Numerical evaluation of multi-loop integrals by sector decomposition

Abstract

In a recent paper we have presented an automated subtraction method for divergent multi-loop/leg integrals in dimensional regularisation which allows for their numerical evaluation, and applied it to diagrams with massless internal lines. Here we show how to extend this algorithm to Feynman diagrams with massive propagators and arbitrary propagator powers. As applications, we present numerical results for the master 2-loop 4-point topologies with massive internal lines occurring in Bhabha scattering at two loops, and for the master integrals of planar and non-planar massless double box graphs with two off-shell legs. We also evaluate numerically some two-point functions up to 5 loops relevant for beta-function calculations, and a 3-loop 4-point function, the massless on-shell planar triple box. Whereas the 4-point functions are evaluated in non-physical kinematic regions, the results for the propagator functions are valid for arbitrary kinematics.

15 pages latex, 11 eps figures included

References in corpus (37)

• An automatized algorithm to compute infrared divergent multi-loop integrals
• Connection between Feynman integrals having different values of the space-time dimension
• Analytical Result for Dimensionally Regularized Massless On-shell Double Box
• Non-planar massless two-loop Feynman diagrams with four on-shell legs
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The non-planar topologies
• Two-loop QCD corrections to gluon-gluon scattering
• Two-Loop QCD Helicity Amplitudes for $e^{+}{e}^{-}\to 3$~Jets
• The Two-Loop QCD Matrix Element for $e^{+}{e}^{-}\to 3$ Jets
• Two-Loop Amplitudes for Gluon Fusion into Two Photons
• A Two-Loop Four-Gluon Helicity Amplitude in QCD
• Two-loop QCD corrections to the scattering of massless distinct quarks
• QCD and QED Corrections to Light-by-Light Scattering
• Two-loop QCD corrections to massless quark-gluon scattering
• High-Energy Approximation of One-Loop Feynman Integrals
• Two-Loop Correction to Bhabha Scattering
• Two-loop QCD corrections to massless identical quark scattering
• Two-loop QED and QCD corrections to massless fermion-boson scattering
• Results and Techniques of Multi-Loop Calculations
• All-Purpose Numerical Evaluation of One-Loop Multi-Leg Feynman Diagrams
• Two-loop electroweak angular-dependent logarithms at high energies
• An Approach Toward the Numerical Evaluation of Multi-Loop Feynman Diagrams
• High-precision e-expansions of massive four-loop vacuum bubbles
• Two-loop QCD helicity amplitudes for massless quark-massless gauge boson scattering
• Two-loop corrections to Light-by-Light scattering in Supersymmetric QED
• A numerical method for NNLO calculations
• Two-loop amplitudes with nested sums: Fermionic contributions to e+ e- --> q qbar g
• A numerical evaluation of the scalar hexagon integral in the physical region
• Analytical Result for Dimensionally Regularized Massless On-Shell Planar Triple Box
• Analytical Result for Dimensionally Regularized Massless Master Double Box with One Leg off Shell
• Algebraic algorithms for multiloop calculations. The first 15 years. What's next?
• Algebraic-Numerical Evaluation of Feynman Diagrams: Two-Loop Self-Energies
• Analytical Result for Dimensionally Regularized Massless Master Non-planar Double Box with One Leg off Shell
• Towards systematic near-threshold calculations in perturbative QFT
• Reduction and evaluation of two-loop graphs with arbitrary masses
• Evaluating double and triple (?) boxes

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