that we fix the $\rm{SL}(2,\mathds{C})$ redundancy by deleting three ${\mathrm{d}}\sigma$’s and three delta-functions ([*e.g.*]{} both chosen to be $i,j,k$) with a compensation factor $(\sigma_{i,j}\sigma_{j,k}\sigma_{k,i})^2$.

We will refer to as CHY formula for amplitudes in a given theory, where $\mathcal{\bf I}_n$ is the “CHY integrand" of the theory that can generally also depends on momenta and polarizations. Note that for to be well defined, the CHY integrand must transform covariantly, with opposite weight as ${\mathrm{d}}{\bm \mu}$, under a $\rm{SL}(2,\mathds{C})$ transformation (here $\alpha \delta-\beta\gamma=1$): \[trans\] \_a : \_n \_[a=1]{}\^n(\_a +)\^[-4]{} \_n \_n \_[a=1]{}\^n(\_a +)\^4[**I**]{}\_n({\_a}), and we will refer to this as the fact that ${\bf I}_n$ has weight 4. For most theories that admit CHY representations, the CHY integrand factorizes into two parts $\mathcal{\bf I}_n=\mathcal{\bf I}_n^{(L)}\,\mathcal{\bf I}_n^{(R)}$ where each of them transforms as in with weight 2 and we will refer to ${\bf I}_n^{(L)}$ and ${\bf I}_n^{(R)}$ as “half integrands".

The simplest function with this transformation property is probably the so-called [*Parke-Taylor*]{} (PT) factor. Given an ordering of $n$ labels, $\alpha:=(\alpha(1), \alpha(2), \cdots, \alpha(n))$ we define $$\label{pt}
{\rm{\bf PT}}(\alpha):=\frac{1}{\sigma_{\alpha(1),\alpha(2)}\sigma_{\alpha(2),\alpha(3)}\cdot \cdot \cdot \sigma_{\alpha(n),\alpha(1)}}\,.$$ It turns out that such Parke-Taylor factors play an important role in CHY formula for various theories (with ordering), and 