Bethe-Salpeter Meson Masses Beyond Ladder Approximation

The effect of quark-gluon vertex dressing on the ground state masses of the u/d-quark pseudoscalar, vector and axialvector mesons is considered with the Dyson-Schwinger equations. This extends the ladder-rainbow Bethe-Salpeter kernel to 2-loops. To render the calculations feasible for this exploratory study, we employ a simple infrared dominant model for the gluon exchange that implements the vertex dressing. The resulting model, involving two distinct representations of the effective gluon exchange kernel, preserves both the axial-vector Ward-Takahashi identity and charge conjugation symmetry. Numerical results confirm that the pseudoscalar meson retains its Goldstone boson character. The vector meson mass, already at a very acceptable value at ladder level, receives only 30 MeV of attraction from this vertex dressing. For the axial-vector states, which are about 300 MeV too low in ladder approximation, the results are mixed: the $1^{+-}$ state receives 290 MeV of repulsion, but the $1^{++}$ state is lowered further by 30 MeV. The exotic channels $0^{--}$ and $1^{-+}$ are found to have no states below 1.5 GeV in this model.