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Forward-backward correlations and the multiplicity distribution in high-energy collisions
Nuclear Physics B110 (1976) 488-492 © North-Holland Publishing Company
FORWARD-BACKWARD CORRELATIONS AND THE MULTIPLICITY DISTRIBUTION IN HIGH-ENERGY COLLISIONS J. BENECKE Max-Planck-Institut fiir Physik und A strophysik, Munich A. BIALAS Institute of Physics, Jagellonian University and Institute of Nuclear Physics, Cracow S. POKORSKI * Institute o f Physics, Warsaw University
Received 24 May 1976
Simple expressions are derived which relate correlations between (a) the leading particles and (b) the forward and backward multiplicities (in the c.m.s.) to the overall multiplicity distribution. These correlations are positive and should be present at asymptotic energies. Their magnitude is estimated.
In a recent paper [ 1] a multicomponent model of particle production was developed, as a generalization o f the bremsstrahlung mechanism [2]. It was shown that this model, which by construction has Koba-Nielsen-Olesen (KNO) scaling [3], describes reasonably well the leading particle spectrum d a / d x and the associated multiplicity ~(x) of produced mesons (x is the scaled longitudinal c.m.s, m o m e n t u m of the observed leading particle). In the present paper we continue the study of that model. We discuss in particular correlations between the leading particles and between the multiplicities in the forward and backward c.m.s, hemispheres. It is shown that the model implies fairly strong correlations, at least at asymptotic energies. Our result on the two-particle spectrum should be contrasted to the one obtained from the triple-Regge ansatz [4], where long-range correlations between the leading particles are a consequence of the superposition o f PPP and PPM terms and, therefore, vanish asymptotically as s-1/2. The data on the inclusive correlation function between two leading protons, which are to be expected from the CERN-ISR within the next months, may thus serve as a
* Supported in part by the US National Science Foundation under the Grant Nr. 42060.
J. Benecke et al. / Forward-backward correlations
489
crucial test for the various ideas on the origin and the nature of the long-range correlations in the presently known data. Let us start by recalling briefly the main assumptions of our model. It is proposed that the observed particle spectra are superpositions of spectra from different components, each component being characterized by (i) the absence of correlations between clusters produced in the collision and (ii) a rapidity plateau for the inclusive one-cluster spectrum. The absence of correlations implies that in each component the clusters are produced with a Poisson distribution, the average multiplicity being ~(X) = X~,
(1)
where X is a parameter characterizing a given component and ~ is the average cluster multiplicity in all events. The parameter X may be thought of as characterizing the "radiation strength" associated with a given collision. Summing all possible energy fluctuations of produced clusters yields the leading particle spectrum in a given component, do0t)/dx = o(X)X(1 - Ix [)x- 1 ,
(2)
where o(X) is the cross section for producing the component X, i.e. 1
o(x) = f dxdo(X)/dx. 0 The overall leading particle spectrum do/dx is obtained by integrating eq. (2) over X, with
o(x)/o = ~(x),
(3)
where ~(X) is the KNO scaling function *, and o = f~dXa(X) is the inelastic cross section. Eq. (3) was derived in ref. [1 ] from the requirement of KNO scaling for the overall multiplicity distribution. In dealing with the inclusive two-proton spectrum, we now formulate our basic assumption which, from the bremsstrahlung point of view, is a very natural one. Namely, within each component we assume the two leading particles in the two opposite hemispheres to be uncorrelated, i.e. do0t)/dXBdXF = o(X)X2(1 _ ix B i)x- 1(1 _ XF)h- 1 ,
(4)
(x B ( 0, XF > 0). The overall inclusive spectrum for the two leading particles is obtained by integrating over 3., again using (3). It will be demonstrated in a subsequent paper that our generalized bremsstrahlurlg calculations for inclusive spectra (2) and (4), both integrated over X, follow also from the ansatz of an independent jet model (with factorized matrix elements for n-cluster production and appropriate weights a n (s) for the various multiplicities). * For all numerical calculations we use M~ller's parametrization of the KNO function [5,1].
490
J. Benecke et a L / Forward-backward correlations
0.5
0.1 0
I
I
--+
I
0.1
I
I
I
I
I
0.5
1
~ x
Fig. 1. Normalized correlation R ( x B, xf) beiween symmetrical leading particles, - x B = x F -- x. For explanation see the text.
Using the latter ansatz one convinces oneself that the situation of both leading particles moving in the same hemisphere is suppressed by a factor s - 1 as compared to (4). In order to exhibit long-range correlations, we consider the normalized correlation function do R(XB,XF)
= 0 dX~F
d/q
do
1,
(5)
l aXB d x v
which in our model is given by
f
dN~j(X)X2(1 - IX[B)X(1 - XF) x
0 R(XB,XF)
-
= __
f 0
dX~tX)(1 - I x B I)x f 0
dX'~(X')(1
1.
(6)
xF) x'
The plot o f R ( x B, XF) in fig. 1 for the case of symmetrical leading particles, - x B = x F = x, leads us to expect fairly strong positive correlations at ISR energies: the observation of a proton at a given Ixl in one hemisphere increases the probability to find another proton with the same Ix l in the other hemisphere. It should also be stressed that this prediction is asymptotic, i.e. R does not vanish in the high-energy limit. This is in contrast to the triple-Regge model [4]. Let us now discuss correlations between the clusters emitted in opposite hemispheres. Consider first one component with average multiplicity fi(X). The numbers of clusters in the forward and the backward hemisphere are denoted by n F and n B.
J. Benecke et al. / Forward-backward correlations
491
Remembering that the multiplicity is distributed according to Poisson's law, we get the probability PX(nB, nF) to find n B and n F = n - n B clusters in a given component, Px(nB' nF) = e - n(x)
n!
.
(~) B (~)
nB
F
= e-h(X)/2 (n(X)/2)nB e-~(x)/2 (-n(X)/2)nF . nB] nF!
(7)
The overall probability P(n B , nF) is obtained by integrating Ph(nB, nF) over X with the weight function if(X). This integration introduces positive correlations which increase with rising energy until an asymptotic limit is reached. Indeed, the Poisson distributions in eq. (7) become more and more narrow, reaching finally the 6 function limit. It is customary to express the correlations between clusters in opposite hemispheres by the quantity gF(nB), i.e. the average number of clusters in the forward hemisphere observed when the number of clusters in the backward hemisphere is fixed to be n B. In our model ~F(nB) is given by
BF(nB) .
j
dX~(X)e_h(x)/2(_~(X)/2)nt~+ 1
0 .
.
.
.
.
.
.
.
(8)
f dXq,(X)e-h(x)/2(-n(X)/2)na o (remember ~(X) -- X~). In the asymptotic limit ~ ~ 0% nB fixed, we thus obtain the simple relation ~F(nB)_ =
nB + 1 + a,
(9)
if ~(X) ~x Xc~ near X ; 0. In order to give an idea of how the asymptotic limit is approached, we evaluated formula (8) for different values of the average cluster multiplicity B. The results are shown in fig. 2. It is seen that the correlations are quite appreciable already at low average multiplicities. In the ISR energy region (~ ~ 5 clusters) we expect sizeable effects. Preliminary data at x/s = 22 and 53 GeV [6] give values for the slope of ~F(nB) which agree very well with our expectation; in particular, it should be noted that the experimental slope increases with rising x/s. We end with the following comments: (a) Near x = 1, R ( - x , x) rises as II
1 1 l+a
R(-x, x) x~l~X n i - ~ 3
,
492
J. Benecke et aL / Forward-backward correlations
fiF { I ~ B }
~
fi = 10
10
5n=2 0 0
I
I
I
I
5
I0
15
20
~ nB
Fig. 2. Average multiplicity of clusters in the forward c.m.s. ~aemisphereplotted versus the number of clusters in the backward hemisphere. Different curves correspond to different values for the average total multiplicity of clusters.
if ~b(X) ~x ha near X = 0 (c~ > - 1 ) . M~iler's fit [5] has a = 0.886. (b) The predicted forward-backward multiplicity correlations may be affected by fluctuations induced by the cluster decay. This effect should not be dramatic, however. Indeed, it can provide only short-range correlations and thus can be neglected in the asymptotic limit. (c) The forward-backward multiplicity correlations described in this paper were derived from the (generalized) bremsstrahlung model. But they exist under more general conditions. Such correlations should appear in any multicomponent model consistent with KNO scaling where each component has a flat rapidity distribution for produced particles and has short-range order. In conclusion, we have shown that our simple multicomponent model implies strong forward-backward correlations, although the basic interaction at high energy may be assumed to be a bremsstrahlung-like production of approximately independently emitted clusters. The link for combining both features is KNO scaling. The measurement of correlations between the two leading protons at the ISR will be crucial for testing our picture against the triple-Regge idea. It will also test the conjecture that the KNO function ff(?~) retains an essentially energy-independent width at very high energies. References [1 ] J. Benecke, A. Biat'as and E.H. de Groot, Phys. Letters 57B (1975) 447. [2] L. Stodolsky, Phys. Rev. Letters 28 (1972) 60. [3] Z. Koba, H.B. Nielsen and P. Olesen, Nucl. Phys. B40 (1972) 317. [4] J. Kwiecifiski and R.G. Roberts, Phys. Letters 57B (1975) 349. [5] R. M¢ller, Nucl. Phys. B74 (1974) 145. [6] S. Uhlig, private communication from the Aachen-CERN-Heidelberg-Miinchen Streamer Chamber Collaboration.
FORWARD-BACKWARD CORRELATIONS AND THE MULTIPLICITY DISTRIBUTION IN HIGH-ENERGY COLLISIONS J. BENECKE Max-Planck-Institut fiir Physik und A strophysik, Munich A. BIALAS Institute of Physics, Jagellonian University and Institute of Nuclear Physics, Cracow S. POKORSKI * Institute o f Physics, Warsaw University
Received 24 May 1976
Simple expressions are derived which relate correlations between (a) the leading particles and (b) the forward and backward multiplicities (in the c.m.s.) to the overall multiplicity distribution. These correlations are positive and should be present at asymptotic energies. Their magnitude is estimated.
In a recent paper [ 1] a multicomponent model of particle production was developed, as a generalization o f the bremsstrahlung mechanism [2]. It was shown that this model, which by construction has Koba-Nielsen-Olesen (KNO) scaling [3], describes reasonably well the leading particle spectrum d a / d x and the associated multiplicity ~(x) of produced mesons (x is the scaled longitudinal c.m.s, m o m e n t u m of the observed leading particle). In the present paper we continue the study of that model. We discuss in particular correlations between the leading particles and between the multiplicities in the forward and backward c.m.s, hemispheres. It is shown that the model implies fairly strong correlations, at least at asymptotic energies. Our result on the two-particle spectrum should be contrasted to the one obtained from the triple-Regge ansatz [4], where long-range correlations between the leading particles are a consequence of the superposition o f PPP and PPM terms and, therefore, vanish asymptotically as s-1/2. The data on the inclusive correlation function between two leading protons, which are to be expected from the CERN-ISR within the next months, may thus serve as a
* Supported in part by the US National Science Foundation under the Grant Nr. 42060.
J. Benecke et al. / Forward-backward correlations
489
crucial test for the various ideas on the origin and the nature of the long-range correlations in the presently known data. Let us start by recalling briefly the main assumptions of our model. It is proposed that the observed particle spectra are superpositions of spectra from different components, each component being characterized by (i) the absence of correlations between clusters produced in the collision and (ii) a rapidity plateau for the inclusive one-cluster spectrum. The absence of correlations implies that in each component the clusters are produced with a Poisson distribution, the average multiplicity being ~(X) = X~,
(1)
where X is a parameter characterizing a given component and ~ is the average cluster multiplicity in all events. The parameter X may be thought of as characterizing the "radiation strength" associated with a given collision. Summing all possible energy fluctuations of produced clusters yields the leading particle spectrum in a given component, do0t)/dx = o(X)X(1 - Ix [)x- 1 ,
(2)
where o(X) is the cross section for producing the component X, i.e. 1
o(x) = f dxdo(X)/dx. 0 The overall leading particle spectrum do/dx is obtained by integrating eq. (2) over X, with
o(x)/o = ~(x),
(3)
where ~(X) is the KNO scaling function *, and o = f~dXa(X) is the inelastic cross section. Eq. (3) was derived in ref. [1 ] from the requirement of KNO scaling for the overall multiplicity distribution. In dealing with the inclusive two-proton spectrum, we now formulate our basic assumption which, from the bremsstrahlung point of view, is a very natural one. Namely, within each component we assume the two leading particles in the two opposite hemispheres to be uncorrelated, i.e. do0t)/dXBdXF = o(X)X2(1 _ ix B i)x- 1(1 _ XF)h- 1 ,
(4)
(x B ( 0, XF > 0). The overall inclusive spectrum for the two leading particles is obtained by integrating over 3., again using (3). It will be demonstrated in a subsequent paper that our generalized bremsstrahlurlg calculations for inclusive spectra (2) and (4), both integrated over X, follow also from the ansatz of an independent jet model (with factorized matrix elements for n-cluster production and appropriate weights a n (s) for the various multiplicities). * For all numerical calculations we use M~ller's parametrization of the KNO function [5,1].
490
J. Benecke et a L / Forward-backward correlations
0.5
0.1 0
I
I
--+
I
0.1
I
I
I
I
I
0.5
1
~ x
Fig. 1. Normalized correlation R ( x B, xf) beiween symmetrical leading particles, - x B = x F -- x. For explanation see the text.
Using the latter ansatz one convinces oneself that the situation of both leading particles moving in the same hemisphere is suppressed by a factor s - 1 as compared to (4). In order to exhibit long-range correlations, we consider the normalized correlation function do R(XB,XF)
= 0 dX~F
d/q
do
1,
(5)
l aXB d x v
which in our model is given by
f
dN~j(X)X2(1 - IX[B)X(1 - XF) x
0 R(XB,XF)
-
= __
f 0
dX~tX)(1 - I x B I)x f 0
dX'~(X')(1
1.
(6)
xF) x'
The plot o f R ( x B, XF) in fig. 1 for the case of symmetrical leading particles, - x B = x F = x, leads us to expect fairly strong positive correlations at ISR energies: the observation of a proton at a given Ixl in one hemisphere increases the probability to find another proton with the same Ix l in the other hemisphere. It should also be stressed that this prediction is asymptotic, i.e. R does not vanish in the high-energy limit. This is in contrast to the triple-Regge model [4]. Let us now discuss correlations between the clusters emitted in opposite hemispheres. Consider first one component with average multiplicity fi(X). The numbers of clusters in the forward and the backward hemisphere are denoted by n F and n B.
J. Benecke et al. / Forward-backward correlations
491
Remembering that the multiplicity is distributed according to Poisson's law, we get the probability PX(nB, nF) to find n B and n F = n - n B clusters in a given component, Px(nB' nF) = e - n(x)
n!
.
(~) B (~)
nB
F
= e-h(X)/2 (n(X)/2)nB e-~(x)/2 (-n(X)/2)nF . nB] nF!
(7)
The overall probability P(n B , nF) is obtained by integrating Ph(nB, nF) over X with the weight function if(X). This integration introduces positive correlations which increase with rising energy until an asymptotic limit is reached. Indeed, the Poisson distributions in eq. (7) become more and more narrow, reaching finally the 6 function limit. It is customary to express the correlations between clusters in opposite hemispheres by the quantity gF(nB), i.e. the average number of clusters in the forward hemisphere observed when the number of clusters in the backward hemisphere is fixed to be n B. In our model ~F(nB) is given by
BF(nB) .
j
dX~(X)e_h(x)/2(_~(X)/2)nt~+ 1
0 .
.
.
.
.
.
.
.
(8)
f dXq,(X)e-h(x)/2(-n(X)/2)na o (remember ~(X) -- X~). In the asymptotic limit ~ ~ 0% nB fixed, we thus obtain the simple relation ~F(nB)_ =
nB + 1 + a,
(9)
if ~(X) ~x Xc~ near X ; 0. In order to give an idea of how the asymptotic limit is approached, we evaluated formula (8) for different values of the average cluster multiplicity B. The results are shown in fig. 2. It is seen that the correlations are quite appreciable already at low average multiplicities. In the ISR energy region (~ ~ 5 clusters) we expect sizeable effects. Preliminary data at x/s = 22 and 53 GeV [6] give values for the slope of ~F(nB) which agree very well with our expectation; in particular, it should be noted that the experimental slope increases with rising x/s. We end with the following comments: (a) Near x = 1, R ( - x , x) rises as II
1 1 l+a
R(-x, x) x~l~X n i - ~ 3
,
492
J. Benecke et aL / Forward-backward correlations
fiF { I ~ B }
~
fi = 10
10
5n=2 0 0
I
I
I
I
5
I0
15
20
~ nB
Fig. 2. Average multiplicity of clusters in the forward c.m.s. ~aemisphereplotted versus the number of clusters in the backward hemisphere. Different curves correspond to different values for the average total multiplicity of clusters.
if ~b(X) ~x ha near X = 0 (c~ > - 1 ) . M~iler's fit [5] has a = 0.886. (b) The predicted forward-backward multiplicity correlations may be affected by fluctuations induced by the cluster decay. This effect should not be dramatic, however. Indeed, it can provide only short-range correlations and thus can be neglected in the asymptotic limit. (c) The forward-backward multiplicity correlations described in this paper were derived from the (generalized) bremsstrahlung model. But they exist under more general conditions. Such correlations should appear in any multicomponent model consistent with KNO scaling where each component has a flat rapidity distribution for produced particles and has short-range order. In conclusion, we have shown that our simple multicomponent model implies strong forward-backward correlations, although the basic interaction at high energy may be assumed to be a bremsstrahlung-like production of approximately independently emitted clusters. The link for combining both features is KNO scaling. The measurement of correlations between the two leading protons at the ISR will be crucial for testing our picture against the triple-Regge idea. It will also test the conjecture that the KNO function ff(?~) retains an essentially energy-independent width at very high energies. References [1 ] J. Benecke, A. Biat'as and E.H. de Groot, Phys. Letters 57B (1975) 447. [2] L. Stodolsky, Phys. Rev. Letters 28 (1972) 60. [3] Z. Koba, H.B. Nielsen and P. Olesen, Nucl. Phys. B40 (1972) 317. [4] J. Kwiecifiski and R.G. Roberts, Phys. Letters 57B (1975) 349. [5] R. M¢ller, Nucl. Phys. B74 (1974) 145. [6] S. Uhlig, private communication from the Aachen-CERN-Heidelberg-Miinchen Streamer Chamber Collaboration.