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Weights and recursion relations for $ϕ^p$ tree amplitudes from the positive geometry

Recently, the accordiohedron in kinematic space was proposed as the positive geometry for planar tree-level scattering amplitudes in the $ϕ^p$ theory \cite{Raman:2019utu}. The scattering amplitudes are given as a weighted sum over canonical forms of some accordiohedra with appropriate weights. These weights were determined by demanding that the weighted sum corresponds to the scattering amplitudes. It means that we need additional data from the quantum field theory to compute amplitudes from the geometry. It has been an important problem whether scattering amplitudes are completely obtained from only the geometry even in this $ϕ^p$ theory. In this paper, we show that these weights are completely determined by the factorization property of the accordiohedron. It means that the geometry of the accordiohedron is enough to determine these weights. In addition to this, we study one-parameter recursion relations for the $ϕ^p$ amplitudes. The one-parameter "BCFW"-like recursion relation for the $ϕ^3$ amplitudes was obtained from the triangulation of the ABHY-associahedron \cite{Arkani-Hamed:2017tmz}. After this, a new recursion relation was proposed from the projecting triangulation of the generalized ABHY-associahedron in \cite{Arkani-Hamed:2019vag, Yang:2019esm}. We generalize these one-parameter recursion relations to the $ϕ^p$ amplitudes and interpret as triangulations of the accordiohedra.