Light tetraquarks and mesons in a DSE/BSE approach
Part I:Bound states and their properties are an inherent non-perturbative feature of QCD. Moreover, QCD is a confining theory so that instead of the elementary quarks and gluons themselves, only colourless bound states formed of these elementary particles are directly measurable. One non-perturbative framework to describe QCD are the ... Dyson-Schwinger equations, which interrelate all Green functions of the theory by an infinite tower of integral equations, and the corresponding Bethe-Salpeter equations that define the bound states of the theory. To reduce the infinite tower to a tractable form, the equations have to be truncated. In this thesis the so-called rainbow ladder´ truncation was used that reduces the quark-gluon vertex to the bare vertex and replaces the gluon by an effective modeled one so that the only Green function that has to be solved, is the quark propagator. This truncation preserves the important axial Ward-Takahashi-identity and the Gell-Mann-Oakes-Renner relation.For the effective gluon the Maris-Tandy interaction was used, modeled to reproduce the pion mass and decay constant. Starting from this well-established truncation, the four-body tetraquark Bethe-Salpeter equation was constructed. To solve the tetraquark Bethe-Salpeter equation, a fully covariant basis for the tetraquark amplitude is necessary. Additionally, the basis has to reflect the quantum numbers of the tetraquark and has to fulfill the Pauli principle. The construction of such a basis was performed for all parts of the amplitude: The Dirac-tensor structure, the phase space, the colour and the flavour tensor structure.Upon solving the tetraquark bound state equation, dynamical pion poles in the tetraquark amplitude phase space appeared, reflecting the actual physics that determines the tetraquark:The tetraquark is dominated by two-body correlations which manifest themselves as poles in the phase space. It is especially noteworthy that these two-body correlations in form of poles are of a dynamical nature and are not put in by hand. Additionally, these two-body poles in the four-body equation can be interpreted as connection between the more fundamental four-body picture, where four quarks bind together, and the two-body picture, where the tetraquark is pictured as a bound state of two mesons and/or diquarks. In accordance with previous studies in a two-body framework, the pion-pion correlations are found to be much more dominant than the diquark-diquark correlations.Guided by the result that the tetraquark is dominated by poles in the phase space, an explicit pole ansatz for the amplitude was constructed, improving the numerical stability considerably. Subsequently, the Bethe-Salpeter equation was solved for tetraquarks with the quantum numbers 0++.For physical u/d-quark masses, the masses of the sigma (0.35 GeV), the kappa (0.64 GeV) and the f0/a0 (0.89 GeV) were calculated, with the corresponding masses given in brackets. Compared with the values of the experimental candidates, the masses are generically too low, probably caused by truncation artifacts. Nonetheless, according to the success of the Maris-Tandy model to describe ground state properties of mesons and baryons, the result is a strong indication that the lowest scalar nonet has indeed a considerable tetraquark component. Investigating the quark mass dependence of the sigma, candidates for an all strange tetraquark around 1.6 GeV and an all charm tetraquark around 5.7 GeV were found. These findings agree qualitatively with former results from a two-body approach. Additionally, the mass curve features an interesting cusp at a quark mass of about 0.65 GeV. Such cusps are known in the literature to be related to whether the T-matrix pole corresponds to a bound-state, a resonance or a virtual state.Part II:Following the time-honored concept of taking functional derivatives to obtain an interaction kernel, this technique is extended to vertex models which explicitly depend on the quark propagator and it´s dressing functions. This enables one to derive closed expressions for the interaction kernelbeyond the rainbow-ladder approximation. This technique is very general, and in principle applicable to any vertex that is given in terms of quark dressing functions. As an improvement over previous approaches this technique allows one to determine not only the masses of the bound-states but also their Bethe-Salpeter wave functions. As examples, this technique was applied to two type of vertices, the Ball-Chiu vertex and the Munczek vertex that both respect the constraints due to the vector Ward-Takahashi identity but contain additional structures related to spin-orbit forces. Upon solving the BSE for pseudo-scalar, scalar, vector and axial-vector mesons it was found that these structures alone are not sufficient to generate a physical spectrum of light mesons while keeping the pion properties intact.