Factorization of glue and mass terms in $\mathrm{SU}(N)$ gauge theories

In this paper we investigate the structure of the glue in Zwanziger’s gauge invariant expansion for the $A^2$-type mass term in Yang-Mills theory. We show how to derive this expansion, in terms of the inverse covariant Laplacian, and extend it to higher orders. In particular, we give an explicit expression, for the first time, for the next to next to leading order term. We further show that the expansion is not unique and give examples of the resulting ambiguity.