In this work, we use functional methods, namely the formalism of Dyson-Schwinger
and Bethe-Salpeter equations (DSEs and BSEs) and the n-particle irreducible
effective action formalism, to describe properties of quarks and mesons as bound
states of quarks and antiquarks. We explore how truncations of these equations
relate to the phenomenon of dynamical chiral symmetry breaking in quantum
chromodynamics (QCD). Within the rainbow-ladder truncation, we explore, how
different effective running couplings give rise to interaction potentials and how they
relate to Regge behaviour in the resulting meson spectra. In particular, we explore
spectra of light mesons, kaons, ss-states, charmonia and bottomonia with total
angular momenta up to J ≤ 5 within this truncation.
Furthermore, we use a universal kernel-first truncation to construct a quark
self-energy from a quark-antiquark scattering kernel, such that the axialvector
Ward-Takahashi identity and thus the effects of dynamical chiral symmetry breaking
are conserved. In this truncation we solve the equations of motion for the quark
propagator and quark-photon vertex to investigate, whether the vector Ward-
Takahashi identity is fulfilled. Starting from a three particle irreducible effective
action, we use this method to extract exploratory spectra of light mesons, kaons
and ss-states with total angular momentum up to J ≤ 4 and sketch a way to
expand this framework to heavier quarkonia and heavy-light mesons.
