the simplest example is the so-called bi-adjoint $\phi^3$ theory [@Cachazo:2013iea]. This is a theory with scalars in the adjoint of two flavor groups, [*e.g.*]{} U(N) $\times$ U(N’), with a cubic vertex $\sim f^{abc} f^{a'b'c'} \phi_{a, a'} \phi_{b, b'} \phi_{c, c'}$. By doing trace-decomposition in both groups, the so-called double-partial amplitude, $m(\alpha|\beta)$ for orderings $\alpha$ and $\beta$, is given by the sum of scalar Feynman diagrams (cubic tree graphs with $n$ external legs) that are compatible with both $\alpha$ and $\beta$: \[mab\] m(|)=(-1)\^[[flip]{}(|)]{}\_[g T() T() ]{} \_[I P(g)]{} 1 [s\_I]{}, where $T(\alpha)$ denotes the set of cubic tree graphs compatible with ordering $\alpha$ (similarly for $T(\beta)$), and for each graph $g$ we have the product of $n{-}3$ propagators labelled by $I$ (the collection of all poles/propagators of a Feynman diagram $g$ is denoted as $P(g)$) [^1].Although this $\phi^3$ theory is simple, it is remarkable that $m(\alpha|\beta)$ is given by the simplest CHY formula, with two PT factors: m(|)=\_n [**PT**]{}() [**PT**]{}(), which is a rather non-trivial mathematical identity first proposed and shown in [@Cachazo:2013iea]. In particular, if we choose $\alpha=\beta$ the CHY formula can be viewed as a map from a half integrand, [**PT**]{}($\alpha$), to the collection of Feynman diagrams that are 