Convex Geometry Perspective to the (Standard Model) Effective Field Theory Space

TL;DR

The paper casts SMEFT coefficient space, specifically the second s-derivatives of forward 2→2 amplitudes, as a convex cone whose extremal rays correspond to irreducible SM representations and encode how heavy UV states can generate the EFT. Using symmetry to build projectors and an extremal representation, the authors show tree-level UV completions arise from exchanges in a single irrep, and any EFT point is a positive sum of such ERs, revealing UV content from low-energy data. Through concrete examples (Higgs, W-boson, fermions), they derive new bounds that can be stronger than elastic positivity and, in some cases, originate from scattering of entangled states. The work highlights the special role of dim-8 operators in reverse-engineering UV physics and provides a practical geometric toolkit to constrain and potentially identify UV states from SMEFT measurements.

Abstract

We present a convex geometry perspective to the Effective Field Theory (EFT) parameter space. We show that the second $s$ derivatives of the forward EFT amplitudes form a convex cone, whose extremal rays are closely connected with states in the UV theory. For tree-level UV-completions, these rays are simply theories with all UV particles living in at most one irreducible representation of the symmetries of the theory. In addition, all the extremal rays are determined by the symmetries and can be systematically identified via group theoretical considerations. The implications are twofold. First, geometric information encoded in the EFT space can help reconstruct the UV-completion. In particular, we will show that the dim-8 operators are important in reverse-engineering the UV physics from the Standard Model EFT, and thus deserve more theoretical and experimental investigations. Second, theoretical bounds on the Wilson coefficients can be obtained by identifying the boundaries of the cone and are in general stronger than the current positivity bounds. We show explicit examples of these new bounds, and demonstrate that they originate from the scattering amplitudes corresponding to entangled states.

Figures (2)

• Figure 1: A cross section of the Higgs triangular cone with the PERs, taken to be perpendicular to the direction $E_{1}+E_{1S}+E_{1A}$.
• Figure 2: A slice of the 4D fermion cone, taken to be perpendicular to the direction $(1,1,0,1)$. The three axes are taken to be $(1,-1,0,0)$, $(0,0,1,0)$ and $(-1,-1,0,2)$.