Transformations of triangle ladder diagrams

TL;DR

This paper presents a method to transform triangle ladder diagram integrals into half-diamond diagrams in dual space, simplifying the derivation of relations without relying on known four-point diagram results.

Contribution

It introduces a new diagrammatic transformation technique that relates triangle ladder diagrams to half-diamonds without external input from four-point diagrams.

Findings

Transformation from triangle ladder to half-diamonds demonstrated

Derivation of relations without using known four-point diagram results

Intermediate use of diamond diagram results in the transformation process

Abstract

It is shown how dual space diagrammatic representation of momentum integrals corresponding to triangle ladder diagrams with an arbitrary number of rungs can be transformed to half-diamonds. In paper arXiv:0803.3420 [hep-th] the half-diamonds were related by conformal integral substitution to the diamonds which represent the dual space image of four-point ladder integrals in the four-dimensional momentum space. Acting in the way described in the present paper we do not need to use the known result for diamond (four-point) diagrams as an external input in deriving relations of arXiv:0803.3420 [hep-th], however, that result for the diamond diagram arises in the present proof as an intermediate consequence in a step-by-step diagrammatic transformation from the triangle ladder diagram to the half-diamond diagrams.