The isospin, spin and parity dependent potential of a pair of $B$ mesons is computed using Wilson twisted mass lattice QCD with two flavors of degenerate dynamical quarks. The $B$ meson is addressed in the static-light approximation, i.e. the $b$ quarks are infinitely heavy. From the results of the $BB$ meson-meson potentials, a simple rule can be deduced stating which isospin, spin and parity combinations correspond to attractive and which to repulsive forces. We provide fits to the ground state potentials in the attractive channels and discuss the potentials in the repulsive and excited channels. The attractive channels are the most important since they can possibly lead to a bound four-quark state, i.e. a $\overline{b}\overline{b}ud$ tetraquark. Using these attractive potentials in the Schrödinger equation, we find an indication for such a tetraquark state of two static bottom antiquarks and two light $u/d$ quarks with mass extrapolated down to the physical value.

The interpretation of the Landau gauge lattice gluon propagator as a massive type bosonic propagator is investigated. Three different scenarios are discussed: i) an infrared constant gluon mass; ii) an ultraviolet constant gluon mass; iii) a momentum dependent mass. We find that the infrared data can be associated with a massive propagator up to momenta $\sim 500$ MeV, with a constant gluon mass of 723(11) MeV, if one excludes the zero momentum gluon propagator from the analysis, or 648(7) MeV, if the zero momentum gluon propagator is included in the data sets. The ultraviolet lattice data is not compatible with a massive type propagator with a constant mass. The scenario of a momentum dependent gluon mass gives a decreasing mass with the momentum, which vanishes in the deep ultraviolet region. Furthermore, we show that the functional forms used to describe the decoupling like solution of the Dyson-Schwinger equations are compatible with the lattice data with similar mass scales.

We combine lattice QCD results for the potential of two static antiquarks in the presence of two quarks $qq$ of finite mass and quark model techniques to study possible existing $qq\overline{b}\overline{b}$ tetraquarks within the Born-Oppenheimer approximation. While there is strong indication for a bound four-quark state for $qq=(ud-du)/\sqrt{2}$, i.e., isospin $I=0$, which is stable with respect to QCD, i.e., can only decay weakly, we find clear evidence against the existence of corresponding tetraquarks with $qq\in \{uu,(ud+du)/\sqrt{2},dd\}$, i.e., isospin $I=1$, $qq=ss$, and $qq=cc$.

Utilizing lattice QCD results for the potential of two static antiquarks and two dynamical quarks as well as quark model techniques for the dynamics of two heavy antiquarks in a cloud of two light quarks, we are provided with an accurate framework for the study of possibly existing heavy-heavy-light-light tetraquarks. Among the possible quantum numbers of such a system, we find binding in only one channel the scalar isosinglet. Solving the Schr\"odinger equation for the displacement of the heavy antiquarks and taking systematic errors into account, we find an antibottom-antibottom-light-light bound state with a confidence level of around $1.8$\sigma${}\char21{}3.0$\sigma${}$ and binding energy of approximately 30\char21{}57 MeV.

We investigate spin effects in four-quark systems consisting of two heavy antibottom quarks and two light up/down quarks. To this end, we use the Born-Oppenheimer approximation. We utilize potentials of two static antiquarks in the presence of two quarks of finite mass computed via lattice QCD and solve a coupled-channel Schrödinger equation for the antibottom-antibottom separation. Without taking heavy quark spins into account, this approach predicted a $ud\overline{b}\overline{b}$ tetraquark bound state with quantum numbers $I(J^P)=0(1^{+})$. We now extend this Born-Oppenheimer approach with coupled-channel Schrödinger equations, allowing us to incorporate effects due to the heavy $\overline{b}$ spins. We confirm the existence of the $ud\overline{b}\overline{b}$ tetraquark.

In this work we assume that quarks are described by Dirac spinors, with current masses which eventually can be set to zero, interacting through a confining and chirally invariant potential. Other than the strength of the interquark potential and the current masses of the quarks we have no free parameters. ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ quark-antiquark vacuum condensation is allowed and the mass gap equation is solved for the chosen potential. The solution, and even the mere existence of it, depends quite sharply on the chosen potentials. Vacuum condensation is shown to be responsible for partial conservation of axial-vector current and for the constituent scale. The mass gap equation also ensures us that quark annihilation is obtained in a consistent way. In our formalism quarks and antiquarks appear explicitly, which greatly simplifies the derivation of both the Salpeter equations for meson bound states and the resonating-group-method equations for meson decays and scattering.