TOWARDS THE TWO-LOOP Lcc VERTEX IN LANDAU GAUGE

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.

Alternative method of Reduction of the Feynman Diagrams to a set of Master Integrals

“…where the notation notation [12] = (x 1 − x 2 ) 2 of Ref. [2] is used. This property has been proven via Mellin-Barnes transformation, that is, [12] [31]…”

“…Integral relation (2) has been proven via Barnes lemmas in [16]. Going back from such a type of integral relations to diagrammatic relations we may construct the diagrammatic relations similar to the relation depicted in Fig.…”

“…x is a measure of the integration in the position space. This measure is useful in massless theories [2]. The result for the diagram (c) from Ref.…”

“…In Refs. [1,2,3,4] the calculation of the two-loop correction to this auxiliary double ghost vertex for N = 4 super-Yang-Mills (SYM) theory has been done in the limit ε → 0. The contributing diagrams are shown in Fig.1.…”

Gluon Self-Interaction in the Position Space in Landau Gauge

“…UV-finiteness in the momentum space can be checked for the Fourier transform of integral (4) for each subgraph and for both momentum integrations together, according to BPHZ R-operation. IR-finiteness of the double integrals can be checked directly for integral (4), for each integration separately and for both integrations together in the area |y| → ∞, |z| → ∞, in complete analogy with R-operation of the momentum space:…”