Softness, polynomial boundedness and amplitudes' positivity

Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences Beijing 100190, China and School of Physical Sciences, University of Chinese Academy of Sciences - No. 19A Yuquan Road Beijing 100049, China

^(a) Present address: School of Physics, Nanjing University - Nanjing, 210093, China; dbai@itp.ac.cn

Received: 25 September 2017
Accepted: 12 December 2017

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 in the IR and could be embedded into UV completions satisfying unitarity, analyticity, crossing symmetry and polynomial boundedness (), with M and N integers, we show that the inequality must hold, where 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.