New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams

TL;DR

This work advances the epsilon-expansion program for a broad class of one-, two-, and three-loop Feynman diagrams in dimensional regularization by developing analytic continuation rules, organizing constants into even/odd log-sine and Nielsen-polylogarithm bases, and introducing a new weight-five constant χ5. It delivers explicit epsilon-expansions for key master integrals, often in hypergeometric form (2F1, 3F2, 4F3) and via Mellin–Barnes representations, with PSLQ used to identify transcendental structures and connect to known constants such as Clausen and log-sine integrals. The results support higher-loop calculations and illuminate deep connections between Feynman diagram values, polylogarithms, and knot theory. Overall, the paper provides systematic techniques and concrete expansions that improve analytic control over multi-loop contributions in quantum field theory.

Abstract

For certain dimensionally-regulated one-, two- and three-loop diagrams, problems of constructing the epsilon-expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the epsilon-expansion can be calculated. For more complicated cases, only a few higher terms in epsilon are obtained. Apart from the one-loop two- and three-point diagrams, the examples include two-loop (mainly on-shell) propagator-type diagrams and three-loop vacuum diagrams. As a by-product, some new relations involving Clausen function, generalized log-sine integrals and certain Euler--Zagier sums are established, and some useful results for the hypergeometric functions of argument 1/4 are presented.

Figures (2)

• Figure 1: One-loop three-point function with masses $m_i$ and momenta $p_i$ ($p_1 + p_2 + p_3 = 0$)
• Figure 2: Two- and three-loop diagrams considered in the paper. Bold and thin lines correspond to massive and massless propagators, respectively.