Massive Inflationary Amplitudes: New Representations and Degenerate Limits

TL;DR

This work introduces a new variable representation for massive inflationary tree amplitudes with bilinear mixings, enabling straightforward folded and partial-energy analyses. It demonstrates that folded (degenerate) kinematics systematically reduce the transcendental weight of the resulting hypergeometric functions, and expresses general tree amplitudes as completely inhomogeneous solutions plus cuts (CIS+cuts) built from massive family trees. The degenerate-star analysis yields compact closed forms in terms of Lauricella and Kampé de Fériet functions, and the results are directly applied to the inflaton bispectrum with single, double, and triple massive exchanges, revealing distinct squeezed-limit signatures. The framework provides efficient computational strategies and highlights deep connections between cosmological collider signals and multivariate hypergeometric structures, with potential extensions to spins, chemical potentials, and loop amplitudes.

Abstract

The particle model building of cosmological collider physics often involves boost-breaking bilinear mixing between a heavy particle and the nearly massless inflaton mode. In cosmological correlators, such a mixing is obtained by taking a folded limit of a generic tree graph, which is a special case of degenerate kinematics. In this work, we continue our exploration of massive inflationary amplitudes with a focus on degenerate kinematics. With a suitable change of variables, we derive new differential equations and full analytical solutions for generic tree graphs, making it trivial to take the folded limit and partial-energy limit at a vertex. Our result shows that folded tree graphs generally involve functions of smaller transcendental weights than their nondegenerate counterparts. In particular, the inflaton bispectrum with triple massive exchanges can be expressed in terms of a trivariate Kampé de Fériet function and simpler hypergeometric functions.

Figures (8)

• Figure 1: The 4-point (left) and 3-point (right) functions with a single massive exchange. The lower graphs show our convention for drawing inflation correlators: 1) All bulk-to-boundary propagators are removed; 2) A bilinear-mixing vertex is denoted by a large crossed circle.
• Figure 2: An illustration of the differential equation (\ref{['eq_DEinUvar']}).
• Figure 3: An illustration of the differential equation (\ref{['eq_DEinUvar']}).
• Figure 4: The momentum configuration of a folded limit (colinear limit) at a folded leaf (Site $i$). Here, we are considering an example with 3 external lines attached to Site $i$ with momenta $\bm k_i$ ($i=1,2,3$). Thus we have $E_i=|\bm k_1|+|\bm k_2|+|\bm k_3|$, $K_i=|\bm k_1+\bm k_2+\bm k_3|$, and $E_i\rightarrow K_i$ in the folded limit.
• Figure 5: The degenerate star graph with $N$ mixed propagators.