Tetraquark bound states and resonances in the unitary and microscopic triple string flip-flop quark model, the light-light-antiheavy-antiheavy $q q \bar Q\bar Q$ case study

We address $q q \bar Q\bar Q$ exotic tetraquark bound states and resonances with a fully unitarized and microscopic quark model. We propose a triple string flip-flop potential, inspired in lattice QCD tetraquarks static potentials and fluxtubes, combining meson-meson and tetraquark potentials. Our potential goes up to the color excited potential, but neglects spin-tensor potentials. To search for bound states and resonances, we first solve the two-body mesonic problem. Then we develop fully unitary techniques to address the four-body tetraquark problem. We fold the four-body Shcr\"odinger equation with the mesonic wavefunctions, transforming it into a two-body meson-meson problem with non-local potentials. We find bound states for some quark masses numbers, including the one reported in lattice QCD. Moreover, we also find resonances and calculate their masses and widths, by computing the $\mbox{T}$ matrix and finding it's pole positions in the complex energy plane, for some quantum numbers. However a detailed analysis of the quantum numbers where binding exists shows a discrepancy with recent lattice QCD results for the $l l \bar b \bar b$ tetraquark bound states. We conclude that the string flip-flop models need further improvement.