June, 2021

Three-point functions of higher-spin spinor current multiplets in
${\mathcal{N}}=1$ superconformal theory

Evgeny I. Buchbinder, Jessica Hutomo and Sergei M. Kuzenko

Department of Physics M013, The University of Western Australia
35 Stirling Highway, Crawley W.A. 6009, Australia

Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
In this paper, we study the general form of three-point functions of conserved
current multiplets $S_{\alpha(k)}=S_{(\alpha_{1}\dots\alpha_{k})}$ of
arbitrary rank in four-dimensional ${\mathcal{N}}=1$ superconformal theory. We
find that the correlation function of three such operators
$\langle\bar{S}_{\dot{\alpha}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{\dot{\gamma}(l)}(z_{3})\rangle$
is fixed by the superconformal symmetry up to a single complex coefficient
though the precise form of the correlator depends on the values of $k$ and
$l$. In addition, we present the general structure of mixed correlators of the
form
$\langle\bar{S}_{\dot{\alpha}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3})\rangle$
and
$\langle\bar{S}_{\dot{\alpha}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma\dot{\gamma}}(z_{3})\rangle$,
where $L$ is the flavour current multiplet and $J_{\gamma\dot{\gamma}}$ is the
supercurrent.

###### Contents

  1. 1 Introduction
  2. 2 Superconformal building blocks
    1. 2.1 Infinitesimal superconformal transformations
    2. 2.2 Two-point structures
    3. 2.3 Three-point structures
    4. 2.4 Correlation functions of primary superfields
  3. 3 Correlator ${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}$
  4. 4 Correlator ${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}$
  5. 5 Correlator ${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}$

##  1 Introduction

It is well known that in (super)conformal field theory the general form of
two- and three-point functions of conserved currents is fully determined by
the (super)conformal symmetry and conservation laws up to finitely many
independent coefficients. In the non-supersymmetric case, the systematic study
of the two- and three-point functions of conserved currents was presented in
[1, 2] building on the earlier results obtained in Refs. [3, 4, 5, 6, 7, 8, 9,
10, 11, 12].111Parity violating structures in three-dimensional conformal
field theory were not considered in [1, 2]. They were later found in [13]. The
approach of [1, 2] was later extended to superconformal field theories in
diverse dimensions in [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

The most important conserved currents in conformal field theory are the
energy-momentum tensor and vector currents. In the supersymmetric case, they
are embedded in special multiplet of conserved currents. The energy-momentum
tensor is, therefore, replaced with the supercurrent [25] (see also [26, 27,
28, 29, 30]) and vector currents are replaced with flavour current multiplets
[31]. Three-point functions of these current multiplets have been extensively
studied. In general, (super)conformal field theories also possess higher-spin
conserved currents. In the case of three-dimensional conformal field theory,
it was proven (under certain assumptions) by Maldacena and Zhiboedov in [32]
that all correlation functions of higher-spin currents are equal to the ones
in a free theory. The theorem of Maldacena and Zhiboedov was later generalised
by Stanev [33] and by Alba and Diab [34, 35] to the four- (and higher-)
dimensional case. One can view these results as the analog of the Coleman-
Mandula theorem [36] for conformal field theories.

We believe that the analysis of [32, 33, 34, 35] has some limitations. First,
the authors of [32, 33, 34, 35] considered only bosonic symmetric traceless
currents. However, in supersymmetric theories conserved currents form
supermultiplets consisting of both bosonic and fermionic component currents.
Second, the results of [32, 33, 34, 35] are proven under certain assumptions,
the main being the existence of only one conserved current of spin two which
is the energy-momentum tensor. In [32] it was shown that in three-dimensional
conformal field theory the existence of a half-integer higher-spin conserved
current implies the existence of one more conserved current of spin two. This
means that in supersymmetric conformal field theory possessing higher-spin
currents the assumptions of [32] might be violated. It is likely that the same
conclusion also holds in four dimensions. Therefore, in the supersymmetric
case it is unclear if higher-spin current multiplets exist only in free
theory. In any case, regardless whether a theory is free or not, it is
interesting to understand the general structure of their correlation
functions.

In the non-supersymmetric case, the general structure of the three-point
functions of conserved bosonic, vector currents of arbitrary spin was
determined by Stanev [37] and Zhiboedov [38], see also [39] for similar
results in the embedding formalism. The aim of this paper is to make first
steps towards determining the three-point functions of conserved higher-spin
currents in the ${\cal N}=1$ supersymmetric case in four dimensions. In fact,
we will be interested in multiplets which do not contain vector currents. To
start with, it is worth reminding the reader of the known conformal current
multiplets.

In the case of ${\cal N}=1$ supersymmetry, there are three types of conformal
current multiplets depending on the corresponding superfield Lorentz type
$(k/2,l/2)$. These multiplets were explicitly described in flat [40] and
curved [41] backgrounds.

  * •

Given two positive integers $k$ and $l$, the conformal current multiplet
$J_{\alpha(k){\dot{\alpha}}(l)}:=J_{\alpha_{1}\dots\alpha_{k}{\dot{\alpha}}_{1}\dots\dot{\alpha}_{l}}=J_{(\alpha_{1}\dots\alpha_{k})({\dot{\alpha}}_{1}\dots\dot{\alpha}_{l})}$
obeys the conservation equations

$\displaystyle
D^{\beta}J_{\alpha(k-1)\beta{\dot{\alpha}}(l)}=0~{},\qquad\bar{D}^{\dot{\beta}}J_{\alpha(k){\dot{\alpha}}(l-1){\dot{\beta}}}=0~{}.$
(1.1)

This superfield has dimension $\big{(}2+\frac{1}{2}(k+l)\big{)}$ and its
$\mathsf{U}(1)_{R}$ charge is equal to $\frac{1}{3}(k-l)$, see [41] for the
technical details. If $k=l$, the supercurrent $J_{\alpha(k){\dot{\alpha}}(k)}$
is neutral with respect to the $R$-symmetry group $\mathsf{U}(1)_{R}$, and
therefore it is consistent to restrict $J_{\alpha(k){\dot{\alpha}}(k)}$ to be
real. The $k=l=1$ case corresponds to the ordinary conformal supercurrent
[25]. The case $k=l>1$ was first discussed in [42].

  * •

If $k>0$ and $l=0$, the conformal current multiplet $S_{\alpha(k)}$ obeys the
the conservation equation

$\displaystyle
D^{\beta}S_{\beta\alpha_{1}\dots\alpha_{k-1}}=0\,,\qquad\bar{D}^{2}S_{\alpha(k)}=0~{}.$
(1.2)

The case $k=1$ was first considered in [19], where it was shown that the
spinor supercurrent $S_{\alpha}$ naturally originates from the reduction of
the conformal ${\cal N}=2$ supercurrent [43] to ${\cal N}=1$ superspace.

  * •

Finally, the $k=l=0$ case corresponds to the flavour current multiplet [31],
$L=\bar{L}$, constrained by

$\displaystyle D^{2}L=0~{},\qquad\bar{D}^{2}L=0~{}.$ (1.3)

Do all of these multiplets occur in superconformal field theories?

The conformal supercurrent $J_{\alpha{\dot{\alpha}}}$ exists in every ${\cal
N}=1$ superconformal field theory. Flavour current multiplets exists in every
superconformal field theory possessing an internal symmetry Lie group. The
spinor current multiplet $S_{\alpha}$ exists in every ${\cal N}=2$
superconformal field theory realised in terms of ${\cal N}=1$ superfields.
Explicit realisations in terms of free conformal scalar multiplets are known
for the conformal current higher-spin multiplets
$J_{\alpha(k){\dot{\alpha}}(k)}$ and $J_{\alpha(k+1){\dot{\alpha}}(k)}$, with
$k>1$, both in Minkowski [44] and anti-de Sitter [45] superspace.

Higher-spin current multiplets $J_{\alpha(k){\dot{\alpha}}(l)}$ and
$S_{\alpha(k)}$ may be realised in terms of the on-shell chiral field
strengths $W_{\alpha(k)}$ and massless antichiral scalar $\bar{\Phi}$
constrained as

$\displaystyle\bar{D}_{\dot{\beta}}W_{\alpha(k)}$ $\displaystyle=$
$\displaystyle 0~{},\qquad D^{\beta}W_{\beta\alpha(k-1)}=0~{},$ (1.4a)
$\displaystyle D_{\beta}\bar{\Phi}$ $\displaystyle=$ $\displaystyle
0~{},\qquad\bar{D}^{2}\bar{\Phi}=0~{}.$ (1.4b)

These realisations are as follows [45, 46, 47]:

$\displaystyle
J_{\alpha(k){\dot{\alpha}}(l)}=W_{\alpha(k)}\bar{W}_{{\dot{\alpha}}(l)}~{},\qquad
S_{\alpha(k)}=W_{\alpha(k)}\bar{\Phi}~{}.$ (1.5)

Here $W_{\alpha}$, $W_{\alpha(2)}$ and $W_{\alpha(3)}$ are the gauge-invariant
field strength describing the on-shell vector, gravitino and linearised
supergravity multiplets, respectively. The chiral superfields $W_{\alpha(k)}$
for $k>3$ are the on-shell gauge-invariant field strengths corresponding to
the massless higher-spin gauge multiplets [48, 49] (see section 6.9 in [50]
for a review). Choosing $k=l=1$ in (1.5) gives the supercurrent of the free
${\cal N}=1$ vector multiplet,
$J_{\alpha{\dot{\alpha}}}=W_{\alpha}\bar{W}_{{\dot{\alpha}}}$, and the spinor
supercurrent of the free ${\cal N}=2$ vector multiplet,
$S_{\alpha}=W_{\alpha}\bar{\Phi}$.

The off-shell gauge-invariant models for massless higher-spin multiplets
proposed in [48, 49] are not superconformal, although the corresponding on-
shell field strengths $W_{\alpha(k)}$ constrained by (1.4a) furnish
irreducible representations of the superconformal group, see [50] for the
technical details. At the moment we are not aware of explicit realisations of
the current multiplets $S_{\alpha(k)}$ for $k>1$ in superconformal field
theories. Nevertheless, it is of interest to understand the most general
structure of three-point correlation functions of such current multiplets,
which are compatible with the superconformal symmetry and conservation laws.

In this paper we will restrict ourselves to four-dimensional ${\cal N}=1$
superconformal field theory and to higher-spin current multiplets carrying
only undotted or dotted indices, $S_{\alpha(k)}$ and its conjugate
${\bar{S}_{{\dot{\alpha}}(k)}}$. We will refer to them as “higher-spin spinor
current multiplets”. For $k=1$ the multiplet $S_{\alpha}$ is just a spinor
current multiplet. This case was studied in our previous work [51]. Three-
point correlation functions of more general higher-spin current multiplets
require additional study and will be explored elsewhere.

The main result of this paper is the most general structure of the three-point
function

${{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}$
(1.6)

for arbitrary integers $k$ and $\ell$. We showed that in all cases eq. (1.6)
is fixed by the superconformal symmetry and the conservation equations (1.2)
up to a single overall coefficient. However, we found that one has to
distinguish two cases depending on values of $k$ and $\ell$, see section 5 for
details. Additionally, we also studied the mixed correlators

${{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}}\,,\qquad{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}~{},$
(1.7)

where $L$ is the flavour current multiplet and $J_{\gamma{\dot{\gamma}}}$ is
the supercurrent.

The paper is organised as follows. A brief review of the two- and three-point
building blocks for correlation functions is given in section 2. In section 3
we study a three-point function of higher-spin spinor current multiplets with
the flavour current multiplet. We show that this three-point function is fixed
by the superconformal symmetry up to two independent real coefficients. In the
special case $k=1$, our result coincides with that found in our previous work
[51]. In section 4 we consider a three-point function of higher-spin spinor
current multiplets with the supercurrent. In this case the general form of the
correlation function is fixed up to three independent real coefficients. In
the special case $k=1$, our result coincides with that found in [51]. Finally,
in section 5 we compute three-point correlator involving just the conserved
higher-spin spinor current multiplets with arbitrary number of dotted and
undotted indices. We found that its general structure is fixed up to a single
complex coefficient.

##  2 Superconformal building blocks

This section contains a concise summary of two and three-point superconformal
building blocks in 4D ${\cal N}=1$ superspace, which are important for our
subsequent analysis. These superconformal structures were introduced in [14,
15], and later generalised to arbitrary ${{\cal N}}$ in [16] (see also [19]
for a review). A review of the general structure of two- and three-point
correlation functions of primary operators is also given. The presentation of
this section closely follows [51]. Our 4D notation and conventions are those
of [50].

###  2.1 Infinitesimal superconformal transformations

We denote the ${\cal N}=1$ Minkowski superspace by ${\mathbb{M}}^{4|4}$. It is
parametrised by the local coordinates
$z^{A}=(x^{a},\theta^{\alpha},{\bar{\theta}}_{\dot{\alpha}})$, where
$a=0,1,\cdots 3,~{}\alpha,{\dot{\alpha}}=1,2$. Infinitesimal superconformal
transformations

$\delta z^{A}=\xi z^{A}~{},$ (2.1)

are generated by conformal Killing real supervector fields [50, 19]

$\xi={\overline{\xi}}=\xi^{a}(z)\partial_{a}+\xi^{\alpha}(z)D_{\alpha}+{\bar{\xi}}_{\dot{\alpha}}(z){\bar{D}}^{\dot{\alpha}}$
(2.2)

obeying the equation

$[\xi\,,\,D_{\alpha}]~{}\propto~{}D_{\beta}~{}.$ (2.3)

As a result, the spinor parameters are expressed in terms of the vector ones

$\xi^{\alpha}=-\frac{\rm
i}{8}{\bar{D}}_{{\dot{\beta}}}\xi^{{\dot{\beta}}\alpha}\;,\qquad{\bar{D}}_{{\dot{\beta}}}\xi^{\alpha}=0~{}.$
(2.4)

The latter satisfy

$D_{(\alpha}\xi_{\beta){\dot{\beta}}}={\bar{D}}_{({\dot{\alpha}}}\xi_{\beta{\dot{\beta}})}=0~{},$
(2.5)

which leads to the standard conformal Killing equation

$\partial_{a}\xi_{b}+\partial_{b}\xi_{a}=\frac{1}{2}\,\eta_{ab}\,\partial_{c}\xi^{c}\;.$
(2.6)

The general solution to eq. (2.5) was given in [50] for ${\cal N}=1$ and in
[16] for ${\cal N}>1$. These conformal Killing supervector fields span a Lie
superalgebra which is isomorphic to $\rm su(2,2|1)$. For the purpose of this
paper, it suffices to consider the relation

$[\xi\;,\;D_{\alpha}]=-(D_{\alpha}\xi^{\beta})D_{\beta}=\hat{\omega}_{\alpha}{}^{\beta}(z)D_{\beta}-\left(2{\bar{\sigma}}(z)-\sigma(z)\right)D_{\alpha}~{}.$
(2.7)

The superfield parameters $\hat{\omega}_{\alpha\beta}(z)$ and $\sigma(z)$ are
expressed in terms of
$\xi^{A}=(\xi^{a},\xi^{\alpha},\bar{\xi}_{{\dot{\alpha}}})$ as follows

$\displaystyle\hat{\omega}_{\alpha\beta}(z)=-D_{(\alpha}\xi_{\beta)}\;,\qquad\sigma(z)=\frac{1}{6}\big{(}D_{\alpha}\xi^{\alpha}+2{\bar{D}}^{\dot{\alpha}}{\bar{\xi}}_{{\dot{\alpha}}}\big{)}$
(2.8)

and can be explicitly found using the components of the conformal Killing
supervector, see Refs. [50, 16, 19] for detail. Due to their action on the
covariant derivative (2.7), the $z$-dependent parameters
$\hat{\omega}_{\alpha\beta}(z)$ and $\sigma(z)$ can be thought of as the
parameters of special local Lorentz and scale transformations, respectively.
These $z$-dependent parameters, along with $\xi$, appear in the superconformal
transformation law of a primary tensor superfield, see subsection 2.4.

###  2.2 Two-point structures

Let $z_{1}$ and $z_{2}$ be two different points in superspace. In 4D
superconformal theories, all building blocks for the two- and three-point
correlation functions are composed of the two-point structures:

$\displaystyle x^{a}_{\bar{1}2}$ $\displaystyle=$
$\displaystyle-x^{a}_{2\bar{1}}=x^{a}_{1-}-x^{a}_{2+}+2{\rm
i}\,\theta_{2}\sigma^{a}{\bar{\theta}}_{1}~{},$ (2.9a)
$\displaystyle\theta_{12}$ $\displaystyle=$
$\displaystyle\theta_{1}-\theta_{2}~{},\qquad{\bar{\theta}}_{12}={\bar{\theta}}_{1}-{\bar{\theta}}_{2}~{},$
(2.9b)

where $x^{a}_{\pm}=x^{a}\pm{\rm i}\theta\sigma^{a}{\bar{\theta}}$. In spinor
notation, we write

$\displaystyle({x}_{\bar{1}2})^{{\dot{\alpha}}\alpha}$ $\displaystyle=$
$\displaystyle(\tilde{\sigma_{a}})^{{\dot{\alpha}}\alpha}x^{a}_{\bar{1}2}~{},$
(2.10a) $\displaystyle({x}_{2\bar{1}})_{\alpha{\dot{\alpha}}}$
$\displaystyle=$
$\displaystyle(\sigma_{a})_{\alpha{\dot{\alpha}}}\,x^{a}_{2\bar{1}}=-(\sigma_{a})_{\alpha{\dot{\alpha}}}\,x^{a}_{\bar{1}2}=-\varepsilon_{\alpha\beta}\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}({x}_{\bar{1}2})^{{\dot{\beta}}\beta}~{},$
(2.10b) $\displaystyle({x}_{\bar{1}2}^{\,\dagger})^{{\dot{\alpha}}\alpha}$
$\displaystyle=$ $\displaystyle-({x}_{\bar{2}1})^{{\dot{\alpha}}\alpha}~{}.$
(2.10c)

Note that
$({x}_{\bar{1}2})^{{\dot{\alpha}}\alpha}(x_{2\bar{1}})_{\alpha{\dot{\beta}}}=x_{\bar{1}2}{}^{2}\,\delta^{{\dot{\alpha}}}{}_{{\dot{\beta}}}$.
We sometimes employ matrix-like conventions of [15, 19] where the spinor
indices are not explicitly written:

$\displaystyle\psi=(\psi^{\alpha})~{},\quad\tilde{\psi}=(\psi_{\alpha})~{},\quad\bar{\psi}=(\bar{\psi}^{{\dot{\alpha}}})~{},\quad\tilde{\bar{\psi}}=(\bar{\psi}_{{\dot{\alpha}}})~{},$
(2.11a) $\displaystyle
x=(x_{\alpha{\dot{\alpha}}})~{},\quad\tilde{x}=(x^{{\dot{\alpha}}\alpha})~{}.$
(2.11b)

Since $x^{2}\equiv x^{a}x_{a}=-\frac{1}{2}{\rm tr}(\tilde{x}x)$, it follows
that $\tilde{x}^{-1}=-x/x^{2}$. The notation ’$\tilde{x}_{\bar{1}2}$’ means
that $\tilde{x}_{\bar{1}2}$ is antichiral with respect to $z_{1}$ and chiral
with respect to $z_{2}$. That is,

$D_{(1)\alpha}\tilde{x}_{\bar{1}2}=0\,,\qquad\bar{D}_{(2)\bar{\alpha}}\tilde{x}_{\bar{1}2}=0~{},$
(2.12)

where $D_{(1)\alpha}$ and $\bar{D}_{(1){\dot{\alpha}}}$ are the superspace
covariant spinor derivatives acting on the point $z_{1}$. Similarly,
$D_{(2)\alpha}$ and $\bar{D}_{(2){\dot{\alpha}}}$ act on the point $z_{2}$.
Explicitly, they are given by

$\displaystyle D_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}+{\rm
i}(\sigma^{a})_{\alpha{\dot{\alpha}}}\bar{\theta}^{{\dot{\alpha}}}\frac{\partial}{\partial
x^{a}}~{},\qquad\bar{D}_{{\dot{\alpha}}}=-\frac{\partial}{\partial\bar{\theta}^{{\dot{\alpha}}}}-{\rm
i}{\theta}^{\alpha}(\sigma^{a})_{\alpha{\dot{\alpha}}}\frac{\partial}{\partial
x^{a}}~{}.$ (2.13)

The superconformal transformation laws of the two-point structures are given
by

$\displaystyle\delta x^{{\dot{\alpha}}\alpha}_{{\bar{1}}2}$ $\displaystyle=$
$\displaystyle-\Big{(}\hat{{\bar{\omega}}}^{\dot{\alpha}}{}_{\dot{\beta}}(z_{1})-\delta^{\dot{\alpha}}{}_{\dot{\beta}}\,{\bar{\sigma}}(z_{1})\Big{)}x^{{\dot{\beta}}\alpha}_{{\bar{1}}2}-x^{{\dot{\alpha}}\beta}_{{\bar{1}}2}\Big{(}\hat{\omega}_{\beta}{}^{\alpha}(z_{2})-\delta_{\beta}{}^{\alpha}\,\sigma(z_{2})\Big{)}$
(2.14a) $\displaystyle\delta\theta^{\alpha}_{12}$ $\displaystyle=$
$\displaystyle
2\left({\bar{\sigma}}(z_{1})-\sigma(z_{1})\right)\theta^{\alpha}_{12}-{\rm
i}\,\hat{{\bar{\eta}}}_{{\dot{\beta}}}(z_{1})x^{{\dot{\beta}}\alpha}_{{\bar{1}}2}$
(2.14b)
$\displaystyle-\theta^{\beta}_{12}\left(\hat{\omega}_{\beta}{}^{\alpha}(z_{2})-\delta_{\beta}{}^{\alpha}\,\sigma(z_{2})\right)~{},$

with $\hat{\eta}_{\alpha}{}(z):=\frac{1}{2}D_{\alpha}\sigma(z)$. Let us define

$\displaystyle{\cal
I}_{\alpha{\dot{\alpha}}}(x_{2\bar{1}})=\frac{(x_{2\bar{1}})_{\alpha{\dot{\alpha}}}}{(x_{\bar{1}2}{}^{2})^{\frac{1}{2}}}\in{\rm
SL(2,\mathbb{C})}~{}.$ (2.15)

With the aid of (2.14a), it follows that $x_{\bar{1}2}{}^{2}$ and ${\cal
I}_{\alpha{\dot{\alpha}}}(x_{2\bar{1}})$ transform covariantly under
superconformal transformations:

$\displaystyle\delta x_{\bar{1}2}{}^{2}$ $\displaystyle=$ $\displaystyle
2\big{(}\bar{\sigma}(z_{1})+\sigma(z_{2})\big{)}x_{\bar{1}2}{}^{2}~{},$
(2.16a) $\displaystyle\delta{\cal I}_{\alpha{\dot{\alpha}}}(x_{2\bar{1}})$
$\displaystyle=$ $\displaystyle{\cal
I}_{\alpha{\dot{\gamma}}}(x_{2\bar{1}})\,\hat{\bar{\omega}}^{{\dot{\gamma}}}{}_{{\dot{\alpha}}}(z_{1})+\hat{\omega}^{\gamma}{}_{\alpha}(z_{2})\,{\cal
I}_{\gamma{\dot{\alpha}}}(x_{2\bar{1}})~{}.$ (2.16b)

We also note several useful differential identities:

$\displaystyle D_{(1)\alpha}(x_{\bar{2}1})^{{\dot{\beta}}\beta}=4{\rm
i}\delta_{\alpha}{}^{\beta}\bar{\theta}_{12}^{{\dot{\beta}}}~{},\qquad\bar{D}_{(1){\dot{\alpha}}}\,(x_{\bar{1}2})^{{\dot{\beta}}\beta}=4{\rm
i}\delta_{{\dot{\alpha}}}{}^{{\dot{\beta}}}\theta_{12}^{\beta}~{},$ (2.17a)
$\displaystyle
D_{(1)\alpha}\bigg{(}\frac{1}{x_{\bar{2}1}{}^{2}}\bigg{)}=-\frac{4{\rm
i}}{x_{\bar{2}1}{}^{2}}(\tilde{x}_{\bar{2}1}{}^{-1})_{\alpha{\dot{\beta}}}\bar{\theta}_{12}^{{\dot{\beta}}}~{},\quad\bar{D}_{(1){\dot{\alpha}}}\bigg{(}\frac{1}{x_{\bar{1}2}{}^{2}}\bigg{)}=-\frac{4{\rm
i}}{x_{\bar{1}2}{}^{2}}(\tilde{x}_{\bar{1}2}{}^{-1})_{\alpha{\dot{\alpha}}}{\theta}_{12}^{\alpha}~{},~{}~{}~{}~{}~{}$
(2.17b)

Here and throughout, we assume that the space points are not coincident,
$x_{1}\neq x_{2}$.

###  2.3 Three-point structures

Given three superspace points $z_{1},z_{2}$ and $z_{3}$, we have the following
three-point structures $\mbox{\boldmath$Z$}_{1},\mbox{\boldmath$Z$}_{2}$ and
$\mbox{\boldmath$Z$}_{3}$, with
$\mbox{\boldmath$Z$}_{1}=(\mbox{\boldmath$X$}_{1}^{a},\mbox{\boldmath$\Theta$}_{1}^{\alpha},\mbox{\boldmath$\bar{\Theta}$}_{1}^{{\dot{\alpha}}})$:

$\displaystyle\mbox{\boldmath$X$}_{1}=\tilde{x}^{-1}_{\bar{2}1}\tilde{x}_{\bar{2}3}\tilde{x}^{-1}_{\bar{1}3}~{},\,\,\,\tilde{\mbox{\boldmath$\Theta$}}_{1}={\rm
i}\big{(}\tilde{x}^{-1}_{\bar{2}1}\bar{\theta}_{12}-\tilde{x}^{-1}_{\bar{3}1}\bar{\theta}_{13}\big{)}~{},\,\,\,\tilde{\bar{\mbox{\boldmath$\Theta$}}}_{1}={\rm
i}\big{(}{\theta}_{12}\tilde{x}^{-1}_{\bar{1}2}\
-{\theta}_{13}\tilde{x}^{-1}_{\bar{1}3}\big{)}~{},~{}~{}~{}~{}~{}$ (2.18a)
$\displaystyle\mbox{\boldmath$X$}_{2}=\tilde{x}^{-1}_{\bar{3}2}\tilde{x}_{\bar{3}1}\tilde{x}^{-1}_{\bar{2}1}~{},\,\,\,\tilde{\mbox{\boldmath$\Theta$}}_{2}={\rm
i}\big{(}\tilde{x}^{-1}_{\bar{3}2}\bar{\theta}_{23}-\tilde{x}^{-1}_{\bar{1}2}\bar{\theta}_{21}\big{)}~{},\,\,\,\tilde{\bar{\mbox{\boldmath$\Theta$}}}_{2}={\rm
i}\big{(}{\theta}_{23}\tilde{x}^{-1}_{\bar{2}3}\
-{\theta}_{21}\tilde{x}^{-1}_{\bar{2}1}\big{)}~{},~{}~{}~{}~{}~{}$ (2.18b)
$\displaystyle\mbox{\boldmath$X$}_{3}=\tilde{x}^{-1}_{\bar{1}3}\tilde{x}_{\bar{1}2}\tilde{x}^{-1}_{\bar{3}2}~{},\,\,\,\tilde{\mbox{\boldmath$\Theta$}}_{3}={\rm
i}\big{(}\tilde{x}^{-1}_{\bar{1}3}\bar{\theta}_{31}-\tilde{x}^{-1}_{\bar{2}3}\bar{\theta}_{32}\big{)}~{},\,\,\,\tilde{\bar{\mbox{\boldmath$\Theta$}}}_{3}={\rm
i}\big{(}{\theta}_{31}\tilde{x}^{-1}_{\bar{3}1}\
-{\theta}_{32}\tilde{x}^{-1}_{\bar{3}2}\big{)}~{}.~{}~{}~{}~{}~{}$ (2.18c)

Since (2.18b) and (2.18c) are obtained through cyclic permutations of
superspace points, it suffices to study the properties of (2.18a). Let us also
define

$\displaystyle\bar{\mbox{\boldmath$X$}}_{1}=\mbox{\boldmath$X$}_{1}^{\dagger}$
$\displaystyle=$
$\displaystyle-\tilde{x}_{\bar{3}1}{}^{-1}\tilde{x}_{\bar{3}2}\tilde{x}_{\bar{1}2}{}^{-1}~{}.$
(2.19)

Similar relations hold for
$\bar{\mbox{\boldmath$X$}}_{2},\bar{\mbox{\boldmath$X$}}_{3}$.

The structures $\mbox{\boldmath$Z$}_{i}$ transform as tensors at the point
$z_{i},\,i=1,2,3$. For instance, the transformation law of (2.18a) reads

$\displaystyle\delta\mbox{\boldmath$X$}_{1\alpha{\dot{\alpha}}}$
$\displaystyle=$
$\displaystyle\Big{(}\hat{\omega}_{\alpha}\,^{\beta}(z_{1})-\delta_{\alpha}\,^{\beta}\sigma(z_{1})\Big{)}\mbox{\boldmath$X$}_{1\beta{\dot{\alpha}}}+\mbox{\boldmath$X$}_{1\alpha{\dot{\beta}}}\Big{(}\hat{\bar{\omega}}^{{\dot{\beta}}}\,_{{\dot{\alpha}}}(z_{1})-\delta^{{\dot{\beta}}}\,_{{\dot{\alpha}}}\bar{\sigma}(z_{1})\Big{)}~{},$
(2.20a) $\displaystyle\delta{\mbox{\boldmath$\Theta$}}_{1\,\alpha}$
$\displaystyle=$
$\displaystyle\hat{\omega}_{\alpha}{}^{\beta}(z_{1}){\mbox{\boldmath$\Theta$}}_{1\,\beta}+\big{(}\sigma(z_{1})-2{\bar{\sigma}}(z_{1})\big{)}{\mbox{\boldmath$\Theta$}}_{1\,\alpha}~{}.$
(2.20b)

We list several properties of $Z$’s which will be useful later (see [15] for
details):

$\displaystyle\mbox{\boldmath$X$}_{1}^{2}=\frac{x_{\bar{2}3}{}^{2}}{x_{\bar{2}1}{}^{2}x_{\bar{1}3}{}^{2}}~{},\qquad\mbox{\boldmath$\bar{X}$}_{1}^{2}=\frac{x_{\bar{3}2}{}^{2}}{x_{\bar{3}1}{}^{2}x_{\bar{1}2}{}^{2}}~{},$
(2.21a)
$\displaystyle\mbox{\boldmath$\bar{X}$}_{1\alpha{\dot{\alpha}}}=\mbox{\boldmath$X$}_{1\alpha{\dot{\alpha}}}+{\rm
i}\mbox{\boldmath$P$}_{1\alpha{\dot{\alpha}}}~{},\quad\mbox{\boldmath$P$}_{1\alpha{\dot{\alpha}}}=-4\mbox{\boldmath$\Theta$}_{1\alpha}\mbox{\boldmath$\bar{\Theta}$}_{1{\dot{\alpha}}}~{},\quad\mbox{\boldmath$P$}_{1}^{2}=-8\mbox{\boldmath$\Theta$}_{1}^{2}\mbox{\boldmath$\bar{\Theta}$}_{1}^{2}~{},$
(2.21b)
$\displaystyle\frac{1}{\mbox{\boldmath$\bar{X}$}_{1}^{2}}=\frac{1}{\mbox{\boldmath$X$}_{1}^{2}}-2{\rm
i}\frac{(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath$X$}_{1})}{(\mbox{\boldmath$X$}_{1}^{2})^{2}}~{},\quad(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath$X$}_{1})=-\frac{1}{2}\mbox{\boldmath$P$}_{1}^{{\dot{\alpha}}\alpha}\mbox{\boldmath$X$}_{1\alpha{\dot{\alpha}}}~{},$
(2.21c)

and hence, $\bar{X}$ is not an independent variable for it can be expressed in
terms of
$\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\mbox{\boldmath$\bar{\Theta}$}$.
The variables $Z$ with different labels are related to each other via the
identities

$\displaystyle\tilde{x}_{\bar{1}3}\mbox{\boldmath$X$}_{3}\tilde{x}_{\bar{3}1}$
$\displaystyle=$
$\displaystyle-\bar{\mbox{\boldmath$X$}}_{1}^{-1}=\frac{\tilde{\bar{\mbox{\boldmath$X$}}}_{1}}{\bar{\mbox{\boldmath$X$}}_{1}^{2}}~{},\qquad\tilde{x}_{\bar{1}3}\bar{\mbox{\boldmath$X$}}_{3}\tilde{x}_{\bar{3}1}=-\mbox{\boldmath$X$}_{1}^{-1}=\frac{\tilde{\mbox{\boldmath$X$}}_{1}}{\mbox{\boldmath$X$}_{1}^{2}}~{},$
(2.22a)
$\displaystyle\frac{x_{\bar{3}1}{}^{2}}{x_{\bar{1}3}{}^{2}}\tilde{x}_{\bar{1}3}\tilde{\mbox{\boldmath$\Theta$}}_{3}$
$\displaystyle=$
$\displaystyle-\mbox{\boldmath$X$}_{1}^{-1}\tilde{\mbox{\boldmath$\Theta$}}_{1}~{},\qquad\frac{x_{\bar{1}3}{}^{2}}{x_{\bar{3}1}{}^{2}}\tilde{\bar{\mbox{\boldmath$\Theta$}}}_{3}\tilde{x}_{\bar{3}1}=\tilde{\bar{\mbox{\boldmath$\Theta$}}}_{1}\bar{\mbox{\boldmath$X$}}_{1}^{-1}~{}.$
(2.22b)

Eqs. (2.20) and (2.22) imply that

$\displaystyle\frac{\mbox{\boldmath$X$}_{1}^{2}}{\mbox{\boldmath$\bar{X}$}_{1}^{2}}=\frac{\mbox{\boldmath$X$}_{2}^{2}}{\mbox{\boldmath$\bar{X}$}_{2}^{2}}=\frac{\mbox{\boldmath$X$}_{3}^{2}}{\mbox{\boldmath$\bar{X}$}_{3}^{2}}~{},$
(2.23)

and this combination is a superconformal invariant.

###  2.4 Correlation functions of primary superfields

A tensor superfield ${\cal O}^{{\cal A}}(z)$ is called primary, if it is
characterised by the following infinitesimal superconformal transformation law

$\displaystyle\delta\,{\cal O}^{\cal A}(z)$ $\displaystyle=-\xi\,{\cal
O}^{\cal
A}(z)+(\hat{\omega}^{\alpha\beta}(z)M_{\alpha\beta}+\hat{\bar{\omega}}^{\dot{\alpha}\dot{\beta}}(z)\bar{M}_{\dot{\alpha}\dot{\beta}})^{\cal
A}{}_{\cal B}\,{\cal O}^{\cal B}(z)$ (2.24)
$\displaystyle\quad-2\left(q\,\sigma(z)+\bar{q}\,\bar{\sigma}(z)\right){\cal
O}^{\cal A}(z)~{}.$

In the above, $\xi$ is the superconformal Killing vector, while
$\hat{\omega}^{\alpha\beta}(z)$ and $\sigma(z)$ are the $z$-dependent
parameters associated with $\xi$, see eq. (2.8). The superscript ‘${\cal A}$’
collectively denotes the undotted and dotted spinor indices on which the
Lorentz generators $M_{\alpha\beta}$ and
$\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}$ act. The weights $q$ and $\bar{q}$ are
such that $(q+\bar{q})$ is the scale dimension and $(q-\bar{q})$ is
proportional to the $\rm U(1)$ $R$-symmetry charge of the superfield ${\cal
O}^{{\cal A}}$.

Various primary superfields, including conserved current multiplets, are
subject to certain differential constraints. These constraints need to be
taken into account when computing correlation functions. It proves beneficial
to make use of these conformally covariant operators [15]: ${\cal
D}_{\bar{A}}=(\partial/\partial{\bf X}^{a}_{3},{\cal D}_{\alpha},\bar{{\cal
D}}^{{\dot{\alpha}}})$ and ${\cal Q}_{\bar{A}}=(\partial/\partial{\bf
X}^{a}_{3},{\cal Q}_{\alpha},\bar{\cal Q}^{{\dot{\alpha}}})$ defined by

$\displaystyle{\cal
D}_{\alpha}=\frac{\partial}{\partial{\mbox{\boldmath$\Theta$}}^{\alpha}_{3}}-2{\rm
i}\,(\sigma^{a})_{\alpha{\dot{\alpha}}}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\alpha}}_{3}\frac{\partial}{\partial{\bf
X}^{a}_{3}}~{},\qquad\bar{{\cal
D}}^{{\dot{\alpha}}}=\frac{\partial}{\partial\bar{\mbox{\boldmath$\Theta$}}_{3\,{\dot{\alpha}}}}~{},$
(2.25) $\displaystyle{\cal
Q}_{\alpha}=\frac{\partial}{\partial{\mbox{\boldmath$\Theta$}}^{\alpha}_{3}}~{},\qquad\bar{\cal
Q}^{{\dot{\alpha}}}=\frac{\partial}{\partial\bar{\mbox{\boldmath$\Theta$}}_{3{\dot{\alpha}}}}+2{\rm
i}\,{\mbox{\boldmath$\Theta$}}_{3\,\alpha}(\tilde{\sigma}^{a})^{{\dot{\alpha}}\alpha}\frac{\partial}{\partial{\bf
X}^{a}_{3}}~{},$ $\displaystyle[{\cal D}_{\bar{A}},{\cal
Q}_{\bar{B}}\\}~{}=~{}0~{},$

from which we can derive these anti-commutation relations

$\displaystyle\\{{\cal D}^{\alpha},\bar{{\cal D}}^{{\dot{\alpha}}}\\}=2{\rm
i}(\tilde{\sigma}^{a})^{{\dot{\alpha}}\alpha}\frac{\partial}{\partial\mbox{\boldmath$X$}^{a}}~{},\qquad\\{{{\cal
Q}}^{\alpha},\bar{{\cal Q}}^{{\dot{\alpha}}}\\}=-2{\rm
i}\,(\tilde{\sigma}^{a})^{{\dot{\alpha}}\alpha}\frac{\partial}{\partial\mbox{\boldmath$X$}^{a}}~{}.$
(2.26)

One can further prove the following differential identities:

$\displaystyle
D_{(1)\alpha}\,t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{x_{\bar{1}3}{}^{2}}(x_{1\bar{3}})_{\alpha{\dot{\alpha}}}\bar{{\cal
D}}^{{\dot{\alpha}}}t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$
(2.27a)
$\displaystyle\bar{D}_{(1){\dot{\alpha}}}\,t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{x_{\bar{3}1}{}^{2}}(x_{3\bar{1}})_{\alpha{\dot{\alpha}}}{{\cal
D}}^{\alpha}t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$
(2.27b) $\displaystyle
D_{(2)\alpha}\,t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{x_{\bar{2}3}{}^{2}}(x_{2\bar{3}})_{\alpha{\dot{\alpha}}}\bar{{\cal
Q}}^{{\dot{\alpha}}}t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$
(2.27c)
$\displaystyle\bar{D}_{(2){\dot{\alpha}}}\,t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{x_{\bar{3}2}{}^{2}}(x_{3\bar{2}})_{\alpha{\dot{\alpha}}}{{\cal
Q}}^{\alpha}t({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{}.$
(2.27d)

In accordance with the general prescription of [15, 14, 16], the two-point
function of a primary superfield ${\cal O}^{{\cal A}}$ with its conjugate
$\bar{{\cal O}}^{{\cal B}}$ is expressed in terms of ${\cal I}$, see eq.
(2.15):

$\langle{\cal O}^{{\cal A}}(z_{1})\;\bar{{\cal O}}^{{\cal
B}}(z_{2})\rangle~{}=~{}C_{{\cal O}}\;\frac{{\cal I}^{{\cal A}{\cal
B}}({{x}_{1\bar{2}}},{{x}_{2\bar{1}}})}{(x_{\bar{1}2}{}^{2})^{\bar{q}}(x_{\bar{2}1}{}^{2})^{q}}~{},$
(2.28)

with $C_{{\cal O}}$ being a normalisation constant.

The three-point function of primary superfields $\Phi^{{\cal
A}_{1}},\Psi^{{\cal A}_{2}}$ and $\Pi^{{\cal A}_{3}}$ has the following
general expression [15, 14, 16]:

$\displaystyle\langle\Phi^{{\cal A}_{1}}(z_{1})\,\Psi^{{\cal
A}_{2}}(z_{2})\,\Pi^{{\cal A}_{3}}(z_{3})\rangle=\frac{{\cal I}^{{\cal
A}_{1}{\cal B}_{1}}({{x}_{1\bar{3}}},{{x}_{3\bar{1}}})\,{\cal I}^{{\cal
A}_{2}{\cal
B}_{2}}({{x}_{2\bar{3}}},{{x}_{3\bar{2}}})\,}{(x_{\bar{1}3}{}^{2})^{\bar{q}_{1}}(x_{\bar{3}1}{}^{2})^{q_{1}}(x_{\bar{2}3}{}^{2})^{\bar{q}_{2}}(x_{\bar{3}2}{}^{2})^{q_{2}}}H_{{\cal
B}_{1}{\cal B}_{2}}{}^{{\cal
A}_{3}}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},~{}~{}~{}~{}~{}~{}~{}~{}$
(2.29)

where the functional form of the tensor $H_{{\cal B}_{1}{\cal B}_{2}}{}^{{\cal
A}_{3}}$ is highly constrained by the superconformal symmetry:

  * (i)

It possesses the homogeneity property

$\displaystyle H_{{\cal B}_{1}{\cal B}_{2}}{}^{{\cal
A}_{3}}(\lambda\bar{\lambda}\,{\mbox{\boldmath$X$}},\lambda\,{\mbox{\boldmath$\Theta$}},\bar{\lambda}\bar{\mbox{\boldmath$\Theta$}})=\lambda^{2a}\bar{\lambda}^{2\bar{a}}H_{{\cal
B}_{1}{\cal B}_{2}}{}^{{\cal
A}_{3}}({\mbox{\boldmath$X$}},{\mbox{\boldmath$\Theta$}},\bar{\mbox{\boldmath$\Theta$}})~{},$
(2.30) $\displaystyle
a-2\bar{a}=\bar{q}_{1}+\bar{q}_{2}-q_{3}~{},\qquad\bar{a}-2a=q_{1}+q_{2}-\bar{q}_{3}~{},$

This condition guarantees that the correlation function has the correct
transformation law under the superconformal group. By construction, eq. (2.29)
has the correct transformation properties at the points $z_{1}$ and $z_{2}$.
The above homogeneity property implies that it also transforms correctly at
$z_{3}$. The tensor $H_{{\cal B}_{1}{\cal B}_{2}}{}^{{\cal A}_{3}}$ has
dimension $(a+\bar{a})$.

  * (ii)

If any of the superfields $\Phi$, $\Psi$ and $\Pi$ obey differential equations
(such as conservation laws in the case of conserved current multiplets), then
$H_{{\cal B}_{1}{\cal B}_{2}}{}^{{\cal A}_{3}}$ is constrained by certain
differential equations too. The latter may be derived using (2.27).

  * (iii)

If any (or all) of the superfields $\Phi$, $\Psi$ and $\Pi$ coincide, then
$H_{{\cal B}_{1}{\cal B}_{2}}{}^{{\cal A}_{3}}$ obeys certain constraints, as
a consequence of the symmetry under permutations of superspace points. As an
example,

$\langle\Phi^{{\cal A}}(z_{1})\Phi^{{\cal B}}(z_{2})\Pi^{{\cal
C}}(z_{3})\rangle=(-1)^{\epsilon(\Phi)}\langle\Phi^{{\cal
B}}(z_{2})\Phi^{{\cal A}}(z_{1})\Pi^{{\cal C}}(z_{3})\rangle~{},$ (2.31)

where $\epsilon(\Phi)$ denotes the Grassmann parity of $\Phi^{{\cal A}}$. Note
that under permutations of any two superspace points, the three-point building
blocks transform as

$\displaystyle{\mbox{\boldmath$X$}}_{3\,\alpha{\dot{\alpha}}}$
$\displaystyle\stackrel{{\scriptstyle 1\leftrightarrow
2}}{{\longrightarrow}}-\bar{\mbox{\boldmath$X$}}_{3\,\alpha{\dot{\alpha}}}\,,\hskip
28.45274pt{\mbox{\boldmath$\Theta$}}_{3\,\alpha}\stackrel{{\scriptstyle
1\leftrightarrow
2}}{{\longrightarrow}}-{\mbox{\boldmath$\Theta$}}_{3\,\alpha}\,,$ (2.32a)
$\displaystyle{\mbox{\boldmath$X$}}_{3\,\alpha{\dot{\alpha}}}$
$\displaystyle\stackrel{{\scriptstyle 2\leftrightarrow
3}}{{\longrightarrow}}-\bar{\mbox{\boldmath$X$}}_{2\,\alpha{\dot{\alpha}}}\,,\hskip
28.45274pt{\mbox{\boldmath$\Theta$}}_{3\,\alpha}\stackrel{{\scriptstyle
2\leftrightarrow
3}}{{\longrightarrow}}-{\mbox{\boldmath$\Theta$}}_{2\,\alpha}\,,$ (2.32b)
$\displaystyle{\mbox{\boldmath$X$}}_{3\,\alpha{\dot{\alpha}}}$
$\displaystyle\stackrel{{\scriptstyle 1\leftrightarrow
3}}{{\longrightarrow}}-\bar{\mbox{\boldmath$X$}}_{1\,\alpha{\dot{\alpha}}}\,,\hskip
28.45274pt{\mbox{\boldmath$\Theta$}}_{3\,\alpha}\stackrel{{\scriptstyle
1\leftrightarrow
3}}{{\longrightarrow}}-{\mbox{\boldmath$\Theta$}}_{1\,\alpha}\,.$ (2.32c)

The above conditions fix the functional form of $H_{{\cal B}_{1}{\cal
B}_{2}}{}^{{\cal A}_{3}}$ (and, therefore, the three-point function under
consideration) up to a few arbitrary constants.

The goal of this paper is to study how ${\cal N}=1$ superconformal symmetry
imposes constraints on the general structure of two- and three-point
correlation functions involving a higher-spin conserved spinor current
multiplet $S_{\alpha(k)}$ and its conjugate $\bar{S}_{{\dot{\alpha}}(k)}$.
Here the complex superfield
$S_{\alpha(k)}=S_{\alpha_{1}\dots\alpha_{k}}=S_{(\alpha_{1}\dots\alpha_{k})}$
is a symmetric rank-$k$ spinor, subject to the following conservation
conditions [41]

$\displaystyle D^{\beta}S_{\beta\alpha_{1}\dots\alpha_{k-1}}$ $\displaystyle=$
$\displaystyle 0$ (2.33a) $\displaystyle\bar{D}^{2}S_{\alpha(k)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (2.33b)

Making use of the transformation law (2.24), it can be shown that eqs. (2.33)
are consistent with superconformal invariance provided $S_{\alpha(k)}$ is a
primary superfield with weight $(q,\bar{q})=(1+\frac{k}{2},1)$ and dimension
$2+\frac{k}{2}$.

Let us first consider the two-point correlation function of $S_{\alpha(k)}$
and its conjugate $\bar{S}_{{\dot{\alpha}}(k)}$. Adapting the general
prescription (2.28) to this case leads to

$\displaystyle{\langle}S_{\alpha(k)}(z_{1})\bar{S}_{{\dot{\alpha}}(k)}(z_{2}){\rangle}={\rm
i}^{k}A\frac{(x_{1\bar{2}})_{(\alpha_{1}({\dot{\alpha}}_{1}}\dots(x_{1\bar{2}})_{\alpha_{k}){\dot{\alpha}}_{k})}}{x_{\bar{1}2}{}^{2}(x_{\bar{2}1}{}^{2})^{k+1}}~{},$
(2.34)

where $A$ is a real coefficient. Using the identities (2.17), along with

$\displaystyle
D_{(1)}^{\alpha}\bigg{[}\frac{(x_{1\bar{2}})_{\alpha{\dot{\alpha}}}}{(x_{\bar{2}1}{}^{2})^{p}}\bigg{]}=-4{\rm
i}(p-2)\frac{\bar{\theta}_{12{\dot{\alpha}}}}{(x_{\bar{2}1}{}^{2})^{p}}~{},$
(2.35)

one may verify that the correlator (2.34) respects the conservation conditions

$\displaystyle
D_{(1)}^{\alpha_{1}}{\langle}S_{\alpha(k)}(z_{1})\bar{S}_{{\dot{\alpha}}(k)}(z_{2}){\rangle}=\bar{D}^{2}_{(1)}{\langle}S_{\alpha(k)}(z_{1})\bar{S}_{{\dot{\alpha}}(k)}(z_{2}){\rangle}=0~{},$
$\displaystyle\bar{D}_{(2)}^{{\dot{\alpha}}_{1}}{\langle}S_{\alpha(k)}(z_{1})\bar{S}_{{\dot{\alpha}}(k)}(z_{2}){\rangle}={D}^{2}_{(2)}{\langle}S_{\alpha(k)}(z_{1})\bar{S}_{{\dot{\alpha}}(k)}(z_{2}){\rangle}=0~{},$
(2.36)

at non-coincident points $z_{1}\neq z_{2}$.

In the case of $k=1$, the expression (2.34) is in agreement with [15, 19].
Various three-point functions involving $S_{\alpha}$ and its conjugate
$\bar{S}_{{\dot{\alpha}}}$ have been studied in detail in [51].

##  3 Correlator
${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}$

In ${\cal N}=1$ superconformal field theory, the $\rm U(1)$ flavour current
multiplet is a primary real superfield $L=\bar{L}$, subject to the
conservation equation

$\displaystyle D^{2}L=\bar{D}^{2}L=0~{}.$ (3.1)

Its superconformal transformation law is

$\delta L=-\xi L-2(\sigma+\bar{\sigma})L~{},$ (3.2)

which means that $L$ has weights $(q,\bar{q})=(1,1)$ and dimension 2.

In our recent work [51], we found that the three-point correlator with two
spinor current insertions and a flavour current multiplet has two linearly
independent functional structures with real coefficients $c_{1}$ and $d_{1}$:

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}}(z_{1})S_{\beta}(z_{2})L(z_{3}){\rangle}=\frac{(x_{3\bar{1}})_{{\dot{\alpha}}}{}^{\alpha}(x_{2\bar{3}})_{\beta}{}^{{\dot{\beta}}}}{(x_{\bar{1}3}{}^{2})^{2}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{2}}H_{\alpha{\dot{\beta}}}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$
(3.3a) where $H_{\alpha{\dot{\beta}}}={\rm
i}c_{1}\frac{\mbox{\boldmath$X$}_{\alpha{\dot{\beta}}}}{\mbox{\boldmath$X$}^{4}}+\frac{d_{1}}{2\mbox{\boldmath$X$}^{6}}\Big{(}\mbox{\boldmath$X$}^{2}\mbox{\boldmath$P$}_{\alpha{\dot{\beta}}}-4(\mbox{\boldmath$P$}\cdot\mbox{\boldmath$X$})\mbox{\boldmath$X$}_{\alpha{\dot{\beta}}}\Big{)}~{}.$
(3.3b)

Here we adopt the notation in which
$\mbox{\boldmath$X$}^{k}\equiv(\mbox{\boldmath$X$}^{2})^{k/2}$ .

A natural extension is to consider the three-point function
${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}$,
with $k=1,2,\dots$.222We can also consider
${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(\ell)}(z_{2})L(z_{3}){\rangle}$
with $k\neq\ell$. However, this correlator carries a non-trivial $R$-symmetry
charge and, hence, vanishes. In accordance with (2.29), we start with the
ansatz

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}$
$\displaystyle=\frac{(x_{3\bar{1}})_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\dots(x_{3\bar{1}})_{{\dot{\alpha}}_{k})}{}^{\beta_{k}}(x_{2\bar{3}})_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\dots(x_{2\bar{3}})_{\alpha_{k})}{}^{{\dot{\beta}}_{k}}}{(x_{\bar{1}3}{}^{2})^{k+1}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{k+1}}$
(3.4) $\displaystyle\times
H_{\beta_{1}\dots\beta_{k}{\dot{\beta}}_{1}\dots\dot{\beta}_{k}}\big{(}{\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}}\big{)}~{},$

where the tensor
$H_{\beta_{1}\dots\beta_{k}{\dot{\beta}}_{1}\dots\dot{\beta}_{k}}$ has the
symmetry property
$H_{(\beta_{1}\dots\beta_{k})({\dot{\beta}}_{1}\dots\dot{\beta}_{k})}=H_{\beta(k){\dot{\beta}}(k)}$.
It is also characterised by the homogeneity property

$\displaystyle
H_{\beta(k){\dot{\beta}}(k)}(\lambda\bar{\lambda}\mbox{\boldmath$X$},\lambda\mbox{\boldmath$\Theta$},\bar{\lambda}\bar{\mbox{\boldmath$\Theta$}})=\lambda^{-(k+2)}\bar{\lambda}^{-(k+2)}H_{\beta(k){\dot{\beta}}(k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})~{},$
(3.5)

and hence, its dimension is $-(k+2)$. Taking the complex conjugate of the
correlator (3.4),

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}^{*}={\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{2})S_{\alpha(k)}(z_{1})L(z_{3}){\rangle}~{},$
(3.6)

leads to the reality constraint on $H_{\beta(k){\dot{\beta}}(k)}$:

$\bar{H}_{\beta(k){\dot{\beta}}(k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})=H_{\beta(k){\dot{\beta}}(k)}(-\mbox{\boldmath$\bar{X}$},-{\mbox{\boldmath$\Theta$}},-\bar{\mbox{\boldmath$\Theta$}})~{}.$
(3.7)

The conservation equations of the higher-spin spinor current and flavour
current multiplets result in

$\displaystyle\bar{D}^{{\dot{\alpha}}_{1}}_{(1)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}}$
$\displaystyle=$ $\displaystyle 0~{},\qquad
D^{2}_{(1)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}}=0~{},$
(3.8a) $\displaystyle
D^{\alpha_{1}}_{(2)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}}$
$\displaystyle=$ $\displaystyle
0~{},\qquad\bar{D}^{2}_{(2)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}}=0~{},$
(3.8b) $\displaystyle
D^{2}_{(3)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}}$
$\displaystyle=$ $\displaystyle
0~{},\qquad\bar{D}^{2}_{(3)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}}=0~{}.$
(3.8c)

With the use of identities (2.27), conditions (3.8a) and (3.8b) are translated
to

$\displaystyle{\cal
D}^{\beta_{1}}H_{\beta_{1}\dots\beta_{k}{\dot{\beta}}(k)}=0~{},$ (3.9a)
$\displaystyle\bar{\cal D}^{2}H_{\beta(k){\dot{\beta}}(k)}=0~{},$ (3.9b)
$\displaystyle\bar{\cal
Q}^{{\dot{\beta}}_{1}}H_{\beta(k){\dot{\beta}}_{1}\dots\dot{\beta}_{k}}=0~{},$
(3.9c) $\displaystyle{\cal Q}^{2}H_{\beta(k){\dot{\beta}}(k)}=0~{}.$ (3.9d)

Imposing differential constraints (3.8c) is more complicated. We will take
care of (3.8c) at the end.

The problem of computing higher-spin correlator (3.4) is thus reduced to
determining the most general form of $H_{\beta(k){\dot{\beta}}(k)}$ satisfying
the constraints (3.5), (3.7), (3.8c) and (3.9).

Since $H_{\beta(k){\dot{\beta}}(k)}$ is Grassmann even, the most general
expansion we can write is

$\displaystyle
H_{\beta(k){\dot{\beta}}(k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})$
$\displaystyle=$ $\displaystyle
A^{(1)}_{\beta(k){\dot{\beta}}(k)}(\mbox{\boldmath$X$})+A^{(2)}_{\beta(k){\dot{\beta}}(k)}(\mbox{\boldmath$X$}){\mbox{\boldmath$\Theta$}}^{2}+A^{(3)}_{\beta(k){\dot{\beta}}(k)}(\mbox{\boldmath$X$})\bar{\mbox{\boldmath$\Theta$}}^{2}$
(3.10)
$\displaystyle+A^{(4)}_{\beta(k){\dot{\beta}}(k)}(\mbox{\boldmath$X$}){\mbox{\boldmath$\Theta$}}^{2}\bar{\mbox{\boldmath$\Theta$}}^{2}+B_{\beta(k){\dot{\beta}}(k),\,\gamma\dot{\gamma}}(\mbox{\boldmath$X$}){\mbox{\boldmath$\Theta$}}^{\gamma}\bar{\mbox{\boldmath$\Theta$}}^{{\dot{\gamma}}}\,.$

Constraints (3.9b) and (3.9d) immediately tell us that

$A^{(2)}_{\beta(k){\dot{\beta}}(k)}=A^{(3)}_{\beta(k){\dot{\beta}}(k)}=A^{(4)}_{\beta(k){\dot{\beta}}(k)}=0~{},$
(3.11)

which leaves us with

$H_{\beta(k){\dot{\beta}}(k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})=A_{\beta(k){\dot{\beta}}(k)}(\mbox{\boldmath$X$})+B_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}(\mbox{\boldmath$X$}){\mbox{\boldmath$\Theta$}}^{\gamma}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\gamma}}\,,$
(3.12a) where we have redenoted
$A_{\beta(k){\dot{\beta}}(k)}=A^{(1)}_{\beta(k){\dot{\beta}}(k)}$. Since
$A_{\beta(k){\dot{\beta}}(k)}$ and
$B_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}$ can be constructed using
only tensors $\mbox{\boldmath$X$}_{\alpha{\dot{\alpha}}},\
\varepsilon_{\alpha\beta},\ \varepsilon_{{\dot{\alpha}}{\dot{\beta}}}$, it is
not hard to list all possible independent structures consistent with the
homogeneity property (3.5) by performing irreducible decompositions into
symmetric and antisymmetric parts in dotted and undotted indiices:
$\displaystyle A_{\beta(k){\dot{\beta}}(k)}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{\mbox{\boldmath$X$}^{2k+2}}\mbox{\boldmath$X$}_{(\beta_{1}({\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}~{},$
(3.12b) $\displaystyle B_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}$
$\displaystyle=$
$\displaystyle\frac{b_{1}}{\mbox{\boldmath$X$}^{2k+4}}\mbox{\boldmath$X$}_{(\gamma({\dot{\gamma}}}\mbox{\boldmath$X$}_{\beta_{1}{\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}$
(3.12c)
$\displaystyle+\frac{b_{2}}{\mbox{\boldmath$X$}^{2k+2}}\varepsilon_{\gamma(\beta_{1}}\varepsilon_{{\dot{\gamma}}({\dot{\beta}}_{1}}\mbox{\boldmath$X$}_{\beta_{2}{\dot{\beta}}_{2}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}~{}.$

Here $a_{1},b_{1},b_{2}$ are arbitrary complex coefficients. To continue it is
convenient to introduce auxiliary commuting complex variables
$(u^{\alpha},\bar{w}^{{\dot{\alpha}}})$, with the property
$u^{\alpha}u_{\alpha}=\bar{w}_{{\dot{\alpha}}}\bar{w}^{{\dot{\alpha}}}=0$.
Given a tensor superfield $T_{\alpha(k){\dot{\alpha}}(l)}$, we can associate
to it the following index-free superfield

$\displaystyle T_{(k,l)}(u,\bar{w}):=u^{\alpha_{1}}\dots
u^{\alpha_{k}}\bar{w}^{{\dot{\alpha}}_{1}}\dots\bar{w}^{{\dot{\alpha}}_{l}}T_{\alpha_{1}\dots\alpha_{k}{\dot{\alpha}}_{1}\dots\dot{\alpha}_{l}}~{},$
(3.13)

which is homogeneous of degree $(k,l)$ in the variables
$u^{\alpha},\bar{w}^{{\dot{\alpha}}}$. Making use of these auxiliary variables
and their corresponding partial derivatives $(\partial/\partial
u^{\alpha},\partial/\partial\bar{w}^{{\dot{\alpha}}})$ allows us to also
convert the conformally covariant derivatives into index-free operators, for
example:

$\displaystyle{\cal D}_{(-1,0)}$ $\displaystyle:=$ $\displaystyle{\cal
D}^{\alpha}\frac{\partial}{\partial u^{\alpha}}~{},\qquad{\cal
D}_{(-1,0)}^{2}=0~{},$ (3.14) $\displaystyle\bar{\cal Q}_{(0,-1)}$
$\displaystyle:=$ $\displaystyle\bar{\cal
Q}^{{\dot{\alpha}}}\frac{\partial}{\partial\bar{w}^{{\dot{\alpha}}}}~{},\qquad\bar{\cal
Q}_{(0,-1)}^{2}=0~{}.$ (3.15)

These nilpotent operators decrease the degree of homogeneity in $u^{\alpha}$
and $\bar{w}^{{\dot{\alpha}}}$.

Using the notation introduced, eq. (3.12) turns into

$\displaystyle
H_{(k,k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{w})=\frac{a_{1}}{\mbox{\boldmath$X$}^{2k+2}}\mbox{\boldmath$X$}^{k}_{(1,1)}+\frac{1}{2(k+1)}\bigg{\\{}\big{(}b_{1}-(k+1)b_{2}\big{)}\frac{(\mbox{\boldmath$P$}\cdot\mbox{\boldmath$X$})\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\big{(}kb_{1}+(k+1)b_{2}\big{)}\frac{(\mbox{\boldmath$P$}\cdot\mbox{\boldmath${\cal
K}$})\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}\bigg{\\}}~{},$
(3.16)

where we have defined

$\displaystyle\mbox{\boldmath$X$}_{(1,1)}$ $\displaystyle:=$ $\displaystyle
u^{\alpha}\bar{w}^{{\dot{\alpha}}}\mbox{\boldmath$X$}_{\alpha{\dot{\alpha}}}~{},\qquad\quad\quad\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}:=u^{\alpha}\bar{w}^{{\dot{\alpha}}}\mbox{\boldmath$X$}_{\alpha{\dot{\gamma}}}\mbox{\boldmath$X$}_{\gamma{\dot{\alpha}}}~{},$
(3.17) $\displaystyle(\mbox{\boldmath$P$}\cdot\mbox{\boldmath$X$})$
$\displaystyle:=$ $\displaystyle
2\mbox{\boldmath$\Theta$}^{\alpha}\mbox{\boldmath$\bar{\Theta}$}^{{\dot{\alpha}}}\mbox{\boldmath$X$}_{\alpha{\dot{\alpha}}}~{},\qquad(\mbox{\boldmath$P$}\cdot\mbox{\boldmath${\cal
K}$}):=2\mbox{\boldmath$\Theta$}^{\alpha}\mbox{\boldmath$\bar{\Theta}$}^{{\dot{\alpha}}}\mbox{\boldmath${\cal
K}$}_{\alpha{\dot{\alpha}}}~{}.$ (3.18)

Throughout the paper, we will also often employ the following shorthand
notation

$\displaystyle(u\cdot\mbox{\boldmath$\Theta$})$
$\displaystyle=u^{\alpha}\mbox{\boldmath$\Theta$}_{\alpha}~{},\qquad\qquad\,(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})=\bar{w}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\Theta$}}_{{\dot{\alpha}}}~{},$
$\displaystyle(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})$
$\displaystyle=\mbox{\boldmath$\Theta$}^{\alpha}\mbox{\boldmath$X$}_{\alpha{\dot{\alpha}}}\bar{w}^{{\dot{\alpha}}}~{},\qquad(u\mbox{\boldmath$X$}\bar{\mbox{\boldmath$\Theta$}})=u^{\alpha}\mbox{\boldmath$X$}_{\alpha{\dot{\alpha}}}\bar{\mbox{\boldmath$\Theta$}}^{{\dot{\alpha}}}~{}.~{}~{}~{}~{}~{}$
(3.19)

The differential constraints (3.9a) and (3.9c) now read

$\displaystyle{\cal D}_{(-1,0)}H_{(k,k)}$ $\displaystyle=$ $\displaystyle
0~{},$ (3.20) $\displaystyle\bar{\cal Q}_{(0,-1)}H_{(k,k)}$ $\displaystyle=$
$\displaystyle 0~{}.$ (3.21)

A useful observation is that

$\displaystyle{\cal
D}_{(-1,0)}\bigg{(}\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}\bigg{)}=\bar{\cal
Q}_{(0,-1)}\bigg{(}\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}\bigg{)}=0~{}.$
(3.22)

This means that $a_{1}$ is an independent coefficient, since the first term in
(3.16) already satisfies conservation equations. It can be explicitly checked
that

$\displaystyle{\cal
D}_{(-1,0)}H_{(k,k)}=-(k+1)b_{2}\,\bigg{\\{}(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}+2{\rm
i}\bar{\mbox{\boldmath$\Theta$}}^{2}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}\bigg{\\}}~{},$
(3.23)

and

$\displaystyle\bar{{\cal
Q}}_{(0,-1)}H_{(k,k)}=-(k+1)b_{2}\,\bigg{\\{}(u\cdot\mbox{\boldmath$\Theta$})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}+2{\rm
i}{\mbox{\boldmath$\Theta$}}^{2}(u\mbox{\boldmath$X$}\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}\bigg{\\}}~{},$
(3.24)

which imply that the conservation laws (3.20) and (3.21) hold provided we set

$\displaystyle b_{2}=0~{}.$ (3.25)

In deriving (3.23) and (3.24), the following easily verified identities have
been used:

$\displaystyle{\cal
D}^{\alpha}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}=-4{\rm
i}\,\delta^{\alpha}{}_{\gamma}\bar{\mbox{\boldmath$\Theta$}}_{{\dot{\gamma}}}~{},\qquad\quad\quad~{}\bar{{\cal
Q}}^{{\dot{\alpha}}}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}=-4{\rm
i}\,\delta^{{\dot{\alpha}}}{}_{{\dot{\gamma}}}\mbox{\boldmath$\Theta$}_{\gamma}~{},$
(3.26a) $\displaystyle{\cal D}^{\alpha}\mbox{\boldmath$X$}^{k}_{(1,1)}=-4{\rm
i}k(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})u^{\alpha}\mbox{\boldmath$X$}^{k-1}_{(1,1)}~{},$
(3.26b) $\displaystyle\bar{{\cal
Q}}^{{\dot{\alpha}}}\mbox{\boldmath$X$}^{k}_{(1,1)}=-4{\rm
i}k(u\cdot\mbox{\boldmath$\Theta$})\bar{w}^{{\dot{\alpha}}}\mbox{\boldmath$X$}^{k-1}_{(1,1)}~{},$
(3.26c) $\displaystyle{\cal D}_{(-1,0)}\mbox{\boldmath$X$}^{k}_{(1,1)}=-4{\rm
i}k(k+1)(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\mbox{\boldmath$X$}^{k-1}_{(1,1)}~{},$
(3.26d) $\displaystyle\bar{\cal
Q}_{(0,-1)}\mbox{\boldmath$X$}^{k}_{(1,1)}=-4{\rm
i}k(k+1)(u\cdot\mbox{\boldmath$\Theta$})\mbox{\boldmath$X$}^{k-1}_{(1,1)}~{},$
(3.26e) $\displaystyle{\cal
D}^{\alpha}\Big{(}\frac{1}{\mbox{\boldmath$X$}^{2p}}\Big{)}=-4{\rm
i}p\frac{\bar{\mbox{\boldmath$\Theta$}}_{{\dot{\alpha}}}\mbox{\boldmath$X$}^{{\dot{\alpha}}\alpha}}{\mbox{\boldmath$X$}^{2p+2}}~{},\qquad\bar{\cal
Q}^{{\dot{\alpha}}}\Big{(}\frac{1}{\mbox{\boldmath$X$}^{2p}}\Big{)}=-4{\rm
i}p\frac{{\mbox{\boldmath$\Theta$}}_{\alpha}\mbox{\boldmath$X$}^{{\dot{\alpha}}\alpha}}{\mbox{\boldmath$X$}^{2p+2}}~{},$
(3.26f) $\displaystyle{\cal
D}^{\alpha}\mbox{\boldmath$P$}_{\gamma{\dot{\gamma}}}=4\,\delta^{\alpha}{}_{\gamma}\bar{\mbox{\boldmath$\Theta$}}_{{\dot{\gamma}}}~{},\qquad\qquad\quad\bar{{\cal
Q}}^{{\dot{\alpha}}}\mbox{\boldmath$P$}_{\gamma{\dot{\gamma}}}=4\,\delta^{{\dot{\alpha}}}{}_{{\dot{\gamma}}}\mbox{\boldmath$\Theta$}_{\gamma}~{}.$
(3.26g)

At this stage, we are left with two unconstrained complex parameters, $a_{1}$
and $b_{1}$. The next task is to check if the flavour current conservation
equations (3.8c) are satisfied. Checking conservation laws on $z_{3}$ requires
more work as there are no identities that would allow differential operators
acting on the $z_{3}$ dependence to pass through the prefactor of (3.4).
Following the procedure carried out in [51], let us first express (3.4) as

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}=\frac{1}{k_{1}}{\cal
I}_{\beta(k){\dot{\alpha}}(k)}(x_{3\bar{1}})\,{\cal
I}_{\alpha(k){\dot{\beta}}(k)}(x_{2\bar{3}})H^{{\dot{\beta}}(k)\beta(k)}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},~{}~{}~{}~{}~{}$
(3.27) $\displaystyle
k_{1}:=(x_{\bar{1}3}{}^{2})^{(k+2)/2}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{(k+2)/2}~{}.$

Here the ${\cal I}$-operators are the higher-spin extensions of (2.15).
Specifically, we define

$\displaystyle{\cal
I}_{\alpha(k){\dot{\alpha}}(k)}(x_{3\bar{1}}):=\frac{1}{(x_{\bar{1}3}{}^{2}){}^{k/2}}(x_{3\bar{1}})_{(\alpha_{1}({\dot{\alpha}}_{1}}\dots(x_{3\bar{1}})_{\alpha_{k}){\dot{\alpha}}_{k})}~{}.$
(3.28)

Its inverse will be denoted by

$\displaystyle\bar{\cal
I}^{{\dot{\alpha}}{(k)}\alpha(k)}(\tilde{x}_{\bar{1}3}):=\frac{1}{(x_{\bar{1}3}{}^{2}){}^{k/2}}(x_{\bar{1}3})^{({\dot{\alpha}}_{1}(\alpha_{1}}\dots(x_{\bar{1}3})^{{\dot{\alpha}}_{k})\alpha_{k})}~{}.$
(3.29)

These operators satisfy

$\displaystyle{\cal I}_{\alpha(k){\dot{\alpha}}(k)}(x_{3\bar{1}})\bar{\cal
I}^{{\dot{\alpha}}{(k)}\gamma(k)}(\tilde{x}_{\bar{1}3})=\delta_{(\alpha_{1}}^{(\gamma_{1}}\dots\delta_{\alpha_{k})}^{\gamma_{k})}~{},$
(3.30a) $\displaystyle\bar{\cal
I}^{{\dot{\gamma}}{(k)}\gamma(k)}(\tilde{x}_{\bar{1}3}){\cal
I}_{\gamma(k){\dot{\alpha}}(k)}(x_{3\bar{1}})=\delta_{({\dot{\alpha}}_{1}}^{({\dot{\gamma}}_{1}}\dots\delta_{{\dot{\alpha}}_{k})}^{{\dot{\gamma}}_{k})}~{}.$
(3.30b)

It should be kept in mind that the spinor indices can be raised or lowered in
accordance with (2.10b). This allows us to write

$\displaystyle{\cal
I}_{{\dot{\alpha}}(k)}{}^{\alpha(k)}(x_{3\bar{1}})=\varepsilon^{\alpha_{1}\beta_{1}}\dots\varepsilon^{\alpha_{k}\beta_{k}}{\cal
I}_{\beta(k){\dot{\alpha}}(k)}(x_{3\bar{1}})~{},$ (3.31)
$\displaystyle\bar{{\cal
I}}_{\alpha(k)}{}^{{\dot{\alpha}}(k)}(\tilde{x}_{\bar{1}3})=\varepsilon_{\alpha_{1}\beta_{1}}\dots\varepsilon_{\alpha_{k}\beta_{k}}\bar{{\cal
I}}^{{\dot{\alpha}}(k)\beta(k)}(\tilde{x}_{\bar{1}3})~{},$ (3.32)

By rearranging the operators in the three-point function and taking into
account their Grassmann parities, one may express the left-hand side of (3.27)
as

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}=(-1)^{k}{\langle}S_{\alpha(k)}(z_{2})L(z_{3})\bar{S}_{{\dot{\alpha}}(k)}(z_{1}){\rangle}$
$\displaystyle=$ $\displaystyle\frac{(-1)^{k}}{k_{2}}{\cal
I}_{\alpha(k){\dot{\gamma}}(k)}(x_{2\bar{1}})\widetilde{H}^{{\dot{\gamma}}(k)}{}_{{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})~{},$
$\displaystyle
k_{2}:=(x_{\bar{1}2}{}^{2})^{(k+2)/2}x_{\bar{2}1}{}^{2}x_{\bar{1}3}{}^{2}x_{\bar{3}1}{}^{2}~{},$

for some function
$\widetilde{H}^{{\dot{\gamma}}(k)}{}_{{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})$.
Comparing eqs. (3.27) and (3) allows us to relate $H$ and $\widetilde{H}$

$\displaystyle\widetilde{H}^{{\dot{\gamma}}(k)}{}_{{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})$
$\displaystyle=$ $\displaystyle\frac{(-1)^{k}k_{2}}{k_{1}}\bar{{\cal
I}}^{{\dot{\gamma}}(k)\beta(k)}(\tilde{\bar{\mbox{\boldmath$X$}}}_{1}){\cal
I}_{\beta(k){\dot{\beta}}(k)}(x_{1\bar{3}}){\cal
I}_{\alpha(k){\dot{\alpha}}(k)}(x_{3\bar{1}})$ (3.34)
$\displaystyle\quad\times
H^{{\dot{\beta}}(k)\alpha(k)}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$

where we have used the identity

$\displaystyle\bar{{\cal
I}}^{{\dot{\alpha}}(k)\alpha(k)}(\tilde{x}_{\bar{1}2}){\cal
I}_{\alpha(k){\dot{\beta}}(k)}(x_{2\bar{3}})=\bar{{\cal
I}}^{{\dot{\alpha}}(k)\alpha(k)}(\tilde{\bar{\mbox{\boldmath$X$}}}_{1}){\cal
I}_{\alpha(k){\dot{\beta}}(k)}(x_{1\bar{3}})~{},$ (3.35a) with
$\displaystyle\bar{{\cal
I}}^{{\dot{\alpha}}(k)\alpha(k)}(\tilde{\bar{\mbox{\boldmath$X$}}})=\frac{1}{\bar{\mbox{\boldmath$X$}}^{k}}\bar{\mbox{\boldmath$X$}}^{({\dot{\alpha}}_{1}(\alpha_{1}}\dots\bar{\mbox{\boldmath$X$}}^{{\dot{\alpha}}_{k})\alpha_{k})}~{}.$
(3.35b)

In order to compute the expression (3.34), we further note that

$\displaystyle{\cal
I}_{{\dot{\alpha}}(k)}{}^{\alpha(k)}(x_{3\bar{1}})\bar{{\cal
I}}_{\beta(k)}{}^{{\dot{\beta}}(k)}(\tilde{x}_{\bar{3}1})\mbox{\boldmath$X$}_{3(\alpha_{1}({\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{3\alpha_{k}){\dot{\beta}}_{k})}=(-1)^{k}\frac{\bar{\mbox{\boldmath$X$}}_{1(\beta_{1}({\dot{\alpha}}_{1}}\dots\bar{\mbox{\boldmath$X$}}_{1\beta_{k}){\dot{\alpha}}_{k})}}{\bar{\mbox{\boldmath$X$}}_{1}^{2k}(x_{\bar{1}3}{}^{2}x_{\bar{3}1}{}^{2})^{k/2}}~{},~{}~{}~{}~{}~{}~{}$
(3.36a)
$\displaystyle(\mbox{\boldmath$P$}_{3}\cdot\mbox{\boldmath$X$}_{3})=\frac{(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath$X$}_{1})}{\mbox{\boldmath$X$}_{1}^{2}\bar{\mbox{\boldmath$X$}}^{2}_{1}x_{\bar{1}3}{}^{2}x_{\bar{3}1}{}^{2}}~{},\qquad\frac{1}{\mbox{\boldmath$X$}_{3}^{2}}=\bar{\mbox{\boldmath$X$}}^{2}_{1}x_{\bar{1}3}{}^{2}x_{\bar{3}1}{}^{2}~{}.$
(3.36b)

Direct computation yields

$\displaystyle\widetilde{H}^{{\dot{\gamma}}(k)}{}_{{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\mbox{\boldmath$\bar{\Theta}$}_{1})=\frac{a_{1}}{\mbox{\boldmath$X$}_{1}^{2}}\delta_{({\dot{\alpha}}_{1}}^{({\dot{\gamma}}_{1}}\dots\delta_{{\dot{\alpha}}_{k})}^{{\dot{\gamma}}_{k})}$
$\displaystyle\qquad\quad+\frac{b_{1}}{\mbox{\boldmath$X$}_{1}^{4}}\Big{(}\mbox{\boldmath$\Theta$}_{1}^{\lambda}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\lambda}}_{1}\mbox{\boldmath$X$}_{1\lambda\dot{\lambda}}\delta_{({\dot{\alpha}}_{1}}^{({\dot{\gamma}}_{1}}\dots\delta_{{\dot{\alpha}}_{k})}^{{\dot{\gamma}}_{k})}-\frac{k}{k+1}\mbox{\boldmath$\Theta$}_{1}^{\lambda}\bar{\mbox{\boldmath$\Theta$}}^{\dot{(\gamma_{1}}}\mbox{\boldmath$X$}_{1\lambda({\dot{\alpha}}_{1}}\delta^{{\dot{\gamma}}_{2}}_{{\dot{\alpha}}_{2}}\dots\delta^{{\dot{\gamma}}_{k})}_{{\dot{\alpha}}_{k})}~{}\Big{)}~{},$
(3.37)

which can again be written in index-free notation as

$\displaystyle\widetilde{H}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\mbox{\boldmath$\bar{\Theta}$}_{1};\,\bar{v},\bar{w})=\bar{v}^{{\dot{\alpha}}_{1}}\dots\bar{v}^{{\dot{\alpha}}_{k}}\bar{w}_{{\dot{\gamma}}_{1}}\dots\bar{w}_{{\dot{\gamma}}_{k}}\widetilde{H}^{{\dot{\gamma}}(k)}{}_{{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\mbox{\boldmath$\bar{\Theta}$}_{1})$
$\displaystyle=\frac{a_{1}}{\mbox{\boldmath$X$}_{1}^{2}}(\bar{v}\cdot\bar{w})^{k}+\frac{b_{1}}{2\mbox{\boldmath$X$}_{1}^{4}}(\bar{v}\cdot\bar{w})^{k-1}\Big{\\{}(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath$X$}_{1})(\bar{v}\cdot\bar{w})+\frac{k}{2(k+1)}\mbox{\boldmath$P$}_{1}^{{\dot{\alpha}}\alpha}\mbox{\boldmath$X$}_{1\alpha{\dot{\beta}}}\bar{v}^{{\dot{\beta}}}\bar{w}_{{\dot{\alpha}}}\Big{\\}}~{},~{}~{}~{}~{}~{}$
(3.38)

with
$(\bar{v}\cdot\bar{w})=\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}\bar{v}^{{\dot{\alpha}}}\bar{w}^{{\dot{\beta}}}$.
We now observe that

$\displaystyle{\langle}S_{\alpha(k)}(z_{2})L(z_{3})\bar{S}_{{\dot{\alpha}}(k)}(z_{1}){\rangle}=[\,\text{relabel}\,z_{2}\rightarrow
z_{1},\,z_{3}\rightarrow z_{2},\,z_{1}\rightarrow z_{3}]$
$\displaystyle={\langle}S_{\alpha(k)}(z_{1})L(z_{2})\bar{S}_{{\dot{\alpha}}(k)}(z_{3}){\rangle}=\frac{1}{k_{3}}{\cal
I}_{\alpha(k){\dot{\gamma}}(k)}(x_{1\bar{3}})\widetilde{H}^{{\dot{\gamma}}(k)}{}_{{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\mbox{\boldmath$\bar{\Theta}$}_{3})~{},$
(3.39) $\displaystyle
k_{3}:=(x_{\bar{3}1}{}^{2})^{(k+2)/2}x_{\bar{3}2}{}^{2}x_{\bar{2}3}{}^{2}x_{\bar{1}3}{}^{2}~{}.$

As a consequence of (2.27), the conservation conditions on the flavour current
multiplet are equivalent to

$\displaystyle{\cal
Q}^{2}\widetilde{H}(\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\mbox{\boldmath$\bar{\Theta}$}_{3};\,\bar{v},\bar{w})=0~{},\qquad\bar{\cal
Q}^{2}\widetilde{H}(\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\mbox{\boldmath$\bar{\Theta}$}_{3};\,\bar{v},\bar{w})=0~{}.$
(3.40)

It is straightforward to show that (3.38) indeed satisfies (3.40) for
arbitrary $a_{1}$ and $b_{1}$.

The final step is to impose the reality constraint, (3.7), which requires

$\displaystyle\bar{a}_{1}=(-1)^{k}a_{1}~{},\qquad\bar{b}_{1}=(-1)^{k+1}b_{1}~{},$
$\displaystyle\implies a_{1}={\rm i}^{k}A~{},\qquad b_{1}={\rm i}^{k-1}B~{},$
(3.41)

where $A$ and $B$ are real. Thus, the higher-spin correlator (3.4) is fixed by
the ${\cal N}=1$ superconformal symmetry up to two independent, real
coefficients $A$ and $B$.

As a result, the final form of (3.4) proves to be

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}=\frac{(x_{3\bar{1}})_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\dots(x_{3\bar{1}})_{{\dot{\alpha}}_{k})}{}^{\beta_{k}}(x_{2\bar{3}})_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\dots(x_{2\bar{3}})_{\alpha_{k})}{}^{{\dot{\beta}}_{k}}}{(x_{\bar{1}3}{}^{2})^{k+1}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{k+1}}$
$\displaystyle\hskip 142.26378pt\times
H_{\beta_{1}\dots\beta_{k}{\dot{\beta}}_{1}\dots\dot{\beta}_{k}}\big{(}{\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}}\big{)}~{},$
(3.42a) where the tensor
$H_{\beta_{1}\dots\beta_{k}{\dot{\beta}}_{1}\dots\dot{\beta}_{k}}\big{(}{\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}}\big{)}$
has the compact expression $\displaystyle
H_{(k,k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{w})=u^{\beta_{1}}\dots
u^{\beta_{k}}\bar{w}^{{\dot{\beta}}_{1}}\dots\bar{w}^{{\dot{\beta}}_{k}}H_{\beta(k){\dot{\beta}}(k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})$
$\displaystyle={\rm i}^{k-1}\bigg{(}{\rm
i}A\,\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}+\frac{B}{2(k+1)}\bigg{\\{}\frac{(\mbox{\boldmath$P$}\cdot\mbox{\boldmath$X$})\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}+k\frac{(\mbox{\boldmath$P$}\cdot\mbox{\boldmath${\cal
K}$})\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}\bigg{\\}}\bigg{)}~{}.$
(3.42b)

Setting $k=1$, direct comparison with (3.3) shows that we have an agreement
provided the parameters are related in the following way:

$c_{1}=A~{},\qquad d_{1}=-\frac{B}{4}~{}.$ (3.43)

In a more general setting, one may also consider

$\displaystyle{\langle}\bar{S}^{\prime}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})L(z_{3}){\rangle}~{},$
(3.44)

in which case the only difference with (3.42a) is that the superfield
$\bar{S}^{\prime}_{{\dot{\alpha}}(k)}$ is not the complex conjugate of
$S_{\alpha(k)}$, i.e.
$(S_{\alpha(k)})^{*}\neq\bar{S}^{\prime}_{{\dot{\alpha}}(k)}$. It still obeys
the conservation conditions
$\bar{D}^{{\dot{\alpha}}_{1}}\bar{S}^{\prime}_{{\dot{\alpha}}_{1}\dots\dot{\alpha}_{k}}=D^{2}\bar{S}^{\prime}_{{\dot{\alpha}}(k)}=0$.
Since the reality constraint (3.7) no longer holds, the correlator (3.44) is
determined up to two complex coefficients, $a_{1}$ and $b_{1}$.

##  4 Correlator
${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}$

We turn to constructing the three-point function of the higher-spin spinor
current multiplets with the supercurrent, namely
${{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}$.333The
correlator
${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(\ell)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}$
with $k\neq\ell$ vanishes since it carries a non-trivial $R$-symmetry charge.

The ${\cal N}=1$ supercurrent [25] is described by a primary real superfield
$J_{\alpha{\dot{\alpha}}}=\bar{J}_{\alpha{\dot{\alpha}}}$ obeying the
conservation law

$\displaystyle
D^{\alpha}J_{\alpha{\dot{\alpha}}}=\bar{D}^{{\dot{\alpha}}}J_{\alpha{\dot{\alpha}}}=0~{}.$
(4.1)

Its superconformal transformation is

$\delta J_{\alpha{\dot{\alpha}}}=-\xi
J_{\alpha{\dot{\alpha}}}+(\hat{\omega}_{\alpha}\,^{\beta}\delta_{{\dot{\alpha}}}^{\,{\dot{\beta}}}+\bar{\hat{\omega}}_{{\dot{\alpha}}}\,^{{\dot{\beta}}}\delta_{\alpha}^{\,\beta})J_{\beta{\dot{\beta}}}-3(\sigma+\bar{\sigma})J_{\alpha{\dot{\alpha}}}~{}.$
(4.2)

As a consequence of the reality constraint and conservation law,
$J_{\alpha{\dot{\alpha}}}$ has weights $(q,\bar{q})=(\frac{3}{2},\frac{3}{2})$
and dimension $3$.

The three-point function of the type
${\langle}\bar{S}_{{\dot{\alpha}}}(z_{1})S_{\alpha}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}$
was constructed in [51]. It is determined up to three independent, real
coefficients:

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}}(z_{1})S_{\alpha}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}$
$\displaystyle=$
$\displaystyle\frac{(x_{3\bar{1}})_{\beta{\dot{\alpha}}}(x_{2\bar{3}})_{\alpha{\dot{\beta}}}}{(x_{\bar{1}3}{}^{2})^{2}(x_{\bar{3}2}{}^{2})^{2}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}}H^{{\dot{\beta}}\beta}\,_{\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$
(4.3a) with $\displaystyle
H^{{\dot{\beta}}\beta}\,_{\gamma{\dot{\gamma}}}(\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\mbox{\boldmath$\bar{\Theta}$})=\frac{1}{2}(d_{2}+d_{3})\frac{1}{\mbox{\boldmath$X$}^{2}}\delta^{\beta}_{\gamma}\,\delta^{{\dot{\beta}}}_{{\dot{\gamma}}}+\frac{1}{2}(d_{3}+d_{4})\frac{1}{(\mbox{\boldmath$X$}^{2})^{2}}\mbox{\boldmath$X$}^{{\dot{\beta}}\beta}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}$
$\displaystyle\qquad\quad+\frac{1}{2}d_{4}\frac{{\rm
i}}{(\mbox{\boldmath$X$}^{2})^{2}}\mbox{\boldmath$X$}^{{\dot{\beta}}\beta}\mbox{\boldmath$P$}_{\gamma{\dot{\gamma}}}+\frac{1}{2}d_{3}\frac{{\rm
i}}{(\mbox{\boldmath$X$}^{2})^{2}}\mbox{\boldmath$P$}^{{\dot{\beta}}\beta}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}$
$\displaystyle\qquad\quad-d_{2}\frac{{\rm
i}}{(\mbox{\boldmath$X$}^{2})^{2}}(\mbox{\boldmath$P$}\cdot\mbox{\boldmath$X$})\delta^{\beta}_{\gamma}\,\delta^{{\dot{\beta}}}_{{\dot{\gamma}}}-d_{3}\frac{{\rm
i}}{(\mbox{\boldmath$X$}^{2})^{3}}(\mbox{\boldmath$P$}\cdot\mbox{\boldmath$X$})\mbox{\boldmath$X$}^{{\dot{\beta}}\beta}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}~{}.$
(4.3b)

Let us consider
${{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}$.
Using the prescription (2.29), the correlator takes the form

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}$
$\displaystyle=$
$\displaystyle\frac{(x_{3\bar{1}})_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\dots(x_{3\bar{1}})_{{\dot{\alpha}}_{k})}{}^{\beta_{k}}(x_{2\bar{3}})_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\dots(x_{2\bar{3}})_{\alpha_{k})}{}^{{\dot{\beta}}_{k}}}{(x_{\bar{1}3}{}^{2})^{k+1}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{k+1}}$
(4.4) $\displaystyle\times
H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{}.$

Following similar analysis as in section 3, the tensor
$H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}$ is required to satisfy
these properties:

  * •

Homogeneity:

$\displaystyle
H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}(\lambda\bar{\lambda}\mbox{\boldmath$X$},\lambda\mbox{\boldmath$\Theta$},\bar{\lambda}\bar{\mbox{\boldmath$\Theta$}})=\lambda^{-(k+1)}\bar{\lambda}^{-(k+1)}H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})~{},$
(4.5)

  * •

Reality:

$\displaystyle{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}^{*}={\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{2})S_{\alpha(k)}(z_{1})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}~{},$
$\displaystyle\implies\bar{H}_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})=H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}(-\mbox{\boldmath$\bar{X}$},-{\mbox{\boldmath$\Theta$}},-\bar{\mbox{\boldmath$\Theta$}})~{}.$
(4.6)

  * •

Conservation conditions:

$\displaystyle{\cal
D}^{\beta_{1}}H_{\beta_{1}\dots\beta_{k}{\dot{\beta}}(k),\gamma{\dot{\gamma}}}=0~{},\qquad\bar{\cal
D}^{2}H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}=0~{},$ (4.7a)
$\displaystyle\bar{\cal
Q}^{{\dot{\beta}}_{1}}H_{\beta(k){\dot{\beta}}_{1}\dots\dot{\beta}_{k},\gamma{\dot{\gamma}}}=0~{},\qquad{\cal
Q}^{2}H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}=0~{}.$ (4.7b)

Furthermore, the conservation equations of the supercurrent, eq. (4.1), demand
that the following be satisfied:

$D^{\gamma}_{(3)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}=0~{},\qquad\bar{D}^{{\dot{\gamma}}}_{(3)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}=0~{}.$
(4.8)

First of all, the fact that
$H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}$ is Grassmann even and
obeys the differential constraints $\bar{\cal
D}^{2}H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}=0$ and ${\cal
Q}^{2}H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}=0$, imply that it has
the general form

$H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})=A_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}(\mbox{\boldmath$X$})+B_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}},\delta\dot{\delta}}(\mbox{\boldmath$X$}){\mbox{\boldmath$\Theta$}}^{\delta}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}~{}.$
(4.9a) We then write all possible independent structures consistent with
(4.5). This is done by decomposing
$A_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}$ and
$B_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}},\delta\dot{\delta}}$ in
terms of their irreducible components $\displaystyle
A_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}=\frac{a_{1}}{\mbox{\boldmath$X$}^{2k+2}}\mbox{\boldmath$X$}_{(\gamma({\dot{\gamma}}}\mbox{\boldmath$X$}_{\beta_{1}{\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}$
$\displaystyle+\frac{a_{2}}{\mbox{\boldmath$X$}^{2k}}\varepsilon_{\gamma(\beta_{1}}\varepsilon_{{\dot{\gamma}}({\dot{\beta}}_{1}}\mbox{\boldmath$X$}_{\beta_{2}{\dot{\beta}}_{2}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}~{},$
(4.9b) $\displaystyle
B_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}},\delta\dot{\delta}}=\frac{b_{1}}{\mbox{\boldmath$X$}^{2k+4}}\mbox{\boldmath$X$}_{(\gamma({\dot{\gamma}}}\mbox{\boldmath$X$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{\beta_{1}{\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}$
$\displaystyle+\frac{b_{2}}{\mbox{\boldmath$X$}^{2k+2}}\varepsilon_{\gamma(\delta}\varepsilon_{{\dot{\gamma}}(\dot{\delta}}\mbox{\boldmath$X$}_{\beta_{1}{\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}+\frac{b_{3}}{\mbox{\boldmath$X$}^{2k+2}}\varepsilon_{\delta(\gamma}\varepsilon_{\dot{\delta}({\dot{\gamma}}}\mbox{\boldmath$X$}_{\beta_{1}{\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}$
$\displaystyle+\frac{b_{4}}{\mbox{\boldmath$X$}^{2k+2}}\varepsilon_{\gamma\delta}\varepsilon_{{\dot{\gamma}}\dot{\delta}}\mbox{\boldmath$X$}_{(\beta_{1}({\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}+\frac{b_{5}}{\mbox{\boldmath$X$}^{2k+2}}\varepsilon_{\delta(\gamma}\varepsilon_{{\dot{\gamma}}(\dot{\delta}}\mbox{\boldmath$X$}_{\beta_{1}{\dot{\beta}}_{1}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}$
$\displaystyle+\frac{b_{6}}{\mbox{\boldmath$X$}^{2k}}\varepsilon_{\gamma(\beta_{1}}\varepsilon_{|\delta|\beta_{2}}\varepsilon_{{\dot{\gamma}}({\dot{\beta}}_{1}}\varepsilon_{|\dot{\delta}|{\dot{\beta}}_{2}}\mbox{\boldmath$X$}_{\beta_{3}{\dot{\beta}}_{3}}\dots\mbox{\boldmath$X$}_{\beta_{k}){\dot{\beta}}_{k})}~{},$
(4.9c)

where $a_{1},a_{2}$ and $b_{1},\dots b_{6}$ are arbitrary complex
coefficients. Before imposing conservation conditions (4.7), it is useful to
contract the indices of $H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}$
with auxiliary variables. We thus introduce

$\displaystyle
H_{(k,k),\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{w}):=u^{\beta_{1}}\dots
u^{\beta_{k}}\bar{w}^{{\dot{\beta}}_{1}}\dots\bar{w}^{{\dot{\beta}}_{k}}H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}$
$\displaystyle=\frac{\tilde{a}_{1}}{\mbox{\boldmath$X$}^{2k+2}}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}^{k}_{(1,1)}+\frac{\tilde{a}_{2}}{\mbox{\boldmath$X$}^{2k+2}}\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}^{k-1}_{(1,1)}$ (4.10)
$\displaystyle+\frac{\mbox{\boldmath$\Theta$}^{\delta}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}}{\mbox{\boldmath$X$}^{2k+4}}\Big{\\{}\tilde{b}_{1}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{(1,1)}^{k}+\tilde{b}_{2}\mbox{\boldmath$X$}_{\gamma\dot{\delta}}\mbox{\boldmath$X$}_{\delta\dot{\gamma}}\mbox{\boldmath$X$}_{(1,1)}^{k}+\tilde{b}_{3}\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{(1,1)}^{k-1}$
$\displaystyle\qquad\qquad+\tilde{b}_{4}\mbox{\boldmath${\cal
K}$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{\gamma\dot{\gamma}}\mbox{\boldmath$X$}_{(1,1)}^{k-1}+\tilde{b}_{5}\mbox{\boldmath${\cal
K}$}_{\delta{\dot{\gamma}}}\mbox{\boldmath$X$}_{\gamma\dot{\delta}}\mbox{\boldmath$X$}_{(1,1)}^{k-1}+\tilde{b}_{6}\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath${\cal
K}$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{(1,1)}^{k-2}\Big{\\}}~{},$

where $\mbox{\boldmath${\cal K}$}_{\gamma{\dot{\gamma}}}$ is defined the same
way as in (3.17), $\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}=u^{\alpha}\bar{w}^{{\dot{\alpha}}}\mbox{\boldmath$X$}_{\alpha{\dot{\gamma}}}\mbox{\boldmath$X$}_{\gamma{\dot{\alpha}}}$.
In this basis, the (tilde) coefficients are related to those in (4.9c) by the
rule

$\displaystyle\tilde{a}_{1}=\frac{1}{k+1}a_{1}-a_{2}~{},\qquad\tilde{a}_{2}=\frac{k}{k+1}a_{1}+a_{2}~{},$
(4.11a)
$\displaystyle\tilde{b}_{1}=-\frac{k-1}{(k+1)(k+2)}b_{1}-\frac{2k+1}{(k+1)^{2}}(b_{2}+b_{3})-b_{4}+\frac{k^{2}+k+1}{(k+1)^{2}}b_{5}+b_{6}~{},$
(4.11b)
$\displaystyle\tilde{b}_{2}=\frac{1}{k+2}b_{1}+\frac{1}{k+1}(b_{2}+b_{3})+b_{4}-b_{5}~{},$
(4.11c)
$\displaystyle\tilde{b}_{3}=\frac{2k}{(k+1)(k+2)}b_{1}+\frac{2k}{(k+1)^{2}}b_{2}-\frac{k^{2}}{(k+1)^{2}}(b_{3}+b_{5})-b_{6}~{},$
(4.11d)
$\displaystyle\tilde{b}_{4}=\frac{2k}{(k+1)(k+2)}b_{1}-\frac{k^{2}}{(k+1)^{2}}b_{2}+\frac{2k}{(k+1)^{2}}b_{3}-\frac{k^{2}}{(k+1)^{2}}b_{5}-b_{6}~{},$
(4.11e) $\displaystyle\tilde{b}_{5}=\frac{k(k+2)}{(k+1)^{2}}b_{5}~{},$ (4.11f)
$\displaystyle\tilde{b}_{6}=\frac{k(k-1)}{(k+1)^{2}}\Big{(}\frac{k+1}{k+2}b_{1}+b_{2}+b_{3}+b_{5}\Big{)}+b_{6}~{}.$
(4.11g)

As in the previous section, the first order constraints in (4.7) are
equivalent to

$\displaystyle{\cal
D}_{(-1,0)}H_{(k,k),\,\gamma{\dot{\gamma}}}=0~{},\qquad\bar{{\cal
Q}}_{(0,-1)}H_{(k,k),\,\gamma{\dot{\gamma}}}=0~{}.$ (4.12)

Let us first look at the condition ${\cal
D}_{(-1,0)}H_{(k,k),\,\gamma{\dot{\gamma}}}=0$. With the help of identities
(3.26) and considering only the terms linear in
$\bar{\mbox{\boldmath$\Theta$}}_{{\dot{\alpha}}}$, we find the following
constraints on the coefficients

$\displaystyle k\tilde{a}_{1}+\tilde{a}_{2}=-\frac{{\rm
i}}{4}(k\tilde{b}_{1}+\tilde{b}_{3}-\tilde{b}_{4}+\tilde{b}_{6})~{},\qquad
k\tilde{a}_{1}=-\frac{{\rm
i}}{4}(-k\tilde{b}_{2}+\tilde{b}_{3}+\tilde{b}_{5}+\tilde{b}_{6})~{},$
$\displaystyle\qquad\qquad\qquad\qquad(k-1)\tilde{a}_{2}=-\frac{{\rm
i}}{4}((k-1)\tilde{b}_{3}-2\tilde{b}_{6})~{}.$ (4.13)

After some lengthy calculations, it can be shown that the terms cubic in the
Grassmann variables (that is,
$\sim\bar{\mbox{\boldmath$\Theta$}}^{2}\mbox{\boldmath$\Theta$}^{\delta}$)
result in

$\displaystyle
k(\tilde{b}_{1}+\tilde{b}_{2})+\tilde{b}_{3}+\tilde{b}_{5}+\tilde{b}_{6}=0~{},\qquad(k-1)\big{(}2\tilde{b}_{3}+\tilde{b}_{4}+2\tilde{b}_{5}\big{)}+(k-3)\tilde{b}_{6}=0~{}.$
(4.14)

In a similar way, one can compute $\bar{{\cal
Q}}_{(0,-1)}H_{(k,k),\,\gamma{\dot{\gamma}}}$ and show that the terms linear
in $\Theta$ lead to the requirements

$\displaystyle k\tilde{a}_{1}+\tilde{a}_{2}=-\frac{{\rm
i}}{4}(k\tilde{b}_{1}+\tilde{b}_{3}-\tilde{b}_{4}-k\tilde{b}_{5}+\tilde{b}_{6})~{},\qquad
k\tilde{a}_{1}=-\frac{{\rm
i}}{4}(-k\tilde{b}_{2}+\tilde{b}_{3}-k\tilde{b}_{5}+\tilde{b}_{6})~{},$
$\displaystyle\qquad\qquad\qquad\qquad(k-1)\tilde{a}_{2}=-\frac{{\rm
i}}{4}\big{(}(k-1)(\tilde{b}_{3}+\tilde{b}_{5})-2\tilde{b}_{6}\big{)}~{},$
(4.15)

while the terms proportional to
$\mbox{\boldmath$\Theta$}^{2}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}$
give

$\displaystyle
k(\tilde{b}_{1}+\tilde{b}_{2})+\tilde{b}_{3}+\tilde{b}_{6}=0~{},\qquad(k-1)\big{(}2\tilde{b}_{3}+\tilde{b}_{4}+\tilde{b}_{5}\big{)}+(k-3)\tilde{b}_{6}=0~{}.$
(4.16)

The system of equations (4.13)–(4.16) turn out to be consistent and can be
solved in terms of three independent coefficients which we choose to be
$\tilde{a}_{2},\tilde{b}_{1}$ and $\tilde{b}_{3}$:

$\displaystyle\tilde{b}_{5}=0~{},$
$\displaystyle\tilde{a}_{1}=-\frac{1}{4k}\Big{(}4(k-1)\tilde{a}_{2}+{\rm
i}\big{(}k\tilde{b}_{1}+(k+1)\tilde{b}_{3}\big{)}\Big{)}~{},$ (4.17)
$\displaystyle\tilde{b}_{2}=\frac{1}{2k}\Big{(}4{\rm
i}(k-1)\tilde{a}_{2}-2k\tilde{b}_{1}-(k+1)\tilde{b}_{3}\Big{)}~{},\qquad\tilde{b}_{4}=2{\rm
i}(k-3)\tilde{a}_{2}-\frac{1}{2}(k+1)\tilde{b}_{3}~{},$
$\displaystyle\tilde{b}_{6}=\frac{1}{2}(k-1)(\tilde{b}_{3}-4{\rm
i}\tilde{a}_{2})~{}.$

Keeping in mind the relations (4.11), one may rewrite (4.17) in the basis
(4.9). Choosing the three independent coefficients to be $b_{1},b_{2},b_{4}$,
we find that

$\displaystyle b_{5}=0~{},\qquad b_{6}=0~{},$ $\displaystyle a_{1}=\frac{{\rm
i}}{4}\Big{(}\frac{k+3}{k+1}b_{1}+\frac{k+2}{(k+1)^{2}}b_{2}+b_{4}\Big{)}~{},\qquad
a_{2}=-\frac{{\rm i}k}{4(k+1)}b_{4}~{},$ (4.18) $\displaystyle
b_{3}=\frac{k+3}{k+2}b_{1}+\frac{1}{k+1}b_{2}~{}.$

It remains to impose conservation equations on the supercurrent
$J_{\gamma{\dot{\gamma}}}$, eq. (4.8). As in section 3, this can be done by
appropriately rearranging the operators in the correlator and transforming the
tensor $H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}$. Let us rewrite
(4.4) in terms of the ${\cal I}$-operators:

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}=\frac{1}{k_{1}}{\cal
I}_{\beta(k){\dot{\alpha}}(k)}(x_{3\bar{1}}){\cal
I}_{\alpha(k){\dot{\beta}}(k)}(x_{2\bar{3}})H^{{\dot{\beta}}(k)\beta(k)}\,_{\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$
$\displaystyle
k_{1}:=(x_{\bar{1}3}{}^{2})^{(k+2)/2}(x_{\bar{3}2}{}^{2})^{(k+2)/2}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}~{}.$
(4.19)

On the other hand, the same correlator can be written as

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}=(-1)^{k}{\langle}{S}_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3})\bar{S}_{{\dot{\alpha}}(k)}(z_{1}){\rangle}$
$\displaystyle=\frac{(-1)^{k}}{k_{2}}{\cal
I}_{\alpha(k)\dot{\rho}(k)}(x_{2\bar{1}}){\cal
I}_{\gamma\dot{\delta}}(x_{3\bar{1}}){\cal
I}_{\delta\dot{\gamma}}(x_{1\bar{3}})\widetilde{H}^{\dot{\rho}(k),\,\dot{\delta}\delta}{}_{{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})~{},$
(4.20) $\displaystyle
k_{2}:=(x_{\bar{1}2}{}^{2})^{(k+2)/2}(x_{\bar{1}3}{}^{2})^{(k+2)/2}(x_{\bar{3}1}{}^{2})^{(k+2)/2}x_{\bar{2}1}{}^{2}~{}.$

Thus, knowing $H$, the tensor $\tilde{H}$ can be computed using the formula

$\displaystyle\widetilde{H}^{\dot{\mu}(k)}{}_{{\dot{\alpha}}(k),\,\delta\dot{\delta}}\,(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})$
(4.21) $\displaystyle=-\frac{k_{2}}{k_{1}}\,\bar{{\cal
I}}^{\dot{\mu}(k)\mu(k)}(\tilde{\bar{\mbox{\boldmath$X$}}}_{1})\Big{[}{\cal
I}_{{\dot{\alpha}}(k)}{}^{\beta(k)}(x_{3\bar{1}})\bar{{\cal
I}}_{\mu(k)}{}^{{\dot{\beta}}(k)}(\tilde{x}_{\bar{3}1}){{\cal
I}}_{\dot{\delta}}{}^{\gamma}({x}_{3\bar{1}})\bar{{\cal
I}}_{\delta}{}^{{\dot{\gamma}}}(\tilde{x}_{\bar{3}1})\Big{]}$
$\displaystyle\quad\times
H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$

where the identity (3.35a) has been used.

We now substitute the explicit form of
$H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}$ into (4). This is given
by (4.9), but with $b_{5}=b_{6}=0$, in accordance with the constraints (4).
After some tedious calculations and making use of (3.36), along with the
identities

$\displaystyle{\cal
I}_{\alpha{\dot{\alpha}}}(x_{1\bar{3}})\bar{\mbox{\boldmath$\Theta$}}_{3}^{{\dot{\alpha}}}=\bigg{(}\frac{x_{\bar{3}1}{}^{2}}{\bar{\mbox{\boldmath$X$}}_{1}^{2}}\bigg{)}^{\frac{1}{2}}\frac{\mbox{\boldmath$\bar{\Theta}$}_{1\alpha}^{I}}{x_{\bar{1}3}{}^{2}}~{},\qquad\mbox{\boldmath$\Theta$}_{3}^{\alpha}{\cal
I}_{\alpha{\dot{\alpha}}}(x_{3\bar{1}})=\bigg{(}\frac{x_{\bar{1}3}{}^{2}}{\mbox{\boldmath$X$}_{1}^{2}}\bigg{)}^{\frac{1}{2}}\frac{\mbox{\boldmath$\Theta$}_{1{\dot{\alpha}}}^{I}}{x_{\bar{3}1}{}^{2}}~{},$
(4.22a)
$\displaystyle\mbox{\boldmath$\Theta$}_{1{\dot{\alpha}}}^{I}:=\mbox{\boldmath$\Theta$}_{1}^{\alpha}{\cal
I}_{\alpha{\dot{\alpha}}}(-\mbox{\boldmath$X$}_{1})=-\frac{\mbox{\boldmath$\Theta$}_{1}^{\alpha}\mbox{\boldmath$X$}_{1\alpha{\dot{\alpha}}}}{(\mbox{\boldmath$X$}_{1}^{2})^{\frac{1}{2}}}~{},\quad\bar{\mbox{\boldmath$\Theta$}}_{1\alpha}^{I}:={\cal
I}_{\alpha{\dot{\alpha}}}(\bar{\mbox{\boldmath$X$}}_{1})\bar{\mbox{\boldmath$\Theta$}}_{1}^{{\dot{\alpha}}}=\frac{\bar{\mbox{\boldmath$X$}}_{1\alpha{\dot{\alpha}}}}{(\bar{\mbox{\boldmath$X$}}_{1}^{2})^{\frac{1}{2}}}\bar{\mbox{\boldmath$\Theta$}}_{1}^{{\dot{\alpha}}}~{},~{}~{}~{}~{}~{}$
(4.22b)

we obtain the following expression

$\displaystyle\widetilde{H}_{\delta\dot{\delta}}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\mbox{\boldmath$\bar{\Theta}$}_{1};\,\bar{v},\bar{w})=\bar{v}^{{\dot{\alpha}}_{1}}\dots\bar{v}^{{\dot{\alpha}}_{k}}\bar{w}_{\dot{\mu}_{1}}\dots\bar{w}_{\dot{\mu}_{k}}\widetilde{H}^{\dot{\mu}(k)}{}_{{\dot{\alpha}}(k),\,\delta\dot{\delta}}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\mbox{\boldmath$\bar{\Theta}$}_{1})$
$\displaystyle=\frac{(\bar{v}\cdot\bar{w})^{k-2}}{4(k+1)^{2}}\,\bigg{\\{}{\rm
i}\Big{[}(k+1)(k+3)b_{1}+(k+2)b_{2}+(k+1)^{2}b_{4}\Big{]}(\bar{v}\cdot\bar{w})^{2}\,\frac{\mbox{\boldmath$X$}_{1\delta\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{4}}$
$\displaystyle+{\rm
i}\Big{[}k(k+3)b_{1}+\frac{k(k+2)}{k+1}b_{2}\Big{]}(\bar{v}\cdot\bar{w})\,\frac{\mbox{\boldmath${\cal
V}$}_{1\delta\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{4}}+\Big{[}(k-1)b_{1}+\frac{k(k+2)}{k+1}b_{2}\Big{]}(\bar{v}\cdot\bar{w})^{2}\,\frac{\mbox{\boldmath$P$}_{1\delta\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{4}}$
$\displaystyle+4(k+1)b_{1}(\bar{v}\cdot\bar{w})^{2}\,\frac{(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath$X$}_{1})\mbox{\boldmath$X$}_{1\delta\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{6}}+\Big{[}8kb_{1}+\frac{4k(k+2)}{k+1}b_{2}\Big{]}(\bar{v}\cdot\bar{w})\,\frac{(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath$X$}_{1})\mbox{\boldmath${\cal
V}$}_{1\delta\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{6}}$
$\displaystyle-\Big{[}2k(k-1)b_{1}-\frac{2k(k+2)}{k+1}b_{2}\Big{]}(\bar{v}\cdot\bar{w})\,\frac{(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath${\cal
V}$}_{1})\mbox{\boldmath$X$}_{1\delta\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{6}}$
$\displaystyle+\Big{[}k(k+3)b_{1}+\frac{k(k+2)}{k+1}b_{2}\Big{]}(\bar{v}\cdot\bar{w})\,\frac{\mbox{\boldmath$P$}_{1}^{{\dot{\gamma}}\gamma}\mbox{\boldmath${\cal
V}$}_{1\delta{\dot{\gamma}}}\mbox{\boldmath$X$}_{1\gamma\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{6}}$
$\displaystyle-\Big{[}4k(k-1)b_{1}+\frac{2k(k-1)(k+2)}{k+1}b_{2}\Big{]}\frac{(\mbox{\boldmath$P$}_{1}\cdot\mbox{\boldmath${\cal
V}$}_{1})\mbox{\boldmath${\cal
V}$}_{1\delta\dot{\delta}}}{\mbox{\boldmath$X$}_{1}^{6}}\bigg{\\}}~{}.$ (4.23)

We recall that $b_{1},b_{2}$ and $b_{4}$ are independent. We have also defined
$\mbox{\boldmath${\cal
V}$}_{\alpha{\dot{\alpha}}}:=\mbox{\boldmath$X$}_{\alpha{\dot{\beta}}}\,\bar{v}_{{\dot{\alpha}}}\bar{w}^{{\dot{\beta}}}~{}.$

Going back to eq. (4.20) and relabelling superspace points, we see that

$\displaystyle{\langle}S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3})\bar{S}_{{\dot{\alpha}}(k)}(z_{1}){\rangle}=[\,\text{relabel}\,z_{2}\rightarrow
z_{1},\,z_{3}\rightarrow z_{2},\,z_{1}\rightarrow z_{3}]$
$\displaystyle={\langle}S_{\alpha(k)}(z_{1})J_{\gamma{\dot{\gamma}}}(z_{2})\bar{S}_{{\dot{\alpha}}(k)}(z_{3}){\rangle}$
(4.24) $\displaystyle=-\frac{1}{k_{3}}{\cal
I}_{\alpha(k)\dot{\mu}(k)}(x_{1\bar{3}})\bar{{\cal
I}}_{\gamma}{}^{\dot{\delta}}(\tilde{x}_{\bar{3}2}){{\cal
I}}_{\dot{\gamma}}{}^{\delta}({x}_{3\bar{2}})\widetilde{H}^{\dot{\mu}(k)}{}_{{\dot{\alpha}}(k),\,\delta\dot{\delta}}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})~{},$
$\displaystyle
k_{3}:=(x_{\bar{3}1}{}^{2})^{(k+2)/2}(x_{\bar{3}2}{}^{2})^{(k+2)/2}(x_{\bar{2}3}{}^{2})^{(k+2)/2}x_{\bar{1}3}{}^{2}~{}.$

By virtue of eq. (2.27), the conservation law of $J_{\gamma{\dot{\gamma}}}$
requires that the following equations must hold:

$\displaystyle{\cal
Q}^{\delta}\widetilde{H}_{\delta\dot{\delta}}(\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\mbox{\boldmath$\bar{\Theta}$}_{3};\,\bar{v},\bar{w})=0~{},\qquad\bar{\cal
Q}^{\dot{\delta}}\widetilde{H}_{\delta\dot{\delta}}(\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\mbox{\boldmath$\bar{\Theta}$}_{3};\,\bar{v},\bar{w})=0~{}.$
(4.25)

It is not hard to verify that (4.23) satisfies (4.25) for an arbitrary choice
of complex coefficients $b_{1},b_{2},b_{4}$.

The last step is to impose the reality condition (4.6):

$\displaystyle\bar{H}_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})=H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}(-\mbox{\boldmath$\bar{X}$},-{\mbox{\boldmath$\Theta$}},-\bar{\mbox{\boldmath$\Theta$}})~{}.$
(4.26)

Here
$\bar{H}_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})$
means that we are taking the complex conjugate of the expression (4.9). This
implies that

$\displaystyle b_{1}={\rm i}^{k}A~{},\qquad b_{2}={\rm i}^{k}B~{},\qquad
b_{4}={\rm i}^{k}C~{},$ (4.27)

where $A,B,C$ are real coefficients.

The correlator (4.4) is thus fixed up to three real coefficients. It is given
by

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}}=\frac{(x_{3\bar{1}})_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\dots(x_{3\bar{1}})_{{\dot{\alpha}}_{k})}{}^{\beta_{k}}(x_{2\bar{3}})_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\dots(x_{2\bar{3}})_{\alpha_{k})}{}^{{\dot{\beta}}_{k}}}{(x_{\bar{1}3}{}^{2})^{k+1}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{k+1}}$
$\displaystyle\hskip 142.26378pt\times
H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}\big{(}{\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}}\big{)}~{}.$
(4.28a) For convenience, here we rewrite explicitly the form of
$H_{\beta(k){\dot{\beta}}(k),\,\gamma{\dot{\gamma}}}\big{(}{\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}}\big{)}$
in the basis (4) $\displaystyle
H_{(k,k),\gamma{\dot{\gamma}}}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{w}):=u^{\beta_{1}}\dots
u^{\beta_{k}}\bar{w}^{{\dot{\beta}}_{1}}\dots\bar{w}^{{\dot{\beta}}_{k}}H_{\beta(k){\dot{\beta}}(k),\gamma{\dot{\gamma}}}$
$\displaystyle=\frac{\tilde{a}_{1}}{\mbox{\boldmath$X$}^{2k+2}}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}^{k}_{(1,1)}+\frac{\tilde{a}_{2}}{\mbox{\boldmath$X$}^{2k+2}}\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}^{k-1}_{(1,1)}$ (4.28b)
$\displaystyle+\frac{\mbox{\boldmath$\Theta$}^{\delta}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}}{\mbox{\boldmath$X$}^{2k+4}}\Big{\\{}\tilde{b}_{1}\mbox{\boldmath$X$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{(1,1)}^{k}+\tilde{b}_{2}\mbox{\boldmath$X$}_{\gamma\dot{\delta}}\mbox{\boldmath$X$}_{\delta\dot{\gamma}}\mbox{\boldmath$X$}_{(1,1)}^{k}+\tilde{b}_{3}\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath$X$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{(1,1)}^{k-1}$
$\displaystyle\qquad\qquad+\tilde{b}_{4}\mbox{\boldmath${\cal
K}$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{\gamma\dot{\gamma}}\mbox{\boldmath$X$}_{(1,1)}^{k-1}+\tilde{b}_{6}\mbox{\boldmath${\cal
K}$}_{\gamma{\dot{\gamma}}}\mbox{\boldmath${\cal
K}$}_{\delta\dot{\delta}}\mbox{\boldmath$X$}_{(1,1)}^{k-2}\Big{\\}}~{},$ where
$\displaystyle\tilde{a}_{1}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}^{k+1}}{4(k+1)^{3}}\Big{(}(k+1)(k+3)A+(k+2)B+(k+1)^{3}C\Big{)}~{},$
$\displaystyle\tilde{a}_{2}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}^{k+1}k}{4(k+1)^{3}}\Big{(}(k+1)(k+3)A+(k+2)B\Big{)}~{},$
$\displaystyle\tilde{b}_{1}$ $\displaystyle=$ $\displaystyle-\frac{{\rm
i}^{k}}{(k+1)^{3}}\Big{(}(k+1)(3k+1)A+(2k+1)(k+2)B+(k+1)^{3}C\Big{)}~{},$
$\displaystyle\tilde{b}_{2}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}^{k}}{(k+1)^{2}}\Big{(}2(k+1)A+(k+2)B+(k+1)^{2}C\Big{)}~{},$ (4.28c)
$\displaystyle\tilde{b}_{3}$ $\displaystyle=$ $\displaystyle-\frac{{\rm
i}^{k}k}{(k+1)^{3}}\Big{(}(k^{2}-1)A-(k+2)B\Big{)}~{},$
$\displaystyle\tilde{b}_{4}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}^{k}}{(k+1)^{3}}\Big{(}4k(k+1)A-k(k+2)(k-1)B\Big{)}~{},$
$\displaystyle\tilde{b}_{6}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}^{k}k(k-1)}{(k+1)^{3}}\Big{(}2(k+1)A+(k+2)B\Big{)}~{}.$

Setting $k=1$ and comparing (4.28) with (4.3), we see that we have an
agreement with the parameters related by

$d_{2}=-\frac{3B}{16}~{},\qquad d_{3}=-\frac{A}{2}~{},\qquad
d_{4}=-\frac{1}{8}(3B+4(A+C))~{}.$ (4.29)

In a more general setting, one may also consider

$\displaystyle{\langle}\bar{S}^{\prime}_{{\dot{\alpha}}(k)}(z_{1})S_{\alpha(k)}(z_{2})J_{\gamma{\dot{\gamma}}}(z_{3}){\rangle}~{}.$
(4.30)

The only difference with (4.28a) is that now
$(S_{\alpha(k)})^{*}\neq\bar{S}^{\prime}_{{\dot{\alpha}}(k)}$. Since one can
no longer impose the reality constraint (4.6), the correlator (4.30) is
determined up to three complex coefficients, i.e. $b_{1},b_{2}$ and $b_{4}$.

##  5 Correlator
${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}$

We consider a three-point function of the higher-spin spinor current
multiplets in the form
${{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}$.444The
correlator
${\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(m)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}$
with $m\neq k+\ell$ vanishes because it carries a non-trivial $R$-symmetry
charge. Its general form is given by

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}$
$\displaystyle=$
$\displaystyle\frac{(x_{3\bar{1}})_{({\dot{\alpha}}_{1}}{}^{\alpha_{1}}\dots(x_{3\bar{1}})_{{\dot{\alpha}}_{k})}{}^{\alpha_{k}}(x_{2\bar{3}})_{(\beta_{1}}{}^{{\dot{\beta}}_{1}}\dots(x_{2\bar{3}})_{\beta_{k+l})}{}^{{\dot{\beta}}_{k+l}}}{(x_{\bar{1}3}{}^{2})^{k+1}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{k+l+1}}$
(5.1) $\displaystyle\times
H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$

with $k,\,l=1,2,\dots$. The tensor
$H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}$ is subject to the
following constraints:

  * •

Homogeneity:

$\displaystyle
H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}(\lambda\bar{\lambda}\mbox{\boldmath$X$},\lambda\mbox{\boldmath$\Theta$},\bar{\lambda}\bar{\mbox{\boldmath$\Theta$}})=\lambda^{-(k+2)}\bar{\lambda}^{-(k+2)}H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})~{},$
(5.2)

  * •

Symmetry under $z_{1}\leftrightarrow z_{3}$:

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}=(-1)^{k+l+kl}{\langle}\bar{S}_{{\dot{\gamma}}(l)}(z_{3})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\alpha}}(k)}(z_{1}){\rangle}$
$\displaystyle=(-1)^{k+l+kl}{\langle}\bar{S}_{{\dot{\alpha}}(l)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(k)}(z_{3}){\rangle}\bigg{|}_{z_{1}\leftrightarrow
z_{3},~{}{\dot{\alpha}}\leftrightarrow{\dot{\gamma}}}~{},$ (5.3)

where we have taken into account the Grassmann parities of the superfields.

  * •

Conservation conditions:

$\displaystyle{\cal
D}^{\alpha_{1}}H_{\alpha_{1}\dots\alpha_{k}{\dot{\beta}}(k+l),{\dot{\gamma}}(l)}=0~{},\qquad\bar{\cal
D}^{2}H_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}=0~{},$ (5.4a)
$\displaystyle\bar{\cal
Q}^{{\dot{\beta}}_{1}}H_{\alpha(k){\dot{\beta}}_{1}\dots\dot{\beta}_{k+l},{\dot{\gamma}}(l)}=0~{},\qquad{\cal
Q}^{2}H_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}=0~{}.$ (5.4b)

Upon imposing (5.4), it follows from the symmetry property (5.3) that the
conservation conditions at $z_{3}$ are automatically satisfied,

$\bar{D}^{{\dot{\gamma}}_{1}}_{(3)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}=0~{},\quad
D^{2}_{(3)}{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}=0~{}.$
(5.5)

The constraints $\bar{\cal
D}^{2}H_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}=0$ and ${\cal
Q}^{2}H_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}=0$ tell us that
$H_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}$, being Grassmann even, can
be expressed in the form

$H_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}})=A_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}(\mbox{\boldmath$X$})+B_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l),\delta\dot{\delta}}(\mbox{\boldmath$X$}){\mbox{\boldmath$\Theta$}}^{\delta}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}~{}.$
(5.6a) Let us then contract the indices using auxiliary variables and list all
possible independent structures consistent with (5.2). The independent tensor
structures can be constructed by performing decomposition of
$A_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l)}(\mbox{\boldmath$X$})$ and
$B_{\alpha(k){\dot{\beta}}(k+l),{\dot{\gamma}}(l),\delta\dot{\delta}}(\mbox{\boldmath$X$})$
into the irreducible components. We obtain $\displaystyle
H_{(k,k+l,l)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{v},\bar{w})$
$\displaystyle=u^{\alpha_{1}}\dots
u^{\alpha_{k}}\bar{w}^{{\dot{\beta}}_{1}}\dots\bar{w}^{{\dot{\beta}}_{k+l}}\bar{v}^{{\dot{\gamma}}_{1}}\dots\bar{v}^{{\dot{\gamma}}_{l}}\bigg{\\{}\frac{a_{1}}{\mbox{\boldmath$X$}^{2k+2}}\varepsilon_{{\dot{\gamma}}_{1}{\dot{\beta}}_{1}}\dots\varepsilon_{{\dot{\gamma}}_{l}{\dot{\beta}}_{l}}\mbox{\boldmath$X$}_{\alpha_{1}{\dot{\beta}}_{l+1}}\dots\mbox{\boldmath$X$}_{\alpha_{k}{\dot{\beta}}_{k+l}}$
$\displaystyle+\frac{\mbox{\boldmath$\Theta$}^{\delta}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}}{\mbox{\boldmath$X$}^{2k+2}}\Big{(}\frac{b_{1}}{\mbox{\boldmath$X$}^{2}}~{}\varepsilon_{{\dot{\gamma}}_{l}\dot{\delta}}\varepsilon_{{\dot{\gamma}}_{1}{\dot{\beta}}_{1}}\dots\varepsilon_{{\dot{\gamma}}_{l-1}{\dot{\beta}}_{l-1}}\mbox{\boldmath$X$}_{(\delta{\dot{\beta}}_{l}}\mbox{\boldmath$X$}_{\alpha_{1}{\dot{\beta}}_{l+1}}\dots\mbox{\boldmath$X$}_{\alpha_{k}){\dot{\beta}}_{k+l}}$
$\displaystyle\qquad\qquad+\frac{b_{2}}{\mbox{\boldmath$X$}^{2}}~{}\varepsilon_{{\dot{\beta}}_{l}\dot{\delta}}\varepsilon_{{\dot{\gamma}}_{1}{\dot{\beta}}_{1}}\dots\varepsilon_{{\dot{\gamma}}_{l-1}{\dot{\beta}}_{l-1}}\mbox{\boldmath$X$}_{(\delta{\dot{\gamma}}_{l}}\mbox{\boldmath$X$}_{\alpha_{1}{\dot{\beta}}_{l+1}}\dots\mbox{\boldmath$X$}_{\alpha_{k}){\dot{\beta}}_{k+l}}$
$\displaystyle\qquad\qquad+\,b_{3}~{}\varepsilon_{\delta\alpha_{1}}\varepsilon_{\dot{\delta}{\dot{\beta}}_{1}}\varepsilon_{{\dot{\gamma}}_{1}{\dot{\beta}}_{2}}\dots\varepsilon_{{\dot{\gamma}}_{l}{\dot{\beta}}_{l+1}}\mbox{\boldmath$X$}_{\alpha_{2}{\dot{\beta}}_{l+2}}\dots\mbox{\boldmath$X$}_{\alpha_{k}{\dot{\beta}}_{k+l}}\Big{)}\bigg{\\}}$
(5.6b)
$\displaystyle=a_{1}(\bar{v}\cdot\bar{w})^{l}\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}+b_{1}(\bar{v}\cdot\bar{w})^{l-1}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}$
$\displaystyle+b_{2}(\bar{v}\cdot\bar{w})^{l-1}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{v})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}$
$\displaystyle+\Big{(}b_{3}-\frac{k}{k+1}b_{2}\Big{)}(\bar{v}\cdot\bar{w})^{l}(u\cdot\mbox{\boldmath$\Theta$})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}~{}.$

All the coefficients above are complex. Looking at the conformally covariant
structures in (5.6b) after contraction with the auxiliary spinors, it might
appear that there are more admissible structures which can be added to the
list, namely

$\displaystyle(\bar{v}\cdot\bar{w})^{l}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(u\mbox{\boldmath$X$}\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}~{},\qquad(\bar{v}\cdot\bar{w})^{l}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}~{},$
$\displaystyle\qquad\qquad(\bar{v}\cdot\bar{w})^{l-1}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})(u\mbox{\boldmath$X$}\bar{v})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}~{}.$
(5.7)

However, these structures prove to be linearly dependent of the others. More
precisely, one can prove that

$\displaystyle(\bar{v}\cdot\bar{w})^{l}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(u\mbox{\boldmath$X$}\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}=(\bar{v}\cdot\bar{w})^{l-1}\Big{[}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})-(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{v})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\Big{]}\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(\bar{v}\cdot\bar{w})^{l}(u\cdot\mbox{\boldmath$\Theta$})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}~{},$
(5.8a)
$\displaystyle(\bar{v}\cdot\bar{w})^{l}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}=(\bar{v}\cdot\bar{w})^{l-1}\Big{[}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})-(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{v})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\Big{]}\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}~{},$
(5.8b)
$\displaystyle(\bar{v}\cdot\bar{w})^{l-1}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})(u\mbox{\boldmath$X$}\bar{v})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}$
$\displaystyle=(\bar{v}\cdot\bar{w})^{l-1}\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}\Big{[}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{v})\frac{\mbox{\boldmath$X$}_{(1,1)}}{\mbox{\boldmath$X$}^{2}}-(\bar{v}\cdot\bar{w})(u\cdot\mbox{\boldmath$\Theta$})\Big{]}(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})~{}.$
(5.8c)

Next, we impose conservation laws (5.4). The first condition, ${\cal
D}^{\alpha_{1}}H_{\alpha_{1}\dots\alpha_{k}{\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}=0$,
is equivalent to requiring

$\displaystyle{\cal
D}_{(-1,0,0)}H_{(k,k+l,l)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{v},\bar{w})=0~{},\qquad{\cal
D}_{(-1,0,0)}:={\cal D}^{\alpha}\frac{\partial}{\partial u^{\alpha}}~{}.$
(5.9)

Keeping in mind the relations (3.26) and further noting that

$\displaystyle{\cal
D}_{(-1,0,0)}\bigg{(}\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}\bigg{)}={\cal
D}_{(-1,0,0)}\bigg{(}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}\bigg{)}=0~{},$
(5.10)

it can be easily checked that (5.9) gives rise to

$\displaystyle b_{3}=0~{}.$ (5.11)

The second conservation condition reads

$\displaystyle\bar{{\cal Q}}_{(0,-1,0)}H_{(k,k+l,l)}=0~{},\qquad\bar{\cal
Q}_{(0,-1,0)}:=\bar{\cal
Q}^{{\dot{\alpha}}}\frac{\partial}{\partial\bar{w}^{{\dot{\alpha}}}}~{}.$
(5.12)

Using the fact that $b_{3}=0$, we obtain

$\displaystyle 0$ $\displaystyle=\bar{\cal Q}_{(0,-1,0)}H_{(k,k+l,l)}$ (5.13)
$\displaystyle=\big{[}4{\rm i}l(k+1)a_{1}-(k+1)b_{1}-(k+l+1)b_{2}\big{]}$
$\displaystyle\quad\times\bigg{\\{}(\bar{v}\cdot\bar{w})^{l-1}(\mbox{\boldmath$\Theta$}\mbox{\boldmath$X$}\bar{v})\frac{\mbox{\boldmath$X$}^{k}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+4}}-\frac{k}{k+1}(\bar{v}\cdot\bar{w})^{l}(u\cdot\mbox{\boldmath$\Theta$})\frac{\mbox{\boldmath$X$}^{k-1}_{(1,1)}}{\mbox{\boldmath$X$}^{2k+2}}\bigg{\\}}$
$\displaystyle\implies b_{2}=\frac{k+1}{k+l+1}\big{(}4{\rm
i}\,la_{1}-b_{1}\big{)}~{}.$

Imposing conservation laws thus leaves us with only two independent
coefficients, which we choose to be $a_{1}$ and $b_{1}$.

We turn to analysing the algebraic constraint (5.3). Under the exchange
$z_{1}\leftrightarrow z_{3}$, the three-point function possesses the property

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}=(-1)^{k+l+kl}{\langle}\bar{S}_{{\dot{\gamma}}(l)}(z_{3})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\alpha}}(k)}(z_{1}){\rangle}~{}$
$\displaystyle=(-1)^{k+l+kl}{\langle}\bar{S}_{{\dot{\alpha}}(l)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(k)}(z_{3}){\rangle}\bigg{|}_{z_{1}\leftrightarrow
z_{3},~{}{\dot{\alpha}}\leftrightarrow{\dot{\gamma}}}~{}.$ (5.14)

As per usual, upon expressing the first line of (5.14) in terms of the ${\cal
I}$-operators, we have

$\displaystyle\frac{(-1)^{k+l}}{k_{1}}{\cal
I}_{{\dot{\alpha}}(k)}{}^{\alpha(k)}(x_{3\bar{1}})\bar{{\cal
I}}_{\beta(k+l)}{}^{{\dot{\beta}}(k+l)}(\tilde{x}_{\bar{3}2})H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})$
(5.15) $\displaystyle=(-1)^{kl}\,\frac{1}{k_{2}}{\cal
I}_{{\dot{\gamma}}(l)}{}^{\rho(l)}(x_{1\bar{3}})\bar{{\cal
I}}_{\beta(k+l)}{}^{\dot{\rho}(k+l)}(\tilde{x}_{\bar{1}2})\widetilde{H}_{\rho(l)\dot{\rho}(k+l),\,{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})~{},$
$\displaystyle
k_{1}:=(x_{\bar{1}3}{}^{2})^{(k+2)/2}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{(k+l+2)/2}~{},\qquad
k_{2}:=x_{\bar{1}3}{}^{2}(x_{\bar{3}1}{}^{2})^{(l+2)/2}x_{\bar{2}1}{}^{2}(x_{\bar{1}2}{}^{2})^{(k+l+2)/2}~{},~{}$

which then yields the relation

$\displaystyle\widetilde{H}_{\rho(l)\dot{\rho}(k+l),\,{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})$
$\displaystyle=(-1)^{kl+l}\,\frac{k_{2}}{k_{1}}{\cal
I}_{\beta(k+l)\dot{\rho}(k+l)}({\bar{\mbox{\boldmath$X$}}}_{1})$ (5.16)
$\displaystyle\quad\times\,{\cal
I}_{{\dot{\alpha}}(k)}{}^{\alpha(k)}(x_{3\bar{1}})\bar{{\cal
I}}^{{\dot{\beta}}(k+l)\beta(k+l)}(\tilde{x}_{\bar{3}1})\bar{{\cal
I}}_{\rho(l)}{}^{{\dot{\gamma}}(l)}(\tilde{x}_{\bar{3}1})H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{}.~{}~{}$

On the other hand, from the second line of (5.14), it follows that

$\displaystyle\widetilde{H}_{\rho(l)\dot{\rho}(k+l),\,{\dot{\alpha}}(k)}(\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}})=H_{\rho(l)\dot{\rho}(k+l),\,{\dot{\alpha}}(k)}(-\bar{\mbox{\boldmath$X$}},-\mbox{\boldmath$\Theta$},-\bar{\mbox{\boldmath$\Theta$}})~{},$
(5.17)

where (2.32c) has been used.

By virtue of eqs. (5.6b), (5.11) and (5.13), the right-hand side of (5.17)
takes the form

$\displaystyle
H_{(l,k+l,k)}(-\bar{\mbox{\boldmath$X$}},-\mbox{\boldmath$\Theta$},-\bar{\mbox{\boldmath$\Theta$}};\,u,\bar{v},\bar{w})=u^{\rho(l)}\bar{w}^{\dot{\rho}(k+l)}\bar{v}^{{\dot{\alpha}}(k)}H_{\rho(l)\dot{\rho}(k+l),\,{\dot{\alpha}}(k)}(-\bar{\mbox{\boldmath$X$}},-\mbox{\boldmath$\Theta$},-\bar{\mbox{\boldmath$\Theta$}})$
$\displaystyle=(-1)^{l}\bigg{\\{}a_{1}(\bar{v}\cdot\bar{w})^{k}\frac{\bar{\mbox{\boldmath$X$}}^{l}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+2}}-b_{1}(\bar{v}\cdot\bar{w})^{k-1}(\mbox{\boldmath$\Theta$}\bar{\mbox{\boldmath$X$}}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\bar{\mbox{\boldmath$X$}}^{l}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+4}}$
$\displaystyle+\frac{(\bar{v}\cdot\bar{w})^{k-1}}{k+l+1}(b_{1}-4{\rm
i}ka_{1})\bigg{[}(l+1)(\mbox{\boldmath$\Theta$}\bar{\mbox{\boldmath$X$}}\bar{v})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\bar{\mbox{\boldmath$X$}}^{l}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+4}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-l\,(\bar{v}\cdot\bar{w})(u\cdot\mbox{\boldmath$\Theta$})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\bar{\mbox{\boldmath$X$}}^{l-1}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+2}}\bigg{]}\bigg{\\}}~{}.$
(5.18)

Direct computation of (5) gives rise to

$\displaystyle\widetilde{H}_{(l,k+l,k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{v},\bar{w})=u^{\rho(l)}\bar{w}^{\dot{\rho}(k+l)}\bar{v}^{{\dot{\alpha}}(k)}~{}\frac{(-1)^{kl+l}}{\mbox{\boldmath$X$}^{2}\bar{\mbox{\boldmath$X$}}^{2l}}$
$\displaystyle\times\bigg{\\{}a_{1}\varepsilon_{{\dot{\alpha}}_{1}\dot{\rho}_{1}}\dots\varepsilon_{{\dot{\alpha}}_{k}\dot{\rho}_{k}}\bar{\mbox{\boldmath$X$}}_{\rho_{1}\dot{\rho}_{k+1}}\dots\bar{\mbox{\boldmath$X$}}_{\rho_{l}\dot{\rho}_{k+l}}$
$\displaystyle+\frac{b_{1}}{\mbox{\boldmath$X$}^{2}}\mbox{\boldmath$\Theta$}^{\delta}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}\varepsilon_{{\dot{\alpha}}_{1}\dot{\rho}_{1}}\dots\varepsilon_{{\dot{\alpha}}_{k}\dot{\rho}_{k}}\mbox{\boldmath$X$}_{|\delta|\dot{\rho}_{k+1}}\bar{\mbox{\boldmath$X$}}_{\rho_{1}|\dot{\delta}|}\bar{\mbox{\boldmath$X$}}_{\rho_{2}\dot{\rho}_{k+2}}\dots\bar{\mbox{\boldmath$X$}}_{\rho_{l}\dot{\rho}_{k+l}}$
$\displaystyle-\frac{1}{{\mbox{\boldmath$X$}}^{2}}\frac{(4{\rm
i}la_{1}-b_{1})}{k+l+1}\bigg{[}(k+1)\varepsilon_{{\dot{\alpha}}_{1}\dot{\rho}_{1}}\dots\varepsilon_{{\dot{\alpha}}_{k}\dot{\rho}_{k}}\mbox{\boldmath$X$}^{\dot{\delta}\delta}\mbox{\boldmath$\Theta$}_{|\delta|}\bar{\mbox{\boldmath$\Theta$}}_{\dot{\rho}_{k+1}}\bar{\mbox{\boldmath$X$}}_{\rho_{1}\dot{\rho}_{k+2}}\dots\bar{\mbox{\boldmath$X$}}_{\rho_{l-1}\dot{\rho}_{k+l}}\bar{\mbox{\boldmath$X$}}_{\rho_{l}|\dot{\delta}|}$
$\displaystyle\qquad\qquad-k\bar{\mbox{\boldmath$X$}}_{\rho_{1}\dot{\rho}_{1}}\dots\bar{\mbox{\boldmath$X$}}_{\rho_{l}\dot{\rho}_{l}}\mbox{\boldmath$\Theta$}^{\delta}\bar{\mbox{\boldmath$\Theta$}}^{\dot{\delta}}\mbox{\boldmath$X$}_{|\delta|{\dot{\alpha}}_{1}}\varepsilon_{|\dot{\delta}|\dot{\rho}_{l+1}}\varepsilon_{{\dot{\alpha}}_{2}\dot{\rho}_{l+2}}\dots\varepsilon_{{\dot{\alpha}}_{k}\dot{\rho}_{k+l}}\bigg{]}\bigg{\\}}~{}.~{}~{}~{}$
(5.19)

In order to compare this result with eq. (5.18) we will use eq. (2.21) to
express $X$ in terms of $\bar{\mbox{\boldmath$X$}}$. This gives

$\displaystyle\widetilde{H}_{(l,k+l,k)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{v},\bar{w})$
$\displaystyle=(-1)^{kl+l}\bigg{\\{}a_{1}(\bar{v}\cdot\bar{w})^{k}\frac{\bar{\mbox{\boldmath$X$}}^{l}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+2}}+(4{\rm
i}a_{1}+b_{1})(\bar{v}\cdot\bar{w})^{k-1}(\mbox{\boldmath$\Theta$}\bar{\mbox{\boldmath$X$}}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\bar{\mbox{\boldmath$X$}}^{l}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+4}}$
$\displaystyle-\frac{(\bar{v}\cdot\bar{w})^{k-1}}{k+l+1}\big{(}b_{1}+4{\rm
i}(k+1)a_{1}\big{)}\bigg{[}(l+1)(\mbox{\boldmath$\Theta$}\bar{\mbox{\boldmath$X$}}\bar{v})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\bar{\mbox{\boldmath$X$}}^{l}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+4}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad-l\,(\bar{v}\cdot\bar{w})(u\cdot\mbox{\boldmath$\Theta$})(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{\bar{\mbox{\boldmath$X$}}^{l-1}_{(1,1)}}{\bar{\mbox{\boldmath$X$}}^{2l+2}}\bigg{]}\bigg{\\}}~{}.$
(5.20)

Equating (5.18) and (5.20), it can be seen that the coefficients are
constrained by

$\displaystyle a_{1}=(-1)^{kl}a_{1}~{},$ $\displaystyle
b_{1}=(-1)^{kl+1}(4{\rm i}a_{1}+b_{1})~{},\qquad b_{1}-4{\rm
i}ka_{1}=(-1)^{kl+1}(b_{1}+4{\rm i}(k+1)a_{1})~{}.~{}~{}~{}~{}$ (5.21)

As a result, we have two scenarios to consider. The first case is when both
$k$ and $l$ are odd. Here $a_{1}=0$ and $b_{1}$ is an arbitrary complex
parameter. The other case is when at least one of them is even, for which both
parameters are now non-vanishing but related by $a_{1}=\dfrac{{\rm
i}b_{1}}{2}$.

The three-point correlator of the higher-spin spinor current multiplets is
thus determined up to a single complex coefficient $b_{1}\equiv B$. It has the
following structure

$\displaystyle{{\langle}\bar{S}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}}$
$\displaystyle=$
$\displaystyle\frac{(x_{3\bar{1}})_{({\dot{\alpha}}_{1}}{}^{\alpha_{1}}\dots(x_{3\bar{1}})_{{\dot{\alpha}}_{k})}{}^{\alpha_{k}}(x_{2\bar{3}})_{(\beta_{1}}{}^{{\dot{\beta}}_{1}}\dots(x_{2\bar{3}})_{\beta_{k+l})}{}^{{\dot{\beta}}_{k+l}}}{(x_{\bar{1}3}{}^{2})^{k+1}x_{\bar{3}1}{}^{2}x_{\bar{2}3}{}^{2}(x_{\bar{3}2}{}^{2})^{k+l+1}}$
(5.22a) $\displaystyle\times
H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}({\mbox{\boldmath$X$}_{3},\mbox{\boldmath$\Theta$}_{3},\bar{\mbox{\boldmath$\Theta$}}_{3}})~{},$
where the functional form of
$H_{\alpha(k){\dot{\beta}}(k+l),\,{\dot{\gamma}}(l)}$ depends on the values of
$k$ and $l$. There are two different cases to consider:

  * •

$\mbox{\boldmath$(k,l)$}=$ (odd, odd)

$\displaystyle
H_{(k,k+l,l)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{v},\bar{w})=B(\bar{v}\cdot\bar{w})^{l-1}\bigg{\\{}(\mbox{\boldmath$\Theta$}{\mbox{\boldmath$X$}}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{{\mbox{\boldmath$X$}}^{k}_{(1,1)}}{{\mbox{\boldmath$X$}}^{2k+4}}$
$\displaystyle+\frac{(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})}{k+l+1}\bigg{[}(k+1)(\mbox{\boldmath$\Theta$}{\mbox{\boldmath$X$}}\bar{v})\frac{{\mbox{\boldmath$X$}}^{k}_{(1,1)}}{{\mbox{\boldmath$X$}}^{2k+4}}-k\,(\bar{v}\cdot\bar{w})(u\cdot\mbox{\boldmath$\Theta$})\frac{{\mbox{\boldmath$X$}}^{k-1}_{(1,1)}}{{\mbox{\boldmath$X$}}^{2k+2}}\bigg{]}\bigg{\\}}~{}.~{}~{}~{}~{}~{}~{}~{}~{}$
(5.22b)

  * •

$\mbox{\boldmath$(k,l)$}=$ (even, even), (odd, even), (even, odd)

$\displaystyle
H_{(k,k+l,l)}({\mbox{\boldmath$X$},\mbox{\boldmath$\Theta$},\bar{\mbox{\boldmath$\Theta$}}};\,u,\bar{v},\bar{w})$
$\displaystyle=B(\bar{v}\cdot\bar{w})^{l-1}\bigg{\\{}\frac{{\rm
i}}{2}(\bar{v}\cdot\bar{w})\frac{{\mbox{\boldmath$X$}}^{k}_{(1,1)}}{{\mbox{\boldmath$X$}}^{2k+2}}+(\mbox{\boldmath$\Theta$}{\mbox{\boldmath$X$}}\bar{w})(\bar{v}\cdot\bar{\mbox{\boldmath$\Theta$}})\frac{{\mbox{\boldmath$X$}}^{k}_{(1,1)}}{{\mbox{\boldmath$X$}}^{2k+4}}$
$\displaystyle+\frac{2l+1}{k+l+1}(\bar{w}\cdot\bar{\mbox{\boldmath$\Theta$}})\bigg{[}(k+1)(\mbox{\boldmath$\Theta$}{\mbox{\boldmath$X$}}\bar{v})\frac{{\mbox{\boldmath$X$}}^{k}_{(1,1)}}{{\mbox{\boldmath$X$}}^{2k+4}}-k\,(\bar{v}\cdot\bar{w})(u\cdot\mbox{\boldmath$\Theta$})\frac{{\mbox{\boldmath$X$}}^{k-1}_{(1,1)}}{{\mbox{\boldmath$X$}}^{2k+2}}\bigg{]}\bigg{\\}}~{}.~{}~{}~{}~{}~{}~{}~{}~{}$
(5.22c)

In a more general setting, one may also consider

$\displaystyle{\langle}\bar{S}^{\prime}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}^{\prime\prime}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}~{},$
(5.23)

where the correlator now does not satisfy (5.3) under the exchange
$z_{1}\leftrightarrow z_{3}$. As a result, one has to explicitly check the
conservation equation at $z_{3}$, that is

$\displaystyle\bar{D}_{(3)}^{{\dot{\gamma}}_{1}}{\langle}\bar{S}^{\prime}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}^{\prime\prime}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}=0~{},\quad
D^{2}_{(3)}{\langle}\bar{S}^{\prime}_{{\dot{\alpha}}(k)}(z_{1})S_{\beta(k+l)}(z_{2})\bar{S}^{\prime\prime}_{{\dot{\gamma}}(l)}(z_{3}){\rangle}=0~{}.~{}~{}~{}~{}~{}$
(5.24)

Making use of the relation (5.15) and the expression of
$\widetilde{H}_{\rho(l)\dot{\rho}(k+l),\,{\dot{\alpha}}(k)}(\mbox{\boldmath$X$}_{1},\mbox{\boldmath$\Theta$}_{1},\bar{\mbox{\boldmath$\Theta$}}_{1})$
which has been computed previously in (5.20), it can be shown that (5.24)
holds. Hence, the correlator (5.23) is determined up to two complex
coefficients, $a_{1}$ and $b_{1}$.

## Acknowledgements

The work is supported in part by the Australian Research Council, project No.
DP200101944.

## References

  * [1] H. Osborn and A. C. Petkou, “Implications of conformal invariance in field theories for general dimensions,” Annals Phys. 231 (1994), 311-362 [arXiv:hep-th/9307010 [hep-th]].
  * [2] J. Erdmenger and H. Osborn, “Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions,” Nucl. Phys. B 483 (1997), 431-474 [arXiv:hep-th/9605009 [hep-th]].
  * [3] A. M. Polyakov, “Conformal symmetry of critical fluctuations,” JETP Lett. 12 (1970), 381-383.
  * [4] E. J. Schreier, “Conformal symmetry and three-point functions,” Phys. Rev. D 3 (1971), 980-988.
  * [5] A. A. Migdal, “On hadronic interactions at small distances,” Phys. Lett. B 37 (1971), 98-100.
  * [6] A. A. Migdal, “Conformal invariance and bootstrap,” Phys. Lett. B 37 (1971), 386-388.
  * [7] S. Ferrara, A. F. Grillo and R. Gatto, “Manifestly conformal-covariant expansion on the light cone,” Phys. Rev. D 5 (1972), 3102-3108.
  * [8] S. Ferrara, A. F. Grillo and R. Gatto, “Tensor representations of conformal algebra and conformally covariant operator product expansion,” Annals Phys. 76 (1973), 161-188.
  * [9] K. Koller, “The significance of conformal inversion in quantum field theory,” Commun. Math. Phys. 40, 15-35 DESY-74-8.
  * [10] G. Mack, “Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory,” Commun. Math. Phys. 53 (1977), 155.
  * [11] Y. S. Stanev, “Stress - energy tensor and U(1) current operator product expansions in conformal QFT,” Bulg. J. Phys. 15 (1988), 93-107.
  * [12] E. S. Fradkin and M. Y. Palchik, “Recent developments in conformal invariant quantum field theory,” Phys. Rept. 44 (1978), 249-349.
  * [13] S. Giombi, S. Prakash and X. Yin, JHEP 07 (2013), 105 [arXiv:1104.4317 [hep-th]].
  * [14] J. H. Park, “N=1 superconformal symmetry in four-dimensions,” Int. J. Mod. Phys. A 13 (1998), 1743-1772 [arXiv:hep-th/9703191 [hep-th]].
  * [15] H. Osborn, “N=1 superconformal symmetry in four-dimensional quantum field theory,” Annals Phys. 272 (1999), 243-294 [arXiv:hep-th/9808041 [hep-th]].
  * [16] J. H. Park, “Superconformal symmetry and correlation functions,” Nucl. Phys. B 559 (1999), 455-501 [arXiv:hep-th/9903230 [hep-th]].
  * [17] J. H. Park, “Superconformal symmetry in six-dimensions and its reduction to four-dimensions,” Nucl. Phys. B 539 (1999), 599-642 [arXiv:hep-th/9807186 [hep-th]].
  * [18] J. H. Park, “Superconformal symmetry in three-dimensions,” J. Math. Phys. 41 (2000), 7129-7161 [arXiv:hep-th/9910199 [hep-th]].
  * [19] S. M. Kuzenko and S. Theisen, “Correlation functions of conserved currents in N=2 superconformal theory,” Class. Quant. Grav. 17 (2000), 665-696 [arXiv:hep-th/9907107 [hep-th]].
  * [20] A. A. Nizami, T. Sharma and V. Umesh, “Superspace formulation and correlation functions of 3d superconformal field theories,” JHEP 07 (2014), 022 [arXiv:1308.4778 [hep-th]].
  * [21] E. I. Buchbinder, S. M. Kuzenko and I. B. Samsonov, “Superconformal field theory in three dimensions: Correlation functions of conserved currents,” JHEP 06 (2015), 138 [arXiv:1503.04961 [hep-th]].
  * [22] E. I. Buchbinder, S. M. Kuzenko and I. B. Samsonov, “Implications of $\mathcal{N}=4$ superconformal symmetry in three spacetime dimensions,” JHEP 08 (2015), 125 [arXiv:1507.00221 [hep-th]].
  * [23] S. M. Kuzenko and I. B. Samsonov, “Implications of $\mathcal{N}$ = 5, 6 superconformal symmetry in three spacetime dimensions,” JHEP 08 (2016), 084 [arXiv:1605.08208 [hep-th]].
  * [24] E. I. Buchbinder and B. J. Stone, “Mixed three-point functions of conserved currents in three-dimensional superconformal field theory,” Phys. Rev. D 103 (2021) no.8, 086023 [arXiv:2102.04827 [hep-th]].
  * [25] S. Ferrara and B. Zumino, “Transformation properties of the supercurrent,” Nucl. Phys. B 87 (1975), 207.
  * [26] S. J. Gates Jr., M. T. Grisaru, M. Roček and W. Siegel, Superspace, or One Thousand and One Lessons in Supersymmetry, Benjamin/Cummings (Reading, MA), 1983, hep-th/0108200.
  * [27] M. Magro, I. Sachs and S. Wolf, “Superfield Noether procedure,” Annals Phys. 298 (2002), 123-166 [arXiv:hep-th/0110131 [hep-th]].
  * [28] Z. Komargodski and N. Seiberg, “Comments on supercurrent multiplets, supersymmetric field theories and supergravity,” JHEP 07 (2010), 017 [arXiv:1002.2228 [hep-th]].
  * [29] S. M. Kuzenko, “Variant supercurrent multiplets,” JHEP 04 (2010), 022 [arXiv:1002.4932 [hep-th]].
  * [30] S. M. Kuzenko, “Variant supercurrents and Noether procedure,” Eur. Phys. J. C 71 (2011), 1513 [arXiv:1008.1877 [hep-th]].
  * [31] S. Ferrara, J. Wess and B. Zumino, “Supergauge multiplets and superfields,” Phys. Lett. B 51 (1974), 239.
  * [32] J. Maldacena and A. Zhiboedov, “Constraining conformal field theories with a higher spin symmetry,” J. Phys. A 46 (2013), 214011 [arXiv:1112.1016 [hep-th]].
  * [33] Y. S. Stanev, “Constraining conformal field theory with higher spin symmetry in four dimensions,” Nucl. Phys. B 876 (2013), 651-666 [arXiv:1307.5209 [hep-th]].
  * [34] V. Alba and K. Diab, “Constraining conformal field theories with a higher spin symmetry in d=4,” [arXiv:1307.8092 [hep-th]].
  * [35] V. Alba and K. Diab, “Constraining conformal field theories with a higher spin symmetry in $d>3$ dimensions,” JHEP 03 (2016), 044 [arXiv:1510.02535 [hep-th]].
  * [36] S. R. Coleman and J. Mandula, “All possible symmetries of the S Matrix,” Phys. Rev. 159 (1967), 1251-1256.
  * [37] Y. S. Stanev, “Correlation functions of conserved currents in four dimensional conformal field theory,” Nucl. Phys. B 865 (2012), 200-215 [arXiv:1206.5639 [hep-th]].
  * [38] A. Zhiboedov, “A note on three-point functions of conserved currents,” [arXiv:1206.6370 [hep-th]].
  * [39] E. Elkhidir, D. Karateev and M. Serone, “General three-point functions in 4D CFT,” JHEP 01 (2015), 133 [arXiv:1412.1796 [hep-th]].
  * [40] A. Ceresole, G. Dall’Agata, R. D’Auria and S. Ferrara, “Spectrum of type IIB supergravity on AdS(5) x T**11: Predictions on N=1 SCFT’s,” Phys. Rev. D 61 (2000), 066001 [arXiv:hep-th/9905226 [hep-th]].
  * [41] S. M. Kuzenko and E. S. N. Raptakis, “Symmetries of supergravity backgrounds and supersymmetric field theory,” JHEP 04 (2020), 133 [arXiv:1912.08552 [hep-th]].
  * [42] P. S. Howe, K. S. Stelle and P. K. Townsend, “Supercurrents,” Nucl. Phys. B 192 (1981), 332-352.
  * [43] M. F. Sohnius, “The multiplet of currents for $N=2$ extended supersymmetry,” Phys. Lett. B 81 (1979), 8-10.
  * [44] S. M. Kuzenko, R. Manvelyan and S. Theisen, “Off-shell superconformal higher spin multiplets in four dimensions,” JHEP 07 (2017), 034 [arXiv:1701.00682 [hep-th]].
  * [45] E. I. Buchbinder, J. Hutomo and S. M. Kuzenko, “Higher spin supercurrents in anti-de Sitter space,” JHEP 09 (2018), 027 [arXiv:1805.08055 [hep-th]].
  * [46] I. L. Buchbinder, S. J. Gates and K. Koutrolikos, “Conserved higher spin supercurrents for arbitrary spin massless supermultiplets and higher spin superfield cubic interactions,” JHEP 08 (2018), 055 [arXiv:1805.04413 [hep-th]].
  * [47] S. J. Gates and K. Koutrolikos, “Progress on cubic interactions of arbitrary superspin supermultiplets via gauge invariant supercurrents,” Phys. Lett. B 797 (2019), 134868 [arXiv:1904.13336 [hep-th]].
  * [48] S. M. Kuzenko, A. G. Sibiryakov and V. V. Postnikov, “Massless gauge superfields of higher half integer superspins,” JETP Lett. 57 (1993), 534-538.
  * [49] S. M. Kuzenko and A. G. Sibiryakov, “Massless gauge superfields of higher integer superspins,” JETP Lett. 57 (1993), 539-542.
  * [50] I. L. Buchbinder and S. M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace, IOP, Bristol, 1995 (Revised Edition: 1998).
  * [51] E. I. Buchbinder, J. Hutomo and S. M. Kuzenko, “Correlation functions of spinor current multiplets in ${\mathcal{N}}=1$ superconformal theory,” [arXiv:2103.09472 [hep-th]].