Labelled tree graphs, Feynman diagrams and disk integrals

TL;DR

This paper introduces Cayley functions, a new class of half integrands in CHY formulas linked to labelled tree graphs, providing combinatorial and algebraic tools to analyze Feynman diagrams and disk integrals in string theory.

Contribution

It defines Cayley functions as a new basis for half integrands, linking tree graphs to Feynman diagrams and disk integrals, with explicit reduction formulas and combinatorial polytopes.

Findings

Cayley functions generalize Parke-Taylor factors and form a new basis.

A simple graphic rule derives polytopes from labelled trees.

Cayley functions simplify the computation of Feynman diagrams.

Abstract

In this note, we introduce and study a new class of "half integrands" in Cachazo-He-Yuan (CHY) formula, which naturally generalize the so-called Parke-Taylor factors; these are dubbed Cayley functions as each of them corresponds to a labelled tree graph. The CHY formula with a Cayley function squared gives a sum of Feynman diagrams, and we represent it by a combinatoric polytope whose vertices correspond to Feynman diagrams. We provide a simple graphic rule to derive the polytope from a labelled tree graph, and classify such polytopes ranging from the associahedron to the permutohedron. Furthermore, we study the linear space of such half integrands and find (1) a nice formula reducing any Cayley function to a sum of Parke-Taylor factors in the Kleiss-Kuijf basis (2) a set of Cayley functions as a new basis of the space; each element has the remarkable property that its CHY formula with a given Parke-Taylor factor gives either a single Feynman diagram or zero. We also briefly discuss applications of Cayley functions and the new basis in certain disk integrals of superstring theory.