Abstract
Tree-level Feynman diagrams in a cubic scalar theory can be given a metric such that each edge has a length. The space of metric trees is made out of orthants joined where a tree degenerates. Here we restrict to planar trees since each degeneration of a tree leads to a single planar neighbor. Amplitudes are computed as an integral over the space of metrics where edge lengths are Schwinger parameters. In this work we propose that a natural generalization of Feynman diagrams is provided by what are known as metric tree arrangements. These are collections of metric trees subject to a compatibility condition on the metrics. We introduce the notion of planar col lections of Feynman diagrams and argue that using planarity one can generate all planar collections starting from any one. Moreover, we identify a canonical initial collection for all n. Generalized k = 3 biadjoint amplitudes, introduced by Early, Guevara, Mizera, and one of the authors, are easily computed as an integral over the space of metrics of planar collections of Feynman diagrams.
Highlights
In physical applications only the contribution of a tree T to an amplitude is of importance
In this work we propose that a natural generalization of Feynman diagrams is provided by what are known as metric tree arrangements
We introduce the notion of planar collections of Feynman diagrams and argue that using planarity one can generate all planar collections starting from any one
Summary
The space of metric trees with n leaves is closely related to the space of tropical two-planes, i.e., Trop G(2, n) [9, 10]. In order to motivate this, one can start with Trop G(3, 5) which is isomorphic to Trop G(2, 5) This means that whatever generates the tropical Plücker vectors πabc for Trop G(3, 5) must somehow be related to the metric trees governing Trop G(2, 5). We show the resulting collection in figure 2 This is one the simplest examples of what is known as an abstract tree arrangement in the mathematical literature. In the first tree which has leaves {2, 3, 4, 5} the internal edge’s length is given by f (1) = 1 4 d(315) + d(314) + d(215) + d(214) − 2d(415) − 2d(213). D(bac) = d(abc) = d(acb) and it motivates the introduction of a completely symmetric object πabc := d(bac) This gives the dual Plücker vector in Trop G(3, 5). In appendix A we give more details on the definition and relation to Trop G(3, n)
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