Meson scattering and tetraquarks in two-dimensional QCD

Two-quark--two-antiquark systems with four different quark flavors are considered in the framework of two-dimensional QCD, in the light-cone gauge, at leading orders of the $1/N_c^{}$ expansion. Introducing basis functions with color-singlet mesonic clusters, integral equations are established for the Green's functions and the related scattering amplitudes involved in the sectors of the direct and quark-exchange channels. The problem of infrared divergences is dealt with a systematic use of an infrared regulator cutoff introduced in the gluon propagator. It is shown that the on-mass shell scattering amplitudes are free of infrared divergences up to order $1/N_c^2$. In the limit of vanishing of the infrared cutoff, they can be represented by effective four-meson contact-type interaction terms with unitarity loops, calculable in terms of the meson wave functions and propagators. The results, obtained at order $1/N_c^2$, are summed with the constraint of unitarization. The unitarized scattering amplitudes are continued in the total mass-squared variable below the two-meson thresholds, leading to a tetraquark bound state equation, which generally has one solution. Spectroscopic applications of the above results are sketched and discussed.