Cluster Algebras for Feynman Integrals

We bootstrap the three-point form factor of the chiral part of the stresstensor supermultiplet in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, “extended-Steinmann-like” conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as gg → Hg and γ* → $$ q\overline{q}g $$ q q ¯ g . We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar $$ \mathcal{N} $$ N = 4 three-point form factor can be placed in an even smaller space of functions, with no independent ζ values at weights two and three.

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“…The same adjacency condition was also observed in ref [89],. which appeared on the arXiv on the same day as this paper.…”

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confidence: 83%

A three-point form factor through five loops

Dixon

McLeod

Wilhelm

2021

J. High Energ. Phys.

106

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“…pb ] with that of D 3 A 3 cluster algebra, and we do so in two ways. First, similar to [48], we find the bi-rational change of variables…”

Section: Now We Identify the Alphabet A[i (L)mentioning

confidence: 55%

“…To identify an alphabet with certain finite-type cluster algebra, it is convenient to parametrize the coordinates of the latter in a nice way. For example, for type A d and D d , we have the following N = d(d + 3)/2 and N = d 2 letters respectively [48]:…”

Section: Review Of Cluster Algebra and Cluster Configuration Spacementioning

confidence: 99%

“…One can bootstrap Feynman integrals [45,46] based on such knowledge (see also [47]). Very recently, the authors of [48] have argued that cluster algebra structures appear for rather general Feynman integrals which go way beyond planar N = 4 SYM. They have provided strong evidence that four-point Feynman integrals with an off-shell leg is controlled by a C 2 cluster algebra, and found cluster-algebra alphabets for various one-loop integrals, as well as the general five-particle alphabet.…”

Section: Introduction and Reviewmentioning

confidence: 99%

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Notes on cluster algebras and some all-loop Feynman integrals

Yang

2021

J. High Energ. Phys.

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We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

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“…The connection between cluster algebras and symbol alphabets is of course an interesting question also beyond SYM theory; see for example[9].…”

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confidence: 99%

Symbol alphabets from plabic graphs II: rational letters

Mago

Schreiber

Spradlin

et al. 2021

J. High Energ. Phys.

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Symbol alphabets of n-particle amplitudes in $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory are known to contain certain cluster variables of G(4, n) as well as certain algebraic functions of cluster variables. The first paper arXiv:2007.00646 in this series focused on n = 8 algebraic letters. In this paper we show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving matrix equations of the form C Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of G+(n−4, n).

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Cluster Algebras for Feynman Integrals

Cited by 72 publications

References 64 publications

A three-point form factor through five loops

A three-point form factor through five loops

Notes on cluster algebras and some all-loop Feynman integrals

Symbol alphabets from plabic graphs II: rational letters

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