Lectures on multiloop calculations
A. G. Grozin
TL;DR
This work surveys systematic strategies for evaluating multiloop propagator diagrams across massless, HQET, and massive on-shell regimes. It centers on reducing complex integrals via integration-by-parts to a finite basis, then computing master integrals with Gegenbauer techniques, hypergeometric representations, or Mellin–Barnes methods, followed by ε-expansions tied to multiple zeta values. The text details concrete bases and inversion relations for 1–3 loop cases, provides explicit master-integral formulas (including many in terms of {}_3F_2 and Γ-functions), and describes algorithms for ε-expansion that underpin high-precision perturbative results. Collectively, the lectures offer a practical framework for high-order propagator calculations and connect physical diagrams to rich mathematical structures like hypergeometric functions and multiple zeta values. The methods enable precise renormalization-group analyses and HQET matching, with broad applicability in quantum field theory computations.
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.