States and IR divergences in factorization algebras

In field theory, one can consider a variety of states. Within the framework of factorization algebras, one typically works with the natural augmentation state $⟨-{⟩}_{\mathrm{a}\mathrm{u}\mathrm{g}}$. In physics, however, other states arise naturally, such as the compactification state $⟨-{⟩}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{f}}$ or the Schwartz state $⟨-{⟩}_{\mathrm{S}\mathrm{c}\mathrm{h}}$, 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.