Excited $K$ meson, $K_c(4180)$, with hidden charm as a $D\bar{D}K$ bound state

Abstract

Motivated by the recent discovery of two new states in the $B^{+}\to D^{+}{D}^{-}{K}^{+}$ decay by the LHCb Collaboration, we study the $D\bar{D}K$ three-body system by solving the Schrödinger equation with the Gaussian Expansion Method. We show that the $D\bar{D}K$ system can bind with quantum numbers $I(J^P)=\frac{1}{2}(0^{-})$ and a binding energy of $B_3(D\bar{D}K)=48.9_{-2.4}^{+1.4}$ MeV. It can decay into $J/\ddot{I}ˆK$ and $D_s{\bar{D}}^{\ast }$ via triangle diagrams, yielding a partial decay width of about 1 MeV. As a result, if discovered, it will serve as a highly nontrivial check on the nature of the many exotic hadrons discovered so far and on non-perturbative QCD as well. Assuming heavy quark spin symmetry, the same formalism is applied to study the $D{\bar{D}}^{\ast }K$ system, which is shown to also bind with quantum numbers $I(J^P)=\frac{1}{2}(1^{-})$ and a binding energy of $B_3(D{\bar{D}}^{\ast }K)\simeq 77.3_{-6.6}^{+3.1}$ MeV, consistent with the results of previous works.

6 pages, 3 figures