Constraints on a Gravitational Higgs Mechanism

PHYSICAL REVIEW D 100, 084017 (2019)

Constraints on a gravitational Higgs mechanism

† ‡ James Bonifacio,1,* Kurt Hinterbichler,1, and Rachel A. Rosen2, 1CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA 2Center for Theoretical Physics, Department of Physics, Columbia University, New York, New York 10027, USA

(Received 29 April 2019; published 10 October 2019; corrected 25 November 2019)

We show that it is impossible to improve the high-energy behavior of the tree-level four-point amplitude of a massive spin-2 particle by including the exchange of any number of scalars and vectors in four spacetime dimensions. This constrains possible weakly coupled ultraviolet extensions of massive gravity, ruling out gravitational analogues of the Higgs mechanism based on particles with spins less than two. Any tree-level ultraviolet extension that is Lorentz invariant and unitary must involve additional massive particles with spins greater than or equal to two, as in Kaluza-Klein theories and string theory.

DOI: 10.1103/PhysRevD.100.084017

I. INTRODUCTION leading to an amplitude which does not grow with energy, and thus raising the strong coupling scale all the way to The Higgs mechanism is a central feature of the standard infinity. model, the theory of superconductivity, and countless other A natural question is whether a similar mechanism exists more speculative scenarios. The mechanism is often con- for the gravitational field, i.e., for a spin-2 particle. From ceptualized in terms of spontaneous symmetry breaking: a the symmetry breaking point of view, this would be a gauge symmetry is broken by the vacuum expectation value mechanism in which a lower-spin Higgs field gets a of some scalar Higgs field, and the massless gauge fields vacuum expectation value which breaks the diffeomor- “eat” some components of the Higgs field to become phism symmetry of the massless graviton. The graviton massive, leaving behind physical scalars. Looking only at the S-matrix, we may think about the would then eat some of the Higgs field, becoming a Higgs mechanism differently: it is a method of raising the massive graviton and leaving some other lower-spin physi- ultraviolet (UV) strong coupling scale of an effective theory cal fields left over. Given that the global symmetry which is of self-interacting massive spin-1 particles by adding gauged to diffeomorphism symmetry is Poincaré sym- weakly coupled scalars to the theory. For example, in metry, one might expect this gravitational Higgs mecha- the low-energy effective theory of W and Z0 massive nism to spontaneously break Poincaré symmetry. Indeed, vector bosons, the four-point amplitude of the longitudinal the ghost condensate can be understood along these modes grows at high energies as ∼E2=v2, and violates lines [1]. perturbative unitarity when the center-of-mass energy E However, despite this intuition, we would like to know if becomes of order v ¼ 246 GeV. If this unitarity violation is there is a fully Poincaré-invariant gravitational Higgs to be cured while remaining weakly coupled, then another mechanism. This question is an old one, and there are – particle must enter before a scale of order v and contribute many previous proposals and studies, see e.g., [2 19].In to the tree amplitude in such a way as to cancel the bad terms of the S-matrix, the question is whether there is a high-energy growth. The physical Higgs scalar is the method of raising the UV strong coupling scale of an simplest particle that accomplishes this cancellation, effective field theory of self-interacting massive spin-2 bosons while remaining weakly coupled. In analogy to the spin-1 case, we might expect that this can be done by *[email protected] adding lower-spin massive particles to the theory. † [email protected] The goal of this paper is to determine in complete ‡ [email protected] generality whether it is possible to introduce additional particles with spins less than two into the effective field Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. theory of a single massive spin-2 particle so as to improve Further distribution of this work must maintain attribution to the high-energy behavior of the tree amplitudes and thus the author(s) and the published article’s title, journal citation, raise the strong coupling scale of the low-energy theory. and DOI. Funded by SCOAP3. We know that the four-point tree-level amplitude in any

2470-0010=2019=100(8)=084017(13) 084017-1 Published by the American Physical Society BONIFACIO, HINTERBICHLER, and ROSEN PHYS. REV. D 100, 084017 (2019)

[41,42]. Our conclusions imply that for massive gravity and bigravity, any tree-level UV extension must contain additional massive particles with spins greater than or equal to two. This conclusion is consistent with the arguments concerning full weakly coupled UV completions, but is a stronger statement as it concerns a UV extension and as FIG. 1. Schematic depiction of the four-point tree amplitude for we need no assumptions about the asymptotic behavior of a massive spin-2 particle. The sum is over exchanged states, X, the full S-matrix. Explicit examples of theories with which we allow to include the massive spin-2 particle, a graviton, massive spin-2 particles and improved UV behavior are and any number of scalars and vectors. Kaluza-Klein theories, string theory and large-N QCD. In each of these examples, an infinite tower of massive higher- spin particles (i.e., s ≥ 2) appear in the theory with masses effective theory of a massive spin-2 particle scales at least that are parametrically close to that of the spin-2 particle. ∼ 6 as badly as E at high energies [20], corresponding to a Conventions: We work in four spacetime dimensions Λ 2 1=3 strong coupling scale of 3 ¼ðm MpÞ . We will thus be and use the mostly plus metric signature. The four- asking whether this high-energy behavior can be softened dimensional epsilon symbol is defined with ε0123 ¼ 1. at all without sacrificing Lorentz invariance or unitarity. Conventions on kinematics and polarizations are detailed This is a weaker requirement than asking for a full UV in the Appendix A. completion, for which the amplitude would be bounded at high energies, so we will say that we are looking for a II. MASSIVE GRAVITY COUPLED TO extension weakly coupled UV . SCALARS AND VECTORS Our approach is to study in a model-independent way the high-energy behavior of the tree-level four-point amplitude We begin in this section by showing that the best high- energy behavior of the four-point amplitude in a theory of of a massive spin-2 particle, allowing for the exchange 6 1 ∼ of various other particles, as depicted in Fig. 1. Our massive gravity coupled to scalars and vectors is E .In “ ” conclusion will be that there is no way to improve the using the words massive gravity, we are assuming that the high-energy behavior of the four-point amplitude by massive spin-2 interactions have the form of the Einstein- exchanging any finite number of spin-0 and spin-1 particles Hilbert kinetic term plus a potential. This is not the most in four spacetime dimensions.2 This remains true even if we general case, but it will serve as a good warm-up for the include a massless spin-2 particle in the spectrum. general argument presented in Sec. III since we will be able Our results apply to massive gravity and bigravity to show intermediate steps of the calculation and more theories. Finding UV completions of these theories remains easily visualize what is going on in terms of a Lagrangian. an important open problem in the field. Important clues about possible UV completions of massive gravity or A. Interactions bigravity come from positivity and causality constraints We consider a massive gravity Lagrangian given by, [23–33]. These constraints pertain to full weakly coupled UV completions that, according to conventional wisdom, M2 ffiffiffiffiffiffi 1 L p p− − 2 would require introducing infinitely many new particles mg ¼ 2 g R 4 m Vðg; hÞ ; ð2:1Þ with arbitrarily high spins (see, e.g., Ref. [23] for an explicit argument using the Froissart bound). Ghostfree theories of where the metric is gμν ¼ ημν þ hμν and m is the graviton massive gravity [34–38], bigravity [39], and multigravity mass. The potential Vðg; hÞ can be expanded as [40] have improved high-energy behavior compared to 2 2 3 2 3 generic ghostly theories of massive spin-2 particles, becom- Vðg; hÞ¼hh i − hhi þ c1hh iþc2hh ihhiþc3hhi 2 1=3 ing strongly coupled around the scale Λ3 ¼ðm M Þ p ð2:2Þ

1 4 3 2 2 A similar calculation, but restricting to operators with þ d1hh iþd2hh ihhiþd3hh i dimensions ≤ 4 and bounded amplitudes, was presented in 2 2 4 Ref. [21]. þ d4hh ihhi þ d5hhi þ …; ð2:3Þ 2 An argument against UV extending massive gravity up to Mp with a Higgs mechanism is given in Ref. [22], namely that at high where angled brackets denote traces of matrix products energies the massive graviton’s longitudinal mode does not μν 2 μν λρ couple to its tensor modes with the interactions dictated by with indices raised using g , e.g., hh i¼g g hμλhνρ. The the equivalence principle. However, as pointed out in Ref. [23], quadratic term is fixed to the Fierz-Pauli form. The the equivalence principle constraints do not apply straightfor- canonically normalized massive spin-2 field is given by wardly in the massless limit, since departures from masslessness can be important due to factors of the inverse mass occurring in ˆ 2 interactions. hμν ¼ hμν=Mp: ð2:4Þ

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All the terms up to fourth order in hμν are shown, with graphs that contribute to the tree-level four-point ampli- arbitrary coefficients in front of each. Only the terms tude of massive gravitons are those that exchange one of proportional to c1, c2, d1, and d3 contribute to the four- the new particles. These involve cubic interactions of the ˆ 2 ˆ 2 point scattering amplitude. form h ϕj and h Aj, which can be both parity even and We now add to the massive gravity Lagrangian a parity odd since we do not assume that parity is ϕ collection of scalar fields j and vector fields Aj;μ.We conserved. The most general interactions of this form consider both massive and massless scalars with masses are given by mϕj and massive vectors with masses mAj . The only new

2 X m ˆ ˆ μν −2 ˆ ν ˆ μλ −4 ˆ μ ν ˆ λρ Lˆ ˆ ϕ ¼ ðc1;l;jhμνh þ c2;l;jm ∂λhμν∂ h þ c3;l;jm ∂λ∂ρhμν∂ ∂ h h h j 2 Mp l≥0 −2 μνλρ ˆ ˆ σ −4 μνλρ σ ˆ γ ˆ −2l l þ c˜1;l;jm ε ∂μhλσ∂νhρ þ c˜2;l;jm ε ∂μ∂ hλγ∂ν∂ hρσÞm □ ϕj; ð2:5Þ m m X Aj −1 ˆ μ ˆ νλ −3 ˆ μ ν ˆ ρλ Lˆ ˆ ¼ ðd1 m hμν∂ h þ d2 m ∂ρhμν∂ ∂ h h hAj ;l;j ;l;j Mp l≥0 ˜ −1 μνρλ ˆ ˆ σ ˜ −3 μνρσ ˆ λ ˆ γ −2l l þ d1;l;jm ε ∂μhνσhρ þ d2;l;jm ε ∂μ∂γhρ ∂νhσ Þm □ Aj;λ m X Aj ˆ ˆ μν −2∂ ˆ ∂ν ˆ μλ −4∂ ∂ ˆ ∂μ∂ν ˆ λρ þ 2 ðd3;l;jhμνh þ d4;l;jm λhμν h þ d5;l;jm λ ρhμν h Mp l≥0 ˜ −2 μνλρ ˆ ˆ σ ˜ −4 μνλρ σ ˆ γ ˆ −2l l λ þ d3;l;jm ε ∂μhλσ∂νhρ þ d4;l;jm ε ∂μ∂ hλγ∂ν∂ hρσÞm □ ∂ Aj;λ; ð2:6Þ ˜ ˜ where ci;l;j, ci;l;j, di;l;j, and di;l;j are real dimensionless coupling constants and the factors of mAj , MP extracted out front are to simplify later expressions. To obtain these interactions we modified the procedure for finding all on-shell cubic vertices, ˆ μ described in Appendix A, to allow the particle of lowest spin to be off shell. This amounts to ignoring terms involving hμ μ ˆ and ∂ hμν, and any terms that can be brought to this form by integration by parts, since these do not contribute when the massive spin-2 particle is an external leg. However, since the particles on the internal leg are off shell, we include λ interactions containing ∂ Aj;λ and powers of □ acting on the lower-spin fields, which are equivalent to higher-order contact terms under a field redefinition and may contribute to the massive spin-2 four-point amplitude. The total Lagrangian we consider is thus X 1 1 1 1 L L − ∂ϕ 2 − 2 ϕ2 L − j μν − 2 2 L … ¼ mg þ ð jÞ mϕ þ ˆ ˆ ϕ FμνF m A þ ˆ ˆ þ ; ð2:7Þ 2 2 j j h h j 4 j 2 Aj j h hAj j

j j j where Fμν ≡∂μAν − ∂νAμ and the terms not shown do not have the best possible high-energy behavior for fixed-angle contribute to the four-point amplitude with external mas- scattering. sive spin-2 particles. We assume that the total number of Consider the amplitudes with h1 ¼ h3 and h2 ¼ h4. derivatives in the interactions is bounded above by 2N for In massive gravity these amplitudes grow with energy at some integer N>1, so the index l in Eqs. (2.5) and (2.6) worst like has a finite range. This means that we do not consider the possibility of having infinitely many derivatives that resum A ∼ 10−2ðjh1jþjh2jÞ h1h2h1h2 E : ð2:8Þ into a function with soft high-energy behavior. The scalar interactions with 2n derivatives or vector B. Amplitudes interactions with 2n − 1 derivatives produce exchange We now calculate the four graviton tree amplitude from amplitudes that generically grow with energy like this Lagrangian. For this calculation we use helicity polar- A ∼ 4nþ6−2ðjh1jþjh2jÞ izations and work in the center-of-mass frame (kinematic h1h2h1h2 E : ð2:9Þ details and conventions are reviewed in Appendix A). A ∈ We denote this amplitude by h1h2h3h4, where hj By comparing these, we see that the leading amplitudes f0; 1; 2g denotes the helicity of particle j. Our aim produced by the scalar and vector interactions with more is to fix the coupling constants so that the tree amplitudes than two derivatives must cancel between themselves,

084017-3 BONIFACIO, HINTERBICHLER, and ROSEN PHYS. REV. D 100, 084017 (2019) otherwise the high-energy behavior would be as bad as or With these constraints enforced, the helicity-2 amplitude is worse than in massive gravity. As we will see, this 1 X condition forces these higher-derivative interactions to 2N−1 2 2 A2222 ¼ − s ð4c þ 4d vanish. 4 2 4N−4 1;N;j 3;N−1;j Mpm j By an explicit calculation, we find the scattering ampli- ˜2 2˜ − ˜ 2 tude for helicity-0 massive gravitons to be þ c1;N−1;j þð d1;N−1;j d3;N−2;jÞ Þþ; 1 ð2:18Þ 2Nþ3 2Nþ3 2Nþ3 A0000 ¼ − ðs þ t þ u Þ 576M2 m4Nþ4 X p which is again a sum of squares, giving us the further 2 constraints × ðð4c1;N;j þ 2c2;N−1;j þ c3;N−2;jÞ j ˜ ˜ c1;N;j ¼ d3;N−1;j ¼ c˜1;N−1;j ¼ 0; d3;N−2;j ¼ 2d1;N−1;j: þð4d1;N−1;j þ 2d2;N−2;j − 4d3;N−1;j 2 ð2:19Þ − 2d4;N−2;j − d5;N−3;jÞ Þþ; ð2:10Þ 2 − 1 where here and below we display only the leading term for To constrain the remaining ( N )-derivative interactions, we need to look at amplitudes with more than one high-energy fixed-angle scattering, i.e., terms with the 3 highest combined power of s and t for s, t ≫ 1. helicity type. The amplitude for helicity-1 and helicity-0 Considering the leading terms in the amplitude (2.10), scattering is we see that the couplings combine into a sum of squares 2N−1 2N−1 X with the same sign coefficients. This property follows from A suðs þ u Þ 2 2 2 1010 ¼ 2 4 2 m ðð d1;N−1;j þ d2;N−2;jÞ 192 Nþ Aj the Goldstone equivalence theorem and unitarity of scalar Mpm j amplitudes. Thus each term in the sum must separately ˜ 2 þ 4d1 −1 Þþ: ð2:20Þ cancel to improve the high-energy growth, since unitarity ;N ;j implies that the couplings are all real. For each j we thus get Setting this to zero gives the constraints the constraints ˜ d2;N−2;j ¼ −2d1;N−1;j; d1;N−1;j ¼ 0: ð2:21Þ c3;N−2;j ¼ −4c1;N;j − 2c2;N−1;j; ð2:11Þ Lastly, we look at the amplitude for helicity-2 and helicity-1 d5 −3 ¼ 4d1 −1 þ 2d2 −2 − 4d3 −1 − 2d4 −2 : ;N ;j ;N ;j ;N ;j ;N ;j ;N ;j scattering, ð2:12Þ 1 2N−2 A2121 s u With these constraints imposed, the helicity-1 amplitude ¼ 2 4N−2 32Mpm is now X 2 ˜ 2 2 × m ðd2 −2 þ 4d Þþ; ð2:22Þ Aj ;N ;j 1;N−1;j 1 X j 2Nþ1 2 A1111 ¼ − s ð4ð4c1 þ c2 −1 Þ 256M2 m4N ;N;j ;N ;j p j which gives the constraints 2 þ 4ð2d1;N−1;j − 4d3;N−1;j − d4;N−2;jÞ ˜ 0 ˜ ˜ 2 d2;N−2;j ¼ d1;N−1;j ¼ : ð2:23Þ þð2c1;N−1;j þ c2;N−2;jÞ ˜ ˜ ˜ ˜ 2 þð4d1;N−1;j þ 2d2;N−2;j − 2d3;N−2;j − d4;N−3;jÞ Þ The above argument shows that all of the highest- þ: ð2:13Þ derivative interactions have to vanish, otherwise the high-energy behavior is at least as bad as in massive This is again a sum of squares, so enforcing that this gravity. Note that it was important that the leading parts vanishes gives the additional constraints of the amplitudes we considered were not contaminated by contributions from lower-derivative terms. We can thus repeat this argument for the next highest-derivative inter- c2;N−1;j ¼ −4c1;N;j; ð2:14Þ actions, and so on, until only interactions with two or fewer derivatives remain. These remaining interactions contribute d4;N−2;j ¼ 2d1;N−1;j − 4d3;N−1;j; ð2:15Þ

3 c˜2;N−2;j ¼ −2c˜1;N−1;j; ð2:16Þ The leading terms of the two amplitudes we consider next arise at an order E2 lower than the power-counting estimate (2.9), ˜ 4˜ 2˜ − 2˜ but they are still more divergent than the corresponding massive d4;N−3;j ¼ d1;N−1;j þ d2;N−2;j d3;N−2;j: ð2:17Þ gravity terms so have to independently cancel for N>1.

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X at the same order as the pure massive graviton terms, so we 1 2 A0000 stu 128d1 − 115 − 6 c ; next need to check whether these can cancel against ¼ 2 4 1;0;j þ 72Mpm each other. j The helicity-0 amplitude is now ð2:32Þ 5 2 2 2 A − stuðs þ t þ u Þ 2 6 4 − 1 2 and cancelling this gives 0000 ¼ 2 8 ð c1 þ c2 Þ 864Mpm 1 X X 115 6 2 2 2 d1 ¼ þ c1;0;j : ð2:33Þ þ ð3ð2c1 1 þ c2 0 Þ þ 12ðd1 0 − d3 0 Þ Þ 128 ; ;j ; ;j ; ;j ; ;j j j þ; ð2:24Þ With the conditions determined so far, many but not all of the ∼E6 terms of the amplitudes vanish. One of the 10 which grows like ∼E . We see that the contributions from surviving amplitudes is scalar and vector exchange cannot cancel the pure massive gravity contribution, so setting this to zero gives the X 1 2 A2000 ¼ pffiffiffi stu 1 þ 2 c þ: ð2:34Þ constraints 32 6 2 4 1;0;j Mpm j 1 3 − c2 ¼ 4 2 c1; ð2:25Þ There is no way to set this to zero with real couplings, so we conclude that it is impossible to improve the high-energy

c2;0;j ¼ −2c1;1;j; ð2:26Þ behavior for Lagrangians of the form (2.7). This implies that there is no tree-level UV extension of massive gravity with only spin-0 and spin-1 particles. d3;0;j ¼ d1;0;j: ð2:27Þ

Imposing these constraints, the new leading part of the III. MODEL INDEPENDENT NO-GO RESULT helicity-0 amplitude is In the previous section we assumed a particular form for

1 2 2 2 2 the massive spin-2 part of the Lagrangian, namely that it A0000 ¼ − ðs þ t þ u Þ ð16d1 þ 32d3 − 3Þ 144M2 m6 was the Einstein-Hilbert term plus a general potential. Now p we relax this assumption and prove that there is no way to þ: ð2:28Þ improve the high-energy behavior of the tree-level four- point amplitude for any theory with a massive spin-2 Setting this to zero further constrains the coefficients in the particle coupled to scalars and vectors. In addition, we graviton potential, allow for the presence of a single massless spin-2 particle, which covers the case of bigravity models. 3 d1 d3 ¼ − : ð2:29Þ Our approach here is somewhat different than in the 32 2 previous section. We bypass the Lagrangian and directly write down the most general four-point amplitude with a The constraints we have found on c2 and d3 are the given high-energy behavior that is consistent with Lorentz conditions defining the on-shell de Rham-Gabadadze- invariance, locality, unitarity, crossing symmetry, and a Tolley (dRGT) potential up to this order [34].Nowwe bounded number of derivatives. We follow the procedure of look at the helicity-1 amplitude, whose new leading part is Refs. [20,43], which we review in Appendix A.In particular, we construct the amplitudes using general on- 1 3 2 A1111 ¼ − s 24ðc1 − 1Þ shell cubic and quartic vertices. This encompasses the 64M2 m4 p Lagrangian approach of the previous section as a special X case, since any cubic interactions in the Lagrangian that 4 2 4 2 ˜2 4˜ 2 þ ð c1;1;j þ d1;0;j þ c1;0;j þ d1;0;jÞ þ: vanish on-shell are equivalent to higher-point interactions j under a field redefinition. ð2:30Þ The result we derive is stronger than the one from the previous section, but it is also less transparent since we Requiring that this vanishes, we get the further constraints cannot include the lengthy output from the intermediate steps. In Appendix B we consider the simpler example of a ˜ c1;1;j ¼ d1;0;j ¼ c˜1;0;j ¼ d1;0;j ¼ 0;c1 ¼ 1: ð2:31Þ single spin-1 particle coupled to scalars and see that improving the high-energy behavior leads to the Abelian The remaining helicity-0 amplitude is then Higgs model, as expected.

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A. On-shell vertices than two that can contribute to the four-point amplitude with We again consider the coupling of a massive spin-2 external massive spin-2 particles. In addition, we now also include couplings to a single massless spin-2 particle, γμν. particle hμν to arbitrary numbers of spin-0 particles ϕj with ≥ 0 We now list all the relevant on-shell cubic and quartic masses mϕj and arbitrary numbers of massive spin-1 μ 2 vertices with these d.o.f. Details of how to classify these particles A with masses m > 0. There are no on-shell h A j Aj vertices are given in Appendix A. cubic interactions between a massless spin-1 particle and a single real massive spin-2 particle, thus massless spin-1 1. Cubic vertices particles cannot contribute. In addition, fermions of any spin cannot be exchanged by external bosons due to angular Let us start with the cubic vertices. The most general momentum conservation. Thus the particles we consider are cubic self-interactions of a massive spin-2 particle are all of the possible degrees of freedom (d.o.f.) with spins less described by the following vertex:

2 2 2 2 2 2 V 3 ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ h ¼ ia1ð 1 · 2Þð 1 · 3Þð 2 · 3Þþia2ðð 2 · 3Þ ð 1 · p2Þ þð 1 · 3Þ ð 2 · p3Þ þð 1 · 2Þ ð 3 · p1Þ Þ

þ ia3ððϵ1 · ϵ3Þðϵ2 · ϵ3Þðϵ1 · p2Þðϵ2 · p3Þþðϵ1 · ϵ2Þðϵ2 · ϵ3Þðϵ1 · p2Þðϵ3 · p1Þþðϵ1 · ϵ2Þðϵ1 · ϵ3Þðϵ2 · p3Þðϵ3 · p1ÞÞ

þ ia4ðϵ1 · p2Þðϵ2 · p3Þðϵ3 · p1Þððϵ1 · ϵ2Þðϵ3 · p1Þþðϵ2 · ϵ3Þðϵ1 · p2Þþðϵ1 · ϵ3Þðϵ2 · p3ÞÞ 2 2 2 þ ia5ðϵ1 · p2Þ ðϵ2 · p3Þ ðϵ3 · p1Þ þ ia˜ 1ððϵ1 · ϵ3Þðϵ2 · ϵ3Þεðp1p2ϵ1ϵ2Þ − ðϵ1 · ϵ2Þðϵ2 · ϵ3Þεðp1p2ϵ1ϵ3Þ

þðϵ1 · ϵ2Þðϵ1 · ϵ3Þεðp1p2ϵ2ϵ3ÞÞ þ ia˜ 2ðϵ1 · p2Þðϵ2 · p3Þðϵ3 · p1Þððϵ3 · p1Þεðp1p2ϵ1ϵ2Þ − ðϵ2 · p3Þεðp1p2ϵ1ϵ3Þ

þðϵ1 · p2Þεðp1p2ϵ2ϵ3ÞÞ; ð3:1Þ

where ai and a˜ i are (in general dimensionful) coupling without loss of generality, which from the Lagrangian point constants and εð·Þ denotes the contraction of the antisym- of view corresponds to the vanishing of the Gauss-Bonnet metric tensor with the enclosed vectors. Due to dimen- term in four dimensions. sionally dependent identities, we can set The general h2γ interactions between a massive spin-2 particle and a massless spin-2 particle are described by the 0 a4 ¼ ð3:2Þ following vertex:

2 2 V 2 ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ h γ ¼ ib1ð 1 · 2Þ ð 3 · p1Þ þ ib2ð 1 · 2Þð 3 · p1Þðð 2 · 3Þð 1 · p2Þþð 1 · 3Þð 2 · p3ÞÞ þ ib3ðð 2 · 3Þð 1 · p2Þ 2 þðϵ1 · ϵ3Þðϵ2 · p3ÞÞ þ ib4ðϵ1 · p2Þðϵ2 · p3Þðϵ3 · p1Þððϵ2 · ϵ3Þðϵ1 · p2Þþðϵ1 · ϵ3Þðϵ2 · p3ÞÞ 2 2 2 2 þ ib5ðϵ1 · ϵ2Þðϵ1 · p2Þðϵ2 · p3Þðϵ3 · p1Þ þ ib6ðϵ1 · p2Þ ðϵ2 · p3Þ ðϵ3 · p1Þ ˜ ˜ þ ib1ððϵ2 · ϵ3Þðϵ1 · p2Þþðϵ1 · ϵ3Þðϵ2 · p3ÞÞεðp3ϵ1ϵ2ϵ3Þþib2ðϵ1 · ϵ2Þðϵ3 · p1Þεðp3ϵ1ϵ2ϵ3Þ ˜ ˜ 2 þ ib3ðϵ1 · p2Þðϵ2 · p3Þðϵ3 · p1Þεðp3ϵ1ϵ2ϵ3Þþib4ðϵ1 · p2Þðϵ2 · p3Þðϵ3 · p1Þ εðp1p2ϵ1ϵ2Þ; ð3:3Þ

˜ 4 where the bi, bi are coupling constants and particle 3 is 2b1 ¼ b2 ¼ : ð3:5Þ massless. Using dimensionally dependent identities, we Mp can set 0 2 2 b4 ¼ ð3:4Þ The general h ϕj and h Aj cubic interactions between the massive spin-2 particle and the particles with spin 0 and without loss of generality. If the massless spin-2 particle spin 1 are described by the following vertices: self interacts via the cubic Einstein-Hilbert interaction, then gauge invariance implies that4 2 V 2 ic1 ϵ1 · ϵ2 h ϕj ¼ ;jð Þ 4 This follows from gauge invariance of the amplitude for þ ic2 ðϵ1 · ϵ2Þðϵ1 · p2Þðϵ2 · p3Þ graviton Compton scattering off a massive spin-2 particle, as ;j 2 2 reviewed in the Appendix of Ref. [43], or from consistent þ ic3;jðϵ1 · p2Þ ðϵ2 · p3Þ factorization of the amplitude in massive spinor-helicity variables [44]. The constraint (3.5) can be violated in theories with linear − ic˜1;jðϵ1 · ϵ2Þεðp1p2ϵ1ϵ2Þ gauge symmetry [45], but here we assume that the spin-2 gauge − ˜ ϵ ϵ ε ϵ ϵ symmetry is nonlinear. ic2;jð 1 · p2Þð 2 · p3Þ ðp1p2 1 2Þ; ð3:6Þ

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B. Results V 2 m d1 ϵ1 · ϵ2 ϵ2 · ϵ3 ϵ1 · p2 h Aj ¼ Aj ½ ;jð Þðð Þð Þ With the complete list of on-shell vertices in hand, we can − ϵ ϵ ϵ ð 1 · 3Þð 2 · p3ÞÞ follow the procedure outlined in Appendix A to determine if

þ d2;jðϵ1 · p2Þðϵ2 · p3Þððϵ2 · ϵ3Þðϵ1 · p2Þ there exists a four-point amplitude with improved high- energy behavior compared to massive gravity.5 The output of − ϵ ϵ ϵ ˜ ϵ ϵ ε ϵ ϵ ϵ ð 1 · 3Þð 2 · p3ÞÞ þ d1;jð 1 · 2Þð ðp1 1 2 3Þ this procedure is a system of polynomial equations in the ˜ − εðp2ϵ1ϵ2ϵ3ÞÞ − d2;jεðp1p2ϵ1ϵ2Þððϵ2 · ϵ3Þðϵ1 · p2Þ cubic coupling constants and mass ratios. These equations − ϵ ϵ ϵ are sum rules that must be satisfied for the high-energy ð 1 · 3Þð 2 · p3ÞÞ; ð3:7Þ amplitudes to grow more slowly than ∼E6, similar to what we found in the previous section. ˜ ˜ where ci;j, ci;j, di;j, and di;j are coupling constants that are We now show that these equations have no real solutions. real in a unitary theory and particle 3 is the low-spin First we look at the constraints that depend only on the particle. These correspond to the on-shell vertices produced coupling constants a , a˜ , b , and b˜ , which define the self- 0 i i i i by the interactions with l ¼ in Eq. (2.5) and the first two interactions and gravitational interactions of the massive lines of Eq. (2.6) (up to factors of m and Mp). The fields spin-2 particle. Two of these constraints are given by can be assigned definite parity if some of the couplings ˜ 2 − 2 0 ˜ 2 2 0 vanish, e.g., a pseudoscalar would have zero ci;j and b3 þðb5 b6Þ ¼ ; b4 þ b6 ¼ ; ð3:10Þ nonzero c˜ . However, we consider the general case where i;j ˜ ˜ 0 parity is not necessarily conserved. so we conclude that b3 ¼ b4 ¼ b5 ¼ b6 ¼ . The remaining constraints of this type are

2. Quartic vertices ða1 þ 2a2Þa5 ¼ð2a1 − a5Þa5 ¼ 6a˜ 1a˜ 2 þ a1a5 2 We also need the quartic contact term for identical spin-2 ¼ 9a˜ 2 þ 2a1a5 ¼ 0; ð3:11Þ particles. This can be written as 3 4 a1a5 − ˜ 2 − 4 2 − 2 0 X201 a2a3 þ 8 a1 a2 a3 ¼ a3a5 ¼ : ð3:12Þ V 4 T ϵ h ¼ i fIðs; tÞ Ið ;pÞ; ð3:8Þ I¼1 The only real solution to these equations is where fIðs; tÞ are polynomials in the Mandelstam invar- a3 ¼ 2a2;a5 ¼ a˜ 1 ¼ a˜ 2 ¼ 0; ð3:13Þ iants that we assume have bounded degree. The tensor structures T Iðϵ;pÞ encode the different ways of contracting which correspond to the cubic couplings in dRGT massive polarization tensors and are invariant under the group of gravity. Substituting this solution into the remaining permutations that preserve the Mandelstam invariants, Πkin, equations, four of them reduce to X X X X which is defined in Eq. (A19). For example, the six zero- 2 2 2 2 2 ˜ 2 c3 ¼ c˜2 ¼ m d2 ¼ m d2 ¼ 0; derivative tensor structures are ;j ;j Aj ;j Aj ;j j j j j 2 2 T 1ðϵ;pÞ¼ðϵ1 · ϵ2Þ ðϵ3 · ϵ4Þ ; ð3:9aÞ ð3:14Þ

2 2 so we conclude that T 2ðϵ;pÞ¼ðϵ1 · ϵ3Þ ðϵ2 · ϵ4Þ ; ð3:9bÞ ˜ ˜ 0 2 2 c3;j ¼ c2;j ¼ d2;j ¼ d2;j ¼ : ð3:15Þ T 3ðϵ;pÞ¼ðϵ1 · ϵ4Þ ðϵ2 · ϵ3Þ ; ð3:9cÞ Next we substitute these solutions into the remaining T 4ðϵ;pÞ¼ðϵ1 · ϵ2Þðϵ1 · ϵ3Þðϵ2 · ϵ4Þðϵ3 · ϵ4Þ; ð3:9dÞ equations and look for linear combinations that depend only on the couplings b1, b2 and b3. This gives the T 5ðϵ;pÞ¼ðϵ1 · ϵ2Þðϵ1 · ϵ4Þðϵ2 · ϵ3Þðϵ3 · ϵ4Þ; ð3:9eÞ constraints ˜ ˜ T 6ðϵ;pÞ¼ðϵ1 · ϵ3Þðϵ1 · ϵ4Þðϵ2 · ϵ3Þðϵ2 · ϵ4Þ: ð3:9fÞ ð2b1 − b2Þb3 − b1b2 ¼ðb1 − b2 þ b3Þb1 ˜ 2 ¼ðb1 − b3Þb3 − b1 Only 97 of the 201 tensor structures we use are independent 2 2 ˜ 2 in four dimensions because of dimensionally dependent ¼ b1 þ 3b1b3 − b2 − b2 ¼ 0: ð3:16Þ identities, but finding a basis is difficult and not needed for our calculation. We also consider only the parity-even 5An alternative procedure is to construct the fully symmetric quartic vertex, corresponding to four-point amplitudes contact terms up to some given number of derivatives. We have where the sum of transversities of the external particles checked that this gives identical results when including terms is even, since this is sufficient to prove our result. with up to 14 derivatives.

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The S-matrix equivalence principle further implies that IV. DISCUSSION b2 ¼ 2b1, as in Eq. (3.5). Enforcing this condition and We have shown that it is impossible to improve the high- finding the real solutions to the constraints (3.16) gives energy behavior of massive spin-2 tree amplitudes by ˜ ˜ coupling to particles with spins less than 2, even in the b2 ¼ 2b1 ¼ 2b3; b1 ¼ b2 ¼ 0: ð3:17Þ presence of ordinary massless gravity. This implies that any tree-level UV extension of massive gravity or bigravity By taking appropriate linear combinations of the remaining must include additional massive particles with spins 2 or equations, we get the following additional constraints: higher. This has consequences for many proposed models, X e.g., the proposed UV completion of massive gravity with 2 2 2 an additional scalar field of Ref. [19]. Indeed, various 2a1 þ 3 ðc2 þ 4d1 Þ ;j ;j Lorentz-invariant extensions of massive gravity that j X 2 2 2 2 2 2 2 ˜ 2 include extra scalars, such as quasidilaton [46,47] and ¼ 4b1 þ a2 − a1 þ ðc˜1 − 2m d1 þð4 þ 6m Þd1 Þ ;j Aj ;j Aj ;j galileon-extended models [48,49], all have high-energy j behavior that is the same as in pure massive gravity. ¼ 0: ð3:18Þ Our result also has consequences for the supersymmetric (SUSY) case. It might be thought that SUSY could help with 6 These imply that the UV behavior of massive gravity. The N ¼ 1 massive spin-2 SUSY multiplet contains a massive spin-2 particle, a ˜ massive vector and two massive spin-3=2 fermions [52–56]. a1 ¼ a2 ¼ b1 ¼ c2 ¼ d1 ¼ c˜1 ¼ d1 ¼ 0: ð3:19Þ ;j ;j ;j ;j Since the fermions can never appear in the internal line of the tree-level graviton four-point amplitude, we can restrict Finally, substituting the solutions obtained so far into the attention to only the additional massive vector, which then remaining equations gives falls under the assumptions of our result. X Examples of theories containing massive spin-2 particles 2 0 c1;j ¼ ; ð3:20Þ that do have improved high-energy behavior come from j Kaluza-Klein dimensional reduction. For example, dimensionally reducing 5D general relativity on a single compact so we must also have c1;j ¼ 0. This shows that all of the extra dimension gives a lower-dimensional theory containing cubic couplings must vanish and the only surviving a tower of complex massive spin-2 particles coupled to amplitude is the trivial one. gravity, a massless spin-1 graviphoton, and a scalar radion. The above argument shows that there are no nontrivial By 5D momentum conservation, the four-point amplitude of unitary amplitudes that grow more slowly than ∼E6 for one of these massive spin-2 particles receives contributions high-energy fixed-angle scattering with the d.o.f. we from the exchange of the massless fields and a massive spin- considered. This rules out the existence of a unitary and 2 particle with twice the mass. With these additional fields, Lorentz-invariant tree-level UV extension of any theory cancellations of the worst high-energy parts of the amplitude Λ with a massive spin-2 particle coupled to gravity and occur, resulting in a raised strong coupling scale of 3=2 ¼ 2 1=3 7 particles with spin less than 2 in four dimensions. Any ðmMpÞ [42]. Another example is string theory, which such UV extension must contain additional massive par- achieves soft high-energy amplitudes with the exchange of ticles with spins 2 or higher. an infinite number of massive higher-spin particles.8 Both of

6Supersymmetry does help in the analogous case of a massive spin-3=2 particle. The highest strong coupling scale for a 1=2 single massive spin-3=2 particle coupled to gravity is Λ2 ¼ðmMpÞ [50]. By including a light scalar and pseudoscalar, this can be raised to Mp, as realized by broken N ¼ 1 supergravity with a chiral supermultiplet [51]. 7 In contrast, dimensionally reducing higher-dimensional massive gravity cannot give a theory with a strong coupling scale above Λ3 [57]. 8For example, in open bosonic string theory without Chan-Paton factors, the massive spin-2 particle on the leading Regge trajectory 2 0 has cubic interactions of the form 2-2-s with all of the even spin-s states on the leading trajectory with masses ms ¼ðs − 1Þ=α [58],

s 2−1 s−4 2 2 V2;2;s ¼ ig2 ðϵ3 · p1Þ ½ðs − 3Þ4ðϵ1 · ϵ3Þ ðϵ2 · ϵ3Þ þ 4ðs − 2Þ3ðϵ1 · ϵ3Þðϵ2 · ϵ3Þððϵ2 · ϵ3Þðϵ1 · p2Þ

þðϵ1 · ϵ3Þðϵ2 · p3ÞÞðϵ3 · p1Þþ4ðs − 1Þsð2ðϵ1 · ϵ2Þðϵ1 · ϵ3Þðϵ2 · ϵ3Þ 2 2 2 2 2 þðϵ2 · ϵ3Þ ðϵ1 · p2Þ þ 4ðϵ1 · ϵ3Þðϵ2 · ϵ3Þðϵ1 · p2Þðϵ2 · p3Þþðϵ1 · ϵ3Þ ðϵ2 · p3Þ Þðϵ3 · p1Þ 3 þ 16sððϵ2 · ϵ3Þðϵ1 · p2Þþðϵ1 · ϵ3Þðϵ2 · p3ÞÞððϵ1 · ϵ2Þþðϵ1 · p2Þðϵ2 · p3ÞÞðϵ3 · p1Þ 2 2 2 4 þ 8ððϵ1 · ϵ2Þ þ 4ðϵ1 · ϵ2Þðϵ1 · p2Þðϵ2 · p3Þþ2ðϵ1 · p2Þ ðϵ2 · p3Þ Þðϵ3 · p1Þ ; ð4:1Þ

2 0 where g is the string coupling constant and we have set m2 ¼ 1=α ¼ 1.

084017-8 CONSTRAINTS ON GRAVITATIONAL HIGGS MECHANISMS PHYS. REV. D 100, 084017 (2019) these examples contain infinite towers of massive particles A massive spin-1 particle has three independent polari- with masses that are not parametrically separated. An zation vectors. The standard helicity polarizations used in obvious further question is whether there can be a weakly Sec. II are defined by coupled UV completion of massive gravity with a para- 1 metrically large gap between the mass of the graviton and the ð1Þ ϵμ ðpjÞ¼pffiffiffi ð0; ∓ cos θ ; −i; sin θ Þ; ðA4Þ scale of new physics, or whether any UV extension exists 2 j j using only a finite number of higher-spin particles. 1 Finally, we assumed Poincaré invariance throughout, but ð0Þ j ϵμ ðp Þ¼ ðp; E sin θj; 0;Ecos θjÞ; ðA5Þ another approach to the low strong coupling scale is to m consider backgrounds that break Poincaré invariance. For where j labels the external particle. These are transverse, example, in ghost-free massive gravity there are Poincaré orthonormal, and complete. They describe states that have violating backgrounds with higher strong coupling scales definite values of spin projected in their direction of motion. [59], and in AdS the strong coupling scale is raised and new – To simplify the implementation of crossing symmetry in Higgs-like mechanisms are possible [60 78]. It would be Sec. III, we instead use a basis of polarizations that semi- interesting to extend our S-matrix based arguments to AdS diagonalize the crossing matrix, the so-called transversity by studying the dual CFT correlators. basis [29,79,80]. For particle j, these are given by

ACKNOWLEDGMENTS ð1Þ j ϵμ ðp Þ We would like to thank Brando Bellazzini and Clifford ffiffiffii θ θ 0 θ ∓ θ Cheung for helpful conversations. K. H. and J. B. would ¼ p ðp; E sin j im cos j; ;Ecos j im sin jÞ; 2m like the thank the University of Amsterdam for hospitality while this work was completed. K. H. and J. B. acknowl- ðA6aÞ edge support from DOE Grant No. DE-SC0019143 and ð0Þ j Simons Foundation Grant No. 658908. R. A. R. is sup- ϵμ ðp Þ¼ð0; 0; 1; 0Þ: ðA6bÞ ported by DOE Grant No. DE-SC0011941 and Simons Foundation Grant No. 555117 and by NASA Grant These are transverse, orthonormal, and complete, and No. NNX16AB27G. describe states with definite spin projection in the direction transverse to the scattering plane. A massive spin-2 particle has five polarization tensors. A APPENDIX A: DETAILS AND CONVENTIONS basis for these can be written in terms of the vector In this Appendix we collect various details and con- polarizations as ventions used in our calculations. ð2Þ ð1Þ ð1Þ ϵμν ¼ ϵμ ϵν ; ðA7aÞ 1. Kinematics ð1Þ 1 ð1Þ ð0Þ ð0Þ ð1Þ ϵμν ¼ pffiffiffi ðϵμ ϵν þ ϵμ ϵν Þ; ðA7bÞ Here we specify the kinematics used to calculate four- 2 point scattering amplitudes. We consider center-of-mass scattering of identical particles of mass m in the xz-plane ð0Þ 1 ð1Þ ð−1Þ ð−1Þ ð1Þ ð0Þ ð0Þ ϵμν ¼ pffiffiffi ðϵμ ϵν þ ϵμ ϵν þ 2ϵμ ϵν Þ: ðA7cÞ with particle 1 incoming along the þzˆ direction and 6 particles 3 and 4 outgoing. The momenta can be written as These are transverse, traceless, orthonormal, and complete. j pμ ¼ðE; p sin θj; 0;pcos θjÞ; ðA1Þ A general polarization can be written as a linear combi- nation of these, where j labels the external particle, E2 ¼ p2 þ m2 and j j ð2Þ j ð1Þ j ð0Þ j ð−1Þ j ð−2Þ θ1 ¼ 0, θ2 ¼ π, θ3 ¼ θ, θ4 ¼ θ − π. The Mandelstam ϵμν ¼ α2ϵμν þ α1ϵμν þ α0ϵμν þ α−1ϵμν þ α−2ϵμν ; variables are defined by ðA8Þ − 2 − − 2 − − 2 s ¼ ðp1 þ p2Þ ;t¼ ðp1 p3Þ ;u¼ ðp1 p4Þ : where ðA2Þ j 2 j 2 j 2 j 2 j 2 jα2j þjα1j þjα0j þjα−1j þjα−2j ¼ 1: ðA9Þ These are related to the center-of-mass energy E and the scattering angle θ by The propagator for a spin-0 particle with mass m is

2t −i s ¼ 4E2; cos θ ¼ 1 − : ðA3Þ : ðA10Þ 4m2 − s p2 þ m2 − iϵ

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Defining the projector for i ¼ 1; …;n.Ifηi ¼ 0 for all i, then we drop the εð·Þ factor, otherwise we also require pμpν Πμν ¼ ημν þ ; ðA11Þ m2 2Xn−1 Xn ηi ¼ d; ηi > 0: ðA18Þ the propagator for a spin-1 particle with mass m>0 is i¼1 i¼1

−iΠμν To get a general vertex, each tensor structure is multiplied : ðA12Þ p2 þ m2 − iϵ by a function of the independent contractions of momenta. For a tree-level contact vertex this function is a polynomial. The propagator for a spin-2 particle with mass m>0 is When d ≤ 2n − 2 there can be nonlinear Gram identities, which reduce the number of independent tensor structures.9 2 The number of independent tensor structures can be i Πμ ν Πμ ν þ Πμ ν Πμ ν − 3 Πμ μ Πν ν − 1 1 2 2 1 2 2 1 1 2 1 2 : ðA13Þ obtained using the representation theory of stabilizer 2 p2 þ m2 − iϵ groups and is equal to the number of independent helicity The massless spin-2 propagator (in de Donder gauge) is amplitudes [82,83]. When n ≤ 4 there can also be fewer independent tensor i ημ ν ημ ν þ ημ ν ημ ν − ημ μ ην ν structures due to permutation symmetries that interchange − 1 1 2 2 1 2 2 1 1 2 1 2 2 : ðA14Þ identical particles without changing the Mandelstam invar- 2 p − iϵ iants, which are called kinematic permutations [82].For n ¼ 3, the symmetry group consists of all permutations of the identical particles, which is the symmetric group Sk if 2. Classifying vertices k ≤ 3 particles are identical. For n ¼ 4, if all external Here we review the classification of on-shell vertices. particles are identical then the kinematic permutations are 2 Consider an n-point vertex in d dimensions where particle i given by a Z2 subgroup of S4 [82], has integer spin si and mass mi. We write the symmetric μ1…μ kin ϵ si Π ¼fI; ð12Þð34Þ; ð13Þð24Þ; ð14Þð23Þg; ðA19Þ polarization tensor i formally as a product of vectors μ μ ϵ 1 ϵ si . The vertex can then be written as a polynomial i i where I is the identity element. If there are two pairs of in the Lorentz-invariant contractions ϵi · ϵj, ϵi · pj, and identical particles then the symmetry group is Z2.We pi · pj, possibly also multiplied by a contraction of the ε ϵ’ ’ ≤ 2 − 1 always work with tensor structures that are invariant under antisymmetric tensor ð·Þ with s and p sifd n . the kinematic permutations. These contractions are not independent due to the on-shell Amplitudes with massless external particles must also be conditions gauge invariant. If particle j is massless then cubic vertices Xn should be invariant under ϵ · p ¼ 0; ϵ · ϵ ¼ 0;p· p ¼ −m2; p ¼ 0: i i i i i i i i ϵ → ϵ ξ i¼1 j j þ pj; ðA20Þ ðA15Þ to first order in ξ. For n>3, the total amplitude must be gauge invariant. Moreover, the amplitude must be linear in each polarization tensor. The tensor structures encoding the possible contractions of polarizations are thus built from the following 3. Four-point amplitudes building blocks [81]: Here we briefly review our procedure for obtaining the general four-point amplitude with a given high-energy Yn Yn η η η 1 η scaling, following Refs. [20,43]. ε ϵ 1 …ϵ n nþ … 2n−1 ϵ ϵ nij ϵ mij ð 1 n p1 pn−1 Þ ð i · jÞ ð i · pjÞ ; Denote the four-point tree amplitude for identical exter- i;j¼1 i;j¼1 ≠ 1 A τ i

084017-10 CONSTRAINTS ON GRAVITATIONAL HIGGS MECHANISMS PHYS. REV. D 100, 084017 (2019) where the ambiguity of such a split is unimportant for us. APPENDIX B: MASSIVE SPIN-1 EXAMPLE To calculate the most general Aτ τ τ τ with a given high- 1 2 3 4 In this Appendix we apply the procedure of Sec. III to the energy scaling ∼En, we go through the following steps: exchange simple example of massive spin-1 scattering with scalar iAτ τ τ τ (1) Calculate 1 2 3 4 using the general cubic vertex exchange, verifying that we find the expected Abelian for each exchanged particle. Higgs model. We compute the four-point amplitude where Acontact (2) Construct an ansatz for that factors out the all external particles have spin 1. For a single spin-1 kinematical singularities [80,84,85], particle, the best nontrivial high-energy behavior of this ffiffiffiffiffiffiffi ∼ 4 contact p contact amplitude is E , so we look for amplitudes that grow contact aτ1τ2τ3τ4 ðs; tÞþi stubτ1τ2τ3τ4 ðs; tÞ Aτ τ τ τ ðs; tÞ¼ P ; more slowly than this. 1 2 3 4 2 j τ j=2 ðs − 4m Þ i i First we need to write down all the relevant vertices. Our d.o.f. are a single massive vector, Aμ, with mass mA and a ðA22Þ ϕ collection of real scalars, j, with masses mϕj . The general contact contact on-shell cubic vertex between Aμ and ϕj that contributes to where aτ1τ2τ3τ4 ðs; tÞ and bτ1τ2τ3τ4 ðs; tÞ are polynomials to be determined. the four-point spin-1 amplitude is (3) Constrain the above polynomials by the requirement 2 V 2ϕ ¼ im g1 ϵ1 · ϵ2 þ ig2 ðϵ1 · p2Þðϵ2 · p3Þ that they cancel the exchange terms when the total A j A ;j ;j amplitude is expanded at high energies, down to þ ig˜1;jεðp1p2ϵ1ϵ2Þ; ðB1Þ whatever assumed high-energy scaling is taken as ˜ input. Replace the products of cubic couplings and where g1;j, g2;j, and g3;j are real coupling constants. There masses with new variables so that the equations are is no on-shell cubic self-interaction for a single spin-1 linear. particle. The general quartic vertex with external spin-1 (4) Impose crossing symmetry on the contact terms particles is [29,79]: X17 P V 4 T ϵ A ¼ i fIðs; tÞ Ið ;pÞ; ðB2Þ iðπ−χ Þ τ Acontact t j j Acontact I¼1 τ1τ2τ3τ4 ðs; tÞ¼e −τ1−τ3−τ2−τ4 ðt; sÞ;

ðA23Þ where fiðs; tÞ are polynomials in the Mandelstam variables T ϵ Z2 P and Ið ;pÞ are 2-invariant tensor structures. A basis for π−χ τ contact ið uÞ j contact these structures can be found in Appendix A of Ref. [20]. Aτ τ τ τ ðs; tÞ¼e j A−τ −τ −τ −τ ðu; tÞ; 1 2 3 4 1 4 3 2 Applying the procedure outlined in A3, we find that it is ðA24Þ possible to reduce the high-energy behavior of the amplitude to ∼E2. However, with the appropriate contact terms where added, this gives no constraints on the cubic couplings. We pffiffiffiffiffiffiffi can further improve the high-energy behavior if the − − 2 − χ st im stu following sum rules are satisfied: e i t ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; − 4 2 − 4 2 X sðs m Þtðt m Þ 2 2 pffiffiffiffiffiffiffi g2;jðg2;jðmϕ − 2m Þþ2g1;jÞ¼0; ðB3Þ − 2 j A − χ su þ im stu j e i u ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðA25Þ − 4 2 − 4 2 X sðs m Þuðu m Þ 2 ˜2 0 ðg2;j þ g1;jÞ¼ : ðB4Þ These can be cast as linear equations in the param- j eters by equatingpffiffiffiffiffiffiffi the coefficients of the monomials The only real solution to these equations is in s, t, and stu on each side.

(5) Impose little group covariance by enforcing that g2;j ¼ g˜1;j ¼ 0: ðB5Þ Acontact i τ1τ2τ3τ4 matches a covariant quartic vertex evalu- ated at four-dimensional kinematics. This corresponds to the Abelian Higgs theory. The remain- (6) Solve the nonlinear equations relating the products ing amplitudes are then bounded at high energies by of cubic couplings and masses to the linear variables constants depending on the cubic couplings g1;j and masses

defined earlier. The parameters from the contact mϕj . Perturbative unitarity implies that these constants terms appear linearly and are easily eliminated, so cannot be too large, so there are further constraints on the result is a system of polynomial equations in the the masses of the spin-0 particles, as in the Lee-Quigg- cubic coupling constants and mass ratios. Thacker bound on the Higgs mass [86].

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