---
author:
- Xiangrui Gao
- ', Song He'
- ',  Yong Zhang'
bibliography:
- 'mybibliography.bib'
title: |
    Labelled tree graphs, Feynman diagrams\
    and disk integrals
---

Invitation: a new class of CHY half integrands {#invitation}
==============================================

In 2013, F. Cachazo, E. Y. Yuan and one of the authors found a new formulation for tree-level S-matrices for a large variety of massless theories in arbitrary dimensions [@Cachazo:2013iaa; @Cachazo:2013gna; @Cachazo:2013hca; @Cachazo:2013iea] (for extension to more theories, see [*i.e.*]{} [@Cachazo:2014nsa; @Cachazo:2014xea; @He:2016iqi; @He:2016dol; @Cachazo:2016njl]). The key ingredient of the formulation is the so-called [*scattering equations*]{}, which link kinematics of $n$ massless particles to points in the moduli space of $n$-punctured Riemann spheres, ${\cal M}_{0,n}$ [@Cachazo:2013iaa; @Cachazo:2013gna]:

$$\label{se}
  \sum_{b\neq a}\frac{k_a\cdot k_b}{\sigma_a-\sigma_b}=0,\quad a\in\{1,2,...,n\},$$

where $\sigma_a$ denotes the position of the $a^{th}$ puncture on the Riemann sphere. The tree-level S-matrix can be compactly formulated as an integral over ${\cal M}_{0,n}$ localized on solutions of the scattering equations  [@Cachazo:2013hca]: \[sm\] [M]{}\^[tree]{}\_n ({k, })=\_n[**I**]{}\_n ({k, , }), \_n:=\_\^n ( \_a(\_[ba]{} ))(\_[i,j]{}\_[j,k]{}\_[k,i]{})\^2, where $\sigma_{a,b}:=\sigma_a-\sigma_b$ and we have included delta functions imposing in the measure ${\mathrm{d}}{\bm \mu}$. Note that both the moduli space ${\cal M}_{0,n}$ and the scattering equations have an $\rm{SL}(2,\mathds{C})$ redundancy; our definition of ${\mathrm{d}}{\bm \mu}$ means 