High-precision epsilon expansions of single-mass-scale four-loop vacuum bubbles

TL;DR

The paper develops and applies a high-precision numerical framework to compute the $\epsilon$-expansions of single-mass-scale, four-loop vacuum master integrals using difference equations (Laporta method). By constructing and solving linear difference equations with factorial-series, and supplementing with a Laplace-transform treatment for a non-convergent case, the authors obtain extensive $\epsilon$-series up to $\epsilon^{10}$ for many masters, while cross-validating against known analytic results at lower loops. They provide both numerical results and analytic expressions for several 1–3-loop integrals, and compile a coherent analytic reference for single-mass-scale vacuum integrals up to four loops. The work enhances precision in perturbative calculations within QED, QCD, and finite-temperature field theory, where these master integrals act as building blocks for high-order corrections.

Abstract

In this article we present a high-precision evaluation of the expansions in $\e=(4-d)/2$ of (up to) four-loop scalar vacuum master integrals, using the method of difference equations developed by S. Laporta. We cover the complete set of `QED-type' master integrals, i.e. those with a single mass scale only (i.e. $m_i\in\{0,m\}$) and an even number of massive lines at each vertex. Furthermore, we collect all that is known analytically about four-loop `QED-type' masters, as well as about {\em all} single-mass-scale vacuum integrals at one-, two- and three-loop order.