1 Introduction

Among the Swampland conjectures [1], one of the most popular and best tested is probably the weak gravity conjecture (WGC). Its simplest formulation [2] considers the case of a D-dimensional U(1) gauge theory, with a coupling constant g, and requires the existence of at least one state of mass m and charge q which satisfies:

$$\begin{aligned} gq\ge \sqrt{\frac{D-3}{D-2}}\kappa _D m, \end{aligned}$$
(1.1)

where \(\kappa _D\) is defined as \(\kappa ^2_D=8\pi G_D=\frac{1}{M_{P,D}^{D-2}}\) with \(M_{P,D}\) the reduced Planck mass in D dimensions. This inequality implies, among others, that in the non-relativistic limit, the Newton force is not stronger than the Coulomb force. The particular states for which the equality in (1.1) is satisfied are said to saturate the WGC. In this work we will be interested in a particular case of them.

The present work is dedicated to the study of two different generalizations of the WGC: one that arises when the gauge interaction is complemented by a dilaton interaction [3, 4], and another [5,6,7] that broadly requires the dominance of scalar interactions with respect to gravity in some scattering processes depending on the specific theory. We are interested in the modes that propagate in an extra dimension forming a tower of KK excitations [8,9,10,11]. We will explicitly show that these modes undergo gravitational and non-gravitational interactions of equal intensity, which allows us to use them as probes for the conjectured inequalities generalizing the one mentioned above. They will also be useful to investigate the behavior of the scalar WGC under compactification.

Obviously, the KK excitations considered here saturate the inequalities conjectured only at the classical level, to which our study will be limited, since both terms of these inequalities are in general corrected by quantum effects. However, one has in mind that extending the theory with enough supersymmetries, the KK modes can be BPS states which saturate them even at the quantum level.

The fact that KK modes saturate the inequalities of the various conjectures is a known property, but we will give a derivation of it here in a simple form that we have not found in the existing literature. Our derivation of the various inequalities will be based on amplitude calculations, not for example on the conditions for decay of extremal black holes, and some of the explicit expressions for the amplitudes needed to make the comparisons seem to be either missing or scattered and hard to find, so we hope that presenting them altogether here might be useful.

This work is organized as follows. Section 2 reviews the well-known reduction of KK from \(D+1\) to D dimensions of the Hilbert–Einstein action and a massless scalar. It allows us to introduce our notations, presents the Lagrangian expansion needed to extract the Feymann rules for calculating amplitudes, and compute the numerical factor in the total derivative term, often misquoted in the literature, which will be useful in Sect. 5. The dilatonic WGC inequality is derived in Sect. 3, where we also calculate various KK pair production amplitudes. In Sect. 4, we consider adding a mass term for the scalar in \(D+1\) dimensions and we find our form of the scalar WGC. A non-minimal coupling to gravity is considered in Sect. 5. The interactions due to the presence of higher dimensional gauge fields are discussed in Sect. 6. Our conclusions are presented in Sect. 7. Finally, some technical details about our calculations are gathered in appendices.

2 Expansion to second order in the gravitational field

We work with the signature \((+,-,...,-)\). The \(D+1\) dimensional quantities will be denoted with a hat. We use Latin and Greek letters for the D+1 and D-dimensional coordinates, respectively. We denote by x the D non-compact and by \(z \equiv z + 2 \pi L\) the compact coordinates. We recall the steps of the simple dimensional reduction of a free real massless scalar field \({\hat{\Phi }}\) coupled to General Relativity:

$$\begin{aligned} {\mathcal {S}}^{(D+1)}={\mathcal {S}}_{EH}^{(D+1)}+{\mathcal {S}}_{\Phi ,0}^{(D+1)}, \end{aligned}$$
(2.1)

where

$$\begin{aligned} {\mathcal {S}}_{EH}^{(D+1)}=\frac{1}{2\hat{\kappa }^2}\int \textrm{d}^{D+1}x \sqrt{(-1)^D\hat{g}}\,\hat{R}, \end{aligned}$$
(2.2)

and

$$\begin{aligned} {\mathcal {S}}_{\Phi ,0}^{(D+1)}=\int \textrm{d}^{D+1}x\,\,\sqrt{(-1)^{D}\hat{g}}\,\,\frac{1}{2}\hat{g}^{MN}\partial _M {\hat{\Phi }} \partial _N {\hat{\Phi }} \end{aligned}$$
(2.3)

The Ricci scalar \(\hat{R}\) is computed from the metric \(\hat{g}_{MN}\). In the simplest compactification from \(D+1\) to D dimensions it takes the form

$$\begin{aligned} \hat{g}_{MN}=\begin{pmatrix} e^{2\alpha \phi }g_{\mu \nu }-e^{2\beta \phi }A_{\mu }A_{\nu } &{} e^{2\beta \phi }A_{\mu } \\ e^{2\beta \phi }A_{\nu } &{} -e^{2\beta \phi } \end{pmatrix} \end{aligned}$$
(2.4)

with \(\phi \), \(A_{\mu }\) and \(g_{\mu \nu }\) D-dimensional fields independent of the z coordinate:

$$\begin{aligned}&{\mathcal {S}}_{EH}^{(D+1)}=\frac{1}{2\hat{\kappa }^2} \int \textrm{d}^{D+1}x\, \sqrt{(-1)^{D-1}g} \,\, e^{((D-2)\alpha +\beta )\phi }\bigg \{R\nonumber \\&\quad -\big [2(1-D)\alpha -2\beta \big ]\Box \phi \nonumber \\&\quad - \left[ (D-2)(1-D)\alpha ^2+2\beta \big ((2-D)\alpha -\beta \big )\right] (\partial \phi )^2 \nonumber \\&\quad -\frac{1}{4}e^{2(\beta -\alpha )\phi }F^2\bigg \}. \end{aligned}$$
(2.5)

where g is the determinant of the D-dimensional metric.

A canonical D-dimensional Einstein–Hilbert action is obtained for

$$\begin{aligned} (D-2)\alpha +\beta =0. \end{aligned}$$
(2.6)

and the canonical dilaton kinetic term fixes the constant \(\alpha \) to be:

$$\begin{aligned} \alpha ^2=\frac{1}{2(D-1)(D-2)}. \end{aligned}$$
(2.7)

Since all fields are independent of z, we can perform the integration over this coordinate to obtain, keeping only the zero modes,Footnote 1

$$\begin{aligned} {\mathcal {S}}_{0,0}^{(D)}= & {} \frac{2\pi L}{2\hat{\kappa }^2}\int \textrm{d}^Dx\sqrt{(-1)^{D-1}g}\left[ R+2\alpha \Box \phi +\frac{1}{2}(\partial \phi )^2\right. \nonumber \\{} & {} \left. -\frac{1}{4}e^{2(1-D)\alpha \phi }F^2\right] . \end{aligned}$$
(2.8)

We define the D-dimensional constant \(\kappa \) in terms of the \((D+1)\)-dimensional \(\hat{\kappa }\) as

$$\begin{aligned} \frac{1}{\kappa ^2}=\frac{2\pi L}{\hat{\kappa }^2} \Longrightarrow M_P^{D-2}=2\pi L\,{\hat{M}}_P^{D-1} \end{aligned}$$
(2.9)

In (2.4), the \(\phi \) and \(A_{\mu }\) fields are dimensionless. Dimensional fields, that we denote \({\tilde{\phi }}\) and \(\tilde{A}_{\mu }\), can be written as

$$\begin{aligned} {\tilde{\phi }}=\frac{\phi }{\sqrt{2}\kappa }; \,\,\,\, \,\,\,\, \tilde{A}_{\mu }=\frac{A_{\mu }}{\sqrt{2}\kappa } \end{aligned}$$
(2.10)

The action of the D-dimensional gauge and scalar fields, denoted as the graviphoton and the dilaton, respectively, reads:

$$\begin{aligned} {\mathcal {S}}_{0,0}^{(D)}= & {} \int \textrm{d}^Dx\sqrt{(-1)^{D-1}g}\left[ \frac{R}{2\kappa ^2}+2\alpha \Box {\tilde{\phi }}+\frac{1}{2}(\partial {\tilde{\phi }})^2\right. \nonumber \\{} & {} \left. -\frac{1}{4}e^{2\sqrt{2}(1-D)\alpha \kappa {\tilde{\phi }}}{\tilde{F}}^2\right] . \end{aligned}$$
(2.11)

In the following, with the exception of Sect. 5, the second term in (2.11), being a total derivative, will be discarded and, for notational simplicity, we remove the tilde in our notation.

For simplicity, we restrict to the simplest case where the field \({\hat{\Phi }}\) is periodic and single-valued on the compact dimension

$$\begin{aligned}{} & {} {\hat{\Phi }}(x,z+2\pi L)={\hat{\Phi }}(x,z),\nonumber \\{} & {} {\hat{\Phi }}(x,z)={\frac{1}{\sqrt{2}\pi L}}\sum _{n=-\infty }^{+\infty }\varphi _n(x)e^{\frac{inz}{L}}, \end{aligned}$$
(2.12)

which leads to

$$\begin{aligned} \mathcal {S}&=\int \textrm{d}^Dx\sqrt{(-1)^{D-1}g}\Bigg \{\frac{R}{2\kappa ^2}+\frac{1}{2}(\partial \phi )^2\nonumber \\&\quad -\frac{1}{4}e^{-2\sqrt{\frac{D-1}{D-2}\kappa {\phi }}F^2+\frac{1}{2}}\partial _\mu \varphi _0\partial ^\mu \varphi _0 \nonumber \\&\quad +\sum _{n=1}^{\infty }\left( \partial _\mu \varphi _n\partial ^\mu \varphi _n^*-\frac{n^2}{L^2}e^{2\sqrt{\frac{D-1}{D-2}}\kappa {\phi }}\varphi _n\varphi _n^*\right) \nonumber \\&\quad +\sum _{n=1}^\infty \left( i\sqrt{2}\kappa \frac{ n}{L } A^{\mu }\left( \partial _{\mu }\varphi _n \varphi _n^*-\varphi _n\partial _\mu \varphi _n^*\right) \right. \nonumber \\&\quad \left. + {2}\kappa ^2 \frac{n^2}{L^2}A_\mu A^\mu \varphi _n\varphi _n^* \right) \Bigg \}, \end{aligned}$$
(2.13)

where we have chosen in (2.7) the positive root for \(\alpha \). The complex scalars \(\varphi _n\) form the Kaluza–Klein (KK) tower and appear minimally coupled to the graviphoton. Around a generic background value \(\phi _0\) for the dilaton, the gauge coupling g is given by

$$\begin{aligned} g^2=e^{2\sqrt{\frac{D-1}{D-2}}\kappa {\phi _0}}. \end{aligned}$$
(2.14)

For each KK mode, the mass and charge read

$$\begin{aligned} gq_n=\sqrt{2}\kappa \frac{n}{L}e^{\sqrt{\frac{D-1}{D-2}}\kappa {\phi _0}} \; \qquad m_n=\frac{n}{L}e^{\sqrt{\frac{D-1}{D-2}}\kappa {\phi _0}}. \end{aligned}$$
(2.15)

This shows that they are related through

$$\begin{aligned} (gq_n)^2=2\kappa ^2m_n^2, \end{aligned}$$
(2.16)

saturating the dilatonic WGC condition. This is expected as all the interactions unify to descend from the unique gravitational interaction of a free scalar field in higher dimensions. Useful for the rest of the manuscript is to derive this result proceeding instead with the expansion of the metric (2.4) to second order:

$$\begin{aligned} {\hat{g}}_{MN}={\hat{\zeta }}_{MN}+2{\hat{\kappa }}{\hat{h}}_{MN}+4{\hat{\kappa }}^2{\hat{f}}_{MN}+o({\hat{\kappa }}^3) \end{aligned}$$
(2.17)

where:

$$\begin{aligned} {\hat{\zeta }}_{MN}= \begin{pmatrix} e^{2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}} \eta _{\mu \nu } &{} 0 \\ 0 &{} -e^{2\sqrt{2}\beta {\hat{\kappa }}{\phi _0}} \end{pmatrix}. \end{aligned}$$
(2.18)

is the background metric and \({\hat{\kappa }}^2{\hat{f}}_{MN}\ll {\hat{\kappa }} {\hat{h}}_{MN}\ll 1\), for all M, N. We write the perturbation as

$$\begin{aligned} {\left\{ \begin{array}{ll} {\hat{g}}_{M N}={\hat{\zeta }}_{M N}+2 \kappa {\hat{h}}_{M N}+4\kappa ^2 {\hat{f}}_{M N}+{\mathcal {O}}(\kappa ^3) \\ {\hat{g}}^{M N}={\hat{\zeta }}^{M N}+2 \kappa {\hat{t}}^{M N}+4\kappa ^2 {\hat{l}}^{M N}+{\mathcal {O}}(\kappa ^3). \end{array}\right. } \end{aligned}$$
(2.19)

The relation \({\hat{g}}_{M P}{\hat{g}}^{P N}\equiv \delta _{M}^{N}\) reads

$$\begin{aligned} {\left\{ \begin{array}{ll} {\hat{t}}^{M N}=-{\hat{h}}^{M N} \\ {\hat{l}}^{M N}+ {\hat{f}}^{M N}= {\hat{h}}^M_P {\hat{h}}^{P N}, \end{array}\right. } \end{aligned}$$
(2.20)

where it is understood that the indices are raised and lowered with the background metric \({\hat{\zeta }}\), then

$$\begin{aligned}&\sqrt{(-1)^D{\hat{g}}}{\mathcal {L}}_\Phi = \sqrt{(-1)^D{\hat{\zeta }}}\left[ \frac{1}{2}\partial _M {\hat{\Phi }}\partial ^M {\hat{\Phi }}-\frac{{\hat{\kappa }}'}{2}{\hat{h}}^{MN}\right. \nonumber \\&\quad \times \left( \partial _M {\hat{\Phi }}\partial _N {\hat{\Phi }}-\frac{1}{2}{\hat{\zeta }}_{MN}\partial _P {\hat{\Phi }}{\partial ^P} {\hat{\Phi }}\right) \nonumber \\&\quad \left. +\frac{{\hat{\kappa }}'^2}{2}\left( {\hat{l}}^{MN}-\frac{1}{2}{\hat{h}}^{MN}{\hat{h}}^P_{\,P}\right) \partial _M{\hat{\Phi }}\partial _N{\hat{\Phi }}\right. \nonumber \\&\quad \left. +\frac{{\hat{\kappa }}'^2}{4}\left( {\hat{f}}^P_{\,P}-\frac{1}{2}{\hat{h}}_{MP}{\hat{h}}^{PM}+\frac{1}{4} ({\hat{h}}^P_{\,P})^2\right) \partial _M{\hat{\Phi }}\partial ^M{\hat{\Phi }}\right] . \end{aligned}$$
(2.21)

where \({\hat{\kappa }}'\equiv 2{\hat{\kappa }}\). With:

$$\begin{aligned} {\hat{h}}^{MN}=\frac{1}{\sqrt{2\pi L}}\begin{pmatrix} e^{-2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}}\left( \sqrt{2}\alpha \phi \,\eta ^{\mu \nu }+h^{\mu \nu }\right) &{} -e^{-2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}} \frac{A^\mu }{\sqrt{2}} \\ -e^{-2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}} \frac{A^\nu }{\sqrt{2}} &{} -e^{-2\sqrt{2}\beta {\hat{\kappa }}{\phi _0}} \sqrt{2}\beta \phi \end{pmatrix},\nonumber \\ \end{aligned}$$
(2.22)

and using \(\sqrt{(-1)^D{\hat{\zeta }}}=e^{\sqrt{2}(D\alpha +\beta ) {\hat{\kappa }}{\phi _0}}\), this leads to the coupling between the leading order fluctuations \({{\hat{h}}^{MN}}\) of the metric and the stress–energy–momentum of the scalar field \({\hat{T}}^{{\hat{\Phi }}}_{MN}\):

$$\begin{aligned} {{\mathcal {L}}}_{int}^{(1)}&= -{\hat{\kappa }} {\hat{h}}^{MN} {\hat{T}}^{{\hat{\Phi }}}_{MN}=- {\hat{\kappa }} h^{\mu \nu } T^{(\varphi _0,\varphi _n)}_{\mu \nu }\nonumber \\&\quad -i \sqrt{2} {\hat{\kappa }} A^\mu \sum _{n=1}^{\infty }\frac{n}{L}\left( \partial _\mu \varphi _n\,\varphi _n^*-\varphi _n\,\partial _\mu \varphi _n^*\right) \nonumber \\&\quad -2\sqrt{\frac{D-1}{D-2}} {\hat{\kappa }} e^{2\sqrt{\frac{D-1}{D-2}}{\hat{\kappa }} \phi _0} \phi \sum _{n=1}^{\infty }\frac{n^2}{L^2}\varphi _n\varphi _n^*. \end{aligned}$$
(2.23)

Next, we identify \({\hat{f}}_{MN}\) from the metric decomposition at second order:

$$\begin{aligned} \hat{f}_{MN}=\frac{1}{2\pi L}\begin{pmatrix} e^{2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}}\bigg ( \alpha ^2\phi ^2\eta _{\mu \nu }+\sqrt{2}\alpha \phi \,h_{\mu \nu }+f_{\mu \nu }\bigg ) -\frac{1}{2}e^{2\sqrt{2}\beta {\hat{\kappa }}{\phi _0}} A_\mu A_\nu &{} e^{2\sqrt{2}\beta {\hat{\kappa }}{\phi _0}} \beta \phi A_\mu \\ e^{2\sqrt{2}\beta {\hat{\kappa }}{\phi _0}} \beta \phi A_\nu &{} -e^{2\sqrt{2}\beta {\hat{\kappa }}{\phi _0}} \beta ^2\phi ^2 \end{pmatrix}. \end{aligned}$$
(2.24)

With this result, \({\hat{l}}^{MN}\) in (2.21) is given by

$$\begin{aligned} {\hat{l}}^{MN}={\hat{h}}^{MP}{\hat{h}}_{\,P}^N-{\hat{f}}^{MN} \end{aligned}$$
(2.25)

Using (2.24) and (2.22) one obtains

$$\begin{aligned} {\hat{l}}^{MN} =\frac{1}{2\pi L}\begin{pmatrix} e^{-2\sqrt{2}\alpha {\hat{\kappa }} {\phi _0}}\bigg ( \alpha ^2\phi ^2\eta ^{\mu \nu }+\sqrt{2}\alpha \phi \,h^{\mu \nu }+l^{\mu \nu }\bigg ) &{} -e^{-2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}} \left( \alpha \phi A^\mu +\frac{1}{\sqrt{2}}h^{\mu \rho }A_\rho \right) \\ -e^{-2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}} \left( \alpha \phi A^\nu +\frac{1}{\sqrt{2}}h_\rho ^\nu A^\rho \right) &{} -e^{-2\sqrt{2}\beta {\hat{\kappa }}{\phi _0}} \beta ^2\phi ^2+e^{-2\sqrt{2}\alpha {\hat{\kappa }}{\phi _0}}\frac{1}{2}A_\rho A^\rho \end{pmatrix}. \end{aligned}$$
(2.26)

We define \(J_{\mu ,n}=(\varphi _n\partial _{\mu } \varphi _n^*-\varphi _n^*\partial _{\mu } \varphi _n)\), then the second order interaction in the Lagrangian is given by

$$\begin{aligned} {{\mathcal {L}}}_{int}^{(2)}&= \frac{1}{2} \partial _\mu \varphi _0 \partial _\nu \varphi _0\left[ \left( \frac{f^\rho _{\;\rho }}{2}-\frac{h^{\rho \sigma }h_{\rho \sigma }}{4}+\frac{(h^\rho _{\;\rho })^2}{8}\right. \right. \nonumber \\&\quad \left. \left. +\frac{1}{2}\left( D^2\alpha ^2+2D\beta \alpha +\beta ^2-4D\alpha ^2-4\beta \alpha +4\alpha ^2\right) \phi ^2 \right. \right. \nonumber \\&\quad \left. \left. +\frac{1}{2}((D-2)\alpha +\beta )\phi h^\rho _{\;\rho }\right) \eta ^{\mu \nu }+l^{\mu \nu }-\frac{1}{2}h^\rho _{\;\rho }h^{\mu \nu }\right] \nonumber \\&\quad +\sum _{n=1}^{\infty }\partial _\mu \varphi _n\partial _\nu \varphi _n^*\left[ \left( \frac{f^\rho _{\;\rho }}{2}-\frac{h^{\rho \sigma }h_{\rho \sigma }}{4}+\frac{(h^\rho _{\;\rho })^2}{8}\right. \right. \nonumber \\&\quad \left. \left. +\frac{1}{2}\left( D^2\alpha ^2+2D\beta \alpha +\beta ^2-4D\alpha ^2-4\beta \alpha +4\alpha ^2\right) \phi ^2 \right. \right. \nonumber \\&\quad \left. \left. +\frac{1}{2}((D-2)\alpha +\beta )\phi h^\rho _{\;\rho }\right) \eta ^{\mu \nu }+l^{\mu \nu }-\frac{1}{2}h^\rho _{\;\rho }h^{\mu \nu }\right] \nonumber \\&\quad -\sum _{n=1}^{\infty }\frac{n^2}{L^2}|\varphi _n|^2\left[ -A^2e^{\sqrt{2}(D\alpha -\beta )\kappa {\phi _0}}\left( \frac{f^\rho _{\;\rho }}{2}-\frac{h^{\rho \sigma }h_{\rho \sigma }}{4}\right. \right. \nonumber \\&\quad \left. \left. +\frac{(h^\rho _{\;\rho })^2}{8}+\left( \frac{1}{2}(D\alpha +\beta )-\beta \right) \phi h^\rho _{\;\rho }\right. \right. \nonumber \\&\quad \left. \left. +\frac{1}{2}(D^2\alpha ^2+2D\alpha \beta +\beta ^2-4D\alpha \beta -4\beta ^2+4\beta ^2)\phi ^2\right) \right] \nonumber \\&\quad -\sum _{n=1}^{\infty }i\frac{n}{L}h^{\rho \sigma }A_{\rho }J_{\sigma ,n}+i\frac{n}{L}A^{\rho }J_{\rho ,n}\nonumber \\&\quad \times \left( -\frac{h^\sigma _{\;\sigma }}{2}-((D-2)\alpha +\beta )\phi \right) \end{aligned}$$
(2.27)

This expression simplifies using the relation between \(\beta \) and \(\alpha \) (2.6). In particular, the coefficients of \(\phi ^2\) and \(\phi \, h_{\rho }^{\rho }\) vanish. One obtains

$$\begin{aligned}&{{\mathcal {L}}}_{int}^{(2)}= \, \, \frac{1}{2} \partial _\mu \varphi _0 \partial _\nu \varphi _0\left[ \left( \frac{f^\rho _{\;\rho }}{2}-\frac{h^{\rho \sigma }h_{\rho \sigma }}{4}+\frac{(h^\rho _{\;\rho })^2}{8}\right) \eta ^{\mu \nu }\right. \nonumber \\&\quad \left. +l^{\mu \nu }-\frac{1}{2}h^\rho _{\;\rho }h^{\mu \nu }\right] +\sum _{n=1}^{\infty }\partial _\mu \varphi _n\partial _\nu \varphi _n^*\left[ \left( \frac{f^\rho _{\;\rho }}{2}\right. \right. \nonumber \\&\quad \left. \left. -\frac{h^{\rho \sigma }h_{\rho \sigma }}{4}+\frac{(h^\rho _{\;\rho })^2}{8}\right) \eta ^{\mu \nu }+l^{\mu \nu }-\frac{1}{2}h^\rho _{\;\rho }h^{\mu \nu }\right] \nonumber \\&\quad -\sum _{n=1}^{\infty }\frac{n^2}{L^2}|\varphi _n|^2e^{{2\sqrt{2}(D-1)\alpha \kappa \phi _0}}\left( \frac{f^\rho _{\;\rho }}{2}-\frac{h^{\rho \sigma }h_{\rho \sigma }}{4}+\frac{(h^\rho _{\;\rho })^2}{8}\right) \nonumber \\&\quad -\sum _{n=1}^{\infty }\frac{n^2}{L^2}|\varphi _n|^2\left[ -A^2+e^{{2\sqrt{2}(D-1)\alpha \kappa \phi _0}}\left( 2(D-1)^2\alpha ^2\phi ^2\right. \right. \nonumber \\&\quad \left. \left. +(D-1)\alpha \phi h^\rho _{\;\rho }\right) \right] \nonumber \\&\quad +i\frac{n}{L}A^{\rho }\frac{h^\sigma _{\;\sigma }}{2}J_{\rho ,n}-i\frac{n}{L}h^{\rho \sigma }A_{\rho }J_{\sigma ,n} \end{aligned}$$
(2.28)

which shows how the gauge invariance of the graviphoton is recovered in this expansion at second order in \({\hat{\kappa }}\) and exhibits the minimal coupling of the graviphoton to the tower of scalars in \({\hat{\kappa }}\).

3 Scattering amplitudes and weak gravity conjectures

In this section, we will compute diverse \(2 \rightarrow 2\) amplitudes in the simple model defined above and compare two sets to be identified, one denoted as gravitational and the other as non-gravitational mediated interactions.

We expand the dilaton around its background value \(\phi _0\) as \(\phi _0+\phi \) in the action (2.13) to obtain:

$$\begin{aligned} {\mathcal {S}}_f&=\int \textrm{d}^Dx\sqrt{(-1)^{D-1}g}\left\{ \frac{R}{2\kappa ^2}+\frac{1}{2}(\partial \phi )^2-\frac{1}{4}e^{-2\sqrt{\frac{D-1}{D-2}\kappa \phi _0}} \right. \nonumber \\&\quad \times \left. \sum _{m=0}^\infty \left( -2\sqrt{\frac{D-1}{D-2}}\kappa \right) ^m\frac{\phi ^m}{m!}F^2 \right. \nonumber \\&\quad \left. +\frac{1}{2}\partial _\mu \varphi _0\partial ^\mu \varphi _0 +\sum _{n=1}^{\infty } \partial _\mu \varphi _n\partial ^\mu \varphi _n^* \right. \nonumber \\&\quad \left. -\sum _{n=1}^{\infty }\left( \frac{n^2}{L^2}e^{2\sqrt{\frac{D-1}{D-2}}\kappa {\phi _0}}\sum _{m=0}^\infty \left( 2\sqrt{\frac{D-1}{D-2}}\kappa \right) ^m\frac{\phi ^m}{m!}\varphi _n\varphi _n^*\right) \right. \nonumber \\&\quad \left. +\sum _{n=1}^\infty \left( i{\sqrt{2}}\kappa \frac{n}{L}A^{\mu }\left( \partial _{\mu }\varphi _n \varphi _n^*-\varphi _n\partial _\mu \varphi _n^*\right) \right. \right. \nonumber \\&\quad \left. \left. +{2}\kappa ^2 \frac{n^2}{L^2}A_\mu A^\mu \varphi _n\varphi _n^* \right) \right\} \end{aligned}$$
(3.1)

where diverse interactions can be identified. For instance:

  • 3 and 4-point vertices for minimally-coupled scalars to graviphotons appear in the last line. We can identify the KK electric charges

    $$\begin{aligned} gq_n={\sqrt{2}}\kappa \frac{n}{L} e^{\sqrt{\frac{D-1}{D-2}}\kappa \, {\phi _0}}. \end{aligned}$$
    (3.2)

  • In the third line, the m-th term (\(m\ne 0\)) in the sum gives a \((2+m)\)-point interaction with m dilatons and two KK scalars with coupling

    $$\begin{aligned} -i\left( 2\sqrt{\frac{D-1}{D-2}}\kappa \right) ^m\, \frac{n^2}{L^2} \, e^{2\sqrt{\frac{D-1}{D-2}}\kappa {\phi _0}}. \end{aligned}$$
    (3.3)

  • The m-th term in the sum in front of \(F^2\) in the first line gives a coupling of m dilatons with two gauge fields

    $$\begin{aligned} -i\left( -2\sqrt{\frac{D-1}{D-2}}\kappa \right) ^m \left( p_1\cdot p_2 \,\eta _{\mu \nu }-p_{1\,\nu }p_{2\,\mu }\right) . \end{aligned}$$
    (3.4)

Expansion of the metric around flat space-time \(g_{\mu \nu }=\eta _{\mu \nu }+2\kappa h_{\mu \nu }\) gives the usual minimal couplings to gravity for both the matter fields (\(\varphi _0\), \(\varphi _n\)) and the massless mediators (\(\phi \), \(A_\mu \)).

3.1 The dilatonic WGC

Consider the tree-level \(2\rightarrow 2\) scatteringFootnote 2\(^,\)Footnote 3\(\varphi _n(p_1)\varphi _n(p_2) \rightarrow \varphi _n(p_3) \varphi _n(p_4)\):

$$\begin{aligned} i{\mathcal {M}}&=ig^2q_n^2\left( \frac{(p_1+p_3)\cdot (p_2+p_4)}{t}+\frac{(p_1+p_4)\cdot (p_2+p_3)}{u}\right) \nonumber \\&\quad -4i\frac{D-1}{D-2}\kappa ^2 m^4_n\left( \frac{1}{t}+\frac{1}{u}\right) \nonumber \\&\quad -\frac{\kappa ^2}{4}\Bigg [\Big (p_{1\mu }p_{3\nu }+p_{3\mu }p_{1\nu }-\eta _{\mu \nu }\left( p_1\cdot p_3-m_n^2\right) \Big )\nonumber \\&\quad \times \frac{i{\mathcal {P}}^{\mu \nu \alpha \beta }}{t}\Big (p_{2\alpha }p_{4\beta }+p_{4\alpha }p_{2\beta }-\eta _{\alpha \beta }\left( p_2\cdot p_4-m_n^2\right) \Big )\nonumber \\&\quad +(t,p_3,p_4)\leftrightarrow (u,p_4,p_3) \Bigg ] \end{aligned}$$
(3.5)

where \({\mathcal {P}}\) is the usual massless spin-2 projector

$$\begin{aligned} {\mathcal {P}}^{\alpha \beta \rho \sigma }=\frac{\eta ^{\alpha \rho }\eta ^{\beta \sigma }+\eta ^{\alpha \sigma }\eta ^{\beta \rho }}{2}-\frac{\eta ^{\alpha \beta }\eta ^{\rho \sigma }}{D-2} \end{aligned}$$
(3.6)

and we have separated the contributions from the exchanges of the gauge boson, the dilaton and the graviton, respectively.

Taking the non-relativistic (NR) limit

$$\begin{aligned} \frac{s-4m_n^2}{m_n^2}\rightarrow 0,\qquad \frac{t}{m_n^2}\rightarrow 0,\qquad \textrm{and} \quad \frac{u}{m_n^2} \rightarrow 0 \end{aligned}$$
(3.7)

and expressing the charge in terms of the mass we obtain

$$\begin{aligned}{} & {} i{\mathcal {M}} \rightarrow i{\mathcal {M}}_{NR} =4im_n^2\left[ g^2q_n^2-\kappa ^2m_n^2\left( \frac{D-1}{D-2}+\frac{D-3}{D-2}\right) \right] \nonumber \\{} & {} \qquad \times \left( \frac{1}{t}+\frac{1}{u}\right) =0. \end{aligned}$$
(3.8)

The relation between the charge and the mass (2.16) ensures the cancellation between the three forces.

Fig. 1
figure 1

Feynman diagrams for the non-gravitational production of a pair of matter states \(\varphi _n,\varphi _n^*\) from two photons (first line), two dilatons (second line) and a dilaton and a photon (third line)

Full size image

It is straightforward to generalize this to see that dominance of the gauge interaction requires that a state with charge q and mass m satisfying the relation

$$\begin{aligned} g^2q^2\ge \left( \frac{\alpha ^2}{2}+\frac{D-3}{D-2}\right) \kappa ^2 m^2, \end{aligned}$$
(3.9)

where \(\alpha \) is the dilatonic coupling of the form \(e^{2\sqrt{2}\alpha \kappa \phi }F^2\), exists. We have therefore recovered in this explicit amplitude computation the dilatonic weak gravity conjecture that was derived in [3] (see also [4] for its generalization) from the study of the extremal Einstein–Maxwell-dilaton black hole solutions. In the absence of the massless dilaton field \(\alpha =0\), one trivially retrieves the original WGC condition

$$\begin{aligned} g^2q^2\ge \frac{D-3}{D-2}\kappa ^2 m^2. \end{aligned}$$
(3.10)

3.2 Amplitudes for pair production

Consider the production of a pair of matter states, here scalar KK states, of momenta \(p_3,\, p_4\) from massless particles of momenta \(p_1,\,p_2\). We can split the production processes into two sets:

  • Non-gravitational production: a pair of KK scalar modes \({|{\varphi _n, \varphi _n^*}\rangle }\) can arise from a pair of photons \({\langle {\gamma , \gamma }|}\), a pair of dilatons \({\langle {\phi ,\phi }|}\), or a dilaton and a photon \({\langle {\phi , \gamma }|}\).

  • Gravitational production: this includes the presence of a graviton G in initial states as \({\langle {G, G}|}\), \({\langle {G, \gamma }|}\) or \({\langle {G, \phi }|}\), but also gravitons as intermediate states in the production from \({\langle {\gamma , \gamma }|}\) or \({\langle {\phi , \phi }|}\). For later convenience, we further divide the gravitational production processes into purely gravitational (the \({\langle {G,G}|}\) production) and mixed (all the others).

3.2.1 Non gravitational amplitudes

The production from photons \(\gamma \gamma \rightarrow \varphi _n \varphi _n^*\) occurs through the coupling to the U(1) gauge boson plus an s-channel term mediated by the dilaton, as depicted in the first line of Fig. 1. These give:

$$\begin{aligned}&i{\mathcal {M}}_{\gamma \gamma }=ig^2q_n^2\, \, \epsilon _\mu (p_1)\epsilon _\nu (p_2)\left( \frac{(2 p_3^\mu -p_1^\mu ) (2 p_4^\nu -p_2^\nu )}{t-m_n^2}\right. \nonumber \\&\quad \left. +\frac{(2 p_4^\mu -p_1^\mu )(2 p_3^\nu -p_2^\nu )}{u-m_n^2}+2\eta ^{\mu \nu }\right) \nonumber \\&\quad -2i g^2q_n^2\frac{D-1}{D-2}\, \, \, \epsilon _\mu (p_1)\epsilon _\nu (p_2)\frac{p_1\cdot p_2 \eta ^{\mu \nu }-p_1^\nu p_2^\mu }{s}. \end{aligned}$$
(3.11)

We are interested in the threshold limit

$$\begin{aligned} \frac{s-4m_n^2}{m_n^2}\rightarrow 0,\qquad \frac{t+m_n^2}{m_n ^2}\rightarrow 0,\qquad \frac{u+m_n^2}{m_n^2}\rightarrow 0, \end{aligned}$$
(3.12)

leading to

$$\begin{aligned}&\left| {\mathcal {M}}_{\gamma \gamma }\right| ^2 \xrightarrow [\textrm{Threshold}] {} \frac{4}{(D-2)^2}\left[ (D-2) -\frac{3}{4}\frac{(D-1)^2}{(D-2)}\right. \nonumber \\&\quad \left. +\frac{D-1}{D-2}\right] g^4q_n^4=\left( \frac{D-3}{D-2}\right) ^2\frac{g^4q_n^4}{D-2} \end{aligned}$$
(3.13)

We note that, in a U(1) gauge theory with no dilaton, the amplitude would be given by the first line of (3.11) only, that means in the threshold limit \(4 g^4q^4/(D-2) \) for a state of charge q.

Fig. 2
figure 2

Feynman diagrams for pair production, gravitationally mediated, from photons and dilatons

Full size image

The production from a dilation pair \(\phi \phi \rightarrow \varphi _n \varphi _n^*\) (second line of Fig. 1) is immediately recognized to give a null result in the limit of interest:

$$\begin{aligned} i{\mathcal {M}}_{\phi \phi }&= - 4i\kappa ^2\frac{D-1}{D-2}m_n^4\left( \frac{1}{t-m_n^2}+\frac{1}{u-m_n^2}\right) \nonumber \\&\quad -4i\kappa ^2\frac{D-1}{D-2}m_n^2 \qquad \xrightarrow [\textrm{Threshold}] {} 0. \end{aligned}$$
(3.14)

Finally, the production from the pair photon-dilaton \(\phi \gamma \rightarrow \varphi _n \varphi _n^*\) receives contributions from the three \(s,t\, \textrm{and}\, u\)-channels (see the third line of Fig. 1)

$$\begin{aligned}&i{\mathcal {M}}_{\gamma (p_1)\phi (p_2)}\nonumber \\&\quad =\epsilon _\mu (p_1) \left\{ -2\sqrt{\frac{D-1}{D-2}}\kappa gq_n\left( p_1\cdot (p_1+p_2)g^{\mu \rho }\right. \right. \nonumber \\&\quad \left. \left. -p_1^\rho (p_1+p_2)^\mu \right) (p_3-p_4)_{\rho }\frac{i}{s} +2i\sqrt{\frac{D-1}{D-2}}\kappa gq_nm_n^2 \right. \nonumber \\&\quad \times \left. \left( \frac{(2p_3-p_1)^\mu }{t-m_n^2}-\frac{(2p_4-p_1)^\mu }{u-m_n^2}\right) \right\} , \end{aligned}$$
(3.15)

and this is easily verified to give a null contribution in the threshold limit.

3.2.2 Mixed amplitudes

We consider now the “mixed gravitational” processes: we start by computing the graviton s-channel mediation for \(\gamma \gamma \) and \(\phi \phi \) initial states, then the amplitudes with initial states \(\gamma \, G\) and \(\phi \, G\). We present hereafter the results for the particular case \(D=4\). When it will be of interest, we will show the results for a generic number of dimensions D.

The additional contribution to the \(\gamma \gamma \) and \(\phi \phi \) productions described in Fig. 2 respectively read

$$\begin{aligned}&i{\mathcal {M}}_{\gamma \gamma }^{\textrm{G}}=-\kappa ^2 \Big \{(p_1\cdot p_2)(\epsilon _{1\,\alpha }\epsilon _{2\,\beta }+\epsilon _{1\,\beta }\epsilon _{2\,\alpha })\nonumber \\&\quad +(p_{1\,\alpha }p_{2\,\beta }+p_{1\,\beta }p_{2\,\alpha })(\epsilon _1\cdot \epsilon _2)\nonumber \\&\quad -(\epsilon _{1\,\alpha }p_{2\,\beta }+p_{2\,\alpha }\epsilon _{1\,\beta })(p_1\cdot \epsilon _2) \nonumber \\&\quad -(\epsilon _{2\,\alpha }p_{1\,\beta }+p_{1\,\alpha }\epsilon _{2\,\beta })(p_2\cdot \epsilon _1)\nonumber \\&\quad -\eta _{\alpha \beta }(p_1\cdot p_2\, \epsilon _1\cdot \epsilon _2-\epsilon _1\cdot p_2\,\epsilon _2\cdot p_1)\Big \}\frac{i{\mathcal {P}}^{\alpha \beta \rho \sigma }}{s}\nonumber \\&\quad \times \Big \{p_{3\,\rho }p_{4\,\sigma }+p_{3\,\sigma }p_{4\,\rho }-\eta _{\rho \sigma }(p_3\cdot p_4+m^2)\Big \}, \end{aligned}$$
(3.16)

and

$$\begin{aligned}&i{\mathcal {M}}_{\phi \phi }^{\textrm{G}}= -\kappa ^2 \Big \{p_{1\,\alpha }p_{2\,\beta }+p_{1\,\beta }p_{2\,\alpha }-\eta _{\alpha \beta }p_1\cdot p_2\Big \}\nonumber \\&\quad \times \frac{i{\mathcal {P}}^{\alpha \beta \rho \sigma }}{s}\Big \{p_{3 \,\rho }p_{4 \,\sigma }+p_{3 \,\sigma }p_{4\, \rho }-\eta _{\rho \sigma }(p_3\cdot p_4+m^2)\Big \}, \end{aligned}$$
(3.17)

where \(\epsilon _{i}=\epsilon (p_i)\). For the \(\gamma \gamma \rightarrow \varphi _n \varphi _n^*\) amplitude, a (simpler) way to compute this is through projecting onto a specific basis for the polarizations \(\epsilon \) (see Appendix B).

Working in the center of mass frame for the massive particles, we obtain the different components of the graviton mediated \(\gamma \gamma \rightarrow \varphi _n \varphi _n^*\) as follows

$$\begin{aligned}&i{\mathcal {M}}^{\textrm{G}}_{+,+}=i{\mathcal {M}}^{\textrm{G}}_{-,-}\nonumber \\&=-i\frac{\kappa ^2}{s}\left[ tu-m_n^4+(m_n^2-u)^2+su \right] = 0\nonumber \\&i{\mathcal {M}}^{\textrm{G}}_{+,-}=i{\mathcal {M}}^{\textrm{G}}_{-,+}=i\frac{\kappa ^2}{s}\left[ tu-m_n^4 \right] , \end{aligned}$$
(3.18)

where the ± sign refers to the helicities of the incoming gauge bosons. In the threshold limit the graviton mediated contribution vanishes for both components.

In D dimensions, the whole \({\mathcal {M}}_{\gamma \gamma }\) amplitude reads

$$\begin{aligned} \left| {\mathcal {M}}_{\gamma \gamma }\right| ^2 \xrightarrow [\textrm{Threshold}] {}\frac{\left( 2 (D-2) (gq)^2+(D-4) \kappa ^2 m^2\right) ^2}{(D-2)^3}\nonumber \\ \end{aligned}$$
(3.19)

for a generic U(1) gauge theory (i.e. when the dilaton is put to zero) and

$$\begin{aligned} \left| {\mathcal {M}}_{\gamma \gamma }\right| ^2 \xrightarrow [\textrm{Threshold}] {}\frac{\left( (D-3) (gq_n)^2+(D-4) \kappa ^2 m_n^2\right) ^2}{(D-2)^3}\nonumber \\ \end{aligned}$$
(3.20)

in the dilatonic theory we are studying here. Both the results for the U(1) and dilatonic theory ((3.13) and discussion below) are recovered in the limit \(\kappa \rightarrow 0\). It is instructive to note, from these equations, that the vanishing of the graviton mediated contribution to the production from a photon pair is specific to the case of \(D=4\) dimensions, and in \(D\ne 4\) dimensions mixed terms of the form \(g^2q^2\times \kappa ^2m^2\) are generated.

For the \(\phi \phi \rightarrow \varphi _n \varphi _n^*\) the amplitude reads

$$\begin{aligned} i{\mathcal {M}}_{\phi \phi }^{\textrm{G}}=-i\frac{\kappa ^2}{s}\left[ m_n^4-ut-m_n^2 s \right] . \end{aligned}$$
(3.21)

This results in a non vanishing contribution in the limit of interest such that

$$\begin{aligned} i{\mathcal {M}}_{\phi \phi }^{\textrm{G}} = i\kappa ^2m_n^2. \end{aligned}$$
(3.22)
Fig. 3
figure 3

Feynman diagrams for the mixed pair production from a graviton and a photon

Full size image
Fig. 4
figure 4

Feynman diagrams for the mixed pair production from a graviton and a dilaton

Full size image

Concerning the mixed initial states, we have both \(\gamma \, G\rightarrow \varphi _n \varphi _n^*\) (see Fig. 3) and \(\phi \, G\rightarrow \varphi _n \varphi _n^*\) (see Fig. 4). Each of these two processes receive contributions from four diagrams.

Starting with the graviton-photon production, the amplitude \(G(p_1)\gamma (p_2)\rightarrow \varphi _n \varphi _n^*\) takes the form

$$\begin{aligned} i{\mathcal {M}}^{\mathrm {mix.}}_{G\gamma }&=i\kappa gq_n\Bigg (\frac{4(\epsilon _1\cdot p_3)^2\epsilon _2\cdot p_4}{t-m_n^2}-\frac{4(\epsilon _1\cdot p_4)^2\epsilon _2\cdot p_3}{u-m_n^2}\nonumber \\&\quad +2\epsilon _1\cdot \epsilon _2 \epsilon _1\cdot (p_3-p_4)\nonumber \\&\quad -\frac{(p_1+p_2)\cdot p_2(2\epsilon _1\cdot \epsilon _2 \epsilon _1\cdot (p_3-p_4)}{s}\Bigg ) \end{aligned}$$
(3.23)

and so for the different choices of graviton and photon helicities:

$$\begin{aligned} {\left\{ \begin{array}{ll} i{\mathcal {M}}^{\mathrm {mix.}}_{++,+}=- i{\mathcal {M}}^{\mathrm {mix.}}_{--,-}\\ \quad =-i\kappa gq_n\sqrt{2\frac{tu-m_n^4}{s}}\left( \frac{m_n^4-tu}{(t-m_n^2)(u-m_n^2)}+3\right) \\ i{\mathcal {M}}^{\mathrm {mix.}}_{++,-}=- i{\mathcal {M}}^{\mathrm {mix.}}_{--,+}=i\kappa gq_n\sqrt{2\frac{tu-m_n^4}{s}}\left( \frac{m_n^4-tu}{(t-m_n^2)(u-m_n^2)}\right) . \end{array}\right. }\nonumber \\ \end{aligned}$$
(3.24)

It is immediately verified that all these contributions vanish in the threshold limit where \(t\rightarrow -m_n^2\) and \(u\rightarrow -m_n^2\).

The same vanishing limit at threshold holds for the mixed graviton-dilaton production, where the amplitude is

$$\begin{aligned} i{\mathcal {M}}^{\mathrm {mix.}}_{G\phi }=-2i\kappa \mu _n\left( \frac{(\epsilon _1\cdot p_3)^2}{t-m_n^2}+\frac{(\epsilon _1\cdot p_4)^2}{u-m_n^2}\right) \end{aligned}$$
(3.25)

with \(\mu _n=\sqrt{6}\kappa m_n^2\) the three-point \(\phi \varphi _n\varphi _n^*\) \(D=4\) coupling, and finally

$$\begin{aligned} i{\mathcal {M}}^{\mathrm {mix.}}_{++}= i{\mathcal {M}}^{\mathrm {mix.}}_{--}=i\kappa \mu _n \frac{tu-m_n^4}{(t-m_n^2)(u-m_n^2)}. \end{aligned}$$
(3.26)

From the explicit results presented in Appendix B, it is also immediate to realize that the mixed contributions vanish at threshold for all D.

Fig. 5
figure 5

Feynman diagrams for the production of a pair of matter states from two gravitons

Full size image

3.2.3 Gravitational production amplitudes

Finally, we discuss the purely gravitational production. The starting point for the expression of the amplitude is rather long. It receives in fact contribution from the four diagrams of Fig. 5, each one with vertices determined from a two-derivative interacting term (some details about two-derivative interactions are discussed in Appendix A). We prefer to give here a more compact expression that is obtained after some algebra:

(3.27)

The complete results for each one of the four diagrams contributing to the amplitude are presented in Appendix B, together with the description of the helicity method. Using now the specific basis for \(D=4\) dimensions, we find

$$\begin{aligned} i{\mathcal {M}}_{++,++}=i{\mathcal {M}}_{--,--}&=i\kappa ^2 \left( \frac{\left( m_n^4-t u\right) m^2}{(t-m_n^2)(u-m_n^2)}+ m_n^2\right) \nonumber \\ i{\mathcal {M}}_{++,--}=i{\mathcal {M}}_{--,++}&=i\kappa ^2 \frac{\left( m_n^4-t u\right) ^2}{s \left( t-m_n^2\right) \left( u-m_n^2\right) }, \end{aligned}$$
(3.28)

Comparing this result with the one obtained from the \(\gamma \gamma \) production in the case with no dilaton, we verify the factorization

$$\begin{aligned} {\mathcal {M}}^{(GG)}_{++,++}&=\frac{\kappa ^2}{4 (gq)^4}\frac{\left( t-m_n^2\right) \left( u-m_n^2\right) }{s} {\mathcal {M}}^{(\gamma \gamma )}_{+,+}\nonumber \\ {\mathcal {M}}^{(GG)}_{++,--}&=\frac{\kappa ^2}{4 (gq)^4}\frac{\left( t-m_n^2\right) \left( u-m_n^2\right) }{s} {\mathcal {M}}^{(\gamma \gamma )}_{+,-}. \end{aligned}$$
(3.29)

The corresponding factorization for the comparison between the gravitational Compton scattering \(G\varphi \rightarrow G\varphi \) (with \(\varphi \) a generic scalar field) and the usual Compton scattering was found in [12, 13] (see also [14]).

From the above results, in the threshold limit we have

$$\begin{aligned} \left| {\mathcal {M}}_{GG}\right| ^2= & {} \frac{1}{4}\left( \left| {\mathcal {M}}_{++,++}\right| ^2+\left| {\mathcal {M}}_{++,--}\right| ^2+\left| {\mathcal {M}}_{--,++}\right| ^2\right. \nonumber \\{} & {} \left. +\left| {\mathcal {M}}_{--,--}\right| ^2\right) \rightarrow \frac{\kappa ^4 m_n^4}{2}. \end{aligned}$$
(3.30)

Note that the result \( \left| {\mathcal {M}}_{GG}\right| ^2\rightarrow \kappa ^4\,m^4/2\), and more generally the “purely gravitational” pair production, is independent from the presence of the dilaton. This is easily generalized to the case of generic D (see again Appendix B for details) and leads in the threshold limit to

$$\begin{aligned} \left| {\mathcal {M}}_{GG}\right| ^2\rightarrow \frac{1}{D-2}\kappa ^4 m_n^4. \end{aligned}$$
(3.31)

3.2.4 Gravitational vs gauge amplitudes

When the dilaton is put to zero, the requirement

$$\begin{aligned} \left| {\mathcal {M}}_{\gamma \gamma }\right| ^2\underset{\textrm{Threshold}}{\ge }\left| {\mathcal {M}}_{GG}\right| ^2 \end{aligned}$$
(3.32)

gives the original \(U(1),\, D=4\) WGC bound \(\sqrt{2} gq\ge \kappa m\).

Using cross-symmetry on the results of [12,13,14], the authors of [7] observed that (3.32) leads to the WGC relation and proposed (3.32) as a possible alternative formulation of the WGC. In [7], the graviton-mediated diagram was not taken into account in the \(\gamma \) amplitude. Our calculation shows that in the threshold limit, the contribution of this additional diagram disappears. Therefore, in the four-dimensional U(1) gauge theory, we can safely compare, as in the (3.32), the \(\gamma \gamma \) and GG productions without having to neglect any contribution.

Our calculation also shows that in \(D=4\) dimensions, the KK states saturate (3.32). In fact, we emphasize again that the gravitational amplitude \({\mathcal {M}}_{GG}\), here, does not care about the presence of the dilaton: whether the theory is a simple U(1) or a dilatonic U(1), the result for \({\mathcal {M}}_{GG}\) is unchanged. On the other hand, the amplitude \({\mathcal {M}}_{\gamma \gamma }\) receives an additional contribution which changes the numerical coefficient in front of \(g^4q^4\) from 2 to 1/8. Since the \({\mathcal {M}}_{\gamma \phi }\) and \({\mathcal {M}}_{\phi \phi }\) amplitudes both vanish in the threshold limit, the comparison of the pair production processes in this KK theory leads to

$$\begin{aligned} \frac{g^4 q_n^4}{8}\ge \frac{\kappa ^4 m_n^4}{2}\Longrightarrow gq\ge \sqrt{2}\kappa m \end{aligned}$$
(3.33)

and (2.16) shows that KK states saturate it.

However, if, in the presence of the dilaton, we consider gravitationally mediated diagrams for \(\gamma \gamma \) and \(\phi \phi \) amplitudes, there is a non-vanishing contribution that comes from \({\mathcal {M}}_{\phi \phi }^{\mathrm G}\) in (3.22), and this would clearly spoil the saturation observed for the KK states. The inclusion of the mixed production channels \(G\gamma \) (3.23) and \(G\phi \) (3.25) cannot restore the saturation property, since both do not contribute in the limit of interest. The dilatonic WGC will be recovered only if the contributions from graviton exchanges in \(\gamma \gamma \) and \(\phi \phi \) amplitudes are not included.

Note also that the pairwise production comparison does not reproduce the constraints of WGCs in more than 4 dimensions. The \({\mathcal {M}}_{\gamma \gamma }\) and \({\mathcal {M}}_{GG}\) amplitudes lead, for any D, to compare \(\sqrt{2} gq\) and \(\kappa m\). For the case of a simple theory U(1), setting as quoted above the dilaton to zero in our calculations, the result for the production from a photon pair in D dimensions in the threshold limit is

$$\begin{aligned} \left| {\mathcal {M}}_{\gamma \gamma }\right| ^2=\frac{4}{D-2}(gq)^4. \end{aligned}$$
(3.34)

In Appendix B we learn that the purely gravitational production of pairs gives, in the same limit of interest,

$$\begin{aligned} \left| {\mathcal {M}}_{GG}\right| ^2=\frac{1}{D-2}(\kappa m)^4. \end{aligned}$$
(3.35)

By comparing (3.34) and (3.35), it is immediate to observe that requiring \(\left| {\mathcal {M}}_{\gamma }\right| ^2 \ge \left| {\mathcal {M}}_{GG}\right| ^2\), one does not reproduce the WGC bound

$$\begin{aligned} gq\ge \sqrt{\frac{D-3}{D-2}}\kappa m. \end{aligned}$$
(3.36)

Similarly, the comparison of purely gravitational pair production and purely non-gravitational pair production in the KK theory we consider here amounts to a comparison of the results

$$\begin{aligned}{} & {} \left| {\mathcal {M}}_{\gamma \gamma }\right| ^2\rightarrow \frac{(D-3)^2}{(D-2)^3}g^4q_n^4,\; \left| {\mathcal {M}}_{\phi \phi }\right| ^2\rightarrow 0,\nonumber \\{} & {} \left| {\mathcal {M}}_{GG}\right| ^2 \rightarrow \frac{1}{D-2}\kappa ^4 m_n^4. \end{aligned}$$
(3.37)

Using (2.16), it is immediate to realize that the KK states saturate the (3.32) (or an equivalent generalization of it to include the \({\mathcal {M}}_{\phi \phi }\) contribution which disappears here) only for \(D=4\). The results of Sect. 3.2.2 show that the addition of mixed contributions does not change this.

4 Massive and self-interacting scalars

We next consider the presence of mass and self-interacting terms in the higher dimensional scalar theory. The KK scalar modes are no more extremal states of the WGC, but this set-up will allow us to retrieve scalar weak gravity conjectures which are postulated to constrain the relative strength of the additional terms.

We will consider the simple extension of (2.1)

$$\begin{aligned} {\mathcal {S}}_{int}=\int \textrm{d}^{D+1}x \sqrt{(-1)^D{\hat{g}}}\, \left[ -\frac{1}{2}{\hat{m}}^2{\hat{\Phi }}^2+\frac{{\hat{\mu }}}{3!}{\hat{\Phi }}^3-\frac{{\hat{\lambda }}}{4!}{\hat{\Phi }}^4 \right] .\nonumber \\ \end{aligned}$$
(4.1)

Here, \({\hat{m}}\) has mass dimension one, \({\hat{\mu }}\) has dimension \(3-\frac{D+1}{2}\) and \(\lambda \) has dimension \(4-(D+1)\). Using the ansatz (2.12), it is straightforward to see that the action takes the form

$$\begin{aligned} {\mathcal {S}}&={\mathcal {S}}_f+{\mathcal {S}}_{int} \nonumber \\&=\int \textrm{d}^Dx\sqrt{(-1)^{D-1}g}\left\{ \frac{R}{2\kappa ^2}+ \frac{1}{2}(\partial \phi )^2-\frac{1}{4}e^{-2\sqrt{\frac{D-1}{D-2}\kappa {\phi }}}F^2\right. \nonumber \\&\quad \left. +\frac{1}{2}\partial _\mu \varphi _0\partial ^\mu \varphi _0 -\frac{1}{2}e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi }}{\hat{m}}^2\varphi _0^2 \right. \nonumber \\&\quad \left. +\sum _{n=1}^{\infty } \partial _\mu \varphi _n\partial ^\mu \varphi _n^* -\sum _{n=1}^{\infty }\left( e^{2\sqrt{\frac{D-1}{D-2}}\kappa {\phi }} \frac{n^2}{L^2} +e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi }}{\hat{m}}^2\right) \varphi _n\varphi _n^* \right. \nonumber \\&\quad +\sum _{n=1}^\infty \left[ i{\sqrt{2}}\kappa \frac{n}{L}A^{\mu }\left( \partial _{\mu }\varphi _n \varphi _n^*-\varphi _n\partial _\mu \varphi _n^*\right) +{2}\kappa ^2 \frac{n^2}{L^2}A_\mu A^\mu \varphi _n\varphi _n^* \right] \nonumber \\&\quad +e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi }} \Bigg [ \frac{\mu }{3!}\varphi _0^3-\frac{\lambda }{4!}\varphi _0^4 +\mu \varphi _0\sum _{n=1}^\infty \varphi _n\varphi _n^*-\frac{\lambda }{2}\varphi _0^2\sum _{n=1}^\infty \varphi _n\varphi _n^* \nonumber \\&\quad -\frac{\lambda }{2}\varphi _0\sum _{m,n=1}^{\infty }\left( \varphi _m\varphi _m\varphi ^*_{n+m} +\varphi _m^*\varphi _n^*\varphi _{n+m}\right) \nonumber \\&\quad +\frac{\mu }{2}\sum _{n,m=1}^\infty \left( \varphi _n\varphi _m\varphi _{n+m}^*+\varphi _n^*\varphi _m^*\varphi _{n+m}\right) \nonumber \\&\quad -\frac{\lambda }{3!}\sum _{m,n,p=1}^\infty \left( \varphi _m\varphi _n\varphi _p\varphi ^*_{m+n+p}+\varphi ^*_m\varphi ^*_n\varphi ^*_p\varphi _{m+n+p}\right) \nonumber \\&\quad -\frac{\lambda }{2}\sum _{n=1}^{\infty }\varphi _n\varphi _n^*\sum _{m=1}^{\infty }\varphi _m\varphi _m^*-\frac{\lambda }{4}\sum _{\underset{m\ne p,n\ne p ; m+n>p}{m,n,p=1}}^\infty \varphi _m\varphi _n\varphi _p^*\varphi _{n+m-p}^*\Bigg ]\Bigg \}, \end{aligned}$$
(4.2)

where we have kept the notation compact, but, in our perturbative analysis, the dilaton will again be expanded around a background value \(\phi _0\) as above. The couplings constants \(\mu \) and \(\lambda \) are defined, from their higher dimensional counterpart, as

$$\begin{aligned} \mu =\frac{{\hat{\mu }}}{\sqrt{2\pi L}}, \qquad \lambda =\frac{{\hat{\lambda }}}{2\pi L}. \end{aligned}$$
(4.3)

The tree-level masses for the zero mode \(\varphi _0\) and the KK excitations are given by:

$$\begin{aligned} m^2_0= & {} e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}{\hat{m}}^2, \qquad m_n^2= e^{2\sqrt{\frac{D-1}{D-2}}\kappa {\phi _0}}\frac{n^2}{L^2}\nonumber \\{} & {} +e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}{\hat{m}}^2. \end{aligned}$$
(4.4)

4.1 The scalar weak gravity conjecture

Fig. 6
figure 6

Feynman diagrams for the \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) scattering when a potential for the higher dimensional scalar, “parent” of \(\varphi _0\), has been turned o.n

Full size image

We start by computing the \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) amplitude. The diagrams intervening in the scattering are presented in the Fig. 6. The non-relativistic limit of the tree-level amplitude reads

$$\begin{aligned}&i{\mathcal {M}}=ie^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}\left[ e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}\frac{5}{3}\frac{\mu ^2}{m_0^2}-\lambda \right] \nonumber \\&\quad -\frac{i}{(D-1)(D-2)}\kappa ^2 m_0^2\nonumber \\&\quad -4\frac{i}{(D-1)(D-2)}\kappa ^2 m_0^4\left( \frac{1}{t}+\frac{1}{u}\right) \nonumber \\&\quad +i\frac{D-1}{D-2}\kappa ^2 m_0^2-4i\frac{D-3}{D-2}\kappa ^2 m_0^4\left( \frac{1}{t}+\frac{1}{u}\right) , \end{aligned}$$
(4.5)

where the different lines correspond to the contributions from the self-interaction, dilaton and graviton exchanges, respectively.

Following [6], we compare the contributions to the amplitude at the energy scale given by the (massive) external states at rest. In the non-relativistic limit, we can further split (4.5) into contributions from short and long range interactions. We can identify an effective contact interaction:

$$\begin{aligned}&i{\mathcal {M}}_{CT}^{(D)}= ie^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}\left( \frac{5}{3}\frac{\mu ^2}{{\hat{m}}^2}-\lambda -\frac{1}{(D-1)(D-2)}\kappa ^2 {\hat{m}}^2\right. \nonumber \\&\quad \left. +\frac{D-1}{D-2}\kappa ^2 {\hat{m}}^2\right) =i\frac{e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}}{2\pi L} \nonumber \\&\quad \times \left( \frac{5}{3}\frac{{\hat{\mu }}^2}{{\hat{m}}^2}-{\hat{\lambda }}+2\pi L\frac{D}{D-1}\kappa ^2 {\hat{m}}^2\right) . \end{aligned}$$
(4.6)

where in the first line we can identify the contributions from the scalar interaction for the first two terms, then from the dilaton and graviton, respectively. Using (2.9) and the \((D+1)\)-gravitational coupling \({\hat{\kappa }}=\sqrt{2\pi L}\,\kappa \), the last term is recognized to be the gravitational s-channel contribution to the \({\hat{\Phi }}{\hat{\Phi }}\rightarrow {\hat{\Phi }}{\hat{\Phi }}\) scattering in \(D+1\) dimensions:

$$\begin{aligned} i{\mathcal {M}}_{CT}^{(D+1)}= & {} i\frac{e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}}{2\pi L}\nonumber \\{} & {} \times \left( \frac{5}{3}\frac{{\hat{\mu }}^2}{{\hat{m}}^2}-{\hat{\lambda }}+ \frac{(D+1)-1}{(D+1)-2}{\hat{\kappa }}^2 {\hat{m}}^2\right) . \end{aligned}$$
(4.7)

The above equation illustrates the fact that constraining the scalar interactions of the field \({\hat{\Phi }}\) to be dominant with respect to gravity in \(D+1\) dimensions is enough to ensure that the scalar interactions of the zero mode \(\varphi _0\) are dominant with respect to the combination of gravitational and dilatonic contributions in D dimensions. In other words, the effective (tree-level) non-relativistic four-point function of the zero mode \(\varphi _0\) that emerges in the reduced-dimensional theory is the same as the effective non-relativistic four-point coupling for the “parent” field \({\hat{\Phi }}\) in the higher-dimensional theory. Requiring that in such a contact term, the contributions of the \({\hat{\Phi }}\) self-interactions are the dominant ones in the \(D+1\) dimensions automatically ensures that the same property holds for the \(\varphi _0\) self-interactions with respect to the set of interactions that appear in the D dimensional theory.

It is interesting to observe that the higher dimensional result is recovered here thanks to a cancellation, rather than an addition, between the graviton and dilaton mediated diagrams. This is dictated by the form of the D-dependent coefficient \(\gamma _s(D) \equiv (D-1)/(D-2)\) appearing in front of the graviton-mediated amplitude in the s-channel which decreases with D: \(\gamma _s(D+1)<\gamma _s(D)\). The dimension-dependent factor appearing in the t and u-channels, \(\gamma _{t,u}(D) \equiv (D-3)/(D-2)\) vary in the opposite direction. In other words, the peculiar feature is that, for the contact terms, the spin-2 and spin-0 bosonic mediators give opposite contributions. This feature will also appear in the amplitudes computed with the non minimal coupling to gravity. As a consequence of particular interest in the case of a massive dilaton the higher dimensional sub-dominance of gravity does not imply that gravity by itself (i.e. without the dilaton) is subdominant in the lower dimensional theory too. This violation happens in the parametric region

$$\begin{aligned} \frac{D}{D-1}{\hat{\kappa }}^2{\hat{m}}^2\le \left| \frac{5}{3}\frac{{\hat{\mu }}^2}{{\hat{m}}^2}-{\hat{\lambda }}\right| \le \frac{D-1}{D-2} {\hat{\kappa }}^2{\hat{m}}^2, \end{aligned}$$
(4.8)

which is an interval of lenght \({\hat{\kappa }}^2{\hat{m}}^2/(D-1)(D-2)\) inversely proportional to the dimension D.

Fig. 7
figure 7

Feynman diagrams for the \(\varphi _n\varphi _n\rightarrow \varphi _n\varphi _n\) scattering in the t-channel

Full size image

The amplitude \(\varphi _n\varphi _n\rightarrow \varphi _n\varphi _n\) provides a generalization in the presence of self-interacting terms of the computation done in Sect. 3.1. The scattering amplitude receives contributions from gauge bosons, dilatons, gravitons in the t and u-channels, \(\varphi _0\) exchange, from the s-channel exchange of a \(\varphi _{2n}\) particle and from a 4-point contact term. These are the diagrams that are presented in Fig. 7 and lead to

$$\begin{aligned}&i {\mathcal {M}}= -ie^{\frac{4}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}\mu ^2\left( \frac{1}{s-m_{2n}^2}+\frac{1}{t-m_0^2}+\frac{1}{u-m_0^2}\right) \nonumber \\&\quad -i\lambda e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0} +i\left( \frac{1}{t}+\frac{1}{u}\right) \nonumber \\&\quad \times \left( 4g^2q_n^2m_n^2-4\frac{D-3}{D-2}\frac{m_n^4}{M_P^{D-2}}-(\partial _\phi m_n^2)^2\right) \end{aligned}$$
(4.9)

with

$$\begin{aligned} \partial _\phi m_n^2= & {} \frac{1}{M_P^{(D-2)/2}}\left( \frac{2}{\sqrt{(D-1)(D-2)}}e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}{\hat{m}}^2\right. \nonumber \\{} & {} \left. +2\sqrt{\frac{D-1}{D-2}}e^{2\sqrt{\frac{D-1}{D-2}}\kappa {\phi _0}}\frac{n^2}{L^2}\right) . \end{aligned}$$
(4.10)

4.2 Massive dilatons

Let us consider for our illustrative discussion a simple potential for the dilaton in a polynomial expansion of the form

$$\begin{aligned} V(\phi )=\frac{1}{2}m^2_\phi \phi ^2-\frac{\mu _\phi }{3!}\phi ^3+\frac{\lambda _\phi }{4!} \phi ^4. \end{aligned}$$
(4.11)

In the \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) scattering amplitude (4.5), the addition of a dilaton mass gives in the non-relativistic limit

$$\begin{aligned}&i{\mathcal {M}}(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0)=ie^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}\nonumber \\&\quad \times \left[ e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}\frac{5}{3}\frac{\mu ^2}{m_0^2}-\lambda \right] \nonumber \\&\quad -4\frac{i}{(D-1)(D-2)}\kappa ^2 m_0^4\frac{1}{s-m^2_\phi }\nonumber \\&\quad -4\frac{i}{(D-1)(D-2)}\kappa ^2 m_0^4\left( \frac{1}{t-m^2_\phi }+\frac{1}{u-m^2_\phi }\right) \nonumber \\&\quad +i\frac{D-1}{D-2}\kappa ^2 m_0^2-4i\frac{D-3}{D-2}\kappa ^2 m_0^4\left( \frac{1}{t}+\frac{1}{u}\right) , \end{aligned}$$
(4.12)

where the limit still needs to be implemented in the dilaton propagators according to its mass. We can thus follow the evolution of \({\mathcal {M}}\) with respect to \(m_\phi \) to better expand it.

For the \(\varphi _n\varphi _n\rightarrow \varphi _n\varphi _n\) case, the scattering amplitude with the massive dilaton reads

$$\begin{aligned}&i{\mathcal {M}}(\varphi _n\varphi _n\rightarrow \varphi _n\varphi _n)= -ie^{\frac{4}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}}\mu ^2\nonumber \\&\quad \times \left( \frac{1}{s-m_{2n}^2}+\frac{1}{t-m_0^2}+\frac{1}{u-m_0^2}\right) -i\lambda e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\phi _0}} \nonumber \\&\quad -i(\partial _\phi m_n^2)^2\left( \frac{1}{t-m^2_\phi }+\frac{1}{u-m^2_\phi }\right) \nonumber \\&\quad +i\left( \frac{1}{t}+\frac{1}{u}\right) \left( 4g^2q_n^2m_n^2-4\frac{D-3}{D-2}\kappa ^2 m_n^4\right) . \end{aligned}$$
(4.13)

Putting all the analysis for both the \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) and \(\varphi _n\varphi _n\rightarrow \varphi _n\varphi _n\) scattering amplitudes together, we give a brief overview of the results here.

When the mass \(m_\phi \) of the dilaton is less than that of the zero mode, \(m_0\), its mass can be neglected to first order in an expansion, in powers of \(m_\phi \) over the exchanged momentum, and requiring that the self-interactions of a scalar field dominate in \(D+1\) dimensions is sufficient to ensure that the same property is verified by its zero mode in D dimensions; a result that follows from the studies of the previous sections. As soon as the mass of the dilaton is comparable to that of the 0-mode, the massless dilaton approximation is no longer adequate and an appropriate discussion must be made for different denominators involving \(m_\phi \), \(m_0\), \(m_n\) and \(m_{2n}\). The analysis can be done easily but it is cumbersome and not really illuminating. In short, there is no easy way to relate combinations appearing in D dimensions in this case with quantities already constrained, by assumption, in \(D+1\) dimensions.

5 \({\hat{\Phi }}^2 R\) interaction

Let us consider now the effect on the different D-dimensional amplitudes of the presence of a non-minimal coupling to gravity of the form

$$\begin{aligned} S_{(\xi )}=\int d^{D+1}x \sqrt{(-1)^D{\hat{g}}}\, \frac{\xi }{2} {\hat{\Phi }}^2 {\hat{R}}, \end{aligned}$$
(5.1)

with \({\hat{R}}\) the Ricci scalar (see for example [15]). We assume here that \({\langle {{\hat{\Phi }}}\rangle } = 0\) as a non-vanishing vev would correspond to a redefinition of the Planck mass and a shift of the canonical fields. After compactification, one gets:

(5.2)

This leads to new three-point couplings. First, using the linear expansion of the metric \(g_{\mu \nu }=\eta _{\mu \nu }+2\kappa h_{\mu \nu }\), the R term gives the new coupling \(\kappa (\partial _\mu \partial _\lambda h^{\mu \lambda }-\Box h^\lambda _{\,\lambda })\left( \varphi _0^2+2\sum _{n=1}^{\infty }\varphi _n\varphi _n^*\right) \) of the graviton to the scalar matter fields. Then, the \(\nabla _\mu \partial ^\mu \phi \) term, that we discard in previous sections as it takes the form of a total derivative, gives an additional three-point vertex between the dilaton and the matter fields and can enter, for example, in the computation of the dilatonic force in the non-relativistic limit. At first order in \(\kappa \), we can write \(\kappa {\nabla _\mu \partial ^\mu \phi }=\kappa {\partial _\mu \partial ^\mu \phi }+\mathcal {\mathcal {O}}(\kappa ^2)\), the Christoffel symbols starting themselves at order \(\kappa \).

The \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) amplitude resulting from the action (5.2), receives a contribution from the dilaton exchange (see Appendix A for some details on the Feynman rules for two-derivative vertices)

$$\begin{aligned} i{\mathcal {M}}_\phi= & {} -i\frac{4}{(D-1)(D-2)}\xi ^2\kappa ^2(s+t+u)\nonumber \\= & {} -i \frac{16}{(D-1)(D-2)}\xi ^2\kappa ^2m_0^2. \end{aligned}$$
(5.3)

and one from the graviton

$$\begin{aligned} i{\mathcal {M}}_G=4i\frac{D-1}{D-2}\xi ^2\kappa ^2(s+t+u)= 16i\frac{D-1}{D-2}\xi ^2\kappa ^2m_0^2.\nonumber \\ \end{aligned}$$
(5.4)

Their sum gives

$$\begin{aligned} i{\mathcal {M}}_{(non{\text {-}}minimal)}= & {} i{\mathcal {M}}_\phi +i{\mathcal {M}}_G\nonumber \\= & {} 4i\frac{D}{D-1}\xi ^2\kappa ^2(s+t+u)\nonumber \\= & {} 16i\frac{D}{D-1}\xi ^2\kappa ^2m_0^2. \end{aligned}$$
(5.5)

This matches the result one would obtain for the \({\hat{\Phi }}{\hat{\Phi }}\rightarrow {\hat{\Phi }}{\hat{\Phi }}\) scattering in \(D+1\) dimensions.

At this point, we have computed tree-level four point amplitudes where both vertices arise either from minimal or non-minimal couplings to gravity in D+1 dimensions. In order to compute the total \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) amplitude we need to compute the contribution from “mixed” diagrams involving one minimal and one non-minimal vertices. This mixed gravitational diagrams give in the s-channel

$$\begin{aligned}&i{\mathcal {M}}^{\text {G-mix.}}_{\text {s-channel}}=-2i\xi \frac{\kappa ^2}{s}\Bigg (2\,p_1\cdot (p_1+p_2)\,p_2\cdot (p_1+p_2)\nonumber \\&\quad -(p_1+p_2)^2(p_1\cdot p_2+m_0^2)\nonumber \\&\quad +\frac{2}{D-2}(p_1+p_2)^2(p_1\cdot p_2)\nonumber \\&\quad -\frac{D}{D-2}(p_1+p_2)^2(p_1\cdot p_2+m_0^2)\Bigg ) \end{aligned}$$
(5.6)

and in the t-channel

$$\begin{aligned}&i{\mathcal {M}}^{\text {G-mix.}}_{\text {t-channel}}=2i\xi \frac{\kappa ^2}{t}\Bigg (2\,p_1\cdot (p_1-p_3)\,p_3\cdot (p_1-p_3)\nonumber \\&\quad -(p_1-p_3)^2(p_1\cdot p_3-m_0^2)+\frac{2}{D-2}(p_1-p_3)^2(p_1\cdot p_3)\nonumber \\&\quad -\frac{D}{D-2}(p_1-p_3)^2(p_1\cdot p_3-m_0^2)\Bigg ), \end{aligned}$$
(5.7)

while the u channel can be obtained through the replacements \(t \leftrightarrow u \) and \(p_3 \leftrightarrow p_4\). After some simple algebra, their sum reads

$$\begin{aligned}&i{\mathcal {M}}^{\text {G-mix.}}_{\text {s-channel}}=i\xi \kappa ^2\left( s+\frac{4 m_0^2}{D-2}\right) ; \, i{\mathcal {M}}^{\text {G-mix.}}_{\text {t-channel}}\nonumber \\&\quad =i\xi \kappa ^2\left( t+\frac{4 m_0^2}{D-2} \right) ; \, i{\mathcal {M}}^{\text {G-mix.}}_{\text {u-channel}}=i\xi \kappa ^2\left( u+\frac{4m_0^2}{D-2} \right) \nonumber \\&\quad \Longrightarrow i{\mathcal {M}}^{\text {G-mix.}}=i\xi \kappa ^2 \left( s+t+u+\frac{12}{D-2}m_0^2\right) \nonumber \\&\quad =4i\xi \kappa ^2\frac{D+1}{D-2} m_0^2. \end{aligned}$$
(5.8)

The computation of the similar mixed diagrams with dilaton exchange gives

$$\begin{aligned} i{\mathcal {M}}^{\phi }{\text {-mix.}}=-12i\xi \kappa ^2\frac{m_0^2}{(D-1)(D-2)}, \end{aligned}$$
(5.9)

where each channel contributes the same amount.

Summing up all the contributions, the final result for the amplitude is

$$\begin{aligned} i{\mathcal {M}}^{\mathrm {mix.}}= 4i\xi \kappa ^2 \frac{D+2}{D-1}m_0^2, \end{aligned}$$
(5.10)

as it is expected from the higher dimensional Lagrangian. Again, the higher dimensional gravitational contribution is obtained after a cancellation between the effective spin-2 and spin-0 mediators. From the two results obtained above, we see that the direct non-minimal coupling to gravity (5.1) contributes with a constant term in the \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) amplitude. If one takes the non-minimal coupling into account from the start and modifies the SWGC in D generic dimensions requiring

$$\begin{aligned} \left| \frac{5}{3}\frac{{\hat{\mu }}^2}{ {\hat{m}}^2}-\lambda \right| \ge \left( \frac{D-1}{D-2}+4\xi \frac{D+1}{D-2}+16\xi ^2\frac{D-1}{D-2}\right) {\hat{\kappa }}^2 {\hat{m}}^2,\nonumber \\ \end{aligned}$$
(5.11)

the same property will be respected by the zero mode \(\varphi _0\) in \(D-1\) dimensions with the replacement of hatted by unhatted quantities \({{\hat{\mu }}^2}, \ldots \rightarrow \mu ^2, \ldots \).

In the \(\varphi _0\varphi _0\rightarrow \varphi _0\varphi _0\) scattering, the four point amplitudes appear as a sum of the three channels s, t, u whose coefficients add-up to a factor \(s+t+u= 4m_0^2\). Therefore, the total amplitude does not increase with the exchanged momentum. This is not always the case as for example in the two examples of the \(\varphi _n\varphi _n\rightarrow \varphi _n\varphi _n\) or \(\varphi _n\varphi _n^*\rightarrow \varphi _n\varphi _n^*\) scattering amplitudes. The computation of the available channels, t and u in the first case, s and t in the second, proceeds as in the \(\varphi _0\) case described above, but these contributions with two or one non minimal vertex do not close the sum \(s+t+u\), as was the case in (5.4) and (5.8).

6 Higher dimensional gauge theory

So far, we have considered gravitational and scalar interactions in the higher dimensional theory. We will discuss now the case with gauge interactions. We consider a charged scalar \({\hat{\Phi }}\) of charge q and mass \({\hat{M}}\) minimally coupled to a U(1) gauge field \({\hat{B}}_M\) with gauge coupling \({\hat{g}}\) in \(D+1\) dimensions

$$\begin{aligned} {\mathcal {S}}_{EH,\Phi ,H}^{(D+1)}= & {} \int \textrm{d}^{D+1}x\,\,\sqrt{(-1)^D\hat{g}}\,\, \left\{ \frac{{\hat{R}}}{2{\hat{\kappa }}^2}+{\hat{D}}_M {\hat{\Phi }} {\hat{D}}^M {\hat{\Phi }}^*\right. \nonumber \\{} & {} \left. -{\hat{M}}^2{\hat{\Phi }}{\hat{\Phi }}^* -\frac{1}{4}{\hat{H}}_{M\,N}{\hat{H}}^{M\,N}\right\} , \end{aligned}$$
(6.1)

where \({\hat{H}}\) is the field strenght for the gauge field \({\hat{B}}\) and \({\hat{D}}_M\) the \(D+1\) dimensional covariant derivative \({\hat{D}}_M\equiv \partial _M-i{\hat{g}}' q {\hat{B}}_M\), with \({\hat{g}}'\) the gauge coupling. For simplicity, we choose the following periodicities for the fields

$$\begin{aligned}&{\hat{B}}_M(x,z+2\pi L)={\hat{B}}_M(x,z),\nonumber \\&{\hat{B}}_M(x,z)=\frac{1}{\sqrt{2\pi L}}\sum _{n=-\infty }^{+\infty }B_{(n) M}(x) e^{\frac{inz}{L}} \nonumber \\&{\hat{\Phi }}(x,z+2\pi L)=e^{i2\pi q_\Phi }{\hat{\Phi }}(x,z),\nonumber \\&{\hat{\Phi }} (x,z)=\frac{1}{\sqrt{2\pi L}}\sum _{n=-\infty }^{+\infty } \varphi _n(x)e^{i(n+q_\Phi )\frac{z}{L}}, \end{aligned}$$
(6.2)

where \(q_\Phi \) is a putative charge of \({\hat{\Phi }}\) under an internal symmetry. The compactification of the (kinetic term of the) gauge field gives the lagrangian

$$\begin{aligned}&{\mathcal {L}}^{(D)}_H=-e^{-2\alpha \phi }\left( \frac{H_0^{\,2}}{4}+\sum _{n=1}^\infty \frac{ |H_{(n)}|^{\,2}}{2}\right) \nonumber \\&\quad +e^{-2\beta \phi }\left( \frac{(\partial h_0)^2}{2}+\sum _{n=1}^\infty \left| \partial h_n-i\frac{n}{L}B_{(n)}\right| ^2\right) \nonumber \\&\quad +e^{-2\alpha \phi } A^\mu \left( -H_{(0) \mu \nu }\,\partial ^\nu h_0+\sum _{n=1}^{\infty }H_{(n) \mu \nu }\left( \partial ^\nu h_n^*-i\frac{n}{L}B_{(n)}^{*\, \nu }\right) \right. \nonumber \\&\quad \left. +H^*_{(n) \mu \nu }\left( \partial ^\nu h_n-i\frac{n}{L}B_{(n)}^{\, \nu }\right) \right) \nonumber \\&\quad +e^{-2\alpha \phi }\Bigg [A^2\left( \frac{\left( \partial h_0\right) ^2}{2}+\sum _{n=1}^{\infty }\left| \partial h_n-i\frac{n}{L}B_{(n)}\right| ^2\right) \nonumber \\&\quad +A^\mu A^\nu \left( \partial _\mu h_0\partial _\nu h_0+2\sum _{n=1}^\infty \left( \partial _\mu h_n-i\frac{n}{L}B_{(n) \mu }\right) \right. \nonumber \\&\quad \times \left. \left( \partial _\nu h_{n}-i\frac{n}{L}B_{(n) \nu }\right) ^*\right) \Bigg ], \end{aligned}$$
(6.3)

where \(h_0\equiv B_{(0) z}\) is a real scalar corresponding to the zero mode of the gauge field \({\hat{B}}_M\) component along the compact dimension z and \(h_n\equiv B_{(n) z}\) are the complex scalars forming the KK tower of the same field. From the above action, each field \(h_n\) is seen to generate a mass for the KK excitations \(B_{(n) \mu }\) of the non-compact components of the gauge field, that are then complex massive vectors, and to behave as the Goldstones in the Higgs mechanism (or in a Stuckelberg mechanism). Note that the relations \(B_{(-n) \mu }=B_{(n) \mu }^*\) and \(h_{-n}=h_n^*\) are valid, although the same cannot be said for the Fourier modes of the complex field \({\hat{\Phi }}\).

The D-dimensional lagrangian obtained from the kinetic and mass term of the scalar field \({\hat{\Phi }}\) reads

$$\begin{aligned}&{\mathcal {L}}^{(D)}_\Phi =\sum _{n=-\infty }^{+\infty }\left| D\varphi _n\right| ^2\nonumber \\&\quad -\left( e^{2\alpha \phi }{\hat{M}}^2+e^{-2(\beta -\alpha )\phi }\left[ \frac{n+q_\Phi }{L}-g'qh_0\right] ^2\right) \left| \varphi _n\right| ^2\nonumber \\&\quad +g'q\sum _{\underset{n\ne 0}{n,p=-\infty }}^{+\infty }\Bigg [iB^\mu _{(n)}\left( \varphi _p\partial _\mu \varphi _{n+p}^*-\partial _\mu \varphi _p\varphi ^*_{n+p}\right) \nonumber \\&\quad -2g'qB_{(0)\mu }B_{(n)}^\mu \varphi _p\varphi _{n+p}^* -g'q\sum _{\underset{m\ne 0}{m=-\infty }}^{+\infty }B_{(n)\mu }B_{(m)}^\mu \varphi _p\varphi ^*_{n+m+p}\Bigg ] \nonumber \\&\quad +g'q A^\mu \Bigg (\sum _{\underset{n\ne 0}{n,p=-\infty }}^{+\infty }i h_n\left( \partial _\mu \varphi _p\,\varphi ^*_{n+p}-\varphi _p\partial _\mu \varphi ^*_{n+p}\right) \nonumber \\&\quad -2\sum _{n,p=-\infty }^{+\infty }\frac{n+p+q_\varphi }{L}\,B_{(n) \mu } \varphi _p\varphi ^*_{n+p}\nonumber \\&\quad +2 g'q h_0\sum _{n,p=-\infty }^{+\infty }B_{(n) \mu }\varphi _p\varphi ^*_{n+p}\nonumber \\&\quad +2 g'q \sum _{\underset{m\ne 0}{n,m,p=-\infty }}^{+\infty }h_m B_{(n) \mu }\varphi _p\varphi ^*_{n+p}\Bigg ) +\left( A^2+e^{-2(\beta -\alpha )\phi }\right) \nonumber \\&\quad \times \left( 2g'q\sum _{\underset{n\ne 0}{n,p=-\infty }}^{+\infty }\left[ \frac{n+p+q_\varphi }{L}-g'qh_0\right] h_n\varphi _p\varphi ^*_{n+p} \right. \nonumber \\&\quad \left. -g'^2q^2\sum _{\underset{m\ne 0}{n,m,p=-\infty }}^{+\infty }h_n h_m \varphi _p\varphi ^*_{n+m+p}\right) , \end{aligned}$$
(6.4)

where \(g'q\equiv {\hat{g}}' q/\sqrt{2\pi L}\) and when acting on \(\varphi _n\)

$$\begin{aligned} D_\mu \equiv \partial _\mu -ig'qB_{(0) \mu }-ig\left[ \left( \frac{n+q_\Phi }{L}-g'qh_0\right) \right] A_\mu ,\nonumber \\ \end{aligned}$$
(6.5)

from which one can read the charge under the graviphoton. The \(h_0\) term in this expression is a manifestation of the Aharonov–Bohm effect for the Wilson line of \(B_z\), \(\oint _z B_z\).

Here we are interested in comparing the different gravitational and non-gravitational long range classical interactions, which can be obtained from the t-channel amplitudes. The t-channel contribution to the \(\varphi _n(p_1)\varphi _n(p_2)\rightarrow \varphi _n(p_3)\varphi _n(p_4)\) scattering amplitude is

$$\begin{aligned}&i{\mathcal {M}}_n=\frac{i}{t}\left( g'^2q^2e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}+ 2 \kappa ^2\left( \frac{n+q_\Phi }{L}\right. \right. \nonumber \\&\quad \left. \left. -g' qe^{-\sqrt{\frac{D-2}{D-1}}\kappa \phi _0}{\bar{h}}_0\right) ^2e^{2\sqrt{\frac{D-1}{D-2}\kappa \phi _0}}\right) \left( p_1+p_3\right) \cdot \left( p_2+p_4\right) \nonumber \\&\quad -\frac{i}{t}\left[ 4g'^2q^2\left( g'q{\bar{h}}_0 e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}-\frac{n+q_\Phi }{L}e^{\frac{D}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}\right) ^2\right. \nonumber \\&\quad \left. +\left( 2\sqrt{\frac{D-1}{D-2}}\kappa \left( \frac{n+q_\Phi }{L}-g'q{\bar{h}}_0e^{-\sqrt{\frac{D-2}{D-1}\kappa \phi _0}}\right) ^2e^{2\sqrt{\frac{D-1}{D-2}\kappa \phi _0}}\right. \right. \nonumber \\&\quad \left. \left. +\frac{2}{\sqrt{(D-1)(D-2)}}\kappa {\hat{M}}^2 e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}\right) ^2\right] , \end{aligned}$$
(6.6)

where we have omitted writing the gravitational contribution, to avoid lengthy expressions, only to reinsert it in the next step when we perform the non-relativistic limit. The mass of the nth KK state can be read from the first line of the action in (6.4)

$$\begin{aligned} m_n^2&=\left( \frac{n+q_\Phi }{L}-g' qe^{-\sqrt{\frac{D-2}{D-1}}\kappa \phi _0}{\bar{h}}_0\right) ^2e^{2\sqrt{\frac{D-1}{D-2}\kappa \phi _0}}\nonumber \\&\quad +{{\hat{M}}}^2 e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}\ \end{aligned}$$
(6.7)

Let us first consider the simplest case where \(q_\Phi ={\bar{h}}_0={\hat{M}}=0\). In the non-relativistic limit, for \(n\ne 0\), the coefficient of \(\frac{1}{t}\) in the t-channel amplitude takes the form

$$\begin{aligned} {\mathcal {M}}_n^{\mathrm {t-pole}}&=\left( g'^2q^2 e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}+2 \kappa ^2m_n^2\right) 4m_n^2\nonumber \\&\quad -4g'^2q^2m_n^2e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0} \nonumber \\&\quad -4\frac{D-1}{D-2}\kappa ^2m_n^4-4\frac{D-3}{D-2}\kappa ^2m_n^4 = 0, \end{aligned}$$
(6.8)

where \(m_n^2\) in this case is simply \(m_n^2=e^{2\sqrt{\frac{D-1}{D-2}}\kappa \phi _0}n^2/L^2\) and the gravitational scattering has been reinserted. The vanishing amplitude results from the (expected) two by two cancellation of interactions for the massive KK modes: namely gravitational vs dilatonic and D-dimensional gauge vs scalar from the (D+1)-direction gauge field component. The \(n=0\) amplitude is different as the zero mode is massless with our specific choice. The non gravitational amplitude reads

$$\begin{aligned} i{\mathcal {M}}_0^{relativistic}= & {} \frac{i}{t}g'^2q^2e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}(p_1+p_3)\nonumber \\{} & {} \cdot (p_2+p_4). \end{aligned}$$
(6.9)

Let us now consider the case \(q_\Phi \ne 0\). The zero mode is massive

$$\begin{aligned} m_0^2=e^{\sqrt{\frac{D-1}{D-2}}\kappa \phi _0}\frac{q_\Phi ^2}{L^2}, \end{aligned}$$
(6.10)

and the corresponding four-point amplitude is given by (again, we do not include here the gravitational contribution whose expression for generic exchanger momenta is long and not very illuminating)

$$\begin{aligned} i{\mathcal {M}}_{0}&=\frac{i}{t}\left( g'^2q^2e^{\frac{2}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}+ 2 \kappa ^2\frac{q^2_\Phi }{L^2}e^{2\sqrt{\frac{D-1}{D-2}\kappa \phi _0}}\right) \left( p_1+p_3\right) \nonumber \\&\quad \cdot \left( p_2+p_4\right) \nonumber \\&\quad -\frac{i}{t}\left[ 4g'^2q^2\frac{q_\Phi ^2}{L^2}e^{2\frac{D}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}\right. \nonumber \\&\quad \left. +4\frac{D-1}{D-2}\kappa ^2\frac{q_\Phi ^2}{L^2}e^{2\sqrt{\frac{D-1}{D-2}\kappa \phi _0}}\right] . \end{aligned}$$
(6.11)

In the non-relativistic limit, the total amplitude obtained by adding the gravitational contribution to (6.11), cancels. The non-periodicity, which makes the zero mode massive, also generates couplings at \(h_0\) and \(\phi \), whose exchanges cancel, respectively, the gauge and gravitational amplitudes of the zero mode. This is to be expected since integer values of \(q_\Phi \) reshuffle the KK states; what was the zero mode becomes one of the massive modes for which we have seen that the total amplitude disappears. It is immediate to verify that the same is true for generic \(n\ne 0\), \({\mathcal {M}}_n\) remains null, and the same thing happens if one turns on \({\bar{h}}_0\), as can be easily verified.

We can now study the general case. It is immediately verified that, after some algebra, in the non-relativistic limit the scattering amplitude (6.6) simplifies to

$$\begin{aligned}&i{\mathcal {M}}_{NR}^{(D)}=4i e^{\frac{4}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}{\hat{M}}^2\left( g'^2q^2-\frac{D-2}{D-1}\kappa ^2{\hat{M}}^2\right) \nonumber \\&\quad =4i \frac{e^{\frac{4}{\sqrt{(D-1)(D-2)}}\kappa \phi _0}}{2\pi L}{{\hat{M}}^2}\left( {\hat{g}}'^2q^2-\frac{(D+1)-3}{(D+1)-2}{\hat{\kappa }}^2{\hat{M}}^2\right) \nonumber \\&\qquad \propto i{\mathcal {M}}_{NR}^{(D+1)} \end{aligned}$$
(6.12)

where one recognizes in the combination inside the parenthesis the \(D+1\) dimensional corresponding dependence. The \(q_\Phi \) and \({\bar{h}}_0\) dependences cancel out to leave this simple expression only in terms of the higher dimensional mass and charge. We conclude that the requirement that the state in \(D+1\) dimensions feels a repulsive long range force ensures that the KK modes in D dimensions also feel a repulsive long range force.

The mapping of the \(D+1\) dimensional U(1) WGC into the D dimensional form of the conjecture with gauge and scalar fields was discussed in [3] from the requirement of extremal black holes and black p-branes decays, leading to the establishment of the dilatonic WGC, and in [16] for the special case of a five to four dimensional circle compactification retaining only the zero modes. The analysis presented here generalizes, from the standpoint of scattering amplitudes, the connection between these different forms of the conjecture to the case with several gauge and scalar fields with reasonings involving the whole Kaluza–Klein tower.

6.1 Effective potential for \(h_0\)

Finally, we comment on the confrontation of the effective one-loop potential for the Wilson line with the scalar WGC of [6]. The potential is generated by the integration of the KK excitations.Footnote 4 In the case of a circle compactification from five to four dimensions, the potential takes the simple form

$$\begin{aligned}{} & {} V_{\textrm{eff}}(h_0)=-\frac{3}{64\pi ^6L^4}\sum _{n=1}^\infty \frac{\cos \left( 2\pi n g'qh_0 L\right) }{|n|^5}\nonumber \\{} & {} \quad =-\frac{3 \left( \text {Li}_5\left( e^{-2 \pi i g' q h_0 L}\right) +\text {Li}_5\left( e^{2 \pi i g' q h_0 L }\right) \right) }{128 \pi ^6 L^4}, \end{aligned}$$
(6.13)

where the symbols \(\text {Li}_n\) denote the usual Polylogarithm functions defined as

$$\begin{aligned} \text {Li}_n(x)=\sum _{k=1}^\infty \frac{x^k}{k^n}. \end{aligned}$$
(6.14)

For the Wilson line to satisfy the Scalar WGC inequality of [6] around a generic background value \({\bar{h}}_0\) (we indicate with \(\eta \) the excitations around it, \(h_0={\bar{h}}_0+\eta \)), one then needs

$$\begin{aligned} L^2\ge \frac{3\kappa ^2}{ 2\pi ^2 g'^2q^2}\left[ \frac{\text {Li}_3\left( e^{i x}\right) +\text {Li}_3\left( e^{-i x}\right) }{\left| \frac{20}{9}\frac{ \left( \text {Li}_2\left( e^{i x}\right) -\text {Li}_2\left( e^{-i x}\right) \right) ^2}{\text {Li}_3\left( e^{i x}\right) +\text {Li}_3\left( e^{-i x}\right) }-\log \left( 2-2 \cos x\right) \right| }\right] .\nonumber \\ \end{aligned}$$
(6.15)

where x is defined to be \(x\equiv 2\pi g'q {\bar{h}}_0 L\), to be respected for \(m^2_\eta >0\), while the inequality is trivially verified for \(m^2_\eta <0\), but this case is of no interest. In the inequality (6.15), the factor inside the square parenthesis on the right hand side is periodic and reaches a maximal value around \(0.6-0.7\) in the regions of parameters where \(m^2_\eta > 0\). Taken to be approximately an order one, the gravitational sub-dominance is then realized around any background value \({\bar{h}}_0\) ifFootnote 5

$$\begin{aligned} L^2\ge \frac{3{\hat{\kappa }}^2}{2\pi ^2 {\hat{g}}'^2q^2}=\frac{3}{2\pi ^2 g'^2q^2}\frac{1}{M_P^2}, \end{aligned}$$
(6.16)

which means that the compactification length cannot be parametrically smaller than the Planck’s one as expected.

From (6.3) and (6.4), it is immediate to observe that the self-couplings induced by radiative corrections are not the only ones that can appear in the 4-point function \(\eta \eta \rightarrow \eta \eta \). A first contribution may come from the kinetic term of \(h_0\), coupled to the dilaton as in (6.3). This gives a two derivative vertex that would then induce contributions to the four point function proportional to the scalar product of external momenta (\(p_1\cdot p_2\times p_3\cdot p_4\) in the s-channel, and so on). For the effective four point non relativistic coupling, this only accounts for a shift of the gravitational contribution, the second term in (6.16). In particular, the numerical coefficient 3/2 should be changed with 5 in (6.16) and all the subsequent inequalities.

7 Conclusions

An extra dimension for our space-time was originally introduced to unify gravity with electromagnetism: [8,9,10,11]. From the point of view of a lower dimensional observer, this unification makes the KK modes undergo attractive gravitational plus scalar interactions and repulsive electric interactions with the same intensity. This motivated the use of the KK states interactions in this work to extract the form of the inequalities that appear when one is interested in comparing gravitational interactions to other types of interactions.

Taking into account the scalar interaction due to the presence of a dilaton, the calculation of four-point amplitudes allowed us to find the inequalities of the Dilatonic WGC. Our observations go further, with the extension of the construction to include interactions in the higher dimension, and we have shown how the Scalar WGC is found as well as the behavior of these conjectures under dimensional reduction. Meanwhile, we have also computed a number of scattering amplitudes for the pair production of KK states and have been able to compare the contributions of the different channels for spacetime dimensions \(D \ge 4\).