Abstract
Using Hopf-algebraic structures as well as diagrammatic techniques for determining the Slavnov-Taylor identities for QCD we construct the relations for the triple and quartic gluon vertices at one loop. By making the longitudinal projection on an external gluon of a Green's function we show that the gluon self-energy of that leg is consistently replaced by a ghost self-energy. The resulting identities are then studied by evaluating all the graphs for an off-shell non-exceptional momentum configuration. In the case of the 3-point function this is for the most general momentum case and for the 4-point function we consider the fully symmetric point.
Highlights
One of the cornerstones of quantum field theory is the accommodation of spin-1 gauge fields in the Lagrangian of a theory in such a way that the core properties of the gauge field are retained
The main goal is to clarify how gauge symmetry can be expressed on the level of renormalized
In the previous section we considered the completely off-shell 3-point identity and it is apparent in Fig. 6 that these functions play a role
Summary
One of the cornerstones of quantum field theory is the accommodation of spin-1 gauge fields in the Lagrangian of a theory in such a way that the core properties of the gauge field are retained. To study the Slavnov-Taylor identities afresh we return to basics and apply modern algebraic and diagrammatic methods to construct the identities of the various relevant 3- and 4-point functions These will involve the triple gluon and ghost-gluon vertices and both the pure gluon and ghost-gluon 4-point functions. While the 3- and 4-point ghost-gluon vertex functions have been studied in earlier work we have to carry out a new evaluation here This is because in the standard construction of the Slavnov-Taylor identity the vertex connecting to one of the external ghost fields is not the standard one derived using the Faddeev-Popov method [1,2,3]. 4-point function calculation with the final appendix giving explicit expressions for the purely gluonic 3- and 4-point functions
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