# Superconformal blocks for stress-tensor and chiral operators for 4D
$\mathcal{N}=2$ superconformal field theories

Subhadeep Rakshit and Subir Mukhopadhyay
srakshit<EMAIL_ADDRESS>
( Department of Physics, Sikkim University, 6th Mile, Gangtok 737102. )

###### Abstract

We have computed superconformal partial wave for the mixed correlators
involving $J$, $\phi$ and $\phi^{\dagger}$, where $J$ is the superconformal
primary of 4D ${\mathcal{N}}=2$ stress-tensor multiplet, $\phi$, and
$\phi^{\dagger}$ are chiral and anti-chiral scalars respectively. We have used
the superembedding formalism and computed conformal partial wave corresponding
to the exchange of long multiplet using supershadow approach. This study
provides part of the ingredients necessary for the study of conformal
bootstrap for mixed correlator in 4D ${\mathcal{N}}=2$ superconformal field
theories.

###### Contents

  1. 1 Introduction
  2. 2 Revisit $\mathcal{N}=2$ superconformal algebra and selection rule for OPEs
    1. 2.1 Elements of 4D $\mathcal{N}=2$ superconformal algebra
    2. 2.2  Three point functions in $\mathcal{N}=2$ superspace
  3. 3 Review of superembedding space formalism
    1. 3.1 Superembedding space
    2. 3.2 Superconformal invariants
    3. 3.3 Superconformal tensor structures
    4. 3.4 Supershadow approach
  4. 4 Three point function and OPE coefficient
    1. 4.1 Three point function $\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle$ for long multiplet $\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}^{\Delta}$
    2. 4.2 The three point function $\langle\Phi\Phi^{\dagger}\mathcal{O}\rangle$ for long multiplet $\mathcal{A}_{0,0,(\frac{l}{2},\frac{l}{2})}^{\Delta}$
  5. 5 Computation of superconformal partial waves
    1. 5.1 Result for $\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}$ with $l$ odd
    2. 5.2 Result for $\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}$ with $l$ even
  6. 6 $\mathcal{N}=2$ superconformal block decompositions
  7. 7 Discussion
  8. A Some invariants and tensor structures
    1. A.1 Superconformal invariants and its derivatives
    2. A.2 Tensor structures and its derivatives
  9. B List of Conformal Integrations
    1. B.1 Integration related to $\mathcal{N}_{l}$
    2. B.2 Conformal integration with (P0, P1, P2, P3) and (R0, R1, R2, R3)

##  1 Introduction

In recent years, the conformal bootstrap program [1, 2, 3] has emerged as a
useful techinque[4], which puts strong constraints on conformal field theories
(CFT) in dimensions $D\geq 3$ [5]. By exploiting the general consistency
conditions of the theories, like crossing symmetry and unitarity, one can
impose constraints on the conformal theories. This technique can be used even
though the Lagrangian formulation is not available. As one of the major
successes, the critical exponent of the 3D ising model has been obtained by
using the conformal bootstrap method [6, 7]. It is expected that the
application of this technique in other theories will deepen our understanding
of CFTs.

####

Four dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) have
played an important role in theoretical physics. Such theories can be
constructed using the conventional Lagrangian approach in which one starts
from gauge theories with proper matter representations which leads to the
vanishing of the beta function of all gauge couplings. A large number of SCFTs
can also be obtained from the class $\mathcal{S}$ constructions [8, 9] and
they correspond to a web of dualities. Conformal bootstrap techniques could
substantially contribute towards the classification of these theories.

####

One important component of conformal bootstrap is the decomposition of four
point correlators in terms of conformal blocks or superconformal blocks
corresponding to the exchange of superconformal primary operators and all of
their descendants. The results for four dimensional superconformal blocks for
${\mathcal{N}}=1,2,4$ have appeared in [10, 11, 12, 13, 14, 15]. The four
point correlator of moment map operator and chiral operators for general $4D$
$\mathcal{N}=2$ superconformal field theories was studied in [16], which was
further carried out in Argyres-Douglas fixed point in [17, 18].

####

Stress-tensor operator is universal in any local conformal field theories and
hence it is a natural candidate to bootstrap. For $3D$ the stress-tensor four
point function without the supersymmetry has been studied in [19]. The
supersymmetric stress-tensor for $4D$ $\mathcal{N}=4$ and $6D$ $(2,0)$ SCFTs
has been bootstraped in [20, 21] and [22] respectively. The selection rule for
$\mathcal{J}\times\mathcal{J}$ has been studied in [23, 24]. In [25] the
superconformal partial wave for $\langle JJJJ\rangle$ has been studied, though
the corresponding bootstrap analysis has not appeared yet. So far the mixed
correlators are concerned, in [26] superconformal partial wave has been
computed for $\langle JJJJ\rangle$ and $\langle JJ\phi\phi^{\dagger}\rangle$
for $4D$ $\mathcal{N}=1$ SCFTs. Another work for mixed correlator bootstrap
for $4D$ $\mathcal{N}=1$ has been carried out in [27] for chiral-antichiral
and chiral-real operators. Correlators of moment map and chiral operator in
$4D$ ${\mathcal{N}}=2$ superconformal theories have been studied in [28].
However, the mixed four point correlator $\langle JJ\phi\phi^{\dagger}\rangle$
in case of $4D$ $\mathcal{N}=2$ has not been studied yet and in this work, we
will present the superconformal partial wave for this mixed four point
correlator.

####

The full-fledged conformal bootstrap depends on the conformal partial waves
[29] or CPWs which describes the contribution of conformal four point function
[30] with the exchange of a primary operator and its descendants. The
consistency conditions of the four point correlator are imposed depending on
the conformal partial wave. The conformal blocks for the external scalars are
studied in [30, 29, 12] and are expressed in terms of conformal invariant
cross-ratios. An account of various methods used to compute conformal blocks
can be obtained in [4].

The superconformal blocks are composed of several conformal blocks and the
explicit form depends on the particular superconformal algebra and the
representations. In the case of $\mathcal{N}=1$ SCFTs superconformal blocks
can be obtained from the analysis of exchanged supermultiplets in terms of
conformal primary families [10, 15, 27]. However, this method becomes quite
cumbersome for ${\mathcal{N}}=2$ or higher.

In the case of extended supersymmetry, such as for $\frac{1}{2}$-BPS external
operators, the superconformal blocks can be obtained by using superconformal
Ward identities [31, 12, 14]. Superconformal Casimir approach is more general
and can be applied in any dimension. Here conformal partial waves are
identified as eigenfunctions of quadratic Casimir operators of the
superconformal group. The eigenvalue is related to the quantum number of the
representation. The superconformal blocks for the multiplet
${\mathcal{E}}_{r(0,0)}$ has been computed in [32]. This approach can also be
used to compute the superconformal blocks of shortened operators as solutions
of the superconformal Casimir eigen-equation [33, 34, 32]. However, for the
general case, there are more nilpotent superconformal invariants, for which
this approach is not very suitable.

In the shadow approach, one writes the superembedding coordinates in terms of
supertwisters, which transform naturally under the superconformal group. Then
one writes a manifestly invariant projector onto an irreducible representation
of the superconformal group. The partial waves can then be written as
manifestly invariant supertwistor integrals. However, this approach is more
involved and complex.

####

In this work, we have used the shadow approach to compute superconformal
partial waves. As mentioned above, this involves a supersymmetric
generalization of the embedding formalism, in which the conformal
transformations are realized linearly. This has been developed in [32, 35] and
the superconformal blocks for the general external operator have been computed
in [35, 27]. The relation between the superembedding space and the superspace
has been studied in [36, 37, 38, 39]. In the superspace formalism, study of 4D
$\mathcal{N}=2$ have been carried out [23, 40, 39, 41]. As mentioned in [25],
the three point correlations $\langle{\mathcal{J}JO}\rangle$ gives rise to two
linearly independent superconformal invariants [40] and new tensor structures
[25]. In our analysis, we have used the supershadow approach to project to the
contribution of exchange multiplets.

####

We have organized the paper as follows. In section 2 we review the $4D$
$\mathcal{N}=2$ superconformal algebra, its representation, and the OPE
selection rules for $\mathcal{J}\times\mathcal{J}$ and
$\Phi\times\Phi^{\dagger}$. In the next section, we discuss the superembedding
space formalism with $\mathcal{N}=2$ supersymmetry, the superconformal
invariants, and the tensor structures. In section 4 we discuss the three point
function of $\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle$ and
$\langle\Phi\Phi^{\dagger}\mathcal{O}\rangle$ for the long multiplet. We
computed the superconformal partial wave for the four point correlation
function $\langle{JJ}\phi\phi^{\dagger}\rangle$ in section 5. We discuss our
results with reference to the decomposition of $\mathcal{N}=2$ supermultiplet
into several $\mathcal{N}=1$ multiplets in section 6, which provide us the
nontrivial consistency check for odd $l$. In section 7 we conclude. Section A
and Section B are a small sampling of computations of superconformal
invariants and conformal invariants.

##  2 Revisit $\mathcal{N}=2$ superconformal algebra and selection rule for
OPEs

###  2.1 Elements of 4D $\mathcal{N}=2$ superconformal algebra

In this subsection, we review essential ingredients of 4D $\mathcal{N}=2$
superconformal algebra following [25, 23]. The $\mathcal{N}=2$ superconformal
algebra forms the supergroup $SU(2,2|2)$. This group is a supersymmetric
extension of the conformal group $SU(2,2)\sim SO(4,2)$. The conformal group
generators are
$\left\\{\mathcal{P}_{\alpha,\dot{\alpha}},\mathcal{K}^{\alpha,\dot{\alpha}},\mathcal{M}_{\alpha}^{\beta},\bar{\mathcal{M}}_{\dot{\beta}}^{\dot{\alpha}},D\right\\}$,
with the Lorentz indices $\alpha=\pm$ and $\dot{\alpha}=\pm$. The genrators
for $SU(2)_{R}\times U(1)_{r}$ R-symmery group are
$\left\\{\mathcal{R}^{i}_{j},r\right\\}$ with $SU(2)_{R}$ indices $i,j=1,2$.
These are the bosonic generator of the superconformal group and the fermionic
poincaré and conformal supercharges,
$\left\\{\mathcal{Q}^{i}_{\alpha},\bar{\mathcal{Q}}_{i\dot{\alpha}},\mathcal{S}^{\alpha}_{i},\bar{\mathcal{S}}^{i\dot{\alpha}}\right\\}$.

####

The representation of the superconformal group $SU(2,2|2)$ of a general
multiplet can be obtained from the highest weight for superconformal primaries
with quantum numbers $(\Delta,j,\bar{j},R,r)$ where $(\Delta,j,\bar{j})$ are
the Dynkin labels of the conformal group and $(R,r)$ are the R-symmetry
groups. The conformal supercharges annihilate the highest weights and the
multiplets are constructed using the Poincarè supercharges. A generic
supermultiplet is called a long multiplet and denoted by
$\mathcal{A}^{\Delta}_{R,r,(j,\bar{j})}$ as per the convention of [42]. In
table 1 the unitary irreducible representation of the $\mathcal{N}=2$
superconformal algebra has been shown [23, 25].

Shortening | Quantum Number Relations | Multiplet
---|---|---
$\oslash$ | $\Delta\geq 2+2j+2R-(-1)^{i}r,i=1,2$ | $\mathcal{A}^{\Delta}_{R,r,j_{1},j_{2}}$
$\mathcal{B}^{1}$ |  $\Delta=2R+r$ $j_{1}=0$ | $\mathcal{B}_{R,r(0,j_{2})}$
$\bar{\mathcal{B}}_{2}$ |  $\Delta=2R-r$ $j_{2}=0$ | $\bar{\mathcal{B}}_{R,r(j_{1},0)}$
$\mathcal{B}^{1}\cap\mathcal{B}^{2}$ |  $\Delta=r$ $R=0$ | $\mathcal{E}_{r(0,\bar{j})}$
$\mathcal{B}_{1}\cap\mathcal{B}_{2}$ |  $\Delta=-r$ $R=0$ | $\bar{\mathcal{E}}_{r(j,0)}$
$\mathcal{B}^{1}\cap\mathcal{B}^{2}$ |  $\Delta=-r$ $R=0$ | $\bar{\mathcal{E}}_{r(j,0)}$
$\mathcal{B}^{1}\cap\mathcal{B}_{2}$ |  $\Delta=2R$ $j=\bar{j}=r=0$ | $\hat{\mathcal{B}}_{r(j,0)}$
$\mathcal{C}^{1}$ | $\Delta=2+2j_{1}+2R+r$ | $\mathcal{C}_{R,r(j_{1},j_{2})}$
$\bar{\mathcal{C}}^{2}$ | $\Delta=2+2j_{2}+2R+r$ | $\bar{\mathcal{C}}_{R,r(j_{1},j_{2})}$
$\mathcal{C}^{1}\cap\mathcal{C}^{2}$ |  $\Delta=2+2j_{1}+r$ $R=0$ | $\mathcal{C}_{0,r(j_{1},j_{2})}$
$\bar{\mathcal{C}}_{1}\cap\bar{\mathcal{C}}_{2}$ |  $\Delta=2+2j_{2}-r$ $R=0$ | $\bar{\mathcal{C}}_{0,r(j_{1},j_{2})}$
$\bar{\mathcal{C}}^{1}\cap\bar{\mathcal{C}}_{2}$ |  $\Delta=2+2R+j_{1}+j_{2}$ $r=j_{2}-j_{1}$ | $\hat{\mathcal{C}}_{R(j_{1},j_{2})}$
$\mathcal{B}^{1}\cap\bar{\mathcal{C}}_{2}$ |  $\Delta=1+j_{2}+2R$ $r=j_{2}+1$ | $\mathcal{D}_{R(0,j_{2})}$
$\bar{\mathcal{B}}_{2}\cap\mathcal{C}^{1}$ |  $\Delta=1+j_{1}+2R$ $-r=j_{1}+1$ | $\bar{\mathcal{D}}_{R(j_{1},0)}$
$\mathcal{B}^{1}\cap\mathcal{B}^{2}\cap\bar{\mathcal{C}}_{2}$ |  $\Delta=r=1+j_{2}$ $R=0$ | $\mathcal{D}_{0(0,j_{2})}$
$\mathcal{C}^{1}\cap\bar{\mathcal{B}}_{1}\cap\bar{\mathcal{B}}_{2}$ |  $\Delta=-r=1+j_{1}$ $R=0$ | $\bar{\mathcal{D}}_{0(0,j_{2})}$
Table 1: Classification of the unitary irreducible representations of the
$\mathcal{N}=2$ superconformal algebra

For generic long multiplet, the unitarity bound is,

$\Delta\geq 2+2j_{i}+2R-(-1)^{i}r,\hskip 139.41832ptj_{i}\neq 0$ (2.1)
$\Delta=2j_{1}+2R-(-1)^{i}r\hskip 5.69046pt\text{or}\hskip 5.69046pt\Delta\geq
2+2j_{i}+2R-(-1)^{i}r,\hskip 14.22636ptj_{i}=0$ (2.2)

On the other hand, for $\hat{\mathcal{C}}_{0(j_{1},j_{2})}$,
$\Delta=2+j_{1}+j_{2}$ saturates unitary bound, as seen from the table. If the
highest weight is annihilated by some combination of $\mathcal{Q}$ and
$\mathcal{\bar{Q}}$ the multiplet undergoes shortening. There are several
kinds of shortening conditions [42, 43] depending on the Lorentz and
$SU(2)_{R}$ quantum numbers of the charges that annihilate the highest weight.
Following [42, 43], we call them $\mathcal{B}$ type and $\mathcal{C}$ type
shortening conditions, and the conditions are :

$\mathcal{B}^{i}:\mathcal{Q}^{i}_{\alpha}\mathcal{O}=0,\hskip 105.2751pt$
(2.3)
$\bar{\mathcal{B}}^{i}:\bar{\mathcal{Q}}_{i\dot{\alpha}}\mathcal{O}=0,\hskip
105.2751pt$ (2.4)
$\mathcal{C}^{i}:\begin{cases}\epsilon^{\alpha\beta}\mathcal{Q}^{i}_{\alpha}\mathcal{O}_{\beta}=0,\hskip
36.98866ptj_{1}\neq 0\\\
\epsilon^{\alpha\beta}\mathcal{Q}^{i}_{\alpha}\mathcal{Q}^{i}_{\beta}\mathcal{O}=0,\hskip
28.45274ptj_{1}=0,\end{cases}$ (2.5)
$\bar{\mathcal{C}}^{i}:\begin{cases}\epsilon^{\dot{\alpha}\dot{\beta}}\bar{\mathcal{Q}}_{i{\dot{\alpha}}}\mathcal{O}_{\dot{\beta}}=0,\hskip
36.98866ptj_{2}\neq 0\\\
\epsilon^{\dot{\alpha}\dot{\beta}}\bar{\mathcal{Q}}_{i\dot{\alpha}}\bar{\mathcal{Q}}_{i\dot{\beta}}\mathcal{O}=0,\hskip
28.45274ptj_{2}=0,\end{cases}$ (2.6)

The $\mathcal{B}$ type shortening conditions are called short and the
$\mathcal{C}$ types are called semi-short. In the above table:1, all the
possible shortening conditions for $\mathcal{N}=2$ superconformal algebra have
been presented. In this table, the shortest multiplet $\mathcal{E}_{r}$
satisfies two $\mathcal{B}$ types of shortening conditions. They are called
$\frac{1}{2}$-bps multiplets because they are annihilated by half of the
Poincaré supercharges. The vacuum expectation values of the superprimary of
the multiplet $\mathcal{E}_{r}$ parametrize the Coulomb branch.

###  2.2  Three point functions in $\mathcal{N}=2$ superspace

Three point function $\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle$ and OPE
selection rules:
In this subsection, we will discuss the three point functions in terms of
variables of superspace. We will begin with the three point function
consisting of two identical stress tensor primary operators $J$ and a third
arbitrary operator. These have been discussed in [25, 23] and we will briefly
review it.

The superfield representing stress-tensor multiplet is given by,

$\mathcal{J}(x,\theta,\bar{\theta})=J(x)+J^{i}_{\alpha\dot{\alpha}j}\theta^{\alpha}_{i}\bar{\theta}^{j\dot{\alpha}}+.....$
(2.7)

This superfield satisfies the following reality condition and the conservation
equations

$\mathcal{J}=\bar{\mathcal{J}},\qquad D^{\alpha
i}D^{j}_{\alpha}\mathcal{J}=0,\qquad\bar{D}^{i}_{\dot{\alpha}}\bar{D}^{j\dot{\alpha}}\mathcal{J}=0.$
(2.8)

In order to obtain the $\mathcal{N}=2$ SCFT two point and three point
correlators we introduce superconformal covariant coordinates in superspace,
$z^{I}=(\tilde{x}^{\dot{\alpha}a},\theta_{i},\bar{\theta}^{i})$. Following
[23] we use
$x_{\alpha\dot{\alpha}}=x^{\mu}(\sigma_{\mu})_{\alpha\dot{\alpha}}$ and
$\tilde{x}^{\dot{\alpha}\alpha}=x^{\mu}(\tilde{\sigma}_{\mu})^{\dot{\alpha}\alpha}$.
The chiral combinations are given by
$\tilde{x}^{\dot{\alpha}\alpha}_{\pm}=\tilde{x}^{\dot{\alpha}\alpha}\mp
2i\theta^{\alpha}_{i}\bar{\theta}^{\dot{\alpha}i}$.

Using two points $(z_{1},z_{2})$ we can construct the following variables

$\tilde{x}^{\dot{\alpha}\alpha}_{\bar{1}2}=\tilde{x}^{\dot{\alpha}\alpha}_{1-}-\tilde{x}^{\dot{\alpha}\alpha}_{2+}-4i\theta^{\alpha}_{2i}\bar{\theta}^{\dot{\alpha}i}_{1}=(\tilde{x}_{12})_{-}^{\dot{\alpha}\alpha},\qquad
z^{I}_{12}=z^{I}_{1}-z^{I}_{2}$ (2.9)

Using three points $(z_{1},z_{2},z_{3})$ we construct superconformally
covariant coordinates $Z_{3}=(X_{3},\Theta_{3},\bar{\Theta}_{3},)$ as follows,

$X_{3\alpha\dot{\alpha}}=\frac{x_{3\bar{1}\alpha\dot{\beta}}\tilde{x}^{\dot{\beta}\beta}_{\bar{1}2}x_{2\bar{3}\beta\dot{\alpha}}}{(x_{3\bar{1}})^{2}(x_{2\bar{3}})^{2}},\qquad\bar{X}_{3\alpha\dot{\alpha}}=X^{\dagger}_{3\alpha\dot{\alpha}}=-\frac{x_{3\bar{2}\alpha\dot{\beta}}\tilde{x}_{\bar{2}1}^{\dot{\beta}\beta}x_{1\bar{3}\beta\dot{\alpha}}}{(x_{3\bar{2}})^{2}(x_{1\bar{3}})^{2}}$
(2.10)
$\Theta^{i}_{3\alpha}=i(\frac{x_{\bar{2}3\alpha\dot{\alpha}}}{x^{2}_{\bar{2}3}}\bar{\theta}_{32}^{\dot{\alpha}i}-\frac{x_{\bar{1}3\alpha\dot{\alpha}}}{x^{2}_{\bar{1}3}}\bar{\theta}^{\dot{\alpha}i}_{31}),\qquad\bar{\Theta}_{3\dot{\alpha}i}=i(\theta^{\alpha}_{32i}\frac{x_{\bar{3}2\alpha\dot{\alpha}}}{x^{2}_{\bar{3}2}}-\theta^{\alpha}_{31i}\frac{x_{\bar{3}1}\alpha\dot{\alpha}}{x^{2}_{\bar{3}1}}).$
(2.11)

We also have

$X_{3}^{2}=\frac{x_{\bar{1}2}}{x_{\bar{1}3}^{2}x_{\bar{3}2}^{2}},\qquad\bar{X}_{3}^{2}=\frac{x_{\bar{2}1}}{x_{\bar{3}1}^{2}x_{\bar{2}3}^{2}}$
(2.12)

$\bar{X}$ and $X$ are related through the following relation

$\bar{X}_{3\alpha\dot{\alpha}}=X_{3\alpha\dot{\alpha}}-4i\Theta_{3{\alpha}}^{i}\bar{\Theta}_{3\dot{\alpha}i}.$
(2.13)

By cyclically permutating $z_{1}$, $z_{2}$ and $z_{3}$ one can construct
$Z_{1}$ and $Z_{2}$ in a similar manner.

Using $Z_{1}$, $Z_{2}$ and $Z_{3}$ we can obtain a superconformal invariant
given by

${\bf
u}=\frac{X_{1}^{2}}{\bar{X}_{1}^{2}}=\frac{X_{2}^{2}}{\bar{X}_{2}^{2}}=\frac{X_{3}^{2}}{\bar{X}_{3}^{2}}.$
(2.14)

This variable appears only in the superconformal theories and is absent in
non-supersymmetric conformal theories [36]. In addition, ${\mathcal{N}}=2$
theories involve one more superconformal invariant given by

$-{\bf w}^{\prime}=\frac{X_{1}.\bar{X}_{1}}{\sqrt{X_{1}^{2}\bar{X}_{1}^{2}}}.$
(2.15)

In $\mathcal{N}=2$ superspace, the three point function can be written as

$\langle\mathcal{J}(z_{1})\mathcal{J}(z_{2})\mathcal{O}(z_{3})\rangle=\frac{1}{(x_{\bar{1}3})^{2}(x_{\bar{3}1})^{2}(x_{\bar{2}3})^{2}(x_{\bar{3}}2)^{2}}H(Z_{3}).$
(2.16)

The function $H$ satisfies the scaling condition,

$H({\lambda\bar{\lambda}X_{3},\lambda\Theta_{3},\bar{\lambda}\bar{\Theta}_{3}})=\lambda^{2a}\bar{\lambda}^{2\bar{a}}H(X_{3},\Theta_{3},\bar{\Theta}_{3})$
(2.17)

with $a-2\bar{a}=2-q$ and $\bar{a}-2a=2-q$, where $\triangle=q+\bar{q}$ and
$r=q-\bar{q}$.

An additional restriction is being imposed on $H$ coming from the conservation
equations of $\mathcal{J}$. These imply,

$\frac{\partial^{2}}{{\partial\Theta}_{3{\alpha}}^{i}\partial{\Theta}^{\alpha
j}_{3}}H(Z_{3})=0,\qquad\frac{\partial^{2}}{\partial\bar{\Theta}_{3i}^{\dot{\alpha}}\partial\bar{\Theta}_{3\alpha
j}}H(Z_{3})=0,$ (2.18) $D^{\alpha}_{i}D_{\alpha
j}H(Z_{3})=0,\qquad\tilde{D}^{\dot{\alpha}i}\tilde{D}^{j}_{\dot{\alpha}}H(Z_{3})=0$
(2.19)

where

$D_{\alpha i}=\frac{\partial}{\partial\Theta_{3}^{\alpha
i}}+4i\bar{\Theta}^{\dot{\alpha}}_{3i}\frac{\partial}{\partial
X_{3}^{\dot{\alpha}\alpha}},\qquad\tilde{D}^{\dot{\alpha
i}}=\frac{\partial}{\partial\bar{\Theta}_{3\dot{\alpha}i}}-4i\Theta^{i}_{3\alpha}\frac{\partial}{\partial
X_{3\alpha\dot{\alpha}}}.$ (2.20)

The correlator also needs to be invariant under $z_{1}\leftrightarrow z_{2}$,
which implies
$(X_{3},\Theta_{3},\bar{\Theta}_{3})\leftrightarrow(-\bar{X}_{3},-\Theta_{3},-\bar{\Theta}_{3})$
Using the above restrictions one can write down expressions of the three point
correlators for various operators $\mathcal{O}$. In particular, the three
point function
$\langle\mathcal{J}(z_{1})\mathcal{J}(z_{2})\mathcal{J}(z_{3})\rangle$ has
been written by [39] . Once we know which operators can appear in the
$\mathcal{J}\times\mathcal{J}$ operator product expansion from the selection
rule, we can obtain an expression for ${H}$.

The stress-tensor multiplet is denoted by $\hat{\mathcal{C}}_{0(0,0)}$, which
contains spin two conserved current[25, 23]. All the possible multiplets
$\mathcal{O}$ with equantum number
$(\frac{l}{2},\frac{l}{2},\frac{\Delta}{2}\frac{\Delta}{2})$ that appear in
$\mathcal{J}\times\mathcal{J}$ OPE has been studied by Liendo et.al[23]. The
selection rule of the OPE is

$\begin{split}\hat{\mathcal{C}}_{0(0,0)}\times\hat{\mathcal{C}}_{0(0,0)}\sim\mathcal{I}+\hat{\mathcal{C}}_{0(\frac{l}{2},\frac{l}{2})}+\hat{\mathcal{C}}_{1(\frac{l}{2},\frac{l}{2})}+\mathcal{C}_{\frac{1}{2},\frac{3}{2}(\frac{l}{2},\frac{l+1}{2})}+\\\
\mathcal{C}_{0,3(\frac{l}{2},\frac{l+2}{2})}+\mathcal{C}_{0,0(\frac{l+2}{2},\frac{l}{2})}+\mathcal{C}_{0,0(\frac{l+4}{2},\frac{l}{2})}+\mathcal{A}^{\Delta}_{0,0(\frac{l}{2},\frac{l}{2})}+\mathcal{A}^{\Delta}_{0,0(\frac{l+2}{2},\frac{l}{2})}+\mathcal{A}^{\Delta}_{0,0(\frac{l+4}{2},\frac{l}{2})}+c.c...\end{split}$
(2.21)

The three point correlators have been studied in [25] for the long multiplets
$\mathcal{A}^{\Delta}_{0,0(\frac{l}{2},\frac{l}{2})}$,
$\mathcal{A}^{\Delta}_{0,0(\frac{l+2}{2},\frac{l}{2})}$ and
$\mathcal{A}^{\Delta}_{0,0(\frac{l+4}{2},\frac{l}{2})}$.

$\Phi\times\Phi^{\dagger}$ OPE selection rules for $\mathcal{N}=2$ superspace:

Similar considerations can be used to determine the second three point
function of our interest, which is given by
$\langle\Phi\Phi^{\dagger}\mathcal{O}\rangle$. It takes the following form in
superspace covariant coordinates [15]

$\langle\Phi(z_{1})\Phi^{\dagger}(z_{2})\mathcal{O}(z_{3})\rangle\propto\frac{1}{x_{\bar{3}1}^{2q}x^{2q}_{\bar{2}3}}t(X_{3},\Theta_{3},\bar{\Theta}_{3})$
(2.22)

In this case, the chirality condition will impose the additional constraints
[32]. The chirality condition $\bar{D}^{\dot{\alpha}}_{1}t=0$ implies $t$ is a
function of $\bar{X}_{3}$ and $\bar{\Theta}_{3}$ and other constraint will
come from $D_{2}^{\alpha}$, which reads

$D_{2}^{\alpha}t(\bar{X}_{3},\bar{\Theta}_{3})=i\frac{x_{2\bar{3}}^{\dot{\alpha}\alpha}}{x^{2}_{\bar{2}3}}\frac{\partial}{\partial\bar{\Theta}_{3}^{\dot{\alpha}}}t(\bar{X}_{3},\bar{\Theta}_{3})=0$
(2.23)

From the above, we can conclude that the three point function
$\langle\Phi(z_{1})\Phi^{\dagger}(z_{2})\mathcal{O}(z_{3})\rangle$ depends
only on $\bar{X}_{3}$. The scaling conditions further fix it to,

$\langle\Phi(z_{1})\Phi^{\dagger}(z_{2})\mathcal{O}^{(l)}(z_{3})\rangle\propto\frac{1}{\langle\bar{3}1\rangle^{q}\langle\bar{2}3\rangle^{q}}{\bar{X}^{\Delta_{\mathcal{O}}-2\Delta_{\Phi}-l}_{3}\bar{X}_{3}^{\mu_{1}}...\bar{X}_{3}^{\mu_{l}}}$
(2.24)

The chiral and anti-chiral operators are the scalars of $\mathcal{E}_{r}$
multiplets [16]. The operator $\mathcal{O}(x_{3})$ appears in their OPE if
they are $SU(2)_{R}$ singlet, $r_{\mathcal{O}}=0$ and has spin
$j_{1}=j_{2}=j$. The OPE selection rule is as follows [16],

$\mathcal{E}_{r_{0}}\times\bar{\mathcal{E}}_{-r_{0}}\sim\mathbf{1}+\hat{\mathcal{C}}_{0,(\frac{l}{2},\frac{l}{2})}+\mathcal{A}^{\Delta}_{0,0(\frac{l}{2},\frac{l}{2})}$
(2.25)

####

Comparing the selection rules for OPE of $\mathcal{J}\times\mathcal{J}$ and
$\Phi\times\Phi^{\dagger}$ as given in (2.21) and (2.25) we find the common
multiplets are $\hat{\mathcal{C}}_{0(\frac{l}{2},\frac{l}{2})}$,
$\mathcal{A}^{\Delta}_{0,0(\frac{l}{2},\frac{l}{2})}$ and identity.

Let us consider the multiplet $\hat{\mathcal{C}}_{0(j,\bar{j})}$ which is a
special version of the semi-short multiplet $\hat{\mathcal{C}}_{R(j,\bar{j})}$
and can be obtained by setting the $R$ charge $0$. The multiplet
$\hat{\mathcal{C}}_{0(j,\bar{j})}$ obeys the shortening condition
$\mathcal{C}^{1}\cap\mathcal{C}^{2}\cap\bar{\mathcal{C}}_{1}\cap\bar{\mathcal{C}}_{2}$
and it contains conserved higher spin currents. A single higher spin current
indicates that there is the existence of an infinite number of high spin
currents. As explained in [44, 45] the existence of this multiplet implies the
theory is a free theory. So in the case of interacting theory, we need not
consider the multiplet $\hat{\mathcal{C}}_{0(j,\bar{j})}$, except
${\mathcal{J}}$.

####

##  3 Review of superembedding space formalism

###  3.1 Superembedding space

In superembedding space the coordinates transform linearly under
superconformal transformations. The local operators of a conformal field
theory can be uplifted to respective superembedding space and in terms of them
one can compute the superconformal correlators. Superembbeding space formalism
has been discussed in this context [46, 47, 32, 36, Alba:2015upa]. For our
purpose we will review superembedding formalism for $\mathcal{N}=2$ SCFT in
this section following [25]. In this work, we have used the notation and
convention of [32, 26].

####

The basic constituents of the superembedding space are supertwistors

$\mathbb{Z}_{A}=\begin{pmatrix}Z_{\alpha}\\\ Z^{\dot{\alpha}}\\\
Z_{i}\end{pmatrix}\in\mathbb{C}^{4|\mathcal{N}}$ (3.1)

and its dual is

$\bar{\mathbb{Z}}_{A}=\begin{pmatrix}\bar{Z}_{\alpha}&\bar{Z}^{\dot{\alpha}}&\bar{Z}_{i}\end{pmatrix}\in\mathbb{C}^{4|\mathcal{N}}$
(3.2)

Here $Z_{\alpha}$,$Z^{\dot{\alpha}}$ are the four bosonic components and
$Z_{i}$ are the $\mathcal{N}$ fermionic components. The superconformal group
$SU(2,2|\mathcal{N})$ preserves the dot product

$\langle\mathbb{Z}_{1},\mathbb{Z}_{2}\rangle=\mathbb{Z}_{1}^{\dagger}\Omega\mathbb{Z}_{2},\hskip
28.45274pt\Omega=\begin{pmatrix}0&\delta^{\dot{\beta}}_{\dot{\alpha}}&0\\\
\delta_{\beta}^{\alpha}&0&0\\\ 0&0&\delta^{i}_{j}\end{pmatrix}.$ (3.3)

The superspace is spanned by two supertwistors $\mathbb{Z}_{A}^{a}$ where
$a\in\left\\{1,2\right\\}$ and its dual $\bar{\mathbb{Z}}^{\dot{a}A}$ where
$\dot{a}\in$$\left\\{1,2\right\\}$, subject to the constraint

$\bar{\mathbb{Z}}^{\dot{a}A}\mathbb{Z}^{a}_{A}=0,\hskip
28.45274pta,\dot{a}\in\left\\{1,2\right\\}$ (3.4)

####

The two planes in the supertwistor space suffer from the following gauge
redundancies

$\mathbb{Z}^{a}_{A}\sim\mathbb{Z}_{A}^{b}M^{a}_{b}\hskip 28.45274ptM\in
GL(2,\mathbb{C})$ (3.5)

which changes the basis and the same for the two planes in the dual
supertwistor space. In order to get rid of this $GL(2,\mathbb{C})\times
GL(2,\mathbb{C})$ redundancies we can adopt gauge fixing by choosing the
Poinacaré section. In that case, the supertwistors reduce to their Poincaré
sections given as follows

$\mathbb{Z}_{A}^{a}=\begin{pmatrix}\delta_{A}^{a}\\\
i\tilde{x}_{+}^{\dot{\alpha}a}\\\ 2\theta^{a}_{i}\end{pmatrix}\hskip
28.45274pt\bar{\mathbb{Z}}^{\dot{a}A}=\begin{pmatrix}-i\tilde{x}_{-}^{\dot{a}\alpha}&\delta^{\dot{a}}_{\dot{\alpha}}&2{\bar{\theta}}^{\dot{a}i}\end{pmatrix}.$
(3.6)

In terms of the Poincaré section the constraint (3.4) becomes

$\tilde{x}_{+}^{\dot{\alpha}a}-\tilde{x}_{-}^{\dot{\alpha}a}-4i\bar{\theta}^{\dot{a}i}\theta^{a}_{i}=0,$
(3.7)

where $\tilde{x}_{\pm}$ are the usual chiral and antichiral bosonic
coordinates and $\theta,\bar{\theta}$ are the fermionic coordinates on the
superspace. However, the Poincare section gauge fixing actually breaks the
covariance of the supertwistor and one way to avoid this is to introduce bi-
supertwistors. Bi-supertwistors are the physical quantities that are invariant
under $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$, and its gauge redundancies
are fixed up to scaling. The bi-supertwistors are given by

$\chi_{AB}\equiv\mathbb{Z}^{a}_{A}\mathbb{Z}^{b}_{B}\epsilon_{ab},\hskip
28.45274pt\bar{\chi}^{AB}\equiv\bar{\mathbb{Z}}^{\dot{a}A}\bar{\mathbb{Z}}^{\dot{b}B}\epsilon_{\dot{a}\dot{b}}$
(3.8)

and they satisfy a constraint in the projective null space, which is

$\bar{\chi}^{AB}\chi_{BC}=0$ (3.9)

The space described by the bi-supertwistors $\chi$ and $\bar{\chi}$, is called
superembedding space.

###  3.2 Superconformal invariants

In superembedding space, the superconformal invariants can be expressed in
terms of supertraces of the products of $\chi$ and $\bar{\chi}$ [48, 49, 40].
The supertrace of bi-supertwistors is denoted as,

$\langle\bar{2}1\rangle\equiv\bar{\chi}_{2}^{AB}\chi_{1BA}$ (3.10)
$\langle\bar{4}3\bar{2}1\rangle\equiv\bar{\chi}_{4}^{AB}\chi_{3BC}\bar{\chi}_{2}^{CD}\chi_{1DA}(-1)^{p_{C}}$
(3.11)

For example supertrace $\langle\bar{i}j\rangle$ gives the two point product

$\langle\bar{i}j\rangle=-2x_{\bar{i}j}^{2}$ (3.12)

This can be generalized to products of more bi-supertwistors. These invariants
are chiral in unbarred coordinates and antichiral in barred coordinates.

The superconformal invariant u can be constructed in superembedding space by
using three points $(1,2,0)$ as follows [50] ,

${\textbf{u}}=\frac{\langle\bar{1}2\rangle\langle 2\bar{0}\rangle\langle
0\bar{1}\rangle}{\langle 2\bar{1}\rangle\langle 0\bar{2}\rangle\langle
1\bar{0}\rangle}.$ (3.13)

Using u we can obtain a nilpotent superconformal invariant $z$, given by,

$z=\frac{1-\textbf{u}}{1+\textbf{u}}=\frac{\langle\bar{1}2\rangle\langle\bar{2}0\rangle\langle\bar{0}1\rangle-\langle\bar{2}1\rangle\langle
1\bar{0}\rangle\langle\bar{0}2\rangle}{\langle\bar{1}2\rangle\langle\bar{2}0\rangle\langle\bar{0}1\rangle+\langle\bar{2}1\rangle\langle
1\bar{0}\rangle\langle\bar{0}2\rangle}$ (3.14)

$z$ is antisymmteric under the ${\mathbb{Z}}_{2}$ that corresponds to
$1\leftrightarrow 2$ interchange . For $\mathcal{N}=2$ theories one can check
that $z$ is nilpotent and $z^{5}$ and higher orders vanish. With the same
spirit, we can construct supertraces of more points. We can obtain a new
independent structure, $\langle\bar{1}2\bar{0}1\bar{2}0\rangle$, and the rest
of the other structures either vanish or degenerate to 2-traces. Using these
6-traces we obtain another superconformal invariant $\textbf{w}^{\prime}$
given by,

$\textbf{w}^{\prime}=\frac{4\langle\bar{1}2\bar{0}1\bar{2}0\rangle}{(\langle\bar{1}2\rangle\langle\bar{2}1\rangle\langle\bar{1}0\rangle\langle\bar{0}1\rangle\langle\bar{0}2\rangle\langle\bar{2}0\rangle)^{\frac{1}{2}}}$
(3.15)

Unlike $z$, $\textbf{w}^{\prime}$ is symmetric under $1\leftrightarrow 2$
exchange. For non-supersymmetric theories where $\theta=\bar{\theta}=0$,
$\textbf{w}^{\prime}=-1$. We define

$\textbf{w}=\textbf{w}^{\prime}+1$ (3.16)

In $\mathcal{N}=1$ theories the invariant w is proportional to $z^{2}$ and
hence it does not give any new independent superconformal invariant. But for
$\mathcal{N}=2$ theories there are two independent superconformal invariants
in three point function [40] and the same for $\mathcal{N}\geq 2$.

###  3.3 Superconformal tensor structures

In superembedding space, a general superfield with spin lifts to a multi-
twistor operator

$\phi^{\dot{\beta}_{1}...\dot{\beta_{\bar{j}}}}_{\alpha_{1}...\alpha_{j}}\rightarrow\Phi_{B_{1}...B_{\bar{j}}}^{A_{1}...A_{j}}(\chi,\bar{\chi}).$
(3.17)

The field $\Phi$ has gauge redundancies in each index,

${\Phi_{B_{1},..\bar{B}_{\bar{j}}}}^{A_{1},..A_{j}}(\chi,\bar{\chi})\sim{\Phi_{B_{1},..B_{\bar{j}}}}^{A_{1},..A_{j}}(\chi,\bar{\chi})+\chi_{B_{1},C}{\Lambda_{B_{1},..B_{\bar{j}}}}^{CA_{1}..A_{j}}$
(3.18)

So it is convenient to make the field $\Phi$ index free. In order to make it
index free we introduce auxiliary twistors $\mathcal{S}_{A}$ and
$\bar{\mathcal{S}}^{A}$, which absorb the indices of the superembedding field.
They are defined by

$\Phi(\chi,\bar{\chi},\mathcal{S},\bar{\mathcal{S}})\equiv\bar{\mathcal{S}}^{B_{\bar{j}}}...\bar{\mathcal{S}}^{B_{\bar{1}}}{\Phi_{B_{1}...B_{\bar{j}}}}^{A_{1}...A_{j}}\mathcal{S}_{A_{j}}...\mathcal{S}_{A_{1}}$
(3.19)

The gauge redundancy of $\Phi$ put restrictions on
$\mathcal{S}$,$\bar{\mathcal{S}}$ to be transverse and null

$\bar{\chi}\mathcal{S}=0\hskip 28.45274pt\bar{\mathcal{S}}\chi=0\hskip
28.45274pt\bar{\mathcal{S}}\mathcal{S}=0$ (3.20)

The superfield can be recovered by

$\phi^{\dot{\beta}_{1}..\dot{\beta}_{\bar{j}}}_{\alpha_{1}...\alpha_{j}}=\frac{1}{j!}\frac{1}{\bar{j}!}\left(\bar{\chi}\overrightarrow{\partial}_{\bar{\mathcal{S}}}\right)^{\dot{\beta}_{1}}...\left(\bar{\chi}\overrightarrow{\partial}_{\bar{\mathcal{S}}}\right)^{\dot{\beta}_{\bar{j}}}\Phi(\chi,\bar{\chi},\mathcal{S},\bar{\mathcal{S}})\left(\overleftarrow{\partial}_{\mathcal{S}}\chi\right)_{\alpha_{1}}...\left(\overleftarrow{\partial}_{\mathcal{S}}\chi\right)_{\alpha_{j}}\mid_{\text{Poincare}}$
(3.21)

The correlators are functions of tensor structures and from two points $(1,2)$
one can construct

$S\equiv\frac{\bar{\mathcal{S}}1\bar{2}\mathcal{S}}{\langle
1\bar{2}\rangle},\hskip 28.45274ptS_{*}=S|_{1\leftrightarrow
2}\equiv\frac{\bar{\mathcal{S}}2\bar{1}\mathcal{S}}{\langle 2\bar{1}\rangle},$
(3.22)

where
$\bar{\mathcal{S}}1\bar{2}\mathcal{S}=\bar{\mathcal{S}}^{A}\chi_{1AC}\bar{\chi}_{2}^{CD}\mathcal{S}_{D}$

Similarly for the points $(3,4)$ one obtains,

$T\equiv\frac{\bar{\mathcal{T}}3\bar{4}\mathcal{T}}{\langle
3\bar{4}\rangle},\hskip 28.45274ptT_{*}=T|_{3\leftrightarrow
4}\equiv\frac{\bar{\mathcal{T}}4\bar{3}\mathcal{T}}{\langle 4\bar{3}\rangle}.$
(3.23)

With (3.22) and (3.23) we construct

$S_{\pm}=\frac{1}{2}(S\pm S_{*})$ (3.24)

and

$T_{\pm}=\frac{1}{2}(T\pm T_{*})$ (3.25)

$S_{+}$ is nilpotent and vanishes for $\theta_{i}=0$. For $\mathcal{N}=1$
superembedding space, $S_{+}^{2}$ becomes $0$ and for $\mathcal{N}=2$
superembedding space $S_{+}^{3}$ vanishes so the tensor structures terminate
at $S_{+}^{2}$. For the points (1,2) we have the tensor structures,

$S_{-}^{l}\sim\frac{1}{2}\left(S^{l}+(-1)^{l}S_{*}^{l}\right),$ (3.26)
$S_{+}S_{-}^{l-1}=\frac{1}{2l}\left(S^{l}-(-1)^{l}S_{*}^{l}\right),$ (3.27)
$S_{+}^{2}S_{-}^{l-2}=\frac{1}{4(l-1)}\left(S^{l}+S^{l-1}S_{*}+(-1)^{l}SS_{*}^{l-1}+(-1)^{l}S_{*}^{l}\right)$
(3.28)

and for the tensor structures associated with the points $(3,4)$ are:

$T_{-}^{l}\sim\frac{1}{2}\left(T^{l}+(-1)^{l}T_{*}^{l}\right),$ (3.29)
$T_{+}T_{-}^{l-1}=\frac{1}{2l}\left(T^{l}-(-1)^{l}T_{*}^{l}\right),$ (3.30)

The parity of the tensor structure has been taken care of by $(-1)^{l}$ under
the $1\leftrightarrow 2$ exchange. $S_{-}^{l}$ and $S_{+}^{2}S_{-}^{l-2}$ have
same parity while parity of $S_{-}^{l}$ and $S_{+}S_{-}^{l-1}$ are completly
opposite to each other under the exchange $1\leftrightarrow 2$. Similarly,
under the coordinate exchange of $3\leftrightarrow 4$, $T_{-}^{l}$ and
$T_{+}T_{-}^{l-1}$ have opposite parities, as follows from (3.29) and (3.30).

###  3.4 Supershadow approach

We strictly follow [32, 26] for supershadow formalism. For the
nonsupersymmetric case the shadow formalism was introduced in [51]. The main
idea behind this approach is that given an operator
$\mathcal{O}:(\frac{l}{2},\frac{\bar{l}}{2},q,\bar{q})$ one can construct
another operator
$\tilde{\mathcal{O}}:(\frac{\bar{l}}{2},\frac{l}{2},1-q,1-\bar{q})$ which has
the same eigenvalue of the superconformal Casimir as that of
$\mathcal{O}:(\frac{l}{2},\frac{\bar{l}}{2},q,\bar{q})$. The operator
$\tilde{\mathcal{O}}$ is called the shadow operator of the operator
$\mathcal{O}$.

For a given multiplet $\mathcal{O}$ of superconformal wieghts $(q,\bar{q})$ of
spin $(j_{1},j_{2})$ the shadow operator [32] is given by

$\tilde{\mathcal{O}}(1,\bar{1},\mathcal{S},\mathcal{\bar{S}})=\int
D[2,\bar{2}]\frac{\mathcal{O}^{\dagger}(2,\bar{2},2\bar{\mathcal{S}},\bar{2}\mathcal{S})}{\langle
1\bar{2}\rangle^{2-\mathcal{N}-q+j_{1}}\langle\bar{1}2\rangle^{2-\mathcal{N}-\bar{q}+j_{2}}}.$
(3.31)

In the above equation the integration measure is in superembedding space and
the operator ${\mathcal{O}^{\dagger}}$ of spin $(j_{2},j_{1})$ is the Lorentz
conjugate of the operator $\mathcal{O}$. Since the operator
$\tilde{\mathcal{O}}$ and $\mathcal{O}^{\dagger}$ are related through a non-
local, linear transformation, they share the same Lorentz indices.

Considering that a general superfield $\varphi$ in $\mathcal{N}=2$
superembedding space of quantum numbers $(\Delta,j_{1},j_{2},R=0,r)$ satisfies
the homogeneity given by,

$\varphi(\lambda\chi,\bar{\lambda}\bar{\chi})=\lambda^{-q-j_{1}}\bar{\lambda}^{-\bar{q}-j_{2}}\varphi(\chi,\bar{\chi}).$
(3.32)

where $q$ and $\bar{q}$ are the superconformal weights of the operator,

$q\equiv\frac{1}{2}\left(\Delta+\frac{3}{2}r\right),\hskip
28.45274pt\bar{q}\equiv\frac{1}{2}\left(\Delta-\frac{3}{2}r\right).$ (3.33)

one can check the homogeneity of the shadow operator.

The homogeneity of the shadow operator, as per (3.31) is given by,

$\tilde{\mathcal{O}}(\lambda\chi,\bar{\lambda}\bar{\chi},a\mathcal{S},\bar{a}\bar{\mathcal{S}})=\lambda^{-(2-\mathcal{N}-q+\bar{j_{1}})}\bar{\lambda}^{-(2-\mathcal{N}-q+\bar{j_{2}})}a^{2j_{1}}\bar{a}^{2j_{2}}\tilde{\mathcal{O}}(\chi,\bar{\chi},\mathcal{S},\bar{\mathcal{S}}).$
(3.34)

Thus it transforms as a superconformal multiplet with quantum numbers
$(j_{1},j_{2},2-\mathcal{N}-q,2-\mathcal{N}-\bar{q})$.

Given an operator $\mathcal{O}$ and the corresponding shadow operator
$\tilde{\mathcal{O}}$ one can construct a conformal projector

$\big{|}\mathcal{O}\big{|}=\frac{1}{(2j_{1})!^{2}(2j_{2})!^{2}}\int
D[0,\bar{0}]\ket{\mathcal{O}(0,\bar{0},\mathcal{S},\bar{\mathcal{S}})}\left(\overleftarrow{\partial_{\mathcal{S}}}0\overrightarrow{\partial_{\mathcal{T}}}\right)\left(\overleftarrow{\partial_{\bar{\mathcal{S}}}}0\overrightarrow{\partial_{\bar{\mathcal{T}}}}\right)\bra{\tilde{\mathcal{O}}(0,\bar{0},\mathcal{T},\bar{\mathcal{T}})}\Big{|}_{M}$
(3.35)

In the above equation, $M$ refers to monodromy projection [51]. In general,
the expression would involve a contribution from the shadow operators. As
explained in [51], by restricting to specific monodromy projection one can
restrict the region of integration and avoid erroneous shadow contributions.
Given a four-point correlator, this projector can be used to project the
correlation function into the superconformal partial waves corresponding to
operator $\mathcal{O}$. In particular, considering the present correlator
$\langle{\mathcal{J}J}\Phi\Phi^{\dagger}\rangle$ can be projected to the
partial waves,

$\mathcal{W}_{\mathcal{O}}\propto\langle\mathcal{J}\mathcal{J}|\mathcal{O}|\Phi\Phi^{\dagger}\rangle\sim\int
D[0,\bar{0}]\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle\overleftrightarrow{\mathcal{D}}_{j_{1},j_{2}}\langle\tilde{\mathcal{O}}\Phi\Phi^{\dagger}\rangle$
(3.36)

where,

$\overleftrightarrow{\mathcal{D}}_{j_{1},j_{2}}=\frac{1}{(2j_{1})!^{2}(2j_{2})!^{2}}(\partial_{\mathcal{S}}0\partial_{\mathcal{T}})^{2j_{1}}(\partial_{\bar{\mathcal{S}}}\bar{0}\partial_{\bar{\mathcal{T}}})^{2j_{2}}.$
(3.37)

We will compute the partial waves using this projector.

##  4 Three point function and OPE coefficient

Our objective is to obtain an expression for a mixed four-point correlator
$\langle J(x_{1})J(x_{2})\phi(x_{3})\phi^{\dagger}(x_{4})\rangle$ using the
techniques of shadow formalism. The procedure, as we have already explained,
involves two kinds of three point functions. The first one consists of two
scalar operators of stress tensor multiplet with a third arbitrary operator
whereas the second kind consists of chiral-antichiral operators with a third
arbitrary operator. In this section, we will briefly sketch how to determine
these three point functions.

According to the OPE selection rule the third arbitrary operator $\mathcal{O}$
appearing in the three point functions is invariant under R-symmetry
$SU(2)_{R}$. As we have already mentioned, it consists of the long multiplets
$\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}^{\Delta}$. The $\mathcal{C}$ type
multiplets either disappear or contain a unique conformal block whose
solutions are closely related to long multiplets. So therefore in this present
work, we are interested in the long multiplets.

####

The procedures to obtain the three point functions
$\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle$ and
$\langle\Phi\Phi^{\dagger}\mathcal{O}\rangle$ has been discussed in [23]. It
involves writing down the most general ansatz consistent with the
superconformal symmetry and the relevant equation. Then imposing
$z_{1}\leftrightarrow z_{2}$, one can determine the coefficients. A similar
procedure is adopted in superembedding space [32, 25], which consists of
following two steps :

$\bullet$ Write the general ansatz using homogeneity and reflection symmetry.

$\bullet$ Solve the coefficients by imposing the relevant equations.

The fundamental constituents of a three point function in superembedding space
are the superconformal invariants $z$, $w$ and the tensor structures
$S_{-}^{l}$, $S_{+}S_{-}^{l-1}$, $S_{+}^{2}S_{-}^{l-2}$. In what follows we
will discuss the three point functions in terms of these tensor structures.

###  4.1 Three point function
$\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle$ for long multiplet
$\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}^{\Delta}$

The three point function $\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle$ in
terms of the tensor structures has been discussed in [25] for different
multiplets and we review the computation for exchange of long multiplet. These
are determined using superconformal symmetry, reflection symmetry, and the
relevant equation [25]. It turns out that they are different for odd/even
values of spin due to reflection symmetry. For odd spin long multiplet the
most general expression for the three point function, in terms of the tensor
structures, consistent with the reflection symmetry is given by

$\langle\mathcal{J}(1,\bar{1})\mathcal{J}(2,\bar{2})\mathcal{O}(0,\bar{0})\rangle=\frac{S^{l}_{-}(\lambda_{1}z+\lambda_{3}z^{3})+S_{+}S_{-}^{l-1}(\lambda_{0}+\lambda_{2}z^{2})}{(\langle\bar{1}2\rangle\langle\bar{2}1\rangle)^{1-\frac{1}{4}(\Delta+l)}(\langle\bar{0}1\rangle\langle\bar{1}0\rangle\langle\bar{0}2\rangle\langle\bar{2}0\rangle)^{\frac{1}{4}(\Delta+l)}}.$
(4.1)

The parity of $S_{-}^{l}$ and $z$ are odd under $1\leftrightarrow 2$ exchange.
There can be some extra contribution of the tensor structure
$S_{+}^{2}S_{-}^{l-2}z$. However, they will not give any new independent term
and the other superconformal invariant w does not appear in the above
equation. Application of the conservation equation will impose additional
restrictions on the three point function ansatz and the OPE coefficients
$\lambda_{i}$ are fixed to the following [25],

$\vec{\lambda}_{\mathcal{O}}=\lambda_{\mathcal{O}}\left(1,-\frac{\Delta+l}{2(\Delta-2)},\frac{(\Delta+l)^{2}(2l^{2}-2l-8+6\Delta+l\Delta-\Delta^{2})}{8(\Delta-2)},-\frac{(\Delta+l)(5l^{2}-8+16\Delta+2l\Delta-3\Delta^{2})}{48(\Delta-2)}\right)$
(4.2)

Expanding the equation (4.1) with the OPE coefficients above it will be
consistent with the result of [23]. The coefficients are multiplied by an
overall factor $(4+l-\Delta)$ comparing with the expressions in [23] to avoid
the unphysical poles in the above coefficients. There is another pole at
$\Delta=2$ but this is below the unitary bounds of any multiplet of odd spin.

####

In the case of even $l$, the ansatz for the three point function from
supersymmetry and reflection symmetry is

$\langle\mathcal{J}(1,\bar{1})\mathcal{J}(2,\bar{2})\mathcal{O}(0,\bar{0})\rangle=\frac{S^{l}_{-}(\lambda_{0}+\lambda_{2}z^{2}+\lambda_{3}\textbf{w}+\lambda_{4}z^{4})+S_{+}S_{-}^{l-1}\lambda_{1}z+S^{2}_{+}S_{-}^{l-2}\lambda_{5}}{(\langle\bar{1}2\rangle\langle\bar{2}1\rangle)^{1-\frac{1}{4}(\Delta+l)}(\langle\bar{0}1\rangle\langle\bar{1}0\rangle\langle\bar{0}2\rangle\langle\bar{2}0\rangle)^{\frac{1}{4}(\Delta+l)}}.$
(4.3)

Unlike the case of odd $l$, the coefficients $\lambda_{i}$s, in this case,
cannot be fixed up to an overall term due to the presence of two independent
superconformal invariants $z$ and w. Rather the conservation equation leads to
two independent solutions, which are given by [25],

$\vec{\lambda}^{(1)}_{\mathcal{O}}=\lambda^{(1)}_{\mathcal{O}}\left(1,\frac{(\Delta+l)(2-\Delta)}{2},\frac{(\Delta+l)^{2}}{8},0,\frac{((\Delta+l)(64-3l^{3}+(\Delta-4)^{2})\Delta-l^{2}(5\Delta-8)-l(\Delta^{2}-48))}{384},\\\
\frac{(2+l-\Delta)(2-l-\Delta)}{2}\right),$ (4.4)

and

$\vec{\lambda}^{(2)}_{\mathcal{O}}=\lambda_{\mathcal{O}}^{(2)}(0,\frac{(\Delta+l)(6+l-3\Delta)}{2},\frac{(\Delta+l)(l+3\Delta)}{8},-\frac{(\Delta+l)(4+l-\Delta)}{4},\\\
\frac{(\Delta+l)(3(4+3\Delta)-l(1-2\Delta+2l(3+l+\Delta)))}{96},\frac{(3(\Delta-2)^{2}-2l-l^{2})}{2}).$
(4.5)

These two solutions once again are rescaled by multiplying a factor of
$(4+l-\Delta)$ to avoid unphysical poles in the coefficients. The three point
function $\langle\mathcal{J}\mathcal{J}\mathcal{J}\rangle$ has been studied in
superspace [39]. They have determined the restrictions on the general
structure of the three point function imposed by the $\mathcal{N}=2$
superconformal symmetry, which admits two independent structures. In section
5.2 we see that the conformal blocks will appear with two different sets of
$\lambda^{(i)}$s.

###  4.2 The three point function
$\langle\Phi\Phi^{\dagger}\mathcal{O}\rangle$ for long multiplet
$\mathcal{A}_{0,0,(\frac{l}{2},\frac{l}{2})}^{\Delta}$

The chiral-antichiral three point function is given in [32].

$\langle\Phi(3,\bar{3})\Phi^{\dagger}(4,\bar{4})\mathcal{O}(\chi_{0},\chi_{\bar{0}},\mathcal{T},\bar{\mathcal{T}})\rangle\propto\frac{(\bar{\mathcal{T}}3\bar{4}\mathcal{T})^{l}}{\langle{3\bar{4}}\rangle^{q_{\phi}+\frac{1}{4}(\Delta-l)}\langle{4\bar{3}}\rangle^{{\frac{1}{4}(\Delta-l)}}(\langle\bar{0}3\rangle\langle\bar{3}0\rangle\langle\bar{0}4\rangle\langle\bar{4}0\rangle)^{\frac{1}{4}(\Delta-l)}}$
(4.6)

For the purpose of computation of the four point correlator, we require the
three point function with the shadow operator
$\langle\tilde{\mathcal{O}}\Phi{\Phi}^{\dagger}\rangle$, which can be obtained
by setting $\Delta\rightarrow-\Delta$.

As mentioned in [26], in order to express the above expression in terms of the
tensor structures and superconformal invariants mentioned above, we have used
the identity

$\left(\frac{\langle 3\bar{4}\rangle}{\langle 3\bar{0}\rangle\langle
0\bar{4}\rangle}\right)^{2\delta}=\left(\frac{\langle 3\bar{4}\rangle\langle
4\bar{3}\rangle}{\langle 3\bar{0}\rangle\langle 0\bar{4}\langle
4\bar{0}\rangle\langle
0\bar{3}\rangle}\right)^{\delta}(1-2\delta\tilde{z}+2\delta^{2}{\tilde{z}}^{2}-\frac{2\delta}{3}(2\delta^{2}+1){\tilde{z}}^{3}+\frac{2\delta^{2}}{3}(\delta^{2}+2){\tilde{z}}^{4}).$
(4.7)

We are truncating after the order 4 of the superconformal invariant $z$, since
the terms of higher order in $z$ vanish because of the number of $\theta$
contains. Hence the most general ansatz we obtained for spin $l$

$\begin{split}\langle\Phi(3,\bar{3})\Phi^{\dagger}(4,\bar{4})\mathcal{O}(0,\bar{0})\rangle&=\frac{(T^{l}_{-}+l~{}T_{+}T_{-}^{l-1})(\tilde{\lambda}_{0}+\tilde{\lambda}_{1}\tilde{z}+\tilde{\lambda}_{2}\tilde{z}^{2}+\tilde{\lambda}_{3}\tilde{z}^{3}+\tilde{\lambda}_{4}\tilde{z}^{4})}{\langle
3\bar{4}\rangle^{\frac{1}{4}(\Delta-l)}\langle
4\bar{3}\rangle^{\frac{1}{4}(\Delta-l)}(\langle\bar{0}3\rangle\langle\bar{3}0\rangle\langle\bar{0}4\rangle\langle\bar{4}0\rangle)^{\frac{1}{4}(\Delta-l)}}.\end{split}$
(4.8)

The above three point function is consistent with superconformal symmetry and
reflection symmetry. The superconformal blocks
$\mathcal{G}_{\Delta,l}^{\mathcal{N}=2|JJ;\phi\phi^{\dagger}}$ we obtained
using the above three point function, follows the $\mathcal{N}=1$
decompositions as mentioned in section 6. The OPE coefficients are

$\begin{split}\vec{\tilde{\lambda}}_{\mathcal{O}}&={\tilde{\lambda}}_{\mathcal{O}}\left(1,-\frac{(\Delta+l)}{2},\frac{(\Delta+l)^{2}}{8},-\frac{(\Delta+l)}{6}(1+\frac{(\Delta+l)^{2}}{8}),\frac{(\Delta+l)^{2}}{24}(2+\frac{(\Delta+l)^{2}}{16})\right).\end{split}$
(4.9)

For shadow three point function
$\langle\tilde{\mathcal{O}}\Phi{\Phi}^{\dagger}\rangle$ the OPE coefficients
will be rewritten by replacing $\Delta$ to $-\Delta$.

##  5 Computation of superconformal partial waves

In this section we calculate superconformal partial waves
$\mathcal{W}_{\mathcal{O}}$ for the mixed correlation function
$\langle
J(1,\bar{1})J(2,\bar{2})\phi(3,\bar{3})\phi^{\dagger}(4,\bar{4})\rangle$,
where $J$, $\phi$ and $\phi^{\dagger}$ are the external operator representing
superconformal primary operators of stress-tensor, chiral and anti-chiral
multiplets respectively and are given by

$J(x)=\mathcal{J}(x,\theta,\bar{\theta})\bigg{|}_{\theta_{i}=\bar{\theta}^{i}=0}\quad\text{and}\quad\phi(x)=\Phi(x,\theta,\bar{\theta})\bigg{|}_{\theta_{i}=\bar{\theta}^{i}=0}.$

and similarly for $\phi^{\dagger}$.

We use the method of projectors where the superconformal partial wave
$\mathcal{W}_{\mathcal{O}}$ under the exchange of a real superfield
$\mathcal{O}$ with quantum numbers
$(\frac{l}{2},\frac{l}{2},\frac{\Delta}{2},\frac{\Delta}{2})$ for the
$s$-channel or the direct channel is given by

$\mathcal{W}_{\mathcal{O}}\propto\langle\mathcal{J}\mathcal{J}|\mathcal{O}(0,\bar{0})|\Phi\Phi^{\dagger}\rangle\sim\int
D[0,\bar{0}]\langle\mathcal{J}\mathcal{J}\mathcal{O}(0,\bar{0},\mathcal{S},\bar{\mathcal{S}})\rangle\overleftrightarrow{\mathcal{D}}_{j_{1},j_{2}}\langle\tilde{\mathcal{O}}(0,\bar{0},\mathcal{S},\bar{\mathcal{S}})\Phi\Phi^{\dagger}\rangle$
(5.1)

where,

$\overleftrightarrow{\mathcal{D}}_{j_{1},j_{2}}=\frac{1}{(2j_{1})!^{2}(2j_{2})!^{2}}(\partial_{\mathcal{S}}0\partial_{\mathcal{T}})^{2j_{1}}(\partial_{\bar{\mathcal{S}}}\bar{0}\partial_{\bar{\mathcal{T}}})^{2j_{2}}.$
(5.2)

The superintegrand consists of two three-point functions
$\langle\mathcal{J}\mathcal{J}\mathcal{O}\rangle$ and the supershadow of
$\langle\Phi\Phi^{\dagger}\mathcal{O}\rangle$. The superconformal integral was
presented in [32].

###  5.1 Result for $\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}$ with $l$ odd

We begin with the long multiplet with odd $l$ as the exchange operator, and as
follows from (5.1) the superconformal partial wave is,

$\begin{split}{\mathcal{W}}_{\mathcal{O}}&\propto\langle{\mathcal{J}}(1,\bar{1}){\mathcal{J}}(2,\bar{2})|{\mathcal{O}}|\Phi(3,\bar{3})\Phi^{\dagger}(4,\bar{4})\rangle\\\
&\propto\int
D[0,\bar{0}]\langle{\mathcal{J}}(1,\bar{1})\mathcal{J}(2,\bar{2})\mathcal{O}(0,\bar{0},\mathcal{S},\mathcal{\bar{S}})\rangle\overleftrightarrow{\mathcal{D}}_{l}\langle\tilde{\mathcal{O}}(0,\bar{0},{\mathcal{T}},\tilde{\mathcal{T}})\Phi(3,\bar{3})\Phi^{\dagger}(4,\bar{4})\rangle.\end{split}$
(5.3)

The three point functions in terms of the tensor structures have already been
given in (4.1)) and (4.8). Substituting these expressions in the equation
above, the superconformal partial wave (for odd $l$ ) can be written as

$\mathcal{W}_{\mathcal{O}}\propto\frac{1}{(\langle 1\bar{2}\rangle\langle
2\bar{1}\rangle)^{1-\frac{1}{4}(\Delta+l)}(\langle 3\bar{4}\rangle\langle
4\bar{3}\rangle)^{1-\frac{1}{4}(l-\Delta)}}\times\\\ \int
D[0,\bar{0}]\frac{\mathcal{N}^{full}_{l}}{(\langle 1\bar{0}\rangle\langle
2\bar{0}\rangle\langle 0\bar{1}\rangle\langle
0\bar{2}\rangle)^{\frac{1}{4}(l+\Delta)}(\langle 3\bar{0}\rangle\langle
4\bar{0}\rangle\langle 0\bar{3}\rangle\langle
0\bar{4}\rangle)^{\frac{1}{4}(l-\Delta)}}$ (5.4)

where

$\mathcal{N}_{l}^{Full}=\mathcal{N}_{l}^{(1)}+\mathcal{N}_{l}^{(2)}+\mathcal{N}_{l}^{(3)}+\mathcal{N}_{l}^{(4)}$
(5.5)

Here,

$\begin{split}\mathcal{N}_{l}^{(1)}&=({\lambda_{1}}\tilde{\lambda}_{1}z\tilde{z}+{\lambda_{1}}\tilde{\lambda}_{3}z\tilde{z}^{3}+\lambda_{3}\tilde{\lambda}_{1}z\tilde{z})S_{-}^{l}\overleftrightarrow{\mathcal{D}_{l}}T_{-}^{l}\\\
\mathcal{N}_{l}^{(2)}&=l({\lambda_{1}}\tilde{\lambda}_{0}z+{\lambda}_{1}\tilde{\lambda}_{2}z\tilde{z}^{2}+{\lambda}_{3}\tilde{\lambda}_{0}z^{3})S_{-}^{l}\overleftrightarrow{\mathcal{D}_{l}}T_{+}T_{-}^{l-1}\\\
\mathcal{N}_{l}^{(3)}&=({\lambda_{0}}\tilde{\lambda}_{1}\tilde{z}-{\lambda}_{0}\tilde{\lambda}_{3}\tilde{z}^{3}+\lambda_{2}\tilde{\lambda}_{1}z^{2}\tilde{z})S_{+}S_{-}^{l-1}\overleftrightarrow{\mathcal{D}_{l}}T_{-}^{l}\\\
\mathcal{N}_{l}^{(4)}&=l({\lambda_{0}}\tilde{\lambda}_{0}+{\lambda_{0}}\tilde{\lambda}_{2}\tilde{z}^{2}+\lambda_{0}\tilde{\lambda}_{4}\tilde{z}^{4}+{\lambda}_{2}\bar{\lambda}_{0}z^{2}+{\lambda}_{2}\bar{\lambda}_{2}z^{2}\tilde{z}^{2})S_{+}S_{-}^{l-1}\overleftrightarrow{\mathcal{D}_{l}}T_{+}T_{-}^{l-1}\\\
\end{split}$ (5.6)

and the tensor structures are given by [25],

$S_{-}^{l}\overleftrightarrow{\mathcal{D}_{l}}T_{-}^{l}=\left(\frac{\mathcal{N}_{l}}{4\langle
12\rangle^{l}\langle 34\rangle^{l}}+(-1)^{l}(1\leftrightarrow
2)+(-1)^{l}(3\leftrightarrow 4)\right)$ (5.7)
$S_{-}^{l}\overleftrightarrow{\mathcal{D}_{l}}T_{+}T_{-}^{l-1}=\left(\frac{\mathcal{N}_{l}}{4l\langle
12\rangle^{l}\langle 34\rangle^{l}}+(-1)^{l}(1\leftrightarrow
2)-(-1)^{l}(3\leftrightarrow 4)\right)$ (5.8)
$S_{+}S_{-}^{l-1}\overleftrightarrow{\mathcal{D}_{l}}T_{-}^{l}=\left(\frac{\mathcal{N}_{l}}{4l\langle
12\rangle^{l}\langle 34\rangle^{l}}-(-1)^{l}(1\leftrightarrow
2)+(-1)^{l}(3\leftrightarrow 4)\right)$ (5.9)
$S_{+}S_{-}^{l-1}\overleftrightarrow{\mathcal{D}_{l}}T_{+}T_{-}^{l-1}=\left(\frac{\mathcal{N}_{l}}{4l^{2}\langle
12\rangle^{l}\langle 34\rangle^{l}}-(-1)^{l}(1\leftrightarrow
2)-(-1)^{l}(3\leftrightarrow 4)\right)$ (5.10)

In the above equations, the tensor structures are expanded in terms of
$\mathcal{N}_{l}$, where

$\mathcal{N}_{l}=\frac{1}{(l!)^{2}}(\partial_{\mathcal{S}}0\partial_{\mathcal{T}})^{l}(\mathcal{S}\bar{2}1\bar{0}3\bar{4}\mathcal{T})^{l}=(-1)^{l}s^{\frac{l}{2}}C^{(1)}_{l}(t)$

in which

$t\equiv\frac{\langle\bar{2}1\bar{0}3\bar{4}0\rangle}{2\sqrt{y}}\hskip
28.45274pts\equiv\frac{1}{2^{16}}\langle\bar{0}1\rangle\langle\bar{2}0\rangle\langle\bar{0}3\rangle\langle\bar{4}0\rangle\langle\bar{2}1\rangle\langle\bar{4}3\rangle.$

We are interested in the case where the superfields in the four point function
are restricted to their lowest component fields. We refer to these superfields
as external fields in contrast with the exchanged operator ${\mathcal{O}}$.
Setting the fields to their lowest component amounts to setting their
grassmann coordinates to be zero i.e.
$\theta_{\text{ext}}=\bar{\theta}_{\text{ext}}=0$, $\mathcal{S}$ becomes $S$
and $\mathcal{T}$ becomes $T$ and the bi-supertwistors $\chi_{\alpha\beta}$
and $\bar{\chi}^{\alpha\beta}$ becomes bi-twistors $X_{\alpha\beta}$ and
$\bar{X}^{\alpha\beta}$ which are $4\times 4$ anti-symmetric matrices with
twistor indices $\alpha,\beta$ [26, 25]. The bi-twistors $X_{\alpha\beta}$ and
$\bar{X}^{\alpha\beta}$ can be defined in terms of the vectors $X_{m}$ of the
conformal group as follows

$X^{\alpha\beta}=\frac{1}{2}X_{m}\Gamma^{m\alpha\beta}\hskip
28.45274pt\bar{X}_{\alpha\beta}=\frac{1}{2}X^{m}\tilde{\Gamma}_{m\alpha\beta}$

and the supertraces $\langle i\bar{j}\rangle$ reduces to the inner products of
the 6D vectors $X_{i}$ and $X_{j}$, i.e.,

$\langle i\bar{j}\rangle=-X_{i}.X_{j}=\frac{1}{2}X_{ij}$

. After setting the external $\theta$s to zero the tensor structure becomes

$\mathcal{N}_{l}\equiv N_{l}=(-1)^{l}s_{0}^{\frac{l}{2}}C_{l}^{(1)}(t_{0})$

where,

$s\rightarrow s_{0}=\frac{1}{2^{12}}X_{10}X_{20}X_{30}X_{40}X_{12}X_{34}$
$t\rightarrow
t_{0}=-\frac{X_{13}X_{20}X_{40}}{2\sqrt{X_{10}X_{20}X_{30}X_{40}X_{12}X_{34}}}-(-1)^{l}(1\leftrightarrow{2})-(-1)^{l}(3\leftrightarrow{4}).$

In eq.5.4, it is convenient to integrate over the fermionic components of
$(0,\bar{0})$ at first [25], which leads to an integration in embedding space
and the superconformal integration becomes

$\mathcal{W}_{\mathcal{O}}|_{\theta_{\text{ext}}=0}\propto\frac{1}{X_{12}^{2-\frac{1}{2}(\Delta+l)}X_{34}^{2-\frac{1}{2}(\l-\Delta)}}\int
D^{4}X_{0}\partial_{\bar{0}}^{4}\frac{\mathcal{N}_{l}}{D_{l}}|_{\bar{0}=0}$
(5.11)

where $\frac{1}{D_{l}}$ is given by,

$\frac{1}{D_{l}}\equiv\frac{1}{(X_{1\bar{0}}X_{0\bar{1}}X_{2\bar{0}}X_{0\bar{2}})^{\frac{1}{4}(l+\Delta)}(X_{3\bar{0}}X_{0\bar{3}}X_{4\bar{0}}X_{0\bar{4}})^{\frac{1}{4}(l-\Delta)}}$
(5.12)

####

$\mathcal{N}^{Full}_{l}$ contains several terms with $z$ and $\bar{z}$. One
can carry out the derivatives ($\partial_{\bar{0}}^{4}$) in a straightforward
manner and necessary formulas are collected in Appendix A, (A.3-A.14).
However, the derivatives of the terms corresponding to that in (6.2) lead to
long expressions for each of the terms. To avoid clutter we are not including
all the expressions. As one example, we are giving the result for the
following term,

$\lambda_{1}\tilde{\lambda}_{1}z\tilde{z}\frac{1}{{D}_{l}}S_{-}^{l}\overleftrightarrow{\mathcal{D}_{l}}T_{-}^{l}$.

Integrating out the fermionic variables we obtain the conformal integration in
embedding space:

$(\partial^{2}_{\bar{0}}\partial^{2}_{\bar{0}}z\tilde{z})\frac{1}{D_{l}}N_{l}+(\partial^{n}_{\bar{0}}\partial^{2}_{\bar{0}}z\tilde{z})(\partial_{\bar{0}n}\frac{1}{D_{l}})N_{l}+2(\partial^{2}_{\bar{0}}z\tilde{z})({\partial}^{2}_{\bar{0}}\frac{1}{D_{l}})N_{l}+4(\partial^{n}\partial^{m}z\tilde{z})(\partial_{n}\partial_{m}\frac{1}{D_{l}})N_{l}+4(\partial^{n}_{\bar{0}}\partial^{2}_{\bar{0}}z\tilde{z})\frac{1}{D_{l}}(\partial_{\bar{0}n}N_{l})+8(\partial^{n}\partial^{m}z\tilde{z})(\partial_{n}\frac{1}{D_{l}}\partial_{m}N_{l})+4(\partial^{2}z\tilde{z})(\partial_{n}\frac{1}{D_{l}}\partial_{m}N_{l})+4(\partial^{n}\partial^{m}z\tilde{z})\frac{1}{D_{l}}\partial_{n}\partial_{m}N_{l}+(-1)^{l}(1\leftrightarrow{2})+(-1)^{l}(3\leftrightarrow{4}).$
(5.13)

After carrying out the partial derivatives, we set $\theta_{ext}=0$ and the
above expression becomes,

$2\bigg{(}\Omega_{-}\Big{(}\big{(}(\large\Delta+l\large)^{2}+8\big{)}\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}+\big{(}(\Delta-l)^{2}+8\big{)}\frac{X_{34}}{X_{3\bar{0}}X_{4\bar{0}}}\Big{)}\\\
+(2+l-\Delta)(2+l+\Delta)(\Omega_{A}\Omega_{B}+\Omega_{-}\Omega_{+})\bigg{)}+\frac{\Delta+l+2}{2}\frac{l}{D_{l}}\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}(\Omega_{B}P1-\Omega_{-}P3)-\frac{\Delta-l-2}{2}\frac{l}{D_{l}}\frac{X_{34}}{X_{3\bar{0}}X_{4\bar{0}}}(\Omega_{A}R1-\Omega_{-}R3)+\frac{l(l-1)}{8}\frac{1}{D_{l}}\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}\frac{X_{34}}{X_{3\bar{0}}X_{4\bar{0}}}(P0R3+P1R1+P2R2+P3R0).$
(5.14)

where the expressions for $\Omega_{\pm}$, $\Omega_{A}$, $\Omega_{B}$,
$P0,P1,P2,P3$ and $R0,R1,R2,R3$ are presented in Appendics B.

The next step is to carry out the conformal integrations. Once again it is
straightforward and the necessary formula are collected in Appendix B.
Suppressing the kinematic factors, the partial wave of the four point function
$\langle
J(1,\bar{1})J(2,\bar{2})\phi(3,\bar{3})\phi^{\dagger}(4,\bar{4})\rangle$ for
general long multiplet $\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}^{\Delta}$
with odd spin $l$ is expressed as given below,

$\mathcal{G}_{\Delta,l,odd}^{\mathcal{N}=2|JJ;\phi\phi^{\dagger}}=\lambda_{\tilde{\mathcal{O}}\mathcal{O}}\lambda_{\phi\phi^{\dagger}}\bigg{[}-\frac{1}{2(\Delta+l+1)(\Delta+l+3)}g_{\Delta+3,l+1}\\\
+\frac{8(\Delta+1)(\Delta-l)}{(\Delta-1)(\Delta-l-2)(\Delta+l)(\Delta+l+2)}g_{\Delta+1,l+1}-\frac{8(\Delta+1)(l+2)}{l(\Delta-1)(\Delta+l)^{2}}g_{\Delta+1,l-1}+\frac{(\Delta-l)(\Delta-l-2)(l+2)}{2l(\Delta-l+1)(\Delta+l)(\Delta+l+2)(\Delta-l-1)}g_{\Delta+3,l-1}\bigg{]}$
(5.15)

Here we have presented the conformal partial wave in terms of conformal blocks
and as one can observe, (5.15) is quite compact for the odd $l$.

####

We can compare the above result of partial waves with that obtained for
$\mathcal{N}=1$. The latter has been derived in [32]. The partial wave
representing the four point correlation function in terms of the conformal
blocks for odd spin is given by,

$\mathcal{G}_{\Delta,l,odd}^{\mathcal{N}=1|JJ;\phi\phi^{\dagger}}=\lambda_{\mathcal{O}}\lambda_{\tilde{\mathcal{O}}\phi\phi^{\dagger}}\bigg{[}-\frac{1}{2(\Delta+l+1)}g_{\Delta+1,l+1}+\frac{(l+2)(\Delta-l-2)}{2l(\Delta+l)(\Delta-l-1)}g_{\Delta+1,l-1}\bigg{]}$
(5.16)

while for even spin the expression becomes,

$\mathcal{G}_{\Delta,l,even}^{\mathcal{N}=1|JJ;\phi\phi^{\dagger}}=\lambda^{(0)}_{\mathcal{J}\mathcal{J}\mathcal{O}}\lambda_{\tilde{\mathcal{O}}\phi\phi^{\dagger}}\left[g_{\Delta,l}-\frac{1}{16}\frac{(\Delta-2)(\Delta+l)(\Delta-l-2)}{\Delta(\Delta+l+1)(\Delta-l-1)}g_{\Delta+2,l}\right].$
(5.17)

in (5.15) the ratio of the blocks $g_{\Delta+3,l+1}$ and $g_{\Delta+1,l+1}$
are nearly close to the of $\mathcal{N}=1$ even blocks. The ratio is not
exactly matching with $\mathcal{N}=1$ because the number of supercharges is
more in our case.

###  5.2 Result for $\mathcal{A}_{0,0(\frac{l}{2},\frac{l}{2})}$ with $l$ even

In equation 4.3 we see that one of the three point functions contains two
different kinds of superconformal invariants. Because of that, the OPE
coefficients have two types of solutions and the three point function contains
more tensor structures compared to its counterpart in the odd $l$. Inserting
the superconformal projector we obtain the superconformal partial wave for the
even spin as,

$\mathcal{W}_{\mathcal{O}}\propto\int
D[0,\bar{0}]\langle\mathcal{J}(1,\bar{1})\mathcal{J}(2,\bar{2})\mathcal{O}(0,\bar{0},\mathcal{S},\tilde{\mathcal{S}})\rangle\overleftrightarrow{\mathcal{D}}\langle\tilde{\mathcal{O}}(0,\bar{0},\mathcal{T},\tilde{\mathcal{T}})\Phi(3,\bar{3})\Phi^{\dagger}(4,\bar{4})\rangle\\\
=\frac{1}{(\langle 1\bar{2}\rangle\langle
2\bar{1}\rangle)^{1-\frac{1}{4}(\Delta+l)}(\langle 3\bar{4}\rangle\langle
4\bar{3}\rangle)^{1-\frac{1}{4}(l-\Delta)}}\times\int
D[0,\bar{0}]\frac{\mathcal{N^{\prime}}^{full}_{l}}{(\langle 1\bar{0}\langle
2\bar{0}\rangle\langle 0\bar{1}\rangle\langle
0\bar{2}\rangle)^{\frac{1}{4}(l+\Delta)}(\langle 3\bar{0}\rangle\langle
4\bar{0}\rangle\langle 0\bar{3}\rangle\langle
0\bar{4}\rangle)^{\frac{1}{4}(l-\Delta)}}$ (5.18)

here,

$\mathcal{N^{\prime}}^{full}_{l}=\mathcal{N^{\prime}}^{(1)}_{l}+\mathcal{N^{\prime}}^{(2)}_{l}+\mathcal{N^{\prime}}^{(3)}_{l}+\mathcal{N^{\prime}}^{(4)}_{l}+\mathcal{N^{\prime}}^{(5)}_{l}+\mathcal{N^{\prime}}^{(6)}_{l}$

According to (5.18)

$\begin{split}\mathcal{N^{\prime}}^{(1)}_{l}&=S_{-}^{l}\overleftrightarrow{\mathcal{D}}_{l}T_{-}^{l}(\lambda_{0}\tilde{\lambda}_{0}+\lambda_{2}\tilde{\lambda}_{0}z^{2}+\lambda_{3}\tilde{\lambda}_{0}\textbf{w}+\tilde{\lambda}_{0}\lambda_{4}z^{4}+\lambda_{0}\tilde{\lambda}_{2}\tilde{z}^{2}+\lambda_{2}\tilde{\lambda}_{2}z^{2}\tilde{z}^{2}+\lambda_{3}\tilde{\lambda}_{2}\textbf{w}\tilde{z}^{2}+\lambda_{0}\tilde{\lambda}_{4}\tilde{z}^{4})\\\
\mathcal{N^{\prime}}^{(2)}_{l}&=lS_{-}^{l}\overleftrightarrow{\mathcal{D}}T_{+}T_{-}^{l-1}(\lambda_{0}\tilde{\lambda}_{1}\tilde{z}+\lambda_{0}\tilde{\lambda}_{3}\tilde{z}^{3}+\lambda_{2}\tilde{\lambda}_{1}z^{2}\tilde{z}+\lambda_{3}\tilde{\lambda}_{1}\textbf{w}\tilde{z})\\\
\mathcal{N^{\prime}}^{(3)}_{l}&=S_{+}S_{-}^{l-1}\overleftrightarrow{\mathcal{D}}_{l}T_{-}^{l}(\lambda_{1}\tilde{\lambda}_{0}z+\lambda_{3}\tilde{\lambda}_{0}{\textbf{w}}+\lambda_{1}\tilde{\lambda}_{2}z\tilde{z}^{2})\\\
\mathcal{N^{\prime}}^{(4)}_{l}&=lS_{+}S_{-}^{l-1}\overleftrightarrow{D}_{l}T_{+}T_{-}^{l-1}(\lambda_{1}\tilde{\lambda}_{1}z\tilde{z}+\lambda_{1}\tilde{\lambda}_{3}z\tilde{z}^{3})\\\
\mathcal{N^{\prime}}^{(5)}_{l}&=S_{+}^{2}S_{-}^{l-2}\overleftrightarrow{D}_{l}T_{-}^{l}(\lambda_{5}\tilde{\lambda}_{0}+\lambda_{5}\tilde{\lambda}_{2}\tilde{z}^{2}+\lambda_{5}\tilde{\lambda}_{4}\tilde{z}^{4})\\\
\mathcal{N^{\prime}}^{(6)}_{l}&=lS_{+}^{2}S_{-}^{l-2}\overleftrightarrow{D}_{l}T_{+}T_{-}^{l-1}(\lambda_{5}\tilde{\lambda}_{1}\tilde{z}+\lambda_{5}\tilde{\lambda}_{3}\tilde{z}^{3})\end{split}$
(5.19)

The additional tensor structures are

$S_{+}^{2}S_{-}^{l-2}\overleftrightarrow{D}_{l}T_{-}^{l}=\frac{\mathcal{N}_{l}+\mathcal{M}_{l}}{8(l-1)\langle
1\bar{2}\rangle^{l}\langle 3\bar{4}\rangle^{l}}+(-1)^{l}(1\leftrightarrow
2)+(-1)^{l}(3\leftrightarrow 4),$ (5.20)
$S_{+}^{2}S_{-}^{l-2}\overleftrightarrow{D}_{l}T_{+}T_{-}^{l-1}=\frac{\mathcal{N}_{l}+\mathcal{M}_{l}}{8l(l-1)\langle
1\bar{2}\rangle^{l}\langle 3\bar{4}\rangle^{l}}+(-1)^{l}(1\leftrightarrow
2)-(-1)^{l}(3\leftrightarrow 4),$ (5.21)

where,

$\mathcal{M}_{l}=\frac{1}{l!^{2}}(\partial_{\mathcal{S}}0\partial_{\mathcal{T}})^{l}(\mathcal{S}\bar{2}1\bar{0}3\bar{4}\mathcal{T})^{l-1}\mathcal{S}\bar{1}2\bar{0}3\bar{4}\mathcal{T}$

All the above tensor structures will vanish when we set
$\theta_{ext}=\bar{\theta}_{ext}=0$ but after the action of the partial
derivatives $\partial_{\bar{0}}$ they may have non-zero contributions. As in
the case of odd $l$, after taking the derivatives the next step is to carry
out the conformal integration. The necessary formulas for the integration are
given in Appendix B. Unlike the case of odd $l$, here the conformal
integrations are mathematically more involved due to the presence of the
derivative of the tensor structures. The final expression for the partial wave
in terms of the conformal blocks for the even spin can be summarised as
follows,

$\mathcal{G}_{\Delta,l,even}^{\mathcal{N}=2|JJ;\Phi\Phi^{\dagger}}=c_{0}g_{\Delta,l}+c_{1}g_{\Delta+2,l+2}+c_{2}g_{\Delta+2,l}+c_{3}g_{\Delta+2,l-2}+c_{4}g_{\Delta+4,l}.$
(5.22)

Unlike the odd $l$ case, it involves five conformal blocks and the
coefficients are quite long, which we have given below.

$c_{0}={\lambda}^{(1)}_{\mathcal{O}}\lambda_{\tilde{\mathcal{O}}\phi\phi^{\dagger}}\left(4(\Delta-l)(\Delta+l+2)(\Delta^{2}-2l-l^{2})\right)$
$c_{1}=\lambda_{\tilde{\mathcal{O}}\phi\phi^{\dagger}}\left(\lambda_{\mathcal{O}}^{(1)}\frac{(\Delta-l)(\Delta+l)(\Delta+l+2)^{2}(\Delta^{2}-2l-l^{2}-2)}{4(\Delta+l+1)(\Delta+l+3)}+\lambda_{\mathcal{O}}^{(2)}\frac{(\Delta-l-1)(\Delta-l)(\Delta+l)(\Delta+l+2)^{2}}{2(\Delta+l+3)}\right)$
$c_{2}=\lambda_{\tilde{\mathcal{O}}\phi\phi^{\dagger}}\left(-{\lambda}_{\mathcal{O}}^{(1)}\frac{(\Delta-l-2)(\Delta-l)(\Delta+l)(\Delta+l+2)(\Delta+\Delta
l+1)}{l(\Delta+1)}-{\lambda}_{\mathcal{O}}^{(2)}\frac{(\Delta-l-2)(\Delta-l)(\Delta+l)(\Delta+l+2)(3+3\Delta+2l+\Delta
l)}{l(\Delta+1)}\right)$
$c_{3}=\lambda_{\tilde{\mathcal{O}}\phi\phi^{\dagger}}\left(\lambda_{\mathcal{O}}^{(1)}\frac{(\Delta-l-2)(\Delta-l)^{2}(\Delta+l+2)(4\Delta^{2}-10l+3\Delta^{2}l-12l^{2}+\Delta^{2}l^{2}-5l^{3}-l^{4}-4)}{4(\Delta-l-1)(\Delta-l+1)(l-1)l}+\lambda_{\mathcal{O}}^{(2)}\frac{(\Delta-l-2)(\Delta-l)^{2}(l+2)(l+3)(\Delta+l+1)(\Delta+l+2)}{2(\Delta-l+1)(l-1)l}\right)$
$c_{4}=\lambda_{\tilde{\mathcal{O}}\phi\phi^{\dagger}}\left(\lambda_{\mathcal{O}}^{(1)}\frac{(\Delta-l-2)(\Delta-l)^{2}(\Delta+l)(\Delta+l+2)^{2}(4\Delta+2\Delta^{2}-\Delta^{3}+\Delta^{4}-4l+2\Delta
l-2\Delta^{2}l-2l^{2}+\Delta
l^{2}-\Delta^{2}l^{2})}{64(\Delta+1)(\Delta+2)(\Delta-l-1)(\Delta-l+1)(\Delta+l+1)(\Delta+l+3)}+\lambda_{\mathcal{O}}^{(2)}\frac{\Delta(\Delta-l-2)(\Delta-l)^{2}(\Delta+l)(\Delta+l+2)^{2}}{16(\Delta+1)(\Delta+2)(\Delta-l+1)(\Delta+l+3)}\right)$

As we have already mentioned, the contribution from
$\langle\mathcal{J}\mathcal{J}{\mathcal{O}}\rangle$ involves two unknown
overall coefficients, namely, ${\lambda}^{(1)}_{\mathcal{O}}$ and
${\lambda}^{(2)}_{\mathcal{O}}$ while the three point function
$\langle\Phi\Phi^{\dagger}\mathcal{O}\rangle$ involves one unknown coefficient
$\lambda_{\Phi\Phi^{\dagger}\mathcal{O}}$. As has been discussed in [25] the
supershadow OPE coefficient $\lambda_{\tilde{\mathcal{O}}\Phi\Phi^{\dagger}}$
is related to $\lambda_{\Phi\Phi^{\dagger}\mathcal{O}}$. From the above
expression, one can see $c_{2},c_{3}$ and $c_{4}$ will vanish at unitarity
bound i.e $\Delta=l+2$ and $c_{0},c_{1}$ will be non zero. This vanishing may
be a result of the fact that at unitarity bound general long multiplets splits
into

$\mathcal{A}^{2+l}_{0,0(\frac{l}{2},\frac{l}{2})}=\hat{\mathcal{C}}_{0(\frac{l}{2},\frac{l}{2})}+\hat{\mathcal{C}}_{\frac{1}{2}(\frac{l-1}{2},\frac{l}{2})}+\hat{\mathcal{C}}_{\frac{1}{2}(\frac{l}{2},\frac{l-1}{2})}+\hat{\mathcal{C}}_{1(\frac{l-1}{2},\frac{l-1}{2})}$

Out of these multiplets only $\hat{\mathcal{C}}_{0(\frac{l}{2},\frac{l}{2})}$
will contribute to the four point correlator. Hence the conformal blocks
$g_{\Delta,l}$ and $g_{\Delta+2,l+2}$ are the contribution from the multiplet
$\hat{\mathcal{C}}_{0(\frac{l}{2},\frac{l}{2})}$. This is consistent with the
decomposition of a $\hat{\mathcal{C}}_{0(\frac{l}{2},\frac{l}{2})}$ as given
in [42].

####

We close this section with the following comments regarding analytic
continuation of the result obtained above for general long multiplet
$\mathcal{A}^{\Delta}_{0,0(\frac{l}{2}{\frac{l}{2}})}$. When we analytically
continue the superconformal block
$\mathcal{G}_{\Delta,l,even}^{\mathcal{N}=2|JJ;\Phi\Phi^{\dagger}}$ to
$\Delta=2,l=0$ however, as explained in [25] the supershadow transformation
for $\Delta=2,l=0$ may be pathological and unphysical term may arise from
analytical continuation. It is not clear if there are unphysical terms in the
analytical continuation $\Delta\rightarrow 2+l$ of the conformal block for
$l\geq 0$. It may be interesting to investigate the issue further.

##  6 $\mathcal{N}=2$ superconformal block decompositions

The decomposition of $\mathcal{N}=2$ superconformal blocks into
$\mathcal{N}=1$ helps us to check the consistency of our results. The
decomposition occurs due to the decomposition of $\mathcal{N}=2$ multiplets
into $\mathcal{N}=1$ multiplets. This relation has been used in [26, 23, 15,
10] for the consistency check of the $\mathcal{N}=1$ blocks. This agrees with
the $\mathcal{N}=1$ decomposition of the $\mathcal{N}=2$ superconformal
considering the operators are global symmetry conserved current [12]. Our
result also satisfies the nontrivial consistency checks which we will discuss
below.

####

The decomposition of a general $\mathcal{N}=2$ superconformal long multiplet
$\mathcal{A}^{\Delta}_{0,0(\frac{l}{2},\frac{l}{2})}$ has been solved in [23],
which shows:

$\mathcal{A}^{\Delta}_{0,0(\frac{l}{2},\frac{l}{2})}\rightarrow
A^{\Delta}_{r^{\prime}=0(\frac{l}{2},\frac{l}{2})}+A^{\Delta+1}_{r^{\prime}=0(\frac{l-1}{2},\frac{l-1}{2})}+A^{\Delta+1}_{r^{\prime}=0(\frac{l+1}{2},\frac{l+1}{2})}+A^{\Delta+2}_{r^{\prime}=0(\frac{l}{2},\frac{l}{2})}+A^{\Delta+1}_{r^{\prime}=0(\frac{l-1}{2},\frac{l+1}{2})}+A^{\Delta+1}_{r^{\prime}=0(\frac{l+1}{2},\frac{l-1}{2})},$
(6.1)

In this above equation $\mathcal{A}$ is the $\mathcal{N}=2$ multiplet and $A$
are the $\mathcal{N}=1$ multiplets. The subindex
$r^{\prime}$=$\frac{2}{3}(2R+r)$ is the $U(1)_{r^{\prime}}$ charge for
$\mathcal{N}=1$ multiplets. We have not mentioned additional terms in (6.1)
with non-zero $r^{\prime}$, as they do not contribute to the correlator
$\langle JJ\Phi\Phi^{\dagger}\rangle$. For odd $l$ the last two terms of the
above equation (6.1) do not contribute to the correlator $\langle
JJ\Phi\Phi^{\dagger}\rangle$. Considering the first four terms the
decomposition implies,

$\mathcal{G}^{\mathcal{N}=2|JJ\Phi\Phi^{\dagger}}_{\Delta,l,odd}=a_{0}\mathcal{G}^{\mathcal{N}=1}_{\Delta,l,odd}+a_{1}\mathcal{G}^{\mathcal{N}=1}_{\Delta+1,l-1,even}+a_{2}\mathcal{G}^{\mathcal{N}=1}_{\Delta+1,l+1,even}+a_{3}\mathcal{G}^{\mathcal{N}=1}_{\Delta+2,l,odd},$
(6.2)

Here $\mathcal{G}^{\mathcal{N}=1}_{\Delta,l,odd/even}$ are the $\mathcal{N}=1$
superconformal blocks. Using equation (6.2), we expand the $\mathcal{N}=2$
superconformal blocks in terms of the bosonic conformal blocks as follows:

$\mathcal{G}^{\mathcal{N}=2}_{\Delta,l,odd}\propto
b_{0}g_{\Delta+1,l+1}+b_{1}g_{\Delta+1,l-1}+b_{2}g_{\Delta+3,l+1}+b_{3}g_{\Delta+3,l-1}.$
(6.3)

Comparing with (5.15) we can obtain the expressions of
$\\{b_{0},b_{1},b_{2},b_{3}\\}$ upto a constant. Expanding
$\mathcal{G}^{\mathcal{N}=1}$ in terms of the conformal blocks $g$ and
comparing that with (6.3) we can establish a linear relation (up to an overall
multiplicative constant) between $\\{a_{0},a_{1},a_{2},a_{3}\\}$ and
$\\{b_{0},b_{1},b_{2},b_{3}\\}$, as ${\mathbf{a}}.N={\mathbf{b}}$. A
straightforward computation shows that the matrix $N$ has a null eigenvector
and that gives rise to a relation among the coefficients
$\\{b_{0},b_{1},b_{2},b_{3}\\}$. Therefore, as in [25] our coefficients
$b_{i}$, satisfy a constraint, which is given as follows.

$\frac{16(\Delta+1)(\Delta+l+1)}{(\Delta-1)(\Delta+l)}\left[\frac{(l+2)(\Delta+l+3)}{l(\Delta+l+2)}b_{2}+\frac{(\Delta-l+1)}{(\Delta-l)}b_{3}\right]+\frac{(\Delta-l-2)(l+2)(\Delta+l+1)}{(\Delta-l-1)l(\Delta+l)}b_{0}+b_{1}=0,$
(6.4)

here $b_{i}$ are the coefficients of the conformal blocks presented in (5.15)
and the above equation (6.4) provides a nontrivial consistency check of our
result for odd spin in equation (5.15).

####

For even spin, we can carry out a similar analysis. As given by [26], the
$\mathcal{N}=2$ superconformal block can be expanded in terms of
$\mathcal{N}=1$ blocks as follows,

$\mathcal{G}^{\mathcal{N}=2|JJ\Phi\Phi^{\dagger}}_{\Delta,l,even}=a^{\prime}_{0}\mathcal{G}^{\mathcal{N}=1}_{\Delta,l,even}+a^{\prime}_{1}\mathcal{G}^{\mathcal{N}=1}_{\Delta+1,l-1,odd}+a^{\prime}_{2}\mathcal{G}^{\mathcal{N}=1}_{\Delta+1,l+1,odd}+a^{\prime}_{3}\mathcal{G}^{\mathcal{N}=1}_{\Delta+2,l,even},$
(6.5)

Substituting the expressions for the $\mathcal{N}=1$ blocks in terms of the
conformal blocks we can write

$\mathcal{G}^{\mathcal{N}=2}_{\Delta,l,even}\propto
b^{\prime}_{0}g_{\Delta,l}+b^{\prime}_{1}g_{\Delta+2,l+2}+b^{\prime}_{2}g_{\Delta+2,l}+b^{\prime}_{3}g_{\Delta+2,l-2}+b^{\prime}_{4}g_{\Delta+4,l}.$
(6.6)

Clearly,
$\\{b^{\prime}_{0},b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3},b^{\prime}_{4}\\}$
can be identified with $\\{c_{0},c_{1},c_{2},c_{3},c_{4}\\}$ given in (5.22)
upto an overall constant of proportionality. These five parameters, $c_{i}$,
being linear combinations of four parameters
$\\{a^{\prime}_{0},a^{\prime}_{1},a^{\prime}_{2},a^{\prime}_{3}\\}$ satisfy
the following constraint:

$\begin{split}&\frac{\Delta-l-2}{\Delta-l-1}\left[\frac{(\Delta-2)(\Delta+l)}{16\Delta(\Delta+l+1)}c_{0}+\frac{(l+3)(\Delta+l+3)}{(l+1)(\Delta+l+2)}c_{1}\right]+c_{2}\\\
&+\frac{\Delta-l+1}{\Delta-l}\left[\frac{(l-1)(\Delta+l)}{(l+1)(\Delta+l+1)}c_{3}+\frac{16(\Delta+2)(\Delta+l+3)}{\Delta(\Delta+l+2)}c_{4}\right]=0.\end{split}$
(6.7)

The three point correlator $\langle\mathcal{J}\mathcal{J}{\mathcal{O}}\rangle$
invovles $\lambda_{\mathcal{O}}^{(1)}$ and $\lambda_{\mathcal{O}}^{(2)}$,
which are independent of each other. The expressions $c_{i},i=0,1,2,3,4$ are
linear in $\lambda_{\mathcal{O}}^{(1)}$ and $\lambda_{\mathcal{O}}^{(2)}$ and
after putting the values of different $c_{i}$s the above consistency condition
satisfies for each of the $\lambda_{\mathcal{O}}^{(i)}$s.

##  7 Discussion

In this work, we have evaluated the superconformal partial wave $\langle
JJ\phi\phi^{\dagger}\rangle$ using the superembedding space. The three point
correlators $\langle\mathcal{J}\mathcal{J}{\mathcal{O}}\rangle$ and
$\langle\Phi\Phi^{\dagger}{\mathcal{O}}\rangle$ can be expanded in terms of
the superconformal invariants and the tensor structures. The coefficients are
determined using the relevant equations and ${\mathbb{Z}}_{2}$ symmetry. Then
using shadow formalism we obtain the four point correlator as a sum over
conformal blocks. Unlike $\langle JJJJ\rangle$, this mixed correlator has non-
zero contribution only for the general long multiplet. Compared with the
${\mathcal{N}}=1$ superconformal blocks, we can have a consistency check for
odd $l$. For even $l$, however, due to presence of two parameters,
$\lambda_{\mathcal{O}}^{(1)}$ and $\lambda_{\mathcal{O}}^{(2)}$, the
comparison provides a relation between them.

The present analysis sets up the necessary ingredients for ${\mathcal{N}}=2$
bootstrap analysis for mixed correlator. The power of mixed correlator in the
context of three dimensional bootstrap has been elaborated in [7]. Numerical
study of the bootstrap of mixed four-point correlator for chiral and real
scalars in D=4 ${\mathcal{N}}=1$ has been done in [27]. The present model
provides the part of the ingredients necessary for analysis of a mixed
correlator in D=4, ${\mathcal{N}}=2$ theory.

It has been found that the critical three dimensional Ising model lives at the
kink of the bound on the CFT data, [52, 6, 7]. Plausibly, a CFT containing
minimum spectrum of operators to satisfy the crossing relation lives at kink.
When one approaches the kink along the boundary of the allowed region, squared
OPE coefficients of certain operators vanish, which can be interpreted as two
different solution branches meet at the kink [4, 6, 53] and leads to
decoupling of certain operators. One may explore the possibility of a similar
scenario for the mixed correlator of 4D ${\mathcal{N}}=2$. Since stress tensor
operator is universal in any local CFT it is natural to consider correlators
involving stress tensor multiplet. However, It is not clear which theories are
relevant for this possibility. But as mentioned in [16], 4D ${\mathcal{N}}=2$
superconformal QCD with gauge group $SU(2)$, $N_{f}=4$ may deserve further
study 54.

A further generalization of the present work is to analyze the mixed
correlators involving the vector and higher spin currents in superconformal
field theories. A study of higher-spin supercurrents in D=4 ${\mathcal{N}}=1$
superconformal theory has been done in [55]. We hope to report on some of
these in the future.

## Acknowledgement

A preliminary version of this work was presented at XXV DAE-BRNS High Energy
Physics Symposium 2022, IISER Mohali. SR is thankful to the local organizing
committee for the hospitality during the stay at IISER Mohali.

##  Appendix A Some invariants and tensor structures

###  A.1 Superconformal invariants and its derivatives

Two independent invariants in $\mathcal{N}=2$ superembedding space are $z$ and
w which are constructed by taking the 2-point traces of three points
$(1,2,0)$:

$z=\frac{\langle\bar{1}2\rangle\langle\bar{2}0\rangle\langle\bar{0}1\rangle-\langle\bar{2}1\rangle\langle\bar{1}0\rangle\langle\bar{0}2\rangle}{\langle\bar{1}2\rangle\langle\bar{2}0\rangle\langle\bar{0}1\rangle+\langle\bar{2}1\rangle\langle\bar{1}0\rangle\langle\bar{0}2\rangle}$
(A.1)

and the six point trace will produce

$\textbf{w}=\frac{4\langle\bar{1}2\bar{0}1\bar{2}0\rangle}{(\langle\bar{1}2\rangle\langle\bar{2}1\rangle\langle\bar{1}0\rangle\langle\bar{0}1\rangle\langle\bar{0}2\rangle\langle\bar{2}0\rangle)^{\frac{1}{2}}}+1.$
(A.2)

Exchanging the coordinates by $1\leftrightarrow 3$ and $2\leftrightarrow 4$ we
obtain other superconformal invariants in $\mathcal{N}=2$ superembedding space
$\bar{z}$ and $\bar{\textbf{w}}$. All the superconformal invariants are
nilpotent and vanish after setting external Grassmann variables to zero, which
reads

$\displaystyle z^{5}=\textbf{w}^{3}=0$ $\displaystyle
z|_{\theta_{ext}=0}=\textbf{w}|_{\theta_{ext}}=0.$

$z$ or $w$ will vanish unless the derivatives $\partial_{\bar{0}}$ act on
them. The derivatives on $z$ are as follows:

$\partial_{\bar{0}}^{m}z=(z^{2}-1)\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}-\frac{X_{2}^{m}}{X_{2\bar{0}}}\right),\hskip
28.45274pt\partial_{\bar{0}}^{2}z=2z(z^{2}-1)\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}$
(A.3)
$\partial_{\bar{0}}^{m}z^{2}=2z(z^{2}-1)\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}-\frac{X_{2}^{m}}{X_{2\bar{0}}}\right),\hskip
28.45274pt\partial_{\bar{0}}^{2}z^{2}=(6z^{2}-2)(z^{2}-1)\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}$
(A.4)
$(\partial_{\bar{0}}z).(\partial_{\bar{0}}\tilde{z})|_{\bar{0}=0}=-\frac{1}{2}\left(\frac{X_{13}}{X_{1\bar{0}}X_{3\bar{0}}}-(1\leftrightarrow
2)-(3\leftrightarrow 4)\right)$ (A.5)
$\partial_{\bar{0}}^{m}\partial_{\bar{0}}^{n}z=2(z^{2}-1)\left(z\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}-\frac{X_{2}^{m}}{X_{2\bar{0}}}\right)\left(\frac{X_{1}^{n}}{X_{1\bar{0}}}-\frac{X_{2}^{n}}{X_{2\bar{0}}}\right)+\left(\frac{X_{1}^{m}X_{1}^{n}}{X_{1\bar{0}}X_{1\bar{0}}}-\frac{X_{2}^{m}X_{2}^{n}}{X_{2\bar{0}}X_{2\bar{0}}}\right)\right)$
(A.6)

These are also anti-symmetric under the coordinate interchange
$1\leftrightarrow 2$. The higher order derivatives on $z$ can be obtained
using the same procedure and also for the $z^{2},z^{3}\text{and}z^{4}$. If the
degree of $z$ is larger than the order of the derivative the ultimate
contribution after setting $\theta=\bar{\theta}=0$ will become zero. The same
will be applicable for w. The derivatives on w are as follows:

$\partial_{\bar{0}}^{m}\textbf{w}=\frac{16\langle\bar{1}2\bar{\Gamma}^{m}1\bar{2}0\rangle}{(X_{1\bar{0}}X_{0\bar{1}}X_{2\bar{0}}X_{0\bar{2}}X_{2\bar{1}}X_{1\bar{2}})^{\frac{1}{2}}}+(\textbf{w}-1)\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}+\frac{X_{2}^{m}}{X_{2\bar{0}}}\right),$
(A.7)

Unlike $z$, w is symmetric under the permutation of $1\leftrightarrow 2$ and
the higher order derivatives are :

$\partial_{\bar{0}}^{2}\textbf{w}=-(\textbf{w}-1)\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}},$
(A.8)
$\partial_{\bar{0}}^{m}\partial_{\bar{0}}^{n}\textbf{w}=\partial_{\bar{0}}^{m}\textbf{w}\left(\frac{X_{1}^{n}}{X_{1\bar{0}}}+\frac{X_{2}^{n}}{X_{2\bar{0}}}\right)+\partial_{\bar{0}}^{n}\textbf{w}\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}+\frac{X_{2}^{m}}{X_{2\bar{0}}}\right)+(\textbf{w}-1)\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}-\frac{X_{2}^{m}}{X_{2\bar{0}}}\right)\left(\frac{X_{1}^{n}}{X_{1\bar{0}}}-\frac{X_{2}^{n}}{X_{2\bar{0}}}\right),$
(A.9)
$\partial_{\bar{0}}^{m}\partial_{\bar{0}}^{2}\textbf{w}=-\partial_{\bar{0}}^{m}\textbf{w}\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}-2(\textbf{w}-1)\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}+\frac{X_{2}^{m}}{X_{2\bar{0}}}\right)$
(A.10)
$\partial_{\bar{0}}^{2}\partial_{\bar{0}}^{2}\textbf{w}=9(\textbf{w}-1)\left(\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}\right)^{2}$
(A.11)

The bare w always gives $0$ after setting $\theta_{ext}=0$. To get the non-
zero values the order of w must be less than or equal to the degree of the
partial derivatives. Setting $\theta_{ext}=0$ we obtain,

$\partial_{\bar{0}}^{m}\textbf{w}|_{\theta_{ext}=0}=-\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}X_{0}^{m}.$
(A.12)

On the other hand, $D_{l}$ is invariant under the exchanges of
$1\leftrightarrow 2$ and $3\leftrightarrow 4$ and the derivatives reads

$\partial_{\bar{0}}^{m}\frac{1}{D_{l}}=\frac{1}{D_{l}}\left[\left(\frac{\Delta+l}{2}\right)\left(\frac{X_{1}^{m}}{X_{1\bar{0}}}+\frac{X_{2}^{m}}{X_{2\bar{0}}}\right)-\left(\frac{\Delta-l-2}{2}\right)\left(\frac{X_{3}^{m}}{X_{3\bar{0}}}+\frac{X_{4}^{m}}{X_{4\bar{0}}}\right)\right],$
(A.13)
$\partial_{\bar{0}}^{2}\frac{1}{D_{l}}=-\frac{1}{D_{l}}\left[\left(\frac{\Delta+l}{2}\right)^{2}\frac{X_{12}}{X_{1\bar{0}}X_{2\bar{0}}}+\left(\frac{\Delta-l}{2}\right)^{2}\frac{X_{34}}{X_{3\bar{0}}X_{4\bar{0}}}-\left(\frac{\Delta+l}{2}\right)\left(\frac{\Delta-l}{2}\right)\left(\Omega_{+}\right)\right]$
(A.14)

where

$\Omega_{+}=\left(\frac{X_{13}}{X_{1\bar{0}}X_{3\bar{0}}}+\text{even
permutations}\right).$

###  A.2 Tensor structures and its derivatives

In this section we will deal with the single order partial derivatives and
second order partial derivatives on tensor structures, because of different
permutations of $\mathcal{N}_{l}$ and $\mathcal{L}_{l}$ the first order
derivatives on the tensor structures will give rise to new tensor structures
[50] which consists P0, P1, P2, P3, R0, R1, R2, R3.
In the equation we can write them as,

$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{10}S234T+X_{20}S134T+X_{10}S243T+X_{20}S143T)$
(A.15)
$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{10}S234T+X_{20}S134T-X_{10}S243T-X_{20}S143T)$
(A.16)
$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{10}S234T-X_{20}S134T+X_{10}S243T-X_{20}S143T)$
(A.17)
$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{10}S234T-X_{20}S134T-X_{10}S243T+X_{20}S143T).$
(A.18)

As per the convention

$\displaystyle P0=(X_{10}S234T+X_{20}S134T+X_{10}S243T+X_{20}S143T)$
$\displaystyle P1=(X_{10}S234T+X_{20}S134T-X_{10}S243T-X_{20}S143T)$
$\displaystyle P2=(X_{10}S234T-X_{20}S134T+X_{10}S243T-X_{20}S143T)$
$\displaystyle P3=(X_{10}S234T-X_{20}S134T-X_{10}S243T+X_{20}S143T)$

Similarly, we reduce the tensors structures to compact forms proportional to
$N_{l}$,

$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{30}S214T+X_{40}S213T+X_{30}S124T+X_{40}S123T)$
(A.19)
$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{30}S214T+X_{40}S213T-X_{30}S124T-X_{40}S123T)$
(A.20)
$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{30}S214T-X_{40}S213T+X_{30}S124T-X_{40}S123T)$
(A.21)
$\displaystyle(\partial_{S}0\partial_{T})^{l}(S21034T)^{l-1}(X_{30}S214T-X_{40}S213T-X_{30}S124T+X_{40}S123T).$
(A.22)

Again as per convention,

$\displaystyle R0=(X_{30}S214T+X_{40}S213T+X_{30}S124T+X_{40}S123T)$
$\displaystyle R1=(X_{30}S214T+X_{40}S213T-X_{30}S124T-X_{40}S123T)$
$\displaystyle R2=(X_{30}S214T-X_{40}S213T+X_{30}S124T-X_{40}S123T)$
$\displaystyle R3=(X_{30}S214T-X_{40}S213T-X_{30}S124T+X_{40}S123T)$

The tensor structure related to $SPT$ and $SRT$ are :

$P_{l}=\frac{1}{l!^{2}}(\partial_{\mathcal{S}}0\partial_{\mathcal{T}})^{l}(S\bar{2}1\bar{0}3\bar{4}T)^{l-1}\times(X_{10}S\bar{2}3\bar{4}T+X_{20}S\bar{1}3\bar{4}T-X_{10}S\bar{2}4\bar{3}T-X_{20}S\bar{1}4\bar{3}T),$
(A.23)

and,

$R_{l}=\frac{1}{l!^{2}}(\partial_{\mathcal{S}}0\partial_{\mathcal{T}})^{l}(S\bar{2}1\bar{0}3\bar{4}T)^{l-1}\times(X_{30}S\bar{2}1\bar{4}T+X_{40}S\bar{2}1\bar{3}T-X_{30}S\bar{1}2\bar{4}T-X_{40}S\bar{1}2\bar{3}T).$
(A.24)

##  Appendix B List of Conformal Integrations

###  B.1 Integration related to $\mathcal{N}_{l}$

In this section we will discuss how to perform the conformal integration. We
will provide the formulas used in this work to evaluate the conformal
integrations.The conformal integration related to $N_{l}$, or Gegenbauer
polynomial $C^{(1)}_{l}(t_{0})$ are given by,

$\int_{M}D^{4}X_{0}\frac{(-1)^{l}C^{(1)}_{l}(t_{0})}{X_{10}^{\frac{\Delta+r}{2}}X_{20}^{\frac{\Delta-r}{2}}X_{30}^{\frac{\tilde{\Delta}+\tilde{r}}{2}}X_{40}^{\frac{\tilde{\Delta}-\tilde{r}}{2}}}=\xi_{\Delta,\tilde{\Delta},\tilde{r},l}\bigg{(}\frac{X_{14}}{X_{34}}\bigg{)}^{\frac{\tilde{r}}{2}}\bigg{(}\frac{X_{24}}{X_{14}}\bigg{)}^{\frac{r}{2}}X_{12}^{-\frac{\Delta}{2}}X_{34}^{-\frac{\tilde{\Delta}}{2}}g_{\Delta,l}^{r,\tilde{r}}(u,v).$

In which $r=\Delta_{1}-\Delta_{2}$,$\tilde{r}=\Delta_{3}-\Delta_{4}$ and

$\xi_{\Delta,\tilde{\Delta},\tilde{r},l}=\frac{\pi^{2}\Gamma(\Delta+l-1)\Gamma(\frac{\Delta-\tilde{r}+l}{2})\Gamma(\frac{\Delta+\tilde{r}+l}{2})}{(2-\Delta)\Gamma(\Delta+l)\Gamma(\frac{\tilde{\Delta}-\tilde{r}+}{2})\Gamma(\frac{\tilde{\Delta}+\tilde{r}+l}{2})}$

The closed form structure of the conformal block
$g^{r,\tilde{r}}_{\Delta,l}(u,v)$ is,

$g^{r,\tilde{r}}_{\Delta,l}(u,v)=\frac{\rho\bar{\rho}}{\rho-\bar{\rho}}[k_{\Delta+l}(\rho)k_{\Delta-l-2}(\bar{\rho})-(\rho\leftrightarrow{\bar{\rho}})]$

and

$k_{\beta}(x)={x^{\frac{\beta}{2}}}_{2}F_{1}\bigg{(}\frac{\beta-r}{2},\frac{\beta+\tilde{r}}{2},\beta,x\bigg{)},$

where the conformal invariants $u,v$ are $\rho\bar{\rho}$ and
$(1-\rho)(1-\bar{\rho})$ respectively.

$\int
D^{4}X_{0}\frac{X_{12}}{X_{10}X_{20}}\frac{N_{l}}{D_{l}}\bigg{|}_{\bar{0}=0}=\frac{2^{-6l}\xi_{\Delta+2,2-\Delta,0,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}g_{\Delta+2,l}$
(B.1) $\int
D^{4}X_{0}\frac{X_{34}}{X_{30}X_{40}}\frac{N_{l}}{D_{l}}\bigg{|}_{\bar{0}=0}=\frac{2^{-6l}\xi_{\Delta,4-\Delta,0,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}g_{\Delta,l}$
(B.2) $\int
D^{4}X_{0}\frac{N_{l-1}}{D_{l}}\bigg{|}_{\bar{0}=0}=\frac{2^{-6(l-1)}\xi_{\Delta+1,3-\Delta,0,l-1}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}g_{\Delta+1,l-1}$
(B.3)

####

Some conformal integrations related to $\Omega_{x}$: Before listing out the
integration with $\Omega_{x}$ $(x\in+,-,A,B)$, we mention

$\Omega_{+}=\frac{X_{13}}{X_{10}X_{30}}+\frac{X_{23}}{X_{20}X_{30}}+\frac{X_{14}}{X_{10}X_{40}}+\frac{X_{24}}{X_{20}X_{40}}$
(B.4)
$\Omega_{-}=\frac{X_{13}}{X_{10}X_{30}}-\frac{X_{23}}{X_{20}X_{30}}-\frac{X_{14}}{X_{10}X_{40}}+\frac{X_{24}}{X_{20}X_{40}}$
(B.5)
$\Omega_{A}=\frac{X_{13}}{X_{10}X_{30}}-\frac{X_{23}}{X_{20}X_{30}}+\frac{X_{14}}{X_{10}X_{40}}-\frac{X_{24}}{X_{20}X_{40}}$
(B.6)
$\Omega_{A}=\frac{X_{13}}{X_{10}X_{30}}+\frac{X_{23}}{X_{20}X_{30}}-\frac{X_{14}}{X_{10}X_{40}}-\frac{X_{24}}{X_{20}X_{40}}$
(B.7)

$\Omega_{x}$ are anti-symmetric under coordinate exchange of $1\leftrightarrow
2$ and $3\leftrightarrow 4$. The integrations are listed below.

$\int D^{4}X_{0}\frac{X_{12}}{X_{10}X_{20}}\Omega_{A}\frac{N_{l}}{D_{l}}=\int
D^{4}X_{0}\frac{X_{12}}{X_{10}X_{20}}\Omega_{B}\frac{N_{l}}{D_{l}}=0$ (B.8)
$\int D^{4}X_{0}\frac{X_{34}}{X_{30}X_{40}}\Omega_{A}\frac{N_{l}}{D_{l}}=\int
D^{4}X_{0}\frac{X_{34}}{X_{30}X_{40}}\Omega_{B}\frac{N_{l}}{D_{l}}=0$ (B.9)
$\int
D^{4}X_{0}\Omega_{+}\frac{N_{l}}{D_{l}}=\frac{2^{-6l}\xi_{\Delta+1,3-\Delta,1,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}\bigg{[}4g_{\Delta,l}+\frac{(\Delta+l)(\Delta-l-2)}{4(\Delta+l+1)(\Delta-l-1)}g_{\Delta+2,l}\bigg{]}$
(B.10) $\int
D^{4}X_{0}\Omega_{-}\frac{N_{l}}{D_{l}}=\frac{2^{-6l}\xi_{\Delta+1,3-\Delta,1,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}\bigg{[}\frac{\Delta+l}{\Delta+l+1}g_{\Delta+1,l+1}+\frac{(\Delta-l-2)}{(\Delta-l-1)}g_{\Delta+1,l-1}\bigg{]}$
(B.11) $\int
D^{4}X_{0}\Omega^{2}_{-}\frac{N_{l}}{D_{l}}=\frac{2^{-6l}\xi_{\Delta+2,2-\Delta,2,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}\left(\frac{16((\Delta-l)(\Delta+l)-2l)}{(\Delta+l+2)(\Delta-l)}g_{\Delta,l}+\frac{2(\Delta-l-2)(\Delta+l)}{(\Delta+l+2)(\Delta-l)}g_{\Delta+2,l}\\\
+\frac{((\Delta-l)(\Delta+l)-2l)(\Delta-l-2)(\Delta+l)}{16(\Delta+l+3)(\Delta+l+1)(\Delta-l+1)(\Delta-l-1)}g_{\Delta+4,l}-\frac{\Delta+l}{(\Delta+l+3)(\Delta-l)(\Delta+l+1)}g_{\Delta+2,l+2}\\\
-\frac{(\Delta-l-2)}{(\Delta-l-1)(\Delta-l+1)(\Delta+l+2)}g_{\Delta+2,l-2}\right)$
(B.12) $\int
D^{4}X_{0}\Omega_{-}^{2}\frac{N_{l}}{D_{l}}=\frac{2^{-6l}\xi_{\Delta+2,2-\Delta,2,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}\left(\frac{2(\Delta-l-2)(\Delta+l)}{(\Delta+l+2)(\Delta-l)}g_{\Delta+2,l}\\\
+\frac{(\Delta+l)(\Delta-l-1)}{(\Delta+l+3)(\Delta-l)}g_{\Delta+2,l+2}+\frac{(\Delta-l-2)(\Delta+l+1)}{(\Delta-l+1)(\Delta+l+2)}g_{\Delta+2,l-2}\right)$
(B.13) $\int
D^{4}X_{0}\Omega_{A}^{2}\frac{N_{l}}{D_{l}}=\frac{2^{-6l}\xi_{\Delta+2,2-\Delta,2,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}\left(-\frac{\Delta(\Delta-l-2)(\Delta+l)}{8(\Delta+l+3)(\Delta+l+1)(\Delta-l+1)(\Delta-l-1)}g_{\Delta+4,l}+\frac{\Delta+l}{(\Delta+l+3)(\Delta-l)}g_{\Delta+2,l+2}+\frac{\Delta-l-2}{(\Delta-l+1)(\Delta+l+2)}g_{\Delta+2,l-2}\right)$
(B.14) $\int
D^{4}X_{0}\Omega_{B}^{2}\frac{N_{l}}{D_{l}}=\frac{2^{-6l}\xi_{\Delta+2,2-\Delta,2,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}\left(\frac{32\Delta}{(\Delta+l+2)(\Delta-l)}g_{\Delta,l}-\frac{(\Delta-l-1)(\Delta+l)}{(\Delta+l+3)(\Delta-l)(\Delta+l+1)}g_{\Delta+2,l+2}-\frac{(\Delta-l-2)(\Delta+l+1)}{(\Delta-l-1)(\Delta-l+1)(\Delta+l+2)}g_{\Delta+2,l-2}\right)$
(B.15) $\int
D^{4}X_{0}(\Omega_{A}\Omega_{B}+\Omega_{+}\Omega_{-})\frac{N_{l}}{D_{l}}=\frac{2^{-6l}\xi_{\Delta+2,2-\Delta,2,l}}{X^{\frac{\Delta-l}{2}}_{12}X^{-\frac{\Delta+l}{2}}_{34}}\left(\frac{4(\Delta+l)}{\Delta+l+2}g_{\Delta+1,l+1}+4\frac{\Delta-l-2}{\Delta-l}g_{\Delta+1,l-1}+\frac{(\Delta-l-2)(\Delta+l)^{2}}{4(\Delta-l)(\Delta+l+1)(\Delta+l+3)}g_{\Delta+3,l+1}+\frac{(\Delta-l-2)^{2}(\Delta+l)}{4(\Delta-l+1)(\Delta-l-1)(\Delta+l+2)}g_{\Delta+3,l-1}\right)$
(B.16)

###  B.2 Conformal integration with (P0, P1, P2, P3) and (R0, R1, R2, R3)

For $\mathcal{N}=2$ theories the action of double derivatives
$\partial_{0}^{n}\partial_{0}^{m}$ on the tensor structures will lead to
higher order tensor structures. For higher order tensor structures, conformal
integrations are not very straightforward. One way to simplify the higher
order tensor structures is to reduce them to first order tensor structures,
wherever possible. In fact, one can show that $P3=R3=8S21034T$, while $P2$ and
$R2=0$ vanish (see [50] for $P0$, $R0$ and other details). The tensor
structures $P1R1$, $P1^{2}$ and $R1^{2}$ cannot be reduced further. We have
closely followed [25] for computing our results and mostly the integrations
are available in the appendix of the above mentioned paper.

$\int
D^{4}X_{0}\frac{X_{12}}{X_{10}X_{20}}\Omega_{A}\frac{P_{l}}{D_{l}}=\frac{2^{3-6l}\xi_{\Delta+3,1-\Delta,0,l-1}}{X_{12}^{\frac{1}{2}(\Delta-l)}X_{34}^{-\frac{1}{2}(\Delta+l)}}\left(\frac{4(\Delta-l+1)}{\Delta
l(l-\Delta)}g_{\Delta+2,l}-\frac{4(\Delta+1)(l+1)}{\Delta
l(\Delta+l)}g_{\Delta+2,l-2}\\\
+\frac{\Delta+l+2}{4(\Delta+l+1)(\Delta+l+3)}g_{\Delta+4,l}\right)$ (B.17)
$\int
D^{4}X_{0}\frac{X_{12}}{X_{10}X_{20}}\Omega_{B}\frac{P_{l}}{D_{l}}=\frac{2^{3-6l}\xi_{\Delta+3,1-\Delta,0,l-1}}{X_{12}^{\frac{1}{2}(\Delta-l)}X_{34}^{-\frac{1}{2}(\Delta+l)}}\left(-\frac{1}{\Delta
l}g_{\Delta+3,l-1}-\frac{16(\Delta+1)(l+1)(\Delta-l+1)}{\Delta
l(\Delta+l)(\Delta-l)}g_{\Delta+1,l-1}\\\
+\frac{(\Delta+l+2)(\Delta-l+1)}{(\Delta-l)(\Delta+l+1)(\Delta+l+3)}g_{\Delta+3,l+1}\right)$
(B.18) $\int
D^{4}X_{0}\frac{X_{34}}{X_{30}X_{40}}\Omega_{A}\frac{R_{l}}{D_{l}}=\frac{2^{3-6l}\xi_{\Delta+1,3-\Delta,0,l-1}}{X_{12}^{\frac{1}{2}(\Delta-l)}X_{34}^{-\frac{1}{2}(\Delta+l)}}\left(-\frac{\Delta+l}{(\Delta+l+1)(\Delta-l-2)}g_{\Delta+1,l+1}\\\
+\frac{1}{\Delta l}g_{\Delta+1,l-1}\\\
+\frac{(\Delta-1)(l+1)(\Delta+l)(\Delta-l)}{16\Delta
l(\Delta+l+1)(\Delta-l-1)(\Delta-l+1)}g_{\Delta+3,l-1}\right)$ (B.19) $\int
D^{4}X_{0}\frac{X_{34}}{X_{30}X_{40}}\Omega_{B}\frac{R_{l}}{D_{l}}=\frac{2^{3-6l}\xi_{\Delta+1,3-\Delta,0,l-1}}{X_{12}^{\frac{1}{2}(\Delta-l)}X_{34}^{-\frac{1}{2}(\Delta+l)}}\left(-\frac{4}{\Delta-l-2}g_{\Delta,l}+\frac{\Delta+l}{4\Delta
l(\Delta+l+1)}g_{\Delta+2,l}\\\ +\frac{(\Delta-1)(l+1)(\Delta-l)}{4\Delta
l(\Delta-l-1)(\Delta-l+1)}g_{\Delta+2,l-2}\right)$ (B.20)

The conformal integration involving $P1^{2}$ and $R1^{2}$ are[25]

$\int
D^{4}X_{0}\left(\frac{X_{12}}{X_{10}X_{20}}\right)^{2}\frac{1}{D_{l}}\frac{l(l-1)}{(l!)^{2}}(\partial_{S}0\partial_{T})^{l}(S\bar{2}1\bar{0}3\bar{4}T)^{l-2}P1^{2}=\frac{2^{6(l-1)}\xi_{\Delta+2,2-\Delta,0,l-2}}{X_{12}^{\frac{1}{2}(\Delta-l)}X_{34}^{-\frac{1}{2}(\Delta+l)}}\left(\frac{l(l+1)}{\Delta+l+1}g_{\Delta+2,l+2}\\\
-\frac{l(l-1)\Delta(\Delta+l)(\Delta+l+2)}{16(\Delta+2)(\Delta+l+1)^{2}(\Delta+l+3)}g_{\Delta+4,l}\right)$
(B.21) $\int
D^{4}X_{0}\left(\frac{X_{34}}{X_{30}X_{40}}\right)^{2}\frac{1}{D_{l}}\frac{l(l-1)}{(l!)^{2}}(\partial_{S}0\partial_{T})^{l}(S\bar{2}1\bar{0}3\bar{4}T)^{l-2}R1^{2}=\frac{2^{6(l-1)}\xi_{\Delta+2,2-\Delta,0,l-2}}{X_{12}^{\frac{1}{2}(\Delta-l)}X_{34}^{-\frac{1}{2}(\Delta+l)}}\left(l(l-1)\frac{\Delta+l+2}{(\Delta-l-2)(\Delta+l-1)}g_{\Delta,l}-l(l+1)\frac{(\Delta-2)(\Delta-l)(\Delta+l-2)}{16\Delta((l-\Delta)^{2}-1)(\Delta+l-1)}g_{\Delta+2,l-2}\right)$
(B.22)

The conformal integrations related to the tensor structures $P1\times R1$ will
vanish.

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