Causality constraints on corrections to Einstein gravity

TL;DR

The work develops a dispersive, S-matrix–based program to constrain modifications of Einstein gravity within four-dimensional, weakly-coupled EFTs. By combining analyticity, unitarity, crossing symmetry, and Regge behavior, the authors construct impact-parameter functionals and improved sum rules to obtain two-sided bounds on higher-derivative gravitational couplings in terms of the mass M of new higher-spin states, revealing that gravity must fade as $G\to0$ and that $g_{R^{(3)}}$ and $g_{R^{(4)}}$ scale with M as expected from dimensional analysis, up to an infrared logarithm. They connect these bounds to the CEMZ time-delay argument and show light spin-0/2 matter cannot lower the cutoff, while heavy-spin completions are constrained by positivity and spectral data; they also explore implications for potential near-future tests of GR and collider visibility of higher-spin states. The results significantly sharpen the allowed region for gravitational EFTs and provide a framework for translating high-energy spectral information into precise low-energy constraints on gravity.

Abstract

We study constraints from causality and unitarity on $2\to2$ graviton scattering in four-dimensional weakly-coupled effective field theories. Together, causality and unitarity imply dispersion relations that connect low-energy observables to high-energy data. Using such dispersion relations, we derive two-sided bounds on gravitational Wilson coefficients in terms of the mass $M$ of new higher-spin states. Our bounds imply that gravitational interactions must shut off uniformly in the limit $G \to 0$, and prove the scaling with $M$ expected from dimensional analysis (up to an infrared logarithm). We speculate that causality, together with the non-observation of gravitationally-coupled higher spin states at colliders, severely restricts modifications to Einstein gravity that could be probed by experiments in the near future.

Figures (14)

• Figure 1: $2$-to-$2$ scattering amplitudes of gravitons within the low-energy effective theory. We include at tree-level both the graviton exchange and (higher-derivative) contact diagrams, as well as exchanges of possible light spin-0 and spin-2 particles. Other spins are forbidden by angular momentum conservation.
• Figure 2: Contour deformation which gives rise to sum rules eq. \ref{['sum rule schematic']}. The final contour relates low-energy EFT data along the arcs to heavy discontinuities along the branch cuts.
• Figure 3: Contour plots which confirm non-negativity of the functional $\mathcal{F}_{g_3}$ giving the upper bound \ref{['g3 bound']}, in units where $M=1$. We scaled the functional by $m^{10}$ to make scaling limits more manifest. The sampling points are detailed in the text. The lower-left corner is blank in the second figure due to the selection rule $J\geq4$.
• Figure 4: Similar to figure \ref{['fig: functional_g3']}: Contour plots which confirm non-negativity of $\mathcal{F}_{g_4}$, establishing the bound \ref{['g4 bound']}.
• Figure 5: Plots of functionals $\mathcal{F}_{g_3}$ at the scaling limit $J,m\rightarrow\infty$ where $b=2J/m$. (a) confirms the positivity at finite $b$ where the cutoff $m_{\rm IR}$ makes no difference. (b) displays the large $b$ behavior for both $m_{\rm IR}=0$ (blue curve) and $m_{\rm IR}/M=10^{-6}$ (orange curve). There is a negative plateau in the range $m_{\rm IR}^{-2/3} \mathrel{\hbox{$\sim$ $<$}} b\mathrel{\hbox{$\sim$ $<$}} m_{\rm IR}^{-1}$ as detailed in the text.