Combinatorial origins and generalizations of tree-level scattering amplitudes: triangulations, associahedra and the SU(n) root lattice

by Nick Early (MPI for Physics, Munich)

Main/0-174 - Auditorium (Main) (MPI for Physics, Munich)

Main/0-174 - Auditorium (Main)

MPI for Physics, Munich


Abstract: We explain how the tree-level Feynman diagram expansion of the scattering amplitude for the cubic scalar is carved into the local structure of an integer lattice inside $\mathbb{Z}^{\times n}$, the root lattice for the group SU(n). This observation has several immediate consequences: first, it leads to a recursion relation which constructs the set of Feynman diagrams (planar trees with cyclically labeled leaves) by triangulating a certain lattice polytope, the root polytope. Second, it suggests a generalization of the worldsheet associahedron which has a rich combinatorial structure, as well as an unexpected generalization of the SU(n) root lattice and polytope. The usual associahedron is known to govern tree-level Feynman diagrams for the cubic scalar theory for a fixed planar order. Do there exist, in parallel, generalized scattering amplitudes, and if so what are the physical lessons that we can extract from them? This is based on 2106.07142 and draws on joint works with Cachazo, Guevara and Mizera (1903.08904, 2010.09708).