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\begin{document}

\title{ BFKL Pomeron and production amplitudes  in
$N=4$ SUSY}
%\footnote{Talk at the International Conference "Landau-100"}
\classification{11.30.Pb}
\keywords{Pomeron, supersymmetry}
\author{L.N.Lipatov}{address={Petersburg Nuclear Physics Institute, Russia}}
\begin{abstract}
Theoretical
approaches to the problem
of the high energy
hadron-hadron scattering in the
Regge kinematics are reviewed.
It is shown, that
the gluon in QCD is
reggeized and the
Pomeron is a two gluon composite
state. Further, the equation for the multi-gluon composite
states is integrable at $N_c\rightarrow \infty$.
Due to the
AdS/CFT correspondence in $N=4$ SUSY
the BFKL Pomeron is equivalent
to the reggeized graviton.
The important properties of the
maximal transcendentality
and integrability are realized
in this model. Multi-gluon
scattering amplitudes are investigated
in the Regge limit. The BDS ansatz for them  
is not valid beyond one loop due to the presence
of the Mandelstam cuts. 
The hamiltonian for the corresonding  reggeon states
coincides with the hamiltonian of
an integrable open Heisenberg spin chain.
\end{abstract}
%\date{}
\maketitle





\section{Regge approach to high energy interactions}

Hadron-hadron scattering
in the Regge kinematics 
\beq
s=(p_A+p_B)^2 =(2E)^2 >> \vec{q}^2=-(p_{A'}-p_A)^2 \sim m^2
\eeq
is usually described in terms of a $t$-channel exchange of the
Reggeon



\begin{eqnarray}
&&A_p(s,t)=\xi _p(t)\,g(t)\,s^{j_p(t)}\,g(t)\,,\,\,j_p(t)\nonumber \\
&&=j_0+\alpha   
't\,,\,\,\xi_p(t)=\frac{e^{-i\pi j_p(t)}+p}{\sin (\pi j_p)}\,,
\end{eqnarray}
where $j_p(t)$ is the Regge trajectory which is assumed to be linear,
$j_0$ and $\alpha '$ are its
itercept and slope, respectively. The signature factor $\xi_p$ is a
complex quantity depending on the Reggeon signature $p=\pm 1$. Around
40 years ago V.N. Gribov constructed the phenomenological field theory
for all possible Reggeon interactions.
A special Reggeon
- Pomeron with the vacuum quantum numbers and positive signature was 
introduced 
to explain 
an approximately constant behavior
of total cross-sections at high energies and a fullfillment of the Pomeranchuck
theorem $\sigma _{h\bar{h}}/\sigma _{hh}\rightarrow 1$. 







%\[ 
%s=4E^2\gg -t=|q|^2 \approx E^2\,\theta ^2\,
%\]
In the Born approximation of QCD the elastic amplitude
for two colored particle scattering is factorized 
\begin{eqnarray}
&&M_{AB}^{A^{\prime }B^{\prime }}(s,t)|_{Born}=\Gamma _{A^{\prime
}A}^{c}\,\frac{2s}{t}\,\Gamma _{B^{\prime }B}^{c}\,,\,\,
\Gamma _{A^{\prime }A}^{c}\nonumber \\
&&=g\,T_{A^{\prime }A}^{c}\,\delta _{\lambda
_{A^{\prime }}\lambda _{A}}\,,
\end{eqnarray}
where $T^c$ are the generators of the color group $SU(N_c)$
in the corresponding
representation and $\lambda _r$ are helicities of the
colliding and final state particles. In the leading logarithmic
approximation (LLA)
this amplitude has the Regge form~\cite{BFKL}
\beq
M_{AB}^{A^{\prime }B^{\prime }}(s,t)=M_{AB}^{A^{\prime
}B^{\prime }}(s,t)|_{Born}\,s^{\omega (t)},\,\,\alpha _{s}\ln s
\sim 1 \,,
\eeq
where the gluon Regge trajectory with the use of the dimensional 
regularization can be written as follows

\begin{eqnarray}
&&\omega (-|q|^2)=-\frac{\alpha _s N_c}{(2\pi ) ^{2-2\epsilon}}\,|q^2|\int
\frac{\mu^{2\epsilon}\,d^{2-2\epsilon}k}{|k|^2|q-k|^2}\nonumber \\
 &&\approx
-a\,\left(\ln \frac{|q^2|}{\mu ^2}-\frac{1}{\epsilon}\right)\,,\,\,
a=\frac{\alpha _s\,N_c}{2\pi}\,\left(4\pi\,e^{-\gamma}\right)^\epsilon
\end{eqnarray}
and $\gamma$ is the Euler constant $\gamma =-\psi (1)$.
This Regge trajectory
was calculated also in two-loop approximation in QCD~\cite{trajQCD} and in
supersymmetric gauge theories~\cite{trajN4}.

Further, the scattering amplitude in the multi-Regge
kinematics for produced gluons with momenta $k_r$ 
\begin{eqnarray}
&&s\gg s_1\,,\,\,s_2\,,...\,,\,\,s_{n+1}\gg
t_1\,,\,\,t_2\,,...\,,\,\,t_{n+1}\,;\,\nonumber \\
&&\,s_r=(k_{r-1}+k_r)^2\,,\,\,
t_r=-|q_r|^2
\end{eqnarray}
has the form~\cite{BFKL}   
\begin{eqnarray}
&&M_{2\rightarrow 1+n}
=2s\,\Gamma _{A^{\prime
}A}^{c_1}\, \frac{s_1^{\omega (-|q_1|^2)}}{|q_1|^2}
\,gT_{c_2c_1}^{d_1}C(q_2,q_1)\, \nonumber \\
&&\times\frac{s_2^{\omega
(-|q_2|^2)}}{|q_2|^2}...C(q_{n},q_{n-1})\, \frac{s_{n}^{\omega
(-|q_n|^2)}}{|q_{n}|^2}\,\Gamma _{B^{\prime
}B}^{c_n}\,.
\end{eqnarray}
Here $C$ are the Reggeon-Reggeon-gluon vertices 
\beq
C(q_2,q_1)=\frac{q_2\,q_1^*}{q_2^*-q_1^*}
\eeq
and we used the complex notations $k=k_x+ik_y$ for the 
transverse components
of momenta.









\section{BFKL equation}
Because the production amplitudes
in QCD are   
factorized, one can write a Bethe-Salpeter-type equation
for the total cross-section $\sigma _t$. It is
governed by the Pomeron exchange. The Pomeron 
wave function satisfies the equation of
Balitsky, Fadin, Kuraev and Lipatov (BFKL)~\cite{BFKL}

\beq
E\,\Psi (\vec{\rho}_{1},\vec{\rho}_{2})=H_{12}\,\Psi (\vec{\rho}_{1},\vec{%
\rho}_{2})\;,\,\,\Delta =-\frac{\alpha
_{s}N_{c}}{2\pi }\,E\,,
\eeq
where $\sigma _t \sim s^{\Delta _{max}}$. The BFKL Hamiltonian
in the coordinate representation $\rho$ is
\begin{eqnarray}
&&H_{12}=\ln \,|p_1p_2|^{2}+
\frac{1}{p_1p_2^*}(\ln |\rho _{12}|^{2})p_1p_2^*\nonumber \\
&&+\frac{1}{p_1^*p_2}(\ln |\rho _{12}|^{2})p_1^*p_2
-4\psi (1)\,,
\end{eqnarray}
where $\rho _{12}=\rho _1-\rho _2$.
It is invariant under the M\"obius transformations~\cite{int1, moeb}
\beq
 \rho _k \rightarrow \frac{a\rho _k+b}{c\rho _k+d}
\eeq
and has the property of the holomorphic separability
\begin{eqnarray}
&&H_{12}=h_{12}+h^*_{12}\,,\,\,h_{12}=\ln (p_1p_2)+
\frac{1}{p_1}\ln (\rho _{12}) \,p_1\nonumber \\
&&+\frac{1}{p_2}\ln (\rho _{12}) \,p_2
-2\psi (1)\,.
\label{pairh}
\end{eqnarray}
Here we used the complex notations $\rho _r =x_r+iy_r, \,p_r=i\partial _r$ for 
two-dimensional 
transverse coordinates and
their canonically conjugated momenta.
For the principal series of
unitary representations of the M\"{o}bius group the conformal weights are
\beq
m=\gamma +n/2\,,\,\,\widetilde{m}=\gamma -n/2\,,\,\,\gamma
=1/2+i\nu \,,
\eeq
where $\gamma$ is the anomalous dimension of the twist-2 operators
and $n=0,\pm 1, \pm 2,...$ is the conformal spin.


The Bartels-Kwiecinski-Praszalowicz (BKP) equation
for colorless  composite states of several reggeized gluons has the 
form~\cite{BKP}

\begin{eqnarray}
&&E\,\Psi (\vec{\rho}_{1},...\vec{\rho}_n)=
H\,\Psi
(\vec{\rho}_{1},...\vec{\rho}_n )\;,\,\,H\nonumber \\
&&=\sum _{k<l}
\frac{\vec{T}_k\vec{T}_l}{-N_c}\,H_{kl}\,,
\end{eqnarray}
where $H_{kl}$ is the BFKL hamiltonian.
Apart from the M\"{o}bius invariance its wave function
in the multi-color QCD ($N_c \rightarrow \infty$) has the property of the
holomorphic factorization~\cite{separ}

\beq
\Psi (\vec{\rho} _1,...,\vec{\rho}_n)=\sum _{r,s}
a_{r,s}\,\Psi _r(\rho _1,...,\rho _n)\,
\Psi _s (\rho _1^*,...,\rho _n^*)\,,
\eeq
where the sum is performed over a degenerate set of solutions for the corresponding
holomorphic and anti-holomorphic equations. The BKP equation has also the
duality symmetry~\cite{dual} 
\begin{equation}
p_k\rightarrow \rho _{k,k+1}\rightarrow p_{k+1}
\end{equation}
and $n$
integrals of motion $q_r,\,q_r^*$~\cite{int}. The corresponding hamiltonians
$h$ and $h^*$ are local hamiltonians of the integrable Heisenberg spin model, in which
spins are generators of the M\"{o}biuos group~\cite{LiFK}. We can introduce
the transfer ($T$) and monodromy ($t$) matrices according to the definitions~\cite{int}
\beq
T(u)=Tr\,t(u)=\sum_{r=0}^{n}u^{n-r}\,q_{r}\,,\,\,t(u)=
L_{1}(u)L_{2}(u)...L_{n}(u)\,,
\eeq
\begin{eqnarray}
&&L_{k}(u)=\left(
\begin{array}{cc}
u+\rho _{k}\,p_{k} & p_{k} \\
-\rho _{k}^{2}\,p_{k} & u-\rho _{k}\,p_{k}\end{array}\right)\,,\,\,\nonumber \\
&&t(u)=\left(
\begin{array}{cc}
A(u) & B(u) \\
C(u) & D(u)\end{array}\right)\,.
\label{ABCD}
\end{eqnarray}
The matrix elements of $t(u)$ satisfy some bilinear commutation relations
following from the
Yang-Baxter equation~\cite{int}
\begin{eqnarray}
&&t_{r_{1}^
{\prime
}}^{s_{1}}(u)\,t_{r_{2}^{\prime
}}^{s_{2}}(v)\,\hat{l}_{r_{1}r_{2}}^{r_{1}^{\prime }r_{2}^{\prime
}}(v-u)\nonumber \\
&&=\hat{l}_{s_{1}^{\prime }s_{2}^{\prime
}}^{s_{1}s_{2}}(v-u)\,t_{r_{2}}^{s_{2}^{\prime
}}(v)\,t_{r_{1}}^{s_{1}^{\prime }}(u)\,,\,\,\nonumber \\
&&\hat{l}(u)=u\,
\hat{1}+i\,\hat{P}\,,
\end{eqnarray}
where $\hat{l}(u)$ is the monodromy matrix for the usual Heisenberg spin model
and $\hat{P}$ is the permutation operator.
This equation can be solved with the use of the Bethe ansatz and the
Baxter-Sklyanin approach~{\cite{Veg, DKM}}.

\section{Pomeron in $N=4$ SUSY}



One can calculate the integral kernel for the BFKL equation
also in
two loops~\cite{FL}. Its eigenvalue can be written as follows
\beq
\omega =4\,\hat{a}\,\,\chi (n,\gamma )+4\,\,\hat{a}^{2}\,\Delta
(n,\gamma )\,,\,\,\hat{a}=g^{2}N_{c}/(16\pi ^{2})\,,
\eeq
where
\begin{eqnarray}
 &&\chi
(n,\gamma )=2\Psi (1)-\Psi (\gamma +|n|/2)-\Psi (1-\gamma +|n|/2)\,,\,\,\nonumber \\
&&\Psi (x)=\Gamma '(x)/\Gamma (x)\,.
\end{eqnarray}
 The one-loop correction
$\Delta (n, \gamma )$ in QCD contains the non-analytic terms -
the Kroniker symbols
$\delta _{|n|,0}$ and $\delta _{|n|,2}$,
but in $N=4$ SUSY they are cancelled and
we obtain for $\Delta (n, \gamma )$ the following result~\cite{trajN4, KL}
\begin{eqnarray}
&&\Delta (n,\gamma )=\phi (M)+\phi (M^{\ast })-\frac{
\rho (M)+ \rho (M^{\ast })}{2\hat{a}/\omega}\,,\, \nonumber \\
&&M=\gamma
+\frac{|n|}{2}\,,
\end{eqnarray}
\begin{eqnarray}
&&\rho (M)=\beta ^{\prime
}(M)+\frac{1}{2}%
\zeta (2)\,,\,\beta ^{\prime }(z)\nonumber \\
&&=\frac{1}{4}\Biggl[\Psi ^{\prime
}\Bigl(\frac{z+1}{2}\Bigr)-\Psi ^{\prime
}\Bigl(\frac{z}{2}\Bigr)\Biggr]\,.
\end{eqnarray}
It is interesting, that all functions entering in these expressions
have the property of the maximal transcendentality~\cite{KL}.
In particular,
$\phi (M)$ can be written in the form
\beq
\phi (M)=3\zeta (3)+\Psi ^{^{\prime \prime }}(M)-2\Phi (M)+ 2\beta
^{^{\prime }}(M)\Bigl(\Psi (1)-\Psi (M)\Bigr),
\eeq
\beq
\Phi (M)=\sum_{k=0}^{\infty } \frac{(-1)^{k}}{k+M}\left( \Psi ^{\prime
}(k+1)~-
~\frac{%
\Psi (k+1)-\Psi (1)}{k+M}\right)\,.
\eeq
Here $\Psi (M)$ has the transcedentality equal to 1, its differentiation
$\Psi ^{(n)}$ increases it to $n+1$, 
the special number $\zeta (3)$ has the transcendality 3, the additional 
poles in
the sum over $k$ add the transcedentality of the
function $\Phi (M)$ up to 3.
The maximal transcendentality
hypothesis is
valid also for the anomalous dimensions of twist-2 -operators in
$N=4$ SUSY~\cite{KLV, KLOV} contrary to the case of QCD~\cite{VMV}.

The eigenvalue of the BFKL kernel in the diffusion approximation 
is written
below~\cite{BFKL}
\beq
j=2-\Delta -D\,\nu ^2\,,
\eeq
where $\nu$ is related to the anomalous dimension $\gamma$ of
the twist-2 operators as follows~\cite{FL}
\beq
\gamma =1+\frac{j-2}{2}+i\nu \,.
\eeq
The parameters $\Delta$ and $D$ are functions of the coupling constant
$\hat{a}$ and are known up to two loops. Higher order perturbative
corrections can be obtained with the use of the effective
action~\cite{eff, last}.
For large coupling constants one can expect, that the leading
Pomeron singularity in $N=4$ SUSY is moved to the point $j=2$ and
asymptotically the Pomeron coincides with the graviton Regge pole.
This assumption
is related to the AdS/CFT correspondence, formulated in the framework of
the Maldacena hypothesis claiming, that $N=4$ SUSY is equivalent 
to the superstring
model living on the 10-dimensional
anti-de-Sitter space~\cite{Malda, GKP, W}. Therefore it is
natural to impose on the BFKL equation in the diffusion approximation
the physical condition, that for the conserved energy-momentum tensor
$\theta _{\mu \nu}(x)$ having $j=2$ the anomalous dimension $\gamma$ is
zero. As a result, we obtain, that
the parameters $\Delta$ and $D$ coincide~\cite{KLOV} and
\beq
\gamma
=(j-2)\left(\frac{1}{2}-\frac{1/\Delta}{1+\sqrt{1+(j-2)/\Delta}}\right)\,.
\eeq

Using the dictionary developed in the framework of the AdS/CFT
correspondence~\cite{GKP}, one can rewrite the eigenvalue relation
for the BFKL kernel in the form
of the graviton Regge trajectory~\cite{KLOV}
\beq
j=2+\frac{\alpha '}{2}\,t\,,\,\,t=E^2/R^2\,,\,\,\alpha
'=\frac{R^2}{2}\,\Delta \,.
\eeq
On the other hand, Gubser, Klebanov and Polyakov predicted the following
asymptotics of the anomalous dimension at large $\hat{a}$ and
$j$~\cite{GKP2}
\beq
\gamma _{|\hat{a},j \rightarrow \infty} = -\sqrt{2\pi j}\, \hat{a}^{1/4}\,.
\eeq
As a result, one can obtain the explicit expression for the Pomeron intercept
at large coupling constants~\cite{KLOV, Polch}
\beq
j=2-\Delta \,,\,\,\Delta =\frac{1}{2\pi} \,\hat{a}^{-1/2}\,.
\eeq
%$$
%\frac{\pi ^{2}}{6}z=-\widetilde{b}+
%\frac{1}{2}\widetilde{b}^2\,,\,\, b=\gamma '(2)=-\frac{\pi



In Ref.~\cite{L4} it was argued, that for $N=4$
SUSY the evolution equations for anomalous dimensions of
quasi-partonic operators are integrable in LLA. Later such
integrability was generalized to other operators~\cite{MZ}
and to higher loops~\cite{BS}. Using additionally the maximal
transcendentality hypothesis the integral equation
for the so-called casp anomalous dimension was constructed
in all orders of perturbation theory~\cite{ES,BES}. Further,
the anomalous dimension of twist-2 operators in four loops 
was calculated~\cite{KLRSV}, but due to the absence of the so-called
wrapping contributions in the asymptotic Bethe anzatz the obtained 
results do not agree with the BFKL predictions~\cite{trajN4, KL}.

\section{Bern-Dixon-Smirnov scattering amplitudes in $N=4$ SUSY}
To calculate higher order corrections to the BFKL equation 
in QCD and supersymmetric models one should
know production amplitudes in higher orders of perturbation theory. 
Several years ago Bern, Dixon and Smirnov suggested a simple anzatz for 
the multi-gluon scattering amplitude with the maximal helicity violation
in the planar limit 
$\alpha N_c \sim 1$ for the $N=4$ super-symmetric gauge theory~\cite{BDS}.
It turns out, that this amplitude is proportional to its Born expression.
The proportionality coefficient $M_n$ for $n$ external particles 
is a function of relativistic
invariants and can be expressed at $\epsilon =(4-D)/2\rightarrow 0$ 
in terms of an infraredly divergent factor and an expression depending
on three functions $\gamma (a), \beta (a)$ and $\delta (a)$, which are known
up to a rather  large order of perturbation theory.  
In particular, $\gamma (a)$ is the 
so-called cusp anomalous dimension which was calculated 
in all orders~\cite{ES, 
BES}
 
In Ref.~\cite{BLS} the BDS anzatz was investigated in the Regge kinematics
(see also Ref.~\cite{BNST}). 
In particular, the elastic amplitude has the Regge asymptotics
\begin{eqnarray}
M_{2\rightarrow 2}&=&\Gamma (t)\,\left(\frac{-s}{\mu ^2}\right)^{\omega (t)}\,
\Gamma (t)\, \left(1+ {\cal O}(\epsilon)\right),
\label{M2a2Regge}
\end{eqnarray}
where $\mu ^2$ is the renormalization point. The quantity
\[
\omega (t)=-\frac{\gamma (a)}{4}\,\ln \frac{-t}{\mu ^2}+\int _0^a
\frac{da'}{a'}\left(\frac{\gamma (a')}{4\epsilon}+\beta (a')\right)
\]
\begin{equation}
=\left(-\ln \frac{-t}{\mu ^2}+\frac{1}{\epsilon}\right)a+
\left[\zeta _2\left(\ln \frac{-t}{\mu ^2}-\frac{1}{2\epsilon}\right)-
\frac{\zeta _3}{2}\right]a^2 +...
\label{gluontrajectory}
\end{equation}
is the all-order gluon Regge trajectory obtained from the BDS
formula~\cite{BLS} and
\begin{eqnarray}
&&\ln \Gamma (t)=\ln  \frac{-t}{\mu ^2}\,\int _0^a
\frac{da'}{a'}\left(\frac{\gamma (a')}{8\epsilon}+\frac{\beta (a')}{2}\right)\nonumber \\
&&+\frac{C(a)}{2}+\frac{\gamma (a)}{2}\,\zeta _2 
- \int _0^a \frac{da'}{a'}
\ln \frac{a}{a'}\,\left(\frac{\gamma (a')}{4\epsilon ^2}\right.\nonumber \\
&&\left.+\frac{\beta (a')}{\epsilon} +\delta (a')\right),
\label{ReggeonParticlevertex}
\end{eqnarray}  
is the vertex for the Reggeized gluon coupling to the external
particles. Note that the perturbative expansion for $\omega (t)$ is 
in an agreement with its direct calculations performed initially in the
$\overline{MS}$-scheme~\cite{trajN4}.

One can 
verify that in all physical regions the BDS amplitude for one gluon
production
in the multi-Regge kinematics can be obtained with the use of
an analytic continuation from the expression~\cite{BLS}
\begin{eqnarray}
&&\frac{M_{2\rightarrow 3}}{\Gamma (t_1)\Gamma (t_2)} ~=~
\left(\frac{-s_1}{\mu^2}\right)^{\omega (t_1)-\omega (t_2)}
\left(\frac{-s\kappa }{\mu^4}\right)^{\omega (t_2)}c_1\nonumber \\
&&+
\left(\frac{-s_2}{\mu^2}\right)^{\omega (t_2)-\omega (t_1)}
\left(\frac{-s\kappa }{\mu^4}\right)^{\omega (t_1)}c_2\,,
\label{Mdosatres}
\end{eqnarray}
where  $\kappa =s_1s_2/s=|k_\perp|^2$ and the coefficients $c_i$ are real 
\begin{eqnarray}
c_1(\kappa )&=&|\Gamma (t_2,t_1, \ln {-\kappa} )|\,
\frac{\sin \pi (\omega (t_1)-\phi _\Gamma )}{\sin \pi (\omega (t_1)-\omega (t_2))}\,,
\label{ces1}\\
c_2(\kappa )&=&|\Gamma (t_2,t_1, \ln {-\kappa} )|\,
\frac{\sin \pi (\omega (t_2)-\phi _\Gamma )}{\sin \pi (\omega (t_2)-\omega (t_1))}\,.
\label{ces2}
\end{eqnarray}
Here $\phi _\Gamma$ is the phase of the Reggeon-Reggeon-gluon vertex
$\Gamma$, {\it i.e.}
\begin{equation}
\Gamma (t_2,t_1, \ln \kappa -i\pi )=|\Gamma (t_2,t_1, \ln -\kappa )|\,
e^{i\pi \phi _\Gamma}\,,
\label{phigamma}
\end{equation}
\begin{eqnarray}
&&\ln \Gamma (t_2,t_1, \ln -\kappa )=-\frac{\gamma (a)}{16}\,
\ln ^2\frac{-\kappa }{\mu ^2}\nonumber \\
&&-\frac{1}{2}\int _0^a \frac{da'}{a'}
\ln \frac{a}{a'}\,\left(\frac{\gamma (a')}{4\epsilon ^2}\right.\nonumber \\
&&\left.+\frac{\beta (a')}{\epsilon}
+\delta (a')\right)\nonumber\\
&&\hspace{-4cm} -\frac{\gamma (a)}{16}\ln ^2\frac{-t_1}{-t_2}-
\frac{\gamma (a)}{16}\zeta _2-\frac{1}{2}
\left(\omega (t_1)+\omega (t_2)-
\int _0^a\frac{da'}{a'}\,\right.\nonumber \\
&&\left.\left(\frac{\gamma (a')}{4\epsilon}+\beta(a')\right)\right)
\ln \frac{-\kappa }{\mu ^2}
\,.
\label{Gammavertex}
\end{eqnarray}



In a similar way two gluon production amplitude in the multi-Regge kinematics
almost in all physical regions can be obtained by an analytic continuation
from a dispersion-like representation containing 5 contributions.
However, in the physical kinematical region, where $s,s_2>0$ but
$s_1,s_3<0$ the Regge factorization for the BDS amplitude is broken~\cite{BLS} 
\begin{eqnarray}
&&\frac{M_{2\rightarrow 4}}{\Gamma (t_1)\Gamma (t_3)} = 
C \left(\frac{-s_1}{\mu ^2}\right)^{\omega (t_1)}
\Gamma (t_2,t_1,\ln -\kappa _{12} )\nonumber \\
&&\times\left(\frac{-s_2}{\mu ^2}\right)^{\omega (t_2)}
\Gamma (t_3,t_2, \ln -\kappa _{23} )
\left(\frac{-s_3}{\mu^2}\right)^{\omega (t_3)},
\label{24mixedregion}
\end{eqnarray}
where the coefficient $C$ is given below
\begin{equation}
C=\exp \left[\frac{\gamma _K(a)}{4} \,i\pi \,\left(
\ln \frac{\vec{q}_1^2\vec{q}_3^2}{(\vec{k}_1+\vec{k}_2)^2\mu ^2}   
-\frac{1}{\epsilon}\right)\right].
\label{coeffC}
\end{equation}
Similarly for the BDS amplitude describing the transition
$3\rightarrow 3$ in the physical region, where $s,s_2=t'_2>0$ but
$s_1,s_3<0$ we obtain the result
\begin{eqnarray}
&&\frac{M_{3\rightarrow 3}}{\Gamma (t_1)\Gamma (t_3)} =
C' \, \left(\frac{-s_1}{\mu ^2}\right)^{\omega (t_1)}
\Gamma (t_2,t_1,\ln -\kappa _{12} )\nonumber \\
&&\times\left(\frac{-s_2}{\mu ^2}\right)^{\omega (t_2)}
\Gamma (t_2,t_1, -\ln \kappa _{23} )
\left(\frac{-s_3}{\mu^2}\right)^{\omega (t_3)},
\end{eqnarray}
where the phase factor $C'$ is
\begin{equation}
C'=\exp \left[\frac{\gamma _K(a)}{4} \,(-i\pi ) \,
\ln \frac{(\vec{q}_1-\vec{q}_2)^2\,
(\vec{q}_2-\vec{q}_3)^2}{(\vec{q}_1+\vec{q}_3-\vec{q}_2)^2\,\vec{q}_2^2}
\right],
\label{coeffC'}
\end{equation}
which also contradicts the Regge factorization. The reason for these drawbacks
is that just in these kinematical regions the amplitudes $A_{2\rightarrow 4}$ and
$A_{3\rightarrow 3}$ should contain the Mandelstam cuts in the $j$-pane of the 
$t_2$-channel~\cite{BLS}. 
Therefore the BDS amplitudes for these processes are not correct beyond 1 loop.
  
\section{Mandelstam cuts in the adjoint representation at LLA}

The Mandelstam cuts in the elastic amplitude appear only in the non-planar 
diagrams because the integrals over the Sudakov variables $\alpha =2kP_A/s$ 
and $\beta =2kp_B$ for the reggeon momenta $k$ and $q-k$ should have the
singularities above and below the corresponding integration contours. 
For the case of planar diagrams this Mandelstam condition is fulfilled
only for inelastic amplitudes starting from six external particles in
the kinematical region where $s,s_2>0$ and $s_1,s_3<0$. Two reggeons in the
$t_2$-channel with an adjoint representation of the gauge group $SU(N_c)$ 
can also scatter each from 
another. The corresponding contribution to the  imaginary part in the 
$s_2$-channel for the amplitude $A_{2\rightarrow 4}$ can be written 
as follows~\cite{BLS}
\begin{eqnarray}
&&\frac{1}{\pi}\Im _{s_2} M_{2\rightarrow 4}=s_2^{\omega (t_2)}\,
\int _{\sigma -i\infty}^{\sigma +i\infty}\nonumber \\
&&\times\frac{d\omega}{2\pi i}
\left(\frac{s_2}{\mu ^2}\right)^\omega \,
\widetilde{f}_2(\omega )\,
\end{eqnarray}
The reduced partial wave $\widetilde{f}_2(\omega)$ is given by
\begin{eqnarray}
\label{f-reduced}
&&\widetilde{f}_2(\omega )=\hat{\alpha}_\epsilon \,
|q_2|^2\int
d^{2-2\epsilon}k\,d^{2-2\epsilon}k'\,\nonumber \\
&&\times\Phi _1 (k,q_2,q_1)\,
G_{\omega}(k,k',q_2)\,
\Phi_3 (k',q_2,q_3)\,
\end{eqnarray}
where $\Phi _{1,3}$ are the impact factors
\begin{eqnarray}
\label{phi3}
&&\Phi_1(k,q_2,q_1)= \frac{k_1^* 
(q_2-k)^*}{q_2^*(k+k_1)^*}\,,\,\,\nonumber \\
&&\Phi_3(k',q_2,q_3)=\frac{k_2(k'-q_2)}{q_2(k'-k_2)}\,.
\end{eqnarray}

The Green's function $G_{\omega}(k,k',q_2)$
satisfies the BFKL-type equation
\begin{eqnarray}  
&&\omega G_{\omega}^{(8_A)}(k,k',q_2) = \frac{(2\pi)^3
\delta^{(2)}(k-k')}{|k|^2 |k+q_2|^2}\nonumber \\
&&+ \frac{1}{|k|^2 |k+q_2|^2} 
\left(K^{(8_A)} \otimes G_{\omega}^{(8_A)} \right)
(k,k',q_2)\,,
\label{BFKLoctet}
\end{eqnarray}
where
\begin{eqnarray}
&&K^{(8_A)}(k,k';q_2)\nonumber \\
&&= \delta^{(2)}(k-k') \left( \omega(-|k|^2) +
\omega(-|q_2-k|^2)-2\omega (-|q|^2)
\right) \nonumber \\
&&+  \frac{a}{2} \frac{k^* (q_2-k) k'(q_2-k')^* + c.c.}{|k-k|^2}\,.
\end{eqnarray}

The infrared divergencies are extracted from $M_{2\rightarrow 4}$
in the form of the Regge factor 
$s_2^{\omega (t_2)}$ and coincide with those of the BDS amplitude, as it
should be. The partial wave $\widetilde{f}_2(\omega)$ contains the 
infrared divergency only in
one loop
\begin{eqnarray}
\label{C-phase}
&&\hat{\alpha}_\epsilon \,
|q_2|^2\int
d^{2-2\epsilon}k\,
\frac{k^*q_1^*}{q_2^* (k+k_1)^*}\,\,\,
\frac{1}{|k|^2 |q_2 -k|^2}\,\,\,
\frac{k q_3}{q_2 (k-k_2)} \nonumber \\
&&=
\frac{a}{2} \left( \ln \frac{|q_1|^2 |q_3|^2}{|k_1+k_2|^2 \mu^2}
- \frac{1}{\epsilon}\right)\,,
\end{eqnarray}
which is also compatible with the BDS result. But in upper loops
the iteration of the above equation leads to terms which are
absent in the BDS amplitude. For example, in two loops we
obtain for the imaginary part of $A_{2\rightarrow 4}$ in the 
$s_2$-channel the more complicated expression~\cite{BLS2} 
\beq
A_{s_2}=\frac{a^2}{2}\,\ln s_2 \,\ln \frac{|q_1-q_3|^2|q_2|^2}{|q_1|^2|k_2|^2}\,
\ln \frac{|q_1-q_3|^2|q_2|^2}{|q_3|^2|k_1|^2}\,.
\eeq
It is symmetric with respect to the simultaneous transmutation of momenta
\beq
k_1 \leftrightarrow k_2\,,\,\,
q_1 \leftrightarrow -q_3\,.
\eeq
The same expression is valid also for the imaginary part in the 
$s$-channel.

In a similar way we can calculate the $s$-channel imaginary part of
the amplitude
for the transition $3\rightarrow 3$
\beq
A_{s}^{3\rightarrow 3}=\frac{a^2}{2}\,\ln t'_2\,\ln \frac{|q_2-q_1-q_3|^2|q_2|^2}{|k_1|^2|k_2|^2}\,
\ln \frac{|q_2-q_1-q_3|^2|q_2|^2}{|q_3|^2|q_1|^2}\,.
\eeq
Moreover, the BFKL equation for the state with adjoint quantum numbers can be
solved exactly and we obtain for the imaginary part 
in $s_2$-channel~\cite{BLS2}
\begin{eqnarray}
&&\Im M_{2\rightarrow 4}\sim s_2^{\omega (t_2)}\sum _{n=-\infty}^\infty
\int _{-\infty}^\infty \frac{d\nu }{\nu ^2+\frac{n^2}{4}}\,\left(\frac{q_3^*k^*_1}{k^*_2q_1^*}\right)^{i\nu -\frac{n}{2}}\,  \nonumber \\
&&
\times\left(\frac{q_3k_1}{k_2q_1}\right)^{i\nu +\frac{n}{2}}\,
\exp \left(\omega (\nu , n)\,\ln s_2\right)\,,
\end{eqnarray}
where the eigenvalue of the reduced BFKL kernel for the adjoint
representation is
\beq
\omega (\nu , n) =-a\left( \psi (i\nu +\frac{|n|}{2})+
\psi (-i\nu +\frac{|n|}{2})-2\psi (1)\right)\,.
\eeq
It turns out, that the leading singularity of the $t_2$-partial wave
corresponds to $n=1$ and is situated at 
\[
j-1=\omega (t_2)+a(4\ln 2-2).
\]

\section{Multi-reggeon Mandelstam cuts}

Let us consider now the Mandelstam cuts constructed from
several reggeons~\cite{intBDS}. The non-vanishing contribution from
the exchange of $n+1$ reggeons appears in the planar diagrams only if the
number of external lines is $r\ge 2n+4$. 
In the case of production of $2n$ gluons with the same helicity
the amplitude in $N=4$ SUSY is proportional to the Born expression.
In the leading logarithmic approximation 
for the $n+1$-reggeon contribution to the $s_{n+1}$-channel the 
proportionality factor has the form~\cite{intBDS}
\begin{eqnarray}
&&f_{LLA}^{2\rightarrow 2+2n}=\left(i\,\frac{g^2\,N_c}{4\pi}\right)^{n}\,  
Q^*\widetilde{Q}\nonumber \\
&&\int \prod _{l=1}^n\frac{\mu ^{2\epsilon}d^{2-2\epsilon }
p_l}{(2\pi)^{1-2\epsilon}}\,\frac{\mu ^{2\epsilon}d^{2-2\epsilon }
p'_l}{(2\pi)^{1-2\epsilon}}\,\nonumber \\
&&\times\prod _{l=1}^{n}
\frac{k^*_lk_{2r-l}}{|p_l|^2}\,\frac{G(p,p';s_{n+1})}{|p_{n+1}|^2}\Phi _1\,
\Phi_2\,,
\end{eqnarray}
where $Q$ and $\widetilde{Q}$ are momentum transfers between momenta
$p_A,p_{A'}$ and $p_B,p_{B'}$. The impact factors are
\begin{equation*}
\Phi _1(\vec{p_1},...,\vec{p}_{n+1})=
\end{equation*}
\begin{eqnarray*}
\prod_{l=1}^n
\frac{p^{*}_{l+1}}{(Q^*-\sum _{s=1}^lp^{ *}_s-\sum 
_{s=1}^{l-1}k^*_s)(Q^*-\sum _{t=1}^{l+1}p^{*}_t
-\sum _{t=1}^{l-1}k_t^*)},
\end{eqnarray*}
\begin{equation*}
\Phi _2(\vec{p}'_1,...,\vec{p}'_{n+1})=
\end{equation*}
\begin{eqnarray*}
\prod _{l=1}^n\frac{p'_{l+1}}{(\widetilde{Q}+\sum _{s=1}^lp'_s-
\sum _{s=1}^{l-1}k_{2n -s+1})(\widetilde{Q}+\sum _{t=1}^{l+1}p^{\prime 
}_t
-\sum _{t=1}^{l-1}k_{2n-t+1})}.
\end{eqnarray*}
The
multi-reggeon Green function satisfies the equation~\cite{intBDS}
\begin{eqnarray}
&&\frac{\partial}{\partial \ln s_{n+1}}\,
G(\vec{p},\vec{p}';s_{n+1})\nonumber \\
&&=
K\,G(\vec{p},\vec{p}';s_{n+1})\,,\,\,\nonumber \\
&&G(\vec{p},\vec{p}';0)
=
\prod _{l=1}^n\frac{(2\pi )^{1-2\epsilon}}{\mu ^{2\epsilon}}\,
\delta ^{2-2\epsilon}(p_{l}-p'_{l})\,.
\end{eqnarray}
Here the kernel $K$ in LLA can be expressed in terms of
the infraredly stable Hamiltonian $H$ 
\beq
K=\omega (t) -\frac{g^2N_c}{16 \pi ^2}\,H\,,\,\,
\omega (t)=a\left(\frac{1}{\epsilon}-\ln \frac{-t}{\mu _2}\right)\,,
\,\,t=-|q|^2\,,
\eeq
\beq
H=\ln \frac{|p_1|^2|p_{n+1}|^2}{|q|^4}+\sum _{l=1}^{n}H_{l,l+1}\,,
\eeq
where the pair Hamiltonians are
\begin{eqnarray}
&&H_{l,l+1}=\ln |p_l|^2+\ln |p_{l+1}|^2\nonumber \\
&&+
p_l\,p_{l+1}^*\,\ln |\rho _{l,l+1}|^2\,\frac{1}{p_l\,p_{l+1}^*}\nonumber \\
&&+
p_l^*\,p_{l+1}\,\ln |\rho _{l,l+1}|^2\,\frac{1}{p_l^*\,p_{l+1}}\,.
\end{eqnarray}

\section{Integrable open Heisenberg spin chain}
The Hamiltonian for the $n+1$-gluon composite state 
in the ajoint representation has the property of the holomorphic
separability~\cite{intBDS}
\beq
H=h+h^*\,,\,\,h=\ln \frac{p_1\,p_{n+1}}{q^2}+
\sum _{l=1}^{n}h_{l,l+1}\,,
\eeq
where
\beq
h_{l,l+1}=\ln p_l+\ln p_{l+1}+
p_l\,\ln \rho _{l,l+1}\,\frac{1}{p_l}+
p_{l+1}\,\ln \rho _{l,l+1}\,\frac{1}{p_{l+1}}\,.
\eeq


Using the duality transformations (cf.~\cite{dual})
\begin{equation}
p_1=z_{0,1}\,,\,\,p_r=z_{r-1,r}\,,\,\,q=z_{0,n}\,,\,\,\rho _{r,r+1}=
i\frac{\partial}{\partial z_r}=i\partial _r\,,
\end{equation}
the holomorphic hamiltonian can be rewritten in the form invariant 
under the M\"{o}bius
transformations
\begin{equation}
z_k\rightarrow \frac{az_k+b}{cz_k+d}\,.
\end{equation}
Therefore we can put
\begin{equation}
z_0=0\,,\,\,z_n=\infty \,,
\end{equation}
Further, by regrouping the terms one can present it in another 
form~\cite{intBDS}
\begin{equation}
\ln (z_{1}^2\partial _1)+
\ln (\partial _{n-1})+2\gamma +
\sum_{r=1}^{n-2}\,h'_{r,r+1}\,,
\end{equation}
where
\[
h'_{r,r+1}=\ln (z_{r,r+1}^2\partial _r)+
\ln (z_{r,r+1}^2\partial _{r+1})-2\ln z_{r,r+1}+
2\gamma
\]
\begin{equation}
=\ln (\partial _r)+\ln (\partial _{r+1})
+\frac{1}{\partial _r}\,\ln z_{r,r+1}\,\partial _r+
\frac{1}{\partial _{r+1}}\,\ln z_{r,r+1}\,\partial _{r+1}
+2\gamma \,.
\end{equation}
The pair hamiltonian $h'_{r,r+1}$ coincides in fact with
the expression (\ref{pairh}) in the coordinate representation
acting on the wave function with 
non-amputated propagators.

The remarkable property of $h$ is 
its commutativity with the matrix element $D(u)$ of the monodromy matrix 
(\ref{ABCD}) introduced above for the description of integrability 
of the BKP equations in the multi-color QCD~\cite{intBDS}
\beq
[D(u),h]=0\,.
\eeq
Therefore if we write $D(u)$ as a polynomial in $u$
\begin{equation}
D(u)=\sum _{k=0}^{n-1}u^{n-1-k}\,q'_k\,,
\end{equation}
then the differential operators
\begin{equation}
q'_k=-\sum _{0<r_1<r_2<...<r_{k}<n}z_{r_1}\,
\prod _{s=1}^{k-1}z _{r_s,r_{s+1}}\,
\prod _{t=1}^k i\partial _{r_t}
\end{equation}
are independent integrals of motion with the properties
\beq
[q'_k,h]=[q'_k,q'_t]=0\,.
\eeq 
It turns out, that $h$
coincides with the local hamiltonian
of the open integrable Heisenberg model in which spins
are generators of the M\"{o}bius group. 

To solve this model one can use the algebraic Bethe anzatz.
In this case it is convenient to go to the transposed space,
where there exists the pseudo-vacuum state $\Psi _0$
\beq
\Psi _0=\prod_{r=1}^{n-1}z_r^{-2}\,,
\eeq
satisfying the equation
\beq
C^t(u)\Psi _0=0\,.
\eeq
Here $C^t(u)$ is the transposed matrix element $C(t)$ of the
monodromy matrix (\ref{ABCD}). The eigenvalues of the hamiltonian and 
the integral of motion $D(u)$ are constructed by applying
the product of its matrix elements $B^t(u)$ to the pseudovacuum
state
\beq
\Psi _k=\prod _{r=1}^k B^t(u_r)\,\Psi _0\,.
\eeq
For such eigenfunctions the spectral parameters $u_r$ should obey 
the Bethe equations. Instead one can introduce the Baxter function
which is the generating function of the Bethe roots
\beq
Q(u)=\prod _{k=1}^\infty (u-u_k)\,.
\eeq 
Generally the number of the roots $u_k$ is infinite. The Baxter
function satisfies the Baxter equation which is reduced to the simple
recurrent relation for our open spin chain
\begin{equation}
\Lambda (u)\,Q(u)=(u+i)^{n-1}\,Q(u+i)\,,
\end{equation}
where $\Lambda (u)$ is an eigenvalue of the integral of motion $D(u)$
and can be written in terms of its roots
\begin{equation}
D(u)\,\Psi _{a_1,a_2,...,a_{n-1}}=\Lambda (u)\,\Psi _{a_1,a_2,...,a_{n-1}}
\,,\,\, \Lambda (u)=\prod _{r=1}^{n-1}(u-ia_r)\,.
\end{equation}
As a result, the solution of the Baxter equation can be found
in the form~\cite{intBDS}
\beq
Q(u)=\prod _{r=1}^{n-1}\frac{\Gamma (-iu-a_r)}{\Gamma (-iu+1)}
\eeq
up to a possible factor being a periodic function of $-iu$.

The Regge trajectory of the composite state  of $n-1$ gluons has the 
additivity property
\begin{equation}
\omega _n(t)=\omega (t)-\frac{a}{2}\,E\,,\,\,
E=
\sum _{r=1}^{n-1}\epsilon (a_r)+
\sum _{r=1}^{n-1}\epsilon (\widetilde{a}_r)\,,
\end{equation}
where 
\beq
\epsilon (a)=\psi (a)+\psi (1-a)-2\psi (1)\,,\,\,
a_r=i\nu _r+\frac{n_r}{2}\,.
\eeq

\section{Three gluon composite state}

The wave function of the 
three gluon composite state in the adjoint representation can be
constructed as a bilinear combination of eigenfunctions of
the integrals of motion $D(u)$ and $D^*(u)$ having the property
of single-valuedness in the coordinate space~\cite{intBDS}   
\begin{eqnarray*}
&&\Psi \sim z^{a_1+a_2}_2\,(z^*_2)^{\widetilde{a_1}+\widetilde{a_2}}\,\\
&&\int \frac{d^2y}{|y|^2}\,
y^{-a_2}(y^*)^{
-\widetilde{a_2}} 
\left(\frac{y-1}{y-x}\right)^{a_1}\,
\left(\frac{y^*-1}{y^*-x^*}\right)^{\widetilde{a_1}},\\
&&x=\frac{z_2}{z_1}.
\end{eqnarray*}

One can perform its Fourie transformation to the momentum space
\begin{equation*}
\Psi ^t(\vec{p}_1,\vec{p}_2 )=(p_1+p_2)^{-a_1-a_2}
(p_1^*+p_2^*)^{-\widetilde{a}_1-\widetilde{a}_2}
\,\phi (\vec{y})\,,\,\,y=\frac{p_2}{p_1}\,,
\end{equation*}
where
\begin{equation*}
\phi (\vec{y})=\int d^2t\,
\left(\frac{1}{t\,y}+1\right)^{a_1}\,
\left(\frac{1}{t^*\,y^*}+1\right)^{\widetilde{a}_1}\,
(1-t)^{a_2-1}\,(1-t^*)^{\widetilde{a}_2-1}\,.
\end{equation*}
This function can be presented in terms of its Mellin transformation
\begin{eqnarray*}
&&\Psi ^t(\vec{p}_1,\vec{p}_2 )=(p_1+p_2)^{-a_1-a_2}
(p_1^*+p_2^*)^{-\widetilde{a}_1-\widetilde{a}_2}
\\
&&\int d^2u\,
\phi (u,\widetilde{u})\,\left(\frac{p_1}{p_2}\right)^{-iu}\,
\left(\frac{p_1^*}{p_2^*}\right)^{-i\widetilde{u}},
\end{eqnarray*}
where
\begin{equation*}
-iu=i\nu _u+\frac{N_u}{2}\,,
\,\,-i\widetilde{u}=i\nu _u-\frac{N_u}{2}\,\,,\,\,\,
\int d^2u\equiv \int _{-\infty}^{\infty} d\nu _u \sum
_{N_u=-\infty}^{\infty} \,.
\end{equation*}
and
\begin{eqnarray*}
&&\phi (u,\widetilde{u})=
\frac{\pi ^2\Gamma (1+\widetilde{a}_1)
\Gamma (a_2)}{\Gamma (-a_1)\,\Gamma (1-\widetilde{a}_2)}\,
\frac{\Gamma (iu)\Gamma (1+i\widetilde{u})}{\Gamma (-iu)\,
\Gamma (1-i\widetilde{u})}\\
&&\frac{\Gamma (-iu-a_1)  
\,\Gamma (-iu-a_2)}{
\Gamma (1+i\widetilde{u}+\widetilde{a}_1)
\Gamma (1+i\widetilde{u}+\widetilde{a}_2)}.
\end{eqnarray*}
Really the last form of $\Psi^t$ corresponds to the 
Baxter-Sklyanin representation~\cite{Veg},
because the function $\phi$ is a product of the pseudovacuum
state and the Baxter function~\cite{intBDS}
\begin{equation*}
\phi (u,\widetilde{u})=u\,\widetilde{u}\,Q(u,\widetilde{u})\,,
\end{equation*}
where
\begin{eqnarray*}
&&Q(u,\widetilde{u})\sim
\frac{\Gamma (iu)\Gamma (i\widetilde{u})}{\Gamma (1-iu)\,
\Gamma (1-i\widetilde{u})}\\
&&\frac{\Gamma (-iu-a_1)
\Gamma (-iu-a_2)}{
\Gamma (1+i\widetilde{u}+\widetilde{a}_1)
\Gamma (1+i\widetilde{u}+\widetilde{a}_2)}.
\end{eqnarray*}
  




\section{Discussion of obtained results}

It was demonstated, that Pomeron in QCD is a composite state of reggeized
gluons.
The BFKL dynamics is integrable in LLA. In the next-to-leading
approximation in $N=4$ SUSY the equation for the Pomeron
wave function has remarkable properties including the analyticity in
the conformal spin $n$ and the maximal transcendentality. In this model the
BFKL Pomeron coincides with the reggeized graviton.
The BDS ansatz for scattering amplitudes in $N=4$ SUSY does not agree with 
the BFKL approach in the multi-Regge kinematics. The reason for this drawback 
is the absence of the Mandelstam cuts. The BFKL-like equation for the
composite state of two reggeized gluons with adjoint quantum numbers is 
explicitely solved. It is shown, that the equation for the composite
state of an arbitrary number of reggeized gluons in the adjoint representation
is equivalent to the Schr\"{o}dinger equation for an integrable open Heisenberg
spin chain. The wave function for three gluon composite state is constructed in the 
Baxter-Sklyanin  representation. 

\begin{theacknowledgments}
I thank L.D. Faddeev, J. Bartels and A. Sabio Vera for
helpful discussions.
\end{theacknowledgments}
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\end{thebibliography}

\end{document}


