Abstract
In this paper we study the nonperturbative structure of the SU(3) four-gluon vertex in the Landau gauge, concentrating on contributions quadratic in the metric. We employ an approximation scheme where "one-loop" diagrams are computed using fully dressed gluon and ghost propagators, and tree-level vertices. When a suitable kinematical configuration depending on a single momentum scale $p$ is chosen, only two structures emerge: the tree-level four-gluon vertex, and a tensor orthogonal to it. A detailed numerical analysis reveals that the form factor associated with this latter tensor displays a change of sign (zero-crossing) in the deep infrared, and finally diverges logarithmically. The origin of this characteristic behavior is proven to be entirely due to the masslessness of the ghost propagators forming the corresponding ghost-loop diagram, in close analogy to a similar effect established for the three-gluon vertex. However, in the case at hand, and under the approximations employed, this particular divergence does not affect the form factor proportional to the tree-level tensor, which remains finite in the entire range of momenta, and deviates moderately from its naive tree-level value. It turns out that the kinematic configuration chosen is ideal for carrying out lattice simulations, because it eliminates from the connected Green's function all one-particle reducible contributions, projecting out the genuine one-particle irreducible vertex. Motivated by this possibility, we discuss in detail how a hypothetical lattice measurement of this quantity would compare to the results presented here, and the potential interference from an additional tensorial structure, allowed by Bose symmetry, but not encountered within our scheme.
Highlights
From the point of view of lattice simulations, the situation is simpler, in the sense that, to the best of our knowledge, no simulations of the four-gluon vertex have been performed, for any kinematic configuration
In this paper we study the nonperturbative structure of the SU(3) four-gluon vertex in the Landau gauge, concentrating on contributions quadratic in the metric
A detailed numerical analysis reveals that the form factor associated with this latter tensor displays a change of sign in the deep infrared, and diverges logarithmically. The origin of this characteristic behavior is proven to be entirely due to the masslessness of the ghost propagators forming the corresponding ghost-loop diagram, in close analogy to a similar effect established for the three-gluon vertex
Summary
The 1-PI four-gluon vertex will be denoted by the expression (all momenta entering). ΓAaμAbν AcρAdσ (p1, p2, p3, p4) = −ig2Γaμbνcρdσ(p1, p2, p3, p4). It turns out that, within the one-loop dressed approximation and the kinematical configuration that we will employ (see figure 2 for the 18 diagrams appearing in this case), the color tensors reduce to the two structures appearing in the conventional one-loop calculation of this vertex (for N = 3), namely the tree-level tensor Γ(0) defined in eq (2.2), and the totally symmetric (both in Minkowski and color space) tensor. It is useful to briefly review some of their IR features that are most relevant to the present work Both large-volume lattice simulations and a plethora of continuous nonperturbative studies, carried out both in SU(2) and in SU(3), converge to the conclusion that the function ∆(q2) reaches a finite (nonvanishing) value in the IR. Qualitatively distinct compared to the three-gluon case, is that, at least within the approximation scheme that we employ, this particular divergence does not manifest itself in the part proportional to Γ(0), but rather in the orthogonal combination G
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