Understanding zeros and splittings of ordered tree amplitudes via Feynman diagrams

In this paper, we propose new understandings for recently discovered hidden zeros and novel splittings, by utilizing Feynman diagrams. The study focus on ordered tree level amplitudes of three theories, which are $\mathrm{T}\mathrm{r}({\phi }^3)$, Yang-Mills, and non-linear sigma model. We find three universal ways of cutting Feynman diagrams, which are valid for any diagram, allowing us to separate a full amplitude into two/three pieces. As will be shown, the first type of cuttings leads to hidden zeros, the second one gives rise to $2$-splits, while the third one corresponds to $3$-splits called smooth splittings. Throughout this work, we frequently use the helpful auxiliary technic of thinking the resulting pieces as in orthogonal spaces. However, final results are independent of this auxiliary picture.