Notes on Resonances and Unitarity from Celestial Amplitudes

TL;DR

This paper explores the celestial amplitude framework for the $O(N)$ sigma model in three dimensions, revealing how resonance phenomena influence unitarity, crossing symmetry, and factorization in a UV-complete setting.

Contribution

It introduces a celestial amplitude analysis of the $O(N)$ sigma model, highlighting new IR/UV properties, resonance effects, and a novel optical theorem for unitarity in this context.

Findings

Resonances are crucial for four-point function factorization.

Celestial correlators exhibit IR/UV softness and metastable states.

A form of the optical theorem relates imaginary parts to resonances.

Abstract

We study the celestial description of the $O(N)$ sigma model in the large $N$ limit as introduced by Coleman, Jackiw and Politzer. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand unitarity, we propose a form of the optical theorem that controls the imaginary part of the correlator based solely on the presence of these resonances. We also explicitly analyze the conformal block expansions and factorization of four-point functions into three-point functions. We find that summing over resonances is key for these factorization properties to hold. We end with some topics for future study.