-1cm

 

\

[**Abstract**]{}

The paper presents the $QCD$ description of the hard and semihard processes in the framework of the Wilson operator product expansion. The smooth transition between the cases of the soft and hard Pomerons is obtained.

The recent measurements of the deep-inelastic (DIS) structure function (SF) $F_2$ by the $H1$ [@1] and $ZEUS$ [@2] collaborations open a new kinematical range to study proton structure. The new $HERA$ data show the strong increase of $F_2$ with decresing $x$. However, the data of the $NMC$ [@3] and $E665$ collaboration [@4] at small $x$ and smaller $Q^2$ is in the good agreement with the standard Pomeron or with the Donnachie-Landshoff picture where the Pomeron intercept: $\alpha_p = 1.08$, is very close to standard one. The interpritation of the fast changing of the intercept in the region of $Q^2$ between $Q^2=1GeV^2$ and $Q^2=10GeV^2$ (see Fig.3 in [@5]) is yet absent. There are the arguments in favour of that is one intercept (see [@6]) or the superposition of two different Pomeron trajectories, one having an intercept of $1.08$ and the one of $1.5$ (see Fig.4 in [@5]).

The aim of this article is the possible “solution” of this problem in the framework of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [@6.5]. It is good known (see, for example, [@7]), that in the double-logarithmical approximation the DGLAP equation solution is the Bessel function, or $\exp{\sqrt{\phi (Q^2) ln(1/x)}}$, where $\phi (Q^2)$ is known $Q^2$-dependent function[^1]. However, we will seek the “solution”[^2] of DGLAP equation in the Regge form (we use the parton distributions (PD) multiplied by $x$ and neglect the nonsinglet quark distribution at small $x$): $$\begin{aligned}
f_a (x,Q^2) \sim x^{-\delta} \tilde
f_a(x,Q^2), ~~~~(a=q,g)~~(\alpha_p \equiv 1+\delta ) \label{1}
 \end{aligned}$$ where $\tilde f_a(x,Q^2)$ is nonsingular at $x \to 0$ and $\tilde
f_a(x,Q^2) \sim (1-x)^{\nu}$ at $x \to 1$[^3]. The similar investigations were already done and the results are good known (see [@8], [@11]-[@15])[^4]. The aim of this letter is to expand these results to the range where $\delta \sim 0$ (and $Q^2$ is not large) following to the observed early (see [@12; @13])[^5] method to replace the Mellin convolution by a simple product. Of course, we understand that the Regge behaviour (\[1\]) is not in the agreement with the double-logarithmic solution, however the range, where $\delta \sim 0$ and the $Q^2$ values are nonlarge, is really the Regge regime and a “solution” of DGLAP equation in the form of (\[1\]) would be worthwhile. This “solution” may be understand as the solution of DGLAP equation together with the condition of its Regge asymptotic at $x \to 0$.

Consider DGLAP equation and apply the method from [@13] to the Mellin convolution in its r.h.s. (in contrast with standard case, we use below $\alpha(Q^2)=\alpha_s(Q^2)/(4\pi)$): $$\begin{aligned}
{&&\hspace*{-1cm}}\frac{d}{dt}f_a (x,t)~=~- \frac{1}{2} \sum_{i=a,b}
\hat \gamma_{ai}(\alpha,x) \otimes f_a(x,t)~~~(a,b)=(q,g) \nonumber \\
{&&\hspace*{-1cm}}=~- \frac{1}{2} \sum_{i=a,b}
\tilde \gamma_{ai}(\alpha,1+\delta)f_a(x,t)~+~O(x^{1-\delta})~~~
\Bigl( \gamma_{ab}(\alpha,n)=\alpha \gamma_{ab}^{(0)}(n)+\alpha^2
\gamma_{ab}^{(1)}(n)+...\Bigr)
, \label{2}
 \end{aligned}$$ where $t=ln(Q^2/\Lambda ^2)$. The $\hat \gamma_{ab}(\alpha,x)$ are the spliting functions corresponding to the anomalous dimensions (AD) $\gamma_{ab}(\alpha,n) = \int^1_0 dx x^{n-1} \hat \gamma_{ab}(\alpha,x)$. Here the functions $\gamma_{ab}(\alpha,1+\delta)$ are the AD $\gamma_{ab}(\alpha,n)$ expanded from the integer argument “$n$” to the noninteger one “$1+\delta$”. The functions $\tilde \gamma_{ab}(\alpha,1+\delta)$ (marked lower as AD, too) can be obtained from the functions $\gamma_{ab}(\alpha,1+\delta)$ replacing the term $1/\delta$ by the one $1/\tilde \delta$:

$$\begin{aligned}
\frac{1}{\delta} \to \frac{1}{\tilde \delta}~=~\frac{1}{\delta}
\Bigl( 1 - \varphi(x,\delta)x^{\delta} \Bigr)
 \label{3}
 \end{aligned}$$

This replacement (\[3\]) is appeared very naturally from the consideration the Mellin convolution at $x \to
0$ (see [@13]) and preserves the smooth and nonsingular transition to the case $\delta =0$, where

$$\begin{aligned}
 \frac{1}{\tilde \delta}~=~ln\frac{1}{x} - \varrho(x)
 \label{4}
 \end{aligned}$$

The concrete form of the functions $\varphi(x,\delta)$ and $\varrho(x)$ depends strongly on the type of the behaviour of the PD $f_a(x,Q^2)$ at $x \to 0$ and in the case of the Regge regime (\[1\]) they are (see [@12; @13]):

$$\begin{aligned}
\varphi(x,\delta)~=~ \frac{\Gamma(\nu +1)\Gamma(1-\delta)}{\Gamma(\nu
  +1-\delta)}~~ \mbox{ and }~~ \varrho(x)~=~\Psi(\nu+1)-\Psi(1),
 \label{5}
 \end{aligned}$$

where $\Gamma(\nu+1)$ and $\Psi(\nu+1)$ are the Eulerian $\Gamma$- and $\Psi$-functions, respectively. As it can be seen, there is the correlation with the PD behaviour at large $x$.

If $\delta$ is not small (i.e. $x^{\delta}>>1$), we can replace $1/ \tilde \delta$ to $1/\delta$ in the r.h.s. of Eq.(\[2\]) and obtain its solution in the form (hereafter $t_0=t(Q^2=Q^2_0)$):

$$\begin{aligned}
 \frac{f_a(x,t)}{f_a(x,t_0)}~=~ \frac{M_a(1+\delta,t)}{M_a(1+\delta,t_0)},
 \label{6}
 \end{aligned}$$

where $M_a(1+\delta,t)$ is the analytical expansion of the PD moments $M_a(n,t) = \int^1_0 dx x^{n-1} f_a(x,t)$ to the noninteger value “$n=1+\delta$”.

This solution is good known one (see [@12] for the first two orders of the perturbation theory, [@14] for the first three orders and [@15] containing a resummation of all orders, respectively). Note that recently the fit of $HERA$ data was done in [@17] with the formula for PD $f_q(x,t)$ very close [^6] to (\[6\]) and the very well agreement (the $\chi^2$ per degree of freedom is $0.85$) is found at $\delta = 0.40 \pm 0.03$. There are also the fits [@17.5] of the another group using equations which are similar to (\[6\]) in the LO approximation.

The news in our investigations are in the follows. Note that the $Q^2$-evolution of $M_a(1+\delta,t)$ contains the two: “+” and “$-$” components, i.e. $M_a(1+\delta,t)= \sum_{i= \pm} M_a^i(1+\delta,t)$, and in principle the every component evolves separately and may have the independent (and not equal) intercept. Here for the simplicity we restricte ourselves to the LO analysis and give NLO formulae lower without large intermediate equations.

[**1.**]{} Consider DGLAP equation for the “+” and “$-$” parts (hereafter $s=ln(lnt/lnt_0)$):

$$\begin{aligned}
 \frac{d}{ds} f_a^{\pm}(x,t)~=~- \frac{1}{2\beta_0}
\tilde
\gamma_{\pm}(\alpha,1+\delta_{\pm})f_a^{\pm}(x,t)~+~O(x^{1-\delta}),
 \label{7} \end{aligned}$$

where $$\gamma_{\pm}~=~ \frac{1}{2}
\biggl[
\Bigl(\gamma_{gg}+\gamma_{qq} \Bigr)~\pm ~
\sqrt{ {\Bigl( \gamma_{gg}- \gamma_{qq} \Bigr)}^2
  ~+~4\gamma_{qg}\gamma_{gq}}
\biggr]$$ are the AD of the “$\pm $” components (see, for example, [@18])

The “$-$” component $\tilde \gamma_{-}(\alpha,1+\delta_-)$ does not contain the singular term (see [@12; @14] and lower) and its solution have the form:

$$\begin{aligned}
 \frac{f_a^-(x,t)}{f_a^-(x,t_0)}~=~e^{-d_-(1+\delta_-)s}, \mbox{ where }
d_{\pm}=\frac{\gamma_{\pm}(1+\delta_{\pm})}{2\beta_0}
 \label{8}
 \end{aligned}$$

The “+” component $\tilde \gamma_{+}(\alpha,1+\delta_+)$ contains the singular term and $f_a^+(x,t)$ have the solution similar (\[8\]) only for $x^{\delta_+}>>1$:

$$\begin{aligned}
 \frac{f_a^+(x,t)}{f_a^+(x,t_0)}~=~e^{-d_+(1+\delta_+)s}, \mbox{ if }
 x^{\delta_+}>>1
 \label{9}
 \end{aligned}$$

The both intersepts $1+\delta_+$ and $1+\delta_-$ are unknown and should be found, in principle, from the analysis of the experimental data. However there is the another way. From the small $Q^2$ (and small $x$) data of the $NMC$ [@3] and $E665$ collaboration [@4] we can conclude that the SF $F_2$ and hence the PD $f_a(x,Q^2)$ have the flat asymptotics for $x \to 0$ and $Q^2 \sim (1\div2)GeV^2$. Thus we know that the values of $\delta_+$ and $\delta_-$ is approximately zero at $Q^2 \sim 1GeV^2$.

Consider Eqs.(\[7\]) with $\delta_{\pm}=0$ and with the boundary condition $f_a(x,Q^2_0)=A_a$ at $Q^2_0=1GeV^2$. For the “$-$” component we already have the solution: the Eq.(\[8\]) with $\delta_-=0$ and $d_-(1)=16f/(27\beta_0$), where $f$ is the number of the active quarks and $\beta_i$ are the coefficients in the $\alpha$-expansion of QCD $\beta$-function. For its “+” component Eq.(\[7\]) can be rewritten in the form (hereafter the index $1+\delta $ will be omitted in the case $\delta \to 0$):

$$\begin{aligned}
 ln(\frac{1}{x})\frac{d}{ds}\delta_+(s)~+~
 \frac{d}{ds} ln(A_a^+) ~=~- \frac{1}{2\beta_0}
\biggr[ \hat
\gamma_{+}
\Bigl( ln(\frac{1}{x}) -\varrho(\nu)  \Bigr) ~+~ \overline \gamma_+
\biggl]
 \label{10} \end{aligned}$$

where $\hat\gamma_{+}$ and $\overline \gamma_+$ are the coefficients of the singular and regular parts at $\delta \to 0$ of AD $\gamma^+(1+\delta)$: $$\gamma^+(1+\delta)~=~\hat\gamma^+ \frac{1}{\delta} ~+~
\overline\gamma^+,~~~\hat\gamma^+=-24,~\overline\gamma^+=22+
\frac{4f}{27}$$

The solution of Eq.([10]{}) is

$$\begin{aligned}
 f_a^+(x,t)~=~A^+_a~x^{\hat d_+s}e^{-\overline d_+s},
 \label{11}
 \end{aligned}$$

where $$\hat d_+ \equiv \frac{\hat \gamma^+}{2\beta_0} \simeq -
\frac{4}{3},~~
\overline d_+ \equiv \frac{1}{2\beta_0}
\Bigl( \overline \gamma_+  ~-~ \hat \gamma_+ \varrho(\nu)\Bigr)
 \simeq \frac{4}{3} \varrho(\nu) + \frac{101}{81}$$ Herefter the symbol $\simeq $ marks the case $f=3$.

As it can be seen from (\[11\]) the flat form $\delta_+=0$ of the “+”-component of PD is very nonstable from the (perturbative) viewpoint, because $d(\delta_+)/ds \neq 0$, and for $Q^2 > Q_0^2$ we have already the nonzero power of $x$ (i.e. pomeron intercept $\alpha_p >1$). This is in the agreement with the experimental data. Let us note that the power of x is positive for $Q^2<Q^2_0$ that is in principle also supported by the $NMC$ [@3] data, but the use of this analysis to $Q^2<1GeV^2$ is open the question.

Thus, we have the DGLAP equation solution for the “+” component at $Q^2$ is close to $Q^2_0=1GeV^2$, where Pomeron starts in its movement to the subcritical (or Lipatov [@19.5; @19.6]) regime and also for the large $Q^2$, where pomeron have the $Q^2$-independent intercept. In principle, the general solution of (\[7\]) should contain the smooth transition between these pictures but this solution is absent [^7]. We introduce the some “critical” value of $Q^2$: $Q^2_c$, where the solution (\[9\]) is replaced by the solution (\[11\]). The exact value of $Q^2_c$ may be obtained from the fit of experimental data. Thus, we have in the LO of the perturbation theory:

$$\begin{aligned}
{&&\hspace*{-1cm}}f_a(x,t)~=~ f_a^-(x,t)~+~ f_a^+(x,t) \nonumber  \\
{&&\hspace*{-1cm}}f_a^-(x,t)~=~A^-_a~\exp{(- d_-s)} \nonumber  \\
{&&\hspace*{-1cm}}f_a^+(x,t)~=~
\left\{
\begin{array}{ll} A^+_a
x^{\hat d_+s}\exp{(-\overline d_+s)}, & \mbox{ if } Q^2 \leq Q^2_c \\
f_a^+(x,t_c)
\exp{\Bigl(-d_+(1+\delta_c)(s-s_c)\Bigr)},
& \mbox{ if } Q^2>Q^2_c
\end{array} \right.
 \label{12}
 \end{aligned}$$

where $$\begin{aligned}
{&&\hspace*{-1cm}}t_c~=~t(Q^2_c),~~s_c~=~s(Q^2_c) \nonumber  \\
{&&\hspace*{-1cm}}A^+_q~=~(1- \overline \alpha )A_q ~+~ \tilde \alpha A_g,~~
A^+_g~=~ \overline \alpha A_g ~-~ \varepsilon A_q \nonumber \\
{&&\hspace*{-1cm}}\mbox{and } A_a^-~=~A_a ~-~ A_a^+
 \label{13}
 \end{aligned}$$ and the values of the coefficients $\overline \alpha$, $\tilde \alpha$ and $\varepsilon$ may be found, for example, in [@18].

Using the concrete AD values at $\delta =0$ and $f=3$, we have

$$\begin{aligned}
 {&&\hspace*{-1cm}}A^+_q~ \approx ~\frac{1}{27}\frac{4A_q+9A_g}{ln(\frac{1}{x})-\varrho
  (\nu) - \frac{85}{108}}  \nonumber \\
{&&\hspace*{-1cm}}A^+_g~ \approx ~A_g~+~\frac{4}{9}A_q ~-~
\frac{4}{27}\frac{9A_g-A_q}{ln(\frac{1}{x})-\varrho
  (\nu) - \frac{85}{108}}
 \label{14}
 \end{aligned}$$

Thus, the value of the “+”component of the quark PD is suppressed logarithmically that is in the qualitative agreement with the $HERA$ parametrizations of SF $F_2$ (see [@20.5; @21]) (in the LO $F_2(x,Q^2)~=~(2/9)f_q(x,Q^2)$ for $f=3$), where the magnitude connected with the factor $x^{-\delta}$ is $5 \div 10 \%$ from the flat (for $x \to
                                        0$) magnitude.

[**2.**]{} By analogy with the subsection [**1**]{} and knowing the NLO $Q^2$-dependence of PD moments, we obtain the following equations for the NLO $Q^2$-evolution of the both: ”+” and “$-$” PD components (hereafter $\tilde s=ln(\alpha(Q^2_0)/\alpha(Q^2)), p=\alpha(Q^2_0)-\alpha(Q^2)$):

$$\begin{aligned}
{&&\hspace*{-1cm}}f_a(x,t)~=~ f_a^-(x,t)~+~ f_a^+(x,t) \nonumber  \\
{&&\hspace*{-1cm}}f_a^-(x,t)=~\tilde A^-_a~\exp{(- d_-\tilde s -d_{--}^ap)}
 \nonumber  \\
{&&\hspace*{-1cm}}f_a^+(x,t)=
\left\{
\begin{array}{ll} \tilde A^+_a
x^{(\hat d_+\tilde s + \hat d_{++}^a p)}\exp{(-\overline d_+\tilde s
  -\overline d_{++}^ap)}, & \mbox{if } Q^2 \leq Q^2_c \\
f_a^+(x,t_c)
\exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde
  s_c)-d_{++}^a(1+\delta_c)(p-p_c) \Bigr) },
& \mbox{if } Q^2>Q^2_c
\end{array} \right.
 \label{15}
 \end{aligned}$$

where $$\begin{aligned}
 {&&\hspace*{-1cm}}\tilde s_c ~=~ \tilde s(Q^2_c),~
p_c~=~p(Q^2_c),~\alpha_0~=~\alpha(Q^2_0)
,~\alpha_c~=~\alpha(Q^2_c) \nonumber \\ {&&\hspace*{-1cm}}\tilde A^{\pm}_a~=~\Bigl(1~-~\alpha_0 K^a_{\pm} \Bigr)
A^{\pm}_a ~+~ \alpha_0 K^a_{\pm} A^{\mp}_a \nonumber \\  {&&\hspace*{-1cm}}d_{++}^a~=~ \hat d_{++}^a
\Bigl( ln(\frac{1}{x}) - \varrho(\nu) \Bigl) ~+~ \overline d_{++}^a, ~~
d^a_{++}~=~ \frac{\gamma_{\pm \pm}}{2\beta_0} ~-~
\frac{\gamma_{\pm} \beta_1}{2\beta^2_0} ~-~ K^a_{\pm}  \nonumber \\
{&&\hspace*{-1cm}}\mbox{and }~~ K^q_{\pm}~=~ \frac{\gamma_{\pm \mp}}{2\beta_0 +
  \gamma_{\pm} - \gamma_{\mp}},~~ K^g_{\pm}~=~ K^q_{\pm}
\frac{\gamma_{\pm}-\gamma^{(0)}_{qq}}{\gamma_{\mp}-\gamma^{(0)}_{qq}}
 \label{16}
 \end{aligned}$$

The NLO AD of the “$\pm$” components are connected with the NLO AD $\gamma^{(1)}_{ab}$. The corresponding formulae can be found in [@18].

Using the concrete values of the LO and NLO AD at $\delta =0$ and $f=3$, we obtain the following values for the NLO components from (\[15\]),(\[16\]) (note that we remail only the terms $\sim O(1)$ in the NLO terms)

$$\begin{aligned}
 {&&\hspace*{-1cm}}d^q_{--}~=~ \frac{16}{81} \Big[ 2\zeta (3) + 9 \zeta (2) -
\frac{779}{108}  \Big] \approx 1.97, ~~
d^g_{--}~=~ d^q_{--}~+~ \frac{28}{81} \approx 2.32 \nonumber \\ {&&\hspace*{-1cm}}\hat d^q_{++}~=~ \frac{2800}{81} , ~~
\overline d^q_{++}~=~ 32 \Big[ \zeta (3) + \frac{263}{216}\zeta (2) -
\frac{372607}{69984}  \Big] \approx -67.82    \nonumber \\ {&&\hspace*{-1cm}}\hat d^g_{++}~=~ \frac{1180}{81} , ~~
\overline d^g_{++}~=~ \overline d^q_{++}~+~ \frac{953}{27} -12\zeta
(2) \approx -52.26
 \label{17}
 \end{aligned}$$

and

$$\begin{aligned}
 {&&\hspace*{-1cm}}\tilde A^+_q~ \simeq ~\frac{20}{3} \alpha_0
\Bigl[ A_g + \frac{4}{9} A_q \Bigr] ~+~
\frac{1}{27}\frac{4A_q(1-7.67 \alpha_0)+9A_g(1-8.71
  \alpha_0)}{ln(\frac{1}{x})-\varrho
  (\nu) - \frac{85}{108}}  \nonumber \\
{&&\hspace*{-1cm}}\tilde A^+_g~ \simeq ~ \Bigl(A_g ~+~\frac{4}{9}A_q \Bigr)
 \Bigl(1-\frac{80}{9}\alpha_0 \Bigr)
 ~-~
\frac{4}{27}\frac{9A_g-A_q}{ln(\frac{1}{x})-\varrho
  (\nu) - \frac{85}{108}} \Bigl( 1+ \frac{692}{81}\alpha_0)
 \nonumber \\
{&&\hspace*{-1cm}}\mbox{and } \tilde A_a^-~=~A_a ~-~ \tilde A_a^+
 \label{18}
 \end{aligned}$$

It is useful to change in Eqs.(\[15\])-(\[18\]) from the quark PD to the SF $F_2(x,Q^2)$, which is connected in NLO approximation with the PD by the following way (see [@18]):

$$\begin{aligned}
F_2(x,Q^2)~=~ \Bigl(  1+\alpha(Q^2)B_q(1+\delta) \Bigr) \delta^2_s
f_q(x,Q^2) ~+~ \alpha(Q^2)B_g(1+\delta) \delta^2_s f_g(x,Q^2),
 \label{19}
 \end{aligned}$$

where $\delta ^2_s = \sum_{i=1}^f/f \equiv <e_f^2>$ is the average charge square of the active quarks: $\delta ^2_s$ = (2/9 and 5/18) for $f$ = (3 and 4), respectively. The NLO corrections lead to the appearence in the r.h.s. of Eqs.(\[15\]) of the additional terms $\Bigl(  1+\alpha B_{\pm} \Bigr)/\Bigl(  1+\alpha_0 B_{\pm} \Bigr)$ and the necessarity to transform $\tilde A^{\pm}_q$ to $C^{\pm} \equiv F_2^{\pm}(x,Q^2)$ into the input parts. The final results for $F_2(x,Q^2)$ are in the form:

$$\begin{aligned}
{&&\hspace*{-1cm}}F_2(x,t)~=~ F_2^-(x,t)~+~ F_2^+(x,t) \nonumber  \\
{&&\hspace*{-1cm}}F_2^-(x,t)~=~ C^-~\exp{(- d_-\tilde s -d_{--}^qp)}
(1+\alpha B^-)/(1+\alpha_0 B^-)
 \nonumber  \\
{&&\hspace*{-1cm}}F_2^+(x,t)~=~
\left\{
\begin{array}{ll} C^+
x^{(\hat d_+\tilde s + \hat d_{++}^q p)}\exp{(-\overline d_+\tilde s
  -\overline d_{++}^qp)}(1+\alpha B^+)/(1+\alpha_0 B^+)
, & \mbox{ if } Q^2 \leq Q^2_c \\
F_2^+(x,t_c)
\exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde
  s_c)-d_{++}^q(1+\delta_c)(p-p_c) \Bigr) } &  \\
\biggl(1+
\alpha B^+(1+\delta_c) \biggr)/
\biggl(1+
\alpha_c B^+(1+\delta_c) \biggr),
& \mbox{ if } Q^2>Q^2_c
\end{array} \right.
 \label{20}
 \end{aligned}$$

where $$B^{\pm}~=~B_q ~+~ \frac{\gamma_{\pm}}{\gamma^{(0)}_{qg}}B_g,~~
C^{\pm}~=~\tilde A^{\pm}_q (1+\alpha_0 B^{\pm})$$ with the substitution of $A_q$ by $C \equiv F_2(x,Q^2_0)$ into Eq.(\[18\]) $\tilde
A^{\pm}_q$ according

$$\begin{aligned}
 {&&\hspace*{-1cm}}C~=~ \Bigl(  1+\alpha_0 B_q \Bigr) \delta^2_s
A_q ~+~ \alpha_0 B_g \delta^2_s A_g,
 \label{21}
 \end{aligned}$$

For the gluon PD the situation is more simple: in Eq.(\[18\]) it is necessary to replace $A_q$ by $C$ according (\[21\]).

For the concrete values of the LO and NLO AD at $\delta =0$ and $f=3$, we have for $Q^2$-evolution of $F_2(x,Q^2)$ and the gluon PD:

$$\begin{aligned}
{&&\hspace*{-1cm}}F_2(x,t)~=~ F_2^-(x,t)~+~ F_2^+(x,t),~~
f_g(x,t)~=~ f_g^-(x,t)~+~ f_g^+(x,t) \nonumber  \\
{&&\hspace*{-1cm}}F_2^-(x,t)~=~ C^-~\exp{(- \frac{32}{81} \tilde s
  -1.97p)}(1-\frac{8}{9} \alpha )/(1-\frac{8}{9} \alpha_0 )
 \nonumber  \\
{&&\hspace*{-1cm}}F_2^+(x,t)~=~
\left\{
\begin{array}{ll} C^+
x^{(-\frac{4}{3} \tilde s + \frac{2800}{81}
  p)}
\exp{\Bigl(- \frac{4}{3}(\varrho(\nu)+\frac{101}{108}) \tilde s
  +(\frac{2800}{81} \varrho(\nu)-67.82)p \Bigr)}  &  \\
\Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha \Bigr)/
\Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha_0 \Bigr)
, & \mbox{if } Q^2 \leq Q^2_c \\
F_2^+(x,t_c)
\exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde
  s_c)-d_{++}^q(1+\delta_c)(p-p_c) \Bigr) }  &  \\
\biggl(1+
\alpha B^+(1+\delta_c) \biggr)/
\biggl(1+
 \alpha_c B^+(1+\delta_c) \biggr),
& \mbox{if } Q^2>Q^2_c
\end{array} \right.
 \label{22}  \\
{&&\hspace*{-1cm}}f_g^-(x,t)~=~ A_g^-~\exp{(- \frac{32}{81} \tilde s
  -2.32p)}(1-\frac{8}{9} \alpha )/(1-\frac{8}{9} \alpha_0 )
 \nonumber  \\
{&&\hspace*{-1cm}}f_g^+(x,t)~=~
\left\{
\begin{array}{ll} A_g^+
x^{(-\frac{4}{3} \tilde s + \frac{1180}{81}
  p)}\exp{\Bigl(- \frac{4}{3}(\varrho(\nu)+\frac{101}{108}) \tilde s
  +(\frac{1180}{81} \varrho(\nu)-52.26)p \Bigr)} &  \\
\Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha \Bigr)/
\Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha_0 \Bigr)
, & \mbox{if } Q^2 \leq Q^2_c \\
f_g^+(x,t_c)
\exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde
  s_c)+d_{++}^a(1+\delta_c)(p-p_c) \Bigr) } &   \\
\biggl(1+
\alpha B^+(1+\delta_c) \biggr)/
\biggl(1+
 \alpha_c B^+(1+\delta_c) \biggr),
& \mbox{ if } Q^2>Q^2_c
\end{array} \right.
 \label{23}
 \end{aligned}$$

where

$$\begin{aligned}
 {&&\hspace*{-1cm}}\tilde C^+~ \simeq ~\frac{2}{27}
\Biggl(  26\alpha_0
\Bigl[ A_g + 2C \Bigr] ~+~
\frac{A_g(1-9.74 \alpha_0)+2C(1-7.82
  \alpha_0)}{ln(\frac{1}{x})-\varrho
  (\nu) - \frac{85}{108}} \Biggr) \nonumber \\
{&&\hspace*{-1cm}}\mbox{and }  C^-~=~C
{}~-~ C^+
 \label{24}  \\
{&&\hspace*{-1cm}}\tilde A^+_g~ \simeq ~A_g \Bigl(1-\frac{28}{3}\alpha_0 \Bigr)
{}~+~2C ~-~
\frac{2}{27}\frac{2A_g(1+ \frac{590}{81}\alpha_0)-C(1+
\frac{572}{81}\alpha_0)}{ln(\frac{1}{x})-\varrho
  (\nu) - \frac{85}{108}}
 \nonumber \\
{&&\hspace*{-1cm}}\mbox{and } \tilde A_g^-~=~A_g ~-~ \tilde A_g^+
 \label{25}
 \end{aligned}$$

Let us give some conclusions following from Eqs.(\[24\])-(\[25\]). It is clearly seen that the NLO corrections reduce the LO contributions. Indeed, the value of the subcritical Pomeron intercept, which increases as $ln(\alpha_0/\alpha)$ in the LO, obtaines the additional term $ \sim (\alpha_0 - \alpha)$ with the large (and opposite in sign to the LO term) numerical coefficient. Note that this coefficient is different for the quark and gluon PD, that is in the agreement with the recent $MRS(G)$ fit in [@19] and the data analysis by $ZEUS$ group (see [@20]). The intercept of the gluon PD is larger then the quark PD one (see also [@19; @20]). However, the effective reduction of the quark PD is smaller (that is in the agreement with W.-K. Tung analysis in [@21]), because the quark PD part increasing at small $x$ obtains the additional ($ \sim
\alpha_0$ but not $ \sim 1/lnx $) term, which is important at very small $x$.

Note that there is the fourth quark threshold at $Q^2_{th} \sim 10
GeV^2$ and the $Q^2_{th}$ value may be larger or smaller to $Q^2_c$ one. Then, either the solution in the r.h.s. of Eqs. (\[20\],\[22\],\[23\]) before the critical point $Q^2_c$ and the one for $Q^2 > Q^2_c$ contain the threshold transition, where the values of all variables are changed from ones at $f=3$ to ones at $f=4$. The $\alpha(Q^2)$ is smooth because $\Lambda^{f=3}_{\overline{MS}} \to  \Lambda^{f=4}_{\overline{MS}}$ (see also the recent experimental test of the flavour independence of strong interactions into [@22]).

For simplicity here we suppose that $Q^2_{th} = Q^2_c$ and all changes initiated by threshold are done authomatically: the first (at $Q^2
\leq Q^2_c$) solutions contain $f=3$ and second (at $Q^2 > Q^2_c$) ones have $f=4$, respectively. For the “$-$” component we should use $Q^2_{th}=Q^2_c$, too.

Note only that the Pomeron intercept $\alpha_p = 1~-~(d_+ \tilde s +
\hat d^q_{++}p)$ increases at $Q^2=Q^2_{th}$, because

$$\alpha_p ~-~1 ~=~
\left\{
\begin{array}{ll}
\frac{4}{3} \tilde s(Q^2_{th},Q^2_0)~-~ \frac{2800}{81} p(Q^2_{th},Q^2_0)
 , & \mbox{ if } Q^2 \leq Q^2_c \\
1.44 \tilde s(Q^2_{th},Q^2_0)~-~ 38.11 p(Q^2_{th},Q^2_0)
,& \mbox{ if } Q^2>Q^2_c
\end{array} \right.$$ that agrees with results [@23] obtained in the framework of dual parton model. The difference $$\bigtriangleup \alpha_p ~=~ 0.11 \tilde s(Q^2_{th},Q^2_0) - 3.55
p(Q^2_{th},Q^2_0)$$ dependes from the values of $Q^2_{th}$ and $Q^2_0$. For $Q^2_{th}=10GeV^2$ and $Q^2_0=1GeV^2$ it is very small: $$\bigtriangleup \alpha_p ~=~ 0.012$$

[**3.**]{} Let us resume the obtained results. We have got the DGLAP equation “solution” having the Regge form (\[1\]) for the two cases: at small $Q^2$ ($Q^2 \sim 1GeV^2$), where SF and PD have the flat behaviour at small $x$, and at large $Q^2$, where SF $F_2(x,Q^2)$ fastly increases when $x \to 0$. The behaviour in the flat case is nonstable with the perturbative viewpoint because it leads to the production of the subcritical value of pomeron intercept at larger $Q^2$ and the its increase (like $4/3~ ln(\alpha (Q^2_0)/\alpha(Q^2)$ in LO) when the $Q^2$ value increases[^8]. The solution in the Lipatov Pomeron case corresponds to the well-known results (see [@12; @14; @17]) with $Q^2$-independent Pomeron intercept. The general “solution” should contains the smooth transition between these pictures. Unfortunately, it is impossible to obtain it in the case of the simple approximation (\[1\]), because the r.h.s. of DGLAP equation (\[7\]) contains the both: $\sim x^{-\delta}$ and $\sim
Const$, terms. As a result, we used two above “solutions” gluing in some point $Q^2_c$.

Note that our “solution” is some generation (or a application) of the solution of DGLAP equation in the momentum space. The last one have two: ”+” and “$-$” components. The above our conclusions are related to the “+” component, which is the basic Regge asymptotic. The Pomeron intercept corresponding to “$-$” component, is $Q^2$-independent and this component is the subasymptotical one at large $Q^2$. However, the magnitude of the “+” is suppressed like $1/ln(1/x)$ and $\alpha (Q^2_0)$, and the subasymptotical “$-$” component may be important. Indeed, it is observed experimentally (see [@20.5; @20]). Note, however, that the suppression $\sim
\alpha(Q^2_0)$ is really very slight if we choose a small value of $Q^2_0$.

Our “solution” in the form of Eqs.(\[22\])-(\[25\]) is in the very well agreement with the recent $MRS(G)$ fit [@19] and with the results of [@17] at $Q^2=15GeV^2$. As it can be seen from Eqs.(\[22\]),(\[23\]), in our formulae there is the dependence on the PD behaviour at large $x$. Following to [@25] we choose $\nu =5$ that agrees in the gluon case with the quark counting rule [@26]. This $\nu$ value is also close to the values obtained by $CCFR$ group [@27] ($\nu = 4$) and in the last $MRS(G)$ analysis [@19] ($\nu =6$). Note that this dependence is strongly reduced for the gluon PD in the form

$$f_g(x,Q^2_0)~=~A_g(\nu)(1-x)^{\nu},$$ if we suppose that the proton’s momentum is carred by gluon, is $\nu$-independent. We used $A_g(5)=2.1$ and $F_2(x,Q^2_0)=0.3$ when $x \to 0$.

For the quark PD the choise $\nu =3$ is more preferable, however the use of two different $\nu$ values complicates the analysis. Because the quark contribution to the “+” component is not large, we put $\nu =5$ to both: quark and gluon cases. Note also that the variable $\nu (Q^2)$ have (see [@28]) the $Q^2$-dependence determinated by the LO AD $\gamma^{(0)}_{NS}$. However this $Q^2$-dependence is proportional $s$ and it is not important in our analysis.

Starting from $Q^2_0=1GeV^2$ (by analogy with [@23.6]) and from $Q^2_0=2GeV^2$, and using two values of QCD parameter $\Lambda$: more standard one ($\Lambda^{f=4}_{\overline {MS}}$ = 200 $MeV$) and ($\Lambda^{f=4}_{\overline {MS}}$= 255 $MeV$) obtained in [@19], we have the following values of the quark and gluon PD “intercepts” $\delta_a ~=~-(d_+ \tilde s +\hat d^q_{++}a)$ (here $\Lambda^{f=4}_{\overline {MS}}$ is marked as $\Lambda$):

if $Q^2_0$ = 1 $GeV^2$

  ------- ------------------- ------------------- ------------------- -----------------
  $Q^2$   $\delta_q(Q^2)$     $ \delta_g(Q^2)$    $\delta_q(Q^2)$     $\delta_g(Q^2)$

          $\Lambda =200MeV$   $\Lambda =200MeV$   $\Lambda =255MeV$   $\Lambda
                                                                          =255MeV$

  4       0.191               0.389               0.165               0.447

  10      0.318               0.583               0.295               0.659

  15      0.367               0.652               0.345               0.734
  ------- ------------------- ------------------- ------------------- -----------------

if $Q^2_0$ = 2 $GeV^2$

  ------- ------------------- ------------------- ------------------- -----------------
  $Q^2$   $\delta_q(Q^2)$     $ \delta_g(Q^2)$    $\delta_q(Q^2)$     $\delta_g(Q^2)$

          $\Lambda =200MeV$   $\Lambda =200MeV$   $\Lambda =255MeV$   $\Lambda
                                                                          =255MeV$

  4       0.099               0.175               0.097               0.198

  10      0.226               0.368               0.227               0.410

  15      0.275               0.438               0.278               0.486
  ------- ------------------- ------------------- ------------------- -----------------

Note that these values of $\delta_a $ are above the ones from [@19]. Because we have the second (subasymptotical) part, the effective our “intercepts” have the smaller values.

As a conclusion, we note that BFKL equation (and thus the value of Lipatov Pomeron intercept) was obtained in [@19.5] in the framework of perturbative QCD. The large-$Q^2$ $HERA$ experimental data are in the good agreement with Lipatov’s trajectory and thus with perturbative QCD. The small $Q^2$ data agrees with the standard Pomeron intercept $\alpha_p=1$ or with Donnachie-Landshoff pisture: $\alpha_p=1.08$. Perhaps, this range requires already the knowledge of nonperturbative QCD dynamics and perturbative solutions (including BFKL one) should be not applied here directly and are corrected by some nonperturbative contributions.

In our analysis Eq.(\[1\]) can be considered as the nonperturbative (Regge-type) input at $Q^2_0 \sim 1GeV^2$. Above $Q^2_0$ the PD behaviour obeys DGLAP equation, Pomeron moves to the subcritical regime and tends to its perturbative value. After some $Q^2_c$, where its perturbative value was already attained, Pomeron intercept saves the permanent value. The application of this approach to analyse small $x$ data invites futher investigation.

[99]{} H1 Collab.: T.Ahmed et al., $DESY$ preprint 95-006 (1995). ZEUS Collab.: M.Derrick et al., $DESY$ preprint 94-143 (1994). NM Collab.: P.Amadrus et al., [*Phys.Lett.*]{} [**B295**]{}, (1992) 159, [**B309**]{}, (1993) 222. E665 Collab.: in the B.Badelek’s report “Low $Q^2$, low $x$ in electroproduction. An overview.”. In Proceeding de Moriond on QCD and high energy hadron interactions (1995) Les Arc. A.Levy, $DESY$ preprint 95-003 (1995). J.D. Bjorken, In Proceeding of the International Workshop on DIS, Eilat, Izrael, Feb.1994. V.N.Gribov and L.N.Lipatov, [*Sov.J.Nucl.Phys.*]{} [**18**]{}, (1972) 438; L.N.Lipatov, [*Yad.Fiz.*]{} [**20**]{}, (1974) 181; G.Altarelli and G.Parisi, [*Nucl.Phys.*]{} [**B126**]{}, (1977) 298; Yu.L.Dokshitzer, [*ZHETF*]{} [**46**]{} (1977) 641. V.N.Gribov, E.M.Levin and M.G.Ryskin, [*Phys.Rep.*]{} [**100**]{} (1983) 1; E.M.Levin and M.G.Ryskin, [*Phys.Rep.*]{} [**189**]{} (1990) 267. V.I.Vovk, A.V.Kotikov and S.J.Maximov, [*Teor.Mat.Fiz.*]{} [**84**]{} (1990) 101; A.V.Kotikov, S.I.Maximov and I.S. Parobij, Preprint ITP-93-21E (1993) Kiev, [*Teor.Mat.Fiz.*]{} (1995) in press. M.Virchaux and A.Milsztain, [*Phys.Lett.*]{} [**B274**]{} (1992) 221. A.V.Kotikov, work in progress L.L.Enkovszky, A.V.Kotikov and F.Paccanoni, [ *Yad.Fiz.*]{} [**55**]{} (1993) 2205. A.V.Kotikov, [*Yad.Fiz.*]{} [**56**]{} (1993) N9, 217. A.V.Kotikov, [*Yad.Fiz.*]{} [**57**]{} (1994) 142; [*Phys.Rev.*]{} [**D49**]{} (1994) 5746. R.K.Ellis, E.Levin and Z.Kunst, [*Nucl.Phys.*]{} [**420B**]{} (1994) 514. R.K.Ellis, F.Hautmann and B.R.Webber, [*Phys.Lett.*]{} [**B348**]{} (1995) 582. N.N.Nikolaev, B.G.Zakharov and V.R.Zoller, [ *Phys.Lett.*]{} [**B328**]{}, (1994) 486; N.N.Nikolaev and B.G.Zakharov, [*Phys.Lett.*]{} [**B327**]{}, (1994) 149. C.Lopez and F.J.Yndurain, [*Nucl.Phys.*]{} [**171B**]{} (1980) 231; [**183B**]{} (1981) 157; A.M.Cooper-Sarkar, G.Ingelman, K.R.Long, R.G.Roberts and D.H.Saxon, [*Z.Phys.*]{} [**C39**]{} (1988) 281; A.V.Kotikov, $JINR$ preprints P2-88-139, E2-88-422 (1988) Dubna (unpublished). G.M.Frichter, D.W.McKay and J.P.Ralston, [*Phys.Rev.Lett.*]{} [**74**]{} (1995) 1508. M.Bertini, P.Desgrolard, M.Giffon, L.Jenkovszky and F.Paccanoni, Preprint LYCEN/9366 (1993). A.J.Buras, [*Rev.Mod.Phys.*]{} [**52**]{} (1980) 149. E.A.Kuraev, L.N.Lipatov and V.S.Fadin, [*ZHETF*]{} [**53**]{} (1976) 2018, [**54**]{} (1977) 128; Ya.Ya.Balitzki and L.N.Lipatov, [*Yad.Fiz.*]{} [**28**]{} (1978) 822; L.N.Lipatov, [*ZHETF*]{} [**63**]{} (1986) 904. M.Giafaloni, [*Nucl.Phys.*]{} [**B296**]{}, (1987) 249; S.Catani, F.Fiorani and G.Marchesini, [*Phys.Lett.*]{} [**B234**]{} (1990) 389, [*Nucl.Phys.*]{} [**B336**]{} (1990) 18; S.Catani, F.Fiorani, G.Marchesini and G.Oriani, [*Nucl.Phys.*]{} [**B361**]{} (1991) 645; G.Wolf, $DESY$ preprint 94-022 (1994). ZEUS Collab.: M.Derrick et al., [*Phys.Lett.*]{} [**B345**]{}, (1995) 576. A.D.Martin, W.S.Stirling and R.G.Roberts, Preprint RAL-95-021, DTP/95/14 (1995). W.K.Tung, [*Nucl.Phys.*]{} [**B315**]{} (1989) 378. SLD Collab.: K.Abe et al., preprint $SLAC$-PUB-6687 (1995), submitted to [*Phys.Rev.Lett.*]{}. A.Capella, U.Sukhatme,C.-I.Tan and J.Tran Thanh Van, [*Phys.Rep.*]{} [**236**]{} (1993) 225; [*Phys.Rev.*]{} [**D36**]{} (1987) 109. A.Capella, A.Kaidalov, C.Merino,and J.Tran Thanh Van, [*Phys.Lett.*]{} [**B337**]{} (1994) 358; M.Bertini, M.Giffon and E.Predazzi, Preprint LYCEN/9504 (1995). R.D.Ball and S.Forte, [*Phys.Lett.*]{} [**B336**]{} (1994) 77; preprints CERN-TH-7422-94 (1994), CERN-TH-95-1(1995). A.Donnachie and P.V.Landshoff, [*Nucl.Phys.*]{} [**B303**]{} (1988) 634. S.Brodsky and G.Farrar, [*Phys.Rev.Lett.*]{} [**31**]{} (1973) 1153; V.Matveev, R.Muradyan and A.Tavkhelidze, [ *Lett. Nouvo Cim.*]{} [**7**]{} (1973) D654.. CCFR Collab.: R.Z.Quintas et al., [*Phys.Rev.Lett.*]{} [**71**]{} (1993) 1307. D.J.Gross, [*Phys.Rev.Lett.*]{} [**32**]{} (1974) 1071.

[^1]: More correctly, $\phi$ is $Q^2$-dependent for the solution of DGLAP equation with the boundary condition: $f_a(x,Q^2_0)= Const$ at $x \to 0$. In the case of the boundary condition: $f_a(x,Q^2_0) \sim
      \exp{\sqrt{ ln(1/x)}}$, $\phi$ is lost (see [@8]) its $Q^2$-dependence

[^2]: We use the termin “solution” because we will work in the leading twist approximation in the range of $Q^2$: $Q^2>1GeV^2$, where the higher twist terms may give the sizeable contribution (see, for example, [@9]). Moreover, our “solution” is the Regge asymptotic with unknown parameters rather then the solution of DGLAP equation. The parameters are found from the agreement of the r.h.s. and l.h.s. of the equation.

[^3]: Consideration of the more complicate behaviour in the form $x^{-\delta}(ln(1/x))^b
      I_{2g}(\sqrt{\phi ln(1/x)})$ is given in [@8] and will be considered in this content in the forthcomming article [@10]

[^4]: In the double-logarithmical approximation the similar results were obtained in [@15.5]

[^5]: The method is based on the earlier results [@16]

[^6]: The used formula (Eq.(2) from [@17]) coincides with (\[6\]) in the leading order (LO) approximation, if we save only $f_g(x,Q^2)$ in the r.h.s. of (\[2\]) (or put $\gamma_{qq}=0$ and $\gamma_{qg}=0$ formally). Eq.(\[6\]) and Eq.(\[2\]) from [@17] have some differences in the next-to-leading order (NLO), which are not very important because they are corrections to the $\alpha$-correction.

[^7]: The form $\exp \Bigl({ -s \tilde
        \gamma_+(1+\delta)/(2\beta_0)} \Bigr)$ coincides with the both solution: Eq.(\[9\]) if $x^{\hat d_+} >>1$ and Eq.(\[11\]) when $\delta
      =0$ but it is not the solution of DGLAP equation.

[^8]: The Pomeron intercept value increasing with $Q^2$ was obtained also in [@23.5].
