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Multiplicity distributions in high energy collisions
,hmm~'a.;L'~.1[qla PROCEEDINGS SUFPLEME~,ITS
Nuclear Physics B (Proc. Suppl.) 25B (1992) 119-123 Nartb.-ilol!a~d
MULTIPLICITY DIST~UTIOr~S
A. Giovannini*,
S. Lupia*,
~,.'" n t ~ , r t ~z,,r~ttI= x ~ u . , b t ~ I u N ~
R. Ugoccioni
Dipartimento di Fisica Teorica, Universit~ di Torino and INFN, Sezione di Torino, Italy
We discuss the important phases in the evolution of our understanding of multiplicity distributions in high energy collisions with particular emphasis to intermittent behavior and shoulder structure problem.
1. M u l t i p a r t i c l e p r o d u c t i o n : m a i n facts
Much of the extensive work done in the last 30 years on multiparticle production in high energy collisions can be regarded as a continued experimental search for correlations among produced particles and a succession of theoretical and experimental attempts to explain them in dynamical terms. A substantial part of this work was devoted to multiplicity distributions (MD) where correlations are revealed by non-Poissoniar behavior. The important phases in the evolution or"our understanding of MD's in Jfigh energy collisions are summarized in the following and can be divided in two eras. The first era was dominated by the study, in the accelerators region up to ISR, of the energy dependence of the total MD in haxlron-hadron collisions. The second era began when UA5 collai~oration decided to analyze MD's in/~3 annihilation in different pseudo-rapidity interv£o ~t c.m. eneigics up to ~ = 900 GeV. First Era:
i) Wr6blewski regularity [1]. It says that D/(n) in full phase space stays approximately constant as the c.m. energy increases; (n) is the average charged multiplicity and D = x/(n 2) - (hi 2 the dispersion for the
reaction under investigation. The occurrence of this behavior was interpreted as an indication that charged particles are independently produced in full phase space (absence of correlation among produced particles).
ii) KNO scaring [2]. The probability of producing n charged pactides in full phase space, P,s, is expected to be n universal function of n/(n} as the energy increases, i.e.
This relation shows that once ~(n/(n)) has been determined from experimental observations, data ou MD's from new experiments should fall on the same curve.
This prediction on P , is very striking; it is related to the validity of Feynman scaling [3], which says that the charged particle multiplicity density per unit of rapidity, dn/dy, should stay constant as the energy increases (the plateau in dn/dy versus y widens with energy).
iii) KNO scaling violation at S/~S collider in full phase space [4]. The experimental observation of the UA5 collaboration has shown that what was considered a well established asymptotic regime in hh collisions was indeed n transition region to a new energy domain:
* The talks presented by A. G. (Multiplicity Distributions in High En-.rgy Co~lislons) and by S. L. (Intermittency and Clan Structure Analysis) converged in a single contribution for practical reasons. Work supported in part by M.U.R.S.T. (Italy) under Grant 1990.
116
A. Giovannini et al./ Multiplicity distributions in high energy collisions
an i~__,portant ~varni_-~gt~ remember t-~.at shouid prevent experts from drawing too m~c~ ~-i-~si.~-s from pre~e~:~ experi--~.-ent*L:-~e+e - an,=iNia~i-n [~]o . . . .
.!
Y3__.
/) Negative Binomial (NB) shape of total MD's and of MD's in symmetric rapidity windows with the related "clan structure", not only for hadronic collisions but also for e+e - annihilation and deep inelastic leptonnucleon collisions [6]. According to NBMD the probability of producing n charged particles, Pn(l~)(~, k), depends on two parameters, the average charged multiplicity ~ and k, which is related to the dispersion D by D2 / ~ 2 = 1/k + 1/f~. p(Ne) is defined by the following relation
p(NB)
(n + 1~ "+-----!1 a + bn -' p ( N B ) =
(2)
with a - ~k/(ft + k) and b = fi/(~ + k). It should be noticed that relation (2) characterizes a class of distributions. In fact for b = 0 it reduces to a Poisson MD; therefore b measures deviations from Poisson behavior. For a = b ene gets the geometric MD; when a --* k and b --* 1 relation (2) leads to a gamma MD and finally for a -* 0 to a logarithmic MD. It is to be pointed out that the occurrence of NB behavior in full phase space was already noticed in the accelerator region and its relevance for a possible early KNO scaling violation emphasized [7]. The interpretation of NB regularity led to the introduction of the concept of 'clan'. Clans are group of particles of common ~ncestry; they are independently produced by assumption and each clan contains at least one particle by definition; particle correlations are exhausted within each clan. In terms of the MD generating function
aNs(~,k;~) = ~ ~"P.(~B)(~,t)
(3)
the average number of clans, ~ , can be defined by GNB(fi,k; z) = e ~[a~(b;z)-l]
(4)
and
is the generating function of a logarithmic MD, which I S *.... h h MI) ol the average clan; tLe average number of particles belonging to an average clan will be of course ~
=
~/~7.
(7)
ii) Occurrence of "NB regularity" in the QCD Parton Shower Models, which were known to give the best description of c+e - annihilation data, both for the fined state hadrons and for the final partons, lea ii~ag to a genereJ;.zcd property of local parton hadron dueeity (GLPIID) [8]. GLPHD says that n-patton and nhadron inclusive rapidity distributions, Q~) and Q~), are related by the following simple equation:
Q(a)(y*,... ,y,~) -~ p " Q ~ ) ( y , , . . . , y , )
(8)
with p ~ 2Qo/(1 GeV), Qo being the parrot, virtuality cut-off at which the shower development is stopped and hadronizatlon takes place. The dependence of the abow, regularities on patton virtueeity cut-off Q0 (Qo = 1 - 2GcV) in the context of the Lund Patton Shower Model was also investigated [9]. It was found that Lund hadronization prescription leads to infrared stable results. Finally it was discovered that the angulax ordering option incorporated in the model affected mean multiplicity only, leaving substantially unchanged the other parameters of the distributions. An instructive result which rises some interesting questions on the relevance of angular ordering on higher moments of the MD. It is interesting t i n t the clan concept can be extended to the parton level, where clans can be understood as bremsstre.hlung gluon jets [8]. In solving the Konishi-Ukawa-Veneziano equations, within the exclusive approach [10], one finds that the MD of gluons generated by a quark jet follows the NB MD with the average number of gluons (ne) q given by
where = - b In(l - b) = k l n ( l + ~ )
(5)
{ns) q = -~(e Ay - 1) ,at
(9)
A. Giovannini et al./Multiplicity distributions in high energy colli:4ons
! !7
colors and c a fixed cut-off to regularize divergences; y
fcrentiy from what happens ~o a jet, From this point of v{ew clan concept is more prim-~-ive tha_-.-1jet con~ . _=~ _:--:--.°a.._:-_ _-:=- :=*. : . . , . d~fi~fi-iem Far thermore parton shower recalls intuitively the cn,rnA.
is the standard evolution p,rameter
ing mechanism which seems to be the main dynamical
where
A
------ ( H ?
......
-o~'-~'-
"~. . . . . . . . . . .
s;-~-'' process g ~-~ 9 + 9 and A ~ -~-e!. . . . . . . . . . . . . . . .
6 [ In(Q2/A2) ] Y = 11Af, - 2Af! In [In(QI/A2) l with Af! number of flavors.
(10)
Q~ is the initial parton
virtuality, Qo2 the infrared cut-off and A the scale parameter. The dispersion can also be calculated: one finds D2
= ~Ae
ay,teAy
-1)
(11)
Accordingly
feature emerging from the Markofllan nature of the solutions of the KUV equations in the exclusive approach [10].
iii) Intermittency in the sense of the continued growth of normalized factorial moments of the MD's in decreasing rapidity windows [11]. Intermittency renewed the interest on measurement and interpretation of 2- and 3-particle correlations, including BoseEinstein interferometry of equal charge mesons, and es-
1
A = ~
and
# = .4y
(12)
In the 'pure bremsstrahlung' limit A --, 0, so that eq. (9) gives
tablished a useful contact between studies of MD's and correlations. Until quite recently there was practically no contact between them. The two topics were brought together when various authors attempted to explain the
which proves our statement.
experimental results on intermittency in terms of the kno,:n 2-particle correlations [12,13], while others noted that in many cases the factorial moments measured in
In conclusion, in the present phase two important facts are worth mentioning.
intermittency studies obey approximately the relations which are implied by NB behavior of the MD's in small
a) The existence of a simple link between the NB regularity for final charged hadrons MD's in full phase
rapidity windows [14]. These recent developments allow for the first time to
space and in symmetric rapidity windows, and the corresponding final parton~ MD %in qq and gg systems obtained by means of a Monte Carlo "experiment" based on the Lund Patton Shower as implemented in the pro-
attempt an integrated description of the overal~ correlation structure of multiparticle production and of its dynamical interpretation in terms of QCD parton shower
gram JETSET 6.3. This link is what we call GLPHD and it has been expressed in its more general form by eq. (8). It is a stronger condition than the standard
scription can only be tentative, but the abundant data becoming available at LEP and soon at HERA should lead to improvements and refinements. Through con-
one on local parton hadron duality.
cation of clans with bremsstrahlung gluon jets or patton showers originated by independent gluons follows.
frontation with these data one can expect the QCDbased description of e+e - annihilation and semileptonic collisions to get tested more severely and corrected where necessary, especially in the non-perturbative areas of soft gluon emission and of final patton hadronization, for which neither analytical methods nor lattice QCD computations are available so far.
We prefer to talk of parton showers in connection with clans, since a clan ~ as in general a parton shower is not affected by cuts in transverse momentum, dig
iv) Appearance of the shoulder structure in p/3 at c.m. enexgy V/~ = 900 GeV in large rapidity windows [15], and in e+e - annihilation at c.m. energy ~/~ = 91
(ng)~ = A3' =: N
(13)
b) The occurrence of NB behavior for glaon population initiated by a quark jet in the early stages of their evolution when perturbative QCD at the leading log level in the axial gauge is still applicable. The identifi-
mechanism with GLPHD. For the moment such a de-
A. Giova~niniet al./ Multiplicity distributionsin high energy collisions
118 ~ey
i~
sVrn,-~,_egr:_,c# ...... ra_Di~i~g
~_vii%=lowR :=~._~_
_ . . . . r~r,e~.2~_=
0o(~ . . . . . . vA:
d . . . . . . . i~, the ~,,~s ~h. . . . ¢ the MD. A.'oparentiv ~'° f
der structn_~ in e+e - ann~ihil_ation_w ~ ,_,nderst_,m-d in terms of 2- 3- 4-jets contributions to the whole sample of data [17]. As a consequence, NB regularity is found
where Q,(y,
in the next two sections.
d,,
f}
,'..
. . .
~Av)
roy
to be valid at a more fundamental level. In view of the present interest on points (iii) and (is) of the second era, we will discuss them mrre extensively
,4,,.
....
,y,) -
1
dqa
-- aia,~ dyl . . .
dy,
(16)
and n is the number of particles in a rapidity interval of finite width 611. In order to introduce correlation functions, we define the unnormalised factorial cumulants i'(~y)"
2. I n t e r m | t t e n e y
i'(6y) = ~ dy,...dy, 0,(~,,,...,~,,)
In the last few years the study of intermittent behavior assumed increasing relevance in Multiparticle Dynamics. Attention has been called on this new topic when three experimental groups [11] found strong local density fluctuations in the rapidity distribution of charged hadrons produced in high energy collisions. In order to give a quantitative characterization of the phenomenon, it was proposed [18] to study the normalized factorial moments of multiplicity distribution in restricted domains of rapidity I/. Intermittency is forreally defined by the relation:
F'(6~) o~ (6~)-*,
for ~ - .
0,
/ , > 0.
04)
So intermittency essentially means that factorial moments are divergent in the limit of high resolution. This power law is natural in self-similar cascading processes. Vvith this analysis, intermittent behavior was found in all multihadron producing reactions. Besides, stealdexd Monte Carlo programs, contrary to what occurs in e+e - annihilation, fail to reproduce the effect in leptonhadron and hadron-hadron collisions (for an experimental review see [19]). In this section we show that particle correlations and m,.ltiplicity distributions, as already stressed, can be brought together by iutermittency. The study of normalized factorial moments gives information on the correlations between final particles. In fact the unnormalif.ed factorial moments ~'g($y) are directly related to the inclusive rapidity distribution
(17)
Y
where Og(~1 . . . . . ~e) is the @-particles correlation function. Factorial cumulants vre simply related to factorial moments via a standard cluster expansion:
(18) -J -' \ / * i ~ 1
So corrdation functions are strictly linked to factorial moments. The power law behavior expected from intermittency requires the presence of s singularity in the correlation functions for small rapidity d~fferences. Intermittent behavior is also related to MD. In order to study MD, we define the exclusive rapidity distribution P,,Oh,... ,Yn) as the probability to find exactly n particles with rapidity y x , . . . , y ~ . The exclusive rapidity distribution P n ( Y l , . . . , Y,) and the inclusive one Q,~(yI,... ,yn) satisfy the following relation [20]:
~'d~,,..-, y.) = Q ~ ( ~ , . . . , ~ . )
+ £ (~ v'-----1
lP'.
f Q,~+,(yt.... , y,~,y~,... ,y~)dy;
d'
--
(19) We can express the unnormalizod factorial moments Fq(6y) in terms of the probability to find n particles in one interval of width 6y, P,~(6y): ~,($,) = £ ~vt=:Q
where
(n +.~)'P~+,~(6y)
(20)
A. Giovannini et al,/Multiplicity distributions in high energy collisions
,,~,~
/ g>ooog>->4> .... ,>)
(2_~)
ig .a =o_¢ very i~_erea_i~g _o notice tha~ e=perln~a~t~.i factorial moments are consistent with factorial moments calculated from NB regularity [14,13]. This fact suggests to extend the clan structure ~n~lysis to restricted domain of rapidity in order to investigate intermittent behavior. In terms of factorial moments generating function
(f.m.g.f.) P6~(u)
119
possible connection, between c'_,ana~aivsis - - d intern~;tte_ncyo =-->~:~"~::~ege:::t e'- v~e:% ~=~::~_'~t:_~:_-.° :: = w=y --= e_-_plore multiparticle production in restricted domMns of phase space and its occurrence in all reactions supports the idea of a unified picture for multihadron production based on selfsimilar cascade processes. 3. T h e s h o u l d e r s t r u c t u r e p r o b l e m The DELPHI collaboration discovered in e+e - an-
defined by:
(23)
nihilation at c.m. energy V~ = 91 GeV that NB diJtrl. bntirn reproducea only the groga ahape of MD's for final charged particles and giv-~s poor fits both in full phase space and in symmetric rapidity windows [y[ < Yc [16]. This fact is due to the presence of a shoulder in the MD's which is more evident in yc = 1.5 and 2.0 and
where P6y(u), Hs~(u) and g6y(u) are reapectively the
similar to that seen by UA5 collaboration at p/~ collider at c.m. energy v/~ = 900 GeV. However by using clan
P~y(u) - ~
Pn(~y)(1 + n)'*
(22)
vt-----S
we can define the d a n structure:
P,,(u) = H,,[g,,(u) - 1]
One can easily extend dam structure analysis in the
structure parameters on the same sample of events the average number of hadronic clans, Nh, was found to be ~pprozimu~ely constan~ wi~h ene~jy for fized rapid-
limit of high resolution with a simple ansatz [21] which
it 9 intereab. The two results seemed in contradiction.
relates the f.m.g.L in one interva/of width ~;y to those in a larger interval Ay with ~y C Ay:
Clapt structure analysis has been proposed in fact as a possible interpretation of the wide occurrence of the NB
f.m.g.f, of p~ticles, clans and particles in an average clan.
{ ll,,(u) -- H~,,(Ou)
g~,(u) ~s(su)
O,S ~
regularity in hh, lh reactions and - - at lower energy - e+e ~ annihilation; it is indeed rather surprising to find
[0,1];
=
(24)
In this way we get:
d~P~(u)/du ~I~=0 Fq(~y) = iap~,(u)/d~l,,=~iq ._- (0[-I')~-,~ + 0 ( 0 ~-')
(25)
w h e r e / t ~ are the unnormalized factorial moments of the cla~ distribution and gq the normalized factorial moments of the particle distribution in an average clan. Assuming that the average number of particles in the interval ~y, ~S~s is linear in 6y, we easily obtain the
out that the most appealing property of clan structure analysis of NB regularity was valid at LEP energy in spite of the violation of the regularity itself. One way out was to remark that the reproduction of the gross shape of the MD's was sumclent in order to guarantee the constancy of/9a with energy for fixed rapidity intervals. Clearly this way is unsatisfactory, we don't see in it any deep physics.
power law (14) of intermittency. With ansatz (24), the clan structure is consistent
A second possibility went back to the discovery by HRS collaboration of NB regularity in e+e - annihilation at c.m. energy v/~ = 29 GeV [22]. The important point to be recalled is that HRS collaboratio~ found NB fits with better X~/NDF for the 2-jet sample of events
with the intermittent behavior for factorial moments. It is still to be investigated how realistic this ansatz is. Here we want simply to focus the attention on the
than for the whole sample. The suspect is that contamination of qq9 events might hide the regularity which shows up fully at the q~ jet level. Accordingly it is more
120
A. Giovannini et al./ Multiplicity distributions in high energy collisions
natural to proceed as the DELPH1 cohaboration did subsequently [17] and to study at L.F,P energy the separaged ~- ~- 4-je~ - ~ p ! e , -f -*-heregu-a--V h ~ - dee~e~ meaning it ~hould be te~ted on the sample of events generated by the elementary subprocesses qq, qq9, qqYg... taken separately and not on the whole sample of events where the overlap of contributions coming from different subprocesses might spoil the regularity. Following this idea we decided to study, some time ago, MD's in Monte Carlo experiments on q~ and 9g systems by using JETSET 6.3 both at partonic and hadronic le~el and we were lead to the formulation of the Generalized Local Parton Hadron Duality (GLPHD) hadronization prescription (8) [8]. The result of DELPHI collaboration is consistent with this view. It shows that after separating with JADE algorithm (lh,.,, = 0.04) 2- 34-jets contributions the NB regularity holds at LEP energy on all separated samples with good x~/NDF. N B regularity e~sults from data more fundamental than it ~as anticipated by Monte Carlo ezpeeiments and theoretical sF.culations as will be discussed in the following.
By comparing 2-jet data sample at c.m. energy vt~ = 29 GeV (HRS) and x/~ = 91 GeV (DELPHI) (see Table I) we find that the average number of hadronic clans ~r/~stays constant within errors for rapidity intervais Ye = 0.5 and F¢ = 1.0 and approzimatei~ constant for larger intervals. This fact is an important one. It tells us that the constancy of frs is more pronounced in the 2-jet sample than in the whole sample. More elementary structures favor an higher degree of simplicity. Following previous studies on q~ and ~9 systems in Monte Carlo experiments by using JETSET 6.3, on lepton-ha&on deep inelastic scattel"ing and oa hadronhadron collisions we propose now to explore via GLPHD the final parton level (label p) in e+c - annihilation at the same above energies in the 2-jet sample start';ng from the final hadron level (labd h). We are aware of the fact that GLPHD was proposed in q,~ system simulated by JETSET 6.3 for c.m. energies larger than 200 GeV. The suggestion comes from the success of our study on EMC data in deep inelastic muon-proton scattering [23l, where the application of GLPHD lead us to discover - - in spite of the relatively low c.m. energy
an higher degree of si:,,p-:.ci~.yat. the par~onic level than al the hadronic level %%Lefound that d~fferentlv from her of ha_dronMc e!ans~ ~h~ is only app~_;m~tely constant with energy for rapidity intervals y¢ = 0.5,1,1.5, at partonic level the corresponding average number of partonic clans, Rp, is constant with energy toithin ermrs in the same intervals: furthermore in the available c.m. energy range of the hadronic system (from 6-8 GeV up to 18-20 GeV) the density of partonic clans per unit of rapidity sta~s constant, i.¢. (A2C~/Ay),p -~ 0.7 for Ay _> 1 and Ay < 3. The application of GLPHD to 2-jet sample of events in e+e - annihilation shows that Np is again constant toith enemy within errors in all explored fixed rapidity intervals. Also the partonic clan density per unit of rapidity remains constant for Ay _> 1; the only difference concerns the m,.merical value of the constant i.e. (ANp/Ay),+e- ~- 1, which is higher by 30% in e+e - annihilation than in deep inelastic pp scattering. In addition as the energy increases from 29 GeV to 91 GeV the plateau tends to widen in y, whereas its height ( A ~ / A y ) stays constant. Once again an higher degree of simplicity is showing up at the partonic level obtained by using GLPHD prescription. We call for test of variations of ( A N a / A y ) for Ay < 1. They might result to be interesting for studying intermittent behavior. The above results are improved by going, in the JADE algorithm, to smaller y,,~i,~. This fact suggests that the validity of the regularity may become a criterion for selecting y,~i,,. The effect of the onset of hard gluons on clan structure parameters is visible in 3- 4-jets sample of events (Table II). It is interesting to point out that the observed constant density of partonic clans agrees with the idea of independent production of clans, which was assumed for interpreting the observed wide occurrence of NB regularity. Secondly, the density of one clan per unit of rapidity seems not to result in the production of a too large number of clans. By inspection of tables we see that the number of partonic clans is not as large as at hadronic level and maybe, in view of that, one should not be so surprised
A. Giovannini et al./ Multip!icity distributions in high energy collisions
by the abunde~nee of v]ans at hedronic level,
Conclusions We have discussed the importavt phases in the evolution of our understanding of MD's in high energy collisions. The need for an integrated description of MD's and correlation phenomena has emerged from the intermittent behavior discovered in small rapidity intervals. The occurrence of NB regularity in 2- 3- 4-jets sample of events in different rapidity intervals in e+e - annihilations as seen by the DELPHI collaboration explains the shoulder structure. The finding that clan density per unit of rapidity is energy independent within errors at the partonic level constructed via Generalized Local Parton Hadron Duality from 2-jet flats sample of events, is striking. Both facts, if confirmed in other reactions and at different energies, point in the direction of a unified description of all collisions in terms of elementary QCD parton showers. In this respect, theoretical and experimental studies of single parton and hadrons showers become of particular interest. In our opinion, experimentalists should be encouraged to concentrate their efforts in disentangling 2- 3- 4-jets uontributions in the total charged hadron MD's whenewr possible in order to analyse data at a more fundamental level. Theorists should provide simplified single parton showers models based on a correct kinematical framework and inspired by essen~iais of QCD (see for an example [24]), in order to provide experimentalists w;th simple tools to analyse the complex structures emerging from their data. Acknowledgements Two of us (A. G. and S. L. ) would like to express their gratitude to the Organizin 8 Committee for the splendid atmosphere which was crested at this meeting.
!21
F_e!%renec~
2. Z. Koba, H. B. Nielsen and P. Olesen, Nucl. Phys. B40 (1972) 317. 3. It. P. Feynman, Phys. Rev. Lett. 23 (1969) 1415. 4. G. J. Alner et al., UA5 Collaboration, Phys. Lett. B138 (1984) 304; G. J. Alner et M., UA5 Collaboration, Phys. Lett. B167 (1986) 476. 5. A. K. Wr6blewski, "Soft hadron physics", plenary talk at XXVth Int. Conf. on High Energy Physics, Singapore, 1990, Warsaw University Preprint IFD[10/1990. 6. N. Schmitz, "Multiparticle Dynamics", in Proceed-
ings of Multiparticle Dynamics, Festschrift for L~ou Van Hove, La Thuile, Italy, Eds. A. Giovannini and W. Kittel, World Scientific, Singapot'e 1990, p. 25, and references quoted therein. 7. A. Giovannini et al., II Nuovo Cimento 24A (1974) 421; A. Giovannini et el., I1 Nuovo Cimento 38A (1977) 38. 8. L. Van Hove and A. Giovannini, Acts Phys. Pol. B19 (1988) 917. 9. M. Garetto, A. Giovannini, T. Sj6strand and L.
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A. Giov,~nnini et al./Multiplicity distributions in high energy collisions
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/1Qfl~
Oflfl.
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_
I l n O O ~ 1• £ ~ o _ o l •
n ~ l
24. R. Ugoccioni and A. Giovannini, "Virtuallty Evolution and Rapidity Structure of a Simplified Parton Shower", DFTT 14/01, to be published in Z. Phys. C, Particles and Fields.
A. Giovannini et al./ Multiplicity distributions in high energy collisions T a b i e L Clan atracr_.ure parameters :-or *he 2-jet s~mpie of events are compare~ e~ ~-eeg_=!: ( :7-=, _ ) l:--:-~ iv
•
- • a
V/~--91GeV 0.5 1.0 1.5 2.0 2.5 3.0 all
1.77 3.45 5.06 6.57 8.22 10.24 16.05
2-jet Ye
V~--91GeV
0.5 1.0 1.5 2.0 2.5 3.0 all
0.94 1.92 2.93 3.93 4.97 6.11 8.76
'~e,S
q'~-29GeV
4- 0.i4 4- 0.13 ± 0.12 ::i: 0.11 4- 0.13 ± 0.17 ::l: 0.59
1.86 4- 0.10 3.58 4- 0.12 5.56 4- 0.19 7.71 4- 0.26 9.96:1:0.62
]~rp v~=29GeV
± 0.08 :::E0.08 ± 0.09 4- 0.08 4- 0.10 -4- 0.13 4- 0.36
0.99 1.95 3.02 4.14 5.20
4-0.06 :t: 0.37 :l: 0 , i l ::t::0.14 4- 0.33
v/~=91GeV 1.15 1.26 1.40 1.54 1.59 1.53 1.21
44444+ 4-
v/~--29GeV
0.14 0.08 0.06 0.05 0.04 0.04 0.07
1.135 1.191 1.196 1.163 1.092
± 0.005 4- 0.005 ::t: 0.004 ± 0.004 :t: 0.004
~,e ,p V~ - 9 1 G e V 1.07 1.14 1.21 1.29 1.31 1.28 1.11
± 0.14 4- 0.07 4- 0.05 4- 0.04 4- 0.03 4- 0.04 4- 0.06
V~--29GeV 1,071 ± 0.004 1.098 :t: 0.003 1.101:1:0.003 1.083 :::E0.003 1.047 :t= 0.003
T a b l e I I . Clan structure parameters for the 3- and 4-jet sample of events are compared at hadronic (Ns, fie,s) and at partonlc (Np, file.p) level. 3-jet !/e 0.5 1.0 1.5 2.0 2.5 3.0 all
3.71 6.96 9.32 11.44 13.85 16.35 19.50
4- 0.14 :f:: 0.17 4- 0.19 ::1:0.21 4- 0.29 + 0.41 :t: 0.73
5.81 11.66 18.01 21.31 24.28 27.16 27.27
-4- 0.61 ± 1.00 4- 2.04 4- 2.27 ± 3.44 ± 5.77 ± 5.81
2.10 4.09 5.63 6.94 8.27 9.49 10.87
4- 0.09 4- 0.12 4- 0.15 4- 0.10 ± 0.19 ± 0.26 ± 0.47
1.33 1.46 1.58 1.60 1.53 1.41 1.27
4- 0.08 :::1::0.06 4- 0.06 4- 0.05 ± 0.05 :t= 0.06 ~ 0.05
1.17 1.24 1.31 1.32 1.28 1.22 1.14
+ 0.07 4- 0.05 4- 0.05 4- 0.05 ~ 0.05 4- 0.05 ~ 0.07
3.37 ::E:0.43 6.87 ± 0.73 10,3 4- 1.4 12.1 ::J: 1.5 13.4 ± 2.1 14.5 4- 3.3 14.6 4- 3.4
1.41 1.48 1.34 1.34 1.24 1.15 1.15
:t:: 0.25 4- 0.21 4- 0.25 4- 0.23 4- 0.28 :i: 0.37 4- 0.37
1.21 1.25 1.18 1.17 1.12 1.08 1.08
4- 0.20 :::E0.21 ± 0.23 ::1::0.22 4- 0.26 ::1:0.35 4- 0.36
4-jet Ye 0.5 1.0 1.5 2.0 2.5 3.0 all
123
Nuclear Physics B (Proc. Suppl.) 25B (1992) 119-123 Nartb.-ilol!a~d
MULTIPLICITY DIST~UTIOr~S
A. Giovannini*,
S. Lupia*,
~,.'" n t ~ , r t ~z,,r~ttI= x ~ u . , b t ~ I u N ~
R. Ugoccioni
Dipartimento di Fisica Teorica, Universit~ di Torino and INFN, Sezione di Torino, Italy
We discuss the important phases in the evolution of our understanding of multiplicity distributions in high energy collisions with particular emphasis to intermittent behavior and shoulder structure problem.
1. M u l t i p a r t i c l e p r o d u c t i o n : m a i n facts
Much of the extensive work done in the last 30 years on multiparticle production in high energy collisions can be regarded as a continued experimental search for correlations among produced particles and a succession of theoretical and experimental attempts to explain them in dynamical terms. A substantial part of this work was devoted to multiplicity distributions (MD) where correlations are revealed by non-Poissoniar behavior. The important phases in the evolution or"our understanding of MD's in Jfigh energy collisions are summarized in the following and can be divided in two eras. The first era was dominated by the study, in the accelerators region up to ISR, of the energy dependence of the total MD in haxlron-hadron collisions. The second era began when UA5 collai~oration decided to analyze MD's in/~3 annihilation in different pseudo-rapidity interv£o ~t c.m. eneigics up to ~ = 900 GeV. First Era:
i) Wr6blewski regularity [1]. It says that D/(n) in full phase space stays approximately constant as the c.m. energy increases; (n) is the average charged multiplicity and D = x/(n 2) - (hi 2 the dispersion for the
reaction under investigation. The occurrence of this behavior was interpreted as an indication that charged particles are independently produced in full phase space (absence of correlation among produced particles).
ii) KNO scaring [2]. The probability of producing n charged pactides in full phase space, P,s, is expected to be n universal function of n/(n} as the energy increases, i.e.
This relation shows that once ~(n/(n)) has been determined from experimental observations, data ou MD's from new experiments should fall on the same curve.
This prediction on P , is very striking; it is related to the validity of Feynman scaling [3], which says that the charged particle multiplicity density per unit of rapidity, dn/dy, should stay constant as the energy increases (the plateau in dn/dy versus y widens with energy).
iii) KNO scaling violation at S/~S collider in full phase space [4]. The experimental observation of the UA5 collaboration has shown that what was considered a well established asymptotic regime in hh collisions was indeed n transition region to a new energy domain:
* The talks presented by A. G. (Multiplicity Distributions in High En-.rgy Co~lislons) and by S. L. (Intermittency and Clan Structure Analysis) converged in a single contribution for practical reasons. Work supported in part by M.U.R.S.T. (Italy) under Grant 1990.
116
A. Giovannini et al./ Multiplicity distributions in high energy collisions
an i~__,portant ~varni_-~gt~ remember t-~.at shouid prevent experts from drawing too m~c~ ~-i-~si.~-s from pre~e~:~ experi--~.-ent*L:-~e+e - an,=iNia~i-n [~]o . . . .
.!
Y3__.
/) Negative Binomial (NB) shape of total MD's and of MD's in symmetric rapidity windows with the related "clan structure", not only for hadronic collisions but also for e+e - annihilation and deep inelastic leptonnucleon collisions [6]. According to NBMD the probability of producing n charged particles, Pn(l~)(~, k), depends on two parameters, the average charged multiplicity ~ and k, which is related to the dispersion D by D2 / ~ 2 = 1/k + 1/f~. p(Ne) is defined by the following relation
p(NB)
(n + 1~ "+-----!1 a + bn -' p ( N B ) =
(2)
with a - ~k/(ft + k) and b = fi/(~ + k). It should be noticed that relation (2) characterizes a class of distributions. In fact for b = 0 it reduces to a Poisson MD; therefore b measures deviations from Poisson behavior. For a = b ene gets the geometric MD; when a --* k and b --* 1 relation (2) leads to a gamma MD and finally for a -* 0 to a logarithmic MD. It is to be pointed out that the occurrence of NB behavior in full phase space was already noticed in the accelerator region and its relevance for a possible early KNO scaling violation emphasized [7]. The interpretation of NB regularity led to the introduction of the concept of 'clan'. Clans are group of particles of common ~ncestry; they are independently produced by assumption and each clan contains at least one particle by definition; particle correlations are exhausted within each clan. In terms of the MD generating function
aNs(~,k;~) = ~ ~"P.(~B)(~,t)
(3)
the average number of clans, ~ , can be defined by GNB(fi,k; z) = e ~[a~(b;z)-l]
(4)
and
is the generating function of a logarithmic MD, which I S *.... h h MI) ol the average clan; tLe average number of particles belonging to an average clan will be of course ~
=
~/~7.
(7)
ii) Occurrence of "NB regularity" in the QCD Parton Shower Models, which were known to give the best description of c+e - annihilation data, both for the fined state hadrons and for the final partons, lea ii~ag to a genereJ;.zcd property of local parton hadron dueeity (GLPIID) [8]. GLPHD says that n-patton and nhadron inclusive rapidity distributions, Q~) and Q~), are related by the following simple equation:
Q(a)(y*,... ,y,~) -~ p " Q ~ ) ( y , , . . . , y , )
(8)
with p ~ 2Qo/(1 GeV), Qo being the parrot, virtuality cut-off at which the shower development is stopped and hadronizatlon takes place. The dependence of the abow, regularities on patton virtueeity cut-off Q0 (Qo = 1 - 2GcV) in the context of the Lund Patton Shower Model was also investigated [9]. It was found that Lund hadronization prescription leads to infrared stable results. Finally it was discovered that the angulax ordering option incorporated in the model affected mean multiplicity only, leaving substantially unchanged the other parameters of the distributions. An instructive result which rises some interesting questions on the relevance of angular ordering on higher moments of the MD. It is interesting t i n t the clan concept can be extended to the parton level, where clans can be understood as bremsstre.hlung gluon jets [8]. In solving the Konishi-Ukawa-Veneziano equations, within the exclusive approach [10], one finds that the MD of gluons generated by a quark jet follows the NB MD with the average number of gluons (ne) q given by
where = - b In(l - b) = k l n ( l + ~ )
(5)
{ns) q = -~(e Ay - 1) ,at
(9)
A. Giovannini et al./Multiplicity distributions in high energy colli:4ons
! !7
colors and c a fixed cut-off to regularize divergences; y
fcrentiy from what happens ~o a jet, From this point of v{ew clan concept is more prim-~-ive tha_-.-1jet con~ . _=~ _:--:--.°a.._:-_ _-:=- :=*. : . . , . d~fi~fi-iem Far thermore parton shower recalls intuitively the cn,rnA.
is the standard evolution p,rameter
ing mechanism which seems to be the main dynamical
where
A
------ ( H ?
......
-o~'-~'-
"~. . . . . . . . . . .
s;-~-'' process g ~-~ 9 + 9 and A ~ -~-e!. . . . . . . . . . . . . . . .
6 [ In(Q2/A2) ] Y = 11Af, - 2Af! In [In(QI/A2) l with Af! number of flavors.
(10)
Q~ is the initial parton
virtuality, Qo2 the infrared cut-off and A the scale parameter. The dispersion can also be calculated: one finds D2
= ~Ae
ay,teAy
-1)
(11)
Accordingly
feature emerging from the Markofllan nature of the solutions of the KUV equations in the exclusive approach [10].
iii) Intermittency in the sense of the continued growth of normalized factorial moments of the MD's in decreasing rapidity windows [11]. Intermittency renewed the interest on measurement and interpretation of 2- and 3-particle correlations, including BoseEinstein interferometry of equal charge mesons, and es-
1
A = ~
and
# = .4y
(12)
In the 'pure bremsstrahlung' limit A --, 0, so that eq. (9) gives
tablished a useful contact between studies of MD's and correlations. Until quite recently there was practically no contact between them. The two topics were brought together when various authors attempted to explain the
which proves our statement.
experimental results on intermittency in terms of the kno,:n 2-particle correlations [12,13], while others noted that in many cases the factorial moments measured in
In conclusion, in the present phase two important facts are worth mentioning.
intermittency studies obey approximately the relations which are implied by NB behavior of the MD's in small
a) The existence of a simple link between the NB regularity for final charged hadrons MD's in full phase
rapidity windows [14]. These recent developments allow for the first time to
space and in symmetric rapidity windows, and the corresponding final parton~ MD %in qq and gg systems obtained by means of a Monte Carlo "experiment" based on the Lund Patton Shower as implemented in the pro-
attempt an integrated description of the overal~ correlation structure of multiparticle production and of its dynamical interpretation in terms of QCD parton shower
gram JETSET 6.3. This link is what we call GLPHD and it has been expressed in its more general form by eq. (8). It is a stronger condition than the standard
scription can only be tentative, but the abundant data becoming available at LEP and soon at HERA should lead to improvements and refinements. Through con-
one on local parton hadron duality.
cation of clans with bremsstrahlung gluon jets or patton showers originated by independent gluons follows.
frontation with these data one can expect the QCDbased description of e+e - annihilation and semileptonic collisions to get tested more severely and corrected where necessary, especially in the non-perturbative areas of soft gluon emission and of final patton hadronization, for which neither analytical methods nor lattice QCD computations are available so far.
We prefer to talk of parton showers in connection with clans, since a clan ~ as in general a parton shower is not affected by cuts in transverse momentum, dig
iv) Appearance of the shoulder structure in p/3 at c.m. enexgy V/~ = 900 GeV in large rapidity windows [15], and in e+e - annihilation at c.m. energy ~/~ = 91
(ng)~ = A3' =: N
(13)
b) The occurrence of NB behavior for glaon population initiated by a quark jet in the early stages of their evolution when perturbative QCD at the leading log level in the axial gauge is still applicable. The identifi-
mechanism with GLPHD. For the moment such a de-
A. Giova~niniet al./ Multiplicity distributionsin high energy collisions
118 ~ey
i~
sVrn,-~,_egr:_,c# ...... ra_Di~i~g
~_vii%=lowR :=~._~_
_ . . . . r~r,e~.2~_=
0o(~ . . . . . . vA:
d . . . . . . . i~, the ~,,~s ~h. . . . ¢ the MD. A.'oparentiv ~'° f
der structn_~ in e+e - ann~ihil_ation_w ~ ,_,nderst_,m-d in terms of 2- 3- 4-jets contributions to the whole sample of data [17]. As a consequence, NB regularity is found
where Q,(y,
in the next two sections.
d,,
f}
,'..
. . .
~Av)
roy
to be valid at a more fundamental level. In view of the present interest on points (iii) and (is) of the second era, we will discuss them mrre extensively
,4,,.
....
,y,) -
1
dqa
-- aia,~ dyl . . .
dy,
(16)
and n is the number of particles in a rapidity interval of finite width 611. In order to introduce correlation functions, we define the unnormalised factorial cumulants i'(~y)"
2. I n t e r m | t t e n e y
i'(6y) = ~ dy,...dy, 0,(~,,,...,~,,)
In the last few years the study of intermittent behavior assumed increasing relevance in Multiparticle Dynamics. Attention has been called on this new topic when three experimental groups [11] found strong local density fluctuations in the rapidity distribution of charged hadrons produced in high energy collisions. In order to give a quantitative characterization of the phenomenon, it was proposed [18] to study the normalized factorial moments of multiplicity distribution in restricted domains of rapidity I/. Intermittency is forreally defined by the relation:
F'(6~) o~ (6~)-*,
for ~ - .
0,
/ , > 0.
04)
So intermittency essentially means that factorial moments are divergent in the limit of high resolution. This power law is natural in self-similar cascading processes. Vvith this analysis, intermittent behavior was found in all multihadron producing reactions. Besides, stealdexd Monte Carlo programs, contrary to what occurs in e+e - annihilation, fail to reproduce the effect in leptonhadron and hadron-hadron collisions (for an experimental review see [19]). In this section we show that particle correlations and m,.ltiplicity distributions, as already stressed, can be brought together by iutermittency. The study of normalized factorial moments gives information on the correlations between final particles. In fact the unnormalif.ed factorial moments ~'g($y) are directly related to the inclusive rapidity distribution
(17)
Y
where Og(~1 . . . . . ~e) is the @-particles correlation function. Factorial cumulants vre simply related to factorial moments via a standard cluster expansion:
(18) -J -' \ / * i ~ 1
So corrdation functions are strictly linked to factorial moments. The power law behavior expected from intermittency requires the presence of s singularity in the correlation functions for small rapidity d~fferences. Intermittent behavior is also related to MD. In order to study MD, we define the exclusive rapidity distribution P,,Oh,... ,Yn) as the probability to find exactly n particles with rapidity y x , . . . , y ~ . The exclusive rapidity distribution P n ( Y l , . . . , Y,) and the inclusive one Q,~(yI,... ,yn) satisfy the following relation [20]:
~'d~,,..-, y.) = Q ~ ( ~ , . . . , ~ . )
+ £ (~ v'-----1
lP'.
f Q,~+,(yt.... , y,~,y~,... ,y~)dy;
d'
--
(19) We can express the unnormalizod factorial moments Fq(6y) in terms of the probability to find n particles in one interval of width 6y, P,~(6y): ~,($,) = £ ~vt=:Q
where
(n +.~)'P~+,~(6y)
(20)
A. Giovannini et al,/Multiplicity distributions in high energy collisions
,,~,~
/ g>ooog>->4> .... ,>)
(2_~)
ig .a =o_¢ very i~_erea_i~g _o notice tha~ e=perln~a~t~.i factorial moments are consistent with factorial moments calculated from NB regularity [14,13]. This fact suggests to extend the clan structure ~n~lysis to restricted domain of rapidity in order to investigate intermittent behavior. In terms of factorial moments generating function
(f.m.g.f.) P6~(u)
119
possible connection, between c'_,ana~aivsis - - d intern~;tte_ncyo =-->~:~"~::~ege:::t e'- v~e:% ~=~::~_'~t:_~:_-.° :: = w=y --= e_-_plore multiparticle production in restricted domMns of phase space and its occurrence in all reactions supports the idea of a unified picture for multihadron production based on selfsimilar cascade processes. 3. T h e s h o u l d e r s t r u c t u r e p r o b l e m The DELPHI collaboration discovered in e+e - an-
defined by:
(23)
nihilation at c.m. energy V~ = 91 GeV that NB diJtrl. bntirn reproducea only the groga ahape of MD's for final charged particles and giv-~s poor fits both in full phase space and in symmetric rapidity windows [y[ < Yc [16]. This fact is due to the presence of a shoulder in the MD's which is more evident in yc = 1.5 and 2.0 and
where P6y(u), Hs~(u) and g6y(u) are reapectively the
similar to that seen by UA5 collaboration at p/~ collider at c.m. energy v/~ = 900 GeV. However by using clan
P~y(u) - ~
Pn(~y)(1 + n)'*
(22)
vt-----S
we can define the d a n structure:
P,,(u) = H,,[g,,(u) - 1]
One can easily extend dam structure analysis in the
structure parameters on the same sample of events the average number of hadronic clans, Nh, was found to be ~pprozimu~ely constan~ wi~h ene~jy for fized rapid-
limit of high resolution with a simple ansatz [21] which
it 9 intereab. The two results seemed in contradiction.
relates the f.m.g.L in one interva/of width ~;y to those in a larger interval Ay with ~y C Ay:
Clapt structure analysis has been proposed in fact as a possible interpretation of the wide occurrence of the NB
f.m.g.f, of p~ticles, clans and particles in an average clan.
{ ll,,(u) -- H~,,(Ou)
g~,(u) ~s(su)
O,S ~
regularity in hh, lh reactions and - - at lower energy - e+e ~ annihilation; it is indeed rather surprising to find
[0,1];
=
(24)
In this way we get:
d~P~(u)/du ~I~=0 Fq(~y) = iap~,(u)/d~l,,=~iq ._- (0[-I')~-,~ + 0 ( 0 ~-')
(25)
w h e r e / t ~ are the unnormalized factorial moments of the cla~ distribution and gq the normalized factorial moments of the particle distribution in an average clan. Assuming that the average number of particles in the interval ~y, ~S~s is linear in 6y, we easily obtain the
out that the most appealing property of clan structure analysis of NB regularity was valid at LEP energy in spite of the violation of the regularity itself. One way out was to remark that the reproduction of the gross shape of the MD's was sumclent in order to guarantee the constancy of/9a with energy for fixed rapidity intervals. Clearly this way is unsatisfactory, we don't see in it any deep physics.
power law (14) of intermittency. With ansatz (24), the clan structure is consistent
A second possibility went back to the discovery by HRS collaboration of NB regularity in e+e - annihilation at c.m. energy v/~ = 29 GeV [22]. The important point to be recalled is that HRS collaboratio~ found NB fits with better X~/NDF for the 2-jet sample of events
with the intermittent behavior for factorial moments. It is still to be investigated how realistic this ansatz is. Here we want simply to focus the attention on the
than for the whole sample. The suspect is that contamination of qq9 events might hide the regularity which shows up fully at the q~ jet level. Accordingly it is more
120
A. Giovannini et al./ Multiplicity distributions in high energy collisions
natural to proceed as the DELPH1 cohaboration did subsequently [17] and to study at L.F,P energy the separaged ~- ~- 4-je~ - ~ p ! e , -f -*-heregu-a--V h ~ - dee~e~ meaning it ~hould be te~ted on the sample of events generated by the elementary subprocesses qq, qq9, qqYg... taken separately and not on the whole sample of events where the overlap of contributions coming from different subprocesses might spoil the regularity. Following this idea we decided to study, some time ago, MD's in Monte Carlo experiments on q~ and 9g systems by using JETSET 6.3 both at partonic and hadronic le~el and we were lead to the formulation of the Generalized Local Parton Hadron Duality (GLPHD) hadronization prescription (8) [8]. The result of DELPHI collaboration is consistent with this view. It shows that after separating with JADE algorithm (lh,.,, = 0.04) 2- 34-jets contributions the NB regularity holds at LEP energy on all separated samples with good x~/NDF. N B regularity e~sults from data more fundamental than it ~as anticipated by Monte Carlo ezpeeiments and theoretical sF.culations as will be discussed in the following.
By comparing 2-jet data sample at c.m. energy vt~ = 29 GeV (HRS) and x/~ = 91 GeV (DELPHI) (see Table I) we find that the average number of hadronic clans ~r/~stays constant within errors for rapidity intervais Ye = 0.5 and F¢ = 1.0 and approzimatei~ constant for larger intervals. This fact is an important one. It tells us that the constancy of frs is more pronounced in the 2-jet sample than in the whole sample. More elementary structures favor an higher degree of simplicity. Following previous studies on q~ and ~9 systems in Monte Carlo experiments by using JETSET 6.3, on lepton-ha&on deep inelastic scattel"ing and oa hadronhadron collisions we propose now to explore via GLPHD the final parton level (label p) in e+c - annihilation at the same above energies in the 2-jet sample start';ng from the final hadron level (labd h). We are aware of the fact that GLPHD was proposed in q,~ system simulated by JETSET 6.3 for c.m. energies larger than 200 GeV. The suggestion comes from the success of our study on EMC data in deep inelastic muon-proton scattering [23l, where the application of GLPHD lead us to discover - - in spite of the relatively low c.m. energy
an higher degree of si:,,p-:.ci~.yat. the par~onic level than al the hadronic level %%Lefound that d~fferentlv from her of ha_dronMc e!ans~ ~h~ is only app~_;m~tely constant with energy for rapidity intervals y¢ = 0.5,1,1.5, at partonic level the corresponding average number of partonic clans, Rp, is constant with energy toithin ermrs in the same intervals: furthermore in the available c.m. energy range of the hadronic system (from 6-8 GeV up to 18-20 GeV) the density of partonic clans per unit of rapidity sta~s constant, i.¢. (A2C~/Ay),p -~ 0.7 for Ay _> 1 and Ay < 3. The application of GLPHD to 2-jet sample of events in e+e - annihilation shows that Np is again constant toith enemy within errors in all explored fixed rapidity intervals. Also the partonic clan density per unit of rapidity remains constant for Ay _> 1; the only difference concerns the m,.merical value of the constant i.e. (ANp/Ay),+e- ~- 1, which is higher by 30% in e+e - annihilation than in deep inelastic pp scattering. In addition as the energy increases from 29 GeV to 91 GeV the plateau tends to widen in y, whereas its height ( A ~ / A y ) stays constant. Once again an higher degree of simplicity is showing up at the partonic level obtained by using GLPHD prescription. We call for test of variations of ( A N a / A y ) for Ay < 1. They might result to be interesting for studying intermittent behavior. The above results are improved by going, in the JADE algorithm, to smaller y,,~i,~. This fact suggests that the validity of the regularity may become a criterion for selecting y,~i,,. The effect of the onset of hard gluons on clan structure parameters is visible in 3- 4-jets sample of events (Table II). It is interesting to point out that the observed constant density of partonic clans agrees with the idea of independent production of clans, which was assumed for interpreting the observed wide occurrence of NB regularity. Secondly, the density of one clan per unit of rapidity seems not to result in the production of a too large number of clans. By inspection of tables we see that the number of partonic clans is not as large as at hadronic level and maybe, in view of that, one should not be so surprised
A. Giovannini et al./ Multip!icity distributions in high energy collisions
by the abunde~nee of v]ans at hedronic level,
Conclusions We have discussed the importavt phases in the evolution of our understanding of MD's in high energy collisions. The need for an integrated description of MD's and correlation phenomena has emerged from the intermittent behavior discovered in small rapidity intervals. The occurrence of NB regularity in 2- 3- 4-jets sample of events in different rapidity intervals in e+e - annihilations as seen by the DELPHI collaboration explains the shoulder structure. The finding that clan density per unit of rapidity is energy independent within errors at the partonic level constructed via Generalized Local Parton Hadron Duality from 2-jet flats sample of events, is striking. Both facts, if confirmed in other reactions and at different energies, point in the direction of a unified description of all collisions in terms of elementary QCD parton showers. In this respect, theoretical and experimental studies of single parton and hadrons showers become of particular interest. In our opinion, experimentalists should be encouraged to concentrate their efforts in disentangling 2- 3- 4-jets uontributions in the total charged hadron MD's whenewr possible in order to analyse data at a more fundamental level. Theorists should provide simplified single parton showers models based on a correct kinematical framework and inspired by essen~iais of QCD (see for an example [24]), in order to provide experimentalists w;th simple tools to analyse the complex structures emerging from their data. Acknowledgements Two of us (A. G. and S. L. ) would like to express their gratitude to the Organizin 8 Committee for the splendid atmosphere which was crested at this meeting.
!21
F_e!%renec~
2. Z. Koba, H. B. Nielsen and P. Olesen, Nucl. Phys. B40 (1972) 317. 3. It. P. Feynman, Phys. Rev. Lett. 23 (1969) 1415. 4. G. J. Alner et al., UA5 Collaboration, Phys. Lett. B138 (1984) 304; G. J. Alner et M., UA5 Collaboration, Phys. Lett. B167 (1986) 476. 5. A. K. Wr6blewski, "Soft hadron physics", plenary talk at XXVth Int. Conf. on High Energy Physics, Singapore, 1990, Warsaw University Preprint IFD[10/1990. 6. N. Schmitz, "Multiparticle Dynamics", in Proceed-
ings of Multiparticle Dynamics, Festschrift for L~ou Van Hove, La Thuile, Italy, Eds. A. Giovannini and W. Kittel, World Scientific, Singapot'e 1990, p. 25, and references quoted therein. 7. A. Giovannini et al., II Nuovo Cimento 24A (1974) 421; A. Giovannini et el., I1 Nuovo Cimento 38A (1977) 38. 8. L. Van Hove and A. Giovannini, Acts Phys. Pol. B19 (1988) 917. 9. M. Garetto, A. Giovannini, T. Sj6strand and L.
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122
A. Giov,~nnini et al./Multiplicity distributions in high energy collisions
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/1Qfl~
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_
I l n O O ~ 1• £ ~ o _ o l •
n ~ l
24. R. Ugoccioni and A. Giovannini, "Virtuallty Evolution and Rapidity Structure of a Simplified Parton Shower", DFTT 14/01, to be published in Z. Phys. C, Particles and Fields.
A. Giovannini et al./ Multiplicity distributions in high energy collisions T a b i e L Clan atracr_.ure parameters :-or *he 2-jet s~mpie of events are compare~ e~ ~-eeg_=!: ( :7-=, _ ) l:--:-~ iv
•
- • a
V/~--91GeV 0.5 1.0 1.5 2.0 2.5 3.0 all
1.77 3.45 5.06 6.57 8.22 10.24 16.05
2-jet Ye
V~--91GeV
0.5 1.0 1.5 2.0 2.5 3.0 all
0.94 1.92 2.93 3.93 4.97 6.11 8.76
'~e,S
q'~-29GeV
4- 0.i4 4- 0.13 ± 0.12 ::i: 0.11 4- 0.13 ± 0.17 ::l: 0.59
1.86 4- 0.10 3.58 4- 0.12 5.56 4- 0.19 7.71 4- 0.26 9.96:1:0.62
]~rp v~=29GeV
± 0.08 :::E0.08 ± 0.09 4- 0.08 4- 0.10 -4- 0.13 4- 0.36
0.99 1.95 3.02 4.14 5.20
4-0.06 :t: 0.37 :l: 0 , i l ::t::0.14 4- 0.33
v/~=91GeV 1.15 1.26 1.40 1.54 1.59 1.53 1.21
44444+ 4-
v/~--29GeV
0.14 0.08 0.06 0.05 0.04 0.04 0.07
1.135 1.191 1.196 1.163 1.092
± 0.005 4- 0.005 ::t: 0.004 ± 0.004 :t: 0.004
~,e ,p V~ - 9 1 G e V 1.07 1.14 1.21 1.29 1.31 1.28 1.11
± 0.14 4- 0.07 4- 0.05 4- 0.04 4- 0.03 4- 0.04 4- 0.06
V~--29GeV 1,071 ± 0.004 1.098 :t: 0.003 1.101:1:0.003 1.083 :::E0.003 1.047 :t= 0.003
T a b l e I I . Clan structure parameters for the 3- and 4-jet sample of events are compared at hadronic (Ns, fie,s) and at partonlc (Np, file.p) level. 3-jet !/e 0.5 1.0 1.5 2.0 2.5 3.0 all
3.71 6.96 9.32 11.44 13.85 16.35 19.50
4- 0.14 :f:: 0.17 4- 0.19 ::1:0.21 4- 0.29 + 0.41 :t: 0.73
5.81 11.66 18.01 21.31 24.28 27.16 27.27
-4- 0.61 ± 1.00 4- 2.04 4- 2.27 ± 3.44 ± 5.77 ± 5.81
2.10 4.09 5.63 6.94 8.27 9.49 10.87
4- 0.09 4- 0.12 4- 0.15 4- 0.10 ± 0.19 ± 0.26 ± 0.47
1.33 1.46 1.58 1.60 1.53 1.41 1.27
4- 0.08 :::1::0.06 4- 0.06 4- 0.05 ± 0.05 :t= 0.06 ~ 0.05
1.17 1.24 1.31 1.32 1.28 1.22 1.14
+ 0.07 4- 0.05 4- 0.05 4- 0.05 ~ 0.05 4- 0.05 ~ 0.07
3.37 ::E:0.43 6.87 ± 0.73 10,3 4- 1.4 12.1 ::J: 1.5 13.4 ± 2.1 14.5 4- 3.3 14.6 4- 3.4
1.41 1.48 1.34 1.34 1.24 1.15 1.15
:t:: 0.25 4- 0.21 4- 0.25 4- 0.23 4- 0.28 :i: 0.37 4- 0.37
1.21 1.25 1.18 1.17 1.12 1.08 1.08
4- 0.20 :::E0.21 ± 0.23 ::1::0.22 4- 0.26 ::1:0.35 4- 0.36
4-jet Ye 0.5 1.0 1.5 2.0 2.5 3.0 all
123