CPHT-RR020.052023, UMN-TH-4212/23, FTPI-MINN-23/06

Testing the Scalar Weak Gravity Conjecture in No-scale Supergravity

Emilian Dudasa, Tony Gherghettab, Keith A. Oliveb,c, and Sarunas Vernerd

aCentre de Physique Théorique, École Polytechnique, CNRS and IP Paris, 91128
Palaiseau Cedex, France
bSchool of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA

cWilliam I. Fine Theoretical Physics Institute, School of Physics and
Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA

dInstitute for Fundamental Theory, Physics Department, University of Florida,
Gainesville, FL 32611, USA

ABSTRACT

We explore possible extensions of the Weak Gravity Conjecture (WGC) to scalar
field theories. To avoid charged black hole remnants, the WGC requires the
existence of a particle with a mass $m<gqM_{P}$, with charge $q$ and U(1)
gauge coupling $g$, allowing the decay to shed the black hole charge. Although
there is no obvious problem that arises in the absence of a U(1) charge, it
has been postulated that gravity must remain the weakest force even when
extended to scalar interactions. Quantifying this conjecture may be done by
comparing scalar and gravitational amplitudes, or as we advocate here by
comparing scattering cross sections. In theories with non-trivial field space
geometries, by working out examples with perturbation theory around arbitrary
field values and performing tadpole resummations, we argue that the conjecture
must be applied only at extrema of the scalar potential (when expressed in
locally canonical coordinates). We consider several toy models in the context
of no-scale supergravity and also consider examples of inflationary models.

May 2023

##  1 Introduction

It has become apparent in recent years that not all effective field theories
can be consistently embedded into a theory of quantum gravity. In particular,
arguments based on black hole physics and string theory give rise to
nontrivial constraints that restrict the observable parameter space of low-
energy quantum field theories. Theories which violate these constraints are
deemed inconsistent when considered part of the larger, fundamental theory of
quantum of gravity and are relegated to what has become known as the
swampland[1, 2]. Various swampland conjectures have been formulated (see [3]
for a review) and used to test the available parameter space of low-energy
quantum field theories.

A nontrivial conjecture about a theory of quantum gravity is the weak gravity
conjecture (WGC) [4], which states that in a U(1) gauge theory with gauge
coupling $g$ coupled to gravity, there must exist a charged particle with
charge $q$ and mass $m$ satisfying $m\leq gqM_{P}$ (where $M_{P}$ is the
(reduced) Planck mass). This bound is also consistent with the absence of
global symmetries in quantum gravity and the requirement of a maximum charge-
to-mass ratio $q/m$, so that extremal mass black holes can decay and leave no
charged remnants. Interestingly, the WGC can also be reformulated as the
absence of a stable weakly-coupled gravitational bound state [5]. In other
words, requiring that the Coulomb repulsion is larger than the gravitational
attraction for the WGC particle with the largest charge-to-mass ratio is
equivalent to the WGC. This is sometimes also referred to as the Repulsive
Force Conjecture [6].

The bound state argument provides one way to generalize the WGC to include
scalar fields. For massless (moduli) fields that have cubic interactions with
the WGC charged particle, this gives rise to a Coulomb-like attractive force
that modifies the WGC condition to be[5]

$m^{2}+g^{ij}\mu_{i}\mu_{j}M_{P}^{2}\leq g^{2}q^{2}M_{P}^{2}\,,$ (1.1)

where $m\mu_{i}=\nabla_{t^{i}}m^{2}/2$ is the cubic coupling of the WGC
particle to moduli $t^{i}$ and $\nabla_{t^{i}}$ is the covariant derivative.
An interesting consequence of (1.1) occurs when the scalar cubic interaction
is tuned to the U(1) gauge coupling. In this case the mass, $m$, must be
hierarchically smaller than the Planck scale suggesting a possible form of
UV/IR mixing [7]. However, these generalizations require the presence of a
charged particle and are not applicable to purely scalar theories. Strictly
speaking, Eq. (1.1) would imply that stable, uncharged massive particles are
forbidden. In the limit that the charge vanishes, the conjecture can be
salvaged by including repulsive scalar interactions and rewriting Eq. (1.1) as
[5]

$m^{2}+g^{ij}\mu_{i}\mu_{j}M_{P}^{2}\leq\frac{1}{2}g^{ij}\nabla_{t^{i}}\nabla_{t^{j}}m^{2}M_{P}^{2}\,.$
(1.2)

Nevertheless, a conjecture solely for massless, scalar fields, partially-
motivated in the $g\rightarrow 0$ limit can be formulated by requiring that
the magnitude of the scalar force, whether attractive or repulsive, is
stronger than gravity [5]. The Scalar Weak Gravity Conjecture (SWGC) can be
expressed as 111Repulsive interactions can also be included in (1.3).

$m^{2}\leq g^{ij}\mu_{i}\mu_{j}M_{P}^{2}\,,$ (1.3)

where $m$ is now the mass of a state that has a cubic coupling to the moduli
$t^{i}$. Unlike the WGC, there are no charged black hole remnant arguments,
and therefore this conjecture remains on a weaker footing since it is not
clear what goes fundamentally wrong if the scalar force is weaker than
gravity. Moreover, this conjecture based on long-range forces only applies to
massless scalar fields.

Instead, another form of the conjecture for scalar fields was put forth in
Ref. [8] based entirely on elevating the statement of gravity as the weakest
force to a general principle. The conjecture applies to any scalar field and
gives rise to a condition on the scalar potential of a canonically-normalized
scalar field that is motivated from four-point scattering amplitudes. The so-
called strong SWGC (sSWGC) is a condition that combines a contact four-point
scalar interaction, which is repulsive and proportional to the Dirac delta
function, with the long range attractive interactions arising from gravity and
the scalar field exchange due to cubic interactions. As written, the condition
on the scalar potential was expressed as

$m^{2}\frac{\partial^{2}}{\partial\phi^{2}}\left(\frac{1}{m^{2}}\right)M_{P}^{2}\geq
1$ (1.4)

or

$2V^{\prime\prime\prime
2}-V^{\prime\prime\prime\prime}V^{\prime\prime}\geq\frac{V^{\prime\prime
2}}{M_{P}^{2}}\,,$ (1.5)

where $V^{\prime}=\partial V/\partial\phi$ and $m^{2}=V^{\prime\prime}$. This
implies one cannot factor out a universal distance dependence and therefore
the sSWGC encapsulates a UV/IR mixing effect. In fact, in the case of one real
scalar field, the condition is saturated for a potential that can be
interpreted as containing both Kaluza-Klein and winding states [8].
Furthermore, it was noticed in [8], that even simple potentials such as
$V(\phi)=\frac{1}{2}m^{2}\phi^{2}+\frac{1}{4}\lambda\phi^{4}$, violate the
conjecture near the minimum, $\phi=0$. This point was further emphasized in
[9] where it was noted that the same potential could be used to describe a
dilute gas of non-relativistic atoms and hence would violate this version of
the conjecture. For other applications of this and related conjectures see
[10, 12, 11, 13, 14].

The sSWGC of Ref. [8] was also scrutinized in Ref. [15], where it was
suggested to consider the non-relativistic limit of four-point scattering
amplitudes and to compare the strength of the scalar and gravitational forces
at the same energy scale in an attempt to make sense of the UV/IR condition.
In particular, this involved comparing the short-range scalar contact
interactions with the $s$-channel contact interaction of gravity at arbitrary
scalar field values. This requires computing the scattering amplitudes for the
scalar field fluctuations. In terms of the potential this modifies Eq. (1.5)
to read as

$\left|\lambda-\frac{5}{3}\mu^{2}\right|\geq\frac{3}{2}\frac{m^{2}}{M_{P}^{2}}\,,$
(1.6)

where $\lambda$ corresponds to the four-point contact interaction, the factor
of $\frac{5}{3}$ accounts for $s,t,$ and $u$-channel diagrams in the non-
relativistic limit and the factor of $\frac{3}{2}$ corresponds to single
graviton exchange in the $s$-channel. These factors are easily understood from
the tree level diagrams shown in Fig. 1. In the nonrelativistic limit, the
scalar exchange diagrams contribute ($\frac{1}{3},-1,-1$) for the
$(s,t,u)$-channel diagrams. The absolute value in (1.6) accounts for either
net repulsive or attractive forces, so long as they are stronger than gravity.
However, if Eq. (1.6) is applied at arbitrary field values, fluctuations about
these values induce generic linear terms in the scalar potential. Away from
the potential extrema, these linear terms can contribute to the potential
scattering, but were not included in the examples considered in Ref. [15].

Figure 1: Tree level diagrams contributing to the four-point amplitude.

A further reformulation of the sSWGC was presented in Ref. [14] where it was
proposed that the pair production rate into charged particles of mass $m$ by
any force mediator (spin 0 or 1) should be larger than the equivalent
gravitational production rate. Inspired by the pair production rate by
massless moduli, Ref [14] also modified the condition (1.5) to

$\left|\xi V^{\prime\prime\prime
2}-V^{\prime\prime\prime\prime}V^{\prime\prime}\right|\geq\frac{V^{\prime\prime
2}}{M_{P}^{2}}\,,$ (1.7)

where absolute values are inserted and $\xi$ is an order one constant. The
pair production argument, which involves an interaction rate rather than an
amplitude, gives $\xi=1$ in contrast to $\xi=2$ from (1.5), whereas the
amplitude for the nonrelativistic scattering of a single scalar field gives
$\xi=\frac{5}{3}$ as argued from (1.6). The approach taken in Ref. [14] is
that $\xi$ is not particularly relevant and (1.7) is thought of as a swampland
condition in the sense that it becomes trivial when $M_{P}\rightarrow\infty$.
However, the issue of whether the simple recipe $m^{2}\rightarrow
V^{\prime\prime}$ is consistent or how to generalize the condition to multiple
scalar fields remain unanswered questions.

In this paper, we further investigate the sSWGC and generalize the
interpretation of the constraint in terms of scattering amplitudes and
interaction rates (or cross sections). To compute the four-point scattering
amplitudes we employ the novel techniques developed in Ref. [16]. In general,
we may have several scalar fields with nontrivial kinetic terms which require
a canonical normalization in order to compute scattering amplitudes [17, 18,
19, 20, 21, 22]. As shown in Ref. [16], we use Riemann normal coordinates
(RNC) to canonically normalize the kinetic terms. This allows us to calculate
scattering amplitudes and test the scalar conjectures for the various examples
in no-scale supergravity [23]. Because of the Kähler geometry inherent in
supergravity theories, the use of Kähler normal coordinates (KNC) was also
advocated in [16]. Indeed in theories with complex fields the use of KNC can
greatly simplify the computation of amplitudes (when supersymmetry is
unbroken). These techniques are reviewed in Section 2.

Furthermore, as discussed in Ref. [16], under RNC (or KNC) field
transformations, the scattering amplitudes are only invariant at the potential
extrema where the linear terms vanish. When the linear terms are present,
tadpole contributions to the four-point scattering must be included, and we
argue that the four-point scattering at an arbitrary field value is then
equivalent to computing the scattering amplitude at one of the potential
extrema. This means that the four-point scattering, and hence the scalar WGC,
only needs to be checked at the scalar potential extrema. In Section 3, we
examine in some detail the role of tadpoles. We first show that in simple
models involving one or two real scalar fields, invariance of the scattering
amplitudes under field redefinitions is lost when tadpoles are ignored. Of
course at potential extrema where tadpole contributions are absent, invariance
is maintained. We also consider a perturbative example, which allows us to
include tadpole contributions consistently in our expansion parameter, and
show that at next-to-leading order the amplitudes computed away from the
minimum are equal to the amplitude at the minimum when tadpoles are included.
Because we are formulating the sSWGC in terms of amplitudes, in general,
unless we have a way of summing up all tadpole contributions, we will restrict
our attention to computing amplitudes at extrema. We adapt an argument from
[24] using the Newton Tangent method to argue this point.

In Section 4, we consider real (and complex) scalar fields minimally coupled
to gravity through the energy momentum tensor. Assuming a Minkowski
background, we compute the $s$-channel gravitational scattering amplitudes
which are then compared to scalar interactions. Thus we further restrict our
calculation to minima of the potential with (near) vanishing vacuum energy. In
general, these scattering amplitudes will contain an angular dependence, and
hence when strictly comparing amplitudes, the conjecture is only satisfied in
particular directions. To avoid this issue, we also integrate over the
scattering angle, effectively calculating a scattering cross section for each
example we consider. The spirit of the sSWGC would then require the total
scalar cross section to be larger than the gravitational cross section. This
captures the sense of the sSWGC reformulation in [14] arguing that
gravitational interaction rates (here given in terms of a cross section) are
smaller than all other competing rates. We label this the $\sigma$SWGC.

In Section 5, we consider several examples that contain one or two complex
scalar fields. The amplitudes for these examples were computed in [16]. Since
we are necessarily dealing with multiple real scalar fields that have non-
trivial kinetic terms, we cannot express the constraint as simply as in Eq.
(1.6). Instead, we can consider

$\left|\mathcal{A}\right|_{\rm scalar}\geq\left|\mathcal{A}\right|_{\rm
gravity}\,,$ (1.8)

which is effectively the general form for Eq. (1.6) for a specific process
(i.e., with specific initial and final states). Then as noted above, we can
integrate these amplitudes (squared), and sum over final states to obtain

$\sigma_{\rm scalar}\geq\sigma_{\rm gravity}\,.$ (1.9)

In all of the simple examples considered, some form of Eq. (1.9) is satisfied
whereas there are instances where (1.8) is not, particularly since there is in
general a dependence on the scattering angle in the scattering amplitude.

In the WGC, it sufficed to have one particle satisfying Eq. (1.1). In the
scalar version, one could similarly argue that it is sufficient to have at
least one initial state and final state interaction which is stronger than
gravity. As will see below this condition ($\sigma_{11}$SWGC) is satisfied in
all of the examples we consider and is not very constraining. A stronger
version of this would require the existence of a scalar field for which the
total inclusive cross section is greater than the gravitational cross section
($\sigma_{1i}$SWGC). This too is satisfied in all of our examples and
$\sigma_{1i}$SWGC implies $\sigma_{11}$SWGC. Still stronger versions of the
conjecture might require the interactions of all scalars to be stronger than
gravity in at least one final state channel ($\sigma_{a1}$SWGC) or summed over
all final states ($\sigma_{ai}$SWGC). The latter is the most restrictive and
$\sigma_{ai}$SWGC implies $\sigma_{a1}$SWGC. In the examples considered, we
will comment on which of these conjectures is satisfied.

In Section 6, we consider two inflationary models motivated from no-scale
supergravity. Both examples exhibit Starobinsky-like inflation [25], but
originate from different superpotentials and as such have different mass
spectra. At the minimum of the potential, both models satisfy the
$\sigma_{ai}$SWGC, (i.e. Eq. (1.9) for inclusive cross sections for all
initial states). However, both models exhibit a limiting case extremum which
is a saddle point as the inflaton takes infinite field values. In one of the
two examples, the $\sigma_{ai}$SWGC is violated at the saddle point, for a
limited range in center of mass energies (though $\sigma_{1i}$SWGC is not),
whereas in the other, the $\sigma_{ai}$SWGC is always satisfied. Our
conclusions are summarized in Section 7.

##  2 Review of Scattering Amplitudes in No-scale Supergravity

As shown in [16], the calculation of scattering amplitudes in $\mathcal{N}=1$
supergravity theories can be greatly simplified by making a field redefinition
to Kähler normal coordinates. Here, we briefly outline that procedure.

Given a Kähler potential, $K(Z^{I},\bar{Z}^{\bar{I}})$, and a superpotential,
$W$, we can write the following Lagrangian for the scalar sector of the theory
as [26]

$\mathcal{L}\;=\;K_{I\bar{J}}(Z,\bar{Z})\partial_{\mu}Z^{I}\partial^{\mu}\bar{Z}^{\bar{J}}-V(Z,\bar{Z})\,,$
(2.1)

where $Z^{I}$ is a complex scalar field and
$K_{I\bar{J}}\equiv\;\partial_{i}\partial_{\,\bar{j}}K$. The scalar potential
can be expressed as

$V(Z,\bar{Z})\;=\;e^{G}\left[G_{I}G_{\bar{J}}K^{I\bar{J}}-3\right]\;=\;K_{I\bar{J}}F^{I}F^{\bar{J}}-3e^{G}\,,$
(2.2)

where

$G\equiv K+\ln|W|^{2}\,$ (2.3)

and

$F^{I}\;=\;-e^{G/2}G^{I\bar{J}}G_{\bar{J}}\,.$ (2.4)

The inverse Kähler metric is expressed with upper indices and
$K^{I\bar{J}}K_{M\bar{J}}=\delta^{I}_{M}$. For a general Kähler metric, the
calculation of scattering amplitudes can be quite cumbersome as the fields
$Z^{I}$ may not have canonical kinetic terms, thus complicating the
expressions for masses and propagators, etc.

As in the general theory of relativity, given a non-trivial space-time metric,
it is always possible to choose a local set of coordinates such that the
metric is Minkowskian. Of course, the necessary coordinate transformation is
space-time dependent. These coordinates are often referred to as Riemann
Normal Coordinates (RNC). For a theory with real scalar fields and a non-
trivial background field metric, we can also transform to RNC by a suitable
field redefinition defined at any point in field space [27, 28, 29, 16]. Here,
we will restrict our attention to theories described by a Kähler geometry and
the appropriate field redefinition involves Kähler Normal Coordinates (KNC).

The transformation to KNC can be applied at any point in field space, and we
take $Z^{I}=w^{I}+z^{I}$, where $z^{I}$ is the fluctuation about a background
field value, $w^{I}$. As shown in Refs. [27, 28], one can introduce a
holomorphic field redefinition of the field fluctuations $z^{I}$

$\displaystyle z^{I}\longrightarrow
z^{\alpha}-\sum_{N=2}^{\infty}\frac{1}{N!}\Gamma^{\alpha}_{~{}\beta_{1}\beta_{2}\ldots\beta_{N}}\Big{|}_{w}z^{\beta_{1}}z^{\beta_{2}}\ldots
z^{\beta_{N}}\,,$ (2.5)

where

$\Gamma^{\alpha}_{~{}\beta_{1}\beta_{2}\ldots\beta_{N}}\;=\;\nabla_{\beta_{1}}\nabla_{\beta_{2}}\ldots\nabla_{\beta_{N-2}}\Gamma^{\alpha}_{~{}\beta_{N-1}\beta_{N}}\,,$
(2.6)

with the covariant derivatives only acting on the lower indices. The capital
Latin letters $I,J,\ldots$ denote the original basis and the Greek letters
$\alpha,\beta,\ldots$ denote the Kähler normal coordinate basis. In (2.5), the
Christoffel symbols evaluated at the background field value should be regarded
as coefficients of the non-linear transformation.

The transformation to Kähler normal coordinates diagonalizes the Kähler metric
and canonically normalizes the complex fields. The additional transformation
$z^{\alpha}\rightarrow K^{-1/2}_{z^{\alpha}z^{\alpha}}z^{\alpha}$ ensures that
$K_{\alpha\bar{\beta}}(w,\bar{w})=\delta_{\alpha\bar{\beta}}(w,\bar{w})$.
Thus, in Kähler normal coordinates, the general two-derivative interaction
Lagrangian with $N$ massive complex scalar fields (2.1) becomes

$\mathcal{L}\;=\;\left(\delta_{\alpha\bar{\beta}}+R_{\alpha\bar{\beta}\gamma\bar{\delta}}z^{\gamma}\bar{z}^{\bar{\delta}}+\ldots\right)\partial_{\mu}z^{\alpha}\partial^{\mu}\bar{z}^{\bar{\beta}}-V(z,\bar{z})\,,$
(2.7)

where in terms of the Kähler potential, the Kähler curvature tensor
$R_{\alpha\bar{\beta}\gamma\bar{\delta}}=K_{\alpha\bar{\beta},\gamma\bar{\delta}}-K^{\mu\bar{\nu}}K_{\mu\bar{\beta},\bar{\delta}}K_{\alpha\bar{\nu},\gamma}$
involves just ordinary partial derivatives.

The kinetic terms in the Lagrangian (2.7) lead to a four-point scattering
amplitude involving two complex and two complex conjugate fields given by

$A_{4,\rm
kin}^{z_{\alpha_{1}}z_{\alpha_{2}}{\bar{z}}_{\bar{\alpha}_{3}}{\bar{z}}_{\bar{\alpha}_{4}}}\;=\;R^{\alpha_{1}\bar{\alpha}_{3}\alpha_{2}\bar{\alpha}_{4}}s_{12}\
,$ (2.8)

along with similar amplitudes obtained by permuting the indices and crossing
symmetry. The scattering amplitudes arising from the potential terms can be
easily obtained by applying the field redefinition (2.5) in the scalar
potential (2.2). Then the amplitudes can be determined by taking ordinary
partial derivatives with respect to the KNC complex fields. For example, one
obtains the following four-point amplitudes involving complex fields

$\displaystyle A_{4,\rm
pot}^{z_{\alpha_{1}}z_{\alpha_{2}}\bar{z}_{\bar{\alpha}_{3}}\bar{z}_{\bar{\alpha}_{4}}}\;=\;$
$\displaystyle-V^{\alpha_{1}\bar{\alpha}_{3}\alpha_{2}\bar{\alpha}_{4}}-V^{\bar{\alpha}_{3}\alpha_{1}\bar{\alpha}_{4}\alpha_{2}}+\frac{1}{2}(V^{\alpha_{1}\alpha_{2}\bar{\alpha}_{3}\bar{\alpha}_{4}}+V^{\bar{\alpha}_{3}\bar{\alpha}_{4}\alpha_{1}\alpha_{2}})$
$\displaystyle-\sum_{\beta}$
$\displaystyle\left(\frac{V^{\alpha_{1}\alpha_{2}\rho_{\beta}}V_{\rho_{\beta}}{}^{\bar{\alpha}_{3}\bar{\alpha}_{4}}}{s_{12}-m_{\beta\bar{\beta}}^{2}}+\frac{V^{\alpha_{2}\bar{\alpha}_{3}\rho_{\beta}}V_{\rho_{\beta}}{}^{\alpha_{1}\bar{\alpha}_{4}}}{s_{14}-m_{\beta\bar{\beta}}^{2}}+\frac{V^{\alpha_{1}\bar{\alpha}_{3}\rho_{\beta}}V_{\rho_{\beta}}{}^{\alpha_{2}\bar{\alpha}_{4}}}{s_{13}-m_{\beta\bar{\beta}}^{2}}\right)\,,$
(2.9)

where the indices are complex $\rho_{\beta}=\beta,{\bar{\beta}}$. Note that
the expressions for the contractions over $\rho_{\beta}$ are shorthand for
example,
$V^{\alpha_{1}\alpha_{2}\rho_{\beta}}V_{\rho_{\beta}}{}^{\alpha_{3}\alpha_{4}}=V^{\alpha_{1}\alpha_{2}\bar{\beta}}V_{\beta}{}^{\alpha_{3}\alpha_{4}}+V^{\alpha_{1}\alpha_{2}\beta}V_{\bar{\beta}}{}^{\alpha_{3}\alpha_{4}}$.
Recall also that in this basis, the mass matrix is diagonal, with
$V_{\alpha\beta}=V_{\bar{\alpha}\bar{\beta}}=0$, and that indices in this
basis can be raised and lowered with a Kronecker delta. For more detail, see
Ref. [16].

It is sometimes more convenient to express the Lagrangian in terms of real
scalar fields. If we use the field expansion
$z^{\alpha}=\frac{1}{\sqrt{2}}\left(\chi^{\alpha}+i\xi^{\alpha}\right)$ and
$\bar{z}^{\bar{\alpha}}=\frac{1}{\sqrt{2}}\left(\chi^{\bar{\alpha}}-i\xi^{\bar{\alpha}}\right)=\bar{z}^{\alpha}=\frac{1}{\sqrt{2}}\left(\chi^{\alpha}-i\xi^{\alpha}\right)$,
the kinetic terms of the Lagrangian become

$\displaystyle\mathcal{K}_{\rm{KNC}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\chi^{\alpha}\partial^{\mu}\chi_{\alpha}+\frac{1}{2}\partial_{\mu}\xi^{\alpha}\partial^{\mu}\xi_{\alpha}$
(2.10)
$\displaystyle+\frac{1}{4}R_{\alpha\bar{\beta}\gamma\bar{\delta}}\left[(\partial_{\mu}\chi^{\alpha}\partial^{\mu}\chi^{\bar{\beta}}+\partial_{\mu}\xi^{\alpha}\partial^{\mu}\xi^{\bar{\beta}})(\chi^{\gamma}\chi^{\bar{\delta}}+\xi^{\gamma}\xi^{\bar{\delta}}-i(\chi^{\gamma}\xi^{\bar{\delta}}-\xi^{\gamma}\chi^{\bar{\delta}}))\right.$
$\displaystyle\left.-(\partial_{\mu}\chi^{\alpha}\partial^{\mu}\xi^{\bar{\beta}}-\partial_{\mu}\xi^{\alpha}\partial^{\mu}\chi^{\bar{\beta}})(\chi^{\gamma}\xi^{\bar{\delta}}-\xi^{\gamma}\chi^{\bar{\delta}}+i(\chi^{\gamma}\chi^{\bar{\delta}}+\xi^{\gamma}\xi^{\bar{\delta}}))\right]~{}.$

Using the above kinetic terms of the Lagrangian, we can write the four-point
scattering amplitudes. For example,

$\displaystyle A_{4,\,\rm
kin}^{\chi_{\alpha_{1}}\chi_{\alpha_{2}}\chi_{\alpha_{3}}\chi_{\alpha_{4}}}$
$\displaystyle=$ $\displaystyle A_{4,\,\rm
kin}^{\xi_{\alpha_{1}}\xi_{\alpha_{2}}\xi_{\alpha_{3}}\xi_{\alpha_{4}}}=\frac{1}{4}\left(R^{\alpha_{1}\bar{\alpha}_{3}\alpha_{2}\bar{\alpha}_{4}}s_{12}+R^{\alpha_{1}\bar{\alpha}_{2}\alpha_{3}\bar{\alpha}_{4}}s_{13}+R^{\alpha_{1}\bar{\alpha}_{2}\alpha_{4}\bar{\alpha}_{3}}s_{14}\right)+\rm{c.c.}$

Other combinations of the real fields $\chi$ and $\xi$ as incoming and
outgoing states are possible and given in [16]. The potential contributions in
this basis are analytically cumbersome (these are also given in full in [16])
but again are easily obtained from the field redefinition and splitting the
complex fields to their real components.

The full scalar amplitude is given by $\mathcal{A}_{\rm
scalar}=\mathcal{A}_{4,\,\rm kin}+\mathcal{A}_{4,\,\rm pot}$ for a specified
set of initial states $\\{\alpha_{1},\alpha_{2}\\}$ and final states
$\\{\alpha_{3},\alpha_{4}\\}$. In the reformulated sSWGC, we then require
$|\mathcal{A}|_{\rm scalar}>|\mathcal{A}|_{\rm gravity}$ which is discussed in
detail in Section 4 below.

##  3 The role of tadpoles

###  3.1 Simple scattering amplitudes and non-linear field redefinitions

It was shown in Ref.[16] that scattering amplitudes in no-scale supergravity
do not remain invariant with respect to field redefinitions (in general non-
linear) whenever tadpole terms are present. That is, whenever amplitudes are
evaluated away from a local extremum. Here we show that the same problem
exists in very simple toy models involving one or two real scalar fields.

Consider first a single, massive real scalar field with quartic self-
interactions described by the Lagrangian

${\cal
L}\;=\;\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m_{\phi}^{2}\phi^{2}-\frac{1}{4!}\lambda_{\phi}\phi^{4}\,.$
(3.1)

The four-point scattering amplitude for fluctuations about the potential
minimum, $\phi=0$ is immediately seen to be simply

${\cal A}_{4\phi}^{(0)}=\lambda_{\phi}\,.$ (3.2)

Next, we perform a non-linear field redefinition $\phi\to\phi+a\phi^{3}$ where
$a$ is a real parameter. The Lagrangian up to quartic interactions becomes

${\cal
L}\;=\;\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m_{\phi}^{2}\phi^{2}+3a\phi^{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{4!}\left(24am_{\phi}^{2}+\lambda_{\phi}\right)\phi^{4}\,.$
(3.3)

Note that the field redefinition was chosen with a linear term in $\phi$ so
that at the quadratic level the Lagrangians (3.1) and (3.3) are identical. The
four-point amplitude derived from the potential terms is now

${\cal A}_{4\phi}^{\rm pot}=24am_{\phi}^{2}+\lambda_{\phi}\,.$ (3.4)

However, kinetic terms of the form
$f\phi^{2}\partial_{\mu}\phi\partial^{\mu}\phi$ also provide a contribution,
$-8fm_{\phi}^{2}$, so that with $f=3a$,

${\cal A}_{4\phi}^{\rm kin}=-24am_{\phi}^{2}\,,$ (3.5)

and the total amplitude is again given by Eq. (3.2). Note that adding a cubic
term to the Lagrangian of the form $(\mu/3!)\phi^{3}$ would not affect the
invariance for fluctuations about the zero background value. The cubic
interaction would add $\phi$ exchange diagrams (as in Fig. 1) to the four-
point amplitude which becomes ${\cal
A}_{4\phi}=\lambda_{\phi}-\frac{5}{3}\mu^{2}/m_{\phi}^{2}$ in the non-
relativistic limit. A simple calculation shows that the same result is
obtained after the non-linear transformation.

Perhaps a more interesting example involves two real scalar fields, $\phi$ and
$\eta$, where we assume a similar Lagrangian for $\eta$ with mass $m_{\eta}$
and self-interaction $\lambda_{\eta}$. Since the fields are decoupled, there
is no non-zero amplitude for $\phi\phi\to\eta\eta$ scattering (or any
permutation of these fields). Performing the transformations
$\phi\to\phi+a\phi^{3}+b\phi\eta^{2}$ and
$\eta\to\eta+c\eta^{3}+d\eta\phi^{2}$, where $a,b,c,d$ are constants, the
Lagrangian, up to quartic interactions, becomes

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+(3a\phi^{2}+b\eta^{2})\partial_{\mu}\phi\partial^{\mu}\phi+2(b+d)\eta\phi\partial_{\mu}\phi\partial^{\mu}\eta$
(3.6)
$\displaystyle+\frac{1}{2}\partial_{\mu}\eta\partial^{\mu}\eta+(d\phi^{2}+3c\eta^{2})\partial_{\mu}\eta\partial^{\mu}\eta$
$\displaystyle-\frac{1}{2}m_{\phi}^{2}(\phi^{2}+2a\phi^{4}+2b\phi^{2}\eta^{2})-\frac{1}{4!}\lambda_{\phi}\phi^{4}$
$\displaystyle-\frac{1}{2}m_{\eta}^{2}(\eta^{2}+2c\eta^{4}+2d\phi^{2}\eta^{2})-\frac{1}{4!}\lambda_{\eta}\eta^{4}\,.$

The amplitude for $\phi\phi\to\phi\phi$ is, as before, ${\cal
A}_{4\phi}=\lambda_{\phi}$, and similarly ${\cal A}_{4\eta}=\lambda_{\eta}$.
For $\phi\phi\to\eta\eta$, the contributions from the potential terms are

${\cal A}_{2\phi 2\eta}^{\rm pot}=4(bm_{\phi}^{2}+dm_{\eta}^{2})\,,$ (3.7)

and from the kinetic terms

${\cal A}_{2\phi 2\eta}^{\rm
kin}=2b(s-2m_{\phi}^{2})+2d(s-2m_{\eta}^{2})-(2b+2d)s=-4(bm_{\phi}^{2}+dm_{\eta}^{2})\,,$
(3.8)

where $s=(p_{1}+p_{2})^{2}$ is the Mandelstam variable. The sum of these
contributions ${\cal A}_{2\phi 2\eta}^{\rm pot}+{\cal A}_{2\phi 2\eta}^{\rm
kin}=0$. Of course, this should not come as a surprise. We are calculating a
scattering amplitude in a well-defined background corresponding to the
potential minimum at $\phi=\eta=0$ and we expect the invariance of the
scattering amplitudes with respect to field redefinitions because the original
and field-transformed Lagrangians are identical at the quadratic level.

Let us now shift the field in Eq. (3.1) away from its minimum by taking
$\phi=v+\varphi$. While the kinetic terms for the shifted field are unchanged
(i.e. they remain minimal), the potential contains additional constant,
linear, quadratic and cubic terms. The nonzero linear terms generated at
generic field values $v\neq 0$, imply that the scattering of the fluctuations
$\varphi$, is no longer occurring at a potential minimum. Clearly, the
background field value is now time dependent and the process is not strictly
equivalent to potential scattering around a stable vacuum. Nevertheless, in a
Fourier expansion of the time-dependent field, we can ignore the time-
dependent modes of the scalar field and just consider the time-independent
zero mode to define a scattering process. This approximation is only valid
where $\dot{v}$ can be neglected compared to the interaction time scale, but
will be sufficient for our purposes.

With these caveats, the scattering amplitude for $v\neq 0$ in the non-
relativistic limit becomes

${\cal
A}_{4\varphi}^{(v)}=\lambda_{\phi}-\frac{5}{3}\frac{\lambda_{\phi}^{2}v^{2}}{m_{\phi}^{2}+\frac{1}{2}\lambda_{\phi}v^{2}}\,,$
(3.9)

where the second term includes contributions from $s$, $t$, and $u$ exchange
diagrams generated by the cubic coupling, $\lambda_{\phi}v$ with a mass
squared, $m_{\phi}^{2}+\frac{1}{2}\lambda_{\phi}v^{2}$, and we have ignored
any contributions from the linear (tadpole) terms. If we now perform the non-
linear redefinition on the fluctuation, $\varphi\to\varphi+a\varphi^{3}$, the
kinetic and potential contributions to the four-point contact term conspire to
still give $\lambda_{\phi}$, however, the $\varphi$-exchange terms in the non-
relativistic limit lead to a total amplitude

${\cal
A}_{4\varphi}^{(v)}=\lambda_{\phi}-\frac{5}{3}\frac{v^{2}(\lambda_{\phi}+6a(m_{\phi}^{2}+\frac{1}{6}\lambda_{\phi}v^{2}))^{2}}{m_{\phi}^{2}+\frac{1}{2}\lambda_{\phi}v^{2}}\,,$
(3.10)

which disagrees with Eq. (3.9) for all $a\neq 0$, when $v\neq 0$. We can only
conclude that the inclusion of the tadpole terms is necessary for the
invariance of the amplitudes.

The latter statement is much easier expressed than calculated. Consider for
example, the above single field model with $m_{\phi}=0$. When we shift the
field to a non-zero vacuum value, we generate a constant, a tadpole term, a
mass term, and a cubic term. Imagine including a single tadpole contribution
to the four-point contact term, as shown in the second diagram in Fig. 2. The
contribution to the amplitude is

$\Delta{\cal
A}_{4\varphi}^{(1ti)}=\lambda_{\phi}\frac{(\lambda_{\phi}v)(\frac{1}{6}\lambda_{\phi}v^{3})}{\frac{1}{2}\lambda_{\phi}v^{2}(p_{i}^{2}-\frac{1}{2}\lambda_{\phi}v^{2})}=\frac{\frac{1}{3}\lambda_{\phi}^{2}v^{2}}{p_{i}^{2}-\frac{1}{2}\lambda_{\phi}v^{2}}\,,$
(3.11)

where $\lambda_{\phi}v$ corresponds to the cubic vertex connecting to the
tadpole, $\frac{1}{6}\lambda_{\phi}v^{3}$ is the linear term and the
denominator accounts for the two propagators with mass
$\frac{1}{2}\lambda_{\phi}v^{2}$ and momentum $p_{i}$ of the external leg
containing the tadpole. Since this is a non-perturbative contribution, we must
also include diagrams with an arbitrary number of tadpoles generated either
through the contact or cubic couplings. Examples of the series of diagrams
needed to be summed are shown in Figs. 2 and 3. In addition, we would have to
include all of the contributions of the tadpoles to the $s$, $t$, and $u$
exchange diagrams generated by the shifted field. We have not found a way to
sum all of these diagrams in complete generality. We provide however a
specific example, where the contribution of the tadpoles can be included
perturbatively.

Figure 2: Corrections to the four-point contact diagram including an infinite
series of tadpole diagrams from cubic vertices.  Figure 3: Additional
correction diagrams involving both cubic and quartic vertices.

###  3.2 A perturbative toy example including tadpoles

Let us start again with the Lagrangian in Eq. (3.1) and shift the field to an
nonzero vacuum value with $\epsilon\equiv v/m_{\phi}\ll 1$. That is, we
consider only background field values close to the potential minimum at
$\phi=0$. The Lagrangian (3.1) becomes

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{1}{2}\epsilon^{2}m_{\phi}^{4}\left(1+\epsilon^{2}\frac{\lambda_{\phi}}{12}\right)-m_{\phi}^{3}\epsilon\left(1+\epsilon^{2}\frac{\lambda_{\phi}}{6}\right)\varphi~{}$
(3.12)
$\displaystyle-\frac{1}{2}m_{\phi}^{2}\left(1+\epsilon^{2}\frac{\lambda_{\phi}}{2}\right)\varphi^{2}-\frac{1}{3!}\epsilon\lambda_{\phi}m_{\phi}\varphi^{3}-\frac{1}{4!}\lambda_{\phi}\varphi^{4}\,.$

Consider the four-point scattering amplitude. To order $\epsilon^{0}$ (i.e.,
ignoring the tadpoles and exchange diagrams), there is only a contact term
contributing

${\cal A}_{4\varphi}^{(0)}=\lambda_{\phi}~{},$ (3.13)

to the total amplitude as in Eq. (3.2). For the next correction to the four-
point amplitude, at order $\epsilon^{2}$, there are $s$, $t$, and $u$ exchange
diagrams which contribute

${\cal
A}_{4\varphi}^{(3)}=\epsilon^{2}\lambda_{\phi}^{2}m_{\phi}^{2}\left(\frac{1}{s-m_{\phi}^{2}}+\frac{1}{t-m_{\phi}^{2}}+\frac{1}{u-m_{\phi}^{2}}\right)=-\frac{5}{3}\epsilon^{2}\lambda_{\phi}^{2}~{},$
(3.14)

where the second equality corresponds to the non-relativistic limit ($s\approx
4m_{\phi}^{2},t\approx 0,u\approx 0$). These arise from the same diagrams
shown in Fig. 1 with $\mu=\epsilon\lambda_{\phi}m_{\phi}$.

Next, it is possible to place a tadpole on one or both of the vertices of the
exchange diagrams. For one tadpole insertion,

${\cal
A}_{4\varphi}^{(31t)}=-2\epsilon^{2}\lambda_{\phi}^{2}m_{\phi}^{2}\left(\frac{1}{s-m_{\phi}^{2}}+\frac{1}{t-m_{\phi}^{2}}+\frac{1}{u-m_{\phi}^{2}}\right)=\frac{10}{3}\epsilon^{2}\lambda_{\phi}^{2}~{},$
(3.15)

where the factor of two in front accounts for inserting the tadpole on either
vertex and the second equality is the nonrelativistic limit. Inserting
tadpoles on each vertex gives

${\cal
A}_{4\varphi}^{(32t)}=\epsilon^{2}\lambda_{\phi}^{2}m_{\phi}^{2}\left(\frac{1}{s-m_{\phi}^{2}}+\frac{1}{t-m_{\phi}^{2}}+\frac{1}{u-m_{\phi}^{2}}\right)=-\frac{5}{3}\epsilon^{2}\lambda_{\phi}^{2}\,,$
(3.16)

where again the nonrelativistic limit has been taken in the second equality.
The three diagrams contributing to Eqs. (3.15) and (3.16) are shown in Fig. 4.
One readily sees that ${\cal A}_{4\varphi}^{(3)}+{\cal
A}_{4\varphi}^{(31t)}+{\cal A}_{4\varphi}^{(32t)}=0$.

Figure 4: Tadpole contributions to $\mathcal{A}_{4\varphi}^{(31t)}$ and
$\mathcal{A}_{4\varphi}^{(32t)}$ in Eqs. (3.15) and (3.16).

This cancellation at order $\epsilon^{2}$ in the four-point amplitude is
already apparent in the cubic vertex at order $\epsilon$. Indeed, the
effective cubic vertex obtained from inserting the tadpole in the contact term
is

${\cal
A}_{3\varphi}^{(1t)}=-\lambda_{\phi}m_{\phi}\epsilon\left(1-\frac{\lambda_{\phi}}{2}\epsilon^{2}+\dots\right)~{}.$
(3.17)

Thus, the effective cubic vertex ${\cal A}_{3\varphi}^{(0)}+{\cal
A}_{3\varphi}^{(1t)}={\cal O}(\epsilon^{3})$ where from the Lagrangian (3.12)
${\cal A}_{3\varphi}^{(0)}=\lambda_{\phi}m_{\phi}\epsilon$.

Furthermore, there are contributions to order $\epsilon^{2}$ from adding
tadpoles to the contact interaction. There is 1) a contact term with a single
tadpole

${\cal
A}_{4\varphi}^{(41t)}=-\lambda_{\phi}\frac{(m_{\phi}^{2}v)(\lambda_{\phi}v)}{m_{\phi}^{2}(p_{i}^{2}-m_{\phi}^{2})}=-\epsilon^{2}\frac{\lambda_{\phi}^{2}m_{\phi}^{2}}{p_{i}^{2}-m_{\phi}^{2}}~{},$
(3.18)

as in the second diagram in Fig. 2. In this expression, $p_{i}$ refers to the
momentum of the leg where the tadpole is placed, which can be any one of the
incoming or outgoing momenta. A pair of tadpoles can also be joined to the
contact coupling on any of the four legs as in the second diagram of Fig. 3,
giving

${\cal
A}_{4\varphi}^{(42t)}=\lambda_{\phi}\frac{\lambda_{\phi}(m_{\phi}^{2}v)^{2}}{m_{\phi}^{4}(p_{i}^{2}-m_{\phi}^{2})}=\epsilon^{2}\frac{\lambda_{\phi}^{2}m_{\phi}^{2}}{p_{i}^{2}-m_{\phi}^{2}}~{},$
(3.19)

so that their sum ${\cal A}_{4\varphi}^{(41t)}+{\cal A}_{4\varphi}^{(42t)}=0$.

Thus to order $\epsilon^{2}$, all tadpole terms are cancelled and the total
amplitude at background field values away from (but close to) the minimum is
exactly equal to the amplitude at the minimum, ${\cal A}_{\rm tot}={\cal
A}_{4\varphi}^{(0)}$. All other diagrams with tadpoles are of higher order in
$\epsilon$. We have also checked that this cancellation is maintained to order
$\epsilon^{4}$. This result, if true to all orders in $\epsilon$ and for
arbitrarily large values of $\epsilon$ (i.e., non-perturbative), would imply
that scattering amplitudes calculated at arbitrary field values are simply
equivalent to the amplitudes calculated at the minimum. If true, this is an
important result as we have already seen, the calculation of amplitudes at
arbitrary field values when ignoring the tadpole contributions leads to
results which are dependent on particular field redefinitions which renders
the notion of an invariant scattering amplitude meaningless.

###  3.3 Tadpole resummations and the Newton tangent method: Feynman versus
Newton

Given that a general scalar potential can have several extrema, an important
question is the following. If a scattering amplitude is computed at an
arbitrary scalar field value, to which extremum will the amplitude converge
to, after tadpole resummations? The answer was provided in [24], where it was
shown that tadpole resummations for scalar VEVs have a remarkably simple
interpretation using Newton’s tangent method, a highly convergent iterative
procedure to find roots of non-linear algebraic equations. In the single field
case, it amounts to considering the function $V^{\prime}(\Phi)$, whose zeroes
are the extrema of the scalar potential. One starts at a scalar field value
$\Phi^{(0)}$ with a “wrong vacuum”, and proceeds with a sequence of iterations
determined by the zeroes of the curves tangent to the derivative of the
potential

$V^{\prime}(\Phi)=V^{\prime}(\Phi^{(n)})+V^{{}^{\prime\prime}}(\Phi^{(n)})\
(\Phi-\Phi^{(n)})\ .$ (3.20)

The subsequent field values are defined recursively from
$V^{\prime}(\Phi=\Phi^{(n+1)})=0$ to give

$\Phi^{(n+1)}=\Phi^{(n)}-\frac{V^{\prime}(\Phi^{(n)})}{V^{{}^{\prime\prime}}(\Phi^{(n)})}\
,$ (3.21)

with $\Phi^{(n)}$ denoting the $n$-th iteration of the initial field value
$\Phi^{(0)}$.

By denoting $V^{\prime}(\Phi^{(0)})\equiv V^{\prime}$,
$V^{{}^{\prime\prime}}(\Phi^{(0)})\equiv V^{{}^{\prime\prime}}$ and similarly
for the higher derivatives, the first two iterations of the field value are
given explicitly by

$\displaystyle\Phi^{(1)}=\Phi^{(0)}-\frac{V^{\prime}}{V^{{}^{\prime\prime}}}\
,$
$\displaystyle\Phi^{(2)}=\Phi^{(1)}-\frac{V^{\prime}(\Phi^{(1)})}{V^{{}^{\prime\prime}}(\Phi^{(1)})}=\Phi^{(0)}-\frac{V^{\prime}}{V^{{}^{\prime\prime}}}\left[1+\frac{\frac{(V^{\prime})V^{{}^{\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{2}}-\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime\prime}}}{6(V^{{}^{\prime\prime}})^{3}}}{1-\frac{V^{\prime}V^{{}^{\prime\prime\prime}}}{(V^{{}^{\prime\prime}})^{2}}+\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{3}}}\right]\
\,,$
$\displaystyle\qquad=\Phi^{(0)}-\frac{V^{\prime}}{V^{{}^{\prime\prime}}}-\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{3}}+\frac{(V^{\prime})^{3}V^{{}^{\prime\prime\prime\prime}}}{6(V^{{}^{\prime\prime}})^{4}}+\cdots\
,$ (3.22)

where in the last line an expansion is made in the number of tadpole
insertions. It can be shown that each term in the tadpole expansion (3.22)
corresponds to a specific Feynman diagram, including its correct combinatorial
factor. The Feynman diagrams for the terms in (3.22) are shown in Figure 5. It
is clear from (3.22) that the tangent method implements an infinite
resummation of diagrams (analogous to Dyson resummation in perturbation
theory), with better convergence properties towards an extremum. In general,
as is well known for the Newton tangent method (NTM) and exemplified in [24]
for some scalar potentials, the iterated field value always ends at an
extremum. If the initial field value is sufficiently close to an extremum,
then the iteration always converges to this extremum. However, for field
values far from an extremum, the limit of the iteration depends on the form of
the scalar potential (in particular, derivatives, extrema and inflection
points), and one can end in a far away extremum. The number of iterations is a
priori infinite, but for special tuned initial field values, the iteration can
end after one or two steps, since an iteration can “shoot” directly onto an
extremum!

Figure 5: The Feynman diagrams corresponding to the first few terms in the
Newton tangent method.

The upshot of the method is that starting from a given initial field value,
one can graphically find the extremum one is driven to upon the tadpole
diagram resummations. If one restricts to a specific initial field range, only
a subset of all extrema can be reached, whereas if one is interested in all
possible initial field values, one can generically end up at any extremum of
the scalar potential.

The tangent method can also be applied to higher derivatives of the scalar
potential, or more generally in the quantum case to the one-point irreducible
(1PI) diagrams. Each iteration of the field value $\Phi^{(n)}$ defines a
corresponding iteration for these diagrams and a posteriori for the whole
effective potential. Let us consider the second derivative. The NTM iteration
in this case is defined by

$V^{{}^{\prime\prime}}(\Phi^{(n+1)})=V^{{}^{\prime\prime}}\left(\Phi^{(n)}-\frac{V^{\prime}(\Phi^{(n)})}{V^{{}^{\prime\prime}}(\Phi^{(n)})}\right)\
.$ (3.23)

The first iteration is given by

$V^{{}^{\prime\prime}}(\Phi^{(1)})=V^{{}^{\prime\prime}}\left(\Phi^{(0)}-\frac{V^{\prime}(\Phi^{(0)})}{V^{{}^{\prime\prime}}(\Phi^{(0)})}\right)\
=\
V^{{}^{\prime\prime}}-\frac{V^{\prime}V^{{}^{\prime\prime\prime}}}{V^{{}^{\prime\prime}}}+\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{2}}+\dots\
,$ (3.24)

corresponding to the diagrams shown in the top panel of Fig. 6. Working out
the numerical factors one finds that the first iteration already gives the
exact result for the second derivative in the correct vacuum, $m^{2}$. The
second iteration starts from

$V^{{}^{\prime\prime}}(\Phi^{(2)})=V^{{}^{\prime\prime}}\left(\Phi^{(1)}-\frac{V^{\prime}(\Phi^{(1)})}{V^{{}^{\prime\prime}}(\Phi^{(1)})}\right)\
=\
V^{{}^{\prime\prime}}(\Phi^{(1)})-\frac{V^{\prime}(\Phi^{(1)})V^{{}^{\prime\prime\prime}}(\Phi^{(1)})}{V^{{}^{\prime\prime}}(\Phi^{(1)})}+\frac{(V^{\prime}(\Phi^{(1)}))^{2}V^{{}^{\prime\prime\prime\prime}}(\Phi^{(1)})}{2(V^{{}^{\prime\prime}}(\Phi^{(1)}))^{2}}+\dots\
.$ (3.25)

Using the first iteration (3.24), one finally obtains

$\displaystyle V^{{}^{\prime\prime}}(\Phi^{(2)})$ $\displaystyle=$
$\displaystyle
V^{{}^{\prime\prime}}(\Phi^{(1)})-\frac{V^{\prime}}{V^{{}^{\prime\prime}}}\frac{\frac{V^{\prime}V^{{}^{\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{2}}-\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime\prime}}}{6(V^{{}^{\prime\prime}})^{3}}}{1-\frac{V^{\prime}V^{{}^{\prime\prime\prime}}}{(V^{{}^{\prime\prime}})^{2}}+\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{3}}}\left(V^{{}^{\prime\prime\prime}}-\frac{V^{\prime}V^{{}^{\prime\prime\prime\prime}}}{V^{{}^{\prime\prime}}}\right)\,,$
(3.26) $\displaystyle\simeq$ $\displaystyle
V^{{}^{\prime\prime}}(\Phi^{(1)})-\Bigg{[}\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{3}}-\frac{(V^{\prime})^{3}V^{{}^{\prime\prime\prime\prime}}}{6(V^{{}^{\prime\prime}})^{4}}\Bigg{]}\left(V^{{}^{\prime\prime\prime}}-\frac{V^{\prime}V^{{}^{\prime\prime\prime\prime}}}{V^{{}^{\prime\prime}}}\right)+\cdots\
,$

corresponding to the Feynman diagrams displayed at the bottom panel of Fig. 6.
We find that the contributions of the diagrams from the second iteration
cancel precisely among themselves. The next iteration only comes from
expanding the denominator (first line of (3.26)) in the second iteration, and
the corresponding diagrams are obtained graphically with additional tadpole
insertions obtained in the second iteration. This leads to a “non-
renormalization” theorem: since the diagrams obtained in the second iteration
exactly cancel amongst themselves, the higher order iterations will also
cancel, and therefore the first iteration already provides the exact 1PI two-
point function at zero-momentum (physical mass) in the correct vacuum.

Figure 6: The Feynman diagrams corresponding to the first iteration given by
Eq. (3.24) (top panel) and the second iteration given by Eq. (3.25) (bottom
panel) using Newton’s tangent method.

It is also instructive to work out the case of the three-point function
$V^{{}^{\prime\prime\prime}}$. The iteration in this case is

$V^{{}^{\prime\prime\prime}}(\Phi^{(n+1)})=V^{{}^{\prime\prime\prime}}\left(\Phi^{(n)}-\frac{V^{\prime}(\Phi^{(n)})}{V^{{}^{\prime\prime}}(\Phi^{(n)})}\right)=V^{{}^{\prime\prime\prime}}(\Phi^{(n)})-\frac{V^{\prime}(\Phi^{(n)})}{V^{{}^{\prime\prime}}(\Phi^{(n)})}V^{{}^{\prime\prime\prime\prime}}(\Phi^{(n)})+\dots\
,$ (3.27)

where $\dots$ refer to higher-order derivatives which vanish for the example
in Section 3.2. The first iteration leads to

$V^{{}^{\prime\prime\prime}}(\Phi^{(1)})=V^{{}^{\prime\prime\prime}}-\frac{V^{\prime}}{V^{{}^{\prime\prime}}}V^{{}^{\prime\prime\prime\prime}}+\dots\
,$ (3.28)

corresponding to the diagram shown in the top panel of Fig. 7. The second
iteration leads to

$\displaystyle V^{{}^{\prime\prime\prime}}(\Phi^{(2)})$ $\displaystyle=$
$\displaystyle
V^{{}^{\prime\prime\prime}}\left(\Phi^{(1)}-\frac{V^{\prime}(\Phi^{(1)})}{V^{{}^{\prime\prime}}(\Phi^{(1)})}\right)\,,$
(3.29) $\displaystyle=$ $\displaystyle
V^{{}^{\prime\prime\prime}}(\Phi^{(1)})-\frac{V^{\prime}}{V^{{}^{\prime\prime}}}\frac{\frac{V^{\prime}V^{{}^{\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{2}}-\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime\prime}}}{6(V^{{}^{\prime\prime}})^{3}}}{1-\frac{V^{\prime}V^{{}^{\prime\prime\prime}}}{(V^{{}^{\prime\prime}})^{2}}+\frac{(V^{\prime})^{2}V^{{}^{\prime\prime\prime\prime}}}{2(V^{{}^{\prime\prime}})^{3}}}V^{{}^{\prime\prime\prime\prime}}+\dots\
,$

corresponding to the diagrams illustrated in the bottom panel of Fig. 7.

Figure 7: The Feynman diagrams corresponding to the first iteration given by
Eq. (3.28) (top panel) and the second iteration given by Eq. (3.29) (bottom
panel) using Newton’s tangent method. Figure 8: Diagrams that do not
contribute to the tadpole resummations in the cubic vertex, since they have a
non-zero momentum flowing in the internal propagator.

Note that an interesting new feature arises in this case. Whereas in the case
of the two-point function the NTM iteration provides all Feynman diagrams with
tadpole insertions in perturbation theory, in the case of three-point
functions there are four additional Feynman diagrams in Fig. 8 that are not
present in the NTM iterations shown in Fig. 7. We find that first of all, the
four diagrams in Fig. 8 cancel amongst themselves. Secondly, by adding momenta
on the external lines, NTM iterations only generate diagrams where internal
propagators have zero momenta, since they are always connected to tadpole
insertions. This is indeed the case for the diagrams in Fig. 7 and are
therefore the analog of 1PI diagrams of the effective action in perturbation
theory. The four different diagrams in Fig. 8 have internal propagators
carrying non-zero momentum, since they are not uniquely connected to tadpole
insertions and therefore are the analog of reducible 1PI diagrams for the
effective action. One can also check that the numerical factors in the NTM
iteration do match the two diagrams in Fig. 7, whereas some diagrams in Fig. 8
have different (compared to the NTM iteration) numerical factors. From this
viewpoint, the case of the four-point function
($V^{{}^{\prime\prime\prime\prime}}$) in the model defined in Eq. (3.12) is
very simple. The NTM iteration in this case is trivial since
$V^{{}^{\prime\prime\prime\prime\prime}}=0$ and therefore all possible Feynman
diagrams with tadpole insertions should be reducible. Indeed, by an explicit
check, one finds that all diagrams with four-external legs and tadpole
insertions have at least one internal propagator with a non-zero momentum,
since they are not uniquely connected to tadpole insertions. Therefore, they
do not contribute to the renormalization of the self-coupling, although they
should be taken into account in computing physical four-point amplitudes.

Physical amplitudes contain both reducible and irreducible diagrams. Since
irreducible diagrams converge by themselves to the result at extrema of the
scalar potential, it is clear that all diagrams, reducible and irreducible,
will do the same. Whereas we checked this explicitly in Section 3.2 for the
example at hand, we believe that the results of this section add strong
evidence that this is a general result for perturbation theory amplitudes
evaluated at an arbitrary field value.

##  4 Gravitational scattering amplitudes

To test the SWGC, we must compare the amplitudes from non-gravitational
interactions with the amplitude induced by $s$-channel graviton exchange. The
gravity $t$ and $u$ channels are not included because they give rise to a
long-range Coulomb interaction and therefore cannot be compared with the
effective short range (Dirac delta function) scalar interaction in the non-
relativistic limit. Only the gravity $s$-channel gives a short range
interaction, which can then be compared with the scalar interaction [15].

We start with the Lagrangian (see e.g., [30]) coupling a real scalar field
$\phi$ to gravity

${\cal L}=-\frac{1}{M_{P}}h_{\mu\nu}T^{\mu\nu}_{\phi}\,,$ (4.1)

where the metric is expanded about Minkowski space-time with
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}/2M_{P}$ and $h_{\mu\nu}$ is the
canonically-normalized perturbation. The energy-momentum tensor for a complex
scalar field takes the standard form

$T_{z}^{\mu\nu}=\left(\partial^{\mu}z^{\alpha}\partial^{\nu}\bar{z}_{\bar{\alpha}}+\partial^{\nu}z^{\alpha}\partial^{\mu}\bar{z}_{\bar{\alpha}}\right)-\eta^{\mu\nu}\left[\partial^{\lambda}z^{\alpha}\partial_{\lambda}\bar{z}_{\bar{\alpha}}-V(z,\bar{z})\right]+\ldots\,,$
(4.2)

where … refers to couplings of $z^{\alpha}$ to additional degrees of freedom
in the supergravity Lagrangian. The scattering amplitude for
$z^{\alpha_{1}}\bar{z}^{\bar{\alpha}_{1}}\to
z^{\alpha_{2}}\bar{z}^{\bar{\alpha}_{2}}$ then takes the form

${\cal
A}^{z_{\alpha_{1}}\bar{z}_{\bar{\alpha}_{1}}z_{\alpha_{2}}\bar{z}_{\bar{\alpha}_{2}}}=\frac{1}{M_{P}^{2}}M_{\mu\nu}^{\,\alpha_{2}\bar{\alpha}_{2}}\,\Pi^{\mu\nu\rho\sigma}\,M_{\rho\sigma}^{\,\alpha_{1}\bar{\alpha}_{1}}\,.$
(4.3)

Here the graviton propagator, in de Donder gauge, is given by

$\Pi^{\mu\nu\rho\sigma}(k)=\frac{1}{2s}\left(\eta^{\rho\nu}\eta^{\sigma\mu}+\eta^{\rho\mu}\eta^{\sigma\nu}-\eta^{\rho\sigma}\eta^{\mu\nu}\right)\,,$
(4.4)

where $s$ is the Mandelstam variable, and

$M_{\mu\nu}^{\,\alpha_{1}\bar{\alpha}_{1}}\;=\;(p_{\mu}{}^{\alpha_{1}}p_{\nu}{}^{\bar{\alpha}_{1}}+p_{\nu}{}^{\alpha_{1}}p_{\mu}{}^{\bar{\alpha}_{1}}-\eta_{\mu\nu}p^{\alpha_{1}}\cdot
p^{\bar{\alpha}_{1}}-\eta_{\mu\nu}V^{\alpha_{1}\bar{\alpha}_{1}})\,,$ (4.5)

with a similar expression for the outgoing states
$M_{\rho\sigma}^{\,\alpha_{2}\bar{\alpha}_{2}}$. The scattering amplitude
(4.3) becomes

${\cal
A}^{z_{\alpha_{1}}\bar{z}_{\bar{\alpha}_{1}}z_{\alpha_{2}}\bar{z}_{\bar{\alpha}_{2}}}\;=\;\frac{2}{sM_{P}}\left[2(m_{\alpha_{1}\bar{\alpha}_{1}}^{4}+m_{\alpha_{2}\bar{\alpha}_{2}}^{4})-2(m_{\alpha_{1}\bar{\alpha}_{1}}^{2}+m_{\alpha_{2}\bar{\alpha}_{2}}^{2})(t+u)-s^{2}+t^{2}+u^{2}\right]\,,$
(4.6)

where $m_{\alpha\bar{\alpha}}^{2}=V_{\alpha\bar{\alpha}}$, and
$s=(p_{1}+p_{2})^{2}$, $t=(p_{1}-p_{3})^{2}$ and $u=(p_{1}-p_{4})^{2}$ are the
Mandelstam invariants.

For real fields, the energy-momentum tensor (4.2) can be expressed as

$\displaystyle
T^{\mu\nu}_{\chi,\,\xi}\;=\;\left(\partial^{\mu}\chi^{\alpha}+i\partial^{\mu}\xi^{\alpha}\right)\left(\partial^{\nu}\chi_{\alpha}-i\partial^{\nu}\xi_{\alpha}\right)-\eta^{\mu\nu}\left[\frac{1}{2}\left(\partial^{\lambda}\chi^{\alpha}+i\partial^{\lambda}\xi^{\alpha}\right)\left(\partial_{\lambda}\chi_{\alpha}-i\partial_{\lambda}\xi_{\alpha}\right)-V(\chi,\xi)\right]\,,$
(4.7)

where $z^{\alpha}=\frac{1}{\sqrt{2}}(\chi^{\alpha}+i\xi^{\alpha})$ and
$\bar{z}^{\bar{\alpha}}=\bar{z}^{\alpha}$. The amplitude for the scattering of
$ii\to ff$, where the initial and final states must be identical particles of
the type $\\{\chi^{\alpha},\xi^{\alpha}\\}$, then takes the form

$\mathcal{A}\;=\;\frac{1}{M_{P}^{2}}M_{\mu\nu}^{f}\Pi^{\mu\nu\rho\sigma}M_{\rho\sigma}^{i}\,,$
(4.8)

where

$M^{i}_{\mu\nu}=\frac{1}{2}(p_{1\mu}p_{2\nu}+p_{1\nu}p_{2\mu}-\eta_{\mu\nu}p_{1}\cdot
p_{2}-\eta_{\mu\nu}V_{ii})\,,$ (4.9)

and a similar expression for the outgoing states $M_{\mu\nu}^{f}$. If we
denote the mass of the initial state scalar field as $m_{i}$ and that of the
final state as $m_{f}$, and we take into account factors of two for identical
particles in the initial and final state, we can evaluate Eq. (4.8) 222In the
limit of negligible masses, this amplitude agrees with that calculated in
[31].

$\mathcal{A}\;=\;\frac{1}{2sM_{P}^{2}}\left[2(m_{i}^{4}+m_{f}^{4})-2(m_{i}^{2}+m_{f}^{2})(t+u)-s^{2}+t^{2}+u^{2}\right]\,,$
(4.10)

which in the non-relativistic limit (for the incoming states) with
$t=u=m_{f}^{2}-m_{i}^{2}$ and $s=4m_{i}^{2}$, is simply

$\mathcal{A}\;=\;-\frac{1}{M_{P}^{2}}\left(m_{i}^{2}+\frac{m_{f}^{2}}{2}\right)\,.$
(4.11)

The square of the amplitude in Eq. (4.10) can be integrated to obtain a
gravitational scattering cross section, $\sigma_{\rm grav}$. Using the
expression

$\sigma=\frac{1}{128\pi sp_{i}^{2}}\int|\mathcal{A}|^{2}dt=\frac{1}{64\pi
s}\frac{p_{f}}{p_{i}}\int_{-1}^{1}|\mathcal{A}|^{2}d(\cos\theta)\,,$ (4.12)

where we have included a factor of 1/2 for identical particles in the final
state, and $p_{i,f}=\frac{1}{2}\sqrt{s-4m_{i,f}^{2}}$, we obtain

$\displaystyle\sigma_{\rm grav}=\frac{1}{960\pi M_{P}^{4}}\frac{p_{f}}{p_{i}}$
$\displaystyle\left(s+2(m_{i}^{2}+m_{f}^{2})+\frac{6}{s}(m_{i}^{4}+4m_{i}^{2}m_{f}^{2}+m_{f}^{4})\right.$
$\displaystyle\left.-\frac{8}{s^{2}}m_{i}^{2}m_{f}^{2}(m_{i}^{2}+m_{f}^{2})+\frac{56}{s^{3}}m_{i}^{4}m_{f}^{4}\right)\,.$
(4.13)

Note that as $\sqrt{s}=E_{CM}\rightarrow\infty$ the cross section (4.13)
diverges like $s$, as expected for the (nonrenormalizable) effective theory of
gravity.

##  5 Testing the Conjecture with simple examples

We are now in a position to test the scalar weak gravity conjecture in some of
the simple models considered in [16] where the supergravity-induced amplitudes
were calculated. Recall that the amplitudes were found to be field-
redefinition invariant only when evaluating the amplitudes in a stable vacuum.
Indeed in [16], it was shown that away from an extremum of the potential, the
amplitudes depend on whether we choose KNC or RNC. Away from the minimum, and
for minima with non-zero vacuum energy density, the calculation of the
gravitational scattering amplitudes is also incorrect as one would necessarily
be forced to consider states in a non-Minkowski background. Thus here, and in
the following section, we restrict our attention to calculating amplitudes in
a supersymmetry preserving minimum with zero vacuum energy density.

In all of the models considered, we will assume a maximally symmetric Kähler
geometry with one or two chiral superfields. No-scale supergravity [23] is the
natural framework for a low-energy field theory derived from string theory
[32]. The simplest $\mathcal{N}=1$ no-scale supergravity models were first
considered in [23, 33], and are characterized by the following Kähler
potential [34]:

$K\;=\;-\,3\alpha\ln(T+\bar{T})\,,$ (5.1)

where $T$ is a complex chiral field that can be identified as the volume
modulus field, and $\alpha$ is a real parameter associated with the curvature
of the Kähler manifold. The minimal no-scale Kähler potential (5.1) describes
a non-compact $\frac{SU(1,1)}{U(1)}$ coset manifold. This can be generalized
by including matter fields $\phi^{i}$ that parametrize, together with the
volume modulus $T$, an $\frac{SU(N,1)}{SU(N)\times U(1)}$ coset space, defined
by the Kähler potential [35]

$K\;=\;-3\alpha\ln(T+\bar{T}-\frac{|\phi^{i}|^{2}}{3})\,.$ (5.2)

We will restrict our attention to examples with $N=1$ or 2. There are other
generalizations based on other non-compact coset spaces, which also appear in
some string models and involve multiple moduli fields.

###  5.1 Single Complex Scalar Field

####  5.1.1 Massless Case

In our first example, we consider a simple single field model with a Kähler
potential given by Eq. (5.1). For $W=0$, the Lagrangian is simply,

${\cal
L}=\frac{3\alpha}{(T+\bar{T})^{2}}\partial_{\mu}T\partial^{\mu}{\bar{T}}\,.$
(5.3)

We choose a complex background field value $w$ and define the fluctuation
about that background as $T(x)=w+t(x)$. Using the transformation (2.5) to KNC,
which in this case can be written as 333Note that an additional rescaling
$\hat{t}\rightarrow\hat{g}_{\hat{t}\hat{\bar{t}}}^{-1/2}\hat{t}$ has been
performed.

$t\;=\;\sqrt{2}v\sum_{n=1}^{\infty}\left(\frac{\hat{t}}{\sqrt{3\alpha}}\right)^{n}\,,$
(5.4)

where $\hat{t}$ is the complex scalar field in the KNC basis and
$v=\sqrt{2}\,{\rm Re}[w]$. In terms of real KNC fields defined by
$\hat{t}=\frac{1}{\sqrt{2}}\left(\hat{\chi}+i\hat{\xi}\right)$, the Lagrangian
up to quartic interactions is

$\mathcal{L}_{\rm
KNC}=\frac{1}{2}(\partial_{\mu}\hat{\chi})^{2}+\frac{1}{2}(\partial_{\mu}\hat{\xi})^{2}+\frac{1}{6\alpha}\left(\hat{\chi}^{2}+\hat{\xi}^{2}\right)\left((\partial_{\mu}\hat{\chi})^{2}+(\partial_{\mu}\hat{\xi})^{2}\right)\,.$
(5.5)

The amplitudes for this simple theory have been calculated [16] and for states
with identical initial and final states they are given by

$\displaystyle A^{\chi\chi\xi\xi}_{4}$
$\displaystyle\;=\;A^{\xi\xi\chi\chi}_{4}$
$\displaystyle\;=\;-\frac{2s}{3\alpha M_{P}^{2}}\,,$ (5.6) $\displaystyle
A^{\chi\chi\chi\chi}_{4}$ $\displaystyle\;=\;A^{\xi\xi\xi\xi}_{4}$
$\displaystyle\;=\;0\,.$ (5.7)

For $W=0$, the fields are massless and $s=4p^{2}$ where $p$ is the momentum of
the incoming (and outgoing) state in the center-of-mass frame.

The gravitational scattering amplitude for this case is easily obtained from
Eq. (4.10) giving

${\cal A}_{\rm grav}=\frac{s}{4M_{P}^{2}}(\cos^{2}\theta-1)\,,$ (5.8)

where $\theta$ is the angle between $p_{1}$ and $p_{3}$. The amplitudes for
$\chi\chi\to\chi\chi$ and $\xi\xi\to\xi\xi$ would appear to violate the sSWGC
for all but forward scattering in the gravitational amplitude. In the
$\chi\chi\to\xi\xi$ and $\xi\xi\to\chi\chi$ channels, the sSWGC is satisfied
provided $\alpha<8/3$.

When we calculate the total inclusive cross section for $\chi\chi$ scattering
(the cross section for $\xi\xi$ scattering is identical in this case), we find

$\sigma_{\rm
scalar}=\sigma_{\chi\chi}=\frac{s}{72\pi\alpha^{2}M_{P}^{4}}>\frac{s}{480\pi
M_{P}^{4}}=\sigma_{\rm gravity}\,,$ (5.9)

for $3\alpha^{2}<20$ and all $\sigma$SWGCs are satisfied. This theory is
surely overly minimal as there are no interactions for $T$, beyond those
generated by the kinetic terms, and the potential for $T$ is completely flat,
the key feature of no-scale models with the full $SU(1,1)/U(1)$ symmetry.

####  5.1.2 Massive Case

In [16], we also considered a model with a single chiral field, but with a
simple quadratic superpotential

$W(T)=m(T-1)^{2}\,.$ (5.10)

The kinetic terms are still given by Eq. (5.5), but there is now a potential

$V_{\rm
KNC}(\hat{\chi},\hat{\xi})=\frac{2m^{2}}{27}(\hat{\chi}^{2}+\hat{\xi}^{2})(6-3\sqrt{6}\hat{\chi}+4\hat{\chi}^{2}+4\hat{\xi}^{2})\,,$
(5.11)

where we have expanded about the potential minimum at $v=\sqrt{2}$. At the
minimum, the two scalar fields are degenerate with masses
$m_{\hat{\chi}}^{2}=m_{\hat{\xi}}^{2}=8m^{2}/9$.

$\displaystyle\mathcal{A}^{\chi\chi\chi\chi}_{4}\;=\;\frac{32m^{2}}{27M_{P}^{2}}\left[-5+81m^{2}\left(\frac{1}{8m^{2}-9s}+\frac{1}{8m^{2}-9t}+\frac{1}{8m^{2}-9u}\right)\right]\,,$
(5.12)
$\displaystyle\mathcal{A}^{\xi\xi\xi\xi}_{4}\;=\;\frac{32m^{2}}{27M_{P}^{2}}\left[-5+9m^{2}\left(\frac{1}{8m^{2}-9s}+\frac{1}{8m^{2}-9t}+\frac{1}{8m^{2}-9u}\right)\right]\,,$
(5.13)
$\displaystyle\mathcal{A}^{\chi\chi\xi\xi}_{4}\;=\;\mathcal{A}^{\xi\xi\chi\chi}_{4}\;=\;\frac{2m^{2}}{27M_{P}^{2}}\left[-16-\frac{9s}{m^{2}}+144m^{2}\left(\frac{3}{8m^{2}-9s}+\frac{1}{8m^{2}-9t}+\frac{1}{8m^{2}-9u}\right)\right]\,,$
(5.14)

where $t=(\frac{16}{9}m^{2}-\frac{s}{2})(1-\cos{\theta})$,
$u=(\frac{16}{9}m^{2}-\frac{s}{2})(1+\cos{\theta})$. Analytic expressions for
the cross sections can be obtained by integrating the amplitudes but are not
reproduced here.

In the low energy limit, the gravitational scattering amplitude from Eq.
(4.10) is

$\mathcal{A}_{\rm grav}=-\frac{4m^{2}}{3M_{P}^{2}}\,,$ (5.15)

which (in absolute value) is indeed smaller than the non-gravitational
amplitudes which were found to be
$(\frac{380}{27},-\frac{100}{27},-\frac{20}{9},-\frac{20}{9})\frac{m^{2}}{M_{P}^{2}}$
for the $\chi\chi\to\chi\chi$, $\xi\xi\to\xi\xi$, $\chi\chi\to\xi\xi$,
$\xi\xi\to\chi\chi$ channels, respectively. Since the current formulation of
the sSWGC is no longer based on the formation of bound states, we may wish to
extend the conjecture to higher energies and require gravitational scattering
at all energies be weaker than non-gravitational scattering. At higher
energies, we show a comparison of the scattering amplitudes in Fig. 9 for each
of the processes in Eqs. (5.12), (5.13), (5.14) for $\cos\theta=0,0.5$, and 1.
These are compared with the gravitational amplitude for the same processes, as
labeled.

Figure 9: The scattering amplitudes given in Eqs. (5.12), (5.13), (5.14), for
three choices of the scattering angle, $\theta$. These are compared with the
gravitational amplitude for the same processes. The fractions written in the
limit $s\to 4m^{2}$, correspond to the non-relativistic limit for each of the
amplitudes found in [16].

As one can see, that although the scattering amplitudes for the non-
gravitational interactions are always larger than the gravitational amplitudes
at low energy, at higher energy, for all but the forward scattering angle, the
amplitude for $\chi\chi\to\chi\chi$ changes sign causing the amplitude to be
smaller than the gravitational amplitude for some range in energy. However, we
see in Fig. 10, when $|\mathcal{A}|^{2}$ is integrated over the scattering
angle as in Eq. (4.12), the total inclusive cross sections for both $\chi\chi$
and $\xi\xi$ scattering are greater than the gravitational cross sections. In
Fig. 10, we show separately the cross sections for $\chi\chi\to\chi\chi$ and
$\chi\chi\to\xi\xi$ relative to the gravitational cross section given in Eq.
(4.13) in the upper panel. While the former ratio is $<1$ for
$s/m_{\chi}^{2}\gtrsim 10$, the total inclusive ratio is always greater than
1, indicating that gravity is the weakest interaction in this case and the
$\sigma_{ai}$SWGC is satisfied. The lower panel of Fig. 10 shows the ratio of
cross sections for $\xi\xi\to\xi\xi$ and $\xi\xi\to\chi\chi$.

Figure 10: The ratio of non-gravitational to gravitational cross sections for
(upper) $\chi\chi\to\chi\chi$ or $\xi\xi$ and (lower) $\xi\xi\to\xi\xi$ or
$\chi\chi$ from the amplitudes in Eqs. (5.12), (5.13), (5.14). Also shown is
the ratio of inclusive cross sections.

###  5.2 KKLT

It is also interesting to consider a second single model based on KKLT
stabilization of the modulus with superpotential [36]

$W=W_{0}+Be^{-bT}\,,$ (5.16)

where $W_{0}$ and $b>0$ are constants. The model has a supersymmetry
preserving (AdS) minimum which relates $W_{0}$ and the expectation value of
$T={\bar{T}}=v/\sqrt{2}$ at the minimum,

$W_{0}\;=\;-Be^{-\frac{bv}{\sqrt{2}}}\left(1+\frac{\sqrt{2}}{3}bv\right)\,.$
(5.17)

The scalar potential can be written in terms of $T$ and $\bar{T}$ as

$V_{\rm
KKLT}=\frac{bB\,e^{-b(T+{\bar{T}})}}{3(T+{\bar{T}})^{2}}\left(3(e^{bT}+e^{b{\bar{T}}})W_{0}+B(6+b(T+{\bar{T}}))\right)\,.$
(5.18)

To obtain a minimum with zero vacuum energy, we uplift the potential with [37]

$\Delta V\approx\frac{1}{3\sqrt{2}v^{\rm AdS}}b^{2}B^{2}e^{-\sqrt{2}bv^{\rm
AdS}}\left(\frac{\sqrt{2}v^{\rm AdS}}{T+{\bar{T}}}\right)^{2}\,,$ (5.19)

where $v^{\rm AdS}$ is the VEV corresponding to the AdS minimum of $V_{\rm
KKLT}$. For the choice, $B=1,b=1$, and $W_{0}=-10^{-12}$, the minimum occurs
at $v=43.4573$ [37] and the scalar masses are given by

$m_{\chi}\;=\;m_{\xi}\;\simeq\;1.2\times 10^{-13}\,M_{P}\,.$ (5.20)

The expression in terms of the KNC redefined real fields, $\chi$ and $\xi$ is
quite cumbersome and we rely on a numerical calculation to obtain the
amplitudes. This is shown in Fig. 11. As in the previous example, the
$\theta$-dependent amplitudes for $\chi\chi\to\chi\chi$ and in this case
$\chi\chi\to\xi\xi$ change sign and therefore drop below the gravitational
amplitudes for certain values of $s$.

Figure 11: As in Fig. 9 for the uplifted KKLT model described by the scalar
potential in Eq. (5.18).

The corresponding scattering cross sections for the uplifted KKLT model are
shown in Fig. 12. In this case, the non-gravitational cross sections are all
significantly larger than the gravitational cross sections.

Figure 12: As in Fig. 10 for the uplifted KKLT model.

###  5.3 Two Complex Scalar Fields

Before moving to testing models of inflation, we first consider a simple model
involving two chiral scalar fields which may be interpreted as a modulus $T$
and matter field, $\phi$. In this case, we begin with the Kähler potential of
the form given in Eq. (5.2). The transformation to the KNC basis was given in
[16]. Once again, we expand the fields about a fixed background where $t(x)$
and $f(x)$ are the field fluctuations and $w,r$ are the field VEVs. From Eq.
(2.5), we find for $r=0$

$\displaystyle t$ $\displaystyle=$ $\displaystyle
2w\sum_{n=1}^{\infty}\left(\frac{\hat{t}}{\sqrt{3\alpha}}\right)^{n}\,,$
(5.21) $\displaystyle f$ $\displaystyle=$
$\displaystyle\sqrt{2w}\hat{f}\sum_{n=1}^{\infty}\frac{1}{\sqrt{\alpha}^{n}}\left(\frac{\hat{t}}{\sqrt{3}}\right)^{n-1}\,,$
(5.22)

leading to kinetic terms

$\displaystyle\mathcal{K}_{\rm{KNC}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\hat{\chi})^{2}+\frac{1}{2}(\partial_{\mu}\hat{\rho})^{2}+\frac{1}{2}(\partial_{\mu}\hat{\xi})^{2}+\frac{1}{2}(\partial_{\mu}\hat{\tau})^{2}~{}$
(5.23) $\displaystyle+$
$\displaystyle\frac{1}{6\alpha}\left(((\partial_{\mu}\hat{\chi})^{2}+(\partial_{\mu}\hat{\xi})^{2})(\hat{\chi}^{2}+\hat{\xi}^{2})+((\partial_{\mu}\hat{\rho})^{2}+(\partial_{\mu}\hat{\tau})^{2})(\hat{\rho}^{2}+\hat{\tau}^{2})\right.~{}$
$\displaystyle\left.\qquad+(\partial_{\mu}\hat{\rho}\partial^{\mu}\hat{\chi}+\partial_{\mu}\hat{\tau}\partial^{\mu}\hat{\xi})(\hat{\rho}\hat{\chi}+\hat{\tau}\hat{\xi})+(\partial_{\mu}\hat{\rho}\partial^{\mu}\hat{\xi}-\partial_{\mu}\hat{\tau}\partial^{\mu}\hat{\chi})(\hat{\rho}\hat{\xi}-\hat{\tau}\hat{\chi})\right)~{}$
$\displaystyle+$
$\displaystyle\frac{1}{12\alpha}\left(((\partial_{\mu}\hat{\chi})^{2}+(\partial_{\mu}\hat{\xi})^{2})(\hat{\rho}^{2}+\hat{\tau}^{2})+((\partial_{\mu}\hat{\rho})^{2}+(\partial_{\mu}\hat{\tau})^{2})(\hat{\chi}^{2}+\hat{\xi}^{2})\right)\,,$

where the complex fields are decomposed as
$\hat{t}=\frac{1}{\sqrt{2}}(\hat{\chi}+i\hat{\xi})$ and
$\hat{f}=\frac{1}{\sqrt{2}}\left(\hat{\rho}+i\hat{\tau}\right)$. This same
expression will be used in computing amplitudes for the two-field models of
inflation discussed in the next section.

####  5.3.1 Massless case

As in the single field example, we can first take $W=0$, and compare the
scattering amplitudes resulting from the kinetic terms with the gravitational
amplitudes. The relevant scattering amplitudes are [16]

$\displaystyle A^{\chi\chi\xi\xi}_{4,\rm kin}\;=\;A^{\xi\xi\chi\chi}_{4,\rm
kin}\;=\;A^{\rho\rho\tau\tau}_{4,\rm kin}\;=\;A^{\tau\tau\rho\rho}_{4,\rm
kin}\;=\;-\frac{2s}{3\alpha M_{P}^{2}}\,,~{}$ $\displaystyle
A^{\chi\chi\rho\rho}_{4,\rm kin}\;=\;A^{\rho\rho\chi\chi}_{4,\rm
kin}\;=\;A^{\chi\chi\tau\tau}_{4,\rm kin}\;=\;A^{\tau\tau\chi\chi}_{4,\rm
kin}\;=\;-\frac{s}{6\alpha M_{P}^{2}}\,.$ (5.24)

As in Eq. (5.7), amplitudes with all four incoming and outgoing identical
states vanish. The gravitational amplitudes in this case are the same as that
given in Eq. (5.8). As before, the amplitudes for $\chi\chi\to\chi\chi$,
$\xi\xi\to\xi\xi$, $\rho\rho\to\rho\rho$, and $\tau\tau\to\tau\tau$ violate
the sSWGC for all but forward scattering in the gravitational amplitude. In
the remaining channels, the sSWGC is always satisfied provided $\alpha<2/3$.
The inclusive cross section for all four initial states ($\chi\chi$, $\xi\xi$,
$\rho\rho$, and $\tau\tau$) are equal giving

$\sigma_{ii}=\frac{s}{64\pi\alpha^{2}M_{P}^{4}}>\frac{s}{240\pi
M_{P}^{4}}=\sigma_{\rm gravity}\,,$ (5.25)

for $\alpha^{2}<15/4$ where $i=\chi,\xi,\rho,\sigma$ satisfying the
$\sigma_{ai}$SWGC. The additional final states increased the scalar cross
section (5.9) by a factor of 9/8 while the gravitational cross section
increased by a factor of 2.

Before moving onto the massive case, it is appropriate to consider the role of
Standard Model fields even for this toy example. In the case of $\chi\chi$ (or
any of our initial states) scattering, the inclusive gravitational cross
section (here again including only s-channel graviton exchange) includes an
additional contribution of $2Ns/960\pi M_{P}^{4}$, where $N=49$ is the number
of Standard Model complex scalar fields (in the Minimimal Supersymmetic
Standard Model). From Eqs. (5.23) and (5.24) we can discern that the scalar
inclusive cross sections would include a contribution of $2Ns/1152\pi
M_{P}^{4}$, giving

$\sigma_{ii}=\frac{s}{32\pi\alpha^{2}M_{P}^{4}}\left(\frac{1}{2}+\frac{2N}{36}\right)>\frac{s}{960\pi
M_{P}^{4}}(4+2N)=\sigma_{\rm gravity}\,,$ (5.26)

for the $\sigma_{ai}$SWGC where the two terms in the parenthesis on each side
of the equation come from the toy model fields, $\phi$ and $T$ and $N$
Standard Model fields respectively. This condition is violated when $N>33$
although the $\sigma_{a1}$SWGC is satisfied as the inequality is satisfied for
a single final state.

At this point, it is worth asking what we mean by gravitational interactions.
In this example, there is no superpotential and all interactions arise from
the supergravity Lagrangian, in this case non-trivial kinetic interactions,
and as one can see are Planck suppressed. Though Planck suppressed, we view
these as a remnant of dimensional reduction to yield the supergravity
Lagrangian. In Eq. (5.26), $\sigma_{\rm gravity}$ includes only the s-channel
graviton exchange. Additional ’scalar’ interactions would arise from a non-
trivial superpotential as discussed in the next subsection.

####  5.3.2 Massive case

Since we are more interested in cases with a non-zero potential, we first
consider the simple example with superpotential

$W(T,\phi)\;=\;m(T-1)^{2}+\lambda\phi^{3}\,.$ (5.27)

The resulting scalar potential in the KNC basis is

$\displaystyle
V_{\rm{KNC}}=\frac{4}{9}m^{2}(\tilde{\chi}^{2}+\tilde{\xi}^{2})+\frac{9}{4}\lambda^{2}(\tilde{\rho}^{2}+\tilde{\tau}^{2})^{2}-\frac{2}{3}\sqrt{\frac{2}{3}}m^{2}(\tilde{\xi}^{2}\tilde{\chi}+\tilde{\chi}^{3})~{}$
$\displaystyle\qquad\qquad+\frac{8}{27}m^{2}(\tilde{\chi}^{2}+\tilde{\xi}^{2})^{2}+\frac{4}{27}m^{2}(\tilde{\chi}^{2}+\tilde{\xi}^{2})(\tilde{\rho}^{2}+\tilde{\tau}^{2})\,.$
(5.28)

The fields $\chi$ and $\xi$ are again massive with
$m_{\hat{\chi}}^{2}=m_{\hat{\xi}}^{2}=8m^{2}/9$ as in the previous example.
However, in the absence of a bilinear term for $\phi$, or supersymmetry
breaking, the matter fields, $\rho$ and $\tau$ remain massless.

For this example, the amplitudes for $\chi\chi\to\chi\chi$, $\xi\xi\to\xi\xi$
and $\chi\chi\to\xi\xi$ are identical to those given in Eqs.
(5.12),(5.13),(5.14). The additional relevant amplitudes including kinetic
contributions for $\alpha=1$ are

$\displaystyle\mathcal{A}^{\chi\chi\rho\rho}_{4}\;=\;\mathcal{A}^{\rho\rho\chi\chi}_{4}\;=\;\mathcal{A}^{\chi\chi\tau\tau}_{4}\;=\;\mathcal{A}^{\tau\tau\chi\chi}_{4}\;=$
$\displaystyle\;-\frac{8m^{2}}{27M_{P}^{2}}-\frac{s}{6M_{P}^{2}}\,,$ (5.29)
$\displaystyle\mathcal{A}^{\xi\xi\rho\rho}_{4}\;=\;\mathcal{A}^{\rho\rho\xi\xi}_{4}\;=\;\mathcal{A}^{\xi\xi\tau\tau}_{4}\;=\;\mathcal{A}^{\tau\tau\xi\xi}_{4}\;=$
$\displaystyle\;-\frac{8m^{2}}{27M_{P}^{2}}-\frac{s}{6M_{P}^{2}}\,,$ (5.30)
$\displaystyle\mathcal{A}^{\rho\rho\rho\rho}_{4}\;=\;\mathcal{A}^{\tau\tau\tau\tau}_{4}\;=$
$\displaystyle\;-54\lambda^{2}\,,$ (5.31)
$\displaystyle\mathcal{A}^{\rho\rho\tau\tau}_{4}\;=\;\mathcal{A}^{\tau\tau\rho\rho}_{4}\;=$
$\displaystyle\;-\frac{2s}{3M_{P}^{2}}-18\lambda^{2}\,.$ (5.32)

These can be compared to the gravitational amplitude in Eq. (5.8).

In Fig. 13, we show the scattering cross sections for $\chi\chi$ (upper panel)
and $\rho\rho$ (lower panel) initial states. As one can see, for $\chi\chi$
initial states, the cross sections to $\chi\chi$ and $\xi\xi$ are the same as
in the previous section, and the cross section to $\rho\rho$ and $\tau\tau$
are small in comparison. Since there are twice as many final states, at large
$s$, the gravitational cross section is twice as large compared to the two-
field example and the ratio of scalar to gravity cross sections is down by a
factor of 2 as seen by comparing Figs. 10 and 13.

Figure 13: As in Fig. 10 for the two-chiral field model with superpotential
(5.27).

The cross sections for $\rho\rho$ (or $\tau\tau$) to any final state are
easily obtained from Eqs. (5.29) - (5.32). They are

$\sigma_{\chi\chi,\xi\xi}=\frac{s}{1152\pi
M_{P}^{4}}\left(1+\frac{2m_{\chi}^{2}}{s}\right)^{2}\sqrt{1-\frac{4m_{\chi}^{2}}{s}}\,,$
(5.33)

for $\rho\rho\to\chi\chi$, $\rho\rho\to\xi\xi$. Also,

$\sigma_{\rho\rho}=\frac{729\lambda^{4}}{8\pi s}\,,$ (5.34)

for $\rho\rho\to\rho\rho$, and

$\sigma_{\tau\tau}=\frac{s}{72\pi
M_{P}^{4}}\left(1+27\lambda^{2}\frac{M_{P}^{2}}{s}\right)^{2}\,,$ (5.35)

for $\rho\rho\to\tau\tau$. These should be compared with the gravitational
cross sections which at large $s$ are simply

$\sigma_{\rm grav}=\frac{s}{960\pi M_{P}^{4}}\,,$ (5.36)

for each channel. The cross section for $\tau\tau$ initial states is the same.
At large $s$, the ratio of the scalar cross sections to gravity for
$\rho\rho\to\chi\chi$ is simply 5/6. While the ratio of the
$\rho\rho\to\rho\rho$ cross sections is large at small $s$, the ratio is $<1$
for $s\gtrsim 300\lambda^{2}M_{P}^{2}$. However at large $s$, the ratio of the
$\rho\rho\to\tau\tau$ cross sections is 40/3 and hence the scalar cross
section always exceeds the gravitational cross section as seen in Fig. 13.

Before concluding this section, we come back to the question of which
interactions are gravitational. Only the processes $\rho\rho\to\rho\rho$ and
$\rho\rho\to\tau\tau$ (and similarly for $\tau\tau$ initial states), have
interactions which are not Planck suppressed. As these cross sections fall
with $1/s$ as $s$ increases, while the gravitational cross sections increase
with $s$. Should the conjecture be applied only at low $s$?

##  6 Confronting Inflation in No-scale Supergravity

Perhaps a more interesting confrontation of supergravity models with the sSWGC
arise in two-field no-scale models [39, 40, 41, 42, 43, 38] which lead to
Starobinsky inflation [25]. In [8, 15], it was argued that single field
Starobinsky inflationary models must be part of the swampland as the
gravitational scattering amplitudes exceeded the inflaton self-scattering
amplitudes at large background field values. At the minimum, however, gravity
was found to be the weakest force. As we have argued above and in [16], away
from the minimum, it is not possible to calculate accurate scattering
amplitudes while neglecting the effect of the tadpole contributions.
Furthermore, at large field values, one can not simply calculate the
gravitational amplitude using the propagator in Eq. (4.4) for which the
gravitational fluctuations, $h_{\mu\nu}$ are expanded in a Minkowski
background.

Nevertheless, it remains an interesting question whether or not the
interactions of the inflaton and related fields derived from no-scale
supergravity satisfy the conjecture. In no-scale supergravity, at least two
chiral fields are necessary to obtain a direction in field space which yields
Starobinsky inflation [40]. Starting with a Kähler potential of the form
(5.2), there are many choices for the superpotential, related by the
SU(2,1)/SU(2)$\times$U(1) symmetry, which result in the same scalar potential
in a given direction [40, 41]. Even the identity of the inflaton (as either
$T$ or $\phi$) is arbitrary up to a field redefinition. This is perhaps not
surprising as (5.2) can also be written as

$K=-3\alpha\ln\left(1-\frac{y_{1}^{2}}{3}-\frac{y_{2}^{2}}{3}\right)\,,$ (6.1)

up to a Kähler transformation of the superpotential. For the inflationary
models considered here, we restrict our analysis to $\alpha=1$.

We will consider two common examples. One in which Re $\phi$ is identified
with the inflaton and the superpotential has a Wess-Zumino-like form [39]

$W=M\left(\frac{1}{2}\phi^{2}-\frac{1}{3\sqrt{3}}\phi^{3}\right)\,,$ (6.2)

where $M$ is the inflaton mass at the minimum. With another choice of
superpotential [46],

$W=\sqrt{3}M\phi\left(T-\frac{1}{2}\right)\,,$ (6.3)

Re $T$ is identified with the inflaton and also leads to Starobinsky-like
inflation. The mass scale, $M$, is fixed by the amplitude of CMB anisotropies
to be $M\approx 1.5\times 10^{-5}M_{P}$. We discuss both of these
possibilities in the next subsections. As we will see, successful inflation in
either case will require a modification of the Kähler potential so that one of
the two fields is stabilized during inflation. This is most easily done by
adding a quartic term in the argument of the logarithm in the Kähler potential
[47, 40, 48] or by imposing a superfield constraint eliminating the
corresponding scalar field.

In the absence of a stabilization term, the transformation to the KNC basis
and the kinetic part of the Lagrangian is given by Eqs. (5.21), (5.22) and
(5.23), respectively. Similarly, the kinetic contributions to the scattering
amplitudes are given by Eq. (5.24). We discuss the modifications to the
transformations and kinetic terms due to stabilization for each of the two
models in their respective subsections. Because the mass spectra of the two
models differ, the derived scattering amplitudes and cross sections differ and
we discuss each of the two models separately.

###  6.1 Wess-Zumino-Like Inflation

Combining the Kähler potential (5.2) with the superpotential (6.2), the scalar
potential along the direction $T={\bar{T}}=\frac{1}{2}$, Im $\phi=0$, is of
the Starobinsky form

$V=\frac{3}{4}M^{2}\left(1-e^{-\sqrt{2/3}x}\right)^{2}\,,$ (6.4)

where along this direction, we define the canonical field
$\tilde{\phi}=\sqrt{3}\tanh^{-1}(\phi/\sqrt{3})$, and
$\tilde{\phi}=(x+iy)/\sqrt{2}$. Indeed, inflation only occurs if the volume
modulus is stabilized (Im $\phi=0$ minimizes the potential). One possibility
for the stabilization of the modulus is achieved by adding quartic couplings
to the argument of the Kähler potential [47, 40, 48] of the form

$K\;=\;-3\,\ln\left(T+{\bar{T}}-\frac{|\phi|^{2}}{3}+\frac{\left(T+{\bar{T}}-1\right)^{4}+d\left(T-{\bar{T}}\right)^{4}}{\Lambda_{T}^{2}}\right)\,,$
(6.5)

where $d$ is a constant and $\Lambda_{T}$ is a mass scale smaller than the
Planck scale $M_{P}$. The first quartic term stabilizes the real part and
leads to ${\rm{Re}}\,T=\frac{1}{2}$, and the second term stabilizes the
potential in the imaginary direction of $T$ with ${\rm{Im}}\,T=0$. For $d=1$,
the strength of the stabilization in the real and imaginary directions is
equal. When supersymmetry is broken, stabilization provides masses for $T$

$m_{{\rm{Re}}\,T}^{2}=144\,\frac{m_{3/2}^{2}}{\Lambda_{T}^{2}};\qquad
m_{{\rm{Im}}\,T}^{2}=144\,d\,\frac{m_{3/2}^{2}}{\Lambda_{T}^{2}},$ (6.6)

which are hierarchically larger than the gravitino mass if $\Lambda_{T}\ll 1$.
During inflation supersymmetry is broken as $W\neq 0$. At the minimum,
supersymmetry is most simply broken by adding a constant, $m_{3/2}$, to the
superpotential. We assume $m_{3/2}=1$ TeV.

The stabilization terms require a modification of the transformation of $t$ in
Eq. (5.21) to express the Lagrangian in terms of KNC. To fourth order in
${\hat{t}}$ this is

$\Delta
t=-\frac{16w^{4}}{\Lambda_{T}}\left[4(1-d)\left(\frac{{\hat{t}}}{\sqrt{3}}\right)^{3}+(5-3d)\left(\frac{{\hat{t}}}{\sqrt{3}}\right)^{4}\right]\,.$
(6.7)

Note that for $d=1$, the lowest order correction is quartic. The effect of the
modified transformations induces additional kinetic terms to be added to Eq.
(5.23)

$\Delta\mathcal{K}_{\rm{KNC}}=\frac{2(1+d)}{\Lambda_{T}^{2}}\left(\chi^{2}+\xi^{2}\right)\left((\partial_{\mu}\chi)^{2}+(\partial_{\mu}\xi)^{2}\right)\,,$
(6.8)

At the potential minimum, as in the previous (two-field) example, our
canonical fields are $\hat{t}=\frac{1}{\sqrt{2}}(\hat{\chi}+i\hat{\xi})$ and
$\hat{f}=\frac{1}{\sqrt{2}}\left(\hat{\rho}+i\hat{\tau}\right)$. The masses of
the fluctuations about the minimum at $T=1/2$ and $\phi=0$ are
$m_{\chi}=4\sqrt{3}m_{3/2}/\Lambda_{T}$,
$m_{\xi}=4\sqrt{3}dm_{3/2}/\Lambda_{T}$, and $m_{\rho}=m_{\tau}=M$. The
amplitudes for $\chi\chi\to\chi\chi$, $\chi\chi\to\xi\xi$ and
$\rho\rho\to\rho\rho$, $\rho\rho\to\chi\chi$ scattering are shown in Fig. 14
for $d=1$ and $\Lambda_{T}=M_{P}$. The partial and inclusive cross sections
for these initial states are shown in Fig. 15. The processes
$\chi\chi\to\rho\rho$ and $\chi\chi\to\tau\tau$ are kinematically forbidden
unless $s$ is sufficiently large (far beyond what is shown in these plots).
The amplitudes and cross sections for $\rho\rho\to\xi\xi$ and
$\rho\rho\to\tau\tau$ as well as $\xi\xi$ and $\tau\tau$ scattering are
similar and therefore not shown.

Figure 14: The scattering amplitudes for the Wess-Zumino-like inflationary
model for three choices of the scattering angle with $d=1$ and
$\lambda_{T}=M_{P}$. These are compared with the gravitational amplitude for
the same processes. Shown are the amplitudes for $\chi\chi\to\chi\chi$ (upper
left), $\chi\chi\to\xi\xi$ (upper right), $\rho\rho\to\chi\chi$ (lower left),
and $\rho\rho\to\rho\rho$ (lower right).

For the $\chi\chi$ initial state (effectively the volume modulus $T$), once
again, we see that the amplitude in one channel ($\chi\chi\to\xi\xi$) becomes
smaller than the gravitational amplitude since it changes sign. The integrated
cross section for this process also falls below the gravitational cross
section and has a minimum at $s\simeq 4.18m_{\chi}^{2}$. However, the
$\chi\chi\to\chi\chi$ cross section is large for all $s$, dominated by the
quartic contact term, and the total inclusive cross section is larger than the
gravitational cross section as seen in the top panel of Fig. 15. Inflaton
scattering is shown here as $\rho\rho$ scattering. While several of the final
state channels have cross sections below the gravitational cross section, the
total scattering cross section is dominated by $\rho\rho\to\rho\rho$
scattering. This channel receives large contributions from $t$ and $u$ channel
$\chi$ exchange. Since $m_{\chi}\ll m_{\rho}$, this cross section is many
orders of magnitude larger than the gravitational cross section as shown in
bottom panel of Fig. 15.

Figure 15: The ratio of the scalar scattering cross section relative to the
gravitational cross section as a function of $s/m^{2}$ for the Wess-Zumino-
like inflationary model. In the upper panel, the ratio of cross sections for
$\chi\chi\to\chi\chi$ and $\xi\xi$ as well as the inclusive cross section
ratio are shown. In the lower panel, the ratio of cross sections for
$\rho\rho\to\rho\rho$, $\tau\tau,\chi\chi$, and $\xi\xi$, as well as the
inclusive cross section are shown.

As discussed earlier, near the minimum, we can be confident that the
scattering amplitudes are given by those at the minimum. We have also argued
that when far from a minimum, the amplitudes should be calculated at the
appropriate extremum determined by the Newton tangent method [24]. Thus for
most if not all of the inflationary trajectory, the appropriate extremum to
calculate a scattering amplitude is at $T=1/2$ and $\phi\to\sqrt{3}$
corresponding to $x\to\infty$. In terms of the KNC fields, this corresponds to
$w=\frac{1}{\sqrt{2}}(v+iu)$ with $v=\frac{1}{\sqrt{2}}$ and $u=0$ and
$r=\frac{1}{\sqrt{2}}(p+it)$ with $p=\sqrt{6}$ and $t=0$. This is an extreme
limit of the field space manifold. Indeed, the argument of the logarithm of
the Kähler potential vanishes in this limit. We note that in the absence of
the stabilization term this limit would not correspond to an extremum as there
would be no minimum in the $T$ direction. For any finite value of
$\Lambda_{T}$, the minimum occurs at approximately $v=\frac{1}{\sqrt{2}}$ \-
the approximation improves as $\Lambda_{T}\to 0$. Thus, any attempt at a
calculation in this limit is approximate since $v=\frac{1}{\sqrt{2}}$ is not
the exact minimum, and therefore there are residual tadpoles which should (but
have not) been accounted for. Numerically, instabilities may occur as the
limit in several divergent quantities may not be well defined. Finally, in
this limit, the background space-time metric is no longer Minkowski but rather
de Sitter. In principle the scattering amplitudes should be recomputed in a de
Sitter background and in particular the gravitational amplitude would be
modified when using the correct graviton propagator. We have not considered
these effects.

In the limit $v\rightarrow\frac{1}{\sqrt{2}}$ and $p\rightarrow\sqrt{6}$, the
masses of $\chi$ and $\tau$ diverge, leaving us with two ‘light’ scalars:
$\xi$ with mass $m_{\xi}=M\sqrt{12M_{P}^{2}+\Lambda_{T}^{2}}/\Lambda_{T}$ and
the inflaton $\rho$, which is massless in this limit. The scattering cross
sections compared to gravity are shown in Fig. 16 with $d=1$ and
$\Lambda_{T}=M_{P}$. Final states for $\xi\xi$ scattering include both
$\xi\xi$ and $\rho\rho$ for any center of mass energy and the cross section is
always larger than the gravitational cross section. In the case of $\rho\rho$
scattering, at low energy, only $\rho\rho$ final states are allowed. As this
cross section decreases with $s$, for $s\gtrsim 0.01m_{\xi}^{2}$, the scalar
cross section violates the $\sigma_{ai}$SWGC until $s>4m_{\xi}^{2}$ when
$\rho\rho\to\xi\xi$ scattering becomes kinematically viable. In this case,
even $\sigma_{a1}$SWGC is violated in the energy interval
$0.01<s/4m_{\xi}^{2}<4$. However, because $\xi\xi\to\rho\rho$ scattering is
greater than its gravitational counterpart, the $\sigma_{1i}$SWGC remains
satisfied.

Note that in this model during the inflationary epoch, along the potential
plateau, the kinetic terms approach a diverging pole, which are also known as
pole-inflation [44], [45]. It was argued in [10] that such models generically
violate the Swampland Distance Conjecture (SDC) [2]. It is tempting to
speculate that the violation of $\sigma_{ai}$SWGC and $\sigma_{a1}$SWGC in
some energy range could be related to the violation of the SDC.

Figure 16: As in Fig. 15 for scattering when the background field values take
the limit, $v\rightarrow\frac{1}{\sqrt{2}}$ and $p\rightarrow\sqrt{6}$.

###  6.2 Cecotti-Like Inflation

Alternatively, the volume modulus, $T$, may play the role of the inflaton if
we combine the Kähler potential (5.2) with the superpotential (6.3). In this
case, the Starobinsky potential is found along the direction $\phi=0$ and
Im$T=0$ for a canonical field, $x$ defined by $T=\frac{1}{2}e^{\sqrt{2/3}x}$.

Once again, inflationary dynamics require stabilization, in this case for
$\phi$. Quartic stabilization is again sufficient and can be achieved with the
following Kähler potential

$K\;=\;-3\alpha\ln(T+\bar{T}-\frac{|\phi|^{2}}{3}+\frac{|\phi|^{4}}{\Lambda_{\phi}^{2}})\,.$
(6.9)

In this case, there are no changes to the field redefinitions given in Eq.
(5.21) and (5.22), and the kinetic terms receive a correction given by

$\Delta\mathcal{K}_{\rm{KNC}}=\frac{3}{\Lambda_{\phi}^{2}}\left(\rho^{2}+\tau^{2}\right)\left((\partial_{\mu}\rho)^{2}+(\partial_{\mu}\tau)^{2}\right)\,.$
(6.10)

At the potential minimum, the masses of all four real scalar fields are equal
to $M$. The amplitudes and cross sections for $\chi\chi$ and $\rho\rho$
scattering at the minimum are shown in Figs. 17 and 18, respectively with
$\Lambda_{\phi}=M_{P}$. In this case all four final states are available in
$\chi\chi$ scattering and both cross sections are larger than the
corresponding gravitational cross section.

Figure 17: As in Fig. 14 for the Cecotti inflationary model.

Figure 18: As in Fig. 15 for the Cecotti inflationary model.

The saddle point in the Cecotti model occurs when $\phi=0$ and $T\to\infty$ or
when $p=0$ and $v\to\infty$. In this limit the masses of $\rho$ and $\tau$
both diverge and the light fields have masses, $m_{\chi}=0$ and $m_{\xi}=M$.
The cross section for $\chi\chi\to\xi\xi$ and the equal reverse process at the
saddle point is shown in Fig. 19. In contrast to the previous inflationary
example, the cross section in this model always satisfies the
$\sigma_{ai}$SWGC.

Note that in this model during the inflationary period, the kinetic terms
approach a zero, instead of a pole seen in the previous Wess-Zumino
inflationary model. It was argued in [10] that such models respect the SDC [2]
and that they have better prospects for string realizations. Other swampland
conjectures related to no-scale supergravity were discussed in [49].

Figure 19: As in Fig. 18 for scattering when the background field values take
the limit, $v\to\infty$ and $p=0$.

##  7 Conclusions

When low-energy effective field theories are embedded into a theory of quantum
gravity, nontrivial constraints can be obtained on the available parameter
space. Based on black hole physics, the WGC provides one of the most
surprising constraints on a U(1) gauge theory when coupled to gravity by
requiring the existence of a WGC charged particle with $m\leq gqM_{P}$.
Interestingly, this constraint can also be understood as the condition that
the Coulomb interaction is stronger than gravity. The problems that arise when
the WGC is violated are evident. This conjecture can be modified to include
scalar fields that couple to the WGC particle in the U(1) gauge theory.
However, when this conjecture is generalized to only include scalar fields
(SWGC), arguments based on black hole physics no longer apply. Indeed, the
absence of a charge would violate the conjecture unless repulsive interactions
are introduced as in Eq. (1.2). In contrast, in the SWGC, the attractive
Coulomb-like forces are moved to the opposite side of the inequality as in Eq.
(1.3), but we lose the obvious problem which arises when the conjecture is
violated. Instead, as advocated in Ref. [8], one can adopt the principle that
gravity should always be the weakest force (sSWGC).

One way to test the sSWGC at arbitrary field values, involves comparing the
four-point scattering amplitudes from scalar interactions with those arising
from gravity (specifically at a particular energy scale). For the case of
multiple scalar fields with noncanonical kinetic terms, which are generic in
no-scale supergravity models, we employ the methods of Ref. [16] to
canonically normalize the scalar fields. This is done by transforming the
scalar fields to the KNC (or RNC) basis, greatly simplifying the calculation
of scattering amplitudes of the asymptotic states. Since the scattering is
computed at arbitrary field values, all terms in the scalar potential
$V(\phi)$ should be included in the calculation of the amplitudes. In
particular, linear terms are generated at generic field values away from the
potential extrema. However, as shown in Ref. [16], the scattering amplitudes
do not remain invariant under a field transformation, if the linear terms are
neglected. Instead, as expected, the amplitudes are invariant at the potential
extrema (where the linear terms vanish). To the extent that a scattering
process can be defined at non-extremal field values (by neglecting the time-
dependence of the background value) we also provide evidence that summing all
tadpole contributions reproduces the scattering that would be obtained at the
potential minima. This suggests that scattering amplitudes only need to be
compared at the extrema of the scalar potential.

The fact that scattering amplitudes are covariant, depending on the energy
scale and angular direction, further suggests one should integrate over the
scattering angle in order to compare the scalar interactions with gravity
($\sigma$SWGC). We consider several examples of real (and complex) scalars
coupled to gravity and sum over the possible final states. This effectively
yields a scattering cross section that can then be compared to gravity,
providing a much more general condition compared to the usual comparison with
nonrelativistic forces. This also mimics the reformulation of the sSWGC in Ref
[14], which argued that the gravitational interaction rates should always be
smaller than the pair production at threshold. Therefore, adopting the
condition that $\sigma_{\rm scalar}>\sigma_{\rm gravity}$ to test the sSWGC,
we find no violations in our toy examples depending on the specific
formulation of the conjecture.

We also applied our methods to two realistic inflationary models motivated
from no-scale supergravity. At the potential minimum, both models satisfied
the $\sigma_{ai}$SWGC. Namely for each of the four real fields in the model,
the inclusive scalar cross section is larger than the inclusive gravitational
cross section. Along one real direction, both models of inflation are
Starobinsky-like. Therefore, each has a saddle point extremum in the limit
that the canonical inflaton tends to infinity. At this extremal limit, we
found a violation of our $\sigma_{ai}$SWGC and $\sigma_{a1}$SWGC conditions in
the Wess-Zumino-like inflation example. Namely, the cross section for inflaton
scattering (the inflaton is massless in this limit) is smaller than the
gravitational cross section for a certain range of center of mass energies.
Indeed, both conjectures are satisfied in both the high and low energy limits.
On the other hand, the $\sigma_{1i}$SWGC is satisfied as it requires only one
initial state inclusive cross section to be larger than the gravitational
cross section. Thus, the model may not be relegated to the swampland. In
addition, we reiterate that we have not accounted for the fact that the
background metric is de Sitter in this limit, whereas we have assumed a
Minkowski background throughout the paper.

Previous attempts at testing scalar field theories using some form of the
SWGC, were based on amplitudes or properties of the scalar potential at all
points in field space. In the latter case, the presence of tadpole terms
(necessarily present away from extrema - by definition) were neglected. In
[16], we provided a procedure for calculating amplitudes for non-trivial field
space manifold utilizing either Riemann or Kähler normal coordinates. The
latter is particularly well suited for calculations in the context of non-
minimal supergravity models such as no-scale supergravity. We also showed that
amplitudes calculated when neglecting tadpole contributions were not field
redefinition invariant. Using a simple example, this point was clarified in
section 3.1, while in section 3.2 we showed that for small deviations from an
extremum, the inclusion of tadpole contributions leads to amplitudes which are
given by their extremal expressions. We also argued more generally that from
the Newton tangent method, amplitudes at arbitrary field values should be
equivalent to the corresponding amplitudes evaluated at the extrema of the
potential.

Nevertheless, we are left with a number of questions. Which (if any) of the
several versions of the $\sigma$SWGC is a true test of whether a scalar field
theory is in the swampland? We have seen in nearly every example, that a
comparison of the scalar and gravitational amplitudes is not a suitable test
because at certain center of mass energies and scattering angles, the scalar
amplitude changes sign and therefore vanishes at some energies. Perhaps this
is avoided in the low energy limit, but it is not clear why this limit would
be appropriate as we do not rely on the existence of bound states (as in the
WGC). Even when comparing amplitudes, it is unclear if this should be applied
for each distinct initial and final state, or if to satisfy the conjecture, it
is sufficient to have a large (low energy) amplitude for a single final state
for any initial state, or if it is necessary to satisfy the inequality for all
initial states and at least one (or all) final state(s).

The problem of scattering angle is removed by calculating and comparing cross
sections. We have outlined several possible forms for the conjecture expressed
in terms of scattering cross sections. The strongest constraint would require
satisfying the inequality for all initial and final states. However, with the
exception of the KKLT model discussed in section 5.2, this version of the
conjecture is violated in every model considered, as there is always a
specific initial and final state where the cross section is smaller than the
gravitational cross section. Finally, it is also not clear which states are
necessary to consider. All of the toy models (including the inflationary
models) consist of one or two chiral fields. However, in addition, there are
all of the Standard Model fields and in a supersymmetric theory would include
at least 98 Standard Model (real) scalars. Should these be included in a
gravitational inclusive cross-section? Should the conjecture be applied only
in the low energy limit, or at all scattering energies? In the context of
supergravity, which interactions should be considered gravitational? It is
clear that a better understanding of the nature of the conjecture as applied
to scalar fields is needed if we wish to discriminate between viable UV
theories and the swampland.

## Acknowledgements

We would like to thank E. Gonzalo, L. Ibanez, M. Luty, and E. Palti for useful
discussions. The work of T.G and K.A.O. was supported in part by DOE grant DE-
SC0011842 at the University of Minnesota. The work of T.G. was performed in
part at the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1607611. The work of S.V. was supported in part by DOE
grant DE-SC0022148.

## References

  * [1] C. Vafa, [arXiv:hep-th/0509212 [hep-th]].
  * [2] H. Ooguri and C. Vafa, Nucl. Phys. B 766, 21-33 (2007) [arXiv:hep-th/0605264 [hep-th]].
  * [3] E. Palti, Fortsch. Phys. 67, no.6, 1900037 (2019) [arXiv:1903.06239 [hep-th]].
  * [4] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, JHEP 0706 (2007) 060 [hep-th/0601001].
  * [5] E. Palti, JHEP 08, 034 (2017) [arXiv:1705.04328 [hep-th]].
  * [6] B. Heidenreich, M. Reece and T. Rudelius, JHEP 10, 055 (2019) [arXiv:1906.02206 [hep-th]].
  * [7] D. Lust and E. Palti, JHEP 02, 040 (2018) [arXiv:1709.01790 [hep-th]].
  * [8] E. Gonzalo and L. E. Ibáñez, JHEP 08, 118 (2019) [arXiv:1903.08878 [hep-th]].
  * [9] B. Freivogel, T. Gasenzer, A. Hebecker and S. Leonhardt, SciPost Phys. 8 (2020) no.4, 058 [arXiv:1912.09485 [hep-th]].
  * [10] M. Scalisi and I. Valenzuela, JHEP 08, 160 (2019) [arXiv:1812.07558 [hep-th]].
  * [11] S. Shirai and M. Yamazaki, Class. Quant. Grav. 38, no.3, 035006 (2021) [arXiv:1904.10577 [hep-th]].
  * [12] A. Kusenko, V. Takhistov, M. Yamada and M. Yamazaki, Phys. Lett. B 804, 135369 (2020) [arXiv:1908.10930 [hep-th]].
  * [13] D. Andriot, N. Cribiori and D. Erkinger, JHEP 07, 162 (2020) [arXiv:2004.00030 [hep-th]].
  * [14] E. Gonzalo and L. E. Ibáñez, JHEP 12, 039 (2020) [arXiv:2005.07720 [hep-th]].
  * [15] K. Benakli, C. Branchina and G. Lafforgue-Marmet, Eur. Phys. J. C 80, no.8, 742 (2020) [arXiv:2004.12476 [hep-th]].
  * [16] E. Dudas, T. Gherghetta, K. A. Olive and S. Verner, [arXiv:2302.05456 [hep-th]].
  * [17] R. Alonso, E. E. Jenkins and A. V. Manohar, Phys. Lett. B 754, 335-342 (2016) [arXiv:1511.00724 [hep-ph]]; R. Alonso, E. E. Jenkins and A. V. Manohar, JHEP 08, 101 (2016) [arXiv:1605.03602 [hep-ph]].
  * [18] T. Cohen, N. Craig, X. Lu and D. Sutherland, JHEP 12, 003 (2021) [arXiv:2108.03240 [hep-ph]].
  * [19] R. Alonso and M. West, Phys. Rev. D 105, no.9, 096028 (2022) [arXiv:2109.13290 [hep-ph]].
  * [20] C. Cheung, A. Helset and J. Parra-Martinez, JHEP 04, 011 (2022) [arXiv:2111.03045 [hep-th]].
  * [21] C. Cheung, A. Helset and J. Parra-Martinez, Phys. Rev. D 106, no.4, 045016 (2022) [arXiv:2202.06972 [hep-th]].
  * [22] A. Helset, E. E. Jenkins and A. V. Manohar, Phys. Rev. D 106, no.11, 116018 (2022) [arXiv:2210.08000 [hep-ph]].
  * [23] E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, Phys. Lett. B 133 (1983) 61; A. B. Lahanas and D. V. Nanopoulos, Phys. Rept. 145 (1987) 1.
  * [24] E. Dudas, G. Pradisi, M. Nicolosi and A. Sagnotti, Nucl. Phys. B 708, 3-44 (2005) doi:10.1016/j.nuclphysb.2004.11.028 [arXiv:hep-th/0410101 [hep-th]].
  * [25] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
  * [26] H. P. Nilles, Phys. Rept. 110 (1984) 1.
  * [27] K. Higashijima and M. Nitta, Prog. Theor. Phys. 105, 243-260 (2001) [arXiv:hep-th/0006027 [hep-th]].
  * [28] K. Higashijima, E. Itou and M. Nitta, Prog. Theor. Phys. 108, 185-202 (2002) [arXiv:hep-th/0203081 [hep-th]].
  * [29] A. Hatzinikitas, [arXiv:hep-th/0001078 [hep-th]].
  * [30] B. R. Holstein, Am. J. Phys. 74, 1002-1011 (2006) [arXiv:gr-qc/0607045 [gr-qc]].
  * [31] S. Clery, Y. Mambrini, K. A. Olive and S. Verner, Phys. Rev. D 105 (2022) no.7, 075005 [arXiv:2112.15214 [hep-ph]].
  * [32] E. Witten, Phys. Lett. 155B (1985) 151.
  * [33] J. R. Ellis, A. B. Lahanas, D. V. Nanopoulos and K. Tamvakis, Phys. Lett. 134B, 429 (1984).
  * [34] J. R. Ellis, C. Kounnas and D. V. Nanopoulos, Nucl. Phys. B 241, 406 (1984).
  * [35] J. R. Ellis, C. Kounnas and D. V. Nanopoulos, Nucl. Phys. B 247 (1984) 373.
  * [36] S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240].
  * [37] A. Linde, Y. Mambrini and K. A. Olive, Phys. Rev. D 85, 066005 (2012) [arXiv:1111.1465 [hep-th]].
  * [38] J. Ellis, M. A. G. Garcia, N. Nagata, D. V. Nanopoulos, K. A. Olive and S. Verner, Int. J. Mod. Phys. D 29, no.16, 2030011 (2020) [arXiv:2009.01709 [hep-ph]].
  * [39] J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. Lett. 111 (2013) 111301 [arXiv:1305.1247 [hep-th]].
  * [40] J. Ellis, D. V. Nanopoulos and K. A. Olive, JCAP 1310 (2013) 009 [arXiv:1307.3537 [hep-th]].
  * [41] J. Ellis, D. V. Nanopoulos, K. A. Olive and S. Verner, JHEP 03, 099 (2019) [arXiv:1812.02192 [hep-th]].
  * [42] S. Ferrara, A. Kehagias and A. Riotto, Fortsch. Phys. 62, 573 (2014) [arXiv:1403.5531 [hep-th]]; S. Ferrara, A. Kehagias and A. Riotto, Fortsch. Phys. 63, 2 (2015) [arXiv:1405.2353 [hep-th]]; R. Kallosh, A. Linde, B. Vercnocke and W. Chemissany, JCAP 1407, 053 (2014) [arXiv:1403.7189 [hep-th]]; K. Hamaguchi, T. Moroi and T. Terada, Phys. Lett. B 733, 305 (2014) [arXiv:1403.7521 [hep-ph]]; J. Ellis, M. A. G. García, D. V. Nanopoulos and K. A. Olive, JCAP 1405, 037 (2014) [arXiv:1403.7518 [hep-ph]]; J. Ellis, M. A. G. Garcia, D. V. Nanopoulos and K. A. Olive, JCAP 08, 044 (2014) [arXiv:1405.0271 [hep-ph]]; J. Ellis, M. A. G. García, D. V. Nanopoulos and K. A. Olive, JCAP 01, 010 (2015) [arXiv:1409.8197 [hep-ph]].
  * [43] R. Kallosh and A. Linde, JCAP 1306 (2013) 028 [arXiv:1306.3214 [hep-th]]; T. Li, Z. Li and D. V. Nanopoulos, JCAP 1404, 018 (2014) [arXiv:1310.3331 [hep-ph]]; C. P. Burgess, M. Cicoli and F. Quevedo, JCAP 1311 (2013) 003 [arXiv:1306.3512 [hep-th]]; F. Farakos, A. Kehagias and A. Riotto, Nucl. Phys. B 876, 187 (2013) [arXiv:1307.1137 [hep-th]]; S. Ferrara, R. Kallosh, A. Linde and M. Porrati, Phys. Rev. D 88 (2013) 8, 085038 [arXiv:1307.7696 [hep-th]]; W. Buchmüller, V. Domcke and C. Wieck, Phys. Lett. B 730, 155 (2014) [arXiv:1309.3122 [hep-th]]; C. Pallis, JCAP 1404, 024 (2014) [arXiv:1312.3623 [hep-ph]]; C. Pallis, JCAP 1408, 057 (2014) [arXiv:1403.5486 [hep-ph]]; I. Antoniadis, E. Dudas, S. Ferrara and A. Sagnotti, Phys. Lett. B 733, 32 (2014) [arXiv:1403.3269 [hep-th]]; T. Li, Z. Li and D. V. Nanopoulos, Eur. Phys. J. C 75, no. 2, 55 (2015) [arXiv:1405.0197 [hep-th]]; W. Buchmuller, E. Dudas, L. Heurtier and C. Wieck, JHEP 1409, 053 (2014) [arXiv:1407.0253 [hep-th]]; T. Terada, Y. Watanabe, Y. Yamada and J. Yokoyama, JHEP 1502, 105 (2015) [arXiv:1411.6746 [hep-ph]]; W. Buchmuller, E. Dudas, L. Heurtier, A. Westphal, C. Wieck and M. W. Winkler, JHEP 1504, 058 (2015) [arXiv:1501.05812 [hep-th]]; A. B. Lahanas and K. Tamvakis, Phys. Rev. D 91, no. 8, 085001 (2015) [arXiv:1501.06547 [hep-th]]; M. C. Romao and S. F. King, JHEP 1707, 033 (2017) [arXiv:1703.08333 [hep-ph]].
  * [44] M. Galante, R. Kallosh, A. Linde and D. Roest, Phys. Rev. Lett. 114, no.14, 141302 (2015) doi:10.1103/PhysRevLett.114.141302 [arXiv:1412.3797 [hep-th]].
  * [45] B. J. Broy, M. Galante, D. Roest and A. Westphal, JHEP 12, 149 (2015) doi:10.1007/JHEP12(2015)149 [arXiv:1507.02277 [hep-th]].
  * [46] S. Cecotti, Phys. Lett. B 190 (1987) 86.
  * [47] J. R. Ellis, C. Kounnas and D. V. Nanopoulos, Phys. Lett. 143B, 410 (1984).
  * [48] J. Ellis, M. A. G. García, D. V. Nanopoulos and K. A. Olive, JCAP 1510, 003 (2015) [arXiv:1503.08867 [hep-ph]]; J. Ellis, D. V. Nanopoulos, K. A. Olive and S. Verner, JCAP 08, 037 (2020) [arXiv:2004.00643 [hep-ph]].
  * [49] I. M. Rasulian, M. Torabian and L. Velasco-Sevilla, Phys. Rev. D 104, no.4, 044028 (2021) [arXiv:2105.14501 [hep-th]].