Scale Invariance in Heavy Hadron Molecules

Abstract

We discuss a scenario in which the $P_c(4450{)}^{+}$ heavy pentaquark is a ${\Sigma }_c{\bar{D}}^{\ast }$-${\Lambda }_c(2595)\bar{D}$ molecule. The ${\Lambda }_{c1}\bar{D}\to {\Sigma }_c{\bar{D}}^{\ast }$ transition is mediated by the exchange of a pion almost on the mass shell that generates a long-range $1/r^2$ 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 $1/r^2$ 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 ${\frac{3}{2}}^{-}$ the attraction is not enough to exhibit discrete scale invariance, but this prospect might very well be realized in a ${\frac{1}{2}}^{+}$ 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 ${\Lambda }_c(2595){\bar{\Xi }}_b-{\Sigma }_c{\bar{\Xi }}_b^{\prime }$ molecule. Independently of this, the $1/r^2$ force is expected to play a very important role in the formation of this type of hadron molecule, which points to the existence of ${\frac{1}{2}}^{+}$ ${\Sigma }_c{D}^{\ast }$-${\Lambda }_c(2595)D$ and $1^{+}$ ${\Lambda }_c(2595){\Xi }_b-{\Sigma }_c{\Xi }_b^{\prime }$ molecules and $0^{+}$/$1^{-}$ ${\Lambda }_c(2595){\bar{\Xi }}_b-{\Sigma }_c{\bar{\Xi }}_b^{\prime }$ baryonia.

6 pages, 1 figure, corresponds to published version

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