Axial-vector mesons and the problem of $\pi a_1$ mixing

In the framework of the Nambu–Jona–Lasinio (NJL) model, a mechanism for eliminating the off-diagonal $\pi a_1$ transitions from the effective meson Lagrangian is investigated. The problem is considered in the oneloop approximation for the quark loops. We employ the minimal superposition linear in the fields, $a_\mu' =a_\mu - c\partial_\mu\pi$, and show that the covariant transformation properties of both the axial-vector and vector physical fields are affected by this substitution. Contrary to some claims found in the literature, we demonstrate that the discussed change in covariant transformation properties of these fields does not result in various violations of chiral symmetry, but rather reduces to a redefinition of the $S$-matrix elements beyond the mass shell. At the same time, the covariant and noncovariant approaches yield identical on-shell results.