Softness, Polynomial Boundedness and Amplitudes' Positivity

In this note, we study the connections between infrared (IR) and ultraviolet (UV) behaviors of scattering amplitudes of massless channels by exploiting dispersion relations and positivity bounds. Given forward scattering amplitudes, which scale as $\mathcal{A}(s)\sim s^M$ in the IR ($s\to0$) and could be embedded into UV completions satisfying unitarity, analyticity, crossing symmetry and polynomial boundedness $|\mathcal{A}(s)|< c\, |s|^N$ ($|s|\to\infty$), with $M$ and $N$ integers, we show that the inequality $2\ceil*{\frac{N}{2}}\ge M \ge \ceil*{\frac{N}{2}}$ must hold, where $\ceil*{x}$ is the smallest integer greater than or equal to $x$. One immediate consequence of the above inequality is the bound on the UV growth of scattering amplitudes in terms of their IR behaviors. Our results could be useful in studies of massless higher spin particles, as well as the program of UV improvement and weakly-coupled UV completion.