Compatible cycles and CHY integrals

• Main Authors: Freddy Cachazo, Karen Yeats, Samuel Yusim
• Format: Article
• Language: English
• Published: SpringerOpen 2019-12-01
• Series: Journal of High Energy Physics
• Subjects: Scattering Amplitudes; Field Theories in Higher Dimensions
• Online Access: https://doi.org/10.1007/JHEP12(2019)105

Abstract The CHY construction naturally associates a vector in ℝ(n−3)! to every 2- regular graph with n vertices. Partial amplitudes in the biadjoint scalar theory are given by the inner product of vectors associated with a pair of cycles. In this work we study the problem of extending the computation to pairs of arbitrary 2-regular graphs. This requires the construction of compatible cycles, i.e. cycles such that their union with a 2-regular graph admits a Hamiltonian decomposition. We prove that there are at least (n − 2)!/4 such cycles for any 2-regular graph. We also find a connection to breakpoint graphs when the initial 2-regular graph only has double edges. We end with a comparison of the lower bound on the number of randomly selected cycles needed to generate a basis of ℝ(n−3)!, using the super Catalan numbers and our lower bound for compatible cycles.