SMEFTs living on the edge: determining the UV theories from positivity and extremality

TL;DR

This work develops a geometric framework to address the SMEFT inverse problem using dim-8 positivity bounds. By treating the allowed Wilson-coefficient space as a convex cone generated by UV-induced vectors, it connects extremal rays to unique UV completions and shows that SMEFTs on or near the boundary can reveal UV particle content with minimal degeneracy. The extremal-positivity approach yields sharp, often model-independent exclusion limits on UV states and clarifies when a dim-8 measurement can uniquely reconstruct the UV theory. It also extends to loop-level completions, demonstrating how loop generators decompose into convex combinations of tree-level generators and how this affects degeneracy. Overall, the paper argues that dim-8 SMEFT encodes substantial information about UV physics, offering a principled path toward reverse engineering the UV from low-energy data, while highlighting practical challenges for collider phenomenology.

Abstract

We study the "inverse problem" in the context of the Standard Model Effective Field Theory (SMEFT): how and to what extend can one reconstruct the UV theory, given the measured values of the operator coefficients in the IR? The main obstacle of this problem is the degeneracies in the space of coefficients: a given SMEFT truncated at a finite dimension can be mapped to infinitely many UV theories. We discuss these degeneracies at the dimension-8 level, and show that positivity bounds play a crucial role in the inverse problem. In particular, the degeneracies either vanish or become significantly limited for SMEFTs that live on or close to the positivity bounds. The UV particles of these SMEFTs, and their properties such as spin, charge, other quantum numbers, and interactions with the SM particles, can often be uniquely determined, assuming dimension-8 coefficients are measured. The allowed region for SMEFTs, which forms a convex cone, can be systematically constructed by enumerating its generators. We show that a geometric notion, extremality, conveniently connects the positivity problem with the inverse problem. We discuss the implications of a SMEFT living on an extremal ray, on a $k$-face, and on the vertex of the positive cone. We also show that the information of the dimension-8 coefficients can be used to set exclusion limits on all individual UV states that interact with the SM, independent of specific model assumptions. Our results indicate that the dimension-8 operators encode much more information about the UV than one would naively expect, which can be used to reverse engineer the UV physics from the SMEFT.

Figures (20)

• Figure 1: Left: the positivity cone in the $\left(C_{1}^{8}, C_{2}^{8}, C_{3}^{8}\right)$ space, which is a pyramid. The edge vectors are labeled in different colors. Right: a cross section of this cone, which is a quadrilateral. The cross section is obtained with a hyperplane $C_{1}+C_{3}=\sqrt{2}$ (also shown in the left figure) intersecting the cone.
• Figure 2: The degeneracy $\Delta$ on a cross section of the positivity pyramid, defined by $C_{1}+C_{3}=\sqrt{2}$. See Eq. (\ref{['eq:2.21']}) for definition of $\Delta$.
• Figure 3: Positivity cone for an EFT with 2-scalars under $\mathrm{SO}(2)$. The arrows represent the generator vectors.
• Figure 4: Positivity cone of an EFT with 2 scalars and the $Z_{2}$ symmetry. We show a cross section of the cone, in two different angles. The axes are $C_{x}=\frac{\sqrt{3}\left(C_{1}-C_{3}\right)}{\sqrt{2}\left(C_{1}+C_{3}+C_{4}\right)}, C_{y}=\frac{\sqrt{3} C_{2}}{C_{1}+C_{3}+C_{4}}$, and $C_{z}=-\frac{C_{1}+C_{3}-2 C_{4}}{\sqrt{C_{1}+C_{3}+C_{4}}} .$ Generators are labeled by the green and blue dots, and the red circle.
• Figure 5: The positivity cone for 4-Higgs operators, with the corresponding generators. Colors represent different irreps. They are only labeled with $\mathrm{SU}(2)$ irrep $(1,3)$ and the exchange symmetry (S,A). The cone is shown in the left plot, while the right plot shows a slice of the cone. The latter can be thought of as intersecting the cone with a hyperplane $2 C_{1}+3 C_{2}+C_{3}=1$.