Differential Equations for Feynman Graph Amplitudes

Abstract

It is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is shown in this paper that the integration by part identities can be further used for obtaining a linear system of first order differential equations for the master integrals themselves. The equations can then be used for the numerical evaluation of the amplitudes as well as for investigating their analytic properties, such as the asymptotic and threshold behaviours and the corresponding expansions (and for analytic integration purposes, when possible). The new method is illustrated through its somewhat detailed application to the case of the one loop self-mass amplitude, by explicitly working out expansions and quadrature formulas, both in arbitrary continuous dimension n and in the n \to 4 limit. It is then shortly discussed which features of the new method are expected to work in the more general case of multi-point, multi-loop amplitudes.

19 pages, Latex, 3 encapsulated Postscript figures, uses epsfig.sty, quotation added

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• Two-parton contribution to the heavy-quark A_FB at NNLO QCD in e+e- collisions
• NNLO mixed EW-QCD corrections to single vector boson production
• Three loop QCD corrections to heavy quark form factors
• Numerator Seagull and Extended Symmetries of Feynman Integrals
• Two-loop amplitude for mixed QCD-EW corrections to $gg\to Hg$
• Analytic auxiliary mass flow to compute master integrals in singular kinematics
• Massive Two-Loop Heavy Particle Diagrams
• Dihadron Angular Correlations in the $e^{+}{e}^{-}$ Collision
• Universal Decomposition of Phase-Space Integrands
• Charge asymmetry in the spectra of bremsstrahlung and pair production
• Evaluating Feynman Integrals through differential equations and series expansions
• Top-Quark Decay at Next-to-Next-to-Next-to-Leading Order in QCD
• One-Loop QCD Corrections to $\bar{u}d\to t\bar{t}W$ at $\mathcal{O}({\epsilon }^2)$
• Two-loop QCD corrections to the vector form factors for the heavy-quark photo-production
• Scalar one-loop 4-point integral with one massless vertex in loop regularization
• Anomalous Ward identities for on-shell amplitudes at the conformal fixed point
• QCD Amplitudes: new perspectives on Feynman integral calculus
• The canonical form has got baubles
• Master integrals for two-loop $C$-odd contribution to $e^{+}{e}^{-}\to {\ell }^{+}{\ell }^{-}$ process
• Sunrise integral in non-relativistic QCD with elliptics
• Analytic calculation of Energy-Energy Correlation in $e^{+}{e}^{-}$ annihilation at NLO
• Two-loop mixed QCD-EW corrections to neutral current Drell-Yan
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
• Towards NNLO corrections to muon-electron scattering
• Analytic results for two-loop planar master integrals for Bhabha scattering
• Co-homology of Differential Forms and Feynman diagrams
• Integral representation for three-loop banana graph
• On the Four-Loop Higgs--Gluon Form Factor in Nonlocal Quantum Field Theory
• Multi-parton contributions to $\bar{B}\to X_s\gamma$ at NLO