Scale invariance in heavy hadron molecules

We discuss a scenario in which the Pc(4450)⁺ heavy pentaquark is a ΣcD̄*-Λc(2595)D̄ molecule. The Λc1D̄→ΣcD̄* transition is mediated by the exchange of a pion almost on the mass shell that generates a long-range 1/r² potential. This is analogous to the effective force that is responsible for the Efimov spectrum in three-boson systems interacting through short-range forces. The equations describing this molecule exhibit approximate scale invariance, which is anomalous and broken by the solutions. If the 1/r² potential is strong enough this symmetry survives in the form of discrete scale invariance, opening the prospect of an Efimov-like geometrical spectrum in two-hadron systems. For a molecular pentaquark with quantum numbers 3/2^- the attraction is not enough to exhibit discrete scale invariance, but this prospect might very well be realized in a 1/2⁺ pentaquark or in other hadron molecules involving transitions between particle channels with opposite intrinsic parity and a pion near the mass shell. A very good candidate is the Λc(2595)Ξb-ΣcΞb^' molecule. Independently of this, the 1/r² force is expected to play a very important role in the formation of this type of hadron molecule, which points to the existence of 1/2⁺ ΣcD*-Λc(2595)D and 1⁺ Λc(2595)Ξb-ΣcΞb^' molecules and 0⁺/1^- Λc(2595)Ξb-ΣcΞb^' baryonia.