JHEP05(2023)122
Published for SISSA by
Springer
Received
:
June 10, 2022
Accepted
:
April 16, 2023
Published
:
May 16, 2023
Causality constraints on corrections to Einstein gravity
Simon Caron-Huot,
a
Yue-Zhou Li,
a
Julio Parra-Martinez
b
and David Simmons-Duffin
b
a
Department of Physics, McGill University,
3600 Rue University, Montréal, H3A 2T8, QC Canada
b
Walter Burke Institute for Theoretical Physics, Caltech,
Pasadena, California 91125, U.S.A.
E-mail:
schuot@physics.mcgill.ca
,
liyuezhou@physics.mcgill.ca
,
jparram@caltech.edu
,
dsd@caltech.edu
Abstract:
We study constraints from causality and unitarity on
2
→
2
graviton scattering
in four-dimensional weakly-coupled effective field theories. Together, causality and unitarity
imply dispersion relations that connect low-energy observables to high-energy data. Using
such dispersion relations, we derive two-sided bounds on gravitational Wilson coefficients
in terms of the mass
M
of new higher-spin states. Our bounds imply that gravitational
interactions must shut off uniformly in the limit
G
→
0
, and prove the scaling with
M
expected from dimensional analysis (up to an infrared logarithm). We speculate that
causality, together with the non-observation of gravitationally-coupled higher spin states
at colliders, severely restricts modifications to Einstein gravity that could be probed by
experiments in the near future.
Keywords:
Classical Theories of Gravity, Effective Field Theories, Models of Quantum
Gravity, Scattering Amplitudes
ArXiv ePrint:
2201.06602
Open Access
,
c
©
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP05(2023)122
JHEP05(2023)122
Contents
1 Introduction
1
2 Review: dispersive sum rules
4
2.1 Helicity amplitudes and low-energy EFT
5
2.2 High energies: partial waves and unitarity
7
2.3 Regge boundedness and all that
9
2.4 Dispersive sum rules
11
2.5 Review of simple bounds from forward limits
14
3 Bounds that relate gravity and higher derivatives
16
3.1 Impact parameter functionals
16
3.2 Example positive functionals involving gravity
17
3.3 Light spin-0 and spin-2 matter fields don’t lower the cutoff
20
3.4 Systematic strategy: improved sum rules
22
3.5 Relation with the CEMZ argument
23
4 Results
26
4.1 Comparison with model amplitudes
26
4.2 Bounds involving
R
3
,
R
4
and gravity
27
4.3 Bounds involving
D
2
R
4
29
4.4 Bounds involving
D
4
R
4
and low spin dominance
30
4.5 Can higher-spin states be hidden from the Standard Model?
33
5 Conclusions
35
A Spinning partial waves and Wigner-d functions
37
B Amplitudes from matter and Kaluza-Klein exchanges
38
C Details on numerics
39
1 Introduction
Einstein’s theory of general relativity (GR) has been extraordinarily successful since its
inception over a century ago. Nevertheless, modifications to GR are often discussed in
relation to various puzzles, such as the nature of dark energy and dark matter, see [
1
–
3
] for
reviews. An important class of modifications add higher-derivative terms to the equations
of motion, which lead to physical effects which grow at short distances. Such corrections,
handily classified in [
4
], arise naturally in string theory, and presumably in any UV-complete
theory of quantum gravity.
In this paper, we consider
2
→
2
scattering of gravitons and ask a simple question:
assuming that graviton scattering respects causality at
all
energies, by how much can the
– 1 –
JHEP05(2023)122
low-energy amplitude differ from the predictions of general relativity? A well-known result
by [
5
,
6
] shows any Lorentz-invariant theory of massless spin-2 particles must reproduce
general relativity at large distances. Our goal will be to bound the corrections to this limit,
assuming that relativistic causality (as we understand it) holds.
Our answer will depend on the spectrum of the theory. Let us describe our setup and
assumptions. At low energies, we assume there exists a massless graviton, together with a
finite number of fields of spin
≤
2
, which can be described by an effective field theory (EFT).
Schematically, the low-energy effective action encodes modifications to Einstein gravity:
S
=
1
16
πG
∫
d
4
x
√
g
(
R
+
g
R
(3)
Riem
3
+
g
R
(4)
Riem
4
+
...
)
+
S
matter
,
(1.1)
where
Riem
n
denotes (possibly non-unique) contractions of products of
n
Riemann tensors.
An important idea is that the sizes of the Wilson coefficients
g
R
(3)
,g
R
(4)
,...
are con-
strained by causality, that is, the notion that signals cannot travel faster than light. For
example in [
7
] it was observed by Camanho-Edelstein-Maldacena-Zhiboedov (CEMZ) that
in the presence of
g
R
(3)
, the two polarization modes of the graviton would move at different
velocities in certain backgrounds, and inevitably one of them moves faster than “light”. By
considering a setup with large enough black holes and mirrors, this effect could lead to
closed timelike curves, with ensuing grandparent-type paradoxes; the conclusion is that a
classical theory with
g
R
(3)
6
= 0
is inconsistent.
1
Ref. [
7
] further pointed out that paradoxes
can be avoided at the quantum level if the graviton couples to higher-spin states. Denoting
the mass of the lightest higher-spin state (spin 4 or higher) by
M
, this led to a parametric
bound:
|
g
R
(3)
|
∼
<
1
M
4
.
Our main goal in this paper will be to quantitatively bound higher-derivative corrections
in
(1.1)
in terms of the mass
M
of higher-spin states. This mass
M
provides a UV-cutoff
scale for the low-energy EFT in (
1.1). We will assume a large hierarchy between the Plank
scale and this cutoff
M
2
M
2
pl
,
(1.2)
or, equivalently,
GM
2
1
, so that gravity is weakly-interacting below the cutoff. Our
methods will test causality of graviton scattering with arbitrary center-of-mass energies,
although physically the most important region for us will be near the cutoff
M
.
The notion of causality is subtle in gravitational EFTs because there is no globally
well-defined lightcone in nontrivial backgrounds. As discussed recently in [
10
,
11
], one
should contrast “asymptotic causality,” which exploits a fixed causal structure at large
distances, with “infrared causality” which compares local time delays between species [
11
].
We will use asymptotic causality, but crucially, imposed at all energy scales and not only
within the EFT regime. This leads to sharp mathematical statements involving crossing
symmetry, analyticity, and Regge boundedness of scattering amplitudes [
12]. Many works
have examined how these conditions constrain EFTs and their UV completions, see e.g. [
13
–
37
] and references therein. In particular, these conditions give rise to dispersion relations
that relate high and low energies. In some cases, dispersion relations can be interpreted as
1
A related effect is that the classical initial-value problem is generically ill-posed when higher-derivative
terms dominate [
8, 9 ].
– 2 –
JHEP05(2023)122
expressing the commutativity of coincident shockwaves [
38
]. Initially, the use of dispersion
relations in gravitational EFTs was hindered by divergences related to the forward limit of
the low-energy graviton amplitude. These technical issues were recently partially overcome
in [
15
] by studying dispersion relations in impact parameter space; hence our renewed
interest in this problem.
There is a natural motivation to study scattering events which have center-of-mass
energies above the “cutoff”
M
. In any scenario where a higher-derivative correction to GR
might be observed in an astrophysical or other large-distance process, the suppression scale
would have to be very low,
M
1
TeV
= (2
×
10
−
19
m)
−
1
: energies above such a cutoff
are routinely probed at colliders. However, collider experiments have not yet reported any
higher-spin particles of the type suggested above. Is it at all possible to modify GR in a
way that simultaneously: (1) satisfies collider constraints, (2) is relevant at large scales, and
(3) respects causality as we understand it?
In this paper we take a modest step toward answering this question, by quantitatively
relating higher-derivative corrections to the mass
M
of higher-spin states, using methods
from [
15
]. Our main results will be that dimensionless ratios, of the schematic form
g
R
(3)
M
4
or
g
R
(4)
M
6
, are bounded by order-unity constants, times an infrared logarithmic divergence
log
(
M/m
IR
)
. Alternatively, given a measurement of such couplings, we constrain the mass
M
of new states and their couplings to gravitons. The task of bounding their couplings to
Standard Model fields, as well as and potential collider signals, is left to future work.
The operational definition of “gravity” in this paper is a force which grows linearly with
energy at high speeds, corresponding in particle physics language to exchange of a spin-2
particle. We stress that static long-range forces, which could come from direct interactions
between matter and light spin-0 or spin-1 particles (also sometimes called fifth forces), are
unconstrained by our arguments.
The infrared logarithmic divergence in our bounds is related to the divergence of
the eikonal phase in four dimensions. In the context of scalar scattering in AdS/CFT,
this logarithmic divergence gets regulated by the AdS curvature scale, yielding rigorous,
finite bounds proportional to
log
MR
AdS
[
39
]. We expect the same mechanism to apply
to graviton scattering as well. Thus, our bounds can be interpreted as finite bounds on
gravitational Wilson coefficients in AdS
4
. The key feature of our bounds is the absence
of power-law infrared divergences; eventually, we hope that infrared logarithms can be
removed by studying suitable IR finite observables. In our view, an incredibly conservative
assumption would be to replace
m
IR
with the Hubble scale, which for
M
∼
1
TeV
, multiplies
our bounds by a factor of only
log
M/m
IR
∼
100
. Note that this would still strongly rule
out a modification of GR that simultaneously satisfies (1), (2), and (3) above.
This paper is organized as follows. In section
2 , we state our assumed axioms encoding
causality of graviton scattering, review dispersive sum rules, and some of their known
implications, notably positivity bounds from forward limits. In section
3 , we review the
impact parameter approach of [
15
] and provide example positive functionals which prove
upper bounds on gravitational EFT coefficients. We explain why the bounds are only
weakly affected by possible light matter fields. We then describe our numerical strategy to
systematically search for optimal bounds, and how CEMZ-like bounds are automatically
included. In section
4 , we report the bounds obtained with this method and comment on
– 3 –
JHEP05(2023)122
their relations with known theories. In a more conjectural part
4.5 we speculate about the
possibility of modifications associated with a low scale
M
. We summarize in section
5 . In
appendix
A , we review partial waves for graviton scattering amplitudes. In appendix
B we
present graviton amplitudes from light exchanges of spin-0 and spin-2 particles. Finally, we
record our numerical set-ups in appendix
C .
2 Review: dispersive sum rules
In this section, we describe our key physical assumptions and tools: effective field theory
at low energies (section
2.1 ), unitarity (section
2.2 ), causality (section
2.3 ), and Kramers-
Kronig-type dispersion relations and some of their consequences (section
2.4 ). Most of this
material is standard, except perhaps for our discussion of Regge boundedness in section
2.3 ,
using compact-support wavefunctions.
Let us first briefly motivate our assumptions, before we state them technically.
By unitarity, we mean that initial and final states of scattering processes are elements
of positive-definite Hilbert spaces, whose norms are conserved by time evolution. This repre-
sents the idea that the probabilities of all possible events are positive and add up to one. The
reason we assume this is obvious: we wouldn’t know how to interpret negative probabilities.
By causality, we refer to the notion that “signals can’t travel faster than light”. We
assume causality because of its tremendous explanatory power and past successes: by
forbidding instantaneous action at a distance, it qualitatively explains why electromagnetic
and gravitational waves
must
exist, why forces in nature are mediated by particles, why
antiparticles exist, how their interactions are quantitatively related [
40
], and so much
more. Abandoning causality without a good replacement principle seems akin to opening
Pandora’s box.
2
There is an interesting interplay between unitarity and causality, as displayed by
quantum fields with wrong-sign kinetic terms, sometimes called “ghosts”. In one quantization,
positive-frequency modes propagate forward in time but have negative norms. In an
alternative quantization choice, norms are fine but positive-frequency modes propagate
backward in time. Lee and Wick famously proposed that the ensuing acausality could be
made unobservably small if one treats backward-moving modes as resonances which decay to
normal forward-moving modes [
41
], as is indeed seen at low orders in perturbation theory [
42
,
43
]. A problem is that, as soon as interactions with normal matter are included, negative-
frequency modes make the vacuum unstable against resonant particle production. This
instability can be nicely discussed in connection with a classical theorem by Ostrogradsky [
44
];
it seems incompatible with a long-lived Universe [
45].
3
2
If one were not worried about instantaneous action at a distance, any many-body Hamiltonian such as
H
=
∑
i
~p
i
2
2
m
i
+
∑
i<j
V
ij
with potential
V
ij
=
−
Gm
i
m
j
|
~x
i
−
~x
j
|
, or others, would trivially define a “quantum theory of gravity”.
3
As was pointed out with much deference to the original authors, the Lee-Wick prescription is ambiguous
and “an additional prescription would be needed to completely define the theory” [
46
]. In our view, the
Lee-Wick idea fails to address the vacuum stability issue for the simple reason that the timescales in the
relevant vacuum diagram are shorter than the decay time of Lee-Wick quanta.
– 4 –
JHEP05(2023)122
In short, we assume unitarity and causality because
we see no alternatives
. It is possible
that Nature does not conform to these principles as we understand them, but resulting
bounds can be viewed as tests of these principles.
2.1 Helicity amplitudes and low-energy EFT
Four-dimensional gravitons possess two helicity states. A complete set of independent
amplitudes for graviton-graviton scattering is
M
(1
+
2
−
3
−
4
+
) =
〈
23
〉
4
[14]
4
f
(
s,u
)
,
(2.1)
M
(1
+
2
+
3
+
4
−
) = ([12][13]
〈
14
〉
)
4
g
(
s,u
)
,
(2.2)
M
(1
+
2
+
3
+
4
+
) =
[12]
2
[34]
2
〈
12
〉
2
〈
34
〉
2
h
(
s,u
)
,
(2.3)
and permutations thereof. At tree-level in Einstein’s theory, only the maximal-helicity-
violating (MHV) amplitude
f
(
s,u
)
is nonvanishing, and a large fraction of our results will
be derived by studying only this amplitude. Here, we use spinor-helicity variables (see [
47
])
and we have introduced the Mandelstam invariants
s
=
−
(
p
1
+
p
2
)
2
, t
=
−
(
p
2
+
p
3
)
2
, u
=
−
(
p
1
+
p
3
)
2
,
(2.4)
with
s
+
t
+
u
= 0
. The functions
f
(
s,u
)
,
g
(
s,u
)
, and
h
(
s,u
)
are analytic in the upper-half
plane with
Im
s >
0
, and crossing symmetric:
f
(
s,u
) =
f
(
u,s
)
, g
(
s,u
) =
g
(
t,s
) =
g
(
u,t
)
, h
(
s,u
) =
h
(
t,s
) =
h
(
u,t
)
.
(2.5)
Other helicity amplitudes may be obtained by complex conjugation and Schwarz reflection,
for example the
−−−
+
amplitude is
g
(
s,u
) = (
g
(
s
∗
,u
∗
))
∗
.
At low energies, we assume a spectrum comprising massless gravitons together with
possible light particles of spin
≤
2
. These can be described by an effective field theory
(EFT) of the generic form
(1.1)
, with higher derivative terms encoding modifications to
Einstein gravity generated by physics above the EFT cutoff
M
. We assume
M
M
pl
and thus neglect loops within the EFT. States with spin greater than two are genuinely
gravitational, and assumed to have mass above the cutoff,
m > M
.
By contrast, particles of spin two or less and mass
m
`
M
can be interpreted as
additional states in the Standard Model of particle physics and its extensions, or states
arising from Kaluza-Klein reduction of massless gravity in higher-dimensions. Angular
momentum conservation forbids the decay of a state of half-integer or odd spin to a pair of
gravitons, so only matter fields of spins 0 and 2 can affect graviton scattering at tree level.
We will refer to both as “matter”, even though this nomenclature is slightly unconventional
for spin 2 fields.
The best way to enumerate EFT couplings is to list how they modify graviton scattering
amplitudes. On-shell three-particle vertices are determined by Lorentz invariance up to
overall parameters
M
(1
+
,
2
+
,
3
−
) =
√
8
πG
[12]
6
[13]
2
[23]
2
,
M
(1
+
,
2
+
,
3
+
) =
̂
g
3
2
√
8
πG
([12][13][23])
2
.
(2.6)
– 5 –
JHEP05(2023)122
Figure 1
.
2
-to-
2
scattering amplitudes of gravitons within the low-energy effective theory. We
include at tree-level both the graviton exchange and (higher-derivative) contact diagrams, as well
as exchanges of possible light spin-0 and spin-2 particles. Other spins are forbidden by angular
momentum conservation.
The tree-level four-particle amplitudes
(2.3)
may then be written in terms of exchange
diagrams, plus a sum of contact interactions, which are simply polynomials with the
symmetry (
2.5):
f
low
(
s,u
) =
8
πG
stu
+
2
πGsu
t
|
̂
g
3
|
2
+
g
4
+
g
5
t
+
g
6
t
2
−
g
′
6
su
+
...
+
f
matter
(
s,u
)
+
O
(loops)
,
(2.7a)
g
low
(
s,u
) =
4
πG
stu
̂
g
3
+
1
2
̂
g
′′
6
+
...
+
g
matter
(
s,u
) +
O
(loops)
,
(2.7b)
h
low
(
s,u
) = 40
πG
̂
g
3
stu
+
1
2
̂
g
4
(
s
2
+
t
2
+
u
2
)
2
+ 2
̂
g
5
stu
(
s
2
+
t
2
+
u
2
)
+
̂
g
6
(
s
2
+
t
2
+
u
2
)
3
+
̂
g
′
6
s
2
t
2
u
2
+
...
+
h
matter
(
s,u
) +
O
(loops)
,
(2.7c)
where hatted couplings are complex (the real and imaginary part representing parity-even
and parity-odd couplings, respectively). The subscript “low” emphasizes that this expansion
is used only for
|
s
|
< M
2
. The signs on the first line have been chosen so that our couplings
relate simply to those in [
48
].
4
The matter contributions
f
matter
(
s,u
)
,
g
matter
(
s,u
)
, and
h
matter
(
s,u
)
are recorded in appendix
B .
It is straightforward to write down Lagrangians that give rise to the above amplitudes.
Before doing so, it is important to note that Lagrangian densities are only defined modulo
field redefinitions (which change contact interactions by equation of motions) and total
derivatives. In particular, any higher-derivative term involving the Ricci tensor
R
μν
or scalar
R
is removable, so only powers of the Riemann curvature
R
μνσρ
must be kept.
5
Furthermore,
numerous identities relate various contractions of Riemann tensors and derivatives. This is
the reason why we do not include
R
2
:
R
2
-terms can be recast into the Gauss-Bonnet term,
which is topological in
d
= 4
. In contrast, the amplitudes (
2.7) are unambiguous.
4
The conversion is simply:
{
g
4
, g
5
, g
6
, g
′
6
}
here
=
{
a
0
, a
1
, a
2
,
0
, a
2
,
1
}
there
.
In our notation the subscript always denotes half the number of derivatives in the contact interaction.
5
It is well-known for example that
f
(
R
)
gravity is equivalent to standard Einstein gravity minimally
coupled to a scalar field with a specific potential. From our perspective,
f
(
R
)
gravity thus does not constitute
a higher-derivative correction to Einstein’s gravity. Instead, it is a specific choice of matter sector.
– 6 –
JHEP05(2023)122
With this being said, it is straightforward to list a minimal set of irreducible higher-
dimension operators and map them to the amplitudes
(2.7)
by computing the resulting
tree-level amplitudes. For example, the parity-even sector of cubic gravity contains
10
different operators, but field redefinitions and various identities leave us with only one
independent operators [
49]. Up to dimension eight, our effective action is
S
=
1
16
πG
∫
d
4
x
√
−
g
[
R
−
1
3!
(
α
3
R
(3)
+ ̃
α
3
̃
R
(3)
)
+
1
4
(
α
4
(
R
(2)
)
2
+
α
′
4
(
̃
R
(2)
)
2
+ 2 ̃
α
4
R
(2)
̃
R
(2)
)
+
...
]
+
S
matter
,
(2.8)
where we defined
R
(2)
=
R
μνρσ
R
μνρσ
,
̃
R
(2)
=
R
μνρσ
̃
R
μνρσ
,
̃
R
μνρσ
≡
1
2
μν
αβ
R
αβρσ
,
R
(3)
=
R
μν
ρσ
R
ρσ
αβ
R
αβ
μν
,
̃
R
(3)
=
R
μν
ρσ
R
ρσ
αβ
̃
R
αβ
μν
.
(2.9)
It is then straightforward to expand
g
μν
=
η
μν
+
√
32
πGh
μν
and apply the standard Feynman
techniques to evaluate scattering amplitudes and compare with eqs. (
2.7):
̂
g
3
=
α
3
+
i
̃
α
3
, g
4
= 8
πG
(
α
4
+
α
′
4
)
,
̂
g
4
= 8
πG
(
α
4
−
α
′
4
+
i
̃
α
4
)
.
(2.10)
Note that we absorbed a factor of
8
πG
in three-point couplings but not in four-point
couplings.
2.2 High energies: partial waves and unitarity
We will assume that graviton scattering remains sensible even at center-of-mass energies
that exceed the EFT cutoff
M
(where the parametrization
(2.7)
no longer applies). Our
minimal assumptions are that the amplitude remains causal (that is, analytic) and unitary,
and that the spectrum is relativistic so that it can be organized in terms of mass,
m
2
, and
spin,
J
.
In other words, the amplitude admits a partial wave expansion of the form [
50]
M
(1
h
1
2
h
2
3
h
3
4
h
4
) = 16
π
∑
J
(2
J
+ 1)
a
{
h
}
J
(
s
)
d
J
h
12
,h
34
(
1 +
2
t
s
)
.
(2.11)
Here,
d
J
α,β
(
x
)
are the well-known Wigner-
D
functions, which are explicitly written in
appendix
A , and
h
ij
=
h
i
−
h
j
. The partial wave coefficients,
a
{
h
}
J
(
s
)
, encode all dynamical
information. Without boost invariance, partial waves would be more complicated, but study
of causality constraints has been initiated in [
51, 52 ].
Unitarity of the S-matrix
S
= 1 +
i
M
imposes crucial positivity properties on the
“absorptive part” of the amplitude, through the familiar relation:
i
(
M
†
−M
) =
M
†
M
. The
matrix structure will be important and is in contrast to the scalar case studied in [
13
–
19
].
In terms of partial waves,
i
[(
a
−
h
4
,
−
h
3
,
−
h
2
,
−
h
1
J
(
s
)
)
∗
−
a
h
1
,h
2
,h
3
,h
4
J
(
s
)
]
=
∑
X
(
a
−
h
3
,
−
h
4
→
X
J
(
s
)
)
∗
a
h
1
,h
2
→
X
J
(
s
)
,
(2.12)
– 7 –
JHEP05(2023)122
where
X
runs over intermediate states. The right-hand-side is a positive semi-definite
matrix. To capture its positive properties we adopt an abbreviated notation from [
53
] and
omit the
X
sum, writing the right-hand-side simply as
2(
c
−
h
4
,
−
h
3
)
∗
c
h
1
h
2
. Specializing to
the MHV amplitude, we have
Im
a
+
−
+
−
J
(
s
) =
|
c
+
−
J,s
|
2
,
Im
a
++
−−
J
(
s
) =
|
c
++
J,s
|
2
,
(2.13)
where
c
+
−
J,s
is real. In particular, the quantities in (
2.13) are positive.
6
Note this would not
be so for the permutation
a
+
−−
+
J
= (
−
1)
J
a
+
−
+
−
J
. Partial waves admit two-sided bounds,
which follow from applying the same argument to the unitary matrix
−
S
= 1 +
i
(2
i
−M
)
.
In particular, we have
7
0
≤
Im
a
+
−
+
−
J
(
s
)
≤
1
,
and
0
≤
Im
a
++
−−
J
(
s
)
≤
2
.
(2.14)
Positivity of the spectral density is key for establishing bounds on the low-energy EFT
Wilson coefficients [
13–18].
Explicitly, the MHV amplitude
f
(
s,u
)
has distinct discontinuities in the
s
- and
t
- channels:
s
=
m
2
>
0 :
Im
f
(
m
2
,
−
p
2
) =
16
π
m
8
∑
J
≥
4
(2
J
+ 1)
|
c
+
−
J,m
2
|
2
̃
d
J
4
,
4
(
1
−
2
p
2
m
2
)
,
(2.15a)
t
=
m
2
>
0 :
Im
f
(
−
p
2
,p
2
−
m
2
) =
16
π
m
8
∑
J
≥
0
even
(2
J
+ 1)
|
c
++
J,m
2
|
2
̃
d
J
0
,
0
(
1
−
2
p
2
m
2
)
,
(2.15b)
where
̃
d
J
α,β
are Wigner-
D
functions with stripped helicity factors, see appendix
A for more
details. The overall
m
−
8
originates from the prefactor in (
2.3).
For other helicity configurations, we have similar relations, except that the “imaginary
part” gets replaced by the discontinuity
̃
Im
a
≡
[
a
(
s
+
i
0)
−
a
(
s
−
i
0)]
/
(2
i
)
, and the
right-hand-sides are now complex numbers:
̃
Im
a
+++
−
J
(
s
) =
c
++
J,s
c
+
−
J,s
,
̃
Im
a
++++
J
(
s
) = (
c
++
J,s
)
2
.
(2.16)
The corresponding partial wave expansions are
̃
Im
g
(
m
2
,
−
p
2
)
∣
∣
∣
s
=
m
2
=
16
π
m
12
∑
J
≥
4
even
(2
J
+ 1)
c
++
J,s
c
+
−
J,s
̃
d
J
4
,
0
(
1
−
2
p
2
m
2
)
,
(2.17)
̃
Im
h
(
m
2
,
−
p
2
)
∣
∣
∣
s
=
m
2
= 16
π
∑
J
≥
0
even
(2
J
+ 1)(
c
++
J,s
)
2
̃
d
J
0
,
0
(
1
−
2
p
2
m
2
)
.
(2.18)
6
Even though
c
+
−
J,s
is real, we nonetheless write the absolute value sign
|
c
+
−
J,s
|
2
throughout to emphasize
that its square is positive.
7
In more detail: in the even spin sector, there are three incoming states for the helicities
|
h
1
h
2
〉
: namely
|
++
〉
,
|−−〉
, and
1
√
2
(
|
+
−〉
+
|−
+
〉
)
. The corresponding diagonal elements of the
S
-matrix are
1+
ia
−−
++
,
1+
ia
++
−−
, and
1+2
ia
+
−
+
−
. In the odd-spin sector, the
S
-matrix is a
1
×
1
matrix with element
1+2
ia
+
−
+
−
.
By unitarity, each of these diagonal elements must have real part in
[
−
1
,
1]
, which leads to (
2.14).
– 8 –
JHEP05(2023)122
2.3 Regge boundedness and all that
Low and high energies are related by Kramers-Kronig-type dispersion relations. It will be
crucial that we can predict beforehand which dispersion relations converge.
Typically one assumes a Froissart-Martin-like bound at fixed momentum transfer and
large complex energies:
lim
|
s
|→∞
M
/s
2
→
0
at fixed
t <
0
(
not
what we’ll assume)
.
(2.19)
For example, in tree-level string theory,
M∼
s
2+
α
′
t
< s
2
. However the validity of this bound
is not generally established in an abstract theory of quantum gravity. Martin’s original
proof of the Froissart-Martin bound in axiomatic field theory [
54
] does not apply to gravity,
due to the absence of a mass gap. For holographic theories it has been argued that the
behavior
(2.19)
holds for physical kinematics as a consequence of the chaos bound [
55
,
56
].
The constraint we will use is in fact weaker than
(2.19)
: we will only assume that it
holds after integrating in
t
against suitable wavefunctions. We also assume analyticity in
the upper-half
s
-plane, at least for
s
large, as well as sub-exponential growth for
|
s
|→∞
,
as described more precisely below. This is equivalent to assuming that the unsmeared
amplitude satisfies dispersion relations with a finite number of subtractions, which is a weak
form of UV locality.
The difficulty with
(2.19)
is a physical one and not merely technical: to bound amplitudes
at large complex energies, one must generally combine analyticity with some boundedness
property on the real axis, as we do shortly. The difficulty is that analyticity holds at
fixed momentum, while boundedness holds at fixed impact-parameter; these two spaces are
related by a Fourier transform which is not easy to control. Namely, it is not straightforward
to estimate large-impact-parameter contributions in the absence of a mass gap or of
an explicit model of the dynamics. Thankfully, large-impact-parameter physics however
seems immaterial for bounding EFT couplings at the scale
M
. The intuition, stressed
in [
14
,
16
], is that EFT parameters at the scale
M
satisfy sum rules saturated by impact
parameters
b
∼
M
−
1
.
Let us explain how we sidestep
(2.19)
by adapting a recent method from [
39
], which
showed that the conclusions from flat space sum rules apply to quantum gravity in AdS
(defined as a CFT with large but finite central charges and single-trace gap). The method
is simple: we integrate scattering amplitudes against wavepackets that have finite support
in momentum space and decay rapidly at large impact parameters
b
. Formally, for a
wavefunction
Ψ(
p
)
, we define the smeared amplitude:
M
Ψ
( ̃
s
) =
∫
M
0
dp
Ψ(
p
)
M
(
̃
s
+
1
2
p
2
,
−
p
2
)
.
(2.20)
It is apparent that for
|
̃
s
|
>
1
2
M
2
, all amplitudes on the right-hand-side are in the physical
region where the partial wave expansion
(2.11)
applies. (The offset of
s
by
1
2
p
2
is not essential
but ensures that
s
↔
u
crossing symmetry is simply reflection of
̃
s
.) Furthermore, thanks
to compactness of the integral,
M
Ψ
(
̃
s
)
inherits the analyticity properties of the original
– 9 –
JHEP05(2023)122
amplitude: our fundamental assumption is that a crossing path exists which connects the
two points
̃
s
=
±
1
2
M
2
, and that the amplitude is analytic outside of that arc.
Fast decay in
b
requires
Ψ(
p
)
to be smooth and to vanish rapidly enough at the
endpoints; the precise condition is detailed below (see
(3.16)
). The upshot is that if the
decay sets in at some
b > b
∗
, then the spin sum in
(2.11)
is effectively limited to
J
≤
√
sb
∗
.
Since individual
a
J
are bounded (see (
2.14)), one trivially gets the bound
|M
Ψ
(
s
)
|≤
s
×
constant
(
|
s
|
>
1
2
M
2
,
real
)
.
(2.21)
We thus have an analytic function which is bounded on the real axis. Unless this func-
tion grows exponentially at complex energies (which would imply blatant time advances
when Fourier transformed to the time domain, a behavior which was not seen in theo-
ries of quantum gravity in AdS realized by unitary CFTs [
39
]), it must be bounded in
all complex directions by a version of the maximum principle called Phragmén-Lindelöf
principle (see [
55]):
|M
Ψ
(
s
)
|≤|
s
|×
constant
(
complex
s
outside
s
∼
M
2
arc
)
.
(2.22)
The results presented in this paper rely only on the above properties of smeared amplitudes
M
Ψ
, and not on
(2.19)
. It is intriguing that this reasoning produces an exponent in
(2.22)
that is formally stronger than that in
(2.19)
:
|M
Ψ
/s
|
bounded versus
|M
Ψ
/s
2
|→
0
. The
extra power will not be exploited in the present paper, but it would be interesting to see if
it has any implications. We will only use that
lim
|
s
|→∞
|M
Ψ
(
s
)
|
/
|
s
|
2
= 0
, which is easily
implied by (
2.22) and is strictly weaker than (
2.19).
8
To be fully explicit, since smearing is technically involved, in this paper we will only
use smeared amplitudes when strictly required by the
t
→
0
singularities of amplitudes.
For higher-subtracted dispersion relations, that do not suffer from a graviton pole, we will
effectively use an assumption similar to
(2.19)
, namely
|M
/s
4
|→
0
for
t <
0
, as detailed in
section
3.4 . This is similar to the assumption made in ref. [
48].
9
Now that our mathematical assumptions (analyticity and boundedness) have been
stated precisely, we would like to explain why we believe these are conservative and physically
relevant notions directly related to causality and unitarity. Analyticity is related to the
notion that “signals cannot move faster than light” in the context of signals that are waves.
For example for vacuum two-point functions in quantum field theory, well-known arguments
show that the vanishing of spacelike commutators requires the existence of antiparticles. This
happens because waves components of different frequencies can only interfere destructively
in the spacelike region if they are related to each other in a very specific way, namely by an
analytic continuation in the upper-half energy plane, which relates positive and negative
frequencies. The same interpretation applies to Kramers-Kronig dispersion relations.
8
After the first arXiv version of this article appeared, the paper [
57
] analyzed the high-energy behavior of
smeared amplitudes and independently confirmed
(2.22)
for a suitable class of smearing functions in
d
≥
5
where there are no infrared divergences.
9
Note that the spin-2 dispersion relations that we use are nonperturbative statements that need not be
satisfied order-by-order, for example by the individual one-loop diagrams studied in [
48].
– 10 –
JHEP05(2023)122
We interpret S-matrix dispersion relations in a similar way: for (asymptotic) mea-
surements at space-like separated points
A
and
B
to commute (in the presence of other
particles), the amplitude for a particle moving from
A
to
B
must be related, by analytic
continuation, to that for an antiparticle moving the other way. In this way, causality is
entwined with analyticity and crossing symmetry [
58
]. While we find this picture intuitive
and compelling, we should note that analyticity is nontrivial to prove mathematically even
in quantum field theory (for recent discussions see [
59
,
60
]). In quantum gravity, of course,
no axioms are established and rigorous proofs are impossible. That our assumptions are
compatible with gravity is nonetheless supported by the recent work [
39
], which showed
that well-established CFT axioms imply that graviton scattering in AdS space satisfies
dispersion relations.
The implications of
(2.22)
depend on the helicity of scattered particles. Recalling that
for physical kinematics
〈
ij
〉
=
±
[
ij
]
∗
, and
|〈
12
〉|
=
|〈
34
〉|
=
√
|
s
|
,
|〈
23
〉|
=
|〈
14
〉|
=
√
|
t
|
,
|〈
13
〉|
=
|〈
24
〉|
=
√
|
u
|
,
(2.23)
we find that for the component amplitudes
f
,
g
and
h
from
(2.3)
, the condition
(2.22)
yields:
lim
|
s
|→∞
f
(
s,
−
p
2
)
≤
Cs
−
3
,
lim
|
s
|→∞
f
(
s,p
2
−
s
)
≤
Cs,
lim
|
s
|→∞
g
(
s,
−
p
2
)
≤
Cs
−
3
,
lim
|
s
|→∞
h
(
s,
−
p
2
)
≤
Cs,
(after smearing in
p
)
.
(2.24)
The first line gives respectively the fixed-
u
and fixed-
t
Regge limits of the MHV amplitude.
Notice that certain limits enjoy improved behavior
∼
s
−
3
: when amplitudes are normalized
so that contact interactions are polynomial (see eq.
(2.7)
), they vanish in some high-energy
limits. This phenomenon is known as superconvergence and is the main reason why we will
find stronger constraints on graviton contact interactions than for scalars.
10
Superconvergence is also related to the observation of [
62
] that a very limited number
of graviton contact interactions obey (or more precisely, saturate) the classical bound
(2.19)
.
Although it is simpler to prove, the bound
(2.22)
is stronger and is not satisfied by any
individual graviton contact interaction.
2.4 Dispersive sum rules
We are now ready to write dispersive sum rules for the amplitudes
f
(
s,u
)
,
g
(
s,u
)
, and
h
(
s,u
)
. We begin with the MHV amplitude
f
. From the behavior
(2.24)
, we get two types
of constraints: from fixed-
u
and fixed-
t
. For fixed-
u
we can separate
f
into combinations
10
In general, superconvergence occurs in scattering of particles of spins
J
1
and
J
2
whenever
J
1
+
J
2
−
1
> J
0
,
where
J
0
is the Regge intercept of the theory, see e.g. [
38
,
61
]. The bound
(2.22)
amounts to
J
0
≤
1
but all
we ultimately use in this paper is
J
0
<
2
.
– 11 –
JHEP05(2023)122
that are even/odd under
s
↔
t
and obtain the following basis of sum rules, for integer
k
:
B
(1)
k
(
p
2
) =
∮
∞
ds
4
πi
2
s
−
p
2
[
s
(
s
−
p
2
)]
k
−
2
2
[
f
(
s,
−
p
2
) +
f
(
p
2
−
s,
−
p
2
)
]
= 0
(
k
≥
2
even
)
,
(2.25a)
B
(1)
k
(
p
2
) =
∮
∞
ds
4
πi
1
[
s
(
s
−
p
2
)]
k
−
3
2
[
−
f
(
s,
−
p
2
) +
f
(
p
2
−
s,
−
p
2
)
]
= 0
(
k
≥
3
odd
)
,
(2.25b)
where the integrals are along a large circle at infinity. We additionally have three fixed-
t
dispersion relations, which also integrate to zero for
k
≥
2
even:
{
B
(2)
k
,B
(3)
k
,B
(4)
k
}
(
p
2
) =
∮
∞
ds
4
πi
(2
s
−
p
2
)
{
f
(
s,p
2
−
s
)
[
s
(
s
−
p
2
)]
k
+2
2
,
g
(
s,p
2
−
s
)
[
s
(
s
−
p
2
)]
k
−
2
2
,
h
(
s,p
2
−
s
)
[
s
(
s
−
p
2
)]
k
+2
2
}
.
(2.26)
To avoid confusion between different channels, we always write the fixed momentum transfer
as
p
. These sum rules become strictly valid after the
p
-dependence is integrated against
appropriate wavepackets as in eq.
(2.20)
. In the limit
|
s
|→∞
the subtractions only give
an inverse power of
s
, so the convergence of the smeared sum rules in eqs.
(2.25)
–
(2.26)
directly follows from that of the amplitudes in eq. (
2.24).
The subscript
k
indicates the Regge spin of a sum rule. This concept is closely related,
but distinct, from the “number of subtractions” or power of
1
/s
inserted to improve high-
energy convergence. For example,
B
(1)
2
has fewer subtractions than
B
(2)
2
(and is even
“anti-subtracted” since it has no denominator!), yet they possess the same convergence
properties. The nomenclature is motivated by the fact that exchange of a single
t
-channel
particle of spin
J
yields an amplitude that grows like
M∼
s
J
: we say that a sum rule has
spin
k
if it converges on exchanges with
J < k
(and marginally diverges on spin
k
).
Regge spin is more important than subtraction-counting because the Regge growth
(2.22)
translates into the simple convergence criterion
k >
1
. This is the same criterion as
convergence of the Froissart-Gribov formula which extracts partial waves of spin
J >
1
,
or of the analogous Lorentzian inversion formula [
63
–
65
] which extracts CFT data for
spin
J >
1
.
Sum rules are obtained by deforming the contour towards the real axis but avoiding
the low-energy region: the contour in figure
2 relates low-energy data at the scale
M
and
heavy data above
M
:
−
B
(
i
)
k
(
p
2
)
∣
∣
∣
low
=
B
(
i
)
k
(
p
2
)
∣
∣
∣
high
(2.27)
∮
u
=
M
2
s
=
M
2
ds
4
πi
(
···
) =
∫
∞
M
2
ds
π
Im (
···
)
.
(2.28)
Note that the
s
and
u
channel cuts contribute identically due to symmetry of eqs.
(2.25)
,
so we included only the right cut. (The contour on the left is in reality the union of upper
and lower half-circles, separated by the branch cut of the amplitude.)
– 12 –
JHEP05(2023)122
−
t
s
0
M
2
−
M
2
−
t
−→
−
t
M
2
s
0
−
M
2
−
t
Figure 2
. Contour deformation which gives rise to sum rules eq.
(2.27)
. The final contour relates
low-energy EFT data along the arcs to heavy discontinuities along the branch cuts.
Let us focus on the first sum rule for simplicity. At tree-level we find only two residues,
from
s
= 0
and
u
= 0
, which contribute the same amount:
−
B
(1)
k
(
p
2
)
∣
∣
∣
low
=
Res
s
=0
(
2
s
−
p
2
[
s
(
s
−
p
2
)]
k
−
2
2
[
f
(
s,
−
p
2
) +
f
(
p
2
−
s,
−
p
2
)
]
)
(tree-level)
.
(2.29)
Substituting in the low-energy amplitude
(2.7a)
, only the exchange graphs contribute
for
k
= 2
,
3
:
−
B
(1)
2
(
p
2
)
∣
∣
∣
low
=
16
πG
p
2
+ 2
πG
|
̂
g
3
|
2
p
6
+
O
(
matter and loops
)
,
(2.30a)
−
B
(1)
3
(
p
2
)
∣
∣
∣
low
=
−
2
πG
|
̂
g
3
|
2
p
4
+
O
(
matter and loops
)
.
(2.30b)
The absence of contact term contributions is a hallmark of superconvergent sum rules.
Examples that probe contact interactions include:
−
B
(1)
4
(
p
2
)
∣
∣
∣
low
= 2
g
4
+ (4
πG
|
̂
g
3
|
2
+
g
5
)
p
2
+ (
g
6
+
g
′
6
)
p
4
+
... ,
(2.31a)
−
B
(1)
5
(
p
2
)
∣
∣
∣
low
=
g
5
+ (
g
6
−
g
′
6
)
p
2
+
... ,
(2.31b)
−
B
(2)
2
(
p
2
)
∣
∣
∣
low
= 2
πG
|
̂
g
3
|
2
1
p
2
+
g
′
6
+
... .
(2.31c)
A salient feature is that the same couplings appear in multiple sum rules: this reflects
crossing symmetry. Another feature is the appearance of the cubic self-coupling in
B
(1)
4
: this
is due to the rapid growth with
t
of the
t
-channel exchange diagram with derivatives. This
rapid energy growth at zero impact parameter will turn out to be a powerful mechanism to
bound
̂
g
3
, as was proposed in section 7 of [
62
]; this mechanism is distinct from the spin-2
growth at large impact parameter that was exploited by CEMZ [
7].
At high energies
s
≥
M
2
, the amplitudes are beyond our knowledge. We can never-
theless evaluate the contribution to the dispersive sum rule, by inserting the partial wave
– 13 –