Abstract
In this paper we test an approximate method that is often used in lattice studies of the Landau gauge three-gluon vertex. The approximation consists in describing the lattice correlator with tensor bases from the continuum theory. With the help of vertex reconstruction, we show that this "continuum" approach may lead, for general kinematics, to significant errors in vertex tensor representations. Such errors are highly unwelcome, as they can lead to wrong quantitative estimates for vertex form factors and related quantities of interest, like the three-gluon running coupling. As a possible solution, we demonstrate numerically and analytically that there exist special kinematic configurations for which the vertex tensor structures can be described exactly on the lattice. For these kinematics, the dimensionless tensor elements are equal to the continuum ones, regardless of the details of the lattice implementation. We ran our simulations for an $SU(2)$ gauge theory in two and three spacetime dimensions, with Wilson and $\mathcal{O}(a^2)$ tree-level improved gauge actions. Our results and conclusions can be straightforwardly generalised to higher dimensions and, with some precautions, to other lattice correlators, like the ghost-gluon, quark-gluon and four-gluon vertices.
Highlights
The primitively divergent vertex functions of quantum chromodynamics (QCD) and its quenched version, the pure Yang-Mills theory, have been the subject of numerous nonperturbative investigations in the past two decades
With the help of vertex reconstruction, we show that this “continuum” approach may lead, for general kinematics, to significant errors in vertex tensor representations
In this paper we have introduced the method of vertex reconstruction as a means of checking the fidelity of various tensor representations of lattice vertex functions
Summary
The primitively divergent vertex functions of quantum chromodynamics (QCD) and its quenched version, the pure Yang-Mills theory, have been the subject of numerous nonperturbative investigations in the past two decades. Due to the breaking of rotational symmetry, the continuum tensor bases cannot be applied in discretized spacetime, at least not for general kinematics This has been explicitly demonstrated for the lattice gluon propagator in Landau gauge [53]. As an alternative to continuum bases, some authors have used tree-level tensor elements from lattice perturbation theory [41,42,43,44,45], which, do not provide a complete representation for most vertex functions. We apply the method to the lattice Landau gauge gluon propagator and three-gluon vertex, and demonstrate that, for general kinematics, these functions are described relatively poorly by the continuum tensor bases. The important technical details have been relegated to Appendixes A and B, while Appendix C contains some of our results for vertex dressing functions
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