Yangian symmetry applied to Quantum chromodynamics

R<EMAIL_ADDRESS>
Institut für Theoretische Physik, Universität Leipzig,
PF 100 920, D-04009 Leipzig, Germany

###### Abstract

We review applications of Yangain symmetry to high-energy QCD phenomenology.
Some basic facts about high-energy QCD are recalled, in particular the spinor-
helicity form of scattering amplitudes, the scale evolution equations of deep-
inelastic scattering structure functions and the high-energy asymptotics of
scattering. As the working tool the Yangian symmetric correlators are
introduced and constructed in the framework of the Yangian algebra of
$g\ell(n)$ type. We present the application to the tree scattering amplitudes
and their iterative relation, to the parton splitting amplitudes and the
kernels of the scale evolution equations of structure functions and to the
equations describing the high-energy asymptotics of scattering.

keywords: Non-abelian gauge field theory, Quantum chromodynamics, scattering
amplitudes, Bjorken asymptotics, Regge asymptotics, Yang-Baxter relations,
Yangian algebra, Yangian symmetric correlators.

###### Contents

  1. 1 Gauge field theories and Quantum integrable systems
  2. 2 High-energy scattering in QCD
    1. 2.1 Scattering amplitudes
    2. 2.2 The Bjorken asymptotics
    3. 2.3 The Regge asymptotics
  3. 3 Yangian symmetry
    1. 3.1 Monodromy matrix operators and symmetric correlators
    2. 3.2 YSC construction methods
      1. 3.2.1  The convolution method
      2. 3.2.2  Cyclicty
      3. 3.2.3  The R operator method
    3. 3.3 The Yang-Baxter operator
    4. 3.4 The helicity representation
    5. 3.5 Examples of YSC constructions
      1. 3.5.1 Two and three points
      2. 3.5.2 Four points
      3. 3.5.3 Crossing
    6. 3.6 Integral operators from symmetric correlators
      1. 3.6.1  The point permutation operator
      2. 3.6.2  The parameter pair permutation operators
      3. 3.6.3  The complete Yang-Baxter operator
      4. 3.6.4  The operators of the crossed YSC
  4. 4 Application to scattering amplitudes
    1. 4.1 The three-point YSC and the three-point amplitudes
    2. 4.2  The convolution construction and BCFW
    3. 4.3 4-point YSC and parton tree amplitudes
  5. 5 Application to the Bjorken asymptotics
    1. 5.1 Three-point YSC and the parton splitting amplitude
    2. 5.2 Four-point YSC and $RLL$ relations
    3. 5.3 Four-point YSC and evolution kernels in positions
    4. 5.4 4-point YSC and evolution kernels in momenta
  6. 6 Application to the Regge asymptotics
    1. 6.1 Three-point YSC and the triple reggeon vertex
    2. 6.2 The factorized $R$ operator and the BFKL hamiltonian
    3. 6.3 4-point YSC and the dipole form of the BFKL kernel
  7. 7 Conclusions

##  1 Gauge field theories and Quantum integrable systems

The history of Quantum integrable systems started with simple models for
demonstrating basic physics, spin chains [1], scattering in one dimension [2],
two-dimensional lattices [3]. In the first period the application to Quantum
field theory was restricted to exceptional and sophisticated models in 1+1
dimensions.

The basic methodical concepts, Yang-Baxter relations, Bethe ansatz,
established in the first basic publications, have been developed and extended
to the Quantum inverse scattering method [4, 5, 6, 7, 8]. A new chapter of
mathematics, the Quantum algebra, has been opened [9, 10], which is actively
developing.

Quantum chromodynamics (QCD) is now well established as the gauge field theory
of hadronic interactions, relying on basic concepts as non-abelian gauge field
theory [11], its quantization [12, 13], renormalizability [14], asymptotic
freedom [15, 16], and the quark model [17]. After the period of experimental
confirmation the phenomenology of high-energy scattering is now oriented on
finding signals beyond the Standard Model and on understanding the detailed
structures of hadronic interaction, where extensive perturbative QCD
calculations serve as a working tool.

The first application of Quantum integrable systems to high-energy QCD was
pointed out by Lipatov [18]. Treating the Regge asympototics [19, 20, 21] in
perturbative gauge theory [22, 23, 24] he showed that the multiple exchange of
reggeized gluons can be treated as a Heisenberg quantum spin chain with the
spin $\frac{1}{2}$ representations at its sites replaced by infinite-
dimensional representations.

It was clear immediately, that the Bjorken asymptotics [25, 26, 27, 28, 29,
30] and the composite operator renormalization have similar features of
integrability [31]. In both cases the Yangian symmetry underlying
integrability is based on conformal symmetry, which is broken by loop
corrections in the case of QCD. The breaking is suppressed by supersymmetry
and is absent in $\mathcal{N}=4$ super Yang-Mills theory. Much work has been
devoted in the last decades to super Yang-Mills theory under this aspect.
Quantum integrability works in the composite operator renormalization in the
planar limit to all orders [32] and in the computation of scattering
amplitudes and Wilson loops [33, 34, 35].

Describing the relations of high-energy QCD to Yangian symmetry we do not
intend to give a review of all relevant literature. Only a few references will
be given. We try to include sufficient details of explanations in order to
make the text readable without consulting these papers and books.

We shall recollect the subjects of high-energy QCD, scattering amplitudes in
helicity form, Bjorken asymptotics and Regge asymptotics, where the
applications are to be addressed. We shall formulate well-known relations,
emphasizing the conformal symmetry, in order to prepare the comparison with
the results of the Yangian symmetry. [36], [37] provide a broader introduction
with a similar orientation.

We shall not attempt to give a complete introduction with general and rigorous
definitions to the basics of Quantum algebra. For an introduction to the
particular subject of Yangian algebra we refer to [38]. We focus on the items
needed to define the Yangian symmetric correlators (YSC), to derive and to
explain the relations used for their construction. We shall introduce the
Yangian algebra notions in a way well understandable by physicists. The
algebra generators are constructed from underlying Heisenberg algebra
generators like in Quantum mechanics the operators of observables are
constructed from the basic position and momentum operators.

Our approach uses in part ideas and methods developed for the the scattering
amplitude computations, [35] Unlike the majority of related publications we do
not approach the amplitudes directly and do not work in the frame of super
Yang-Mills.

The YSC are obtained as N-point functions for Yangians of type $g\ell(n)$ in
terms of expressions depending on $n$ and a set of parameters. In the
applications $n$ will be specified to $4$ for amplitudes or to $2$ for
evolution kernels of structure functions and for reggeon interaction kernels.
The representation parameters are related in particular to homogeneity
weights, which allow to describe the scattering states by substituting
particular values. The dependence of the amplitudes or kernels on the particle
type and helicity is derived from the analytic parameter dependence of the YSC
by approaching particular values. The construction methods of YSC are related
to the method of on-shell graphs in the amplititude construction [39]. In the
original form of the latter method [35] no parameters appeared. In a number of
papers the deformation of amplitude expressions by parameters has been studied
and their eventual role for regularization of loop divergencies has been
discussed [40, 41, 42, 43]. The parameters characterizing the monodromy
defining the YSC play the essential role in our approach.

##  2 High-energy scattering in QCD

###  2.1 Scattering amplitudes

We shall consider the scattering amplitudes of quarks and gluons contributing
to high-energy hadronic processes, e.g. inclusive jet production, $AB\to
jet+X$, at large values of $s=(p_{A}+p_{B})^{2}$.

The advantages of using helicity states in the perturbative calculation of
scattering amplitudes in gauge theories were known for long time [44] and
widely used in the detailed calculations for collider phenomenology [45]. Both
the momenta and the polarization states of the scattering quarks and gluons
are expressed in terms of helicity spinors. The spinor components of massless
momenta appear as products of the components of a left and right spinor,

$k_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\sigma_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}^{\mu}k_{\mu}=\lambda_{\alpha}\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}$
(2.1)

The solutions of the massless Dirac equation can be expressed in terms of the
Weyl spinors and they represent the helicity states of the scattering quark.
The polarization vectors characterizing the helicity states of the gluons can
be composed with the help of reference spinors
$\mu_{\alpha},\bar{\mu}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}$, the
factors of the light-like gauge vector
$q_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\mu_{\alpha}\bar{\mu}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}$.

$\epsilon^{+}_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\frac{\lambda_{\alpha}\bar{\mu}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}}{[\bar{\mu}\bar{\lambda}]},\
\ \
\epsilon^{-}_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\frac{\mu_{\alpha}\bar{\alpha}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}}{<\mu\lambda>}.$
(2.2)

The spinor indices $\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}$ take two
values and the inner product of left-handed (undotted) spinors is abbreviated
by angular brackets and the one of right-handed (dotted) by square brackets,

$<\mu\lambda>=\mu_{1}\lambda_{2}-\mu_{2}\lambda_{1},\ \ \
[\bar{\mu}\bar{\lambda}]=\bar{\mu}_{1}\bar{\lambda}_{2}-\bar{\mu}_{2}\bar{\lambda}_{1}.$
(2.3)

At high energies the massless case can be adopted as a first approximation,
and then conformal symmetry holds at tree level. The action of infinitesimal
conformal transformations, originally representing the actions on space-time
coordinates, the shift $P_{\mu}$, the Lorentz $M_{\mu\nu}$, the special-
conformal $K_{mu}$ and the dilatation $D$ transformations, appear in the
spinor helicity representation as [46]

$P_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\lambda_{\alpha}\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}},\
\ M_{\alpha\beta}=\lambda_{\alpha}\partial_{\beta},\ \
\bar{M}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}\stackrel{{\scriptstyle\cdot}}{{\beta}}}=\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\partial}_{\stackrel{{\scriptstyle\cdot}}{{\beta}}},$
$K_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\partial_{\alpha}\bar{\partial}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}},\
\ \
D=\frac{1}{2}(\lambda_{\alpha}\partial^{\alpha}+\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\partial}^{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}+2).$
(2.4)

The Lorentz vector components are transformed into the spinor components as in
the particular case of the momentum above (2.1).

The eigenvalues of the dilatation in action on
$\epsilon^{\pm}_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}},\lambda_{\alpha},\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}$
are the canonical scaling dimensions of the corresponding fields in the QCD
action, namely $1$ or $\frac{3}{2}$ for gluons or quarks. (Here $D$ measures
the momentum dimension as $+1$,
$[D,P_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}]=P_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}$).
The scattering states are characterized by the helicities, the eigenvalues of
the helicity operator $\hat{h}$,

$2\hat{h}=\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\partial}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}-\lambda_{\alpha}\partial_{\alpha},$
(2.5)

commuting wth all conformal transformations (2.4).

The spinor-helicity presentation of the conformal symmetry generators does not
have the Jordan-Schwinger form. This form is obtained by elementary canonical
transformations on the left spinor variables,

$\begin{pmatrix}\partial_{\alpha}\\\
\lambda_{\alpha}\end{pmatrix}\rightarrow\begin{pmatrix}x_{\alpha}\\\
-\partial_{x\ \alpha}\end{pmatrix}.$ (2.6)

The two canonical pairs of the right spinors are not changed, but we prefer to
rename them as

$\begin{pmatrix}\bar{\partial}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\\\
\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\end{pmatrix}\rightarrow\begin{pmatrix}\partial_{x\
2+\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\\\
x_{2+\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\end{pmatrix}.$ (2.7)

In terms of the resulting Witten twistor canonical pairs $x_{a},\partial_{a}$
the conformal symmetry generators are written in the Jordan-Schwinger form as

$M_{ab}=x_{a}\partial_{b}.$ (2.8)

The indices take 4 values, $a,b=1,...,4$. The helicity operator (2.5) is
included as the trace of the $4\times 4$ matrix of generators.

$2\hat{h}\to
x_{1}\partial_{1}+x_{2}\partial_{2}+\partial_{3}x_{3}+\partial_{4}x_{4},\ \ \
2\hat{h}=(x\partial)+2.$ (2.9)

The momenta of the incoming particles $A\ B$ lead us to prefer in the center-
of-mass frame the basis of light-like vectors $n_{+},n_{-},n_{*},n_{o}$,

$p_{A}={\frac{\sqrt{s}}{2}}n_{+}+\mathcal{O}(\frac{M^{2}}{s}),\ \ \
p_{B}={\frac{\sqrt{s}}{2}}n_{-}+\mathcal{O}(\frac{M^{2}}{s}).$ (2.10)

$n_{*},n_{o}$ span the transverse plane. The normalization is
$n_{+}n_{-}=2,n_{*}n_{o}=-2$. A generic Lorentz vector $k$ is expanded as

$k=k^{-}n_{-}+k^{+}n_{+}+\kappa n_{*}+\kappa^{*}n_{o}.$ (2.11)

If it is light-like, $k^{+}k^{-}=\kappa\kappa^{*}$, then it can be factorised
as in (2.1),
$(\lambda)=(\lambda_{1},\lambda_{2}),(\bar{\lambda})=(\bar{\lambda}_{1},\bar{\lambda}_{2})$
with

$(\lambda)=(\sqrt{k^{+}},\frac{\kappa}{\sqrt{k^{+}}}),\ \ \
(\bar{\lambda})=(\sqrt{k^{+}},\frac{\kappa^{*}}{\sqrt{k^{+}}}).$ (2.12)

The polarisation vectors of a scattering gluons of momentum $k$ are

$\epsilon^{+}(k)=\bar{E}(k)(\frac{n_{o}}{\sqrt{2}}+\frac{k^{-}}{\sqrt{2}\kappa^{*}}n_{-}),\
\
\epsilon^{-}(k)=E(k)(\frac{n_{*}}{\sqrt{2}}+\frac{k^{-}}{\sqrt{2}\kappa}n_{-})$
(2.13)

with $n_{-}$ chosen as the gauge vector and $\bar{E}k)E(k)=1$. Their spinor
form factorizes as written above (2.2), e.g.

$\epsilon_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}^{+}(k)=const\mu^{-}_{\alpha}\epsilon^{+}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}},\
\ (\mu)=(n_{+})=(1,0),\ \ (\epsilon^{+}(k))=E(k)(\frac{k^{-}}{\kappa^{*}},1)$
(2.14)

A way to calculate the lowest order perturbative contributions starts from the
light-cone gauge form of the action. Being aware of the peculiarities of this
non-covariant form we see the advantage that the physical degrees of freedom
of the scattering partons are described by the transverse components $A^{*},A$
of the gauge field potential and projections $f^{*},f$ of the fermion fields
in the QCD action after imposing the gauge $n_{-}A=0$ and integrating in the
path integral over the redundant components $n_{+}^{\mu}A_{\mu}$ and
$n_{+}^{\mu}\gamma_{\mu}\psi$. Here we consider the Dirac fermion field $\psi$
decomposed with respect to a basis of Majorana spinors $u_{a,b},a=\pm,b=o,*$
obeying $\gamma^{\pm}u_{\pm,b}=0,\gamma^{*}u_{a,o}=\gamma^{o}u_{a,*}=0$, with
the vector of the Dirac gamma matrices decomposed in analogy to the momentum
vector (2.11). $f^{*},f$ represent the two helicities of a quark of particular
chirality and flavor. The field degrees of freedom of the quarks and gluons
are directly represented by the fields, $A^{*},A$, the gluons of helicities
$+1,-1$, and $f^{*},f$, the quarks of a particular flavor and particular
chirality of helicities $+\frac{1}{2},-\frac{1}{2}$.

We write the kinetic part of the resulting light cone action for later
reference,

$S_{2}=\int d^{4}y\mathcal{L}_{2}(y),\ \ \
\mathcal{L}_{2}(y)=-2A^{B*}(y)(\partial_{+}\partial_{-}-\partial\partial^{*})A^{B}(y)+if^{b*}\partial_{+}^{-1}(\partial_{+}\partial_{-}-\partial\partial^{*})f^{b}(y).$
(2.15)

$b$ runs over the range of the fundamental gauge group representation
$b=1,...,N_{c}$ and $B$ over the range of the adjoint representation,
$B=1,...,N_{c}^{2}-1$ and
$\partial_{\pm}=\frac{1}{2}(\partial_{0}\pm\partial_{3}),\ \
\partial^{*}=\frac{1}{2}(\partial_{1}+i\partial_{2})$.

The dependence of the amplitudes on the the gauge group (color ) indices can
be separated [45]. Here we focus on the the amplitude factors carrying the
dependence on momenta and helicities.

We label the parton scattering states by their helicities and assume that the
position $I$ in the array of arguments stands for the corresponding momentum
$k_{I}$ and helicity spinors $\lambda_{I},\bar{\lambda}_{I}$. We assume all
momenta ingoing, unifying in this way amplitudes related by crossing. In the
planar partial amplitudes the dependence on the scattering states is cyclic.

In the case of physical (on-shell) momentum values
$\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}$ is complex
conjugated to $\lambda_{\alpha}$. In course of a calculation we allow for
complex momentum values. In particular modifications of values depending on a
complex parameter $c$, preserving the momentum conservation condition, can be
considered. The analytic structure of the $c$ dependence of the amplitudes
$M(c)$ is determined by the physical unitarity. Tree amplitudes result in
meromorphic functions of $c$ and as a consequence one obtains the BCFW
iteration relations [47].

Consider the $N$ particle helicity amplitude $M_{1,..,N}$ and deform the
helicity variables as

$\bar{\lambda}_{1}\to\bar{\lambda}_{1}-c\bar{\lambda}_{N},\lambda_{N}\to\lambda_{N}+c\lambda_{1}$
(2.16)

and $k_{1}(c)=\lambda_{1}(\bar{\lambda}_{1}-c\bar{\lambda}_{N})$,
$k_{N}(c)=(\lambda_{N}+c\lambda_{1})\bar{\lambda}_{N}$. The deformed amplitude
$M_{1^{*},2,...,N^{*}}(c)$ has single poles at $c=c_{l}$ corresponding to the
one-particle intermediate states, where

$P_{l}(c)=P_{l}-c\lambda_{1}\bar{\lambda}_{N},P_{l}=k_{1}+...+k_{l}$

is light-like, $P_{l}(c_{l})^{2}=0$. This leads to the BCFW interative
relation

$M_{1,..,N}=\sum_{l}\sum_{h}M^{h}_{1,...,l,P}(c_{l})\frac{1}{P_{l}^{2}}M^{-h}_{P_{l},l+1,...,N}(c_{l}).$

$M^{h}_{1,...,l,P}(c_{l})$ is the tree amplitude of $l+1$ particles of momenta
$k_{1}(z_{l}),k_{2},...,k_{l},P_{l}(c_{l})$ and $h$ is the helicity of the
last one. $M^{-h}_{P_{l},l+1,...,N}(z_{l})$ is the tree amplitude of $N-l+1$
particles with momenta $-P_{l}(c_{l}),k_{l+1},...,k_{N}(c_{l})$ and helicity
$-h$ of the first one. Here it is assumed that the deformed amplitude vanishes
at $c\to\infty$, which holds e.g. in the gluon case for helicity values $-1$
for particle 1 and $+1$ for particle $N$.

The deformation (2.16) can be written by shift operators as

$M_{1,...,N}(c)=e^{-c\bar{\lambda}_{N}\bar{\partial}_{1}}e^{c\lambda_{1}\partial_{N}}M_{1,...,N},\
\ \ \
\bar{\partial}_{1}^{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\frac{\partial}{\partial\bar{\lambda}_{1\stackrel{{\scriptstyle\cdot}}{{\alpha}}}},\partial_{N}^{\alpha}=\frac{\partial}{\partial\lambda_{N\alpha}},$

and in terms of the amplitudes with energy-momentum delta included,
$\widetilde{M}_{1,...,l}=M_{1,...,l}\delta(\lambda_{1\alpha_{1}}\bar{\lambda}_{1,\stackrel{{\scriptstyle\cdot}}{{\alpha}}_{1}}+...+\lambda_{l\alpha_{l}}\bar{\lambda}_{l,\stackrel{{\scriptstyle\cdot}}{{\alpha}}_{l}})$,
the particular contribution to $\widetilde{M}_{1,...,N}$ of the channel
$1,..,l\to l+1,...,N$ obtains the form

$\int_{\mathcal{C}_{l}}\frac{dc}{c}\exp({-c\bar{\lambda}_{N\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\partial}_{1}^{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}})\exp({c\lambda_{1\alpha}\partial_{N}^{\alpha}})\int
d^{4}P\delta(P^{2})\widetilde{M}^{h}_{1,...,l,P}\widetilde{M}^{-h}_{P,l+1,...,N}.$
(2.17)

$\mathcal{C}_{l}$ denotes the small circle around $c_{l}$.

The 3-point amplitudes can be used as the building blocks to construct all
tree amplitudes. In the Parke-Taylor form [48] we have e.g.

$M(-1,-1,+1)=\frac{<12>^{4}}{<12><23><31>},\ \ \
M(-\frac{1}{2}.-1,+\frac{1}{2})=\frac{<12>^{3}<23>}{<12><23><31>}.$

The amplitudes with all helicities $h_{i}$ reversed are obtained by replacing
the brackets $<...>$ by $[...]$ (2.3 ). Further 3-point amplitudes are
obtained from the above by cyclicity, the remaining ones vanish. All cases can
be summarized as

$M(h_{1},h_{2},h_{2})=<12>^{2\eta h_{3}+1}<23>^{2\eta h_{1}+1}<31>^{2\eta
h_{2}+1},\ \ \ h_{1}+h_{2}+h_{3}=\eta,\eta=\pm 1.$ (2.18)

###  2.2 The Bjorken asymptotics

Deep-inelastic electron - proton scattering $eP\to eX$ ($X$ denotes multi-
hadron states ) is the prominent example of processes described by the Bjorken
asymptotics of QCD. The electron scatters at large momentum transfer $q$,
$-q^{2}=Q^{2}\gg M^{2}$. The data on this process allow to resolve the
structure of the hadron $P$ into quasi-free quarks and gluons [25]. This
heuristic view is confirmed by the analysis of the relevant Feynman graphs in
the leading $\ln Q^{2}$ approximation [26, 27, 28].

The inclusive cross section factorizes into the leptonic and hadronic tensor
factors. The latter can be considered as the s-channel discontinuity of the
virtual Compton scattering amplitude, $\gamma^{*}P\to\gamma^{*}P$. It expands
into Lorentz structures multiplied by structure functions
$F(z,Q^{2}),z=\frac{Q^{2}}{2(pq)}$.

In the perturbative expansion of the hadronic tensor we see (Fig. 1 ) the
virtual Compton scattering off a quark convoluted with the structure function
of the hadron $F(z,Q^{2})$. The Bjorken variable $z=\frac{Q^{2}}{2(pq)}$ is
the fraction of the proton momentum $p$ carried by the quark and $F(z,Q^{2})$
can be interpreted as the probability distribution for finding in the target
the particular parton with momentum fraction $z$.

Figure 1: Virtual Compton scattering, contribution of the structure function
F.

The asymptotics is governed by the Feynman graphs contributing a factor $\ln
Q^{2}$ with each loop, i.e. the leading logarithmic approximation applies. In
the light cone gauge the leading graphs simplify to ladders with self-energy
and vertex corrections inserted. The leading contribution arises from the
integration region of the loop momenta
$k_{i}=k^{-}_{i}n_{-}+k^{+}_{i}n_{+}+\underline{\kappa}_{i}$ of strongly
ordered transverse components

$Q^{2}\gg\kappa_{N}^{2}\gg...\gg\kappa_{1}^{2}\gg M^{2}.$ (2.19)

The loop integral reduces to the one over $k^{+}_{i}$, proportional to the
momentum fraction of the proton, and the integration over the remaining
components reduces to $\int\frac{d^{2}\kappa}{\kappa^{2}}$ to produce the
required factor $\ln Q^{2}$.

The ladder rungs represent the partons hadronizing to the final hadronic state
X. The sum of ladder contributions to the structure function leads to the
equation for the structure function [26, 30, 29].

The elementary building block of the ladder is reduced in the limit (2.19) to
the 4-point function $W_{ABCD}$ depending on the $k^{+}$ momentum components
only, $k^{+}=\frac{2}{\sqrt{s}}z$. The sum of the leading ladder contributions
including the self-energy and vertex corrections leads to the equation for the
scale dependence of the structure function. $A,B,C,D$ denote the parton type
and helicity ($\pm 1$ for gluons, $\pm\frac{1}{2}$ for quarks or anti-quarks).
In the case of deep inelastic scattering $B$ is related to $A$ and also $C$ to
$D$ by conjugation, $W_{A,\bar{A};C,\bar{C}}\to W_{A;C}$.

$\frac{d}{d\xi}F_{A}(z;Q^{2})=-W_{A}F_{A}(z;Q^{2})+\sum_{C}\int_{z}^{1}\frac{dz^{\prime}}{z^{\prime}}W_{C,A}(\frac{z}{z^{\prime}})F_{C}(z^{\prime},Q^{2}).$
(2.20)

The scale dependence is formulated in terms of
$\xi(Q^{2})=-\frac{2N_{C}}{b}\ln\frac{\alpha(Q^{2})}{\alpha(M^{2})},$
depending on the one-loop QCD coupling
$\alpha(Q^{2})=\frac{\alpha(M^{2})}{1+b\frac{\alpha(M^{2})}{2\pi}\ln\frac{Q^{2}}{M^{2}}},b=\frac{11}{3}N_{C}-\frac{2}{3}N_{F}$.
$N_{c}$ stands for the number of colors and $N_{F}$ for the number of quark
flavors .

This form is the one of a branching process with $W_{A,B}(z)$ being the rate
of the parton $A$ to decay into a parton $B$ carrying the momentum fraction
$z$ of $A$ and

$W_{A}=\sum_{B}\int_{0}^{1}dzzW_{BA}(z).$

The structure function $F_{A}(z;Q^{2})$ is proportional to the density of
partons $A$ in the target hadron. Actually $W_{A,B}(z)$ is singular at $z=1$
and correspondingly $W_{A}$, and this branching picture becomes meaningful
with a regularization at $z=1$ only.

With the definition of the evolution kernel as the well-defined distribution

$P_{BA}(z)=W_{B,A}(z)-\delta_{A,B}\delta(1-z)W_{A}$ (2.21)

we have the Altarelli-Parisi form [29],

$\frac{d}{d\xi}F_{A}(z;Q^{2})=\sum_{C}\int_{z}^{1}\frac{dz^{\prime}}{z^{\prime}}P_{C,A}(\frac{z}{z^{\prime}})F_{C}(z^{\prime},Q^{2}).$
(2.22)

It describes the scale $Q^{2}$ dependence of the structure functions and is
usually referred to as DGLAP evolution equation.

The spectra of the operators defined by such kernels are obtained by
calculating the moments

$\gamma_{AB}(n)=\int_{0}^{1}dzz^{n-1}P_{AB}(z)$ (2.23)

and diagonalizing the resulting matrices in the parton type indices $A,B$.

The hadronic tensor factor can be formulated also as the hadron states matrix
element of the product of electromagnetic currents,

$\int d^{4}xe^{iqx}<P^{\prime}|Tj_{\mu}(0)j_{\nu}(x)|P>.$ (2.24)

$T$ stands for time-ordering. The treatment of the scale dependence can be
formulated in terms of the operator product expansion and renormalization
group [51, 52].

In the case at hand both hadron states coincide, the general case appears in
semi-exclusive hard processes. In the general case in the Lorentz expansion of
the hadronic tensor factor more structures contribute and the $n_{+}$
component of the momentum transfer appears as an additional variable in the
structure functions, called generalized parton distributions [53, 54, 55, 56].

We see in both formulations of the hadronic tensor factor described that at
the asymptotics the dynamics becomes one-dimensional. In the operator product
form (2.24) the dominant integration region is $x^{2}\sim\frac{1}{Q^{2}}$. The
leading contributions in the operator product expansion are products of two
light cone components of quarks ($f^{*}(x),f(x)$) or gluons ($A^{*},A$)
located at the light ray. Let us write such quasi-parton operator
contributions as

$F_{AB}(x_{1},x_{2};Q^{2})=<P^{\prime}|f^{*}_{A}(x_{1})f_{B}(x_{2})|P>,$
(2.25)

where$A,B$ denote the parton type and helicity ($\pm 1$ for gluons,
$\pm\frac{1}{2}$ for quarks or anti-quarks) and $x_{1},x_{2}$ are points on
the light ray. The angular brackets stand for hadronic states.

For the field modes producing the leading contributions the QCD light cone
action reduces to an one-dimensional effective action. In particular from the
kinetic terms (2.15) we get schematically

$S_{2Bj}=\int d^{1}x\mathcal{L}_{2Bj},\ \
\mathcal{L}_{2Bj}=2A^{*}A+if^{*}\partial^{-1}f.$ (2.26)

The scaling dimensions of the effective parton fields are $\ell=-\frac{1}{2}$
for gluons and $\ell=-1$ for quarks.

By renormalization group the operator product contributions depend on momentum
scale as

$\frac{d}{d\xi}F_{AB}(x_{1},x_{2};Q^{2})=-\frac{1}{2}(W_{A}+W_{B})\delta_{AC}\delta_{BD}\delta(x_{1}-x_{1}^{\prime})\delta(x_{2}-x_{2}^{\prime})+$
(2.27) $\sum_{CD}\int
dx_{1}^{\prime}dx_{2}^{\prime}W_{CD,AB}(x_{1}^{\prime},x_{2}^{\prime};x_{1},x_{2})F_{CD}(x_{1}^{\prime},x_{2}^{\prime};Q^{2})$

The evolution equation (2.20) is the light-cone momentum representation of the
latter equation restricted to the case of vanishing momentum transfer. The
position form is well suited for the case of generalized parton distributions.
The eigenvalues of the operators defined by the kernels in light-cone position
(2.27) and momentum representation (2.20), (2.22) coincide. The eigenvalues
are the anomalous scaling dimensions of the corresponding linear combinations
of local operator products obtained by expansion of the bi-local operators of
(2.25) in powers of the positions $x_{1},x_{2}$.

The dynamics reduced in the Bjorken asymptotics to one dimension preserves the
conformal symmetry at the tree level. The only relevant momentum component
$k^{+}$ can be written as factorised into one-component variables, in analogy
to the two-component spinor representation of light-like Lorentz vectors,
$k^{+}=\lambda\bar{\lambda}$. The second spinor components (2.12) vanish with
the transverse momenta in the asymptotics, the collinear limit.

The basic QCD three-point amplitudes (2.18) result in this limit in the parton
splitting amplitudes $M_{{split}}$ [57]. They lead to the kernels $W_{AB}$ in
analogy to the construction of the scattering amplitudes from the basic three-
point amplitudes.

Consider the spinors (2.12) in terms of light-cone momentum components (2.11),
substitute $k^{+}=\frac{\sqrt{s}}{2}z$ and choose
$z_{1}=1,\kappa_{1}=0,z_{2}=z,\kappa_{2}=\kappa,z_{3}=1-z,\kappa_{3}=-\kappa$.
The parton splitting amplitude $1\to 2+3$ is then

$\lim_{\kappa\to
0}\frac{1}{<23>}M(h_{1},h_{2},h_{3})=\frac{z^{\frac{1}{2}}(1-z)^{\frac{1}{2}}}{z^{\eta
h_{2}}(1-z)^{\eta h_{2}}}.$

We obtain the symmetric from of the parton splitting amplitude,
$z_{i}=\frac{2}{\sqrt{s}}k^{+}_{i},\sum z_{1}+z_{2}+z_{3}=0$,
$h_{1}+h_{2}+h_{3}=\eta$,

$M_{{split}}(z_{1},z_{2},z_{3};h_{1},h_{2},h_{3})=z_{1}^{\frac{1}{2}-\eta
h_{1}}z_{2}^{+\frac{1}{2}-\eta h_{2}}z_{3}^{\frac{1}{2}-\eta h_{3}}.$ (2.28)

The splitting amplitude results in the kernels of the above scale evolution
equation as (changing here from the convention of ingoing momenta to the
normal physical one)

$W_{AB}(z)=\sum_{C}W_{ABC}(z),\ \
W_{ABC}(z)=|M_{split}(1,-z,z-1,h_{A},-h_{B}-h_{C})|^{2}.$ (2.29)

We describe the action of the conformal symmetry in the Bjorken limit.
Consider as generators $M_{ab}=x_{a}\partial_{b}$ (2.8), but now with the
indices restricted to $a,b,=1,2$ and also the analogy of the elementary
canonical transformation (2.6), (2.7) to the helicity form . Now
$\alpha,\stackrel{{\scriptstyle\cdot}}{{\alpha}}$ take only one value so that
the indices will be omitted. We have here (compare (2.9) )

$2\hat{h}=(\underline{x}\underline{\partial})+1,\ \
(\underline{x}\underline{\partial})=x_{a}\partial_{a}.$

Let the generators, $M_{ab}=x_{a}\partial_{b}$, act on functions of $x_{a}$.
This representation is reducible to homogeneous functions of degree $2\ell$,
$\phi(x)(x_{2})^{2\ell},\ \ x=\frac{x_{1}}{x_{2}}$. $2\ell$ is the eigenvalue
of the dilatation $(\underline{x}\underline{\partial})$, and is related to the
helicity as

$2h=2\ell+1,$ (2.30)

and the conformal symmetry generators act on $\phi(x)$ as

$(M_{ab})=\begin{pmatrix}x\partial+\ell&\partial\\\
-x(x\partial+2\ell)&x\partial-\ell\end{pmatrix}.$ (2.31)

For $2\ell=0$ the Möbius transformations on the line parametrized by $x$ are
generated.

This ratio variable $x$ is the one to represent the light-ray positions
appearing above in the operator products and $k=\lambda\bar{\lambda}$ is to
represent the light-cone momenta.

###  2.3 The Regge asymptotics

The Regge asymptotics of the scattering $AB\to A^{\prime}B^{\prime}$ is
approached in the kinematics $s\gg t$, where
$s=(p_{A}+p_{B})^{2},t=(p_{A}-p_{A^{\prime}})^{2}=q^{2}$. Tree amplitudes with
charges and (physical) helicities of $A$ and $B$ coinciding with the ones of
$A^{\prime}$ and $B^{\prime}$, correspondingly, have a leading contribution
behaving as $s$, where gluons are exchanged in the $t$ channel. Amplitudes
with $n_{q}$ quarks exchanged behave asymptotically as
$s^{1-\frac{1}{2}n_{q}}$.

It is convenient to consider the Mellin transform of the amplitude,

$M(s,t)=\int_{-i\infty+\delta}^{+i\infty+\delta}d\omega
F(\omega,q)\left(\frac{s}{M^{2}}\right)^{1+\omega}.$ (2.32)

Here the details about signature are suppressed. The Mellin amplitude
$F(\omega,q)$, also called partial wave, appears as a convolution by
integration over transverse momenta $\kappa_{i}$ of impact factors
$\Phi_{A}(\underline{\kappa})$, coupling to the particles $A$ and
$A^{\prime}$, and $\Phi_{B}(\underline{\kappa}^{\prime},)$, coupling to the
particles $B$ and $B^{\prime}$, with the reggeon Green function
$G(\omega,\underline{\kappa},\underline{\kappa}^{\prime})$, illustrated
schematically in Fig. 2,

$F(\omega,q)=\int
d\underline{\kappa}d\underline{\kappa}^{\prime}\Phi_{A}(\underline{\kappa})G(\omega,\underline{\kappa},\underline{\kappa}^{\prime})\Phi_{B}(\underline{\kappa}^{\prime})$
(2.33)

Actually there is a sum over contributions of multiple reggeon exchanges and
$\underline{\kappa}$ stands for the set of reggeon transverse momenta. We
shall use also the convolution with one of the impact factors, in the case of
two reggeons

$f(\omega,\kappa_{1},\kappa_{2})=\int
d\kappa_{1}^{\prime}d\kappa_{2}^{\prime}G(\omega,\kappa_{1},\kappa_{2},\kappa_{1}^{\prime},\kappa_{2}^{\prime})\Phi_{B}(\kappa_{1}^{\prime},\kappa_{2}^{\prime}).$

Figure 2: Mellin amplitude.

The asymptotics in $s$ is determined by the right-most singularities in
$\omega$ of the reggeon Green functions
$G(\omega,\underline{\kappa},\underline{\kappa}^{\prime})$ .

The reggeon Green functions corresponding to the single gluon or quark
exchange are, correspondingly,

$\frac{1}{|\kappa|^{2}}\frac{1}{\omega},\ \ \ \
\frac{1}{\kappa}\frac{1}{\omega+\frac{1}{2}},$ (2.34)

or the complex conjugate for the opposite quark helicity. The transverse
momentum dependence can be described by the propagators of the effective
reggeon fields in two transverse dimensions,

$S_{2R}=\int d^{2}x\mathcal{L}_{2R}(x),\ \
\mathcal{L}_{2R}(x)=-\mathcal{A}_{+}(x)\partial\partial^{*}\mathcal{A}_{-}(x)-\chi_{+}(x)\partial\chi_{-}(x)-\widetilde{\chi}_{+}(x)\partial\widetilde{\chi}_{-}(x).$
(2.35)

The reggeon fields with subcript $+$ are to couple to the impact factor
$\Phi_{A}$ and to particles flying almost in the direction $p_{A}$ and the
fields with subscript $-$ are to couple to particles flying almost in the
direction $p_{B}$.

In the leading logarithmic approximation the conformal symmetry is preserved,
acting by linear fractions (Möbius transformations) on the transverse
positions, separately for $x$ and $x^{*}$. We have in particular scale
symmetry $x\to\lambda x,x^{*}\to\bar{\lambda}x^{*}$ for the corresponding
transformation of the fields as appearing in the kinetic terms (2.35) with the
scale exponents $(\ell,\bar{\ell})$ being $(0,0)$ for gluons and
$(0,-\frac{1}{2})$ or $(-\frac{1}{2},0)$ for quarks.

In the leading logarithmic approximation the Feynman graph contributions
proportional to $\ln s$ for each loop are summed. In the light-cone gauge the
leading graphs simplify to ladders with the t-channel exchanged gluons or
quarks being reggeized by loop corrections. The kinematics of the leading
ladder contributions are characterized by the strong ordering conditions on
the loop momentum components $k^{+},k^{-}$,

$k^{+}_{1}\gg...\gg k^{+}_{N},\ \ \ k^{-}_{1}\ll...\ll k^{-}_{N}.$ (2.36)

The integration over these components reduces to the logarithmic form
$\sim\int\frac{dk^{+}}{k^{+}}$ and, consequently, the dynamics reduces to the
two transverse dimensions.

The elementary building block of the ladder and the reggeization can be
understood in terms of the effective production vertex [82]. Consider the
scattering of two partons with production of a parton. The sum of the
bremsstrahlung off the scattered parton Fig. 3a and the emission from the
exchanged reggeon Fig. 3b is represented by the emission with the effective
production vertex $V_{gg}$ Fig. 3c.

Figure 3: $2\to 3$ scattering. a) Bremsstrahlung, b) Direct emission, c)
Effective production vertex.

For the gluonic reggeons we have in transverse momenta written in complex
numbers the effective production vertex

$V_{gg}(\kappa,\kappa^{\prime})=\frac{\kappa^{*}\kappa^{\prime}}{\kappa-\kappa^{\prime}}$
(2.37)

or as an effective action term to be added to (2.35)

$\mathcal{L}_{p}(x)=g(\partial\mathcal{A}_{+}(x)\partial^{*}\mathcal{A}_{-}(x))a(x).$

$a(x)$ stands for the emitted gluon. Its momentum range is close to the mass
shell. $\mathcal{A}_{\pm}$ are the exchanged gluonic reggeons with momenta in
the range $k^{+}k^{-}\ll\kappa\kappa^{*}$.

Instead of the argument in terms of graphs Fig. 3 the effective production
term can be obtained by integrating out (elimination by equation of motion)
the modes far off shell, $k^{+}k^{-}\gg\kappa\kappa^{*}$, represented by the
fat line in Fig. 3a.

The contribution of the elementary building block of the ladder Fig. 4a is
obtained from this effective production vertex (black dot in the figure) as

$K^{(0)}(\kappa_{1},\kappa_{2},\kappa_{1}^{\prime},\kappa_{2}^{\prime})=V_{gg}(\kappa_{1},\kappa^{\prime}_{1})V^{*}_{gg}(\kappa_{2},\kappa^{\prime}_{2})+c.c.$
(2.38)

The one-loop contribution to the gluon reggeisation can be written as

$\omega_{g}(\kappa)=\frac{1}{2(2\pi)^{3}}|\kappa|^{2}\int\frac{d^{2}\kappa^{\prime}}{|\kappa^{\prime}|^{2}|\kappa-\kappa^{\prime}|^{2}}.$
(2.39)

Including this contribution modifies the Green function of the reggeized gluon
from (2.34) to

$\frac{1}{|\kappa|^{2}}\ \frac{1}{\omega+g_{r}^{2}\omega_{g}(\kappa)},\ \ \
g_{r}^{2}=\frac{g^{2}N_{c}}{2(2\pi)^{3}}.$ (2.40)

The Green function of two non-interacting reggeised gluons is

$G^{(0)}(\omega,\kappa_{1},\kappa_{2},\kappa_{1}^{\prime},\kappa_{2}^{\prime})=|\kappa_{1}|^{-2}|\kappa_{2}|^{-2}\delta(\kappa_{1}-\kappa_{1}^{\prime})\delta(\kappa_{2}-\kappa_{2}^{\prime})f^{(0)}(\omega,\kappa_{1},\kappa_{2}),$
$f^{(0)}(\omega,\kappa_{1},\kappa_{2})=\frac{1}{\omega+g_{r}^{2}\omega_{g}(\kappa_{1})+g_{r}^{2}\omega_{g}(\kappa_{2})}$

The sum of the ladder graph contributions of the two-reggeon interaction is
obtained by solving the equation

$G(\omega,\kappa_{1},\kappa_{2},\kappa_{1}^{\prime},\kappa_{2}^{\prime})=G^{(0)}(\omega,\kappa_{1},\kappa_{2},\kappa_{1}^{\prime},\kappa_{2}^{\prime})+g_{r}^{2}\int\frac{d^{2}\kappa_{1}^{\prime}d^{2}\kappa_{2}^{\prime\prime}}{|\kappa_{1}^{\prime\prime}|^{2}|\kappa_{2}^{\prime\prime}|^{2}}K^{(0)}(\kappa_{1}^{\prime\prime},\kappa_{2}^{\prime\prime},\kappa_{1},\kappa_{2})G(\omega,\kappa_{1}^{\prime\prime},\kappa_{2}^{\prime\prime},\kappa_{1}^{\prime},\kappa_{2}^{\prime}).$

The reggeization contribution $\omega_{g}(\kappa)$ (2.39) becomes well defined
with regularization only and the bare kernel $K^{(0)}$ defines an operator
only with regularization. We rewrite the equation in terms of the BFKL kernel
resulting in a well defined operator at vanishing regularization parameter
[22, 23, 24],

$K(\kappa_{1}^{\prime},\kappa_{2}^{\prime},\kappa_{1},\kappa_{2})=K^{(0)}(\kappa_{1}^{\prime},\kappa_{2}^{\prime},\kappa_{1},\kappa_{2})-(\omega_{g}(\kappa_{1})+\omega_{g}(\kappa_{2}))\delta(\kappa_{1}-\kappa_{1}^{\prime})\delta(\kappa_{2}-\kappa_{2}^{\prime}),$
$\omega
f(\omega,\kappa_{1},\kappa_{2})=g_{r}^{2}\int\frac{d^{2}\kappa_{1}^{\prime}d^{2}\kappa_{2}^{\prime}}{|\kappa_{1}^{\prime}|^{2}|\kappa_{2}^{\prime}|^{2}}K(\kappa_{1}^{\prime},\kappa_{2}^{\prime},\kappa_{1},\kappa_{2})f(\omega,\kappa_{1}^{\prime},\kappa_{2}^{\prime}).$
(2.41)

The kernel can be reconstructed from elementary reggeon triple vertices [80],
obtained from the transverse part of the induced vertex obtained from the
bremsstrahlung contribution Fig. 3a.
$\mathcal{V}^{+--}(\kappa_{1},\kappa_{2})$ is the interaction vertex of a
reggeon $\mathcal{A}_{+}$ with two reggeons $\mathcal{A}_{-}$,
$\kappa_{1},\kappa_{2}$ are the momenta of the latter.

$\mathcal{V}^{-++}(\kappa_{1},\kappa_{2})=\mathcal{V}^{+--}(\kappa_{1},\kappa_{2})=|\kappa_{1}+\kappa_{2}|^{2}.$
(2.42)

Figure 4: Elementary ladder block, composed by effective production vertices
(a) or by reggeon triple vertices (b).

Fig. 4b shows the graphs with these vertices resulting in the bare BFKL kernel
$K^{(0)}$ (2.38)), standing term by term for

$\frac{|\kappa_{2}|^{2}|\kappa_{1}^{\prime}|^{2}}{|\kappa_{1}-\kappa_{1}^{\prime}|^{2}}+\frac{|\kappa_{1}|^{2}|\kappa_{2}^{\prime}|^{2}}{|\kappa_{1}-\kappa_{1}^{\prime}|^{2}}-|\kappa_{1}+\kappa_{2}|^{2}=K^{(0)}(\kappa_{1}^{\prime},\kappa_{2}^{\prime},\kappa_{1},\kappa_{2}),$
(2.43)

and Fig. 5 shows the loop resulting in the reggeization $\omega_{g}$ (2.39).
In Fig. 4b and Fig. 5 the edges of a triple reggeon vertex (open circle)
pointing upwards represent $\mathcal{A}_{-}$ and the edges pointing downwards
represent $\mathcal{A}_{+}$.

Figure 5: Reggeization loop composed of reggeon triple vertices.

The conformal symmetry of the equation (2.41) becomes evident after the
transformation to transverse positions. The Fourier transformation is to be
done with care because of divergencies. Schematically the result expressed as
an operator in terms of the canonical pairs $x_{I},\partial_{I},I=1,2$ and
their complex conjugates $x^{*}_{I},\partial^{*}_{I}$ can be understood
applying to the expression of the BFKL kernel (2.41) the substitution rules

$|\kappa_{1}-\kappa_{1}^{\prime}|\to-\ln|x_{12}|^{2},\omega(\kappa_{1})\to\psi(1)-\ln(\partial_{1}\partial_{1}^{*}),\
\ \kappa_{1}\to\partial_{1}^{*},\ \kappa_{2}^{*}\to\partial_{2}.$ (2.44)

This leads to the equation for the Fourier transform
$\widetilde{f}(\omega,x_{1},x_{2})$

$\omega\widetilde{f}(\omega,x_{1},x_{2})=\frac{g^{2}N_{c}}{8\pi^{2}}\hat{H}_{g}\
\widetilde{f}(\omega,x_{1},x_{2})$ (2.45)

with the BFKL hamiltonian $\hat{H}_{g}$ decomposing into a holomorphic and an
anti-holomrophic parts, the first involving $x_{I},\partial_{I},I=1,2$ only
and the second $x^{*}_{I},\partial^{*}_{I},I=1,2$ only.

$\hat{H}_{g}=(H_{g}+H^{*}_{g}),$ $H_{g}=2\psi(1)-\partial_{1}^{-1}\ln x_{12}\
\partial_{1}-\partial_{2}^{-1}\ln x_{12}\
\partial_{2}-\ln\partial_{1}-\ln\partial_{2}.$ (2.46)

Further formal operator manipulations lead to the expression in terms of the
tensor Casimir operator

$C_{12}=(M^{(1)}+M^{(2)})_{ab}(M^{(1)}+M^{(2)})_{ba},$ (2.47)

where $M^{(I)}_{ab}$ are the operators in the matrix elements of (2.31) with
$x,\partial$ replaced by $x_{I},\partial_{I}$ and $\ell$ by $\ell_{I}$. The
symmetry algebra consists of two copies of $g\ell(2)$, generated by $M_{ab}$
acting on $x$ but not on $x^{*}$ and $M^{*}_{ab}$ acting on $x^{*}$.
Correspondingly we have two labels $(\ell,\bar{\ell})$ for the
representations, $\ell$ referring to the first and $\bar{\ell}$ referring to
the second.

In the case of two gluonic reggeons considered the representation parameters
are $(\ell_{1},\bar{\ell}_{1}),(\ell_{2},\bar{\ell}_{2})=(0,0),(0,0)$. This
corresponds to the above remark after (2.35) about the scale dimensions of the
reggeon fields. The formulation generalizes to more than two reggeons and to
the inclusions of fermionic reggeons. The latter are described by the
representations $(-\frac{1}{2},0)$ or $(0,-\frac{1}{2})$ depending on helicity
and chirality [81].

The colour-dipole model [58, 59, 60, 61] of high-energy scattering leads to
the integral form of the Hamiltonian

$\hat{H}_{g}\widetilde{f}(\omega,x_{1},x_{2})=\frac{1}{\pi}\int
d^{2}x_{0}\frac{|x_{12}|^{2}}{|x_{10}|^{2}|x_{20}|^{2}}\left(\widetilde{f}(\omega,x_{1},x_{2})-\widetilde{f}(\omega,x_{1},x_{0})-\widetilde{f}(\omega,x_{0},x_{2})\right).$
(2.48)

$\widetilde{f}(\omega,x_{1},x_{2})$ can be regarded as the Mellin transform of
the dipole distribution with respect to their end-points in the transverse
plane and the equation (2.45) describes its evolution with rapidity $Y\sim\ln
s$.

We present the arguments of [62] leading to this form. By the formal operator
relations

$\partial\ln x\ \partial^{-1}=\ln x+x^{-1}\partial^{-1},\ \ x\ln\partial\
x^{-1}=\ln\partial-\partial^{-1}x^{-1}$ (2.49)

the Hamiltonian (2.46) is rewritten as

$\hat{H}_{g}=4\psi(1)-2\ln|x_{12}|^{2}-2|x_{12}|^{-2}\ln|\partial_{1}\partial_{2}|^{2}|x_{12}|^{2}.$
(2.50)

Regularizing the kernel in (2.48) we have

$\int\frac{d^{2}x_{0}}{\pi}\frac{|x_{12}|^{2}}{(|x_{10}|^{2}+\delta^{2})(|x_{20}|^{2}+\delta^{2})}=\int_{0}^{1}d\alpha\frac{|x_{12}|^{2}}{(\alpha(1-\alpha)|x_{12}|^{2}+\delta^{2}}=2\ln{\frac{|x_{12}|^{2}}{\delta^{2}}}+\mathcal{O}(\delta^{2})$
$-\int\frac{d^{2}x_{0}}{\pi}\frac{|x_{12}|^{2}}{(|x_{10}|^{2}+\delta^{2})(|x_{20}|^{2}+\delta^{2})}\widetilde{f}(\omega,x_{1},x_{0})=|x_{12}|^{2}(\ln(\delta^{2}|dd_{2}|^{2}-2\psi(1))|x_{12}|^{-2}\widetilde{f}(\omega,x_{1},x_{2})+\mathcal{O}(\delta^{2})$

This shows that the integral operator (2.48) acts in the limit of vanishing
$\delta$ as the operator (2.50).

The eigenvalues of the operator defined by the kernel
$K(\kappa_{1}^{\prime},\kappa_{2}^{\prime},\kappa_{1}\kappa_{2})$ (2.41) are
easily calculated in the transverse momentum form in the forward limit,
$\kappa_{1}+\kappa_{2}=-\kappa_{1}^{\prime}-\kappa_{2}^{\prime}=0$. In this
case the eigenfunctions are

$E^{(n,\nu)}(\kappa)=|\kappa|^{\frac{1}{2}+i\nu}\left(\frac{\kappa^{*}}{|\kappa|}\right)^{n},$

a basis of functions on the complex plane for integer $n$ and
$\nu\in\hbox{\bbd R}^{1}$. The eigenvalues can be written as

$\Omega_{g}(n,\nu)=\chi_{0}(m)+\chi_{0}(\bar{m}),\chi_{0}(m)=2\psi(1)-\psi(m)-\psi(1-m),$
(2.51) $m=\frac{1}{2}+i\nu+\frac{1}{2}n,\ \
\bar{m}=\frac{1}{2}+i\nu-\frac{1}{2}n.$

$\chi_{0}(m)$ is actually a function of $m(1-m)$, the eigenvalue of the
Casimir operator (2.47). In the form of transverse positions the
eigenfunctions are the conformal three-point functions.

$E^{(n,\nu)}(x_{1},x_{2};x_{0})=E_{m}(x_{1},x_{2};x_{0})E_{\bar{m}}(x^{*}_{1},x^{*}_{2};x^{*}_{0}),$
(2.52)
$E_{m}(x_{1},x_{2};x_{0})=\left(\frac{x_{12}}{x_{10}x_{20}}\right)^{m},$

obeying the completeness relation

$\delta^{(2)}(x_{1}-x_{1}^{\prime})\delta^{(2)}(x_{2}-x_{2}^{\prime})=\int
d^{2}x_{0}\int_{-\infty}^{+\infty}d\nu\sum_{n=-\infty}^{+\infty}$
$\frac{\nu^{2}+\frac{n^{2}}{4}}{\pi^{4}|x_{11^{\prime}}|^{2}x_{22^{\prime}}|^{2}}E^{(n,\nu)*}(x_{1},x_{2},x_{0})E^{(n,\nu)}(x^{\prime}_{1},x^{\prime}_{2},x_{0}).$

The orthogonality relation expresses the fact that this basis is twofold over-
complete [63].

The spectral properties are the basis for giving the above forms of the BFKL
operator and the unconventional steps of transformations (2.44), (2.49) well
defined meanings.

The renormalized coupling, depending on the scale $Q^{2}$, enters the Bjorken
limit evolution equations at leading order. In the BFKL equation, describing
the Regge asymptotics, the bare coupling enters at leading order. Its
renormalization is included at next-to-leading order only [64],[65], [66].

For the region of overlap of both asymptotics combinations of both types of
equations have to be considered, including the BFKL contributions as
corrections to the DGLAP equation [69] or improving the BFKL equation by the
renormalization group [63]. In the overlap region the contributions of muti-
reggeon exchange become important. They can be described by non-linear terms
to be included into the BFKL equation [67, 68]. In the dipole picture the
multi-reggeon contributions appear as creation of more dipoles [60, 61]. In
the ultimate asymptotics a regime of saturation emerges [70, 71] allowing the
application of thermodynamic concepts. Characteristics of the thermodynamic
limit of models directly related to the integrable spin chains, where the BFKL
hamiltonian describes the pair interaction, have been proposed to be applied
here [72, 73].

##  3 Yangian symmetry

###  3.1 Monodromy matrix operators and symmetric correlators

We start with the formulation of the $g\ell(n)$ Lie algebra relations as the
fundamental Yang-Baxter relation

$\mathcal{R}(u-v)(L(u)\otimes I)(I\otimes L(v))=(I\otimes L(v))(L(u)\otimes
I)\mathcal{R}(u-v).$ (3.1)

Here $\mathcal{R}(u-v)$ stands for Yang’s $R$ matrix. Its size is $n^{2}\times
n^{2}$ in our case and composed of the unit matrix and the matrix representing
the permutation of the factors in the tensor product of the $n$-dimensional
fundamental representation spaces, $\mathcal{R}(u)=uI_{n^{2}\times n^{2}}+P$.
The $L$ matrices are of size $n\times n$ and linear in the spectral parameter
$u$, $L(u)=uI_{n\times n}+L$. $L_{ab},a,b,=1,...,n$ are the generators of the
Lie algebra $g\ell(n)$. In index notation we have

$L_{ab}(u)=u\delta_{ab}+L_{ab},\ \ \
\mathcal{R}^{a_{1}a_{2}}_{b_{1}b_{2}}(u)=u\delta^{a_{1}}_{b_{1}}\delta^{a_{2}}_{b_{2}}+\delta^{a_{1}}_{b_{2}}\delta^{a_{2}}_{b_{1}}.$
(3.2)

The Yangian algebra can be introduced by the matrix operator $T(u)$ obeying
the above Yang-Baxter relation (3.1) with $L(u)$ substituted by $T(u)$ and
where now the dependence on $u$ is not restricted as above. Then the
generators of the extended Yangian algebra [9], [38] appear in the expansion
of $T(u)$ in inverse powers of $u$ and the extended Yangian algebra relations
follow from the Yang-Baxter relation (3.1).

We shall deal with evaluations of the $g\ell(n)$ Yangian of finite order $N$
where $T(u)$ is constructed from the $L$ matrices in the way well know in the
context of integrable models. Let the matrix elements of $L_{I}(u_{I})$ act on
the representation space $V_{I}$ and consider the matrix product defining the
monodromy matrix

$T(u,\delta_{1},\dots,\delta_{N})=T(\mathbf{u})=L_{1}(u_{1})\cdots
L_{N}(u_{N}),\quad u_{I}=u+\delta_{I},$ (3.3)

acting on the tensor product $V_{1}\otimes\cdots\otimes V_{N}$.

We consider the case of representations by homogeneous functions of
$\mathbf{x}_{I}=(x_{i,1},...,x_{I,n})$ and the weight $2\ell_{I}$ is the
degree of homogeneity,

$\psi(\mathbf{x}_{I})=x_{i,n}^{2\ell_{I}}\phi(\mathbf{x}^{\prime}_{I}),\ \ \
\mathbf{x}_{I}^{\prime}=(\frac{x_{I,1}}{x_{I,n}},...,\frac{x_{I,n-1}}{x_{I,n}}).$
(3.4)

The elements of the tensor product $V_{1}\otimes\cdots\otimes V_{N}$ are
functions of $N$ points in the $n$ dimensional space.

We consider the Lie algebra generators of the Jordan-Schwinger (JS) form,

$L^{+}_{I,ab}=\partial_{I,a}x_{I,b},\ \ \
L^{-}_{I,ab}=-x_{I,a}\partial_{I,b},$ (3.5)

built from $n$ canonical pairs $x_{I,a},\partial_{I,a}$ at each point $I$,

$[\partial_{I,a},x_{J,b}]=\delta_{IJ}\delta_{ab}.$ (3.6)

Notice that the elementary canonical transformation

$\begin{pmatrix}x_{I,a}\\\
\partial_{I,a}\end{pmatrix}\rightarrow\begin{pmatrix}\partial_{I,a}\\\
-x_{I,a}\end{pmatrix}$ (3.7)

transforms $L_{I}^{+}(u)$ into $L_{I}^{-}(u)$.

We have the matrix inversion relations (for any point $I$)

$\left(\frac{L^{+}(u)}{u}\right)^{-1}=\frac{L^{+}(-u-1-(\underline{x}\underline{\partial}))}{-u-1-(\underline{x}\underline{\partial})},\
\ \
\left(\frac{L^{-}(u)}{u}\right)^{-1}=\frac{L^{-}(-u+1+(\underline{x}\underline{\partial}))}{-u+1+(\underline{x}\underline{\partial})},$
(3.8)

where $(\underline{x}\underline{\partial})=\sum_{a=1}^{n}x_{a}\partial_{a}$.

The two forms of JS generators are also related by matrix transposition,
$(L^{t})_{ab}=L_{ba}$,

$(L^{+}(u))^{t}=-L^{-}(-u-1).$ (3.9)

This transposition operation and the inversion act depending on ordering,

$(L^{+}(u))^{t\ -1}=(L^{+}(u+n))^{-1\ t}.$ (3.10)

Related to integrals over the positions $x_{a}$ we consider another
transposition operation acting on the canonical pairs as

$(x_{a})^{T}=x_{a},\ \ \partial_{a}^{T}=-\partial_{a},$

leaving indices unchanged but reversing the order in products of such
operators, similar to the matrix transposition, at the same point e.g.
$(x_{I,a}\partial_{I,b}\partial_{J,c})^{T}=(\partial_{I,b})^{T}(x_{I,a})^{T}\partial_{J,c}^{T}$.

$(L^{+}(u))^{T}=(L^{-}(u))^{t}=-L^{+}(-u-1),\ \ \
(\underline{x}\underline{\partial})^{T}=-n-(\underline{x}\underline{\partial}).$
(3.11)

We intend to consider representations by functions of $x_{I,a}$ homogeneous at
each point $I$ (3.4), i.e. we restrict to the subspace of fixed eigenvalues of
the dilatation operators, $(\underline{x}_{I}\underline{\partial}_{I})\to
2\ell_{I}$. The corresponding projector will be denoted by
$\Pi_{I}(2\ell_{I})$.

The projection of the $L$ operator depends on this degree of homogeneity,

$\Pi_{I}(2\ell_{I})L_{I}^{+}(u)\Pi_{I}(2\ell_{I})=L_{I}^{+}(u_{I}^{+},u_{I}),\
u_{I}^{+}=u+\delta_{I}+2\ell_{I}$ (3.12)

The action of the inversion and the transpositions $t$ and $T$ on the
projected $L$ obey

$(L^{+}(u^{+},u))^{-1}=\frac{L^{+}(-u-1,-u^{+}-1)}{u(-u^{+}-1)},$ (3.13)
$(L^{+}(u^{+},u))^{T}=-L^{+}(-u^{+}-1-n,-u-1),\ \ \ (L^{+}(u^{+},u))^{t\
-1}=(L^{+}(u^{+}+n,u+n))^{-1\ t},$ (3.14) $(L^{+}(u^{+},u))^{T\
-1}=\frac{L^{+}(u,u^{+}+n)}{(u+1)(u^{+}+n)},\ \ \ (L^{+}(u^{+},u))^{-1\
T}=\frac{L^{+}(u-n,u^{+})}{u(u^{+}+1)}.$ (3.15)

The projected $L$ obey the fundamental Yang-Baxter relation (3.1). In the
following we prefer the form $L^{+}$ and omit the superscript $+$.

The fundamental Yang-Baxter relation (3.1) is fulfilled by the monodromy
composed as the ordered matrix product of $L_{I}(u_{I})$ (3.3) as well as of
the projected $L_{I}(u_{I}^{+},u_{I})$.

$T(\mathbf{u})=L_{1}(u_{1}^{+},u_{1})...L_{N}(u_{N}^{+},u_{N}),\ \
\mathbf{u}=\begin{pmatrix}u_{1}&...,u_{N}\\\
u_{1}^{+}&...,u^{+}_{N}\end{pmatrix}$ (3.16)

$N$-point functions $\Phi$ homogeneous at each point obeying the condition

$T(\mathbf{u})\ \Phi=IE(\mathbf{u})\ \Phi$ (3.17)

are called Yangian symmetric correlators (YSC) [78]. On l.h.s an operator-
valued matrix is acting and on r.h.s. there is the unit matrix $I$ and the
eigenvalue depending on the set of parameters $\mathbf{u}$. The dependence of
$\Phi$ on the parameters is invariant with respect to shifts of the spectral
parameter, $u_{I}\to u_{I}+v,u_{I}^{+}\to u_{I}^{+}+v$. The YSC are invariants
in the corresponding representation of the Yangian algebra.

As solutions of (3.17) we shall accept also expressions involving
distributions, extending the representation space $V_{1}\otimes\cdots\otimes
V_{N}$. The $\delta$-distributions are to be understood in Dolbeault sense
(cf. [74]), their arguments must not be restricted to real values.

By elementary canonical transformations of the underlying canonical pairs
(3.7), where this transfomation may be applied to a subset of the index values
$I,a$, the monodromies and the symmetric correlators acquire other
representation forms. In the applications to be considered the helicity
representation plays an important role. It will be desribed in subsection 3.4.

In the next subsection we outline methods of construction of YSC. The Yang-
Baxter operators and the corresponding $RLL$ relations, being the working tool
in one of them, will be discussed in the subsection 3.3.

Examples of YSC constructions will be presented in subsection 3.5. The role of
YSC as kernels of symmetric integral operators will be considered in
subsection 3.6.

###  3.2 YSC construction methods

We find trivial examples of YSC in the case of one point, $N=1$.

$L(u,u)\cdot 1=(u+1)1,\ \ \
L(u-n,u)\cdot\delta(\mathbf{x})=u\delta^{(n)}(\mathbf{x}).$

The monodromy $T_{1,2..,N}$ can be considered as the product
$T_{1,2..,N}=T_{1,2,...,N_{1}}T_{N_{1}+1,...,N}$, and consequently a product
of YSC,

$\Phi_{1,..,N_{1}}\cdot\Phi_{N_{1},...,N},$ (3.18)

the first obeying (3.17) with $T_{1,2,...,N_{1}}$ and the second obeying
(3.17) with $T_{N_{1}+1,...,N}$, is a YSC obeying (3.17).

####  3.2.1  The convolution method

In the product correlator (3.18) the points with the labels $1,...,N_{1}$ and
$N_{1}+1,...,N$ are not connected. A way to get connected YSC from a product
goes by identifying some points of the first and the second factor and
integrating over these identified points. We illustrate this in the example of
the convolution of two 3-point correlators and derive the conditions on the
representation parameters for obeying the YSC condition of the result.

We derive the conditions for a one-point convolution of YCS to be YSC.
Consider the convolution of two 3-point YSC

$\Phi_{1,2,3,4}=\int
d\mathbf{x}_{0}\Phi_{1,2,0}(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{0})\bar{\Phi}_{0,3,4}(\mathbf{x}_{0},\mathbf{x}_{3},\mathbf{x}_{4}),$

each obeying (3.17)

$L_{1}(u_{1}^{+},u_{1})L_{2}(u_{2}^{+},u_{2})L_{0}(u_{0}^{+},u_{0})\Phi_{1,2,0}=E\
\Phi_{1,2,0},$
$L_{0}(v_{0}^{+},v_{0})L_{3}(u_{3}^{+},u_{3})L_{4}(u_{4}^{+},u_{4})\bar{\Phi}_{0,3,4}=\bar{E}\
\bar{\Phi}_{0,3,4}.$

We calculate the action of the order 4 monodromy on $\Phi_{1,2,3,4}$

$L_{1}(u_{1}^{+},u_{1}))L_{2}(u_{2}^{+},u_{2})L_{3}(u_{3}^{+},u_{3})L_{4}(u_{4}^{+},u_{4})\Phi_{1,2,3,4}=$
$L_{1}(u_{1}^{+},u_{1}))L_{2}(u_{2}^{+},u_{2})\int
d\mathbf{x}_{0}\Phi_{1,2,0}\
\bar{E}(L_{0}(v_{0}^{+},v_{0}))^{-1}\bar{\Phi}_{0,3,4}.$

Integration by parts moves $L_{0}^{-1}$ acting on $\bar{\Phi}$ to
$L_{0}^{-1T}$ acting on $\Phi$. The result is proportional to $\Phi_{1234}$ if

$(L_{0}(v_{0}^{+},v_{0}))^{-1T}=constL_{0}(u^{+}_{0},u_{0}).$

We evaluate the condition

$(L_{0}(v_{0}^{+},v_{0}))^{-1}=const(L_{0}(u_{0}^{+},u_{0}))^{T},$

using (3.13) and (3.14)

$L_{0}(-u_{0}^{+}-1-n,-u_{0}-1)=constL_{0}(-v_{0}-1,-v_{0}^{+}-1),$

and obtain the wanted conditions on the representation parameters

$u_{0}=v_{0}^{+},\ \ \ u_{0}^{+}=v_{0}-n.$ (3.19)

####  3.2.2  Cyclicty

If $\Phi$ obeys the YSC condition (3.17) with $T(\mathbf{u})$ then it obeys
this condition also with $T(\mathbf{u^{\prime}})$ with the change of the
sequence of points from $1,2,...,N$ to $2,...,N,1$ and of the parameters by
the cyclicity rule

$\mathbf{u^{\prime}}=\begin{pmatrix}u_{2}&...&u_{N}&u_{1}-n\\\
u_{2}^{+}&...&u^{+}_{N}&u_{1}^{+}-n\end{pmatrix}.$ (3.20)

The cyclicity operation does not affect the weights associated with the points
$I$. We formulate and proof the cyclicity rule (3.20) in the situation before
the weight projection (3.12) is done.

$L_{1}(u_{1})L_{2}(u_{2})...L_{N}(u_{N})\Phi=E\Phi$

implies by inversion

$L_{2}(u_{2})...L_{N}(u_{N})\Phi=EL_{1}^{-1}(u_{1})\Phi$

and by matrix transposition (3.10)

$L^{t}_{N}(u_{N})...L^{t}_{2}(u_{2})=E(L_{1}^{-1}(u_{1}))^{t}\Phi=E(L_{1}^{t}(u_{1}-n))^{-1}\Phi.$

Multiplying by $L_{1}^{t}(u_{1}-n)$ and applying the matrix transposition once
more we obtain

$L_{2}(u_{2})...L_{N}(u_{N})L_{1}(u_{1}-n)\Phi=E\Phi.$

In this way we have proven (3.20).

The inversion (3.13) results in YSC with the reversed order of points.

####  3.2.3  The R operator method

If $\Phi$ obeys the YSC condition (3.17) with $T(\mathbf{u})$ then we shall
find operators $R_{I,I+1}(u_{I}-u_{I+1})$ and
$R_{I+1,I}(u^{+}_{I}-u^{+}_{I+1})$ such that their action on $\Phi$ leads to a
YSC with monodromies of parameter sets obtained from $\mathbf{u}$ by permuting
adjacent entries, $u_{I}$ and $u_{I+1}$ of the upper row in the first case and
$u^{+}_{I}$ and $u^{+}_{I+1}$ of the lower row in the second. By such $R$
operations we can construct more connected YSC. The weights associated with
the points $I,I+1$ change from
$2\ell_{I}=u_{I}^{+}-u_{I},2\ell_{I+1}=u_{I+1}^{+}-u_{I+1}$ by the action of
$R_{I,I+1}$ to
$2\ell_{I}^{\prime}=u_{I}^{+}-u_{I+1},2\ell_{I+1}^{\prime}=u_{I+1}^{+}-u_{I}$
or by the action of $R_{I+1,I}$ to
$2\ell_{I}^{\prime\prime}=u_{I+1}^{+}-u_{I},2\ell_{I+1}^{\prime\prime}=u_{I}^{+}-u_{I+1}$.

###  3.3 The Yang-Baxter operator

In the fundamental Yang-Baxter relation (3.1) the $R$ matrix intertwines the
tensor product of fundamental $n$ dimensional representations. We shall obtain
a more general $R$ operator intertwining the tensor product of JS type
representations. It should obey the Yang-Baxter relation of the form

$R_{12}(u-v)L_{1}(u)L_{2}(v)=L_{1}(v)L_{2}(u)R_{12}(u-v),$ (3.21)
$(L_{1}(u))_{ab}=\delta_{ab}u+\partial^{1}_{a}x_{b}^{1},\ \ \
(L_{2}(u))_{ab}=\delta_{ab}u+\partial^{2}_{a}x_{b}^{2}.$

The relation involves the matrix product $L_{1}(u)L_{2}(v)$. $R_{12}$ is not a
matrix, i.e. does not act on the fundamental representation indices, but acts
rather on the two sets of canonical pairs $x_{a}^{1},\partial^{1}_{a}$ and
$x_{a}^{2},\partial^{2}_{a}$.

We read (3.21) as the defining relation of $R_{12}$ and solve it using the
ansatz

$R_{12}(u)=\int\mathrm{d}c\,\phi(c)e^{-c(\underline{x}_{1}\underline{\partial}_{2})}.$

We calculate the action of the shift operator on the product of $L$ matrices:

$e^{-c(\underline{x}_{1}\underline{\partial}_{2})}\underline{\partial}_{1}=(\underline{\partial}_{1}+c\underline{\partial}_{2})e^{-c(\underline{x}_{1}\underline{\partial}_{2})},\qquad
e^{-c(\underline{x}_{1}\underline{\partial}_{2})}\mathbf{x}_{2}=(\mathbf{x}_{2}-c\mathbf{x}_{1})\
e^{-c(\underline{x}_{1}\underline{\partial}_{2})},$
$e^{-c(\underline{x}_{1}\underline{\partial}_{2})}(L_{1}^{+}(u)L_{2}^{+}(v))_{ab}e^{c(\underline{x}_{1}\underline{\partial}_{2})}=\big{(}L_{1ac}^{+}(u)+c\partial^{2}_{ad}x^{1}_{dc}\big{)}\big{(}L_{2cb}^{+}(v)-c\partial^{2}_{cd}x^{1}_{db}\big{)}=$
$\big{(}L_{1}^{+}(v)L_{2}^{+}(u)\big{)}_{ab}+\big{(}L_{2}^{+}(0)-L_{1}^{+}(0)\big{)}_{ab}\\{[u-v+c(\underline{x}_{1}\cdot\underline{\partial}_{2})]-c[u-v-1+c(\underline{x}_{1}\underline{\partial}_{2})]\\}.$

The condition that the contribution of the second term vanishes implies a
differential equation on $\phi(c)$, because

$\int\mathrm{d}c\,\phi(c)\cdot
c\cdot(\underline{x}_{1}\underline{\partial}_{2})e^{-c(\underline{x}_{1}\underline{\partial}_{2})}=-\int\mathrm{d}c\,\phi(c)\cdot
c\cdot\partial_{c}\,e^{-c(\underline{x}_{1}\underline{\partial}_{2})}=\int\mathrm{d}c\,\partial_{c}(c\phi(c))\,e^{-c(\underline{x}_{1}\underline{\partial}_{2})}.$

In the last step we have assumed that the integration by parts is done without
boundary terms as it holds for closed contours. Thus the condition on $\phi$
is

$\partial_{c}(c\phi(c))+(u-v)\phi(c)=0,$

and it is solved by

$\phi(c)=\frac{1}{c^{1+u-v}}.$

The condition of vanishing of the third term can be written as

$0=\partial_{c}(c^{2}\phi(c))+c(u-v-1)\phi(c)=c[\partial_{c}(c\phi(c))+(u-v)\phi(c)].$

We see that it does not imply a further condition on $\phi(c)$. Thus we have
proved that

$R_{12}(u)=\int\frac{\mathrm{d}c}{c^{1+u}}e^{-c(\underline{x}_{1}\underline{\partial}_{2})}$
(3.22)

obeys the Yang-Baxter relation provided the simple rule of integration by
parts. If the latter rule is different then the indicated procedure leads to
the appropriate modification.

Now we derive two Yang-Baxter relations for the projected $L$ operators
(3.12). We notice that the shift operation included in the above solution of
$R_{12}$ affects the dilatation operators

$e^{-c(\underline{x}_{1}\underline{\partial}_{2})}(\underline{x}_{1}\underline{\partial}_{1})e^{c(\underline{x}_{1}\underline{\partial}_{2})}=(\underline{x}_{1}\underline{\partial}_{1})+c(\underline{x}_{1}\underline{\partial}_{2}),\
\
e^{-c(\underline{x}_{1}\underline{\partial}_{2})}(\underline{x}_{2}\underline{\partial}_{2})e^{c(\underline{x}_{1}\underline{\partial}_{2})}=(\underline{x}_{2}\underline{\partial}_{2})-c(\underline{x}_{1}\underline{\partial}_{2}).$

By partial integration we obtain

$\int
dc\phi(c)c(\underline{x}_{1}\underline{\partial}_{2})e^{-c(\underline{x}_{1}\underline{\partial}_{2})}=-\int
dc\phi(c)c\partial_{c}e^{-c(\underline{x}_{1}\underline{\partial}_{2})}=$
$\int
dc(\partial_{c}c\varphi(c))e^{-c(\underline{x}_{1}\underline{\partial}_{2})}=(u-v)\int
dc\varphi(c)e^{-c(\underline{x}_{1}\underline{\partial}_{2})}.$

In this way we have

$R_{12}(u-v)\left(v+(\underline{x}_{1}\underline{\partial}_{1})\right)R_{12}^{-1}(u-v)=u+(\underline{x}_{1}\underline{\partial}_{1}),\
\
R_{12}(u-v)\left(u+(\underline{x}_{2}\underline{\partial}_{2})\right)R_{12}^{-1}(u-v)=v+(\underline{x}_{2}\underline{\partial}_{2})$

or

$R_{12}(u-v)\Pi_{1}(2\ell_{1})=\Pi_{1}(2\ell_{1}+u-v)R_{12}(u-v),\ \ \
R_{12}(u-v)\Pi_{2}(2\ell_{2})=\Pi_{2}(2\ell_{2}-u+v)R_{12}(u-v).$ (3.23)

This implies that (3.21) results for the projected $L$ in a similar relation
with the permutation of $u$ and $v$ but coincidence of the additional
arguments on both sides.

$R_{12}(u-v)L_{1}(u^{+},u)L_{2}(v^{+},v)=L_{1}(u^{+},v)L_{2}(v^{+},u)R_{12}(u-v).$
(3.24)

Now we apply the inversion relation of the projected $L$ (3.13) to obtain the
Yang-Baxter relation with permutation of the first arguments. We have from the
last relation (3.24)

$R_{12}(u-v)(L_{2}(v^{+},v))^{-1}(L_{1}(u^{+},u))^{-1}=(L_{2}(v^{+},u))^{-1}(L_{1}(u^{+},v))^{-1}R_{12}(u-v).$

The inversion relation (3.13) implies

$R_{12}(u-v)L_{2}(-v-1,-v^{+}-1)L_{1}(-u-1,-u^{+}-1)=$
$L_{2}(-u-1,-v^{+}-1)L_{1}(-v-1,-u^{+}-1)R_{12}(u-v).$

We interchange the labels $1\leftrightarrow 2$,

$R_{21}(u-v)L_{1}(-v-1,-v^{+}-1)L_{2}(-u-1,-u^{+}-1)=$
$L_{1}(-u-1,-v^{+}-1)L_{2}(-v-1,-u^{+}-1)R_{21}(u-v),$

and substitute the representation parameters as

$u^{\prime}=-v^{+}-1,v^{\prime}=-u^{+}-1,{u^{\prime}}^{+}=-v-1,{v^{\prime}}^{+}=-u-1$

to obtain

$R_{21}({u^{\prime}}^{+}-{v^{\prime}}^{+})L_{1}({u^{\prime}}^{+},u^{\prime})L_{2}({v^{\prime}}^{+},v^{\prime})=L_{1}({u^{\prime}}^{+},v^{\prime})L_{2}({v^{\prime}}^{+},u^{\prime})R_{21}({u^{\prime}}^{+}-{v^{\prime}}^{+}).$
(3.25)

Now the primes can be omitted. We have obtained the Yang-Baxter relations for
the $L$ operators projected to definite weights in the convenient form
characterized by the permutation of the first or the second arguments. This
results in the $R$ operation rule for the YSC above.

The repeated application of this rule allows to construct YSC starting from
the trivial products of delta distributions in $K$ of the $N$ points and
constant functions in the other points, e.g.

$\Phi_{K,N;0}(\mathbf{x}_{1},...,\mathbf{x}_{N})=\Pi_{L=1}^{K}\delta((\mathbf{x}_{L}).$

The expression of the $R$ operators in terms of a contour integral over a
shift operator (3.22) leads to the general form

$\Phi_{K,N}(\mathbf{x}_{1},...,\mathbf{x}_{N})=\int\varphi(c)\Pi_{L=1}^{K}dc_{L,K+1}...dc_{L,N}\delta(\mathbf{x}_{L}-c_{L,K+1}\mathbf{x}_{K+1}-c_{L,N}\mathbf{x}_{N}).$
(3.26)

The integration variables can be regarded as coordinates on the Graßmann
variety $\mathcal{M}_{K,N}$. The integrand function $\varphi(c)$ contains the
information about the particular YSC. The starting YSC $\Phi_{K,N;0}$, where
the points are disconnected, can be written in this form too with $\varphi(c)$
being a product of delta distributions in the $c_{L,M}$ . If the points are
maximally connected, then the rectangular matrix with the elements
$c_{L,M},L=1,...K;M=1,...,N$ has maximal rank, the integration extends over
the maximal Schubert cell of $\mathcal{M}_{K,N}$.

###  3.4 The helicity representation

We have noticed that the two forms $L^{+},L^{-}$ of the JS presentation of the
$L$ operators (3.5) are related by the elementary canonical transformation
(3.7). There the transformation acts uniformly for the index values
$a=1,...n$. Now we restrict ourselves to the case of even $n=2m$. We split the
index range into the parts $1,...,m$ and $m+1,...,2m$ and label the first by
$\alpha=1,..,m$ and the second by
$\stackrel{{\scriptstyle\cdot}}{{\alpha}}=1,...,m$. With a transformation
modified in this respect, i.e. non-trivial for the index value
$a=\alpha=1,...,m$ only,

$\begin{pmatrix}x_{I,\alpha}\\\
\partial_{I,\alpha}\end{pmatrix}\rightarrow\begin{pmatrix}\partial^{\lambda}_{I,\alpha}\\\
-\lambda_{I,\alpha}\end{pmatrix},\ \ \
\begin{pmatrix}x_{I,m+\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\\\
\partial_{I,m+\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\end{pmatrix}\rightarrow\begin{pmatrix}\bar{\lambda}_{I,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\\\
\bar{\partial}^{\lambda}_{I,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\end{pmatrix},$
(3.27)

applied to $L^{+}_{I}$ (3.5) we are lead to the helicity form
$L^{\lambda}_{I}$ appearing in $m\times m$ blocks (for any $I$ )

$L^{\lambda}_{\alpha,\beta}=-\lambda_{\alpha}\partial^{\lambda}_{\beta},\ \ \
L^{\lambda}_{\alpha,\stackrel{{\scriptstyle\cdot}}{{\beta}}}=-\lambda_{\alpha}\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\beta}}},$
$L^{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}},\beta}=\bar{\partial}^{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\partial^{\lambda}_{\beta},\
\ \
L^{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}},\stackrel{{\scriptstyle\cdot}}{{\beta}}}=\bar{\partial}^{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\beta}}}.$

The transformation applied to the dilatation operator
$(\underline{x}\underline{\partial})$ results in

$(\underline{x}\underline{\partial})\to-\partial^{\lambda}_{\alpha}\lambda^{\alpha}+\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\partial}^{\lambda\
\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=-m+2\hat{h}$

We notice that (3.27) is for $m=2$ the inversion of (2.6), (2.7). In view of
the application to scattering we call $\hat{h}$ helicity. The relation of its
eigenvalues to the dilatation weight is

$2h=m+2\ell.$ (3.28)

The expression for $R^{\lambda}_{12}$ is obtained from the one for $R_{12}(u)$
(3.22) by substituting according to the canonical transformation (3.27)

$(\underline{x}_{1}\underline{\partial}_{2})\to-\partial^{\lambda}_{1\alpha}\lambda_{2}^{\alpha}+\bar{\lambda}_{1\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\partial}^{\lambda\stackrel{{\scriptstyle\cdot}}{{\alpha}}}_{2}.$
(3.29)

The YSC condition (3.17) implies in particular

$\sum_{I}L^{\lambda}_{I\alpha\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\
\Phi^{\lambda}=\sum_{I}\lambda_{I\alpha}\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\
\Phi^{\lambda}=0.$ (3.30)

The transformation of $\Phi_{K,N}$ (3.26) to the helicity representation
results in

$\Phi^{\lambda}_{K,N}(\lambda_{1},\bar{\lambda}_{1}...,\lambda_{N},\bar{\lambda}_{N})=\int\varphi(c)\Pi_{L=1}^{K}dc_{L,K+1}...dc_{L,N}$
$\delta^{(m)}(\bar{\lambda}_{L}-c_{L,K+1}\bar{\lambda}_{K+1}-c_{L,N}\bar{\lambda}_{N})\Pi_{M=K+1}^{N}\delta^{(m)}(\lambda_{M}+c_{1,M}\lambda_{1}+...+c_{K,M}\lambda_{K}).$
(3.31)

Instead of $K$ factors of $n=2m$ dimensional delta distributions we have now
$N$ factors of $m$ dimensional ones. The integrand function $\varphi(c)$ is
not changed. (3.30) follows from the linear equations implied by the arguments
of the delta distributions. This means, the factor
$\delta^{(2m)}(\sum_{I}\lambda_{I\alpha}\bar{\lambda}_{I\stackrel{{\scriptstyle\cdot}}{{\alpha}}})$
can be separated.

###  3.5 Examples of YSC constructions

####  3.5.1 Two and three points

By one $R$ operation we obtain the 2-point YSC

$\Phi^{-+}_{12}(\mathbf{x}_{1},\mathbf{x}_{2})=R_{21}(u_{1}^{+}-u_{2})\delta(\mathbf{x}_{1})=\int\frac{dc_{12}}{c_{12}^{1+u_{1}^{+}-u_{2}}}\delta(\mathbf{x}_{1}-c_{12}\mathbf{x}_{2}).$

Let us abbreviate in $\Phi$ the dependence on the points $\mathbf{x}_{I}$ by
subscripts $I$.

We construct 3-point correlators starting from the trivial ones

$\Phi_{123,0}^{--+}=\delta(\mathbf{x}_{1})\delta(\mathbf{x}_{2}),\ \ \
\Phi_{123,0}^{-++}=\delta(\mathbf{x}_{1}).$

The corresponding monodromy is characterized by the set of parameters
$\mathbf{u}$ conveniently displayed by rows of $u_{I}$ and $u^{+}_{I}$ (3.16).
We abbreviate here parameters by their position indices $I$ eventually
supplemented by a superscript $+$ for indicating the shift by $-n$. In case
$\Phi_{123,0}^{--+}$ we have

$\mathbf{u}^{--+(0)}=\begin{pmatrix}1&2&3&\\\ 1^{+}&2^{+}&3\end{pmatrix},$

and in the case $\Phi_{123,0}^{-++}$

$\mathbf{u}^{-++(0)}=\begin{pmatrix}1&2&3&\\\ 1^{+}&2&3\end{pmatrix}.$

By one $R$ operation we obtain

$\Phi_{123,1}^{--+}=R_{31}(u_{1}^{+}-u_{3})\Phi_{123,0}^{--+}=\int\frac{c_{13}}{c_{13}^{1+u^{+}_{1}-u_{3}}}\delta(\mathbf{x}_{1}-c_{13}\mathbf{x}_{3})\delta(\mathbf{x}_{2})$

with

$\mathbf{u}^{--+(1)}=\begin{pmatrix}1&2&3&\\\ 3&2^{+}&1^{+}\end{pmatrix}.$

By a second $R$ operation we obtain the completely connected correlator.

$\Phi_{123,2}^{--+}=R_{32}u^{+}_{2}-u^{+}_{1})R_{31}(u_{1}^{+}-u_{3})\Phi_{123,0}^{--+}=\int\frac{dc_{13}}{c_{13}^{1+u^{+}_{1}-u_{3}}}\frac{dc_{23}}{c_{23}^{1+u^{+}_{2}-u^{+}_{1}}}\delta(\mathbf{x}_{1}-c_{13}\mathbf{x}_{3})\delta(\mathbf{x}_{2}-c_{23}\mathbf{x}_{3})$
(3.32)

with

$\mathbf{u}^{--+(2)}=\begin{pmatrix}1&2&3&\\\ 3&1^{+}&2^{+}\end{pmatrix}.$
(3.33)

For the other case we have

$\Phi_{123,1}^{-++}=R_{31}(u_{1}^{+}-u_{3})\delta(\mathbf{x}_{1})=\int\frac{dc_{13}}{c_{13}^{1+u_{1}^{+}-u_{3}}}\delta(\mathbf{x}_{1}-c_{13}\mathbf{x}_{3})$

with

$\mathbf{u}^{-++(1)}=\begin{pmatrix}1&2&3&\\\ 3&2&1^{+}\end{pmatrix}.$

The completely connected correlator is obtained in the next step as

$\Phi_{123,2}^{-++}=R_{21}(u_{3}-u_{2})R_{31}(u_{1}^{+}-u_{3})\delta(\mathbf{x}_{1})=\int\frac{dc_{13}}{c_{13}^{1+u_{1}^{+}-u_{3}}}\frac{dc_{12}}{c_{12}^{1+u_{3}-u_{2}}}\delta(\mathbf{x}_{1}-c_{12}\mathbf{x}_{2}-c_{13}\mathbf{x}_{3})$
(3.34)

with

$\mathbf{u}^{-++(2)}=\begin{pmatrix}1&2&3&\\\ 2&3&1^{+}\end{pmatrix}.$ (3.35)

####  3.5.2 Four points

We illustrate the construction of a connected 4-point YSC in two ways, first
by R operations on the product of 2-point YSC and second by the convolution of
two 3-point YSC followed by a R operation.

We start from the product of 2-point YSC,

$\Phi_{1234,2}=\Phi_{12}\Phi_{34}=R_{21}(u_{1}^{+}-u_{2})R_{43}(u_{3}^{+}-u_{4})\delta(\mathbf{x}_{1})\delta(\mathbf{x}_{3})=$
$\int\frac{dc_{12}dc_{34}}{c_{12}^{1+u_{1}^{+}-u_{2}}c_{34}^{1+u_{3}^{+}-u_{4}}}\delta(\mathbf{x}_{1}-c_{12}\mathbf{x}_{2})\delta(\mathbf{x}_{3}-c_{34}\mathbf{x}_{4}).$

The corresponding monodromy is characterised by the set of parameters

$\mathbf{u}^{(2)}=\begin{pmatrix}1&2&3&4\\\ 2&1^{+}&4&3^{+}\end{pmatrix}.$
(3.36)

To obtain a completely connected 4-point YSC we act by $R_{23}R_{32}$. The
appropriate arguments are read off from (3.36). The action by $R_{32}$ results
in

$\Phi_{1234,3}=R_{32}(u_{1}^{+}-u_{4})\Phi_{1234,2}=$
$\int\frac{dc_{23}dc_{12}dc_{34}}{c_{23}^{1+u_{1}^{+}-u_{4}}c_{12}^{1+u_{1}^{+}-u_{2}}c_{34}^{1+u_{3}^{+}-u_{4}}}\delta(\mathbf{x}_{1}-c_{12}\mathbf{x}_{2}-c_{12}c_{23}c_{34}\mathbf{x}_{4})\delta(\mathbf{x}_{3}-c_{34}\mathbf{x}_{4}).$

The substitution $c_{14}=c_{12}c_{23}c_{34}$ leads to

$\Phi_{1234,3}=\int\frac{dc_{23}dc_{12}dc_{34}}{c_{14}^{1+u_{1}^{+}-u_{4}}c_{12}^{1+u_{4}-u_{2}}c_{34}^{1+u_{3}^{+}-u_{1}^{+}}}\delta(\mathbf{x}_{1}-c_{12}\mathbf{x}_{2}-c_{14}\mathbf{x}_{4})\delta(\mathbf{x}_{3}-c_{34}\mathbf{x}_{4}).$

The monodromy parameters are

$\mathbf{u}^{(3)}=\begin{pmatrix}1&2&3&4\\\ 2&4&1^{+}&3^{+}\end{pmatrix}.$
(3.37)

This correlator is still not completely connected. The action by $R_{23}$
leads to the completely connected correlator

$\Phi_{1234,4}=\int\frac{dc_{32}dc_{23}dc_{12}dc_{34}}{c_{32}^{1+u_{2}-u_{3}}c_{14}^{1+u_{1}^{+}-u_{4}}c_{12}^{1+u_{4}-u_{2}}c_{34}^{1+u_{3}^{+}-u_{1}^{+}}}\delta(\mathbf{x}_{1}-c_{12}\mathbf{x}_{2}-c_{14}\mathbf{x}_{4})\delta(\mathbf{x}_{3}-c_{32}\mathbf{x}_{2}-c_{34}\mathbf{x}_{4}),$

with

$\mathbf{u}^{(4)}=\begin{pmatrix}1&3&2&4\\\ 2&4&1^{+}&3^{+}\end{pmatrix}.$
(3.38)

Now we show how the same result is obtained by the convolution method.

We consider the convolution of 3-point YSC (3.34) and (3.32),

$\int
d\mathbf{x}_{0}\Phi_{120,2}^{-++}(\mathbf{u})\Phi_{034,2}^{--+}(\mathbf{v}).$

The condition on the spectral parameters is obtained from (3.19) taking into
account the parameter sets (3.35) and (3.33),

$v_{0}-n=u_{1}-n,u_{0}-n=v_{4}-n.$ (3.39)

Thus we have

$\int\frac{dc_{12}dc_{10}dc_{04}dc_{34}}{c_{12}^{1+v_{4}-u_{2}}c_{10}^{1+u_{1}^{+}-v_{4}}c_{04}^{1+u_{1}-n-v_{4}}c_{34}^{1+u_{2}^{+}-u_{1}^{+}}}\delta(\mathbf{x}_{1}-\widetilde{c}_{12}\mathbf{x}_{2}-\widetilde{c}_{10}c_{04}\mathbf{x}_{4})\delta(\mathbf{x}_{3}-c_{34}\mathbf{x}_{4}).$

The substitution $c_{14}=c_{10}c_{04}$ leads to

$\int\frac{dc_{04}}{c_{04}}\int\frac{dc_{12}dc_{14}dc_{34}}{c_{12}^{1+v_{4}-u_{2}}c_{14}^{1+u_{1}^{+}-v_{4}}c_{34}^{1+u_{2}^{+}-u_{1}^{+}}}\delta(\mathbf{x}_{1}-\widetilde{c}_{12}\mathbf{x}_{2}-\widetilde{c}_{14}\mathbf{x}_{4})\delta(\mathbf{x}_{3}-c_{34}\mathbf{x}_{4}).$

The first integral can be regarded as an irrelevant factor. The result can be
represented as generated directly by three $R$ operators as in the next-to-
last step of $R$ operations above

$R_{32}(u_{1}^{+}-u_{4})R_{21}(u_{1}^{+}-u_{2})R_{43}(u_{3}^{+}-u_{4})\delta(\mathbf{x}_{1})\delta(\mathbf{x}_{3})$

with the substitution $c_{14}=c_{12}c_{23}c_{34}$, i.e. it coincides with
$\Phi_{1234,3}$ and the parameter set with (3.37). As above the completely
connected correlator (3.38) is obtained by the action of $R_{23}$. Thus the
reconstruction of this 4-point YSC by convolution of two 3-point YSC can be
summarized as

$\Phi_{1234,4}=R_{23}(u_{2}-u_{3})\int
d\mathbf{x}_{0}\Phi_{120,2}^{-++}(\mathbf{u})\Phi_{034,2}^{--+}(\mathbf{v})|_{v|u},$
(3.40)

where $v|u$ advises the substitution (3.39).

The points of the correlators seem to be of two types marked by signature
$\pm$. Those $\mathbf{x}_{I}$ entering the initial trivial correlator
$\Phi_{...,0}$ by $\delta(\mathbf{x}_{I})$ we have marked by $-$ and the
others by $+$. In particular, our examples above of 4-point YSC carry the
signature $-+-+$. In completely connected YSC this signature assignment is
inessential. This holds in general, we show this in the case of 4 points.

The standard $-+-+$ form

$\Phi_{1234}^{-+-+}=\int
dc\varphi^{-+-+}(c_{12},c_{14},c_{32},c_{34})\delta(\mathbf{x}_{1}-c_{12}\mathbf{x}_{2}-c_{14}\mathbf{x}_{4})\delta(\mathbf{x}_{3}-c_{32}\mathbf{x}_{2}-c_{34}\mathbf{x}_{4})$

can be rewritten as e.g.

$\Phi_{1234}^{--++}=\int
dc^{\prime}\varphi^{--++}(c^{\prime})\delta(\mathbf{x}_{1}-c^{\prime}_{13}\mathbf{x}_{3}-c^{\prime}_{14}\mathbf{x}_{4})\delta(\mathbf{x}_{2}-c^{\prime}_{23}\mathbf{x}_{3}-c^{\prime}_{24}\mathbf{x}_{4})$

where

$\varphi^{--++}(c^{\prime}_{13},c^{\prime}_{14},c^{\prime}_{23},c^{\prime}_{24})=c_{23}^{\prime
n-4}\varphi^{-+-+}(\frac{c^{\prime}_{13}}{c^{\prime}_{23}},\frac{c^{\prime}_{14}c^{\prime}_{23}-c^{\prime}_{24}c^{\prime}_{13}}{c^{\prime}_{23}},\frac{1}{c^{\prime}_{23}},-\frac{c^{\prime}_{24}}{c^{\prime}_{23}}).$
(3.41)

Therefore the maximally connected correlator (3.38) transforms to the $--++$
form as

$\Phi_{1234,4}^{--++}=\int\frac{dc^{\prime}_{13}dc^{\prime}_{14}dc^{\prime}_{23}dc^{\prime}_{24}}{(c^{\prime}_{13})^{1+u_{4}-u_{2}}(c^{\prime}_{24})^{1+u_{3}^{+}-u_{1}^{+}}(c^{\prime}_{14}c^{\prime}_{23}-c^{\prime}_{24}c^{\prime}_{13})^{1+u_{1}^{+}-u_{4}}}$
(3.42)
$\delta(\mathbf{x}_{1}-c^{\prime}_{13}\mathbf{x}_{3}-c^{\prime}_{14}\mathbf{x}_{4})\
\delta(\mathbf{x}_{2}-c^{\prime}_{23}\mathbf{x}_{3}-c^{\prime}_{24}\mathbf{x}_{4})$

Both expressions $\Phi_{1234,4}^{-+-+}$ and $\Phi_{1234,4}^{--++}$ are
equivalent, in particular they obey the YSC condition (3.17) with the same
monodromy of the parameters (3.38). The latter expression (3.42) can be
obtained directly from
$\Phi_{1234,0}^{--++}=\delta(\mathbf{x}_{1})\delta(\mathbf{x}_{2})$ by $R$
operations

$\Phi_{1234,4}^{--++}=R_{12}(v_{1}-v_{2})R_{21}(v_{4}-v_{3})R_{32}(v_{2}^{+}-v_{3})R_{41}(v_{1}^{+}-v_{4})\
\delta(\mathbf{x}_{1})\delta(\mathbf{x}_{2})$ (3.43)

and corresponds to the monodromy with the parameters $\mathbf{v}$ abbreviated
by the pattern

$\mathbf{v}^{(4)}=\begin{pmatrix}2&1&3&4\\\ 3&4&2^{+}&1^{+}\end{pmatrix}.$
(3.44)

The parameters $\mathbf{u}^{(4)}$ and $\mathbf{v}^{(4)}$ are related as
$u_{1}=v_{2},u_{2}=v_{3},u_{3}=v_{1},u_{4}=v_{4}$. Indeed, r.h.s of (3.43)
reads explicitly

$\int\frac{dc_{21}dc_{12}dc_{23}dc_{14}}{c_{21}^{1+v_{1}-v_{2}}c_{12}^{1+v_{4}-v_{3}}c_{23}^{1+v_{1}^{+}-v_{3}}c_{14}^{1+v_{1}^{+}-v_{4}}}$
$\delta(\mathbf{x}_{1}-c_{12}c_{23}\mathbf{x}_{3}-c_{14}\mathbf{x}_{4})\delta(\mathbf{x}_{2}-(1+c_{12}c_{21}+1)c_{23}\mathbf{x}_{3}-c_{21}c_{14}\mathbf{x}_{4})$

The transformation to the standard $--++$ form (3.41 results in

$\Phi_{1234,4}^{--++}=\int\frac{dc_{13}^{\prime}dc_{14}^{\prime}dc_{23}^{\prime}dc_{24}^{\prime}}{c_{13}^{\prime
1+v_{4}-v_{3}}c_{24}^{\prime
1+v_{1}-v_{2}}(c^{\prime}_{14}c^{\prime}_{23}-c^{\prime}_{24}c^{\prime}_{13})^{1+v_{2}^{+}-v_{4}}}$
(3.45)
$\delta(\mathbf{x}_{1}-c^{\prime}_{13}\mathbf{x}_{3}-c^{\prime}_{14}\mathbf{x}_{4})\
\delta(\mathbf{x}_{2}-c^{\prime}_{23}\mathbf{x}_{3}-c^{\prime}_{24}\mathbf{x}_{4})$

with the monodromy parameters (3.44). We shall use $\Phi_{1234;4}^{--++}$
(3.43) and the intermediate YSC of this R operation construction
$\Phi_{1234;2}^{--++}$ and $\Phi_{1234;3}^{--++}$.

####  3.5.3 Crossing

We recall that the parameters $u^{+}_{I},u_{I}$ are related to the dilatation
weight of the YSC at the point $I$ as $2\ell_{I}=u_{I}^{+}-u_{I}$. The set of
weights in a correlator $\Phi_{K,N}$ obeys the constraint

$\sum_{I=1}^{N}2\ell_{I}+nK=0.$ (3.46)

The weights of the 4-point YSC $\Phi_{1234,4}$ constructed above are more
restricted,

$2\ell_{1}+2\ell_{3}+n=0,\ \ 2\ell_{2}+2\ell_{4}+n=0.$ (3.47)

We intend to construct a 4-point YSC where such constraints connect the points
$1,4$ and $2,3$ instead.

For this we apply additional $R$ operators to the YSC (3.44), (3.45),

$R_{43}(v_{2}-v_{1})R_{34}(v_{3}-v_{4})\Phi_{1234;4}^{--++4}.$

The action affects the arguments of the delta distributions (3.45),

$\delta(\mathbf{x}_{1}-(c_{13}-c_{14}\bar{c}_{42})\mathbf{x}_{3}-(c_{14}(1+\bar{c}_{43}\bar{c}_{34})-c_{13}\bar{c}_{34})\mathbf{x}_{4})\
\delta(\mathbf{x}_{2}-(c_{23}-c_{24}\bar{c}_{42})\mathbf{x}_{3}-(c_{24}(1+\bar{c}_{43}\bar{c}_{34})-c_{23}\bar{c}_{34})\mathbf{x}_{4}).$

The standard form (3.26) is recovered by the transformation of the variables
$c_{I3},c_{I4}$ to $\widetilde{c}_{I3},\widetilde{c}_{I4}$ ($I=1,2$ ),

$\begin{pmatrix}\widetilde{c}_{I3}\\\ \widetilde{c}_{I4}\\\
\end{pmatrix}=\begin{pmatrix}1&-\bar{c}_{43}\\\
-\bar{c}_{34}&1+\bar{c}_{43}\bar{c}_{34}\\\
\end{pmatrix}\begin{pmatrix}c_{I3}\\\ c_{I4}\\\ \end{pmatrix},$

by the matrix depending on the variables $\bar{c}_{34},\bar{c}_{43}$,
corresponding to the additional $R_{43},R_{34}$ respectively.

We transform the correlator integrand $\varphi(c)$ of $\Phi_{1234,4}^{--++}$
(3.45) to the new variables

$(\varphi(c)^{-1}\cdot\bar{c}_{34}^{1+v_{2}-v_{1}}\bar{c}_{43}^{1+v_{3}-v_{4}}=((1+\bar{c}_{43}\bar{c}_{34})\widetilde{c}_{13}\bar{c}_{43}\widetilde{c}_{14})^{1+v_{4}-v_{3}}\
(\bar{c}_{34}\widetilde{c}_{23}+\widetilde{c}_{24})^{1+v_{1}-v_{2}}$
$(\widetilde{c}_{13}\widetilde{c}_{24}-\widetilde{c}_{23}\widetilde{c}_{14})^{1+v_{2}^{+}-v_{4}}\cdot\bar{c}_{34}^{1+v_{2}-v_{1}}\bar{c}_{43}^{1+v_{3}-v_{4}}=$
$(\widetilde{c}_{13}\widetilde{c}_{24}-\widetilde{c}_{23}\widetilde{c}_{14})^{1+v_{2}^{+}-v_{4}}\bar{c}_{34}^{2}(\widetilde{c}_{23}+\frac{\widetilde{c}_{24}}{\bar{c}_{34}})^{1+v_{1}-v_{2}}\
\
\bar{c}_{43}^{2}(\widetilde{c}_{14}+\frac{\widetilde{c}_{13}}{\bar{c}_{43}}(1+\bar{c}_{43}\bar{c}_{34}))^{1+v_{4}-v_{3}}.$

We do the integrals over $\bar{c}_{43},\bar{c}_{34}$.

$\int\frac{d\bar{c}_{43}}{\bar{c}_{43}^{2}(\widetilde{c}_{14}+\frac{\widetilde{c}_{13}}{\bar{c}_{43}}(1+\bar{c}_{43}\bar{c}_{34}))^{1+v_{4}-v_{3}}}=\widetilde{c}_{14}^{-1-v_{4}+v_{3}}\int\frac{dc^{\prime}_{43}}{(1+\frac{\widetilde{c}_{13}}{\widetilde{c}_{14}}\bar{c}_{34}+c^{\prime}_{34}\frac{\widetilde{c}_{13}}{\widetilde{c}_{14}})^{1+v_{4}-v_{3}}}.$

We have substituted $c^{\prime}_{43}=\bar{c}_{43}^{-1}$ and in the next step
we substitute
$c_{43}^{\prime\prime}=c^{\prime}_{43}\frac{\widetilde{c}_{13}}{\widetilde{c}_{14}}+1+\bar{c}_{34}\frac{\widetilde{c}_{13}}{\widetilde{c}_{14}}$
and obtain

$\frac{1}{\widetilde{c}_{13}\widetilde{c}_{14}^{v_{4}-v_{3}}}\int\frac{c_{43}^{\prime\prime}}{c_{43}^{\prime\prime
1+v_{4}-v_{3}}}.$

Similarly, we substitute
$c_{34}^{\prime}=\frac{\widetilde{c}_{24}}{\widetilde{c}_{23}\bar{c}_{34}}$,

$\int\frac{d\bar{c}_{34}}{\bar{c}_{34}^{2}(\widetilde{c}_{23}+\frac{\widetilde{c}_{24}}{\bar{c}_{34}})^{1+v_{1}-v_{2}}}=\frac{1}{\widetilde{c}_{24}\widetilde{c}_{23}^{v_{1}-v_{2}}}\int\frac{c_{34}^{\prime}}{c_{43}^{\prime
1+v_{1}-v_{2}}}.$

The integrals over $c_{43}^{\prime\prime},c_{34}^{\prime}$ result in undefined
but irrelevant factors. As the result the correlator
$R_{43}(v_{2}-v_{1})R_{34}(v_{3}-v_{4})\Phi^{--++,4}$ has the standard form
(3.26) with the integrand $\widetilde{\varphi}$,

$(\widetilde{\varphi}(c))^{-1}=c_{13}c_{14}^{v_{4}-v_{3}}c_{24}c_{23}^{v_{1}-v_{2}}(c_{23}c_{14}-c_{13}c_{24})^{1+v_{2}^{+}-v_{4}},$
(3.48)

and the parameters of the $L$ operators in the monodromy
$L_{1}L_{2}L_{3}L_{4}$ are abbreviated by

$\mathbf{v}^{(4)c}=\begin{pmatrix}2&1&4&3\\\ 3&4&1^{+}&2^{+}\end{pmatrix}.$
(3.49)

Now the weight restriction reads

$2\ell_{1}+2\ell_{4}+n=0,\ \ \ 2\ell_{2}+2\ell_{3}+n=0.$ (3.50)

If one would like to have the YSC with the related weights to be adjacent the
cyclicity relation (3.20) is to be applied. The same correlator obeys the YSC
relation with the monodromy $L_{2}L_{3}L_{4}L_{1}$ where the parameter
arguments are read off from

$\mathbf{v}^{(4)cc}=\begin{pmatrix}1&4&3&2-n\\\
4&1^{+}&2^{+}&3-n\end{pmatrix}.$ (3.51)

Here $-n$ in the last column means that the last factor with the explicit
arguments is $L_{1}(v_{3}-n,v_{2}-n)$.

###  3.6 Integral operators from symmetric correlators

We consider integral operators with YSC as kernels. Such operators act in a
highly symmetric way, defining homomorphisms not only of the $g\ell(n)$ Lie
algebra action on the functions (global symmetry) but of the related Yangian
algebra. Let us illustrate this in the case of operators on functions of two
points with 4-point YSC as kernels [79].

We define the operator $Q$ in action on $\psi(\mathbf{x}_{1},\mathbf{x}_{2})$
as

$Q\psi(\mathbf{x}_{1},\mathbf{x}_{2})=\int
d\mathbf{x}^{\prime}_{1}d\mathbf{x}^{\prime}_{2}\psi(\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2})\Phi(\mathbf{x}^{\prime}_{2},\mathbf{x}^{\prime}_{1},\mathbf{x}_{2},\mathbf{x}_{1})$
(3.52)

with the YSC condition (3.17) ( relabelling $1234\to 2^{\prime}1^{\prime}21$)
on the kernel

$L_{2^{\prime}}(u_{2}^{\prime+},u_{2}^{\prime})L_{1^{\prime}}(u_{1}^{\prime+},u_{1}^{\prime})L_{2}(u_{2}^{+},u_{2})L_{1}(u_{1}^{+},u_{1})\Phi(\mathbf{x}^{\prime}_{2},\mathbf{x}^{\prime}_{1},\mathbf{x}_{2},\mathbf{x}_{1})=E(\mathbf{u})\Phi(\mathbf{x}^{\prime}_{2},\mathbf{x}^{\prime}_{1},\mathbf{x}_{2},\mathbf{x}_{1}).$

The involved monodromy has the parameter pattern

$\mathbf{u}^{\prime}=\begin{pmatrix}2^{\prime}&1^{\prime}&2&1\\\
2^{\prime+}&1^{\prime+}&2^{+}&1^{+}\end{pmatrix}.$ (3.53)

We show that $Q$ obeys

$L_{2}(u_{2}^{+},u_{2})L_{1}(u_{1}^{+},u_{1})Q=F((\mathbf{u}^{\prime})QL_{1}(u_{1}^{\prime}-n,u_{1}^{\prime+})L_{2}(u_{2}^{\prime}-n,u_{2}^{\prime+}),$
(3.54)
$F(\mathbf{u}^{\prime})=\frac{E(\mathbf{u}^{\prime})}{u_{1}^{\prime}(u_{1}^{\prime+}+1)u_{2}^{\prime}(u_{2}^{\prime+}+1)}.$

Indeed by the YSC condition on the kernel,

$L_{2}(u_{2}^{+},u_{2})L_{1}(u_{1}^{+},u_{1})Q=E(\mathbf{u})\int
d\mathbf{x}^{\prime}_{1}d\mathbf{x}^{\prime}_{2}\psi(\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2})L^{-1}_{1^{\prime}}(u_{1}^{\prime+},u_{1}^{\prime})L^{-1}_{2^{\prime}}(u_{2}^{\prime+},u_{2}^{\prime})\Phi(\mathbf{x}^{\prime}_{2},\mathbf{x}^{\prime}_{1},\mathbf{x}_{2},\mathbf{x}_{1}).$

We integrate by parts, assuming that the integration region is such that no
boundary contributions appear and the relations (3.11, 3.14) apply.

$E(\mathbf{u}^{\prime})\int
d\mathbf{x}^{\prime}_{1}d\mathbf{x}^{\prime}_{2}(L^{-1}_{1^{\prime}}(u_{1}^{\prime+},u_{1}^{\prime}))^{T}(L^{-1}_{2^{\prime}}(u_{2}^{\prime+},u_{2}^{\prime}))^{T}\psi(\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2})\Phi(\mathbf{x}^{\prime}_{2},\mathbf{x}^{\prime}_{1},\mathbf{x}_{2},\mathbf{x}_{1})=$
$\frac{E(\mathbf{u}^{\prime})}{u_{1}^{\prime}(u_{1}^{\prime+}+1)u_{2}^{\prime}(u_{2}^{\prime+}+1)}\int
d\mathbf{x}^{\prime}_{1}d\mathbf{x}^{\prime}_{2}L_{1^{\prime}}(u_{1}^{\prime}-n,u_{1}^{\prime+})L_{2^{\prime}}(u_{2}^{\prime}-n,u_{2}^{\prime+})\psi(\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2})\Phi(\mathbf{x}^{\prime}_{2},\mathbf{x}^{\prime}_{1},\mathbf{x}_{2},\mathbf{x}_{1}).$

The relation proven for arbitrary functions implies the operator relation
(3.54).

If $\Phi_{2^{\prime}1^{\prime}21}$ in (3.52 ) is constructed by $R$ operations
from
$\Phi_{2^{\prime}1^{\prime}21;0}=\delta(\mathbf{x}^{\prime}_{1})\delta(\mathbf{x}^{\prime}_{2})$
then its parameters $\mathbf{v}$ are permutations of

$\mathbf{v}^{(0)}=\begin{pmatrix}2^{\prime}&1^{\prime}&2&1\\\
2^{\prime+}&1^{\prime+}&2&1\end{pmatrix}.$

and
$E(\mathbf{u})=E(\mathbf{v}^{(0)})=v_{1}^{\prime}v_{2}^{\prime}(v_{1}+1)(v_{2}+1)$.

####  3.6.1  The point permutation operator

In the case of $\Delta=\Phi_{2^{\prime}1^{\prime}21;2}^{--++}$ (3.43) we have

$\Delta(\mathbf{x}_{2}^{\prime},\mathbf{x}_{1}^{\prime},\mathbf{x}_{2},\mathbf{x}_{1})=R_{22^{\prime}}(v_{1^{\prime}}^{+}-v_{2})R_{12^{\prime}}(v_{2^{\prime}}^{+}-v_{1})\delta(\mathbf{x}^{\prime}_{1})\delta(\mathbf{x}^{\prime}_{2}),$
$\mathbf{v}^{(2)}=\begin{pmatrix}2^{\prime}&1^{\prime}&2&1\\\
1&2&1^{\prime}&2^{\prime}\end{pmatrix}.$

$\mathbf{u}^{\prime}=\mathbf{v}^{(2)}$ means
$v_{1^{\prime}}=u_{1^{\prime}},v_{2^{\prime}}=u_{2^{\prime}},v_{1}=u_{1},v_{2}=u_{2}$,
$u_{2^{\prime}}^{+}=v_{1},u_{1^{\prime}}^{+}=v_{2},u_{2}^{+}={v_{2^{\prime}}}^{\prime},u_{1}^{+}={v_{2^{\prime}}}^{+}$
and this implies $F(\mathbf{u}^{\prime})=1$. We obtain as the specification of
(3.54) with the kernel in (3.52) substituted by $\Delta$ and
$\mathbf{u}^{\prime}=\mathbf{v}^{(2)}$ that the corresponding operator
$\hat{\Delta}$ defined with this kernel obeys

$\hat{\Delta}L_{1}(v_{1^{\prime}}-n,v_{2})L_{2}(v_{2^{\prime}}-n,v_{1})=L_{2}(v_{1^{\prime}}-n,v_{2})L_{1}(v_{2^{\prime}}-n,v_{1})\hat{\Delta}.$
(3.55)

Thus choosing $\Delta=\Phi_{2^{\prime}1^{\prime}21;2}^{--++}$ (3.43) as the
kernel in (3.52) we obtain the representation of the operator of permutation
$P_{12}$ acting as
$P_{12}({x}_{1,a},\partial_{1,a})P_{12}=({x}_{2,a},\partial_{2,a})$.

####  3.6.2  The parameter pair permutation operators

We show that the kernel
$R^{++}_{1^{\prime}2^{\prime}}(v_{1}-v_{2})\Delta=\Phi^{--++}_{2^{\prime}1^{\prime}21;3}$
defines by (3.52) the operator $\hat{R}_{12}=P_{12}R_{12}$, where $R_{12}$
obeys (3.24). Indeed, the parameter permutation pattern is

$\mathbf{v}^{(3)}=\begin{pmatrix}2^{\prime}&1^{\prime}&2&1\\\
2&1&{1^{\prime}}^{+}&{2^{\prime}}^{+}\end{pmatrix}.$

We see that $F(\mathbf{u}^{\prime})=1$ and the relation (3.54) specifies with
$\mathbf{u}^{\prime}=\mathbf{v}^{(3)}$ to

$\hat{R}_{12}(v_{1}-v_{2})L_{1}(v_{1^{\prime}}^{+},v_{1})L_{2}(v_{1^{\prime}}^{+},v_{2})=L_{2}(v_{1^{\prime}}^{+},v_{2})L_{1}(v_{2^{\prime}}^{+},v_{1})\hat{R}_{12}(v_{1}-v_{2}).$
(3.56)

The relation is to be compared with (3.24). The parameters $v_{1},v_{2}$ and
the underlying canonical pairs are permuted. and the parameters
$\widetilde{v}_{1}^{+}=v_{2^{\prime}}^{+}$ and
$\widetilde{v}_{2}^{+}=v_{1^{\prime}}^{+}$ remain.

Next we show that the kernel
$R_{2^{\prime}1^{\prime}}(v_{2^{\prime}}-v_{1^{\prime}})\Delta$ defines by
(3.52) the operator $\hat{R}_{21}=P_{12}R_{21}$, where $R_{21}$ obeys (3.25).
Indeed, the permutation pattern is

$\mathbf{v}^{(3^{\prime})}=\begin{pmatrix}1^{\prime}&2^{\prime}&2&1\\\
1&2&{1^{\prime}}^{+}&{2^{\prime}}^{+}\end{pmatrix}.$

Again $F(\mathbf{u}^{\prime})=1$. The relation (3.54) specifies with
$\mathbf{u}^{\prime}=\mathbf{v}^{(3^{\prime})}$ to

$\hat{R}_{21}(v_{2^{\prime}}-v_{1^{\prime}})L_{1}(v_{2^{\prime}}^{+},v_{2})L_{2}(v_{2^{\prime}}^{+},v_{1})=L_{2}(v_{1^{\prime}}^{+},v_{2})L_{1}(v_{2^{\prime}}^{+},v_{1})\hat{R}_{21}(v_{2^{\prime}}-v_{1^{\prime}}).$
(3.57)

This relation is to be compared with (3.25). The parameters
$\widetilde{v}_{1}^{+}=v_{2^{\prime}}^{+},\widetilde{v}_{2}^{+}=v_{1^{\prime}}^{+}$
are permuted together with the underlying canonical pairs and the parameters
$v_{1},v_{2}$ remain.

####  3.6.3  The complete Yang-Baxter operator

The kernel
$R_{2^{\prime}1^{\prime}}(v_{2^{\prime}}-v_{1^{\prime}})R_{1^{\prime}2^{\prime}}(v_{1}-v_{2})\Delta=\Phi^{--++}_{2^{\prime}1^{\prime}21;4}$
(3.43) defines by (3.52) the complete Yang-Baxter operator $\hat{\hbox{\bbd
R}}_{12}$. Indeed, the permutation pattern of parameters is

$\mathbf{v}^{(4)}=\begin{pmatrix}1^{\prime}&2^{\prime}&2&1\\\
2&1&{1^{\prime}}^{+}&{2^{\prime}}^{+}\end{pmatrix}.$

We find $F(\mathbf{u}^{\prime})=1$ and the relation (3.54) results with
$\mathbf{u}^{\prime}=\mathbf{v}^{(4)}$ in

$\hat{\hbox{\bbd
R}}_{12}(\widetilde{v}_{1}^{+}-\widetilde{v}_{2}^{+},v_{1}-v_{2})L_{1}(\widetilde{v}_{1}^{+},v_{1})L_{2}(\widetilde{v}_{2}^{+},v_{2})=L_{2}(\widetilde{v}_{2}^{+},v_{2})L_{1}(\widetilde{v}_{1}^{+},v_{1})\hat{\hbox{\bbd
R}}_{12}(\widetilde{v}_{1}^{+}-\widetilde{v}_{2}^{+},v_{1}-v_{2}).$ (3.58)

where
$\widetilde{v}_{1}^{+}=v_{2^{\prime}}-n,\widetilde{v}_{2}^{+}=v_{1^{\prime}}-n$.
Both parameters and the underlying canonical pairs are permuted. The weights
corresponding to the points of the correlator
$R_{2^{\prime}1^{\prime}}(v_{2^{\prime}}-v_{1^{\prime}})R_{1^{\prime}2^{\prime}}(v_{1}-v_{2})\Delta=\Phi^{--++}_{2^{\prime}1^{\prime}21;4}$
are calculated from $\mathbf{v}^{(4)}$ as
$2\ell_{2^{\prime}}=-2\ell_{2}-n=v_{2}-v_{1^{\prime}}$,
$2\ell_{1^{\prime}}=-2\ell_{1}-n=v_{1}-v_{2^{\prime}}$. The weights in the $L$
operators $L_{1}$ and $L_{2}$ are $\widetilde{v}_{1}^{+}-v_{1}=2\ell_{1}$ and
$\widetilde{v}_{2}^{+}-v_{2}=2\ell_{2}$. The operator $\hbox{\bbd R}_{12}$
depends on these weights and the spectral parameter difference $v_{1}-v_{2}$.

####  3.6.4  The operators of the crossed YSC

We have constructed a completely connected correlator by crossing the former
one (3.48),
$R_{12}(v_{1^{\prime}}^{+}-v_{2^{\prime}}^{+})R_{23}(v_{2}-v_{1})\Phi^{--++}_{2^{\prime}1^{\prime}21;4}$.
Its parameter pattern is obtained from $\mathbf{v}^{(4)}$ (3.44) by permuting
the last two columns and becomes (3.49),

$\mathbf{v}^{(4)c}=\begin{pmatrix}1^{\prime}&2^{\prime}&1&2\\\
2&1&{2^{\prime}}^{+}&{1^{\prime}}^{+}\end{pmatrix}.$

Whereas in $\Phi^{--++}_{2^{\prime}1^{\prime}21;4}$ the weights of the next-to
nearest points are related, $2\ell_{2^{\prime}}+2\ell_{2}+n=0$, we have here
$2\ell_{2^{\prime}}+2\ell_{1}+n=0$. Now the relation (3.54) results with
$\mathbf{u}^{\prime}=\mathbf{v}^{(4)c}$ in

$\hbox{\bbd
Q}L_{1}(v_{2^{\prime}}^{+},v_{1})L_{2}(v_{1^{\prime}}^{+}-v_{2})=L_{2}(v_{2^{\prime}}^{+},v_{1})L_{1}(v_{1^{\prime}}^{+}-v_{2})\hbox{\bbd
Q}$ (3.59)

It compares with the relation (3.55) for $\hat{\Delta}$. Both act on the
monodromy $L_{1}L_{2}$ like the permutation of points $P_{12}$, but ℚ is
depends not only on the weights but additionally on $v_{1}-v_{2}$.

The YSC obtained from the latter by the cyclic permutation has the property
that the weights of the adjacent points are related, because now the sequence
of points is $2^{\prime},1,2,1^{\prime}$. The parameter permutation pattern is

$\mathbf{v}^{(4)cc}=\begin{pmatrix}2^{\prime}&1&2&1^{\prime}-n\\\
1&{2^{\prime}}^{+}&{1^{\prime}}^{+}&2-n\end{pmatrix}.$

Now the relation (3.54) results with $\mathbf{u}^{\prime}=\mathbf{v}^{(4)cc}$
in

$\hbox{\bbd
Q}^{c}L_{1}(v_{1}^{+},v_{2^{\prime}}^{+})L_{2}(v_{2^{\prime}}^{+},v_{1})=L_{2}(v_{1^{\prime}}^{+},v_{2})L_{1}(v_{2}^{+},v_{1^{\prime}}^{+})\hbox{\bbd
Q}^{c}.$ (3.60)

The weights of the left monodromy are related and the weights of the right
monodromy are related as well, but the weights of $L_{1},L_{2}$ on the left
hand side are not related to the weights appearing on the right. Unlike the
previous cases the parameter arguments of the $L$ operators on the left hand
side are not permutations of the ones on the right hand side. The operator
depends besides of the weights additionally on $v_{1}-v_{2}$.

##  4 Application to scattering amplitudes

###  4.1 The three-point YSC and the three-point amplitudes

We consider the completely connected 3-point YSC in the helicity form (3.4),
in particular (3.34).

$\Phi_{123,2}^{\lambda-++}=\int\frac{dc_{13}dc_{12}}{c_{13}^{1+u_{1}^{+}-u_{3}}c_{12}^{1+u_{3}-u_{2}}}\delta^{(m)}(\bar{\lambda}_{1}-c_{12}\bar{\lambda}_{2}-c_{13}\bar{\lambda}_{3})\delta^{(m)}(\lambda_{2}+c_{12}\lambda_{1})\delta^{(m)}(\lambda_{3}+c_{13}\lambda_{3}))$

The monodromy parameters are

$\begin{pmatrix}1&2&3\\\ 2&3&1^{+}\end{pmatrix}$

and result in the weights

$2\ell_{1}=u_{2}-u_{1},2\ell_{2}=u_{3}-u_{1},2\ell_{3}=u_{1}^{+}-u_{3}.$ (4.1)

The conditions in the delta arguments lead to

$\lambda^{1}_{\alpha}\bar{\lambda}^{1}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}+\lambda^{2}_{\alpha}\bar{\lambda}^{2}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}+\lambda^{3}_{\alpha}\bar{\lambda}^{3}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=0.$
(4.2)

Indeed, multiply the condition in the first delta by $\lambda^{1}_{\alpha}$,

$\bar{\lambda}^{1}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\lambda^{1}_{\alpha}-c_{12}\bar{\lambda}^{2}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\lambda^{1}_{\alpha}-c_{13}\bar{\lambda}^{3}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\lambda^{1}_{\alpha}=0$

We use the other deltas to substitute
$c_{12}\lambda^{1}_{\alpha}=\lambda^{2}_{\alpha},c_{13}\lambda^{1}_{\alpha}=\lambda^{3}_{\alpha}$
and obtain the momentum condition (4.2).

One non-trivial projection of the second and third conditions are sufficient
for fixing the integration variables:

$c^{*}_{12}=-\frac{<2\mu>}{<1\mu>},\
c^{*}_{13}=-\frac{<3\mu^{\prime}>}{<1\mu^{\prime}>}.$

We may choose $\mu=|3>$ and $\mu^{\prime}=|2>$ . The result of the integration
over $c_{12},c_{13}$ is

$\left(\frac{<1\mu^{\prime}>}{<3\mu^{\prime}>}\right)^{u_{1}^{+}-u_{3}}\left(\frac{<1\mu>}{<2\mu>}\right)^{u_{3}-u_{2}}$

up to Jacobi factors not depending on the parameters. The latter can be fixed
by comparing the scaling of the expression under
$\lambda_{I}\to\mu_{I}^{-1}\lambda_{I}$ with the scaling law of the YSC in the
helicity form, $\mu_{I}^{2h_{I}}$.

For $m=2$ we obtain, using (4.1) and (3.28),

$\Phi^{\lambda-++}=<12>^{2h_{3}+1}<13>^{2h_{2}+1}<23>^{2h_{1}+1}\delta^{(4)}(\bar{\lambda}^{1}\lambda^{1}+\bar{\lambda}^{2}\lambda^{2}+\bar{\lambda}^{3}\lambda^{3})$

The expression (2.18) is reproduced for the case $\eta=+1$. The case $\eta=-1$
is obtained in the analogous way from $\Phi_{123,2}^{--+}$ (3.32).

Thus we have shown that the 3-point amplitudes are 3-point YSC.

###  4.2  The convolution construction and BCFW

In sect. 3.2.1 and 3.5.2 we have shown in the particular case of three points
that the symmetric convolution of YSC allows to construct further YSC. The
generalization to higher point YSC is obvious. The convolution has been
formulated in terms of the canonical pairs $x_{a},\partial_{a},a=1,...,n$, in
particular the convolution means integration over the coordinates $x_{a}$ of
an identified point of the two correlators. The convolution construction can
be reformulated for YSC in other representation forms, in particular in the
helicity form if $n=2m$. Then the integration is over the helicity coordinates
$\lambda_{\alpha},\bar{\lambda}_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}},\alpha=1,...,m,\stackrel{{\scriptstyle\cdot}}{{\alpha}}=1,...,m$
of the identified point. In the application to scattering amplitudes of QCD
($n=2m=4$) this integration can be rewritten in terms of the on-shell momentum
components of the fused legs
$P_{\alpha\stackrel{{\scriptstyle\cdot}}{{\alpha}}}=\lambda_{\alpha}\
\lambda_{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}$, $d^{4}P\delta(P^{2})$ and
a phase. Thus we obtain a form compatible with the second factor in (2.17).

The energy-momentum delta distribution included in each correlator results in
the overall energy-momentum delta distribution of the resulting correlator.

In sect. 3.5.2 we have seen that the result of the convolution of two
completely connected 3-point YSC is a 4-point YSC, but not a completely
connected one. A particular $R$ operation (3.40) leads to the completely
connected YSC. The $R$ operator can be expressed by a contour integral over a
shift operator (3.22). In the helicity representation it appears by (3.29) as

$R_{IJ}(u)=\int_{\mathcal{C}}\frac{dc}{c^{1+u}}\exp(-c\bar{\lambda}_{J\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\bar{\partial}_{I}^{\stackrel{{\scriptstyle\cdot}}{{\alpha}}}+c\lambda_{I\alpha}\partial_{J}^{\alpha}).$

We observe that the first factor in (2.17) is a particular case of the latter
$R$ operator expression.

From this comparison we conclude that the convolution construction of YSC
supplemented by $R$ operations generalizes the BCFW construction of QCD tree
amplitudes.

###  4.3 4-point YSC and parton tree amplitudes

We consider the four-point YSC at $n=4$ with $K=2$ in the helicity
representation

$\Phi^{--++\lambda}=\int\mathrm{d}^{4}c\,\varphi(c)\,\delta^{(2)}(\bar{\lambda}_{1}-c_{13}\bar{\lambda}_{3}-c_{14}\bar{\lambda}_{4})\delta^{(2)}(\bar{\lambda}_{2}-c_{23}\bar{\lambda}_{3}-c_{24}\bar{\lambda}_{4})\delta^{(2)}(\lambda_{3}+c_{13}\lambda_{1}+c_{23}\lambda_{2})$
(4.3) $\times\delta^{(2)}(\lambda_{4}+c_{14}\lambda_{1}+c_{24}\lambda_{2}).$

and do the integrals over the $c$ variables with the result

$\Phi^{--++\lambda}=\varphi(c^{*})\ \ \delta^{(4)}(\sum k_{i}),$ (4.4)
$c_{13}^{*}=\frac{[14]}{[34]},\ c_{14}^{*}=\frac{[13]}{[43]},\
c_{23}^{*}=\frac{[24]}{[34]},\ c_{24}^{*}=\frac{[23]}{[43]},\
c_{13}^{*}c_{24}^{*}-c_{14}^{*}c_{23}^{*}=\frac{[12]}{[34]}.$

Here we denote
$(k_{i})_{\stackrel{{\scriptstyle\cdot}}{{\alpha}},\alpha}=\bar{\lambda}_{i,\stackrel{{\scriptstyle\cdot}}{{\alpha}}}\lambda_{i,\alpha}$,
$[ij]=\bar{\lambda}_{i,1}\bar{\lambda}_{j,2}-\bar{\lambda}_{i,2}\bar{\lambda}_{j,1}$.

The gluon and quark helicities $h_{i}$ are related to the weights as
$2h_{i}=2\ell_{i}+2$. The YSC $\Phi_{X}$ (3.45) and $\Phi_{||}$ (3.48) depend
on two independent helicities and one extra parameter
$\varepsilon=v_{3}-v_{4}$. In the case of $\Phi_{X}$ the helicities are
related by $h_{1}=-h_{3},\ \ h_{2}=-h_{4}$. With these conditions only 4 of
the 6 helicity configurations of the $2\to 2$ helicity amplitudes are
accessible. With $\Phi_{||}$ we cover the remaining cases. Here the helicities
are related by $h_{1}=-h_{4},\ \ h_{2}=-h_{3}$. With these conditions again 4
helicity configuration cases can be covered, two of them doubling some of the
above correlator.

From (4.4) with $\varphi(c)$ from (3.45) and (3.48) we obtain the explicit
expressions as functions of the independent helicities and the extra parameter
$\varepsilon$,

$\varphi_{X}(c^{*};h_{1},h_{2},\varepsilon)=\left(\frac{[14][23]}{[12][34]}\right)^{\varepsilon}\
\frac{[12]^{1+2h_{1}}[34]^{1-2h_{2}}}{[14][23]^{1+2h_{1}-2h_{2}}},$

$\varphi_{||}(c^{*};h_{1},h_{2},\varepsilon)=\left(\frac{[13][24]}{[12][34]}\right)^{\varepsilon}\
\frac{[12]^{1+2h_{1}}[34]^{1-2h_{2}}}{[14][23][24]^{2h_{1}-2h_{2}}}.$

We observe that the gluon tree amplitudes in spinor-helicity form [45, 75] are
reproduced by these two YSC as

$M(1,1,-1,-1)=\varphi_{X}(c^{*},1,1,0),\ \
M(-1,-1,1,1)=\varphi_{X}(c^{*},-1,-1,0),\ \ $
$M(1,-1,-1,1)=\varphi_{X}(c^{*},1,-1,4),M(-1,1,1,-1)=\varphi_{X}(c^{*},-1,1,0),$

$M(1,1,-1,-1)=\varphi_{||}(c^{*},1,1,0),\ \
M(-1,-1,1,1)=\varphi_{||}(c^{*},-1,-1,0),\ \ $
$M(1,-1,1,-1)=\varphi_{||}(c^{*},1,-1,4),M(-1,1,-1,1)=\varphi_{||}(c^{*},-1,1,0).$

The quark tree amplitudes are reproduced as

$M(\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2})=\varphi_{X}(c^{*},\frac{1}{2},\frac{1}{2},0),$
$M(\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2})=\varphi_{X}(c^{*},\frac{1}{2},-\frac{1}{2},3),\
\
M(-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2})=\varphi_{X}(c^{*},-\frac{1}{2},\frac{1}{2},1),\
\ $
$M(\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2})=\varphi_{||}(c^{*},\frac{1}{2},\frac{1}{2},0),$
$M(\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2})=\varphi_{||}(c^{*},\frac{1}{2},-\frac{1}{2},3),\
\
M(-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2})=\varphi_{||}(c^{*},-\frac{1}{2},\frac{1}{2},1).$

The gluon-quark tree amplitudes are reproduced as

$M(1,\frac{1}{2},-1,-\frac{1}{2})=\varphi_{X}(c^{*},1,\frac{1}{2},1),\ \
M(1,-\frac{1}{2},-1,\frac{1}{2})=\varphi_{X}(c^{*},1,-\frac{1}{2},3),$
$M(1,\frac{1}{2},-\frac{1}{2},-1)=\varphi_{||}(c^{*},1,\frac{1}{2},1),\ \
M(1,-\frac{1}{2},\frac{1}{2},-1)=\varphi_{||}(c^{*},1,-\frac{1}{2},3).$

To obtain the tree amplitudes the monodromy parameters of the YSC are fixed by
the values of the helicities besides of the parameter $\varepsilon$. The
physical criterion for choosing $\varepsilon$ is to allow only single poles in
$s=|<12>|^{2}=|<34>|^{2},t=|<14>|^{2}=|<23>|^{2}$ or
$u=|<13>|^{2}=|<24>|^{2}$.

##  5 Application to the Bjorken asymptotics

###  5.1 Three-point YSC and the parton splitting amplitude

We consider the 3-point YSC (3.34), (3.32) in the case $n=2,m=1$ converted to
the helicity form according to (3.4). We start with (3.34).

$\Phi_{123,2}^{-++,\lambda}=\int\frac{dc_{13}dc_{12}}{c_{13}^{1+u_{1}^{+}-u_{3}}c_{12}^{1+u_{3}-u_{2}}}\delta(\bar{\lambda}_{1}-c_{12}\bar{\lambda}_{2}-c_{13}\bar{\lambda}_{3})\delta(\lambda_{2}+c_{12}\lambda_{1})\delta(\lambda_{3}+c_{13}\lambda_{1}).$

The delta distributions fix the values of $c_{12},c_{13}$.

$\Phi_{123,2}^{-++,\lambda}=\lambda_{1}^{1+u_{1}^{+}-u_{2}}\lambda_{2}^{-1-u_{3}+u-2}\lambda_{3}^{-1-u_{1}+u_{3}}\delta(\lambda_{1}\bar{\lambda}_{1}+\lambda_{2}\bar{\lambda}_{2}+\lambda_{3}\bar{\lambda}_{3}).$

From the monodromy parameters (3.35) we obtain the weights as
$2\ell_{1}=u_{2}-u_{1},2\ell_{2}=u_{3}-u_{2},2\ell_{2}=u_{1}^{+}-u_{3}$. We
recall the relation to the helicity (3.28), (2.30), $2h=1+2\ell$, and express
the exponents in terms of helicities constrained by $2h_{1}+2h_{2}+2h_{3}=+1$,

$\Phi_{123,2}^{-++,\lambda}=\lambda_{1}^{-h_{1}}\lambda_{2}^{-h_{2}}\lambda_{3}^{-h_{3}}\delta(\lambda_{1}\bar{\lambda}_{1}+\lambda_{2}\bar{\lambda}_{2}+\lambda_{3}\bar{\lambda}_{3}).$

We treat (3.32) in analogy

$\Phi_{123,2}^{--+,\lambda}=\int\frac{dc_{13}dc_{23}}{c_{13}^{1+u_{1}^{+}-u_{3}}c_{23}^{1+u_{2}^{+}-u_{1}^{+}}}\delta(\bar{\lambda}_{1}-c_{13}\bar{\lambda}_{3})\delta(\bar{\lambda}_{1}-c_{12}\bar{\lambda}_{2})\delta(\lambda_{3}++c_{13}\lambda_{1}+c_{23}\lambda_{2})$

and obtain

$\Phi_{123,2}^{--+,\lambda}=\bar{\lambda}_{1}^{h_{1}}\bar{\lambda}_{2}^{h_{2}}\bar{\lambda}_{3}^{h_{3}}\delta(\lambda_{1}\bar{\lambda}_{1}+\lambda_{2}\bar{\lambda}_{2}+\lambda_{3}\bar{\lambda}_{3}).$

The monodromy parameters of this correlator lead to the constraint on the
weights $2\ell_{1}+2\ell_{2}+2\ell_{3}+4=0$ and therefore the helicity
parameters obey $2h_{1}+2h_{2}+2h_{3}=-1$.

If $k_{I}=\lambda_{I}\bar{\lambda}_{I}$ represent light-cone components of
momenta and therefore take real values, we rewrite both results by separating
the phase and the momentum conservation factors as

$\Phi_{123,2}^{\lambda}=e^{i\alpha}M(k_{1},k_{2},k_{3},h_{1},h_{2},h_{3})\delta(k_{1}+k_{2}+k_{3}).$

With the notation

$2h_{1}+2h_{2}+2h_{3}=\eta,\ \ \ \eta=\pm 1$ (5.5)

we write for both cases

$M(k_{1},k_{2},k_{3},h_{1},h_{2},h_{3})=k_{1}^{-h_{1}\eta}k_{2}^{-h_{2}\eta}k_{3}^{-h_{3}\eta}.$

We define the collinear parton triple vertex differing by a factor related to
the propagators in the convolutions.

$M_{3}(k_{1},k_{2},k_{3};h_{1},h_{2},h_{3})=(k_{1}k_{2}k_{3})^{\frac{1}{2}}M(k_{1},k_{2},k_{3},h_{1},h_{2},h_{3})=k_{1}^{\frac{1}{2}-h_{1}\eta}k_{2}^{\frac{1}{2}-h_{2}\eta}k_{3}^{\frac{1}{2}-h_{3}\eta}.$
(5.6)

In the case of the scattering amplitudes, sect. 4, the helicity parameters in
the related YSC are identified with the parton helicities (up to sign for the
all-ingoing convention). There the YSC helicities obey in the case of 3 points
$2h_{1}+2h_{3}+2h_{3}=2\eta$, because $n=2m=4$. Here $n=2m=2$ and therefore
the helicity parameters obey (5.5).

We observe that the collinear parton triple vertex coincides with the parton
splitting amplitude (2.28), resulting in the parton splitting probability
(2.29) $A\to BC$ by substituting for $B,C$ the momentum fractions $z,1-z$ and
the physical parton helicities with signs regarding the all-ingoing
convention.

$W_{ABC}(z)=|M_{3}(1,-z,z-1;h_{B}+h_{B}-\frac{1}{2}\eta,-h_{B},-h_{C})|^{2}.$

For comparison we write some results for the splitting kernels (up to
contributions proportional to $\delta(1-z)$) as calculated from the splitting
amplitude by (2.29). Recall that the labels at $W_{AB}$ refer to the physical
helicities of the involved quarks ($\pm\frac{1}{2}$) and gluons ($\pm 1$).

$W_{-1,-1}(z)=W_{+1,+1}(z)=W_{+1,+1,-1}(z)+W_{+1,+1,+1}(z)=\frac{1}{z(1-z)}+\frac{z^{3}}{(1-z)},$
$W_{-\frac{1}{2},-\frac{1}{2}}(z)=W_{+\frac{1}{2},+\frac{1}{2}}(z)=\frac{z}{1-z},$
(5.7) $W_{-1,1}=W_{+1,-1}(z)=\frac{(1-z)^{3}}{z},\ \ \
W_{1,-\frac{1}{2}}(z)=(1-z)^{2},\ \ \ W_{1,+\frac{1}{2}}(z)=z^{2}.$

###  5.2 Four-point YSC and $RLL$ relations

We have shown in sec. 3.6 that four-point YSC define as kernels operators
obeying RLL type relations involving the products of two $L$ operators. In
particular, the integral operator with the YSC (3.45) obeys the Yang-Baxter
relation (3.21), (3.58). Here we specify the $L$ operators to the case
$n=2m=2$ and use properties derived from the explicit form in normal
coordinates.

The reduction of the $L$ operators to the subspace of eigenvalue $2\ell$ is
done for the action on functions of the coordinate components by the
restriction to a definite degree of homogeneity. We specify (3.4), (3.12) to
the present case.

$\psi_{2\ell}(\mathbf{x})=(x_{2})^{2\ell}\phi(x),\ \
L^{+}(u)(x_{2})^{2\ell}\phi(x)=(x_{2})^{2\ell}L_{x}(u^{+}+1,u+1)\phi(x),$
$\mathbf{x}=(x_{1},x_{2}),\ \ \ x=\frac{x_{1}}{x_{2}}.$ (5.8)

We change to the canonical pair $x,\partial$, where $x$ is the normal
coordinate ratio and $\partial$ the corresponding derivative operator, using
$x_{1}\partial_{1}\to x\partial,\
x_{2}\partial_{2}=(\mathbf{x}\mathbf{p})-x_{1}\partial_{1}\to
2\ell-x\partial$. Explicitly we have

$L^{+}(u)=\begin{pmatrix}u+1+x_{1}\partial_{1}&x_{2}\partial_{1}\\\
x_{1}\partial_{2}&u+1+x_{2}\partial_{2}\end{pmatrix},\ $
$L_{x}(u^{+}+1,u+1)=\begin{pmatrix}u+1+x\partial&\partial\\\
x(-x\partial+2\ell)&u+1+2\ell-x\partial\end{pmatrix},$

where $u^{+}=u+2\ell$ as above. A more symmetric definition of the parameters
is to substitute $v=u+\ell$ and to write

$L_{x}(u^{+},u)=L(v^{(1)},v^{(2)}),\ \ v^{(1)}=v+\ell,v^{(2)}=v-\ell-1.$ (5.9)

The result for $L$ compares with (2.31),
$L_{ab}(v^{(1)},v^{(2)})=v\delta_{ab}+M_{ab}$. The $L$ matrix and the
$s\ell(2)$ generators are related as

$L(v^{(1)},v^{(2)})=\begin{pmatrix}v+S^{0}&S^{-}\\\
S^{+}&v-S^{0}\end{pmatrix},$ $S^{-}=\partial,\ \ S^{+}=x(-x\partial+2\ell),\ \
S^{0}=x\partial-\ell.$ (5.10)

We continue with the formulation in terms of the normal coordinates.
Considering more than one point we label now by the index at $x$ the point
$I$, i.e. in the following $x_{1},x_{2}$ are not the components of
$\mathbf{x}$ as in (5.8). Let $\psi(x_{1},x_{2})$ obey the eigenvalue relation

$\mathrm{R}_{21}(w)\psi(x_{1},x_{2})=\lambda(w)\psi(x_{1},x_{2}).$

The Yang-Baxter relation (3.21) implies

$\mathrm{R}_{21}(w)L_{2}(v)L_{1}(v-w)\psi(x_{1},x_{2})=\lambda(w)L_{1}(v-w)L_{2}(v)\psi(x_{1},x_{2}).$
(5.11)

We recall the known fact that the RLL relation implies the eigenvalues of the
operator $R_{21}$. For this we do not use its integral operator form, which
will be written explicitly in the next subsection.

We abbreviate the relation (5.11) as
$\mathrm{R}\mathcal{L}\psi=\lambda\mathcal{R}\psi$ and expand in powers of
$v$. At $v^{2}$ we have a trivial relation, at $v^{1}$ we find: If $\psi$ is
an eigenfunction then it is also $(S_{1}^{a}+S_{2}^{a})\psi$ with the same
eigenvalue, i.e. we have $s\ell(2)$ irreducible representation subspaces of
degeneracy. The conditions at $v^{0}$ are related by this global symmetry.
Therefore it is sufficient to analyze the condition related to one matrix
element, e.g. $12$.

$\mathcal{R}_{12}=(S_{1}^{0}-\delta)S_{2}^{-}-S_{1}^{-}S_{2}^{0},\qquad\mathcal{L}_{12}=S_{2}^{0}S_{1}^{-}-S_{2}^{-}(S_{1}^{0}+\delta).$

We consider the function $\psi_{n}^{(0)}=(x_{1}-x_{2})^{n}$ being a lowest
weight vector

$(S_{1}^{-}+S_{2}^{-})\psi_{n}^{(0)}=0,\quad(S_{1}^{0}+S_{2}^{0})\psi_{n}^{(0)}=-\mu_{n}\psi_{n}^{(0)},\quad\mu_{n}=-\ell_{1}-\ell_{2}+n.$

We calculate

$\mathcal{R}_{12}\psi_{n}^{(0)}=\frac{1}{2}\mu_{n}\mathrm{S}^{-}\psi_{n}^{(0)}+\frac{1}{2}(1+\delta)\mathrm{S}^{-}\psi_{n}^{(0)},\quad\mathcal{L}_{12}\psi_{n}^{(0)}=-\frac{1}{2}\mu_{n}\mathrm{S}^{-}\psi_{n}^{(0)}-\frac{1}{2}(1-\delta)\mathrm{S}^{-}\psi_{n}^{(0)},$

where $\mathrm{S}^{a}=S_{1}^{a}-S_{2}^{a}$. We obtain by (5.11)

$\lambda_{n}[\mu_{n}+1+\delta]\,\mathrm{S}^{-}\psi_{n}^{(0)}=[-\mu_{n}+\delta-1]\,\mathrm{R}_{21}\,\mathrm{S}^{-}\psi_{n}^{(0)}.$

We have marked the eigenvalue of the representation generated from the lowest
weight vector $\psi_{n}^{(0)}$ by the index $n$. We find that also
$\mathrm{S}^{-}\psi_{n}^{(0)}$ is an eigenfunction of $\mathrm{R}_{21}$ with
another eigenvalue,

$\mathrm{R}_{21}\
\mathrm{S}^{-}\psi_{n}^{(0)}=\lambda_{n-1}\mathrm{S}^{-}\psi_{n}^{(0)}.$

By comparison we obtain a recurrence relation for $\lambda_{n}$. It is easy to
see that the constant function is an eigenfunction and moreover a lowest
weight vector. Thus as the result of the analysis we find all eigenfunctions
and eigenvalues. The latter obey the iterative relation

$\lambda_{n+1}=-\lambda_{n}\frac{n-\ell_{1}-\ell_{2}+w}{n-\ell_{1}-\ell_{2}-w}$

solved by

$\lambda_{n}(w)=(-1)^{n}\frac{\Gamma(n-\ell_{1}-\ell_{2}+w)}{\Gamma(n-\ell_{1}-\ell_{2}-w)}\frac{\Gamma(-\ell_{1}-\ell_{2}-w)}{\Gamma(-\ell_{1}-\ell_{2}+w)}.$
(5.12)

###  5.3 Four-point YSC and evolution kernels in positions

We specify the general integral form of YSC (3.26) to $N=4,K=2,n=2m=2$.

$\Phi^{--++}=\int\mathrm{d}^{4}c\,\varphi(c)\,\delta^{(2)}(\mathbf{x}_{1}-c_{13}\mathbf{x}_{3}-c_{14}\mathbf{x}_{4})\delta^{(2)}(\mathbf{x}_{2}-c_{23}\mathbf{x}_{3}-c_{24}\mathbf{x}_{4}),$
(5.13)

We have four delta distributions. The vanishing of the arguments of the first
two results in

$c_{13}^{(0)}=\frac{\langle 14\rangle}{\langle 34\rangle},\ \ \
c_{14}^{(0)}=\frac{\langle 13\rangle}{\langle 43\rangle}.$

We use the notations $\langle IJ\rangle=x_{I,1}x_{J,2}-x_{I,2}x_{J,1},\ \
x_{I}=\frac{x_{I,1}}{x_{I,2}}$. The last two result in

$c_{23}^{(0)}=\frac{\langle 24\rangle}{\langle 34\rangle},\ \ \
c_{24}^{(0)}=\frac{\langle 23\rangle}{\langle 43\rangle}.$

Further,

$c^{(0)}_{13}c^{(0)}_{24}-c^{(0)}_{14}c^{(0)}_{23}=\frac{\langle
12\rangle}{\langle 34\rangle}.$

The Jacobi determinant is

$\frac{\partial(x_{1,1}^{\prime},x_{1,2}^{\prime},x_{2,1}^{\prime},x_{2,2}^{\prime})}{\partial(c_{13},c_{14},c_{23},c_{24})}=\langle
34\rangle^{2}.$

$x_{1,1}^{\prime},x_{1,2}^{\prime},x_{2,1}^{\prime},x_{2,2}^{\prime}$
abbreviate the arguments of the delta distributions in (5.13).

The result is

$\Phi_{1234}^{--++}=\langle 34\rangle^{-2}\varphi^{--++}(c^{(0)}).$ (5.14)

We change from the homogeneous to the normal coordinates as in the previous
subsection,

$\langle IJ\rangle=x_{I,2}x_{J,2}x_{IJ},\ \ x_{IJ}=x_{I}-x_{J},$

and extract the scale factor

$\Phi_{1234}=x_{1,2}^{2\ell_{1}}x_{2,2}^{2\ell_{2}}x_{3,2}^{2\ell_{3}}x_{4,2}^{2\ell_{4}}\phi_{1234}.$
(5.15)

$\phi_{1234}$ is obtained from $\Phi_{1234}$ by replacing $\langle IJ\rangle$
by $x_{IJ}$.

In the case of the 4-point YSC ( 3.45) we obtain by (5.14), (5.15)

$\phi_{X}(x_{1},x_{2},x_{3},x_{4})=\frac{x_{34}^{1+v_{1}^{+}-v_{3}}}{x_{23}^{1+v_{1}-v_{2}}x_{14}^{1+v_{4}-v_{3}}x_{12}^{1+v_{2}^{+}-v_{4}}}=\frac{x_{12}^{1+2\ell_{1}-\varepsilon}}{x_{23}^{1+2\ell_{1}-2\ell_{2}-\varepsilon}x_{14}^{1-\varepsilon}x_{24}^{1+2\ell_{2}+\varepsilon}}$

where $2\ell_{1}=v_{3}-v_{1},2\ell_{2}=v_{4}-v_{1},\varepsilon=v_{3}-v_{4}$.
We substitute $1,2,3,4,\to 2^{\prime},1^{\prime},2,1$ and obtain the explicit
form of the kernel of the $R$ operator $R(\varepsilon)$ obeying the RLL
relation (3.21), (3.58) for the case $n=2m=2$ in the normal coordinate form.

$\phi_{X}(x_{2^{\prime}},x_{1^{\prime}},x_{2},x_{1})=x_{12^{\prime}}^{2h_{1}-2h_{2}-1}x_{1^{\prime}2}^{-1}x_{1^{\prime}2^{\prime}}^{-2h_{1}}x_{12}^{2h_{2}}\left(\frac{x_{1^{\prime}2^{\prime}}x_{12}}{x_{1^{\prime}2}x_{12^{\prime}}}\right)^{\varepsilon}$
(5.16)

We define the integral operator as in (3.52) specifying $\mathbf{x}_{I}$ to
$x_{I}$ and fix the integration in $x_{1^{\prime}},x_{2^{\prime}}$ as
Pochhammer contours about the points $x_{1},x_{2}$. The functions $x_{12}^{n}$
are eigenfunctions with the eigenvalues in terms of the Euler Beta function,

$B(-\varepsilon,n+1-2h_{1}+\varepsilon)B(1+n-2h_{2},2h_{1}-2h_{2}-\varepsilon).$
(5.17)

Comparing the dependence on $n$ with (5.12) we confirm that this operator with
the kernel (5.16) represents the $R$ operator obeying the $RLL$ relation
(3.21), where the spectral parameter difference as the argument of $R$ as in
(3.21) (5.11) is $w=u-v=\varepsilon-h_{1}+h_{2}$.

For real values of the positions $x_{1},x_{2}$ and for generic values of the
parameters the contours can be contracted to the interval between
$x_{1},x_{2}$. Then we can write the kernel in the simplex integral form.

$\int_{0}^{1}\mathrm{d}^{3}\alpha\,\delta(\alpha_{1}+\alpha_{2}+\alpha_{3}-1)\chi(\alpha)\delta(x_{11^{\prime}}-\alpha_{1}x_{12})\delta(x_{22^{\prime}}+\alpha_{2}x_{12})=J_{\chi}.$
(5.18)

This form is the convenient one for comparison with the evolution kernels of
generalized parton distribution as given in [49], [50]. It is also convenient
because the Fourier transformation is easily done.

We notice that the weights of the parton fields as they appear in the action
(2.26) correspond by (3.28) to the negative helicity value for both physical
states. The substitution rule for the helicities in the kernels has the
peculiarity, that the choice of the helicity sign depends of the form of the
integration in the action of the operator. The symmetry demands that the
helicities of the points to be integrated over, e.g. $x_{1^{\prime}}$, in both
factors are opposite, $h_{1^{\prime}},-h_{1^{\prime}}$, but there are two
options. If the choice is oriented on the weights in the action, then the
negative value is to be taken in $\psi$ irrespective of the physical helicity
state.

We consider first the case $h_{1}=h_{2}=-h$ for $h=1,\frac{1}{2}$ and show
that the evolution kernels for gluon $\to$ gluon and quark $\to$ quark without
helicity flip are reproduced.

In the limit $\varepsilon\to 0$ this $R$ operator kernel (5.16) is
proportional to $\delta(x_{12^{\prime}})\delta(x_{21^{\prime}})$ representing
the operator of point permutation $P_{12}$. The operator
$P_{12}R_{12}(\varepsilon)$ has the kernel obtained from the above (5.16) by
the permutation of the points $x_{1}$ and $x_{2}$.

The next term in the expansion of the latter operator in $\varepsilon$ is the
relevant contribution $J^{(2h)}_{11^{\prime}}+J^{(2h)}_{22^{\prime}}$,
represented as the simplex integral (5.18) with the integrand

$\chi_{11^{\prime}}(\alpha,h)+\chi_{22^{\prime}}(\alpha),\ \ \
\chi_{11^{\prime}}=(1-\alpha_{2})^{2h}\delta(\alpha_{1})$

The next relevant contribution is obtained at $\varepsilon=-1$ as $J_{11,2h}$,
represented as the simplex integral with the integrand

$\chi_{11h}(\alpha)=\alpha_{3}^{2h-1}.$

In the gluon case the third relevant contribution is obtained at
$\varepsilon=-2$, $J_{221}$, represented as the simplex integral with the
integrand

$\chi_{221}(\alpha)=\alpha_{1}\alpha_{2}.$

.

Thus the evolution kernel is obtained in the gluon case as

$W_{1-11-1}(x_{2^{\prime}},x_{1^{\prime}},x_{2},x_{1})=a_{0}(J^{(2)}_{11^{\prime}}+J^{(2)}_{22^{\prime}})+a_{1}J_{112}+a_{2}J_{221},$

and in the quark case as

$W_{\frac{1}{2}-\frac{1}{2}\frac{1}{2}-\frac{1}{2}}(x_{2^{\prime}},x_{1^{\prime}},x_{2},x_{1})=b_{0}(J^{(1)}_{11^{\prime}}+J^{(1)}_{22^{\prime}})+b_{1}J_{111}.$

The coefficients are not fixed by the symmetry, we can get the information
from the comparison with the parton splitting results as discussed in the next
section. The eigenvalues are calculated from (5.12), (5.17) with the
corresponding substitutions and the resulting dependence on $n$ is compatible
with the one of the anomalous dimensions calculated by (2.23) from the
corresponding parton splitting probabilities $W_{11}$ and
$W_{\frac{1}{2},\frac{1}{2}}$.

In the case $h_{1}=-h_{2}=-h$ we may substitute the negative value for both
and obtain $(J^{(2h)}_{11^{\prime}}+J^{(2h)}_{22^{\prime}})$ as the first non-
trivial term in the expansion in $\varepsilon$ at $\varepsilon=0$, as expected
for the kernels of the transversity parton distributions
$W_{1111},W_{\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}}$ from the results
of the Feynman rule calculations [49], [50].

The transition kernels are not obtained as specifications of the YSC (3.45),
for these we have to start with the crossed YSC (3.48). The anlogous steps
that lead above to (5.16) result in

$\phi_{||}(x_{2^{\prime}},x_{1^{\prime}},x_{2},x_{1})=x_{12}^{-1}x_{1^{\prime}2^{\prime}}^{-1}x_{11^{\prime}}^{2h_{1}-2h_{2^{\prime}}}x_{12^{\prime}}^{2h_{2^{\prime}}}x_{1^{\prime}2}^{-2h_{1}}\left(\frac{x_{12^{\prime}}x_{1^{\prime}2}}{x_{11^{\prime}}x_{22^{\prime}}}\right)^{\varepsilon},$
(5.19)

$h_{1}=-h_{2},h_{1^{\prime}}=-h_{2^{\prime}}$. We specify the contours and the
parameter $\varepsilon$ to reproduce the transition kernels.

We fix $\varepsilon=-2h_{2^{\prime}}-1$ and let the contour of
$x_{2^{\prime}}$ be a circle around $x_{1}$. Then the kernel reduces to

$\delta(x_{12^{\prime}})x_{12}^{2h_{2^{\prime}}}x_{1^{\prime}2^{\prime}}^{2h_{1}}x_{1^{\prime}2}^{-1-2h_{1}-2h_{2^{\prime}}}.$

The action on functions of $x_{1},x_{2}$ is defined as the Pochhammer contour
about $x_{1},x_{2}$ resulting on the basic eigenfunctions $x_{12}^{n}$ in the
eigenvalues proportional to

$B(1+n+2h_{1},-2h_{1}-2h_{2^{\prime}}).$

By comparing the $n$ dependence with the one of the anomalous dimensions
calculated from $W_{AB}(z)$ we see that for
$h_{1}=-1,h_{2^{\prime}}=-\frac{1}{2}$ and $h_{1}=-1,h_{2^{\prime}}=-1$ the
kernels for the transition gluon - quark $W_{1-1-\frac{1}{2}\frac{1}{2}}$ and
gluon - gluon $W_{1-1-11}$ with spin flip are obtained. For
$h_{2^{\prime}}=-1,h_{1}=-\frac{1}{2}$ the transition kernel quark - gluon
non-flip $W_{\frac{1}{2}-\frac{1}{2}1-1}$ is obtained. Other similar choices
of the contours and the parameter $\varepsilon$ lead to transition kernels
with the same or opposite helicities.

###  5.4 4-point YSC and evolution kernels in momenta

The helicity form of the 4-point YSC reads at $n=2m=2$

$\Phi^{--++\lambda}=\int\mathrm{d}^{4}c\,\varphi(c)\,\delta(\bar{\lambda}_{1}-c_{13}\bar{\lambda}_{3}-c_{14}\bar{\lambda}_{4})\delta(\bar{\lambda}_{2}-c_{23}\bar{\lambda}_{3}-c_{24}\bar{\lambda}_{4})\delta(\lambda_{3}+c_{13}\lambda_{1}+c_{23}\lambda_{2})\\\
\times\delta(\lambda_{4}+c_{14}\lambda_{1}+c_{24}\lambda_{2}).$

We have again four delta distributions but they are not removing the four $c$
integrations. We see the dependence between the related linear equations by
multiplying the first by $\lambda_{1}$, the second by $\lambda_{2}$, the third
by $\bar{\lambda}_{3}$, the fourth by $\bar{\lambda}_{4}$, and adding them
with the result
$\lambda_{1}\bar{\lambda}_{1}+\lambda_{2}\bar{\lambda}_{2}+\lambda_{3}\bar{\lambda}_{3}+\lambda_{4}\bar{\lambda}_{4}=0.$

We use the first three $\delta$-distributions to do the integrations over
$c_{13},c_{14},c_{23}$. Their values are fixed at

$c_{13}^{(0)}=\frac{\lambda_{1}\bar{\lambda}_{1}+\lambda_{4}\bar{\lambda}_{4}+c_{24}\bar{\lambda}_{4}\lambda_{2}}{\bar{\lambda}_{3}\lambda_{1}},\
\ \ c_{14}^{(0)}=\frac{-\lambda_{4}-c_{24}\lambda_{2}}{\lambda_{1}},$
$c_{23}^{(0)}=\frac{\bar{\lambda}_{2}-c_{24}\bar{\lambda}_{4}}{\bar{\lambda}_{3}},\
\ \
c_{13}^{(0)}c_{24}-c_{14}^{(0)}c_{23}^{(0)}=\frac{\bar{\lambda}_{2}\lambda_{4}+c_{24}(\lambda_{1}\bar{\lambda}_{1}+\lambda_{2}\bar{\lambda}_{2})}{\bar{\lambda}_{3}\lambda_{1}}.$

We change to $c=c_{24}\lambda_{2}\bar{\lambda}_{4}$ and introduce
$k_{I}=\lambda_{I}\bar{\lambda}_{I}$ in order to separate the one-dimensional
momentum dependence.

$c_{13}^{*}=\lambda_{1}\bar{\lambda}_{3}c_{13}^{(0)}=k_{1}+k_{4}+c,\ \ \
c_{14}^{*}=\lambda_{1}\bar{\lambda}_{4}c_{14}^{(0)}=k_{1}+k_{2}+k_{3}-c,$
$c_{23}^{*}=\lambda_{2}\bar{\lambda}_{3}c_{23}^{(0)}=k_{2}-c,\ \ \ \
\lambda_{1}\lambda_{2}\bar{\lambda}_{3}\bar{\lambda}_{4}(c_{13}^{(0)}c_{24}-c_{14}^{(0)}c_{23}^{(0)})=k_{2}k_{4}+c(k_{1}+k_{2})$

The result of the correlator in helicity form is

$\Phi_{1234}^{--++\lambda}=\lambda_{1}^{-2h_{1}}\lambda_{2}^{-2h_{2}}\bar{\lambda}_{3}^{-2h_{1}}\bar{\lambda}_{4}^{-2h_{2}}\int\mathrm{d}c\,\varphi(c^{*})\
\ \delta(k_{1}+k_{2}+k_{3}+k_{4}).$ (5.20)

The kernel of the $R$ operator obeying the RLL relation (3.21), (3.58) is
obtained by substituting $\varphi$ from (3.45) and relabeling the points
$1234\to 2^{\prime}1^{\prime}21$. We omit the momentum conservation delta and
the phase factor.

$\phi_{X2^{\prime}1^{\prime}21,4}^{--++}=(k_{1^{\prime}}k_{1})^{h_{1}}(k_{2^{\prime}}k_{2})^{h_{2}}\int
dc(k_{2}^{\prime}+k_{1}+c)^{-1-\varepsilon
2h_{1}-2h_{2}}c^{-1-\varepsilon}(k_{1^{\prime}}k_{1}+c(k_{2}^{\prime}+k_{1}^{\prime}))^{\varepsilon-2h_{1}}.$
(5.21)

The question about choosing the contour of the $c$ integral simplifies in the
forward case $k_{2}^{\prime}+k_{1}^{\prime}=0$. Then a non trivial kernel is
obtained with choosing the Pochhammer contour about the branch points related
to the first two factors in the integrand. We obtain the kernel

$w_{X}(k_{1},k_{1^{\prime}};h_{1},h_{2},\varepsilon))=(k_{1}k_{1^{\prime}})^{1-h_{1}+h_{2}+\varepsilon}\
\ \ (k_{1}-k_{1^{\prime}})^{2h_{1}-2h_{2}-1-2\varepsilon}.$ (5.22)

In the case $h_{1}=h_{2}=h$ this reproduces with the substitution
$k_{1}=z,k_{1^{\prime}}=1$ contributions to the kernels $W_{AB}(z)$ of the
particular physical helicities $h_{A}=h_{B}=\pm
1=h,h_{A}=h_{B}=\pm\frac{1}{2}=h$ by choosing the particular values
$\varepsilon=0,-1,-2$.

$W_{11}(z)=W_{-1-1}(z)=2zw_{X}(z,1;1,1,0)+4zw_{X}(z,1;1,1,-1)+zw_{X}(z,1;1,1,-2)=$
$2zw_{X}(z,1;-1,-1,0)+4zw_{X}(z,1;-1,-1,-1)+zw_{X}(z,1;-1,-1,-2),$
$W_{\frac{1}{2}\frac{1}{2}}(z)=W_{-\frac{1}{2}-\frac{1}{2}}(z)=2zw_{X}(z,1;\frac{1}{2},\frac{1}{2},0)+zw_{X}(z,1;\frac{1}{2},\frac{1}{2},-1)=$
$2zw_{X}(z,1;-\frac{1}{2},-\frac{1}{2},0)+zw_{X}(z,1;-\frac{1}{2},-\frac{1}{2},-1).$

The extra factor $z$ is included because the convolution integral is usually
written as $\frac{dz}{z}$.

Similar to the scattering amplitudes the admitted values of $\varepsilon$ can
be understood from the condition to have as singularities in the scattering
channels only simple poles. But here we see no argument for the coefficients
of these contributions.

In the case $h_{1}=-h_{2}=h$ (5.22) reproduces contributions to the evolution
kernels of the anti-parallel helicity configuration (projection $P_{T}$ in
[49] ),

$w(z;1,-1,2)=w(z;\frac{1}{2},-\frac{1}{2},1)=\frac{z}{1-z},$

which determine the scale dependence of transversity structure functions and
which do not fit into the parton branching picture.

Similar to the scattering amplitudes not all parton splitting kernels can be
reproduced with the 4-point YSC (3.42). It takes the crossed YSC (3.48) to
reproduce $W_{AB}$ with helicity flip or transition from gluon to quark.

$\phi_{||2^{\prime}1^{\prime}21,4}^{--++}=(k_{1}k_{2})^{-h_{1}}(k_{1^{\prime}}k_{2^{\prime}})^{h_{1^{\prime}}}\int
dc(k_{2}+c)^{-\varepsilon+2h_{1}-2h_{2^{\prime}}}c^{-1}(k_{1}+k_{2}+c)^{-1}(k_{1^{\prime}}k_{1}+c(k_{2}^{\prime}+k_{1}))^{\varepsilon+2h_{2^{\prime}}}.$
(5.23)

In the forward case, $k_{1}+k_{2}=0$, parton splitting kernels are reproduced
e.g. by fixing $\varepsilon=-2h_{2^{\prime}}-1$ and the contour as a circle
around the pole of the last factor in (5.23). We choose as independent
variables $k_{1},k_{1}^{\prime},h_{1},h_{1^{\prime}}$ and obtain

$w_{||}(k_{1},k_{1^{\prime}},h_{1},h_{1^{\prime}})=(k_{1}-k_{1^{\prime}})^{2h_{1^{\prime}}-2h_{1}-1}k_{1^{\prime}}^{-2h_{1}^{\prime}}k_{1}^{2h_{1}}.$

Then $zw(1,z;h_{1},h_{1^{\prime}})$ reproduces with
$h_{1}=-1,h_{1^{\prime}}=+1$ the gluon-gluon helicity flip and with
$h_{1}=-1,h_{1^{\prime}}=+\frac{1}{2}$ the gluon - quark helicity flip kernel.
$zw(z,1;h_{1},h_{1^{\prime}})$ with $h_{1}=+\frac{1}{2},h_{1^{\prime}}=+1$
reproduces the quark - gluon non-flip kernel.

Summarizing, we have seen that the evolution kernels of the generalized parton
distributions are obtained from 4-point YSC. In the momentum representation
their explicit form appears involved and simplifies in the forward limit. In
this respect the representation in the light-ray positions discussed in the
previous subsection is more convenient.

##  6 Application to the Regge asymptotics

###  6.1 Three-point YSC and the triple reggeon vertex

The BFKL equation summing the leading contributions to the exchange of two
reggeized gluons is formulated in the transverse momenta for which the complex
number notation is convenient. The kernel of this equation has been shown to