Symmetry preserving truncations of the gap and Bethe-Salpeter equations

Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing stringent relationships between the kernels of the one- and two-body problems, which must be preserved in any veracious treatment of mesons as bound states. In this connection, one may view the dressed gluon-quark vertex, Γ_μ^a , as fundamental. We use a novel representation of Γ_μ^a , in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quark-antiquark Bethe-Salpeter kernel, K , that is symmetry consistent with a given quark gap equation. A strength of the scheme is its ability to expose and capitalize on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of H -diagrams in K , which are two-particle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of Γ_μ^a in a Bethe-Salpeter kernel nor expressed as a member of the class of crossed-box diagrams. Thus, there are no general circumstances under which the WGT identities essential for a valid description of mesons can be preserved by a Bethe-Salpeter kernel obtained simply by dressing both gluon-quark vertices in a ladderlike truncation; and, moreover, adding any number of similarly dressed crossed-box diagrams cannot improve the situation.