CERN-TH-2021-035

# Gravitational Effective Field Theory Islands,
Low-Spin Dominance, and the Four-Graviton Amplitude

Zvi Berna, Dimitrios Kosmopoulosa and Alexander Zhiboedovb
a Mani L. Bhaumik Institute for Theoretical Physics
UCLA Department of Physics and Astronomy
Los Angeles, CA 90095, USA
b CERN, Theoretical Physics Department
CH-1211 Geneva 23, Switzerland

###### Abstract

We analyze constraints from perturbative unitarity and crossing on the leading
contributions of higher-dimension operators to the four-graviton amplitude in
four spacetime dimensions, including constraints that follow from distinct
helicity configurations. We focus on the leading-order effect due to exchange
by massive degrees of freedom which makes the amplitudes of interest infrared
finite. In particular, we place a bound on the coefficient of the $R^{3}$
operator that corrects the graviton three-point amplitude in terms of the
$R^{4}$ coefficient. To test the constraints we obtain nontrivial effective
field-theory data by computing and taking the large-mass expansion of the one-
loop minimally-coupled four-graviton amplitude with massive particles up to
spin 2 circulating in the loop. Remarkably, we observe that the leading EFT
coefficients obtained from both string and one-loop field-theory amplitudes
lie in small islands. The shape and location of the islands can be derived
from the dispersive representation for the Wilson coefficients using crossing
and assuming that the lowest-spin spectral densities are the largest. Our
analysis suggests that the Wilson coefficients of weakly-coupled gravitational
physical theories are much more constrained than indicated by bounds arising
from dispersive considerations of $2\to 2$ scattering. The one-loop four-
graviton amplitudes used to obtain the EFT data are computed using modern
amplitude methods, including generalized unitarity, supersymmetric
decompositions and the double copy.

###### Contents

  1. I Introduction
  2. II Construction of one-loop four-graviton scattering amplitudes
  3. III Amplitudes in the low-energy effective field theory
  4. IV Properties of gravitational amplitudes
  5. V Deriving bounds: elastic amplitude
  6. VI Deriving bounds: multiple polarizations
  7. VII Conclusions
  8. A Minimal Coupling
  9. B Tree-level string amplitudes
  10. C Bounding the coupling space at $k=6$
  11. D Amplitude with an accumulation point in the spectrum
  12. E Wigner d-matrices
  13. F Explicit values of one-loop four-graviton amplitudes
  14. G Values of one-loop integrals
  15. H High-order expansion of the one-loop four-graviton amplitudes in the large-mass limit

##  I Introduction

Remarkably, systematic bounds can be placed on possible corrections to
Einstein gravity causality ; BiaHigherOrder ; Tolley:2020gtv ; Caron-
Huot:2020cmc ; Arkani-Hamed:2020blm ; Sinha:2020win ; Caron-Huot:2021rmr .
Such corrections naturally appear due to the presence of heavy particles in
the theory. To leading order in Newton’s constant $G$, such particles can be
exchanged at tree-level, as in string theory, or at one-loop, as in the case
of matter minimally coupled to gravity. By expanding such amplitudes at low
energies and matching to a low-energy effective field theory one finds an
infinite series of higher-derivative corrections to Einstein gravity. The
coefficients in front of these higher-derivative operators, or Wilson
coefficients, satisfy various bounds due to unitarity and causality of the
underlying amplitude causality ; BiaHigherOrder . In this paper, we focus on
the leading corrections to Einstein gravity.111In particular, higher-loop
effects do not affect the discussion in this paper since by assumption gravity
is weakly coupled and we are focusing on the leading-order effect. A central
question, which we investigate in this paper, is to understand if there are
principles that can greatly restrict the values of physically allowed Wilson
coefficients.

Consistency bounds on the Wilson coefficients received a lot of attention
recently in the context of $2\to 2$ scattering, which is also the subject of
our paper. The basic tool to derive such bounds is given by dispersion
relations which express low-energy Wilson coefficients as weighted sums of the
discontinuity of the amplitude. Unitarity constrains the form of the
discontinuity of the amplitude which can be further used to derive the bounds.
The simplest examples of this type constrain the sign of Wilson coefficients.
More interesting bounds arise when one accommodates constraints coming from
crossing symmetry. Including those leads to the two-sided bounds on the Wilson
coefficients Tolley:2020gtv ; Caron-Huot:2020cmc ; Arkani-Hamed:2020blm ;
Sinha:2020win . In this way the ultraviolet (UV) complete theories form
bounded regions in the space of couplings. Ref. Arkani-Hamed:2020blm also
analyzed a few examples of physical EFTs in the context of scattering of
scalars and noted that they lie near the boundaries of the allowed region due
to the importance of low-spin contributions to the partial-wave expansions
(see Sect. 10.3 and Appendix D of Ref. Arkani-Hamed:2020blm ).

In the context of physical theories, especially gravitational ones, it is then
natural to ask the following question:

Is it possible that the Wilson coefficients of physical theories live in much
smaller regions than the bounds coming from considerations of $2\to 2$
scattering suggest?

By physical theories in this paper we mean perturbatively consistent
$S$-matrices that satisfy unitarity, causality, and crossing for any $n\to m$
scattering processes. Constructing such $S$-matrices is far beyond the scope
of bootstrap methods that focus on $2\to 2$ scattering, but such examples are
provided to us by string theory and matter minimally coupled to
gravity.222Perturbatively consistent $S$-matrices occupy a somewhat
intermediate position between fully non-perturbatively consistent quantum
gravities (often referred to as landscape) and consistency of $2\to 2$
scattering studied by bootstrap methods. We can then imagine that consistency
of the full $S$-matrix is reflected back on the $2\to 2$ scattering through
more stringent constraints on Wilson coefficients that one would naively have
found by analyzing $2\to 2$ scattering. In this paper we present data
extracted from field-theory and string-theory $2\to 2$ scattering amplitudes
that suggest that the above assertion is indeed true and we identify a
principle behind it. This principle is low-spin dominance (LSD), which, if
fundamentally correct, might be traced back to the consistency of the full
gravitational $S$-matrix, beyond $2\to 2$ scattering. However, demonstrating
this is beyond the scope of the present paper.

The universal nature of gravity together with the strict consistency
requirements that graviton scattering obeys make such an assertion plausible.
It is a well-known fact that scattering of massless spinning particles is very
constrained Benincasa:2007xk . In fact, massless particles of spin larger than
two do not admit a non-trivial $S$-matrix Weinberg . Gravitons, being massless
spin-two particles, are thus expected to have an especially constrained
$S$-matrix, and the assertion made in the previous paragraph is thus
particularly plausible for graviton scattering which is the subject of the
present paper.

Firstly, we use the techniques of Ref. Arkani-Hamed:2020blm to derive bounds
between low-energy couplings of the same dimensionality in gravitational
scattering.333It would be very interesting to generalize our analysis to
include bounds that relate couplings of different dimensionality along the
lines of Refs. Tolley:2020gtv ; Caron-Huot:2020cmc ; Caron-Huot:2021rmr . We
focus on the first few corrections to Einstein gravity. We then ask where do
the Wilson coefficients obtained from string theory and from the low-energy
limit of the one-loop minimally-coupled amplitudes land in the space allowed
by the general bounds. Remarkably, in all cases studied here we find that both
the string and field-theory coefficients land on a small theory island, which
to a good approximation is a thin line segment in the space of EFT
coefficients. (See, for example, Fig. 9 and Fig. 12 in Sect. V).444In Ref.
Huang:2020nqy the string-theory island was interpreted in terms of unitarity
constraints coupled with world-sheet monodromy constraints. The location of
this island can be found by assuming that lowest-spin partial waves dominate
the dispersive representation of the low-energy couplings, which is the LSD
principle. See Sect. V.2 for the precise mathematical formulation. More
generally, we show how one can combine an assumed hierarchy among the spectral
densities of various spins with crossing symmetry to systematically derive
stronger bounds on the Wilson coefficients. We impose crossing symmetry via
the use of null constraints Tolley:2020gtv ; Caron-Huot:2020cmc .

The idea that LSD is a true property of physical theories can be traced back
to causality, or the statement that the amplitude cannot grow too fast in the
Regge limit. Otherwise we could have simply added a tree-level exchange by a
large-spin particle which would contribute to a given spin partial wave. Due
to causality we cannot do this (see e.g. Ref. Camanho:2014apa ). The situation
is particularly dramatic in gravity. In this case the only particle that can
be exchanged at tree-level in graviton scattering without violating causality
is the graviton itself. Moreover, its self-coupling has to be the one of
Einstein gravity Ref. Camanho:2014apa ; Meltzer:2017rtf ; Belin:2019mnx ;
Simons-Duffin-Zhiboedov . Alternatively, particles of all spins have to be
exchanged at tree level to preserve causality, which is the mechanism realized
in string theory. It is important to emphasize that at the level of $2\to 2$
scattering LSD does not follow from causality and we do not prove it in this
paper, rather we use it as a principle to organize the known data, and suggest
that it may hold more generally. It would be interesting to understand if it
follows from considerations on the consistency of $n\to m$ graviton
scattering.

Alternatively, it is also possible that our finding of LSD could be special
for the models considered here and bears little significance for more general
gravitational models. This possibility, which we cannot exclude, would be
still very interesting. Indeed, as we demonstrate, any such violation is an
indication for non-stringy, non-weakly-coupled-matter physics. For example, it
would be very interesting to see if one can somehow violate LSD by making the
matter sector strongly coupled, e.g. by considering large-$N$ QCD coupled to
gravity Kaplan:2020tdz .

Curiously, the phenomenon of LSD generates hierarchies between different
Wilson coefficients in the absence of any symmetry. We call this phenomenon
hierarchy from unitarity and it is something that could have puzzled an
unassuming low-energy physicist. We find that specific combinations of Wilson
coefficients whose dispersive representation does not involve the lowest-spin
partial waves can be much smaller than their counterparts that do have them in
their definitions.

Secondly, we apply the dispersive sum rules Caron-Huot:2020adz ; Caron-
Huot:2020cmc to amplitudes with various helicity configurations of the
external gravitons.555Flat space superconvergence considered in Ref. Simons-
Duffin-Zhiboedov is a particular example of these more general sum rules. We
derive various bounds on the inelastic scattering (the one in which the final
and initial state gravitons have different helicities) in terms of the elastic
one (see e.g. Ref. Trott:2020ebl ). We also place a precise bound on the
$R^{3}$ coefficient in terms of the $R^{4}$ coefficient (see (204)). Such a
bound translates the problem of making the analysis of Ref. Camanho:2014apa
quantitatively precise to the problem of the bounding the leading $R^{4}$
contact coefficient in terms of the gap of the theory. This has been recently
done in Ref. Caron-Huot:2021rmr in a similar perturbative setting for $D=10$
maximal supergravity; see also Ref. Guerrieri:2021ivu for the nonperturbative
analysis of the same problem. It would, of course, be very interesting to
generalize these studies to more general cases of graviton scattering.

In order to provide data for checking and understanding the derived
constraints, we first compute the one-loop four-graviton scattering amplitude
with the gravitons minimally coupled to massive matter up to spin 2.
Amplitudes corresponding to the ones discussed here, but with massless
particles circulating in the loop were obtained a while ago in Ref.
Dunbar:1994bn and corresponding gauge-theory amplitudes with massive particles
in the loop were computed in Ref. BernMorgan . We use the same type of
organization of the amplitude in terms of supersymmetric multiplets as applied
in the earlier calculations, since they naturally group contributions
according to their analytic properties.

To evaluate the amplitudes, we make use of standard tools including the
unitarity method Unitarity and the Bern-Carrasco-Johansson (BCJ) BCJ ;
BCJReview double copy, which gives gravity integrands in terms of
corresponding gauge-theory ones. We build on the $D$-dimensional version of
the unitarity method of Ref. BernMorgan in order to fix the rational terms in
the amplitudes. At four points gauge-theory tree-level amplitudes
automatically satisfy the duality between color and kinematics, so the
associated double-copy relations also hold automatically on the unitarity
cuts. We use this to express the cuts of the gravity loop integrands directly
in terms of the corresponding gauge-theory ones. By using the double copy our
computation parallels the corresponding gauge-theory one BernMorgan allowing
us to import many of the same steps into the gravitational amplitude
calculations.

A complication with massive amplitudes is that there is a class of terms that
depend on the mass but do not have branch cuts in any kinematic variable. This
makes their construction tricky in the context of the unitarity method. Ref.
BrittoMirabella introduced an approach to this problem. Here we instead solve
the problem differently by making use of a special property of the scattering
amplitudes under study that exploits their simple dependence on the mass of
the particle circulating in the loop. In our case (i.e. a single mass
circulating in the loop) we instead use knowledge of the ultraviolet
properties of the amplitudes to fix all remaining functions in the amplitude
not determined by unitarity. This procedure is greatly aided by arranging the
amplitude in terms of integrals that have no mass or spacetime dependence in
their coefficients. To ensure the veracity of our amplitudes we perform a
number of nontrivial checks on the mass dependence, and infrared and
ultraviolet properties. Related to this, we also note a simple relation
between ultraviolet divergences of appropriate spacetime dimension shifts of
the amplitudes and the terms in the large-mass expansion in four dimensions
(see Eq. (65)).

We analyze our amplitudes in the large-mass limit and match to a low-energy
effective field theory. In this way we systematically obtain corrections to
Einstein gravity due to the presence of a heavy spinning particle. These
corrections are organized in inverse powers of the particle’s mass. As already
noted not only are our results for the Wilson coefficients fully consistent
with the general analysis of bounds on gravitational scattering, but are
restricted to small islands.

Since we focus on the leading effect due to heavy particles in the weakly
coupled setting neither IR divergences, nor logarithms due to the loops of
massless particles make an appearance in our analysis. Taking these into
consideration is an important task which we leave for the future.

Our paper naturally consists of two parts: In the first part we explain in
detail the construction of the one-loop massive amplitudes used to provide
theoretical data that we interpret in the second part in terms of bounds on
coefficients of gravitational EFTs. Readers who are interested in the EFT
constraints can skip Sect. II on the construction of the one-loop amplitudes.
Particularly important plots that illustrate the theory islands and the
concept of low-spin dominance in the partial-wave expansion are given in Figs.
9-12.

In more detail, the sections are organized as follows: In Sect. II we describe
our construction of the one-loop four-graviton amplitude with massive matter
up to spin 2 in the loop. In Sect. III we compute graviton scattering in a
general low-energy effective theory. By expanding our amplitudes in the low-
energy limit, we extract the Wilson coefficients of the effective field
theory. In Sect. IV we describe the general properties of the gravitational
amplitudes stemming from unitarity and causality. In Sect. V we derive two-
sided bounds on Wilson coefficients that follow from a single helicity
configuration that describes elastic scattering; comparing to known data from
string theory and our computed one-loop amplitude, we show that the results
fall into small islands. We trace the position of these islands using low-spin
dominance of partial waves. In Sect. VI we obtain bounds that arise from
considering multiple helicities. We bound the low-energy expansion
coefficients of inelastic amplitudes in terms of elastic ones. We also derive
a bound for the coefficient of the $R^{3}$ operator in terms of the $R^{4}$
coefficient. Finally, we provide our concluding remarks in Sect. VII. We
include various appendices. In Appendix A we describe in some detail our
definition of minimal coupling of gravity to a massive spinning particle. In
Appendix B we collect tree-level graviton four-point amplitudes in various
string theories. In Appendix C we present details on the derivation of some
low-energy bounds that are not listed in the main text of the paper. In
Appendix D we analyze an amplitude function with an accumulation point in the
spectrum that partially violates low-spin dominance, but show that the
corresponding low-energy coefficients still land on the small islands.
Appendix E collects the Wigner d-matrices used throughout the paper. In
Appendix F we present our results for the one-loop amplitudes. We give the
expressions for one-loop integrals in terms of which the amplitudes are
expressed in Appendix G. Finally, in Appendix H we expand these results to
high orders in the large-mass expansion.

##  II Construction of one-loop four-graviton scattering amplitudes

In this section we describe the construction of the one-loop four-graviton
amplitudes with massive matter up to spin 2 in the loop. We collect the
results in Appendix F. We first briefly review the methods used to obtain the
amplitudes. Then, following the generalized-unitarity method we build the
integrand-level generalized-unitarity cuts. We describe a natural and
efficient organization of the unitarity cuts and the amplitudes motivated by
supersymmetry. This organization also meshes well with the double-copy
construction which we use to obtain gravitational unitarity cuts from gauge-
theory ones. Having obtained the unitarity cuts we describe the necessary
integral reduction and cut merging into the amplitudes. This process fixes all
but a few pieces of the amplitudes, which we obtain by exploiting the known
ultraviolet properties of the amplitudes. After calculating the amplitudes, we
comment on some interesting ultraviolet properties we observe. Finally, we
conclude this section by listing the consistency checks we performed on our
calculation.

###  II.1 Basic methods

Spinor helicity

We use the spinor-helicity method SpinorHelicity to describe the external
graviton states of amplitudes (for reviews see Refs. Mangano:1990by ). The
natural quantities in this formalism are two component Weyl spinors

$(\lambda_{i})_{\alpha}\equiv[u_{+}(k_{i})]_{\alpha}\,,\hskip
36.98866pt(\tilde{\lambda}_{i})_{\alpha}\equiv[u_{-}(k_{i})]_{\alpha}\,,$ (1)

which we write in a ‘bra’ and ‘ket’ notation as

$|k_{i}^{+}\rangle=|\,i\,\rangle=\lambda_{i}\,,\hskip
22.76228pt|k_{i}^{-}\rangle=|\,i\,]=\tilde{\lambda}_{i}\,,\hskip
22.76228pt\langle k_{i}^{-}|=\langle\,i\,|=\lambda_{i}\,,\hskip
28.45274pt\langle k_{i}^{+}|=[\,i\,|=\tilde{\lambda}_{i}\,.$ (2)

where $k_{i}^{\mu}$ refers to the null momentum of the $i$-th external
particle, while the ‘$\pm$’ superscript refers to the helicity of the
corresponding state. The spinor inner products are defined using the
antisymmetric tensors $\varepsilon^{\alpha\beta}$ and
$\varepsilon^{\dot{\alpha}\dot{\beta}}$,

$\langle k_{i}^{-}|k_{j}^{+}\rangle=\langle
ij\rangle=\varepsilon^{\alpha\beta}(\lambda_{i})_{\alpha}(\lambda_{j})_{\alpha}\,,\qquad\langle
k_{i}^{+}|k_{j}^{-}\rangle=[ij]=-\varepsilon^{\dot{\alpha}\dot{\beta}}(\tilde{\lambda}_{i})_{\dot{\alpha}}(\tilde{\lambda}_{i})_{\dot{\alpha}}\,.$
(3)

These spinor products are antisymmetric in their arguments and we choose a
convention where they satisfy $\langle ij\rangle[ij]=2k_{i}\cdot k_{j}$.

In order to construct amplitudes with external gravitons, our starting point
is the corresponding ones with external gluons. For calculations involving
external gluons the helicity polarization vectors are defined as

$\varepsilon_{\mu}^{+}(k_{i};q_{i})=\frac{\langle
q_{i}|\gamma_{\mu}|i]}{\sqrt{2}\langle iq_{i}\rangle}\,,\hskip
56.9055pt\varepsilon_{\mu}^{-}(k_{i};q_{i})=\frac{[q_{i}|\gamma_{\mu}|i\rangle}{\sqrt{2}[iq_{i}]}\,,$
(4)

where $q_{i}$ are arbitrary null ‘reference momenta’ which drop out of the
final gauge-invariant amplitudes. Note that we do not use a shorthand notation
for the spinors corresponding to the reference momenta. The polarization
tensors for gravitons are simply given in terms of products of gluon
polarization vectors,

$\varepsilon_{\mu\nu}^{+}(k;q)=\varepsilon_{\mu}^{+}(k;q)\,\varepsilon_{\nu}^{+}(k;q)\,,\hskip
56.9055pt\varepsilon_{\mu\nu}^{-}(k;q)=\varepsilon_{\mu}^{-}(k;q)\,\varepsilon_{\nu}^{-}(k;q)\,,\hskip
56.9055pt$ (5)

which automatically satisfy the graviton tracelessness condition, due to the
Fierz identity. When these polarization vectors are contracted into external
momenta $k_{i}^{\mu}$ or loop momenta $\ell^{\mu}$ we define,

$k_{1}\cdot\varepsilon_{2}^{+}=\frac{\langle
q_{2}|\not{k}_{1}|2]}{\sqrt{2}\langle q_{2}2\rangle}\equiv\frac{\langle
q_{2}|1|2]}{\sqrt{2}\langle q_{2}2\rangle}\,,\hskip
42.67912pt\ell\cdot\varepsilon_{2}^{+}=\frac{\langle
q_{2}|\not{\ell}|2]}{\sqrt{2}\langle q_{2}2\rangle}\equiv\frac{\langle
q_{2}|\ell|2]}{\sqrt{2}\langle q_{2}2\rangle}\,,\quad\text{etc.},$ (6)

where we also use the abbreviation
$\varepsilon_{2}^{+}\equiv\varepsilon^{+}(k_{2};q_{2})$.

We note that, in general, for loop calculations some care is needed when using
dimensional regularization. To take advantage of the spinor-helicity
formulation in a one-loop calculation we need to choose an appropriate version
of dimensional regularization. Specifically, instead of taking the external
polarization tensors and momenta to be ($4-2\epsilon$)-dimensional as in
conventional dimensional regularization Collins , we use the so called four-
dimensional helicity (FDH) scheme FDHScheme ; FDHExamples where both external
and loop state counts are kept in four dimensions and only the loop momentum
is continued to $4-2\epsilon$ dimensions. Because the massive one-loop
amplitudes that we obtain here are neither ultraviolet nor infrared divergent,
the precise distinction between the different versions of dimensional
regularization drops out from the final results for the amplitudes. We do,
however, need to regularize intermediate steps because individual loop
integrals are ultraviolet divergent, with the divergence canceling in final
results.

Generalized unitarity

In order to construct the loop integrands we use the generalized-unitarity
method Unitarity . This method systematically builds complete loop-level
integrands using as input on-shell tree-level amplitudes. A central advantage
is that simplifications and features of the latter are directly imported into
the former. Reviews of the generalized-unitarity methods are found in Refs.
UnitarityReview1 ; UnitarityReview2 .

In general, the task of computing an amplitude is to reduce it to a linear
combination of known scalar integrals. Using standard integral-reduction
techniques (see e.g. Refs. IntegralReduction ; Smirnov:2019qkx ) any four-
point one-loop amplitude can be written as a linear combination of box,
triangle, bubble and tadpole integrals,

$\mathcal{M}_{4}^{\text{1-loop}}=\Bigl{(}d_{s,t}I_{4}(s,t)+c_{s}I_{3}(s)+b_{s}I_{2}(s)+\hbox{perms.}\Bigr{)}+b_{0}I_{2}(0)+a_{0}I_{1}\,,$
(7)

where the permutations run over distinct relabelings of the integrals. At the
four-point level there are a total of 11 coefficients. These coefficients
depend on polarization vectors, momenta, masses and the dimensional-
regularization parameter $\epsilon$. We define the basis integrals appearing
in Eq. (7) by

$\displaystyle
I_{4}(s,t)=\int\frac{d^{D}L}{(2\pi)^{D}}\frac{-i(4\pi)^{D/2}}{(L^{2}-m^{2})((L+k_{1})^{2}-m^{2})((L+k_{1}+k_{2})^{2}-m^{2})((L-k_{4})^{2}-m^{2})}\,,$
$\displaystyle
I_{3}(s)=\int\frac{d^{D}L}{(2\pi)^{D}}\frac{i(4\pi)^{D/2}}{(L^{2}-m^{2})((L+k_{1})^{2}-m^{2})((L+k_{1}+k_{2})^{2}-m^{2})}\,,$
$\displaystyle
I_{2}(s)=\int\frac{d^{D}L}{(2\pi)^{D}}\frac{-i(4\pi)^{D/2}}{(L^{2}-m^{2})((L+k_{1}+k_{2})^{2}-m^{2})}\,,$
$\displaystyle
I_{1}=\int\frac{d^{D}L}{(2\pi)^{D}}\frac{i(4\pi)^{D/2}}{(L^{2}-m^{2})}\,,$ (8)

where $D=4-2\epsilon$, $s=(k_{1}+k_{2})^{2}$, $t=(k_{2}+k_{3})^{2}$ and
$u=(k_{1}+k_{3})^{2}$. We obtain the remaining integrals in Eq. (7) by
permuting the external legs. The unitarity method efficiently targets the
coefficients of the integrals in Eq. (7). The integrals $I_{2}(0)$ and $I_{1}$
are respectively bubble on external leg and tadpole contributions, and are
independent of kinematic variables. As we discuss below, because they lack
dependence on kinematic variables, these latter integrals require special
treatment to determine their coefficients.

Figure 1: The (a) $s$-, (b) $t$\- and (c) $u$-channel two-particle cuts of a
one-loop four-point amplitude. The exposed lines are all on shell and the
blobs represent tree-level amplitudes.

Traditionally, unitarity of the scattering matrix is implemented at the
integrated level via dispersion relations (see e.g. Ref. AnalyticSMatrixBook
). However, for our purposes, it is much more convenient to use an integrand-
level version of unitarity Unitarity . This is based on the concept of a
generalized-unitarity cut that reduces an integrand to a sum of products of
tree-level amplitudes. For example, for the $s$-channel cut displayed in Fig.
1(a),

$\mathcal{M}_{4}^{\text{1-loop}}\Big{\rvert}_{s\rm\hbox{-}cut}\\!\\!=\\!\int\\!\\!\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}\frac{i}{L_{1}^{2}-m^{2}}\frac{i}{L_{2}^{2}-m^{2}}\sum_{\rm
states}\\!\mathcal{M}_{4}^{\rm tree}(1,2,L_{1},L_{2})\mathcal{M}_{4}^{\rm
tree}(-L_{1},-L_{2},3,4)\Big{\rvert}_{s\rm\hbox{-}cut},$ (9)

where $L_{1}=L$ and $L_{2}=-L-k_{1}-k_{2}$ represent the two cut legs, and
$\mathcal{M}_{4}^{\rm tree}$ denote the tree-level amplitudes. The sum runs
over all intermediate physical states that contribute for a given set of
external states. The three generalized-unitarity cuts of the one-loop four-
point amplitude are shown in Fig. 1. In this figure the exposed lines are all
on shell and the blobs represent on-shell tree-level amplitudes.

To obtain the full one-loop amplitude we must combine the unitarity cuts. One
possibility is to carry this out prior to integration by finding a single
integrand with the correct unitarity cuts in all channels Bern:2004cz . Some
non-trivial examples where this approach was implemented are high-loop
computations in super-Yang-Mills and supergravity (see e.g. Refs.
SuperHighLoopExamples ). On the other hand, in high-multiplicity QCD
calculations (see e.g. Ref. BlackHat ) the cuts are usually combined after
reducing to a basis of integrals. We apply the latter approach here. We do so
by promoting each cut propagator to a Feynman propagator, and each cut to a
Feynman integral. We then use FIRE6 Smirnov:2019qkx to reduce each Feynman
integral to the scalar integrals appearing in Eq. (7). In each cut channel we
only determine coefficients of basis integrals with cuts in that channel. By
systematically evaluating each cut we determine all coefficients except for
those of integrals without kinematic dependence, i.e. $I_{2}(0)$ and $I_{1}$.
In the case of gauge theory, the corresponding coefficients are determined by
imposing the known ultraviolet behavior of the amplitudes BernMorgan . Below,
we describe an analogous procedure for the case of gravitational amplitudes.

Double copy

To efficiently obtain the unitarity cuts of the four-graviton amplitude, we
use the double-copy construction Kawai:1985xq ; BCJ which expresses
gravitational scattering amplitudes directly in terms of gauge-theory ones.
Here we use the BCJ form of the double-copy relations BCJ ; BCJReview , which
is more natural when organizing expressions in terms of diagrams.

Figure 2: Example of a color or numerator relation for the one-loop four-point
amplitudes. Here the diagram represent either color or kinematic numerators.
We use these relations on the generalized-unitarity cuts, as indicated by the
dashed lines. The relations are effectively tree-level ones except that the
state and color sums on the cut legs are carried out.

To apply the BCJ double copy, we start by writing a four-point one-loop gauge-
theory amplitude in the following form:

$\mathcal{A}_{4}^{\text{1-loop}}=g^{4}\sum_{i}\int\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}\frac{1}{S_{i}}\frac{n_{i}c_{i}}{D_{i}}\,.$
(10)

The sum runs over all distinct four-point one-loop graphs with trivalent
vertices. We denote the gauge-theory coupling constant by $g$. We label each
graph by an integer $i$. The $S_{i}$ are the symmetry factors of the graphs.
The color factor $c_{i}$ of each graph is obtained by dressing each vertex by
a structure constant $\tilde{f}^{abc}$, since we take all particles to be in
the adjoint representation. Our normalization of the structure constants
follows that of Ref. BCJ . The denominator $D_{i}$ contains the propagators of
each graph. Finally, we capture all non-trivial kinematic dependence by the
numerator $n_{i}$.

The color factors obey color-algebra relations of the type

$c_{i}-c_{j}-c_{k}=0\,,$ (11)

where $i$, $j$ and $k$ are some graphs. These relations follow from the Jacobi
identity obeyed by the structure constants $\tilde{f}^{abc}$. For a
representation obeying color-kinematics duality, the numerators satisfy the
same Jacobi relations, i.e.

$\tilde{n}_{i}-\tilde{n}_{j}-\tilde{n}_{k}=0\,.$ (12)

The tildes on the numerators signify that these numerators do not have to be
the same as the ones appearing in Eq. (10), but as noted in Ref. BCJ these can
be kinematic numerators from a different theory. Given such a representation
we may obtain the corresponding gravitational amplitude simply by replacing
the color factor with the corresponding kinematic numerator,

$c_{i}\rightarrow\tilde{n}_{i}\,,$ (13)

so that

$\mathcal{M}_{4}^{\text{1-loop}}=\left(\frac{\kappa}{2}\right)^{4}\sum_{i}\int\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}\frac{1}{S_{i}}\frac{n_{i}\tilde{n}_{i}}{D_{i}}\,,$
(14)

where $\kappa$ is the gravitational coupling constant, which is given in terms
of Newton’s constant by,

$\kappa^{2}=32\pi G\,.$ (15)

The matter content of the resulting gravitational theory is determined by the
choice of the numerators $n_{i}$ and $\tilde{n}_{i}$. We use this to control
the type of particle circulating in the loop.

While the BCJ double copy is usually formulated at the level of the full
integrand, since we extract the final answer directly from the cuts it is more
convenient to use it at the level of generalized-unitarity cuts. In Fig. 2 we
depict an example of a color relation in terms of cut graphs. In this way we
can ignore duality relations that contain diagrams without support on the
given cut. For duality relations with support on the cut, the particles of
each tree-level amplitude entering the cut remain on the same side of the cut
for all three diagrams, as is the case in Fig. 2. Effectively this amounts to
using the duality relations for the two tree-level amplitudes on each side of
the cut (see e.g. Fig. 3). The tree-level relations are sufficient to ensure
that for the cut diagram, the double-copy replacement formula (13) holds.

Figure 3: The tree-level four-point color or numerator Jacobi identity. This
can be used to set the numerator of one of the diagrams to zero.

We also take advantage of a property of four-point tree-level amplitudes noted
in the original BCJ paper BCJ , which states that we can effectively set one
of the three numerators to zero. In this way the duality relation implies that
the other two numerators must be equal (up to a possible sign). In order to
achieve that, we absorb the propagator of the diagram whose numerator is set
to zero into the numerators of the other two diagrams. Specifically, consider
the four-point all-gluon tree-level amplitude,

$\mathcal{A}_{4}^{\text{tree}}=g^{2}\Bigl{(}\frac{n_{s}c_{s}}{s}+\frac{n_{t}c_{t}}{t}+\frac{n_{u}c_{u}}{u}\Bigr{)}\,.$
(16)

Using the color Jacobi identity depicted in Fig. 3, $c_{s}=c_{t}-c_{u}$, we
can eliminate $c_{s}$ in favor of the other two color factors, so that

$\mathcal{A}_{4}^{\text{tree}}=g^{2}\Bigl{(}\frac{n_{t}^{\prime}c_{t}}{t}+\frac{n_{u}^{\prime}c_{u}}{u}\Bigr{)}\,,$
(17)

where

$n_{t}^{\prime}=n_{t}+n_{s}\frac{t}{s}\,,\hskip
56.9055ptn_{u}^{\prime}=n_{u}-n_{s}\frac{u}{s}\,.$ (18)

Demanding that the numerators $n_{t}^{\prime}$ and $n_{u}^{\prime}$ (and
$n_{s}^{\prime}=0$) satisfy the duality relation of Fig. 3 then implies that

$n_{t}^{\prime}=n_{u}^{\prime}\,.$ (19)

The analysis is identical in the presence of matter.

###  II.2 Setup of the calculation

Our goal is to obtain the four-point one-loop amplitude with external
gravitons and with minimally-coupled massive spinning particles up to spin 2
circulating in the loop. Following the generalized-unitarity method we first
build the integrand-level generalized-unitarity cuts in Eq. (9). For the one-
loop four-point amplitude there are three independent cuts, labeled by the
Mandelstam invariant that can be build out of the external graviton momenta on
the tree-level amplitudes, i.e. $s$-, $t$\- and $u$-channel cuts. We consider
all spins up to spin 2 for the massive particles and we denote their mass by
$m$. While these masses can be different for each particle since only a single
particle at a time circulates in the loop there is no need to put an index
labeling the massive particle. We take the massive particles to be real.

We note that there is an ambiguity in the definition of the minimal coupling
of a spin 2 particle to gravity. In this paper, we fix this ambiguity by
demanding that we recover pure gravity in the appropriate massless limit. This
choice also preserves tree-level unitarity treeLevelUnitarity and causality
Bonifacio:2017nnt . We discuss this ambiguity and the choice we make in this
paper in Appendix A.

For the amplitude in question there exist three independent helicity
configurations. Specifically, we calculate

$\mathcal{M}_{4}^{\text{1-loop}}(1^{+},2^{-},3^{-},4^{+})\,,\hskip
28.45274pt\mathcal{M}_{4}^{\text{1-loop}}(1^{+},2^{+},3^{-},4^{+})\,,\hskip
28.45274pt\mathcal{M}_{4}^{\text{1-loop}}(1^{+},2^{+},3^{+},4^{+})\,.$ (20)

We refer to these configurations respectively as double-minus, single-minus
and all-plus. All other amplitudes are related to these via relabelings and
parity.

Figure 4: The (a) $s$-, (b) $t$\- and (c) $u$-channel two-particle cuts of a
one-loop four-point amplitude with two negative- and two positive-helicity
external gluons or gravitons (double-minus configuration). The internal lines
represent massive spinning particles. The exposed lines are all on shell and
the blobs represent tree-level amplitudes.

We build the generalized-unitarity cuts from four-point tree-level gravity
amplitudes. The double copy implies that these tree-level amplitudes can be
directly obtained from the corresponding gauge-theory ones, which can be
described by the three diagrams shown in Fig. 3. As we discussed in the
previous section we can use the color Jacobi identity in the gauge-theory case
to remove one diagram, at the expense of other diagrams obtaining numerators
that are nonlocal in the external kinematic invariants. The net effect is that
after multiplying and dividing by appropriate propagators every contribution
to the cut can be assigned to cut box diagrams. Moreover, on the generalized
cuts the four-point tree-level BCJ numerator relations set the remaining
numerators equal to each other, as noted in Eq. (19).

For example, for the four-point gauge-theory amplitude the $s$-channel cut in
Fig. 4(a) is of the form,

$\mathcal{A}_{4}\Big{\rvert}_{s\rm\hbox{-}cut}=g^{4}\int\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}\,n_{\text{g},s}\left(\frac{c_{1234}}{D_{1234}}+\frac{c_{1342}}{D_{1342}}\right)\bigg{\rvert}_{s\rm\hbox{-}cut}\,.$
(21)

The box color factor (see Fig. 5) is given by

$c_{abcd}=\tilde{f}^{e_{a}g_{1}g_{4}}\tilde{f}^{e_{b}g_{2}g_{1}}\tilde{f}^{e_{c}g_{3}g_{2}}\tilde{f}^{e_{d}g_{4}g_{3}}\,,$
(22)

where $abcd$ takes in the values indicated in Eq. (21). As usual, repeated
indices are summed. The denominators are given by products of the usual
Feynman propagators,

$D_{abcd}=\Big{(}L^{2}-m^{2}\Big{)}\Big{(}(L+k_{a})^{2}-m^{2}\Big{)}\Big{(}(L+k_{a}+k_{b})^{2}-m^{2}\Big{)}\Big{(}(L+k_{a}+k_{b}+k_{c})^{2}-m^{2}\Big{)}\,,$
(23)

where the $k_{a}$ are external momenta. Finally, $n_{\text{g},s}$ is a gauge-
theory kinematic factor common to both box diagrams.

Figure 5: Generic box diagram whose color factor and denominator are given by
Eqs. (22) and (23) respectively. The external momenta are taken incoming while
the direction of the loop momentum is indicated by the arrow.

The gravitational cuts are similar. For example, the $s$-channel cut is of the
form,

$\mathcal{M}_{4}\Big{\rvert}_{s\rm\hbox{-}cut}=\left(\frac{\kappa}{2}\right)^{4}\int\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}\,n_{\text{GR},s}\left(\frac{1}{D_{1234}}+\frac{1}{D_{1342}}\right)\bigg{\rvert}_{s\rm\hbox{-}cut}\,,$
(24)

where $n_{\text{GR},s}$ is the single $s$-channel gravitational kinematic
numerator. As we noted in Sect. II.1, the gravity amplitudes follow from
replacing the color factor with a gauge-theory kinematic numerator. As we
describe in more detail below, the particle content circulating in the loop is
determined by the choice of the gauge-theory numerators.

As can be seen from Figs. 4(a) and 4(c), for the indicated helicity
configuration, the $u$-channel cut is obtained by relabeling the momenta and
spinors in the $s$-channel cut: $k_{2}\leftrightarrow k_{3}$ and $\langle
2|\leftrightarrow\langle 3|$. For the single-minus and all-plus
configurations, all three cuts are given by appropriate relabelings of a
single cut.

###  II.3 Supersymmetric decompositions

We are interested in the problem of minimally-coupled massive matter
circulating in the loop. A convenient way to organize this calculation is by
following the massless case where each particle can be described as a linear
combination of supersymmetric multiplets circulating in the loop. Our
organization is in direct correspondence to this supersymmetric decomposition
SusyDecomposition ; Dunbar:1994bn ; Bern:1993tz . This allows us to recycle
the results of the calculation of the lower-spin particles circulating in the
loop into contributions for the higher-spin particles. It also has the
advantage of grouping together terms that contain integrals of the same tensor
rank. For the gauge-theory case such a decomposition has already been used to
organize the contributions of massive spin $0$, $1/2$ and $1$ particles
circulating in the loops Bern:1993tz ; BernMorgan . The double-copy
construction will allow us to import this into minimally-coupled gravitational
theories with up to spin-2 massive particles.

Gauge theory

We start by examining the corresponding amplitudes in gauge theory. For the
case of one-loop four-point amplitudes with external gluons and massive matter
circulating in the loop, Ref. Bern:1993tz showed that666We combine the ‘gauge
boson’ and ‘scalar’ contributions of Ref. Bern:1993tz into the massive $S=1$
result.

$\displaystyle\begin{split}\mathcal{A}_{4}^{S=0}&=\mathcal{A}_{4}^{\\{0\\}}\,,\\\
\mathcal{A}_{4}^{S=1/2}&=-2\mathcal{A}_{4}^{\\{0\\}}+\mathcal{A}_{4}^{\\{1/2\\}}\,,\\\
\mathcal{A}_{4}^{S=1}&=3\mathcal{A}_{4}^{\\{0\\}}-4\mathcal{A}_{4}^{\\{1/2\\}}+\mathcal{A}_{4}^{\\{1\\}}\,.\end{split}$
(25)

where the notation $\mathcal{A}_{4}^{\\{\it S\\}}$ denotes the new piece we
need to calculate at spin-$S$. In this way we express the amplitudes with a
given spinning particle circulating in the loop in terms of the simpler-to-
calculate new pieces. Inverting the above equations, we may think of the new
pieces as amplitudes with multiplets circulating in the loop. These massive
multiplets have the same degrees of freedom as the corresponding massless
ones. Hence, in terms of on-shell supersymmetric representation theory they
satisfy the BPS condition susyRepReviews . For recent calculations involving
massive supersymmetric multiplets in gauge theory see Refs. n4symCoulombBranch
; Alday:2009zm .

Before turning to the corresponding gravitational decomposition it is useful
to first look at the massless limit. For the gauge theory case we have that as
$m\rightarrow 0$,

$\displaystyle\begin{split}\mathcal{A}_{4}^{S=0}&\rightarrow\mathcal{A}_{4}^{S=0,\,\textit{m}=0}\,,\\\
\mathcal{A}_{4}^{S=1/2}&\rightarrow\mathcal{A}_{4}^{S=1/2,\,\textit{m}=0}\,,\\\
\mathcal{A}_{4}^{S=1}&\rightarrow\mathcal{A}_{4}^{S=1,\,\textit{m}=0}+\mathcal{A}_{4}^{S=0,\,\textit{m}=0}\,,\end{split}$
(26)

where we see that the $S=1$ case is nontrivial, which follows from the
mismatch in the number of degrees of freedom in a massive and massless vector
boson.

Background field gauge

A nice way to understand the above supersymmetric decomposition is in terms of
background field gauge UnitarityReview1 . While we do not use background field
gauge to compute the amplitudes, it does offer useful insight into the
structure of the amplitude. For the different particles circulating in the
loop we can write the effective action as

$\displaystyle\Gamma_{S=0}[A]$
$\displaystyle=\ln\det{}_{[0]}^{-1/2}\bigl{(}D^{2}+m^{2}\bigr{)}\,,$
$\displaystyle\Gamma_{S=1/2}[A]$
$\displaystyle=\ln\det{}_{[1/2]}^{1/4}\bigl{(}D^{2}\,\mathbb{I}+m^{2}\,\mathbb{I}-g\frac{1}{2}\sigma^{\mu\nu}F_{\mu\nu})\,,$
(27) $\displaystyle\Gamma_{S=1}[A]$
$\displaystyle=\ln\det{}_{[1]}^{-1/2}\bigl{(}D^{2}\mathbb{I}+m^{2}\mathbb{I}-g\Sigma^{\mu\nu}F_{\mu\nu}\bigl{)}+\ln\det{}_{[0]}\bigl{(}D^{2}+m^{2}\bigr{)}+\ln\det{}_{[0]}^{-1/2}\bigl{(}D^{2}+m^{2}\bigr{)}\,,$

where $\mathbb{I}$ is the identity matrix, $\sigma_{\mu\nu}/2$ is the
spin-$1/2$ Lorentz generator and $\Sigma_{\mu\nu}$ is the spin-1 Lorentz
generator. Ignoring the $F_{\mu\nu}$ terms and focusing on the $(D^{2}+m^{2})$
term, each state corresponds to a power to which the determinant is raised:
$-1/2$ for a bosonic state and $+1/2$ for a fermionic state. For a massive
real scalar there is precisely one bosonic state corresponding the $-1/2$
power to which the first determinant is raised. For the $S=1/2$ fermion the
determinant is raised to the $1/4$ power to account for the $4\times 4$ Dirac
determinant, effectively leaving a single power that corresponds to the two
states of a Majorana fermion. Similarly, for the $S=1$ vector, the determinant
is over a $4\times 4$ Lorentz generator space so the exponent of $-1/2$ in the
first term in $\Gamma_{S=1}[A]$ corresponds to 4 bosonic states. This is then
reduced to two states by the ghost determinant corresponding to the second
term and increased by one state from the scalar longitudinal degree of freedom
required by a massive vector boson corresponding to the third term. This extra
degree of freedom is incorporated into Eq. (25) as well. To make the
supersymmetric cancellations more manifest we rewrite the Dirac determinant as
a product of determinants so that the similarity to the bosonic case is clear,

$\displaystyle\det{}_{[1/2]}^{1/2}({\raise 0.91666pt\hbox
to8.55695pt{\hss/\hss}\hbox to0.0pt{\hss$D$}}+im)\det{}_{[1/2]}^{1/2}({\raise
0.91666pt\hbox to8.55695pt{\hss/\hss}\hbox to0.0pt{\hss$D$}}-im)$
$\displaystyle=\det{}_{[1/2]}^{1/4}({\raise 0.91666pt\hbox
to8.55695pt{\hss/\hss}\hbox to0.0pt{\hss$D$}}^{2}+m^{2})$
$\displaystyle=\det{}_{[1/2]}^{1/4}(D^{2}+m^{2}-g\frac{1}{2}\sigma^{\mu\nu}F_{\mu\nu})\,,$
(28)

where ${\raise 0.91666pt\hbox to8.55695pt{\hss/\hss}\hbox
to0.0pt{\hss$D$}}^{2}=\frac{1}{2}\\{{\raise 0.91666pt\hbox
to8.55695pt{\hss/\hss}\hbox to0.0pt{\hss$D$}},{\raise 0.91666pt\hbox
to8.55695pt{\hss/\hss}\hbox to0.0pt{\hss$D$}}\\}+\frac{1}{2}[{\raise
0.91666pt\hbox to8.55695pt{\hss/\hss}\hbox to0.0pt{\hss$D$}},{\raise
0.91666pt\hbox to8.55695pt{\hss/\hss}\hbox to0.0pt{\hss$D$}}]$. This
corresponds to using the second order formalism for fermions described in Ref.
SecondOrderFermions . The fact that the mass enters into Eq. (27) in such a
simple manner can also be understood in terms of a Kaluza-Klein reduction of
the massless case from five dimensions, truncated to keeping only the lowest
massive state in the loop.

The effective-action determinants (27) can be straightforwardly applied to
show that the supersymmetric decomposition organizes contributions in terms of
power counting of the resulting diagrams. The terms with leading power of loop
momenta come from the $D^{2}$ terms in Eq. (27), because $F_{\mu\nu}$ contains
only the external gluon momenta. If we set the $F_{\mu\nu}$ terms to zero then
in all supersymmetric combinations the balance between the bosons and fermions
implies the leading powers of loop momentum cancel. Subleading terms in
supersymmetric combinations come from using one or more factors of
$F_{\mu\nu}$ when generating a graph; each $F_{\mu\nu}$ reduces the maximum
power of momentum by one. Terms with a lone $F_{\mu\nu}$ vanish, thanks to
${\rm Tr}(\sigma^{\mu\nu})={\rm Tr}(\Sigma^{\mu\nu})=0$. This reduces the
leading power in an $m$-point one-particle-irreducible diagram from $\ell^{m}$
down to $\ell^{m-2}$. For $\mathcal{A}_{m}^{\\{1\\}}$, a comparison of the
traces of products of two and three $\sigma^{\mu\nu}$ and $\Sigma_{\mu\nu}$
shows that further cancellations reduce the leading behavior all the way down
to $\ell^{m-4}$. In gauges other than Feynman background field gauge, these
cancellations would be more more obscure.

Gravity

Now consider the gravitational case. In the $m\rightarrow 0$ limit we again
have a mismatch in the number of degrees of freedom for $S\geq 1$ between the
massive and massless cases,

$\displaystyle\begin{split}\mathcal{M}_{4}^{S=0}&\rightarrow\mathcal{M}_{4}^{S=0,\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=1/2}&\rightarrow\mathcal{M}_{4}^{S=1/2,\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=1}&\rightarrow\mathcal{M}_{4}^{S=1,\,\textit{m}=0}+\mathcal{M}_{4}^{S=0,\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=3/2}&\rightarrow\mathcal{M}_{4}^{S=3/2,\,\textit{m}=0}+\mathcal{M}_{4}^{S=1/2,\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=2}&\rightarrow\mathcal{M}_{4}^{S=2,\,\textit{m}=0}+\mathcal{M}_{4}^{S=1,\,\textit{m}=0}+\mathcal{M}_{4}^{S=0,\,\textit{m}=0}\,.\end{split}$
(29)

In the massless case with spinning particles circulating in the loop we can
again decompose the amplitudes in terms of ones with supermultiplets
circulating in the loop Dunbar:1994bn ,

$\displaystyle\begin{split}\mathcal{M}_{4}^{S=0,\,\textit{m}=0}&=\mathcal{M}_{4}^{\\{0\\},\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=1/2,\,\textit{m}=0}&=-2\mathcal{M}_{4}^{\\{0\\},\,\textit{m}=0}+\mathcal{M}_{4}^{\\{1/2\\},\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=1,\,\textit{m}=0}&=2\mathcal{M}_{4}^{\\{0\\},\,\textit{m}=0}-4\mathcal{M}_{4}^{\\{1/2\\},\,\textit{m}=0}+\mathcal{M}_{4}^{\\{1\\},\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=3/2,\,\textit{m}=0}&=-2\mathcal{M}_{4}^{\\{0\\},\,\textit{m}=0}+9\mathcal{M}_{4}^{\\{1/2\\},\,\textit{m}=0}-6\mathcal{M}_{4}^{\\{1\\},\,\textit{m}=0}+\mathcal{M}_{4}^{\\{3/2\\},\,\textit{m}=0}\,,\\\
\mathcal{M}_{4}^{S=2,\,m=0}&=2\mathcal{M}_{4}^{\\{0\\},\,\textit{m}=0}-16\mathcal{M}_{4}^{\\{1/2\\},\,\textit{m}=0}+20\mathcal{M}_{4}^{\\{1\\},\,\textit{m}=0}-8\mathcal{M}_{4}^{\\{3/2\\},\,\textit{m}=0}\\\
&\hskip 14.22636pt+\mathcal{M}_{4}^{\\{2\\},\,\textit{m}=0}\,.\end{split}$
(30)

The $\\{S\\}$ pieces in each case are in direct correspondence to the
supermultiplets circulating in the loop, as defined in Ref. Dunbar:1994bn
777We use a real scalar while Ref. Dunbar:1994bn used a complex one. Hence
there is relative factor of 1/2 for that contribution.. For example,

$\mathcal{M}_{4}^{\\{1/2\\},\,\textit{m}=0}=\mathcal{M}_{4}^{\mathcal{N}=1,\,\textit{m}=0},$
(31)

where $\mathcal{N}=1$ denotes the chiral multiplet consisting of a Weyl
fermion and two real scalars.

Using the relation between the massive and massless amplitudes in Eq. (29) and
the massless supersymmtric decomposition in Eq. (30), we can organize our
computation in a similar way as for gauge theory in Eq. (25). Specifically, we
have

$\displaystyle\begin{split}\mathcal{M}_{4}^{S=0}&=\mathcal{M}_{4}^{\\{0\\}}\,,\\\
\mathcal{M}_{4}^{S=1/2}&=-2\mathcal{M}_{4}^{\\{0\\}}+\mathcal{M}_{4}^{\\{1/2\\}}\,,\\\
\mathcal{M}_{4}^{S=1}&=3\mathcal{M}_{4}^{\\{0\\}}-4\mathcal{M}_{4}^{\\{1/2\\}}+\mathcal{M}_{4}^{\\{1\\}}\,,\\\
\mathcal{M}_{4}^{S=3/2}&=-4\mathcal{M}_{4}^{\\{0\\}}+10\mathcal{M}_{4}^{\\{1/2\\}}-6\mathcal{M}_{4}^{\\{1\\}}+\mathcal{M}_{4}^{\\{3/2\\}}\,,\\\
\mathcal{M}_{4}^{S=2}&=5\mathcal{M}_{4}^{\\{0\\}}-20\mathcal{M}_{4}^{\\{1/2\\}}+21\mathcal{M}_{4}^{\\{1\\}}-8\mathcal{M}_{4}^{\\{3/2\\}}+\mathcal{M}_{4}^{\\{2\\}}\,.\end{split}$
(32)

The massive multiplets circulating in the loop are ‘short’, i.e. they have the
same degrees of freedom as the corresponding massless ones. Hence, they obey
the BPS condition susyRepReviews . While we have not tried directly embedding
these amplitudes into supergravity theories, it is an interesting question to
do so. Here we use the relation to supermultiplets for a more modest aim of
reorganizing the contributions, so that as the spin increases the new pieces
become simpler. Examples of supergravity calculations involving massive
multiplets are given in Ref. massiveSugra .

We note that in general, care is required when using dimensional
regularization in conjunction with helicity methods and supersymmetric
decompositions. To allow for different choices of regularization scheme, we
would need to correct the last line of Eq. (25) to be Bern:1993tz

$\mathcal{A}_{4}^{S=1}=(3-2\delta_{R}\epsilon)\mathcal{A}_{4}^{\\{0\\}}-4\mathcal{A}_{4}^{\\{1/2\\}}+\mathcal{A}_{4}^{\\{1\\}},$
(33)

where $\delta_{R}=0$ in the FDH scheme Bern:1991aq and $\delta_{R}=1$ in
either conventional dimensional regularization Collins or the ’t Hooft-Veltman
scheme tHooft:1972tcz . One may then propagate this correction to the
gravitational amplitudes through the double copy. While the correction is of
$\mathcal{O}(\epsilon)$, it can interfere with infrared or ultraviolet
singularities to give nontrivial contributions. However, for the massive
amplitudes that we are computing here, the distinction between different
schemes is not important because the four-graviton amplitudes with massive
matter in the loop are both ultraviolet and infrared finite (see Sects. II.6
and II.8).

We also note that the coefficient of the scalar ($\mathcal{M}_{4}^{\\{0\\}}$)
counts the degrees of freedom of the particle in question, modulo a minus sign
for the fermions. Recall that all particles are taken to be real. Also, given
the general setup of our amplitudes (Eqs. (21) and (24)), we get a similar
decomposition for the corresponding numerators $n_{\text{g},\alpha}$ and
$n_{\text{GR},\alpha}$.

Finally, the supersymmetric decomposition is simplified in the case of the
single-minus and all-plus configurations. In these cases, supersymmetric Ward
identities SWI show that only the $\mathcal{M}_{4}^{\\{0\\}}$ piece gives a
nonzero contribution for each spin particle in the loop. Using this
observation, Eq. (32) becomes

$\displaystyle\begin{split}\mathcal{M}_{4}^{S=0}(1^{\pm},2^{+},3^{+},4^{+})&=\mathcal{M}_{4}^{\\{0\\}}(1^{\pm},2^{+},3^{+},4^{+})\,,\\\
\mathcal{M}_{4}^{S=1/2}(1^{\pm},2^{+},3^{+},4^{+})&=-2\mathcal{M}_{4}^{\\{0\\}}(1^{\pm},2^{+},3^{+},4^{+})\,,\\\
\mathcal{M}_{4}^{S=1}(1^{\pm},2^{+},3^{+},4^{+})&=3\mathcal{M}_{4}^{\\{0\\}}(1^{\pm},2^{+},3^{+},4^{+})\,,\\\
\mathcal{M}_{4}^{S=3/2}(1^{\pm},2^{+},3^{+},4^{+})&=-4\mathcal{M}_{4}^{\\{0\\}}(1^{\pm},2^{+},3^{+},4^{+})\,,\\\
\mathcal{M}_{4}^{S=2}(1^{\pm},2^{+},3^{+},4^{+})&=5\mathcal{M}_{4}^{\\{0\\}}(1^{\pm},2^{+},3^{+},4^{+})\,.\end{split}$
(34)

Therefore for these two helicity configurations, it is sufficient to calculate
the $S=0$ amplitude where only a scalar circulates in the loop.

###  II.4 Kinematic numerators through the double copy

Following the double-copy construction we can directly obtain the
gravitational unitarity-cut numerators from gauge-theory ones. We may express
the double copy in terms of spinning particles or new pieces (Eq. (32))
circulating in the loop. In the latter case, we find an especially compact
representation for the numerators.

Taking the product of gauge-theory kinematic numerators we decompose them into
cut numerators of the gravitational case. In terms of spin we have

$\displaystyle\begin{split}n_{\text{g},\alpha}^{\tilde{S}}n_{\text{g},\alpha}^{S=0}&=n_{\text{GR},\alpha}^{\tilde{S}}\,,\qquad\hbox{for
$\tilde{S}=0,1/2,1$}\,,\\\
n_{\text{g},\alpha}^{S=1/2}n_{\text{g},\alpha}^{S=1/2}&=n_{\text{GR},\alpha}^{S=1}+n_{\text{GR},\alpha}^{S=0}\,,\\\
n_{\text{g},\alpha}^{S=1}n_{\text{g},\alpha}^{S=1/2}&=n_{\text{GR},\alpha}^{S=3/2}+n_{\text{GR},\alpha}^{S=1/2}\,,\\\
n_{\text{g},\alpha}^{S=1}n_{\text{g},\alpha}^{S=1}&=n_{\text{GR},\alpha}^{S=2}+n_{\text{GR},\alpha}^{S=1}+n_{\text{GR},\alpha}^{S=0}\,,\end{split}$
(35)

where $\alpha$ denotes the cut under consideration. These are in direct
correspondence to the Clebsch-Gordan decomposition. In terms of the new pieces
in Eq. (32),

$\displaystyle\begin{split}n_{\text{g},\alpha}^{\\{\it
S\\}}n_{\text{g},\alpha}^{\\{0\\}}&=n_{\text{GR},\alpha}^{\\{\it
S\\}}\,,\qquad\hbox{for $S=0,1/2,1$ },\\\
n_{\text{g},\alpha}^{\\{1/2\\}}n_{\text{g},\alpha}^{\\{1/2\\}}&=n_{\text{GR},\alpha}^{\\{1\\}}\,,\\\
n_{\text{g},\alpha}^{\\{1\\}}n_{\text{g},\alpha}^{\\{1/2\\}}&=n_{\text{GR},\alpha}^{\\{3/2\\}}\,,\\\
n_{\text{g},\alpha}^{\\{1\\}}n_{\text{g},\alpha}^{\\{1\\}}&=n_{\text{GR},\alpha}^{\\{2\\}}\,.\end{split}$
(36)

Observe that the gauge-theory numerator $n_{\text{g},\alpha}^{S=0}$ along with
either $n_{\text{g},\alpha}^{S=1/2}$ or $n_{\text{g},\alpha}^{S=1}$ are
sufficient to construct all the gravitational numerators up to spin 2\. We
explicitly verified that both constructions yield the same result. Refs.
Bern:1993tz ; BernMorgan calculated the corresponding amplitudes
$\mathcal{A}_{4}^{S=0}$, $\mathcal{A}_{4}^{S=1/2}$ and
$\mathcal{A}_{4}^{S=1}$, from which we may extract the desired kinematic
numerators. As a consistency check, we match $\mathcal{A}_{4}^{\\{1\\}}$,
which was calculated in Ref. Alday:2009zm .

From the above construction we find a remarkably simple form for the kinematic
numerators. For the double-minus gauge-theory numerators we have,

$n_{\text{g},\alpha}^{\\{\it S\\}}=\big{(}[14]\langle
23\rangle\big{)}^{2}(\psi_{\alpha})^{2-2S}\,,$ (37)

while for the corresponding gravity numerators we have,

$n_{\text{GR},\alpha}^{\\{\it S\\}}=\big{(}[14]\langle
23\rangle\big{)}^{4}(\psi_{\alpha})^{4-2S}\,,$ (38)

where

$\psi_{s}\equiv\frac{\langle 2|\ell|1]\langle 3|\ell|4]}{s[14]\langle
23\rangle}\,,\hskip 56.9055pt\psi_{t}\equiv\frac{(m^{2}+\mu^{2})}{t}\,,\hskip
56.9055pt\psi_{u}\equiv\frac{\langle 3|\ell|1]\langle 2|\ell|4]}{u[14]\langle
32\rangle}\,.$ (39)

For the single-minus and all-plus configurations it is sufficient to calculate
the numerators for a scalar circulating in the loop. For the single-minus
configuration for gauge theory and gravity we find

$n_{\text{g},s}^{S=0}=\frac{(m^{2}+\mu^{2})}{s}\frac{[12]}{\langle
12\rangle}\langle 3|\ell|4]^{2},\hskip
42.67912ptn_{\text{GR},s}^{S=0}=\frac{(m^{2}+\mu^{2})^{2}}{s^{2}}\frac{[12]^{2}}{\langle
12\rangle^{2}}\langle 3|\ell|4]^{4},$ (40)

while for the all-plus configuration we have

$n_{\text{g},s}^{S=0}=(m^{2}+\mu^{2})^{2}\frac{[12][34]}{\langle
12\rangle\langle 34\rangle}\,,\hskip
42.67912ptn_{\text{GR},s}^{S=0}=(m^{2}+\mu^{2})^{4}\Bigg{(}\frac{[12][34]}{\langle
12\rangle\langle 34\rangle}\Bigg{)}^{2}\,.$ (41)

For these two configurations, we obtain the numerators for the $t$\- and
$u$-channel cuts by appropriate relabelings.

Following the conventions of Ref. BernMorgan , we break the
$(4-2\epsilon)$-dimensional loop momentum $L$ into a four-dimensional part
$\ell$ and a $(-2\epsilon)$-dimensional part $\mu$. We write $L=(\ell,\mu)$.
Using this notation we have for example

$\ell^{2}=m^{2}+\mu^{2}\,,$ (42)

when the cut condition $L^{2}=m^{2}$ is satisfied. We take $\epsilon<0$ so
that we can break the loop momentum in this fashion. Further, whenever a four-
dimensional vector $v\equiv(v,0)$ is contracted with the
$(4-2\epsilon)$-dimensional loop momentum, the $(-2\epsilon)$-dimensional part
is projected out,

$v\cdot L=v\cdot\ell\,.$ (43)

Ref. Arkani-Hamed:2017jhn (Eqs. (5.20) and (5.36)) calculated Compton
amplitudes for a massive particle in four dimensions of up to spin 1 in gauge
theory and up to spin 2 in gravity. We find similar spin dependence in our
double-minus numerators as for these Compton amplitudes.

###  II.5 Integral reduction and cut merging

In this subsection we use standard integration-by-parts (IBP) methods to
reduce the generalized-unitarity cuts we calculated in the previous section in
terms of a basis of master integrals. This allows us to fix all integral
coefficients in Eq. (7) other than those of the tadpole and the bubble on
external leg. We discuss these remaining pieces in Sect. II.6. We show details
for the double-minus configuration with a scalar in the loop; the remaining
helicity configurations and particle-in-the-loop contributions are similar. In
order to keep the expressions concise, we do not include the helicity labels.

The general strategy for constructing the full amplitudes is to evaluate the
cuts one by one in terms of the master integrals appearing in Eq. (7). If a
given integral has a generalized cut in the channel being evaluated then that
channel fully determines its coefficient. By stepping through the three
channels in Fig. 1 we obtain the coefficients of all master integrals except
$I_{2}(0)$ and $I_{1}$. The box integrals each have cuts in two channels so
consistency requires that they give the same coefficient.

We start with the $s$-channel cut of the $S=0$ double-minus helicity
configuration defined in Eq. (20). The discussion for the $u$-channel cut
follows in the same way, since it is given by a relabeling of the $s$-channel
one. We have

$\displaystyle\mathcal{M}_{4}^{S=0}\bigg{\rvert}_{s\text{-cut}}=\left(\frac{\kappa}{2}\right)^{4}(\langle
23\rangle[14])^{4}\int\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}\left(\frac{\langle
2|\ell|1]\langle 3|\ell|4]}{s[14]\langle
23\rangle}\right)^{4}\left(\frac{1}{D_{1234}}+\frac{1}{D_{1342}}\right)\bigg{\rvert}_{s\text{-cut}}.$
(44)

We define

$v_{1}^{\mu}=\langle 2|\gamma^{\mu}|1]\,,\,\hskip 56.9055ptv_{2}^{\mu}=\langle
3|\gamma^{\mu}|4]\,,$ (45)

which live in the four-dimensional subspace so that the $v_{i}$ effectively
project out the $(4-2\epsilon)$-dimensional components. This implies that
$v_{i}\cdot\ell=v_{i}\cdot L$, as follows from the prescription BernMorgan
that $\epsilon<0$ . Next, we lift the cut condition and regard our expression
as part of the full integrand that we would have obtained by Feynman rules,

$\mathcal{M}_{4}^{S=0,\,s\text{-channel}}=\left(\frac{\kappa}{2}\right)^{4}(\langle
23\rangle[14])^{4}\int\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}\left(\frac{\langle
2|L|1]\langle 3|L|4]}{s[14]\langle
23\rangle}\right)^{4}\left(\frac{1}{D_{1234}}+\frac{1}{D_{1342}}\right),$ (46)

keeping in mind that it is only valid for contributions that have an
$s$-channel cut. We apply IBP identities IBP in $4-2\epsilon$ dimensions to
reduce the above integrals to the master integrals in Eq. (7), using the
software FIRE6 Smirnov:2019qkx . Upon reducing to master integrals we remove
the ones that have no $s$-channel cut.

Next, we turn to the $t$-channel cut. We have

$\mathcal{M}_{4}^{S=0}\bigg{\rvert}_{t\text{-cut}}=\left(\frac{\kappa}{2}\right)^{4}(\langle
23\rangle[14])^{4}\int\frac{d^{4}\ell}{(2\pi)^{4}}\frac{d^{-2\epsilon}\mu}{(2\pi)^{-2\epsilon}}\left(\frac{(m^{2}+\mu^{2})}{t}\right)^{4}\left(\frac{1}{D_{1234}}+\frac{1}{D_{1423}}\right)\bigg{\rvert}_{t\text{-cut}},\quad$
(47)

where we subdivide the integration into four- and $(-2\epsilon)$-dimensional
parts

$\int\frac{d^{4-2\epsilon}L}{(2\pi)^{4-2\epsilon}}=\int\frac{d^{4}\ell}{(2\pi)^{4}}\frac{d^{-2\epsilon}\mu}{(2\pi)^{-2\epsilon}}\,.$
(48)

As for the $s$ channel, we lift the cut conditions and regard our expression
as part of the full integrand,

$\mathcal{M}_{4}^{S=0,\,t\text{-channel}}=\left(\frac{\kappa}{2}\right)^{4}(\langle
23\rangle[14])^{4}\int\frac{d^{4}\ell}{(2\pi)^{4}}\frac{d^{-2\epsilon}\mu}{(2\pi)^{-2\epsilon}}\left(\frac{(m^{2}+\mu^{2})}{t}\right)^{4}\left(\frac{1}{D_{1234}}+\frac{1}{D_{1423}}\right).$
(49)

As we discuss in Appendix G.2, the integrals with the
$(-2\epsilon)$-dimensional components of loop-momentum $\mu$ in the numerators
can be expressed directly in terms of the master integrals defined in Eq. (8).
After reducing our expression we eliminate master integrals that vanish on the
$t$-channel cut.

As noted above, an important consistency condition arises from the fact that
the box integrals have cuts in two channels, so we can determine their
coefficients from either channel. We explicitly verified that the coefficients
we obtain for the box integrals from looking at the different channels are the
same. In this way we are able to extract all coefficients of Eq. (7) other
than $a_{0}$ and $b_{0}$, since the corresponding integrals have no cuts in
any channel; we obtain these remaining two integral coefficients in Sect.
II.6.

###  II.6 Ultraviolet behavior and rational pieces

Analyzing the different generalized-unitarity cuts we obtain all the
coefficients in Eq. (7) other than $a_{0}$ and $b_{0}$. Tadpoles and bubbles
on external legs (see Fig. 6) vanish on any cut, therefore their coefficients
are not accessible through generalized unitarity. In this subsection we use
the known UV properties of the amplitude to obtain these coefficients.
Similarly to Sect. II.5, we discuss the double-minus configuration as a
specific example. The other configurations follow in the same manner.

First, we observe that simply neglecting these two integrals leads to an
inconsistent answer. Expanding to leading order in $\epsilon$ we get

$\mathcal{M}_{4}^{S=0}\bigg{\rvert}_{a_{0}\rightarrow 0,\,b_{0}\rightarrow
0}=(\langle
23\rangle[14])^{4}\frac{m^{2}\tilde{Q}(s,t,m)}{\epsilon}+\mathcal{O}(\epsilon^{0})\,,$
(50)

where $\tilde{Q}(s,t,m)$ is some rational function. We note that we expect no
UV divergence to appear in the four-graviton amplitude since the only
counterterm that we could write to absorb it is the Gauss-Bonnet term, which
is evanescent in four dimensions tHooft:1974toh . Also, the UV divergence not
coming out local hints that we neglected to include some integrals.

Figure 6: The tadpole and bubble-on-external-leg integrals.

We use the vanishing of the UV divergence to obtain the remaining coefficients
$a_{0}$ and $b_{0}$. Note that since $a_{0}$ and $b_{0}$ are rational
functions of $\epsilon$, it seems that we do not have enough information to
fully fix them. We surpass this difficulty by realizing that our problem
admits a second integral basis where the integral coefficients do not depend
on $\epsilon$ and $m$. This basis is overcomplete in that it contains
($4-2\epsilon$)- and higher-dimensional integrals, while the one introduced in
Eq. (7) only includes ($4-2\epsilon$)-dimensional integrals. We discuss this
basis in detail and provide an algorithmic way of switching between the two
bases in Appendix G.2.

We start by bringing the quantity we get from cut merging to the basis
containing higher-dimensional integrals. In doing so, we remove any integrals
that vanish on all cuts. We refer to this piece expressed in this basis as the
‘on-shell-constructable’ piece. Then, we need to figure out which integrals
without unitarity cuts we need to include in our expression, since in
principle one could add infinitely many higher-dimensional integrals.

We consider the pieces that would arise in a Feynman-rules calculation. For
the amplitude at hand we would find integrals up to quartically divergent.
After we IBP reduce them and express them in the basis that contains the
higher-dimensional integrals, we would in principle find all scalar integrals
up to quartically divergent appearing. Hence we need to consider:

$I_{1},\qquad I_{2}(k^{2})\,,\qquad I_{1}^{6-2\epsilon}\,,\qquad
I_{2}^{6-2\epsilon}(k^{2})\,,\qquad I_{2}^{8-2\epsilon}(k^{2})\,,$ (51)

where we imagine the limiting case $k^{2}\rightarrow 0$. For our definition of
higher-dimensional integrals see Eq. (279). These are all the integrals that
have no unitarity cuts and are up to quartically divergent. The reason why we
consider the above limiting case is because the coefficients of these
integrals might have a $1/k^{2}$ pole.

Then, the unknown piece in our amplitude up to normalization takes the form

$\lim_{k^{2}\rightarrow
0}\Big{(}\beta_{1}I_{1}+\beta_{2}I_{2}(k^{2})+\beta_{3}I_{1}^{6-2\epsilon}+\beta_{4}I_{2}^{6-2\epsilon}(k^{2})+\beta_{5}I_{2}^{8-2\epsilon}(k^{2})\Big{)}=\gamma_{1}I_{1}+\gamma_{2}I_{2}(0)+\gamma_{3}I_{1}^{6-2\epsilon}\,,$
(52)

where we use

$I_{2}^{6-2\epsilon}(k^{2})=I_{1}+\frac{k^{2}}{6}I_{2}(0)+\mathcal{O}(k^{4})\,,\quad\quad
I_{2}^{8-2\epsilon}(k^{2})=I_{1}^{6-2\epsilon}+\frac{k^{2}}{6}I_{1}+\mathcal{O}(k^{4})\,,$
(53)

following Appendix B of Ref. BernMorgan . The unknown coefficients
$\gamma_{i}$ are rational functions of the kinematics that do not contain
$\epsilon$ or $m$ dependence. In writing this expression we assume that the
coefficients of the integrals are at worse divergent as $1/k^{2}$ and that the
expression is finite in the $k^{2}\rightarrow 0$ limit when a massive particle
is circulating in the loop.888We drop ill-defined ‘cosmological constant’
tadpole diagrams with a $1/\left(\sum_{i=1}^{4}k_{i}\right)^{2}$ propagator.

Upon these considerations, our amplitude takes the form

$\mathcal{M}_{4}^{S=0}=\mathcal{M}_{4}^{S=0}\bigg{\rvert}_{\text{on-shell-
constr.}}+\frac{1}{(4\pi)^{2-\epsilon}}\left(\frac{\kappa}{2}\right)^{2}\mathcal{M}^{\text{tree}}_{4}\big{(}\gamma_{1}I_{1}+\gamma_{2}I_{2}(0)+\gamma_{3}I_{1}^{6-2\epsilon}\big{)}\,,$
(54)

where the $\gamma_{i}$ are coefficients are determined from the requirement
that ultraviolet divergences cancel, the integrals $I_{1}$ and $I_{2}$ are
given in Eqs. (280) and (282), and

$\mathcal{M}^{\text{tree}}_{4}\equiv\mathcal{M}^{\text{tree}}_{4}(1^{+},2^{-},3^{-},4^{+})=\left(\frac{\kappa}{2}\right)^{2}\frac{(\langle
23\rangle[14])^{4}}{stu}\,.$ (55)

The Mandelstam variables are defined below Eq. (8). Note that the little-group
scaling for the unknown terms is fixed to be $(\langle 23\rangle[14])^{4}$.
Since the $\gamma_{i}$ are rational functions of the kinematics, we are free
to multiply and divide by $(stu)$ to introduce
$\mathcal{M}^{\text{tree}}_{4}$.

Next, demanding that the amplitude has no UV divergence and that all three
$\gamma_{i}$ are independent of the mass $m$ uniquely fixes $\gamma_{1}$ and
$\gamma_{3}$ to nonzero values while setting $\gamma_{2}=0$. The results for
the amplitudes collected in Appendix F the values of the $\gamma_{i}$ are all
chosen to make the amplitudes UV finite.

As a simple consistency check, we repeat this analysis adding integrals that
have no unitarity cuts and are divergent worse than quartically (namely
($8-2\epsilon$)- and higher-dimensional tadpoles). We verify that the answer
is the same, i.e. the coefficients of these new integrals are set to zero.

We now briefly comment on the $m\rightarrow 0$ limit. In this limit, both
$I_{1}$ and $I_{2}(0)$ vanish, and the amplitude has no UV divergence, as can
be seen from Eq. (50). Hence our amplitude has the correct UV behavior in this
limit as well. Moreover, all integrals that have no unitarity cuts become
scaleless and hence zero in dimensional regularization. Therefore, there are
no unfixed coefficients and our construction based on the unitarity cuts
captures the full amplitude.

Finally, we want to clarify why we chose the overcomplete basis with higher-
dimensional integrals. The coefficients $a_{0}$ and $b_{0}$ are in principle
arbitrary rational functions of $\epsilon$ and $m$, and of the kinematic
variables $s$ and $t$. The existence of a basis where the integral
coefficients do not contain $\epsilon$ or $m$ significantly restricts the
functional dependence of $a_{0}$ and $b_{0}$ on $\epsilon$ and $m$. The
existence of such a basis is a nontrivial fact that may be understood via
analyzing the calculation in a covariant gauge, as we explain in Appendix G.2.
In this way the two integral bases are not completely equivalent, since the
latter one is exploiting more of the specific properties of the problem at
hand.

###  II.7 Further ultraviolet properties

Quadratic and quartic divergence

Next, we want to analyze the quadratic and quartic divergence of our
amplitude. In dimensional regularization, the signature of these divergences
are poles around $\epsilon=1$ and $\epsilon=2$ in the final answer. Since we
compute our amplitude to all order in $\epsilon$ we may probe these poles. We
tackle this question in the basis that contains only
($4-2\epsilon$)-dimensional integrals. In the basis that includes higher-
dimensional integrals, many of the integrals are quadratically or quartically
divergent, which obscures the analysis. We discuss the double-minus
configuration for a spin-0 particle in the loop; the other cases are similar.
As we demonstrate, our amplitude has no quadratic or quartic divergences.

We start with the quadratic divergence. In the chosen basis the only
quadratically divergent integral is the tadpole. However, the coefficient of
the tadpole is linear in $(\epsilon-1)$, hence there is no contribution to the
quadratic divergence from it. It then suffices to expand all coefficients
around $\epsilon=1$ and only keep the divergent piece. We get

$\displaystyle\mathcal{M}_{4}^{S=0}$
$\displaystyle=-\frac{1}{(4\pi)^{2-\epsilon}}\left(\frac{\kappa\hbox{}}{2}\right)^{2}\mathcal{M}^{\text{tree}}_{4}\frac{su}{(\epsilon-1)t^{6}}(3s^{2}u^{2}+10m^{2}stu+6m^{4}t^{2})\Big{(}\iota_{2}(s)+\iota_{2}(u)\Big{)}$
$\displaystyle\hskip
85.35826pt\hbox{}+\mathcal{O}\Big{(}(\epsilon-1)^{0}\Big{)}\,,$ (56)

where we introduced

$\iota_{2}(s)=s{}\big{(}2m^{2}I_{3}(s)-I_{2}(s)\big{)}\,.$ (57)

Note that $I_{n}\equiv I_{n}^{4-2\epsilon}=I_{n}^{2-2(\epsilon-1)}$. It
appears that our amplitude has a quadratic divergence which is non local due
to the $1/t^{6}$ factor. However, using Eq. (289) with $D=2-2(\epsilon-1)$ we
get

$I_{3}^{4-2(\epsilon-1)}(s)=\frac{1}{2(\epsilon-1)}\big{(}-2m^{2}I_{3}^{2-2(\epsilon-1)}(s)+I_{2}^{2-2(\epsilon-1)}(s)\big{)}\,,$
(58)

which gives

$\iota_{2}(s)=-2s{}(\epsilon-1)I_{3}^{4-2(\epsilon-1)}{(s)}\,.$ (59)

Since $I_{3}^{4-2(\epsilon-1)}(s)$ is finite as $\epsilon\rightarrow 1$, we
see that there is no quadratic divergence.

The analysis of the quartic divergence follows in a similar manner. In this
basis there are no quartically-divergent integrals. We expand our result
around $\epsilon=2$ to get

$\displaystyle\mathcal{M}_{4}^{S=0}=\frac{1}{(4\pi)^{2-\epsilon}}\left(\frac{\kappa\hbox{}}{2}\right)^{2}\mathcal{M}^{\text{tree}}_{4}\frac{su}{(\epsilon-2)t^{5}}(5su+6m^{2}t)\Big{(}\iota_{4}(s)+\iota_{4}(u)\Big{)}+\mathcal{O}\Big{(}(\epsilon-2)^{0}\Big{)}\,,$
(60)

where

$\iota_{4}(s)=2sI_{1}+s{}(s-6m^{2})I_{2}(s)+4m^{4}sI_{3}(s)\,.$ (61)

Using Eq. (289) two consecutive times we get

$\iota_{4}(s)=-4s{}(\epsilon-2)I_{3}^{4-2(\epsilon-2)}{(s)}\,,$ (62)

which shows that there is no quartic divergence.

In our problem we demonstrated that the coefficients of the integrals which
have no unitarity cuts in Eq. (54) contain no $m$ dependence. This property
was crucial in order to fix them using the vanishing of the (logarithmic) UV
divergence. In a more general situation we expect this property to no longer
be true. In such a scenario, analyzing the higher UV divergences offers an
alternative method for obtaining these coefficients. For example, if we demand
the vanishing of the quadratic and quartic divergences along with the
logarithmic one for our problem, we may fix the coefficients $\gamma_{i}$ to
the values we found above, without needing to impose that they do not contain
$m$ dependence.

Ultraviolet divergences in higher dimensions

We can also inspect the ultraviolet properties in higher dimensions. This is
straightforward because we obtain expressions for the amplitudes valid to all
orders in $\epsilon$. In the calculations we keep the external kinematics and
helicities fixed in four dimensions. In addition, we use the FDH scheme
FDHScheme which keeps the number of physical states circulating in the loop
fixed at their four-dimensional values. However, we can still analytically
continue the loop momentum to any dimension and study the divergence
properties. We can use this as a rather nontrivial check on our expressions
and also to point out a simple relation between appropriately defined
divergences in higher dimensions and coefficients of terms in the large-mass
expansion. We discuss the double-minus configuration with a scalar in the loop
for concreteness, however our results hold for all cases considered in this
paper.

Since our expressions are valid to all orders in the dimensional-
regularization parameter $\epsilon$, we may shift the spacetime dimension of
the loop momentum by $2\sigma$ via $\epsilon\rightarrow\epsilon-\sigma$. For
example for $\sigma=1$ the spacetime dimension is shifted to $D=6-2\epsilon$.
In the shifted dimension we define the coefficients of the ultraviolet
divergences as

$\mathcal{M}_{4}^{S=0}\big{|}_{\epsilon\rightarrow\epsilon-\sigma}=\frac{1}{(4\pi)^{\sigma}}\frac{1}{(\sigma-1)!}\frac{1}{\epsilon}\sum_{n=0}^{\sigma-1}\left(m^{2}\right)^{n}\delta_{\sigma,\,n}^{UV}(s,t)+\mathcal{O}\left(\epsilon^{0}\right),$
(63)

where we choose the normalization in a particular way to account for the
angular loop-integration in different dimensions. Note that there is no UV
divergence for $\sigma=0$, corresponding to $D=4-2\epsilon$, due to the lack
of a corresponding counterterm matrix element. For $\sigma=1$ the counterterm
is an $R^{3}$-type operator. In this case, up to a numerical factor the
coefficient of the divergence matches the tree amplitude generated from an
$R^{3}$ insertion, which we describe in Sect. III. For convenience we restrict
ourselves to even dimensions (for $\epsilon\rightarrow 0$) because in these
cases the dimension-shifting formulas (Eq. (289)) bring the higher-dimensional
integrals back to four dimensions. For $2\leq\sigma\leq 6$ we have explicitly
confirmed that the divergences are local and therefore correspond to
appropriate derivatives of $R^{4}$-type operators, as expected. We note that
because individual integrals have nonlocal coefficients the fact that the
divergences are local provides a rather nontrivial check on our expressions.
Furthermore, starting at $D=8-2\epsilon$ all the integrals are divergent, so
their coefficients feed into this check.

Finally, we point out a relation between the ultraviolet divergences in Eq.
(63) and terms in the large-mass expansion in four dimensions. Specifically,
defining

$\mathcal{M}_{4}^{S=0}=\sum_{n=1}^{\infty}\frac{\delta^{IR}_{n}(s,t)}{m^{2n}}\,,$
(64)

we have explicitly checked through $\sigma=4$ or equivalently $D=12-2\epsilon$
that

$\delta^{UV}_{\sigma,0}=\delta^{IR}_{\sigma}\,.$ (65)

Similarly, we find that $\delta^{UV}_{\sigma,n}$ is proportional to
$\delta^{UV}_{\sigma-n,0}=\delta^{IR}_{\sigma-n}$ with a common
proportionality constant for all multiplets. For this correspondence to hold
it is important to keep both the external and internal states at their four
dimensional value; only the loop momentum is analytically continued to higher
spacetime dimensions. It is quite remarkable that such simple relations exist
between the coefficients of the divergences in higher dimensions and the terms
in the large-mass expansion of the amplitudes in four dimensions.

###  II.8 Consistency checks

We have carried out a variety of checks on our amplitudes. Basic self-
consistency checks are that ultraviolet and infrared or mass singularities be
of the right form.

Ultraviolet singularities must be local. In general this is nontrivial and
happens only after the ultraviolet singularities are combined. In our case,
the coefficient of the divergences vanishes. We also verified that the
$1/(\epsilon-1)$ pole cancels, consistent with the fact that the Ricci-scalar
counterterm vanishes by the equation of motion. Similarly for the
$1/(\epsilon-2)$ pole, the expression is not only local but it vanishes.
Further, we verified that the divergences obtained by analytically continuing
the loop momentum to higher dimensions while keeping the state counts and
external kinematics to their four-dimensional values are also local.

Another nontrivial check comes from looking at the $m\rightarrow 0$ limit.
Since the internal lines are massive, there is no IR divergence for our
amplitude. However, we may regard the mass of the internal lines as an
infrared regulator for the corresponding massless amplitude and study the IR
divergence as $m\rightarrow 0$. In gravity the infrared singularities are
quite simple since there are only soft singularities and no collinear or mass
singularities GravIR . The soft singularities arise only from gravitons
circulating in the loop. This implies that as $m\rightarrow 0$ all
contributions must be infrared finite except for the
$\mathcal{M}_{4}^{\\{2\\}}$ piece (corresponding to $\mathcal{N}=8$
supergravity in the massless limit), since this is the only piece that has a
graviton circulating in the loop in this limit.

To carry out this check we start with the on-shell-constructable piece. We
start from the ($4-2\epsilon$)-dimensional integral basis, where only the
boxes and the triangles have IR singularities. The triangles only contain
simple logarithms (see Eq. (284)) while the boxes contain both logarithms and
dilogarithms (see Eq. (285)). To simplify the check we use Eq. (289) to trade
the boxes for triangles and ($6-2\epsilon$)-dimensional boxes. Importantly,
the latter have no IR singularities. In this form the infrared singularities
are all pushed into triangle integrals, hence it suffices to verify that their
coefficients vanish as $m\rightarrow 0$, which we confirm for all but the
$\mathcal{M}_{4}^{\\{2\\}}$ piece. Next we look at the contributions with no
unitarity cuts. These pieces are zero if we take $m\rightarrow 0$ before we
expand in $\epsilon$. On the other hand, divergent terms appear if we first
expand in $\epsilon$ and then take $m\rightarrow 0$. However, given that the
ultraviolet divergence vanishes in four-dimensions these infrared divergent
$\log(m)$ pieces also cancel among themselves. For the
$\mathcal{M}_{4}^{\\{2\\}}$ piece there is indeed a infrared singularity that
develops as $m\rightarrow 0$. In this case, we recover the known infrared
divergence of pure Einstein gravity GravIR .

As another check, we have explicitly verified that as $m\rightarrow 0$ our
massive results match the massless ones given in Ref. Dunbar:1994bn , up to an
overall sign in $\mathcal{M}_{4}^{\\{1/2\\}}$, as noted in Ref. Abreu:2020lyk
. One simple way to implement this check is to start with the expressions for
the amplitudes in the $(4-2\epsilon)$-dimensional integral basis, set
$m\rightarrow 0$ in the integral coefficients, and replace the massive
integrals with massless ones.

Finally, we note that contributions from individual integrals do not decay at
large mass as required by decoupling, while the amplitudes have the required
property. This involves nontrivial cancellation between the pieces, providing
yet another check.

##  III Amplitudes in the low-energy effective field theory

In this section we study four-graviton scattering in a general parity-even
low-energy EFT. Such EFTs start from the Einstein-Hilbert action and extend to
systematically include higher-dimension operators. We include a massless
scalar field to our analysis, corresponding to the dilaton found in string
theory.

We match this EFT to the one-loop amplitudes determined in Sect. II and
collected in Appendix F. In this context, the EFT is valid for energies
significantly smaller than the mass of the spinning particle in the loop. In
this way we determine the modification to the low-energy theory of gravity due
to the presence of a heavy particle. We take this as a nontrivial model of
ultraviolet physics feeding into low-energy physics.

For the lowest-dimension operators, we calculate the four-point tree-level
amplitudes in this EFT and compare them to the expansion of our one-loop
amplitudes in the large-mass limit in order to obtain their Wilson
coefficients. More generally, since we later put bounds on the coefficients
appearing in the amplitudes themselves, there is no need to relate these back
to a Lagrangian. For comparison to the bounds derived in subsequent sections
we also present the one-loop scattering amplitudes expanded in the large-mass
limit. In Appendix H we present the expansions to much higher orders, which
should be useful for further studies of the bounds.

Finally, we obtain the Regge limits of our one-loop amplitudes. These are
useful later in analyzing low-energy coefficients via dispersion relations.

###  III.1 Setup of the effective field theory

The first few terms of the EFT describing low-energy gravitational scattering
are

$S_{\rm EFT}=\int
dx^{4}\sqrt{-g}\,\biggl{[}-\frac{2}{\kappa^{2}}R+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{2\beta_{\phi}}{\kappa^{2}}\phi\,C+\frac{8}{\kappa^{3}}\frac{\beta_{R^{3}}}{3!}\,R^{3}+\frac{2\beta_{R^{4}}}{\kappa^{4}}C^{2}+\frac{2\tilde{\beta}_{R^{4}}}{\kappa^{4}}\tilde{C}^{2}+\ldots\biggr{]}\,,$
(66)

where $R$ is the Ricci scalar, $\kappa$ is given in terms of Newton’s constant
$G$ via $\kappa^{2}=32\pi G$, and the metric is
$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$ in terms of the graviton field
$h_{\mu\nu}$. To describe the most general parity-even theory that captures
low-energy four-graviton scattering, we include the massless scalar field
$\phi$. The factors of ${1\over\kappa}$ are chosen to normalize the kinetic
term canonically and to remove the factors of $\kappa$ that would appear in
the three-point tree-level amplitudes built out of a single insertion of
$\phi\,C$ or $R^{3}$ and the four-graviton tree-level amplitudes built out of
a single insertion of $C^{2}$ or $\tilde{C}^{2}$. The $\beta_{\phi}$,
$\beta_{R^{3}}$, $\beta_{R^{4}}$ and $\tilde{\beta}_{R^{4}}$ are Wilson
coefficients that depend on the details of the UV physics. The composite
operators are defined by

$\displaystyle R^{3}\equiv
R^{\mu\nu\kappa\lambda}R_{\kappa\lambda\alpha\gamma}R^{\alpha\gamma}_{\quad\mu\nu}\,,$
$\displaystyle\qquad C\equiv
R^{\mu\nu\kappa\lambda}R_{\mu\nu\kappa\lambda}\,,\qquad\tilde{C}\equiv\frac{1}{2}R^{\mu\nu\alpha\beta}\epsilon_{\alpha\beta}^{\quad\gamma\delta}R_{\gamma\delta\mu\nu}\,,$
(67)

where $R_{\lambda\mu\nu\kappa}$ is the Riemann tensor and $R_{\mu\nu}$ is the
Ricci tensor. One can systematically add higher-dimension operators; we choose
not to do so here since our later analysis is at the amplitude level, so the
mapping back to Lagrangian coefficients is not necessary.

In writing the effective action (66) we apply the equations of motion and
integrate by parts to reduce the number of operators to a minimum independent
set. In particular, we do not include $R^{2}$-type terms because the Ricci
scalar and tensor, $R$ and $R_{\mu\nu}$, vanish due to the equations of
motion, while the contraction of two Riemann tensors can be traded for the
Gauss-Bonnet contribution $C$, which is equal to a total derivative in four
dimensions tHooft:1974toh . Furthermore, we do not include operators that our
calculation is not sensitive to. Specifically, we do not consider the other
possible contraction of three Riemann tensors, since it does not contribute to
four-graviton scattering vanNieuwenhuizen:1976vb ; Broedel:2012rc . We
restrict ourselves to parity-even interactions and neglect the parity-odd
operator $C\tilde{C}$ Sennett:2019bpc ; Endlich:2017tqa . The possible parity-
even contractions of four Riemann tensors were obtained in Ref. Fulling:1992vm
. Recent studies that use similar Lagrangians are found in Refs.
Endlich:2017tqa ; Dunbar:2017szm ; Brandhuber:2019qpg ;
AccettulliHuber:2020oou .

###  III.2 Scattering amplitudes in the effective field theory

To describe the amplitudes it is useful to extract overall dependence on the
spinors, leaving only functions of $s,u$ with simple crossing properties.
Specifically, for the independent helicity configurations we define,

$\displaystyle\mathcal{M}_{4}(1^{+},2^{-},3^{-},4^{+})\,$
$\displaystyle=(\langle 23\rangle[14])^{4}\,f(s,u)\,,$
$\displaystyle\mathcal{M}_{4}(1^{+},2^{+},3^{-},4^{+})\,$
$\displaystyle=\left([12][14]\langle 13\rangle\right)^{4}\,g(s,u)\,,$
$\displaystyle\mathcal{M}_{4}(1^{+},2^{+},3^{+},4^{+})$
$\displaystyle=\left(\frac{[12][34]}{\langle 12\rangle\langle
34\rangle}\right)^{2}\,h(s,u)\,,$ (68)

corresponding to the double-minus, single-minus and all-plus helicity
configurations. As usual we do not include the overall $i$ that normally would
appear in Feynman diagrams. All other amplitudes are given by permutations and
complex conjugation,

$\mathcal{M}_{4}(1^{h_{1}},2^{h_{2}},3^{h_{3}},4^{h_{4}})=\mathcal{M}_{4}^{*}(1^{-h_{1}},2^{-h_{2}},3^{-h_{3}},4^{-h_{4}})\,.$
(69)

where we define the complex conjugation to not act on the $i\epsilon$
prescription.999More precisely, given complex conjugation $\gamma:z\mapsto
z^{*}$, we define $f^{*}(z)$ as $f^{*}\equiv\gamma\circ
f\circ\gamma:z\mapsto(f(z^{*}))^{*}$. For identical scalar particles (69)
becomes the familiar hermitian analyticity. In parity-preserving theories
complex conjugation acts only on the spinors swapping the positive and
negative helicity spinors,

$\mathcal{M}_{4}(1^{h_{1}},2^{h_{2}},3^{h_{3}},4^{h_{4}})=\mathcal{M}_{4}(1^{-h_{1}},2^{-h_{2}},3^{-h_{3}},4^{-h_{4}})\Bigr{|}_{\lambda_{i}\leftrightarrow\tilde{\lambda}_{i}}\,,$
(70)

so that the $f$, $g$ and $h$ functions are unaltered.

In general relativity the leading-order results for the amplitudes above take
the form

$\displaystyle f_{\text{GR}}(s,u)$
$\displaystyle=\left(\frac{\kappa\hbox{}}{2}\right)^{2}{1\over
stu}+\ldots,\,,$ (71) $\displaystyle g_{\text{GR}}(s,u)$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa\hbox{}}{2}\right)^{4}\frac{1}{s^{2}t^{2}u^{2}}\frac{s^{2}+t^{2}+u^{2}}{360}+\ldots\,,$
(72) $\displaystyle h_{\text{GR}}(s,u)$
$\displaystyle=-\frac{1}{(4\pi)^{2}}\left(\frac{\kappa\hbox{}}{2}\right)^{4}\frac{s^{2}+t^{2}+u^{2}}{120}+\ldots\,.$
(73)

Below we do not consider loop effects in general relativity itself (due to
gravitons circulating in the loops) but focus on the properties of the higher-
derivative operators in the gravitational EFT generated by integrating out
massive degrees of freedom. For the same reason IR divergences are not an
issue for our analysis since the corrections of interest are manifestly IR
finite.

Permutation symmetry of the amplitudes plays a crucial role in our analysis
and manifests itself in the following crossing relations

$\displaystyle f(s,u)$ $\displaystyle=f(u,s)\,,$ (74) $\displaystyle g(s,u)$
$\displaystyle=g(u,s)=g(s,t)\,,$ (75) $\displaystyle h(s,u)$
$\displaystyle=h(u,s)=h(s,t)\,.$ (76)

where $s+t+u=0$.

With our normalizations the three-point amplitude arising from the Einstein
term is101010We implicitly use complex momenta so that the three-point
amplitude does not vanish from kinematic constraints.

$\mathcal{M}_{3}^{\rm
GR}(1^{+},2^{+},3^{-})=\frac{\kappa}{2}\biggl{(}\frac{[12]^{3}}{[23][31]}\biggr{)}^{2}\,.$
(77)

The three-points amplitudes with an insertion of the $\phi\,C$ or the $R^{3}$
operator are Broedel:2012rc

$\mathcal{M}_{3}^{\phi\,C}(1^{+},2^{+},3^{\phi})=\beta_{\phi}[12]^{4}\,,~{}~{}~{}\mathcal{M}_{3}^{R^{3}}(1^{+},2^{+},3^{+})=\beta_{R^{3}}\Big{(}[12][23][31]\Big{)}^{2}\,,$
(78)

where $\beta_{\phi}$ and $\beta_{R^{3}}$ are the Wilson coefficient for these
operators appearing in Eq. (66).

At four points $\phi\,C$ contributes to the double-minus and all-plus
configurations,

$\displaystyle\mathcal{M}_{4}^{\phi\,C}(1^{+},2^{+},3^{+},4^{+})$
$\displaystyle=-3(\beta_{\phi})^{2}stu\left(\frac{[12][34]}{\langle
12\rangle\langle 34\rangle}\right)^{2}\,,$
$\displaystyle\mathcal{M}_{4}^{\phi\,C}(1^{+},2^{-},3^{-},4^{+})$
$\displaystyle=-(\beta_{\phi})^{2}\frac{\bigl{(}\langle
23\rangle[14]\bigr{)}^{4}}{t}\,.$ (79)

On the other hand, there are two independent helicity configurations, the all-
plus and the single-minus configurations, that contain a single insertion of
$R^{3}$. We obtain these amplitudes following Ref. Broedel:2012rc . We find

$\displaystyle\mathcal{M}^{R^{3}}_{4}(1^{+},2^{+},3^{-},4^{+})$
$\displaystyle=\beta_{R^{3}}\Bigl{(}\frac{\kappa}{2}\Bigr{)}\frac{1}{stu}\left([12][14]\langle
13\rangle\right)^{4}\,,$ (80)

and

$\displaystyle\mathcal{M}^{R^{3}}_{4}(1^{+},2^{+},3^{+},4^{+})$
$\displaystyle=10\beta_{R^{3}}\Bigl{(}\frac{\kappa}{2}\Bigr{)}stu\left(\frac{[12][34]}{\langle
12\rangle\langle 34\rangle}\right)^{2}\,,$ (81)

which is slightly rearranged compared to Ref. Broedel:2012rc . We build a
double-minus contribution out of two insertions of $R^{3}$ BiaHigherOrder ,

$\mathcal{M}_{4}^{R^{3}}(1^{+},2^{-},3^{-},4^{+})=(\beta_{R^{3}})^{2}\frac{su}{t}\bigl{(}\langle
23\rangle[14]\bigr{)}^{4}\,.$ (82)

For the $R^{4}$-type operators the amplitudes are AccettulliHuber:2020oou

$\displaystyle\begin{split}\mathcal{M}_{4}^{R^{4}}(1^{+},2^{+},3^{+},4^{+})&=\beta_{R^{4}}^{-}\frac{(s^{2}+t^{2}+u^{2})^{2}}{2}\left(\frac{[12]\,[34]}{\langle
12\rangle\,\langle 34\rangle}\right)^{2}\,,\\\
\mathcal{M}_{4}^{R^{4}}(1^{+},2^{-},3^{-},4^{+})&=\beta_{R^{4}}^{+}\bigl{(}\langle
23\rangle[14]\bigr{)}^{4}\,,\end{split}$ (83)

where by $\mathcal{M}_{4}^{R^{4}}$ we refer to the amplitudes build out of
both $C^{2}$ and $\tilde{C}^{2}$, with

$\beta_{R^{4}}^{\pm}\equiv\beta_{R^{4}}\pm\tilde{\beta}_{R^{4}}\,,$ (84)

Using these four-point amplitudes we may extract the coefficients
$\beta_{R^{3}}$, $\beta_{R^{4}}$ and $\tilde{\beta}_{R^{4}}$ by matching to
our one-loop calculation in the large-mass limit. Since we did not include the
massless scalar field in the construction of our one-loop amplitudes, we have
$\beta_{\phi}=0$ in this case.

Next, we bring our one-loop amplitudes in a form suitable to compare to the
EFT amplitudes listed above. We start with the double minus amplitude. As
usual, we organize the contributions according to the supersymmetric
decomposition (32),

$\mathcal{M}^{\\{\it S\\}}_{4}(1^{+},2^{-},3^{-},4^{+})=(\langle
23\rangle[14])^{4}\ f^{\\{\it S\\}}(s,u)\,.$ (85)

In the large-mass limit, for the double-minus amplitudes we have,

$\displaystyle f$
${}^{\\{0\\}}=\mathcal{K}\biggl{(}\frac{1}{6300m^{4}}+\frac{t}{41580m^{6}}+\frac{81(s^{2}+u^{2})+155su}{15135120m^{8}}+\frac{\big{(}161(s^{2}+u^{2})+324su\big{)}t}{151351200m^{10}}+\cdots\biggr{)}\,,$
$\displaystyle f$
${}^{\\{1/2\\}}=\mathcal{K}\biggl{(}\frac{1}{1120m^{4}}+\frac{t}{8400m^{6}}+\frac{15(s^{2}+u^{2})+28su}{554400m^{8}}+\frac{\big{(}153(s^{2}+u^{2})+313su\big{)}t}{30270240m^{10}}+\cdots\biggr{)}\,,$
$\displaystyle f$
${}^{\\{1\\}}=\mathcal{K}\biggl{(}\frac{1}{180m^{4}}+\frac{t}{1680m^{6}}+\frac{22(s^{2}+u^{2})+39su}{151200m^{8}}+\frac{\big{(}20(s^{2}+u^{2})+43su\big{)}t}{831600m^{10}}+\cdots\biggr{)}\,,$
$\displaystyle f$
${}^{\\{3/2\\}}=\mathcal{K}\biggl{(}\frac{1}{24m^{4}}+\frac{t}{360m^{6}}+\frac{9(s^{2}+u^{2})+14su}{10080m^{8}}+\frac{\big{(}8(s^{2}+u^{2})+21su\big{)}t}{75600m^{10}}+\cdots\biggr{)}\,,$
$\displaystyle f$
${}^{\\{2\\}}=\mathcal{K}\biggl{(}\frac{1}{2m^{4}}+\frac{s^{2}+su+u^{2}}{120m^{8}}+\frac{stu}{504m^{10}}+\cdots\biggr{)}\,,\hskip
14.22636pt$ (86)

where

$\mathcal{K}=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}\,.$ (87)

For the all-plus and single-minus configurations it suffices to give the
result for the spin-0 contribution since we obtain the remaining amplitudes
via Eq. (34). For the single-minus configuration we have

$\mathcal{M}_{4}^{S=0}(1^{+},2^{+},3^{-},4^{+})=\left([12]\langle
13\rangle[14]\right)^{4}\,g(s,u)\,,$ (88)

where

$g(s,u)=\mathcal{K}\bigg{(}\frac{1}{5040m^{2}stu}+\frac{1}{6306300m^{8}}+\frac{(s^{2}+su+u^{2})}{441080640m^{12}}+\ldots\bigg{)}\,.$
(89)

Finally, for the all-plus configuration we have,

$\mathcal{M}_{4}^{S=0}(1^{+},2^{+},3^{+},4^{+})=\left(\frac{[12][34]}{\langle
12\rangle\langle 34\rangle}\right)^{2}\,h(s,u)\,,$ (90)

where

$\displaystyle h(s,u)$
$\displaystyle=\mathcal{K}\biggl{(}\frac{stu}{504m^{2}}+\frac{(s^{2}+su+u^{2})^{2}}{3780m^{4}}+\frac{(s^{2}+su+u^{2})stu}{7920m^{6}}$
$\displaystyle\hskip
28.45274pt+\frac{75(s^{6}+u^{6})+225(s^{5}u+su^{5})+559(s^{4}u^{2}+s^{2}u^{4})+743s^{3}u^{3}}{7207200m^{8}}$
$\displaystyle\hskip
28.45274pt+\frac{3(s^{2}+su+u^{2})^{2}stu}{400400m^{10}}+\ldots\biggr{)}\,.$
(91)

The fact that the highest power of $m$ appearing in the large-mass expansion
is $m^{-2}$ may be contrasted to the high powers of the $m$ in the
coefficients of the integrals in the ($4-2\epsilon$)-dimensional integral
basis (see Eqs. (268), (273) and (274)). It is a nontrivial consistency check
that our amplitudes vanish in the large-mass limit, as expected from
decoupling. Indeed, the above large-mass behavior hinges on nontrivial
cancellations between all pieces of the amplitude.

Now consider the matching and extraction of the Wilson coefficients
$\beta_{R^{3}}$, $\beta_{R^{4}}$ and $\tilde{\beta}_{R^{4}}$ (or,
equivalently, $\beta_{R^{3}}$ and $\beta_{R^{4}}^{\pm}$). Since the relation
between the Wilson coefficients and the amplitudes is linear, the Wilson
coefficients satisfy the supersymmetric decomposition (Eq. (32)). Hence, we
may organize our results in terms of the multiplets circulating in the loop.
One may then assemble the corresponding coefficients for any spinning particle
circulating in the loop using Eq. (32).

Since the all-plus and single-minus amplitudes are nonzero only for the
$\\{0\\}$ piece, we have

$\displaystyle(\beta_{R^{3}})^{\\{0\\}}=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{3}\frac{1}{m^{2}}\frac{1}{5040}\,,\qquad(\beta_{R^{4}}^{-})^{\\{0\\}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}\frac{1}{m^{4}}\frac{1}{7560}\,,$
$\displaystyle(\beta_{R^{3}})^{\\{\it S\\}\neq 0}=(\beta_{R^{4}}^{-})^{\\{\it
S\\}\neq 0}$ $\displaystyle=0\,.$ (92)

where our notation $(\beta_{X})^{\\{S\\}}$ means the value of $\beta_{X}$ as
determined by the data for the new piece for a given spin $S$. The double-
minus configuration is nonzero for any multiplet circulating in the loop. We
find

$\displaystyle(\beta_{R^{4}}^{+})^{\\{0\\}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}\frac{1}{m^{4}}\frac{1}{6300}\,,\hskip
42.67912pt(\beta_{R^{4}}^{+})^{\\{1/2\\}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}\frac{1}{m^{4}}\frac{1}{1120}\,,$
$\displaystyle(\beta_{R^{4}}^{+})^{\\{1\\}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}\frac{1}{m^{4}}\frac{1}{180}\,,\hskip
48.36958pt(\beta_{R^{4}}^{+})^{\\{3/2\\}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}\frac{1}{m^{4}}\frac{1}{24}\,,$
$\displaystyle(\beta_{R^{4}}^{+})^{\\{2\\}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}\frac{1}{m^{4}}\frac{1}{2}\,.$
(93)

###  III.3 Regge limits of the amplitudes

For our analysis of the amplitudes with dispersion relations in the next
sections, we need the behavior of the amplitudes for $t\rightarrow\infty$ and
$s\in\mathbb{R}$ fixed, with $|s|<4m^{2}$ and for $s\rightarrow\infty$ and
$t\in\mathbb{R}$ fixed, with $|t|<4m^{2}$. We extract these directly from the
explicit values of the amplitudes in Appendix H.

We start with the $t\rightarrow\infty$ limit. For the new-pieces in the
supersymmetric decomposition we have

$\displaystyle f^{\\{\it S\\}}(s,u)$
$\displaystyle\sim\frac{\log(t)}{t^{2}}\,,\qquad$ $\displaystyle
S=0,\,1/2,\,1,$ $\displaystyle f^{\\{3/2\\}}(s,u)$
$\displaystyle\sim\frac{\log^{2}(t)}{t^{2}}\,,\qquad$ $\displaystyle
f^{\\{2\\}}(s,u)\sim\frac{1}{t}\,,$ $\displaystyle g^{\\{0\\}}(s,u)$
$\displaystyle\sim\frac{1}{t^{2}}\,,\qquad$ $\displaystyle
h^{\\{0\\}}(s,u)\sim t^{2}\,,$ (94)

where the $f,g$, and $h$ function are related to the amplitude via Eq. (68).
Using Eq. (32) we assemble the contributions for each particle of a given
spin,

$\displaystyle f^{{S}}(s,u)$ $\displaystyle\sim\frac{\log(t)}{t^{2}}\,,\qquad$
$\displaystyle S=0,\,1/2,\,1,$ $\displaystyle f^{{3/2}}(s,u)$
$\displaystyle\sim\frac{\log^{2}(t)}{t^{2}}\,,\qquad$ $\displaystyle
f^{{2}}(s,u)\sim\frac{1}{t}\,,$ $\displaystyle g^{S}(s,u)$
$\displaystyle\sim\frac{1}{t^{2}}\,,\qquad$ $\displaystyle h^{S}(s,u)\sim
t^{2}\,,\qquad S=0,\,1/2,\,1,\,3/2,\,2.$ (95)

For the $f^{{S}}$ functions corresponding to the case of no helicity flips
between incoming and incoming states the spin 2 contribution dominates as
expected.

Next, we consider the $s\rightarrow\infty$ limit. We find

$\displaystyle f^{\\{\it S\\}}(s,u)$ $\displaystyle\sim\frac{1}{s}\,,\qquad$
$\displaystyle S=0,\,1/2,\,1,\,3/2,\,2,$ $\displaystyle g^{\\{0\\}}(s,u)$
$\displaystyle\sim\frac{1}{s^{2}}\,,\qquad$ $\displaystyle
h^{\\{0\\}}(s,u)\sim s^{2}\,,$ (96)

which gives

$f^{S}(s,u)\sim\frac{1}{s}\,,\qquad g^{S}(s,u)\sim\frac{1}{s^{2}}\,,\qquad
h^{S}(s,u)\sim s^{2}\,,\hskip 42.67912ptS=0,\,1/2,\,1,\,3/2,\,2.$ (97)

Note that the limits of the functions $g^{S}$ and $h^{S}$ in Eqs. (96) and
(97) follow from those in Eqs. (94) and (95), since these functions are
crossing symmetric.

##  IV Properties of gravitational amplitudes

We now turn to the properties of the low-energy effective field theory,
arising from taking the large-mass expansion of the one-loop four-graviton
amplitudes calculated in Sect. II. Here we do not consider loop effects in
general relativity itself (due to gravitons circulating in the loops) but
focus on the properties of the leading-order higher-derivative operators in a
weakly-coupled gravitational EFT generated by integrating out massive degrees
of freedom. For the same reason IR divergences are not an issue for our
analysis since the corrections of interest are manifestly IR finite. We also
note that we do not need to deal with UV divergences, since gravity is one-
loop UV finite tHooft:1974toh .

To make the analysis more complete we also include the example of tree-level
graviton scattering in string theory (see Appendix B). As recently discussed
in Chowdhury:2019kaq , in this case the scattering amplitudes have a great
degree of universality; e.g. to leading order considered here they do not
depend on the details of the string compactification.

A general question we ask in this paper is the following: where do physical
theories land in the space of couplings that satisfy the bounds from
causality, unitarity and crossing? In this section we review general
properties of the gravitational amplitudes relevant for the derivation of the
bounds. In subsequent sections we then proceed with the derivation of various
bounds on the Wilson coefficients, following the recent developments of Refs.
Tolley:2020gtv ; Caron-Huot:2020cmc ; Arkani-Hamed:2020blm ; Alberte:2020bdz ;
Sinha:2020win . Finally, we check that the results presented in the paper
satisfy the expected bounds and analyze the region in the space of couplings
covered by known theories.

###  IV.1 Low-energy expansion

The functions $f$, $g$ and $h$ defined in Eq. (68) correspond to the
independent helicity configurations. We consider their low-energy
expansion111111Here we allowed for parity-odd and parity-violating effects
which render certain coefficients in the low-energy expansion complex.

$\displaystyle f(s,u)$
$\displaystyle=\left(\frac{\kappa\hbox{}}{2}\right)^{2}{1\over
stu}+|\beta_{R^{3}}|^{2}{su\over t}-|\beta_{\phi}|^{2}{1\over
t}+\sum_{i=0}^{\infty}f_{2i,i}s^{i}u^{i}+\sum_{i=1}^{\infty}\sum_{j=0}^{[{i\over
2}]}f_{i,j}(s^{i-j}u^{j}+s^{j}u^{i-j})\,,$ $\displaystyle g(s,u)$
$\displaystyle=\Bigl{(}\frac{\kappa}{2}\Bigr{)}{\beta_{R^{3}}\over
stu}+\sum_{p,q=0}^{\infty}g_{2p+3q,q}\sigma_{2}^{p}\sigma_{3}^{q}\,,$
$\displaystyle h(s,u)$
$\displaystyle=[10\Bigl{(}\frac{\kappa}{2}\Bigr{)}\beta_{R^{3}}-3\beta_{\phi}^{2}]stu+\sum_{p,q=0,2p+3q\geq
4}^{\infty}h_{2p+3q,q}\sigma_{2}^{p}\sigma_{3}^{q}\,,$ (98)

where $[x]$ means the integer part of $x$, and we introduced

$\sigma_{k}\equiv(-1)^{k}{s^{k}+t^{k}+u^{k}\over k}.$ (99)

In Eq. (98) we explicitly write the massless exchange poles and assume that
the rest of the amplitude admits a simple low-energy expansion. This is a
structure expected from integrating out massive degrees of freedom and indeed,
the amplitudes analyzed in the present paper are of this type. It does not
however capture correctly the structure of the amplitude once the loops
involving massless particles are included. For example, consider the case of
the one-loop correction due to gravitons circulating in the loop; see Eq. (71)
and Appendix F.4. In this case, we see that for the all-plus amplitude one-
loop Einstein gravity generates a non-zero $h_{2,0}$. Similarly, for the
single-minus amplitude $g(s,u)$ the one-loop result in Einstein gravity has
poles in each of the Mandelstam variables. Finally, the double-minus amplitude
$f(s,u)$, see Eq. (277), contains IR divergences, logarithms, as well as
higher-order singularities in $1/t$ compared to the formulas above. In the
present paper we focus on the effect produced by integrating out massive
degrees of freedom and do not analyze the effects from one-loop massless
exchanges. Mathematically, this is simply due to the fact that at leading
order that we are interested in, the two effects lead to additive
contributions to the scattering amplitude and can be analyzed separately.
Moreover, the corrections to graviton scattering due to integrating out
massive degrees of freedom satisfy all the basic properties that we discuss
later in the section and as such the low-energy expansion generated in this
way satisfies consistency bounds. Physically, integrating out massive degrees
of freedom leads to effects that are localized in the impact parameter space
$b\lesssim{1/m_{\rm gap}}$, which are encoded via higher-derivative operators
in a gravitational EFT, whereas the one-loop effect due to graviton exchange
contributes at any impact parameter, which also manifests itself through the
fact that such corrections do not admit the representation (98). It would be
very interesting to develop a systematic and unified approach to treat both
effects in the context of gravitational scattering, but this is beyond the
scope of the present paper and we leave it for future work.

In writing Eq. (98) we take into account the crossing relations (74). Note
that $\sigma_{3}=-stu$ for $u=-s-t$. In the expansion (98) $f_{i,j}$ are real.
For parity-preserving theories $h_{k,j}$ and $g_{k,j}$ are real as well. In
the formulas above $\beta_{R^{3}}$ encodes the unique non-minimal correction
to the three-point amplitude of gravitons as defined in Eq. (78). Similarly
$\beta_{\phi}$ encodes the non-minimal coupling of a scalar to two gravitons.
In parity-preserving theories these are real.

For completeness we take into account the possibility of non-minimal coupling
to a massless scalar in the amplitude above with the three-point amplitude
given in Eq. (78). Curiously the non-minimal correction to the three-point
graviton amplitude and the non-minimal coupling to a massless scalar mix in
the first term in the low-energy expansion of $h(s,u)$ (98). This point was
discussed in detail in Ref. Broedel:2012rc .

We list the explicit results for these functions obtained for the amplitudes
considered in the present paper as an expansion in large mass in Eqs. (86),
(89) and (91), as well as in Appendix H. The exact form of the amplitudes is
found in Appendix F.

###  IV.2 Unitarity constraints

Here we consider the constraints that arise from unitarity. Since the
gravitational EFTs of interest are weakly coupled, we limit ourselves to
perturbative unitarity as discussed for example in Refs. Arkani-Hamed:2017jhn
; Arkani-Hamed:2020blm . It would be of course be very interesting to
implement unitarity nonperturbatively along the lines of Refs. Hebbar:2020ukp
; KelianMaster . In four dimensions this would also require understanding the
constraints of unitarity at the level of IR-finite observables; see e.g. Refs.
Strominger:2017zoo ; Arkani-Hamed:2020gyp for a recent discussion. Luckily,
for our purpose of investigating the leading-order corrections to the
gravitational EFT these subtle but important issues are irrelevant. Here we
primarily follow the discussion of Ref. Arkani-Hamed:2020blm but modify it to
account for differing helicity configurations.

To discuss the constraints coming from unitarity we note that the general
incoming two-graviton state is a superposition of amplitudes with different
helicities. In total there are four choices for the incoming and four choices
for the outgoing states. We therefore consider the following matrix of
possible amplitudes

$\displaystyle\begin{pmatrix}\mathcal{M}_{(+,-,-,+)}&\mathcal{M}_{(+,-,-,-)}&\mathcal{M}_{(-,-,-,+)}&\mathcal{M}_{(-,-,-,-)}\\\
\mathcal{M}_{(+,-,+,+)}&\mathcal{M}_{(+,-,+,-)}&\mathcal{M}_{(-,-,+,+)}&\mathcal{M}_{(-,-,+,-)}\\\
\mathcal{M}_{(+,+,-,+)}&\mathcal{M}_{(+,+,-,-)}&\mathcal{M}_{(-,+,-,+)}&\mathcal{M}_{(-,+,-,-)}\\\
\mathcal{M}_{(+,+,+,+)}&\mathcal{M}_{(+,+,+,-)}&\mathcal{M}_{(-,+,+,+)}&\mathcal{M}_{(-,+,+,-)}\end{pmatrix},$
(100)

where we labeled the helicities of the gravitons using an all-incoming
convention.

Consider now scattering in the physical $t$-channel $14\to 23$. To describe
this situation we consider the center-of-mass frame and choose helicity
spinors as follows Arkani-Hamed:2017jhn ; Arkani-Hamed:2020blm

$\displaystyle\lambda_{1}$ $\displaystyle=t^{1/4}\begin{pmatrix}1\\\
0\end{pmatrix},~{}~{}~{}\lambda_{4}=t^{1/4}\begin{pmatrix}0\\\
1\end{pmatrix},$ $\displaystyle\lambda_{2}$
$\displaystyle=it^{1/4}\begin{pmatrix}\cos{\theta\over 2}\\\ \sin{\theta\over
2}\end{pmatrix},~{}~{}~{}\lambda_{3}=it^{1/4}\begin{pmatrix}\sin{\theta\over
2}\\\ -\cos{\theta\over 2}\end{pmatrix}.$ (101)

Since we consider particles $1$ and $4$ to be incoming we take
$\tilde{\lambda}_{1,4}=\lambda_{1,4}^{*}$. Particles $2$ and $3$ are outgoing,
therefore $\tilde{\lambda}_{2,3}=-\lambda_{2,3}^{*}$. With this choice we get
$t=\langle 14\rangle[14]$, $s=\langle 12\rangle[12]=-t\sin^{2}{\theta\over
2}$, so that $\cos\theta=1+{2s\over t}$. Evaluating the matrix (100) for this
kinematics we obtain

$\displaystyle\mathcal{M}(s,t)\equiv\begin{pmatrix}t^{4}f(s,u)&s^{2}t^{2}u^{2}g^{*}(s,u)&s^{2}t^{2}u^{2}g^{*}(s,u)&h^{*}(s,u)\\\
s^{2}t^{2}u^{2}g(s,u)&u^{4}f(s,t)&s^{4}f(t,u)&s^{2}t^{2}u^{2}g^{*}(s,u)\\\
s^{2}t^{2}u^{2}g(s,u)&s^{4}f(t,u)&u^{4}f(s,t)&s^{2}t^{2}u^{2}g^{*}(s,u)\\\
h(s,u)&s^{2}t^{2}u^{2}g(s,u)&s^{2}t^{2}u^{2}g(s,u)&t^{4}f(s,u)\end{pmatrix}.$
(102)

Unitarity restricts the form of the discontinuity of various amplitudes. We
introduce the $t$-channel discontinuity as

$\displaystyle{\rm
Disc}_{t}\mathcal{M}(s,t)\equiv{\mathcal{M}(s,t+i\epsilon)-\mathcal{M}(s,t-i\epsilon)\over
2i}\,.$ (103)

Through the optical theorem or unitarity, the discontinuity (103) is related
to the square of the $2\to n$ amplitude where we insert a complete set of
intermediate states. It is convenient to decompose intermediate states into
the irreducible representation of the Poincare group, which are therefore
labeled by the total energy $\sqrt{t}$, spin $J$ and potentially other quantum
numbers $i$. The simplest example is when we have an exchange by a single
particle of mass $\sqrt{t}$ and spin $J$. In this case (103) produces the
square of the corresponding three-point couplings multiplied by a kinematical
polynomial.121212In the case of external scalars these are familiar Legendre
polynomials. For spinning particles the analogous polynomials in $D=4$ are
Wigner d-functions, which are written explicitly below. A slightly more
general situation is when we have an exchange by multiple particles of the
same mass and spin. In this case the discontinuity produces (103) a sum over
the products of the corresponding three-point couplings. Finally, we can also
have a multi-particle state as an intermediate state of total energy
$\sqrt{t}$ and spin $J$. It is convenient to think about it again as a single-
particle state with the continuous label for the species (which corresponds to
the distribution of the total energy among the constituent particles). The
result is always the same: we can write Eq. (103) as a sum of kinematical
polynomials multiplied by various spectral densities $\rho_{J}(t)$ which
encode the sum over the products of couplings to the intermediate states of
energy $\sqrt{t}$ and $J$ in a given theory. What we just said is simply a
restatement of the standard partial-wave expansion in a language which is
perhaps slightly more intuitive. For a more complete and detailed derivation
see Refs. Hebbar:2020ukp ; KelianMaster . In this way we can write the
discontinuity in terms of the spectral density. For example,

${\rm
Disc}_{t}\mathcal{M}_{(+,-,-,+)}(s,t)=\sum_{J=0}^{\infty}{1+(-1)^{J}\over
2}\rho_{J}^{++}(t)d_{0,0}^{J}(x)\,.$ (104)

Applying this to all helicity configurations in the matrix of amplitudes (102)
we get

$\displaystyle{\rm Disc}_{t}\mathcal{M}(s,t)$
$\displaystyle=\sum_{J=0}^{\infty}{1+(-1)^{J}\over
2}\begin{pmatrix}\rho_{J}^{++}(t)d_{0,0}^{J}(x)&0&0&\left[\tilde{\rho}_{J}^{++}(t)\right]^{*}d_{0,0}^{J}(x)\\\
0&0&0&0\\\ 0&0&0&0\\\
\tilde{\rho}_{J}^{++}(t)d_{0,0}^{J}(x)&0&0&\rho_{J}^{++}(t)d_{0,0}^{J}(x)\end{pmatrix}$
$\displaystyle+\sum_{J=4}^{\infty}\begin{pmatrix}0&0&0&0\\\
0&\rho_{J}^{+-}(t)d_{4,4}^{J}(x)&\rho_{J}^{+-}(t)(-1)^{J}d_{4,-4}^{J}(x)&0\\\
0&\rho_{J}^{+-}(t)(-1)^{J}d_{4,-4}^{J}(x)&\rho_{J}^{+-}(t)d_{4,4}^{J}(x)&0\\\
0&0&0&0\end{pmatrix}$ $\displaystyle+\sum_{J=4}^{\infty}{1+(-1)^{J}\over
2}\begin{pmatrix}0&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)&0\\\
\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&0&0&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)\\\
\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&0&0&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)\\\
0&\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&0\end{pmatrix},$
(105)

where we introduced $x\equiv\cos\theta=1+{2s\over t}$ and we used Wigner
d-matrices $d_{\lambda,\lambda^{\prime}}^{J}(x)$ which we list explicitly in
Appendix E. The formula above can be derived by explicitly analyzing the
effect of an exchange by a particle, or equivalently an irreducible
representation, of given mass $\sqrt{t}$ and spin $J$. The relevant three-
point amplitudes are fixed up to a number and their products are encoded in
the various spectral densities $\rho_{J}(m^{2})$. The indices $++$ and $+-$
denote the helicities of the corresponding incoming gravitons.

To discuss dispersion relations we also need to understand the properties of
the $u$-channel discontinuity which is defined as

$\displaystyle{\rm
Disc}_{u}\mathcal{M}(s,t)\equiv{\mathcal{M}(s,-s-t-i\epsilon)-\mathcal{M}(s,-s-t+i\epsilon)\over
2i}.$ (106)

By the same argument it takes the following form

$\displaystyle{\rm Disc}_{u}\mathcal{M}(s,t)$
$\displaystyle=\sum_{J=0}^{\infty}{1+(-1)^{J}\over
2}\begin{pmatrix}0&0&0&\left[\tilde{\rho}_{J}^{++}(t)\right]^{*}d_{0,0}^{J}(x)\\\
0&\rho_{J}^{++}(t)d_{0,0}^{J}(x)&0&0\\\
0&0&\rho_{J}^{++}(t)d_{0,0}^{J}(x)&0\\\
\tilde{\rho}_{J}^{++}(t)d_{0,0}^{J}(x)&0&0&0\end{pmatrix}$
$\displaystyle+\sum_{J=4}^{\infty}\begin{pmatrix}\rho_{J}^{+-}(t)d_{4,4}^{J}(x)&0&0&0\\\
0&0&\rho_{J}^{+-}(t)(-1)^{J}d_{4,-4}^{J}(x)&0\\\
0&\rho_{J}^{+-}(t)(-1)^{J}d_{4,-4}^{J}(x)&0&0\\\
0&0&0&\rho_{J}^{+-}(t)d_{4,4}^{J}(x)\end{pmatrix}$
$\displaystyle+\sum_{J=4}^{\infty}{1+(-1)^{J}\over
2}\begin{pmatrix}0&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)&0\\\
\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&0&0&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)\\\
\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&0&0&[\tilde{\rho}_{J}^{+-}(t)]^{*}d_{4,0}^{J}(x)\\\
0&\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&\tilde{\rho}_{J}^{+-}(t)d_{4,0}^{J}(x)&0\end{pmatrix},$
(107)

where the only effect of crossing is the different form of the diagonal matrix
elements.

The spectral densities that enter the unitarity relation are given in terms of
the product of the couplings to the corresponding intermediate states. The
diagonal terms being given by the absolute value square of the couplings are
nonnegative

$\displaystyle\rho_{J}^{++}(t)$ $\displaystyle\geq
0,~{}~{}~{}\rho_{J}^{+-}(t)\geq 0\,.$ (108)

The off-diagonal terms satisfy simple Cauchy-Schwartz inequalities131313In
terms of couplings to the intermediate state of energy $\sqrt{t}$ and spin $J$
these inequalities simply state that
$\left|\sum_{i}\lambda_{++i}\lambda_{+-i}^{*}\right|^{2}\leq(\sum_{i}|\lambda_{++i}|^{2})(\sum_{i}|\lambda_{+-i}|^{2})$
and $|\sum_{i}\lambda_{++i}^{2}|\leq\sum_{i}|\lambda_{++i}|^{2}$ respectively.

$\displaystyle|\tilde{\rho}_{J}^{+-}(t)|^{2}$
$\displaystyle\leq\rho_{J}^{++}(t)\rho_{J}^{+-}(t),~{}~{}~{}|\tilde{\rho}_{J}^{++}(t)|\leq\rho_{J}^{++}(t)\,.$
(109)

Below we discuss bounds on the possible form of the low-energy coefficients
stemming from unitarity and growth of the amplitude at infinity. In
particular, it is convenient for us to consider combinations of amplitudes
that have nonnegative semi-definite discontinuities both in the $t$\- and
$u$-channel. We consider the following convenient choices that are sufficient
for our purposes

$\displaystyle\mathcal{M}_{h}(s,t)$
$\displaystyle\equiv\begin{pmatrix}t^{4}f(s,u)+u^{4}f(s,t)-s^{4}f(t,u)&h^{*}(s,u)\\\
h(s,u)&t^{4}f(s,u)+u^{4}f(s,t)-s^{4}f(t,u)\end{pmatrix},$ (110)
$\displaystyle\mathcal{M}_{g}(s,t)$
$\displaystyle\equiv\begin{pmatrix}t^{4}f(s,u)+u^{4}f(s,t)&2s^{2}t^{2}u^{2}g^{*}(s,u)\\\
2s^{2}t^{2}u^{2}g(s,u)&t^{4}f(s,u)+u^{4}f(s,t)\end{pmatrix}.$ (111)

These matrices are $t$–$u$ crossing symmetric

$\displaystyle\mathcal{M}_{h,g}(s,t)=\mathcal{M}_{h,g}(s,u)\,.$ (112)

Using the form of the discontinuity dictated by unitarity above together with
the Cauchy-Schwartz inequalities (109) one can check that

$\displaystyle\partial_{s}^{n}{\rm
Disc}_{t}\mathcal{M}_{h,g}|_{s=0}=\partial_{s}^{n}{\rm
Disc}_{u}\mathcal{M}_{h,g}|_{s=0}\succeq 0,~{}~{}~{}~{}~{}n\geq 0\,,$ (113)

where the notation ‘$\succeq 0$’ means that the matrix is positive semi-
definite.141414A hermitian $n\times n$ matrix $M$ is called positive semi-
definite if $z^{*}Mz\geq 0$ for all $z\in\mathbb{C}^{n}$. We obtain these
inequalities from the following properties of the Wigner d-matrices together
with (109)

$\displaystyle\partial_{s}^{n}d_{0,0}^{J}(x)|_{s=0}\geq 0\,,$ (114)
$\displaystyle\partial_{s}^{n}\left(d_{4,4}^{J}(x)-(-1)^{J}d_{4,-4}^{J}(x)\right)|_{s=0}\geq
0\,,$ (115)
$\displaystyle\partial_{s}^{n}\left(\rho_{J}^{++}(t)d_{0,0}^{J}(x)+\rho_{J}^{+-}(t)d_{4,4}^{J}(x)-2|\tilde{\rho}_{J}^{+-}(t)|d_{4,0}^{J}(x)\right)|_{s=0}\geq
0\,,$ (116)

where $s=0$ corresponds to $x=1$ (see definition of $x$ below Eq. (105)). The
first property (114) is a well-known property of Legendre polynomials. We have
not attempted to prove (115) and (116) that rely on the properties of the
relevant $d_{\lambda,\lambda^{\prime}}^{J}(x)$, however we checked them
explicitly up to $J=30$. We also checked using formulas from Appendix E the
conditions above for any $J$ and $n=0,1,2$; these are the cases that we
consider below in more detail. For forward scattering a closely related
discussion can be found in Ref. Trott:2020ebl .

The matrices above admit simple eigenvectors in the parity-preserving case,
where we have $g(s,u)=g^{*}(s,u)$ and $h(s,u)=h^{*}(s,u)$. In this case the
matrices above have eigenvalues

$\displaystyle\mathcal{M}_{h,\pm}(s,t)$
$\displaystyle=u^{4}f(s,t)+t^{4}f(s,u)-s^{4}f(t,u)\pm h(s,u)\,,$
$\displaystyle\mathcal{M}_{g,\pm}(s,t)$
$\displaystyle=u^{4}f(s,t)+t^{4}f(s,u)\pm 2s^{2}t^{2}u^{2}g(s,u)\,.$ (117)

Unitarity (113) then becomes the statement about nonnegativity of the
discontinuity of Eq. (IV.2). In this case we can apply dispersion relations
directly to the functions (IV.2).

###  IV.3 Causality

Causality and unitarity put constraints on the high-energy behavior of the
amplitude. The corresponding bounds are well known in the case of gapped QFTs
Martin:1965jj , but are on a less-rigorous footing in gravitational theories.

At tree level the gravitational amplitudes are expected to satisfy

$\displaystyle\lim_{|t|\to\infty}|\mathcal{M}_{\text{tree}}(s,t)|\leq
t^{2}\,,\hskip 42.67912pts<0\,.$ (118)

This result is very intuitive but hard to establish rigorously. It naturally
emerges from various considerations Camanho:2014apa ; Maldacena:2015waa ;
Chandorkar:2021viw .

The situation is less clear nonperturbatively, however the simple qualitative
picture of high-energy scattering in gravity together with unitarity and
causality again imply that a similar bound exists. The bound is usually
assumed to be

$\displaystyle\lim_{|t|\to\infty}|\mathcal{M}_{\text{full}}(s,t)|<t^{2}\,,\hskip
42.67912pts<0\,.$ (119)

This can be used to write dispersion relations in $D\geq 5$ where the
amplitudes are IR finite and make sense nonperturbatively. In $D=4$ the
situation is less clear due to the IR divergences, but presumably a similar
bound exists for the IR safe observables. It is also possible to satisfy (119)
at tree level but this requires an infinite number of particles of arbitrary
high spin to be exchanged. A famous example of this type is given by the tree-
level string amplitudes.

In this paper, we are interested in particular in the properties of the one-
loop scattering amplitudes in $D=4$. These do not have to satisfy (118) or
(119), therefore we do not use them. Indeed, let us imagine that the bound
(118) is saturated at tree-level. This corresponds to exchange of a particle
of spin $2$. When we go to one-loop we can exchange a pair of spin-$2$
particles and the resulting amplitude will grow like $t^{3}$. Therefore it is
natural that the following bound holds at one loop

$\displaystyle\lim_{|t|\to\infty}|\mathcal{M}_{1-\text{loop}}(s,t)|\leq
t^{3},\hskip 42.67912pts<0\,.$ (120)

The explicit amplitudes computed in the paper indeed satisfy Eq. (120). Below,
in writing dispersion relations we assume the Regge bound to be as given in
Eq. (120) and choose the number of subtractions accordingly. Note that since
Eq. (120) is weaker than Eq. (118), bounds derived in this way apply both to
the tree and one-loop amplitudes. They also apply to full nonperturbative
amplitudes in $D\geq 5$.

It is a special feature of gravitational amplitudes that an infinite number of
particles has to be exchanged at tree-level in order for the amplitudes to
satisfy (118). This follows from the fact that such exchanges at tree-level
correspond to non-minimal couplings, which are known to lead to violations of
Eq. (118) Camanho:2014apa ; Belin:2019mnx ; Simons-Duffin-Zhiboedov . The only
exception to this rule is general relativity, which corresponds to the minimal
self-coupling of the graviton.

The tree-level Regge growth bound (118) is the reason why it is so hard to
construct tree-level gravitational amplitudes different from the ones of
general relativity. This famous achievement of string theory leads to causal
and unitarity amplitudes with infinitely many poles that corresponds to
exchanges of infinitely many particles of arbitrarily large spin. In fact,
every such modification must contain strings Caron-Huot:2016icg
.151515Curiously, as explained in some detail in Appendix D, if we allow for
accumulation points in the spectrum we can consider much simpler amplitude
functions. These functions, however, contain an infinite number of particles
of a given mass.

###  IV.4 Dispersive sum rules

Given the gravitational amplitudes that satisfy unitarity and the Regge bounds
we can consider various dispersion relations. One class of dispersion
relations that we find useful recasts the vanishing of the (subtracted)
amplitude in the Regge limit in terms of its discontinuity.

Such relations are known as superconvergence relations Simons-Duffin-Zhiboedov
, or dispersive sum rules Caron-Huot:2020adz . We follow the latter
terminology and consider the following integrals

$\displaystyle B_{k}^{+}(s)$ $\displaystyle=\oint_{\infty}{dt\over 2\pi
i}\mathcal{M}(s,t){1\over t}{1\over(t(s+t))^{k}}=0\,,\hskip 42.67912ptk\geq
2\,,$ (121)

where the condition $k\geq 2$ originates from the Regge bound (120) and
guarantees that the arc at infinity produces a vanishing contribution.

By deforming the contour we get the following formula

$\displaystyle B_{k}^{+}(s)$ $\displaystyle:\oint_{t_{0}}{dt\over 2\pi
i}\mathcal{M}(s,t){1\over t}{1\over(t(s+t))^{k}}$ (122) $\displaystyle\hskip
56.9055pt\hbox{}=\int_{m_{\rm
gap}^{2}}^{\infty}{dt\over\pi}{1\over(t(s+t))^{k}}\left({1\over t}{\rm
Disc}_{t}\mathcal{M}(s,t)+{1\over s+t}{\rm Disc}_{u}\mathcal{M}(s,t)\right),$

where the integrals are depicted in Fig. 7.

Figure 7: The vanishing of the arc integral at infinity on the left panel, cf.
(121), can be restated as an equality of the integrals on the right panel (see
(122).)

We evaluate the LHS of Eq. (122) using the low-energy expansion, and we use
the RHS to establish some nontrivial properties that the coefficients of the
low-energy expansion should satisfy. More precisely, we consider the expansion
of (122) around $s=0$.

A few comments are in order. Firstly, note that the minimal tree-level
graviton-exchange term in $f(s,u)\sim{1\over stu}$ does not contribute to the
sum rules of interest here, $B_{k\geq 2}^{+}(s)$. In particular, we are able
to expand the sum rules in powers of $s$. Assuming a more stringent Regge
bound, $B^{+}_{1}(s)$ can be studied Caron-Huot:2021rmr and used to bound
various coefficients in terms of $G$. We do not consider this sum rule in the
present paper. Secondly, the non-minimal coupling $\beta_{R^{3}}$ contributes
to $B_{2}^{+}$ and we utilize this fact in Sect. VI.2 in order to bound
$\beta_{R^{3}}$ in terms of other EFT data.

###  IV.5 The theory islands

What is the space of gravitational EFTs that admit a consistent UV completion?
Answering this question nonperturbatively is a formidable task, but we can
consider the simpler question of understanding perturbatively in $G$ possible
consistent UV completions of gravity. It is in this spirit that in this paper
we focus on the leading-order-in-$G$ correction to general relativity.

Imagine that we label all possible perturbatively consistent theories of
gravity by an index $i$ so that the four-graviton scattering amplitudes in a
given theory are given by the matrix $\mathcal{M}_{i}$ (see Eq.
(100)).161616For simplicity we keep the index discrete, but of course it can
be continuous, e.g. denoting the mass of the particle exchanged in the loop.
It does not affect the discussion. It is clear then that by taking a
superposition of such amplitudes with non-negative coefficients we again get
an amplitude that satisfies all the constraints discussed in the sections
above

$\displaystyle\mathcal{M}_{\text{theory}}=\sum_{i}c_{i}\mathcal{M}_{i},~{}~{}~{}~{}~{}c_{i}\geq
0\,.$ (123)

By expanding such amplitudes at low energies, as in Sect. IV.1, we get a set
of higher-derivative corrections. The low-energy Wilson coefficients then form
a convex cone which we can think of as the theory space,171717A convex cone is
a subset of vector space closed under linear combinations with positive
coefficients. which is generated by vectors
$\mathcal{M}_{i}$.181818Nonperturbatively in $G$, Eq.(123) is not necessarily
consistent since is can violate nonperturbative unitarity $|S_{J}|\leq 1$.
Since we are interested in the leading-order correction in the regime when
gravity is weakly coupled, this issue is irrelevant.

It has been recently established that causality, crossing and unitarity
constrain Wilson coefficients both from above and from below Tolley:2020gtv ;
Caron-Huot:2020cmc ; Arkani-Hamed:2020blm ; Sinha:2020win . In the space of
couplings formed by weakly-coupled theories it is thus natural to talk about
the theory island, namely the region spanned by known perturbative UV
completions of gravity. By definition we have

$\displaystyle\text{Theory island}\subseteq\text{EFThedron},$ (124)

where by the EFThedron we call the set of amplitudes satisfying the
constraints reviewed in this section (in particular we only impose
perturbative unitarity). It is interesting to ask to what extent we can
populate the space of allowed couplings by known perturbative UV completions.
Recall that by a perturbative UV completion we call an $S$-matrix that
satisfies unitarity, causality and crossing perturbatively in Newton’s
constant $G$ and for any process $m\to n$.

There are two classes of perturbative UV completions we consider here: tree-
level corrected theories and one-loop corrected theories. In tree-level
corrected theories the leading-order correction to general relativity enters
at tree-level, in other words the higher-derivative corrections in the
Lagrangian are suppressed by a new scale, e.g. string scale, which can be much
lower than the Planck scale. Perturbative string theories are famous examples
of this type. It is a well-known fact that gravitational amplitudes in string
theory have a great degree of universality, as discussed recently in Ref.
Chowdhury:2019kaq . In Appendix B we review the cases of superstring,
heterotic and bosonic strings at tree level.191919Strictly speaking the
bosonic string is not part of the theory space due to the presence of the
tachyon in the spectrum, but it is useful to keep it to check various
formulas. Alternatively, we can consider theories where the higher-derivative
operators come with an extra power of $G$. The one-loop amplitudes computed in
the present paper are precisely of this type. By choosing the mass spectrum
and spins of the particles propagating in the loop we can get various
amplitudes.

Combining these two examples we consider the following set of amplitudes in
the present paper