  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.
