Beyond Amplitudes' Positivity and the Fate of Massive Gravity

TL;DR

This work develops dispersion-relations-based bounds that go beyond traditional positivity by incorporating calculable IR cross-sections in EFTs. The method links low-energy EFT data to high-energy consistency and is applied to Galileon theories and ghost-free massive gravity (dRGT), yielding sharper constraints on symmetry-breaking terms and graviton parameters. In particular, the bounds force non-negligible Galileon symmetry-breaking terms and impose stringent, testable limits on the graviton mass and the ratio $g_*=(\Lambda/\Lambda_3)^3$ in massive gravity, often ruling out MG as a viable GR replacement unless new UV physics raises the cutoff. These results highlight the need for UV completion or alternative IR modifications to recover GR phenomenology while remaining consistent with fundamental S-matrix principles and fifth-force tests.

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.

Figures (4)

• Figure 1: Exclusion region for massive gravity in the plane of $(g_*,m)$, where $g_*= (\Lambda/\Lambda_3)^3$ is the hierarchy between the physical cutoff $\Lambda$ and the strong coupling scale $\Lambda_3$, and $m$ is the graviton mass. The gray region is theoretically excluded by our lower bound Eq. (\ref{['FVSmaxrelation']}), with accuracy either $\delta=1\%$ (dark) or $\delta=5\%$ (light), irrespectively of the values for $(c_3,d_5)$ in the massive graviton potential. Colored lines show the physical cutoff length: solid lines correspond to $\Lambda$ in Eq. (\ref{['barecutoff']}), while dashed lines correspond to $\Lambda_\oplus$, obtained after assuming ad-hoc a Vainshtein redressing of $\Lambda$ due to the gravitational field on the Earth's surface, Eq. (\ref{['VainCutoff']}). Either cutoff, and with it the domain of predictivity of massive gravity, increases with $g_*$ and $m$, at odds with our theoretical constraint and the experimental upper bounds on the graviton mass. The black horizontal line is a representative of the latter, corresponding to $m=10^{-32}$ eV.
• Figure 2: Integration contours in the complex $s$-plane at fixed $t=0$, with poles at $s_1=M^2$ and $s_2=4m^2-M$. The point $s=\mu^2$ is on the real axis between the branch-cuts shown in red.
• Figure 3: Exclusion plots in the $(c_3,d_5)$ plane for ghost-free massive gravity, for fixed accuracy $\delta= 1\%$, mass $m=10^{-32}$ eV, and coupling $g_* = 3 \, (5) \cdot 10^{-10}$ in the left (right) panel. The two plots illustrate how the region allowed by our bounds (green region inside the solid line) shrinks to the point of disappearing as the coupling is increased above $4.5\cdot 10^{-10}$. The yellow region is allowed by the standard positivity constraints, Eqs. (\ref{['sresidues']}, \ref{['yellowregionupper']}), whose optimized version from Ref. Cheung:2016yqr is delimited by the dotted black line. The other regions are the ones consistent with our new bounds, Eq. (\ref{['boundmassivegr1']}), the different colors corresponding to each of the $F_i$ in Eq. (\ref{['ffunctions']}), as specified in the legend. On the dash-dotted red (dashed black) line, $F_{VV}$ ($F_{SS}$) vanishes, and so it does the corresponding bound. On the red dot $(c_3, d_5)=(1/4,-9/256)$ the vector and scalar modes decouple from the tensors, but not from each other, and on the black dot $(c_3, d_5)=(1/6,-1/48)$ the scalar mode decouples from the tensor mode and itself.
• Figure 4: Exclusion plots in the $(c_3,d_5)$ plane for ghost-free massive gravity, for fixed accuracy $\delta= 1\%$, mass $m=10^{-32}$ eV, and coupling $g_* = 3 \cdot 10^{-10}$, using inelastic channels. See the caption of Fig. \ref{['fig:MGbound']} for other information about the figure. For couplings larger than $g_* \approx 4.4 \cdot 10^{-10}$ the green island disappears, for the same value of the mass, and the model is ruled out.