  We derive positivity bounds on low energy effective field theories which
admit gapped, analytic, unitary, Lorentz invariant, and possibly non-local UV
completions, by considering 2 to 2 scatterings of Jaffe fields whose
Lehmann-K\"{a}ll\'{e}n spectral density can grow exponentially. Several
properties of S-matrix, such as analyticity properties, are assumed in our
derivation. Interestingly, we find that some of the positivity bounds obtained
in the literature, such as sub-leading order forward-limit bounds, must be
satisfied even when UV completions fall into non-localizable theories in
Jaffe's language, unless momentum space Wightman functions grow too rapidly at
high energy. Under this restriction on the growth rate, such bounds may provide
IR obstructions to analytic, unitary, and Lorentz invariant UV completions.

High Energy Physics - Theory;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology