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\begin{document} \title{MPI08 \\ Proceedings }
\author{B. Blok$^1$\protect\footnote{\ \ speaker},
 L. Frankfurt$^2$, M. Strikman$^3$}
\institute{$^1$Physics department, Technion, Haifa, Israel\\
$^2$Institute School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel,\\
$^3$Physics Department, Pensilvania State University, College
Station, USA. }
\title{ Increase with energy of parton transverse momenta in the
fragmentation region in  DIS and related phenomena. } \maketitle
\begin{abstract}
We demonstrate the fundamental
property of pQCD: smaller is the size of the colorless quark-gluon
configurations, more rapid is the increase of its interaction with
energy. In the limit of fixed $Q^2$ and $x\to 0$ we find the
increase with the energy of the transverse momenta of the
quark(antiquark) within the $q\bar{q}$ pair produced in the
fragmentation region by the strongly virtual photon.
 Practical consequences of discovered effects is
that the ratio of DVCS to DIS amplitudes should very
slowly tend to one at very large collision energies,
that a rapid projectile has the biconcave shape, which is different from the expectations of the preQCD parton model
where a fast hadron has a pancake shape. We  found
dominance of different phases of chiral and conformal
symmetries in the central and peripheral pp, pA, and AA
collisions.
\end{abstract}
\maketitle \setcounter{page}{1}
\section{Introduction.}
\par

 A leading order dipole approximation Ref. \cite{BFGMS,AFS,FMS,BBFS,CFS},  provides the solution of
the equations of QCD in the kinematics of fixed and not too small
$x=Q^2/\nu$ but $Q^2\to \infty$. The characteristic feature of
this solution is the approximate Bjorken  scaling for the
structure functions of DIS, i.e. the two dimensional conformal
invariance for the moments of the structure functions. In this
approximation as well as within the leading $\log(x_0/x)$
approximation, the transverse momenta of quarks within the
dipole  produced by the local  electroweak current are restricted
by the virtuality of the external field:
\begin{equation}
\Lambda^2 \le p_t^2 \le Q^2/4.
\end{equation}
Here $\Lambda\equiv \Lambda_{QCD}=300 $ Mev is a QCD scale. It
follows from the QCD factorization theorem proved in Refs.
\cite{CH,CE} that within this kinematical range the smaller
transverse size $d$ of the configuration
 (the transverse distance between the constituents of the dipole) corresponds to a
 more rapid increase of its interaction with the collision energy:
\begin{equation}
\sigma= \alpha_s(c/d^2)F^2\frac{\pi^2}{4} d^2
xG_{T}(x,c/d^2),\label{1.3}
\end{equation}
here $F^2=4/3$ or $9/4$ depends whether the dipole consists of
color triplet or color octet constituents,  $G_T$ is an integrated
gluon distribution function and $c$ is a parameter $c=4\div 9$. It
is well known in the DGLAP approximation that the structure
function $G_T(x,Q^2)$ increases more rapidly with $1/x$ at larger
$Q^2$. This property agrees well with the recent HERA data.
  The aim of the present  talk is to demonstrate  that
the transverse  momenta of the (anti)quark of the   $q\bar q$ pair
produced by a local current  increase  with the energy and become
larger than $Q^2/4$ at sufficiently large energies. In other words
the characteristic transverse momenta in the fragmentation region
increase with the energy. Technically this effect follows from the
more rapid increase with the energy of the pQCD interaction
 for smaller dipole and the $k_t$ factorization theorem.
 \par
It is worth  noting  that this  kinematics  is very different from the central rapidity kinematics where
the increase of  $p_t^2$
  was  found in the leading $\alpha_s\log(x_0/x)$ BFKL approximation\cite{BFKL}: $\log^2(p_{t}^2/p_{t 0}^2)
\propto \log(s/s_0)$. Indeed, the latter rapid increase is absent
in a fixed order of perturbation theory, and is the property of
the ladder: the further we go along the ladder, the larger are
characteristic transverse momenta, i.e. we have a diffusion in the
space of transverse momenta \cite{BFKL}. On the other hand the
property we are dealing here with is the property of a
characteristic transverse momenta in the wave function of the
projectile.
\par
The dipole  approximation provides the target rest frame description which is   equivalent to the Infinite
Momentum Frame (IMF) description of DIS in LO DGLAP and BFKL  approximations. To achieve equivalence with the
 IMF description in the NLO  approximation it
 is necessary to calculate radiative corrections to cross section in the fragmentation  region,
  i.e. to take into account the increase of the number of constituents  and related  renormalization
  of the dipole wave function.  Recent calculations \cite{Fadin,SP} suggest that these corrections
   are small. Consequently in the talk we will neglect  these corrections.

\par
Our main result  is that the median transverse momenta $k^2_t$ and
invariant masses of the leading $q\bar q$ pair in the
fragmentation region grow as
\begin{eqnarray}
k^2_t&\sim& a(Q^2)/(x/x_0)^{\lambda (Q^2)},\nonumber\\[10pt]
M^2&\sim& b(Q^2)/(x/x_0)^{\lambda_{\rm M} (Q^2)}.\nonumber\\[10pt]
\label{main}
\end{eqnarray}
Here $k^2_t$ and $M^2$ are the median squared transverse momentum
and invariant mass of the quark-antiquark pair in the
fragmentation region.  (The median  means that  the configurations
with the momentum/masses less than the median one contribute half
of the total crosssection). The exponential factors
 $\lambda$ and $\lambda_{\rm M}$ are both  approximately
 $\sim 0.1$.  These  factors are
  weakly dependent on the external virtuality $Q^2$. The
exact values  also depend on the details of the process, i.e.
whether we consider the DIS process
 with longitudinal or
transverse photons, as well as on the model and approximation
used. The exact form of $\lambda (Q^2)$, and $\lambda_{\rm M}
(Q^2)$ are given below.

The rapid increase of the characteristic transverse scales in the
fragmentation region has been found first in Refs.
\cite{FGMS,FSW,Rogers1,Rogers2},
 but within the black disk   regime (BDR). Our
new result is the prediction of the increase with energy of the
jet transverse momenta in the fragmentation region/the rise of the
transverse momenta in the impact factor with the energy, in
 the kinematical domain where methods  of pQCD are still applicable. This effect could be considered as
  a precursor of the black disk regime indicating the possibility of the smooth matching between two regimes.
\par
Our results can be applied to a number of  processes. First we
consider the deeply
 virtual Compton scattering (DVCS)
process, i.e. $\gamma +p\to \gamma^*+p$.
 \par We also find that
 at sufficiently large energies
\begin{equation}
\sigma_L(x,Q^2)/\sigma_T(x,Q^2)\propto (Q^2/4p_t^2)\propto
(1/x)^{\lambda}.
\end{equation}
Hence the $\sigma_L/\sigma_T$ ratio should decrease as the  power
of energy instead of being $O(\alpha_s)$.
\par Our results have the implication for the space structure of the
wave packet describing a rapid hadron. In the classical
multiperipheral picture of Gribov a hadron has a shape of a
pancake of the longitudinal size $1/\mu$ (where $\mu $ is the
scale of soft QCD) which does
 not depend on the incident energy \cite{Gribovspace-time}. On the contrary, we find in
   section 5 the biconcave shape for the rapid hadron in pQCD with the minimal
longitudinal length (that corresponds to small impact
 parameter $b$) decreasing with increase of energy and being smaller for nuclei than for the nucleons.
\par
Finally, in the last section  we discuss the possible applications
of our results to pp, pA collisions at the LHC.


\section{The target rest frame description.}
\par
Within the LO approximation the QCD factorization theorem allows
to express the total cross section of
 the scattering of  the longitudinally
polarized photon with virtuality $-Q^2 \gg \Lambda_{QCD}^2$ off a
hadron target as the convolution of the square of the virtual
photon wave function calculated in the dipole approximation and
the cross section of the dipole scattering off a hadron
\cite{BFGMS,FS,FRS}. In the target rest frame the
cross section for the scattering of longitudinally polarized
photon has the form :
\begin{equation}
\sigma(\gamma_L^{*}+T\to X)={e^2\over 12\pi^2} \int d^2 p_t dz
\left<\psi_{\gamma_L^{*}}(p_t,z)\right|
 \sigma(s,p_t^2) \left|\psi_{\gamma_L^{*}}(p_t,z)\right>.
\label{crosssection1}
\end{equation}
Here $\sigma$ is the dipole crosssection operator:
\begin{equation}
\sigma= F^2\cdot \pi^2 \alpha_s(4p_t^2)(-\vec \Delta_t ) \cdot
xG(\tilde{x}= (M^2+Q^2)/s, 4p_t^2),
 \label{crosssection}
\end{equation}
here $\vec \Delta_t$ is the  two dimensional Laplace operator in
the space of the transverse momenta, and $
M^2=(p_t^2+m_q^2)/z(1-z)$ is the invariant mass squared of the
dipole. In the coordinate representation $\sigma$ is just a number
function, and not a differential operator as in the momentum
representation.
 \par  Integrating by parts over $p_t$ it is easy
to rewrite Eq. ~\ref{crosssection1} with the LO accuracy in the
form where the integrand is
 explicitly positive:
\begin{equation}
\sigma(\gamma_L^{*}+T\to X)={e^2\over 12\pi^2} \int
\alpha_s (4p_t^2)  d^2 p_t dz  \left< \nabla{\psi_{\gamma_L^{*}}}(p_t,z)\right| f(s, z, p_t^2)
\left|\nabla \psi_{\gamma_L^{*}}(p_t,z)\right>,
\label{crosssection2}
\end{equation}
here
\begin{equation}
f=(4\pi^2/3)\alpha_s(4p_t^2)
xG(\tilde{x}, 4p_t^2).
\end{equation}
In the derivation we use  the boundary conditions that follow from
the fact that the photon wave function  decreases rapidly
 in the  $p_t^2\to \infty$ limit
 and that the
contribution of small $p_t$ is the higher twist effect.

\par
Eq. \ref{crosssection2} can be  explicitly rewritten in terms of
integration in $k^2_t$ and $z$ as
 \beq
\sigma_L (x,Q^2)=6\pi\frac{\pi\alpha_{\rm e.m.}\sum
e^2_qF^2Q^2}{12} \int dk^2_t \alpha_s (4k^2_t)z^2(1-z)^2
\frac{k_t^2}{(k^2_t+Q^2z(1-z))^4}\cdot g(\tilde{x},4k^2_t).
\label{r10a} \eeq where $\tilde x$ is given by
$(k^2_t/((z(1-z)+Q^2)/s$. Here we take into account explicitly the
(rather weak) $z$-dependence of the integrand.

The similar derivation can be made for the scattering of
 transverse photon in configurations of spatially small size.
 In this case the contribution of small $p_t$ region (Aligned Jet Model contribution) is  comparable
 to the pQCD one.  The main interest in this paper is in the
 region of high energies (HERA and beyond) i.e.  sufficiently small
 $\tilde{x}$, and small $Q^2$,
 where pQCD contribution
dominates because of the rapid increase
 of the gluon distribution with the decrease of $x$. We include a
 contribution of the
aligned jet configurations by
  imposing a cutoff in transverse
 momenta (see below for the details).
 \par
 The pQCD contribution into the total cross section
 initiated by the transverse photon  has the form:
 \begin{eqnarray}
 \sigma_T&=& 6\pi\frac{\pi\alpha_{\rm e.m.}\sum
e^2_q F^2}{12} \nonumber\\[10pt]
&\times &\int^1_0dz \int dk_t^2\alpha_s(4k^2_t)(z^2+(1-z)^2)
\frac{(k_t^4+Q^4z^2(1-z)^2)}{(k^2+Q^2z(1-z))^4}\cdot g(\tilde{x},
4k^2_t).\nonumber\\[10pt]\label{r12}
\end{eqnarray}
In the numerical calculations using Eq. \ref{r12} we introduced a
cutoff in the space transverse momenta $M^2z(1-z)\ge u, u\sim
0.35$ GeV$^2$. The contribution of smaller $k^2_t$ in the total
crosssection was calculated using the AJM model.




\section{The characteristic transverse momenta in hard
fragmentation processes in LO approximation.}
\par

Here we carry out the calculations for realistic energies and
realistic structure functions. The numerical results indicate that
the effects discussed above are manifest even at the energies of
the order $s\sim 10^5\div 10^7$ GeV$^2$. We want to draw attention
that our main qualitatively new result-the increase of the parton
transverse momenta in the current fragmentation region should be
valid in NLO, NNLO approximations as well because its derivation
uses specific property of DGLAP
 approximation to pQCD -a larger virtuality leads to a more rapid  increase of amplitude with energy.
We will also consider the extrapolation of  our results to energies of the
order $s\sim 10^7 $GeV$^2$. These energies are unattainable  at existing facilities.
The proposed e-p collider at LHC may reach the invariant energies  of order $10^6$ GeV$^2$.
 However these results are interesting  from the theoretical point of
view- probing the limits of the
pQCD. The relation of our results to the processes at the LHC will be discussed in the last section.
\par
Challenging and unresolved problem  is how to use resummation
methods at extremely small $x$ \cite{ABF,Ciafaloni}
 to evaluate dependence on energy of parton distribution in the current fragmentation region.
At $x$ achieved at HERA account of the  energy-momentum conservation restricts the number of possible
 gluon emissions by one-two.   Such emissions are correctly accounted for within NLO, NNLO  DGLAP approximation.
   One can substantiate this point by  evaluation of the number of radiated gluons in the multiRegge
kinematics \cite{FSW}. At extremely small x where number
 of gluon  radiations would be sufficiently large and therefore essential impact parameters would exceed radius
  of a nucleon  the intercept of pQCD Pomeron  may become independent on $Q^2$ as a result of
diffusion in the space of transverse momenta. This interesting problem is beyond the scope of this paper.



\subsection{The longitudinal photons.}
\par
In the case of longitudinal photons we have considered the
characteristic median/average transverse momenta scale, that
corresponds to the half of the total crosssection $\sigma_L$. This
scale is determined from Eq. \ref{r10a} by first integrating over
z for given $k_t$, and then analyzing the corresponding jet
distribution.   In Figure \ref{Longitudinal} we present the
characteristic graphs for the ratio \beq R(k^2_t)=\frac{\sigma(
k^2_t)}{\sigma_L},\label{1000}\eeq where $\sigma (k^2_t)$
corresponds to the result of integration of Eq.\ref{r10a} over
transverse momenta
 $\le k^2_t$. We see from Fig. \ref{Longitudinal}, that  for fixed $k_t$
 $R(k_t)$ slowly increases with the increase of the energy.
 The results based on using CTEQ5 parametrization
are qualitatively similar, although the increase of median $k^2_t$
with the energy is more rapid. The energy dependence
 of median $k^2_t$ can be described
with a very good accuracy  by an approximate formula
$(x/0.01)^{0.04+0.025\log(Q^2/Q_0^2)}.$
 Here $Q^2_0=10$ GeV$^2$,$x_0\sim 0.01$. The power increases from
$\sim 0.04$ at $Q^2\sim 5$ GeV$^2$, to $0.09$ at $Q^2\sim 100$
GeV$^2$.
  For CTEQ5 this power increases to 0.1 at $Q^2=100 GeV^2$ instead of $0.09$. This is
consistent with the enhanced rate of the increase of CTEQ5
structure functions as compared to the CTEQ6 ones (see below).
\par
These results allow us to estimate the scales, where one  expects
the appearance of the new QCD regime, i.e. one has to use the
$k_t$ factorization approach. Indeed, the DGLAP approximation is
based on the strong ordering in all rungs of the ladder, in
particular in the first rung (the impact factor in the $4k_t$
factorization language ) we must have $4\Lambda_{\rm QCD}^2\le
4k^2_t\le Q^2$. It is clear, this ordering can not hold,  once the
median $4k^2_{tm}$ becomes of order $Q^2$. Then we obtain the
condition (using CTEQ6 distribution functions): \beq
4a(Q^2)/(x/0.01)^{0.04+0.025\log(Q^2/Q_0^2)}\sim Q^2.
\label{condition1} \eeq Here the function $a$  corresponds to the
transverse momenta at $x=0.01$.
\par
The numerical calculations show   that for $Q^2=5$ GeV$^2$ one
gets from eq. \ref{condition1} $x\sim 10^{-4}$, for $Q^2=10$
GeV$^2$ one gets $x\sim 10^{-6}$, which may be reached at LeHC.
For larger $Q^2$ we are however beyond the realistic energies: say
for $Q^2\sim 20$ GeV$^2$ we need $x\sim 10^{-9}$. The use of CTEQ5
gives qualitatively the same results (for $Q^2=30 $ GeV$^2$ we
obtain $x\sim 10^{-8}$.  Thus we may hope to observe the onset
 of the new regime for  the $k_t$ dependence analyzing
  small $x$ jet distributions at LeHC/LHC.
rations.

\subsection{Transverse photons}
\par
We performed the numerical analysis for the transverse photons
using eqs. \ref{r10a},\ref{r12} in the same fashion as for the
longitudinal photons. In Figure \ref{Transverse} we depicted the
characteristic function $R(k^2_t)$ given by Eq. \ref{1000} that
gives the characteristic momenta as a function of x for several
different values of $Q^2$. The characteristic energy dependence
for median $k^2_t$ is $(x/0.01)^{0.09+0.014\log(Q^2/Q_0^2)}$ where
$x_0= 0.01, Q^2_0=10$ GeV$^2$.  The curves in Fig.
\ref{Transverse} clearly show that the characteristic momenta
increase with the increase of $1/x$, as the corresponding curves
slowly shift to the right.
\par We see  that the average transverse
momenta for longitudinal photons is significantly larger than for
transverse photons. On the other hand, the invariant masses for
transverse photons are always significantly larger than $4k^2_t$.
This is due to the large contribution of the AJM type
configurations with $z\sim 0,1$ ($z$ is the fraction of the total
momentum of the dipole carried by one of its constituents). Since
$M^2=k^2_t/(z(1-z))$,   a more slow increase  of $M^2$ than
of $k^2_t$ is consistent with the slow increase of average $z$
towards 1/2, i.e. the symmetric  configurations
become dominant, but only at asymptotically large energies.
\par Once again, we can estimate the boundary of the region
where the direct DGLAP approach stops being self-consistent.
Assuming $k^2_t\sim Q^2/4$, we obtain that the boundary for $Q^2 =
3, 5 , 10\,\, \mbox{GeV}^2$ is reached at $x \sim 10^{-3},
10^{-4}, 10^{-6}$. For higher $Q^2$ this boundary lies at
unrealistically high energies. The use of the CTEQ5
parametrization gives qualitatively the same results.
\par
So far we considered only perturbative QCD contribution, and the
median transverse momentum was determined relative to the total
perturbative crosssection, i.e. the one starting from the cut off
$u=0.35$ GeV$^2$. It is well known that even at HERA energies the
contribution of AJM into the total crosssection is significant.
The corresponding AJM contribution to the total crosssection is
given in fig. 3a. Note that the median $k^2_t$ at small
virtualities at HERA energies significantly decreases if we
calculate it using the crosssection that includes both the pQCD
and soft (AJM) contributions. For example, at $Q^2\sim 10$ GeV$^2$
the median transverse momentum squared  decreases by almost a
factor of two down to
 $k^2_t\sim 0.65$ GeV$^2$.



\section{Deeply virtual Compton scattering.}

\par
As the application of the formulae obtained in this paper we shall
consider the DVCS processes $\gamma+p\to \gamma^*+p$. We shall
show that the slow increase  in the median transverse momenta
leads to the slow decrease of the ratio $R=A_{\rm DVCS}/A_{\rm
CS}$ with energy to the limiting value  equal one.
\par
The DCVS amplitude is described in pQCD by the same formula
\ref{r12}  as the amplitude describing total cross section of DIS
at given  $x,Q^2$ but with the substitution in
Eq.\ref{crosssection2} of the wave function of virtual photon by
wave function of a real photon,  i.e. $Q^2=0$.

As a  result in  pQCD $R$ has the form :
\beq
R_{\rm pQCD}=\displaystyle{ \frac{\int^1_0 dz \int dM^2 \alpha_s (M^2z(1-z)) (1/(M^2+Q^2)^2)\cdot g(\tilde{x},M^2).}{\int^1_0 dz
\int dM^2 \alpha_s (M^2z(1-z)) ((M^4+Q^4)/(M^2+Q^2)^4))\cdot
g(\tilde{x},M^2).}} \label{DCVS} \eeq
\par
Let us note that strictly speaking, we must use the generalized parton distributions  (GPD) in Eq. \ref{DCVS}.
 However the difference  between gluon GPD and gluon
pdf is not large in this case because
fractions carried by gluons in GPD differ by the factor
$\approx$  two  at moderate x and tend to one at extremely large energies as
the consequence of increase of parton momenta with energy. (In
fact most of the  non-diagonal effect in this approach is included  in the wave functions  of the initial
 and final photons.)
 As a result we may neglect the difference
between GPD and distribution functions in the considered
kinematics.
 The numerical analysis of Eq. \ref{DCVS} shows
 that indeed the ratio $R$ very slowly decreases with the increase of energy
due to  a slow increase of a ratio $M^2/Q^2$ discussed in the
previous section, and  $R\sim 1.6$ for HERA energies.
\par
The result Eq. \ref{DCVS} is however not complete since we
neglected the contribution of the AJ configurations. In this paper
we take them into account using the AJM model \cite{FS2} (and
references therein, see also Appendix B of this paper). Indeed as
we see from  Fig. 3a, the AJ configurations
 give a substantial
contribution to the  total crosssection of the DIS of the
transverse photons. We refer the reader to appendix B and ref.
\cite{FFS} for the discussion of main properties of the AJM. We
see that  the AJM contribution to the total crosssection is of
order $70\%$ at $Q^2\sim 1$ GeV$^2$, $x\sim 0.01$.
\par
 Rough estimate gives
\beq R_{\rm AJM}\approx 2, \label{DVCS2} \eeq since the major
difference in the amplitudes describing  total cross section of
DIS and DCVS is in the difference between the wave functions of
the  virtual and real photons-the factor
$\frac{Q^2+M^2}{Q^2}\approx 2$. But in the essential region of
integration $M^2\approx Q^2$. In the framework of the AJM model
the ratio of amplitudes of the DVCS to DIS can be calculated
within the leading twist approximation as: \beq R_{\rm AJM}=
\frac{Q^2+m_0^2}{Q^2}\log(1+\frac{Q^2}{m^2_0}).\label{DVCS1}\eeq
Here the parameter $m^2_0=0.3-0.5$ GeV$^2$ is the cut off
parameter $m^2_0\le m^2_\rho$, $m_\rho$ is the $\rho$ meson mass.



\par
Combining the pQCD and AJ model
 contributions we have
 \beq
 R=\frac{R_{\rm pQCD}\sigma_T+R_{\rm AJM}\sigma_{\rm
 AJM}}{\sigma_T+\sigma_{\rm AJM}}.
 \label{DVCS2a}
 \eeq
 \par
 Here the pQCD contribution into the total crosssection
 $\sigma_T$ is  given by Eq. \ref{r12} and the contribution of AJ to the  total crosssection is
  given by AJM - Eq. \ref{DVCS1}. The results of numerical calculation   as a function of $x$ for
several values of $Q^2$ are depicted in Fig. 3b.
 The ratio R is close to 2 at HERA energies and
increases with $Q^2$ (from 5 to 100 GeV$^2$ by $\sim 40\%$). This
result is in a good agreement with the analysis of the H1 and ZEUS data in Ref. \cite{S}  (see in particular Table 4
in Ref. \cite{S}). Our main prediction is that the ratio R should decreases with the rise of energy. It
tends to one at asymptotically large energies
 in agreement with the result for the BDR \cite{PRL}.  However the onset of this regime is very slow.
This prediction can be checked experimentally in the study of DVCS
processes at LHeC.
\par
Our conclusion on the important role of AJM contribution
in DVCS  at HERA energies is  in the qualitative agreement with the recent experimental data
\cite{LatestHera} that shows the important role of soft QCD in the diffractive processes in DIS at HERA.
\par
We want to draw attention that agreement between experimental
results and theoretical prediction is rather good. This is due to
the fact that the interaction of dipole effectively includes the
NLO corrections since parton distributions were obtained by
fitting the experimental data. Consequently one may hope that NLO
corrections to impact factors are relatively small.
\par
Let us stress that the current calculation is preliminary.  More
detailed calculation should account for the contribution of
c-quark, and study in detail the dependence of R on the AJM
parameters).


\section{The shape of the fast nucleon and nuclei.}
\par
The coherence length $l_c$ corresponds to the life-time of the
dipole fluctuation at a given energy in the rest frame of the target.
Within the parton model approximation  the coherence
length  is  $l_c\sim 1/2m_Nx$  \cite{Ioffe} i.e. it linearly increases
with energy. In  pQCD as a result of QCD evolution coherence length increases with energy more slowly \cite{KS,BFcoh}:
\begin{equation}
l_c=(1/2m_Nx)(s_0/s)^{\lambda}.
\end{equation}
Such energy dependence of the coherence length shows that  the wave function of a fast hadron differs in QCD from that in the Gribov picture \cite{Gribovspace-time} .

\par
Let us consider the longitudinal distribution of the partons in a
fast hadron.   In the parton model the longitudinal spread of the
gluonic cloud  is $L_z\sim 1/\mu$ for  the wee partons (where
$\mu $ is the soft scale) and it is much larger than for harder
partons, with $L_z \sim 1/xP_h$ for partons carrying a  finite $x$  fraction of the hadron momentum \cite{Gribovspace-time}.
The picture is changed qualitatively in the
 limit of very high energies when interactions
reach BD regime for $k_t \gg \mu$. In this case the smallest possible characteristic momenta in the frame where
 hadron is fast are of the order $k_t (BDR)$ which is a function of both initial energy  and transverse coordinate, $b$ of the hadron.  Correspondingly, the longitudinal size is $\sim 1/ k_t (BDR) \ll 1/\mu$.
 There is always a tail to the  much smaller momenta all the way down to  $k_t \sim \mu$ which corresponds to the partons with much larger longitudinal size  (a pancake of soft gluons corresponding to the Gribov's picture).  However at large  energies at the proximity of the unitarity limit the contribution of the  gluons with $k_t<k_{\rm tb}$ is strongly suppressed . In the BDR this tail is suppressed by a factor
$ k_t^2/k_t (BDR)^2$  \cite{Mueller02,FGMS}. In the color glass
condensate model the suppression is  exponential
\cite{Dumitru:2002qt}.

Since the gluon parton density  decreases with the increase of $b$
the longitudinal size of the hadron is larger for large $b$, so a
hadron has a shape of biconcave lens, see Figs. \ref{S9},\ref{S10}

In the numerical calculation we took
\begin{equation}
\left| l_z \right|= 1/k_t(BDR),
\end{equation}
neglecting overall  factors of the order  of one (typically in the
Fourier transform one finds $\left<z\right> \sim {\pi \over
\left<p_z\right>}$).  We calculated $k_t(BDR)$ for fixed external
virtuality $Q_0^2 \sim 40 GeV^2$. Our results are not sensitive to
the value of $Q_0^2$, as the value of $Q^2$  only enters in the
combination  $x'= (Q_0^2+M^2)/s$, and the $k_t ^2$ we found were
comparable or larger than $Q_0^2/4$. Indeed, the direct
calculation shows that for small b the change of $1/k_t$ if we go
between external virtualities of $60$ and  $5$ GeV$^2$ is less
than $5\%$. Such weak dependence continues almost to the boundary
of the picture Fig. 4a, where $k_t\sim 1$ GeV. Near the boundary
the uncertainty increase to $\sim 25\%$, meaning that for large b
(beyond those depicted in Fig. 4a) the nucleon once again becomes
a pancake and there is  a smooth transition between two pictures (
biconcave lens and pancake). We want to emphasize here that the
discussed above weak dependence of  $k_t(BDR)$ on the resolution
scale indicates that the shape of the wave function for small x is
almost insensitive to the scale of the probe.
\par
We depict the typical transverse quark structure of the fast
nucleon in Fig. 4a.  We see that it is drastically different from
the naive picture of a fast moving nucleon as a flat narrow disk
with small constant thickness. (Similar plot for the gluon
distribution is even more narrow). %%
Note also that for the discussed  small x  range
$k_t \ge 1 {\mbox GeV/c}$ for $b\le 1 fm$. Since the spontaneous chiral  symmetry breaking corresponds to
quark virtuality $\mu^2 \le 1 GeV^2$, probably $\sim 0.7 {\mbox GeV}^2$ \cite{DP}, corresponding to
$k_t \sim \sqrt{{2\over 3 } \mu^2} \sim 0.7 {\mbox GeV/c}$ the chiral symmetry should be restored for a
large range of $b$ in the proton wave function for small x.
%%
\par
Let us consider  the DIS on the nuclei for
the case of external virtualities of the order
of several GeV. In this case the  shadowing effects  to the large
extent   cancel  the factor $A^{1/3}$
in the gluon density of a nucleus  for a central
impact parameters, $b$ \cite{FGS}, and
 the gluon density in the nuclei is comparable to that in a single nucleon for $b\sim 0$.
Consequently over the large range of the impact parameters the
nucleus longitudinal size is approximately the
 same as in the nucleon at $b\sim 0$.
 \par
 However for very small $x$
 we find large $k_t(BDR)$ corresponding to $4 k_t^2(BDR) \ge  40$
GeV$^2$.  This is a self consistent value as indeed for such $Q^2$
the leading twist shadowing is small.

  Accordingly we calculated the shape of the nucleus for the external virtuality  $Q^2 \ge  40$ GeV$^2$.
   We should emphasize here that taking  a smaller virtuality would not significantly
 change our result for $k_t(BDR)$ (at the same time LT nuclear shadowing reduces a low momentum tail of
  the $k_t$ distribution
 as compared to the nucleon case).


   In the discussed limit of the small leading twist
 shadowing,  the corresponding gluon density
unintegrated over b is given by a product of a nucleon gluon density and the nuclear profile function:
\beq
T(b)=\int dz \rho(b,z),\label{bor1}
\eeq
where the nuclear three-dimensional
density is normalized to A. We use standard Fermi step
parametrization  \cite{Bohr}
 \beq
 \rho(r)=C(A)\frac{A}{1+\exp((r-R_A)/a)},\,\,\, R_A=
 1.1A^{1/3} {\rm fm}, a=0.56 {\rm fm}.\label{improved}\eeq Here  $r=\sqrt{z^2+b^2}$,
 and A is the atomic number.
C(A) is a  normalization factor, that can be
calculated numerically from the condition
$\int d^3r \rho(r)=A$.  At the zero
impact parameter $T(b) \approx 0.5 A^{1/3}$ for large A.
\par The dependence of the thickness of
a fast  nucleus as a function of the transverse size is depicted
in Fig. 4b for a typical high energy $s=10^7$ GeV$^2$, $Q^2=40$
GeV$^2$. We see that the nuclei also has a form of a biconcave
lens instead of a flat disk. The dependence on the external
virtuality for the nuclei is qualitatively very similar to the
case of the nucleon. For small b the dependence is very weak (of
order $5\%$) and increases only close to the boundary of the
biconcave lens region where it is of order $20\%$ ( and $k_t\sim
1$ GeV). For larger b we smoothly return to the pancake picture.
\par
Note that this picture is very counterintuitive:
the thickness of a nucleus is  smaller than of a nucleon
in spite of
$\sim A^{1/3}$ nucleons at the same impact parameter.
The resolution of the paradox in the BD regime is quite simple:  the soft fields of
 individual nucleons destructively interfere  cancelling each other.  Besides for a given impact
parameter $b $, the longitudinal size of a heavy nucleus
$1/ k_t^{(A)}(BDR) < 1/ k_t ^{(p)}(BDR)$ since the gluon distribution  function in
the nuclei $G_A(x,b) > G_N(x,b)$. So a naive classical  picture of a system build of the constituents being
 larger than  each of the constituents is grossly violated.
 The higher density of partons leads to the restoration of the chiral symmetry in a broad b range and much
 larger x range than in the nucleon case.
%%





\section{Experimental consequences.}

\par
The current calculations of  the cross sections of the hard
processes at the LHC are based on the use of the DGLAP parton
distributions and the application of the factorization theorem.
Our results imply that
in the kinematical region of sufficiently small $x$ it is necessary
to use the $k_t$ factorization and the dipole model, instead of
the direct use of DGLAP.

A similar analysis must  be made for the $pp$ collisions at LHC.
It has been understood long ago that  the probability of pp
collisions at central impact parameter is close to 100\% (total
$\Gamma$ is close to 1) even for soft QCD, i.e. at lesser energies
than those necessary to achieve BDR for the hard interactions. The
compatibility of probability conservation with  the rapid increase
of hard interactions with energy , predicted by QCD, requires the
decrease of importance of soft QCD contribution with energy
\cite{FSZ}. As a result the hadronic state emerged in pp, pA, AA
collisions at sufficiently large energies consists of two phases.
Central collisions would be dominated by the strong interaction
with small coupling constant - the phase with unbroken chiral and
conformal symmetries. On the contrary, the  peripheral collisions
are dominated by the more familiar phase with broken chiral and
conformal symmetries.
 At these energies the QCD phase at central
 collisions - with the unbroken chiral and conformal symmetries -will be different from that
for the peripheral collisions.  This new phenomenon may appear
especially important for the central heavy
 ion collisions at LHC and at RHIC. Quantitative analysis
of this problem will be presented elsewhere.

\par
The hard processes initiated by the real photon can be directly
observed in the ultrapheripheral collisions \cite{Mark1}. The
 processes where a real photon scatters on a target,
 and creates two jets with an invariant mass $M^2$, can be analyzed in the dipole model by formally putting
 $Q^2=0$, while $M^2$ is an invariant mass of the jets. In this case  with a  good accuracy the spectral density
discussed above will give the spectrum of jets in the
fragmentation region. Our results show that
 the jet distribution over the transverse momenta will be broad
 with the maximum moving towards larger
transverse momenta with increase of the energy and centrality of
the $\gamma A$ collision.
\par We have seen that our results can also describe DCVS
processes. The ratio R of DCVS $\gamma^*\rightarrow \gamma^*$ and
forward amplitudes at $t=0$ is of order 2 at HERA energies at
small external virtualities, and rapidly growing with $Q^2$. This
ratio slowly decreases with the decrease of $x$.
\par Finally, our results can be checked directly, if and when the
LHeC facility will be built at CERN.
\par More detailed version of this work can be found in Ref.
\cite{Blok}
\par One of us, B.Blok, thanks S.Brodsky for the
useful discussions of the results obtained in the paper. This work
was supported in part by the US DOE Contract Number DE-
FG02-93ER40771 and BSF.
------------------------------------------------------------------------------
%       Bibliography
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\begin{figure}[htp]
  \begin{center}
    \subfigure[The ratio $R(k^2_t)$  for $Q^2=10$ GeV$^2$ for
longitudinal photons. The three curves correspond to x=$10^{-3}$
(upper one), $10^{-5}$ (middle one) and $10^{-7}$ (lower
one).]{\label{S1}\includegraphics[scale=0.5]{mfig3c.eps}}\qquad
    \subfigure[The ratio $R(k^2_t)$  for $Q^2=40$ GeV$^2$ for
longitudinal photons. The three curves correspond to x=$10^{-3}$
(upper one), $10^{-5}$ (middle one) and $10^{-7}$ (lower one).]{\label{S2}\includegraphics[scale=0.5]{mfig4c.eps}} \\
    \subfigure[The ratio $R(k^2_t)$  for $Q^2=80$ GeV$^2$ for
longitudinal photons. The three curves correspond to x=$10^{-3}$
(upper one), $10^{-5}$ (middle one) and $10^{-7}$ (lower
one).]{\label{S3}\includegraphics[scale=0.5]{mfig5c.eps}}
  \end{center}
  \caption{The ratio $R(k^2_t)$   for
longitudinal photons for different values of $Q^2$ and $x$.}
  \label{Longitudinal}
\end{figure}

\begin{figure}[htp]
  \begin{center}
    \subfigure[The ratio $R(k^2_t)$  for $Q^2=10$ GeV$^2$ for
transverse photons. The three curves correspond to x=$10^{-3}$
(upper one), $10^{-5}$ (middle one) and $10^{-7}$ (lower
one).]{\label{S4}\includegraphics[scale=0.5]{mfig6c.eps}}\qquad
    \subfigure[The ratio $R(k^2_t)$  for $Q^2=40$ GeV$^2$ for
transverse photons. The three curves correspond to x=$10^{-3}$
(upper one), $10^{-5}$ (middle one) and $10^{-7}$ (lower one).]{\label{S5}\includegraphics[scale=0.5]{mfig7c.eps}} \\
    \subfigure[The ratio $R(k^2_t)$  for $Q^2=80$ GeV$^2$ for
transverse photons. The three curves correspond to x=$10^{-3}$
(upper one), $10^{-5}$ (middle one) and $10^{-7}$ (lower
one).]{\label{S6}\includegraphics[scale=0.5]{mfig8c.eps}}
  \end{center}
  \caption{The ratio $R(k^2_t)$   for
transverse photons for different values of $Q^2$ and $x$.}
  \label{Transverse}
\end{figure}

\begin{figure}[htp]
  \begin{center}
    \subfigure[The contribution R of pQCD to the total crosssection,
that is a sum of pQCD and AJM model contributions. The cut off of
the AJM model is 0.35 GeV$^2$, for $Q^2=5$ GeV$^2$ (lower curve),
$20$ GeV$^2$ (middle curve), 40 GeV$^2$, the x axis corresponds to
$\log_{10}(1/x)$]{\label{S7}\includegraphics[scale=0.5]{mfig1c.eps}}\qquad
    \subfigure[The ratio R of the DVCS crosssection to total transverse
crosssection for different values of $Q^2$ as a function of $x_B$.
for $Q^2=5$ GeV$^2$ (lower curve), $20$ GeV$^2$ (middle curve), 60
GeV$^2$ (upper curve),the x axis corresponds to $\log_{10}(1/x)$]{\label{S8}\includegraphics[scale=0.75]{mfig2c.eps}} \\
  \end{center}
  \caption{The contribution of the AJM to the total crosssection (a),
  and the the ratio of the DVCS crosssection to a total crossection.}
  \label{DVCS}
\end{figure}
\begin{figure}[htp]
  \begin{center}
   \subfigure {\label{S9}
 \includegraphics[scale=0.4]{nucleon.pdf}}
% \qquad
    \subfigure {\label{S10}\includegraphics[scale=0.4]{gold.pdf}} \\
  \end{center}
  \caption{The shape of the fast nucleon (a) and gold nucleus (b) in QCD at $s=10^7  \,
\mbox{GeV}^2 $ and the resolution scale $Q^2 \le  40
\mbox{GeV}^2$.}
  \label{shape}
\end{figure}
\end{document}
\begin{figure*}
%
\begin{tabular}{cc}
\includegraphics[width=8cm,height=8cm]{nucleon.pdf}
&
\includegraphics[width=8cm,height=8cm]{gold.pdf}
&
\end{tabular}
\caption[]{The shape of the fast nucleon (a) and gold nucleus (b) in QCD at $s=10^7  \,
\mbox{GeV}^2 $ and the resolution scale $Q^2 \le  40
\mbox{GeV}^2$.} 
\label{fig_pb}
\end{figure*}


\end{document}
