11institutetext: Mani L. Bhaumik Institute for Theoretical Physics,
UCLA Department of Physics and Astronomy, Los Angeles, CA 90095,
USA22institutetext: Institute for Gravitation and the Cosmos,
Pennsylvania State University, University Park, PA 16802, USA

# Effective Field Theory Islands from Perturbative and Nonperturbative Four-
Graviton Amplitudes

Zvi Bern<EMAIL_ADDRESS>1 Enrico Herrmann<EMAIL_ADDRESS>1 Dimitrios
Kosmopoulos<EMAIL_ADDRESS>2 Radu Roiban<EMAIL_ADDRESS>
###### Abstract

Theoretical data obtained from physically sensible field and string theory
models suggest that gravitational Effective Field Theories (EFTs) live on
islands that are tiny compared to current general bounds determined from
unitarity, causality, crossing symmetry, and a good high-energy behavior. In
this work, we present explicit perturbative and nonperturbative $2\to 2$
graviton scattering amplitudes and their associated low-energy expansion in
spacetime dimensions $D\geq 4$ to support this notion. Our new results include
a first nonperturbative example consisting of a $D=4$, $\mathcal{N}=1$
supersymmetric field theory that is coupled weakly to gravity. We show that
this nonperturbative model lies on the same islands identified using four-
dimensional perturbative models based on string theory and minimally-coupled
matter circulating a loop. Furthermore, we generalize the previous four-
dimensional perturbative models based on string theory and minimally-coupled
massive spin-0 and spin-1 states circulating in the loop to $D$ dimensions.
Remarkably, we again find that the low-energy EFT coefficients lie on small
islands. These results offer a useful guide towards constraining possible
extensions of Einstein gravity.

##  1 Introduction

The language of effective field theory (EFT) is a widely accepted framework in
which to formulate the physical laws at a certain energy scale (often referred
to as the infrared (IR) scale). Typically, this language is used in the
context of current and near-future high-energy physics experiments
Buchmuller:2001dc ; Brivio:2017vri , but has also found applications in a
variety of topics including hydrodynamics Dubovsky:2011sj , inflation
Cheung:2007st , the large scale structure of the Universe Carrasco:2012cv ,
and the description of binary motion in general relativity Goldberger:2004jt ,
among many others. In EFTs, the relevant physics is parameterized by
independent local operators that capture all relevant physical degrees of
freedom and are consistent with the known symmetries of the problem. Examples
of such symmetries include Lorentz invariance and possibly gauge or global
symmetries. The unknown physics at high energy or ultraviolet (UV) physics is
then systematically parameterized by successively including higher-dimension
operators that capture corrections to low-energy observables. Naively, the
(Wilson) coefficients of such higher-dimension operators are undetermined and
can take on arbitrary values. However, desirable properties of the underlying
theory such as causality (analyticity) and unitarity impose nontrivial
constraints or bounds on the allowed values of the low-energy couplings
Adams:2006sv ; Camanho:2014apa . A way to expose these bounds is to study the
connection of the $2\to 2$ scattering amplitude in the IR and the UV by means
of _dispersion relations_ which relate low-energy Wilson coefficients to the
discontinuities of the UV amplitude by a contour deformation subject to
certain assumptions about the Regge growth of amplitudes at large energies in
the complex plane, see e.g. Refs. Froissart:1961ux ; Martin:1962rt ;
Maldacena:2015waa ; Haring:2022cyf . In recent years this basic philosophy has
been systematized to extract various nontrivial constraints Bellazzini:2020cot
; Arkani-Hamed:2020blm ; Bern:2021ppb ; Caron-Huot:2021enk ; Caron-
Huot:2022ugt ; Chiang:2022jep ; Chiang:2022ltp .

It is critical to understand the full implications of these constraints and
whether sensible physical theories must necessarily lie in small regions of
the EFT parameter space. Ref. Bern:2021ppb  observed that the Wilson
coefficients of two distinct classes of gravitational effective field theories
derived from models of UV physics populate small _theory islands_ in the
larger space allowed by current dispersive arguments. The first class are
string theories, which are ultraviolet complete, and the second class comes
from integrating out minimally coupled matter circulating in loops. In both
instances gravity is assumed to be weakly coupled so that only the leading
order contributions are required, corresponding to tree level in string theory
and one loop in the field-theory models. While the field-theory models are not
full UV completions, they can be interpreted as intermediate-scale theories,
which satisfy all the assumptions used to derive bounds on the EFT
coefficients. The fact that such dissimilar models land on the same small
theory island suggests that sensible theories should obey much stronger
constraints than have been found as yet from the general arguments. The
observed small islands were interpreted as being related to _low spin
dominance_ —essentially the property that the spectral density in these models
is dominated by the lowest spin partial waves.

While suggestive, an obvious question is whether the appearance of small
islands is an artifact of the special theories that were considered or whether
they are generic for physically sensible theories. Here we provide evidence
towards the latter by obtaining data from two new classes of theories. The
first is a nonperturbative strongly coupled $\mathcal{N}=1$ supersymmetric
gauge theory which is then weakly coupled to gravity and the second is matter
minimally coupled to gravity in $D>4$ spacetime dimensions. Specifically, we
present explicit results in $D=6,10$ dimensions with further data and
evaluation routines available in the ancillary files. The well known string-
theory amplitudes in $D>4$ provide a third class of EFT data. We use this data
to support the notion that small theory islands are not a special feature of
$D=4$ perturbative examples, but indeed generalize beyond the cases analyzed
in Ref. Bern:2021ppb . It remains a challenge to find the tightest bounds that
physically sensible EFTs must satisfy. Some recent progress on improving
bounds is found in Refs. Caron-Huot:2022ugt ; Chiang:2022jep .

One of the key lessons of the modern scattering amplitudes program is to focus
on gauge- and field-redefinition-invariant quantities. In this spirit, as in
Refs. Bellazzini:2020cot ; Arkani-Hamed:2020blm ; Bern:2021ppb ; Caron-
Huot:2021enk ; Caron-Huot:2022ugt , we focus on the low-energy expansions of
scattering amplitudes directly, rather than Wilson coefficients in a
Lagrangian that are subject to field-redefinition and integration-by-parts
ambiguities. Assuming that gravity couples weakly, we can work to tree-level
accuracy in the EFT. There is then a one-to-one map between S-matrix elements
in the IR and Wilson coefficients in any given basis of operators, see e.g.
Shadmi:2018xan . In this way, the low-energy amplitude can be schematically
expanded in the form111Below, we denote amplitude coefficients by their
monomial term, e.g. $a[s^{k-q}t^{q}]$. For the sake of compactness, here we
simply use $a_{k,q}$ which will have a different meaning for a particular $4D$
helicity amplitude.

$\displaystyle\mathcal{M}_{{\rm IR}}(s,t)\sim\text{light exchange}+\sum_{k\geq
q\geq 0}a_{k,q}\ s^{k-q}t^{q}\,,$ (1)

where the Mandelstam invariants are $s=(p_{1}+p_{2})^{2}$,
$t=(p_{1}+p_{4})^{2}$, and $u=(p_{1}+p_{3})^{2}$. As usual, for massless
external states, they satisfy the relation $u=-s-t$. The terms denoted by
“light exchange” correspond to low-energy _poles_ from massless or light
(relative to the scale of Mandelstam invariants) exchange of states that are
within the low-energy EFT. Finally, the $a_{k,q}$ parameterize new four-point
contact interactions graded by mass dimension $k$.

In the ultraviolet, it is convenient to parameterize the unknown physics in
terms of the partial-wave expansion of the amplitude, involving the spectral
density and some characteristic polynomial of the scattering angle
$\cos\theta=1+2t/s$ (for massless external states in the $s$-channel center of
mass) that encodes the Poincaré-invariance properties akin to conformal
partial waves in conformal-field-theory (CFT) correlation functions
Ferrara:1972kab ; Ferrara:1974ny ; Dolan:2000ut ; Dolan:2003hv . For external
scalars, these are the Gegenbauer polynomials (see e.g. Arkani-Hamed:2020blm )
in general $D$ and the Wigner-$d$ matrices for spinning external states in
$D=4$, see e.g. Refs. Itzykson:1980rh ; Hebbar:2020ukp .

The simplest incarnation of the bounds on the coefficients $a_{k,q}$ in Eq.
(1) is relatively easy to understand. In the presence of some elastic channel
where the ‘out’ state is the same as the ‘in’ state, in the forward limit
(i.e. $t\rightarrow 0$) the discontinuity of the amplitude becomes an absolute
square which then implies positivity constraints on EFT amplitude coefficients
Adams:2006sv :

$\displaystyle a_{k,0}\sim\langle{\rm in}|T^{\dagger}T|{\rm
in}\rangle=\big{|}T|{\rm in}\rangle\big{|}^{2}\geq 0\ .$ (2)

Such bounds have first appeared in the context of chiral Lagrangians and pion
scattering Pham:1985cr ; Ananthanarayan:1994hf ; Pennington:1994kc , before
experiencing a revival inspired by the seminal works of Refs. Adams:2006sv ;
Camanho:2014apa . Recently, similar bounds Nicolis:2009qm ; Bellazzini:2015cra
; deRham:2017avq ; deRham:2017zjm ; Bellazzini:2020cot ; Sinha:2020win ;
Chowdhury:2021ynh ; Bellazzini:2021oaj  were organized into a novel geometric
structure termed the EFT-hedron Arkani-Hamed:2020blm  (see also Ref.
Chiang:2021ziz ; Chiang:2022jep ; Bern:2021ppb ), related to the Weak Gravity
Conjecture Cheung:2014ega ; Cheung:2014vva ; Arkani-Hamed:2021ajd , the
analytic bootstrap in AdS/CFT Caron-Huot:2020adz ; Caron-Huot:2021enk , and
applied to the Standard Model EFT and pion scattering Distler:2006if ;
Manohar:2008tc ; Remmen:2019cyz ; Remmen:2020uze . Furthermore, these bounds
were refined away from the forward limit Caron-Huot:2020cmc ; Caron-
Huot:2021rmr ; Caron-Huot:2022ugt  in order to handle cases with gravitational
couplings where the $t$-channel graviton exchange causes difficulties with
some of the naive forward limit bounds. Cases with different external helicity
configurations, which individually cannot be considered as elastic scattering,
were also considered in Refs. Bern:2021ppb ; Caron-Huot:2022ugt ;
Chiang:2022jep , in a spirit similar to Refs. Cheung:2016yqr ; Zhang:2020jyn ;
Li:2021lpe . A key feature, common to all presently known bounds is the
appearance of the demarcation of allowed and disallowed regions in the space
of low-energy couplings $a_{k,q}$.

As already noted above, in previous four-dimensional studies, explicit string-
and field-theory data suggest that physical EFTs live on small theory islands
Bern:2021ppb . In contrast to the four-dimensional case, and for the
$D$-dimensional scattering of scalar particles, the spinning partial-wave
decomposition that enter the UV part of the dispersion relations are not
presently analyzed for the scattering of $D$-dimensional spinning states (see
however our “note added” below and the upcoming work of Ref. Caron-
Huot:2022toAppear ). Nevertheless, independently of the availability of
precise bounded regions, we can ask where do explicit data lie in order to
guide further explorations. Here, we address the question of whether similar
islands are observed for more general models than the ones considered in
Bern:2021ppb . We do so by obtaining new explicit examples of UV models,
including nonperturbative matter and cases outside of four dimensions, from
which we extract the low-energy expansion coefficients for gravitational
scattering amplitudes. Our analysis further supports the notion that small
theory islands are a robust feature of gravitational EFTs. Our example of
nonperturbative matter in gravitational $2\to 2$ scattering opens up a new
class of possible theories to analyze in the future. Higher dimensions are
interesting for various reasons, including that they allow for analyses of
bounds that avoid complications with IR singularities Caron-Huot:2021rmr  and
because 10 dimensions is natural for addressing the question of where does
string theory lie in the space of possible UV completions Guerrieri:2021ivu .

The remainder of this paper is organized as follows: In section 2 we summarize
our kinematic conventions for $D$-dimensional graviton scattering including a
parametrization of the center-of-mass momenta and the polarization states of
the external gravitons. In section 3, we use supersymmetric arguments to
generate a first four-dimensional example of nonperturbative matter for
gravitational $2\to 2$ scattering. In section 4 we discuss the straightforward
case of tree-level four-graviton string-theory amplitudes in general spacetime
dimensions. Our results for one-loop minimally-coupled graviton amplitudes in
$D$ dimensions, from which we extract nontrivial EFT data, is presented in
section 5. We consider the cases of minimally-coupled massive spin-0 and
spin-1 matter circulating in the loop. We also briefly summarize the well-
known amplitudes techniques such as generalized unitarity and integration
tools used to evaluate and manipulate such expressions. In section 6 we give a
summary of our data by plotting some of the obtained low-energy amplitude
coefficients that should serve as a useful guide for any near-future attempts
to place dispersive bounds on higher-dimensional graviton scattering. We
supply the relevant amplitudes and their low-energy expansion in a computer-
readable form as ancillary files to this paper. We close with conclusions and
a future outlook in section 7.

##  2 External kinematics

To describe $2\to 2$ graviton scattering in $D$ dimensions we introduce
external momenta $p_{i}$, where $i=1,\ldots,4$ labels the external graviton in
question, and work in an all-incoming convention. To capture scattering of all
possible external states, we use formal polarization tensors for the gravitons

$\displaystyle\varepsilon^{\mu\nu}_{i}=\varepsilon^{\mu}_{i}\varepsilon^{\nu}_{i}\,,\hskip
5.69046pt\text{with }\hskip 5.69046ptp_{i}\cdot\varepsilon_{i}=0\,\hskip
5.69046pt\text{and }\hskip 5.69046pt\varepsilon_{i}\cdot\varepsilon_{i}=0\,.$
(3)

We express gravitational polarization tensors in terms of transverse, null
polarization vectors $\varepsilon^{\mu}_{i}$. Note that there are $D-2$ such
independent null vectors in $D$ spacetime dimensions, while there are
$D(D-3)/2$ independent symmetric traceless tensors. The expressions in terms
of the above tensors capture the entire space of states for the
$D$-dimensional gravitons. Indeed, a generic polarization tensor
$E^{\mu\nu}\equiv\varepsilon^{\mu}\tilde{\varepsilon}^{\nu}+\tilde{\varepsilon}^{\mu}\varepsilon^{\nu}$
may always be written in terms of linear combinations of factorized tensors,
e.g. Boels:2009bv ; Chowdhury:2019kaq

$\displaystyle\varepsilon^{\mu}\tilde{\varepsilon}^{\nu}+\tilde{\varepsilon}^{\mu}\varepsilon^{\nu}=(\varepsilon+\tilde{\varepsilon})^{\mu}(\varepsilon+\tilde{\varepsilon})^{\nu}-\varepsilon^{\mu}\varepsilon^{\nu}-\tilde{\varepsilon}^{\mu}\tilde{\varepsilon}^{\nu}\,,$
(4)

where we take $\varepsilon\cdot\tilde{\varepsilon}=0$ to ensure tracelessness.

To study specific examples, we introduce explicit momenta and polarization
tensors. We consider scattering in the center-of-mass frame, where

$\displaystyle p^{\mu}_{1}$
$\displaystyle=\sqrt{\frac{s}{2}}\begin{pmatrix}+1\\\ -1\\\ 0\\\
\vec{0}_{D{-}3}\end{pmatrix},\hskip 42.67912pt$ $\displaystyle p^{\mu}_{2}$
$\displaystyle=\sqrt{\frac{s}{2}}\begin{pmatrix}+1\\\ +1\\\ 0\\\
\vec{0}_{D{-}3}\end{pmatrix},$ $\displaystyle\hskip 42.67912ptp^{\mu}_{3}$
$\displaystyle=\sqrt{\frac{s}{2}}\begin{pmatrix}-1\\\ -\cos\theta\\\
-\sin\theta\\\ \vec{0}_{D{-}3}\end{pmatrix},$ $\displaystyle p^{\mu}_{4}$
$\displaystyle=\sqrt{\frac{s}{2}}\begin{pmatrix}-1\\\ \cos\theta\\\
\sin\theta\\\ \vec{0}_{D{-}3}\end{pmatrix},$ (5)

and the scattering angle $\theta$ is related to the Mandelstam invariants via
$\cos\theta=1+\frac{2t}{s}$, with $s>0$, and $-s<t<0$ for physical $s$-channel
scattering. In all examples analyzed in this paper, we consider external
polarization tensors of the factorized form in Eq. (3). Different cases can
also be obtained straightforwardly as explained above. Focusing on even $D$,
given a set of spatial unit vectors $e_{a}^{\mu}=\delta_{a}^{\mu}$, with
$a=1,\ldots,D$, we define (see e.g. Ref. Boels:2009bv )

$\displaystyle\varepsilon_{1,2n^{\pm}}^{\mu}=\frac{1}{\sqrt{2}}(e^{\mu}_{2n-1}\pm\mathrm{i}\,e^{\mu}_{2n})\,,\hskip
28.45274ptn=2,\ldots\frac{D}{2}\,.$ (6)

We obtain the polarization vectors for the other three gravitons using
appropriate rotations. More details on the polarization choices are included
in the ancillary files with explicit evaluation code for all Lorentz products
for graviton polarizations similar to the ones discussed here. Note that for
$D=4$ ($n=2$) the polarizations in Eq. (6) describe helicity states. Indeed,
for this choice our results reproduce the ones obtained in Ref. Bern:2021ppb
using spinor-helicity methods.

##  3 Non-perturbative data

To gain some direct indication on nonperturbative low-energy effective
actions, we consider a matter-coupled ${\cal N}=1$ supersymmetric gauge theory
which we couple to gravity. This gauge theory confines in flat space at some
scale $\Lambda$, whose specific relation to the high-energy couplings will not
be important. We assume that $\Lambda$ is relatively high and that the the
low-energy theory is described by the glueball superfield which is much
lighter than the confinement scale. This can be arranged by adjusting the
couplings of the high-energy theory. We then focus our discussion on energies
below the mass $m_{s}$ of the glueball superfield. Thus we can ignore terms
${\cal O}(p^{2}/\Lambda^{2})$, but we cannot ignore terms ${\cal
O}(p^{n}/m_{S}^{n})$.

The Wilsonian effective action below the confinement scale for an ${\cal N}=1$
supersymmetric gauge theory with (holomorphic) tree-level superpotential
$W_{\text{tree}}(\phi,Q,{\bar{Q}})$ depending on some chiral superfields
$\phi$ in the adjoint representation and other superfields $Q,{\bar{Q}}$ in
the fundamental representation, has the standard form

${\cal L}=\int d^{4}\theta\,K_{\text{eff}}(S,{\bar{S}},G,{\bar{G}})+\int
d^{2}\theta\,W_{\text{eff}}(S,G)+\text{h.c.}\,,$ (7)

where $S$ is the glueball (chiral) superfield, $G$ is the Weyl superfield and
$K_{\text{eff}}$ and $W_{\text{eff}}$ are the effective Kähler potential and
superpotential respectively. The latter is completely nonperturbative
Grisaru:1979wc , while the former receives both perturbative and
nonperturbative contributions. The Weyl superfield $G$, capturing the induced
coupling of the effective theory with gravity, carries left-handed spinor
indices and its first two components are the self-dual gravitino field
strength and the self-dual Riemann tensor respectively.

While the effective Kähler potential is largely unconstrained, the effective
superpotential is stringently constrained by symmetry and holomorphy arguments
and instanton calculations Seiberg:1994bz ; Intriligator:1994jr ;
Affleck:1983vc . Refs. Dijkgraaf:2002dh ; Dijkgraaf:2002fc  argued that the
effective superpotential has the form

$W_{\text{eff}}=\frac{\partial{\cal F}_{0}}{\partial
S}+\frac{\partial^{2}{\cal F}_{0}}{\partial
S^{2}}G^{\alpha\beta\gamma}G_{\alpha\beta\gamma}+{\cal
O}\big{(}(G^{\alpha\beta\gamma}G_{\alpha\beta\gamma})^{2}\big{)}\,,$ (8)

where ${\cal F}_{0}$ is a holomorphic function of the glueball superfield
$S$.222We do not include a detailed expression of the higher-order terms
because they contribute only to higher-point gravitational amplitudes, so we
do not need them. Given a tree-level superpotential $W_{\text{tree}}$, the
nonperturbative effective potential $W_{\text{eff}}$ can be computed
algorithmically, including its gravitational couplings, either via a symmetry
and holomorphy analysis Seiberg:1994bz ; Intriligator:1994jr , or through
matrix-model methods Dijkgraaf:2002dh ; Dijkgraaf:2002fc ; Bena:2002kw ;
Dijkgraaf:2002xd ; Dijkgraaf:2003sk ; Cachazo:2002ry ; Cachazo:2002zk ;
Cachazo:2003yc . For our purpose here we do not need its detailed form.

In the latter approach most terms in $W_{\text{eff}}$ are evaluated
perturbatively, and the glueball superfield $S$ enters initially in the form
of the gauge invariant bilinear $S\propto{\rm Tr}[W^{\alpha}W_{\alpha}]$,
where $W$ is the vector superfield, whose lowest component is the gluino. The
nonperturbative nature of the superpotential comes from interpreting this
superfield as a fundamental field, and from the inclusion in ${\cal F}_{0}$ of
the Veneziano-Yankielovicz superpotential Veneziano:1982ah , which accounts
for the chiral anomaly.

The critical point of the $G$-independent part of the effective
superpotential, which is a solution of $\partial^{2}{\cal F}_{0}/\partial
S^{2}=0$, fixes a vacuum expectation value, $S=S_{*}$, of the glueball
superfield. This breaks chiral symmetry and determines the non-normalized mass
of this superfield as ${\tilde{m}}_{S}=\partial^{3}{\cal F}_{0}/\partial
S^{3}|_{S=S_{*}}$.333The mass $m_{S}$ depends on the details of the Kähler
potential. On dimensional and holomorphy grounds one may expect that up to
numerical factors, $m_{S}\propto S_{*}^{4/3}{\tilde{m}}_{S}$. It also implies
that among the terms with two Weyl-superfields there exists a linear coupling
to $(S-S_{*})$, $m_{S}(S-S_{*})G^{\alpha\beta\gamma}G_{\alpha\beta\gamma}$,
which is also proportional to the mass of these fields. Evaluating the
integral over Grassmann variables in Eq. (8), integrating out the the
auxiliary fields and normalizing the quadratic term for $\varphi$ leads to

$\displaystyle\int d^{2}\theta
W_{\text{eff}}+h.c.=|{m}_{S}|^{2}\bar{\varphi}\varphi+M\,\varphi\,R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}+M^{*}\,{\bar{\varphi}}\,R_{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}R^{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}+\text{fermions}\
,$ (9)

where
$M={\tilde{m}}_{S}/(\partial_{S}\partial_{\bar{S}}K_{\text{eff}}|_{S=S_{*}})^{1/2}$,
$\varphi=(S-S_{*})|_{\theta=0}$ is the scalar in the glueball superfield and
$R^{\alpha\beta\gamma\delta}$ and
$R^{\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta}}$ are the self-dual and
anti-self-dual parts of the Riemann tensor, containing the negative- and
positive-helicity gravitons respectively. Thus, the nonperturbative
superpotential couples the fluctuation $\varphi$ of the glueball scalar around
its expectation value $S_{*}$ with two gravitons of the same helicity,

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Setting aside the exchange of gravitons, the leading-order terms in the four-
graviton amplitude come from tree-level exchange of $\varphi$. There are three
possible contributions: two vertices from the superpotential, two vertices
from the Kähler potential, and a contact term from the Kähler potential. Since
extracting gravitons out of the $G$ superfield requires only one Grassmann
derivative, it follows that the third field in a vertex with two gravitons is
acted upon by two Grassmann derivatives, so it must be the auxiliary field of
the glueball superfield. Thus, the Kähler potential can contribute only
contact terms at tree level. We graphically denote superpotential
contributions by circles and Kähler potential contributions by boxes. The
corresponding graphs are:

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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
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\par{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@rect{-5.69055pt}{-5.69055pt}{11.3811pt}{11.3811pt}\pgfsys@fillstroke\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\,.$ (11)

We first focus on the superpotential contributions and argue later that under
certain circumstances the Kähler potential contributions do not affect the
conclusions.

Four-graviton tree-level diagrams with vertices from the superpotential are
very simple. Eq. (9) implies that, for a fixed graviton helicity
configuration, the amplitude receives contributions from a single exchange
diagram. For example, if gravitons 1 and 2 have negative helicity and
gravitons 3 and 4 have positive helicity, the only diagram that contributes
is:

$\displaystyle\mathcal{M}(1^{-},2^{-},3^{+},4^{+})\quad\sim\vbox{\hbox{\scalebox{1.0}{
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This diagram depends on the scalar-field propagator, which in turn depends on
the Kähler potential.444A dimensional-analysis-based suggestion was put forth
in Ref. Veneziano:1982ah , $K(S)\propto({\bar{S}}S)^{1/3}$. The Källén-Lehmann
representation of massive two-point functions identifies this propagator as
some function $f$ of the momentum with a simple pole at $p^{2}=m_{S}^{2}$
where $m_{S}$ is the physical mass of the field $\varphi$. In particular, this
implies that it has a regular expansion around vanishing momentum, i.e.

$f(s)=\sum_{n\geq 0}a_{n}s^{n}\ .$ (13)

Then, the amplitude ${\cal M}(1^{-},2^{-},3^{+},4^{+})$ is given by

$\displaystyle{\cal M}(1^{-},2^{-},3^{+},4^{+})$ $\displaystyle={\langle
12\rangle^{4}[34]^{4}}\,f(s)\ ,$ (14)

with other helicity configurations, ${\cal M}(1^{-},2^{+},3^{+},4^{-})$ and
${\cal M}(1^{-},2^{+},3^{-},4^{+})$, obtained by relabeling. This implies that

$\displaystyle\frac{a_{n,j}}{a_{n,0}}=0\,,\ \text{ for all }j>0\,,$ (15)

which puts the nonperturbative contributions from the superpotential on the
small islands of physical EFTs and the low-spin-dominance lines in $D=4$
discussed in Ref. Bern:2021ppb .

Consider now briefly the consequences of including Kähler-potential
contributions to leading-order gravitational amplitudes. As we saw earlier,
these contributions can come only from contact terms which arise either from
terms of the type $\int d^{4}\theta f(S,{\bar{S}})G^{2}{\bar{G}}^{2}$ in
$K_{\text{eff}}$ in Eq. (7), or from terms with two Weyl multiplets in
$K_{\text{eff}}$ upon integrating out the auxiliary field in the glueball
superfield.

Nonperturbative contributions to $K_{\text{eff}}$ necessarily depend on the
confinement scale $\Lambda$; moreover, if they are dependent on momenta, then
dimensional analysis suggests that they depend on $p^{2}/\Lambda^{2}$ which,
according to our initial assumption, is negligible at the energy scale
$p^{2}\ll\Lambda^{2}$ that we are focusing on. Thus, the contributions of such
Kähler potential terms to amplitudes are essentially constant (up to the
helicity-dependent factor), and therefore they affect only a limited number of
$a_{n,j}$ coefficients, perhaps only $a_{0,0}$, leading to effectively no
changes to Eq. (15).

One may wonder if the expansion around the nonvanishing expectation value for
the glueball superfield $S$ may enhance these terms suppressed by the ratio
$p^{2}/\Lambda^{2}$. The expectation of a smooth limit in which the high-
energy theory is trivial (i.e. that the tree-level superpotential is zero)
suggests that the momentum dependence cannot be enhanced for small values of
the parameters of the high-energy theory. While this argument suggests that
the nonperturbative momentum dependence could be enhanced at large values of
these parameters, naively of the order of $\Lambda^{2}/p^{2}$, there remains a
comfortable range of parameters of the high-energy theory for which inclusion
of the purely nonperturbative terms in the Kähler potential does not affect
Eq. (15).

Extension of this discussion to the perturbative and mixed part of the Kähler
potential is difficult because of their detailed dependence on the parameters
of the high-energy theory. Using however the identity

$\displaystyle
x\leq\frac{a^{(1)}_{n,j}}{a^{(1)}_{n,0}},\,\frac{a^{(2)}_{n,j}}{a^{(2)}_{n,0}}\leq
y\quad\Longrightarrow\quad
x\leq\frac{a^{(1)}_{n,j}+a^{(2)}_{n,j}}{a^{(1)}_{n,0}+a^{(2)}_{n,0}}\leq y\ ,$
(16)

if all $a_{n,j}>0$ together with the results of Ref. Bern:2021ppb  and those
discussed in later sections, we expect that the complete nonperturbative four-
graviton amplitude in the class of theories discussed here belongs to the EFT
island for $p^{2}/\Lambda^{2}\ll 1$. Further study and explicit calculations
are necessary to fully settle this issue and to extend our analysis to scales
$p^{2}\lesssim\Lambda^{2}$ in which all nonperturbative contributions to the
Wilsonian effective action become important.

##  4 Tree-level graviton amplitudes in string theory

Having analyzed the first nonperturbative data for four-dimensional graviton
scattering via supersymmetric arguments, we now move on to more traditional
perturbative amplitudes, although beyond the commonly considered four-
dimensional setup. We expect that this data will provide useful guidance for
any attempts to place dispersive bounds on graviton scattering in higher
dimensions.

To this end, we collect the available tree-level results for the scattering of
four external gravitons in superstring (ss), heterotic-string (hs), and
bosonic-string (bs) theory. These closed-string amplitudes are determined by
Kawai-Lewellen-Tye (KLT) relations Kawai:1985xq  in terms of open-string ones.
We take the mass of the first excited string level to be
$m^{2}_{s}=4/\alpha^{\prime}$ for all string theories and express the string-
theory amplitudes in terms of gauge-invariant tensor structures $\mathcal{T}$.
We need two such structures,

$\displaystyle\begin{split}\mathcal{T}_{\text{sYM}}&=-s\,t\,(\varepsilon_{1}\cdot\varepsilon_{3})(\varepsilon_{2}\cdot\varepsilon_{2})+\cdots\,,\\\
\mathcal{T}_{\text{bos}}&=-\frac{s\,u}{(1+\alpha^{\prime}t/4)}(\varepsilon_{1}\cdot\varepsilon_{4})(\varepsilon_{2}\cdot\varepsilon_{3})+\cdots\,,\end{split}$
(17)

which are normalized to have mass-dimension four. The full expression for
these structures is provided in the ancillary file. We may now write the four-
graviton amplitudes in a form that uniformly applies to the three string
theories,

$\displaystyle\begin{split}\mathcal{M}^{(\beta)}={-}\left(\frac{\kappa}{2}\right)^{2}\\!\\!\left(\frac{\alpha^{\prime}}{4}\right)^{3}\\!\\!\mathcal{T}_{\beta}\,\frac{\Gamma\\!\left[{-}\frac{\alpha^{\prime}s}{4}\right]\Gamma\\!\left[{-}\frac{\alpha^{\prime}t}{4}\right]\Gamma\\!\left[{-}\frac{\alpha^{\prime}u}{4}\right]}{\Gamma\\!\left[1{+}\frac{\alpha^{\prime}s}{4}\right]\Gamma\\!\left[1{+}\frac{\alpha^{\prime}t}{4}\right]\Gamma\\!\left[1{+}\frac{\alpha^{\prime}u}{4}\right]},\end{split}$
(18)

where $\beta\in\\{\text{ss, hs, bs}\\}$ and

$\mathcal{T}_{\text{ss}}=\mathcal{T}^{2}_{\text{sYM}}\,,\qquad\mathcal{T}_{\text{hs}}=\mathcal{T}_{\text{sYM}}\,\mathcal{T}_{\text{bos}}\,,\qquad\mathcal{T}_{\text{bs}}=\mathcal{T}^{2}_{\text{bos}}\,.$
(19)

Newton’s constant $G$ is related to $\kappa$ via $\kappa^{2}=32\pi G$. As a
consistency check, by specializing the polarization tensors to describe four-
dimensional helicity states we recover the results collected in Appendix B of
Ref. Bern:2021ppb .

The bosonic string-theory amplitude contains the tachyon exchange. Following
Ref. Bern:2021ppb , we define a modified bosonic string-theory amplitude by
subtracting this exchange and present our data within this definition. In four
dimensions, this amplitude is consistent with the generic bounds and hence we
expect the same to be true in higher dimensions as well. Let us, however, note
that none of the conclusions drawn in the present paper change if we chose to
drop the bosonic string-theory amplitude, given the undesired appearance of
the tachyon.

We may also identify the massless exchanges in all three sting theories.
Specifically, the superstring amplitude only contains the minimal-coupling
graviton exchange. The heterotic string-theory amplitude contains both the
minimal-coupling and Gauss-Bonnet graviton exchanges, as well as the dilaton
exchange. Finally, the bosonic string-theory amplitude contains all the
heterotic string-theory amplitude exchanges as well as those from an
$R^{3}$-type coupling. Subtracting these exchange contributions is possible
but not necessary in order for the amplitudes to be physical. We do not
perform such a subtraction in the present work.

Furthermore, we point out that there is an inherent ambiguity in separating
the non-analytic from the analytic part (or equivalently the non-local from
the local part) of the amplitude. This ambiguity is reflecting part of the
freedom in writing a Lagrangian. When there are no $R^{3}$-type contributions,
there is no such ambiguity. In these cases, a simple powercounting argument
shows that dilaton and Gauss-Bonnet contributions do not mix with
$D^{n}R^{4}$-type (contact) operators. In contrast, when $R^{3}$-type
operators are present, terms of the form $R^{2}\times R^{3}$ and $R^{3}\times
R^{3}$ contribute at the same order as $R^{4}$-type and $D^{2}R^{4}$-type
operators respectively. Hence, in order to obtain the coefficients of the
latter, one needs to calculate the low-energy amplitudes starting from a
Lagrangian (or define in a different way a scheme for separating the non-
analytic from the analytic part of the amplitude). For the complete analysis
of the inherent ambiguity in mapping amplitude coefficients to Lagrangian
coefficients one has to also include massless loop effects Bellazzini:2020cot
; Bellazzini:2021oaj . In this work we neglect massless loop effects, and plot
coefficients of $D^{n}R^{4}$-type operators with $n>2$ which do not mix with
the massless exchange contributions.

##  5 Loop-level graviton amplitudes in quantum field theory

Besides the tree-level string theory amplitudes discussed in the previous
section, we are also interested in field-theory models where we allow massive
states circulating in the loop as a $D$-dimensional extension of the four-
dimensional analysis of Ref. Bern:2021ppb . Such models should not be viewed
as proper UV completions, but should instead be viewed as intermediate-energy
theories. Because they satisfy all input assumptions used to derive bounds on
the low-energy EFTs they provide useful guidance on where physically sensible
theories live. Here we opt to construct the full amplitudes for a simple
reason: In the derivation of EFT bounds one needs the Regge behavior to
determine the validity of bounds which rely on knowing (or assuming) the high-
every behavior. With the exact amplitudes in hand it is straightforward to
extract the high-energy behavior. A side benefit in having the full one-loop
amplitudes computed is that they may be useful for purposes other that
studying low-energy EFTs.

###  5.1 Maximally supersymmetric massive matter in the loop

The simplest loop-level data we can consider originates from a massive
deformation of the maximally supersymmetric gravity amplitude at one-loop,
where we give a common mass to the supermultiplet circulating in the loop.
This is accomplished by taking the original massless one-loop four-point
amplitude written in terms of scalar box integrals Green:1982sw ,

$\displaystyle\mathcal{M}^{(\mathcal{N}{=}8)}=\left(\frac{\kappa}{2}\right)^{4}\mathcal{T}_{\text{ss}}\left[I^{(D)}_{{\rm
box}}(s,t)+I^{(D)}_{{\rm box}}(s,u)+I^{(D)}_{{\rm box}}(u,t)\right]\,.$ (20)

and replacing the loop propagators by massive ones. This may be interpreted as
dimensionally-reducing a higher-dimensional maximal supergravity and
integrating out a Kaluza-Klein mode whose mass is the extra-dimensional
momentum. The polarization information is encoded in the tree-level
superstring tensor $\mathcal{T}_{\text{ss}}$ and the integrals are defined
e.g. as

$\displaystyle I^{(D)}_{{\rm
box}}(s,t)=\int\frac{d^{D}\ell}{(2\pi)^{D}}\prod_{n=0}^{3}\frac{1}{\left(\ell+\sum_{i=1}^{n}p_{i}\right)^{2}-m^{2}}\,.$
(21)

The UV divergences that are present for $D\geq 8$ are straightforward to
extract. In fact, the UV expansion of the amplitude and its large-mass
expansion are intimately related: both are equivalent to an expansion around
small external momenta yielding massive tadpoles, as can be easily seen from
the interpretation of the mass as Kaluza-Klein momentum. We will return
shortly to the large-mass expansion.

###  5.2 Non-supersymmetric massive matter in the loop

We obtained the four-graviton amplitude with a maximally supersymmetric
massive multiplet circulating in the loop by simply replacing formerly
massless propagators by massive ones in the known representation of the one-
loop amplitude Green:1982sw . We are also interested in $D$-dimensional one-
loop amplitudes with massive scalars and massive spin-1 fields running in the
loop. We do not explicitly consider massive fermions, gravitinos, or massive
spin-2 states here for the sake of brevity even though there is no conceptual
problem to construct these amplitudes as well. The workflow of our
construction uses a number of modern scattering-amplitude methods.

We start by constructing the amplitude’s loop integrand via generalized
unitarity Bern:1994zx ; Bern:1994cg ; Britto:2004nc  from the knowledge of
tree-level amplitudes. We find it most convenient to directly match two-
particle cuts

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}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
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}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@curveto{11.38092pt}{17.36917pt}{6.28558pt}{19.91684pt}{0.0pt}{19.91684pt}\pgfsys@curveto{-6.28558pt}{19.91684pt}{-11.38092pt}{17.36917pt}{-11.38092pt}{14.22638pt}\pgfsys@curveto{-11.38092pt}{11.08359pt}{-6.28558pt}{8.53592pt}{0.0pt}{8.53592pt}\pgfsys@curveto{6.28558pt}{8.53592pt}{11.38092pt}{11.08359pt}{11.38092pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
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where the external states are gravitons (denoted by wiggly lines) and the
massive states propagating inside the loop are generically denoted by a solid
line (which can represent either massive spin-0 or spin-1 states). Since we
are interested in expressions that are valid in arbitrary spacetime dimension
$D$, we utilize tree-level gravitational Compton amplitudes (grey ‘blobs’ in
(22)) in terms of formal (traceless symmetric) polarization tensors
$\varepsilon^{\mu\nu}_{i}=\varepsilon^{\mu}_{i}\varepsilon^{\nu}_{i}$ of the
gravitons that can be found e.g. in Ref. Bern:2020buy . The unitarity cuts
involve a sum over physical states. For the massive spin-1 circulating in the
loop, the relevant states are selected out by inserting the physical state
projector $\Pi^{\mu\nu}(p,m)=\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{m^{2}}$ for
each cut leg. From the resulting expressions, we extract the kinematic
numerators of the cubic diagrams

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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@curveto{8.5359pt}{4.71431pt}{4.71431pt}{8.5359pt}{0.0pt}{8.5359pt}\pgfsys@curveto{-4.71431pt}{8.5359pt}{-8.5359pt}{4.71431pt}{-8.5359pt}{0.0pt}\pgfsys@curveto{-8.5359pt}{-4.71431pt}{-4.71431pt}{-8.5359pt}{0.0pt}{-8.5359pt}\pgfsys@curveto{4.71431pt}{-8.5359pt}{8.5359pt}{-4.71431pt}{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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which are functions of the following Lorentz invariants: $\varepsilon_{i}\cdot
p_{j}$, $s,t$, $p_{i}\cdot\ell$, and $\varepsilon_{i}\cdot\ell$. For the
massive spin-1 exchange, it is convenient to split the numerators into a
‘spin-0’ part and a ‘spin-1’ remainder akin to the supersymmetric
decomposition of the four-dimensional amplitudes in Ref. Bern:2021ppb :

$\displaystyle
N^{\text{spin}-1}=\overline{N}^{\text{spin}-1}+(D_{s}-1)N^{\text{spin}-0}\,,$
(24)

where $D_{s}=\eta^{\mu}_{\mu}$ is the state-counting parameter. This
separation effectively eliminates the terms of highest degree in the loop
momentum $\ell$ from $N^{\text{spin}-1}$.

At one loop, all contractions between the loop momentum $\ell$ and the
external momenta $p_{i}$ can be written in terms of inverse propagators, see
e.g. Smirnov:2012gma , but contractions of the loop-momentum with polarization
vectors $\varepsilon_{i}$ require the evaluation of tensor integrals. Such
integrals are well known from e.g. Refs. Tarasov:1996bz ; Tarasov:1997kx
where one converts tensor integrals into dimension-shifted Tarasov:1996br ;
Lee:2009dh ; Lee:2010wea  scalar integrals, see also Ref. Anastasiou:2005cb .
The relevant dimension shifts can be derived algorithmically with the help of
integration-by-parts relations Tkachov:1981wb ; Chetyrkin:1981qh ;
Laporta:1996mq ; Laporta:2001dd , implemented in modern computer codes such as
FIRE Smirnov:2008iw ; Smirnov:2019qkx .

We find it convenient to organize the resulting amplitudes in a special basis
of scalar integrals where all integral coefficients are independent of the
mass of the state in the loop and of the spacetime dimension $D$. This closely
follows the discussion of the four-dimensional amplitude construction of Ref.
Bern:2021ppb  and can be likewise achieved by judiciously using dimension-
shifting relations Tarasov:1996br ; Lee:2009dh ; Lee:2010wea .

The attentive reader might have noticed that our amplitudes construction via
the two-particle cut in Eq. (22) is not yet complete as we could be missing
contributions from bubble integrals with a single massless line on one side
(sometimes referred to a ‘snail integrals’), or from tadpole integrals. These
contributions are determined along the lines of Ref. Bern:1995db  by demanding
that the amplitudes considered here remain IR-finite in the massless limit and
by requiring that the UV divergences are mass-independent.555The latter
requirement is equivalent to demanding that the corresponding counterterm has
a local expression in terms of Riemann tensors. At the end of our
construction, we find that all one-loop four-point amplitudes due to massive
spin-$\rm S$ exchange considered in this work can be expressed in terms of 42
independent (possibly dimension shifted) integrals that are multiplied by
gauge-invariant tensors involving only the Mandelstam invariants and
contractions between polarization vectors and external momenta,

$\displaystyle\mathcal{M}^{({\rm S})}=\sum_{k}\mathcal{T}^{({\rm
S})}_{k}(\varepsilon,p)\ I_{k}(s,t,D,m^{2})\,.$ (25)

We include the explicit expressions of the amplitudes in the representation of
(25) in computer readable form with explicit rules for the $\mathcal{T}^{({\rm
S})}_{k}$ in terms of Lorentz products of the external data.

In view of our goal to provide explicit expressions for the low-energy
gravitational EFT amplitudes once the massive state is integrated out, it is
highly advantageous to organize the amplitudes as in Eq. (25). Indeed, since
only the scalar (dimension-shifted) integrals, $I_{k}(s,t,D,m^{2})$, depend on
the mass, we only need to find their large-mass expansion. This can be
achieved by dimension shifting all integrals back to $D=4$ and using the known
polylogarithmic expressions, conveniently collected in Ref. Ellis:2007qk , and
expanding them for $m^{2}\gg|s|,|t|$. Alternatively, the same expressions can
be obtained by expanding the integrand along the lines of the ‘method of
regions’ Beneke:1997zp  for $|\ell|\sim m\ll|p_{i}|$, leading effectively to
tadpole integrals which are known exactly in $D$. We find the latter method
computationally much more efficient. We provide the expansion of all one-loop
integrals that appear in (25) to sufficiently high order in $1/m^{2}$ in the
attached ancillary file.

As a further consistency check on our computation, we can evaluate our
$D$-dimensional scattering amplitudes for $D=4-2\epsilon$ and for specific
choices of polarization vectors $\varepsilon^{\mu}_{i}$ that correspond to
four-dimensional helicity states, thereby reproducing all earlier results from
Ref. Bern:2021ppb , e.g.

$\displaystyle\mathcal{M}^{(0)}(1_{4^{+}}2_{4^{+}}3_{4^{+}}4_{4^{+}})=\frac{stu}{504m^{2}}+\frac{(s^{2}+st+t^{2})^{2}}{3780m^{4}}+\ldots\,,$
(26)

where $1_{4^{+}}$ refers to the polarization vector
$\varepsilon_{1,4^{+}}^{\mu}$, etc. (See our conventions in Eq. (6)).

Using our scattering amplitudes we may now probe the richer space of states of
the graviton in $D>4$. We consider examples of elastic amplitudes of the form
$\mathcal{M}(1_{a^{-}}2_{b^{-}}3_{b^{+}}4_{a^{+}})$ for various values of $a$
and $b$.

As mentioned above, generally, in order to ensure the validity of EFT bounds
it is crucial to know the behavior of our amplitudes in the Regge limit,
$|s|\gg-t,\,|s|\gg m^{2}$. Since our field theory data are only stand-in
models for a intermediate UV completion of gravitational scattering, their
behavior is generically worse than expected from quantum Regge bounds
Maldacena:2015waa ; Chandorkar:2021viw ; Haring:2022cyf . The exact behavior
of the amplitudes in the Regge limit depends on the graviton polarizations,
but using the explicit expressions for the amplitudes we have checked that the
worst behavior is saturated by the spin-2 exchange (inside the $\mathcal{N}=8$
amplitude) with a scaling (in even spacetime dimension $D$) of the form

$\displaystyle\mathcal{M}\sim s^{D-1}\,,$ (27)

which recovers the $s^{3}$ behavior in $D=4$ explored in Ref. Bern:2021ppb .

Similarly to the string-theory cases, the non-supersymmetric examples we
consider here contain massless exchanges that would need to be taken into
account in mapping the amplitude coefficients to Lagrangian coefficients.

##  6 Data summary and plots

Having discussed the relevant computations of the new explicit models of UV
completions, we now proceed to explore the associated values of the low-energy
couplings in the large-mass expansion of the amplitudes. Our analysis supports
the notion that physical theories lie on small islands. Interestingly, if we
choose the external states to be $4_{\pm}$ (which may be thought of as
polarization tensors restricted to a four-dimensional subspace), we find that
the projective data points remain on the same four-dimensional islands
independent of the spacetime dimension.

###  6.1 Sample data in $D=4$

To confirm our $D$-dimensional setup, we first reproduce the data points in
Fig. 10 of Ref. Bern:2021ppb  by specializing to $D=4$. To do so we select
four-dimensional helicity states for the external gravitons corresponding to
$\mathcal{M}(1_{4^{-}}2_{4^{-}}3_{4^{+}}4_{4^{+}})=s^{4}f(t,u)$ (see Eq. (6)
for our polarization conventions), where $f(t,u)$ admits the low-energy
expansion

$\displaystyle f(t,u)=\sum_{k\geq q\geq 0}a_{k,q}\,s^{k-q}t^{q}\,.$ (28)

Additionally, as explained in section 3, we add a new data point for our
nonperturbative results for the four-graviton scattering generated from the
effective superpotential of a $\mathcal{N}=1$ matter-coupled supersymmetric
gauge theory further coupled to gravity. In this special case, $f$ is a
function of $s$ only so that the sum truncates and the only nonzero
coefficients are the $a_{k,0}$.

Figure 1:  The $D=4$ EFT data for various models for ${a_{4,1}/a_{4,0}}$ and
${a_{4,2}/a_{4,0}}$. A line with slope 3/2 is added. The data points do not
land perfectly on this line.

In Fig. 1 and the following, our labeling conventions are as follows: NP
Matter $\equiv$ nonperturbative matter-coupled $\mathcal{N}=1$ supersymmetric