Permutation in the CHY formulation

• Main Authors: Rijun Huang, Fei Teng, Bo Feng
• Format: Article
• Language: English
• Published: Elsevier 2018-07-01
• Series: Nuclear Physics B
• Online Access: http://www.sciencedirect.com/science/article/pii/S055032131830138X

The CHY-integrand of bi-adjoint cubic scalar theory is a product of two PT-factors. This pair of PT-factors can be interpreted as defining a permutation. We introduce the cycle representation of permutation in this paper for the understanding of cubic scalar amplitude. We show that, given a permutation related to the pair of PT-factors, the pole and vertex information of Feynman diagrams of corresponding CHY-integrand is completely characterized by the cycle representation of permutation. Inversely, we also show that, given a set of Feynman diagrams, the cycle representation of corresponding PT-factor can be recursively constructed. In this sense, there exists a deep connection between cycles of a permutation and amplitude. Based on these results, we have investigated the relations among different independent pairs of PT-factors in the context of cycle representation as well as the multiplication of cross-ratio factors.