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Moment analysis, multiplicity distributions and correlations in high energy processes: nucleus-nucleus collisions
4 September 1997
PHYSICS LETTERS 6 Physics Letters B 407 (1997) 335-340
Moment analysis, multiplicity distributions and correlations in high energy processes: nucleus-nucleus collisions J. Dias de Dew ‘, C. Pajares, C.A. Salgado Departumento de Fisica de Partikdas, Universidade de Santiago de Composfelu, 157WSantiago
de Compostela, Spain
Received 27 February 1997; revised manuscript received 26 May 1997
Editor: R. Gatto
Abstract Cum&ant oscillations, or H,, moment oscillations, appear if the KNO multiparticle distribution decreases at large z, r zz n/(n), faster than the exponential, exp( -DzJ‘), with p > 1. In nucleus-nucleus interactions this behaviour is related to the limitation in the average number of elementary central collisions (or average number of strings centrally produced), due to the finite number of nucleons involved. Colour deconfinement, via percolating string fusion, will drastically decrease the fraction of centrally produced strings and increase the cut-off parameter p: Moment oscillations will be displaced to smaller q and the width of the KNO ~s~~tion and foray-backwa~ particle correlations will become smaller. @ 1997 Elsevier Science B.V. PACS: 25.75.D~; 12.38.Mh; 13.87.Ce; 24.8S.+p
In the framework oretically predicted,
of perturbative QCD it was thesometime ago, that the factorial cumulants K, (or, equivalently, the moments Hq E K4/F9, K, being the factorial moments) of the multiple distribution should present oscillations in sign as a function of q [ 11. The predictions turned out to be confirmed by experiment in e+e- annihilations [ 21, in pp interactions [ 31 and in hadron-nucleus and nucleusnucleus collisions [4]. It is now clear that such oscillations are present in all known high energy processes. Recently, in [S], it was shown that a necessary condition for a particle distribution P( n, (n) ), (n) being the average multiplicity, with positive two particle correlation, K2 > 0, to have oscillations in K4
’ Iherdrofavisiting professor. On leave of absence from Institute Superior Tecnico, 1096 LISBOA codex, Portugal.
is to exist, asymptotically, a RN0 distribution q (z ), 161, of the form (n)W,
(n),
Wz) (n)-+cXJ, .?rn/(n)=const.
Z;,w[-Dzpl,
(1)
z E n/(n) being the scaling variable, D and Al.positive parameters, with p>
1.
(2)
Most of the existing popular parametrizations (of the Negative Binomial Distribution family) do not satisfy ( 1) and (2). They behave exponentially at large z, P = 1, and, not surprisingly, they do not show oscillations in q: rU, > 0 for all values of q. The importance of the large z behaviour of the distributions to generate the oscillations can be easily
0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00677-l
336
1. Dias de Deus et ai. /Physics Letters B 407 (1997) 33.5-340
seen from the expression relating K, to F4 ( F4 s (n(n-l)...(n-q+l))/(n)“))andK~,~
Kq= F, -
c
P&z,_.. f&L
1
(3)
T,J?,...
with q + r2 + . . . = 4, Ti 2 1, ~i+l 2 li and r-1 < 4. 3-k Pr,,r2r... are the positive combinato~~ factors associated to the partition of an integer q into integers rz, t-2, . . .. The meaning of (3) is straightforward. To obtain the q particle cumulant Kq one has to substract from the q particle factorial moment F4 (integrated q particle inclusive density) all the q - 1, q - 2, . . . particle cumulants (integrated inclusive particle correlations) in all clustering combinations. We have, in particular (note that we are using normalized moments, KI = FI = 1),
argue here that in nucleus-nucleus collisions, at least, the oscillations have also a physical origin. Succesful models that attempt to explain hadronhadron, h-h, hadron-nucleus, h-A, and nucleusnucleus, A-B, collisions, making use, or not, of basic information from efe- annihilations, are multiple scattering models. We shall take as reference the Dual Parton Model (DPM), [9]_ but most of our results do not depend on detailed features of a particular model. In any multiple scattering model the distribution P(n) of produced particles is the result of the superposition of the contributions from elementary inelastic collisions. At each elementary collision particles (via string formation, for instance) are emitted with a given dis~ibution. There is a certain probability of Y elementary collisions to occur. In general, we can write P(n> = x
K2 = F2 - 1
24
t:
SP(v)p(n1 )p(nz)
. . .P(G)
f
Ill....*&
(5)
K3 = F3 - 3Kz - 1 K4 = F4 - 4K3-3K,2-6K2-3 KS = Fs - 5K4 - 10K3Kz - lOK3 - 15K; - lOK2 - 1, (4)
If the distribution if of exponential type the F4 grow very fast with q, essentially as q!, and all the terms in (3) and (5) remain positive (Kz is assumed positive). If at large z the distribution decreases faster than the exponential, p > 1, the )rq grow slower and at some value of q, Kq start becoming negative. But this negative Kq enter in the future steps, q -i- 1, q + 2, . . . giving positive contibutions to the right hand side of (3) and (5). At some stage the X4 become again positive, etc.: the oscillations start. It can be argued that the required supression at large z is not really physical, in the sense that at finite energy and limited experimental acceptance, distributions are always cut at large z [7]. In e+e- annihilations there are theoretical reasons to belive that the oscillations are physical [ I]. For intance, in Ref. [ 81 the oscillatory behaviour of the ratio of factorial cummulants over factorial moments as a function of q are repr~u~ in terms of a weighted superposition of two multiplicity distribution associated to two and multi-jet production. The first one is well described in terms of two negative binomial distributions, associated to the contribution of b& and light flavoured events, respectively. We shall
with nl + n2 -I-. . . A- n, = n, p( ni) being the particle distribu~on from the i-th elementary collision, qf Y) the probability distribution for v elementary collisions. The parameter v represents, as well, the number of intermediate produced objects: pairs of strings, in DPM. In (5) we have assumed that the formed strings emit inde~ndently and that fluctuactions in the size of the strings are negligeable, or can be reabsorbed in the distribution (o(v) . Let us now intr~u~ the generating function G( z >, G(z) =x(1
-tz)“P(n),
(6)
n=o
with G(0) = 1, such that 1 dqG(z)
I
Fq=(n)4-&F- z&Y
(71
and 1 dQlogG(z) Kq = 0” d.@
1z=o*
(8)
By combining (5) and (6) we obtain G(z) =t]:/p(4g(z)‘, V=l
(9)
where g( z ) is the generating function for the elementary process, Knowing the elementary generating
.I. Dius de Dew et al. /Physics Lettms B 407 (19971335-340
Table 1 The factorial cumulants Kq for q = 2, 3 and 4, obtained from (9) and (8). Wben the fluctuations in the number of strings am negligible the curnu~~ts are the cumulants resuiting from (Y) independent sources: Ka = Kq/(w)q-‘. When string fluctuations dominate, as in heavy nuclei collisions, only the terms inside square brackets are important. In that limit, the factorial cumulants Kq are given by the cumulants of the string distribution. In that same limit the factorial moments Fq are given by the moments (vQ)/{v)q of the string distribution. For definition of moments see, for instance, { 171.
function g( z, ) (from d”e-, string model. Poisson approximation, etc.) and the elementary collision distribution q(v) (from multiple scattering combinatorics, impact parameter integrations, etc) the full generating function C( z ) can be constructed and the moments compute, (9>, (7) and (8). A summ~ of results is contained in Table 1. For the average multiplicity (R) and the normalized KNO dispersion D/(n), where 13’ = (n2) - (n)* we have
(n) = (Y)ii
(10)
o2 _ V> - H2 + (1) d2, m*
w2
Y ii2
(11)
where A and d are the average multiplicity and the dispersion of the elementary process, respectively. Concerning relation ( 11) an important observation can be made. If fuctuactions in the effective number of strings were negligeable, i.e., ( (v2) - (v)~)/(v)~ ~0, the second term in the right hand side of ( 11) should dominate. The KNO dispersion, as (v) > 1, should then be smaller than the normalized dispersion of the elementary process (say, e+e- or Poisson distribution). This is against experiment: D2/(n)2 increases with the complexity of the systems involved, from 0.09 in e+e- annihilations [ IO], to 0.25-0.30 in pp collisions [ 111, to larger values in h-A processes, and to 0.8- 1 in A-B collisions [ 12,131. The conclusion is that
337
in ( 11) the first term in the right hand side, counting the fluctuations in the effective number of strings, is do~n~t. This is p~cul~ly true in nucleus-nucleus collisions where, for large A and B, (v} N 102-103, thus making the second term in the right hand side of ( 11) completely negligeable. What we have shown for D/(n) applies to all the cumulants Kq (see Table 1) . At least in nucleus-nucleus collisions the KNO particle distribution function (n~~(n, (n)) must be very close to the KNO string dis~ibution function (P)~(Y, (v)), with scaling variable z = n/(n) CTV(V). It should by now be clear that oscillations in the factorial cumulants & (or in the Hq moments), at least in nucleus-nucIeus collisions, have nothing to do with the elementary interactions and cannot be related in a simple way to some perturbativeQCD calculations (which may apply to the elementary process). We shall try next to explain the origin of the oscillations in A-B collisions. In nucleus-nucleus A-B collisions, if instead of measuring the unconstrained multiplicity distributions P(n) one measures the distribution at a fixed impact parameter b (in particular in very central collisions, at b = 0) the obtained distribution if totaly different. While in the unconstr~n~ situation the KNO distribution is wide (~/~n) 21 1) and roughly inde~ndent of the nuclei, the central KNO distribution is very narrow (D/(n) << 1) and strongly dependent on the nuclei envolved, specially on the lightest nucleus [ 121. These differences can be easily understood from (11). If the impact parameter is fixed the fluctuation in the number of strings is drastically reduced, ((v2) - ‘+>2,/1~)2 --+ 0, and the second term in the right hand side of ( 11f dominates:
(12) where (Y)~ is the average number of central elementary collisions. One sees that the KNO distribution has to be very narrow (d/E is almost Poisson-like and (Y)~ may be very large), and the width depends on (Y>~.The value of (vjc increases with the atomic weight of the nuclei, being limited by the atomic weight of lightest nucleus, Independently of A and B one roughly has [ 123 +)I(+
= 114.
(13)
J. Dias de Deus et al. /Physics Letters B 407 (1997) 335-340
338
A good way of parameterising multiplicity distributions, from e+e- annihilations to A-B collisions, is by using the generalized gamma function for the asymptotic RN0 function,
1I’(z) =
r(K
&[
f
l/,Sut
r(K)
“’ I
(14) with
and, in particular, (z*) = (z’) = 1. The parameters K and p have to be fixed. We shall now impose the constant that the large IZ behaviour of the central collision particle distribution must be the same as the behaviour of the unconstrained distribution:
If the central distribution is also parametrized as ( 14), with parameters pidcand IC,, it is clear, from ( 161, that ,uC= ,u. However, IC, is very different from K, as the central distribution is very narrow. This means, see (15) for q = 2, (z2) 5~ 1 or K, > 1. We can finally impose (16) in a stronger way, see ( 14) r(K + l//-‘u) r(K)
+‘)
(17)
=(v)c’
In order to have estimates for K and p we introduce the ad~tion~ relations, see ( IS), r(K)
r(K i- 2/~) w
f vh)
r(K -k b$d
=2+1, (nj2
(18)
and make use of experimental information on D/(n). Reasonable values for K and p in agreement with [ 181 and [ 193, (v)/(Y)~ z l/4 and D/(n) N 0.9, are K”O.o.1,
pcf5.
(19)
A few remarks can now be made: l)As,from(l9),K~< l,~e~Ofunctiond~s not turn to zero as z --+ 0, in agreement with experimental data on multiplicity and transverse energy distributions [ 13,141,
Fig. 1. KNO negative particle distributions in heavy nucleus-nucleus collisions. Continuous curve: present energy situation, K = 0.1, p = 5 in Eq. ( 14). Dashed curve: the same distribution for percolating string fusion, with K = 0.1, p = 10, corresponding to about 50% reduction in the density of central strings.
2) The parameter ,Uis very large, which means that the particle dis~bution is drasticaly cut at large n, in agreement with multiple scattering models 112-l 51 and data: 3) As ,u is large, oscillations occur in the cumulants K4’ In Fig. 1 we show a plot of the KNO distribution which agrees with experimental data. In Fig. 2 we present Hq as a function of q. In both cases K and p were fixed at the values ( 19). Let us next suppose that the road to the fo~ation of the quark-gluon plasma in nucleus-nucleus collisions is by string fusion [ 161, mainly occuring in central collisions, where the density of strings is higher. The net result is that the large n tail of the multiplicity distribution is further cut at large n, or, in other words, ,CLis further increased. This can directly be seen in (17): an increase of (v)/(Y)~, due to string fusion, is translated in an increase in P. In Figs. 1 and 2 we also show the KNO multiplicity dis~bution and moments H, as function of q when the ratio (~)/(~)~ is increased from roughly 114 to l/2. The oscillations tend to start earlier. The same kind of effect occurs if one is triggering
.I. Dias de Lfeus et al. f PhysicsLetrers B 407 (I9971 33.5-340
339
(231
and, see (1 l),
where the factor 2 in the second term in the right hand side of (24) corresponds to particles being independently emitted in rapidity in each elementary collision. It is clear &ha&in an unconstrain~ nucleusnucleus, A-A, collision, from (20), (21) and (221, the forward-backward correlation parameter b is essentially 1. If we trigger in a central collision, (v2) - {v)~,/(v)~ -+ 0, as seen before, and the parameter b drops to zero. For instance, in &hecase of percolating fusion [IS], as ~~)/{~)~ 21 l/2, if one triggers on forward events with Fig. 2. The moments Hq z Kq/Fq for q = 2, 3, 4, 5, 6, 7 in the case of nucleus-nucleus collisions without fusion (p = 10, open circles). Also represented with crosses (squares) the Hy deduced from the muItjpIie~ty distributions of S-U collisions at fi = 19.4 GeV when a J/Y (a D&l-Yan pair) is triggered.
a heavy particle like the J/q. The n-tail of the multiplicity distribution is further cut and therefore the oscillations tend to start earlier, as it is seen in Fig. 2 where it is plotted Hq for the mul&~plicitydistribution in S-U collisions at fi = 19.4 GeV when a J/q is triggered. Also, it is shown the result when Drell-Yan pairs are triggered instead of f/P. We look now to the foxed-backw~d co~elations. If one fixes the number of particles forward, and studies the distribution in the backward hemisphere, one approximately has
nF
2
2{nF)
+
(25)
should already obtain b 2 0. This agrees with the results of [ 2 I 1, but as we have here strong fusion the net effect is more spectacular. one
on
b zz D&,,/D; 9 && = ~~~~~~-
(211 ~~~~~~F~
,
(221
and L$$being the variance in the forward hemisphere. Within the spirit of our appraximation of keeping only fluctuations in the number of elementary collisions, we assume that particles are produced symme~ic~ly in center of mass rapidity, see [ 191, to obtain
We thank CICYT of Spain for ~nancial support and one of us (C.A.S.) to Xunta de Galicia for a fellowship. We also thank Prof. M.A. Braun who participated in the initial stage of this research.
References f I ] I.M. Dremin, Pbys. Lett. B 313 ( 1993) 209; I.M. Dremin and V.A. Nechitaiho, JETP Lett. 51 (1993) 881. lZ] J.C. Zhou, Preprint SLAB-PUB-66~ ( 1994). [3] I.M. Dremin, V. Arena, G. Boca et al., Phys. I.&t. B 336 (1994) 119. [4] A. Capeila, I.M. Dremin, V.A. Neehita~~o and J. Tran Thanh Van, 2. Phys. C. to appear. [S] S. Hegyi, Phys. I.&t. B 387 ( 1996) 642. 161 Z. Koba, H.B. NieIsen and P, Olesen, Nucl. Phys. B 40 (1972) 317. 171 R. Ugoccioni, A. Giovannini and S. Lupia, Phys. I&t. B 342 (1994) 387. IS] A. Giov~nin~, S. Lupia and R. Ugoccioni, Phys. Lett. B 374 (1996) 231 and Phys. L&t. B 388 (1996) 639. 191 A. Capella, J. Tran Thanh Van, 2. Phys. C 23 ( 1484) 165, {IO] I? Abreu, A. De Angelis, R. Henriques and M. Pimenta+ Preprint DELPHI 96-144 PHYS 642 f 1996).
340
f. Dias de Deeuset al. /~h~sics Letters 3 407 (19971335-340
f 111 A. Wroblewski, Acta. Phys. Pal. B 4 (1974) 857, P Malhotra, Nucl. Phys. B 46 (1973) 559. [ 121 A. Capella, C. Pajares, A.V. Ramallo, Nucl. Phys. B 241 (1984) 75. [ 131 A. Capella, J.A. Casado, C. Pajares, A.V. Ramallo and J. Tran Thaah Van, Phys. Rev. D 35 (1987) 292. [ 141 K. Werner, Phys. Reports 232 (1993) 87. f 151 B. Andersson, G. Gustafson and B. Nilsson-Ahnquist, Nucl. Phys. B 281 (1987) 289. [ 161 N.S. Amelin, M.A. Braun and C. Pajares, Phys. L&t. B 306 (1993) 312, Z. Phys. C 63 (1994) 507.
1171 P Carmthers and C.C. Shih, Int. Mod. Phys. A 2 ( 1987) 1447. f 181 A. Capella, A. Kouider Akil and J. Tran Thanh Van, Phys. J&t. B 30.5 ( 1993) 302. [ 191 A. Capella and A. Krzywicky, Phys. Rev. D 18 (1978) 4120; A. Capella and J. Tran Thanh Van, Z. Phys. C 18 ( 1983) 85; Phys. Rev. D 29 (1984) 2512. f20] N. Armesto, M.A. Bmun, E.G. Fern&o and C. Pajares, Phys. Rev. Len. 77 ( 1996) 3736. [Zl] N. Amelin, N. Armesto, M.A. Braun, E.G. Ferreiro and C. Pajares, Phys. Rev. L&t. 73 (1994) 2813.
PHYSICS LETTERS 6 Physics Letters B 407 (1997) 335-340
Moment analysis, multiplicity distributions and correlations in high energy processes: nucleus-nucleus collisions J. Dias de Dew ‘, C. Pajares, C.A. Salgado Departumento de Fisica de Partikdas, Universidade de Santiago de Composfelu, 157WSantiago
de Compostela, Spain
Received 27 February 1997; revised manuscript received 26 May 1997
Editor: R. Gatto
Abstract Cum&ant oscillations, or H,, moment oscillations, appear if the KNO multiparticle distribution decreases at large z, r zz n/(n), faster than the exponential, exp( -DzJ‘), with p > 1. In nucleus-nucleus interactions this behaviour is related to the limitation in the average number of elementary central collisions (or average number of strings centrally produced), due to the finite number of nucleons involved. Colour deconfinement, via percolating string fusion, will drastically decrease the fraction of centrally produced strings and increase the cut-off parameter p: Moment oscillations will be displaced to smaller q and the width of the KNO ~s~~tion and foray-backwa~ particle correlations will become smaller. @ 1997 Elsevier Science B.V. PACS: 25.75.D~; 12.38.Mh; 13.87.Ce; 24.8S.+p
In the framework oretically predicted,
of perturbative QCD it was thesometime ago, that the factorial cumulants K, (or, equivalently, the moments Hq E K4/F9, K, being the factorial moments) of the multiple distribution should present oscillations in sign as a function of q [ 11. The predictions turned out to be confirmed by experiment in e+e- annihilations [ 21, in pp interactions [ 31 and in hadron-nucleus and nucleusnucleus collisions [4]. It is now clear that such oscillations are present in all known high energy processes. Recently, in [S], it was shown that a necessary condition for a particle distribution P( n, (n) ), (n) being the average multiplicity, with positive two particle correlation, K2 > 0, to have oscillations in K4
’ Iherdrofavisiting professor. On leave of absence from Institute Superior Tecnico, 1096 LISBOA codex, Portugal.
is to exist, asymptotically, a RN0 distribution q (z ), 161, of the form (n)W,
(n),
Wz) (n)-+cXJ, .?rn/(n)=const.
Z;,w[-Dzpl,
(1)
z E n/(n) being the scaling variable, D and Al.positive parameters, with p>
1.
(2)
Most of the existing popular parametrizations (of the Negative Binomial Distribution family) do not satisfy ( 1) and (2). They behave exponentially at large z, P = 1, and, not surprisingly, they do not show oscillations in q: rU, > 0 for all values of q. The importance of the large z behaviour of the distributions to generate the oscillations can be easily
0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00677-l
336
1. Dias de Deus et ai. /Physics Letters B 407 (1997) 33.5-340
seen from the expression relating K, to F4 ( F4 s (n(n-l)...(n-q+l))/(n)“))andK~,~
Kq= F, -
c
P&z,_.. f&L
1
(3)
T,J?,...
with q + r2 + . . . = 4, Ti 2 1, ~i+l 2 li and r-1 < 4. 3-k Pr,,r2r... are the positive combinato~~ factors associated to the partition of an integer q into integers rz, t-2, . . .. The meaning of (3) is straightforward. To obtain the q particle cumulant Kq one has to substract from the q particle factorial moment F4 (integrated q particle inclusive density) all the q - 1, q - 2, . . . particle cumulants (integrated inclusive particle correlations) in all clustering combinations. We have, in particular (note that we are using normalized moments, KI = FI = 1),
argue here that in nucleus-nucleus collisions, at least, the oscillations have also a physical origin. Succesful models that attempt to explain hadronhadron, h-h, hadron-nucleus, h-A, and nucleusnucleus, A-B, collisions, making use, or not, of basic information from efe- annihilations, are multiple scattering models. We shall take as reference the Dual Parton Model (DPM), [9]_ but most of our results do not depend on detailed features of a particular model. In any multiple scattering model the distribution P(n) of produced particles is the result of the superposition of the contributions from elementary inelastic collisions. At each elementary collision particles (via string formation, for instance) are emitted with a given dis~ibution. There is a certain probability of Y elementary collisions to occur. In general, we can write P(n> = x
K2 = F2 - 1
24
t:
SP(v)p(n1 )p(nz)
. . .P(G)
f
Ill....*&
(5)
K3 = F3 - 3Kz - 1 K4 = F4 - 4K3-3K,2-6K2-3 KS = Fs - 5K4 - 10K3Kz - lOK3 - 15K; - lOK2 - 1, (4)
If the distribution if of exponential type the F4 grow very fast with q, essentially as q!, and all the terms in (3) and (5) remain positive (Kz is assumed positive). If at large z the distribution decreases faster than the exponential, p > 1, the )rq grow slower and at some value of q, Kq start becoming negative. But this negative Kq enter in the future steps, q -i- 1, q + 2, . . . giving positive contibutions to the right hand side of (3) and (5). At some stage the X4 become again positive, etc.: the oscillations start. It can be argued that the required supression at large z is not really physical, in the sense that at finite energy and limited experimental acceptance, distributions are always cut at large z [7]. In e+e- annihilations there are theoretical reasons to belive that the oscillations are physical [ I]. For intance, in Ref. [ 81 the oscillatory behaviour of the ratio of factorial cummulants over factorial moments as a function of q are repr~u~ in terms of a weighted superposition of two multiplicity distribution associated to two and multi-jet production. The first one is well described in terms of two negative binomial distributions, associated to the contribution of b& and light flavoured events, respectively. We shall
with nl + n2 -I-. . . A- n, = n, p( ni) being the particle distribu~on from the i-th elementary collision, qf Y) the probability distribution for v elementary collisions. The parameter v represents, as well, the number of intermediate produced objects: pairs of strings, in DPM. In (5) we have assumed that the formed strings emit inde~ndently and that fluctuactions in the size of the strings are negligeable, or can be reabsorbed in the distribution (o(v) . Let us now intr~u~ the generating function G( z >, G(z) =x(1
-tz)“P(n),
(6)
n=o
with G(0) = 1, such that 1 dqG(z)
I
Fq=(n)4-&F- z&Y
(71
and 1 dQlogG(z) Kq = 0” d.@
1z=o*
(8)
By combining (5) and (6) we obtain G(z) =t]:/p(4g(z)‘, V=l
(9)
where g( z ) is the generating function for the elementary process, Knowing the elementary generating
.I. Dius de Dew et al. /Physics Lettms B 407 (19971335-340
Table 1 The factorial cumulants Kq for q = 2, 3 and 4, obtained from (9) and (8). Wben the fluctuations in the number of strings am negligible the curnu~~ts are the cumulants resuiting from (Y) independent sources: Ka = Kq/(w)q-‘. When string fluctuations dominate, as in heavy nuclei collisions, only the terms inside square brackets are important. In that limit, the factorial cumulants Kq are given by the cumulants of the string distribution. In that same limit the factorial moments Fq are given by the moments (vQ)/{v)q of the string distribution. For definition of moments see, for instance, { 171.
function g( z, ) (from d”e-, string model. Poisson approximation, etc.) and the elementary collision distribution q(v) (from multiple scattering combinatorics, impact parameter integrations, etc) the full generating function C( z ) can be constructed and the moments compute, (9>, (7) and (8). A summ~ of results is contained in Table 1. For the average multiplicity (R) and the normalized KNO dispersion D/(n), where 13’ = (n2) - (n)* we have
(n) = (Y)ii
(10)
o2 _ V> - H2 + (1) d2, m*
w2
Y ii2
(11)
where A and d are the average multiplicity and the dispersion of the elementary process, respectively. Concerning relation ( 11) an important observation can be made. If fuctuactions in the effective number of strings were negligeable, i.e., ( (v2) - (v)~)/(v)~ ~0, the second term in the right hand side of ( 11) should dominate. The KNO dispersion, as (v) > 1, should then be smaller than the normalized dispersion of the elementary process (say, e+e- or Poisson distribution). This is against experiment: D2/(n)2 increases with the complexity of the systems involved, from 0.09 in e+e- annihilations [ IO], to 0.25-0.30 in pp collisions [ 111, to larger values in h-A processes, and to 0.8- 1 in A-B collisions [ 12,131. The conclusion is that
337
in ( 11) the first term in the right hand side, counting the fluctuations in the effective number of strings, is do~n~t. This is p~cul~ly true in nucleus-nucleus collisions where, for large A and B, (v} N 102-103, thus making the second term in the right hand side of ( 11) completely negligeable. What we have shown for D/(n) applies to all the cumulants Kq (see Table 1) . At least in nucleus-nucleus collisions the KNO particle distribution function (n~~(n, (n)) must be very close to the KNO string dis~ibution function (P)~(Y, (v)), with scaling variable z = n/(n) CTV(V). It should by now be clear that oscillations in the factorial cumulants & (or in the Hq moments), at least in nucleus-nucIeus collisions, have nothing to do with the elementary interactions and cannot be related in a simple way to some perturbativeQCD calculations (which may apply to the elementary process). We shall try next to explain the origin of the oscillations in A-B collisions. In nucleus-nucleus A-B collisions, if instead of measuring the unconstrained multiplicity distributions P(n) one measures the distribution at a fixed impact parameter b (in particular in very central collisions, at b = 0) the obtained distribution if totaly different. While in the unconstr~n~ situation the KNO distribution is wide (~/~n) 21 1) and roughly inde~ndent of the nuclei, the central KNO distribution is very narrow (D/(n) << 1) and strongly dependent on the nuclei envolved, specially on the lightest nucleus [ 121. These differences can be easily understood from (11). If the impact parameter is fixed the fluctuation in the number of strings is drastically reduced, ((v2) - ‘+>2,/1~)2 --+ 0, and the second term in the right hand side of ( 11f dominates:
(12) where (Y)~ is the average number of central elementary collisions. One sees that the KNO distribution has to be very narrow (d/E is almost Poisson-like and (Y)~ may be very large), and the width depends on (Y>~.The value of (vjc increases with the atomic weight of the nuclei, being limited by the atomic weight of lightest nucleus, Independently of A and B one roughly has [ 123 +)I(+
= 114.
(13)
J. Dias de Deus et al. /Physics Letters B 407 (1997) 335-340
338
A good way of parameterising multiplicity distributions, from e+e- annihilations to A-B collisions, is by using the generalized gamma function for the asymptotic RN0 function,
1I’(z) =
r(K
&[
f
l/,Sut
r(K)
“’ I
(14) with
and, in particular, (z*) = (z’) = 1. The parameters K and p have to be fixed. We shall now impose the constant that the large IZ behaviour of the central collision particle distribution must be the same as the behaviour of the unconstrained distribution:
If the central distribution is also parametrized as ( 14), with parameters pidcand IC,, it is clear, from ( 161, that ,uC= ,u. However, IC, is very different from K, as the central distribution is very narrow. This means, see (15) for q = 2, (z2) 5~ 1 or K, > 1. We can finally impose (16) in a stronger way, see ( 14) r(K + l//-‘u) r(K)
+‘)
(17)
=(v)c’
In order to have estimates for K and p we introduce the ad~tion~ relations, see ( IS), r(K)
r(K i- 2/~) w
f vh)
r(K -k b$d
=2+1, (nj2
(18)
and make use of experimental information on D/(n). Reasonable values for K and p in agreement with [ 181 and [ 193, (v)/(Y)~ z l/4 and D/(n) N 0.9, are K”O.o.1,
pcf5.
(19)
A few remarks can now be made: l)As,from(l9),K~< l,~e~Ofunctiond~s not turn to zero as z --+ 0, in agreement with experimental data on multiplicity and transverse energy distributions [ 13,141,
Fig. 1. KNO negative particle distributions in heavy nucleus-nucleus collisions. Continuous curve: present energy situation, K = 0.1, p = 5 in Eq. ( 14). Dashed curve: the same distribution for percolating string fusion, with K = 0.1, p = 10, corresponding to about 50% reduction in the density of central strings.
2) The parameter ,Uis very large, which means that the particle dis~bution is drasticaly cut at large n, in agreement with multiple scattering models 112-l 51 and data: 3) As ,u is large, oscillations occur in the cumulants K4’ In Fig. 1 we show a plot of the KNO distribution which agrees with experimental data. In Fig. 2 we present Hq as a function of q. In both cases K and p were fixed at the values ( 19). Let us next suppose that the road to the fo~ation of the quark-gluon plasma in nucleus-nucleus collisions is by string fusion [ 161, mainly occuring in central collisions, where the density of strings is higher. The net result is that the large n tail of the multiplicity distribution is further cut at large n, or, in other words, ,CLis further increased. This can directly be seen in (17): an increase of (v)/(Y)~, due to string fusion, is translated in an increase in P. In Figs. 1 and 2 we also show the KNO multiplicity dis~bution and moments H, as function of q when the ratio (~)/(~)~ is increased from roughly 114 to l/2. The oscillations tend to start earlier. The same kind of effect occurs if one is triggering
.I. Dias de Lfeus et al. f PhysicsLetrers B 407 (I9971 33.5-340
339
(231
and, see (1 l),
where the factor 2 in the second term in the right hand side of (24) corresponds to particles being independently emitted in rapidity in each elementary collision. It is clear &ha&in an unconstrain~ nucleusnucleus, A-A, collision, from (20), (21) and (221, the forward-backward correlation parameter b is essentially 1. If we trigger in a central collision, (v2) - {v)~,/(v)~ -+ 0, as seen before, and the parameter b drops to zero. For instance, in &hecase of percolating fusion [IS], as ~~)/{~)~ 21 l/2, if one triggers on forward events with Fig. 2. The moments Hq z Kq/Fq for q = 2, 3, 4, 5, 6, 7 in the case of nucleus-nucleus collisions without fusion (p = 10, open circles). Also represented with crosses (squares) the Hy deduced from the muItjpIie~ty distributions of S-U collisions at fi = 19.4 GeV when a J/Y (a D&l-Yan pair) is triggered.
a heavy particle like the J/q. The n-tail of the multiplicity distribution is further cut and therefore the oscillations tend to start earlier, as it is seen in Fig. 2 where it is plotted Hq for the mul&~plicitydistribution in S-U collisions at fi = 19.4 GeV when a J/q is triggered. Also, it is shown the result when Drell-Yan pairs are triggered instead of f/P. We look now to the foxed-backw~d co~elations. If one fixes the number of particles forward, and studies the distribution in the backward hemisphere, one approximately has
nF
2
2{nF)
+
(25)
should already obtain b 2 0. This agrees with the results of [ 2 I 1, but as we have here strong fusion the net effect is more spectacular. one
on
b zz D&,,/D; 9 && = ~~~~~~-
(211 ~~~~~~F~
,
(221
and L$$being the variance in the forward hemisphere. Within the spirit of our appraximation of keeping only fluctuations in the number of elementary collisions, we assume that particles are produced symme~ic~ly in center of mass rapidity, see [ 191, to obtain
We thank CICYT of Spain for ~nancial support and one of us (C.A.S.) to Xunta de Galicia for a fellowship. We also thank Prof. M.A. Braun who participated in the initial stage of this research.
References f I ] I.M. Dremin, Pbys. Lett. B 313 ( 1993) 209; I.M. Dremin and V.A. Nechitaiho, JETP Lett. 51 (1993) 881. lZ] J.C. Zhou, Preprint SLAB-PUB-66~ ( 1994). [3] I.M. Dremin, V. Arena, G. Boca et al., Phys. I.&t. B 336 (1994) 119. [4] A. Capeila, I.M. Dremin, V.A. Neehita~~o and J. Tran Thanh Van, 2. Phys. C. to appear. [S] S. Hegyi, Phys. I.&t. B 387 ( 1996) 642. 161 Z. Koba, H.B. NieIsen and P, Olesen, Nucl. Phys. B 40 (1972) 317. 171 R. Ugoccioni, A. Giovannini and S. Lupia, Phys. I&t. B 342 (1994) 387. IS] A. Giov~nin~, S. Lupia and R. Ugoccioni, Phys. Lett. B 374 (1996) 231 and Phys. L&t. B 388 (1996) 639. 191 A. Capella, J. Tran Thanh Van, 2. Phys. C 23 ( 1484) 165, {IO] I? Abreu, A. De Angelis, R. Henriques and M. Pimenta+ Preprint DELPHI 96-144 PHYS 642 f 1996).
340
f. Dias de Deeuset al. /~h~sics Letters 3 407 (19971335-340
f 111 A. Wroblewski, Acta. Phys. Pal. B 4 (1974) 857, P Malhotra, Nucl. Phys. B 46 (1973) 559. [ 121 A. Capella, C. Pajares, A.V. Ramallo, Nucl. Phys. B 241 (1984) 75. [ 131 A. Capella, J.A. Casado, C. Pajares, A.V. Ramallo and J. Tran Thaah Van, Phys. Rev. D 35 (1987) 292. [ 141 K. Werner, Phys. Reports 232 (1993) 87. f 151 B. Andersson, G. Gustafson and B. Nilsson-Ahnquist, Nucl. Phys. B 281 (1987) 289. [ 161 N.S. Amelin, M.A. Braun and C. Pajares, Phys. L&t. B 306 (1993) 312, Z. Phys. C 63 (1994) 507.
1171 P Carmthers and C.C. Shih, Int. Mod. Phys. A 2 ( 1987) 1447. f 181 A. Capella, A. Kouider Akil and J. Tran Thanh Van, Phys. J&t. B 30.5 ( 1993) 302. [ 191 A. Capella and A. Krzywicky, Phys. Rev. D 18 (1978) 4120; A. Capella and J. Tran Thanh Van, Z. Phys. C 18 ( 1983) 85; Phys. Rev. D 29 (1984) 2512. f20] N. Armesto, M.A. Bmun, E.G. Fern&o and C. Pajares, Phys. Rev. Len. 77 ( 1996) 3736. [Zl] N. Amelin, N. Armesto, M.A. Braun, E.G. Ferreiro and C. Pajares, Phys. Rev. L&t. 73 (1994) 2813.