Internal gauge invariance and inconsistency of the ladder approximation in QCD

The validity of the ladder approximation (LA) in QCD and QED in the context of the corresponding Schwinger–Dyson (SD) equations and Slavnov–Taylor (ST) and Ward–Takahashi (WT) identities is investigated. In contrast to QED, in QCD because of color degrees of freedom the summation of the ladder diagrams within the Bethe–Salpeter (BS) integral equation for the quark-gluon vertex at zero momentum transfer on account of the corresponding ST identity does provide an addition constraint on the quark SD equation itself. Moreover, the solution of the constraint equation requires the full quark propagator should be almost trivial (free-type) one, i.e. there is no nontrivial quark propagator in QCD in the LA. This triviality results in the fact that the standard LA ignores the self-interaction between gluons caused by color charges (non-Abelian character of QCD).