Towards the two-loop Lcc vertex in Landau gauge

TL;DR

This paper computes a specific two-loop contribution to the Lcc vertex in Yang-Mills theory in position space, revealing scale independence and potential insights into the theory's conformal structure.

Contribution

It presents the first calculation of one of five two-loop contributions to the Lcc vertex in Landau gauge Yang-Mills theory, using position space techniques.

Findings

Result is scale-independent, depending only on ratios of distances.

Expressed in terms of logarithms and Davydychev integral J(1,1,1).

Utilizes Gegenbauer polynomial and uniqueness methods.

Abstract

We are interested in the structure of the Lcc vertex in the Yang-Mills theory, where c is the ghost field and L the corresponding BRST auxiliary field. This vertex can give us information on other vertices, and the possible conformal structure of the theory should be reflected in the structure of this vertex. There are five two-loop contributions to the Lcc vertex in the Yang-Mills theory. We present here calculation of the first of the five contributions. The calculation has been performed in the position space. One main feature of the result is that it does not depend on any scale, ultraviolet or infrared. The result is expressed in terms of logarithms and Davydychev integral J(1,1,1) that are functions of the ratios of the intervals between points of effective fields in the position space. To perform the calculation we apply Gegenbauer polynomial technique and uniqueness method.