States and IR divergences in factorization algebras

TL;DR

This paper explores different states in factorization algebras in field theory, introduces a new compactification state, and demonstrates the equivalence of these states, providing a method to handle infrared divergences.

Contribution

It defines the compactification state within factorization algebras and establishes the equivalence of natural, compactification, and Schwartz states, along with a method for infrared divergence management.

Findings

Defined the compactification state in factorization algebras.

Proved the equivalence of three states in massive and massless theories.

Provided a method to handle infrared divergences in massless theories.

Abstract

In field theory, one can consider a variety of states. Within the framework of factorization algebras, one typically works with the natural augmentation state $\langle-\rangle_{\rm aug}$. In physics, however, other states arise naturally, such as the compactification state $\langle-\rangle_{\rm cptf}$ or the Schwartz state $\langle-\rangle_{\rm Sch}$, defined by imposing Schwartz boundary conditions. At first sight, the relation among these three states is not obvious. This paper gives a definition of the compactification state in factorization algebras and provides a method for handling infrared divergences in the massless theory. We then prove that the three states are equivalent in both the massive and massless cases.