On the existence of heavy pentaquarks: The large $N_c$ and heavy quark limits and beyond

We present a very general argument that the analogue of a heavy pentaquark (a state with the quantum numbers of a baryon combined with an additional light quark and a heavy antiquark $\overline{Q}$) must exist as a particle stable under strong interactions in the combined heavy quark and large $N_c$ limits of QCD. Moreover, in the combined limit these heavy pentaquark states fill multiplets of $\mathrm{SU}(4)\times \mathrm{O}(8)\times \mathrm{SU}(2)$. We explore the question of whether corrections in the combined $1/N_c$ and $1/m_Q$ expansions are sufficiently small to maintain this qualitative result. Since no model-independent way is known to answer this question, we use a class of realistic hadronic models in which a pentaquark can be formed via nucleon-heavy-meson binding through a pion-exchange potential. These models have the virtue that they necessarily yield the correct behavior in the combined limit, and the long-distance parts of the interactions are model independent. If the long-distance attraction in these models were to predict bound states in a robust way (i.e., largely insensitive to the details of the short-range interaction), then one could safely conclude that heavy pentaquarks do exist. However, in practice the binding does depend very strongly on the details of the short-distance physics, suggesting that the real world is not sufficiently near the combined large $N_c$, $m_Q$ limit to use it as a reliable guide. Whether stable heavy pentaquarks exist remains an open question.