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

Motivated by the recent discovery of two new states in the $B^+\rightarrow D^+D^-K^+$ decay by the LHCb Collaboration, we study the $D\bar{D}K$ three-body system by solving the Schr\"odinger 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 $B_3(D\bar{D}K)=48.9^{+1.4}_{-2.4}$ MeV. It can decay into $J/\psi K$ and $D_s\bar{D}^*$ 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 $D\bar{D}^*K$ system, which is shown to also bind with quantum numbers $I(J^P)=\frac{1}{2}(1^-)$ and a binding energy of $B_3(D\bar{D}^*K)\simeq 77.3^{+3.1}_{-6.6}$ MeV, consistent with the results of previous works.