Three-gluon vertex in arbitrary gauge and dimension

Abstract

One-loop off-shell contributions to the three-gluon vertex are calculated, in arbitrary covariant gauge and in arbitrary space-time dimension, including quark-loop contributions (with massless quarks). It is shown how one can get the results for all on-shell limits of interest directly from the general off-shell expression. The corresponding general expressions for the one-loop ghost-gluon vertex are also obtained. They allow for a check of consistency with the Ward-Slavnov-Taylor identity.

Causal Construction of the Massless Vertex Diagram

“…This retarded distribution∆ ret The vertex function will be constructed below by performing the following calculational steps. The Fourier transform∆ 3 …”

“…Of course, the massless vertex functions plays also a prominent role in applied quantum field theories like QCD, where it emerges in the three-gluon vertex [3], or in perturbative models of quantum gravity. In this paper, the Fourier transform of the massless vertex distribution (also called vertex function in the sequel)…”

“…A typical example which is used in this paper as a model theory is the massless Φ 3 -theory, where the interaction Hamiltonian density is given by the normal ordered third order monomial of a free uncharged massless scalar field and a coupling constant λ H int (x) = λ 3! : Φ(x) 3 :,…”

“…The result for the massless case can be easily obtained by setting the electron mass equal to zero, since no infrared problems are present in∆ ± 3 and∆ 3 . For p ∈ V + and q ∈ V − ,∆ − 3 (p, q) vanishes and one can finally write down in a 'sloppy' style the result that can be found in the literature [3,6,14] …”

Non-perturbative Features of the Three-gluon Vertex in Landau Gauge

“…General tensor decompositions consistent with the SlavnovTaylor identities of Yang-Mills theory were considered in [2,3]. There are extensive studies within perturbation theory [4,5], but also non-perturbative calculations of infrared critical exponents for scaling [6,7] and decoupling [8] solutions. Together with lattice results for a particular projection of the vertex [9,10], this knowledge is sufficient for the construction of vertex models used in Dyson-Schwinger equation (DSE) based calculations of the gluon propagator [11].…”