\documentclass[twocolumn,showpacs,fleqn,nobibnotes]{revtex4}
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\newcommand{\xgrv}{x_{\text{GRV}}}
\newcommand{\cms}[1]{#1^{(CMS)}}

\begin{document}

\title{DGLAP evolution extends the triple pole pomeron fit}
\pacs{11.55.-m, 13.60.-r}
%\author{J.-R. Cudell}
%\email{jr.cudell@ulg.ac.be}
\author{G. Soyez}
\email{g.soyez@ulg.ac.be}
\affiliation{Inst. de Physique, B\^{a}t. B5, Universit\'{e} de Li\`{e}ge, Sart-Tilman, B4000 LI\`{e}ge, Belgium}

\begin{abstract}
We show that, using the triple pole pomeron model \cite{CMS} as an initial condition for a DGLAP evolution \cite{DGLAP}, we are able to produce a fit to high $Q^2$ experimental DIS data. Since the triple pole pomeron model only applies in the Regge domain, i.e. for $x$ sufficiently small, we used the GRV98 parametrisation \cite{GRV98} at large $x$.
\end{abstract}

\maketitle

\section{Introduction}

We have shown in a previous paper \cite{CMS} that it is possible to fit experimental data for $F_2^p$ with a double or triple pole pomeron model in the region
\begin{equation}\label{eq:cmsdomain}
\begin{cases}
2\nu \ge 49\:\text{GeV}^2, \\
\cos(\theta_t) = \frac{\sqrt{Q^2}}{2xm_p} \ge \frac{49}{2m_p^2}, \\
Q^2 \le 150\:\text{GeV}^2, \\
x \le 0.3.
\end{cases}
\end{equation}
We have also shown that it was possible to extend the usual $t$-channel unitarity relations \cite{GP} to the case of multiple thresholds and multiple poles. This allowed us to predict $F_2^{\gamma}$ from $F_2^p$ and the $pp$ total cross-section. In the latter case, we have shown that all processes have the same singularity structure at least at $Q^2=0$. 

However, in the usual parton distribution sets, each distribution presents its own singularities. As example, in the MRST2001 parametrization \cite{MRST2001}, we have
\[
xq(x, Q_0^2) = A (1+B\sqrt{x}+Cx)(1-x)^{\eta_q}x^{\varepsilon_q},
\]
with $\varepsilon_\text{sea}=-0.26$, $\varepsilon_g^{(1)}=-0.33$, $\varepsilon_g^{(1)}=0.09$. In fact, these singularities do not correspond to any singularity present in hadronic cross sections and, inversely, cross section singularities are not present in parton distributions. There must therefore exist a mechanism through which the singularities in partonic distributions disappear and cross section singularities arise when $Q^2$ goes to zero. Such a mechanism is unknown and seems forbidden by Regge theory. In this framework, a singularity structure common both to parton distibutions and to hadronic cross sections is the most natural choice.

At that level, one may ask if the Regge fit in \cite{CMS} is compatible with pQCD and if it is possible to have the same singularities in all parton distributions. Actually, although Regge theory \cite{Regge:mz,Regge:1960zc} and DGLAP \cite{DGLAP} evolution both provide well-known descriptions of the structure functions, the connection between the two approaches is badly known. In this paper, we will confront the triple-pole parametrisation, for each parton distribution, with the DGLAP evolution. This is done by fixing the initial distribution at $Q_0^2$ in order to reproduce the $F_2^p$ value obtained from the global QCD fit. We will see that we are able to produce a fit to experimental data which is compatible both with Regge theory and with the DGLAP equation. This comparison of two aspects of the theory will allow us to split the $F_2$ structure function in smaller contributions and to predict the density of gluons, which is generally not accessible directly from Regge fits to $F_2$. 

By varying the initial scale $Q_0^2$, we can predict a scale where perturbative QCD breaks down.

However, due to the application domain \eqref{eq:cmsdomain} of the global fit, this constraint is not valid at large $x$. We solve the problem by using the GRV98 parton distributions \cite{GRV98} at large $x$ ($x>\xgrv$). We will also argue that the results is not significantly dependent on the choice of the large $x$ parametrisation. Since we will use leading order (LO) DGLAP evolution, one can choose any of the usual PDF sets to extend our fits to large $x$.

We will show that, within a reasonable region of $Q_0^2$ and $\xgrv$, using the triple-pole pomeron model as the initial condition for LO DGLAP evolution reproduces the experimental data. The scale $Q_0^2$ should be considered as the minimal scale where perturbative QCD can be applied.

Since a good precision on the gluon density is of primary importance for the LHC, it is also very interesting to look at the prediction of this model for the density of gluons. We will see that the densities we obtain are of the same order of magnitude as in the usual DGLAP fits. This work also  allow us to separate the various contributions to $F_2^p$.

One should mention that such an extension of a triple pole Regge fit by a DGLAP evolution has already been introduced in \cite{Csernai}. However, as we will see, our approach here is slightly different: our parametrisation is much more constrained, we are able to extract a gluon distribution and all the distributions we use have the same singularity structure. There are also some less important differences in the treatment of the large-$x$ domain.

\section{Perturbative QCD and Regge theory}

The main goal of this paper is to compare perturbative QCD (pQCD) and Regge theory predictions. This comparison can be useful to estimate the scale $Q_0^2$ beyond which we may apply pQCD.

\subsection{Perturbative QCD}

In pQCD, the high $Q^2$ behaviour of Deep Inelastic Scattering (DIS) is given by the DGLAP evolution equations \cite{DGLAP}. These equations introduce the {\em parton distribution functions} $q_i(x, Q^2)$, $\bar q_i(x, Q^2)$ and $g(x, Q^2)$, which represent the probability of finding, in the proton, respectively a quark, an anti-quark or a gluon with virtuality less than $Q^2$ and with  longitudinal momentum fraction $x$. When $Q^2 \to \infty$, the $Q^2$ evolution of these densities (at fixed $x$) are given by the DGLAP equations
\begin{eqnarray}\label{eq:DGLAP}
\lefteqn{ Q^2\partial_{Q^2}
\begin{pmatrix}q_i(x,Q^2)\\\bar q_i(x,Q^2)\\g(x,Q^2)\end{pmatrix}}\\
& = &\frac{\alpha_s}{2\pi} \int_x^1 \frac{d\xi}{\xi}
\left.\begin{pmatrix}
 P_{q_iq_j} & . & P_{q_ig}\\
 . & P_{q_iq_j} & P_{q_ig}\\
 P_{gq} & P_{gq} & P_{gg}
\end{pmatrix}\right|_{\frac{x}{\xi}}
\begin{pmatrix} q_j(\xi,Q^2)\\\bar q_j(\xi,Q^2)\\ g(\xi, Q^2) \end{pmatrix},\nonumber
\end{eqnarray}
at leading order. Using these definitions, we have 
\begin{equation}\label{eq:F2}
F_2(x,Q^2) = x \sum_i e_{q_i}^2 \left[q_i(x,Q^2) + \bar q_i(x,Q^2) \right],
\end{equation}
where the sum runs over all quark flavours.

\subsection{Regge theory}

Beside the predictions of pQCD, we can study DIS through its analytical properties. In Regge theory \cite{Regge:mz,Regge:1960zc}, we consider amplitudes in complex angular momentum space ${\cal A}(j,t)$ by performing a Sommerfeld-Watson transform. In that formalism, we choose a singularity structure in $j$-plane for the amplitudes and the residues of the singularities is a function of $t$. This technique can be applied to the domain $\cos(\theta_t) \gg 1$. For example, we can fit the DIS data or the photon structure function at large $\nu$ (small $x$), and the total cross sections at large $s$. 

Most of the models based on the Regge theory use a pomeron term, reproducing the rise of the structure function (cross sections) at small $x$ (at large $s$), and reggeon contributions reproducing the meson trajectories ($a$, $f$ and $\omega$ trajectories). 
%The structure of the pomeron contribution depends on the model but some parametrisations seems to be preferred:
%\begin{itemize}
%\item the Donnachie-Landshoff two pomerons model \cite{DL}
%\[
%a(Q^2)\nu^\epsilon + b(Q^2)\nu^{\epsilon'},
%\]
%where $\epsilon\approx 0.4$ and $\epsilon'\approx 0.08$. In $j$-plane, this corresponds to a simple pole at $j=1+\epsilon$ (hard pomeron) and a simple pole at $j=1+\epsilon'$ (soft pomeron).
%\item the double pole pomeron model \cite{CMS}
%\[
%a(Q^2)\ln\left[1+\Lambda(Q^2)\nu^\delta\right] + b(Q^2),
%\]
%which becomes
%\[
%\frac{\delta a}{(j-1)^2} + \frac{\ln(\Lambda)+b}{j-1}+\sum_{n=1}^\infty\frac{\Lambda^{-n}}{n(j-1-n\delta)}
%\]
%in Mellin space. We thus have a double pole at $j=1$ and a series of simple poles.
%\item the triple pole pomeron model \cite{CMS}
We shall consider here the following parametrisation for the pomeron term
\begin{equation}\label{eq:triple}
a(Q^2)\ln^2\left[\nu/\nu_0(Q^2)\right] + c(Q^2),
\end{equation}
corresponding to a triple pole in $j$-plane \cite{CMS}
\[
\frac{a}{(j-1)^3}-\frac{2a\ln(\nu_0)}{(j-1)^2}+\frac{a\ln^2(\nu_0)+c}{j-1}.
\]
%\end{itemize}
This seems to be the preferred phenomenological choice \cite{compete}.

Note that the upper expression, given in terms of $\nu$ and $Q^2$, can be rewritten in terms of $Q^2$ and $x=Q^2/(2\nu)$. Hence, the large $\nu$ limit coresponds to the small $x$ one.

%Another property one can obtain from Regge theory is the well-known factorisation of the residues, obtained by Gribov and Pomeranchuk \cite{GP} using $t$-channel unitarity of the $S$-matrix.
%If we we parametrise proton and pion amplitudes with simple poles
%\[
%A_{ij}(t) = \frac{R_{ij}(t)}{j-\varepsilon}\qquad i,j=\pi,p.
%\]
%we have, for $4m_\pi^2 \le t \le 9m_\pi^2$ (2 thresholds), 
%\[
%R_{pp}R_{\pi\pi}=R_{p\pi}^2.
%\]
In a previous paper \cite{CMS}, we have also shown from unitarity constraints that we can extend the Gribov-Pomeranchuk argument about factorisation of residues to any number of thresholds and to any type of singularities. Hence, if we parametrise the $pp$ and the $\gamma^{(*)}p$ cross sections, one can predict a value for the $\gamma^{(*)}\gamma^{(*)}$ cross-section using the $t$-channel unitarity ($t$CU) relation
\[
A_{\gamma\gamma}(j,Q_1^2,Q_2^2) = \frac{A_{\gamma p}(j,Q_1^2)A_{\gamma p}(j,Q_2^2)}{A_{pp}(j)}+\text{finite terms}
\]
for the amplitudes in $j$-plane. This relation proves the universality of the singularities, in other words, all singularities present in $\gamma p$ interaction also appear in $\gamma \gamma$ interactions. We applied the $t$CU rules to the case of double and triple pole pomeron models in the region \eqref{eq:cmsdomain}. Therefore, if our fit in this paper keeps consistency with the fits in \cite{CMS} for $Q^2\le Q_0^2$, it can also be used to reproduce the $\gamma^{(*)}\gamma^{(*)}$ experimental results.

\section{Initial distributions}

%We want to confront the prediction of the DGLAP evolution with our triple pole parametrisation in the Regge domain \eqref{eq:cmsdomain}. We have not used the two-pomeron model due to the fact that it violates the $t$CU relations, and we will see that we encounter some practical problems for the double pole case.

The main problem of our approach is that the DGLAP evolution will generate an essential singularity at $j=1$, and the essential singularity will arise at the scale where we start the evolution. It is thus quite tedious to make a direct comparison of the essential singularity behaviour with our triple pole. The easiest way to solve this problem is to say that we have two regimes: for $Q>Q_0^2$, we have a perturbative DGLAP evolution with an essential singularity, while for $Q^2\le Q_0^2$, the soft QCD fit applies, and $F_2$ behaves like a triple pole at small $x$. Physically, this comes from the fact that, below $Q_0^2$, perturbation theory needs further resumation (or even breaks down) and we expect the result to behave like a triple pole at high energy \cite{cgc}.

Due to the fact that the domain \eqref{eq:cmsdomain} does not extend up to $x=1$, we have used the GRV98 \cite{GRV98} parametrisation at large $x$, i.e. for $x > \xgrv$. It is worth mentioning that, in the DGLAP equation \eqref{eq:DGLAP}, the evolution for $x > \xgrv$ does not depend on the distributions below $\xgrv$. This means that the evolution of the GRV98 distribution functions for $x > \xgrv$ is not influenced by the parametrisation we will impose for $x \le \xgrv$.

Since we want to test our fit to $F_2$ using a triple-pole pomeron \eqref{eq:triple}, for $x\le\xgrv$, we want to have an initial distribution of the form ($Q_0^2$ is the scale at which we start the DGLAP evolution)
\begin{equation}\label{eq:initF2}
F_2(x, Q_0^2) = a \log^2(1/x) + b \log(1/x) + c + d x^\eta,
\end{equation}
i.e. described with a triple pole pomeron and an $f$-reggeon trajectory component ($\eta=0.31$ as given in \cite{CMS}). Once we have that initial distribution, we can evolve it with DGLAP and compare with experimental data.

However, the DGLAP equation \eqref{eq:DGLAP} does not allow us to compute $F_2$ directly. But, performing linear combinations in \eqref{eq:DGLAP}, one can easily check that the minimal set of densities needed to obtain $F_2$ from the DGLAP equation is given by
\begin{eqnarray}
T      & = & x\left\lbrack (u^+ +c^++t^+)-(d^++s^++b^+)\right\rbrack,\\
\Sigma & = & x\left\lbrack (u^+ +c^++t^+)+(d^++s^++b^+)\right\rbrack,\\
G      & = & x g,
\end{eqnarray}
where $q^+ = q+\bar q$ for $q=u,d,s,c,t,b$.
The evolution equations for these distributions turn out to be
\begin{eqnarray*}
Q^2\partial_{Q^2} T(x,Q^2) & = & \frac{\alpha_s}{2\pi} \int_x^1 \frac{xd\xi}{\xi^2}
P_{qq}\left(\frac{x}{\xi}\right) T(\xi,Q^2),\\
Q^2\partial_{Q^2}
\begin{pmatrix}\Sigma\\G\end{pmatrix} 
  & = &\frac{\alpha_s}{2\pi} \int_x^1 \frac{xd\xi}{\xi^2}
\begin{pmatrix}
 P_{qq} & 2n_fP_{qg}\\
 P_{gq} & P_{gg}
\end{pmatrix}
\begin{pmatrix} \Sigma\\ G \end{pmatrix}
\end{eqnarray*}
and $F_2$ is then given by
\[
F_2=\frac{5\Sigma+3T}{18}.
\]
This clearly shows that, if we want to use \eqref{eq:initF2} as the initial condition for a DGLAP evolution, we need to split it into $T$ and $\Sigma$ contributions, but we also need to introduce a gluon density. In this way, using \eqref{eq:initF2} as the initial condition for the evolution allows us to predict the gluon distribution function.

Since, below $Q_0^2$, we do not use singularities of order larger than 3, we expect this behaviour to be valid for the $T$ and $\Sigma$ distributions. The natural way of separating the initial $F_2$ value given by \eqref{eq:initF2} is thus to consider both $T$ and $\Sigma$ as a sum of a triple pole pomeron and a $f$-reggeon. The gluon distribution, beeing coupled to $\Sigma$, should also not contain singularities of higher order.
Thus, we can write
\begin{eqnarray}
T(x,Q_0^2) & = & a_T \log^2(1/x) + b_T \log(1/x) + c_T + d_T x^\eta,\nonumber\\
\Sigma(x,Q_0^2) & = & a_\Sigma \log^2(1/x) + b_\Sigma \log(1/x) + c_\Sigma + d_\Sigma x^\eta,\nonumber\\
G(x,Q_0^2) & = & a_G \log^2(1/x) + b_G \log(1/x) + c_G + d_G x^\eta.\nonumber
\end{eqnarray}
%It is worth noting that although in most of the usual DGLAP fits, $T$ , $\Sigma$ and $G$ are not constrained to have the same singularities, such an universality can be expected. Actually, from the point of view of $QCD$, gluons can fluctuate into sea quarks and $q\bar q$ pairs can produce gluons.

Most of the 12 parameters in these expressions are constrained. First of all, since the triple-pole pomeron should describe the high energy interactions, it should not be sensitive to the quarks flavours. This means that, at high energy, one expects $T \to 0$. Therefore, we set $a_T=b_T=c_T=0$. Then, since we connect our parametrisation with GRV's at $\xgrv$, we want the distribution functions to be continuous over the whole $x$ range. Continuity of the $T$ distribution fixes the $d_T$ parameter and we finally have
\[
T(x,Q_0^2) = T^{(GRV)}(\xgrv,Q_0^2) \left(\frac{x}{\xgrv}\right)^\eta.
\]

Moreover, we want to fix $F_2(Q_0^2)$ to be equal to $\cms{F_2}$ obtained from our previous global fit (each quantity with a superscript $^{(CMS)}$ refers to the corresponding quantity obtained from the global QCD fit in \cite{CMS}). Since $T$ is entirely known, this constraint fixes all the $\Sigma$ parameters through the relation
\begin{equation}\label{eq:formrel}
\phi_\Sigma = \frac{18\cms{\phi}-3\phi_T}{5},\quad \phi=a,b,c,d.
\end{equation}

At this level, only the gluon distribution parameters are free. But, since the $f$-reggeon trajectory is expected to be mainly constituted of quarks and due to to the fact that the $f$-reggeon should not be coupled to the Pomeron, we may exclude its contribution from the gluon density. Thus, $d_G=0$. Finally, we used continuity of the gluon density with the GRV distribution at $\xgrv$ to fix $c_G$.

We are finally left with only 2 free parameters: $a_G$ and $b_G$.

%Before going to the next section, one may ask why we have chosen the triple pole model instead of the generalised double pole. The reason is that, if we write
%\[
%F_2 = a(Q^2)\ln\left[1+\Lambda(Q^2)x^{-\delta}\right] + b(Q^2) + c(Q^2) x^\eta,
%\]
%we may not simply use the same kind of expression for $T$ and $\Sigma$ because we have no relation like \eqref{eq:formrel} for $\Lambda$.

\section{Fit}

We will fit the DIS data coming from H1\cite{H1-1,H1-2,H1-3}, ZEUS\cite{ZEUS-1,ZEUS-2}, BCDMS\cite{BCDMS}, E665\cite{E665}, NMC\cite{NMC} and SLAC\cite{SLAC}. In this paper, we will only consider data for $F_2^p$. We have not included data from $F_2^d$, $F^{\nu N}$, Drell-Yan proccesses and Tevatron Jets for the following results
\begin{itemize}
\item for many experiments, most of the data points are at large $x$ or at low $Q^2$. Thus, they do not constrain our fit much.
\item some experiments allow to measure the valence quark distributions. We do not need them here since we only want the $T$, $\Sigma$ and gluon distributions.
%\item among all these data, our previous paper, used to constraint the $F_2$ initial value, does only include $F_2^p$ measurement experiments.
\end{itemize}

Since we want to test the domain common to Regge theory and to the DGLAP evolution, we only keep the experimental points verifying
\begin{equation}\label{eq:domain}
\begin{cases}
\cos(\theta_t) \le \frac{49}{2m_p^2},\\
Q_0^2 \le Q^2 \le 3000\:\text{GeV}^2,\\
x\le \xgrv.
\end{cases}
\end{equation}
The choice of the initial scale $Q_0^2$ depends on how far we want to apply perturbative QCD. We have tried several values around 5 GeV$^2$. Given an initial scale, the Regge limit on $\cos(\theta_t)$ translates into a natural value for $\xgrv$
\begin{equation}
\xgrv^{(0)} = \frac{m_p\sqrt{Q_0^2}}{49}.
\end{equation}
A graph of that limit is presented in Fig. \ref{fig:xgrv}.
\begin{figure}[ht]
\includegraphics{xgrv.eps}
\caption{Natural value of $\xgrv$ as a function of the scale}
\label{fig:xgrv}
\end{figure}
However, since the $x$ limit (at a given $Q^2$) grows with $Q^2$, we also tried some higher values for $\xgrv$.

\section{Results}

The results of the fits are given in table \ref{tab:chi2} as a function of $Q_0^2$ and $\xgrv$. We can see that this 2-parameters fit reproduces very well the experimental points in \eqref{eq:domain} for $Q_0^2\ge 3$ GeV$^2$ and $\xgrv \le 0.1$. The values of the fitted parameters, as well as the constrained parameters are given in table \ref{tab:param}

\begin{table}
\begin{tabular}{|l||c|c|c||c|c|c||c|c|c|}
\hline
\multicolumn{1}{|r||}{$\xgrv$} & \multicolumn{3}{c||}{$\xgrv^{(0)}$} & \multicolumn{3}{c||}{0.1} & \multicolumn{3}{c|}{0.2} \\
\hline
$Q_0^2$ & $\chi^2$ & $n$ & $\chi^2/n$ & $\chi^2$ & $n$ & $\chi^2/n$ & $\chi^2$ & $n$ & $\chi^2/n$ \\
\hline\hline
%10.0 & 483.765 & 515 & 0.939 & 557.302 & 577 & 0.966 & 633.129 & 591 & 1.071 \\
10.0 & 484 & 515 & 0.939 & 561 & 581 & 0.966 & 774 & 639 & 1.212 \\
\hline
%5.0  & 561.211 & 581 & 0.966 & 676.021 & 686 & 0.985 & 740.724 & 735 & 1.008 \\
5.0  & 557 & 577 & 0.966 & 676 & 686 & 0.985 & 862 & 744 & 1.159 \\
\hline
%3.0  & 774.256 & 639 & 1.212 & 862.188 & 744 & 1.159 &    -    &  -  &   -   \\
3.0  & 633 & 591 & 1.071 & 741 & 735 & 1.008 &    -    &  -  &   -   \\
\hline
\end{tabular}
\caption{$\chi^2$ for various values of $Q_0^2$ and $\xgrv$. ($n$ is the number of experimental point satisfying \eqref{eq:domain})}
\label{tab:chi2}
\end{table}

\begin{table}
\begin{tabular}{|l||c|c||c|c||c|c|}
\hline
$Q_0^2$       & \multicolumn{2}{c||}{3.0}     & \multicolumn{2}{c||}{5.0}     & \multicolumn{2}{c|}{10.0}    \\
\hline
\hline
$a_{\gamma p}$& \multicolumn{2}{c||}{0.00541}& \multicolumn{2}{c||}{0.00644}& \multicolumn{2}{c|}{0.00780}\\
$b_{\gamma p}$& \multicolumn{2}{c||}{0.0712} & \multicolumn{2}{c||}{0.0990} & \multicolumn{2}{c|}{0.142}  \\
$c_{\gamma p}$& \multicolumn{2}{c||}{0.00541} & \multicolumn{2}{c||}{0.0064} & \multicolumn{2}{c|}{0.00780} \\
$d_{\gamma p}$& \multicolumn{2}{c||}{0.890}   & \multicolumn{2}{c||}{1.06}   & \multicolumn{2}{c|}{1.27}   \\
\hline
\hline
$\xgrv$       & $\xgrv^{(0)}$ &     0.1      & $\xgrv^{(0)}$ &     0.1      & $\xgrv^{(0)}$ &     0.1      \\
\hline
\hline
$d_T$         &   -0.0722    &    0.167    &   -0.0478    &    0.166    &    0.0101    &    0.165    \\
$a_G$  &\textbf{0.147}&\textbf{0.00617}&\textbf{0.0908}&\textbf{0.0271}&\textbf{0.158}&\textbf{0.131}\\
$b_G$  & \textbf{-0.852}&\textbf{0.718}&\textbf{0.193} &\textbf{0.822} &\textbf{0.178}&\textbf{0.419}\\
$c_G$         &    3.45      &   -0.495    &    0.595     &   -0.851    &    0.0299    &   -0.463    \\
\hline
\end{tabular}
\caption{Values of the parameters for $\le Q_0^2 \le 10$ GeV$^2$ and $\xgrv\le 0.1$. Only $a_G$ and $b_G$ are fitted, while the other parameters are constrained.}
\label{tab:param}
\end{table}

We show the initial distributions and the $F_2^p$ plot for $Q_0^2=5$ GeV$^2$ and $\xgrv = 0.1$ in Fig. \ref{fig:distrib} and Figs. \ref{fig:F2fit-low},\ref{fig:F2fit-high} respectively.

%It is quite interesting to take $\xgrv > \xgrv^{(0)}$ since at large $Q^2$ (typically, $Q^2 \ge 400$ GeV$^2$), most of the experimental points are in the region $x \ge 0.01$. So, if we really want to test the high $Q^2$ behaviour of the evolution, we need to extend the initial parametrisation for $x$ larger than the natural limit.

In Fig. \ref{fig:gluons}, we have compared the gluon distribution we obtain in our fit with some well known DGLAP fits like GRV\cite{GRV98}, CTEQ\cite{CTEQ6} and MRST\cite{MRST2001}.
One can see that the gluon distribution is of the same order of magnitude as the distributions obtained in most DGLAP fit to DIS data.

It is also interesting to check whether or not our results depend on the choice of the large $x$ parametrisation. Actually, since the DGLAP evolution equation couples the small $x$ distributions to the large $x$ ones, so, at first sight, our results may depend on such a choice. However, looking at the study of the PDF uncertainties, it can be seen that the large $x$ behaviour of the $T$ and $\Sigma$ distributions does nearly not depend on the chosen fit down to $x\approx 0.1$. Moreover, in the large $x$ limit, the splitting matrix can be written
\[
\begin{pmatrix} P_{qq} & P_{qg} \\ P_{gq} & P_{gg} \end{pmatrix}
\approx \frac{1}{(1-x)_+}\begin{pmatrix} 2C_F & . \\ . & 2C_A \end{pmatrix}.
\]
Thus, in the large $x$ region, the gluon distribution and the sea are not coupled. Since, in our method, both $T$ and $\Sigma$ are fixed, we study the influence of the gluon distribution on the $F_2$ prediction. Due to the fact these are not coupled at large $x$, we expect that our fit does not depend on the large $x$ behaviour of the distributions.

Unfortunately, it is quite hard to determine a unique scale $Q_0^2$ or $\xgrv$ from the fit. From Table \ref{tab:chi2}, it is clear that $\xgrv$ can be taken to be 0.1 but can not be pushed up to 0.2. But, as we have argued, for such values of $Q_0^2$ and such high $\xgrv$, we are outside the domain \eqref{eq:cmsdomain} and we may not ensure that Regge theory will still stand at $x=0.1$ and $Q^2=Q_0^2$. We can thus adopt two different points of view:
\begin{enumerate}
\item we stay in the domain \eqref{eq:cmsdomain}. We have thus $\xgrv = \xgrv^{(0)}$ and we can take $Q_0^2$ down to 3 GeV$^2$. The problem is that as $Q_0^2$ goes down, $\xgrv$ goes down too. And, since high $Q^2$ experimental points have large $x$ values, we do not test $pQCD$ over a large range. This does not really allows us to predict a ``best value'' for $Q_0^2$.
\item we extrapolate the Regge fit outside the domain~\eqref{eq:cmsdomain}. In such a case, depending on our confidence in this extrapolation, we can consider that pQCD applies down to 3 GeV$^2$ or 5 GeV$^2$ and taking $\xgrv=0.1$. This value is compatible with the HERA predictions as well as with the Donnachie-Landshoff prediction \cite{DLclose}.
\end{enumerate}

\begin{figure}[ht]
\includegraphics{distrib.eps}
\caption{Initial distributions for $Q_0^2=5$ GeV$^2$ and $\xgrv = 0.1$. $q_{2/3} = x (u^+ + c^+ + t^+)$ and $q_{-1/3} = x(d^++s^++b^+)$}
\label{fig:distrib}
\end{figure}

\begin{figure}[ht]
\includegraphics{ug.ps}
\caption {Fitted gluon distribution compared with some well known parton distributions}
\label{fig:gluons}
\end{figure}

\begin{figure*}[ht]
\includegraphics{f2-low.ps}
\caption{$F_2^p$ fit for $Q_0^2=5$ GeV$^2$ and $\xgrv \le 0.1$ (low $Q^2$ values).}
\label{fig:F2fit-low}
\end{figure*}

\begin{figure*}[ht]
\includegraphics{f2-high.ps}
\caption{$F_2^p$ fit for $Q_0^2=5$ GeV$^2$ and $\xgrv \le 0.1$ (high $Q^2$ values).}
\label{fig:F2fit-high}
\end{figure*}

\section{Conclusion}
In this paper, we have shown that it is possible to use a very simple analytic form, namely a triple-pole pomeron and an $f$ reggeon, as initial condition for DGLAP evolution. Applying the constraint from a global QCD fit obtained in a previous paper \cite{CMS} as well as some expected properties of the parton distribution functions, we have shown that we can fit the DIS data in the domain $Q_0^2\ge 3$ GeV$^2$, $x\le 0.1$ and $\cos(\theta_t)\ge 49/(2m_p^2)$. This fit has only 2 free parameters in the gluon distribution. 

The most interesting thing here is that our fit is at the interplay between Regge theory and pQCD. We have thus proven that Regge theory can be used to extend QCD down to the non-perturbative domain. From the fit, we can also say that the scale down to which we can apply pQCD is of order 3-5 GeV$^2$.

Moreover, we have seen that our approach can be used to split $F_2$ in $T$ and $\Sigma$-components with precise physical properties. In this way, it is of prime importance to point out that all the initial distributions have the same singularity structure, which is rarely the case of usual parton sets. Since $\Sigma$ is coupled to the gluon distribution, the latter can also be predicted. We have shown that the fitted gluon distribution is of the same order of magnitude as the gluon distributions obtained by the usual DGLAP fits to DIS data like MRST, CTEQ or GRV.

By requiring the same singularities in each distribution, we have seen that we were able to construct a full model both for DGLAP evolution and Regge theory in the case of a triple-pole pomeron model. It should be interesting, in the future, to test if we can apply the same method to the case of double pole pomeron or Donnachie-Landshoff two-pomeron model.

In the future, it should also be interesting to see if it is possible to adapt this point of view, in order to derive the triple pole pomeron form factors.

\begin{acknowledgments}
I would like to thanks J.-R. Cudell for useful suggestions. This work is supported by the National Fund for Scientific Research (FNRS), Belgium.
\end{acknowledgments}

\begin{thebibliography}{99}
\bibitem{CMS} J.-R. Cudell, E. Martynov and G. Soyez, .

\bibitem{GP} V.N. Gribov and I. Ya. Pomeranchuk,
\textit{Phys. Rev. Lett.} \textbf{8}, 4343
(1962).

\bibitem{MRST2001}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne, \textit{Eur. Phys. J.} \textbf{C23} (2002) 73-87

% Regge Theory
\bibitem{Regge:mz} T.~Regge, \textit{Nuovo Cim.} \textbf{14} (1959) 951.
\bibitem{Regge:1960zc} T.~Regge, \textit{Nuovo Cim.}  \textbf{18} (1960) 947.

\bibitem{DGLAP} V.N. Gribov and L.N. Lipatov, \textit{Sov. J. Nucl. Phys.} \textbf{15} (1972) 438.
G. Altarelli and G. Parisi, \textit{Nucl. Phys.} \textbf{B126} (1977) 298.
Yu.L. Dokshitzer, \textit{Sov. Phys. JETP} \textbf{46} (1977) 641.

% GRV parametrisation
\bibitem{GRV98} M. Gluck, E. Reya and A. Vogt, \textit{Eur. Phys. J.} \textbf{C5} (1998) 461-470, or see http://cpt19.dur.ac.uk/hepdata/grv.html.

\bibitem{Csernai} L. Csernai \textit{et al.}, \textit{Eur. Phys. J.} \textbf{C24} (2002) 205.

% DL parametrisation
%\bibitem{DL} A. Donnachie and P.V. Landshoff, \textit{Phys. Lett.} \textbf{B 437} (1998) 408.

\bibitem{compete} J.-R.~Cudell, V.~V.~Ezhela, P.~Gauron, K.~Kang, Yu.~V.~Kuyanov, S.~B.~Lugovsky, B.~Nicolescu, and N.~P.~Tkachenko, \textit{Phys. Rev.} \textbf{D65} (2002) 075024 ; see also 2002 Review of Particle Physics, K.~Hagiwara \textit{et al.}, \textit{Phys. Rev.} \textbf{D66} (2002) 010001-9.

\bibitem{cgc} E.~Iancu, A.~Leonidov and L.~McLerran, Lectures given at the NATO Advanced Study Institute ``QCD perspectives on hot and dense matter'', August 6--18, 2001, in Carg\`ese, Corsica, France.

% Experimental papers
% H1
\bibitem{H1-1} H1 Collaboration: C.~Adloff \textit{et al.}, \textit{Eur. Phys. J.} \textbf{C13} (2000) 609.
\bibitem{H1-2} H1 Collaboration: C.~Adloff \textit{et al.}, \textit{Eur. Phys. J.} \textbf{C19} (2001) 269.
\bibitem{H1-3} H1 Collaboration: C.~Adloff \textit{et al.}, \textit{Eur. Phys. J.} \textbf{C21} (2001) 33.
 
% ZEUS
\bibitem{ZEUS-1} ZEUS Collaboration: J.~Breitweg \textit{et al.}, \textit{Eur. Phys. J.} \textbf{C12} (2000) 35.
\bibitem{ZEUS-2} ZEUS Collaboration: S.~Chekanov \textit{et al.}, \textit{Eur. Phys. J.} \textbf{C21} (2001) 443.
 
% BCDMS
\bibitem{BCDMS}
BCDMS Collaboration: A.~C.~Benvenuti \textit{et al.}, \textit{Phys. Lett. B} \textbf{223} (1989) 485.
 
% E665
\bibitem{E665} E665 Collaboration: Adams \textit{et al.}, \textit{Phys. Rev.} \textbf{D54} (96) 3006.
 
% NMC
\bibitem{NMC} NMC Collaboration: M.~Arneodo \textit{et al.}, \textit{Nucl. Phys.} \textbf{B483} (1997) 3; \textit{Nucl. Phys.} \textbf{B487} (1997) 3.
 
% SLAC
\bibitem{SLAC} SLAC Experiments: Whitlow, PL \textbf{B282} (92) 475 and SLAC-357 (1990).

\bibitem{CTEQ6} J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, P. Nadolsky, W.K. Tung, \textit{JHEP.} \textbf{0207} (2002) 012.

\bibitem{DLclose}
A.~Donnachie and P.~V.~Landshoff, \textit{Phys.Lett.} \textbf{B533} (2002) 277-284

\end{thebibliography}
\end{document}

