We develop a covariant approach to describe the low-lying scalar, pseudoscalar, vector, and axial-vector mesons as quark-antiquark bound states. This approach is based on an effective interaction modeling the nonperturbative structure of the gluon propagator that enters the quark Schwinger-Dyson and meson Bethe-Salpeter equations. We consistently treat these integral equations by precisely implementing the quark propagator functions that solve the Schwinger-Dyson equations into the Bethe-Salpeter equations in the relevant kinematical region. We extract the meson masses and compute the pion and kaon decay constants. We obtain a quantitatively correct description for pions, kaons, and vector mesons while the calculated spectra of scalar and axial-vector mesons suggest that their structure is more complex than being quark-antiquark bound states.

We compare numerical solutions to the full field equations to simplified approaches based on implementing three collective coordinates for kink-antikink interactions within the ${\phi }^4$ and ${\varphi }^6$ models in one time and one space dimensions. We particularly pursue the question whether the collective coordinate approximation substantiates the conjecture that vibrational modes are important for resonance structures to occur in kink-antikink scattering.