Into the EFThedron and UV constraints from IR consistency

TL;DR

The paper identifies the EFThedron as the convex hull of a product of two moment curves, casting the space of UV-complete EFTs into a bi-variate moment problem whose infinite-dimensional boundary is captured by a moment matrix $M(a)\ge0$. Through a GL rotation, the $s$- and full EFThedrons are shown to be descriptions of the same underlying geometry, with truncations governed by positive extension theorems that converge to SDP results. Crossing symmetry in the IR imposes non-trivial UV constraints, notably requiring an infinite tower of higher spins for identical scalars and yielding sharp bounds on spin-spectral weights and on the masses of high-spin states. These geometric null constraints translate into concrete UV-spectrum requirements, with explicit bounds that extend to electromagnetism and gravity EFTs, and suggest deep connections to string theory and holography for future exploration.

Abstract

Recently it was proposed that the theory space of effective field theories with consistent UV completions can be described as a positive geometry, termed the EFThedron. In this paper we demonstrate that at the core, the geometry is given by the convex hull of the product of two moment curves. This makes contact with the well studied bi-variate moment problem, which in various instances has known solutions, generalizing the Hankel matrices of couplings into moment matrices. We are thus able to obtain analytic expressions for bounds, which perfectly match numerical results from semi-definite programing methods. Furthermore, we demonstrate that crossing symmetry in the IR imposes non-trivial constraints on the UV spectrum. In particular, permutation invariance for identical scalar scattering requires that any UV completion beyond the scalar sector must contain arbitrarily high spins, including at least all even spins $\ell\le28$, with the ratio of spinning spectral functions bounded from above, exhibiting large spin suppression. The spinning spectrum must also include at least one state satisfying a bound $m^2_{J}<M^2 \frac{( J^2-12) ( J^4 - 32 J^2 +204)}{8 (150 - 43 J^2 + 2 J^4)}$, where $J^2=\ell(\ell+1)$, and $M^2$ is the mass of the heaviest spin 2 state in the spectrum.

Figures (12)

• Figure 1: Deriving $s$- and $su$-EFThedrons from $a$-geometry
• Figure 2: ($\tilde{g}_{3,1}$, $\tilde{g}_{4,0}$) space with the $k=4$ null constraint, geometry vs SDPB
• Figure 3: 1. $(a_{4,2}/a_{4,0},a_{4,1}/a_{4,0})$ polytope and $a_{4,2}=8a_{4,1}$ symmetry plane
• Figure 4: $(\tilde{a}_{3,1},\tilde{a}_{4,1},\tilde{a}_{4,2})$ space intersected with $a_{4,2}=8a_{4,1}$ symmetry plane, and the resulting $(\tilde{a}_{3,1},\tilde{a}_{4,1})$ slice
• Figure 5: Geometry bound and SDPB bound for $(\tilde{g}_{3,1},\tilde{g}_{3,0})$ space. Including higher order couplings and constraints improves the geometry bound. The region with the maximal mismatch is zoomed in for clarity.