aainstitutetext: Department of Theoretical Physics, Tata Institute of
Fundamental Research, Homi Bhabha Rd, Mumbai 400005, Indiabbinstitutetext:
Department of Physics, Jai Hind College (Autonomous), A Rd, Churchgate, Mumbai
400020, India

# Bounds on Regge growth of flat space scattering from bounds on chaos

Deeksha<EMAIL_ADDRESS>a,2 Subham Dutta
<EMAIL_ADDRESS>a,3 Suman
<EMAIL_ADDRESS>a,4 Shiraz Minwalla
<EMAIL_ADDRESS>
###### Abstract

We study four-point functions of scalars, conserved currents, and stress
tensors in a conformal field theory, generated by a local contact term in the
bulk dual description, in two different causal configurations. The first of
these is the standard Regge configuration in which the chaos bound applies.
The second is the ‘causally scattering configuration’ in which the correlator
develops a bulk point singularity. We find an expression for the coefficient
of the bulk point singularity in terms of the bulk S matrix of the bulk dual
metric, gauge fields and scalars, and use it to determine the Regge scaling of
the correlator on the causally scattering sheet in terms of the Regge growth
of this S matrix. We then demonstrate that the Regge scaling on this sheet is
governed by the same power as in the standard Regge configuration, and so is
constrained by the chaos bound, which turns out to be violated unless the bulk
flat space S matrix grows no faster than $s^{2}$ in the Regge limit. It
follows that in the context of the AdS/CFT correspondence, the chaos bound
applied to the boundary field theory implies that the S matrices of the dual
bulk scalars, gauge fields, and gravitons obey the Classical Regge Growth
(CRG) conjecture.

††preprint: TIFR/TH/21-2

##  1 Introduction

It has recently been conjectured that the tree level S matrix of any
‘consistent’ 555That is a causal classical theory whose energy is bounded from
below (and perhaps is also required to obey other constraints of a similar
general nature). See Chowdhury:2019kaq ; Chakraborty:2020rxf  for a discussion
and Chowdhury:2020ddc  for a generalization to gluons. classical theory never
grows faster with $s$ than $s^{2}$ in the Regge limit. Arguments supporting
this so-called Classical Regge Growth (CRG) conjecture were advanced in the
CEMZ paper Camanho:2014apa , the chaos bound paper Maldacena:2015waa  and also
in discussions of the inversion formula Caron-Huot:2017vep  in large $N$
theories. (see e.g. Turiaci:2018dht ).

The authors of Chowdhury:2019kaq  demonstrated that the classical Einstein S
matrix is the only CRG consistent local tree-level 666That is an S matrix that
is the sum of a polynomial in momenta plus a finite number of physical
particle exchange poles. gravitational S matrix in $D\leq 6$. This striking
result suggests that if the CRG conjecture indeed holds then the set of
consistent classical gravitational S matrices is highly constrained by simple
general considerations 777See in particular the introduction of
Chowdhury:2019kaq , and in particular Conjectures 1-3 presented in that
introduction, for a detailed discussion., and motivates a more detailed
investigation of the CRG conjecture.

We are aware of three arguments in support of the CRG conjecture. The first
(and weakest) of these arguments is the simple observation that the CRG
conjecture is indeed obeyed in every classical theory that is known for
certain to be consistent. For instance, all two derivative classical theories
always obey the CRG conjecture Chowdhury:2019kaq . Moreover, the classical
Einstein S matrix saturates CRG growth 888Demonstrating the impossibility of a
tighter than $s^{2}$ bound on Regge growth. and the Type II and Heterotic
analogues of the Virasoro Shapiro amplitude of string theory temper this
growth to a power that is strictly less than two at every physical finite
value of $\alpha^{\prime}t$ (see Chowdhury:2019kaq  for some more details).

A second reason to believe the CRG conjecture is contained in the analysis of
Camanho:2014apa . The authors of Camanho:2014apa  used a ‘signal model’ (see
Appendix D of that paper) to argue that the the function

${\cal T}(\delta,s)=1+i\frac{S(\delta,s)}{s}$ (1)

should obey the inequality

$|{\cal T}(\delta,s)|\leq 1$ (2)

everywhere in the upper half complex $s$ plane for every nonzero physical
value of $\delta$. Here $S(\delta,s)$ is the transition amplitude - atleast
roughly speaking- for a particle passing through a shock at impact parameter
$\delta$. Roughly, $S(\delta,s)$ can be thought of as the invariant amplitude
for the scattering in impact parameter space (see Kologlu:2019bco  for related
discussions). In the classical limit the tree amplitude is parametrically
small 999This is why loops are sub dominant compared to trees. and (2) reduces
to the condition

${\rm Im}\left(\frac{S(\delta,s)}{s}\right)\geq 0$ (3)

If the large $|s|$ fixed $\delta$ expression for the classical S matrix is

$S(\delta,s)=B(\delta)s^{A(\delta)}$

and $s=|s|e^{i\phi}$ then (3) implies

$T(\delta)\sin\left((A(\delta)-1)\phi\right)\geq
0,~{}~{}~{}~{}~{}0\leq\phi\leq\pi$ (4)

When $A(\delta)>2$ the LHS of (4) switches sign as $\phi$ varies over the
range $[0,\pi]$ so (4) cannot be obeyed for all values of $\phi$ in this
range. 101010When $|A(\delta)|\leq 2$, the $\sin$ function that appears in (4)
is always positive, and (4) is obeyed when $B(\delta)>0$.. It follows that the
condition (2) can only be satisfied if $A(\delta)\leq 2$, i.e if the CRG
conjecture is obeyed. As the passage from the impact space S matrix
$S(\delta,s)$ to the usual momentum space S matrix $S(t,s)$ is, roughly
speaking a Fourier Transform in $t$ at fixed $s$, this argument suggests that
$S(t,s)$ obeys the CRG conjecture. The weakness of this beautiful argument
lies in the fact it contains elements that are intuitive rather than
completely precise. For instance the transition from ‘impact parameter space’
to ‘momentum space’ may have subtleties related to $\delta$ function terms in
impact parameter space 111111We thank S. Caron Huot for very useful
discussions on this point.

The third (and possibly the strongest) reason to believe in the correctness of
the CRG conjecture is its connection to the chaos bound in holographic
theories. A relationship between the chaos bound and the CRG conjecture was
already suggested in the original paper Maldacena:2015waa  (see Appendix A for
a discussion and of the relevant comments in that paper) and has been somewhat
elaborated upon since (see e.g. Turiaci:2018dht ). The key strength of the
chaos-bound based argument for the CRG conjecture is that the starting point
for this argument is a theorem, at least by physicists’ standards. One
weakness of this argument lies in the fact that the connection between the
chaos bound and the CRG conjecture has never (to our knowledge) been carefully
argued through 121212See however the papers Costa:2017twz ; Afkhami-
Jeddi:2016ntf ; Afkhami-Jeddi:2017rmx ; Afkhami-Jeddi:2018apj ; Afkhami-
Jeddi:2018own ; Kologlu:2019bco ; Kulaxizi:2018dxo ; Kulaxizi:2017ixa ;
Meltzer:2017rtf  which put use the chaos bound (both the sign constraint and
the growth constraint part of this bound) or closely related causality
constraints Hofman:2008ar ; Hartman:2015lfa ; Hartman:2016dxc ; Hofman:2016awc
; Komargodski:2016gci ; Hartman:2016lgu ; Chowdhury:2017vel ; Cordova:2017zej
; Cordova:2017dhq ; Meltzer:2018tnm ; Cordova:2017dhq ; Chowdhury:2018uyv ;
Kologlu:2019mfz ; Belin:2019mnx  to obtain many interesting results that are
in the same broad universality class as this paper. In particular the analysis
of Afkhami-Jeddi:2018apj ; Perlmutter:2016pkf  use such arguments to
demonstrate the unphysicality of dual bulk theories with a finite number of
higher spin fields. Analogous causality bounds for CFTs (not necessarily large
$N$) have been studied in Kundu:2020gkz ; Hartman:2016lgu ; Afkhami-
Jeddi:2018own . (see Appendix A for some discussion). 131313Another weakness
is the fact that it uses an elaborate theoretical framework - namely that of
holography and consistency of the quantum structure of the boundary field
theory - to establish a fact about scattering in the simpler theoretical
structure of classical field theories. The goal of this paper is to fill in
this gap for the case that bulk particles are either scalars, gauge fields, or
the metric (i.e. all particles relevant to the analysis of Chowdhury:2019kaq
).

In particular, in this paper we demonstrate that the four-point function
generated holographically by a local bulk contact term for scalars, gauge
fields, or the metric necessarily violates the chaos bound whenever the flat
space S matrix generated by the same contact term violates the CRG conjecture.
In other words, the classical holographic dual of a consistent unitary
boundary field theory necessarily obeys the CRG conjecture.

Our argument for a sharp connection between the chaos bound and the CRG
conjecture consists largely of stitching together well known results from the
now classic papers Heemskerk:2009pn  and Gary:2009ae ; Penedones:2010ue ;
Okuda:2010ym ; Maldacena:2015iua  (see also Paulos:2016fap  141414See
Komatsu:2020sag ; Hijano:2019qmi  for recent progress regarding external
massive scalar states.). In section 2 we study time-ordered four-point
functions in a holographic conformal field theory of four operators of
arbitrary spin inserted on the two-parameter family of points (11). 151515The
same kinematic configurations were studied in section 6.1 of Maldacena:2015iua
. Our four-point function is normalized by dividing with a product of two two-
point functions (see (31)). As the insertion parameters $\theta$ and $\tau$
run over the range of study (12), the conformal cross ratios $\sigma$ and
$\rho$ (or equivalently $z$ and ${\bar{z}}$ -see around (16) for definitions)
range over three different sheets in the complex cross ratio space
corresponding to three distinct causal configurations (13) of the boundary
points. The first of these is the principal sheet which we refer to as the
Causally Euclidean sheet through this paper. The second sheet -which we refer
to as the Causally Regge sheet through this paper - is reached starting from
the Causally Euclidean sheet by circling the branch point at ${\bar{z}}=1$ in
a counter-clockwise manner. The third sheet - which we refer to as the
Causally Scattering sheet through this paper- is obtained starting from the
Causally Regge sheet by circling counter-clockwise around the branch point at
$z=0$.

Our argument proceeds by focusing attention on two special one parameter
limits of our two-parameter set of insertion points (11). The first of these
is the much-studied ‘Regge limit’ (19) in which $\sigma$ is taken to be small
but $\rho$ is allowed to be arbitrary. Here and throughout this paper, $\rho$
and $\sigma$ are conformal cross ratios, related by the more familiar cross
ratios $z$ and ${\bar{z}}$ by the relations $z=\sigma e^{\rho}$ and
${\bar{z}}=\sigma e^{-\rho}$. In several studies of conformal field theory,
the Regge limit is studied on the Causally Regge sheet (13) of cross ratio
space. The Regge limit studied in section 3 of this paper, however, straddles
across both the Causally Regge and the Causally scattering configuration
161616One reason for this is that our ‘coordinate patch’ (11) is more flexible
than the $\rho$ coordinate patch (LABEL:labn) often employed in CFT studies.
As we explain in Appendix E, that part of the two-parameter insertion space
(11) which lies on the Causally Scattering sheet has no image into time-
ordered correlators on the $\rho$ plane (LABEL:labn).  In section 3 below we
use use a small variant of the analysis of section 5.2 of Heemskerk:2009pn  to
demonstrate that to leading order in this limit (i.e. to leading order in
small $\sigma$ ) the $\sigma$ and $\rho$ dependence of our correlator takes
the form

$\frac{g_{CS}(e^{2\rho})}{\sigma^{A^{\prime}-1}}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}~{}\frac{g_{CR}(e^{2\rho})}{\sigma^{A^{\prime}-1}}$ (5)

respectively in the Causally Scattering and Causally Regge regimes, where
$A^{\prime}$ is a fixed but as yet unknown number. $g_{CS}(e^{2\rho})$ and
$g_{CR}(e^{2\rho})$ are as yet unknown functions of the cross ratio $\rho$.
The crucial point here, however, is that they are not completely independent
of each other. There exists a function ${\tilde{H}}(z)$ which is analytic
apart away from the branch cut at $z=0$. $g_{CR}(e^{2\rho})$ equals
${\tilde{H}}(z)$ evaluated on $z=e^{2\rho}$ on one sheet of this function,
while $g_{CR}(e^{2\rho})$ equals ${\tilde{H}}(z)$ evaluated on $z=e^{2\rho}$
on a second sheet. 171717On the configurations studied in this paper
$e^{2\rho}$ is real and lies in the interval $(0,1)$. This fact makes it
impossible for $g(e^{2\rho})$ to vanish identically in the Causally Regge
branch if it is nontrivial on the Causally Scattering branch. 181818The point
$\tau=\theta$ in the configuration (11) maps to $e^{2\rho}=z=0$, the branch
point of the correlator. As a consequence the functions $g_{CS}(\rho)$ and
$g_{CR}(e^{2\rho})$ are not directly smoothly connected but are more
indirectly related, as explained in this paragraph. It follows, in other
words, that in the small $\sigma$ limit, correlators scale with the same power
of $\sigma$ in the Causally Scattering and Causally Regge sheets of cross
ratio space.

In section (4) we turn to the study the limit $\rho\to 0$ on the Causally
Scattering sheet; this is the bulk point limit of Gary:2009ae ;
Penedones:2010ue ; Okuda:2010ym ; Maldacena:2015iua . In section (4) we
generalize the discussion of section 6 of Maldacena:2015iua  and
Penedones:2010ue ; Okuda:2010ym  to demonstrate that the four point function
is given, to leading order in this limit, by the expression

$\begin{split}G_{\rm
sing}&\propto-\frac{1}{4\sqrt{\sigma(1-\sigma)}}\int_{\mathbb{H}_{D-2}}d^{D-2}X\int
d\omega\omega^{\Delta-4}e^{i\omega P.X}{\cal S}\\\ \end{split}$ (6)

(see (132) for the proportionality factor that makes this equation precise)
where ${\cal S}$ is the classical flat space S matrix of four specified waves
with momenta (88), polarizations (135), $\omega$ is the energy of each of
these waves, $\Delta$ is the sum of the scaling dimensions of the four
operators and the $\mathbb{H}_{D-2}$ is the part of $AdS_{D+1}$ space that is
orthogonal to the four boundary points $P_{i}$ at $\tau=0$ (see subsection
2.6).191919In other words it consists of the points $X$ in the embedding space
$\mathbb{R}^{D,2}$ which obey the equations $P_{i}.X=0$ for all $i$ together
with $X^{2}=-1$. See Appendix B.3 for details.

Whenever the bulk $S$ matrix is generated by a local contact term, ${\cal S}$
grows like a non-negative power of $\omega$. In this situation the integral
over $\omega$ in (6) receives its dominant contributions from large values of
$\omega$. The phase factor $e^{i\omega P.X}$ cuts off a would-be large
$\omega$ divergence in this integral and turns it into a power-law
singularity, of the form $\frac{1}{\rho^{a}}$, in $\rho$ 202020When ${\cal S}$
grows like $\omega^{r}$,  $a=\Delta+r-3.$ ; this is the famous bulk point
singularity of Maldacena:2015iua (see Section 4 for details). (6) allows one
to compute the coefficient of this singularity; we find that it is
proportional to a simple known function of $\sigma$ times an integral of flat
space S matrix over $\mathbb{H}_{D-2}$, with the scattering angle $\theta$ and
scattering polarizations determined in terms of boundary cross ratios and
polarization in a simple way. 212121Specifically, $\theta$ is identified with
the conformal cross ratio $\sigma$ according
$\sin^{2}\frac{\theta}{2}=\sigma$, while the transverse polarizations are
identified with the parallel transport or boundary polarizations to the bulk
point $X$ on $\mathbb{H}_{D-2}$ along the unique null geodesic that connects
the boundary points $P_{a}$ to $X$. Our final result for the coefficient of
the singularity $1/\rho^{a}$ in terms of the flat space S matrix, presented in
(136), is a generalization of the results of Gary:2009ae  to the study of
holographic correlators of arbitrary spin, and reduces to the results of
Gary:2009ae  in the special case that all particles have spin-zero.

Recall that (5) applies when $\sigma$ is small while (6) applies when $\rho$
is small. We will now extract information from the fact that these two
expressions must hold simultaneously when $\rho$ and $\sigma$ are both small.
When we specialize the small $\rho$ discussion ((6) and surrounding) to small
values of $\sigma$ it turns out that we find the following simple universal
result. If the flat space $S$ matrix ${\cal S}$ scales at large $s$ but fixed
$t$ like

$S\propto s^{A}$ (7)

then the small $\sigma$ limit of the $\rho\to 0$ limit of the Greens function
- normalized as in (31) - is proportional to

$G\propto\frac{1}{\rho^{a}\sigma^{A-1}}$ (8)

(8) tells us how fast our correlator grows with $\sigma$ in the $\rho\to 0$
limit. On the other hand (5) captures the fastest growth of the correlator at
any value of $\rho$. In subsection 5 we present a detailed study of the inter
relation between the small $\sigma$ and small $\rho$ expansions, and establish
in particular that

$A^{\prime}\geq A$ (9)

Recall that the chaos bound theorem of Maldacena:2015waa  implies (see the
Appendix of that paper and section 6 for a brief review) that the correlator
for a well behaved (unitary etc) boundary theory cannot grow faster than
$\frac{1}{\sigma}$ in the small $\sigma$ limit on the causally Regge sheet. As
$A^{\prime}$ in (5) equals $A$ even on the causally Regge sheet, it follows as
a consequence of the chaos bound as

$A\leq 2.$ (10)

In words, the flat space $S$ matrix cannot grow faster than $s^{2}$ in the
Regge limit. Restated, the chaos bound applied to boundary correlators of a
unitary theory implies that the bulk dual of that theory obeys the CRG
conjecture.

##  2 Kinematics

###  2.1 Insertion locations

In this paper we study the four point function of boundary operators in a
holographic field theory. Following section 6 of Maldacena:2015iua , we study
correlators of operators inserted at the following two parameter set of
boundary points of $AdS_{D+1}$:

$\begin{split}&P_{1}=(\cos\tau,\sin\tau,1,0,{\vec{0}})\\\
&P_{3}=(\cos\tau,\sin\tau,-1,0,{\vec{0}})\\\
&P_{2}=(-1,0,-\cos\theta,-\sin\theta,{\vec{0}})\\\
&P_{4}=(-1,0,\cos\theta,\sin\theta,{\vec{0}})\\\ \end{split}$ (11)

$P_{1}~{}~{}t=\tau$$P_{3}$$t=\pi$$t$$P_{4}$$P_{2}$$\theta$ Figure 1: Insertion
points in global AdS

where we parameterize boundary points in $AdS_{D+1}$ by null rays in embedding
space with the first two coordinates timelike and the remaining $D$
coordinates space-like (see Appendix (B) for notations and conventions and a
brief review of the embedding space formalism), ${\vec{0}}$ is the zero vector
in $D-2$ dimensional Euclidean space. Points $P_{1}$ and $P_{3}$ are inserted
at global time $\tau$, and points $P_{2}$ and $P_{4}$ are inserted at global
time $\pi$. We will focus our attention on the range of parameters

$0\leq\tau\leq\pi,~{}~{}~{}~{}0\leq\theta\leq\frac{\pi}{2}$ (12) 222222The
restriction to $\theta$ in the range $0\leq\theta\leq\frac{\pi}{2}$ rather
than $0\leq\theta\leq\pi$ is a matter of convenience; the flip
$2\leftrightarrow 4$ maps the smaller to the bigger range.

###  2.2 Causal relations and cross ratio sheets

The causal relations between the points $P_{a}$ are given by 232323See around
Fig. 8 ofMaldacena:2015iua . The argument is as follows. The pairs
$(P_{1},P_{3})$ and $(P_{2},P_{4})$ are each inserted at equal global times;
it follows that elements of the same pair are always space-like separated from
each other. The time difference between the insertion points of the second and
first pair is $\pi-\tau$. The angular separation between $1$ and $3$ (or $2$
and $4$) is $\pi-\theta$, and the angular separation between $1$ and $4$ (or
$2$ and $3$) is $\theta$. Using the fact that two points are time-like
separated only if the time difference between them exceeds their angular
separation gives (13).

$\begin{split}&\tau>\pi-\theta~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
Causally~{}Euclidean},\\\
&\pi-\theta>\tau>\theta~{}~{}~{}~{}~{}~{}{\rm~{}~{}Causally~{}Regge}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(P_{4}>P_{1},~{}{\rm
and}~{}P_{2}>P_{3})\\\
&\tau<\theta~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm~{}Causally~{}Scattering}~{}~{}~{}~{}~{}~{}~{}~{}~{}(P_{4},P_{2})>(P_{1},P_{3})\\\
\end{split}$ (13)

where we have used the notation $A>B$ to denote that $A$ is in the causal
future of $B$ and it is understood that two points are space-like separated
with respect to each other if their causal ordering is not specified242424For
complete clarity, all four points are space-like separated from each other in
the Euclidean configuration, $P_{4}$ is in the future light-cone of $P_{1}$,
$P_{2}$ is the the future light-cone of $P_{3}$ while each of the pair
$P_{1},P_{4}$ is space-like separated with each of the pair $P_{3},P_{2}$ in
the Regge configuration. On the other hand $P_{1}$ and $P_{3}$ are space-like
separated with respect to each other, $P_{2}$, and $P_{4}$ are also space-like
separated with respect to each other, but each of $P_{4},P_{2}$ lie in the
future light-cones of each of $P_{1},P_{3}$ in the causally scattering
regions. 252525We use the name Causally Regge because this is the causal
relationship between points in the well studied Regge limit of CFT on the
cross ratio sheet on which the chaos bound constrains the small $\sigma$
behavior of correlators. Note that the three distinct causal relations (13)
correspond to three distinct sheets in complex cross ratio space.

We end this subsection with an aside. We see from (13) that when
$\tau>\theta$, $\tau=\theta$ or $\tau<\theta$, the pairs of points
$(P_{2},P_{1})$ and $(P_{4},P_{3})$ are, respectively both space-like, null or
time-like separated. The fact that the switch of causal relations for the
pairs $(P_{2},P_{1})$ and $(P_{4},P_{3})$ happens simultaneously indicates
that the two parameter set of coordinates (11) are rather special especially
in the neighbourhood of $\tau=\theta$. If we allowed for general variations of
all four boundary points, in the neighbourhood of the configuration (11) with
$\tau=\theta$ we would find points with $P_{2}>P_{1}$ but $(P_{4},P_{3})$
space-like and vice versa. The configurations (11) do not include any such
causal configurations, and so are non generic in the neighbourhood of
$\tau=\theta$. See Appendix C for a detailed analysis in the case that $\tau$
and $\theta$ are both small.

###  2.3 Conformal cross ratios

The conformal cross ratios (see Appendix B.5) associated with these four
points are given by

$\begin{split}&z{\bar{z}}\equiv\frac{(P_{2}.P_{1})(P_{3}.P_{4})}{(P_{2}.P_{4})(P_{3}.P_{1})}=\frac{(\cos\tau-\cos\theta)^{2}}{4}\equiv
A\\\
&(1-z)(1-{\bar{z}})\equiv\frac{(P_{4}.P_{1})(P_{2}.P_{3})}{(P_{2}.P_{4})(P_{3}.P_{1})}=\frac{(\cos\tau+\cos\theta)^{2}}{4}\\\
&\implies z+{\bar{z}}=1-\cos\tau\cos\theta\equiv 2B\\\ &z^{2}-2Bz+A=0\\\
&z=B\pm\sqrt{B^{2}-A}\\\ &z=\frac{1-\cos\left(\tau-\theta\right)}{2},\\\
&{\bar{z}}=\frac{1-\cos\left(\tau+\theta\right)}{2},\end{split}$ (14)

Note that both $z$ and ${\bar{z}}$ lie in the interval

$z\in[0.1],~{}~{}~{}~{}{\bar{z}}\in[0,1]$ (15)

As usual we define the cross ratios $\sigma$ and $\rho$ by the relations
262626See Appendix B.5 for more about various cross ratios.

$z=\sigma e^{\rho},~{}~{}~{}~{}{\bar{z}}=\sigma e^{-\rho}$ (16)

With these definitions

$\begin{split}&\sigma^{2}=\frac{(\cos\theta-\cos\tau)^{2}}{4}\\\
&\sinh^{2}\rho=\frac{\sin^{2}\theta\sin^{2}\tau}{(\cos\theta-\cos\tau)^{2}}\end{split}$
(17)

To end this subsection we specialize (14) in the two parametric limits that
are of particular interest in this paper.

####  2.3.1 The small $\tau$ limit

In the small $\tau$ limit at fixed $\theta$, (14) simplifies to

$\begin{split}z&=\sin\frac{\theta}{2}\left(\sin\frac{\theta}{2}-\tau\cos\frac{\theta}{2}\right)+{\cal
O}(\tau^{2})\\\
{\bar{z}}&=\sin\frac{\theta}{2}\left(\sin\frac{\theta}{2}+\tau\cos\frac{\theta}{2}\right)+{\cal
O}(\tau^{2})\\\ \sigma&=\sin^{2}\frac{\theta}{2}+{\cal O}(\tau^{2})\\\
\rho&=-\tau\cot\frac{\theta}{2}+{\cal O}(\tau^{3})\\\ \end{split}$ (18)

Note, in particular, that $\rho$ approaches zero (from the negative side) as
$\tau\to 0$.

####  2.3.2 The Regge limit

On the other hand in the Regge limit

$\tau\to 0,~{}~{}~{}~{}\theta\to 0,~{}~{}~{}~{}\frac{\tau}{\theta}=a={\rm
fixed}$ (19)

(25) simplifies to

$\begin{split}&z=\frac{(\theta-\tau-i\epsilon)^{2}}{4}=\frac{\theta^{2}}{4}(1-a-i\epsilon)^{2}+{\cal
O}(\theta^{4})\\\
&{\bar{z}}=\frac{(\theta+\tau+i\epsilon)^{2}}{4}=\frac{\theta^{2}}{4}(1+a+i\epsilon)^{2}+{\cal
O}(\theta^{4})\\\ &\sigma^{2}=\frac{\theta^{4}(1-a^{2})}{16}+{\cal
O}(\theta^{6})\\\
&e^{2\rho}=\left(\frac{1-a-i\epsilon}{1+a+i\epsilon}\right)^{2}+{\cal
O}(\theta^{2})\end{split}$ (20)

(in (19) we have presented $i\epsilon$ corrected formulae using the method of
subsection 2.5 below).

Note that the Regge limit explores the neighbourhood of the boundary point
(11) obtained by setting $\tau=\theta=0$, i.e. the point

$\begin{split}&P_{1}=(1,0,1,0,{\vec{0}})\\\ &P_{3}=(1,0,-1,0,{\vec{0}})\\\
&P_{2}=(-1,0,-1,0,{\vec{0}})\\\ &P_{4}=(-1,0,1,0,{\vec{0}})\\\ \end{split}$
(21)

In this paper we study only that part of the neighbourhood of this point that
we can be reached by turning on small values of $\theta$ and $\tau$ in (11).
As an aside we note that this two parameter set of points do not give a
complete cover of the neighbourhood of (21) modulo conformal transformations.
In addition to the points obtained from (11) at small values of $\theta$ and
$\tau$, there are additional infinitesimal deformations of (21) in which the
insertion points enjoy different causal relations from any of those listed in
(13). This fact (which we will never use anywhere else in this paper) is
explained in some detail in Appendix C.

####  2.3.3 Overlap between small $\tau$ and Regge

Note that the small $\theta$ limit of the small $\tau$ limit (18) overlaps
with the small $a$ limit of the Regge limit (20). In particular if we expand
the RHS of (18) at leading order in $\theta$ we obtain

$\begin{split}\sigma&=\frac{\theta^{2}}{4}\\\ \rho&=-\frac{2\tau}{\theta}\\\
\end{split}$ (22)

On the other hand if we expand (20) to leading order in $a$ we obtain

$\begin{split}\sigma&=\frac{\theta^{2}}{4}\\\ \rho&=-2a\\\ \end{split}$ (23)

It follows from (19) that (20) and (18) are equivalent.

Notice that the $\tau\to 0$ limit places us in the Causally Scattering regime
of (13). On the other hand, the Regge limit lies in the causally scattering
regime when $a<1$ but in the Causally Regge regime for $a>1$. The fact that
the Regge limit straddles two distinct causal regimes will be of central
importance to this paper.

###  2.4 A path in cross ratio space

It is useful to track the evolution of $z$, ${\bar{z}}$, $\sigma$ and $\rho$
as we keep $\theta$ fixed and adiabatically decrease $\tau$ from $\tau=\pi$ to
$\tau=0$. When $\tau=\pi$, $z={\bar{z}}=\frac{1+\cos\theta}{2}$ and our
configuration is Euclidean. As we decrease $\tau$, $z$ decreases while
${\bar{z}}$ increases. ${\bar{z}}$ reaches its maximum value, namely unity
when $\tau=\pi-\theta$ (i.e. at the boundary between the Causally Euclidean
and Causally Regge regimes) . As $\tau$ is further decreased, $z$ continues to
decrease, but now ${\bar{z}}$ also begins to decrease. Once $\tau$ reaches
$\theta$ (the boundary between the causally Regge and causally scattering
regime) $z$ reaches its minimum value namely zero. As $\tau$ is decreased even
further ${\bar{z}}$ continues to decrease but $z$ now starts to increase.
Finally, at $\tau=0$ we have $z={\bar{z}}=\frac{1-\cos\theta}{2}$. 272727It
follows (see (207)) that the starting point $\tau=\pi$ and the end point
$\tau=0$ of our paths each have $\rho=0$.

In the previous paragraph, we have described a one parameter path in
configuration space. The evolutions of various conformal cross ratios along
this path is depicted in the graphs Figs 2, 3 and 4 below in which we have
displayed graphs of the cross ratios $z$, ${\bar{z}}$, $\sigma$ $\rho$ versus
$\tau$ (note the arrows in those graphs track the journey from Causally
Euclidean configurations, starting at $\tau=\pi$, to Causally Scattering
configurations, ending at $\tau=0$.)

Figure 2: The evolution of $z$ and ${\bar{z}}$ as $\tau$ is decreased from
$\pi$ down to $0$ at fixed $\theta$. Note ${\bar{z}}$ touches its maximum
value unity at $\tau=\pi-\theta$ and $z$ touches its minimum value, $z=0$ at
$\tau=\theta$. Figure 3: The evolution of the cross ratio $\sigma$ as $\tau$
is decreased from $\pi$ down to $0$ at fixed $\theta$. Note that $\sigma$
touches its minimum value, $\sigma=0$ at $\tau=\theta$, when $z$ vanishes.
Figure 4: The evolution of the cross ratio $e^{2\rho}=\frac{z}{\bar{z}}$ as
$\tau$ is decreased from $\pi$ down to $0$ at fixed $\theta$. Note that
$e^{2\rho}$ touches its minimum value, $e^{2\rho}=0$ at $\tau=\theta$, when
$z$ vanishes.

We end this subsection with a remark. Several investigations of CFT work in
the so-called ‘$\rho$ plane’. 282828To forestall confusion we emphasize that
in this paragraph and in parts of Appendix E \- but nowhere else in this paper
- $\rho$ refers to the insertion coordinate in (LABEL:labn) and not to the
conformal cross ratio paired with $\sigma$. In this plane the four CFT
operators are inserted in $\mathbb{R}^{1,1}$ at the locations (LABEL:labn). A
question that might occur to the reader is the following: how does the path
described in this subsection (fixed $\theta$, $\tau$ lowered from $\pi$ to
zero) map to the $\rho$ plane? This question is addressed in detail in
Appendix E. Here we only make a simple qualitative point. In the $\rho$
coordinate system (LABEL:labn) the insertion points $P_{3}$ and $P_{4}$ are
fixed and unmoving; in particular, these two points are spacelike separated
from each other. It follows immediately that the part of the trajectory of
this subsection that lies in the Causally scattering region ($\tau<\theta$)
has no image in the $\rho$ plane. On the other hand, the part of the
trajectory of this subsection that lies on the Causally Euclidean and Causally
Regge sheets has a faithful map onto the $\rho$ plane, as we describe in
Appendix E.

###  2.5 The same path on the complex cross ratio sheets

As conformal correlators have branch points at ${\bar{z}}=1$ and $z=0$, and as
the trajectories described in the previous subsection all ‘touch’ these branch
points, it follows that the description of the paths presented in the previous
subsection is ambiguous and needs to be improved. 292929Along the path of the
previous subsection, for instance, ${\bar{z}}$ increases to unity and then
retreats back. We need to know whether the retreat occurs on a different sheet
from the onward trajectory, and if so which one. A similar question arises for
the part of the trajectory that touches $z=0$. The true paths traversed in
cross ratio space are given by making the replacements
$\tau\rightarrow\tau-i\epsilon\tau$ (see Appendix (255)). The $i\epsilon$
corrected insertion points are

$\begin{split}&P_{1}=(\cos(\tau-i\epsilon\tau),\sin(\tau-i\epsilon\tau),1,0)\\\
&P_{3}=(\cos(\tau-i\epsilon\tau),\sin(\tau-i\epsilon\tau),-1,0)\\\
&P_{2}=(\cos(\pi-i\pi\epsilon),\sin(\pi-i\pi\epsilon),-\cos\theta,-\sin\theta)\\\
&P_{4}=(\cos(\pi-i\pi\epsilon),\sin(\pi-i\pi\epsilon),\cos\theta,\sin\theta)\\\
\end{split}$ (24)

with $\epsilon>0$. The $i\epsilon$ corrected cross ratios are given by

$\begin{split}&z{\bar{z}}\equiv\frac{(P_{2}.P_{1})(P_{3}.P_{4})}{(P_{2}.P_{4})(P_{3}.P_{1})}=\frac{(\cos\theta-\cos\left[\tau+i\epsilon(\pi-\tau)\right])^{2}}{4}\equiv
A\\\
&(1-z)(1-{\bar{z}})\equiv\frac{(P_{4}.P_{1})(P_{2}.P_{3})}{(P_{2}.P_{4})(P_{3}.P_{1})}=\frac{(\cos\theta+\cos\left[\tau+i\epsilon(\pi-\tau)\right])^{2}}{4}\\\
&\implies
z+{\bar{z}}=1-\cos\theta\cos\left[\tau+i\epsilon(\pi-\tau)\right]\equiv 2B\\\
&z^{2}-2Bz+A=0\\\ &z=B\pm\sqrt{B^{2}-A}\\\
&z=\frac{1}{2}(1-\cos(\theta-\tau-i(\pi-\tau)\epsilon)),\\\
&{\bar{z}}=\frac{1}{2}(1-\cos(\theta+\tau+i(\pi-\tau)\epsilon)),\\\
&\sigma^{2}=\frac{(\cos\theta-\cos[\tau+i\epsilon(\pi-\tau)])^{2}}{4}\\\
&\sinh^{2}\rho=\frac{\sin^{2}\theta\sin[\tau+i\epsilon(\pi-\tau)]^{2}}{(\cos\theta-\cos[\tau+i\epsilon(\pi-\tau)])^{2}}\end{split}$
(25)

####  2.5.1 The neighbourhood of ${\bar{z}}=1$

To examine the route traced by the path of subsection 2.4 in complex cross
ratio space, we first examine (25) in the neighbourhood of ${\bar{z}}=1$ by
setting $\tau=\pi-\theta+\delta\tau$, assuming $\delta\tau={\cal O}(\epsilon)$
and expanding to second order in $\epsilon$. We find that the formula for
${\bar{z}}$ reduces to

$\begin{split}&{\bar{z}}-1=-\frac{(\delta\tau+i{\tilde{\epsilon}})^{2}}{4}\\\
&|{\bar{z}}-1|=\frac{\delta\tau^{2}+{\tilde{\epsilon}}^{2}}{4}\\\ &{\rm
Arg}({\bar{z}-1})=-\pi+2\tan^{-1}\left(\frac{{\tilde{\epsilon}}}{\delta\tau}\right),~{}~{}~{}~{}~{}~{}\tan^{-1}(x)\in[0,\pi)\\\
\end{split}$ (26)

where ${\tilde{\epsilon}}=(\pi-\tau)\epsilon$

It follows immediately from (26) that as $\tau$ is lowered from just above
$\pi-\theta$ to just below $\pi-\theta$, our path in cross ratio space circles
around the branch point at ${\bar{z}}=1$ in a counter-clockwise manner. The
much-studied passage from the Euclidean to standard ‘Regge’ behavior -
involves traversing precisely the same path in cross ratio space (see e.g.
subsection 5.1 of Kravchuk:2018htv  ). It follows that the Causally Regge
sheet of (13) is the sheet encountered in studies of the CFT in the Regge
limit303030As far as we are aware, this limit was first studied in the context
of $AdS/CFT$ in Cornalba:2006xm ; Cornalba:2006xk  and later extended to more
general CFTs in Cornalba:2007fs ; Costa:2012cb . See Afkhami-Jeddi:2016ntf ;
Caron-Huot:2017vep ; Simmons-Duffin:2017nub ; Kravchuk:2018htv  for more
recent use of this kinematic limit in recent times., i.e. the sheet on which
CFT correlators are constrained by the chaos bound of Maldacena:2015waa .

####  2.5.2 The neighbourhood of $z=0$

In the similar fashion we examine the behaviour of our path in the
neighbourhood of $z=0$ by setting $\tau=\theta+\delta\tau$. Once again we take
${\cal O}(\delta\tau)={\cal O}(\epsilon)$ and work to second order in
$\epsilon$ to obtain

$\begin{split}&{z}=\frac{(\delta\tau+i{\tilde{\epsilon}})^{2}}{4}\\\
&|z|=\frac{\delta\tau^{2}+{\tilde{\epsilon}}^{2}}{4}\\\ &{\rm
Arg}(z)=2\tan^{-1}\left(\frac{{\tilde{\epsilon}}}{\delta\tau}\right),~{}~{}~{}~{}~{}~{}\tan^{-1}(x)\in[0,\pi)\\\
\end{split}$ (27)

It follows that as $\tau$ is lowered from just above $\theta$ to just below
$\theta$, our path circles round the branch point at $z=0$ in a counter-
clockwise manner (note ${\tilde{\epsilon}}>0)$ (see Fig. 6 of Heemskerk:2009pn
).

####  2.5.3 Summary of the trajectory in the complex plane

In summary, as we lower $\tau$ from $\pi$ to zero at constant $\theta$ we move
along the trajectories in (complex) cross ratio space depicted schematically
in Fig 5 (the vertical scale in these graphs is highly exaggerated; all curves
hug the real axis except when ${\bar{z}}$ is in the neighbourhood of unity or
when $z$ is in the neighbourhood of zero). In other words, our path starts on
the principal or Causally Euclidean sheet when $\tau=\pi$. As $\tau$ is
lowered below $\pi-\theta$ our path circles around the branch cut at
${\bar{z}}=1$ counter-clockwise, bringing us onto the Causally Regge sheet. As
$\tau$ is further lowered past $\theta$, the path circles counter-clockwise
around the branch cut at $z=0$, taking us to the Causally Scattering sheet
(see Fig. 8 of Maldacena:2015iua  for a very closely related discussion).

Figure 5: The path traversed in the complex plane by the variables $z$
(purple) and ${\bar{z}}$ (green) as we lower $\tau$ from $\pi$ to $0$ at fixed
$\theta$. The vertical scale in these graphs is greatly exaggerated to make
them visible. The actual curves should be thought of as hugging the real axis
except in the neighbourhood of the branch points which they circle in the
manner shown in this Figure.

###  2.6 Intersections of boundary lightcones

When do the lightcones emanating out of the four points $P_{1},P_{2},P_{3}$
and $P_{4}$ have a common intersection point in the bulk? For $\tau\neq 0$ the
four vectors $P_{1}$, $P_{2}$, $P_{3}$ and $P_{4}$ span a four dimensional
subspace of $\mathbb{R}^{D,2}$ and this subspace is metrically
$\mathbb{R}^{2,2}$. It follows from the analysis of Appendix B.4 313131In the
language Appendx B.4(11) is a Case 1 configuration of boundary points with
${\hat{Q}}=4$ . that the lightcones that emanate out of these four points have
no common intersection for $\tau\neq 0$. 323232In this generic situation the
subalgebra of the conformal algebra that stabilizes the collection of points
$P_{1}$, $P_{2}$, $P_{3}$ and $P_{4}$ \- and so the vector space
$\mathbb{R}^{2,2}$ of embedding space vectors spanned by $P_{1}\ldots P_{4}$\-
is $SO(D-2)$.

When $\tau=0$, on the other hand (i.e. at the edge of the parameter range
(12), i.e.the limit (18) in which $\rho\to 0$) 333333There are actually two
distinct limits of the insertions (11) in which $\rho\to 0$. The first of
these is the limit $\tau\to 0$ of interest to this paper. The second is the
simpler limit $\tau\to\pi$ in which, once again, $\rho\to 0$. These two limits
are physically distinct because they lie on different sheets on the $z$ and
${\bar{z}}$ complex plane. In the second simpler limit the correlators lie on
the principal or Euclidean sheet (i.e. the Causally Euclidean region of (13)).
Correlation functions at nonzero $\theta$ are manifestly non-singular in this
second $\rho\to 0$ limit (see under equation 6.21 of Maldacena:2015iua ). This
non-singular limit is of no interest to this paper and will never again be
considered here. the vectors $P_{1}$, $P_{2}$, $P_{3}$ and $P_{4}$ are
linearly dependent. In particular, at $\tau=0$

$P_{1}+P_{2}+P_{3}+P_{4}=0$ (28)

In this case these vectors span a three dimensional subspace of
$\mathbb{R}^{D,2}$. This subspace is metrically $\mathbb{R}^{2,1}$. In the
language Appendx B.4, (11) is a Case 2 configuration of boundary points with
${\hat{Q}}=3$. The lightcones that emanate out of these four points intersect
on an $\mathbb{H}_{D-2}$ , and the subalgebra of the conformal algebra that
stabilizes the four point is $SO(D-2,1)$.

In the special case that $\tau=0$ and $\theta=0$ (i.e. the Regge limit (19)),
it is easy to check that (28) breaks up into two equations (see (21))

$\begin{split}&P_{1}+P_{2}=0\\\ &P_{3}+P_{4}=0\\\ \end{split}$ (29)

It follows that the subspace of $\mathbb{R}^{D,2}$ spanned by the vectors
$P_{1},P_{2},P_{3},P_{4}$ in this limit is two dimensional; infact it is the
$\mathbb{R}^{1,1}$ spanned by

$(a,0,b,0,\ldots 0)$

for arbitrary $a$ and $b$. The space orthogonal to this $\mathbb{R}^{1,1}$ is
$\mathbb{R}^{D-1,1}$ spanned by coordinates of the form

$(0,y_{0},0,y_{1},y_{i})$ (30)

and the subgroup of the conformal group that stabilizes the collection of
points $P_{1},P_{2},P_{3},P_{4}$ is $SO(D-1,1)$. 343434Moving beyond the
consideration of the specific boundary configurations studied in this paper,
the reader may wonder there exists a general interplay between the
$\mathbb{R}^{p,q}$ classification of boundary points, their possible causal
structures and the nature of their cross ratios. The answer to this general
question (that has no application to the current paper) is clearly in the
affirmative and we present a preliminary investigation of this point in
Appendix B.7. For example we demonstrate that whenever the four boundary
points are in an $\mathbb{R}^{3,1}$ configuration (this can happen for a
number of different causal relations between the points), the $\rho$ cross
ratio corresponding to these points is always imaginary. On the other hand
$\rho$ could be either real or imaginary when the four points span an
$\mathbb{R}^{2,2}$.

Although this fact will have no implications for the main flow of this paper,
we note as an aside that the neighbourhood of the Regge point (21) includes
points whose span in embedding space is an $\mathbb{R}^{3,1}$ (in addition to
the points obtained from small values of $\theta$ and $\tau$ in (11) whose
span in embedding space make up an $\mathbb{R}^{2,2}$ or an
$\mathbb{R}^{2,1}$); this point is explained in some detail in Appendix C.

###  2.7 Scaling limits of the two point function

In this paper, we will be interested in studying the normalized four-point
function of operators inserted at the positions $P_{1}\ldots P_{4}$ defined by

$G^{\rm norm}=\frac{G}{G_{12}G_{34}}$ (31)

where $G$ is the simple four-point function, and $G_{12}$ and $G_{34}$ are
two-point functions. We will be particularly interested in the scaling of the
correlator (31) in the Regge and small $\tau$ limits. In the subsequent two
sections, we present a study of the four-point function - the numerator of
(31) - in these two scaling limits. In this brief subsection, we present the
much simpler analysis of the scaling of the denominator of (31) in these
limits.

To end this section we study the scaling of the two-point function of spin $J$
operators inserted at the locations $P_{1}$ and $P_{2}$ in the Regge and
$\tau\to 0$ limits.

As we have reviewed in Appendix B.10, the boundary to boundary two point
function for a spin $J$ field is given by

$G_{ij}=\mathcal{C}_{\Delta,J}\frac{\left(Z_{i}.Z_{j}~{}P_{i}.P_{j}-{Z_{i}.P_{j}~{}Z_{j}.P_{i}}\right)^{J}}{(P_{i}.P_{j})^{\Delta+J}}$
(32)

where $Z_{i}$ is the boundary polarization vector for the $i^{th}$ particle
(see under (195) for a discussion of boundary tangent vectors in the embedding
space formalism) and

$\mathcal{C}_{\Delta,J}=\frac{(J+\Delta-1)\Gamma(\Delta)}{2\pi^{d/2}(\Delta-1)\Gamma(\Delta+1-h)}$
(33)

Note that on the configurations (11)

$P_{1}.P_{2}=P_{3}.P_{4}=\cos\tau-\cos\theta$ (34)

####  2.7.1 Regge scaling

In this limit (34) simplifies to

$P_{1}.P_{2}=P_{3}.P_{4}=\frac{\theta^{2}-\tau^{2}}{2}=\frac{\theta^{2}}{2}(1-a^{2})$
(35)

Also

$\begin{split}&Z_{1}.P_{2}=Z_{1}.(P_{2}+P_{1})=\theta\left(Z_{1}.(0,a,0,-1,{\vec{0}})\right)+{\cal
O}(\theta^{2})\leq{\cal O}(\theta)\\\
&Z_{2}.P_{1}=Z_{2}.(P_{2}+P_{1})=\theta\left(Z_{2}.(0,a,0,-1,{\vec{0}})\right)+{\cal
O}(\theta^{2})\leq{\cal O}(\theta)\\\ &Z_{1}.Z_{2}\leq{\cal O}(1)\end{split}$
(36)

where we have used $Z_{1}.P_{1}=Z_{2}.P_{2}=0$ (see around (195)). It follows
from (36) that

$\left(Z_{1}.Z_{2}~{}P_{1}.P_{2}-{Z_{1}.P_{2}~{}Z_{2}.P_{1}}\right)^{J}\leq{\cal
O}(\theta^{2J})$ (37)

Generic polarizations saturate the inequality (37). Through this paper and we
restrict attention to such correlators; In other words in this paper we
restrict attention to generic polarizations - those that obey

$\left(Z_{1}.Z_{2}~{}P_{1}.P_{2}-{Z_{1}.P_{2}~{}Z_{2}.P_{1}}\right)^{J}={\cal
O}(\theta^{2J})$ (38)

(similar comments apply to $G_{34}$.) 353535 This restriction allows for
greater simplicity of presentation and does not really result in loss of
generality. Our assumption is maximally violated when the LHS of (38)
vanishes, in which situation (31). See section 6 for a discussion of how the
omission of such correlators does not result in a lack of generality.

It follows immediately that in the Regge limit, the two point function
$G_{12}$ scales like

$\begin{split}G_{12}&=\frac{1}{\sigma^{\Delta_{1}}}\\\
G_{34}&=\frac{1}{\sigma^{\Delta_{3}}}\\\ \end{split}$ (39)

####  2.7.2 Small $\tau$ limit

In the limit in which $\tau$ is taken to zero

$P_{1}.P_{2}=P_{3}.P_{4}=1-\cos\theta$ (40)

If we now also take $\theta$ to be small

$P_{1}.P_{2}=P_{3}.P_{4}=\frac{\theta^{2}}{2}$ (41)

in agreement with (35) at $a=0$.

At generic $\theta$ we generically have

$\left(Z_{1}.Z_{2}~{}P_{1}.P_{2}-{Z_{1}.P_{2}~{}Z_{2}.P_{1}}\right)^{J}={\cal
O}(1)$ (42)

When we take $\theta$ small, however, we recover the $a\to 0$ limit of (38).
While at generic $\theta$ the $G_{12}$ and $G_{34}$ are of order unity, at
small $\theta$ (39) applies.

##  3 The $\sigma\to 0$ Regge limit

Consider a correlation function with four operators $O_{1},O_{2},O_{3},O_{4}$
inserted at the points $P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$ given in (11) In all
the concrete formulae presented in this paper we will assume that $O_{i}$ are
primary operators of the conformal group $SO(D,2)$ that have some dimension
$\Delta_{i}=w_{i}+J_{i}$ and transform in the traceless symmetric
representation with $J_{i}$ indices of $SO(D)$ 363636The extension of this
discussion to general representations of $SO(D)$ seems conceptually
straightforward but is not considered here as it is notationally cumbersome..

In this section we study the correlator described above holographically, and
in the Regge limit (19), i.e. the limit in which the conformal cross ratios
are given by (20). We emphasize that, in the terminology of this paper, the
Regge limit straddles the Causally Regge and Causally Scattering sheets (see
the discussion around (20)). The discussion in this section is inspired by and
has considerable overlap with the analysis of section 5.2 of Heemskerk:2009pn
. The slight novelty of our presentation lies in two aspects. First our
emphasis on the fact that the analysis of this section applies both to the
Causally Regge as well as the Causally Scattering sheets, and in fact allows
for smooth interpolation between the two. Second in our study of the structure
of the systematic expansion (beyond simply leading order) of the correlator in
the small $\theta$ limit, an expansion we will return to in Section 5.

###  3.1 Scaling with $\theta$

We wish to study a four point function generated by a bulk contact term in the
Regge limit. Any such four point function is given by the sum over expressions
of the schematic form

$\int
d^{D+1}X\frac{N}{(-2P_{1}.X+i\epsilon)^{{\tilde{a}}_{1}}(-2P_{2}.X+i\epsilon)^{{\tilde{a}}_{2}}(-2P_{3}.X+i\epsilon)^{{\tilde{a}}_{3}}(-2P_{4}.X+i\epsilon)^{{\tilde{a}}_{4}}}$
(43)

where $N$ is a numerator function; $N=N(Z_{i},X)$, $Z_{i}$ are boundary
polarization vectors and ${\tilde{a}}_{i}$ are positive numbers (not
necessarily integers). Bellow we will have use for the symbol

$2{\tilde{B}}=\sum_{i}{\tilde{a}}_{i}.$ (44)

As we have explained around (29), in the strict Regge limit the four points
$P_{1}\ldots P_{4}$ span an $\mathbb{R}^{1,1}$. The space orthogonal to this
$\mathbb{R}^{1,1}$ is $\mathbb{R}^{D-1,1}$ spanned by vectors of the form
(30). It is thus useful to parameterize the general bulk point in this limit
as

$X=(\frac{u+v}{2},y_{0},\frac{v-u}{2},y_{1},y_{i})$ (45)

with

$-(y^{\mu})^{2}+uv=1$ (46)

When $u=v=0$ and in the strict Regge limit the points (45) are orthogonal to
all $P_{a}$, and so all denominators in (43) vanish when $u$, $v$, $\theta$,
$\tau$ all equal zero. Recall that in the Regge limit $\theta$ and $\tau$ are
of the same order of smallness. It will turn out that the leading contribution
to the integral in (43) comes from values of $u$ and $v$ that are also of
order $\theta$ (see Heemskerk:2009pn ). It is thus useful to expand the
integrand in (43) in a power series expansion in the four small variables
($\theta,\tau,u,v$) with all treated as being of the same order in smallness,
with all other quantities ($y_{i}$ and $Z_{i}$) held fixed 373737The $D-1$
variables $y_{i}$ are the coordinates on the $\mathbb{H}_{D-1}$ below. It is
convenient not to view $y_{0}$ as an independent variable, but instead to use
(46) to solve for $y_{0}$ in a power series expansion in $uv$. It follows that
the coefficients of the power series expansion in $\theta,\tau,u,v$ are all
functions of $y_{i}$ and $Z_{i}$.  Let us suppose that the numerator $N$ is of
order $\theta^{M}$ plus subleading in this expansion. We define

$2B=2{\tilde{B}}-M.$ (47)

It follows that the integrand of (43) scales like $\frac{1}{\theta^{2B}}$ in
the limit under consideration.

At leading order

$\begin{split}&-2P_{1}.X=u+\tau~{}{\tilde{y}}_{0}\equiv D_{1}^{0}\\\
&-2P_{2}.X=-u+\theta~{}y_{1}\equiv D_{2}^{0}\\\
&-2P_{3}.X=v+\tau~{}{\tilde{y}}_{0}\equiv D_{3}^{0}\\\
&-2P_{4}.X=-v-\theta~{}y_{1}\equiv D_{4}^{0}\\\
&y_{0}=\sqrt{1+y_{i}^{2}-uv}\\\
&{\tilde{y}}_{0}=\sqrt{1+y_{i}^{2}}\end{split}$ (48)

Note that ${\tilde{y}}_{0}$ is simply $y_{0}$ at leading order in the small
$\theta$ expansion - recall the monomial $uv$ is of order $\theta^{2}$.

As we have mentioned above, in order to generate the small $\theta$ expansion
we proceed to Taylor expand the integrand of (43) in a Taylor series in small
quantities. Notice that the quantities $D_{i}^{0}$ are a collection of four
linearly independent combinations of these small quantities. As $D_{i}^{0}$
will play a distinguished role in our perturbative expansion, we find it
useful to use them rather than (see (48)) ) as the the basic monomials for our
Taylor series expansion. One can easily pass between $D_{i}^{0}$ and
$u,v,\theta,\tau$ using (48) and the inverse relations (49)

$\begin{split}u&=\frac{1}{2}\left(D_{0}^{1}-D_{0}^{2}-D_{0}^{3}-D_{0}^{4}\right)\\\
v&=-\frac{1}{2}\left(D_{0}^{1}+D_{0}^{2}-D_{0}^{3}+D_{0}^{4}\right)\\\
\theta&=\frac{1}{2y_{1}}\left(D_{0}^{1}+D_{0}^{2}-D_{0}^{3}-D_{0}^{4}\right)\\\
\tau&=\frac{1}{2\sqrt{1+y_{1}^{2}+y_{\perp}^{2}}}\left(D_{0}^{1}+D_{0}^{2}+D_{0}^{3}+D_{0}^{4}\right)\end{split}$
(49)

In (49) we have split up the $D-1$ $y_{i}$ variables into $y_{1}$ and the
remaining $D-2$ variables $y_{\perp}$.

To proceed we now expand both the numerator and each denominator in (43) in a
power series expansion in $D^{i}_{0}$. Let us first consider the numerator
$N$. This quantity is a polynomial in $Z_{i}$ and $X$ and so can be expanded
in a power series in $D^{i}_{0}$. The Taylor expansion of the numerator takes
the schematic form

$\sum_{n_{i}}a_{n_{1},n_{2},n_{3},n_{4}}(Z_{i},y_{i})\left(D_{0}^{1}\right)^{n_{1}}\left(D_{0}^{2}\right)^{n_{2}}\left(D_{0}^{3}\right)^{n_{3}}\left(D_{0}^{4}\right)^{n_{4}}$
(50)

where $M$ is the smallest homogeneity (i.e. smallest value of
$n_{1}+n_{2}+n_{3}+n_{4}$) that appears in this expression)

In a similar manner each term in the denominator of (43) can also be expanded.
For instance

$\frac{1}{(-P_{1}.X+i\epsilon)^{{\tilde{a}}_{1}}}=\sum_{n=0}^{\infty}E_{n}$
(51)

where $E_{n}$ takes the form

$E_{n}=\sum
Q_{n_{1},n_{2},n_{3},n_{4}}(y_{i})\frac{(D^{2}_{0})^{n_{2}}(D^{3}_{0})^{n_{3}}(D_{0}^{4})^{n_{4}}}{(D^{1}_{0})^{{\tilde{a}}_{1}+n_{1}}}$
(52)

where all $n_{i}$ are integers, $n_{2},n_{3},n_{4}$ are all positive, $n_{1}$
is either positive or negative and

$n_{2}+n_{3}+n_{4}-n_{1}=n$ (53)

Putting it all together, it follows that the integrand in (43) admits an
expansion of the form

$\sum_{a_{i}}\frac{1}{(u+\tau~{}y_{0}+i\epsilon)^{a_{1}}(-u+\theta~{}y_{1}+i\epsilon)^{a_{2}}(v+\tau~{}y_{0}+i\epsilon)^{a_{3}}(-v-\theta~{}y_{1}+i\epsilon)^{a_{4}}}N_{\\{a_{i}\\}}(y_{i},Z_{i})$
(54)

where

$\left(\sum_{i=1}^{4}a_{i}\right)\leq 2B$ (55)

where, for each $i$, $a_{i}-{\tilde{a}}_{i}$ is a (positive or negative)
integer

The summation in (54) includes terms at all orders in the $\theta$ expansion.
Terms of leading order in this expansion (those which scale like
$\frac{1}{\theta^{2B}}$)are those for which

$\sum_{i}a_{i}=2B$

On the other hand terms at $n^{th}$ subleading order in the small $\theta$
expansion (those which scale like $\frac{1}{\theta^{2B-n}}$) are those for
which

$\sum_{i}a_{i}=2B-n$

.

(54), (55) in particular express the fact that the integrand in (43) is of
order $\frac{1}{\theta^{2{\tilde{B}}-M}}=\frac{1}{\theta^{2B}}$ in the small
$\theta$ limit (the $M$ arises from the scaling of the numerators) and that
there are additional power series corrections to this leading small $\theta$
scaling behaviour.

We now turn to the bulk integral over $u$ and $v$. It is convenient to break
up this integral into two regions; the first $R_{1}$ being disk of radius
(say) A in the $u,v$ plane and the second, $R_{2}$, which is the complement of
$R_{1}$. $A$ is a fixed number, independent of $\theta$, and is chosen to be
smaller than the radius of convergence of the power series expansions for $u$
and $v$. 383838This expansion has a finite radius of convergence because it
arises from the expansion of simple rational functions of $u$ and $v$. The
point of this split is the following; in order to generate a $\sigma$
expansion of our result, it is useful to use the Taylor expansion (54).
However this expansion is only valid at small enough $u$, $v$. 393939In
particular if we illegally used the expansion at large $u$ and $v$, terms that
appear at high enough orders in this integrand would have divergent integrals
over $u$ and $v$., We are allowed to use this expansion only in the region
$R_{1}$ but not in $R_{2}$.

Indeed, the appropriate Taylor expansion of the integrand in $R_{2}$ is in a
power series in $\theta$ and $\tau$, but with $u$, $v$ being treated of order
unity, i.e. $u$ and $v$ dependence being dealt with exactly and not in
expansion. Performing this expansion we immediately find that the integral
(43) in $R_{2}$ is of order unity (in $\theta$, $\tau$ smallness or smaller.
This will be parametrically smaller than the $\theta$ dependence we will
obtain from region $R_{1}$, and so the integral over $R_{2}$ will be
irrelevant for the determination of the behavior of the integral at leading
order in $\theta$. The region $R_{2}$ is also everywhere finitely separated
from the ‘bulk point singularity’ $H_{D-2}$ of section 4 (this is the surface
$u=v=y_{1}=0$, see around (104)) and so the integral over this region is also
non-singular in the bulk point $\rho\to 0$ $(a\to 0)$ limit. For this reason
the integral over $R_{2}$ will play absolutely no role in the analysis of this
paper, and will be ignored both in the rest of this section and also all
through section 5.

For the integral over $R_{1}$ we employ the expansion (54) and perform the
integral term by term. Upto corrections of order unity or smaller (that come
from the fact that the region of integration in $R_{1}$ is bounded)404040As
mentioned above these corrections are non-singular at small $\rho$ and so play
no role - and will be ignored - through the rest of this paper. we can perform
the integral as follows. We make the variable change $u=\theta U$ and
$v=\theta V$ to obtain

$\sum_{i}\theta^{\left(2-\sum_{m}a_{m}\right)}N_{\\{a_{i}\\}}(y_{\mu},Z_{i})f_{\\{a_{m}\\}}(a,y_{0},y_{1})$
(56)

where

$f_{\\{a^{i}\\}}(a,y_{0},y_{i})=\int\frac{dUdV}{(U+a~{}y_{0}+i\epsilon)^{a_{1}}(-U+~{}y_{1}+i\epsilon)^{a_{2}}(V+a~{}~{}y_{0}+i\epsilon)^{a_{3}}(-V-~{}y_{1}+i\epsilon)^{a_{4}}}$
(57)

Recall $a=\frac{\tau}{\theta}$. The integration (57) is performed over the
full real line for $U$ and $V$; the correction from the finite integration
range is of order unity or smaller, as we have explained above. In the
integral above $y_{0}=\sqrt{1+y_{i}^{2}}$. The integrals over in (57) are
easily evaluated using Schwinger parameters; one obtains Heemskerk:2009pn

$\begin{split}f_{\\{a^{i}\\}}(a,y_{0},y_{i})&=\frac{C_{a_{1},a_{2},a_{3},a_{4}}}{\left(a~{}y_{0}+y_{1}+i\epsilon\right)^{a_{1}+a_{2}-1}\left(a~{}y_{0}-y_{1}+i\epsilon\right)^{a_{3}+a_{4}-1}}\\\
&C_{a_{1},a_{2},a_{3},a_{4}}=\frac{\Gamma\left(a_{1}+a_{2}-1\right)\Gamma\left(a_{3}+a_{4}-1\right)}{\Gamma\left(a_{1}\right)\Gamma\left(a_{2}\right)\Gamma\left(a_{3}\right)\Gamma\left(a_{4}\right)}\end{split}$
(58)

In the small $\theta$ limit the dominant term in (56) is of order
$\frac{1}{\theta^{2B-2}}\sim\frac{1}{\sigma^{B-1}}$. The contribution of this
term to the full integral (43) is

$\theta^{-2B+2}H(a,Z_{i})$ (59)

where

$H(a,Z)=\int_{\mathbb{H}_{D-1}}\sum_{i}N_{i}(y_{i},Z_{i})f_{i}(a,\sqrt{1+y_{i}^{2}},y_{1})$
(60)

where the summation in (60) runs only over those terms for which
$\sum_{i}a^{i}_{m}=2B$.

The integral in (60) is taken over the hyperboloid $y_{0}^{2}-y_{i}^{2}=1$,
more precisely on its branch in which

$y^{0}=+\sqrt{1+y_{i}^{2}},~{}~{}~{}~{}~{}0\leq\tau\leq\pi$ (61) 414141In
embedding space the curve $y_{0}^{2}-y_{i}^{2}=1$ has two branches
$y^{0}=\pm\sqrt{1+y_{i}^{2}}$ Each of these branches maps into an infinite
number of branches in covering space. The branch (61) is the one of relevance
to this paper. This is the branch on which the bulk point singularity lies,
and is also the branch on which the $i\epsilon$ assignment in (57) is correct.

###  3.2 Analyticity in $a$

$H(a,Z_{i})$ is a function of $a$ as well as the boundary polarizations
$Z_{i}$. In what follows we allow these polarizations to be a function of $a$,
but demand that this function is chosen to be analytic. For any such choice
$H(a,Z_{i})={\tilde{H}}(a)$ and our four-point function in the Regge limit
takes the form

$\theta^{-2B+2}{\tilde{H}}(a)$ (62)

While the function ${\tilde{H}}(a)$ is in general complicated, in this paper,
we are concerned only with its analytic properties, which are easy to
understand.

${\tilde{H}}(a)$ has two singularities; the power-law ‘bulk point’ singularity
at $a=0$ and a lightcone singularity at $a=1$. In section 5 we study the bulk
point $a\to 0$ singularity in great detail. In this section, we focus our
attention on the lightcone singularity at $a=1$.

The main concern we address in this section is the following. Given that
${\tilde{H}}(a)$ has a singularity at $a=1$, one might worry that it is
possible for ${\tilde{H}}(a)\propto\theta(1-a)$. If ${\tilde{H}}(a)$ behaved
in this manner then the effective value of $B$ in (59) could be smaller for
$a>1$ than it is for $a<1$. In this subsection we explain that this cannot
happen.

Our result follows almost immediately once we analytically continue the
function ${\tilde{H}}(a)$ to complex values of $a$. The fact that such an
analytic continuation is clear from the integral representations (60) and (58)
(see Appendix F for a class of worked examples). It is also expected on
general grounds; the continuation to complex $a$ is effectively a continuation
to complex cross ratios (see (20)), and conformal correlators are, of course,
famously analytic function of cross ratios with branch cuts at lightcone
singularities (the branch cut nature of the complex singularity is explicitly
displayed in a class of simple examples in Appendix F). Elsewhere the
correlators are analytic functions (see Appendix (G) for a review).

Indeed the $i\epsilon$ prescription effectively tells us that we need to
perform an analytic continuation even to work at physical values of
parameters. In particular, we see from (58) 424242And using the fact that
$y_{0}>0$ everywhere on the integration domain in (60)., the ${\tilde{H}}(a)$
is actually a function of the combination of variables

${\tilde{a}}=a+i{\tilde{\epsilon}}$ (63)

for an appropriate definition of ${\tilde{\epsilon}}>0$. It is useful to view
${\tilde{H}}$ as a function of ${\tilde{a}}$. With this convention, the
function ${\tilde{H}}$ has its branch cut singularity exactly at
${\tilde{a}}=1$. 434343The analytic function
${\tilde{H}}^{\tilde{\epsilon}}(a)$ is most conveniently defined to have a
branch cut running from $a=1$ down to $a=0$.. Physics instructs us to work at
real values of $a$, and so at slightly complex values of ${\tilde{a}}$, more
particularly to evaluate the function ${\tilde{H}}({\tilde{a}})$ on a contour
that passes just above the real axis.

Restated, both for $a>1$ and $a<1$, ${\tilde{H}}(a)$ is the restriction of the
analytic function ${\tilde{H}}({\tilde{a}})$ on the real axis, with the limit
to the real axis being taken from above (i.e. from positive values of ${\rm
Im}({\tilde{a}})$). It follows that ${\tilde{H}}(a)$ for $a<1$ is related to
${\tilde{H}}(a)$ for $a>1$ via analytic continuation, the continuation being
taken in the upper half complex ${\tilde{a}}$ plane. Despite the singularity
at $a=1$ 444444When we take $\epsilon\to 0$, therefore, it follows that
${\tilde{H}}(a)$ for $a>1$ and $a<1$ are analytic continuations of each other
. In particular if ${\tilde{H}}(a)=0$ for all $a>1$ then it follows that
${\tilde{H}}(a)$ also vanish for all $a<1$, and so in the limit $a\to 0$.
Restated, if $H(a)$ does not vanish at small $a$, it cannot vanish identically
all $a>1$.

Figure 6: The analytic continuation that takes one from $a>1$ to $a<1$.

###  3.3 Analyticity in the cross ratio $e^{2\rho}$

The discussion above about the analytic properties of the function $H(a)$ can
also be understood in what may be more familiar terms when worded in terms of
cross ratio function $e^{2\rho}=\frac{z}{{\bar{z}}}$ . Recall from (20) that

$\begin{split}&e^{2\rho}=\left(\frac{1-a-i\epsilon}{1+a+i\epsilon}\right)^{2}\end{split}$
(64)

Or in other words

$\begin{split}&e^{2\rho}=\left(\frac{{\tilde{a}}-1}{1+{\tilde{a}}}\right)^{2}\end{split}$
(65)

with ${\tilde{a}}$ given in (63). If we ensure that ${\tilde{a}}$ has a small
positive imaginary part and we take the real part of ${\tilde{a}}$ from
greater than one less than one then the variable $\zeta=e^{2\rho}$ effectively
circles counter-clockwise round the branch cut at $\zeta=0$ 454545To see this
note that that as we pass from ${\rm Re}({\tilde{a}})>1$ to ${\rm
Re}({\tilde{a}}<1)$ with ${\rm Im}(a)=\epsilon>0$, the argument of
${\tilde{a}}-1$ changes from $0$ to $\pi$. It follows, therefore, that when
this happens the argument of $({\tilde{a}}-1)^{2}$ increase from $0$ to
$2\pi$.  In other words, the passage from $a>1$ to $a<1$ (roughly speaking a
$\pi$ rotation in the complex $a$ plane) corresponds to winding counter-
clockwise around the branch point at zero of the cross ratio variable
$e^{2\rho}$ (roughly speaking performing a $2\pi$ rotation in the counter
clockwise direction in the $e^{2\rho}$ plane). The analytic continuation that
takes us from $a>1$ to $a<1$ is simply the analytic continuation that takes us
from just above to just below the branch cut of the correlator when viewed as
a function of the cross ratio variable $e^{2\rho}$. 464646 Noting that
$z={\bar{z}}e^{2\rho}$ and recalling that ${\bar{z}}$ is fixed and finite at
$a=1$ we see that this is basically the same as the fact, established in the
previous section, that passing from $\tau>\theta$ to $\tau<\theta$ corresponds
to circling counter-clockwise round the branch cut singularity at $z=0$. The
impossibility of the function ${\tilde{H}}(a)$ being nontrivial for $a<1$ but
vanishing for $a>1$ now follows as in the previous subsection.

Figure 7: The path traversed in the complex plane by the cross ratio
$e^{2\rho}$ as $a$ moves from greater to one to less than one. $e^{2\rho}$
(and so $z$) circles round the branch point at zero in a counter-clockwise
direction.

All the analytic properties described above are explicitly illustrated in
Appendix G in the context of a simple example; the correlator generated by a
$\phi^{4}$ interaction for four identical bulk scalars dual to boundary
operators of dimension $\Delta$ in Appendix F.

###  3.4 Regge scaling of normalized correlators

In this paper our interest lies mainly in the normalized correlator (31)
(because this is the correlator that is constrained by the chaos bound).
Recall this normalized correlator is only defined when $\Delta_{3}=\Delta_{1}$
and $\Delta_{4}=\Delta_{2}$. Defining

$A^{\prime}=B-\Delta_{1}-\Delta_{2}$ (66)

it follows immediately from (62) and (39) that the leading behavior of
normalized four point function (31) in the Regge limit takes the form

$\frac{H(a)}{\theta^{2A^{\prime}-2}}$ (67)

The fact that (67) holds uniformly - i.e. with the same value of $A$ both for
$a>1$ and $a<1$, i.e. with the same value of $A$ in both the Causally
Scattering and the Causally Regge sheets - is the a central result of this
section.

##  4 The $\rho\to 0$ ($\tau\to 0$) limit

As we have mentioned in the introduction, it is possible to show on very
general grounds that the correlation function studied in this paper is an
analytic function of the insertion points away from $\tau=1$ (see the previous
section) and $\tau=0$ (see Appendix G following Maldacena:2015iua ). In this
section, we will study the singularity at $\tau=0$ more closely, and in
particular, determine its precise structure including its coefficient. The
analysis of this section follows section 6 of Maldacena:2015iua , and also
overlaps with Gary:2009ae ; Penedones:2010ue ; Okuda:2010ym .

###  4.1 Modulated propagators

Recall that every boundary to bulk propagator is a polynomial times

$\frac{1}{(-2P.X+i\epsilon)^{A}}$ (68)

for some value of $A$. In this section we adopt the following strategy. We
choose an arbitrary function $f(a)$ with the following properties

$\begin{split}f_{A}(a)&=1-a^{A}P(a),~{}~{}~{}a\ll 1\\\
P(a)&=\sum_{m=0}^{\infty}c_{m}a^{2m}\\\ \lim_{a\to\infty}&f_{A}(a)=0\\\
\end{split}$ (69)

In words, the function $f$ vanishes at large values of its argument. It tends
to unity at $a=0$, and the deviations from unity take a very particular form.
An example of an $f$ function with these properties is

$f_{A}(a)=1-\frac{a^{A}}{(1+a^{2})^{\frac{A}{2}}}$ (70)

Given any such $f_{A}$ function we next use the trivial identity

$\frac{1}{(x+i\epsilon)^{A}}=\frac{1-f_{A}(\frac{x}{L})}{(x+i\epsilon)^{A}}+\frac{f_{A}(\frac{x}{L})}{(x+i\epsilon)^{A}}$
(71)

to split every propagator into two pieces. At the moment $L$ is an arbitrary
constant. We will, choose its value so that

$\tau\ll L\ll 1$ (72)

(this is possible in the small $\tau$ limit, the regime of interest to this
section).

A key point is that the first term on the RHS of (71) is non-singular at
$x=0$; we call this the smooth piece. On the other hand, the second term in
(71) continues to be singular at $x=0$; indeed this term (which we call the
modulated term) is essentially indistinguishable from (68) for $x\ll L$.

Now bulk contact interaction contribution to any correlator is given by the
product of four boundaries to bulk propagators (to the bulk point $X$) sewn
together by the bulk interaction and integrated over the bulk. Our strategy is
to replace each of the four bulk to boundary propagators by the sum of two
terms (the regular part of the propagator and the singular part of the
propagator) in the integrand above using the identity (71). The integrand now
breaks up into $2^{4}=16$ terms, depending on whether we retain the regular or
modulated part of each propagator. Upon performing the integral over $X$, it
follows immediately from the analysis of Appendix G that the integral of any
term in which even one of the propagators is replaced by its regular part
evaluates to an expression that is non-singular at $\tau=0$. The singularity
in our correlator - the object of study of the current section - comes
entirely from the integrand with every propagator replaced by the
corresponding modulated propagator.

Let us summarize. We have demonstrated that the singular term in the
correlation is unaffected if we make the replacement

$\frac{1}{(-2P.X+i\epsilon)^{A}}\rightarrow\frac{f_{A}(\frac{-2P.X}{L})}{(-2P.X+i\epsilon)^{A}}$
(73)

in every propagator that appears in the integrand of our correlation
functions, i.e. if we replace every propagator by a modulated propagator.

All through this section, we will make the replacement (73) on every
propagator in our integrand. The utility of this manoeuvre is the following.
As the envelope function, $f_{A}(P.X)$ decays away from the lightcone of the
boundary point $P$, once we make the replacement (73), it follows immediately
that the singularity of our correlator receives contributions only from an
envelope of width $L$ around the common intersections of the lightcones of the
boundary points $P_{i}$, a fact that simplifies our analysis below.

###  4.2 Wave representation for propagators

The next step in our procedure is to express the modulated propagators as a
sum over propagating waves (a sort of Fourier transform). This may be
accomplished (Okuda:2010ym ; Maldacena:2015iua ) using the identity

$\frac{1}{(-2P.X+i\epsilon)^{A}}=\frac{e^{-\frac{i\pi
A}{2}}}{2^{A}\Gamma(A)}\int_{0}^{\infty}d\omega~{}\omega^{A-1}e^{i\omega(-P.X+i\epsilon)}$
(74)

It follows that the modulated spin $J$ propagator can be written as

$G_{J}(P,Z,X,W)=\frac{\mathcal{C}^{D}_{\Delta,J}e^{-\frac{i\pi(\Delta+J)}{2}}}{2^{\Delta}\Gamma(\Delta+J)}f(\frac{P.X}{L})\left(-Z.W~{}P.X+{Z.X~{}W.P}\right)^{J}\int
d\omega~{}\omega^{\Delta+J-1}e^{i\omega(-P.X+i\epsilon)}$ (75)

where

$\mathcal{C}^{D}_{\Delta,J}=\frac{(J+\Delta-1)\Gamma(\Delta)}{2\pi^{\frac{D}{2}}(\Delta-1)\Gamma(\Delta-\frac{D}{2}+1)}$
(76)

(75) represents the propagator as a sum over waves of momenta $\omega P_{M}$.
These plane waves are multiplied by the envelope function $f$ polynomial
dressings. However these factors (as well as the curvature of the underlying
AdS background) all vary on scales ranging from $L$ to unity. On the other
hand the singularity of our correlator will turn out to have its origin in
$\omega$ of order $\frac{1}{\tau}\gg\frac{1}{L}\gg 1$. At distance scales
$\tau$ the multiplying factors are effectively constant, and $AdS_{D+1}$ space
is effectively flat, so at these scales (75) is literally a Fourier Transform
representation of the propagator.

Following Maldacena:2015iua  we will now explain how the wave representation
of propagators, (75), can be used to find an expression for the singularity in
$\tau$ of the correlators. We begin by considering very simple situations but
building up to more complicated ones.

###  4.3 Scalar $\phi^{4}$ correlator in $AdS_{3}$

Let us first study the correlation function induced by a
$\phi_{1}\phi_{2}\phi_{3}\phi_{4}$ bulk interaction between 4 scalars of
dimension $\Delta_{1}$, $\Delta_{2}$, $\Delta_{3}$ and $\Delta_{4}$ in $D=2$
(i.e. 2 boundary or 3 bulk dimensions).

We need to evaluate the integral (75) is

$G_{\rm
sing}=\left(\prod_{a}\frac{\mathcal{C}_{\Delta_{a}}e^{-\frac{i\pi\Delta_{a}}{2}}}{2^{\Delta_{a}}\Gamma(\Delta_{a})}\right)\int
dX\left(\prod_{a}d\omega_{a}\omega_{a}^{\Delta_{a}-1}f_{\Delta_{a}}(P_{a}.X/L)\right)e^{-iX.\left(\sum_{a}\omega_{a}P_{a}\right)}$
(77)

where $\mathcal{C}_{\Delta_{a}}$ is short notation for
$\mathcal{C}_{\Delta_{a},J=0}$.

Recall that there are no bulk points $X$ that solve the equation $P_{i}.X=0$
for all $i$ except when $\tau=0$. When $\tau=0$, the unique bulk point that
obeys this equation in $D=2$ (and also has global time between $0$ and $\pi$)
is

$X=X_{0}=(0,1,0,0).$ (78)

It follows that when $\tau\gg L$ the envelopes around the four lightcones (the
lightcones of $P_{1},P_{2},P_{3},P_{4}$) never intersect and the integral in
(77) is very small. On the other hand when $\tau\ll L$ the four envelope
functions overlap in a spacetime region in $AdS_{3}$ of ‘size’ $L$ (volume
$L^{3}$) centred around $X_{0}$. We refer to this region as ‘the elevator’.

It is useful to set

$X=X_{0}+x$ (79)

As only values of $x$ with $x$ less than or of order $L$ lie in the ‘elevator’
(and so contribute significantly to the integral) and as $L\ll 1$, $x$
effectively a tangent space coordinate about the point $X_{0}$ in $AdS_{3}$.
It follows that the space parameterized by the coordinate $x$ is effectively
flat. Now

${\cal P}_{X_{0}}^{MN}=\eta^{MN}+X_{0}^{M}X_{0}^{N}$ (80)

is the projector in $\mathbb{R}^{D,2}$ that projects vectors in this space
orthogonal to $X_{0}$. Expanding the equation $(X_{0}+x)^{2}=-1$ to first
order in $x$, we conclude that

${\cal P}_{X_{0}}^{MN}x_{M}=x_{N}$ (81)

Now the integral (77) evaluates to

$G_{\rm
sing}=\left(\prod_{a}\frac{\mathcal{C}_{\Delta_{a}}e^{-\frac{i\pi\Delta_{a}}{2}}}{2^{\Delta_{a}}\Gamma(\Delta_{a})}\right)\int\left(\prod_{a}d\omega_{a}\omega_{a}^{\Delta_{a}-1}\right)e^{-i\sum_{a}\omega_{a}P_{a}.X_{0}}(2\pi)^{3}{\tilde{\delta}}^{3}(\sum_{a}k_{a})$
(82)

where

$k_{a}^{M}=\omega_{a}({\cal P}_{X_{0}})^{M}_{N}P_{a}^{N}$

${\tilde{\delta}}$ is a $\delta$ function broadened in a Gaussian manner over
scale of order $\delta\omega\sim\frac{1}{L}$. 474747The precise form of this
function can be systematically evaluated in a saddle point approximation. For
our purposes (i.e. leading order computations in the small $\tau$ limit) , the
only thing about about this function that is relevant is that it integrates to
unity, and is nonzero only over a range of of order $\frac{1}{\delta L}$ of
its arguments.

In terms of the vector

$P=\sum_{a}\omega_{a}P_{a}$ (83)

(82) can be recast as

$G_{\rm
sing}=(2\pi)^{3}\left(\prod_{a}\frac{\mathcal{C}_{\Delta_{a}}e^{-\frac{i\pi\Delta_{a}}{2}}}{2^{\Delta_{a}}\Gamma(\Delta_{a})}\right)\int\left(\prod_{a}d\omega_{a}\omega_{a}^{\Delta_{a}-1}\right)e^{-iP.X_{0}}{\tilde{\delta}}^{3}\left(({\cal
P}_{X_{0}})^{M}_{N}P_{N}\right)$ (84)

The ${\tilde{\delta}}$ function in (84) clicks when $({\cal
P}_{X_{0}})^{M}_{N}P_{N}$ vanishes. This happens on a one-parameter set of
$\omega^{a}$. 484848The counting is the following. The space in which the
vectors $k_{a}^{M}$ varies is the tangent space of the point $X_{0}$ and so is
3 dimensional. Any four vectors in 3-dimensional space are linearly dependent
and so obey an equation of the form $\zeta_{a}k_{a}=0$. It follows that the
$\delta$ function in (84) clicks for $\omega_{a}=\omega\zeta_{a}$. Note all
this is true at generic $\tau$, despite the fact that the four 4 dimensional
vectors $P_{a}$ are linearly independent unless $\tau=0$. The values of
$\omega^{a}$ at which this happens is easy to deduce. Expanding to first order
in $\tau$ we find

$P_{1}+P_{2}+P_{3}+P_{4}=2\tau X_{0}$ (85)

so that

$\sum_{a}k_{a}=0$ (86)

It follows that the ${\tilde{\delta}}$ function is nonzero if and only if for
every value of $a$

$\omega_{a}=\omega$ (87) 494949As the delta function has a spread of order
$1/L$, the solution (87) also has the same fuzz. This fuzz will not be
important in what follows and we ignore it.

With these values of $\omega_{a}$ we find

$\begin{split}&k_{1}=\omega(1,0,1,0)\\\ &k_{3}=\omega(1,0,-1,0)\\\
&k_{2}=-\omega(1,0,\cos\theta,\sin\theta)\\\
&k_{4}=-\omega(1,0,-\cos\theta,-\sin\theta)\end{split}$ (88)

Now it is easy to check that

$\int
d\omega_{1}d\omega_{2}d\omega_{3}d\omega_{4}~{}{\tilde{\delta}}^{3}\left(({\cal
P}_{X_{0}})^{M}_{N}P_{N}\right)=-\frac{1}{2\sin\theta}\int d\omega$ (89)

(where $\omega$ is defined in (87)) and so (84) reduces to

$G_{\rm
sing}=-\frac{4\pi^{3}e^{-\frac{i\pi\Delta}{2}}}{\sin\theta}\left(\prod_{a}\frac{\mathcal{C}_{\Delta_{a}}}{2^{\Delta_{a}}\Gamma(\Delta_{a})}\right)\int_{0}^{\infty}d\omega\omega^{\Delta-4}e^{-2i\omega\tau}$
(90)

where,

$\Delta=\sum_{a}\Delta_{a}$

Note that (90) would have diverged at large $\omega$ in the absence of the
phase factor $e^{-i\omega\tau}$. The phase factor regulates this divergence at
$\omega\sim\frac{1}{\tau}$, converting it into a singularity at $\tau=0$. As
promised the singularity has its origin in waves with

$\omega\sim 1/\tau$

and so is a UV effect at small $\tau$. The precise form of the singularity is

$\begin{split}G_{\rm
sing}&=-\frac{4\pi^{3}e^{-\frac{i\pi\Delta}{2}}}{\sin\theta}\left(\prod_{a}\frac{\mathcal{C}_{\Delta_{a}}}{2^{\Delta_{a}}\Gamma(\Delta_{a})}\right)\left(-ie^{-\frac{i\pi\Delta}{2}}\Gamma(\Delta-3)\frac{1}{(2\tau)^{\Delta-3}}\right)\\\
&=\frac{4i\pi^{3}e^{-i\pi\Delta}}{\sin\theta}\left(\Gamma(\Delta-3)\prod_{a}\frac{\mathcal{C}_{\Delta_{a}}}{2^{\Delta_{a}}\Gamma(\Delta_{a})}\right)\frac{1}{(2\tau)^{\Delta-3}}\end{split}$
(91) 505050The final result (91) is correct only at leading order in the limit
$\tau\to 0$. The corrections to this answer come from the fact that the
${\tilde{\delta}}$ function is not a completely sharp but has a width $\delta
w$. It follows, for instance, that the integrand in (90) is actually a power
series in $\omega$ of the schematic form
$\omega^{a}\rightarrow\omega^{a}+\omega^{a-1}\delta\omega+\omega^{a-2}\delta\omega^{2}+\ldots$
As every factor of $\omega$ is effectively of order $\frac{1}{\tau}$ while
$\delta\omega$ is independent of $\tau$, these correction give corrections to
the singularity in (91) of order ${\cal
O}(1/\tau^{\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-m})$ where $m$ is an
integer.

###  4.4 General scalar contact correlator in $AdS_{3}$

The generalization of the discussion of the previous subsubsection to the
study of the singularity of a scalar correlator resulting from a more general
bulk contact interaction than $\phi^{4}$ (i.e. some number of derivatives
acting on the $\phi$ fields) is completely straightforward.

The analogue of (82) in this case is

$G_{\rm sing}=\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\int
d\omega_{a}\omega_{a}^{\Delta_{a}-1}\right)S(k_{a})e^{-i\sum_{a}\omega_{a}P_{a}.X_{0}}(2\pi)^{3}{\tilde{\delta}}^{3}(\sum_{a}k_{a})$
(92)

where,

$\begin{split}\tilde{\mathcal{C}}_{\Delta,J}&=\frac{\mathcal{C}_{\Delta,J}e^{-\frac{i\pi(\Delta+J)}{2}}}{2^{\Delta}\Gamma(\Delta+J)}\\\
\text{and,
}\tilde{\mathcal{C}}_{\Delta}&=\tilde{\mathcal{C}}_{\Delta,0}\end{split}$ (93)

and, $S(k_{a})$ is the contact interaction evaluated on the waves
$k_{a}=\omega P_{a}$. Evaluating the integral over $\omega_{a}$, as in the
previous subsection, gives the analogue of (90)

$G_{\rm
sing}=-\frac{4\pi^{3}}{\sin\theta}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\int
d\omega\omega^{\Delta-4}S(\omega)e^{-2i\omega\tau}$ (94)

where $S(\omega)$ is now the interaction contact term evaluated on the waves
(95), i.e. the invariant transition amplitude of the S matrix generated in
flat space by the contact term in question for the scattering momenta (95).

The vectors on which $S(\omega)$ is evaluated are (88) where they are
displayed as momenta in the embedding space $\mathbb{R}^{2,2}$. All the
vectors in (88), however, are orthogonal to $X_{0}$ and so lie in its tangent
space. This tangent space consists of the collection of $\mathbb{R}^{2,2}$
vectors with second component set to zero. Effectively these vectors lie in
the $\mathbb{R}^{2,1}$ obtained by deleting the second component of all
vectors in (88). With this convention

$\begin{split}&k_{1}=\omega(1,1,0)\\\ &k_{3}=\omega(1,-1,0)\\\
&k_{2}=-\omega(1,\cos\theta,\sin\theta)\\\
&k_{4}=-\omega(1,-\cos\theta,-\sin\theta)\end{split}$ (95) 515151For complete
clarity we reiterate that the first coordinate of each of the vectors here is
timelike, the second and third coordinates are spacelike. The vectors in (95)
are obtained from those in (88) by deleting the zero in the second component
of all vectors in (88).

Note that $k_{1}$ and $k_{3}$ are 3 momenta with positive energy, and so
particles $1$ and $3$ are initial states. On the other hand the momenta
$k_{2}$ and $k_{4}$ have negative energy, so that particles $2$ and $4$ are
the final states. The 3 momenta (95) are conserved (this is, of course, a
consequence of the delta function in (92)). And $S(\omega)$ in (94) is simply
the S matrix 525252Or more precisely the invariant transition amplitude, the S
matrix with the momentum conserving delta function deleted. for the scattering
process with the momenta (88). The Mandlestam variables for this scattering
process are given by

$\begin{split}&s\equiv-(k_{1}+k_{3})^{2}=4\omega^{2},\\\
&t\equiv-(k_{1}+k_{2})^{2}=-2\omega^{2}(1-\cos\theta)\\\
&u\equiv(k_{1}+k_{4})^{2}=-2\omega^{2}(1+\cos\theta)\end{split}$ (96)

From (18) we see that $\sin\theta=2\sqrt{\sigma(1-\sigma)}$ so that (94) can
be rewritten more invariantly as

$G_{\rm
sing}=-\frac{2\pi^{3}}{\sqrt{\sigma(1-\sigma)}}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\int
d\omega\omega^{\Delta-4}S(\omega)e^{-2i\omega\tau}$ (97)

In the case of the scattering of four scalars, the S matrix is a function only
of $s$ and $t$. Let the invariant amplitude as a function of $s$ and $t$ be
denoted by $T(s,t)$. It follows that

${\cal S}(\omega)=iT\left(4\omega^{2},-2\omega^{2}(1-\cos\theta)\right)$ (98)

It follows that (94) can be rewritten in terms of the invariant transition
amplitude $T(s,t)$ generated by the contact interaction as

$\begin{split}G_{\rm
sing}&=-\frac{i2\pi^{3}}{\sqrt{\sigma(1-\sigma)}}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\int
d\omega\omega^{\Delta-4}e^{-2i\omega\tau}T\left(4\omega^{2},-2\omega^{2}(1-\cos\theta)\right)\end{split}$
(99)

Let us suppose that the contact term in question is of $r^{th}$ order in
spacetime derivatives. Then

$T\left(4\omega^{2},-2\omega^{2}(1-\cos\theta)\right)=\omega^{r}T\left(4,-2(1-\cos\theta)\right)$
(100)

so that

$\begin{split}G_{\rm
sing}&=-\frac{2\pi^{3}}{\sqrt{\sigma(1-\sigma)}}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\int
d\omega~{}\omega^{\Delta+r-4}e^{-2i\omega\tau}(iT\left(4,-2(1-\cos\theta)\right))\\\
&=-\frac{2\pi^{3}(iT\left(4,-2(1-\cos\theta)\right))}{\sqrt{\sigma(1-\sigma)}}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\int
d\omega~{}\omega^{\Delta+r-4}e^{-2i\omega\tau}\\\
&=-\frac{2\pi^{3}(iT\left(4,-2(1-\cos\theta)\right))}{\sqrt{\sigma(1-\sigma)}}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\left(-ie^{-\frac{i\pi(\Delta+r)}{2}}\Gamma(\Delta+r-3)\frac{1}{(2\tau)^{\Delta+r-3}}\right)\end{split}$
(101)

which simplifies to,

$\begin{split}G_{\rm
sing}&=\frac{2i\pi^{3}e^{-\frac{i\pi(\Delta+r)}{2}}}{\sqrt{\sigma(1-\sigma)}}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\Gamma(\Delta+r-3)\frac{(iT\left(4,-2(1-\cos\theta)\right))}{(2\tau)^{\Delta+r-3}}\end{split}$
(102)

Using the relationship (18), (102) can be rewritten as

$\begin{split}G_{\rm
sing}&=2i\pi^{3}\frac{e^{-\frac{i\pi(\Delta+r)}{2}}}{2^{\Delta+r-3}}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\Gamma(\Delta+r-3)~{}~{}~{}\frac{\sqrt{1-\sigma}^{\left(\Delta+r-4\right)}(iT\left(4,-4\sigma\right))}{\sigma^{\frac{\Delta+r-2}{2}}(-\rho)^{\Delta+r-3}}\end{split}$
(103)

(103) is a completely explicit expression for the leading singularity at small
$\rho$ of the four point function generated by a particular contact term in
$AdS_{3}$ in terms of the flat space S matrix generated by the same contact
term. (112) agrees perfectly with the expression for the S-matrix in terms of
the coefficient of singularity of the CFT four point function as derived using
a slightly different method (consisting of producing localized bulk waves by
smearing boundary insertion points) in Gary:2009ae  (see eq (3.37)).

As an aside let us highlight an initially puzzling aspect of (103). As
$\Delta$ is not necessarily an integer, the correlator (103) at fixed $\sigma$
has a branch cut around $\rho=0$ in the complex $\rho$ plane. This is
puzzling, and branch cuts in correlation functions in conformal field theories
are usually associated with operator ordering ambiguities across lightcones;
however, there is no causal significance to the point $\rho=0$ (the boundary
separation between no two points changes from spacelike to timelike across
$\rho=0$). We present a brief discussion of this question in Appendix H.

###  4.5 General contact scalar interaction in $AdS_{D+1}$

The new element in this story when $D>2$ is that the boundary lightcones
intersect over a manifold ($\mathbb{H}_{D-2}$) rather than at a point.

The integrand in the analogue of the expression (77) is now nonzero everywhere
in an envelope around an $\mathbb{H}_{D-2}$. To describe this hyperboloid it
is useful to choose coordinates so that

$X_{M}=(V^{0},Y^{0},V^{1},V^{2},Y^{i}),~{}~{}~{}~{}i=1\ldots D-2$ (104) 535353
Let us recall that the space spanned by the $P_{a}$ is $\mathbb{R}^{2,2}$
(preserving the symmetry group $SO(D-2)$) when $\tau\neq 0$, but is
$\mathbb{R}^{2,1}$ (preserving the symmetry group $SO(D-2,1)$) when $\tau=0$.
The $SO(D-2,1)$ is the subgroup that rotates the $Y_{\mu}$ coordinates into
each other in the usual Minkowskian fashion. The $SO(D-2)$ rotates the $Y^{i}$
into each other.

The hyperboloid over which boundary lightcones intersect is located at
$V^{\mu}=0$. Points on the hyperboloid are parameterized by the coordinates
$Y^{\mu}$, subject to the relation

$Y^{2}=-1.$

It is useful to proceed as follows. We break up the $\mathbb{H}_{D-2}$ into
little cells of volume $V_{D-2}$. Like the variable $L$ earlier in this
section, these cells are chosen to be large compared to $\tau$ but small
compared to unity. Restricting ourselves to a particular cell and performing
the integral over $x$ we find

$G_{\rm
sing}=\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\int\left(\prod_{a}d\omega_{a}\omega_{a}^{\Delta_{a}-1}\right)S(k_{a})e^{i\sum_{a}\omega_{a}P_{a}.X_{0}}(2\pi)^{(D+1)}{\tilde{\delta}}^{D+1}(\sum_{a}k_{a})$
(105)

Note that (105) is the same as (92) except for the replacement

${\tilde{\delta}}^{3}\left(({\cal
P}_{X_{0}})^{M}_{N}P_{N})\right)\rightarrow{\tilde{\delta}}^{D+1}\left(({\cal
P}_{X_{0}})^{M}_{N}P_{N})\right)$ (106)

The manipulations leading up (99) continue to work, except that the integral
over 3 of the four $\omega_{a}$ variables (see (82)) are sufficient to use up
only 3 of the $D-1$ ${\tilde{\delta}}$ functions on the RHS of (106). The
analogue of (89) is

$(2\pi)^{(D+1)}\int
d\omega_{1}d\omega_{2}d\omega_{3}d\omega_{4}~{}{\tilde{\delta}}^{D+1}\left(({\cal
P}_{X_{0}})^{M}_{N}P_{N}\right)=-\frac{(2\pi)^{(D+1)}\delta^{D-2}(0)}{2\sin\theta}\int
d\omega=-(2\pi)^{3}\frac{V_{D-2}}{2\sin\theta}\int d\omega$ (107)

We now need to sum the results (107) over all the cells in $\mathbb{H}_{D-2}$.
Note that this summation is weighted by the volume of those cells. As the size
of each cell is small compared to the $AdS$ radius (unity in our units) the
summation is well approximated by an integral and (99) turns into

$G_{\rm
sing}=-\frac{2\pi^{3}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)}{\sqrt{\sigma(1-\sigma)}}\int_{\mathbb{H}_{D-2}}\sqrt{g_{D-2}}~{}d^{D-2}X\int
d\omega\omega^{\Delta-4}e^{i\omega P.X}{\cal S}\left(\omega\right)$ (108)

where $\sqrt{g_{D-2}}d^{D-2}X$ is the volume element on $\mathbb{H}_{D-2}$
where $S(k)$ is the S matrix (more precisely invariant transition amplitude)
for four scalars with momenta

$\begin{split}&k_{1}=\omega(1,0,1,0,{\vec{0}})\\\
&k_{3}=\omega(1,0,-1,0,{\vec{0}})\\\
&k_{2}=-\omega(1,0,\cos\theta,\sin\theta,{\vec{0}})\\\
&k_{4}=-\omega(1,0,-\cos\theta,-\sin\theta,{\vec{0}})\end{split}$ (109)

Note that all the vectors $k_{a}^{M}$ have vanishing components in all $Y_{M}$
directions. It follows that the vectors $k_{a}$ lie in the tangent space of
the $\mathbb{H}_{D-2}$ at every value of $Y^{\mu}$. It follows that (96) and
(98) apply without modification to this $D+1$ dimensional scattering process.
Note, in particular, that in this case $S(k)$ is independent of the point
$Y_{\mu}$ on the hyperboloid. For this reason the integral over the
hyperboloid in (108) is easily performed (the same will not be true for
scattering of spinning particles, as we will see in the next section). To do
this is is useful to parameterize points on $\mathbb{H}_{D-2}$ by

$Y_{i}=\sinh\zeta{\hat{n}}_{i},~{}~{}~{}Y_{0}=\cosh\zeta$ (110)

where ${\hat{n}}_{i}$ is a unit vector in $\mathbb{R}^{D-2}$. It is easily
verified that

$d^{D-2}X=d\zeta~{}\sinh^{D-3}\zeta~{}d\omega_{D-3}$

. It follows from (85) that

$X.P=-2\tau\cosh\zeta$

The integral over angles produce the volume of the unit $D-3$ sphere,
$\Omega_{D-3}$. It remains to perform the integrals over $\zeta$ and $\omega$.
While we can perform these integrals in any order, it is easier to perform the
(elementary) integral over $\omega$ first. Assuming that it is of $r^{th}$
order in derivatives, we find

$G_{\rm
sing}=i(2\pi^{3})\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\Omega_{D-3}e^{-\frac{i(\Delta+r)}{2}}\frac{\Gamma(\Delta+r-3)}{2^{\Delta+r-3}}\frac{\sqrt{1-\sigma}^{\left(\Delta+r-4\right)}~{}(i{T}\left(4,-4\sigma\right))}{\sigma^{\frac{\Delta+r-2}{2}}(-\rho)^{\Delta+r-3}}\int
d\zeta\frac{\sinh^{D-3}\zeta}{\cosh\zeta^{\Delta+r-3}}$ (111)

The integral over $\zeta$ is now simply a finite number. Our final answer is

$\begin{split}G_{\rm
sing}&=\left[i(2\pi^{3})N_{D,\Delta}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a}}\right)\Omega_{D-3}e^{-\frac{i(\Delta+r)}{2}}\frac{\Gamma(\Delta+r-3)}{2^{\Delta+r-3}}\right]\frac{\sqrt{1-\sigma}^{\left(\Delta+r-4\right)}~{}(i{T}\left(4,-4\sigma\right))}{\sigma^{\frac{\Delta+r-2}{2}}(-\rho)^{\Delta+r-3}},\\\
N_{D,\Delta}&=\int_{0}^{\infty}d\zeta\frac{\sinh^{D-3}\zeta}{\cosh\zeta^{\Delta+r-3}}=\frac{\Gamma\left(\frac{D-2}{2}\right)\Gamma\left(\frac{1}{2}(\Delta+r-D)\right)}{2~{}\Gamma\left(\frac{1}{2}(\Delta+r-2)\right)},\qquad\Omega_{n}=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}\\\
\end{split}$ (112)

As in the case $D=2$ (112) agrees exactly with the expression for the S-matrix
in terms of the coefficient of singularity of the CFT four point function
derived in Gary:2009ae  (see eq (3.37)) using slightly different methods. The
method of derivation employed in this paper will allow for easy generalization
to the scattering of massless spinning particles in the rest of this section.

As in the previous subsection, it might seem odd to the reader that (112)
displays a branch cut type singularity at $\rho=0$, a point at which no causal
relations get changed. We investigate this question in Appendix H.2.

###  4.6 Contact interactions involving gauge bosons and gravitons

The analysis of the previous subsection goes through when studying photons (or
non-abelian gauge bosons) and gravitons with a few extra twists. From the
point of view of the analysis of this section, the main difference between
gauge bosons, gravitons, and scalars lies in their boundary to bulk
propagators.

####  4.6.1 The spin one propagator

Let us denote the propagator of a spin $J$ particle of dimension $\Delta$ by
$(G_{\Delta})_{A_{1}\ldots A_{J}}^{M_{1}\ldots M_{J}}$ where $A_{i}$ are the
boundary indices and $M_{J}$ are the bulk indices. Then (see Appendix B.9 and
Costa:2014kfa )

$Z^{A_{1}}\ldots Z^{A_{J}}(G_{\Delta})_{A_{1}\ldots A_{J}}^{M_{1}\ldots
M_{J}}W_{M_{1}}\ldots W_{M_{J}}=\frac{\mathcal{C}_{\Delta,J}}{e^{-i\pi
J}2^{\Delta}}\frac{\left(Z.W~{}P.X-{Z.X~{}W.P}\right)^{J}}{(-P.X)^{\Delta+J}}$
(113)

where $Z^{A}$ and $W_{M}$, respectively, are arbitrary boundary and bulk
tangent vectors that obey $Z_{A}Z^{A}=W_{M}$ and the explicit form of
$\mathcal{C}_{\Delta,J}$ is given by (76). When $J=1$, the bulk gauge field at
position $X$, sourced by ‘current’ with polarization $Z^{M}$, is given by

$Z^{A}(G_{\Delta})_{AM}=\frac{-\mathcal{C}_{\Delta,1}(D-1)}{(-2)^{\Delta+1}}\left(\frac{Z_{M}}{(P.X)^{\Delta}}-\frac{(Z.X)P_{M}}{(P.X)^{\Delta+1}}\right)$
(114)

where $Z^{A}$ is an

Here $A$ is a bulk index which is understood to be projected orthogonal to
$X_{A}$. It is useful to make the projection more explicit. Let

$\left({\cal P}_{X}\right)^{M}_{N}\equiv\delta^{M}_{N}+X^{M}X_{N}$ (115)

denote the projector orthogonal to $X_{0}$. Given any vector ‘field $A^{M}$
let us also define

$A_{M}^{\perp}\equiv\left({\cal P}_{X}\right)^{M}_{N}A^{N}$ (116)

It follows that the gauge field at the point $X$ is given by

$Z^{A}(G_{\Delta})_{AM}=\frac{-\mathcal{C}_{\Delta,1}(D-1)}{(-2)^{\Delta+1}}\left(\frac{Z_{M}^{\perp}}{(P.X)^{\Delta}}-\frac{(Z.X)P^{\perp}_{M}}{(P.X)^{\Delta+1}}\right)$
(117)

This expression can further be manipulated to545454The boundary tangent vector
$Z_{A}$ has $D$ independent components, which is the same as the number of
degrees of freedom in a massive vector field propagating in $D+1$ bulk
dimensions. (118) makes clear that - roughly speaking - the $D-1$ components
in $Z^{\perp}$ parameterize the transverse degrees of freedom while $Z.X$
labels the longitudinal degree of freedom.

$Z^{A}(G_{\Delta})_{AM}=\frac{-\mathcal{C}_{\Delta,1}(D-1)}{(-2)^{\Delta+1}}\left(\left(1-\frac{1}{\Delta}\right)\frac{Z_{M}^{\perp}}{(P.X)^{\Delta}}+\nabla_{M}\left(\frac{Z.X}{\Delta(P.X)^{\Delta}}\right)\right)$
(118)

where the $AdS_{D+1}$ covariant derivative is defined by

$\nabla_{M}=({\cal P}_{X})^{N}_{M}\partial_{N}.$

It follows that

$(G_{\Delta})_{AM}=(G_{\Delta}^{1})_{AM}+(G_{\Delta}^{2})_{AM}$ (119)

with

$\begin{split}(G_{\Delta}^{1})_{AM}&=\frac{-\mathcal{C}_{\Delta,1}(D-1)}{(-2)^{\Delta+1}}\left(1-\frac{1}{\Delta}\right)\frac{\eta_{AM}+X_{A}X_{M}}{(P.X)^{\Delta}},\\\
(G_{\Delta}^{2})_{AM}&=\nabla_{M}(\zeta_{\Delta})_{A},~{}~{}~{}\zeta_{\Delta}^{A}=\frac{-\mathcal{C}_{\Delta,1}(D-1)}{(-2)^{\Delta+1}}\frac{X^{A}}{\Delta(P.X)^{\Delta}},\end{split}$
(120)

It is easily verified (see Appendix B.11) that

$\nabla^{A}(G_{\Delta}^{1})_{AM}=0$ (121) 555555On the other hand
$\nabla^{A}\left(G^{2}_{AM}\right)=\nabla_{M}\nabla^{A}\zeta_{A}=\nabla_{M}\left(\frac{1}{(P.X)^{\Delta+1}}\right)$
(122)

It follows that (119) decomposes the boundary propagator into two pieces, the
first of which is conserved (divergenceless) on the boundary while the second
of which is a gradient in the bulk. 565656 It follows that the contribution of
$G^{1}_{AM}$ to Witten diagrams yields a term in the boundary correlator that
is identically conserved. On the other hand the contribution of $G^{2}_{AM}$
to Witten diagrams yields a term in the bulk correlator that is generically
non conserved. However this second contribution sometimes vanishes. This
happens, for instance, when all bulk interactions happen to be ‘gauge
invariant’ - i.e. built only out of field strengths of the bulk vector field
and their derivatives. For general $\Delta$ there is no reason for bulk
interactions to be gauge invariant. The case of $\Delta=D-1$ is special and
will be dealt with in detail below.

So far we have worked at arbitrary values of the dimension, $\Delta$ of our
spin one operator. In this paper, we will focus our attention on the special
case $\Delta=D-1$. In this case, the bulk gauge field enjoys invariance under
bulk gauge transformations. In this case, the second terms in (118) and (119)
are pure gauge and can be dropped (as their contribution to any correlator
vanishes). It follows in this case that, effectively,

$\begin{split}&Z^{A}(G_{D-1})_{AM}=\frac{\mathcal{C}_{D-1,1}(D-1)}{2^{D}}\left(\frac{D-2}{D-1}\right)\frac{Z_{M}^{\perp}}{(-P.X)^{D-1}}\\\
&(G_{D-1})_{AM}=\frac{\mathcal{C}_{D-1,1}(D-1)}{2^{D}}\left(\frac{D-2}{D-1}\right)\frac{\eta_{AM}+X_{A}X_{M}}{(-P.X)^{D-1}}\end{split}$
(123)

The ‘plane wave’ representation of this propagator is

$\begin{split}Z^{A}(G_{D-1})_{AM}&=\frac{\mathcal{C}_{D-1,1}\left(D-2\right)}{2^{D}i^{D-1}\Gamma(D-1)}\int_{0}^{\infty}d\omega~{}\omega^{D-2}Z_{M}^{\perp}~{}e^{-i\omega
P.X-\epsilon\omega}\end{split}$ (124)

####  4.6.2 The spin two propagator

In the case of the spin two propagator, similar manipulations to those
performed in the previous subsubsection yield the analogue of (117)

$\begin{split}Z^{A_{1}}Z^{A_{2}}(G_{\Delta})_{A_{1}A_{2}M_{1}M_{2}}&=\frac{\mathcal{C}_{\Delta,2}(D^{2}-1)}{2(-2)^{\Delta}}\left(\frac{Z_{M_{1}}^{\perp}Z_{M_{2}}^{\perp}}{(P.X)^{\Delta}}-\frac{(Z.X)(P_{M_{1}}^{\perp}Z_{M_{2}}^{\perp}+Z_{M_{1}}^{\perp}P_{M_{2}}^{\perp})}{(P.X)^{\Delta+1}}\right.\\\
&\left.+\frac{(Z.X)^{2}}{(P.X)^{\Delta+2}}P_{M_{1}}^{\perp}P_{M_{2}}^{\perp}\right)\end{split}$
(125)

and of (118)

$\begin{split}Z^{A_{1}}Z^{A_{2}}(G_{\Delta})_{A_{1}A_{2}M_{1}M_{2}}=&\frac{\mathcal{C}_{\Delta,2}}{(-2)^{\Delta}}\left(-\frac{\left(D^{2}-1\right)(X.P)^{-\Delta}\left((X.Z)^{2}\left(\eta_{M_{1}M_{2}}+X_{M_{1}}X_{M_{2}}\right)-(\Delta-1)Z^{\perp}_{M_{1}}Z^{\perp}_{M_{2}}\right)}{2(\Delta+1)}\right.\\\
&\left.+\left(\text{D}^{2}-1\right)\left(\nabla_{M_{1}}\xi^{\perp}_{M_{2}}+\nabla_{M_{1}}\xi^{\perp}_{M_{2}}\right)\right)\end{split}$
(126)

where,

$\xi^{\perp}_{A}=\frac{1}{2(\Delta+1)}\frac{Z.X}{(P.X)^{\Delta}}\left(-Z^{\perp}_{A}+\frac{1}{2}\frac{Z.X}{P.X}P^{\perp}_{A}\right)$
(127)

Note that when $\Delta=D$, the second term in (126) is pure gauge. As in the
previous subsection it follows that the propagator can be decomposed into two
pieces

$\displaystyle Z^{A_{1}}Z^{A_{2}}(G^{1}_{\Delta})_{A_{1}A_{2}M_{1}M_{2}}$
$\displaystyle=$
$\displaystyle\frac{\mathcal{C}_{\Delta,2}}{(-2)^{\Delta}}\left(-\frac{\left(D^{2}-1\right)(X.P)^{-\Delta}\left((X.Z)^{2}\left(\eta_{M_{1}M_{2}}+X_{M_{1}}X_{M_{2}}\right)-(\Delta-1)Z^{\perp}_{M_{1}}Z^{\perp}_{M_{2}}\right)}{2(\Delta+1)}\right)$
$\displaystyle Z^{A_{1}}Z^{A_{2}}(G^{2}_{\Delta})_{A_{1}A_{2}M_{1}M_{2}}$
$\displaystyle=$
$\displaystyle\frac{\mathcal{C}_{\Delta,2}}{(-2)^{\Delta}}\left(\left(\text{D}^{2}-1\right)\left(\nabla_{M_{1}}\xi^{\perp}_{M_{2}}+\nabla_{M_{1}}\xi^{\perp}_{M_{2}}\right)\right)$
(128)

It is then easily verified (see Appendix B.11) that for $\Delta=D$

$\nabla^{A_{1}}Z_{2}^{A_{2}}(G^{1}_{D})_{A_{1}A_{2}M_{1}M_{2}}=0$ (129)

We will focus on the special case $\Delta=D$. In this case $G_{D}^{2}$ is pure
gauge and can be ignored. Effectively

$\begin{split}Z^{A_{1}}Z^{A_{2}}(G_{D})_{A_{1}A_{2}M_{1}M_{2}}&=-\frac{\mathcal{C}_{\Delta,2}\left(D-1\right)}{2(-2)^{D}(X.P)^{D}}\left((X.Z)^{2}\left(\eta_{M_{1}M_{2}}+X_{M_{1}}X_{M_{2}}\right)-(D-1)Z^{\perp}_{M_{1}}Z^{\perp}_{M_{2}}\right)\\\
\end{split}$ (130)

The ‘plane wave’ representation of this propagator is

$\begin{split}&Z^{A_{1}}Z^{A_{2}}(G_{D})_{A_{1}A_{2}M_{1}M_{2}}\\\
&=\frac{\mathcal{C}_{D,2}}{i^{D}\Gamma(D)}\left(\frac{(D-1)^{2}}{2^{D+1}}\right)\int_{0}^{\infty}d\omega~{}\omega^{D-1}\left(Z^{\perp}_{M_{1}}Z^{\perp}_{M_{2}}-\frac{(X.Z)^{2}\left(\eta_{M_{1}M_{2}}+X_{M_{1}}X_{M_{2}}\right)}{D-1}\right)~{}e^{-i\omega
P.X-\epsilon\omega}\end{split}$ (131)

###  4.7 Singularity in terms of scattering

It follows immediately from (124) and (131) that the singular part of
correlator involving boundary scalars, conserved current and conserved stress
tensor operators is given by the following generalization of (108)

$G_{\rm
sing}=-\frac{2\pi^{3}\left(\prod_{a}\tilde{\mathcal{C}}_{\Delta_{a},J}\right)}{\sqrt{\sigma(1-\sigma)}}\int_{\mathbb{H}_{D-2}}\sqrt{g_{D-2}}~{}d^{D-2}X\int
d\omega\omega^{\Delta-4}e^{i\omega P.X}{\cal S}_{X}\left(\omega\right)$ (132)

with

$\displaystyle\tilde{\mathcal{C}}_{\Delta,0}=\frac{\mathcal{C}_{\Delta,0}e^{-\frac{i\pi(\Delta)}{2}}}{2^{\Delta}\Gamma(\Delta)},\qquad\tilde{\mathcal{C}}_{D-1,1}=\frac{\mathcal{C}_{D-1,1}\left(D-2\right)}{2^{D}i^{D-1}\Gamma(D-1)},\qquad\tilde{\mathcal{C}}_{D,2}=\frac{\mathcal{C}_{D,2}}{i^{D}\Gamma(D)}\left(\frac{(D-1)^{2}}{2^{D+1}}\right)$

for scalars, photons and gravitons respectively and ${\cal S}_{X}$ is the S
matrix for the scattering of the waves

$\begin{split}\phi&=e^{ik.x}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{Scalar}\\\
A_{M}&=Z^{\perp}_{M}~{}e^{ik.x}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{Vector}\\\
h_{MN}&=\left(Z^{\perp}_{M_{1}}Z^{\perp}_{M_{2}}-\frac{(X.Z)^{2}\left(\eta_{M_{1}M_{2}}+X_{M_{1}}X_{M_{2}}\right)}{D-1}\right)~{}~{}~{}~{}~{}~{}{Graviton}\\\
\end{split}$ (134)

Here $Z_{M}$ is the boundary polarization of the current or stress tensor
operator, and $k$ is the momentum for the appropriate particle listed in
(109)).

Notice that while all four scattering momenta, given by (109), are independent
of the point $X$, the polarizations listed in (134) depend on the point $X$ on
the hyperboloid (over which the integral in (132) is performed) because

$Z_{M}^{\perp}=({\cal P}_{X})^{N}_{M}Z_{N}$ (135)

(see (115) for the definition of the projector $({\cal P}_{X})$). It follows
that the $S$ matrix ${\cal S}_{X}(\omega)$ that appears in (132) depends on
$X$, and so we cannot trivially perform the integral over the $S^{D-3}$ in
(132) (as we were able to do in subsection (4.5)).

Performing the integral over $\omega$ in (132) we find the analogue of (111):

$G_{\rm
sing}=i\left(2\pi^{3}\left(\tilde{\mathcal{C}}_{\Delta,J}\right)^{4}\right)\Gamma(\Delta+r-3)e^{-\frac{i(\Delta+r)}{2}}~{}~{}~{}\frac{\sqrt{1-\sigma}^{\left(\Delta+r-4\right)}~{}}{\sigma^{\frac{\Delta+r-2}{2}}\rho^{\Delta+r-3}}\int
d\Omega_{D-3}d\zeta\frac{\sinh^{D-3}\zeta}{\cosh\zeta^{\Delta+r-3}}\left(\frac{{\cal
S}_{X}(\omega)}{\omega^{r}}\right)$ (136)

where we have assumed, as in the subsection (4.5), that the interaction term
is of order $r$ in derivatives so that the S matrix scales with overall energy
scale like ${\cal S}(\omega)\sim\omega^{r}$ so that the quantity

${\tilde{\cal S}}(\omega)=\left(\frac{{\cal
S}_{X}(\omega)}{\omega^{r}}\right)$ (137)

that appears in (136) is actually independent of $\omega$. 575757Note that the
RHS of (136) involves an integral over the hyperboloid $\mathbb{H}_{D-2}$.
$\zeta$ and the angles on the $D-3$ sphere $\Omega_{D-3}$ are coordinates on
this hyperboloid. $d\Omega_{D-3}d\zeta{\sinh^{D-3}\zeta}$ is the usual volume
element on the hyperboloid. The factor of $\cosh\zeta^{\Delta+r-3}$ in the
denominator is a result of doing the $\omega$ integral, and is a consequence
of the fact that the phase factor $e^{iP.X}$ breaks the $SO(D-2,1)$ isometry
of the hyperboloid.

###  4.8 Transforming to standard graviton polarizations

While the formulae presented in the previous subsection are all accurate, the
flat space graviton polarization that appears in (134) is presented in an
unusual gauge. In this subsection, we will explain this fact and also gauge
transform to a more standard gauge.

A linearized onshell graviton in flat space always obeys the massless onshell
condition $k^{2}=0$. In addition, the linearized Einstein equations also
impose the following condition on $h_{MN}$

$k^{M}h_{MN}-\frac{k_{N}}{2}h=0$ (138)

where $h=h^{M}_{M}$ is the trace of $h_{MN}$.

When studying gravitational scattering, it is conventional to work in a
Lorentz type gauge in which

$k^{M}h_{MN}=0$ (139)

If we impose this condition, it follows immediately from (138) that

$h=0$ (140)

Let us now turn to the graviton wave presented in (134). It is easy to verify
that $h_{MN}$ that appears in (134) obeys

$k^{M}h_{MN}=-\frac{k_{N}\left(X.Z\right){}^{2}}{D-1},~{}~{}~{}~{}~{}h=-\frac{2\left(X.Z\right){}^{2}}{D-1}$
(141)

It follows that $h_{MN}$ listed in (134) obeys (138), even though it does not
obey the equations (139) or (140) individually.

It is, of course, possible to gauge transform $h_{MN}$ listed in (134) to
ensure that it obeys (139) (and so, automatically, (140)). Under an
infinitesimal gauge transformation

${\tilde{h}}_{MN}=h_{MN}+\zeta_{M}k_{N}+\zeta_{N}k_{M}$ (142)

It follows from (141) that the transfrormed ${\tilde{h}}_{MN}$ obeys both
(139) and (140) provided

$k.\zeta=\frac{\left(X.Z\right){}^{2}}{D-1}$ (143)

Of course (143) is one condition on $D+1$ variables, and so does not
completely determine $\zeta$. If we want a concrete particular formula for the
transformed $h_{MN}$ we need to impose $D$ additional conditions; these
conditions are arbitrary and can be chosen as per convenience. 585858The
reason for this ambiguity is the following. In $D+1$ spacetime dimensions the
number of metric components is $\frac{(D+2)(D+1)}{2}$ while the number of
independent graviton polarizations is $\frac{D(D-1)}{2}-1$. The difference
between these two numbers is $2(D+1)$. If we impose the $D+1$ conditions
(139), we get the one additional condition (140) free. This still leaves us
with $D$ more parameters in $h_{MN}$ than the number of physical gravitons. We
need to fix these additional parameters by imposing an additional (arbitrary)
gauge condition; these are the $D$ undetermined parameters in $\zeta$. The
Maxwell analogue of the discussion of this footnote is simply the fact that
Lorentz gauge does not completely fix photon polarizations; the remaining
ambiguity is $\epsilon_{M}\rightarrow\epsilon_{M}+k_{M}$. A physically natural
additional gauge condition - the one we will choose to adopt - is to demand
that all gravitons polarizations are transverse to the centre of mass momentum
$k_{1}+k_{3}=-(k_{2}+k_{4})$. Since we have already demanded that $h^{a}_{MN}$
is transverse to $k_{a}$, this additional requirement requires us only to
impose the conditions

$(k_{3})^{M}h^{1}_{MN}=(k_{1})^{M}h^{3}_{MN}=(k_{4})^{M}h^{2}_{MN}=(k_{2})^{M}h^{4}_{MN}=0$
(144)

Despite first appearances, (144) imposes $D$ (rather than $D+1$) conditions on
each graviton as one of each of the four groups of $D+1$ equations in (144) is
automatic from (139).595959For the graviton inserted at the point $P_{1}$, for
instance, the condition $k_{3}^{M}h^{1}_{MN}k_{1}^{N}$ is already ensured by
(139).

If we choose to impose both (139) as well as (144) on our scattering
gravitons, the scattering graviton waves reported in (134) are modified to

$\begin{split}h^{1}_{MN}&=\left(Z^{1\perp}_{M}Z^{1\perp}_{N}-\frac{(X.Z^{1})^{2}\left(\eta_{MN}+X_{M}X_{N}\right)}{D-1}\right)+\frac{(X.Z^{1})^{2}\left(k^{3}{}_{M}k^{1}_{N}+k^{1}_{M}k^{3}_{N}\right)}{(D-1)k^{1}.k^{3}}\\\
&-\frac{k^{3}.Z^{1}\left(k^{1}_{N}Z^{1\perp}_{M}+k^{1}_{M}Z^{1\perp}_{N}\right)}{k^{1}.k^{3}}+\frac{k^{1}_{M}k^{1}_{N}\left(k^{3}.Z^{1}\right)^{2}}{\left(k^{1}.k^{3}\right)^{2}}\\\
h^{2}_{MN}&=\left(Z^{2\perp}_{M}Z^{2\perp}_{N}-\frac{(X.Z^{2})^{2}\left(\eta_{MN}+X_{M}X_{N}\right)}{D-1}\right)+\frac{(X.Z^{2})^{2}\left(k^{4}_{M}k^{2}_{N}+k^{2}_{M}k^{4}_{N}\right)}{(D-1)k^{2}.k^{4}}\\\
&-\frac{k^{4}.Z^{2}\left(k^{2}_{N}Z^{2\perp}_{M}+k^{2}_{M}Z^{2\perp}_{N}\right)}{k^{2}.k^{4}}+\frac{k^{2}_{M}k^{2}_{N}\left(k^{4}.Z^{2}\right)^{2}}{\left(k^{2}.k^{4}\right)^{2}}\\\
h^{3}_{MN}&=\left(Z^{3\perp}_{M}Z^{3\perp}_{N}-\frac{(X.Z^{3})^{2}\left(\eta_{MN}+X_{M}X_{N}\right)}{D-1}\right)+\frac{(X.Z^{3})^{2}\left(k^{1}{}_{M}k^{3}_{N}+k^{3}_{M}k^{1}_{N}\right)}{(D-1)k^{1}.k^{3}}\\\
&-\frac{k^{1}.Z^{3}\left(k^{3}_{N}Z^{3\perp}_{M}+k^{3}_{M}Z^{3\perp}_{N}\right)}{k^{1}.k^{3}}+\frac{k^{3}_{M}k^{3}_{N}\left(k^{1}.Z^{3}\right)^{2}}{\left(k^{1}.k^{3}\right)^{2}}\\\
h^{4}_{MN}&=\left(Z^{4\perp}_{M}Z^{4\perp}_{N}-\frac{(X.Z^{4})^{2}\left(\eta_{MN}+X_{M}X_{N}\right)}{D-1}\right)+\frac{(X.Z^{4})^{2}\left(k^{4}_{M}k^{2}_{N}+k^{2}_{M}k^{4}_{N}\right)}{(D-1)k^{2}.k^{4}}\\\
&-\frac{k^{2}.Z^{4}\left(k^{4}_{N}Z^{4\perp}_{M}+k^{4}_{M}Z^{4\perp}_{N}\right)}{k^{2}.k^{4}}+\frac{k^{4}_{M}k^{4}_{N}\left(k^{2}.Z^{4}\right)^{2}}{\left(k^{2}.k^{4}\right)^{2}}\\\
\end{split}$ (145)

In final summary, the S matrix that appears in (136) is the S matrix for the
scalars and vectors reported in (134) and either the gravitons reported in
(134) or the gravitons reported in (145). The two sets of gravitons are gauge
related and have equal S matrices. The advantage of expressions presented in
(134) is that the polarization for the $i^{th}$ particle does not refer to any
other particle. Its disadvantage is that it appears in an unfamiliar gauge.
These advantages and disadvantages are reversed in the polarizations (145).
The reader is free to choose either of these gauges (or any other) according
to her convenience.

###  4.9 Regge scaling

In (136), the small $\rho$ behavior of the four-point function is expressed in
terms of the integral over flat space S matrices over an $\mathbb{H}_{D-2}$.
As we have emphasized above, all the S matrices that appear in this formula
have the same effective scattering momenta, but the effective scattering
polarizations depend on the scattering point $X$.

The Regge scaling of the S matrices ${\cal S}_{X}$ that appear in (136) is not
necessarily the same for all values of $X$. Recall, however, that the
dependence of the S matrix on $X$ is very simple; it arises entirely through
the dependence on the scattering polarizations on $X$. The dependence of the S
matrix is very simple. In the case of photons, it is a linear function of
polarizations (separately in the polarization of every particle) while in the
case of gravitons, it is a bilinear function of polarizations. It follows from
this fact that if the S matrix displays a certain Regge growth at some point
$X$ on the hyperboloid, it must grow at least as fast at all but possibly
(measure zero sets of) isolated points on the hyperboloid. Ignoring these
isolated points that make no significant contribution to the integral, it
follows that all the S matrices that appear in (136) scale in the same manner
in the Regge limit.

Let us suppose that in the small angle limit, this generic S matrix behaves
like

${\hat{\cal S}}\propto\theta^{r-2A}$ (146)

Using the fact that our ${\cal S}$ matrix scales with energy like
$\omega^{r}$, and that in the small angle limit $t\sim\theta^{2}\omega^{2}$,
it follows from (146) that the Regge (i.e. fixed $t$) scaling of ${\cal S}$ is

${\cal S}\sim\omega^{2A}\sim s^{A}$ (147)

Plugging this estimate into (136) we find that in the small $\sigma$ limit

$G_{\rm
sing}\propto\frac{1}{\sigma^{\frac{\Delta}{2}}}\times\frac{1}{\sigma^{A-1}}$
(148)

Note in particular that the scaling (148) is independent of $r$.

###  4.10 Regge scaling of normalized correlators

In applications related to the chaos bound it is useful to study the
normalized four point correlator (31). Recall that in order for the
denominator of (31) to be nonvanishing, $\Delta_{1}=\Delta_{2}$ and
$\Delta_{3}=\Delta_{4}$. It follows that

$\Delta=2\Delta_{1}+2\Delta_{2}$

Combining (39) and (148), it follows that

$G^{\rm norm}_{\rm sing}\propto\frac{1}{\sigma^{A-1}}$ (149)

###  4.11 Massive higher spin scattering

In this subsection, we have restricted our attention to the study of the
massless spin one and spin two fields (dual to the correlators of conserved
currents and the stress tensor on the boundary). While the generalization to
the study of massive higher spin particles should certainly be possible, such
a generalization involves new complications, which will be briefly outlined in
this subsection (and discuss in much greater detail in Appendix I).

Consider the case of a massive vector field. Because the bulk interactions of
such a field are not necessarily ‘gauge invariant’, in this case we are forced
to deal with the full propagator (118) rather than a simplified propagator
analogous to (123). As a consequence, the scattering modes include an
additional polarization; a longitudinal polarization (which is pure gauge in
the massless theory). In equations, at leading order, the wave representation
of his propagator, tells us that the effective scattering wave of the massive
spin one particle is given by

$\begin{split}A^{i}_{M}&=\epsilon^{i}_{M}e^{i{k}_{i}.x}\\\
\epsilon^{i}_{M}&=\left(1-\frac{1}{\Delta_{i}}\right)(Z_{i}^{\perp})_{M}+\frac{ik^{i}_{M}}{\Delta^{i}}(Z_{i}.X_{0})\end{split}$
(150)

($k^{i}_{M}$ are as listed in (95)). The second term in (150) is the new
longitudinal polarization.

Notice that the coefficient of the longitudinal mode in (150) is of order
$\omega$, while that of the transverse mode is of order unity. Recall also
that increasing the number of powers of $\omega$ in the S matrix increases the
degree of singularity in $\rho$ (in formulae like (136)). It follows that the
analog of the formula (136) involves multiple terms on the RHS with different
inverse powers of $\rho$. The coefficient of the maximally singular term in
$\rho$ is the S matrix involving the maximal number of longitudinal
polarizations. The coefficients of subleading singularities in $\rho$ include
the S matrices of modes with less than the maximal number of longitudinal
polarizations - but also receive contributions from subleading effects from,
e.g., the scattering of the maximal number of longitudinal modes. These
subleading effects - which presumably involve corrections to the simple flat
space S matrix (resulting from the fact that the relevant propagators are not
precisely plane waves and from the curvature of $AdS_{D+1}$) - complicate the
analysis. While we believe that these complications are tractable, and the
main result of this paper - namely that scattering amplitudes that violate the
CRG conjecture lead to correlators that violate the chaos bound - likely also
holds for these massive modes, the proof of this claim needs more care in
these cases, and we leave this to future work. See Appendix I for more
discussion of this case.

##  5 Inter relationship between the small $\theta$ and small $a$ expansion

In section (3) we studied Regge expansion (the small $\theta$ fixed $a$)
expansion of our correlator. In section 4, on the other hand, we have studied
the small $a$ fixed $\theta$ expansion of our correlator on the Causally
scattering sheet. In this section, we will explore the interrelationship
between these two expansions. We will do this by returning to the expansion of
section (3), first performing the small $\theta$ fixed $a$ expansion, and then
examining how each coefficient in this expansion scales with $a$ in the $a\to
0$ limit. 606060We developed the material recorded in this section in part to
address questions posed to us by A. Gadde and A. Zhiboedov. We would like to
thank them for their probing questions which helped us understand that one of
the arguments we presented in an earlier version of this paper was incorrect,
and spurred us to better understand the structure of the small $\theta$
perturbation theory (see the material in this section) to correctly argue for
the same conclusion.

As we explained in subsection 3.1, the correlator under study in this paper
can be expanded in a systematic power series expansion in $\theta$ in the
Regge limit. In particular we demonstrated that the (unnormalized) correlator
takes the form

$G=\frac{1}{\theta^{2B-2}}\left(\sum_{n=0}^{\infty}\theta^{n}H_{n}(a)\right)$
(151)

where $B$ was defined in equations (44) and (47). ($H(a)$ of subsection 3.2 is
the same as $H_{0}(a)$ in (151)). In this section, we return to the expansion
of subsection 3.1 to investigate the small $a$ scaling of the functions
$H_{n}(a)$.

Recall that the expansion of subsection 3.1 was obtained starting with
integral (43) and Taylor expanding the integrand in the variables $u$ $v$,
$\theta$ and $\tau$ with $y_{i}$ and $y_{1}$ unexpanded (i.e. treated as order
unity). As we explained in subsection 3.1 it is actually most convenient to
use the variables $D_{0}^{i}$ as the expansion variables for the power series
expansion; the expression for $u$ $v$, $\tau$ and $\theta$ in terms of
$D_{0}^{i}$ is given in (49). In subsection 3.1 $H_{n}(a)$ was obtained by
expanding the integrand of (43) in a power series expansion in the variables
$D_{0}^{i}$ with $y_{1}$ and $y_{i}$ regarded as order unity, and then
performing the integrals, first over $u$ and $v$ and then over $y_{1}$ and
$y_{i}$.

###  5.1 Statement of the $a$ counting rule

We are interested in the scaling of the functions $H_{n}(a)$ in the small $a$
limit. The functions $H_{n}(a)$ are obtained after performing the integration
in (43) over $u$, $v$, $y_{1}$ and $y_{i}$.

As in the case of the small $\theta$ expansion, it is possible to find an
effective small $a$ scaling assignment for the integrand in (43). This rule is
devised to obey the following property. If any particular term in the
integrand is assigned scaling $a^{n}$, then the integral of this term will
scale like $a^{n}$.

In this subsection, we simply state this ‘small $a$ integrand scaling rule’.
The justification for this rule is provided later in this section.

The ‘small $a$ integrand scaling rule’ goes as follows.

  * •

$\tau$, $u$ $v$ and $y_{1}$ are each of order $a$, while $\theta$ and $y_{i}$
are of order unity.

  * •

Working with the variables $D^{0}_{i}$ , $y_{1}$ and $y_{i}$ we conclude from
(49) that all four $D_{0}^{i}$ and $y_{1}$ are of order $a$ while $y_{i}$ are
of order unity.

###  5.2 Consequence of the $a$ scaling rule

Assuming the small $a$ integrand scaling rule of the previous subsection, it
is easy to bound the small $a$ behaviour of each $H_{n}(a)$. All we need to do
is count the $a$ power of every term in the expansion of the integrand of (43)
that contributes to $H_{n}(a)$. It is easy to convince oneself that every such
terms scales in the small $a$ limit like

$\frac{P(a)}{a^{{\tilde{a}}_{1}+{\tilde{a}}_{2}+{\tilde{a}}_{3}+{\tilde{a}}_{4}}}$
(152)

(see (43) for the definition of ${\tilde{a}}_{i}$) where $P(a)$ is an
expression that has a power series expansion in terms of order $a$. It follows
that the integral of this quantity over $u$, $v$ and $y_{1}$ scales like

$\frac{P^{\prime}(a)}{a^{{\tilde{B}}-3}}=\frac{P^{\prime}(a)}{a^{2B+M-3}}$
(153)

(see (44) for the definition of ${\tilde{B}}$) where $P^{\prime}(a)$ is a
function that admits a power series expansion in $a$. (153) follows from (152)
upon integrating. (152) follows from the following facts

  * •

1\. Each $D^{i}_{0}$ is of order $a$.

  * •

2\. If we write true denominators in (43) as $D^{i}=D^{i}_{0}+\delta D_{i}$
(where $\delta D_{i}$ is the collection of all terms that are subleading to
$D_{i}^{0}$ in the $\theta$ expansion, i.e. $\delta D_{i}$ consists of terms
that are of $\theta^{2}$ or smaller) then each $\delta D_{i}$ is of order
$aP(a)$ where $P(a)$ is a function that starts out at order unity (for
$i=2,4$) or order $a^{2}$ (for $i=1,3$) and has a power series expansion in
$a$. (see around (156).)

  * •

3\. While the numerator in (43) starts at order $\theta^{M}$ in the small
$\theta$ expansion, it starts either at order unity (this is the generic
expectation) or higher (possible but non generic) in the small $a$ expansion.
In other words the function in (43) has a power series expansion in terms of
order $a$.

Note in particular that the leading possible singularity in $a$ in (152) is
independent of $n$. In other words, the leading small $a$ scaling of distinct
coefficients of the small $\theta$ expansion does not increase with the order
to the expansion; it generically also does not decrease with the order of the
expansion.

The reason that coefficients of higher-order in the $\theta$ expansion do not
start out at higher orders in the $a$ expansion is the following. While each
of $u$, $v$, $\tau$, and $\theta$ are all of the same order of smallness in
the Regge expansion, only the first three of these terms are treated as small
for the small $a$ expansion. Consequently a term which is $\theta^{n}$
616161In this sentence $\theta^{n}$ means the $n^{th}$ power of the variable
$\theta$ rather than just a term that is of $n^{th}$ order in the small
$\theta$ expansion. times the leading order contribution to the expansion
would be counted as a contribution to $H_{n}(a)$ (as it is $n^{th}$ order
suppressed in the small $\theta$ expansion) but would be of leading order in
the small $a$ expansion (as $\theta$ is of order unity in the small $a$
expansion).

It is also nontrivial that terms at successively higher-order in the $\theta$
are not increasingly singular in the $a$ expansion. Such a situation could
have arisen as follows. As we have reviewed above, the leading small $\theta$
approximation to the denominators, $D^{0}_{i}$, are all of order $a$ in the
small $a$ expansion. Now as in item 2 above, let us consider the true
denominator $D_{i}$, and let $D_{i}=D_{i}^{0}+\delta D_{i}$ where $\delta
D_{i}$ is of order $\theta^{2}$ or higher in the small $\theta$ expansion. If
had turned out that $\delta D_{i}$ had included a piece that was of order
unity in the small $a$ expansion - e.g. a term proportional to $\theta^{r}$
with no additional small $a$ suppression factors - then higher orders in the
small $\theta$ expansion of the denominator (which would have included a power
series in $\frac{\delta D_{i}}{D_{i}^{0}}$) would have included terms with
increasingly high singularities at small $a$. As we have mentioned in item 2,
the explicit expressions for $P_{i}.X$ make it clear that this never happens.
Every term in $\delta D_{i}$ is always of order $a$ or smaller in the small
$a$ expansion.

As the point made in the previous paragraph is a crucial link in our
derivation of the CRG conjecture from the chaos bound, we pause to explain
this point in more detail with the aid of equations. Let us use the notation

$D_{i}=-2P_{i}.X$ (154)

It is easy to explicitly evaluate $D_{i}$ in terms of $\theta,\tau,u,v$ and
then reexpress the result in terms of $D^{0}_{i}$. We find

$\begin{split}D_{1}&=\frac{1}{2}\left(-\left(D_{2}^{0}+D_{4}^{0}\right)\cos\tau+\left(D_{1}^{0}-D_{3}^{0}\right)+2y_{0}\sin\tau\right)\\\
D_{2}&=\frac{1}{2}\left(\left(D_{3}^{0}-D_{1}^{0}\right)\cos\left(\frac{D_{1}^{0}+D_{2}^{0}-D_{3}^{0}-D_{4}^{0}}{2y_{1}}\right)+D_{2}^{0}+D_{4}^{0}\right)+y_{1}\sin\left(\frac{D_{1}^{0}+D_{2}^{0}-D_{3}^{0}-D_{4}^{0}}{2y_{1}}\right)\\\
D_{3}&=\frac{1}{2}\left(-\left(D_{2}^{0}+D_{4}^{0}\right)\cos\tau-
D_{1}^{0}+D_{3}^{0}+2y_{0}\sin\tau\right)\\\
D_{4}&=\frac{1}{2}\left(\left(D_{1}^{0}-D_{3}^{0}\right)\cos\left(\frac{D_{1}^{0}+D_{2}^{0}-D_{3}^{0}-D_{4}^{0}}{2y_{1}}\right)+D_{2}^{0}+D_{4}^{0}\right)-y_{1}\sin\left(\frac{D_{1}^{0}+D_{2}^{0}-D_{3}^{0}-D_{4}^{0}}{2y_{1}}\right)\end{split}$
(155)

where,

$\tau(D_{1}^{0},D_{2}^{0},D_{3}^{0},D_{4}^{0})=\frac{D_{1}^{0}+D_{2}^{0}+D_{3}^{0}+D_{4}^{0}}{\sqrt{\left(D_{1}^{0}-D_{2}^{0}-D_{3}^{0}-D_{4}^{0}\right)\left(D_{1}^{0}+D_{2}^{0}-D_{3}^{0}+D_{4}^{0}\right)+4\left(1+y_{1}^{2}+y_{\perp}^{2}\right)}}$

and

$y_{0}(D_{1}^{0},D_{2}^{0},D_{3}^{0},D_{4}^{0})=\sqrt{1+y_{1}^{2}+y_{\perp}^{2}+\frac{1}{4}\left(D_{1}^{0}-D_{2}^{0}-D_{3}^{0}-D_{4}^{0}\right)\left(D_{1}^{0}+D_{2}^{0}-D_{3}^{0}+D_{4}^{0}\right)}$

Using the explicit relations (155) it is easy to verify that the small
$\theta$ expansion of each of the denominator factors $D_{i}$ takes the form

$D_{i}=\sum_{r=0}^{\infty}D_{i}^{r}$ (156)

where $D_{i}^{r}$ is of order $2r+1$ in the small $\theta$ expansion. Again
using these explicit expressions it is easy to convince oneself that the small
$a$ expansion of $D_{1}^{r}$ and $D_{3}^{r}$ starts at order $a^{2r+1}$
626262This is a consequence of the fact that $\tau$ and $uv$ are
simultaneously of order $\theta^{2}$ and $a^{2}$, while $y_{1}^{2}$ is of
order $a^{2}$ but order unity in the $\theta$ expansion.. On the other hand
$D_{2}^{r}$ and $D_{4}^{r}$ respectively each start at order $a$ in the small
$a$ expansion. 636363This is a consequence of the fact that the argument of
the cosine and sine in these expressions is of order $\theta$ in the Regge
expansion but of order unity in the small $a$ expansion. Note its crucial that
the pre factor of both the cosine and sine in these expressions is of order
$a$ in the small $a$ expansion. These facts are illustrated by the explicit
expressions for the first subleading corrections

$\begin{split}&D_{1}^{1}=-\frac{(D^{0}_{1}-2D^{0}_{2}+D^{0}_{3}-2D^{0}_{4})(D^{0}_{1}+D^{0}_{2}+D^{0}_{3}+D^{0}_{4})^{2}}{48\left(1+y_{\perp}^{2}+y_{1}^{2}\right)}\\\
&D_{2}^{1}=\frac{(2D^{0}_{1}-D^{0}_{2}-2D^{0}_{3}+D^{0}_{4})(D^{0}_{1}+D^{0}_{2}-D^{0}_{3}-D^{0}_{4})^{2}}{48~{}y_{1}^{2}}\\\
&D_{3}^{1}=-\frac{(D^{0}_{1}-2D^{0}_{2}+D^{0}_{3}-2D^{0}_{4})(D^{0}_{1}+D^{0}_{2}+D^{0}_{3}+D^{0}_{4})^{2}}{48\left(1+y_{\perp}^{2}+y_{1}^{2}\right)}\\\
&D_{4}^{1}=-\frac{(2D^{0}_{1}-D^{0}_{2}-2D^{0}_{3}+D^{0}_{4})(D^{0}_{1}+D^{0}_{2}-D^{0}_{3}-D^{0}_{4})^{2}}{48~{}y_{1}^{2}}\\\
\end{split}$ (157)

Note that $D_{1}^{1}$ and $D_{3}^{1}$ each start at order $a^{3}$ while
$D_{2}^{1}$ and $D_{4}^{1}$ each start at order $a$, in agreement with the
general rule spelt out above.

###  5.3 Derivation of the $a$ counting rule

In this subsection, we explain why the counting rule of subsubsection 5.1 is
correct.

Recall from subsection (3.1) that the final expression for $H_{n}(a)$ is given
by

$H_{n}(a)=\sum_{\\{a_{i}\\}}\int
dy_{i}dy_{1}\frac{N_{\\{a_{i}\\}}(y_{1},y_{i},a)}{\left(a~{}y_{0}+y_{1}+i\epsilon\right)^{a_{1}+a_{2}-1}\left(a~{}y_{0}-y_{1}+i\epsilon\right)^{a_{3}+a_{4}-1}}$
(158) 646464If we had expanded the integrand in (43) in a power series
expansion in $\tau$, $\theta$, $u$ and $v$ then the coefficients of this
expansion would have been independent of $a$. However we instead chose to
expand the integrand in a power series in $D_{0}^{i}$. The linear
transformation to the new variables (49) involves $a$. This is why the
function $N_{\\{\alpha_{i}\\}}$ depends on $a$.

where the sum is taken over terms with

$a_{1}+a_{2}+a_{3}+a_{4}=2{B}-n$ (159)

(${\tilde{B}}$ was defined in (44)) and all factors of
$C_{a_{1},a_{2},a_{3},a_{4}}$ (see (58)) have been absorbed into the function
$N_{\\{a_{i}\\}}(y_{1},y_{i},a))$.

Now the small $a$ singularity in (158) is a consequence of the pinch in the
integral over $y_{1}$ in the denominators in the integrand of that expression.
It is easy to convince oneself (see Appendix F, especially F.3 for some more
detail) that if the numerator function scales like

$N_{\\{a_{i}\\}}(\lambda y_{1},y_{i},\lambda a)\sim\lambda^{Y_{N}}$ (160)

at small $\lambda$ then

$H_{n}(a)\sim a^{Y_{N}+3-a_{1}-a_{2}-a_{3}-a_{4}}=a^{Y_{N}+3-2B+n}$ (161)

where in the last equality we have used (159). (161) follows from the
observation that $y_{1}$ is of order $a$ in the two poles that participate in
the pinch in (158); as a consequence the singularity of the integral is
determined by small $y_{1}$ expansion of the numerator in (158). The result
(161) then follows from the change of variables $y_{1}=ay_{1}^{\prime}$.

The counting rule described in the previous paragraph follows immediately from
the observation that each additional factor of $D^{i}_{0}$ and $y_{1}$ in the
integrand of (158) gives an additional factor of $a$ in (161). The fact that
$\theta$ is of order unity then follows from (49). 656565In particular, the
fact that $y_{1}$ is of order $a$ follows simply from the fact that the pole
values of $y_{1}$ are each of order $a$ in the integrand in (158).

To end this subsection we briefly outline how the RHS of (161) explicitly
reproduces (153). Infact (153) follows from (161) from the observation that
$Y_{N}$ is bounded from below according to

$Y_{N}\geq-n-M$ (162)

Inserting (162) in (161) gives (153).

(162) follows because terms in the integrand of (43) which contribute to
$H_{n}(a)$ are of order $\theta^{M+n}$ compared to the leading contribution to
the denominator in (43). It is possible for some - or all- of these additional
small factors to consist of powers of $\theta$ (rather than $\tau$ or $u$ or
$v$). As $\theta$ is of order unity rather than $a$ in the small $a$ limit,
every insertion of $\theta$ leads to a factor of $D_{i}^{0}$ or $\tau$
(already accounted for above) divided by a factor of $a$ or $y_{1}$ (see (49))
which have to be accounted for additionally, leading to (162).

###  5.4 Strength of the $a$ singularity

In order to make quantitative contact with the small $\rho$ analysis presented
in section 4, we now specialize to the case of a correlators generated by the
interaction of four bulk fields dual to operators of dimension $\Delta_{i}$
($i=1\ldots 4$) interacting via a bulk interaction vertex that is of $r^{th}$
order in derivatives. As in the earlier parts of this section we restrict
attention to the case that the bulk particles are either gravitons dual to the
stress tensor of dimension $D$, photons dual to a conserved current of
dimension $D-1$ or massive scalars dual to scalar operators of arbitrary mass.
In this context the largest possible value of the
${\tilde{a}}_{1}+{\tilde{a}}_{2}+{\tilde{a}}_{3}+{\tilde{a}}_{4}$ in (152) is

${\tilde{a}}_{1}+{\tilde{a}}_{2}+{\tilde{a}}_{3}+{\tilde{a}}_{4}=\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}+r$
(163)

(this maximum is obtained when all bulk derivatives are allowed to hit the
denominators of propagators in the bulk). It follows from (153) that the most
singular possible small $a$ behaviour from such a bulk interaction is

$\frac{1}{a^{\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}+r-3}}$ (164)

in perfect agreement with the power of the $\rho$ singularity in (136).

###  5.5 Order of limits

In this section, we have established that the correlator under study has the
following simple analytic structure in the limit that $\theta$ and $a$ are
both small.

$G=\frac{1}{\theta^{2B-2}a^{\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}+r-3}}H(\theta,a)$
(165)

where the function $H(\theta,a)$ admits a double power series expansion in $a$
and $\theta$. We also know from (136) that the function $H(\theta,a)$ is