Positive moments for scattering amplitudes

We find the complete set of conditions satisfied by forward $2\to2$ scattering amplitude in unitarity and causal theories. These are based on an infinite energy dependent quantities -- arcs which dispersively expressed as moments a positive measure defined at (arbitrarily) higher energies. identify optimal finite subsets constraints, suitable to bound Effective Field Theories (EFTs), any order expansion. At tree-level one-to-one correspondence with Wilson coefficients. establish under this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss range validity both couplings energy. also extend our results case small but finite~$t$. A consequence study is that some regime grows faster than $E^6$ cannot be UV-completed.