Dimensionally reducing the classical Regge growth conjecture

We explore the classical Regge growth conjecture 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 Kaluza-Klain 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 (treelevel) S-matrix 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 lim s→∞, t<0 fixed A(s, t) where by s → ∞ we mean Λ ≫ s ≫ |t|, with Λ 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]. 1his 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 [2,[8][9][10][11][12][13][14] 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.Using the duality between Antide Sitter space (AdS) and conformal field theories, 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. 2In 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 R 1,D−1 trivially satisfies the CRG conjecture, the 2 → 2 scattering of a GR graviton scales with s in the Regge limit at most as A ∼ s 2 .The point is that when GR is compactified to 4d (we go from R 1,D−1 to R 1,3 × 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 constraints 3 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 [23,24].
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 → 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.• 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. 4 Symmetries forbid interactions between two identical massive spin-2 particles and one massless spin-1 field or one fermion.

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.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 where the c ii k,j are arbitrary constants and the f j h i , h i , h k 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 µν , h i µν } only interacts with one different massive spin-2 state {h k µν }.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 Generalising [1] to include any number of massive spin-2 fields would correspond to take K ii = arbitrary and c ii k,j = 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 [23,31], 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 with M D the D-dimensional Plank mass.In 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 Kaluza-Klain massive spin-2 copies of the graviton and compute its 2 → 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.

A. Dimensionally reduced theory
We will study the Lagrangian (4) in the direct product space M D = R 1,3 × X D−4 , with metric ds 2 = G AB dX A dX B = η µν dx µ dx ν + g ab dy a dy b , (5) where A = 1, . . ., D, µ = 1, . . .4, a = D − 4, . . ., D and we require X D−4 to be a closed, smooth, connected, orientable Ricci-flat 5 Riemannian manifold.To obtain the interactions in the lower dimensional description, one first needs to expand the metric around the background ḠAB 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 where and the {ψ n , Y n,i } satisfy being R ab the internal Ricci curvature.We refer again to [23,31] 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.

Spectrum
From the quadratic terms one can see that the fourdimensional theory contains: • One massless graviton, h 0 µν .• A tower of massive spin-2 particles h n µν with squared masses m 2 n > 0. They come from the eigenfunctions of the scalar Laplacian on X D−4 .
• A tower of spin-1 fields A i µ with squared masses m 2 i ≥ 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 ϕ 0 controlling the internal volume.
There are more scalar fields in the spectrum coming from the terms omitted in the decomposition of δG ab .We will ignore them since they will not play any role in our computation -see appendix B 1 for the details-.
Finally, let us remember that the relation between the higher dimensional (M D ) and the lower dimensional (M d ) Planck mass is given by

Cubic interactions
We are interested in the (classical) scattering h ni h nj → h n k h n l .Therefore, we will only need the three-point functions involving two (on-shell) 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: A (h n1 , h n2 , h n3 ).
• Two massive spin-2 particles and a scalar field: A (h n1 , h n2 , ϕ) A pictorial representation of the exchange contributions to the h ni h ni → h ni h ni scattering can be seen in figure 2. It is also worthy to define in this section the triple FIG. 2. Exchange contributions to the 2 → 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 nj ̸ = ni.
overlap integrals g n1n2n3 -the ψ n1 introduced in expression (7a) -which will be used later on.

Contact terms
Finally, to compute any h ni h nj → h n k h n l scattering, we need the (on-shell) 4-point interactions.Repeating the previous game, one has to insert the decompositions introduced in section IV A 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 A contact (h n1 , h n2 , h n3 , h n4 ), in appendix B 2. For posterior uses, we define here the quartic overlap integrals which, as discussed in [23], can be written in terms of the cubic overlap integrals

B. 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. Compute the tree level h ni h nj → h n k h n l scattering.Using the language presented in the previous section, it reads where, to construct the exchange diagrams, we take two sets of the three-point functions introduced in section IV A 2, "remove" the exchanged leg, and connect them through the correspondent propagator.
2. Expand the total amplitude in the limit s ≫ t, where s and t are the usual Mandelstam variables In appendix C we recall the definition of the Mandelstam variables and set the conventions for the kinematics.
We will do this for any of the 5 4 = 625 choices of polarisation of the scattered spin-2 particles. 6aking n i = n j = n k = n l equation ( 15) requires which can also be written, using expansion ( 13), as 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 [23,24] -actually, in [23] they were even able to find stronger conditions-. 7rom ( 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 ni there must exist some h n l to which the h ni couples (that is, g ninin l ̸ = 0) and such that Taking m ni = m n1 , this equation tells us that there must exist another spin-2 particle with mass m n2 > m n1 in the spectrum.We can then apply the same strategy to m n2 and conclude that the spectrum must contain a third spin-2 state with mass m n3 > m n2 .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 see8 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, ∀h ni there must exist some h n k to which the h ni couples (that is, g ninin k ̸ = 0) and such that constraining the mass ratios of consecutive spin-2 KK particles. 9 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 Λ of the lightest integrated particle should satisfy m, E << Λ EFT << m Λ where m is the mass scale of the particles kept in the theory, E is the energy at which the theory is being probed and Λ EFT is the cutoff 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. Either there is none or an infinite number of massive spin-2 modes.
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 ni h ni → h ni h ni amplitude, so one could wonder about the more general h ni h nj → 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. 10efore 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 [23,24] -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 Ddimensional description 11 , 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 [23,24].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 B. 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.

TABLE III .
Lagrangians describing parity-even three-point amplitudes for two identical, one different massive spin-2 particles