Geometry of effective field theory positivity cones

Positivity bounds are theoretical constraints on the parameters of physically meaningful models in particle physics. It was observed in [1] that the task of deriving (forward scattering) positivity bounds can be reformulated as the geometric problem of finding the extremal rays of a convex cone $\mathcal{C}_W$ of four-tensors with certain symmetries. More precisely, $\mathcal{C}_W$ consists of all positive semi-definite tensors in $W=\{S\in Sym^2(Sym^2 V^*)\oplus Sym^2(\Lambda^2 V^*) : \tau S=S\}\subset Sym^2 (V^*\otimes V^*)$, where $\tau$ denotes the transposition in the second and fourth tensor factor and $V\cong \mathbb{R}^n$.
In this talk, we will give an idea of what positivity bounds are and why they can be described by extremal elements of $\mathcal{C}_W$. Then we will classify all extremal elements of $\mathcal{C}_W$ in the case where $V$ is at most 3-dimensional. This gives rise to new constraints that improve the known (forward scattering) positivity bounds from the literature. Based on arXiv:2508.18165 (to appear in Annales Henri Poincaré).

[1] Li, Xu, Yang, Zhang and Zhou. (2021). Positivity in Multifield Effective Field Theories. Physical Review Letters. 127. 10.1103/PhysRevLett.127.121601.