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\begin{center}
{\bf
SEARCH FOR $Z$-SCALING VIOLATION\\[0.35cm]
IN $p-p$ AND $p-A$ COLLISIONS
 AT HIGH ENERGIES}

% TITLE OF THE ARTICLE (IN CAPITAL LETTERS)}

\vskip 5mm

 D. Toivonen $^{\S}$ and  M. Tokarev $^{\natural}$


{\small
\vskip 0.3cm
{\it Laboratory of High Energies,\\
Joint Institute for Nuclear Research,\\
141980, Dubna, Moscow region, Russia}

$^{\S}$ {\it  E-mail: {toivonen@sunhe.jinr.ru}}

$^{\natural}$ {\it  E-mail: {tokarev@sunhe.jinr.ru}}
}
\end{center}

\vskip  0.5cm

%\newpage


\begin{center}
\begin{minipage}{150mm}
\centerline{\bf Abstract}
New analysis of experimental data on hadron
$(\pi^{\pm}, K^{\pm}, \bar p) $ production at high-$p_T$
in $p-p$ and  $p-D$ collisions in $z$-presentation
is performed. Data on inclusive cross sections of
cumulative particles produced in backward hemisphere
in $p-A$ collisions  are analyzed as well. The scaling function for
nuclear targets $(Li, Be, C, Al, Cu$ and $ Ta)$ are constructed
and compared with high-$p_T$ data $z$-presentation.
The hypothesis on $z$-scaling violation as a signature
of new physics phenomena is discussed.
\\
{\bf Key-words:}
high energy, hard proton-proton and proton-nucleus collisions,
cumulative particles, scaling
\end{minipage}
\end{center}

%\begin{center}
%    Submitted to "International Journal of Modern Physics A"
%\end{center}


  {\section{Introduction}}

  The numerous results obtained from data analysis for high-$p_T$
  particle production in $p-p$ and $p-A$ collisions at high energy show
  that $z$-scaling
\cite{Z96}-\cite{Z00}
%\cite{Z96,Z99,Dedovich,Rog1,Z01}
 \footnote{See contributions
 presented by I.Zborovsk\'{y}, G.\v{S}koro and M.Tokarev}
 reflect general  properties of particle formation.
 Data $z$-presentation can be obtained
  using the experimental observables, the momentum and
  the inclusive cross section $Ed^3\sigma/dq^3$ of produced particle
  at given energy $\sqrt s$ and the multiplicity density
  of charged particles  $dN/d{\eta}|_{\eta=0}=\rho(s)$.

  As argued in  \cite{Z96,Z99,Dedovich,Rog1,Z01}  the scaling function
  $\psi(z)$ describes the  probability density to form a particle
  with formation length $z$. The scaling variable $z$ reveals the property
  of fractal measure. It means that the formation length increases with
  the scale resolution.

  Therefore the violation of $z$-scaling  is of interest to search for
  new phonomena in particle and nuclear physics.

  In the paper experimental data \cite{Jaffe}
  for charged hadrons ($\pi^{\pm}, K^{\pm}, \bar p$)
  produced in $p-p$ and $p-D$ at
  $p_{lab}=400$ and $800~GeV/c $ obtained at Fermilab are analyzed
  and compared with results obtained at  $p_{lab}=70, 200, 300$
  and $400~GeV/c $ before.


  Data sets \cite{Nikif} for cumulative particle
  production in $p-A$ collisions
  are analyzed in the $z$-presentation as well.
  Assuming  that the shape of the scaling curve $\psi(z)$
  is the same as  for high-$p_T$ data
  \cite{Jaffe,Cronin,Protvino}
  the angular  dependence of multiplicity particle density
  for backward hemisphere  particles production \cite{Nikif} is found.

  The obtained results allowed us to compare different data sets
   and point out the kinematical regions interested for search for
  new physics phenomena.



%  {\section{ $Z$-scaling}}


\vskip 0.5cm
{\section{ Hadron production in $p-p$ and $p-D$ at high-$p_T$}}

   Experimental data sets  \cite{Cronin,Protvino} and \cite{Jaffe}
   of inclusive cross sections for $\pi^{\pm}, K^{\pm}, \bar p$ hadrons
   produced  in $p-p$ and $p-D$ collisions at high transverse
   momentum $p_T$  are analyzed.
   The experimental data \cite{Protvino} includes the cross
   sections for $p-p$ and  $p-D$ collisions for
   $\pi^{\pm}, K^{\pm}, p^{\pm}$ hadrons produced at
   $70~GeV/c$  over the  range $1.5<p_T<4.0~GeV/c$.
   The measurements were made at an angle corresponding
   to approximately  $90^0$ in the nucleon-nucleon center-of-mass frame.
   Data  \cite{Jaffe} used in our
   new analysis corresponds to cross sections for
   $\pi^{\pm}, K^{\pm}, p^{\pm}$ hadron
   production in $p-p$  and $p-D$ collisions at $p_{lab} = 400$ and
   $ 800~GeV/c$.
   The produced particles  were registered over the
   transverse momentum  range of $p_{T} = 4.0-10.0~GeV/c$
   and at $\theta_{cm}^{N N}\simeq 90^0$.



% Note that the cross section  data  \cite{Protvino,Cronin,Jaffe}
% demonstrate a non-exponential behavior as a function of $p_{T}$ and
% the dependence on the incident proton momentum $p_{lab}$ is visible.



  \vskip 0.5cm
{\subsection{$p-p$ collisions}}

 In this section  we study the properties of $z$-scaling
 for hadrons produced in $p-p$  collisions.
% General formulas (\ref{eq:r20}) and (\ref{eq:r28}) are
% used to calculate the scaling function $\psi(z)$ and the variable
% $z$. The quantities $\rho (s,\eta) $ and $\sigma_{inel}$ are the
% average charged particle multiplicity density and the inelastic
% cross section of $p-p$-collisions, respectively.
%We use in our analysis of data $z$-presentation
%the experimentally measured dependence of the average charged particle
%multiplicity density  for $p-p$ and $\bar p-p$ collisions
%on collision energy $\sqrt s$.
%   The results presented in the paper were
%   obtained at $m=1~GeV$, $\delta_N=0.5$,
%   $\delta_A=\delta_N \cdot A$,  where $A$ is the atomic weight.
%   The values of the parameter
%   $m_2$ are defined by the internal conservation laws
%   for baryon number, electric charge, isospin, strangeness
%   of corresponding exclusive process
%   $p + p \rightarrow  h + X$  and are taken to be 0., 0.140., 0., 0.178,
%   0.494, 0.938 for $ \pi^+ , \pi^- , \pi^0 , K^+ , K^-$ and $\bar p
%   $, respectively.  The dimension of the quantity $m_2$ is $GeV$.
%\vskip 1.cm
%{\subsection {Energy independence  of $\psi(z)$}}
% Let us study the energy  dependence of the scaling function
% $\psi(z)$ of hadrons produced in $p-p$ collisions.  For
% analysis, we use the sets of cross section data
% \cite{Protvino,Cronin}  and  \cite{Jaffe}.
% It should be  noted that a strong dependence of the cross
% section on energy  $\sqrt s$ for \cite{Jaffe} was experimentally found.
% The similar feature was also observed for hadron production in
% $p-p$ collisions \cite{Cronin,Protvino}.
% The effect increases with transverse momentum and is about 3 orders
%for $p_{lab}=200$ and $800~GeV$ at $p_T=4~GeV/c$.
%
We verify the hypothesis of energy scaling for data
$z$-presentation for  hadron  production
in $p-p$ collisions using the available experimental data
 \cite{Cronin,Protvino}  and  \cite{Jaffe}.
The energy scaling for data  $z$-presentation means that the
shape of the scaling curve is  independent of the collision energy
$\sqrt s$.

 Figures 1(a) and 2(a)  show the dependence of the cross section
 $Ed^3\sigma/dq^3$  of $\pi^{+}$ and $K^{-}$ produced in $p-p$
 on transverse momentum $p_{T}$ at $p_{lab} = 70, 200, 300, 400$
 and $800~GeV/c$ and the angle $\theta_{cm}$ near $90^0$.
 Note that the data cover the wide transverse momentum
 range, $p_{T}=1-10~GeV/c$.

 The hadron spectra have a power behavior and
 demonstrate the strong energy dependence
 increasing with the transverse momentum.

 Figures 1(b) and 2(b) show $z$-presentation of the same data sets.
 Taking into account the experimental errors we can conclude that
 the scaling function $\psi(z)$ demonstrates an energy
 independence over a wide energy and transverse momentum
 range at $\theta_{cm} \simeq 90^0$.

 We found that new data \cite{Jaffe} included in the
 analysis give no indications on $z$-scaling violation.
 The result is the new confirmation of energy independence of data
 $z$-presentation.


%\vskip 1cm
%{\subsection {Angular dependence  of $\psi(z)$}}

% To analyze the angular dependence
% of the scaling function $\psi(z)$ of charged hadrons
% $\pi^{\pm}, K^{\pm}, \bar p$ produced
% in $\pi^--p$ collisions  we use the data set obtained
% at Protvino \cite{turch93}. The data set includes the results
% of measurements of the invariant cross section
% $Ed^3\sigma/dq^3$  at the pion incident momentum
% $p_{lab} = 40~GeV$ over the momentum and angular ranges of
% $p_{T}=1.05-3.75~GeV/c$ and $\theta_{cm}^{\pi N} = 49^0-93^0$.
% A dependence of the cross section on the angle of the
% produced hadrons  was experimentally found.
% The general feature of a spectrum angular dependence is
% the decreasing of the cross section with an angle.
% However we would like to note that the experimental errors  of data
% \cite{turch93} are large enough and do not allow us to make
% strong statement about the angular dependence of $\psi(z)$.
% The points corresponding to
% the  transverse momentum $p_T > 2~GeV/c$ and the incident momentum
% $p_{lab}=40~GeV/c $ dispose upper than the
% corresponding points at $p_{lab}=200$ and $300~GeV/c$. Moreover the
% difference between  points corresponding
% to the different $p_{lab}$ increases with a transverse momentum.
% From our point of view  it means the
% existence of the considerable systematic errors of cross section
%  measurements over a high $p_T$ range.
%
% Thus the obtained results show that more high accuracy data on the
% cross section angular dependence are necessary to verify carefully the
% angular independence of the scaling function $\psi(z)$ of hadrons
% produced in $\pi^--p$ collisions  as a function
% of energy $\sqrt s $ and  transverse momentum $p_T$.
%


\vskip 0.5cm
{\subsection{$p-D$ collisions}}

The study of the energy independence of data $z$-presentation
for $p-D$ collisions is especially interest. It is connected with
the possibility to investigate the influence of nuclear
matter on particle formation. We assume that in the high-$p_T$
range the effects can give direct information
on the state of nuclear medium which determines the
properties of particle formation.
Moreover as shown in \cite{Z01}  due to
A-dependence of data $z$-presentation
the scaling functions
for different nuclei from $D$ to $Pb$ are coincide each other
if the scaling transformation
$ z \rightarrow \alpha (A) \cdot z,$  \
$ \psi \rightarrow \alpha^{-1} (A) \cdot \psi $
depending on the single parameter, the atomic weight $A$ is used.
It means that the lightest nucleus, deuterium, can be a good target
to search for a signature of nuclear phase transition at high-$p_T$.
Let us remind that the quantitative measure of $z$-scaling
violation and consequently one of the possible signatures
of new physics phenomena
is the change of the anomalous fractal dimension $\delta$.

Figures 3(a) and 4(a) show $p_T$-presentation of inclusive cross section
$Ed^3\sigma/dq^3$ of $\pi^{+}$ and $K^{-}$ produced in $p-D$
at $p_{lab} = 70$ and $400~GeV/c$
and the angle $\theta_{cm}^{NN}\simeq 90^0$.
Note that the data \cite{Jaffe} are in a good agreement with
the data obtained by Cronin group \cite{Cronin} at $p_{lab}=400~GeV/c$.

As  seen  from  Figures  3(b) and 4(b)  $z$-presentation
for  all  data  sets \cite{Cronin,Protvino} and \cite{Jaffe}
demonstrate the energy independence.
It means  that $A$-dependence of $z$-presentation
is correctly described  by the function $\alpha(A)=0.9A^{0.15}$
established in \cite{Z01}.

Thus we can conclude that
data \cite{Jaffe} for $p-D$ collisions included in the
new analysis do not give any indications on $z$-scaling
violation up $z\simeq 20$.


\vskip 0.5cm
{\subsection{$z-p_T$ plot}}

To determine the kinematical region preferable for searching for the
scaling violation $z-p_T$ plot is suggested  to use.

Figure 5  shows  the  dependence of $z$ as a function of transverse
momentum $p_T$  for  $p-p$ (a)  and $p-D$ (b)  collisions at
$p_{lab} = 70, 200, 400$ and $800~GeV/c$ and
$\theta_{cm} \simeq 90^0$.
%As seen from Figure 5  the constant value of $z$ determines the
%kinematical boundary for value of pion transverse momentum.
The results of present analysis allow us to conclude that
the systematic experimental study of high-$p_T$ particle spectra
for $p-p$ and $p-D$ collisions at $z > 10 $ is necessary to determine
the asymptotic behavior of the scaling function $\psi(z)$.
The kinematical region $ z > 10$ is more preferable to search for
$z$-scaling violation.


\vskip 0.5cm
{\section{Cumulative particle production in $p-A$}}

Cumulative particles are called particles produced in the
kinematical region forbidden for free nucleon-nucleon interaction
\cite{Stavinsky,Zolin}-\cite{ABaldin}.
Such particles can be only produced in the processes with participation
of nuclei (in hadron-nucleus, nucleus-nucleus and lepton-nucleus collisions).

 Inclusive cross sections $Ed^3\sigma/dq^3$  for
 $\pi^{\pm}, K^{\pm}, p^{\pm}$ hadron
 production in backward hemisphere in  $p-A$
 collisions at $p_{lab} = 400~GeV/c$ and
 at the angle  $\theta_{lab}$ of $ 70^0,
 90^0, 118^0$ and $160^0$ are presented in \cite{Nikif}.
 The measurements were performed over the  momentum
 range $0.2<p<1.25~GeV/c$. Nuclear targets, $Li, Be, C, Al,Cu$
 and $Ta$ were used.
%The strong  dependence of cross sections
% is observed as $\theta_{lab}$ changes from $70^0$ to $160^0$.
 The data cover in particular the kinematical range
 forbidden for particle production in nucleon-nucleon collisions.

In the paper we restrict ourselves of the analysis of data
\cite{Nikif} obtained at Batavia. More complete analysis of
cumalative data sets will be presented elsewhere.


Figures 6(a) and 7(a) present the inclusive cross sections for
$\pi^{+}$ mesons  produced in backward hemisphere in $p-Be$ and $p-Ta$
collisions at $p_{lab} = 400~GeV/c$ and
 the angle  $\theta_{lab}$ of $ 70^0,
90^0, 118^0$ and $160^0$.
As seen from Figures 6(a) and 7(a)
the strong  dependence of cross sections
as $\theta_{lab}$ changes from $70^0$ to $160^0$
is observed for all nuclear targets, $Li, Be, C, Al, Cu, Ta$.



We assume that the shape of the scaling curve will be the same
as for data points corresponding to the high-$p_T$ region.
At present there are not experimental data on
the angular dependence of  $\rho(s,\eta, A)$ for particles
produced in backward hemisphere in order to construct directly
the scaling function.
Therefore we study the possibility to describe the shape of the $\psi(z)$
found from the analysis of high-$p_T$ data sets
\cite{Protvino,Cronin,Jaffe}   using data points \cite{Nikif}.
The function  $\rho(s,\eta, A)$ is parameterized in the form
$\rho(s,\eta, A)= \rho(s,A)|_{\eta=0} \cdot \chi(\theta, A)$, where
the angular dependence is described by  $\chi(\theta, A)$.
The latter is shown in Figure 8(a).
As seen from Figure the ratio
$\chi_{Ta}/\chi_{Li}$  decreases from 3.5 to 1.5 as
the angle $\theta_{lab}$ increases from $70^0$ to $160^0$.
The $A$-dependence of the ratio demonstrates the saturation
reached for backward particle production.


Figures 6(b) and 7(b)  show the $z$-presentation of data
\cite{Nikif}. One can see that the curve found
for nuclei  $ Be$ and $ Ta$ is in a good
agreement  with  high-$p_T$  data $z$-presentation for D.
Note that the angular dependence  of
$\chi(\theta, A)$  shown in Figure  8 (a)
describes  both non- and cumulative data points
\cite{Nikif}.
Thus we found that there is possibility to combine
the scaling functions corresponding to
the cumulative and  high-$p_T$ data.
As seen from Figures 6(b) and 7(b)
all points \cite{Nikif} are out of the asymptotic region.
Therefore it is of interest to determine the kinematical region
for backward pion production where $z_{cum} >  z_{hard} \simeq 20$.


\vskip 0.5cm
{\subsection { $z-p$ plot}}

 The dependence of $z$ on momentum $p$ ( $z-p$  plot)
 as a function of an atomic weight $(A)$, an angle of produced
 particle $(\theta)$ and a collision energy $\sqrt s $
 can be used  to select the domain where
 the scaling can be violated and  new physical
 phenomena can be found.

  Figure  8(b)  shows a $z-p$ plot for  $p-Ta$
  collisions at $p_{lab} = 70, 200, 400 $ and $800~GeV/c$ and
  $\theta_{lab}=160^0$.
  The value $z=20$
  for $p-Ta$ collisions  corresponds to the values of the
  momentum $p$ of 1.9, 2.1, 2.2 and $2.25~GeV/c$, at
  $p_{lab} = 70, 200, 400$ and $800~GeV/c$, respectively.


\vskip  0.5cm
{\section{Conclusions}}

To  search for $z$-scaling violation
the analysis of the scaling features of
 hadrons produced in $p-p$ and $p-A$
collisions at high energies in
terms of $z$-presentation are performed.
The experimental high-$p_T$  \cite{Jaffe,Cronin,Protvino}
and cumulative  \cite{Nikif} data  sets on the inclusive
cross sections are used in the analysis.


%The momentum of incident proton beam $p_{lab}$ changes
%from 70 to $800~GeV/c$.
%For central rapidity range $\theta_{cms}\simeq 90^0$
%the high transverse momentum
% $p_T$ changes from 1 to $10~GeV/c$.
% For particle production in backward hemisphere
% ($\theta_{lab}=70^0-160^0$) pion momentum covers the range
% $p=0.2-1.2~GeV/c$.

  Data $z$-presentation for high-$p_T$ and cumulative data
  is constructed. It is expressed via the
  experimental observables, momenta and masses of
  colliding and produced particles,
  the invariant inclusive cross section
  $Ed^3\sigma/dq^3$  and the multiplicity  density of charged
  particles.

 A new confirmation of $z$-scaling for particle production
 in high-$p_T$ range is obtained.
  It is shown that available experimental high-$p_T$ data
  \cite{Protvino,Cronin,Jaffe} on hadron production in $p-p$ and $p-D$
  collisions give no indications on $z$-scaling violation.
  The shape of the scaling curve does not change with  the collision
  energy $\sqrt s$.



  The found angular dependence of particle multiplicity density
  for backward  pion  production  in $p-A$  reproduces the
  shape of the scaling curve obtained from high-$p_T$  data.

%  The $A$-dependence  of data  $z$-presentation  for $p-A$ collisions
%  is established  to describe by the
%  function  $\alpha = \alpha(A)$  depending on the single parameter,
%%  atomic weight,  both for high-$p_T$ and cumulative data.


 %The symmetry transformations of the function  $\psi$ and its
 % argument,  $\psi \rightarrow \alpha^{-1}(A)  \psi $ and $z\rightarrow
 % \alpha (A) z$, are used to compare $\psi(z)$  of different nuclei
 % $(A = Be, Cu, W)$.

 New data \cite{Jaffe} included in the present analysis
 point out the asymptotic regime of the scaling function,
 $\psi(z)\sim  z^{-\beta}$,  at $z>4$.


The kinematical ranges for high-$p_T$ and backward hemisphere
particle production preferable  for search for $z$-scaling violation
in $p-p$ and $p-A$ collisions are found using the $z-p_T$ and $z-p$ plots.

% Using the properties of $z$-scaling, the inclusive cross
% sections of $\pi^+$-mesons produced in $p-p, p-D, p-Be$
% and $p-Ta$ collisions in backward hemisphere
% at high energy $p_{lab} = 70, 200, 400$ and $800~GeV$ are predicted.

% The presented results  show  that data $z$-presentation
% can be used for data analysis of particle production in new
% kinematical
% regions and  for comparison with  data obtained previously.
% A new confirmation of $z$-scaling for particle production
% in high-$p_T$ range is obtained.


% We consider that the properties of $z$-presentation reflect
% general symmetry properties of the internal particle structure,
% interaction of their constituents and formation mechanism
% such as self-similarity, locality, scale relativity and fractality.

Analysis of data on cumulative particle production has
shown a possibility to use the class of events to search for
$z$-scaling violation. It is assumed  that the properties
of nuclear matter in the
cumulative range should drastically change the mechanism of particle
formation due to multiple interactions of elementary constituents.
Therefore the systematic study of particle spectra at high-$z$
and multiplicity particle density in the backward hemisphere
is of interest.



% As one can assume the properties
% reflect through the anomalous dimension $\delta_{\pi}$
% the features of elementary constituent substructure too.
%


% \vskip 1cm
%\acknowledgments
%
%The authors would like to thank S.Gerasimov
%for useful discussions of the present work
%and M.Zielinski from the E706 Collaboration
%for providing one of the authors (M.T.)
%with the experimental data.

\vskip 1.5cm
{\large \bf Acknowledgments}

\vskip 0.5cm
The authors would like to thank I.Zborovsky and O.Rogachevsky
for useful discussions of the present work.

%This work has been partially supported
%by Grant of the Czech Academy of Sciences No. 1048703.
%
%\vskip 1cm
%

%\newpage
{\small
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 {\it    Proceedings of the 15th International Conference
 on Ultra-Relativistic Nucleus-Nucleus Collisions},
 Long Island, New York, USA, 2001, edited by T.J.Hallman et al.;
  Nucl. Phys.  {\bf A698}  (2002).


 \bibitem{Zolin}  O.P. Gavrishchuk  et al.,
 Nucl. Phys. {\bf A523} (1991) 589.
 %I.M.Beliaev et al.,  Yad. Fiz. {\bf 56} (1993) 135.


\bibitem{ITEP}  S.V. Boyarinov   et al.,
 Sov.\ J.\ Nucl.\ Phys.\
%Yad. Fiz.
% {\bf 57} (1994) 1452; ibid
% {\bf 46} (1987) 1473; ibid
% {\bf 50} (1989) 1605; ibid
% {\bf 54} (1991) 119; ibid
% {\bf 56} (1993) 125.


\bibitem{Leksin}
 G.A. Leksin, Report No. ITEP-147, 1976 (unpublished) ;
 G.A. Leksin, in {\it Proceedings of the XVIII International
Conference  on High Energy Physics}, Tbilisi, Georgia, 1976,
 edited by N.N. Bogolubov {\it et al.} (JINR Report No. D1,2-10400,
 Tbilisi, 1977), p. A6-3.


 \bibitem{Strikman}  L.L. Frankfurt, M.I. Strikman,
 Phys. Rep. {\bf 160} (1988) 235.

%\bibitem{Schukraft} J. Schukraft, CERN-PPE/91-04, 16 January, 1991;\\
%H.R.Schmidt, J.Schukraft,  J.Phys. G: Nucl.Phys. {\bf 19} (1993) 1705.


 \bibitem{Bondarev}  V.K. Bondarev,
 Physics of Elementary Particles and Atomic Nuclei {\bf 28} (1997) 13.

 \bibitem{ABaldin}  A.A. Baldin, A.M. Baldin,
 % JINR Preprint P2-97-309, Dubna, 1997;
 Physics of Elementary Particles and Atomic Nuclei
 {\bf 29} (1998) 577.


%\bibitem{Carroll}
% J. Carroll {\it et al.}, Phys. Lett. B {\bf 80}, 319 (1979).

%\bibitem{Hij1}
% X.N. Wang and M. Gyulassy, Phys. Rev. D {\bf 44}, 3501 (1991).
%
%\bibitem{Hij2}
% X.N. Wang and M. Gyulassy, Phys. Rev. D {\bf 45}, 844 (1992).


%
%\bibitem{Eides} D.I.Diakonov, M.I.Eides, in: Materials of the
%XVI Winter School of LNPI, Leningrad (1981) 123.
%

%\bibitem{Wang1}  X.-N. Wang,  22 July, 1999.
%
%\bibitem{PYTHIA}
% T. Sjostrand, Computer Physics Communications, {\bf 82}, 74  (1994).


% \bibitem{NA22}
% M. Adamus et al., IHEP preprint 88-121, Serpukhov, 1988.
%
% \bibitem{Lund}
% B. Andersson, G. Gustafson, G. Ingelman, and  T. Sjostrand,
% Phys. Rep. {\bf 97}, 31 (1983);
%% B. Andersson, The Lund Model, Cambridge University Press, Cambridge, 1998;
% B. Andersson, In: Proc. XXII International Symposium on Multiparticle Dynamics, p.428
% Edited by C.Pajares, Santiago de Compostela, Spain, July 13-17, 1992, World Scientific Publishing
% Co. Pte. Ltd.


% \bibitem{MCSFM}
% N.S. Amelin, M. Braun, and C.Pojares,
% Z. Phys. C {\bf 63}, 507 (1994).
% %In: Proc. XXII International Symposium
% % on Multiparticle Dynamics, p.482,
% % Edited by C.Pajares, Santiago de Compostela, Spain, July 13-17, 1992,
% % World Scientific Publishing
% % Co. Pte. Ltd.
%

\end{thebibliography}

% **************************************************************
%%************************  1(a,b)  2(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig1a.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig1b.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 1.}
 (a) Dependence of  the
 inclusive cross section of $\pi^+$-meson  production in  $p-p$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70, 200,300,400$ and
 $800~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Jaffe,Cronin,Protvino}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig2a.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig2b.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 2.}
 (a) Dependence of  the
 inclusive cross section of $K^-$-meson  production
 in  $p-p$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70, 200,300,400$ and
 $800~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Jaffe,Cronin,Protvino}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

% **************************************************************
%%************************  3(a,b)  4(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig3a.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig3b.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 3.}
 (a) Dependence of  the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-D$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70$ and
 $400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Jaffe,Cronin,Protvino}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig4a.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig4b.eps,width=7.0cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 4.}
 (a) Dependence of  the
 inclusive cross section of $K^-$-meson  production
 in  $p-D$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70$ and
 $400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Jaffe,Cronin,Protvino}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}


% **************************************************************
%%************************  5(a,b)  6(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig5a.eps,width=7.0cm}}
\vskip -5.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig5b.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 5.}
 $z-p_T$ plot for $\pi^+$-meson production in $p-p$ (a) and
$p-D$ (b)
    collisions  at
    $p_{lab} = 70, 200, 400$ and $800~GeV/c$
and $\theta_{cm} \simeq 90^0$. }
\end{figure}


%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig6a.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig6b.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption

{{\bf Figure 6.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Be$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and
\cite{Jaffe,Cronin,Protvino}.
 }
\end{figure}

% **************************************************************
%%************************  7(a,b)  8(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig7a.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig7b.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 7.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Ta$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Jaffe,Cronin,Protvino}.  }
\end{figure}


%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=fig8a.eps,width=7.0cm}}
\vskip -6.8cm
\hspace*{4cm}
\centerline{\epsfig{file=fig8b.eps,width=7.cm}}
\vskip 1.cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 8.}
 (a)
  Angular dependence $\chi(s, \theta, A)$ of
     particle multiplicity density for particle production
in $p-A$ collisions  at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
(b)  $z-p$ plot for $\pi^+$-meson production in $p-Ta$
   collisions  at  $p_{lab} = 70, 200, 400$ and $800~GeV/c$
and $\theta_{lab} =160^0$. }
\end{figure}


\end{document}


% **************************************************************
% **************************************************************
% **************************************************************
% **************************************************************
% **************************************************************
% **************************************************************
% **************************************************************
%%************************  3(a,b)  4(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=ppkp2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=ppkp1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 3.}
 (a) Dependence of  the
 inclusive cross section of $K^+$-meson  production
 in  $p-p$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70, 200,300,400$ and
 $800~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=ppkm2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=ppkm1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 4.}
 (a) Dependence of  the
 inclusive cross section of $K^-$-meson  production
 in  $p-p$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70, 200,300,400$ and
 $800~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}



% **************************************************************
%%************************  5(a,b)  6(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=ppantip2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=ppantip1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 5.}
 (a) Dependence of  the
 inclusive cross section of $\bar p$  production
 in  $p-p$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70, 200,300,400$ and
 $800~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=ppkm2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=ppkm1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 6.}
 (a) Dependence of  the
 inclusive cross section of $K^-$-meson  production
 on transverse momentum $p_{T}$ at $p_{lab} = 70, 200,300,400$ and
 $800~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$
 in  $p-p$ collisions.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe,Nikif}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}


% **************************************************************
%%************************  7(a,b)  8(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pdpip2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pdpip1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 7.}
 (a) Dependence of  the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-D$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70$ and
 $400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pdpim2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pdpim1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 8.}
 (a) Dependence of  the
 inclusive cross section of $\pi^-$-meson  production
 in  $p-D$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70$ and
 $400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}


% **************************************************************
%%************************  8(a,b)  9(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pdkp2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pdkp1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 9.}
 (a) Dependence of  the
 inclusive cross section of $K^+$-meson  production
 in  $p-D$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70$ and
 $400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pdkm2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pdkm1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 10.}
 (a) Dependence of  the
 inclusive cross section of $K^-$-meson  production
 in  $p-D$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70$ and
 $400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}






% **************************************************************
%%************************  10(a,b)  11(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pdantip2.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pdantip1.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 11.}
 (a) Dependence of  the
 inclusive cross section of $\bar p$ production
 in  $p-D$ collisions
 on transverse momentum $p_{T}$ at $p_{lab} = 70$ and
 $400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=zpt_pp.eps,width=7.0cm}}
\vskip -5.8cm
\hspace*{4cm}
\centerline{\epsfig{file=zpt_pd.eps,width=7.cm}}
\vskip 1.cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 12.}
 $Z-p_T$ plot for $\pi^+$-meson production in $p-p$ (a) and
$p-D$ (b)
    collisions  at
    $p_{lab} = 70, 200, 400$ and $800~GeV/c$
and $\theta_{cm} \simeq 90^0$. }
\end{figure}






% **************************************************************
%%************************  9(a,b)  10(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=70kina.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=400kina.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 13.}
Kinematical boundary ($q_z-q_T$ plot)  for $\pi^+$-meson
produced in $p-p, p-D$  and $p-Li$  collisions at $p_{lab}=70$
and $400~GeV/c$.}
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=ptapip1.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=ptapip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 14.}
 (a) Dependence of  the
 inclusive cross section of $\pi^-$-meson  production
 on  momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{cm} \simeq 90^{0}$
 in  $p-Li$ collisions.
 Experimental data are taken from
 \cite{Cronin,Protvino,Jaffe,Nikif}.
 (b) The corresponding scaling function $\psi(z)$. }
\end{figure}



% **************************************************************
%%************************  9(a,b)  10(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=plipip1.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=plipip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 15.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Li$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Cronin,Protvino,Jaffe}. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pbepip1.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pbepip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 16.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Be$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Cronin,Protvino,Jaffe}. }
\end{figure}




% **************************************************************
%%************************  9(a,b)  10(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pcpip1.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pcpip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 17.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-C$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Cronin,Protvino,Jaffe}. }
\end{figure}

%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=palpip1.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=palpip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption

{{\bf Figure 18.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Al$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Cronin,Protvino,Jaffe}. }
\end{figure}



% **************************************************************
%%************************  9(a,b)  10(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=pcupip1.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pcupip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 19.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Cu$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Cronin,Protvino,Jaffe}. }
\end{figure}


%\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=ptapip1.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=ptapip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption

{{\bf Figure 20.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Ta$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Cronin,Protvino,Jaffe}. }
\end{figure}


% **************************************************************
%%************************  9(a,b)  10(a,b) *********************
% **************************************************************
\newpage

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=ugol.eps,width=7.0cm}}
\vskip -7.8cm
\hspace*{4cm}
\centerline{\epsfig{file=pcupip2.eps,width=7.cm}}
\vskip 0.5cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 9.}
  Angular dependence $\chi(s, \theta, A)$ of
     particle multiplicity density for particle production
in $p-A$ collisions
 at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.}
\end{figure}


\vskip  1cm

\begin{figure}[htb]

\hspace*{-4cm}
\centerline{\epsfig{file=zpt_pp.eps,width=7.0cm}}
\vskip -5.8cm
\hspace*{4cm}
\centerline{\epsfig{file=zpt_pd.eps,width=7.cm}}
\vskip 1.cm
\hspace*{4.cm} a) \hspace*{8.cm} b)\\[0.0cm]
%\caption
{{\bf Figure 22.}
 (a) Dependence of the
 inclusive cross section of $\pi^+$-meson  production
 in  $p-Ta$ collisions
 on momentum $p$ at $p_{lab} = 400~GeV/c$
 and $\theta_{lab}=70^0, 90^0, 118^0, 160^0$.
 Experimental data are taken from \cite{Nikif}.
 (b) The corresponding $z$-presentation  of data sets
\cite{Nikif} and \cite{Cronin,Protvino,Jaffe}. }
\end{figure}






\end{document}



%%%% **********************************************
%%%% **********************************************
%%%% **********************************************
%%%% **********************************************
%  {\section{Method. Properties of $z$-scaling}}

  In this section, we would like to remind the basic ideas of $z$-scaling
  dealing with the investigation of  the inclusive process

  \begin{equation}
  P_{1}+P_{2} \rightarrow q + X.
  \label{eq:r1}
  \end{equation}
  The momenta and masses of colliding nuclei and  inclusive particles are
  denoted by  $P_1,P_2,q$ and $M_{1},
  M_{2}, m_1$, respectively. The gross features of the inclusive particle
  distributions for reaction
  (\ref{eq:r1}) at high energies are assumed to be described in terms of
  the corresponding kinematic
  characteristics of the exclusive subprocess written in the symbolic
   form \cite{Stavinsky},

  \begin{equation}
  (x_{1}M_{1}) + (x_{2}M_{2}) \rightarrow m_{1} +
  (x_{1}M_{1}+x_{2}M_{2} + m_{2}).
  \label{eq:r2}
  \end{equation}
  The parameter $m_{2}$ is introduced  to satisfy the  internal conservation
  laws (for isospin, baryon number,
  and strangeness). The $x_{1}$ and $x_{2}$ are the scale-invariant
  fractions of the incoming four-momenta
  $P_{1}$ and $P_{2}$ of colliding objects. The cross section of process
  (\ref{eq:r1}) is assumed to be
  expressed via the cross section of the corresponding parton subprocesses
  depending on a minimum energy, which
  is necessary for the production of the secondary particle with mass $m_1$
  and  four-momentum $q$.

  \vskip 0.5cm
  {\subsection{Fractions $x_1$ and $x_2$}}

  The elementary parton-parton collision is considered  as a binary
  sub-process which satisfies the condition

  \begin{equation}
  (x_{1}P_{1} + x_{2}P_{2} - q)^{2} = (x_{1}M_{1} + x_{2}M_{2} +
  m_{2})^{2}.
  \label{eq:r5}
  \end{equation}
  The equation reflects minimum recoil mass hypothesis in the
  elementary sub-process. To connect kinematic and structural
  characteristics of the interaction, the coefficient
  $\Omega$ is introduced. It is chosen in the form
  \begin{equation}
  \Omega(x_1,x_2) = m(1-x_{1})^{\delta_1}(1-x_{2})^{\delta_2},
  \label{eq:r8}
  \end{equation}
  where $m$ is a mass constant and $\delta_1$ and $\delta_2$
  are factors relating to the fractal structure of
  the colliding objects \cite{Z99}\footnote{The anomalous dimensions
  are found to be  $\delta_{1,2} = \delta_h$ and
  $\delta_1 = \delta_h$, $\delta_2 = \delta_A = \delta_N \cdot A$
  for $h-h $ and $h-A$ collisions, respectively.}.
  The fractions $x_{1}$ and
  $x_{2}$  are determined  to maximize the value of $\Omega(x_1,x_2)$,
  simultaneously fulfilling condition (\ref{eq:r5})
  \begin{equation}
  {d\Omega(x_1,x_2)/ dx_1}|_{x_2=x_2(x_1)} = 0.
  \label{eq:r9}
  \end{equation}
  Expressions for
  $x_{1}$ and $x_{2}$ as a function of the momenta and masses of
  the colliding and produced particles are given in \cite{Z99}.
  The variables
  $x_{1,2}$ are equal to unity along the phase space limit and
  cover the full phase space accessible at any
  energy.


%The threshold condition for the process
% (\ref{eq:r1})  can be written as follows
% \begin{equation}
% (M_1+M_2+m_2)^2 +E^2-m_1^2 \leq (\sqrt {s_A} -E)^2.
%\label{eq:r10}
%\end{equation}
% Here $\sqrt {s_A}$ is the total center-of-mass energy.
% The inequality  bounds kinematically the maximum
%% energy $E$ of the inclusive particle $m_1$ in the
% c.m.s. of the reaction (\ref{eq:r1}).


\vskip 0.5cm
{\subsection{Scaling function $\psi(z)$ and variable $z$}}


%  In accordance with the self-similarity principle, we search for
%  the solution depending on a single scaling
%  variable $z$ in the form
%
% \begin{equation}
% \psi (z)  \equiv
% \frac{1}{<N>\sigma_{inel}} \frac{d\sigma}{dz}.
% % \frac{1}{\sigma_{inel}} \frac{d\sigma}{dz}.
% \label{eq:r17}
% \end{equation}
% Here, $\sigma_{inel}$ is the inelastic cross section and $<N>$ is the average
%  multiplicity of charged
% particles. The function  $\psi(z)$ has to be dependent of the  scaling
%  variable $z$.
%We would like to note
% that the existence  of such a solution is not evident in advance. All the
%  quantities refer to  $p-A$
% interactions.
%

 The self-similarity principle in high energy physics states
 that some physical quantities can depend on dimensionless variables.
 The Feynman and Bjorken variables are very well known quantity.
 We search the solution depending on a single scaling
 variable. The existence  of such a solution is not evident in advance.
 As shown in \cite{Z99} the solution exists.
 The function $\psi$ expressed via the invariant differential
 cross section for the production of
 the inclusive particle $m_{1}$  is introduced as follows
 \begin{equation}
 \psi(z) = - \frac{\pi s_A}{\rho_A(s, \eta) \sigma_{inel}}J^{-1}
 E\frac{d\sigma}{dq^{3}}.
 \label{eq:r20}
 \end{equation}
 Here,
 $\sigma_{inel}$ is the inelastic cross section and $<N>$ is the average
 multiplicity of charged  particles.
$s_A \simeq s \cdot A$ and $ s $ are the center-of-mass energy
  squared of the corresponding $ h-A $
 and $ h-N $ systems and  $A$ is the atomic weight. The factor $J$ is
 a known function of kinematic variables
 \cite{Z99}. The expression (\ref{eq:r20}) relates the inclusive differential
  cross section and the average
 multiplicity density $\rho_A(s,\eta) = d<N>/d\eta$ to the scaling function $\psi(z)$.
 %The combination $y = 0.5
 %\ln ({\lambda_2}/{\lambda_1})$ is approximated to (pseudo)rapidity $\eta$ at high energies.

%The function  $\psi(z)$ has to be
%dependent on the scaling variable $z$.
%All the  quantities refer to  $pp$ interactions.
%The expression (\ref{eq:r17}) allows us to give
%the physical meaning of $\psi(z)$ as the probability density
%to form a particle with the formation length $z$.
The invariant differential cross section for the production of
inclusive particle is normalized as
\begin{equation}
\int_{z_{min}}^{\infty} \psi(z) dz = 1.
\label{eq:b6}
\end{equation}
According to the choice of
${\hat s_{\bot}}^{1/2}$ described below we have $z_{min}= 0$.
The equation allow us to give the physical meaning
of the scaling function $\psi$ as a probability density to form
a particle  with a corresponding value of the variable $z$.

   Here we would like to argue that the variable $z$ can be interpreted
   as a particle formation length. We choose
   $z$ as a physically meaningful variable that  could reflect
   self-similarity (scale invariance) as a general
    pattern of hadron production in accordance with the ansatz
    suggested in \cite{Z99}

\begin{equation}
z = \frac{ \sqrt{ {\hat s}_{\bot} }} {\Omega \cdot \rho_A(s) },
\label{eq:r28}
\end{equation}
where ${\hat{s}}^{1/2}_{\bot}$
 is the transverse kinetic energy
of subprocess (\ref{eq:r2}), defined  by the  expression
${\hat{s}}^{1/2}_{\bot}
%\hat{E}^{kin}_{\bot}
= \hat{s}^{1/2}_{\lambda} +
{\hat{s}}^{1/2}_{\chi} - m_1 - (M_1x_1 + M_2x_2 + m_2)$;
$\Omega$ is the measure given by (\ref{eq:r8}) and
$\rho_A(s) = \rho_A(s, \eta=0)$.

The transverse energy consists of two parts
\begin{equation}
\hat{s}^{1/2}_{\lambda} = \sqrt{(\lambda_1P_1 + \lambda_2P_2)^{2}}
, \ \ \ \ \
\hat{s}^{1/2}_{\chi} = \sqrt{(\chi_1P_1 + \chi_2P_2)^2},
\label{eq:r30}
\end{equation}
 which represent the transverse energy of the inclusive particle
 and its recoil, respectively\footnote{The fractions
 $\chi_i$ and  $\lambda_i$ are known functions of
 momenta $P_1, P_2, q$ and masses $M_1, M_2, m_1, m_2$
 of initial and produced particles \cite{Z99}.}.
 %The sharing of the energy in the elementary constituent
 %interaction is discussed in subsection 2.3.
 We would like to note that the form of $z$ determines its variation range.
 The boundaries of the range are 0
 and $\infty$, as defined by (\ref{eq:r28})  and (\ref{eq:r8}).
 These values are scale independent and
 kinematically accessible at any energy.

%***********************************

  Thus, one of the features of the procedure described above
 is the joint use of
 the experimental observables characterizing hard and soft processes.

  The scaling variable $z$ and the scaling function  $\psi(z)$
  are expressed via experimental quantities,
  inclusive cross section and multiplicity
  particle density.  The first one describes the hard
  and the second one describes the soft regimes of particle formation.
  The physical features of hard and soft processes
  are very different. Therefore there is a real problem
  for the theoretical description of $z$-scaling in the
  framework of perturbative QCD. We would like to note that
  $z$-construction is not direct mathematical consequence of
  parton model of strong interaction but this is a new
  data presentation motivated by parton-parton and string-like
  scenarios of particle interactions.


%***********************************

    Let us consider the definition of the variable
  $z=\sqrt{\hat s_{\bot}}/(\Omega\rho_A)$  more closely
   and clarify its physical meaning.
  %Here we would like to argue that $z$ is the hadron formation length.
  We assume that the gross features of the inclusive
  particle distribution for the reactions (\ref{eq:r1})
  at high energies can be described  via the
  corresponding characteristics of exclusive  subprocess (\ref{eq:r2}).
  The value $ \sqrt {\hat s_{\bot}}$ is the minimal transverse energy of
  colliding constituents necessary to produce a real hadron in the reaction
  (\ref{eq:r1}). It is assumed that two point-like and massless
  elementary constituents interact each other in the initial state and
  convert into real hadrons in the final state. The conversion is not
  instant process and is usually called  hadronization.
  Space-time microscopic picture of the process is not
  understood enough at present time.
  We assume that a number of hadrons produced
  in the hard interaction of constituents
  is proportional to $\rho_A$. Therefore the value
   $ \sqrt {\hat s_{\bot}}/\rho_A$ corresponds to the energy density
  per one hadron produced in the subprocess.
  The factor $\Omega \simeq M_1+M_2 - x_1 M_1- x_2 M_2  $
  is the missing mass for the process (\ref{eq:r1})
   expressed via the kinematic characteristics of the
  subprocess (\ref{eq:r2}). It corresponds to the energy
  consumed on creation
  of the associative particle multiplicity and characterizes
  the property of hadron environment such
  as energy tension of nuclear matter.
  Taking into account the qualitative scenario of hadron formation
  as a conversion of a point-like constituent into a real hadron
  we interpreted the value $z$ as a hadron formation length.
  It increases with $p_{\bot}$ at fixed $\sqrt s$ and  takes into account
  kinematic and dynamic features of hadron formation in
  the elementary subprocess (\ref{eq:r2}).

  \vskip 0.5cm
  {\subsection{Fractality and scale-relativity}}

  Fractality in particle and nuclear physics concerns the internal
  structure of particles and their interactions. It is manifested
  by their self-similarity on any scale.
  This general principle is described by power law dependencies of
  the corresponding quantities  \cite{Nottale,Z99}.
  The equation (\ref{eq:r5}) written in the form
  $x_1 x_2 - x_1 \lambda_2 - x_2 \lambda_1 = \lambda_0 $,
  does not change under the scale transformation
  \begin{equation}
  \lambda_{1,2} \rightarrow \rho_{1,2} \cdot \lambda_{1,2},\ \ \
  x_{1,2} \rightarrow \rho_{1,2} \cdot x_{1,2}, \ \ \
  \lambda_0 \rightarrow  \rho_1 \cdot  \rho_2 \cdot  \lambda_0.
  \label{eq:r50}
  \end{equation}
   The transformation with the scale parameters
   $\rho_{1,2}$ allows us to consider the collisions of the complex
   objects in terms of suitable sub-processes of the
   interacting elementary constituents. It is reasonable to use
   $\rho_1=1$, $\rho_2=1$ and $\rho_1=1$, $\rho_2=A$ for
  the description of $p-p$ and  $p-A$ interactions,
  respectively. Here, $A$ is the  corresponding atomic weight.
 The coefficient $\Omega$, given by (\ref{eq:r8}),
 connects the kinematic and fractal characteristics of
 the interaction.
 The factors $\delta_1$ and $\delta_2$ are
 anomalous fractal dimensions of the colliding objects.
 The fractal structure itself is defined by the structure
 of the interacting constituents, which is not an elementary one either.
 In this scheme, high energy hadron-hadron, hadron-nucleus and
 nucleus-nucleus interactions are considered as interactions of fractals.


  For the collisions of protons  with nuclei we have
  $\delta_1<<\delta_2=\delta_A$, what is demonstrated
  on the experimental data below.
  In this case, the factor $\Omega$ can be approximated as
  follows
  \begin{equation}
   \Omega = m
    (1- {x_1})^{\delta_1}
    (1- {x_2})^{\delta_2} \simeq m(1-{x}_2)^{{\delta}_2}.
  \label{eq:r52}
  \end{equation}
In the beam and target fragmentation  regions at high collision energy
we have the relations $ x_1\rightarrow 1, x_2<<1 $ and
$x_1<<1, x_2\rightarrow 1 $, respectively.
The factor $\Omega$ is not sensitive to structure of one of the
interacting   objects (particle or nuclei). The condition
 $ x_2\rightarrow 1 $  determines the kinematical region for
cumulative particle production. For double cumulative region
we have  $ x_1\rightarrow 1 $  and $ x_2\rightarrow 1 $.
The last condition can be only satisfied in nucleus-nucleus collisions.
   When introducing the notation
   $z_0=\sqrt{{\hat s}_{\bot}}/\rho_A(s)$,
and  $V=( 1-x_1)^{\delta_1/\delta}( 1-x_2)^{\delta_2/\delta}$
   the scaling variable $z$ can be written in the form
  \begin{equation}
   z=z_0\cdot V^{-\delta}.
  \label{eq:r51}
  \end{equation}
   The variable has character of a fractal measure
   where $\delta$ is the anomalous fractal dimension describing
   the intrinsic structure of the interaction constituents
   revealed at high energies.
   We interpret the factor $V$ as the relative part of the full phase-space
   volume corresponding to such parton-parton
   collisions in which the inclusive particle can be produced.
   The fractal property of the collision reveals itself so that
   only the part of all multi-scattering corresponding to the phase space
   $V^{\delta}$ produces the inclusive particle.


