Integration-by-parts reductions from unitarity cuts and algebraic geometry

Larsen, Kasper J.; Zhang, Yang

doi:10.1103/physrevd.93.041701

Cited by 168 publications

(176 citation statements)

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“…A key problem for generating unitarity-compatible IBP relations is finding these special IBP-generating vectors. One solution is to solve "syzygy equations" using computational algebraic geometry [8,14,17,18]. This is often time-consuming for the more complicated multi-loop integrals, and produces lengthy and unenlightening results.…”

Section: Introductionmentioning

confidence: 99%

Dual conformal symmetry, integration-by-parts reduction, differential equations, and the nonplanar sector

et al. 2017

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We show that dual conformal symmetry, mainly studied in planar N = 4 super-Yang-Mills theory, has interesting consequences for Feynman integrals in nonsupersymmetric theories such as QCD, including the nonplanar sector. A simple observation is that dual conformal transformations preserve unitarity cut conditions for any planar integrals, including those without dual conformal symmetry. Such transformations generate differential equations without raised propagator powers, often with the right hand side of the system proportional to the dimensional regularization parameter ǫ. A nontrivial subgroup of dual conformal transformations, which leaves all external momenta invariant, generates integration-by-parts relations without raised propagator powers, reproducing, in a simpler form, previous results from computational algebraic geometry for several examples with up to two loops and five legs. By opening up the two-loop three-and four-point nonplanar diagrams into planar ones, we find a nonplanar analog of dual conformal symmetry. As for the planar case this is used to generate integration-by-parts relations and differential equations. This implies that the symmetry is tied to the analytic properties of the nonplanar sector of the two-loop four-point amplitude of N = 4 super-Yang-Mills theory.

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Section: Introductionmentioning

confidence: 99%

Dual conformal symmetry, integration-by-parts reduction, differential equations, and the nonplanar sector

et al. 2017

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“…We find it convenient to adopt the strategy suggested in Ref. [9] to first use dimension-shifting identities and then generate IBP identities in the same spacetime dimension by solving syzygy equations [8]. We omit the technical details and refer the reader to the literature.…”

Section: Maximal Cut Integration Check In D = 22/5mentioning

confidence: 99%

“…The A tree m (ρ) are called color-ordered partial amplitudes., The terminology ordered refers to the fact that all graphs contributing to any given A tree m (ρ) have the same ordering or external legs as the cyclic ordering of the color trace Tr(ρ). We can write the color-ordered amplitudes in terms of graphs via 8) where Γ ρ refers to the graphs with cubic vertices where the legs are ordered following the color ordering.…”

Section: Ordered Partial Amplitudesmentioning

confidence: 99%

“…If we start from the BCJ double-copy construction, and as for the gauge-theory case suspend enforcing the Jacobi relations, the variation of the double-copy cut under the gauge transformation is then, 8) where cut conditions are imposed as in the gauge-theory case. Thus, to restore the linearized diffeomorphism invariance we must add terms whose gauge transformation cancels δM naive cut to the naive double copy.…”

Section: Contact Terms From Bcj Dualitymentioning

confidence: 99%

“…The second is the discovery of the Bern-Carrasco-Johansson (BCJ) color-kinematics duality and associated doublecopy procedure [3,4]. The third is the progress in loop integration methods, specifically integration-by-parts (IBP) reduction [5][6][7][8][9], which has been critical to extracting ultraviolet information, as in Refs. [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Five-loop four-point integrand of N=8 supergravity as a generalized double copy

Bern¹,

Carrasco

Chen³

et al. 2017

Phys. Rev. D

117

127

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We use the recently developed generalized double-copy procedure to construct an integrand for the five-loop four-point amplitude of N = 8 supergravity. This construction starts from a naive double copy of the previously computed corresponding amplitude of N = 4 super-Yang-Mills theory. This is then systematically modified by adding contact terms generated in the context of the method of maximal unitarity cuts. For the simpler generalized cuts, whose corresponding contact terms tend to be the most complicated, we derive a set of formulas relating the contact contributions to the violations of the dual Jacobi identities in the relevant gauge-theory amplitudes.For more complex generalized unitarity cuts, which tend to have simpler contact terms associated with them, we use the method of maximal cuts more directly. The five-loop four-point integrand is a crucial ingredient towards future studies of ultraviolet properties of N = 8 supergravity at five loops and beyond. We also present a nontrivial check of the consistency of the integrand, based on modern approaches for integrating over the loop momenta in the ultraviolet region.

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Top-Down Decomposition: A Cut-Based Approach to Integral Reductions

Frellesvig¹

2021

Texts &Amp; Monographs in Symbolic Computation

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Integration-by-parts reductions from unitarity cuts and algebraic geometry

Abstract: We show that the integration-by-parts reductions of various two-loop integral topologies can be efficiently obtained by applying unitarity cuts to a specific set of subgraphs and solving associated polynomial (syzygy) equations.

Cited by 168 publications

References 24 publications

Dual conformal symmetry, integration-by-parts reduction, differential equations, and the nonplanar sector

Dual conformal symmetry, integration-by-parts reduction, differential equations, and the nonplanar sector

Five-loop four-point integrand of N=8 supergravity as a generalized double copy

Top-Down Decomposition: A Cut-Based Approach to Integral Reductions

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