thus we can remove all $\sigma_n$-dependence in ${\rm SL}(2,{\mathbb C})$-fixed ${\mathrm{d}}{ \mu}_n$ and ${\cal I}_n$. For example, we define the ${\rm SL}(2,{\mathbb C})$-fixed PT factor as (since there are $(n{-}1)!$ $\alpha$’s we can always choose $n$ in the end) \[PT\] [PT]{}((1), , (n[-]{}1), n)=. From now on, we will mostly be using this ${\rm SL}(2,{\mathbb C})$-fixed form of CHY formulas and integrands, and only switch back to the covariant (boldface) form when necessary.

Now we can define our new half integrands in this $\sigma_n\to \infty$ form. Given any labelled tree graph with with points $1,2,\ldots, n{-}1$ (also called $(n{-}1)$-pt Cayley tree graph), we define Calyley function as the product of $n{-}2$ $\frac{1}{\s_i-\s_j}$, one for each edge $(i,j)$ of the tree $(1\leq i<j\leq n-1)$ [^2]. $$\label{C}
C_n(\{i,j\}):=\prod_{a=1}^{n{-}2} \frac 1 {\sigma_{i_a, j_a}}\,,$$ where equivalently we can say that no cycle is formed with these $n{-}2$ pairs $\{i,j\}$. For example, for $n=4$ we can have the following three labelled trees , see figure \[sijk123\] ,where [*e.g.*]{} Cayley function for the first one is $C_4(\{1,2\}, \{2,3\})=\frac 1 {\sigma_{12} \sigma_{23}}$.

For $n=5$ there are 16 labelled trees, and here we give two examples of $C_5$ for the two topologies , see figure \[sijkl123\] C\_5 ({1,2}, 