Kinematic Flow for Banana Loops and Unparticles

TL;DR

This paper extends kinematic flow to loop-level cosmological correlators involving banana loops and unparticles, providing a systematic framework with new combinatorial rules and a basis of master integrals.

Contribution

It introduces a novel kinematic flow framework for unparticle exchanges in cosmology, with a basis constructed from graph tubings and new combinatorial rules.

Findings

Correlators are described by a finite set of master integrals obeying differential equations.

The basis is constructed from tubings of marked graphs with nested tubes and arborescence ordering.

New combinatorial rules induce richer mixing and introduce new kinematic letters.

Abstract

We extend kinematic flow to momentum-integrated loop-level cosmological correlators, focusing on banana loops of conformally coupled scalars in power-law cosmologies and, in de Sitter, on arbitrary mixtures of massless and conformally coupled scalars. Exploiting their dual description as tree-level exchanges of unparticles, we show that the associated correlators are described by a finite set of master integrals obeying a first-order system of differential equations. The corresponding basis is constructed from tubings of marked graphs and is distinguished by the appearance of nested tubes and an arborescence ordering of the vertices. We derive the connection matrices from four combinatorial rules -- activation, merger, swap, and copy. The last two are unique to unparticle exchanges: they induce richer mixing among basis functions and introduce new kinematic letters. Our framework extends systematically to arbitrarily complicated configurations, including necklace diagrams, and establishes unparticle exchange as a distinct class of kinematic flow.