A definition of “dynamical generation,” a hotly debated topic at present, is proposed and its implications are discussed. This definition, in turn, leads to a method allowing one to distinguish, in principle, tetraquark and molecular states. The different concept of “dynamical reconstruction” is also introduced and applies to the generation of preexisting mesons (quark-antiquark, glueballs, …) via unitarization methods applied to low-energy effective Lagrangians. Large-$N_c$ arguments play an important role in all these investigations. A simple toy model with two scalar fields is introduced to elucidate these concepts. The large-$N_c$ behavior of the parameters is chosen in order that the two scalar fields behave as quark-antiquark mesons. When the heavier field is integrated out, one is left with an effective Lagrangian with the lighter field only. A unitarization method applied to the latter allows one to “reconstruct” the heavier “quarkoniumlike” field, which was previously integrated out. It is shown that a Bethe-Salpeter analysis is capable of reproducing the preformed quark-antiquark state, and that the corresponding large-$N_c$ behavior can be brought in agreement with the expected large-$N_c$ limit; this is a subtle and interesting issue on its own. However, when only the lowest term of the effective Lagrangian is retained, the large-$N_c$ limit of the reconstructed state is not reproduced: Instead of the correct large-$N_c$ quarkonium limit, it fades out as a molecular state would do. Implications of these results are presented: It is proposed that axial-vector, tensor, and (some) scalar mesons just above 1 GeV, obtained via the Bethe-Salpeter approach from the corresponding low-energy, effective Lagrangian in which only the lowest term is kept, are quarkonia states, in agreement with the constituent quark model, although they might fade away as molecular states in the large-$N_c$ limit.

• Received 2 April 2009

DOI:https://doi.org/10.1103/PhysRevD.80.074028