Evaluating Feynman Integrals through differential equations and series expansions

Abstract

We review the method of the differential equations for the evaluation of multi-loop Feynman integrals. In particular, we focus on the series expansion approach for solving the system of differential equation and we discuss how to perform the analytical continuation of the result to entire (complex) phase-space. This approach allow us to consider arbitrary internal complex masses. This review is based on a lecture given by the author at the "Advanced School and Workshop on Multiloop Scattering Amplitudes" held in NISER, Bhubaneswar (India) in January 2024.

21 pages, 9 figures

References in corpus (18)

• Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC
• Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC
• LiteRed 1.4: a powerful tool for the reduction of the multiloop integrals
• Algorithm FIRE -- Feynman Integral REduction
• Kira - A Feynman Integral Reduction Program
• HPL, a Mathematica implementation of the harmonic polylogarithms
• Integral Reduction with Kira 2.0 and Finite Field Methods
• Automatic Integral Reduction for Higher Order Perturbative Calculations
• pySecDec: a toolbox for the numerical evaluation of multi-scale integrals
• A novel approach to integration by parts reduction
• AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• Electroweak Radiative Corrections for Collider Physics
• Reconstructing Rational Functions with $\text{FireFly}$
• Extension of HPL to complex arguments
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• On the Generalized Harmonic Polylogarithms of One Complex Variable
• FIRE 6.5: Feynman Integral Reduction with New Simplification Library