of 16
Graviton partial waves and causality in higher dimensions
Simon Caron-Huot and Yue-Zhou Li
Department of Physics, McGill University, 3600 Rue University, Montr ́
eal H3A 2T8, Quebec, Canada
Julio Parra-Martinez and David Simmons-Duffin
Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, California 91125, USA
(Received 26 July 2022; accepted 6 June 2023; published 17 July 2023)
Do gravitational interactions respect the basic principles of relativity and quantum mechanics? We show
that any graviton
S
-matrix that satisfies these assumptions cannot significantly differ from General
Relativity at low energies. We provide sharp bounds on the size of potential corrections in terms of the mass
M
of new higher-spin states, in spacetime dimensions
D
5
where the
S
-matrix does not suffer from
infrared ambiguities. The key novel ingredient is the full set of SO
ð
D
1
Þ
partial waves for this process,
which we show how to efficiently compute with Young tableau manipulations. We record new bounds on
the central charges of holographic conformal theories.
DOI:
10.1103/PhysRevD.108.026007
I. INTRODUCTION
Relativity and quantum mechanics lie at the heart of
particle physics. Notions such as relativistic causality
(
signals cannot move faster than light
) naturally lead
to the concepts of waves, fields, and particles as force
carriers
[1]
. Gravity challenges this unification; for example
the precise meaning of causality in a fluctuating spacetime
remains unclear. In this paper we study a situation where
causality can be unambiguously stated, and is in principle
experimentally testable.
Our setup is
2
2
scattering between initially well-
separated objects in a flat Minkowski-like region of
spacetime. A notion of causality is inherited from the flat
background, and encoded in the mathematically precise
axioms of scattering (
S
-matrix) theory. It can be used to
constrain gravity itself. Consider higher-derivative correc-
tions to Einstein
s gravity at long distances,
S
¼
Z
d
D
x
ffiffiffiffiffiffi
g
p
16
π
G

R
þ
α
2
4
C
2
þ
α
4
12
C
3
þ
α
0
4
6
C
0
3
þ
...

;
ð
1
Þ
wwhere
C
2
;C
3
;C
0
3
are higher-curvature terms
defined below. Weinberg famously argued that any theory
of a massless spin-two boson must reduce to GR
at long distances
[2]
. This was significantly extended
in
[3]
, who argued that the parameters
α
i
must be
parametrically suppressed by the mass
M
of new
higher-spin states. In parallel,
S
-matrix dispersion rela-
tions have been used to constrain signs and sizes of certain
corrections
[4
6]
.
Recently, by combining these methods we showed how
to bound dimensionless ratios of the form
j
α
i
M
i
j
in any
scenario where
M
M
pl
, such that corrections are larger
than Planck suppressed. However, these bounds featured
the infrared logarithms that are well-known to plague
massless
S
-matrices in four dimensions.
In this paper we present rigorous bounds in higher-
dimensional gravity, where infrared issues are absent. We
overcome significant technical hurdles regarding the partial
wave decompositions of higher-dimensional amplitudes.
The resulting bounds have interesting applications to
holographic conformal field theories.
II. FOUR-POINT GRAVITY AMPLITUDES
A. Four-point
S
-matrices and local module
We treat the graviton as a massless particle of spin 2. The
amplitude for
2
2
graviton scattering depends on the
energy-momentum
p
μ
j
and polarization
ε
μ
j
of each. It can be
written generally as a sum over Lorentz-invariant poly-
nomials times scalar functions,
M
¼
X
ð
i
Þ
Poly
ð
i
Þ
ðf
p
j
;
ε
j
×
M
ð
i
Þ
ð
s; t
Þ
:
ð
2
Þ
We use conventions in which all momenta are outgoing and
Mandelstam invariants, satisfying
s
þ
t
þ
u
¼
0
, are
Published by the American Physical Society under the terms of
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Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
s title, journal citation,
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3
.
PHYSICAL REVIEW D
108,
026007 (2023)
2470-0010
=
2023
=
108(2)
=
026007(16)
026007-1
Published by the American Physical Society
s
¼
ð
p
1
þ
p
2
Þ
2
;t
¼
ð
p
2
þ
p
3
Þ
2
;u
¼
ð
p
1
þ
p
3
Þ
2
:
ð
3
Þ
In kinematics where
p
1
,
p
2
are incoming,
s
and
t
are,
respectively, the squares of the center-of-mass energy and
momentum transfer.
The allowed polynomials in
(2)
are restricted by the fact
that graviton polarizations are transverse traceless and
subject to gauge redundancies
[7]
,
p
j
·
p
j
¼
p
j
·
ε
j
¼
ε
j
·
ε
j
¼
0
;
ε
j
ε
j
þ
#
p
j
:
ð
4
Þ
Depending on the choice of spanning polynomials, the
functions
M
ð
i
Þ
ð
s; t
Þ
may develop spurious singularities
which would complicate their use. As explained in
[8]
,
there exist special generators of the
local module
such
that any amplitude that is polynomial in polarizations and
momenta leads to
M
ð
i
Þ
s that are polynomial in
s
and
t
.
These can be simply presented using gauge- and Lorentz-
invariant building blocks:
H
12
¼
F
μ
1
ν
F
ν
2
μ
;H
123
¼
F
μ
1
ν
F
ν
2
σ
F
σ
3
μ
;
H
1234
¼
F
μ
1
ν
F
ν
2
σ
F
σ
3
ρ
F
ρ
4
μ
;V
1
¼
p
4
μ
F
μ
1
ν
p
ν
2
;
ð
5
Þ
where
F
μ
i
ν
¼
p
μ
i
ε
i
ν
ε
μ
i
p
i
ν
is proportional to the field
strength. We define
H
s with other indices by permutation,
and
V
i
by cyclic permutations.
In this notation, any
S
-matrix involving four photons
(thus homogeneous of degree 1 in each of the vectors
ε
μ
j
)
can be written as a sum of seven terms, involving three
basic functions
[8]
,
M
4
γ
¼½
H
14
H
23
M
ð
1
Þ
4
γ
ð
s; u
Þþ
X
1243
M
ð
2
Þ
4
γ
ð
s; u
Þþ
cyclic

þ
S
M
ð
3
Þ
4
γ
ð
s; t
Þ
:
ð
6
Þ
Here, we introduced the shorthands
X
and
S
,
X
1234
¼
H
1234
1
4
H
12
H
34
1
4
H
13
H
24
1
4
H
14
H
23
;
S
¼
V
1
H
234
þ
V
2
H
341
þ
V
3
H
412
þ
V
4
H
123
:
ð
7
Þ
Thanks to Bose symmetry, all basic functions
M
ð
i
Þ
4
γ
ð
a; b
Þ
are symmetrical in their two arguments, while the third one
is further invariant under all permutations of
s
,
t
,
u
, since
S
is fully permutation symmetric. The combination
X
enjoys
improved Regge behavior (discussed below).
The general four-graviton amplitude
M
can now be
written using all products of the photon structures, sup-
plemented by the element
G
equal to the determinant of all
dot products between
ð
p
1
;p
2
;p
3
;
ε
1
;
ε
2
;
ε
3
;
ε
4
Þ
. The result-
ing 29 generators organize under permutations as two
singlets, seven cyclic triplets, and one sextuplet
[8]
:
singlets
GM
ð
1
Þ
ð
s;u
Þ
;S
2
M
ð
10
Þ
ð
s;u
Þ
;
triplets
H
2
14
H
2
23
M
ð
2
Þ
ð
s;u
Þ
;H
12
H
13
H
24
H
34
M
ð
3
Þ
ð
s;u
Þ
;
H
14
H
23
ð
X
1243
X
1234
X
1324
Þ
M
ð
4
Þ
ð
s;u
Þ
;
X
2
1243
M
ð
6
Þ
ð
s;u
Þ
;X
1234
X
1324
M
ð
7
Þ
ð
s;u
Þ
;
H
14
H
23
S
M
ð
8
Þ
ð
s;u
Þ
;X
1243
S
M
ð
9
Þ
ð
s;u
Þ
;
sextuplet
H
12
H
34
X
1243
M
ð
5
Þ
ð
s;u
Þ
:
ð
8
Þ
These constitute a basis in generic spacetime dimension
(
D
8
); lower dimensions are reviewed in Appendix
A
.
B. Regge limit and dispersive sum rules
At low energies, the effect of quartic self-interactions
in the effective theory
(1)
is to add polynomials in
Mandelstam invariants to the amplitudes
M
ð
i
Þ
: this is a
defining property of the local module
[9]
. We would like to
use the assumption that graviton scattering remains sensible
at all energies to constrain the size of these interactions.
Our axioms are best stated using smeared amplitudes,
M
Ψ
ð
s
Þ
Z
M
0
dp
Ψ
ð
p
Þ
M
ð
s;
p
2
Þ
:
ð
9
Þ
The traditional statement of causality in
S
-matrix theory,
alluded to in the introduction, is that amplitudes are analytic
in the upper-half
s
-plane. We will specifically assume that
the smeared amplitude is analytic in the upper-half-plane
for
s
large,
j
s
j
M
2
. As argued in
[10
13]
, combined with
unitarity on the real axis, this implies boundedness along
any complex direction for suitable wavefunctions
Ψ
:
j
M
Ψ
ð
s
Þj
s
s
× constant
:
ð
10
Þ
The essential conditions on
Ψ
ð
p
Þ
are: finite support in
p
(required for analyticity of
M
Ψ
), and normalizability at
large impact parameters (ensuring boundedness).
The bound
(10)
is assumed for polarizations that do not
grow with energy. The behavior of the scalar functions
M
ð
i
Þ
can be deduced from the Regge scaling of the
polarization structures they multiply; leading growth rates
are recorded in Table
I
. An important observation is that the
leading terms are not all linearly independent, for example
while both
X
1234
;X
1324
s
2
, their difference grows more
slowly. The coefficients of these structures inherit the
opposite behavior. For example, the (smeared) photon
amplitudes
M
ð
2
Þ
4
γ
ð
s; t
Þ
M
ð
2
Þ
4
γ
ð
u; t
Þ
are bounded by con-
stants times
s
1
and
s
0
, respectively.
We say that a dispersive sum rule has Regge spin
k
if it
converges assuming that
M
=s
k
0
; our axioms above
state that sum rules with
k>
1
converge. As can be seen
from
(8)
and Table
I
,
M
s
k
implies
M
ð
3
Þ
s
k
4
, ensur-
ing convergence of the following integral at fixed
t
¼
p
2
(with
u
¼
p
2
s
):
SIMON CARON-HUOT
et al.
PHYS. REV. D
108,
026007 (2023)
026007-2
B
½
1

k
ð
p
2
Þ¼
I
ds
4
π
i

ð
s
u
Þ
M
ð
3
Þ
ð
s; u
Þ
ð
su
Þ
k
2
2

0
ð
k
2
even
Þ
:
ð
11
Þ
This identity yields a Kramers-Kronig type relation
between scattering at low and high energies, by a standard
contour deformation argument. Namely, one finds a low-
energy contribution at the scale
M
M
pl
which is effective
field theory (EFT) computable by assumption, plus a
discontinuity at high energies
s
M
2
(see Ref.
[12]
for more detail). See Appendix
D
for the low-energy
amplitudes.
A salient feature of graviton scattering is that many sum
rules, like
B
½
1

2
above, have no denominator: only the poles
of
M
contribute at low energies. Acting on the low-energy
amplitude [see
(D4)
], it yields,
8
π
G

1
2
p
2
þ
α
2
2
2
α
4
16
p
2

¼
Z
M
2
ds
π
ð
s
u
Þ
Im
M
ð
3
Þ
ð
s; u
Þ
:
ð
12
Þ
The dependence on
p
is exact up to EFT-computable
contributions from other light poles (such as light Kaluza-
Klein modes), which we account for in our analysis below,
and Planck-suppressed loop corrections, which we neglect
since
M
M
pl
. Thus
(12)
constitutes an infinite number of
sum rules involving two EFT parameters
α
i
. This
super-
convergence
phenomenon is related to the graviton
s spin
and gauge invariance, which led to the energy growth of
structures in
(8)
. For other sum rules we construct improved
combinations
B
imp
k
ð
p
2
Þ
which are designed to probe finite
sets of EFT couplings. Our complete set of sum rules is
detailed in Appendix
A1
.
III. CONSTRUCTION OF PARTIAL WAVES
Our assumptions about the right-hand-side of
(12)
and
similar relations are minimal; Lorentz symmetry and
unitarity with respect to the asymptotic states. The inter-
mediate states that can appear in a scattering process in
D
¼
d
þ
1
dimensions form representations
ρ
under SO
ð
d
Þ
rotations in the center-of-mass frame. Thus, the
S
-matrix
can be written as a sum over projectors onto each
representation. As far as the
2
2
S
-matrix is concerned,
unitarity is simply the statement that
j
S
ρ
j
1
for the
coefficient of each projector.
The main technical complication in
D>
4
is that many
intermediate representations can appear. Furthermore,
multiple index contractions can exist for a given repre-
sentation. Listing them is equivalent to enumerating on
shell three-point vertices between two massless and one
massive particle. We introduce here an efficient method to
construct structures and projectors in arbitrary
D
.
A. Partial wave expansion
Concretely, the partial wave expansion for a
2
2
graviton scattering amplitude takes the form
M
¼
s
4
D
2
X
ρ
n
ð
D
Þ
ρ
X
ij
ð
a
ρ
ð
s
ÞÞ
ji
π
ij
ρ
;
ð
13
Þ
where
ρ
runs over finite-dimensional irreps of SO
ð
d
Þ
, and
the normalization
n
ð
D
Þ
ρ
is in
(C10)
. For completeness, a
derivation of this formula is presented in Appendix
C
.
The partial waves
π
ij
ρ
are functions of polarizations and
momenta that transform in the representation
ρ
under the
little group SO
ð
d
Þ
preserving
P
μ
¼
p
μ
1
þ
p
μ
2
. We build
them by gluing vertices
v
i;a
ð
n; e
1
;e
2
Þ
, where
a
is an
SO
ð
d
Þ
-index for
ρ
,
i
labels linearly-independent vertices,
and
n
μ
p
μ
2
p
μ
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
p
1
p
2
Þ
2
p
;e
μ
i
ε
μ
i
p
μ
i
ε
i
·
P
p
i
·
P
;
ð
14
Þ
are natural vectors orthogonal to
P
. Note that
n
2
¼
1
, and
the
e
i
are gauge invariant, null, and orthogonal to
n
,
n
·
e
i
¼
e
2
i
¼
0
:
ð
15
Þ
In the center-of-mass frame,
n
and
e
i
are simply the orienta-
tion and polarizations of incoming particles. Defining an
outgoing orientation similarly,
n
0
μ
ð
p
4
p
3
Þ
μ
,partial
waves are defined by summing over intermediate indices,
π
ij
ρ
ð
v
i
;v
j
Þ
v
i;a
ð
n
0
;e
3
;e
4
Þ
g
ab
v
j;b
ð
n; e
1
;e
2
Þ
;
ð
16
Þ
where
g
ab
is an SO
ð
d
Þ
-invariant metric on
ρ
,and
̄
f
denotes
Schwarz reflection
̄
f
ð
x
Þ¼ð
f
ð
x

ÞÞ

.
Unitarity of
S
implies that the matrix
S
ρ
ð
s
Þ
1
þ
ia
ρ
ð
s
Þ
satisfies
j
S
ρ
ð
s
Þj
1
, which implies
0
Im
a
ρ
2
(where
an inequality of matrices is interpreted as positive-
semidefiniteness of the difference). We illustrate these
concepts in some examples in Appendix
C
.
Our method uses the two sides of the unitarity inequality
in different ways: the non-perturbative upper bound ensures
convergence of sum rules through
(10)
, whereas positivity
(
0
Im
a
ρ
) constrains the sign of otherwise unknown
heavy state contributions to these sum rules.
TABLE I. Behavior in the fixed-
t
Regge limit of polarization
structures, omitting some simple permutations, i.e.
H
34
H
12
.
H
12
H
13
H
14
X
1234
X
1324
X
1243
X
1234
X
1324
S
G
s
1
s
1
s
0
s
2
s
2
s
1
s
1
s
2
s
2
GRAVITON PARTIAL WAVES AND CAUSALITY IN HIGHER
...
PHYS. REV. D
108,
026007 (2023)
026007-3
B. Review of orthogonal representations
A finite-dimensional irrep of SO
ð
d
Þ
is specified by a
highest weight
ρ
¼ð
m
1
;
...
;m
n
Þ
,where
n
¼b
d=
2
c
, see e.g.,
[14,15]
.The
m
s are integers for bosonic representations and
half-integers for fermionic representations, satisfying
m
1

m
n
1
j
m
n
j
:
ð
17
Þ
For tensor representations,
j
m
i
j
are the row lengths of the
Young diagram for
ρ
. Note that
m
n
must be positive in odd-
d
,
but can be negative in even-
d
the sign of
m
n
indicates the
chirality of the representation. We omit vanishing
m
s from
the end of the list, for instance denoting a spin-
J
traceless
symmetric tensor by (
J
).
To manipulate tensors, we represent them as index-free
polynomials in polarization vectors
w
1
;
...
;w
n
C
d
, one
for each row. The traceless and symmetry properties of a
given irrep are captured by taking these to be orthogonal
and defined modulo gauge redundancies
[16]
:
w
2
i
¼
w
i
·
w
j
¼
0
;w
j
w
j
þ
#
w
i
for
j>i:
ð
18
Þ
The latter means that allowed functions of
w
must be
annihilated by
w
1
·
w
2
, etc. Three-point vertices are then
simply SO
ð
d
Þ
-invariant polynomials
v
i
ð
w
1
;
...
;w
n
;
n;
e
1
;e
2
Þ
where the
w
s play the same role for a massive
particle that the
ε
s play for gravitons.
Polynomials satisfying the gauge condition can be
easily constructed by inscribing vectors in the boxes of a
Young tableau, where each column represents an anti-
symmetrized product with
w
s. For example, given vectors
a
μ
;
...
;e
μ
C
d
, we can define a tensor in the (3,2)
representation via
ð
19
Þ
Any tableau defines a valid tensor. Tableaux are not unique,
since we can permute columns. Also, antisymmetrizing all
the boxes in one column with another box (of not higher
height) yields a vanishing polynomial, e.g.,
ð
20
Þ
C. Vertices with two massless and one heavy state
With this technology, we can straightforwardly write all
three-point vertices between two gravitons and an arbitrary
massive state. Here we focus on generic dimensions
D
8
, relegating special cases in lower dimensions to
Appendix
B
. All we can write are the dot product
e
1
·
e
2
and Young tableaux in which each box contains either
n
,
e
1
or
e
2
. Evidently, no tableau can have more than three rows,
by antisymmetry.
As a warmup, consider two nonidentical massless
scalars. Two-particle states form traceless symmetric ten-
sors of rank
J
, i.e., single-row tableaux. The only possible
SO
ð
d
Þ
-invariant vertex involving
n
is then
ð
21
Þ
Denoting by
an arbitrary (possibly zero) number of
boxes containing
n
, the most general coupling between two
scalars and a heavy particle is thus simply
.
Moving on to two spin-1 particles, one must add one
power of each of
e
1
,
e
2
. These can appear either as
e
1
·
e
2
or
inside a tableau, giving the exhaustive list:
ð
22
Þ
A potential tableau
was removed since it is redundant
thanks to
(20)
. Thus, there are six possible vertices. If the
two particles are identical, e.g., photons, we get additional
restrictions on the parity in
n
for example the number of
boxes in the first two structures must be even.
The analogous basis of couplings for gravitons in generic
dimension
D
8
are shown in Table
II
. This basis agrees
with
[17]
. Changes in lower dimensions are listed in
Appendix
B
.
TABLE II. The 20 graviton-graviton-massive couplings in generic dimension (
D
8
). Cells collect structures that can be in the same
representation.
stands for an arbitrary (possibly zero) even number of
n
boxes;
S
flips
n
and swaps
e
1
and
e
2
.
SIMON CARON-HUOT
et al.
PHYS. REV. D
108,
026007 (2023)
026007-4
D. Gluing vertices using weight-shifting operators
To glue vertices into partial waves we need to sum over
intermediate spin states. This can be achieved efficiently
using weight-shifting operators
[18]
. A general weight-
shifting operator
D
a
is an SO
ð
d
Þ
-covariant differential
operator that carries an index
a
for some finite-dimensional
representation of SO
ð
d
Þ
, such that acting on a tensor in the
representation
ρ
it gives a tensor in the representation with
shifted weights
ρ
þ
δ
. We will be particularly interested in
the operator
D
ð
h
Þ
μ
that removes one box at height
h
from a
Young diagram with height
h
,
D
ð
h
Þ
μ
ρ
¼ð
m
1
;
...
;m
h
Þ
ð
m
1
;
...
;m
h
1
Þ
ρ
0
:
ð
23
Þ
Conceptually,
D
ð
h
Þ
μ
is a Clebsch-Gordon coefficient for
: this ensures its existence and uniqueness up to
normalization. Explicitly,
D
ð
h
Þ
μ
is given by
[19]
D
ð
h
Þ
μ
0
¼

δ
μ
0
μ
1
w
μ
0
1
N
ð
h
Þ
1
w
μ
1
1

δ
μ
1
μ
2
w
μ
1
2
N
ð
h
Þ
2
w
μ
2
2


×

δ
μ
h
1
μ
h
w
μ
h
1
h
N
ð
h
Þ
h
1
w
μ
h
h

w
h
μ
h
;
ð
24
Þ
where
N
ð
h
Þ
i
¼
d
1
þ
m
i
þ
m
h
i
h
. Notice the shift by
1 in the last parenthesis:
1
=
ð
N
ð
h
Þ
h
1
Þ
. The
h
¼
1
case of
(24)
is the familiar Todorov/Thomas operator that acts on
traceless symmetric tensors
[21]
.
For the definition
(24)
to be consistent, the following
properties must hold:
(i)
D
ð
h
Þ
μ
preserves the gauge constraints: for all
i<j
,
w
i
·
w
j
D
ð
h
Þ
μ
X
¼
0
if
X
satisfies the same.
(ii)
D
ð
h
Þ
μ
sends traces to traces. By
traces
we mean
index contractions in strictly gauge-invariant poly-
nomials (
not
just products
w
2
·
w
3
)
for example,
the following expression where
μ
denotes a unit
vector in the
μ
direction:
ð
25
Þ
These properties are nontrivial and determine
D
ð
h
Þ
μ
up to an
overall constant, which can be fixed by considering traces
on height-
h
columns. For example, consider adjacent gauge
transformations
w
i
·
w
i
þ
1
. Commuting across the
i
th and
(
i
þ
1
)th parentheses one finds an unwanted term propor-
tional to
ð
N
ð
h
Þ
i
m
i
Þ
ð
N
ð
h
Þ
i
þ
1
m
i
þ
1
þ
1
Þ
, whose vanish-
ing recursively determines all
N
s in terms of
N
ð
h
Þ
h
as stated
below
(24)
.
Effectively,
D
ð
h
Þ
recovers indices from index-free poly-
nomials and enables one to evaluate the pairing
(16)
recursively in terms of simpler pairings, for example
ð
26
Þ
Such a formula holds for any choice of a column of
maximal height
h
on the left factor, giving
1
=m
h
times a
sum with alternating sign over the boxes it contains,
see
(F1)
. In practice, since
D
ð
h
Þ
sends tableaux to tableaux,
it can be elegantly implemented as a combinatorial oper-
ation, as discussed in Appendix
F
.
By repeatedly applying
(26)
and its generalization
(F1)
,
any pairing can be reduced to a pairing between single-row
tableaux of length
m
1
¼
J
:
ð
27
Þ
This can be computed by taking derivatives with respect to
n
and
n
0
of the scalar partial wave (see also
[22]
):
ð
28
Þ
where
P
J
ð
x
Þ
is a Gegenbauer polynomial [see
(C14)
] and
ð
a
Þ
n
is the Pochhammer symbol. Thus,
(26)
and
(28)
allow
us to glue the vertices from Table
II
into partial wave
expressions which hold for arbitrary
J
¼
m
1
, involving
derivatives of
P
J
ð
x
Þ
times dot products between graviton
polarizations
e
j
and directions
n
,
n
0
. This procedure can be
straightforwardly and efficiently automated on a computer.
To limit the size of final expressions, we use the
Gegenbauer equation
ð
x
2
1
Þ
2
x
P
J
ð
x
Þþ
...
¼
0
to remove
any monomial of the form
x
a
P
ð
b
Þ
J
ð
x
Þ
with
a
,
b
2
. We then
insert a set of linearly independent polarizations to project
onto the generators
(8)
of the local module and extract
M
ð
i
Þ
s that are polynomials in
x
. Finally, we use the Gram-
Schmidt method to find orthonormal combinations of
vertices according to
(C15)
. As a consistency check on
our results, we verified that our partial waves are eigen-
vectors of the SO
ð
d
Þ
quadratic Casimir.
IV. RESULTS AND INTERPRETATION
Dispersive sum rules like
(12)
express low-energy EFT
parameters as sums of high-energy partial waves, times
unknown positive couplings [through
(13)
with
0
Im
a
ρ
].
The
bootstrap
game consists in finding linear combinations
such that all unknowns contribute with the same sign. Such
combinations yield rigorous inequalities that EFT parameters
must satisfy if a causal and unitary UV completion exists.
To obtain optimal inequalities in a gravitational setting,
we follow the numerical search strategy of
[10,12]
.
GRAVITON PARTIAL WAVES AND CAUSALITY IN HIGHER
...
PHYS. REV. D
108,
026007 (2023)
026007-5
Because of the graviton pole, it is not legitimate to expand
around the forward limit; rather our trial basis consists of
the improved sum rules
B
imp
k
ð
p
2
Þ
integrated against wave
packets
ψ
i
ð
p
Þ
with
j
p
j
M
. We ask for a positive action on
every state of mass
m
M
and arbitrary SO
ð
d
Þ
irrep, as
well as on light exchanges of spin
J
2
and any mass. Full
details of our implementation are given in Appendix
E
.
Figure
1
displays our main result; the allowed region for
the dimensionless parameters
ð
α
2
M
2
;
α
4
M
4
Þ
which control
the leading corrections to the action
(1)
, in terms of the
mass
M
of higher-spin states. For the purposes of illus-
tration, we show the results for
D
¼
5
, 7, 10; other
dimensions
D
lead to qualitatively similar plots. The
parameters are defined more precisely in
(D1)
, and enter
the on shell three-graviton vertex
(D3)
. It would be
interesting to compare these bounds with the explicit
values of Wilson coefficients in
theory islands
arising
from known UV completions
[23]
.
The
M
scaling of the bounds is significant: it implies that
higher-derivative corrections can never parametrically com-
pete with the Einstein-Hilbert term, within the regime of
validity of a gravitational EFT. As soon as corrections
become significant, new particles must be around the corner.
Since we assume
M
M
pl
, graviton scattering is still weak
at the cutoff. In gravity, unlike in other low-energy theories,
the leading (Einstein-Hilbert) interactions cannot be tuned to
zero without setting all other interactions to zero.
What happens at the scale
M
? Since we allowed for
exchanges of arbitrary light states of low spins,
M
is
associated with the mass of
J
3
states. The importance of
higher-spin states was anticipated in
[3]
. In general, higher-
spin states must come in towers that include all spins
[24]
.
For instance,
M
could signal the beginning of a tower of
higher-spin particles (as in weakly coupled string theory),
that each couple to two gravitons with strength
M
2
ffiffiffiffi
G
p
.
Alternatively,
M
could be the energy at which loops
representing a large number
N
M
2
D
=G
of two-particle
states that couple with weaker strength
M
D
þ
2
2
G
to two
gravitons, become non-negligible
[25,26]
. Either way,
graviton scattering must be profoundly modified at the
scale
M
and above, while remaining weak.
Our flat-space bounds have implications in curved space-
times. As explained in
[11]
, since the scattering processes
under consideration take place in a region of small size
1
=M
, flat-space dispersive bounds uplift in anti
de Sitter
(AdS) to rigorous bounds on holographic CFTs, up to
corrections suppressed by
1
=
ð
MR
AdS
Þ¼
1
=
Δ
gap
.
Focusing on
D
¼
5
(the AdS
5
=
CFT
4
correspondence),
stress-tensor two- and three-point functions are character-
ized by three parameters, including the central charges
a
and
c
that enter the conformal anomaly
[27]
. Their relation
to higher-derivative couplings is particularly simple when
the EFTaction is expressed in terms of Weyl tensors, so that
renormalization of the AdS radius is avoided. Using the
field redefinition invariant formulas from
[28]
we find
a
¼
π
2
R
3
AdS
8
π
G
;
a
c
a
¼
2
α
2
R
2
AdS
:
ð
29
Þ
Figure
1
thus implies a sharp central charge bound,




a
c
c




23
Δ
2
gap
þ
O
ð
1
=
Δ
4
gap
Þð
AdS
5
=
CFT
4
Þ
;
ð
30
Þ
which could potentially be improved at the
5%
level. In
holographic theories, this result is stronger than the
conformal collider bound
1
3
a
c
31
18
[29]
and establishes
the parametric scaling anticipated in
[3,30,31]
. We stress
that since
Δ
gap
is the dimension of the lightest higher-spin
(nondouble-trace) operator, the bound holds even in the
presence of light Kaluza-Klein modes (as in AdS
5
×S
5
)
and is generally independent of the geometry of the internal
manifold. The sign of (
a
c
) is significant
[32]
; our results
do not exclude either sign.
The leading contact interaction in
D
7
is the
6-derivative
third Lovelock term
, which is related to
α
0
4
in
(1)
. Our bounds for this coefficient depend only
weakly on its sign and on
α
2
,
α
4
, and yield the absolute
limits in e.g.,
D
¼
7
, 10:
j
α
0
4
M
4
j
56
ð
D
¼
7
Þ
;
j
α
0
4
M
4
j
25
ð
D
¼
10
Þ
:
ð
31
Þ
In analogy with scalar EFTs
[24,33
37]
and four-
dimensional gravitons and photons
[12,38
40]
, we expect
this method to yield two-sided bounds on all higher-
derivative interactions that can be probed by four-graviton
scattering, and on many derivative couplings involving
matter fields.
FIG. 1. Allowed region for couplings
α
2
and
α
4
in
D
¼
5
, 7 and
10 spacetime dimensions, in units of the mass
M
of higher-spin
states.
SIMON CARON-HUOT
et al.
PHYS. REV. D
108,
026007 (2023)
026007-6
ACKNOWLEDGMENTS
We thank Cyuan-Han Chang, Clifford Cheung, Yanky
Landau, Petr Kravchuk, and Sasha Zhiboedov for discus-
sions. D. S. D., S. C. H., and Y. Z. L. are supported by the
Simons Foundation through the Simons Collaboration on
the Nonperturbative Bootstrap. DSD is also supported by a
DOE Early Career Award under Grant No. DE-SC0019085.
S. C. H. is also supported by the Canada Research Chair
program, reference number CRC-2021-0042 and the Sloan
Foundation. J. P. M. is supported by the DOE under Grant
No. DE-SC0011632. The computations presented here were
conducted in the Resnick High Performance Computing
Center, a facility supported by Resnick Sustainability
Institute at the California Institute of Technology. This
research was enabled in part by support provided by
Calcul Qu ́
ebec and Compute Canada (Narval and
Graham clusters).
APPENDIX A: LOCAL MODULE AND SUM
RULES IN VARIOUS DIMENSIONS
1. Sum rules in
D
8
In
D
8
, there are 19 independent sum rules with even
spin
k
2
that can be constructed from applying dispersion
relations to coefficients of the local basis with independent
Regge limits
(8)
,
B
k
ð
p
2
Þ¼
I
ds
4
π
i

ð
s
u
Þ
M
ð
3
;
10
Þ
ð
s; u
Þ
ð
su
Þ
k
2
2
;
ð
s
u
Þ
M
ð
2
;
5
;
8
;
9
Þþ
ð
s; t
Þ
ð
su
Þ
k
2
2
;
ð
s
u
Þð
M
ð
6
Þþ
ð
s; t
Þþ
M
ð
7
Þ
ð
s; u
ÞÞ
ð
su
Þ
k
2
2
;
M
ð
4
;
6
;
9
Þ
ð
s; t
Þ
ð
su
Þ
k
2
2
;
×
M
ð
5
Þ
ð
s; u
Þ
ð
su
Þ
k
2
2
;
ð
s
u
Þ
M
ð
1
;
6
;
7
;
8
Þ
ð
s; u
Þ
ð
su
Þ
k
2
;
ð
s
u
Þ
M
ð
3
Þþ
ð
s; t
Þ
ð
su
Þ
k
2
;
ð
s
u
Þð
M
ð
5
Þþ
ð
t; s
Þ
2
M
ð
4
Þ
ð
s; u
ÞÞ
ð
su
Þ
k
2
;
M
ð
5
Þ
ð
t; u
Þ
ð
su
Þ
k
2
;
×
ð
s
u
Þ
M
ð
2
Þ
ð
s; u
Þ
ð
su
Þ
k
þ
2
2
¼
0
;
ð
A1
Þ
where
M

M
s
u
Þ
and
t
¼
p
2
¼
s
u
is held fixed. We use multiple superscripts
M
ð
i
1
;
...
;i
k
Þ
to indicate a
sequence of similar expressions involving the amplitudes
M
ð
i
1
Þ
:
...
;
M
ð
i
k
Þ
. Forodd
k>
1
, there are 10 independent sum rules,
B
k
ð
p
2
Þ¼
I
ds
4
π
i

M
ð
2
;
5
;
8
Þ
ð
s; t
Þ
ð
su
Þ
k
3
2
;
M
ð
3
;
7
Þ
ð
s; t
Þ
ð
su
Þ
k
1
2
;
ð
s
u
Þ
M
ð
4
;
7
Þþ
ð
s; t
Þ
ð
su
Þ
k
1
2
;
ð
s
u
Þ
M
ð
5
Þþ
ð
s; u
Þ
ð
su
Þ
k
1
2
;
×
ð
s
u
Þ
M
ð
9
Þ
ð
s; u
Þ
ð
su
Þ
k
1
2
;
ð
s
u
Þ
M
ð
4
Þ
ð
s; u
Þ
ð
su
Þ
k
þ
1
2
¼
0
:
ð
A2
Þ
The Regge bound
(10)
implies that these sum rules converge for
k>
1
.
2. Sum rules in lower dimensions
In lower dimensions
D
7
, there are two novelties for local modules as noted in
[8]
. First, new identities can reduce the
number of parity-even generators of the local module. This does not occur in
D
¼
7
.However,in
D
¼
6
the generator
G
does not exist, thus we must remove the parity-even sum rules involving
M
ð
1
Þ
ð
s; u
Þ
. Similarly in
D
¼
5
, we simply remove
the parity-even sum rules involving
M
ð
1
;
6
;
7
Þ
ð
s; u
Þ
.
The second novelty in lower dimensions is that new parity-odd structures appear. Following
[8]
, we organize them into
multiplets under permutations. In
D
¼
7
, there is one parity-odd singlet and two parity-odd triplets:
singlets
iS
ε
ð
ε
1
;
ε
2
;
ε
3
;
ε
4
;p
1
;p
2
;p
4
Þ
M
ð
13
Þ
ð
s; u
Þ
;
triplets
iH
14
H
23
ε
ð
ε
1
;
ε
2
;
ε
3
;
ε
4
;p
1
;p
2
;p
4
Þ
M
ð
11
Þ
ð
s; u
Þ
;
ð
D
¼
7
Þ
iX
1243
ε
ð
ε
1
;
ε
2
;
ε
3
;
ε
4
;p
1
;p
2
;p
4
Þ
M
ð
12
Þ
ð
s; u
Þ
:
ð
A3
Þ
Correspondingly, we can construct more sum rules
B
k
ð
p
2
Þ¼
I
ds
4
π
i

M
ð
11
Þ
ð
s; t
Þ
ð
su
Þ
k
2
2
;
ð
s
u
Þ
M
ð
12
Þ
ð
s; u
Þ
ð
su
Þ
k
2
¼
0
ð
even
k; D
¼
7
Þ
;
B
k
ð
p
2
Þ¼
I
ds
4
π
i

M
ð
12
Þ
ð
s; t
Þ
ð
su
Þ
k
1
2
;
ð
s
u
Þ
M
ð
11
;
12
Þþ
ð
s; t
Þ
ð
su
Þ
k
1
2
;
ð
s
u
Þ
M
ð
13
Þ
ð
s; u
Þ
ð
su
Þ
k
1
2
¼
0
ð
odd
k; D
¼
7
Þ
:
ð
A4
Þ
In
D
¼
6
, there are three parity-odd triplets:
GRAVITON PARTIAL WAVES AND CAUSALITY IN HIGHER
...
PHYS. REV. D
108,
026007 (2023)
026007-7