Single-scale diagrams and multiple binomial sums

TL;DR

The paper studies the $\varepsilon$-expansion of two-loop, single-scale self-energy diagrams with different thresholds and one mass, showing that on-shell results reduce to multiple binomial sums that can be evaluated analytically. By combining a semi-analytic method, PSLQ, and a basis aligned with sixth roots of unity, the authors express the on-shell results in terms of zeta values, log-sine integrals, and Euler–Zagier sums, with the transcendental content governed by diagram topology and the presence of specific mass thresholds. They provide explicit on-shell results for three diagrams, reveal the need for new constants when two massive cuts are present, and present hypergeometric and integral representations that facilitate continuation and higher-order expansions. The findings illuminate how topology constrains transcendental structures in massive single-scale diagrams and offer practical tools for their analytic evaluation in higher-loop renormalization contexts.

Abstract

The $ε$-expansion of several two-loop self-energy diagrams with different thresholds and one mass are calculated. On-shell results are reduced to multiple binomial sums which values are presented in analytical form.

Figures (1)

• Figure 1: Bold and thin lines correspond to massive and massless propagators, respectively.