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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>1</a:mn><a:mo>/</a:mo><a:msub><a:mi>N</a:mi><a:mi>c</a:mi></a:msub></a:math> 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 via 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mn>1</c:mn><c:mo>/</c:mo><c:msubsup><c:mi>N</c:mi><c:mi>c</c:mi><c:mn>2</c:mn></c:msubsup></c:math>. 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>1</e:mn><e:mo>/</e:mo><e:msubsup><e:mi>N</e:mi><e:mi>c</e:mi><e:mn>2</e:mn></e:msubsup></e:math>, 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.