MUON PAIR PRODUCTION IN PROTON-ANTIPROTON % INTERACTIONS AT INTERMEDIATE ENERGIES.
\noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %1234567890123456789012345678901234567890123456789012345678901234567890 The process of lepton pair production in antiproton-proton collisions at PANDA experiment is simulated for studying the distributions of energy, transverse momentum and angle variables of individual leptons created in a antiquark-quark annihilation subprocess $q\bar{q} \rightarrow l^{+}l^{-}$. The correlations between mentioned above physical variables and the corresponding distributions of the number of events versus the values of these variables are presented. A special attention is paid to the distributions that describe the produced $l^{+}l^{-}$-system as a whole system. It is shown that the spectrum of the total energy of $l^{+}l^{-}$ pair may spread from 1 GeV up to 12 GeV and in about a half of events its energy is higher than 5 GeV, i.e. the produced pairs are rather energetic. It is argued that the measurement of the total transverse momentum of a lepton pair as a whole may provide an important information about an intrinsic transverse momentum $<k_{T}>$ that appear due to the Fermi motion of quarks inside the nucleon. The problems due to the presence of fake leptons that appear from meson decays are discussed also. It is shown that in muon pair production case about of $40\%$ of events may not contain charged pions at all in their final state, i.e. they would be free of fake muons. In the remaining part of events, the information about the muon production vertex and the moderate $PT^{\mu}$-cuts from below may be also useful for discrimination of fake muons. The contribution from the background processes that may contain leptons is discussed also. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PAN140605.tex


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\begin{document}
        
%\title{MUON PAIR PRODUCTION IN PROTON-ANTIPROTON
%            INTERACTIONS AT INTERMEDIATE ENERGIES.}

\title{Monte-Carlo simulation  of lepton pair production 
%       in  "$\bar{p}p \rightarrow \mu^{+}\mu^{-} + X$"
       in  "$\bar{p}p \rightarrow l^{+}l^{-} + X$"
%       events at $E_{cm}=5.3$ GeV.}
       events at $E_{beam}=14$ GeV.}

%%%    $P_{t}^{Jet}=461.1$ GeV
%%%    $\eta -\phi$ 
%%%    $2\rightarrow 2$
%%%    $p\bar p$ 
%%%    ${q\bar{q} \to \gamma + g}$
%%%    $(\chi^{2}/dof$
%%%    $6\%$
%%%    $P_{t}=90$ GeV 
%%%    {GeV/c}
%%%    ${GeV^2/c^2}$
%%%    $ p\bar{p} \rightarrow l\bar{l} + X $ 
%%%    $\mu^{+}\mu^{-}$
%%%    $Pt^{\gamma}$
%%%    $Pt^{\mu}$
%%%    $E^{\mu}$ 
%%%    $< M_{630}>=2.54 \pm 0.03$ 
%%%    ${\hat{p}_{\perp}^{\;min}}$
%%%    $E_{beam}$=14 GeV
%%%    ${q\bar{q} \rightarrow \gamma^{*} \rightarrow $\mu^{+}\mu^{-}$}$
%%%    $ < {\theta^{\mu}_{fast}} > = 16.5^{o}$
%%%    $p\bar p$ 
%%%    $p\bar p$ 
%%%    $ p\bar{p} \rightarrow l\bar{l} + X $ 
 

\author{ A.N.~Skachkova, N.B.~Skachkov,\\
%\author{ N.B.~Skachkov,\\
%(on behalf of D0 Collaboration) \\
Joint Institute for Nuclear Research,
Dubna, 141980, Russia\\
E-mail:Anna.Skachkova@cern.ch, skachkov@jinr.ru}


\maketitle



\begin{abstract}
\noindent 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%1234567890123456789012345678901234567890123456789012345678901234567890

      The process of lepton pair production in antiproton-proton 
    collisions at PANDA experiment is simulated for studying the 
    distributions of energy, transverse momentum and angle variables
    of individual leptons created in a antiquark-quark  annihilation 
    subprocess  $q\bar{q} \rightarrow l^{+}l^{-}$. The 
    correlations  between mentioned above physical variables and 
    the corresponding distributions of the number of events 
    versus the values of these  variables  are presented.

       A special attention is paid to the distributions that 
    describe the produced $l^{+}l^{-}$-system as a whole 
    system. It is shown that the spectrum of the total energy of
    $l^{+}l^{-}$ pair may spread from 1 GeV up to 12 GeV and 
    in about a half of events its  energy is higher than 5 GeV,
    i.e. the produced pairs are rather energetic. It is argued
    that the measurement of the total 
    transverse momentum  of a lepton pair as a whole may provide
    an important information about an intrinsic transverse 
    momentum $<k_{T}>$ that appear due to the Fermi motion
    of quarks inside the nucleon.  

      The problems due to the presence of fake  leptons that
    appear from meson decays  are discussed also. It is 
    shown that in muon pair production case about  of $40\%$ 
    of events may  not contain charged pions at all in their 
    final state, i.e. they  would be free of fake muons. In the 
    remaining part of events,  the information about the muon
    production vertex and the moderate  $PT^{\mu}$-cuts from
    below may be also useful for discrimination of fake muons.
    The contribution from the background processes that may 
    contain leptons is discussed also.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{abstract}



%#########################################################
\section{Introduction.}
\label{intro}



%%%%%%#######################################################


~~~~ Quark-antiquark annihilation subprocess of the hadron-hadron 
     collision, mentioned in the abstract (see \cite{Matv} and 
     \cite{Drell}), may  provide an interesting information 
     about the quark dynamics inside the hadron. 
     The results of measuring of lepton pairs, produced
     in these collisions at high energies \cite{DYexper}, were
     already used to enlarge  our knowledge about the structure 
     functions that describe  the parton distributions in hadrons 
     \cite{MRST}.

%\bibitem{MRST}

    Due to a limited size of a range of avaliable total
   energy PANDA would 
   be rather an intermediate than a high energy experiment. 
   In this energy region the important role plays the
   physics of bound states and  resonances, which is strongly
   connected with the quark confinement problem.

    At the same time it is 
    well known from the previous $p\bar{p}$ experiments, 
    done  at low energies (like the $p\bar{p}$ annihilation 
    experiments done at LEAR at CERN), that many features 
    of the resonance  decays may be also quite well explained, 
    basing on the simple phenomenological 
    rules, like the so called quark line rule \cite{Okubo}.

% \cite{Okubo} !! \bibitem{S.Okubo, Phys.Lett. B 5, 165 (1963);
%                  G.Zweig, CERN Report no.8419/th 412 (1964) ;
%                  I.Izuka, Prog. Theor. Phys (suppl 37) 38 21 
%                  (1966);  

     Recently, the OBELIX Collaboration (CERN) has published 
     the results  of a detailed  analysis of three reactions 
     of the  annihilation at rest with the three mesons
     production  in the final state:
     $p\bar{p} \rightarrow \pi^{+}\pi^{-}\pi^{0},
     K^{+}K^{-}\pi^{0},  \pi^{\pm}\pi^{\mp}K^{0}
     $ \cite{OBELIX1}, \cite{OBELIX2}.
     The  analysis done in these articles also does not 
     exclude the possibility  that some  features of the
     observed  resonant structures, or enhancements, as 
     well as the observation of some resonant structures 
     suppression in the S- and P-waves of the $p\bar{p}$
     system, which  undergo the annihilation process at rest, 
     may need  the
     $SU_{3}$ model  application for their explanation.

        From here we may conclude that if we shall consider a
     case of the antiproton scattering off the proton target 
     in a case of the antiproton  beam of  14 GeV energy 
     \footnote{this value of energy was used also in 
               our  previous Note  \cite{Note1}, devoted to 
               muon pair production study.}
     (that corresponds  to the center of mass energy of the
      $p\bar{p}$  system  $E_{cm}= 5.3 GeV$), then it would be
     quite natural to use a quark parton model to describe 
     the $\bar{p}p \rightarrow l^{+}l^{-} + X$ process.      
     To this reason 
     \footnote{other arguments may be found in \cite{Note1}.}
     we have  used here the PYTHIA event generator \cite{Sjost}, 
     which is based on the ideas of the quark parton model 
     and is well tested  and widely used for the simulation
     of high energy hadron-hadron interactions.  Some 
     parameters of the generator were set here, following 
     \cite{Note1}, to those values that  allow fast and 
     reliable simulation in  the case of $E_{beam}$=14 GeV.
 

%$q\bar{q} \rightarrow \mu^{+}\mu^{-} + X$
%~~~~~The  measurements of lepton pair production in hadron-hadron 
%   interactions (see  \cite{Matv} and \cite{Drell}) have
%   demonstrated  already their great potential for studying the 
%   properties of elementary particles. It is enough to mention
%   the facts of the discoveries  of charmed  J
%   (also J/$\Psi$) meson
%%  \cite{Ting}[Ting, et al.],
%   as well as of beauty $\Upsilon$ meson which were confirmed 
%   latter on in  e+e- experiments. Dilepton events serve also as
%%  \cite{SLAC}[SLAC].
%   a powerful tool to get out the information about the parton
%   distribution  functions (PDF) in hadrons as it was already
%   shown  in a number 
%   of high energy experiments \cite{DYexper} and the
%   theoretical papers, devoted to the data  analysis in the 
%   framework of QCD. 
%  \cite{DYtheor}.


%   \footnote {More arguments in a favor of studying 
%   of this process may be found also in a number 
%   of recent proposals for experimental program at HESR.}

%   In this connection it is worth to mention that, nevertheless, 
%   up to now  there is no good theoretical understanding of
%   the dynamics and physics of  lepton pair production process
%   in hadron-hadron collisions. As for the illustration of the
%   situation let us remind as an example, a  widely  used 
%   practice of 
%   applying of such a "theoretical method"  of  elimination of 
%   a quite a noticeable  discrepancy among the existing lepton
%   pair production data and  theory predictions
%   \footnote {based on strait-forward quark-parton  model of 
%   hard interactions  and its QCD LO and NLO extensions.} as a 
%   simple multiplication of theory predictions by the so
%   called  ``phenomenological K-factors" (which  values are
%   found  from fitting of theory to data and they are mostly
%   varying  in the interval from 1.5 to 2.5).


%      Intermediate energy experiments may play an important role as
%   they allow to study the range where the  perturbative methods
%   of QCD (pQCD) come into interplay with a rich physics of bound 
%   states and  resonances.  The physics of hadron resonances
%   formation and decay  is strongly 
%   connected with the confinement problem , i.e. with 
%   the QCD dynamics at large distances. Thus, detailed and  high
%   precision  experimental study at PANDA  would allow to
%   discriminate  between  a large variety of non-perturbative 
%   approaches and other  models that are already proposed or 
%   are under  development now. 
 
%      Therefore, for future studies it may be instructive  to 
%   define  a boundary among the areas of  application of the
%   perturbative theory and  non-perturbative approaches.
%   For this reason we have chosen the well known PYTHIA 
%   event generator  \cite{Sjost} , which is widely and 
%   successfully used in high energy experiments.
%
 


%  Our simulation would be  based on perturbative QCD
%   amplitudes used in PYTHIA what allow a proper account

   It is important to note that PYTHIA simulation is based
   on the use of the amplitudes of the  relativistic quantum
   field theory. In our case it is the perturbative QCD
   amplitude of the $q\bar{q} \rightarrow l^{+}l^{-}$
   subprocess, implemented into the PYTHIA package.
   This allows  a proper account of the relativistic  
   kinematics while simulation of 
   the distributions of  different physical variables, 
   connected with  the  $\mu^{+}\mu^{-}$- 
   (or  $e^{+}e^{-}$-) pairs produced in event.

%   in PYTHIA what allow a proper account
%    
%   of relativistic  kinematics while studying the  distributions 
%   of  different physical variables connected with 
%   $\mu^{+}\mu^{-}$- (or  $e^{+}e^{-}$-) pairs
%   production in the range of future PANDA experiment 
%   energies. This kind of generation may provide an
%   experience for setting  the low energy limits for pQCD 
%   application as well as it  may be also useful in future
%   as a reference point for  comparison of pQCD  predictions
%   with  the predictions ( present and future) based on
%   the non-perturbative  approaches.


%%%    $P_{t}^{Jet}=461.1$ Rev
%%%    $\eta -\phi$ 
%%%    $2\rightarrow 2$
%%%    $p\bar p$ 
%%%    ${q\bar{q} \to \gamma + g}$
%%%    $(\chi^{2}/dof$
%%%    $6\%$
%%%    $P_{t}=90$ GeV 
%%%    {GeV/c}
%%%    ${GeV^2/c^2}$
%%%    $ p\bar{p} \rightarrow l\bar{l} + X $ 


      It is worth to mention also that in any  case 
%   any another then perturbative QCD model 
   the dominant mechanism of the lepton-anti-lepton pair 
   production would be  one and the same in the most
   of another than perturbative QCD approaches or models.
   Really,  in this special case of 
   $ p\bar{p} \rightarrow l\bar{l} + X $ process  the dominant
   contribution  to the process  amplitude in any model  would come 
   from one and the same   amplitude of quark pair  annihilation
   ${q\bar{q} \to l\bar{l}}$ in the chosen range of beam energy
   value. 
%  (see Figure 1 with the corresponding 
%  LO Feynman graphs,''---> it is not included yet into attachment
%   that contains only distributions of different observable 
%   variables'').
%   PYTHIA includes this amplitude as well as provides a 
%   proper account of relativistic kinematics.

%      In this  Note we shall present the results of simulation
%   the scattering of the  anti-protons  of $14 GeV$ energy  on the 
%   proton target,  supposed to be at the rest frame. 
%  (i.e. neglecting, for the first approximation, the 
%  transversal move of proton drops \cite{PandaLOI} ).

%   This kind  of simulation based on the QCD amplitude of 
%   ${q\bar{q} \to l\bar{l}}$ process, implemented into PYTHIA,
%   of course,  does not  take into account   the effects 
%   connected with the real detector conditions 
%   (what would be the subject of the following papers).
%


%   It is clear that the muon physical
%   variables would be different after account  
%   as well as particle misidentification and showering effects
%   in the detector. 

   It should be mentioned that PYTHIA generates the events
   in which  the final state  particles are produced in a
    result of a rather long  chain of 
   interactions. This chain  includes the primary partons, 
   which  are produced in a hard  perturbative $2 \to 2$ 
   quark-parton  fundamental  subprocess and latter on do
   pass the stage of interaction between themselves (which is
   described by the LUND string fragmentation model
   \cite{And79} - \cite{And83} used in PYTHIA). After
   this the 
   quarks, diquarks  and strings, produced at this stage,
   do pass the hadronization phase. The variety of  the 
   produced elementary  particles then undergo  a stage of 
   decays in a case of the  presence of the nonstable 
   particles or resonances among them. At the end  of the 
   simulation there are  left only free  particles in the
   final state. Therefore, while using the PYTHIA, 
   we may get, as the result of simulation,  
   only the  distributions of free particles, unbiased by 
   any detector effects like, for example, by a magnetic 
   field. The account  of the influence of the detector
   effects would be a subject of  the following papers.
    
   
%   The present simulation allows to get a 
%  picture of "unbiased distributions" of  produced  particles
%   ( i.e. ``free distributions'' , which are  not biased  by
%   the detector effects like the magnetic field influence,
%   showering in calorimeters  as well as of 
%   particle misidentification ).

%   as well as the corresponding 
%   quark - parton distributions and allows, in principle,
%   to estimate  the size  of the corrections needed to 
%   reconstruct the original  input parton distributions. 
%   Let us mention that in what follows  we choose 
%   MRST4 \cite{MRST} parameterization of structure functions , 
%   which  is one from the list of proton parton-distributions 
%   set of PYTHIA6.  


      In the Section 2, devoted to lepton distributions, the results 
   of the previous  Note \cite{Note1} will  be added by a
   number of new 3- and 2-dimensional plots that show different
   kind of correlations (like Energy-Energy and Angle-Angle 
   correlations)  between the physical variables 
   of the leptons, produced in the quark level subprocess
   $q\bar{q} \rightarrow l^{+}l^{-}$.  All plots of this Note
   are collected in the Appendix (Section 7:"Figures").

     Let us mention  that the main results  of this Note (Sections 2,
   3 and 4) are  valid for a  case of muon pair  as well as for the 
   electron-positron pair production.  To this  reason in these three
   Sections we shall speak about the distributions 
   obtained for the lepton  case because muon distributions are
   practically do not
   differ from the electron ones (so, the word ``muon'' is equal
   to the ``electron '' in the Sections 2,3 and 4).  The main 
   difference of  $e^{+}e^{-}$ case from that one
   of  ${\mu^{+}\mu^{-}}$ pair production  would appear
   only  while considering the ``background'' contribution
   from the fake muons (see Sections 5 and 6). 

     The next important new addition, as comparing to \cite{Note1},  
   is presented  in the Section 3, where the distributions 
   of some  global  variables, that describe  the produced muon
   or electron pair as a whole system, like for instance, the 
   invariant mass of the signal lepton  pair.  Among them 
   of most interest is the total transverse momentum of
   a  lepton pair,  which value is connected with the intrinsic 
   transverse velocity of a quark inside the proton. Recent
   results of the experiment E706 on the direct photon production, 
   performed at Fermilab \cite{E706}, have shown that
   the averaged value of the quark transverse momentum 
   in the proton may be of $<k_{T}> \approx 1.4 $ GeV order. 
   Two sources of a possible appearance of quark transverse 
   momentum in the proton, i.e. the intrinsic Fermi
   motion of quark inside the proton  and the influence 
   of an initial state radiation,  are studied in Section 3.
   It is pointed out that due to the fact that the transverse
   motion  of quark in the nucleon does not depend on the value
   of the boost along the beam line, there is a unique 
   possibility for PANDA to measure the value of
   quark $<k_{T}>$, which is so important for a precise
   setting of the absolute scale of jet energy 
   \cite{SBKLHC} - \cite{BSTevat} and thus
   for a precise determination of particles mass (like
   of top quark and may be of Higgs boson and SUSY 
   particles) in the experimental measurements at  
   Tevatron and LHC.


%\cite{SBKLHC}, \cite{BKSLHC}, \cite{BSTevat}

%\bibitem{E706}
 

      Let us noted that each generator has its
    own internal restrictions. In  present generation we had,
    due  to the internal PYTHIA  cuts, a restriction on the  
    value of the invariant mass, produced in a fundamental
    $2\rightarrow 2$  hard QCD subprocess. 
    In practice, the value of  the invariant mass of a muon 
    (electron) pair,
%  $ M_{inv}^{\mu^{+}\mu^{-}}= {\hat{s}_{\;min}}$,
    produced  in  a signal event, turned out to be restricted  
    by the  limit from below:
    $ M_{inv}^{l^{+}l^{-}} >  1$ GeV. The influence of
    this cut on the spectrum of the signal lepton pair energy
    will  be discussed in the Sections  3 and 4.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%% Zdes' bylo  Mesto dlya abzaca o e+e- i mumu sxodstve,
%%%% kotoroe ja perenes vyshe.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    The problem of fake muons  background is 
   discussed in  the last Section 5. We do not consider here 
   a  case of a contribution from the  events, defined 
   by another than  $q\bar{q} \rightarrow \mu^{+}\mu^{-}$
   subprocess, because, as far as we have an aim to study 
   the muon pair production (i.e., a case of  muon pair which has 
   a common vertex) the appearance of this pair from some
   vector mesons decays would not be so large. Nevertheless
   we plan to study it in the nearest future. The main
   problem is  caused by the presence of the fake muons 
   from pions decays in the same $p\bar{p}$ process
   (based on the annihilation
   $q\bar{q} \rightarrow \mu^{+}\mu^{-}$ subprocess).
     \footnote{ or from fake electrons in a case of
              $q\bar{q} \rightarrow \e^{+}\e^{-}$ process. 
              It is clear that the  appearance of this kind 
              of fake electrons in the same  signal annihilation
              $\bar{p}p \rightarrow \e^{+}\e^{-}$  process  
              has a negligibly small probability  as comparing
              with the case of fake muons from pion decays.}
  
   As the first step
   the number of events that would not contain at all the
   charged  pions in the final state  is estimated in Section 5.
   The simulation has shown that the fraction  of such events
   without fake decay muons may turn out to be not a small one. 
   The distribution of the number of fake muons  versus their PT 
   value, presented in Section 5, provides an information for
   the proper choice of an effective additional cut for their
   discrimination. 
   Section 5 includes  also the distributions of  x-,y- and 
   z-components of the position of the vertex of the fake muons
   production, which may be also  useful for their separation.
   In Section 6 the spectra of background leptons, mainly 
   electrons, produced  in other than
   $q\bar{q} \rightarrow l^{+}l^{-}$ QCD subprocesses, are
   presented.  
      
%   The detailed  study of background  contributions would be
%   presented in the forthcoming papers.
 
% vertex  position 
       

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

%%%%%%%%%%%%%%%    Nachalo 1-oi' Sekcii po PANDA


%==================================================================
%
\section{Muon/electron distributions in  $p\bar p$ collisions.}


%
%============================================================
%
%


%   ${q\bar{q} \to \gamma + g}$

%   $ p\bar{p} \rightarrow l\bar{l} + X $ 

%   ${q\bar{q} \rightarrow \gamma^{*} \rightarrow $\mu^{+}\mu^{-}$}$

 
   
    We have done the generation of 100 000 \footnote{this number 
   allows an easy  estimation of  
   the number  of events that shall pass any momentum or angle 
   cuts  by a help of the correlation plots which are obtained 
   in  ROOT "TEXT"  format \cite{ROOT} and do contain the
   information in the numerical form.}
   "$\bar{p}p \rightarrow l^{+}l^{-} + X$" events with 
   $2\rightarrow 2$  QED 
   ${q\bar{q} \to \gamma^{*} \to l^{+}l^{-}}$ subprocess.
   In what follows these events should be called as ``signal
   events'', and, respectively, the muons (electrons) that
   are  produced in  hard QED  ${q\bar{q} \to l^{+}l^{-}}$
   subprocess  would be called as the "signal" leptons.
   Those leptons that  appear in events  due to hadron (mainly
   mesons) decays  will be called as "decay"  leptons.

%  Some of Feynman diagrams , corresponding to
%   switched on initial state radiation are shown in Figure 2.

   The distributions of the number of generated signal
   events versus the values of the energy $E^{l^{(+/-)}}$
   of the signal leptons,
   as well  as versus  the modulus of the transverse
   momentum  $PT^{l^{(+/-)}}$ 
      \footnote {In  all plots of Section 7 it is supposed
       that the  energy and the PT scales are presented in GeV 
       units (c=1 system is used) while for the angle scales
       the degrees measure is used.}
   and of  the polar angles
   $\theta^{l^{(+/-)}}$ , measured from the beam direction
   ( they are denoted in plots as $TETA^{l}$)
   are shown (from top to bottom, respectively) 
   in the Figure 1  of the Section 7 (Appendix: "Figures"), which 
   contains all plots and  figures. The  left  hand column in
   the Figure 1 is for  $l^{-}$ distributions and the right
   one is for $l^{+}$. No difference is seen between 
   $l^{+}$  and  $l^{-}$ distributions.


   One can see from the top raw of Figure 1 that the energy of
   leptons may vary in the  interval $ 0 < E^{l} < 10 GeV$ with
   a mean value $<E^{l}> = 2.6 $ GeV and its spectrum has
   a peak at a 
   rather small value of about  $E^{l}_{peak}$ =0.4 GeV. 
      
   The $PT^{l}$   spectrum (see the  middle raw of Figure 1) 
   has an analogous peak position at $PT^{l}_{peak} = 0.5$ GeV.
   Behind their  peaks both $E^{l}$  and $PT^{l}$
   spectra falls rather steeply but the spectrum of 
   $PT^{l}$ is confined in a more narrow interval 
   $0 < PT^{l} < 2$ GeV.

   The number of events ($N_{event}$) spectrum versus the polar 
   angle  ${\theta^{l}}$ ( see bottom raw) has a peak around 
   ${\theta^{l}} =10^{o} $  and the mean  value
   $ <\theta^{l}> =28.4^{o}$. We see that 
   while the most of signal leptons are scattered into the 
   forward direction ($\theta^{l}<90^{o}$) still there is a
   small number of them which may  fly even into the backward
   hemisphere ($\theta^{l} > 90^{o} $).
%  We shall discuss  this point in more details latter on.

      It is useful to have the set of plots 
   done separately for the signal leptons having the largest
   energy ( we shall call them as  "fast" leptons in what
   follows ) in the lepton pair, produced in an  event, and,  
   correspondingly,  for those, having a  smaller energy 
   ("slow" lepton in a pair).  This set of plots for the signal
   leptons  is given in Figure 2 of the Appendix, where the
   left hand column shows  the distributions  of "slow" leptons
   while  the right one is for the "fast" leptons distribution.  

%   This set
%   of plots for the signal muons with highest value of the 
%   $PT$ in the pair,  i.e. $PT^{\mu}_{fast}$, and that one
%   with the smallest one, i.e.  $PT^{\mu}_{slow}$ , are shown
%   in Figure 3.

     One can see from Figure 2 that  the energy spectrum 
   of the fast signal leptons (top raw) grows up rather fast 
   from the point  $E^{l}_{fast}=0.5 GeV $
   (more than  $90\%$ of fast leptons have
   $E^{l}_{fast} > 1 GeV $) up to the peak position  at 
   the point $E^{l}_{fast}$ = $2.5 GeV$  and  then 
   practically vanishes at  $ E^{l}_{fast}$ = $10 GeV $.

     In contrast to this picture the spectrum of the  less
   energetic  signal leptons (left raw) has a peak around the
   point  $E^{l}_{slow}$=0.4 GeV (where the spectrum of  
   fast leptons only starts) and then it vanishes at the point, 
   which practically do corresponds to the mean value of the 
   most energetic  signal leptons, i.e. at
   $ < E^{l}_{fast} >$ = 3.8 GeV. One may see
   that the spectrum of slow leptons in a pair is 
   a very different from that one of fast leptons.

    The difference between  the  $PT^{l}$ spectra (see the
   middle raw of  Figure 2) of  fast  and slow  leptons 
   seams not to be so large. It results only in about of
   340 MeV shift to the left  of the peak position
   of slow leptons spectrum  as well as in the  corresponding 
   end point shift to the left. Both 
   of these spectra do demonstrate that the main part of slow 
   as well as of fast leptons have  $PT^{l} > 0.2$ GeV.

     This similarity of $PT^{l}$ spectra of fast and slow 
   leptons results (due to a large difference of their energy 
   spectra, i.e. of $P^{l}_{z}$ components)  in a 
   large difference of their  polar angle ${\theta^{l}}$ 
   distributions (see bottom raw in Figure 2). Really, the
   ${\theta^{l}_{slow}}$  spectrum is shifted to the 
   right to the higher values, as comparing to that one of 
   ${\theta^{l}_{fast}}$. Note also that their mean value
   $ < {\theta^{l}_{slow}} > $ =  $40^{o}$ is a more
   than two times higher than the analogous mean value of 
   the fast leptons: $ < {\theta^{l}_{fast}} > = 16^{o}$.

     Another and the most important difference that is seen from 
   these angular distributions is that all fast leptons  fly
   in the forward  direction 
   (${\theta^{l}_{fast}} < 90^{o}$) and their spectrum
   practically finishes at ${\theta^{l}_{fast}} = 60^{o}$,
   while about of $17\%$ of slow leptons  have 
   ${\theta^{l}_{slow}} > 60^{o}$.  It is worth to mention
   that  about of  $5\%$ of slow leptons  may scatter into  the 
   back hemisphere.
   
%%%   {\footnote { 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

%     The angular distribution of muons having a higher $PT$
%   (see Figures 15-17, shown at page 2 of Attached .ps file)
%   where $E^{\mu}_{PTmax}$, $PT^{\mu}_{PTmax}$
%   and  $TETA^{\mu}_{PTmax}$ distributions do correspond to those 
%   muons that have the maximum value of $PT$ in the produced muon
%   pair in event) or the  smaller values of $PT$ 
%   in $\mu^{+}\mu^{-}$ pair 
%   (see Figures 18-20,  shown at page 2 of Attached .ps file)
%   where $E^{\mu}_{PTmin}$, $PT^{\mu}_{PTmin}$
%   and  $TETA^{\mu}_{PTmin}$ distributions do correspond to those 
%   muons that have the minimum value of $PT$ in the produced muon
%   pair in event)  do not show this kind of difference
%   what follows naturally from their definition:
%   higher $PT$  do correspond to higher values of polar angles. 
%   To this reason  we shall use in what follows the energy value 
%   to discriminate between two signal muons in 
%   $\mu^{+}\mu^{-}$ pair produced in event.}.
   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

     The number (or percentage) of signal leptons, scattered
   into the forward or back hemispheres,  may be easily found 
   from the 3-dimensional "Angle - Energy" correlation plots, 
   presented in Figures 3,4  and 5 (see Section 7, Appendix)
   done by help of ROOT \cite{ROOT}). 
   The left hand side of Figure 3
   contains the ``LEGO1'' ROOT plot with the distribution 
   of the number of  generated  events versus: \\

    $\bullet$ the  energy  $E^{l}_{slow}$  of the less energetic
     leptons in $l^{+}l^{-}$ pair (the energy axis is on 
     the right hand side of this plot and its scale  is 
     given in GeV units), as well as versus: \\

    $\bullet$ the polar angles  ${\theta^{l}_{slow}}$ (the angle 
     axis is on the left hand side of a plot and the scale
     on it is given in a degree measure units).\\

   The right  hand column contains the analogous "Angle-
   Energy"  correlation plots with the  distributions of
   the  number of events versus the energy $E^{l}_{fast}$ of 
   the  most  energetic  leptons in a $l^{+}l^{-}$ pair,
   as  well as  versus their polar  angles ${\theta^{l}_{fast}}$. 
   These plots show the  correlation between the angle and 
   the energy of slow and fast leptons in a $l^{+}l^{-}$ pair
   in event, as well as the distribution of the number of events for
   the correlated values of these variables. It becomes much 
   more easy to see these numbers after  making the projection 
   of these surfaces onto a plane.      
 
   The projections of these 3-dimensional distributions onto the  
   2-dimensional ${\theta}/E$ -planes are shown in Figures 4 and
   5. The correspondence between the color  contours in Figure 4 
   (SURF2Z plots) and the number of events per unit of a  bin
   square is defined by the vertical color scale, included into
   the right hand side of  each plot. For a convenience of
   counting of the number of events in each bin square of the 
   2-dimensional  ${\theta^{l}}$/$E^{l}$  planes we present 
   in the Figure 5 the ``TEXT'' ROOT analogs of plots from Figure
   4. The exact numbers in each square bin may be seen by a help 
   of the zoom application to this place of .ps file plots.
        \footnote{or use the central plate on PC mouse to click
         on the place you are interested with in the .ps file
         of the Section 7 (Appendix) and choose the zoom grade.}  
    From these plots one can see that the most of slow leptons
   that fly into the  back  hemisphere, i.e.
   ${\theta^{l}_{slow}} > 90^{o}$, (on total it is about 
   about of 5.5{\%} of events with  $l^{+}l^{-}$ pairs), 
   do have the energy which is less than
   $E^{l}_{slow}=0.5$ GeV.   

     The number of events distribution versus the energies of
   leptons in a pair (slow and fast leptons in  a pair),
   i.e.   that come  from pions decays.  the "Energy-Energy"  
   correlation,  as well  as 
   versus the angles of both leptons in a pair, i.e. the
   "Angle-Angle" correlation, are shown in the  3-dimensional
   ``LEGO1'' ROOT plots in Figure 6  of Section 7 (Appendix).
   The  projections  of these 3-dimensional plots  onto the
   Energy-Energy  $E^{l}_{slow}/E^{l}_{fast}$ plane 
   (left hand side),
   as well as onto the Angle-Angle 
   ${\theta^{l}_{slow}}/{\theta^{l}_{fast}}$ plane
   (right hand side),  are shown in Figure 7 (as for the
    ROOT ``SURF2Z'' color contour plots) and in  Figure 8 
   (as for the ROOT ``TEXT''  numerical plots). 

     From Figures 7 and 8 we may conclude that among the 
   signal events with $l^{+}l^{-}$ pairs there are :\\
 
     $\bullet$ about of  $14\%$ of them  have lepton pairs with
      the polar angles belonging to the square,

~ defined  by the  following bounds:\\
  
  
 ~~~~~${\theta^{l}_{slow}}, {\theta^{l}_{fast}} < 20^{o}$; \\
   

     $\bullet$ about of $40\%$ of events do have lepton pairs which angles are
      contained in a square: \\


 ~~~~~~${\theta^{l}_{slow}}, {\theta^{l}_{fast}} < 40^{o}$ . \\
   

   
     $\bullet$ about of $72\%$ of events do have lepton pairs which
      angles fit into  a more wide square: \\
   
   
~~~~~~~${\theta^{l}_{slow}}, {\theta^{l}_{fast}} < 60^{o}$. \\

     
      $\bullet$ and there is about of $94\%$ of events, which angles are 
       restricted by the relations: \\
   
~~~~~~ ${\theta^{l}_{slow}}, {\theta^{l}_{fast}} < 90^{o}$. \\
   
      
   It means that in case, when the lepton system would cover
   the angle region $\theta^{l} \leq 90^{o}$,
   then less than  $6\%$ of events, which do contain the
      $l^{+}l^{-}$ signal pairs,  would be lost.\\

%in case when  muon system would cover the angle region:

   From the Figure 2 and the  Energy-Energy correlation plots
   shown in the Figures 7 and 8  one may see that:\\
   
     $\bullet$ if the following  loose cut from below  would be applied to
      the energy of both (fast

~    and  slow)  leptons in a pair (recall about the peak at
    $E^{l}_{slow}=0.4 GeV$) \\


~~~~~~   $ E^{l}_{fast}, E^{l}_{slow}   >  0.5 GeV$ , \\

       then more than $25\%$ of  simulated signal events with
    lepton pairs would be lost, \\

     $\bullet$ but in a case of setting of a more strict cut: \\

~~~~~   $ E^{l}_{fast}, E^{l}_{slow}  > 1 GeV$ , \\

     then about of $45\%$ of  simulated events with signal lepton
     pairs would  be lost.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%   !!!!!!!!!!!!!!   END of PANDA Section: "MUON DISTRIBUTIONS"



%%%%%%%%%%----------------------------------------------------------
\section{
    Global variables for   
    ${q\bar{q} \to \gamma^{*} \to l^{+}l^{-}}$ 
    subprocess.\\
    u- and  d- quarks components of a proton \\
    structure function .}  
  
 
% and the role of $k_{T}$- and ISR- effects.}
   

%%%%%%%%%%----------------------------------------------------------
%
%    
    ~~~So far  we considered  the distributions
    of physical variables that describe the individual features 
    of leptons belonging to a $l^{+}l^{-}$ pair. 
    Here we shall discuss a set of variables 
    which characterize the lepton pair system (and 
    the quark-antiquark system) as a whole.

   Figure 9 of Section 7 (Appendix) contains the  plots that 
   present  the distributions of the number of generated events 
   versus the  most typical  physical variables that may describe 
   the initial and final states of two particle systems which
   appear in the annihilation subprocess
   ${q\bar{q} \to \gamma^{*} \to l^{+}l^{-}}$ .
   Let us firstly mention those of them which include different
   components  of the total  4-momentum  
   $P^{l{+}l{-}}_{\alpha}$  (${\alpha}={0,1,2,3}$)
   of a lepton pair : 
%    ( $\mu\mu \equiv  \mu^{+}\mu{-}$)  \\

    
    
%\begin{equation}
%%%\bean
%M_{\pi^0\to\gamma e^+ e^-}(q_1,q_2|P)=\frac{F_{\pi^0\to\gamma e^+ e^-}
%(q^2_1)}{q_1^2} V_{\mu\nu}(q_2|P) (e)  j_{V}^{\mu}(p_+,p_-) e^{\nu}
%(q_2) \, \, ,
%%%\eean
%\end{equation}
  

\begin{equation}
%\bean
{P^{l{+}l{-}}_{\alpha}= P^{l^{+}}_{\alpha} + P^{l^{-}}_{\alpha}}.
%\eean
\end{equation}
The distribution of the total energy of a lepton system 

\begin{equation}
{E^{l^{+}l^{-}} = E^{l^{+}} + E^{l^{-}}} ,
\end{equation}
is shown  in the right hand plot of the middle raw
of  Figure 9 of  Appendix 
( and noted there as the ``E-l+l- distribution'' ).  
The longitudinal component distribution

\begin{equation}
{P^{l^{+}l^{-}}_{z} = P^{l^{+}}_{z} + P^{l^{-}}_{z}}  
\end{equation}
(noted there as ``Pz-l+l- distribution'') is included 
into the left hand side plot of the same middle raw.
These two plots look very similar. The explanation of this fact 
follows from the form of the plot for the transversal component 

%%%  VPT56   P^{q\bar{q}}_{T}  $P^{q\bar{q}}_{T}\equiv PT-qqbar-tot$

%\vec{P}^{q\bar{q}}_{T}=\vec{P}^{q}_{T} + \vec{P}^{\bar{q}}_{T} 


\begin{equation}
{\vec{P}^{l^{+}l^{-}}_{T} =
 {\vec{P}^{l^{+}}_{T}} +{\vec{P}^{l^{-}}_{T}}},
\end{equation}
 namely, of its modulus 
\begin{equation}
  P^{l^{+}l^{-}}_{T} = |\vec{P}^{l^{+}l^{-}}_{T}| =
   |\vec{P}^{l^{+}}_{T} + \vec{P}^{l^{-}}_{T}|,
\end{equation}
which is presented as 
% $P^{l^{+}l^{-}}_{T} \equiv ``PT-l+l-distribution''$
$``PT-l+l-distribution'' (\equiv P^{l^{+}l^{-}}_{T}) $
in the left bottom corner  of Figure 9. From here one may
see that the transversal component distribution is, really, much
more narrower (its mean value is about 1 GeV) than those of
${P^{l^{+}l^{-}}_{z}}$ and ${E^{l^{+}l^{-}}}$
distributions. 
Thus, the main contribution to the 
value of the lepton pair energy comes from the
longitudinal component ${P^{l^{+}l^{-}}_{z}}$.
It is worth to mention that both of
${P^{l^{+}l^{-}}_{z}}$ and ${E^{l^{+}l^{-}}}$
distributions
do not start exactly from the zero values.  The starting 
point of the total energy  $E^{l^{+}l^{-}}$ spectrum 
is around 1 GeV.
         \footnote { This value is in an  accordance 
          with the shape  of the 
           $M_{inv}^{l^{+}l{-}}$- and 
           $M_{inv}^{ q\bar{q} }$ - distributions that
           are shown in the upper raw of Figure 9 .
           See also the Introduction and the discussion below.
           Note also that, according to the top raw plots of
           the Figure 1,  the energy spectra of individual
           leptons in a pair may start, practically, from the
           zero values.}
It is important to mention that the spectrum of a lepton pair
energy in $\bar{p}p$ collision has the mean value around 5 GeV
(as well as the spectrum of the 
${P^{l^{+}l^{-}}_{z}}$ component).

It means that on average the lepton pair caries a bit 
more than 1/3 of the total energy (in the rest frame of 
the proton) of the $\bar{p}p$-system. From the same
right hand plot of the middle raw of Figure 9 it is
clearly seen that in about a half of events the energy
of a lepton pair may be  higher than 5 GeV and it may 
spread up to the value of 12 GeV.
 
%{\mu^{+}\mu{-}}
%
%sqrt(( P^{\mu^{+}\mu{-}})^2)
%
%{\mu^{+}\mu{-}}

   Another important variable for describing the partonic subprocess 
${q\bar{q} \to \gamma^{*} \to l^{+}l^{-}}$
is the  modulus of the ${P^{l^{+}l^{-}}}$
4-vector, which we shall denote as ``Q''  and  
%which is defined as follows:
which is defined as follows:

\begin{equation}
{Q^2=( P^{l^{+}}_{\alpha} + P^{l^{-}}_{\alpha})^2 }.
\end{equation}


 It is clear from (1) that Q
%is clear from (1) that Q
 has the meaning of 
the invariant mass of two signal leptons pair, i.e.
\begin{equation}
 M_{inv}^{l^{+}l{-}}=\sqrt( P^{l^{+}l{-}} )^2=Q=\sqrt(Q^2) .
\end{equation}
The left hand side plot in the upper raw of Figure 9 includes
the distribution of the number of events versus the values of
this variable  (it is denoted there as
$``M_{inv}-l+l- distribution''~~(\equiv M_{inv}^{l^{+}l{-}}) $).

Let us look now at the set of  variables that describe
the initial quark-antiquark state of the annihilation
subprocess ${q\bar{q} \to \gamma^{*} \to l^{+}l^{-}}$.
In analogy with (1) one  may consider the 4-momentum of
${q\bar{q}}$ system
\begin{eqnarray}
%\vec{P}^{q\bar{q}}=\vec{P}^q + \vec{P}^{\bar{q}} 
P^{q\bar{q}}_{\alpha} =P^{q}_{\alpha} + P^{\bar{q}}_{\alpha},
\end{eqnarray}
and construct with its help the variable of the invariant
mass of quark-antiquark system  $M_{inv}^{q\bar{q}}$
(in PYTHIA it is  named  as ``$\hat{m}$''). Thus,

\begin{equation}
 M_{inv}^{ q\bar{q} }=\sqrt( P^{q\bar{q}})^2=\hat{m} .
\end{equation}
Its distribution is shown in the right hand plot of 
the same first raw of Figure 9
%($ M_{inv}^{ q\bar{q} } \equiv ``M_{inv}-qqbar$ distribution'' ).
$(``M_{inv}-qqbar~~distribution''~~(\equiv M_{inv}^{q\bar{q}})) $.

%i.e. $\hat{m}= M_{inv}^{ q\bar{q} }=\sqrt( P^{q\bar{q}})^2 $.


   Both of  the  considered here invariant mass distributions,
   i.e. $ M_{inv}^{ q\bar{q} }$= $\hat{m}$ and 
   $ M_{inv}^{ q\bar{q} }=Q$ ,
   look very similar. From the  plots of Figure 9  it
   is seen  that the most of events are concentrated near 
   the starting  point  $M^{l^{+}l{-}}_{inv} = 1$ GeV
   and there are, practically, no events  after the point
   $M^{l^{+}l{-}}_{inv}$ = 2.5 GeV.
   So, the spectra of the invariant masses  of produced
   quark-antiquark system as well as of lepton  pair are 
   rather narrow. The appearance of the left boundary 
   points 
%  $\hat{m_{min}}=Q_{min}=1$ GeV
   $M_{inv}^{l^{+}l{-}}/min$ = $M_{inv}^{q\bar{q}}/min$ =1 GeV
   stems from the  internal PYTHIA restriction
   (see the discussion above and the Introduction) for the
   lowest value of  invariant mass of two body system produced
   in  fundamental quark-parton level $2 \to 2$ subprocess.  


%    ~~~ Up to now we have discussed the results obtained
%    without any other kinematic cuts than those one 
%    implemented  onto internal PYTHIA variables (what was  
%    mentioned in the Introduction) and needed to run  this 
%    generator at as low as possible values of beam energy.
%
%
%    As it was mentioned in the Introduction, in our generation
%    we had the value of the
%    invariant mass of $\mu^{+}\mu^{-}$ signal pair, produced
%    in event, to be restricted
%    by the inequality  $ M_{inv}^{\mu^{+}\mu^{-}} >  1$ GeV.

%%%     as well as well as the value of  ${\hat{p}_{\perp}^{\;min}}$
%    (see the discussion in Section III).

    The distributions of Bjorken x-variables  are shown in
    Figure 10  for up- and antiup- quarks (top raw) and for
    down- and antidown- quarks (bottom raw) correspondingly.
    They represent the  up- and antiup- quarks components
    of the proton structure function.
%    (see pages 11 and 10 of present Attachment
%    The dotted lines in these Figures show the original shape


%      In what follows we shall refer to these distributions
%    as to the "unbiased" ones because the influence of 
%    different cuts  onto muons energy $E^{\mu}_{cut}$ as 
%    well as the angle  cuts (see Sections 2 and 5), connected  
%    with the possible different choices of the size of muon 
%    system would be presented in future publications.
%    As it can be easily seen from these plots
%    the obtained distributions do not start from the point
%    x=0, what take place for the parton distribution
%    parameterizations used in PYTHIA. Such a difference at low
%    values of x appears due to the mentioned above cut on the
%    value of produced  muon pair invariant mass
%     $ M_{inv}^{\mu^{+}\mu^{-}} >  1$ GeV .


    The number of entries in  these plots reflects the valence 
    quark  flavor structure of the proton (the analogous
    distributions and the number of entries for the antiquarks,
    given in Figure 10, show the absolute similarity of quark 
    and antiquarks distributions, obtained from the generated
    PYTHIA samples).




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%----------------------------------------------------------
\section{ The remarks on the possibility to measure the 
          size of the intrinsic quark transverse momentum
          in the proton and on the role of the initial
          state radiation.}

                               
%%%%%%%%%%----------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    Now let us consider, following \cite{SBKLHC}-\cite{BSTevat}
and in a complete analogy with the equations (4)and (5),  
 the vector sum of quark and anti-quark
transverse momentum vectors
\begin{eqnarray}
%\vec{PT}^{q\bar{q}}=\vec{PT}^q + \vec{PT}^{\bar{q}}
\vec{P}^{q\bar{q}}_{T}=\vec{P}^{q}_{T} + \vec{P}^{\bar{q}}_{T}. 
\end{eqnarray}
%\vspace{-7mm}

and the modulus of this vector sum 
\begin{eqnarray}
 P^{q\bar{q}}_{T} = |\vec{P}^{q\bar{q}}_{T}| =
|\vec{P}^q_{T} + \vec{P}^{\bar{q}}_{T} | .
%\vec{Pt}^{5=6}=|\vec{Pt}^q + \vec{Pt}^{\bar{q} } |
%\vec{Pt}^{q\bar{q}}=|\vec{Pt}^q + \vec{Pt}^{\bar{q} } |
\end{eqnarray}
The last variable has a  distribution  (see right hand side 
bottom corner of  the Figure 9 where it is denoted as 
 ``PT-qqbar-$ distribution'' ( \equiv P^{q\bar{q}}_{T}$)
which shape  practically  coincides (in  accordance  with the
transverse momentum conservation law) with the  shape of the 
distribution of the number of events versus the lepton pair 
transverse momentum   $P^{l^{+}l^{-}}_{T}$, shown in the 
left bottom corner.

   The variable  $P^{q\bar{q}}_{T}$
is of a special  interest because
it contains an important information about two  physical 
features of quark dynamics inside the hadron \footnote {see 
\cite{SBKLHC}-\cite{BSTevat} (it was denoted there as VPT56).}.
Really,
in our case when a beam antiproton moves along the z-axis
and scatters  off the proton
fixed target (we neglect the effect of target drops 
transverse motion ) the only sources of  transverse motion
of quarks \footnote {the transverse motion of the constituents
of a target proton (which is at rest)  as well as of those 
inside a beam antiproton (which moves along z-axis) is
invariant under boosts transformations along the z-axis.}
may be as follows: 
 
       a) quark internal Fermi-motion (with some transverse  
          velocity) inside a \\
          proton, i.e. the  so called " $k_{T}-effect$" ;

       b) initial state radiation (ISR) of gluons (or 
          photons) from quarks before \\
          hard quark-antiquark annihilation.
 

      The importance of these two  mentioned above  effects 
    was recently discussed
    in a number of papers devoted to the study of "${\gamma +
    jet}$" events at LHC  \cite{SBKLHC},  \cite{BKSLHC}
    and Tevatron \cite{BSTevat}. A lot of
    attention was  paid to this problem also  in \cite{Huston}
    in connection to the interpretation of direct photons
    production  study in the experiment E706 at Fermilab
    \cite{E706}.

       In the absence of these two sources the initial state
    quark and antiquark (from the colliding  antiproton and 
    proton, respectively) should have zero  transverse  
    momenta.   Here we have used the  possibilities of 
    PYTHIA generator which allow to switch  ``on/off'' each of  
    ``a)''-  and ``b)''-  effects separately as well as to
    switch them ``off/on'' simultaneously. 
      Let us remind (see Section 2) that both plots in the 
    third  raw of Figure 9 are done for a case when  ``a)-effect'' 
    (i.e. ``Fermi motion'' or "$k_{T}$-effect") as well as 
    ``b)-effect'' (or ISR) were switched  ``on''(i.e., we
    have chosen the parameters $MSTP(91)=MSTP(61)=1$).
     \footnote { Let us note also
    that sometimes in the literature the both of these
    features  appear  under one and the same name of 
    "$k_{T}$-effect"  (see, for instance, \cite{Huston}).} .

        Firstly we have checked that in a case when  both sources
    a) and b) were switched  ``off''  the corresponding plots
    for  
    ${P^{l^{+}l^{-}}_{T}}(\equiv''PT-l+l-$ distribution``)
    and
    $P^{q\bar{q}}_{T} (\equiv ``PT-qqbar-$ distribution'')
    really show zero in their distributions (the initial state 
    quark  and antiquark
    have zero transverse momenta  and thus they may produce 
    a lepton-antilepton  pair in which the leptons may have either 
    zero or  back-to-back  transverse  momenta).
    Therefore, any deviation from zero of the
    \textit {total transverse momentum of a lepton pair},
%    \large {total transverse momentum of lepton pair},
    produced in a hard quark-antiquark 
    annihilation, may appear only due to the presence of 
    one of  the two mentioned  above features ``a)'' or 
    ``b)' of quark dynamics, or as the simultaneous
     result of  both of them.
     
    To study the influence of ISR-effect we have switched it ``off'' 
    (i.e. we set MSTP(61)=0) while keeping the
    parameter of Fermi  motion, i.e.  "$k_{T}$-effect", to be 
    switching ``on'',  (i.e.  MSTP(91)=1). No difference (in
    the shapes  as well as in the mean values) was found between
    the plots for $P^{l^{+}l^{-}}_{T} (``PT-l+l- 
    distribution'')$  and $P^{q\bar{q}}_{T} (``PT-qqbar- 
    distribution'')$,  obtained in this case (when only the effect
    of  Fermi  motion or "$k_{T}$-effect" was taken into account)
    with the previously discussed  plots shown in the lowest
    raw of Figure 9 done with the account of both effects
    (MSTP(91)=MSTP(61)=1). \footnote{ This  may  mean that 
    there is not enough  volume of phase space left for the
    initial state radiation  at the chosen value of beam energy.}  
        
      So, one may conclude that, within PYTHIA simulation,  
    switching ``on/off'' of the initial state radiation does
    not produce any effect on the total lepton  pair transverse
    momentum  at the chosen beam energy (14 GeV). 
    This result  may indicate (still keeping in mind
    that this feature is based on the ISR model used
    in PYTHIA, i.e., it has a  model nature)  that the
    phase  space limitations at the chosen value of beam
    energy $E_{beam}=14$ GeV may allow 
    to measure at PANDA the Fermi motion effect separately 
    from the ISR effect. Further study in this direction is
    necessary, of course. Thus, future measurements  of the  
    total transverse momentum  $P^{l^{+}l^{-}}_{T}$
    of the  lepton pairs at PANDA energies  may be of a big
    interest for  a proper treatment of many QCD  results
    from LHC (and Tevatron) as well as by itself.

%    because
%    in any case the ISR  contribution  would be  naturally 
%    restricted  by the of a final state energy range 
%    available at PANDA .

%      Another argument in favor of measuring this distribution
%    at PANDA is also based on the natural existence of an
%    upper limit on the total energy of the produced final
%    state due to the fixed and limited value of the beam energy,
%    what should  result  in a reasonably limited number 
%    of the final state particles.  Really, in  such a case the   
%    application of  a simple picture of valence quarks 
%    interactions in $P\bar{P}$-collision is quite justified.
%    Thus,  within the Born  approximation,  
%    the process $ p\bar{p} \rightarrow 2\pi + l\bar{l} $
%    may be defined  by a contribution of the quantum field
%    theory (QFT) diagram 
%    that would include a subprocesses in which  two valence
%    quarks  (that have  not taken part in the QCD annihilation
%    (see the corresponding Feynman diagrams at Figures...''-->
%    --> they are not yet ready  for this draft `` ),
%    subprocess  $q\bar{q} \rightarrow \mu^{+}\mu^{-}$ ) 
%    do combine with the remaining two antiquarks 
%    from the antiproton to produce two pions.
%    The QFT diagram of such a process would  not include
%    any gluon line. Hence, the corresponding process may
%    by described at the leading order by a purely quark
%    model diagram (i.e. without an inclusion of  any 
%    gluon  emission lines) and therefore 
%    may have nothing to do with the ISR processes.
%   in difference with that one
%    shown in Figure ........( which has to include first
%    two diagrams from the first page of draft ).
%

      To this reason we may suppose that the fraction of 
   events that have the final state consisting of
   a $l\bar{l}$-pair  and a pair of two charged pions
   (see also the discussion  in the following Section)
   may be the most suitable for the  measuring of the primordial
    "$k_{T}$-effect" (caused by  the intrinsic
   Fermi motion of quarks inside the proton) by  measuring 
   of the total transverse momentum of the lepton-antilepton
   pair in the final state.
% 
%
%
%%%%%%%%%%%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%\section{ Structure functions: \\ ~~~~u- and  d- quark distributions.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%  II. SRUCTURE FUNCTIONS AND u- AND d- QUARK DISTRIBUTIONS.
%    ---------------------------------------------------------          
%
%%%%  ${\hat{p}_{\perp}^{\;min}}$
%
%    ~~~ Up to now we have discussed the results obtained without 
%    any other kinematic cuts than that one implemented  onto internal
%    PYTHIA variables (and mentioned in Introduction) and needed to
%    run  this generator at as low as possible values of beam energy.
%    As it was mentioned in the Introduction, in our generation
%    we had the value of the
%    invariant mass of $\mu^{+}\mu^{-}$ signal pair, produced
%    in event, to be restricted
%    by the inequality  $ M_{inv}^{\mu^{+}\mu^{-}} >  1$ GeV.
%
%%%     as well as well as the value of  ${\hat{p}_{\perp}^{\;min}}$
%    (see the discussion in Section III). 
%
%    The distributions of Bjorken x-variables  are shown in
%    Figure 10  for up- and antiup- quarks (top raw) and for
%    down- and antidown- quarks (bottom raw) correspondingly.
%    (see pages 11 and 10 of present Attachment
%    The dotted lines in these Figures show the original shape
%    of the distributions implemented into PYTHIA  and allow
%   to see the influence of internal PYTHIA cuts on them.
%
%       In what follows we shall refer to these distributions  
%    as to the "unbiased" ones because the influence of different cuts
%    onto muons energy $E^{\mu}_{cut}$ as well as the angle cuts
%    (see Sections 2 and 5),  connected  with possible different
%    choices of sizes of muon system would be 
%   presented bellow.
%    studied latter. As it can be easily seen from these plots
%
%    the obtained distributions do not start from the point
%    x=0, what take place for the parton distribution
%    parameterizations used in PYTHIA. Such a difference at low 
%    values of x appears due to the mentioned above cut on the
%    value of produced  muon pair invariant mass
%     $ M_{inv}^{\mu^{+}\mu^{-}} >  1$ GeV .
%      
%
%    The number of entries at these plots reflects the valence quark 
%    flavor structure of the proton (the analogous distributions 
%    and the number of entries for the antiquarks show the absolute 
%    similarity of quark and antiquarks distributions).
%
%    As can be seen from the plots     
%    The plots of Feynman variable $x_{F}=x_{1}-x-{2}$ distributions (see
%    Figure 31 at page 9  of present Attachment) show  the difference
%    in momenta of quarks that participate in the annihilation subprocess
%    ${q\bar{q} \to \gamma^{*} \to \mu^{+}\mu^{-}}$. 
%    is rather small in the most of events. 
%
%    ${q\bar{q} \rightarrow \gamma^{*} \rightarrow $\mu^{+}\mu^{-}$}$
%    
%       The influence of the set of the polar angular cuts
%     of the signal muons produced in one and the same event :\\
%
%
%      ${\theta^{\mu}_{slow}}$, ${\theta^{/mu}_{fast}} < 40^{o}$;
%
%      ${\theta^{\mu}_{slow}}$, ${\theta^{/mu}_{fast}} < 60^{o}$; 
%
%
%      ${\theta^{\mu}_{slow}}$, ${\theta^{/mu}_{fast}} < 90^{o}$; 
%    onto the x-distributions is shown in Figures ....... 
%    ( not yet finished , would be presented later)
%    correspondingly.  From here we can conclude ....... 
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%*****************************************************************



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{  Fake muons from hadron decays in signal events.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%     IV.  FAKE MUONS FROM MESON DECAYS


%    ~~~ All what was said above may be to a good approximation
%    applied also to a case of electron-positron pair production
%    in the final state. The most difference of  $e^{+}e^{-}$ 
%    case with that one of ${\mu^{+}\mu^{-}}$ pair production
%    appears, as it was already mentioned in the Introduction,
%    when one turns to the problem of background or fake leptons
%    appearing in the same signal lepton pair production process
%    from meson decays.

      The signal events, defined by 
    $q\bar{q} \rightarrow l^{+}l^{-}$ subprocess, should  
    contain also some hadrons in the final state. The pions,
    produced  directly or in the  decays cascades of other
    hadrons,  may decay  in the  detector volume and thus
    serve as a main source of background  muons that may fake
    the signal muons, produced in the annihilation subprocess.
    
     Figure 11 includes (like in the  Figures 1 and 2)   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
%    (smotri  stranicu  13 v  DY_PANDA.ps ) !!!!!!
% 
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    the distributions (from top to bottom) of the  number of 
    events versus the energy $E_{PI}$= $E_{\pi}$,  the transverse
    momentum $ PT_{PI}$ = ${PT_{\pi}}$  and versus the polar angle  
    $TETA_{PI}$= $\theta_{\pi}$ of produced pions.

    The right hand side bottom plot contains  
    the distribution of the total number (NPI)  of charged
    ${\pi}$-mesons in the signal events. One may see from this 
    plot that there is a big number of signal events (about $42\%$)
    which do not contain any charged pions at all (their final 
    states mostly include the nucleon-antinucleon pairs).
 
      This result is rather interesting. Let us underline that 
    PYTHIA provides a good (at least one of the bests if not 
    the best one) but still a model approximation 
    \footnote{due to the fact of up to now absence of a complete
    physical and theoretical understanding of parton to hadrons 
    fragmentation processes.}
    to the hadronization effects that take part in a reality. 
    The right hand side plot of Figure 11 indicates that it may
    happen so that about of $40\%$  of events with signal muon
    pairs may appear practically  without any additional fake
    decay  muons content. 
%    (it does not yet means that the muon
%    channel may be quite competitive to the $e^{+}e^{-}$ one).
    
    From the same right hand side plot we also see that  about  
    of  $25\%$ of events have only one charged  pion in the
    final state (appearing mostly from  ${\Delta}$ resonance 
    decays) and a bit less than  $25\%$ of events may have 
    two charged pions.  It demonstrates also  that about of 
    $5\%$ of events may  have 3 charged pions and there may 
    be about of $1.5\%$ of events with 4 final state 
    charged  pions. 


    Figure 12 contains the plots with the fake muon distributions
    in signal muon events.
    Its  left column includes, respectively, from top to bottom, 
    the energy  $E^{\mu}_{dec }$, transverse momentum
    $PT^{\mu}_{dec }$  and  polar angle $\theta^{\mu}_{dec }$
    distributions for  muons that appear from hadrons (mostly charged  
    ${\pi}$ -mesons) decays. The shapes of distributions follow the 
    analogues spectra of parent pions, presented in Figure 11.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%   (It should be 
%    noted that while the simulation we have used, as the 
%    first approximation, the existing PYTHIA option to 
%    restrict the  decay volume inside the detector by the
%    cylinder with  R=3m and L=8m.      
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    By comparing of these plots with those from signal muon 
    pairs (see Figure 1), one may conclude that the 
    mean  value of the  energy of signal  muons 
    $< E^{mu} > $ =2.6 GeV  corresponds to the point in 
    the energy distribution of fake decay muons where the
    contribution of fake  muons (in the same signal events) is 
    already very low. Analogously the mean value of  the PT-
    distribution of signal muons $PT^{mu}_{signal} = 0.7$ GeV
    (see Figure 1) do  corresponds to the point where the
    spectrum of $PT^{mu}_{dec}$ practically vanishes .
    
      The E- and PT- spectra of decay fake muons are rather similar
    to those of ``slow '' muons from Figure 2, but a  bit softer
    than the last ones. They are  shifted by 
    300 MeV (i.e. by $30\%)$ to the left (see their mean values)  
    as comparing  with  those of signal ``slow'' muons. Analogously
    the fake decay muons polar angle  spectrum is 
    shifted also to the left by  $30\%$, i.e. by $16^{o}$.
    Still,  due to a big similarity of "decay" and ``slow''
    signal  muons the former may appear as a serious background.
       
      Therefore  a reasonable cut upon the muons energy
    $E^{mu}_{dec}$ (see discussion in Section 2) as well as 
    on the  $PT^{mu}_{dec}$  value may lead to an essential 
    reduction of the decay mesons  contribution and allow to 
    keep the main part of signal events. Thus, a cut 
    $PT^{mu}_{dec}> 0.2$ GeV may allow to get rid of a half
    of decay mesons in the signal events.
     
      Another way to discriminate up to a some level the signal 
    muons from 
    the "decay"  ones may be connected with the  use of the 
    information about the position of the  fake muon production 
    vertex. The right hand side column in Figure 12 contains (from
    top to bottom) the Vx-, Vy- and Vz- (i.e. along the beam 
    direction)  components of the 3-vector that defines the position 
    of the fake muon production vertex, obtained, as everything at 
    the PYTHIA generator level of simulation, for a space that is
    free of the detector equipment. The distances at these
    distribution plots  are  given in millimeters (mm). Therefore,
    one  may see that the tails of the Vz-distribution may expand 
    up  to 100 meters ($82\%$ of events have $Vz < 100$ m)
              \footnote{the ``Overflow = 1.828E+04'' shows
               that in about of $18\%$ of events the decay 
               muons may appear 
               even at more larger than 100 m distances.}, 
    while those that  are along x- and  y- axises 
    practically do finish at the 20 meters distance. From these
    right hand plots  it is  clearly seen that in a case of 
    the signal events that do include  the "decay"  fake muons 
    ($58\%$ of all events) the account of the real size of the  
    detector ``decay volume''  may strongly reduce the number of 
    charged ${\pi}$-mesons, that potentially may produce muons
    in the decays,
         \footnote{ because at large distances the most of 
                    parent pions should interact with the 
                    material of the detector components.} 

    To this reason, one may expect that in the  real
    experimental  conditions the situation with the contribution  
    of the  additional  "decay" fake muons in signal events may 
    turn out to be even much more easier. Only $25\%$  of 
    events  have the position of Vz-component (as it may be seen
    from the right bottom plot of Figure 12)  within the  PANDA 
    detector volume, i.e. $V_{z} < 10$ m. At the same time it is
    also seen that in about of  $5\%$ of events the Vz componet 
    may be  very close to zero. The corresponding fake muons
    produced near the interaction point may pose the most
    difficult background.
    Thus the detailed GEANT 
    simulation, which is in our nearest plans, should allow
    to get more definite predictions about the decay mesons
    contribution to the background in signal events. The 
    contribution of the background muons from the other
    type of events (QCD mainly) should be discussed in the 
    following Section.

%    Figure 38 at  page 6 of present Attachment )
%    shows that the number of background decay muons
%    flying in background  hemisphere is smaller then the 
%    analogous number of slow muons (see Fig. 14 at the  page 3 
%    of present Attachment ) . 

%    The information about muon
%    production vertex may be also quite useful for separating
%    of background  meson decays muons.
%    (see Figures .... at Future Attachment).



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{ Lepton distributions in QCD background events.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




     In this Section we consider the distributions of leptons
   that may be produced in other than QED
   $q\bar{q} \rightarrow l^{+}l^{-}$ subprocesses. For this aim 
   the generation of 100000 events was done with the use of 
   more than 20 QCD  subprocesses (including the signal   
   $q\bar{q} \rightarrow l^{+}l^{-}$ subprocess ) that exist
   in PYTHIA. The  results have shown that the main contributions
   come from  the events  that  include the following (presented
   in the decreasing order of  $p\bar{p} \to X$ ) 
   partonic subprocesses:\\

~~~~~  $\bullet$  $q + g \rightarrow q + g$ \\ 

~~~~~   (gives $50\%$  of events and the corresponding value of cross section 
        $\sigma= 4.88$ mb); \\

~~~~~  $\bullet$  $g + g \rightarrow g + g$ \\ 

~~~~~   (gives $30\%$  of events and the corresponding value of cross section 
        $\sigma= 2.96$ mb); \\

~~~~~  $\bullet$   $q + q' \rightarrow q + q'$ \\

~~~~~   (gives $18\%$  of events and the corresponding value of cross section  
        $\sigma= 1.75$ mb);\\

~~~~~~  $\bullet$  $q + q' \rightarrow g + g'$ \\

~~~~~   (gives $0.6\%$  of events and the  value of cross section 
        $\sigma= 5.89E-02$ mb); \\

   The signal  $p\bar{p} \rightarrow l^{+}l^{-} + X$ processes,
   that  includes the QED partonic subprocess: \\

~~~~~~  $\bullet$ $q + \bar{q} \rightarrow l^{+} + l^{-}$ \\

~~~~~ has the value  of $\sigma= 1.72E-05$ mb what explains its  
    very small contribution into the total number of generated
    events: there is only one event of this kind has appeared 
    among 100 000 of generated events.

      The shapes of the distributions of muons, produced  in 
   the  mentioned above generated background QCD events (see left 
   column of Figure 13 in Appendix) are not so much differ (what
   is natural) from those of  fake ``decay'' muons produced 
   in the signal  $p\bar{p} \rightarrow l^{+}l^{-}$ processes,
   shown in Figure 12. The main 
   difference appear  in the number of entries: for the 100 000
   of generated background events there are 3.3 times more
   entries  for  muons (335372), i.e. there are, on 
   average,  about of  3 muons (that mostly appear from pions
   decays)  per one QCD background event. It means that the
   probability of appearing  of a muon pair with the different
   signs of their charges (like in signal events) is a rather 
   high one in QCD background events. So, they may fake quite
   well the  signal events.
  
   The distribution plots for the production vertices of 
   background muons are shown in the left column of Figure 13.
   From the left column bottom plot of Figure 13 it is seen 
   that  in about of $75\%$  of events there are $V_{z}>10$
   meters.
            \footnote{see ``Overflow is 2.449E+05'' in the
             right bottom plot of Figure 13.}
   Thus, in
   these events the potential parent hadrons definitely should
   not produce muons within the volume of PANDA detector.
   At the same time one may see that 
   for a lot of events the muon  production vertices  
   are rather close to the interaction  point: in  about of 
   $0.5\%$ of events the values of Vz are practicaly equal 
   to zero and on total about of $5\%$ of events have
   $V_{z} < 1$ metre and $25\%$ of events  have  $V_{z} < 10$
   metres. So, these fake decay muons may be separated from the 
   signal muons latter on at the level of full GEANT simualtion 
   with account of real detector conditions if, firstly, 
   the information  from the tracker
       \footnote{the ideal situation would be if the tracker
                 should allow to discriminate between 
                 the pion or muon tracks. It is also supposed
                 that the problem of matching the
                 information  from the tracker with the
                 information from the muon system would be
                 solved.}
   or latter on (and what is more likely
   may happen) the information from 
   the calorimeter should allow to identify the muon parent
   particles  as hadrons. 
       \footnote{ the parent hadrons of  muons, which 
                  production  vertices may  enter the 
                  detector components volume, may give 
                  the hadron  showers in a detector material
                  before they may decay into muons.
                  Thus, all the tracks that are pointed in
                  the direction of fired calorimeter cells
                  should be excluded.}
   So, it seams that the separation of QED signal and QCD
   background events would be a rather hard problem 
   for  the muon channel and a more detailed 
   GEANT simulation of  muon processes with an  account
   of the real detector conditions (material volume and the
   magnetic field) would be needed.   


     Now let us turn to the situation with the  background for
   the  electron  production  processes. The distributions for
   the electrons, produced in 100 000 of generated  signal events
   based on the  $q\bar{q} \to  e^{+}e^{-}$ subprocess, are presented
   in the Figure 14 of Appendix. From the number of entries one may
   see that, on average, the enters number of fake electrons is 
   105283, i.e. close to  muon case (see Figure 12). The  analysis 
   of the generated event listings has shown that about of $50\%$ 
   of generated signal events have no other electrons
   than those produced in  signal  $e^{+}e^{-}$ pairs. 
   In the most of other events the electrons are produced
   in  muon decays. Some electrons do appear also  as the  
   Dalitz pairs, i.e. from the process of neutral pion decays 
   $\pi^{0}  \rightarrow  e^{+}e^{-}\gamma$. 
%      \footnote{this observation explains the two times exeed
%                of decay electrons in signal events as compared
%                with the number of analogouse decay muons
%                (produced mostly as single ones in $\pi$- meson
%                 decays) as seen from Figure 12.} 
   These pions in their turn may
   appear from the decays of $\eta$ and  $\omega$ mesons
   or from  more heavy  mesons or barions, produced as the resonance
   states according  to the  LUND fragmentation model.
%   \cite{And79},
%   \cite{And80},  \cite{And81},  \cite{And81a},  \cite{And82},
%   \cite{And83}.
   Due to the fact that the life time of these resonance states is 
   rather short, this kind of electron pairs are  produced practically
   at  the interaction point,  what is  reflected  in the right hand
   side column of Figure 14, that contains the values of the 
   components of the vector of the production  vertex position.
   This fact shows some disadvantage of e+e- channel as comparing
   with muon one where the  most of background muons were coming
   from the  light charged pions decaying in flight at rather large
   distances  from the interaction point. But it does not mean that
   the vertex position information turns not be so 
   efficient for separation of signal and background electron 
   events as  comparing to the  background muons case.

   Really, as it is seen from the right hand side bottom plot
   for the Vz-component of the vector of the fake electron vertex 
   position, only less than $2\%$ of fake electrons are produced
   near the interaction point position $V_{z}=0$. About of $97\%$ 
   of events have the
   value  $V_{z} > 10$ metres (``Overflow = 1.001E+05''), i.e.
   most likely are produced from the decays of fast muons (less
   energetic muons may give about of $0.6\%$ of fake electrons
   inside the  $V_{z} < 10$ m volume of PANDA detector,  as
   it is seen from the ``Integral'' value in this right corner
   plot).

      Let us tern now to the case of electrons produced in the
   non-signal events.     
   The corresponding  distributions, obtained from the same sample
   of  100 000 generated QCD background events, are shown  in
   Figure 15 of Appendix. From the number of entries
   one may see that, on average,  the fraction of fake  electrons 
   in QCD background events (5487) is about of 60 times less than 
   the analogous muons fraction (335372) (see Figure 13).
   This is an obvious advantage of the electron channel.
   From the left hand side column, that contains the values of the 
   components of background electron vertex production  position,
   we see that the QCD background
   electrons  are also produced mostly at the origin, i.e. from
   Dalitz decays of neutral mesons like pion, $\eta$, $\omega$ 
   and others. The number of  ``Overflow'' 
   events (443) may be connected with the fast  muon decays
%   (or with the suppressed charged  pion  decays
%    $\pi \to e + \nu$ and other channels)
   out of the detector size, (i.e. with  $V_{z} > 10$ m). 

   Now, taking into account that, as it became clear from the 
   PYTHIA event listings, in QCD events the decay electrons, that
   are produced within the PANDA volume $V_{z} < 10$ m, do appear 
   mostly in Dalitz  pairs, we may estimate the fraction of QCD 
   background  events, that really may pose the background 
   $e+e-$-pairs, as of about of $3\%$ order. 
     \footnote{ or about of 2 orders less as comparing to the decay
                muon background in muon channel}
%   Thus, the upper limit on the total 
%   number of events which  may contain the electon  pairs is 
%   of  $5\%$ order ( $2.5\%$ for muons).

      As it is seen from the background electron  Energy- and 
   PT-distributions, shown  in the left hand side of Figure 15,
   these spectra fall much more steeply (as pion energy in Dalitz 
   process have to be shared between 3 particles)  than the 
   analogous background  muon spectra of Figures 12 and 13.
   To this reason the energy cut $E^{\mu} < 2 $ GeV or the PT cut
   $PT^{\mu} < 0.4$ GeV may kill a lot of background events.

     Nevertheless, these cuts may not be sufficient for the
   compensation  of five orders difference in the rates of  
   signal  and background  events  for their clear separation. 

    A possible way to solve this task (taking into account the
   discussed above  suppression  of electron background in QCD events
   down to only  $3\%$ content in generated events) may be connected
   with the detector (a tracker and an 
   electromagnetic calorimeter) possibilities  of the identification 
   of those electron pairs  that may  be associated with their parent
   decayed mesons by the reconstruction of the invariant mass of the
   system composed of the  electron pair and of one of the registered
   photons, which may appear  from the Dalitz decay process
   $\pi^{0}  \rightarrow  e^{+}e^{-}\gamma$.
   This way, combined with the imposing of the cut from bellow
   on electron energy and its transverse momentum,  may be even much 
   more efficient in a case if the electromagnetic calorimeter 
   possibilities would also allow to reconstruct the  masses  
   of the neutral pions from the other photon pair signals in event. 
  

   So, one may conclude from here that a more detailed study,  based 
   on full GEANT reconstruction with account of the detector
   conditions, would be  very important  for further study of electron
   pair production  channel as well as for the muon one.      



%================================================================
%
\section{Conclusion.}
%
%=================================================================


    The sample of $ p\bar{p} \rightarrow l\bar{l} + X $
  events , simulated  by help of PYTHIA
  event generator with the use of the hard partonic 
  $q\bar{q} \rightarrow l^{+}l^{-}$  subprocess, is
  used to make  a set of 50 plots which give a reasonable, 
  but approximate, physical picture of the lepton pair 
  production in a case  when the antiproton beam of 
  intermediate energy  $E_{beam}=14$ GeV would scatter 
  off the  target proton.
%  The simulation was done with the restriction imposed 
%  (due to some internal PYTHIA cuts) onto the value of 
%  the lepton pair invariant  mass:
%  $ M_{inv}^{l^{+}l^{-}} >  1$ GeV. Also some tuning 
%  of PYTHIA default parameters was done to make it possible
%  to  run it effectively at the chosen value of beam energy.

    The distributions of the most of essential kinematic
   variables of individual leptons, produced in the parton 
   subprocess  $q\bar{q} \rightarrow l^{+}l^{-}$, are 
   described in the Section 2 and are  presented  in the
   Figures 1 and 2  of the Section 7 (Appendix). They show
   that the energy and the angle spectra of the fast (most 
   energetic) leptons in a pair  are very different from 
   those of slow leptons: the mean values 
   $ < {E^{l}_{fast}} > = 3.86$ GeV are about three 
   times higher than those  of slow ones
   $ < {E^{l}_{slow}} > = 1.36$ GeV. The simulation
   has shown also a tendency, which may be rather a 
   general one: the fast leptons do fly predominantly at
   smaller angles $<{\theta^{l}_{fast}}> = 16.5^{o}$
   as  comparing  to the angles of the slow 
   ones $<{\theta^{l}_{slow}}>$= $38.2^{o}$. It is 
   worth to mention that about of $6\%$ of events may 
   have slow leptons in $l^{+}l^{-}$ pairs that may
   scatter into the back
   hemisphere, i.e., ${\theta^{l}_{slow}} > 90^{o}$.
     
     The energy-energy and angle-angle correlations among 
   a slow and a fast lepton in one and the same lepton pair
   in event are also described in Section 2 and are
   presented in the plots of Figures 3-8 in Section 7 
   together with the corresponding distributions of the 
   number of events versus the corresponding lepton energies
   and angles.

%   These plots demonstrate that the bulk of 
%   events would belong to the following regions:\\
%
%   $\bullet$ in energy-energy plane, Figure 7, a "color" countur
%      within the boundaries defined as follows:\\ 
%~~~~~~ $0<{E^{l}_{fast}}<4$ GeV; $0<{E^{l}_{slow}}<2$ GeV;\\
%     
%    $\bullet$ in angle-angle plane, Figure 7, a "color" countur
%      within a square with the boundaries:\\        
%
% ~~~~~~     $0<{\theta^{l}_{fast}}<26^{o}$;
%            $0<{\theta^{l}_{slow}}<60^{o}$.       
%    
%

   It should be noted that the
   present simulation, based on PYTHIA, does not take 
   into account, of course,  the detector effects (like
   the  magnetic field, for example). Hence, the obtained plots, 
   which describe 
   the unbiased distributiond of free produced particles,
   may be useful as for the preliminary estimations mainly. 
   Nevertheless, there is  a big probability that the 
   main  features of  the  process would be similar to
   those presented at these plots. Thus, from the Figures 
   2 and 8 it is clear  that   one may expect that 
   the cut $E^{l}>0.5$ GeV, which is a very reasonable 
   due to the effect of the detector  material,
   may  lead to a loss of $25\%$ of events with signal 
   lepton pairs, mainly due to the presence of the
   peak in the slow lepton energy spectrum in this
   region.  The data from the generated  samples are planned 
   to be used for a further GEANT  simulation of these 
  processes within the detector conditions.     

    The study of the 4-momentum components variables of the 
   lepton pair system as a whole, done in the Section 3, has 
   shown that the spectrum of the invariant mass of the lepton 
   pair, which starts (due to the already discussed in
   Introduction restriction 
   onto the value of the lepton pair invariant  mass)
   at the value of  $M^{l^{+}l{-}}_{inv}=1$ GeV, decreases 
   rather fast and do vanishes at 
   $M^{l{+}l{-}}_{inv} = 2.5$ GeV.  At the same time  
   the lepton pairs total energy   $E^{l^{+}l^{-}}$ 
   spectrum starts at around 1 GeV and spreads  up to the 
   value of 12 GeV. In Section 3 it was shown also that about
   a  half of events do have the  energy of a lepton pair
   higher than 5 GeV. So, in about a half of events the lepton
   pairs are rather energetic and they take away  of more 
   than 1/3 of the total energy of the produced
   $l^{+}l^{-}$ system.

   In Section 4 it was pointed out that the total transverse 
  momentum  $P^{q\bar{q}}_{T}$ of the annihilating 
  $q\bar{q}$-pair may be caused, in principle, by two sources.
  It may appear  due to the intrinsic  transversal Fermi 
  motion of quarks inside the nucleon or in a result of 
  the initial state radiation  of gluons.
  The results of the simulation have shown that, at least 
  within the framework of PYTHIA simulation, 
  the initial state radiation (ISR) gives a negligible 
  contribution at the chosen value of energy. 
  Therefore,  it was  argued that the study of the spectrum
  of the total transverse momentum of ${l^{+}l{-}}$ pair
  (which coincides with the spectrum of the total transverse
  momentum  $P^{q\bar{q}}_{T}$ of the annihilating 
  $q\bar{q}$-pair due to the 4-momentum conservation law)
  may allow to measure the intrinsic Fermi motion of quarks
  inside the nucleon.  The results of such a  measurement  
  may be very useful for an interpretation of many QCD 
  processes  at future experiments at LHC as well as of the 
  data collected at Tevatron. 
%  It is also  pointed out that
%  the events, in which  only two charged pions would be 
%  produced in addition to a lepton-antilepton  pair, would be most
%  useful for measuring of the Fermi motion "$k_{T}$-effect".
  All what was said above and discussed in the Sections 2-4
  may be to a good approximation  applied both to the case of
  $\mu^{+}\mu^{-}$ as well as to the case of $e^{+}e^{-}$ pair
  production in the final state
    
   An important question of a background is considered 
  in Sections 5 and 6. In Section 5 we  have  concentrated 
  on the additional muons that may appear from meson decays
  in the same signal process, defined by the mentioned above
  quark-antiquark annihilation to muon pair. The  problem is 
  that the spectra of ``slow ``muons, which are present in 
  the signal muon pair, are very similar to those of fake 
  ``decay'' muons. The most important observation,  done in 
  the Section 5, is that PYTHIA predicts  that there would 
  be about of $42\%$ of events that would not contain at
  all the charged pions in the final state and thus they
  would be free of fake muons. It is also
  shown  there that the information about the coordinate 
  position of the muon production vertex, as well as the
  imposing of the cut on the values of $E^{mu}$ or of 
  $PT^{mu}$, may be  useful for the discrimination  
  of the fake muons. 

    In Section 6 the study of background lepton events is
  presented.  It has been shown that in both muon and electron
  cases the  separation  of signal and muon background should 
  not be a  simple task and it needs more efforts. It was
  pointed out that the electron pair production channel  has 
  an evident advantage as been compared with the muon one. It is 
  argued that  the  detailed GEANT simulation  with an account
  of the detector design (and based on the simulated and
  described here PYTHIA event sample)
  may be very useful. The results of this simulation should 
  be a subject of our following publications.   
 
       The authors are grateful to G.D.Alexeev for the
  suggestion of this topic for study, the interest to the work
  and multiple stimulating discussions of the questions concerned.
    


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

\begin{thebibliography}{50}

%%%%%%%%%%%%%...........................................................

%\section{PANDA  References.}


%%%%%%%%%%%%%...........................................................


\bibitem{Matv}
V.A.~Matveev, R.M.~Muradian, A.N.~Tavkhelidze,
JINR P2-4543, JINR, Dubna, 1969;
SLAC-TRANS-0098, JINR R2-4543, Jun 1969; 27p. 

\bibitem{Drell}
S.D.~Drell, T.M.~Yan, SLAC-PUB-0755, Jun 1970, 12p.;  Phys.Rev.Lett.
{\bf 25}(1970)316-320, 1970.

%\bibitem{Ting}
 
%S.~Ting
%
%\bibitem{SLAC}
%S.~SLAC

\bibitem{DYexper} 
CERN UA1 Collaboration, C.~Albajar et al., 
Phys. Lett., { \bf B209} (1988) 397;
FNAL E772 Collaboration, P.L.~Gaughey at al.
Phys. Rev. { \bf D50} (1994) 3038;


\bibitem{MRST}
A.D.~Martin {\it et al.}, Eur. Phys. J. {\bf C4} (1998) 463.

%\bibitem{DYtheor}
%D.Y.~Theor

%\bibitem{PandaLOI}
%http://www.

\bibitem{Okubo}
S.Okubo,
Phys.Lett. B 5, 165 (1963);
G.Zweig, CERN Report 8419/th 412 (1964) ;
I.Izuka, Prog. Theor. Phys (suppl 37) 38 21 (1966);


\bibitem{OBELIX1} 
M.Bargiotti et al., EPJdirect A 2,1  (2002);

\bibitem{OBELIX2}
M.Bargiotti et al., Eur. Phys. J. C 35, 177 (2004);

\bibitem{Note1}
A.N.Skachkova, N.B.Skachkov hep-ph/0412279, 2004.

\bibitem{Sjost}
T.Sjostrand, Computer Phys. Commun.{\bf 39} (1986) 347,
T.Sjostrand and M.Bengtsson, Computer Phys. Commun.{\bf 43}
(1987 )367.

%\bibitem{MRST}
%A.D.~Martin {\it et al.}, Eur. Phys. J. {\bf C4} (1998) 463.

\bibitem{ROOT}
R.~Brun and F. Rademakers, 
ROOT - An Object Oriented Data Analysis Framework, 
Proceedings AIHENP '96 Workshop, Lausanne, Sep. 1996,
Nucl. Inst. \& Meth. in Phys. Res. 
{ \bf A389} (1997) 81.
See also http://root.cern.ch/.


\bibitem{E706}
Fermilab E706 Collab., L. Apanasevich et al. Phys.Rev.D70:092009
,2004, hep-ex/0407011;  Phys.Rev.Lett.81:2642-2645,1998,  
hep-ex/9711017.

\bibitem{SBKLHC}
 
N.B.~Skachkov, V.F.~Konoplyanikov D.V.~Bandourin,
Third Annual RDMS CMS Collaboration Meeting.
CMS-Document, 1997--168. CERN, December 16-17, 1997,
p.139-153.


\bibitem{BKSLHC}
D.V. Bandurin, V.F. Konoplyanikov, N.B. Skachkov, hep-ex/0207028;

\bibitem{BSTevat}
 D.V. Bandurin, N.B. Skachkov, D0-NOTE-3948,FNAL, Mar. 2002, 
hep-ex/0203003, 2002;  Phys.Part.Nucl.35:66-106, 2004
(Fiz.Elem.Chast.Atom.Yadra 35:113-177,2004), hep-ex/0304010; 
%

\bibitem{Huston}
L. Apanasevich et al., Phys.Rev.D59:074007,1999; hep-ph/9808467;\\
J. Huston; Int.J.Mod.Phys.A16S1A:205-208,2001; 

%  { \bf B209}

\bibitem{And79}
B. Andersson, G. Gustafson and C. Peterson,
Z. Phys.  { \bf C1} (1979) 105;

B. Andersson, G. Gustafson, 
Z. Phys.  { \bf C3} (1980) 22;

B. Andersson, G. Gustafson and T. Sjostrand,
Z. Phys.  { \bf C6} (1980) 235; Z. Phys.  { \bf C12} (1982) 49;

 
\bibitem{And80} 
B. Andersson, G. Gustafson  and T. Sjostrand,
Phys. Lett. { \bf B94} (1980) 211;

\bibitem{And81}
B. Andersson, G. Gustafson, I. Holgersson and O. Mansson,
Nucl. Phys. { \bf B178} (1981) 242;

\bibitem{And81a}
B. Andersson, G. Gustafson, G.Ingelman and T. Sjostrand,
Z. Phys.  { \bf C9} (1981) 233;

\bibitem{And82}
B. Andersson, G. Gustafson and T. Sjostrand,
Nucl. Phys. { \bf B197} (1982) 45;

\bibitem{And83}
B. Andersson, G. Gustafson, G. Ingelman and T. Sjostrand,\\
Phys. Rep.  { \bf 97} (1983) 31;

%
%
%
\end{thebibliography}



%%%%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@




%%%%%%%  \end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   \section{Appendix: Figures.} 
\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig1.eps}
\vskip 5mm
\caption{\hspace*{0.0cm}  Signal leptons energy (in GeV) 
    $E^{l^{(+/-)}}$ (top raw), the modulus of 
    the transverse momentum  (in GeV)  $PT^{l^{(+/-)}}$
    (middle raw) and  the  polar angle $\theta^{l^{(+/-)}}$
    (bottom raw)  distributions. Left column is for  $l^{-}$ 
    and right one for  $l^{+}$.}
\label{fig:1}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig2.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The analogous to Figure 1. signal 
    distributions for leptons with the
    largest energy ("fast" leptons) in the lepton pair 
    $E^{l}_{fast}$ (left column) and the smaller energy 
    ("slow" leptons)  $E^{l}_{slow}$ (right column).
    Top raw includes their energies, in middle raw are $PT^{l}$ and
    in bottom raw are $\theta^{l}$.}
\label{fig:2}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig3.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The ``LEGO'' ROOT plots for the the number
    of  events distribution versus angle (in degrees)) and energy
   (in GeV) of leptons.
    It  shows the  angle/energy  correlations of signal leptons in
    events. Left hand plot shows the  correlation for ``slow'' leptons
    in event, while the right hand  one for ``fast'' leptons.}    
\label{fig:3}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig4.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The ``SURF2Z'' ROOT plots show the same as in 
    Figure 3  angle/energy correlations but as been projected
    onto $\theta^{l}$/$E^{l}$ plane. In each plot the right hand
    vertical scale shows the one-to-one correspondence between the
    color and  the number of events per square bin (see Fig.3 "LEGO" 
    and Fig.5 ``TEXT'' plots).}    
\label{fig:4}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig5.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The ``TEXT'' ROOT plots show the same as in 
    Figure 4 leptons angle/energy correlations but with the number 
    of events per square bin (see Fig.3 "LEGO" plot and compare
    with Fig.4). Exact numbers in each square may be seen by help
    of zoom applied to .ps file plots.}    
\label{fig:5}
\end{figure}
\end{center}

\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig6.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The ``LEGO'' ROOT plots for the the number
    of events distribution versus  angle and energy of leptons.
    It shows the  the signal leptons
    $E^{l}_{slow}$/$E^{l}_{fast}$ (left side) and
    $\theta^{l}_{slow}$/$\theta^{l}_{fast}$ (right side)
    correlations in generated events.}    
\label{fig:6}
\end{figure}
\end{center}



\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig7.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The ``SURF2Z'' ROOT plots show the same as in 
    Figure 6   $E^{l}_{slow}$/$E^{l}_{fast}$ (left side)
    and  $\theta^{l}_{slow}$/$\theta^{l}_{fast}$ (right side)
    correlations but as been projected onto 
    $E^{l}_{slow}$/$E^{l}_{fast}$ 
    and  $\theta^{l}_{slow}$/$\theta^{l}_{fast}$ planes.
    The right hand vertical scale in each plot
    shows the one-to-one correspondence between the color and
    the number of events per square bin (see Fig.6 "LEGO" and Fig.8
    ``TEXT'' plots).}    
\label{fig:7}
\end{figure}
\end{center}



\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig8.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The ``TEXT'' ROOT plots show the same as in 
    Figure 6 and Figure 7 leptons energy/energy and angle/angle
    correlations but presented as the number  of events per each 
    square bin (see Fig.6 "LEGO" plot and compare with Fig.7). 
    The exact numbers  in each square bin may be seen by help of zoom 
    applied to .ps file plots.}    
\label{fig:8}
\end{figure}
\end{center}




\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig9.eps}
\vskip 3mm
\caption{\hspace*{0.0cm} The upper raw includes the distributions of 
              invariant masses of lepton pair $M_{inv}^{l^{+}l{-}}$ 
              (left column) and  of quark-antiquark system 
              $ M_{inv}^{ q\bar{q} }$ (right column) . The middle 
              raw contains the distributions of Pz- and energy-
              components of $l^{+}l{-}$ system .
              In the bottom raw the total transversal momenta
              of lepton system (left corner) and of
              of $q\bar{q}$ system (right side) are shown. }             
\label{fig:9}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig10.eps}
\vskip 5mm
\caption{\hspace*{0.0cm}  Top raw shows the x distributions of
    valence up -quarks and up-anti-quarks, while bottom raw 
              includes the analogous distributions of
              down-quarks and down-anti-quarks.}
\label{fig:10}
\end{figure}
\end{center}



\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig11.eps}
\vskip 5mm
\caption{\hspace*{0.0cm} The left column  includes the distributions
              (from top to bottom, respectively) of number of
              events versus the energy $E_{PI}$=$E_{\pi}$, the
              transverse momentum  $PT_{PI}$= $PT_{\pi}$  and 
              versus the polar angle $TETA_{PI}$= ${\theta_{\pi}}$
              of produced pions. The right hand  plot shows the
              distribution of the total number (NPI)  of charged
               ${\pi}$-mesons in the signal events.}
\label{fig:11}
\end{figure}
\end{center}



\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig12.eps}
\vskip 3mm
\caption{\hspace*{0.0cm} The left column  includes the distributions
    (from top to bottom, respectively) of number of events versus the 
    energy $E_{dec }^{\mu}$, the transverse momentum $PT_{dec }^{\mu}$ 
    and  the polar angle  ${\theta_{dec }^{\mu}}$ of decay muons, 
    produced  in hadron decays in signal events that include the
    $q\bar{q} \rightarrow \mu^{+}\mu^{-}$ subprocess. The right column 
    (from top to bottom)  shows the  distributions (in mm)
    of $V_{x}$-, $V_{y}$- and  $V_{z}$- components of the position
    vector of a decay electron production vertex.}
\label{fig:12}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig13.eps}
\vskip 3mm
\caption{\hspace*{0.0cm} The left column  includes the distributions
    (from top to bottom, respectively) of number of events versus the 
    energy $E_{dec }^{\mu}$, the transverse momentum $PT_{dec }^{\mu}$ 
    and  the polar angle  ${\theta_{dec }^{\mu}}$ of  muons, produced 
    in QCD background events. The right column (from top to bottom) 
    shows the  distributions (in mm) of  the $V_{x}$-, $V_{y}$- and  
    $V_{z}$- components of the position vector of a decay electron
    production vertex.}
\label{fig:13}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig14.eps}
\vskip 3mm
\caption{\hspace*{0.0cm} The left column  includes the distributions
    (from top to bottom, respectively) of number of events versus the 
    energy $E_{dec }^{e}$, the transverse momentum $PT_{dec }^{e}$ 
    and  the polar angle  ${\theta_{dec }^{e}}$ of electrons,
    produced in hadron decays in signal events that include the
    $q\bar{q} \rightarrow e^{+}e^{-}$ subprocess. The right column 
    (from top to bottom) shows the  distributions (in mm) 
    of $V_{x}$-, $V_{y}$- and  $V_{z}$- components of the position
    vector of a decay electron production vertex.}
\label{fig:14}
\end{figure}
\end{center}


\begin{center}
\begin{figure}[h]
\vskip -5mm
%\vspace*{-30mm}
\hspace*{-15mm} \includegraphics[width=18cm,height=20cm]{fig15.eps}
\vskip 3mm
\caption{\hspace*{0.0cm} The left column  includes the distributions
    (from top to bottom, respectively) of number of events versus the 
    energy $E_{dec }^{e}$, the transverse momentum $PT_{dec }^{e}$ 
    and  the polar angle  ${\theta_{dec }^{e}}$ of electrons produced 
    in QCD background events. The right column (from top to bottom)
    shows the distributions (in mm) of  $V_{x}$-, $V_{y}$- and  
    $V_{z}$- components of the position vector of a decay electron
    production vertex.}
\label{fig:15}
\end{figure}
\end{center}







\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%################################################################
%%%%%%%%%%%%################################################################
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{140mm}\hspace*{-15mm}\special{psfile=fig2.eps voffset=50 vscale=50 hscale=70 hoffset=-9 angle=0}
\vspace*{-30}
%\vspace*{5}


   Figure 2.: Signal muons distributions for muons with the
    largest energy ("fast" muons) in the muon pair 
    $E^{\mu}_{fast}$ (left column) and the smaller energy 
    ("slow" muons)  $E^{\mu}_{slow}$ (right column).
    Top raw includes their energies, in middle raw are PT and
    in bottom raw are $\theta^{\mu}$.\\


\vspace*{140mm}\hspace*{-15mm}\special{psfile=fig3.eps voffset=50 vscale=50 hscale=70 hoffset=-9 angle=0}
\vspace*{-30}
%\vspace*{5}


   Figure 3.: Top raw shows the distributions of valence
              up -quarks and anti-quarks, while bottom raw 
              includes the analogous distributions of
              down-quarks and anti-quarks. \\


\vspace*{140mm}\hspace*{-15mm}\special{psfile=fig4.eps voffset=50 vscale=50 hscale=70 hoffset=-9 angle=0}
\vspace*{-30}
%\vspace*{5}


   Figure 4.: The left column  includes the distributions
              ( from top to bottom, respectively ) of number of
              events versus the energy $E_{PI}$=$E_{\pi}$, the
              transverse momentum  $PT_{PI}$= $PT_{\pi}$  and 
              versus the polar angle $TETA_{PI}$= ${\theta_{\pi}}$
              of produced pions. The right hand  plot shows the
              distribution of the total number (NPI)  of charged
               ${\pi}$-mesons in the signal events. \\


\vspace*{140mm}\hspace*{-15mm}\special{psfile=fig5.eps voffset=50 vscale=50 hscale=70 hoffset=-9 angle=0}
\vspace*{-30}
%\vspace*{5}

   Figure 5.: The left column  includes the distributions
              ( from top to bottom, respectively ) of number of
              events versus the energy $E_{PI}$=$E_{\pi}$, the
              transverse momentum $PT_{PI}$= $PT_{\pi}$  and 
              versus the polar angle $TETA_{PI}$= ${\theta_{\pi}}$
              of produced pions. The right column (from top to 
              bottom)  shows the distributions of  x-, y- and z-
              components of decay muon vertex position. \\   

\vspace*{140mm}\hspace*{-15mm}\special{psfile=fig5.eps voffset=50 vscale=50 hscale=70 hoffset=-9 angle=0}
\vspace*{-30}
%\vspace*{5}

   Figure 5.: The left column  includes the distributions
              ( from top to bottom, respectively ) of number of
              events versus the energy $E_{PI}$=$E_{\pi}$, the
              transverse momentum $PT_{PI}$= $PT_{\pi}$  and 
              versus the polar angle $TETA_{PI}$= ${\theta_{\pi}}$
              of produced pions. The right column (from top to 
              bottom)  shows the distributions of  x-, y- and z-
              components of decay muon vertex position. \\   

\vspace*{140mm}\hspace*{-15mm}\special{psfile=fig6.eps voffset=50 vscale=50 hscale=70 hoffset=-9 angle=0}
\vspace*{-30}
%\vspace*{5}

   Figure 6.: The left column  includes the distributions
              ( from top to bottom, respectively ) of number of
%              events versus the energy $E_{PI}$=$E_{\pi}$, the
%              transverse momentum $PT_{PI}$= $PT_{\pi}$  and 
%              versus the polar angle $TETA_{PI}$= ${\theta_{\pi}}$
%              of produced pions. The right column (from top to 
%              bottom)  shows the distributions of  x-, y- and z-
%              components of decay muon vertex position. \\   



%\end{document}



\begin{eqnarray}
\frac{d\sigma}{d\eta_1d\eta_2d\Pt^2} = \sum\limits_{a,b}\,x_a\,f_a(x_a,Q^2)\,
x_b\,f_b(x_b,Q^2)\frac{d\sigma}{d\hat{t}}(a\,b\rightarrow 3\,4),
\label{gl:4}
\end{eqnarray}



hep-ph0506139.extracted_bib
\begin{thebibliography}{50}

%%%%%%%%%%%%%...........................................................

%\section{PANDA  References.}


%%%%%%%%%%%%%...........................................................


\bibitem{Matv}
V.A.~Matveev, R.M.~Muradian, A.N.~Tavkhelidze,
JINR P2-4543, JINR, Dubna, 1969;
SLAC-TRANS-0098, JINR R2-4543, Jun 1969; 27p. 

\bibitem{Drell}
S.D.~Drell, T.M.~Yan, SLAC-PUB-0755, Jun 1970, 12p.;  Phys.Rev.Lett.
{\bf 25}(1970)316-320, 1970.

%\bibitem{Ting}
 
%S.~Ting
%
%\bibitem{SLAC}
%S.~SLAC

\bibitem{DYexper} 
CERN UA1 Collaboration, C.~Albajar et al., 
Phys. Lett., { \bf B209} (1988) 397;
FNAL E772 Collaboration, P.L.~Gaughey at al.
Phys. Rev. { \bf D50} (1994) 3038;


\bibitem{MRST}
A.D.~Martin {\it et al.}, Eur. Phys. J. {\bf C4} (1998) 463.

%\bibitem{DYtheor}
%D.Y.~Theor

%\bibitem{PandaLOI}
%http://www.

\bibitem{Okubo}
S.Okubo,
Phys.Lett. B 5, 165 (1963);
G.Zweig, CERN Report 8419/th 412 (1964) ;
I.Izuka, Prog. Theor. Phys (suppl 37) 38 21 (1966);


\bibitem{OBELIX1} 
M.Bargiotti et al., EPJdirect A 2,1  (2002);

\bibitem{OBELIX2}
M.Bargiotti et al., Eur. Phys. J. C 35, 177 (2004);

\bibitem{Note1}
A.N.Skachkova, N.B.Skachkov hep-ph/0412279, 2004.

\bibitem{Sjost}
T.Sjostrand, Computer Phys. Commun.{\bf 39} (1986) 347,
T.Sjostrand and M.Bengtsson, Computer Phys. Commun.{\bf 43}
(1987 )367.

%\bibitem{MRST}
%A.D.~Martin {\it et al.}, Eur. Phys. J. {\bf C4} (1998) 463.

\bibitem{ROOT}
R.~Brun and F. Rademakers, 
ROOT - An Object Oriented Data Analysis Framework, 
Proceedings AIHENP '96 Workshop, Lausanne, Sep. 1996,
Nucl. Inst. \& Meth. in Phys. Res. 
{ \bf A389} (1997) 81.
See also http://root.cern.ch/.


\bibitem{E706}
Fermilab E706 Collab., L. Apanasevich et al. Phys.Rev.D70:092009
,2004, hep-ex/0407011;  Phys.Rev.Lett.81:2642-2645,1998,  
hep-ex/9711017.

\bibitem{SBKLHC}
 
N.B.~Skachkov, V.F.~Konoplyanikov D.V.~Bandourin,
Third Annual RDMS CMS Collaboration Meeting.
CMS-Document, 1997--168. CERN, December 16-17, 1997,
p.139-153.


\bibitem{BKSLHC}
D.V. Bandurin, V.F. Konoplyanikov, N.B. Skachkov, hep-ex/0207028;

\bibitem{BSTevat}
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\end{thebibliography}