Origin of codimension-dependent singular behaviour in triangle correlator integrals

Determine the origin of the observed distinction in singular behaviour of the reduced triangle integral for one-loop cosmological correlators, explaining why intersections of singular and boundary hypersurfaces with codimension zero (as subvarieties of the boundary A+) produce logarithmic divergences, whereas codimension-one intersections yield only a discontinuity in the first derivative of the imaginary part while the real part remains finite and continuous.

Background

In the paper’s Landau and contour-pinching analysis of the reduced triangle integral, the integration cycle Γ has a boundary A+ comprised of hyperplanes T1–T4, while the integrand is singular on L1=0, L2=0, and the quadratic curve Ω=0. The authors classify candidate singularities by how these hypersurfaces intersect on A+ and observe two distinct behaviours: some loci generate true divergences, while others appear only as regular branch points.

Empirically, the distinction correlates with the codimension of the intersection on A+: codimension-zero intersections lead to logarithmic divergences, whereas codimension-one intersections yield non-divergent branch points with a kink in the imaginary part. The authors note that they currently lack a complete understanding of why this codimension rule governs the nature of the singularity.