Mode-sum construction of the two-point functions for the Stueckelberg vector fields in the Poincar\'e patch of de Sitter space

We perform canonical quantization of the Stueckelberg Lagrangian for massive vector fields in the conformally flat patch of de Sitter space in the Bunch-Davies vacuum and find their Wightman two-point functions by the mode-sum method. We discuss the zero-mass limit of these two-point functions and their limits where the Stueckelberg parameter $\xi$ tends to zero or infinity. It is shown that our results reproduce the standard flat-space propagator in the appropriate limit. We also point out that the classic work of Allen and Jacobson for the two-point function of the Proca field and a recent work by Tsamis and Woodard for that of the transverse vector field are two limits of our two-point function, one for $\xi \to \infty$ and the other for $\xi \to 0$. Thus, these two works are consistent with each other, contrary to the claim by the latter authors.