# Multiplicity distribution and entropy of produced gluons in deep inelastic
scattering at high energies

Eugene Levin<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Particle Physics, Tel Aviv University, Tel Aviv 69978, Israel  Departemento de
Física, Universidad Técnica Federico Santa María, and Centro Científico-
Tecnológico de Valparaíso, Avda. Espana 1680, Casilla 110-V, Valparaíso, Chile

###### Abstract

In this paper we found the multiplicity distribution of the produced gluons in
deep inelastic scattering at large
$z=\ln\left(Q^{2}_{s}/Q^{2}\right)\,\,\gg\,\,1$ where $Q_{s}$ is the
saturation momentum and $Q^{2}$ is the photon virtuality. It turns out that
this distribution at large $n>\bar{n}$ almost reproduces the KNO scaling
behaviour with the average number of gluons
$\bar{n}\propto\exp\left(z^{2}/2\kappa\right)$, where $\kappa=4.88$ in the
leading order of perturbative QCD. The KNO function
$\Psi\left(\frac{n}{\bar{n}}\right)=\exp\left(-\,n/\bar{n}\right)$. For
$n<\bar{n}$ we found that
$\sigma_{n}\propto\Big{(}z-\sqrt{2\,\kappa\,\ln(n-1)}\Big{)}/(n-1)$. Such
small $n$ determine the value of entropy of produced gluons
$S_{E}=0.3\,z^{2}/(2\,\kappa)$ at large $z$. The factor $0.3$ stems from the
non-perturbative corrections that provide the correct behaviour of the
saturation momentum at large $b$.

###### pacs:

13.60.Hb, 12.38.Cy

###### Contents

  1. I Introduction
  2. II  Generating functional for multipartical production processes
    1. II.1  Generating functional for the scattering amplitude
    2. II.2  AGK cutting rules and generating functional for the production of gluons
    3. II.3  The master equation for $\sigma_{n}$
    4. II.4 Solutions for Pomeron calculus in zero transverse dimension
  3. III  The cross sections of gluon production deep in the saturation region
    1. III.1  The total cross section
    2. III.2  The diffraction production
    3. III.3  $\sigma_{1}\left(Y,r,b\right)$
    4. III.4  $\sigma_{2}\left(Y,r,b\right)$
    5. III.5  $\sigma_{3}\left(Y,r,b\right)$ and $\sigma_{k}\left(Y,r,b\right)$
  4. IV  Entropy of produced gluons
  5. V Conclusions

##  I Introduction

Deep inelastic scattering (DIS) processes play a unique role in developing the
theoretical understanding of the high energy interaction in QCD. Indeed, the
main ideas of Colour Glass Condensate(CGC)/saturation approach (see Ref.KOLEB
for a review) : the saturation of the dipole density and the new dimensional
scale ($Q_{s}$), which increases with energy, have become the common language
for discussing the high energy scattering in QCD. All these ideas are rooted
in the theoretical description of DISBFKL ; LIP ; LIREV ; GLR ; MUQI ; MUDI ;
MV .

We are going to discuss the multiplicity distribution of the produced gluons
in DIS. The multiplicity distributions and especially the entropy of produced
gluons have become a hot subject during the past several years KUT ; PES ;
KOLU1 ; PESE ; KHLE ; BAKH ; BFV ; HHXY ; KOV1 ; GOLE1 ; GOLE2 ; KOV2 ; NEWA ;
LIZA ; FPV ; TKU ; KOV3 ; KOV4 ; DVA1  and we hope that this paper will look
at these problems at different angle. However, before discussing the
multiplicity distribution it looks reasonable to summarize our theoretical
achievements in DIS.

First and most importantly is that the scattering amplitude of the colourless
dipole with the size $x_{01}$ which determines the DIS cross section,
satisfies the Balitsky-Kovchegov (BK) non-linear equationBK :

$\frac{\partial N_{01}}{\partial
Y}\,=\,\bar{\alpha}_{S}\int\frac{d^{2}\,x_{02}}{2\pi}\frac{x^{2}_{01}}{x^{2}_{02}\,x^{2}_{12}}\Big{\\{}N_{02}+N_{12}-N_{02}N_{12}-N_{01}\Big{\\}}$
(1)

where $N_{ik}=N\left(Y,\bm{x}_{ik},\bm{b}\right)$ is the scattering amplitude
of the dipoles with size $x_{ik}$ and with rapidity $Y$ at the impact
parameter $\bm{b}$.

It has been shown that Eq. (1) leads to a new dimensional scale: saturation
momentumGLR  which has the following $Y$ dependenceGLR ; MUT ; MUPE :

$Q^{2}_{s}\left(Y,b\right)\,\,=\,\,Q^{2}_{s}\left(Y=Y_{0},b\right)\,e^{\bar{\alpha}_{S}\,\kappa\,Y\,-\,\,\frac{3}{2\,\gamma_{cr}}\ln
Y}$ (2)

where $Y_{0}$ is the initial value of rapidity and $\kappa$ and $\gamma_{cr}$
are determined by the following equations111$\chi\left(\gamma\right)$ is the
BFKL kernelBFKL  in anomalous dimension ($\gamma$) representation.$\psi$ is
the Euler psi -function (see Ref.RY  formula 8.36). :

$\kappa\,\,\equiv\,\,\frac{\chi\left(\gamma_{cr}\right)}{1-\gamma_{cr}}\,\,=\,\,-\frac{d\chi\left(\gamma_{cr}\right)}{d\gamma_{cr}}~{}~{}~{}\,\,\,\mbox{and}\,\,\,~{}~{}~{}\chi\left(\gamma\right)\,=\,\,2\,\psi\left(1\right)\,-\,\psi\left(\gamma\right)\,-\,\psi\left(1-\gamma\right)$
(3)

In Ref.MUT  (see also Re,MUPE ) it is shown that in the vicinity of the
saturation scale the scattering amplitude takes the following form:

$N_{01}\left(z\right)\,\,\,=\,\,\,\mbox{Const}\left(x^{2}_{10}\,Q^{2}_{s}\left(Y\right)\right)^{\bar{\gamma}}$
(4)

with $\bar{\gamma}=1-\gamma_{cr}$. In Eq. (4) we introduce a new variable $z$,
which is equal to:

$z\,\,=\,\,\ln\left(x^{2}_{01}\,Q^{2}_{s}\left(Y,b\right)\right)\,\,\,=\,\,\,\,\bar{\alpha}_{S}\,\kappa\,\left(Y\,-\,Y_{A}\right)\,\,+\,\,\xi$
(5)

where $\xi=\ln x^{2}_{01}$. Note that we neglected the term
$\frac{3}{2\,\gamma_{cr}}\ln Y$ in Eq. (2).

It turns out that inside the saturation region:
$x^{2}_{01}\,Q^{2}_{s}\left(Y\right)\,>\,1$ the scattering amplitude shows the
geometric scaling behaviour, being a function of one variable
$x^{2}_{01}\,Q^{2}_{s}\left(Y\right)$ :

$N_{01}\left(Y,x_{01},b\right)=N\left(x^{2}_{01}\,Q^{2}_{s}\left(Y,b\right)\right)$
(6)

It is instructive to note, that this behaviour has been proven on general
theoretical groundsBALE  and has been seen in the experimental data on DISGS .

Finally, in Ref.LETU  the solution to Eq. (1) was found deep into saturation
region for $z\,\gg\,1$ which has the following form:

$N_{01}\left(z\right)\,\,\,=\,\,1\,\,\,-\,\,\,\mbox{Const}\,\exp\Big{(}-\frac{z^{2}}{2\,\kappa}\Big{)}$
(7)

where ${\rm Const}$ is a smooth function of $z$.

We have defined the saturation region as
$x^{2}_{10}\,Q^{2}_{s}\left(Y,b\right)\,>\,1$. However , in Refs.GOST ; BEST
it has been noted that actually for very large $x_{10}$ the non-linear
corrections become small and we have to solve linear BFKL equation. This
feature can be seen directly from the eigenfunction of this equation. Indeed,
the eigenfunction has the following form LIP

$\phi_{\gamma}\left(\bm{r},\bm{R},\bm{b}\right)\,\,\,=\,\,\,\left(\frac{r^{2}\,R^{2}}{\left(\bm{b}+\frac{1}{2}(\bm{r}-\bm{R})\right)^{2}\,\left(\bm{b}-\frac{1}{2}(\bm{r}-\bm{R})\right)^{2}}\right)^{\gamma}\,\,\xrightarrow{b\,\gg\,r,rR}\,\,\left(\frac{R^{2}\,r^{2}}{b^{4}}\right)^{\gamma}\,\,\equiv\,\,e^{\gamma\,\xi}~{}~{}\mbox{with}\,\,0\,<\,Re\gamma\,<\,1$
(8)

for any kernel which satisfies the conformal symmetry. In Eq. (8) $R$ is the
size of the initial dipole at $Y=0$ while $r\equiv x_{10}$ is the size of the
dipole with rapidity $Y$.

One can see that for $r=x_{10}\,>\,min[R,b]$, $\phi_{\gamma}$ starts to be
small and the non-linear term in the BK equation could be neglected. In this
paper we wish to discuss the DIS for $Q^{2}\geq 1\,GeV^{2}$ and in the region
of small $x$. In other words we consider $x_{10}<R$, where $R$ is the radius
of a hadron. Bearing this in mind we can replace Eq. (8) by the following one:

$\phi_{\gamma}\left(\bm{r},\bm{R},\bm{b}\right)\,\,\,=\,\,\,\left(\frac{r^{2}\,R^{2}}{\left(\bm{b}-\frac{1}{2}\bm{R}\right)^{2}\,\left(\bm{b}+\frac{1}{2}\bm{R})\right)^{2}}\right)^{\gamma}=\left(r^{2}Q_{s}\left(Y=0,b,R\right)\right)^{\gamma}$
(9)

Therefore, we can absorb all impact parameter dependence in the dependence of
the saturation scale on $b$.

The main goal is to find the multiplicity distributions and the entropy of
produced gluons in the kinematical region of Eq. (7). In other words, we wish
to decipher the saturated amplitude in terms of produced gluons. Fortunately,
the equations for the cross section of production of $n$ gluons have been
derived in the framework of CGC (see Ref.KLP ). The next section is a brief
review of the results of Ref.KLP . In this section we emphasize the two main
ingredients on which the derivation of the equations is based: the AGK cutting
rulesAGK  and the BFKL equationBFKL  , which gives the production of gluons in
a particular kinematic region typical for leading log(1/x) approximation of
perturbative QCD.

In section 3 we discuss solutions to the equations for the total cross section
$\sigma_{tot}$, for the cross section of diffraction production $\sigma_{sd}$
and the cross section for productions of $n$ gluons $\sigma_{n}$. We show that
both the total cross section and the cross section of the diffraction
production reaches the unitarity limit: $\sigma_{tot}(b)=2$ and
$\sigma_{sd}=1$, while $\sigma_{n}$ are small and proportional to
$\,\frac{1}{\bar{n}(z)}\exp\left(-n/\bar{n}(z)\right)$ with
$\bar{n}(z)\,\,\propto\,\,\exp\Big{(}\frac{z^{2}}{2\,\kappa}\Big{)}$. In
section 4 we estimate the value of the entropy of produced gluons. In
conclusions we summarize our results and discussed related problems.

##  II  Generating functional for multipartical production processes

###  II.1  Generating functional for the scattering amplitude

The scattering amplitude, given by BK equation (see Eq. (1)) , can be viewed
as a sum of the "fan" BFKL Pomeron diagramsGLR ; MUQI ; BART ; BRN ; BRAUN ;
KOLU (see Fig. 1). The value of the triple Pomeron vertex is determined by the
non-linear term of the BK equation. However, it has been shown in Ref/MUDI
that the BK equation has a quite different interpretation in which the BFKL
kernel

Figure 1:  Fig. 1-a:The fan diagrams in the BFKL Pomeron calculus that
describe the BK equation (see Eq. (1)).Fig. 1-b: The unitarity constraints for
the exchange of the BFKL Pomeron: the structure of the cut Pomeron. This
constraints determines the cut Pomerons that are shown in Fig. 2 by the wavvy
lines which are crossed by the dashed ones.

$\frac{\bar{\alpha}_{S}}{2\,\pi}K\left(x_{01}|x_{02},x_{12}\right)\,\,=\,\,\frac{\bar{\alpha}_{S}}{2\,\pi}\frac{x^{2}_{01}}{x^{2}_{02}\,x^{2}_{12}}$
(10)

gives the probability for decay of one dipole with size $x_{10}$ to two
dipoles with sizes $x_{02}$ and $x_{12}$.

The simplest and the most transparent technique to incorporate this decay is
the generating functional which allows us to reduce the calculation of the
high energy elastic amplitude to consideration of a Markov process. The
generating functional is defined as MUDI ; MUSA ; Salam ; LELU

$Z_{0}\left(Y;\\{u\\}\right)\,\,\equiv\,\,\sum_{n=1}\,\int\,\,P_{n}\left(Y;r_{1},\dots,r_{n}\right)\,\,\prod^{n}_{i=1}\,u(r_{i})d^{2}r_{i}$
(11)

where $u(r_{i})$ is an arbitrary function of $r_{i}$ and $b_{i}$. $P_{n}$ is
the probability density to find $n$ dipoles with sizes $r_{1},\dots,r_{n}$ at
rapidity $Y$. 222The dipole $(x_{i},y_{i})$ with coordinates $x_{i}$ for quark
and $y_{i}$ for antiquark can be characterized by the dipole size
$\bm{r}_{i}=\bm{x}_{i}-\bm{y}_{i}$ and
$\bm{b}_{i}=\frac{1}{2}(\bm{x}_{i}+\bm{y}_{i})$ . For simplicity we suppress
in Eq. (11) and below the coordinate $b_{i}$. For the scattering with the
nuclear target we can consider that impact parameters of all dipoles are the
same $b_{i}=b$ (see Ref. BK ). The alternative notations are
$\bm{r}_{i}=\bm{r}_{ik}=|\bm{x}_{i}-\bm{y}_{k=i+1}|$. For functional of Eq.
(11) we have two obvious conditions:

initial conditions: $\displaystyle
Y=0\,\,P_{n}\,=\,0\,\,\mbox{for\,\,\,}n\,>\,1\,\,\mbox{and}\,\,P_{1}=\delta(\bm{r}\,-\,\bm{r}_{1})\delta(\bm{b}-\bm{b}_{1});Z_{0}\left(Y=0;\\{u\\}\right)\,\,=\,\,u(r)\,\,;$
(12a) boundary conditions: $\displaystyle
u=1\,\,Z_{0}\left(Y;\\{u\\}\right)|_{u=1}\,\,=\,\,1;$ (12b)

Eq. (12b) follows from the physical meaning of $P_{n}$ and represents the
conservation of the total probability, while Eq. (12a) indicates that we are
considering the interaction of one dipole with the target.

The Markov process can be described as the following equation for the
generating functional:

$\frac{dZ_{0}\left(Y;\\{u\\}\right)}{dY}=\,\frac{\bar{\alpha}_{S}}{2\pi}\,\int\,\,d^{2}r\;d^{2}r^{\prime}\,K\left(r|r^{\prime},|\bm{r}-\bm{r^{\prime}}|\right)\left\\{\,u(r)\,-\,u(r^{\prime})\,u(|\bm{r^{\prime}}-\bm{r}|)\right\\}\,\frac{\delta}{\delta
u(r)}\,\,Z_{0}\left(Y-y;\\{u\\}\right)$ (13)

Two terms of this equation has simple meaning of increase of the probability
to find $n$-dipoles due to decay of one dipole to two dipoles (birth terms,
the second term in Eq. (13)) and of decrease of the probability since one of
$n$-dipoles can decay (death term, the first term in Eq. (13))

Eq. (13) is general and can be used for arbitrary initial condition. In the
case of Eq. (12a) it can be rewritten as the non-linear equation. Eq. (13)
being a linear equation with only first derivative has a general solution of
the form $Z_{0}\left(Y-y;\\{u\\}\right)\equiv Z_{0}\left(\\{u(Y-y)\\}\right)$.
Inserting this solution and using the initial condition of Eq. (12a) we obtain
the non-linear equation

$\frac{dZ_{0}\left(Y;\\{u\\}\right)}{dY}=\frac{\bar{\alpha}_{S}}{2\pi}\int
d^{2}r^{\prime}\,K\left(r|r^{\prime},|\bm{r}-\bm{r^{\prime}}|\right)\left\\{Z_{0}\left(Y,\\{u\\}\right)-Z^{2}_{0}\left(Y,\\{u\\}\right)\right\\}$
(14)

It is easy to see that Eq. (14) can be re-written as the Balitsky-Kovchegov
equation BK  for the scattering amplitude (see for example Ref.KOLEB ).

###  II.2  AGK cutting rules and generating functional for the production of
gluons

Based on this probability interpretation of the sum of "fan” diagrams in
Ref.KLP  an attempt was made to develop the probability approach for the
production of the gluon using the AGK cutting rulesAGK . It is worthwhile
mentioning that such kind of approach was successfully applied for
interactions of two BFKL cascades in the simple case with zero transverse
dimension MUSA ; Salam ; LEPRSM . The AGK cutting rules use that the BFKL
PomeronBFKL  gives the cross section of produced gluons in the specific
kinematic region typical for leading log(1/x) approximation of perturbative
QCD. These cutting rules allow us to expand the contribution of the exchange
of $n$ BFKL Pomerons as a definite sum of the final states with the average
number of produced gluon $k\Delta Y$, where $k\,\leq\,n$ and $\Delta\,Y$ is
the mean multiplicity for one BFKL Pomeron exchange. Actually, the AGK cutting
rules introduce three different triple Pomeron vertices (see Fig. 2 ).

Figure 2:  Three vertices for gluon production accordingly to AGK cutting
rules for fan diagrams in the BFKL Pomeron calculus. The wavy lines denote the
BFKL Pomerons. The wavy lines crossed by the dashed ones show the cut
Pomerons.

The structure of the production processes is determined by the s-channel
unitarity constraints for the BFKL Pomeron:

$2N^{BFKL}\left(Y;r,b\right)\,\,=\,\,G^{BFKL}_{in}\left(Y;r,b\right)$ (15)

where $N^{BFKL}\left(Y;r,b\right)$ is the contribution to the scattering
amplitude by the BFKL Pomeron $G^{BFKL}_{in}\left(Y;r,b\right)$ is the cross
section of produced gluons, which was actually calculated in Ref.BFKL  and
which is shown in Fig. 1-b. In framework of the AGK cutting rules it is called
the cut Pomeron which is shown in Fig. 2 by the wavy lines which are crossed
by the dashed ones.

In Ref.KLP  it is introduced a generalization of Eq. (11):

$\displaystyle
Z\left(Y;\\{u\\},\\{v\\}\right)\,\,\equiv\,\,\sum^{\infty}_{n=0,m=0}\,\int\,\,P^{m}_{n}\left(y;r_{1},\dots,r_{n};r_{1},\dots,r_{m}\right)\,\,\prod^{n}_{i=1}\,u(r_{i})\prod^{m}_{k=1}\,v(r_{k})d^{2}r_{i}\,d^{2}r_{k}$
(16)

where $P^{m}_{n}$ is the probability to find (i) $n$ dipoles in the wave
function of the fast dipole, which interact with the target at time $\tau=0$;
and (ii) $m$ dipoles which can be detected at $\tau=\infty$. In other words,
$P^{m}_{n}$ is the probability to have $n$ dipoles with sizes
$r_{1},\dots,r_{n}$ at $\tau=0$ at rapidity $Y$, which do not survive until
$\tau=\infty$ and cannot be measured, while the dipoles with sizes
$r_{1},\dots,r_{m}$ reach $\tau=\infty$ and can be caught by detectors.

The initial conditions of Eq. (12a) has to be replaced by

$Z\left(Y=0;\\{u\\},\\{v\\}\right)\,\,=\,\,v(r)$ (17)

and we have two boundary conditions:

$(1)~{}~{}Z\left(Y;\\{u\\},\\{v\\}\right)|_{u=1,v=1}\,\,=\,\,1;~{}~{}~{}~{}~{}(2)~{}~{}Z\left(Y;\\{u\\},\\{v\\}\right)|_{v=2u-1}=2\;Z_{0}\left(Y;\\{u\\}\right)\,\,-1;$
(18)

The first one comes from the conservation of probabilities while the second
stems from the s-channel unitarity:

$2N\left(Y,r,b\right)\,\,=\,\,\sigma_{sd}\left(Y,r,b\right)\,\,+\,\,\sigma_{in}\left(Y,r,b\right)$
(19)

where $\sigma_{sd}$ and $\sigma_{in}$ are the single diffraction and inelastic
cross sections at fixed impact parameter $b$.

In Ref.KLP  the AGK cutting rules for the ”fan” diagrams were reduced to the
following linear equation for $Z\left(Y;\\{u\\},\\{\zeta=2\,u-v\\}\right)$:

$\displaystyle\frac{\partial\tilde{Z}\left(Y;\\{u\\},\\{\zeta\\}\right)}{\partial
Y}\,\,=\,\,\frac{\bar{\alpha}_{S}}{2\pi}\,\int\,d^{2}r_{2}\,K\left(r_{10}|r_{12},r_{02}\right)\times$
(20)
$\displaystyle\left\\{\left(u(r_{12})\,u(r_{02})\,-\,u(r_{10})\right)\frac{\delta\tilde{Z}\left(y;\\{u\\},\\{\zeta\\}\right)}{\delta
u(r_{10})}\,\,+\,\,\left(\zeta(r_{12})\,\zeta(r_{02})\,-\,\zeta(r_{10})\right)\,\frac{\delta\tilde{Z}\left(y;\\{u\\},\\{\zeta\\}\right)}{\delta\zeta(r_{10})}\right\\}$

Each term in Eq. (20) has the same transparent probabilistic interpretation as
in Eq. (13).

###  II.3  The master equation for $\sigma_{n}$

Eq. (20) can be used for an arbitrary initial condition. For DIS scattering
the initial condition of Eq. (17) results in the non-linear equation. This
equation takes the most elegant form for the generating functional which is
defined as

$\displaystyle
M\left(Y-Y_{0};r,b\right)\,\,=\,\,1\,\,\,-\,\,Z\left(Y-Y_{0};\\{u\\},\\{v\\}\right)$
(21)
$\displaystyle=\,\sum^{\infty}_{n=1,m=0}\,\,\frac{(-1)^{n+m+1}}{n!\,m!}\,\,\left\\{\int\,\prod^{n}_{i=1}\,\,d^{2}r_{i}\,\int\,\prod^{m}_{k=1}\,\,d^{2}r_{k}\,\gamma(r_{i})\,\gamma_{in}(r_{k})\,\,\frac{\delta}{\delta
u(r_{i})}\,\,\frac{\delta}{\delta
v(r_{k})}\right\\}\,Z\left(Y-Y_{0});\\{u\\},\\{v\\}\right)\Big{|}_{u(r)=1,v(r)=1}$

where $u(r)=1-\gamma(r)$ and $v(r)=1-\gamma_{in}(r)$.

For $M\left(Y;r,b\right)$ the non-linear equation is obtained from Eq. (20) in
Ref.KLP :

$\displaystyle\frac{\partial M(Y;r_{10},b)}{\partial
Y}\,=\,\,\,\frac{\bar{\alpha}_{S}}{2\pi}\,\int\,d^{2}r_{2}\,K\left(r_{10}|r_{12},r_{02}\right)\left\\{M(Y;r_{12},b)\,+\,\scriptsize\footnotesize
M(Y;r_{20},b)\,-\,M(Y;r_{10},b)\,\right.$ (22)
$\displaystyle\left.+M(Y;r_{12},b)M(Y;r_{20},b)\,-\,2\,M(Y;r_{12},b)N(Y;r_{20},b)-\,2\,N(Y;r_{12},b)M(Y;r_{20},b)\,+\,2N(Y;r_{12},b)N(Y;r_{20},b)\right\\}$

The first check of this equation: $M(y;r_{10},b)$ for $\gamma_{in}=0$ gives
the cross section of diffraction production
$M(y;r_{10},b)|_{\gamma_{in}=0}=\sigma_{sd}$. This equation coincides with the
equation for the diffraction production of Ref.KOLE .

If we want to find a cross section with the $k$-produced gluons in dipole
nucleus (or proton) interaction we need to calculate

$\sigma_{k}\left(Y,r;b\right)\,\,\,=\,\,\frac{1}{k!}\,\prod^{k}_{i=1}\,\gamma_{in}(r_{i})\left(\frac{\delta}{\delta\gamma_{in}(r_{i})}\,M\left(Y;\\{\gamma\\},\\{\gamma_{in}\\}\right)\right)\Big{|}_{\gamma_{in}=0}$
(23)

where $\gamma(r)$ is the low energy elastic amplitude and
$\gamma_{in}(r)=2\gamma(r)$ at low energy ($Y_{0}$).

###  II.4 Solutions for Pomeron calculus in zero transverse dimension

For the Pomeron calculus in zero transverse dimension Eq. (20) and Eq. (22)
were solved in Ref.LEPRSM . The solutions for the generating functions take
the forms:

$\displaystyle Z_{0}\left(Y,u\right)$ $\displaystyle=$
$\displaystyle\,\frac{ue^{-\Delta\,Y}}{1-u\left(1\,-\,e^{-\Delta\,Y}\right)};$
(24a) $\displaystyle Z\left(Y,u,v\right)$ $\displaystyle=$
$\displaystyle\frac{2ue^{-\Delta\,Y}}{1-u\left(1\,-\,e^{-\Delta\,Y}\right)}\,-\,\frac{(2u-v)e^{-\Delta\,Y}}{1-(2u-v)\left(1\,-\,e^{-\Delta\,Y}\right)};$
(24b)

where $\Delta$ is the intercept of the BFKL Pomeron. From Eq. (23) we obtain:

$\sigma_{k}\,\,=\,\,\frac{e^{\Delta\,Y}\,\left(e^{\Delta\,Y}\,-\,1\right)^{k-1}}{\left(1-2\,\gamma\left(e^{\Delta\,Y}\,-\,1\right)\right)^{k+1}}\,\gamma_{in}^{k}$
(25)

where $\gamma$ and $\gamma_{in}$ are the scattering amplitude and [production
cross section of two dipoles at low energy. From Eq. (15)
$\gamma_{in}=2\gamma$.

Using Eq. (24a) we can calculate the scattering amplitude, which has the
following form:

$N\left(Y\right)\,\,=\,\,\frac{\gamma}{1-(1-\gamma)\left(1\,\,-\,\,e^{-\Delta\,Y}\right)}\,\,=\,\,\frac{\tilde{\gamma}e^{\Delta\,Y}}{\tilde{\gamma}e^{\Delta\,Y}\,-\,1}$
(26)

where $\tilde{\gamma}=\gamma/(1-\gamma)$.

Considering $e^{\Delta\,Y}$ as Pomeron Green’s function and using the AGK
cutting rules for the scattering amplitude of Eq. (26) we reproduce at large
values of $Y$ the same $\sigma_{k}$ which follow from Eq. (25). We believe
that this observation is a strong argument for that our probabilistic
treatment of gluon production is correct.

##  III  The cross sections of gluon production deep in the saturation region

The main goal of this paper is to find $\sigma_{k}$ solving Eq. (22) in the
kinematic region where we have the scattering amplitude of Eq. (7). In other
words, we are looking for the solutions for the scattering anplitude of the
dipole with size $r\equiv r_{01}$ and rapidity $Y$ in the kinematic region
where $r^{2}Q^{2}_{s}(Y,b)\,\,\gg\,\,1$.

###  III.1  The total cross section

The total cross section can be obtained considering $M\left(Y,r,b\right)$ for
$\gamma_{in}(r_{i})=2\gamma(r_{i})$. From the unitarity constraints of Eq.
(19) we expect that in our kinematic region $\sigma_{tot}\left(Y,r,b\right)$
approaches the unitarity limit of $\sigma_{tot}\to 2$. Introducing
$\sigma_{tot}\left(Y,r,b\right)\,\,=\,\,2\,\,-\,\,\Delta_{tot}\left(Y,r,b\right)$
and $N\left(Y,r,b\right)\,\,=\,\,1\,\,-\,\,\Delta_{0}\left(Y,r,b\right)$ , we
can rewrite Eq. (22) in the following form. neglecting the contribution of
$\Delta_{in}^{2}$ or/and $\Delta_{0}^{2}$:

$\displaystyle\frac{\partial\Delta_{tot}(Y;r_{01},b)}{\partial Y}\,\,$
$\displaystyle=$
$\displaystyle\,\,\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\left(\Delta_{tot}(Y;r_{02},b)\,\,+\,\,\Delta_{tot}(Y;r_{12},b)\,\,-\,\,\Delta_{tot}(Y;r_{01},b)\right)$
(27) $\displaystyle-\,$ $\displaystyle
2\,\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\left(\Delta_{0}(Y;r_{02},b),\,+\,\,\Delta_{0}(Y;r_{12},b)\right)$

One can see that Eq. (27) has solution:
$\Delta_{tot}(Y;r_{01},b)\,=\,2\,\Delta_{0}(Y;r_{01},b)$. Indeed, substituting
this solution into Eq. (27) we obtain the equation for
$\Delta_{0}(Y;r_{01},b)$:

$\frac{\partial\Delta_{0}(Y;r_{01},b)}{\partial
Y}\,\,=\,\,-\,\left(\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\right)\,\Delta_{0}(Y;r_{01},b)\,\,=\,\,-z\,\,\Delta_{0}(Y;r_{01},b)$
(28)

which is the equation for $\Delta_{0}$, that has been discussed in Ref.LETU ,
and from which follows the asymptotic behaviour of Eq. (7).

###  III.2  The diffraction production

As we have discussed the master equation (see Eq. (22)) coincides with the
equation of Ref.KOLE , which has been derived in a quite different way than it
is done here. For the cross section of diffraction production we need to
consider $M(Y;r_{10},b)$ of Eq. (21) at $\gamma_{in}\left(r\right)=0$. From
Eq. (19) we expect that $\sigma_{in}(Y;r_{10},b)$ approaches 1 at high
energies. Bearing this in mind we introduce
$\sigma_{sd}\left(Y,r,b\right)\,\,=\,\,1\,\,-\,\,\Delta_{sd}\left(Y,r,b\right)$
and $N\left(Y,r,b\right)\,\,=\,\,1\,\,-\,\,\Delta_{0}\left(Y,r,b\right)$.
Plugging these expressions in Eq. (22) we see that

$\frac{\partial\Delta_{sd}(Y;r_{01},b)}{\partial
Y}\,\,=\,\,-\,\left(\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\right)\,\Delta_{sd}(Y;r_{01},b)\,\,=\,\,-z\,\,\Delta_{sd}(Y;r_{01},b).$
(29)

In the variable $z$ of Eq. (5) Eq. (29) takes the form:

$\kappa\frac{d\,\Delta_{sd}(z)}{d\,z}\,\,=\,\,-z\,\,\Delta_{sd}(z)$ (30)

with the solution:

$\Delta_{sd}(z)\,\,=\,\,C(z)\exp\left(-\frac{z^{2}}{2\,\kappa}\right)$ (31)

where $\kappa$ is given by Eq. (3) and $C(z)$ is a smooth function of $z$.
Therefore, $\sigma_{sd}\left(Y,r,b\right)$ at large values of $Y$ has the form
of Eq. (7) and shows the geometric scaling behaviour as it has been discussed
in Ref.KOLE .

It is instructive to note that
$\sigma_{tot}\left(Y,r,b\right)\,\,=\,\,2\,\,-\,\,\Delta_{in}\left(Y,r,b\right)$
and
$\sigma_{sd}\left(Y,r,b\right)\,\,=\,\,1\,\,-\,\,\Delta_{sd}\left(Y,r,b\right)$
are the solutions of the same equation. We believe that this observation is
the important check that the main ideas of Ref.KLP , which result in deriving
Eq. (22), are correct.

###  III.3  $\sigma_{1}\left(Y,r,b\right)$

Using Eq. (23) we can obtain the equations for $\sigma_{k}$ from the master
equation by differentiating this equation. First taking
$\frac{\delta}{\delta\gamma_{in}(r_{i})}$ from both parts of Eq. (22) and
putting $\gamma_{in}(r_{i})=0$ we obtain the following equation for
$\sigma_{1}\left(Y,r,b\right)$:

$\displaystyle\frac{\partial\sigma_{1}(Y;r_{01},b)}{\partial Y}\,\,$
$\displaystyle=$
$\displaystyle\,\,\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\Bigg{\\{}\sigma_{1}\left(Y,r_{02},b\right)\,+\,\sigma_{1}\left(Y,r_{12},b\right)\,-\,\sigma_{1}\left(Y,r_{01},b\right)\,\,$
$\displaystyle+$
$\displaystyle\,\,\sigma_{1}\left(Y,r_{02},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)\,+\,\sigma_{1}\left(Y,r_{12},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)$
$\displaystyle-$
$\displaystyle\,2\,\sigma_{1}\left(Y,r_{02},b\right)\,N\left(Y,r_{02},b\right)\,-\,2\,\sigma_{1}\left(Y,r_{12},b\right)\,N\left(Y,r_{02},b\right)\Bigg{\\}}$
(32)

The high energy asymptotic behaviour we get from Eq. (III.3) substituting
$\sigma_{sd}\left(Y,r_{i,i+1},b\right)\,=\,1-\Delta_{sd}\left(z\right)$ and
$N\left(Y,r,b\right)\,\,=\,\,1\,\,-\,\,\Delta_{0}\left(Y,r,b\right)$, Eq.
(III.3) takes the following form:

$\displaystyle\frac{\partial\sigma_{1}(Y;r_{01},b)}{\partial Y}\,\,$
$\displaystyle=$
$\displaystyle\,\,-\,\left(\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\right)\Bigg{\\{}\,\sigma_{1}\left(Y,r_{01},b\right)\,-\,\,\sigma_{1}\left(Y,r_{02},b\right)\,\Delta_{in}\left(Y,r_{02},b\right)\,$
(33)
$\displaystyle-\,\,\sigma_{1}\left(Y,r_{12},b\right)\,\Delta_{in}\left(Y,r_{02},b\right)\Bigg{\\}}\,=\,\,-z\,\sigma_{1}\left(Y,r_{01},b\right)$

where
$\Delta_{in}\left(Y,r_{0i},b\right)=2\Delta_{0}\left(Y,r_{0i},b\right)\,-\,\Delta_{sd}\left(Y,r_{0i},b\right)$.
The total inelastic cross section has the form:
$\sigma_{in}=1-\Delta_{in}\left(Y,r_{0i},b\right)$ and the above equation for
$\Delta_{in}$ follows from the unitarity constraints (see Eq. (19)).

Neglecting the second and the third terms in the curly bracket one can see
that the geometric scaling solution of Eq. (33) has the same form as Eq. (31)
for $\Delta_{sd}(z)$. Hence

$\kappa\frac{d\,\sigma_{1}(z)}{d\,z}\,\,=\,\,-z\,\,\sigma_{1}(z)~{}~{}~{}\mbox{with
the
solution}~{}~{}~{}\sigma_{1}(z)\,\,=\,\,C(z)\exp\left(-\frac{z^{2}}{2\,\kappa}\right)$
(34)

We will discuss below the influence of the neglected terms on the solution of
Eq. (34)

It turns out that it is convenient to discuss the solutions in the momentum
representation, where Eq. (34) takes the form of convolution. Adding and
subtracting the gluon reggeization term in the momentum representation we
obtain the following equation for the geometric scaling solution:

$\kappa\frac{d\,\sigma_{1}(\tilde{z})}{d\,\tilde{z}}\,\,=\,\,\,\,\int\frac{d^{2}k^{\prime}_{T}}{(2\pi)^{2}}K\left(\bm{k}_{T},\bm{k}^{\prime}_{T}\right)\,\,\sigma_{1}\left(\tilde{z}^{\prime}\right)\,\,\,-\,\,\tilde{z}\,\sigma_{1}\left(\tilde{z}\right)\,\,+\,\,\Delta_{in}\left(\tilde{z}\right)\sigma_{1}\left(\tilde{z}\right)$
(35)

where $K\left(\bm{k}_{T},\bm{k}^{\prime}_{T}\right)$ is the BFKL kernel in
momentum representation:

$K\left(\bm{k}_{T},\bm{k}^{\prime}_{T}\right)\,\,\sigma_{1}\left(Y,\bm{k}^{\prime}_{T},b\right)\,=\,\frac{1}{\left(\bm{k}_{T}-\bm{k}^{\prime}_{T}\right)^{2}}\,\,\,\sigma_{1}\left(Y,\bm{k}^{\prime}_{T},b\right)\,\,-\,\,\frac{k^{2}_{T}}{\left(\bm{k}_{T}-\bm{k}^{\prime}_{T}\right)^{2}\,\left(\left(\bm{k}_{T}-\bm{k}^{\prime}_{T}\right)^{2}\,+\,k^{\prime
2}_{T}\right)}\,\,\sigma_{1}\left(Y,\bm{k}_{T},b\right)$ (36)

$\tilde{z}$ is a new scaling variable:

$\tilde{z}\,\,\,=\,\,\kappa\,\bar{\alpha}_{S}\,Y\,\,-\,\,\ln
k^{2}_{T}\,\,=\,\,\ln\left(\frac{Q^{2}_{s}\left(Y,b\right)}{k^{2}_{T}}\right)$
(37)

Using the Mellin transform:

$\sigma_{1}\left(\tilde{z},b\right)\,\,=\,\,\int\limits^{\epsilon+i\infty}_{\epsilon-i\infty}\frac{d\gamma}{2\,\pi\,i}e^{\gamma\,\tilde{z}}\,\sigma_{1}\left(\gamma,b\right)$
(38)

we can rewrite Eq. (35) in the form:

$\left(\kappa\,\gamma\,\,-\,\,\chi\left(\gamma\right)\right)\sigma_{1}\left(\gamma,b\right)=\frac{d\sigma_{1}\left(\gamma,b\right)}{d\,\,\gamma}$
(39)

with the solution

$\sigma_{1}\left(\gamma,b\right)\,\,=\,\,\sigma^{init}_{1}\left(b\right)\exp\left(\frac{1}{2}\kappa\,\gamma^{2}\,\,+\,\int^{\gamma}_{\frac{1}{2}}\chi\left(\gamma^{\prime}\right)d\gamma^{\prime}\right)$
(40)

where $\sigma^{init}_{1}$ has to be found from the initial conditions.
Plugging Eq. (40) into Eq. (38) we can take the integral over $\gamma$ using
the method of steepest decent. The equation for the saddle point takes the
form:

$\tilde{z}+\kappa\gamma_{SP}\,+\,\chi\left(\gamma_{SP}\right)=0$ (41)

Since $\chi\left(\gamma_{SP}\right)\sim\ln\gamma_{SP}$ the value of
$\gamma_{SP}=-\tilde{z}/\kappa$. Hence

$\sigma_{1}\left(\tilde{z}\right)\,\,=\,\,C\left(\tilde{z}\right)e^{-\frac{\tilde{z}^{2}}{2\,\kappa}}$
(42)

with $C\left(\tilde{z}\right)$ being a smooth function of $\tilde{z}$.
Therefore, at large values of $Y$ the solutions in coordinate and momentum
representations have the same form in $z$ and $\tilde{z}$, respectively. The
last term in Eq. (35) leads to replacement of
$\tilde{z}\to\tilde{z}-\int\limits^{\infty}_{0}d\tilde{z}^{\prime}\,\Delta_{in}\left(\tilde{z}\right)$
in Eq. (42). One can see this solving the simplified equation:

$\kappa\frac{d\,\sigma_{1}(\tilde{z})}{d\,\tilde{z}}\,\,=\,\,\,\,\,\,\,-\,\,\tilde{z}\,\sigma_{1}\left(\tilde{z}\right)\,\,+\,\,\Delta_{in}\left(\tilde{z}\right)\sigma_{1}\left(\tilde{z}\right)$
(43)

with the solution:

$\displaystyle\sigma_{1}\left(\tilde{z}\right)={\rm
C}\exp\left(-\frac{\tilde{z}^{2}}{2\,\kappa}+\frac{1}{\kappa}\int^{\tilde{z}}_{0}d\tilde{z}^{\prime}\,\Delta_{in}\left(\tilde{z}^{\prime}\right)\right)$
$\displaystyle={\rm
C}\exp\left(-\frac{\tilde{z}^{2}}{2\,\kappa}+\frac{1}{\kappa}\int^{\infty}_{0}d\tilde{z}^{\prime}\,\Delta_{in}\left(\tilde{z}^{\prime}\right)-\underbrace{\frac{1}{\kappa}\int^{\infty}_{\tilde{z}}d\tilde{z}^{\prime}\,\Delta_{in}\left(\tilde{z}^{\prime}\right)}_{\ll
1}\right)={\rm
C^{\prime}}\exp\left(-\frac{\left(\tilde{z}-\int^{\infty}_{\tilde{z}}d\tilde{z}^{\prime}\,\Delta_{in}\left(\tilde{z}^{\prime}\right)\right)^{2}}{2\,\kappa}\right)$
(44)

###  III.4  $\sigma_{2}\left(Y,r,b\right)$

Applying
$\frac{1}{2}\frac{\delta}{\delta\gamma_{in}(r_{1})}\,\frac{\delta}{\delta\gamma_{in}(r_{2})}$
to the both parts of the master equation we obtain:

$\displaystyle\frac{\partial\sigma_{2}(Y;r_{01},b)}{\partial Y}\,\,$
$\displaystyle=$
$\displaystyle\,\,\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\Bigg{\\{}\sigma_{2}\left(Y,r_{02},b\right)\,+\,\sigma_{2}\left(Y,r_{12},b\right)\,-\,\sigma_{2}\left(Y,r_{01},b\right)\,\,$
$\displaystyle+$
$\displaystyle\,\,\sigma_{2}\left(Y,r_{02},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)\,+\,\sigma_{1}\left(Y,r_{12},b\right)\,\sigma_{1}\left(Y,r_{02},b\right)\,+\,\sigma_{2}\left(Y,r_{12},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)$
$\displaystyle-$
$\displaystyle\,2\,\sigma_{2}\left(Y,r_{02},b\right)\,N\left(Y,r_{02},b\right)\,-\,2\,\sigma_{2}\left(Y,r_{12},b\right)\,N\left(Y,r_{02},b\right)\Bigg{\\}}$
(45)

Replacing at large $Y$
$\sigma_{sd}\left(Y,r,b\right)\,\,=\,\,1\,\,-\,\,\Delta_{sd}\left(Y,r,b\right)$
and $N\left(Y,r,b\right)\,\,=\,\,1\,\,-\,\,\Delta_{0}\left(Y,r,b\right)$ we
reduce Eq. (33) to the form:

$\displaystyle\frac{\partial\sigma_{2}(Y;r_{01},b)}{\partial Y}\,\,$
$\displaystyle=$
$\displaystyle\,\,-\,\left(\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\right)\,\sigma_{2}\left(Y,r_{01},b\right)$
$\displaystyle+$
$\displaystyle\,\,\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\,\Big{\\{}\,\sigma_{1}\left(Y,r_{02},b\right)\,\sigma_{1}\left(Y,r_{12},b\right)$
$\displaystyle+$
$\displaystyle\,\sigma_{2}\left(Y,r_{02},b\right)\left(2\,\Delta_{0}\left(Y,r_{12},b\right)\,-\,\Delta_{sd}\left(Y,r_{12},b\right)\right)\,\,+\,\,\sigma_{2}\left(Y,r_{12},b\right)\left(2\,\Delta_{0}\left(Y,r_{02},b\right)\,-\,\Delta_{sd}\left(Y,r_{02},b\right)\right)$

In momentum representation in the region with the geometric scaling behaviour
Eq. (III.4) takes the form:

$\kappa\frac{d\,\sigma_{2}\left(\tilde{z}\right)}{d\,\tilde{z}}\,\,=\,\,-\tilde{z}\,\sigma_{2}\left(\tilde{z}\right)\,\,+\,\,2\,\left(\,\underbrace{2\,\Delta_{0}\left(\tilde{z}\right)\,-\,\Delta_{sd}\left(\tilde{z}\right)}_{\Delta_{in}\left(\tilde{z}\right)}\right)\sigma_{2}\left(\tilde{z}\right)\,+\,\,\sigma^{2}_{1}\left(\tilde{z}\right)$
(47)

where $\Delta_{in}\left(\tilde{z}\right)$ is the moment representation of
$\Delta_{in}\left(z\right)$ with
$\sigma_{in}\left(z\right)=1-\Delta_{in}\left(z\right)$.

The general solution to Eq. (47) is a sum of the solution to homogeneous part
of the equation and the particular solution of the non-homogeneous one. The
homogeneous solution has the form:

$\sigma_{2}\left(\tilde{z}\right)\,\,=\,\,\sigma_{1}\left(\tilde{z}\right)\exp\left(-2\int\limits^{\infty}_{\tilde{z}}\Delta_{in}\left(\tilde{z}^{\prime}\right)d\tilde{z}^{\prime}\right)$
(48)

where $\sigma_{1}\left(\tilde{z}\right)$ is given by Eq. (42).

We can find the initial conditions for $\sigma_{n}$ by assuming that for
$\tilde{z}\,<\,0$ only the BFKL Pomeron exchange contribute to $\sigma_{1}$,
while all $\sigma_{k}=0$ in this kinematic region. In other words, we assume
that we can neglect the "fan" BFKL Pomeron diagrams dealing only with the BFKL
Pomeron exchange for $\tilde{z}<0$. For this initial conditions the solution
to Eq. (47) can be written as:

$\sigma_{2}\left(\tilde{z}\right)\,\,\,=\,\,\sigma_{1}\left(\tilde{z}\right)\exp\left(-2\int\limits^{\infty}_{\tilde{z}}\Delta_{in}\left(\tilde{z}^{\prime}\right)d\tilde{z}^{\prime}\right)\int\limits^{\tilde{z}}_{0}\sigma_{1}\left(\tilde{z}^{\prime}\right)\frac{d\tilde{z}^{\prime}}{\kappa},$
(49)

if we neglect the contributions of the order of $\sigma_{1}^{3}$.

###  III.5  $\sigma_{3}\left(Y,r,b\right)$ and $\sigma_{k}\left(Y,r,b\right)$

Using Eq. (23) we obtain for $\sigma_{3}(Y;r_{01},b)$:

$\displaystyle\frac{\partial\sigma_{3}(Y;r_{01},b)}{\partial Y}\,\,$
$\displaystyle=$
$\displaystyle\,\,\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\Bigg{\\{}\sigma_{3}\left(Y,r_{02},b\right)\,+\,\sigma_{3}\left(Y,r_{12},b\right)\,-\,\sigma_{3}\left(Y,r_{01},b\right)\,\,$
(50) $\displaystyle+$
$\displaystyle\,\,\sigma_{3}\left(Y,r_{02},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)\,\,+\,\sigma_{3}\left(Y,r_{12},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)$
$\displaystyle+$
$\displaystyle\sigma_{2}\left(Y,r_{02},b\right)\,\sigma_{1}\left(Y,r_{12},b\right)\,\,+\,\,\,\sigma_{2}\left(Y,r_{12},b\right)\,\sigma_{1}\left(Y,r_{02},b\right)$
$\displaystyle-$
$\displaystyle\,2\,\sigma_{3}\left(Y,r_{02},b\right)\,N\left(Y,r_{02},b\right)\,-\,2\,\sigma_{3}\left(Y,r_{12},b\right)\,N\left(Y,r_{02},b\right)\Bigg{\\}}$

Using the asymptotic expansion for $\sigma_{sd}$ and $N$ we obtain for the
solutions with geometric scaling behaviour :

$\kappa\frac{d\,\sigma_{3}\left(\tilde{z}\right)}{d\,\tilde{z}}\,\,=\,\,-\tilde{z}\,\sigma_{3}\left(\tilde{z}\right)\,\,+\,\,2\,\Delta_{in}\left(\tilde{z}\right)\sigma_{3}\left(\tilde{z}\right)\,+\,\,2\sigma_{2}\left(\tilde{z}\right)\,\sigma_{1}\left(\tilde{z}\right)$
(51)

The solution to Eq. (51) has the form:

$\sigma_{3}\left(\tilde{z}\right)\,\,\,=\,\,\sigma_{1}\left(\tilde{z}\right)\exp\left(-2\int\limits^{\infty}_{\tilde{z}}\Delta_{in}\left(\tilde{z}^{\prime}\right)d\tilde{z}^{\prime}\right)\Bigg{(}\int\limits^{\tilde{z}}_{0}\sigma_{1}\left(\tilde{z}^{\prime}\right)\
\frac{d\tilde{z}^{\prime}}{\kappa}\Bigg{)}^{2}$ (52)

The general equation for $\sigma_{k}\left(Y,r,b\right)$ reads as follows:

$\displaystyle\frac{\partial\sigma_{k}(Y;r_{01},b)}{\partial Y}\,\,$
$\displaystyle=$
$\displaystyle\,\,\frac{\bar{\alpha}_{S}}{2\pi}\\!\\!\\!\\!\\!\int\limits_{\begin{subarray}{l}r_{02}\,\gg\,1/Q_{s}(Y)\\\
r_{12}\,\gg\,1/Q_{s}(Y)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d^{2}r_{2}\,K\left(r_{01}|r_{12},r_{02}\right)\Bigg{\\{}\sigma_{k}\left(Y,r_{02},b\right)\,+\,\sigma_{k}\left(Y,r_{12},b\right)\,-\,\sigma_{k}\left(Y,r_{01},b\right)\,\,$
(53) $\displaystyle+$
$\displaystyle\,\,\sigma_{k}\left(Y,r_{02},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)\,\,+\,\sigma_{k}\left(Y,r_{12},b\right)\,\sigma_{sd}\left(Y,r_{02},b\right)+\sum_{j=1}^{k-1}\sigma_{k-j}\left(Y,r_{02},b\right)\,\sigma_{j}\left(Y,r_{12},b\right)$
$\displaystyle-$
$\displaystyle\,2\,\sigma_{k}\left(Y,r_{02},b\right)\,N\left(Y,r_{02},b\right)\,-\,2\,\sigma_{k}\left(Y,r_{12},b\right)\,N\left(Y,r_{02},b\right)\Bigg{\\}}$

We can check that solution to Eq. (53) has the following form:

$\sigma_{k}\left(\tilde{z}\right)\,\,\,=\,\,\sigma_{1}\left(\tilde{z}\right)\exp\left(-2\int\limits^{\infty}_{\tilde{z}}\Delta_{in}\left(\tilde{z}^{\prime}\right)d\tilde{z}^{\prime}\right)\Bigg{(}\int\limits^{\tilde{z}}_{0}\sigma_{1}\left(\tilde{z}^{\prime}\right)\
\frac{d\tilde{z}^{\prime}}{\kappa}\Bigg{)}^{k-1}$ (54)

Note that in term
$\sum_{j=1}^{k-1}\sigma_{k-j}\left(Y,r_{02},b\right)\,\sigma_{j}\left(Y,r_{12},b\right)$
in Eq. (53) we can replace the factor
$\exp\left(-2\int\limits^{\infty}_{\tilde{z}}\Delta_{in}\left(\tilde{z}^{\prime}\right)d\tilde{z}^{\prime}\right)\,=\,1$
keeping accuracy of the order of $\sigma^{k}_{1}$.

In Eq. (54) we have two smooth functions that have to be determined from the
initial conditions: $C(z)$ of Eq. (42) and function $D\left(\tilde{z}\right)$
in the expression for
$2\Delta_{in}\left(\tilde{z}\right)=D(\tilde{z})\exp\left(-\frac{\tilde{z}^{2}}{2\,\kappa}\right)$.
We believe that the only condition that we need is

$\int\limits^{\infty}_{0}C\left(\tilde{z}^{\prime}\right)\,\frac{d\tilde{z}^{\prime}}{\kappa}\,=\,\,1$
(55)

In this case the unitary constraints of Eq. (19) or/and Eq. (27):

$\sigma_{in}\left(\tilde{z}\right)\,\,=\,\,\sum^{\infty}_{k=1}\,\sigma_{k}\left(\tilde{z}\right)\,\,\propto\,\,\tilde{z}$
(56)

which provides that $\sigma_{in}\left(z\right)$ in the coordinate
representation reaches the unitarity limit $\sigma_{in}\left(z\right)=1$. The
sum itself has the form:

$\sigma_{in}\left(\tilde{z}\right)\,\,=\,\,\frac{e^{-\tilde{z}^{2}/2\kappa}C\left(\tilde{z}\right)\exp\left(-\int\limits^{\infty}_{\tilde{z}}D\left(\tilde{z}^{\prime}\right)e^{-\tilde{z}^{\prime
2}/2\kappa}\frac{d\tilde{z}^{\prime}}{\kappa}\right)}{\int\limits^{\infty}_{\tilde{z}}C\left(\tilde{z}^{\prime}\right)e^{-\tilde{z}^{\prime
2}/2\kappa}\frac{d\tilde{z}^{\prime}}{\kappa}}$ (57)

while

$\sigma_{k}\left(\tilde{z}\right)\,\,=\,\,e^{-\tilde{z}^{2}/2\kappa}C\left(\tilde{z}\right)\exp\left(-\int\limits^{\infty}_{\tilde{z}}D\left(\tilde{z}^{\prime}\right)e^{-\tilde{z}^{\prime
2}/2\kappa}\frac{d\tilde{z}^{\prime}}{\kappa}\right)\left(1\,\,-\,\,\int\limits^{\infty}_{\tilde{z}}C\left(\tilde{z}^{\prime}\right)e^{-\tilde{z}^{\prime
2}/2\kappa}\frac{d\tilde{z}^{\prime}}{\kappa}\right)^{k-1}$ (58)

Neglecting
$\int\limits^{\infty}_{\tilde{z}}D\left(\tilde{z}^{\prime}\right)e^{-\tilde{z}^{\prime
2}/2\kappa}\frac{d\tilde{z}^{\prime}}{\kappa}\,\sim
e^{-\tilde{z}^{\prime}2/2\kappa}\,\ll\,1$ and denoting
$\int\limits^{\infty}_{\tilde{z}}C\left(\tilde{z}^{\prime}\right)e^{-\tilde{z}^{\prime
2}/2\kappa}\frac{d\tilde{z}^{\prime}}{\kappa}$ by
$1/\bar{n}\left(\tilde{z}\right)$ we see that

$\sigma_{k}\left(\tilde{z}\right)\,\,=\,\,e^{-\tilde{z}^{2}/2\kappa}C\left(\tilde{z}\right)\left(1\,\,-\,\,\frac{1}{\bar{n}\left(\tilde{z}\right)}\right)^{k-1}\xrightarrow{\bar{n}\left(\tilde{z}\right)\,\,\gg\,\,1}e^{-\tilde{z}^{2}/2\kappa}C\left(\tilde{z}\right)\exp\Big{(}-\frac{k-1}{\bar{n}\left(\tilde{z}\right)}\Big{)}$
(59)

Eq. (59) as well as all equations in this paper, are written at fixed impact
parameter $b$. Hence we need to integrate $\sigma_{k}\left(\tilde{z}\right)$
over $b$ to obtain the cross section of produced $k$-gluons. The
$b$-dependence enters our formulae through the saturation momentum (see Eq.
(5), Eq. (9) and Eq. (37)). In perturbative QCD the variable $\xi$ in Eq. (5)
is determined by the form of the eigenfunction for the BFKL equation (see Eq.
(8)). However, it has been shown KW1 ; KW2 ; KW3 ; FIIM  that the typical $b$
that contribute in the integrals for cross sections turn out to be of the
order of $re^{\lambda Y}$ and they lead to the violation of the Froissart
theoremFROI . Therefore, we need to introduce the non-perturbative corrections
at large $b$. Eq. (59) is written in the convenient form to introduce such
corrections to the saturation momentum:
$Q^{2}_{s}\left(Y,b\right)=Q^{2}_{s}\left(Y,b=0\right)\exp\left(-\mu\,b\right)$.
, where $\mu$ is a new non-perturbative parameter. The multiplicity
$\bar{n}\left(\tilde{z}\right)$ takes the form

$\bar{n}\left(\tilde{z}(b)\right)\,\,=\,\,\Bigg{(}\frac{C\left(\infty\right)}{\tilde{z}(b=0)-\mu\,b}\exp\left(-\frac{\left(\tilde{z}(b=0)-\mu
b\right)^{2}}{2\kappa}\right)\Bigg{)}^{-1}\,\,\,=\,\,\frac{1}{{\cal
A}}\,\exp\left(\frac{\left(\tilde{z}(b=0)-\mu b\right)^{2}}{2\kappa}\right)$
(60)

where ${\cal A}$ is a smooth function of $\tilde{z}$.

Figure 3:  $\sigma_{k}\left(\tilde{z},b\right)$ in Eq. (62) versus
$t=\tilde{z}(b=0)-\mu\,b$ at different values of $k$ . $\kappa=4.88,{\cal
A}=2$.

The region of integration over $b$ is $0<b<\tilde{z}(b=0)$ since for larger
$b$ the scattering amplitude is small and cannot be treated in our approach.
Introducing a new variable $t=\tilde{z}(b=0)-\mu\,b$ one can see that for the
integral has a maximum at $t_{SP}=\sqrt{2\,\kappa\,\ln\left({\cal
A}(k-1)\right)}$(see Fig. 3).. Therefore, for for $t_{SP}\,\leq\tilde{z}(b=0)$
the main contribution is determined by the saddle point contribution which
gives

$\displaystyle\sigma_{k}=$
$\displaystyle\frac{t_{SP})(2\,\pi)^{3/2}\kappa}{e\,\mu^{2}}\frac{1}{k-1}\frac{1}{t_{SP}}\,\Big{(}\tilde{z}(b=0)-t_{SP}\Big{)}=\frac{(2\,\pi)^{3/2}\kappa}{e\,\mu^{2}}\frac{1}{(k-1)}\Bigg{(}\tilde{z}(b=0)-\sqrt{2\,\kappa\,\ln\left({\cal
A}(k-1)\right)}\Bigg{)}$

These $\sigma_{k}$ mildly depend on $\tilde{z}(b=0)$ for all
$k-1\leq\,\bar{n}\left(\tilde{z}(b=0)\right)$. For larger $k$ the main
contributions stem from $b\to 0$. Estimating the slope at b=0 by the method of
steepest decent we obtain for $k-1\,>\,\bar{n}\left(\tilde{z}(b=0)\right)$ :

$\sigma_{k}\left(\tilde{z}(b=0),t=0\right)\,\,=\,\,\,\int\limits_{0<b<\tilde{z}(b=0)}\\!\\!\\!\\!\\!\\!\\!\\!\sigma_{k}\left(\tilde{z}(b)\right)d^{2}b\,\,=\,\,\frac{2\,\pi}{\mu^{2}}\,\,\frac{\kappa^{2}}{\tilde{z}(b=0)\,\,\bar{n}\left(\tilde{z}(b=0)\right)}\frac{\bar{n}^{2}\left(\tilde{z}(b=0)\right)}{(k-1)^{2}}\,\,\exp\Big{(}-\frac{k-1}{\bar{n}\left(\tilde{z}(b=0)\right)}\Big{)}$
(62)

The notation indicates that the cross section is calculated for the forward
scattering.

Using Eq. (III.5) and Eq. (62) we can estimate the value of the total
inelastic cross section:

$\displaystyle\sigma_{in}\left(\tilde{z}(b=0),t=0\right)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\sigma_{k}\left(\tilde{z}(b=0)\right)\,\,=\,\,\int\limits^{\bar{n}\left(\tilde{z}(b=0)\right)}_{1}\\!\\!\\!\\!\\!\\!dk\,\sigma_{k}\left({Eq.~{}(\ref{XSKI0})}\right)\,\,+\,\,\int\limits_{\bar{n}\left(\tilde{z}(b=0)\right)}^{\infty}\\!\\!\\!\\!\\!\\!dk\,\sigma_{k}\left({Eq.~{}(\ref{XSKI1})}\right)$
(63) $\displaystyle=$
$\displaystyle\frac{(2\,\pi)^{3/2}}{3\,e\,\mu^{2}}\tilde{z}^{3}(b=0)\,\,+\,\,\frac{2\,\pi}{\mu^{2}}\frac{\kappa^{2}}{\tilde{z}(b=0)}$

From Eq. (63) one can see that the main contribution to the total inelastic
cross section comes from $k-1\leq\,\bar{n}\left(\tilde{z}(b=0)\right)$.

Hence, the probability to emit $k$ gluons at large $\tilde{z}$ is equal to

$\displaystyle
p_{k}\,\,=\,\,\frac{\sigma_{k}\left(\tilde{z}(b=0)\right)}{\sigma_{in}\left(\tilde{z}(b=0),t=0\right)}\,\,=\,\,\left\\{\begin{array}[]{l}\,\,\frac{2\,\kappa}{\tilde{z}^{3}(b=0)}\frac{\Big{(}\tilde{z}(b=0)-\sqrt{2\,\kappa\,\ln(k-1)}\Big{)}}{(k-1)}\,\,\,\,\,\,\,\,\,\,\mbox{for}\,\,\,k-1\leq\bar{n}\left(\tilde{z}(b=0)\right);\\\
\\\
\,\,\frac{2\,e\,\kappa^{2}}{\tilde{z}^{4}(b=0)}\frac{1}{\bar{n}\left(\tilde{z}(b=0)\right)}\frac{\bar{n}^{2}\left(\tilde{z}(b=0)\right)}{(k-1)^{2}}\,\,\exp\Big{(}-\frac{k-1}{\bar{n}\left(\tilde{z}(b=0)\right)}\Big{)}\,\,\,\,\,\mbox{for}\,\,\,k-1>\bar{n}\left(\tilde{z}(b=0)\right);\\\
\end{array}\right.$ (67)

##  IV  Entropy of produced gluons

From Eq. (67) we can find the value of von Neumann entropy of produced gluons:

$S_{E}\,\,=\,\,-\sum_{k=1}^{\infty}p_{k}\ln
p_{k}\,\,\,=\int\limits^{\bar{n}\left(\tilde{z}(b=0)\right)}_{1}\\!\\!\\!\\!\\!dk\,p_{k}\ln
p_{k}\,\,+\,\,\int\limits_{\bar{n}\left(\tilde{z}(b=0)\right)}^{\infty}\\!\\!\\!\\!\\!dk\,p_{k}\ln
p_{k}$ (68)

From Eq. (67) one can see that (i) the largest contribution stems from the
first sum in Eq. (68), since the second term is suppressed as
$1/\tilde{z}^{4}(b=0)$ (see Eq. (63)); and (ii) in $\ln p_{k}$ we can restrict
ourselves by the term: $-\ln(k-1)$. All other contributions are smaller.
Taking the integral we obtain for large values of $\tilde{z}(b=0)$:

$S_{E}\,\,=\,\,0.3\,\frac{\tilde{z}^{2}(b=0)}{2\,\kappa}\,\,$ (69)

In Eq. (69) we see two strange results. First,
$S_{E}\propto\frac{\tilde{z}^{2}(b=0)}{2\,\kappa}$. At large $Y$
$\frac{\tilde{z}^{2}(b=0)}{2\,\kappa}$ can be written as
$S_{E}\propto\frac{1}{2}\tilde{z}\,\bar{\alpha}_{S}\,Y$ which is in two times
smaller the entropy for the BFKL cascade estimated in Ref.KHLE . However, it
turns out that actually extra $\frac{1}{2}$ stems from our assumption about
geometric scaling behaviour of the cross sections. In Ref.KHLE  as well as in
Ref.GOLEE , it is assumed that the cross sections are functions of $Y$ and
$\tilde{z}$ but not only $\tilde{z}$. Coming back to Eq. (33) one can see that
for $\sigma_{1}\left(Y,\tilde{z}\right)$ we obtain the solution:
$\sigma_{1}\left(Y,\tilde{z}\right)\propto\exp\left(-\,\bar{\alpha}_{S}Y\tilde{z}\right)$
and
$\bar{n}\left(\tilde{z}(b=0)\right)\propto\,\,\exp\left(\bar{\alpha}_{S}Y\tilde{z}\right)$.
Finally, $S_{E}\propto\alpha_{S}Y\tilde{z}$ in agreement with Ref.KHLE .
Second, factor $0.3$ in Eq. (69) stems from the integration over $b$. Indeed,
if we estimate the entropy at fixed $b$ using Eq. (59), we obtain
$S_{E}\,\,=\,\frac{\tilde{z}^{2}(b)}{2\,\kappa}$. Therefore, we can state that
non -perturbative corrections that determine the large $b$ behaviour of the
saturation momentum, lead to a decrease of the entropy if we believe in
estimate of Refs.KHLE ; GOLEE  for the entropy of the BFKL cascade at the
moment of interaction with the target.

##  V Conclusions

In this paper we found the multiplicity distributions for produced gluons
based on the equations derived in Ref.KLP . These equations follow from the
generating functional approach to Color Glass Condensate effective theory and
from AGK cutting rulesAGK . AGK cutting rules allow us to use the principal
property of the BFKL Pomeron: it gives the cross section of produced gluons in
the particular kinematic region of leading log(1/x) approximation of
perturbative QCD.

We solve the equations in the deep saturation region where the scattering
amplitude has the form of Eq. (7). It turns out that the multiplicity
distribution for $k-1\,<\,\bar{n}\propto\exp\left(z^{2}/2\kappa\right)$ is
proportional to $(1/(k-1))\Big{(}z-\sqrt{2\,\kappa\,\ln(k-1)}\Big{)}$ . In
this equation $\bar{n}\propto\exp\left(z^{2}/2\kappa\right)$ is the average
number of produced gluons with $z=\kappa\,\bar{\alpha}_{S}Y-\ln Q^{2}$ where
$Q^{2}$ is the virtuality of the photon and $Y=\ln(1/x)$. For
$k-1\,\geq\,\bar{n}\propto\exp\left(z^{2}/2\kappa\right)$ the multiplicity
distribution almost reproduces the KNO scaling behaviour with the average
number of gluons $\bar{n}\propto\exp\left(z^{2}/2\kappa\right)$. The value of
$\kappa$ is given by Eq. (3). The KNO function
$\Psi\left(\frac{k}{\bar{n}}\right)=\exp\left(-\,k/\bar{n}\right)$.

We estimated the value of the entropy of produced gluons which has been a
subject of hot recent discussionsKUT ; PES ; KOLU1 ; PESE ; KHLE ; BAKH ; BFV
; HHXY ; KOV1 ; GOLE1 ; GOLE2 ; KOV2 ; NEWA ; LIZA ; FPV ; TKU ; KOV3 ; KOV4 ;
DVA1 . We obtain $S_{E}=0.3\,\frac{z^{2}}{2\kappa}$. First factor
$\frac{z^{2}}{2\kappa}$ is twice smaller the entropy of the BFKL cascade
estimated in Ref.KHLE  (see also Ref.GOLEE )333 In Refs.KOLE ; GOLEE  we cut
the divergent entropy of the BFKL cascade by the saturation momentum. We
believe that the saturation momentum provides the natural scale for
interactions with the target. The result of this paper supports this
assumption..

As we have discussed in the previous section, we believe that this difference
stems from our assumption that all cross sections should have geometric
scaling behaviour. Second, factor $0.3$ in the equation for the entropy stems
from the non-perturbative corrections, which provide the correct behaviour of
the saturation momentum at large $b$. Therefore, we claim that the non-
perturbative corrections can decrease the entropy if we believe in the
estimates of Refs.KHLE ; GOLEE  for the entropy of the BFKL cascade. We would
like to stress that we deliberately calculated the entropy of the produced
dipoles, which can be measured and has been measured in DISENTRDIS : we
evaluated the cross sections for $n$ produced dipoles, estimate the
probability to produce $n$ dipoles ($p_{n}$) and determine the entropy
$S_{E}=-\sum_{n}p_{n}\ln p_{n}$. It should be noted that the entropy at fixed
impact parameter is equal to
$S_{E}\,\,=\,\,\ln\left(\bar{n}\left(z\right)\right)=\frac{z(b)^{2}}{2\,\kappa}$
and coincides with the entropy of the parton cascade estimated in Refs.KHLE ;
GOLEE . Our results can be trusted at large values of $z$ and it is difficult
to compare it with the experimental data of Ref.ENTRDIS , since it turns out
that for all $x$ in this paper $z\leq 1$ with the saturation momentum from
Ref.RESH .

It should be stressed that we consider DIS in the region of large $z$ where
the scattering amplitude has the geometric scaling behaviour. It means that we
can discuss DIS with proton at small $x$, but we have to be very careful with
DIS on nuclear target. Indeed, for the nuclear target the geometric scaling
behaviour is valid only in the limited region even at large $z$ LTHI ; KLT ;
CLM . In the vast part of the saturation region we do not expect the geometric
scaling behaviour for the cross section of produced gluons and have to solve
the equation for $\sigma_{k}$ with the initial conditions:

$\sigma_{k}\left(\xi,Y=Y_{A}\right)=\frac{1}{k!}\Omega^{k}\left(Y_{A},r,b\right)\exp|\left(-\,\Omega\left(Y_{A},r,b\right)\right)$
(70)

where $\Omega\left(Y_{A},r,b\right)=\sigma_{\mbox{ dipole-
proton}}\left(Y_{A},r\right)\,T_{A}\left(b\right)$ . $\sigma_{\mbox{ dipole-
proton}}\left(Y_{A},r\right)=2\int d^{2}bN_{0}\left(Y_{A},r]\right)$ and
$N_{0}$ is the initial condition for the scattering amplitude of the dipole
with the proton. $Y_{A}=\ln A^{1/3}$ and A is the number of the nucleons in a
nucleus. $T_{A}$ is the optical width of nucleus, which gives the number of
nucleons at given value of impact parameter $b$. To find the multiplicity
distribution for DIS with nucleus naturally is our next problem to solve.

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