Exploratory study of three-point Green’s functions in Landau-gauge Yang-Mills theory

Abstract

Green's functions are a central element in the attempt to understand non-perturbative phenomena in Yang-Mills theory. Besides the propagators, 3-point Green's functions play a significant role, since they permit access to the running coupling constant and are an important input in functional methods. Here we present numerical results for the two non-vanishing 3-point Green's functions in 3d pure SU (2) Yang-Mills theory in (minimal) Landau gauge, i.e. the three-gluon vertex and the ghost-gluon vertex, considering various kinematical regimes. In this exploratory investigation the lattice volumes are limited to 20 3 and 30 3 at β = 4.2 and β = 6.0. We also present results for the gluon and the ghost propagators, as well as for the eigenvalue spectrum of the Faddeev-Popov operator. Finally, we compare two different numerical methods for the evaluation of the inverse of the Faddeev-Popov matrix, the point-source and the plane-wave-source methods.

Nonperturbative study of the four gluon vertex

“…We conclude this section by commenting on certain issues that appear when the quantities defined on the lattice for studying vertices [15,16] (to date, only three-point functions) are extended to the case of the four-gluon vertex.…”

“…This choice, in addition to simplifying the algebraic structures considerably, corresponds precisely to the contributions that would survive on the lattice, if one were to consider the standard quantities employed in the simulations of vertices [15,16] (we will return to this point in section 4.3).…”

“…This is to be contrasted with the corresponding status of all other vertices, namely the quark-gluon, the ghost-gluon, and three-gluon vertex, which have been studied on the lattice, at least for some special choices of their momenta [11][12][13][14][15][16].…”

“…In particular, after the replacements mentioned above, the integrals in table 1 can be expressed in terms of I(M 2 ) as follows 15) where the ellipses in the last two expressions indicate linear combinations of the integrals 5 m 2…”

“…It is likely, however, that this particular property will not persist in a complete nonperturbative computation, as the one provided by lattice simulations. Therefore, under the assumption that such a structure might eventually emerge, we describe how to express the complete set of form factors characterizing Γ abcd µνρσ in terms of the standard lattice ratios R, used in studies of the three-gluon vertex [15,16].…”