%Paper: 
%From: Boris Levtchenko <boris@sgi.npi.msu.su>
%Date: Wed, 19 Oct 1994 15:12:49 +0300 (MSK)

\documentstyle[fleqn,epsfig,12pt]{article}
\textheight 210mm
\textwidth 145mm
\topmargin -0.5cm
\rightmargin 2.0cm
\leftmargin -3.0cm
\def\preprint{}
%\setcounter{page}{3}
\setlength{\unitlength}{1mm}\thicklines
\newcommand{\nl}{\nonumber \\}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}


\renewcommand\baselinestretch{0.91}
\renewcommand\refname{\center \bf References}
% ************************************************************
% Changes in TheBibliography command
% ************************************************************
\def\thebibliography#1{\centerline{\bf References}
\list
 {\arabic{enumi}.}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
 \advance\leftmargin\labelsep
 \usecounter{enumi}}
 \def\newblock{\hskip .11em plus .33em minus .07em}
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\relax}
\let\endthebibliography=\endlist
% ************************************************************

\begin{document}
\begin{center}

\begin{flushright}
 INP MSU 94 -- 16/338\\
May 1994\\
\end{flushright}


\vskip 2cm


{\bf
WHAT ONE CAN LEARN ABOUT  THE QCD PARTON CASCADES
STUDYING THE MULTIPLICITY DISTRIBUTIONS AT HERA ?\footnote{Talk given
at the "Joint
International Workshop" on High Energy Physics
and Quantum Field Theory, September 16-22, 1993, Zvenigorod, Russia.
To be published in Proc. of the Workshop, MSU, 1994, Moscow }}

\vskip 1.5cm

{\large  Boris Levtchenko}
\vskip 0.7cm
{Institute of Nuclear Physics, Moscow State University,\\ 119899 Moscow,
 Russia}
\vskip 3cm
\end{center}

\begin{abstract}
In a brief form
new peculiarities of the multiplicity distributions predicted by higher
order QCD and their physical origin are discussed.
Several problems which can be studied at HERA  are formulated.
\end{abstract}
%\noindent{\small $^{1}$ E.mail address:boris@npi.msu.su\\

\newpage


\section{Definitions}


Multiplicity distributions (MD) of particles produced in high energy
collisions are the most typical and widely discussed characteristics
of the interaction dynamics. In a condensed form MD provide
information about the fluctuations of energy spent for multiple particle
production during collisions. The goal of the present paper is to review
briefly  the new features of the multiplicity distributions predicted
by higher order QCD.

 There are two complementary ways of dealing
with  multiplicity fluctuations:

-- studying the distribution
$P_{n}=\sigma_{n}/\sigma$ of events over the number of produced particles;

-- measuring the inclusive multiplicity correlators.

In practice, one uses often the normalized factorial moments
 $F_q$ and cumulants $K_q$ (for review see \cite{DeWolf}) defined as
\vspace{-2mm}
\ba
F_q &=& \sum_{n=0}^{\infty}n(n-1)...(n-q+1)P_{n}/\langle n\rangle ^q \nl
 &=& \frac{<n(n-1)...(n-q+1)>}{\langle n\rangle ^q}, \label{1}
\ea
\begin{equation}
K_q = F_q  - \sum_{m=1}^{q-1}C_{q-1}^{m}K_{q-m}F_{m}.  \label{2}
\end{equation}
\vspace{-2mm}
\noindent Here  $C_{q}^{m} = \frac{q!}{m!(q-m)!}$ are the
binomial coefficients and $F_{0}=F_{1}=K_{1}=1$.

These moments have an important advantage over the ordinary
moments.  The average shown in (1) implies mean value of the corresponding
expressions over the available set of experimental events.
In experiment, this averaging takes into account both statistical and
dynamical effects. If one assumes that random fluctuations due to  limited
number of detected particles are described by the Poissonian distribution
then the total average of factorial moments is equivalent to the dynamical
average of usual moment \cite {Bial}. Thus, the usage of factorial moments
 suppresses
statistical fluctuations due to finite number of particles in a given
region of the phase space.

In the Feynman
diagram's language, $F_q$ correspond to the set of all graphs while
cumulants $K_q$ describe the connected graphs only. The cumulants provide
the knowledge about the "true" correlations, non-reducible to the product
of the correlations of lower order.
At asymptotically
high energies the normalized factorial moments (as well as the ordinary
ones) do not depend on energy and are the functions of their rank only.
The higher is the rank  of the moment, the more sensitive it is to the "tail"
of the distributions at large $n$.  The steeper decrease of the
distribution at large $n$ leads to smaller values of high rank factorial
moments.

For  a theoretical analysis the study of the numerical series $P_{n}$
is convenient to replace by an
analysis of function "generating" it, namely the generating function (GF).
$F_q$ and $K_q$  are easily calculated if the
generating function  $G(u)$ is known
\begin{equation}
G(u) = \sum_{n=0}^{\infty}P_{n}(1+u)^n.
\label{3}
\end{equation}
\noindent Then
\begin{equation}
P_n = \left.\frac{1}{n!}\frac{d^{n}G(u)}{du^n}\right|_{u=-1}   \label{4}
\end{equation}
\begin{equation}
F_q = \left.\frac{1}{\langle n\rangle ^q}\frac{d^{q}G(u)}{du^q}\right|_{u=0}
\label{5}
\end{equation}
\begin{equation}
K_q = \left.\frac{1}{\langle n\rangle ^q}
\frac{d^{q}\ln G(u)}{du^q}\right|_{u=0}.
\label{6}
\end{equation}
\noindent Thus, the knowledge of  GF
gives us a possibility to calculate both the multiplicity distribution
and cumulant and factorial moments
i.e., (3)-(6) demonstrate mathematical equivalence of description of
multiplicity distributions by functions $P_n, F_q$ and $K_q$.
In \cite{Dr} it has been proposed to use the ratio of
cumulant to factorial moments
$H_q \equiv K_{q}/F_{q}$ which behaves in a qualitatively different way
for various distributions and is more sensitive to a special
form of  $P_n$ which one is unable to see clearly when just plotting
$P_n$ or even $F_q$ .
 That way, in a definite situation, either one of them or another can
be preferred due to the stronger sensitivity to the specific details.


\section{ Some properties of the multiplicity distributions on  experiment}


In pre-QCD time  Koba, Nielsen and
Olesen  published paper \cite{KNO} with a hypothesis about the scaling
properties of the multiplicity distributions at asymptotic energies
(KNO scaling).  If $z$ is the scaled multiplicity $z=n/\langle n\rangle$,
 then KNO scaling implies a universal form
\ba
\psi(z) = \langle n\rangle\,P_n \nonumber
\ea
for the multiplicity distribution.
During last 30 years the KNO-like behaviour of the
multiplicity distributions was experimentally confirmed in various types
 of  high
energy particle production processes except the data on
pro\-ton-anti\-pro\-ton interactions at the highest energies  $\sqrt{s} =
546$ and 900 GeV obtained by $UA5$ collaboration \cite{UA5} in CERN.

The negative binomial distribution (NBD)
\ba
G(u)&=& \left(1\,-\,\frac{u\langle n\rangle }{k}\right)^{-k} \nl
P_n &=& \frac{(n+k-1)!}{n!(k-1)!}\left(
\frac{\langle n\rangle /k}{1+\langle n\rangle /k}\right)^{n}
\,\left(1+\frac{\langle n\rangle }{k}\right)^{-k} \\
F_q &=& \frac{(k+1)\cdots (k+q-1)}{k ^{q-1}}\,,\ \ \
K_q \,=\, \frac{(q-1)!}{k ^{q-1}} \nl
H_q &=& \frac{(q-1)!}{(k+1)\cdots (k+q-1)}
\label{7*}
\ea
is another example of the distribution which
fits experimental ones pretty well both in full phase space and in smaller
phase space domains. It depends  on two parameters,
the average multiplicity $\langle n\rangle $ and a positive parameter
 $k$ describing
the shape of the distribution\footnote{ As it follows  from (\ref{7*})
\ $H_q$ is ever positive and tends to zero at
high ranks as $q^{-k}$. For the Poisson
distribution $H_q$ is identically equal to zero (except of $H_{1}=1$ ).}.
There are many ways to come to NBD. Here we
will mention only two classes of mechanisms proposed to generate NBD,
(partial) stimulated emission \cite{CS,GVH} and cascading \cite{GVH}.


\section{ What does QCD  tell us about the multiplicity
distributions? }

The KNO hypothesis was strongly
 supported by QCD when the equations for generating functions were solved
 in the so-called double logarithmic approximation (DLA).
DLA happens
 to be too crude however for making reasonable predictions even for
asymptotically high energies: the predicted KNO shape of the distribution
appeared to be much wider than the experimental one.  On the qualitative
level, DLA can be thought to overestimate cascading processes, ignoring
completely the ener\-gy-mo\-men\-tum balance since the energy of the
radiating
particles remains unchanged after a  soft gluon emission. Therefore it
seemingly
overestimates gluon multiplicities, the characteristic energy of partons
that multiplicate more actively etc.  The account of higher order
perturbative corrections must lead to a more accurate control over
 the parton splitting processes and energy conservation.

Such a program has been realized  (see \cite{BCM}, \cite{QCD}) in
the framework of the modified LLA (MLLA) by a generalization of the
standard LLA scheme  following the logic of the famous
Gri\-bov-Li\-pa\-tov-Alta\-relli-Pa\-risi
approach and including the exact angular ordering (AO) (instead of the strong
AO within DLA). Thus the system of the MLLA
integro-diffe\-rential equations
for quark and gluon GF has been derived.

A recent series of publication \cite{Dr}, \cite{YDok}-\cite{DrHwa} was
 devoted to solving
of these equations in the case of $e^+e^-$-collisions  with account
of different next-to-next-to leading (NNL) effects.
Corresponding
corrections can be looked upon \cite{CuyTes} as due to a more
accurate account of energy
conservation in the course of parton splitting.
For instance,
an approximation used in \cite{YDok} which allowed
to derive in the framework of gluodynamics  the analytical expressions
for the asymptotic behaviour of factorial moments and of the KNO function
improves the agreement with experiment by reducing substantially the width
of the theoretical distribution.
Cumulant and factorial
moments of the multiplicity distribution in perturbative gluodynamics
have been calculated in \cite{Dr}, \cite{DrN}.
The account of quark degrees of freedom \cite{DrLN} does not change
the essential
qualitative features of $F_q$, $K_q$ and they influence $H_q$ rather
weakly. The exact solution of the QCD equations for quark and
gluon GF in the case of fixed coupling has been obtained in \cite{DrHwa}.

One has noticed already that the ratio $H_q$ is more sensitive to a special
form of $P_n$ at large $n$ in comparison with $F_q$ (see Fig. 1).
\begin{figure}[t]
  \label{Fq_Hq}
   \begin{center}
   \vspace*{-1.cm}
   \mbox{\epsfig{file=lfig1u.ps,width=15cm,height=10.2cm}}
%   \mbox{\epsfig{file=lfig1.ps,width=9.5cm,height=10.2cm}}
   \end{center}
   \vspace*{-.5cm}
  \caption{  a) The factorial moments for QCD distributions
         [3] (solid line),  [10] (dotted line)
and the negative binomial distribution with $k=7.6$ (short-dashed line).
b) The ratio $H_q$ for the same distributions as in a). }
\end{figure}
\begin{figure}[htb]
\vspace{5. cm}
\caption{The ratio $H_q$ predicted by QCD [11].}
\label{fig:figure2}
\end{figure}
It is
seen \cite{DrLN} that all curves in Fig. 1a with qualitatively the same
behaviour are very close to each other for $q\leq 10$  but $H_q$
(Fig. 1b) demonstrate much stronger sensitivity to the assumptions used.
The most typical feature of the ratio $H_q$ predicted by QCD \cite{DrN}
is its  quasi-oscillating form  with variations the sign (Fig. 2).
Such oscillating behaviour of $H_q$ is a specific property of higher
order QCD.
Less complete account of nonlinearities in the equations for GF leads
 \cite{Dr}, \cite{DrLN}
only to one minimum with a very small value of $H_q$
\footnote{Recall that for NBD $H_q$ is  monotone declining,
ever positive and tending to zero at large ranks $q$ (Fig. 1b).}.

\begin{figure}[p]
\vspace{15.5 cm}
\caption{Experimental data [15] on $H_q$ for  a)--d) $e^{+}e^{-}$ \ \
      ($\surd s =$ 29, 34.8, 43.6, 91 GeV) and  e)--h)
     $hh$ ($\surd s=$ 62.2, 200, 546, 900 GeV) collisions. Lines are
 to guide the  eye.}
\label{fig:figure3}
\end{figure}
\begin{figure}[p]
  \label{QCDMC}
   \begin{center}
   \vspace*{-1.cm}
   \mbox{\epsfig{file=lfig4u.ps,width=25.cm,height=18.cm}}
%   \mbox{\epsfig{file=lfig4.ps,width=14.cm,height=14.cm}}
    \hskip -5cm
   \end{center}
   \vspace*{-.5cm}
  \caption{ The ratio $H_q$  due to the QCD MC codes (see the text).
Lines are to guide the eye.}
\end{figure}

The results of \cite{Dr} have initiated a search for
the peculiarities of $H_q$ from the experimental data.
According to calculations of $H_q$ from experimental data on
$e^{+}e^{-}$ - interactions in the energy range from 22 to 91 GeV
and on $hh$--collisions in the energy range from 24 to 900 GeV done in
\cite{Gian}, its behavior corresponds to the predictions of higher order QCD
and does not satisfy NBD-requirements.( A few examples are presented
in Fig. \ref{fig:figure3}\footnote{ It is surprising that the
theoretical results, which have been obtained for hard processes at
asymptotically high energies, are qualitatively well reproduced in
very wide range of energies not only for $e^{+}e^{-}$ processes but for
soft hadronic collisions as well.}.)
Therefore, the  conclusion that NBD is not appropriate for
description  of tiny features of multiplicity distributions in
particle production processes has been claimed in \cite{DrLN}, \cite{Gian}.

\section{ Monte Carlo Generators}


All the Monte Carlo (MC) generators for high energy physics \cite{Sjo}
 and, in particular,
those which simulate deep inelastic scattering (DIS)\cite{HERA} are
 based on the
leading logarithm (LL) picture with two body parton splitting
$a\rightarrow b\,+\,c$. However, as one mentioned in the previous section,
higher orders in the perturbative QCD are necessary for a proper description
of multiproduction at  high energies.

At present this can only be achieved through approximate  methods
such as different kinds of QCD cascade models for multiple parton emission
implemented in the MC generators (the Lund parton shower (PS) and the color
dipole model). LLA used in PS does not give a proper
treatment of hard emissions and a method was therefore developed to let the
single hard emission be controlled  by the exact $O(\alpha_{s})$ or
$O(\alpha_{s}^{2})$ \ QCD matrix elements and then add softer radiation
using the parton shower technique.

One can ask a question:\\
Are the above-mentioned improvements of the MC models
enough for a proper description of
 $H_q$ ? The answer one would think is obvious:
since LLA  is a base of PS one should not expect
the oscillatory behavior of $H_q$. However,
according to calculations of the correlators
  with the MC codes JETSET 7.3, ARIADNE 4.4
($e^{+}e^{-}$, $\sqrt{s}=91\,GeV,\, 10^3$ events)
and PYTHIA 5.5 ($e^{-}p,\ \sqrt{s}=314\,GeV,\ 10^3$ events)  $H_q$ has,
nevertheless, an  oscillating form (Fig. 4 ).
%%%%\begin{figure}[t]
An explanation
of such a phenomenon can be found immediately if one recalls two facts:
1) each MC code takes special care about both local (in the course of parton
splitting) and global energy-momentum conservation in the collision;
2) the limitation of the total energy is a physical origin of large
$O(\alpha_{s})$ corrections \cite{CuyTes}. Thus, LLA completed by
accounting of  the energy-momentum conservation in the MC models
imitate  {\it  some} part of higher order corrections and this can lead to
the oscillation of $H_{q}$. But it is still a question how large is that part?

\section{Phenomenological examples }

In favour of the suggested interpretation testify also the following
 remarkable fact
\cite{BBL}. Formally, according to (7) NBD has an infinite "tail"
 at finite energy of collision (finite $\langle n\rangle$).
This circumstance due to (3) results  in positive $K_q$ and monotone
declining $H_q$ (8). On the other hand, an infinite "tail" of MD is
 possible only either
for production of  massless particles
or neglecting of the energy conservation during a reaction.  Account of
these factors leads to a truncation of the MD "tail"  at some finite
multiplicity $n_{tr}$. As a result,
$H_q$ calculated for the truncated NBD starts oscillate around
the curve $q^{-k}$ with alternation the sigh.
An amplitude of the oscillation  tends fast to zero with
increasing of $n_{tr}$ and $H_{q}^{(tr)}$ transforms to $H_{q}^{(NBD)}$.
The same property of $H_q$ has been found also for the truncated
Poisson distribution.

Another example is a behaviour of $H_q$ in soft $p \bar{p}$ collisions
at $Sp \bar{p} S$ and Tevatron energies calculated \cite{LevShum} in the
framework of the Dual Parton Model \cite{CapTr}. As it has been found
\cite{LevShum}
the shape of $H_q$ ( an amplitude of the oscillation, position of minima
and maxima) is very sensitive to the number of cut Pomerons.



\section{ What can be done at HERA ?}

Among the high energy reactions where the oscillation of $H_q$
 \cite{Gian} has been found
\cite{Gian} $ep$ collisions are missed. A new data from the $ep$
 collider HERA will be able to fill in this gap.
The invariant mass $W$ of the hadronic final state in deep
inelastic scattering at HERA extends, with significant cross sections,
to the phase space limit ($\sqrt{s}=314\, GeV$).
This circumstance allows us to formulate several problems related with
properties of MD which can de studied with the H1 and ZEUS detectors:

1. Detailed studies of MD as a function of $z=n/\langle n\rangle $
 over the whole
kinematical region at $W$ up to the highest available ones.
Does the KNO scaling violate for large values of $W$ ?

2. High precision measurement of the ratio $H_{q}=K_{q}/F_{q}$
of cumulant and factorial moments both for the full phase space
and restricted rapidity windows, for events with one, two,... jets, etc.
 Does $H_q$  as
a function of the order $q$ show an oscillation around $H_{q}=0$ ?
If so, confronting the data with predictions of the MC models we
would learn more about the quality
 accounting for the higher order effects in the improved MC models.

3. Measurements of $W$ dependence of $H_q$ would allow us to make
a conclusion about an influence of the finite energy effects on the
shape of $H_q$.

\vskip 2mm
{\it Acknowledgments.} This work was supported in part by the International
Science Foundation under Grant N5V000 and DESY.
\vskip 2mm
\begin{thebibliography}{99}
\bibitem{DeWolf}
 E.A.~DeWolf, I.M.~Dremin, W.~Kittel, Usp. Fiz. Nauk. 163 (1993) 3.
\bibitem {Bial}
A.Bialas, R.Peschanski, Nucl. Phys. B273 (1986) 703.
\bibitem {Dr}
I.M. Dremin,\, Phys.Lett.\,B\,313 (1993) 209.
\bibitem {KNO}
Z.\,Koba, H.B.\,Nielsen, P.\,Olesen, Nucl.\,Phys. B\,40 (1972)\,317.
\bibitem{UA5}
 G.J. Alner et al. (UA5), Phys.\,Rep., 154\,(1987)\,247.
\bibitem {CS}
P.\,Carruthers, C.C.\,Shih, Phys. Lett., 127B (1983)\,242.
\bibitem {GVH}
A.\,Giovannini, L.\, Van\,Hove, Z.\,Phys.\,C30(1986)\,391.
\bibitem {BCM}
A. Basseto, M. Ciafoloni and G Marchesini, Phys. Rep. 100 (1983) 202.
\bibitem {QCD}
Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller, S.I.~Troyan, Basics
of Perturbative QCD. Ed. J. Tran Thanh Van, Editions Frontieres, 1991.
\bibitem {YDok}
Yu.L. Dokshitzer, Phys. Lett. B305 (1993) 295.
\bibitem {DrN}
I.M. Dremin, V.A. Nechitailo,\, JETP Lett. 58 (1993) 945;
Preprint OITS 520, FIAN TD-23, September 1993.
\bibitem {DrLN}
I.M. Dremin,\,B.B. Levtchenko, V.A. Nechitailo, Nucl. Phys.(Russia) 57
(1994) 1091; Preprint Univ. of Oregon, OITS 524, FIAN TD-25, November 1993.
\bibitem {DrHwa}
I.M. Dremin, R.C. Hwa, Preprint Univ. of Oregon, OITS 525, FIAN TD-26,
 November 1993; Preprint Univ. of Oregon, OITS 531, FIAN TD-27, December 1993.
\bibitem {CuyTes}
F.\,Cuypers, K.\,Teshima, Z.\,Phys. C54 (1992) 87.
\bibitem {Gian}
G. Gianini, in: {\it Proc. of "Multiparticle Dynamics-93"},
Aspen, 1993 (to be published by World Scientific, Singapore).
\bibitem {Sjo}
T. Sj\"{o}strand, Monte Carlo Event Generation for LHC,
Preptint CERN-TH.6275/91.
\bibitem {HERA}
Physics at HERA, {\it Proc. of the Workshop},
Eds. W. Buchm\"{u}ller and G. Ingelman, Hamburg, 1991, v.3.
\bibitem {BBL}
B.B. Levtchenko, to be published.
\bibitem {LevShum}
B.B. Levtchenko, A.V. Shumilin, to be published.
\bibitem {CapTr}
A. Capella, J. Tran Thanh Van, Z. Phys. C23 (1984) 165.
\end{thebibliography}


\end{document}

