Softness, Polynomial Boundedness and Amplitudes' Positivity

TL;DR

This paper establishes bounds relating the infrared and ultraviolet behaviors of massless scattering amplitudes using dispersion relations and positivity bounds, providing constraints on their growth and implications for UV completions.

Contribution

It derives a new inequality linking IR and UV scaling of massless scattering amplitudes, advancing understanding of their behavior under fundamental physical principles.

Findings

Derived inequality: $2\lceil N/2 \rceil \ge M \ge \lceil N/2 \rceil$ for amplitude scaling

Bound on UV growth based on IR behavior of amplitudes

Implications for massless higher spin particles and UV completion

Abstract

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.