Ladder and zig-zag Feynman diagrams,
operator formalism and conformal triangles

S.E. Derkachov${}^{a}$, A.P. Isaev${}^{b,c}$
and L.A. Shumilov${}^{a}$

${}^a$ St.Petersburg Department of the Steklov Mathematical

of Russian Academy of Sciences,Fontanka 27, 191023, St.Petersburg,

${}^b$ Bogoliubov  Laboratory of Theoretical Physics,

Joint Institute for Nuclear Research,
141980 Dubna, Russia

${}^c$ Lomonosov Moscow State University,

Physics Faculty, Russia

<EMAIL_ADDRESS><EMAIL_ADDRESS>
We develop an operator approach to the evaluation of
multiple integrals for multiloop Feynman massless diagrams.
A commutative family of graph building operators $H_\alpha$
for ladder diagrams is constructed and
investigated. The complete set of eigenfunctions and
the corresponding
eigenvalues for the operators $H_\alpha$ are found.
This enables us to explicitly express a wide
class of four-point ladder diagrams and a general
two-loop propagator-type master diagram (with
arbitrary indices on the lines) as
Mellin-Barnes-type integrals. Special cases  of
these integrals are explicitly evaluated.
A certain class of zig-zag four-point and
two-point planar Feynman diagrams (relevant to
the bi-scalar $D$-dimensional
”fishnet” field theory and to the calculation of the
in $\phi^4$-theory) is considered.
The graph building operators and convenient integral representations
for these Feynman diagrams are obtained.
The explicit form of the
eigenfunctions for the graph building operators
of the zig-zag diagrams is fixed by
conformal symmetry and these eigenfunctions coincide
with the 3-point correlation functions in $D$-dimensional
conformal field theories.
By means of this approach, we exactly evaluate
the diagrams of the zig-zag series in special cases.
In particular, we find a fairly simple derivation of the values for
the zig-zag multi-loop two-point diagrams for $D=4$.
The role of conformal symmetry in this approach,
especially a connection of the considered graph
building operators with conformal
invariant solutions of the Yang-Baxter equation
is investigated in detail.

§ INTRODUCTION

In multiloop calculations, the number of diagrams
in quantum field theories grows
faster than $(2L+1)!!$ for the order of
perturbations $L \to \infty$
(see [1] for estimates of this number growth in various
Since numerical calculations give errors that
increase drastically
with the number of diagrams, analytical evaluation become important.
Up to now, there are no universal methods
for analytical evaluation of higher loop
diagrams (see however the method of differential equations
for master integrals, [2, 3, 4], which is based on the
”integration by parts” reduction method, Refs. [5, 6] ;
see also an approach to Feynman period computations [7]).
Thus, it becomes important to propose methods to compute integrals
for a certain infinite class of special
diagrams with an arbitrary number of loops.
The well known examples of such classes of diagrams
are ladder diagrams
in $\phi^3_D$-theory [8, 9, 10],
their generalizations [11],
zig-zag diagrams for $\phi^4_D$-theory
[12, 14, 16, 15, 48] and Basso-Dixon
fishnet diagrams [17, 18, 19, 20, 21, 22].

In this paper, we
present an effective method for the analytical
evaluation of multi-loop
massless Feynman diagrams, which in
particular gives us an easy way to evaluate the ladder
and zig-zag series of diagrams.
The impressive fact is that our approach to the analytical
evaluation of perturbative
multiple Feynman integrals uses the full power
of the methods of $D$-dimensional conformal field theories
[23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].

Let us itemize the main ideas.

* In special cases, the expression for the Feynman diagram
can be interpreted as integral kernel of some integral operator.
In this case, the most natural representation is the spectral
representation for the corresponding operator instead of
the initial coordinate or momentum representation.

* Special examples are diagrams of iterative type containing some
elementary building block (a similar effect occurs in the study of
the Bethe-Salpeter equations [35]).
This elementary building block deserves a
special name and is usually called the graph building operator
(see e.g. [39]).
The Feynman diagram containing convolution of $n$ building blocks
corresponds to
the $n$-th power of eigenvalues
of the graph building operator. This means that the spectral
representation for the graph building
operator allows one to write the general
expression for an infinite number of Feynman diagrams obtained by
iteration of the single graph building operator.
We should note that the so-called fishnet conformal field theories
[36], [37], [38], [39], [40],
[41], [42], [43]
produce many examples of nontrivial graph building operators.

* Of course, the possibility of a constructive use
of the spectral representation for graph building operator heavily
depends on the effective construction of the
eigenfunction and the statements
of orthogonality and completeness.
This can be done in interesting cases of ladder diagrams and more
sophisticated zig-zag diagrams and Basso-Dixon diagrams.

* In interesting cases, one obtains a family of
commuting graph building operators with a nontrivial
symmetry group.

* All this is very close in spirit to the operator approach
to the evaluation of Feynman diagrams initiated by one
of the authors [10], [44], [45].

The plan of the paper is the following.
In Section <ref> we recall the basic facts about
the operator method [10], [44]  and [47]
of multiloop evaluations,
that will be used in this paper.
Section <ref> is devoted to the simplest case,
where a special family of commuting graph building operators
$H_\alpha$: $H_\alpha H_\beta = H_\beta H_\alpha$ is
considered. The operators $H_\alpha$ are constructed
from the generators of one copy of the
D-dimensional Heisenberg algebra and should be
equivalently  understood as integral
operators acting in the space of functions of one vector
$x \in \mathbb{R}^D$
with coordinates $x^\mu$ $(\mu=1, \dots, D)$.
The main examples of the diagrams in this situation are
$2$-point and $4$-point
ladder diagrams and in particular the general two-loop master
propagator type diagram with arbitrary indices on the lines.

In Section <ref>, we consider a special class of
2-point and 4-point zig-zag diagrams in
D-dimensional $\phi^4$ quantum field theory.
The 2-point zig-zag diagrams give considerable contributions to
the renormalization group $\beta$-function  in
$\phi^4_{D=4}$  theory. An intriguing history of the
evaluation of renormalization group functions (up to seven loops)
in $\phi^4_{D=4}$ theory  and
the 2-point zig-zag diagrams (for $D=4$)
is outlined in [12] and in [16],
[48] (see also references therein). Here, in Section
<ref>, the graph building operator is constructed
for zig-zag diagrams as an element of the product of two copies
of the D-dimensional Heisenberg algebras,
or equivalently it is an integral
operator acting on the space of functions of two vector arguments
$x_1,x_2 \in \mathbb{R}^D$.
It is interesting that a repeating block in the case of
zig-zag diagrams can be represented as a square of
a simpler operator $\hat{Q}_{12}$.
It is a particular example of a more general phenomenon.
As we discuss in Section <ref>, the
initial repeating block is associated with
the $R$-operator, which is a conformally
invariant solution of the famous Yang-Baxter
equation and its factorization in the product
of two operators $\hat{Q}_{12}$ is a particular example
of the factorization of the $R$-operator [77, 76].

We consider integrals for massless Feynman diagrams,
which possess invariance under conformal transformations.
In fact, due to conformal invariance of the graph building operators
the orthogonality and completeness of their
eigenfunctions are well known
from $D$-dimensional conformal field theory.
In Section <ref>, we discuss
the needed properties of the
corresponding conformal triangles in detail and
then demonstrate bydirect calculation
that conformal triangles are eigenfunctions
of graph building operators and calculate the corresponding
In this section, we also try to present all the
needed proofs and
perform all calculations directly step by step to demonstrate how it

In Section <ref>, we use the worked technique
to calculate the general 2-point and 4-point
zig-zag diagrams and
to prove (outlined in [48]) of the
Broadhurst and Kreimer conjecture [12]
for the zig-zag diagrams in four-dimensional $\phi^4$ quantum field
theory. Another proof was given in [15].

The last Section <ref> of the main part
of the paper is devoted to a discussion
of the role of conformal symmetry and the
connection of the considered graph
building operators with the $R$-operator, the solution of the
Yang-Baxter equation.
In the previous sections, we tried to carry
out all the calculations explicitly
and combined in a reasonable way the integral identities and
their translation to the
corresponding operator forms. In the last section, we use mainly the
compact operator formalism, which is set out in the second section.

In Sections <ref> – <ref> (Appendixes),
we give proofs of some statements and provide
the technical details of the derivation of useful
formulas that we use in the main text of the paper.

§ OPERATOR FORMALISM AND DIAGRAM TECHNIQUE

In this section, we
briefly recall the result of
[10], [46] and [47] which will be used

Consider a $D$-dimensional Euclidean space
with coordinates $x^\mu$ where $\mu=1, \dots, D$ and
denote $x^{2\alpha} = (\sum_\mu \, x^\mu \, x^\mu)^{\alpha} $.
Let $\{ \q^\mu , \; \p^\nu \}$
be hermitian generators of the $D$-dimensional
Heisenberg algebra ${\cal H}$:
[^μ,   ^ν] = i   δ^μν   .
The algebra ${\cal H}$ acts in the space $V$ and
we introduce two sets
of basis states $|x \rangle$ and $|k \rangle$ in $V$
which respectively diagonalize the
operators $\q^\mu$ and $\p^\nu$
^μ|x ⟩= x^μ|x ⟩  ,
^μ|k ⟩= k^μ|k ⟩  .
We also introduce the dual states $\langle x |$
and $\langle k |$ such that the orthogonality and
completeness conditions are valid
⟨x | x' ⟩= δ^D (x-x')  ,
⟨k | k' ⟩= δ^D (k-k')   ,
∫d^D x  |x ⟩⟨x | = I =
∫d^D k  |k ⟩⟨k |   .
Relations (<ref>), (<ref>),
(<ref>) are consistent if we have

k^\mu \langle x | k \rangle =
\langle x | \p^\mu |k \rangle = -i \frac{\partial}{\partial x^\mu}
\langle x | k \rangle \;\;\; \Rightarrow \;\;\;
\langle x | k \rangle = \frac{1}{(2\pi)^{D/2}}
e^{i k^\mu x^\mu} \; ,

where the normalization constant $(2\pi)^{-D/2}$ is fixed by

Define the inversion operator ${\cal I}$ such that (see [10])

I^2 = 1   ,      I^†= I ·^2D
,       ⟨x | I = ⟨1/x |
,       I | x ⟩= x^-2D | 1/x ⟩          (1/x : = x^μ/x^2)   ,
I ·^μ·I = ^μ/^2   ,
I ·^μ·I  =
^2   ^μ- 2   ^μ  (  ) ≡K^μ
I ·(  ) ·I  = - (  )   ,

I ·^2α ·I = ^-2α   ,

I ·^2α ·I =
^2(α+D/2) ·^2α ·^2(α-D/2)   ,

where $(\q \, \p) := \q^\mu \, \p^\mu$.
Below we also use the shifted inversion
I_Δ:= I   ^2Δ   ,
for which
formulas in (<ref>)
are modified (we will need these formulas in Subsect.

I_Δ^ 2 = 1   ,
(I_Δ)^†= I_D-Δ   ,
^μ/^2   ,
I_Δ| x ⟩=
x^2(Δ-D) | 1/x ⟩  ,
⟨x | I_Δ=
x^-2Δ ⟨1/x|       ⇒      ⟨x | I_Δ| Φ⟩=
x^-2Δ   ⟨1/x|Φ⟩  ,
I_Δ·^2α ·I_Δ= ^-2α   ,
I_Δ·^2α ·I_Δ=
^2(α+D/2-Δ) ·^2α ·^2(α-D/2+Δ)   ,

and for generators of special conformal transformations
we deduce
K^(Δ)_μ: =
^2   ^μ- 2   ^μ  (  ) +
2 i Δ  ^μ  .
Here we also need a special case of
${\cal I}_\Delta$, i.e.
${\cal I}' := {\cal I}_{\Delta}|_{\Delta = \frac{D}{2}}
= {\cal I}\, \q^{2(\frac{D}{2})}$
that is Hermitian ${\cal I}^{\, \prime \dagger} =
{\cal I}^{\, \prime}$
with respect to (<ref>)
(I^')^2 = 1 ,
⟨x |   I' =
x^-2(D 2)   ⟨1/x |
,        I'   | x ⟩=
x^-2(D 2)   | 1/x  ⟩,       I' ·^2α ·I' =
^2α ·^2α ·^2α     ,
where the last formula
in (<ref>), (<ref>) is simplified.
Then, from the obvious identity
${\cal I}' \cdot \p^{2(\alpha + \beta)} \cdot {\cal I}' =
{\cal I}' \cdot \p^{2\alpha} \cdot
({\cal I}')^2 \cdot \p^{2\beta} \cdot {\cal I}'$,
we deduce the operator version [10] of
the star-triangle relation
^2α   ^2(α+ β)   p^2β =
^2β   ^2(α+ β)   q^2α

(∀   α,β)    .

We stress here that the operators $\q^{2\alpha}$ and
$\p^{2\beta}$ with noninteger $\alpha$ and $\beta$ are
understood as integral operators defined
via their integral kernels. Namely,
the operators $\q^{2\alpha}$ and
$\p^{2\beta}$ act on the states $|\psi \rangle$ in
the coordinate representation as
$\langle x | \q^{2\alpha} |\psi \rangle =
\int d^D y \langle x | \q^{2\alpha} | y \rangle
\langle y |\psi \rangle = x ^{2\alpha}
\langle x | \psi \rangle$ and

\langle x | \p^{-2\alpha} |\psi \rangle =
\int d^D y \; \langle x | \p^{-2\alpha} |y \rangle
\langle y |\psi \rangle =
a(\alpha) \; \int d^D y
\frac{\langle y | \psi \rangle }{(x-y)^{2(D/2-\alpha)}} \; ,

where we applied the well known formula for Fourier transformation

⟨x| ^-2α |y⟩=
∫d^D k ⟨x| ^-2α |k ⟩⟨k  |y⟩=
∫  d^D k/(2 π)^D
e^i k ( x - y)/k^2 α =
a(α)/(x-y)^2 α'    ,
1/2^2 α   π^D/2
Γ(α') / Γ(α)    ,
a(α)   a(α') = (2π)^-D
,         α' := D/2 - α  .

Then, we write the operator
identity (<ref>) as

\begin{array}{c}
\int d^D z \;
\langle y| \p^{-2\alpha'} \, | z \rangle\,
\langle z| \,  \q^{-2(\alpha' + \beta')}
\, p^{-2\beta'} | x \rangle =
\langle y| \q^{-2\beta'} \, \p^{-2(\alpha' + \beta')}
\, \q^{-2\alpha'} | x \rangle \; ,
\end{array}

and, taking into account eq. (<ref>), we represent it
in the familiar form of the star-triangle relation
[55, 25, 56, 57, 58, 82]
∫d^D z/(y-z)^2α
z^2γ   (z-x)^2β   =
a(γ)/a(α^  ') a(β^  ')  ·
1/y^2β'  1/(x-y)^2γ^  '
1/x^2α^  '    ,
where $(\alpha + \beta + \gamma) = D$ and $\frac{a(\gamma)}{
a(\alpha^{\, \prime})a(\beta^{\, \prime})} = \pi^{D/2}
\frac{\Gamma(\alpha')\Gamma(\beta')\Gamma(\gamma')}{
\Gamma(\alpha)\Gamma(\beta)\Gamma(\gamma)}$. Analogously from
the identity

\p^{-2\alpha'} \p^{-2\beta'} = \p^{-2(\alpha' + \beta')}
\;\; \Rightarrow
\;\;
\int d^D z \;
\langle y| \p^{-2\alpha'} \, | z \rangle\,
\langle z| \,
p^{-2\beta'} | x \rangle =
\langle y| \p^{-2(\alpha' + \beta')} | x \rangle \; ,

we deduce the chain relation
∫d^D z   1/(x-z)^2α  (z-y)^2β =
a(α')  a(β') ·1/(x-y)^2(α+β-D/2)   ,
and the coefficients in the right-hand sides of (<ref>)
and (<ref>) are the same.

Remark 1.  Of course, the coincidence of the coefficients
has a simple explanation – the point is that
relations (<ref>) and (<ref>) are equivalent [82].
Indeed, after inversion of all variables $x\to 1/x,y\to 1/y$ and $z\to
in (<ref>) one obtains relation (<ref>).
On the other hand, after shifting all the variables
$x\to x+w,y\to y+w$ and $z\to z+w$ in (<ref>)
and then sending $w \to \infty$, one obtains (<ref>).

Integral kernels and relations (<ref>), (<ref>)
can be depicted as Feynman graphs.
The Feynman rules that will be used in this paper
are shown in Fig. <ref>.

(a) - Propagator; (b) - Delta function; (c) - Bold
vertex depict integration over the whole space $\mathbb{R}^D$.

Using the language of Feynman graphs, we can easily represent different
calculations. For example, the chain relation (<ref>) is shown on
Fig. <ref>.

\pi^{D/2} \frac{\Gamma(\alpha')\Gamma(\beta')\Gamma(\gamma')}{
\Gamma(\alpha)\Gamma(\beta)\Gamma(\gamma)}\;\;\; \cdot$

(13.5,1.1)$\alpha \! + \! \beta \! - \! \frac{D}{2}$

Scalar chain rule

Remark 2.
In the framework of the dimensional regularization scheme
we have the following identity [47]:
∫  d^D x   1/x^2(D/2 + i α) = π  Ω_D
δ(α)   ,
Ω_D = 2 π^D/2/Γ(D/2)   ,
where $\alpha$
is a real number, $\delta(\alpha)$ is the standard delta-function
and $\Omega_D$ is the volume
of the unit sphere $\mathbb{S}^{D-1}$. For applications,
it turns out to be convenient to use formula
(<ref>) in the form
. ∫  d^D x   f(z)/x^ 2z
|_Re(z) = D/2= π  Ω_D
f(D/2)    δ(Im(z))   ,

where $z$ is a complex number and $f(z)$ is
a function analytic in $z=0$. This form permits one [47]
to bring the evaluation of propagator-type
perturbative integrals
to the evaluation
of vacuum perturbative integrals
(it also permits to search their symmetries).
Further, breaking any of the lines (the propagators)
in the corresponding vacuum diagram,
one can obtain another propagator type integral and thus
deduce many remarkable nontrivial
relations between the propagator type integrals
in $D$ dimensions.
Sometimes these relations are called
the ”glue-and-cut” symmetry.
For details see [47], [50])
(for $D=2$ such relations were used in [61]).
We apply relation (<ref>) in the next section
(see e.g. (<ref>)).

§ OPERATORS $H_\ALPHA$, TWO-LOOP MASTER
DIAGRAM AND LADDER DIAGRAMS

§.§ Definition of the operators $H_\alpha$
and two-loop master diagram

We note that the star-triangle relation (<ref>)
can be written as the commutativity condition
[^2β   ^2β,
^2(α+β)   ^2(α+β)]=0
[H_α,   H_α+β] = 0   ,
∀α,β  ,
H_α:= ^2α   ^2α   .
This section will focus on applying the
operators $H_\alpha$, which will be used
for analytical evaluation of the
two loop master-diagram and the $L$-loop ladder diagrams.
The two loop master-diagram and related diagram
for the 3-point function are respectively depicted
in Fig. <ref> and Fig. <ref>.

(6.5,2.2)${\displaystyle \int \!
\frac{d^D y \;\; d^D z}{(x-y)^{2\alpha_1} \, y^{2\alpha_2} \,
(y-z)^{2\alpha_5} \, z^{2\alpha_3} \, (z-x)^{2\alpha_4}}
\, \equiv \, \frac{C(\alpha_1,\alpha_2,\alpha_6;
\alpha_3,\alpha_4,\alpha_5)}{x^{2(D/2-\alpha_6)}}}$ ,

(14,0.5)$\alpha_6 := (3D/2 - \alpha_{1...5})$.

Two-loop master diagram.

\frac{1}{a(\alpha'_1)\,  a(\alpha'_4)
\, a(\alpha'_5)} \;
\langle x| \p^{-2\alpha_1'} \q^{-2\alpha_2}
\p^{-2\alpha_5'} \q^{-2\alpha_3} \p^{-2\alpha_4'} |y \rangle$ ,

The 3-point diagram.

The functions $a(\beta)$
in Fig. <ref> are defined in (<ref>).
We note that
the function for the master-diagram in Fig. <ref> is obtained
from the 3-point function
in Fig. <ref> by fixing $y=x$.
The tetrahedron vacuum diagram (which is related to the master
two-loop diagram in Fig. <ref>) is presented in Fig.<ref>.
All diagrams in Fig.<ref>–Fig.<ref>
are understood as diagrams in the configuration space and boldface
vertices (see Fig.<ref>)
denote integration over $\mathbb{R}^D$.

\int \,
\frac{d^D x}{a(\alpha'_1) a(\alpha'_4)
a(\alpha'_5)} \,
\langle x| \p^{-2\alpha_1'} \q^{-2\alpha_2}
\p^{-2\alpha_5'} \q^{-2\alpha_3} \p^{-2\alpha_4'}
\q^{-2\alpha_6} |x \rangle$ .

The tetrahedron vacuum diagram.

According toidentity (<ref>) the integral in
the right-hand side of equality in Fig.<ref>
(after the substitution of the relation in Fig.<ref>)
is not equal to zero only in the case when
α_6   =   α^0_6   :=   3D/2 -α_1...5
α_1'+α_5'+α_4'=α_2 +
α_3+α_6   .
Here we use the convenient notation
$\alpha_{b_1... b_k} := \sum_{i=1}^k \alpha_{b_i}$. In particular,
the $D$-dimensional ”glue-and-cut” symmetry [47]
(mentioned in Remark in previous Section
<ref>), which follows
from the vacuum diagram in Fig.<ref>,
gives the tetrahedral symmetry
[47] of the
two-loop master diagram in Fig.<ref>:

C(α_1,α_2,α_6; α_3,α_4,α_5)     =
C(α_6,α_1,α_2; α_5,α_3,α_4)   ,

=   C(α_6,α_3,α_4; α_5,α_1,α_2)
=   C(α_1,α_6,α_2; α_3,α_5,α_4)

Here the first two symmetries
are rotations of the tetrahedron in Fig.<ref>
with respect of the vertices $[\alpha_3,\alpha_4,\alpha_5]$ and
and the last symmetry is a
reflection of the tetrahedron in Fig.<ref> with
respect to the plane perpendicular to the
edge $[\alpha_1]$ (which is reduced to
the exchange of  vertices $[\alpha_1,\alpha_4,\alpha_6]$
and $[\alpha_1,\alpha_2,\alpha_5]$)  in Fig.<ref>.
All symmetries (<ref>) are elements of
the tetrahedral group $S_4$ which is
the permutation group of four vertices
of the tetrahedron in Fig. <ref>.
The whole $S_6 \times \mathbb{Z}_2$ symmetry of
the coefficient function $C(\alpha_1,\alpha_2,\alpha_6;
\alpha_3,\alpha_4,\alpha_5)$, which is discovered in [52],
[53] (see also [10]), is generated by
transformations (<ref>) and by additional symmetry

C(α_1,α_2,α_6; α_3,α_4,α_5) =

= a(α_1'-α_2+α_5' )
α_2+α_3- α_5',α_4,α_2')   ,

which is produced by the identity

\p^{-2\alpha_1'} \q^{-2\alpha_2}
\p^{-2\alpha_5'} \q^{-2\alpha_3} \p^{-2\alpha_4'}  \q^{-2\alpha_6} =
\p^{-2(\alpha_1'-\alpha_2+\alpha_5')} \q^{-2\alpha_5'}
\p^{-2\alpha_2} \q^{-2(\alpha_2 -\alpha_5' +\alpha_3)}
\p^{-2\alpha_4'}
\q^{-2\alpha_6} \; ,

following from the star-triangle relation (<ref>).

Now we note that the operators $H_\alpha $ (<ref>)

H_α^†= ^2α   ^2α
≡I' ·H_α·I'
H_α^†≡(H_-α)^-1   ,
and expressions, shown in
Fig.<ref>, could be expressed
in terms of $H_\alpha$.
the product of the operators $\p^{2\alpha}$ and $\q^{2\beta}$
in the right-hand side of the equality in Fig.<ref>
is written  as

\begin{array}{c}
\p^{-2\alpha_1'} \, \q^{-2\alpha_2} \,
\p^{-2\alpha_5'} \, \q^{-2\alpha_3} \, \p^{-2\alpha_4'} \,
\q^{-2\alpha^{0}_6}  \,
\q^{-2(\alpha_6-\alpha^{0}_6)} =  \\ [0.2cm]
= \bigl( H_{\alpha_1-\frac{D}{2}}
H_{\alpha_{125}-D} H^\dagger_{D-\alpha_{1235}}
H_{\alpha_{12345}-\frac{3D}{2}} \bigr) \,
\q^{-2(\alpha_6-\alpha^{0}_6)} \; ,
\end{array}

where $\alpha^{0}_6$ is defined in (<ref>).
So we get the following
expression for the diagram in Fig.<ref>

\begin{equation}\label{twoloop}
\int \,
\frac{d^D x}{a(\alpha'_1) a(\alpha'_4)
a(\alpha'_5)} \,
\langle x|H_{\alpha_1-\frac{D}{2}}
H_{\alpha_{125}-D} H^\dagger_{D-\alpha_{1235}}
H_{\alpha_{12345}-\frac{3D}{2}} \;|x \rangle
\, x^{-2(\alpha_6-\alpha^{0}_6)} \; .
\end{equation}

§.§ The L-loop ladder diagrams

Moreover, the L-loop ladder diagrams
can also be expressed as an integral kernel of the products
of operators $H_\alpha$.
We start with dimensionaly
and analytically regularized massless integrals
D_L (p_0, p_L+1, p) =
[ ∏_k=1^L ∫d^D p_k/p^2α_k_k   (p_k - p)^2β_k ]
∏_m=0^L 1/(p_m+1 - p_m)^2γ_m
which correspond to the diagram
depicted in Fig.<ref> ($x = p_0$, $y = p_{L+1}$, $z =p$).
The dual to this diagram for the case
$\alpha_k = \beta_k = \gamma_k =1$
is the $L$-loop ladder diagram for massless $\phi^3$ theory
presented in Fig.<ref>.

Graphical representation of the integral

(0.5,2.8)$p_0$ - $p$
(2.2,2.8)$p_1$ - $p$
(5.7,2.8)$p_{_{L+1}}$ - $p$

(8.5,1.8)$(p_{mk} := p_m - p_k)$

The $L$-loop ladder diagram in momentum
space for massless $\phi^3$ theory.

The integral (<ref>) is written
in the following operator form [10]
D_L (x, y, z; α, β, γ) =
( ∏_k=0^L   1/a(γ_k') )
⟨x| 1/^2γ_0' ( ∏_k=1^L
1/^2α_k 1/(-z)^2β_k
1/^2γ_k' )  |y⟩  .
Let the indices on the lines in the diagram in Fig.6 satisfy
the conformal condition

\gamma_{k-1} + \alpha_k + \beta_k + \gamma_k = D \; , \;\;\;\;\;
k=1,..., L \; .

Then by using (<ref>) and relation ${\cal I}'(\q-z)^2 =
\frac{z^2}{\q^2}(\q-\frac{1}{z})^2 {\cal I}'$ we obtain for
the matrix element in the right-hand side of (<ref>):

\langle x| ({\cal I}')^2 \frac{1}{\p^{2\gamma_0'}} \left( \prod_{k=1}^L
\,
\frac{1}{\q^{2\alpha_k}} \frac{1}{(\q-z)^{2\beta_k}}
\frac{1}{\p^{2\gamma_k'}} \right)  |y\rangle =

= \langle x| {\cal I}' \; \q^{-2\gamma_0'} \,
\frac{1}{\p^{2\gamma_0'}} \left( \prod_{k=1}^L \,
\frac{z^{-2\beta_k}}{(\q-\frac{1}{z})^{2\beta_k}}
\frac{1}{\p^{2\gamma_k'}} \right) \; \q^{-2\gamma_L'}
\; {\cal I}' |y\rangle =

= x^-2γ_0 y^-2γ_L z^-2β_1...L
⟨1/x |
1/^2γ_0' ( ∏_k=1^L
1/^2γ_k' )  | 1/y
= x^-2γ_0 y^-2γ_L z^-2β_1...L
1/^2γ_0' ( ∏_k=1^L
1/^2γ_k' )
| w⟩  .

where $u = \frac{1}{x} -\frac{1}{z}$, $w = \frac{1}{y} -\frac{1}{z}$
and $(\frac{1}{x})_i = \frac{x_i}{x^2}$ etc. Thus, to evaluate
$L$-loop ladder diagrams (<ref>), we need to calculate the matrix
1/^2γ_0' ( ∏_k=1^L
1/^2γ_k' )
| w⟩=
⟨u|   H_-γ_0' H^†_γ_0' - β_1
H_β_1 - γ'_01

⋯H_β_1...L - γ_01...L'
^2 δ_1...L
| w⟩  ,
where $\delta_{1...L} := \gamma_{01...L}' - \beta_{1...L}
\equiv \gamma_{0}' +\gamma_1' + ...+\gamma_L' - \beta_{1}-
\beta_{1}-...-\beta_{L}$.
Now, in view of relations (<ref>)
and (<ref>),
the problem of evaluating
the two-loop master-diagram and L-loop ladder
diagrams is reduced to the spectral problem of
the operators $H_\alpha$.

§.§ Diagonalization of operator $H_\alpha$

As we mentioned above,
the star-triangle relation (<ref>) is written as
a commutativity condition (<ref>),
and this means that the operators $H_\alpha$ form a commutative set
$[H_\alpha, \; H_\beta] = [H_\alpha, \; H^\dagger_\beta] =
[H^\dagger_\alpha, \; H^\dagger_\beta]=0$ for all parameters
To investigate the spectrum of operators
(<ref>), (<ref>),
we firstly note that the elements $\q^2$, $\p^2$ and dilatation

H := \frac{i}{2}\bigl(  (\q \, \p) + (\p \, \q) \bigr)
=i \, (\q \, \p) + D/2 \; ,

generate the $s\ell(2)$ Lie algebra:
[^2,   ^2] = 4 H   ,
H  ^2 = ^2   (H +2)  ,
H  ^2 = ^2   (H -2)   .
For the dilatation operator we have
$H \cdot f_{k,\ell}(\p,\q) = f_{k,\ell}(\p,\q) \cdot (H +\ell -k)$,
where $f_{k,\ell}(\p,\q)$ is a monomial of order $k$ in $\p_\mu$
and of order $\ell$ in $\q_\nu$. Thus,
the dilatation operator $H=-H^\dagger$ (the Cartan element for
the Lie algebra $s\ell(2)$) commutes with
all operators (<ref>), (<ref>).
The quadratic Casimir operator
for the $s\ell(2)$ algebra (<ref>):

C_(2) : = 1/2 (^2   ^2 + ^2   ^2) +  H^2
= (^2 ^2 + 2H + H^2)  = (H_1 + (H + 1)^2 - 1)   ,

is related to the
operator $H_1$ from the set (<ref>).
Further we need the following useful relations
which generalize (<ref>):
H  ^2α = ^2α    (H +2α)  ,

H  ^2α = ^2α   (H -2α)   ,
^2 β   ^2 =  ^2   ^2β -
4  β  ( H + β- 1)  ^2(β-1)   ,

^2 β   ^2 = ^2   ^2 β +
4  β  ( H - β+ 1)   ^2(β-1)   .

$|\psi_{j,\nu} \rangle$ be common eigenvectors
of the operators $H_\beta$ (<ref>):
H_β  |ψ_j,ν ⟩=
τ_j,ν(β)   |ψ_j,ν ⟩  ,
where $\tau_{j,\nu}(\beta)$ are the corresponding eigenvalues.
We numerate  $|\psi_{j,\nu} \rangle$
by two real numbers $\nu$ and $j$ which are respectively
fixed by the eigenvalues of the Cartan element $H=-H^\dagger$
and quadratic Casimir operator $C_{(2)}$:
H   |ψ_j,ν ⟩= -2i ν|ψ_ν,j ⟩  ,

C_(2)   |ψ_j,ν ⟩= 4j(j - 1)  |ψ_j,ν ⟩
Then, the eigenvalues $\tau_{j,\nu}(\beta)$ in
(<ref>) are
τ_j,ν(β) = 4^β
Γ(j+β- iν)  Γ(j +iν)/
Γ(j-β+ iν)  Γ(j - iν)   .

Proof. Relations (<ref>), (<ref>) are
written as identities

H_β= (H_1  - 4 (β- 1) (H +β) )
H_β-1     ⇒
H_β= (C_2 -H^2-2H - 4 (β- 1) (H +β) )
H_β-1   ,

and the last identity gives the functional equation
for the eigenvalues $\tau_{j,\nu}(\beta)$ of the operators $H_\beta$.
Indeed, from (<ref>), (<ref>) and identity
(<ref>) we deduce the equation
τ_j,ν(β) = 4 (j +  i ν- β)
(j-1  + β- i ν) τ_j,ν(β-1)   ,
which is solved as

\tau_{j,\nu}(\beta) = 4^\beta \frac{\Gamma(j+\beta - i\nu)}{
\Gamma(j-\beta +i\nu)} \; c(j,\nu) \; ,

where $c(j,\nu)$ is a function independent of $\beta$.
This function is fixed by obvious condition $\tau_{j,\nu}(0)=1$
and we finally obtain (<ref>).

Equation (<ref>) gives for $\beta=1$ the equality
τ_j(1) = 4 (j - i ν) (j +  i ν- 1 )   ,
which is used below.

As it is seen from the examples of subsections
<ref>, <ref>,
we deal with the
algebra ${\cal U}$ of polynomials which are generated by the operators
$H$, $\q^{2\alpha}$ and $\p^{2\beta}$, i.e. the basis elements
in ${\cal U}$ are
E_α_1,...,α_k;β_1,...,β_k =
⋯^2α_k^2β_k   ,       k=0,1,2,3,...
and any element of ${\cal U}$ is written as a formal
a = ∑_k=0^∞ϕ_k(H)
E_α_1,...,α_k;β_1,...,β_k   ,
where $\phi_{k}(H)$ are functions in $H$. The
operator $C_2$ defined in (<ref>)
commutes with any element (<ref>) in ${\cal U}$.
Indeed, the Casimir operator $C_2$ commutes
with an arbitrary function $\phi_{k}(H)$ of the Cartan element $H$
as well as with the operators $\q^{2\alpha}$ and $\p^{2\beta}$
$(\forall \alpha,\beta)$.
Note that the basis elements (<ref>)
are written in the form

E_α_1,...,α_k;β_1,...,β_k =
H_α_1   H_β_1-α_1^†  H_α_12-β_1   H_β_12-α_12^†⋯H_α_1...k-β_1...(k-1)
^2(β_1...k-α_1...k)  =
= H_α_1   H_α_1-β_1^-1
H_α_12-β_1   H_α_12-β_12^-1
^2(β_1...k-α_1...k)   ,

where the commutative operators $H_\alpha$ and
$H_\beta^\dagger$ are defined in (<ref>),
(<ref>), and we use concise notation
$\alpha_{1...\ell}= \alpha_1 + ... +\alpha_\ell$.
Equation (<ref>) means that for $\beta_{1...k}=\alpha_{1...k}$
an alternating product of
$\q^{2\alpha}$ and $\p^{2\beta}$ in (<ref>)
commutes with any element of the set (<ref>),

In the paper [46], a complete set of orthogonal common
eigenvectors $| \psi^{\mu_1 ... \mu_n}_\nu \rangle$ for
the operators (<ref>),
(<ref>) was explicitly constructed:
⟨x | ψ^μ_1 ... μ_n_ν⟩=
x^μ_1 ... μ_n/x^2(D/4+n/2+i ν)    .

These eigenfunctions satisfy the following relations
∑_n=0^∞  μ(n)
∫_-∞^+∞ dν
| ψ^μ_1 ... μ_n_ν⟩⟨ψ^μ_1 ... μ_n_ν|   =   I   ,
2^n-1Γ(D/2+n)/π^D/2+1   n!  ,

^2α   ^2α ·x^μ_1 ... μ_n/x^2β = 4^α
Γ(D/2-β+α+n)  Γ(β)/
Γ(β-α)   Γ(D/2-β+n)
x^μ_1 ... μ_n/x^2β

H_α  | ψ^μ_1 ... μ_n_ν⟩=
τ_n,ν(α)    | ψ^μ_1 ... μ_n_ν⟩  ,

H_α^†  | ψ^μ_1 ... μ_n_ν⟩=
τ_n,-ν(α)    | ψ^μ_1 ... μ_n_ν⟩  ,

H   | ψ^μ_1 ... μ_n_ν⟩= - 2iν
| ψ^μ_1 ... μ_n_ν⟩  ,
I'   | ψ^μ_1 ... μ_n_ν⟩=
| ψ^μ_1 ... μ_n_-ν ⟩  ,

τ_n,ν(α) = 4^αa_n(D/4 + n/2-α+iν)/
a_n(D/4 + n/2 + i ν)     ,
a_n(w) := Γ(D/2-w+n)/Γ(w)   ,
⟨ψ^μ_1 ... μ_n_ν|
ψ^ν_1 ... ν_m_ν' ⟩=
π^D/2+1 n!/2^n-1 Γ(D/2+n)
δ(ν- ν') δ_nm
P^μ_1 ... μ_n_ν_1 ... ν_m    ,
where $x^{\mu_1 ... \mu_n} = P^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_m}
\; x^{\nu_1} \cdots x^{\nu_n}$ is the traceless
symmetric homogeneous polynomial in $x^{\mu}$ and
$P^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_m}$ is
the projector on such polynomials. We stress
that the key spectral
formulas (<ref>) are operator versions of
the well known relation [82, 81]

\begin{equation}\label{dchain}
\int d^{D} x \dfrac{1}{\left(y - x\right)^{2\alpha}}
\dfrac{x^{\mu_1\ldots\mu_n}}{x^{2\beta}} =
\pi^{D/2}\dfrac{a_n\left(\beta\right)a_0\left(\alpha\right)}{
a_n{\left(\alpha + \beta - D/2\right)}}\dfrac{y^{\mu_1\ldots\mu_n}}{
y^{2\left(\alpha + \beta - D/2\right)}},
\end{equation}

where $a_n\left(\alpha\right)$ is defined in (<ref>).
The detailed information
about the projector
$P^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_m}$ and the proof of
the orthogonality condition (<ref>) for
the eigenvectors $| \psi^{\nu_1 ... \nu_m}_{\nu} \rangle$
are given in Appendix A (see Section <ref>).
The convolution product of
the symmetrized polynomials has the form
(see [46] and Appendix A)
x^μ_1 ...μ_nx^μ_1 ...μ_n =
2^n Γ(n+D/2-1)Γ(D-2) x^2n   .
We stress that the orthogonality condition (<ref>)
is consistent with the resolution of unity in

Note that
for the quadratic Casimir operator (<ref>) we have
(cf. (<ref>))

C_{(2)} \; | \psi^{\mu_1 ... \mu_n}_\nu \rangle =
\frac{1}{4} (D + 2n) (D + 2n-4)
\; | \psi^{\mu_1 ... \mu_n}_\nu \rangle
\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;
j=\frac{D}{4} + \frac{n}{2} \; ,

and, for these $j$, the eigenvalue $\tau_{n,\nu}(\beta)$ of $H_\beta$,
given in (<ref>),
is equal to the function (<ref>) from Proposition <ref>.
This fact indicates that vectors (<ref>)
could form a complete set since the statement of
Proposition <ref>
gives the general form for any
possible eigenvalues for $H_\beta$. The proof of completeness is
given in [46].
Then, we note that there are useful identities for
functions (<ref>):
a_n(u +n/2)   a_n(u' +n/2) = 1
⇒     τ_n,ν(u) =  1/τ_n,-ν(-u)
H_u ·H^†_-u = 1   ,
where $u' = D/2-u$, and for $u,\nu \in \mathbb{R}$ we have
$\tau_{n,-\nu}(u) = \tau_{n,\nu}^*(u)$.

Remark 1. The second equation in (<ref>)
follows from identities (<ref>).
The second relation in (<ref>) is
proven as

\begin{array}{c}
\langle x |\,  {\cal I}^{\prime} \,| \psi^{\mu_1 ... \mu_n}_\nu \rangle
x^{2(-\frac{D}{2})} \; \langle \frac{1}{x} |
\psi^{\mu_1 ... \mu_n}_\nu \rangle = \displaystyle
\frac{x^{\mu_1 ... \mu_n}}{x^{2(D/4+n/2-i \nu)}} =
\langle x | \psi^{\mu_1 ... \mu_n}_{-\nu} \rangle \; .
\end{array}

Remark 2. The Gegenbauer polynomial technique [49]
(see also [51]) is based on the identity [46]

⟨x | ^-2α' | y ⟩=
∫_-∞^+∞ dν⟨x | ^2α'   H^†_-α'
| ψ^μ_1 ... μ_n_ν⟩⟨ψ^μ_1 ... μ_n_ν| y ⟩=
= ∑_n=0^∞μ(n) ∫_-∞^+∞ dν
τ_n,-ν(α-D/2)   x^2α'
⟨x | ψ^μ_1 ... μ_n_ν⟩⟨ψ^μ_1 ... μ_n_ν| y ⟩=
= 2^2α- D
∫_-∞^+∞ dν
x^μ_1 ...μ_n   y^μ_1 ...μ_n/x^2(n/2+iν+α-D/4) y^2(D/4 + n/2-iν)   ,

where in the right-hand side of this chain of relations
one can make a shift of the coordinates
$x,y \to (x-z),(y-z)$
with an arbitrary vector  $z \in \mathbb{R}^D$ since the
left-hand side of (<ref>) is invariant under this shift.

Remark 3. Below we also need the following matrix element

⟨ψ^μ_1 ... μ_n_ν|
^2(3D/2 - α_1...6)
| ψ^μ_1 ... μ_n_ν⟩=
∫d^Dx ⟨ψ^μ_1 ... μ_n_ν|
^2(3D/2 - α_1...6)
|x ⟩⟨x | ψ^μ_1 ... μ_n_ν⟩=

= ∫d^Dx
x^μ_1 ... μ_n x^μ_1 ... μ_n/
x^2(3D/2 - α_1...6) =
2^n Γ(n+D/2-1)Γ(D-2)
∫d^Dx 1/x^2(α_1...6 - D) =
= Γ(n+D-2)Γ(D/2-1)πΩ_D/
2^n Γ(n+D/2-1)Γ(D-2)
δ(α_1...6 - 3D/2)
= Γ(n+D-2)π^D/2+1/
2^n-2 Γ(n+D/2-1)Γ(D-1)
δ(α_1...6 - 3D/2)   .

The evaluation of (<ref>)
illustrates the application of the formula (<ref>). Here
we also use (<ref>) and
identity (<ref>).

§.§ Answer for the two loop master-diagram

Expression (<ref>)
for the vacuum diagram in Fig.<ref> is represented
in the form

\frac{1}{a(\alpha'_1) a(\alpha'_4)
a(\alpha'_5)} \sum_{n=0}^\infty \mu(n)
\int\limits_{-\infty}^{+\infty} d\nu \,
\langle \psi^{\mu_1 ... \mu_n}_\nu |
\p^{-2\alpha_1'} \q^{-2\alpha_2}
\p^{-2\alpha_5'} \q^{-2\alpha_3} \p^{-2\alpha_4'}
\q^{-2\alpha^{0}_6}
\q^{-2(\alpha_6 -\alpha^{0}_6)}
| \psi^{\mu_1 ... \mu_n}_\nu  \rangle =

\begin{array}{c}
= \frac{1}{a(\alpha'_1) a(\alpha'_4)
a(\alpha'_5)} \sum\limits_{n=0}^\infty
\mu(n) \int\limits_{-\infty}^{+\infty} d\nu \,
\tau_{n,\nu}(\alpha_1-\frac{D}{2})
\tau_{n,\nu}^*(\frac{D}{2}-\alpha_{12})
\tau_{n,\nu}(\alpha_{125} - D)
\tau_{n,\nu}^*(D-\alpha_{1235})
\tau_{n,\nu}(-\alpha^0_{6}) \cdot \\ [0.2cm]
\cdot \; \langle \psi^{\mu_1 ... \mu_n}_\nu |
\q^{2(\frac{3D}{2} - \alpha_{1...6})}
| \psi^{\mu_1 ... \mu_n}_\nu  \rangle  =
\end{array}

= 1/a(α'_1) a(α'_4)
a(α'_5)     ∑_n=0^∞μ(n)
2^n Γ(n+D/2-1)Γ(D-2)    ×
×    ∫_-∞^+∞ dν
τ_n,ν(α_125 - D)  τ_n,ν(-α^0_6)/
δ(α_1...6 - 3D/2)    ,

where $\mu(n)$ is defined in (<ref>);
in the last equality, we use identities (<ref>),
(<ref>) and $\tau_{n,-\nu}(u) = \tau_{n,\nu}^*(u)$.
Comparing the expression in Fig.<ref> with
eq. (<ref>), we deduce the formula

α_3,α_4,α_5) =
Γ(D/2-1)/Γ(D-2)/a(α'_1) a(α'_4)
a(α'_5) ∑_n=0^∞  μ(n)
2^n Γ(n+D/2-1)   ×
×  ∫_-∞^+∞ dν
τ_n,ν(α_125 - D)  τ_n,ν(-α^0_6)/
τ_n,ν(α_1235-D)    .

We note that the infinite sum (<ref>)
of the Mellin-Barnes integrals is invariant under
the $S_6 \times \mathbb{Z}_2$ symmetry discovered in [52],
[53] (see also [10]). This $S_6 \times \mathbb{Z}_2$
is generated by the tetrahedral symmetry (<ref>)
and by additional symmetry (<ref>) which is produced by
the star-triangle relation.

The example of calculation of the
coefficient function $C(\alpha_1,\alpha_2,\alpha_6;
\alpha_3,\alpha_4,\alpha_5)$ in (<ref>)
for a special case of indices

\alpha_1 = \alpha_4 = \alpha_5 = D/2 -1 \; , \;\;\;\;
\alpha_2 = \beta \; , \;\;\;\;
\alpha_3 = \alpha \; ,

is given in Appendix B (see Section <ref>).
Analytical answer for
the coefficient function
(this result was firstly obtained in [72]
by means of the integration-by-parts method)
is related to the case of the dual master two-loop diagram,
Fig.<ref>) when three indices on the
lines are fixed
$\alpha_1=\alpha_2=\alpha_5 =1$ and indices
$\alpha_3,\alpha_4$ are arbitrary.

Remark 4. One can obtain from eq. (<ref>)
the known results
for 2-loop master diagrams in Fig.<ref>,
when  $\alpha_1=\alpha_2=\alpha_3=\alpha_4=1$
(see e.g. [51], [81] and references therein),
or say $\alpha_3=\alpha_4=\alpha_5=1$,
which is exactly evaluated by the
integration-by-parts method and represented as
a finite sum (see [5], [6]).
A rather general result for the 2-loop master integral (<ref>)
was obtained in the paper [60].

For physical applications one has to know the
explicit expansion of
function (<ref>) in the limit $\epsilon \to 0$,
when $D=4-2\epsilon$,
$\alpha_i = n_i + m_i \epsilon$ $(n_i,m_i \in \mathbb{Z})$.

§.§ Answer for the L-loop ladder diagram

To evaluate
$L$-loop ladder diagrams (<ref>), we need to calculate the matrix
element in (<ref>), (<ref>):

\bigl\langle u\bigr| \,
\frac{1}{\p^{2\gamma_0'}} \left( \prod_{k=1}^L \,
\frac{1}{\q^{2\beta_k}}
\frac{1}{\p^{2\gamma_k'}} \right) \,
\bigl| w\bigr\rangle =
\bigl\langle u\bigr| \, H_{-\gamma_0'} H^\dagger_{\gamma_0' - \beta_1}
H_{\beta_1 - \gamma'_{01}} H^\dagger_{\gamma_{01}' - \beta_{12}}
\cdots H_{\beta_{1...L} - \gamma_{01...L}'}
\q^{2(\gamma_{01...L}' - \beta_{1...L})} \,
\bigl| w\bigr\rangle =

= w^{2(\gamma_{01...L}' - \beta_{1...L})}
\sum_{n=0}^\infty \mu(n) \int\limits_{-\infty}^{+\infty} d\nu
\bigl\langle u\bigr| \, H_{-\gamma_0'} H^\dagger_{\gamma_0' - \beta_1}
H_{\beta_1 - \gamma'_{01}}
%H^\dagger_{\gamma_{01}' - \beta_{12}}
\cdots H_{\beta_{1...L} - \gamma_{01...L}'}
| \psi^{\mu_1 ... \mu_n}_\nu \rangle
\langle \psi^{\mu_1 ... \mu_n}_\nu
\bigl| w\bigr\rangle =

= w^2(γ_01...L' - β_1...L)
∑_n=0^∞μ(n) ∫_-∞^+∞ dντ_n,ν(-γ_0')  τ_n,ν(β_1 -
τ_n,ν(β_1-γ_0')  τ_n,ν(β_12-γ_01')
( u^μ_1 ... μ_n w^μ_1 ... μ_n/
u^2(D/4+n/2+iν) w^2(D/4+n/2-iν))    ,
where $\gamma'_{01...k} = \gamma'_0 + \gamma'_1 + ... +\gamma'_k =
(k+1) \frac{D}{2} - \gamma_{01...k}$ and $\mu(n)$
is defined in (<ref>).
For physical applications one has to know the
infinite sum of
Mellin-Barnes type integrals (<ref>) for $D=4-2\epsilon$,
$\alpha_i = n_i + m_i \epsilon$ $(n_i,m_i \in \mathbb{Z})$
and an explicit expansion of (<ref>)
in the limit $\epsilon \to 0$.

In the case $\gamma_k' = \beta_k = \beta$,
we write the generating function for matrix elements in
the left-hand side of (<ref>) as follows:

\begin{array}{c}
\bigl\langle u\bigr| \, \frac{1}{\p^{2\beta} -
g \, \q^{-2\beta}} \,
\bigl| w\bigr\rangle = \bigl\langle u\bigr| \q^{2\beta} \,
\frac{1}{\p^{2\beta} \q^{2\beta}-
g \, } \, \bigl| w\bigr\rangle = u^{2\beta} \,
\bigl\langle u\bigr| \frac{1}{H_{\beta}- g \, } \,
\bigl| w\bigr\rangle = \\ [0.2cm]
\displaystyle
= u^{2\beta} \,
\sum\limits_{n=0}^\infty \mu(n)
\int\limits_{-\infty}^{+\infty} d\nu
\bigl\langle u\bigr| \frac{1}{(H_{\beta}- g)} \,
| \psi^{\mu_1 ... \mu_n}_\nu \rangle
\langle \psi^{\mu_1 ... \mu_n}_\nu
\bigl| w\bigr\rangle = \\ [0.2cm]
\displaystyle = u^{2\beta} \,
\sum\limits_{n=0}^\infty \mu(n)
\int\limits_{-\infty}^{+\infty} d\nu
\frac{1}{(\tau_{n,\nu}(\beta)- g)} \, \frac{
u^{\mu_1 ... \mu_n}\, w^{\mu_1 ... \mu_n}}{
(u^2\, w^2)^{(D/4+n/2)}(u^2/w^2)^{i\nu}} =  \\ [0.2cm]
\displaystyle =
u^{2\beta} \,
\sum\limits_{n=0}^\infty \mu(n) \, \frac{
u^{\mu_1 ... \mu_n}\, w^{\mu_1 ... \mu_n}}{
(u^2\, w^2)^{(D/4+n/2)}}
\int\limits_{-\infty}^{+\infty} d\nu
\, \frac{(w^2/u^2)^{i\nu}}{(\tau_{n,\nu}(\beta)- g)}
\end{array}

Then for $\beta=1$ we have Green's
function for conformal quantum mechanics

\bigl\langle u\bigr| \, \frac{1}{\p^{2} -
g \, \q^{-2}} \,\bigl| w\bigr\rangle =
u^{2} \,
\sum\limits_{n=0}^\infty \mu(n) \, \frac{
u^{\mu_1 ... \mu_n}\, w^{\mu_1 ... \mu_n}}{
(u^2\, w^2)^{(D/4+n/2)}}
\int\limits_{-\infty}^{+\infty} d\nu
\, \frac{(w^2/u^2)^{i\nu}}{(\tau_{n,\nu}(1)- g)} =

= u^2   ∑_L=0^∞  g^L ∑_n=0^∞μ(n)
u^μ_1 ... μ_n  w^μ_1 ... μ_n/
(u^2  w^2)^(D/4+n/2)
∫_-∞^+∞ dν
(w^2/u^2)^iν/(τ_n,ν(1))^L+1   ,
where we expand the function $1/(\tau_{n,\nu}(1)- g)$ over $g$.
Now we take into account relation (<ref>):
$\tau_{n,\nu}(1) =
4(\frac{D}{4}+\frac{n}{2}-i \nu)(\frac{D}{4}+\frac{n}{2}+i \nu-1)$
and write
expansion (<ref>) over $g$ in the form
⟨u|   1/(^2 - g/^2)   |w ⟩=
1/L! (g 4 )^L   Φ_L(u,w)
where for $\Phi_L(u,w)$ we have the representation
Φ_L(u,w) = u^2 L!/4
u^μ_1 ... μ_n  w^μ_1 ... μ_n/
(u^2  w^2)^(D/4+n/2)
dν   (w^2/u^2)^iν/
((D/4+n/2-i ν)
(D/4+n/2+i ν-1))^L+1   .

Note that the function $\Phi_L(u,w)$ is related to the
4-point $L$-loop ladder integral (<ref>),
(<ref>) with fixed indices
$\alpha_k=\beta_k=\gamma_k'=1$ of the propagators.
This relation is given by the formula
(see (<ref>) and (<ref>))
D_L(x,y,z;1,1,D/2-1)   =   (x^2y^2)^(1-D/2)z^-2L/
L!  (4   a(1))^L
1/y-1/z )   ,
where $x=p_0$, $y=p_{L+1}$, $z=p$. We recall
that in [10] the expansion
(<ref>) (for any $D$) was also considered and
another integral representation
for the coefficients $\Phi_L(u,w)$ was deduced:
Φ_L(u,w) = a(1)/(L!)
∫^∞_0   dt   t^L
[ log( u^2/w^2) + t]^L
∂_t ( e^-t   /
(u -  e^-tw)^2)^(D 2-1)   ,
Then we reproduced [10] by means of
(<ref>) for $D=4$  the famous  result [8]:

\begin{equation}
\label{d4}
\Phi_{L}(u,w) = \dfrac{1}{4\pi^2}\dfrac{1}{u^2}
\dfrac{1}{(z - \bar{z})}\sum_{k = 0}^{L}
\dfrac{(-1)^k(2L - k)!}{k!(L -
k)!}\operatorname{Log}^k(z\bar{z})\left[\operatorname{Li}_{2L -
k}\left(z\right) - \operatorname{Li}_{2L -
k}\left(\bar{z}\right)\right]  \; ,
\end{equation}

z + \bar{z} = \dfrac{2\, u\cdot w}{u^2 } \; , \;\;\;\;\;
z\bar{z} = \dfrac{w^2}{u^2} \; ,

which is needed, in view of (<ref>),
for explicit evaluation of ladder diagrams.
Formula (<ref>) was
intensively applied to the evaluation of planar
amplitudes in $D=4$ conformal field theories and,
in particular, in the $N=4$ supersymmetric Yang-Mills theory
(see, e.g. [62],[63],[32]
and references therein).

By making use (<ref>),
we easily find the first coefficient
$\Phi_0=a(1) (u-w)^{2(1-D/2)}$ (it is
instructive for $D=4$ to deduce this coefficient
from (<ref>))
and find the symmetry of $\Phi_L(u,w)$ for all $L$

\Phi_L(u,w) = \Phi_L(w,u) =  (u^2 w^2)^{(1 - {D \over 2})} \,
\Phi_L \Bigl({1 \over u}, {1 \over w}\Bigr)  \; .

Here we show, as an example of the
effectiveness of the proposed methods, that the
integral in (<ref>) gives the same result (<ref>).
The integral in (<ref>)
is calculated by residues. By condition $u^2 > w^2$ the contour
can be closed in a lower half-plane and it remains to calculate
at the $(L+1)$-order pole $i\nu = D/4 + n/2$

\begin{align*}
& Res =
\dfrac{1}{L!}\left(\dfrac{d}{d\nu}\right)^L\left[\dfrac{(u^2/w^2)^{-i\nu}}{(\nu
- i(D/4 + n/2 - 1))^{L +1}}\right]\bigg|_{\nu = -i(D/4 + n/2)} = \\
%&\dfrac{1}{L!}\sum_{k = 0}^{L}\binom{L}{k}(-i)^k
%(u^2/w^2)^{D/4 + n/2}}(-1)^{L - k}
%\dfrac{(2L - k)!}{L!}\dfrac{i^{2L - k + 1}}{
%(D/2 + n - 1)^{2L - k + 1}} = \\
& = \dfrac{i}{L!}\sum_{k = 0}^{L}\binom{L}{k}\dfrac{(2L -
k)!}{L!}(u^2/w^2)^{-D/4 - n/2}
\dfrac{\operatorname{Log}^k(u^2/w^2)}{
(D/2 + n - 1)^{2L - k + 1}} \; .
\end{align*}

In this way we obtain the following representation for the function
$\Phi_L(u, w)$:

\begin{equation}
\Phi_L(u, w) = 2\pi\dfrac{u^2}{4}\sum_{n =
w^{\mu_1\ldots\mu_n}}{(u^2)^{D/2 + n}}\sum_{k = 0}^{L}\dfrac{(2L -
k)!}{k!(L - k)!}\dfrac{\operatorname{Log}^k(u^2/w^2)}{
(D/2 + n - 1)^{2L - k + 1}}
\end{equation}

which is transformed to

\begin{multline}
\Phi_L(u, w) = \pi^{-D/2}\dfrac{u^2}{4(u^2)^{D/2}}
\sum_{n = 0}^{\infty}\dfrac{\Gamma(D/2 + n)\Gamma(D/2 -
1)}{\Gamma(D/2 - 1 +
\sum_{k = 0}^{L}\dfrac{(2L - k)!}{k!(L -
k)!}\dfrac{\operatorname{Log}^k(u^2/w^2)}{(D/2 + n - 1)^{2L - k +
\end{multline}

where we use the explicit formula (<ref>) for
the measure $\mu(n)$ and
the formula with the Gegenbauer polynomials

\begin{equation}
u^{\mu_1\ldots\mu_n}w^{\mu_1\ldots\mu_n} = \dfrac{n!\Gamma\left(D/2
- 1\right)}{2^n\Gamma\left(n + D/2 -
\left(u^2w^2\right)^{n/2} .
\end{equation}

In the case $D = 4$, we have a
simple expression for the Gegenbauer polynomials

\begin{equation}
C_n^{(1)}(\hat{u}\hat{w}) = \dfrac{\sin((n +
\end{equation}

where $\theta$ is the angle between the vectors $u$ and $w$.
Then, after a shift $n \to (n-1)$ in the summation we have

\begin{align}
\nonumber
&\Phi_L(u, w) =
\dfrac{1}{(u^2w^2)^{\frac{1}{2}}4 \pi^2}\sum_{n = 1}^{\infty}n
\dfrac{\sin(n\theta)}{(u^2/w^2)^{n/2}\sin\theta}\sum_{k =
0}^{L}\dfrac{(2L - k)!}{k!(L -
k)!}\dfrac{\operatorname{Log}^k(u^2/w^2)}{n^{2L - k + 1}}\; ,
\end{align}

where the sum over $n$ can be expressed in terms of polylogs

{\rm Li}_k(z) = \sum\limits_{n=0}^\infty \frac{z^n}{n^k}
\;\;\; \Rightarrow \;\;\;
\sum_{n = 1}^{\infty}\dfrac{e^{in\theta}}{
(u^2/w^2)^{n/2}n^{2L - k}} = \operatorname{Li}_{2L -
k}\left(\dfrac{e^{i\theta}}{(u^2/w^2)^{1/2}}\right) \; .

As a result, we deduce

\begin{multline}
\label{d4p}
\Phi_{L}(u, w) = \dfrac{1}{(u^2w^2)^{1/2}}
\dfrac{1}{4 \pi^2}\dfrac{1}{(e^{i\theta} - e^{-i\theta})} \sum_{k =
0}^L\dfrac{(-1)^k(2L - k)!}{k!(L -
k)!}\operatorname{Log}^k\left(\dfrac{w^2}{u^2}\right)\times \\
\times\left[\operatorname{Li}_{2L -
k}\left(\dfrac{e^{i\theta}}{(u^2/w^2)^{1/2}}\right) -
\operatorname{Li}_{2L -
\end{multline}

We use the parametrization

\begin{equation}
z = \dfrac{e^{i\theta}}{(u^2/w^2)^{1/2}}, ~~~~~ \bar{z} =
\dfrac{e^{-i\theta}}{(u^2/w^2)^{1/2}} \;\;\;\;~~
\Rightarrow  \;\;\;\;~~
z + \bar{z} = \dfrac{2\, u\cdot w}{u^2 } \; , \;\;\;\;
z\bar{z} = \dfrac{w^2}{u^2} \; ,
\end{equation}

and finally obtain from (<ref>) the known result (<ref>).

Remark 5. In view of eq. (<ref>),
after some renormalization and
change of variables, the Green's function (<ref>)
is related to the sum of the 4-point ladder integrals
For $D=4$ this sum  was investigated
in [9] and then was written
in another form
of a single integral over the Bessel function $J_0$ with
a trigonometric measure (see eqs. (14), (15) in [9]).

§ OPERATOR FORMALISM AND ZIG-ZAG DIAGRAMS

Below we consider a special class of Feynman perturbative
4-point $G_4^{(M)} (x_1, x_{2}; y_1, y_{2})$
and 2-point $G_2^{(M)} (x_{2}, y_1)$
massless integrals

\begin{align}
\label{zgzg44}
G_4^{(2N)} (x_1, x_{2}; y_1, y_{2}) &= (y_1 - y_2)^{2\beta}\,
\int \left[ \prod_{i=1}^{N} d^D z_i \,
d^D \widetilde{z}_i \right]\,
\prod\limits_{j=1}^{N}
\frac{1}{z_{j,j+1}^{2\beta'} \,
\widetilde{z}_{j,j+1}^{2\beta'} \,
z_{j,\,\widetilde{j+1}}^{2\beta} \,
z_{j+1,\widetilde{j+1}}^{2\beta}}\,, \\
\label{zg44}
G_4^{(2N+1)} (x_1, x_{2}; y_1, y_{2}) &=
\int d^Dz \,
\frac{G_4^{(2N)} (x_1, x_{2}; y_2, z)}{
(z - y_1)^{2\beta'} (z - y_2)^{2\beta}}\,,\\
\label{zgzg22}
G^{(M)}_2(x_2,y_1) &= \int d^D x_1 \; d^D y_2 \;
\dfrac{G^{(M)}_4(x_1,x_2;y_1,y_2)} {
\end{align}

where $z_0=x_1$, $\widetilde{z}_0=x_2$,
$z_{N+1}=y_1$, $\widetilde{z}_{N+1}=y_2$ and we use compact notation
$z_{ik} = z_i - z_k\,, \widetilde{z}_{ik} =
\widetilde{z}_i - \widetilde{z}_k\,,
z_{i\widetilde{k}} = z_i - \widetilde{z}_k$ and $\beta' = D/2-\beta$.

The integrals (<ref>) – (<ref>)
are called zig-zag integrals and are
visualized below as Feynman diagrams (zig-zag diagrams)
in Figs. <ref>,  <ref> and  <ref>,

§.§ Definition of the
graph building operator $\hat{Q}_{12}$

The first step is to obtain
operator expressions for the zig-zag integrals
(<ref>) – (<ref>)
using the operator formalism discussed in Sections
<ref> and <ref>.
In these Sections we used only one $D$-dimensional
Heisenberg algebra ${\cal H}$ with defining relations
(<ref>). For the purposes of evaluation
of zig-zag integrals, it would be convenient to consider a
direct product of several Heisenberg algebras (<ref>).
Consider the algebra $\mathcal{H}^{(n)}$
consisting of $n$ copies of the $D$-dimensional Heisenberg algebras
$\mathcal{H}_i \equiv \mathcal{H}$ $(i=1,...,n)$, so
$\mathcal{H}^{(n)} = \otimes_{i=1}^n \mathcal{H}_i$,
with the commutation relations of generators

\begin{equation}
\lb{deff}
[\hat{q}^{\mu}_i, \hat{q}^{\nu}_j] = 0 =
[\hat{p}^{\mu}_i, \hat{p}^{\nu}_j] \, , \;\;\;\;\;\;\;\;
[\hat{q}^{\mu}_i, \hat{p}^{\nu}_j] =
i\delta^{\mu\nu}\delta_{ij} \; , \;\;\;\; i,j = 1,...,n
\; , \;\;\;\; \mu,\nu = 1,...,D \; .
\end{equation}

Further we introduce states $|x_i \rangle$ and $|k_i \rangle$
which respectively diagonalize $\q^\mu_i$
and $\p^\nu_i$ as in (<ref>) for each copy of the algebras
Such states form a basis in the space $V_i$, where the subalgebra
$\mathcal{H}_i$ acts. The whole algebra $\mathcal{H}^{(n)}$ acts in
the space
$V_1 \otimes \cdots \otimes V_n$
with the basis vectors

\begin{equation}
|x_1,..., x_n \rangle : = |x_1\rangle \otimes
\cdots \otimes |x_n\rangle.
\end{equation}

To obtain
operator expressions for the zig-zag integrals
(<ref>) – (<ref>),
we need only the
algebra $\mathcal{H}^{(2)}= \mathcal{H}_1 \otimes \mathcal{H}_2$
consisting of two copies of the $D$-dimensional Heisenberg algebra.
The entire subsequent procedure of building the diagram technique that
uses the operator formalism in the algebra $\mathcal{H}^{(2)}$ repeats
the methods of Section <ref>, except for the fact that all
operators carry an additional index $i=1,2$.

We want to obtain, in the case of zig-zag diagrams,
expressions similar to (<ref>) and (<ref>),
so we introduce analogs of the operators $H_\alpha =
\p^{2\alpha} q^{2\alpha}$, which
we call graph building operators. The original problem of
calculating zig-zag diagram was formulated in $D = 4$, so firstly we
will consider the
four-dimensional case and then its generalization to any $D$.

Let us define the four-dimensional
graph-building operator $\hat{Q}_{12}$ in the following form:

\begin{equation}
\label{q-4}
\hat{Q}_{12} =
\end{equation}

where we use the convenient notation
$\hat{q}^{\mu}_{ij} = \hat{q}^{\mu}_i - \hat{q}^{\mu}_j$ and
$\mathcal{P}_{12}$ is a permutation operator
P_12   _1 = _2   P_12   ,
P_12   _1 = _2   P_12   ,      ⟨x_1,x_2 |P_12| Ψ⟩=
⟨x_2,x_1 | Ψ⟩  ,      (P_12)^2 = I

For the calculation we need to know the
integral kernel of the operator $\hat{Q}_{12}$:

\begin{equation}\label{q-kern}
\langle x_1,x_2 |
\hat{Q}_{12}  | y_1,y_2 \rangle \, =
\, (2\pi)^2 \, \cdot \, \langle x_1,x_2 |\;
{\cal P}_{12} \,\p_1^{-2} \,
\hat{q}_{12}^{-2} \, | y_1,y_2 \rangle \, =
\frac{\delta^4(x_1-y_2)}{(x_2-y_1)^2 \, (y_1-y_2)^2} \; .
\end{equation}

The visualization of the integral
kernel (<ref>) is given in Fig. <ref>

(1,2.8)$\langle x_1,x_2 |
\hat{Q}_{12}  | y_1,y_2 \rangle \; =
\; {\cal P}_{12} \; \cdot$

(15,2.8)$ =$

The diagram that represents the integral kernel $\langle x_1,x_2 |
\hat{Q}_{12}  | y_1,y_2 \rangle$ (we do not
indicate the index $1$ on the solid lines).

To illustrate why
$\hat{Q}_{12}$ is  the graph-building operator for the zig-zag
we consider the operator $\bigl(\hat{Q}_{12}\bigr)^2$ and its integral

\begin{equation}
\bigl(\hat{Q}_{12}\bigr)^2 =
\hat{p}_1^{-2}\hat{q}_{12}^{-2}
\end{equation}

\begin{equation}
\langle x_1, x_2| \bigl(\hat{Q}_{12}\bigr)^2|y_1,y_2\rangle =
\dfrac{1}{(x_1 - y_1)^2(x_2 - y_2)^2(x_1 - y_2)^2(y_1 - y_2)^2}
\end{equation}

The visualization of the
evaluation of the integral kernel for
$\bigl(\hat{Q}_{12}\bigr)^2$ is given in
Fig. <ref>. The integral operator
(<ref>) was
considered in [39] and denoted there as $H_{1\!\!1}$.

(0,5.8)$\langle x_1,x_2 |
\hat{Q}_{12}\; \hat{Q}_{12} | y_1,y_2 \rangle
\;\;\;\;\; =$

(-1.2,4)$\overbrace{I=\int d^D z_1 d^D z_2 | z_1,z_2 \rangle
\langle z_1,z_2 |}$

(2.6,1.3)$= \;\; \int d^D z_1 \; d^D z_2
\; {\cal P}_{12} \; \cdot$

(14,1.3)$\cdot \; {\cal P}_{12} \; \cdot $

The diagram that represents the integral kernel  $\langle x_1,x_2
\bigl(\hat{Q}_{12}\bigr)^2  | y_1,y_2 \rangle$.
Two operators ${\cal P}_{12}$
turn over the first graph for
$\langle x_1,x_2 |
\hat{p}_1^{-2}\hat{q}_{12}^{-2} | z_1,z_2 \rangle$ and
the result of the calculation is
obtained by shrinking dashed lines
($\delta$-functions) into points.

The definition of the graph-building
operator (<ref>) sufficient for the calculation
of the standard zig-zag series of diagrams,
arised in the calculation of the $\beta$-function in four-dimensional
$\varphi^4$ theory, but this approach can be extended for the
calculation of a more general series of zig-zag diagrams with
non-unit indices on the lines in the case
of an arbitrary $D$. Consider the generalized version of
the operator (<ref>) (for  any dimension $D$)

\begin{equation}
\label{q-d}
\hat{Q}_{12}^{(\beta)} : = \dfrac{1}{a(\beta)} \;
{\cal P}_{12} \; (\p_1)^{-2\beta} \; (\hat{q}_{12})^{-2\beta} \; ,
\end{equation}

where $\beta$ is an arbitrary parameter
and the function $a(\beta)$ is defined in
The integral kernel of this operator is

\begin{equation}
\langle x_1,x_2 |
\hat{Q}_{12}^{(\beta)}  | y_1,y_2 \rangle \, = \;\;
\frac{ \delta^D(x_1-y_2)}{(x_2-y_1)^{2\beta'} \, (y_1-y_2)^{2\beta}},
\end{equation}

and we can build a zig-zag series
of diagrams with arbitrary parameters $\beta$
and $\beta'=D/2-\beta$
on the lines by using the kernel of the operator
$\left(\hat{Q}^{(\beta)}_{12}\right)$ as a building block. Indeed,
consider four point planar zig-zag Feynman graphs:

for even loops[The number
of loops coincides
with the number of bold face vertices.]

(8,2)$. \; .\; . \; . \; .\; . $

(17,2)$= \;\;\;
\langle x_1,x_2 | (\hat{Q}_{12}^{(\beta)})^{2N}| y_1,y_2 \rangle
(y_1-y_2)^{2\beta} = $

(14.5,3)$. \; .\; . \; . \; .\; . $

for odd loops

(10,2)$. \; .\; . \; . $

(20, 2)$= \;
%\put(4,-2){\footnotesize $= \;
\langle x_1,x_2 |
(\hat{Q}_{12}^{(\beta)})^{2N+1}| y_1,y_2 \rangle
(y_1-y_2)^{2\beta} =$

(14.5,3)$. \; .\; . \; . \; .\; . $

Thus, to find the corresponding four point
zig-zag correlation function in $(M-2)$-loop order,
we need to calculate

\begin{equation}
\label{g-4}
G^{(M)}_4(x_1,x_2;y_1,y_2) = \langle x_1,x_2 |
\bigl(\hat{Q}_{12}^{(\beta)}\bigr)^{M}| y_1,y_2 \rangle
\end{equation}

In eqs. (<ref>), (<ref>)
we present the spiral visualization of the
zig-zag diagrams which are specific for the
conformal biscalar fishnet
theories [36], [38].

The two-point zig-zag Feynman graphs
and the corresponding integrals are
defined via the 4-point
functions (<ref>) as follows:

for even loops

(9.8,2)$. \; .\; . \; . \; .\; . $

(18.5, 2)

$\;\; = \;\; \int d^D x_1 d^D y_2
\dfrac{\langle x_1,x_2 | (\hat{Q}_{12}^{(\beta)})^{2N}| y_1,y_2
\rangle} {(x_1-x_2)^{2\beta}}\;$ ,

for odd loops

(10,2)$. \; .\; . \; . $

$\; = \;\; \int d^D x_1 d^D y_2
\dfrac{\langle x_1,x_2 | (\hat{Q}_{12}^{(\beta)})^{2N+1}| y_1,y_2
\rangle} {(x_1-x_2)^{2\beta}}\,$ ,

and the corresponding two point correlation
function in $M$-loop order is given by

\begin{equation}\label{g-2}
G^{(M)}_2(x_2,y_1) = \int d^D x_1 d^D y_2
\dfrac{\langle x_1,x_2 | (\hat{Q}_{12}^{(\beta)})^{M}| y_1,y_2
\rangle} {(x_1-x_2)^{2\beta}}\,.
\end{equation}

As we will see
in the following sections,  the
problem of calculating the
diagrams of the zig-zag series
is reduced to the
construction of a complete
set of eigenvectors (and the corresponding
eigenvalues) for the graph building
operator $Q_{12}^{(\beta)}$. In other
words, we need to investigate the
spectral problem
for the operator $Q_{12}^{(\beta)}$.
So our method for evaluating zig-zag
diagrams will almost repeat
the method for evaluating
ladder diagrams applied in Sect. <ref>.

§ EIGENFUNCTIONS FOR THE
GRAPH BUILDING OPERATOR $\HAT{Q}_{12}$.
SCALAR PRODUCT AND COMPLETENESS

§.§ Conformal triangles

To find a complete set of eigenvectors for the
graph building operator (<ref>) we
firstly consider the 3-point correlation function
of three conformal
fields ${\cal O}_{\Delta_1}$, ${\cal O}_{\Delta_2}$
and ${\cal O}_{\Delta}^{\mu_1...\mu_n}$
in the $D$-dimensional conformal field theory $(D>2)$.
Here  ${\cal O}_{\Delta_1}$ and ${\cal O}_{\Delta_2}$
are two scalar fields with conformal dimensions
$\Delta_1$ and $\Delta_2$, while
${\cal O}_{\Delta}^{\mu_1...\mu_n}$
is the rank-$n$ tensor field with conformal dimension
$\Delta$. Recall that the rank-$n$ tensor
${\cal O}_{\Delta}^{\mu_1...\mu_n}$
is symmetric and traceless. The single conformally-invariant 3-point
correlation function is completely fixed
(up to a normalization constant
factor) and is represented as
[27], [68],

\begin{multline}
\lb{cor01}
\langle  {\cal O}_{\Delta_1}(x_1) \;
{\cal O}_{\Delta_2}(x_2) \; {\cal O}_{\Delta}^{\mu_1...\mu_n}(x)
\rangle = \\
\frac{\left(\frac{x-x_1}{(x-x_1)^2}-\frac{x-x_2}{
(x-x_2)^{\Delta_2-\Delta_1+\Delta-n}}\; ,
\end{multline}

where $x_1, x_2, x \in \mathbb{R}^D$ and

$x^{\lambda} := (x^2)^{\frac{\lambda}{2}}$.
The index structure is the same as in (<ref>),
so $x^{\mu_1\ldots\mu_n}$ denotes the symmetric and traceless part of
$x^{\mu_1}\ldots x^{\mu_n}$.

As we will see in Section <ref> for generic
$\Delta_1, \Delta_2 \in \mathbb{R}$ and a special choice
of the parameter
Δ= D/2 + 2iν ,      ν∈ℝ   ,
system of functions

\begin{align}
\lb{conf-tr}
\Psi^{\mu_1\cdots\mu_n}_{\nu,x}(x_1\,,x_2) =
\frac{\left(\frac{x-x_1}{(x-x_1)^2}-\frac{x-x_2}{
\end{align}

is orthogonal and complete with respect to some
natural scalar product on the space of functions
of two variables $x_1,x_2$.
For the sake of simplicity, we
will sometimes not indicate the
dependence on the parameters $\Delta_1$ and
$\Delta_2$ in the notation

In Sections <ref>,
<ref>, we show that the eigenvectors of
the operator $\hat{Q}^{(\beta)}_{12}$
are given by a special form of the 3-point function (<ref>)
when three parameters $\Delta_1,\Delta_2,\Delta$ are expressed
in terms of dimension $D$
and two arbitrary numbers
$\alpha \in \mathbb{C}\,,\beta \in \mathbb{R}$
Δ_1 = D/2   ,     Δ_2 = D/2-β  ,     Δ= D-2α-β+n   ,
so that we define
Ψ^μ_1...μ_n_α,β,x(x_1 ,x_2)    : =
( x-x_1/(x-x_1)^2 -
x-x_2/(x-x_2)^2 )^μ_1...μ_n/(x_1-x_2)^2α
where $\alpha' := \frac{D}{2} - \alpha$ and
$(\alpha+ \beta)' = \frac{D}{2} - (\alpha + \beta)$.

Note that (<ref>) is a formal eigenfunction for an arbitrary
complex parameter $\alpha$, but there exists some special choice
of the parameter $\alpha$ when the system of eigenfunctions is
orthogonal and complete. This choice was indicated
in (<ref>) and for this choice we deduce
the relations

Δ= D - 2α- β+n = D/2 + 2iν     ⇒
α= 1/2(n + D/2 - β) - i ν .
Finally, the orthogonal and complete system of eigenfunctions
of the operator $\hat{Q}^{(\beta)}_{12}$ is given by

\begin{equation}\label{d-eigenf}
\Psi^{\mu_1...\mu_n}_{\nu,\beta,x}(x_1\,,x_2)
{(x_1-x_2)^{D/2 - \beta + n - 2i\nu}
(x-x_1)^{D/2 + \beta - n + 2i\nu}
(x-x_2)^{D/2 - \beta - n + 2i\nu}}\,.
\end{equation}

In (<ref>) we consider
$x \in \mathbb{R}^D$ as a parameter, while $x_1$ and $x_2$ are related
to the Heisenberg algebras
${\cal H}_1$ and ${\cal H}_2$.
In other words, we interpret
the wave function (<ref>) as

\begin{equation}
\label{wavef}
\Psi^{\mu_1...\mu_n}_{\nu,x}(x_1\,,x_2)
= \langle x_1, x_2 |
\Psi^{\mu_1...\mu_n}_{\nu,x}\rangle \;
\end{equation}

and similarly

\begin{equation}
\label{wavefab}
\Psi^{\mu_1...\mu_n}_{\alpha,\beta,x}(x_1\,,x_2)
= \langle x_1, x_2 |
\Psi^{\mu_1...\mu_n}_{\alpha,\beta,x}\rangle \ \ \ ;\ \ \
\Psi^{\mu_1...\mu_n}_{\nu,\beta,x}(x_1\,,x_2)
= \langle x_1, x_2 |
\Psi^{\mu_1...\mu_n}_{\nu,\beta,x}\rangle \; .
\end{equation}

Following the paper [57], we call
the functions (<ref>),(<ref>) and
which are a special form of the three-point correlators (<ref>),
conformal triangles.
In the next section we show that the conformal triangles (<ref>)
(3-point correlation functions (<ref>)
with a special choice of the conformal
dimensions) are the eigenfunctions of the operator
$\hat{Q}^{(\beta)}_{12}$ that
was defined in (<ref>). So it is possible to write a spectral
decomposition of the operator $\hat{Q}^{(\beta)}_{12}$ and
evaluate[When we say analytically
multiple integrals of the
type (<ref>) – (<ref>),
we mean reducing these multiple
integrals to a
single integral and/or a single infinite sum
and then expressing the answer in terms of known special functions.]
integrals (<ref>) and (<ref>).

Before performing such calculations, we
first need to explore the properties of functions (<ref>),
The set of functions (<ref>) forms a
complete orthogonal system (see [26],[27]
and references therein).
In Section <ref>, we present our direct explicit proof of
orthogonality of these functions with a special measure and derive
from them a possible form of the completeness identity.

§.§ Diagonalization of the operator

First we
prove that the functions (<ref>)
are eigenfunctions for the operator $\hat{Q}^{(\beta)}_{12}$.

The vectors
$| \Psi_{\alpha,\beta,y}^{\mu_1...\mu_n} \rangle \in V_1 \otimes V_2$,
in (<ref>) and (<ref>):

(0,2)$\langle y_1,y_2 |
\Psi_{\alpha, \beta , y}^{\mu_1...\mu_n} \rangle \;\;
= \;$

(9,2)$\Bigl( \frac{y-y_1}{(y-y_1)^2} -
\frac{y-y_2}{(y-y_2)^2} \Bigr)^{\mu_1...\mu_n} =
\frac{\bigl( \frac{y-y_1}{(y-y_1)^2} -
\frac{y-y_2}{(y-y_2)^2} \bigr)^{\mu_1...\mu_n}}{(y_1-y_2)^{2\alpha}
(y-y_1)^{2\alpha'}(y-y_2)^{2(\alpha+\beta)'}}\, ,$

for any parameters $\alpha$
and $\beta$, are eigenvectors for the
graph building operator (<ref>)
| Ψ_α,β,y^μ_1...μ_n ⟩=
| Ψ_α,β,y^μ_1...μ_n ⟩  ,
with the eigenvalue
τ(α,β,n) = (-1)^n
π^D/2 Γ(β) Γ(α)
Γ(α'-β+n)/ Γ(β') Γ(α'+n)
Γ(α+β)    .

Proof. The statement of this
Proposition follows from the chain
of equalities:

\begin{multline*}
\int d^D y_1 d^D y_2 \langle x_1,x_2 |
\hat{Q}^{(\beta)}_{12}  | y_1,y_2 \rangle \langle y_1,y_2 |
\Psi_{\alpha, \beta, y}^{\mu_1...\mu_n} \rangle = \\
\int d^D y_1 \;
\frac{1}{(x_2-y_1)^{2\beta'} (y_1-x_1)^{2(\alpha+\beta)}}
\frac{\bigl( \frac{y-y_1}{(y-y_1)^2} -
\frac{y-x_1}{(y-x_1)^2} \bigr)^{\mu_1...\mu_n}}{
\stackrel{(\ref{int04mu})}{=}
\\
\frac{\pi^{D/2} \Gamma(\beta) \Gamma(\alpha) \Gamma(\alpha'-\beta+n)}{
\Gamma(\beta') \Gamma(\alpha'+n)
\Gamma(\alpha+\beta)}\frac{1}{(x_1-x_2)^{2\alpha}}
\frac{\bigl( \frac{y-x_2}{(y-x_2)^2} -
\frac{y-x_1}{(y-x_1)^2}
\bigr)^{\mu_1...\mu_n}}{(y-x_2)^{2(\alpha'-\beta)}
(y-x_1)^{2\alpha'}} = \\
(-1)^n \,
\frac{\pi^{D/2} \Gamma(\beta) \Gamma(\alpha) \Gamma(\alpha'-\beta+n)}{
\Gamma(\beta') \Gamma(\alpha'+n) \Gamma(\alpha+\beta)}
\langle x_1,x_2 | \Psi_{\alpha, \beta, y}^{\mu_1...\mu_n} \rangle
\end{multline*}

The proof is based on the tensor generalization of the
star-triangle identity:
∫d^D z      ( y-z/(y-z)^2 -
y-x_1/(y-x_1)^2 )^μ_1...μ_n/(z-x_2)^2β'
(z-x_1)^2(α+β)(z-y)^2α' =
y-x_2/(y-x_2)^2 -
(the standard star-triangle identity (<ref>) is obtained for
The derivation of this identity is given
in Appendix C.
We note that
identity (<ref>) was presented in
[25] (see there eq. (A3.2) in Appendix 3).

Remark 1.  In the scalar version of
the star-triangle relation (<ref>) we
have an equivalent but simpler
counterpart, the chain relation (<ref>).
Of course, there exists a similar relative of
relation (<ref>)
∫d^D z      ( y-z/(y-z)^2 -
y-x/(y-x)^2 )^μ_1...μ_n/(z-x)^2(α+β)(z-y)^2α' =
(x-y)^μ_1...μ_n/(x-y)^2(n+β) .
In full analogy with the scalar case this relation
is obtained from (<ref>) by sending $x_2 \to \infty$ and
changing the notation $x_1\to x$.
Relation (<ref>) can be rewritten in an equivalent compact form
as follows:
∫d^D z
⟨z,x |Ψ^μ_1⋯μ_n_α,β,y⟩/
(z-x)^2β = τ(α,β,n)
and it will be used later. Note that relation (<ref>)
generalizes the identity (<ref>).

Remark 2.  The analog of
Proposition <ref>
for fixed $D=4$ and $\beta=1$ was proven in [39].

§.§ Properties of the graph building operator and
modified scalar product in $V_1 \otimes V_2$

Here we list properties of the graph building operator
Note that with respect to the standard Hermitian
scalar product in $V_1 \otimes V_2$
⟨Ψ| Φ⟩=
∫d^D x_1   d^D x_2
⟨Ψ| x_1,x_2 ⟩⟨x_1,x_2| Φ⟩=
∫d^D x_1 d^D x_2   Ψ^*(x_1,x_2)  Φ(x_1,x_2)   ,
the operator (<ref>)
is Hermitian, for $\beta \in \mathbb{R}$,  up to the equivalence
1/a(β)   (q̂_12)^-2β    (_1)^-2β
P_12 = U   Q̂_12^(β)   U^-1  ,
U:= P_12   (q̂_12)^-2β=
(q̂_12)^-2β   P_12   .
It means that the operator $\hat{Q}^{(\beta)}_{12}$
is Hermitian with respect to the modified
scalar product [48]

\begin{align}
\lb{scprod}
\langle \overline{\Psi}|\Phi\rangle :=
\langle \Psi| \, U \, |\Phi\rangle =
\int d^D x_1 d^D x_2\frac{\Psi^{*}(x_2\,,x_1) \,\Phi(x_1\,,x_2)}{
(x_1-x_2)^{2\beta}} \; ,
\end{align}

where $\langle \Psi|$ is the standard Hermitian conjugation of the
vector $|\Psi \rangle$, and we introduce new conjugation

\begin{equation}
\label{nconj}
\langle \overline{\Psi} | :=  \langle \Psi|U = \langle \Psi
\; .
\end{equation}

The operator $U$ in (<ref>),
(<ref>) plays the role of
the metric in the space $V_1 \otimes V_2$.
Since the operator $\hat{Q}^{(\beta)}_{12}$
is Hermitian with respect to the
scalar product (<ref>),
the eigenfunctions of $\hat{Q}^{(\beta)}_{12}$ with
different eigenvalues should be orthogonal (we use this below).

It is evident that the graph building operator
(<ref>) commutes with the dilatation operator:
D̂ = i/2 ∑_a=1^2 (_a _a + _a _a)
+ 1/2(y^μ  ∂_y^μ
+ ∂_y^μ   y^μ) - β,
which acts on the eigenvector
$| \Psi_{\alpha,\beta,y}^{\mu_1...\mu_n}\rangle$ as following
| Ψ_α,β,y^μ_1...μ_n⟩=
(2α+β-1/2 D -n )
| Ψ_α,β,y^μ_1...μ_n⟩  ,
and, for $\beta \in \mathbb{R}$, satisfies
$\hat{\sf D}^\dagger = - U \, \hat{\sf D}\, U^{-1}$
(to prove formula (<ref>), one needs to act
on both sides of (<ref>) by the vector $\langle x_1,x_2|$).
the operator $\hat{\sf D}$ is anti-Hermitian
with respect to the scalar product (<ref>), and
the corresponding condition on its eigenvalue gives (cf.
2(α^* + α) = 2n +D-2β     ⇒
α= 1/2(n + D/2 - β) - i ν  ,

ν∈ℝ   .
It is a remarkable fact that under this condition,
the eigenvalue (<ref>) is real
τ(α,β,n) =  (-1)^n
π^D/2 Γ(β)   Γ(D/4 + n/2 -
β/2 + i ν)   Γ(D/4 + n/2 -
β/2 - i ν)/ Γ(β')
Γ(D/4 + n/2 +β/2 + i ν)
Γ(D/4 + n/2 +β/2 - i ν)   ,
and the parameter $\Delta$ in (<ref>)
acquires the form $\Delta = \frac{D}{2} + 2 i \nu$ (see

Note that Proposition
<ref> holds for an arbitrary parameter $\alpha$, but the
set of functions (<ref>) forms a complete orthogonal system
only in case
when $\Delta = D/2 + 2i\nu$, which imposes restrictions on the
parameter $\alpha$.

§.§ Orthogonality and completeness
for the eigenfunctions $| \Psi_{\nu,x}^{\mu_1...\mu_n}\rangle$

Since the eigenvalue (<ref>) is real (it is invariant under
the transformation $\nu \to - \nu$), two eigenvectors
$| \Psi_{\nu,\beta,x}^{\mu_1...\mu_n}\rangle$ and
$| \Psi_{\lambda,\beta,y}^{\nu_1...\nu_m}\rangle$,
having different eigenvalues (<ref>)
(e.g. $n \neq m$ and $\lambda \neq \pm \nu$), should be
orthogonal to each other with respect to
the scalar product (<ref>).

It is indeed the fact and moreover,
we have the following more general statement –
the orthogonality condition
for two general conformal triangles (see, e.g.,
[26], [27], [68], [39]).
The scalar product in this more general situation is defined
in a similar to (<ref>) way

\begin{equation}
\lb{Cscprod}
\langle \overline{\Psi}|\Phi\rangle :=
\langle \Psi| \, U \, |\Phi\rangle =
\int d^D x_1 d^D x_2\frac{\Psi^{*}(x_2\,,x_1) \,\Phi(x_1\,,x_2)}{
(x_1-x_2)^{2(D-\Delta_1-\Delta_2)}} \; ,
%(\beta \equiv D-\Delta_1 - \Delta_2) \; ,
\end{equation}

where $\langle \Psi|$ is the standard Hermitian conjugation of the
vector $|\Psi \rangle$ and we introduce new conjugation

\begin{equation}
\label{Cnconj}
\langle \overline{\Psi} | :=  \langle \Psi|U = \langle \Psi
\; .
\end{equation}

Eigenfunctions (<ref>) form an
orthogonal system of functions and the following orthogonality relation

\begin{multline}
\label{trig-prod}
\langle \overline{\Psi_{\lambda,y}^{\nu_1...\nu_m}}|
\Psi_{\nu,x}^{\mu_1...\mu_n}\rangle =
\int \! d^D x_1 \, d^D x_2
\frac{\Bigl(\Psi^{\nu_1...\nu_m}_{\lambda,y}(x_2\,,x_1)
\Bigr)^{*} \Psi^{\mu_1...\mu_n}_{\nu,x}(x_1\,,x_2)}
{(x_1-x_2)^{2(D-\Delta_1-\Delta_2)}} = \\ C_1(n\,,\nu)\,\delta_{n
m}\,\delta(\nu -\lambda)\,\delta^D(x-y)\,P^{\mu_1 ... \mu_n}_{\nu_1 ...
\nu_n} +
C_2(n\,,\nu)\,\delta_{n m}\,\delta(\nu +\lambda)\,
\frac{S^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}(x,y)}{(x-y)^{D+4i\nu}},
\end{multline}

where $P^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}$ is the projector on
the symmetric traceless tensors
(see (<ref>) Appendix A) and

\begin{align*}
S^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}(x,y) =
P^{\mu_1 ... \mu_n}_{\alpha_1 ... \alpha_n}
\left(\delta^{\alpha_1}_{\beta_1} -
\left(\delta^{\alpha_n}_{\beta_n} -
P^{\beta_1 ... \beta_n}_{\nu_1 ... \nu_n}.
\end{align*}

The explicit form of the coefficients $C_1$ and $C_2$ is

\begin{align}\label{C1}
C_1(n\,,\nu)  =
\frac{(-1)^n\,2^{1-n}\,\pi^{3D/2 + 1}\,n!\,
\Gamma\left(2i\nu\right)\Gamma\left(-2i\nu\right)}{
\Gamma\left(\frac{D}{2} + n\right)
\left(\left(\frac{D}{2}+n-1\right)^2 + 4\nu^2\right)\,
\Gamma\left(\frac{D}{2}+2i\nu-1\right)\,
\Gamma\left(\frac{D}{2}-2i\nu-1\right)}\\
\label{C2}
C_2(n\,,\nu) =
\pi^{D+1}\,\frac{n!}{2^{n-1}}\,C_{\Delta_1\Delta_2}(n\,,\nu)\,
\frac{\Gamma\left(2i\nu\right)\Gamma\left(\frac{D}{2}+2i\nu-1+n\right)}
\Gamma\left(\frac{D}{2}+2i\nu-1\right)
\Gamma\left(\frac{D}{2}+n\right)}\,,
\end{align}

where for simplicity we use special notation
for the following combination of $\Gamma$-functions

\begin{align}
\label{C}
C_{\Delta_1\Delta_2}(n\,,\nu) =
\frac{\Gamma\left(\frac{D}{4}-\frac{\Delta_1-\Delta_2}{2}+
\frac{n}{2}-i\nu\right)}
\frac{n}{2}+i\nu\right)}\,
\frac{\Gamma\left(\frac{D}{4}+\frac{\Delta_1-\Delta_2}{2}+
\frac{n}{2}-i\nu\right)}
\frac{n}{2}+i\nu\right)}\,
\end{align}

Proof. The proof of this statement is very technical and
and we move it to Appendix D.

Remark 1. Formula (<ref>) for the scalar product
was obtained and discussed without its detailed proof in many papers
(see e.g.  [64], [26, 27, 68], [39]).
That is why we decided
to give here our direct
derivation of this formula
(see Appendix D).
For example, the two-dimensional analog of (<ref>) is
given by L.Lipatov in [64] and in a completely
different context of the representation theory
of $SL(2,\mathbb{C})$ [66] by M.Naimark in [65].
Our consideration in Appendix D is very similar to
the two-dimensional derivation given in [61].

Remark 2.
The general conformal triangle
$| \Psi_{\nu,x}^{\mu_1...\mu_n}\rangle$  (<ref>)
depends on two arbitrary real parameters $\Delta_1$ and $\Delta_2$.
The eigenfunction $| \Psi_{\nu,\beta,x}^{\mu_1...\mu_n}\rangle$
is obtained from the general conformal triangle
$| \Psi_{\nu,x}^{\mu_1...\mu_n}\rangle$  by reduction
when the parameters
are fixed in a special way
Δ_1 = D/2   ,     Δ_2 = D/2-β .
The choice of a specific scalar product for the functions $|
\Psi_{\nu,\beta,x}^{\mu_1...\mu_n}\rangle$ was motivated in
Section <ref> (see also
Section <ref>  below), and indeed due to the relation
$D - \Delta_1 - \Delta_2 = \beta$ it coincides with the
scalar product from Proposition <ref>.
Thus, the scalar product of two eigenfunctions
$| \Psi_{\nu,\beta,x}^{\mu_1...\mu_n}\rangle$ is  taken
as in the left-hand side of (<ref>).
We also note that the coefficient $C_1$ does not depend
on $\Delta_1,\Delta_2$ and plays an important role
as the inverse of the Plancherel measure used
in the completeness condition (see below). In contrast to this,
the coefficient $C_2$ in (<ref>)
depends on $\Delta_1,\Delta_2$, but the explicit form for $C_2$
will not be important for us.

Now we turn over to a consideration
of the completeness relation. Since
the orthogonality condition (<ref>)
is known, the natural conjecture
for the completeness relation
(resolution of unity) has the
following form

\begin{equation}
\label{complet2}
\sum_{n = 0}^{\infty}\int\limits_{-\infty}^{+\infty} d\nu \mu(n,
\nu) \int d^Dx |\Psi_{\nu,x}^{\mu_1\ldots\mu_n}\rangle\langle
\overline{\Psi_{\nu,x}^{\mu_1\ldots\mu_n}}| = \operatorname{I},
\end{equation}

where $\operatorname{I}$ is the unity operator
in $V_1 \otimes V_2$, and the conjugated vector
$\langle\overline{\Psi}|$ is defined in (<ref>).

The integration measure $\mu(n,\nu)$
in (<ref>) has the following form:

\begin{equation}
\label{measure}
\mu(n\,,\nu)= \frac{1}{2}\frac{1}{C_1(n\,,\nu)},
\end{equation}

where $C_1$ is a constant (<ref>).

Proof. Let us rewrite relation (<ref>)
in terms of integral kernels applying the
vectors $\langle x_1, x_2|$ from the left
and $|x_3, x_4 \rangle$ from the right

\begin{multline*}
\delta^D(x_{13})\delta^D(x_{24}) = \sum_{n=0}^{\infty}
\int_{-\infty}^{\infty} d \nu \mu(n\,,\nu)
\frac{1}{x_{12}^{2\left(D-\Delta_1-\Delta_2\right)}}\,
\int d^D x\, \Psi^{\mu_1...\mu_n}_{n,-\nu,x}(x_1\,,x_2)\,
\Psi^{\mu_1...\mu_n}_{n, \nu,x}(x_4\,,x_3),
\end{multline*}

where we used $\langle \Psi^{\mu_1...\mu_n}_{\nu,x} |x_1,x_2\rangle=
(\Psi^{\mu_1...\mu_n}_{\nu, x}(x_1, x_2))^* =
\Psi^{\mu_1...\mu_n}_{-\nu, x}(x_1, x_2)$. For further convenience, we
change the
variables and represent the previous relation
in a slightly different form

\begin{multline*}
\delta^D(x_{13})\delta^D(x_{24}) = \sum_{m=0}^{\infty}
\int_{-\infty}^{\infty} d \lambda\, \mu(m\,,\lambda)
\frac{1}{x_{12}^{2\left(D-\Delta_1-\Delta_2\right)}}\,
\int d^D y \,\Psi^{\nu_1...\nu_m}_{m,-\lambda,y}(x_1\,,x_2)\,
\Psi^{\nu_1...\nu_m}_{m,\lambda,y}(x_4\,,x_3)
\end{multline*}

Then we multiply both sides of this relation by
integrate over $x_3$ and $x_4$ and use formula
(<ref>) for the scalar product of two
conformal triangles written in the form

\begin{multline*}
\langle\overline{
\Psi^{\nu_1...\nu_m}_{-\lambda,y}}|
\Psi^{\mu_1...\mu_n}_{\nu,x}\rangle =
C_1(n\,,\nu)\,\delta_{n m}\,\delta(\nu +\lambda)\,\delta^D(x-y)\,
P^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n} + \\
C_2(n\,,\nu)\,\delta_{n m}\,\delta(\nu -\lambda)\,
\frac{S^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}(x,y)}
\end{multline*}

and obtain

\begin{multline*}
\Psi^{\mu_1...\mu_n}_{\nu,x}(x_1\,,x_2) =
\int_{-\infty}^{\infty} d \lambda\, \mu(n\,,\lambda) \int d^D y
\Psi^{\nu_1...\nu_n}_{-\lambda,y}(x_1\,,x_2)\,\\
\left(
C_1(n\,,\nu)\,\delta(\nu +\lambda)\,\delta^D(x-y)\,
P^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n} +
C_2(n\,,\nu)\,\delta(\nu -\lambda)\,
\frac{S^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}(x,y)}
{(x-y)^{2\left(\frac{D}{2}+2i\nu\right)}}\right) = \\
\mu(n\,,-\nu)\,C_1(n\,,\nu)\,\Psi^{\mu_1...\mu_n}_{n,\nu,x}(x_1\,,x_2)
\int d^D y\, \Psi^{\nu_1...\nu_n}_{-\nu,y}(x_1\,,x_2)\,
\frac{S^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}(x,y)}
\end{multline*}

Now we need relation (<ref>) with change $\nu \to -\nu$

\begin{multline*}
\int \mathrm{d}^D\, z\,
\frac{S^{\mu_1...\mu_n}_{\nu_1...\nu_n}(x,z)}
\Psi^{\nu_1...\nu_n}_{-\nu,z}(x_1\,,x_2) = \\
\Psi^{\mu_1...\mu_n}_{\nu,x}(x_1\,,x_2)
\frac{\Gamma(-2i\nu)\Gamma\left(\frac{D}{2}-2i\nu+n-1\right)}
\end{multline*}

so that

\begin{multline*}
\Psi^{\mu_1...\mu_n}_{\nu,x}(x_1\,,x_2) =
\mu(n\,,-\nu)\,C_1(n\,,\nu)\,\Psi^{\mu_1...\mu_n}_{n,\nu,x}(x_1\,,x_2)
\, + \\
\mu(n\,,\nu)\,C_2(n\,,\nu)\,\Psi^{\mu_1...\mu_n}_{n,\nu,x}(x_1\,,x_2)
\frac{\Gamma(-2i\nu)\Gamma\left(\frac{D}{2}-2i\nu+n-1\right)}
= \\
= \mu(n\,,-\nu)\,C_1(n\,,\nu)\,\Psi^{\mu_1...\mu_n}_{\nu,x}(x_1\,,x_2)
\end{multline*}

where at the last step we used the consistency relation (<ref>).
First of all, note that $C_1(n\,,-\nu) = C_1(n\,,\nu)$.
Secondly, we should have
$\mu(n\,,-\nu) = \mu(n\,,\nu)$ due to the fact that the convolution of
conformal triangles inside the completeness relation
is invariant under the change $\nu \to -\nu$.
The last statement should be checked with the help of two relations
(<ref>) and (<ref>).
Finally, we obtain relation (<ref>)
and arrive at the completeness relation in the form (<ref>).

Remark 1. Using the
above mentioned statement that the convolution of conformal triangles
inside the completeness relation is invariant
under the change $\nu \to -\nu$, it is possible to rewrite
the completeness relation in an equivalent form
(see e.g. [26], [27], [68], [39])

\begin{multline}\label{complet1}
\delta^D(x_{13})\delta^D(x_{24}) = \sum_{n=0}^{\infty}
\int_{0}^{\infty} \frac{d \nu}{C_1(n\,,\nu)}
\frac{1}{x_{12}^{2\left(D-\Delta_1-\Delta_2\right)}}\,
\int d^D x\, \Psi^{\mu_1...\mu_n}_{-\nu,x}(x_1\,,x_2)\,
\Psi^{\mu_1...\mu_n}_{\nu,x}(x_4\,,x_3)
\end{multline}

Remark 2.
The concise version of
the completeness condition (<ref>)
(or resolution of unity $I$)
for the basis eigenfunctions (<ref>) is written as (cf.

I = ∑_n=0^∞
∫_0^∞ d ν/C_1(n ,ν) ∫d^D x
| Ψ^μ_1...μ_n_ν,x⟩⟨Ψ^μ_1...μ_n_ν,x  |
= ∑_n=0^∞
∫_0^∞ d ν/C_1(n ,ν) ∫d^D x
| Ψ^μ_1...μ_n_ν,x⟩⟨Ψ^μ_1...μ_n_ν,x
|  U   .

Remark 3. Due to the degeneracy $\nu \to -\nu$,
it is possible to restrict everything to the case
of positive $\nu$ from the very beginning.
In this case, the term with $C_2(n\,,\nu)$ disappears from the
orthogonality relation (<ref>) and the form of the measure in
completeness relation (<ref>) can be fixed using the
orthogonality relation for the positive $\nu$. We used the
orthogonality relation  for
arbitrary $\nu$ for the sake of completeness
of the presentation and for additional
nontrivial crosschecks of the derived constants
$C_1(n,\nu)$ and $C_2(n,\nu)$ and the consistency relation (<ref>).

Remark 4. Note that we have fixed the measure
in the completeness relation but the whole completeness
relation needs independent proof which is given in
[26, 27, 68].
In the two-dimensional case, the completeness is proved in the context
of representation theory of $SL(2,\mathbb{C})$ in
[66, 65].
The simple and direct proof of the completeness in the two-dimensional
case is presented in [67].

Let us point out the four-dimensional case,
i.e. $D = 4, \beta = 1$. In this case,
a complete orthogonal set of functions has parameters $\alpha =
\frac{n + 1}{2} - i\nu$ and the following function is shown in

Set of eigenfunctions in case $D = 4$ and $\beta = 1$

For these functions we have

\begin{equation}
\hat{Q}_{12}|\Psi_{\nu,x}^{\mu_1...\mu_n}\rangle =
\tau(\nu, n)|\Psi_{\nu,x}^{\mu_1...\mu_n}\rangle,
\end{equation}

\begin{equation}\label{4-egvalue}
\tau(\nu, n) = (-1)^n\dfrac{4\pi^2}{(1+n)^2 + 4\nu^2}.
\end{equation}

Integration measure (<ref>) in the completeness relation is

\begin{equation}\label{4-measure}
\dfrac{1}{\mu(n, \nu)} = \dfrac{\pi^5}{2^{n + 2}(n +
1)\nu^2}\,\tau(\nu, n)
\end{equation}

and we can write spectral decomposition for the operator

\begin{equation}
\hat{Q}_{12} = \sum_{n =
d\nu\,\mu(n, \nu)\,\tau(\nu, n)\,\int d^4x\,
\overline{\Psi^{\mu_1...\mu_n}_{\nu,x}}|
\end{equation}

§ FOUR-POINT AND TWO-POINT CORRELATION
FUNCTIONS FOR ZIG-ZAG DIAGRAMS

Substitution of the resolution of unity (<ref>) into
expressions (<ref>), (<ref>) for zig-zag 4-point
Feynman graphs gives

\begin{multline}\label{cor09}
G^{(M)}_4(x_1,x_2;y_1,y_2) =
\langle x_1,x_2 | \bigl(\hat{Q}_{12}^{(\beta)}\bigr)^{M}| y_1,y_2
\rangle
(y_1-y_2)^{2\beta} = \\
= \sum\limits_{n=0}^{\infty}
\int\limits_{0}^{\infty} \frac{d \nu}{C_1(n\,,\nu)} \int d^D x\,
\langle x_1,x_2 | \bigl(\hat{Q}_{12}^{(\beta)}\bigr)^{M}|
\Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x}\rangle
\langle \Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x} |\, U  | y_1,y_2 \rangle
(y_1-y_2)^{2\beta}  = \\
= \sum\limits_{n=0}^{\infty}
\int\limits_{0}^{\infty} d \nu \,
\frac{\bigl(\tau(\alpha,\beta,n)\bigr)^M}{
C_1(n\,,\nu)}  \int d^D x\,
\langle x_1,x_2 |\Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x}\rangle
\langle \Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x} |y_2,y_1 \rangle \, ,
\end{multline}

where the integral over $x$ in the right-hand side
of (<ref>) is evaluated
in terms of conformal blocks [32], [33],
[68]
(in the four-dimensional case, this integral was considered in detail
in [39]).

Making use of the standard relations between the
4-point zig-zag functions $G^{(M)}_4(x_1,x_2;y_1,y_2)$ constructed
in (<ref>)
and 2-point zig-zag functions
$G^{(M)}_2(x_2,y_1)$ (the graphs for these functions are presented
in (<ref>) – (<ref>)),
we write explicit expressions
for the 2-point $M$-loop zig-zag diagrams as follows:

\begin{multline}\label{cor12}
= \sum\limits_{n=0}^{\infty}
\int\limits_{0}^{\infty} d \nu
\frac{(\tau(\alpha,\beta,n))^M}{C_1(n\,,\nu)}
\, \int d^D x_1 d^D y_2 \, d^D x\,
\frac{\langle x_1,x_2 |\Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x}\rangle
\langle \Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x} |y_2,y_1 \rangle}{
(x_1-x_2)^{2\beta}(y_1-y_2)^{2\beta}} = \\
= \frac{1}{(x_2-y_1)^{2\beta}} \frac{\Gamma(D/2-1)}{\Gamma(D-2)}
\sum\limits_{n=0}^{\infty}
\frac{(-1)^n \Gamma(n+D-2)}{2^{n} \Gamma(n+D/2-1)}
\int\limits_{0}^{\infty} d \nu \;
\frac{\tau^{M+3}(\alpha,\beta,n)}{C_1(n\,,\nu)} \, ,
\end{multline}

where we apply the two-point master integral

\begin{multline}\label{cor13}
\int d^D x_1 d^D y_2 \, d^D x\,
\frac{\langle x_1,x_2 |\Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x}\rangle
\langle \Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x} |y_2,y_1 \rangle}{
(x_1-x_2)^{2\beta}(y_1-y_2)^{2\beta}} = \\
= \frac{(-1)^n \Gamma(n+D-2)\Gamma(D/2-1)}{2^{n}
\Gamma(n+D/2-1)\Gamma(D-2)}
\, \frac{\tau^3(\alpha,\beta,n)}{(x_2-y_1)^{2\beta}} \; .
\end{multline}

To calculate this integral, we apply
identity (<ref>) in the form
∫d^D x_1
⟨x_1,x_2 |Ψ^μ_1⋯μ_n_ν,β,x⟩/
(x_1-x_2)^2β = τ(α,β,n)
and the corresponding complex conjugate relation
∫d^D y_2
⟨Ψ^μ_1⋯μ_n_ν,β,x |y_2,y_1 ⟩/
(y_1-y_2)^2β = τ(α,β,n)
(y_1-x)^μ_1...μ_n/(y_1-x)^2(n+α̅')   .
where we take into account $\langle x_1,x_2
|\Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x}\rangle^* =
\langle \Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x} |x_1,x_2\rangle$
and $\tau(\bar{\alpha},\beta,n) = \tau(\alpha,\beta,n)$.
Finally, for the left-hand side of (<ref>) we obtain

\begin{array}{c}
\int d^D x_1 d^D y_2 \, d^D x\,
\frac{\langle x_1,x_2 |\Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x}\rangle
\langle \Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x} |y_2,y_1 \rangle}{
(x_1-x_2)^{2\beta}(y_1-y_2)^{2\beta}} =
\tau^2(\alpha,\beta,n) \; \int  d^D x\,
\frac{(x-x_2)^{\mu_1...\mu_n}}{(x-x_2)^{2(n+\alpha')}}
\frac{(x-y_1)^{\mu_1...\mu_n}}{(x-y_1)^{2(n+\bar{\alpha}')}}
\; \stackrel{(\ref{zig04})}{=} \\ [0.3cm]
= \tau^2(\alpha,\beta,n)
\frac{\pi^{D/2} \Gamma(\alpha)\Gamma(\bar{\alpha})\Gamma(\beta)}{
\Gamma(\alpha'+n)\Gamma(\bar{\alpha}'+n)\Gamma(\beta')} \cdot
\frac{\Gamma(n+D-2)\Gamma(D/2-1)}{2^{n} \Gamma(n+D/2-1)\Gamma(D-2)}
\cdot \frac{1}{(x_2-y_1)^{2\beta}} \stackrel{(\ref{cor04})}{=}
\\ [0.3cm]
= (-1)^n \tau^3(\alpha,\beta,n) \cdot
\frac{\Gamma(n+D-2)\Gamma(D/2-1)}{2^{n} \Gamma(n+D/2-1)\Gamma(D-2)}
\cdot \frac{1}{(x_2-y_1)^{2\beta}}
\end{array}

where $\alpha = \frac{1}{2}(n+\frac{D}{2}-\beta)+i\nu$ and
we have used the identity (<ref>)
derived in Appendix C.

The integral over $\nu$ in the right-hand side of (<ref>)
for $\beta=1$ and even $D >2$ can be evaluated explicitly
and gives a linear combination of $\zeta$-values with
rational coefficients.
We will publish the explicit formula for (<ref>) elsewhere.
Here we consider only one special case
$\beta=1$ and $D=4$, which is needed to prove the
Broadhurst and Kreimer conjecture [12]
for zig-zag diagrams (we use the
approach of the paper [48]).
In this case we have
and the master integral (<ref>)
is simplified

∫d^4 x_1 d^4 y_2   d^4 x
⟨x_1,x_2 |Ψ^μ_1⋯μ_n_ν,x⟩⟨Ψ^μ_1⋯μ_n_ν,x |y_2,y_1 ⟩/
=  (-1)^n (n+1)/2^n   τ^3(ν,n)
1/(x_2-y_1)^2   ,

$\langle x_1,x_2 |\Psi^{\mu_1\cdots\mu_n}_{\nu,x}\rangle :=
\Psi^{\mu_1\cdots\mu_n}_{\nu,\beta,x}(x_1,x_2)
|_{D=4,\beta=1}$ and (see (<ref>))
τ(ν,n) := . τ(α,β,n)|_D=4,β=1 =
(-1)^n (2π)^2/(1+n)^2+4ν^2   .
The coefficient $C_1$ in (<ref>) for $D=4$ and $\beta=1$
is reduced to
C_1(n,ν) =
π^5/2^n+3(1+n)  ν^2  τ(ν,n)    .
Finally we substitute (<ref>) – (<ref>)
into (<ref>) for $D=4$ and obtain

\begin{multline}\label{cor17}
%=\frac{1}{(x_2-y_1)^2} \frac{2^3}{\pi^5}
%\sum\limits_{n=0}^{\infty}(-1)^n\, (n+1)^2\,
%\int\limits_{0}^{\infty} d \nu \,\nu^2\, (\tau(\nu,n))^{M+2}
= \\
= \frac{(2\pi)^{2(M+2)}}{(x_2-y_1)^2} \frac{2^3}{\pi^5}
\sum\limits_{n=0}^{\infty}(-1)^{n(M+1)} (n+1)^2
\int\limits_{0}^{\infty} d \nu \,
\frac{\nu^2}{\left((1+n)^2+4\nu^2\right)^{M+2}}  = \\
= \frac{4 \pi^{2M}}{(x_2-y_1)^2} \, {\sf C}_M  \,
%\frac{4 \pi^{2M}}{(M+1)}\binom{2M}{M}
\sum\limits_{n=0}^{\infty}(-1)^{n(M+1)}
\frac{1}{(n+1)^{2M-1}} \; ,
%= \\= \frac{4 \pi^{2M}}{(x_2-y_1)^2} C_M
%\frac{1}{p^{2(M+1)-3}} \; ,
\end{multline}

where ${\sf C}_M = \frac{1}{(M+1)} \binom{2M}{M}$ is the Catalan
In the last equality in (<ref>) we used the integral

\begin{align}
\int_{0}^{+\infty} d\nu\,
\frac{\nu^2}{\left(4\nu^2+\left(1+n\right)^2\right)^{M+2}} =
\frac{1}{2^5(n+1)^{2M+1}}\,
\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(M+\frac{1}{2}\right)}
\end{align}

and applied the identity
= \frac{(2M)!\; \pi}{2^{2M} M!(M+1)!}$.

Thus, we get the result (<ref>) for the two point correlation
function related to the zig-zag series,
which can be written in the form
G^(M + 1)_2(x,y) = π^2M/(x-y)^2   Z(M+1)   ,

\begin{align}\label{BK}
Z(M+1) &= 4 {\sf C}_M
\sum\limits_{p=1}^{\infty}
\frac{(-1)^{(p-1)(M+1)}}{p^{2(M+1)-3}} =\footnotesize
\left\{
\begin{array}{l}
4 \, C_M \, \zeta_{2M-1} \;\; {\rm for} \;\; M =2N+1 \, ,\\ [0.2cm]
4\, C_M \, (1 - 2^{2(1-M)}) \, \zeta_{2M-1} \;\;
{\rm for} \;\; M=2N \, .
\end{array}
\right.
\end{align}

Finally we note
that D.Broadhurst and D.Kreimer fixed
in their paper [12]
the loop measure for each integration over
loop momenta $k$ as $\frac{d^4 k}{\pi^2}$.
Expression (<ref>) is related
to the $M$ loop zig-zag diagram
(it corresponds to the $n=(M+1)$ loop contribution to
the $\beta$-function of $\phi^4_{D=4}$ theory). Therefore,
we have to divide our answer in (<ref>)
by $(\pi^{2})^M$.
In this case our result (<ref>) justifies the
normalization factor $(\pi^{2})^M$ in relation (<ref>)
which together with (<ref>)
states the Broadhurst and Kreimer conjecture [12].

§ GENERAL GRAPH BUILDING OPERATOR AND YANG-BAXTER

§.§ Conformal algebra and scalar conformal

In this Subsection we prove that the modified
scalar product (<ref>) is needed
also for the hermitian property
of the Casimir operator (see the definition below) for
the conformal Lie algebra $\textsf{conf}(\mathbb{R}^{D})$
in the representation which acts in the space $V_1 \otimes V_2$.
For this purpose, we
summarize some known facts about the
$D$-dimensional conformal Lie algebra
and its field representations [78]
(see also
[75] and references therein).
Here we denote by $\textsf{conf}(\mathbb{R}^{D}) = so(1,D+1)$
the Lie algebra
of the conformal group in $\mathbb{R}^{D}$ with the basis elements
$T_{AB} = - T_{BA}$ $(A,B = 0,1,\dots,D,D+1)$,
and use the notation:

T_{\mu \nu} = L_{\mu \nu} \; , \;\;\;
T_{0 \mu} = \frac{1}{2}(P_{\mu}+K_{\mu})\; , \;\;\;
T_{D+1,\mu} = \frac{1}{2}(P_{\mu}-K_{\mu}) \; , \;\;\;
{\cal D} = T_{D+1,0} \; ,

where $\mu,\nu = 1,\dots,D$ and $\mathbb{R}^{1,D+1}$-metric is
$\eta_{00}=\eta_{11} =...=\eta_{_{DD}}=-\eta_{_{D+1,D+1}}=1$.
The defining relations
for $\textsf{conf}(\mathbb{R}^{D})$ are:

[ {\cal D}\, ,\, P_{\mu} ] = i\, P_{ \mu }\ \,, \ \ \  [ {\cal D}\, ,\,
K_{\mu} ]  = - i\, K_{\mu}\ \,,
\ \ \  [ L_{\mu \nu}\, ,\, L_{\rho \sigma} ]  = i\,( \delta_{\nu
\rho}\, L_{\mu
\sigma} + \delta_{\mu \sigma}\, L_{\nu \rho} - \delta_{\mu \rho}\,
\sigma} - \delta_{\nu \sigma}\, L_{\mu \rho} )

\begin{equation}
\label{cnfA}
[ K_{\rho}\, ,\, L_{\mu \nu} ] = i\,( \delta_{\rho \mu}\, K_{\nu} -
\delta_{\rho
\nu}\, K_{\mu} ) \; , \;\;\;  [ P_{\rho}\, ,\, L_{\mu \nu} ]  = i\, (
\delta_{\rho
\mu}\, P_{\nu} - \delta_{\rho \nu}\, P_{\mu} ) \; ,
\end{equation}

[ K_{\mu}\, ,\, P_{\nu} ] = 2 i\, ( \delta_{\mu \nu}\, {\cal D} -
L_{\mu \nu} ) \; , \;\;\;
[ P_{\mu}\, ,\, P_{\nu} ] = 0 \; , \;\;\; [ K_{\mu}\, ,\, K_{\nu} ] = 0
\; , \;\;\;
[ L_{\mu \nu}\, ,\, {\cal D} ] = 0 \; .

Note that the elements $L_{\mu \nu}$ generate the Lie
subalgebra $so(D)$ in the conformal algebra
$\textsf{conf}(\mathbb{R}^{D}) = so(1,D+1)$.
The quadratic Casimir operator for
$\textsf{conf}(\mathbb{R}^{D}) = so(1,D+1)$ is

\begin{equation}
\label{cnfA2b}
\widehat{C}_2  = \frac{1}{2} T_{AB} \, T^{AB} =
\frac{1}{2}\left( L_{\mu \nu} L^{\mu \nu} +  P_{\mu}
K^{\mu} + K_{\mu} P^{\mu} \right) - {\cal D}^2  \; .
\end{equation}

The standard
realization $\rho$ of the elements
$T_{AB}=\{L_{\mu \nu} , P_{\mu},K_{\mu},{\cal D} \}$ of the algebra
(<ref>) is [78]:

\begin{equation}
\label{cnfA1}
\begin{array}{c}
\rho(P_{\mu}) =
%- i \partial_{x_\mu}  \equiv
\hat{p}_{\mu} \; , \;\;\;\;
\rho({\cal D}) = \q^{\mu} \, \hat{p}_{\mu} - i \Delta \; ,
\;\;\;
\rho(L_{\mu \nu}) = \hat{\ell}_{\mu \nu} + S_{\mu \nu}   \; ,
\\ [0.2cm]
\rho(K_{\mu}) = 2 \, \q^{\nu} \, (\hat{\ell}_{\nu \mu}
+ S_{\nu \mu})
+ \q^2 \,  \hat{p}_{\mu}
- 2 i \Delta \, \q_{\mu} \; = \;
2 \, \q^{\nu} \, S_{\nu \mu} + 2 \q_\mu (\q^\nu \p_\nu)
- \q^2 \, \p_{\mu} - 2i \Delta \, \q_\mu \; ,
\end{array}
\end{equation}

\hat{\ell}_{\mu \nu} := (\q_{\nu} \hat{p}_{\mu} -  \q_{\mu}
\hat{p}_{\nu}) \; ,

$\q_\mu$, $\p_\mu$ are the generators of the
Heisenberg algebra (<ref>),
$\Delta \in \mathbb{R}$ is the conformal parameter,
$S_{\mu \nu}=-S_{\nu \mu}$ are the spin generators
of $so(D)$ with the same
commutation relations as for the generators
$L_{\mu \nu}$ (see (<ref>)):

\begin{equation}
\lb{spin}
[ S_{\mu \nu} , S_{\rho \sigma} ] =
i( \delta_{\nu \rho} S_{\mu \sigma}
+ \delta_{\mu \sigma} S_{\nu \rho} - \delta_{\mu \rho} S_{\nu \sigma}
\delta_{\nu \sigma} S_{\mu \rho} )  \; ,
\end{equation}

and $[ S_{\mu \nu} , \q_\rho] = 0 = [ S_{\mu \nu} , \hat{p}_\rho]$.
Note that the quadratic
Casimir operator (<ref>)
in the representation (<ref>)  acquires the form:

\begin{equation}
\label{cnfA2z}
\rho(\widehat{C}_2) =
\frac{1}{2} \left( S_{\mu \nu} \, S^{\mu \nu}
- \hat{\ell}_{\mu \nu} \, \hat{\ell}^{\mu \nu} \right) +
C_{(2)} + \frac{1}{4}(4-D)D+\Delta(\Delta - D) =
\frac{1}{2}  S_{\mu \nu}S^{\mu \nu} +\Delta(\Delta - D) \; ,
\end{equation}

where we use

\frac{1}{2} \hat{\ell}_{\mu \nu} \, \hat{\ell}^{\mu \nu} =
\left(\q^2 \, \p^2 + i \,  (D-2) (\q^\nu \, \p_\nu) -
(\q^\nu \, \p_\nu)^2 \right) =  C_{(2)}
+ \frac{1}{4}(4-D)D \; ,

and the operator $C_{(2)}$ was introduced
in (<ref>).
We will only use the
scalar representations $S_{\mu \nu} = 0$ of
$\textsf{conf}(\mathbb{R}^{D}) = so(1,D+1)$.
In this case, we have
ρ(C_2) = Δ(Δ- D)   I_ρ  ,
and $I_\rho$ is the unit operator in the space of
the representation $\rho|_{S_{\mu\nu}=0}$
(we will omit such unit operators below).
Since we deal with a scalar case of the
conformal algebra
representation $\rho$ parameterized by only one parameter,
conformal dimension $\Delta$, we denote the representation by
$\rho^{\Delta}:=  \rho|_{S_{\mu\nu}=0}$.

We recall that
the scalar representation $\rho^{\Delta}$ of the conformal
group acts in the space of the scalar
conformal fields $\Phi_\Delta(x) =
\langle x | \Phi_\Delta \rangle$. An infinitesimal form
of this action is $\delta \Phi_\Delta(x) =
\omega^{AB} \langle x | \rho^{\Delta}(T_{AB})
\Phi_\Delta \rangle$, where $\omega^{AB}$ are the parameters and

\begin{array}{c}
\langle x | \rho^{\Delta}(P_{\mu})
\Phi_\Delta \rangle = - i \partial_{x_\mu} \Phi_\Delta(x)
\; , \;\;\;\;
\langle x | \rho^{\Delta}({\cal D})
\Phi_\Delta \rangle
= - i (x^{\mu} \, \partial_{\mu} +\Delta)
\Phi_\Delta(x) \; , \\ [0.2cm]
\langle x | \rho^{\Delta}(L_{\mu \nu})
\Phi_\Delta \rangle = i (x_\mu \partial_{\nu}-
x_\nu \partial_{\mu})\Phi_\Delta(x) \; ,
\\ [0.2cm]
\langle x | \rho^{\Delta}(K_{\mu})
\Phi_\Delta \rangle  \; = \;
- 2 i \bigl(x_\mu (x^\nu \partial_{\nu})
- \frac{1}{2} x^2 \, \partial_{\mu} + \Delta \, x_\mu\bigr)
\Phi_\Delta(x) \; ,
\end{array}

where $\partial_{\mu} = \frac{\partial}{\partial x_\mu}$.

Let us introduce the operator
C_12 :=
( Δ(C_2) -
C_2 ⊗1 -1 ⊗C_2 )
= T_AB ⊗T^AB   ,
${\bf \Delta}$ is the comultiplication
in the universal enveloping algebra of
$\textsf{conf}(\mathbb{R}^{D}) = so(1,D+1)$, i.e.
${\bf \Delta}(T_{AB}) = 1 \otimes T_{AB} +
T_{AB} \otimes 1$.
Operator (<ref>) is called split (or polarized) Casimir
and plays an important role in the representation theory
of Lie algebras and Lie groups (see e.g. [75]).
It is instructive to present the explicit
form of (<ref>) and (<ref>) in the
representation $\rho^{\Delta_1} \otimes \rho^{\Delta_2}$.
We have[In view of (<ref>),  in the
representation $\rho^{\Delta_1} \otimes \rho^{\Delta_2}$,
the operator
(<ref>) is equal to (<ref>)
up to an additional constant.]

(ρ^Δ_1 ⊗ρ^Δ_2)   C_12 =
(ρ^Δ_1 ⊗ρ^Δ_2)
(ℓ̂_μν ⊗ℓ̂^μν +
P_μ⊗K^μ+ K_μ⊗P^μ- 2 D ⊗D  ) =
= -(_12)^2 (_1_2) +
2 _12^μ   _12^ν  _1μ  _2ν
+2iΔ_2  (_12   _1) -
2iΔ_1  (_12   _2) + 2 Δ_1 Δ_2   ,

where we use a more compact
notation $\q^{\mu}_{12} := \left(\q_1-\q_2\right)^{\mu}$,
$\;\left(\q_{12}\right)^{2\alpha} :=
(\q_{12}^\mu\, \q_{12\, \mu})^\alpha$,
$\;(\q_{12} \, \p_{1}):= \q_{12}^\mu\, \p_{1\, \mu}$, etc,
and subscripts $1$ and $2$ indicate the
generators of the first and second Heisenberg algebras in
the product ${\cal H} \otimes {\cal H}$.
Now we introduce the notation

\begin{align}
\lb{hcaz5}
\widehat{C}_{\Delta_1\Delta_2} :=
(\rho^{\Delta_1} \otimes \rho^{\Delta_2})\,
{\bf \Delta}(\widehat{C}_2) \equiv
(\rho^{\Delta_1} \otimes \rho^{\Delta_2})\,\widehat{C}_{12}
+ \Delta_1\,(\Delta_1-D) + \Delta_2\,(\Delta_2-D) \; ,
\end{align}

and, using (<ref>),
we write the operator $\widehat{C}_{\Delta_1\Delta_2}$
in the equivalent form

(_12)^-Δ_1-Δ_2[ -(_12)^2 (_1_2)
2 _12^μ  _12^ν   _1 μ   _2  ν
] (_12)^Δ_1+Δ_2   -
-   i(Δ_1-Δ_2)_12^μ
(_1  μ+_2 μ)

The equivalence can be checked by direct calculations
with the help of the main formula

(_12)^-2α   D_12
(_12)^2α = D_12 +
2i α _12^μ
(_1  μ-_2 μ)+
2α(2α-D)   ,
D_12 := -(_12)^2  (_1 _2) +
2 _12^μ _12^ν  _1 μ _2 ν
≡( -(_12)^2  δ^μν +
2 _12^μ _12^ν)
_1 μ _2 ν   .

Using identity (<ref>) for $\alpha = D$,
we obtain
[_1 μ _2 ν ,
(-δ^μν (_12)^2 +
2 _12^μ  _12^ν)(_12)^-2D] = 0
D_12^†= (_12)^-2D   D_12  (_12)^2D   ,
where $\dagger$ is the standard Hermitian conjugation with
respect  to the scalar product (<ref>).
Then, we calculate with the help of (<ref>)
and obvious relation
$[\left(\p_1+\p_2\right)^{\nu}, \, f(\q_{12})] = 0$
the conjugation of the operator

C_Δ_1Δ_2^  † =
D_12^†  _12^-Δ_1-Δ_2   +
i(Δ_1-Δ_2)(_1+_2)_μ _12^μ
= _12^Δ_1+Δ_2-2D   D_12
_12^2 D -Δ_1-Δ_2
+i(Δ_1-Δ_2) _12^μ (_1+_2)^μ =

= _12^-2(D-Δ_1-Δ_2) ( P_12
C_Δ_1Δ_2   P_12)
_12^2(D-Δ_1-Δ_2)   .

Thus, the operator $\widehat{C}_{\Delta_1\Delta_2}$
is Hermitian for $\Delta_1\,,\Delta_2 \in \mathbb{R}$
up to the equivalence transformation (cf. (<ref>)):

\begin{align}
\lb{Chermc}
\widehat{C}_{\Delta_1\Delta_2}^\dagger =
U \, \widehat{C}_{\Delta_1\Delta_2} \, U^{-1}\, ,
\;\;\;\;\;  U:= {\cal P}_{12} \,
(\hat{q}_{12})^{-2(D-\Delta_1-\Delta_2)} \, {\cal P}_{12} \, .
\end{align}

It means that the operator $\widehat{C}_{\Delta_1\Delta_2}=
(\rho^{\Delta_1} \otimes \rho^{\Delta_2})\,
{\bf \Delta}(\widehat{C}_2)$
is Hermitian with respect
to the modified scalar product (<ref>).

§.§ General graph building operator

The generalization of the graph building operator
(<ref>) is:
Q_12^(ζ,κ,γ)  : =
1/a(κ)a(γ)  P_12
q̂_12^-2ζ   _1^-2κ   _2^-2γ
q̂_12^-2β    ,
where the parameters are restricted by only one
condition[This condition guarantees
the scale invariance of (<ref>) under transformations
$\q_i \to \lambda \, \q_i$, $\p_i \to \lambda^{-1} \, \p_i$.]
$\zeta + \beta = \kappa + \gamma$ and
we have $\hat{Q}_{12}^{(\beta)} =
\bigr|_{\zeta,\gamma=0}$ .
We depict the integral kernel of the
$D$-dimensional operator
$Q_{12}^{(\zeta,\kappa,\gamma)} $
as follows ($\kappa':=D/2-\kappa$,

$= \; \langle x_1,x_2 |
Q_{12}^{(\zeta,\kappa,\gamma)}  | y_1,y_2 \rangle \, =$

$= \, \frac{1}{a(\kappa)a(\gamma)}
\cdot  \langle x_1,x_2 |\;
{\cal P}_{12} \,\hat{q}_{12}^{-2\zeta} \; \p_1^{-2\kappa}
\,\p_2^{-2\gamma} \,
\hat{q}_{12}^{-2\beta} \, | y_1,y_2 \rangle \, = $

= \;
\frac{1}{(x_1-x_2)^{2\zeta}
(x_2-y_1)^{2\kappa\,'} \, (x_1-y_2)^{2\gamma\,'} \,
(y_1-y_2)^{2\beta}} \; .

Thus, the operator $Q_{12}^{(\zeta,\kappa,\gamma)}$
is the graph building operator for the ladder diagrams
(these diagrams are Fourier dual to the diagrams in Fig.<ref>)

(9.8,2)$. \; .\; . \; . \; .\; . $

$= \;\; (x_1-x_2)^{2\zeta} \,
\langle x_1,x_2 | (\hat{Q}_{12}^{(\zeta,\kappa,\gamma)})^{2N}
| y_1,y_2 \rangle (y_1-y_2)^{2\beta}$,

where four parameters on the lines are restricted
by the condition $\kappa'+\gamma'+\beta+\zeta=D$.

The eigenfunction for the operator
$Q_{12}^{(\zeta,\kappa,\gamma)}$ is
given by the conformal 3-point correlation
function (<ref>)

(0,2)$\langle y_1,y_2 | \Psi_{\delta,\rho}^{\mu_1\cdots\mu_n}(y)
\rangle = $

(10,2)$\Bigl( \frac{y-y_1}{(y-y_1)^2} -
\frac{y-y_2}{(y-y_2)^2} \Bigr)^{\mu_1\cdots\mu_n}\equiv$

where we depict the nontrivial rank-$n$ tensor numerator
as arrows on the lines (the rank is fixed by the modification
of indices on the lines: $\rho \to (\rho,n)$, etc) and denote
2α= Δ_1 + Δ_2 -(Δ-n)   ,
2δ= Δ_1 - Δ_2 +(Δ-n)   ,
2ρ= Δ_2 - Δ_1 +(Δ-n)   ,
i.e., conformal dimensions $\Delta,\Delta_1,\Delta_2$ are
arbitrary parameters in this case. Thus, we have
| Ψ_δ,ρ^μ_1⋯μ_n(y)⟩=
| Ψ_δ,ρ^μ_1⋯μ_n(y) ⟩  ,
where the parameters of (<ref>) are
connected to the parameters of $Q_{12}^{(\zeta,\kappa,\gamma)}$
as follows:
α+ρ= κ' + ζ  ,
α+δ=γ' + ζ  ,
and $\bar{\tau}(\kappa,\gamma;\delta,\alpha;n)$
is the eigenvalue

τ̅(κ,γ;δ,α;n) = (-1)^n ·τ(δ',κ,n)·τ((ρ+κ)',γ,n)    ,

τ(α,β,n) = (-1)^n
π^D/2 Γ(β) Γ(α) Γ(α'-β+n)/
Γ(β') Γ(α'+n) Γ(α+β)    .

Proof. The proof is based on the tensor
generalization of the star-triangle identity
(<ref>) that is depicted as

(7.5,2.3)$= \; \tau(\alpha,\beta,n)$

where the function $\tau(\alpha,\beta,n)$ is defined
in (<ref>) and (<ref>).
We act by the operator (<ref>) on the
wave function (<ref>) and deduce

(6.9,2)$= \tau(\delta',\kappa,n)\cdot$

(15.7,1.9)$\; = \;
\tau(\delta',\kappa,n)\cdot
\tau\bigl((\rho+\kappa)',\gamma,n\bigr)\cdot$

where we apply the star-triangle identity (<ref>) twice
(to the vertices $y_1$ and $y_2$)
and fix the index $\alpha$ on the line $(x_1,x_2)$
in the right-hand side of (<ref>) to obtain
the eigenfunction (<ref>), which
give the conditions

α+β+δ-κ=D/2   ,
ρ+κ=δ+γ  ,
α+ρ+κ-ζ=D/2        ⟹
β+ζ=κ+γ  ,
γ- ζ= D/2 - α- δ  ,
κ- ζ= D/2 - α- ρ  .
The first two relations in (<ref>)
are the conditions for applying (<ref>), and
the third relation in (<ref>) is the result
of fixing $\alpha$ in the right-hand side of (<ref>).
We can see that (<ref>) is the eigenfunction of (<ref>)
only if $\beta+\zeta=\kappa+\gamma$, which was
chosen at the beginning,
and the two last relations in (<ref>) are equivalent to
(<ref>). An additional
factor $(-1)^n$ appears in (<ref>)
since we have to change the arrows to the opposite
in the right-hand side of  (<ref>).
Thus, we have proved (<ref>)
and (<ref>).

Remark 1. We multiply both sides of
(<ref>) by $a(\gamma)$ and take the limit
$\zeta,\gamma \to 0$; as a result,
we reproduce (<ref>).

Remark 2. Substitution of (<ref>) in
(<ref>) gives

\beta - \zeta = D -\Delta_1 - \Delta_2 \; , \;\;\;
\gamma - \zeta = D/2 -\Delta_1 \; , \;\;\;
\kappa - \zeta = D/2 -\Delta_2 \; ,

and after introducing a new notation
$\beta + \zeta = - 2 u$, we deduce
β= 1/2 (D - Δ_1 -Δ_2) - u   ,
ζ= - 1/2 (D - Δ_1 -Δ_2) - u   ,
κ= 1/2(Δ_1 -Δ_2) - u   ,
γ= 1/2(Δ_2 -Δ_1) - u    .
In this case, we obtain the expression for (<ref>)

Q_12^(ζ,κ,γ)  =
1/a(κ)a(γ)   P_12
q̂_12^2(u+ D - Δ_1 -Δ_2/2)
_1^2(u+Δ_2 -Δ_1/2)
_2^2(u+Δ_1 -Δ_2/2)
q̂_12^2(u+Δ_1 +Δ_2-D/2)   .

§.§ Conformal R-operator

We multiply
the graph building operator (<ref>) by
the factor $a(\kappa)a(\gamma)$ and introduce a set of
the $so(D)$-invariant operator functions
R_Δ_iΔ_k(u) = P_ik
that are expressed in terms of the generators of two Heisenberg
algebras ${\cal H}_i$ and ${\cal H}_k$ (see (<ref>)).
A remarkable fact is that the operators (<ref>)
satisfy the Yang-Baxter equation

\begin{align}
\lb{RRR}
R_{\Delta_1\Delta_3}(u) R_{\Delta_2\Delta_3}(v) =
R_{\Delta_1\Delta_3}(u) R_{\Delta_1\Delta_2}(u-v)
\end{align}

Indeed, in [76] we constructed
the $so(D)$-invariant operator

Ř_12^(Δ_1Δ_2)(u)  =
q̂_12^2(u+ D - Δ_1 -Δ_2/2)
_1^2(u+Δ_2 -Δ_1/2)
_2^2(u+Δ_1 -Δ_2/2)
q̂_12^2(u+Δ_1 +Δ_2-D/2) =
P_12   R_Δ_1Δ_2(u)    ,

a solution of the Yang-Baxter equation
Ř^(Δ_2Δ_3)_23(v) =
Ř^(Δ_1Δ_2)_12(u-v)   .
Substitution of the right-hand side of (<ref>)
into (<ref>) gives (<ref>). There are other
forms of the $R$-operator (<ref>), which are useful  in many

Ř_12^(Δ_1Δ_2)(u)      =
q̂_12^2(u - Δ_-)
_1^2(u+ Δ_+')
q̂_12^2(u+ Δ_-)    _2^2Δ_2' =
= _2^-2Δ_1'
q̂_12^2(u + Δ_-)
_2^2(u+ Δ_+')
q̂_12^2(u- Δ_-)    _1^2Δ_1'   ,

where $\Delta_{\pm} = \frac{1}{2}(\Delta_1 \pm \Delta_2)$
and $\Delta_i'=D/2 - \Delta_i$.
These forms demonstrate the
symmetry $\check{R}_{12}^{(\Delta_1\Delta_2)}(u)  =
\check{R}_{21}^{(\Delta_2\Delta_1)}(u)$ and can be
deduced from (<ref>) by means
of the generalization
of the star-triangle identity (<ref>):

\p_i^{2\alpha} \, \q_{ij}^{2(\alpha + \beta)}\, \p_i^{2\beta} =
\q_{ij}^{2\beta}\, \p_i^{2(\alpha + \beta)}\, \q_{ij}^{2\alpha} \; .

The operators (<ref>) are conformal invariant,
i.e. they are invariant under the adjoint action of the
group $\textsf{Conf}(\mathbb{R}^{D}) = SO(1,D+1)$
in the representation $\rho^{\Delta_i} \otimes \rho^{\Delta_k}$.

Proof. The
operator (<ref>) is evidently invariant under translations
$\hat{q}_j^\mu \to \hat{q}_j^\mu + a^\mu$,
$\hat{p}_j^\mu \to \hat{p}_j^\mu$,
$SO(D)$-rotations and dilatations
$\hat{q}_j^\mu \to \lambda \hat{q}_j^\mu$,
$\hat{p}_j^\mu \to \frac{1}{\lambda} \hat{p}_j^\mu$,
which are generated by the adjoint action of translation, rotation
and dilatation elements of
$\textsf{Conf}(\mathbb{R}^{D}) = SO(1,D+1)$.
Thus, it remains to prove the invariance of (<ref>)
under inversions

\begin{align}
\lb{RIIR}
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2}\, =
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2}\,
R_{\Delta_1\Delta_2}(u) \; ,
\end{align}

where ${\cal I}^{(k)}_{\Delta_i}
:= {\cal I}_k\,\q_k^{2\Delta_i}$ are the shifted
inversion operators (<ref>)
for the Heisenberg algebras ${\cal H}_k$.
The shifted inversion
operator ${\cal I}_\Delta$ is interpreted
as the $\rho^\Delta$-representation of the inversion
element in the conformal group. In particular,
special conformal generators $K_\mu^{(\Delta)}$
in this representation
are given in (<ref>) and they are the same as
$- \rho(K_{\mu})$ in (<ref>)
for $S_{\mu\nu}=0$.

To prove (<ref>), we note that
the operator $R_{\Delta_1\Delta_2}(u)$ admits a
factorization in the product of simpler operators [77, 76]

\begin{align}
\lb{QQ5}
R_{\Delta_1\Delta_2}(u) = Q_{_{\Delta_1\Delta_2}}^{-1}(-u)\;
{\cal P}_{12}\; Q_{_{\Delta_1\Delta_2}}(u)
\end{align}

Q__Δ_1Δ_2(u) =
Operator $Q_{_{\Delta_1\Delta_2}}(u)$ has the following transformation

\begin{align}
\label{confR}
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2}\, =
{\cal I}^{(1)}_{\frac{\Delta_1+\Delta_2}{2}+u}\,
{\cal I}^{(2)}_{\frac{\Delta_1+\Delta_2}{2} -u}
\,Q_{_{\Delta_1\Delta_2}}(u) \; ,
\end{align}

or equivalently

\begin{align}
\lb{QIIQ}
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2}\; Q_{_{\Delta_1\Delta_2}}^{-1}(u) =
{\cal I}^{(1)}_{\frac{\Delta_1+\Delta_2}{2}+u}\,
{\cal I}^{(2)}_{\frac{\Delta_1+\Delta_2}{2} -u} \; .
\end{align}

The proof of (<ref>)
is based on the following identities
I^(2)_β  q̂_12^2λ =
I^(2)_β+λ  ,

p̂_k^2α = p̂_k^2α
I^(k)_D/2-α  ,
which follow from (<ref>).
Then we have

\begin{align*}
{\cal I}^{(1)}_{\frac{\Delta_1+\Delta_2}{2}+u}\,
{\cal I}^{(2)}_{\frac{\Delta_1+\Delta_2}{2} -u}\,
Q_{_{\Delta_1\Delta_2}}(u) & =
{\cal I}^{(1)}_{\frac{\Delta_1+\Delta_2}{2}+u}\,
{\cal I}^{(2)}_{\frac{\Delta_1+\Delta_2}{2}
\p_1^{2\left(u+\frac{\Delta_2-\Delta_1}{2}\right)} \;
\hat{q}_{12}^{2\left(u+\frac{\Delta_1+\Delta_2-D}{2}\right)}
\\[0.2cm]
%& = \hat{q}_{12}^{2\left(\frac{D}{2}-\Delta_1\right)}
%{\cal I}^{(1)}_{\frac{D}{2}+\frac{\Delta_2-\Delta_1}{2}+u}\,
%{\cal I}^{(2)}_{\frac{D}{2}+\frac{\Delta_2-\Delta_1}{2}-u}\,
%\p_1^{2\left(u+\frac{\Delta_2-\Delta_1}{2}\right)} \;
%& = \hat{q}_{12}^{2\left(\frac{D}{2}-\Delta_1\right)}\,
%{\cal I}^{(1)}_{\frac{D}{2}-\frac{\Delta_2-\Delta_1}{2}-u}\,
%{\cal I}^{(2)}_{\frac{D}{2}+\frac{\Delta_2-\Delta_1}{2}-u}\,
& = \hat{q}_{12}^{2\left(\frac{D}{2}-\Delta_1\right)}\,
\p_1^{2\left(u+\frac{\Delta_2-\Delta_1}{2}\right)}\,
\hat{q}_{12}^{2\left(u+\frac{\Delta_1+\Delta_2-D}{2}\right)}\,
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2} = Q_{_{\Delta_1\Delta_2}}(u)\,
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2} \; .
\end{align*}

Now it is not difficult to check conformal invariance
of the whole operator $R_{\Delta_1\Delta_2}(u)$:

\begin{align*}
Q_{_{\Delta_1\Delta_2}}^{-1}(-u)\,{\cal P}_{12}\,
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2}\, =
Q_{_{\Delta_1\Delta_2}}^{-1}(-u)\,{\cal P}_{12}\, {\cal
{\cal I}^{(2)}_{\frac{\Delta_1+\Delta_2}{2} -u}\,
Q_{_{\Delta_1\Delta_2}}(u) = \\
{\cal I}^{(1)}_{\frac{\Delta_1+\Delta_2}{2}-u}\,
{\cal I}^{(2)}_{\frac{\Delta_1+\Delta_2}{2} + u}\,
{\cal P}_{12}\,Q_{_{\Delta_1\Delta_2}}(u) =
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2}\,
Q_{_{\Delta_1\Delta_2}}^{-1}(-u)\,{\cal P}_{12}
\,Q_{_{\Delta_1\Delta_2}}(u)\,  ,
\end{align*}

which is equivalent to (<ref>).

Remark 1. It is instructive to prove (<ref>) by using
the representations of the $R$-operators
(<ref>) in terms of the integral kernels.
The proof of the main identity (<ref>)
in this representation is given in
Appendix E.

Remark 2. Since the
operator $R_{\Delta_1\Delta_2}(u)$ is conformal
invariant, i.e. commutes with all elements  of the conformal group
in the representation $\rho^{\Delta_1}\otimes\rho^{\Delta_2}$,
it is natural to consider $R_{\Delta_1\Delta_2}(u)$
as an operator acting in the tensor product of two spaces
$V_{\Delta_1} \otimes V_{\Delta_2}$
of scalar conformal fields with conformal dimensions
$\Delta_1$ and $\Delta_2$.
Moreover, the conformal
invariance of the operator $R_{\Delta_1\Delta_2}(u)$
guarantees its commutativity
with the quadratic Casimir operator
{\bf \Delta} (\widehat{C}_2)$
acting in $V_{\Delta_1} \otimes V_{\Delta_2}$.
This means that the operator $R_{\Delta_1\Delta_2}(u)$
(given in (<ref>)) and
quadratic Casimir operator (<ref>) for the
conformal algebra ${\sf conf}(\mathbb{R}^D)$
in the representation $\rho^{\Delta_1}\otimes\rho^{\Delta_2}$
(see (<ref>))
have a common set of eigenvectors.

In particular, this Remark 2
explains  why conformal 3-point correlators (conformal
should be eigenfunctions of the general
graph building operator (<ref>).
Indeed, the
eigenvectors of the Casimir operator $\widehat{C}_2$ acting in
are conformal 3-point
correlation functions [26],[23],[24]
which we considered in Subsection <ref>.
This system of functions (<ref>) is defined by
$\Delta_1, \Delta_2 \in \mathbb{R}$ and  $\Delta$

\begin{equation}
\lb{cnftri}
\langle x_1\,,x_2 |
\Psi^{\mu_1\cdots\mu_n}_{\Delta_1,\Delta_2,\Delta,x}
\rangle =
\frac{\left(\frac{x-x_1}{(x-x_1)^2}-\frac{x-x_2}{
\end{equation}

(here we restore the dependence of $\Psi$
in $\Delta_1, \Delta_2$)
and for the special choice (<ref>)
of the parameter $\Delta$ this system
is orthogonal (<ref>)
and complete (<ref>) with respect to the
modified scalar product (<ref>) which we defined
in the space of functions of two variables $x_1,x_2$.
The form of the scalar product (<ref>)
is dictated  by the
requirement that the graph building operator
(see Subsection <ref>) and the
Casimir operator $\widehat{C}_2$ (defined in (<ref>))
in the representation $(\rho^{\Delta_1}\otimes\rho^{\Delta_2})$
(see (<ref>), (<ref>),
Subsection <ref>) are Hermitian with
respect to this scalar product.

At the end of this subsection we show that the factorization
of the eigenvalue (<ref>) of the
general graph building operator (or $R$-matrix)
(<ref>), (<ref>)
follows from the factorization (<ref>) of the $R$-matrix.

Due to the special conformal
transformation properties, the operator
converts the conformal triangle (<ref>)
to another conformal triangle

\begin{align}
\label{qtpsi}
|\Psi^{\mu_1\cdots\mu_n}_{\Delta_1\,,\Delta_2,\,\Delta}\rangle =
\Big|\Psi^{\mu_1\cdots\mu_n}_{\frac{\Delta_1+\Delta_2}{2} +
u\,,\frac{\Delta_1+\Delta_2}{2} - u,\,\Delta} \Big\rangle
\end{align}

where we omit $x$ in the notation
of $|\Psi\rangle$ for simplicity and define

\begin{align}
\delta = \frac{1}{2}(\Delta_1-\Delta_2+\Delta-n) \
\ \ \ , \ \ \
\kappa := \frac{1}{2}(\Delta_1 -\Delta_2) - u \; , \;\;\;\;
\delta' = D/2 - \delta \; .
\end{align}

Proof. The integral kernel for the
building operator (<ref>) is

⟨x_1,x_2| Q__Δ_1Δ_2 | y_1,y_2 ⟩=
⟨x_1,x_2| (_12)^2(γ-ζ)
(_1)^-2κ (_12)^-2β | y_1,y_2 ⟩=

= a(κ) δ^D(x_2-y_2)/
(x_1-x_2)^2(ζ-γ) (x_1-y_1)^2κ'

where we use the parameters (<ref>)
to make the notation concise. Here we use the diagram techniques
developed in the proof of Proposition <ref>
to deduce  (<ref>). The action of the
building operator (<ref>) to the function (<ref>)
gives (cf. (<ref>))

(7.3,2)$= \; \tau(\delta',\kappa,n)\; \cdot$

(17.3,2)$= \; \tau(\delta',\kappa,n)\; \cdot \;
\widetilde{\Delta}_2,\,\widetilde{\Delta}}\rangle$   ,

where, according to the definition (<ref>), we fix
the parameters
δ= 1/2(Δ_1-Δ_2+Δ-n)   ,
ρ= 1/2(Δ_2-Δ_1+Δ-n)   ,
α= 1/2(Δ_1+Δ_2-Δ+n)   ,
(α+β)' =
+Δ-n)   ,
ρ+ κ=
+Δ-n)   ,
ζ-γ+δ' =
-Δ+n)   .
Substitution of (<ref>) and (<ref>)
in equations (<ref>) leads to relations

\widetilde{\Delta}_1 = \frac{\Delta_1 + \Delta_2}{2} +u \, , \;\;\;
\widetilde{\Delta}_2 = \frac{\Delta_1 + \Delta_2}{2} -u \, , \;\;\;
\widetilde{\Delta} = \Delta \; ,

that finish the proof of (<ref>).

Remark 3. Another derivation of
(<ref>) is based on conformal transformations.
We are going to demonstrate that relation
is dictated by the commutation relations of the operator
with transformations from the conformal group and it remains
to prove that the coefficient $\tau(\delta',\kappa,n)$
is given by formula (<ref>).

The system of functions (<ref>),
(<ref>) is defined by
$\Delta_1, \Delta_2 \in \mathbb{R}$ and
by the parameter $\Delta$ and can
be represented in the following form:

\begin{align}
\Psi^{\mu_1\cdots\mu_n}_{\Delta_1,\Delta_2,\Delta,x}
(x_1\,,x_2) =
\frac{\left(\frac{x_1}{x_1^2}-
\frac{x_2}{x_2^2}\right)^{\mu_1\cdots\mu_n}}
x_2^{\Delta_2-\Delta_1+\Delta-n}} = \\
e^{-i x(\p_1+\p_2)}\,
{\cal I}^{(1)}_{\Delta_1}\,
{\cal I}^{(2)}_{\Delta_2}\circ
\frac{\left(x_1-x_2\right)^{\mu_1\cdots\mu_n}}
{(x_1-x_2)^{\Delta_1+\Delta_2-\Delta+n}}\, ,
\end{align}

where the circle $\circ$ denotes the action of
operators on the wave function and we apply formulas (<ref>).
Now we use the commutation relations (<ref>) and
$[Q_{_{\Delta_1\Delta_2}}(u), \,\left(\p_1+\p_2\right)] =0$
to obtain

(x_1 ,x_2)  = Q__Δ_1Δ_2(u)
I^(2)_Δ_2∘(x_12)^μ_1⋯μ_n/(x_12)^Δ_1+Δ_2-Δ+n =

= e^-ix(_1+_2)
I^(2)_Δ_1+Δ_2/2 -u
Q__Δ_1Δ_2(u)∘(x_12)^μ_1⋯μ_n/(x_12)^Δ_1+Δ_2-Δ+n  ,

$x_{12}^\mu :=x_{1}^\mu - x_{2}^\mu$.
Thus, the problem is reduced to the action of the operator
$Q_{_{\Delta_1\Delta_2}}(u)$ on the function
In view of the spectral relation (<ref>),
any function of the form $\frac{x_{12}^{\mu_1\cdots\mu_n}}
{x_{12}^{2\sigma}}$ must be an eigenfunction
of the operator $Q_{_{\Delta_1\Delta_2}}(u)$.
Indeed, we obtain

Q__Δ_1Δ_2(u)∘x_12^μ_1⋯μ_n/x_12^2σ =
_12^2(u+Δ_1+Δ_2-D/2)∘x_12^μ_1⋯μ_n/x_12^2σ =
= e^-x_2∂_1x_1^2(D/2-Δ_1)
( _1^2(u+Δ_2-Δ_1/2)
x_1^2(σ+D/2-Δ_1) =
q_Δ_1Δ_2(u, σ)
x_12^μ_1⋯μ_n/x_12^2σ   ,

where, according to (<ref>), the eigenvalue is
q_Δ_1Δ_2(u, σ) =
Γ(D/2-χ-κ+n)  Γ(χ)/
Γ(χ+κ)   Γ(D/2-χ+n)   ,
κ:= Δ_1-Δ_2/2 - u   ,
χ:= σ+D/2-Δ_1   .
Substitution of (<ref>) into (<ref>) gives

(x_1 ,x_2)  =

= q_Δ_1Δ_2(u,σ)  e^-x(∂_1+∂_2)
I^(2)_Δ_1+Δ_2/2 -u∘(x_1-x_2)^μ_1⋯μ_n/(x_1-x_2)^Δ_1+Δ_2-Δ+n =

= q_Δ_1Δ_2 (u,σ)
(x_1 ,x_2)

where $\sigma = \frac{1}{2}(\Delta_1+\Delta_2-\Delta+n)$,
and we have $\chi :=
\frac{1}{2}(\Delta_2-\Delta_1-\Delta+n)+\frac{D}{2}$.
Thus, the explicit form of the eigenvalue
(<ref>) is

\begin{array}{c}
\displaystyle
q_{\Delta_1\Delta_2}\,(u) = q_{\Delta_1\Delta_2}\,(u,\sigma ) =
4^{-\kappa} \,
\frac{\Gamma(D/2-\chi-\kappa+n)\,  \Gamma\left(\chi\right)}{
\Gamma(\chi+\kappa)\,  \Gamma(D/2-\chi+n)} = \\ [0.4cm]
\displaystyle
= 4^{\frac{\Delta_2-\Delta_1}{2} + u} \,
\frac{\Gamma\left(\frac{1}{2}(\Delta + n) +u\right)\,
\Gamma\left(\frac{1}{2}(D+n-\Delta+\Delta_2-\Delta_1)\right)}{
\Gamma\left(\frac{1}{2}(D + n-\Delta)-u \right)\,
\Gamma\left(\frac{1}{2}(\Delta + n+\Delta_1-\Delta_2)\right)}
\; .
\end{array}

where we omit for simplicity $\sigma$ in the notation
for the eigenvalue $q_{\Delta_1\Delta_2}\,(u) =
q_{\Delta_1\Delta_2}\,(u,\sigma )$.
It is easy to check that
$q_{\Delta_1\Delta_2}\,(u) =
so that this expression is compatible to (<ref>).

Finally we
note that relation (<ref>) is equivalently written as

\begin{align}
\nonumber
|\Psi^{\mu_1\cdots\mu_n}_{\Delta_1\,,\Delta_2}\rangle =
|\Psi^{\mu_1\cdots\mu_n}_{\frac{\Delta_1+\Delta_2}{2} +
u\,,\frac{\Delta_1+\Delta_2}{2} - u}\rangle \; ,
\end{align}

Now for the whole R-operator one obtains

\begin{multline*}
= Q_{_{\Delta_1\Delta_2}}^{-1}(-u)\,{\cal P}_{12}\,
|\Psi^{\mu_1\cdots\mu_n}_{\Delta_1\,,\Delta_2}\rangle = \\
\,{\cal P}_{12}\,|
\Psi^{\mu_1\cdots\mu_n}_{\frac{\Delta_1+\Delta_2}{2} +
u\,,\frac{\Delta_1+\Delta_2}{2} - u}\rangle = \\
\Psi^{\mu_1\cdots\mu_n}_{\frac{\Delta_1+\Delta_2}{2} -
u\,,\frac{\Delta_1+\Delta_2}{2} + u}\rangle =
(-1)^n\, \frac{q_{\Delta_1\Delta_2}\,(u)}
\end{multline*}

so that the eigenvalue $r_{\Delta,n}(u)$ of the operator
$R_{\Delta_1\Delta_2}(u)$ can be represented in a
compact form

\begin{align}
\label{eigR}
r_{\Delta,n}(u) = (-1)^n\, \frac{q_{\Delta_1\Delta_2}\,(u)}
{q_{\Delta_1\Delta_2}\,(-u)} = (-1)^n\,4^{2u}\,
\frac{\Gamma\left(\frac{\Delta+n}{2}+u\right)}{\Gamma\left(\frac{\Delta+n}{2}-u\right)}
\frac{\Gamma\left(\frac{D-\Delta+n}{2}+u\right)}{\Gamma\left(\frac{D-\Delta+n}{2}-u\right)}
\end{align}

One can check directly that this eigenvalue is equal to
the eigenvalue (<ref>), (<ref>)
up to the factor $a(\kappa)a(\gamma)$ which appears
in the relation of the graph building operator
$Q_{12}^{(\zeta,\kappa,\gamma)}$, given in (<ref>),
and the $R$-operator (<ref>)

Q_{12}^{(\zeta,\kappa,\gamma)} =
\frac{1}{a(\kappa)a(\gamma)} R_{\Delta_1,\Delta_2}(u) \; ,

where the parameters $\zeta$, $\kappa$, $\gamma$
are expressed via the parameters $\Delta_1$, $\Delta_2$, $u$
by means of equations (<ref>).

§ CONCLUSION

In this paper, we have proposed an operator approach
to evaluating
multiple integrals for special classes of
the multiloop Feynman massless diagrams.

We have considered classes of
diagrams of iterative type containing
some repeating elementary building blocks. These blocks are
as special operators (graph-building operators)
being elements of the
direct product of several Heisenberg algebras.
For the  ladder diagrams, we have identified
(see also [10]) a set of
commutative graph building operators $H_\alpha$. Here
the parameters $\alpha$ are related to the indices of the
massless propagators. Then the
multiple integrals for $L$-loop ladder diagrams are given
by the product of $L$ commuting operators $H_{\alpha_i}$
and therefore expressed as a product of $L$ eigenvalues
of $H_{\alpha_i}$. A single complete set of eigenfunctions
and corresponding eigenvalues for all commuting
operators $H_\alpha$ $(\forall \alpha)$ were found.
This enables us to explicitly express a wide
class of $2$- and $4$-point ladder diagrams  as
Mellin-Barnes type integrals. Special cases of
these integrals are explicitly evaluated, that demonstrated
the advantages of the method.

As the second class of repeating type diagrams, we considered
the zig-zag 2-point and
4-point planar Feynman graphs relevant to
the bi-scalar $D$-dimensional ”fishnet” field theory
[36], [38].
The graph building operators for zig-zag diagrams were identified
and convenient operator representations
for these Feynman diagrams were obtained. An amusing fact was that
the complete set of eigenfunctions for
zig-zag graph building operators was given
by special 3-point correlation functions
in $D$-dimensional conformal field theories. By making
use of all these facts, we were enable to exactly evaluate
the multiple integrals for the special zig-zag diagrams.
In particular, we found a fairly simple derivation of values for
zig-zag $L$-loop two-point diagrams for $D=4$, that was the
rationale of the Broadhurst-Kreimer conjecture [12].

The role of conformal symmetry in our approach, and
especially the
surprising appearance of 3-point conformal correlators as
eigenfunctions for the graph building operators were
explained by the connection of the graph
building operators with the conformal
invariant solution $R_{\Delta_1,\Delta_2}(u)$
of the Yang-Baxter equation.
The operator $R_{\Delta_1,\Delta_2}(u)$ is interpreted as an
intertwining operator
of two spaces $V_{\Delta_1}$ and
$V_{\Delta_2}$ of scalar conformal fields with
conformal dimensions $\Delta_1$ and $\Delta_2$.

Finally, we have to mention the relation of the
graph building operator (<ref>)
(and $R$-matrix (<ref>), (<ref>))
to the BFKL equation which describes compound states of two
reggeized gluons [64],[83],
(for recent review about BFKL equation see [87]).
Indeed, the special case
$\Delta_1=\Delta_2 \equiv \Delta_+$
of the $R$-operator (<ref>), (<ref>)
underlies the Lipatov integrable
model of the high-energy asymptotics of
multicolor QCD

\begin{array}{c}
\check{R}_{12}^{(\Delta_1,\Delta_2)}(u)
\;\; \stackrel{u \to 0}{\rightarrow} \;\;
1 + u \, H^{(\Delta_+)}_{12} + u^2 \dots \, , \;\;\;\;\;\;
H^{(\Delta_+)}_{12} = 2 \ln q_{12}^2 +
\hat{q}_{12}^{2\Delta_+'} \ln(  \p_1^{2} \;
\p_2^{2}) \; \hat{q}_{12}^{-2\Delta_+'} \, \equiv  \\ [0.3cm]
\equiv \ln \p_1^{2} +
\ln \p_2^{2} + \p_{1}^{-2\Delta_+'} \ln  (\hat{q}_{12}^{2})
\p_{1}^{2\Delta_+'} + \p_{2}^{-2\Delta_+'} \ln  (\hat{q}_{12}^{2})
\p_{2}^{2\Delta_+'} \; ,
\end{array}

where as usual $\Delta_+' = D/2 -\Delta_+$
and $H^{(\Delta_+)}_{12}$ gives the Hamiltonian of the
In two dimensions $D=2$ and in the special case
$\Delta_1=\Delta_2=\Delta_+= 0$
we obtain

\begin{align*}
H^{(\Delta_+)}_{12}|_{\Delta_+=0,D=2} = 2 \ln q_{12}^2 +
\hat{q}_{12}^{2} \ln(  \p_1^{2} \;
\p_2^{2}) \; \hat{q}_{12}^{-2} = \ln \p_1^{2} +
\ln \p_2^{2} + \p_{1}^{-2} \ln  (\hat{q}_{12}^{2})
\p_{1}^{2} + \p_{2}^{-2} \ln  (\hat{q}_{12}^{2})
\p_{2}^{2}\,.
\end{align*}

There exists (in complex coordinates $\hat{q}_{12}^{2} \to
z_{12} \, \bar{z}_{12}$)
a holomorphic factorization when the initial
operator can be represented as the sum of operators acting on
and antiholomorphic coordinates separately

\begin{align*}
H^{(\Delta_+)}_{12}|_{\Delta_+=0,D=2} \to H_{12} + \bar{H}_{12}\,,\\
H_{12} = \ln {\sf p}_{1} + \ln {\sf p}_{2} +
{\sf p}^{-1}_{1}\,\ln(z_{12})\, {\sf p}_{1} +
{\sf p}_{2}^{-1}\,\ln(z_{12})\, {\sf p}_{2} \; ,
\end{align*}

where ${\sf p}_k = -i\frac{\partial}{\partial z_k}$,
and the same story
occurs with the
Casimir operator (<ref>) in two dimensions

\begin{align*}
\widehat{C}_{\Delta_1\Delta_2}|_{\Delta_k=0,D=2}=
-\left(\q_{12}\right)^2 (\p_1^\mu \, \p_{2\, \mu}) +
2 \q_{12}^{\mu}\, \q_{12}^{\nu} \, \p_{1\,\mu} \, \p_{2\, \nu} =
- 2 \bigl( z_{12}^2\,\partial_{z_1}\partial_{z_2}+
\bar{z}_{12}^2\,\partial_{\bar{z}_1}\partial_{\bar{z}_2}\bigr) \; .
\end{align*}

The operator $\mathcal{H}_{12} \equiv
coincides (up to an additional constant)
with the local (pair) Hamiltonian
of the Lipatov model [64].
For the Lipatov model, the BFKL equation takes the form of the
Schrödinger equation
$\mathcal{H}_{12} \, \psi(x_1,x_2) = E \, \psi(x_1,x_2)$,
where two-dimensional vectors $x_1,x_2$ are the
eigenvalues of the position operators $\q_1,\q_2$.
The ground state eigenvalue $E$ is related
to the pomeron intercept and
eigenvalues $E$ can be extracted from
(<ref>) in the limit $u \to 0$.
The complete and orthogonal set of eigenfunctions
[64] is given by (<ref>).
In two dimensions the
symmetric traceless tensor $x^{\mu_1\cdots\mu_n}$
has two independent components which,
in complex coordinates, are: $z^n$ and $\bar{z}^n$.
The $z^n$-component of the function (<ref>) written in
complex coordinates (and under the conditions
$\Delta_1 =\Delta_2 \to 0$ and
$\Delta = \frac{D}{2}+2 i\nu \to 1+2 i\nu$)
is reduced to the form [64]

\begin{multline*}
\langle x_1\,,x_2 |
\Psi^{\mu_1\cdots\mu_n}_{\Delta_1,\Delta_2,\Delta,x}
\rangle \to \frac{\left(\frac{z-z_1}{|z-z_1|^2}-\frac{z-z_2}{
|x-x_1|^{1-n+2 i\nu}
|x-x_2|^{1-n+2 i\nu}} = \\
\left(\frac{\bar{z}_1-\bar{z}_2}{(\bar{z}-\bar{z}_1)(\bar{z}-\bar{z}_2)}\right)^n
\left(\frac{|x_1-x_2|}{|x-x_1||x-x_2|}\right)^{1-n+2i\nu}
\end{multline*}

The compound states of $N$
reggeized gluons are described by the Lipatov model with the
Hamiltonian ${\cal H} = \sum_{i=1}^N H_{i,i+1}$ with the periodic
boundary condition $x_{i+N}=x_i$. The Hamiltonian ${\cal H}$
is deduced by the standard method [84],[86] from the
transfer-matrix which is constructed
as the product of $N$ copies of
$R$-operators (<ref>), (<ref>).

§ ACKNOWLEDGMENT

We thank G. Arutyunov, K. Chetyrkin, M. Kompaniets, A. Kotikov,
A. Kataev and V. Kazakov for valuable discussions.
API is grateful to the L. Euler
International Mathematical Institute
in Saint Petersburg for kind hospitality
during his visit at the end of 2021
when this work was started.
The work of SED and LAS was supported by the Theoretical Physics and
Advancement Foundation «BASIS».

§ APPENDIX A

The projector $P^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}
\equiv (\Pi_{1\to n})^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}$ in
$(\mathbb{R}^D)^{\otimes n}$,
which was introduced in (<ref>), satisfies

\begin{array}{c}
\Pi_{1 \to n} P_{k,r} = \Pi_{1 \to n} =  P_{k,r} \Pi_{1 \to n} \;,
\;\;\;
\Pi_{1 \to n} K_{k,r} = 0 = K_{k,r} \, \Pi_{1 \to n} \;, \;\;\;\;\;
\forall k < r \leq n \; ,
\end{array}

(P_{kr})^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n} =\delta^{\mu_1}_{\nu_1}
\cdots  \delta^{\mu_k}_{\nu_k} \!\!\!\!\!\!\!\! \diagup \;\;
\cdots \delta^{\mu_r}_{\nu_r} \!\!\!\!\!\!\!\! \diagup \;\;
\cdots \delta^{\mu_n}_{\nu_n}\, P^{\mu_k\mu_r}_{\nu_k\nu_r} \, ,
\;\;\; (K_{kr})^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_n}
\cdots  \delta^{\mu_k}_{\nu_k} \!\!\!\!\!\!\!\! \diagup \;\,
\cdots \delta^{\mu_r}_{\nu_r} \!\!\!\!\!\!\!\! \diagup \;\,
\cdots \delta^{\mu_n}_{\nu_n}\, K^{\mu_k\mu_r}_{\nu_k\nu_r} \, ,

and $P^{\mu_1\mu_2}_{\nu_1\nu_2} =
\delta^{\mu_1}_{\nu_2}  \delta^{\mu_2}_{\nu_1}$ is the permutation
while $K^{\mu_1\mu_2}_{\nu_1\nu_2} =
\delta^{\mu_1\mu_2}  \delta_{\nu_1\nu_2}$ is the rank-1 matrix;
$P$ and $K$ are the operators in $(\mathbb{R}^{D})^{\otimes 2}$.
The projector $\Pi_{1 \to n}$ can be defined
recurrently [13] and we write this definition in the form
Π_1→n = Π_1→n-1 ·(y_n +1)·(y_n +ω+ n-3)/n ·(2n -4+ω) ·Π_1→n-1   ,
where $\omega$ is a parameter (in our case $\omega=D$) and
$y_n = \sum\limits_{k=1}^{n-1} (P_{k,n} - K_{k,n})$ is the
commuting set of Jucyc-Murphy elements.  Relation
(<ref>) is also written as
(cf. Remark 3.8 in [69]; [71])

Π_1→n = Π_1→n-1 ·R_n-1(n-1) ·Π_1→n-1
= Π_1 →n-1 ·R_n-1(n-1) ⋯R_2(2) ·R_1(1)

= R_1(1) ·R_2(2) ⋯R_n-1(n-1) ·Π_1→n-1

$R_{k}(u) = I^{\otimes k-1} \otimes R(u) \otimes I^{\otimes n-k-1}$

R(u) = \frac{1}{1+u}(I + u P + \frac{u}{1-D/2-u}K) \;
\;\;\;\;\; \Bigl(R(u) \cdot R(-u) =  I \Bigr)\; ,

is the $so(D)$-invariant
Zamolodchikov solution of the Yang-Baxter equation.
The last expression in (<ref>) is simplified as

Π_1 →n = S_1 →n (1 - n-1/D  + 2(n-2)
K_n-1,n ) Π_1 →n-1 =
S_1 →n (1 - 1/D  + 2(n-2)
(∑_i=1^n-1 K_in))
Π_1→n-1 =
= S_1 →n (1 - 1/D  + 2(n-2)
∑_i=1^n-1 K_in) ⋯(1 - 1/D  + 2
∑_i=1^2 K_i3)
(1 - 1/D  K_12))
∈  End( (ℝ^D)^⊗n)    ,

where $S_{1\to n}$ is the standard symmetrizer
(see e.g. [13]) in
the group algebra $\mathbb{C}[S_n]$ of the symmetric group $S_n$.
In fact, the projector (<ref>), (<ref>)
is the image of the complete symmetrizer in the Brauer algebra
(see details in [69], [13]).
From the first equality in
(<ref>) we deduce

Tr_n(Π_1 →n) = Π_1→n-1 ·Tr_n( R_n-1(n-1))  ·Π_1→n-1 =
(D+2n-2)(D+n-3)/n(D+2n-4) Π_1 →n-1   ⇒
Tr_1...n(Π_1 →n) =
(D+2n-2)  Γ(D+n-2)/n!  Γ(D-1) .

It is clear that ${\rm Tr}_{1...n}(\Pi_{1 \to n})$ is equal to
the dimension of the totally symmetric irreducible representation
$[n]={\scriptsize  \begin{array}{|c|c|c|c|}
\hline
\,   &    \,  & $\dots$ &  \,   \\ [0.1cm]
\hline
\end{array} }$ of $so(D)$.
Now we define the
generating function  of the projector (<ref>), (<ref>):
x_μ_1 ⋯x_μ_n
(Π_n)^μ_1 ... μ_n_   ν_1 ... ν_n
u^ν_1 ⋯u^ν_n ≡x^μ_1 ... μ_n u^μ_1 ... μ_n   ,
where we use the convenient notation
$x^{\mu_1 ... \mu_n} := x^{\nu_1} \cdots  x^{\nu_n}
\; (\Pi_{n})^{\mu_1 ... \mu_n}_{\;\; \nu_1 ... \nu_n}$.
After substitution of (<ref>) in (<ref>),
this generating function is written
as[For details see, e.g., an
analogous statement for the Behrends-Fronsdal
projector in [70], [71], where it is necessary to
a shift
$D \to D+1$ to obtain (<ref>);
see also [46] and references therein.]
Π_n(x;u) =  ∑_A=0^[n/2] a^(n)_A
u^2A   x^2A   (ux)^n-2A   ,
a^(n)_A  := (-1)^A   n!/(n-2A)!  A!
Γ(n+D/2-1-A)/Γ(n+D/2-1)   .
We take $u=x$ and use the first equality in (<ref>),
which gives an
equation for the generating function $\Pi_{n}(x;x) = F(n) \, x^{2n}$:

\Pi_n(x;x)  = \frac{(D  + n-3)}{D  + 2(n-2)}\;
x^2 \,  \Pi_{n-1}(x;x) \;\;\;\; \Rightarrow \;\;\;\;
F(n) = \frac{(D  + n-3)}{D  + 2(n-2)} F(n-1) \; /

This equation is immediately solved as

x^{\mu_1 ... \mu_n} x^{\mu_1 ... \mu_n} \equiv \Pi_n(x;x)  =
\frac{\Gamma(D +n-2)\Gamma(D /2-1)}{
2^{n}\Gamma(D -2)\Gamma(D /2+n-1)} \; x^{2n} \; .

Finally, we prove the orthogonality
relation (<ref>). For the left-hand side of (<ref>)
we have
∫d^D x x^μ_1 ... μ_n/x^2(D/4+n/2-i v)
x_ν_1 ... ν_m/x^2(D/4+m/2+i v') =
δ_n,m C_n Π^μ_1 ... μ_n_ν_1 ... ν_n
∫d^D x 1/x^2(D/2+i (v'-v)) =
= π  Ω_D   C_n   Π^μ_1 ... μ_n_ν_1 ... ν_n
δ(v'-v) δ_n,m    ,
where in the  last equality we use (<ref>).
The constant $C_n$ in (<ref>) is found if we put $m=n$,
contract indices $\mu_1 ... \mu_n$ with $\nu_1 ... \nu_n$
and use relations (<ref>) and (<ref>).
As a result we deduce $C_n = \frac{n! \Gamma(D/2)}{2^n
\Gamma(D/2+n)}$.
Substitution of this constant into (<ref>) gives

§ APPENDIX B

For the coefficient function $C(\alpha_1,\alpha_2,\alpha_6;
\alpha_3,\alpha_4,\alpha_5)$ of the
two-loop master diagram in Fig.<ref>
we deduced formula (<ref>).  We write the  integrand in
(<ref>) in terms of the $\Gamma$-functions

Γ(α_1...5-5D/4 + n/2-iν)
Γ(α_125-3D/4 + n/2-iν)
Γ(α_1-D/4 + n/2-iν)/
Γ(α_1235-3D/4 + n/2-iν)
Γ(α_12-D/4 + n/2-iν)
Γ(D/4 + n/2-iν)   ×
Γ(5D/4 - α_1...5 + n/2+iν)
Γ(3D/4-α_12 + n/2+iν)
Γ(D/4 + n/2+iν)/
Γ(7D/4 - α_1...5 + n/2 + iν)
Γ(5D/4 - α_125 + n/2+iν)
Γ(3D/4 - α_1 + n/2+iν).

Zeros and poles of expression (<ref>)
in the upper half-plane
are defined respectively by the $\Gamma$-functions
in the denominator and numerator in (<ref>).
Thus, in the upper half-plane we have 3 sets of poles

ν= i (D/4 +n/2+k)   ;

ν= i (3D/4 -α_12 +n/2+k)   ;

ν= i (5D/4 -α_1235 +n/2+k)  ;

and 3 sets of zeros

ν= i (3D/4 -α_1 +n/2+k)   ;

ν= i (5D/4 -α_125 +n/2+k)   ;

ν= i (7D/4 -α_1...5 +n/2+k)   ;

where $k \in \mathbb{Z}_{>0}$.

For the Chetyrkin-Tkachov diagram we have

\alpha_1 = \alpha_4 = \alpha_5 = D/2 - 1 \; , \;\;\;\;
\alpha_3 = \alpha\; , \;\;\;\; \alpha_2 = \beta \; ,

and the sets of poles (<ref>)
and zeros (<ref>) are written respectively as

ν= i (D/4 +n/2+k)   ;

ν= i (D/4 +1 -β+n/2+k)   ;

ν= i (D/4 +2 -α-β+n/2+k)   ;

ν= i (D/4 +1 +n/2+k)   ;

ν= i (D/4 +2 -β+n/2+k)   ;

ν= i (D/4 + 3 -α-β+n/2+k)   ,

where $k \in \mathbb{Z}_{>0}$. We see that poles and zeros cancel each
other out, except the three poles for every $n \in \mathbb{Z}_{>0}$.
Thus, we can easily evaluate the integral (<ref>) for
the Chetyrkin-Tkachov diagram by closing the contour in the upper
half-plane. Summing the residues, we obtain

\begin{align}\label{ChT-sum}
ChT(\alpha, \beta) &= \dfrac{\pi^{D/2}}{(\Gamma(D/2 - 1))^2\Gamma(D
- 2)}\dfrac{1}{(\alpha - 1)(\beta - 1)(\alpha + \beta - 2)}\sum_{n
= 0}^{\infty}\dfrac{\Gamma(n + D - 2)}{n!}\times\nonumber \\
\times&\left(\dfrac{\beta - 1}{(\alpha - D/2 - n)(\alpha + \beta -
1 - D/2 - n)} + \dfrac{\alpha - 1}{(D/2 - 2 + \beta + n)(D/2 - 3 +
\alpha + \beta + n)} + \nonumber\right.\\ + &\left.\dfrac{\alpha +
\beta - 2}{(\beta - D/2 - n)(D/2 - 2 + \alpha + n)}\right),
\end{align}

where we use the notation $ChT(\alpha, \beta) = C(D/2 - 1, \beta, 3 -
\alpha - \beta; \alpha, D/2 - 1, D/2 - 1)$.

The sum over $n$ can be calculated analytically. Let us give an
of how one can calculate the third term in (<ref>)

\begin{equation}
\sum_{n = 0}^{\infty}\dfrac{\Gamma(n + D - 2)}{n!}\dfrac{1}{(D/2 -
\beta + n)(D/2 - 2 + \alpha + n)}.
\end{equation}

First of all, we write the summand in the form

\begin{align}\label{how-to-sum-3}
\dfrac{1}{(D/2 - \beta +   n)(D/2 - 2 + \alpha + n)}& =
\dfrac{1}{\alpha + \beta - 2}\times\nonumber \\
&\times\left(\dfrac{1}{D/2 - \beta + n} - \dfrac{1}{\alpha + D/2 -2
+ n}\right).
\end{align}

Each term represents the definition of the
hypergeometric function, so we have

\begin{align}\label{how-to-sum-2}
\sum_{n = 0}^{\infty}\dfrac{\Gamma(n + D - 2)}{n!}\dfrac{\Gamma(D/2
- \beta + n)}{\Gamma(D/2 - \beta + n + 1)}& =  \dfrac{\Gamma(D -
2)\Gamma(D/2 - \beta)}{\Gamma(D/2 - \beta + 1)}\times\nonumber \\
&\times {}_2F_1(D - 2, D/2 - \beta;D/2 - \beta + 1; 1).
\end{align}

For a special value of the
hypergeometric function at the point $1$, we have the relation

\begin{equation}
{}_2F_1(D - 2, D/2 - \beta;D/2 - \beta + 1; 1) = \dfrac{\Gamma(D -
2)\Gamma(D/2 - \beta)\Gamma(3 - D)}{\Gamma(3 - D/2 -
\beta)\Gamma(1)}.
\end{equation}

All other terms can be calculated in the same way, which lead us to the
well-known answer

\begin{align}\label{cht-final}
ChT(\alpha,\beta) = \dfrac{\pi^{D/2}a_0(D - 2)}{\Gamma(D/2 -
1)}&\left(\dfrac{a_0(\alpha)a_0(2 - \alpha)}{(1 - \beta)(\alpha +
\beta - 2)} + \dfrac{a_0(\beta)a_0(2 - \beta)}{(1 - \alpha)(\alpha
+ \beta - 2)} +\right.\nonumber \\ &\left. + \dfrac{a_0(\alpha +
\beta - 1)a_0(3 - \alpha - \beta)}{(\alpha - 1)(\beta -
\end{align}

§ APPENDIX C

In this Appendix we collect some formulae
which we have used in the main text.
These formulae are useful in calculations so that for completeness