The simplest

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:

$$gq\geq\sqrt{\frac{D-3}{D-2}}\kappa_{D}m,$$

(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 Section 5. The dilatonic WGC inequality is derived in Section 3, where we also calculate various KK pair production amplitudes. In Section 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 section 5. The interactions due to the presence of higher dimensional gauge fields are discussed in section 6. Our conclusions are presented in section 7. Finally, some technical details about our calculations are gathered in appendices.

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:

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

(2.1)

where

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

(2.2)

and

$$\mathcal{S}_{\Phi,0}^{(D+1)}=\int\mathrm{d}^{D+1}x\,\,\sqrt{(-1)^{D}\hat{g}}\,\,\frac{1}{2}\hat{g}^{MN}\partial_{M}\hat{\Phi}\partial_{N}\hat{\Phi}$$

(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

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

(2.4)

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

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

where $g$ is the determinant of the $D$-dimensional metric. A canonical $D$-dimensional Einstein-Hilbert action is obtained for

$$(D-2)\alpha+\beta=0.$$

(2.6)

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

$$\alpha^{2}=\frac{1}{2(D-1)(D-2)}.$$

(2.7)

Since all fields are independent of $z$, we can perform the integration over this coordinate to obtain, keeping only the zero modes,¹¹1The factor in front of the D’Alambertian operator, $2\alpha$, corrects the expression sometimes found in the literature, $(D-3)\alpha$. As long as only minimal coupling to gravity is considered, the difference is harmless.

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

(2.8)

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

$$\frac{1}{\kappa^{2}}=\frac{2\pi L}{\hat{\kappa}^{2}}\Longrightarrow M_{P}^{D-2}=2\pi L\,\hat{M}_{P}^{D-1}$$

(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

$$\tilde{\phi}=\frac{\phi}{\sqrt{2}\kappa};\,\,\,\,\,\,\,\,\tilde{A_{\mu}}=\frac{A_{\mu}}{\sqrt{2}\kappa}$$

(2.10)

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

$$\mathcal{S}_{0,0}^{(D)}=\int\mathrm{d}^{D}x\sqrt{(-1)^{D-1}g}\left[\frac{R}{2\kappa^{2}}+\frac{\sqrt{2}\alpha}{\kappa}\Box\tilde{\phi}+\frac{1}{2}(\partial\tilde{\phi})^{2}-\frac{1}{4}e^{2\sqrt{2}(1-D)\alpha\kappa\tilde{\phi}}\tilde{F}^{2}\right].$$

(2.11)

In the following, with the exception of section 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

$$\hat{\Phi}(x,z+2\pi L)=\hat{\Phi}(x,z),\qquad\hat{\Phi}(x,z)=\frac{1}{\sqrt{2\pi L}}\sum_{n=-\infty}^{+\infty}\varphi_{n}(x)e^{\frac{inz}{L}},$$

(2.12)

which leads to

	$\displaystyle\mathcal{S}=\int\mathrm{d}^{D}x\sqrt{(-1)^{D-1}g}\Bigg{\{}$	$\displaystyle\frac{R}{2\kappa^{2}}+\frac{1}{2}(\partial\phi)^{2}-\frac{1}{4}e^{-2\sqrt{\frac{D-1}{D-2}}\kappa{\phi}}F^{2}+\frac{1}{2}\partial_{\mu}\varphi_{0}\partial^{\mu}\varphi_{0}$
		$\displaystyle+\sum_{n=1}^{\infty}\left(\partial_{\mu}\varphi_{n}\partial^{\mu}\varphi_{n}^{*}-\frac{n^{2}}{L^{2}}e^{2\sqrt{\frac{D-1}{D-2}}\kappa{\phi}}\varphi_{n}\varphi_{n}^{*}\right)$
		$\displaystyle+\sum_{n=1}^{\infty}\left(i\sqrt{2}\kappa\frac{n}{L}A^{\mu}\left(\partial_{\mu}\varphi_{n}\varphi_{n}^{*}-\varphi_{n}\partial_{\mu}\varphi_{n}^{*}\right)+{2}\kappa^{2}\frac{n^{2}}{L^{2}}A_{\mu}A^{\mu}\varphi_{n}\varphi_{n}^{*}\right)\Bigg{\}},$		(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

$$g^{2}=e^{2\sqrt{\frac{D-1}{D-2}}\kappa{\phi_{0}}}.$$

(2.14)

For each KK mode, the mass and charge read

$$gq_{n}=\sqrt{2}\kappa\frac{n}{L}e^{\sqrt{\frac{D-1}{D-2}}\kappa{\phi_{0}}}\;\qquad m_{n}=\frac{n}{L}e^{\sqrt{\frac{D-1}{D-2}}\kappa{\phi_{0}}}.$$

(2.15)

This shows that they are related through

$$(gq_{n})^{2}=2\kappa^{2}m_{n}^{2},$$

(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:

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

(2.17)

where:

$$\hat{\zeta}_{MN}=\begin{pmatrix}e^{2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\eta_{\mu\nu}&0\\ 0&-e^{2\sqrt{2}\beta\hat{\kappa}{\phi_{0}}}\end{pmatrix}.$$

(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

$\displaystyle\begin{cases}&\hat{g}_{MN}=\hat{\zeta}_{MN}+2\kappa\hat{h}_{MN}+4\kappa^{2}\hat{f}_{MN}+\mathcal{O}(\kappa^{3})\\ &\hat{g}^{MN}=\hat{\zeta}^{MN}+2\kappa\hat{t}^{MN}+4\kappa^{2}\hat{l}^{MN}+\mathcal{O}(\kappa^{3}).\end{cases}$

(2.19)

The relation $\hat{g}_{MP}\hat{g}^{PN}\equiv\delta_{M}^{N}$ reads

$$\begin{cases}\hat{t}^{MN}=-\hat{h}^{MN}\\ \hat{l}^{MN}+\hat{f}^{MN}=\hat{h}^{M}_{P}\hat{h}^{PN},\end{cases}$$

(2.20)

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

	$\displaystyle\sqrt{(-1)^{D}\hat{g}}\mathcal{L}_{\Phi}=$	$\displaystyle\sqrt{(-1)^{D}\hat{\zeta}}\left[\frac{1}{2}\partial_{M}\hat{\Phi}\partial^{M}\hat{\Phi}-\frac{\hat{\kappa}^{\prime}}{2}\hat{h}^{MN}\left(\partial_{M}\hat{\Phi}\partial_{N}\hat{\Phi}-\frac{1}{2}\hat{\zeta}_{MN}\partial_{P}\hat{\Phi}\partial^{P}\hat{\Phi}\right)\right.$
		$\displaystyle\left.+\frac{\hat{\kappa}^{\prime 2}}{2}\left(\hat{l}^{MN}-\frac{1}{2}\hat{h}^{MN}\hat{h}^{P}_{\,P}\right)\partial_{M}\hat{\Phi}\partial_{N}\hat{\Phi}\right.$
		$\displaystyle\left.+\frac{\hat{\kappa}^{\prime 2}}{4}\left(\hat{f}^{P}_{\,P}-\frac{1}{2}\hat{h}_{MP}\hat{h}^{PM}+\frac{1}{4}(\hat{h}^{P}_{\,P})^{2}\right)\partial_{M}\hat{\Phi}\partial^{M}\hat{\Phi}\right].$		(2.21)

where $\hat{\kappa}^{\prime}\equiv 2\hat{\kappa}$. With:

$$\hat{h}^{MN}=\frac{1}{\sqrt{2\pi L}}\begin{pmatrix}e^{-2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\left(\sqrt{2}\alpha\phi\,\eta^{\mu\nu}+h^{\mu\nu}\right)&-e^{-2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\frac{A^{\mu}}{\sqrt{2}}\\ -e^{-2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\frac{A^{\nu}}{\sqrt{2}}&-e^{-2\sqrt{2}\beta\hat{\kappa}{\phi_{0}}}\sqrt{2}\beta\phi\end{pmatrix},$$

(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 $\hat{T}^{\hat{\Phi}}_{MN}$:

	$\displaystyle{\mathcal{L}}_{int}^{(1)}=-\hat{\kappa}\hat{h}^{MN}\hat{T}^{\hat{\Phi}}_{MN}=$	$\displaystyle-\hat{\kappa}h^{\mu\nu}T^{(\varphi_{0},\varphi_{n})}_{\mu\nu}$		(2.23)
		$\displaystyle-i\sqrt{2}\hat{\kappa}A^{\mu}\sum_{n=1}^{\infty}\frac{n}{L}\left(\partial_{\mu}\varphi_{n}\,\varphi_{n}^{*}-\varphi_{n}\,\partial_{\mu}\varphi_{n}^{*}\right)-2\sqrt{\frac{D-1}{D-2}}\hat{\kappa}e^{2\sqrt{\frac{D-1}{D-2}}\hat{\kappa}\phi_{0}}\phi\sum_{n=1}^{\infty}\frac{n^{2}}{L^{2}}\varphi_{n}\varphi_{n}^{*}.$

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

$$\hat{f}_{MN}=\frac{1}{2\pi L}\begin{pmatrix}e^{2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\bigg{(}\alpha^{2}\phi^{2}\eta_{\mu\nu}+\sqrt{2}\alpha\phi\,h_{\mu\nu}+f_{\mu\nu}\bigg{)}-\frac{1}{2}e^{2\sqrt{2}\beta\hat{\kappa}{\phi_{0}}}A_{\mu}A_{\nu}&e^{2\sqrt{2}\beta\hat{\kappa}{\phi_{0}}}\beta\phi A_{\mu}\\ e^{2\sqrt{2}\beta\hat{\kappa}{\phi_{0}}}\beta\phi A_{\nu}&-e^{2\sqrt{2}\beta\hat{\kappa}{\phi_{0}}}\beta^{2}\phi^{2}\end{pmatrix}.$$

(2.24)

With this result, $\hat{l}^{MN}$ in (2) is given by

$$\hat{l}^{MN}=\hat{h}^{MP}\hat{h}_{\,P}^{N}-\hat{f}^{MN}$$

(2.25)

Using (2.24) and (2.22) one obtains

$$\hat{l}^{MN}=\frac{1}{2\pi L}\begin{pmatrix}e^{-2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\bigg{(}\alpha^{2}\phi^{2}\eta^{\mu\nu}+\sqrt{2}\alpha\phi\,h^{\mu\nu}+l^{\mu\nu}\bigg{)}&-e^{-2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\left(\alpha\phi A^{\mu}+\frac{1}{\sqrt{2}}h^{\mu\rho}A_{\rho}\right)\\ -e^{-2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\left(\alpha\phi A^{\nu}+\frac{1}{\sqrt{2}}h_{\rho}^{\nu}A^{\rho}\right)&-e^{-2\sqrt{2}\beta\hat{\kappa}{\phi_{0}}}\beta^{2}\phi^{2}+e^{-2\sqrt{2}\alpha\hat{\kappa}{\phi_{0}}}\frac{1}{2}A_{\rho}A^{\rho}\end{pmatrix}.$$

(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

	$\displaystyle{\mathcal{L}}_{int}^{(2)}=\frac{1}{2}\partial_{\mu}\varphi_{0}\partial_{\nu}\varphi_{0}$	$\displaystyle\left[\left(\frac{f^{\rho}_{\;\rho}}{2}-\frac{h^{\rho\sigma}h_{\rho\sigma}}{4}+\frac{(h^{\rho}_{\;\rho})^{2}}{8}+\frac{1}{2}\left(D^{2}\alpha^{2}+2D\beta\alpha+\beta^{2}-4D\alpha^{2}-4\beta\alpha+4\alpha^{2}\right)\phi^{2}\right.\right.$		(2.27)
		$\displaystyle\left.\left.+\frac{1}{2}((D-2)\alpha+\beta)\phi h^{\rho}_{\;\rho}\right)\eta^{\mu\nu}+l^{\mu\nu}-\frac{1}{2}h^{\rho}_{\;\rho}h^{\mu\nu}\right]$
	$\displaystyle+\sum_{n=1}^{\infty}\partial_{\mu}\varphi_{n}\partial_{\nu}\varphi_{n}^{*}$	$\displaystyle\left[\left(\frac{f^{\rho}_{\;\rho}}{2}-\frac{h^{\rho\sigma}h_{\rho\sigma}}{4}+\frac{(h^{\rho}_{\;\rho})^{2}}{8}+\frac{1}{2}\left(D^{2}\alpha^{2}+2D\beta\alpha+\beta^{2}-4D\alpha^{2}-4\beta\alpha+4\alpha^{2}\right)\phi^{2}\right.\right.$
		$\displaystyle\left.\left.+\frac{1}{2}((D-2)\alpha+\beta)\phi h^{\rho}_{\;\rho}\right)\eta^{\mu\nu}+l^{\mu\nu}-\frac{1}{2}h^{\rho}_{\;\rho}h^{\mu\nu}\right]$
	$\displaystyle-\sum_{n=1}^{\infty}\frac{n^{2}}{L^{2}}|\varphi_{n}|^{2}$	$\displaystyle\left[-A^{2}e^{\sqrt{2}(D\alpha-\beta)\kappa{\phi_{0}}}\left(\frac{f^{\rho}_{\;\rho}}{2}-\frac{h^{\rho\sigma}h_{\rho\sigma}}{4}+\frac{(h^{\rho}_{\;\rho})^{2}}{8}+\left(\frac{1}{2}(D\alpha+\beta)-\beta\right)\phi h^{\rho}_{\;\rho}\right.\right.$
		$\displaystyle\left.\left.+\frac{1}{2}(D^{2}\alpha^{2}+2D\alpha\beta+\beta^{2}-4D\alpha\beta-4\beta^{2}+4\beta^{2})\phi^{2}\right)\right]$
	$\displaystyle-\sum_{n=1}^{\infty}i\frac{n}{L}h^{\rho\sigma}A_{\rho}$	$\displaystyle J_{\sigma,n}+i\frac{n}{L}A^{\rho}J_{\rho,n}\left(-\frac{h^{\sigma}_{\;\sigma}}{2}-((D-2)\alpha+\beta)\phi\right)$

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

	$\displaystyle{\mathcal{L}}_{int}^{(2)}=\,\,$	$\displaystyle\frac{1}{2}\partial_{\mu}\varphi_{0}\partial_{\nu}\varphi_{0}\left[\left(\frac{f^{\rho}_{\;\rho}}{2}-\frac{h^{\rho\sigma}h_{\rho\sigma}}{4}+\frac{(h^{\rho}_{\;\rho})^{2}}{8}\right)\eta^{\mu\nu}+l^{\mu\nu}-\frac{1}{2}h^{\rho}_{\;\rho}h^{\mu\nu}\right]$
	$\displaystyle+$	$\displaystyle\sum_{n=1}^{\infty}\partial_{\mu}\varphi_{n}\partial_{\nu}\varphi_{n}^{*}\left[\left(\frac{f^{\rho}_{\;\rho}}{2}-\frac{h^{\rho\sigma}h_{\rho\sigma}}{4}+\frac{(h^{\rho}_{\;\rho})^{2}}{8}\right)\eta^{\mu\nu}+l^{\mu\nu}-\frac{1}{2}h^{\rho}_{\;\rho}h^{\mu\nu}\right]$
	$\displaystyle-$	$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{L^{2}}|\varphi_{n}|^{2}e^{{2\sqrt{2}(D-1)\alpha\kappa\phi_{0}}}\left(\frac{f^{\rho}_{\;\rho}}{2}-\frac{h^{\rho\sigma}h_{\rho\sigma}}{4}+\frac{(h^{\rho}_{\;\rho})^{2}}{8}\right)$
	$\displaystyle-$	$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{L^{2}}|\varphi_{n}|^{2}\left[-A^{2}+e^{{2\sqrt{2}(D-1)\alpha\kappa\phi_{0}}}\left(2(D-1)^{2}\alpha^{2}\phi^{2}+(D-1)\alpha\phi h^{\rho}_{\;\rho}\right)\right]$
		$\displaystyle+i\frac{n}{L}A^{\rho}\frac{h^{\sigma}_{\;\sigma}}{2}J_{\rho,n}-i\frac{n}{L}h^{\rho\sigma}A_{\rho}J_{\sigma,n}$		(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 $\hat{\kappa}$.

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) to obtain:

	$\displaystyle\mathcal{S}_{f}=\int\mathrm{d}^{D}x\sqrt{(-1)^{D-1}g}\Bigg{\{}$	$\displaystyle\frac{R}{2\kappa^{2}}+\frac{1}{2}(\partial\phi)^{2}-\frac{1}{4}e^{-2\sqrt{\frac{D-1}{D-2}}\kappa\phi_{0}}\sum_{m=0}^{\infty}\left(-2\sqrt{\frac{D-1}{D-2}}\kappa\right)^{m}\frac{\phi^{m}}{m!}F^{2}$
		$\displaystyle+\frac{1}{2}\partial_{\mu}\varphi_{0}\partial^{\mu}\varphi_{0}+\sum_{n=1}^{\infty}\partial_{\mu}\varphi_{n}\partial^{\mu}\varphi_{n}^{*}$
		$\displaystyle-\sum_{n=1}^{\infty}\left(\frac{n^{2}}{L^{2}}e^{2\sqrt{\frac{D-1}{D-2}}\kappa{\phi_{0}}}\sum_{m=0}^{\infty}\left(2\sqrt{\frac{D-1}{D-2}}\kappa\right)^{m}\frac{\phi^{m}}{m!}\varphi_{n}\varphi_{n}^{*}\right)$
		$\displaystyle+\sum_{n=1}^{\infty}\left(i{\sqrt{2}}\kappa\frac{n}{L}A^{\mu}\left(\partial_{\mu}\varphi_{n}\varphi_{n}^{*}-\varphi_{n}\partial_{\mu}\varphi_{n}^{*}\right)+{2}\kappa^{2}\frac{n^{2}}{L^{2}}A_{\mu}A^{\mu}\varphi_{n}\varphi_{n}^{*}\right)\Bigg{\}}$		(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

  $$gq_{n}={\sqrt{2}}\kappa\frac{n}{L}e^{\sqrt{\frac{D-1}{D-2}}\kappa\,{\phi_{0}}}.$$

  (3.2)

  
  
• •

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

  $$-i\left(2\sqrt{\frac{D-1}{D-2}}\kappa\right)^{m}\,\frac{n^{2}}{L^{2}}\,e^{2\sqrt{\frac{D-1}{D-2}}\kappa{\phi_{0}}}.$$

  (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

  $$-i\left(-2\sqrt{\frac{D-1}{D-2}}\kappa\right)^{m}\left(p_{1}\cdot p_{2}\,\eta_{\mu\nu}-p_{1\,\nu}p_{2\,\mu}\right).$$

  (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 ($\phi$, $A_{\mu}$).