The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals

Abstract

The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the Standard Model and effective field theories, also with classical applications. These calculations play a central role in the analysis of precision measurements at present and future colliders to obtain ultimate information for fundamental physics.

Evaluation of Feynman integrals with arbitrary complex masses via series expansions

“…Also the solution of the second equation is plugged, together with the solution of the first one, into the third equation, that in turn becomes a first-order non-homogeneous differential equation as well. We proceed in this way up to the complete solution of the system 4 . Each first-order differential equation can be solved using the procedure outlined in section 2.1, i.e.…”

“…Comprehensive reviews on the Feynman integrals, covering the various strategies can be found in Refs. [1,2,3,4].…”

The SAGEX review on scattering amplitudes Chapter 12: Amplitudes and collider physics

“…Then a reduced set of integrals appear, but the difficult problem remains of carrying out the integrals themselves, in terms of known functions. In recent years, this has generated a fascinating dialogue between pure mathematicians and theoretical physicists, and more details can be found in references [43][44][45] (chapters 3-5 of this review [46]). What is clear, however, is that there is a clear direction of travel of new computational techniques on average.…”

The SAGEX review on scattering amplitudes Chapter 5: Analytic bootstraps for scattering amplitudes and beyond

“…All explicit calculations to date, as well as an analysis at the level of the integrand [74] (note however the subtleties pointed out in [75]), suggest that at least for k = 0, 1, E (L) n,k can be expressed in terms of generalized or Goncharov or multiple polylogarithms (MPLs) [76][77][78] of weight m = 2L. These functions are also mentioned in chapters 3 [79] and 4 [80] of the SAGEX review [42], and in addition they are discussed in great detail in the recent textbook [81]. Let us briefly collect here the definitions and properties that will be useful for our purposes.…”