Likelihood degenerations

Agostini, Daniele; Brysiewicz, Taylor; Fevola, Claudia; Kühne, Lukas; Sturmfels, Bernd; Telen, Simon

doi:10.1016/j.aim.2023.108863

Cited by 12 publications

(16 citation statements)

References 15 publications

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“…Prominent among these are the exploration of generalized soft theorems [29,37], the discernment of the semi-locally triple splitting amplitudes [38], and the unveiling of certain factorization channels [39,40]. Furthermore, comprehensive characterizations of the scattering equations have been presented in [34,[41][42][43][44][45] within the original CEGM integrals in configuration spaces over - k 1 . Delving into the intricate relationships between the poles and these inherent properties of the amplitudes promises to be a captivating avenue of research.…”

Section: Discussionmentioning

confidence: 99%

Planar matrices and arrays of Feynman diagrams: poles for higher k

Guevara,

Zhang

2024

Commun. Theor. Phys.

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Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enables the computation of biadjoint amplitudes m^{(k)}_n for k>2 . In this follow-up work we investigate the poles of m^{(k)}_n from the perspective of such arrays. For general k we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude based solely on the knowledge of poles, whose number is drastically less than the number of full arrays. As an example we first provide all the poles for the cases (k,n)=(3,7),(3,8),(3,9),(3,10),(4,8) and (4,9) in terms of their planar arrays of degenerate Feynman diagrams. We then implement a simple compatibility criteria together with an addition operation between arrays, and recover the full collections/arrays for such cases. Along the way we implement hard and soft kinematical limits, which provide a map between poles in kinematic space and their combinatoric arrays. We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in (k,n) and (n-k,n). We also outline the relation to boundary maps of the hypersimplex \Delta_{k,n} and rays in the tropical Grassmannian \textrm{Tr}(k,n).

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Section: Discussionmentioning

confidence: 99%

Planar matrices and arrays of Feynman diagrams: poles for higher k

Guevara,

Zhang

2024

Commun. Theor. Phys.

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“…One can then write a k = 3 CHY formula as a product over n k = 2 CHY integrals linked by the 'compatibility constraints' imposing that the absolute value of |jk| (i) , |ij| (k) , |ik| ( j) all be equal. Owing to the techniques developed in [11,[27][28][29][30], many non-trivial k = 3, 4 CHY formulas have been verified to match the partial amplitudes obtained by using planar collections or matrices of Feynman diagrams, but general analysis such as we present here is still needed to understand the most general cases.…”

Section: Future Directionsmentioning

confidence: 99%

Planar matrices and arrays of Feynman diagrams

Cachazo,

Guevara,

Umbert

et al. 2024

Commun. Theor. Phys.

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Very recently planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes. Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition. In this work, we introduce planar matrices of Feynman diagrams as the objects that compute k=4 biadjoint amplitudes. These are symmetric matrices of metric trees satisfying compatibility conditions. We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections. As applications of the first, we find all 693, 13 612, and 346 710 collections for (k,n)=(3,7), (3,8), and (3,9) respectively. As applications of the second kind, we find all 90 608 and 30 659 424 planar matrices that compute (k,n)=(4,8) and (4,9) biadjoint amplitudes respectively. As an example of the evaluation of matrices of Feynman diagrams, we present the complete form of the (4,8) and (4,9) biadjoint amplitudes. We also start the study of higher dimensional arrays of Feynman diagrams, including the combinatorial version of the duality between (k,n) and (n-k,n) objects.

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“…An equivalent definition which parallels more closely that of generalized Feynman diagrams is the following. 1) , σ (2) , . .…”

Section: Generalized Color Orderingsmentioning

confidence: 99%

“…M n ({k i , a i , ãi }) = α,β∈Sn/Zn tr (T a α(1) T a α (2) • • • T a α(n) ) × tr T ãβ (1) T ãβ (2) • • • T ãβ(n) m n (α, β).…”

Section: Introductionmentioning

confidence: 99%

“…We first concentrate on k = 3, and define a k = 3 color ordering as a collection of n, k = 2 orderings on n − 1 labels which can be derived from a generic arrangement of projective lines on RP 2 . For example, Σ := σ (1) , σ (2) , σ (3) , σ (4) , σ (5) = ((2345), ( 1345), ( 1245), ( 1235), ( 1234)) (1.3) is one of the twelve possible color orderings for (k, n) = (3, 5) amplitudes.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

Cachazo,

Early,

Zhang

2024

SIGMA

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The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde N}\big)$. Amplitudes are ''color'' decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of ''local planarity'' into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$. We present results for $n\leq 9$ generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the ''colorless'' generalized scalar amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$, discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.

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Likelihood degenerations

Cited by 12 publications

References 15 publications

Planar matrices and arrays of Feynman diagrams: poles for higher k

Planar matrices and arrays of Feynman diagrams: poles for higher k

Planar matrices and arrays of Feynman diagrams

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

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