# On correlation functions of higher-spin currents in arbitrary dimensions
$d>3$

Melik Karapetyan  Ruben Manvelyan  Karapet Mkrtchyan

###### Abstract

We revisit the problem of classification and explicit construction of the
conformal three-point correlation functions of currents of arbitrary integer
spin in arbitrary dimensions. For the conserved currents, we set up the
equations for the conservation conditions and solve them completely for some
values of spins, confirming the earlier counting of the number of independent
structures matching them with the higher-spin cubic vertices in one higher
dimension. The general solution for the correlators of conserved currents we
delegate to a follow-up work.

##  1 Introduction

The holographic duality [1, 2] remains one of the most promising approaches to
Quantum Gravity. Particular interest is attracted by Higher-Spin (HS) Gravity
[3, 4, 5] as the AdS dual candidate [6, 7, 8, 9] of the simplest CFT — $O(N)$
vector model [10, 11]. Lagrangian formulation of Vasiliev’s HS Gravity is not
available so far. However, the classification of interaction vertices between
symmetric HS fields in arbitrary dimensions has been an impressive collective
effort. See [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68] for some key references.

The holographic dictionary relates interaction vertices in AdS space-time to
the conformal correlators on the boundary. Massless HS fields in AdS
correspond to conserved currents on the boundary. The classification of the
correlators of the (conserved) currents of arbitrary spin has been an
independent parallel program. See [69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99, 100, 101] for some key references.

Generally, in conformal field theory, two and three-point correlation
functions are fixed by conformal symmetry leaving no functional freedom. While
the two-point function is fixed up to a normalization constant for any spin
conformal operator (or traceless current of any rank) the three-point function
depends on several constants for each triplet of currents. It is natural to
expect that the number of independent structures here should match the number
of independent vertices of cubic interaction in the bulk AdS gravity, via
AdS/CFT dictionary. Moreover, the cubic vertices in AdS are uniquely
determined from the flat space cubic vertices, by adding curvature corrections
fixed by the requirement of AdS covariance [32, 33, 34, 45, 46, 52, 56, 59].
Hence, there should be a one-to-one correspondence between cubic vertices in
$d+1$-dimensional Minkowski space and conformal corellators in $d$ dimensions.
At least, the number of structures on both sides should match. This one-to-one
correspondence between three-point correlators of conserved currents of
arbitrary spin in $d>3$ dimensions and cubic vertices of massless symmetric
fields in $d+1$ dimensional Minkowski space [25, 38] was conjectured and
elaborated upon in [89] (see also [90, 94]).

Four-dimensional bulk spacetime corresponding to three-dimensional CFT has
some peculiarities (see, e.g., [57, 88, 98]), while similar correspondence has
been established in $d=2$ (with three-dimensional bulk) not only at cubic
order but also for arbitrary higher-order interactions [102] with the help of
the full classification of cubic [103, 104] and higher-order [105] independent
vertices involving massless bosonic HS fields.

The holographic reconstruction of HS Gravity has also progressed in the last
decades: see [106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118]
for some key references.

In this work, we revisit the construction and investigation of two and three-
point correlation functions for HS conformal currents in arbitrary dimensions
via Osborn-Petkou general formulation [77]. In Appendix A we briefly review
this formulation adopted for higher spin case. But here we would like to note
that the main advantage of formulation developed in [77] is the reduction of
the problem to construction instead of correlation function depending on three
space-time points to the tensor depending on three sets of symmetrized indices
but depending only on one variable which is roughly the difference of two
coordinates inverted around the third point. In this way, we have a much
simpler object for investigation depending on one variable polynomially with
certain symmetry properties and satisfying conservation conditions.

In this work, we present a general Ansatz for the local object that defines
the correlation functions***We work with symmetric currents in arbitrary
dimensions and do not consider lower-dimensional aspects like Schouten
identities (relevant in $d\leq 3$) and parity-odd correlators (relevant in
$d\leq 4$). of arbitrary-spin currents. This Ansatz is a sum of the most
general tensorial polynomials in one space-time variable and Kronecker
symbols. Then we apply the symmetry conditions described in [77] (see also
Appendix A) for general three-point correlation function with different spins
$s_{1},s_{2},s_{3}$. Natural triangle inequalities stem from the locality of
our Ansatz. The solution of the latter is not simple, as expected (the
approach of [77] is known to lead to complications). However, we present the
general solution in Section 3, reproducing all low spin examples presented in
[77].

Then in the next section (Section $4$) we derive conservation conditions for
our general ansatz. This allows investigation by computer calculation of the
rank of an equivalent linear system of equations for getting independent
parameters of the ansatz. One obtains general restriction on the number of
independent parameters of the three-point function. Our results align with
those of [89] (establishing one-to-one correspondence with the Minkowski
vertices of massless fields [25, 38]): The number of independent parameters of
the parity-even three-point function of three conserved currents depends only
on the minimal spin of the involved currents and is equal to:
$\text{min}(s_{1},s_{2},s_{3})+1.$

We further formulate the conservation condition in the form of a differential
equation on the generating function of the correlators instead of a recursion
relation for coefficients of the ansatz. We leave the full solution of these
relations to future work.

Some technical details and derivations are delegated to Appendices.

##  2 General setup and two-point function

We present very shortly the key points of our technical setup and construction
of the two-point function as a preliminary exercise before our main task: the
three-point function. As customary when dealing with HS fields, we introduce
auxiliary vector variables $a_{\mu},b_{\mu},\dots$ to handle an arbitrary
number of symmetrized indices. As usual, we utilize instead of symmetric
tensors such as $h^{(s)}_{\mu_{1}\mu_{2}...\mu_{s}}(x)$ the homogeneous
polynomials in a vector $a^{\mu}$ of degree $s$ at the base point $x$:

$h^{(s)}(x;a)=h^{(s)}_{\mu_{1}\mu_{2}...\mu_{s}}(x)a^{\mu_{1}}a^{\mu_{2}}\dots
a^{\mu_{s}}.$ (2.1)

Then the symmetrized gradient, divergence, and trace operations are given
as†††To distinguish easily between “a” and “x” spaces we introduce the
notation $\nabla_{\mu}$ for space-time derivatives $\frac{\partial}{\partial
x^{\mu}}$.

$\displaystyle
Grad:h^{(s)}(x;a)\Rightarrow(Grad\,h)^{(s+1)}(x;a)=(a\nabla)h^{(s)}(x;a)\,,$
(2.2) $\displaystyle
Div:h^{(s)}(x;a)\Rightarrow(Div\,h)^{(s-1)}(x;a)=\frac{1}{s}(\nabla\partial_{a})h^{(s)}(x;a)\,,$
(2.3) $\displaystyle
Tr:h^{(s)}(x;a)\Rightarrow(Tr\,h)^{(s-2)}(x;a)=\frac{1}{s(s-1)}\Box_{a}h^{(s)}(x;a)\,.$
(2.4)

Moreover we introduce the notation $*_{a},*_{b},\dots$ for a full contraction
of $s$ symmetric indices:

$\displaystyle*^{(s)}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{(s!)^{2}}\prod^{s}_{i=1}\overleftarrow{\partial}^{\mu_{i}}_{a}\overrightarrow{\partial}_{\mu_{i}}^{a}.$
(2.5)

These operators, together with their duals‡‡‡It is easy to see that the
operators $(a\partial_{b}),a^{2},b^{2}$ are dual (or adjoint) to
$(b\partial_{a}),\Box_{a},\Box_{b}$ with respect to the “star” product of
tensors with two sets of symmetrized indices (2.5)
$\displaystyle\frac{1}{n}(a\partial_{b})f^{(m-1,n)}(a,b)*_{a,b}g^{(m,n-1)}(a,b)=f^{(m-1,n)}(a,b)*_{a,b}\frac{1}{m}(b\partial_{a})g^{(m,n-1)}(a,b),\quad\quad$
$\displaystyle
a^{2}f^{(m-2,n)}(a,b)*_{a,b}g^{(m,n)}(a,b)=f^{(m-2,n)}(a,b)*_{a,b}\frac{1}{m(m-1)}\Box_{a}g^{(m,n)}(a,b).\quad\quad$
In the same fashion gradients and divergences are dual with respect to the
full scalar product in the space $(x,a,b)$, where we allow for integration by
parts:  $\displaystyle(a\nabla)f^{(m-1,n)}(x;a,b)*_{a,b}g^{(m,n)}(x;a,b)$
$\displaystyle=$
$\displaystyle-f^{(m-1,n)}(x;a,b)*_{a,b}\frac{1}{m}(\nabla\partial_{a})g^{(m,n)}(x;a,b).$
Analogous equations can be formulated for the operators $b^{2}$ or $b\nabla$.
will be the building blocks of the correlation functions of higher spin
currents. As it was mentioned before, we use the formulation of [77] reviewed
in Appendix A. Here we just extend this formulation of the two-point
correlation function for the case of general spin-$s$ conformal conserved
(traceless-transverse) currents.

First of all, we construct the traceless projector for rank $s$ symmetric
tensors:

$\displaystyle T^{(s)}_{traceless}(a)$ $\displaystyle={\cal
E}^{(s)}(a,b)*^{(s)}_{b}T^{(s)}(b)$ (2.6)

Starting from the ansatz

$\displaystyle{\cal
E}^{(s)}(a,b)=\sum^{s/2}_{p=0}\lambda_{p}(ab)^{s-2p}(a^{2}b^{2})^{p},\quad\lambda_{0}=1$
(2.7)

and solving the tracelessness condition

$\displaystyle\Box_{a}{\cal E}^{(s)}(a,b)=\Box_{b}{\cal E}^{(s)}(a,b)=0$ (2.8)

we arrive at a set of coefficients $\\{\lambda_{p}\\}^{s/2}_{p=0}$ which are
the object of the recursion equation:

$\displaystyle\lambda_{p}=-\frac{(s-2p+2)(s-2p+1)}{4p(d/2+s-p-1)}\lambda_{p-1}$
(2.9)

with solution corresponding to the initial condition from (2.7):

$\displaystyle\lambda_{p}=\frac{(-1)^{p}[s]_{2p}}{2^{2p}p![d/2+s-2]_{p}}$
(2.10)

Here we use notations $[a]_{n}$ for falling factorials (Phochhammer symbols):

$\displaystyle[a]_{n}=\frac{a!}{(a-n)!}=\frac{\Gamma(a+1)}{\Gamma(a-n+1)}$
(2.11)

Then it is easy to construct spin $s$ representation for inversion matrix
given by:

$I(a,b;x)=(ab)-2(a{\hat{x}})(b{\hat{x}}),\quad{\hat{x}}_{\mu}=\frac{x_{\mu}}{\sqrt{x^{2}}}$
(2.12)

To do that we just take the traceless part of the $s$-th power of the
inversion matrix:

$\displaystyle{\cal I}^{(s)}(a,b;x)$ $\displaystyle=$
$\displaystyle\big{(}I(a,c;x)\big{)}^{s}*^{s}_{c}{\cal E}^{(s)}(c,b)={\cal
E}^{(s)}(a,c)*^{s}_{c}(I(c,b;x))^{s}$ (2.13) $\displaystyle\Box_{a,b}{\cal
I}^{(s)}(a,b;x)$ $\displaystyle=$ $\displaystyle 0$ (2.14)

The result is easy to handle

$\displaystyle{\cal
I}^{(s)}(a,b;x)=\sum^{s/2}_{p=0}\lambda_{p}\big{(}I(a,b;x)\big{)}^{s-2p}(a^{2}b^{2})^{p}\,,\quad\lambda_{0}=1\,.$
(2.15)

Then we search for two point function of conformal conserved currents with
spin $s$:

$\displaystyle{\cal J}^{(s)}(a;x)={\cal
J}^{(s)}_{\mu_{1}\mu_{2}\dots\mu_{s}}(x)a^{\mu_{1}}a^{\mu_{2}}\dots
a^{\mu_{s}}$ (2.16) $\displaystyle(\nabla\partial_{a}){\cal J}^{(s)}(a;x)=0$
(2.17) $\displaystyle\Box_{a}{\cal J}^{(s)}(a;x)=0$ (2.18)

The natural proposal is

$\big{<}{\cal J}^{(s)}(a;x_{1}){\cal J}^{(s)}(b;x_{2})\big{>}=\frac{C_{{\cal
J}}}{(x_{12}^{2})^{\Delta_{(s)}}}{\cal I}^{(s)}(a,b;x_{12})$ (2.19)

This expression is traceless by construction due to (2.14). The scaling number
$\Delta_{(s)}$ we can obtain from conservation condition (2.17) applied to
(2.19) :

$\displaystyle 0=(\nabla_{1}\partial_{a})\frac{{\cal
I}^{(s)}(a,b;x_{12})}{(x_{12}^{2})^{\Delta_{(s)}}}$
$\displaystyle=\frac{2(\Delta_{(s)}-s-d+2)}{(x_{12}^{2})^{\Delta_{(s)}+1}}\sum_{k=0}^{s/2-1}\lambda_{k}(s-2k)\big{(}I(a,b;x_{12})\big{)}^{s-2k-1}(b{\hat{x}}_{12})(a^{2}b^{2})^{k}$
(2.20)

So we see that we should choose for the conformal dimension of spin $s$ field
standard value:

$\displaystyle\Delta_{(s)}=s+d-2$ (2.21)

Equivalently we can say that the conservation of the two-point function (2.19)
comes from the following relation :

$\displaystyle\big{[}(\nabla_{x}\partial_{a})-2\frac{({\hat{x}}\partial_{a})}{\sqrt{x^{2}}}\big{]}{\cal
I}^{(s)}(a,b;x)$ (2.22)

The interesting point here is that if we start with expression (2.19), where
we take the correct conformal dimension (2.21) but in expression (2.15)
undefined general set of coefficients $\lambda_{k}$ then after implementation
of conservation condition we arrive to the same recursion (2.15) for set
$\lambda_{k}$ which we obtained before from the tracelessness condition (2.9)
or equivalently (2.14).

For the odd spin case, the generalization is straightforward: we should just
replace $s/2$ in summation limit by integer part $[s/2]$, which means that the
highest trace, in this case, produces a vector instead of a scalar.

##  3 Three-point function: the structure of the ansatz

For the construction of the three-point function we should investigate
structure, symmetry, and conservation condition for object
$t^{j_{1}j_{2}i_{3}}(X)$, which lives in three different representations of
different spins but depends locally from one point in space-time (see [77] or
Appendix A for details). New important restrictions on the correlators enter
the game for conserved currents: the corresponding conservation conditions
should be implemented independently, restricting the correlators further.
These we consider in the next section.

First note that restricting our structure to the

$\displaystyle t^{i_{1}i_{2}i_{3}}(X)=t^{i_{1}i_{2}i_{3}}({\hat{X}}),$ (3.1)

where

$\displaystyle{\hat{X}}_{\mu}=\frac{X_{\mu}}{\sqrt{X^{2}}}\,,\qquad
X_{12\mu}=-X_{21\mu}=\frac{x_{13\mu}}{x_{13}^{\,2}}-\frac{x_{23\mu}}{x_{23}^{\,2}}\,,$
(3.2)

is unit vector, we have $q=0$ in (A.6) and (A.8)-(A.10). Taking into account
that the nonsingular, tensorial part of the two-point function is given by the
inversion matrix which is a function of the same unit vector (A.11), we see
from (A.5), (A.6) that the scaling behavior of conformal correlators depends
on dimensions of fields only.

Now we formulate a general three-point function for the case of the
correlation functions of three different higher-spin traceless currents.
Rewriting the (A.5) for different spins $s_{1},s_{2},s_{3}$, we get:

$\displaystyle\langle{\cal J}^{(s_{1})}(a;x_{1})\,{\cal
J}^{(s_{2})}(b;x_{2})\,{\cal J}^{(s_{3})}(c;x_{3})\rangle=$
$\displaystyle=\frac{{\cal I}^{(s_{1})}(a,a^{\prime};x_{13}){\cal
I}^{(s_{2})}(b,b^{\prime};x_{23})*^{(s_{1})}_{a^{\prime}}*^{(s_{2})}_{b^{\prime}}t^{(s_{3})}(a^{\prime},b^{\prime};c;{\hat{X}}_{12})}{x_{12}^{\Delta_{(s_{1})}+\Delta_{(s_{2})}-\Delta_{(s_{3})}}x_{23}^{\Delta_{(s_{2})}+\Delta_{(s_{3})}-\Delta_{(s_{1})}}x_{31}^{\Delta_{(s_{1})}+\Delta_{(s_{3})}-\Delta_{(s_{2})}}}$
(3.3)

where for $t^{(s_{3})}(a,b;c;{\hat{X}}_{12})$ we should propose a general
ansatz. For that we note that this object is traceless in all three sets of
symmetrized indices, therefore we can define it as a “kernel” object
$\tilde{t}^{(s_{3})}(a,b;c;{\hat{X}})$ enveloped by three traceless projectors

$t^{(s_{3})}({\tilde{a}},{\tilde{b}};{\tilde{c}};{\hat{X}})={\cal
E}^{(s_{1})}({\tilde{a}},a)*_{a}{\cal
E}^{(s_{2})}({\tilde{b}},b)*_{b}\tilde{t}^{(s_{3})}(a,b;c;{\hat{X}})*_{c}{\cal
E}^{(s_{3})}(c,{\tilde{c}})$ (3.4)

Then for $\tilde{t}^{(s_{3})}(a,b;c;{\hat{X}})$ we propose the following
ansatz:

$\displaystyle\tilde{t}^{(s_{3})}(a,b;c;{\hat{X}})=I^{s_{3}}(c,c^{\prime};{\hat{X}})*_{c^{\prime}}{\tilde{H}}(a,b,c^{\prime};{\hat{X}})$
(3.5)

where

$\displaystyle{\tilde{H}}(a,b,c;{\hat{X}})$
$\displaystyle=\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\in\mathcal{A}\end{subarray}}{\tilde{C}}_{\ell_{1}\ell_{2}\ell_{3}}({\hat{X}}a)^{\ell_{1}}({\hat{X}}b)^{\ell_{2}}({\hat{X}}c)^{\ell_{3}}(ab)^{\alpha}(bc)^{\beta}(ca)^{\gamma}$
(3.6)

To define scope of indices $\mathcal{A}$ we note that natural restriction:

$\displaystyle\alpha+\gamma+\ell_{1}=s_{1}$
$\displaystyle\alpha+\beta+\ell_{2}=s_{2}$
$\displaystyle\gamma+\beta+\ell_{3}=s_{3}$ (3.7)

completely fix $\alpha,\beta,\gamma$ for any choice of
$\ell_{1},\ell_{2},\ell_{3}$:

$\displaystyle 2\alpha$
$\displaystyle=s_{1}+s_{2}-s_{3}+\ell_{3}-\ell_{1}-\ell_{2}$ (3.8)
$\displaystyle 2\beta$
$\displaystyle=s_{2}+s_{3}-s_{1}+\ell_{1}-\ell_{2}-\ell_{3}$ (3.9)
$\displaystyle 2\gamma$
$\displaystyle=s_{1}+s_{3}-s_{2}+\ell_{2}-\ell_{1}-\ell_{3}$ (3.10)
$\displaystyle 2(\alpha$ $\displaystyle+\beta+\gamma)=\sum s_{i}-\sum\ell_{i}$
(3.11)

So introducing:

$\displaystyle n_{i}=s_{i}-\ell_{i},\quad i=1,2,3.$ (3.12)

we have:

$\displaystyle 2\alpha$ $\displaystyle=n_{1}+n_{2}-n_{3}$ (3.13)
$\displaystyle 2\beta$ $\displaystyle=n_{2}+n_{3}-n_{1}$ (3.14) $\displaystyle
2\gamma$ $\displaystyle=n_{1}+n_{3}-n_{2}$ (3.15)

and therefore from positiveness of $\alpha,\beta,\gamma$ we have triangle
inequalities:

$\displaystyle n_{i}+n_{j}\geq n_{k},\quad i\neq j\neq k.$ (3.16)

These inequalities completely fix the scope of $\ell_{i}$ and define the
number of nonzero independent parameters in our ansatz (3.6). For general
conformal dimensions of our currents, these are the only restrictions on the
number of structures. The short representations, corresponding to
(partially-)conserved currents, will be discussed later.

We analyzed the inequalities given above for arbitrary triplets of spins and
were able to guess the analytical expressions for the number of terms in the
ansatz. Interestingly, this number is not a smooth function of spins, which
manifests itself by gaps when some spins coincide and different dependence of
even and odd spins. We will use the step function in the following:

$\displaystyle\eta(s)=\frac{1-(-1)^{s}}{2}$ (3.17)

Then the solution for numbers of allowed monomials in the case when all spins
are the same $s_{1}=s_{2}=s_{3}=s$ is

$\displaystyle N_{sss}=\frac{1}{24}(s+2-\eta(s))(s+3)(s+4+\eta(s))$ (3.18)

Then we turn to the case when two out of three spins are equal. There is a
special point in this case: $s_{1}=s_{2}=s,s_{3}=2s$. The number of structures
in this case is:

$\displaystyle N_{ss2s}=\frac{1}{6}(s+1)(s+2)(s+3)$ (3.19)

There are two cases beyond this point:

  * •

$s_{3}>s=s_{1}=s_{2}$

$\displaystyle
N^{s_{3}>s}_{sss_{3}}=\frac{1}{6}(s+1)(s+2)(s+3)-\frac{1}{24}p(p+2)(p+4)-\frac{1}{8}(p+2)\eta(p)$
(3.20)

where $p=2s-s_{3}$, and

  * •

$s_{1}<s=s_{2}=s_{3}$

$\displaystyle
N^{s_{1}<s}_{s_{1}ss}=\frac{1}{8}[(s_{1}+2)^{2}-\eta(s_{1})](2s-s_{1}+2)$
(3.21)

Then the next observation from computer calculation is for the case
$s_{1}+s_{2}=s_{3}$:

$\displaystyle
N^{s_{1}+s_{2}=s_{3}}_{s_{1}s_{2}s_{3}}=\frac{1}{2}(s_{1}+1)(s_{1}+2)(s_{2}-\frac{1}{3}(s_{1}-3))$
(3.22)

And finally the last observation is about numbers of monomials for the case
with just general ordering $s_{1}<s_{2}<s_{3}$:

$\displaystyle
N^{s_{1}<s_{2}<s_{3}}_{s_{1}s_{2}s_{3}}=N^{s_{1}+s_{2}=s_{3}}_{s_{1}s_{2}s_{3}}-\frac{1}{24}P(P+2)(2P+5)-\frac{1}{8}\eta(P)$
(3.23) $\displaystyle P=s_{1}+s_{2}-s_{3}$

So we see that (3.18)-(3.23) completely cover all scope of indices
$\mathcal{A}$ and we have analytic formula for number of all monomials in our
ansatz with indices satisfying triangle inequalities. The last question
remains, what happens when in our different spin case the greatest one stops
to satisfy triangle inequality $s_{3}>s_{1}+s_{2}$ ? The answer is that number
of monomials in this case stabilized with latest one satisfying triangle
inequality
$N^{s_{1}+s_{2}<s_{3}}_{s_{1}s_{2}s_{3}}=N^{s_{1}+s_{2}=s_{3}}_{s_{1}s_{2}s_{3}}$.

Finalizing this consideration we present some geometric arguments for cubic
behaviour and discontinuities in points with coincident spins. Let us rewrite
our inequality (3.16) in the form of equations introducing three new
nonnegative variables $\lambda_{i}$

$\displaystyle n_{i}+n_{j}=n_{k}+\lambda_{k},\quad i\neq j\neq k$ (3.24)

then summing any pair of these equations we come to the important relation:

$\displaystyle\lambda_{i}+\lambda_{j}=2n_{k},\quad\quad i\neq j\neq k$ (3.25)
$\displaystyle n_{i}\in[0,1,\dots s_{i}]$ (3.26)

So replacing r.h.s. with maximal value we see that the scope of allowed
indices is integer numbers with the following restrictions:

  * •

From (3.25) we see that allowed $\lambda_{i}$ are all even or odd, so we have
separate even or odd lattice.

  * •

these even or odd pairs restricted by positiveness and inequality

$\displaystyle\lambda_{i}+\lambda_{j}\leq 2s_{k}\quad\quad i\neq j\neq k.$
(3.27)

$\lambda_{i}$$\lambda_{j}$$2s_{k}$ $2s_{k}$ Figure 1: Area of
$\lambda_{i}+\lambda_{j}\leq 2s_{k}$

The important point here is that allowed points fulfill all vertexes of our
lattice triangle in Figure 1 completely. So the number of these points should
be proportional to the area of this triangle. To get the general picture of
the numbers of allowed monomials in our ansatz, we should expand our discrete
triangle in the third direction in the form of a triangle prism with a hight
in the third direction. Then the full solution will be intersection of three
different prisms constructed on planes $(\lambda_{1},\lambda_{2})$ ,
$(\lambda_{2},\lambda_{3})$ and $(\lambda_{3},\lambda_{1})$ with corresponding
legs $2s_{3},2s_{1},2s_{2}$ of right triangle bases (See Figure 2). This
picture explains everything about the non-smooth behavior of our formulas
above because of an irregular intersection of these prisms for different spins
$s_{1},s_{2},s_{3}$.

Figure 2: Intersection of Prisms in the case $s_{1}\leq s_{2}\leq s_{3}$

Then we can understand that in coincident cases the geometrical figures we get
as a result of intersections of our prisms are more symmetric. We illustrate
this for the cases $s_{1}=s_{2}\leq s_{3}$ and $s_{1}\leq s_{2}=s_{3}$ (see
Figure 3) and the most symmetric case $s_{1}=s_{2}=s_{3}$ (Figure 4).

Figure 3: Intersection in the case $s_{1}=s_{2}\leq s_{3}$ and $s_{1}\leq
s_{2}=s_{3}$ Figure 4: Intersection of Prisms in the case $s_{1}=s_{2}=s_{3}$

So we see that something like “phase transitions” happen in our formulas. On
the other hand, this geometrical three-dimensional picture with previous
consideration of Figure 1 leads to the understanding that the full number of
monomials allowed by triangle inequalities is proportional to the volume of
our intersection and therefore should be a cubic function of spins.

In the end we note that all the examples considered in [77] can be exactly
produced from our general formulas (3.4)-(3.6) with corresponding choice of
the value of spins and solution of the triangle inequality. For illustration,
we discuss the important case of coinciding spins in Appendix B.

##  4 Three-point function: conservation condition

Now we turn to the investigation of the conservation condition for higher spin
three point function. To formulate it for higher spin case we first introduce
short notation for combinations of dimensions:

$\displaystyle\Delta_{12}$ $\displaystyle=$
$\displaystyle\Delta_{(s_{1})}+\Delta_{(s_{2})}-\Delta_{(s_{3})}$ (4.1)
$\displaystyle\Delta_{23}$ $\displaystyle=$
$\displaystyle\Delta_{(s_{2})}+\Delta_{(s_{3})}-\Delta_{(s_{1})}$ (4.2)
$\displaystyle\Delta_{31}$ $\displaystyle=$
$\displaystyle\Delta_{(s_{3})}+\Delta_{(s_{1})}-\Delta_{(s_{2})}$ (4.3)
$\displaystyle\Delta_{(s_{i})}$ $\displaystyle=$ $\displaystyle
d+s_{i}-2,\quad\quad i=1,2,3$ (4.4)

The latter expressions are the dimensions of conserved currents. Then
redirecting readers for details of derivation to the last part of Appendix A,
we can write conservation condition

$\displaystyle(\nabla_{x_{1}}\partial_{a})\langle{\cal
J}^{(s_{1})}(a;x_{1})\,{\cal J}^{(s_{2})}(b;x_{2})\,{\cal
J}^{(s_{3})}(c;x_{3})\rangle=0$ (4.5)

in the form:

$\displaystyle(\nabla_{X}\partial_{a})t^{(s_{3})}(a,b;c;X)$
$\displaystyle=\Delta_{12}\frac{(X\partial_{a})}{X^{2}}t^{(s_{3})}(a,b;c;X)$
(4.6)

The last one is the equation for structural tensor object
$t^{(s_{3})}(a,b;c;X)$ which is completely equivalent to the conservation
condition for the three-point function. Then, separating the traceless
projector from the “kernel” part of (3.4) (see also Appendix A for details)
and introducing the $k-$th trace of our ansatz:

$\displaystyle\square^{k}_{a}\tilde{t}^{(s)}(a,b,c;{\hat{X}})=\sum_{\begin{subarray}{c}\ell_{1}\in[2k,\dots
s_{1}];\ell_{2},\ell_{3}\in[0,\dots s_{2},s_{3}]\\\ \\{\ell_{i}\\}\in{\cal
A}\end{subarray}}T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}\left[\genfrac{}{}{0.0pt}{}{\ell_{1}-2k,\ell_{2},\ell_{3}}{\alpha;\beta,\gamma}\right]$
(4.7)

where we shortened the formulas using the notation:

$\displaystyle\left[\genfrac{}{}{0.0pt}{}{\ell_{1},\ell_{2},\ell_{3}}{\alpha;\beta,\gamma}\right]=({\hat{X}}a)^{\ell_{1}}({\hat{X}}b)^{\ell_{2}}({\hat{X}}c)^{\ell_{3}}(ab)^{\alpha}I^{\beta}(b,c;{\hat{X}})I^{\gamma}(c,a;{\hat{X}})$
(4.8)

and $T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}$ is $k-$th trace map of
${\tilde{C}}_{\ell_{1}\ell_{2}\ell_{3}}$ from (3.6). In this way using
important formula (A.31) and expression (4.7) after long manipulations we
write conservation condition (4.6) in terms of equations on
$T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}$:

$\displaystyle(\ell_{1}-2k)(s_{3}-s_{2})T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}$
$\displaystyle+$
$\displaystyle(\alpha+1)(2\ell_{3}-2k-d-2s_{2}+2)T^{(k)}_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}+(\gamma+1)(2\ell_{2}-2k-d-2s_{3}+2)T^{(k)}_{\ell_{1}-1,\ell_{2},\ell_{3}-1}$
$\displaystyle+$
$\displaystyle(\alpha+1)(\ell_{3}+1)T^{(k)}_{\ell_{1}-1,\ell_{2},\ell_{3}+1}+(\gamma+1)(\ell_{2}+1)T^{(k)}_{\ell_{1}-1,\ell_{2}+1,\ell_{3}}$
$\displaystyle+$
$\displaystyle\frac{1}{d+2s_{1}-2k-4}\left[2(\ell_{2}-\ell_{3})T^{(k+1)}_{\ell_{1},\ell_{2},\ell_{3}}+2(\beta+1)\big{(}T^{(k+1)}_{\ell_{1}+1,\ell_{2},\ell_{3}-1}+T^{(k+1)}_{\ell_{1}+1,\ell_{2}-1,\ell_{3}}\big{)}\right.$
$\displaystyle-$
$\displaystyle\left.(\ell_{2}+1)T^{(k+1)}_{\ell_{1}+1,\ell_{2}+1,\ell_{3}}-(\ell_{3}+1)T^{(k+1)}_{\ell_{1}+1,\ell_{2},\ell_{3}+1}\right]=0$
(4.9)

where the traces themselves satisfy the following recursion relation:

$\displaystyle
T^{(k+1)}_{\ell_{1},\ell_{2},\ell_{3}}=(\ell_{1}-2k)(\ell_{1}-2k-1)T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}+2(\alpha+1)(\gamma+1)T^{(k)}_{\ell_{1}-2,\ell_{2},\ell_{3}}$
$\displaystyle+2(\alpha+1)(\ell_{1}-2k-1)T^{(k)}_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}-2(\gamma+1)(\ell_{1}-2k-1)T^{(k)}_{\ell_{1}-1,\ell_{2},\ell_{3}-1}$
(4.10)

That is not the whole story. The bad news here is that the equation (4.9)
should be supplemented by a conservation condition for the second current in
the correlation function when the latter is also conserved. This can be done
in (4.6) by replacements of $s_{1}\leftrightarrow s_{2}$ and
$x_{1}\leftrightarrow x_{2}$ and $a_{\mu}\leftrightarrow b_{\mu}$, or directly
in (4.9), (4.10) replacing $s_{1}\leftrightarrow
s_{2},\ell_{1}\leftrightarrow\ell_{2}$.

The good news here is that we do not need to solve all recursion equations
(4.9) for all $T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}\,(k=0,1\dots[s_{1}/2])$.
In fact, we need to solve only the first conservation condition for $k=0$, all
others will be satisfied automatically because they are higher ($k-$th) traces
of the first one with $k=0$.

Using the helpful ansatz-normalization:

$\displaystyle
T^{(0)}_{\ell_{1},\ell_{2},\ell_{3}}=\frac{(-1)^{\ell_{3}}}{\alpha!\beta!\gamma!}C_{\ell_{1},\ell_{2},\ell_{3}}$
(4.11) $\displaystyle
T^{(1)}_{\ell_{1},\ell_{2},\ell_{3}}=\frac{(-1)^{\ell_{3}}}{\alpha!\beta!\gamma!}\Big{[}\ell_{1}(\ell_{1}-1)C_{\ell_{1},\ell_{2},\ell_{3}}+2\beta
C_{\ell_{1}-2,\ell_{2},\ell_{3}}$
$\displaystyle+2(\ell_{1}-1)C_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}+2(\ell_{1}-1)C_{\ell_{1}-1,\ell_{2},\ell_{3}-1}\Big{]}=\frac{(-1)^{\ell_{3}}}{\alpha!\beta!\gamma!}T_{\ell_{1},\ell_{2},\ell_{3}}$
(4.12)

we obtain effective conservation condition:

$\displaystyle\ell_{1}(s_{3}-s_{2})C_{\ell_{1},\ell_{2},\ell_{3}}$
$\displaystyle+$
$\displaystyle(2\ell_{3}-d-2s_{2}+2)C_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}-(2\ell_{2}-d-2s_{3}+2)C_{\ell_{1}-1,\ell_{2},\ell_{3}-1}$
$\displaystyle+$
$\displaystyle(\ell_{2}+1)C_{\ell_{1}-1,\ell_{2}+1,\ell_{3}}-(\ell_{3}+1)C_{\ell_{1}-1,\ell_{2},\ell_{3}+1}$
$\displaystyle+$
$\displaystyle\frac{1}{d+2s_{1}-4}\left[2(\ell_{2}-\ell_{3})T_{\ell_{1},\ell_{2},\ell_{3}}+2(\beta+1)\big{(}T_{\ell_{1}+1,\ell_{2},\ell_{3}-1}+T_{\ell_{1}+1,\ell_{2}-1,\ell_{3}}\big{)}\right.$
$\displaystyle-$
$\displaystyle\left.(\ell_{2}+1)T_{\ell_{1}+1,\ell_{2}+1,\ell_{3}}-(\ell_{3}+1)T_{\ell_{1}+1,\ell_{2},\ell_{3}+1}\right]=0$
(4.13)

which we should amend with the same type of equation but now for $s_{2}$, if
the second current is also conserved:

$\displaystyle\ell_{2}(s_{3}-s_{1})C_{\ell_{1},\ell_{2},\ell_{3}}$
$\displaystyle+$
$\displaystyle(2\ell_{3}-d-2s_{1}+2)C_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}-(2\ell_{1}-d-2s_{3}+2)C_{\ell_{1},\ell_{2}-1,\ell_{3}-1}$
$\displaystyle+$
$\displaystyle(\ell_{1}+1)C_{\ell_{1}+1,\ell_{2}-1,\ell_{3}}-(\ell_{3}+1)C_{\ell_{1},\ell_{2}-1,\ell_{3}+1}$
$\displaystyle+$
$\displaystyle\frac{1}{d+2s_{2}-4}\left[2(\ell_{1}-\ell_{3})\bar{T}_{\ell_{1},\ell_{2},\ell_{3}}+2(\gamma+1)\big{(}\bar{T}_{\ell_{1},\ell_{2}+1,\ell_{3}-1}+\bar{T}_{\ell_{1}-1,\ell_{2}+1,\ell_{3}}\big{)}\right.$
$\displaystyle-$
$\displaystyle\left.(\ell_{1}+1)\bar{T}_{\ell_{1}+1,\ell_{2}+1,\ell_{3}}-(\ell_{3}+1)\bar{T}_{\ell_{1},\ell_{2}+1,\ell_{3}+1}\right]=0$
(4.14)

where $T_{\ell_{1},\ell_{2},\ell_{3}},\bar{T}_{\ell_{1},\ell_{2},\ell_{3}}$
are corresponding trace maps:

$\displaystyle
T_{\ell_{1},\ell_{2},\ell_{3}}=\Big{[}\ell_{1}(\ell_{1}-1)C_{\ell_{1},\ell_{2},\ell_{3}}+2\beta
C_{\ell_{1}-2,\ell_{2},\ell_{3}}$
$\displaystyle+2(\ell_{1}-1)C_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}+2(\ell_{1}-1)C_{\ell_{1}-1,\ell_{2},\ell_{3}-1}\Big{]}$
(4.15)
$\displaystyle\bar{T}_{\ell_{1},\ell_{2},\ell_{3}}=\Big{[}\ell_{2}(\ell_{2}-1)C_{\ell_{1},\ell_{2},\ell_{3}}+2\gamma
C_{\ell_{1},\ell_{2}-2,\ell_{3}}$
$\displaystyle+2(\ell_{2}-1)C_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}+2(\ell_{2}-1)C_{\ell_{1},\ell_{2}-1,\ell_{3}-1}\Big{]}$
(4.16)

We do not yet have a full solution for this system of equations. But we
analyzed these equations using a computer program and investigated the rank of
this linear system for different triplets of spins using our ansatz
(3.4)-(3.6) and normalization (4.11), (4.12). This system of linear equations
for $C_{\ell_{1},\ell_{2},\ell_{3}}$ has a number of independent parameters
satisfying triangle inequalities described in the previous section. Then
computing the rank of the corresponding system for multiple cases we obtain a
universal answer: the rank of the system (4.13), (4.14) depends only on the
minimal spin:

  * •

The number of independent parameters of the three-point function (or linearly
independent correlators) of conserved currents with spins $s_{1},s_{2},s_{3}$
is equal to

$N_{s_{1},s_{2},s_{3}}=min\\{s_{1},s_{2},s_{3}\\}+1\,.$

We refer to Appendix B for some details on the special case of coincident
spins.

##  5 Conservation condition as a differential equation

In this section, we first construct differential equations for the correlators
of conserved currents in the case of coincident spins and then generalize them
to the cases with different spins. First, we transform our recursion equation
(B.20) to a differential equation multiplying it by the following powers of
formal variables $x^{\ell_{1}-1}y^{\ell_{2}}z^{\ell_{3}}$ and summing on all
possible values of $\ell^{i}$

$\displaystyle D(\partial_{x},\partial_{y},\partial_{z};C(x,y,z))$
$\displaystyle=\sum_{\\{\ell_{i}\\}}D_{\ell_{1}\ell_{2}\ell_{3}}x^{\ell_{1}-1}y^{\ell_{2}}z^{\ell_{3}}=0$
(5.1)

in other words we should obtain differential equation for the functions

$C(x,y,z)=\sum_{\\{\ell_{i}\\}}C_{\ell_{1}\ell_{2}\ell_{3}}x^{\ell_{1}}y^{\ell_{2}}z^{\ell_{3}}$
(5.2)

and

$T(x,y,z)=\sum_{\\{\ell_{i}\\}}T_{\ell_{1}\ell_{2}\ell_{3}}x^{\ell_{1}-2}y^{\ell_{2}}z^{\ell_{3}}$
(5.3)

In all these equations $\\{\ell_{i}\\}$ means value of indeces
$\ell_{i},i=1,2,3$ satisfying the triangle inequality

$s+\ell_{i}\geq\ell_{j}+\ell_{k},\quad i\neq j\neq k$ (5.4)

Comparing (5.3) with (B.21)we obtain:

$\displaystyle
T(x,y,z)=[\partial^{2}_{x}+(x+2y+2z)\partial_{x}-y\partial_{y}-z\partial_{z}+s+2]C(x,y,z)$
(5.5)

Then we can obtain differential equation version of our conservation equation
(B.20):

$\displaystyle D(\partial_{x},\partial_{y},\partial_{z};C(x,y,z))$
$\displaystyle=\Big{[}(\Delta_{s}+s)(z-y)+\frac{1}{2}(s+1-4yz+x\partial_{x}-y\partial_{y}-z\partial_{z})(\partial_{y}-\partial_{z})\Big{]}C(x,y,z)$
$\displaystyle+\frac{1}{d+2s-4}\Big{[}(2x+y+z+\frac{1}{2}[\partial_{y}+\partial_{z}])(y\partial_{y}-z\partial_{z})$
$\displaystyle+(s-x\partial_{x})(y-z-\frac{1}{2}[\partial_{y}-\partial_{z}])\Big{]}T(x,y,z)=0$
(5.6)

We see that our differential operator is antisymmetric in $z$ and $y$ although
the functions $C(x,y,z)$ and $T(x.y.z)$ are symmetric. A generalization to
different spins is straightforward: instead of (5.6) we have an equation
obtained with the same scheme from the recursion equation (4.13):

$\displaystyle
D^{(s_{1},s_{2},s_{3})}(\partial;C(x,y,z))=\big{[}(s_{3}-s_{2})\partial_{x}+(\Delta_{s_{3}}+s_{3})z-(\Delta_{s_{2}}+s_{2})y\big{]}C(x,y,z)$
$\displaystyle+\frac{1}{2}(s_{2}+s_{3}-s_{1}+1-4yz+x\partial_{x}-y\partial_{y}-z\partial_{z})(\partial_{y}-\partial_{z})C(x,y,z)$
$\displaystyle+\frac{1}{d+2s_{1}-4}\Big{[}(2x+y+z+\frac{1}{2}[\partial_{y}+\partial_{z}])(y\partial_{y}-z\partial_{z})$
$\displaystyle+(s_{1}-x\partial_{x})(y-z-\frac{1}{2}[\partial_{y}-\partial_{z}])+\frac{1}{2}(s_{3}-s_{2})[y+z+\partial_{y}+\partial_{z}]\Big{]}T(x,y,z)=0$

where $T(x,y,z)$ in this case is

$\displaystyle
T(x,y,z)=[\partial^{2}_{x}+(x+2y+2z)\partial_{x}-y\partial_{y}-z\partial_{z}+s_{2}+s_{3}-s_{1}+2]C(x,y,z)\quad$
(5.8)

The equation (LABEL:3.7) should be supplemented by a conservation condition
for the second current, when the latter is conserved. This can be obtained
from (LABEL:3.7) and (5.8) by replacements $s_{1}\leftrightarrow s_{2}$ and
$x\leftrightarrow y$. The solution to these general equations for the
correlators of conserved currents will be addressed in an upcoming work.

##  6 Conclusions

We have established a general ansatz for the tensorial structure of the
conformal three-point function for general spins and general dimensions. This
allows us to calculate the exact numbers of conformal structures corresponding
to all cases of AdS dual bulk interaction vertices. We present explicit
formulas for three-point functions of conformal correlators of three non-
conserved currents, corresponding to massive fields in the bulk. The number of
structures for non-conserved currents is equivalent to the number of vertices
with massive fields in the bulk, counting the number of contractions of three
symmetric fields of ranks $s_{1},s_{2},s_{3}$ with each other and derivatives
acting on them, with a condition that the traces and divergences are excluded,
and the derivatives do not contract between themselves (this latter condition,
stemming from field-redefinition freedom, limits the possible Lorentz scalars
to a finite number: see, e.g., [25, 48, 104]).

The special cases of (partially) conserved currents, corresponding to the
short representations or (partially-)massless fields in the bulk, will be
studied elsewhere: the extra constraints on the correlators stemming from the
conservation of the currents imply non-trivial differential equations, for
which the general solutions will be treated in future work. However, we worked
out and further studied the structure of the constraints in the case of the
conserved currents, both as differential equations and as recursion relations
on the coefficients of the ansatz. The latter form allowed us to tackle a
large number of cases numerically. Our results confirm the expectation from
earlier works [89, 90, 92, 94] about the number of structures in the
correlators of conserved currents, which, in turn, coincides with the number
of massless vertices in the bulk [25, 38]. We hope to solve analytically the
conservation conditions to fully classify the correlators of
(partially-)conserved currents and make a match with the $AdS$ vertices
involving (partially-)massless fields [53]. The case of all massive fields is
fully covered by our ansatz in one-to-one correspondence with the vertices in
the bulk [25, 48].

The correlation functions of three conserved currents were derived earlier
using different approaches in [92, 94]. In even dimensions, they were
described by the correlators in free theories of so-called singletons —
conformal fields describing the short conformal representations described by
the (self-dual) multi-forms, corresponding to rectangular Young diagrams of
the half-maximal height of the massless little group in even dimensions (see,
e.g., [119]). In four dimensions, these are the spin-s massless fields, which
are representations of the conformal algebra $SO(4,2)$ despite the lack of
conformal symmetry in their standard off-shell descriptions (see, e.g., [120,
121]).§§§ Explicit descriptions of the singleton theories in terms of
covariant Lagrangians are so far only well-studied for the spin-one case (see
[122] for a review).  The situation is different in the odd dimensions [94],
where the singletons are missing or, presumably, correspond to some
generalized free field theories lacking locality: free field equations
containing square root of d’Alambertian operator (see, e.g., [123]).

The formulation [77] and our generalization for higher spins are also suitable
for the investigation of the singular part of the correlation function to get
a route to the trace anomaly structure in the higher-spin case. We leave this
to future investigations.

## Acknowledgements

R. M. would like to thank Stefan Theisen, Rubik Poghossian and Aleksey Isaev
for many valuable discussions during long period of preparation of this paper,
and special gratitude to Ruben Mkrtchyan for productive and focused on result
discussions. R. M. and M. K. where supported by the Science Committee of RA,
in the frames of the research project # 21AG-1C060. K. M. was supported by the
European Union’s Horizon 2020 Research and Innovation Programme under the
Marie Skłodowska-Curie Grant No. 844265, UKRI and STFC Consolidated Grant
ST/T000791/1.

##  Appendix A Short review of Osborn-Petkou formulation and adaptation to
higher spin case

In this appendix, we present a short review of useful formulas and
constructions proposed in article [77] (see also [78, 79]).

### Conformal Transformations

The conformal transformations (combination of translation, rotation, scale
transformation, and special conformal boosts) are diffeomorphisms preserving
metric up to a local scale factor:

$\displaystyle x_{\mu}\rightarrow x^{\prime}_{\mu}(x),\quad
g_{\mu\nu}dx^{\prime\mu}dx^{\prime\nu}\rightarrow\Omega(x)^{-2}g_{\mu\nu}dx^{\mu}dx^{\nu}$
(A.1)

Combining this transformation with local dilatation we arrive at local
rotations:

$\displaystyle R_{\mu}^{\;\alpha}(x)=\Omega(x)\frac{\partial
x^{\prime}_{\mu}}{\partial x_{\alpha}},\quad
R_{\mu}^{\;\alpha}(x)R_{\alpha}^{\;\nu}(x)=\delta_{\mu}^{\nu}\,.$ (A.2)

Adding inversion to this picture :

$\displaystyle
x^{\prime}_{\mu}=\frac{x_{\mu}}{x^{2}},\quad\Omega(x)=x^{2},\quad
R_{\mu\nu}(x)=I_{\mu\nu}(x)=\delta_{\mu\nu}-2\frac{x_{\mu}x_{\nu}}{x^{2}}$
(A.3)

we see that the rotation operator, in this case, is the Inversion matrix
$I_{\mu\nu}$. A combination of inversion, rotation, and translation can
describe any conformal transformation.

We will show below how the conformal symmetry fixes the form of the two and
three-point correlation functions for arbitrary quasi-primary fields
$\mathcal{O}^{i}(x)$, where $i$ is an index counting corresponding
representation of the rotation group $O(d)$ (see [77] for details). The
symmetric representation of the conformal group is defined by two quantum
numbers: the spin and the conformal dimension. The two-point function of two
operators is fixed by conformal symmetry up to an overall constant:

$\displaystyle<\mathcal{O}^{i}(x_{1})\mathcal{\bar{O}}_{j}(x_{2})>=\frac{C_{\mathcal{O}}}{(x_{12}^{2})^{\eta}}D^{i}_{j}(I(x_{12})),\quad
x_{12\mu}=x_{1\mu}-x_{2\mu}$ (A.4)

Here $\mathcal{\bar{O}}_{j}(x)$ is conjugate representation for
$\mathcal{O}^{i}(x)$ with the same conformal dimension. Another important
object here is $D(I(x_{12}))$ which is corresponding representation for the
inversion matrix $I_{\mu\nu}(x)=\delta_{\mu\nu}-2x_{\mu}x_{\nu}/x^{2}$ .

### Three point function

Since conformal transformations transform any three points into any others,
the three-point function is also essentially defined in general dimension $d$.
Our discussion for arbitrary representations for the fields ${\cal
O}_{1},{\cal O}_{2},{\cal O}_{3}$ with dimensions $\eta_{1},\eta_{2},\eta_{3}$
is based on the following formula from [77]

$\displaystyle\langle{\cal O}_{1}^{i_{1}}(x_{1})\,{\cal
O}_{2}^{i_{2}}(x_{2})\,{\cal O}_{3}^{i_{3}}(x_{3})\rangle$
$\displaystyle=\frac{1}{(x_{12}^{\,2})^{\delta_{12}}\,(x_{23}^{\,2})^{\delta_{23}}\,(x_{31}^{\,2})^{\delta_{31}}}$
$\displaystyle\times
D_{1}^{\,i_{1}}{}_{\\!j_{1}}(I(x_{13}))D_{2}^{\,i_{2}}{}_{\\!j_{2}}(I(x_{23}))\,t^{j_{1}j_{2}i_{3}}(X_{12})\,,$
(A.5)

where $t^{i_{1}i_{2}i_{3}}(X)$ is a tensor living in three different spin
representations in general case. This object transforms in a proper way with
respect to local rotation and dilatations.

$\displaystyle
D_{1}^{\,i_{1}}{}_{\\!j_{1}}(R)D_{2}^{\,i_{2}}{}_{\\!j_{2}}(R)D_{3}^{\,i_{3}}{}_{\\!j_{3}}(R)\,t^{j_{1}j_{2}j_{3}}(X)=t^{i_{1}i_{2}i_{3}}(RX)\
\hbox{for all}\ R\in O(d)\ ,$ $\displaystyle t^{i_{1}i_{2}i_{3}}(\lambda
X)={}\lambda^{q}t^{i_{1}i_{2}i_{3}}(X)$ (A.6)

and

$\displaystyle
X_{12\mu}=-X_{21\mu}=\frac{x_{13\mu}}{x_{13}^{\,2}}-\frac{x_{23\mu}}{x_{23}^{\,2}}\,,\quad
X_{12}^{\,2}=\frac{x_{12}^{\,2}}{x_{13}^{\,2}x_{23}^{\,2}}$ (A.7)

The scaling dimensions of the fields should satisfy the following expressions

$\displaystyle\delta_{12}=\frac{1}{2}(\eta_{1}+\eta_{2}-\eta_{3}+q)\,,$ (A.8)
$\displaystyle\delta_{23}=\frac{1}{2}(\eta_{2}+\eta_{3}-\eta_{1}-q)\,,$ (A.9)
$\displaystyle\delta_{31}=\frac{1}{2}(\eta_{3}+\eta_{1}-\eta_{2}-q)\,.$ (A.10)

So we see that for the construction of the two-point function for spin $s$
currents, we should realize the construction of the representation of the
inversion matrix $D(I(x_{12}))$ where:

$I_{\mu\nu}(x_{12})=\delta_{\mu\nu}-2{\hat{x}}_{12\mu}{\hat{x}}_{12\nu},\quad{\hat{x}}_{12}=\frac{x_{12}}{\sqrt{x^{2}_{12}}}$
(A.11)

which is more or less obvious and known. Another important property of this
formulation is that in the three-point function we can rearrange all three
representations due to the following important properties [77] of structural
function ($q=0$):

$\displaystyle
D_{1}^{\,i_{1}}{}_{\\!j_{1}}(I({\hat{x}}_{13}))D_{2}^{\,i_{2}}{}_{\\!j_{2}}(I({\hat{x}}_{23}))\,t^{j_{1}j_{2}i_{3}}({\hat{X}}_{12})$
$\displaystyle=D_{1}^{\,i_{1}}{}_{\\!j_{1}}(I({\hat{x}}_{12}))D_{3}^{\,i_{3}}{}_{\\!j_{3}}(I({\hat{x}}_{32}))\,{\tilde{t}}^{\,j_{1}i_{2}j_{3}}({\hat{X}}_{13})=D_{2}^{\,i_{2}}{}_{\\!j_{2}}(I({\hat{x}}_{21}))D_{3}^{\,i_{3}}{}_{\\!j_{3}}(I({\hat{x}}_{31}))\,{\hat{t}}^{\,i_{1}j_{2}j_{3}}({\hat{X}}_{32})\,,$
$\displaystyle{\tilde{t}}^{\,i_{1}i_{2}i_{3}}({\hat{X}})=D_{1}^{\,i_{1}}{}_{\\!j_{1}}(I({\hat{X}}))\,t^{j_{1}i_{2}i_{3}}({\hat{X}}),\quad{\hat{t}}^{\,i_{1}i_{2}i_{3}}({\hat{X}})=D_{2}^{\,i_{2}}{}_{\\!j_{2}}(I({\hat{X}}))\,t^{i_{1}j_{2}i_{3}}({\hat{X}})\,.$
(A.12)

It follows then, that in the case when all three representations are the same
(i.e. same spin currents) and the three-point function is symmetric for all
fields ${\cal O}_{1},\,{\cal O}_{2},\,{\cal O}_{3},$ then:

$t^{i_{2}i_{1}i_{3}}(X)=t^{i_{1}i_{2}i_{3}}(-X)\,,\quad
D^{\,i_{1}}{}_{\\!j_{1}}(I(X))\,{t}^{j_{1}i_{2}i_{3}}(X)=t^{i_{3}i_{1}i_{2}}(-X).$
(A.13)

The first relation contains $-X$ in r.h.s. because this object depends on the
space-time coordinates through the difference between the inversions of the
first and second coordinates around the third point (A.7), and when we replace
the first two operators we also exchange $x_{1}$ with $x_{2}$. The importance
of the minus sign in the second relation we consider in detail during the
investigation of our ansatz for $t^{i_{1}i_{2}i_{3}}(X)$. Then for irreducible
representations, for which the two-point functions are fixed as (A.4), we see
consistent scaling behavior and covariance with respect to inversions,
rotations, and translations. All these mean that $D(I(x_{12}))$ behaves as a
parallel transport transformation between two space-time points for local
conformal rotations. This fact is very important for understanding an
analogous formula for three-point functions. The important property of
conformal transformations is that one can map any three points into any other
three points. This leads to an essentially (almost) unique three-point
function in general dimension $d$. The general form of the three-point
function is considered in [77] and presented here in (A.5). The original point
of this consideration is that the three-point function is described through
the homogeneous tensor $t^{i_{1}i_{2}i_{3}}(X)$ satisfying (A.6) and (A.12).
More details can be found in [77],[78] and [79], here we just note that if we
restrict ourselves to the polynomial function of unit vector:

$\displaystyle t^{i_{1}i_{2}i_{3}}(X)=t^{i_{1}i_{2}i_{3}}({\hat{X}}),$ (A.14)

where

$\displaystyle{\hat{X}}_{\mu}=\frac{X_{\mu}}{\sqrt{X^{2}}},$ (A.15)

then in (A.8)-(A.10) we have

$\displaystyle q=0$ (A.16)

and instead of

$\displaystyle
I_{\mu\alpha}(x_{23}){\hat{X}}_{12\,\alpha}=\frac{x^{2}_{12}}{x^{2}_{13}}{\hat{X}}_{13\,\mu}\,,\quad
I_{\mu\alpha}(x_{13}){\hat{X}}_{12\,\alpha}=\frac{x^{2}_{12}}{x^{2}_{23}}{\hat{X}}_{32\,\mu}\,,$
(A.17)

we have

$\displaystyle
I_{\mu\alpha}(x_{23}){\hat{X}}_{12\,\alpha}={\hat{X}}_{13\,\mu}\,,\quad
I_{\mu\alpha}(x_{13}){\hat{X}}_{12\,\alpha}={\hat{X}}_{32\,\mu}\,,$ (A.18)

and we see that inversion operators $I_{\mu\alpha}(x_{ij}),i\neq j,i,j=1,2,3$
really rotate from one direction to other unit inverted vectors
${\hat{X}}_{ij}$. This leads to the familiar expression for the three-point
function:

$\displaystyle\langle{\cal O}_{1}^{i_{1}}(x_{1})\,{\cal
O}_{2}^{i_{2}}(x_{2})\,{\cal O}_{3}^{i_{3}}(x_{3})\rangle={}$
$\displaystyle\frac{1}{(x_{12}^{\,2})^{\delta_{12}}\,(x_{23}^{\,2})^{\delta_{23}}\,(x_{31}^{\,2})^{\delta_{31}}}$
$\displaystyle\times
D_{1}^{\,i_{1}}{}_{\\!j_{1}}(I(x_{13}))D_{2}^{\,i_{2}}{}_{\\!j_{2}}(I(x_{23}))\,t^{j_{1}j_{2}i_{3}}({\hat{X}}_{12})\,,$
(A.19)

where $t^{i_{1}i_{2}i_{3}}({\hat{X}})$ is a homogeneous and dimensionless
tensor satisfying

$\displaystyle
D_{1}^{\,i_{1}}{}_{\\!j_{1}}(R)D_{2}^{\,i_{2}}{}_{\\!j_{2}}(R)D_{3}^{\,i_{3}}{}_{\\!j_{3}}(R)$
$\displaystyle\,t^{j_{1}j_{2}j_{3}}({\hat{X}})=t^{i_{1}i_{2}i_{3}}(R{\hat{X}})\
\hbox{for all}\ R\ ,$ (A.21) $\displaystyle
t^{i_{1}i_{2}i_{3}}(\lambda{\hat{X}})={}$ $\displaystyle
t^{i_{1}i_{2}i_{3}}({\hat{X}})$ (A.22)

and

${\hat{X}}_{12\mu}=-{\hat{X}}_{21\mu}=\sqrt{\frac{x_{13}^{\,2}x_{23}^{\,2}}{x_{12}^{\,2}}}\left[\frac{x_{13\mu}}{x_{13}^{\,2}}-\frac{x_{23\mu}}{x_{23}^{\,2}}\right]$
(A.23)

The scaling dimensions of the fields for $q=0$ are

$\displaystyle\delta_{12}=$
$\displaystyle\frac{1}{2}(\eta_{1}+\eta_{2}-\eta_{3})\,,$
$\displaystyle\delta_{23}=$
$\displaystyle\frac{1}{2}(\eta_{2}+\eta_{3}-\eta_{1})\,,$
$\displaystyle\delta_{31}=$
$\displaystyle\frac{1}{2}(\eta_{3}+\eta_{1}-\eta_{2})\,.$ (A.24)

### Conservation condition

For the derivation of the conservation conditions, we note that:

$\displaystyle(\nabla_{x_{1}}\partial_{a})\langle{\cal
J}^{(s_{1})}(a;x_{1})\,{\cal J}^{(s_{2})}(b;x_{2})\,{\cal
J}^{(s_{3})}(c;x_{3})\rangle$
$\displaystyle=\nabla_{x_{1}^{\mu}}\left[\frac{1}{x_{12}^{\Delta_{12}}x_{23}^{\Delta_{23}}x_{31}^{\Delta_{31}}}\partial_{a^{\mu}}{\cal
I}^{(s_{1})}(a,a^{\prime};x_{13})\right]{\cal
I}^{(s_{2})}(b,b^{\prime};x_{23})*^{(s_{1})}_{a^{\prime}}*^{(s_{2})}_{b^{\prime}}t^{(s_{3})}(a^{\prime},b^{\prime};c;{\hat{X}}_{12})$
$\displaystyle+\frac{1}{x_{12}^{\Delta_{12}}x_{23}^{\Delta_{23}}x_{31}^{\Delta_{31}}}\partial_{a^{\mu}}{\cal
I}^{(s_{1})}(a,a^{\prime};x_{13}){\cal
I}^{(s_{2})}(b,b^{\prime};x_{23})*^{(s_{1})}_{a^{\prime}}*^{(s_{2})}_{b^{\prime}}\nabla_{x_{1}^{\mu}}t^{(s_{3})}(a^{\prime},b^{\prime};c;{\hat{X}}_{12})$
(A.25)

Using the following relations:

$\displaystyle\nabla_{x_{1}^{\mu}}\frac{1}{x_{12}^{\Delta_{12}}x_{31}^{\Delta_{31}}}=-\frac{1}{x_{12}^{\Delta_{12}}x_{31}^{\Delta_{31}}}\left[\frac{\Delta_{12}x_{12\mu}}{x^{2}_{12}}+\frac{\Delta_{31}x_{13\mu}}{x^{2}_{13}}\right]$
$\displaystyle=-\frac{1}{x_{12}^{\Delta_{12}}x_{31}^{\Delta_{31}}}\left[\Delta_{12}X_{32\mu}+(\Delta_{12}+\Delta_{31})\frac{x_{13\mu}}{x^{2}_{13}}\right]$
$\displaystyle=-\frac{1}{x_{12}^{\Delta_{12}}x_{31}^{\Delta_{31}+2}}\left[\Delta_{12}I_{\mu\alpha}(x_{13})\frac{X^{\alpha}_{12}}{X^{2}_{12}}+(\Delta_{12}+\Delta_{31})\frac{x_{13\mu}}{x^{2}_{13}}\right]$
(A.26) $\displaystyle(\nabla_{x_{1}}\partial_{a}){\cal
I}^{(s_{1})}(a,a^{\prime};x_{13})=2(d+s_{1}-2)\frac{(x_{13}\partial_{a})}{x_{13}^{2}}{\cal
I}^{(s_{1})}(a,a^{\prime};x_{13})$ (A.27)
$\displaystyle\nabla_{x_{1}^{\mu}}t^{(s_{3})}(a,b;c;X_{12})=\nabla_{X^{\alpha}_{12}}t^{(s_{3})}(a,b;c;X_{12})\frac{\partial
X^{\alpha}_{12}}{\partial{x^{\mu}_{1}}}$
$\displaystyle=\nabla_{X^{\alpha}_{12}}t^{(s_{3})}(a,b;c;X_{12})\frac{I_{\mu}^{\alpha}(x_{13})}{x^{2}_{13}}$
(A.28)

we see that the conservation condition is satisfied when:

$\displaystyle\Delta_{12}+\Delta_{31}$ $\displaystyle=$ $\displaystyle
2\Delta_{(s_{1})}=2(d+s_{1}-2)$ (A.29)

and:

$\displaystyle(\nabla_{X}\partial_{a})t^{(s_{3})}(a,b;c;X)$
$\displaystyle=\Delta_{12}\frac{(X\partial_{a})}{X^{2}}t^{(s_{3})}(a,b;c;X)$
(A.30)

This is the equation for structural tensor object $t^{(s_{3})}(a,b;c;X)$ which
we use in the second section. The equation (A.30) (or (4.6)) is equivalent to
the conservation condition for the first current in the three-point function.

Now we can separate the traceless projector from “kernel” part and write
(A.30) in the following form:

$\displaystyle\Big{(}\nabla^{\mu}-\Delta_{12}\frac{{\hat{X}}^{\mu}}{\sqrt{X^{2}}}\Big{)}\partial^{a}_{\mu}{\cal
E}^{(s_{1})}(a,a^{\prime})*^{s_{1}}_{a^{\prime}}\tilde{t}^{(s_{3})}(a^{\prime},b,c;{\hat{X}})*^{s_{2}}_{b}*^{s_{3}}_{c}{\cal
E}^{(s_{2})}(b,{\tilde{b}}){\cal E}^{(s_{3})}(c,{\tilde{c}})$
$\displaystyle=\scalebox{0.9}{${1\over
s_{1}!}\sum^{s_{1}/2-1}_{k=0}(s_{1}-2k)!\lambda^{s_{1}}_{k}[a^{2}]^{k}\Big{[}\Big{(}(\nabla\partial^{a})-\Delta_{12}\frac{({\hat{X}}\partial^{a})}{\sqrt{X^{2}}}\Big{)}\square^{k}_{a}$}$
$\displaystyle-\scalebox{0.9}{$\frac{1}{d+2s_{1}-2k-4}\Big{(}(a\nabla)-\Delta_{12}\frac{(a{\hat{X}})}{\sqrt{X^{2}}}\Big{)}\square^{k+1}_{a}\Big{]}\tilde{t}^{(s_{3})}(a,b,c;{\hat{X}})*^{s_{2}}_{b}*^{s_{3}}_{c}{\cal
E}^{(s_{2})}(b,{\tilde{b}}){\cal E}^{(s_{3})}(c,{\tilde{c}})$}$ (A.31)

where $\tilde{t}^{(s_{3})}(a,b,c;{\hat{X}})$ now is:

$\displaystyle\tilde{t}^{(s_{3})}(a,b,c;{\hat{X}})=I^{s_{3}}(c,c^{\prime};{\hat{X}})*_{c^{\prime}}{\tilde{H}}^{(s_{123})}(a,b,c^{\prime};{\hat{X}})$
$\displaystyle=\sum_{\begin{subarray}{c}s_{i}\in[0,\dots s_{i}]\\\
\\{s_{i}\\}\in{\cal
A}\end{subarray}}(-1)^{\ell_{3}}{\tilde{C}}_{\ell_{1}\ell_{2}\ell_{3}}({\hat{X}}a)^{\ell_{1}}({\hat{X}}b)^{\ell_{2}}({\hat{X}}c)^{\ell_{3}}(ab)^{\alpha}I^{\beta}(b,c;{\hat{X}})I^{\gamma}(c,a;{\hat{X}}).$
(A.32)

Then we compute the $p$-th trace as:

$\displaystyle\Phi^{k}(a;b,c;{\hat{X}};\alpha,\gamma)=\Box^{k}_{a}(ab)^{\alpha}(ac)^{\gamma}({\hat{X}}a)^{\ell_{1}}$
$\displaystyle=\sum_{{p,q,n\atop p+q+n\leq
k}}\rho\left(\genfrac{}{}{0.0pt}{}{k;p,q,n}{\alpha,\gamma,\ell_{1}}\right)(ab)^{\alpha-k+n+q}(ac)^{\gamma-k+n+p}(bc)^{k-n-
p-q}({\hat{X}}a)^{\ell_{1}-2n-p-q}({\hat{X}}b)^{p}({\hat{X}}c)^{q},$ (A.33)

where we neglected all terms of type $O(b^{2},c^{2})$. From the equation

$\Phi^{k+1}(a;b,c;{\hat{X}};\alpha,\gamma)=\Box_{a}\Phi^{k}(a;b,c;{\hat{X}};\alpha,\gamma)$
(A.34)

we get the following recursion relation

$\displaystyle\rho\left(\genfrac{}{}{0.0pt}{}{k+1;p,q,n}{\alpha,\gamma,\ell_{1}}\right)$
$\displaystyle=2\rho\left(\genfrac{}{}{0.0pt}{}{k;p,q,n}{\alpha,\gamma,\ell_{1}}\right)(\alpha-k+n+q)(\gamma-k+n+p)$
$\displaystyle+2\rho\left(\genfrac{}{}{0.0pt}{}{k;p-1,q,n}{\alpha,\gamma,\ell_{1}}\right)(\alpha-k+n+q)(\ell_{1}-2n-p-q+1)$
$\displaystyle+2\rho\left(\genfrac{}{}{0.0pt}{}{k;p,q-1,n}{\alpha,\gamma,\ell_{1}}\right)(\gamma-k+n+p)(\ell_{1}-2n-p-q+1)$
$\displaystyle+\rho\left(\genfrac{}{}{0.0pt}{}{k;p,q,n-1}{\alpha,\gamma,\ell_{1}}\right)(\ell_{1}-2n-p-q+2)(\ell_{1}-2n-p-q+1)$
(A.35)

This equation after substitution

$\rho\left(\genfrac{}{}{0.0pt}{}{k;p,q,n}{\alpha,\gamma,\ell_{1}}\right)=2^{k-n}[\alpha]_{k-n-q}[\gamma]_{k-n-p}[\ell_{1}]_{2n+p+q}\hat{\rho}(k;p,q,n)$
(A.36)

goes to Pascal’s identity for multinomial:

$\displaystyle\hat{\rho}(k+1;p,q,n)$
$\displaystyle=\hat{\rho}(k;p,q,n)+\hat{\rho}(k;p,q,n-1)$
$\displaystyle+\hat{\rho}(k;p-1,q,n)+\hat{\rho}(k;p,q-1,n)$ (A.37)

with obvious solution

$\hat{\rho}(k;p,q,n)=\frac{[k]_{n+p+q}}{p!q!n!}=\binom{k}{p,q,n}$ (A.38)

Then we can easily derive the $k$th trace of our ansatz:

$\displaystyle\square^{k}_{a}\tilde{t}^{(s)}(a,b,c;{\hat{X}})=\sum_{\begin{subarray}{c}\ell_{1}\in[2k,\dots
s_{1}];\ell_{2},\ell_{3}\in[0,\dots s_{2},s_{3}]\\\ \\{\ell_{i}\\}\in{\cal
A}\end{subarray}}T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}\left[\genfrac{}{}{0.0pt}{}{\ell_{1}-2k,\ell_{2},\ell_{3}}{\alpha;\beta,\gamma}\right]$
(A.39)

where we introduced notation:

$\displaystyle\left[\genfrac{}{}{0.0pt}{}{\ell_{1},\ell_{2},\ell_{3}}{\alpha;\beta,\gamma}\right]=({\hat{X}}a)^{\ell_{1}}({\hat{X}}b)^{\ell_{2}}({\hat{X}}c)^{\ell_{3}}(ab)^{\alpha}I^{\beta}(b,c;{\hat{X}})I^{\gamma}(c,a;{\hat{X}})$
(A.40)

and $T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}$ is $k$th trace map of
${\tilde{C}}_{\ell_{1}\ell_{2}\ell_{3}}$

$\displaystyle
T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}=(-1)^{\ell_{3}}\sum_{{p,q,n\atop
p+q+n\leq
k}}{\tilde{C}}_{\ell_{1}-2k+2n+p+q,\ell_{2}-p,\ell_{3}-q}\,\,\rho\left(\genfrac{}{}{0.0pt}{}{k;p,q,n}{\alpha,\gamma,\ell_{1}}\right)$
(A.41)

In this way substituting (A.39) in (A.31) one can straightforwardly derive the
conservation condition on $T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}$ given in
(4.9).

##  Appendix B Examples

### Coincident spins $s_{1}=s_{2}=s_{3}=s$

We present examples for the most symmetric case of equal spins
$s_{1}=s_{2}=s_{3}=s$. It is enough to write a “kernel” term with the
following symmetry properties:

$\displaystyle\tilde{t}^{(s)}(a,b;c;{\hat{X}})=\tilde{t}^{(s)}(b,a;c;-{\hat{X}})$
(B.1) $\displaystyle
I^{s}(a,a^{\prime};{\hat{X}})*_{a^{\prime}}\tilde{t}^{(s)}(a^{\prime},b;c;{\hat{X}})=\tilde{t}^{(s)}(c,a;b;-{\hat{X}})$
(B.2)

From these conditions, we derive the most general polynomial ansatz for
$t^{(s)}(a,b;c;{\hat{X}})$:

$\displaystyle\tilde{t}^{(s)}(a,b;c;{\hat{X}})=$ $\displaystyle
I^{s}(c,c^{\prime};{\hat{X}})*_{c^{\prime}}{\tilde{H}}^{(s)}(a,b,c^{\prime};{\hat{X}})$
(B.3) $\displaystyle\tilde{t}_{1}^{(s)}(a,b;c;{\hat{X}})=$
$\displaystyle\big{[}{\tilde{H}}^{(s)}(a,b,c;{\hat{X}})+I^{s}(a,a^{\prime};{\hat{X}})*_{a^{\prime}}{\tilde{H}}^{(s)}(a^{\prime},b,c;-{\hat{X}})$
$\displaystyle+$ $\displaystyle
I^{s}(b,b^{\prime};{\hat{X}})*_{b^{\prime}}{\tilde{H}}^{(s)}(a,b^{\prime},c;{\hat{X}})\big{]}$
(B.4)

where the main object, ${\tilde{H}}$, is given by

$\displaystyle{\tilde{H}}^{(s)}(a,b,c;{\hat{X}})$
$\displaystyle=\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\in[0,\dots
s]\\\ \\{\ell_{i}\\}\in\bar{{\cal
A}}\end{subarray}}{\tilde{C}}_{\ell_{1}\ell_{2}\ell_{3}}({\hat{X}}a)^{\ell_{1}}({\hat{X}}b)^{\ell_{2}}({\hat{X}}c)^{\ell_{3}}(ab)^{\alpha}(bc)^{\beta}(ca)^{\gamma}\,.$
(B.5)

Here $\bar{{\cal A}}$ is the range of indices defined by the following natural
restrictions:

$\displaystyle\alpha+\gamma+\ell_{1}=s$ (B.6)
$\displaystyle\alpha+\beta+\ell_{2}=s$ (B.7)
$\displaystyle\gamma+\beta+\ell_{3}=s$ (B.8)

These also can be resolved fixing $\alpha,\beta,\gamma$ for any choice of
$\ell_{1},\ell_{2},\ell_{3}$:

$\displaystyle 2\alpha$ $\displaystyle=s+\ell_{3}-\ell_{1}-\ell_{k}$ (B.9)
$\displaystyle 2\beta$ $\displaystyle=s+\ell_{1}-\ell_{2}-\ell_{3}$ (B.10)
$\displaystyle 2\gamma$ $\displaystyle=s+\ell_{2}-\ell_{1}-\ell_{3}$ (B.11)

The positiveness of $\alpha,\beta,\gamma$ for coincident spins leads to the
triangle inequalities:

$\displaystyle s+\ell_{i}\geq\ell_{j}+\ell_{k}\,\quad i\neq j\neq k$ (B.12)

Another special point in consideration of equal spins is that conditions (B.1)
and (B.2) force the coefficients $T^{(0)}_{\ell_{1}\ell_{2}\ell_{3}}$ to be
completely symmetric with respect to $\ell_{1},\ell_{2},\ell_{3}$. Then:

$\displaystyle
I^{s}(a,a^{\prime};{\hat{X}})*_{a^{\prime}}I^{s}(b,b^{\prime};{\hat{X}})*_{b^{\prime}}$
$\displaystyle{\tilde{H}}^{(s)}(a^{\prime},b^{\prime},c;{\hat{X}})=(-1)^{\ell_{1}+\ell_{2}+\ell_{3}}I^{s}(c,c^{\prime};{\hat{X}})*_{c^{\prime}}{\tilde{H}}^{(s)}(a,b,c^{\prime};{\hat{X}})$
$\displaystyle=I^{s}(c,c^{\prime};{\hat{X}})*_{c^{\prime}}{\tilde{H}}^{(s)}(a,b,c^{\prime};-{\hat{X}})$
(B.13)

and we get even (odd) sum of $\ell$’s for even (odd) spin $s$:

$\sum_{i=1,2,3}\ell_{i}=3s-2(\alpha+\beta+\gamma)\,.$ (B.14)

The relation (B.13) helps to explain the minus sign in condition (B.2) and we
can make the following simple derivation showing that (B.4) is equivalent to
(B.3):

$\displaystyle{\tilde{H}}^{(s)}(a,b,c;{\hat{X}})+I^{s}(a,a^{\prime};{\hat{X}})*_{a^{\prime}}{\tilde{H}}^{(s)}(a^{\prime},b,c;-{\hat{X}})+I^{s}(b,b^{\prime};{\hat{X}})*_{b^{\prime}}{\tilde{H}}^{(s)}(a,b^{\prime},c;{\hat{X}})$
$\displaystyle=I^{s}(c,c^{\prime};{\hat{X}})*_{c^{\prime}}\Big{[}I^{s}(c^{\prime},c^{\prime\prime};{\hat{X}})*_{c^{\prime\prime}}H^{(s)}(a,b,c^{\prime\prime};{\hat{X}})+I^{s}(b,b^{\prime};{\hat{X}})*_{b^{\prime}}{\tilde{H}}^{(s)}(a,b^{\prime},c^{\prime};{\hat{X}})$
$\displaystyle+I^{s}(a,a^{\prime};{\hat{X}})*_{a^{\prime}}{\tilde{H}}^{(s)}(b,a^{\prime},c^{\prime};{\hat{X}})\Big{]}=I^{s}(c,c^{\prime};{\hat{X}})*_{c^{\prime}}\bar{{\tilde{H}}}^{(s)}(a,b,c^{\prime};{\hat{X}})$
(B.15)

where

$\displaystyle\bar{{\tilde{H}}}^{(s)}(a,b,c^{\prime};{\hat{X}})$
$\displaystyle=\sum_{\begin{subarray}{c}\bar{\ell}_{1},\bar{\ell}_{2},\bar{\ell}_{3}\in[0,\dots
s]\\\
\bar{\ell}_{1}+\bar{\ell}_{2}+\bar{\ell}_{3}=\textnormal{even}\end{subarray}}\bar{T}^{(0)}_{\bar{\ell}_{1}\bar{\ell}_{2}\bar{\ell}_{3}}({\hat{X}}a)^{\bar{\ell}_{1}}({\hat{X}}b)^{\bar{\ell}_{2}}({\hat{X}}c)^{\tilde{\ell}_{3}}(ab)^{\bar{\alpha}}(bc)^{\bar{\beta}}(ca)^{\bar{\gamma}}\,,$
(B.16)
$\displaystyle\bar{T}^{(0)}_{\bar{\ell}_{1}\bar{\ell}_{2}\bar{\ell}_{3}}$
$\displaystyle=\bar{T}^{(0)}({\bar{\ell}_{1}|\bar{\ell}_{2}\bar{\ell}_{3}})+\bar{T}^{(0)}({\bar{\ell}_{2}|\bar{\ell}_{3}\bar{\ell}_{1}})+\bar{T}^{(0)}({\bar{\ell}_{3}|\bar{\ell}_{1}\bar{\ell}_{2}})\,,$
(B.17)

where (symmetric in all $\bar{\ell}_{i};i=1,2,3$) coefficients
$\bar{T}^{(0)}_{\bar{\ell}_{1}\bar{\ell}_{2}\bar{\ell}_{3}}$ are constructed
as a cyclic permutation (B.17) of the object that is symmetric in two indices
only:

$\displaystyle\bar{T}^{(0)}({\bar{\ell}_{1}|\bar{\ell}_{2}\bar{\ell}_{3}})$
$\displaystyle=(-1)^{\bar{\ell}_{1}}\sum_{n_{2},n_{3}}^{\bar{\ell}_{2},\bar{\ell}_{3}}2^{n_{2}+n_{3}}T^{(0)}_{\bar{\ell}_{1}-n_{2}-n_{3},\bar{\ell}_{2}-n_{2}\bar{\ell}_{3}-n_{3}}\binom{\bar{\alpha}+n_{2}}{\bar{\alpha}}\binom{\bar{\gamma}+n_{3}}{\bar{\gamma}}$
(B.18)

The most general ansatz in this case is (B.3), with traceless projectors
written as:

$t^{(s)}({\tilde{a}},{\tilde{b}};{\tilde{c}};{\hat{X}})={\cal
E}^{(s)}({\tilde{a}},a)*_{a}\tilde{t}^{(s)}(a,b;c;{\hat{X}})*_{b}*_{c}{\cal
E}^{(s)}(b,{\tilde{b}}){\cal E}^{(s)}(c,{\tilde{c}}).$ (B.19)

### Conservation condition for coincident spins

When all spins coincide, we need only one equation for fully symmetric
coefficients:

$\displaystyle(\alpha+1)(2\ell_{3}-2k-\Delta_{(s)}-s)T^{(k)}_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}+(\gamma+1)(2\ell_{2}-2k-\Delta_{(s)}-s)T^{(k)}_{\ell_{1}-1,\ell_{2},\ell_{3}-1}$
$\displaystyle+$
$\displaystyle(\alpha+1)(\ell_{3}+1)T^{(k)}_{\ell_{1}-1,\ell_{2},\ell_{3}+1}+(\gamma+1)(\ell_{2}+1)T^{(k)}_{\ell_{1}-1,\ell_{2}+1,\ell_{3}}$
$\displaystyle+$
$\displaystyle\frac{1}{d+2s-2k-4}\left[2(\ell_{2}-\ell_{3})T^{(k+1)}_{\ell_{1},\ell_{2},\ell_{3}}+2(\beta+1)\big{(}T^{(k+1)}_{\ell_{1}+1,\ell_{2},\ell_{3}-1}+T^{(k+1)}_{\ell_{1}+1,\ell_{2}-1,\ell_{3}}\big{)}\right.$
$\displaystyle-$
$\displaystyle\left.(\ell_{2}+1)T^{(k+1)}_{\ell_{1}+1,\ell_{2}+1,\ell_{3}}-(\ell_{3}+1)T^{(k+1)}_{\ell_{1}+1,\ell_{2},\ell_{3}+1}\right]=0$
(B.20)

where

$\displaystyle
T^{(k+1)}_{\ell_{1},\ell_{2},\ell_{3}}=(\ell_{1}-2k)(\ell_{1}-2k-1)T^{(k)}_{\ell_{1},\ell_{2},\ell_{3}}+2(\alpha+1)(\gamma+1)T^{(k)}_{\ell_{1}-2,\ell_{2},\ell_{3}}$
$\displaystyle+2(\alpha+1)(\ell_{1}-2k-1)T^{(k)}_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}-2(\gamma+1)(\ell_{1}-2k-1)T^{(k)}_{\ell_{1}-1,\ell_{2},\ell_{3}-1}\,,$
(B.21)

and we need to solve only the first conservation condition for $k=0$ (the rest
follow from tracelessness). Using the helpful ansatz (4.11), (4.12) we arrive
to the following recursion for (symmetric in $\ell_{1},\ell_{2},\ell_{3}$)
expressions $C_{\ell_{1},\ell_{2},\ell_{3}}$ and
$T_{\ell_{1},\ell_{2},\ell_{3}}$

$\displaystyle
D_{\ell_{1},\ell_{2},\ell_{3}}=(2\ell_{3}-\Delta_{(s)}-s)C_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}-(2\ell_{2}-\Delta_{(s)}-s)C_{\ell_{1}-1,\ell_{2},\ell_{3}-1}$
$\displaystyle+$
$\displaystyle\beta(\ell_{2}+1)C_{\ell_{1}-1,\ell_{2}+1,\ell_{3}}-\beta(\ell_{3}+1)C_{\ell_{1}-1,\ell_{2},\ell_{3}+1}$
$\displaystyle+$
$\displaystyle\frac{1}{d+2s-4}\left[2(\ell_{2}-\ell_{3})T_{\ell_{1},\ell_{2},\ell_{3}}+2\big{(}\gamma
T_{\ell_{1}+1,\ell_{2}-1,\ell_{3}}-\alpha
T_{\ell_{1}+1,\ell_{2},\ell_{3}-1}\big{)}\right.$ $\displaystyle+$
$\displaystyle\left.\gamma(\ell_{3}+1)T_{\ell_{1}+1,\ell_{2},\ell_{3}+1}-\alpha(\ell_{2}+1)T_{\ell_{1}+1,\ell_{2}+1,\ell_{3}}\right]=0$
(B.22)

where

$\displaystyle
T_{\ell_{1},\ell_{2},\ell_{3}}=\ell_{1}(\ell_{1}-1)C_{\ell_{1},\ell_{2},\ell_{3}}+2\beta
C_{\ell_{1}-2,\ell_{2},\ell_{3}}$
$\displaystyle+2(\ell_{1}-1)C_{\ell_{1}-1,\ell_{2}-1,\ell_{3}}+2(\ell_{1}-1)C_{\ell_{1}-1,\ell_{2},\ell_{3}-1}$
(B.23)

Computer-assisted solutions have $s+1$ independent parameters as they should.

### Spin 2 case: energy-momentum tensor and connection with (B.3), (B.5)

First we review construction in the case of spin two following [77]. For three
point function of energy-momentum tensors we have:

$\displaystyle\langle
T_{\mu\nu}(x_{1})\,T_{\sigma\rho}(x_{2})\,T_{\alpha\beta}(x_{3})\rangle={}$
$\displaystyle{1\over
x_{12}^{\,d\vphantom{\eta}}\,x_{13}^{\,d\vphantom{\eta}}\,x_{23}^{\,d\vphantom{\eta}}}\,{\cal
I}_{\mu\nu,\mu^{\prime}\nu^{\prime}}(x_{13}){\cal
I}_{\sigma\rho,\sigma^{\prime}\rho^{\prime}}(x_{23})\,t_{\mu^{\prime}\nu^{\prime}\sigma^{\prime}\rho^{\prime}\alpha\beta}(X_{12})\,,$
(B.24)

with $t_{\mu\nu\sigma\rho\alpha\beta}(X)$ homogeneous of degree zero in $X$,
symmetric and traceless on each pair of indices $\mu\nu,\ \sigma\rho$ and
$\alpha\beta$ and from satisfying

$\displaystyle
t_{\mu\nu\sigma\rho\alpha\beta}(X)={}t_{\sigma\rho\mu\nu\alpha\beta}(X)\,.$
(B.25) $\displaystyle{\cal
I}_{\mu\nu,\mu^{\prime}\nu^{\prime}}(X)t_{\mu^{\prime}\nu^{\prime}\sigma\rho\alpha\beta}(X)={}t_{\alpha\beta\mu\nu\sigma\rho}(X)\,,$
(B.26)

The conservation equations require just

$\Bigl{(}\partial_{\mu}-d\,{X_{\mu}\over
X^{2}}\Bigl{)}t_{\mu\nu\sigma\rho\alpha\beta}(X)=0$ (B.27)

Defining

$\displaystyle h^{1}_{\mu\nu}({\hat{X}})$ $\displaystyle=$
$\displaystyle{\hat{X}}_{\mu}{\hat{X}}_{\nu}-{1\over
d}\,\delta_{\mu\nu}\,,\quad{\hat{X}}_{\mu}={X_{\mu}\over\sqrt{X^{2}}}$ (B.28)
$\displaystyle h^{2}_{\mu\nu\sigma\rho}({\hat{X}})$ $\displaystyle=$
$\displaystyle{}{\hat{X}}_{\mu}{\hat{X}}_{\sigma}\delta_{\nu\rho}+(\mu\leftrightarrow\nu,\sigma\leftrightarrow\rho)$
(B.29) $\displaystyle-$ $\displaystyle{4\over
d}{\hat{X}}_{\mu}{\hat{X}}_{\nu}\delta_{\sigma\rho}-{4\over
d}{\hat{X}}_{\sigma}{\hat{X}}_{\rho}\delta_{\mu\nu}+{4\over
d^{2}}\delta_{\mu\nu}\delta_{\sigma\rho}\qquad$ $\displaystyle
h^{3}_{\mu\nu\sigma\rho}$ $\displaystyle=$
$\displaystyle{}\delta_{\mu\sigma}\delta_{\nu\rho}+\delta_{\mu\rho}\delta_{\nu\sigma}-{2\over
d}\,\delta_{\mu\nu}\delta_{\sigma\rho}=2{\cal E}_{\mu\nu,\sigma\rho}$ (B.30)
$\displaystyle h^{4}_{\mu\nu\sigma\rho\alpha\beta}({\hat{X}})$
$\displaystyle=$
$\displaystyle{}h^{3}_{\mu\nu\sigma\alpha}{\hat{X}}_{\rho}{\hat{X}}_{\beta}+(\sigma\leftrightarrow\rho,\alpha\leftrightarrow\beta)$
(B.31) $\displaystyle-$ $\displaystyle{2\over
d}\,\delta_{\sigma\rho}^{\vphantom{h}}h^{2}_{\mu\nu\alpha\beta}({\hat{X}})-{2\over
d}\,\delta_{\alpha\beta}^{\vphantom{h}}h^{2}_{\mu\nu\sigma\rho}({\hat{X}})-{8\over
d^{2}}\,\delta_{\sigma\rho}^{\vphantom{h}}\delta_{\alpha\beta}^{\vphantom{h}}h^{1}_{\mu\nu}({\hat{X}})\,,$
$\displaystyle h^{5}_{\mu\nu\sigma\rho\alpha\beta}$ $\displaystyle=$
$\displaystyle{}\delta_{\mu\sigma}^{\vphantom{h}}\delta_{\nu\alpha}^{\vphantom{h}}\delta_{\rho\beta}^{\vphantom{h}}+(\mu\leftrightarrow\nu,\sigma\leftrightarrow\rho,\alpha\leftrightarrow\beta)$
(B.32) $\displaystyle-$ $\displaystyle{4\over
d}\,\delta_{\mu\nu}^{\vphantom{h}}h^{3}_{\sigma\rho\alpha\beta}-{4\over
d}\,\delta_{\sigma\rho}^{\vphantom{h}}h^{3}_{\mu\nu\alpha\beta}-{4\over
d}\,\delta_{\alpha\beta}^{\vphantom{h}}h^{3}_{\mu\nu\sigma\rho}-{8\over
d^{2}}\,\delta_{\mu\nu}^{\vphantom{h}}\delta_{\sigma\rho}^{\vphantom{h}}\delta_{\alpha\beta}^{\vphantom{h}}\,,$

a general expansion for $t_{\mu\nu\sigma\rho\alpha\beta}(X)$ has the form

$\displaystyle t_{\mu\nu\sigma\rho\alpha\beta}(X)={}$ $\displaystyle
a\,h^{5}_{\mu\nu\sigma\rho\alpha\beta}+b\,h^{4}_{\alpha\beta\mu\nu\sigma\rho}({\hat{X}})+b^{\prime}\bigl{(}h^{4}_{\mu\nu\sigma\rho\alpha\beta}({\hat{X}})+h^{4}_{\sigma\rho\mu\nu\alpha\beta}({\hat{X}})\bigl{)}$
(B.33)
$\displaystyle+c\,h^{3}_{\mu\nu\sigma\rho}h^{1}_{\alpha\beta}({\hat{X}})+c^{\prime}\bigl{(}h^{3}_{\sigma\rho\alpha\beta}h^{1}_{\mu\nu}({\hat{X}})+h^{3}_{\mu\nu\alpha\beta}h^{1}_{\sigma\rho}({\hat{X}})\bigl{)}$
(B.34)
$\displaystyle+e\,h^{2}_{\mu\nu\sigma\rho}({\hat{X}})h^{1}_{\alpha\beta}({\hat{X}})+e^{\prime}\bigl{(}h^{2}_{\sigma\rho\alpha\beta}({\hat{X}})h^{1}_{\mu\nu}({\hat{X}})+h^{2}_{\mu\nu\alpha\beta}({\hat{X}})h^{1}_{\sigma\rho}({\hat{X}})\bigl{)}$
(B.35)
$\displaystyle+f\,h^{1}_{\mu\nu}({\hat{X}})h^{1}_{\sigma\rho}({\hat{X}})h^{1}_{\alpha\beta}({\hat{X}})\,.$
(B.36)

From the symmetry condition (B.25), (B.26) we have

$\displaystyle b+b^{\prime}=-2a\,,\quad c^{\prime}=c\,,\quad
e+e^{\prime}=-4b^{\prime}-2c\,,$ (B.37)

so that $a,b,c,e,f$ may be regarded as independent. Then using conservation
condition (B.23) we have two additional constraints:

$\displaystyle d^{2}a+2(b+b^{\prime})-(d-2)b^{\prime}-dc+e^{\prime}={}$
$\displaystyle 0\,,$ (B.38) $\displaystyle
d(d+2)(2b^{\prime}+c)+4(e+e^{\prime})+f={}$ $\displaystyle 0\,.$ (B.39)

Therefore, we have three undetermined independent coefficients, say, $a,b,c$,
which are the free parameters of the three-point function (in arbitrary
dimension $d$):

$\displaystyle f=(d+4)(d-2)(4a+2b-c),$ (B.40) $\displaystyle
e^{\prime}=-(d+4)(d-2)a-(d-2)b+dc,$ (B.41) $\displaystyle e=(d+2)(da+b-c).$
(B.42)

Now we can compare these with our general formulation in the case of spin two.
We should look at ansatz (B.3) and (B.5) for the case of $s=2$. First of all
putting $s=2$ in corresponding number of solution of triangle inequality
(3.18) we obtain $N_{222}=5$ which is correct number of parameters after
applying symmetry constraints (B.3) then investigating these independent five
terms in ansatz (B.5), identifying with (B.36) and using notation

${\tilde{C}}_{\ell_{1},\ell_{2},\ell_{3}}=(-1)^{\ell_{3}}C_{\ell_{1},\ell_{2},\ell_{3}}$
(B.43)

where $C_{\ell_{1},\ell_{2},\ell_{3}}$ is symmetric in
$\ell_{1},\ell_{2},\ell_{3}$. we obtain the following connections between
coefficients:

$\displaystyle a={C_{000}\over 8};\quad b={C_{110}\over 8};\quad
b^{\prime}=-{C_{000}\over 4}-{C_{110}\over 8};$ (B.44) $\displaystyle
c=c^{\prime}={C_{200}\over 2};\quad e=C_{000}+C_{110}+{C_{112}\over 4};$
(B.45) $\displaystyle e^{\prime}=-{C_{110}\over 2}-{C_{112}\over
4}-C_{200};\quad f=4C_{110}+4C_{112}+8C_{200}+C_{222}.$ (B.46)

So we see that these 8 coefficients
$a,b,b^{\prime},c,c^{\prime},e,e^{\prime},f$ from [77] expressed through the
five coefficients from our ansatz $C_{000},C_{110},C_{200},C_{112},C_{222}$.
Because triangle inequality and symmetricity of
$C_{\ell_{1},\ell_{2},\ell_{3}}$ lead to the solution (B.25), (B.26) in
general case. Then we can investigate conservation condition (B.27). taking
into account that our normalization here slightly differ and we should insert
in (B.5)

${\tilde{C}}_{\ell_{1},\ell_{2},\ell_{3}}=\alpha!\beta!\gamma!C_{\ell_{1},\ell_{2},\ell_{3}}$
(B.47)

we see that for $s=2$ we have only two nonzero independent equations:

$\displaystyle
D_{1,1,0}\sim(8-d^{2}-2d)C_{000}+(6-d)C_{110}+(4d+8)C_{200}+2C_{112}=0$ (B.48)
$\displaystyle
D_{1,2,1}\sim(d^{2}-12)C_{110}-12C_{112}-2d(d-2)C_{200}-4C_{222}=0$ (B.49)

Now we see that it is possible to express $C_{112}$ and $C_{222}$ through the
remaining three arbitrary parameter $C_{000}$, $C_{110}$ and $C_{200}$ and
these free parameters from (B.48), (B.49) are exactly equivalent to $a,b,c$
(see (B.44) and (B.45)). Moreover after some straightforward manipulation we
can see that all relations (B.37)-(B.42) are also satisfied exactly.

### Spin 3 case: solution of the conservation condition (B.22)

Finalizing this Appendix we just present solution of the conservation
condition for spin three case. Here we have eight different parameters in our
ansatz and conservation equations expressed four from them through the four
independent :

$\displaystyle
C_{3,0,0}=\frac{1}{9(d+2)}\Big{[}(d-2)(d+8)C_{1,0,0}+(d-14)C_{2,1,0}-2C_{1,1,1}-2C_{2,2,1}\Big{]}$
(B.50) $\displaystyle
C_{3,1,1}=\frac{1}{6(d+2)}\Big{[}(d+8)(d-2)^{2}C_{1,0,0}+(d(d+2)+8)C_{1,1,1}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-4(d(d+8)-4)C_{2,1,0}+8C_{2,2,1}\Big{]}$
(B.51) $\displaystyle
C_{3,2,2}=\frac{1}{12(d+2)}\Big{[}-(d+6)(d+8)(d-2)^{2}C_{1,0,0}-2(d(d+10)+32)C_{1,1,1}$
$\displaystyle\hskip
71.13188pt+2(d^{3}+24d^{2}+60d-96)C_{2,1,0}+2(d(d-12)-44)C_{2,2,1}\Big{]}$
(B.52) $\displaystyle
C_{3,3,3}=\frac{1}{54(d+2)}\Big{[}-(d+8)(d-2)^{2}(d^{2}-10d-60)C_{1,0,0}$
$\displaystyle+(640-d^{4}+2d^{3}-12d^{2}-200d)C_{1,1,1}+4(d^{4}+3d^{3}-124d^{2}-300d+480)C_{2,1,0}$
$\displaystyle+(3d^{3}-16d^{2}+180d+736)C_{2,2,1}\Big{]}$ (B.53)

## References

  * [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998), 231; [hep-th/9711200].
  * [2] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998), 253; [hep-th/9802150].
  * [3] M. A. Vasiliev, “Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions,” Phys. Lett. B 243 (1990), 378.
  * [4] M. A. Vasiliev, “Nonlinear equations for symmetric massless higher spin fields in (A)dS(d),” Phys. Lett. B 567 (2003), 139; [hep-th/0304049].
  * [5] V. E. Didenko and E. D. Skvortsov, “Elements of Vasiliev theory,” [1401.2975].
  * [6] E. Sezgin and P. Sundell, “Massless higher spins and holography,” Nucl. Phys. B 644 (2002), 303 [erratum: Nucl. Phys. B 660 (2003), 403-403]; [hep-th/0205131].
  * [7] I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical O(N) vector model,” Phys. Lett. B 550 (2002), 213; [hep-th/0210114].
  * [8] S. Giombi, “Higher Spin — CFT Duality,” New Frontiers in Fields and Strings, p. 137 (2017); [1607.02967].
  * [9] S. Giombi, I. R. Klebanov and Z. M. Tan, “The ABC of Higher-Spin AdS/CFT,” Universe 4 (2018), 18; [1608.07611].
  * [10] K. Lang and W. Rühl, “The Critical O(N) sigma model at dimension 2 $<$ d $<$ 4 and order 1/n**2: Operator product expansions and renormalization,” Nucl. Phys. B 377 (1992), 371.
  * [11] K. Lang and W. Rühl, “The Critical O(N) sigma model at dimensions 2 $<$ d $<$ 4: A List of quasiprimary fields,” Nucl. Phys. B 402 (1993), 573.
  * [12] A. K. H. Bengtsson, I. Bengtsson and L. Brink, “Cubic Interaction Terms for Arbitrary Spin,” Nucl. Phys. B 227 (1983), 31.
  * [13] F. A. Berends, G. J. H. Burgers and H. Van Dam, “On Spin Three Selfinteractions,” Z. Phys. C 24 (1984), 247.
  * [14] F. A. Berends, G. J. H. Burgers and H. van Dam, “On the Theoretical Problems in Constructing Interactions Involving Higher Spin Massless Particles,” Nucl. Phys. B 260 (1985), 295.
  * [15] E. S. Fradkin and M. A. Vasiliev, “On the Gravitational Interaction of Massless Higher Spin Fields,” Phys. Lett. B 189 (1987), 89.
  * [16] E. S. Fradkin and M. A. Vasiliev, “Cubic Interaction in Extended Theories of Massless Higher Spin Fields,” Nucl. Phys. B 291 (1987), 141.
  * [17] A. K. H. Bengtsson, I. Bengtsson and N. Linden, “Interacting Higher Spin Gauge Fields on the Light Front,” Class. Quant. Grav. 4 (1987), 1333.
  * [18] E. S. Fradkin and R. R. Metsaev, “A Cubic interaction of totally symmetric massless representations of the Lorentz group in arbitrary dimensions,” Class. Quant. Grav. 8 (1991), L89.
  * [19] R. R. Metsaev, “Poincare invariant dynamics of massless higher spins: Fourth order analysis on mass shell,” Mod. Phys. Lett. A 6 (1991), 359.
  * [20] R. R. Metsaev, “S matrix approach to massless higher spins theory. 2: The Case of internal symmetry,” Mod. Phys. Lett. A 6 (1991), 2411.
  * [21] M. A. Vasiliev, “Cubic interactions of bosonic higher spin gauge fields in AdS5,” Nucl. Phys. B 616 (2001), 106-162 [erratum: Nucl. Phys. B 652 (2003), 407-407]; [hep-th/0106200].
  * [22] K. B. Alkalaev and M. A. Vasiliev, “N=1 supersymmetric theory of higher spin gauge fields in AdS(5) at the cubic level,” Nucl. Phys. B 655 (2003), 57; [hep-th/0206068].
  * [23] R. Manvelyan and W. Rühl, “Conformal coupling of higher spin gauge fields to a scalar field in AdS(4) and generalized Weyl invariance,” Phys. Lett. B 593 (2004), 253; [hep-th/0403241].
  * [24] X. Bekaert, N. Boulanger and S. Cnockaert, “Spin three gauge theory revisited,” JHEP 01 (2006), 052; [hep-th/0508048].
  * [25] R. R. Metsaev, “Cubic interaction vertices of massive and massless higher spin fields,” Nucl. Phys. B 759 (2006), 147; [hep-th/0512342].
  * [26] X. Bekaert, N. Boulanger, S. Cnockaert and S. Leclercq, “On killing tensors and cubic vertices in higher-spin gauge theories,” Fortsch. Phys. 54 (2006), 282; [hep-th/0602092].
  * [27] N. Boulanger and S. Leclercq, “Consistent couplings between spin-2 and spin-3 massless fields,” JHEP 11 (2006), 034; [hep-th/0609221].
  * [28] D. Francia, J. Mourad and A. Sagnotti, “Current Exchanges and Unconstrained Higher Spins,” Nucl. Phys. B 773 (2007), 203; [hep-th/0701163].
  * [29] A. Fotopoulos, N. Irges, A. C. Petkou and M. Tsulaia, “Higher-Spin Gauge Fields Interacting with Scalars: The Lagrangian Cubic Vertex,” JHEP 10 (2007), 021; [0708.1399].
  * [30] R. R. Metsaev, “Cubic interaction vertices for fermionic and bosonic arbitrary spin fields,” Nucl. Phys. B 859 (2012), 13; [0712.3526].
  * [31] A. Fotopoulos and M. Tsulaia, “Gauge Invariant Lagrangians for Free and Interacting Higher Spin Fields. A Review of the BRST formulation,” Int. J. Mod. Phys. A 24 (2009), 1; [0805.1346].
  * [32] Y. M. Zinoviev, “On spin 3 interacting with gravity,” Class. Quant. Grav. 26 (2009), 035022; [0805.2226].
  * [33] N. Boulanger, S. Leclercq and P. Sundell, “On The Uniqueness of Minimal Coupling in Higher-Spin Gauge Theory,” JHEP 08 (2008), 056; [0805.2764].
  * [34] R. Manvelyan and K. Mkrtchyan, “Conformal invariant interaction of a scalar field with the higher spin field in AdS(D),” Mod. Phys. Lett. A 25 (2010), 1333; [0903.0058].
  * [35] R. Manvelyan, K. Mkrtchyan and W. Rühl, “Off-shell construction of some trilinear higher spin gauge field interactions,” Nucl. Phys. B 826 (2010), 1; [0903.0243].
  * [36] X. Bekaert, E. Joung and J. Mourad, “On higher spin interactions with matter,” JHEP 05 (2009), 126; [0903.3338].
  * [37] R. Manvelyan, K. Mkrtchyan and W. Rühl, “Direct Construction of A Cubic Selfinteraction for Higher Spin gauge Fields,” Nucl. Phys. B 844 (2011), 348; [1002.1358].
  * [38] R. Manvelyan, K. Mkrtchyan and W. Rühl, “General trilinear interaction for arbitrary even higher spin gauge fields,” Nucl. Phys. B 836 (2010), 204; [1003.2877].
  * [39] A. Sagnotti and M. Taronna, “String Lessons for Higher-Spin Interactions,” Nucl. Phys. B 842 (2011), 299; [1006.5242].
  * [40] Y. M. Zinoviev, “Spin 3 cubic vertices in a frame-like formalism,” JHEP 08 (2010), 084; [1007.0158].
  * [41] A. Fotopoulos and M. Tsulaia, “On the Tensionless Limit of String theory, Off - Shell Higher Spin Interaction Vertices and BCFW Recursion Relations,” JHEP 11 (2010), 086; [1009.0727].
  * [42] R. Manvelyan, K. Mkrtchyan and W. Rühl, “A Generating function for the cubic interactions of higher spin fields,” Phys. Lett. B 696 (2011), 410; [1009.1054].
  * [43] D. Polyakov, “Higher Spins and Open Strings: Quartic Interactions,” Phys. Rev. D 83 (2011), 046005; [1011.0353].
  * [44] W. Rühl, “Solving Noether’s equations for gauge invariant local Lagrangians of N arbitrary higher even spin fields,” [1108.0225].
  * [45] M. A. Vasiliev, “Cubic Vertices for Symmetric Higher-Spin Gauge Fields in $(A)dS_{d}$,” Nucl. Phys. B 862 (2012), 341; [1108.5921].
  * [46] E. Joung and M. Taronna, “Cubic interactions of massless higher spins in (A)dS: metric-like approach,” Nucl. Phys. B 861 (2012), 145; [1110.5918].
  * [47] P. Dempster and M. Tsulaia, “On the Structure of Quartic Vertices for Massless Higher Spin Fields on Minkowski Background,” Nucl. Phys. B 865 (2012), 353; [1203.5597].
  * [48] E. Joung, L. Lopez and M. Taronna, “On the cubic interactions of massive and partially-massless higher spins in (A)dS,” JHEP 07 (2012), 041; [1203.6578].
  * [49] I. L. Buchbinder, T. V. Snegirev and Y. M. Zinoviev, “Cubic interaction vertex of higher-spin fields with external electromagnetic field,” Nucl. Phys. B 864 (2012), 694; [1204.2341].
  * [50] M. Henneaux, G. Lucena Gómez and R. Rahman, “Higher-Spin Fermionic Gauge Fields and Their Electromagnetic Coupling,” JHEP 08 (2012), 093; [1206.1048].
  * [51] E. Joung, L. Lopez and M. Taronna, “Solving the Noether procedure for cubic interactions of higher spins in (A)dS,” J. Phys. A 46 (2013), 214020; [1207.5520].
  * [52] R. Manvelyan, R. Mkrtchyan and W. Rühl, “Radial Reduction and Cubic Interaction for Higher Spins in (A)dS space,” Nucl. Phys. B 872 (2013), 265; [1210.7227].
  * [53] E. Joung, L. Lopez and M. Taronna, “Generating functions of (partially-)massless higher-spin cubic interactions,” JHEP 01 (2013), 168; [1211.5912].
  * [54] N. Boulanger, D. Ponomarev and E. D. Skvortsov, “Non-abelian cubic vertices for higher-spin fields in anti-de Sitter space,” JHEP 05 (2013), 008; [1211.6979].
  * [55] M. Henneaux, G. Lucena Gómez and R. Rahman, “Gravitational Interactions of Higher-Spin Fermions,” JHEP 01 (2014), 087; [1310.5152].
  * [56] E. Joung and M. Taronna, “Cubic-interaction-induced deformations of higher-spin symmetries,” JHEP 03 (2014), 103; [1311.0242].
  * [57] E. Conde, E. Joung and K. Mkrtchyan, “Spinor-Helicity Three-Point Amplitudes from Local Cubic Interactions,” JHEP 08 (2016), 040; [1605.07402].
  * [58] A. K. H. Bengtsson, “Investigations into Light-front Quartic Interactions for Massless Fields (I): Non-constructibility of Higher Spin Quartic Amplitudes,” JHEP 12 (2016), 134; [1607.06659].
  * [59] D. Francia, G. L. Monaco and K. Mkrtchyan, “Cubic interactions of Maxwell-like higher spins,” JHEP 04 (2017), 068; [1611.00292].
  * [60] M. Taronna, “On the Non-Local Obstruction to Interacting Higher Spins in Flat Space,” JHEP 05 (2017), 026; [1701.05772].
  * [61] R. Roiban and A. A. Tseytlin, “On four-point interactions in massless higher spin theory in flat space,” JHEP 04 (2017), 139; [1701.05773].
  * [62] C. Sleight and M. Taronna, “Higher-Spin Gauge Theories and Bulk Locality,” Phys. Rev. Lett. 121 (2018), 171604; [1704.07859].
  * [63] C. Sleight and M. Taronna, “Feynman rules for higher-spin gauge fields on AdSd+1,” JHEP 01 (2018), 060; [1708.08668].
  * [64] M. Karapetyan, R. Manvelyan and R. Poghossian, “Cubic interaction for higher spins in AdSd+1 space in the explicit covariant form,” Nucl. Phys. B 950 (2020), 114876; [1908.07901].
  * [65] E. Joung and M. Taronna, “A note on higher-order vertices of higher-spin fields in flat and (A)dS space,” JHEP 09 (2020), 171; [1912.12357].
  * [66] S. Fredenhagen, O. Krüger and K. Mkrtchyan, “Restrictions for $n$-Point Vertices in Higher-Spin Theories,” JHEP 06 (2020), 118; [1912.13476].
  * [67] M. V. Khabarov and Y. M. Zinoviev, “Massless higher spin cubic vertices in flat four dimensional space,” JHEP 08 (2020), 112; [2005.09851].
  * [68] M. Karapetyan, R. Manvelyan and G. Poghosyan, “On special quartic interaction of higher spin gauge fields with scalars and gauge symmetry commutator in the linear approximation,” Nucl. Phys. B 971 (2021), 115512; [2104.09139].
  * [69] A. M. Polyakov, “Conformal symmetry of critical fluctuations,” JETP Lett. 12 (1970), 381.
  * [70] E. J. Schreier, “Conformal symmetry and three-point functions,” Phys. Rev. D 3 (1971), 980.
  * [71] A. A. Migdal, “Conformal invariance and bootstrap,” Phys. Lett. B 37 (1971), 386.
  * [72] S. Ferrara, A. F. Grillo and R. Gatto, “Tensor representations of conformal algebra and conformally covariant operator product expansion,” Annals Phys. 76 (1973), 161.
  * [73] W. Rühl, “Field representations of the conformal group with continuous mass spectrum,” Commun. Math. Phys. 30 (1973), 287.
  * [74] W. Rühl, “On conformal invariance of interacting fields,” Commun. Math. Phys. 34 (1973), 149.
  * [75] K. Koller, “The Significance of Conformal Inversion in Quantum Field Theory,” Commun. Math. Phys. 40, 15; DESY-74-8.
  * [76] G. Mack, “Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory,” Commun. Math. Phys. 53 (1977), 155.
  * [77] H. Osborn and A. C. Petkou, “Implications of conformal invariance in field theories for general dimensions,” Annals Phys. 231 (1994), 311; [hep-th/9307010].
  * [78] H. Osborn, “Implications of conformal invariance for quantum field theories in d $>$ 2,” [hep-th/9312176].
  * [79] J. Erdmenger and H. Osborn, “Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions,” Nucl. Phys. B 483 (1997), 431; [hep-th/9605009].
  * [80] J. H. Park, “N=1 superconformal symmetry in four-dimensions,” Int. J. Mod. Phys. A 13 (1998), 1743; [hep-th/9703191].
  * [81] H. Osborn, “N=1 superconformal symmetry in four-dimensional quantum field theory,” Annals Phys. 272 (1999), 243; [hep-th/9808041].
  * [82] J. H. Park, “Superconformal symmetry and correlation functions,” Nucl. Phys. B 559 (1999), 455; [hep-th/9903230].
  * [83] D. Anselmi, “Higher spin current multiplets in operator product expansions,” Class. Quant. Grav. 17 (2000), 1383; [hep-th/9906167].
  * [84] S. M. Kuzenko and S. Theisen, “Correlation functions of conserved currents in N=2 superconformal theory,” Class. Quant. Grav. 17 (2000), 665; [hep-th/9907107].
  * [85] J. H. Park, “Superconformal symmetry in three-dimensions,” J. Math. Phys. 41 (2000), 7129; [hep-th/9910199].
  * [86] S. Giombi and X. Yin, “Higher Spin Gauge Theory and Holography: The Three-Point Functions,” JHEP 09 (2010), 115; [0912.3462].
  * [87] S. Giombi and X. Yin, “Higher Spins in AdS and Twistorial Holography,” JHEP 04 (2011), 086; [1004.3736].
  * [88] S. Giombi, S. Prakash and X. Yin, “A Note on CFT Correlators in Three Dimensions,” JHEP 07 (2013), 105; [1104.4317].
  * [89] M. S. Costa, J. Penedones, D. Poland and S. Rychkov, “Spinning Conformal Correlators,” JHEP 11 (2011), 071; [1107.3554].
  * [90] M. S. Costa, J. Penedones, D. Poland and S. Rychkov, “Spinning Conformal Blocks,” JHEP 11 (2011), 154; [1109.6321].
  * [91] J. Maldacena and A. Zhiboedov, “Constraining Conformal Field Theories with A Higher Spin Symmetry,” J. Phys. A 46 (2013), 214011; [1112.1016].
  * [92] Y. S. Stanev, “Correlation Functions of Conserved Currents in Four Dimensional Conformal Field Theory,” Nucl. Phys. B 865 (2012), 200; [1206.5639].
  * [93] I. Todorov, “Conformal field theories with infinitely many conservation laws,” J. Math. Phys. 54 (2013), 022303; [1207.3661].
  * [94] A. Zhiboedov, “A note on three-point functions of conserved currents,” [1206.6370].
  * [95] V. Alba and K. Diab, “Constraining conformal field theories with a higher spin symmetry in d=4,” [1307.8092].
  * [96] M. S. Costa and T. Hansen, “Conformal correlators of mixed-symmetry tensors,” JHEP 02 (2015), 151; [1411.7351].
  * [97] V. Alba and K. Diab, “Constraining conformal field theories with a higher spin symmetry in $d>3$ dimensions,” JHEP 03 (2016), 044; [1510.02535].
  * [98] E. Skvortsov, “Light-Front Bootstrap for Chern-Simons Matter Theories,” JHEP 06 (2019), 058; [1811.12333].
  * [99] E. I. Buchbinder, J. Hutomo and G. Tartaglino-Mazzucchelli, “Three-Point Functions of Higher-Spin Supercurrents in 4D N=1${{\cal N}}=1$ Superconformal Field Theory,” Fortsch. Phys. 70 (2022), 2200133; [2208.07057].
  * [100] E. I. Buchbinder and B. J. Stone, “Three-point functions of conserved currents in 3D CFT: General formalism for arbitrary spins,” Phys. Rev. D 107 (2023) no.4, 046007; [2210.13135].
  * [101] E. I. Buchbinder and B. J. Stone, “Three-point functions of conserved currents in 4D CFT: general formalism for arbitrary spins,” [2307.11435].
  * [102] S. Fredenhagen, O. Krüger and K. Mkrtchyan, “Constraints for Three-Dimensional Higher-Spin Interactions and Conformal Correlators,” Phys. Rev. D 100 (2019), 066019; [1812.10462].
  * [103] K. Mkrtchyan, “Cubic interactions of massless bosonic fields in three dimensions,” Phys. Rev. Lett. 120 (2018), 221601; [1712.10003].
  * [104] P. Kessel and K. Mkrtchyan, “Cubic interactions of massless bosonic fields in three dimensions II: Parity-odd and Chern-Simons vertices,” Phys. Rev. D 97 (2018), 106021; [1803.02737].
  * [105] S. Fredenhagen, O. Krüger and K. Mkrtchyan, “Vertex-Constraints in 3D Higher Spin Theories,” Phys. Rev. Lett. 123 (2019), 131601; [1905.00093].
  * [106] T. Leonhardt, W. Ruhl and R. Manvelyan, “The Group approach to AdS space propagators: A Fast algorithm,” J. Phys. A 37 (2004), 7051 [hep-th/0310063].
  * [107] R. Manvelyan and W. Ruhl, “The Quantum one loop trace anomaly of the higher spin conformal conserved currents in the bulk of AdS(4),” Nucl. Phys. B 733 (2006), 104; [hep-th/0506185].
  * [108] R. Manvelyan and W. Ruhl, “The Structure of the trace anomaly of higher spin conformal currents in the bulk of AdS(4),” Nucl. Phys. B 751 (2006), 285; [hep-th/0602067].
  * [109] R. Manvelyan and W. Ruhl, “The Off-shell behaviour of propagators and the Goldstone field in higher spin gauge theory on AdS(d+1) space,” Nucl. Phys. B 717 (2005), 3; [hep-th/0502123].
  * [110] R. Manvelyan, K. Mkrtchyan and W. Ruhl, “Ultraviolet behaviour of higher spin gauge field propagators and one loop mass renormalization,” Nucl. Phys. B 803 (2008), 405; [0804.1211].
  * [111] M. S. Costa, V. Gonçalves and J. Penedones, “Spinning AdS Propagators,” JHEP 09 (2014), 064; [1404.5625].
  * [112] M. Beccaria and A. A. Tseytlin, “Higher spins in AdS5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT,” JHEP 11 (2014), 114; [1410.3273].
  * [113] X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, “Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange,” JHEP 03 (2015), 170; [1412.0016].
  * [114] X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, “Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory,” JHEP 11 (2015), 149; [1508.04292].
  * [115] C. Sleight and M. Taronna, “Higher Spin Interactions from Conformal Field Theory: The Complete Cubic Couplings,” Phys. Rev. Lett. 116 (2016), 181602; [1603.00022].
  * [116] D. Ponomarev and A. A. Tseytlin, “On quantum corrections in higher-spin theory in flat space,” JHEP 05 (2016), 184; [1603.06273].
  * [117] C. Sleight, “Interactions in Higher-Spin Gravity: a Holographic Perspective,” J. Phys. A 50 (2017), 383001; [1610.01318].
  * [118] D. Ponomarev, “A Note on (Non)-Locality in Holographic Higher Spin Theories,” Universe 4 (2018), 2; [1710.00403].
  * [119] X. Bekaert and M. Grigoriev, “Manifestly conformal descriptions and higher symmetries of bosonic singletons,” SIGMA 6 (2010), 038; [0907.3195].
  * [120] E. Joung and K. Mkrtchyan, “Notes on higher-spin algebras: minimal representations and structure constants,” JHEP 05 (2014), 103; [1401.7977].
  * [121] G. Barnich, X. Bekaert and M. Grigoriev, “Notes on conformal invariance of gauge fields,” J. Phys. A 48 (2015), 505402; [1506.00595].
  * [122] O. Evnin and K. Mkrtchyan, “Three approaches to chiral form interactions,” Differ. Geom. Appl. 89 (2023), 102016; [2207.01767].
  * [123] E. Joung and K. Mkrtchyan, “Partially-massless higher-spin algebras and their finite-dimensional truncations,” JHEP 01 (2016), 003; [1508.07332].