Are There Tetraquarks at Large $N_c$ in QCD(F)?

Weinberg recently pointed out a flaw in the standard argument that large $N_c$ QCD with color-fundamental quarks [QCD(F)] cannot yield narrow tetraquark states. In particular, he observed that the argument does not rule out narrow tetraquarks associated with the leading-order connected diagrams; such tetraquarks would have a width scaling as $N_c^{-1}$. It is shown here, however, that while the standard analysis of tetraquarks does not rule them out, a more thorough analysis rules out quantum-number exotic tetraquarks associated with the leading-order connected diagrams. This analysis is based entirely on conventional assumptions used in large $N_c$ physics applied to the analytic properties of meson-meson scattering. Our result implies that one of three possibilities must be true: i) quantum-number exotic tetraquarks do not exist at large $N_c$; ii) quantum-number exotic tetraquarks exist, but are associated with subleading connected diagrams and have anomalously small widths that scale as $N_c^{-2}$ or smaller; or iii) the conventional assumptions used in large $N_c$ analysis are inadequate.