Massive Gravity is not Positive

We derive new positivity bounds at finite momentum transfer, assuming a large separation between the mass $m$ of the lightest particle in the effective theory and the mass gap $M$ to new heavy states. Massive gravity parametrically violates these bounds unless the cutoff is within one order of magnitude of the graviton mass $M\lesssim O(10)m$. Non-gravitational effective theories of massive spin-2 particles are similarly bounded.


I. INTRODUCTION
The principles of causality, unitarity, crossing symmetry, and Lorentz invariance enforce non-trivial constraints on otherwise healthy-looking effective field theories (EFT), which describe the emergent infrared (IR) dynamics relevant to low energy observers.The simplest of these constraints take the form of inequalities among scattering amplitudes or Wilson coefficients, and are therefore known as positivity bounds.These have found interesting applications in particle physics and cosmology, allowing to discern EFTs that can have a consistent ultraviolet (UV) description-the EFT landscape-from theories that do not-the EFT swampland.
Positivity bounds shape the space of EFT amplitudes, by constraining the structure of higher derivative interactions.These have played a particularly important role in our understanding of possible departures from Einstein gravity, see e.g. .
In this article, we turn our attention towards the theory of dRGT massive gravity [26][27][28].This is a (generally covariant) EFT of a single massive spin-2 particle, whose mass m is experimentally constrained to be of the order of the smallest energy scale in our universe, the Hubble parameter: m ∼ H 0 (see e.g.[29] and references therein).Our goal is to understand whether this EFT can be used to describe physics at parametrically larger scales (distances much shorter than the Hubble radius H −1 0 ), as relevant for any practical and cosmological application.
Positivity has already been employed in this context [30][31][32].Bounds restricted to forward-scattering imply that the two free parameters of dRGT mas-sive gravity, c 3 and d 5 , must live in a certain finite compact region [30], and that the ultimate energy-cutoff M of the theory is smaller than M ≪ (m 3 m Pl ) 1/4 [31,33], with m Pl the reduced Planck mass. 1 This scale corresponds to distances M −1 roughly larger than the size of our solar system.
In this work we extend massive gravity positivity bounds to regimes of large momentum transfer, |t| ≫ m2 , exploiting two basic observations.Firstly, inelastic matrix elements are bounded by elastic ones; so finite-t dispersive integrals must be smaller than forward ones.Secondly, for the unknown part of dispersive integrals with c.o.m. energy squared s ≥ M 2 ≫ |t|, crossing symmetry is simple and resembles near-forward or massless scattering.
The positivity bounds emerging from this analysis lead to a much stronger condition on the regime of validity of dRGT.We find that the cutoff of massive gravity is parametrically close to its mass, and tied to it by the linear relation, independently of all the other parameters in the theory.This conclusion can not be avoided by the mechanism of Vainshtein screening [34][35][36], as all Vainshtein radii from compact sources are smaller than the size of the universe.
In section II we lay out our assumptions, and derive a simple version of positivity constraints that is suitable to study massive higher-spin scattering.We apply these bounds to dRGT in section III.In section IV we show that the bounds apply also to general deformations from dRGT with higher-derivative interactions, detuned potentials, and non-gravitational theories.Finally in section V we summarize our results and discuss future directions.

II. POSITIVITY
We study the 2 → 2 scattering of massive gravitons in flat spacetime.In what follows we list our assumptions and show how they can be efficiently employed to obtain parametric bounds of the form of Eq. (1).i) Unitarity of the S-matrix, for physical energies s ≥ 4m 2 .This equation, evaluated on any complete set of states, is ultimately responsible for positivity.In practical applications only truncated sets of states can be considered (e.g.finite number of partial waves, states of definite helicity, etc. . .), and each of these sets accesses different partial information.We work with generic initial |1 λ1 2 λ2 ⟩ and final |3 λ3 4 λ4 ⟩ 2-particle states of arbitrary momentum and helicity λ i .Here This has a simple, but powerful, physical interpretation: inelastic M † M matrix elements must be smaller than elastic ones.By Eq. ( 2) the same statement holds for (M − M † )/i.When reduced to equal helicities (λ 1 , λ 2 ) = (λ 3 , λ 4 ), Eq. ( 3) implies that M † M in non-forward scattering must be smaller than in forward one (generalising [37,38] to all helicities).When limited to the forward limit, instead, it implies that scattering of inelastic helicity must be suppressed w.r.t. the elastic one.Since much of our understanding of dispersion relations relies on elastic scattering, Eq. ( 3) provides an intuitive way of readily extending previous results to inelastic scattering.

ii) Causality/Analyticity
The center of mass scattering matrix elements 4) are analytic functions in the complex (Mandelstam) s plane at fixed t ≤ 0, except for branch cuts and poles located on the real axis (in fact, our results rely only on analyticity for large enough |s| > R(t) with |R(t)| < M 2 at small enough |t| < M 2 , as proven in Ref. [39]).These discontinuities are associated with physical thresholds (intermediate states exchanged in the s or u channels) as well as kinematic singularities.The latter, classified long ago [40], originate either from the fact that helicity states are ill-defined when the momenta vanish, at s = 4m 2 in the c.o.m frame, or from angular-momentum selection rules. 2 For instance, the elastic-helicity scattering amplitude in the theory of a massive spin-2 particle, exhibits only simple dynamical poles at s = m 2 and 3m 2 − t, and a kinematic higher order pole at s = 4m 2 , see Fig. 1.

Re s
Im s frames.In the forward or massless limit, it involves exchanging any two legs of an amplitude.At finite momentum exchange and mass, however, an additional boost must be performed to bring back the amplitude into the c.o.m frame.In a crossing transformation that takes one particle in the in/out state into an anti-particle of the out/in state, the resulting Wigner rotations generically mix all helicities, with S the spin of the particle, and X the crossing matrix-writable in terms of a string of Wigner-d matrices [40][41][42][43], where for later convenience we have defined λi ≡ −λ i .
In this work, we exploit the fact that for large center of mass energy m 2 , −t ≪ s, the structure of X greatly simplifies, providing a |t|m/s suppression for any helicity change from the λ 1 λ4 → λ 1 λ2 (high-energy) configuration.So, for elastic helicity, Eq. ( 6) becomes, where in the first term of the second line we sum over the 8 inelastic amplitudes λ ′ 1 λ ′ 2 λ ′ 3 λ ′ 4 with only one ±1 helicity change w.r.t λ 1 λ2 λ 1 λ2 .Moreover, c's in (8) are all known and bounded, c ≤ √ 6 for spin-2 particles.Notice that some of the inelastic amplitudes on the r.h.s. of Eq. ( 8) are further suppressed by powers of t due to angular momentum conservation close to the forward limit, see footnote 2. iv) Hermitian Analyticity Amplitudes in the upper and lower s-plane are related by complex conjugation, e.g.
for s and t real.

v) Polynomial Boundedness
The amplitude at fixed t ≤ 0 is polynomially bounded in s; in particular the Froissart-Martin bound [44,45] for a gapped theory implies that, A similar bound has been extended recently to massless gravity in various dimensions [46].

vi) EFT separation of scales
We assume we are dealing with a relativistic EFT where m ≪ M , so that one can systematically calculate amplitudes in the low energy window, to any desired accuracy, provided one works at sufficiently high loop order and includes operators of sufficiently large dimension.In the context of massive gravity, at sufficiently small energy, the EFT is well described by a Lagrangian comprised of the Einstein-Hilbert term and the dRGT potential.Furthermore, because the theory is weakly coupled all the way to the cutoff (see footnote 1), we assume that it is possible to neglect the effects of IR loops; these can systematically be taken into account; see Refs.[37,[47][48][49][50][51].Now the goal is to show for what values of the ratio m/M the above assumptions are compatible with each other, in the context of dRGT.Because of the simple analytic structure and the simple behaviour under crossing discussed in ii) and iii) respectively, we focus on elastic-helicity amplitudes.We introduce the integral, along a contour C in s ∈ C running around the origin at 4m 2 ≪ |s| ≪ M 2 , so that it avoids the amplitude poles while remaining within the region of validity of the EFT, as shown in Fig. 1.Then, A λ1λ2 can be calculated explicitly in terms of the free parameters of the EFT: c 3 and d 5 in the case of dRGT. 3ecause of analyticity ii), C can be deformed to run along the branch cuts and a big circle at infinity, which vanishes due to Eq. ( 10) in v).Hermitian analyticity iv) puts A λ1λ2 in the form of a dispersive integral of (M − M † )/i, and by crossing symmetry iii) it can be rewritten as a single integral over the physical values of the Mandelstam variable s.
The EFT scale separation vi), allows us to work at m 2 ≪ |t| ≪ M 2 , so that crossing symmetry within the integral in |s| ≥ M 2 takes the simple approximate form (8). Using unitarity i) we rewrite (M − M † )/i as in Eq. ( 2) to obtain the following UV representation for A λ1λ2 , where ∆ λ1λ2 captures departures from elastic crossing in Eq. ( 8), for s ≥ M 2 , and is bounded by a known linear function of other A λiλj (0), see appendix B.
The positivity bounds follow directly from the UV representation of A λ1λ2 in Eq. ( 13).In the forward with the equal sign obtained only in the free theory.
For t ̸ = 0 instead, we use the fact that the matrix elements of M † M are smaller than those at t = 0, see Eq. ( 3), and obtain, The term O( |t|m/M 2 ) stems from ∆ λ1λ2 in Eq. ( 13) and, as discussed in appendix B, is bounded by the sum of the 8 known IR terms, ))/2A λ1λ2 (0), summed as described below Eq. ( 8), and where we defined cλ1λ2 In this way, Eq. ( 15) can be used to formulate positivity bounds with complete control of terms of order |t|m/M 2 .
We remark that in the general case of scattering identical massless particles of arbitrary spin, we can write the exact inequality, similarly to the massless scalar case of Ref. [37].
The problem of finding all positivity constraints for massive spin-2 particles is quite complex, since crossing symmetry mixes hundreds of different amplitudes with each other, producing a nested network of positivity relations.These can in principle be solved with the methods of e.g.[10,37,47,48,[52][53][54], but the advantage of working at leading order in |t|m/M 2 is captured by the simplicity of Eq. ( 15), which singles out 6 independent inequalities for the elastic helicities (where we denote by 0, +, and = | , the longitudinal, transverse and transverse-transverse helicities, with other elastic configurations related to these ones by accidental parity, time-reversal and crossing in dRGT).The inequalities in Eq. ( 15), via the IR representation Eq. ( 12), will be sufficient to constrain the parameter space of dRGT in the next section.

III. POSITIVITY IN DRGT
Scattering amplitudes in dRGT massive gravity are suppressed by m 2 in the forward limit and, for some helicities, grow rapidly at large |t|.For |t| ≫ m 2 this behaviour is incompatible with Eq. ( 15), for |t|/M 2 small enough.
Of the six elastic-helicity configurations at our disposal, the strongest bounds will come from λ 1 λ 2 = 00, 0+, ++, which give (from the dRGT action reported in Appendix A), while amplitudes involving the transverse polarisations do not grow with |t|.This has to be contrasted with the values in the forward limit, Now, an EFT with a large range of validity can, by definition, be used at energies much larger than the particle mass, m 2 ≪ |t| ≪ M 2 .In this limit, the bounds from applying Eq. ( 15) to (17)(18)(19)(20)(21)(22), would converge to three lines in the (c 3 , d 5 ) plane, corresponding to the vanishing of ( 17), ( 18) and (19).These three lines have no common intersection, as illustrated in the left panel of Fig. 2.
This implies that in dRGT massive gravity, the cutoff of the theory cannot be arbitrarily large compared to the mass.To quantify this, we run a bootstrap algorithm for the ratio m 2 /M 2 , assuming only the existence of a range |t| ≪ M 2 for which dRGT is a valid description of massive spin-2 scattering.For each value of m 2 /M 2 , we determine the set of points (c 3 , d 5 ) that are compatible with the finite-t bound in Eq. (15); if the set is not empty then we lower m 2 /M 2 and repeat; if the set is empty, the value is inconsistent with the assumptions i)-vi) and is discarded.The results of this algorithm are illustrated in the right panel of Fig. 2.
In this way, we find that the cutoff scales linearly with the mass, and is limited to being parametrically close to it, We have presented the bound in this way to highlight the fact that it becomes stronger as the theory is evaluated at larger energies |t|/M 2 , closer and closer to the cutoff, while still being described by dRGT.Since m 2 /M 2 is small, for −t/M 2 ≤ 0.1 it implies m √ −t/M 2 ≤ 0.01 and we have checked that the error described below Eq. ( 15) is indeed negligible.

IV. BEYOND DRGT: HIGHER ORDERS IN THE ENERGY EXPANSION
In the previous section we have assumed that dRGT accurately describes massive spin-2 scattering within the EFT.In general, there might be terms of higher order in the energy expansion, beyond those of dRGT, that also contribute to the scattering amplitudes via terms with more powers of the energy.These enter A λ1λ2 (t) as higher powers in t.As long as these terms are suppressed by powers of M , and are controlled by coefficients ∼ O(1) w.r.t dRGT, our arguments are modified only by higher powers of the small ratio |t|/M 2 .In this section, we relax this assumption, and study the possibility that above some intermediate scale E * , with m ≪ E * < M , dRGT transitions into a different theory, controlled by the most general higher-derivative EFT.Can such a theory exist?
If such a theory is dominated by large coefficients in just a few terms of higher-order in the energy expansion, we could apply the same arguments as in section III to A λ1λ2 (E 2 * ≪ |t| ≪ M 2 )/A λ1λ2 (0) (and to the generalisations of A λ1λ2 (t) to more subtractions) and exclude it, since a fixed order polynomial in t/E 2 * would quickly exceed 1 in ( 15).a. Detuned potential.An example of such a theory is provided by detuning the graviton potential from its dRGT values, i.e. of all the dRGT relations in Appendix A we only keep the Fierz-Pauli mass tuning b 1 = −b 2 .Then the 2-to-2 amplitudes depend on 4 parameters, (c 1 , c 2 , d 1 , d 3 ), which we can constrain as follows.
The amplitude for λ 1 λ 2 = 00 at 0-th order in the mass grows as ∝ (c 2 + 3c 1 /2 − 1/4)s4 t.We can therefore define the analog of Eq. ( 12) with 4 subtractions A (4) λ1λ2 (i.e.exponent 5 at denominator), that also satisfies Eq. ( 15), and implies after imposing the c 2 tuning).These are the dRGT relations spanned by (c 1 , d 1 ) and deliver the Λ 3 -theory, reproducing Ref. [32] and further showing that detuned massive gravity is also incompatible with positivity bounds as soon as m 2 and M 2 are parametrically separated. 4 b.General higher derivative terms.Rather than a normal EFT with a finite number of leading higher-energy terms, we can even address the case of a theory with a tower of infinitely many higher derivative terms with large coefficients, arranged such that their contributions to A λ1λ2 resum to a small function of t/M .To answer this question we provide an alternative derivation of the bound that led to the non-intersecting lines in the left panel of Fig. 2 (for which we used |t| ≫ m 2 in section III).In this derivation we will not assume that A λ1λ2 is at most linear in t, as in dRGT, but allow for arbitrary powers of t with arbitrary coefficients.On the other hand, in this derivation, we will work at zeroth order in m, keeping Λ 3 fixed (this is known as the de-FIG.2. The (c 3 , d 5 ) parameter space of dRGT massive gravity, and a comparison with the forward-only positivity bounds from Ref. [30] which carve the region inside the closed black line.LEFT: In the |t| ≫ m 2 limit, each elastic helicity reduces the parameter space to a line (corresponding to the vanishing of (17) in blue, (18) in green, and (19) in orange).In this limit, the lines do not intersect, and the theory is ruled out.RIGHT: A close-up of the same figure, for finite values of |t|/m 2 (we have used the exact amplitudes, rather than the ones expanded at large |t|/m reported in the main text).Different shadings correspond to the allowed parameter space for different values of M/m, represented at fixed |t|/M 2 = 0.1.As the ratio between the cutoff and the mass increases, the parameter space shrinks and eventually disappears, hence providing Eq. ( 23).
coupling limit, in which the transverse polarizations decouple).
At this order, besides the simplification of crossing symmetry discussed in iii), the EFT amplitudes also simplify because the theory effectively reduces to that of a massless shift-symmetric scalar, a photon, and a graviton.At high energy we are thus able to write all-orders Ansätze, the relevant ones being, where we have factored out little group scalings, and H and G λ1λ2 are functions that contain only dynamical singularities.Moreover, within the decoupling limit and within the EFT range of validity, they are also analytic functions, since none of the 3-pt functions between one neutral Goldstone boson and the gauge boson give rise to on-shell poles.Crossing symmetry implies that G 0+ (s, t) = G 0+ (u, t), G +− (s, t) = G +− (t, s), while H is fully s − t − u crossing symmetric.Therefore, their most general tree-level low-energy expressions are, Refs. [10,47,52,53] have derived bounds for all the coefficients in the most general EFT for scalars.These can be readily applied to the ratios h i /h 0 , constraining them from above and below in appropriate units of M , independently of the value of all the other coefficients.Similarly, Ref. [19,55,56] derived twosided bounds for spin-1 particles, which can be read in terms of f i /f 0 .
In Appendix C we perform a similar analysis, for amplitudes involving both spin-1 and spin-0 particles.
We exploit the fact that, again because of crossing symmetry, the form factors in Eqs.(24)(25)(26) also control other amplitudes, namely, . This allows us to study inelastic channels to find lower and upper bounds on the g i 's.
A simple-albeit non-optimal-subset of these bounds reads, and holds regardless of higher derivative terms, which are also similarly bounded.
In a theory that reduces to dRGT at low ener-gies, and departs from it only by higher derivative terms, the most relevant terms h 0,1 , f 0,1 and g 0,1 must match with dRGT, i.e.
The coefficients h 0 , g 0 , and f 0 are mass-suppressed and thus vanish at the order O(m 0 ) that we assume in this section.Therefore, combining these explicit expressions with the bounds in Eqs.(30)(31)(32) leads to exactly the same situation as in the left panel of Fig. 2, but this time, independently of all higher derivative terms.This holds also for detuned massive gravity with higher energy terms, here with different expressions for h 1 , g 1 , f 1 in terms of (c 1 , d 1 ).c. Non-gravitational spin-2 theories.It is interesting to extend our bounds to non-gravitational massive spin-2 theories, i.e.
theories without diffeomorphism-invariance, that would describe the dynamics of e.g.glueballs in gauge theories.One practical way to define these theories is by starting with a gravitational theory such as massive gravity with a generic potential (but keeping the Fierz-Pauli mass tuning) and then add diffeomorphism hard-breaking terms.The lowest-dimensional hardbreaking interaction for a parity even massive spin-2 particle corresponds to a single on-shell 3-point function with 2 momentum insertions, schematically ∂ 2 h 3 , different from the one in the Einstein-Hilbert term.This new vertex can be chosen to be [57][58][59], (parity odd spin-2 particles admit another trilinear with an ϵ tensor [57]).Restricting to parity-even for simplicity, the new trilinear dramatically changes the m ≪ E ≪ M behaviour of scattering amplitudes 5 .
In particular for the transverse-transverse polarizations (denoted | = and =) the leading energy behavior m ≪ E ≪ M reads e.g.
5 An interesting limit is obtained by decoupling the Einstein-Hilbert term, m Pl → ∞, while keeping the other interactions ζ/m Pl , c i /m Pl and d i /m 2 Pl finite, and where the strong-coupling scale of longitudinal modes is raised to Λ 3 by performing the aforementioned limit of the dRGTtunings.In this corner of parameter space the theory becomes invariant under linear diffeomorphisms [58], and its (in)consistency has been already studied in Ref. [60].
and likewise for the mixed transverse-longitudinal, e.g.M = | + = −s 2 (s + t)ζ 2 /Λ 6  3 , or for only longitudinal modes, M 0 + = ζst 2 (s + t)/6Λ 6  3 m 2 , . . ., (after the scalar-sector tuning as required by positivity Scattering with such a fast leading-energy scaling is in fact inconsistent with (15) (and as well as with the results of Ref. [15]) unless ζ is strongly suppressed or the cutoff is parametrically close to the mass.Indeed, for |t| ≫ m 2 we have, whereas the values in the forward limits are masssuppressed: after enforcing the scalar-sector tuning, 6  3 .Analogously for the scalar-vector, where 6  3 m 2 whereas the forward arc A 0+ (0) is mass suppressed, thus demanding again |ζ| ≪ 1.
However, as soon as |ζ| ≪ 1, e.g. by setting ζ = ζm 2 /M 2 for a finite ζ, this new coupling essentially drops out from the constraints (15) for the longitudinal-only polarizations λ 1 λ 2 = 00, ++.It survives in the 0+ but only proportionally to a t/M 2 factor which is of the same order of other higher derivative corrections, hence irrelevant.Indeed, the bounds (30), (31), and (32) still apply on the h 1 , f 1 , and g 1 , regardless of the higher orders in t that affect instead (h, f, g) i≥2 .All in all, for such a small ζ, we simply recover the same gravitational constraints for the longitudinal modes that we have already studied.See them explicitly in footnote 4.
In conclusion, theories of massive spin-2 particles cannot have a parametric separation of scales M/m, independently of how they are modified at high energy.

V. CONCLUSIONS AND OUTLOOK
The EFT of massless gravitons is a priori consistent from the smallest energy scale in the universe H 0 ∼ 10 −42 GeV, to the largest one m Pl ∼ 10 18 GeV, i.e. over about 60 orders of magnitude.The results presented in this paper show that consistency of the EFT of a massive graviton is instead confined into a narrow energy window, spanning from the graviton mass by at most one order of magnitude.This constitutes an improvement of 15 orders of magnitude w.r.t.previous bounds.
We devised new and simple positivity bounds based on an approximate crossing symmetry that is valid in weakly coupled EFTs with a hierarchy between mass m and cutoff M .The simple relations we obtain can be employed within dispersion relations (based on unitarity and causality) to study complex problems, such as massive higher-spin scattering.They lead to Eq. ( 15), which bounds the energy growth of elastic-helicity amplitudes to lie within a certain envelop.With this, we found that massive gravity can not sustain a parametrically large mass hierarchy, see Eq. ( 23) and Fig. 2, as it would fail our positivity bounds.This conclusion is robust w.r.t. the inclusion of arbitrary number of higher derivative terms, as well as higher order corrections to our version of simplified crossing symmetry.A similar conclusion holds for non-gravitational massive spin-2 particles.
While our results exclude massive gravity with just the graviton and nothing else in the spectrum below O(10)m, they do not exclude theories with no parametrically large separation of scales, such as KK gravitons that arise from the compactification of extra dimensions, or theories that do not fulfil our assumptions i)-vi).Moreover, the quantitative bound in Eq. ( 23) becomes inaccurate if one pushes it to the regime m 2 ∼ t ∼ M 2 , where it seems to become stronger.In the context of gravity, however, more stringent bounds would be incompatible with the inherently flat-space formulation of the dispersive approach, as curvature corrections can no longer be neglected for M ∼ m ∼ H 0 .The extension of positivity bounds to theories in non-flat backgrounds is very interesting [61,62], albeit rather subtle [63].
It would be interesting to apply the techniques developed in this paper to theories with massive higher spin J ≥ 3 states, along the lines of [64].It was shown there that, contrary to massive gravity [30], forwardonly positivity bounds were already sufficient to rule out theories dominated by the most relevant interactions, but didn't exclude the possibility that an EFT with a parametric separation between m and M could exist, if dominated by less relevant interactions.The all-derivative order argument presented in section IV should be sufficient to close this door, and exclude any EFT for a single massive higher-spin particle.
Another interesting direction is to extend the anal-ysis beyond our approximation and derive a version of Eq. ( 23) that remains valid even for m ∼ M [65].While this is not motivated in the framework of massive gravity, as discussed above, it would teach us about the properties of spin-2 resonances, such as glueballs in gauge theories and QCD.(x) are Jacobi polynomials and the kernels K n λ1λ2 are given by,
Likewise, it is easy to prove (31) and the lower bound of (32).Although not optimised, these relations are conservative.
the positive quadratic polynomial (a + bJ 2 /s) 2 , we find positive definiteness of the Hankel matrix,