Four-Loop Tadpoles: Applications in QCD
R. Boughezal, M. Czakon, T. Schutzmeier
TL;DR
The paper reviews four-loop tadpole problems and their applications, focusing on current-current correlators and the vacuum polarization function. It explains how traditional reduction methods become impractical for high-order expansions and introduces a differential-equation framework that maps propagator integrals to a system in z that can be solved with linear complexity, using vacuum integrals as boundary conditions. The authors demonstrate the method by generating 30 terms for the three-loop vacuum polarization and discuss ongoing work to extend reductions to four loops. They also position this approach alongside other techniques (sector decomposition, Mellin-Barnes) and emphasize potential impacts on precise determinations of quark masses and electroweak observables. Overall, the work offers a promising route to high-precision, multi-loop tadpole calculations in QCD.
Abstract
Recent applications of single-scale four-loop tadpoles are briefly reviewed. An algorithm for the evaluation of current correlators based on differential equations is described and applied to obtain high moments of the vacuum polarization function at O(alphas^2) as a preparation of O(alphas^3) calculations.