Note on the Labelled tree graphs

TL;DR

This paper explores the structure of Labelled tree graphs in CHY formalism, organizing Feynman diagrams into effective diagrams and generalizing pole-picking methods for CHY-integrands.

Contribution

It introduces a new organization of cubic Feynman diagrams into effective diagrams and extends pole-picking techniques to general CHY-integrands with simple poles.

Findings

Effective Feynman diagrams reveal the pole structure more clearly

Generalized pole-picking applies to a broader class of CHY-integrands

Enhanced understanding of Labelled tree graphs in scalar theories

Abstract

In the CHY-frame for the tree-level amplitudes, the bi-adjoint scalar theory has played a fundamental role because it gives the on-shell Feynman diagrams for all other theories. Recently, an interesting generalization of the bi-adjoint scalar theory has been given in arXiv:1708.08701 by the "Labelled tree graphs", which carries a lot of similarity comparing to the bi-adjoint scalar theory. In this note, we have investigated the Labelled tree graphs from two different angels. In the first part of the note, we have shown that we can organize all cubic Feynman diagrams produces by the Labelled tree graphs to the "effective Feynman diagrams". In the new picture, the pole structure of the whole theory is more manifest. In the second part, we have generalized the action of "picking pole" in the bi-adjoint scalar theory to general CHY-integrands which produce only simple poles.