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Tetraquark resonances with the triple flip-flop potential, decays in the cherry in a broken glass approximation

We develop a unitarized formalism to study tetraquarks using the triple flip-flop potential, which includes two meson-meson potentials and the tetraquark four-body potential. This can be related to the Jaffe-Wilczek and to the Karliner-Lipkin tetraquark models, where we also consider the possible open channels, since the four quarks and antiquarks may at any time escape to a pair of mesons. Here we study a simplified two-variable toy model and explore the analogy with a cherry in a glass, but a broken one where the cherry may escape from. It is quite interesting to have our system confined or compact in one variable and infinite in the other variable. In this framework we solve the two-variable Schrödinger equation in configuration space. With the finite difference method, we compute the spectrum, we search for localized states and we attempt to compute phase shifts. We then apply the outgoing spherical wave method to compute in detail the phase shifts and and to determine the decay widths. We explore the model in the equal mass case, and we find narrow resonances. In particular the existence of two commuting angular momenta is responsible for our small decay widths.