Bethe-Salpeter equation and a nonperturbative quark-gluon vertex

A Ward-Takahashi identity preserving Bethe-Salpeter kernel can always be calculated explicitly from a dressed-quark-gluon vertex whose diagrammatic content is enumerable. We illustrate that fact using a vertex obtained via the complete resummation of dressed-gluon ladders. While this vertex is planar, the vertex-consistent kernel is nonplanar and that is true for any dressed vertex. In an exemplifying model the rainbow-ladder truncation of the gap and Bethe-Salpeter equations yields many results; e.g., π- and ρ-meson masses, that are changed little by including higher-order corrections. Repulsion generated by nonplanar diagrams in the vertex-consistent Bethe-Salpeter kernel for quark-quark scattering is sufficient to guarantee that diquark bound states do not exist.A. Bender, W. Detmold, A. W. Thomas and C. D. Robert