# Spinors and conformal correlators

Marc Gillioz

(_SISSA, via Bonomea 265, 34136 Trieste, Italy_)

###### Abstract

In conformal field theory, momentum eigenstates can be parameterized by a pair
of real spinors, in terms of which special conformal transformations take a
simpler form. This well-known fact allows to express 2-point functions of
primary operators in the helicity basis, exposing the consequences of
unitarity. What is less known is that the same pair of spinors can be used,
together with a pair of scalar quantities, to parameterize 3-point functions.
We develop this formalism in 3 dimensions and show that it provides a simple
realization of the operator product expansion (OPE) for scalar primary
operators acting on the vacuum.

###### Contents

  1. 1 Introduction
  2. 2 2-point functions
    1. 2.1 Spinors and helicity eigenstates
    2. 2.2 Conformal Ward identities
    3. 2.3 Correlation function
  3. 3 3-point functions
    1. 3.1 From momenta to spinors
    2. 3.2 Conformal symmetry
    3. 3.3 Solutions
    4. 3.4 Correlation function
  4. 4 The OPE for higher-point functions
  5. 5 Conclusions
  6. A Conventions for spinors
  7. B Generalized hypergeometric series

##  1 Introduction

Conformal field theory is remarkable in the sense that it admits a non-
perturbative definition in terms of CFT data characterizing 2- and 3-point
correlation functions only. The problem of constructing all these 2- and
3-point functions in the position-space representation is in some sense
completely solved: since conformal Ward identities are first-order
differential equations in position space, it essentially reduces to the
classification of all tensors transforming covariantly [1, 2, 3].

A related problem is the construction of all 2- and 3-point correlation
functions in the momentum-space representation. One could expect this problem
to be solved as well, since the two representations are related by a Fourier
transform. But this Fourier transform is very difficult to perform explicitly
— so difficult in fact that most results in momentum space do not rely on it
in an essential way [4, 5].

In this work we introduce a new approach for the construction of 2- and
3-point conformal correlation function in momentum space, making use of the
spinor formalism. This method works specifically in Minkowski space-time and
with Wightman functions, i.e. with correlators that are not time-ordered.
Unlike other recent works that use similar methods [6, 7], we do not focus on
(higher-spin) conserved currents but treat all conformal primary operators on
the same footing. To keep the argument simple, our results are limited to 3
space-time dimensions and, in the case of 3-point functions, to correlators
involving two scalar operators. The generalization to more complicated
correlators and to 4 space-time dimensions (or higher) will be presented
elsewhere. As a consequence, the correlators computed here are not new and can
in fact be found in the literature [8, 9].

Nevertheless, this work is more than an academic exercise: the use of spinors
in Minkowski space-time allows to encode both the momentum and the
polarization information of the operators [10, 11], and therefore to test the
Hilbert space structure of the theory, providing a simple understanding of
unitarity bounds on the scaling dimensions of primary operators [12, 11, 13,
14], and of reality conditions on OPE coefficients. Moreover, the method
developed here will serve as a basis for future studies of generic conformal
correlators involving primary operators carrying arbitrary spin
representations.

Our results are presented as follows. Section 2 describes the construction of
2-point functions. Generic CFT states carry massive momenta parameterized by a
pair of real spinors. We show that the Ward identity for special conformal
transformations takes a simple form in the spinor formalism, particularly when
projected onto helicity eigenstates. This is essentially a review of known
facts, but it sets the stage for the computations of section 3. There, we show
that the two independent momenta characterizing a 3-point function can be
parameterized by the same two spinors, together with a pair of scalar
quantities related to ratios of momenta squared.111This contrasts with
standard on-shell methods for massive scattering amplitudes, in which _every_
massive momentum is parameterized by a pair of spinors [15]. In our case a
_single_ pair of spinors is sufficient. Conformal Ward identities are derived
and solved, yielding familiar results involving the Appell $F_{4}$
hypergeometric function, as well as a novel closed-form expression in the
general case. The results of sections 2 and 3 are finally put together in
section 4, leading to the formulation of an operator product expansion for
scalar operators acting on the vacuum. Some conclusions are drawn in section
5. Conventions for spinors in 3 space-time dimensions are given in appendix A.

##  2 2-point functions

We begin by considering states created by a single local primary operator
acting on the vacuum, and work in the momentum-space representation such that

$P^{\mu}\mathcal{O}(p)\left|0\right\rangle=p^{\mu}\mathcal{O}(p)\left|0\right\rangle,$
(2.1)

where $P^{\mu}$ is the generator of translations and $p^{\mu}$ its eigenvalue.
$\left|0\right\rangle$ is the unique vacuum state of the theory, invariant
under conformal transformations. We shall also consider the inner product of
such states, which by translation symmetry is proportional to a delta
function,

$\left\langle
0\right|\mathcal{O}(p^{\prime})\mathcal{O}(p)\left|0\right\rangle=(2\pi)^{3}\delta^{3}(p^{\prime}+p)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}(-p)\mathcal{O}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(2.2)

The object inside double brackets is a function of a single 3-momentum
$p^{\mu}$.

###  2.1 Spinors and helicity eigenstates

Standard assumptions of quantum field theory require states of the form of eq.
(2.1) to only have support in the forward light cone. In other words, the
function in double bracket in eq. (2.2) vanishes unless

$p^{2}=(p^{0})^{2}-\mathbf{p}^{2}\geq 0,\qquad\qquad p^{0}\geq 0.$ (2.3)

Any momentum of this sort can be parameterized by a pair of real spinors, or
equivalently by a unique complex spinor $\lambda^{a}$ and its complex
conjugate $\bar{\lambda}^{a}$ through

$p^{\mu}=\sigma^{\mu}_{ab}\,\lambda^{a}\bar{\lambda}^{b}.$ (2.4)

Note that this parameterization is redundant: it uses a complex spinor with 4
components (two real and two imaginary) for a real 3-vector. This redundancy
is encoded in the little group symmetry

$\lambda^{a}\to e^{i\theta}\lambda^{a},\qquad\qquad\bar{\lambda}^{a}\to
e^{-i\theta}\bar{\lambda}^{a},$ (2.5)

under which $p$ transforms trivially. We say that $\lambda^{a}$ carries
helicity $+\frac{1}{2}$ and $\bar{\lambda}^{a}$ helicity $-\frac{1}{2}$. In
the rest frame in which $\mathbf{p}=0$, this corresponds to spatial rotations.

Spinors are particularly powerful when dealing with operators $\mathcal{O}$
that transform in non-trivial representations of the Lorentz group. In 3
space-time dimensions, all such operators are symmetric tensors in spinor
indices. We shall denote with $2j$ the number of spinor indices, $j$ taking
either integer or half-integer values. A complete and orthogonal basis of
polarization tensors is given by

$\varepsilon^{a_{1}\ldots
a_{2j}}_{m}=\lambda^{a_{1}}\cdots\lambda^{a_{j+m}}\bar{\lambda}^{a_{j+m+1}}\cdots\bar{\lambda}^{a_{2j}},$
(2.6)

where $m$ takes values between $-j$ and $+j$ in integer steps. There are
$2j+1$ such polarization tensors for each spin $j$, and all of them are
eigenstates of helicity $m$, i.e. they transform as $\varepsilon^{a_{1}\ldots
a_{2j}}_{m}\to e^{2im\theta}\varepsilon^{a_{1}\ldots a_{2j}}_{m}$ under the
little group scaling (2.5).

To simplify the notation even further, we introduce an auxiliary spinor
$\zeta^{a}$ and define the index-free polarization

$\varepsilon_{j,m}(\zeta)=(\zeta\cdot\lambda)^{j+m}(\zeta\cdot\bar{\lambda})^{j-m}.$
(2.7)

The explicit form (2.6) of the polarization tensor can be unequivocally
recovered from this expression acting with spinor derivatives.222This
contrasts with the use of polarization _vectors_ that require subtraction of
trace terms [2, 16]; there are no such trace terms in the spinor formalism.
The same auxiliary spinor can be used to express any operator carrying spinor
indices in the index-free notation

$\mathcal{O}_{j}(p,\zeta)\left|0\right\rangle\equiv\zeta^{a_{1}}\cdots\zeta^{a_{2j}}\mathcal{O}_{a_{1}\ldots
a_{2j}}(p)\left|0\right\rangle.$ (2.8)

The completeness of the basis of polarization tensors implies that we can
further decompose this operator into helicity eigenstates as

$\mathcal{O}_{j}(p,\zeta)\left|0\right\rangle=\sum_{m=-j}^{j}\varepsilon_{j,m}(\zeta)\mathcal{O}_{j,m}(p)\left|0\right\rangle.$
(2.9)

Note that these last two equations involve a primary operator $\mathcal{O}$
acting on the vacuum, as it is important to realize that they are only valid
when $p$ takes values inside the forward light cone. The decomposition does
not make sense at the level of operators.

###  2.2 Conformal Ward identities

The norm of these states is computed by the 2-point correlation function

$\left\langle
0\right|\mathcal{O}_{j}(p^{\prime},\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\left|0\right\rangle=(2\pi)^{3}\delta^{3}(p^{\prime}+p)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}},$
(2.10)

where we are now making use of two distinct auxiliary polarization spinors
$\zeta$ and $\zeta^{\prime}$ to encode the information about indices. The
object in double brackets is a correlation function in its own right: taking
the Fourier transform of this equation with respect to $p^{\prime}$, one gets

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=e^{ip\cdot
x}\left\langle
0\right|\widetilde{\mathcal{O}}_{j}(x,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\left|0\right\rangle.$
(2.11)

This mixed position/momentum representation is particularly well-suited to
examine the constraints imposed by conformal symmetry: when evaluated at
$x=0$, the primary operator $\widetilde{\mathcal{O}}(0,\zeta^{\prime})$ has
trivial transformation properties under the conformal group. This implies that
conformal Ward identities for this correlation function can be obtained in the
form of differential equations with respect to the momentum $p^{\mu}$. For
instance, for scale and Lorentz symmetry, we have

$\displaystyle\left[\widehat{D}-2\Delta+3\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
$\displaystyle=0,$ (2.12)
$\displaystyle\left[\widehat{M}_{\mu\nu}-\widehat{\Sigma}_{\mu\nu}(\zeta^{\prime})-\widehat{\Sigma}_{\mu\nu}(\zeta)\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
$\displaystyle=0,$ (2.13)

where

$\widehat{D}=p^{\mu}\frac{\partial}{\partial
p^{\mu}},\qquad\qquad\widehat{M}_{\mu\nu}=p_{\nu}\frac{\partial}{\partial
p^{\mu}}-p_{\mu}\frac{\partial}{\partial p^{\nu}}.$ (2.14)

$\Delta$ is the scaling dimension of $\mathcal{O}$, and
$\widehat{\Sigma}_{\mu\nu}$ the spin operator acting on its Lorentz indices:
in terms of the auxiliary spinor $\zeta$, it can be written as

$\widehat{\Sigma}_{\mu\nu}(\zeta)=-\frac{1}{2}\varepsilon_{\mu\nu\rho}\sigma^{\rho}_{ab}\,\zeta^{a}\frac{\partial}{\partial\zeta_{b}}.$
(2.15)

It is a simple exercise to verify using the relation (2.4) and the chain rule
for derivatives that the scale and Lorentz generators can be written in terms
of spinors as

$\widehat{D}=\frac{1}{2}\left(\lambda^{a}\frac{\partial}{\partial\lambda_{a}}+\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}_{a}}\right)$
(2.16)

and

$\widehat{M}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho}\sigma^{\rho}_{ab}\left(\lambda^{a}\frac{\partial}{\partial\lambda_{b}}+\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}_{b}}\right).$
(2.17)

These results can be obtained enumerating the handful of operators that are
first-order in derivatives, invariant under the little group scaling (2.5),
and that transform respectively as a scalar and as a vector333In 3 dimensions
$\widehat{M}_{\mu\nu}$ is dual to a Lorentz vector, which is obvious in eq.
(2.17). We use the two-index antisymmetric form to adhere to the standard
notation for conformal generators valid in arbitrary space-time dimension and
to match the definition (2.14) in terms of $p^{\mu}$. under Lorentz symmetry.
Note that the polarization tensors (2.6) are eigenvectors of the scale
generator,

$\widehat{D}\,\varepsilon_{j,m}(\zeta)=j\,\varepsilon_{j,m}(\zeta),$ (2.18)

and moreover that they transform covariantly under Lorentz transformation,
since they satisfy

$\widehat{M}_{\mu\nu}\varepsilon_{j,m}(\zeta)=\widehat{\Sigma}_{\mu\nu}(\zeta)\varepsilon_{j,m}(\zeta).$
(2.19)

The same strategy can be applied to determine the constraints imposed by
special conformal symmetry, encoded in the Ward identity

$\widehat{K}_{\mu}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=0,$
(2.20)

where444The fact that eq. (2.21) does not depend on the auxiliary spinor
$\zeta^{\prime}$ and looks therefore asymmetric is due to the representation
(2.11) in which special conformal transformation act trivially on one of the
two operators, but not on the other. Alternatively, this Ward identity could
be obtained commuting the generator of special conformal transformation with
the delta function imposing momentum conservation [17].

$\widehat{K}_{\mu}=\frac{1}{2}p_{\mu}\frac{\partial^{2}}{\partial
p^{\nu}\partial p_{\nu}}-p^{\nu}\frac{\partial^{2}}{\partial p^{\nu}\partial
p^{\mu}}+(\Delta-3)\frac{\partial}{\partial
p^{\mu}}+\widehat{\Sigma}_{\mu\nu}(\zeta)\frac{\partial}{\partial p_{\nu}}.$
(2.21)

In this case the expression for $\widehat{K}_{\mu}$ in terms of spinors is
more complicated because the Ward identity involves second-order derivatives
in the momenta. Nevertheless, it can be determined straightforwardly using

$\frac{\partial}{\partial
p_{\mu}}=\frac{\sigma^{\mu}_{ab}}{4\lambda\cdot\bar{\lambda}}\left(\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}_{b}}-\lambda^{a}\frac{\partial}{\partial\lambda_{b}}\right),$
(2.22)

together with the identity

$\displaystyle\frac{1}{2}p^{\mu}\frac{\partial^{2}}{\partial p^{\nu}\partial
p_{\nu}}-p^{\nu}\frac{\partial^{2}}{\partial p^{\nu}\partial
p_{\mu}}-\frac{\partial}{\partial
p_{\mu}}=-\frac{1}{2}\sigma^{\mu}_{ab}\frac{\partial^{2}}{\partial\lambda_{a}\partial\bar{\lambda}_{b}},$
(2.23)

in which we note that the combination of second-order derivatives precisely
matches that of eq. (2.21).

This is sufficient to write down a differential equation for the correlation
function in terms of spinors only. But the effectiveness of the spinor
formalism goes further, as it also implies that we can decompose this equation
into 3 components of definite helicity, writing

$\widehat{K}^{\mu}=\frac{\sigma^{\mu}_{ab}}{2(\lambda\cdot\bar{\lambda})^{2}}\left(\lambda^{a}\bar{\lambda}^{b}\widehat{K}_{0}+\lambda^{a}\lambda^{b}\widehat{K}_{-}+\bar{\lambda}^{a}\bar{\lambda}^{b}\widehat{K}_{+}\right).$
(2.24)

To give explicit expressions for the differential operators $\widehat{K}_{0}$
and $\widehat{K}_{\pm}$, it is convenient to construct all scalar, first-order
derivative operators in spinors. There are four independent operators: one is
the scale generator (2.16), the other three can be written

$\widehat{H}_{0}=\frac{1}{2}\left(\lambda^{a}\frac{\partial}{\partial\lambda^{a}}-\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}^{a}}\right),\qquad\qquad\widehat{H}_{+}=\lambda^{a}\frac{\partial}{\partial\bar{\lambda}^{a}},\qquad\qquad\widehat{H}_{-}=\bar{\lambda}^{a}\frac{\partial}{\partial\lambda^{a}}.$
(2.25)

$\widehat{H}_{0}$ counts the helicity, while $\widehat{H}_{+}$ and
$\widehat{H}_{-}$ are ladder operator that raise and lower the helicity by one
unit. Together, they form an $\text{su}(2)$ algebra

$\left[\widehat{H}_{0},\widehat{H}_{\pm}\right]=\pm\widehat{H}_{\pm},\qquad\qquad\left[\widehat{H}_{+},\widehat{H}_{-}\right]=2\widehat{H}_{0}.$
(2.26)

Note that $\widehat{H}_{+}$ and $\widehat{H}_{-}$ are complex-conjugate to
each other, and also that all three commute with $\widehat{D}$. Any
differential operator can then be expressed in terms of these: this is the
case of the Lorentz generator

$\varepsilon^{\mu\nu\rho}\widehat{M}_{\nu\rho}=\frac{\sigma^{\mu}_{ab}}{\lambda\cdot\bar{\lambda}}\left(2\lambda^{a}\bar{\lambda}^{b}\widehat{H}_{0}+\bar{\lambda}^{a}\bar{\lambda}^{b}\widehat{H}_{+}-\lambda^{a}\lambda^{b}\widehat{H}_{-}\right),$
(2.27)

and of the derivative with respect to $p$

$\frac{\partial}{\partial
p_{\mu}}=\frac{\sigma^{\mu}_{ab}}{2(\lambda\cdot\bar{\lambda})^{2}}\left(-2\lambda^{a}\bar{\lambda}^{b}\widehat{D}+\bar{\lambda}^{a}\bar{\lambda}^{b}\widehat{H}_{+}+\lambda^{a}\lambda^{b}\widehat{H}_{-}\right).$
(2.28)

To express the components of $\widehat{K}^{\mu}$ in this manner, one also
needs to decompose the spin operator $\widehat{\Sigma}_{\mu\nu}$ into scalar
components. This can be done defining

$\widehat{H}^{\zeta}_{+}=\frac{(\zeta\cdot\lambda)\lambda^{a}}{\lambda\cdot\bar{\lambda}}\frac{\partial}{\partial\zeta^{a}},\qquad\widehat{H}^{\zeta}_{-}=-\frac{(\zeta\cdot\bar{\lambda})\bar{\lambda}^{a}}{\lambda\cdot\bar{\lambda}}\frac{\partial}{\partial\zeta^{a}},\qquad\widehat{H}^{\zeta}_{0}=-\frac{(\zeta\cdot\bar{\lambda})\lambda^{a}+(\zeta\cdot\lambda)\bar{\lambda}^{a}}{2\,\lambda\cdot\bar{\lambda}}\frac{\partial}{\partial\zeta^{a}}.$
(2.29)

These 3 operators obey the same $\text{su}(2)$ algebra as the
$\widehat{H}_{i}$, but act only on functions of the spinor $\zeta$. A set of
differential operators acting on the auxiliary spinor $\zeta^{\prime}$ can be
defined similarly, denoted $\widehat{H}^{\zeta^{\prime}}_{i}$. With these
tools, the Ward identity (2.13) for Lorentz transformations can be projected
onto components of definite helicity, leading to the three simple equations

$\left[\widehat{H}_{i}-\widehat{H}^{\zeta}_{i}-\widehat{H}^{\zeta^{\prime}}_{i}\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=0,\qquad\qquad
i\in\\{0,+,-\\}.$ (2.30)

The same can be done with the components of $\widehat{K}_{\mu}$ defined in eq.
(2.24). We find

$\displaystyle\widehat{K}_{0}$
$\displaystyle=\widehat{H}_{+}\widehat{H}_{-}-\widehat{H}_{+}\widehat{H}^{\zeta}_{-}-\widehat{H}_{-}\widehat{H}^{\zeta}_{+}+\widehat{D}(\widehat{D}-2\Delta+3)-\widehat{H}_{0}(\widehat{H}_{0}+1),$
$\displaystyle\widehat{K}_{+}$
$\displaystyle=\widehat{H}_{+}(-\widehat{D}-\widehat{H}_{0}+\widehat{H}_{0}^{\zeta}+\Delta-2)+\widehat{H}^{\zeta}_{+}(\widehat{D}+1),$
(2.31) $\displaystyle\widehat{K}_{-}$
$\displaystyle=\widehat{H}_{-}(-\widehat{D}+\widehat{H}_{0}-\widehat{H}_{0}^{\zeta}+\Delta-2)+\widehat{H}^{\zeta}_{-}(\widehat{D}+1).$

This is a major improvement compared to the momentum-space Ward identity
(2.21): on the one hand, the helicity projection disentangles the equation
into three independent components; on the other hand, it is very simple to
apply the elementary differential operators in eq. (2.31) to _any_ function of
the spinors. The scalar quantity $p^{2}=-(\lambda\cdot\bar{\lambda})^{2}$ is
for instance annihilated by all $\widehat{H}_{i}$. Moreover, the polarization
tensors (2.6) have simple transformation rules, being eigenstates of
$\widehat{H}_{0}$,

$\widehat{H}_{0}\,\varepsilon_{j,m}(\zeta)=m\,\varepsilon_{j,m}(\zeta),$
(2.32)

and satisfying ladder identities

$\widehat{H}_{+}\,\varepsilon_{j,m}(\zeta)=(j-m)\,\varepsilon_{j,m+1}(\zeta),\qquad\qquad\widehat{H}_{-}\,\varepsilon_{j,m}(\zeta)=(j+m)\,\varepsilon_{j,m-1}(\zeta).$
(2.33)

The numerical factors in these last two equations guarantee that
$\widehat{H}_{\pm}$ annihilate the highest-helicity polarizations $m=\pm j$
respectively. The other differential operators $\widehat{H}^{\zeta}_{i}$ and
$\widehat{H}^{\zeta^{\prime}}_{i}$ act likewise, but only on polarization
tensors defined in terms of the corresponding auxiliary spinor.

###  2.3 Correlation function

Now that the conformal Ward identities have been split into simple equations
in terms of elementary ladder operators, it is straightforward to obtain the
most general solution for the 2-point correlator (2.10). Note that it is the
overlap of two states that can each be decomposed into helicity eigenstates
following eq. (2.9). Moreover, since the correlation function (2.10) transform
covariantly under the little group, its total helicity is zero. This implies
that the decomposition into helicity eigenstates is diagonal, and that one can
write

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=\sum_{m=-j}^{j}\varepsilon_{j,-m}(\zeta^{\prime})\varepsilon_{j,m}(\zeta)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j,-m}(-p)\mathcal{O}_{j,m}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(2.34)

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j,-m}(-p)\mathcal{O}_{j,m}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
is a scalar function of the spinors $\lambda^{a}$ and $\bar{\lambda}^{a}$,
which means that it must necessarily be a function of
$p^{2}=-(\lambda\cdot\bar{\lambda})^{2}$ only. Its overall scaling dimension
is determined by the Ward identity (2.12) to be $2\Delta-2j-3$. We obtain
immediately that

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j,-m}(-p)\mathcal{O}_{j,m}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=(p^{2})^{\Delta-j-3/2}C_{\Delta,j,m},$
(2.35)

where $C_{\Delta,j,m}$ is a numerical coefficient that remains to be
determined. Note that unitarity requires this coefficient to be non-negative.
Together, eqs. (2.34) and (2.35) form the most general ansatz for the 2-point
function that is consistent with Poincaré and scale symmetry, i.e. that
satisfies the Ward identities (2.12) and (2.30).

In fact, the zero-helicity component $\widehat{K}_{0}$ of the special
conformal Ward identity (2.20) is also readily satisfied by this ansatz. This
is not the case of the components $\widehat{K}_{\pm}$, however: using

$\displaystyle\widehat{K}_{+}$
$\displaystyle\left[(p^{2})^{\Delta-j-3/2}\varepsilon_{j,-m}(\zeta^{\prime})\varepsilon_{j,m}(\zeta)\right]=(p^{2})^{\Delta-j-3/2}$
$\displaystyle\times\left[(j+m)(\Delta-m-1)\varepsilon_{j,-m}(\zeta^{\prime})\varepsilon_{j,m-1}(\zeta)-(j-m)(\Delta+m-1)\varepsilon_{j,-m-1}(\zeta^{\prime})\varepsilon_{j,m}(\zeta)\right],$
(2.36)

or a similar identity for $\widehat{K}_{-}$ related to this one by complex
conjugation, one obtains the condition

$\displaystyle\sum_{m=-j}^{j-1}$
$\displaystyle\varepsilon_{j,-m-1}(\zeta^{\prime})\varepsilon_{j,m}(\zeta)$
$\displaystyle\times\left[(j+m+1)(\Delta-m-2)C_{\Delta,j,m+1}-(j-m)(\Delta+m-1)C_{\Delta,j,m}\right]=0.$
(2.37)

For the Ward identity to be satisfied, all individual terms in the sum must
vanish separately. This gives a recursion relation for the coefficient
$C_{\Delta,j,m}$, solved by

$C_{\Delta,j,m}=\frac{(2j)!}{(j-m)!(j+m)!}\frac{\Gamma(\Delta+m-1)\Gamma(\Delta-m-1)}{\Gamma(\Delta+j-1)\Gamma(\Delta-j-1)}\,C_{\Delta,j}.$
(2.38)

$C_{\Delta,j}$ is a normalization constant that is arbitrary up to its sign:
since $C_{\Delta,j,\pm j}=C_{\Delta,j}$ corresponds to the norm of the
eigenstate of helicity $\pm j$, it must therefore be positive. Despite the
appearance of gamma functions in eq. (2.38), the $C_{\Delta,j,m}$ are in fact
rational functions of $\Delta$ multiplying this normalization constant: they
can alternatively be written using the Pochhammer symbol as

$C_{\Delta,j,m}=\frac{(2j)!}{(j-m)!(j+m)!}\frac{(\Delta-j-1)_{j-m}}{(\Delta+m-1)_{j-m}}\,C_{\Delta,j}.$
(2.39)

Requiring that all helicity eigenstates have positive norm, one obtains the
unitarity bound

$\Delta\geq j+1\qquad\qquad\text{for all}\quad j>\tfrac{1}{2}.$ (2.40)

This is a well-known result, particularly transparent in this formalism. When
the bound is saturated, i.e. when $\Delta=j+1$, the operator only admits
transverse polarizations with helicity $m=\pm j$: it is then a conserved
current transforming in a short representation of the conformal group.555The
conservation condition follows simply from the identity
$p_{a_{1}a_{2}}\varepsilon^{a_{1}\ldots a_{2j}}_{\pm j}=0$. In all other cases
the operator admits $2j+1$ polarizations, both transverse and longitudinal,
and there is no shortening condition.

There are also well-known unitarity bounds for scalar ($j=0$) and spinor
($j=\frac{1}{2}$) operators, but their origin is somewhat different. One way
to understand them is to normalize the operator such that the position-space
correlation function obeys, for space-like separated points,

$\left\langle
0\right|\widetilde{\mathcal{O}}_{j}(x^{\prime},\zeta^{\prime})\widetilde{\mathcal{O}}_{j}(x,\zeta)\left|0\right\rangle=\frac{\left[i\sigma^{\mu}_{ab}\zeta^{\prime
a}\zeta^{b}(x^{\prime}_{\mu}-x_{\mu})\right]^{2j}}{\left[-(x^{\prime}-x)^{2}\right]^{\Delta+j}},\qquad\quad(x^{\prime}-x)^{2}<0.$
(2.41)

Performing the Fourier transform of this expression, one recovers the 2-point
correlation function (2.10) with normalization666The case of integer spin is
treated in ref. [8]. For half-integer spins, we refer the reader to the
excellent discussion of ref. [18].

$C_{\Delta,j}=\frac{2\pi^{2}}{(\Delta+j-1)\Gamma(2\Delta-2)}.$ (2.42)

This is positive for all $\Delta\geq\frac{1}{2}$ for a scalar, and for all
$\Delta\geq 1$ for a spin-$\frac{1}{2}$ operator.

#### Distinct operators.

We focused so far on 2-point functions of identical operators, but the methods
presented in this section can also be used to examine the correlation function
of two distinct operators. Based on Poincaré and scale symmetry, we can make
the following ansatz:

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{\prime}_{j^{\prime}}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\stackrel{{\scriptstyle?}}{{=}}\sum_{m}C_{\Delta,j^{\prime},j,m}\varepsilon_{j^{\prime},-m}(\zeta^{\prime})\varepsilon_{j,m}(\zeta)(p^{2})^{(\Delta^{\prime}+\Delta-j^{\prime}-j-3)/2}.$
(2.43)

$\Delta^{\prime}$ and $\Delta$ are the scaling dimensions of
$\mathcal{O}^{\prime}$ and $\mathcal{O}$ respectively. The sum over $m$ runs
over $\left|m\right|\leq\min(j^{\prime},j)$, and in this way invariance under
the little group is also satisfied. When it comes to special conformal
transformation, however, this ansatz is constrained further: the zero-helicity
component $\widehat{K}_{0}$ of the Ward identity (2.20) now gives

$\displaystyle\widehat{K}_{0}$
$\displaystyle\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{\prime}_{j^{\prime}}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
(2.44)
$\displaystyle=\big{[}(\Delta^{\prime}-\Delta)(\Delta^{\prime}+\Delta-3)+(j^{\prime}-j)(j^{\prime}+j+1)\big{]}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{\prime}_{j^{\prime}}(-p,\zeta^{\prime})\mathcal{O}_{j}(p,\zeta)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$

For identical spins $j^{\prime}=j$, special conformal symmetry requires that
the two operators have either identical scaling dimension
$\Delta^{\prime}=\Delta$, or that their scaling dimensions add up as
$\Delta^{\prime}+\Delta=3$. This latter condition is only compatible with the
unitarity bounds for scalar operators. When the spins are not equal,
$j^{\prime}\neq j$, the other Ward identities generated by $\widehat{K}_{\pm}$
require the highest-helicity coefficient with $m=\pm\min(j^{\prime},j)$ to
vanish, and all other coefficients must therefore vanish as well (again
assuming that the unitarity bounds are enforced). This means that 2-point
functions of distinct operators generically vanish in conformal field
theory.777A possible exception concerns scalar operators with scaling
dimensions adding up to 3, which is usually not realized in physical theories.
The other exception concerns theories that have multiple operators in the same
conformal representation: it is usually possible to diagonalize the basis of
such operators, but not always [19]. We shall not consider this possibility
further.

This concludes the discussion on 2-point correlations functions. The results
obtained here are not new: they have recently appeared in the bootstrap
literature as an efficient tool for the construction of conformal blocks in
position space [18], and a similar construction in 4 space-time dimensions
dates back to early days of conformal field theory half a century ago [10,
11].888See also refs. [20, 21] for a recent use of the spinor-helicity
formalism combined with conformal symmetry in the context of free theories.
The probable reason why this approach is absent from modern reviews on the
topic is that it is too remote from the position-space representation in which
the OPE is usually formulated, and in terms of which conformal blocks can be
simply expressed as functions of invariant cross-ratios. The goal of this
paper is precisely to show that the spinor formalism can be carried through
the whole construction of conformal blocks, and that it is a valuable tool in
doing so.

##  3 3-point functions

Let us consider next 3-point correlation functions, specifically those
involving two scalar operators, which we will denote by $\phi_{1}$ and
$\phi_{2}$, and one arbitrary tensor operator $\mathcal{O}$ (only integer-spin
operators will in fact appear, as explained below). As before, translation
symmetry implies that we can write

$\left\langle 0\right|\mathcal{O}_{a_{1}\ldots
a_{2j}}(-p)\phi_{2}(p_{2})\phi_{1}(p_{1})\left|0\right\rangle=(2\pi)^{3}\delta^{3}(p-p_{2}-p_{1})\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{a_{1}\ldots
a_{2j}}(-p)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(3.1)

One way to understand this Wightman function is as the overlap of two states,
one obtained as the product of two operators acting successively on the
vacuum,

$\phi_{2}(p_{2})\phi_{1}(p_{1})\left|0\right\rangle,$ (3.2)

and the other of the form discussed in the previous section,

$\mathcal{O}_{a_{1}\ldots a_{2j}}(p_{1}+p_{2})\left|0\right\rangle.$ (3.3)

Both are momentum-eigenstates, with eigenvalue $p=p_{1}+p_{2}$ that we can
assume to be in the forward light cone. We will also see that both can be
expanded into helicity eigenstates: for the state (3.3) the helicity labels
the spin of the operator, carried by its spinor indices, while for the state
(3.2) it corresponds to angular momentum. Before discussing the constraints of
conformal symmetry that will determine this 3-point function up to a single
OPE coefficient, we need to introduce a unified language that can account for
these two distinct manifestations of helicity.

###  3.1 From momenta to spinors

Once the delta function imposing momentum conservation has been taken out of
its definition, the correlator in eq. (3.1) is a function of two 3-momenta
$p_{1}$ and $p_{2}$ only. The spectral condition on their sum is such that one
can always parameterize it with a complex spinor as

$p^{\mu}=p_{1}^{\mu}+p_{2}^{\mu}=\sigma^{\mu}_{ab}\,\lambda^{a}\bar{\lambda}^{b},$
(3.4)

in complete analogy with eq. (2.4). However, the 3-point function does not
only depend on the sum $p_{1}+p_{2}$, but also on the difference
$p_{1}-p_{2}$. Depending on the kinematics, this difference can either be
space-like, time-like, or even light-like. It is therefore not immediately
well-suited to be expressed in terms of spinors. Instead, let us consider the
vector999In the particle-physics literature, and particularly in the context
of bound states of 2-particles, $q$ is sometimes called the _relative momentum
in the center-of-mass frame_. It is purely spatial ($q^{0}=0$) in a Lorentz
frame in which $\mathbf{p}_{1}+\mathbf{p}_{2}=0$. Moreover, if the
“constituent masses” $p_{1}^{2}$ and $p_{2}^{2}$ are equal, then
$\mathbf{q}\propto\mathbf{p}_{1}-\mathbf{p}_{2}$. Eq. (3.5) is simply the
Lorentz-covariant definition of this quantity.

$q^{\mu}=\frac{\big{[}(p_{1}+p_{2})\cdot
p_{2}\big{]}p_{1}^{\mu}-\big{[}(p_{1}+p_{2})\cdot
p_{1}\big{]}p_{2}^{\mu}}{\sqrt{(p_{1}\cdot p_{2})^{2}-p_{1}^{2}p_{2}^{2}}}.$
(3.5)

This is the unique combination of $p_{1}$ and $p_{2}$ (up to a sign) that is
orthogonal to $p$, i.e. $p\cdot q=0$, and that has the same norm in absolute
value, $q^{2}=-p^{2}$ ($q$ is always space-like). These two conditions are
sufficient to give it a parameterization in terms of the same complex spinor:

$q^{\mu}=\tfrac{1}{2}\sigma^{\mu}_{ab}\left(\lambda^{a}\lambda^{b}+\bar{\lambda}^{a}\bar{\lambda}^{b}\right).$
(3.6)

$q$ is a real vector, but unlike $p$, it is not invariant under the little
group (2.5).

Figure 1: Given a pair of momenta $p_{1}$ and $p_{2}$ that add up to $p$
inside the forward light cone, shown here in the center-of-mass frame
($\mathbf{p}=0$), one can always define two null momenta $q_{1}$ and $q_{2}$
that are the projections of $p$ onto the light cone in the plane spanned by
$p_{1}$ and $p_{2}$. The space-like vector $q=q_{1}-q_{2}$ satisfies $p\cdot
q=0$ and $q^{2}=-p^{2}$.

A different perspective of the same parameterization goes as follows: there is
a unique pair of null vectors $q_{1}$ and $q_{2}$ pointing in the forward
direction that is formed of linear combinations of $p_{1}$ and $p_{2}$ and
adds up to $p$, namely (up to the exchange of the two)

$\displaystyle q_{1}^{\mu}$
$\displaystyle=\frac{1}{2}\left(1+\frac{p_{2}\cdot(p_{1}+p_{2})}{\sqrt{(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}}}\right)p_{1}^{\mu}+\frac{1}{2}\left(1-\frac{p_{1}\cdot(p_{1}+p_{2})}{\sqrt{(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}}}\right)p_{2}^{\mu},$ (3.7) $\displaystyle
q_{2}^{\mu}$
$\displaystyle=\frac{1}{2}\left(1-\frac{p_{2}\cdot(p_{1}+p_{2})}{\sqrt{(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}}}\right)p_{1}^{\mu}+\frac{1}{2}\left(1+\frac{p_{1}\cdot(p_{1}+p_{2})}{\sqrt{(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}}}\right)p_{2}^{\mu}.$

This is illustrated in figure 1. Each of these two null vectors can then be
written using a real spinor,

$q_{1}^{\mu}=\tfrac{1}{2}\sigma^{\mu}_{ab}\lambda_{1}^{a}\lambda_{1}^{b},\qquad\qquad
q_{2}^{\mu}=\tfrac{1}{2}\sigma^{\mu}_{ab}\lambda_{2}^{a}\lambda_{2}^{b}.$
(3.8)

Both the parameterization (3.4) for $p=q_{1}+q_{2}$ and (3.6) for
$q=q_{1}-q_{2}$ are recovered upon identifying
$\lambda^{a}=\frac{1}{\sqrt{2}}\left(\lambda_{1}^{a}+i\lambda_{2}^{a}\right)$.
Note that this procedure does not depend whether $p_{1}$ and $p_{2}$ are
space-like or time-like (even though the spectral condition also requires
$p_{1}$ to be in the forward light cone). The only requirement is that the two
vectors are not colinear, so that $(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}>0$.

It is important to realize that $q_{1}$ and $q_{2}$ are _projections_ of
$p_{1}$ and $p_{2}$ onto the light cone, and as such the map is not
invertible: to reconstruct $p_{1}$ and $p_{2}$ from of the massless vectors
$q_{1}$ and $q_{2}$, we also need information about their length. To keep
track of these, we introduce the dimensionless scalar quantities $x$ and $y$,
defined by the equations

$x(1-y)=\frac{p_{1}^{2}}{(p_{1}+p_{2})^{2}},\qquad\qquad(1-x)y=\frac{p_{2}^{2}}{(p_{1}+p_{2})^{2}},$
(3.9)

in terms of which we can write

$\displaystyle p_{1}^{\mu}$
$\displaystyle=\tfrac{1}{2}\sigma^{\mu}_{ab}\left[(1+x-y)\lambda^{a}\bar{\lambda}^{b}+\tfrac{1}{2}\,(1-x-y)\left(\lambda^{a}\lambda^{b}+\bar{\lambda}^{a}\bar{\lambda}^{b}\right)\right],$
(3.10) $\displaystyle p_{2}^{\mu}$
$\displaystyle=\tfrac{1}{2}\sigma^{\mu}_{ab}\left[(1+x-y)\lambda^{a}\bar{\lambda}^{b}-\tfrac{1}{2}\,(1-x-y)\left(\lambda^{a}\lambda^{b}+\bar{\lambda}^{a}\bar{\lambda}^{b}\right)\right].$

This is the central observation of this work: any 3-point function can be
parameterized with a _unique_ complex spinor, like the 2-point function,
together with a pair of scalar quantities. The map

$\left\\{\lambda^{a},\bar{\lambda}^{a},x,y\right\\}\qquad\to\qquad\left\\{p_{1}^{\mu},p_{2}^{\mu}\right\\}$
(3.11)

is surjective, with both sides described by six real parameters. It is also
regular as long as $p_{1}$ and $p_{2}$ are not colinear, i.e. for $(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}=\frac{1}{4}(\lambda\cdot\bar{\lambda})^{2}(1-x-y)^{2}\neq
0$. If we restrict the range of $x$ and $y$ to the half-plane $x+y<1$, then
the map (3.11) is in fact one-to-one.101010The parameterization in terms of
$x$ and $y$ provides in fact a double cover of the real kinematic domain in
terms of $p_{1}^{2}/(p_{1}+p_{2})^{2}$, $p_{2}^{2}/(p_{1}+p_{2})^{2}$, but it
has the advantage of avoiding the square root that is otherwise present in the
definition (3.5) of $q$. The problem of expressing the conformal Ward
identities in this new set of parameters can therefore essentially be reduced
to the computation of a Jacobian.

Before moving forward, let us emphasize once more that this construction is
very specific to Minkowski space-time (spinors parameterize null vectors), and
also to Wightman functions (the spectral condition is needed to guarantee that
at least one momentum sits inside the forward light cone). It is not at all
obvious how this could be extended to Euclidean space, or even to time-ordered
correlation functions in Minkowski space-time.

###  3.2 Conformal symmetry

One of the great features of the map (3.11) is that the two quantities $x$ and
$y$ are _scalar_ and _dimensionless_. Only the spinor $\lambda^{a}$ and its
complex conjugate are dimensionful and transform under Lorentz symmetry. This
implies that it is straightforward to construct objects that transforms
covariantly under these symmetries, up to arbitrary functions of $x$ and $y$.

In particular, all dependence on spinor indices can be expressed in terms of
the polarization tensors (2.6), or their index-free form (2.7). This also
implies that the decomposition (2.9) into helicity eigenstates is still valid,
and that one can write

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p,\zeta)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=\sum_{m=-j}^{j}\varepsilon_{j,-m}(\zeta)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j,m}(-p)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}},$
(3.12)

where the correlator on the right-hand side is now a function of scalar
quantities only, namely $x$, $y$ and $\lambda\cdot\bar{\lambda}$, equivalent
to the momenta-squared $p_{1}^{2}$, $p_{2}^{2}$ and $p^{2}$. Moreover, since
the overall scaling dimension of the correlator is known, we can further
deduce that

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j,m}(-p)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=(p^{2})^{(\Delta_{1}+\Delta_{2}+\Delta-j-6)/2}F_{j,m}(x,y),$
(3.13)

where $\Delta_{1}$, $\Delta_{2}$ and $\Delta$ are respectively the scaling
dimensions of $\phi_{1}$, $\phi_{2}$ and $\mathcal{O}$, and $F_{j,m}(x,y)$ is
an function that is not constrained by Poincaré or scale symmetry. This ansatz
satisfies the Ward identities111111Note that these identities involve the same
generators (2.16) and (2.17) as in section 2. They can be expressed in terms
of the momenta $p_{1}$ and $p_{2}$ as shown in eq. (A.28).

$\displaystyle\left[\widehat{M}_{\mu\nu}-\widehat{\Sigma}_{\mu\nu}(\zeta)\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p_{1}-p_{2},\zeta)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
$\displaystyle=0,$ (3.14)
$\displaystyle\left[\widehat{D}-\Delta-\Delta_{1}-\Delta_{2}+6\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p_{1}-p_{2},\zeta)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
$\displaystyle=0.$ (3.15)

In fact, each term in the sum (3.12) individually satisfies these identities.

On the contrary, the Ward identity for special conformal transformations
relates different helicity eigenstates, as we saw in section 2, and it also
constrains the functions of $x$ and $y$ entering the ansatz (3.13). In terms
of the momenta $p_{1}$ and $p_{2}$, the correlation functions must satisfy the
second-order differential equations

$\widehat{K}^{\mu}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j}(-p_{1}-p_{2},\zeta)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=0,$
(3.16)

where

$\widehat{K}^{\mu}=\sum_{i=1}^{2}\left[\frac{1}{2}p_{i}^{\mu}\frac{\partial^{2}}{\partial
p_{i}^{\nu}\partial p_{i\nu}}-p_{i}^{\nu}\frac{\partial^{2}}{\partial
p_{i}^{\nu}\partial p_{i\mu}}+(\Delta_{i}-3)\frac{\partial}{\partial
p_{i\mu}}\right].$ (3.17)

Note that this equation does not depend on the scaling dimension $\Delta$ or
spin $j$ of the operator $\mathcal{O}$: as with the 2-point function (2.11),
translation symmetry can be used to place $\mathcal{O}$ at the origin of the
space-time coordinate system where special conformal transformations act
trivially. Nevertheless, solving this Ward identity is not an easy task.
Working with spinors once again proves useful: the 3 components of the vector
operator $\widehat{K}^{\mu}$ can be equivalently encoded into 3 scalar
operators of helicity $-1$, $0$ and $+1$, as in eq. (2.24). After computing
the Jacobian of the map (3.11) (details are given in appendix A), one finds

$\displaystyle\widehat{K}_{0}$ $\displaystyle=2\widehat{A}^{\,\rm
I}_{\Delta_{1},\Delta_{2};6-\Delta_{1}-\Delta_{2},0}-\frac{(1-x+y)(1+x-y)}{(1-x-y)^{2}}\,\widehat{H}_{0}^{2}+\frac{1}{2}(\widehat{H}_{+}\widehat{H}_{-}+\widehat{H}_{-}\widehat{H}_{+})$
(3.18)
$\displaystyle\quad+\widehat{D}^{2}+(9-2\Delta_{1}-2\Delta_{2})\widehat{D}$
$\displaystyle\widehat{K}_{+}$ $\displaystyle=\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}-\widehat{H}_{+}(\widehat{D}+\widehat{H}_{0}+5-\Delta_{1}-\Delta_{2})$
$\displaystyle\quad-\frac{1}{1-x-y}\bigg{[}2x(1-x)\frac{\partial}{\partial
x}-\left(\Delta_{1}-2-\tfrac{1}{2}\widehat{H}_{0}\right)(1-x+y)$
$\displaystyle\hskip 85.35826pt-2y(1-y)\frac{\partial}{\partial
y}+\left(\Delta_{2}-2-\tfrac{1}{2}\widehat{H}_{0}\right)(1+x-y)\bigg{]}\widehat{H}_{0}$
$\displaystyle\widehat{K}_{-}$ $\displaystyle=\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}-\widehat{H}_{-}(\widehat{D}-\widehat{H}_{0}+5-\Delta_{1}-\Delta_{2})$
$\displaystyle\quad+\frac{1}{1-x-y}\bigg{[}2x(1-x)\frac{\partial}{\partial
x}-\left(\Delta_{1}-2+\tfrac{1}{2}\widehat{H}_{0}\right)(1-x+y)$
$\displaystyle\hskip 85.35826pt-2y(1-y)\frac{\partial}{\partial
y}+\left(\Delta_{2}-2+\tfrac{1}{2}\widehat{H}_{0}\right)(1+x-y)\bigg{]}\widehat{H}_{0}$

where we have defined a pair of second-order differential operators in $x$ and
$y$

$\displaystyle\widehat{A}^{\,\rm I}_{\Delta_{1},\Delta_{2};\Delta,m}$
$\displaystyle=x(1-x)\frac{\partial^{2}}{\partial
x^{2}}+y(1-y)\frac{\partial^{2}}{\partial y^{2}}$
$\displaystyle\quad+\left[\left(\tfrac{5}{2}-\Delta_{1}\right)(1-x)-\left(\tfrac{5}{2}-\Delta_{2}\right)x-(2m+1)\frac{x(1-x)}{1-x-y}\right]\frac{\partial}{\partial
x}$
$\displaystyle\quad+\left[\left(\tfrac{5}{2}-\Delta_{2}\right)(1-y)-\left(\tfrac{5}{2}-\Delta_{1}\right)y-(2m+1)\frac{y(1-y)}{1-x-y}\right]\frac{\partial}{\partial
y}$
$\displaystyle\quad-\frac{1}{2}(6-\Delta_{1}-\Delta_{2}-\Delta+m)(3-\Delta_{1}-\Delta_{2}+\Delta+m)$
(3.19)

and

$\displaystyle\widehat{A}^{\,\rm II}_{\Delta_{1},\Delta_{2}}$
$\displaystyle=x(1-x)\frac{\partial^{2}}{\partial
x^{2}}+\left[\left(\tfrac{5}{2}-\Delta_{1}\right)(1-x)-\left(\tfrac{5}{2}-\Delta_{2}\right)x\right]\frac{\partial}{\partial
x}$ (3.20) $\displaystyle\quad-y(1-y)\frac{\partial^{2}}{\partial
y^{2}}-\left[\left(\tfrac{5}{2}-\Delta_{2}\right)(1-y)-\left(\tfrac{5}{2}-\Delta_{1}\right)y\right]\frac{\partial}{\partial
y}.$

These are differential operators of the Appell $F_{4}$ type (see below). A
remarkable property of the $\widehat{K}_{i}$ is that they take a compact form
when applied to eigenfunctions of the scale and helicity operators
$\widehat{D}$ and $\widehat{H}_{0}$: using the fact that
$\widehat{D}=\Delta_{1}+\Delta_{2}+\Delta-6$ and $\widehat{H}_{0}=m$ for each
term in the ansatz (3.12), one can write

$\displaystyle\widehat{K}_{0}$ $\displaystyle=2(1-x-y)^{\pm
m}\widehat{A}^{\,\rm I}_{\Delta_{1},\Delta_{2};\Delta,\pm m}(1-x-y)^{\mp
m}+\frac{1}{2}(\widehat{H}_{+}\widehat{H}_{-}+\widehat{H}_{-}\widehat{H}_{+})\mp
m,$ $\displaystyle\widehat{K}_{+}$
$\displaystyle=(1-x-y)^{-m}\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}(1-x-y)^{m}-(\Delta+m-1)\widehat{H}_{+},$ (3.21)
$\displaystyle\widehat{K}_{-}$ $\displaystyle=(1-x-y)^{m}\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}(1-x-y)^{-m}-(\Delta-m-1)\widehat{H}_{-},$

where for $\widehat{K}_{0}$ either sign is possible. It is important to note
that the differential operators $\widehat{A}^{{\,\rm I},{\,\rm II}}$ act to
the right, meaning that they do not commute with the power of $(1-x-y)$. As a
reminder, $\widehat{H}_{\pm}$ raise or lower the helicity of the polarization
tensors $\varepsilon_{j,m}(\zeta)$, as in eq. (2.33), and it acts trivially on
$p^{2}=-(\lambda\cdot\bar{\lambda})^{2}$. This implies that the Ward identity
(3.16) yields the following conditions on the ansatz (3.13): one set of
equations involving the function $F_{j,m}(x,y)$ only,

$\widehat{A}^{\,\rm I}_{\Delta_{1},\Delta_{2};\Delta,\pm m}\left[(1-x-y)^{\mp
m}F_{j,m}(x,y)\right]=\frac{(j\mp m)(j\pm m+1)}{2}\,(1-x-y)^{\mp
m}F_{j,m}(x,y)$ (3.22)

and two recursion relations involving functions with consecutive values of
$m$,

$\widehat{A}^{\,\rm II}_{\Delta_{1},\Delta_{2}}\left[(1-x-y)^{\mp
m}F_{j,m}(x,y)\right]=(j\pm m+1)(\Delta\mp m-2)(1-x-y)^{\mp m}F_{j,m\pm
1}(x,y).$ (3.23)

These equations apply to all functions $F_{j,m}$ with $m$ between $-j$ and
$+j$. There are no functions with $|m|=j+1$ in the sum (3.12): when $m=+j$ or
$m=-j$, instead of eq. (3.23), one gets

$\widehat{A}^{\,\rm II}_{\Delta_{1},\Delta_{2}}\left[(1-x-y)^{\mp j}F_{j,\pm
j}(x,y)\right]=0.$ (3.24)

In the same cases, the right-hand side of eq. (3.22) vanishes with the
appropriate choice of sign, and thus

$\widehat{A}^{\,\rm I}_{\Delta_{1},\Delta_{2};\Delta,\pm j}\left[(1-x-y)^{\mp
j}F_{j,\pm j}(x,y)\right]=0.$ (3.25)

Eqs. (3.22) to (3.25) incorporate all the constraints that conformal symmetry
imposes on the 3-point correlator. Eq. (3.23) is particularly important as it
relates all functions $F_{j,m}$ with distinct $m$ in an unequivocal way: if
any of these functions is known, then all the others can be obtained
recursively applying the simple differential operator (3.20).

###  3.3 Solutions

In addition to this recursion relation, the two equations (3.24) and (3.25)
taken together are _nearly_ sufficient to determine the functions $F_{j,\pm
j}$ completely. As shown in detail in appendix B, the system of partial
differential equation generated by $\widehat{A}^{\,\rm
I}_{\Delta_{1},\Delta_{2};\Delta,m}$ and $\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}$ is of the Appell $F_{4}$ type. It admits as a
solution the double hypergeometric series

$F_{4}\left(\tfrac{6-\Delta_{1}-\Delta_{2}-\Delta+m}{2},\tfrac{3-\Delta_{1}-\Delta_{2}+\Delta+m}{2},\tfrac{5}{2}-\Delta_{1},\tfrac{5}{2}-\Delta_{2};x(1-y),(1-x)y\right),$
(3.26)

defined in eq. (B.5). Note that the arguments of this function are precisely
$p_{1}^{2}/(p_{1}+p_{2})^{2}$ and $p_{2}^{2}/(p_{1}+p_{2})^{2}$, as defined by
eq. (3.9). This is the unique solution that is analytic around $(x,y)=(0,0)$
(up to an overall multiplicative constant), but there are in fact other
solutions that cannot be expressed as Taylor series in $x$ and $y$. For
generic values of the scaling dimensions, the most general solution to the
system (3.24)–(3.25) is a linear combination of the following four functions
[22]:

$\displaystyle
w_{1}^{\Delta_{1}-3/2}w_{2}^{\Delta_{2}-3/2}f_{\Delta_{1},\Delta_{2},\Delta,j,j}(x,y),$
(3.27) $\displaystyle
w_{1}^{\Delta_{1}-3/2}f_{\Delta_{1},3-\Delta_{2},\Delta,j,j}(x,y),$
$\displaystyle
w_{2}^{\Delta_{2}-3/2}f_{3-\Delta_{1},\Delta_{2},\Delta,j,j}(x,y),$
$\displaystyle f_{3-\Delta_{1},3-\Delta_{2},\Delta,j,j}(x,y),$

where we have used the shorthand notation $w_{1}=x(1-y)$ and $w_{2}=y(1-x)$,
corresponding to the ratios of momenta squared, and defined the function

$\displaystyle f_{\Delta_{1},\Delta_{2},\Delta,j,j}(x,y)$
$\displaystyle=(1-x-y)^{j}$ $\displaystyle\quad\times
F_{4}\left(\tfrac{\Delta_{1}+\Delta_{2}-\Delta+j}{2},\tfrac{\Delta_{1}+\Delta_{2}+\Delta-3+j}{2},\Delta_{1}-\tfrac{1}{2},\Delta_{2}-\tfrac{1}{2};x(1-y),(1-x)y\right).$
(3.28)

The fact that the conformal Ward identities are solved by Appell $F_{4}$
functions in terms of the ratios of momenta squared was first observed in the
scalar case $j=0$ [4, 5]. Here we have just shown that the highest-helicity
components of the 3-point function must equally be of the Appell $F_{4}$ type,
up to a multiplicative factor of $(1-x-y)^{j}$. This fact was already observed
in refs. [9, 23].

The advantage of working in the helicity basis is that the recursion relation
can be solved exactly. The result is not particularly pleasing for the eye,
but it is a function of the generalized hypergeometric type that can be
evaluated efficiently at any value of $x$ and $y$: as explained in appendix B,
we find

$f_{\Delta_{1},\Delta_{2},\Delta,j,m}(x,y)=\sum_{n=0}^{j-m}(1-x-y)^{j-n}\sum_{i=0}^{n}\chi_{\Delta_{1},\Delta_{2},\Delta,j,m}^{(n,i)}f_{\Delta_{1},\Delta_{2},\Delta,j}^{(n-i,i)}(x,y)$
(3.29)

where $f^{(a,b)}$ is a generalization of the Appell $F_{4}$ series, defined by
the infinite series

$\displaystyle
f_{\Delta_{1},\Delta_{2},\Delta,j}^{(a,b)}(x,y)=\sum_{r=0}^{\infty}$
$\displaystyle\frac{\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+j}{2}\right)_{r}\left(\frac{\Delta_{1}+\Delta_{2}+\Delta-3+j}{2}\right)_{r}\left(j+\frac{1}{2}\right)_{r}}{r!\left(\Delta_{1}-\frac{1}{2}+a\right)_{r}\left(\Delta_{2}-\frac{1}{2}+b\right)_{r}}x^{a+r}y^{b+r}$
$\displaystyle\times{}_{2}F_{1}\left(\tfrac{\Delta_{1}+\Delta_{2}-\Delta+j}{2}+r,\tfrac{\Delta_{1}+\Delta_{2}+\Delta-3+j}{2}+r,\Delta_{1}-\tfrac{1}{2}+a+r;x\right)$
$\displaystyle\times{}_{2}F_{1}\left(\tfrac{\Delta_{1}+\Delta_{2}-\Delta+j}{2}+r,\tfrac{\Delta_{1}+\Delta_{2}+\Delta-3+j}{2}+r,\Delta_{2}-\tfrac{1}{2}+b+r;y\right),$
(3.30)

and the coefficients of the linear combination are

$\displaystyle\chi_{\Delta_{1},\Delta_{2},\Delta,j,m}^{(n,i)}$
$\displaystyle=\frac{(-1)^{j+m+n+i}}{i!(n-i)!}\frac{4^{j-m}(m+1)_{j-m}}{(2m+1)_{j-m-n}}\frac{\left(\frac{\Delta+j-\Delta_{12}}{2}-n+i\right)_{n-i}\left(\frac{\Delta+j+\Delta_{12}}{2}-i\right)_{i}}{\left(\Delta_{1}-\frac{1}{2}\right)_{n-i}\left(\Delta_{2}-\frac{1}{2}\right)_{i}(\Delta+m-1)_{j-m}}$
$\displaystyle\times$
$\displaystyle\sum_{k=0}^{j-m-n}\frac{(-1)^{k}\left(m+\frac{1}{2}\right)_{j-m-
n-k}\left(m+\frac{1}{2}\right)_{k}}{k!(j-m-
n-k)!}\left(\tfrac{\Delta-j-1+\Delta_{12}}{2}\right)_{j-m-
i-k}\left(\tfrac{\Delta-j-1-\Delta_{12}}{2}\right)_{i+k},$ (3.31)

where we have denoted $\Delta_{12}=\Delta_{1}-\Delta_{2}$. This is the result
of applying the recursion relation (3.23) starting from the highest helicity
solution given in eq. (3.28). The other solutions can be obtained from this
one applying the “shadow transform” $\Delta_{i}\leftrightarrow 3-\Delta_{i}$
to any of the two scaling dimensions $\Delta_{1}$ and $\Delta_{2}$, and
multiplying with the appropriate power of $w_{1}$ and $w_{2}$ as in eq.
(3.27).

Some important properties of the 3-point function can be deduced from this
general solution:

  * $\circ$

If $j$ is half-integer, the series generated by the recursion relation never
terminates, which is inconsistent with our ansatz (3.12). It is of course well
known that only integer spin operators enter the OPE of two scalars, so that
this pathological situation is never realized. With integer $j$, one can
verify that once the recursion relation passes $m=0$, the linear combination
involves fewer terms again until it finally collapses back to a single term at
$m=-j$. In fact, the symmetry of eq. (3.23) around $m=0$ implies that

$f_{\Delta_{1},\Delta_{2},\Delta,j,-m}(x,y)=f_{\Delta_{1},\Delta_{2},\Delta,j,m}(x,y).$
(3.32)

Therefore, all functions can be expressed in terms of eq. (3.29) with $m\geq
0$.

  * $\circ$

The formula (3.29) involves terms with every positive power of $(1-x-y)$
between $m$ and $j$. In the _colinear limit_ $p_{1}\propto p_{2}$, we have
$1-x-y=0$ and therefore all functions $f$ vanish identically except for the
one with $m=0$. This is because there is no angular momentum in this case, and
therefore the vanishing spin of the state created by the two scalar operators
$\phi_{1}$ and $\phi_{2}$ must be matched by the spin $m$ of the state created
by the operator $\mathcal{O}$. Eq. (B.21) in the appendix gives the simpler
form of the solution in that limit.

  * $\circ$

Since the differential operator $\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}$ involved in the recursion relation (3.23) is odd
under the simultaneous exchange $\Delta_{1}\leftrightarrow\Delta_{2}$ and
$x\leftrightarrow y$, the solution must satisfy

$f_{\Delta_{2},\Delta_{1},\Delta,j,m}(y,x)=(-1)^{j-m}f_{\Delta_{1},\Delta_{2},\Delta,j,m}(x,y)$
(3.33)

This is indeed what is observed: the generalization (3.30) of the Appell
$F_{4}$ function has this symmetry upon exchanging $a\leftrightarrow b$, and
similarly the coefficients $\chi$ transform into each other under
$i\leftrightarrow n-i$. A consequence of this is that when
$\Delta_{1}=\Delta_{2}$, for instance when the two scalar operators are
identical, then the operator $\mathcal{O}$ can only have even spin. This is
again a well-known property of the OPE of two identical scalar operators.

  * $\circ$

Finally, it should be noted that all coefficients $\chi$ with $m<j$ vanish if
$\Delta_{1}=\Delta_{2}$ and $\Delta=j+1$. This corresponds to the case in
which $\mathcal{O}$ is a conserved current, transforming in a short
representation of the conformal group. The vanishing of the Wightman 3-point
function is consistent with our findings of section 2 in which we observed
that $\mathcal{O}$ only creates states of helicity $m=\pm j$.

###  3.4 Correlation function

We are now armed with a set of functions that satisfy all constraints imposed
by conformal symmetry. But what linear combination of solutions corresponds to
the Wightman correlation function?

Conformal Ward identities alone are not enough to answer this question. In
fact, for generic (non-integer) value of the scaling dimensions, different
correlation functions obey the same set of identities: Wightman functions,
time-ordered products, and retarded products can all be expressed as linear
combinations of the solutions that we just found. Additional information is
needed to distinguish between these various cases. It can be obtained using
analyticity properties that follow from the micro-causality condition [24], or
in the specific case of Wightman functions, studying some OPE limits [9]. We
refer the reader to the relevant work for more details, and focus here on the
result.

In the case where both $p_{1}$ and $p_{2}$ are inside the forward light cone,
one finds

$\displaystyle F_{j,m}(x,y)$
$\displaystyle=4\pi\mathcal{N}\,\frac{\sin\left[\pi\frac{\Delta_{1}-\Delta_{2}-\Delta+j+3}{2}\right]}{\Gamma\left(\Delta_{1}-\frac{1}{2}\right)}w_{1}^{\Delta_{1}-3/2}$
$\displaystyle\quad\times\bigg{[}\Gamma\left(\tfrac{3}{2}-\Delta_{2}\right)\Gamma\left(\tfrac{\Delta_{1}+\Delta_{2}-\Delta+j}{2}\right)\Gamma\left(\tfrac{\Delta_{1}+\Delta_{2}+\Delta+j-3}{2}\right)w_{2}^{\Delta_{2}-3/2}f_{\Delta_{1},\Delta_{2},\Delta,j,m}(x,y)$
$\displaystyle\quad\qquad+\Gamma\left(\Delta_{2}-\tfrac{3}{2}\right)\Gamma\left(\tfrac{\Delta_{1}-\Delta_{2}-\Delta+j+3}{2}\right)\Gamma\left(\tfrac{\Delta_{1}-\Delta_{2}+\Delta+j}{2}\right)f_{\Delta_{1},3-\Delta_{2},\Delta,j,m}(x,y)\bigg{]}.$
(3.34)

This is the unique linear combination of solutions that is on the one hand
proportional to $w_{1}^{\Delta_{1}-3/2}$ as $p_{1}^{2}\to 0$ (the other two
solutions are analytic in $w_{1}$), in accordance with the normalization of
the state $\phi_{1}(p_{1})\left|0\right\rangle$ found in section 2, and on the
other hand that is also analytic in $1-w_{2}$ as $w_{2}\to 1_{-}$, as required
by the existence of the limit $p_{1}^{\mu}\to 0$. The constant $\mathcal{N}$
is related to the OPE coefficient, and it is therefore arbitrary unless we
adopt some specific convention. Following one standard choice detailed in ref.
[23] in which the OPE coefficient $\lambda_{12\mathcal{O}}$ is defined from
the position-space correlation function, we have

$\mathcal{N}=\frac{(4\pi)^{d}(-i)^{j}(\Delta-1)_{j}\,\lambda_{12\mathcal{O}}}{2^{\Delta_{1}+\Delta_{2}+\Delta}\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+j}{2}\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+j}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+j}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+j-3}{2}\right)}.$
(3.35)

When $p_{2}$ is instead space-like, we have

$\displaystyle F_{j,m}(x,y)$
$\displaystyle=4\pi^{2}\mathcal{N}\,\frac{1}{\Gamma\left(\Delta_{1}-\frac{1}{2}\right)}w_{1}^{\Delta_{1}-3/2}$
$\displaystyle\quad\times\bigg{[}\frac{\Gamma\left(\frac{3}{2}-\Delta_{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+j-3}{2}\right)}{\Gamma\left(1-\frac{\Delta_{1}+\Delta_{2}-\Delta+j}{2}\right)}(-w_{2})^{\Delta_{2}-3/2}f_{\Delta_{1},\Delta_{2},\Delta,j,m}(x,y)$
$\displaystyle\quad\qquad+\frac{\Gamma\left(\Delta_{2}-\frac{3}{2}\right)\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+j}{2}\right)}{\Gamma\left(1-\frac{\Delta_{1}-\Delta_{2}-\Delta+j+3}{2}\right)}f_{\Delta_{1},3-\Delta_{2},\Delta,j,m}(x,y)\bigg{]}.$
(3.36)

This only differs from eq. (3.34) by a ratio of phases in the first term in
square brackets. In this case, the highest-helicity function with $m=+j$ takes
a particularly simple form thanks to a property of the Appell $F_{4}$
function: it can be written

$\displaystyle F_{j,j}(x,y)$
$\displaystyle=4\pi^{2}\mathcal{N}\,\frac{\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+j}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+j-3}{2}\right)}{\Gamma\left(\Delta_{1}-\frac{1}{2}\right)\Gamma\left(\Delta-\frac{1}{2}\right)}$
$\displaystyle\quad\times
w_{1}^{\Delta_{1}-3/2}(-w_{2})^{-(\Delta_{1}-\Delta_{2}+\Delta+j)/2}f_{\Delta_{1},\Delta,\Delta_{2},j,j}\left(\frac{x}{x-1},\frac{1}{y}\right),$
(3.37)

in terms of a unique Appell $F_{4}$ hypergeometric function in which the roles
of $\Delta_{2}$ and $\Delta$ have been exchanged. In this case the correlation
function is still proportional to $w_{1}^{\Delta_{1}-3/2}$ as $p_{1}^{2}\to
0$, but it has also the scaling limit of a state created by the operator
$\mathcal{O}$ when $(p_{1}+p_{2})^{2}\to 0$. Unfortunately, the type of
transformation property relating series expansions around $(x,y)=(0,0)$ and
$(x,y)=(0,-\infty)$ is special to the Appell $F_{4}$ function and does not
carry on in a simple manner to its generalization (3.30). For this reason, the
problem of finding a closed form expression for the function $F_{j,m}$ with
$m\neq\pm j$ when $p_{2}$ is outside the forward light cone will be treated
elsewhere.121212In fact, solving the conformal Ward identities in the case
where all 3 operators carry spin will contribute to answering this question,
as one can then always choose the reference momentum — the one expressed in
terms of spinors as in eq. (2.4) — to be the largest so that $x$ and $y$
remain small. Our approach in this work relies on the contrary on taking the
momentum of the _spinning_ operator as a reference, and therefore it is not
general enough to obtain a simple solution. Note that eq. (3.36) gives a
partial answer, but it involves the function (3.30) whose definition as a
series expansion is only convenient in a neighborhood of $(x,y)=(0,0)$.
Nevertheless, even in the absence of closed-form solution, the recursion
relation (3.23) can still be used directly onto eq. (3.37).

The results presented in this section are the main output of this work. We
have shown how the parameterization of the 3-point function with spinors helps
express the conformal Ward identities in terms of simple ladder operators, how
the various helicity eigenstates are related by the action of a differential
operator, and we even arrived at a closed-form solution valid when $p_{1}^{2}$
and $p_{2}^{2}$ are small compared to $(p_{1}+p_{2})^{2}$. Before concluding,
we want to illustrate in the next section how this 3-point function can be
used as elementary building block for the construction of higher-point
Wightman functions.

##  4 The OPE for higher-point functions

Our results for the 2- and 3-point correlation functions implicitly define a
momentum-space OPE for conformal primary operators acting on the vacuum, and
the goal of this short section is to illustrate its conceptual simplicity.

The existence of an operator product expansion in quantum field theory is
intrinsically related to the completeness of the Hilbert space: if all the
states created by a single local operator acting on the vacuum span the entire
Hilbert space, then the product of two (or more) operators must also be
expressible in that basis. In other words, we must be able to write

$\phi_{2}(p_{2})\phi_{1}(p_{1})\left|0\right\rangle\approx\sum_{\mathcal{O}}C_{12\mathcal{O}}\mathcal{O}(p_{1}+p_{2})\left|0\right\rangle,$
(4.1)

where $C_{12\mathcal{O}}$ is an object that depends on the momenta $p_{1}$ and
$p_{2}$. This statement can be made sharp in conformal field theory thanks to
radial quantization in Euclidean position space.131313Another salient feature
of radial quantization is that the convergence rate of the OPE is controlled
by the separation of operators in space, and this can be used to show that the
OPE is absolutely convergent over most of the configuration space. There is no
such argument in momentum space, and therefore we generically only expect
convergence in the sense of distributions (see however ref. [25] for a
counterexample in 2 dimensions). In this case however the sum does not only
involve primary operators on the right-hand side, but also descendants. This
inconvenience disappears when working in momentum space, as there is a unique
momentum eigenstate in each conformal family that can appear on the right-hand
side of eq. (4.1).

The other difficulty in defining the OPE has to do with spin: even when the
operators appearing on the left-hand side of eq. (4.1) are scalars, the fact
that they are inserted at separated points — or come with distinct momenta —
means that the state carries angular momentum, and thus that the operator on
the right-hand side might have spin. This is where the spinor-helicity
formalism becomes extremely useful, as it allows to further decompose the OPE
(4.1) into helicity eigenstates.141414Strictly speaking the spinor-helicity
formalism is not necessary to do so: when the two operators are scalars, the
total spin is given by the angular momentum, which means that the polarization
tensors can be constructed out of the momenta $p_{1}$ and $p_{2}$ only. This
fact has been used in ref. [23] to write down the OPE for a scalar 4-point
function in general space-time dimension without making use of spinors. The
decomposition (2.9) can be used to absorb the polarization tensors into the
object $C_{12\mathcal{O}}$ and write

$\phi_{2}(p_{2})\phi_{1}(p_{1})\left|0\right\rangle=\sum_{\mathcal{O}}\sum_{m=-j}^{j}C^{(j,m)}_{12\mathcal{O}}\mathcal{O}_{j,m}(p_{1}+p_{2})\left|0\right\rangle.$
(4.2)

$C_{12\mathcal{O}}$ is now a mere function of the momenta $p_{1}$ and $p_{2}$,
which can be computed taking the overlap of both sides of this equation with a
helicity eigenstate $\mathcal{O}_{j,m}(p)\left|0\right\rangle$. We find

$C^{(j,m)}_{12\mathcal{O}}=\frac{\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j,m}(-p)\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}}{\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}_{j,m}(-p)\mathcal{O}_{j,m}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}}=(p^{2})^{(\Delta_{1}+\Delta_{2}-\Delta+j-3)/2}C_{\Delta,j,m}^{-1}F_{j,m}(x,y),$
(4.3)

where the coefficient $C_{\Delta,j,m}$ is given in eq. (2.38), the function
$F_{j,m}$ in eq. (3.34), and $x$, $y$ are defined in terms of the momenta
$p_{1}$ and $p_{2}$ in eq. (3.9). The dynamical OPE coefficient is contained
in the definition of $F_{j,m}$.

As it is, this result is valid when the helicity of the operator
$\mathcal{O}_{j,m}$ on the right-hand side of eq. (4.2) is defined with
respect to the same spinors $\lambda$ and $\bar{\lambda}$ used to
parameterized the momenta $p_{1}$ and $p_{2}$ as in eq. (3.10). This is not
necessary, however, and in fact it is not wanted in the perspective of
iterating the OPE. Thankfully, the freedom to choose a different basis of
spinors is easy to implement in eq. (4.2): any other basis will be related to
$\\{\lambda^{a},\bar{\lambda}^{a}\\}$ by a little group transformation (2.5).
If we define this new basis by

$\lambda^{\prime a}=e^{i\theta}\lambda^{a},\qquad\qquad\bar{\lambda}^{\prime
a}=e^{-i\theta}\bar{\lambda}^{a},$ (4.4)

then we can again compute the overlap of eq. (4.2) with an eigenstate of
helicity given in the original basis. This time, the computation involves
contracting the indices of two polarization tensors of the form (2.6) that
satisfy

$\varepsilon^{\prime}_{-m^{\prime}}\cdot\varepsilon_{m}=\delta_{m^{\prime}m}e^{i\theta
m}(-1)^{j}(p^{2})^{j}.$ (4.5)

In other words, the change of basis only accounts for a phase $e^{i\theta m}$
in the coefficient (4.3), which is trivial to implement in a computation.

To give a concrete example, this OPE can be used to decompose the 4-point
correlation function of scalar primary operators into conformal partial waves.
We find immediately

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\phi_{4}(p_{4})\phi_{3}(p_{3})\phi_{2}(p_{2})\phi_{1}(p_{1})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=s^{(\Delta_{\Sigma}-9)/2}\sum_{\mathcal{O}}\sum_{m=-j}^{j}C_{\Delta,j,m}^{-1}e^{im\theta}F_{j,-m}(x^{\prime},y^{\prime})^{*}F_{j,m}(x,y)$
(4.6)

where $s=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2}$ is the square of the center-of-
mass energy, the variables $x$, $y$, $x^{\prime}$ and $y^{\prime}$ are defined
by

$x(1-y)=\frac{p_{1}^{2}}{s},\qquad(1-x)y=\frac{p_{2}^{2}}{s},\qquad(1-x^{\prime})y^{\prime}=\frac{p_{3}^{2}}{s},\qquad
x^{\prime}(1-y^{\prime})=\frac{p_{4}^{2}}{s},$ (4.7)

and $\theta$ is the angle formed by the space-like vector $q$ defined in terms
of momenta $p_{1}$ and $p_{2}$ in eq. (3.5) and a similar vector defined in
terms of $p_{3}$ and $p_{4}$. We have also denoted
$\Delta_{\Sigma}=\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}$. Note that
because of the symmetry relation (3.32) the phase always appears in the linear
combination $e^{im\theta}+e^{-im\theta}=2\cos(m\theta)$, which is (twice) the
Chebyshev polynomial $T_{m}(\cos\theta)$ corresponding to spherical harmonics
in 3 space-time dimensions. This result precisely matches the conformal
partial waves obtained in ref. [23] in the special case $d=3$. A significant
improvement with respect to that work is that the 3-point functions are now
known exactly, whereas they were given by a recursion relation there.151515The
two-step recursion relation for the 3-point function given in ref. [23] is a
linear combination of the two equations corresponding to both signs in eq.
(3.23).

##  5 Conclusions

This work develops a new formalism for conformal correlators in terms of
spinor variables. We have emphasized the usefulness of spinors to decompose
the states created by a single conformal primary operator acting on the vacuum
into helicity eigenstates, and extended this formalism to the product of two
operators acting on the vacuum. While the 3-point functions involved in these
computation are not completely new, the novel formalism that we used makes
their derivation particularly clear when compared with other methods.

This work is also an important milestone in a larger research program: the
construction of all conformal Wightman functions in Minkowski momentum space.
The state-of-the-art in this area can be summarized as follows:

  1. 1.

Wightman 2-point functions in momentum space have been known since the early
days of conformal field theory, and used in the constructions of all unitary
representations [11]. They recently re-appeared as an efficient way of
computing Euclidean conformal blocks in ref. [18]. For tensor operators, a
construction valid in any space-time dimension has been presented in ref. [8].

  2. 2.

The Wightman 3-point function of scalar operators was constructed for the
first time in closed-form in ref. [9]. An integral representation particularly
useful for holographic applications is also available [26].

  3. 3.

The 3-point function of two scalars and one symmetric tensor operator was also
presented in ref. [9], but its projection onto spin eigenstates only appeared
in ref. [23]. This allowed for a convenient decomposition of the 4-point
function into partial waves in the same work.161616An interesting feature of
all these results is that they are analytic in the space-time dimension $d$.
This is a peculiarity of working only with tensor representations of the
Lorentz group that exist in any $d$.

The obvious next step is to compute Wightman 3-point functions of operators
carrying arbitrary Lorentz representations. This would not only grant access
to partial wave expansions for generic 4-point functions, but also for any
higher-point function through the momentum-space OPE.171717The current
formalism only allows for the computation of higher-point partial waves in
which one of the intermediate operator is a scalar.

The present work does not yet present results for these generic 3-point
function, but it takes an important step in this direction. The derivation of
conformal Ward identities in terms of helicity ladder operators in not
conceptually different from what has been presented here. It is however vastly
more complicated to solve the resulting partial differential equations, and
for this reason the results will be presented in a separated work. The goal of
the present paper is therefore to introduce the method, and illustrate its
simplicity when applied to 3-point correlation functions involving two scalar
operators. For the same reason, we have chosen to work here exclusively in 3
space-time dimensions. The results will be generalized to conformal field
theory in 4 and higher dimensions elsewhere.

The informed reader will also wonder what is the physical purpose of this
program, besides its mathematical interest. There is in fact already a large
body of literature on the construction of correlation functions that have the
isometries of the Euclidean conformal group $\text{SO}(d+1,1)$, both for CFT
in flat space, of for QFT in AdS. Many works have been specifically dedicated
to the study of operators transforming in various non-trivial representations
[27, 28, 29, 30]. In fact, the problem has essentially been solved by the
introduction of weight-shifting operators [31, 32, 33, 34]. One aspect in
which our approach differs from the standard literature is that it relies on
the Lorentzian conformal group $\text{SO}(d,2)$. However, one might argue that
the two are substantially equivalent thanks to the simple analytic
continuation that is possible between Euclidean correlators and Wightman
functions in Minkowski space-time [35].

Nevertheless, there are at least three directions of research that will
actively benefit from the development of this program. The first one is the
study of higher-point correlation functions, both in CFT and in AdS. The
computation of conformal blocks for functions of more than 4 points is quite
difficult, and various methods have been used [36, 37, 38, 39, 40, 41, 42,
43]. It is obvious that the factorization observed in the Minkowski momentum
space OPE present important advantages, even though it only applies to
Wightman functions. The second direction of research is Hamiltonian truncation
in infinite volume [44, 45], where one of the computational bottlenecks is the
evaluation of matrix elements that correspond to a particular limit of the
Wightman 3-point functions [46]. Finally, the last area in which this work can
have an impact is quantum field theory in de Sitter space-time. Many recent
works have exploited the Euclidean conformal symmetry of late-time correlators
[47, 48, 49, 50, 51, 52], but the Lorentzian conformal group also plays an
important role in the in-in formalism recently put forward [53, 54, 55, 56].

### Acknowledgments

The author would like to thank Brian Henning and Matthew Walters for
stimulating discussions, as well as the University of Zürich for their
hospitality during the completion of this work.

##  Appendix A Conventions for spinors

We work in 3-dimensional Minkowski space-time with the mostly-minus
metric181818While the mostly-plus metric is arguably the best choice in
position space, as it makes the analytic continuation between Euclidean and
Minkowski spaces obvious, the mostly-minus metric is definitely more
convenient to work with in momentum space.
$\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1)_{\mu\nu}$, where $\mu=0,1,2$. Let
$z^{\mu}$ be a null vector pointing in the forward direction,

$z^{2}=(z^{0})^{2}-(z^{1})^{2}-(z^{2})^{2}=0,\qquad\qquad z^{0}>0.$ (A.1)

$z^{\mu}$ is conveniently parameterized in terms of a two-component real
spinor $\zeta^{a}$ as

$z^{\mu}=\tfrac{1}{2}\,\sigma^{\mu}_{ab}\zeta^{a}\zeta^{b},$ (A.2)

where repeated indices $a,b=1,2$ are summed over, and $\sigma^{\mu}$ are the
$2\times 2$ symmetric matrices

$\sigma^{0}_{ab}=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)_{ab},\qquad\sigma^{1}_{ab}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)_{ab},\qquad\sigma^{2}_{ab}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)_{ab}.$ (A.3)

We may sometimes also use the 2-index notation for a vector,
$z^{\mu}\equiv\tfrac{1}{2}\,\sigma^{\mu}_{ab}z^{ab}$. Introducing

$\varepsilon_{ab}=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)_{ab},$ (A.4)

we have the identity

$\eta_{\mu\nu}\sigma^{\mu}_{ab}\sigma^{\nu}_{cd}=\varepsilon_{ac}\varepsilon_{bd}+\varepsilon_{ad}\varepsilon_{bc}.$
(A.5)

Lorentz-invariant quantities can therefore be constructed out of spinors with
upper indices contracted with the matrix $\varepsilon$ with lower indices: for
instance, the antisymmetric scalar product of two spinors is defined by

$\zeta\cdot\xi\equiv\varepsilon_{ab}\zeta^{a}\xi^{b}=-\xi\cdot\zeta.$ (A.6)

For $z^{\mu}$ defined in eq. (A.2) and another vector
$x^{\mu}=\tfrac{1}{2}\,\sigma^{\mu}_{ab}\xi^{a}\xi^{b}$, we have

$z\cdot
x=\eta_{\mu\nu}z^{\mu}x^{\nu}=\tfrac{1}{4}\,\eta_{\mu\nu}\sigma^{\mu}_{ab}\sigma^{\nu}_{cd}\zeta^{a}\zeta^{b}\xi^{c}\xi^{d}=\tfrac{1}{2}\,\varepsilon_{ac}\varepsilon_{bd}\zeta^{a}\zeta^{b}\xi^{c}\xi^{d}=\frac{1}{2}(\zeta\cdot
x)^{2}.$ (A.7)

It is sometimes convenient to introduce the contravariant spinor

$\zeta_{a}\equiv\varepsilon_{ab}\zeta^{b},$ (A.8)

in terms of which the scalar product can be written

$\zeta\cdot\xi=\zeta^{a}\xi_{a}=-\zeta_{a}\xi^{a}.$ (A.9)

A lower spinor index can be raised with the inverse tensor
$\varepsilon^{ab}=-\varepsilon^{ba}$ satisfying

$\varepsilon^{ac}\varepsilon_{cb}=\varepsilon_{bc}\varepsilon^{ca}=\delta^{a}_{b},$
(A.10)

so that $\varepsilon^{ab}\zeta_{b}=\zeta^{a}$. Note that both the (inverse)
metric $\eta^{\mu\nu}$ and the Levi-Civita symbol $\varepsilon^{\mu\nu\rho}$
can be expressed in terms of $\sigma^{\mu}$ through the identity

$(\sigma^{\mu}\cdot\sigma^{\nu})_{ab}\equiv-\sigma^{\mu}_{ac}\varepsilon^{cd}\sigma^{\nu}_{db}=\eta^{\mu\nu}\varepsilon_{ab}+\varepsilon^{\mu\nu\rho}(\sigma_{\rho})_{ab}.$
(A.11)

#### Complex spinors.

Since spinors have two components, any pair of linearly independent spinors
can be used as a basis. We shall take the real spinors $\lambda_{1}^{a}$ and
$\lambda_{2}^{a}$ to form such a basis, or more conveniently the complex
spinors $\lambda^{a}$ and $\bar{\lambda}^{a}$ that are conjugate to each
other, defined by

$\lambda^{a}=\frac{\lambda_{1}^{a}+i\lambda_{2}^{a}}{\sqrt{2}},\qquad\qquad\bar{\lambda}^{a}=\frac{\lambda_{1}^{a}-i\lambda_{2}^{a}}{\sqrt{2}}.$
(A.12)

This means that we can write

$\varepsilon^{ab}=\frac{\bar{\lambda}^{a}\lambda^{b}-\lambda^{a}\bar{\lambda}^{b}}{\lambda\cdot\bar{\lambda}},$
(A.13)

or express any other spinor in this basis through

$\zeta^{a}=\frac{(\zeta\cdot\bar{\lambda})\lambda^{a}-(\zeta\cdot\lambda)\bar{\lambda}^{a}}{\lambda\cdot\bar{\lambda}}.$
(A.14)

These two spinors can be used to parameterize a momentum inside the forward
light cone as in eq. (2.4),

$p^{\mu}=\sigma^{\mu}_{ab}\,\lambda^{a}\bar{\lambda}^{b}=\tfrac{1}{2}\sigma^{\mu}_{ab}\left(\lambda_{1}^{a}\lambda_{1}^{b}+\lambda_{2}^{a}\lambda_{2}^{b}\right).$
(A.15)

Note that the energy
$p^{0}=\left|\lambda_{1}\right|^{2}+\left|\lambda_{2}\right|^{2}$ is positive,
and likewise

$p^{2}=-(\lambda\cdot\bar{\lambda})^{2}=(\lambda_{1}\cdot\lambda_{2})^{2}>0.$
(A.16)

With our conventions the scalar product $\lambda\cdot\bar{\lambda}$ is purely
imaginary.

#### Spinor derivatives.

The spinor derivative is defined by the equation

$\frac{\partial\zeta^{b}}{\partial\zeta^{a}}=\delta_{a}^{b},$ (A.17)

or equivalently

$\frac{\partial\zeta^{b}}{\partial\zeta_{a}}=\varepsilon^{ab},$ (A.18)

which implies for the scalar product

$\frac{\partial}{\partial\zeta^{a}}(\zeta\cdot\xi)=\xi_{a},\qquad\frac{\partial}{\partial\xi^{a}}(\zeta\cdot\xi)=-\zeta_{a}.$
(A.19)

When applied to a function of $z^{\mu}$, the chain rule gives

$\frac{\partial
f(z)}{\partial\zeta^{a}}=\sigma^{\mu}_{ab}\zeta^{b}\frac{\partial
f(z)}{\partial z^{\mu}}.$ (A.20)

Using the identity (A.11), we can form the combination

$\sigma^{\mu}_{ab}\zeta^{a}\frac{\partial
f(z)}{\partial\zeta_{b}}=\varepsilon^{\mu\nu\rho}z_{\nu}\frac{\partial
f(z)}{\partial z^{\rho}}.$ (A.21)

This is precisely the derivative operator appearing in the definition (2.15)
of the spin operator: when applied to a correlation function in which Lorentz
indices are contracted with an auxiliary vector $z^{\mu}$, the spin operator
takes the familiar form [2, 16]

$\widehat{\Sigma}_{\mu\nu}(z)=z_{\mu}\frac{\partial}{\partial
z^{\nu}}-z_{\nu}\frac{\partial}{\partial z^{\mu}}.$ (A.22)

#### Derivatives with respect to momenta.

Expressing derivatives with respect to momenta in terms of spinors is the key
element in the computation of the conformal Ward identities for 2- and 3-point
functions. In the presence of a single momentum $p$, the form of this
derivative was given in eq. (2.22), and its projection onto helicity
components in eq. (2.28). This led to the representations (2.16) and (2.17)
for the generators of scale and Lorentz transformations. For special conformal
transformations, the operator $\widehat{K}^{\mu}$ defined in eq. (2.21) is
given by

$\displaystyle\widehat{K}^{\mu}=\tfrac{1}{2}\sigma^{\mu}_{ab}\bigg{[}\,$
$\displaystyle-\frac{\partial^{2}}{\partial\lambda_{a}\partial\bar{\lambda}_{b}}+\frac{\Delta-2}{\lambda\cdot\bar{\lambda}}\left(\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}_{b}}-\lambda^{a}\frac{\partial}{\partial\lambda_{b}}\right)$
$\displaystyle+\frac{1}{4\lambda\cdot\bar{\lambda}}\left(\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}_{c}}+\bar{\lambda}^{c}\frac{\partial}{\partial\bar{\lambda}_{a}}-\lambda^{a}\frac{\partial}{\partial\lambda_{c}}-\lambda^{c}\frac{\partial}{\partial\lambda_{a}}\right)\left(\zeta^{b}\frac{\partial}{\partial\zeta^{c}}+\zeta_{c}\frac{\partial}{\partial\zeta_{b}}\right)\bigg{]}.$
(A.23)

Its helicity components have been reported in eq. (2.31).

For functions of the two momenta $p_{1}^{\mu}$ and $p_{2}^{\mu}$ defined in
eq. (3.10), the algebra is slightly more complicated but the logic is the
same. One can show that derivatives with respect to $p_{1}$ and $p_{2}$
satisfy

$\displaystyle\frac{\partial}{\partial
p_{1\mu}}=\tfrac{1}{2}\sigma^{\mu}_{ab}\bigg{[}$
$\displaystyle\frac{1}{\lambda\cdot\bar{\lambda}}\left(\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}_{b}}-\lambda^{a}\frac{\partial}{\partial\lambda_{b}}\right)-\frac{\lambda^{a}\lambda^{b}-\bar{\lambda}^{a}\bar{\lambda}^{b}}{(\lambda\cdot\bar{\lambda})^{2}}\frac{1-x+y}{1-x-y}\,\widehat{H}_{0}$
$\displaystyle-2\frac{p_{1}^{ab}}{(\lambda\cdot\bar{\lambda})^{2}}\frac{1}{1-x-y}\left((1-x)^{2}\frac{\partial}{\partial
x}+y^{2}\frac{\partial}{\partial y}\right)$ (A.24)
$\displaystyle+2\frac{p_{2}^{ab}}{(\lambda\cdot\bar{\lambda})^{2}}\frac{1}{1-x-y}\left(x(1-x)\frac{\partial}{\partial
x}+y(1-y)\frac{\partial}{\partial y}\right)\bigg{]},$
$\displaystyle\frac{\partial}{\partial
p_{2\mu}}=\tfrac{1}{2}\sigma^{\mu}_{ab}\bigg{[}$
$\displaystyle\frac{1}{\lambda\cdot\bar{\lambda}}\left(\bar{\lambda}^{a}\frac{\partial}{\partial\bar{\lambda}_{b}}-\lambda^{a}\frac{\partial}{\partial\lambda_{b}}\right)+\frac{\lambda^{a}\lambda^{b}-\bar{\lambda}^{a}\bar{\lambda}^{b}}{(\lambda\cdot\bar{\lambda})^{2}}\frac{1+x-y}{1-x-y}\,\widehat{H}_{0}$
$\displaystyle+2\frac{p_{1}^{ab}}{(\lambda\cdot\bar{\lambda})^{2}}\frac{1}{1-x-y}\left(x(1-x)\frac{\partial}{\partial
x}+y(1-y)\frac{\partial}{\partial y}\right)$ (A.25)
$\displaystyle-2\frac{p_{2}^{ab}}{(\lambda\cdot\bar{\lambda})^{2}}\frac{1}{1-x-y}\left(x^{2}\frac{\partial}{\partial
x}+(1-y)^{2}\frac{\partial}{\partial y}\right)\bigg{]},$

where we have used the operator $\widehat{H}_{0}$ defined in eq. (2.25). They
can equivalently be rewritten in the helicity basis as

$\displaystyle\frac{\partial}{\partial
p_{1\mu}}=\frac{\sigma^{\mu}_{ab}}{2(\lambda\cdot\bar{\lambda})^{2}}\bigg{[}$
$\displaystyle-2\lambda^{a}\bar{\lambda}^{b}\left(\widehat{D}+(1-x)\frac{\partial}{\partial
x}-y\frac{\partial}{\partial y}\right)$
$\displaystyle+\bar{\lambda}^{a}\bar{\lambda}^{b}\left(\widehat{H}_{+}-(1-x)\frac{\partial}{\partial
x}-y\frac{\partial}{\partial y}+\frac{1-x+y}{1-x-y}\widehat{H}_{0}\right)$
(A.26)
$\displaystyle+\lambda^{a}\lambda^{b}\left(\widehat{H}_{-}-(1-x)\frac{\partial}{\partial
x}-y\frac{\partial}{\partial
y}-\frac{1-x+y}{1-x-y}\widehat{H}_{0}\right)\bigg{]},$
$\displaystyle\frac{\partial}{\partial
p_{2\mu}}=\frac{\sigma^{\mu}_{ab}}{2(\lambda\cdot\bar{\lambda})^{2}}\bigg{[}$
$\displaystyle-2\lambda^{a}\bar{\lambda}^{b}\left(\widehat{D}-x\frac{\partial}{\partial
x}+(1-y)\frac{\partial}{\partial y}\right)$
$\displaystyle+\bar{\lambda}^{a}\bar{\lambda}^{b}\left(\widehat{H}_{+}+x\frac{\partial}{\partial
x}+(1-y)\frac{\partial}{\partial y}-\frac{1+x-y}{1-x-y}\widehat{H}_{0}\right)$
(A.27)
$\displaystyle+\lambda^{a}\lambda^{b}\left(\widehat{H}_{-}+x\frac{\partial}{\partial
x}+(1-y)\frac{\partial}{\partial
y}+\frac{1+x-y}{1-x-y}\widehat{H}_{0}\right)\bigg{]}.$

Note that when they are applied to functions that are independent of $x$ and
$y$ and that carry zero total helicity, both differential operators reduce to
$\partial/\partial p_{\mu}$ given in eq. (2.22). Moreover, one can form some
linear combinations of the two operators that do not involve derivatives with
respect to $x$ or $y$, and recover in this way the generators of scale and
Lorentz transformations:

$\sum_{i=1}^{2}\,p_{i}^{\mu}\frac{\partial}{\partial
p_{i}^{\mu}}=\widehat{D},\qquad\qquad\sum_{i=1}^{2}\left[p_{i\nu}\frac{\partial}{\partial
p_{i}^{\mu}}-p_{i\mu}\frac{\partial}{\partial
p_{i}^{\nu}}\right]=\widehat{M}_{\mu\nu}.$ (A.28)

This shows that the 3-point correlation function obeys the simple Ward
identities (3.14) and (3.15) that are independent of $x$ or $y$, as expected
since the latter are dimensionless scalars.

On the contrary, the Ward identity for special conformal transformations
involves derivatives with respect to $x$ and $y$. If we denote with
$\widehat{k}^{\mu}$ the part of the differential operator (3.17) that is
second-order in derivatives of $p_{i}$, we can write

$\widehat{k}^{\mu}\equiv\sum_{i=1}^{2}\left[\frac{1}{2}p_{i}^{\mu}\frac{\partial^{2}}{\partial
p_{i}^{\nu}\partial p_{i\nu}}-p_{i}^{\nu}\frac{\partial^{2}}{\partial
p_{i}^{\nu}\partial
p_{i\mu}}\right]=\frac{\sigma^{\mu}_{ab}}{2(\lambda\cdot\bar{\lambda})^{2}}\left(\lambda^{a}\bar{\lambda}^{b}\widehat{k}_{0}+\bar{\lambda}^{a}\bar{\lambda}^{b}\widehat{k}_{+}+\lambda^{a}\lambda^{b}\widehat{k}_{-}\right),$
(A.29)

and we find

$\displaystyle\widehat{k}_{0}$ $\displaystyle=2\widehat{A}^{\,\rm
I}_{3,3;0,0}+\frac{1}{2}(\widehat{H}_{+}\widehat{H}_{-}+\widehat{H}_{-}\widehat{H}_{+})+\widehat{D}(\widehat{D}-3)-\frac{(1-x+y)(1+x-y)}{(1-x-y)^{2}}\widehat{H}_{0}^{2},$
$\displaystyle\widehat{k}_{+}$ $\displaystyle=\widehat{A}^{\,\rm
II}_{3,3}-\widehat{H}_{+}(\widehat{D}+\widehat{H}_{0}-1)$
$\displaystyle\quad-\frac{1}{1-x-y}\left[2x(1-x)\frac{\partial}{\partial
x}-2y(1-y)\frac{\partial}{\partial
y}+(x-y)(2-\widehat{H}_{0})\right]\widehat{H}_{0}$
$\displaystyle\widehat{k}_{-}$ $\displaystyle=\widehat{A}^{\,\rm
II}_{3,3}-\widehat{H}_{-}(\widehat{D}-\widehat{H}_{0}-1)$
$\displaystyle\quad+\frac{1}{1-x-y}\left[2x(1-x)\frac{\partial}{\partial
x}-2y(1-y)\frac{\partial}{\partial
y}+(x-y)(2+\widehat{H}_{0})\right]\widehat{H}_{0}.$ (A.30)

$\widehat{A}^{\,\rm I}$ and $\widehat{A}^{\,\rm II}$ are differential
operators in $x$ and $y$, corresponding to the definitions (3.19) and (3.20)
but with special values of the parameters $\Delta_{1}=\Delta_{2}=3$ and
$\Delta=m=0$. When combined with the derivatives in eqs. (A.26) and (A.27), we
recover the operators (3.18).

The computations reported in this appendix can all be performed by hand, but
the proliferation of spinors and indices can easily lead to errors. For this
reason, a computer algebra implementation of this formalism has been developed
and used. It is made freely available in the form of a Mathematica notebook at
the following address:

https://github.com/gillioz/Conformal_spinors

##  Appendix B Generalized hypergeometric series

This appendix discusses in some detail the algebra of partial differential
equations generated by the operators $\widehat{A}^{\,\rm I}$ and
$\widehat{A}^{\,\rm II}$ defined in eqs. (3.19) and (3.20), and whose indices
we omit for compactness of the notation. For simplicity, we will also use

$\displaystyle\alpha$
$\displaystyle=\frac{6-\Delta_{1}-\Delta_{2}-\Delta+j}{2},$
$\displaystyle\gamma_{1}$ $\displaystyle=\frac{5}{2}-\Delta_{1},$
$\displaystyle\beta$
$\displaystyle=\frac{3-\Delta_{1}-\Delta_{2}+\Delta+j}{2},$
$\displaystyle\gamma_{2}$ $\displaystyle=\frac{5}{2}-\Delta_{2}.$ (B.1)

The first observation is that, when acting on a function of the two variables

$w_{1}=x(1-y),\qquad\qquad w_{2}=(1-x)y,$ (B.2)

the differential operators $\widehat{A}^{\,\rm I}$ and $\widehat{A}^{\,\rm
II}$ satisfy

$\displaystyle\frac{1}{2}\bigg{(}\widehat{A}^{\,\rm I}$
$\displaystyle+\frac{1-x+y}{1-x-y}\widehat{A}^{\,\rm
II}\bigg{)}f(w_{1},w_{2})$ $\displaystyle=\bigg{[}$ $\displaystyle
w_{1}(1-w_{1})\frac{\partial^{2}}{\partial
w_{1}^{2}}-2w_{1}w_{2}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}-w_{2}^{2}\frac{\partial^{2}}{\partial w_{2}^{2}}$ (B.3)
$\displaystyle+\left(\gamma_{1}-(\alpha+\beta+1)w_{1}\right)\frac{\partial}{\partial
w_{1}}-(\alpha+\beta+1)w_{2}\frac{\partial}{\partial
w_{2}}-\alpha\beta\bigg{]}f(w_{1},w_{2}),$
$\displaystyle\frac{1}{2}\bigg{(}\widehat{A}^{\,\rm I}$
$\displaystyle-\frac{1+x-y}{1-x-y}\widehat{A}^{\,\rm
II}\bigg{)}f(w_{1},w_{2})$ $\displaystyle=\bigg{[}$ $\displaystyle
w_{2}(1-w_{2})\frac{\partial^{2}}{\partial
w_{2}^{2}}-2w_{1}w_{2}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}-w_{1}^{2}\frac{\partial^{2}}{\partial w_{1}^{2}}$ (B.4)
$\displaystyle+\left(\gamma_{2}-(\alpha+\beta+1)w_{2}\right)\frac{\partial}{\partial
w_{2}}-(\alpha+\beta+1)w_{1}\frac{\partial}{\partial
w_{1}}-\alpha\beta\bigg{]}f(w_{1},w_{2}).$

This is the system of Appell $F_{4}$ differential equations in its standard
form. It is solved by the double hypergeometric series

$f(w_{1},w_{2})=\sum_{i,j=0}^{\infty}\frac{(\alpha)_{i+j}(\beta)_{i+j}}{i!j!(\gamma_{1})_{i}(\gamma_{2})_{j}}w_{1}^{i}w_{2}^{j}\equiv
F_{4}(\alpha,\beta,\gamma_{1},\gamma_{2};w_{1},w_{2}).$ (B.5)

Alternatively, if one did not know about the Appell $F_{4}$ series, a solution
to this system of differential equations could be obtained in the following
way. Note that $\widehat{A}^{\,\rm I}$ and $\widehat{A}^{\,\rm II}$ look very
much like the sum and difference of the differential operators generating the
ordinary hypergeometric series in $x$ and $y$ respectively: if one denotes
with

$\widehat{Q}(\alpha,\beta,\gamma;x)=x(1-x)\frac{\partial^{2}}{\partial
x^{2}}+\left[\gamma-(\alpha+\beta+1)x\right]\frac{\partial}{\partial
x}-\alpha\beta$ (B.6)

the operator that annihilate the ordinary hypergeometric function,

$\widehat{Q}(\alpha,\beta,\gamma;x)\,{}_{2}F_{1}(\alpha,\beta,\gamma;x)=0,$
(B.7)

then we have

$\displaystyle\widehat{A}^{\,\rm I}$
$\displaystyle=\widehat{Q}(\alpha,\beta,\gamma_{1};x)+\widehat{Q}(\alpha,\beta,\gamma_{2};y)$
$\displaystyle\quad-(\alpha+\beta-\gamma_{1}-\gamma_{2}+1)\left[\frac{1-x+y}{1-x-y}\frac{\partial}{\partial
x}+\frac{1+x-y}{1-x-y}\frac{\partial}{\partial y}\right]$ (B.8)

and

$\widehat{A}^{\,\rm
II}=\widehat{Q}(\alpha,\beta,\gamma_{1};x)-\widehat{Q}(\alpha,\beta,\gamma_{2};y)+(\alpha+\beta-\gamma_{1}-\gamma_{2}+1)\left[x\frac{\partial}{\partial
x}-y\frac{\partial}{\partial y}\right].$ (B.9)

If it were not for the last term that involves both $x$ and $y$, these
operators would simply generate decoupled systems of hypergeometric equations
in $x$ and $y$. Therefore, this suggests looking for a solution that can be
written as a product of ${}_{2}F_{1}$ functions in $x$ and $y$. If we
introduce

$f_{r}(x,y)=x^{r}y^{r}{}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{1}+r;x){}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{2}+r;y),$
(B.10)

then we can use properties of the ${}_{2}F_{1}$ function to show that

$\displaystyle\widehat{A}^{\,\rm I}f_{r}(x,y)$
$\displaystyle=(\alpha+r)(\alpha+\beta-\gamma_{1}-\gamma_{2}+1+r)f^{\,\rm
I}_{r}(x,y)$
$\displaystyle\quad-\frac{r(\gamma_{1}+r-1)(\gamma_{2}+r-1)}{\beta+r-1}f^{\,\rm
I}_{r-1}(x,y),$ (B.11) $\displaystyle\widehat{A}^{\,\rm II}f_{r}(x,y)$
$\displaystyle=(\alpha+r)(\alpha+\beta-\gamma_{1}-\gamma_{2}+1+r)f^{\,\rm
II}_{r}(x,y)$
$\displaystyle\quad-\frac{r(\gamma_{1}+r-1)(\gamma_{2}+r-1)}{\beta+r-1}f^{\,\rm
II}_{r-1}(x,y),$ (B.12)

where we have denoted

$\displaystyle f^{\,\rm I}_{r}(x,y)=x^{r}y^{r}\bigg{[}$
$\displaystyle\frac{2}{1-x-y}{}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{1}+r;x){}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{2}+r;y)$
$\displaystyle-\frac{1-x+y}{1-x-y}{}_{2}F_{1}(\alpha+r+1,\beta+r,\gamma_{1}+r;x){}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{2}+r;y)$
$\displaystyle-\frac{1+x-y}{1-x-y}{}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{1}+r;x){}_{2}F_{1}(\alpha+r+1,\beta+r,\gamma_{2}+r;y)\bigg{]}$
$\displaystyle f^{\,\rm II}_{r}(x,y)=x^{r}y^{r}\bigg{[}$
$\displaystyle{}_{2}F_{1}(\alpha+r+1,\beta+r,\gamma_{1}+r;x){}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{2}+r;y)$
$\displaystyle-{}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{1}+r;x){}_{2}F_{1}(\alpha+r+1,\beta+r,\gamma_{2}+r;y)\bigg{]}.$
(B.13)

Since eqs. (B.11) and (B.12) are differences of consecutive terms in $r$, the
infinite sum

$f(x,y)=\sum_{r=0}^{\infty}\frac{(\alpha)_{r}(\beta)_{r}(\alpha+\beta-\gamma_{1}-\gamma_{2}+1)_{r}}{r!(\gamma_{1})_{r}(\gamma_{2})_{r}}f_{r}(x,y),$
(B.14)

satisfies

$\widehat{A}^{\,\rm I}f(x,y)=\widehat{A}^{\,\rm II}f(x,y)=0.$ (B.15)

The series (B.14) is in fact a well-known representation of the Appell $F_{4}$
function, completely equivalent to the definition (B.5).

The reason why we discuss this representation here is that it is a convenient
starting point to solve the recursion relation (3.23). Upon acting on the
series (B.14) with $\widehat{A}^{\,\rm II}$, one encounters a generalization
of that series, which we denote

$\displaystyle f^{(a,b)}(x,y)=\sum_{r=0}^{\infty}$
$\displaystyle\frac{(\alpha)_{r}(\beta)_{r}(\alpha+\beta-\gamma_{1}-\gamma_{2}+1)_{r}}{r!(\gamma_{1}+a)_{r}(\gamma_{2}+b)_{r}}x^{a+r}y^{b+r}$
$\displaystyle\times{}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{1}+a+r;x){}_{2}F_{1}(\alpha+r,\beta+r,\gamma_{2}+b+r;y).$
(B.16)

This is an infinite series in $x$ and $y$ that begins at order $x^{a}y^{b}$.
The case $a=b=0$ corresponds to the Appell $F_{4}$ series,
$f^{(0,0)}(x,y)=f(x,y)$. The general solution to the recursion equation could
be guessed after examining the first few steps, and then verified to hold at
higher orders. It is given by the linear combination

$f(x,y)=\sum_{n=0}^{j-m}(1-x-y)^{j-n}\sum_{i=0}^{n}\chi^{(n,i)}f^{(n-i,i)}(x,y),$
(B.17)

with coefficients

$\displaystyle\chi^{(n,i)}$
$\displaystyle=\frac{(-1)^{i}}{i!(n-i)!}\frac{4^{j-m}(m+1)_{j-m}}{(2m+1)_{j-m-n}}\frac{(\gamma_{1}-\beta)_{n-i}(\gamma_{2}-\beta)_{i}}{(\gamma_{1})_{n-i}(\gamma_{2})_{i}(\gamma_{1}+\gamma_{2}-2\beta)_{j-m}}$
$\displaystyle\quad\times\sum_{k=0}^{j-m-n}\frac{(-1)^{k}\left(m+\frac{1}{2}\right)_{j-m-
n-k}\left(m+\frac{1}{2}\right)_{k}}{k!(j-m-n-k)!}(\gamma_{1}-\alpha)_{j-m-
i-k}(\gamma_{2}-\alpha)_{i+k}.$ (B.18)

These are precisely the results reported in eqs. (3.29), (3.30), and (3.31).

This construction so far is specific to the solution of the system of
differential equations that is analytic in $x$ and $y$ around $(x,y)=(0,0)$.
However, the other solutions can easily be extracted from it, noting that the
differential operators $\widehat{A}^{\,\rm I}$ and $\widehat{A}^{\,\rm II}$
obey

$\displaystyle\widehat{A}^{\,\rm
I}_{\Delta_{1},\Delta_{2},\Delta,m}\left[w_{1}^{\Delta_{1}-3/2}f(x,y)\right]$
$\displaystyle=w_{1}^{\Delta_{1}-3/2}\widehat{A}^{\,\rm
I}_{3-\Delta_{1},\Delta_{2},\Delta,m}f(x,y)$ (B.19)
$\displaystyle\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}\left[w_{1}^{\Delta_{1}-3/2}f(x,y)\right]$
$\displaystyle=w_{1}^{\Delta_{1}-3/2}\widehat{A}^{\,\rm
II}_{3-\Delta_{1},\Delta_{2}}f(x,y)$

and

$\displaystyle\widehat{A}^{\,\rm
I}_{\Delta_{1},\Delta_{2},\Delta,m}\left[w_{2}^{\Delta_{2}-3/2}f(x,y)\right]$
$\displaystyle=w_{2}^{\Delta_{2}-3/2}\widehat{A}^{\,\rm
I}_{\Delta_{1},3-\Delta_{2},\Delta,m}f(x,y)$ (B.20)
$\displaystyle\widehat{A}^{\,\rm
II}_{\Delta_{1},\Delta_{2}}\left[w_{2}^{\Delta_{2}-3/2}f(x,y)\right]$
$\displaystyle=w_{2}^{\Delta_{2}-3/2}\widehat{A}^{\,\rm
II}_{\Delta_{1},3-\Delta_{2}}f(x,y)$

This explains immediately why the four functions (3.27) are related by the
“shadow transform” $\Delta_{i}\leftrightarrow 3-\Delta_{i}$ for $i=1,2$. But
since the recursion relation (3.23) also simply depends on $\widehat{A}^{\,\rm
II}$, the construction of the solution follows the same rule, and the result
is directly given by the linear combination (B.17) in which the transformation
$\Delta_{i}\leftrightarrow 3-\Delta_{i}$ is applied both to the functions
$f^{(n-i,i)}(x,y)$ and to the coefficients $\chi^{(n,i)}$.

#### Colinear limit.

Note that in the colinear limit in which $1-x-y=0$, the function $f(x,y)$
vanishes unless $m=0$. Moreover, at $m=0$, only the terms with $n=j$
contribute to the sum (B.17). This means that we have

$f(x,y)=\sum_{i=0}^{j}\chi^{(j,i)}f^{(j-i,i)}(x,y),$ (B.21)

with coefficients taking the simpler form

$\chi^{(j,i)}=\frac{(-1)^{i}4^{j}j!}{i!(j-i)!}\frac{(\gamma_{1}-\alpha)_{j-i}(\gamma_{1}-\beta)_{j-i}(\gamma_{2}-\alpha)_{i}(\gamma_{2}-\beta)_{i}}{(\gamma_{1})_{j-i}(\gamma_{2})_{i}(\gamma_{1}+\gamma_{2}-2\beta)_{j}}.$
(B.22)

#### Light-cone limit.

Finally, it should be noted that our formulation of the solution is
particularly well-suited to examine either of the limits $x\to 0$ or $y\to 0$,
corresponding respectively to $p_{1}^{2}\to 0$ or $p_{2}^{2}\to 0$. Let us
describe the latter limit, $y\to 0$. Since the function $f^{(a,b)}(x,y)$ is a
series of non-negative powers of $y$ starting with $y^{b}$, only the functions
with $b=0$ contribute to that limit. Moreover, only the leading term with
$r=0$ contributes in the sum, so that we find

$F_{j,m}(x,0)=\sum_{n=0}^{j-m}(1-x)^{j-n}\chi^{(n,0)}x^{n}{}_{2}F_{1}(\alpha,\beta,\gamma_{1}+n;x).$
(B.23)

Using the properties of the coefficient $\chi^{(n,0)}$, this can be rewritten

$\displaystyle F_{j,m}(x,0)=\sum_{n=0}^{j-m}$
$\displaystyle\frac{(-1)^{n}4^{j-m}j!\left(m+\frac{1}{2}\right)_{j-m-n}(\Delta-j-1+n)_{j-m-n}\left(\frac{\Delta-j-1-\Delta_{1}+\Delta_{2}}{2}\right)_{n}}{m!n!(j-m-n)!(2m+1)_{j-m-n}(\Delta+m-1)_{j-m}}$
$\displaystyle\times(1-x)^{j-n}{}_{2}F_{1}\left(\tfrac{6-\Delta_{1}-\Delta_{2}-\Delta+j}{2}-n,\tfrac{3-\Delta_{1}-\Delta_{2}+\Delta+j}{2},\tfrac{5}{2}-\Delta_{1};x\right).$
(B.24)

This matches the result obtained in arbitrary space-time dimension in ref.
[23].

## References

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