Massless two-loop "master" and three-loop two point function in NDIM

TL;DR

This paper demonstrates the use of the Negative Dimensional Integration Method (NDIM) for calculating complex multi-loop Feynman diagrams, successfully deriving known two-loop and three-loop two-point functions with improved techniques.

Contribution

It introduces a simplified NDIM approach for triangle diagrams that enables accurate loop-by-loop calculations of multi-loop Feynman integrals.

Findings

Validated the NDIM method against known two-loop results.

Extended the method to compute a three-loop two-point function.

Confirmed the consistency of the new approach with existing results.

Abstract

NDIM (Negative Dimensional Integration Method) is a technique for evaluating Feynman integrals based on the concept of analytic continuation. The method has been successfully applied to many diagrams in covariant and noncovariant gauge field interactions and has shown its utility as a powerful technique to handle Feynman loop integrations in quantum field theories. In principle NDIM can handle any loop calculation; however, in practical terms, the resulting multiseries with several variables in general cannot be summed up conveniently and its analytic properties are generally unknown. The alternative then is to use order by order (loop by loop) integration in which the first integral is of the triangle diagram type. However, the na\"{i}ve momentum integration of this leads to wrong results. Here we use the shortened version for the triangle in NDIM that is suitable for a loop by loop calculation and show that it leads (after appropriate analytic continuation to positive dimension) to agreement with the known result for the two-loop master diagram. From it, a three-loop is then calculated and shown again its consistency with the already published result for such a diagram.