Generalization of the 1-loop flat-space limit derivation to higher loops/sites

Ascertain whether the dimensional-regularization-based derivation of the flat-space limit for one-loop de Sitter correlators extends straightforwardly to higher loop orders or multiple loop-site diagrams, accounting for the required \(\mathcal{O}(\delta^2)\) and higher corrections to de Sitter mode functions.

Background

The authors derive the flat-space limit for the full 1-loop correlator by relating leading powers of the de Sitter propagators to flat-space propagators, using an expansion of the d-dimensional modes about $\delta=0$.

They caution that this approach relies on commuting a series expansion with loop integrals and only includes $\mathcal{O}(\delta)$ corrections, and it is not clear whether the same logic works when $\mathcal{O}(\delta^2)$ and higher terms are needed at higher loop order or with multiple loop sites.