The EFT-Hedron

TL;DR

<3-5 sentence high-level summary>The EFThedron paper probes how causality, analyticity, and unitarity constrain the full tower of higher-dimension operators in low-energy 2→2 scattering. By recasting EFT coefficients as residues and dispersive data, the authors reveal a hidden totally positive structure: forward-limit data form Hankel- and Gegenbauer-based convex geometries that combine into an EFThedron, a high-dimensional polytope-like region cut out by infinitely many linear and nonlinear inequalities. They develop a systematic boundary analysis, mapping EFT coefficients to convex hulls of moment-curve and Gegenbauer-derivative vectors, and illustrate the framework with scalars, photons, and gravitons, including explicit string-theory examples. The results provide rigorous UV-to-IR constraints on EFTs, illuminate how UV spectra (low-spin dominance, gaps) shape EFTs, and offer a geometric lens for understanding the space of consistent UV completions of gravity and gauge theories.

Abstract

We re-examine the constraints imposed by causality and unitarity on the low-energy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden "totally positive" structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra. This forces the infinite tower of higher-dimension operators to lie inside a new geometry we call the "EFThedron". We initiate a systematic investigation of the boundary structure of the EFThedron, giving infinitely many linear and non-linear inequalities that must be satisfied by the EFT expansion in any theory. We illustrate the EFThedron geometry and constraints in a wide variety of examples, including new consistency conditions on the scattering amplitudes of photons and gravitons in the real world.

Figures (19)

• Figure 1: Different origins for the EFT: (I) Integrating away massive states in tree exchanges, for example the Higgs for the Sigma model and the infinite tower of higher spin states in string amplitudes, (II) or massive states in the loop, for example the $\varphi X^2$ coupling.
• Figure 2: An operator of four fields will contribute to the four-point amplitude as a polynomial, and the six-point amplitude as a rational term.
• Figure 3: We define the low energy couplings through a contour integral on the complex $s$-plane, where the contour $\mathcal{C}_0$ encircles the origin. On the complex plane, if the amplitude only has singularities on the real-$s$ axes, either poles or branch points, then we can deform to contour $\mathcal{C}_\infty$.
• Figure 4: The convex hull of these three vectors encloses the origin, and hence trivially covers the entire two-dimensional plane.
• Figure 5: The $s$-channel geometry at fixed $k$. The vector $\mathbf{a}_k$ must live on the intersection between the cyclic plane $\mathbf{X}_{\rm cyc}$ with the unitary polytope $\textbf{U}_k$.