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Article

Beyond Positivity Bounds and the Fate of Massive Gravity

BELLAZZINI, Brando, et al.

Abstract

We constrain effective field theories by going beyond the familiar positivity bounds that follow from unitarity, analyticity, and crossing symmetry of the scattering amplitudes. As interesting examples, we discuss the implications of the bounds for the Galileon and ghost-free massive gravity. The combination of our theoretical bounds with the experimental constraints on the graviton mass implies that the latter is either ruled out or unable to describe gravitational phenomena, let alone to consistently implement the Vainshtein mechanism, down to the relevant scales of fifth-force experiments, where general relativity has been successfully tested. We also show that the Galileon theory must contain symmetry-breaking terms that are at most one-loop suppressed compared to the symmetry-preserving ones. We comment as well on other interesting applications of our bounds.

Reference

BELLAZZINI, Brando, et al. Beyond Positivity Bounds and the Fate of Massive Gravity. Physical Review Letters, 2018, vol. 120, no. 16

DOI : 10.1103/PhysRevLett.120.161101 arxiv : 1710.02539

Available at: http://archive-ouverte.unige.ch/unige:104094

Disclaimer: layout of this document may differ from the published version.

1 / 1 PHYSICAL REVIEW LETTERS 120, 161101 (2018)

Beyond Positivity Bounds and the Fate of Massive Gravity

Brando Bellazzini,1,2 Francesco Riva,3,4 Javi Serra,4,5 and Francesco Sgarlata6 1Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France 2Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, I-35131 Padova, Italy 3Départment de Physique Théorique, UniversitédeGenève, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland 4Theory Division, CERN, CH-1211 Geneva 23, Switzerland 5Physik-Department, Technische Universität München, 85748 Garching, Germany 6SISSA International School for Advanced Studies and INFN Trieste, via Bonomea 265 34136, Trieste, Italy (Received 15 December 2017; published 17 April 2018)

DOI: 10.1103/PhysRevLett.120.161101

The idea that physics at low energy can be described in IR effective Lagrangian. This UV-IR connection can be terms of light degrees of freedom alone, which goes under used to show that coefficients with the “wrong” sign cannot the name of effective field theory (EFT), is one of the most be generated by a Lorentz invariant, unitary, causal, and satisfactory organizing principles in physics. The effect of local UV completion [5]: the corresponding EFT, even if ultraviolet (UV) dynamics is systematically accounted for compatible with the symmetries of the system, is thrown to in the resulting infrared (IR) EFT by integrating out heavy the “swampland.” Positivity bounds have found several degrees of freedom, which generate an effective Lagrangian applications, including the proof of the a theorem [6,7]; the made of infinitely many local operators. Since the sym- study of chiral perturbation theory [8] and WW scattering; metries of the underlying UV theory are retained in the IR, and theories of composite Higgs [9–14], quantum gravity EFTs are predictive even when the UV dynamics is [15], massive gravity [16–18], Galileons [18–21], inflation unknown: only a finite number of symmetric operators [22,23], the weak gravity conjecture [24,25], and conformal contribute, at a given accuracy, to observable quantities. field theory [26–28]. The approach has been recently Remarkably, extra information about the UV can always extended to particles of arbitrary spin [18], leading to a be extracted if the underlying Lorentz invariant micro- general no-go theorem on the leading energy-scaling scopic theory is unitary, causal, and local. These principles behavior of the IR amplitudes, with applications to massive are encoded in the fundamental properties of the S matrix gravity [16] and Goldstini [29–31]. References [21,32,33] such as unitarity, analyticity, crossing symmetry, and extended this technique beyond the forward limit. polynomial boundedness [1,2]. These imply a UV-IR In this Letter, we show that qualitatively new bounds, connection in the form of dispersion relations that link stronger than standard positivity constraints, can be derived the (forward) amplitudes in the deep IR with the disconti- by taking into account the irreducible IR cross sections nuity across the branch cuts integrated all the way to infinite under the dispersive integral, which are calculable within energy [3,4]. Unitarity ensures the positivity of such the EFT. We discuss for what models our bounds can be discontinuities, and in turn the positivity of (certain) important and focus explicitly on two relevant applications: Wilson coefficients associated with the operators in the the EFT for a weakly broken Galileon [34,35], and the ghost-free massive gravity theory [36,37]. Figure 1 gives a preview of our results for the latter. Published by the American Physical Society under the terms of Let us consider the center-of-mass 2-to-2 scattering the Creative Commons Attribution 4.0 International license. amplitude Mz1z2z3z4 ðs; tÞ, where the polarization functions Further distribution of this work must maintain attribution to ’ are labeled zi. The Mandelstam variables are defined by the author(s) and the published article s title, journal citation, 2 2 2 and DOI. Funded by SCOAP3. s ¼ −ðk1 þ k2Þ , t ¼ −ðk1 þ k3Þ , u ¼ −ðk1 þ k4Þ and

0031-9007=18=120(16)=161101(7) 161101-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 120, 161101 (2018) Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ∞ ds m2 Σz1z2 1 − 4 IR ¼ 4m2 π s X sσz1z2→XðsÞ sσ−z¯1z2→XðsÞ × − μ2 3 þ − 4 2 μ2 3 ; ð3Þ ðs Þ ðs m þ Þ IR wherewe use crossing symmetry relating the s and u channels in the forward limit even for particles with spin [18]:inthe helicity basis, Mz1z2 ðsÞ¼M−z¯1z2 ðu¼−sþ4m2Þ, and for particles that are their own antiparticles, z¯ ¼ z.Wealso invoke the optical theorem to relate the imaginary parts of the amplitudes (across the branch cuts) to the cross section. FIG. 1. Exclusion region for massive gravity in the plane of For any theory where particles 1 and 2 are interacting, as ≡ Λ Λ 3 Λ 2 2 graviton mass m and coupling g ð = 3Þ , with being the long as 0 < μ < 4m , the right-hand side (rhs) of Eq. (3) is Λ 2 1=3 physical cutoff and 3 ¼ðm mPlÞ the strong coupling scale. always positive, and one obtains the rigorous positivity Our theoretical bound Eq. (14) excludes with accuracy δ ¼ 1% Σz1z2 0 Σz1z2 bound, IR > . Since IR is calculable in the IR in terms the darkest gray region, and with 10%, 30% the gradually lighter Λ of the Wilson coefficients, this provides a nontrivial gray regions. Solid lines mark the fixed cutoff , whereas the constraint on the EFT. dashed black line shows the upper experimental bound −32 We can in fact extract more than positivity bounds by m ¼ 10 eV. Our constraint gives rise to a tension between high Λ and a small m: the graviton mass can only be below the noticing that the total cross section on the rhs of the experimental bound at the expense of a premature breakdown of dispersion relation Eq. (3) contains an irreducible contri- the EFT at macroscopically large distances. bution from IR physics, which is also calculable within the EFT, by construction. The other contributions, e.g., those from the UV, are incalculable with the EFT but 4 2 satisfy s þ t þ u ¼ m , where m is the mass of the are nevertheless always strictly positive, by unitarity. scattered particles. Our arguments will require finite Moreover, each final-state X in the total cross section ≠ 0 m , yet they hold even for some massless theories contributes positively too. Therefore, an exact inequality 1 2 1 [scalars, spin- = fermions, and softly broken Uð Þ gauge follows from truncating the rhs of Eq. (3) at some energy theories], which have a smooth limit m → 0. We call ≡ 2 ≪ Λ2 Λ smax E below the cutoff of the EFT. To leading order (LO) in powers of ðE=ΛÞ2 and ðm=EÞ2 [hence also 2 Mz1z2 ðsÞ ≡ Mz1z2z1z2 ðs; t ¼ 0Þð1Þ ðμ=EÞ ], the bound in Eq. (3) becomes Z X E2 ds 0 Σz1z2 > ½σz1z2→XðsÞþσz1−z¯2→XðsÞ the forward elastic amplitude at t ¼ and study the IR;LO πs2 IR;LO analyticity properties of Mz1z1 ðsÞ=ðs − μ2Þ3, integrating X 2 2 along a closed contour Γ in the complex s plane, enclosing 1 m E × þ O þ O Λ ; ð4Þ all the physical IR poles si associated with stable light E degrees of freedom entering the scattering (or its crossed 2 where the subscript IR highlights the fact that both sides of symmetric process), together with the point s ¼ μ lying on the equation are computable within the EFT; the main the real axis between s ¼ 0 and s ¼ 4m2. We define source of error for small masses is the truncation of the tower of higher-dimensional operators. Choosing E at or I X 1 Mz1z2 ðsÞ Mz1z2 ðsÞ slightly below the cutoff Λ gives just an order-of-magnitude Σz1z2 ≡ IR ds 2 3 ¼ Res 2 3 ; ð2Þ estimate for the bound [18,21], as originally suggested in 2πi Γ ðs − μ Þ s¼s ;μ2 ðs − μ Þ i Ref. [19]. A rigorous bound can instead be obtained by choosing a sufficiently small ðE=ΛÞ2: percent accuracy can which is calculable within the EFT. Using Cauchy’s be achieved already with E=Λ ≈ 1=10. Σz1z2 integral theorem, the contour can be deformed into a The IR must therefore be not only positive but strictly new contour that runs around the s-channel and u-channel larger than something which is itself positive and calcu- branch cuts, and goes along a big circle eventually sent to lable within the EFT. Moreover, we can include any infinity. The boundary contribution at infinity vanishes, due final states X, elastic or inelastic: the more channels and to the Froissart-Martin asymptotic bound jMðs → ∞Þj < information are retained, the more refined the bound will const × s log2 s, which is always satisfied in any local be. Notice that the 2-to-2 cross section retained on the rhs is massive QFT [38,39]. This leads to a dispersion relation obtained by integrating over t; thus, effectively, our bounds that connects the IR, Eq. (2), to an integral (UV) of the total capture as well the behavior of the amplitude away from the cross section forward limit.

161101-2 PHYSICAL REVIEW LETTERS 120, 161101 (2018) Z pffiffiffiffiffiffi m2 m2 m2 The implications of our bound Eq. (4) are particularly S ¼ d4x −g Pl R − Pl Vðg; hÞ ; ð8Þ interesting in theories where the elastic forward amplitude 2 8 Mz1z2 , appearing on the left-hand side (lhs), is parametri- 8π −1=2 where mPl ¼ð GÞ is the reduced Planck mass, gμν ¼ cally suppressed compared to the nonforward or inelastic η ones appearing on the rhs. The Galileon, massive gravity, μν þ hμν is an effective metric written in terms of the η the dilaton, and WZW-like theories, as well as other models Minkowski metric μν (with mostly þ signature) and a where 2 → 2 is suppressed while 2 → 3 is not, are other spin-2 graviton field hμν in the unitary gauge, R is the Ricci simple examples of theories that get nontrivial constraints, scalar for gμν, and on the couplings and/or masses of the corresponding EFTs, 2 2 3 2 3 Vðg; hÞ¼b1hh iþb2hhi þ c1hh iþc2hh ihhiþc3hhi that include and go beyond the positivity of Σ . Even in IR 4 3 2 2 2 2 situations without parametric suppression, our bound þ d1hh iþd2hh ihhiþd3hh i þ d4hh ihhi carries important information: it links elastic and inelastic 4 þ d5hhi cross sections that might depend on coefficients of the EFT. μν 2 Galileon.—The Lagrangian for the weakly broken is the soft graviton potential, with hhi ≡ hμνg , hh i ≡ μν ρσ Galileon [34,35], g hνρg hσμ, etc. Absence of ghosts implies that the coefficients of this potential depend on just two parameters, 1 2 c3 c4 2 2 c3 and d5; see Ref. [37,42]. L ¼ − ð∂μπÞ 1 þ □π þ (ð□πÞ − ð∂μ∂νπÞ ) 2 Λ3 Λ6 Note that, though tempting, the results obtained above 2 for the Galileon cannot be directly interpreted in the context λ 2 2 m 2 c5 … ∂π − π ; þ ð Þ þ 4Λ4 ½ð Þ 2 ð5Þ of massive gravity (even if the Galileon is the longitudinal component of the massive graviton), since the IR dynamics with physical cutoff Λ, has suppressed symmetry-breaking is different: for example, in the scattering of the Galileon 2 2 2 scalar mode, the helicity-2 mode exchanged in the t channel terms λ ≪ c3, c4 and m ≪ Λ . Forward 2 → 2 scattering 2 contributes as much as the scalar mode. is controlled at Oðs Þ by the symmetry-breaking inter- Since the graviton is its own antiparticle, it is convenient Σ λ Λ4 2 2 2Λ6 actions: the lhs of Eq. (4) is IR ¼ð = Þþðc3m = Þ. to express Eq. (4) in terms of linear polarizations [15,16,18]: On the other hand, the hard scattering is controlled Z ππ→ππ X 2 E2 by the symmetry-preserving interactions, σ ≈ z1z2 ds → Σ > ½σz1z2 XðsÞ : ð9Þ 3 2 − 2 2 5 5120πΛ12 IR;LO π 2 IR;LO ðc3 c4Þ s =ð Þ. X s In the massless limit, or more generally for c2m2=Λ2 ≪λ 3 We adopt the basis of polarizations of Ref. [16] to calculate (a natural hierarchy, given that λ preserves a shift symmetry 2 λ the amplitudes for different initial- and final-state configu- while m does not), the bound Eq. (4) shows not only that Σz1z2 ∼ 2 Λ6 rations, finding that, generally, IR m = 3 is suppressed must be positive, but (parametrically) at most one loop Λ ≡ 2 1=3 2 by the small graviton mass, with 3 ðm mPlÞ the strong factor away from ðc − 2c4Þ=4: 3 coupling scale [43]. For instance, 2 2 8 2 2 2 3 c − 2c4 E c m 2m λ ð 3 Þ 3 ≪ λ ΣSS 7 − 6 1 3 48 0 > 2 for 2 : ð6Þ IR ¼ 6 ð c3ð þ c3Þþ d5Þ > ; 640 16π Λ Λ 9Λ3 2 λ ≠ 0 VV m 2 For a massive Galileon with negligible and c3 , one Σ 5 72c3 − 240c > 0; IR ¼ 16Λ6 ð þ 3Þ gets a lower bound on the mass, 3 2 VS m 2 3 − 2 2 8 2 2 Σ ¼ ð91 − 312c3 þ 432c þ 384d5Þ > 0: ð10Þ 2 2 ðc3 c4=c3Þ E c3m IR 48Λ6 3 m > Λ for ≫ λ; ð7Þ 3 320 × 16π2 Λ Λ2 In contrast, the hard-scattering limits of the amplitudes that 2 where ðE=ΛÞ8 ≈ 10−2 for a 30% accuracy. Therefore, the enter the rhs of Eq. (9) are unsuppressed. For s, t ≫ m , e.g., Galileon symmetry-breaking terms cannot be arbitrarily elastic amplitudes read 2 suppressed, the general lesson being that Oðs Þ terms in the st s t MSS ð þ Þ 1 − 4 1 − 9 64 amplitude cannot be too suppressed compared to the O s3 ¼ 6 ð c3ð c3Þþ d5Þ; ð Þ 6Λ3 terms. The results of Eqs. (6) and (7) hold when loop effects 9st s t MVV ð þ Þ 1 − 4 2 are included, as they simply generate terms that are ¼ 6 ð c3Þ ; ∼ Λ 6 4 2 16π2Λ6 Σ 32Λ3 subleading, e.g., ðm= Þ c3m = in IR (recall that m ≪ E ≪ Λ). 3t 5s2 5st − 9t2 MVS 1 − 2 2 − 2 − þ Massive gravity.—The action for ghost-free massive ¼ 6 c3ð c3Þðs þ st t Þ : 4Λ3 72 gravity, also known as Λ3 or dRGT massive gravity, is [36,37] (for reviews, see Refs. [40,41]) ð11Þ

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−33 At this point, we choose the energy scale E in Eq. (9) below m ≃ H0 ≃ 10 eV, as is customary in cosmology, require ≪ Λ the cutoff, E , so that the EFT calculation of the cross an even smaller coupling. For a given mass, only as g is sections is trustworthy, and above the mass, E ≫ m, so that lowered sufficiently according to Eq. (14), a region allowed the amplitudes Eq. (11) dominate such cross sections. We by our bounds eventually materializes inside the positivity define δ ≡ ðE=ΛÞ2, that controls the accuracy of the EFT region. calculation, and obtain At this point, the crucial question is what the physical meaning of g is, and if it can be arbitrarily small, 4 −10 4πm g 6 ≲ 10 Pl δ g [18]. To our knowledge, most literature of Fz1z2 ðc3;d5Þ > ; ð12Þ m 4π massive gravity has so far taken Λ ¼ Λ3,orΛ ≫ Λ3, corresponding to g ≳ 1. These values are now grossly ≡ Λ Λ 3 where g ð = 3Þ and, e.g., excluded by our bounds. From a theoretical point of view, Λ Λ ℏ 7 − 6 1 3 48 3=2 and 3 scale differently with , so that their ratio actually 960 c3ð þ c3Þþ d5 FSS ¼ 2 ; changes when units are changed, in such a way that g ð1 − 4c3ð1 − 9c3Þþ64d5Þ indeed scales like a coupling constant. This is analogous to 2 3=2 2560 5 þ 72c3 − 240c3 the difference between a vacuum expectation value v and FVV ¼ 4 ; ∼ 27 ð1 − 4c3Þ the mass of a particle coupling × v (e.g., the W-boson ∼ Λ 2 3=2 mass mW gv). The point, then, is that the cutoff is a 80640ð91 − 312c3 þ 432c3 þ 384d5Þ physical scale, which differs from Λ3 that instead does FVS ¼ 1975 − 29808 1 − 2 1 − 4 8 2 : c3ð c3Þð c3 þ c3Þ not have the right dimension to represent a cutoff. Since Λ−1 ≈ 320 10−32 −2=3 ð13Þ 3 kmðm= eVÞ , a very small coupling g translates into a very low cutoff (large in units of distance), Σz1z2 Mz1z2 A more complete set of IR , , as well as the resulting 1=3 2=3 5 −1 g m inequalities involving the Fz1z2 ðc3;d5Þ functions, are Λ ≃ 4 1 10 ð . × kmÞ −10 −32 : reported in the Supplemental Material [44]. 4.5 × 10 10 eV The inequalities following from Eq. (12) are the main ð15Þ result of this discussion and can be read in several ways: as constraints on the plane of the graviton potential This is clearly problematic. For example, let us consider the parameters ðc3;d5Þ for a given graviton mass m and ratio experimental tests of massive gravity in the form of bounds Λ Λ 3 ≡ ð = 3Þ g, as a constraint on g for fixed m at a given on fifth forces from the precise measurements of the point in the (c3, d5) region allowed by positivity, or Earth-Moon precession δϕ [41,53,54]. Due to the equivalently as a bound on the graviton mass for fixed Vainshtein screening [55,56], which is generically domi- coupling at that point. nated by the Galileon cubic interactions in the (c3, d5) An important aspect of Eq. (12) is that it is possible to region allowed by our bounds, the force mediated by the find an absolute maximum value of g above which our scalar mode compared to the standard gravitational one is ∼ 3=2 4π 1=3Λ−1 bounds do not allow for a solution: we write Eq. (12) as FS=FGR ðr=rVÞ , where rV ¼ðM= mPlÞ 3 ¼ ∝ 1 4π 2 2 1=3 m>mmin =Fz1z2 ðc3;d5Þ and note that at each point ðM= m mPlÞ is the Vainshtein radius associated with

ðc3;d5Þ the bound is determined by the smallest Fz1z2 .Now, the (static and spherically symmetric) source under the positivity constraints Eq. (10) provide a compact consideration, in this case the Earth, M ¼ M⊕. Before allowed region in the (c3, d5) plane, within which the our bound, one would find that at lunar distances, ≈ 3 8 105 (continuous) function minfFz1z2 gðc3;d5Þ has a maximum. r ¼ r⊕L . × km, the ratio of forces, and thus also ˆ δϕ ∼ π cˆ3; d5 ≈ 0 18; −0 017 F ≈ the precession ðFS=FGRÞ, even if very small for This corresponds to ð Þ ð . . Þ and VS −32 4 6 106 m ¼ 10 eV, would be borderline compatible with the . × ; thus, the most conservative bound −11 very high accuracy of present measurements ∼10 .Now 4 δ 6 10−32 g our result in Eq. (15) shows that the EFT is not valid already m> eV −10 : ð14Þ ∼ 1 Λ 4.5 × 10 1% for r = >r⊕L. This implies that the Vainshtein screening should receive important corrections before reaching −32 Taking m ¼ 10 eV as a benchmark experimental upper the (inverse) cutoff 1=Λ, and moreover, it means that new bound on the graviton mass (see Ref. [52] for a critical degrees of freedom should become active at that scale: two ≳ 4 5 10−10 discussion), any value g . × is excluded, irre- effects that likely impair the fifth-force suppression and spectively of the values of (c3, d5), a situation that we hinder the agreement with the precise measurement of summarize in Fig. 1. Slightly stronger bounds can be the Earth-Moon precession. (Note that the Vainshtein obtained by working with the nonelastic channels, while if redressing [57] deep inside the Vainshtein region, we were to admit a slightly larger uncertainty, e.g., δ ¼ 5%, Λ3 → zΛ3 with z ≫ 1, does not generically extend to Λ the upper bound on g would increase by 1 order of the physical cutoff , since the rescaling of the kinetic magnitude. Smaller values of the graviton mass, e.g., term in a background does not affect the extra derivative

161101-4 PHYSICAL REVIEW LETTERS 120, 161101 (2018) terms that come without extra field insertions. In other discussions. We thank Duccio Pappadopulo, Massimo words, a potentially large local kinetic term translates into a Porrati, and Gabriele Trevisan for useful comments. We small local coupling, but it does not change the location of thank Claudia de Rham, Scott Melville, Andrew Tolley, poles and thresholds associated with the UV degrees of and Shuangyong Zhou for correspondence concerning the freedom). Vainshtein mechanism. B. B. thanks Cliff Cheung and In summary, our theoretical bound Eq. (14) leads to a Grant Remmen for correspondence. B. B. is supported in tension between direct limits on the graviton mass and fifth- part by MIUR-FIRB Grant No. RBFR12H1MW “Anew force experiments. This either rules out Λ3 massive gravity strong force, the origin of masses and the LHC.” B. 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