A note on tree factorization and no particle production

TL;DR

This paper proves a fundamental factorization property of tree diagrams in quantum field theory, applies it to scalar field models, and clarifies the conditions leading to specific integrable models.

Contribution

It establishes the factorization of the generating functional for connected tree diagrams as a Legendre transform of the action and links no particle production to known integrable models.

Findings

Factorization of the generating functional as a Legendre transform.

Application to 2D scalar field theories.

Identification of models with no particle production as sine-Gordon or Bullough-Dodd.

Abstract

We prove factorization of the generating functional of connected tree diagrams by exploring that it is the Legendre transform of the action. This theorem is then applied to the example of a massive real scalar field theory in 2D. In the process we streamline the proof that the assumption of no particle production leads to either the sin(h)-Gordon or the Bullough-Dodd model.