Large-distance behavior of the massless vector two-point function in de Sitter spacetime

We study the long-distance behaviour of the massless vector propagator in $n$-dimensional de Sitter spacetime, where $n \geq 4$. Specifically, we consider the massless limit of the vector propagator in the Stueckelberg theory, which is an extension of Proca theory, with an additional gauge-fixing term. We work to leading order in the de Sitter-invariant distance $Z$ to show that, in the large $Z$ limit, this propagator tends to a gauge dependent constant, where the gauge worked in is described by the Stueckelberg parameter $\xi$. In the Landau gauge, where $\xi=0$, this constant is found to be 0. This result is in agreement with the 4 dimensional case discussed by Youssef.