QMUL-PH-22-37

Celestial Twistor Amplitudes

 Graham R. Browna,b, Joshua Gowdya,b, Bill Spencea

(a) Centre for Theoretical Physics
Department of Physics and Astronomy
Queen Mary University of London
Mile End Road, London E1 4NS, United Kingdom

(b) Kavli Institute for Theoretical Physics
University of California, Santa Barbara
CA 93106, USA

Abstract

We show how to formulate celestial twistor amplitudes in Yang-Mills (YM) and
gravity. This is motivated by a refined holographic correspondence between the
twistor transform and the light transform in the boundary Lorentzian CFT. The
resulting amplitudes are then equivalent to light transformed correlators on
the celestial torus. Using an ambidextrous basis of twistor and dual twistor
variables, we derive formulae for the three and four-point YM and gravity
amplitudes. The four-point amplitudes take a particularly simple form in terms
of elementary functions, with a striking correspondence between the YM and
gravity expressions. We derive celestial twistor BCFW recursion relations and
show how these may be used to generate the four-point YM amplitude,
illuminating the structure it inherits from the three-point amplitude and
paving the way for the calculation of higher multiplicity light transformed
correlators. Throughout our calculations we utilise the unique properties of
the boundary structure of split signature, and in order to properly motivate
and highlight these properties we first develop our methodology in Lorentzian
signature. This also allows us to prove a holographic correspondence between
Fourier transforms in Lorentzian signature and shadow transforms in the
Euclidean boundary CFT.

{graham.brown, j.k.gowdy<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 A Prelude on Quotients and Representations
  3. 3 Lorentzian Signature Spacetime
    1. 3.1 Little Group
    2. 3.2 Banerjee’s Extended Little Group
    3. 3.3 Chiral Mellin Transform
  4. 4 Shadow Transformed Basis
  5. 5 From Fourier to Shadow
  6. 6 Fourier Amplitudes
  7. 7 Split Signature Spacetime
    1. 7.1 Little Group
    2. 7.2 Banerjee’s Extended Little Group
    3. 7.3 Chiral Mellin Transform
  8. 8 Light Transformed Basis
  9. 9 From Twistor to Light
  10. 10 Celestial Twistor Amplitudes
    1. 10.1 Pure Yang-Mills
      1. 10.1.1 Three Points: Mellin Transform of a Sgn Function
      2. 10.1.2 Four Points: Mellin Transform of a Product of Sgn Functions
    2. 10.2 Pure Gravity
      1. 10.2.1 Three Points: Mellin Transform of a Mod Function
      2. 10.2.2 Four Points: Mellin Transform of a Product of Mod Functions
  11. 11 BCFW Recursion
  12. 12 Conclusions
  13. A All But One Leg Celestial Twistor Amplitudes
  14. B From Shadow to Fourier
  15. C Bulk Conformal Generators
  16. D Proofs of $S^{2}=Id$ and $L^{2}=Id$
  17. E All Leg Fourier Gluon Amplitudes in Split Signature
  18. F Complexified Spacetime
  19. G Inverse Chiral Mellin Transform in Homogeneous Coordinates
  20. H Spinor Conventions

##  1 Introduction

Celestial holography seeks to describe the properties of fundamental theories
in holographic terms from a conformal field theory perspective defined on the
celestial sphere of asymptotically flat spacetimes. This utilises insights and
techniques from the well-established AdS/CFT correspondence, as well as from
the infrared structure of gauge and gravity theories in flat space directly.
Major progress has been made in this field in the past few years and there are
now a number of excellent reviews of the subject - see, for example, [1, 2, 3,
4, 5, 6]. An important part of this work is the study of the scattering
amplitudes of particles on the celestial sphere, which can be interpreted as
conformal correlators. Considerable work has been done on this at tree-level
[7, 8, 9, 10, 11, 12, 13, 14, 15], and in particular general formulae for
tree-level MHV and NMHV $n$-point amplitudes have been derived in [16]. At
loop level, results are now being obtained [17, 18, 19, 20], but in general
the celestial amplitudes found so far take rather implicit or relatively
complex forms, notably in comparison with the simplicity of some standard
results for spacetime amplitudes, such as the $n$-point Parke-Taylor formula
for tree-level MHV amplitudes in four-dimensional spacetime. A simple way to
obtain celestial amplitudes is to perform a Mellin transform in the energy of
each external particle, which turns momentum eigenstates into boost
eigenstates [10, 21]. However, massless four-point amplitudes obtained in this
way contain a delta function enforcing momentum conservation from the bulk,
which complicates their interpretation as conformal correlators. Thus, we are
motivated to investigate other spaces and transformations which describe flat
space amplitudes in a conformal way, in particular twistor space.

Witten’s pioneering paper on amplitudes in twistor space [22] described a
geometry underlying the spinor formulations of amplitudes, leading to many new
insights. For $N=4$ super Yang-Mills (YM) amplitudes, for example, the “half-
Fourier” transformation to the twistor space formulation of an amplitude was
developed in detail (see [23] and references therein). Somewhat later, in [24]
ambidextrous twistor amplitudes in $(2,2)$ signature spacetime were shown to
take remarkably simple forms, with this work prompting a variety of new
developments in amplitudes research. Much has happened since - for a recent
brief overview of amplitudes research see [25], and for detailed reviews of
different areas see the SAGEX papers [26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 6, 37, 38, 39, 40]. The ambitwistor string has played a key role in a
number of developments (c.f. [32]). This has recently been applied to derive
operator expressions for a broad class of celestial amplitudes in [15] and
develop celestial OPEs in [41]. The half-Fourier transformation to twistor
space has a closely related ambidextrous light transform analogue; this was
explored in [42], providing insights into the light-transformed basis of
states, as well as a concise integral formulation for $n$-point celestial
amplitudes, localising on positive energy regions of the Grassmannian. All-
order formulae in the MHV sector have been derived using the twistor string
very recently in [43].

Given the insights that twistor theory has provided in amplitudes research,
and recent progress in this field, it is then natural to ask if the study of
celestial twistor amplitudes might provide a route to further progress in the
celestial programme. In this paper we will develop a formalism to derive these
celestial twistor amplitudes, and show how this gives explicit results for
Yang-Mills and gravity.

The outline of the paper is as follows. In the first half of the paper,
Sections 2 to 9, we develop the general formalism required to discuss
celestial amplitudes in Lorentzian or split signature spacetime and the
various bases we have available to describe them. While the second half,
Sections 10 and 11, is devoted to celestial twistor amplitudes and the BCFW
recursion relations they satisfy.

We begin in Section 2 by outlining the structure of quotient spaces and
representations that will arise in subsequent sections. Then in Section 3 we
will review a general framework for defining the asymptotic states used to
build scattering amplitudes in asymptotically flat spacetimes with signature
$(1,3)$, beginning in momentum space before shifting focus onto conformal
primary states which live on the celestial sphere at the null boundary of
spacetime. This includes a discussion of the extended little group and the
chiral Mellin transform introduced in [12]. In Section 4 we will discuss the
the basis of conformal primaries which are constructed via the shadow
transform, and its relation to the Fourier basis. Following from this, in
Section 5 we bring these results together, showing how the Fourier, shadow and
Mellin transforms are related within a commuting diagram, and in Section 6 we
describe the Fourier-transformed amplitudes. Then, in Section 7 we use the
same methodology to explore how these results are modified in $(2,2)$
signature spacetime. Here the structure of the extended little group implies
that celestial states have imaginary helicity and also discrete parity indices
labelling even and odd representations. Once again, we discuss how the chiral
Mellin transform in split signature constructs celestial states. In Section 8
we give the light transformed basis, and finally in Section 9 we show how the
half-Fourier, light and Mellin transforms are related by a commuting diagram.

We then turn in Section 10 to the study and derivation of celestial twistor
amplitudes. With insights from the earlier analysis, we begin with
analytically continued $(2,2)$ signature amplitudes, mapping these to Mellin
space via a chiral Mellin transform. We use this approach to find the three-
point Yang-Mills amplitudes, taking care to identify the even and odd states
under the $\mathbb{Z}_{2}$ parity subgroups of the extended little group. We
discuss the regularisation needed to evaluate the integrals, basing this on
the key requirement that the Mellin transform is invertible on a suitably
defined strip of definition.

Next, we derive four-point amplitudes using the same general approach.
Beginning with the $(+-+-)$ helicity amplitude, we show that there are eight
separate non-vanishing cases, with different parities. The integrals require
careful regularisation but lead to a very simple expression based on four
rational functions of the cross-ratios and weights, from which the different
parity cases can be found via (anti)symmetrisation. The $(++--)$ amplitude is
related to this result by a certain derivative operator, and we see that this
has a simple action, leading to a similarly concise form of the amplitude.

Following this, we apply these methods to derive the three and four-point
gravity amplitudes. The three-point amplitudes are found to be analogues of
the YM cases, differing just by changes from sgn functions to mod functions
and the doubling of the number of poles, as might be expected. The four-point
amplitudes are also found to be strikingly similar to the YM case, exhibiting
the same basic correspondence mentioned at three-points.

Finally, using Mellin transforms we derive a celestial analogue of the BCFW
twistor recursion relations in Section 11 which enables an iterative
construction of celestial twistor amplitudes. Furthermore, we find an
equivalent but formally simpler recursion relation involving sub-amplitudes
with the glued internal legs in the Mellin basis and not light transformed. To
illustrate the application of these results we use them to rederive the four-
point amplitudes presented earlier. This explains the origin of the remaining
key features in the four point amplitude not fixed by conformal covariance, a
conclusion which we expect to apply in more general cases.

After concluding remarks in Section 12, summarising our results and
highlighting further possible extensions of our work, we give appendices
covering: the all but one leg celestial twistor amplitudes required in the
four-point BCFW (Appendix A), details of the map between Fourier to shadow
transforms and back (Appendix B), formulae for the bulk conformal generators
(Appendix C), proofs that the shadow and light transforms are self-inverse
(Appendix D), the result of all-leg shadow transformed gluon amplitudes in
split signature (Appendix E), a discussion of transforms in complexified
spacetime and slices thereof (Appendix F), the inverse chiral Mellin transform
(Appendix G) and our spinor conventions (Appendix H).

##  2 A Prelude on Quotients and Representations

Before diving into the myriad asymptotic states and transforms of the
celestial sphere/torus, we will discuss a few guiding principles of harmonic
analysis and representation theory that appear frequently (see for example
[44, 45] for a more detailed exposition on the ideas explained here).
Repeatedly we will encounter spaces and their quotient spaces, related to each
other by “dividing out” by some symmetry group. Our goal here will be to make
this notion precise and, in particular, to explain how functions living on
spaces and their quotients are related. We shall do this through two simple
examples that will appear later in this work, before stating the general
framework.

For the first example consider the space of real numbers without zero:
$\operatorname{\mathbb{R}}_{*}$. This space has a $\mathbb{Z}_{2}$ symmetry
which maps points $x\mapsto-x$. We can quotient this space using this symmetry
by ‘gluing’ points together with the equivalence relation $x\sim-x$, giving
us: $(\operatorname{\mathbb{R}}_{*}/\sim)\cong\operatorname{\mathbb{R}}_{+}$.
The map that takes us from the original space to the quotient space is the
projection map $p$, which in this case is simply the absolute value
$p(x)=\lvert x\lvert$. Diagrammatically it is common to write these spaces
vertically, with the quotient space below the original one

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The structure we have just described is actually a trivial principal
$\mathbb{Z}_{2}$-bundle. In fact, all the examples of quotient spaces we will
consider are principal $G$-bundles, for some Abelian group $G$, however, we
will try to avoid using such language whenever possible.

Now let us consider a function living on the original space,
$f:\operatorname{\mathbb{R}}_{*}\rightarrow\operatorname{\mathbb{R}}$. A
natural question is how we can encode this function in the quotient space
without losing any information. The answer is that the single function $f$ on
the original space becomes two functions $\tilde{f}_{0}$ and $\tilde{f}_{1}$
on the quotient space, labelled by the trivial and fundamental representations
of $\mathbb{Z}_{2}$. These are of course just the even ($\tilde{f}_{0}$) and
odd ($\tilde{f}_{1}$) parts of the function $f$, but their interpretation in
terms of representations generalises to more complicated examples. These even
and odd pieces are constructed from the original function $f$ in the usual
way, by taking even and odd linear combinations of the function, but let us
write this in a compact and suggestive form

$\tilde{f}_{s}(x)=\frac{1}{2}\sum_{c\in\mathbb{Z}_{2}=\\{+1,-1\\}}c^{s}f(cx)\,,$
(2.2)

for $s=0,1$. In other words, we construct the $\tilde{f}_{s}$ by summing over
the group elements in $\mathbb{Z}_{2}$ while letting these elements act on the
argument of $f$, and include a phase factor depending on the representation.
This is a recipe that we will use for all of the other groups in this work, be
they continuous or discrete. Note here that the functions $\tilde{f}_{s}(x)$
can be considered as functions on either $\operatorname{\mathbb{R}}_{+}$ or
the original space $\operatorname{\mathbb{R}}_{*}$, since their even and odd
properties let us extend their definition from $x>0$ to $x<0$. This is a
freedom we will often use in other examples. In particular, this lets us
recover the original function on $\operatorname{\mathbb{R}}_{*}$ by simply
summing over the representations

$f(x)=\tilde{f}_{0}(x)+\tilde{f}_{1}(x)\,.$ (2.3)

Thus, even though the projection map $p$ is not invertible, we can go between
functions on the full space and the quotient space at will.

As a second example consider the space of real numbers
$\operatorname{\mathbb{R}}$ with the symmetry group of translations $x\mapsto
x+c$, also denoted by $\operatorname{\mathbb{R}}$. If we quotient the space
$\operatorname{\mathbb{R}}$ by gluing points together related under
translations ($x\sim x+c$ for any $c\in\operatorname{\mathbb{R}}$) then the
quotient space is simply a point, which we denote as
$\\{0\\}\cong\operatorname{\mathbb{R}}/\sim$. The projection map is then
trivial: $p(x)=0$ for all $x\in\operatorname{\mathbb{R}}$, and we can write
the spaces diagrammatically as

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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (2.4)

Now again let us consider a function on the original space
$f:\operatorname{\mathbb{R}}\rightarrow\operatorname{\mathbb{R}}$. To describe
this function on the quotient space we follow the same recipe as the previous
example: one function on the space $\operatorname{\mathbb{R}}$ will become a
family of functions on the quotient space $\\{0\\}$, labelled by
representations of the group of translations $\operatorname{\mathbb{R}}$. The
irreducible representations of the translations (plus the trivial
representation) are labelled by a number $k\in\operatorname{\mathbb{R}}$, so
we write the functions in the quotient space as $\tilde{f}(k,x)$. They are
again constructed from the original function by letting the group act on $f$
and summing over the group, with a phase corresponding to the representation

$\tilde{f}(k,x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dc\,e^{ikc}f(x+c)\,.$
(2.5)

As before, the $\tilde{f}(k,x)$ can be thought of as living at the point $x=0$
(or any other point) or as a function on the original space
$\operatorname{\mathbb{R}}$. This is due to the property

$\tilde{f}(k,x+c)=e^{ikc}\tilde{f}(k,x)\,.$ (2.6)

Thus if you are given the function $\tilde{f}(k,0)$, and the property above,
you can extend it to the whole real line. Once again this property lets us
recover the original function by summing over the representations

$\int_{-\infty}^{\infty}dk\,\tilde{f}(k,x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dk\int_{-\infty}^{+\infty}dc\,e^{-ikc}f(x+c)=f(x)\,.$
(2.7)

The integral transforms (2.5) and (2.7) above are, of course, Fourier
transforms. However, normally one would work exclusively with $\tilde{f}(k,0)$
and consider $k$ and $x$ as conjugate variables in separate spaces. There are
benefits to each one of these interpretations and we shall use both in the
rest of the paper.

Let us briefly summarise the general story - for a generic space $P$ and a
symmetry group $G$ that acts on points $x\in P$ as $x.g$, the quotient space
$P/G$ is the set of equivalence classes under the relation $x\sim x.g$ for all
$g\in G$. The projection map $p:P\rightarrow P/G$ then maps points in $P$ to
their equivalence class

$\leavevmode\hbox to57.17pt{\vbox
to49.47pt{\pgfpicture\makeatletter\hbox{\hskip 28.58568pt\lower-24.7361pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 32.21002pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.21005pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 40.58566pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${P/G}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
16.5857pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{11.99998pt}{4.22366pt}\pgfsys@lineto{11.99998pt}{-12.97643pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{11.99998pt}{-13.1764pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.35275pt}{-5.40276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{p}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (2.8)

For the spaces that we will consider in this work, a function $f$ on the
original space $P$ is encoded in the quotient space by a family of functions
$\tilde{f}_{r}$, labelled by the irreducible representations (and trivial
representation) $r$ of $G$. These $\tilde{f}_{r}$ are always constructed using
the same recipe:

$\tilde{f}_{r}(x)=\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{g\in
G}K(r,g)f(x.g)\,,$ (2.9)

where $K(r,g)$ is a function (often simply a phase) of the group element $g$
and the representation $r$. This kernel is “covariant” under the action of $G$
in the sense that it satisfies111Note that, as previously mentioned, every
group $G$ we will consider here is Abelian, so the order of multiplication
does not matter.

$K(r,gh)=K(r,g)K(r,h)\,.$ (2.10)

for all $g,h\in G$. This implies that the $\tilde{f}_{r}(x)$ has the following
homogeneity property under the action of $g\in G$

$\tilde{f}_{r}(x.g)=K(r,g^{-1})\tilde{f}_{r}(x)$ (2.11)

Thus, the $\tilde{f}_{r}$ can be thought of as living either on $P$ or the
quotient space $P/G$ since the above property lets us restrict or extend their
definition. Finally, to recover the original function $f$ we simply sum over
all of the $\tilde{f}_{r}$

$f(x)=\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{r}\tilde{f}_{r}(x)\,.$
(2.12)

In this work the space $P$ will be a space of spinors, either in Lorentzian or
split signature. We shall then quotient this space by the (extended) little
group to obtain the null cone and celestial sphere/torus. We can also quotient
by translations of spinors to obtain twistor space etc. The discussion above
concerned functions on these spaces, but applies equally well to the
asymptotic states and amplitudes that we will be interested in. Thus to find
maps between states in each space we can simply follow the general formulae
(2.9) and (2.12). These maps will be the Mellin, Fourier transforms etc. and
their respective inverses.

In the next section we will begin with an analysis of the Lorentzian story,
discussing split signature in Section 7. One could also begin with general
independent complex spinors and then take various slices; we comment on this
approach in Appendix F.

##  3 Lorentzian Signature Spacetime

In this section, we will review a general framework for defining asymptotic
states in flat spacetimes. We will begin in Lorentzian momentum space
($(+,-,-,-)$ signature) before shifting focus to the celestial sphere at null
infinity. We call the space of states on the celestial sphere Mellin space
since it is obtained by a Mellin transform of momentum eigenstates. These
Mellin space states transform as conformal primaries under
$\operatorname{\text{SL}(2,\mathbb{C})}$, and hence amplitudes built from
these states transform as conformal correlators. There are diverse approaches
taken in defining these conformal primary states; in this section and what
follows, we will find it very useful to follow Banerjee’s extended little
group approach [46].

Here and throughout the rest of the paper we will write everything in terms of
spinor-helicity variables $\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}}$
such that massless momenta is given by
$p_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}$ (we
will often suppress the indices on the spinors in the discussion). We always
regard $\lambda,\tilde{\lambda}$ as homogeneous coordinates for a particular
projective space - be it the null cone of massless momenta, or the celestial
sphere/torus etc. Homogeneous coordinates make our formulae considerably
tidier and make manifest the links between the momentum space and Mellin space
representations. From the perspective of the holographic CFT, the use of
spinor-helicity variables $\lambda,\tilde{\lambda}$ is simply a version of the
embedding space formalism specific to a two-dimensional CFT.

###  3.1 Little Group

The little group plays a central role in defining asymptotic states. We begin
in momentum space where asymptotic particle states are labelled by a complex
spinor $\operatorname{\lambda}$ corresponding to momentum
$p_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}$. We
denote the complex conjugate spinor by $\operatorname{\tilde{\lambda}}$

$\operatorname{\tilde{\lambda}}=\epsilon\lambda^{*}\,,$ (3.1)

where $\epsilon=\pm 1$ for either incoming or outgoing momentum. Thus in
$(1,3)$ signature the space of on-shell spinors is
$(\lambda,\epsilon)\in\operatorname{\mathbb{C}}_{*}^{2}\times\mathbb{Z}_{2}$.

This parameterisation of a null momentum in terms of spinors is not unique,
since the following little group transformations leave the momentum invariant

$\lambda_{\alpha}\mapsto
e^{i\theta}\,\lambda_{\alpha},\quad\tilde{\lambda}_{\dot{\alpha}}\mapsto
e^{-i\theta}\tilde{\lambda}_{\dot{\alpha}}.$ (3.2)

Hence, the little group of massless momentum in (1,3) spacetime is $U(1)$,
corresponding to rotations about the direction of the null momentum222Here we
are ignoring, as usual, the two ”continuous spin” generators of the massless
little group. This is so we recover the usual notion of helicity.. The freedom
to transform spinors according to (3.2) implies the spinor variables act as
homogeneous coordinates for the space of massless momenta, and the projection
is exactly $(\operatorname{\lambda},\operatorname{\tilde{\lambda}})\mapsto
p_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}$. In
other words, we quotient the space of spinors by little group transformations
to obtain the space of massless momenta, given by
$p_{\alpha\dot{\alpha}}\in\operatorname{\mathbb{R}}_{+}\times\mathbb{CP}^{1}\times\mathbb{Z}_{2}$.

Asymptotic particle states are homogeneous functions of the spinor variables
and transform in representations of the little group. For the little group of
null momenta, $U(1)$ representations are labelled by a half-integer helicity
$J\in\mathbb{Z}/2$. Thus, we write asymptotic particle states in Lorentzian
spacetime as

$|\operatorname{\lambda},\operatorname{\tilde{\lambda}};J\rangle\,,$ (3.3)

and under little group transformations we have

$\boxed{|e^{i\theta}\operatorname{\lambda},e^{-i\theta}\operatorname{\tilde{\lambda}};J\rangle=\left(e^{i\theta}\right)^{-2\,J}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};J\rangle\,.}$
(3.4)

The universal cover of the Lorentz group in (1,3) is
$\operatorname{\text{SL}(2,\mathbb{C})}$ (since
$\text{SO}^{+}(1,3)=\operatorname{\text{SL}(2,\mathbb{C})}/\mathbb{Z}_{2}$)
and the transformation of these states under this group is encoded in the
spinors through

$\ket{\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}};J}\rightarrow\ket{M_{\alpha}^{\,\,\beta}\lambda_{\beta},\bar{M}_{\dot{\alpha}}^{\,\,\dot{\beta}}\tilde{\lambda}_{\dot{\beta}};J}\,,$
(3.5)

where $M_{\alpha}^{\,\,\beta}$ is an $\operatorname{\text{SL}(2,\mathbb{C})}$
matrix.

The process we have described above is our first example of the general story
laid out in Section 2. We have a group action on the space of spinors which we
then quotient out. We then define states in the quotient space as functions of
spinors (regarded as projective coordinates on the space) labelled by
representations of the group and transforming homogeneously under the group
action.

###  3.2 Banerjee’s Extended Little Group

The little group transformations described above exactly correspond to
rotations about the direction of the null momentum, and so stabilise
$p_{\alpha\dot{\alpha}}$. On the celestial sphere however, we are only
interested in preserving the direction the momentum is pointing. Thus, one
could consider an extended little group which only preserves the null
direction. This is the approach presented in [46], which extends the little
group to include boosts.

In $(1,3)$ signature these boosts simply scale the momentum by a positive real
number, $b\in\operatorname{\mathbb{R}}_{+}$

$p_{\alpha\dot{\alpha}}\rightarrow b\,p_{\alpha\dot{\alpha}}\,.$ (3.6)

Note that there is no $\operatorname{\text{SL}(2,\mathbb{C})}$ transformation
acting on the spinors which changes the sign of the energy of the momentum,
i.e., there is no boost in $\text{SO}^{+}(1,3)$ which turns an incoming state
into an outgoing one. This is a crucial difference with $(2,2)$ signature
which we will discuss in Section 7.2 333One could, however, still quotient by
$\mathbb{Z}_{2}$ in Lorentzian signature and consider even and odd
combinations of ingoing and outgoing states. Such even combinations were used
in [47].. We can write the new boost scalings (3.6) as a re-scaling of the
spinors

$\begin{split}\operatorname{\lambda}\rightarrow\sqrt{b}\operatorname{\lambda},\quad\operatorname{\tilde{\lambda}}\rightarrow\sqrt{b}\operatorname{\tilde{\lambda}}.\end{split}$
(3.7)

Combining the positive real boost of the spinors with the usual $\text{U}(1)$
little group gives us the extended little group
$\operatorname{\mathbb{C}}_{*}=\text{U}(1)\times\operatorname{\mathbb{R}}_{+}$
in Lorentzian signature, which we write as

$\operatorname{\lambda}\rightarrow
y\operatorname{\lambda},\quad\operatorname{\tilde{\lambda}}\rightarrow\bar{y}\operatorname{\tilde{\lambda}}\,,$
(3.8)

for any $y\in\operatorname{\mathbb{C}}_{*}$. Again following the general
recipe, we then quotient the space of on-shell spinors by the complex re-
scalings (3.8) to define the celestial sphere
$\mathbb{CP}^{1}\coloneqq\operatorname{\mathbb{C}}_{*}^{2}/\operatorname{\mathbb{C}}_{*}$.
Next we build conformal primaries from celestial states in Lorentzian
signature, with discrete helicity $J$ labeling the representation of
$\text{U}(1)$, and continuous conformal dimension $\Delta$ labeling the
representation of $\operatorname{\mathbb{R}}_{+}$

$|\operatorname{\lambda},\operatorname{\tilde{\lambda}};J,\Delta\rangle=|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h}\rangle\,,$
(3.9)

where we have $h:=\frac{1}{2}(\Delta+J)$ and $\bar{h}:=\frac{1}{2}(\Delta-J)$.
These states (3.9) have the following homogeneity property

$\boxed{|y\operatorname{\lambda},\bar{y}\operatorname{\tilde{\lambda}};h,\bar{h}\rangle=y^{-2h}\bar{y}^{-2\bar{h}}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h}\rangle\,,}$
(3.10)

and transform under $\operatorname{\text{SL}(2,\mathbb{C})}$ like conformal
primaries in a two-dimensional _Euclidean_ CFT

$\begin{split}|\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}};h,\bar{h}\rangle&\rightarrow|M_{\alpha}^{\,\,\beta}\lambda_{\beta},\bar{M}_{\dot{\alpha}}^{\dot{\beta}}\tilde{\lambda}_{\dot{\beta}};h,\bar{h}\rangle,\\\
\ket{\begin{pmatrix}z\\\ 1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}\\\
1\end{pmatrix};h,\bar{h}}&\rightarrow(cz+d)^{-2h}(\bar{c}\bar{z}+\bar{d})^{-2\bar{h}}\ket{\begin{pmatrix}z^{\prime}\\\
1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}^{\prime}\\\
1\end{pmatrix};h,\bar{h}}\,,\end{split}$ (3.11)

where the transformed coordinates $z^{\prime},\bar{z}^{\prime}$ are given by a
Möbius transformation. In the above we have defined the affine coordinate
$z=\frac{\operatorname{\lambda}_{1}}{\operatorname{\lambda}_{2}}$ (in the
patch $\operatorname{\lambda}_{2}\neq 0$) and used the homogeneity property
which implies

$\ket{\begin{pmatrix}z\\\ 1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}\\\
1\end{pmatrix};h,\bar{h}}=\operatorname{\lambda}_{2}^{2h}\bar{\operatorname{\lambda}}_{2}^{2\bar{h}}|\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}};h,\bar{h}\rangle.$
(3.12)

It is only when we consider affine coordinates on the celestial sphere that
the dependence on the incoming/outgoing parameter $\epsilon$ from (3.1) is
made manifest. In (3.12) we can only scale by
$\bar{\operatorname{\lambda}}_{2}$ (the complex conjugate of
$\operatorname{\lambda}_{2}$) and so $\epsilon$ appears explicitly. Note here
that (extended-)little group scalings can be applied to every leg of a
celestial amplitude individually. In other words, we can choose to scale one
leg using (3.10) and leave the others alone, in contrast to the
$\operatorname{\text{SL}(2,\mathbb{C})}$ transformations which are applied to
every leg. This clean separation between
$\operatorname{\text{SL}(2,\mathbb{C})}$ and the little group is lost once we
use the use the affine coordinates $(z,\bar{z})$, as can be seen in (3.11).

Our job now is to find an explicit construction to go between the usual
momentum eigenstates and these celestial states. There are, in fact, two
distinct but interrelated bases for celestial conformal primaries in a
Euclidean CFT - the Mellin basis and the shadow basis - and each has its own
integral transform. In the following section we will first study the Mellin
transform and only later will we study the shadow basis. We will again use
homogeneous coordinates to make a number of expressions more compact. In
addition we will find that the chiral Mellin transform discussed in [12] is
often more natural than the usual non-chiral Mellin transform over just the
particle energy. Finally, before moving on let us summarise the structure of
the spaces we have just discussed diagrammatically

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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{/U(1)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{11.99998pt}{-14.83272pt}\pgfsys@lineto{11.99998pt}{-32.0328pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{11.99998pt}{-32.23279pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.35275pt}{-25.38274pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{/\operatorname{\mathbb{R}}_{+}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (3.13)

###  3.3 Chiral Mellin Transform

As explained in Section 3.2, states on the celestial sphere can be expressed
in terms of homogeneous conformal primaries. Constructing these homogeneous
states can be achieved with a Mellin transform and, as discussed in Section 2,
the use of integral transforms to build homogeneous functional representations
of groups is very general. In a recent paper by the current authors and
Brandhuber and Travaglini [12], a “chiral” form of the Mellin transform in
affine coordinates was introduced (we note here that this chiral Mellin
transform has appeared in the literature previously, see for example [45]). In
this section we will recap this formalism, but working in homogeneous
coordinates from the start and then showing how to recover the affine
expressions.

We define (1,3) celestial states in terms of the following (1,3) chiral Mellin
transform

$\boxed{|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h}\rangle:=\frac{1}{2\pi
i}\int_{\mathbb{C}_{*}}\frac{d\bar{u}}{\bar{u}}\wedge\frac{du}{u}\,u^{2h}\bar{u}^{2\bar{h}}|u\operatorname{\lambda},\bar{u}\operatorname{\tilde{\lambda}}\rangle,}$
(3.14)

Note that this transform is clearly in the general form of $\eqref{eq:
repDecombPrelude}$. The celestial state (3.14) is homogeneous under complex
rescalings of the spinors with weights $h,\bar{h}$ and thus, using (3.11), is
also a conformal primary with the same weights.

Since we are interested in building celestial conformal correlators for a
particular theory, be it Yang-Mills or gravity, we will always Mellin
transform a function which is already homogeneous under the little group with
some helicity $l$. Thus the definition above is equivalent to the usual non-
chiral Mellin transform over the particle energy [21], once we integrate out
the little group degree of freedom. This integral over the compact $U(1)$
takes the form of a discrete Fourier transform and we recover a Kronecker
delta

$|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h},l\rangle=\delta_{J,l}\int_{0}^{\infty}\\!d\omega\,\omega^{\Delta-1}|\sqrt{\omega}\operatorname{\lambda},\sqrt{\omega}\operatorname{\tilde{\lambda}};l\rangle,$
(3.15)

which identifies the bulk helicity $l$ with the holographic helicity
$J:=h-\bar{h}$. A consequence of this is that the only freely varying weight
of a celestial state is the conformal dimension $\Delta$444The chiral Mellin
transform may act on a general Lorentz covariant function of a spinor and by
design will build a function which transforms as a conformal primary with
weights $\Delta,J$, where now these are both free parameters.. To make our
notation more compact, we will often drop the explicit label $l$ with the
understanding that the Kronecker delta imposes $h-\bar{h}=l$.

The non-compact integral is a Mellin transform over the
$\operatorname{\mathbb{R}}_{+}$ scale and maps to modes labelled by a
continuous complex weight $\Delta$. The convergence of the Mellin transform
depends on the behaviour of the function
$|\sqrt{\omega}\operatorname{\lambda},\sqrt{\omega}\operatorname{\tilde{\lambda}}\rangle$
for large and small values of $\omega$. In general, a Mellin transform
converges for some ‘strip of definition’ [48]. For the $\omega$ Mellin
transform with $\Delta=a+i\operatorname{\mathbb{R}}$ we then have convergence
for real $a$ in some interval $(a_{1},a_{2})$. Assuming asymptotic behaviour
$\mathcal{O}(\absolutevalue{\lambda_{\alpha}}^{-p})$ for large
$\absolutevalue{\lambda_{\alpha}}\sim\absolutevalue{\tilde{\lambda}_{\dot{\alpha}}}$
and $\mathcal{O}(\absolutevalue{\lambda_{\alpha}}^{-q})$ for small
$\absolutevalue{\lambda_{\alpha}}\sim\absolutevalue{\tilde{\lambda}_{\dot{\alpha}}}$
then the strip of definition $(a_{1},a_{2})$ must lie within the strip
$(q,p)$. Then the Mellin inversion theorem, see [48] for example, guarantees
that we can invert by integrating $\Delta$ along a contour within the strip of
definition $(a_{1},a_{2})$; this ensures that at least $q<a<p$. Mellin
inversion is demonstrated directly in Appendix G for a celestial state in
homogeneous coordinates and is only slightly different from the usual inverse
Mellin transform in affine coordinates.

The generators of the conformal group of the four-dimensional bulk spacetime
are naturally written in terms of spinors
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$, for example
generating the conformal symmetry of Yang-Mills. As such they can also be
defined in Mellin space using
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$ as homogeneous
coordinates. A full list of such generators can be found in Appendix C. While
the conformal generators in affine coordinates are discussed at length in [12]
they take a much simpler form in homogeneous coordinates, as shown below. We
define the generators in Mellin space by simply commuting them with the chiral
Mellin transform. As an example we can consider the spinor derivative
$\partial_{\alpha}$ and act with a chiral Mellin transform which scales
$\operatorname{\lambda}\rightarrow u\operatorname{\lambda}$

$\int_{\mathbb{C}}\frac{d\bar{u}}{\bar{u}}\wedge\frac{du}{u}\,u^{2h}\bar{u}^{2\bar{h}}\frac{1}{u}\partial_{\alpha}\ket{u\operatorname{\lambda},\bar{u}\operatorname{\tilde{\lambda}}}=\partial_{\alpha}e^{-\frac{1}{2}\partial_{h}}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h}}\,.$
(3.16)

Thus the spinor derivative acting in Mellin space is simply
$\partial_{\alpha}e^{-\frac{1}{2}\partial_{h}}$. To compare this with the
affine coordinate version, using (3.12) we have

$\begin{split}\partial_{\alpha}e^{-\frac{1}{2}\partial_{h}}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h}}=&\partial_{\alpha}e^{-\frac{1}{2}\partial_{h}}\operatorname{\lambda}_{2}^{-2h}\bar{\operatorname{\lambda}}_{2}^{-2\bar{h}}\ket{\begin{pmatrix}z\\\
1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}\\\ 1\end{pmatrix};h,\bar{h}}\\\
=&\frac{1}{\lambda_{2}}\begin{pmatrix}\lambda_{2}\frac{\partial}{\partial\lambda_{2}}-z\partial_{z}\\\
-\partial_{z}\end{pmatrix}e^{-\frac{1}{2}\partial_{h}}\lambda_{2}^{-2h}\bar{\operatorname{\lambda}}_{2}^{-2\bar{h}}\ket{\begin{pmatrix}z\\\
1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}\\\ 1\end{pmatrix};h,\bar{h}}\\\
=&\lambda_{2}^{-2h}\bar{\operatorname{\lambda}}_{2}^{-2\bar{h}}\begin{pmatrix}-2h+1-z\partial_{z}\\\
-\partial_{z}\end{pmatrix}e^{-\frac{1}{2}\partial_{h}}\ket{\begin{pmatrix}z\\\
1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}\\\
1\end{pmatrix};h,\bar{h}}\,,\end{split}$ (3.17)

where we changed variables
$\\{\lambda_{1},\lambda_{2}\\}\rightarrow\\{z,\lambda_{2}\\}$ with
$z=\lambda_{1}/\lambda_{2}=-\lambda^{2}/\lambda^{1}$, recovering the
expression for the spinor derivative found in [12].

##  4 Shadow Transformed Basis

We have described celestial states that are built from a Mellin transform, but
this is only one of the bases for conformal primaries. In an Euclidean CFT
(corresponding to a Lorentzian signature bulk spacetime) there is another
basis of conformal primaries which is constructed via the shadow transform see
[49, 50]. On the other hand, we will see, in Section 8, that in a Lorentzian
CFT (split signature bulk) there are also bases built by performing a light
transform on either of the spinors $\operatorname{\lambda}$ or
$\operatorname{\tilde{\lambda}}$.

This network of conformal primary bases, and the integral transforms that
build them, has been explored in $d$-dimensions for Euclidean CFTs in [51] and
for Lorentzian CFTs in [52]. Importantly, the conformal primaries in these
bases transform in _equivalent_ representations of the conformal group,
meaning that there exist intertwining operators which map between them. These
intertwining operators are exactly the shadow and light transforms, and their
existence is controlled by the restricted Weyl group. The restricted Weyl
group is a finite group of reflection transformations of the weights $\Delta$
and $J$ such that the eigenvalues of the Casimir operators of the conformal
group are left invariant, _and_ continuous and discrete weights are not mixed
[52]. The quadratic Casimir operators of the two-dimensional conformal group
of the celestial sphere/torus are given by those of the Lorentz group of the
bulk. These are555See, for example, Appendix F of [53].

$\begin{split}&C_{2}^{+}:=\frac{1}{4}(m_{\alpha\beta}m^{\alpha\beta}+\tilde{m}_{\dot{\alpha}\dot{\beta}}\tilde{m}^{\dot{\alpha}\dot{\beta}})\sim
Tr(M^{2}),\\\
&C_{2}^{-}:=\frac{1}{4}(m_{\alpha\beta}m^{\alpha\beta}-\tilde{m}_{\dot{\alpha}\dot{\beta}}\tilde{m}^{\dot{\alpha}\dot{\beta}})\sim\epsilon_{\mu\nu\rho\sigma}M^{\mu\nu}M^{\rho\sigma},\end{split}$
(4.1)

where we have employed the notation used in [12] for the Lorentz generators
and, at the level of the generators of the Lorentz algebra, there is no
distinction between (1,3) or (2,2) signature. Note that the existence of the
parity-odd quadratic Casimir $C_{2}^{-}$ is a particular aspect of the four-
dimensional Lorentz algebra (and so specific to two-dimensional CFTs).

Using the explicit form of the Lorentz generators in Appendix C we find that
the eigenvalues of the above Casimir operators acting on a conformal primary
are given by

$\begin{split}&c_{2}^{+}=2h(1-h)+2\bar{h}(1-\bar{h})=\Delta(2-\Delta)-J^{2},\\\
&c_{2}^{-}=2h(1-h)-2\bar{h}(1-\bar{h})=2\,J(1-\Delta).\end{split}$ (4.2)

In the case of a Euclidean CFT the spin $J$ is a discrete, half-integer weight
and so cannot mix with the continuous weight $\Delta$; as such we can only map
$\Delta\rightarrow 2-\Delta$ and $J\rightarrow-J$. Thus, the restricted Weyl
group is $\mathbb{Z}_{2}$ [52] and is generated by the shadow transform.

The Lorentzian CFT case, that is split signature bulk, is dealt with in
Section 8 where the new feature is that the spin $J$ is continuous and so the
restricted Weyl group is larger with new elements which mix $\Delta$ and $J$.
This gives rise to the light transform. We will also discuss unifying the
Euclidean CFT and Lorentzian CFT pictures in Appendix F by complexifying
spacetime.

For now we focus on the Euclidean CFT case, where the shadow transform in
affine coordinates is given by [49, 54, 55, 50]

$\displaystyle\begin{split}&S\bigg{\\{}\ket{\begin{pmatrix}z\\\
1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}\\\
1\end{pmatrix};h,\bar{h}}\bigg{\\}}\\\
&\qquad=\frac{i^{2h-2\bar{h}}\Gamma(2-2\bar{h})}{2\pi
i\,\Gamma(2h-1)}\int_{\mathbb{C}\mathbb{P}^{1}}d\bar{z}\wedge
dz\,(w-z)^{2h-2}(\bar{w}-\bar{z})^{2\bar{h}-2}\ket{\begin{pmatrix}z\\\
1\end{pmatrix},\epsilon\begin{pmatrix}\bar{z}\\\
1\end{pmatrix};h,\bar{h}}\,.\end{split}$ (4.3)

where the normalisation of the above is chosen such that $S^{2}=\text{Id}$, as
shown in Appendix D.

The shadow transform (4.3) can be written in homogeneous coordinates, in which
case we integrate over the complex projective spinor $\operatorname{\lambda}$.
We define a shadowed celestial state as666Note that in our spinor bracket
conventions, listed in Appendix H,
$\langle\operatorname{\lambda}d\operatorname{\lambda}\rangle\wedge[\operatorname{\tilde{\lambda}}d\operatorname{\tilde{\lambda}}]=-dz\wedge
d\bar{z}=2i\,d(\text{Re}(z))d(\text{Im}(z))$ which explains the additional
normalisation by a factor $\frac{1}{2i}$ in the definition of the shadow
transform when compared to [50].

$\boxed{\ket{\mu,\tilde{\mu};1-h,1-\bar{h}}=\frac{i^{2h-2\bar{h}}\Gamma(2-2\bar{h})}{2\pi
i\,\Gamma(2h-1)}\int_{\mathbb{CP}^{1}}\langle\operatorname{\lambda}d\operatorname{\lambda}\rangle\wedge[\operatorname{\tilde{\lambda}}d\operatorname{\tilde{\lambda}}]\,\langle\operatorname{\lambda}\mu\rangle^{2h-2}[\tilde{\mu}\operatorname{\tilde{\lambda}}]^{2\bar{h}-2}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h}}.}$
(4.4)

where we have used homogeneous coordinates $\mu,\tilde{\mu}$ to describe the
shadowed celestial conformal primary777We have also departed slightly from the
historical notation conventions appearing in the literature where the
conjugate of $\lambda_{\alpha}$ is labelled with a tilde,
$\tilde{\mu}_{\alpha}$. These previous conventions mean that a tilde always
denotes a helicity $+\frac{1}{2}$ object, with our choice this is not the
case. Instead, a tilde denotes an object with index $\dot{\alpha}$
transforming with a complex conjugate $\operatorname{\text{SL}(2,\mathbb{C})}$
matrix.. It is now trivial to check that the integral over
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$ is projectively well-
defined and the shadowed state has weights $h\rightarrow
1-h,\,\bar{h}\rightarrow 1-\bar{h}$ or $\Delta\rightarrow
2-\Delta,\,J\rightarrow-J$ under rescalings of the $\mu,\tilde{\mu}$ spinors -
thanks to the economy of the homogeneous coordinates this is enough to show
that it transforms as an $\operatorname{\text{SL}(2,\mathbb{C})}$ conformal
primary. Furthermore, the conjugate $\mu,\tilde{\mu}$ spinors take the same
incoming/outgoing prescription as the
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$ and this ensures that
the integration kernel and measure are independent of $\epsilon$.

The shadow transform above can be combined with the chiral Mellin transform to
give a single transform which takes momentum eigenstates and maps them to
shadowed conformal primaries on the celestial sphere

$\begin{split}\ket{\mu,\tilde{\mu};1-h,1-\bar{h}}=&\frac{i^{2h-2\bar{h}}\Gamma(2-2\bar{h})}{2\pi
i\,\Gamma(2h-1)}\int_{\mathbb{CP}^{1}}\langle\operatorname{\lambda}d\operatorname{\lambda}\rangle\wedge[\operatorname{\tilde{\lambda}}d\operatorname{\tilde{\lambda}}]\,\langle\operatorname{\lambda}\mu\rangle^{2h-2}[\tilde{\mu}\operatorname{\tilde{\lambda}}]^{2\bar{h}-2}\\\
&\qquad\qquad\qquad\qquad\times\frac{1}{2\pi
i}\int_{\mathbb{C}_{*}}\frac{d\bar{u}}{\bar{u}}\wedge\frac{du}{u}\,u^{2h}\bar{u}^{2\bar{h}}|u\operatorname{\lambda},\bar{u}\operatorname{\tilde{\lambda}}\rangle\\\
=&\frac{i^{2h-2\bar{h}}\Gamma(2-2\bar{h})}{(2\pi)^{2}\Gamma(2h-1)}\int_{\mathbb{C}_{*}^{2}}d^{2}\operatorname{\tilde{\lambda}}\wedge\,d^{2}\operatorname{\lambda}\,\langle\operatorname{\lambda}\mu\rangle^{2h-2}[\tilde{\mu}\operatorname{\tilde{\lambda}}]^{2\bar{h}-2}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}},\end{split}$
(4.5)

where we have used that the measure
$d^{2}\operatorname{\tilde{\lambda}}\wedge\,d^{2}\operatorname{\lambda}$ is
given by
$-u\,\bar{u}\,\langle\operatorname{\lambda}d\operatorname{\lambda}\rangle\wedge[\operatorname{\tilde{\lambda}}d\operatorname{\tilde{\lambda}}]\wedge
d\bar{u}\wedge du$.

The shadow transformed state is labelled by homogeneous coordinates
$\mu_{\alpha},\tilde{\mu}_{\dot{\alpha}}$ to emphasise that it can also be
regarded as living on an entirely different dual space to the usual momentum
space and its associated celestial sphere. We will explore this point of view
in the next section where we show that the shadow transformed state can be
recovered by performing a chiral Mellin transform on a Fourier transformed
state. This adds a new (1,3) signature version to the existing duality between
light transformed conformal primaries and twistor eigenstates found in [42].

##  5 From Fourier to Shadow

The expression for the shadow transform above (4.5) has many similarities with
a Fourier transform acting on the spinors. In fact we can make this completely
concrete in terms of a commuting diagram of integral transforms, shown in
Figure 1.

Momentum BasisFourier BasisMellin BasisShadow Basis

$\scriptstyle F$

$\scriptstyle C_{h,\bar{h}}$

$\scriptstyle C_{1-h,1-\bar{h}}$

$\scriptstyle S$

Figure 1: We denote the chiral Mellin transform with weights $h,\bar{h}$ by
$C_{h,\bar{h}}$, Fourier transform by $F$ and shadow transform by $S$.

We now prove the closure of this diagram by showing that a Fourier transform
in spinor space followed by a chiral Mellin transform is exactly the shadow
(4.5). The Fourier transform we will consider is the following

$\boxed{|\mu,\tilde{\mu}\rangle_{=}\frac{1}{4\pi^{2}}\int_{\mathbb{C}^{2}}\\!d^{2}\lambda\wedge
d^{2}\tilde{\lambda}\,\,e^{i(\langle\lambda\mu\rangle+[\tilde{\mu}\operatorname{\tilde{\lambda}}])}|\operatorname{\lambda},\operatorname{\tilde{\lambda}}\rangle\,.}$
(5.1)

Here we work in Minkowski signature so the spinors brackets are complex, and
the integration kernel above can be written as
$\exp(2i\real\langle\operatorname{\lambda}\mu\rangle)$888In our conventions,
listed in Appendix H,
$\langle\lambda\mu\rangle^{*}=[\tilde{\mu}\operatorname{\tilde{\lambda}}]$..
As before, the conjugate $\mu,\tilde{\mu}$ spinors take the same
incoming/outgoing prescription as the
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$.

The transform (5.1) differs from the more conventional null Fourier integral

$\int d^{4}p\,\delta^{+}(p^{2})e^{-ip.k}\,.$ (5.2)

However, such a transform is little group invariant and we have seen that the
little group is crucial in controlling the holographic behaviour of scattering
states. Our Fourier transform is the combination of two ‘twistor’ or half-
Fourier transforms - it has the same kernel, $iZ\cdot W$, as the full Fourier
transform from twistor to dual twistor space but is integrated over an
orthogonal region of the phase space
$\operatorname{\lambda},\operatorname{\tilde{\lambda}},\mu,\tilde{\mu}$. We
can show by direct calculation that the Fourier transform (5.1) is self-
inverse $F^{2}=\text{Id}$. This is different from the usual Fourier transform
property of $F^{2}=P$ where $P$ is a parity or ‘time’ reversal operator. In
our case the angle/square bracket products are skew symmetric and this ensures
that $F$ is exactly self-inverse.

A familiar consequence of the little group covariant Fourier transform is that
the spinor $\mu$ now has helicity $+1/2$ and $\tilde{\mu}$ has helicity
$-1/2$, which is the opposite to the spinors $\operatorname{\lambda}$ and
$\operatorname{\tilde{\lambda}}$. This can be seen by commuting the helicity
operator through the Fourier transform

$\displaystyle\operatorname{\mathcal{J}}:=$
$\displaystyle-\frac{1}{2}\lambda^{\alpha}\partial_{\alpha}+\frac{1}{2}\tilde{\lambda}^{\dot{\alpha}}\tilde{\partial}_{\dot{\alpha}}\sim\frac{1}{2}\mu^{\alpha}\frac{\partial}{\partial\mu^{\alpha}}-\frac{1}{2}\tilde{\mu}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\mu}^{\dot{\alpha}}}=:-\operatorname{\mathcal{J}}_{F},$
(5.3)

where $\sim$ here means ‘conjugate to’ under transform (5.1).

The eigenvalue of the operator $\operatorname{\mathcal{J}}$ remains unchanged,
which is guaranteed by simply commuting $\operatorname{\mathcal{J}}$ through
the Fourier transform. However, we have also defined the operator
$\operatorname{\mathcal{J}}_{F}$, which generates a little group
transformation of the $\mu_{\alpha},\tilde{\mu}_{\dot{\alpha}}$ spinors based
on their $\operatorname{\text{SL}(2,\mathbb{C})}$ holomorphicity. That is,

$\operatorname{\mathcal{J}}_{F}=\operatorname{\mathcal{J}}|_{\operatorname{\lambda}\rightarrow\mu,\operatorname{\tilde{\lambda}}\rightarrow\tilde{\mu}}\,,$
(5.4)

and so performs a little group transformation in a way that exactly mimics
that of the spinors $\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}}$

$|\mu,\tilde{\mu}\rangle\rightarrow|e^{i\theta}\mu,e^{-i\theta}\tilde{\mu}\rangle\,.$
(5.5)

Thus if the eigenvalue of $\operatorname{\mathcal{J}}$ is $J$ then the
eigenvalue of $\operatorname{\mathcal{J}}_{F}$ is $-J$. This is reminiscent of
the action of the shadow transform and we can see this even more clearly with
the dilatation operator

$\displaystyle
d:=\frac{1}{2}\lambda^{\alpha}\partial_{\alpha}+\frac{1}{2}\tilde{\lambda}^{\dot{\alpha}}\tilde{\partial}_{\dot{\alpha}}+1\sim-\frac{1}{2}\mu^{\alpha}\frac{\partial}{\partial\mu^{\alpha}}-\frac{1}{2}\tilde{\mu}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\mu}^{\dot{\alpha}}}-1=:-d_{F},$
(5.6)

where again we have defined the operator $d_{F}$ which is defined in Fourier
space and generates a dilatation of the $\mu,\tilde{\mu}$ spinors.

Now we transform to the celestial sphere by performing a chiral Mellin
transform _on a Fourier state_ (5.1) with weights $k,\bar{k}$. Then relations
(5.3) and (5.6) become

$\begin{split}\operatorname{\mathcal{J}}=h-\bar{h}\sim-(k-\bar{k})=-\operatorname{\mathcal{J}}_{F},\\\
\quad d=-h-\bar{h}+1\sim k+\bar{k}-1=-d_{F},\end{split}$ (5.7)

so we must identify $k=1-h$ and $\bar{k}=1-\bar{h}$ which are the same shifts
generated by a shadow transform.

The Mellin transformed Fourier state is given by

$\begin{split}|\mu,\tilde{\mu};k,\bar{k}\rangle=&\frac{1}{2\pi
i}\int_{\mathbb{C}^{*}}\frac{d\bar{t}}{\bar{t}}\wedge\frac{dt}{t}t^{2k}\bar{t}^{2\bar{k}}|t\mu,\bar{t}\tilde{\mu}\rangle\\\
=&\frac{1}{2\pi
i}\int_{\mathbb{C}^{*}}\frac{d\bar{t}}{\bar{t}}\wedge\frac{dt}{t}t^{2k}\bar{t}^{2\bar{k}}\frac{1}{4\pi^{2}}\int_{\mathbb{C}^{2}}\\!d^{2}\operatorname{\lambda}\wedge
d^{2}\tilde{\operatorname{\lambda}}\,e^{it\langle\operatorname{\lambda}\mu\rangle}e^{i\bar{t}[\tilde{\mu}\tilde{\operatorname{\lambda}}]}|\operatorname{\lambda},\tilde{\operatorname{\lambda}}\rangle\,,\end{split}$
(5.8)

which we will now show is a shadowed conformal primary. First we change
variables $\sigma=t\langle\operatorname{\lambda}\mu\rangle$,
$\bar{\sigma}=\bar{t}[\tilde{\mu}\tilde{\operatorname{\lambda}}]$ to obtain

$|\mu,\tilde{\mu};k,\bar{k}\rangle=\frac{1}{2\pi
i}\int_{\mathbb{C}^{*}}\frac{d\bar{\sigma}}{\bar{\sigma}}\wedge\frac{d\sigma}{\sigma}\sigma^{2k}\bar{\sigma}^{2\bar{k}}e^{i(\sigma+\bar{\sigma})}\int_{\mathbb{C}^{2}}\\!d^{2}\operatorname{\lambda}\wedge
d^{2}\tilde{\operatorname{\lambda}}\,\langle\operatorname{\lambda}\mu\rangle^{-2k}[\tilde{\mu}\tilde{\operatorname{\lambda}}]^{-2\bar{k}}|\operatorname{\lambda},\tilde{\operatorname{\lambda}}\rangle\,.$
(5.9)

We can write this in a form similar to the shadow transform if we encode the
expected shifts of the weights as $k=1-h$, $\bar{k}=1-\bar{h}$

$|\mu,\tilde{\mu};1-h,1-\bar{h}\rangle=\frac{\mathcal{I}}{(2\pi
i)(4\pi^{2})}\int_{\mathbb{C}^{2}}\\!d^{2}\operatorname{\lambda}\wedge
d^{2}\tilde{\operatorname{\lambda}}\,\langle\operatorname{\lambda}\mu\rangle^{2h-2}[\tilde{\mu}\tilde{\operatorname{\lambda}}]^{2\bar{h}-2}|\operatorname{\lambda},\tilde{\operatorname{\lambda}}\rangle\,,$
(5.10)

where we have defined

$\mathcal{I}:=\int_{\mathbb{C}^{*}}d\bar{\sigma}\wedge
d\sigma\,\sigma^{1-2h}\bar{\sigma}^{1-2\bar{h}}e^{i(\sigma+\bar{\sigma})}\,.$
(5.11)

The final, although nontrivial, step is to evaluate the integral $\mathcal{I}$
and compare equation (5.10) with the shadow transform (4.5). The evaluation of
$\mathcal{I}$ is contained in Appendix B and we find that

$\mathcal{I}=2\pi i\,i^{-2J}\frac{\Gamma(2-2h)}{\Gamma(2\bar{h}-1)}=2\pi
i\,i^{2J}\frac{\Gamma(2-2\bar{h})}{\Gamma(2h-1)}.$ (5.12)

To compare to the shadow transform we use the second form above. Plugging this
expression for $\mathcal{I}$ back in to (5.10) we find

$\begin{split}|\mu,\tilde{\mu};1-h,1-\bar{h}\rangle=\frac{i^{2h-2\bar{h}}\Gamma(2-2\bar{h})}{4\pi^{2}\Gamma(2h-1)}\int_{\mathbb{C}^{2}}\\!d^{2}\operatorname{\lambda}\wedge
d^{2}\tilde{\operatorname{\lambda}}\,\langle\operatorname{\lambda}\mu\rangle^{2h-2}[\tilde{\mu}\tilde{\operatorname{\lambda}}]^{2\bar{h}-2}|\operatorname{\lambda},\tilde{\operatorname{\lambda}}\rangle\,,\end{split}$
(5.13)

which agrees exactly with the shadow transform (4.5) hence proving that the
diagram of integral transforms commutes. In Appendix B, we also prove this
equivalence in the other direction, by performing an inverse chiral Mellin
transform on a shadowed conformal primary to recover a Fourier transformed
momentum eigenstate.

##  6 Fourier Amplitudes

Having established the commuting diagram in Figure 1, we now have a new method
of computing shadow transformed celestial amplitudes. One can take an
amplitude in momentum space and Fourier transform any number of the external
legs with (5.1). Then, after a chiral Mellin transform, these legs become
shadow transformed operators in a conformal correlator. We can choose how many
legs to shadow transform, for example in the literature single leg shadowed
gluon amplitudes have been considered in [47, 56] while in [57] both single
leg and all-leg shadow transforms were computed for amplitudes involving
massless and massive scalars.

To illustrate the logic above, we examine a simple amplitude: the tree-level
four-point amplitude from massless $\phi^{4}$ theory

$A_{4}^{\phi^{4}}=g\,\delta^{(4)}\bigg{(}\sum_{i=1}^{4}\operatorname{\lambda}_{i\alpha}\operatorname{\tilde{\lambda}}_{i\dot{\alpha}}\bigg{)}\,.$
(6.1)

where $g$ is the dimensionless coupling. This theory has been considered
before in the celestial context at loop level in [58]. If we simply Mellin
transform this amplitude we will run into the usual delta function constraint
$\delta(z-\bar{z})$ of four-point kinematics, where $z$ is the conformal
cross-ratio on the celestial sphere [10]. This constraint demands that the
external legs lie on a great circle, which is formed from the intersection of
the scattering plane and the celestial sphere. To remove this constraint one
might hope that performing various shadow transforms will “smear out” this
singularity. As demonstrated above, we can do this by first performing Fourier
transforms on, say, the first $m$ legs

$\begin{split}A_{4}^{m,\phi^{4}}\coloneqq\frac{1}{(4\pi^{2})^{m}}\int\prod_{i=1}^{m}\left(d^{2}{\operatorname{\lambda}_{i}}\wedge
d^{2}\\!\operatorname{\tilde{\lambda}}_{i}\,e^{i(\langle\lambda\mu\rangle+[\tilde{\mu}\operatorname{\tilde{\lambda}}])}\right)A_{4}^{\phi^{4}}\,,\end{split}$
(6.2)

and then performing Mellin transforms on all legs. We can rewrite the momentum
conserving delta function appearing in the amplitude as an integral

$\delta^{(4)}\bigg{(}\sum_{j=1}^{4}\operatorname{\lambda}_{j}\operatorname{\tilde{\lambda}}_{j}\bigg{)}=\frac{1}{(2\pi)^{4}}\int
d^{4}x\,e^{i\sum_{j=1}^{4}[\operatorname{\tilde{\lambda}}_{j}\lvert
x\lvert\operatorname{\lambda}_{j}\rangle}\,.$ (6.3)

Substituting this into (6.2) and exchanging the order of integration we have

$A_{4}^{m,\phi^{4}}=\frac{g}{(2\pi)^{2m+4}}\int
d^{4}x\int\prod_{i=1}^{m}d^{2}{\operatorname{\lambda}_{i}}\wedge
d^{2}\\!\operatorname{\tilde{\lambda}}_{i}\,\exp(i\sum_{j=1}^{4}[\operatorname{\tilde{\lambda}}_{j}\lvert
x\lvert\operatorname{\lambda}_{j}\rangle+\sum_{i=1}^{m}i(\langle\lambda\mu\rangle+[\tilde{\mu}\operatorname{\tilde{\lambda}}]))\,.$
(6.4)

The $\operatorname{\lambda}_{i}$ integrals are just complex Gaussian integrals
which we can perform by completing the square

$A_{4}^{m,\phi^{4}}=\frac{g}{(2\pi)^{4}}\int
d^{4}x\frac{1}{(x^{2})^{m}}\exp(-i\sum_{i=1}^{m}[\tilde{\mu}_{i}|x^{-1}|\mu_{i}\rangle+i\sum_{j=m+1}^{4}[\operatorname{\tilde{\lambda}}_{j}|x|\operatorname{\lambda}_{j}\rangle)\,.$
(6.5)

Generically performing this final $x$ integral is difficult, however, if we
choose to Fourier transform all legs we find

$A_{4}^{4,\phi^{4}}=\frac{g(-1)^{4}}{(2\pi)^{4}}\int
d^{4}(x^{-1})\,\exp(-i\sum_{i=1}^{n}[\tilde{\mu}_{i}|x^{-1}|\mu_{i}\rangle)=g\,\delta^{(4)}\bigg{(}\sum_{j=1}^{4}\mu_{j}\tilde{\mu}_{j}\bigg{)}.$
(6.6)

The Fourier transformed amplitude we have obtained above is identical in form
to the original amplitude. However, the delta function now imposes “special
conformal conservation” since the generator of special conformal
transformations $k_{\alpha\dot{\alpha}}$ in the Fourier basis acts
multiplicatively. Indeed, the momentum and special conformal generators swap
roles under the action of the Fourier transform

$k_{\alpha\dot{\alpha}}=\frac{\partial}{\partial\operatorname{\lambda}^{\alpha}}\frac{\partial}{\partial\operatorname{\tilde{\lambda}}^{\dot{\alpha}}}\xrightarrow[]{F}-\mu_{\alpha}\tilde{\mu}_{\dot{\alpha}}\,,\quad
p_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}\xrightarrow[]{F}-\frac{\partial}{\partial\mu^{\alpha}}\frac{\partial}{\partial\tilde{\mu}^{\dot{\alpha}}}$
(6.7)

This can be seen by simply pulling the usual momentum space generator through
the Fourier transform, see Appendix C for more details. In fact, we could have
predicted the delta function in (6.6) since tree-level amplitudes in
$\phi^{4}$ theory are conformally invariant. Note that the all-leg Fourier
amplitude (6.6) has mass dimension +4 which is the opposite of the original
amplitude and this trend continues at $n$-points. This flipping of the mass
dimension, which is due to the mass dimension of the measure of (5.1), is
reminiscent of the action of conformal inversion. In Appendix C we argue that
the variable $k_{\alpha\dot{\alpha}}=-\mu_{\alpha}\tilde{\mu}_{\dot{\alpha}}$
is the dual momentum associated to conformally inverted position space, and so
the action of Fourier transforming induces an active conformal inversion
transformation in position space 999That Fourier conjugation is directly
related to bulk conformal inversion is not too surprising, given that
conformal inversion on the boundary variables $z,\bar{z}$ is related to the
non-identity element of the Weyl group, that is the shadow transform [59]..
That an active conformal inversion should be associated with exchanging
$\operatorname{\lambda}$ and $\mu$ spinors was also noted in [23].

If we now perform a Mellin transform on every leg of the Fourier transformed
amplitude (6.6) we will obtain the all-leg shadowed celestial amplitude.
However, this will simply give us the original celestial amplitude but with
$\operatorname{\lambda}\rightarrow\mu$ and
$\operatorname{\tilde{\lambda}}\rightarrow\tilde{\mu}$ and, in particular,
there will still be a delta function of the conformal cross-ratios! Thus
although the shadow transform can ‘smear out’ delta function singularities
this is clearly not always the case - the all-leg shadow can re-introduce
delta functions of its own.

One would now like to study Fourier amplitudes of gravitons and gluons,
however the integrals required become significantly more involved than the
above example, even at low points. Thus, here we will simply make some
comments on Fourier transformed pure gluon amplitudes and leave the further
study of these ideas to future work. As discussed already, and explored more
in Appendix C, under the Fourier transform (5.1) the conformal generators are
related according to an automorphism of the algebra. The generators
$d,J,m,\bar{m}$ are sent to minus their equivalents in $\mu,\tilde{\mu}$ while
the momentum and special conformal transformations are exchanged. In short, a
Fourier transformed gluon amplitude inherits conformal symmetry and is
annihilated by identical differential operators, but now written in terms of
$\mu,\tilde{\mu}$. It should then have significant similarities with the
momentum space amplitude. For example, special conformal symmetry is generated
by the multiplicative operator $\mu_{\alpha}\tilde{\mu}_{\dot{\alpha}}$
(summed over all legs) and hence the gluon amplitude with all legs Fourier
transformed should have a delta function which enforces special conformal
invariance.

The difficulty of actually performing Fourier transforms (5.1) on amplitudes
is in part due to the fact that the spinors
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$ are complex conjugates
in Lorentzian signature. The integrals above become more tractable if instead
we work in split signature spacetime where the spinors
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$ are real and
independent. Indeed, this will allow us to show in Appendix E that the
analogous all-leg Fourier transformed tree-level gluon amplitude in split
signature takes the same form as the original amplitude. This extends the
result shown here in $\phi^{4}$ theory to pure Yang-Mills, but only in this
particular spacetime signature.

Thus, in the following section and for the rest of the paper we will work in
split signature. We begin by first repeating the analysis of the
(extended-)little group for null momentum which is qualitatively different in
(2,2) signature. Our main purpose in using split signature is that this will
allow us to construct “half” versions of the the Fourier and Mellin transforms
which only act on $\operatorname{\lambda}$ or $\operatorname{\tilde{\lambda}}$
and, as was shown in [42, 60], these are related to the “half” version of the
shadow transform: the light transform. Using these new transformations, and
the relations between them, we will see in Section 10 that certain light
transformed celestial amplitudes take a particularly simple form.

##  7 Split Signature Spacetime

We now shift our focus away from Lorentzian spacetime and consider split
signature spacetime instead. As we shall see this has a number of benefits
including allowing connections to twistor space, see for example [22, 23, 61].
Our aim is to study celestial twistor amplitudes in (2,2) signature and link
these to light transformed correlators. Celestial twistor amplitudes are
defined, as one might expect, through a chiral Mellin transform of an
amplitude in twistor space. In order to land on the correct form of the chiral
Mellin transform we follow an analogous path to that of Section 3 \- we start
by examining the little group in (2,2) signature.

###  7.1 Little Group

In split signature spacetime, $(+,-,+,-)$, we require that null momenta
$p_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}$ be
real. This is achieved by making $\operatorname{\lambda}$ and
$\operatorname{\tilde{\lambda}}$ independent and _real_ two component spinors

$\operatorname{\lambda}=\operatorname{\lambda}^{*},\quad\operatorname{\tilde{\lambda}}=\operatorname{\tilde{\lambda}}^{*}\,.$
(7.1)

In $(2,2)$ spacetime the null boundary has only one component, thus asymptotic
states are not labelled by an incoming/outgoing parameter. In fact, whether
momentum is future or past pointing is not a Lorentz invariant notion in
$(2,2)$ since there exist $\text{SO}^{+}(2,2)$ transformations which take
$p_{\alpha\dot{\alpha}}$ to $-p_{\alpha\dot{\alpha}}$, as has been noted
before in [62, 63, 60] for example. The space of spinors in split signature is
thus
$(\lambda,\tilde{\lambda})\in\operatorname{\mathbb{R}}_{*}^{2}\times\operatorname{\mathbb{R}}_{*}^{2}$.

In $(2,2)$ the little group is
$\operatorname{\mathbb{R}}_{*}=\operatorname{\mathbb{R}}_{+}\times\mathbb{Z}_{2}$
and its action on the spinors is given by

$\lambda_{\alpha}\mapsto
c\,\lambda_{\alpha},\quad\tilde{\lambda}_{\dot{\alpha}}\mapsto
c^{-1}\tilde{\lambda}_{\dot{\alpha}},$ (7.2)

for any non-zero real number $c$. In order to land on the space of massless
momenta in $(2,2)$ signature we then quotient by
$\operatorname{\mathbb{R}}_{*}$ to give locally
$p_{\alpha\dot{\alpha}}\in\operatorname{\mathbb{R}}_{*}^{2}\times\mathbb{RP}^{1}$.

The representations of the little group
$\operatorname{\mathbb{R}}_{*}=\operatorname{\mathbb{R}}_{+}\times\mathbb{Z}_{2}$
are labelled by two numbers:

  1. 1.

A continuous imaginary helicity $J\in i\operatorname{\mathbb{R}}$ which labels
a representation $\operatorname{\mathbb{R}}_{+}$,

  2. 2.

A discrete ‘helicity’ $s_{J}\in{0,1}$ labeling the two representations of
$\mathbb{Z}_{2}$.

The asymptotic particle states are then denoted as

$|\operatorname{\lambda},\operatorname{\tilde{\lambda}};J,s_{J}\rangle\,,$
(7.3)

and transform homogeneously under the little group

$\boxed{|c\operatorname{\lambda},c^{-1}\operatorname{\tilde{\lambda}};J,s_{J}\rangle=\absolutevalue{c}^{-2\,J}\text{sgn}(c)^{-s_{J}}|\operatorname{\lambda},\operatorname{\tilde{\lambda}},J,s_{J}\rangle\,.}$
(7.4)

The appearance of absolute values and sgn functions is a completely general
feature of split signature spacetime objects and is a consequence of the real
disconnected little group
$\operatorname{\mathbb{R}}_{*}=\operatorname{\mathbb{R}}_{+}\times\mathbb{Z}_{2}$.

The ‘helicity’ $s_{J}$ denotes states which are even $s_{J}=0$ and odd
$s_{J}=1$ under the little group transformation which flips the sign of both
spinors

$|{-}\operatorname{\lambda},{-}\operatorname{\tilde{\lambda}};J,s_{J}\rangle=({-}1)^{-s_{J}}|\operatorname{\lambda},\operatorname{\tilde{\lambda}},J,s_{J}\rangle\,.$
(7.5)

The fact that $s_{J}$ controls even and odd symmetry in (2,2) suggests we
should associate $s_{J}=0$ with bosons and $s_{J}=1$ with fermions101010In
Appendix F we demonstrate, by considering complexified spacetime, that $s_{J}$
is directly associated to the (1,3) helicity $J_{1,3}$, giving $s_{J}=0$ for
integer $J_{1,3}$ and $s_{J}=1$ for half-integer $J_{1,3}$.. Interestingly,
this “Bose-Fermi symmetry” is completely independent of the continuous
helicity $J$ of the particles. Split signature is generally discussed as an
analytic continuation of theories in $(1,3)$ signature with half-integer
helicity, and we shall follow this approach here as well. Bootstrapping
theories exclusively in $(2,2)$ signature using its unique little group is an
interesting problem in its own right. In the context of CFTs, continuous
helicity has appeared multiple times, see for example [52, 64, 65].

Finally, in $(2,2)$ signature the universal cover of the Lorentz group is
$\operatorname{\text{SL}(2,\mathbb{R})}\times\operatorname{\text{SL}(2,\mathbb{R})}$
and it acts on states through

$\ket{\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}};J,s_{J}}\rightarrow\ket{M_{\alpha}^{\,\,\beta}\lambda_{\beta},\tilde{M}_{\dot{\alpha}}^{\,\,\dot{\beta}}\tilde{\lambda}_{\dot{\beta}};J,s_{J}}\,,$
(7.6)

where $M_{\alpha}^{\,\,\beta}$ and
$\tilde{M}_{\dot{\alpha}}^{\,\,\dot{\beta}}$ are independent
$\operatorname{\text{SL}(2,\mathbb{R})}$ matrices.

###  7.2 Banerjee’s Extended Little Group

The extended little group in (2,2) is given by independent rescalings of
either spinor by a non-zero real number. It is the group
$\operatorname{\mathbb{R}}_{*}\times\operatorname{\mathbb{R}}_{*}=\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
and acts according to the transformation

$\operatorname{\lambda}\rightarrow
y\operatorname{\lambda},\quad\operatorname{\tilde{\lambda}}\rightarrow\bar{y}\operatorname{\tilde{\lambda}}\,,$
(7.7)

where $y$ and $\bar{y}$ are non-zero real numbers. On top of the usual little
group (7.2) this corresponds to an extension which includes boost rescalings
of the momentum by any non-zero real number

$p\rightarrow bp,$ (7.8)

where $b$ may be negative. The existence of Lorentz transformations which flip
the sign of the momenta is a special feature of (2,2) signature. As mentioned
previously, there is a single null asymptotic boundary in split signature and
all scattering states are defined on this space. This leads to the novel
situation of a scattering vector which combines with a vector of asymptotic
particle states to compute the amplitude for a particle ‘process’ to occur in
the bulk [62].

We conclude that in (2,2) signature the spinors
$\operatorname{\lambda},\operatorname{\tilde{\lambda}}$ serve as real
homogeneous coordinates for the projective space
$\mathbb{RP}^{1}\times\mathbb{RP}^{1}=\operatorname{\mathbb{R}}_{*}^{2}\times\operatorname{\mathbb{R}}_{*}^{2}/(\operatorname{\mathbb{R}}_{*}\times\operatorname{\mathbb{R}}_{*})$,
which is the celestial torus. Conformal primaries on the celestial torus are
built from (2,2) signature celestial states with imaginary helicity $J$ and a
discrete helicity $s_{J}$ as well as two new weights:

  1. 1.

a complex conformal dimension $\Delta\in 1+i\operatorname{\mathbb{R}}$
labelling representations of the positive real boosts,

  2. 2.

a discrete weight $s_{h}\in\\{0,1\\}$ which labels the $\mathbb{Z}_{2}$
symmetry under $\operatorname{\lambda}\rightarrow-\operatorname{\lambda}$.
Correspondingly, we have $s_{\bar{h}}\in\\{0,1\\}$ which labels the
$\mathbb{Z}_{2}$ symmetry under
$\operatorname{\tilde{\lambda}}\rightarrow-\operatorname{\tilde{\lambda}}$.

Hence, (2,2) signature celestial states are denoted as

$|\operatorname{\lambda},\operatorname{\tilde{\lambda}};J,s_{J},\Delta,s_{h}\rangle\equiv|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,s_{h},\bar{h},s_{\bar{h}}\rangle\,,$
(7.9)

where we have also defined $h:=\frac{1}{2}(\Delta+J)$111111We use $h,\bar{h}$
for the weights in both (1,3) and (2,2) signature despite them being distinct
quantities. Which of the two is intended should be clear from context. and
$\bar{h}:=\frac{1}{2}(\Delta-J)$. We have taken a slightly altered route in
defining (2,2) signature states in this way.

The transformation law of the states (7.9) under the extended little group re-
scalings is

$\boxed{|y\operatorname{\lambda},\bar{y}\operatorname{\tilde{\lambda}};h,s_{h},\bar{h},s_{\bar{h}}\rangle=\absolutevalue{y}^{-2h}\text{sgn}(y)^{-s_{h}}\absolutevalue{\bar{y}}^{-2\bar{h}}\text{sgn}(\bar{y})^{-s_{\bar{h}}}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,s_{h},\bar{h},s_{\bar{h}}\rangle\,,}$
(7.10)

where once again we have appearance of absolute values and sgn functions which
is due to the unique topological properties of
$\operatorname{\mathbb{R}}^{2,2}$ 121212An excellent explanation of these
properties of split signature is given throughout [61].. Note that only
positive numbers, like $\absolutevalue{y}$, are raised to a complex power,
while $\text{sgn}(y)$ is raised to either the power zero or
one131313Similarly, in (1,3) signature wherever we have a phase it is also
always raised to integer powers $2J$ while the modulus of a complex number is
raised to a complex power $\Delta$.. This has the benefit of removing branch
cut ambiguities arising from raising a negative number to a complex power.

The (2,2) signature celestial states transform under
$\operatorname{\text{SL}(2,\mathbb{R})}\times\operatorname{\text{SL}(2,\mathbb{R})}$
like conformal primaries in a two-dimensional _Lorentzian_ CFT

$\begin{split}&|\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}};h,s_{h},\bar{h},s_{\bar{h}}\rangle\rightarrow|M_{\alpha}^{\beta}\lambda_{\beta},\tilde{M}_{\dot{\alpha}}^{\dot{\beta}}\tilde{\lambda}_{\dot{\beta}};h,s_{h},\bar{h},s_{\bar{h}}\rangle\,\\\
&\ket{\begin{pmatrix}z\\\ 1\end{pmatrix},\begin{pmatrix}\bar{z}\\\
1\end{pmatrix};h,s_{h},\bar{h},s_{\bar{h}}}\\\
&\rightarrow\absolutevalue{cz+d}^{-2h}\text{sgn}(cz+d)^{-s_{h}}\absolutevalue{\tilde{c}\bar{z}+\tilde{d}}^{-2\bar{h}}\text{sgn}(\tilde{c}\bar{z}+\tilde{d})^{s_{\bar{h}}}\ket{\begin{pmatrix}z^{\prime}\\\
1\end{pmatrix},\begin{pmatrix}\bar{z}^{\prime}\\\
1\end{pmatrix};h,s_{h},\bar{h},s_{\bar{h}}}\,.\end{split}$ (7.11)

It is worth pausing here briefly to explain the relationship between the
states described above and others appearing in the celestial literature [42,
60]. By including the the sign transformations
$\operatorname{\lambda}\rightarrow-\operatorname{\lambda}$ and
$\operatorname{\tilde{\lambda}}\rightarrow-\operatorname{\tilde{\lambda}}$ we
are quotienting the space
$\operatorname{\mathbb{R}}_{*}^{2}\times\operatorname{\mathbb{R}}_{*}^{2}$, by
$\operatorname{\mathbb{R}}_{*}\times\operatorname{\mathbb{R}}_{*}$ and not
just $\operatorname{\mathbb{R}}_{+}\times\operatorname{\mathbb{R}}_{+}$. This
means that instead of considering two patches on the celestial torus we
consider only one. To see this consider just one chirality of spinor, say
$\operatorname{\lambda}$. If one uses only positive rescalings in the little
group, then the two affine patches:

$\begin{pmatrix}z\\\ 1\end{pmatrix},\quad\begin{pmatrix}-z\\\
-1\end{pmatrix}\,,$ (7.12)

are distinct, and together cover one circle of the celestial torus. By
including negative rescalings we glue these two patches on top of each other,
see Figure 2. As discussed in Section 2 we do not lose information by doing
this, instead we simply have more asymptotic states labelled by the
representations $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$: $s_{h}$ and
$s_{\bar{h}}$. In the two patch approach states are labelled by the patch they
belong to, but can move from one patch to the other under the action of
$SL(2,\operatorname{\mathbb{R}})\times SL(2,\operatorname{\mathbb{R}})$. In
the one patch approach however, states are labelled by their symmetry
properties. These symmetries can be used to constrain the corresponding
celestial amplitudes as we shall see explicitly in Section 10.

Figure 2: The quotienting of the celestial torus with two patches (first
diagram) by $\operatorname{\lambda}\rightarrow-\operatorname{\lambda}$ and
$\operatorname{\tilde{\lambda}}\rightarrow-\operatorname{\tilde{\lambda}}$
results in a smaller single patch torus (second diagram). Intuitively one
obtains the second diagram by folding the edges in along the dotted lines. The
blue arrows here indicate that the opposite edges are identified.

###  7.3 Chiral Mellin Transform

Based on the split signature homogeneity property (7.10), celestial states are
homogeneous functions of a pair of real spinors with weights
$(h,s_{h},\bar{h},s_{\bar{h}})$, and as usual we build homogeneous functions
via Mellin transforms. A celestial state in (2,2) signature is then defined by
the following chiral Mellin transform acting on a momentum eigenstate

$\displaystyle\boxed{|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,s_{h},\bar{h},s_{\bar{h}}\rangle:=\frac{1}{4}\int_{\mathbb{R}_{*}\times\mathbb{R}_{*}}\\!\frac{d\bar{u}}{\absolutevalue{\bar{u}}}\wedge\frac{du}{\absolutevalue{u}}\,\absolutevalue{u}^{2h}\,\absolutevalue{\bar{u}}^{2\bar{h}}\,\text{sgn}(u)^{s_{h}}\text{sgn}(\bar{u})^{s_{\bar{h}}}|u\operatorname{\lambda},\bar{u}\operatorname{\tilde{\lambda}}\rangle.}$
(7.13)

Once again we note that this transform is in the general form of (2.9), and
again we denote a chiral Mellin with weights $h,\bar{h},s_{h},s_{\bar{h}}$ as
$C_{h,\bar{h}}$. In (7.13) we integrate over $\operatorname{\mathbb{R}}_{*}$
scalings for each spinor, which includes both positive and negative
‘energies’. Since in $(2,2)$ signature the spinors are real and independent
the integrals over $u,\bar{u}$ are separable and each constitutes a Mellin
transform on its own - this was not the case in $(1,3)$ signature. It can be
easily checked (for example by checking the homogeneity or expressing it in
affine coordinates $z,\bar{z}$) that the above state transforms as a conformal
primary on the celestial torus.

Due to the altered (2,2) signature chiral Mellin transform, the four-
dimensional _bulk_ conformal symmetry generators in (2,2) signature receive
slight modifications to take into account the fact that they also carry the
discrete weights $s_{h},s_{\bar{h}}$. These generators are described in
Appendix C.

We can see the even and odd states explicitly by writing our states in terms
of a chiral Mellin transform over just positive energies

$\displaystyle\begin{split}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,s_{h},\bar{h},s_{\bar{h}}\rangle&:=\int_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\\!\frac{d\bar{u}}{\bar{u}}\wedge\frac{du}{u}\,u^{2h}\,\bar{u}^{2\bar{h}}\,|u\operatorname{\lambda},\bar{u}\operatorname{\tilde{\lambda}};s_{h},s_{\bar{h}}\rangle\,,\end{split}$
(7.14)

where we regard the above as performing two of the usual
$\operatorname{\mathbb{R}}^{+}$-Mellin transforms over
$\lambda,\operatorname{\tilde{\lambda}}$ on the even/odd projection states
given below

$\displaystyle\begin{split}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};s_{h},s_{\bar{h}}\rangle&\coloneqq\frac{1}{4}\sum_{\epsilon=\pm
1}\sum_{\bar{\epsilon}=\pm
1}\epsilon^{\,s_{h}}\bar{\epsilon}^{\,\,s_{\bar{h}}}|\epsilon\,\operatorname{\lambda},\bar{\epsilon}\,\operatorname{\tilde{\lambda}}\rangle\,.\end{split}$
(7.15)

Clearly the projection to an even/odd state loses information about the
original state unless we consider all the above combinations in
tandem141414The method of summing over incoming/outgoing parameters for (2,2)
celestial amplitudes has been used in [66, 63], and was even used in (1,3)
signature in [47]. In these only a fully symmetrised sum was considered and so
only a part of the amplitude was studied.. The Mellin transform over
$\operatorname{\mathbb{R}}^{+}$ preserves information about the state since it
is invertible. We can recover the original state from the following inverse
Mellin transform

$\displaystyle\begin{split}|\operatorname{\lambda},\operatorname{\tilde{\lambda}}\rangle=\int_{a-i\infty}^{a+i\infty}\frac{d(2h)}{2\pi
i}\int_{\tilde{a}-i\infty}^{\tilde{a}+i\infty}\frac{d(2\bar{h})}{2\pi
i}\,\,\sum_{s_{h}=0,1}\sum_{s_{\bar{h}}=0,1}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,s_{h},\bar{h},s_{\bar{h}}\rangle\,.\end{split}$
(7.16)

This is again simply a sum over representations and follows the general form
of (2.12). The (2,2) signature chiral Mellin transform (7.13) has inverse
(7.16) since it is just a symmetrised sum of products of two independent
Mellin transforms and the proof is analogous to that contained in Appendix G.

As with (1,3) signature celestial states, in (2,2) signature we will often act
with the chiral Mellin transform on a momentum eigenstate which already
transforms with some helicity under the little group. For a chiral Mellin
transform of a state in (2,2) signature with imaginary bulk helicity $l$ and
discrete helicity $s_{l}$ we can simplify the symmetrisation sum in (7.15)
using a little group transformation

$\displaystyle\begin{split}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};s_{h},s_{\bar{h}},l,s_{l}\rangle&\coloneqq\frac{1}{4}(1+(-1)^{s_{h}+s_{\bar{h}}+s_{l}})\bigg{(}|\operatorname{\lambda},\operatorname{\tilde{\lambda}},l,s_{l}\rangle+(-1)^{s_{h}}\,|{-}\operatorname{\lambda},\operatorname{\tilde{\lambda}},l,s_{l}\rangle\bigg{)}\\\
&=\delta_{s_{J},s_{l}}\frac{1}{2}\sum_{\epsilon=\pm
1}\epsilon^{s_{h}}|\epsilon\,\operatorname{\lambda},\operatorname{\tilde{\lambda}},l,s_{l}\rangle\,.\end{split}$
(7.17)

Furthermore, for such a state, the integral over the little group degrees of
freedom in (7.16) can also be done directly

$\displaystyle|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,s_{h},\bar{h},s_{\bar{h}},l,s_{l}\rangle=\pi
i\,\delta(J-l)\delta_{s_{J},s_{l}}\int_{0}^{\infty}\\!d\omega\,\omega^{\Delta-1}\,\sum_{\epsilon=\pm
1}\epsilon^{s_{h}}|\epsilon\,\sqrt{\omega}\operatorname{\lambda},\sqrt{\omega}\operatorname{\tilde{\lambda}},l,s_{l}\rangle\,.$
(7.18)

We have recovered a Dirac delta from the non-compact integral, which follows
from the important identity

$\displaystyle\int_{0}^{\infty}dx\,x^{\alpha-1}=2\pi i\,\delta(\alpha)\,.$
(7.19)

This holds for $\alpha=i\gamma$ pure imaginary for which we can write
$\delta(\alpha)=-i\delta(\gamma)$. In the above case (7.18) this is true by
default when we consider representations of the little group which have
imaginary helicity151515It is often necessary to relax this condition and
allow $\alpha$ to be any complex number. In such cases we recover a
generalised delta function [67]. This would have been crucial when we (2,2)
Mellin transform analytically extended amplitudes from (1,3) signature which
have half-integer helicity, in fact we sidestep this by performing only the
half-Mellin transforms of [42].. The constant function $f(x)=1$ is Mellin-dual
to a delta function, which is the analogue of the fact that the Fourier-dual
of $f(y)=1$ is also a delta function. Namely, we can show (7.19) by making the
change of variable $x=e^{y}$ and using the representation of the delta
function as

$\displaystyle\delta(\gamma)=\frac{1}{2\pi}\,\int_{-\infty}^{\infty}dy\,e^{i\gamma
y}\,.$ (7.20)

Since $f(x)=1$ is always order $\mathcal{O}(1)$ the ‘strip of definition’
where the Mellin transform is only marginally convergent is just the imaginary
axis. We hence require a regularisation scheme given by

$\displaystyle\frac{1}{2\pi}\,\int_{-\infty}^{\infty}dy\,e^{i\gamma
y}=\frac{1}{2\pi}\,\int_{-\infty}^{\infty}dy\,e^{i\gamma
y-\epsilon\absolutevalue{y}}=\frac{1}{2\pi}\,\bigg{(}\frac{1}{i\gamma+\epsilon}-\frac{1}{i\gamma-\epsilon}\bigg{)}=\delta(\gamma)\,,$
(7.21)

where $\epsilon$ is a small positive number. Regularisation schemes like the
one above will be ubiquitous in the rest of the paper as we will often come
across marginally convergent Mellin transforms. Regularisation schemes are
chosen such that the Mellin transform has an inverse. All schemes used in the
remainder of this paper essentially boil down to the use of the above identity
whose validity is guaranteed by the Fourier inversion theorem.

Given states or amplitudes already transforming under the little group, an
economical alternative to the chiral Mellin transform is the _half-Mellin
transform_ , first introduced in [42], where we only integrate over the scale
of one spinor

$\displaystyle\begin{split}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,s_{h},l,s_{l}\rangle&:=\frac{1}{2}\int_{\operatorname{\mathbb{R}}_{*}}\frac{du}{\absolutevalue{u}}\absolutevalue{u}^{2h}\text{sgn}(u)^{-s_{h}}|u\operatorname{\lambda},\operatorname{\tilde{\lambda}};l,s_{l}\rangle\\\
&=\frac{1}{2}\int_{\operatorname{\mathbb{R}}^{+}}\frac{du}{u}u^{2h}\sum_{\epsilon=\pm
1}\epsilon^{s_{h}}|\epsilon
u\,\operatorname{\lambda},\operatorname{\tilde{\lambda}};l,s_{l}\rangle\,.\end{split}$
(7.22)

Note that the above state is equivalent to (7.18) with the Dirac and Kronecker
deltas stripped off. Indeed one can check that (7.22) is homogeneous under
rescalings of $\operatorname{\tilde{\lambda}}$ with weight $\bar{h}=h-l$ and
$s_{\bar{h}}=s_{h}-s_{l}\mod 2$. Additionally, the half-Mellin transformed
state where we instead integrate over the scale of
$\tilde{\lambda}_{\dot{\alpha}}$ is equal to (7.22). This can be shown by
writing $\bar{u}=u$ in the first line of (7.22) and using a little group
transformation.

As explored at length in Section 10, we can also take amplitudes from (1,3)
signature with half-integer helicity $l$ and analytically continue them into
(2,2) signature - we will always use the half-Mellin transform in such cases
for its practical benefits. This entails using the above formula (7.22) but
with $l$ half-integer, in which case $s_{l}$ satisfies $s_{l}=2l\mod 2$.

The new features of split signature go beyond just the changes to the
(extended) little group; as mentioned at in Section 4 the collection of
different bases for conformal primaries on the celestial sphere expands when
we consider the Lorentzian boundary CFT. In the next section we describe these
new light transformed bases for conformal primaries.

##  8 Light Transformed Basis

We now turn to the question of what other bases of conformal primaries exist
on the celestial torus. Once again we can use the restricted Weyl group to
indicate the existence of intertwining operators which map to new bases.

In (2,2) signature, the quadratic Casimir operators are still given by (4.1)
and since the Lorentz generators take the same form as in (1,3) signature
(c.f. Appendix C), then the Casimir eigenvalues are also given by (4.2), which
we repeat here

$\begin{split}&c_{2}^{+}=2h(1-h)+2\bar{h}(1-\bar{h})=\Delta(2-\Delta)-J^{2},\\\
&c_{2}^{-}=2h(1-h)-2\bar{h}(1-\bar{h})=2\,J(1-\Delta).\end{split}$ (8.1)

In a two-dimensional Lorentzian CFT, the helicity $J$ is continuous and so can
mix with $\Delta$. It is for this reason that there exists additional
intertwining operators that map between representations equivalent to the
Lorentzian principal continuous series. The restricted Weyl group thus expands
to $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ with the new elements corresponding to
the reflections mixing $\Delta$ and $J$

$\begin{split}h\rightarrow 1-h\iff\Delta\rightarrow 1-J,\,\,\,J\rightarrow
1-\Delta,\\\ \bar{h}\rightarrow 1-\bar{h}\iff\Delta\rightarrow
1+J,\,\,\,J\rightarrow\Delta-1,\end{split}$ (8.2)

which are the light and dual light transforms. The shadow transform in a
Lorentzian two-dimensional CFT is simply the product of these commuting
elements, giving the interpretation of the light transforms as the ‘half-
shadow transforms’.

With the extended little group as a guiding principle, we will now define
explicit formulae for the light transforms. Before we begin, note that the
discrete weights $s_{h}$ and $s_{\bar{h}}$ do not appear in the Casimir
eigenvalues, and so representations with differing values of $s_{h}$ and
$s_{\bar{h}}$ could in principle be equivalent. The only possible Weyl
reflection would be to relate the even and odd $\mathbb{Z}_{2}$
representations so that $s_{h}\rightarrow s_{h}+1$, however, it is a trivial
fact that the even and odd representations of $\mathbb{Z}_{2}$ are not
equivalent. Simply put, the lack of an intertwining operator for even/odd
representations corresponds to the fact that we cannot anti-symmetrise a
function which is already even. A consequence of this is that the light
transforms do not change the weights $s_{h}$ and $s_{\bar{h}}$.

The light transform in homogeneous coordinates is given by a projective
integral over the spinor $\operatorname{\lambda}$. We define the light
transformed celestial state as

$\boxed{\begin{split}\ket{\mu,\operatorname{\tilde{\lambda}};1-h,\bar{h},s_{h},s_{\bar{h}}}&=i^{-s_{h}}\frac{\Gamma(2-2h)}{\Gamma(\frac{3}{2}-h+\frac{s_{h}}{2})\Gamma(h-\frac{s_{h}}{2}-\frac{1}{2})}\\\
&\qquad\qquad\qquad\times\int_{\mathbb{RP}^{1}}\langle\operatorname{\lambda}d\operatorname{\lambda}\rangle\,\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2h-2}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{s_{h}}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h},s_{h},s_{\bar{h}}}.\end{split}}$
(8.3)

We will denote this light transform operation symbolically as $L$. It can be
easily checked that the new state is an
$\operatorname{\text{SL}(2,\mathbb{R})}\times\operatorname{\text{SL}(2,\mathbb{R})}$
conformal primary with shifted weights $1-h,\bar{h}$, while the discrete
weights $s_{h},s_{\bar{h}}$ are unchanged as expected. This formula for the
light transform is adapted to the novel (2,2) chiral Mellin transform which
builds states homogeneous under $\operatorname{\mathbb{R}}_{*}$ scalings.
Note, however, that since (8.3) leaves the discrete weights
$s_{h},s_{\bar{h}}$ invariant, its group theoretic behaviour is equivalent to
the light transforms presented elsewhere in the literature for example
[60]161616In [60], a definition of the light transform was given which acts on
states living in two patches of a celestial torus labelled by an incoming and
outgoing parameter. This is not the same as the above definition but contains
completely equivalent information, just not separated into even and odd
parts..

The definition (8.3) features a factor of $i^{-s_{h}}$ despite the fact that
$s_{h}$ is only defined mod 2. Nevertheless, the entire formula is invariant
when we shift $s_{h}\rightarrow s_{h}+2$ thanks to compensation from the gamma
function factors
$[\Gamma(\frac{3}{2}-h+\frac{s_{h}}{2})\Gamma(h-\frac{s_{h}}{2}-\frac{1}{2})]^{-1}$.
In fact, the normalisation in (8.3) is chosen such that $L^{2}=1$, as shown in
Appendix D.

We can recast the light transform in a compact form by combining it with a
half-Mellin transform which, as explained above equation (7.22), acts on a
momentum eigenstate with (2,2) helicity $l$ and $s_{l}$

$\begin{split}\ket{\mu,\operatorname{\tilde{\lambda}};1-h,s_{h},l,s_{l}}&=i^{-s_{h}}\frac{\Gamma(2-2h)}{\Gamma(\frac{3}{2}-h+\frac{s_{h}}{2})\Gamma(h-\frac{s_{h}}{2}-\frac{1}{2})}\int_{\mathbb{RP}^{1}}\langle\operatorname{\lambda}d\operatorname{\lambda}\rangle\,\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2h-2}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{s_{h}}\\\
&\qquad\times\frac{1}{2}\int_{\mathbb{R}_{*}}\\!\frac{du}{\absolutevalue{u}}\,\absolutevalue{u}^{2h}\,\text{sgn}(u)^{s_{h}}|u\operatorname{\lambda},\operatorname{\tilde{\lambda}};l,s_{l}\rangle\\\
=&\frac{i^{-s_{h}}}{2}\frac{\Gamma(2-2h)}{\Gamma(\frac{3}{2}-h+\frac{s_{h}}{2})\Gamma(h-\frac{s_{h}}{2}-\frac{1}{2})}\int_{\mathbb{R}^{2}}d^{2}\operatorname{\lambda}\,\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2h-2}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{s_{h}}|\operatorname{\lambda},\operatorname{\tilde{\lambda}};l,s_{l}\rangle,\end{split}$
(8.4)

where we have used the fact that the measure $d^{2}\\!\operatorname{\lambda}$
breaks up into
$\absolutevalue{u}\,du\,\langle\operatorname{\lambda}d\operatorname{\lambda}\rangle$.
This measure is in fact the only thing responsible for the weight shift
behaviour $h\rightarrow 1-h$ of the light transform, and its independence of
$\text{sgn}(u)$ is the reason why $s_{h}$ is unchanged.

The light transformed state is labelled with a pair spinors
$\mu,\operatorname{\tilde{\lambda}}$ much like a twistor. Indeed under
rescalings of both spinors $\mu,\operatorname{\tilde{\lambda}}$ we have the
weight $-2(1-h+\bar{h})=-2(1-l)$ which is the usual scaling of a twistor. Of
course, unlike a twistor the celestial state (8.3) is homogeneous under
rescalings of each spinor independently. As one might expect, the bulk
conformal symmetry generators in the light transformed basis in Appendix C
also have many similarities with those in twistor variables, for example all
generators are first order differential operators.

The connection between light and twistor transforms was made concrete in [42]
and we will spend some time refining this relation in the following Section 9.

For completeness we record the formula for the dual light transformed state

$\boxed{\begin{split}\ket{\operatorname{\lambda},\tilde{\mu};h,1-\bar{h},s_{h},s_{\bar{h}}}&=i^{-s_{\bar{h}}}\frac{\Gamma(2-2\bar{h})}{\Gamma(\frac{3}{2}-\bar{h}+\frac{s_{\bar{h}}}{2})\Gamma(\bar{h}-\frac{s_{\bar{h}}}{2}-\frac{1}{2})}\\\
&\qquad\qquad\qquad\times\int_{\mathbb{RP}^{1}}[\operatorname{\tilde{\lambda}}d\operatorname{\tilde{\lambda}}]\,\absolutevalue{[\tilde{\mu}\operatorname{\tilde{\lambda}}]}^{2\bar{h}-2}\text{sgn}([\tilde{\mu}\operatorname{\tilde{\lambda}}])^{s_{\bar{h}}}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}};h,\bar{h},s_{h},s_{\bar{h}}}.\end{split}}$
(8.5)

We will also refer to this dual light transform operation as $\bar{L}$.

##  9 From Twistor to Light

The half-Fourier or twistor transform was introduced in the seminal paper [22]
as a means to directly map amplitudes in (2,2) signature (analytically
continued from (1,3) signature) directly to either twistor or dual twistor
space in four dimensions. The transform to twistor space takes the form

$\ket{\mu,\operatorname{\tilde{\lambda}}}=\frac{1}{2\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\,e^{i\langle\operatorname{\lambda}\mu\rangle}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}\,,$
(9.1)

while the dual twistor transform takes the form

$\ket{\operatorname{\lambda},\tilde{\mu}}=\frac{1}{2\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\tilde{\lambda}}\,e^{i[\tilde{\mu}\operatorname{\tilde{\lambda}}]}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}\,.$
(9.2)

We will again define the symbolic shorthand $T$ to denote the half-Fourier
transform (9.1) and $\bar{T}$ to denote its dual (9.2). The above half-Fourier
transforms are both self inverse since, like the Fourier transform (5.1), they
are themselves Fourier transforms with a skew-symmetric kernel.

The relationship between the twistor and the Fourier transform defined in
(5.1) is that in the former we only transform half of the phase space. This is
reminiscent of the relationship between light and shadow transforms.
Furthermore we can see this relationship already at the level of the
dilatation and helicity operators written in twistor variables

$\begin{split}\operatorname{\mathcal{J}}\sim\frac{1}{2}\mu^{\alpha}\frac{\partial}{\partial\mu^{\alpha}}+\frac{1}{2}\tilde{\lambda}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\lambda}^{\dot{\alpha}}}+1=:d_{T},\\\
d\sim-\frac{1}{2}\mu^{\alpha}\frac{\partial}{\partial\mu^{\alpha}}+\frac{1}{2}\tilde{\lambda}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\lambda}^{\dot{\alpha}}}=:\operatorname{\mathcal{J}}_{T},\end{split}$
(9.3)

which, upon Mellin transforming with weights $k,\bar{k}$, give the light
transform relations $k=1-h,\bar{k}=\bar{h}$. We have also defined the distinct
operators $\operatorname{\mathcal{J}}_{T},d_{T}$ generating helicity and
dilatation operations of the variables
$\mu_{\alpha},\operatorname{\tilde{\lambda}}_{\dot{\alpha}}$ based purely on
their transformation properties in either the left or right hand copy of
$\operatorname{\text{SL}(2,\mathbb{R})}\times\operatorname{\text{SL}(2,\mathbb{R})}$.
That is $\operatorname{\mathcal{J}}_{T},d_{T}$ are not just the operators
$\operatorname{\mathcal{J}},d$ in the twistor basis but are distinct and are
related to $\operatorname{\mathcal{J}},d$ by the naive replacement
$\operatorname{\lambda}\rightarrow\mu$.

Similar considerations hold for the dual twistor basis

$\begin{split}\operatorname{\mathcal{J}}\sim-\frac{1}{2}\lambda^{\alpha}\frac{\partial}{\partial\lambda^{\alpha}}-\frac{1}{2}\tilde{\mu}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\mu}^{\dot{\alpha}}}-1=:-d_{\bar{T}},\\\
d\sim\frac{1}{2}\lambda^{\alpha}\frac{\partial}{\partial\lambda^{\alpha}}+\frac{1}{2}\tilde{\mu}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\mu}^{\dot{\alpha}}}=:-\operatorname{\mathcal{J}}_{\bar{T}},\end{split}$
(9.4)

which upon Mellin transforming give the dual light transform relations
$k=h,\bar{k}=1-\bar{h}$. In Appendix C we give all the conformal symmetry
generators $\operatorname{\mathcal{J}},d,p,k,m$ in the twistor and dual
twistor basis. Just as for the original generators and the ‘Fourier’ ones
$d_{F},\operatorname{\mathcal{J}}_{F},\ldots$ etc. we find that the relation
between $T$ and $\bar{T}$ conformal generators is described by the same
automorphism of the conformal algebra, but now generated by the full Fourier
transform which maps from twistor space to dual twistor space.

Momentum BasisTwistor BasisMellin BasisLight Basis

$\scriptstyle T$

$\scriptstyle C_{h,\bar{h}}$

$\scriptstyle C_{1-h,\bar{h}}$

$\scriptstyle L$

Figure 3: We denote the chiral Mellin transform with weights
$h,\bar{h},s_{h},s_{\bar{h}}$ by $C_{h,\bar{h}}$, half-Fourier by $T$ and
light transform by $L$. Momentum BasisDual Twistor BasisMellin BasisDual Light
Basis

$\scriptstyle\bar{T}$

$\scriptstyle C_{h,\bar{h}}$

$\scriptstyle C_{h,1-\bar{h}}$

$\scriptstyle\bar{L}$

Figure 4: We denote the chiral Mellin transform with weights
$h,\bar{h},s_{h},s_{\bar{h}}$ by $C_{h,\bar{h}}$, dual half-Fourier by
$\bar{T}$ and dual light transform by $\bar{L}$.

The precise relationship between twistor and light transforms is via the
commuting diagrams in Figures 3 and 4 which state that the (dual) light
transformed conformal primary is the same as the chiral Mellin transformed
(dual) twistor eigenstate. These diagrams were first established in [42], but
there the diagrams commuted with the proviso that twistor transformed states
corresponded to a linear combination of light transforms with different
‘incoming/outgoing parameters’. One of the goals of this paper has been to
make it clear that we do not need to introduce any incoming or outgoing
parameters in split signature since there is no sense of the past or future.
In Section 8 we made some crucial modifications to the light transforms to
ensure they act properly on states which are homogeneous under
$\operatorname{\mathbb{R}}_{*}$ scalings. These modifications lead to an exact
relation between twistor and light transforms according to Figure 3, which we
now prove. The proof for the dual light and dual twistor case is completely
analogous.

Since the (2,2) chiral Mellin transform is given by two independent half-
Mellin transforms, the transform on the unchanged spinor
$\operatorname{\tilde{\lambda}}$ passes through as usual and so we just focus
on half-Mellin transforming the $\mu$ spinor

$\begin{split}\ket{\mu,\operatorname{\tilde{\lambda}};k,s_{k}}&=\frac{1}{2}\int_{\operatorname{\mathbb{R}}_{*}}\frac{dt}{\absolutevalue{t}}\absolutevalue{t}^{2k}\text{sgn}(t)^{-s_{k}}\ket{t\mu,\operatorname{\tilde{\lambda}}}\\\
&=\frac{1}{2}\int_{\operatorname{\mathbb{R}}_{*}}\frac{dt}{\absolutevalue{t}}\absolutevalue{t}^{2k}\text{sgn}(t)^{-s_{k}}\frac{1}{2\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\,e^{it\langle\operatorname{\lambda}\mu\rangle}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}\\\
&=\frac{1}{4\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\,\mathcal{I}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}.\end{split}$
(9.5)

We now perform the half-Mellin integral

$\mathcal{I}:=\int_{\operatorname{\mathbb{R}}_{*}}\frac{dt}{\absolutevalue{t}}\absolutevalue{t}^{2k}\text{sgn}(t)^{-s_{k}}e^{it\langle\operatorname{\lambda}\mu\rangle}=\int_{0}^{\infty}dt\,t^{2k-1}\left(e^{it\langle\operatorname{\lambda}\mu\rangle}+(-1)^{-s_{k}}e^{-it\langle\operatorname{\lambda}\mu\rangle}\right),$
(9.6)

and changing variables $t\rightarrow
t\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-1}$ we have

$\begin{split}\mathcal{I}=&\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-2k}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{-s_{k}}\int_{0}^{\infty}dt\,t^{2k-1}\left(e^{it}+(-1)^{-s_{k}}e^{-it}\right)\\\
=&\begin{cases}\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-2k}2\int_{0}^{\infty}dt\,t^{2k-1}\\!\cos(t)&\text{for
$s_{k}=0$},\\\
\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-2k}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)\,2i\int_{0}^{\infty}dt\,t^{2k-1}\\!\sin(t)&\text{for
$s_{k}=1$}.\end{cases}\end{split}$ (9.7)

Now as usual we need to regularise the weights, so we shift
$k\in\frac{1}{2}+i\operatorname{\mathbb{R}}$171717Since $\Delta\in
1+i\operatorname{\mathbb{R}}$ and $J$ is pure imaginary the weights
$h,\bar{h}\in\frac{1}{2}+i\operatorname{\mathbb{R}}$ and similarly
$k,\bar{k}\in\frac{1}{2}+i\operatorname{\mathbb{R}}$. by a small negative
parameter $-\delta$ such that $-1<\text{Re}(2(k-\delta)-1)<0$. Then sending
$\delta\rightarrow 0_{+}$ we recover

$\begin{split}\mathcal{I}=&\begin{cases}2\,\Gamma(2k)\,\sin(k\pi+\frac{\pi}{2})\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-2k}&\text{for
$s_{k}=0$},\\\
2i\,\Gamma(2k)\,\sin(k\pi)\,\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-2k}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)&\text{for
$s_{k}=1$},\end{cases}\\\
=&2\,i^{s_{k}}\Gamma(2k)\sin\bigg(\bigg{(}k-\frac{s_{k}}{2}+\frac{1}{2}\bigg{)}\pi\bigg{missing})\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-2k}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{-s_{k}}.\end{split}$
(9.8)

Now we use the Euler reflection formula for gamma functions to yield

$\begin{split}\mathcal{I}=&2\pi\,i^{s_{k}}\frac{\Gamma(2k)}{\Gamma(k-\frac{s_{k}}{2}+\frac{1}{2})\Gamma(\frac{1}{2}-k+\frac{s_{k}}{2})}\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{-2k}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{-s_{k}}.\end{split}$
(9.9)

Finally we set $k=1-h$ and $s_{k}=-s_{h}$ (equivalently $s_{k}=s_{h}$) and
conclude that from a chiral Mellin transform of a Fourier conjugate state we
recover the light transformed state (8.4)

$\ket{\mu,\operatorname{\tilde{\lambda}};1-h,s_{h}}=\frac{i^{-s_{h}}}{2}\frac{\Gamma(2-2h)}{\Gamma(\frac{3}{2}-h+\frac{s_{h}}{2})\Gamma(h-\frac{s_{h}}{2}-\frac{1}{2})}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\,\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2h-2}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{s_{h}}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}.$
(9.10)

Alternatively, we can check the commuting diagram in Figure 3 by inverse
Mellin transforming (8.4) to recover a half-Fourier transform. That is we
compute,

$\begin{split}\int_{a-i\infty}^{a+i\infty}\frac{dh}{2\pi
i}\sum_{s_{h}=0,1}\frac{i^{-s_{h}}}{2}\frac{\Gamma(2-2h)}{\Gamma(\frac{3}{2}-h+\frac{s_{h}}{2})\Gamma(h-\frac{s_{h}}{2}-\frac{1}{2})}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\,\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2h-2}\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)^{s_{h}}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}},\end{split}$
(9.11)

which gives the two terms

$\begin{split}=\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\int_{a-i\infty}^{a+i\infty}\frac{dh}{2\pi
i}\frac{1}{2\pi}\Gamma(2-2h)\bigg{(}\sin(h\pi-\frac{\pi}{2})+i\sin(h\pi)\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)\bigg{)}\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2h-2}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}},\end{split}$
(9.12)

which we evaluate the same way by chosing $a<1$ and closing the contour to
pick up the poles of the gamma function

$\begin{split}=&\frac{1}{2\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\bigg{(}\sin(\pi\frac{n+1}{2})+i\sin(\pi\frac{n+2}{2})\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)\bigg{)}\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{n}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}\\\
&=\frac{1}{2\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\bigg{(}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(2m)!}\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2m}+i\,\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(2m+1)!}\absolutevalue{\langle\operatorname{\lambda}\mu\rangle}^{2m+1}\bigg{)}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}\\\
&=\frac{1}{2\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}\bigg{(}\cos(\absolutevalue{\langle\operatorname{\lambda}\mu\rangle})+i\,\text{sgn}(\langle\operatorname{\lambda}\mu\rangle)\sin(\absolutevalue{\langle\operatorname{\lambda}\mu\rangle})\bigg{)}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}\\\
&=\frac{1}{2\pi}\int_{\operatorname{\mathbb{R}}^{2}}d^{2}\operatorname{\lambda}e^{i\langle\operatorname{\lambda}\mu\rangle}\ket{\operatorname{\lambda},\operatorname{\tilde{\lambda}}}.\end{split}$
(9.13)

Hence we have recovered a half-Fourier transform.

##  10 Celestial Twistor Amplitudes

In this section we will show how to compute celestial twistor amplitudes in
Yang-Mills and gravity, building on earlier work and the results and formalism
developed above. In what follows, we espouse the viewpoint that the most
direct way to produce light transformed conformal correlators is to perform a
Mellin transform of a twistor amplitude and so traverse the commuting diagrams
in Figures 3 and 4 clockwise. Light transformed correlators have the nice
property that they no longer have singular support coming from the momentum
conserving delta function in the bulk \- see [42, 66, 63, 68, 69] for
discussions. These recent works have used the methodology of traversing the
commuting diagrams counterclockwise, that is Mellin transforming first in
momentum space and then light transforming the resultant celestial amplitude.
The recent paper [63] in particular derived the correlators of two holomorphic
and two anti-holomorphic light-ray operators, from light transforms of the
four gluon celestial amplitude in $(2,2)$ signature spacetime. The results
were written in terms of integrals of products of Gaussian hypergeometric
functions, which when evaluated led to certain special functions which arise
in Mellin-Barnes integrals. In our derivation of four-point celestial twistor
amplitudes below we find somewhat analogous, albeit simpler, interim
integrals.

So far we have only studied asymptotic scattering states and classified the
different bases one can take for these. Since an amplitude is a multi-particle
object one can ask the important question: what is the most useful
prescription for assigning bases to represent each particle leg? In principle,
we can use any of the momentum space, twistor, dual twistor or Fourier bases
for each leg and we can choose this for each leg independently. In the work of
[61] the choice made was to uniformly associate to each leg a dual twistor.
This led to many compact expressions for twistor amplitudes, including
$n$-particle MHV, $\overline{\text{MHV}}$ and up to eight particle
$\text{N}^{2}\text{MHV}$ amplitudes in $N=4$ SYM, as well as MHV,
$\overline{\text{MHV}}$ expressions in $N=8$ supergravity. The authors also
developed a BCFW recursion relation in dual twistor space which resulted in a
simple BCFW shift of the dual twistor. We leave the task of Mellin
transforming amplitudes from dual twistor space to future work and instead we
focus on twistor amplitudes in the ambidextrous bases of [24].

As demonstrated in [24], performing ambidextrous transforms to twistor space
leads to remarkably simple objects - for example, for pure Yang-Mills the
three and four-point ambidextrous twistor amplitudes take the constant value
equal to one or minus one depending on the particular external kinematics
probed. Note that ‘ambidextrously’ here means that we can transform either
$\lambda$ or $\tilde{\lambda}$ for each leg independently and in general the
basis chosen will not be uniform. At low particle multiplicity we will use a
twistor for a plus helicity leg and a dual twistor for a minus helicity leg.
This choice leads to particularly simple integrals which decouple into
products of sgn functions.

In the end we are interested in physical $(1,3)$ signature scattering
amplitudes. The motivation to consider $(2,2)$ signature comes from the fact
that we can analytically continue expressions for scattering amplitudes in
$(1,3)$ signature into $(2,2)$ signature and consider new representations of
those amplitudes in terms of twistors. This is the technique used in [61] and
[24] to build seed amplitudes to be used in twistor BCFW to build higher point
amplitudes and we also follow this methodology. Such analytically continued
$(2,2)$ signature amplitudes do not transform in the expected representations
of the $(2,2)$ little group. Rather, since they originated in $(1,3)$
signature, they are homogeneous under a little group transformation with half-
integer helicity.

Assuming we start with analytically continued (2,2) signature amplitudes with
half-integer helicity (denoted simply as $l$ since $s_{l}=2l\mod 2$), we can
map to Mellin space in the most practical manner by performing just a half-
Mellin transform over the scale of either of the spinors. This method was
introduced used in [42] and is particularly useful when applied to twistor