On the Numerical Integration of One-Loop Cosmological Collider Signals

Abstract

In a broad class of inflationary models, the leading non-Gaussian bispectrum arises from one-loop rather than tree-level processes. However, realistic one-loop contributions remain largely unexplored for phenomenologically relevant masses beyond the simplest bubble diagrams. This is insufficient for observational purposes because, for generic masses and couplings, triangle contributions need not be suppressed relative to their bubble counterparts. We present a numerical method for evaluating scalar one-loop in-in diagrams contributing to the inflationary bispectrum at fully general external momenta and masses. First, the Witten-Feynman parameterization reduces de Sitter loop integrals to generalized Euler-Mellin integrals governed by Symanzik graph polynomials. Unfortunately, the exponents of the polynomials in the integrand become complex for sufficiently heavy masses, leading to sign problems when evaluating the integral using standard numerical techniques. To remedy this problem, we introduce the reduced Schwinger method, which evaluates a highly oscillatory subintegral analytically to obtain a Gauss hypergeometric kernel, leaving the remaining integrals to standard numerical quadrature. We validate the algorithm by reproducing known analytic results for the tree-level bispectrum in terms of ${}_3F_2$ functions, and then apply it to the one-loop bubble and triangle contributions at general kinematics. This yields the first direct numerical evaluation of the complete one-loop bispectrum, valid across the full kinematic range. Using these numerical results, we construct bispectrum templates and compare them with CMB data.