2020
DOI: 10.1007/jhep11(2020)164
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Generalized planar Feynman diagrams: collections

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 metr… Show more

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Cited by 26 publications
(53 citation statements)
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“…The notion of degenerate scattering diagrams has applications beyond planar gauge theories, specifically higher loop integrands of φ 3 . However, it is instead the cluster polytope picture that is more interesting for studying higher loop integrands of φ 3 [128,129] and generalized scattering amplitudes [130][131][132][133][134][135][136]. Both the higher loop integrands of φ 3 and generalized scattering amplitudes can be identified with the canonical rational function of the (degenerate) cluster polytopes discussed in section 4 [111].…”
Section: Jhep07(2021)049mentioning
confidence: 99%
“…The notion of degenerate scattering diagrams has applications beyond planar gauge theories, specifically higher loop integrands of φ 3 . However, it is instead the cluster polytope picture that is more interesting for studying higher loop integrands of φ 3 [128,129] and generalized scattering amplitudes [130][131][132][133][134][135][136]. Both the higher loop integrands of φ 3 and generalized scattering amplitudes can be identified with the canonical rational function of the (degenerate) cluster polytopes discussed in section 4 [111].…”
Section: Jhep07(2021)049mentioning
confidence: 99%
“…Recently, in [46][47][48][49] it has been proposed that both of these obstructions can be overcome by considering the tropical version of the configuration space Gr(k, n) [50,51], or equivalently its dual geometric object [52]. The relevance of tropical Grassmannians for scattering processes was first established in the context of tree-level amplitudes of generalized biadjoint scalar theory [53,54], see also [55][56][57][58][59][60][61] and references therein for recent progress in this direction, as well as [62][63][64][65][66][67][68] for work on further connections between (duals of) tropical Grassmannians, cluster algebras, and scattering amplitudes. In essence, the tropical version of the configuration space is obtained by replacing addition with the minimum and multiplication by addition in the polynomials parameterising (the totally positive) Gr(k, n).…”
Section: Introductionmentioning
confidence: 99%
“…Similar analysis can be carried out for k > 2. We have explicitly evaluated the reduced determinant for all (k, n) up to and including (5,10) finding the expected result (5.1).…”
Section: Evaluation Of the Reduced Determinantmentioning
confidence: 61%
“…An expansion of m (k) n (I, I) in terms of generalized planar Feynman diagrams was introduced by Borges and one of the authors for k = 3 [5], building on the beautiful work of Herrmann, Jensen, Joswig and Sturmfels [28], and later extended to arrays of Feynman diagrams to all k by Cachazo, Guevara, Umbert and Zhang (CGUZ) [8].…”
Section: Introductionmentioning
confidence: 99%