Into the EFThedron and UV constraints from IR consistency.

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 product moment curves. This makes contact with the well studied bi-variate moment problem, which in various instances has known solutions, generalizing the Hankel matrices into moment matrices. In the infinite derivative limit, this identification leads to a complete description of all boundaries. In the finite case, the boundary structure is more complicated, and we compare the analytic description we obtain to that given by semi-definite programing. 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 mass of spinning spectrum must also allow for at least one state satisfying a bound $m^2_{\ell}

Keywords:

• Space (mathematics)
• Theoretical physics
• Scalar (mathematics)
• Moment problem
• Spin-½
• Moment (mathematics)
• Crossing
• UV completion
• Physics
• Product (mathematics)

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