Nuclear Physics BI40 (1978) 179-188
© North-Holland Publishing Company
HOW TO SEE INDEPENDENT EMISSION IN HADRONIC PRODUCTION
J. BENECKE and J. KI3HN
Max-Planck-Institut fiir Physik und Astrophysik, Munich, Fed. Rep. of Germany
Received 28 April 1978
Long-range correlations, as observed at the ISR, can easily be described by a picture
of overlapping independent emission (OIE). In this picture, hadronic production is
viewed as an incoherent superposition of elementary bremsstrahlung components. By
selecting events with fixed values of the multiplicity in one hemisphere, one arrives at
samples with strongly reduced long range correlations. At very high energies, these
samples represent elementary bremsstrahlung components whose factual existence corroborates the OIE model.
1. Introduction
It has been repeatedly suggested that multiproduction at high energy may be
described by a bremsstrahlung-like production of approximately independently
emitted objects ("clusters") [ 1]. Recently it has been observed that particle production in e+e - and deep inelastic scattering is in the form of jets and that these
jets appear to be similar to those observed in hadron reactions (the latter are the
fragmentation jets and those at high transverse momentum) (for a review and references see ref. [2]). This observation corroborates the bremsstrahlung picture
since bremsstrahlung naturally leads to universal jet formation (irrespective of the
underlying interaction which defines the parent and the axis of the jet). One major
difficulty, however, seems to stand in the way of this simple approach: in hadronic
reactions, the multiplicity distribution is a universal and roughly energy-independent function of the scaled multiplicity n/g (KNO scaling) [3]. Such a multiplicity
distribution, although very simple, is totally unlike what would be expected from
the elementary Poisson distribution resulting from independent emission. For
example, Wr6blewski's observation [4] (which is incorporated in KNO scaling) that
the dispersion Of the multiplicity distribution grows as g is contrary to what results
from independent emission where it must, on quite general grounds, grow as x/g.
The recent data from the ISR streamer chamber [5], with x/s ranging from 24 to
63 GeV, brilliantly confirm Wr6blewski's linear fit (d = 0.575 (g - 0.946)) to data
at lower energies. From this we infer that KNO scaling is likely to remain an essential feature of hadronic production at asymptotic energies.
179
J. Beneeke, 3. Kfihn / Independent emission
180
There is a simple way in which elementary independent emission leads to KNO
scaling: the emission process is thought as a sum of Poisson distributions [6]. The
multiplicity distribution of clusters is then
PN(~:
?
d), 5~N(X/V) ~(X),
(1)
o
where 5~2v(/~/) is the Poisson function from elementary independent emission, and
~(X) is the KNO function *. This automatically gives KNO scaling, and yet has
many of the features of independent emission. One may attempt to give a physical
interpretation to ~ as, for example, assuming that it results from different g being
produced at different impact parameters [6], but this is not essential for a formal
application of the idea.
As for the multiplicity distribution, the idea of the simple sum of Poissonian
components is nearly a triviality (since N ~ N ( . ~ ) -+ ~ ( N / N - 1) in the KNO limit **).
But we arrive at non-trivial statements when we apply the superposition rule to
inclusive spectra and to correlation functions. It has been shown [9] that the leading proton spectrum in pp collisions can be represented in this way: the proton
spectrum, as previously calculated in the elementary independent emission picture
[10], needs only be averaged with the weight function ~b in order to agree with the
peak-plateau structure of the experimental proton spectrum.
In addition, forward-backward multiplicity correlations have been studied:
whereas there are no such correlations in the elementary independent emission picture, they are present (also at asymptotic energies) in the overlapping independent
emission model. There is qualitative agreement between the prediction [ 11 ] of such
correlations and the recent data from the ISR streamer chamber experiment [ 1'2].
In the following, we shall demonstrate that the agreement is also satisfactory in a
quantitative way, with no free parameters.
The aim of our analysis is to test the overlapping independent emission (OIE)
* The KNO function ¢ is defined as l i m ~ nPn(n-) = ~ (nine, with n/n fixed. The normalization f0oo dk ~ (h) = 1 = f0~ d~. h~ (k) is a-. consequence of this definition. For our calculations,
we use M611er'sfit [7] to pp and np data,
¢(~.) = 1.38 h°'886 exp(-0.758 h 1"886) .
The capital letters N,.N stand for the multiplicity and the average multiplicity of clusters,
whereas n and K refer to charged particles (dominantly n-+).Since clusters can be assumed
to decay on the average into two charged particles, and since the cluster decay distribution
appears to be rather narrow (probably narrower than Poisson) [8], the property of KNO
scaling applies to clusters as well, and the functional form of qJclusters(N/N} is, at very high
energies, practically the same as that of ¢(n/~. Therefore, we shall not distinguish between
and ~Pclusters.The Poisson distribution in the KNO form, i.e. N~,v(kN) = N(kN)N
× exp(-k/V)/N!, tends to 6 (k - N/N) as N ~ ~, N/N fixed.
See previous footnote.
J. Benecke, J. Kfihn / Independent emission
181
model. If the picture of an incoherent sum of elementary bremsstrahlung components makes sense then we should be able to undo the sum and select single components (or a restricted sum of them). We shall show that this is indeed possible by
applying certain cuts to the data. The selection criteria turn out to sharpen as the
energy is increased, thus enabling us to pick subsamples of events which, at very
high energy, represent elementary independent emission of clusters.
2. The overlapping independent emission model
In the elementary bremsstrahlung picture, clusters are produced uniformly in
rapidity. The plateau height is proportional to the coupling strength X, and the
average multiplicity of clusters for a given value of X is
BT(X)=XAT,
(2)
with/Vthe average cluster multiplicity for all events. For timed X, the emission of
clusters is uncorrelated. In the OIE model, long-range correlations are introduced
by the incoherent superposition of various bremsstrahlung components, each characterized by its coupling strength X. The probability density for the superposition
was shown to be the KNO function tk(X) and can therefore be determined from
experiment. At this stage, the OIE model is completely fixed, except for an ansatz
for the cluster decay distribution.
3. The biased KNO function
For the complete or unbiased sample of events, X is a hidden variable. To what
extent can we select subsamples that represent elementary independent emission
of clusters? In order to answer this question, we consider forward-backward multiplicity correlations and address ourselves to the following problem. We are given an
incoherent superposition of components, characterized by the probability density
~(X). For a ffLxedcomponent (f'Lxed X), the multiplicity distribution of charged particles in the backward hemisphere (in the c.m.s.) is given by pn(X); note that n is
the number of particles and not of clusters. What is the distribution fin(X) of components in those events where we actually observed n particles in the backward
hemisphere? We find the answer from the theorem of Bayes [13]:
q/n(x) = pn(X) ~(X)
_ pn(X) O(X)
fdX @(X) Pn(X)
Pn
(3)
This is our main formula which gives the "biased KNO function" ~n(X). At that
point no assumption is made about the specific form ofpn(X ). From eq. (3) we get
18 2
J. Benecke, J. Kiihn /Independent emission
the multiplicity distribution p(n) for m particles in the forward hemisphere, depending on the number n of observed particles in the backward hemisphere:
p (m
n)
f d X ~n (X) Pm (X).
(4)
Instead of considering p(mn) itself, we shall just concentrate on its first two moments,
i.e. the average particle number mn and the dispersion d n in the forward hemisphere
as a function of n *"
mn = f d X G ( X ) m(X),
dg : f a x
(5a)
G ( X ) mZ(X) - ( f a x
~n(x) m(x))2
.
(5b)
Similarly, the leading particle spectrum in the forward hemisphere for events
with n particles in the backward hemisphere is given by
1
don
On
dx
- fdX G(X) X(1 -
x) x-I
(6)
whereas for elementary independent emission of clusters (i.e. at fixed X), the leading
particle spectrum is independent of n and is given by [ 10]
1
do(X)
o(x)
dx
- X(1
-
x)
x-1
.
For a comparison with the data of ref. [12], we want to evaluate eqs. (5a,b).
Since •(X) = Zm repro(X), and since pro(X) refers to particles (not to clusters) we ,
have to make specific assumptions about the cluster decay. We shall consider two
alternatives which allow an easy calculation o f pm (X):
(a) every cluster decays always into 2 charged particles,
(b) the decay distribution IIi of clusters is Poisson with an average value of 2
charged particles.
It will turn out that our results for r~n and d n are rather insensitive to the choice of
(a) or (b). The experimental information on short-range correlations, analyzed in
terms of the cluster model [8], determines the cluster decay distribution to be
narrower than Poisson. Hence, our choices (a) and (b) set bounds to a more realistic
calculation. We shall neglect short-range correlations for the moment. They are important if one considers small rapidity intervals. In our case of forward-backward
correlations, however, they give only a minor contribution.
From now on, n or m are understood to be the numbers o f produced charged
particles per hemisphere (produced via cluster decay). In the case of pp collisions,
* Eq. (5a) has the same structure as eq. (8) of ref. [ 11 ], but there is an important practical
difference: in ref. [ 11 ], n was the number of clusters in the backward hemisphere whereas
here it is the number of charged particles.
J. Benecke, J. Kfihn / Independen t emission
18 3
these numbers are equal to the actual number of charged particles/hemisphere
minus one (for the leading particle); ~ is their average value. Choosing alternative
(a), we trivially get
{~(XB/k)n/ke-Xfi/kforneven
pn(X) =
for n odd.
(7)
k is the number of charged particles from the decay of a single cluster which we
fixed to be k = 2. For rnn and dZn we find
d~ = ~2x~ + k ~ n - ,~2ff,n)2,
(8)
where
×~ = f d X X G ( X ) ,
X~ = f d X X2 G ( X ) ,
(9)
and ~n(X) is given by (3) and (7). We use a smooth interpolation for odd values of
Y/.
In the case of an arbitrary decay distribution, we have to evaluate the infinite
sum
k(probabilityto } /probabilityor c,ust<
=
produce N clusters
[to decay into n particles J
oo
= ~
~N(XK0
N=0
~
11,, ... II,N.
il+...+iN=n
(10)
Considering the special case (b), we can evaluate this sum explicitly, thus avoiding
Monte Carlo calculations. We find
1 e-XNkn ( d )
n
Pn(X) = n.W
- ~exp(XiV e - g )
n! e-XNkn exp(XN e - k ) ~
i=0
a(i)(xlV) i e -ik ,
(11)
o
where/V = if~k, and the coefficients a(ni) are defined recursively:
•
~(i--1)
a(ni) = ilg(~i)_
1 +Un_
1
(n > 0 ) ,
(12)
with a(0°) = 1, and a(ni) = 0 for i < 0 or i > n.
The results of our calculations for N,, and d,, in the two cases (a) and (b) are
plotted in fig. 1. As far as rHn is concerned, there is obviously not much of a difference between the two curves, which are both in reasonable agreement with the data.
For the dispersion dn, the data seem to favour a decay distribution which is narrower
184
J. Benecke, J. K£ihn /Independent emission
"~ss = 2z, G e V
=
31 G e V
r~ n
Vs=
53 G e V
Vs = 63 GeV
/
5
3
~ s = 1,5 G e V
.......................
,,r .... , .... ~,.,
5
10
15
5
10
15
S
10
15
5
~0
15
20
i .....................
0
5
I0
15
20
5
,S
dn
2
0
.... i .... i .... E"'
5
10
15
. . . . i . . . . i . . . . i,,,
5
10
15
. . . . f . .r. . l , . .,. . ,
5
10
15
..................
S
10
15
1 ....................
20
0
5
10
15
20
1
Fig. 1. Average value mn and dispersion d n of the multiplicity distribution p(mn) for charged
particles in the forward hemisphere, depending on the actual number n of charged particles in
the backward hemisphere. The data points axe from ref. [12]. The curves represent our calculations for two different assumptions about the cluster decay. Full curve: every cluster decays
always into 2 charged particles (case (a)). Dashed curve: Poisson distribution for the decay,
with an average of 2 charged particles (case (b)).
than Poisson. However, we have not taken into account that some clusters may have
decay products in both hemispheres. This fluctuation reduces the dispersion a bit (it
will also increase the slope of mn by a small amount). The relative error caused by
this effect becomes negligible as x/s increases. Since we did not correct for it, no
conclusion should be drawn about the detailed form of the cluster decay. Poisson
decay gives larger values of d n than the fixed decay since the decay width contributes according to
a2n :
D2n~2 + ~ . n ( k 2 _
~2).
(13)
This formula holds for an arbitrary decay distribution with average k and dispersion
(k 2 _ ~2)1/2 ;k~n and D n are the average and dispersion of the cluster distribution
in the forward hemisphere.
As was stated in ref. [11], the function mn in the OIE model tends to a straight
line with slope 1 for s -+ oo. From fig. 1 we see that at ISR energies the slope is
around 0.3, and it is rising with energy. The rise is solely due to the increase of
with s *.
From the energy dependence, we can easily understand why the broken lines in fig. 1 for
mn have a smaller slope than the full ones. Consider a certain large value of n. The Poisson
fluctuations will provide high decay multiplicities in the backward hemisphere such that the
associated cluster multiplicity is smaller than it is for the fixed decay. This leads to smaller
values of mn and also to a smaller slope. The latter follows from the observation that the
slope decreases as the average cluster multiplicity (or the energy) decreases [ 1 1 ].
J. Benecke, J. Kfihn / Independent emission
185
4. Reduction of long-range correlations
It is now rather obvious that long-range correlations are strongly reduced in those
subsamples of events which are selected by demanding a certain value of n (= multiplicity of charged particles in the backward hemisphere). From the agreement of
our calculations with the experimental data in fig. 1, we infer that the OIE model
and the relations (3) and (4) for the biased KNO function ~n(X) and the multiplicity
distribution p(mn) are physically meaningful. (Note that the calculations were done
without any free parameters.) We shall illustrate the reduction of long-range correlations explicitly by taking our formulae for the fixed cluster decay (case (a)) as a
representation of the data and performing the limit fi ~ oo.
First, we consider the KNO limit, i.e. fi ~ oo with n/fi fixed, where we get
~n(~) =
l~l,~n()t) _--S.----+(~(~.
-- "~),
7= n/~ ,
n--.~oo
from the property of the Poisson distribution which was mentioned in the first
1 0
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•
"..
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o.s
.
/
-
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I
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//
~'..
\'..
:""
s
\
:
\
0.3
/
/i
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/-."'"
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~
~
t
,~
X"D
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,,
I
I
I
i
I
5
10
15
20
25
30
Fig. 2. Full curve: K N O f u n c t i o n qJ(h), as given in the first f o o t n o t e in sect. l. T h e other t w o
curves s h o w the biased K N O f u n c t i o n Cn(h) w i t h n/ff = 2; dashed: n-= 5, dotted: ff = 10. N o t e
that if= 5 c o r r e s p o n d s to ~/s = 53 GeV. T h e averages are 1.5 and 1.6, the root m e a n squares
0.47 and 0.42 for n-= 5 and if= 10, respectively. T h e r.m.s, for t~(h) is 0.59.
J. Benecke, J. Kfihn /Independent emission
186
footnote in sect. 1. As a consequence,
fi,
X ~
,
(14)
and, from eqs. (8),
lim t n n = n ,
KNO
lim dn2 = k n = k m n .
KNO
(15)
The second of eqs. (15) illustrates that, in the KNO limit, the subsamples approach
elementary bremsstrahlung components. (The factor k is due to the cluster decay;
by putting k = 1, one gets the analogous relation for clusters itself.) In order to give
an idea of how the asymptotic limit is approached, we show in fig. 2 fin(X) with
n/fi = 2 for two values of h-. (if(X) is also plotted for comparison.) One observes that
the width of fin(X) is less than that of @(X) and is decreasing as n rises; the decrease
of the width means that long-range correlations become weaker. The averages Xn are
seen to rise towards the limiting value n/fi (= 2 here); this rise is related to that of
the slope of mn in fig. 1.
Next, we calculate ~.n in the limit ff+oo, with n fixed. Inserting (7) into (3) and
(9), we find
xn
-->
k r(~+2+n/k)_n
g P(~+l+n/k)
n
k(~+l)
+- fi
'
(16)
having approximated ~(X) by its linear term, t~(X) ~ Xc~ (X ~ 0), since large values
of X are suppressed by the factor exp(-Xff/k) from the Poisson distribution. Similarly,
X2 -->
2 +-k _n_(2c~+ 3) +
[(u+ 1)(c~+ 2)] .
(17)
n n
From (8), (16) and (17) we derive
n~,
lim
dn2 = 2kfftn,
n fixed
(18)
which is essentially the relation for elementary independent emission of clusters,
differing from it only by tire factor of 2.
Remember that KNO scaling, or Wr6blewski's relation, predicts that d 2 grows
quadratically with the average multiplicity. In the limit of fi-+ oo where the Poisson
distribution approaches 6(n - ~ , Wr6blewski's relation is supposed to hold also for
each hemisphere separately, i.e.
d = 0.575 ( f i - 0.473),
(19)
where d and fi refer now to a hemisphere. At ISR energies, the experimental values
o f d are even larger than those given by (19). For comparison with fig. 1, we apply
relation (19) to dn and mn at high values o f n where r~n is large. The values ofdn
J. Benecke, J. Kfihn / Independent emission
187
which follow from (19) would lie distinctly higher than the data or our calculated
curves. Thus we conclude that, at ISR energies, long-range correlations are strongly
reduced in the subsamples with a given value of n. From the discussion of the limit
fi -~ oo in eqs. (15) and (18), we expect that our selection criterion sharpens as x/s
rises, thus enabling us to see elementary bremsstrahlung in hadronic production.
5. Rapidity distributions
The analysis of multiplicity correlations can easily be carried over to rapidity
distributions. Since Xn v~ n/g for not too large values of h-, we expect a step in the
semi-inclusive rapidity distribution at y = 0 (y = c.m.s, rapidity) which is decreasing
in height as the energy rises, i.e., we would expect
d m n _ dr7 .[Xn
dy
fory >0
(20)
dy (n/fi f o r y < 0 ,
provided there were no short-range correlations. In reality, the cluster decay has the
effect that the step a t y = 0 gets rounded off. We convinced ourselves that the data
of the ISR streamer chamber experiment [14] are in reasonable agreement with this
picture. Expressing our observation in words: the semi-inclusive "plateau" height of
the ISR data in the forward hemisphere follows the variations of n/fi, and approaches
(n/fi) d~/dy with increasing energy.
6. Summary
We have demonstrated that the OIE model describes quantitatively, with no free
parameters, the long-range correlations that were observed at the ISR [12]. Strength
and energy dependence of forward-backward correlations are determined by the
following independently measured quantities: (i) the shape of the KNO function,
(ii) the energy dependence of the average multiplicity, and (iii) the average cluster
decay multiplicity. The details of the cluster decay (e.g. the width of the decay distribution) are of minor importance.
Furthermore, we have shown that we can select certain components by choosing
subsamples of events. At very high energy, these components are pure independent
emission (bremsstrahlung).
We would like to thank R. Meinke, H. Preissner and S. Uhlig for helping us with
the data of the ISR streamer chamber experiment, and K. Krause for additional
help in programming. We are indebted to S. Uhlig for catching an error in an earlier
version of the paper. It is also a pleasure to acknowledge discussions with L. Stodolsky
188
J. Benecke, J. Kfihn /Independent emission
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