Newton versus Coulomb for Kaluza-Klein modes Karim Benaklia Carlo Branchinaband Ga etan Lafforgue-Marmetc Sorbonne Universit e CNRS Laboratoire de Physique Th eorique et Hautes Energies

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Newton versus Coulomb for Kaluza-Klein modes

Karim Benaklia, Carlo Branchinaband Ga¨etan Laﬀorgue-Marmetc

Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes Energies,

LPTHE, F-75005 Paris, France.

Abstract

We consider a set of elementary compactiﬁcations of D+ 1 to Dspacetime dimensions on a circle:

ﬁrst for pure general relativity, then in the presence of a scalar ﬁeld, ﬁrst free then with a non minimal

coupling to the Ricci scalar, and ﬁnally in the presence of gauge bosons. We compute the tree-level

amplitudes in order to compare some gravitational and non-gravitational amplitudes. This allows

us to recover the known constraints of the U(1), dilatonic and scalar Weak Gravity Conjectures in

some cases, and to show the interplay of the diﬀerent interactions. We study the KK modes pair-

production in diﬀerent dimensions. We also discuss the contribution to some of these amplitudes of

the non-minimal coupling in higher dimensions for scalar ﬁelds to the Ricci scalar.

akbenakli@lpthe.jussieu.fr

bcbranchina@lpthe.jussieu.fr

cglm@lpthe.jussieu.fr

arXiv:2210.00477v2 [hep-th] 26 Jul 2023

Contents

1 Introduction 2

2 Expansion to Second Order in the Gravitational Field 3

3 Scattering Amplitudes and Weak Gravity Conjectures 7

3.1 TheDilatonicWGC ..................................... 8

3.2 Amplitudes for Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Non gravitational amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.2 Mixedamplitudes .................................. 11

3.2.3 Gravitational production amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.4 Gravitational vs gauge amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Massive and Self-interacting Scalars 16

4.1 The Scalar Weak Gravity Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Massivedilatons ....................................... 19

5ˆ

Φ2Rinteraction 20

6 Higher dimensional gauge theory 22

6.1 Eﬀective potential for h0................................... 25

7 Conclusions 26

A Lagrangians with derivative interactions 27

A.1 Interactions with derivatives of a gauge ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . 27

A.2 Toy model for the two-derivative interaction of the non-minimal coupling . . . . . . . 29

B Helicity basis and Mandelstam variables 30

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 qwhich satisﬁes:

gq ≥rD−3

D−2κDm, (1.1)

where κDis deﬁned as κ2

D= 8πGD=1

MD−2

P,D

with MP,D the reduced Planck mass in Ddimensions.

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 [5–7]

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 [8–11]. 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 Ddimensions 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.

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 xthe Dnon-compact and by z≡z+ 2πL the compact coordinates. We recall the steps of

the simple dimensional reduction of a free real massless scalar ﬁeld ˆ

Φ coupled to General Relativity:

S(D+1) =S(D+1)

EH +S(D+1)

Φ,0,(2.1)

where

S(D+1)

EH =1

2ˆκ2ZdD+1xq(−1)Dˆgˆ

R, (2.2)

and

S(D+1)

Φ,0=ZdD+1xq(−1)Dˆg1

2ˆgMN ∂Mˆ

Φ∂Nˆ

Φ (2.3)

The Ricci scalar ˆ

Ris computed from the metric ˆgMN . In the simplest compactiﬁcation from D+ 1

to Ddimensions it takes the form

ˆgMN = e2αϕgµν −e2βϕAµAνe2βϕAµ

e2βϕAν−e2βϕ !(2.4)

with ϕ,Aµand gµν D-dimensional ﬁelds independent of the zcoordinate:

S(D+1)

EH =1

2ˆκ2ZdD+1xq(−1)D−1g e((D−2)α+β)ϕR−2(1 −D)α−2β□ϕ

−(D−2)(1 −D)α2+ 2β(2 −D)α−β(∂ϕ)2

−1

4e2(β−α)ϕF2.(2.5)

where gis the determinant of the D-dimensional metric. A canonical D-dimensional Einstein-Hilbert

action is obtained for

(D−2)α+β= 0.(2.6)

and the canonical dilaton kinetic term ﬁxes the constant αto be:

α2=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,1

S(D)

0,0=2πL

2ˆκ2ZdDxq(−1)D−1gR+ 2α□ϕ+1

2(∂ϕ)2−1

4e2(1−D)αϕF2.(2.8)

We deﬁne the D-dimensional constant κin terms of the (D+ 1)-dimensional ˆκas

κ2=2πL

ˆκ2=⇒MD−2

P= 2πL ˆ

MD−1

P(2.9)

In (2.4), the ϕand Aµﬁelds are dimensionless. Dimensional ﬁelds, that we denote ˜

ϕand ˜

Aµ, can be

written as

ϕ=ϕ

√2κ;˜

Aµ=Aµ

√2κ(2.10)

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

respectively, reads:

S(D)

0,0=ZdDxq(−1)D−1g"R

2κ2+√2α

κ□˜

ϕ+1

2(∂˜

ϕ)2−1

4e2√2(1−D)ακ ˜

ϕ˜

F2#.(2.11)

1The factor in front of the D’Alambertian operator, 2α, corrects the expression sometimes found in the literature, (D−3)α.

As long as only minimal coupling to gravity is considered, the diﬀerence is harmless.

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 ˆ

Φ is periodic and single-valued on

the compact dimension

Φ(x, z + 2πL) = ˆ

Φ(x, z),ˆ

Φ(x, z) = 1

√2πL

+∞

n=−∞

φn(x)einz

L,(2.12)

which leads to

S=ZdDxq(−1)D−1g(R

2κ2+1

2(∂ϕ)2−1

4e−2qD−1

D−2κϕF2+1

2∂µφ0∂µφ0

+∞

n=1 ∂µφn∂µφ∗

n−n2

L2e2qD−1

D−2κϕφnφ∗

n

+∞

n=1 i√2κn

LAµ(∂µφnφ∗

n−φn∂µφ∗

n)+2κ2n2

L2AµAµφnφ∗

n),(2.13)

where we have chosen in (2.7) the positive root for α. The complex scalars φnform the Kaluza-Klein

(KK) tower and appear minimally coupled to the graviphoton. Around a generic background value

ϕ0for the dilaton, the gauge coupling gis given by

g2=e2qD−1

D−2κϕ0.(2.14)

For each KK mode, the mass and charge read

gqn=√2κn

LeqD−1

D−2κϕ0mn=n

LeqD−1

D−2κϕ0.(2.15)

This shows that they are related through

(gqn)2= 2κ2m2

n,(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:

ˆgMN =ˆ

ζMN + 2ˆκˆ

hMN + 4ˆκ2ˆ

fMN +o(ˆκ3) (2.17)

where:

ζMN = e2√2αˆκϕ0ηµν 0

0−e2√2βˆκϕ0!.(2.18)

is the background metric and ˆκ2ˆ

fMN ≪ˆκˆ

hMN ≪1, for all M, N . We write the perturbation as

(ˆgMN =ˆ

ζMN + 2κˆ

hMN + 4κ2ˆ

fMN +O(κ3)

ˆgMN =ˆ

ζMN + 2κˆ

tMN + 4κ2ˆ

lMN +O(κ3).(2.19)

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NewtonversusCoulombforKaluza-KleinmodesKarimBenaklia,CarloBranchinabandGa¨etanLafforgue-MarmetcSorbonneUniversit´e,CNRS,LaboratoiredePhysiqueTh´eoriqueetHautesEnergies,LPTHE,F-75005Paris,France.AbstractWeconsiderasetofelementarycompactificationsofD+1toDspacetimedimensionsonacircle:firstforpuregenera...

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Newton versus Coulomb for Kaluza-Klein modes Karim Benaklia Carlo Branchinaband Ga etan Lafforgue-Marmetc Sorbonne Universit e CNRS Laboratoire de Physique Th eorique et Hautes Energies

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