Labelled tree graphs, Feynman diagrams and disk integrals

TL;DR

This work introduces Cayley functions as a new class of CHY half-integrands that extend Parke–Taylor factors by associating each to a labelled tree. The authors show that squaring Cayley-half integrands yields sums of cubic Feynman diagrams and can be organized as polytopes whose vertices encode the diagrams, with a graphic rule linking trees to polytopes and a spectrum of polytopes ranging from the associahedron to the permutohedron. They derive a closed-form reduction of any Cayley function to the Kleiss–Kuijf Parke–Taylor basis and construct a new basis consisting of $C^{\rm single}$ and $C^{\rm kernel}$ elements, where $C^{\rm single}$ maps to a single diagram while $C^{\rm kernel}$ maps to zero under CHY with a fixed PT. The paper further develops the connection to disk integrals in open string theory, proposing a leading-alpha-prime structure for these Z-integrals determined by the non-crossing subgraphs, and discusses implications for loop generalizations and deeper geometric interpretations via graph associahedra.

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.

Figures (19)

• Figure 1: Cayley functions for $n=4$
• Figure 2: Two topologies for $n=5$
• Figure 4: $s_{i,j,k}$
• Figure 5: two topologies of subgraphs for $s_{i,j,k,l}$
• Figure 10: Dual graph

Theorems & Definitions (2)

• Theorem 1
• Theorem 2