Notes on cluster algebras and some all-loop Feynman integrals

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Multi-loop scattering amplitudes/null polygonal Wilson loops in $$ \mathcal{N} $$ N = 4 super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an 1 + 1 dimensional subspace of Minkowski spacetime (or boundary of the AdS3 subspace). Since the edges of a 2n-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of G(2, n)/T ∼ An−3. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping A2 functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters v, 1 + v, w, 1 + w of A1 × A1, there are two letters v − w, 1 − vw mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping A2 functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to 2n-gons in terms of A2 functions and beyond.

Section: Introduction and Reviewsupporting

confidence: 79%

Section: Introduction and Reviewmentioning

confidence: 99%

Section: Introduction and Reviewmentioning

confidence: 99%

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Bootstrapping octagons in reduced kinematics from A2 cluster algebras

Tang

et al. 2021

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“…At strong coupling, holographic calculations provide non-trivial predictions for the behavior of N = 4 pSYM amplitudes in the form of the BDS-ansatz [51][52][53] and its generalizations [30,54]. Other formalisms motivated by the duality between Wilson loops and scattering amplitudes have also emerged [55][56][57][58][59][60][61][62].…”

Section: Jhep07(2021)049mentioning

confidence: 99%

Algebraic branch points at all loop orders from positive kinematics and wall crossing

Herderschee

2021

There is a remarkable connection between the boundary structure of the positive kinematic region and branch points of integrated amplitudes in planar $$ \mathcal{N} $$ N = 4 SYM. A long-standing question has been precisely how algebraic branch points emerge from this picture. We use wall crossing and scattering diagrams to systematically study the boundary structure of the positive kinematic regions associated with MHV amplitudes. The notion of asymptotic chambers in the scattering diagram naturally explains the appearance of algebraic branch points. Furthermore, the scattering diagram construction also motivates a new coordinate system for kinematic space that rationalizes the relations between algebraic letters in the symbol alphabet. As a direct application, we conjecture a complete list of all algebraic letters that could appear in the symbol alphabet of the 8-point MHV amplitude.

“…Apart from connection to cluster algebras, knowledge of alphabet (and further information) can be used for bootstrapping Feynman integrals [46,53] (see also [54]). In [55], we identified cluster algebras and certain adjacency conditions for a class of finite, dual conformal invariant (DCI) [56,57] Feynman integrals to high loops, based on recently-proposed Wilson-loop d log representation [58] (see [47] for a closely-related Feynman-parameter representation). For ladder integrals with possible "chiral pentagons" on one or both ends (without any square roots), we find a sequence of cluster algebras D 2 , D 3 , • • • , D 6 for their alphabets, depending on n and the kinematic configurations.…”

Section: Jhep12(2021)110mentioning

confidence: 99%

Truncated cluster algebras and Feynman integrals with algebraic letters

Yang

2021

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We propose that the symbol alphabet for classes of planar, dual-conformal-invariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries of (compactifications of) G+(4, n)/T for the n-particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with n = 7, 8, 9, we find finite cluster algebras D4, D5 and D6 respectively, in accordance with previous result on alphabets of these integrals. As the main example, we consider hexagon kinematics with two massive corners on opposite sides and find a truncated affine D4 cluster algebra whose polytopal realization is a co-dimension 4 boundary of that of G+(4, 8)/T with 39 facets; the normal vectors for 38 of them correspond to g-vectors and the remaining one gives a limit ray, which yields an alphabet of 38 rational letters and 5 algebraic ones with the unique four-mass-box square root. We construct the space of integrable symbols with this alphabet and physical first-entry conditions, whose dimension can be reduced using conditions from a truncated version of cluster adjacency. Already at weight 4, by imposing last-entry conditions inspired by the n = 8 double-pentagon integral, we are able to uniquely determine an integrable symbol that gives the algebraic part of the most generic double-pentagon integral. Finally, we locate in the space the n = 8 double-pentagon ladder integrals up to four loops using differential equations derived from Wilson-loop d log forms, and we find a remarkable pattern about the appearance of algebraic letters.