Exploring the quark-gluon vertex with Slavnov–Taylor identities and lattice simulations

The soft gluon limit of the longitudinal part of the quark-gluon vertex is studied by resorting to non-perturbative approaches to quantum chromodynamics (QCD). Based on a Slavnov–Taylor identity (STI), the longitudinal form factors is expressed in terms of the quark-ghost kernel, the quark self energy and the quark wave function. An exact relation between the non-vanishing longitudinal form factors is derived for the soft gluon limit and explored to understand the behaviour of the vertex. Within a Ball–Chiu vertex, the form factor $$\lambda _1$$λ1 was analysed using recent lattice simulations for full QCD for the soft gluon limit. The lattice data shows that the gluon propagator resumes the momentum dependence of such component of the vertex. This connection is understood via a fully dressed one-loop Bethe–Salpeter equation. The behaviour of the remaining longitudinal form factors $$\lambda _2(p^2)$$λ2(p2) and $$\lambda _3(p^2)$$λ3(p2) is investigated combining both the information of lattice simulations and the derived relations based on the STI.