  SMEFT Wilson coefficients are subject to various positivity bounds in order
to be consistent with the fundamental principles of S-matrix. Previous bounds
on dimension-8 SMEFT operators have been obtained using the positivity part of
UV partial wave unitarity and form a (projective) convex cone. We derive a set
of linear UV unitarity conditions that go beyond positivity and are easy to
implement in an optimization scheme with dispersion relations in a multi-field
EFT. Using Higgs scattering as an example, we demonstrate how to obtain closed
bounds in the space of the three relevant dimension-8 coefficients, making use
of the UV unitarity conditions as well as so-called null constraints that arise
from full crossing symmetry. Specifically, we show that they are bounded by
inequalities schematically going like $C<O\left((4\pi)^2\right)$. We compare
the newly obtained upper bounds with the traditional perturbative unitarity
bounds from within the EFT, and discuss some phenomenological implications of
the two-sided positivity bounds in the context of experimental probes of Vector
Boson Scattering.
