was crucial for this argument. It is therefore true that, ifP(n) is computed exactly, using (33), the resulting function has exactly the correct moments. In particular, settingy = 0 we find dnfi(n) = e~(0) = 1.

(67)

0 A crucial ingredient here is the good asymptotic behaviour of exp D ( - x ) , which is guaranted by our previous arguments. Note, however, that when we derived KNO scaling, we split up exp D ( - x ) into a sum of terms :

1 [exp (c~((n) x))] 7 ((n) x) + . . . . : exp (a((n) x)) + (~-

(68)

Although exp n ( - - x ) has good asymptotic behaviour, there is no guarantee that, when it is split up into pieces as in (68), each piece separately will have good asymptotic behaviour. Hence, even though exp (ct((n) x)) = 1 when x = 0, the KNO scaling

S.P. Auerbach, KNO scaling

235

function is not guaranted to have unit area. In the specific case of Mueller's gas model, the piece of exp ( D ( - x ) ) which gives the KNO function is (see (46)) exp (tel X + ~ f2 X2) whose asymptotic behaviour is sufficiently bad that we cannot close the contour to the right, as was needed to prove (66). We are thus not guaranteed that the area (or any other moment integral) of the KNO function is what we would naively expect. Another illustration of this phenomenon is connected with constraints placed on the fk by positivity. We want to require that all the Pn are positive and so if Iz[ < 1, we must have co

oo

By (3) we therefore have that, for [z[ ~< 1, Re k __~1~-. ( z - l ) k < 0.

(70)

If all moments but f! and ]'2 vanish, then, succesively setting z equal to 1 - e , exp (/e), and - 1 , where e is a small positive number, we find that positivity requires f l ~> 0, f l + f2 ~> 0, and fl - f2 ~> 0. Hence bc2/fl I < l, which is clearly inconsistent with the assumptions which led to KNO scaling (fl -* 0% f2 ~f12) • W,e are thus faced with a somewhat paradoxical situation. Assumingf 2 ~ f 2 and that all other f k vanish, positivity must be violated for some Pn" The alternative, that 1 2 ZPn > 1, is ruled out by setting z = 1 in "2znPn = exp (fl(z - 1) + 5-f2(z-1) )). However, the KNO scaling function ~ computed for this case is a gaussian, which is positive everywhere. The apparent paradox is that, although the moments of g2 surely o b e y f 2 ~ f~, ~2 is positive everywhere. The resolution of this conflict is that the higher moments fk, k > 3, of f2 do not vanish even though the higher moments of the exact probability distribution do vanish. In fact, all of the moments of ~ are different from what one would naively expect, although the difference is just a few percent f o r f l and f2, if the values quoted in section IV are used. All of this'is an illustration of the fact that the moments of the scaling curve are not constrained to be equal to the moments of the probability distribution at finite (n). The above discussion of positivity might make the reader a bit uneasy about our illustrative example of Mueller's gas model with f2 ~ f l2, since if the Pn were calculated exactly in the model, some would necessarily be negative. However, it can be shown that the sum of the leading and first correction term in (49) is positive, at least for the observed values at 400 G e V f l = 9,f2 = 13.5. So, for the sum of these terms, there is nothing wrong with the model. The model could therefore be modified by adding higher moments f3, f4, • • which, for example, are all proportional to (n), which would make the exact Pn positive, and yet not alter the scaling and first correction term in (49).

236

SoP. Auerbach, KNO sealing

I would like to thank my colleagues, Drs. Carl Rosenzweig and M.R. Pennington for numerous stimulating and enjoyable discussions and Dr. R. Sugar for several interesting conversations.

References [ 1 ] T. Ferbel, Particle multiplicities and correlations at NAL energies, Rochester preprint COO-3065-61. [2] Z. Koba, H.B. Nielsen and P. Olesen, Nucl. Phys. B40 (1972) 317. [3] A.H. Mueller, Phys. Rev. D4 (1971) 150. [4] T. Ferbel, Recent results from NAL Rochester preprint COO-3065-41. [5] A. Chodos, M. Rubin and R.L. Sugar, Phys. Rev. D8 (1973) 1620. [6] W.I. Weisberger, Phys. Rev. D8 (1973) 138. [7] Y. Tomozawa, Nucl. Phys. B62 (1973) 539. [8] D. Weingarten, Nucl. Phys. B70 (1974) 501. [9] Handbook of mathematical functions, ed. M. Abramowitz and I. Stegun, (U.S. Government Printing Office, Washington D.C., 1968). [10] P. Slattery, Phys. Rev. D7 (1973) 2073.