Permutation in the CHY-Formulation

Authors: Rijun Huang, Fei Teng and Bo Feng

Preprint Number: UUITP-03/18

Abstract: 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 introduced the cycle representation of permutation in this paper for the understanding of cubic scalar amplitude. We showed 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 showed that, given a set of Feynman diagrams, the cycle representation of corresponding PT-factor can be recursively constructed from three point ones. 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.