Modern summation technologies and the calculation of 3-loop diagrams
Carsten Schneider, Jakob Ablinger, Johannes Blümlein A, M. Round · 2013
Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in 4 + ε-dimensional Minkowski space, can be transformed to multi-integrals or multisums over hyperexponential and/or hypergeometric functions depending on a discrete parameter n.Given such a specific representation, we utilize an enhanced version of the multivariate Almkvist-Zeilberger algorithm (for multi-integrals) and a common summation framework of the holonomic and difference field approach (for multi-sums) to calculate recurrence relations in n.Finally, solving the recurrence we can decide efficiently if the first coefficients of the Laurent series expansion of a given Feynman integral can be expressed in terms of indefinite nested sums and products; if yes, the all n solution is returned in compact representations, i.e., no algebraic relations exist among the occurring sums and products.