# The Form Factors of the Gauge-Invariant Three-Gluon Vertex

**Michael Binger**

The gauge-invariant three-gluon vertex obtained from the pinch technique is characterized by thirteen nonzero form factors, which are given in complete generality for unbroken gauge theory at one loop. The results are given in d dimensions using both dimensional regularization and dimensional reduction, including the effects of massless gluons and arbitrary representations of massive gauge bosons, fermions, and scalars. We find interesting relations between the functional forms of the contributions from gauge bosons, fermions, and scalars. These relations hold only for the gauge-invariant pinch technique vertex and are d-dimensional incarnations of supersymmetric nonrenormalization theorems which include finite terms. The form factors are shown to simplify or N = 1, 2, and 4 supersymmetry in various dimensions. In four-dimensional non-supersymmetric theories, eight of the form factors have the same functional orm for massless gluons, quarks, and scalars, when written in a physically motivated tensor basis. For QCD, these include the tree-level tensor structure which has prefactor beta0 = (11Nc-2Nf )/3, another tensor with prefactor 4Nc-Nf , and six tensors with Nc-Nf . In perturbative calculations our results lead naturally to an effective coupling for the three-gluon vertex, alpha_tilde(k1^2, k2^2, k3^2), which depends on three momenta and gives rise to an effective scale Qeff^2 (k1^2, k2^2, k3^2) which governs the behavior of the vertex. The effects of nonzero internal masses M are important and have a complicated threshold and pseudo-threshold structure. A three-scale effective number of flavors Nf (k1^2/M^2, k2^2/M^2, k3^2/M^2) is defined. The results of this paper are an important part of a gauge-invariant dressed skeleton expansion and a related multi-scale analytic renormalization scheme. In this approach the scale ambiguity problem is resolved since physical kinematic invariants determine the arguments of the couplings.