Binding of the three-hadron $DD^{*}K$ system from the lattice effective field theory

We employ the nuclear lattice effective field theory (NLEFT), an efficient tool for nuclear ab initio calculations, to solve the asymmetric multihadron systems. We take the $DD^*K$ three-body system as an illustration to demonstrate the capability of the method. Here the two-body chiral interactions between $D$, $D^*$, and $K$ are regulated with a soft lattice regulator and calibrated with the binding energies of the $T_{cc}^+$, $D^{*}_{s0}(2317)$, and $D_{s1}(2460)$ molecular states. We then calculate the three-body binding energy using the NLEFT and analyze the systematic uncertainties due to the finite volume effects, the sliding cutoff, and the leading-order three-body forces. Even when the three-body interaction is repulsive (even as large as the infinite repulsive interaction), the three-body system has a bound state unambiguously with binding energy no larger than the $D_{s1}(2460)D$ threshold. To check the renormalization group invariance of our framework, we extract the first excited state. We find that when the ground state is fixed, the first excited states with various cutoffs coincide with each other when the cubic size goes larger. In addition, the standard angular momentum and parity projection technique is implemented for the quantum numbers of the ground and excited states. We find that both of them are $S$-wave states with quantum number $J^{P}=1^-$. Because the three-body state contains two charm quarks, it is easier to be detected in the Large Hadron Collider.