Rationalizing loop integration

Bourjaily, Jacob L.; McLeod, Andrew J.; Hippel, Matt von; Wilhelm, Matthias

doi:10.1007/jhep08(2018)184

Cited by 76 publications

(126 citation statements)

References 93 publications

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“…Remarkably, each of the branch cuts in these amplitudes ends at the vanishing locus of some cluster coordinate on Gr(4,n) [2,[11][12][13], and-even more strikingly-their iterated discontinuities vanish unless sequentially taken in coordinates that appear together in a cluster of Gr(4,n) [14]. All six-and seven-particle next-to-MHV (NMHV) amplitudes that have currently been computed in this theory share these remarkable properties [6][7][8][15][16][17][18][19][20][21], as do certain classes of Feynman integrals [14,[22][23][24], some of which have been computed to all loop orders [25]. While this collection of amplitudes and integrals represents the simplest this theory has to offer, it remains suggestive that cluster algebras combinatorially realize these salient aspects of their analytic structure, thereby encoding locality in a non-obvious way.…”

Section: Introductionmentioning

confidence: 77%

“…While amplitudes in planar N = 4 are not generically expected to have this property (and indeed, certain Feynman integrals have been computed that do not [23,24]), an infinite class of amplitudes do-namely, all two-loop MHV amplitudes [9], and all six-and seven-particle amplitudes computed to date [6][7][8][15][16][17][18][19][20][21]. The significance of this property is illustrated by the two-loop, six-particle remainder function, which encodes the MHV amplitude.…”

Section: Grassmannian Cluster Algebras and Planar N = 4 Sym Theorymentioning

confidence: 99%

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Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

Golden

McLeod

2019

J. High Energ. Phys.

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We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar N = 4 supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory's two-loop MHV amplitudes-considered as functions, symbols, and at the level of their Lie cobracket-and recount how the 'nonclassical' part of these amplitudes can be decomposed into specific functions evaluated on the A 2 or A 3 subalgebras of Gr(4, n). We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the D 5 and A 5 subalgebras of Gr(4, 7), and that these decompositions are themselves decomposable in terms of the same A 4 function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra. arXiv:1810.12181v3 [hep-th] 5 Sep 2019 (2) 7 52 6 Conclusion 54 A Counting Subalgebras of Finite Cluster Algebras 56 B Cobracket Spaces in Finite Cluster Algebras 59 -1 --6 -

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Section: Introductionmentioning

confidence: 77%

Section: Grassmannian Cluster Algebras and Planar N = 4 Sym Theorymentioning

confidence: 99%

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

Golden

McLeod

2019

J. High Energ. Phys.

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“…which was referred to as 'octagon K' in [46], where the particular challenges to its direct integration were described at some length (see also [58]). This integral is in fact the most difficult integral topology required for any eight-point amplitude at two loops for the simple reason that it is the only topology that depends on eight dual-momentum points.…”

Section: The Simplest Nmhv Octagon Component Amplitudementioning

confidence: 99%

“…defined by sequential two-fold rotations r 2 :z i → z i+2 . As described in [46], any rational parameterization of momentum twistors will be free of square roots associated with six-dimensional Gramians, and any rational point in momentum-twistor space can be accessed rationally in any cluster coordinate chart. And so the question of whether or not algebraic letters arise can be answered at any rational point in momentum-twistor space.…”

Section: The Simplest Nmhv Octagon Component Amplitudementioning

confidence: 99%

Rooting out letters: octagonal symbol alphabets and algebraic number theory

Bourjaily

McLeod

Vergu

et al. 2020

J. High Energ. Phys.

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It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Recent advances in loop integration technology have made this an 'experimentally testable' hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is-surprisingly-rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.

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“…refs. [32,42,53]). Most Feynman integrals are not expressible in terms of multiple polylogarithms at all [54], and therefore cannot admit any linearly reducible parametrization.…”

Section: Feynman Parameter Integrationmentioning

confidence: 99%

Manifestly dual-conformal loop integration

2019

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Local, manifestly dual-conformally invariant loop integrands are now known for all finite quantities associated with observables in planar, maximally supersymmetric Yang-Mills theory through three loops. These representations, however, are not infrared-finite term by term and therefore require regularization; and even using a regulator consistent with dual-conformal invariance, ordinary methods of loop integration would naïvely obscure this symmetry. In this work, we show how any planar loop integral through at least two loops can be systematically regulated and evaluated directly in terms of strictly finite, manifestly dual-conformal Feynmanparameter integrals. We apply these methods to the case of the two-loop ratio and remainder functions for six particles, reproducing the known results in terms of individually regulated local loop integrals, and we comment on some of the novelties that arise for this regularization scheme not previously seen at one loop.

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Rationalizing loop integration

Cited by 76 publications

References 93 publications

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

Rooting out letters: octagonal symbol alphabets and algebraic number theory

Manifestly dual-conformal loop integration

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