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LPNHE/LPTPE 2709-00-12

LPT-ORSAY 00-76

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\vspace{.7CM}

{\Large \bf  Soft gluon cascades and BFKL equation \\}
\vskip1.7cm
{\Large Andrei Shuvaev}\\
\vskip.7cm
{\it St. Petersburg Nuclear Physics Institute \\
188350, Gatchina, St. Petersburg district, Russia}
\vskip.3cm
{\tt e-mail: shuvaev@thd.pnpi.spb.ru}

\vskip1.7cm
{\Large Samuel Wallon}\\
 \vskip.7cm
{\it LPNHE, Universit\'e P. \& M. Curie, 4 Place Jussieu \\
 75252 Paris Cedex 05, 
France}\\
{\rm and} \\
{\it Laboratoire de Physique Th\'eorique \\
Universit\'e Paris XI, Centre d'Orsay, b\^at. 211 \\
91405 Orsay Cedex, France}
\vskip.3cm
{\tt e-mail: Samuel.Wallon@th.u-psud.fr}

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\vspace*{1.5cm}

\begin{abstract}
In this paper we deal with high energy scattering in the Regge limit.
In the leading logarithmic
approximation of perturbative QCD, we show that BFKL equation and its 
generalizations 
can be simply recovered
as an evolution equation for gluon density in soft gluons cascade.
Comparison is performed with respect to QED case, and the running of
the coupling is discussed.
\end{abstract}


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\section{Introduction}

Investigation of hard scattering amplitude in the kinematics where
 invariant energy $s$ is large while transfered momentum $t$ is fixed is
 an important problem of QCD. The solution of this problem in the
 Leading Logarithmic Approximation (LLA), collecting terms of the form
 $(\alpha_S \, \ln s)^n\, ,$ was proposed about twenty five years ago,
in the BFKL equation \cite{BFKL}. These original papers where based on
 an effective triple gluons vertex and bootstrap condition in
 $t-$channel.
This approach shew the relation between perturbative QCD and reggeon
 calculus 
which was proposed a decade before. The main feature of this work
is the reggeization of gluon, which appears not to be elementary
but composite, as being a pole in complex momentum plane, with color octet
 quantum number.  Another 
important result is that Pomeron occurs as a bound state of two
 reggeized
gluons in singlet color channel.
Although BFKL equation looks like a ladder type equation, it effectively
sums up an infinite number of $t-$channel gluons.

A few years ago a dipole picture was proposed 
\cite{mueller94,muellerpatel,muellerchen,muellerunitarite,nik,nikital}. It is
 based on subsequent emission of soft gluons, which turns out to iterate 
in the large $N_c$ limit. In this approach elementary degrees of freedom
 are made of colored dipole. The equation for dipole density in this soft cascade
turns out to be identical to the BFKL equation, although it was 
found in the multicolor limit, while original BFKL was derived for
 finite $N_c.$

The aim of this paper is to show that BFKL results can be simply recovered
only using this classical soft emission vertex used in the dipole
model. 
We deal only with gluons for finite $N_c$ rather than with dipoles
at large $N_c.$ We obtain BFKL equation
as an evolution equation for the gluon density in the cascade. This
equation
is similar to DGLAP \cite{GL,AP,GAltarelli,dok} except that the evolution variable is $\ln 1/x$
rather than $\ln Q^2\,.$


Our paper goes as follow. After some preliminary recall about linear classical 
cascades properties, we generalize it to incorporate non linearity.
The main idea is that despite the complicated internal structure of the cascade,
the amplitude of soft emission is the same as the one off a single gluon,
except for some kind of color charge dressing. Using this idea we
rederive BFKL equation for both forward and non forward
cases. Generalization
to the multi-gluon $t-$channel case is straightforward. The same
technique, when applied to QED, gives the same equation except for
reggeization.
\\

Consider the scattering of two particles.
 The asymptotic behavior of the
cross section is determined by the parton cascade from incoming particles
or, in other words, by their wavefunction on the light cone.
For large invariant energy $s$ (the so called
 Regge limit), the amplitude is dominated by
$t$-channel gluons rather than quarks, because of their largest
spin. As a consequence gluons dominates in every channel, and quark
 contributions will be neglected below.

We shall consider a semi-hard region where gluons can be treated to be
 soft with respect to the large invariant energy $s,$ although still
 perturbative. It is known that in any massless field theory scattering
 amplitudes are dominated by IR contributions of two types, namely
 collinear and
soft singularities. This semi-hard region reveals soft gluons
 contributions, which sum up into LLA.



The physics
behind soft resummation is that the field of
ultrarelativistic source can be treated as a state of almost real particles
while the vertex of the real or virtual soft gluon emission can be taken
to be the same. 

Let us denote 
$p_A$ and $p_B$ the momenta of colliding particles $A$ and $B.$ 
Neglecting their masses in the high energy limit we can take
$p_A^2\,=\,p_B^2\,=0$ so that the invariant energy reads $s\,=\,2p_A\cdot p_B$.
Vectors $p_A$, $p_B$ can be taken for Sudakov decomposition,
$k=\alpha p_A + \beta p_B + k_\perp$. It is more convenient sometimes
to deal with dimensionless vectors $n_{A,B}=\sqrt{2/s}\,p_{A,B}$, for which
$n_A^2\,=\,n_B^2\,=0$, $n_A\cdot n_B=1$ and $k = k_-n_A + k_+n_B +
k_\perp$,
with 
$k_-=k\cdot n_B$ and  $k_+=k\cdot n_A$.

Each particle develops a parton cascade, so the scattering of the incoming
particles $A$ and $B$ could be in principle reduced to elementary parton
processes. However the complete description of the cascade is a very
complicated problem involving all perturbative as well as nonperturbative
effects. There are certain limits, where the cascade looks more simple.
A well known example is  deep inelastic scattering. For large
virtuality $Q^2$ the parton density obeys the DGLAP evolution equation
collecting the powers of $\alpha_S \, \ln Q^2$. Although predictions
based on the DGLAP
evolution yield good agreement with the present experimental data,
this approach fails for parametrically small Bjorken variables $x$, when
the powers of $\alpha_S \, \ln 1/x$ dominate. This is the region of Regge kinematics
where the total invariant energy is much greater than the other invariants,
including the virtuality of the deeply virtual photon. The leading $\ln 1/x$
behavior of the hard amplitude is given in this region by the BFKL theory.

An important feature of the Regge kinematics is that the partons are soft there
in a sense that the main contribution appears for small longitudinal
variables, $\alpha, \beta \ll 1$. The vertex for emitting a soft particle is rather
universal and determined by the classical current proportional to the momentum
of a source. For emission off the particle $B$ (see
Fig.\ref{figsoftvertex}) we have

% Fig.1
\begin{figure}[htb]
\begin{picture}(5000,5000)(-12000,2000)
\drawline\gluon[\E\REG](500,1000)[8]
\global\advance \pfrontx by -1000
\global\advance \pfronty by -1000
\global\advance \pbackx by 300
\global\advance \pbacky by -1000
\put(\pbackx,\pbacky){$a$}
\put(\pfrontx,\pfronty){$a^\prime$}
\drawline\gluon[\NE\REG](\pmidx,\pmidy)[4]
\global \advance \pbacky by 500
\global\advance \pmidx by 1500
\global\advance \pmidy by -600
\put(\pbackx,\pbacky){$c$}
\put(\pmidx,\pmidy){$k$}
\end{picture}
\vskip 0.5cm
\caption{Soft vertex}
\label{figsoftvertex}
\end{figure}
\vskip 0.5cm
\begin{equation}
\label{sv}
\Gamma^\lambda_c(k)\,=\,g\,\frac{p_B\cdot \varepsilon^\lambda(k)}{p_B k +
i\delta}\,T_c\,=\, g\,\frac{p_B\cdot \varepsilon^\lambda(k)}{\alpha \frac s2 +
i\delta}\,T_c,
\end{equation}
where $\varepsilon^\lambda(k)$ is the helicity vector of the soft emitted or
absorbed gluon carrying momentum $k$ and color index $c$.
The matrix $T_c$ depends on the re\-pre\-sen\-ta\-tion of the color group.
For gluon like object it is expressed through the structure constants,
$T^c_{a^\prime a} = i f_{a^\prime c a}$, the indices $c,a^\prime,a$ being
in the adjoint representation.


% Fig.2
\begin{figure}[htb]
\begin{picture}(5000,5000)(-6000,2000)
\thicklines
\drawline\fermion[\E\REG](500,1000)[6000]
\thinlines
\global\advance \pfrontx by -1000
\global\advance \pfronty by -1000
\put(\pfrontx,\pfronty){$a^\prime$}
\drawline\gluon[\NE\REG](\pbackx,\pbacky)[4]
\global \advance \pbacky by 500
\global\advance \pmidx by 1000
\global\advance \pmidy by -600
\put(\pbackx,\pbacky){$c_3$}
\put(\pmidx,\pmidy){$k_3$}
\thicklines
\drawline\fermion[\E\REG](\fermionbackx,\fermionbacky)[6000]
\thinlines
\drawline\gluon[\NE\REG](\pbackx,\pbacky)[4]
\global \advance \pbacky by 500
\global\advance \pmidx by 1000
\global\advance \pmidy by -600
\put(\pbackx,\pbacky){$c_2$}
\put(\pmidx,\pmidy){$k_2$}
\thicklines
\drawline\fermion[\E\REG](\fermionbackx,\fermionbacky)[6000]
\thinlines
\drawline\gluon[\NE\REG](\pbackx,\pbacky)[4]
\global \advance \pbacky by 500
\global\advance \pmidx by 1000
\global\advance \pmidy by -600
\put(\pbackx,\pbacky){$c_1$}
\put(\pmidx,\pmidy){$k_1$}
\thicklines
\drawline\fermion[\E\REG](\fermionbackx,\fermionbacky)[6000]
\thinlines
\global\advance \pbackx by 300
\global\advance \pbacky by -1000
\put(\pbackx,\pbacky){$a$}
\end{picture}
\vskip 0.5 cm
\caption{Amplitude for $n$ gluon emission or absorption}
\label{wilson}
\end{figure}
The amplitude for emission or absorption of $n$ gluons (see
Fig.\ref{wilson}) is given by the product of the elementary vertices,
\begin{equation}
\label{nga}
\Gamma^{\mu_1,\ldots \mu_n}_{c_1,\ldots c_n}(k_1,\ldots
k_n)\,A_{\mu_1,c_1}(k_1) \cdots \,A_{\mu_n,c_n}(k_n)\,=\,(T^{c_1}\cdots
T^{c_n})_{a a'}
\times
\end{equation}
$$
\times\,g\frac{p_B\cdot A_{c_1}(k_1)}{\alpha_1\frac s2 + i\delta}\,
g\frac{p_B\cdot A_{c_2}(k_2)}{(\alpha_1+\alpha_2)\frac s2 +
i\delta}\, \cdots \,
g\frac{p_B\cdot A_{c_n}(k_n)}{(\alpha_1+\alpha_2+\ldots
+\alpha_n)\frac s2 + i\delta}\,.
$$
Here the field $A$ is the asymptotic free field
$$ 
A_{\mu,c}(x)= \frac{1}{(2 \pi)^{3/2}} \int \frac{d^3 k}{2 k_0} \left [ e^{i
k x} \varepsilon_{\mu}^{\lambda}(k) \, a^+_{\lambda,c}(k) + e^{-i
k x} \varepsilon_{\mu}^{\lambda}(k) \, a_{\lambda,c}(k) \right ] \,,
$$
whose positive and negative frequency part corresponds respectively to
emission and absorption of gluons.
The longitudinal momenta are supposed to be small in the formula
(\ref{nga}),
namely 
$\alpha,\beta \ll 1$, while
\begin{equation}
\label{kin}
\alpha s,\beta s \gg k_\perp^2, \qquad \alpha\beta s\sim k_\perp^2\,.
\end{equation}
$k_\perp$ is a typical transverse momentum scale.
These conditions determine the kinematical region from where the large
infrared contribution comes. 

Using the relation
$$
\frac 1 {\alpha_1\frac s2 + i\delta}\,=\,-i\int_{-\infty}^\infty dz\,
e^{i \alpha \frac s2 z} \theta(z)
$$
we get
$$
\int d^4k_1\cdots d^4k_n\,\Gamma^{\mu_1,\ldots \mu_n}_{c_1,\ldots c_n}(k_1,
\ldots k_n)\,A_{\mu_1,c_1}(k_1)\, \cdots \,A_{\mu_n,c_n}(k_n)\,=
$$
$$
=\,\frac{(-ig)^n}{n!} \int dz_1\cdots dz_n\,{\rm P}\bigl[n_B\cdot A(z_1 n_B)\,\cdots\,
n_B\cdot A(z_n n_B)\bigr]_{a^\prime a},
$$
where
$$
A_\mu(x)\,=\,\int d^4k\,e^{ikx}A_{\mu,c}(k) T^c
$$
and the symbol P means the conventional ordering of the fields. The sum over the
arbitrary number of soft gluons is given by the P-exponent along the momentum
direction of the parent particle:
$$
\sum_n
\int d^4k_1\cdots d^4k_n\,\Gamma^{\mu_1,\ldots \mu_n}_{c_1,\ldots c_n}(k_1,
\ldots k_n)\,A_{\mu_1,c_1}(k_1)\, \cdots \,A_{\mu_n,c_n}(k_n)\,=
$$
$$
=\,{\rm P}\exp\bigg[-ig \int_0^\infty dz\,n_B\cdot A(z n_B)\bigg].
$$

There is a simple modification of this expression allowing to write the
amplitude to emit any number of soft gluons with fixed total transverse
momentum $p_\perp$. To this end we have to multiply the $n$ gluon amplitude
(\ref{nga}) by the appropriate $\delta$-function,
$$
\delta(p_\perp - \sum k_{n \perp})\,=\,\frac 1{(2\pi)^2}\,\int d^2x\,e^{-i x
p_\perp}\, e^{i x \sum k_{n \perp}},
$$
so that each field gains the exponential factor, $A_{\mu,c}(k) \to A_{\mu,c}(k)
e^{i x k_\perp}$. It leads to the final amplitude of the form
\begin{equation}
\label{Pline}
\Gamma(n_B,p_\perp)\,=\,\frac 1{(2\pi)^2}\,\int d^2x\,e^{-i x p_\perp}\,
{\rm P}\exp\bigg[-ig \int_0^\infty dz\,n_B\cdot A(z n_B+x_\perp)\bigg].
\end{equation}
As a consequence of the soft emission any gluon creation operator,
$a^+_\sigma(q)=a^+_{\sigma c}(q)T_c$, is multiplied by the soft factor,
$a^+_\sigma(q) \to a^+_\sigma(q)\Gamma(n_q,p_\perp)$, with the vector
$n_q$ taken along the direction of the parent momentum $q$.

\section{BFKL equation}

The above formulae do not take into account the subsequent decays of the
emitted gluons. To get the total cascade amplitude each field in the $P$
exponent has to be
replaced by the $P$ exponent itself. This turns into a complicate
non-linear equation, which takes into account only tree-like
contributions.
A more difficult problem is to incorporate loop corrections. In the
LLA approximation loop corrections are responsible for reggeization of
the gluon.

So long as we are interested in the soft contribution
we assume that the gluons are emitted "from the ends" by a bremstrahlung like
mechanism.
At the same time the Regge kinematics (\ref{kin}) ensures the soft vertex to be
of the form (\ref{sv}) regardless from where the particular gluon is emitted
off (or where it is absorbed in).  

Indeed, the emission of a gluon with momentum $k_2=\alpha p_A+\beta
p_B+k_{2\perp}$ off a  parent gluon carrying momentum $k_1=a p_A+b p_B
+k_{1\perp}$ is again given by a soft vertex similar to (\ref{sv})
\begin{equation}
\label{softvertex}
\Gamma^\lambda_c(k_2)\,=\,g\,\frac{k_1\cdot
\varepsilon^\lambda(k_2)}{k_1\cdot k_2}\,T_c.
\end{equation}
In the light cone gauge, $p_A\cdot A(x)=0$, and the polarization vector reads
\begin{equation}
\label{eps}
\varepsilon^{\lambda}_{\mu}(k)\,=\,\varepsilon^{\lambda}_A(k)
\cdot p_{A,\mu}\,+\,\varepsilon^{\lambda}_{\perp \mu}(k)
\end{equation}
$$
\sum_\mu\varepsilon^{\lambda}_{\perp \mu}(k)\cdot
\varepsilon^{\lambda^\prime}_{\perp \mu}(k)\,=\,
\delta^{\lambda \lambda^\prime}.
$$
The emitted gluons which dominate the amplitude in the Regge limit are
soft and thus quasi real.  
The transversality condition for these quasi-real gluons reads
$k\cdot\varepsilon = 0$, which
implies for the helicity vector (\ref{eps})
\begin{equation}
\label{keperp}
\varepsilon^{\lambda}_A(k)\,=\,2\,
\frac{k_\perp \cdot \varepsilon^{\lambda}_{\perp }(k)}{\beta s}\,.
\end{equation}
In this soft gluon approximation, $\beta \ll b$ and assuming the typical transverse momenta to
be of the same order, $k_{1\perp}^2 \sim k_{2\perp}^2$,
 the numerator of the soft vertex is approximated as
$$
k_1\cdot \varepsilon^\lambda(k_2)\,=\,b \,\varepsilon^\lambda_A\,
\frac s2-k_{1\perp} \varepsilon^\lambda_\perp\,\approx \, b \,p_B \cdot
\varepsilon^\lambda(k_2).
$$
Using the mass shell conditions, $a=k_{1\perp}^2/b s$,
$\alpha=k_{2\perp}^2/\beta s,$ the denominator also simplifies as
$$
k_1\cdot k_2\,=\,(a \,\beta + b \,\alpha)\,\frac s2 \,-\,k_{1\perp}\cdot
k_{2\perp}\,\approx \, b \, \alpha\,\frac s2 \,.
$$
It follows that the vertex (\ref{softvertex}) can be written as
$$
\Gamma^\lambda_c(k_2)\,=\,g \frac{p_B. \varepsilon^{\lambda}(k_2)}{p_B
. k_2} T_c \,,
$$
which has exactly the universal form (\ref{sv}). It confirms the
physical picture that soft emission is determined by the total current
of the source rather than by its internal structure.
The vector $p_B$ is the total momentum of the cascade.

Thus after imposing the strong ordering of the longitudinal momenta between
parent and daughter partons the soft cascade tree amplitude takes a form
similar to the P-exponent form (\ref{nga}),
\begin{equation}
\label{Gb}
\Gamma^{\mu_1,\ldots, \mu_n}_{c_1,\ldots, c_n}(k_1,\ldots, k_n)\,=\,
\Gamma_{c_1,\ldots, c_n}(k_{1\perp},\ldots, k_{n\perp})\,\frac{p_B^{\mu_1}}{p_B k_1}\,
\cdots\,\frac{p_B^{\mu_n}}{p_B k_n},
\end{equation}
where the functions $\Gamma_{c_1,\ldots, c_n}$ incorporate both
color structures and transverse momentum delta-functions for the cascade
branches.


The longitudinal momenta ordering of the subsequent emissions allows 
to use a kind of parton model for the hard
amplitude in the Regge limit. The gauge $p_A\cdot A=0$ suppresses soft
emission off the particle $A$, and the amplitude looks like the scattering
of the particle $A$ on the soft cascade produced by the particle $B$.

% Fig.3
\begin{figure}[htb]
\begin{picture}(5000,12000)(-10000,0)
\global \Xone = 1000               % radius of the circles
\global \Yone = \Xone
\global \multiply \Yone by 2        % diameter of the circles
\THICKLINES
\drawline\fermion[\NE\REG](500,500)[3000]
\drawarrow[\LDIR \ATBASE](\pmidx,\pmidy)
\THINLINES
\global\advance\pfrontx by -1500
\global\advance\pfronty by -1500
\put(\pfrontx,\pfronty){$p_B+\frac{l_\perp}2$}
\startphantom
\drawline\fermion[\NE\REG](\pbackx,\pbacky)[\Xone]
\stopphantom
\put(\pbackx,\pbacky){\circle{\Yone}}
\startphantom
\drawline\fermion[\NE\REG](\pbackx,\pbacky)[\Xone]
\stopphantom
\drawline\gluon[\NE\REG](\pbackx,\pbacky)[3]
\global\advance\pmidx by -5000
\put(\pmidx,\pmidy){$k+\frac{l_\perp}2$}
\global\advance\pfrontx by 1500
\global\advance\pfronty by 500
\put(\pfrontx,\pfronty){$\mu $}
\startphantom
\drawline\fermion[\NE\REG](\pbackx,\pbacky)[\Xone]
\drawline\fermion[\W\REG](\pbackx,\pbacky)[\Xone]
\stopphantom
\THICKLINES
\drawline\fermion[\W\REG](\pbackx,\pbacky)[4000]
\drawarrow[\E\ATBASE](\pmidx,\pmidy)
\THINLINES
\global\advance\pbackx by -3000
\global\advance\pbacky by 1000
\put(\pbackx,\pbacky){$p_A-\frac{l_\perp}2$}
\startphantom
\drawline\fermion[\NE\REG](\gluonbackx,\gluonbacky)[\Xone]
\stopphantom
\put(\pbackx,\pbacky){\circle{\Yone}} % upper circle
\startphantom
\drawline\fermion[\SE\REG](\pbackx,\pbacky)[\Xone]
\stopphantom
\drawline\gluon[\SE\REG](\pbackx,\pbacky)[3]
\global\advance\pmidx by 1500
\put(\pmidx,\pmidy){$k-\frac{l_\perp}2$}
\startphantom
\drawline\fermion[\NW\REG](\gluonfrontx,\gluonfronty)[\Xone]
\drawline\fermion[\E\REG](\pbackx,\pbacky)[\Xone]
\stopphantom
\THICKLINES
\drawline\fermion[\E\REG](\pbackx,\pbacky)[4000]
\drawarrow[\E\ATBASE](\pmidx,\pmidy)
\THINLINES
\global\advance\pbackx by -1000
\global\advance\pbacky by 1000
\put(\pbackx,\pbacky){$p_A+\frac{l_\perp}2$}
\startphantom
\drawline\fermion[\SE\REG](\gluonbackx,\gluonbacky)[\Xone]
\stopphantom
\put(\pbackx,\pbacky){\circle{\Yone}}
\startphantom
\drawline\fermion[\SW\REG](\pbackx,\pbacky)[\Xone]
\stopphantom
\global \negate \fermionlengthx
\global\advance\gluonlengthx by \fermionlengthx
\global\multiply \gluonlengthx by 2
\drawline\fermion[\W\REG](\pbackx,\pbacky)[\gluonlengthx]
\global\advance\pmidx by -1000
\global\advance\pmidy by 200
\put(\pmidx,\pmidy){$T^{\mu\nu}$}
\drawline\fermion[\W\REG](\gluonbackx,\gluonbacky)[\gluonlengthx]
\startphantom
\drawline\fermion[\SE\REG](\gluonbackx,\gluonbacky)[\Xone]
\drawline\fermion[\SE\REG](\pbackx,\pbacky)[\Xone]
\stopphantom
\THICKLINES
\drawline\fermion[\SE\REG](\pbackx,\pbacky)[3000]
\drawarrow[\LDIR \ATBASE](\pmidx,\pmidy)
\THINLINES
\global\advance\gluonbackx by -500
\global\advance\gluonbacky by -500
\global\advance\pbackx by -1500
\global\advance\pbacky by -1500
\put(\pbackx,\pbacky){$p_B-\frac{l_\perp}2$}
\global\advance\gluonbackx by -1600
\global\advance\gluonbacky by 1000
\put(\gluonbackx,\gluonbacky){$\nu$}
\end{picture}
\vskip 0.5cm
\caption{"Handbag" diagram for high energy scattering}
\label{handbag}
\end{figure}
The amplitude is given by a "handbag" diagram (see Fig.\ref{handbag}) 
with Born terms in the upper
blob convoluted with the part which can be formally presented as a bilocal
field correlator $T_{\mu \nu}$ taken over the particle $B$,
\begin{equation}
\label{corr}
T_{\mu \nu}\,=\,\int d^4x \, d^4y\,e^{-i(k+\frac l2)x+i(k-\frac l2)y}
\langle p_2|\,T\bigl\{A_\mu(x) A_\nu(y)\bigr\}\,|p_1\rangle \,,
\end{equation}
where $t=l^2 = -l_{\perp}^2.$ Here $p_{1,2} = p_B \pm {l_\perp}/2.$
Since Regge kinematics implies the longitudinal momenta to be strictly ordered,
the momentum fraction $\beta$ grows from the upper to the lower part of
the diagram. At the same time the $\alpha$ variable takes its largest value
at the top and vanishes at the bottom. That is why the momenta $\alpha$ of
the gluons incoming from  below and carrying momenta $k\pm l/2$ 
can be neglected in the
upper blob compared to $\alpha \sim 1$ of the initial particle $A$.
As a consequence the lower blob is actually needed only integrated over
$\alpha.$ It means that light cone time $\tau \equiv x_-,$ which is
conjugated
to $\alpha,$ has to be taken at zero.  In that case fields commute since
their arguments are separated by space-like interval, and $T$ product
can be removed.
There is a  direct relation between fields and parton creation and annihilation
operators at zero light cone time, namely
$$
\sqrt{\frac 2s}\,\frac 1{2 \beta}\varepsilon^\lambda_\mu(\beta,k_\perp)\,a_{\lambda c}^+(\beta,k_{\perp})\,=\,
\frac 1{(2\pi)^{\frac 32}}\int d^3x\,e^{-ikx}A_{\mu c}(0,x),
$$
where
\begin{eqnarray}
k_-\,&=&\,\beta p_B \cdot n_A \ge 0 \nonumber \\
k\,\cdot\,x\,&=&\,k_-x_+\,-\,k_\perp \,. x_\perp, \nonumber \\
d^3x\,&\equiv&\,dx_+dx_\perp \nonumber
\end{eqnarray}
(and the complex conjugated expression for annihilation operator).
These light cone  operators 
$a^+_{\lambda c}(\beta,q_\perp)$, $a_{\lambda c}(\beta,q_\perp)=
a^+_{\lambda c}(-\beta,-q_\perp)$ respectively creates (annihilates) a gluon with
 longitudinal momentum fraction $\beta \ge 0$, transverse momentum
$q_\perp$, polarization and color indices $\lambda$ and $c$:
$$
\bigl[a_{\lambda c}(\beta,q_\perp)\,,
\,a^+_{\lambda^\prime c^\prime}(\beta^\prime,q_\perp^\prime)\bigr]\,=\,
2\beta\, \delta_{\lambda \lambda^\prime}\delta_{c c^\prime}\,\delta(\beta-\beta^\prime)
\delta^{(2)}(q_\perp-q_\perp^\prime).
$$
As a result, the correlator (\ref{corr}) is expressed only in terms
of the light cone parton distribution, namely
\begin{equation}
\label{oldcorr}
T_{\mu \nu}\,\sim\,\varepsilon^\lambda_\mu(k_\perp+l_\perp/2)
\varepsilon^{\lambda^\prime}_\nu(k_\perp-l_\perp/2)\,
\langle p_2|\,a_{\lambda}^+(\beta,k_\perp+l_\perp/2)\,
a_{\lambda^\prime}(\beta,k_\perp-l_\perp/2)\,|p_1\rangle \,.
\end{equation}
It is important to note that the absence of time dependence in the
correlator (\ref{oldcorr}) allows to treat it in the old fashioned perturbation
theory on the light cone. In this case the state $|p\rangle$ can be
interpreted as a light cone wavefunction of the cascade including any
numbers of gluons and formally can be presented as
\begin{eqnarray}
\label{hp}
|p\rangle\,&=&\,\sum_n\sum_{\mu, c} \prod_i \int
d\beta_i \, d^2k_{\perp i}
\,\delta(\sum \beta_i - 1)
\delta^{(2)}(l_\perp/2- \sum k_{i\perp})  \\
&& \hspace{-1cm} \times \,\Gamma^{\mu_1,\ldots \mu_n}_{c_1,\ldots
c_n}(p;\beta_1,k_{\perp 1},\ldots, \beta_n,k_{\perp n})\,
:A_{\mu_1, c_1}(\beta_1,k_{\perp 1})\,\cdots\,A_{\mu_n,
c_n}(\beta_n,k_{\perp n}): \,|0\rangle . \nonumber
\end{eqnarray}
Thus the whole process reduces to the elementary scattering on the real parton
(gluon) carrying fixed longitudinal and transverse momenta. Integrating this
with the distribution of partons momenta yields the total amplitude. The only
difference with the conventional parton model is that here we deal with the
distribution of both longitudinal and transverse momenta.

The exact coefficient functions $\Gamma$ in Eq.(\ref{hp}) are the $n$ gluons components
of the total cascade wave function on the light cone. They are very complicated and involve
in principle all perturbative orders as well as the possible nonperturbative
effects. In LLA approximation,  loop perturbative corrections
 only modifies the coefficient $\Gamma_{c_1,\ldots c_n}$
in equation (\ref{Gb}). 
 Indeed, taking into account that the vertex of the soft emission
is determined in this approximation by the classical current
proportional to the particle momentum  the general structure still remains.
\\

Let us derive the evolution equation for the parton distribution.
We start from the case of zero transfered momentum $l.$ 
Consider the gluon cascade for the particle $B$.
The parton density in the cascade, $n(x,q_\perp)$,
is defined as the average value taken over the state $|p \rangle,$
\begin{equation}
\label{me}
\sum_{\sigma c}\langle p|\,\frac 1{2x}\,a^+_{\sigma c}(x,q_\perp)\,
a_{\sigma c}(x,q_\perp)\,|p^\prime\rangle\,=\,
n(x,q_\perp)\,2p_-\,\delta^{(3)}(p-p^\prime).
\end{equation}
where $\delta^{(3)}(p-p^\prime) = \delta(p_- - p'_-) \, 
\delta^{(2)}(p_\perp - p'_\perp)$. 
The function $n(x,q_\perp)$ having been integrated over the transverse
momenta yields the distribution of the parton longitudinal momenta
which coincides with the conventional gluon structure function in the light
cone gauge.

The total number of gluons with a given transverse momentum $q_\perp$
is evidently
\begin{equation}
\label{Q}
Q(q_\perp)\,=\,\int_0^1 dx\,n(x,q_\perp).
\end{equation}
Because of the infrared divergencies at small $x$ it is natural to introduce
a formal lower cut-off $x_\lambda$ for the longitudinal momenta, so
that all
$\beta_i > x_\lambda$ in the amplitude (\ref{hp}). As a result the gluon
number becomes cut-off dependent,
$$
Q(x_\lambda,q_\perp)\,2p_-\,\delta^{(3)}(p-p^\prime)\,=\,\int_{x_\lambda}^1
\frac{dx}{2x}\,
\sum_{\sigma c}\langle p,\,x_\lambda|\,a^+_{\sigma c}(x,q_\perp)\,a_{\sigma
c}(x,q_\perp)\, |p^\prime,\,x_\lambda\rangle,
$$
while the gluon density can be defined as the variation of this value
with respect to the cut-off,
$$
n(x_\lambda,q_\perp)\,=\,-\frac {\partial}{\partial x_\lambda}\,
Q(x_\lambda,q_\perp).
$$

Let us show that the evolution of the truncated gluon number
$Q(x_\lambda,q_\perp)$ is described in LLA by the BFKL equation.


Consider first the lowest order where one single gluon is emitted and
reabsorbed. The $\alpha$ integral which occur in this loop can be closed
around the pole of the emitted gluon propagator. Indeed, using the same
technique which leads to the eikonal type expression (\ref{sv}), the
denominator
of the integrand reads
$
(\alpha - i \delta)(\alpha \beta s - k_{\perp}^2 + i \delta)
$
which singularities in $\alpha$ plane leaves on the opposite
side of the real axis.
In the
numerator of the gluon propagator in light-cone gauge 
$$
d_{\mu\nu}(k)\,=\,-\varepsilon^\lambda_\mu(k)\, \varepsilon^\lambda_\nu(k)
\,-\,\frac{4k^2}{\beta^2s^2}\,p_{A,\mu}\, p_{A,\nu} \,,
$$
the second term drops out since it cancels the $\alpha$ singularity in
the propagator. 

It implies that in soft loops calculations, gluons can be considered as
real. This is in accordance with the fact that there is almost no
difference between real and virtual massless soft particles.

The general form of amplitude 
\begin{equation}
\label{Gperp}
\Gamma^{(n)}\,=\,\Gamma_{c_1,\ldots c_n}(q_{1 \perp},\ldots , \,q_{n \perp})\,
2\frac{q_{1\perp} \cdot \varepsilon^{\lambda_1}_{\perp }(q_{1 \perp})}
{q_{1\perp}^2}\,\cdots \,
2\frac{q_{n\perp} \cdot \varepsilon^{\lambda_n}_{\perp }(q_{n \perp})}
{q_{n\perp}^2}
\end{equation}
is thus valid for both real and virtual contributions. It only involves 
transverse vectors.


With the expression (\ref{Gperp}) one gets
\begin{eqnarray}
Q(x_\lambda,q_\perp)\,&=&\,\frac 1{(2\pi)^3}\sum_n \frac 1{(n-1)!}
\int 2\frac {d\beta_1}{\beta_1}\,
2\frac {d\beta_2}{\beta_2}\frac{d^2k_{2\perp}}{(2\pi)^3 k_{2\perp}^2}
\cdots 2\frac {d\beta_n}{\beta_n}\frac{d^2k_{n\perp}}{(2\pi)^3 k_{n\perp}^2}
\nonumber \\
&\times&\,\Gamma^2_{c_1,\ldots c_n}(\beta_1,q_\perp ; \beta_2,k_{2\perp}; \ldots ;
\beta_n,k_{n\perp})\nonumber \\
&\times&\,\delta(\beta_1+\beta_2+\cdots +\beta_n - 1)
\delta^{(2)}(q_\perp+k_{2\perp}+\cdots + k_{n\perp}), \nonumber
\end{eqnarray}
all longitudinal integrals being taken over the region $\beta_i > x_\lambda$.

Consider what happens when the cut-off value is lowered, that is when we
take $x_\lambda^\prime < x_\lambda$. The new cut-off allows for
the emission of an extra soft particle whoose longitudinal
momentum lies in the interval $x_\lambda^\prime < \beta < x_\lambda$. This
yields the new amplitude $\Gamma_\lambda^\prime = \Gamma_\lambda \cdot
2\frac{k_{\perp} \varepsilon^{\lambda}_{\perp }(k_n)}{k_{\perp}^2}$
and the new value of the gluon number,
\begin{eqnarray}
\label{fp}
Q(x_\lambda^\prime,q_\perp)\,2p_0\,\delta^3(p-p^\prime)\,&=&\,
\sum_{\sigma c}\langle p,\,x_{\lambda^\prime}\,| \,\int_{x_\lambda}^1
\frac{dx}{2x}\,
\,a^+_{\sigma c}(x,q_\perp)\,a_{\sigma c}(x,q_\perp)\, \\
\label{sp}
&+&\,\int_{x_{\lambda^\prime}}^{x_\lambda} \frac{dx}{2x}\,
a^+_{\sigma c}(x,q_\perp)\,a_{\sigma c}(x,q_\perp)\,
|\,p,\,x_{\lambda^\prime}\rangle.
\end{eqnarray}
These two contributions are displayed in Fig.\ref{figcutoff}.
\begin{figure}[htp]
\begin{picture}(10000,5000)(-10000,0)
\global \Xone = 500                % radius of the circle
\global \Yone = \Xone
\global \multiply \Yone by 2       % diameter of the circle
\THICKLINES
\drawline\fermion[\E\REG](500,500)[2000]
\THINLINES
\drawloop\gluon[\N 5](\pbackx,\pbacky)
\put(\loopmidx,\fermionbacky){\circle{\Yone}}
\negate \Xone
\global\advance\fermionbackx by \Xone
\negate \Xone
\THICKLINES
\drawline\fermion[\E\REG](\loopfrontx,\loopfronty)[\fermionbackx]
\drawline\fermion[\W\REG](\gluonbackx,\gluonbacky)[\fermionlengthx]
\drawline\fermion[\E\REG](\gluonbackx,\gluonbacky)[2000]
\THINLINES
\startphantom
\drawline\fermion[\E\REG](\pbackx,\pbacky)[2000]
\stopphantom
\put(\pbackx,\pbacky){$+$}
\startphantom
\drawline\fermion[\E\REG](\pbackx,\pbacky)[2000]
\stopphantom
\THICKLINES
\drawline\fermion[\E\REG](\pbackx,\pbacky)[2000]
\THINLINES
\drawloop\gluon[\N 3](\pbackx,\pbacky)
\global\advance \gluonbackx by \Xone
\put(\gluonbackx,\gluonbacky){\circle{\Yone}}
\global\advance \gluonbackx by \Xone
\drawloop\gluon[\E 3](\gluonbackx,\gluonbacky)
\global\negate\fermionbackx
\global\advance\gluonbackx by \fermionbackx
\global\negate\fermionbackx
\THICKLINES
\drawline\fermion[\E\REG](\fermionbackx,\fermionbacky)[\gluonbackx]
\drawline\fermion[\E\REG](\fermionbackx,\fermionbacky)[2000]
\THINLINES
\end{picture}
\vskip 0.5cm
\caption{Variation of the density with respect to the cut-off. The solid
line stands for the gluon cascade}
\label{figcutoff}
\end{figure}
We start from the first piece (\ref{fp}), for which the longitudinal momenta
in the operator lie above the cut-off, $x > x_\lambda, x_\lambda^\prime$.
In this case the additional gluon line can not be attached to the operator
vertex in the matrix element (\ref{me}), it can be connected only to itself.
Thus the integrand of the matrix element decays into
the product of the extra particle amplitude squared and the
average of the gluon number operator over the rest part of the amplitude
corresponding to the previous cut-off value $x_\lambda$. As the momentum
fraction $\beta$ is small it can be neglected in LLA everywhere in the
integrand except the $1/\beta$ factor.

The rest part of the amplitude is natural to treat as a state analogous
to the initial amplitude with the same total longitudinal momentum $x=1$
but with the recoil transverse momentum $-k_\perp$. The momentum $q_\perp$
in the parton density (\ref{me}) and in the gluon number (\ref{Q})
are defined with respect to the total momentum, $p_B$, in other words,
$Q(x_\lambda,q_\perp)=Q(x_\lambda,(q-p_B)_\perp)$. Now the vector $p_B-k$
plays the total momentum role, that is why the resulting variation is
\begin{equation}
\label{D1}
\Delta_1 Q(x_\lambda,q_\perp)\,=\,2N_c\,\frac{g^2}{(2\pi)^3}
\ln\frac{x_\lambda}{x_\lambda^\prime}\,
\int \frac{d^2k_\perp}{k_\perp^2} Q(x_\lambda,k_\perp+q_\perp).
\end{equation}

Before discussing the second term (\ref{sp}), let us
consider the virtual loop contribution.
In terms of the light cone wavefunction it is accumulated into an overall
normalization factor. Indeed, the normalization $\langle h(p)|h(p^\prime)
\rangle = 2 p_0 \delta^{(3)}(p -p^\prime)$ has to be kept as the cutoff
$x_\lambda$ is changed. An equivalent statement in terms of amplitudes 
is that any distribution
should be divided by the total cross-section.
Combining equations
(\ref{hp}) and (\ref{Gperp}), this normalization reads
\begin{equation}
\label{norma}
N\,=\,\sum_n \int\prod_{i=1}^n\frac{d^2 q_{i\perp}}{(2\pi)^3 q^2_{i\perp}}2
\frac {d\beta_i}{\beta_i}\,\Gamma^2_{c_1\ldots c_n}(q_{1
\perp},\ldots ,\, q_{n \perp})\,
\delta(\sum q_{i\perp})\delta(1-\sum \beta_i)\,=\,1.
\end{equation}

The change of normalization when varying  the cut-off  can be
incorporated in LLA into gluon $Z$-factors. 
To show this we start from the amplitude to emit one soft gluon with given momentum
and color:
$$
\Gamma_1\,=\,2\frac{q_\perp\cdot \varepsilon_\perp^\lambda(q)}{q_\perp^2}\,
a^+_{\lambda c}(x,q_\perp)\,\Gamma_{c,\ldots}(q_\perp,\ldots),
$$
where dots stand for other gluons irrelevant here. After cut-off is
changed this state becomes more complicated. Apart from the single-gluon
term there appears a two-gluon component, which according (\ref{Gperp}) is
\begin{eqnarray}
\Gamma_2\,&=&\,2\frac{(q-k)_\perp\cdot
\varepsilon_\perp^{\lambda_1}(q)}{(q-k)_\perp^2}\,
a^+_{\lambda_1 c_1}(x,q_\perp-k_\perp) \nonumber \\
&\times&\,
2\frac{k_\perp\varepsilon_\perp^{\lambda_2}(k)}{k_\perp^2}\,
a^+_{\lambda_2 c_2}(\beta,k_\perp)\,ig\,f_{c c_1 c_2}
\Gamma_{c,\ldots}(q_\perp-k_\perp,\ldots),\nonumber
\end{eqnarray}
$$
x_\lambda^\prime\,<\,\beta\,<\,x_\lambda,
$$
so the amplitude, or light cone wavefunction,  is now
$\Gamma=\Gamma_1+\Gamma_2$. 

The single gluon term is normalized as
$$
\Gamma_1^2\,=\,\frac 2{(2\pi)^3}\sum_c \int\,\frac{dx}{x}\,\frac{d^2q_\perp}
{q_\perp^2}\,\Gamma^2_{c\ldots}(q,\ldots).
$$
Squaring the two-gluon component we get
\begin{eqnarray}
\Gamma_2^2\,&=&\,\frac 2{(2\pi)^3}\sum_c \int\,\frac{dx}{x}\,\frac{d^2q_\perp}
{q_\perp^2}\,\ln\frac{x_\lambda}{x_\lambda^\prime} \nonumber \\
\label{helicity}
&\times&\,2N_c\frac{g^2}{(2\pi)^3}\,
\int d^2k_\perp\,
\frac{(q-k)_\perp\cdot\varepsilon_\perp^{\lambda_1}(q)}{(q-k)_\perp^2}\,
\frac{(q-k)_\perp\cdot\varepsilon_\perp^{\lambda_2}(q)}{(q-k)_\perp^2}\,
\frac {q_{\perp}^2}{k_\perp^2}\,\delta^{\lambda_1 \lambda_2} \\
&\times&\,\Gamma^2_{c\ldots}(q_\perp-k_\perp,\ldots).\nonumber
\end{eqnarray}

Comparing these two expressions we conclude that the variation of the
amplitude squared can be completely absorbed into a $Z$-factor acquired by
the gluon operator,
\begin{equation}
\label{zf}
Z(q_\perp)\,=\,1\,+\,N_c\frac{g^2}{(2\pi)^3}\,
\ln\frac{x_\lambda}{x_\lambda^\prime}\int d^2k_\perp\,
\frac{q_\perp^2}{k_\perp^2(q-k)_\perp^2}\,\equiv\,1\,+\,
2\ln\frac{x_\lambda}{x_\lambda^\prime}\,\omega(q).
\end{equation}
Additional 1/2 appers here since the $Z$-factor is prescribed separetely to
each of two gluon helicities. Averaging over transverse directions of the
momentum $k$ in $\Gamma_2^2$ (\ref{helicity}) yields 1/2 for the fixed
indices $\lambda_{1,2}$ as there is no selected transverse direction for
other integrals over $q_{\perp i}$.

The properly normalized state $\Gamma$ again looks like a single
gluon despite its composite character. In this sense $Z$-factor arises
as a manifestation of unresolved very soft particles. 
  
The same treatment can be applied to each gluon in the amplitude.
After the extra soft gluon 
is emitted the general amplitude (\ref{Gperp}) takes the same form
for $n+1$ gluons, so we have to make the following replacement for
 the vertex of the $i^{th}$ emission
$$
\frac{q_{i\perp} \cdot \varepsilon^{\lambda_i}_\perp (q_i)}
{q_{i\perp}^2}\, \to \,
\frac{(q_i-k)_\perp \cdot \varepsilon^{\lambda_i}_\perp (q_i-k)}
{(q_i-k)_\perp^2}\,
\frac{k_\perp \cdot \varepsilon^{\sigma}_\perp (k)}
{k_\perp^2},
$$
whereas the coefficient functions $\Gamma^{(n)}$ remain intact.
The gluons are strongly longitudinally ordered in the LLA, so in
the amplitude squared each gluon line is connected only to itself,
resulting into a ladder type diagram:
\begin{eqnarray}
\delta N^{(n)}\,&=&\,2N_c\frac{g^2}{(2\pi)^3}\,
\ln\frac{x_\lambda}{x_\lambda^\prime}\,\sum_i
\int \frac{d^2q_{i\perp}}{(2\pi)^3 q_{i\perp}^2} \, 2
\frac{d\beta_i}{\beta_i} \int d^2k_\perp
\frac{q_{i\perp}}{k_\perp^2(q-k)_\perp^2} 
\nonumber \\
&& \hspace{-.5cm}\times \,\int\prod_{j\not=i}
\frac{d^2q_{j\perp}}{(2\pi)^3 q_{j\perp}^2}\,2 \frac{d\beta_j}{\beta_j}\,
\delta^{(2)}\bigl(\sum_{j=1}^n q_{j\perp} \bigr)\,
\delta\bigl( 1-\sum \beta_j\bigr)\,\Gamma^2_{c_1\ldots c_n}(q_{1
\perp},\ldots \, q_{n \perp})\,.
\nonumber
\end{eqnarray}

Thus, after rescaling the gluon operators
$a_\lambda^\prime(x,q_\perp) = Z^{-1/2}(q_\perp) a_\lambda(x,q_\perp)$
the normalization of the amplitude $\Gamma_\lambda$ is kept 
with the leading logarithmic accuracy, when the cut-off $x_\lambda$ is varied. 
It produces the variation of the gluon density
\begin{equation}
\label{D2}
\Delta_2 Q(x_\lambda,q_\perp)\,=\,\delta Z^{-1}(q_\perp)\,
Q(x_\lambda,q_\perp).
\end{equation}

Putting both variations (\ref{D1}) and
(\ref{D2}) together we arrive at the homogeneous form of the BFKL equation
$$
x_\lambda\frac{\partial}{\partial x_\lambda} Q(x_\lambda,q_\perp)\,=\,-N_c\frac{g^2}{(2\pi)^3}
\int \frac{d^2k_\perp}{(q-k)_\perp^2}\,\biggl[2\,Q(x_\lambda,k_\perp)\,-\,
\frac{q_\perp^2}{k_\perp^2}\,Q(x_\lambda,q_\perp)\biggr].
$$

Let us turn now to the second piece in the equation (\ref{sp}),
corrsponding to the second graph in Fig.\ref{figcutoff}. The operator
vertex ehits in this case the line of the extra soft gluon with longitudinal
momentum $x_{\lambda^\prime} <\beta<x_\lambda$. In this case the
operator is averaged over a single gluon state whereas the rest of the
amplitude is taken at the previous $x_\lambda$ longitudinal
cut-off. This amplitude is normalized to unity so the contribution of
this piece produces the variation
\begin{equation}
\label{D3}
\Delta_3 Q(x_\lambda,q_\perp)\,=\,2N_c\frac{g^2}{(2\pi)^3}\,
\ln \frac{x_\lambda}{x_{\lambda^\prime}}\,\frac 1{q_\perp^2},
\end{equation}
which results into an inhomogeneous term in the BFKL equation:
\begin{eqnarray}
x_\lambda\frac{\partial}{\partial x_\lambda} Q(x_\lambda,q_\perp)\,=\,&-&2N_c\frac{g^2}{(2\pi)^3}\,
\frac 1{q_\perp^2} \nonumber \\
&-&\,N_c\frac{g^2}{(2\pi)^3}
\int \frac{d^2k_\perp}{(q-k)_\perp^2}\,\biggl[2\,Q(x_\lambda,k_\perp)\,-\,
\frac{q_\perp^2}{k_\perp^2}\,Q(x_\lambda,q_\perp)\biggr]. \nonumber
\end{eqnarray}
The inhomogeneous term, which is of lowest order, can be
interpreted as the initial gluon contribution in the absence of cascade.

There is a natural physical value for the IR cut-off $x_\lambda \sim
\mu^2/s$ where $\mu^2$ is some typical scale for tranverse momentum.
This follows from the Regge kinematics (see Eq.(\ref{kin})).

This derivation can be easily generalized to the non-diagonal case,
$t\not =0$. Defining the gluon distribution as
\begin{eqnarray}
&&\hspace{-3 cm}\langle p_2|\,\int_{x_\lambda}^1 \frac{dx}{2 x}\,\sum_{\sigma c}
a^+_{\sigma c}(x,q_\perp+ l_\perp/2)\,
a_{\sigma c}(x,q_\perp - l_\perp/2)\,|p_1\rangle \nonumber \\
&& \hspace{3 cm} = \, Q(x_\lambda,q_\perp,l_\perp)\,2 p_-
\,\delta^3(p-p^\prime),
\nonumber
\end{eqnarray}
with
$$
t\,=\,l^2, \qquad l\,\simeq\, l_\perp
$$
and $p_{1,2} = p_B \pm l_{\perp}/2\,,$
and repeating the steps leading to the equations (\ref{D1}-\ref{D2})
we arrive at the following evolution equation
\begin{eqnarray}
x_\lambda\frac{\partial}{\partial x_\lambda} Q(x_\lambda,q_\perp,l_\perp)\,=\,
&-& 2 N_c \frac{g^2}{(2 \pi)^3} \frac{(q+l/2)_\perp \cdot (q -
l/2)_\perp}{(q+l/2)^2_\perp
 (q - l/2)^2_\perp} \nonumber \\
&-&N_c\frac{g^2}{(2\pi)^3}\int \frac{d^2k_\perp}{k_\perp^2}\,
\biggl[2\,Q(x_\lambda,q_\perp+k_\perp,l_\perp+k_\perp) \nonumber \\
&&\hspace{-1cm} -\,\frac 12 \biggl(\frac{(q+l/2)_\perp^2}{(q+ l/2 -k)_\perp^2}+
\frac{(q-l/2)_\perp^2}{(q-l/2-k)_\perp^2}\biggr)\,Q(x_\lambda,q_\perp,l_\perp)\biggr].
\nonumber
\end{eqnarray}
The first term as the same interpretation as in the forward case.
Similarly, the second term is just the shift of the
momenta in the 
original
gluon distribution, due to the recoil effect after soft gluon emission.
The two last terms are the $Z$-factors corresponding the creation and annihilation
operators respectively.

Another generalization is the many gluon case. Consider the diagram
similar to those in Fig.3 but with $n$ gluons in $t$-channel. It
describes twist  contribution to the scattering equal or larger to $n$. Like in twist 2 case
the amplitude can be expressed as a convolution of the elementary
Born amplitude in the upper blob with the correlator of the fields taken at the zero light
cone time in the lower blob,
$$
T_{\mu_1\ldots \mu_n}\,\sim\,\varepsilon^{\lambda_1}_{\mu_1}(q_{\perp 1})\,\cdots\,
\varepsilon^{\lambda_n}_{\mu_n}(q_{\perp n})\,
\langle p_2|\,a_{\lambda_1}(\beta_1,q_{\perp 1})\,\cdots
a_{\lambda_n}(\beta_n,q_{\perp_n}\,|p_1\rangle \,.
$$ 
Here we do not explicitly distinguish between annihilation and creation operators as
they are specified by the sign of the momenta $\beta$. By convention all
gluons with positive $\beta$ are treated as incoming and are included
in the state $|p_1\rangle,$
and similarly gluons with negative $\beta$ are treated as outgoing and are included
in the state $|p_2\rangle.$

Since we are interested in the high $s$ limit of the amplitude we keep in the
correlator only most singular terms occuring for small $\beta_i$, actually when all of
them are of the order of the natural longitudinal cut-off, $|\beta_i|\sim \mu^2/s$.
Therefore the asymptotics of the amplitude is determined by the small $x_\lambda$ behavior
of the gluon distribution $Q(x_\lambda,q_{\perp 1},\ldots, q_{\perp n})$ whoose
derivative with respect to $\ln 1/x_\lambda$ gives the correlator value at equal 
$\beta_i = x_\lambda$ for any $i.$ 

The evolution equation for this distribution can be obtained like for the $n=2$ case.
The distribution is presented by the blob with $n$ external gluon legs (there were two
external legs in the previous case drawn in Fig.4)
carrying incoming transverse momenta $q_{\perp i}$. When the cut-off $x_\lambda$ is lowered it
allows for emission of the extra soft gluon. When it is emitted off the external line
$i$ and absorbed by the line $j$ it gives the term
\begin{eqnarray}
\Delta Q\,&=&\,T_i T_j\,
\ln \frac{x_\lambda}{x_{\lambda^\prime}}\,
\frac{g^2}{(2\pi)^3}\int \frac{d^2k_\perp}{k_\perp^2}\,2\,
Q(x_\lambda,\ldots, q_{i\perp}+k_\perp,\ldots,q_{j\perp}-k_\perp,\ldots)
\nonumber \\
&\equiv& \,\ln \frac{x_\lambda}{x_{\lambda^\prime}}\,T_i T_j\,h_{ij}Q
\nonumber
\end{eqnarray}
When the gluon is reabsorbed by the same line $i$ the variation results into
multiplication by the $Z$-factor (\ref{zf}). Combining all terms we get a
Schr\"odinger-like equation (which is called the BJKP equation \cite{bBJKP,jBJKP,kpBJKP})
\begin{equation}
\label{mt}
x_\lambda\frac{\partial}{\partial x_\lambda}Q(x_\lambda,q_{1\perp},\ldots,q_{n\perp})\,=\,
-H\,Q(x_\lambda,q_{1\perp},\ldots,q_{n\perp})
\end{equation}
with the $n$-particle hamiltonian
\begin{equation}
\label{BJKP}
H\,=\,\frac 12\,\sum_{i\not = j}T_i T_j\,h_{ij}\,-\,\sum_i \omega(q_{i \perp}).
\end{equation}
For the colorless channel, $\sum T_i=0$, this hamiltonian can be rewritten
as
$$
H\,=\,\frac 12\,\sum_{i\not = j}T_i T_j\,\biggl[h_{ij}\,+\,\frac 1{N_c}
\bigl(\omega(q_{i \perp})+\omega(q_{j \perp})\bigr)\biggr].
$$
Note that equation (\ref{BJKP}) is the homogeneous form of the BJKP
equation. The inhomogeneous term is a sum  of all possible pairings
between gluon fields inside the averaged operator and gluon fields
corresponding to incoming and outgoing states. 
Incoming and outgoing gluon fields carry color indices $c_i$ while gluon
fields inside the operator carry color indices $c'_i.$
This inhomogeneous term then reads
$$
A = -  \left (\sqrt{2} \frac{g}{(2 \pi)^{3/2}} \right )^n \, \sum_\sigma
\prod_{i=1}^{n} \delta_{c_{\sigma(i)}}^{c'_i} \, \prod_{i=1}^{n-1} \frac{q_{\sigma(i) \perp} \cdot q_{\sigma(i+1) \perp}} {q_{\sigma(i) \perp}^2 
\, q_{\sigma(i+1) \perp}^2} \,.
$$ 


\section{QED example}

Consider, as an example, the electrodynamics calculation of the photon
density in an electron of longitudinal momentum $p_B$ and transverse
momentum $r_\perp$. As in the QCD case, in the asymptotic
regime we neglect any electron-positron pairs. 
Since there are no color factors and
no photon self interaction,
 the amplitude to emit off the electron an arbitrary
number of photons with fixed total transverse momentum $p_\perp$ and
longitudinal momenta $\beta>x_\lambda$ is
\begin{eqnarray}
\label{Psilp}
|\Psi(r_\perp,p_\perp)\rangle\,&=&\,\psi^+(r_\perp)\,
\frac 1{(2\pi)^2}\,\int d^2x \,e^{-ixp_\perp} \\
&\times&\,
\exp\left\{\frac{e}{(2\pi)^{3/2}}\,\int_{x_\lambda}^1
\frac{d\beta}\beta\,d^2k_\perp\,
\frac{p_B\cdot\varepsilon^\sigma(k)}{p_B \cdot k}\,
e^{ik_\perp x_\perp}\,a_\sigma^+(k)\right\} \, . \nonumber
\end{eqnarray}
We have used the expression  (\ref{Pline}) for the case of QED where
there
is no color ordering.
$\psi^+(r_\perp)$ is the electron creation operator. Sudakov variable is
defined with respect to the longitudinal momentum of the electron, so
that $x=1$. The exponent
presents the surrounding electron photon cloud. $e$ is the electron
charge.
The wavefunction (\ref{Psilp})
implies that the transverse momentum of the cloud, $p_\perp$, and of the
electron,
$r_\perp$, are fixed separately. We consider the analogous situation of BFKL
diagonal case. The total transverse momentum $p_\perp$ of the cascade is taken to
be zero. We then have for the photon density
\begin{eqnarray}
\label{me1}
&&\hspace{-2cm} n(x,q_\perp,p_\perp) \, 2 p_- \delta^{(3)} (p - p')\nonumber \\
&&= 
\sum_\sigma \langle \,\Psi(r_\perp^\prime,p_\perp^\prime)
\,|\,\frac{1}{2
x} a^+_{\sigma}(x,q_\perp)\,a_{\sigma}(x,q_\perp) \, |
\Psi(r_\perp,p_\perp)\rangle\,\nonumber \\
&&=\,
\frac 1{(2\pi)^2}\,\delta^2(r_\perp-r_\perp^\prime)\,
\delta^2(p_\perp-p_\perp^\prime) \,\int d^2z\,e^{-izp_\perp}\,\Phi(z),
\end{eqnarray}
where
\begin{eqnarray}
\Phi(z)\,&=&\,\frac{\alpha_{em}}{\pi}\,\frac 1x\,\frac 1{q_\perp^2}\,
e^{iq_\perp z}\,\theta(x-x_\lambda)\, \exp\left \{  \frac{\alpha_{em}}{\pi}\,\int
\frac{d\beta}\beta\,d^2k_\perp\, \frac{e^{ik_\perp z}}{k_\perp^2}\right\}
\nonumber \\
&=&\,\frac{\alpha_{em}}{\pi} \,\frac 1x\,\frac 1{q_\perp^2}\,e^{iq_\perp z}\,
\bigl(\mu^2 z^2\bigr)^{-\alpha_{em}\, Y}\,\theta(x-x_\lambda)\,. \nonumber
\end{eqnarray}
Here $\alpha_{em}=e^2/(2\pi)^2$, $Y=\ln 1/x_\lambda$ and $\mu^2$ is the infrared
cut-off in the transverse space. We define $n(x,q_\perp) \equiv n(x,q_\perp,p_\perp=0).$

Let us construct the function 
$$
Q(x_\lambda,q_\perp)\,=\,\int_{x_\lambda}^1 dx \, n(x,q_\perp)\,=\,
\frac{\alpha_{em}}{\pi}\,\frac 1{q_\perp^2}\,Y\,\int d^2z\,\bigl(\mu^2 z^2\bigr)
^{-\alpha_{em}Y}
$$ 
which is proportional to the number of photons with a given
momentum $q_\perp$ (the factor of proportionality will be
discussed later). 
 
The function $Q$ obeys the BFKL-like equation
\begin{equation}
\label{BFKLQED}
\frac{\partial}{\partial Y}Q(Y,q_\perp)\,=\,
2\,\frac{\alpha_{em}}{q_\perp^2}\,\int d^2z\,e^{iqz}\bigl(\mu^2z^2\bigr)
^{-\alpha_{em}Y}\,+\,\frac{\alpha_{em}}{\pi}\,
\int\frac{d^2k}{(q-k)^2}\,Q(Y,k_\perp),
\end{equation}
which, of course, does not include the virtual part of the QCD BFKL
equation since there is no photon
reggeization ($Z$-factor appearing due to soft photons corresponds
to the electron wave fuction renormalization). This is the reason why 
the first term differs from the inhomogeneous term in QCD BFKL. The extra
factor multiplying it is due to the fact that photon state (\ref{Psilp}) is unnormalized
(this is just the norm of the photon state with the total transverse momentum
$-q_\perp$). 
%Since the cascade starts from an electron there is no inhomogeneous
%term in equation (\ref{BFKLQED}). 
For the same reason there is an
explicit IR cut-off in the photon density (\ref{me1}).

BFKL QCD kinematics, in which the total transverse momentum of
the cascade is fixed, is more similar to the QED case
where the total transverse momentum $p_{tot \perp}$ of the electron
plus cloud is given, rather than their separate values.
The corresponding wave function then reads
\begin{eqnarray}
\label{Psitot}
|\Psi(p_{tot \perp})\rangle\,&=&\,\frac 1{(2\pi)^2}\,\int d^2x\,
e^{-ix p_{tot \perp}}\,\int d^2r_\perp \,\psi^+(r_\perp)\,e^{i r_\perp x}\\
&\times&\,
\exp\left\{\frac{e}{(2\pi)^{3/2}}\,\int_{x_\lambda}^1
\frac{d\beta}\beta\,d^2k_\perp\,
\frac{p_B\cdot\varepsilon^\sigma(k)}{p_B \cdot k}\,
e^{ik_\perp x_\perp}\,a_\sigma^+(k)\right\}, \nonumber
\end{eqnarray}
which leads to the analogous $n$ function
\begin{eqnarray}
\label{me2}
&& \hspace{0cm} \sum_\sigma \langle \,\Psi (p_{tot \perp}^\prime)\,|\,\frac{1}{2
x} a^+_{\sigma}(x,q_\perp)\,a_{\sigma}(x,q_\perp) \,
|\Psi(p_{tot \perp})\rangle\, \\
&& = \frac 1{(2\pi)^2}\,
\delta^2(p_{tot \perp}-p_{tot \perp}^\prime)\,\frac{\alpha_{em}}{2 \pi}  \,
\frac 1x\,\frac 1{q_\perp^2} \,\exp\left\{\frac{e^2}{(2\pi)^3}\,
\int_{x_\lambda}^1\frac{d\beta}\beta\,\frac{d^2k_\perp}{k_\perp^2}\,
\right\}\,\theta(x-x_\lambda)\, .\nonumber
\end{eqnarray}

The properly normalized photon density should be divided by the square of
the amplitude. In the second case
$\langle\Psi(p_{tot \perp}^\prime)|\Psi(p_{tot \perp})\rangle$ is just the exponential
factor in (\ref{me2}). In the first case
\begin{equation}
\label{n1}
\langle \,\Psi(l_\perp^\prime,p_\perp^\prime)\,|\,
\Psi(l_\perp,p_\perp)\rangle\,=\,
\frac 1{(2\pi)^2}\,\delta^2(r_\perp-r_\perp^\prime)
\delta^2(p_\perp-p_\perp^\prime)\,\int d^2z\,\bigl(\mu^2 z^2\bigr)^{-\alpha_{em}Y}.
\end{equation}
Introducing the ultraviolet (short distance) cut-off $r_0^2$ in
equations (\ref{me1})
and (\ref{n1}), and making the replacement $z^2 \to z^2+r_0^2$, we get
\begin{eqnarray}
\int d^2z\,\bigl(z^2+r_0^2\bigr)^{-\alpha_{em}Y}e^{iqz}\,&=&\,\frac{2\pi}{\Gamma(\alpha_{em}Y)}\,
\biggl(\frac{|q|}{2r_0}\biggr)^{{\alpha_{em}Y}-1}K_{{\alpha_{em}Y}-1}(|q|r_0),\nonumber \\
\int d^2z\,\bigl(z^2+r_0^2\bigr)^{{-\alpha_{em}Y}}\,&=&\,\frac{\pi}{{\alpha_{em}Y}-1}\,(r_0^2)^{1-{\alpha_{em}Y}},
\nonumber
\end{eqnarray}
where $K_\nu(x)$ is the McDonald function. The infrared cut-off $\mu^2$
drops out from the ratio. Taking the limit $r_0 \to 0$ we finally obtain
the same expression
$$
\overline n(x,q_\perp)\,=\,\frac{\alpha_{em}}{\pi}\,
\frac 1x\,\frac 1{q_\perp^2}
$$
for both cases.

Thus, like for BFKL equation, the infrared divergencies disappear from the photon density after it
has been properly normalized, in agreement with Bloch-Nordsieck theorem.
The obtained result is the well-known
Weiz\"acker-Williams distribution.

\section{Running coupling with respect to longitudinal cut-off}

According to our picture the soft gluons can be really emitted off any parton inside
the cascade. The sum of these elementary
processes results in an amplitude which looks like the emission off a single gluon. Its color and spin
structure do not feel the internal cascade properties and therefore are
independent on the cut-off $x_\lambda$. However the coupling constant
$g$ multiplying the amplitude could in principle be sensitive to the
$x_\lambda$. The actual coupling could differ for a real single gluon state
at the beginning of evolution, $x_\lambda=1$, and for the developed cascade,
$x_\lambda\sim 0$.  To make our approach internally consistent we have in principle to
take into account the infrared loop corrections not only to $Z$-factors
but to the coupling constant as well. This situation is similar to DGLAP equation, where
the coupling constant runs with $Q^2$ which plays the role of the
cut-off in transverse space. 

We define the physical
coupling, $g(x_\lambda)=g(Y)$ (where ($Y = \ln 1/{x_{\lambda}}$),
through the form factor to emit a soft  gluon with the momentum $l$
off the cascade. This form factor stands in Fig.\ref{figcutoff} for the vertex between the
gluon and the solid line corresponding to the cascade.
In the derivation of the evolution equation (\ref{fp},\ref{sp}), this
vertex is
needed only when the emitted or absorpted gluon is the softest among all
the involved gluons. This is the reason why it is self-consistent to
neglect its momentum in the form factor and to take
 $l=0$. Any dependence with respect to $l$
 is beyond LLA approximation since it destroys the form of the
soft vertex and would lead to next-to-leading order corrections. 

In other words, we have
to average the operator
$$
G_a\,=\,if_{cac^\prime}\int\frac{d\beta}{2\beta}d^2k_\perp\,
a^+_{\lambda c}(\beta,k_\perp)\,a_{\lambda c^\prime}(\beta,k_\perp),
$$
over the cascade state $|p,x_\lambda,c\rangle$
$$
\langle p,x_\lambda,c\,|\,g_B\,G_a\,|p^\prime,\,x_\lambda,c^\prime\,\rangle\,=\,
2p_-\delta^{(3)}(p-p^\prime)\,g(Y)\,f_{c a c^\prime}\,
$$
where $c$ labels the total color of the cascade.
The constant $g_B$ in the left hand side is to be treated as a bare
coupling whereas the physical value, $g(Y)$, results in the right hand side.  
It is the dressed constant $g(Y)$ that should enter the evolution equation rather
than $g_B$.

Introducing the color density 
$$
Q_8(Y,q_{\perp})= \int^1_{x_{\lambda}} d \beta \, n_8(\beta,q_\perp)
$$ 
for the octet state 
in the same way as for the singlet one, we have
\begin{equation}
\label{defgY}
g(Y)\,=\,\int d^2q_\perp Q_8(Y,q_\perp).
\end{equation}
This density satisfies BFKL equation for the gluon channel
\begin{eqnarray}
\frac{\partial}{\partial Y} Q_8(Y,q_\perp)\,&=&\,N_c\,\frac{g^2(Y)}{(2\pi)^3}\,
\frac 1{q_\perp^2} \nonumber \\
&+&\,N_c\frac{g^2(Y)}{(2\pi)^3}
\int \frac{d^2k_\perp}{(q-k)_\perp^2}\,\biggl[\,Q_8(Y,k_\perp)\,-\,
\frac{q_\perp^2}{k_\perp^2}\,Q_8(Y,q_\perp)\biggr]. \nonumber
\end{eqnarray} 
Introducing the variable  $\xi = \xi(Y)$
\begin{equation}
\label{defxi}
\frac 1{N_c}\,\frac{(2\pi)^3}{g^2(Y)}\,\frac\partial{\partial Y}\,\equiv\,
\frac\partial{\partial \xi}
\end{equation}
and the function $\varphi(\xi,q_{\perp})$,
$$
Q_8(Y,q_\perp)\,=\,\frac{\varphi(\xi,q_\perp)}{q_\perp^2},
$$
we get
\begin{equation}
\label{BFKL8}
\frac{\partial}{\partial \xi} \varphi(\xi,q_\perp)\,=\,
1+
\int d^2k_\perp\,\frac{q_\perp^2}{(q-k)_\perp^2 k_\perp^2}
\,\biggl[\varphi(\xi,k_\perp)\,-\,\varphi(\xi,q_\perp)\biggr].
\end{equation}

This equation along with the definitions of  $\xi$ (\ref{defxi}) and of
the dressed coupling (\ref{defgY}), which now reads
\begin{equation}
\label{gxi}
g(Y)\,=\,\int\frac{d^2q_\perp}{q_\perp^2}\,\varphi(\xi(Y),q_\perp^2),  
\end{equation}
implicitly determines the function $g(Y)$.  

On the contrary to the singlet case, there is no cancellation of IR
divergencies between virtual and real terms in the equation (\ref{BFKL8}), so we need
to introduce a regularization. We use dimensional regularization,
so we replace transverse dimension $d=2$ by $d=2+\epsilon.$

We seek for the azimutally symmetrical solution of the form 
$$
\varphi(\xi,q_\perp)\,=\,\int\frac{d\nu}{2\pi i}\,\left(\frac{q_\perp^2}{q_0^2}
\right)^\nu \, \varphi(\xi,\nu),
$$
where the contour of integration is a line joining $\nu_0 -i \infty$
and  $\nu_0 +i \infty,$ with $Re \, {\nu_0} \, < 1.$
We then have
\begin{eqnarray}
\frac \partial{\partial\xi}\varphi(\xi,\nu)\,&=&\,\frac 1{\nu-\epsilon}
\nonumber\\
&+&\,
\pi^{1+\epsilon}\left[\frac{\Gamma(1-\nu)\,\Gamma(\nu)}
{\Gamma(1+\epsilon-\nu)\,\Gamma(\epsilon+\nu)}\,-\,\frac{\Gamma(1-\epsilon)}
{\Gamma(2\epsilon)}\,\Gamma(\epsilon)\right]\,\Gamma(\epsilon)\,
\varphi(\xi,\nu-\epsilon),\nonumber
\end{eqnarray}
or, symbolically,
$$
\frac \partial{\partial\xi}\varphi(\xi,\nu)\,=\,\frac 1{\nu-\epsilon}\,+\,
K(\nu)\,\varphi(\xi,\nu-\epsilon).
$$
The function $\varphi(\xi,\nu)$ can be presented as a sum of two terms,
$$
\varphi(\xi,\nu)\,=\,\varphi_0(\nu)\,-\,\varphi_1(\xi,\nu),
$$
with the first term independent on $\xi$,
$$
K(\nu)\,\varphi_0(\nu-\epsilon)\,=\,-\frac {1}{\nu-\epsilon}
$$
and the second one obeying the homogeneous equation
$$
\frac \partial{\partial\xi}\varphi_1(\xi,\nu)\,=\,
K(\nu)\,\varphi_1(\xi,\nu-\epsilon).
$$

The IR divergency in the octet channel manifests itself in the
singularity occurring 
for $\epsilon \to 0$,
$$
K(\nu)\,=\,\pi^(1+\epsilon)\biggl[-\frac 1\epsilon\,-\,3\gamma_E\,-\,
\psi(1-\nu)\,-\,\psi(\nu)\,+\,O(\epsilon)\biggr].
$$
The $\epsilon$ singularity is factorized by looking for a solution of the form 
$$
\varphi_1(\xi,\nu)\,=\,e^{-\frac \pi \epsilon \xi}\varphi_2(\xi,\nu),
$$
where the function $\varphi_2$ is regular for small $\epsilon$,
$$
\left(\frac{1}{\pi} \frac \partial{\partial\xi}\,-\,\frac \partial{\partial\nu}\right)
\varphi_2\,=\,-\,\bigl(\psi(1-\nu)\,+\,\psi(\nu)
\bigr)\,\varphi_2.
$$
(here we redefine $\epsilon$ to include in unimportant numerical
factors, $1/\epsilon \to 1/\epsilon +\ln\pi + 3\gamma_E$).
This equation yields
$$
\varphi_1(\xi,\nu)\,=\,e^{-\frac \pi \epsilon \xi}
\frac{\Gamma(\nu)\,\Gamma(1-\nu-\pi \xi)}{\Gamma(\nu+\pi \xi)\,\Gamma(1-\nu)}\,
f(\nu+\pi \xi),
$$
with the initial condition
\begin{equation}
\label{initial}
\varphi_1(\xi=0,\nu)\,=\,f(\nu) \,.
\end{equation}

Substituting this solution in the expression (\ref{gxi}) we arrive at the
effective coupling of the form
\begin{equation}
\label{gg0}
g(\xi)\,=\,g_0\,-\,e^{-\frac \pi \epsilon \xi}\,F(\pi \xi),
\end{equation}
where $g_0$ results from $\xi$-independent function $\varphi_0$ while
$$
F(\xi)\,=\,\int \frac{d^2q_\perp}{q_\perp^2}\,\varphi_2(\xi,q_\perp).
$$

Note that $\epsilon > 0$ since it is  IR divergency that is dimensionally regularized,
therefore the second term is suppressed, and vanishes in the limit where
$\epsilon \to 0.$ The result is that $g(\xi) = g_0,$ so that there is no
running coupling effect in longitudinal space, and so the constant $g_0$
play the role of physical coupling. This property is based only on IR
singularities and holds independently of the particular structure 
of $F.$ For instance the unknown initial condition (\ref{initial}) is 
unimportant.
This is in complete
agreement with original BFKL formulation  which was proven to collect
all leading logarithmic contributions with fixed coupling constant.
 


There is an even stronger consequence of the equation (\ref{gg0})
governing the
evolution of the coupling.
Differentiating it with respect $Y$ and using the relation following
from Eq.(\ref{defxi})
$$
\frac{d\xi}{dY}\,=\,\frac{N_c}{(2\pi)^3}\,g^2(Y)
$$
we get
$$
\frac{dg(Y)}{dY}\,=\,\frac 1\epsilon \pi\frac{N_c}{(2\pi)^3}\,g^2(Y)\,
e^{-\frac \pi \epsilon \xi}\,F(\pi \xi)\,-\,e^{-\frac \pi \epsilon
\xi}\,F^\prime(\pi \xi)
\pi\frac{N_c}{(2 \pi)^3}\,g^2(Y)
$$
Here we can neglect the derivatives of the function $F$ compared to the large
$1/\epsilon$ contribution. After this we arrive at a rather universal final
equation
\begin{equation}
\label{important}
\frac{dg}{dY}\,=\,\frac 1\epsilon \pi\frac{N_c}{(2\pi)^3}\,g^2\,
\bigl(g_0\,-\,g\bigr).
\end{equation}
From it we recover the LLA solution $g(Y) = g_0.$ Moreover we see that this
solution is stable (for positive $\epsilon$) with respect to any possible deviation 
from $g_0.$ It means that next to leading order corrections, 
which can be treated as a perturbative deviation from LLA,
cannot be large since there growth would be stopped 
by an opposite leading order effect arising from this equation (\ref{important}).

Note at last that there is no straightforward relation between the IR longitudinal
running coupling we have discussed here and the UV transverse running
coupling effect described by the usual QCD $\beta$-function.

\section{Conclusion}

In this paper we have shown 
that the BFKL equation can be derived as an infrared evolution
equation for the soft gluon cascade. Our approach is exactly symetric to
the DGLAP case. In both equations the evolution is written with respect to a
cut-off. 
DGLAP  involves a transverse cut-off $Q^2$ due to collinear
singularities,
while in our case the cut-off is longitudinal, due to soft
singularities.

In DGLAP case the unintegrated parton distributions, which depend both
on the longitudinal and tranverse momenta, are related to the derivative
of the structure functions with respect to 
transverse cut-off.
Similarly we have to consider the derivative of the function 
$Q(x_\lambda,q_\perp),$ which satisfies BFKL equation, with respect to
$x_\lambda$ in order to get the unintegrated distribution.

In DGLAP case the divergencies  occurring at $x \to 1$ are regulated
by the $1/(x-1)_+$ prescription, corresponds to UV $Z$-factors
due to UV renormalization of parton wave functions.
In BFKL case the IR divergencies for $k_\perp \to 0$ disappear because
of virtual corrections which can be treated as IR renormalization of 
the gluon wave function.

Our approach is based on a kind of duality in the description 
of a gluon at high energy.
From the viewpoint of the soft vertex
the gluon state can be treated as a single particle with given momentum and
color. On the other hand the gluon has internal structure, a 
 cloud of many soft gluons surrounding it. It looks like a composite
object with a nontrivial wavefunction. Hence a rather simple interpretation
of reggeization as a manifestation of soft cascades appears in this
picture. The equation (\ref{mt}) can be wieved as describing  $n$
reggeized gluons, or, equivalently,  $n$ gluon  soft cascades.

Let us finish by outlining the relationship between our approach and
the original BFKL derivation based on  multireggeon formula.
In LLA the cascade is dominated by kinematics where the particles are
separated by large rapidity gap, that is when their longitudinal
momenta are strongly ordered, $\beta_1 \gg \beta_2 \gg \ldots \gg \beta_n$.
Actually this configuration is presented by tree diagrams with 
loop corrections incorporated into $Z$-factors. Using Eq.(\ref{zf}) the
variation of the $Z$-factor for the gluon carrying momenta $\beta_i$,
$q_{i\perp}$ with respect to the cut-off can be written as
$$
\delta Z(\beta_i,q_{i\perp})\,=\,\frac{\delta x_\lambda}{x_\lambda}\,
\omega(q_{i\perp})\,Z(\beta_i,q_{i\perp}).
$$
When taking the cut-off to be equal to the parent gluon momentum,
$x_\lambda=\beta_i$, there is no phase space available for
 virtual emission off it, therefore
$Z(x_\lambda=\beta_i,q_{i\perp})=1$. At the same time the momentum of the
next (in the ordering sense) gluon, $\beta_{i+1}$, plays the natural role
of the infrared cut-off for the gluon $i$, so that
$$
Z(\beta_i,q_{i\perp})\,=\,e^{\omega(q_{i\perp})\ln
\frac{\beta_i}{\beta_{i+1}}}.
$$
Since the partial invariant energy for the two gluons is $s_{i,i+1}\simeq
\beta_i/\beta_{i+1}q_{i+1\perp}^2 \,,$ the $n$-gluon amplitude takes a form
similar to the multi-Reggeon asymptotics,
$$
\Gamma^{(n)}\,=\,\Gamma^{(n)}_0\prod_{i=1}^{n-1}\left(\frac{s_{i,i+1}}{s_0}
\right)^{\omega(q_{i\perp})},
$$
where $\Gamma^{(n)}_0$ is the tree amplitude (the product of soft vertices).
As is well known this expression leads to the total cross-section
$\sigma_{\rm tot}\sim \bigl(s/s_0\bigr)^{\omega(t)}$, which again demonstrates
the relation between the Regge trajectory and soft gluon $Z$-factors.

An interesting problem would be to apply this approach to more complicated 
structures like for example the triple Pomeron vertex.

\section*{Acknowledgements}

We thank G. Korchemsky for comments, and A.S is grateful to M. Ryskin
for many useful discussions.

 
We thank 
 J. Bartels  and the II. Instit\"ut f\"ur Theoretische Physik at DESY for support at the
beginning of this work. We acknowledge for support from the IPN, LPTMS
 and LPT (Orsay). 
One of us (S.W) thanks the Alexander von Humboldt Foundation when this
paper was at an 
early stage.
A.S thanks LPNHE for the hospitality, and LPTHE-Steklov Institute
 Visiting Fellowship agreement.
 
\eject


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\end{document}

