BFKL – past and future∗

V.S. Fadin${}^{a,b\,{\dagger}}$

aBudker Institute of Nuclear Physics of SB RAS, 630090 Novosibirsk, Russia

b Novosibirsk State University, 630090 Novosibirsk, Russia

This paper contains my recollection of the creation and development of the so-
called BFKL approach and my ideas about the ways of its further development.

∗Work supported in part by the Ministry of Science and Higher Education of
Russian Federation and in part by RFBR, grant 19-02-00690.
$\begin{array}[]{ll}{}^{{\dagger}}\mbox{{\it e-mail
<EMAIL_ADDRESS>\end{array}$

##  1 BFKL history

The BFKL story goes back to the middle of 70-th of the last century, when Lev
started [1] to study the high energy behaviour of non-Abelian gauge theories
[2]. To avoid infrared singularities, the investigation was performed in the
theories with Higgs mechanism of mass generation [3] \- [5] preserving
renormalizability. More definitely, the model conserving global isotopic
invariance was considered, with local gauge symmetry SU(2) and isotopic
doublet of scalar fields. The paper [1] was devoted to check of Reggeization
of the gauge boson and investigation of the vacuum singularity in this theory.

###  1.1 Historical background

Recall that the notion of Reggeization of elementary particles in perturbation
theory was introduced and study of this phenomenon was started in the series
of papers by Gell-Mann and co-authors [6] \- [9]. In terms of the relativistic
partial wave amplitude $A_{j}(t)$, analytically continued to complex $j$
values, the Reggeization was defined as the disappearance, due to radiative
corrections, of the non-analytic terms of the Born approximation on account of
one-particle exchanges in the $t$ channel. In other words, Reggeization of an
elementary particle with spin $j_{0}$ and mass $m$ means that at large $s$ and
fixed $t$ Born amplitudes with exchange of this particle in the $t$ channel
acquire a factor $s^{j(t)-j_{0}}$, with $j(m^{2})=j_{0}$, as a result of
radiative corrections. Gell-Mann and his collaborators established the
presence of this phenomenon for backward Compton scattering in QED with heavy
photon and thus showed the fermion Reggeization in this theory. They also
investigated scalar electrodynamics and concluded that a scalar is not
Reggeized. In [8] they formulated necessary for Reggeization conditions on the
Born scattering amplitudes of the theory. These conditions were generalized
later in [10]. In [11] it was shown that they are not fulfilled in the vector
channel of massive QED. Non-Reggeization of the heavy photon was confirmed by
direct two-loop calculations in [12].

Note that the term ”Reggeization” can be understood in different ways. If we
assume that it means only the existence of the Regge trajectory on which the
particle lies, then the scalar interacting with the photon is Reggeized [13].
We use this term in a stronger sense: it means not only the existence of the
Regge trajectory of the particle, but that the entire amplitude is given by
the Reggeon with this trajectory.

In non-Abelian gauge theories the Reggeization problem was considered in Refs.
[14] \- [18]. In particular, it was shown in [16], [17] that in the theories
with the Higgs mechanism of mass generation (unlike the theories with mass
terms in original Lagrangian [15]) the necessary for Reggeization conditions
[8] \- [10] are fulfilled for the vector channel. Fulfilment of these
conditions allowed to find [16] the trajectory of the vector meson using the
$t$-channel unitarity:

$j_{V}(t)=1+\frac{8g^{2}}{\pi}(t-m_{V}^{2})\int_{4m_{V}^{2}}^{\infty}\frac{dt^{\prime}}{t^{\prime}-t}\frac{\rho(t^{\prime})}{t^{\prime}-4m_{V}^{2}}~{},$
(1)

where

$\rho(t)=\frac{1}{32\pi}\sqrt{\frac{t-4m_{V}^{2}}{t}}~{}.$ (2)

However, this was only an indication of the possibility of the Reggeization,
and not at all a proof of it. Therefore, it was very important to check the
Reggeization by direct calculations, and the first Lev’s aim was to do it.

The second aim was to investigate the vacuum singularity, i.e. asymptotics of
amplitudes with vacuum quantum numbers in the $t$-channel and positive
signature. In the Regge-Gribov theory of complex angular momenta this
singularity originally was introduced (with intercept equal to one) [19], [20]
to provide constant cross sections at asymptotically high energies. Because of
its fundamental role, this singularity has received a special name: it was
called Pomeron after I. Ya. Pomeranchuk.

This singularity was investigated in QED [21] \- [23] (Lev was one of the main
investigators), and it was shown that in the LLA it is a fixed branch point to
the right of 1 (at $j=1+\frac{11}{32}\pi\alpha^{2}$), i.e. the LLA violates
the Froissart bound [24]. Investigation of the Pomeron in the non-Abelian
gauge theories was especially interesting because of vector boson
Reggeization, unlike photon.

###  1.2 Dispersive approach and the first results

It was important to check the Reggeization by direct calculations, and it was
done in [1] in two loops with the leading logarithmic accuracy (LLA), when in
each order of perturbation theory only terms with the highest powers of $\ln
s$ are kept.

For the calculations Lev used the dispersive approach based on the general
properties of the theory: analyticity, unitarity and renormalizability. To my
mind, it was the first application of the dispersive approach to the non-
Abelian gauge theories. This approach turned out very successful, because it
permits to escape consideration of great number of Feynman diagrams and to
work only with physical particles in the unitarity relations thus avoiding the
use of ghosts by Faddeev-Popov. Now it is widely used, unfortunately, without
any reference to Lev.

Note that the term LLA was used to indicate that only terms with the highest
powers of the logarithm of c.m.s. energy $\sqrt{s}$ are being held in
amplitude discontinuities, not in the amplitudes themselves. Total amplitudes
are obtained from their $s$-channel discontinuities in this approximation by
the substitutions

$is\ln^{n}s\rightarrow-\frac{s}{2\pi(n+1)}\left(\ln^{n+1}(-s-i\epsilon)\pm\ln^{n}(s-i\epsilon)\right)~{},$
(3)

where $+(-)$ sign for negative (positive) signature (symmetry with respect to
the replacement $s\leftrightarrow u\simeq-s$). Therefore the degrees of
logarithms in the real parts of the amplitudes with a negative signature are
one more than the degree in the imaginary parts, whereas for the amplitudes
with a positive signature, the reverse is true, and in the LLA they are purely
imaginary. It is very important, because thanks to this in the LLA (and in the
next to it, as it will be discussed below) only amplitudes with negative
signature must be kept on the right side of the unitarity relations

$\mbox{Im}_{s}{\cal A}^{A^{\prime}B^{\prime}}_{AB}\ =\
\frac{1}{2}\sum_{n=0}^{\infty}\sum_{\\{f\\}}\int{\cal
A}^{\tilde{A}\tilde{B}+n}_{AB}\left({\cal
A}^{\tilde{A}\tilde{B}+n}_{A^{\prime}B^{\prime}}\right)^{*}d\Phi_{\tilde{A}\tilde{B}+n}\,,$
(4)

where $\sum_{\\{f\\}}$ means sum over discrete quantum numbers of intermediate
particles, ${\cal A}^{\tilde{A}\tilde{B}+n}_{AB}$ and $\left({\cal
A}^{\tilde{A}\tilde{B}+n}_{A^{\prime}B^{\prime}}\right)$ are the amplitudes of
$n$-particle production and $d\Phi_{\tilde{A}\tilde{B}+n}$ is the
corresponding phase space element.

As it is seen from (4), calculation of elastic amplitudes in the dispersive
approach requires knowledge of inelastic ones. It was recognized by Lev that
the inelastic amplitudes are necessary in special kinematics. In the LLA it is
so called multi-Regge kinematics (MRK), where produced particles have limited
(not growing with $s$) transverse momenta and are strongly ordered in
rapidity. For two-loop calculations one-particle production amplitudes in the
Born approximation are necessary. Using the $t$-channel unitarity, Lev
obtained a simple factorized form of these amplitudes with famous now
Lipatov’s vertex for production of vector mesons.

The main result of Ref. [1] was that in the LLA two-loop radiative corrections
to elastic scattering amplitudes with the gauge boson quantum numbers in the
$t$-channel and negative signature have the Regge form. As for the vacuum
singularity, it was shown only that it has not a pole form. More definite
conclusions about the nature of this singularity on the basis of two-loop
calculations can not be drawn.

###  1.3 Pre-BFKL

Next important steps in the investigation of the high energy behaviour of non-
Abelian gauge theories was done in [25] \- [27]. Two-particle production
amplitudes were found in the Born approximation and the one-loop corrections
to the one-particle production amplitudes were calculated. It turned out that
the first ones have the factorized form with the same vertices as the one-
particle production amplitudes and that the corrections to the last ones have
the Regge form. Three-loop corrections to elastic amplitudes calculated using
these results proved the vector boson Reggeization in this order. Details of
derivation of all these results and their generalization on colours group
$SU(N)$ with arbitrary $N$ were described in [26].

An extremely important step made on this basis was the hypothesis of vector
meson Reggeization. It was assumed that in the LLA not only elastic, but also
inelastic amplitudes in the MRK with quantum numbers of vector bosons and
negative signatures in all cross-channels are given by the Regge pole
contributions in all orders of perturbation theory. Since only such amplitudes
are important in the unitarity relations (4), it is possible to express
partial waves
$(A_{AB}^{A^{\prime}B^{\prime}})^{T}_{\omega}(t)\;\;\;(\omega=j-1)$ of
amplitudes of the process ${AB}\rightarrow{A^{\prime}B^{\prime}}$ with the
global $SU(2)$ ”isospin” $T$

$(A_{AB}^{A^{\prime}B^{\prime}})^{T}_{\omega}(t)=\int_{s_{0}}^{\infty}\frac{ds}{s}s^{-\omega-1}\mbox{Im}_{s}(A_{AB}^{A^{\prime}B^{\prime}})^{T}(s,t)~{},\;\;$
$(A_{AB}^{A^{\prime}B^{\prime}})^{T}(s,t)=\frac{s}{2\pi
i}\int_{\delta-i\infty}^{\delta+i\infty}\frac{d\omega}{\sin(\pi\omega)}$
$\times\left((-1)^{T}\left(\frac{s}{s_{0}}\right)^{\omega}-\left(\frac{-s}{s_{0}}\right)^{\omega}\right)(A_{AB}^{A^{\prime}B^{\prime}})^{T}_{\omega}(t)~{},$
(5)

through partial waves of amplitude
$(F_{B^{\prime}B})^{T}_{\omega}(k_{\perp},q_{\perp}-k_{\perp})$ of Reggeon
scattering on the particle $B$

$(A_{AB}^{A^{\prime}B^{\prime}})^{T}_{\omega}(t)=\Phi^{T}_{A^{\prime}A}\int\frac{d^{2}{k^{\prime}}_{\perp}\;(F_{B^{\prime}B})^{T}_{\omega}(k^{\prime}_{\perp},q_{\perp}-k^{\prime}_{\perp})}{({k^{\prime}}_{\perp}^{2}-m^{2})({(q-k^{\prime})}_{\perp}^{2}-m^{2})}$
(6)

and to write for them the equation [25]:

$[2+\omega-\alpha(k_{\perp}^{2})-\alpha((q-k)_{\perp}^{2})](F_{B^{\prime}B})^{T}_{\omega}(k_{\perp},q_{\perp}-k_{\perp})=\Phi^{T}_{B^{\prime}B}+\frac{g^{2}}{(2\pi)^{3}}$
$\times\int\frac{d^{2}{k^{\prime}}_{\perp}\;(F_{B^{\prime}B})^{T}_{\omega}(k^{\prime}_{\perp},q_{\perp}-k^{\prime}_{\perp})}{({k^{\prime}}_{\perp}^{2}-m^{2})({(q-k^{\prime})}_{\perp}^{2}-m^{2})}\biggl{[}A_{T}(q_{\perp}^{2})-\left(2-\frac{1}{2}T(T+1)\right)$
$\times\frac{(k_{\perp}^{2}-m^{2})((q-k^{\prime})_{\perp}^{2}-m^{2})+({k^{\prime}}_{\perp}^{2}-m^{2})((q-k)_{\perp}^{2}-m^{2})}{{(k-k^{\prime})}_{\perp}^{2}-m^{2}}\biggr{]}~{},$
(7)

where $t=q^{2}=q_{\perp}^{2}$, $\alpha(t)$ is the vector meson trajectory,

$\alpha(t)=1+\frac{g^{2}}{(2\pi)^{3}}(t-m^{2})\int\frac{d^{2}k_{\perp}}{(k_{\perp}^{2}-m^{2})((q-k)_{\perp}^{2}-m^{2})}~{},\;\;t=q^{2}=q_{\perp}^{2}~{},$
(8)

$\Phi^{T}_{A^{\prime}A}$ and $\Phi^{T}_{B^{\prime}B}$ are the impact factor
for the $A\rightarrow A^{\prime}$ and $B\rightarrow B^{\prime}$ transitions,

$A_{T}(q^{2})=\frac{m^{2}}{2}+\left(2-\frac{1}{2}T(T+1)\right)(t-\frac{1}{2}m^{2})~{}.$
(9)

It is not difficult to see that $\alpha(t)$ is equal to $j_{V}(t)$ defined in
(8).

It was shown in [25] that there is bootstrap in the vector boson channel,
corresponding to $T=1$: the equation (7 ) gives in the $j$-plane only the same
Regge pole which was assumed. Indeed, the solution of Eq. (7) for $T=1$ is

$(F_{B^{\prime}B})^{T}_{\omega}(k_{\perp},q_{\perp}-k_{\perp})\Big{|}_{T=1}=\frac{\Phi^{T}_{B^{\prime}B}}{\omega+1-\alpha(q^{2})}~{},$
(10)
$(A_{AB}^{A^{\prime}B^{\prime}})^{T}_{\omega}(t)\Big{|}_{T=1}={\Phi^{T}_{A^{\prime}A}\Phi^{T}_{B^{\prime}B}}\Big{|}_{T=1}\frac{(\alpha(q^{2})-1)}{g^{2}(t-m^{2})(\omega+1-\alpha(q^{2}))}~{},$
(11)

and since $\Phi^{T}_{P^{\prime}P}\Big{|}_{T=1}=g\Gamma_{P^{\prime}P}$, where
$\Gamma_{P^{\prime}P}$ is the vertex of reggeon-particle interaction,

$(A_{AB}^{A^{\prime}B^{\prime}})(s,t)\Big{|}_{T=1}={\Gamma_{A^{\prime}A}\frac{\left((-u)^{\alpha(q^{2})}-(-s)^{\alpha(q^{2})}\right)}{(t-m^{2})}{\Gamma_{B^{\prime}B}}~{},\;\;u\approx-s}~{},$
(12)

in accordance with the Reggeization hypothesis. Of course, this could in no
way be considered as a proof of a hypothesis, because the hypothesis refers
not only to elastic amplitudes. Nevertheless, it was the first step to the
proof. Later the bootstrap conditions for inelastic amplitudes were formulated
and the proof of the hypothesis was held on their basis [28].

So, a primary Reggeon in this approach (subsequently named BFKL) is the
Reggeized gauge boson. The Pomeron, which corresponds to the rightmost
$j$-plane singularity of the partial wave with $T=0$ in (7) and determines the
high energy behaviour of cross sections, appears as a compound state of two
Reggeized gauge bosons. It was shown [25] that in the Pomeron channel the
leading $j$-plane singularity turns out to be a square root branch point at

$j=1+N\frac{g^{2}}{\pi}\ln 2~{},$ (13)

for the gauge group $SU(N)$. Therefore, the Froissart bound [24] is violated
as well as in QED [22], [23], although the mechanism of this violation differs
from QED: in the non-Abelian theories cross sections for production of any
fixed number of particles decrease with energy due to the vector boson
Reggeization, and the total cross section increases as power of $s$ only due
to increasing number of opening channels. The reason of the violation is that
the $s$-channel unitarity is not fulfilled in the LLA.

Another important observation made in [25] was the growth of the typical
transverse momenta with increasing $s$. It was conjectured that due to this
growth and the asymptotic freedom only Regge poles may appear to the right
from the point $j=1$.

Detailed consideration of the Pomeron channel was done in [27].

###  1.4 Appearance of the BFKL and its development

Appearance of the BFKL should be attributed to the end of the 1970s, when Lev
turned to QCD and with his student Yan Balitsky applied the methods developed
for non-Abelian theories with broken symmetry in [1], [25] \- [27] to
scattering of colourless particles in QCD. In 1978, a famous paper [29] was
published. The results of [25] (see (7), (8)) can be applied in the massless
limit (theory without spontaneous breaking of symmetry) only with some
regularization (in the following dimensional regularization will be used). The
reason is that they were obtained for scattering of coloured particles (we
will call them partons). The key difference of colourless particles from
partons is that due to the gauge invariance impact factors of colourless
particles, describing their interaction with Reggeized gluons, vanish at zero
gluon momenta. It was shown in [29] that in this case the transition to
massless theory is not difficult: the equation (7) with
$\Phi^{T}_{B^{\prime}B}$ corresponding to the impact factor of a colourless
particle is free from the infrared singularities and therefore can be used in
QCD.

From now on, we will talk about gauge theories with unbroken symmetry,
although, as it follows from the above, the approach can be used in theories
with broken symmetry as well. If not specified, QCD will be kept in mind,
although for generality, the gauge group will be taken $SU(N_{c})$, with the
number of colours $N_{c}$.

An approximate theory can only be considered consistent when it is possible to
calculate corrections to the approximation. Development of the BFKL approach
in the next-to-leading logarithmic approximation (NLLA) was started in the
late 1980s [30]. In the NLLA, the same scheme of derivation of the BFKL
equation as in the LLA is applicable. Again only amplitudes with negative
signature must be kept on the right side of the unitarity relations (4),
because leading terms in amplitudes with positive signature are imaginary and
have one less degree of $\ln s$. Moreover, only real parts of the amplitudes
with negative signature are necessary because their imaginary parts are
suppressed by one power of $\ln s$ compared to their real parts. It was
supposed that these amplitudes have the Regge pole form, i.e. are expressed in
terms of the gluon trajectory and the Reggeon vertices. Therefore it was
necessary to calculate one-loop corrections to the gluon trajectory and the
vertices used in the LLA. Also, instead of one gluon two-gluon and quark-
antiquark jets can be produced in the NLLA. Therefore it was necessary to
calculate new Reggeon vertices for the jet production.

The calculations required a lot of effort and time. They started with the two-
gluon jet production vertex [30]. A lot of work was done and a lot of papers
were published (see [30] \- [45]) before everything necessary for obtaining
the NLO BFKL kernel in a colourless (Pomeron) channel at zero momentum
transfer became available. Then this kernel and his eigenvalues were found
[46] (see also [47]). It is necessary to notice here that although talking
about BFKL kernel (or BFKL equation ) they usually mean exactly the kernel for
forward scattering, although the approach is also applicable to scattering
with the transfer of both momentum and colour. To date, a lot of work has been
done and a number of important results on the development of the BFKL as
applied to such processes have been obtained.

##  2 Basics of the BFKL approach

###  2.1 The gluon Reggeization

The BFKL approach is based on a remarkable property of QCD – gluon
Reggeization, which gives a very powerful tool for the description of high
energy processes. The gluon Reggeization determines the form of QCD amplitudes
at large energies and limited transverse momenta. Due to the Reggeization,
dominant amplitudes have a simple factorized form in the next-to-leading
logarithmic approximation (NLLA) as well as in the LLA. The Reggeization
allows to express an infinite number of amplitudes through several effective
vertices and gluon trajectory.

In the Regge kinematic region $s\simeq-u\rightarrow\infty$, $t$ fixed,
amplitudes of the process $A+B\rightarrow A^{\prime}+B^{\prime}$ with a colour
octet $t$-channel exchange and negative signature can be depicted by the
diagram of Fig.1 and have the Regge form

Figure 1:  Schematic representation of the process $A+B\rightarrow
A^{\prime}+B^{\prime}$ with colour octet in the $t$ channel and negative
signature. The zigzag line represents Reggeized gluon.

${{\cal
A}}^{A^{\prime}B^{\prime}}_{AB}={\Gamma}^{c}_{A^{\prime}A}\left[\left({-s\over-t}\right)^{j(t)}-\left({s\over-t}\right)^{j(t)}\right]\Gamma^{c}_{B^{\prime}B}~{},$
(14)

where ${\Gamma}^{c}_{P^{\prime}P}$ are energy-independent particle-particle-
Reggeon (PPR) vertices (or scattering vertices), $c$ is a colour index,
$j(t)=1+\omega(t)$ is the Reggeized gluon trajectory.

As it is seen from (14), in the leading order (LO) PPR vertices are determined
by the Born amplitudes, so that their calculation is trivial assuming the
gluon Reggeization, i.e. the form (14). Expressions for them are also quite
simple. In the helicity basis they have the same form for all partons (quarks
and gluons in QCD):

$\Gamma_{P^{\prime}P}^{c}\ =\
gT^{c}_{P^{\prime}P}\delta_{\lambda_{P^{\prime}}\lambda_{P}}\,,$ (15)

where $T^{c}_{P^{\prime}P}$ are matrix elements of the colour group generators
in the corresponding representations and $\lambda$ are parton helicities.
Except for a common coefficient the vertices (15) can be written down without
calculation, because they are given by forward matrix elements of the
conserved current. Note that Eq. (15) implies a definite choice of the
relative phase of spin wave functions of particles $P^{\prime}$ and $P$.
Evidently, the phase is zero at $t=0$. In (15) the $s$-channel helicity
conservation is exhibited explicitly. Note that for gluons and for massive
quarks it is valid only in the LO.

Again, as it is seen from (14), the LLA (one-loop) trajectory is determined by
the $s$-channel discontinuity of any amplitude. It gives

$\omega(t)=\frac{g^{2}N_{c}t}{2(2\pi)^{D-1}}\int\\!\frac{d^{D-2}q_{1}}{\vec{q}_{1}^{\>2}(\vec{q}-\vec{q}_{1})^{2}}=-g^{2}\frac{N_{c}\Gamma(1-\epsilon)}{(4\pi)^{D/2}}\frac{\Gamma^{2}(\epsilon)}{\Gamma(2\epsilon)}(\vec{q}^{\>2})^{\epsilon}$
$\simeq-g^{2}\frac{N_{c}\Gamma(1-\epsilon)}{(4\pi)^{2+\epsilon}}\frac{2}{\epsilon}(\vec{q}^{\>2})^{\epsilon}\,.$
(16)

In (16) and below the vector sign means transverse to the $p_{A},p_{B}$ plane
components, $D=4-2\epsilon$ is the space-time dimension taken different from 4
to regularize infrared divergencies, which are inevitable in parton
amplitudes. Of course, using the dimensional regularization in (8) at $m=0$
one obtains $\alpha(t)=1+\omega(t)$, where $\omega(t)$ is given by (16) at
$N_{c}=2$.

The Reggeization means also definite (multi-Regge) form of production
amplitudes in the multi-Regge kinematics (MRK). MRK is the kinematics where
all particles have limited (not growing with $s$) transverse momenta and are
combined into jets with limited invariant mass of each jet and large (growing
with $s$) invariant masses of any pair of the jets. This kinematics gives
dominant contributions to cross sections of QCD processes at high energy
$\sqrt{s}$. In each order of perturbation theory dominant (having the largest
$\ln s$ degrees) are amplitudes with gluon quantum numbers and a negative
signature. For the amplitude ${\cal A}_{2\rightarrow n+2}$ of the process
$A+B\rightarrow A^{\prime}+J_{1}+\ldots+J_{n}+B^{\prime}$ of production of $n$
jets with momenta $k_{1},k_{2},\ldots k_{n}$ the MRK means

$s\gg
s_{i}\gg|t_{i}|\simeq\vec{q}^{\;2}_{i},~{}~{}~{}~{}\;\;s\simeq\frac{\prod_{i=1}^{n+1}s_{i}}{\prod_{i=1}^{n}(\vec{k}^{\;2}_{i}+k_{i}^{2})}~{},$
(17)

where

$s=(p_{A}+p_{B})^{2},\;s_{i}=(k_{i-1}+k_{i})^{2},\;\;i=1,\cdot\cdot\cdot
n+1,\;\;k_{0}\equiv P_{A^{\prime}},\;\;k_{n+1}\equiv P_{B^{\prime}},$ (18)
$q_{1}=p_{A}-p_{A}^{\prime},\;q_{j+1}=q_{j}-k_{j},\;j=1,\cdot\cdot\cdot
n,\;\;q_{n+1}=p_{B^{\prime}}-p_{B}~{}.$ (19)

In this region the amplitudes ${\cal A}_{2\rightarrow n+2}$ can be represented
by Fig. 2.

Figure 2:  Schematic representation of the process $A+B\rightarrow
A^{\prime}+J_{1}+\dots+J_{n}+B^{\prime}$ in MRK. The zig-zag lines represent
Reggeized gluon exchange; the black circles denote the Reggeon vertices;
$q_{i}$ are Reggeon momenta.

Multi-particle amplitudes have a complicated analytical structure. They are
not simple even in MRK (see, for instance, [48] \- [32]). Fortunately, only
real parts of these amplitudes are used in the BFKL approach in the NLLA as
well as in the LLA. The reason is that imaginary parts of amplitudes with
negative signature are suppressed by one power of $\ln s$ compared to their
real parts, so that account of them in the right part of the unitarity
relations (4) means loss of two powers of $\ln s$ (as well as account of
amplitudes with positive signature). Restricting ourselves to the real parts
we can write

$\Re{\cal A}_{2\rightarrow
n+2}=2s\Gamma^{{R}_{1}}_{A^{\prime}A}\left(\prod_{i=1}^{n}\frac{1}{t_{i}}\Big{(}\frac{s_{i}}{|\vec{k}_{i-1}||\vec{k}_{i}|}\Big{)}^{\omega(t_{i})}\gamma^{G_{i}}_{R_{i}{R}_{i+1}}\right)$
$\frac{1}{t_{n+1}}\Big{(}\frac{s_{n+1}}{|k_{n}||\vec{q}_{n+1}|}\Big{)}^{\omega(t_{n+1})}\Gamma^{{R}_{n+1}}_{B^{\prime}B}~{}.$
(20)

Here ${\Gamma}^{{R}}_{A^{\prime}A}$ and $\Gamma^{{R}}_{B^{\prime}B}$ are the
same scattering vertices as in (14) and $\gamma^{J_{i}}_{{R}_{i}{R}_{i+1}}$
are the Reggeon-Reggeon-Jet (RRJ) vertices (or production vertices), i.e. the
effective vertices for production of jets $J_{i}$ with momenta
$k_{i}$=$q_{i}-q_{i+1}$ in collisions of Reggeons with momenta $q_{i}$ and
$-q_{i+1}$.

In the LLA only gluons can be produced, $J=G$,

$\gamma^{G}_{{R}_{1}{R}_{2}}=gT^{c}_{c_{1}c_{2}}e^{*}_{\mu}(k)C^{\mu}(q_{2},q_{1})\,,$
(21)

where $k=q_{1}-q_{2}$, $e_{\mu}(k)$ and $c$ are the gluon momentum,
polarization vector and colour index, $c_{1}$ and $c_{2}$ are the colour
indices of the Reggeons ${R}_{1}$ and ${R}_{1}$ respectively, and
$C^{\mu}(q_{1},q_{2})$ is the famous Lipatov’s vertex [1]:

$C_{\mu}(q_{2},q_{1})\ =\
-q_{1\mu}-q_{2\mu}+p_{1\mu}\left(\frac{q_{1}^{2}}{kp_{1}}+2\frac{kp_{2}}{p_{1}p_{2}}\right)-p_{2\mu}\left(\frac{q_{2}^{2}}{kp_{2}}+2\frac{kp_{1}}{p_{1}p_{2}}\right)$
$=\
-q_{1\perp\mu}-q_{2\perp\mu}-\frac{p_{1\mu}}{2(kp_{1})}\left(k_{\perp}^{2}-2q_{1\perp}^{2}\right)+\frac{p_{2\mu}}{2(kp_{2})}\left(k_{\perp}^{2}-2q_{2\perp}^{2}\right).$
(22)

The Reggeon vertices and the gluon trajectory are known now in the next-to-
leading order (NLO), that means the one-loop approximation for the vertices
and the two-loop approximation for the trajectory. It is just the accuracy
which is required for the derivation of the BFKL equation in the NLLA.
Validity of the forms (14) and (20) is proved now in all orders of
perturbation theory in the coupling constant $g$ both in the LLA [28] and in
the NLLA [51, 52, 53, 54, 55].

In the LLA only gluons can be produced. In the NLA one has to consider not
only the amplitudes (14), (20), but also amplitudes obtained from them by
replacement of one of final particles by a jet containing a couple of
particles with fixed (of order of transverse momenta) invariant mass.

###  2.2 The scheme of derivation of the BFKL equation

The Reggeization provides a simple derivation of the BFKL equation both in the
LLA and NLLA. Two-to-two scattering amplitudes with all possible quantum
numbers in the $t$–channel are calculated using the amplitudes (20) in the
$s$-channel unitarity relations (4) and analyticity. The $s$-channel
discontinuities of the amplitude ${\cal A}_{AB}^{A^{\prime}B^{\prime}}$ for
the high energy process $A+B\rightarrow A^{\prime}+B^{\prime}$ at fixed
momentum transfer $\sqrt{-t}$ may be presented by Fig.3,

Figure 3:  Schematic representation of the $s$-channel discontinuities of
amplitudes $A+B\rightarrow A^{\prime}+B^{\prime}$.

where
$q_{1}+q_{2}=q^{\prime}_{1}+q^{\prime}_{2}=q=p_{A}-p_{A^{\prime}}=p_{B^{\prime}}-p_{B}$
and symbolically written as the convolution

$\;\;\;\;\;\;\;\;\Phi_{A^{\prime}A}\;\otimes\;G\;\otimes\;\Phi_{B^{\prime}B}$

where the impact factors $\Phi_{A^{\prime}A}$ and $\Phi_{B^{\prime}B}$
describe transitions $A\rightarrow A^{\prime}$ and $B\rightarrow B^{\prime}$
due to interactions with Reggeized gluons, $G$ is the Green’s function for two
interacting Reggeized gluons with an operator form

${\hat{\cal G}}=e^{Y\hat{\cal{K}}}~{},$ (23)

where $\;Y=\ln(s/s_{0})$, $s_{0}$ is an energy scale, ${\hat{\cal{K}}}$ is the
BFKL kernel. The impact factors and the BFKL kernel are expressed in terms of
the Reggeon vertices and trajectory. Energy dependence of scattering
amplitudes is determined by the BFKL kernel, which is universal (process
independent). The kernel

${\hat{\cal{K}}}={\hat{{\omega}_{1}}}+{\hat{{\omega}}_{2}}+{\hat{\cal{K}}_{r}}$
(24)

is expressed through the Regge trajectories ${\hat{{\omega}}_{1}}$ and
${\hat{{\omega}}_{2}}$ of two gluons and the “real part” ${\hat{\cal{K}}_{r}}$
describing production of particles in their interaction:

${\hat{\cal{K}}_{r}}={\hat{\cal{K}}_{G}}+{\hat{\cal{K}}_{Q\bar{Q}}}+{\hat{\cal{K}}_{GG}}~{}.$
(25)

In the LLA only ${\hat{\cal{K}}_{G}}$ must be kept, because only gluons can be
produced; in the NLLA production of quark-antiquark (${Q\bar{Q}}$) and gluon
($GG$) pairs is also possible. The impact factors $\Phi_{A^{\prime}A}$ and
$\Phi_{B^{\prime}B}$ describing transitions $A\rightarrow A^{\prime}\;\;\;$
and $\;\;\;B\rightarrow B^{\prime}$ depend on properties of scattering
particles. All energy dependence is contained in the Green’s function $G$ for
two interacting Reggeized gluons.

While the contribution of the trajectories to the kernel is diagonal in
colour, the contribution associated with the production of real particles can
change colour states. Using the projection operators $\hat{\cal P}^{R}$ on the
representation $R$ of the colour group, one can perform the decomposition

${\hat{\cal{K}}}=\sum_{R}\hat{\cal P}^{R}\hat{\cal{K}}^{R}~{}.$ (26)

In QCD, possible representations are $\bf{1}$, $\bf{8}_{a}$, $\bf{8}_{s}$,
$\bf{10}$, $\bf{10^{*}}$ and $\bf{27}$. In the following, we will consider
only the singlet representation, corresponding to scattering of colourless
particles, and omit the index $R$.

###  2.3 Forward BFKL kernel

Separate contributions to the kernel (24) are singular. They were calculated
in the dimensional regularization, so that in [46] the kernel also was written
at the space-time dimension $D=4-2\epsilon$, different from the physical one.
To work with this kernel is rather difficult. But the kernel is infrared safe,
i.e. singularities in various parts cancel. It makes possible to perform the
cancellation of the most singular terms and to write the kernel in the form
[56] (here the Reggeon momenta are q and l )

${\cal
K}(\vec{q},\vec{l})=\frac{\alpha_{s}(\mu^{2})N_{c}}{2\pi^{2}}\left[\frac{2}{(\vec{q}-\vec{l})^{2}}-\delta(\vec{q}-\vec{l})\int\frac{d\vec{k}\;\vec{q}^{\;2}}{(\vec{q}-\vec{k})^{2}\vec{k}^{\>2}}\right]$
$\times\left[1+\frac{\alpha_{s}(\mu^{2})N_{c}}{4\pi}\left(\frac{67}{9}-2\zeta(2)-\frac{10}{9}\frac{n_{f}}{N_{c}}\right)\right]$
$+\frac{\alpha^{2}_{s}(\mu^{2})N^{2}_{c}}{4\pi^{3}}\left[\frac{1}{(\vec{q}-\vec{l})^{2}}\left(\frac{\beta_{0}}{N_{c}}\ln\left(\frac{{\mu}^{2}}{(\vec{q}-\vec{l})^{\;2}}\right)-\ln^{2}\left(\frac{\vec{q}^{\;{2}}}{\vec{l}^{\;2}}\right)\right)+f_{1}(\vec{q},\vec{l})\right.$
$\left.+f_{2}(\vec{q},\vec{l})+\delta(\vec{q}-\vec{l})\left(\frac{\beta_{0}}{2N_{c}}\int\frac{d\vec{k}\;\vec{q}^{\;2}}{(\vec{q}-\vec{k})^{2}\vec{k}^{\>2}}\ln\left(\frac{(\vec{q}-\vec{k})^{2}\vec{k}^{\>2}}{{\mu}^{2}\vec{q}^{\;2}}\right)+6\pi\zeta(3)\right)\right]~{},$
(27)

where where $\beta_{0}=11N_{c}/3-2n_{f}/3$ is the first coefficient of the
$\beta$-function,

$f_{1}(\vec{q}_{1},\vec{q}_{2})\ =-\
\frac{2(\vec{q}_{1}^{~{}2}-\vec{q}_{2}^{~{}2})}{\vec{k}^{\;2}(\vec{q}_{1}+\vec{q}_{2})^{2}}\Bigg{(}\frac{1}{2}\ln\left(\frac{{\vec{q}_{1}}^{~{}2}}{{\vec{q}_{2}}^{~{}2}}\right)\ln\left(\frac{{\vec{q}_{1}}^{~{}2}{\vec{q}_{2}}^{~{}2}\vec{k}^{\>4}}{\left({\vec{q}^{\;2}_{1}}+{\vec{q}^{\;2}_{2}}\right)^{4}}\right)+Li_{2}\left(-\frac{{\vec{q}_{1}}^{~{}2}}{{\vec{q}_{2}}^{~{}2}}\right)$
$-Li_{2}\left(-\frac{{\vec{q}_{2}}^{~{}2}}{{\vec{q}_{1}}^{~{}2}}\right)\Bigg{)}-\left(1-{(\vec{q}_{1}^{~{}2}-\vec{q}_{2}^{~{}2})^{2}}/{\vec{k}^{\>2}(\vec{q}_{1}+\vec{q}_{2})^{2}}\right)\left(\int\limits_{0}^{1}-\int\limits_{1}^{\infty}\right)\frac{dz\,\ln\frac{(z{\vec{q}_{1}})^{2}}{({\vec{q}_{2}})^{2}}}{(\vec{q}_{2}-z\vec{q}_{1})^{2}},$
(28) $f_{2}(\vec{q}_{1},\vec{q}_{2})\ =-\
\left(1+\frac{n_{f}}{N_{c}^{3}}\right)\frac{2{\vec{q}_{1}}^{~{}2}{\vec{q}_{2}}^{~{}2}-3({\vec{q}_{1}}{\vec{q}_{2}})^{2}}{16{\vec{q}_{1}}^{~{}2}{\vec{q}_{2}}^{~{}2}}\left(\frac{2}{{\vec{q}_{2}}^{~{}2}}+\frac{2}{{\vec{q}_{1}}^{~{}2}}+\Big{(}\frac{1}{{\vec{q}_{2}}^{~{}2}}-\frac{1}{{\vec{q}_{1}}^{~{}2}}\Big{)}\ln\frac{{\vec{q}_{1}}^{~{}2}}{{\vec{q}_{2}}^{~{}2}}\right)$
$-\
\left(\\!3+\Big{(}1+\frac{n_{f}}{N_{c}^{3}}\Big{)}\\!\left(\\!1-\frac{({\vec{q}_{1}}^{~{}2}+{\vec{q}_{2}}^{~{}2})^{2}}{8{\vec{q}_{1}}^{~{}2}{\vec{q}_{2}}^{~{}2}}-\frac{2{\vec{q}_{1}}^{~{}2}{\vec{q}_{2}}^{~{}2}-3{\vec{q}_{1}}^{~{}4}-3{\vec{q}_{2}}^{~{}4}}{16{\vec{q}_{1}}^{~{}4}{\vec{q}_{2}}^{~{}4}}({\vec{q}_{1}}{\vec{q}_{2}})^{2}\\!\right)\\!\right)$
$\times\
\int\limits_{0}^{\infty}\frac{dx\,\ln\left|\frac{1+x}{1-x}\right|}{{\vec{q}_{1}}^{~{}2}+x^{2}{\vec{q}_{2}}^{~{}2}}~{},$
(29)

where $\vec{k}=\vec{q}_{1}-\vec{q}_{2}~{}.$

This representation greatly simplifies the calculation of the eigenvalues of
the kernel. Strictly speaking, because of the charge renormalization the
eigenfunctions of the LO kernel $\left({\vec{l}^{\>2}}\right)^{\gamma-1}$,
which at $\gamma=1/2+i\nu,\;\;-\infty<\nu<\infty$ form a complete set for the
kernel averaged over angles, are not any more eigenfunctions of the NLO
kernel. But with the NLO accuracy one can write

$\alpha_{s}(\mu^{2})=\alpha_{s}(\vec{q}^{\>2})\left(1+\beta_{0}\frac{\alpha_{s}(\vec{q}^{\>2})}{4\pi}\ln\left(\frac{\vec{q}^{\>2}}{\mu^{2}}\right)\right)~{},$
(30)

and obtain

$\int d^{2}l\;{\cal
K}(\vec{q},\vec{l})\left(\frac{\vec{l}^{\>2}}{\vec{q}^{\>2}}\right)^{\gamma-1}=\omega(\vec{q}^{\,2},\gamma)=\frac{\alpha_{s}(\vec{q}^{~{}2})\,N_{c}\,}{\pi}\chi(\gamma)~{},\;\;\
$
$\chi(\gamma)=\chi_{B}(\gamma)+\frac{\alpha_{s}N_{c}}{\pi}\chi^{(1)}(\gamma)~{}.$
(31)

Here $\chi_{B}(\gamma)$ gives the LO eigenvalues,

$\chi_{B}(\gamma)=2\psi(1)-\psi(\gamma)-\psi(1-\gamma)$ (32)

and the correction $\chi^{(1)}(\gamma)$ is:

$\chi^{(1)}(\gamma)\ =\
-\frac{1}{4}\Bigg{[}\frac{\beta_{0}}{2N_{c}}\left(\chi_{B}^{2}(\gamma)(\gamma)+\chi^{\prime}_{B}(\gamma)-\
6\zeta(3)\right)$
$+\frac{\pi^{2}\cos(\pi\gamma)}{\sin^{2}(\pi\gamma)(1-2\gamma)}\left(3+\left(1+\frac{n_{f}}{N_{c}^{3}}\right)\frac{2+3\gamma(1-\gamma)}{(3-2\gamma)(1+2\gamma)}\right)$
$-\
\left(\frac{67}{9}-\frac{\pi^{2}}{3}-\frac{10}{9}\frac{n_{f}}{N_{c}}\right)\chi_{B}(\gamma)+\
\chi_{B}^{\prime\prime}(\gamma))-\frac{\pi^{3}}{\sin(\pi\gamma)}+4\phi(\gamma)\Bigg{]}~{},$
(33)

where

$\phi(\gamma)=-\int\limits_{0}^{1}\frac{dx}{1+x}\left(x^{\gamma-1}+x^{-\gamma}\right)\int_{x}^{1}\frac{dt}{t}\ln(1-t)$
$=\sum_{n=0}^{\infty}(-1)^{n}\left[\frac{\psi(n+1+\gamma)-\psi(1)}{(n+\gamma)^{2}}+\frac{\psi(n+2-\gamma)-\psi(1)}{(n+1-\gamma)^{2}}\right].$
(34)

This result was firstly obtained in [46].

The representation (27) is useful also for finding all eigenvalues of the
kernel. Defining them as

$\int
d^{2}l\;{K}(\vec{q},\vec{l})\left(\frac{\vec{l}^{\>2}}{\vec{q}^{\>2}}\right)^{\gamma-1}e^{in(\phi_{l}-\phi_{q})}$
$=\frac{\alpha_{s}(\vec{q}^{~{}2})\,N_{c}\,}{\pi}\left(\chi_{B}(\gamma,n)+\frac{\alpha_{s}N_{c}}{\pi}\chi^{(1)}(\gamma,n)\right)~{},$
(35)

where

$\chi_{B}(\gamma,n)=2\psi(1)-\psi(\gamma+\frac{|n|}{2})-\psi(1-\gamma+\frac{|n|}{2})~{},$
(36)

we come to the correction obtained in [57]:

$4\chi^{(1)}\left(\gamma,n\right)\ =\
-\frac{\beta_{0}}{2N_{c}}\left(\chi_{B}^{2}(\gamma,n)+\chi_{B}^{\prime}(\gamma,n)\right)+6\zeta\left(3\right)-\chi_{B}^{\prime\prime}(\gamma,n)$
$+\left(\frac{67}{9}-\frac{\pi^{2}}{3}-\frac{10}{9}\frac{n_{f}}{N_{c}}\right)\chi_{B}(\gamma,n)-2\Phi\left(n,\gamma\right)-2\Phi\left(n,1-\gamma\right)+F\left(n,\gamma\right)~{},$
(37)

where

$F(n,\gamma)=\frac{\pi^{2}\cos\left(\pi\gamma\right)}{\sin^{2}\left(\pi\gamma\right)\left(1-2\gamma\right)}\Biggl{[}\frac{\gamma\left(1-\gamma\right)\left(\delta_{n,2}+\delta_{n,-2}\right)}{2\left(3-2\gamma\right)\left(1+2\gamma\right)}$
$\times\left(1+\frac{n_{f}}{N_{c}^{3}}\right)-\left(\frac{3\gamma\left(1-\gamma\right)+2}{\left(3-2\gamma\right)\left(1+2\gamma\right)}\left(1+\frac{n_{f}}{N_{c}^{3}}\right)+3\right)\delta_{n,0}\Biggr{]}~{},$
(38)

and

$\Phi\left(n,\gamma\right)=\int_{0}^{1}\frac{dt}{1+t}t^{\gamma-1+n/2}\left\\{\frac{\pi^{2}}{12}-\frac{1}{2}\psi^{\prime}\left(\frac{n+1}{2}\right)-{Li}_{2}\left(t\right)\right.$
$-{Li}_{2}\left(-t\right)-\left(\psi\left(n+1\right)-\psi\left(1\right)+\ln\left(1+t\right)+\sum_{k=1}^{\infty}\frac{\left(-t\right)^{k}}{k+n}\right)\ln
t$
$\left.-\sum_{k=1}^{\infty}\frac{t^{k}}{\left(k+n\right)^{2}}\left[1-\left(-1\right)^{k}\right]\right\\}~{}.$
(39)

##  3 Investigation of the BFKL properties

In the wide range of problems that Lev dealt with, the BFKL approach was one
of the first. Lev many times turned to it, and his analysis was always
distinguished by its depth and originality.

###  3.1 Conformal invariance

In 1985 he discovered that for scattering of colourless particles the LO BFKL
equation can be solved in a general form not only for the forward case, but
for general momentum transfer [58]. This striking fact is related to the
remarkable property of the BFKL equation discovered in [58]: for scattering of
colourless particles it can be written in the special representation (which
was called later Möbius representation [59]), which is invariant with respect
to conformal (Möbius) transformations in the impact parameter space
$\vec{r}_{1},\vec{r}_{2}$ of the Reggeized gluons. In this representation the
BFKL kernel $\hat{\cal{K}}_{M}$ operates in the space of functions having the
colour transparency property (turning into zero at $\vec{r}_{1}=\vec{r}_{2}$).
Note that the colourlessness means the Pomeron exchange (while the reverse
statement is not correct) and that it is necessary for writing the BFKL
equation in the Möbius representation. The colourlessness of colliding
particles provides the gauge invariance of impact factors, i.e. their
vanishing at zero Reggeized gluon momenta. Together with an analogous property
of the BFKL kernel, it gives a possibility to restrict the space of functions
where the kernel acts by the functions having the colour transparency.

The conformal invariance of the BFKL equation in the Möbius representation is
extremely important because it permits to classify all solutions of the
homogeneous equation and to find their complete set. Another significant
property is the holomorphic separability of BFKL kernel in this
representation. Actually, this property was used already in [58]; it was
explicitly exhibited and used later [60, 61] for solving the Bartels-
Kwiecinski-Praszalowichz (BKP) equation – the generalization of the BFKL
equation to the case of many Reggeized gluons in the $t$–channel [50, 62]. The
conformal invariance permits to obtain eigenfunctions of the kernel
$\hat{\cal{K}}_{M}$ from eigenfunctions for forward scattering by
transformations of the Möbius group (inversion and shift). In [58] they were
chosen in the form

$E_{n,\nu}(\vec{r}_{10},\vec{r}_{20})=(-1)^{n}\left(\frac{z_{12}}{z_{10}z_{20}}\right)^{\frac{1}{2}+i\nu-\frac{n}{2}}\left(\frac{z^{*}_{12}}{z^{*}_{10}z^{*}_{20}}\right)^{\frac{1}{2}+i\nu+\frac{n}{2}}~{},$
(40)

where $z=x+iy$ for $\vec{r}=(x,y)$, the vector $\vec{r}_{0}$ is introduced for
indexing of the wave functions, $\vec{r}_{ij}=\vec{r}_{i}-\vec{r}_{j}$, $n$ is
integer and $\nu$ is real. The eigenvalues coincide with the eigenvalues of
the forward kernel

$\omega_{B}(\gamma,n)=\frac{\alpha_{s}N_{c}}{\pi}\chi_{B}(\gamma,n)~{},\gamma=1/2+i\nu~{},$
(41)

$\chi_{B}(\gamma,n)$ is given in (36). One can see it noticing that the
eigenvalues do not depend on $\vec{r}_{0}$ and that integration over
$\vec{r}_{0}$ gives

$\int\,d\vec{r}_{0}\,E_{n,\nu}(\vec{r}_{10},\vec{r}_{20})=C_{n,\nu}\,(z_{12})^{\frac{1}{2}-i\nu+\frac{n}{2}}(z^{*}_{12})^{\frac{1}{2}-i\nu-\frac{n}{2}}=C_{n,\nu}\,e^{i\,n\phi_{12}}\vec{r}_{12}^{2(1-\gamma)}~{},$
(42)

where $\phi_{12}$ is the azimuthal angle of the vector $\vec{r}_{12}$. Then
the problem is reduced to the calculations of the integrals for the forward
case. The form of the integral (42) can be easily obtained using the change of
the integration variables $z_{10}=z_{12}z$; the coefficient $C_{n,\nu}$ is
irrelevant for the calculation of the eigenvalues. In fact, it was found in
[58]:

$C_{n,\nu}=\frac{\pi
2^{4\,i\,\nu}}{|n|-2\,i\,\nu}\frac{\Gamma(\kappa)}{\Gamma(\kappa^{*})}\frac{\Gamma(\kappa^{*}-1/2)}{\Gamma(\kappa-1/2)}~{},\;\;\kappa=\frac{1}{2}-i\nu+\frac{|n|}{2}~{}.$
(43)

One can easily show that

$z^{2}_{12}\frac{\partial}{\partial z_{1}}\frac{\partial}{\partial
z_{2}}\,E_{n,\nu}=\lambda_{n,\nu}E_{n,\nu}~{},\;\;z^{*\,2}_{12}\frac{\partial}{\partial
z^{*}_{1}}\frac{\partial}{\partial
z^{*}_{2}}\,E_{n,\nu}=\lambda_{n,-\nu}E_{n,\nu}~{},$
$\lambda_{n,\nu}=\frac{1}{4}-(\frac{n}{2}-i\nu)^{2}~{}.$ (44)

The completeness condition derived in [58] has the form

$4\sum_{n=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\ d\nu\int\
d^{2}r_{0}\frac{n^{2}+4\nu^{2}}{\vec{r}^{2}_{12}\vec{r}^{2}_{1^{\prime}2^{\prime}}}E_{n,\nu}(\vec{r}_{10},\vec{r}_{20})E^{*}_{n,\nu}(\vec{r}_{1^{\prime}0},\vec{r}_{2^{\prime}0})$
$=(2\pi)^{4}\delta(\vec{r}_{11^{\prime}})\delta(\vec{r}_{22^{\prime}})~{}.$
(45)

Using (44) and (45) one obtains for the Green’s function in the impact
parameter space

$\langle\vec{r}_{1},\vec{r}_{2}|\frac{1}{\omega-\hat{\cal{K}}_{M}}\frac{1}{\hat{\vec{q}}_{\,1}^{\,2}\hat{\vec{q}}_{\,2}^{\,2}}|\vec{r}_{\,1}^{\;\prime},\vec{r}_{\,2}^{\;\prime}\rangle$
$=4\sum_{n=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\ d\nu\int\
d^{2}r_{0}\frac{({n^{2}+4\nu^{2}})E_{n,\nu}(\vec{r}_{10},\vec{r}_{20})E^{*}_{n,\nu}(\vec{r}_{1^{\prime}0},\vec{r}_{2^{\prime}0})}{[(n+1)^{2}+4\nu^{2}][(n-1)^{2}+4\nu^{2}](\omega-\omega(\gamma,n))}~{}.$
(46)

For $n=\pm 1$ the integral over $\nu$ here must be taken in the sense of its
principal value.

Later an explicit expression for the kernel ${\cal{K}_{M}}$ in the impact
parameter space was obtained [63]:

$\langle\vec{r}_{1},\vec{r}_{2}|\hat{\cal{K}}_{M}|\vec{r}_{1}^{\;\prime},\vec{r}_{2}^{\;\prime}\rangle$
$=\frac{\alpha_{s}N_{c}}{2\pi^{2}}\int\,{d\vec{r}_{0}}\
\frac{\vec{r}_{12}^{\,2}}{\vec{r}_{10}^{\,2}\vec{r}_{20}^{\;2}}\left(\delta(\vec{r}_{11^{\prime}})\delta(\vec{r}_{02^{\prime}})+\delta(\vec{r}_{22^{\prime}})\delta(\vec{r}_{01^{\prime}})-\delta(\vec{r}_{11^{\prime}})\delta(\vec{r}_{22^{\prime}})\right)~{},$
(47)

where the subscript $M$ denotes the Möbius form. It is worth to note that the
ultraviolet singularities of separate terms in (47) cancel in their sum with
account of the dipole property of the target impact factors.

The transformations of the Möbius group in the two-dimensional space
$\vec{r}=(x,y)$ can be written as

$z\rightarrow\frac{az+b}{cz+d}\;,\;\;\;$ (48)

where $z=x+iy,\;\;a,b,c,d$ are complex numbers, with $ad-bc\neq 0$. Under
these transformations, one has

$z_{1}-z_{2}\rightarrow\frac{z_{1}-z_{2}}{(cz_{1}+d)(cz_{2}+d)}(ad-bc)\;,$
$dzdz^{*}\rightarrow dzdz^{*}\frac{|ad-
bc|^{2}}{\left|(cz+d)^{2}\right|^{2}}\;,$ (49)

so that the conformal invariance of (47) is evident.

It turns out that the Möbius form (47) coincides with the kernel of the colour
dipole model [64]-[67] formulated in the coordinate space. It means that the
colour dipole kernel could have been written by Lev as early as 1985. It was
not done only by chance, because he used the operator form of the kernel
instead of the explicit expression for it in the impact parameter space.

Investigation of the conformal properties of the BFKL kernel in the NLO [68]
\- [71] showed later that the ambiguity of the NLO kernel allows to present
the kernel in the form where the conformal invariance is violated only by
renormalization.

In the same paper [58] Lev performed another great study: account of running
of the coupling constant. Strictly speaking, this account oversteps the limits
of the LLA. Nevertheless, it is not unreasonable to take it into account just
in the LO BFKL kernel in order to understand qualitative effects of the
running. One important effect discovered by Lev is the conversion of the cut
$[0,\omega_{P}^{B}]$ into an infinite series of moving poles with limiting
point at $\omega=0$. The Pomeron trajectories found in [58] are represented in
the form

$\omega_{k}(t)=\frac{c}{k+\eta(q^{2})+1/4}~{},\;\;k=0,\,\pm 1,\,\pm 2\,...,$
(50)

where $c$ is a calculable constant. The function $\eta(q^{2})$ is determined
by large distances, but is limited by the inequalities $-1\leq 4\eta\leq 1$ at
small $|t|$. In the region of large $|t|$ ($\alpha_{s}(|t|)\ll 1$) the family
of poles is approximated by the moving cut with the branch point

$\omega(t)=\frac{4N_{c}}{\pi}\ln 2\,\,\alpha_{s}(|t|)~{}.$ (51)

###  3.2 Relation with DGLAP

The relation between the eigenvalues of BFKL kernel
$\frac{\alpha_{s}N_{c}}{\pi}\chi(\gamma)$ and the anomalous dimension
$\gamma_{\omega}$ of the twist-2 operators with $j=1+\omega$ was first
discovered in the LLA in [72], where it has the form:

$\omega=\frac{\alpha_{s}N_{c}}{\pi}\chi(\gamma_{\omega})~{}.$ (52)

It is a remarkable result because it means that the BFKL approach permits to
obtain the resummation of the most singular at $j=1$ terms in the anomalous
dimension of the twist-two operators.

It was noticed in [46] that in the NLLA this relation must be changed. The
reason is that energy scale $s_{0}$ becomes significant in the NNLLA, and this
scale is different from DGLAP. In the NLA $s_{0}=Q_{A}Q_{B}$, where $Q_{A}$
and $Q_{B}$ are typical momenta for the impact factors $\Phi_{A}$ and
$\Phi_{B}$. Therefore the deep-inelastic moments are defined as

$M_{\omega}(Q^{2})\ =\
\frac{1}{4(2\pi)^{2}}\int\frac{d\vec{q}}{\vec{q}^{2}}\int\frac{d\vec{l}}{\vec{l}^{2}}\Phi_{A}(\vec{q})e^{\omega\ln(Q/Q_{B})}G_{\omega}(\vec{q},\vec{l})\Phi_{B}(\vec{l})\,.$
(53)

As the result, for the anomalous dimension $\gamma_{\omega}(\alpha_{s})$ one
has

$\gamma_{\omega}=\frac{\omega}{2}+\chi^{-1}\left(\frac{\pi\omega}{N_{c}\alpha_{s}}\right)~{},$
(54)

where $\chi^{-1}$ is the inverse function. In other words, the anomalous
dimensions $\gamma_{\omega}(\alpha_{s})$ of the twist-2 operators near the
point $\omega=0$ are determined from the solution of the equation

$\omega\ =\
\frac{\alpha_{s}N_{c}}{\pi}\chi(\gamma-\frac{\omega}{2})\simeq\frac{\alpha_{s}N_{c}}{\pi}\chi(\gamma)-\frac{\alpha^{2}_{s}N^{2}_{c}}{2\pi^{2}}\chi(\gamma)\chi^{\prime}(\gamma)$
$\simeq\frac{\alpha_{s}N_{c}}{\pi}\frac{1}{\gamma}-\frac{\alpha_{s}^{2}N_{c}^{2}}{4\,\pi^{2}}\left(\frac{11+2n_{f}/N_{c}^{3}}{3\,\gamma^{2}}+\
\frac{n_{f}(10+13/N_{c}^{2})}{9\gamma\,N_{c}}\right.$
$\left.+\frac{395}{27}-2\zeta(3)-\frac{11}{3}\frac{\pi^{2}}{6}+\frac{n_{f}}{N_{c}^{3}}\left(\frac{71}{27}-\frac{\pi^{2}}{9}\right)+{\cal
O}(\gamma)\right)$ (55)

for $\gamma\rightarrow 0$. This equation was derived in [46] and used to
reproduce the known results and predict the higher loop correction for
$\omega\rightarrow 0$:

$\displaystyle\gamma$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha_{s}\,N_{c}}{\pi}\left(\frac{1}{\omega}-\frac{11}{12}-\frac{n_{f}}{6N_{c}^{3}}\right)-\left(\frac{\alpha_{s}\,}{\pi}\right)^{2}\frac{n_{f}\,N_{c}}{6\,\omega}\left(\frac{5}{3}+\frac{13}{6N_{c}^{2}}\right)$
(56) $\displaystyle-\
\frac{1}{4\omega^{2}}\left(\frac{\alpha_{s}N_{c}}{\pi}\right)^{3}\left(\frac{395}{27}-2\zeta(3)-\frac{11}{3}\frac{\pi^{2}}{6}+\frac{n_{f}}{N_{c}^{3}}\left(\frac{71}{27}-\frac{\pi^{2}}{9}\right)\right).$

The result for the three-loop correction was confirmed later in [73], [74].

###  3.3 BFKL in N=4 SUSY

Impressive and surprising results connected with BFKL were obtained by Lev in
maximally extended supersymmetric Yang-Mills theory (N=4 SYM). The eigenvalues
of the NLO BFKL kernel were calculated in [57]. They turned out to be much
simpler than in QCD: instead of (37) the one-loop correction takes the form

$4\chi^{(1)}\left(\gamma,n\right)\
=-2\Phi\left(n,\gamma\right)-2\Phi\left(n,1-\gamma\right)+6\zeta(3)+\psi^{\prime\prime}(\gamma+\frac{n}{2})+\psi^{\prime\prime}(\gamma+\frac{n}{2})~{}.$
(57)

A remarkable fact is the disappearance of non analytical in conformal spin $n$
terms. Moreover, all functions entering in (57) have the property of maximal
transcendentality [74]. The maximal transcendentality of an expression means
by definition, that the special functions and numbers with lower complexities
do not contribute to it. By definition $\psi(x)$ has the transcendentality
equal to 1, the transcendentalities of $\psi^{(n)}$ and $\zeta(n+1)$ are $n+1$
and the additional poles in the sum over $k$ increase the transcendentality of
the function $\Phi(M)$ up to 3. Lev put forward a remarkable hypothesis about
growing with order of perturbation theory maximal transcendentality of the
eigenvalues of the BFKL kernel and the anomalous dimensions of twist-2
operators in $N=4$ SUSY. This hypothesis is not disproved and widely used now.
Another of Lev’s remarkable hypothesis is the hypothesis that in $N=4$ SUSY
the Pomeron coincides with the Reggeized graviton, which gives a possibility
to calculate its intercept at large coupling constants. Maximal
transcendentality together with integrability allow one to find the anomalous
dimensions of twist-2 operators in this model up to 4 loops.

##  4 Future development

Evidently, the next step in the development of the BFKL approach should be the
next-to-next-to-leading logarithmic approximation (NNLLA). Unfortunately, in
QCD the BFKL equation in the next-to-next-to-leading approximation is not yet
obtained, although in maximally extended supersymmetric Yang-Mills theory with
large number of colours (planar N=4 SYM) impressive results have been obtained
[75] \- [78].

###  4.1 NNLLA BFKL in planar N=4 SYM

In [75] the next-to-next-to-leading order (NNLO) corrections to the BFKL
Pomeron eigenvalues for conformal spin $n=0$ were obtained. The method of
calculation was quite different from the one used in the BFKL approach, in
which, first of all, the kernel of the BFKL equation is calculated using the
unitarity and analyticity. It was based on the integrability of the planar N=4
SYM and on the observation of L. N. Lipatov and A.V. Kotikov [57] that BFKL
eigenvalues and anomalous dimensions of twist-2 operators are connected in
planar N=4 SYM by analytic continuation. The problem of calculation of the
anomalous dimensions was solved by the quantum spectral curve method (QSC)
[79], [80] giving a finite set of Riemann-Hilbert equations for exact spectrum
of planar N=4 SYM theory. Applicability of this method to calculation of the
LO BFKL Pomeron eigenvalues was demonstrated in [81]. Another significant
assumption, besides the applicability of the QSC, was the possibility to write
the relation between the spin $S$ of the twist-two operator $Tr(ZD_{+}^{S}Z)$
and its full conformal dimension $\Delta=2+S+\gamma$, $\gamma$ is the
anomalous dimension, in the form

$j(\Delta)=2+S(\Delta)=1+\sum\limits_{n=1}^{\infty}(\frac{g^{2}N_{c}}{(4\pi)^{2}})^{n}\left[F_{n}\left(\frac{\Delta-1}{2}\right)+F_{n}\left(\frac{-\Delta-1}{2}\right)\right]~{},$
(58)

where $F_{n}(x)$ are simple linear combinations of the nested harmonic sums
with the same transcendentality $n$

$S_{a_{1},a_{2},\dots,a_{n}}(x)=\sum\limits_{y=1}^{x}\frac{(\textrm{sign}\left(a_{1}\right))^{y}}{y^{|a_{1}|}}S_{a_{2},\dots,a_{n}}(y)\;\;,\;\;S(x)=1\;.$
(59)

The possibility of such representation for the LO and NLO eigenvalues with

$F_{1}=-4S_{1},\frac{F_{2}}{4}=-\frac{3}{2}\zeta_{3}+\pi^{2}\ln
2+\frac{\pi^{2}}{3}S_{1}+2S_{3}+\pi^{2}S_{-1}-4S_{-2,1}$ (60)

was shown in [82]. The three-loop result obtained in [75] is:

$\frac{F_{3}(x)}{256}=-\frac{5S_{-5}}{8}-\frac{S_{-4,1}}{2}+\frac{S_{1}S_{-3,1}}{2}+\frac{S_{-3,2}}{2}-\frac{5S_{2}S_{-2,1}}{4}+\frac{S_{-4}S_{1}}{4}+\frac{S_{-3}S_{2}}{8}$
$+\frac{3S_{3,-2}}{4}-\frac{3S_{-3,1,1}}{2}-S_{1}S_{-2,1,1}+S_{2,-2,1}+3S_{-2,1,1,1}-\frac{3S_{-2}S_{3}}{4}-\frac{S_{5}}{8}$
$+\frac{S_{-2}S_{1}S_{2}}{4}+\pi^{2}\left[\frac{S_{-2,1}}{8}-\frac{7S_{-3}}{48}-\frac{S_{-2}S_{1}}{12}+\frac{S_{1}S_{2}}{48}\right]\\!-\\!\pi^{4}\left[\frac{2S_{-1}}{45}-\frac{S_{1}}{96}\right]$
$+\zeta_{3}\left[-\frac{7S_{-1,1}}{4}+\frac{7S_{-2}}{8}+\frac{7S_{-1}S_{1}}{4}-\frac{S_{2}}{16}\right]$
$+\left[2\text{Li}_{4}\\!\\!\left(\tfrac{1}{2}\right)-\frac{\pi^{2}\log^{2}\\!2}{12}+\frac{\log^{4}\\!2}{12}\right]\left(S_{-1}-S_{1}\right)$
$+\frac{\log^{5}\\!2}{60}-\frac{\pi^{2}\log^{3}\\!2}{36}-\frac{2\pi^{4}\log
2}{45}-\frac{\pi^{2}\zeta_{3}}{24}+\frac{49\zeta_{5}}{32}-2\text{Li}_{5}\\!\\!\left(\tfrac{1}{2}\right)\;.$
(61)

The eigenvalues $\omega(\gamma)=j(i\nu)-1$ at $\gamma=\frac{1}{2}+i\nu$
correspond to complex values of $\Delta$ and require analytical continuation
of the harmonic sums. Corresponding prescriptions can be found in [83] -[86].

The approach used in [76] was also different from the BFKL approach. The NNLO
corrections to the Pomeron eigenvalues for conformal spin $n=0$ were obtained
from the constraints, coming from the six-loop anomalous dimension of twist-2
operators and large-gamma limit. The results agree with [75].

The calculations of the three-loop corrections in [77] is closer to the BFKL
approach since the corrections are calculated firstly to the BFKL kernel (in
the Pomeron channel). But the method of the calculation is quite different
from the dispersive method used in the BFKL approach. Moreover, the kernel is
calculated not in the momentum representation, but in the impact parameter
space. The method used is based on established in [87] connection of the BFKL
logarithms with so-called non-global logarithms [88], [89] the physics of soft
wide-angle radiation.

In [78] a general non-perturbative framework for the BFKL spectrum of planar N
= 4 SYM was developed. As well as in [75], it is based on the QSC method, but
the spectrum was studied at arbitrary values of conformal spin n.

###  4.2 NNLLA in the BFKL approach

The results described above were obtained in $N=4$ SUSY model with unique
properties and concern only the BFKL kernel. Recall that amplitudes are given
by convolution of impact factors of scattering particles and the Green’s
function, which is determined by the kernel, so that to calculate the
amplitudes, it is necessary to know not only the kernel, but also the impact
factors. The BFKL approach gives an algorithm for calculating both components.
Besides this, the amplitudes of both elastic and inelastic processes are
calculated in the original (dispersive) BFKL approach. Therefore, its
development in the NNLLA is very desirable. And finally, it is highly
desirable to develop this approach in QCD. Unfortunately, this development is
facing great difficulties.

Remind that original scheme of derivation of the BFKL equation looks as
follows. Elastic scattering amplitudes are calculated using the $s$-channel
unitarity and analyticity. In the LLA and in the NLLA, only amplitudes having
the pole Regge form contribute to unitarity relations. It permits to present
the $s$-channel discontinuities of elastic processes as the convolutions of
impact factors of scattering particles and the Green’s function for two
interacting Reggeized gluons. All these components are expressed in terms of
the Reggeon vertices and trajectory.

If this scheme were applicable, for derivation of the BFKL equation in the
NNLLA would be sufficient to calculate three-loop corrections to the
trajectory, two-loop corrections to the vertex of one gluon production, one-
loop corrections to the vertices of two gluon and quark-antiquark production
and to find in the Born approximation new vertices: for three gluon and quark-
antiquark-gluon production.

However, this scheme is based on the pole Regge form of amplitudes which are
used in the unitarity relations (4). It can not be used the NNLLA. In this
approximation two large logarithms can be lost in the product of two
amplitudes in the unitarity relations (4). It can be done losing either one
logarithm in each of the amplitudes or both logarithms in one of the
amplitudes.

In the first case it becomes necessary to take into account the imaginary
parts of amplitudes with negative signature as well as amplitudes with
positive signature, which both are suppressed by one degree of the logarithm.
Note that the simple factorized form (20) is valid only for the real part of
the amplitudes and must be corrected to account for the imaginary parts, which
creates certain difficulties. But more difficulties are connected with the
second case. In this case one of the amplitudes in (4) must be taken in the
LLA and the other in the NNLLA. Since the amplitudes in the LLA are real, only
real parts of the NNLLA amplitudes are important in this case. But even for
these parts the pole Regge form becomes inapplicable because of the
contributions of the three-Reggeon cuts which appear in this approximation.

The first observation of the violation the pole Regge form was done [90] in
the high-energy limit of the results of direct two-loop calculations of the
two-loop amplitudes for $gg,gq$ and $qq$ scattering. Then, the terms breaking
the pole Regge form in two and three-loop amplitudes of elastic scattering
were found in [91, 92, 93] using the techniques of infrared factorization.

It is necessary to say that, in general, breaking of the pole Regge form is
not a surprise. It is well known that Regge poles in the complex angular
momenta plane generate Regge cuts. Moreover, in amplitudes with positive
signature the Regge cuts appear already in the LLA. In particular, the BFKL
Pomeron is the two-Reggeon cut in the complex angular momentum plane. But in
amplitudes with negative signature Regge cuts appear only in the NNLLA. It is
natural to expect that the observed violation of the pole Regge form can be
explained by their contributions.

Indeed, all known cases of breaking of the pole Regge form are now explained
by the three-reggeon cuts [94, 95]. Unfortunately, the approaches and the
explanations used in these papers are different. Their results coincide in
three loops but may diverge in more loops. It requires further investigation.

Consideration of three-reggeon cuts in many-particle amplitudes is an even
more complicated problem.

##  5 Conclusion

L.N. Lipatov stood at the origins of the BFKL approach and played a prominent
role in its development. He left a huge legacy to his disciples and followers.
Unfortunately, now they are deprived of his highest skill, and the depth and
originality of his ideas.

## References

  * [1] L. N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642].
  * [2] C. N. Yang and R. L. Mills, Phys. Rev. 96 (1954) 191.
  * [3] P. W. Higgs, Phys. Rev. 145 (1966) 1156.
  * [4] T. W. B. Kibble, Phys. Rev. 155 (1967) 1554.
  * [5] F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321.
  * [6] M. Gell-Mann and M. L. Goldberger, Phys. Rev. Lett. 9 (1962) no.6, 275.
  * [7] M. Gell-Mann, M. L. Goldberger, F. E. Low and F. Zachariasen, Phys. Lett. 4 (1963) no.5, 265.
  * [8] M. Gell-Mann, M. Goldberger, F. Low, E. Marx and F. Zachariasen, Phys. Rev. 133 (1964) no.1B, B145.
  * [9] M. Gell-Mann, M. L. Goldberger, F. E. Low, V. Singh and F. Zachariasen, Phys. Rev. 133 (1964) B161.
  * [10] S. Mandelstam, Phys. Rev. 137 (1965) B949.
  * [11] S. Mandelstam, Conference: C64-08-05, p.376-377. Proceedings, 12th International Conference on High Energy Physics (ICHEP 1964) : Dubna, 1964.
  * [12] A.N. Chester, Phys. Rev. B 140 (1965) B85.
  * [13] H. Cheng and C. C. Lo, Phys. Lett. 57B (1975) 177.
  * [14] E. Abers, R. A. Keller and V. L. Teplitz, Phys. Rev. D 2 (1970) 1757.
  * [15] D. A. Dicus, D. Z. Freedman and V. L. Teplitz, Phys. Rev. D 4 (1971) 2320.
  * [16] M. T. Grisaru, H. J. Schnitzer and H. S. Tsao, Phys. Rev. Lett. 30 (1973) 811;
  * [17] M. T. Grisaru, H. J. Schnitzer and H. S. Tsao, Phys. Rev. D 8 (1973) 4498.
  * [18] M. T. Grisaru, Phys. Rev. D 16 (1977) 1962.
  * [19] V. N. Gribov, Sov. Phys. JETP 14 (1962) 1395 [ Zh. Eksp. Teor. Fiz. 41 (1961) 1962].
  * [20] Chew, G. F. and Frautschi, S. C., Phys. Rev. Lett. 7 (1961) 394.
  * [21] V. N. Gribov, L. N. Lipatov and G. V. Frolov, Sov. J. Nucl. Phys. 12 (1971) 543 [Yad. Fiz. 12 (1970) 994].
  * [22] G. V. Frolov, V. N. Gribov and L. N. Lipatov, Phys. Lett. 31B (1970) 34.
  * [23] H. Cheng and T. T. Wu, Phys. Rev. D1 (1970) 2775.
  * [24] Froissart, M., Phys. Rev. 123 (1961) 1053.
  * [25] V.S. Fadin, E.A. Kuraev, and L.N. Lipatov, Phys. Lett. B 60 (1975) 50-52.
  * [26] E.A. Kuraev, L.N. Lipatov, and V.S. Fadin, Zh. Eksp. Teor. Fiz. 71 (1976) 840-855 [Sov. Phys. JETP 44 (1976) 443-450].
  * [27] E.A. Kuraev, L.N. Lipatov, and V.S. Fadin, Zh. Eksp. Teor. Fiz. 72 (1977) 377-389 [Sov. Phys. JETP 45 (1977) 199-207].
  * [28] Ya.Ya. Balitskii, L.N. Lipatov, and V.S. Fadin,“Regge Processes In Nonabelian Gauge Theories” (in Russian), in Materials of IV Winter School of LNPI (Leningrad, 1979), pp. 109-149.
  * [29] I.I. Balitsky, and L.N. Lipatov, Yad. Fiz. 28 (1978) 1597-1611 [Sov. J. Nucl. Phys. 28 (1978) 822-829].
  * [30] V. S. Fadin and L. N. Lipatov, JETP Lett. 49 (1989) 352 [Yad. Fiz. 50 (1989) 1141] [Sov. J. Nucl. Phys. 50 (1989) 712].
  * [31] V. S. Fadin and R. Fiore, Phys. Lett. B 294 (1992) 286.
  * [32] V. S. Fadin and L. N. Lipatov, Nucl. Phys. B 406 (1993) 259.
  * [33] V. S. Fadin, R. Fiore and A. Quartarolo, Phys. Rev. D 50 (1994) 2265.
  * [34] V. S. Fadin, R. Fiore and A. Quartarolo, Phys. Rev. D 50 (1994) 5893.
  * [35] V. S. Fadin, Phys. Atom. Nucl. 58 (1995) 1762 [Yad. Fiz. 58 (1995) 1839].
  * [36] V. S. Fadin, JETP Lett. 61 (1995) 346 [Pisma Zh. Eksp. Teor. Fiz. 61 (1995) 342].
  * [37] V. S. Fadin, M. I. Kotsky and R. Fiore, Phys. Lett. B 359 (1995) 181.
  * [38] V. S. Fadin,
  * [39] V. S. Fadin, R. Fiore and A. Quartarolo, Phys. Rev. D 53 (1996) 2729
  * [40] V. S. Fadin and L. N. Lipatov, Nucl. Phys. B 477 (1996) 767
  * [41] M. I. Kotsky and V. S. Fadin, Phys. Atom. Nucl. 59 (1996) 1035 [Yad. Fiz. 59N6 (1996) 1080].
  * [42] V. S. Fadin, R. Fiore and M. I. Kotsky, Phys. Lett. B 387 (1996) 593
  * [43] V. S. Fadin, M. I. Kotsky and L. N. Lipatov, hep-ph/9704267.
  * [44] V. S. Fadin, M. I. Kotsky and L. N. Lipatov, Phys. Lett. B 415 (1997) 97.
  * [45] V. S. Fadin, R. Fiore, A. Flachi and M. I. Kotsky, Phys. Lett. B 422 (1998) 287.
  * [46] V. S. Fadin and L. N. Lipatov, Phys. Lett. B 429 (1998) 127.
  * [47] M. Ciafaloni and G. Camici, Phys. Lett. B 430 (1998) 349.
  * [48] J. Bartels, Phys. Rev. D11 (1975) 2977.
  * [49] J. Bartels, Phys. Rev. D11 (1975) 2989.
  * [50] J. Bartels, Nucl. Phys. B175  (1980) 365.
  * [51] V.S. Fadin, R. Fiore, M.G. Kozlov, and A.V. Reznichenko, Phys. Lett. B 639, 74-81 (2006).
  * [52] M.G. Kozlov, A.V. Reznichenko, and V.S. Fadin, Yad. Fiz. 74, 784-796 (2011) [Phys. Atom. Nucl. 74, 758-770 (2011)].
  * [53] M.G. Kozlov, A.V. Reznichenko, and V.S. Fadin, Yad. Fiz. 75, 905-920 (2012) [Phys. Atom. Nucl. 75, 850-863 (2012)].
  * [54] M.G. Kozlov, A.V. Reznichenko, and V.S. Fadin, Yad. Fiz. 75, 529-542 (2012) [Phys. Atom. Nucl. 75, 493-506 (2012).
  * [55] V. S. Fadin, M. G. Kozlov, and A. V. Reznichenko, Phys. Rev. D 92 no.8, 085044 (2015).
  * [56] V. S. Fadin, R. Fiore and A. V. Grabovsky, Nucl. Phys. B 820 (2009) 334.
  * [57] A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B 582 (2000) 19.
  * [58] L. N. Lipatov, Sov. Phys. JETP 63 (1986) 904 [Zh. Eksp. Teor. Fiz. 90 (1986) 1536].
  * [59] J. Bartels, L.N. Lipatov and G.P. Vacca, Nucl. Phys. B706 (2005) 391.
  * [60] L. N. Lipatov, Phys. Lett. B 251 (1990) 284.
  * [61] L. N. Lipatov, Phys. Lett. B 309 (1993) 394.
  * [62] J. Kwiecinski and M. Praszalowicz, Phys. Lett. B 94 (1980) 413.
  * [63] V. S. Fadin, R. Fiore and A. Papa, Nucl. Phys. B 769 (2007) 108
  * [64] N.N. Nikolaev and B.G. Zakharov, Z. Phys. C64 (1994) 631.
  * [65] N.N. Nikolaev, B.G. Zakharov, and V.R. Zoller, JETP Lett. 59 (1994) 6.
  * [66] A.H. Mueller, Nucl. Phys.B415 (1994) 373.
  * [67] A.H. Mueller and B. Patel, Nucl. Phys. B425 (1994) 471.
  * [68] V. S. Fadin, R. Fiore, and A. Papa, Phys. Lett. B647 (2007) 179.
  * [69] V. S. Fadin, R. Fiore, A. V. Grabovsky, and A. Papa, Nucl. Phys. B784, 49 (2007).
  * [70] V. S. Fadin and R. Fiore, Phys. Lett. B661 (2008) 139.
  * [71] V. S. Fadin, R. Fiore and A. V. Grabovsky, Nucl. Phys. B 831 (2010) 248.
  * [72] Jaroszewicz, T., Phys. Lett. B116 (1982) 291.
  * [73] Vogt, A., Moch, S. and Vermaseren, J. A. M., Nucl. Phys. B691 (2004) 129.
  * [74] A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko and V. N. Velizhanin, Phys. Lett. B 595 (2004) 521 Erratum: [Phys. Lett. B 632 (2006) 754]
  * [75] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Phys. Rev. Lett. 115 (2015) no.25, 251601.
  * [76] V. N. Velizhanin, arXiv:1508.02857 [hep-th].
  * [77] S. Caron-Huot and M. Herranen, JHEP 1802 (2018) 058.
  * [78] M. Alfimov, N. Gromov and G. Sizov, JHEP 1807 (2018) 181.
  * [79] N. Gromov, V. Kazakov, S. Leurent and D. Volin, Phys. Rev. Lett. 112 (2014) no.1, 011602.
  * [80] N. Gromov, V. Kazakov, S. Leurent and D. Volin, JHEP 1509 (2015) 187.
  * [81] M. Alfimov, N. Gromov and V. Kazakov, JHEP 1507 (2015) 164.
  * [82] M. S. Costa, V. Goncalves and J. Penedones, “Conformal Regge theory,” JHEP 1212 (2012) 091.
  * [83] A. V. Kotikov and V. N. Velizhanin, hep-ph/0501274.
  * [84] D. I. Kazakov and A. V. Kotikov, Nucl. Phys. B 307 (1988) 721 Erratum: [Nucl. Phys. B 345 (1990) 299].
  * [85] C. Lopez and F. J. Yndurain, Nucl. Phys. B 183 (1981) 157.
  * [86] J. Blumlein, Comput. Phys. Commun. 180 (2009) 2218
  * [87] S. Caron-Huot, JHEP 1803 (2018) 036
  * [88] M. Dasgupta and G. P. Salam, Phys. Lett. B 512 (2001) 323
  * [89] A. Banfi, G. Marchesini and G. Smye, JHEP 0208, 006 (2002).
  * [90] Del Duca V., Glover E.W.N., JHEP. 0110 (2001) 035.
  * [91] Del Duca V., Falcioni G., Magnea L., Vernazza L. Phys. Lett. B. 732  (2014) 233
  * [92] Del Duca V., Falcioni G., Magnea L., Vernazza L., PoS RADCOR 2013 (2013) 046.
  * [93] Del Duca V., Falcioni G., Magnea L., Vernazza L., JHEP 1502 (2015) 029.
  * [94] Fadin V.S., AIP Conf. Proc. 1819 (2017) 060003.
  * [95] Caron-Huot S., Gardi E., Vernazza L., JHEP 1706 ( 2017) 016.