The one-loop setting sun diagram with complex external momenta

We revisit the issue of analytically continuing Feynman integrals from Euclidean to Minkowski signature, allowing for generic complex momenta. Although this is well-known in terms of the Källén–Lehmann representation, we consider potential alternative takes on the same problem and discuss how these are not necessarily equivalent to the Källén–Lehmann integral outcome. We present our analysis for a simple enough case—the setting sun diagram in $d=2$ with a real mass at one-loop—but already with an eye out to the more general case with complex masses which will further complicate matters.