Lectures on multiloop calculations

Abstract

I discuss methods of calculation of propagator diagrams (massless, those of Heavy Quark Effective Theory, and massive on-shell diagrams) up to 3 loops. Integration-by-parts recurrence relations are used to reduce them to linear combinations of basis integrals. Non-trivial basis integrals have to be calculated by some other method, e.g., using Gegenbauer polynomial technique. Many of them are expressed via hypergeometric functions; in the massless and HQET cases, their indices tend to integers at $ε\to0$. I discuss the algorithm of their expansion in $ε$, in terms of multiple $ζ$ values. These lectures were given at Calc-03 school, Dubna, 14--20 June 2003.