# Dimensionally reducing the Classical Regge Growth (CRG) conjecture

Joan Quirant Department of Physics, Ben-Gurion University of the Negev, Beer-
Sheva 84105, Israel

###### Abstract

We explore the Classical Regge Growth conjecture (CRG) in the 4d effective
field theory that results from compactifying $D$-dimensional General
Relativity on a compact, Ricci-flat manifold. While the higher dimensional
description is given in terms of pure Einstein gravity and the conjecture is
automatically satisfied, it imposes several non-trivial constraints in the 4d
spectrum. Namely, there must be either none or an infinite number of massive
spin-2 modes, and the mass ratio between consecutive KK spin-2 replicas is
bounded by the 4d coupling constants.

##  I Introduction

In the recent work [1] -see [2] for related ideas- it was studied if a
gravitational effective field theory (EFT) that includes a massive spin-2
particle in its spectrum was compatible with the Classical Regge Growth
conjecture (CRG) [3]. This conjecture states that the classical (tree-level)
S-matrix $\mathcal{A}(s,t)$ of any consistent theory can never grow faster
than $s^{2}$ in the Regge limit, that is, at large $s$ and fixed and physical
$t$ -with $s$ and $t$ the usual Mandelstam variables-. In terms of equations,
it states

$\displaystyle\lim_{s\rightarrow\infty,\,t<0\text{
fixed}}\frac{\mathcal{A}(s,t)}{s^{3}}=0\,,$ (1)

where by $s\rightarrow\infty$ we mean $\Lambda\gg s\gg|t|$, with $\Lambda$ the
cut-off of the EFT considered.

The main conclusion of [1] was to show the incompatibility of the setup
described above with the CRG conjecture. If the CRG holds, a gravitational EFT
which includes a massive spin-2 particle -and no other higher spin particles-
would be in the swampland [4] .

This result was in line with the so-called spin-2 swampland conjecture [5, 6,
7] and with the recent works studying the (in)consistency of massive gravity
-see [8, 9, 10, 11, 12, 13, 14, 2] for a biased selection and [15, 16] for
reviews-.

Regarding the state of the CRG conjecture, though a complete demonstration is
still lacking, there is strong evidence in favour of it. In the context of
AdS/CFT, it has been proven in [17] that, in the dual picture, the CRG
conjecture follows from the chaos bound of [18]. In flat space, it was shown
in [19] that the scattering of scalar particles in dimensions bigger or equal
to five satisfies it. Here, as we did in [1], we will limit ourselves to
assuming its validity, studying the consequences that derive from it.

This being said, the logical next step after [1] is to consider an EFT with
not only one but any number of massive spin-2 particles in the spectrum. This
is a common ingredient in theories with extra dimensions, where in the 4d EFT
the graviton comes typically accompanied by an infinite tower of massive
spin-2 particles, its Kaluza-Klain (KK) replicas.111When the length of the
internal $l_{\rm internal}$ and external $l_{\rm external}$ dimensions satisfy
$l_{\rm external}\gg l_{\rm internal}$, the mass of the Kaluza-Klain spin-2
replicas usually becomes much bigger than the energy scale proved in the EFT
and the massive spin-2 states can be ignored in the low-energy description. In
this paper, we would like to understand how the CRG conjecture is satisfied in
these scenarios.

To do so, we will focus on a very concrete but general model: we will study
General Relativity (GR) dimensionally reduced to four dimensions. Of course,
GR in $\mathds{R}^{1,D-1}$ trivially satisfies the CRG conjecture, the
$2\rightarrow 2$ scattering of a GR graviton scales with $s$ in the Regge
limit at most as $\mathcal{A}\sim s^{2}$. The point is that when GR is
compactified to 4d (we go from $\mathds{R}^{1,D-1}$ to $\mathds{R}^{1,3}\times
X_{D-4}$), the description is given not only in terms of a graviton but it
includes an infinite tower of massive spin-2 particles. This provides an arena
where the CRG conjecture can be _tested_ in the presence of several massive
spin-2 states.

What we will see in this work is that the CRG conjecture imposes non-trivial
constraints222These constraints will be automatically satisfied once a valid
internal geometry is specified. on the particle content of the 4d description.
They teach us how the CRG requirement can be fulfilled in a 4d EFT containing
massive spin-2 particles. Namely, as we will see in section IV, in the
spectrum of the 4d effective field theory:

  * •

There must be either none or an infinite number of massive spin-2 particles.
This is in line with the absence of consistent finite truncations of the
graviton tower, already discussed in the literature [20, 21, 22, 23].

  * •

The mass ratio between consecutive massive spin-2 replicas is bounded by the
4d couplings constants. Similar results imposing unitarity were derived in
[24, 23].

Before presenting these results, we will start by the beginning, briefly
recalling the work done in [1].

##  II A single massive spin-2 particle

In [1] it was studied the tree-level $2\rightarrow 2$ scattering of a massive
spin-2 particle in a theory containing neither other massive spin-2 states nor
higher-spin particles. We wanted to check if this setup was compatible with
the CRG conjecture. To do so:

  * •

We assumed that the spin-2 particle could couple to a graviton, a (massive or
massless) scalar particle, and a massive spin-1 particle.333Symmetries forbid
interactions between two identical massive spin-2 particles and one massless
spin-1 field or one fermion.

  * •

We considered both parity-even and parity-odd interactions.

  * •

We included all contact terms with an arbitrary but finite number of
derivatives.

Exchange diagrams and contact terms are the two sources of contributions to
any classical two-to-two scattering amplitude. Both can be computed directly
using on-shell methods, in a Lagrangian independent way, as explained in [25].

  * •

Exchange diagrams can be built from the on-shell cubic couplings. First, one
has to list all possible on-shell three-point interactions between two massive
spin-2 particles and the exchanged particle. Then, two sets of these vertices
(multiplied by arbitrary constants) are connected through the correspondent
propagator. In four dimensions, we found 24 independent exchange pieces,
reproducing the results of [26, 27, 28, 29, 30].

  * •

Contact terms are a bit more tricky since, in principle, one can construct
infinitely many of them, introducing more and more derivatives. In [1],
adapting the ideas developed in [28], we included all contact interactions
with an arbitrary, but finite, number of derivatives.

With all these ingredients, we showed in [1] that a gravitational theory of a
single massive spin-2 particle, coupled to any other state of spin $<2$, can
never be made consistent with the CRG conjecture.

##  III Several massive spin-2 particles

We will now explain how to generalise the results of the previous section to
incorporate any number of massive spin-2 particles in the spectrum. As we will
see, the modifications are conceptually quite simple but technically very
involved.

The _only_ novelty concerning the previous computation lies in the number of
the allowed cubic couplings. Besides the 24 previous pieces, we must include
interactions between two identical and one different massive spin-2 particles,
as shown in figure 1.

Figure 1: Cubic interaction between two identical ($h^{i}_{\mu\nu}$ with mass
$m_{i}$), one different ($h^{k}_{\mu\nu}$ with mass $m_{k}\neq m$) massive
spin-2 particles.

In four dimensions there are seven parity-even and nine parity-odd independent
on-shell three-point functions of this kind, some of them already computed in
[23]. We give the complete list with the explicit expressions in appendix A,
where we also derive a Lagrangian basis for the party-even contributions. For
practical proposes, let us denote this set of interactions by

$\displaystyle\mathcal{A}\left(h^{i}(m_{i}),h^{i}(m_{i}),h^{k}(m_{k})\right)\equiv\sum_{j=1}^{16}c^{ii}_{k,j}f_{j}\left(h^{i},h^{i},h^{k}\right)\,,$
(2)

where the $c^{ii}_{k,j}$ are arbitrary constants and the
$f_{j}\left(h^{i},h^{i},h^{k}\right)$ are the ($9$ parity-even and $7$ parity-
odd) on-shell cubic amplitudes, given in appendix A.

This is the first step in accounting for several massive spin-2 particles, but
it is not the end of the story. What we have just described corresponds to a
theory in which any pair of identical massive spin-2 particles
$\\{h^{i}_{\mu\nu},h^{i}_{\mu\nu}\\}$ only interacts with one different
massive spin-2 state $\\{h^{k}_{\mu\nu}\\}$. If we want to include the
possibility that they couple to any number of different massive spin-2
particles, we need to replace (2) with

$\displaystyle\sum_{k=1}^{K^{ii}}\mathcal{A}\left(h^{i}(m_{i}),h^{i}(m_{i}),h^{k}(m_{k})\right)=\sum_{k=1}^{K^{ii}}\sum_{j=1}^{16}c^{ii}_{k,j}f_{j}\left(h^{i},h^{i},h^{k}\right)\,,$
(3)

where the index $k=1,\dots K^{ii}$ describes the coupling with $K^{ii}$
distinguishable ($m_{i}\neq m_{k}$) massive spin-2 particles.

Generalising [1] to include any number of massive spin-2 fields would
correspond to take $K^{ii}=\rm{arbitrary}$ and $c^{ii}_{k,j}=\rm{arbitrary}$,
since this is the most general possibility. Unfortunately, this case is
technically very complicated, and little can be done explicitly. It would
require introducing an arbitrarily large number of new constants in the
equations of [1], which were already very complex.

To understand whether the CRG conjecture can be satisfied in the presence of
several massive spin-2 particles, we find starting with a simpler model more
illuminating. As a proof of concept example, we will study the 4d effective
field theory obtained after dimensionally reducing GR. In this case, the
$c^{ii}_{k,j}$ are not arbitrary: they are completely fixed once the internal
manifold is specified -GR has no free parameters-and there will be relations
among them. Similar ideas studying the unitarity of GR under dimensional
reductions were derived in [23], which we will use in this note.

##  IV Proof of concept: general relativity

We will start by commenting and motivating again why this example is
interesting. The framework described here is a summary of [31, 23], which we
refer the reader for a more detailed discussion -we will only introduce the
minimal ingredients to make the note self-contained-.

Consider the Einstein-Hilbert action in $D>4$ dimensions

$\displaystyle\mathcal{L}=\frac{M_{D}^{D-2}}{2}\sqrt{-G}R(G)\,,$ (4)

with $M_{D}$ the $D$-dimensional Plank mass. In $\mathds{R}^{1,D-1}$ this
theory is, of course, consistent with the CRG conjecture [3]. Dimensionally
reducing it to 4d while keeping all massive modes just means selecting a
different background for the theory. Consequently, one would expect the CRG
conjecture to continue to be satisfied in the 4d picture. The interesting
point is that, while in $D$ dimensions we have a description in terms of pure
(Einstein) gravity, in 4d the dimensional reduction of the graviton _produces_
a graviton but also a tower of massive spin-2, spin-1 and scalar particles. We
can then take any Kauza-Klain massive spin-2 copies of the graviton and
compute its $2\rightarrow 2$ scattering. In a generic theory with no other
massive spin-2 particles, we saw in [1] that this scattering would violate the
CRG bounds. In contrast, we will see below how the CRG conjecture is satisfied
in this set-up, imposing restrictions in the _effective_ spectrum.

###  IV.1 Dimensionally reduced theory

We will study the Lagrangian (4) in the direct product space
$\mathcal{M}_{D}=\mathds{R}^{1,3}\times X_{D-4}$, with metric

$\displaystyle
ds^{2}=G_{AB}dX^{A}dX^{B}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}+g_{ab}dy^{a}dy^{b}\,,$
(5)

where $A=1,\dots,D$, $\mu=1,\dots 4$, $a=D-4,\dots,D$ and we require $X_{D-4}$
to be a closed, smooth, connected, orientable Ricci-flat444This is necessary
to solve the vacuum equations. Riemannian manifold. To obtain the interactions
in the lower dimensional description, one first needs to expand the metric
around the background $\bar{G}_{AB}$

$\displaystyle G_{AB}=\bar{G}_{AB}+\frac{2}{M_{D}^{\frac{D-2}{2}}}\delta
G_{AB}\,,$ (6)

and then expand the fluctuations using the usual Hodge decomposition. Skipping
some field redefinitions and showing only the contributions relevant to our
computations, we have

$\displaystyle\delta G_{\mu\nu}(x,y)$
$\displaystyle=\sum_{n}h^{n}_{\mu\nu}(x)\psi_{n}(y)+\frac{1}{\sqrt{V}}h^{0}_{\mu\nu}(x)\,,$
(7a) $\displaystyle\delta G_{\mu a}(x,y)$
$\displaystyle=\sum_{i}A^{i}_{\mu}(x)Y_{a,i}(y)+\dots\,$ (7b)
$\displaystyle\delta G_{ab}(x,y)$
$\displaystyle=\frac{1}{D-4}\frac{1}{\sqrt{V}}\phi^{0}(x)g_{ab}+\dots\,,$ (7c)

where

$V=\int_{X_{D-4}}\sqrt{-g}dy^{a}$ (8)

and the $\\{\psi_{n},\,Y_{n,i}\\}$ satisfy

$\displaystyle\Delta\psi_{n}$
$\displaystyle\equiv-\square\psi_{i}=m_{n}^{2}\psi_{i}\,,$ (9a)
$\displaystyle\int_{X_{D-4}}\psi_{n}\psi_{m}$
$\displaystyle=\delta_{nm}\,,\quad m_{n}^{2}>0\,;$ (9b) $\displaystyle\Delta
Y_{a,i}$ $\displaystyle\equiv-\square
Y_{a,i}+R_{a}^{b}Y_{b,i}=m_{i}^{2}Y_{a,i}\,,$ (9c)
$\displaystyle\int_{X_{D-4}}Y^{a}_{i}Y_{a,j}$
$\displaystyle=\delta_{ij}\,,\quad m_{i}^{2}\geq 0\,,$ (9d)

being $R_{ab}$ the internal Ricci curvature. We refer again to [31, 23] for a
more detailed discussion of all the quantities and definitions. Plugging all
these expressions into (4) and expanding the action, one can obtain the
spectrum and the interactions of the dimensionally reduced theory. Let us
summarise the main results we will need.

####  IV.1.1 Spectrum

From the quadratic terms one can see that the four-dimensional theory
contains:

  * •

One massless graviton, $h^{0}_{\mu\nu}$.

  * •

A tower of massive spin-2 particles $h^{n}_{\mu\nu}$ with squared masses
$m_{n}^{2}>0$. They come from the eigenfunctions of the scalar Laplacian on
$X_{D-4}$.

  * •

A tower of spin-1 fields $A^{i}_{\mu}$ with squared masses $m_{i}^{2}\geq 0$.
They come from the eigenfunctions of the vector Laplacian on $X_{D-4}$. This
tower includes the killing vectors, which are massless.

  * •

A massless scalar field $\phi^{0}$ controlling the internal volume.

There are more scalar fields in the spectrum coming from the terms omitted in
the decomposition of $\delta G_{ab}$. We will ignore them since they will not
play any role in our computation.

Finally, let us remember that the relation between the higher dimensional
($M_{D}$) and the lower dimensional ($M_{d}$) Planck mass is given by

$\displaystyle M_{d}^{d-2}=VM_{D}^{D-2}\,.$ (10)

####  IV.1.2 Cubic interactions

We are interested in the (classical) scattering $h^{n_{i}}h^{n_{j}}\rightarrow
h^{n_{k}}h^{n_{l}}$. Therefore, we will only need the three-point functions
involving two massive spin-2 fields to construct the exchange diagrams. We
relegate the explicit expressions to appendix B.1, while we list here the
relevant interactions:

  * •

Three massive spin-2 particles:
$\mathcal{A}\left(h^{n_{1}},h^{n_{2}},h^{n_{3}}\right)$.

  * •

Two identical massive spin-2 particles -otherwise this interaction vanishes,
see (• ‣ B.1)- and the graviton:
$\mathcal{A}\left(h^{n_{1}},h^{n_{1}},h^{0}\right)$.

  * •

Two distinct massive spin-2 particles -otherwise this interaction vanishes,
see (• ‣ B.1)- and a spin-1 particle:
$\mathcal{A}\left(h^{n_{1}},h^{n_{2}},A^{i}_{\mu}\right)$.

  * •

Two massive spin-2 particles and a scalar field:
$\mathcal{A}\left(h^{n_{1}},h^{n_{2}},\phi\right)$

A pictorial representation of the exchange contributions to the
$h^{n_{i}}h^{n_{i}}\rightarrow h^{n_{i}}h^{n_{i}}$ scattering can be seen in
figure 2.

Figure 2: Exchange contributions to the $2\rightarrow 2$ scattering of a
massive particle $h^{n_{i}}$. The particle is a part of the KK tower of the
graviton in 4d. In the picture $n_{j}\neq n_{i}$.

It is also worthy to define in this section the triple overlap integrals
$g_{n_{1}n_{2}n_{3}}$

$\displaystyle
g_{n_{1}n_{2}n_{3}}=\int_{X_{D-4}}\psi_{n_{1}}\psi_{n_{2}}\psi_{n_{3}}\,,$
(11)

-the $\psi_{n_{1}}$ introduced in expression (7a) - which will be used later on.

####  IV.1.3 Contact terms

Finally, to compute any $h^{n_{i}}h^{n_{j}}\rightarrow h^{n_{k}}h^{n_{l}}$
scattering, we need the 4-point interactions. Repeating the previous game, one
has to insert the decompositions introduced in section IV.1 into the action
(4) and collect the terms involving four massive spin-2 particles. We write
the explicit form of this interaction, which we denote by
$\mathcal{A}_{\text{contact}}\left(h^{n_{1}},h^{n_{2}},h^{n_{3}},h^{n_{4}}\right)$,
in appendix B.2. For posterior uses, we define here the quartic overlap
integrals

$\displaystyle
g_{n_{1}n_{2}n_{3}n_{4}}=\int_{X_{D-4}}\psi_{n_{1}}\psi_{n_{2}}\psi_{n_{3}}\psi_{n_{4}}\,,$
(12)

which, as discussed in [23], can be written in terms of the cubic overlap
integrals

$\displaystyle
g_{n_{1}n_{2}n_{3}n_{4}}=\sum_{i}g_{n_{1}n_{2}n_{i}}g_{n_{3}n_{4}n_{i}}+\frac{1}{V}\delta(n_{1}n_{2})\delta(n_{3}n_{4})$
$\displaystyle=\sum_{i}g_{n_{1}n_{3}n_{i}}g_{n_{2}n_{4}n_{i}}+\frac{1}{V}\delta(n_{1}n_{3})\delta(n_{2}n_{4})$
$\displaystyle=\sum_{i}g_{n_{1}n_{4}n_{i}}g_{n_{2}n_{3}n_{i}}+\frac{1}{V}\delta(n_{1}n_{4})\delta(n_{2}n_{3})\,.$
(13)

###  IV.2 Results

Having introduced all the necessary ingredients, we are finally in the
position to test the CRG conjecture in the EFT obtained from the dimensional
reduction of GR. Let us briefly recall what are the steps to follow:

  1. 1.

Compute the tree level $h^{n_{i}}h^{n_{j}}\rightarrow h^{n_{k}}h^{n_{l}}$
scattering. Using the language presented in the previous section, it reads

$\displaystyle\mathcal{A}\left(h^{n_{i}},h^{n_{j}},h^{n_{k}},h^{n_{l}}\right)$
$\displaystyle=\mathcal{A}_{\text{contact}}+\mathcal{A}_{\rm exhange}\,,$

where, to construct the exchange diagrams, we take two sets of the three-point
functions introduced in section IV.1.2, “remove” the exchanged leg, and
connect them through the correspondent propagator.

  2. 2.

Expand the total amplitude in the limit $s\gg t$, where $s$ and $t$ are the
usual Mandelstam variables

$\displaystyle\lim_{s\gg
t}\mathcal{A}\left(h^{n_{i}},h^{n_{j}},h^{n_{k}},h^{n_{l}}\right)=\mathcal{A}_{0}(t)s^{0}+\mathcal{A}_{1}(t)s^{1}$
$\displaystyle+\mathcal{A}_{2}(t)s^{2}+\mathcal{A}_{3}(t)s^{3}+\dots\,.$ (14)

In appendix C we recall the definition of the Mandelstam variables and set the
conventions for the kinematics.

  3. 3.

Finally, from the previous expansion we impose that

$\displaystyle\mathcal{A}_{n}(t)=0\,,\\{n\geq 3,\,\forall t\\}\,.$ (15)

We will do this for any of the $5^{4}=625$ choices of polarisation of the
scattered spin-2 particles.555Not all choices are independent, some of them
will be related by crossing symmetry.

Taking $n_{i}=n_{j}=n_{k}=n_{l}$ equation (15) requires

$\displaystyle\forall
n_{i}\,,\,4m_{n_{i}}^{2}g_{n_{i}n_{i}n_{i}n_{i}}-\sum_{k}3m_{n_{k}}^{2}g_{n_{i}n_{i}n_{k}}^{2}=0\,,$
(16)

which can also be written, using expansion (IV.1.3), as

$\displaystyle\forall
n_{i}\,,\,\sum_{k}\left(4m_{n_{i}}^{2}-3m_{n_{k}}^{2}\right)g^{2}_{n_{i}n_{i}n_{k}}+4m_{n_{i}}^{2}V^{-1}=0\,.$
(17)

These relations, which will be automatically satisfied by any valid internal
geometry, have several consequences regarding the 4d spectrum. They are not
completely new since they also appear when one demands the dimensional reduced
theory to be unitary [24, 23] -actually, in [23] they were even able to find
stronger conditions-.666This is because, while the CRG conjecture _cares_
about terms scaling with the energy at order $s^{3}\sim E^{6}$ or higher,
unitarity in this context places conditions on terms scaling with the energy
at order $E^{4}$ or higher.

From (17) it can be deduced that there must be an infinite number of KK modes
in the spectrum. Since the term outside the sum is positive definite, the sum
itself must produce a negative contribution that compensates it. This implies
that for any $h^{n_{i}}$ there must exist some $h^{n_{l}}$ to which the
$h^{n_{i}}$ couples (that is, $g_{n_{i}n_{i}n_{l}}\neq 0$) and such that

$\displaystyle\frac{2}{\sqrt{3}}m_{n_{i}}<m_{n_{l}}\,.$ (18)

Taking $m_{n_{i}}=m_{n_{1}}$, this equation tells us that there must exist
another spin-2 particle with mass $m_{n_{2}}>m_{n_{1}}$ in the spectrum. We
can then apply the same strategy to $m_{n_{2}}$ and conclude that the spectrum
must contain a third spin-2 state with mass $m_{n_{3}}>m_{n_{2}}$. Repeating
this reasoning, we see that any finite truncation of the KK tower of the
graviton is incompatible with the CRG conjecture. This goes in the lines of
[20] -see also [21, 22]\- who first showed the inconsistencies of a truncated
KK spin-2 spectrum by using the breaking of the massive gauge invariances.

On the other side, equation (16) is useful to see777This can also be seen from
(17) since they are equivalent, eq. (16) just gives a cleaner expression. that
the mass ratio of consecutive spin-2 modes is bounded by the 4d couplings.
Since the first term in (16) is positive definite, the second term cannot be
“too negative”. In other words, $\forall h^{n_{i}}$ there must exist some
$h^{n_{k}}$ to which the $h^{n_{i}}$ couples (that is,
$g_{n_{i}n_{i}n_{k}}\neq 0$) and such that

$\displaystyle\frac{m_{n_{k}}^{2}}{m_{n_{i}}^{2}}\leq\frac{4}{3}\frac{g_{n_{i}n_{i}n_{i}n_{i}}}{g_{n_{i}n_{i}n_{k}}^{2}}>1\,,$
(19)

constraining the mass ratios of consecutive spin-2 KK particles.888If
$m_{n_{k}}=m_{n_{i+1}}$ then this is the maximum allowed gap between
consecutive massive spin-2 states. If not, this means that
$m_{n_{k}}>m_{n_{i+1}}$ -since $m_{n_{k}}>m_{n_{i}}$\- and so the gap is even
smaller.

A consequence of the relation (19) is that it seems difficult to generate a
consistent gravitational 4d theory in which part of the graviton KK tower can
be integrated out leaving a finite -bigger than zero- number of massive spin-2
particles in the spectrum. For this to make sense, the mass $m_{\Lambda}$ of
the lightest integrated particle should satisfy
$m,E<<\Lambda_{\rm{EFT}}<<m_{\Lambda}$ where $m$ is the mass scale of the
particles kept in the theory, $E$ is the energy at which the theory is being
proved and $\Lambda_{\rm{EFT}}$ is the cut-off of the EFT. What we learn from
(19) is that the 4d coupling constants bound the gap between the mass of the
spin-2 replicas, so the couplings should be appropriately tuned to achieve the
desired _mass separation_. Once chosen, one should find the concrete geometry
producing these values for the 4d couplings, which can be a very non-trivial
task.

Equation (17) -or its equivalent expression (16)- and its consequences are the
main result of this paper. They teach us that, even if we start with a theory
compatible with the CRG conjecture, as it is GR, when it is dimensionally
reduced to 4d the spectrum of the resulting theory satisfies two non-trivial
constraints:

  1. 1.

Either there is none or an infinite number of massive spin-2 modes.

  2. 2.

The gap in the mass ratio of consecutive KK spin-2 states is bounded by the
coupling constants of the theory.

We derived all these conditions by looking at the
$h^{n_{i}}h^{n_{i}}\rightarrow h^{n_{i}}h^{n_{i}}$ amplitude, so one could
wonder about the more general $h^{n_{i}}h^{n_{j}}\rightarrow
h^{n_{k}}h^{n_{l}}$ interaction. We also studied the CRG conjecture for this
case. Nevertheless, the results and constraints derived from it are less
powerful and interesting than the ones we already presented. In any case, the
reader interested can find an ancillary Mathematica notebook with the code
used.

Before moving to the conclusions, it is worth pausing here for a moment to
make a couple of comments.

As pointed out throughout the section, the constraints (16)-(17), imposed by
the CRG conjecture, had already appeared in the literature. They are also a
requirement for the dimensionally reduced theory to be unitary [24, 23] -which
actually demands more stringent conditions-. On the other hand, a condition
similar to (18) was derived in [20] by studying the gauge invariances of a
dimensionally reduced theory when the KK tower of the graviton is truncated.
The novelty here is that we have derived all these requisites using the CRG
conjecture. This is a non-trivial check of the conjecture: for the first time
it has been tested under dimensional reduction.

We have studied the dimensional reduction of GR on a compact, Riemannian,
Ricci flat internal manifold down to a 4d flat space. A natural question is
thus how the conclusions would change if we modified any of the ingredients:
including matter or higher-derivative corrections, choosing a different
external space… These considerations can be taken into account all at once by
studying the most generic case, discussed in section III, which is a
formidable task. A more doable approach could be to study the changes one by
one, for instance by looking at the scattering of massive spin-2 particles in
AdS or by starting from GR coupled to some matter. Based on the results of [1]
and on the apparent impossibility of constructing truncations with a finite
number of massive spin-2 modes [20, 22], we would expect the conclusions
obtained here to hold in more general scenarios. We leave the exploration of
these ideas for future work.

##  V Conclusions

In this note, we have studied the CRG conjecture in the 4d effective field
theory that results from compactifying $D$-dimensional General Relativity
(with $D>4$) on a closed, Ricci-flat manifold. To do so, we have used the
tools and the framework developed in [23].

Whereas the conjecture is trivially satisfied in the $D$-dimensional
description999The $2\rightarrow 2$ scattering of a GR graviton in
$\mathds{R}^{1,D-1}$ scales with $s$ in the Regge limit as $\mathcal{A}\sim
s^{n}\,,\,n\leq 2$, [3]., the 4d picture consists of a theory of gravity
coupled to an infinite number of massive spin-2 particles. We already saw in
[1] -see [2] for related work- that any gravitational EFT containing a single
massive spin-2 particle cannot be made consistent with the CRG conjecture. The
example studied here, in contrast, serves as an arena to see how the CRG
conjecture is satisfied in the presence of several massive spin-2 states.

The main result of this work is equation (16) -or equivalently equation (17)-
which is required for the CRG conjecture to hold in the 4d framework. Both
conditions are automatically met when choosing a valid internal geometry. From
the 4d perspective, they teach us how the CRG conjecture can be realized in a
theory containing massive spin-2 particles. These expressions are also part of
the conditions for the theory to be unitary [24, 23]. Two consequences follow
from them:

  * •

The 4d spectrum must include either no massive spin-2 fields or an infinite
number of them, a finite truncation would not be possible. This was also
discussed from other points of view in [20, 21, 22, 23].

  * •

The mass ratio between the consecutive KK spin-2 replicas is bounded by the 4d
coupling constants. We concrete this point in section IV.2. Similar
conclusions (actually a bit stronger) were derived in [23] by studying
unitarity in the 4d theory.

We see, then, that even if we start with a theory satisfying the CRG
conjecture in $D$ dimensions, the conjecture imposes non-trivial conditions in
the 4d spectrum. This shows the power of the CRG conjecture to discern between
consistent 4d theories, in this case regarding the ones with a higher-
dimensional embedding, in line with the spirit of the swampland program [4]
-see [7, 32] for reviews-.

It is important to keep in mind that in this work we have focused on the
concrete example of GR dimensionally reduced to a flat 4d background.
Therefore, one could wonder about other possibilities: starting with GR plus
some matter, including higher-derivative corrections, changing the external
space… We explained in section III how to address the most generic situation,
which would simultaneously encode all these possibilities. Unfortunately, this
seems to be a highly complex task, so it may be smarter to add more
ingredients one by one. These are exciting scenarios that for sure deserve
further investigation.

In any case, we actually expect the conclusions presented here to hold in more
general contexts. In light of the results of [1] together with this work, it
seems pretty unlikely that the CRG conjecture could be satisfied in a
gravitational EFT with a finite number of massive spin-2 particles, at least
in flat space. We leave the exploration of these avenues and any other
potential cases of interest for future work.

Acknowledgements

We would like to thank Eran Palti for very useful discussions, collaboration
and comments on the manuscript. This work is supported by the Israel Science
Foundation (grant No. 741/20) and by the German Research Foundation through a
German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”.

##  Appendix A Cubic vertices

In this appendix, we will discuss the possible three-point amplitudes for
three massive spin-2 particles, two of which are identical and different from
the third one. We divide this appendix into two sections. In the first part,
A.1, we list all the parity-even and parity-odd on-shell three-point
interactions. In the second part, A.2, we give a Lagrangian basis for the
parity-even terms.

###  A.1 On-shell amplitudes

Here we list all the possible (parity-even and parity-odd) on-shell three-
point functions between two identical and one different massive spin-2
particles. In the particular case $d=4$, we also discuss the dimensionally
dependent relations, which come from the fact that any set of five or more
vectors is linearly dependent in four dimensions.

Notation: we denote by $\mathcal{M}_{i,j,k}\left(m_{i},m_{j},m_{k}\right)$ the
on-shell three-point amplitude involving three particles of spin
$\\{i,j,k\\}$, mass $\\{m_{i},m_{j},m_{k}\\}$, polarisation matrices
$\\{\epsilon_{1},\epsilon_{2},\epsilon_{3}\\}$ and momentum
$\\{p_{1},p_{2},p_{3}\\}$. We define $A_{ij}\equiv\epsilon_{i}\cdot p_{j}$ and
$B_{ij}=\epsilon_{i}\cdot\epsilon_{j}$. $\varepsilon$ is the Levi-Civita
tensor,
$\varepsilon\left(p_{i},p_{j},\epsilon_{k},\epsilon_{l}\right)\equiv\varepsilon^{\mu\nu\alpha\beta}p_{i\mu}p_{j\nu}\epsilon_{k\alpha}\epsilon_{l\beta}$

Parity even

There are in general $8$ different parity-even on-shell cubic amplitudes,
listed in table 1. Part of this classification was already discussed in [23].

$\mathcal{M}^{\rm even}_{2,2,2}\left(m,m,m_{k}\right)$
---
$B_{12}B_{23}B_{13}=\mathcal{X}_{1}$
$B_{12}^{2}A_{31}^{2}=\mathcal{X}_{2}$
$B_{13}^{2}A_{23}^{2}+B_{23}^{2}A_{12}^{2}=\mathcal{X}_{3}$
$B_{13}B_{23}A_{12}A_{23}=\mathcal{X}_{4}$
$B_{12}B_{23}A_{12}A_{31}+B_{12}B_{13}A_{23}A_{31}=\mathcal{X}_{5}$
$B_{12}A_{12}A_{21}A_{31}^{2}=\mathcal{X}_{6}$
$B_{23}A_{12}^{2}A_{21}A_{31}-B_{13}A_{12}A_{21}^{2}A_{31}=\mathcal{X}_{7}$
$A_{12}^{2}A_{21}^{2}A_{31}^{2}=\mathcal{X}_{8}$
Table 1: All possible parity even on-shell cubic amplitudes for two identical,
one different massive spin-2 particles.

In $d=4$ the Gram matrix of the vectors $\\{p_{i},\epsilon_{i\mu}\\}$ must
vanish, from which we obtain

$\displaystyle(-4m^{2}m_{k}^{2}+m_{k}^{4})\mathcal{X}_{1}+2m_{k}^{2}\mathcal{X}_{2}+2m^{2}\mathcal{X}_{3}$
$\displaystyle+(2m_{k}^{2}-4m^{2})\mathcal{X}_{4}+2m_{k}^{2}\mathcal{X}_{5}+4\mathcal{X}_{6}+4\mathcal{X}_{7}=0\,,$
(20)

which can be used to ignore, for instance, $\mathcal{X}_{7}$.

Parity Odd

Regarding the parity odd terms, in general there are 13 distinct
possibilities, enumerated in table 2.

$\mathcal{M}_{2,2,2}^{\rm odd}\left(m,m,m_{k}\right)$
---
$B_{13}B_{23}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{2}\right)=\tilde{\mathcal{X}}_{1}$
$B_{12}B_{23}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{3}\right)-B_{12}B_{13}\,\varepsilon\left(p_{1},p_{2},\epsilon_{2},\epsilon_{3}\right)=\tilde{\mathcal{X}}_{2}$
$A_{31}B_{12}\,\left(\varepsilon\left(p_{1},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)+\varepsilon\left(p_{2},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)\right)=\tilde{\mathcal{X}}_{3}$
$A_{21}B_{13}\,\varepsilon\left(p_{1},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)-A_{12}B_{23}\,\varepsilon\left(p_{2},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)=\tilde{\mathcal{X}}_{4}$
$A_{21}B_{13}\,\varepsilon\left(p_{2},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)-A_{12}B_{23}\,\varepsilon\left(p_{1},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)=\tilde{\mathcal{X}}_{5}$
$A_{31}^{2}B_{12}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{2}\right)=\tilde{\mathcal{X}}_{6}$
$\left(A_{21}A_{31}B_{13}-A_{12}A_{31}B_{23}\right)\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{2}\right)=\tilde{\mathcal{X}}_{7}$
$A_{31}B_{12}\left(A_{21}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{3}\right)+A_{12}\,\varepsilon\left(p_{1},p_{2},\epsilon_{2},\epsilon_{3}\right)\right)=\tilde{\mathcal{X}}_{8}$
$A_{21}^{2}B_{13}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{3}\right)-A_{12}^{2}B_{23}\,\varepsilon\left(p_{1},p_{2},\epsilon_{2},\epsilon_{3}\right)=\tilde{\mathcal{X}}_{9}$
$A_{12}A_{21}\left(B_{23}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{3}\right)-B_{13}\,\varepsilon\left(p_{1},p_{2},\epsilon_{2},\epsilon_{3}\right)\right)=\tilde{\mathcal{X}}_{10}$
$A_{12}A_{21}A_{31}\,\left(\varepsilon\left(p_{1},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)+\,\varepsilon\left(p_{2},\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)\right)=\tilde{\mathcal{X}}_{11}$
$A_{12}A_{21}A_{31}^{2}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{2}\right)=\tilde{\mathcal{X}}_{12}$
$A_{21}A_{31}A_{12}\,\left(A_{21}\,\varepsilon\left(p_{1},p_{2},\epsilon_{1},\epsilon_{3}\right)+A_{12}\,\varepsilon\left(p_{1},p_{2},\epsilon_{2},\epsilon_{3}\right)\right)=\tilde{\mathcal{X}}_{13}$
Table 2: All possible parity-odd on-shell three-point amplitudes for two
identical, one different massive spin-2 particles.

Dimensional-dependent relations in $d=4$ -see [28] for the details- impose

$\displaystyle
2\left(m_{k}^{4}-4m^{2}m_{k}^{2}\right)\tilde{\mathcal{X}}_{1}+\left(4m^{2}m_{k}^{2}-m_{k}^{4}\right)$
$\displaystyle\tilde{\mathcal{X}}_{2}$
$\displaystyle-\left(m_{k}^{2}-2m^{2}\right){}^{2}\tilde{\mathcal{X}}_{3}+\left(2m^{2}m_{k}^{2}-4m^{4}\right)$
$\displaystyle\tilde{\mathcal{X}}_{4}$
$\displaystyle-4m^{4}\tilde{\mathcal{X}}_{5}+4\left(m^{2}-m_{k}^{2}\right)\tilde{\mathcal{X}}_{6}+4m^{2}\tilde{\mathcal{X}}_{7}$
$\displaystyle=0\,,$ (21a)
$\displaystyle\tilde{\mathcal{X}}_{2}-\tilde{\mathcal{X}}_{3}$
$\displaystyle=0\,,$ (21b)
$\displaystyle\tilde{\mathcal{X}}_{10}-\tilde{\mathcal{X}}_{11}$
$\displaystyle=0\,,$ (21c) $\displaystyle
m_{k}^{2}\tilde{\mathcal{X}}_{7}-m_{k}^{2}\tilde{\mathcal{X}}_{8}-2m^{2}\tilde{\mathcal{X}}_{11}+2\tilde{\mathcal{X}}_{12}$
$\displaystyle=0\,,$ (21d)

which we can use to ignore four of the $\tilde{\mathcal{X}}_{i}$ involved in
(21), writing them as linear combination of the others.

###  A.2 Lagrangian basis

To write a Lagrangian basis we recall the expression for the linearized
version of the Riemann tensor $R_{\alpha\beta\mu\nu}$ for a spin-2 field
$h_{\mu\nu}$

$\displaystyle
R_{\alpha\beta\mu\nu}=\frac{1}{2}\left[\partial_{\mu}\partial_{\beta}h_{\nu\alpha}+\partial_{\nu}\partial_{\alpha}h_{\beta\mu}-\partial_{\nu}\partial_{\beta}h_{\mu\alpha}-\partial_{\mu}\partial_{\alpha}h_{\beta\nu}\right]\,,$
(22)

and define the tensor $F_{\alpha\beta\mu}$ as

$F_{\alpha\beta\mu}\equiv\partial_{\alpha}h_{\beta\mu}-\partial_{\beta}h_{\alpha\mu}\,.$
(23)

Using these two quantities, a Lagrangian basis for the parity-even on-shell
three-point amplitudes introduced in table 1 is given in table 3 below.

Lagrangian basis $\mathcal{L}_{2,2,2}\left(m,m,m_{k}\right)$
---
$h^{\mu}_{1\nu}h^{\nu}_{2\alpha}h^{\alpha}_{3\mu}=\mathcal{L}_{1}$
$F^{1\mu\alpha}_{\beta}F^{2}_{\nu\mu\alpha}h^{3\beta\nu}=\mathcal{L}_{2}$
$F^{1\mu\alpha}_{\beta}h^{2\beta\nu}F^{3}_{\nu\mu\alpha}+h^{1\beta\nu}F^{2\mu\alpha}_{\beta}F^{3}_{\nu\mu\alpha}=\mathcal{L}_{3}$
$h^{1\mu\alpha}h^{2\nu\beta}R^{3}_{\mu\nu\alpha\beta}=\mathcal{L}_{4}$
$h^{1\mu\alpha}R^{2}_{\mu\nu\alpha\beta}h^{3\nu\beta}+R^{1}_{\mu\nu\alpha\beta}h^{2\nu\beta}h^{3\mu\alpha}=\mathcal{L}_{5}$
$R^{1\mu\nu\alpha\beta}F^{2\delta}_{\mu\nu}F^{3}_{\alpha\beta\delta}+F^{1\delta}_{\mu\nu}R^{2\mu\nu\alpha\beta}F^{3}_{\alpha\beta\delta}=\mathcal{L}_{6}$
$F^{1}_{\alpha\beta\delta}F^{2\delta}_{\mu\nu}R^{3\mu\nu\alpha\beta}=\mathcal{L}_{7}$
$R^{1\mu\nu}_{\alpha\beta}R^{2\gamma\delta}_{\mu\nu}R^{3\alpha\beta}_{\gamma\delta}=\mathcal{L}_{8}$
Table 3: Lagrangians describing parity-even three-point amplitudes for two
identical, one different massive spin-2 particles

The relation with table 1 is given by

$\displaystyle\mathcal{X}_{1}$ $\displaystyle=\mathcal{L}_{1}\,,$ (24a)
$\displaystyle\mathcal{X}_{2}$
$\displaystyle=\left(m^{2}-\frac{3}{2}m_{k}^{2}\right)\mathcal{L}_{1}-\mathcal{L}_{2}+\mathcal{L}_{3}-\mathcal{L}_{4}\,,$
(24b) $\displaystyle\mathcal{X}_{3}$
$\displaystyle=\left(m_{k}^{2}-2m^{2}\right)\mathcal{L}_{1}+2\mathcal{L}_{2}-\mathcal{L}_{5}\,,$
(24c) $\displaystyle 2\mathcal{X}_{4}$
$\displaystyle=\left(2m^{2}-m_{k}^{2}\right)\mathcal{L}_{1}-2\mathcal{L}_{2}-\mathcal{L}_{4}+\mathcal{L}_{5}\,,$
(24d) $\displaystyle\mathcal{X}_{5}$
$\displaystyle=m_{k}^{2}\mathcal{L}_{1}-\mathcal{L}_{3}+\mathcal{L}_{4}\,,$
(24e) $\displaystyle 4\mathcal{X}_{6}$
$\displaystyle=3m_{k}^{4}\mathcal{L}_{1}-2m_{k}^{2}(\mathcal{L}_{3}-\mathcal{L}_{4})+\mathcal{L}_{7}\,,$
(24f) $\displaystyle 4\mathcal{X}_{7}$
$\displaystyle=6m_{k}^{2}\left(2m^{2}-m_{k}^{2}\right)\mathcal{L}_{1}-4m_{k}^{2}\mathcal{L}_{2}$
$\displaystyle+2\left(m_{k}^{2}-2m^{2}\right)\mathcal{L}_{3}+4\left(m^{2}-m_{k}^{2}\right)\mathcal{L}_{4}$
$\displaystyle+2m_{k}^{2}\mathcal{L}_{5}+\mathcal{L}_{6}\,,$ (24g)
$\displaystyle 8\mathcal{X}_{8}$
$\displaystyle=5m_{k}^{4}\left(2m^{2}-m_{k}^{2}\right)\mathcal{L}_{1}-2m_{k}^{4}\mathcal{L}_{2}$
$\displaystyle+2m_{k}^{2}\left(m_{k}^{2}-2m^{2}\right)\mathcal{L}_{3}+m_{k}^{2}\left(4m^{2}-3m_{k}^{2}\right)\mathcal{L}_{4}$
$\displaystyle+m_{k}^{4}\mathcal{L}_{5}+m_{k}^{2}\mathcal{L}_{6}+\left(2m^{2}-m_{k}^{2}\right)\mathcal{L}_{7}-\mathcal{L}_{8}\,.$
(24h)

It is important to bear in mind that Lagrangians are off-shell quantities.
Under field redefinitions or integration by parts they give rise to the same
(on-shell) dynamics. This being said, notice that the l.h.s of equation (24)
is defined on-shell. Therefore, the equality only makes sense when the r.h.s
-the linear combination of Lagrangians- is also evaluated on-shell.

##  Appendix B Couplings from GR

###  B.1 Cubic couplings

In this appendix we write the three-point interactions that result from
plugging the decomposition (7) into the Einstein-Hilbert action. These results
were published initially in [31, 23].

To start with, we need to fix the notation. A particle $i$ has momentum
$p_{i}$ and mass $m_{i}$. We denote its polarisation tensor by $\epsilon_{i}$.
This tensor is symmetric and traceless. Formally, when constructing the
interactions, we will write the polarisation matrices as a product of vectors
$\epsilon_{i}=\epsilon_{i\mu\nu}\equiv\epsilon_{i\mu}\epsilon_{i\nu}$. This
does not mean that the polarisations matrices have rank one, it is only a
trick to keep track of the contractions more easily. To simplify the
expressions, we call $A_{ij}\equiv\epsilon_{i}\cdot p_{j}$,
$B_{ij}=\epsilon_{i}\cdot\epsilon_{j}$ and $p_{ij}=p_{i}\cdot p_{j}$.

This being said, the vertices involving (at least) two massive spin-2 are:

  * •

Three massive spin-2 particles $\\{h^{n_{1}},h^{n_{2}},h^{n_{3}}\\}$

$\displaystyle\mathcal{A}\left(h^{n_{1}},h^{n_{2}},h^{n_{3}}\right)=$
$\displaystyle-\frac{g_{n_{1}n_{2}n_{3}}}{4M_{D}^{\frac{D-2}{2}}}B_{23}\left[4A_{13}\left(B_{23}A_{12}-2B_{12}A_{32}\right)\right.$
$\displaystyle\left.+(2p_{12}-m_{n_{1}}^{2})B_{12}B_{23}\right]+\rm{5\,permutations}\,,$
(25)

where we have implicitly assigned the numbers

$\displaystyle\\{1,2,3\\}\equiv\\{h^{n_{1}},h^{n_{2}},h^{n_{3}}\\}\,,$ (26)

-we will also do this for the other interactions- and have introduced the triple overlap integrals

$\displaystyle
g_{n_{1}n_{2}n_{3}}=\int_{X_{D-4}}\psi_{n_{1}}\psi_{n_{2}}\psi_{n_{3}}\,.$
(27)

  * •

Two massive spin-2 particles and the graviton
$\\{h^{n_{1}},h^{n_{2}},h^{0}\\}$

$\displaystyle\mathcal{A}\left(h^{n_{1}},h^{n_{2}},h^{0}\right)=$
$\displaystyle\frac{1}{4M_{d}^{\frac{d-2}{2}}}\delta_{n_{1},n_{2}}\left[8\left(B_{23}A_{12}-B_{13}A_{21}\right)\left(B_{23}A_{12}+B_{12}A_{31}\right)\right.$
$\displaystyle\left.+\left(B_{12}\right)^{2}\left(p_{33}B_{33}-4A_{31}A_{32}\right)\right]+1\leftrightarrow
2\,.$ (28)

  * •

Two massive spin-2 particles and one spin-1 particle
$\\{h^{n_{1}},h^{n_{2}},A^{i_{3}}_{\mu}\\}$

$\displaystyle\mathcal{A}\left(h^{n_{1}},h^{n_{2}},A^{i_{3}}_{\mu}\right)=$
$\displaystyle\frac{\sqrt{2}}{M_{D}^{\frac{D-2}{2}}}g_{n_{1}n_{2}i_{3}}\left(2B_{12}B_{13}A_{21}-B_{12}^{2}A_{31}\right.$
$\displaystyle\left.-\left(1\leftrightarrow 2\right)\right)\,,$ (29)

with

$\displaystyle
g_{n_{1}n_{2}i_{3}}=\int_{X_{D-4}}\partial^{a}\psi_{n_{1}}\psi_{n_{2}}Y_{a,i_{3}}\,,$
(30)

being antisymmetric in the first two indices -so necessarily $n_{1}\neq
n_{2}$-.

  * •

Two massive spin-2 particles and one scalar particle
$\\{h^{n_{1}},h^{n_{2}},\phi\\}$

$\displaystyle\mathcal{A}\left(h^{n_{1}},h^{n_{2}},\phi\right)\propto\phi
B_{12}^{2}\,.$ (31)

As commented in [1], for any polarization of the spin-2 particles this term
scales with $s$ as $s^{n}$, $n\leq 2$: it does not contribute to the CRG
equations. For this reason, for our purposes it is enough to write the part
that gives the dependence on the kinematics.

###  B.2 Quartic couplings

Following the notation introduced in the previous section, the 4-point
interaction between any four massive spin-2 particles
$\\{h^{n_{1}},h^{n_{2}},h^{n_{3}},h^{n_{4}}\\}$ that are part of the KK tower
of the graviton is given by

$\displaystyle\mathcal{A}_{\text{contact}}\left(h^{n_{1}},h^{n_{2}},h^{n_{3}},h^{n_{4}}\right)=$
$\displaystyle\frac{1}{M_{D}^{D-2}}\left[g_{n_{1}n_{2}n_{3}n_{4}}B_{12}B_{34}\left(\left(p_{12}-m_{n_{1}}^{2}\right)B_{14}B_{23}\right.\right.$
$\displaystyle\left.+A_{14}\left[B_{34}A_{23}-B_{24}A_{32}-B_{23}\left(A_{41}+2A_{43}\right)\right]\left.\right)\right.$
$\displaystyle\left.+\sum_{i}\frac{1}{2}m_{n_{i}}^{2}g_{n_{1}n_{2}n_{i}}g_{n_{3}n_{4}n_{i}}B_{12}B_{34}B_{14}B_{23}\right]$
$\displaystyle+$ $\displaystyle\rm{23\,permutations}\,,$ (32)

where

$\displaystyle
g_{n_{1}n_{2}n_{3}n_{4}}=\int_{X_{D-4}}\psi_{n_{1}}\psi_{n_{2}}\psi_{n_{3}}\psi_{n_{4}}\,.$
(33)

##  Appendix C Kinematics

In this appendix we will write the definitions of the variables used to
compute the $2\rightarrow 2$ scattering of section IV.2.

The incoming particles are labelled by $1$ and $2$, and outgoing particles by
$3$ and $4$. Momentum conservation requires

$\displaystyle p_{1}+p_{2}=p_{3}+p_{4}\,,$ (34)

where we take

$\displaystyle
p_{i}^{\mu}=\left(E_{i},p_{i}\sin\theta_{i},0,p_{i}\cos\theta_{i}\right)\,,$
(35)

with $E_{i}^{2}=p_{i}^{2}+m_{i}^{2}$ and $\theta_{1}=0$, $\theta_{2}=\pi$,
$\theta_{3}=\theta$, $\theta_{4}=\theta-\pi$. The Mandelstam variables are

$\displaystyle s=$ $\displaystyle-\left(p_{1}+p_{2}\right)^{2}\,,$
$\displaystyle t$ $\displaystyle=-\left(p_{1}-p_{3}\right)^{2}\,,$
$\displaystyle u$ $\displaystyle=-\left(p_{1}-p_{4}\right)^{2}\,,$ (36)

and we are taking the metric $\eta=\text{diag}\left(-1,1,1,1\right)$. They
satisfy

$\displaystyle s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}\,,$ (37)

with $m_{i}$ the mass of the particle $i$. When $m_{1}=m_{2}=m_{3}=m_{4}=m$
-and $E_{i}=E\,,p_{i}=p$\- the Mandelstam variables have the simple
expressions

$\displaystyle s$ $\displaystyle=4E^{2}\,,$ $\displaystyle
cos\theta=1-\frac{2t}{4m^{2}-s}\,.$ (38)

To construct the polarisation matrices of the spin-2 particles, we first
introduce the polarisation vectors:

$\displaystyle\epsilon_{1}^{\mu}(p_{i})$ $\displaystyle=\left(0,0,1,0\right)$
(39a) $\displaystyle\epsilon_{2}^{\mu}(p_{i})$
$\displaystyle=\left(0,\cos\left(\theta_{i}\right),0,-\sin\left(\theta_{i}\right)\right)$
(39b) $\displaystyle\epsilon_{3}^{\mu}(p_{i})$
$\displaystyle=\frac{1}{m_{i}}\left(p_{i},e_{i}\sin\left(\theta_{i}\right),0,e_{i}\sin\left(\theta_{i}\right)\right)\,,$
(39c)

from which the polarisation matrices can be constructed, adapting the
conventions of [33], as

$\displaystyle\epsilon^{\mu\nu}_{T_{,}1}(p_{i})$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\epsilon_{1}^{\mu}(p_{i})\epsilon_{1}^{\nu}(p_{i})-\epsilon_{2}^{\mu}(p_{i})\epsilon_{2}^{\nu}(p_{i})\right)\,,$
(40a) $\displaystyle\epsilon^{\mu\nu}_{T_{,}2}(p_{i})$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\epsilon_{1}^{\mu}(p_{i})\epsilon_{2}^{\nu}(p_{i})+\epsilon_{2}^{\mu}(p_{i})\epsilon_{1}^{\nu}(p_{i})\right)\,,$
(40b) $\displaystyle\epsilon^{\mu\nu}_{V_{,}1}(p_{i})$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\epsilon_{3}^{\mu}(p_{i})\epsilon_{1}^{\nu}(p_{i})+\epsilon_{1}^{\mu}(p_{i})\epsilon_{3}^{\nu}(p_{i})\right)\,,$
(40c) $\displaystyle\epsilon^{\mu\nu}_{V_{,}2}(p_{i})$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\epsilon_{3}^{\mu}(p_{i})\epsilon_{2}^{\nu}(p_{i})+\epsilon_{2}^{\mu}(p_{i})\epsilon_{3}^{\nu}(p_{i})\right)\,,$
(40d) $\displaystyle\epsilon^{\mu\nu}_{S}(p_{i})$
$\displaystyle=\sqrt{\frac{3}{2}}\left(\epsilon_{3}^{\mu}(p_{i})\epsilon_{3}^{\nu}(p_{i})-\frac{1}{3}\left(\eta^{\mu\nu}+\frac{1}{m_{i}^{2}}p_{i}^{\mu}p_{i}^{\nu}\right)\right)\,,$
(40e)

where $T$, $V$ and $S$ stand for tensor, vector and scalar polarizations,
respectively.

Regarding the propagators, the propagator of a scalar particle with mass $M$
is

$\displaystyle\frac{-i}{p^{2}+M^{2}-i\epsilon}\,.$ (41)

For a massive spin-1 particle, first we need to introduce the projector

$\displaystyle\Pi_{\mu\nu}\left(\tilde{m}\right)=\eta_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{\tilde{m}^{2}}\,,$
(42)

from which one can write the propagator of a massive spin-1 particle with mass
$m_{1}$ as

$\displaystyle
P_{\mu\nu}=\frac{-i\Pi_{\mu\nu}\left(m_{1}\right)}{p^{2}+m_{1}^{2}-i\epsilon}\,.$
(43)

Finally, the propagator of a massive spin-2 particle of mass $m$ is

$\displaystyle
P_{\mu_{1}\mu_{2},\nu_{1}\nu_{2}}=\frac{-i}{2}\frac{\Pi_{\mu_{1}\nu_{1}}\Pi_{\mu_{2}\nu_{2}}+\Pi_{\mu_{1}\nu_{2}}\Pi_{\mu_{2}\nu_{1}}-\frac{2}{3}\Pi_{\mu_{1}\mu_{2}}\Pi_{\nu_{1}\nu_{2}}}{p^{2}+m^{2}-i\epsilon}\,,$
(44)

with $\Pi_{\nu_{1}\nu_{2}}=\Pi_{\nu_{1}\nu_{2}}\left(m\right)$, whereas for a
massless spin-2 (in de Donder gauge) it reads

$\displaystyle\tilde{P}_{\mu_{1}\mu_{2},\nu_{1}\nu_{2}}=\frac{-i}{2}\frac{\eta_{\mu_{1}\nu_{1}}\eta_{\mu_{2}\nu_{2}}+\eta_{\mu_{1}\nu_{2}}\eta_{\mu_{2}\nu_{1}}-\eta_{\mu_{1}\mu_{2}}\eta_{\nu_{1}\nu_{2}}}{p^{2}-i\epsilon}\,.$
(45)

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