text:
\section{Introduction}
In this paper we continue to resurrect the old ideas of Gribov Pomeron
 Calculus, that the Bose
-Einstein correlations lead to strong azimuthal angle
 correlations\cite{PION}, which do not depend on the rapidity
 difference between measured hadrons ( large range rapidity
 (LRR) correlations).  In the framework of QCD, these azimuthal
 correlations stem from the production of two patron showers,
 and have been re-discovered in Refs.\cite{KOWE,KOLUCOR}. In
 Ref.\cite{GLMBE} it  was demonstrated
that  Bose--Einstein correlations  generate $v_n$ with even and odd $n$,
 with values which are close to the experimental values
 \cite{CMSPP,STARAA,PHOBOSAA,STARAA1,CMSPA,CMSAA,ALICEAA,ALICEPA,
ATLASPP,ATLASPA,ATLASAA}.

The goal of this paper is to show that the  Bose--Einstein correlations
 that have been discussed in Refs.\cite{PION,GLMBE},  arise naturally
in the
 perturbative QCD approach, together with ones that have been considered
 in Refs.\cite{KOWE,KOLUCOR}. We believe that the qualitative difference
 between these two approaches  originates   from different sources of
the
 Bose-Einstein correlations: the two parton shower production in
 Refs.\cite{PION,GLMBE} and  one parton shower for  Refs.\cite{KOWE,KOLUCOR}.

  We consider  here the azimuthal correlations for
deuteron-deuteron
 scattering at high energy. It is well known\cite{HBT}, (see also Refs.
\cite{IPCOR}) that Bose-Einstein correlations  provide a possibility
to
 measure the volume of interaction or, in other words, the typical
 sizes of the interaction. Indeed,  the general formula for the Bose-Einstein
 correlations \cite{HBT,IPCOR} takes the form
 \begin{equation} \label{I1}
  \frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \rm identical\,\,
 gluons\right)\,\,\propto\,\,\Big{ \langle}  1\,\,+\,\,e^{i
 r_\mu Q_\mu}\Big{\rangle}
 \end{equation}
 where  averaging $\langle \dots \rangle$ includes the integration
 over $r_\mu = r_{1,\mu} - r_{2,\mu}$.  For the case of $y_1=y_2$,
  $Q_\mu = p_{1,\mu} \,-\,p_{2,\mu}$ simplifies to $\vec{Q}\,\equiv
 \,\vec{p}_{T,12}\,=\,\vec{p}_{T1} \,-\,\vec{p}_{T2}$,

 One can see that \eq{I1} allows us to measure the typical $r_\mu$
 for the interaction. For deuteron-deuteron scattering we expect
 several typical $r$: the size of the deuteron $R_D$, the nucleon
 size  $R_N$, and the typical size, related to the saturation scale
 ($r_{\rm sat}\,=1/Q_s$, where $Q_s$  denotes the saturation
scale\cite{KOLEB}).
 In our calculation we   hope to see  the appearance of these
scales.

It is well known, that the total cross section for the deuteron-deuteron
 scattering can be written in the form: $\sigma_{DD}\,=\,4 \sigma_{NN}
 \,-\,\Delta \sigma_{DD}$, where $\Delta \sigma_{DD}$ is the Glauber
 correction term\cite{GLA} which is proportional to $1/R^2_D$, while
 $\sigma_{NN}$  denotes the total cross section of the
nucleon-nucleon interaction.
   Intuition,  suggests that the correlation radius of the order of
$R_D$,
 stems from the production due to the Glauber correction term (see \fig{ddagk})
\begin{figure}[ht]
 \includegraphics[width=14cm]{BEddAGK.eps}
   \protect\caption{ The two parton showers production that contributes
 to the Glauber correction term for deuteron-deuteron scattering. The
  wavy lines describe the exchange of the BFKL Pomeron. \fig{ddagk}-a
 and \fig{ddagk}-b show two diagrams that can interfer  for
 identical gluons. The dashed lines show the cut BFKL  Pomeron
\cite{AGK}.}
\label{ddagk}
   \end{figure}

 The production of two gluon are shown in \fig{ddagk}-a and \fig{ddagk}-b,
where interference in the case of the generated identical gluon
 leads to
 the correlation function of \eq{I1}. Generally speaking, the
inclusive
 production of two gluons with rapidities $y_1$ and $y_2$ and transverse
 momenta $\vec{p}_{T1}$ and $\vec{p}_{T2}$, takes the form
 \begin{eqnarray} \label{I2}
 &&  \frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \rm
 identical\,\,
 gluons\right)\,\,=\\
 &&  \frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \rm
 different \,\,
 gluons\right) \,\Bigg( \underbrace{1}_{\rm squared\,  of\,
 diagrams}\,\,\,+\,\,\underbrace{C\left( R_c |\vec{p}_{T1}\,-\,\vec{p}_{T2}|
\right)}_{\rm interference\,diagram}\Bigg)\nonumber
 \end{eqnarray}
In \eq{I2}  $R_c$  denotes the correlation radius (correlation length),
and in
 the form of the correlation function, we anticipate that the production
 of two parton showers leads to the double inclusive cross section, that
 does not depend on rapidities $y_1$ and $y_2$.

\section{Born Approximation}

{\boldmath{\subsection{Bose-Einstein correlation function with
 radius $\propto\,\, R_D$}}}
 The simplest contribution in the Born approximation of perturbative
 QCD {\bf} is shown in \fig{ddba}. The second diagram describes the
interference  between  two parton showers,  shown in \fig{ddagk}-b.

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEddBA.eps}
   \protect\caption{ The double inclusive production of two gluons
 with rapidities $y_1$ and $y_2$ and transverse momenta $\vec{p}_{T1}$
 and $\vec{p}_{T2}$, in the Born Approximation of perturbative QCD. The
 interference diagram of \fig{ddba}-b gives the correlation function
 $C\left( R_D |\vec{p}_{T1}\,-\,\vec{p}_{T2}|\right)$ of \eq{I2}.
 The  solid lines denote nucleons in the deuterons,
 which are  specified by double lines}
\label{ddba}
   \end{figure}

 The analytical expressions take the following forms. For the diagram
 of \fig{ddba}-a we have
 \begin{eqnarray} \label{BAD1}
  & &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left(
 \fig{ddba}-a\right)\,\,\propto\,\int d^2 Q_T G^2_D\left( Q_T\right)\nonumber\\
  &\times&  \int d^2k_t\,d^2 l_T\Bigg( \frac{I_P\left( \vec{k}_T, -\vec{k}_T +
 \vec{Q}_T\right)}{\vec{k}^2_T\,\left( \vec{k}-\vec{p}_1\right)^2_T}\Gamma_\mu\left(
 \vec{k}_T, \vec{p}_{T1}\right)\,\Gamma_\mu\left( -\vec{k}_T +\vec{Q}_T,
 \vec{p}_{T1}\right)\,\frac{I_P\left( \vec{k}_T - \vec{p}_{T1}, -\vec{k}_T
  +\vec{p}_{T1}+\vec{Q}_T\right)}{\left( -\vec{k}+\vec{Q}\right)^2_T\,\left(
-\vec{k}+\vec{p}_{1}+\vec{Q}\right)^2_T}\, \Bigg)\nonumber\\
 &\times&
\Bigg( \frac{I_P\left( \vec{l}_T, -\vec{l}_T - \vec{Q}_T\right)}{\vec{l}^2_T\,\left(
 \vec{l}-\vec{p}_2\right)^2_T}\Gamma_\nu\left( \vec{l}_T, \vec{p}_{T2}\right)\,
\Gamma_\nu\left( -\vec{l}_T -\vec{Q}_T, \vec{p}_{T2}\right)\,\frac{I_P\left(
 \vec{l}_T - \vec{p}_{T2}, -\vec{l}_T   +\vec{p}_{T2} - \vec{Q}_T\right)}{\left(
 -\vec{l}-\vec{Q}\right)^2_T\,\left( -\vec{l}+\vec{p}_{2}-\vec{Q}\right)^2_T}\,
 \Bigg)\end{eqnarray}

The interference diagram of \fig{ddba}-b takes the following form
 \begin{eqnarray} \label{BAD2}
  & &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{ddba}-b\right)\,\,\propto\,\int d^2 Q_T G_D\left( Q_T\right) \,G_D\left( \vec{Q}_T\,+\,\vec{p}_{T,12}\right)\nonumber\\
  &\times&  \int d^2k_T\,d^2 l_T\Bigg( \frac{I_P\left( \vec{k}_T, -\vec{k}_T + \vec{Q}_T\right)}{\vec{k}^2_T\,\left( \vec{k}-\vec{p}_1\right)^2_T}\Gamma_\mu\left( \vec{k}_T, \vec{p}_{T1}\right)\,\Gamma_\mu\left( -\vec{l}_T -\vec{Q}_T, \vec{p}_{T1}\right)\,\frac{I_P\left( \vec{k}_T - \vec{p}_{T1}, -\vec{k}_T   +\vec{p}_{T1}+\vec{Q}_T\right)}{\left( -\vec{l}-\vec{Q}\right)^2_T\,\left( -\vec{l}+\vec{p}_{1}-\vec{Q}\right)^2_T}\, \Bigg)\nonumber\\
 &\times&
\Bigg( \frac{I_P\left( \vec{l}_T, -\vec{l}_T - \vec{Q}_T\right)}{\vec{l}^2_T\,\left( \vec{k}-\vec{p}_2\right)^2_T}\Gamma_\mu\left( \vec{l}_T, \vec{p}_{T2}\right)\,\Gamma_\mu\left( -\vec{k}_T +\vec{Q}_T, \vec{p}_{T2}\right)\,\frac{I_P\left( \vec{l}_T - \vec{p}_{T2}, -\vec{l}_T   +\vec{p}_{T2} - \vec{Q}_T\right)}{\left( -\vec{k}+\vec{Q}\right)^2_T\,\left( -\vec{k}+\vec{p}_{2}+\vec{Q}\right)^2_T}\, \Bigg)\end{eqnarray}

 The Lipatov vertices $\Gamma_\mu
 $ have the form (see  reference \cite{KOLEB} for example):

   \begin{equation} \label{LIP}
   \Gamma_\mu\left( k_T, p_{T1}\right)\,\,=\,\,\frac{1}{p_{T1}^2}\left( \,k^2_T\,
\vec{p}_{T1}   \,-\,\vec{k}_T\,p^2_{T1}\right);~~~~~~~~
     \Gamma_\mu\left( k_{T1},p_{T2}\right)\,\,=\,\,\frac{1}{p_{T2}^2}\left( \,k^2_{T1}\,
\vec{p}_{T2}   \,-\,\vec{k}_{T1}\,p^2_{T2}\right);
      \end{equation}

      and

      \begin{equation} \label{A2}
         \Gamma_\mu\left( k_T,p_{T1}\right)\,  \Gamma_\mu\left( k_{T1},p_{T2}\right)
 \,\,=\,\,\frac{k^2_{T1}\,\left( \vec{k}_T - \vec{p}_{T2}\right)^2}{p^2_{T2}}
     \,+\,\frac{k^2_{T}\,\left( \vec{k}_{T1} - \vec{p}_{T1}\right)^2}{p^2_{T1}}
  \,-\,Q^2_T\,-\,p^2_{T,12}\frac{k^2_T\,k^2_{T1}}{p^2_{T1}\,p^2_{T2}}
         \end{equation}

         where $\vec{p}_{T,12} \,=\,\vec{p}_{T1} - \vec{p}_{T2}$
         and  $\vec{k}_{T1} = \vec{k}_T - \vec{Q}_T$. One can see  from \eq{A2} that
         \begin{eqnarray} \label{A21}
            \Gamma_\mu\left( k_T,p_{T1}\right)\,  \Gamma_\mu\left( k_{T1},p_{T2}\right)\,\,&\xrightarrow{k_T \,\ll\,Q_T}\, &     \,\,k^2_T \left( 1 + {\cal O}\left( \frac{Q}{p_{T1}}; \frac{k_T}{p_{T1}}\right)\Rb\nonumber\\
                  \Gamma_\mu\left( k_T,p_{T1}\right)\,  \Gamma_\mu\left( k_{T1},p_{T2}\right)\,\,&\xrightarrow{k_{T1} \,\ll\,Q_T}\,  &    \,\,k^2_{T1} \left( 1 + {\cal O}\left( \frac{Q}{p_{T2}}; \frac{k_{T1}}{p_{T2}}    \right)\Rb;
            \end{eqnarray}

   $G_D\left( Q_T\right)$ is equal to
   \begin{equation} \label{BAD3}
       G_D\left( Q_T\right)\,\,=\,\,\int d^2 r\,e^{i \vec{r} \cdot \vec{Q}_T}|\Psi_D\left( r\right)|^2\,
       \end{equation}
       where $r$ denotes the distance between the proton and the neutron
in the deuteron.
  The impact factors
       ($I_P\left( \vec{k}_T, -\vec{k}_T + \vec{Q}_T\right) $ and others in
 \eq{BAD1} and \eq{BAD2}), determine the interaction of two gluons with
 the nucleon, and their typical momenta   are about $1/R_N$.

       From \eq{BAD3} we can see that typical
       $Q_T \propto\,1/R_D$, where $R_D$ is the deuteron radius. In
 other words, $Q_T$  (and $|\vec{Q}_T \,+\,\vec{p}_{T,12}|$ in
 \eq{BAD2})    turn out to be much smaller than the value of
 $k_T$ and $l_T$, which are determined by the size of the nucleon
 ($R_N$) $ k_T \approx l_T \sim 1/R_N \gg\,Q_T ( |\vec{Q}_T \,+
\,\vec{p}_{T,12}|)  \sim 1/R_D$,  through the impact factors $I_P$
 in \eq{BAD1} and \eq{BAD2}.  The reason is that $R_D\,\,\gg\,\,R_N$.
  Neglecting $Q_T$ and $p_{T,12}$ in comparison with $k_T$ and $l_T$,
       we can simplify \eq{BAD1} and \eq{BAD2}  to the form
      \begin{eqnarray}\label{BAD4}
 & &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left(
 \fig{ddba}-a\right)\,+\,\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1}
 d^2 p_{T2}}\left( \fig{ddba}-b\right)\,\propto\,\\
& & \frac{1}{p^2_{T1} \,p^2_{T2}}\,\int d^2 Q_T \Bigg\{ G^2_D\left(
 Q_T\right) \,+\,\frac{1}{2\left( N^2_c - 1\right)}\,G_D\left( Q_T\right) \,G_D\left(
 \vec{Q}_T\,+\,\vec{p}_{T,12}\right)\Bigg\}\nonumber\\
  &\times& \int d^2k_t\,d^2 l_T
\Bigg( \frac{I_P\left( \vec{k}_T, -\vec{k}_T \right)\,I_P\left( \vec{k}_T -
 \vec{p}_{T1}, -\vec{k}_T   +\vec{p}_{T1}\right)}{\vec{k}^2_T\,\left(
 \vec{k}-\vec{p}_1\right)^2_T}\Bigg)\,\times\,
\Bigg( \frac{I_P\left( \vec{l}_T, -\vec{l}_T \right)\,I_P\left( \vec{l}_T
 - \vec{p}_{T2}, -\vec{l}_T   +\vec{p}_{T2}\right)}{\vec{l}^2_T\,\left(
 \vec{l}-\vec{p}_2\right)^2_T} \Bigg)\nonumber\end{eqnarray}

         In \eq{BAD4} we consider $p_1 = p_2$ for the expressions in
 $\Big(\dots\Big)$.  Note, that the interference diagram
 of \fig{ddba}-b contributes when the polarizations of the produced gluons
 are the same,  this fact is reflected in \eq{BAD2} by the same indices
 of Lipatov vertices. In \eq{BAD4} we replace
      \begin{eqnarray} \label{BAD5}
&&\Gamma_\mu\left( \vec{k}_T, \vec{p}_{T1}\right)\,\Gamma_\mu\left( -\vec{l}_T
 -\vec{Q}_T, \vec{p}_{T1}\right)\,\Gamma_\mu\left( \vec{l}_T, \vec{p}_{T2}\right)\,
\Gamma_\mu\left( -\vec{k}_T +\vec{Q}_T, \vec{p}_{T2}\right) \nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \to~~~\frac{1}{2}
\Gamma_\mu\left( \vec{k}_T, \vec{p}_{T1}\right)   \Gamma_\mu\left( -\vec{k}_T +\vec{Q}_T, \vec{p}_{T2}\right)
\,   \Gamma_\nu\left( \vec{l}_T, \vec{p}_{T2}\right)   \Gamma_\nu\left( -\vec{l}_T -\vec{Q}_T, \vec{p}_{T1}\right)  \\
&&~~~~~~~~~~~~~~~~\xrightarrow{Q_T,p_{T,12} \,\ll\, k_t,l_T}~~~\frac{1}{2}
\Gamma_\mu\left( \vec{k}_T, \vec{p}_{T1}\right)   \Gamma_\mu\left( -\vec{k}_T , \vec{p}_{T1}\right)
\,   \Gamma_\nu\left( \vec{l}_T, \vec{p}_{T2}\right)   \Gamma_\nu\left( -\vec{l}_T , \vec{p}_{T2}\right)  \nonumber\\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=~~~\,\,\,\,\frac{1}{2} \frac{1}{p^2_{T1}\,p^2_{T2}}\,k^2_T\,\left( \vec{k} - \vec{p}_1\right)^2_T\,\,l^2_T\,\left( \vec{l} - \vec{p}_2\right)^2_T \label{BAD51}
\end{eqnarray}
  Factor $1/\left( N^2_c - 1\right)$ in \eq{BAD4} reflects that identical gluons
 have the same colors ($N_c$ is the number of colors).

  Finally, the correlation function $ C\left( R_c |\vec{p}_{T1}\,-\,\vec{p}_{T2}|\right)$ in \eq{I2} is equal to
  \begin{equation} \label{BAD6}
   C\left( R_D\,p_{T,12}\right)\,\,=\,\,\frac{1}{2 \left( N^2_C - 1\right)}\frac{\int d^2 Q_T\,G_D\left( Q_T\right)\,  G_D\left( | \vec{Q}_T
    \,+\,\vec{p}_{T,12}|\right)}{\int d^2 Q_T\,G^2_D\left( Q_T\right)}\,
    \end{equation}

{\boldmath{\subsection{Bose-Einstein correlation function with radius
 $\propto\,\, R_N$: Glauber corrections}}}

In this subsection we  show that the Glauber corrections due to
 interaction of one nucleon with two nucleons of the deuteron, shown
 in \fig{pdba}, lead to a correlation radius of the order of $R_N$.
  In the diagram of \fig{pdba}-a, $Q_T$ is of the order of $1/R_D$ and,
 therefore, it is much smaller that the typical values of $k_T$ and $l_T$,
 which are of the order of $1/R_N$. Hence, the contribution of this diagram
 is similar to \eq{BAD4}: viz.
\begin{eqnarray} \label{BAPD1}
 & &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdba}-a\right)\,\,\propto\,
 \frac{1}{p^2_{T1} \,p^2_{T2}}\,\int d^2 Q_T\,  G_D\left( Q_T\right)\nonumber\\
  &\times& \int d^2 k_T\,d^2 l_T\,I_P\left( \vec{k}_T,\vec{l}_T, -\vec{l}_T , - \vec{k}_T\right)
 \frac{I_P\left( \vec{k}_T - \vec{p}_{T1}, -\vec{k}_T   +\vec{p}_{T1}\right)\,I_P\left( \vec{l}_T - \vec{p}_{T2}, -\vec{l}_T   +\vec{p}_{T2}\right)}{\vec{k}^2_T\,\left( \vec{k}-\vec{p}_1\right)^2_T\,\vec{l}^2_T\,\left( \vec{l}-\vec{p}_2\right)^2_T}\nonumber\end{eqnarray}

\begin{figure}[h]
 \includegraphics[width=16cm]{BEpdBA.eps}
   \protect\caption{ The double inclusive production of two gluons with
 rapidities $y_1$ and $y_2$ and transverse momenta $\vec{p}_{T1}$ and
 $\vec{p}_{T2}$ in the Born Approximation of perturbative QCD. The
 interference diagram of \fig{pdba}-b yields the correlation function
 $C\left( R_N |\vec{p}_{T1}\,-\,\vec{p}_{T2}|\right)$ of \eq{I2}. The
 solid lines denote nucleons in the deuterons, which are  illustated
by
double lines}
\label{pdba}
   \end{figure}

 Unfortunately, we cannot treat the impact factors $I_P$   theoretically
 in the case of  nucleon. The phenomenological approach to $I_P$ has been
 discussed in Refs.\cite{LERE,GLMBE}, and we will
 return to this below.  For the moment we replace the nucleon by the
state
 of  a heavy quark and antiquark (onium), to study the key features of
the
 impact factors in the framework of perturbative QCD (see \fig{if}).
 Introducing, the form factor of the onium in the form
 \begin{equation} \label{ONFF}
 F\left( Q_T\right)\,\,=\,\,\int d^2 r \,e^{i \frac{1}{2}  \vec{Q}_T\cdot \vec{r}}\,| \Psi_{\rm onium}\left( r\right)|^2
 \end{equation}
 we can express the impact factor in the form
 \begin{eqnarray}
 I_P\left( \vec{k}_T,-\vec{k}_T + \vec{Q}_T\right) &=& F\left( \vec{Q}_T\right)\,-\,  F\left( 2 \vec{k}_T + \vec{Q}_T \right);\label{IF1}\\
  I_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T + \vec{Q}_T,-\vec{k}_T+  \vec{Q}_T\right) &=&
  1\,+\, F\left( 2 \vec{Q}_T\right)\,+\, F\left( 2 (\vec{k}_T + \vec{l}_T)\right)\,+\,  F\left( 2 (\vec{k}_T - \vec{l}_T - \vec{Q}_T)\right) \nonumber\\
  &-&  F\left( 2 \vec{k}_T \right) \,-\, F\left( 2 (\vec{k}_T - \vec{Q}_T)\right) \,-\,F\left( 2 \vec{l}_T \right) \,-\,  F\left( 2 (\vec{l}_T + \vec{Q}_T)\right)  \label{IF2} \end{eqnarray}
  In \eq{BAPD1} the impact factors are equal to
  \begin{eqnarray}
 \hspace{-1cm}  I_P\left( \vec{k}_T,-\vec{k}_T\right) &=& 1\,-\,  F\left( 2 \vec{k}_T \right); ~~~~~~
   I_P\left( \vec{l}_T,-\vec{l}_T\right) \,=\, 1\,-\,  F\left( 2 \vec{l}_T \right);\label{IF3}\\
\hspace{-1cm}  I_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T,-\vec{k}_T \right) &=&
 2\,+\, F\left( 2 (\vec{k}_T + \vec{l}_T)\right) +  F\left( 2 (\vec{k}_T - \vec{l}_T )\right) -  F\left( 2 \vec{k}_T \right) \,-\, F\left( 2 (\vec{k}_T )\right) - F\left( 2 \vec{l}_T \right) -  F\left( 2 (\vec{l}_T )\right);  \label{IF4}
   \end{eqnarray}

   The integration over $k_T$ and $l_T$  lead to
 typical values of $k_T \sim 1/R_N$ and $l_T \sim 1/R_N$, and it
 does not generate  azimuthal angle correlations.  Indeed,  this
 is clear from the following features of $I_P$ from \eq{IF4}:
  \begin{eqnarray} \label{BAPD2}  R_N\,k_T \ll 1,  R_N\,l_T \ll 1;~~~~   I_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T,-\vec{k}_T \right)\,&\propto&  \,k^2_T\,l^2_T;\nonumber\\
    R_N\,k_T \ll 1,  R_N\,l_T \gg  1;~~~~   I_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T,-\vec{k}_T \right)\,&\propto&  \,k^2_T; \nonumber\\
     R_N\,k_T \gg 1,  R_N\,l_T \ll 1;~~~~   I_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T,-\vec{k}_T \right)\,&\propto&  l^2_T; \nonumber\\
 R_N\,k_T \gg 1,  R_N\,l_T \gg 1;~~~~   I_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T,-\vec{k}_T \right)\,&\propto&  1;
  \end{eqnarray}

\begin{figure}[ht]
 \includegraphics[width=14cm]{BEddIF.eps}
   \protect\caption{ The impact factors for onium for $R_N k_T> 1$,
 $ R_N l_T> 1$ and $R_N Q_T> 1$ }
\label{if}
   \end{figure}

  In the diagram of \fig{pdba}-b one can see that
 $G_D\left( \vec{Q}_T - \vec{p}_{12}\right)$  regulates that
  $| \vec{Q}_T - \vec{p}_{12}  |$
   is of the order of $1/R_D$. This means that we can put
 $\vec{Q}_T = \vec{p}_{12}$ in all parts of diagrams, since
   the typical values of $k_T$ and $l_T$  are about
 $1/R_N \,\gg\,1/R_D$. Therefore, the diagram of \fig{pdba}-b
 can be reduced to the form
   \begin{eqnarray} \label{BAPD3}
   & &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdba}-b\right)\,\,\propto\,
 \frac{1}{p^2_{T1} \,p^2_{T2}}\,\int d^2 Q_T G_D\left( \vec{Q}_T - \vec{p}_{12}\right)\nonumber\\
  &\times& \int d^2k_T\,d^2 l_T\,I_P\left( \vec{k}_T,\vec{l}_T, -\vec{l}_T - \vec{p}_{12} , - \vec{k}_T +\vec{p}_{12}\right)
  \,I_P\left( \vec{k}_T - \vec{p}_{T1}, -\vec{k}_T   +\vec{p}_{T1}\right)  \,I_P\left( \vec{l}_T - \vec{p}_{T2}, -\vec{l}_T   + \vec{p}_{T2}\right)\nonumber\\
  &\times& \Bigg\{\Bigg(\frac{1}{\left( \vec{k} - \vec{p}_1\right)^2_T\,\left( \vec{l} + \vec{p}_{12}\right)^2_T}\,+\,\frac{1}{k^2_T\,\left( \vec{l} - \vec{p}_2\right)^2_T}\Bigg)\,-\,\frac{\left( \vec{l} - \vec{k} +\vec{p}_{12}\right)^2_T\,p^2_{T,1}}{k^2_T\,\left( \vec{k} - \vec{p}_1\right)^2_T
   \,\left( \vec{l} - \vec{p}_2\right)^2_T \,\left( \vec{l} + \vec{p}_{12}\right)^2}\Bigg\}
  \nonumber\\
   &\times& \Bigg\{\Bigg(\frac{1}{\left( \vec{l} - \vec{p}_2\right)^2_T\,\left( \vec{k} - \vec{p}_{12}\right)^2_T}\,+\,\frac{1}{l^2_T\,\left( \vec{k} - \vec{p}_1\right)^2_T}\Bigg)\,-\,\frac{\left( \vec{l} - \vec{k} +\vec{p}_{12}\right)^2_T\,p^2_{T,2}}{l^2_T\,\left( \vec{l} - \vec{p}_2\right)^2_T
   \,\left( \vec{k} - \vec{p}_1\right)^2_T \,\left( \vec{k} - \vec{p}_{12}\right)^2}\Bigg\}
  \end{eqnarray}
  The largest contributions to the integrals over $k_T$ and $l_T$ lead to the
 logarithmically large terms, which are proportional to $\ln\left( p^2_{T1}
\,R^2_N\right)\,
  \ln\left( p^2_{T2}\,R^2_N\right)$.
 These
  contribution stems from the terms which are  proportional
 to $1/\left(\Lb \vec{k} - \vec{p}_1\right)^2_T\right)^2$, and to  $1/\left(\Lb
 \vec{l} - \vec{p}_2\right)^2_T\right)^2$. We consider the kinematic
 region in the integration over $k_T$ and $l_T$, where $\vec{k}_T
 -  \vec{p}_{T1} \equiv \vec{k}_1 \to 0 $ and $\vec{l}_T -  \vec{p}_{T2}
 \equiv \vec{l}_2 \to 0$. For small $k_1\,\ll\, p_{T2}$  and $l_2 \,\ll\, p_{T1} $ the product of curly brackets  is equal to
  \begin{equation}  \label{BAPD4}
  \frac{1}{p^2_{T1}} \Bigg\{\frac{1}{k^2_1} \,+\,\frac{1}{l^2_1} \,-\,\frac{\left( \vec{k}_1\,-\,\vec{l}_2\right)^2}{k^2_1\,l^2_2}\Bigg\}\,  \frac{1}{p^2_{T2}} \Bigg\{\frac{1}{k^2_1} \,+\,\frac{1}{l^2_1} \,-\,\frac{\left( \vec{k}_1\,-\,\vec{l}_2\right)^2}{k^2_1\,l^2_2}\Bigg\}  =\,\frac{4 \left( \vec{k}_1 \cdot \vec{l}_2\right)^2}{p^2_{T1}\,p^2_{T2}\,k^4_1\,l^4_2}\,\xrightarrow{\rm after \,integration\,over\,angle}\,\frac{2}{p^2_{T1}\,p^2_{T2}\,k^2_1\,l^2_2}
  \end{equation}

  For $ R_N k_1 \ll 1$ $,I_P\left( \vec{k}_T - \vec{p}_{T1}, -\vec{k}_T
   +\vec{p}_{T1}\right)  \propto k^2_1$, and the integral over $k_1$ gives
  a small contribution of the order of $1/(R^2_N p^2_{T1})$. Recall
that
 we can use the perturbative QCD approach only if $R_N p_{T1} \,\gg\,1
 $ and $R_N p_{T2} \,\gg\,1 $. Therefore, the main contribution occurs
 from  the region of integration $(1/R^2_N) \ll k_1 \ll p_{T2}$ and
 $(1/R^2_N ) \ll l_1 \ll p_{T1}$. Integration in this region leads to
 the contribution
  \begin{eqnarray} \label{BAPD5}
\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdba}-b\right)\,\,&\propto\,&
 \frac{1}{p^4_{T1} \,p^4_{T2}}\,\ln\left( p^2_{T1} R^2_N\right)\ln\left( p^2_{T2} R^2_N\right) \int d^2 Q_T G_D\left( \vec{Q}_T - \vec{p}_{12}\right)\nonumber\\
&\times & I_P\left( \vec{p}_{T1},\vec{p}_{T2},  -\vec{p}_{T1} ,- \vec{p}_{T2}\right)
 \end{eqnarray}
 Therefore, from this kinematic region the correlations are determined by
 the impact factor.
   Using the impact factor given in \eq{IF2}, we see that
  \begin{equation} \label{BAPD6}
   I_P\left( \vec{p}_{T1},\vec{p}_{T2},  \vec{p}_{T1} , \vec{p}_{T2}\right)\,=\,
   2 + F\left( 2\left( \vec{p}_{T1} + \vec{p}_{T2}\right)\Rb + F\left( 2 \vec{p}_{T,12}\right) - 2 F\left( 2 \vec{p}_{T1}\right) - 2 F\left( \vec{p}_{T2}\right)
   \end{equation}
   This function is symmetric with respect to $\phi \to
 \pi -\phi$ ($\vec{p}_{T1} \to - \vec{p}_{T1}$),  and with such an
 impact factor, the Born approximation produces only $v_n$ with even $n$,
as
was
 noted in Ref.\cite{KOWE,KOLUCOR}. However, this conclusion is based on the
 impact factor of \eq{IF2}. \eq{BAPD6} shows that this impact factor leads
 to $p_{T,12} \sim 1/R_N$.

\begin{figure}[ht]
 \includegraphics[width=15cm]{BEpdIFFull.eps}
   \protect\caption{ The impact factors for onium for  for $R_N k_T> 1$,
 $R_N l_T> 1$ and $R_N Q_T\leq  1$. The  dashed lines denote the
Coulomb
 gluons that form the onium bound state. }
\label{iff}
   \end{figure}

     Note that the simple expression of \eq{IF2} (see \fig{if})
 is written for sufficiently hard gluons. For small values of $Q_T =
 p_{T,12}$ we need to add the first diagram of \fig{iff}, in which two
 gluons with large transverse momenta (about $p_{T1}$ or $p_{T2}$ ) but
 small $Q_T$. The final expression for the impact factor takes the form

    \begin{equation} \label{IFF}
      I^F_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T + \vec{Q}_T,-\vec{k}_T+  \vec{Q}_T\right)    \,=\,  I_P\left( \vec{k}_T, - \vec{k}_T + \vec{Q}_T   \right)  \,I_P\left( \vec{l}_T, - \vec{l}_T - \vec{Q}_T   \right)   \,+\, \underbrace{  I_P\left( \vec{k}_T,\vec{l}_T, - \vec{l}_T + \vec{Q}_T,-\vec{k}_T+  \vec{Q}_T\right) }_{\rm \eq{IF2}}
      \end{equation}
  The first term in \eq{IFF} generates the correlation function which is
 proportional to $F^2\left( 2 \vec{p}_{T,12}\right) $.

   Summarizing, we see that the Born approximation of perturbative
QCD, generates
 the correlation function which is determined by the impact factor of the
 nucleon, the typical correlation length is about  $R_N$, and even for
the
 unrealistic perturbative model of onium, this correlation function is not
 symmetric with respect to $\phi \to \pi - \phi$. We will consider below the
more realistic case,
  in leading log approximation of perturbative QCD.
  However, we would like to stress  now, that  the correlation function
 stems from the large non-perturbative distances of the order of the
nucleon size.

{\boldmath{\subsection{Bose-Einstein correlation function
 with radius $\propto\,\, R_N$: nucleon-nucleon interaction}}}

 The correlations with $R_c = R_N$ are typical for the
nucleon-nucleon
 interaction ( see \fig{ppba} for the Born approximation of perturbative
 QCD). However, we will consider them below for the general case of
 the production of two parton showers, since we prefer to use a more
 phenomenological and realistic approach for the impact factors
 $I_P$, than we explored  above, replacing the nucleon by the onium
state.

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEppBA.eps}
   \protect\caption{ The double inclusive production of two gluons
 with rapidities $y_1$ and $y_2$ and transverse momenta $\vec{p}_{T1}$
 and $\vec{p}_{T2}$ in the Born Approximation of perturbative QCD for the
 nucleon-nucleon interaction. The interference diagram of \fig{ppba}-b
 yields the correlation function $C\left( R_D
|\vec{p}_{T1}\,-\,\vec{p}_{T2}|\right)$
 of \eq{I2}. The  solid lines denote nucleons in the deuterons, which
 are  illustrated by double lines.}
\label{ppba}
   \end{figure}

   \section{Production of two parton showers}

{\boldmath{\subsection{ $ R_c \,\,\propto\,\, R_D$}}}
In this section we consider the general case of the production of two parton
 showers  shown in \fig{ddagk}.  In  the leading log approximation
 (LLA) of perturbative QCD, the structure of one parton shower is
described
 by the BFKL Pomeron\cite{BFKL,LI}.

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEddscat1.eps}
   \protect\caption{ The double inclusive production of two gluons
 with rapidities $y_1$ and $y_2$ and transverse momenta $\vec{p}_{T1}$
 and $\vec{p}_{T2}$,  for the exchange of two BFKL Pomerons which are
 denoted by  wavy lines. This diagram is the LLA generalization of
 \fig{ddba}-a.
 The  solid lines denote nucleons in the deuterons, which are
 indicated by double lines.}
\label{ddspom}
   \end{figure}

 In the leading log approximation of perturbative
 QCD the Born diagram of \fig{ddba}-a can be generalized to \fig{ddspom}.
 The contribution of  this diagram can be written as follows
\begin{eqnarray} \label{RD1}
& &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{ddspom}\right)\,\,=\,\,\left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2  \frac{1}{p^2_{T,1}\,p^2_{T2}}\int d^2 Q_T  G^2_D\left( Q_T\right) \nonumber\\
&\times&\,\Bigg(\int d^2 k_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{k}_T, -\vec{k}_T + \vec{Q}_T\right)\,\phi^{N}_G\left( y_1; \vec{k}_T- \vec{p}_{T1},-\vec{k}_T+ \vec{p}_{T1} +\vec{Q}_T \right)\Bigg)\nonumber\\
&\times&\,\Bigg(\int d^2 l_{T}\,\,\phi^{N}_G\left( Y - y_2; \vec{l}_T, -\vec{l}_T - \vec{Q}_T\right)\,\phi^{N}_G\left( y_2; \vec{l}_T- \vec{p}_{T2},-\vec{l}_T+ \vec{p}_{T2} -\vec{Q}_T \right)\Bigg)\end{eqnarray}

where $\phi^{N}_G\left( y, \vec{k}_T,-\vec{k}_T + \vec{Q}_T\right)$  denotes
the
 probability to find a gluon with rapidity $ y $   and transverse
momentum
$k_\perp$,
  in the process with momentum transferred  $Q_T$. In
 \eq{RD1} $ \bar{\alpha}_S \,= \,\alpha_S N_c/\pi$ with the number of colours equal to
$N_c$.
  $\phi^{N}_G$ are the solutions of the BFKL  evolution equation
\begin{eqnarray} \label{RD2}
\frac{\partial \phi^N_G\left( y, \vec{k}_T, -\vec{k}_T + \vec{Q}_T \right)}{\partial y}\,&=&\,\bar{\alpha}_S\int \frac{d^2 k'_T}{2 \pi}\,K\left( Q_T; k_T,k'_T\right)\,\phi^N_G\left( y, \vec{k'}_T,-\vec{k'}_T + \vec{Q}_T\right)\,\,\\
&-&\,\,\left( \omega_G\left( \vec{Q}_T - \vec{k}_T\right)\,+\,\omega_G\left( \vec{k}_T\right)\Rb \phi^N_G\left( y, \vec{k}_T,-\vec{k}_T + \vec{Q}_T\right)\nonumber
\end{eqnarray}

where

\begin{eqnarray} \label{RD3}
K\left( Q_T,k_T,k'_T\right)\,&=&\, \frac{1}{\left( \vec{k}_T - \vec{k'}_{T}\right)^2}\Bigg\{ \frac{k^2_T}{k'^2_T} \,+\,\frac{\left( \vec{Q}_T - \vec{k}_T\right)^2}{\left( \vec{Q}_T - \vec{k'}_T\right)^2}\,\,-\,\,\frac{\left( \vec{k}_T - \vec{k'}_{T}\right)^2}{k'^2_T\, \left(\vec{Q}_T - \vec{k'}_T\right)^2}\Bigg\}\\
\omega_G\left(\vec{k}_T\right)&=& \frac{1}{2} \bar{\alpha}_S k^2_T \int \frac{d^2 k'_T}{2 \pi} \frac{1}{k'^2_T\,\left( \vec{k}_T - \vec{k'}_{T}\right)^2 }\nonumber
\end{eqnarray}

The typical momenta in $\phi^N_G$ is about $1/R_N$  or larger,
 (about  $p_{T1}$($p_{T2}$) or $Q_s$, where $Q_s$ denotes   the
saturation
 scale. Bearing this in mind,  and noting that $Q_T \sim 1/R_D
\,\ll\,1/R_N$
 we can put $Q_T  = 0$, in the arguments of $\phi^N_G$.  This simplifies
\eq{RD1}
 reducing it to the following expression
\begin{equation} \label{RD4}
\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left(
 \fig{ddspom}\right)\,\,=\,\,
\frac{d^2 \sigma}{d y_1 \ d^2 p_{T1}}\,\frac{d^2 \sigma}{d y_2
 \ d^2 p_{T2}} \,\times\,\int d^2 Q_T  G^2_D\left( Q_T\right)
\end{equation}

The diagram of \fig{ddba}-b in the LLA, simplifies the expression for the
exchange of two BFKL Pomerons,
but with more complicated vertices. Using \eq{BAD5} and
 considering $\bar{\alpha}_S\left( y_1- y_2 \right) \leq 1$, we can write
 this exchange in the form that is represented in \fig{ddipom},
 and its contribution has the following form

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEddscat2.eps}
   \protect\caption{ The double inclusive production of two gluons
 with rapidities $y_1$ and $y_2$ and transverse momenta $\vec{p}_{T1}$
 and $\vec{p}_{T2}$  for the exchange of two BFKL Pomerons which are
 denoted by  wavy lines. This diagram is the LLA generalization of
\fig{ddba}-b.
 The  solid lines denote nucleons in the deuterons, which are
represented by double lines.}
\label{ddipom}
   \end{figure}

\begin{eqnarray} \label{RD5}
& &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{ddipom}\right)\,\,=\,\,\frac{1}{2} \left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2  \int d^2 Q_T  G_D\left( Q_T\right)\,G_D\left( \vec{Q}_T -  \vec{p}_{T,12}\right) \\
&\times&\,\Bigg(\int d^2 k_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{k}_T, -\vec{k}_T + \vec{Q}_T\right)\,
  \Gamma_\mu\left( k_T, p_{T1}\right)\,  \Gamma_\mu\left( - \vec{k}_{T} + \vec{Q}_T, p_{T2}\right)\,
\phi^{N}_G\left( y_2; \vec{k}_T- \vec{p}_{T1},-\vec{k}_T+ \vec{p}_{T2} +\vec{Q}_T \right)\Bigg)\nonumber\\
&\times&\,\Bigg(\int d^2 l_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{l}_T, -\vec{l}_T - \vec{Q}_T\right)\,  \Gamma_\mu\left( l_T,p_{T1}\right)\,  \Gamma_\mu\left(-\vec{ l}_{T} -  \vec{Q}_T,p_{T2}\right)\phi^{N}_G\left( y_2; \vec{l}_T- \vec{p}_{T2},-\vec{l}_T+ \vec{p}_{T1} -\vec{Q}_T \right)\Bigg)\nonumber\end{eqnarray}

 Since $Q_T \sim 1/R_B \,\ll\,1/R_N$ as well as $|\vec{Q}_T - \vec{p}_{T,12}|
 \sim 1/R_D\,\ll\,1/R_N$, we can take both $Q_T= 0$ and $p_{T,12}=0$, but
it is
 not  sufficient to reduce \eq{RD5} to \eq{BAD6}.  In addition we
need to assume
that
 $\bar{\alpha}_S\left( y_1 - y_2\right)\,\leq\, 1$.   Making this assumption we can
replace
 $y_2$ in  $ \phi^{N}_G\left( y_2; \vec{k}_T- \vec{p}_{T1},-\vec{k}_T+
 \vec{p}_{T2} +\vec{Q}_T \right) $ by $y_1$ and $Y - y_1$, in $\phi^{N}_G\left(
 Y - y_1; \vec{l}_T, -\vec{l}_T - \vec{Q}_T\right) $ by $Y - y_2$. After these
changes
  \eq{RD5} can be reduced to the following expression
 \begin{equation} \label{RD6}
\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{ddipom}\right)\,\,=\,\,
\frac{d^2 \sigma}{d y_1 \ d^2 p_{T1}}\,\frac{d^2 \sigma}{d y_2 \ d^2 p_{T2}} \,\times\,\int d^2 Q_T  G_D\left( Q_T\right)\,G_D\left( \vec{Q}_T + \vec{p}_{T,12}\right)
\end{equation}

 \eq{RD4} and \eq{RD6} lead to the same correlation
 function ($  C\left( R_D\,p_{T,12}\right)$)  \eq{BAD6} as in the Born approximation.

{\boldmath{\subsection{ $ R_c \,\,\propto\,\, R_N$}}}
In LLA the diagrams of the Born approximation of \fig{pdba} can be
 generalized in the same way as has been discussed above. \fig{pdba}-a
 takes the form of \fig{pdspom} while the interference diagram of \fig{pdba}-b
 becomes \fig{pdipom}.

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEpdspom.eps}
   \protect\caption{ The  Mueller diagram\cite{MUDI} for the double
 inclusive production of two gluons with rapidities $y_1$ and $y_2$
 and transverse momenta $\vec{p}_{T1}$ and $\vec{p}_{T2}$,  for the
 exchange of two BFKL Pomerons which are denoted by wavy lines. This
 diagram is the LLA generalization of \fig{pdba}-a.
 The  solid lines denote nucleons in the deuterons, which are
 illustrated by double lines.}
\label{pdspom}
   \end{figure}

 The contribution of  the diagram of \fig{pdspom} can be written as follows
\begin{eqnarray} \label{RN1}
& &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdspom}\right)\,\,=\,\,\left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2  \frac{1}{p^2_{T,1}\,p^2_{T2}}\int d^2 Q_T\,N\left( Q_T\right)\,  G^2_D\left( Q_T\right) \nonumber\\
&\times&\,\Bigg(\int d^2 k_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{k}_T, -\vec{k}_T + \vec{Q}_T\right)\,\phi^{N}_G\left( y_1; \vec{k}_T- \vec{p}_{T1},-\vec{k}_T+ \vec{p}_{T1} +\vec{Q}_T \right)\Bigg)\nonumber\\
&\times&\,\Bigg(\int d^2 l_{T}\,\,\phi^{N}_G\left( Y - y_2; \vec{l}_T, -\vec{l}_T
 - \vec{Q}_T\right)\,\phi^{N}_G\left( y_2; \vec{l}_T- \vec{p}_{T2},-\vec{l}_T+ \vec{p}_{T2} -\vec{Q}_T \right)\Bigg)\end{eqnarray}
where $N\left( Q_T\right)$ denotes the integral over all energies of the
 imaginary part of the Pomeron-nucleon scattering amplitude. This
 amplitude was introduced in Gribov's Pomeron calculus\cite{GRIBPOM},
 but it has been proven that we can use this formalism in LLA of
 perturbative QCD\cite{GLR}.  $N\left( Q_T\right)$  has the following
 general form(see \fig{n})
\begin{equation} \label{N}
N\left( Q_T\right)\,\,=\,\,\underbrace{g^2\left( Q_T\right)}_{\rm elastic\, scattering}\,
+\,\underbrace{\sum^{M_0}_{M_i=m}g^2\left( Q_T; M_i\right)}_{\rm diffraction\, in\,
 low\, masses}\,+\,\underbrace{\int_{M_0} \frac{d M^2}{M^2}\phi^N_G\left( y_M,
 Q_T=0; \{\dots\}\right)\,G_{3 {I\!\!P}}\left( Q_T,\{\dots\}\right)}_{diffraction\, in\,
 high\, masses}\end{equation}
where $G_{3 {I\!\!P}} $ is the triple BFKL Pomeron vertex, and $\{\dots\}$
denotes
 all  transverse momenta which we need to integrate over. $y_M =
 \ln\left( M^2/M^2_0\right)$.

\fig{n}-b shows how all contributions correspond to the onium case,
 where we can use perturbative QCD for theoretical estimates.

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEpdipom.eps}
   \protect\caption{ The Mueller diagram for the double inclusive production
 of two gluons with rapidities $y_1$ and $y_2$ and transverse momenta
 $\vec{p}_{T1}$ and $\vec{p}_{T2}$,  for the exchange of two BFKL Pomerons
 which are denoted by wavy lines. This diagram is the LLA generalization
 of \fig{pdba}-b.
 The  solid lines denote nucleons in the deuterons, which are
illustated
 by double lines.}
\label{pdipom}
   \end{figure}

 Since $Q_T \sim 1/R_D \,\ll\,1.R_N$, and all other transverse momenta
 in \fig{pdspom}  are either of the order of $1/R_N$ or larger ( of
the order
 of  $p_{T1}$,$p_{T2}$ or $Q_s$, where $Q_s$ is the saturation scale), we
 can safely put $Q_T = 0$ and reduce this contribution to the factorized form:
 \begin{equation} \label{RN2}
\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdspom}\right)\,\,=
\frac{d^2 \sigma}{d y_1 \, d^2 p_{T1} }\,\frac{d^2 \sigma}{d y_2  \, d^2 p_{T2}}\,N\left( Q_T=0\right)\,\int d^2 Q_T G_D\left( Q_T\right)
\end{equation}

The contribution of the  relevant diagram, which is shown in
\fig{pdipom}, can be written in the form:
\begin{eqnarray} \label{RN3}
& &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdipom}\right)\,\,=\,\,\frac{1}{2} \left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2  \int d^2 Q_T  N\left( Q_T\right)\,G_D\left( \vec{Q}_T -  \vec{p}_{T,12}\right)\nonumber \\
&\times&\,\Bigg(\int d^2 k_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{k}_T, -\vec{k}_T + \vec{Q}_T\right)\,
  \Gamma_\mu\left( k_T, p_{T1}\right)\,  \Gamma_\mu\left( - \vec{k}_{T} + \vec{Q}_T, p_{T2}\right)\,
\phi^{N}_G\left( y_2; \vec{k}_T- \vec{p}_{T1},-\vec{k}_T+ \vec{p}_{T2} +\vec{Q}_T \right)\Bigg)\nonumber\\
&\times&\,\Bigg(\int d^2 l_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{l}_T, -\vec{l}_T - \vec{Q}_T\right)\,  \Gamma_\mu\left( l_T,p_{T1}\right)\,  \Gamma_\mu\left(-\vec{ l}_{T} -  \vec{Q}_T,p_{T2}\right)\phi^{N}_G\left( y_2; \vec{l}_T- \vec{p}_{T2},-\vec{l}_T+ \vec{p}_{T1} -\vec{Q}_T \right)\Bigg)\end{eqnarray}

 Integration over $Q_T$ leads to $\vec{Q}_T - \vec{p}_{T,12}
 \sim 1/R_D\,\ll\,\,1/R_N$ and, therefore, as in the Born approximation
 we can put $\vec{Q}_T = \vec{p}_{T,12}$. In \eq{RN3} we have two sources
 of $p_{T,12}$ behavior: $N\left( p_{T,12}\right) $ and $\phi^N_G$. Replacing
 $\vec{Q}_T = \vec{p}_{T,12} $ we obtain

 \begin{eqnarray} \label{RN4}
& &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdipom}\right)\,\,=\,\,\frac{1}{2} \left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2  \int d^2 Q_T  N\left( p_{T,12}\right)\,G_D\left( \vec{Q}_T -  \vec{p}_{T,12}\right)\nonumber \\
&\times&\,\Bigg(\int d^2 k_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{k}_T, -\vec{k}_T +\vec{p}_{T,12} \right)\,  \frac{1}{k^2_T\,\left(\vec{k}_T- \vec{p}_{T,12}\right)^2 \left(\vec{k}_T - \vec{p}_{T,1}\right)^4}  \phi^{N}_G\left( y_2; \vec{k}_T- \vec{p}_{T1},-\vec{k}_T+ \vec{p}_{T1} \right)\nonumber\\
&\times&
  \Bigg\{\frac{\left(\vec{k}_T - \vec{p}_{T,12}\right)^2\,\left(\vec{k}_T - \vec{p}_{T,1}\right)^2  }{p^2_{T2}}\,+\,
  \frac{k^2_T\,\left(\vec{k}_T - \vec{p}_{T,1} - \vec{p}_{T,12}\right)^2  }{p^1_{T2}}\,-\,p^2_{T,12}\,-\,p^2_{T,12}  \frac{k^2_T\,\left(\vec{k}_T- \vec{p}_{T,12}\right)^2  }{p^1_{T2} p^2_{T,2}}\Bigg\}  \Bigg)\nonumber\\
&\times&\,\Bigg(\int d^2 l_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{l}_T, -\vec{l}_T -\vec{p}_{T,12}\right)\,\,  \frac{1}{l^2_T\,\left(\vec{l}_T+ \vec{p}_{T,12}\right)^2 \left(\vec{l}_T - \vec{p}_{T,2}\right)^4} \phi^{N}_G\left( y_2; \vec{l}_T- \vec{p}_{T2},-\vec{l}_T+ \vec{p}_{T2} \right)\nonumber\\
&\times&
\Bigg\{\frac{\left(\vec{l}_T + \vec{p}_{T,12}\right)^2\,\left(\vec{l}_T - \vec{p}_{T,1}\right)^2  }{p^2_{T2}}\,+\,
  \frac{l^2_T\,\left(\vec{l}_T - \vec{p}_{T,2} \right)^2  }{p^1_{T2}}\,-\,p^2_{T,12}\,-\,p^2_{T,12}  \frac{l^2_T\,\left(\vec{l}_T+ \vec{p}_{T,12}\right)^2  }{p^1_{T2} p^2_{T,2}}\Bigg\}
\Bigg)\end{eqnarray}
 The products  of $G_\mu G_\mu$  are written in \eq{RN4}
  $\{ \dots\}$ explicitly using \eq{A2}, and $\phi^N_D $ are
 the solution of \eq{RD2}.  Recall that
 $\phi^N_D\left( \vec{k}_T,-\vec{k}_T + \vec{Q}_T\right)$ vanishes
 both at $k_T \to 0$ and $\vec{k}_T - \vec{Q}_T \to 0$. Since
 the products of $G_\mu$ vanish at $\vec{k}_T - \vec{p}_{T1} \,\to\,0$
 or $\vec{l}_T - \vec{p}_{T2} \,\to\,0$, respectively, we can conclude
 that the integrals over $k_T$ and $l_T$ do not have large contributions
 at $k_T$ of the order of $p_{T1}$, and at $l_T$ of the order of $p_{T2}$.

\begin{figure}[h]
 \includegraphics[width=11cm]{BENstr.eps}
   \protect\caption{ The structure of the amplitude $N\left( Q_T\right)$: \fig{n}-a
 for the BFKL Pomeron-nucleon interaction and \fig{n}-b for the BFKL
 Pomeron-onium interactions. The blob shows the triple BFKL Pomeron
 vertex which is the same for both figures. The  dashed vertical
lines
 describe the Coulomb gluons that create the bound state: onium.}
\label{n}
   \end{figure}

In the appendix we show that  the typical value of $Q_T$ for
 the BFKL Pomeron  $\phi^N_G\left( Y; k'_T, k_T,Q_T\right)$ is determined
 by the  smallest  value of transverse momentum $ Q_T \sim
min\{k;_T,k_T\}$.
 In our case  this means that $Q_T =  p_{T,12}\, \sim
1/R_N\,\gg\,p_{T1}$
 and/or $p_{T2}$.

Therefore, we can re-write \eq{RN4} as follows:
\begin{eqnarray} \label{RN5}
\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pdipom}\right)\,&=&\,\,\frac{1}{2} \left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2 \frac{1}{p^2_{T1}\,p^2_{T2}}\, \int d^2 Q_T  N\left( p_{T,12}\right)\,G_D\left( \vec{Q}_T -  \vec{p}_{T,12}\right) \nonumber\\
&\times&\,\Bigg(\int d^2 k_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{k}_T, -\vec{k}_T \right)\,  \phi^{N}_G\left( y_2; \vec{k}_T- \vec{p}_{T1},-\vec{k}_T+ \vec{p}_{T1} \right)\Bigg)\nonumber\\
&\times&\Bigg(\int d^2 l_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{l}_T, -\vec{l}_T \right)\, \phi^{N}_G\left( y_2; \vec{l}_T- \vec{p}_{T2},-\vec{l}_T+ \vec{p}_{T2} \right)\Bigg)
\end{eqnarray}
In \eq{RN5} we introduce $\phi^N_G\left( \vec{k}_t,-\vec{k}_T\right) =
(1/k^2_T)\phi^N_G\left( \vec{k}_t,-\vec{k}_T\right)(\eq{RN4})$. Comparing
 \eq{RN2} and \eq{RN5}, one can see that the correlation function is
 equal to

\begin{equation} \label{RN6}
C\left( R^2_N\,p^2_{T,12}\right)\,\,=\,\,N\left( p^2_{12}\right)
\end{equation}
for $\bar{\alpha}_S(y_1 - y_2) \leq 1$.

The three terms of $N\left( Q_T\right)$ are shown in \fig{n}-a.
 The first contribution $N\left( Q_T\right)\,=\,g^2\left( Q_T\right)$, can
  easily be  evaluated from the differential  elastic cross section,
 which is proportional to $g^4\left( Q_T \right)$. Recall, that the BFKL
 Pomeron does not generate the shrinkage of the diffraction peak
  seen in  the experimental data. This indicates that the
 exchange of the single BFKL Pomeron is not  sufficient to describe
the
 high energy amplitude, and we need to use a more phenomenological
approach
 to describe the elastic contribution to the correlation function (see
 Ref.\cite{GLMBE} in which we tried to describe these correlations
 using a particular model for high energy scattering, which is based
 on CGC/saturation approach).

For the onium, $g\left( Q_T\right)$ can be calculated (see \fig{n}-b and \eq{B5})
in the following way
\begin{eqnarray} \label{RN7}
g\left( Q_T\right)\,=\,V^{\rm onium}\left(  \vec{Q}_T\right)\,&=&\,\int d^2 k'_T I_P\left( \vec{k}_T, - \vec{k'}_T + \vec{Q}_T\right) V^{\rm pr}\left(  \vec{k'}_T, \vec{Q}_T\right)\,\nonumber\\
&=&\,\int d^2 k'_T\left( F\left( Q_T\right) - F\left( 2 \vec{k'}_T - \vec{Q}_T\right)\Rb\, V^{\rm pr}_{\frac{1}{2}}\left( \vec{k'_T},\vec{Q}_T\right)
\end{eqnarray}
where $V^{\rm pr}$ is determined by \eq{B4}.
In \eq{RN7} $k'_T \sim 1/R_N \,\ll\,k_T$.
Assuming that $F\left( Q_T\right)$  of \eq{IF1}  is equal to
 $1/\left( 1 + R^2_N \,Q^2_T\right)$,  we  find that at large $Q_T$,
$g\left( R_N \,Q_T\right)$  decreases as $1/Q_T$.

The second term of \fig{n}-b can be  evaluated from the process of
 diffraction dissociation in the region of small masses. However,
 we need to use a model for this term to be able to extract its $Q_T$
dependence
 from the experimental data. For example, we can replace the sum of
 possible produced diffractive states by one state, as has been done in
 Ref.\cite{GLMBE}. For the onium  state this term has the following form
\begin{equation} \label{RN8}
N_{\rm diff}\left( Q_T\right)\,=\,\int d^2 k'_T \int d^2 l'_T \, I_P\left(
 \vec{k}_T,\vec{l}_T, - \vec{l}_T + \vec{Q}_T,-\vec{k}_T+  \vec{Q}_T\right)
\, V^{\rm pr}\left(  \vec{k'}_T; \vec{Q}_T\right)\,\, V^{\rm pr}\left(   \vec{l'}_T;
 \vec{Q}_T\right)
\end{equation}
where $I_P$ is taken from \eq{IF2}.

Using \eq{B4} we calculate $N_{\rm diff}\left( Q_T\right)$
 which decreases as $1/Q^2_T$ at large $Q_T$.

 ~

{\boldmath{\subsection{ $ R_c \,\,\propto\,\, 1/Q_s$}}}
The last term in \fig{n}-a, gives the contribution of large mass
 production in the diffraction dissociation process. The $Q_T$
 dependence of this term, is determined by the triple BFKL Pomeron
 vertex in perturbative QCD (see \fig{lmdiff}). Therefore,  this
 term generates correlations,  whose  length is determined by the BFKL
 Pomeron structure, and it  is closely related to the typical saturation
 momentum $Q_s$.

\begin{figure}[h]
 \includegraphics[width=11cm]{BEpdenh.eps}
   \protect\caption{ The large mass diffraction contribution to
 $N\left( Q_T\right)$: the source for the correlation length of about
 $1/Q_s$. The blue blob denotes the triple BFKL Pomeron vertex.
 The red  square indicates the contribution of $N\left( Q_T\right)$, which
 includes the integration over rapidity $y'$.}
\label{lmdiff}
   \end{figure}

 Comparing \fig{lmdiff} with \fig{pdspom} and \fig{pdipom}, one can see
 that the difference is only in
 expression for $N\left( Q_T\right) $ which has the following form
 \begin{equation} \label{RQ1}
 N_{\rm large \,mass\,diffraction}\left( Q_T\right)\,\,=\,\,\int \,d\,y'\, \phi^N_G\left( Y - y'; Q_T=0; q_T, q'_T\right) d^2 q'_T
 G_{3 {I\!\!P}}\left( q'_T; k'_T,  l'_T, Q_T\right)
 \end{equation}
 We can obtain the form of $G_{3{I\!\!P}}$ in  momentum space starting from
 the coordinate representation, where the contribution of the triple
 Pomeron diagram of \fig{3p} is known\cite{BRAUN,KLP}:
 \begin{equation} \label{3Pco}
 \bar{\alpha}_S \int \frac{d^2 x_0\,d^2 x_1\,d^2 x_2}{x^2_{01}\,x^2_{02}\,x^2_{21}} N\left( x'_{01}, x_{01}; \vec{b} - \vec{b'}; Y-y'\right) \,N\left( x'_{02}, x_{02}; \vec{b'} - \frac{1}{2} \vec{x}_{21}; y' - y_1\right) \,N\left( x'_{21}, x_{21}; \vec{b'} - \frac{1}{2} \vec{x}_{02}; y' - y_2\right)
 \end{equation}
 Introducing\cite{KOLEB}
 \begin{equation} \label{MOMSP}
 N\left( x_{01},b; Y\right)\,\,=\,\,x^2_{01}\int d^2 k d^2 Q_T \,e^{ i \vec{k}_T\cdot \vec{x}_{01}\,+\,i\,\vec{Q}_T\cdot \vec{b}}  N\left( k_T, Q_T\right)
 \end{equation}
 we see that \eq{3Pco} can be re-written in the form
 \begin{equation} \label{3Pmo}
 \bar{\alpha}_S \int d^2 q'_T \, N\left( q_T,q'_T,Q_T=0, Y-y'\right)\, G_{3 {I\!\!P}}\left( q'_T; k'_T,  l'_T, Q_T\right)
   N\left( k'_T,k_T,Q_T, y'-y_1\right)  \, N\left( l'_T,l_T,-Q_T, y'-y_2
   \right)
 \end{equation}

 with

 \begin{equation} \label{3P}
 G_{3 {I\!\!P}}\left( q'_T; k'_T,  l'_T, Q_T\right)\,\,=\,\,\delta^{(2)}\left( \vec{k'}_T \,-\vec{q'}\, + \frac{1}{2} \vec{Q}_T\right)\, \delta^{(2)}\left( \vec{l'}_T \,-\vec{q'}\,- \frac{1}{2} \vec{Q}_T\right)\,
 \end{equation}

\begin{figure}[h]
 \includegraphics[width=11cm]{BE3P.eps}
   \protect\caption{ The triple BFKL Pomeron vertex in coordinate and
momentum representations. The blue blob denotes the triple Pomeron vertex}
\label{3p}
   \end{figure}

 In \eq{RQ1} we use the following notation for $\phi^N_G\left( Y - y'; Q_T=0;
 k_i, k_f\right)$: $Y-y'$ is the rapidity, $Q_T$ is the momentum transfer
   of the BFKL Pomeron, $k_i$ and $k_f$ are initial and final
transverse
 momenta. Plugging  \eq{3P} in the general expression for the interference
 diagram of \fig{lmdiff}-b, we see that instead of \eq{RN4} we obtain
  \begin{eqnarray} \label{QS1}
& &\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{lmdiff}-b\right)\,\,=\nonumber\\
& &\,\,\frac{1}{2} \left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2  \int d^2 Q_T  G_D\left( \vec{Q}_T -  \vec{p}_{T,12}\right)  \,\,\int \,d\,y'\,\int \phi^N_G\left( Y - y'; Q_T=0; q_T, q'_T\right)\ d^2 q'_T\nonumber\\
&\times&
\Bigg(G^{\rm BFKL}\left( y' - y_1; p_{T,12}; \vec{q'}_T - \frac{1}{2} \vec{p}_{T,12},k_T \right)\, \, \frac{1}{k^2_T\,\left(\vec{k}_T- \vec{p}_{T,12}\right)^2 \left(\vec{k}_T - \vec{p}_{T,1}\right)^4} G^{\rm BFKL}\left( y_2; 0 ;  \vec{k}_T- \vec{p}_{T1},k_N\right)\nonumber\\
&\times&
  \Bigg\{\frac{\left(\vec{k}_T - \vec{p}_{T,12}\right)^2\,\left(\vec{k}_T - \vec{p}_{T,1}\right)^2  }{p^2_{T2}}\,+\,
  \frac{k^2_T\,\left(\vec{k}_T - \vec{p}_{T,1} - \vec{p}_{T,12}\right)^2  }{p^1_{T2}}\,-\,p^2_{T,12}\,-\,p^2_{T,12}  \frac{k^2_T\,\left(\vec{k}_T- \vec{p}_{T,12}\right)^2  }{p^1_{T2} p^2_{T,2}}\Bigg\}  \Bigg)\nonumber\\
&\times&\,\Bigg(\int d^2 l_{T}\,G^{\rm BFKL}\left( y' - y_1; p_{T,12}, \vec{q'}_T + \frac{1}{2} \vec{p}_{T,12}, l_T\right)\,\,  \frac{1}{l^2_T\,\left(\vec{l}_T+ \vec{p}_{T,12}\right)^2 \left(\vec{l}_T - \vec{p}_{T,2}\right)^4} G^{\rm BFKL}\left( y_2; 0;  \vec{l}_T- \vec{p}_{T2}, l_N \right)\nonumber\\
&\times&
\Bigg\{\frac{\left(\vec{l}_T + \vec{p}_{T,12}\right)^2\,\left(\vec{l}_T - \vec{p}_{T,1}\right)^2  }{p^2_{T2}}\,+\,
  \frac{l^2_T\,\left(\vec{l}_T - \vec{p}_{T,2} \right)^2  }{p^1_{T2}}\,-\,p^2_{T,12}\,-\,p^2_{T,12}  \frac{l^2_T\,\left(\vec{l}_T+ \vec{p}_{T,12}\right)^2  }{p^1_{T2} p^2_{T,2}}\Bigg\}
\Bigg)\end{eqnarray}
The main difference  between \eq{QS1} and \eq{RN4}, is that $q'_T $
is larger
 than $q_T \approx\,1/R_N$. Indeed, the typical value of $q'_T = Q_s\left(
 Y - y'\right)\,\sim \left( 1/R^2_N\right)\exp\left( \lambda \left( Y - y'\right)\Rb$, where
 $\lambda = \omega\left( \gamma_{cr},0\right)/(1 - \gamma_{cr}$ with
 $\gamma_{cr}=0.37$ in leading order of perturbative QCD \cite{KOLEB},

From \eq{PHIMY} one can see that each $\phi^N_D ( Y - y')
  \propto e^{\omega\left( \frac{1}{2},0\right)\,\left( Y -  y'\right)}$, $\phi^N_D
 ( y' -  y_1)\, \propto e^{\omega\left( \frac{1}{2},0\right)\,\left(  y' - y_1\right)}$,
 and $\phi^N_D ( y' - y_2) \propto e^{\omega\left( \frac{1}{2},0\right)\,\left(
 y' - y_2\right)}$, since $\gamma  = \frac{1}{2} + i \nu$ with small $\nu$.
 Therefore, integration over $y'$ results in $Y - y' \sim 1/\omega\left(
 \frac{1}{2},0\right)  \propto  1/\bar{\alpha}_S$ while $ y' - y_1$ and $y' - y_2$ are large
 (of the order of $Y$).  Since $Y - y' \ll y' - y_1 $($Y - y' \ll y'
 - y_2$) we can use the factorized formula of \eq{AF} for $\phi^{N}_G\left(
 y' - y_1; p_{T,12}; \vec{q'}_T - \frac{1}{2} \vec{p}_{T,12},k_T \right)$ and for
 $\phi^{N}_G\left( y' - y_1; p_{T,12}, \vec{q'}_T + \frac{1}{2} \vec{p}_{T,12},
 l_T\right)$. Using \eq{AF} we find that $p_{T,12}$ will be determined
 by the lowest momenta in the BFKL Pomeron with $y' - y_1$, and it will
 have the form

\begin{equation} \label{QS2}
C\left( p_{T,12}\right)\,\,\propto\,\,\int d^2 q'_T I_{-\gamma}\left( q'_T\right)\,V_\gamma\left( \vec{q'}_T - \frac{1}{2} \vec{Q}_T, \vec{Q}_T\right)\,V_\gamma\left( \vec{q'}_T + \frac{1}{2} \vec{Q}_T, \vec{Q}_T\right)
\end{equation}
where $V$ is determined by \eq{INTM},  and $Q_T = p_{T,12}$.
 In \eq{QS2} we can put $\gamma\,=\,\frac{1}{2}$,  assuming $y' - y_1$ is
suffiently
 large, that we can neglect $\nu$.

\section{Bose-Einstein correlation function in the nucleon-nucleon
interaction}

In this section we discuss the Bose-Einstein correlations in nucleon-nucleon
 scattering. The Mueller diagrams for the square of the diagrams \fig{ddagk}-a
 and \fig{ddagk}-b, and for the interference diagrams, are shown in
\fig{pppom}.
 This differs from the diagrams that have been discussed above,
only in
 the appearance of the second $N\left( Q_T\right)$, which reflects the fact that
we
 do not have small ( about $1/R_D$) momenta in this process.  Note,
we can
 use perturbative QCD only if $p_{T1} \sim p_{T2} \gg 1/R_N$. Recalling that
 the $Q_T$ dependence of the BFKL Pomeron is determined by the smallest
 transverse momentum, we conclude that in \fig{pppom} the $Q_T$ dependence
 is determined by the function $N\left( Q_T\right)$. For the first two contributions
 to $N\left( Q_T\right)$ (see \fig{n}-a), this   is accurate to the order
of $1/\left(
 R_N\,p_{T1}\right)$. For the third contribution of the large mass
diffraction,
 the  accuracy   is about $Q_s/p_{T1}$, where $Q_s $  denotes
the
saturation momentum
 of the BFKL Pomeron with rapidity $Y - y'$.

 In spite of the fact that we indicate in \fig{n}-a the sources of
 experimental  information on  each contribution, the situation turns
 out to be more complicated. As an example, we discuss the elastic
 contribution. This gives $N\left( Q_T\right) = g^2\left( Q_T\right)$, where $g\left(
Q_T\right)$
 is the Pomeron-hadron vertex. At first sight we can extract this vertex
 directly from the experimental values of $d \sigma_{el}/dt$.  However,
 this is certainly not correct. Indeed, the BFKL Pomeron cannot explain
 the shrinkage of the diffraction peak which is seen experimentally,
 and which gives almost a half of the slope of the elastic cross section
 for the energy range $W = 40 \,-\,7000\,GeV$\cite{TOTEM}. In the only
 model\cite{GLM2CH} for the soft interaction at high energy that is
 based on the BFKL Pomeron and Colour Glass Condensate (CGC)
 approach\cite{BK,JIMWLK}, the effective shrinkage of the
 diffraction peak stems from strong shadowing corrections, which lead to
 an elastic amplitude that is different from  that for the exchange
of the BFKL
 Pomeron. However, it turns out that the most essential shadowing
 corrections  originate from the BFKL Pomeron interaction of
 two scattering hadrons. Such corrections do not contribute to the
 inclusive cross sections, as well as to the correlation due to AGK
 cutting rules\cite{AGK}.

It turns out to be an even more complicated problem to extract from the
 experimental data, the diffraction contribution to $N\left( Q_T\right)$ in the
region
 of small masses. The lack of a theory, as well as insufficient
experimental
 data, especially of the  momentum transfer distribution of the
diffractively
 produced state with fixed mass, lead to the necessity of modeling  this
 process. The two extreme cases of such a modeling
illustrates the difficulties: in our model \cite{GLM2CH} the rich variety
 of the produced states were replaced by a  single state: and in the
constituent
 quark model\cite{CQM} the small mass diffraction stems from production of
 the state of free three constituent quarks. In our model the typical slope
 for $g_{diff}\left( Q_T\right) \propto \exp\left( - B \,Q^2_T\right)$ turns out to
 be 1/4 from the elastic slope, while in the CQM the
 size of the constituent quark
 is very  small.

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEppdd.eps}
   \protect\caption{ The Mueller diagram for the double inclusive
 production of two gluons with rapidities $y_1$ and $y_2$ and transverse
 momenta $\vec{p}_{T1}$ and $\vec{p}_{T2}$
   in  the nucleon-nucleon interaction. The BFKL Pomerons
    are denoted by wavy lines. The first diagram corresponds to the
square
 of the amplitude for two parton showers production, while the second
diagram
 describes the interference.
    }
\label{pppom}
   \end{figure}

 Taking the above into consideration, the uncertainties in the large
mass diffraction term
 look small, and for  the triple BFKL Pomeron vertex, both the value and
 transverse momenta dependence follow directly from the Balitsky-Kovchegov
 equation\cite{BK}.
Bearing this in mind, we can write the expression for the interference
 diagram of \fig{pppom},  for the large mass diffraction contribution
 ( see \fig{n}). As we have discussed in this  case
 $Q_T \sim Q_s\left( Y - y'\right)\,\,\ll\,\, \min\Big\{ p_{T1}(p_{T2}),
 Q_s\left( y' - y_1\right)\Big\} $,  $| \vec{Q}_T - \vec{p}_{T,12}|
 \sim\,Q_s\left( Y - y'\right)\,\,\ll\,\, \min\Big\{ p_{T1}(p_{T2}),
 Q_s\left( y' - y_1\right)\Big\} $, and $k'_T \sim \min\Big\{ p_{T1}(p_{T2}),
 Q_s\left( y' - y_1\right)\Big\}$ ( $l'_T \sim \min\Big\{ p_{T1}(p_{T2}),
 Q_s\left( y' - y_1\right)\Big\}$). Hence, we can use the factorized form
 for $\phi^G_N$ given by \eq{AF} and \eq{PHIMY}.

 Finally, the large mass contribution for the interference diagram
 takes the form

\begin{eqnarray} \label{NN1}
&&\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left( \fig{pppom}, \mbox{interference diagram}\right)\,\,=\,\,\nonumber\\
&&\,\frac{1}{2} \frac{1}{N^2_c - 1}\left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2 \frac{1}{p^2_{T1}\,p^2_{T2}}   \,\,\int \,d\,y'\,\int \phi^N_G\left( Y - y'; Q_T=0; q_N, q'_T\right) \nonumber \\
&& \int  d^2 q'_T\,d^2 k_T
G^{\rm BFKL}\left( y' - y_1; Q_T; \vec{q'}_T - \frac{1}{2} \vec{Q}_{T}, k_T \right)\, \int d y''\, G^{\rm BFKL}\left( y_2  - y''; \vec{Q}_T - \vec{p}_{T,12} ;  \vec{k}_T- \vec{p}_{T1}, \vec{m}'_T - \frac{1}{2}\left( \vec{Q}_T - \vec{p}_{T,12}\right)\Rb\nonumber\\
&&\times\,\,\int d^2 l_{T}\,d^2 m'_T \,G^{\rm BFKL}\left( y' - y_1; Q_{T}, \vec{q'}_T + \frac{1}{2} \vec{Q}_{T}, l_T\right)\,\, G^{\rm BFKL}\left( y_2 - y''; \vec{Q}_T - \vec{p}_{T,12};  \vec{l}_T- \vec{p}_{T2}, \vec{m}'_T + \frac{1}{2}\left( \vec{Q}_T - \vec{p}_{T,12}\right) \right)\nonumber\\
&&\times\,\,
\phi^N_G\left( y'' ; Q_T=0; m_N, m'_T\right)
\end{eqnarray}

In the diagram for the square of the amplitude we can put
 $\vec{p}_{T,12}=0$.  Thus,
the correlation function with the correlation length of the order
 of $1/Q_s\left( Y - y'\right)$, takes the following form

\begin{equation} \label{NN2}
C\left( p_{T,12}/Q_s\right)\,\,=\,\,\frac{1}{2} \frac{1}{N^2_c - 1}\,\frac{N}{D}
\end{equation}
where
\begin{eqnarray} \label{NNN}
&&  N \,=\,  \,\,\int d^2 Q_T\,\int \,d\,y'\,\int \phi^N_G\left( Y - y'; Q_T=0; q_N, q'_T\right) \nonumber \\
& & \,\int  d^2 q'_T\,d^2 k_T
G^{\rm BFKL}\left( y' - y_1; Q_T; \vec{q'}_T - \frac{1}{2} \vec{Q}_{T}, k_T \right)\, \int d y''\,G^{\rm BFKL}\left( y_2  - y''; \vec{Q}_T - \vec{p}_{T,12} ;  \vec{k}_T- \vec{p}_{T1}, \vec{m}'_T - \frac{1}{2}\left( \vec{Q}_T - \vec{p}_{T,12}\right)\Rb\nonumber\\
&\times&\int d^2 l_{T}\,d^2 m'_T \,G^{\rm BFKL}\left( y' - y_1; Q_{T}, \vec{q'}_T + \frac{1}{2} \vec{Q}_{T}, l_T\right)\,\, G^{\rm BFKL}\left( y_2 - y''; \vec{Q}_T - \vec{p}_{T,12};  \vec{l}_T- \vec{p}_{T2}, \vec{m}'_T + \frac{1}{2}\left( \vec{Q}_T - \vec{p}_{T,12}\right) \right)\nonumber\\
&\times&
\phi^N_G\left( y'' ; Q_T=0; m_N, m'_T\right)
\end{eqnarray}
and
\begin{eqnarray} \label{NND}
&&  D \,=\,  \,\,\int d^2 Q_T\,\int \,d\,y'\,\int \phi^N_G\left( Y - y'; Q_T=0; q_N, q'_T\right) \nonumber \\
& & \,\int  d^2 q'_T\,d^2 k_T
\phi^{N}_G\left( y' - y_1; Q_T; \vec{q'}_T - \frac{1}{2} \vec{Q}_{T}, k_T \right)\, \int d y''\, \phi^{N}_G\left( y_2  - y''; \vec{Q}_T - \vec{p}_{T,12} ;  \vec{k}_T- \vec{p}_{T1}, \vec{m}'_T - \frac{1}{2} \vec{Q}_T \right)\nonumber\\
&\times&\int d^2 l_{T}\,d^2 m'_T \,\phi^{N}_G\left( y' - y_1; Q_{T}, \vec{q'}_T + \frac{1}{2} \vec{Q}_{T}, l_T\right)\,\, \phi^{N}_G\left( y_2 - y''; \vec{Q}_T ;  \vec{l}_T, \vec{m}'_T + \frac{1}{2} \vec{Q}_T  \right)\nonumber\\
&\times&
\phi^N_G\left( y'' ; Q_T=0; m_N, m'_T\right)
\end{eqnarray}

The rather long algebraic expression of \eq{NNN} and \eq{NND} can be simplified using \eq{A17}  and they take the following forms
\begin{eqnarray} \label{NNNS}
N\,&=&\,\int d^2 Q_T\,\int d y' e^{ 2 \omega\left(\frac{1}{2},0\right) \,y'}  \,d^2 q'_T \phi^N_G\left( Y - y'; Q_T=0; q_N, q'_T\right)\frac{V_{\frac{1}{2} }\left( \vec{q'_T},\vec{Q}_T - \frac{1}{2} \vec{p}_{T,12} \right)\,V_{\frac{1}{2} }\left( \vec{q'_T},\vec{Q}_T + \frac{1}{2} \vec{p}_{T,12} \right)}{ |\vec{Q}_T - \frac{1}{2} \vec{p}_{T,12} |\,|\vec{Q}_T + \frac{1}{2} \vec{p}_{T,12}|}\nonumber\\
&\times& \,\,\int d y'' e^{ 2 \omega\left(\frac{1}{2},0\right) \,y''}  \,d^2 l'_T \phi^N_G\left(  y''; Q_T=0; l_N, l'_T\right)\frac{V_{\frac{1}{2} }\left( \vec{l'_T},\vec{Q}_T  \right)\,V_{\frac{1}{2} }\left( \vec{l'_T},\vec{Q}_T \right)}{Q^2_T} \label{NNNS}\\
D\,&=&\,\int d^2 Q_T\,\int d y' e^{ 2 \omega\left(\frac{1}{2},0\right) \,y'}  \,d^2 q'_T \phi^N_G\left( Y - y'; Q_T=0; q_N, q'_T\right)\frac{V_{\frac{1}{2} }\left( \vec{q'_T},\vec{Q}_T  \right)\,V_{\frac{1}{2} }\left( \vec{q'_T},\vec{Q}_T  \right)}{ Q^2_T}\nonumber\\
 &\times& \,\,\int d y'' e^{ 2 \omega\left(\frac{1}{2},0\right) \,y''}  \,d^2 l'_T \phi^N_G\left(  y''; Q_T=0; l_N, l'_T\right)\frac{V_{\frac{1}{2} }\left( \vec{l'_T},\vec{Q}_T  \right)\,V_{\frac{1}{2} }\left( \vec{l'_T},\vec{Q}_T \right)}{Q^2_T}  \label{NNDS}\end{eqnarray}

 {\boldmath
\section{$\bar{\alpha}_S \left( y_1 \,-\,y_2\right)\,\,\gg\,\,1$}}
All our previous estimates were performed for  small rapidity difference:
 $\bar{\alpha}_S | y_1 - y_2|\,\leq\,1$. In this section we  discuss
 large rapidity differences ($ \bar{\alpha}_S | y_1 - y_2|\,\geq\,1$). For simplicity,
 we
consider only correlations with the typical length of the order of $R_D$. In
 other words, we discuss the generalization of \fig{ddspom} and \fig{ddipom}
 to the case of large $y_{12} =  | y_1 - y_2|$. This generalization is shown
 in \fig{ddipomlay} for the interference diagrams.  The new features here
are  that
 at rapidity $y'_1 < y_1$, we
need to emit an additional gluon, and integrate over both its rapidity
($y'_1$)
 and its transferred momentum ($p'_{T1}$).  Indeed, without this emission the
 ladder between rapidities $y'_1$ and $y'_2$ in \fig{ddipomlay}-b will be in
 the octet state of  color $SU_3$. The main idea is, that the principle
contribution
 stems from $p'_{T1} \ll p_{T1}$. In this case the BFKL Pomeron between
 rapidities $y_1'$ and $y'_2$ has the momentum transfer which is equal
 to $p_{T1}$. After emission of two extra gluons with rapidities
$y'_2$ and
 $y_2$, we obtain that the lower  BFKL Pomeron, that has momentum transfer
$p_{T,12}$,
 as in \fig{ddipom}-b. In this diagram $Q_T \propto 1/R_D$, and can be put
 equal to zero in all parts of the diagrams, except of $G\left( Q_T\right) $ and
 $G\left( \vec{Q}_T - \vec{p}_{T,12}\right)$.

\begin{figure}[ht]
 \includegraphics[width=16cm]{BEddy12la.eps}
   \protect\caption{ The double inclusive production of two gluons
 with rapidities $y_1$ and $y_2$ in the case of large  $| y_1 - y_2|$
 ( $ \bar{\alpha}_S\,| y_1 - y_2|\,\gg\,1$)  and transverse momenta $\vec{p}_{T1}$
 and $\vec{p}_{T2}$,  for the exchange of two BFKL Pomerons which are
denoted
 by wavy lines. This diagram is the LLA generalization of
\protect\fig{ddipom}.
 The  solid lines denote nucleons in the deuterons, which are
 illustrated by
 double lines. Note, that the ladder, shown in \protect\fig{ddipomlay}-b,
 represents
 the BFKL Pomeron with the momentum transferred $\vec{p}_{T1} + \vec{p}'_{T1}
  \approx \vec{p}_{T1}$ for $p'_{T1} \ll p_{T2}$ (see the text).  }
\label{ddipomlay}
   \end{figure}

First,  we need to integrate over $ p'_{T1}$. The vertex of the emission
 is shown in  \fig{ddnv} which can be written as

\begin{equation} \label{NV1}
\bar{\alpha}_S^2\,\Gamma_\mu\left( \vec{k}_T,\vec{p}_{T1}\right)\,\Gamma_\mu\left( \vec{l}_T,\vec{p}_{T1}\right)\,\frac{1}{k'^2_T\,l'^2_T}
\Gamma_\nu\left( \vec{k}'_T,\vec{p}_{T1}\right)\,\Gamma_\nu\left( \vec{l}'_T,\vec{p}_{T1}\right)\,\frac{1}{k''^2_T\,l''^2_T}\end{equation}
with $\vec{k}'_T =\vec{k}_T - \vec{p}_{T1}$ and $ \vec{k}''_T =\vec{k}'_T - \vec{p}'_{T1} = \vec{k}_T - \vec{p}_{T1} -  \vec{p}'_{T1}$.
Plugging in \eq{LIP},\eq{A2} and\eq{RD3} one can see that \eq{NV1} takes the form
\begin{eqnarray} \label{NV2}
&&\bar{\alpha}_S^2\,\int d^2 p'_{T1} \frac{1}{p^2_{T1}}\Bigg\{ \frac{k^2_T}{k'^2_T}\,+\,\frac{l^2_T}{l'^2_T}\,-\,\frac{p^2_{T1}}{k'^2_T\,l'^2_T}\Bigg\}\,\frac{1}{p'^2_{T1}}\Bigg\{ \frac{k'^2_T}{k''^2_T}\,+\,\frac{l'^2_T}{l''^2_T}\,-\,\frac{p'^2_{T1}}{k''^2_T\,l''^2_T}\Bigg\}\,\\
&&\xrightarrow{p'_{T1} \ll k'_T ( l'_T) }\,2\,\bar{\alpha}_S^2\,\int^{k'_T}\frac{d^2 p'_{T1}}{p'^2_{T1}}\left[ \frac{1}{p^2_{T1}}\Bigg\{ \frac{k^2_T}{k'^2_T}\,+\,\frac{l^2_T}{l'^2_T}\,-\,\frac{p^2_{T1}}{k'^2_T\,l'^2_T}\Bigg\}\right]\,\,=\,\,
2\,\bar{\alpha}_S^2\,\int^{{\rm min} \{k'_T, l'_T\}}\frac{d^2 p'_{T1}}{p'^2_{T1}}\,K\left( \vec{k}_T - \vec{l}_T,\vec{k}_T,\vec{k}'_T\right)  \nonumber
\end{eqnarray}
Note that the term $\Big[\dots\Big]$ is the same as in \fig{ddipom} and
 $K\left( \vec{k}_T - \vec{l}_T,\vec{k}_T,\vec{k}'_T\right)$ is given by \eq{RD3}.

\begin{figure}[ht]
 \includegraphics[width=4cm]{BEddnewvertex.eps}
   \protect\caption{ The part of the diagram of \fig{ddipomlay}-a
 with the vertex of emission of two gluons.  }
\label{ddnv}
   \end{figure}
 Finally,  we obtain the following expression for the interference diagram
 of \fig{ddnv}

 \begin{eqnarray} \label{NV3}
&&\frac{d^2 \sigma}{d y_1 \,d y_2  d^2 p_{T1} d^2 p_{T2}}\left(\fig {ddipomlay}-b \right)\,\,=\,\,2 \left(\frac{2 \pi \bar{\alpha}_S}{C_F}\right)^2  \bar{\alpha}_S^2 \int d^2 Q_T  G_D\left( Q_T\right)\,G_D\left( \vec{Q}_T -  \vec{p}_{T,12}\right)\nonumber \\
&&\times\,\int d^2 k_{T}\,\,\int d^2 l_{T}\,\,\phi^{N}_G\left( Y - y_1; \vec{k}_T, -\vec{k}_T \right)\,
 \,\phi^{N}_G\left( Y - y_1; \vec{l}_T, -\vec{l}_T\right)\, \nonumber\\
 &&
 \int^{{\rm min} \{k'_T, l'_T\}}\int^{y_1} d y'_1 \,\frac{d^2 p'_{T1}}{p'^2_{T1}}\,K\left( \vec{k}_T - \vec{l}_T,\vec{k}_T,\vec{k}'_T\right) \int_{y_2} d y'_2 \,\int^{{\rm min} \{q'_T, m'_T\}} \frac{d^2 p'_{T2}}{p'^2_{T2}}\,K\left( \vec{q}_T - \vec{m}_T,\vec{q}_T,\vec{q}'_T\right)  \nonumber\\
 &&
 \int d^2 q_T \int d^2 m_T\, G^{\rm BFKL}\left( y'_1- y'_2; \vec{p}_{T1}; \vec{k'}_T- \vec{p}'_{T1}, \vec{q}'_T + \vec{p}'_{T2}\right)
  G^{\rm BFKL}\left( y'_1- y'_2; - \vec{p}_{T1}; \vec{l}'_T - \vec{p}'_{T1} , \vec{m}'_T + \vec{p}'_{T2}\right)\nonumber\\
 &&  \phi^{N}_G\left( y_2; \vec{q}_T- \vec{p}_{T1},-\vec{q}_T+ \vec{p}_{T2} \right)
\,\phi^{N}_G\left( y_2; \vec{m}_T, -\vec{m}_T\right)\end{eqnarray}

 In \eq{NV3} we put $Q_T = 0$ everywhere, except  in $ G_D\left( Q_T\right)$
 and $G_D\left( \vec{Q}_T -  \vec{p}_{T,12}\right)$, since $Q_T \,\sim\,1/R_D
 \ll $ all other momenta. At first sight \eq{NV3} gives the cross
section which is suppressed as $\bar{\alpha}_S^2$ in comparison with \eq{RD5}.
 However, the integration over $y'_1$ and $y'_2$ leads to $1/\bar{\alpha}_S^2$
 contributions, resulting in a cross section of the order of $\bar{\alpha}_S^2$.
 One can also see, that the cross section  does not depend on the
rapidity
 difference $y_{12}$ for the large values of this difference.

 The generalization to other cases,  which we have considered above, is
 straightforward, and we not discuss it here.

\section{Conclusions}
 \subsection{Comparison with other estimates in perturbative QCD}

 The first estimate of the azimuthal correlations  due to the
 Bose-Einstein correlation in  perturbative QCD, was performed
 in Ref.\cite{KOWE}
 (see also Ref.\cite{KOLUCOR}). The diagrams, that were considered
 in these papers, are shown in \fig{kov}-a.  The observation is that
 these diagrams give rather strong azimuthal correlations, but they
 are symmetric with respect to $\phi \to \pi  - \phi$, and only generate
$v_n$
 with even $n$. The general origin of this symmetry  was discussed in
 section II-B for  slightly different diagrams.    In
 Refs.\cite{KOWE,KOLUCOR} the   $Q_T$ dependence was neglected leading to
 $\delta$-function contributions, which were smeared out by $Q_T$
dependance,
 with $Q_T \sim 1/R$, where $R$ is the size of the interacting dipoles
 in \fig{kov}-a.

       \begin{figure}[ht]
    \centering
    \begin{tabular}{ccc}
      \includegraphics[width=10cm]{glgraph1.eps} &~~~~~~ &\includegraphics[width=4.2cm]{BEKOV1ps.eps}\\
     \fig{kov}-a & &\fig{kov}-b\\
      \end{tabular}
     \protect\caption{\protect\fig{kov}-a is taken from Ref.\cite{KOWE}
 and describe the correlation in a one parton shower
 (see \protect\fig{kov}-b).}
      \label{kov}
      \end{figure}

  Since \fig{kov}-a describes
  the production of two identical gluons in  the dipole-dipole
amplitude, in the Born approximation of perturbative QCD,
  these diagrams are responsible for the azimuthal correlations in one
 parton cascade shown in \fig{kov}-b. It is worthwhile mentioning that
 the  diagram of \fig{kov}-a leads to a contribution which is
proportional
 to $\exp\left( - \omega\left( \frac{1}{2},0\right)\,y_{12}\right)$ and describes the
correlations
 that decrease for large $y_{12}$.  Therefore, only for $\omega\left( \frac{1}{2},0\right)
 \,y_{12} \ll 1$,  can we consider this diagram as a source of
correlations
 which are independent of $y_{12}$.

\begin{figure}[ht]
 \includegraphics[width=9cm]{BEKOVipom.eps}
   \protect\caption{The  generalized  diagram of
\protect\fig{kov}-a,
taking into account the gluon emission (two parton shower contribution).}
\label{kov1}
   \end{figure}
Taking into account the emission of gluons, we can generalize the
diagram of \fig{kov}-a to the
diagram of \fig{kov1}. We have
considered this diagram above,  and have
shown that there is no symmetry with respect of $\phi \to \pi - \phi$
 in such diagrams. Therefore, we conclude that the symmetry $\phi
 \to  \pi - \phi$, is a feature of  the azimuthal correlations in the one
parton
 shower, in the Born approximation of perturbative QCD.

 \subsection{Summary}

In this paper, we found within the framework of perturbative QCD, that
 the Bose-Einstein correlations due to two parton shower production,
induce  azimuthal angle correlations, with three correlation lengths:
 the size of the deuteron, the proton radius, and the size of the BFKL
 Pomeron which is closely  related to the saturation momentum
 ($R_c \sim 1/Q_s$). These correlations are independent of the values
 of rapidities of produced gluons (long range rapidity correlations),
 and have no symmetry with respect to $\phi \to  \pi  - \phi$ ($ \vec{p}_{T1}
 \to  - \vec{p}_{T1}$). Therefore,
 they give rise to $v_n$ for all values of $n$,  not only even values.

We reproduce the result of Refs.\cite{KOWE,KOLUCOR} which show this
 symmetry in the Born approximation of perturbative QCD. However, even
 in the Born approximation, this symmetry depends  on the amplitude of
 the gluon - nucleon interaction at large distances, of about the
 nucleon size and, therefore, it  inherently has a non-perturbative
 nature. Replacing the nucleon by  an onium: the quark-antiquark
bound
 state of heavy quarks, we see that symmetry $\phi \to \pi
 - \phi$ ($\vec{p}_{T1} \to - \vec{p}_{T1}$),  does not
 hold for distances of the order of the size of the onium.

We demonstrated that the azimuthal correlations with the correlation
 length ($R_c$)  of about the size of the deuteron, and the size of
 nucleon, stem from a non-perturbative contribution, and their estimates
 demand a lot of modeling due to the embryonic state of the theory in the
 non-perturbative region. However, the correlations with $R_c \sim 1/Q_s$
 have a perturbative origin, and can be
evaluated in the framework of the Colour Gluon Condensate (CGC) approach.

We show that the two parton showers contributions, generate long range
 rapidity azimuthal angle correlations, which intuitively have been
expected.  In other words, we demonstrate that the azimuthal
 angle correlations do not depend on $y_{12} = | y_1 - y_2|$
 for large values of $y_{12}$ ($\bar{\alpha}_S \,y_{12} \,\geq \,1$).
 We illustrate that the correlation of Refs.\cite{KOWE,KOLUCOR}, actually
describe
 the correlations in a one parton shower, and can be viewed, as
independent
of the
 rapidity difference, only in the narrow rapidity window $\bar{\alpha}_S y_{12}\ll
1$.

  {\bf Acknowledgements}
   We thank our colleagues at Tel Aviv University and UTFSM for
 encouraging discussions. Our special thanks go to
Carlos Cantreras, Alex Kovner and Michel  Lublinsky for
 elucidating discussions on the
 subject of this paper.

 This research was supported by the BSF grant   2012124, by
 Proyecto Basal FB 0821(Chile) ,  Fondecyt (Chile) grant
 1140842, and by   CONICYT grant PIA ACT1406.
 ~


text:
\section{Introduction}
\setcounter{equation}{0}

Singularity theorems of Hawking and Penrose assert that mild energy conditions
imposed on the energy momentum tensor result in a gravitational collapse
to a singularity. While the final singular state of a collapsing star is
inevitable, not much is known about the dynamical mechanisms leading to the
formations of the singularities. The relevant time dependent exact solutions
to Einstein equations are unknown, and the numerical considerations are made
difficult by the Birkhoff theorem which says that any spherically symmetric
vacuum solution is static, which implies that the metric is Schwarzchild.
To make progress one would need to draw conclusions from
numerical evolution of  a non-spherically
symmetric initial data which is considerably more difficult.

One way to overcome these difficulties is to introduce matter fields to Einstein equations which allows the  study of time evolution, while maintaining the
spherical symmetry. The simplest choice corresponds to the massless scalar
field. Christodoulou has given a complete analysis of this situation
\cite{Ch1}.
Interestingly enough his analysis revealed that certain data on a future null
cone centred at the origin can evolve into a solution with naked singularities
(i.e. singularities not hidden inside an event horizon) in contrary with
the Cosmic Censorship Hypothesis (CCH). This is a mild violation of the CCH,
as the initial data is given on a high co-dimension surface inside the null
cone, and is unstable in a sense that any data away from this surface does not
evolve into naked singularities.

Another way to evade Birkhoff's theorem is to go to more than four
space--time dimensions. This was recently done in \cite{Bizon} where
the following ansatz was made for a (4+1) metric\footnote{In this paper
$g_{(r,s)}$ denotes a real pseudo--Riemannian metric of signature $(r, s)$.}
\begin{equation}
\label{bizon}
g_{(4,1)}=-C e^{-2\delta}\d t^2 +C^{-1}\d r^2+\frac{1}{4}r^2
(e^{2B}(\sigma_1^2+\sigma_2^2)+e^{-4B}\sigma_3^2).
\end{equation}
Here $\sigma_i, i=1,2,3$ are the left invariant one forms on $SU(2)$ satisfying the standard Maurer--Cartan relations, and the functions $B, C, \delta$
depend on $(r, t)$. The authors of \cite{Bizon} have numerically studied the PDEs
for these functions resulting from the Ricci-flatness of (\ref{bizon}), and
have shown that a (4+1) dimensional Schwarzchild black hole
\[
C=1-\frac{\mbox{const}}{\rho^2} ,\qquad B=0, \qquad \delta=0
\]
is formed for a large initial data.
This lead to an explicit numerical profile of settling down to a singularity.

Gibbons \cite{gib_talk}
has pointed out that the gravitational collapse of (\ref{bizon})
is not as inevitable as it may seem from the numerical analysis. A
spherically symmetric star in (4+1) dimensions can also settle into a
soliton, i.e. a non-singular topologically stable solution
of the field equations. The simples case is the Kaluza--Klein
monopole of Gross--Perry and Sorkin \cite{GP, Sor}, where the five--dimensional
metric takes the form
\begin{equation}
\label{KK_1}
g_{(4,1)}=-\d t^2+g_{TAUB-NUT},
\end{equation}
where $g_{TAUB-NUT}$ is the simplest asymptotically locally flat (ALF)
four dimensional gravitational instanton. Any ALF gravitational instanton would
do, so a mild, triaxial, generalisation of the ansatz (\ref{bizon}) admits
a static soliton of the form (\ref{KK_1}) with $g_{TAUB-NUT}$ replaced
by the Atiyah--Hitchin gravitational instanton \cite{AH}, where the
complex structures are rotated by the $SU(2)$ action. Using the
gravitational instantons
can even lead to explicit time dependent solutions. As pointed out in
\cite{GLP05}
exploiting the scaling symmetry in the $A_k$ ALF multi--instanton leads
to the explicit solution of the 4+1 Einstein equations
\begin{equation}
\label{G_H}
g_{(4,1)}=-\d t^2+V\;h_{flat}+V^{-1}(\d\theta+A)^2,
\end{equation}
where
\begin{equation}
\label{time_dep}
V=t+\sum_{i=1}^k\frac{m}{|{\bf r}-{\bf r_k}|}
\end{equation}
is a solution to the three--dimensional Laplace equation $\triangle V=0$
depending on a parameter $t$
and the one--form $A$ satisfies
the monopole equation $\d A=*_3 \d V$, where $*_3$ is
the duality operator of the flat Euclidean metric $h_{flat}$
in three dimensions.
Here $m$ is a constant mass parameter, and ${\bf r}_1, ..., {\bf r}_k$
are positions of fixed points in $\mathbb{R}^3$. The metric appears singular at these
points, but in fact it is not if $\theta$ is taken to be periodic, and
the constant $m$ equals to  half of the period.
In particular choosing $k=1$ and
${\bf r}_1={\bf 0}$ leads to a time--dependent generalisation of the
Taub--NUT Kaluza--Klein monopole
metric which is an explicit solution to the Einstein equations imposed
on (\ref{bizon}).

From the (3+1) dimensional perspective the solutions discussed so far
give rise to solutions of Einstein--Maxwell theory with a dilaton. This is
the standard Kaluza--Klein reduction where the fifths dimension compactifies
to a circle of a small radius. This corresponds to $\theta$ in (\ref{G_H})
being periodic. If the radius is sufficiently
small then low energy experiments will average
over the fifths dimension thus leading to an effective four--dimensional theory
with the Maxwell potential given by $A$, and the dilaton given by
$-(\sqrt{3}/4)\log{V}$.
One even gets one time dependent solution (\ref{time_dep}),
but this seems
to be an isolated case.

 The purpose of this paper is to point out (Proposition \ref{proposition})
that large families of explicit time dependent solutions can be found
in the $(3+1)$ dimensional theory. They will come from  time--like
Kaluza--Klein reductions of pure Einstein equations in $(3+2)$ dimensions.
The five--dimensional metrics $g_{(3, 2)} $ are given by
four--dimensional Ricci--flat metrics $g_{(2,2)}$ of signature $(2, 2)$
\begin{equation}
\label{G32}
g_{(3,2)}=\d z^2 +g_{(2,2)}.
\end{equation}
To make the Kaluza--Klein reduction possible the metric $g_{(2, 2)}$ must admit a
Killing vector, and so be of the form
\begin{equation}
\label{g22}
g_{(2, 2)}=Vh_{(2, 1)}-V^{-1}(\d \theta+A)^2
\end{equation}
where $h_{(2,1)}$ is a metric of signature $(2, 1)$ on the three--dimensional
space of orbits of the Killing vector $\partial/\partial \theta$, and
$(V, A)$ is a function and a one--form on this space.
The physical metric $G_{\mu\nu}$ of signature $(3, 1)$ is then
given from (\ref{G32}) by
\begin{equation}
\label{g31}
g_{(3, 2)}=\exp{(-2\Phi/\sqrt{3})}G_{\mu\nu}\d x^{\mu}\d x^{\nu}
-\exp{(4\Phi/\sqrt{3})}(\d\theta+A)^2.
\end{equation}
This metric has a space--like Killing vector $\partial/\partial z$ but is not stationary
if $h_{(2, 1)}$ does not admit a time--like Killing vector. In the next
section we shall construct explicit examples taking $g_{(2, 2)}$ to
be a Ricci--flat anti--self--dual metric with symmetry. This will imply
that $h_{(2, 1)}$ is a part of the so called Einstein--Weyl
structure \cite{Hi82,JT85}, and can in principle be found explicitly by twistor
methods.

If the reader objects to using the metric $g_{(3,2)}$ of signature
$(3, 2)$ as non-physical, he should regard it as a mathematical trick
for producing interesting Lagrangians in four dimensions. A more serious
objection comes from performing the K-K reduction along the time--like
symmetry. This will result in a change of the relative sign between the
Ricci scalar and the Maxwell term in the four dimensional effective
Lagrangain.  The charges in the resulting electro--vacuum solutions
will therefore attract rather than repel, thus ruling out the
extremality condition. The formalism
is nevertheless well adopted to studying the Einstein anti--Maxwell theory
\cite{GR}.
\section{Anti--Self--Dual Ricci--flat $(2, 2)$ metrics with symmetry}
Consider a $(2, 2)$ signature metric $g_{(2,2)}$ on a four--dimensional
manifold $M$.  The construction outlined
in the introduction demands that it is Ricci flat, but to find explicit
examples we shall also assume that its curvature (when viewed as a two--form)
is anti--self--dual. In the $(2, 2)$ signature the spin group $Spin(2, 2)$
decomposes as a product of two independent copies of $SL(2, \mathbb{R})$, and
the representation space of the spin group splits up into a direct sum of
two real two dimensional vector spaces $S_+$ and $S_-$. In the ASD Ricci flat
case the curvature of the spin connection on $S_+$ is zero,
and the holonomy effectively reduces to
$SL(2, \mathbb{R})$. From the mathematical point of view  $g_{(2,2)}$ is a
pseudo-Riemannian analog of a four dimensional hyper--Kahler structure.
The endomorphisms of the tangent bundle associated to three Kahler structures
satisfy the algebra of pseudo--quaternions.

Any
ASD Ricci--flat $(2, 2)$ metric with a non-null symmetry
is of the form (\ref{g22}) and, by the Jones--Tod correspondence \cite{JT85},
the ASD vacuum  equations reduce down to
the Lorentzian Einstein--Weyl equations in three--dimensional space of
orbits $W$ of the Killing vector $K=\partial/\partial \theta$ (if the symmetry is null
the ASD vacuum equations linearise  \cite{DW06}).
This means that there
exists a torsion--free connection $D$ on $W$ such that the null
geodesics of a
conformal structure $[h]$ defined by $h_{(2,1)}$ are also geodesic of
this connection.
This compatibility condition implies the existence
of a one--form $\omega$ on $W$ such that
\[
D h_{(2, 1)}=\omega\otimes h_{(2, 1)}.
\]
If
we  change this representative by $h\rightarrow \psi^2 h$,
then $\omega\rightarrow \omega +2\d \ln{\psi}$, where $\psi$ is a
non-vanishing function on $W$.

The pair $(D, [h])$ satisfies the conformally invariant Einstein--Weyl
equations which assert that the symmetrized Ricci tensor of $D$
is proportional to $h_{(2, 1)}$.
One can regard $h_{(2, 1)}$ and $\omega$ as the unknowns in these equations.
Once they have been  found, the covariant differentiation w.r.t $D$ is given by
\[
D\chi=\nabla \chi-\frac{1}{2}(\chi\otimes \omega+(1-m)\omega\otimes V-h_{(2, 1)}(\omega,\chi)h_{(2, 1)}),
\]
where $\chi$ is a one--form of conformal weight $m$, and $\nabla$ is
the Levi--Civita connection of $h$.

Given the Einstein--Weyl structure which arises
from and ASD vacuum structure, the metric $g_{(2, 2)}$ is given by
solutions to the generalised monopole equation \cite{JT85}
\begin{equation}
\label{monopole}
*_h(\d V+\frac{1}{2}\omega V)=\d A,
\end{equation}
where $*_h$ is the duality operator in three dimensions corresponding to
$h_{(2, 1)}$,
and the unknowns are the function $V$ and the one--form $A$ on $W$.
The arbitrary solution to the generalised monopole equation would lead
to conformally ASD metric $g_{(2, 2)}$ which is not necessarily vacuum.
One must therefore select a special class of solutions. This problem
has been extensively analysed, and is known to lead to three possibilities
depending
on $(\nabla K)_+$, the self dual derivative of the Killing vector:
\begin{itemize}
\item
If $(\nabla K)_+$ is zero then $K$ preserves the self--dual two
forms on $M$, and the moment map coordinates can be chosen. In this case
If $h_{(2, 1)}$ is flat,
\begin{equation}
\label{22GH}
g_{(2, 2)}=V h_{flat}-V^{-1}(\d \theta+A)^2
\end{equation}
and $(V, A)$ satisfy the 2+1 monopole equation $*\d V=\d A$.
\item
If $(\nabla K)_+$ is a simple two form (i.e.$(\nabla K)_+ =\d p\wedge\d q$
for some functions $(p, q)$ on $M$) then there exist local coordinates such
that \cite{DMT}
\begin{equation}
\label{dkpmet}
h_{(2,1)}=\d y^2-4\d x\d t-4 u\d t^2,
\end{equation}
and
\begin{equation}
\label{HCdKP}
g_{(2, 2)}=\frac{u_x}{2}h_{(2,1)}
-\frac{2}{u_x}(\d \theta-\frac{u_x\d y}{2}-u_y\d t)^2,
\end{equation}
where the function $u=u(x, y, t)$ satisfies the dispersionless
Kadomtsev--Petviashvili equation
\begin{equation}
\label{dKP}
(u_t-uu_x)_x=u_{yy}.
\end{equation}
Note that $V=u_x/2\neq 0$ for (\ref{HCdKP}) to be well defined.
The flat metric $g_{(2,2)}$ corresponds to $u=-x/t$.
\item
Finally if $(\nabla K)_+\wedge (\nabla K)_+\neq 0$ there exist local
coordinates such that
\begin{equation}
\label{Todamet}
{h_{(2, 1)}}^{\pm} =e^u(\d x^2\pm\d y^2)\mp\d t^2,
\end{equation}
and
\begin{equation}
\label{Toda4}
g_{(2, 2)}=\frac{u_t}{2}h_{(2,1)} -\frac{2}{u_t}(\d\theta+A)^2,
\end{equation}
where the one--form $A$ on $W$ is a solution to the linear equation
(\ref{monopole}) with $V=u_t/2$ and $\omega=2u_t\d t$.
The ASD vacuum  equations reduce \cite{W90,L91,T95}
to the $SU(\infty)$ Toda equation
\begin{equation}
\label{BFP}
u_{xx}\pm u_{yy}\mp {(e^u)}_{tt}=0.
\end{equation}
The $\pm$ sign in (\ref{BFP}) correspond to two different Lorentzian slices
$h_{(2, 1)}$.

\end{itemize}
The reader should note that while the case (\ref{G_H}) and (\ref{Todamet}) are
indefinite analogues of positive--definite metrics, the case
(\ref{dkpmet}) is genuinely pseudo--Reimannian, as there are no null two--forms
in the Riemannian case.

\section{Kaluza--Klein reduction}
Let $h_{(2,1)}$ be an Einstein--Weyl given by one of
(\ref{dkpmet}, \ref{Todamet}) or $h_{flat}$,
and let $(V, A)$ be the corresponding solution to the generalised
monopole equation
giving rise  to a vacuum metric (\ref{g22}). Consider the $(3+2)$
dimensional metric $g_{(3, 2)}$ given by (\ref{G32}). It is obviously
Ricci--flat, and it
admits
two commuting Killing vectors $\partial/\partial z$ and $\partial/\partial \theta$. We
preform the Kaluza-Klein reduction with respect to the time like vector
 $\partial/\partial\theta$. The four dimensional theory is invariant under the general coordinate transformations independent of $\theta$. The translation of the fiber
coordinate $\theta\rightarrow \theta+\Lambda(x^{\mu})$ induces
the $U(1)$ transformations of the Maxwell one--form.
The scaling of
\[
\theta\longrightarrow \lambda\theta,
\qquad [g_{(3,2)}]_{\mu\theta}\longrightarrow \lambda^{-2}
[g_{(3,2)}]_{\mu\theta},
\qquad
\]
is spontaneously broken by the Kaluza--Klein
vacuum, since $\theta$ is a coordinate
on a circle with a fixed radius. The scalar field corresponding to this
symmetry breaking is called the dilaton. It is the usual practise
to  conformally rescale the resulting $(3+1)$ dimensional metric, and the
dilaton so that the multiple of the Ricci scalar of $G_{\mu\nu}$ in the reduced Lagrangian is equal to  $\sqrt{|\det(G_{\mu\nu})|}$.
The corresponding Maxwell field is  $F=\d A$, and the physical metric
$G_{\mu\nu}$ in
$(3+1)$ signature is given by (\ref{g31}).
We can summarise our findings in the following proposition
\begin{prop}
\label{proposition}
Let $([h], D, W)$ be an Einstein--Weyl structure in 2+1 dimensions,
and let $V$ be function on $W$ of conformal weight $-1$
which is a solution to the generalised monopole equation {\em(\ref{monopole})}
such that the corresponding
$(++--)$ ASD metric {\em(\ref{g22})} is  Ricci--flat. Then for any
$h_{(2, 1)}\in [h]$ the
triple
\[
G=\sqrt{V}h_{(2,1)}+\frac{1}{\sqrt{V}}\d z^2, \qquad
\Phi=-\frac{\sqrt{3}}{4}\log{V}, \qquad F=*_h D V
\]
satisfies the Einstein--Maxwell--Dilaton equations arising from
the Lagrangian
density
\begin{equation}
\frac{1}{16\pi^2}(R-2(\nabla \Phi)^2+ \frac{1}{4}e^{-2\sqrt{3}\Phi} F_{\mu\nu}F^{\mu\nu}).
\end{equation}
\end{prop}
To understand the unusual sign between the Ricci scalar and the Maxwell term
in this Lagrangian
notice that making the replacements
\[
\theta\longrightarrow i\theta, \qquad A\longrightarrow iA
\]
would lead to a more usual space--like Kaluza--Klein reduction from
$(4+1)$ to $(3+1)$ dimensions, where the relative sign between
the Ricci scalar and  the Maxwell term is negative.

The negative energy of the Maxwell field has peculiar physical consequences.
No static multi--black hole solutions analogous to the Majumdar--Papapetrou
extremal black holes can exist, as both gravity and electromagnetism are
now attractive forces and the cancellations can not take place. Thus the theory
only allows non--extremal black holes which can effectively increase
their masses by radiating photons out! We can also encounter
`tachyonic' solutions invariant under $\mathbb{R}\times SO(2, 1)$.

Proposition (\ref{proposition}) also applies to the ordinary EMD solutions
with positive Maxwell energy if one takes $([h], D, W)$ to be positive
definite Einstein Weyl structure and preforms the usual space-like
Kaluza--Klein
reduction of the product metric $g_{(4)}-\d t^2$, where $g_{(4)}$ is
an ASD Ricci flat metric corresponding to $([h], D, W)$. This will rule out
solutions coming from the dKP equation (\ref{dKP}). There is however
another possible construction which we now outline. Let
\[
h_{(3)}=\d z^2+e^{u}(\d x^2+\d y^2), \qquad \omega=2u_z  \d z
\]
be a positive--definite EW structure corresponding to a solution
$u=u(x, y, z)$ to the elliptic $SU(\infty)$ Toda equation $u_{xx}+u_{yy}+(e^u)_{zz}=0$.
If $(V, A)$ is an arbitrary solution to
the monopole equation (\ref{monopole}) the resulting four--dimensional
Riemannian metric
\[
g_{(4)}= Vh_{(3)}+V^{-1}(\d \theta+A)^2
\]
is scalar--flat and Kahler \cite{L91}. One special solution $V=u_z/2$
makes $g_{(4)}$ Ricci--flat, and the Lorentzian version of this solution
must be used in Proposition (\ref{proposition}). If one instead takes
$V=V_{\Lambda}$, where
\[
V_{\Lambda}=-\frac{1}{3}\Lambda(1-zu_z)
\]
then $g_{(4)}$ is conformal to an Einstein metric $\hat{g}_{(4)}$ with a
cosmological constant $\Lambda$ \cite{T97}:
\begin{equation}
\hat{g}_{(4)}=\frac{V_{\Lambda}}{z^2}h_{(3)}+
\frac{1}{z^2V_{\Lambda}}(\d\theta +A)^2.
\end{equation}
Given $\hat{g}_{(4)}$ we construct a `cosmological' vacuum metric
in 4+1 dimensions given by
\[
g_{(4, 1)}=f(t)^2\hat{g}_{(4)}-\d t^2.
\]
There are several possibilities for $f$:
Choosing $f(t)=L^{-1}\mbox{cosh}{(Lt)}$
yields a regular metric, and $f(t)=L^{-1}\sin{(Lt)}$ gives
a generalised AdS solution with Big Bang and Big Crunch singularities
at $t=0$ and $t=\pi/L$ respectively.
The K-K reduction of this metric (now we need to change the relative
sign between the two terms in (\ref{g31})) gives the following solution
to the EMD equations
\begin{eqnarray*}
G_{(3, 1)}&=&\frac{f}{z}\Big(\sqrt{V_{\Lambda}}\Big(\frac{f}{z}\Big)^2h_{(3)}-
\frac{1}{\sqrt{V_{\Lambda}}} \d t^2\Big),\qquad
\Phi=-\frac{\sqrt{3}}{2}\log{\Big({\frac{f}{z\sqrt{V_{\Lambda}}}}\Big)},\\
F&=&-(V_{\Lambda})_x\d y\wedge \d z-(V_{\Lambda})_y\d z\wedge\d x-
(V_{\Lambda}e^u)_z\d x\wedge\d y.
\end{eqnarray*}
\section{Example}
The simplest illustration of Proposition (\ref{proposition})
corresponds to $g_{(2, 2)}$ being an
analytic continuation of a gravitational instanton in Riemannian signature.
We shall examine the (3+1) solution arising from a $(2, 2)$ analog of the
Taub--Nut gravitational instanton, and emphasise that this example
hints the semi-classical instability of the Einstein--anti Maxwell
theories.

Consider (\ref{22GH}), and take
\begin{equation}
\label{formula_V}
V=\varepsilon+\frac{m}{\sqrt{x^2+y^2-t^2}}, \qquad \varepsilon=1.
\end{equation}

The NUT singularity is absent in $(3+2)$ dimensions. It is a fixed point of the Killing vector $\partial/\partial\theta$ which is regular if $\theta$ is periodic.
The resulting metric (3+1) metric (\ref{g31}) written in  cylindrical polar
coordinates takes the form
\begin{eqnarray}
\label{example1}
\d s^2&=&
\Big(\varepsilon+\frac{m}{\sqrt{\rho^2-t^2}}\Big)^{-1/2}
\d z^2+\Big(\varepsilon+\frac{m}{\sqrt{\rho^2-t^2}}\Big)^{1/2}
(\d \rho^2+\rho^2\d \theta^2-\d t^2),\\
\Phi&=&-\frac{\sqrt{3}}{4}\log{(\varepsilon+\frac{m}{\sqrt{\rho^2-t^2}})},
\qquad A=\Big(\varepsilon+\frac{m}{{\sqrt{\rho^2-t^2}}}\Big)^{-1}\d z\nonumber,
\end{eqnarray}
and we can take $z$ to be periodic\footnote{Another possibility is
to take  $V$ as in (\ref{formula_V}) with
$\varepsilon=z$ which leads to a non-asymptotically flat metric.}.

The initial data on the surface $t=0$ is regular everywhere.
The rescalling of the  three--metric
\[
g_{(3)}=\Big(\varepsilon+\frac{m}{{\rho}}\Big)^{-1/2}\d z^2+
\Big(\varepsilon+\frac{m}{{\rho}}\Big)^{1/2}
(\d \rho^2+\rho^2\d \theta^2)
\]
is regular at $\rho=0$ if $\theta$ is periodic with a period $4\pi$.
This can be seen by setting $\rho=\hat{\rho}^2/m$, so
that around $\hat{\rho}=0$
\[
g_{(3)}\sim \d z^2+4\Big(\d \hat{\rho}^2+\hat{\rho}^2\;\d \Big(\frac{\theta}{2}\Big)^2\Big).
\]
The cylindrical
mass of $g_{(3)}$ can be defined as a deficit angle at infinity, which
is proportional to the integral of the Gaussian curvature of the metric induced
on the surfaces of constant $z$.
This metric evolves to a space-time with naked singularities on the
cone $\rho^2=t^2$. Near this cone the metric behaves like
\[
\sqrt{m}^{-1}e^{-U/2}\d z^2+e^{3U/2}\sqrt{m}(-\d T^2+\mbox{cosh}^2(T)\d\theta^2
+\d U^2), \qquad U\longrightarrow -\infty,
\]
where $\rho=e^U\ \mbox{cosh}(T), t=e^U\ \mbox{sinh}(T)$.
This solution represents a charged particle moving with the along  the $z$ axis. It can be interpreted as a
tachyon in a sense of \cite{GR}, as it is unstable and invariant under $\mathbb{R}\times SO(2, 1)$.

The properties of (\ref{example1}) signal the semi-classical instability
of the vacuum in Einstein--Maxwell--dilaton with attractive electric force.
The argument is analogous to Witten's bubble of nothing \cite{Wit}.
If the coordinate $z$ is periodic the solution is asymptotic to
the flat metric on $\mathbb{R}^3\times S^1$.
The decay of $\mathbb{R}^3\times S^1$ vacuum
is described by the instanton obtained by replacing $t\rightarrow
i\tau$ in the metric (\ref{example1}). This instanton has vanishing action
\cite{GR}, and the probability of the decay is given by the exponential of
the negative action.

\section*{Acknowledgements}
This work resulted from a seminar \cite{gib_talk} given
at the programme Global Problems in Mathematical Relativity
held in Newton Institute  in the Fall 2005. I would like to thank
Gary Gibbons and other participants of the programme for useful comments.

redpajama_set_name: RedPajamaArXiv