Abstract
We build upon the prior works of [1-3] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, where the interaction is given by $\lambda_3\phi^{3}\ +\lambda_4 \phi^{4}$, we show that a specific convex realization of a simple polytope known as the accordiohedron in kinematic space is the positive geometry for this theory. As in the previous cases, there is a unique planar scattering form in kinematic space, associated to each positive geometry which yields planar scattering amplitudes.
Highlights
In [2, 3] the authors tried to locate the positive geometries associated to tree-level planar amplitudes in massless scalar field theories with φp (p ≥ 4) interactions
Λ4φ4 potential and show that the convex realizations of the combinatorial polytopes which belong to the accordiohedron family are the positive geometries for corresponding scattering amplitudes
In [1] and [2], the Feynman diagrams in φ3 theory and in φ4 theory were associated with vertices of the simple polytopes, associahedron and Stokes polytope respectively
Summary
In [1] and [2], the Feynman diagrams in φ3 theory and in φ4 theory were associated with vertices of the simple polytopes, associahedron and Stokes polytope respectively. We define D-accordiohedron to be the simple polytope AC(D) whose vertices are all the D-accordion dissections [11]. Unless otherwise stated we don’t make a distinction between the diagonal (i, j) and the propagator Xij. we can define the canonical projective form associated with the accordiohedron in terms of these planar variables. This corresponds to the channel giving the first term in the foregoing equation For this simple case, the only two compatible dissections of this form are (1 , 3 ) and (2 , 5 ). We present a more nontrivial 6 point amplitude that has two cubic and one quartic vertex
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