PHYSICAL REVIEW D 100, 104033 (2019)

Spin-2 fields and the weak gravity conjecture

† ‡ Claudia de Rham ,1,2,* Lavinia Heisenberg,3, and Andrew J. Tolley1,2, 1Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom 2CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA 3Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland

(Received 19 July 2019; published 18 November 2019)

Recently, it has been argued that application of the weak gravity conjecture (WGC) to spin-2 fields implies a universal upper bound on the cutoff of the effective theory for a single spin-2 field. We point out here that these arguments are largely spurious, because of the absence of states carrying spin-2 Stückelberg Uð1Þ charge, and because of incorrect scaling assumptions. Known examples such as Kaluza-Klein theory that respect the usual WGC do so because of the existence of a genuine Uð1Þ field under which states are charged, as in the case of the Stückelberg formulation of spin-1 theories, for which there is an unambiguously defined Uð1Þ charge. Theories of bigravity naturally satisfy a naive formulation of the WGC, MW massive gravity trivially satisfies this form of the WGC. We also point out that the identification of a massive spin-2 state in a truncated higher derivative theory, such as Einstein-Weyl-squared or its supergravity extension, bears no relationship with massive spin-2 states in the UV completion, contrary to previous statements in the literature. We also discuss the conjecture from a swampland perspective and show how the emergence of a universal upper bound on the cutoff relies on strong assumptions on the scale of the couplings between the spin-2 and other fields, an assumption which is known to be violated in explicit examples.

DOI: 10.1103/PhysRevD.100.104033

I. INTRODUCTION scattering amplitude for massive states which couple to the massive spin-2 particle, then the existence of at least one The prevailing tool of modern physics is that of effective massive spin-2 particle t-channel exchange violates the field theories (EFTs). Given an assumed hierarchy of Froissart bound, and perturbative unitarity can only be scales, EFTs provide a complete description of the low- recovered by including an infinite tower of spins. This is energy dynamics to arbitrary order in an expansion in E=Λ why in all known weakly coupled UV completions, such as where Λ denotes the scale of high-energy physics. For string theory, the spin-2 state is part of an infinite tower of instance, all gravitational quantum field theories should be excitations. However this argument does not in itself forbid a understood as EFTs due to the nonrenormalizable nature of Λ mass-gap between the spin-2 and higher states which would their interactions where is at most MPlanck and may even be lower. Similarly, attempts to give low-energy descrip- be sufficient to derive a low-energy EFT. A long standing question is whether there exists a theory of interacting spins tions of particles with spin-2 or higher are EFTs, since their ≥ 2 interactions are necessarily nonrenormalizable. For massive S which is sufficiently gapped that it is possible to spins, this is easily noted by performing a helicity decom- construct a low-energy effective theory for a finite number of position from which it becomes clear that the interactions of such spins. Stated differently, assuming an EFT for a particle lower helicity states contain increasing numbers of deriv- of spin S and mass m, is the cutoff for that EFT parametri- Λ ≫ atives. In a weakly coupled theory, if we consider the 2–2 cally larger than m,i.e., m? It is well known that in the case of spin-2 particles coupled to gravity the cutoff is at 1 2 1=3 most Λ3 ¼ðm M Þ and this prevents a weakly coupled *[email protected] Pl † description of the limit m → 0 for fixed M . There are [email protected] Pl ‡ [email protected] related conjectures for more general spins [7,8].

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. 1Pushing Λ to larger values requires a resonance of soft modes Further distribution of this work must maintain attribution to [1], or breaking of Lorentz invariance as in [2–5]. Raising of the the author(s) and the published article’s title, journal citation, cutoff via spontaneous breaking of Lorentz invariance was for and DOI. Funded by SCOAP3. instance considered in [6].

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2 2 These specific questions are part of a larger class of force Fg ≤ Fe ∼ q =r , requires the parameters to sat- ≤ questions of when can we embed a given gravitational EFT isfy m=MPl q. into a UV complete theory? A prevailing view is that the In a recent work [14] the interesting question of how EFTs can be divided into two main groups, often referred to the WGC and similar swampland criteria constrain the as the landscape and the swampland. EFTs of the former EFT of a massive spin-2 field was raised. More specifi- group can successfully be embedded into a UV complete cally, it was conjectured that when a spin-2 particle of theory (e.g., string theory), whereas EFTs of the latter mass m with self-interactions at the scale MW is coupled group represent an inhabitable space of theories incompat- to gravity, then there is a universal upper bound on the ible with quantum gravity [9]. From this point of view, it cutoff of the low-energy EFT set by Λ ≤ Λ ≈ mMPl. cutoff m MW has become an indispensable task to comprehend whether a We review the precise arguments of the claim in Sec. II A given EFT coupled to gravity belongs to the swampland or but first discuss the overall logic of the arguments and lies in the safe landscape. In recent years there have been a the relations with the small mass limit of massive spin-2 number of conjectures which define the division between fields. the swampland and landscape. There is some evidence for The arguments of [14] apply the WGC to the Uð1Þ these conjectured criteria from stringy constructions but Stückelberg symmetry that arises in the helicity decom- rigorous proofs are still lacking. Among them, the con- position of a massive spin-2 field in the decoupling limit.In jectures on global symmetries, weak gravity, distance in this article, we investigate the validity of these assumptions moduli space and de-Sitter space have received quite some and the conjecture. In particular we focus our discussion on attention in the literature. Less conjectural are the positivity the following points: bounds which can be proven given a minimal set of (1) The arguments of [14] rely on the crucial assumption assumptions [10]. that the scaling between the helicity modes and the The most well-known folk theorem relating to quantum sources has a smooth massless limit. In practice the gravity is the statement that a consistent gravitationally scaling of the interactions of the helicity modes with coupled theory cannot have global symmetries [11,12]. The sources typically involve negative powers of the motivation for this conjecture arises from the no-hair spin-2 mass. theorem for black holes in standard GR. An EFT with (2) Reference [14] fails to assume a mass gap between global symmetry coupled to gravity would have a spectrum the light spin-2 states and the tower of states in the of charged states. When these charged states are absorbed UV completion. Accounting for this leads to a very into a black hole, a stable charged remnant will form that different conjecture. has no apparent means to decay, violating entropy/infor- (3) Swampland conjectures from emergence lead to mation bounds. Hence, if an EFT possesses a global different conclusions than the conjecture. symmetry, it has to be broken, i.e., at best approximate, (4) The helicity-1 mode of the massive graviton is not a or gauged at some scale. genuine vector field. As a consequence, there are no The weak gravity conjecture (WGC) builds on this global Uð1Þ charges associated with it and no general idea and implies in its strongest form that if an foundations for the spin-2 WGC. This point is EFT with a Uð1Þ symmetry is coupled to gravity, then there explained in Sec. II. should exist a fundamental cutoff for that EFT at the scale2 (5) In Sec. III, we discuss the conjecture from a swamp- Λ ¼ qMPl [13] where q is the fundamental charge asso- land perspective and show how the conjecture gets ciated with the Uð1Þ gauge field. Assuming a weakly violated when diffeomorphism is preserved in the coupled UV completion this implies the existence of a couplings with spin-2 fields, or only weakly broken. particle whose mass m is of order qMPl, possibly accom- (6) In Sec. IV, we discuss examples of interacting spin-2 panied by a tower of higher masses. From an EFT point of theories such as string theory, Kaluza-Klein and view there is naively no problem taking the limit q → 0 for Einstein-Weyl-squared theory and show that they do which the gauge symmetry becomes global, however this not support the conjectured cutoff proposed in [14]. limit will run afoul of the previous folk theorem. The (7) Explicit UV completions of massive gravity (on Λ ¼ – universal cutoff qMPl explains why it is impossible to AdS) such as those considered in [15 17] do not take the global limit. Additional motivation for this con- respect the conjecture. In particular Ref. [18] pro- jecture comes from the fact that all non-BPS black holes poses at alternative conjecture which is consistent should be able to decay. This conjecture may be restated as with standard decoupling limit considerations. the idea that gravitational force should always be the (8) As a byproduct of these considerations, we note weakest [13]. Requiring that the gravitational force Fg ∼ that the identification of the massive spin-2 modes m2=ðM2 r2Þ should be weaker than the repulsive electric (ghosts) in certain higher derivative EFT truncations, Pl with spin-2 states in the UV completion as argued in [19] cannot be trusted. This means in particular 2This is the so-called magnetic version of the conjecture. that it is meaningless to apply spin-2 conjectures to

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ghostly massive spin-2 poles arising from a truncated explicitly how those examples rely almost solely on the expansion. introduction of other ingredients, besides the massive spin- It is well known that a massive spin-2 field in four 2 field and therefore bare little significance to massive dimensions carries five helicity modes corresponding to spin-2 fields in general. We end on more positive notes at two helicity-2 modes, two helicity-1 modes and a the end of Sec. IV by reviewing a potential UV completion helicity-0 mode, while the massless counterpart only of massive gravity and bigravity on AdS and in Sec. V by carries the two helicity-2 modes. If the helicity-1 and discussing powerful constraints that have successfully been 0 modes always entirely decoupled at the linear level from imposed on the cutoff and operators of massive spin-2 the helicity-2 modes and from external sources, one would fields EFTs. Requiring that the S-matrix remains analytic, expect a linear massive spin-2 field theory to smoothly Lorentz invariant, local and unitary at high-energy results continue to a massless spin-2 field theory in the massless in positivity bounds on the scattering amplitudes of the limit m → 0. However this is known not to be the case; considered low-energy EFT. indeed in 1970, van Dam and Veltman [20] and independ- ently Zakharov [21] proved that the small mass limit of a II. WGC AND MASSIVE SPIN-2 FIELDS linear massive spin-2 theory does not smoothly connect A. Helicity-1 mode of a massive spin-2 field with that of a massless spin-2 theory, a discontinuity now well known under the name of vDVZ discontinuity. Only We now review in more depth the precise arguments two years later, Vainshtein understood and proved that for behind the claim of [14] and consider a (not necessarily any massive spin-2 theory, operators that scale as negative gravitational) spin-2 field χμν. The free field action on powers of the mass enter nonlinearly and including those is Minkowski spacetime is the Fierz-Pauli action [26] and it is crucial to the Vainshtein mechanism [22]. The essence of convenient to couple this to a spin-2 source (not necessarily the vDVZ discontinuity, can indeed be phrased as the the stress energy) Tμν at a typical interaction scale MW as a existence of operators that scale like inverse powers of proxy for the interactions of the spin-2 field “ ” mass. This is to be distinguished from the observable 1 1 1 vDVZ discontinuity which would suggest that physical L ¼ χμνEχ − 2ðχ2 − χ2Þþ χ μν ð Þ FP 2 μν 2 m μν μνT ; 2:1 observables in the massless limit of the massive theory MW differ from physical observables in the massless theory. where E is the Lichnerowitz operator, Eχμν ¼ □χμν þ. This observable’ vDVZ discontinuity is an artefact of using In this unitary gauge formulation, the degrees of freedom perturbation theory in a regime where it is no longer valid (d.o.f.) counting is less clear due to the presence of second as first pointed out by Vainshtein [22]. class constraints. A convenient (but by no means required) We expect a typical massive spin-2 field to have a 4 1=5 way to make the d.o.f. more manifest is to introduce cutoff at a scale comparable to Λ5 ¼ðm MWÞ [23], Stückelberg fields Aμ and π via but in principle one can raise that cutoff to the scale 2 1=3 Λ3 ¼ðm MWÞ , which is indeed what is achieved in the 1 1 2 χμν ¼ Hμν þ ∂μAν þ ∂νAμ þ ∂μ∂νπ þ πημν; context of nonlinear massive gravity in [24] or in the m m m2 context of charged spin-2 fields in [25]. The precise way ð2:2Þ the cutoff scales with the spin-2 mass is relevant to understanding precisely how the Vainshtein mechanism for which there is an associated LDiff × Uð1Þ local gets implemented, but in all cases the cutoff scales as a symmetry described by gauge parameters ξμ and χ, positive power of the mass and therefore vanishes in the representing Uð1Þ and LDiff3 respectively, in the form massless limit (unless Vainshtein redressing is taken into account) which is precisely once again the essence of the Hμν → Hμν þ ∂μξν þ ∂νξμ þ mχημν; ð2:3Þ vDVZ discontinuity. It is precisely these interactions that undermine the assumption of a smooth massless limit Aμ → Aμ − mξμ þ ∂μχ; ð2:4Þ which is implicit in the choice of scaling made in [14], as will be discussed in Sec. II. Application of the swamp- π → π − mχ: ð2:5Þ land conjecture to massive spin-2 fields will also be discussed in Sec. III where we show that the conjecture The scales m are introduced so that the kinetic terms for the derived in [14] relies on strong implicit assumptions on different fields are canonical (up to dimensionless factors of how the spin-2 field couple to other sources. order unity). In particular, in the decoupling limit, m → 0, In particular we show how the conjecture gets violated the kinetic term cleanly separates into that for a massless in known examples where diffeomorphism is respected in helicity-2, helicity-1 and helicity-0 mode the coupling of the massive spin-2 with matter sources. In Sec. IV we discuss all the examples that have been 3Linear diffeomorphisms, sometimes known as spin-2 gauge provided as “evidence” for the conjecture and show invariance.

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1 1 1 1 μν 2 2 μν μν m ˜μ limL ¼ H EHμν − Fμν − 3ð∂πÞ þ sources; ð2:6Þ χμνT ¼ HμνT þ AμJ →0 FP m 2 2 MW MW MW 1 1 ˜μ where Fμν ¼ ∂μAν − ∂νAμ is the field strength of the þ πT − π∂μJ : ð2:14Þ M M helicity-1 field. In this decoupling limit, the Uð1Þ gauge W W symmetry disentangles from the LDiff symmetry, and π and ˜μ We further expect a suppression of ∂μJ at least of the form Hμν are both scalars under Uð1Þ ˜μ ˜ ∂μJ ¼ −mK for which in the limit m → 0 (for fixed MW) the source for the helicity-1 mode vanishes Hμν → Hμν þ ∂μξν þ ∂νξμ; ð2:7Þ

1 μν m→0 1 μν 1 → þ ∂ χ ð Þ χμνT ⟶ HμνT þ πT: ð2:15Þ Aμ Aμ μ ; 2:8 MW MW MW

π → π: ð2:9Þ This is the well-known result that the force exchanged between two conserved sources by a massive spin-2 mode The source term on the other hand takes the form after is dominated by the helicity-2 and helicity-0 modes. integration by parts The key argument of [14] is that the smallness of m μ the interaction for the helicity-1 mode AμJ may be MW 2 ∂ μν interpreted as effectively saying that the associated gauge 1 μν 1 μν Aμ νT χμνT ¼ HμνT − coupling is suppressed as gm ∼ m=MW. Assuming the MW MW m μν validity of this, if we now couple this theory to a massless ∂μ∂νT þ πT þ 2π : ð2:10Þ graviton, whose own interaction scale is MPl, then the m2 WGC would seem to claim that the effective cutoff of the theory is at most Since for a massless spin-2 field gauge invariance requires ∂ μν ¼ 0 νT and since the mass term will break this con- Λ ∼ ¼ mMPl ð Þ “ ” ∂ μν m gmMPl : 2:16 servation, then it is natural to assume νT vanishes as MW some positive power of m. This precise behavior will depend on the theory as we elucidate further below. The There are three essential problems with this argument: minimal choice is the one made by Schwinger [27] defined (i) There are no states, neither fundamental nor so that in the limit m → 0 all source interactions are composite particles, charged under this “fake” preserved: Uð1Þ symmetry. Since there is no global limit of the Uð1Þ symmetry, there is no known tension in μν m μ μ coupling this theory to gravity as we shall explain in ∂νT ¼ − J ; ∂μJ ¼ −mðK − TÞðSchwingerÞ 2 details in Secs. II C and II D. μν ð2:11Þ (ii) In general the m scaling of ∂νT is such that interactions blow up as m → 0 for fixed MW, so that the source interactions take the finite form meaning the associated current become large in this limit as we explain in Secs. II E and II F. The ∼ 1 μν μ identification of a gauge coupling gm m=MW is ½HμνT þ AμJ þ πK: ð2:12Þ M largely meaningless. W (iii) The argument itself assumes that the mass scale from In practice as we discuss below, this will require some the interactions is also m as would be the case of the massive spin two state sits in an infinite tower of specific choice of scaling of MW or of the interactions that make up the real sources. For exactly conserved sources, states spaced by the same scale. This assumes in we have K ¼ T and we recover the van Dam-Veltman- advance that the cutoff is of order m due to the Zakharov discontinuity4 [20,21]. existence of these additional states at the scale of By contrast the choice made in [14] is order m.

2 B. Assuming a mass gap m ≪ Λ μν m ˜μ t ∂νT ¼ − J ; ð2:13Þ 2 We will expand on the last point first. In order to consider a low-energy effective theory with at least one massive for which the source term is spin-2 state and a finite number of other states, in a typical UV completion which may contain an infinite number of 4It is clear from Schwinger’s analysis that the essence of the states, it is crucial that the mass of the spin-2 state lies well discontinuity was already understood in [27]. below the lowest scale in the infinite tower that defines the

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UV completion, which we denote as Λt (for “tower”). If a Uð1Þ gauge field. In that decoupling limit we may for this is not the case, then the low-energy effective theory is example introduce some complex spin-0 source Φ charged not under control. Standard Kaluza-Klein theories do not under this gauge field, i.e., described by a Lagrangian provide an example of a UV completion of massive spin-2 2 2 effective theories precisely because there is in general no Lϕ ¼j∂μΦ − iqAμΦj − VðjΦj Þ: ð2:20Þ significant gap, for Kaluza-Klein we quite generically have Λt ∼ m. If we discard WGC considerations and As we move away from the decoupling limit, it is natural to focus entirely on swampland considerations, bounds on ask whether there is a Lagrangian for which Φ would the coupling will come from the infinite tower of states. couple to all of the spin-2 field components Hμν, Aμ and π, It is then more natural to suppose that it is these interactions which would reduce in the decoupling limit to this form. It that will be the ones of concern, i.e., so that it is more is however easy to see that there is no such local natural to identify Lagrangian, precisely because of the modified form of the gauge transformations Aμ → Aμ þ mξμ þ ∂μχ for Λ2 ∂ μν ¼ − t ˜μ ð Þ m ≠ 0. For instance, varying Lϕ we find νT 2 J : 2:17 μ δLϕ ¼ −iqmξ ðΦ ∂μΦ − ∂μΦ ΦÞþ: ð2:21Þ Indeed this is consistent with the argumentation we present in Sec. III. In a standard weakly coupled UV completion, In order to construct a gauge invariant Lagrangian, we it will be this infinite tower whose role is to cure the would need to add at OðmÞ a cubic interaction whose problems of the low-energy effective theory and so we transformation cancels this OðmÞ term. But such an identify Λ ∼ Λ . We then obtain g ∼ Λ3M =M and so m t m t Pl W interaction must necessarily include terms which transform we identify with ξμ without derivatives. There are no such combina- 2 Aμ Λ M tions (other than itself) unless we force ourselves to Λ ¼ Λ ∼ t Pl : ð2:18Þ consider nonlocal ones, which is equivalent to demanding m t mM W the presence of additional fields. Indeed forcing ourselves into nonlocal combinations, the only combination other Hence we infer the cutoff of the low-energy EFT to be μ than Aμ itself that transforms as ξ without derivatives is mM Λ ∼ W : ð2:19Þ 1 1 1 t M ∂ μν − ημν − ∂νπ Pl □ μ H 2 H □ ≪ Λ ≫ Since by assumption m t, then we infer MW MPl, 1 1 1 → ∂ μν − ημν − ∂νπ þ ξν ð Þ which is entirely distinct from the previous argument. Both □ μ H 2 H □ 2:22 arguments are essentially too simplistic as we discuss in Sec. III, since they neglect to account for the fact that in an and we would necessarily be forced into a nonlocal explicit UV completion of a theory with a light spin-2 state, interaction. The appearance of nonlocality should be interactions of the spin-2 with states in the tower will interpreted as arising from having integrated out some necessarily be suppressed by m due to the diffeomorphism/ massless d.o.f., but if such a d.o.f. were introduced, it spin-2 gauge invariance that arises in the limit m → 0. would necessarily give rise to a light ghost. Specifically, we could imagine introducing a vector field Bμ which C. No global Uð1Þ charges transforms as Bμ → Bμ − ξμ. We can construct a local Lagrangian by means of a Lagrange multiplier λμ by adding “ ” ð Þ 1. The case of a fake U 1 A a term With regard to WGC considerations, the first point is that Aμ is not a true Uð1Þ gauge field. If it were, it would be μ μν 1 μν μ λ □Bμ − ∂μ H − η H þ ∂ π ; ð2:23Þ possible to consider states (either fundamental states or 2 composite ones) charged under this Uð1Þ symmetry. If we could construct a black hole carrying such a global charge, and then couple the charged scalar to then we would be led to the usual conundrum that leads to 2 2 the WGC [13]. However in the case considered here, the Lϕ ¼j∂μΦ − iqðAμ − mBμÞΦj − VðjΦj Þ: ð2:24Þ field Aμ introduced in (2.2) is not a gauge field that has been introduced to gauge a global symmetry down to a local one. This is gauge invariant under the full LDiff × Uð1Þ To see why there are no states charged under this fake symmetry but the problem is that the Hamiltonian is ð1Þ ð1Þ U [what we shall call the U A], consider first the unbounded from below in its kinetic structure signaling decoupling limit theory m → 0 for which Aμ transforms as a light ghost. We may also choose to add a separate kinetic

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ð1Þ − ð1Þ term for the true U gauge field Aμ mBμ but in doing so the local U B symmetry was spontaneously broken. we are really introducing a new independent spin-1 field The naive cutoff is Λ. Coupling this to gravity would 0 ¼ − ð1Þ Bμ Aμ mBμ whose U B symmetry is independent of be in tension with the WGC, thus, placing a lower limit 5 the original Stückelberg one. In other words, the only way on q, i.e., requiring the true cutoff of the EFT to be Λ ¼ ðΛ Þ to apply the WGC on massive spin-2 fields in the sense of cutoff Min ;qMPl . We emphasize however that this [14] would be to force an extra sector on the massive spin-2 situation is very different from that considered in the case of ð1Þ ð1Þ field so as to oblige it to carry a genuine U B symmetry the fake U A presented in Sec. II A in the case of massive ð1Þ ð1Þ and be able to constrain that U B sector. Stated yet spin-2 fields, where there is no separate genuine U B, differently, the WGC does not actually constrain the hence no notion of global Uð1Þ charges. ð1Þ massive spin-2 EFT but rather an external U B sector one may choose (or not) to couple to the massive spin-2 D. Absence of charges for static sources EFT. Of course out of any massive spin-2 EFT one can ð1Þ The fact that no matter fields transform under this fake always add on an external U B sector (and sometimes ð1Þ U A symmetry has another important consequence: for a this extra sector enters naturally) as is the case in the static source we necessarily have Q ¼ 0 nonlinearly. Let us examples provided in Sec. IV, but those are genuine define a dimensionless charge Q by implementations of the WGC onto the Uð1Þ sector and B Z Z Z have very little to do with the existence of a massive spin-2 1 3 0 1 3 μ0 d 3 00 field and have certainly nothing to do with the implemen- Q ¼ d xJ ¼ − d x∂μT ¼ − d xT : M M dt tation of the WGC on the massive spin-2 EFT. W W This result is entirely clear in unitary gauge where there ð2:27Þ is no Uð1Þ symmetry, and hence no global remnant of this Uð1Þ symmetry. In this sense the current Jμ cannot be Consider a static source, e.g., that of a point particle for meaningfully associated with any charged state which is which the only nonzero component of the stress energy 00 ¼ δ3ð Þ conserved in a global limit, as would be necessary in order is T Ms x , at zeroth order in m. At leading order to run the WGC arguments. This is not to say that Jμ is Q ¼ 0 since the mass is conserved; however, we might zero, but rather that it arises only from interactions that expect this to be nonzero at higher orders in m due to the specifically come from m ≠ 0 and there is no meaningful nonconservation of the stress energy. This violation of conservation law associated with Jμ in any massless limit. conservation is theory specific but will take the schematic This situation is very different to the spin-1 Stückelberg form theories [28]. In this case, there exists a genuine Uð1Þ ∂ μ0 ¼ 0½∂ χ ð Þ gauge field to which charged particles may couple. μT F a; μν;Tαβ : 2:28 Furthermore there exists a global limit of such a theory. From index contractions, there must be an odd number of ð Þ 0’s on the right-hand side. Given any equation for a tensor 2. The case of a genuine U 1 B χ 0i ¼ 0 The arguments presented above are different from what μν alone, with a source for which T , and given the ∂0 ¼ 0 happens in a theory with a genuine gauge symmetry, even if assumed static nature then at any order in pertur- 0i μ0 it is broken by a mass term. For example, give a bations χ ¼ 0, from which we conclude that ∂μT ¼ 0 at μ Stückelberg theory for spin-1 with some interactions all orders. In other words in order to make a current J out of tensors, we need a vector, but the only vector in play is μ 1 2 1 2 2 2 2 ∂ which has zero timelike component for static sources. L ¼ − Fμν − m Bμ þ λðBμÞ : ð2:25Þ B 4 2 In order to construct a static charge, we need a genuine Uð1Þ symmetry with the phase chosen to be dependent ˜ The Stückelberg procedure amounts to defining Bμ ¼ Bμ þ linearly in time. For instance a complex Uð1Þ scalar field 1 ∂ π 1 −iαt m μ where again the =m is introduced to set normali- ΦðxÞ¼e φðx⃗Þ will give a static charge density and zation. We may choose to couple this to a charged scalar in stress energy since the time dependent phase drops out ˜ 2 2 the same form Lϕ ¼j∂μΦ − iqBμΦj − VðjΦj Þ.Wemay by virtue of the genuine Uð1Þ symmetry. In this way we then consider a decoupling limit m → 0, q → 0, keeping have ∂0 ≠ 0 without violating the static assumption. Λ=m4 ¼ 1=Λ4 fixed for which the Lagrangian becomes Elementary particles acquire charges exactly in this manner through the quantization of fields in the form 1 1 1 − ω 2 2 4 2 2 ΦðxÞ¼e i teik:xa þ. L ¼ − Fμν − ð∂πÞ þ ð∂πÞ þj∂μΦj − VðjΦj Þ: k 4 2 Λ4 Hence, in the case of the Stückelberg helicity-1 field with ð1Þ ð2:26Þ the fake U A symmetry, there are no charged states in the

ð1Þ 5 This gives us a theory with a genuine global U B A more refined and restricted set of conjectures are given for symmetry, for which Jμ is conserved, despite the fact that Stückelberg fields in [28].

104033-6 SPIN-2 FIELDS AND THE WEAK GRAVITY CONJECTURE PHYS. REV. D 100, 104033 (2019) presence of a static source, which is at the heart of the WGC gm ∼ m=MW. The structure of the actual interaction is conjecture. entirely different [29].

E. Scaling the interactions F. Gravity as the weakest force Let us now further clarify the meaning of the assumption A more mundane definition of the WGC is that the force (2.13). To do this, it is helpful to work with a simple induced by anything other than the massless graviton is example. Although the spin-2 field does not have to couple stronger than that of the massless graviton, i.e., gravity is to the stress energy, let us imagine that it does so, and the weakest force. Clearly this statement can only apply couples to that of a massless scalar field φ so that the at distances less than the Compton wavelength of the Lagrangian contains massive states, since in the extreme IR the massless χ graviton is always strongest. Let us consider two well 1 2 μν μν L ⊃− ð∂φÞ þ T ð2:29Þ separated nonlinear configurations A and B for χμν sourced 2 00 MW by for example a delta function stress energy T ¼ M δ3ðr − r ÞþM δ3ðr − r Þ. In general associated with where Tμν ¼ ∂μφ∂νφ − 1 ημνð∂φÞ2. A naive dimensional A A B B 2 the self-interactions scale M is a “Vainshtein” radius r of analysis implies that at finite order the cutoff of this EFT is W V the form rA;B ¼ Λ−1ðM =M Þ1=n, where Λn ¼ M mn−1, at most ðm2M Þ1=3 from the interactions of the form V n s W n W W which sets the scale at which the interactions of the helicity- zero mode of the spin-2 state become significant. The ∂μ∂νπ 1 L ⊃ ð∂μφ∂νφ − ημνð∂φÞ2Þ ð Þ precise values of n depend on the concrete interactions; see 2 2 : 2:30 m MW Ref. [30] for a review on the Vainshtein mechanism. Two well-known cases are the Λ5 theories and Λ3 These cubic interactions can of course be removed with a theories (see [31] for a distinction between these different field redefinition but will survive at higher order unless types of theories). A Λ5 theory of spin-2 particle is one for specifically chosen to cancel. Because of the interaction, which the unitary gauge theory includes self-interactions of the stress energy is no longer conserved and we find the form αβ β 2 β μ ν 1 μν ∂αT ¼ ∂ φ□φ ¼ χμν∂ φ ∂ ∂ φ − η □φ 1 1 2 M 2 L ¼ χμνEχ − 2ðχ2 − χ2Þþ m ð ðχμ3Þ W 2 μν 2 m μν a1Tr ν 2 1 MW þ ∂βφ∂μφ ∂νχ − ∂ χ ð Þ μ2 μ μ 3 μν μ : 2:31 þ a2Trðχν ÞTrðχνÞþa3ðTrðχνÞÞ Þþ; ð2:34Þ MW 2

This is analogous to the covariant conservation equation in where ai are order unity dimensionless coefficients. With μν μν μ αν ν μα GR ∇μT → ∂μT ¼ −ΓμαT − ΓμαT . no special choice of values for ai, introducing Stückelberg Focusing on the leading helicity-zero mode contribution fields this interaction leads to irrelevant interactions at the on the right-hand side we have schematically [29] scale Λ5

1 2 2 αβ m μ m 1 ∂αT ∼ ∂∂∂πT þ: ð2:32Þ ððχ Þ3Þ ∼ ð∂∂π 2Þ3 ∼ ð∂∂πÞ3 ð Þ m2M Tr ν =m 5 : 2:35 W MW MW Λ5 It is apparent from this form that for fixed M this becomes W The associated Vainshtein radius associated with a source arbitrarily large as m → 0. This is non other than the usual of mass Ms is determined by asking when the weak field vDVZ discontinuity [20,21]. The sense in which this term π ∼ ð Þ 2 4 1=5 approximation Ms= MWr breaks down. This is when is m suppressed [29] is for fixed Λ5 ¼ðm MWÞ 1 m2 ð∂πÞ2 ∼ ð∂∂πÞ3 ð Þ ∂ αβ ∼ ∂∂∂π þ ð Þ 5 2:36 αT 5 T : 2:33 Λ5 Λ5

2 1=5 for which Since Λ5 ¼ðm MWÞ is the typical cutoff for the inter- actions of a massive spin-2 particle without any particular 1 1=5 structure, then what is true is if we take a decoupling limit ¼ Ms ð Þ αβ rV Λ : 2:37 MW → ∞, m → 0 keeping Λ5 fixed, ∂αT would be 5 MW suppressed by a single power of m2. Nevertheless it remains the case that there is no mean- The Λ3 theory on the other hand is obtained by tuning the ingful sense in which we can identify a gauge coupling as coefficients ai into the structure

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1 1 L ¼ χμνEχ − 2ðχ2 − χ2Þ whereas for the Λ5 theory it is 2 μν 2 m μν m2 M6m 1=5 þ ˜ϵabcdϵABCDχ χ χ η þ ð Þ ∼ ϵ s ð Þ a aA bB cC dD : 2:38 Q 7 : 2:45 MW MW

Again introducing Stückelberg fields, the leading inter- Demanding the WGC hold in the sense that the force actions are exchanged by the helicity-1 mode is larger than the force m2 1 exchanged by a massless graviton requires 2 ˜ ϵabcdϵABCD π π η a 4 HaA bB cC dD MW m ≳ ð Þ 1 Q Ms=MPl: 2:46 ¼ 2a˜ ϵabcdϵABCDH π π η : ð : Þ Λ3 aA bB cC dD 2 39 3 In the case of a Λ3 theory then this indeed gives

In this case the nonlinearities become important when 3 3 3 ϵ ∂∂π ∼ Λ , i.e., at the Vainshtein radius MPl 3 Ms ≲ ; ð2:47Þ mMW 1 1=3 ¼ Ms ð Þ rV : 2:40 Λ Λ3 MW and in the case of a 5 theory If we consider two such configurations which are 7 MW separated by a distance much larger than r then it is Ms ≳ : ð2:48Þ V mϵ5M5 meaningful to talk about the force between them. When Pl they are closer than r the nonlinearities extend over the V Neither of these estimates bears any resemblance to the region of both particles and it is difficult to infer the force conjecture of (2.16) and both are clearly very sensitive to between them without solving a nonlinear problem. the nonlinear nature of the interactions. The WGC applied As we have already discussed, given these two well- to the force exchanged by the helicity-1 states does not separated configurations, we can associate with them some μ appear to give anything definitive, and neither should it be charge Q and Q induced by the integral of J over their A B since there is no physical content in separating out the separate nonlinear profiles via Z contribution from that mode. 1 In this context it may indeed appear that the repulsive ¼ 3 0 ð Þ Q d xJ : 2:41 force from the helicity-1 exchange will generically be MW weaker than that for massless graviton exchange and indeed If the sources are truly static then Q vanishes and there is no vanishes for static sources since there are no static charges. force to talk about. We can however imagine the force However this small repulsive force cannot in real terms be between two objects, e.g., molecules, galaxies, for which isolated from the much larger attractive force mediated 0 the internal components are moving so for which ∂ ∼ ϵ∂i. by the helicity-2 and helicity-0 exchange. All that is Given then that there must be at least one power of ϵ on the physically relevant is the total force induced by the right-hand side, we can perform simple dimensional exchange of the spin-2 field. Note in particular that the analysis to get a feel for the magnitude of the charges. helicity-zero attractive force 1 πT between two Jμ sources MW From our above estimate will automatically dominate over the helicity-1 repulsion. Z 1 1 As long as MPl >MW, which is the typical assumption, ∼ 3 ∂∂∂π ð Þ Q d x 3 2 T: 2:42 gravity (meaning the force exchanged by the massless MW m MW spin-2) will remain the weakest force. Since the interaction may only be treated perturbatively at distances r ≫ rV, a simple estimate of the charge is III. SWAMPLAND CONJECTURES FOR SPIN-2 obtained by assuming the mass of the sources is evenly FROM EMERGENCE distributed over the Vainshtein radius. Thus we have Related but distinct from the WGC are various “swamp- 1 M M M2 land conjectures” which have been applied to, amongst Q ∼ r3 ϵ s s ∼ ϵ s : ð2:43Þ V m3M2 M r4 r3 m3M3 r4 other things, scalar field theories which have no particular W W V V W V gauge symmetry.6 Since as we have discussed tracking the For the case of a Λ3 theory this corresponds to Uð1Þ gauge symmetry for spin-2 fields is not helpful, it is M2 1=3 Q ∼ ϵ s ; ð2:44Þ 6For a recent review of various swampland conjectures mMW see [32].

104033-8 SPIN-2 FIELDS AND THE WEAK GRAVITY CONJECTURE PHYS. REV. D 100, 104033 (2019) reasonable to ask whether these more general swampland Rather than performing the field redefinition, we may criterion can be brought to bear on this discussion. equivalently ask what is the magnitude of the on-shell 3 point function (continued to complex momenta to allow for A. Dimensionless coupling constants energy-momenta conservation), the helicity-1-scalar-scalar vertex. An alternative way to try to identify a charge, which is At this point we may run arguments analogous to the not wedded to a gauge symmetry per se, is to focus on the swampland conjecture from emergence. In order not to dimensionless coupling constants that arise in the inter- predetermine our answer, we assume that the massive spin- action of the massive spin-2 field and other fields. For 2 states is gapped from the states that define the UV instance let us imagine some matter field, which we model φ completion so that the latter may be modeled (assuming a as a scalar I, of mass mI which is coupled to the massive weakly coupled UV completion) by an near infinite tower spin-2 field in manner ¼ Λ of states whose masses are of the form mI qI t where qI “ ” ≥ 1 1 1 1 are dimensionless charges for which qI . The total − ð∂φ Þ2 − 2φ2 þ χ ð∂μφ Þð∂νφ Þ ð Þ 2 I 2 mI I μν I I ; 3:1 number N of such states is assumed to respect thep speciesffiffiffiffi MW Λ ¼ Λ ¼ bound so that the species cutoff s N t MPl= N i.e., N ¼ðM =Λ Þ2=3. The coupling constants then take the where the mass of the other fields mI should not be Pl t Λ2 confused with the spin-2 mass m. This coupling does form g ¼ q2g where g ¼ t . I I MW m not need to be that of a conserved source a priori. Once Following a standard line of reasoning, loop corrections again, utilizing the helicity decomposition and integrating from the heavy states are expected to contribute to the by parts, this interaction will generate a dimension 6 dimensionless coupling constant g from one-loop matter interaction contributions in the amount7 2 − □φ ð∂μφ Þ ð Þ 1 1 X Λ Aμ I I : 3:2 ∼ þ 4 s ð Þ MWm 2 2 qI ln : 3:8 g gUV I mI We can give this interaction the superficial appearance of a coupling to a current by performing a field redefinition of This calculation is reflecting the renormalization of the the matter field kinetic term for Aμ. Assuming that the UV coupling defined Λ 1 ∼ 0 at the UV scale s satisfies =gUV , i.e., is subdominant 2 μ to the contribution generated from loops, and in turn φ → φ − Aμð∂ φ Þþ; ð3:3Þ I I M m I assuming that the dimensionless charges are spaced in W ∼ the manner qI n then we estimate such that after the redefinition we have an operator of the form 1 M2 5=3 ∼ N5 ∼ Pl : ð3:9Þ g2 Λ2 2 t − 2φ ð∂μφ Þ ð Þ AμmI I I : 3:4 MWm Putting this together we find

μ 3 5 This resembles a current interaction gIAμJ , with current Λ ∼ g = M ; ð : Þ μ μ t Pl 3 10 J ¼ φIð∂ φIÞ and with dimensionless coupling constant which is similar to the standard magnetic WGC relation, m2 I where here Λt which is the lowest mass in the tower of gI ∼ : ð3:5Þ MWm infinite states is correctly (given our assumptions) identi- fied as the cutoff of the low-energy effective theory. More generally we could have started with mixed inter- Substituting in the scale g we find actions in the form 3 3 1 Λ ∼ MWm ð Þ λ χ ð∂μφ Þð∂νφ Þ ð Þ t 5 ; 3:11 IJ μν I J ; 3:6 MPl MW and a similar reasoning would be given for a current in the 7In the usual form of these arguments it is also assumed that the ν ¼ φ ð∂νφ Þ ∼ 2 form JIJ I J and coupling constants charges are spaced like the mass squared gI qIg, i.e., so that qI enters here. This will not substantially change our conclusions, 2 but here we prefer to follow what actually arises from the full mI gIJ ¼ λIJ : ð3:7Þ spin-2 calculation performed in the next section which fixes the MWm scale based on assumed coupling to UV states.

104033-9 DE RHAM, HEISENBERG, and TOLLEY PHYS. REV. D 100, 104033 (2019) which determines the cutoff of the low-energy effective which gives the same result as the previous calculation and ≫ field theory (without the infinite tower of states). This is an would again force us to require MW MPl. By contrast the entirely different scale than the conjectured one of [14].In loop corrections to the coefficient of the kinetic term for the particular, since we require Λt ≫ m given our assumptions, spin-2 field are estimated to be 3 3 ≫ 5 ≫ we need M m M which in turn requires MW MPl. W Pl 1 X Λ Λ2 X Λ We thus conclude that, if this reasoning is to be taken Δ ∼ 2 s ∼ t 2 s Z 2 mI ln 2 qI ln seriously, the infinite tower of states must couple to the MW I mI MW I mI spin-2 field of mass m more weakly than standard gravi- Λ2 M2 ∼ t 3 ∼ Pl ≪ 1 ð Þ tational interactions in order to have the appropriately 2 N 2 : 3:15 gapped effective field theory which would contradict the MW MW general spirit of the WGC. We will return to this point below in Sec. III C. This implies that the kinetic term for the massive spin-2 field cannot itself be viewed as emergent, assuming the validity of the aforementioned scalings. B. Swampland conjectures in unitary gauge The previous argument made use of the swampland C. M ≫ MPl? conjectures from the idea of emergence, entirely at the level W ≫ of the helicity-1 mode of the massive spin-2 state. It is At first sight, the conclusion that MW MPl goes helpful to rerun this argument in unitary gauge to give a against the WGC since it implies that any massive clear understanding of its implications. We again begin spin-2 field couples more weakly than the standard (mass- with a coupling of the form less) graviton. In practice however, it just points to the oversimplification of these arguments. To see why, let us 1 1 1 consider a theory in which the normally massless graviton − ð∂φ Þ2 − 2φ2 þ χ ð∂μφ Þð∂νφ Þ ð Þ 2 I 2 mI I μν I I : 3:12 acquires a mass, but most species of matter are coupled MW ¼ covariantly to the metric so that MW MPl. In other words the action for our matter fields is Integrating out the scalar fields at one-loop and focusing only on the logarithmic divergences (as indeed the above qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 1 −1 −1 μν ημν þ M χμν − ððη þ M χÞ Þ ∂μφ ∂νφ calculation tracks), then we expect corrections to the EFTof Pl 2 Pl I I the schematic form 1 − m2φ2 : ð3:16Þ X Λ X Λ 2 I I ΔL ∼ a1 χ2 4 s þ a2 χ2 4 s 2 μν mI ln 2 mI ln M mI M mI W I W I Expanding this to first order in M−1 gives exactly the type X Λ Pl þ b1 ð∂ χ Þ2 2 s of interaction considered previously. Since this coupling is 2 α μν mI ln M mI covariant, integrating out the matter fields will only W I X Λ generate covariant interactions [33]. The leading nonde- þ b2 ð∂ χÞ2 2 s 2 α mI ln rivative interactions come entirely in the form of a M mI W I cosmological constant, as X Λ b3 μ 2 2 s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð∂ χμνÞ m ln þ ð3:13Þ X Λ M2 I m ΔL ∼− 4 η þ −1χ s ð Þ W I I c mI μν MPl μν ln ; 3:17 I mI where the coefficients ai, bi are of order unity. The ai coefficients encode precisely the running of the “gauge which must be canceled by a counterterm to remove the coupling" g since the kinetic term for the helicity-1 modes tadpole contribution. When this is done the net result is that 2 2 the loop corrections to the mass are zero at one loop comes in unitary gauge from the χμν and χ mass terms With this in mind, the emergence prescription would 2 Δm ∼ 0: ð3:18Þ amount to supposing that the IR mass m is essentially entirely generated by the loop contributions in the form This is of course diffeomorphism invariance at work. Since matter couples covariantly, we cannot generate a mass term 1 X Λ Λ4 X Λ 2 ∼ 4 s ∼ t 4 s which itself breaks diffeomorphisms. What is happening is m 2 mI ln 2 qI ln M mI M mI that in this case the naively order unity coefficients a1 and W I W I X Λ Λ4N5 a2 actually vanish, at least at one-loop order, after the ∼ g2Λ2 q4 ln s ∼ t ; ð3:14Þ tadpole has been canceled. The correct conclusion of the t I m M2 I I W previous estimate should have been that

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ja jΛ4N5 Uð1Þ symmetry which descends from the y diffeomor- m2 ∼ i t ; ð3:19Þ B M2 phism symmetry. Specifically, the form of the metric is W ð1Þ invariant under the U B transformation Λ ≫ which given the requirement t m enforces 1 → − χð Þ ð Þ 2 3 2=3 y y x ; 4:2 j j ≪ m MW ð Þ M4 ai 5 : 3:20 MPl BμðxÞ → BμðxÞþ∂μχðxÞ; ð4:3Þ Thus we are allowed to preserve the naive WGC expect- ≥ χn ð Þ → χn ð Þ 2πinχ=ðLM4Þ ð Þ ation MPl MW provided the breaking of diffeomorphism μν x μν x e ; 4:4 invariance in the coupling of the spin-2 particle to other species is suppressed by powers of MW=MPl, which was under which it is clear that the KK modes transform as shown to be natural [33,34]. Indeed, this is a tuning we charged fields with charges gn ¼ 2πn=ðM4LÞ. The global would expect to be natural in any mechanism whereby the limit of this symmetry χðxÞ¼constant is translation usual massless graviton acquires a mass, such as in the symmetry in the extra dimension. We emphasize that in mechanism proposed in [15–18]. writing (4.2)–(4.4), we have made the diffeomorphism n gauge choice to work in unitary gauge for each χμν, the n IV. EXAMPLE IN INTERACTING reason why the helicity-1 modes Bμ do not appear. But we ð1Þ SPIN-2 THEORIES could have very well reintroduced the LDiff × U A n symmetries by introducing a set of Stückelberg fields Bμ A. Kaluza-Klein theory n n and π for each χμν and we would have then also get In [14] the bound (2.16) is applied to Kaluza-Klein (KK) multiple copies of the symmetry (2.3)–(2.5) that includes theory and it is argued that this gives evidence of its ð1Þ multiple copies of U A. Clearly, as we shall see below validity. In the original KK theory for five dimensions ð1Þ none of those U A play any role for how the WGC compactified on a circle of radius L, there is an infinite applies to Kaluza-Klein theories. tower of massive spin-2 states. However, in addition to Expanding the five-dimensional Einstein-Hilbert action the massive spin-2 field and therefore in addition to the Z Z helicity-1 mode Aμ introduced in (2.2), crucially there are 3 L pffiffiffiffiffiffiffiffi ¼ M5 4 − ½ ð Þ five other zero modes, the massless graviton gμν,a S dy d x g5R g5 ; 4:5 2 0 graviphoton Bμ and a dilaton/radion ϕ. The graviphoton 3 2 Bμ is a genuine massless Uð1Þ gauge field [that we shall with M5 ¼ M4L and integrating over the size L of the extra ð1Þ “ ” designate as U B to distinguish it from the fake dimension, to quadratic order in the KK modes we obtain ð1Þ ð1Þ U A]. The global symmetry for U B is associated with schematically translations in the extra dimension and has nothing to do Z pffiffiffiffiffiffi 2 1 with the helicity-1 modes Aμ of the massive spin-2 field 4 M4 2 S ¼ d x −g R − Fμν Uð1Þ 2 4 tower. The four-dimensional B gauge symmetry is just a remnant of the higher dimensional symmetry. Crucially, X∞ 1 μν n n in this case there does exist a meaningful global limit of this þ χn E ∂μ − i B χμν 2 M4L ð1Þ n¼1 U B gauge field associated with the graviphoton Bμ. X∞ 2 However, this genuine Uð1Þ has nothing to do with the 1 2πn μν B − ðχnχ − jχ j2Þþ ð : Þ fake Uð1Þ that arises in the massive states and therefore 2 μν n n 4 6 A ¼1 L the situation in a Kaluza-Klein theory is dramatically n ð1Þ in different to that of a single massive spin-2. This U B where E½∂μ − B is a Uð1Þ gauge covariant version of M4L B is a completely separate Uð1Þ gauge field under which the the Lichnerowitz operator and Fμν is the strength tensor massive KK modes themselves are charged. Ignoring the graviphoton gauge field Bμ. The gauge coupling is g ¼ dilaton/radion (which is largely irrelevant to the story) we 2π ð Þ¼2π ð Þ3=2 denote the five-dimensional metric as = M4L = M5L and the lightest massive grav- iton state has mass m ¼ 2π=L. In this case the WGC ð Þ 2 2 Bμ x μ applied to the lightest mass state which is charged under the ds ¼ dy þ dx ð1Þ M4 U B gives a cutoff 2 X þ ð Þþ χn ð Þ 2πiny=L μ ν ð Þ 1 1 gμν x μν x e dx dx ; 4:1 Λ ∼ gM4 ∼ M4 ∼ ; ð4:7Þ M4 n≠0 M4L L

n −n where χμν ¼ χμν are the massive KK modes. As is well which is exactly the result we expect since having inte- known, this form of the five-dimensional metric exhibits a grated out the extra dimension, the cutoff of the EFT for the

104033-11 DE RHAM, HEISENBERG, and TOLLEY PHYS. REV. D 100, 104033 (2019) lightest massive spin-2 state will be at the scale of the effective theory for a finite number of massive spin-2 states. next lightest massive spin-2 state. However, this does not To summarize, what this KK setup provides is an explicit provide any evidence for the conjecture of [14] since this example of the standard WGC applied for spin-1 fields. ð1Þ argument only applies because there is a separate U B This is a well-known example and it provides no con- gauge field under which the massive spin-2 states are nections with the actual claims of a WGC for massive charged, whereas the claim of [14] is that this bound should spin-2 fields. apply for uncharged spin-2 particles. Hence, the claimed evidence in [14] is nonexistent. B. Bigravity and massive gravity Five-dimensional diffeomorphisms are realized in four An interesting test case of these theories are the bigravity dimensions as a gauge version of a Kac-Moody-like and massive gravity theories. Since massive gravity arises algebra [35]. At the quadratic level, the Stückelberg fields as a consistent scaling limit of bigravity we may first are made manifest by writing the five-dimensional metric consider the former. Consider a theory with two different perturbation in the form metrics g and f, with associated Planck scales M and M g f X π ðxÞ coupled together by some mass term whose quadratic form ds2 ¼ 1 þ 4 ei2πny=L n dy2 takes the Fierz-Pauli structure. For instance an explicit M4 n nonlinear form known to introduce no new d.o.f. classically X n AμðxÞ is [24] þ 4 ei2πny=L dxμdy  hqffiffiffiffiffiffiffiffiffiffi i hqffiffiffiffiffiffiffiffiffiffi  i n M4 2 −1 −1 −1 2 2 X Uðg fÞ¼4 Tr g f − 1 − Tr g f − 1 : n 2πiny=L μ ν þ ημν þ χμνðxÞe dx dx ; ð4:8Þ ð Þ M4 n≠0 4:12

0 for which Aμ ¼ Bμ. Linear 5D diffeomorphisms are Hence, the bigravity Lagrangian is given by [36] realized as 1 pffiffiffiffiffiffi 1 pffiffiffiffiffiffi L ¼ −gM2 ½ þ −fM2 ½ 2 gR g 2 fR f πn → πn þ mnχn; ð4:9Þ 1 pffiffiffiffiffiffi 2 ˜ 2 −1 n n n − −gm M Uðg fÞþ: ð4:13Þ Aμ → Aμ þ iðmnχμ − ∂μχnÞ; ð4:10Þ 8

n n n n χμν → χμν þ ∂μχν þ ∂νχμ; ð4:11Þ Interactions between the two metrics breaks two copies of diffeomorphisms down to a single copy and the metric with m ¼ 2πn=L the KK masses which for m ≠ 0 eigenstates g and f include an admixture of both the mass n n 9 0 we recognize as the massive spin-2 Stückelberg gauge eigenstates, the massless graviton metric gμν and the n χ symmetry. The generator associated with Aμ is Qn ¼ massive graviton μν. The diagonalization from metric to i2πny=L mass eigenstates is achieved as follows: We define [37] ie ∂y and it is these Qn that in the linear theory are the Uð1Þ ’s associated with the helicity-1 modes of the A 2 ˜ massive KK modes. Nonlinear however these symmetries ¼ 0 þ M χ ð Þ gμν gμν 2 μν; 4:14 are non-Abelian. Individual KK modes are not eigenstates Mg of Qn for n ≠ 0 but they are eigenstates of Q0. The Qn≠0 2 ˜ symmetries are non-Abelian, are nonlinearly realized, and ¼ 0 − M χ ð Þ fμν gμν 2 μν; 4:15 the vacuum is not invariant under them. Mf

1. Modding out the circle where the mass scale M˜ is defined as At this point, one may be tempted to project out the 1 1 1 genuine vector field Bμ by modding out the extra dimen- ¼ þ ð Þ 8 ˜ 2 2 2 : 4:16 sion by a Z2. Such a procedure would indeed remove the M Mg Mf massless boson Bμ, and superficially removes the associ- ated global charge by combining complex n and −n modes From this decomposition it is clear that if Mf ≫ Mp then into real uncharged fields. However the scale of interactions the f metric is “mostly massless” and the g metric is is set by the unprojected theory which itself contains no “mostly massive,” and if Mf ≪ Mp this reverses. mass gap in order to talk about a separate low-energy 9 0 We stress that gμν is not a fixed background metric but 8We would like to thank the authors of [14] for suggesting this the dynamical metric of the massless graviton hμν, i.e., 0 ¼ η þ interesting approach. gμν μν hμν=MPl.

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Then expanding to quadratic order in χ we recover the [39,40] leading to new massive gravity [41]. As noted in action for a massive spin-2 field covariantly coupled to a [39] the new massive gravity action can be obtained from 2 ¼ 2 þ 2 dynamical metric g0 for which MPl Mg Mf, integrating out a second spin-2 field in the form Z pffiffiffiffiffiffiffiffi 1 1 M3 pffiffiffiffiffiffi 1 2 μν ¼ 3 − − − χμν − 2ðχ2 − χ2Þ L ¼ −g0 M R½g0þ χ E½g0χμν S d x g R Gμν m μν ; 2 Pl 2 2 4 1 ð4:20Þ − 2ðχμνχ − χ2Þ þ ð Þ 2 m μν ; 4:17 which exhibits the Fierz-Pauli structure. This in turn can be and the self-interactions of χ are at the scale obtained as a scaling limit of bigravity [40]. In general dimensions, for ghost free bigravity models M2 M2 M M with specific coefficients the leading terms in a derivative ¼ g f ¼ g f ≤ MW Min ˜ ; ˜ Min ; MPl MPl: expansion take the form M M Mf Mg 1 1 ð4:18Þ 2 μνρσ L ¼ M R½gþ WμνρσW þ ð4:21Þ 2 Pl 2 2 gW Crucially, we note that bigravity models always satisfy the ≤ ≈ WGC in the sense that MW MPl. For instance, in the and we are using the fact that in this limit Mg MPl. Taken usual context of massive bigravity models we implicity on its own this truncated Einstein-Weyl-squared action assume a hierarchy of Planck masses, e.g., Mf ≫ Mg.In propagates a massless spin-2 field and a ghostly massive ∼ ˜ ∼ ∼ this hierarchy MPl Mf, M Mg and MW Mg and so spin-2 field [42,43]. The mass of the ghostly spin-2 mode is ≪ m ∼ g M .In[14], the conjecture (2.16) is then we automatically have MW MPl. At the level of the ghost W Pl Lagrangian this arises because every metric contains a part applied to this ghostly massive spin-2 state to give which is the massless graviton, and these parts sum up 2 2 positively, whereas the massive parts add and subtract. mghostMPl gWMPl gWMg Λm ∼ ¼ ¼ ; ð4:22Þ The conjecture of (2.16) would imply an upper bound on MW MW Mf the cutoff of thus defining a WGC for the Einstein-Weyl-squared theory. mMf This identification is simply inconsistent since the massive Λ ≤ : ð4:19Þ Mg spin-2 ghost state that arises in the truncated EFT is unrelated to the physical massive spin-2 state in the UV Given the assumed hierarchy Mf ≫ Mg, even if the theory and in particular there is no reason to identify m with conjecture were true this bound is not necessarily prob- mghost as we show below. lematic since Λ ≫ m and so it still makes sense to talk about the effective theory of the massive spin-2 state 1. Cutoff of truncated expansion vs mass coupled to gravity. Massive gravity is obtained from of integrated mode bigravity in the limit M → ∞ keeping m and M fixed. f g To illustrate why the massive spin-2 ghost in the Thus, in this limit the conjectured upper bound in [14] truncated EFT cannot be directly identified with a physical actually asymptotes to infinity. In other words massive massive spin-2 state in the UV theory that has been gravity automatically would satisfy the WGC in this form. integrated out, let us consider a simple example of a weakly coupled UV completion of a scalar field for which C. Higher derivative gravity the momentum space propagator exhibits a Källen-Lehman ≪ ≈ Let us now consider the limit Mf Mg so that MPl spectral representation with only poles (weakly coupled) Mg and g is the mostly massless metric. At energies well and positive residues below the mass m of the massive spin-2 state we may 1 1 λ2 integrate it out resulting in an covariant theory with an ð Þ¼ þ G k 2 2 2 2 2 infinite number of derivatives whose regime of conver- 1 þ λ m0 þ k M þ k 2 ≪ ∇ ≪ 2 2 2 gence is R m , m. An explicit example of this is M þλ m0 2 2 þ k [38]. Although the theory with an infinite number of higher ¼ 1þλ ð Þ 2 2 2 2 : 4:23 derivatives is formally ghost-free, since the regime of ðm0 þ k ÞðM þ k Þ convergence of the derivative expansion is ∇ ≪ m we must treat it as an EFT with a cutoff Λ ∼ m. Interestingly in The representation is chosen so that λ can take any value three dimensions this procedure may be performed giving without spoiling unitarity and so that Gðk2Þ ∼ 1=k2 at high 2 2 rise to only a finite number of higher derivative terms energies [44]. Assuming M ≫ m0, at energies k ≪ M we

104033-13 DE RHAM, HEISENBERG, and TOLLEY PHYS. REV. D 100, 104033 (2019) can expand the inverse propagator in inverse powers of M2 infinite tower of other states. Reference [19] proposed the to derive the low-energy EFT expansion interesting claim that these stringy massive spin-2 states could be identified with the ghostly massive spin-2 states λ2 that arise in the supergravity version of Einstein-Weyl GðkÞ−1 ¼ð1 þ λ2Þðk2 þ m2Þ 1 − ðk2 þ m2Þ 0 M2 0 squared. However, as we have seen above from the Källen- Lehman spectral representation arguments such an identi- þ Oð1=M4Þ: ð4:24Þ fication cannot be trusted. Both the physical UV spin-2 state and the ghostly one may sit in the same super- Truncating this expansion at this order then we would infer multiplet, but this does not identify them. The Einstein- the existence of the original light pole m0, and a ghostly Weyl-squared supergravity theory should be correctly pole whose mass is m2 ¼ m2 − M2=λ2 ghost 0 interpreted as an EFT for which the Weyl-squared term 1 should only ever be treated perturbatively, i.e., for which GðkÞ ¼ the ghost pole never actually arises. truncated ð1 þ λ2Þð 2 þ 2Þð1 − λ2 ð 2 þ 2ÞÞ k m0 M2 k m0 In order to correctly identify the low-energy EFT 1 1 1 1 associated with the string setup, it is necessary to show ¼ − : 1 þ λ2 m2 þ k2 1 þ λ2 k2 þ m2 that the associated Weyl-squared terms correctly reproduce 0 ghost the leading corrections to the scattering amplitude for the ð4:25Þ massless graviton and the other massless states that propagate within the low-energy EFT. This cannot be We see that the mass of the ghost in the truncated theory achieved by trying to connect unphysical poles with bears no resemblance to an additional massive state in the physical poles in the truncated theory. Indeed we would 2 −∞ UV completion. Indeed mghost is tachyonic and tends to expect all the low excited states to contribute since for as λ → 0 whereas the actual mass M2 remains finite. A example the 2–2 scattering amplitude for massless grav- similar story holds in the spin-2 case, just following from itons can receive exchange contributions for particles of the spin-2 Källen-Lehman spectral representation. Thus, any intermediate spin. we see that even if the spin-2 conjecture of [14] were Once again we can illustrate this point with a simple correct, its application to the ghostly massive spin-2 state example. Suppose we want to compute the 2–2 scattering would be unjustified since the mass of the actual physical amplitude of a massless scalar, e.g., dilaton, in the low- spin-2 state bears no resemblance to that of the spin- energy effective theory. If the UV completion is weakly 2 ghost. coupled, as in the case of string theory with gs ≪ 1, then – ð Þ The actual cutoff of the truncated EFT, or alternatively the UV 2 2 scattering amplitude AUV s; t will take the the regime of convergence for the derivative expansion, is form of a sum of poles whose residues are determined by Λ ¼ ð j jÞ λ 1 Λ ¼ clearly Min M; mghost .If < then M as we the coupling gI;J of the scalar to the excited string states of 10 would expect from having integrated out the massive state. mass mI;J and spin J : If λ > 1 then the cutoff is lower than M since the physical massive spin-2 particle becomes more dominant and its λ2 λ2 A ðs; tÞ¼aðtÞþbðtÞs þ 0 þ 0 negligence at low energies becomes increasingly invalid, UV ð−sÞ ð−uÞ despite its large mass. 2 2t 2 X∞ X∞ g P ð1 þ 2 Þ s I;J J m Indeed more generally, given any truncated higher þ I;J derivative gravity theory, there will be additional ghostly π 4 ð 2 − Þ I¼1 J¼0 mI;J mI;J s spin-2 modes. The ghosts are not a problem since they 2 2t 2 X∞ X∞ g P ð1 þ 2 Þ signify only that the EFT is breaking down (see for example u I;J J m þ I;J : ð : Þ [45]), and so the standard EFT cutoff is at most the mass of π 4 ð 2 − Þ 4 26 ¼1 ¼0 m m u the lightest such ghost but may even be less as in the above I J I;J I;J λ 1 example with > . An intermediate spin-2 particle shows up in this expansion 2 via the Legendre polynomial P2ð1 þ 2t=m Þ. By contrast, D. String theory example I;J the low-energy effective theory will simply give an expan- In string theory, spin-2 states can arise in the infinite sion in powers of s and t tower of excitation of string modes. Recently an interesting string theory UV completion for the supergravity version λ2 λ2 λ2 X 0 0 0 n l A ðs; tÞ¼ þ þ þ c ls t ; ð4:27Þ of the Einstein-Weyl-squared theory was proposed in EFT ð−sÞ ð−uÞ ð−tÞ n; Ref. [19]. The string setup is a 10-dimensional string n;l≥0 theory compactified on R1;3 × T6 with a stack of ND3- branes whose open string sector includes in addition to the 10For a recent application of S-matrix formulas of this type Yang-Mills sector, massive spin-2 excitations as well as an see [46].

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ð Þ ϕa which can be matched to AUV s; t by expanding in inverse limit since is not a vector under the diffeomorphism 2 powers of mI;J. Since every intermediate spin J contributes group but a set of four scalars. Rather they are vectors a term at every power in t up to tJ already from the under an additional global Poincar´e group which acts to ¼ η ∂ ϕa∂ ϕb s-channel, a given coefficient cn;l will receive contributions leave invariant the reference metric fμν ab μ ν . from all spins J ≥ l from the s-channel and all spins from Furthermore π is not a diffeomorphism scalar, and its action the u-channel. A similar argument holds for scattering of is not covariant in the usual sense (e.g., setting Aa to zero external states of arbitrary spin. does not give rise to a covariant action for π). In the present context, this means that in the actual low- There is a profound consequence of this which is that the μ energy EFT related to the string setup, the coefficient of the current J for the vector field receives no direct contribution Weyl-squared term will receive contributions from all excited from matter in the usual theory in which matter couples states of spins J ≥ 2. Thus, purely at the scattering amplitude minimally to the metric g (see [34,45,47] for discussions of level, it is impossible to identify the Weyl-squared term as allowed couplings). In other words, the breaking of diffeo- having come from the massive spin-2 excitation alone. morphisms by the mass term does not in itself lead to a μ Now, let us turn to the application of the conjecture contribution to J directly from matter. Rather it comes (2.16) to the string theory model as proposed in [14]. Since indirectly through the mass term which is itself a nonlinear this model is itself UV complete, we do not expect any function of the Stückelberg fields. problem. However, we can ask what would happen if we In Ref. [14] a stronger spin-2 conjecture is claimed for Λ ∼ integrated all excited string modes other than the massless massive gravity that m m. This is on the grounds that ones and the massive spin-2 supermultiplet. Since the “all the evidence presented holds equally of the lightest pffiffiffi ” spectrum of excited modes takes the form mn ∼ nm spin-2 state is actually not massless. Here the idea is to rerun all the arguments leading to the support of the WGC where m ¼ gsMs, the cutoff of the resulting effective theory will clearly be at most of order m. The conjecture for a massive spin-2 field. However all these arguments fall (2.16) of [14] supposes that the cutoff for the lightest apart in this case. For instance, the nature of black holes in a Λ ∼ theory of pure massive gravity is completely different (for a spin-2 state is mMPl=MW and if we naively identify ¼ Λ ∼ recent discussion see [48,49]). It is already true that the MPl MW then the conjecture is m. In this case the conjecture is in a sense correct because the known cutoff is uniqueness theorems no longer apply, and that there are no at most m; however, its validity here is more of an accident asymptotically flat static solutions. Given this, the usual since it relies on an identification that cannot be trusted. arguments pertaining to black holes carrying global charges cannot be carried through. Furthermore, since the naive Furthermore the identification of MW with MPl cannot be ð 2 Þ1=3 trusted since the open string modes may couple quite unitarity cutoff of these theories m Mg is typically differently than the closed string modes. Crucially though smaller than the inverse Schwarzschild radius, it is not even ð1Þ the fake Stückelberg U A symmetry plays no role here, clear we can talk about black holes in the regime of validity ð1Þ unlike the true U B symmetry in the Kaluza-Klein of this effective theory (to do so implicitly assumes the example. Rather what this string model illustrates is an Vainshtein mechanism redresses the cutoff scale). example of a standard weakly coupled UV completion in Invoking a proper WGC argument on massive spin-2 which the spin-2 state arises in an infinite ungapped tower. fields would require first identifying a global symmetry that is in conflict with coupling to massless gravity. One E. Nonlinear massive gravity possible direction is to work with the global Poincar´e symmetry. We may for instance worry that massive gravity Massive gravity theories are obtained from bigravity by has a conserved global Poincar´e current associated with taking the scaling limit M → ∞. This makes the massless a a a a b f transformations ϕ → ϕ þ c þ Λ bϕ . Coupling this to a graviton infinitely weak, naively consistent with the WGC, massless graviton however immediately turns this global decoupling its fluctuations. The only remaining relevant part symmetry into a local one, i.e., diffeomorphism invariance. of the metric f is to act as a fixed reference metric, which This is very different from an internal gauge symmetry may for example be taken to be Minkowski spacetime to which exists in a global limit coupled to gravity. Indeed preserve a global Poincar´e symmetry. In nonlinear theories another important distinction is that the global Poincar´e of massive gravity, for which the symmetry is spontaneously symmetry is always nonlinearly realized, and there are no broken is the nonlinear Diffeomorphism group, the states in the free theory charged under this symmetry. It Stückelberg fields Aμ and π arise as part of a set of four seems unlikely that an argument along these lines would scalar fields conclude anything other than MW

104033-15 DE RHAM, HEISENBERG, and TOLLEY PHYS. REV. D 100, 104033 (2019) and higher dimensional embeddings have been attempted assume a weakly coupled UV completion, then the cutoff, for more than two decades, either from integrating out i.e., the scale at which new physics comes in, should be conformal fields in AdS with transparent boundary con- comparable or lower than the strong coupling scale Λ Λ ditions [54–58], or by embedding an AdS brane in an AdS cutoff < . For local and Lorentz-invariant massive Karch-Randall extra dimension [59,60], the two pictures spin-2 fields (including but not restricted to massive gravity being holographically dual to each other. and bigravity) this implies that the cutoff should be at Λ ≲ ð 2 Þ1=3 In trying to construct string theory realizations of the most cutoff m MW (unless classical Vainshtein Karch-Randall scenario, in general no clear separation of redressing is invoked). A similar bound also exist for scales between the graviton mass and the AdS curvature charged massive spin-2 fields [25,63]. could be achieved, hence potentially leading to a con- A recently developed powerful constraining tool for struction that would once again carry a cutoff at the scale the low-energy EFTs is the framework of positivity of the graviton mass. However, a very promising new bounds which are independent and complementary to approach has been proposed recently involving highly the Swampland and WGC conjectures. These bounds arise curved Janus throats that connects different AdS regions by exploiting the powerful implications of S-matrix uni- of space [17]. In this UV realization, the graviton mass tarity and analyticity to infer important properties of can be made parametrically smaller than the AdS curvature the low-energy EFT in order for it to have a Lorentz and the cutoff of the theory [18]. An explicit realization invariant local and unitary UV completion. The resulting for bigravity is also discussed in [16]. As is usual, the positivity bounds restrict the signs of the Wilson coef- cutoff of the low-energy effective theory is expected to ficients in the effective Lagrangian and represents crucial go to zero in the limit m → 0. More precisely Ref. [18] theoretical constraints of the parameter space of the 11 conjectures that m ¼ 0 is at an infinite distance in moduli allowed EFT. space, and approaching this point brings down a tower A crucial precondition of this framework is the presence of spin ≥2 excitations with mass spacings vanishing at of a mass gap which can have important consequences Dþ2 2 ðD−2Þ −2 [65]. This theoretical testing ground can also be directly least as Λ where Λ ¼ m M l , where l is Pl AdS AdS applied to massive spin-2 fields [66] and massive gravity the AdS length scale in D dimensions. This is the [31,67–69], placing restrictions on the allowed parameter standard scale expected on anti-de Sitter spacetime space where the theory could admit local, unitary and (see for example [6,52,53]) and the construction of Lorentz invariant UV completion. [16,17] is consistent with this conjecture. Assuming this In its original formulation [10] the positivity bounds construction is indeed completely under control, it successfully constrain the sign or allowed regions for represents an explicit UV completion of massive grav- operators [67] but provide little bounds on the actual ity/bigravity in anti-de Sitter spacetime, and in turn an 12 cutoff. However an extension of these bounds (involving explicit counter example to the arguments of [14]. scattering amplitudes beyond the forward limit) as pre- sented in [70,71] directly involve the cutoff and can be used V. POSITIVITY BOUNDS AND ASYMPTOTIC in certain cases to place an upper bound on the cutoff of the (SUB)LUMINALITY low-energy effective field theory. In their original formulation, the positivity bounds use We have shown that the arguments presented in [14] positivity of the imaginary part of the scattering amplitude do not meaningfully imply any type of weak gravity throughout the physical region (at energies above 2m, conjecture on massive spin-2 fields and cannot be used where m is the mass of the field considered in the low- to impose an upper bound on the cutoff of a massive spin-2 energy EFT). However assuming weak coupling, the EFT. However this is not to say that such bounds do not scattering amplitude can be consistently computed up to exist nor to say that a proper WGC could not in principle the cutoff of the EFT and can be subtracted out from the be adequately derived for massive spin-2 field. First we positivity bounds leading to much more stringent or note that besides WGC considerations, standard EFT weak “improved” positivity bounds [65,69,72] that directly coupling arguments already provide by themselves robust involve an upper bound of the cutoff of the theory. bounds on the cutoff of massive spin-2 fields. Within the context of massive Galileons (which are typical When dealing with EFTs it is standard to associate interactions that arise in self-interacting massive spin-2 the strong coupling scale Λ, i.e., the scale at which Λ fields), the bounds on the cutoff was derived in [69].For perturbative unitarity is broken, with the cutoff cutoff of massive gravity, initial attempts in applying those bounds the theory. This is because in weakly coupled theories new Λ physics should enter at or below the scale to restore 11 unitarity. This strongly relies on the assumption of a For a recent application to EFT corrections to the Standard “ ” Model Lagrangian see [64]. weakly coupled UV completion and in principle unitarity 12Note however that for some EFTs the positivity bounds are could be preserved in a strongly coupled UV completion so powerful at eliminating entire classes of interactions that the without the need for new states [33,61,62]. However if we resulting cutoff can be affected [31].

104033-16 SPIN-2 FIELDS AND THE WEAK GRAVITY CONJECTURE PHYS. REV. D 100, 104033 (2019) were provided13 in [68]. The proper account for the [75–78], which impose constraints on cubic operators of meaning of weak coupling in massive gravity and the massive spin-2 fields. Roughly speaking, these criterion implications of the improved positivity bounds was then demand that the Eisenbud-Wigner scattering time delay given in [69], where it was shown that the improved [79] is positive, encoding the causality of the scattering positivity bounds are satisfied so long as the coupling setup. In [76,77] it is argued that in order to preserve constant g is asymptotic (sub)luminality, the cutoff of the EFT for a spin- 2 field is Λ ∼ m unless the cubic interactions are tuned into m 1=4 a special form that corresponds to a one-parameter family g ≲ ≪ 1; ð5:1Þ MW of the ghost-free massive gravity models [24]. Once this technically natural tuning is made, the cutoff reverts to the where MW is the typical scale of the spin-2 self-interactions standard one as above. and we assume a standard strong coupling at the scale 2 1=3 Λ3 ¼ðm MWÞ (considering interactions for which the VI. DISCUSSION Λ ¼ð 4 Þ1=5 strong coupling scale is 5 m MW would lead to a The existence of a cutoff for EFTs of massive spin-2 much weaker bound on g from the improved positivity particles which acts as a quantum gravity obstruction to ð Þ1=20 bounds alone g < m=MW but such interactions do taking the limit m → 0 is nothing new. In the context of not preserve the positivity bounds beyond the forward ghost-free models of massive gravity, the strong coupling 2 1=3 limit [31]). scale is known to be at most Λ3 ∼ ðm M Þ with possible ¼ Pl When applied to a theory of massive gravity, MW new states arising at scales below this. Since Λ3 → 0 as M and (even though in principle possible) we would → 0 Planck m for fixed MPl we guarantee an obstruction to taking not expect the graviton mass to be parametrically much the massless limit. The crucial question is what is the larger than the Hubble parameter today [73]. Further highest cutoff scale that allows a Lorentz invariant UV demanding for a Vainshtein mechanism to take place while completion, i.e., which does not belong to the swampland. remaining in the weak coupling (5.1) sets a value for the The conjecture of [14] is that this scale is the parametrically cutoff to be [69] Λ ∼ lower scale m mMPl=MW and in the case of a nonlinear theory of massive gravity Λ ∼ m. While these conjectures Λ ≲ 10−5Λ ð Þ m cutoff 3: 5:2 may prove true for other reasons, the arguments and examples given in [14] are themselves spurious and do This is a typical example of how requirements from a not support the conjecture. The application of the WGC to standard UV completion demand new physics to enter at a the helicity-1 mode of the massive spin-2 state does not scale well below the strong coupling scale Λ3 even though stand up to scrutiny since there are no meaningful charged from a pure low-energy approach perturbative unitarity states which source it, and its contribution can never be only starts breaking down at scales close to Λ3 and there isolated from the dominant attractive contributions from the would appear no need for new physics to enter well below helicity-2 and helicity-0 modes. that scale. By contrast, the minimal form of WGC we may actually Within the current state-of-the art the bound (5.2) is the expect to apply for a massive spin-2 particle is simply that most stringent bound one has been able to impose on the the scale MW governing the interactions of the massive cutoff of a massive gravitational theory. Interestingly this spin-2 field be bound relies on the requirement that the UV completion ð Þ remains: MW

104033-17 DE RHAM, HEISENBERG, and TOLLEY PHYS. REV. D 100, 104033 (2019) not yet been exhausted, and a more stringent form of them and interesting discussions. C. d. R. and A. J. T. thank the may further lower the cutoff of these EFTs. A technical Perimeter Institute for Theoretical Physics for its hospitality challenge in considering these bounds for massive states during part of this work and for support from the Simons coupled to gravity is that the massless graviton t-channel Emmy Noether program. The work of C. d. R. and A. J. T. exchange dominates in the vicinity of the forward limit. is supported by STFC Grant No. ST/P000762/1. C. d. R. Related works on asymptotic (sub)luminality explore a thanks the Royal Society for support at ICL through a different regime of the S-matrix [75–78]. A far less well- Wolfson Research Merit Award. C. d. R. is supported by explored possibility is what happens if the UV completion the European Union’s Horizon 2020 Research Council is not weakly coupled. Grant No. 724659 MassiveCosmo ERC-2016-COG and by a Simons Foundation Award No. 555326 under the Simons Foundation’s Origins of the Universe initiative, ACKNOWLEDGMENTS “Cosmology Beyond Einstein’s Theory. A. J. T. thanks the We would like to thank Lasma Alberte, Costas Bachas, Royal Society for support at ICL through a Wolfson Sergio Ferrara, Alex Kehagias, Daniel Klaewer, Dieter Research Merit Award. L. H. is supported by the Swiss Lust, Eran Palti, and Angnis Schmidt-May for very useful National Science Foundation.

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