Positive Signs in Massive Gravity

TL;DR

The paper constrains ghost-free massive gravity by enforcing unitarity and analyticity of scattering amplitudes. By computing tree-level graviton scattering for all polarizations and applying analytic dispersion relations to forward amplitudes, it derives positivity bounds that carve out a small allowed region in the parameter space spanned by $c_3$ and $d_5$ and shows that embedding the Galileon scalar into the full theory remedies its standalone positivity issues. The work reveals that a wide portion of the ghost-free MG parameter space is incompatible with basic S-matrix principles, while a finite region remains viable, consistent with a higher cutoff around $\Lambda_3$. This provides a robust IR constraint on MG theories and informs their potential UV completions and cosmological implications.

Abstract

We derive new constraints on massive gravity from unitarity and analyticity of scattering amplitudes. Our results apply to a general effective theory defined by Einstein gravity plus the leading soft diffeomorphism-breaking corrections. We calculate scattering amplitudes for all combinations of tensor, vector, and scalar polarizations. The high-energy behavior of these amplitudes prescribes a specific choice of couplings that ameliorates the ultraviolet cutoff, in agreement with existing literature. We then derive consistency conditions from analytic dispersion relations, which dictate positivity of certain combinations of parameters appearing in the forward scattering amplitudes. These constraints exclude all but a small island in the parameter space of ghost-free massive gravity. While the theory of the "Galileon" scalar mode alone is known to be inconsistent with positivity constraints, this is remedied in the full massive gravity theory.

Figures (3)

• Figure 1: Diagram of the analytic structure of the forward amplitude in the complex $s$ plane. The simple poles at $s=m^2$ and $3m^2$ and the branch cuts starting at $s=4m^2$ and $0$ correspond to resonances and multi-particle thresholds in the $s$- and $u$-channels, respectively. The scale $\mu^2$ in the dispersion relation is chosen here to be at the symmetric point $\mu^2=2m^2$. The contours $\Gamma$ and $\Gamma'$ referred to in Eqs. (\ref{['eq:fdef']}) and (\ref{['eq:fdef2']}) are also depicted.
• Figure 2: Regions in the $(c_3,d_5)$ parameter space of ghost-free massive gravity excluded by analyticity bounds on scattering of definite-helicity gravitons. The tensor, vector, and scalar modes are denoted by $T$, $V$, and $S$, respectively, and the $\pm$ delineation indicates vector polarizations that are parallel or orthogonal, respectively. Ultimately, by considering indefinite-helicity scattering, we will further restrict the allowed region of parameter space to that within the black curve. The dot marks the parameter choice $(c_3, d_5)=(1/6,-1/48)$, which corresponds to a free scalar sector in the decoupling limit.
• Figure 3: Region of $(c_3,d_5)$ parameter space for ghost-free massive gravity excluded by analyticity bounds on scattering of indefinite-helicity gravitons. Each colored point corresponds to a theory excluded by a scattering process that violates analytic dispersion relations. As noted in text, such violations can be diagnosed by evolving a particular dynamical system that tends toward scattering processes of gravitons of similar polarization. The specific color---plotted in blue, green, and red---corresponds to the power of each polarization in tensors ($\alpha_1^2 + \alpha_2^2$ and $\beta_1^2+\beta_2^2$), vectors ($\alpha_3^2+\alpha_4^2$ and $\beta_3^2 + \beta_4^2$), and scalars ($\alpha_5^2$ and $\beta_5^2$). The allowed region is shown in white and the black dot marks the choice that corresponds to a free Galileon.