#  Non-linear evolution in the re-summed next-to-leading order of perturbative
QCD:
confronting the experimental data

Carlos Contreras<EMAIL_ADDRESS>Departamento de Física, Universidad
Técnica Federico Santa María, Avda. España 1680, Casilla 110-V, Valparaíso,
Chile  Eugene Levin<EMAIL_ADDRESS><EMAIL_ADDRESS>Departamento
de Física, Universidad Técnica Federico Santa María, Avda. España 1680,
Casilla 110-V, Valparaíso, Chile  Centro Científico- Tecnológico de
Valparaíso, Avda. España 1680, Casilla 110-V, Valparaíso, Chile  Department of
Particle Physics, School of Physics and Astronomy, Raymond and Beverly Sackler
Faculty of Exact Science, Tel Aviv University, Tel Aviv, 69978, Israel
Michael Sanhueza<EMAIL_ADDRESS>Departamento de Física,
Universidad Técnica Federico Santa María, Avda. España 1680, Casilla 110-V,
Valparaíso, Chile

###### Abstract

In this paper we compare the experimental HERA data with the next-to-leading
order approach (NLO) of Ref. CLMS . This approach includes the re-summed NLO
corrections to the kernel of the evolution equation, the correct asymptotic
behaviour in the NLO at $\tau=r^{2}Q^{2}_{s}\,\gg\,1$; the impact parameter
dependence of the saturation scale in accord with the Froissarrt theorem as
well as the non-linear corrections. In this paper, we successfully describe
the experimental data with the quality, which is not worse, than in the
leading order fits with larger number of the phenomenological parameters. It
is demonstrated, that the data could be described, taking into account both
the diffusion on $\ln(k_{T})$, which stems from perturbative QCD, and the
Gribov’s diffusion in impact parameters. It is shown an ability to describe
the data at rather large values of $\bar{\alpha}_{S}$ .

BFKL Pomeron, CGC/saturation approach, solution to non-linear equation, deep
inelastic/Users/leving/Library/Containers/com.apple.mail/Data/Library/Mail
Downloads/4CEE7639-2DA9-413A-81BD-72B039E4A25E/Screenshot[1].png structure
function

###### pacs:

12.38.Cy, 12.38g,24.85.+p,25.30.Hm

###### Contents

  1. I  Introduction
  2. II Leading twist approximation for non-linear evolution in the NLO - a recap
    1. II.1  Re-summed anomalous dimensions in NLO and the kernel of linear evolution
    2. II.2 Non-linear equation and the feedback to the kernel of the linear evolution
    3. II.3 NLO BFKL kernel in the saturation domain
      1. II.3.1 The kernel in $\gamma$-representation
      2. II.3.2 The non-linear equation
      3. II.3.3 The solution
  3. III Our approach
    1. III.1 Generalties
    2. III.2 The dipole scattering amplitudes
    3. III.3 Phenomenological input
  4. IV Results of the fits
  5. V Conclusions
  6. VI Acknowledgements

##  I  Introduction

The goal of this paper is to compare with the experimental (HERA) data the
next-to-leading order approach (NLO) of Ref. CLMS . In Ref. CLMS , we develop
the approach in which we include the re-summation procedure, suggested in
Refs. SALAM ; SALAM1 ; SALAM2 , to fix the BFKL kernel in the NLO. In
particular, we introduce the rapidity variable, which plays the role of the
“evolution time", in the same way as in Ref. DIMST . However, we suggest a
different way to account for the non-linear corrections, than in Ref. DIMST ,
which leads to additional change of the NLO kernel of the evolution equation.
The advantage of our kernel of the BFKL equation BFKL ; LIP , is that the
scattering amplitude satisfies the high energy limits, which follows from the
approach of Ref. LETU  (see Refs. CLMP ; XCWZ ) to the NLO Balitsky-Kovchegov
(BK) BK  evolution NLOBK0 ; NLOBK01 ; NLOBK1 ; NLOBK2 ; JIMWLKNLO1 ;
JIMWLKNLO2 ; JIMWLKNLO3 .

We firmly believe that finding the correct NLO approximation for the non-
linear evolution is one of the most important and urgent problem in the
theoretical description of the high energy scattering. Indeed, in the Colour
Glass Condensate(CGC) approach, which is the only candidate for an effective
theory at high energies (see Ref. KOLEB  for a review), the two essential
parameters, that determine the high energy scattering, calculated in leading
order of perturbative QCD BFKL ; GLR ; MUQI ; MV ; JIMWLK1 ; JIMWLK2 ; JIMWLK3
; JIMWLK4 ; JIMWLK5 ; JIMWLK6 ; BK  turns out to be in an apparent
contradiction with the experimental data. The first one is the BFKL Pomeron
BFKL  intercept, which is equal to $\,2.8\,\bar{\alpha}_{S}$ and leads to the
energy behaviour of the scattering amplitude
$N\propto\exp\left(2.8\,\bar{\alpha}_{S}\ln(\frac{1}{x})\right)$. The second
is the energy behaviour of the new dimensional scale: saturation momentum
$Q^{2}_{s}\propto\exp\left(4.88\,\bar{\alpha}_{S}\ln(\frac{1}{x})\right)$.
Both show the increase in the leading order CGC approach, which cannot be
reconciled with the available experimental data. So, the large NLO corrections
appear as the only way out, now as well as two decades ago.

In the next section we will outline the main results of Ref. CLMS  and will
specify our theoretical description of the dipole scattering amplitude.
However, the current stage of our theoretical understanding of non-
perturbative QCD is such, that we have to build a model. We need to take into
account the non-perturbative corrections that will reproduce the correct,
exponentially decreasing at large impact parameters ($b$) scattering
amplitude. It has been demonstrated in Refs. KW1 ; KW2 ; KW3 ; FIIM , that the
CGC equations JIMWLK1 ; JIMWLK2 ; JIMWLK3 ; JIMWLK4 ; JIMWLK5 ; JIMWLK6 ; BK
as well as all other approaches, based on perturbative QCD, lead to the
amplitude that increases as a power of energy, resulting in the violation of
the Froissart theorem FROI 111It should be noted that NLO corrections to the
kernel of CGC evolution equations does not change the power-like behaviour of
the scattering amplitude but improve the situation moving the violation of the
Froissart theorem to larger values of $b$ BEST1 ; BEST2 ; CCM ; BCCM ; CLM ..
Unfortunately, without a theoretical control on non-perturbative QCD we have
to use a phenomenological approach to model the large $b$ behaviour. In this
paper we will exploit two approaches:

  1. 1.

the non-perturbative behaviour of the saturation scale, which we will
parameterize as follows:

$Q^{2}_{s}\left(b,Y\right)\,\propto\,\left(S\left(b,m\right)\right)^{\frac{1}{\bar{\gamma}}}$
(1)

where $S\left(b\right)$ is the Fourier image of
$S\left(Q_{T}\right)=1/\left(1+\frac{Q^{2}_{T}}{m^{2}}\right)^{2}$ and the
value of $\bar{\gamma}$ we will discuss below. In the vicinity of the
saturation scale such $b$ dependance results in the large $b$-dependence of
the scattering amplitude, which is proportional to $\exp\left(-m\,b\right)$ at
$b\gg 1/m$, in accordance of the Froissart theorem FROI . In addition, we
reproduce the large $Q_{T}$ dependence of this amplitude proportional to
$Q^{-4}_{T}$ which follows from the perturbative QCD calculation BRLE .
Theoretically the fact that we can absorb the non-perturbative $b$-dependence
in $Q_{s}(Y,b)$ (see Eq. (1) for example), follows from the semi-classical
approach to BK equation BKL  and has been widely used in all, so called,
saturation models SATMOD0 ; SATMOD1 ; SATMOD2 ; IIM ; SATMOD3 ; SATMOD4 ;
SATMOD5 ; SATMOD6 ; SATMOD7 ; SATMOD8 ; SATMOD9 ; SATMOD10 ; SATMOD11 ;
SATMOD12 ; SATMOD13 ; SATMOD14 ; SATMOD15 ; SATMOD16 ; SATMOD17 ; CLP ; CLMP .

  2. 2.

in Refs. LERY1 ; LERY2 ; LETAN ; QCD2 ; KHLE ; KKL ; BLT ; LEPION ; KAN ;
GOLEB  it is made an attempt to incorporate in the BFKL equation the Gribov’s
diffusion GRIB  in impact parameter ($b$). As the result of this tthe
following formula for the saturation scale was suggested LEPION ; GOLEB :

$Q^{2}_{s}\left(b,Y\right)\,\propto\,\exp\left(-\frac{3}{4}\,{\cal
Z}\right)~{}~{}~{}~{}\mbox{with}~{}~{}~{}~{}{\cal
Z}\,=\,\left(\frac{b^{4}}{4{\alpha^{\prime}}^{2}_{\rm eff}\,Y}\right)^{1/3}$
(2)

where ${\alpha^{\prime}}^{2}_{\rm eff}$ is a new dimensional non-perturbative
parameter which controls the Gribov’s diffusion.

The paper is organized as follows. In section II we will give a brief review
of the approach that we have developed in Ref. CLMS . Next we will discuss the
procedure of obtaining the NLO kernel of the BFKL equation based on the
anomalous dimensions for re-summed NLO corrections, which is suggested in
Refs. SALAM ; SALAM1 ; SALAM2 ; KMRS . Then we will consider the modification
that we need to introduce in the NLO kernel to account for correct behaviour
of the scattering amplitude at high energy and the structure of the non-linear
equation. In section III we specify our approach, which we use for describing
the experimental data. In particular, we introduce the phenomenological
parameters, which have to be calculated in the non-perturbative QCD approach,
and discuss their physical meaning and the range of possible values. In
section IV we collect the results of the fit of the experimental data on DIS.
We summarize our results in the conclusion.

##  II Leading twist approximation for non-linear evolution in the NLO - a
recap

###  II.1  Re-summed anomalous dimensions in NLO and the kernel of linear
evolution

The general solution to the linear equation has the following form:

$N\left(Y,\boldsymbol{r},\boldsymbol{b}\right)\,\,\,=\,\,\int^{\epsilon+i\infty}_{\epsilon-i\infty}\frac{d\gamma}{2\,\pi\,i}e^{\omega\left(\gamma\right)\,Y}\phi_{\gamma}\left(\boldsymbol{r},\boldsymbol{b},\boldsymbol{R}\right)\,\,\phi_{in}\left(\gamma,R\right)\,\,\,=\,\,\int^{\epsilon+i\infty}_{\epsilon-i\infty}\frac{d\omega}{2\,\pi\,i}e^{\omega\,Y}\phi_{\gamma(\omega)}\left(\boldsymbol{r},\boldsymbol{b},\boldsymbol{R}\right)\,\,\phi_{in}\left(\omega,R\right)$
(3)

where $N$ is the scattering amplitude of the dipole with size $r$ at the
impact parameter $b$. $Y$ is the rapidity of the dipole.
$\phi_{\gamma}\left(\boldsymbol{r},\boldsymbol{b}\right)$ is the eigenfunction
of the BFKL equation which has the general form:

$\displaystyle\phi_{\gamma}\left(\boldsymbol{r},\boldsymbol{R},\boldsymbol{b}\right)\,$
$\displaystyle=$
$\displaystyle\,\left(\frac{r^{2}\,R^{2}}{\left(\boldsymbol{b}+\frac{1}{2}(\boldsymbol{r}-\boldsymbol{R})\right)^{2}\,\left(\boldsymbol{b}-\frac{1}{2}(\boldsymbol{r}-\boldsymbol{R})\right)^{2}}\right)^{\gamma}\,\,=\,\,\,e^{\gamma\,\xi}$
(4) $\displaystyle\mbox{with}~{}~{}~{}\xi\,$ $\displaystyle=$
$\displaystyle\,\ln\left(\frac{r^{2}\,R^{2}}{\left(\boldsymbol{b}+\frac{1}{2}(\boldsymbol{r}-\boldsymbol{R})\right)^{2}\,\left(\boldsymbol{b}-\frac{1}{2}(\boldsymbol{r}-\boldsymbol{R})\right)^{2}}\right)$
(5)

where $R$ is the size of the target. $\phi_{in}$ can be found from the initial
condition at $Y=0$.

The eigenvalues $\omega\left(\gamma\right)$ in the NLO has been calculated in
Refs. BFKLNLO ; BFKLNLO1  and have the following form:

$\omega_{\rm
NLO}\left(\bar{\alpha}_{S},\gamma\right)\,\,=\,\,\bar{\alpha}_{S}\,\chi^{LO}\left(\gamma\right)\,\,+\,\,\bar{\alpha}_{S}^{2}\,\chi^{NLO}\left(\gamma\right)$
(6)

The explicit form of $\chi^{NLO}\left(\gamma\right)$ is given in Ref. BFKLNLO
. However, Eq. (6) has singularities at $\gamma\,\to\,1$ which has been re-
summed taking into account the high order corrections in Refs. SALAM ; SALAM1
; SALAM2 ; KMRS . Finally, $\omega_{\rm
NLO}\left(\bar{\alpha}_{S},\gamma\right)$ has the form SALAM ; SALAM1 ; SALAM2
:

$\omega_{\rm
NLO}\left(\bar{\alpha}_{S},\gamma\right)\,=\,\bar{\alpha}_{S}\left(\chi_{0}\left(\omega_{\rm
NLO},\gamma\right)\,+\,\omega_{\rm NLO}\,\frac{\chi_{1}\left(\omega_{\rm
NLO},\gamma\right)}{\chi_{0}\left(\omega_{\rm NLO},\gamma\right)}\right)$ (7)

where

$\chi_{0}\left(\omega,\gamma\right)\,\,=\,\,\chi^{LO}\left(\gamma\right)\,-\,\frac{1}{1\,-\,\gamma}\,+\,\frac{1}{1\,-\,\gamma\,+\,\omega}$
(8)

and

$\displaystyle\chi_{1}\left(\omega,\gamma\right)\,\,=$ (9)
$\displaystyle\,\,\chi^{NLO}\left(\gamma\right)\,+\,F\left(\frac{1}{1-\gamma}\,-\,\frac{1}{1\,-\,\gamma\,+\,\omega}\right)\,+\,\frac{A_{T}\left(\omega\right)\,-\,A_{T}\left(0\right)}{\gamma^{2}}\,+\,\frac{A_{T}\left(\omega\right)-b}{\left(1\,-\,\gamma\,+\,\omega\right)^{2}}\,-\,\frac{A_{T}\left(0\right)-b}{\left(1\,-\,\gamma\right)^{2}}$

Functions $\chi^{NLO}\left(\gamma\right)$ and $A_{T}\left(\omega\right)$ as
well as the constants ($F$ and $b$), are defined in Refs. SALAM ; SALAM1 ;
SALAM2 , while $\chi^{LO}\left(\gamma\right)$ has the following form:

$\omega_{\rm
LO}\left(\bar{\alpha}_{S},\gamma\right)\,\,=\,\,\bar{\alpha}_{S}\,\chi^{LO}\left(\gamma\right)\,\,\,=\,\,\,\bar{\alpha}_{S}\left(2\psi\left(1\right)\,-\,\psi\left(\gamma\right)\,-\,\psi\left(1-\gamma\right)\right)$
(10)

However, in Ref. KMRS  the economic form of
$\chi_{1}\left(\omega,\gamma\right)$ is given, which coincides with Eq. (9) to
within $7\%$:

$\omega^{\rm KMRS}\,=\,\bar{\alpha}_{S}\left(1-\omega^{\rm
KMRS}\right)\left(\frac{1}{\gamma}+\frac{1}{1-\gamma+\omega^{\rm
KMRS}}\,+\,\underbrace{\left(2\psi(1)-\psi\left(2-\gamma\right)-\psi\left(1+\gamma\right)\right)}_{\mbox{
high twist contributions}}\right)$ (11)

One can see that $\gamma(\omega)\to 0$ when $\omega\to 1$ as follows from
energy conservation.

Eq. (7) for $\gamma\to 1$ has the form:

$\omega\,\,=\,\,\frac{\bar{\alpha}_{S}}{1-\gamma+\omega};$ (12)

which leads to

$\omega\left(\gamma\right)\,\,=\,\,\frac{1}{2}\left(-\left(1-\gamma\right)\,+\,\sqrt{4\,\bar{\alpha}_{S}\,+\,\left(1-\gamma\right)^{2}}\right)$
(13)

As it is shown in Ref. CLMS  Eq. (13) corresponds to the kernel of Ref. DIMST
. Resolving Eq. (12) with respect to $\gamma$ ASV ; CLMS  we obtain:

$1\,-\,\gamma\,\,\,=\,\,\,\frac{\bar{\alpha}_{S}}{\omega}\,\,-\,\,\omega$ (14)

Eq. (14) gives the simple equation

$\frac{\partial}{\partial\,\eta}\tilde{N}\left(\xi^{\prime},\eta;b\right)=\bar{\alpha}_{S}\int^{\xi^{\prime}}d\xi^{\prime\prime}\,\tilde{N}\left(\xi^{\prime\prime},\eta;b\right)\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial^{2}}{\partial\,\eta\,\partial\,\xi^{\prime}}\tilde{N}\left(\xi^{\prime},\eta;b\right)\,\,=\,\,\bar{\alpha}_{S}\,\tilde{N}\left(\xi^{\prime},\eta;b\right)\,,$
(15)

for the amplitude
$\tilde{N}\left(\xi^{\prime},\eta;b\right)\,=\,N\left(\xi^{\prime},\eta;b\right)\Big{/}r^{2}$.
$\eta=Y-\xi^{\prime}$ is a new energy variable, which corresponds correct time
ordering in double log approximation (DLA) DIMST . In Eq. (14)
$\xi^{\prime}\,=\,-\,\xi$.

###  II.2 Non-linear equation and the feedback to the kernel of the linear
evolution

The general structure of the non-linear Balitsky-Kovchegov equation BK  has
the following form:

$\displaystyle\frac{\partial}{\partial
Y}N\left(\boldsymbol{x}_{10},\boldsymbol{b},Y;R\right)=$
$\displaystyle\bar{\alpha}_{S}\\!\\!\int\frac{d^{2}\boldsymbol{x}_{2}}{2\,\pi}\,K\left(\boldsymbol{x}_{02},\boldsymbol{x}_{12};\boldsymbol{x}_{10}\right)\Bigg{(}N\left(\boldsymbol{x}_{12},\boldsymbol{b}-\frac{1}{2}\boldsymbol{x}_{20},Y;R\right)+N\left(\boldsymbol{x}_{20},\boldsymbol{b}-\frac{1}{2}\boldsymbol{x}_{12},Y;R\right)-N\left(\boldsymbol{x}_{10},\boldsymbol{b},Y;R\right)$
$\displaystyle-\,\,N\left(\boldsymbol{x}_{12},\boldsymbol{b}-\frac{1}{2}\boldsymbol{x}_{20},Y;R\right)\,N\left(\boldsymbol{x}_{20},\boldsymbol{b}-\frac{1}{2}\boldsymbol{x}_{12},Y;R\right)\Bigg{)}$
(16)

where $\boldsymbol{x}_{ik}\,\,=\,\,\boldsymbol{x}_{i}\,-\,\boldsymbol{x}_{k}$
and $\boldsymbol{x}_{10}\equiv\,\boldsymbol{r}$,
$\boldsymbol{x}_{20}\,\equiv\,\boldsymbol{r}^{\prime}$ and
$\boldsymbol{x}_{12}\,\equiv\,\boldsymbol{r}\,-\,\boldsymbol{r}^{\prime}$. $Y$
is the rapidity of the scattering dipole and $\boldsymbol{b}$ is the impact
factor.
$K\left(\boldsymbol{x}_{02},\boldsymbol{x}_{12};\boldsymbol{x}_{10}\right)$ is
the kernel of the BFKL equation. In our approach we wish to preserve this
form, but include the NLO corrections to the kernel. In particular, we would
like to include the reggeization term in Eq. (II.2), which contribute to the
linear equation, but has been neglected in DLA (see Eq. (15)). It should be
stressed that only keeping this term we can provide the correct asymptotic
behaviour at large $Y$: $N\,\,\to\,\,1$.

Therefore, the linear equation takes the form:

$\displaystyle\frac{\partial}{\partial\eta}N\left(\boldsymbol{x}_{10},\boldsymbol{b},Y;R\right)=$
(17)
$\displaystyle\bar{\alpha}_{S}\\!\\!\int\frac{d^{2}\boldsymbol{x}_{2}}{2\,\pi}\,K\left(\boldsymbol{x}_{02},\boldsymbol{x}_{12};\boldsymbol{x}_{10}\right)\Bigg{\\{}N\left(\boldsymbol{x}_{12},\boldsymbol{b}-\frac{1}{2}\boldsymbol{x}_{20},Y;R\right)+N\left(\boldsymbol{x}_{20},\boldsymbol{b}-\frac{1}{2}\boldsymbol{x}_{12},Y;R\right)-N\left(\boldsymbol{x}_{10},\boldsymbol{b},Y\right)\Bigg{\\}}$
$\displaystyle\xrightarrow{x_{12}\sim
x_{02}\,\,\gg\,\,x_{01}}\,\,\frac{1}{2}\,\bar{\alpha}_{S}\,x^{2}_{01}\,\int_{x^{2}_{01}}\frac{dx^{2}_{12}}{x^{4}_{12}}\Bigg{(}2\,N\left(\boldsymbol{x}_{12},\boldsymbol{b},Y;R\right)\,\,-\,\,N\left(\boldsymbol{x}_{01},\boldsymbol{b},Y;R\right)\Bigg{)}$

and Eq. (14) can be re-written in the form:

$\frac{\partial^{2}}{\partial\eta\,\partial\,\xi^{\prime}}\tilde{N}\left(\xi^{\prime},\eta\right)\,\,=\,\,\bar{\alpha}_{S}\,\tilde{N}\left(\xi^{\prime},\eta\right)\,\,-\,\,\frac{1}{2}\,\bar{\alpha}_{S}\frac{\partial}{\partial\,\xi^{\prime}}\tilde{N}\left(\xi^{\prime},\eta\right)$
(18)

For the dipole amplitude, $N\left(\xi^{\prime},Y\right)$, Eq. (14) takes the
following form:

$\frac{\partial^{2}}{\partial
Y\,\partial\,\xi^{\prime}}N\left(\xi^{\prime},Y\right)\,\,+\,\,\frac{\partial}{\partial
Y}N\left(\xi^{\prime},Y\right)\,\,=\,\,\frac{1}{2}\,\bar{\alpha}_{S}\,N\left(\xi^{\prime},Y\right)\,\,-\,\,\frac{1}{2}\,\bar{\alpha}_{S}\frac{\partial}{\partial\,\xi^{\prime}}N\left(\xi^{\prime},Y\right)$
(19)

which leads to the eigenvalue $\omega(\gamma)$:

$\omega\left(\gamma\right)\,\,=\,\,\frac{1}{2}\,\bar{\alpha}_{S}\,\frac{1\,+\,\gamma}{1\,-\,\gamma}$
(20)

Using Eq. (11) in the vicinity of $\gamma\to 1$ one can see that Eq. (20)
takes the form CLMS :

$\omega\left(\gamma\right)\,\,=\,\,\frac{1}{2}\,\bar{\alpha}_{S}\,\frac{1\,+\,\gamma}{1\,+\,\bar{\alpha}_{S}\,-\,\gamma}$
(21)

Using the general equation to determine the critical anomalous dimension and
the energy behaviour of the saturation scale (see review KOLEB ):

$\lambda_{\eta}\,\,=\,\,\frac{\omega\left(\bar{\gamma}_{\eta}\right)}{\bar{\gamma}_{\eta}}\,\,=\,\,-\,\frac{d\omega\left(\bar{\gamma}_{\eta}\right)}{d\bar{\gamma}_{\eta}};~{}~{}~{}\ln\left(\frac{Q^{2}_{s}\left(Y\right)}{Q^{2}_{s}\left(Y=0\right)}\right)\,\,=\,\,\lambda_{\eta}\,\eta;$
(22)

we obtain:

$\bar{\gamma}_{\eta}\,\,=\,\,\sqrt{2\,+\,\bar{\alpha}_{S}}\,-\,1;~{}~{}~{}~{}\lambda_{\eta}\,\,=\,\,\frac{1}{2}\,\frac{\bar{\alpha}_{S}}{3\,+\,2\,\bar{\alpha}_{S}\,-\,2\sqrt{2\,+\,\bar{\alpha}_{S}}};$
(23)

Re-writing Eq. (23) for the saturation momentum as a function of $Y$ we obtain

$\ln\left(\frac{Q^{2}_{s}\left(Y\right)}{Q^{2}_{s}\left(Y=0\right)}\right)\,\,=\,\,\lambda\,Y~{}~{}~{}\mbox{with}\,\,\lambda\,\,=\,\,\frac{\lambda_{\eta}}{1\,+\,\lambda_{\eta}}~{}~{}\mbox{and}~{}~{}~{}\bar{\gamma}\,=\,\bar{\gamma}_{\eta}\left(1\,+\,\lambda_{\eta}\right).$
(24)

Figure 1:  $\lambda$ and $\bar{\gamma}$ versus $\bar{\alpha}_{S}$. In red we
put $\bar{\gamma}=0.63$ which stems from LO estimates and $\lambda=0.2$ which
is a typical value that is need to describe DIS data (see for example Ref.
SATMOD17 ).

###  II.3 NLO BFKL kernel in the saturation domain

####  II.3.1 The kernel in $\gamma$-representation

The non-linear corrections are essential in the region of small values of $x$
(large $Y$) where $\tau\,\,=\,\,r^{2}Q^{2}_{s}\,\,>\,\,1$. In Ref. LETU  it is
shown that in the BFKL equation we have two types of logarithms, which are
generated by the following kernels

$\displaystyle\chi\left(\gamma\right)\,\,=\,\,\left\\{\begin{array}[]{l}\,\,\,\frac{1}{\gamma}\,\,\,\,\,\,\,\,\,\,\mbox{for}\,\,\,\tau\,=\,rQ_{s}\,>\,1\,\,\,\,\,\,\mbox{summing}\left(\ln\left(rQ_{s}\right)\right)^{n};\\\
\\\
\,\,\,\frac{1}{1\,-\,\gamma}\,\,\,\,\,\mbox{for}\,\,\,\tau\,=\,rQ_{s}\,<\,1\,\,\,\,\,\mbox{summing}\left(\ln\left(1/(r\,\Lambda_{\rm
QCD})\right)\right)^{n};\\\ \end{array}\right.$ (28)

In Ref. CLMS  we suggest to use Eq. (28) instead of the full expression of Eq.
(10).

In the previous section we specified how we changed the kernel in the
perturbative QCD region, taking into account the NLO corrections. In the
region of large $\tau$ we need to find the anomalous dimension at
$\gamma\,\to\,0$, as it follows from Eq. (28). Hence we re-write Eq. (11) in
the vicinity of $\gamma\to 0$, where it has the form:

$\omega\,\,=\,\,\bar{\alpha}_{S}\Bigg{\\{}\frac{1}{\gamma}\,\,-\,\,\omega\Bigg{\\}}$
(29)

Resolving Eq. (29) with respect of $\gamma$ we obtain

$\gamma\,\,=\,\,\frac{\bar{\alpha}_{S}}{1+\bar{\alpha}_{S}}\,\frac{1}{\omega}$
(30)

which differs from the behaviour of the LO kernel, only by replacing
$\bar{\alpha}_{S}\,\to\,\bar{\alpha}_{S}/\left(1\,+\,\bar{\alpha}_{S}\right)$.
Therefore, we can discuss the non-linear equation in the LO, substituting
$\bar{\alpha}_{S}/\left(1\,+\,\bar{\alpha}_{S}\right)$ in place of
$\bar{\alpha}_{S}$, at the final stage.

####  II.3.2 The non-linear equation

In the saturation region where $\tau\,\,>\,\,1$, the logarithms originate from
the decay of a large size dipole, into one small size dipole and one large
size dipole LETU . However, the size of the small dipole is still larger than
$1/Q_{s}$. This observation can be translated in the following form of the
kernel in the LO

$\displaystyle\frac{\bar{\alpha}_{S}}{2\pi}\int\,\displaystyle{K\left(\boldsymbol{x}_{01};\boldsymbol{x}_{02},\boldsymbol{x}_{12}\right)}\,d^{2}x_{02}\,$
$\displaystyle\rightarrow$
$\displaystyle\,\frac{\bar{\alpha}_{S}}{2}\,\int^{x^{2}_{01}}_{1/Q^{2}_{s}(Y,b)}\frac{dx^{2}_{02}}{x_{02}^{2}}\,\,+\,\,\frac{\bar{\alpha}_{S}}{2}\,\int^{x^{2}_{01}}_{1/Q^{2}_{s}(Y,b)}\frac{d|\boldsymbol{x}_{01}-\boldsymbol{x}_{02}|^{2}}{|\boldsymbol{x}_{01}-\boldsymbol{x}_{02}|^{2}}\,\,$
(31) $\displaystyle=$
$\displaystyle\,\,\frac{\bar{\alpha}_{S}}{2}\,\int^{\xi}_{-\xi_{s}}d\xi_{02}\,\,+\,\,\frac{\bar{\alpha}_{S}}{2}\,\int^{\xi}_{-\xi_{s}}d\xi_{12}$

where $\xi_{ik}\,=\,\ln\left(x^{2}_{ik}Q_{s}^{2}(Y=Y_{0})\right)$ and
$\xi_{s}\,=\,\ln\left(Q^{2}_{s}(Y)/Q_{s}^{2}(Y=Y_{0})\right)$.

Inside the saturation region the BK equation of the LO takes the form

$\frac{\partial^{2}\widehat{N}\left(Y,\boldsymbol{x};\boldsymbol{b}\right)}{\partial
Y\,\partial\xi}\,\,=\,\,\bar{\alpha}_{S}\,\left\\{\left(1\,\,-\,\frac{\partial\widehat{N}\left(Y,\xi;\boldsymbol{b}\right)}{\partial\xi}\right)\,\widehat{N}\left(Y,\xi;\boldsymbol{b}\right)\right\\}$
(32)

where
$\widehat{N}\left(Y,\xi;\boldsymbol{b}\right)\,\,=\,\,\int^{\xi}d\xi^{\prime}\,N\left(Y,\xi^{\prime};\boldsymbol{b}\right)$
.

For the NLO kernel of Eq. (30), Eq. (32) takes the form:

$\frac{\partial^{2}\widehat{N}\left(Y,\xi;\boldsymbol{b}\right)}{\partial
Y\,\partial\xi}\,\,=\,\,\frac{\bar{\alpha}_{S}}{1\,+\,\bar{\alpha}_{S}}\,\left\\{\left(1\,\,-\,\frac{\partial\widehat{N}\left(Y,\xi;\boldsymbol{b}\right)}{\partial\xi}\right)\,\widehat{N}\left(Y,\xi;\boldsymbol{b}\right)\right\\}$
(33)

####  II.3.3 The solution

For solving this equation we introduce function
$\Omega\left(Y;\xi,\boldsymbol{b}\right)$ LETU

$N\left(Y,\xi\right)\,\,=\,\,1\,\,-\,\,\exp\left(-\Omega\left(Y,\xi\right)\right)$
(34)

Substituting Eq. (34) into Eq. (33) we reduce it to the form

$\displaystyle\frac{\partial\Omega\left(Y,\xi\right)}{\partial
Y}\,\,=\,\,\frac{\bar{\alpha}_{S}}{1+\bar{\alpha}_{S}}\widetilde{N}\left(Y,\xi\right);~{}~{}~{}\frac{\partial^{2}\Omega\left(Y,\xi\right)}{\partial
Y\,\partial\xi}\,\,\,=\,\,\frac{\bar{\alpha}_{S}}{1+\bar{\alpha}_{S}}\Bigg{(}1-\,\exp\left(-\Omega\left(Y,\xi\right)\right)\Bigg{)};$
(35a)
$\displaystyle~{}\frac{\partial^{2}\Omega\left(\xi_{s};\zeta\right)}{\partial\xi_{s}\,\partial\xi}\,\,\,=\,\,\frac{\bar{\alpha}_{S}}{\lambda(\bar{\alpha}_{S})\,\left(1\,\,+\,\,\bar{\alpha}_{S}\right)}\Bigg{(}1-\,\exp\left(-\Omega\left(\xi_{s};\zeta\right)\right)\Bigg{)}\,\,\equiv\,\,\sigma\Bigg{(}1-\,\exp\left(-\Omega\left(\xi_{s};\zeta\right)\right)\Bigg{)}$
(35b)

where $\lambda$ is given by Eq. (24). The variable $\xi_{s}$ is defined as

$\xi_{s}\,\,=\,\,\ln\left(Q^{2}_{s}\left(Y\right)/Q^{2}_{s}\left(Y=0;\boldsymbol{b},\boldsymbol{R}\right)\right)\,\,=\,\,\lambda\,Y$
(36)

The use of this variable indicates the main idea of our approach in the region
of $\tau\,>\,1$: we wish to match the solution of the non-linear Eq. (35b)
with the solution of linear equation (see Eq. (19)) in the kinematic region
where it has the form MUT :

$N\,\,=\,\,N_{0}\,\exp\left(\bar{\gamma}\,z\right)~{}~{}~{}~{}\mbox{with}~{}~{}~{}~{}~{}z\,\,=\,\,\xi_{s}\,\,+\,\,\xi$
(37)

and $\bar{\gamma}$ is determined by Eq. (24). Eq. (37) leads to the initial
and boundary conditions:

$N\left(r^{2},Y\right)\,\,=\,\,N\left(\tau=1\right)\,\,=\,\,N_{0};~{}~{}~{}~{}~{}~{}~{}\frac{d\ln\left(N\left(r^{2},Y\right)\right)}{d\xi}|_{\tau=1}\,\,=\,\,\bar{\gamma};$
(38)

Eq. (35b) has a traveling wave solution (see formula 3.4.1.1 of Ref. MATH ).
For Eq. (35b) in the canonical form:

$\frac{\partial^{2}\Omega\left(\xi_{s};\tilde{\zeta}\right)}{\partial
t^{2}_{+}}\,\,-\,\,\frac{\partial^{2}\Omega\left(\xi_{s};\tilde{\zeta}\right)}{\partial
t^{2}_{-}}\,\,\,=\,\,\sigma\,\,\Bigg{(}1-\,\exp\left(-\Omega\left(\xi_{s};\tilde{\zeta}\right)\right)\Bigg{)},$
(39)

with $t_{\pm}=\xi_{s}\pm\xi$, the solution takes the form:

$\int^{\Omega}_{\Omega_{0}}\frac{d\Omega^{\prime}}{\sqrt{C_{1}+\frac{2}{(\mu^{2}-\kappa^{2})}\sigma\left(\Omega^{\prime}+\exp\left(-\Omega^{\prime}\right)\right)}}\,\,=\,\,\mu
t_{+}+\kappa t_{-}+C_{2}$ (40)

where all constants have to be determined from the initial and boundary
conditions of Eq. (38). First we see that $C_{2}=0$ and $\kappa=0$. From the
condition $\Omega^{\prime}_{z}/\Omega\,\,=\,\,\bar{\gamma}$ at $t_{+}=0$ we
can find $C_{1}$. Indeed, differentiating Eq. (40) with respect to $t_{+}$ one
can see that at $t_{+}=0$ we have:

$\frac{d\Omega}{dt_{+}}|_{t_{+}=0}\,\frac{1}{\sqrt{C_{1}\,\,+\,\,\frac{2\,\sigma}{\mu^{2}}\left(1\,\,+\,\,\frac{1}{2}\,\Omega^{2}_{0}\right)}}\,\,=\,\,\mu$
(41)

From Eq. (41) one can see that choosing

$C_{1}\,\,=\,\,-\,\frac{2\sigma}{\mu^{2}}\,\,+\,\,\Omega^{2}_{0}\left(1\,\,-\,\,\frac{\sigma}{\mu^{2}}\right);~{}~{}~{}~{}~{}\mbox{and}\,~{}~{}~{}\mu\,\,=\,\,\bar{\gamma}$
(42)

we satisfy the initial condition
$\frac{d\ln\left(\Omega\right)}{dz}|_{t_{+}=0}=\,\bar{\gamma}$ of Eq. (24).

Finally, the solution of Eq. (40) can be re-written in the following form for
$\Omega_{0}\ll 1$:

$\int^{\Omega}_{\Omega_{0}}\frac{d\Omega^{\prime}}{\sqrt{\Omega^{2}_{0}\,\left(1\,\,-\,\,\frac{\sigma}{\bar{\gamma}^{2}}\right)\,\,+\,\,\frac{2\,\sigma}{\bar{\gamma}^{2}}\left(-1\,\,+\,\,\Omega^{\prime}\,\,+\,\,\,e^{-\,\Omega^{\prime}}\right)}}\,\,=\,\,\bar{\gamma}\,z$
(43)

One can see that Eq. (43) gives the geometric scaling solution LETU ; BALE ;
IIML ; SGBK  which depends only on the variable $z$. For $\Omega\to\Omega_{0}$
and if $\Omega_{0}\ll 1$, Eq. (43) can be solved explicitly giving

$\Omega\,\,=\,\,\Omega_{0}\Bigg{\\{}\cosh\left(\sqrt{\sigma}\left(\xi_{s}+\xi\right)\right)\,\,+\,\,\frac{\bar{\gamma}}{\sqrt{\sigma}}\,\sinh\left(\sqrt{\sigma}\left(\xi_{s}+\xi\right)\right)\Bigg{\\}}$
(44)

Eq. (44) gives the solution which depends only on one variable
$z\,\,=\,\,\xi_{s}+\xi$, and satisfies the initial conditions of Eq. (38).

At large $z$ we obtain the solution LETU :

$\Omega\left(z\right)\,\,=\,\,\frac{\sigma}{2}\,z^{2}\,\,+\,\,{\rm Const}$
(45)

or in terms of the amplitude

$N\left(z\right)\,\,=\,\,1\,\,-\,\,{\rm
Const}\,e^{-\,\frac{\sigma}{2}\,z^{2}}$ (46)

We wish to stress that Eq. (46) reproduces the asymptotic solution to the BK
equation in the NLO, which has been derived in Refs. CLMP ; XCWZ , for fixed
$\bar{\alpha}_{S}$.

It should be noted that both solutions of Eq. (44) and Eq. (45) can be derived
directly from Eq. (39) assuming
$1\,-\,\exp\left(-\Omega\right)\,\,\to\,\Omega$ and
$1\,-\,\exp\left(-\Omega\right)\,\,\to\,1$ for small $z$ and large $z$,
respectively.

##  III Our approach

###  III.1 Generalties

The observables in deep inelastic scattering can be expressed through the
following scattering amplitudes (see Fig. 2 and Ref. KOLEB  for the review and
references therein)

$N\left(Q,Y;b\right)\,\,=\,\,\int\frac{d^{2}r}{4\,\pi}\int^{1}_{0}dz\,|\Psi_{\gamma^{*}}\left(Q,r,z\right)|^{2}\,N\left(r,Y;b\right)$
(47)

where $Y\,=\,\ln\left(1/x_{Bj}\right)$ and $x_{Bj}$ is the Bjorken $x$. $z$ is
the fraction of energy carried by quark. $Q$ is the photon virtuality. $b$
denotes the impact parameter of the scattering amplitude.

Eq. (47) shows the main features of the interactions at high energies. They go
in two stages. The first is the decay of virtual photon in quark-antiquark
pair, described by $|\Psi_{\gamma^{*}}\left(Q,r,z\right)|^{2}$ in Eq. (47).
For large $Q^{2}$ ( $Q^{2}\,\geq\,Q^{2}_{0}$ with $Q^{2}_{0}\approx
0.7GeV^{2}$, see Ref. GLMTC ) the wave function is well known (see Ref. KOLEB
and references therein)

$\displaystyle(\Psi^{*}\Psi)_{T}^{\gamma^{*}}$
$\displaystyle=\frac{2N_{c}}{\pi}\alpha_{\mathrm{em}}\sum_{f}e_{f}^{2}\left\\{\left[z^{2}+(1-z)^{2}\right]\epsilon^{2}K_{1}^{2}(\epsilon
r)+m_{f}^{2}K_{0}^{2}(\epsilon r)\right\\},$ (48)
$\displaystyle(\Psi^{*}\Psi)_{L}^{\gamma^{*}}$
$\displaystyle=\frac{8N_{c}}{\pi}\alpha_{\mathrm{em}}\sum_{f}e_{f}^{2}Q^{2}z^{2}(1-z)^{2}K_{0}^{2}(\epsilon
r),$ (49)

where T(L) denotes the polarization of the photon and $f$ is the flavours of
the quarks. $\epsilon^{2}\,\,=\,\,m^{2}_{f}\,\,+\,\,Q^{2}z(1-z)$. However,
even for DIS the non-perturbative corrections become essential and we cannot
use Eq. (48) and Eq. (49) for $Q^{2}\leq Q^{2}_{0}$.

The second is the amplitude $N\left(r,Y;b\right)$ of the interaction of the
dipole with the target. We have discussed this amplitude in the previous
section and will specify the way how we will use the finding of this section
for the practical application to DIS below.

Figure 2: The graphic representation of Eq. (47) for the scattering amplitude.
$Y=\ln\left(1/x_{Bj}\right)$ and $r$ is the size of the interacting dipole.
$z$ denotes the fraction of energy that is carried by one quark. $b$ denotes
the impact parameter of the scattering amplitude.

Using Eq. (47), Eq. (48) and Eq. (49) we can write the main observables in
DIS:

$\displaystyle\sigma_{T,L}$ $\displaystyle\,\,=\,\,$ $\displaystyle 2\int
d^{2}b\,\,N_{T,L}\left(Q,Y,;b\right);$ (50a) $\displaystyle
F_{2}\left(Q,Y\right)$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{4\,\pi^{2}\,\alpha_{\rm
e.m.}}\\{\sigma_{T}\,\,+\,\,\sigma_{L}\\};$ (50b) $\displaystyle
F^{cc}_{2}\left(Q,Y\right)$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{4\,\pi^{2}\,\alpha_{\rm
e.m.}}\\{\sigma^{cc}_{T}\,\,+\,\,\sigma^{cc}_{L}\\};$ (50c) $\displaystyle
F_{L}\left(Q,Y\right)$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{4\,\pi^{2}\,\alpha_{\rm e.m.}}\,\sigma_{L};$ (50d)

$\sigma^{cc}$ in Eq. (50c) are calculated, using Eq. (50a) with
$N_{T,L}\left(Q,Y,;b\right)$, which is calculated using Eq. (48) and Eq. (49)
for $c$-quarks.

###  III.2 The dipole scattering amplitudes

Based on the approach, briefly discussed in the previous section, we can see
three different kinematic regions where we are going to use the scattering
amplitude $N\left(Y,\xi^{\prime};\boldsymbol{b}\right)$ in three different
forms.

  1. 1.

For $Q^{2}_{s}\,r^{2}\,\,\,<\,\,1$ (perturbative QCD region) we suggest to use
the linear evolution equation of Eq. (19), which has been discusssed above.
Recalling that in the derivation of this equation we use the DLA in which both
$\eta\,\,=\,\,Y\,-\,\xi^{\prime}$ and $\xi^{\prime}$ are considered to be
large: $\bar{\alpha}_{S}\,\eta\,\xi\,\gg\,\,1$. For $\eta\,>\,0$ ($Y\,>\,\xi$)
we can use the experimental data as the initial condition for Eq. (19). The
value of the saturation momentum is given by Eq. (24). Since its value turns
out to be much less than $Q_{\rm max}$, which stems from the condition
$\xi_{max}=\ln\left(Q^{2}_{\rm max}/Q^{2}_{0}\right)$ = $\eta$, we can use the
DGLAP evolution equation in the next-to-leading order for
$Q^{2}\,>\,Q^{2}_{\rm max}$.

  2. 2.

For $Q^{2}_{s}\,r^{2}\,\,\,\sim\,\,1$ (vicinity of the saturation scale) we
use the scattering amplitude in the form MUT

$N\left(r,\eta;b\right)\,\,=\,\,N_{0}\,\,\left(Q^{2}_{s}\left(Y,b\right)\,r^{2}\right)^{\bar{\gamma}}$
(51)

with $\bar{\gamma}$ from Eq. (24).

  3. 3.

For $Q^{2}_{s}\,r^{2}\,\,\,\gg\,\,1$ (saturation region), we propose to use
the solution to the non-linear equation of Eq. (43).

For $\tau\,<\,1$ we have to solve the linear equation with some initial
condition which we need to take from the experimental data. This solution will
give us the scattering amplitude in the vicinity of the saturation scale where
the scattering amplitude can be estimated using Eq. (51). We use a different
way to introduce the parameters from the experimental data: we expand Eq. (51)
to the region $\tau\,<\,1$ , replacing $\bar{\gamma}$ by following expression:

$\bar{\gamma}\,\,\longrightarrow\,\,\bar{\gamma}\,\,\,+\,\,\frac{\ln\left(1/\tau\right)}{\kappa\,\lambda\,Y};~{}~{}~{}\mbox{with}\,\,\kappa\,\,=\,\,\frac{\frac{d^{2}\omega\left(\bar{\gamma}_{\eta}\right)}{d\bar{\gamma}_{\eta}^{2}}}{\frac{d\omega\left(\bar{\gamma}_{\eta}\right)}{d\bar{\gamma}_{\eta}}}$
(52)

Eq. (52) is derived in Ref. IIML  and the experience with the saturation
models SATMOD0 ; SATMOD1 ; SATMOD2 ; IIM ; SATMOD3 ; SATMOD4 ; SATMOD5 ;
SATMOD6 ; SATMOD7 ; SATMOD8 ; SATMOD9 ; SATMOD10 ; SATMOD11 ; SATMOD12 ;
SATMOD13 ; SATMOD14 ; SATMOD15 ; SATMOD16 ; SATMOD17 ; CLP ; CLMP  shows that
it is described the experimental data for $x\,\leq\,0.1$ quite well. Hence, in
our approach the values of all phenomenological parameters of Eq. (51) (see
below) should be determined from the experimental data.

The mass of the $c$-quark (about $m_{c}$ = 1.4 GeV) is not small and we took
this into account replacing $x$ in the $c\bar{c}$ scattering by

$x^{c}\,\,=\,\,\left(1\,\,+\,\,\frac{4\,m^{2}_{c}}{Q^{2}}\right)x$ (53)

For $\tau\,\,=\,\,Q^{2}_{s}\,r^{2}\,\,\,\gg\,\,1$ we need to use the solution
of Eq. (43) to the non-linear evolution equation. However, it has been found
in Ref. LEPP  that the following formula:

$N\left(z\right)\,\,=\,\,a\,\Big{(}1\,\,-\,\,\exp\left(-\Omega\left(z\right)\right)\Big{)}\,\,+\,\,\left(1\,-\,a\right)\frac{\Omega\left(z\right)}{1\,\,+\,\,\Omega\left(z\right)}$
(54)

with $\Omega\left(z\right)$ from Eq. (44) and with $a$ = 0.65 describes the
exact solution within accuracy less that 2.5 %. Therefore, we use Eq. (54) in
our attempts to describe the HERA data.

###  III.3 Phenomenological input

As has been mentioned above, we need to set the initial conditions at $Y=0$
for the linear evolution equation for $\tau\,\leq\,1$. In this section we wish
to clarify how we introduce the phenomenological parameters to describe this
conditions. The first one we have considered: $N_{0}$ which determines the
scattering amplitude at $\tau\,=\,1$. Two other parameters describe the
saturation momentum at $Y=0$: the value of $Q_{s}$ at $b$ = 0 and the
behaviour as a function of $b$. collecting everything that we have discussed
about $Q_{s}$ we use two expression for the saturation momentum:

$\displaystyle 1.~{}~{}Q^{(1)2}_{s}\left(Y,b\right)\,\,$ $\displaystyle=$
$\displaystyle\,\,Q^{(1)2}_{s}\left(Y=0,b=0\right)\,\,e^{-m\,b}\,\,e^{\lambda\,Y}\,\,=\,\,Q^{2}_{0}\,e^{-m\,b}\,e^{\lambda\,Y};$
(55a) $\displaystyle 2.~{}~{}Q^{(2)2}_{s}\left(Y,b\right)\,\,$
$\displaystyle=$
$\displaystyle\,\,Q^{(2)2}_{s}\left(Y=0,b=0\right)\,\,e^{-\frac{3}{4}\,{\cal
Z}}\,\,e^{\lambda\,Y}\,\,=\,\,Q^{2}_{0}\,\,e^{\lambda\,Y}\,e^{-\frac{3}{4}\,{\cal
Z}};$ (55b)

where ${\cal Z}$ and $\lambda$ are defined in Eq. (2) and Eq. (24),
respectively. The values of these parameters have to be found from fitting of
the experimental data. The experience with such fitting gives
$Q^{(1)2}_{0}\,=\,Q^{(2)2}_{0}=0.15\div 0.25\,GeV^{2}$. $m$ we can estimate,
assuming that $<b^{2}>\,=\,6/m^{2}\,=\,R^{2}$ where $R$ is the electro-
magnetic radius of the proton. Hence we expect $m\,\approx 0.55\,GeV$ but we
have to remember that $m$ characterizes the distributions of the gluons in the
proton which can be quite different from the quarks222Actually, the above
estimates should be made for the scattering amplitude
$N=N_{0}(r^{2}Q^{2}_{s})^{\bar{\gamma}}$, which leads to
$m\approx\,\,0.34\,GeV$: with this estimates we need to compare the value of
$m$ from the Table I. ${\alpha^{\prime}}^{2}_{\rm eff}$ in ${\cal Z}$ has been
evaluated in Ref. LEPION  with ${\alpha^{\prime}}^{2}_{\rm eff}\approx
0.1\,GeV^{-2}$ for $\bar{\alpha}_{S}=0.2$.

Finally, we introduce three phenomenological parameters from the initial
conditions whose values have to be found from the fit of the experimental
data. The masses of quarks, that determine the wave function of the virtual
photon, determine the infra-red behaviour of the wave function. We take two
sets of them: the current masses and the masses of light quarks are equal to
140 MeV which is the typical infra-red cutoff in our approach.

##  IV Results of the fits

Using the approach, that has been discussed in the previous section, we
attempt to describe the most accurate data for the deep inelastic structure
function $F_{2}$ HERA1 . The implicit parameters of our approach are the
restrictions of the kinematic region of the experimental data that we include
in the fit. We chose: $0.85\,GeV^{2}\leq Q^{2}\leq 27\,GeV^{2}$ and
$x\,\leq\,0.01$. The lower limit of $Q^{2}$ stems from non-perturbative
correction to the wave function of the virtual photon, while the upper limit
originates from two restrictions: $x\,\leq\,0.01$ and contribution of the
additional term in $\bar{\gamma}$ of Eq. (52) is small. The choice of the
largest $x$ is dictated by the needs to have low $x$ for legitimate use of our
theoretical formulae and the practical wish to use in the fit as more data as
possible.

| Dipole amplitude | Wave function | $\chi^{2}/d.o.f.$
---|---|---|---
Set | $\bar{\alpha}_{S}$ | $N_{0}$ | $Q^{2}_{0}(GeV^{2})$ | m(GeV) | $\alpha_{eff}(GeV^{-2})$ |  $m_{u}$(MeV) |  $m_{d}$(MeV) |  $m_{s}$(MeV) |  $m_{c}$(GeV) |  0.85 $\leq Q^{2}\leq$ 27 $GeV^{2}$
1 | 0.091 | 0.236 | 0.998 | 0.612 | n/a | 140 | 140 | 140 | 1.4 | 124.9/133 = 0.93
2 | 0.20 | 0.25 | 1.00 | 0.551 | n/a | 140 | 140 | 140 | 1.4 | 61.99/66 =0.93
3 | 0.096 | 0.448 | 0.921 | 0.840 | n/a | 2.3 | 4.8 | 95 | 1.4 | 117.2/133 = 0.88
4 | 0.20 | 0.343 | 0.999 | 1.300 | n/a | 2.3 | 4.8 | 95 | 1.4 | 91.74/66= 1.39
5 | 0.038 | 0.599 | 1.284 | n/a | 0.216 | 140 | 140 | 140 | 1.4 | 175/133 = 1.31
6 | 0.043 | 0.565 | 1.429 | n/a | 0.149 | 2.3 | 4.8 | 95 | 1.4 | 143.7/133=1.08
Table 1: Parameters of the model. $\bar{\alpha}_{S}$, $N_{0}$,
$m(\alpha_{eff})$ and $Q^{2}_{0}$ are fitted parameters. Two sets of quark
masses are chosen: the current masses (sets 3 and 4) and the masses of light
quarks are equal to $140\,MeV$ (sets 1 and 2), which is the typical infra-red
cutoff in our approach. For the sets 2 and 4 the value of
$\bar{\alpha}_{S}=0.2$ is fixed and only three parameters were chosen from the
fit. Figure 3: Comparison with the experimental data of Refs. HERA1 ; HERA2
of the sets shown in table I. The experimental data as well as the theoretical
estimates are multiplied by factor $2^{i}$ and the values of $i$ are specified
in the figures. In the panel with sets 2 and 4 we use black squares to mark
the experimental data that have been used in the fit.

In Table 1 we show the values of parameters from our fits. It should be noted
that we did not fit the mass of the quarks but use two sets of them: the
masses of light quarks are equal to $m=\,140\,MeV$, which we consider as the
typical infra-red cutoff in our approach (sets 1 and 2); and the current
masses (sets 3 and 4). We used the data for the inclusive $F_{2}$ in the
kinematic region: $0.85\,GeV^{2}\leq Q^{2}\leq 27\,GeV^{2}$ and
$x\,\leq\,0.01$, for finding the parameters while $F^{cc}_{2}$ as well as
$F_{2}$ outside of this kinematic region (for small values of
$Q^{2}\,\leq\,0.85\,GeV^{2}$ or $Q^{2}\,\geq\,27\,GeV^{2}$), are compared with
the experimental data, to demonstrate our ability to describe the experimental
data. The quality of the fit one can see from Fig. 3. In Fig. 4 we show the
data on $F_{2}$ as function of $Q^{2}$ at different values of $x$. This figure
illustrates that we can describe the data for $x\,\leq 0.013$. In addition, we
show in Fig. 3 $F_{2}$ for $27\,\leq\,Q^{2}\,\leq 60\,GeV^{2}$. The quality of
the fit is not very good due to the small experimental errors. Indeed,
$\chi^{2}/d.o.f.$ for the range of $Q^{2}$ from $0.85\,GeV^{2}$ to
$60\,GeV^{2}$ is equal to 2.16 (set 1), 1.93 (set 3) and 3.19 (set 5). The
reason for this stems from the approximate character of Eq. (52). The
contributions of the nonlinear corrections are negligible for such large
photon virtualities and, therefore, our approach coincides with the one of
Ref.DIMST  (see also Ref.IMMST ), which describes the data very well. We
believe, that we demonstrated that our approach can describe the experimental
data at rather small values of $Q^{2}$ where the non-linear corrections give
an essential contribution. For large $Q^{2}$ our approach coincides with the
one of Ref.DIMST  which takes into account the linear evolution in much better
way than weIMMST .

Fig. 5 demonstrates how our fits describe the experimental data on
$F^{cc}_{2}$ while Fig. 6 shows that our fit is able to reproduce the data at
low values of $Q^{2}$. One can see that the agreement with the experimental
data is good even at low $Q^{2}$. It should be stressed that two sets of the
light quark mass give the same description illustrating a possibility to use
the wave function of the virtual photon in perturbative QCD at rather low
values of $Q^{2}$.

In addition to $F_{2}$ and $F^{CC}_{2}$ we compare our approach with the
experimental data on $F_{L}$HERAFL1 ; HERAFL2 . In spite of the fact that the
data have sufficiently large errors one can see from Fig. 7 that we are able
to describe the data quite well.

Sets 2 and 4 need special comments. In these sets we choose
$\bar{\alpha}_{S}=0.2$ to illustrate the ability of our approach to overcome
the difficulties of next-to-leading approaches in describing the experimental
data: the small value of needed $\bar{\alpha}_{S}$. One can see, that we
obtain a good $\chi^{2}/d.o.f.$ but for the specific set of data in the wide
kinematic region $0.85\leq\,Q^{2}\,\leq\,60\,GeV^{2}$ but choosing a
restricted selection of data. Note that these data are marked by the squares
in the panel of Fig. 3 with sets 2 and 4.

The general characteristics of our fits are: (1) the possibility to describe
the data at rather small values of $Q^{2}$, where the non-linear corrections
turn out to be essential, (2) the large value of $Q^{2}_{0}=1\,GeV^{2}$; (3)
rather large value of $m$ in comparison with the estimates from the electro-
magnetic radius of the proton (see above in footnote 2); (4) the possibility
to fit the experimental data, taking into account the Gribov’s diffusion for
impact parameter dependence of the saturation scale (see panel with sets 5 and
6 in Fig. 3); and (5) an ability to describe the data at rather large values
of $\bar{\alpha}_{S}$.

Figure 4: $F_{2}$ versus $Q^{2}$ at fixed $x$ for the set 2 (left panel) and
set 4 (right panel).

Unfortunately, in spite of large numbers of the papers in which the
experimental data have been compared with the saturation models SATMOD0 ;
SATMOD1 ; SATMOD2 ; IIM ; SATMOD3 ; SATMOD4 ; SATMOD5 ; SATMOD6 ; SATMOD7 ;
SATMOD8 ; SATMOD9 ; SATMOD10 ; SATMOD11 ; SATMOD12 ; SATMOD13 ; SATMOD14 ;
SATMOD15 ; SATMOD16 ; SATMOD17 ; CLP ; CLMP  we can confront our approach only
with Refs. CLMP ; CLP  since in all other papers the assumptions were made
that contradict the theoretical informations. For example, in the most models
$Q^{2}_{s}$ is proportional to $\exp\left(-b^{2}/B\right)$ while this
behaviour disagrees with the Froissart theorem FROI ; or/and the behaviour for
$r^{2}Q^{2}_{s}\,\,\gg\,\,1$ contradicts the approach of Ref. LETU . Comparing
our parameters with Ref. CLMP  one can see that the typical value of
$\bar{\alpha}_{S}\approx 0.14-0.15$ in Ref. CLMP  is less than in sets 2 and
4. The value of $Q^{2}_{0}$ is about in three times larger than in this paper.
The value of $m$ in Ref. CLMP  is the same as in the electro-magnetic form
factor of proton, while in this paper $m$ is larger, than in the electro-
magnetic form factor of proton in agreement with the new experimental
information on the behaviour of the two gluon form factors KHAR ; MAZA .
Taking into account that $\chi^{2}/d.o.f.$ are better in this paper than in
Ref. CLMP ; CLP , we believe that the theoretical approach of section II, give
reliable description of the current experimental data on DIS.

Figure 5: Comparison our fits with the experimental data on $F^{cc}_{2}$ (see
Refs. HERA1 ; HERA2 ) of the sets shown in the table I. The experimental data
as well as the theoretical estimates are multiplied by factor $2^{i}$ and the
values of $i$ are specified in the figures.

We wish to draw your attention to the sets 5 and 6, which take into account
the different dependence of the saturation momentum on $b$ (see Eq. (55b)).
From Fig. 3 and Fig. 5 one can see that we are able to describe the
experimental data. Hence we see that the suggested dependence on rapidity of
the impact parameter distribution is in accord with the experimental data.

Figure 6: $F_{2}$ versus $x$ at low $Q^{2}$. Red line for set 2 while the blue
one for set 4.

##  V Conclusions

In this paper we compare with the experimental data the NLO approach of Ref.
CLMS . In this approach we include the re-summation procedure, suggested in
Refs. SALAM ; SALAM1 ; SALAM2 ; DIMST , to fix the BFKL kernel in the NL0, but
we treat differently the non-linear corrections. The advantage of our
treatment is that we reproduce the correct asymptotic behaviour at large
$\tau=r^{2}Q^{2}_{s}$.

Fixing our phenomenological parameters from the fit to the experimental data
on $F_{2}$ DIS structure function in the region: $0.85\leq Q^{2}\leq
27\,GeV^{2}$ and $x\leq 0.013$, we found that can describe these data with
very good $\chi^{2}/d.o.f.$. It should be stressed, that for the first time it
is demonstrated that the impact parameter dependence of the saturation scale,
which is given by Eq. (55b), is an agreement with the data. We believe that
this is an interesting result since Eq. (55b) follows from the first attempt
to take into account both the diffusion on $\ln(k_{T})$, which stems from
perturbative QCD, and the Gribov’s diffusion in $b$ GOLEB .

Figure 7:  Comparison our fits with the experimental data on $F_{L}$ (see Ref.
HERAFL2 ) of the sets 1 and 5. All other sets lead to the curves which are
very close to these two. It is shown the data, extracted from the
unconstrained fit (see Ref.HERAFL2  for explanation). Note that the data for
constrained fit give $F_{L}\,\geq\,0$. All points with negative values of
$F_{L}$ in the constrained fits coincide with 0.

Using the phenomenological parameters, that we found from the fit of $F_{2}$,
we demonstrated that our approach is able to describe the experimental data
both on $F^{cc}_{2}$ and on $F_{2}$ at low values of $Q^{2}=0.4-0.7\,GeV^{2}$.
We also reproduce the experimental data on $F_{L}$.

It should be pointed out that our description in the NLO is better or of the
same quality as the description by LO with more fitting parameters. Two
problems, that we faced in describing the data in the NLO approach, have been
partly healed: we are able to fit the data with not small value of
$\bar{\alpha}_{S}=0.2$ and the value of the saturation momentum at $Y=0$ is
about $Q^{2}_{0}\,\approx\,1\,GeV^{2}$ instead of $3\,GeV^{2}$ of Ref. CLMP .
However, we are aware that $Q_{0}=1\,GeV$ is rather large (see for example
Ref. DKLN ).

Concluding we wish to mentioned that the model that we developed here, is the
only one which includes (i) the re-summed NLO corrections; (2) the correct
theoretical behaviour at large $\tau$; (3) the large $b$ dependence of the
saturation scale in accord with the Froissart theorem as well as the non-
linear evolution. We believe that it will be useful for further discussion of
the high energy scattering in QCD.

##  VI Acknowledgements

We thank our colleagues at Tel Aviv university and UTFSM for encouraging
discussions. Our special thanks go to E. Gotsman for all his remarks and
suggestions on this paper, and to Yuri Ivanov for technical support of the USM
HPC cluster. M.S. thanks M. Arriagada and C. del Valle from Universidad de la
Frontera, and S. Nauto and J. Vidal from Universidad de Playa Ancha, as well
as M. Siddikov from UTFSM for their valuable advices on programming.

This research was supported by ANID PIA/APOYO AFB180002 (Chile), Fondecyt
(Chile) grants 1180118 and 1191434, Conicyt Becas (Chile) and PIIC 009/2021,
DPP, Universidad Técnica Federico Santa María.

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