# Conformal partial waves in momentum space

Marc Gillioz

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

###### Abstract

The decomposition of 4-point correlation functions into conformal partial
waves is a central tool in the study of conformal field theory. We compute
these partial waves for scalar operators in Minkowski momentum space, and find
a closed-form result valid in arbitrary space-time dimension $d\geq 3$
(including non-integer $d$). Each conformal partial wave is expressed as a sum
over ordinary spin partial waves, and the coefficients of this sum factorize
into a product of vertex functions that only depend on the conformal data of
the incoming, respectively outgoing operators. As a simple example, we apply
this conformal partial wave decomposition to the scalar box integral in $d=4$
dimensions.

###### Contents

  1. 1 Introduction
  2. 2 Spin eigenstates and completeness relation
    1. 2.1 Kinematics of the 4-point function
    2. 2.2 Polarization tensors
    3. 2.3 Spin eigenstates and normalization
    4. 2.4 Completeness relation
  3. 3 Vertex functions
    1. 3.1 Conformal Ward identities
    2. 3.2 Highest-spin functions in various orderings
    3. 3.3 Lower spin functions
    4. 3.4 Special kinematic limits
  4. 4 Example: the scalar box integral in 4 dimensions
    1. 4.1 CFT optical theorem
    2. 4.2 IR divergences
    3. 4.3 Conformal partial wave expansion
  5. 5 Conclusions

##  1 Introduction

Conformal field theory (CFT) can be defined non-perturbatively from its 2- and
3-point correlation functions, thanks to a convergent operator product
expansion (OPE) that reduces the computation of higher-point functions to
these elementary building blocks. In particular, the decomposition of 4-point
functions into conformal partial waves is at the heart of the modern conformal
bootstrap program. The bootstrap exploits the self-consistency of the OPE to
put constraints on the _dynamics_ of a theory encoded in the CFT data [1, 2,
3, 4]. The problem of computing the conformal partial waves, i.e. determining
the _kinematics_ underlying 4-point correlators, has been solved long ago [5,
6]. Closed-form results exist for scalar operators in dimensions $d=2$ and
$4$, and various techniques for an efficient computation of the conformal
partial waves in arbitrary dimensions have been developed, both for scalars
[7, 8, 9, 10, 11] and for operators that carry spin [12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].

In this work, we address a different but related problem: we compute the
conformal partial waves in the momentum-space representation of correlation
functions in Minkowski space, as opposed to the ordinary Euclidean position-
space approach. We focus on correlation functions involving 4 scalar primary
operators $\phi_{i}$, each carrying momentum $p_{i}$, in $d\geq 3$ dimensions,
of the form

$\left\langle
0\right|[\phi_{1}(p_{1})\phi_{2}(p_{2})][\phi_{3}(p_{3})\phi_{4}(p_{4})]\left|0\right\rangle\equiv(2\pi)^{d}\delta^{d}(p_{1}+p_{2}+p_{3}+p_{4})\,G(p_{1},p_{2},p_{3}),$
(1.1)

where the square brackets around pairs of operators on the left-hand side can
mean any of the following: no particular ordering (Wightman function), a
commutator (possibly retarded or advanced), a time-ordered product or its
hermitian conjugate. The conformal wave expansion for $G$ takes the form

$G(p_{1},p_{2},p_{3})=s^{(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-3d)/2}\sum_{\mathcal{O}}\lambda_{12\mathcal{O}}\lambda_{\mathcal{O}34}G_{\Delta,\ell}(p_{1},p_{2},p_{3})$
(1.2)

where the sum is over primary operators $\mathcal{O}$ with scaling dimension
$\Delta$ and spin $\ell$, $\lambda_{12\mathcal{O}}$ and
$\lambda_{\mathcal{O}34}$ are OPE coefficients encoding the dynamical
information about the theory, and $s$ is the center-of-mass energy

$s=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2}>0,$ (1.3)

which is strictly positive by the spectral condition on the correlation
function (1.1). The conformal partial waves $G_{\Delta,\ell}$ are completely
fixed by symmetry in terms of $\Delta$ and $\ell$, of the scaling dimensions
$\Delta_{i}$ of the external operators (which may be distinct or not) and of
the space-time dimension $d$ (possibly including non-integer dimensions, as
our results are analytic in $d\geq 3$). We obtain

$\boxed{G_{\Delta,\ell}(p_{1},p_{2},p_{3})=\sum_{m=0}^{\ell}C_{\Delta,\ell,m}\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)V_{\Delta,\ell,m}^{[12]}\left(\frac{p_{1}^{2}}{s},\frac{p_{2}^{2}}{s}\right)V_{\Delta,\ell,m}^{[34]}\left(\frac{p_{3}^{2}}{s},\frac{p_{4}^{2}}{s}\right),}$
(1.4)

where $\theta$ is the scattering angle, defined in terms of the momenta
$p_{i}$ in eq. (2.10), and $\mathcal{C}^{(d-3)/2}_{m}(\cos\theta)$ a
Gegenbauer polynomial of degree $m$ that appears in the ordinary spin partial
wave expansion in $d$ dimensions. The numerical factor $C_{\Delta,\ell,m}\geq
0$ is given in eq. (2.35), and the “vertex functions”
$V_{\Delta,\ell,m}^{[ab]}$ are defined in section 3, depending on which
ordering is understood in the definition (1.1). This result admits a simple
diagrammatic representation given in figure 1. One observes a factorization in
the sense that the dependence on the external CFT data (the scaling dimensions
$\Delta_{i}$) and on the “invariant masses” $p_{i}^{2}$ is entirely contained
in the vertex functions $V_{\Delta,\ell,m}^{[ab]}$. Note that we have used the
evident notation $p_{4}^{2}=(p_{1}+p_{2}+p_{3})^{2}$.

Figure 1: Diagrammatic representation of the conformal partial wave (1.4), in
which each line corresponds to a local primary operator of the CFT carrying a
certain momentum. Note that the vertex functions $V_{\Delta,\ell,m}^{[ab]}$
only depend on the scaling dimensions, on the spin (the total spin $\ell$ and
its projection $m$ onto a reference direction), and on the momenta of the
lines attached to it.

Sections 2 and 3 of this work are dedicated to the derivation of the momentum-
space conformal partial waves (1.4). This derivation does not rely on solving
a Casimir equation as in ordinary position space: in fact, since the conformal
Ward identities are second-order differential equations with respect to the
momenta $p_{i}$, one cannot make use of invariant cross-ratios in momentum
space. Instead, the 4-point function depends on six Lorentz-invariant
quantities (taken here to be all four $p_{i}^{2}$, $s$ and $\cos\theta$), and
solving a differential equation in as many variables appears like a formidable
task. This disadvantage is however counterbalanced by two elements that are
unique to the Minkowski momentum-space representation of correlation
functions:

  * •

The orthogonality of momentum eigenstates, that allows to factorize the
conformal partial wave into a product of 3-point functions, up to the
contraction of Lorentz indices.

  * •

The ability to use a reference frame, the center-of-mass frame, in which the
2-point function of the intermediate operator can be decomposed into
polarizations that transform under irreducible representations of the rotation
group $\text{SO}(d-1)$.

These points have been put forward in a recent derivation of position-space
conformal blocks [24], as well as in Mack’s classification of all the
representations of the Lorentzian conformal group in $d=4$ [30]. The operator
product expansion in general and the conformal partial wave expansion (1.2) in
particular are nothing but the use of a Hilbert space completeness relation,
which in a CFT can be made explicit thanks to the two properties listed above.
In fact, the correlation function (1.1) computes the overlap between two
states of the theory: one state obtained by acting on the vacuum with the
product of operators $[\phi_{3}\phi_{4}]$, and the other state obtained acting
with $[\phi_{1}\phi_{2}]^{\dagger}$. The partial wave (1.4) corresponds to the
projection of this overlap of states onto a conformal family consisting of a
primary operator and of all of its descendants.

A corollary of this observation is that each conformal partial wave is
positive in the forward scattering limit $p_{1}\to-p_{4}$, $p_{2}\to-p_{3}$
provided that $\phi_{1}=\phi_{4}$ and $\phi_{2}=\phi_{3}$. In this case the
vertex functions are complex conjugate of each other,
$V_{\Delta,\ell,m}^{[12]}=\big{(}V_{\Delta,\ell,m}^{[34]}\big{)}^{*}$, and the
Gegenbauer polynomial evaluated at zero scattering angle ($\cos\theta=1$) is
positive, hence $G_{\Delta,\ell}\geq 0$. The positivity of this configuration
and its decomposition into conformal partial waves have actually been used in
the derivation of sum rules for anomaly coefficients [31, 32].

It should be emphasized that the conformal partial wave expansion described in
this work relies heavily on the Hamiltonian evolution being unitary, i.e.
given by $\exp(iHt)$ where $t$ is the Lorentzian time and $H$ a Hamiltonian
operator bounded below. In a Euclidean theory, where this evolution is
replaced by $\exp(-H\tau)$ with Euclidean “time” $\tau$, the OPE only makes
sense for correlation functions that are time-ordered,111If one uses radial
quantization instead, then this applies to any configuration which is
equivalent to a radially-ordered one under a global conformal symmetry
transformation, which in practice extends to most of the space of
configurations. The conformal bootstrap works precisely because it uses
configurations that can be expanded into two or more distinct OPEs by
different choices of quantization surfaces. a property that is lost upon
Fourier transform. For this reason, the conformal partial wave expansion (1.2)
differs at a fundamental level from other decompositions of the momentum-space
4-point function, such as the decomposition into Witten exchange diagrams in
the case of a holographic theory (usually formulated in Mellin space [33, 34,
35]), or the decomposition into symmetric Polyakov blocks expressed directly
in Euclidean momentum space [36, 37].222Instead, there is an interesting
connection between the conformal partial waves expansion described in this
work and the Polyakov-Regge blocks recently introduced [38], through the
generalization in momentum space of the concept of CFT dispersion relations
[39]. We leave the exploration of this venue for future work. For the same
reason, it is difficult to compare our conformal partial wave expansion with
recent results on 4-point correlators in Euclidean momentum space [40, 41, 42,
43, 44].

On the other hand, eqs. (1.2) and (1.4) are immediately suitable for
comparison with results obtained using ordinary Feynman diagram computations
in theories that are both perturbative and conformal. While the goal of this
work is not to study such situations systematically, we present in section 4 a
simple example that involves the real part of the scalar box integral in four
dimensions. This box integral appears in the 4-point function of the composite
operator $\Phi^{2}$ in the theory of a free boson $\Phi$, or in that of the
gauge-invariant operator $\operatorname{tr}(\Phi^{i}\Phi^{i})$ in
$\mathcal{N}=4$ supersymmetric Yang-Mills. The general form of this loop
integral with off-shell momenta is known thanks to its dual conformal [45] or
Yangian invariance [46].

Besides perturbative CFT, the conformal partial waves (1.4) will certainly
find applications in other contexts. One of their salient features is the
presence of double zeroes at scaling dimension of double-trace operators, a
property reminiscent of the “double-twist” functional methods for the
conformal bootstrap [47, 48, 49, 50, 51, 52]. However, it should be emphasized
that the correlator (1.1) is ill-suited for writing a bootstrap equation, as
it does not have crossing symmetry built in. Microcausality in Minkowski
momentum space implies certain analyticity properties of the correlation
functions that might eventually be used to place bounds on the CFT data, but
the relevant methods remain to be developed. Similarly, our conformal partial
wave expansion could also find use in the cosmological bootstrap program [53,
54, 55, 56, 57, 58, 59, 60], even though the connection between correlators in
Minkowski momentum space and observables in de Sitter remains to be
established (see ref. [61] for hints in this direction). For now, we provide
the decomposition (1.2) as a tool and encourage the community to find its own
applications.

##  2 Spin eigenstates and completeness relation

The core of the computation of the conformal partial waves is the formulation
of a Hilbert space completeness relation. In this section, we assemble the
different elements, construct a complete basis of states, and detail its
dependence on the kinematic invariants of the 4-point function. The
computation of the overlap of these states with the pairs of operators
$[\phi_{1}\phi_{2}]$ and $[\phi_{3}\phi_{4}]$ is discussed instead in section
3.

###  2.1 Kinematics of the 4-point function

We begin with a detailed description of the kinematics of the 4-point
function, as it will play a central role in what follows. As seen in eq.
(1.1), momentum-space correlation functions involving $n$ operators are always
proportional to a delta function imposing momentum conservation, and hence
depend non-trivially on $n-1$ momenta only. It is convenient to introduce the
double-bracket notation

$\left\langle
0\right|\phi_{1}(p_{1})\cdots\phi_{n}(p_{n})\left|0\right\rangle\equiv(2\pi)^{d}\delta^{d}(p_{1}+\ldots+p_{n})\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\phi_{1}(p_{1})\cdots\phi_{n}(p_{n})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(2.1)

Whenever this notation is being used, it is understood that one can trade one
of the momenta for the sum of the others, say $p_{n}=-(p_{1}+\ldots+p_{n-1})$.

For the 4-point function, this mean that there are 3 independent momenta, and
hence 6 Lorentz invariants.333We are always working in $d\geq 3$ dimensions.
In $d=2$, three vectors cannot be linearly independent and there is one less
invariant In analogy with scattering amplitudes, one can use the “invariant
masses”

$p_{1}^{2},\quad p_{2}^{2},\quad p_{3}^{2},\quad
p_{4}^{2}=(p_{1}+p_{2}+p_{3})^{2},$ (2.2)

as well as the Mandelstam variables

$s=(p_{1}+p_{2})^{2},\qquad t=(p_{1}+p_{3})^{2},\qquad u=(p_{2}+p_{3})^{2},$
(2.3)

subject to the constraint $s+t+u=\sum p_{i}^{2}$. Unlike scattering
amplitudes, however, we do not require the $p_{i}^{2}$ to take any special
value. In fact, we even want to be agnostic about their sign: the momenta
$p_{i}$ might be space-like, time-like, or even light-like. However, the
specific ordering of the operators in eq. (1.1) does imply a constraint on the
momentum flowing between the pairs $[\phi_{1}\phi_{2}]$ and
$[\phi_{3}\phi_{4}]$: defining

$p\equiv-p_{1}-p_{2}=p_{3}+p_{4},$ (2.4)

the correlation function (1.1) vanishes by the spectral condition whenever $p$
lies outside the forward light cone, i.e. unless

$p^{0}>0,\qquad\text{and}\qquad s=p^{2}>0.$ (2.5)

We will assume henceforth that these two conditions are fulfilled.

With $s$ strictly positive, one can rescale all dimensionful quantities by
$s$, including the 4-point function (1.1) whose overall scaling dimension is
fixed by its operator content and by the space-time dimension. Since we chose
in the definition (1.2) to factor out precisely the appropriate power of $s$,
$G_{\Delta,\ell}$ is dimensionless. It is a function of the rescaled invariant
masses

$w_{i}\equiv\frac{p_{i}^{2}}{s},$ (2.6)

and of a fifth dimensionless variable that we define as follows: we introduce
two new linear combinations of the momenta,

$q_{12}^{\mu}\equiv\frac{(p\cdot p_{2})p_{1}^{\mu}-(p\cdot
p_{1})p_{2}^{\mu}}{\sqrt{(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}}},\qquad\qquad q_{34}^{\mu}\equiv\frac{(p\cdot
p_{3})p_{4}^{\mu}-(p\cdot p_{4})p_{3}^{\mu}}{\sqrt{(p_{3}\cdot
p_{4})^{2}-p_{3}^{2}p_{4}^{2}}},$ (2.7)

which have the property of being both orthogonal to $p$, and normalized in
units of $s$,

$q_{12}\cdot p=q_{34}\cdot p=0,\qquad\qquad q_{12}^{2}=q_{34}^{2}=-s.$ (2.8)

The definitions (2.7) are non-singular as long as neither $p_{1}$ and $p_{2}$,
nor $p_{3}$ and $p_{4}$ are colinear; the special configuration in which some
of these momenta are colinear can be eventually reached as a limit of the
general case. We now define $\theta$ to be the angle between these two
vectors,

$\cos\theta\equiv-\frac{q_{12}\cdot q_{13}}{s},$ (2.9)

or in terms of the Mandelstam invariants

$\boxed{\cos\theta=\frac{s(u-t)-(p_{1}^{2}-p_{2}^{2})(p_{3}^{2}-p_{4}^{2})}{\sqrt{(s-p_{1}^{2}-p_{2}^{2})^{2}-4p_{1}^{2}p_{2}^{2}}\sqrt{(s-p_{3}^{2}-p_{4}^{2})^{2}-4p_{3}^{2}p_{4}^{2}}}.}$
(2.10)

$\theta$ corresponds in fact to the scattering angle in a $2\to 2$ inelastic
process in which the four “particles” have distinct masses (possibly including
negative masses squared). In the center-of-mass frame, working for
definiteness in $d=3$, one can always choose

$\displaystyle p$ $\displaystyle=\sqrt{s}\left(1,0,0\right),$ $\displaystyle
q_{12}$ $\displaystyle=\sqrt{s}\left(0,1,0\right),$ (2.11) $\displaystyle
q_{34}$ $\displaystyle=\sqrt{s}\left(0,\cos\theta,\sin\theta\right),$

which in terms of momenta $p_{i}$ corresponds to

$\displaystyle p_{1}$
$\displaystyle=-\frac{\sqrt{s}}{2}\left(1+w_{1}-w_{2},\Omega_{12},0\right),$
$\displaystyle p_{2}$
$\displaystyle=-\frac{\sqrt{s}}{2}\left(1-w_{1}+w_{2},-\Omega_{12},0\right),$
(2.12) $\displaystyle p_{3}$
$\displaystyle=\frac{\sqrt{s}}{2}\left(1+w_{3}-w_{4},-\Omega_{34}\cos\theta,-\Omega_{34}\sin\theta\right),$

where we have defined

$\Omega_{12}=\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}},\qquad\Omega_{34}=\sqrt{(1-w_{3}-w_{4})^{2}-4w_{3}w_{4}}.$
(2.13)

The quantities under the square roots are non-negative for any configuration
of momenta. Note that $\Omega_{12}$ and $\Omega_{34}$ will play an important
role later. The center-of-mass configuration is illustrated in figure 2.

Figure 2: A possible configuration of momenta in the center-of-mass frame
(energy along the vertical axis). The scattering angle $\theta$ corresponds to
the angle between the planes spanned by $(p_{1},p_{2})$ and $(p_{3},p_{4})$,
or equivalently between the vectors $q_{12}$ and $q_{34}$. In this particular
example $p_{1}$ is time-like (backward directed), while $p_{2}$, $p_{3}$ and
$p_{4}$ are all space-like, as seen by their position relative to the light
cone (dotted lines); the covariant definition (2.10) is valid independently of
the sign of the $p_{i}^{2}$.

###  2.2 Polarization tensors

The Hilbert space completeness relation relies on a basis of momentum
eigenstates that can be constructed out of a single primary operator insertion
acting on the vacuum, for all primaries of the theory. In the case of scalar
external operators, only scalar and traceless symmetric tensor operators enter
the OPE. In order to avoid dealing with the contraction of tensor indices, and
more importantly to obtain the factorized result (1.4), it is necessary to
introduce a complete orthogonal basis of polarization tensors
$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}$, after which one could write

$\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(p)\left|0\right\rangle\equiv\sum_{m}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}\mathcal{O}^{(\ell,m)}(p)\left|0\right\rangle.$
(2.14)

The goal of this section is to provide the explicit construction of such a
basis, using only the momenta at hand, so that its transformation under
special conformal transformations can later be determined explicitly. Such a
construction is most easily done using spinor variables [62], but then it
depends explicitly on the space-time dimension $d$. For scalar external
operators, we will show that a $d$-independent construction can be performed.

The starting point is to realize that, thanks to the ability of going to the
center-of-mass frame (2.11), the polarization tensors
$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}$ can be decomposed into products of
the vector $p$ and of invariant tensors of the group of spatial rotations
$\text{SO}(d-1)$. An orthogonal basis of such polarization tensors is given by

$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}\equiv
s^{-(\ell-m)/2}\left[\varepsilon_{\perp}^{(\mu_{1}\ldots\mu_{m}}p^{\mu_{m+1}}\cdots
p^{\mu_{\ell})}-\text{traces}\right],$ (2.15)

where the indices on the right-hand side are symmetrized, and the tensors
$\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{n}}$ form themselves a basis of
traceless, symmetric tensors orthogonal to $p$,

$p_{\mu_{1}}\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{n}}=0.$ (2.16)

This means that the tensors $\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{n}}$ only
have non-zero entries when all indices are spatial.

In our case, one can even go further and construct explicitly the tensors
$\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{m}}$ out of the momenta at hand. The
reason is that we are only interested in polarization tensors that live in a
plane spanned by a pair of momenta, say $p_{1}$ and $p_{2}$, or equivalently
$p$ and $q_{12}$. Any polarization overlapping with a state created by two
scalar operators acting on the vacuum is of this form. Using the projector
onto the directions transverse to $p$,

$\eta_{\perp}^{\mu\nu}\equiv\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}},$
(2.17)

one can take

$\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{m}}(p,q)\equiv\sum_{n=0}^{\lfloor
m/2\rfloor}\frac{m!}{n!(m-2n)!}\frac{1}{2^{2n}\left(\frac{d-3}{2}+m-n\right)_{n}}\frac{\eta_{\perp}^{(\mu_{1}\mu_{2}}\cdots\eta_{\perp}^{\mu_{2n-1}\mu_{2n}}q^{\mu_{2n+1}}\cdots
q^{\mu_{m})}}{s^{m/2-n}}.$ (2.18)

This is a covariant definition in terms of any pair of momenta $p$ and $q$
that satisfy $p\cdot q=0$, $p^{2}=-q^{2}=s$. Similarly, a covariant definition
of the more general tensors $\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}$ is
directly obtained plugging eq. (2.18) into (2.15). The final result is most
easily expressed with the help of an auxiliary polarization vector $z^{\mu}$
satisfying $z^{2}=0$, in terms of which

$z_{\mu_{1}}\cdots
z_{\mu_{\ell}}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)=\frac{(z\cdot
p)^{\ell-m}(z\cdot
q)^{m}}{s^{\ell/2}}{}_{2}F_{1}\left(-\frac{m}{2},-\frac{m-1}{2};\frac{5-d}{2}-m;\frac{(z\cdot
p)^{2}}{(z\cdot q)^{2}}\right).$ (2.19)

Note that the hypergeometric series terminates both for even and odd $m$
(either one of the first two parameters is a negative integer), so that the
right-hand side is in fact a homogeneous polynomial of degree $\ell$ in
$z\cdot p$ and $z\cdot q$. The hypergeometric form is nevertheless convenient
for its compactness, and it shows explicitly that the definition (2.19) is
analytic in the space-time dimension $d$. It also gives a simple proof of the
identities

$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)=(-1)^{m}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,-q)=(-1)^{\ell-m}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(-p,q)=(-1)^{\ell}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(-p,-q).$
(2.20)

The orthogonality of the basis follows by construction. Its normalization is
given by

$\varepsilon_{m^{\prime}\mu_{1}\ldots\mu_{\ell}}(p,q)\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)=\delta_{m^{\prime}m}(-1)^{m}\mathcal{N}_{\ell,m}.$
(2.21)

where

$\mathcal{N}_{\ell,m}=\frac{m!(\ell-m)!}{\ell!}\frac{(d-2+2m)_{\ell-m}}{2^{\ell-m}\left(\frac{d-2}{2}+m\right)_{\ell-m}}\frac{(d-3)_{m}}{2^{m}\left(\frac{d-3}{2}\right)_{m}}\geq
0.$ (2.22)

Moreover, if one considers two distinct polarization tensors defined with
respect to two reference vectors $q$ and $q^{\prime}$, both orthogonal to the
same vector $p$, then one one obtains the identity

$\varepsilon_{m^{\prime}\mu_{1}\ldots\mu_{\ell}}(p,q)\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q^{\prime})=\delta_{m^{\prime}m}(-1)^{m}\mathcal{N}_{\ell,m}\frac{m!}{(d-3)_{m}}\,\mathcal{C}_{m}^{(d-3)/2}(\cos\theta),$
(2.23)

where $\theta$ is the angle between $q$ and $q^{\prime}$, and
$\mathcal{C}_{m}^{(d-3)/2}$ is a Gegenbauer polynomial. This identity will
play a central role in the computation of the conformal partial waves.

###  2.3 Spin eigenstates and normalization

We show next how these polarization tensors naturally appear in the 2-point
function of primary operators, and how this leads to the construction of an
orthogonal basis of spin eigenstates. The 2-point function of traceless,
symmetric tensor operators in momentum space satisfies [63]

$\displaystyle\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p)\mathcal{O}^{\nu_{1}\ldots\nu_{\ell}}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
$\displaystyle=s^{\Delta-d/2}\Theta(p^{0})\Theta(p^{2})B_{\Delta,\ell}\sum_{n=0}^{\ell}\frac{(-1)^{n}2^{n}\ell!}{n!(\ell-n)!}\frac{\left(\frac{d}{2}-\Delta\right)_{n}}{\left(2-\Delta-\ell\right)_{n}}$
$\displaystyle\quad\qquad\times\left[\frac{p^{(\mu_{1}}p^{\nu_{1}}\cdots
p^{\mu_{n}}p^{\nu_{n}}\eta^{\mu_{n+1}\nu_{n+1}}\cdots\eta^{\mu_{\ell}\nu_{\ell})}}{s^{n}}-\text{traces}\right],$
(2.24)

where

$B_{\Delta,\ell}=\frac{(4\pi)^{(d+2)/2}\left(\Delta-1\right)_{\ell}}{2^{2\Delta+1}\Gamma\left(\Delta-\frac{d-2}{2}\right)\Gamma(\Delta+\ell)}.$
(2.25)

This result is the unique solution (up to a normalization constant) to the
conformal Ward identities for the 2-point function. Alternatively, it can be
obtained from the direct Fourier transform of the position-space 2-point
function in Lorentzian signature. The Heaviside $\Theta$-functions appearing
in eq. (2.24) indicate that the 2-point function vanishes whenever $p$ lies
outside the forward light cone.

To relate eq. (2.24) with the polarization tensors described above, we
consider again its projection onto the plane spanned by $p$ and a vector $q$
orthogonal to it. This amounts to replacing

$\eta^{\mu\nu}\rightarrow\frac{p^{\mu}p^{\nu}-q^{\mu}q^{\nu}}{s}$ (2.26)

in eq. (2.24). A bit of combinatorics allows then to rewrite the 2-point
function (2.24) as

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p)\mathcal{O}^{\nu_{1}\ldots\nu_{\ell}}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=s^{\Delta-d/2}\sum_{m=0}^{\ell}\frac{B_{\Delta,\ell,m}}{\mathcal{N}_{\ell,m}}\,\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(-p,-q)\varepsilon_{m}^{\nu_{1}\ldots\nu_{\ell}}(p,q),$
(2.27)

where

$B_{\Delta,\ell,m}\equiv\frac{(\Delta-\ell-d+2)_{\ell-m}}{(\Delta+m-1)_{\ell-m}}B_{\Delta,\ell}\geq
0.$ (2.28)

Requiring that the 2-point function defines a positive inner product, one
actually recovers the well-known unitarity bounds $\Delta\geq(d-2)/2$ for
scalars, and $\Delta-\ell\geq d-2$ for traceless, symmetric tensors of spin
$\ell$. Note that in the generic case the 2-point function is a sum of
$\ell+1$ polarizations, the exception being spinning operators that saturate
the unitarity bound: when $\Delta-\ell=d-2$, all $B_{\Delta,\ell,m}$ with
$m<\ell$ vanish: conserved currents only have transverse polarizations
$m=\ell$.

This construction suggests to define the spin eigenstates

$\left|\mathcal{O}^{(\ell,m)}(p{\,|\,}q)\right\rangle\equiv\mathcal{N}_{\ell,m}^{-1}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)\mathcal{O}_{\mu_{1}\ldots\mu_{\ell}}(p)\left|0\right\rangle.$
(2.29)

The notation indicates that this is a state carrying momentum $p$, with total
spin $\ell$, and component $m$ along a direction that is defined by a
reference vector $q$. These states form an orthogonal basis,

$\big{\langle}\mathcal{O}^{(\ell^{\prime},m^{\prime})}(-p^{\prime}{\,|\,}-q)\big{|}\mathcal{O}^{(\ell,m)}(p{\,|\,}q)\big{\rangle}=(2\pi)^{d}\delta^{d}(p^{\prime}-p)\delta_{\ell^{\prime}\ell}\delta_{m^{\prime}m}\mathcal{N}_{\ell,m}^{-1}B_{\Delta,\ell,m}s^{\Delta-d/2}.$
(2.30)

They realize the decomposition (2.14) envisioned at the beginning of this
section. Note that an equivalent definition of the spin eigenstates can be
obtained more generically by group-theoretical considerations, according to
which $m$ labels the representation of the state under the little group that
preserves the momentum $p$ [30, 24]. The advantage of our explicit
construction in terms of the momentum $p$ and of the reference vector $q$ is
that it is immediately suited to examine the transformation properties of the
polarization tensors under special conformal transformations, as developed
later in section 3.1.

###  2.4 Completeness relation

We are now equipped with a complete basis of states that can appear in the OPE
of two scalar operators acting on the vacuum. Writing down a completeness
relation amounts to inverting the 2-point function (2.30), and this is now a
trivial matter since the states are readily orthogonal. The null states
corresponding to the longitudinal polarizations ($m<\ell$) of conserved
currents can be safely excluded, as we shall see in section 3.3 that their
overlap with two operators acting on the vacuum is zero. Similarly, it is
sufficient to consider states that carry a momentum $p$ inside the forward
light cone $\mathcal{V}_{+}$, as the states are otherwise null. We write
therefore

$\mathds{1}=\left|0\right\rangle\left\langle
0\right|+\sum_{\mathcal{O}}\sum_{m=0}^{\ell}B_{\Delta,\ell,m}^{-1}\mathcal{N}_{\ell,m}\int\limits_{\mathcal{V}_{+}}\frac{d^{d}k}{(2\pi)^{d}}(k^{2})^{d/2-\Delta}\left|\mathcal{O}^{(\ell,m)}(k{\,|\,}q)\right\rangle\left\langle\mathcal{O}^{(\ell,m)}(-k{\,|\,}-q)\right|,$
(2.31)

where the sum is over all primary operators of the theory. Spin
representations other than traceless symmetric tensors have been omitted, as
they do not appear in the scalar 4-point function. One can also choose
different reference vectors $q$ and $q^{\prime}$ for the two states, in which
case eq. (2.23) leads to the more general completeness relation

$\displaystyle\mathds{1}=\left|0\right\rangle\left\langle
0\right|+\sum_{\mathcal{O}}\sum_{m=0}^{\ell}$ $\displaystyle
B_{\Delta,\ell,m}^{-1}\mathcal{N}_{\ell,m}\frac{m!}{(d-3)_{m}}\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)$
$\displaystyle\times\int\limits_{\mathcal{V}_{+}}\frac{d^{d}k}{(2\pi)^{d}}(k^{2})^{d/2-\Delta}\left|\mathcal{O}^{(\ell,m)}(k{\,|\,}q)\right\rangle\left\langle\mathcal{O}^{(\ell,m)}(-k{\,|\,}-q^{\prime})\right|,$
(2.32)

where $\theta$ is the angle between the vectors $q$ and $q^{\prime}$.

It is important to remember that, in the derivation of these completeness
relations, we made the constraining assumption that the Lorentz indices of the
traceless, symmetric operators only live in the subspace of the
$d$-dimensional Minkowski space spanned by the momenta $p$ and $q$
(respectively $q^{\prime}$). This is sufficient when working with the
kinematics of the scalar 4-point functions, provided that we use $q_{12}$ and
$q_{34}$ defined in eq. (2.7) in place of $q$ and $q^{\prime}$. Plugging the
completeness relation (2.32) in the correlator (1.1), and resolving the
trivial delta functions, one arrives therefore at

$\displaystyle G(p_{1},p_{2},p_{3})$
$\displaystyle=\sum_{\mathcal{O}}s^{d/2-\Delta}\sum_{m=0}^{\ell}C_{\Delta,\ell,m}\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)$
$\displaystyle\qquad\times\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{(\ell,m)}(p{\,|\,}q_{12})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{(\ell,m)}(-p{\,|\,}-q_{34})[\phi_{3}(p_{3})\phi_{4}(p_{4})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(2.33)

Note that the vacuum state does not contribute for $s>0$. This is now
precisely in the form of eqs. (1.2) and (1.4), where the vertex functions are
given by the 3-point correlators, and we have defined

$C_{\Delta,\ell,m}=B_{\Delta,\ell,m}^{-1}\mathcal{N}_{\ell,m}\frac{m!}{(d-3)_{m}},$
(2.34)

or explicitly

$\boxed{C_{\Delta,\ell,m}=\frac{2^{2\Delta-\ell+1}(m!)^{2}(\ell-m)!(d-2+2m)_{\ell-m}\Gamma\left(\Delta-\frac{d-2}{2}\right)\Gamma(\Delta+\ell)}{(4\pi)^{(d+2)/2}\ell!\left(\frac{d-2}{2}+m\right)_{\ell-m}\left(\frac{d-3}{2}\right)_{m}\left(\Delta-1\right)_{m}(\Delta-\ell-d+2)_{\ell-m}}.}$
(2.35)

This coefficient diverges in $d=3$ for $m>0$, but the Gegenbauer polynomial
$\mathcal{C}_{m}^{(d-3)/2}$ vanishes in the same case: the product of the two
is in fact well-defined in the limit $d\to 3$, and it is proportional to the
Chebyshev polynomial $T_{m}(\cos\theta)$.

In summary, using polarization tensors defined in terms of the kinematic
variables, we have been able to write down a decomposition of the 4-point
function into conformal partial waves, and to reduce the computation of these
partial wave to a pair of 3-point functions, which we are going to discuss
next.

##  3 Vertex functions

In this section we describe the computation of the 3-point functions appearing
in eq. (2.33). This computations relies essentially on the results of ref.
[64], but the projection onto spin eigenstates is new. At first this looks
like a major complication, as the standard approach relying on solving
conformal Ward identities requires to determine the transformation of the
polarization tensors under special conformal symmetry. We shall see
nonetheless that the results take a particularly simple form for the highest-
spin function in each conformal family ($m=\ell$), and that the lower spin
functions can then be determined recursively.

###  3.1 Conformal Ward identities

The starting point is to strip the 3-point correlation functions from its
dynamical information, the OPE coefficient, and from an overall power of $s$
fixed by scaling symmetry. We write

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{(\ell,m)}(p{\,|\,}q_{12})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\equiv\lambda_{12\mathcal{O}}\,s^{(\Delta_{1}+\Delta_{2}+\Delta-2d)/2}V_{\Delta,\ell,m}^{[12]}\left(\frac{p_{1}^{2}}{s},\frac{p_{2}^{2}}{s}\right).$
(3.1)

$V_{\Delta,\ell,m}^{[12]}$ on the right-hand side is a function of two
dimensionless variables only. It is precisely the vertex function that enters
in the conformal partial wave (1.4). All information about scale and Poincaré
symmetry is already included in this equation and in the definition (2.19) of
the polarization tensors. What remains to be determined is actually contained
in the action of special conformal transformations, together with a boundary
condition that can be traced back to Euclidean position space. Note that the
vertex function $V_{\Delta,\ell,m}^{[34]}$ in eq. (1.4) is simply related to
$V_{\Delta,\ell,m}^{[12]}$ by hermitian conjugation and a change in the labels
$1\to 4$ and $2\to 3$. We shall therefore focus exclusively on
$V_{\Delta,\ell,m}^{[12]}$ in this whole section.

Working in Minkowski space, there is no choice but to use the infinitesimal
form of special conformal transformation, as their exponentiated form does not
preserve the causal structure of space-time. At the level of correlation
functions, this means solving a partial differential equation: the Ward
identity for special conformal transformations in momentum space is a second-
order differential equation that can be written as [65, 66]

$\widehat{K}_{12}^{\alpha}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p_{1}-p_{2})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=0$
(3.2)

where $\widehat{K}_{12}^{\alpha}$ is the differential operator

$\widehat{K}_{12}^{\alpha}=\sum_{i=1}^{2}\left[p_{i}^{\beta}\frac{\partial^{2}}{\partial
p_{i\alpha}\partial
p_{i}^{\beta}}-\frac{1}{2}p_{i}^{\alpha}\frac{\partial^{2}}{\partial
p_{i\beta}\partial p_{i}^{\beta}}-(\Delta_{i}-d)\frac{\partial}{\partial
p_{i\alpha}}\right].$ (3.3)

Note that this differential equation is valid no matter what the ordering
between the operators $\phi_{1}$ and $\phi_{2}$ is: the difference in
orderings only means different boundary conditions, which will be discussed
later in section 3.2.

The Ward identity (3.2) does not apply directly to the vertex function
$V_{\Delta,\ell,m}^{[12]}$, but rather to the correlator with the primary
operator $\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}$ acting on the vacuum. In
order to include the polarization tensors and the power of $s$ in this
equation, we write, in accordance with eq. (2.14),

$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p_{1}-p_{2})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=\lambda_{12\mathcal{O}}\,s^{(\Delta_{1}+\Delta_{2}+\Delta-2d)/2}\sum_{m=0}^{\ell}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q_{12})V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2}).$
(3.4)

We also introduce an external polarization vector $z^{\mu}$ as in eq. (2.19),
and use the shorthand notation

$\varepsilon_{m}\equiv z_{\mu_{1}}\cdots
z_{\mu_{\ell}}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q_{12}).$ (3.5)

It is straightforward (if tedious) to compute the action of
$\widehat{K}_{12}^{\alpha}$ on the polarization tensors, as they are
explicitly defined in terms of $p_{1}$ and $p_{2}$. In doing so, we encounter
derivatives of the hypergeometric function (2.19), and we find it convenient
to introduce

$\varepsilon^{\prime}_{m}\equiv\frac{\partial\varepsilon_{m}}{\partial(z\cdot
q_{12})},$ (3.6)

which describes a traceless symmetric tensor with $\ell-1$ indices. The Ward
identity (3.2) is now a simple vector equation, whose components can be
resolved taking its scalar product with various reference vectors. For
instance, the Ward identity given by $p\cdot\widehat{K}_{12}$ becomes

$\displaystyle\sum_{m=0}^{\ell}\varepsilon_{m}\bigg{\\{}\Omega_{12}^{m}\left[(1+w_{1}-w_{2})\widehat{D}_{1}+(1-w_{1}+w_{2})\widehat{D}_{2}\right]\Omega_{12}^{-m}$
$\displaystyle+\frac{(\ell-m)(d-2+\ell+m)}{2}$
$\displaystyle\bigg{\\}}V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})=0.$ (3.7)

If one introduce an auxiliary vector $r$ orthogonal to both $p$ and $q_{12}$,
the Ward identity $r\cdot\widehat{K}_{12}$ gives

$\displaystyle\sum_{m=0}^{\ell}\bigg{\\{}\varepsilon^{\prime}_{m}\Omega_{12}^{-1}\widehat{D}_{0}+\varepsilon^{\prime}_{m-1}\frac{m(\Delta-
m-d+2)(d-3+\ell+m)}{(d-3+2m)(d-5+2m)}$
$\displaystyle-\varepsilon^{\prime}_{m+1}\frac{(\ell-m)(\Delta+m-1)}{m+1}$
$\displaystyle\bigg{\\}}V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})=0.$ (3.8)

Finally, the remaining content of the Ward identity can be obtained from the
scalar product $q_{12}\cdot\widehat{K}_{12}$, which after subtraction of eq.
(3.8) leads to

$\displaystyle\sum_{m=0}^{\ell}\bigg{\\{}$
$\displaystyle\varepsilon_{m}\Omega_{12}^{m}\left(\widehat{D}_{1}-\widehat{D}_{2}\right)\Omega_{12}^{-m}$
$\displaystyle-\varepsilon^{\prime}_{m}\frac{z\cdot
p}{\Omega_{12}}(\Delta-\ell-d+2)-\varepsilon^{\prime}_{m-1}\frac{z\cdot
p}{\Omega_{12}^{2}}\frac{m}{d-5+2m}\widehat{D}_{0}\bigg{\\}}V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})=0.$
(3.9)

Several elements of eqs. (3.7), (3.8) and (3.9) must still be explained: the
quantity $\Omega_{12}=\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}}$ was originally
defined in eq. (2.13); $\widehat{D}_{1}$ and $\widehat{D}_{2}$ are elliptic
differential operators of the Appell $F_{4}$ type,

$\displaystyle\widehat{D}_{1}$
$\displaystyle=w_{1}(1-w_{1})\frac{\partial^{2}}{\partial
w_{1}^{2}}-2w_{1}w_{2}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}-w_{2}^{2}\frac{\partial^{2}}{\partial w_{2}^{2}}$
$\displaystyle\quad+\left[c_{1}-(a+b+1)w_{1}\right]\frac{\partial}{\partial
w_{1}}-(a+b+1)w_{2}\frac{\partial}{\partial w_{2}}-ab,$ (3.10)
$\displaystyle\widehat{D}_{2}$
$\displaystyle=w_{2}(1-w_{2})\frac{\partial^{2}}{\partial
w_{2}^{2}}-2w_{1}w_{2}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}-w_{1}^{2}\frac{\partial^{2}}{\partial w_{1}^{2}}$
$\displaystyle\quad+\left[c_{2}-(a+b+1)w_{2}\right]\frac{\partial}{\partial
w_{2}}-(a+b+1)w_{1}\frac{\partial}{\partial w_{1}}-ab,$ (3.11)

with

$\displaystyle a$ $\displaystyle=\frac{d+\Delta-\Delta_{1}-\Delta_{2}+m}{2},$
$\displaystyle c_{1}$ $\displaystyle=1+\frac{d}{2}-\Delta_{1},$ $\displaystyle
b$ $\displaystyle=\frac{2d-\Delta-\Delta_{1}-\Delta_{2}+m}{2},$ $\displaystyle
c_{2}$ $\displaystyle=1+\frac{d}{2}-\Delta_{2};$ (3.12)

the remaining differential operator $\widehat{D}_{0}$ is given by

$\displaystyle\widehat{D}_{0}$
$\displaystyle=(1-w_{1}+w_{2})\left[-2w_{1}\frac{\partial}{\partial
w_{1}}+(\Delta_{1}-d+1)\right]$
$\displaystyle\quad-(1+w_{1}-w_{2})\left[-2w_{2}\frac{\partial}{\partial
w_{2}}+(\Delta_{2}-d+1)\right].$ (3.13)

The interpretation of this system of partial differential equations for
$\ell+1$ functions of 2 variables $w_{1}$ and $w_{2}$ appears difficult at
first. This is worsened by the fact that we have not expressed the conformal
Ward identities in the complete basis $\\{\varepsilon_{m}\\}$ of traceless,
symmetric tensors, but have instead appealed to the additional tensors
$\varepsilon^{\prime}_{m}$. Nevertheless, the system contains a lot of
redundancies, due in particular to the closure of the conformal algebra in
momentum space, and we will see in the next sections how a simple solution
emerges.

###  3.2 Highest-spin functions in various orderings

The first key observation to be made is that the Ward identities (3.7) and
(3.9) contain a closed system of equations for the function
$V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$, i.e. the highest-spin component
$m=\ell$. The tensors $(z\cdot p)\varepsilon^{\prime}_{i}$ appearing in eq.
(3.9) are orthogonal to $\varepsilon_{\ell}$, the latter being totally
transverse while the former not. When projecting against $\varepsilon_{\ell}$,
these two equations give therefore the system

$\displaystyle\left[(1+w_{1}-w_{2})\widehat{D}_{1}+(1-w_{1}+w_{2})\widehat{D}_{2}\right]\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$
$\displaystyle=0,$ (3.14)
$\displaystyle\left(\widehat{D}_{1}-\widehat{D}_{2}\right)\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$
$\displaystyle=0,$ (3.15)

which is equivalent to

$\widehat{D}_{1}\left[\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})\right]=\widehat{D}_{2}\left[\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})\right]=0.$
(3.16)

As already stated, $\widehat{D}_{1}$ and $\widehat{D}_{2}$ form a system of
generalized hypergeometric differential equations of the Appell $F_{4}$ type,
with a particular solution being $F_{4}(a,b;c_{1},c_{2};w_{1},w_{2})$. General
solutions to such systems can be constructed from linear combinations of the
four functions [67]

$\begin{array}[]{r@{\hspace{18mm}}r}w_{1}^{\Delta_{1}-d/2}w_{2}^{\Delta_{2}-d/2}F_{\Delta_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2}),\hskip
51.21495pt&w_{1}^{\Delta_{1}-d/2}F_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2}),\\\
w_{2}^{\Delta_{2}-d/2}F_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2}),\hskip
51.21495pt&F_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2}),\end{array}$
(3.17)

where we have used $\widetilde{\Delta}_{i}=d-\Delta_{i}$ and introduced the
following notation for the Appell $F_{4}$ function:

$F_{\Delta_{a}\Delta_{b};\Delta,\ell}(w_{1},w_{2})\equiv
F_{4}\left(\begin{array}[]{c}\frac{\Delta_{a}+\Delta_{b}-\Delta+\ell}{2},\leavevmode\nobreak\
\frac{\Delta_{a}+\Delta_{b}+\Delta+\ell-d}{2}\\\
\Delta_{a}-\frac{d}{2}+1,\leavevmode\nobreak\
\Delta_{b}-\frac{d}{2}+1\end{array};w_{1},w_{2}\right).$ (3.18)

Note that the function $F$ satisfies the identities

$F_{\Delta_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2})=F_{\Delta_{1}\Delta_{2};\widetilde{\Delta},\ell}(w_{1},w_{2})=F_{\Delta_{2}\Delta_{1};\Delta,\ell}(w_{2},w_{1}).$
(3.19)

An equivalent basis of solutions is obtained after the change of variables
$(w_{1},w_{2})\to(w_{1}/w_{2},1/w_{2})$, in terms of the four functions

$\begin{array}[]{r@{\hspace{8mm}}r}w_{1}^{\Delta_{1}-d/2}w_{2}^{(\Delta_{2}-\Delta_{1}-\Delta-\ell)/2}F_{\Delta_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right),\hskip
22.76219pt&w_{2}^{(\Delta_{1}+\Delta_{2}-\Delta-\ell-d)/2}F_{\widetilde{\Delta}_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right),\\\
w_{1}^{\Delta_{1}-d/2}w_{2}^{(\Delta_{2}-\Delta_{1}+\Delta-\ell-d)/2}F_{\Delta_{1}\widetilde{\Delta};\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right),\hskip
22.76219pt&w_{2}^{(\Delta_{1}+\Delta_{2}+\Delta-\ell-2d)/2}F_{\widetilde{\Delta}_{1}\widetilde{\Delta};\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right).\end{array}$
(3.20)

The highest-spin vertex function $V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$ is
therefore a linear combination of the functions (3.17), or of the functions
(3.20), multiplied by $\Omega_{12}^{\ell}$. What linear combination depends on
the ordering of the operators in the 3-point function (3.1). We examine below
two distinct cases.

#### Wightman function

When $\phi_{1}$ and $\phi_{2}$ are out-of-time-order, the Wightman function
(3.1) has been computed in ref. [64] (see also ref. [68]). The key observation
is that the Euclidean OPE fixes the behavior of the Wightman 3-point function
in a neighborhood of $p_{1}^{2}=p^{2}=0$, i.e. around $w_{1}=0$ and
$w_{2}\to-\infty$. Among the solutions to the conformal Ward identities only
the first function of the list (3.20) satisfies the constraint.

The normalization of the 3-point function can also be determined from the
results of ref. [64], which are given in terms of traceless, symmetric tensors
built out of $p$ and $p_{1}$. The comparison can be made between the term of
order $(z\cdot p_{1})^{\ell}$ in that tensor, and the same power coming from
the term $(z\cdot q_{12})^{\ell}$ in
$\varepsilon_{\ell}^{\mu_{1}\ldots\mu_{\ell}}(p,q_{12})$, using
$q_{12}^{\mu}=-2\Omega_{12}^{-1}p_{1}^{\mu}+\ldots$ We obtain finally444Note
that we replaced the generic label $[12]$ with $12$ for the Wightman function.
Similarly, we use $\operatorname{T}[12]$ for time-ordered product given later
in eq. (3.27).

$\boxed{\begin{aligned}
V_{\Delta,\ell,\ell}^{12}(w_{1},w_{2})&=\frac{i^{\ell}(4\pi)^{d+2}2^{-(\Delta_{1}+\Delta_{2}+\Delta+\ell+2)}(\Delta-1)_{\ell}\left[(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}\right]^{\ell/2}}{\Gamma\left(\Delta_{1}-\frac{d}{2}+1\right)\Gamma\left(\Delta-\frac{d}{2}+1\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right)}\\\
&\quad\times
w_{1}^{\Delta_{1}-d/2}(-w_{2})^{(\Delta_{2}-\Delta_{1}-\Delta-\ell)/2}F_{\Delta_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right).\end{aligned}}$
(3.21)

This expression is valid when $p_{1}$ is time-like and backward-directed (the
Wightman function vanishes otherwise by the spectral condition), and when
$p_{2}$ is space-like, implying $w_{1}>0$ and $w_{2}<0$. The case of time-like
$p_{2}$ can be obtained in terms of the functions in the list (3.17) following
the analysis of ref. [64]. Note that the normalization of OPE coefficients
that we use here is such that the analytic continuation to Euclidean position
space corresponds to the 3-point function

$\displaystyle z_{\mu_{1}}\cdots z_{\mu_{\ell}}$ $\displaystyle\left\langle
0\right|\phi_{1}(x_{1})\phi_{2}(x_{2})\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(0)\left|0\right\rangle$
$\displaystyle=\frac{\lambda_{12\mathcal{O}}\left[x_{2}^{2}(z\cdot
x_{1})-x_{1}^{2}(z\cdot
x_{2})\right]^{\ell}}{|x_{1}|^{\Delta_{1}-\Delta_{2}+\Delta+\ell}|x_{2}|^{\Delta_{2}-\Delta_{1}+\Delta+\ell}|x_{1}-x_{2}|^{\Delta_{1}+\Delta_{2}-\Delta+\ell}}.$
(3.22)

#### Time-ordered product

In the case where the 3-point function (3.1) contains a time-ordered product
of the operators $\phi_{1}$ and $\phi_{2}$, the general structure is more
complicated, as both $p_{1}$ and $p_{2}$ can be either space-like or time-
like. We focus here on the case where they are both space-like, i.e.
$w_{1},w_{2}<0$. In this case, the Euclidean OPE condition only gives
information about the behavior at $p^{2}=0$ ($w_{1},w_{2}\to-\infty$). It
implies that the highest-spin vertex function is a linear combination of the
first two functions in the list (3.20),

$\displaystyle V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(w_{1},w_{2})$
$\displaystyle=(-w_{2})^{(\Delta_{1}+\Delta_{2}-\Delta-\ell-d)/2}\left[(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}\right]^{\ell/2}$
$\displaystyle\quad\times\left[A\,\left(\frac{w_{1}}{w_{2}}\right)^{\Delta_{1}-d/2}F_{\Delta_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right)+B\,F_{\widetilde{\Delta}_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right)\right].$
(3.23)

The coefficients $A$ and $B$ can be fixed by symmetry arguments and by the
relation to the Wightman function. The singularities of the time-ordered
product have been analyzed in ref. [69], where it was found that there is a
branch-point singularity at $w_{1}=0$ when $\Delta_{1}<\frac{d}{2}$, and that
the coefficient of this singularity is related to the Wightman function by a
multiplicative factor, giving

$\lim_{w_{1}\to
0}(-w_{1}-i\epsilon)^{d/2-\Delta_{1}}V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(w_{1},w_{2})=\frac{1}{2i\sin\left[\pi\left(\frac{d}{2}-\Delta_{1}\right)\right]}\lim_{w_{1}\to
0_{+}}w_{1}^{d/2-\Delta_{1}}V_{\Delta,\ell,\ell}^{12}(w_{1},w_{2}).$ (3.24)

This implies

$A=\frac{i^{\ell-1}(4\pi)^{d+1}\Gamma\left(\frac{d}{2}-\Delta_{1}\right)(\Delta-1)_{\ell}}{2^{\Delta_{1}+\Delta_{2}+\Delta+\ell+1}\Gamma\left(\Delta-\frac{d}{2}+1\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right)}.$
(3.25)

The coefficient $B$ can be fixed in turn by imposing the exchange symmetry
between the operators $\phi_{1}$ and $\phi_{2}$, which is built into the time-
ordered product but does not appear obvious in eq. (3.23). This constraint can
be resolved by going from the basis of Appell $F_{4}$ functions (3.20) to the
symmetric basis (3.17). We see that the result is symmetric under the
simultaneous exchange $w_{1}\leftrightarrow w_{2}$ and
$\Delta_{1}\leftrightarrow\Delta_{2}$ if and only if

$B=\frac{\Gamma\left(\Delta_{1}-\frac{d}{2}\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta-\Delta_{1}-\Delta_{2}+\ell+d}{2}\right)}{\Gamma\left(\frac{d}{2}-\Delta_{1}\right)\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+\ell-d}{2}\right)}\,A,$
(3.26)

after which we arrive at

$\boxed{\begin{aligned}
V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(w_{1},w_{2})&=\frac{i^{\ell-1}(4\pi)^{d+1}2^{-(\Delta_{1}+\Delta_{2}+\Delta+\ell+1)}(\Delta-1)_{\ell}\left[(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}\right]^{\ell/2}}{\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+\ell-d}{2}\right)}\\\
&\quad\times\bigg{[}f_{\Delta_{1}\Delta_{2};\Delta,\ell}(-w_{1})^{\Delta_{1}-d/2}(-w_{2})^{\Delta_{2}-d/2}F_{\Delta_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2})\\\
&\quad\qquad+f_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}(-w_{1})^{\Delta_{1}-d/2}F_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2})\\\
&\quad\qquad+f_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}(-w_{2})^{\Delta_{2}-d/2}F_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2})\\\
&\quad\qquad+f_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}F_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2})\bigg{]},\end{aligned}}$
(3.27)

where we have defined

$f_{\Delta_{a}\Delta_{b};\Delta,\ell}=\frac{\Gamma\left(\frac{d}{2}-\Delta_{a}\right)\Gamma\left(\frac{d}{2}-\Delta_{b}\right)\Gamma\left(\frac{\Delta_{a}+\Delta_{b}+\Delta+\ell-d}{2}\right)}{\Gamma\left(1-\frac{\Delta_{a}+\Delta_{b}-\Delta+\ell}{2}\right)}.$
(3.28)

This result is analytic in the scaling dimensions $\Delta_{1},\Delta_{2}$,
with the exception of poles at $\Delta_{1},\Delta_{2}=\frac{d}{2}+n$ with
integer $n$. This means that the result (3.27) is valid in all generality, and
not only in the regime $\Delta_{1}<\frac{d}{2}$ used to fix the coefficient
$A$. The apparent divergences when $\Delta_{1},\Delta_{2}=\frac{d}{2}+n$ are
reminiscent of the anomalies discussed in refs. [70, 71]: they appear
precisely when the powers of $w_{1}$ and $w_{2}$ are integer, in which case
the terms of the different hypergeometric series can be mixed. Resolving these
special cases by analytic continuation in $\Delta_{i}$ (or in $d$), one finds
cancellations between the coefficients $f_{\Delta_{a}\Delta_{b};\Delta,\ell}$,
so that the vertex functions remain in fact finite, however with the
appearance of logarithms of $w_{1}$ and $w_{2}$. An example realizing the case
$\Delta_{1}=\Delta_{2}=\frac{d}{2}$ is presented below in section 4.3.

#### Other orderings

Besides the Wightman function and the time-ordered product, other orderings of
the operators can be envisioned, but they can all be related to the two
results (3.21) and (3.27): ordinary commutators are nothing but the difference
between two Wightman functions; causal commutators (i.e. advanced or retarded)
are similarly related to the sum or difference between the time-ordered
product and the Wightman function, and as such they coincide with the former
when both momenta $p_{1}$ and $p_{2}$ are space-like; the anti-time-ordered
product is complex conjugated to the time-ordered one. In order to obtain a
complete picture of all the possible vertex functions, one needs of course to
go beyond the kinematics assumptions made in deriving eqs. (3.21) and (3.27).
Causal commutators would be a useful tool to do so, as they have well-
understood domains of analyticity in terms of complex momenta. We leave this
task for future work.

One important property that is shared by all orderings is the vanishing of the
highest-spin vertex function when the intermediate operator takes a double-
trace dimension, i.e. when $\Delta=\Delta_{1}+\Delta_{2}+\ell+2n$ with integer
$n$. This ensures the triviality of Gaussian theories in momentum space, and
places on equal footing the contributions of single- and double-trace
operators in large-$N$ theories. It also implies that the conformal partial
wave expansion (1.2) shares the “dispersive” property of the double
discontinuity in position space [38].

###  3.3 Lower spin functions

With the knowledge of the highest-spin vertex function, the computation of all
remaining vertex functions $V_{\Delta,\ell,m}^{[12]}$ with $m<\ell$ does not
require solving more partial differential equations, nor making assumptions
about boundary conditions, as eq. (3.8) gives a simple two-step algebraic
recursion relation: since the tensors $\varepsilon^{\prime}_{m}$ are
orthogonal to each other, we get

$\boxed{\begin{aligned}
V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})&=\frac{m+1}{(\ell-m)(\Delta+m-1)}\\\
&\quad\times\bigg{[}\frac{1}{\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}}}\widehat{D}_{0}V_{\Delta,\ell,m+1}^{[12]}(w_{1},w_{2})\\\
&\quad\qquad+\frac{(m+2)(\Delta-
m-d)(d-1+\ell+m)}{(d-1+2m)(d+1+2m)}\,V_{\Delta,\ell,m+2}^{[12]}(w_{1},w_{2})\bigg{]},\end{aligned}}$
(3.29)

using the first-order differential operator $\widehat{D}_{0}$ defined in eq.
(3.13). The last line must be dropped in the first step of the recursion,
namely when $m=\ell-1$; equivalently, the recursion relation can be defined in
terms of the functions $V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$ computed
above and of $V_{\Delta,\ell,\ell+1}^{[12]}(w_{1},w_{2})=0$. Note that the
operator $\widehat{D}_{0}$ is odd under the permutation $1\leftrightarrow 2$,
which is consistent with the fact that the Gegenbauer polynomials
$\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)$ is similarly odd under
$\cos\theta\to-\cos\theta$ for odd $m$: in this way the exchange symmetry
$1\leftrightarrow 2$ of the time-ordered product is preserved for all $m$.

Derivatives of Appell $F_{4}$ functions can be again expressed in terms of
linear combinations of $F_{4}$ functions, so one could in principle write down
closed-form formulae for the vertex functions
$V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})$ with the different operator orderings.
This exercise is not particularly enlightening, though, and the recursion
relation (3.29) seems simple enough to provide the most efficient
implementation of the vertex functions. In fact, since $\widehat{D}_{0}$ is a
first-order differential operator that only raises powers of $w_{1}$ and
$w_{2}$, the vertex functions with $m<\ell$ inherit the analytic structure of
the highest-spin function.

The case in which the operator $\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}$ is a
conserved current deserves special attention. It turns out that the
differential operator $\widehat{D}_{0}$ annihilates
$V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$ whenever the unitarity bound is
saturated ($\Delta=d-2+\ell$) and the two operators $\phi_{1}$ and $\phi_{2}$
have identical scaling dimensions ($\Delta_{1}=\Delta_{2}$), two conditions
that must be met for $\mathcal{O}$ to be conserved. This implies that
$V_{\Delta,\ell,\ell-1}^{[12]}(w_{1},w_{2})=0$. Moreover, the factor $(\Delta-
m-d)$ in eq. (3.29) ensures that the second step of the recursion is trivial
as well, and so $V_{\Delta,\ell,\ell-2}^{[12]}(w_{1},w_{2})=0$. It follows
immediately that all vertex functions with $m<\ell$ vanish in this case. This
is not a surprise: conserved currents only admit transverse polarizations,
while any longitudinal polarization defines a null state. This property is
essential in the conformal partial wave expansion, as it implies that one does
not need to sum over $m<\ell$ for conserved currents, and hence that the
singularities in the definition (2.35) of the coefficient $C_{\Delta,\ell,m}$
never occurs.

###  3.4 Special kinematic limits

We will finally examine several special kinematic limits in which the vertex
functions admits a simpler form, and compare the conformal partial waves in
these limits with results already existing in the literature.

#### Light-like limit

The first interesting case is the limit $w_{1},w_{2}\to 0$ of the vertex
function with a time-ordered product. As the Appell $F_{4}$ function is
analytic around the point $(w_{1},w_{2})=(0,0)$, one can see from eq. (3.27)
that the limit exists if $\Delta_{1},\Delta_{2}>\frac{d}{2}$ (the opposite
situation is discussed below). In this case, the recursion relation (3.29)
becomes an algebraic equation,

$\displaystyle V_{\Delta,\ell,m}^{\operatorname{T}[12]}(0,0)$
$\displaystyle=\frac{m+1}{(\ell-m)(\Delta+m-1)}\bigg{[}(\Delta_{1}-\Delta_{2})V_{\Delta,\ell,m+1}^{[12]}(0,0)$
$\displaystyle\hskip 71.13188pt+\frac{(m+2)(\Delta-
m-d)(d-1+\ell+m)}{(d-1+2m)(d+1+2m)}\,V_{\Delta,\ell,m+2}^{\operatorname{T}[12]}(0,0)\bigg{]},$
(3.30)

which is solved by

$\displaystyle V_{\Delta,\ell,m}^{\operatorname{T}[12]}(0,0)$
$\displaystyle=V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(0,0)\frac{(-2)^{\ell-m}\ell!}{m!(\ell-m)!}\frac{\left(\frac{\Delta_{2}-\Delta_{1}+\Delta-\ell-d+2}{2}\right)_{\ell-m}}{\left(\Delta-1+m\right)_{\ell-m}}$
$\displaystyle\quad\times{}_{3}F_{2}\left(\begin{array}[]{c}-(\ell-m),\leavevmode\nobreak\
d-\Delta-1+m,\leavevmode\nobreak\ \frac{d-2+2m}{2}\\\
\frac{\Delta_{1}-\Delta_{2}+d-\Delta-\ell+2m}{2},\leavevmode\nobreak\
d-2+2m\end{array};1\right).$ (3.33)

Remarkably, the recursion relation only depends on the difference between the
external scaling dimensions, $\Delta_{1}-\Delta_{2}$, and not on their sum. In
fact, when $\Delta_{1}=\Delta_{2}(\equiv\Delta_{\phi})$, all vertex functions
with odd $\ell-m$ vanish identically, and one obtains

$V_{\Delta,\ell,\ell-2n}^{\operatorname{T}[\phi\phi]}(0,0)=\frac{(-1)^{n}\ell!}{2^{2n}n!(\ell-2n)!}\frac{\left(\frac{\Delta-\ell-d+2}{2}\right)_{n}}{\left(\frac{d-1}{2}+\ell-2n\right)_{n}\left(\frac{3-\Delta-\ell}{2}\right)_{n}}\,V_{\Delta,\ell,\ell}^{\operatorname{T}[\phi\phi]}(0,0).$
(3.34)

The conformal partial waves (1.4) can then be written as

$G_{\Delta,\ell}(w_{i}=0,\cos\theta)=\frac{2^{2\Delta+1}\Gamma\left(\Delta-\frac{d}{2}+1\right)\Gamma(\Delta+\ell)}{(4\pi)^{(d+2)/2}(\Delta-1)_{\ell}}\left[V_{\Delta,\ell,\ell}^{[\phi\phi]}(0,0)\right]^{2}g_{\Delta,\ell}(\cos\theta),$
(3.35)

where $g_{\Delta,\ell}$ is a polynomial of degree $\ell$ in $\cos\theta$,
defined by

$g_{\Delta,\ell}(\cos\theta)=\sum_{n=0}^{\ell/2}\frac{(-1)^{n}\ell!(2n)!}{2^{\ell+2n+1}(n!)^{2}}\frac{d-3+2\ell-4n}{\left(2-\frac{d}{2}-\ell\right)_{n}\left(\frac{d-3}{2}\right)_{\ell-n+1}}\frac{\big{(}\frac{2-\Delta-\ell}{2}\big{)}_{n}\big{(}\frac{2-\widetilde{\Delta}-\ell}{2}\big{)}_{n}}{\big{(}\frac{3-\Delta-\ell}{2}\big{)}_{n}\big{(}\frac{3-\widetilde{\Delta}-\ell}{2}\big{)}_{n}}\mathcal{C}_{\ell-2n}^{(d-3)/2}(\cos\theta).$
(3.36)

These are precisely the polynomials defined in ref. [63], where the light-like
limit of the conformal partial wave expansion was studied. The
$g_{\Delta,\ell}$ have the remarkable property of interpolating continuously
between the Gegenbauer polynomials $\mathcal{C}_{\ell}^{(d-3)/2}(\cos\theta)$
when $\Delta-\ell$ saturates the unitarity bound, and
$\mathcal{C}_{\ell}^{(d-2)/2}(\cos\theta)$ when $\Delta-\ell\to\infty$. Using

$V_{\Delta,\ell,\ell}^{\operatorname{T}[\phi\phi]}(0,0)=\frac{i^{\ell-1}(4\pi)^{d+1}2^{-(2\Delta_{\phi}+\Delta+\ell+1)}\Gamma\left(\Delta_{\phi}-\frac{d}{2}\right)^{2}\Gamma\left(\frac{\Delta+\ell+d}{2}-\Delta_{\phi}\right)(\Delta-1)_{\ell}}{\Gamma\left(\frac{\Delta+\ell}{2}\right)^{2}\Gamma\left(\Delta_{\phi}-\frac{\Delta-\ell}{2}\right)\Gamma\left(\Delta_{\phi}+\frac{\Delta+\ell-d}{2}\right)\Gamma\left(\Delta_{\phi}+\frac{\Delta-\ell}{2}-d+1\right)},$
(3.37)

one can also verify that the multiplicative coefficient in eq. (3.35) also
matches the definition of $G_{\Delta,\ell}(\cos\theta)$ in ref. [63].555Eq.
(1.14) in ref. [63] contains a typo: the numerator is missing a factor of
$\Gamma(\Delta+\ell)$ (corrected in the arXiv version).

#### Scattering amplitude and form factor

An alternative, maybe more interesting situation is the asymptotic limit
$w_{i}\to 0$ in the case $\Delta_{i}<\frac{d}{2}$. The time-ordered vertex
function (3.27) diverges in this case, but one obtains a well-defined limit by
multiplying it with the appropriate powers of $w_{i}$. The peculiarity of this
limit is that it allows to define objects that have an interesting field-
theoretical interpretation: consider

$F(s,t,u)\equiv\left[\prod_{i=1}^{3}\lim_{p_{i}^{2}\to
0_{-}}(-p_{i}^{2})^{d/2-\Delta_{i}}\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\phi_{1}(p_{1})\phi_{2}(p_{2})\phi_{3}(p_{3})\phi_{4}(-p_{1}-p_{2}-p_{3})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}},$
(3.38)

as well as the subsequent limit $p_{4}^{2}=s+t+u\to 0$,

$A(s,t)\equiv\lim_{p_{4}^{2}\to 0_{-}}(-p_{4}^{2})^{d/2-\Delta_{4}}F(s,t,u).$
(3.39)

$F$ and $A$ have been interpreted respectively as a “form factor” and as an
“amplitude” [69]: on the one hand, they are related to the time-ordered
correlation function in eq. (3.38) by a procedure similar to the LSZ reduction
in massive quantum field theory; on the other hand, from a careful analysis of
the singularities in the Fourier transform from position space, one can show
that they admit a conformal partial wave expansion that is reminiscent of the
“unitarity cuts” method relating higher-point amplitudes to lower-point ones.
This latter property follows from the fact that the form factor is related to
the same limit of the partially-time-ordered function,

$F(s,t,u)\sim\left[\prod_{i=1}^{3}\lim_{p_{i}^{2}\to
0_{-}}(-p_{i}^{2})^{d/2-\Delta_{\phi}}\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\operatorname{T}[\phi_{3}(p_{3})\phi_{4}(-p_{1}-p_{2}-p_{3})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}},$
(3.40)

provided that the limit is taken with $p_{1}$, $p_{2}$ backward-directed, and
$p_{3}$ forward-directed. The equivalence is up to a phase involving the
dilatation operator (i.e. it depends on the scaling dimension $\Delta$ of the
intermediate operator in the conformal partial wave). We do not want to re-
derive here the results of ref. [69], but we can directly make use of the
right-hand side of the equivalence (3.40) to provide a simple computation of
the form factor $F$ and of the amplitude $A$ using our conformal partial
waves, up to the aforementioned phase.

The computation involves two distinct vertex functions,
$V_{\Delta,\ell,m}^{\operatorname{T}[12]}$ and the complex conjugate of
$V_{\Delta,\ell,m}^{\bar{\operatorname{T}}[34]}$.
$V_{\Delta,\ell,m}^{\operatorname{T}[12]}$ is evaluated with both $w_{1}$ and
$w_{2}\to 0$. In this case the computation proceeds as in the light-like limit
discussed above: the recursion relation (3.29) for the vertex function is
again algebraic, though it differs from eq. (3.30) by the replacement
$\Delta_{1,2}\to d-\Delta_{1,2}$. One arrives at

$\displaystyle\lim_{w_{1},w_{2}\to
0_{-}}(-w_{1})^{d/2-\Delta_{1}}(-w_{2})^{d/2-\Delta_{2}}V_{\Delta,\ell,m}^{\operatorname{T}[12]}(w_{1},w_{2})$
$\displaystyle\quad=\frac{i^{\ell-1}(4\pi)^{d}\Gamma\left(\frac{d}{2}-\Delta_{1}\right)\Gamma\left(\frac{d}{2}-\Delta_{2}\right)(\Delta-1)_{\ell}\left(\frac{\Delta_{1}-\Delta_{2}+\Delta-\ell-d+2}{2}\right)_{\ell-m}\sin\left[\pi\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right]}{2^{\Delta_{1}+\Delta_{2}+\Delta+\ell-1}\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\left(\Delta-1+m\right)_{\ell-m}}$
$\displaystyle\quad\quad\times\frac{(-2)^{\ell-m}\ell!}{m!(\ell-m)!}{}_{3}F_{2}\left(\begin{array}[]{c}-(\ell-m),\leavevmode\nobreak\
d-\Delta-1+m,\leavevmode\nobreak\ \frac{d-2+2m}{2}\\\
\frac{\Delta_{2}-\Delta_{1}+d-\Delta-\ell+2m}{2},\leavevmode\nobreak\
d-2+2m\end{array};1\right).$ (3.43)

For $V_{\Delta,\ell,m}^{\bar{\operatorname{T}}[34]}$, consider first the
highest-spin vertex function as defined in eq. (3.23), which in the limit
$w_{3}\to 0$ becomes

$\displaystyle\lim_{w_{3}\to 0_{-}}$
$\displaystyle(-w_{3})^{d/2-\Delta_{3}}V_{\Delta,\ell,\ell}^{\bar{\operatorname{T}}[34]}(w_{3},w_{4})$
$\displaystyle=A^{*}(-w_{4})^{(\Delta_{3}+\Delta_{4}-\Delta+\ell-2)/2}(1-w_{4})^{1-\Delta_{3}}$
$\displaystyle\quad\times{}_{2}F_{1}\left(\tfrac{\Delta-\ell-\Delta_{3}-\Delta_{4}+2}{2},\tfrac{\Delta_{4}-\Delta_{3}+\Delta-\ell-d+2}{2};\Delta-\tfrac{d}{2}+1;\frac{1}{w_{4}}\right).$
(3.44)

where $A^{*}$ is the complex conjugate of the coefficient $A$ given in eq.
(3.25), upon replacement $1\to 3$ and $2\to 4$. Applying the recursion
relation (3.29) leads to

$\lim_{w_{3}\to
0_{-}}(-w_{3})^{d/2-\Delta_{3}}V_{\Delta,\ell,m}^{\operatorname{T}[12]}(w_{1},w_{2})=A^{*}(-w_{4})^{(\Delta_{3}+\Delta_{4}-\Delta+\ell-2)/2}(1-w_{4})^{1-\Delta_{3}}v_{\Delta,\ell,m}\left(\frac{1}{w_{4}}\right),$
(3.45)

where we have defined

$\displaystyle v_{\Delta,\ell,m}(z)=\sum_{j=0}^{\ell-m}$
$\displaystyle\frac{2^{\ell-m}\ell!}{m!j!(\ell-
m-j)!}\frac{\left(\frac{d}{2}-1+m\right)_{\ell-
m-j}\left(\Delta-\ell-d+2+j\right)_{\ell-m-j}}{\left(d-2+2m\right)_{\ell-
m-j}}$
$\displaystyle\times\frac{\left(\frac{\Delta-\ell-\Delta_{3}-\Delta_{4}+2}{2}\right)_{j}\left(\frac{\Delta_{4}-\Delta_{3}+\Delta-\ell-d+2}{2}\right)_{j}}{\left(2-\Delta-\ell\right)_{\ell-m}\left(\Delta-\frac{d}{2}+1\right)_{j}}$
$\displaystyle\times
z^{j}\,{}_{2}F_{1}\left(\tfrac{\Delta-\ell-\Delta_{3}-\Delta_{4}+2}{2}+j,\tfrac{\Delta_{4}-\Delta_{3}+\Delta-\ell-d+2}{2}+j;\Delta-\frac{d}{2}+1+j;z\right).$
(3.46)

The conformal partial waves corresponding to the form factor follow simply
from plugging the vertex functions (3.43) and (3.45) in eq. (1.4). This gives
a closed-form expression for distinct $\Delta_{i}$ that was lacking in ref.
[69], as well as a new representation of the result in the case of identical
operators ($\Delta_{i}=\Delta_{\phi}$): the function
$f_{\Delta,\ell}(z,\cos\theta)$ computed in appendix B of ref. [69] using a
method based on the conformal Casimir equations can be written

$\displaystyle f_{\Delta,\ell}(z,\cos\theta)$
$\displaystyle=z^{(\Delta-\ell-2\Delta_{\phi})/2}(1-z)^{1-\Delta_{\phi}}$
$\displaystyle\quad\times\sum_{n=0}^{\ell/2}\frac{(-1)^{n}\ell!}{2^{\ell}n!}\frac{\left(\frac{d-3}{2}+\ell-2n\right)\left(\frac{\Delta-\ell-d+2}{2}\right)_{n}}{\left(\frac{d-3}{2}\right)_{\ell-n+1}\left(\frac{3-\Delta-\ell}{2}\right)_{n}}\mathcal{C}_{\ell-2n}^{(d-3)/2}(\cos\theta)v_{\Delta,\ell,\ell-2n}(z)$
(3.47)

in terms of $z=1/w_{4}=p_{4}^{2}/s$. Similarly, for the scattering amplitude
(3.25) one recovers the polynomials $g_{\Delta,\ell}$ of eq. (3.36) in the
case of identical external operators, as well as a new set of polynomials
obtained by plugging the vertex function (3.43) twice in eq. (1.4) when the
external operators are distinct.

These examples of special kinematics are far from being exhaustive, but they
illustrate how our results can be efficiently used in various specific
configurations. This concludes the derivation of the vertex functions, and we
move next to a complete physical example where the conformal partial wave
expansion are put to good use.

##  4 Example: the scalar box integral in 4 dimensions

We present in this section an example in which the conformal partial wave
expansion can be explicitly compared with a known 4-point function. We take
the theory of a free, complex scalar field $\Phi$, and consider the time-
ordered 4-point function involving the composite operator $\Phi^{2}$ and its
complex conjugate $\overline{\Phi}^{2}$,

$\left\langle
0\right|\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle.$
(4.1)

This correlation function is given in terms of a single Feynman diagram shown
in figure 3. In $d=4$ dimensions, the integral over the loop momentum can be
performed in arbitrary kinematics, and the result is a Bloch-Wigner
dilogarithm [72, 73, 74]. The simplicity of this result can be traced back to
presence of a dual conformal symmetry at the level of the integrand [45, 46].

Figure 3: The scalar box diagram corresponding to the 4-point function (4.1)
in the theory of a free complex scalar field. The solid lines are propagators
of the field $\Phi$, with arrows indicating the charge flow. The external,
dashed lines correspond to the composite operators $\Phi^{2}$ and
$\overline{\Phi}^{2}$. The dual graph in terms of the variables
$x_{i}-x_{i+1}=p_{i}$ is shown in red, with dotted lines.

###  4.1 CFT optical theorem

The 4-point function (4.1) in itself does not admit a decomposition into
conformal partial waves: the time-ordered product runs over all 4 operators,
and as such it does not respect the structure of the correlator (1.1).
However, the real part of that function does, by an identity that is sometimes
referred to as the “CFT optical theorem” [31, 32] (see also ref. [75]): by a
combinatoric identity involving the time-ordered product and its hermitian
conjugate $\operatorname{\overline{T}}$, it can be shown that

$\displaystyle\left\langle
0\right|\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle$
$\displaystyle+\left\langle
0\right|\operatorname{\overline{T}}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle$
$\displaystyle=\left\langle
0\right|\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})]\operatorname{\overline{T}}[\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle,$
(4.2)

provided that the following conditions on the momenta and Mandelstam
invariants are fulfilled:

$p_{i}^{2}<0,\qquad\qquad s>0,\qquad\qquad t,u<0.$ (4.3)

The right-hand side of this optical theorem is precisely in the form of eq.
(1.1), and we can therefore define

$G(s,w_{i},\cos\theta)=\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})]\operatorname{\overline{T}}[\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(4.4)

At the same time, the left-hand side of eq. (4.2) corresponds to (twice) the
real part of the scalar box integral. This can be compared with the known
results that give [74, 45]666Our real part of the box integral differs from
the usual conventions by a multiplicative factor of $2^{10}\pi^{8}$ that has
to do with the use of the standard CFT normalization of operators in position
space: on the one hand, the free field propagator in 4 dimensions is
$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\Phi(p)\overline{\Phi}(-p)]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=4\pi^{2}i/p^{2}$,
accounting for an overall factor of $(4\pi^{2})^{4}$; on the other hand, each
vertex involving the properly-normalized CFT operator
$\frac{1}{\sqrt{2}}\Phi^{2}$ gives rise to a factor of $\sqrt{2}$,
contributing in total another factor of 4 to our box integral.

$G(s,w_{i},\cos\theta)=\frac{(2\pi)^{7}}{su}\frac{\log\left(\frac{1-1/z}{1-1/\bar{z}}\right)}{z-\bar{z}}.$
(4.5)

$z$ and $\bar{z}$ are related to the dual conformal cross ratios, given in
terms of $p_{i}=x_{i}-x_{i+1}$ by the two equations

$z\bar{z}=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}=\frac{p_{1}^{2}p_{3}^{2}}{su},\qquad\qquad(1-z)(1-\bar{z})=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}=\frac{p_{2}^{2}p_{4}^{2}}{su}.$
(4.6)

###  4.2 IR divergences

In spite of the relative simplicity of (4.5) in terms of $z$ and $\bar{z}$,
the dependence on $w_{i}$ and $\cos\theta$ is quite intricate: determining $z$
and $\bar{z}$ requires solving a quadratic equation involving the Mandelstam
invariant $u$, which is itself given by

$\displaystyle u=-\frac{s}{2}\Big{[}\,$ $\displaystyle
1-w_{1}-w_{2}-w_{3}-w_{4}-(w_{1}-w_{2})(w_{3}-w_{4})$
$\displaystyle-\cos\theta\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}}\sqrt{(1-w_{3}-w_{4})^{2}-4w_{3}w_{4}}\Big{]}.$
(4.7)

A comparison with the conformal partial wave expansion at generic kinematics
is quite difficult to achieve. Instead, we will focus on the IR divergences of
this box integral when some or all of the $p_{i}^{2}$ approach zero. In the
simplest case where all “masses” are identical ($w_{i}=w$), we have

$z,\bar{z}=\frac{1}{2}\left(1\pm\sqrt{1+\frac{8w^{2}}{(1-4w)(1-\cos\theta)}}\right),$
(4.8)

and, as $w\to 0$,

$G(s,w_{i}=w,\cos\theta)=-\frac{512\pi^{7}}{s^{2}(1-\cos\theta)}\log\left(\frac{2w^{2}}{1-\cos\theta}\right)+\ldots$
(4.9)

The real part of the box integral has a logarithmic divergence in $w$, and
moreover a pole in the forward limit $\cos\theta\to 1$.

To perform the comparison with the conformal partial waves, one would like to
expand $G$ in spin partial waves first, and write

$G(s,w_{i},\cos\theta)=s^{-2}\sum_{\ell=0}^{\infty}\left(\ell+\tfrac{1}{2}\right)a_{\ell}(w_{i})P_{\ell}(\cos\theta),$
(4.10)

or equivalently, using the orthogonality of Legendre polynomials,

$a_{\ell}(w_{i})=\int_{-1}^{1}d\cos\theta\,s^{2}G(s,w_{i},\cos\theta).$ (4.11)

Note that the asymptotic limit in eq. (4.9) does not commute with the partial
wave expansion, since the pole in $\cos\theta$ is not integrable. One must
work instead with generic $w$, and take the limit $w\to 0$ at the end. To do
so, note that $G$ can be rewritten as

$G(s,w_{i},\cos\theta)=-\frac{64\pi^{7}}{s^{2}\Omega_{12}\Omega_{34}}\frac{\partial}{\partial\cos\theta}\left[\log\left(\frac{1-1/z}{1-1/\bar{z}}\right)\right]^{2}.$
(4.12)

The angular integral can now be performed using integration by parts after
taking the IR limit in the logarithm. For identical $w_{i}=w$, the limit $w\to
0$ gives

$G(s,w_{i}=w,\cos\theta)=-\frac{256\pi^{7}}{s^{2}}\frac{\partial}{\partial\cos\theta}\left[\log\left(\frac{2w^{2}}{1-\cos\theta}\right)+\ldots\right]^{2},$
(4.13)

and thus

$a_{\ell}(w_{i}=w)=1024\pi^{7}[\log(-w)+H_{\ell}]^{2}+\ldots$ (4.14)

where $H_{\ell}$ is the $\ell$-th harmonic number, and the ellipsis indicates
that we have neglected terms of order $w\log(-w)$ and higher. For each
individual partial wave the leading IR divergence is now a double logarithm in
$w$.

To study a slightly more general form, one can also consider the situation in
which the masses are equal two-by-two, say $w_{4}=w_{1}$ and $w_{3}=w_{2}$,
and take the limit $w_{1}\to 0$ while keeping $w_{2}$ generic. Using eq.
(4.12), one arrives at

$a_{\ell}(w_{i})=\frac{256\pi^{7}}{(1-w_{2})^{2}}\left[\log(-w_{1})+\log(-w_{2})-2\log(1-w_{2})+2H_{\ell}\right]^{2}+\ldots,$
(4.15)

neglecting terms that vanish as $w_{1}\to 0$. We shall now see in the next
section how the partial waves (4.14) and (4.15) are matched one-to-one by the
conformal partial waves.

###  4.3 Conformal partial wave expansion

To compute the conformal partial waves, one simply needs to apply the recipe
given in sections 2 and 3 to the 4-point function (4.4). We work with $d=4$,
and with scaling dimensions of the external operator that are all equal,
$\Delta_{i}=2$. Since the theory is non-interacting, it is straightforward to
determine the spectrum of primary operators that enter the OPE of $\Phi^{2}$
and $\overline{\Phi}^{2}$, as these must be composites of the free field.
There are two families of such operators:

  * •

Operators of the schematic form
$[\overline{\Phi}^{2}\square^{n}\partial^{\ell}\Phi^{2}]$, that have spin
$\ell$ and scaling dimensions $\Delta=4+\ell+2n$. As these are precisely
double-trace dimensions, the vertex functions involving them vanish. These
operators do not contribute to the conformal partial wave expansion in
momentum space.

  * •

Operators of the form $[\overline{\Phi}\partial^{\ell}\Phi]$, with spin $\ell$
and scaling dimensions $\Delta=2+\ell$. There is one such operator for every
spin $\ell\geq 0$, and each of them is a conserved current saturating the
unitarity bound, with the exception of the scalar operator
$[\overline{\Phi}\Phi]$.

Therefore, the conformal partial wave decomposition (1.2) is a sum over
integers labeled by the spin $\ell$, with each term given by single
polarization $m=\ell$. Plugging the operator data in the coefficient
$C_{\Delta,\ell,\ell}$ of eq. (2.35), one gets

$G(s,w_{i},\cos\theta)=s^{-2}\sum_{\ell=0}^{\infty}\left(\ell+\tfrac{1}{2}\right)P_{\ell}(\cos\theta)a_{\ell}(w_{i}),$
(4.16)

where now

$a_{\ell}(w_{i})=\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}\frac{2^{3\ell}(\ell!)^{4}}{\pi^{3}(2\ell)!}\,V_{2+\ell,\ell,\ell}^{\operatorname{T}[\phi\phi]}(w_{1},w_{2})V_{2+\ell,\ell,\ell}^{\operatorname{T}[\phi\phi]}(w_{3},w_{4})^{*}.$
(4.17)

The computation of the vertex functions is slightly more complicated, as one
cannot simply plug the relevant scaling dimensions in eq. (3.27): the
coefficients $f_{\Delta_{1}\Delta_{2};\Delta,\ell}$ diverge precisely when
$\Delta_{i}=\frac{d}{2}$. Fortunately, it is straightforward to regularize the
computation of these vertex functions by analytic continuation in $d$: the
free scalar theory can be defined in any $d\neq 4$, and we shall see that the
limit $d\to 4$ ends up to be finite, due to cancellations among the different
terms in eq. (3.27).

In generic $d$, the scaling dimension of $\Phi^{2}$ and $\overline{\Phi}^{2}$
is $\Delta_{i}=d-2$, and the conserved currents entering the OPE have
$\Delta=d-2+\ell$. The coefficients $f_{\Delta_{1}\Delta_{2};\Delta,\ell}$
appearing in eq. (3.27) take the values

$f_{\Delta_{1}\Delta_{2};\Delta,\ell}=\Gamma(d-3+\ell)\Gamma\left(\tfrac{4-d}{2}\right),\quad
f_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}=\ell!\Gamma\left(\tfrac{d-4}{2}\right),\quad
f_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}=f_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}=0.$
(4.18)

Since the combination
$f_{\Delta_{1}\Delta_{2};\Delta,\ell}+f_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}$
vanishes in the limit $d\to 4$, the individual divergences cancel out and the
powers $(-w_{i})^{\Delta_{i}-d/2}$ turn into logarithms. The general structure
of the vertex function is still quite complicated, as it involves Appell
$F_{4}$ functions and derivatives of them with respect to their parameters.
But it is simple to obtain the limit in which one of their argument $w_{i}$ is
small. For instance, as $w_{1}\to 0$, we get

$V_{2+\ell,\ell,\ell}^{\operatorname{T}[\phi\phi]}(w_{1},w_{2})=8\pi^{5}i^{\ell+1}\frac{(2\ell)!}{2^{2\ell}(\ell!)^{3}}\frac{1}{1-w_{2}}\left[\log(-w_{1})+\log\left(\frac{-w_{2}}{(1-w_{2})^{2}}\right)+2H_{\ell}+\ldots\right].$
(4.19)

This function already bears resemblance to the partial wave decomposition
obtained above. Assuming all identical $w_{i}=w$, one gets

$a_{\ell}(w_{i}=w)=256\pi^{7}\frac{(2\ell)!}{2^{\ell}(\ell!)^{2}}\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}\left[\log(-w)+H_{\ell}+\ldots\right]^{2},$
(4.20)

while with $w_{4}=w_{1}$ and $w_{3}=w_{2}$ we have

$a_{\ell}(w_{i})=64\pi^{7}\frac{(2\ell)!}{2^{\ell}(\ell!)^{2}}\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}\left[\log(-w_{1})+\log(-w_{2})-2\log(1-w_{2})+2H_{\ell}+\ldots\right]^{2}.$
(4.21)

These coefficients match precisely eqs. (4.14) and (4.15) provided that

$\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}=\frac{2^{\ell+2}(\ell!)^{2}}{(2\ell)!}.$
(4.22)

This is indeed the known expression for the OPE coefficients in the complex
scalar field theory in $d=4$ dimensions. This example gives therefore a non-
trivial verification of our general results in a specific conformal field
theory.

##  5 Conclusions

In this work, we have presented a computation of the conformal partial waves
in momentum space, with an emphasis on their factorization properties: they
are given as a finite sum over vertex functions and polynomials in the
scattering angle. The terms of this sum are in one-to-one correspondence with
the ordinary spin partial waves. The vertex functions are simple to obtain
using a recursion relation descending from the highest-spin function, and the
latter are expressed in terms of Appell $F_{4}$ double hypergeometric
functions, which are straightforward to evaluate numerically. In some
particular cases where the scaling dimensions of the external operators take
(half-)integer values, the vertex function must be obtained by analytic
continuation: we gave an example where this situation happens in section 4.
The outcome of this work is a new mathematical tool for the study of conformal
field theory, which we hope will be useful in the future. Some directions that
are certainly worth investigating are the possibility to use partial wave
unitarity to put constraints on the CFT data, in the spirit of the S-matrix
bootstrap [76], as well as the prospect of writing CFT dispersion relations
directly in momentum space [39, 38].

We also believe that our computation of the conformal partial waves
illustrates the power of Minkowski momentum-space techniques in CFT. It is
well known that the orthogonality of momentum eigenstates allows to factorize
the computation of correlation functions: after all, this is the reason why
the majority of quantum field theory results are obtained in momentum space.
What is certainly less appreciated is the structural simplicity of the
operator product expansion in momentum space. While 3-point function are
admittedly complicated, the fact that they depend by momentum conservation on
only two momenta is very powerful: they can always be thought to live on a
2-dimensional slice of Minkowski space, and their spin can be projected onto
this subspace.777As already mentioned, this also implies that the ideal
framework to work with momentum-space 3-point functions is in terms of spinor
variables [62]. For this reason, the dependence on the scattering angle can
always be factorized in the operator product expansion in momentum space.

In fact, it is a simple exercise to generalize the computation of conformal
partial waves to correlation functions involving more than four points, at
least as long as one is interested in the comb channel [77, 78, 79, 80, 81].
Similarly, we believe that the inclusion of spinning external operators in the
computation could be potentially simpler in momentum space than it is for
ordinary conformal blocks in position space (see ref. [82], as well as the
connection with the Mellin-space representation in refs. [83, 84]).

Finally, several properties of our conformal partial waves could (and should)
be studied in the future. First and foremost, we have not touched the question
of the convergence of the momentum space OPE. This convergence is actually
guaranteed in a distributional sense, since CFT correlation functions in
Minkowski space are tempered distributions and so are their Fourier
transforms, and the position-space OPE can be shown to converge
distributionally [85]. Nevertheless, the situation can be quite subtle, as
illustrated by the solution to this problem in $d=2$ dimensions [86]:
depending on the kinematics, the OPE might or might not converge in a point-
like manner.

Other interesting properties of the momentum-space conformal partial wave
expansion have to do with the remarkable fact that our results are valid in
any space-time dimension $d\geq 3$. This would allow to study them in the
limit $d\to\infty$ [87, 88], or to examine whether recursion relations between
various space-time dimensions can be found [89, 90, 91]. The analyticity in
$d$ also implies the ability to study momentum-space correlation functions in
non-integer dimensions, where unitarity is known to be broken [92, 93].
Similarly, if one projects the OPE onto ordinary spin partial waves by working
at fixed $m$, then the result is not only analytic in $d$ but also in the spin
$\ell$ of the exchanged operator: in this case one can imagine formulating an
OPE inversion formula in momentum space [94, 95]. It would be interesting in
any case to understand the concept of light-ray operators in the same context
[96, 97, 98]. We leave the study of these interesting properties for future
work.

## Acknowledgments

The author would like to thank the organizers of the workshop “Cosmology Meets
CFT Correlators 2020” at the National Taiwan University for their hospitality,
as well as the workshop’s participants for numerous stimulating discussions.

## References

  * [1] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding scalar operator dimensions in 4D CFT,” JHEP 12 (2008) 031, arXiv:0807.0004 [hep-th].
  * [2] S. Rychkov, EPFL Lectures on Conformal Field Theory in $D\geq 3$ Dimensions.  SpringerBriefs in Physics. 2016.  arXiv:1601.05000 [hep-th].
  * [3] D. Simmons-Duffin, “The Conformal Bootstrap,” in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 1–74.  2017\.  arXiv:1602.07982 [hep-th].
  * [4] D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,” Rev. Mod. Phys. 91 (2019) 015002, arXiv:1805.04405 [hep-th].
  * [5] F. Dolan and H. Osborn, “Conformal four point functions and the operator product expansion,” Nucl. Phys. B 599 (2001) 459–496, arXiv:hep-th/0011040.
  * [6] F. Dolan and H. Osborn, “Conformal partial waves and the operator product expansion,” Nucl. Phys. B 678 (2004) 491–507, arXiv:hep-th/0309180.
  * [7] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, “Solving the 3D Ising Model with the Conformal Bootstrap,” Phys. Rev. D 86 (2012) 025022, arXiv:1203.6064 [hep-th].
  * [8] D. Simmons-Duffin, “Projectors, Shadows, and Conformal Blocks,” JHEP 04 (2014) 146, arXiv:1204.3894 [hep-th].
  * [9] M. Hogervorst and S. Rychkov, “Radial Coordinates for Conformal Blocks,” Phys. Rev. D 87 (2013) 106004, arXiv:1303.1111 [hep-th].
  * [10] F. Kos, D. Poland, and D. Simmons-Duffin, “Bootstrapping the $O(N)$ vector models,” JHEP 06 (2014) 091, arXiv:1307.6856 [hep-th].
  * [11] F. Kos, D. Poland, and D. Simmons-Duffin, “Bootstrapping Mixed Correlators in the 3D Ising Model,” JHEP 11 (2014) 109, arXiv:1406.4858 [hep-th].
  * [12] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning Conformal Blocks,” JHEP 11 (2011) 154, arXiv:1109.6321 [hep-th].
  * [13] J. Penedones, E. Trevisani, and M. Yamazaki, “Recursion Relations for Conformal Blocks,” JHEP 09 (2016) 070, arXiv:1509.00428 [hep-th].
  * [14] M. S. Costa, T. Hansen, J. Penedones, and E. Trevisani, “Radial expansion for spinning conformal blocks,” JHEP 07 (2016) 057, arXiv:1603.05552 [hep-th].
  * [15] L. Iliesiu, F. Kos, D. Poland, S. S. Pufu, D. Simmons-Duffin, and R. Yacoby, “Bootstrapping 3D Fermions,” JHEP 03 (2016) 120, arXiv:1508.00012 [hep-th].
  * [16] L. Iliesiu, F. Kos, D. Poland, S. S. Pufu, D. Simmons-Duffin, and R. Yacoby, “Fermion-Scalar Conformal Blocks,” JHEP 04 (2016) 074, arXiv:1511.01497 [hep-th].
  * [17] A. Castedo Echeverri, E. Elkhidir, D. Karateev, and M. Serone, “Deconstructing Conformal Blocks in 4D CFT,” JHEP 08 (2015) 101, arXiv:1505.03750 [hep-th].
  * [18] A. Castedo Echeverri, E. Elkhidir, D. Karateev, and M. Serone, “Seed Conformal Blocks in 4D CFT,” JHEP 02 (2016) 183, arXiv:1601.05325 [hep-th].
  * [19] H. Isono, “On conformal correlators and blocks with spinors in general dimensions,” Phys. Rev. D 96 no. 6, (2017) 065011, arXiv:1706.02835 [hep-th].
  * [20] P. Kravchuk, “Casimir recursion relations for general conformal blocks,” JHEP 02 (2018) 011, arXiv:1709.05347 [hep-th].
  * [21] G. F. Cuomo, D. Karateev, and P. Kravchuk, “General Bootstrap Equations in 4D CFTs,” JHEP 01 (2018) 130, arXiv:1705.05401 [hep-th].
  * [22] D. Karateev, P. Kravchuk, and D. Simmons-Duffin, “Harmonic Analysis and Mean Field Theory,” JHEP 10 (2019) 217, arXiv:1809.05111 [hep-th].
  * [23] D. Karateev, P. Kravchuk, M. Serone, and A. Vichi, “Fermion Conformal Bootstrap in 4d,” JHEP 06 (2019) 088, arXiv:1902.05969 [hep-th].
  * [24] R. S. Erramilli, L. V. Iliesiu, and P. Kravchuk, “Recursion relation for general 3d blocks,” JHEP 12 (2019) 116, arXiv:1907.11247 [hep-th].
  * [25] V. Schomerus and E. Sobko, “From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models,” JHEP 04 (2018) 052, arXiv:1711.02022 [hep-th].
  * [26] M. Isachenkov and V. Schomerus, “Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory,” JHEP 07 (2018) 180, arXiv:1711.06609 [hep-th].
  * [27] J.-F. Fortin and W. Skiba, “Conformal Bootstrap in Embedding Space,” Phys. Rev. D 93 no. 10, (2016) 105047, arXiv:1602.05794 [hep-th].
  * [28] J.-F. Fortin and W. Skiba, “Conformal Differential Operator in Embedding Space and its Applications,” JHEP 07 (2019) 093, arXiv:1612.08672 [hep-th].
  * [29] J.-F. Fortin, V. Prilepina, and W. Skiba, “Conformal Four-Point Correlation Functions from the Operator Product Expansion,” JHEP 08 (2020) 115, arXiv:1907.10506 [hep-th].
  * [30] G. Mack, “All unitary ray representations of the conformal group SU(2,2) with positive energy,” Commun. Math. Phys. 55 (1977) 1.
  * [31] M. Gillioz, X. Lu, and M. A. Luty, “Scale Anomalies, States, and Rates in Conformal Field Theory,” JHEP 04 (2017) 171, arXiv:1612.07800 [hep-th].
  * [32] M. Gillioz, X. Lu, and M. A. Luty, “Graviton Scattering and a Sum Rule for the c Anomaly in 4D CFT,” JHEP 09 (2018) 025, arXiv:1801.05807 [hep-th].
  * [33] R. Gopakumar, A. Kaviraj, K. Sen, and A. Sinha, “Conformal Bootstrap in Mellin Space,” Phys. Rev. Lett. 118 no. 8, (2017) 081601, arXiv:1609.00572 [hep-th].
  * [34] R. Gopakumar and A. Sinha, “On the Polyakov-Mellin bootstrap,” JHEP 12 (2018) 040, arXiv:1809.10975 [hep-th].
  * [35] J. Penedones, J. A. Silva, and A. Zhiboedov, “Nonperturbative Mellin Amplitudes: Existence, Properties, Applications,” JHEP 08 (2020) 031, arXiv:1912.11100 [hep-th].
  * [36] H. Isono, T. Noumi, and G. Shiu, “Momentum space approach to crossing symmetric CFT correlators,” JHEP 07 (2018) 136, arXiv:1805.11107 [hep-th].
  * [37] H. Isono, T. Noumi, and G. Shiu, “Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension,” JHEP 10 (2019) 183, arXiv:1908.04572 [hep-th].
  * [38] S. Caron-Huot, D. Mazáč, L. Rastelli, and D. Simmons-Duffin, “Dispersive CFT Sum Rules,” arXiv:2008.04931 [hep-th].
  * [39] D. Carmi and S. Caron-Huot, “A Conformal Dispersion Relation: Correlations from Absorption,” JHEP 09 (2020) 009, arXiv:1910.12123 [hep-th].
  * [40] C. Corianò and M. M. Maglio, “On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space,” JHEP 09 (2019) 107, arXiv:1903.05047 [hep-th].
  * [41] A. Bzowski, P. McFadden, and K. Skenderis, “Conformal $n$-point functions in momentum space,” Phys. Rev. Lett. 124 no. 13, (2020) 131602, arXiv:1910.10162 [hep-th].
  * [42] C. Corianò and M. M. Maglio, “The Generalized Hypergeometric Structure of the Ward Identities of CFT’s in Momentum Space in $d>2$,” Axioms 9 no. 2, (2020) 54, arXiv:2001.09622 [hep-th].
  * [43] C. Corianò and M. M. Maglio, “Conformal Field Theory in Momentum Space and Anomaly Actions in Gravity: The Analysis of 3- and 4-Point Functions,” arXiv:2005.06873 [hep-th].
  * [44] A. Bzowski, P. McFadden, and K. Skenderis, “Conformal correlators as simplex integrals in momentum space,” arXiv:2008.07543 [hep-th].
  * [45] L. Corcoran and M. Staudacher, “The Dual Conformal Box Integral in Minkowski Space,” arXiv:2006.11292 [hep-th].
  * [46] L. Corcoran, F. Loebbert, J. Miczajka, and M. Staudacher, “Minkowski Box from Yangian Bootstrap,” arXiv:2012.07852 [hep-th].
  * [47] D. Mazáč, “Analytic bounds and emergence of AdS2 physics from the conformal bootstrap,” JHEP 04 (2017) 146, arXiv:1611.10060 [hep-th].
  * [48] D. Mazáč and M. F. Paulos, “The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices,” JHEP 02 (2019) 162, arXiv:1803.10233 [hep-th].
  * [49] D. Mazáč and M. F. Paulos, “The analytic functional bootstrap. Part II. Natural bases for the crossing equation,” JHEP 02 (2019) 163, arXiv:1811.10646 [hep-th].
  * [50] D. Mazáč, “A Crossing-Symmetric OPE Inversion Formula,” JHEP 06 (2019) 082, arXiv:1812.02254 [hep-th].
  * [51] M. F. Paulos, “Analytic functional bootstrap for CFTs in $d>1$,” JHEP 04 (2020) 093, arXiv:1910.08563 [hep-th].
  * [52] D. Mazáč, L. Rastelli, and X. Zhou, “A Basis of Analytic Functionals for CFTs in General Dimension,” arXiv:1910.12855 [hep-th].
  * [53] N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 [hep-th].
  * [54] H. Lee, D. Baumann, and G. L. Pimentel, “Non-Gaussianity as a Particle Detector,” JHEP 12 (2016) 040, arXiv:1607.03735 [hep-th].
  * [55] N. Arkani-Hamed, D. Baumann, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities,” JHEP 04 (2020) 105, arXiv:1811.00024 [hep-th].
  * [56] C. Sleight, “A Mellin Space Approach to Cosmological Correlators,” JHEP 01 (2020) 090, arXiv:1906.12302 [hep-th].
  * [57] C. Sleight and M. Taronna, “Bootstrapping Inflationary Correlators in Mellin Space,” JHEP 02 (2020) 098, arXiv:1907.01143 [hep-th].
  * [58] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds,” arXiv:1910.14051 [hep-th].
  * [59] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization,” arXiv:2005.04234 [hep-th].
  * [60] H. Isono, H. M. Liu, and T. Noumi, “Wavefunctions in dS/CFT revisited: principal series and double-trace deformations,” arXiv:2011.09479 [hep-th].
  * [61] C. Sleight and M. Taronna, “From AdS to dS Exchanges: Spectral Representation, Mellin Amplitudes and Crossing,” arXiv:2007.09993 [hep-th].
  * [62] M. Gillioz, in preparation.
  * [63] M. Gillioz, “Momentum-space conformal blocks on the light cone,” JHEP 10 (2018) 125, arXiv:1807.07003 [hep-th].
  * [64] M. Gillioz, “Conformal 3-point functions and the Lorentzian OPE in momentum space,” Commun. Math. Phys. 379 no. 1, (2020) 227–259, arXiv:1909.00878 [hep-th].
  * [65] C. Coriano, L. Delle Rose, E. Mottola, and M. Serino, “Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman’s Master Integrals,” JHEP 07 (2013) 011, arXiv:1304.6944 [hep-th].
  * [66] A. Bzowski, P. McFadden, and K. Skenderis, “Implications of conformal invariance in momentum space,” JHEP 03 (2014) 111, arXiv:1304.7760 [hep-th].
  * [67] H. Exton, “On the system of partial differential equations associated with Appell’s function F4,” Phys. A: Math. Gen. 28 (1995) 631.
  * [68] T. Bautista and H. Godazgar, “Lorentzian CFT 3-point functions in momentum space,” JHEP 01 (2020) 142, arXiv:1908.04733 [hep-th].
  * [69] M. Gillioz, M. Meineri, and J. Penedones, “A Scattering Amplitude in Conformal Field Theory,” JHEP 11 (2020) 139, arXiv:2003.07361 [hep-th].
  * [70] A. Bzowski, P. McFadden, and K. Skenderis, “Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies,” JHEP 03 (2016) 066, arXiv:1510.08442 [hep-th].
  * [71] A. Bzowski, P. McFadden, and K. Skenderis, “Renormalised CFT 3-point functions of scalars, currents and stress tensors,” JHEP 11 (2018) 159, arXiv:1805.12100 [hep-th].
  * [72] G. ’t Hooft and M. Veltman, “Scalar One Loop Integrals,” Nucl. Phys. B 153 (1979) 365–401.
  * [73] A. Denner, U. Nierste, and R. Scharf, “A Compact expression for the scalar one loop four point function,” Nucl. Phys. B 367 (1991) 637–656.
  * [74] G. Duplancic and B. Nizic, “IR finite one loop box scalar integral with massless internal lines,” Eur. Phys. J. C 24 (2002) 385–391, arXiv:hep-ph/0201306.
  * [75] D. Meltzer and A. Sivaramakrishnan, “CFT unitarity and the AdS Cutkosky rules,” JHEP 11 (2020) 073, arXiv:2008.11730 [hep-th].
  * [76] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix bootstrap. Part I: QFT in AdS,” JHEP 11 (2017) 133, arXiv:1607.06109 [hep-th].
  * [77] V. Gonçalves, R. Pereira, and X. Zhou, “$20^{\prime}$ Five-Point Function from $AdS_{5}\times S^{5}$ Supergravity,” JHEP 10 (2019) 247, arXiv:1906.05305 [hep-th].
  * [78] S. Parikh, “A multipoint conformal block chain in $d$ dimensions,” JHEP 05 (2020) 120, arXiv:1911.09190 [hep-th].
  * [79] J.-F. Fortin, W. Ma, and W. Skiba, “Higher-Point Conformal Blocks in the Comb Channel,” JHEP 07 (2020) 213, arXiv:1911.11046 [hep-th].
  * [80] S. Hoback and S. Parikh, “Towards Feynman rules for conformal blocks,” arXiv:2006.14736 [hep-th].
  * [81] I. Buric, S. Lacroix, J. A. Mann, L. Quintavalle, and V. Schomerus, “From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks,” arXiv:2009.11882 [hep-th].
  * [82] H. Isono, T. Noumi, and T. Takeuchi, “Momentum space conformal three-point functions of conserved currents and a general spinning operator,” JHEP 05 (2019) 057, arXiv:1903.01110 [hep-th].
  * [83] C. Sleight and M. Taronna, “Spinning Witten Diagrams,” JHEP 06 (2017) 100, arXiv:1702.08619 [hep-th].
  * [84] C. Sleight and M. Taronna, “Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications,” Fortsch. Phys. 66 no. 8-9, (2018) 1800038, arXiv:1804.09334 [hep-th].
  * [85] P. Kravchuk, J. Qiao, and S. Rychkov, “Distributions in CFT. Part I. Cross-ratio space,” JHEP 05 (2020) 137, arXiv:2001.08778 [hep-th].
  * [86] M. Gillioz, X. Lu, M. A. Luty, and G. Mikaberidze, “Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory,” JHEP 03 (2020) 102, arXiv:1912.05550 [hep-th].
  * [87] A. Fitzpatrick, J. Kaplan, and D. Poland, “Conformal Blocks in the Large $D$ Limit,” JHEP 08 (2013) 107, arXiv:1305.0004 [hep-th].
  * [88] A. Gadde and T. Sharma, “Constraining Conformal Theories in Large Dimensions,” arXiv:2002.10147 [hep-th].
  * [89] M. Hogervorst, “Dimensional Reduction for Conformal Blocks,” JHEP 09 (2016) 017, arXiv:1604.08913 [hep-th].
  * [90] A. Kaviraj, S. Rychkov, and E. Trevisani, “Random Field Ising Model and Parisi-Sourlas supersymmetry. Part I. Supersymmetric CFT,” JHEP 04 (2020) 090, arXiv:1912.01617 [hep-th].
  * [91] S. Hoback and S. Parikh, “Dimensional reduction of higher-point conformal blocks,” arXiv:2009.12904 [hep-th].
  * [92] M. Hogervorst, S. Rychkov, and B. C. van Rees, “Unitarity violation at the Wilson-Fisher fixed point in 4-$\epsilon$ dimensions,” Phys. Rev. D 93 no. 12, (2016) 125025, arXiv:1512.00013 [hep-th].
  * [93] S. Giombi, R. Huang, I. R. Klebanov, S. S. Pufu, and G. Tarnopolsky, “The $O(N)$ Model in ${4<d<6}$ : Instantons and complex CFTs,” Phys. Rev. D 101 no. 4, (2020) 045013, arXiv:1910.02462 [hep-th].
  * [94] S. Caron-Huot, “Analyticity in Spin in Conformal Theories,” JHEP 09 (2017) 078, arXiv:1703.00278 [hep-th].
  * [95] D. Simmons-Duffin, D. Stanford, and E. Witten, “A spacetime derivation of the Lorentzian OPE inversion formula,” JHEP 07 (2018) 085, arXiv:1711.03816 [hep-th].
  * [96] P. Kravchuk and D. Simmons-Duffin, “Light-ray operators in conformal field theory,” JHEP 11 (2018) 102, arXiv:1805.00098 [hep-th].
  * [97] M. Kologlu, P. Kravchuk, D. Simmons-Duffin, and A. Zhiboedov, “Shocks, Superconvergence, and a Stringy Equivalence Principle,” JHEP 11 (2020) 096, arXiv:1904.05905 [hep-th].
  * [98] M. Kologlu, P. Kravchuk, D. Simmons-Duffin, and A. Zhiboedov, “The light-ray OPE and conformal colliders,” arXiv:1905.01311 [hep-th].