Automatic Integral Reduction for Higher Order Perturbative Calculations

Abstract

We present a program for the reduction of large systems of integrals to master integrals. The algorithm was first proposed by Laporta; in this paper, we implement it in MAPLE. We also develop two new features which keep the size of intermediate expressions relatively small throughout the calculation. The program requires modest input information from the user and can be used for generic calculations in perturbation theory.

23 pages

Cited by in corpus (228)

• Multiloop integrals in dimensional regularization made simple
• New Developments in FeynCalc 9.0
• FEWZ 2.0: A code for hadronic Z production at next-to-next-to-leading order
• LiteRed 1.4: a powerful tool for the reduction of the multiloop integrals
• FIRE5: a C++ implementation of Feynman Integral REduction
• Combining QCD and electroweak corrections to dilepton production in FEWZ
• Kira - A Feynman Integral Reduction Program
• The third-order QCD corrections to deep-inelastic scattering by photon exchange
• Reduze - Feynman Integral Reduction in C++
• FIRE6: Feynman Integral REduction with Modular Arithmetic
• Integral Reduction with Kira 2.0 and Finite Field Methods
• Lectures on differential equations for Feynman integrals
• Calculation of the quark and gluon form factors to three loops in QCD
• A novel approach to integration by parts reduction
• Mixed QCD-electroweak corrections to Higgs boson production in gluon fusion
• Reduze 2 - Distributed Feynman Integral Reduction
• Fully differential Higgs boson production and the di-photon signal through next-to-next-to-leading order
• D-dimensional unitarity cut method
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes
• Bootstrapping rapidity anomalous dimension for transverse-momentum resummation
• Towards a Basis for Planar Two-Loop Integrals
• Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop
• Two-loop Integrand Decomposition into Master Integrals and Surface Terms
• Integration-by-parts reductions from unitarity cuts and algebraic geometry
• Analytic Results for Virtual QCD Corrections to Higgs Production and Decay
• The Quark Form Factor at Higher Orders
• Two-Loop Quark and Gluon Form Factors in Dimensional Regularisation
• Soft triple-real radiation for Higgs production at N3LO
• Reconstructing Rational Functions with $\text{FireFly}$
• The W boson production cross section at the LHC through O(alpha_s^2)
• FIRE4, LiteRed and accompanying tools to solve integration by parts relations
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• Realistic Composite Higgs Models
• Collider Physics at the Precision Frontier
• Improved ${\alpha }^4$ Term of the Electron Anomalous Magnetic Moment
• Hepta-Cuts of Two-Loop Scattering Amplitudes
• Iterative structure of finite loop integrals
• Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity
• Two-Loop Fermionic Corrections to Heavy-Quark Pair Production: the Quark-Antiquark Channel
• $Apart: A Generalized Mathematica Apart Function
• Toward a NNLO calculation of the B-->X_s gamma decay rate with a cut on photon energy: I. Two-loop result for the soft function
• Maximal Unitarity at Two Loops
• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
• The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity
• A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations
• Interpolation of Dense and Sparse Rational Functions and other Improvements in $\text{FireFly}$
• FeynHelpers: Connecting FeynCalc to FIRE and Package-X
• The gluon jet function at two-loop order
• QCD Corrections to Semi-Inclusive Hadron Production in Electron-Positron Annihilation at Two Loops
• NNLO vertex corrections to non-leptonic B decays: Tree amplitudes
• Two-Loop Four-Gluon Amplitudes with the Numerical Unitarity Method
• Radiation-Reaction in the Effective Field Theory Approach to Post-Minkowskian Dynamics
• Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD
• On the Generalized Harmonic Polylogarithms of One Complex Variable
• The fully differential decay rate of a Higgs boson to bottom-quarks at NNLO in QCD
• Two-Loop Polygon Wilson Loops in N=4 SYM
• Loop amplitudes in gauge theories: modern analytic approaches
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Unitarity Cuts with Massive Propagators and Algebraic Expressions for Coefficients
• Two-Loop Planar Corrections to Heavy-Quark Pair Production in the Quark-Antiquark Channel
• Soft Expansion of Double-Real-Virtual Corrections to Higgs Production at N${}^3$LO
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• The Four-Loop Konishi in N=4 SYM
• Master integrals for massive two-loop Bhabha scattering in QED
• Two-loop QCD correction to differential semi-leptonic b --> u decays in the shape-function region
• Real-Virtual contributions to the inclusive Higgs cross-section at N3LO
• Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
• Analytic calculation of two-loop QCD corrections to b\to s\ell^+\ell^- in the high q^2 region
• The photon energy spectrum in B-> X_s + γin perturbative QCD through O(α_s^2)
• Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
• Feynman integral relations from parametric annihilators
• The electron energy spectrum in muon decay through O(alpha^2)
• Analytic one-loop amplitudes for a Higgs boson plus four partons
• How to choose master integrals
• Differential equations from unitarity cuts: nonplanar hexa-box integrals
• Two-Loop Leading Color Corrections to Heavy-Quark Pair Production in the Gluon Fusion Channel
• Nine-Propagator Master Integrals for Massless Three-Loop Form Factors
• ep-Finite Basis of Master Integrals for the Integration-By-Parts Method
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• Master Integrals For Massless Two-Loop Vertex Diagrams With Three Offshell Legs
• Perturbative Heavy Quark Fragmentation Function through O(α_s^2): Gluon Initiated Contribution
• Two-body non-leptonic heavy-to-heavy decays at NNLO in QCD factorization
• Numerical evaluation of loop integrals
• Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering
• Three-Loop Electroweak Correction to the Rho Parameter in the Large Higgs Mass Limit
• NNLO phase space master integrals for two-to-one inclusive cross sections in dimensional regularization
• The QCD form factor of heavy quarks at NNLO
• Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation
• Master Integrals for Fermionic Contributions to Massless Three-Loop Form Factors
• Doubly-Charged Scalars in the Type-II Seesaw Mechanism: Fundamental Symmetry Tests and High-Energy Searches
• The Sudakov form factor at four loops in maximal super Yang-Mills theory
• Two-Loop Maximal Unitarity with External Masses
• Baikov representations, intersection theory, and canonical Feynman integrals
• Decays of Scalar and Pseudoscalar Higgs Bosons into Fermions: Two-loop QCD Corrections to the Higgs-Quark-Antiquark Amplitude
• Applying Groebner Bases to Solve Reduction Problems for Feynman Integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• The one-loop gluon amplitude for heavy-quark production at NNLO
• Direct Solution of Integration-by-Parts Systems
• The $Hb\bar{b}$ form factor to three loops in QCD
• Master Integrals for Massless Three-Loop Form Factors: One-Loop and Two-Loop Insertions
• Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
• Numerical Loop-Tree Duality: contour deformation and subtraction
• NNLL QCD Contribution of the Electromagnetic Dipole Operator to Gamma(anti-B -> X(s) gamma)
• Calculation of Feynman loop integration and phase-space integration via auxiliary mass flow
• NNLO corrections to B --> X_u l nu in the shape-function region
• Les Houches 2021: Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Pseudo-scalar Form Factors at Three Loops in QCD
• Magnetic dipole operator contributions to the photon energy spectrum in anti-B -> X(s) gamma at O(alpha(s)^2)
• Master integrals for the four-loop Sudakov form factor
• FIRE 6.5: Feynman Integral Reduction with New Simplification Library
• The NNLO gluon fusion Higgs production cross-section with many heavy quarks
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• The R*-operation for Feynman graphs with generic numerators
• Charm quark mass dependence of the electromagnetic dipole operator contribution to anti-B -> X(s) gamma at O(alpha**2(s))
• Two-loop five-point massless QCD amplitudes within the IBP approach
• Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order
• An Integrand Reconstruction Method for Three-Loop Amplitudes
• Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams
• Two-loop QED radiative corrections to the decay pi0 -> e+ e- : The virtual corrections and soft-photon bremsstrahlung
• Two-loop helicity amplitudes for $gg\to ZZ$ with full top-quark mass effects
• Next-to-next-to-leading order calculations for heavy-to-light decays
• Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations
• Large $N_f$ quantum field theory
• Massive planar and non-planar double box integrals for light Nf contributions to gg->tt
• The real radiation antenna function for $S\to Q\bar{Q}q\bar{q}$ at NNLO QCD
• Two-Loop Matching Coefficients for Heavy Quark Currents
• Quantum Field Theory of Fluids
• Differential equations for loop integrals in Baikov representation
• Two-Loop Master Integrals for $q\bar{q}\to VV$: the Planar Topologies
• Gluon-fusion Higgs production at NNLO for a non-standard Higgs sector
• Fermionic Corrections to the Three-Loop Matching Coefficient of the Vector Current
• RI'/SMOM scheme amplitudes for quark currents at two loops
• Finding Linear Dependencies in Integration-By-Parts Equations: A Monte Carlo Approach
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• Two-loop massless QCD corrections to the $g+g\to H+H$ four-point amplitude
• A Numerical Routine for the Crossed Vertex Diagram with a Massive-Particle Loop
• Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation
• Calculation of the Two-Loop Heavy-Flavor Contribution to Bhabha Scattering
• Counting master integrals: integration by parts vs. differential reduction
• Chromomagnetic Dipole-Operator Corrections in Bbar -> X_sgamma at O(beta_0 alpha_s^2)
• Large-m_c Asymptotic Behaviour of O(alpha_s^2) Corrections to B -> X_s gamma
• Maximal Unitarity for the Four-Mass Double Box
• Numerical multi-loop integrals and applications
• Heavy-quark vacuum polarization: first two moments of the O(alphas^3 nf^2) contribution
• NNLO corrections to 2-jet observables in electron-positron annihilation
• Complete Higgs mass dependence of top quark pair threshold production to order $\alpha {\alpha }_s$
• An Algorithm to Construct Groebner Bases for Solving Integration by Parts Relations
• Extension of the functionality of the symbolic program FORM by external software
• On a two-loop crossed six-line master integral with two massive lines
• Kira 1.2 Release Notes
• Konishi Form Factor at Three Loop in $\mathcal{N}=4$ SYM
• Calculation of two-loop QCD corrections for hadronic single top-quark production in the t channel
• Electroweak corrections to double Higgs production at the LHC
• Groebner Bases in Perturbative Calculations
• Three-loop electroweak corrections to the W-boson mass and sin^2 theta_eff in the large Higgs mass limit
• Axinos in Cosmology and at Colliders
• Balancing act: multivariate rational reconstruction for IBP
• Master Integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel
• IBP methods at finite temperature
• Two-loop parameter relations between dimensional regularization and dimensional reduction applied to SUSY-QCD
• The four-dimensional helicity scheme and dimensional reconstruction
• Two-Loop $\mathcal{O}(\alpha {\alpha }_s)$ Corrections to the On-Shell Fermion Propagator in the Standard Model
• Hypergeometric representation of a four-loop vacuum bubble
• S-bases as a tool to solve reduction problems for Feynman integrals
• The diamond rule for multi-loop Feynman diagrams
• Quark and gluon spin-2 form factors to two-loops in QCD
• Complete (O_7,O_8) contribution to B -> X_s gamma at order alpha_s^2
• Global Poles of the Two-Loop Six-Point N=4 SYM integrand
• The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction
• Factorization of denominators in integration-by-parts reductions
• Yukawa Corrections to Higgs Production in Top Partner Models
• HepLib: A C++ Library for High Energy Physics
• Analytic two-loop master integrals for $tW$ production at hadron colliders: I
• Forcer: a FORM program for 4-loop massless propagators
• Two loop QCD amplitudes for di-pseudo scalar production in gluon fusion
• FaRe: a Mathematica package for tensor reduction of Feynman integrals
• Two-Loop QCD Correction to massive spin-2 resonance $\to$ 3 gluons
• Three loop form factors of a massive spin-2 particle with nonuniversal coupling
• On the reduction of Feynman integrals to master integrals
• Manifestly Causal Loop-Tree Duality
• Calculating four-loop massless propagators with Forcer
• Feynman Integrals from Positivity Constraints
• Planar master integrals for the two-loop light-fermion electroweak corrections to Higgs plus jet production
• NNLO QCD corrections to $pp\to {\gamma }^{\ast }{\gamma }^{\ast }$ in the large $N_F$ limit
• Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
• Quark mass effects in two-loop Higgs amplitudes
• Locally finite two-loop amplitudes for electroweak production through gluon fusion
• Twisted Riemann bilinear relations and Feynman integrals
• Constraints on Nonrelativistic-QCD Long-Distance Matrix Elements from $J/\psi$ Plus $W$/$Z$ Production at the LHC
• Feynman integral reduction using Gröbner bases
• Some recent results on evaluating Feynman integrals
• The {\tt Mathematica} package {\tt TopoID} and its application to the Higgs boson production cross section
• NLL QCD contribution of the Electromagnetic Dipole operator to B-> X_s gamma gamma
• Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Developments since Kira 2.0
• Machine Learning Post-Minkowskian Integrals
• Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case
• "Background Field Integration-by-Parts" and the Connection Between One-Loop and Two-Loop Heisenberg-Euler Effective Actions
• Spin-2 Form Factors at Three Loop in QCD
• Methods for the Reduction of Three-Loop QCD Form Factors
• A new reduction strategy for special negative sectors of planar two-loop integrals without Laporta algorithm
• A New Method for Calculating Differential Distributions Directly in Mellin Space
• Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders
• On the two-loop amplitude for $gg\to ZZ$ production with full top-mass dependence
• Computer algebra calculations in supersymmetric electrodynamics
• Reduction of Feynman Integrals in the Parametric Representation
• Rational Function Simplification for Integration-by-Parts Reduction and Beyond
• Evaluation of three-loop self-energy master integrals with four or five propagators
• pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• Multi-loop calculations: numerical methods and applications
• The polarized photon structure function $g_1^{\gamma }(x,Q^2)$ in massive parton model in NLO
• Automated One-loop Computation in Quarkonium Process within NRQCD Framework
• NNLO splitting and coefficient functions with time-like kinematics
• Master Integrals for double real radiation emission in heavy-to-light quark decay
• A new method for finding more symmetry relations of Feynman integrals
• IBP reduction via Gröbner bases in a rational double-shift algebra
• Extended Projection Method for Massive Fermion
• Top-Quark Decay at Next-to-Next-to-Next-to-Leading Order in QCD
• SUSY QCD corrections to electroweak gauge boson production with an associated jet at the LHC
• Evaluating Feynman Integrals through differential equations and series expansions
• Two-parton contribution to the heavy-quark A_FB at NNLO QCD in e+e- collisions
• Evaluating master integrals in non-factorizable corrections to $t$-channel single-top production at NNLO QCD
• Solving recurrence relations for multiloop integrals in the limit of large values of the dimensional regularization parameter
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
• Reduction of planar double-box diagram for single-top production via auxiliary mass flow