Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in $ε$
Roman N. Lee, Alexander V. Smirnov, Vladimir A. Smirnov
TL;DR
The paper presents a method based on dimensional recurrence relations and $d$-analyticity to evaluate high‑order ε‑expansions of master integrals arising in three‑loop massless form factors and related non‑planar propagator diagrams. It delivers analytic finite parts for the previously unknown $A_{9,4}$ and $A_{9,2}$, the $O(\varepsilon^2)$ term for $A_{9,1}$ (weight seven), and a detailed $\varepsilon$‑expansion (up to $O(\varepsilon^5)$) for the non‑planar integral $I_{17}$, expressed in terms of multiple zeta values. The results are cross‑validated against independent calculations and illustrated with attention to homogeneous transcendental weight, PSLQ, and the role of prefactors. The study demonstrates the method’s efficiency and potential for extending analytic ε‑expansions to higher loops and more complex diagrams, aided by computational tools such as IBP, Mellin–Barnes, PSLQ, and sector decomposition.
Abstract
Applications of a method recently suggested by one of the authors (R.L.) are presented. This method is based on the use of dimensional recurrence relations and analytic properties of Feynman integrals as functions of the parameter of dimensional regularization, $d$. The method was used to obtain analytical expressions for two missing constants in the $ε$-expansion of the most complicated master integrals contributing to the three-loop massless quark and gluon form factors and thereby present the form factors in a completely analytic form. To illustrate its power we present, at transcendentality weight seven, the next order of the $ε$-expansion of one of the corresponding most complicated master integrals. As a further application, we present three previously unknown terms of the expansion in $ε$ of the three-loop non-planar massless propagator diagram. Only multiple $ζ$ values at integer points are present in our result.