Evaluating Feynman integrals by Mellin-Barnes representation

• Feynman integrals: basic notation, definitions and properties

Feynman graphs and Feynman integrals

Divergences and regularization

Alpha parameters

Dimensional regularization

One-loop massless integration formula

Analysis of convergence. Cheng-Wu theorem

Feynman parameters

Properties of dimensionally regularized Feynman integrals

A classification of methods of evaluating Feynman integrals

Integration by parts. Reduction to master integrals. Laporta algorithm

Methods to evaluate master integrals

Resolving singularities in Mellin-Barnes representation by two strategies

Basic formula

One-loop examples. One-loop propagator diagram with the masses m and 0. Massless on-shell box diagram.

An example of a twofold MB integral. Strategy #1.

Strategy #2 [Tausk]

Implementation of Strategy #2 in Czakon's code MB.m. One-loop examples. One-loop propagator diagram with the masses m and 0 (.nb and .pdf files). Massless on-shell box diagram (.nb and .pdf files).

Barnes lemmas and their corollaries (.pdf file).

A two-loop example: the massless non-planar 2-loop vertex diagram. The derivation of a twofold MB representation (.pdf file). The evaluation (.nb and .pdf files).

An example where an auxiliary analytic regularization is necessary (.nb and .pdf files).

An algorithm based on Strategy #1 ( .pdf file). An application of the corresponding code (MBresolve.m) in the same examples (.nb and .pdf files).

Evaluating by Mellin-Barnes representation: advanced examples and various additional tools

Massless box diagram with two legs off shell. (.pdf file).

The massless on-shell double-box. The derivation (.pdf file). "Horizontal" and "vertical" checks and the evaluation (.nb and .pdf files). Summing up series, nested sums and HPL (.pdf file).

An example where an auxiliary analytic regularization is necessary: the massless on-shell double-box with a numerator (.nb and .pdf files).

A warning: the loop-by-loop strategy can be problematic for nonplanar diagrams. The massless non-planar 2-loop vertex diagram (.nb and .pdf files).

Using PSLQ for MB integrals (.nb and .pdf files).

Proceeding in 4 loops with a 10fold MB integral (.nb file).

Perspectives.

Software

• MB.m
• MBresolve.m (to appear soon)
• PSLQ
• Packages connected with MB will be soon stored at HepForge