Automatic reduction of four-loop bubbles

TL;DR

This work presents a practical framework for reducing four-loop vacuum bubbles to a minimal set of master integrals using integration-by-parts identities within dimensional regularization, with automation implemented in FORM. It contrasts symbolic general IBP approaches with a brute-force Laporta-type method that relies on a lexicographic ordering to express complex integrals through simpler ones, enabling the extraction of master integrals for both fully massive and QED-like topologies. The authors discuss representations, hierarchical reductions, and basis choices, providing conversion relations between different master sets and addressing issues like spurious poles. The resulting methodology supports high-precision four-loop QCD computations (e.g., the QCD free energy) and is applicable to other massive vacuum problems, including potential re-evaluations of the QCD beta function.

Abstract

We give technical details about the computational strategy employed in a recently completed investigation of the four-loop QCD free energy. In particular, the reduction step from generic vacuum bubbles to master integrals is described from a practical viewpoint, for fully massive as well as QED-type integrals.

Figures (4)

• Figure 1: The 1+1+3+10 generic vacuum topologies up to four loops. The 0+1+2+6 factorized topologies are not shown here.
• Figure 2: The 1+0+2+10 master integrals of QED type, up to four loops. Full lines carry a mass $m$, dotted lines are massless. All numerators are 1, all powers of propagators are 1. Note that there is no two-loop representative needed.
• Figure 3: The 0+0+0+3 fully massive master integrals, in addition to those 1+1+3+10 of Fig. $\!$\ref{['fig:vactopos']}, taken at powers and numerators 1. A dot on a line means it carries an extra power.
• Figure 4: Relations for a basis conversion from the set of massive masters found in laporta4loop to our notation.