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Article Reference 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 Th´eorique, Universit´e Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France 2Dipartimento di Fisica e Astronomia, Universit`a di Padova, Via Marzolo 8, I-35131 Padova, Italy 3D´epartment de Physique Th´eorique, Universit´edeGen`eve, 24 quai Ernest-Ansermet, 1211 Gen`eve 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) 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. 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 gà 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.
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