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\title{\bf A new modified Altarelli-Parisi evolution equation
with parton recombination in proton}

\author{
{\bf Wei Zhu} and {\bf Jianhong Ruan}\\
\normalsize Department of Physics, East China Normal University,
Shanghai 200062, P.R. China 
}
\date{}

\newpage

\maketitle

\vskip 3truecm

\begin{abstract}

	The coefficients of the nonlinear terms in a modified
Altarelli-Parisi evolution equation with parton recombination are
determined in the leading logarithmic ($Q^2$) approximation. The results
are valid in the whole $x$ region and contain the translation 
$GG\rightarrow q\overline q$, which is inhibited in the double leading 
logarithmic approximation. The comparisons of the new evolution equation 
with the Gribov-Levin-Ryskin equation are presented.

\end{abstract}

\newpage

\begin{center}
\section{Introduction}
\end{center}
	
	The inclusion of parton recombination to the proton structure         
function through a modified Altarelli-Parisi equation is an interesting 
subject involving the test of perturbative QCD and the study of new 
effects in small $x$ physics. A traditional tool in this research is the
so-called Gribov-Levin-Ryskin (GLR) equation, presented in
two pioneering papers: one is the idea of shadowing arising from gluon 
recombination, which was proposed by Gribov, Levin, Ryskin [1] based on 
the AGK (Abramovsky, Gribov, Kancheli) cutting rules [2] in the 
double leading logarithmic approximation (DLLA); the other is a   
perturbative calculation of the recombination probabilities in the 
DLLA by Mueller and Qiu [3], which enables the GLR equation to be 
applied phenomenologically.
	
	Unfortunately, a series of questions in the GLR equation 
destroyed the gluon recombination effects.  In our previous work [4], 
one of us (WZ) has proposed that the application of the AGK cutting rules 
is unreasonable in the GLR equation since it sums up the diagrams in which 
the cut lines break the important correlation among the initial gluons. 
For this reason, a new evolution equation including parton recombination was 
established in the leading logarithmic ($Q^2$) approximation (LL($Q^2$)A) 
using time ordered perturbation theory (TOPT) instead of the AGK cutting 
rules [4]. This equation for parton distributions in the proton can be 
generally written as

$$\frac{dG(x_B,Q^2)}{d\ln Q^2}$$
$$=(\frac{1}{RQ})^2
\int_{(x_1+x_2)\geq x_B} G^{(2)}(x_1, x_2,x_1+\Delta, x_2-\Delta,Q^2)
\sum_iP_i^{GG\rightarrow GG}(x_1,x_2,x_3,x_4,\Delta)$$
$$\delta(x_1+x_2-x_3-x_4)
[\delta(x_3-x_B)+\delta(x_4-x_B)]dx_1dx_2dx_3dx_4d\Delta$$
$$-2(\frac{1}{RQ})^2
\int_{x_1\geq x_B} G^{(2)}(x_1, x_2, x_1+\Delta, x_2-\Delta ,Q^2)
\sum_iP_i^{GG\rightarrow GG}(x_1,x_2,x_3,x_4,\Delta)$$
$$\delta(x_1+x_2-x_3-x_4)
[\delta(x_3-x_B)+\delta(x_4-x_B)]dx_1dx_2dx_3dx_4d\Delta, 
\eqno(1a)$$
and

$$\frac{dS(x_B,Q^2)}{d\ln Q^2}$$
$$=(\frac{1}{RQ})^2
\int_{(x_1+x_2)\geq x_B} G^{(2)}(x_1, x_2,x_1+\Delta, x_2-\Delta,Q^2)
\sum_iP_i^{GG\rightarrow q\overline{q}}(x_1,x_2,x_3,x_4,\Delta)$$
$$\delta(x_1+x_2-x_3-x_4)
[\delta(x_3-x_B)+\delta(x_4-x_B)]dx_1dx_2dx_3dx_4d\Delta$$
$$-2(\frac{1}{RQ})^2
\int_{x_1\geq x_B} G^{(2)}(x_1, x_2, x_1+\Delta, x_2-\Delta ,Q^2)
\sum_iP_i^{GG\rightarrow q\overline{q}}(x_1,x_2,x_3,x_4,\Delta)$$
$$\delta(x_1+x_2-x_3-x_4)
[\delta(x_3-x_B)+\delta(x_4-x_B)]dx_1dx_2dx_3dx_4d\Delta, 
\eqno(1b)$$
where we temporarily neglect the linear terms in the Altarelli-Parisi 
equation and $G^{(2)}$ is the correlation function for four-gluon lines. 
The new equation provides the following physical picture for 
the gluon recombination in a QCD evolution process: 
the two-parton-to-two-parton $(2 \to 2)$ amplitudes (Figs. 1a-1b) lead to
positive (antiscreening) effects and the interference
amplitudes between the one-parton-to-two-parton $(1 \to 2)$ and the
three-parton-to two-parton $(3 \to 2)$ (Figs. 1c-1f) amplitudes lead to
negative (screening) effects, respectively.  The TOPT-analysis shows that 
the above two kinds of amplitudes correspond to the same recombination 
function $\sum_iP_i$ but in different kinematic ranges. On the other hand, 
the complete corrections of the gluon recombination to the evolution equation 
should include the contributions from the virtual diagrams corresponding to 
$\delta(x_1-x_B)$ and $\delta(x_2-x_B)$, however, they cancel each
other in the proton [4]. 

	The quantitative predictions of Eq. (1) depend on the recombination
functions. Mueller and Qiu [3] have derived similar transition 
probabilities $\sum_iP_{MQ,i}$ in the DLLA, where only the gluon ladder 
graphs are kept at small $x$. We should remember that the AGK cutting 
rules are applicable only in diagrams which consist of 
gluon ladders [1]. Therefore, the DLLA is a necessary approximation 
for using the AGK cutting rules in the GLR equation. On the other hand, 
in the Mueller-Qiu approach [3], the cut vertex technique [5] was used to 
calculate the transition probabilities, which are entangled in a complex
cut diagram with four-parton propagators [3]. Therefore, the DLLA 
is also a convenient approximation in Ref.~[3]. 

	However, we shall point out that the DLLA leads to a difficulty in 
the GLR equation: the transition of gluon$\rightarrow$ quarks is suppressed 
in the DLLA-manner. Although the special techniques are used to include the 
corrections of gluon recombination to the quark distributions 
in Ref.~[3], however, it seems inconsistent in the theory as we will show
in Sect. 5. 

	The purpose of this paper is to derive the recombination functions 
of Eq. (1) in the LL($Q^2$) approximation. We can generalize the 
recombination function to the whole $x$ region and even include the 
processes with quarks, since Eq. (1) is not restricted by the AGK 
cutting rules [4]. Furthermore, we used TOPT instead of the cut vertex method
in the derivation of Eq. (1) [4]. Thus, we can separately calculate 
the simple two-parton-to-two-parton $(2 \to 2)$ processes in the recombination function. 
Following the above mentioned derivation, we shall complete 
a new modified Altarelli-Parisi evolution equation with parton recombination 
in the LL($Q^2$) approximation, which is valid in the whole $x$ region. 
Momentum conservation is naturally restored due to the coexistence  
of the shadowing- and antishadowing-effects in the new evolution equation. 
We also find that the DLLA is a bad approximation even for the 
recombination function of gluons in the small-$x$ region.

	The outline of the paper is as follows.
In Sect. 2 we shall derive a set of complete recombination functions.
An undetermined factor in Eq. (1) is the correlation function 
$G^{(2)}$, which relates to the nonperturbative structure of the proton. 
We use a simple assumption to model $G^{(2)}$ in Sect. 3.
Combining the results of the above two sections, we establish a modified
Altarelli-Parisi equation including parton recombination in Sect. 4.
In Sect. 5 we compare our new equation with the GLR equation.


\begin{center}
\section{Recombination functions}
\end{center}
	
	The recombination function in Eq. (1) is factorized due to the 
application of TOPT in [4]. All processes of two-parton-to-two-parton 
$(2 \to  2 )$ type can be precisely calculated using the 
standard perturbative QCD, except that the energies of partons are not 
conserved at the vertex connecting with the probe.

	At first, we calculate the recombination function for the
two-gluon-to-two-gluon process. The momenta of all initial and 
final partons are on-shell and they are parametrized as (see Fig. 2),

$$p_1=\left [x_1p,\b{0},x_1p\right ]; \hspace{0.3cm} 
p_2=\left [x_2p,\b{0},x_2p\right ],$$
$$p_1'=\left [x_1'p,\b{0},x_1'p \right ]; \hspace{0.3cm}
p_2'=\left [x_2'p,\b{0},x_2'p \right ],$$
$$k=\left [x_3p+\frac{l^2_{\perp}}{2x_3p},l_{\perp},x_3p \right ];
\hspace {0.3cm}
l'=\left [x_4p+\frac{l^2_{\perp}}{2x_4p},-l_{\perp},x_4p \right ].
\eqno(2)$$

	We take the physical axial gauge and the light-like vector
$n$ fixes the gauge as $n\cdot A=0$, $A$ being the gluon field.
The corresponding recombination function is defined by [4]

$$P_i^{GG\rightarrow GG}(x_1,x_2,x'_1,x'_2,x_3,x_4)dx_4
\frac{dl^2_{\perp}}{l^4_{\perp}}$$
$$=\frac{E_k}{\sqrt{E_{p_1}+E_{p_2}}\sqrt{E_{p'_1}+E_{p'_2}}}
[M(p_1p_2\rightarrow kl')[M(p'_1p'_2\rightarrow kl')]^*]_i
(\frac{1}{E_{p_1}+E_{p_2}-E_k-E_{l'}})^2$$
$$(\frac{1}{2E_k})^2\frac{d^3l'}{(2\pi)^32E_{l'}}$$
$$=\frac{1}{16\pi^2}\frac{x_3x_4}{(x_1+x_2)^3}
[M(p_1p_2\rightarrow kl')[M_(p'_1p'_2\rightarrow kl')]^*]_i
dx_4\frac{dl^2_{\perp}}{l^4_{\perp}}.
\eqno(3)$$

	The index $i$ in Eq. (3) implies the $t$-, $u$-, $s$-channels and 
their interference terms. We begin with the $t$-channel graph (Fig. 3).       
The contribution of this process to the invariant amplitude is 

$$\left\vert M_tM^*_t\right\vert_{x_3=x_B}$$
$$=\frac{g^4}{4}\frac{C_A^2}{N^2-1}\frac{1}{l_L^2}\frac{1}{l_R^2}
g^{\kappa\rho}C^{\kappa\mu\xi}C^{\rho\phi\alpha}g^{\lambda\sigma}
C^{\lambda\eta\nu}C^{\sigma\beta\chi}\delta_{\perp}^{pq}\delta_{\perp}^{rs}
[\delta^{ij}-\frac{k^ik^j}{\left\vert\vec{k}\right\vert^2}]
[\delta^{lm}-\frac{l'^ll'^m}{\left\vert\vec{l'}\right\vert^2}],
\eqno(4)$$
where $(i,j), (l,m), (p,q)$ and $(r,s)$ are the space-indices corresponding
to ($\mu$,$\nu$), ($\alpha$,$\beta$), ($\xi$,$\eta$) and ($\phi$,$\chi$),
respectively. We need to distinguish the probing place, for example,
the transfer momenta $l_L$ and $l_R$ are determined by two down-vertices 
in Fig. 3 where the probing place is $x_3=x_B$, i.e.,

$$l_L=\left [(x_4-x_2)p+\frac{l^2_{\perp}}{2x_4p},-l_{\perp},(x_4-x_2)p \right ],$$
$$l_R=\left [(x_4-x'_2)p+\frac{l^2_{\perp}}{2x_4p},-l_{\perp},(x_4-x'_2)p \right ].
\eqno(5)$$

The algebra in Eq. (4) can be straightforwardly derived without any 
approximation in computer. The contributions from the $u$-channel and 
the interference channels can be similarly obtained by using the interchanges 
of the corresponding momenta.  

	Now we turn to discuss the $s$-channel as shown in Fig. 4.
We should note that the massless partons with the parallel        
momenta can go on-mass-shell simultaneously in the collinear case 
and the collinear singularity may arise in the $s$-channel since 
$l^2=(p_1+p_2)^2=(p_1'+p_2')^2=0$. Fortunately, we now have an useful tool 
[6,7] to pick up the short-distance contributions in the propagator with 
collinear divergence: we use the following special propagators (for quarks)

$$S(l)=\frac{\gamma\cdot n}{2l\cdot n}, \eqno(6)$$
and (for gluons)

$$G^{\kappa\rho}(l)
=\frac{n^\kappa n^\rho}{(l\cdot n)^2}, \eqno(7)$$
to replace the normal Feynman propagators, respectively. Using the 
definitions

$$\overline{n}^{\mu}\equiv (\overline{n}^0, \overline{n}_\perp,
\overline{n}^3)=\frac{1}{\sqrt{2}}(1, 0_\perp,1);\hspace{0.3cm}
n^{\mu}=\frac{1}{\sqrt{2}}(1, 0_\perp, -1),\eqno(8)$$
or equivalently 

$$\overline{n}^{\mu}\equiv(\overline{n}^+,
\overline{n}^-,\overline{n}_\perp)
=(1, 0, 0_\perp);\hspace{0.3cm} n^{\mu}=(0,1, 0_\perp), \eqno(9)$$
we have $\overline{n}^2=0, n^2=0$, and $\overline{n}\cdot n=1$. 
The contribution of the $s$-channel (Fig. 4) to the invariant amplitude 
can then be safely calculated as,

$$\left\vert M_sM^*_s\right\vert$$
$$=\frac{g^4}{4}\frac{C_A^2}{N^2-1}(\frac{1}{l\cdot n})^4
n^{\kappa}C^{\kappa\xi\phi}n^{\lambda}C^{\lambda\eta\chi}
n^{\rho}C^{\rho\alpha\mu}n^{\sigma}C^{\sigma\beta\nu}\delta_{\perp}^{pq}\delta_{\perp}^{rs}
[\delta^{ij}-\frac{k^ik^j}{\left\vert\vec{k}\right\vert^2}]
[\delta^{lm}-\frac{l'^ll'^m}{\left\vert\vec{l'}\right\vert^2}]$$
$$=g^4\frac{C_A^2}{N^2-1}\frac{(x_4-x_3)^2(x_2-x_1)(x_2'-x_1')}{2x^4},
\eqno(10)$$

where we have used the following replacement in the calculations:

$$n^{\kappa}C^{\kappa\xi\phi}=n^{\kappa}(p_1-p_2)^{\kappa}. \eqno(11)$$

Obviously, the contributions given by the $s$-channel and corresponding 
interference channels integrate to zero if the correlation function 
$G^{(2)}(x_1,x_2,x_1',x_2,Q^2)$ is symmetric
under $x_1\leftrightarrow x_2$ and $x_1'\leftrightarrow x_2'$, since these
contributions contain the factor $(x_2-x_1)$ or $(x_2'-x_1')$. Thus, 
we can forget the $s$-channel and its interference terms in $GG\rightarrow GG$.
The contributions of Feynman diagram with four-gluon vertex are neglected 
because they are less singular. The total contributions of $GG\rightarrow GG$ are given 
in Appendix A.

	The recombination function $\sum_iP_i^{GG\rightarrow GG}$ has poles
at $x_1=0$ and $x_2=0$ (see Eq. (A.1)).  However, Eq. (1.a) is really  
infrared safe. One can simply check this conclusion as follows. For example, 
$x_1=0$ implies that $\Delta=0$ since only twist-4 or twist-2 amplitudes 
contribute to an unpolarized amplitude. Summing over the contributions of 
the right-hand side of Eq. (1.a) at the poles, one can find that

$$\int_{1\geq x_1\geq x_B} G^{(2)}(x_1=0, x_2, \Delta=0, Q^2)
\sum_iP_i^{GG\rightarrow GG}(x_1=0,x_2,\Delta=0,x_3,x_4,\Delta)$$
$$\delta(x_1+x_2-x_3-x_4)
[\delta(x_3-x_B)+\delta(x_4-x_B)]dx_2dx_3dx_4$$
$$+\int_{1\geq x_1\geq x_B} G^{(2)}(x_1, x_2=0, \Delta=0, Q^2)
\sum_iP_i^{GG\rightarrow GG}(x_1,x_2=0,\Delta=0,x_3,x_4,\Delta)$$
$$\delta(x_1+x_2-x_3-x_4)
[\delta(x_3-x_B)+\delta(x_4-x_B)]dx_1dx_3dx_4$$
$$-2\int_{1\geq x_1\geq x_B} G^{(2)}(x_1, x_2=0, \Delta=0,Q^2)
\sum_iP_i^{GG\rightarrow GG}(x_1,x_2=0,\Delta=0,x_3,x_4,\Delta)$$
$$\delta(x_1+x_2-x_3-x_4)
[\delta(x_3-x_B)+\delta(x_4-x_B)]dx_1dx_3dx_4=0, 
\eqno(12)$$
where the two positive terms are symmetric under $x_1\leftrightarrow x_2$
and $x_1\ne 0$ in the negative interference term. Therefore, Eq. (1)
is infrared safe. 

Furthermore, unlike the GLR equation, the momentum is conserved
in Eq. (1), i.e.,

$$\frac{d\int^1_0dx_Bx_BG(x_B,Q^2)}{d\ln Q^2}=0, \eqno(13)$$
and

$$\frac{d\int^1_0dx_Bx_Bq(x_B,Q^2)}{d\ln Q^2}=0. \eqno(14)$$

	Now let us return to consider the recombination functions for 
$GG\rightarrow q\overline{q}$. For example, the contribution of the
$t$-channel process in Fig. 5 to the invariant amplitude is  

$$\left\vert M_tM^*_t\right\vert_{x_3=x_B}$$
$$=\frac{g^4}{4}\frac{T_f}{N(N^2-1)} \frac{x_4^2}{x_2x_2'}\frac{1}{l^4_\perp}
Tr[\gamma\cdot k\gamma^{\eta}\gamma\cdot l_R\gamma^{\chi}\gamma\cdot l'
\gamma^{\phi}\gamma\cdot l_L\gamma^{\xi}]
\delta_{\perp}^{pq}\delta_{\perp}^{rs}.
\eqno(15)$$

The final results for $\sum_i^{GG\rightarrow q\overline{q}}$ are listed in 
Appendix A, where the contributions of the $s$-channel and corresponding 
interference channels vanish due to the same symmetry as in 
$\sum_iP_i^{GG\rightarrow GG}$.

We need only consider the gluon recombination at small $x$,
since gluons dominate the small-$x$ behavior of the parton distributions
in proton. However, in principle, our equation can include the 
recombination of quark-quark, quark-antiquark and quark-gluon in the LLA. 
For this end, we list all recombination functions in Appendix B.
We will discuss the properties of our results in Sect. 5.


\begin{center}
\section{Parton correlation functions}
\end{center}

	Equation (1) includes the gluon correlation function 
$G^{(2)}(x_1,x_2;x'_1,x'_2,Q^2)$. In general, the parton density is a 
concept defined at twist-2. The parton correlation function
is a generalization of the parton density beyond the leading twist and 
it has not yet been determined both in theory and experiment.
For comparison with the GLR equation, we shall use a toy model as in
Ref.~[3].  First of all, we assume that
 
$$G^{(2)}(x_1, x_2,x_1+\Delta, x_2-\Delta,Q^2)$$
$$\rightarrow G^{(2)}(x_1, x_2,x_1+\Delta, x_2-\Delta,Q^2)\delta(x_1-x_2)
\delta(\Delta). \eqno(16)$$
One can understand Eq. (16) as follows. Parton recombination is expected to 
be significant only when the actual spatial overlap between the 
fusing partons becomes sufficiently large in the interaction time $\tau_{int}$
of probe-target both in the transverse- and longitudinal-directions. This means that 
(i) the transverse area occupied by partons should become compared to the 
effective transverse area; (ii) only two partons with the same value of 
rapidity $y=\ln{\frac{1}{x}}$ and same impact parameter  
can overlap their wave functions in the longitudinal direction during
$\tau_{int}$, since the initial partons are parallel moving along the 
$z$-direction. Therefore, the fusion between the gluons with different values 
of $y$ is suppressed.  

	We consider that the different branches of the parton cascade evolve 
independently; this is valid in the large $N_c$ limit [8]. Thus, 
$G^{(2)}(x,Q^2)\sim G^2(x,Q^2)$, where $G(x,Q^2)$ is the usual gluon density. 
However, the density $G(x,Q^2)$ can not be normalized since 
$G(x,Q^2)\sim \infty$ when $x\rightarrow 0$.  A more reasonable approach is 
to use the gluon density in rapidity space $xG(x,Q^2)=dn_G/dy$ instead of 
$G(x,Q^2)$. The normalization condition is

$$\sum_a\frac {dn_a}{dy}=1, \eqno(17)$$
due to momentum conservation, where the sum is over all parton flavors. 
Therefore, we can assume the two-gluon distribution per unit area in
proton to be

$$(x_1+x_2)G^{(2)}(x_1,x_2,\Delta, Q^2)=\frac{1}{4\pi}\frac{9}{4\pi}
[x_1G(x_1,Q^2)][x_2G(x_2,Q^2)]
\delta(x_1-x_2)\delta(\Delta), 
\eqno(18)$$
where the first factor is from a normalization factor of the parton 
correlation function [3,6] and the second factor is determined by a simple 
geometric model as in Ref.~[3].  Although Eq. (18) is a phenomenological 
model, however, its $x$-dependence can be checked by experiment.

\begin{center}
\section{Modified Altarelli-Parisi equation}
\end{center}

	In this section we derive a modified Altarelli-Parisi equation 
with gluon recombination. Inputting Eq. (18) with the recombination functions 
to Eq. (1) and adding the linear terms corresponding to the Altarelli-Parisi 
equation, we have

$$\frac{dx_BG(x_B,Q^2)}{d\ln Q^2}$$
$$=\frac{C_A\alpha_s}{\pi}\int^1_{x_B}\frac{dx_1}{x_1}\frac{x_B}{x_1}
P^{G\rightarrow G}_{AP}(x_1,x_B)x_1G(x_1,Q^2)$$
$$+\frac{9}{32\pi^2}(\frac{1}{RQ})^2
\int_{x_B/2}^{1/2}dx_1x_Bx_1G^2(x_1, Q^2)
\sum_iP_i^{GG\rightarrow G}(x_1,x_B)$$
$$-\frac{9}{16\pi^2}(\frac{1}{RQ})^2
\int_{x_B}^{1/2}dx_1x_Bx_1G^2(x_1, Q^2)
\sum_iP_i^{GG\rightarrow G}(x_1,x_B),
\eqno(19.a)$$
where the recombination functions Eqs. (A.1)-(A.2) become simple after 
integral over $x_2$,$\Delta$, $x_3$ and $x_4$ under the assumption (18):

$$\sum_iP_i^{GG\rightarrow G}(x_1,x_B)$$
$$=\frac{3\alpha_s^2}{8}\frac{C_A^2}{N^2-1}
\frac{(2x_1-x_B)(-136x_Bx_1^3-64x_1x_B^3+132x_1^2x_B^2+99x_1^4+16x_B^4)}
{x_Bx_1^5}, \eqno(19.b)$$
which is valid in the whole $x$ region. The second term on the right-hand side 
of Eq. (19.a) is positive due to the contributions of Figs. 1a-1b and
is called as antishadowing effect. The negative term in Eq. (19.a) arises
from the shadowing corrections from the interference diagrams Figs. 1.c-1.f.

	We have a similar result for $GG\rightarrow q\overline {q}$:

$$\frac{dx_BS(x_B,Q^2)}{d\ln Q^2}$$
$$=\frac{2T_f\alpha_s}{\pi}\int^1_{x_B}\frac{dx_1}{x_1}\frac{x_B}{x_1}
P^{G\rightarrow q}_{AP}(x_1,x_B)x_1G(x_1,Q^2)$$
$$+\frac{9}{32\pi^2}(\frac{1}{RQ})^2
\int_{x_B/2}^{1/2}dx_1x_Bx_1G^2(x_1, Q^2)
\sum_iP_i^{GG\rightarrow q}(x_1,x_B)$$
$$-\frac{9}{16\pi^2}(\frac{1}{RQ})^2
\int_{x_B}^{1/2}dx_1x_Bx_1G^2(x_1, Q^2)
\sum_iP_i^{GG\rightarrow q}(x_1,x_B),
\eqno(19.c)$$
where the recombination function with the assumption (18) is

$$\sum_iP_i^{GG\rightarrow q}(x_1,x_B)$$
$$=\alpha_s^2\left[\frac{T_f}{N(N^2-1)}
\frac{(2x_1-x_B)^2(4x_B^2+5x_1^2-6x_1x_B)}{x_1^5}\right.$$
$$\left.+\frac{NT_f}{(N^2-1)^2}\frac{(2x_1-x_B)^2
(4x_B^2-6x_1x_B+4x_1^2)}{x_1^5}\right]. \eqno(19.d)$$

We will discuss the properties of our new modified Altarelli-Parisi 
equation (19) in the next section.

\begin{center}
\section{Discussions}
\end{center}

	In order to compare Eq. (19) with the GLR equation,
we begin with a review of the DLLA. As definited by the Altarelli-Parisi
equation, the DLLA means that in each order in $\alpha_s$, one keeps only 
the $\ln(Q^2/\mu^2)\ln(1/x)$ factor in the solutions of the evolution 
equation, or equivalently, only the terms having $1/z=x_1/x_B$ factor  
in the splitting function are necessary to generate larger logarithms 
in $x_B$ (Fig. 6). We know that the splitting functions
$P_{AP}^{G\rightarrow G}(z)$ and $P_{AP}^{q\rightarrow G}(z)$ have the 
lowest $z$-power ($\sim 1/z$), while $P_{AP}^{q\rightarrow q}(z)$ and 
$P_{AP}^{G\rightarrow q}(z)$ vanish in the DLLA. Therefore, 
the DLLA diagram consists of the gluon ladders, since any transition of 
gluon$\rightarrow$ quark breaks the ladder-structure. Similarly, we also 
find (see the Appendices A and B) that only 
$P_i^{GG\rightarrow G}$, $P_i^{q\overline{q}\rightarrow G}$ 
and $P_i^{qG\rightarrow G}$ give the leading contributions in the DLLA 
since they have the factor $1/z=(x_1+x_2)/x_B$. 
Thus, we can conclude that any transition of $G\rightarrow q$ or
$GG\rightarrow q$ is suppressed in the DLLA-approach.

	Now let us remember the GLR equation, which has following form 
in [3]:

$$\frac{dx_BG(x_B,Q^2)}{d\ln Q^2}$$
$$=\frac{3\alpha_s}{\pi}\int^1_{x_B}dx_1G(x_1,Q^2)$$
$$-5.05(\frac{\alpha_s}{RQ})^2\int_{x_B}^{x_0}
\frac{dx_1}{x_1}[x_1G(x_1, Q^2)]^2, \eqno(18.a)$$
and

$$\frac{dx_BS(x_B,Q^2)}{d\ln Q^2}$$
$$=\frac{\alpha_s}{4\pi}\int^1_{x_B}\frac{dx_1}{x_1}\frac{x_B}{x_1}
x_1G(x_1,Q^2)$$
$$-0.17\frac{\alpha_s^2(Q^2)}{160R^2Q^2}[x_BG(x_B,Q^2)]^2-
0.32\frac{\alpha_s(Q^2)}{Q^2}\int_{x_B}^{x_0}\frac{dx_1}{x_1}\frac{x_B}{x_1}
P_{MQ}^{GG\rightarrow q\overline{q}}x_1H(x_1,Q^2), \eqno(20.b)$$
with

$$\frac{dx_1H(x_1,Q^2)}{d\ln Q^2}$$
$$=-5.05(\frac{\alpha_s}{RQ})^2\int_{x_B}^{x_0}
\frac{dz}{z}[zG(z,Q^2)]^2, \eqno(20.c)$$
and

$$P_{MQ}^{GG\rightarrow q\overline{q}}(z)=-2z+15z^2-30z^3+18z^4. \eqno(20.d)$$

As we have emphasized, the DLLA inhibits any transition of 
$G\rightarrow q$ or $GG\rightarrow q$. Therefore, Ref.~[3] takes some
special treatments to realize the above mentioned transition. For 
example, the last term on the right-hand side 
of Eq. (20.b) with Eq. (20.c) comes from a contribution of the next leading 
logarithms in $Q^2$ and is inconsistent with the DLLA. On the other hand,  
$\delta(x_1-x_B)$ is inserted by hand in the derivation of the second term 
of Eq. (20.b). However, $\delta(x_1-x_B)$ implies a cut line through
an initial gluon line with momentum $x_1$ and is really the contribution of the  
virtual diagram to $(d/d\ln Q^2)x_BG(x_B,Q^2)$; this contribution 
should therefore be canceled in the proton [4].
Even if we forget the above mentioned two questions in Eq. (20),
we still cannot prevent the linear term in Eq. (20) from mixing the 
quark line with the gluon ladder at each $Q^2_i$. In consequence, either the 
evolution from $Q^2_i$ to $Q^2_i+\Delta Q^2$ will stop in the DLLA-manner
due to the inhibition of gluon$\rightarrow$quark, 
or the DLLA as well as the AGK cutting rules, which regard the gluon ladder 
as the pomeron [1], can not be applied in the GLR equation. 

	Unlike Eq. (20), our new equation (19) includes
the transition of gluon$\rightarrow$quark in the whole $x$ region.
Another important difference between Eqs. (19) and (20) is that the positive 
antishadowing effects are separated from the negative shadowing effects in 
Eq. (19) because they have different kinematical domains in $x$.

	We emphasize that the DLLA is an unreasonable approximation even 
in Eq. (19.a). In fact, we take the DLLA and keep the leading term 
$\sim 1/z$ in Eq. (19.a). In this case, this equation can be simplified as 

$$\frac{dx_BG(x_B,Q^2)}{d\ln Q^2}$$
$$=\frac{3\alpha_s}{\pi}\int^1_{x_B}dx_1G(x_1,Q^2)$$
$$+2.4(\frac{\alpha_s}{RQ})^2\int_{x_B/2}^{1/2}\frac{dx_1}{x_1}
[x_1G(x_1, Q^2)]^2-4.8(\frac{\alpha_s}{RQ})^2\int_{x_B}^{1/2}
\frac{dx_1}{x_1}[x_1G(x_1, Q^2)]^2, \eqno(21)$$
where the upper-limit of the integral in the negative interfering terms 
is $1/2$ due to the restriction $x_1+x_2\leq 1$. 
There is such gluon density, for example, $xG(x,Q^2)\sim x^{\lambda_c}
(1-x)^7$, where the net recombination effects disappear
due to the balance of the shadowing- and antishadowing-effects.
The calculation shows that $\lambda_c=1$ and $0.5$ for Eq. (19.a) and 
Eq. (21), respectively. Therefore, we can find that the DLLA obviously 
distorts the recombination effects even in the gluon evolution equation
at small $x$.

	Finally, we estimate the size of the gluon recombination effect.
On can use

$$W=\frac{nonlinear~terms}{linear~terms}\geq \alpha_s, \eqno(22)$$
to determine the kinematic range, where the nonlinear effects can not be 
neglected in Eq. (19). The approximation solutions of Eq. (22) are

$$xG(x,Q^2)\geq\frac{(RQ)^2}{2.4}, ~~~~for~Eq.~(19.a), $$
$$~~~~~~~~~~~~~~~\geq\frac{(RQ)^2}{1.13},~~~~for~Eq.~(19.b). \eqno(23)$$

The HERA data show that $xG(x,Q^2)\sim 2-10$ for $Q^2=(1-10)~GeV^2$ and 
$x=10^{-3}$. Thus, we can expect that the gluon recombination effects will 
be appeared in the proton structure function in the range $Q^2<10~GeV^2$ and 
x$<10^{-3}$ if $R<<1$ fm (i.e., the gluon distribution in the proton 
has the "hot spots"-structure).
	
	In summary, the coefficients of the nonlinear terms in a modified 
Altarelli-Parisi evolution equation with gluon recombination are determined 
in the leading logarithmic ($Q^2$) approximation. The results are valid in 
the whole x region. The comparisons of the new evolution equation with the 
GLR equation are presented.  We expect that the gluon recombination effects
of the new equation can be appeared in the proton structure function 
in the HERA-domain of small $x$ and low $Q^2$ if the "hot spots" structure 
exists in the proton.

\vspace{0.3cm}

\noindent {\bf Acknowledgments}:

We would like to thank Jianwei Qiu for very helpful discussions and 
pointing out Eqs. (6)-(7). 
We would also like to acknowledge D. Indumathi for useful 
comments. This work was supported by National Natural Science Foundation of 
China and `95-Climbing' Plan of China.

\newpage
\begin{center} \Large {\bf Appendix A}
\end{center}
\normalsize
We give the recombination functions in Eq. (1) for the case of $x_1=x_1'$ 
and $x_2=x_2'$ as follows.  

$$\sum_iP_i^{GG\rightarrow G}(x_1,x_2,x_B)$$
$$=\int\sum_iP_i^{GG\rightarrow GG}(x_1,x_2,x_3,x_4)
\delta(x_1+x_2-x_3-x_4)\delta(x_3-x_B)dx_3dx_4$$
$$+\int\sum_iP_i^{GG\rightarrow GG}(x_1,x_2,x_3,x_4)
\delta(x_1+x_2-x_3-x_4)\delta(x_4-x_B)dx_3dx_4$$
$$=2\alpha_s^2\frac{C_A^2}{N^2-1}(x_1+x_2-x_B)(255x_1^2x_2^4
+255x_1^4x_2^2+146x_1x_2^5+146x_1^5x_2+270x_1^3x_2^3$$
$$+64x_1^2x_B^4-132x_1^5x_B-136x_1^3x_B^3+186x_1^4x_B^2-132x_2^5x_B
-136x_2^3x_B^3+186x_2^4x_B^2+64x_2^2x_B^4$$
$$+58x_1^6+58x_2^6-248x_1^2x_2x_B^3+510x_1^2x_2^2x_B^2+351x_1x_2^3x_B^2
-248x_1x_2^2x_B^3+351x_1^3x_2x_B^2$$
$$-401x_1^3x_2^2x_B-283x_1^4x_2x_B-401x_1^2x_2^3x_B-283x_1x_2^4x_B
+64x_1x_2x_B^4)/[8x_Bx_1^2x_2^2(x_1+x_2)^3]. \eqno(A.1)$$

One can get the general result for $x_1\ne x_1'$ and $x_2\ne x_2'$ if
we replace $x_1$ and $x_2$ by $\sqrt{x_1x_1'}$ and $\sqrt{x_2x_2'}$
in Eq. (A.1).

$$\sum_iP_i^{GG\rightarrow q}(x_1,x_2,x_B)$$
$$=\int\sum_iP_i^{GG\rightarrow q\overline{q}}(x_1,x_2,x_3,x_4)
\delta(x_1+x_2-x_3-x_4)\delta(x_3-x_B)dx_3dx_4$$
$$+\int\sum_iP_i^{GG\rightarrow q\overline{q}}(x_1,x_2,x_3,x_4)
\delta(x_1+x_2-x_3-x_4)\delta(x_4-x_B)dx_3dx_4$$
$$=2\alpha_s^2\left\{2\frac{T_f}{N(N^2-1)}\left[
\frac{(x_1+x_2-x_B)^2(4x_B^2+2x_1x_2+x_2^2-2x_2x_B+2x_1^2-4x_1x_B)}
{(x_1+x_2)^3x_2^2}\right.\right.$$
$$\left.+\frac{(x_1+x_2-x_B)^2(4x_B^2+2x_1x_2+x_1^2-2x_1x_B
+2x_2^2-4x_2x_B)}{(x_1+x_2)^3x_1^2}\right]$$
$$\left.+8\frac{NT_f}{(N^2-1)^2}
\frac{(x_1+x_2-x_B)^2(4x_B^2-3x_1x_B-3x_2x_B+x_1^2+2x_2X-1+x_2^2)}
{(x_1+x_2)^3x_2x_1}\right\}, \eqno(A.2)$$
where we don't distinguish quark and antiquark.
In the above mentioned equations, we have not includes the contributions of 
the $s$-channel and corresponding interference terms, since they integrate 
to zero.

\newpage
\begin{center} \Large {\bf Appendix B}
\end{center}
\normalsize
For simplicity, we present the recombination functions of quark-quark
and quark-gluon  
at $x_1=x_2$ and $x_1'=x_2'$, where the contributions of the $s$-channel
are included due to the asymmetry of the two-parton distribution under 
$x_1\leftrightarrow x_2$.

$$\sum_iP_i^{qq\rightarrow q}(x_1,x_B)
=\frac{2}{9}\frac{(2x_1-x_B)^2}{x_1^3}. \eqno(B.1)$$

$$\sum_iP_i^{q\overline{q}\rightarrow q}(x_1,x_B)
=\frac{1}{54}\frac{(2x_1-x_B)(6x_1^2-x_1x_B+12x_Bx_1^2-x_B^2)}{x_1^3}. 
\eqno(B.2)$$

$$\sum_iP_i^{q\overline{q}\rightarrow \overline{q}}(x_1,x_B)
=\frac{1}{54}\frac{(2x_1-x_B)(6x_1^2-x_1x_B+12x_Bx_1^2-x_B^2)}{x_1^3}. 
\eqno(B.3)$$

$$\sum_iP_i^{qG\rightarrow q}(x_1,x_B)
=\frac{7}{144}\frac{(2x_1-x_B)(10x_1-x_B)}{x_1^3}. \eqno(B.4)$$

$$\sum_iP_i^{qG\rightarrow G}(x_1,x_B)
=\frac{1}{144}\frac{(2x_1-x_B)^2(36x_B^2+62x_1^2-39x_Bx_1)}{x_Bx_1^4}. 
\eqno(B.5)$$

$$\sum_iP_i^{q\overline{q}\rightarrow G}(x_1,x_B)
=\frac{-1}{27}\frac{(2x_1-x_B)(8x_1^4-25x_1^2x_B^2-4x_1^3x_B+18x_B^3x_1
-9x_B^4)}{x_Bx_1^5}. \eqno(B.6)$$


\newpage
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\end{thebibliography}

\newpage
\noindent {\bf Figure Captions}

\noindent Fig. 1 The diagrams contributing to a new modified Altarelli-Parisi
equation with gluon recombination. Here the shaded part implies the 
correlation of gluons at short-distance and $"x"$ means the probing place.

\noindent Fig. 2 The two-gluon-to-two-gluon subprocess in recombination 
function.

\noindent Fig. 3 A $t$-channel diagram for $GG\rightarrow G$.

\noindent Fig. 4 A $s$-channel diagram for $GG\rightarrow G$.

\noindent Fig. 5 A $t$-channel diagram for $GG\rightarrow q$.

\noindent Fig. 6 A dominant splitting process in the DLLA, where 
$x_B\ll x_1$ leads to a strong order in $x$.

\end{document}



