Sketching the Bethe-Salpeter Kernel

An exact form is presented for the axial-vector Bethe-Salpeter equation, which is valid when the quark-gluon vertex is fully dressed. A Ward-Takahashi identity for the Bethe-Salpeter kernel is derived therefrom and solved for a class of dressed quark-gluon-vertex models. The solution provides a symmetry-preserving closed system of gap and vertex equations. The analysis can be extended to the vector equation. This enables a comparison between the responses of pseudoscalar and scalar meson masses to nonperturbatively dressing the quark-gluon vertex. The result indicates that dynamical chiral symmetry breaking enhances spin-orbit splitting in the meson spectrum.

Figure 1

Current-quark-mass dependence of pseudoscalar (lower portion) and scalar (upper) meson masses, obtained with $D=1{\mathrm{GeV}}^2$. The Ball-Chiu vertex [BC, Eq. (12)] result is compared with the rainbow-ladder result [RL, Eq. (14)].