Systematic study of $Z^+_c$ family from quark model's perspective

Inspired by the present experimental status of charged charmonium-like states $Z_c^+$, the tetraquark states $[cu][\bar{c}\bar{d}]$ are systematically studied in a color flux-tube model with a multi-body confinement potential. The investigation indicates that charged charmonium-like states $Z_c^+(3900)$ or $Z_c^+(3885)$, $Z_c^+(3930)$, $Z_c^+(4020)$ or $Z_c^+(4025)$, $Z_1^+(4050)$, $Z_2^+(4250)$, and $Z_c^+(4200)$ can be uniformly described as tetraquark states $[cu][\bar{c}\bar{d}]$ with the quantum numbers $n^{2S+1}L_J$ and $J^P$ of $1^{3}S_1$ and $1^+$, $2^{3}S_1$ and $1^+$, $1^5S_2$ and $2^+$, $1^3P_1$ and $1^-$, $1^5D_1$ and $1^+$, and $1^3D_1$ and $1^+$, respectively. The predicted lowest charged tetraquark state $[cu][\bar{c}\bar{d}]$ with $0^+$ and $1^1S_0$ has a energy of $3780\pm10$ MeV in the model. The tetraquark states are compact three-dimensional spatial configurations similar to a rugby ball, the higher orbital angular momentum $L$ between the diquark $[cu]$ and antidiquark $[\bar{c}\bar{d}]$, the more prolate of the states. The multibody color flux-tube, a collective degree of freedom, plays an important role in the formation of those charge tetraquark states. However, the two heavier charged states $Z^+_c(4430)$ and $Z^+_c(4475)$ can not be explained as tetraquark states $[cu][\bar{c}\bar{d}]$ in this model approach.