Analytic derivation of correlator–amplitude equality in the total‑energy limit

Prove analytically that the residue of the one-loop triangle correlator at the total‑energy pole X1+X2+X3=0 equals the off‑shell massless triangle flat‑space loop amplitude I3(P1,P2,P3) expressed via Clausen functions, and clarify the mapping that connects the cosmological correlator’s reduced expression involving Ω(−X3,−X2−X3) to the standard amplitude representation.

Background

The authors show numerically that taking the total‑energy residue of the triangle correlator reproduces the off‑shell massless triangle amplitude I3(P1,P2,P3), where Pi are sums of external four-momenta entering each vertex, and the amplitude is given in terms of Clausen functions with denominator √(−Δ3). They also note that in the correlator, the square roots in the denominator coincide and match √(−Δ3) in the total‑energy limit.

Despite this agreement, the paper states that the analytic origin of this correspondence is unclear. An explicit derivation would elucidate how the correlator’s Whittaker-based contour representation and reduced integrals reorganise into the amplitude’s Clausen-function form, and why the Källén function structure emerges in the residue.