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LoopIntegrals

Compute master integrals using commutative and noncommutative methods from computational algebraic geometry

2024.11-01

9 November 2024

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Robin Brüser
Email: brueser@uni-mainz.de
Homepage: https://www.researchgate.net/scientific-contributions/2138634848_Robin_Brueser
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Tobias Huber
Email: huber@tp1.physik.uni-siegen.de
Homepage: https://www.physik.uni-siegen.de/tp1/research/researchgroups/huber.html
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Jan Piclum
Email: piclum@physik.uni-siegen.de
Homepage: https://www.researchgate.net/scientific-contributions/34408173_Jan_Piclum
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 Loop integrals

1.1 Introduction

  1.2-1 Lorentz vectors
  1.2-2 1 Loop tadpole
  1.2-3 1 Loop bubble
  1.2-4 1 Loop massive on-shell
  1.2-5 1 Loop massive bubble
  1.2-6 1 Loop double massive bubble
  1.2-7 1 Loop box
  1.2-8 Kite
  1.2-9 2 Loop nonplanar ladder
  1.2-10 2 Loop cusp anomalous dimensions
  1.2-11 2 Loop box
  1.2-12 Pentagon with off shell leg

1.3 Global variables

1.3-1 LOOP_INTEGRALS

1.4-1 IsLoopMomentum
1.4-2 IsExternalMomentum

  1.5-1 Dimension
  1.5-2 UnderlyingMatrix
  1.5-3 LoopDiagram
  1.5-4 Components
  1.5-5 LoopMomenta
  1.5-6 ExternalMomenta
  1.5-7 DimensionOfCoefficientsVector
  1.5-8 UnderlyingRing
  1.5-9 RelationsOfExternalMomenta
  1.5-10 RelationsMatrixOfExternalMomenta
  1.5-11 IndependentLorentzInvariants
  1.5-12 Propagators
  1.5-13 Numerators
  1.5-14 ExtraLorentzInvariants
  1.5-15 OriginalIBPGeneratingMatrixOfPropagators
  1.5-16 OriginalIBPGeneratingMatrixOfNumerators
  1.5-17 OriginalTaylorOfPropagators
  1.5-18 MatrixOfMomenta
  1.5-19 IBPGeneratingMatrixOfPropagators
  1.5-20 IBPGeneratingMatrixOfNumerators
  1.5-21 PairOfMatricesOfLoopDiagram
  1.5-22 RingOfExtraLorentzInvariants
  1.5-23 ReductionMatrixOfExtraLorentzInvariants
  1.5-24 RingOfIndependentLorentzInvariants
  1.5-25 ReductionMatrixOfIndependentLorentzInvariants
  1.5-26 RingOfPropagatorsAndNumeratorsAndExtraLorentzInvariants
  1.5-27 RingOfLoopDiagram
  1.5-28 RingOfLoopDiagramWithDimensionShift
  1.5-29 ReductionMatrixOfPropagatorsAndNumeratorsAndExtraLorentzInvariants
  1.5-30 IBPGeneratingMatrixOfPropagatorsInIndependentLorentzInvariants
  1.5-31 IBPGeneratingMatrixOfPropagatorsInPropagators
  1.5-32 IBPGeneratingMatrixOfNumeratorsInPropagators
  1.5-33 PairOfMatricesOfLoopDiagramInIndependentLorentzInvariants
  1.5-34 PairOfMatricesOfLoopDiagramInPropagators
  1.5-35 IBPGeneratingMatrixOfLoopDiagramInPropagators
  1.5-36 MatrixOfIBPRelations
  1.5-37 MatrixOfIBPRelationsWithDimensionShift
  1.5-38 BasisOfIBPRelations
  1.5-39 MatrixOfSpecialIBPRelations
  1.5-40 BasisOfSpecialIBPRelations
  1.5-41 MatrixOfIBPRelationsInWeylAlgebra
  1.5-42 BasisOfIBPRelationsInWeylAlgebra
  1.5-43 MatrixOfSpecialIBPRelationsInWeylAlgebra
  1.5-44 BasisOfSpecialIBPRelationsInWeylAlgebra
  1.5-45 FieldOfCoefficientsOfLoopDiagramInSingular
  1.5-46 FieldOfCoefficientsOfLoopDiagramInMaple
  1.5-47 FieldOfCoefficientsOfLoopDiagramInHecke
  1.5-48 MatrixOfCoefficientsOfParametricIBPs
  1.5-49 SymanzikPolynomials
  1.5-50 AssociatedWeylAlgebra
  1.5-51 GeneratorsOfScalelessSectors
  1.5-52 GeneratorsOfScalelessSectorsInWeylAlgebra

1.6 Constructors

  1.6-1 LorentzVector
  1.6-2 LorentzVector
  1.6-3 LoopDiagram

  1.7-1 SetAbbreviation
  1.7-2 +
  1.7-3 -
  1.7-4 AdditiveInverse
  1.7-5 []
  1.7-6 *
  1.7-7 *
  1.7-8 *
  1.7-9 ExpressInExtraLorentzInvariants
  1.7-10 ExpressInIndependentLorentzInvariants
  1.7-11 ExpressInPropagatorsAndNumeratorsAndExtraLorentzInvariants
  1.7-12 JacobianOfCoefficientsVectorInPropagators
  1.7-13 DivergenceOfCoefficientsVectorOfLoopDiagram
  1.7-14 DoubleShiftAlgebra
  1.7-15 DoubleShiftAlgebraWithDimensionShift
  1.7-16 RationalDoubleShiftAlgebra
  1.7-17 RationalDoubleShiftAlgebraWithDimensionShift
  1.7-18 IBPRelation
  1.7-19 IBPRelation
  1.7-20 MatrixOfIBPRelations
  1.7-21 MatrixOfIBPRelations
  1.7-22 MatrixOfIBPRelations
  1.7-23 MatrixOfSpecialIBPRelations
  1.7-24 IBPRelationInWeylAlgebra
  1.7-25 MatrixOfIBPRelationsInWeylAlgebra
  1.7-26 MatrixOfCoefficientsOfIBPs
  1.7-27 MatrixOfCoefficientsOfIBPs
  1.7-28 MatrixOfCoefficientsOfParametricIBPs
  1.7-29 MatrixOfCoefficientsOfParametricIBPs
  1.7-30 ColumnReversedMatrixOfCoefficientsOfParametricIBPs
  1.7-31 ColumnReversedMatrixOfCoefficientsOfParametricIBPs
  1.7-32 SymanzikPolynomials
  1.7-33 DegreesOfMonomialsOfProductOfSymanzikPolynomials
  1.7-34 IsScalelessLoopIntegral
  1.7-35 GeneratorsOfScalelessSectors
  1.7-36 NormalForm
  1.7-37 NormalForm
  1.7-38 NormalFormWrtInitialIntegral

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