Further study on 5q configuration states in the chiral SU(3) quark model

The structure of the $5q$ configuration states with strangeness ${\cal{S}}=+1$ is further studied in the chiral SU(3) quark model based on our previous work. We calculate the energies of fifteen low configurations of the $5q$ system, four lowest configurations of $J^{\pi}={1/2}^-$ with $4q$ partition $[4]_{orb}(0s^4)[31]^{\sigma f}$, four of $J^{\pi}={1/2}^+$ with $4q$ partition $[31]_{orb}(0s^30p)[4]^{\sigma f}$ and seven of $J^{\pi}={1/2}^+$ with $4q$ partition $[4]_{orb}(0s^30p)[31]^{\sigma f}$. Some modifications are made in this further study, i.e., the orbital wave function is extended as an expansion of 4 different size harmonic oscillator forms; three various forms (quadratic, linear and error function form) of the color confinement potential are considered; the states with $4q$ partition $[4]_{orb}(0s^30p)[31]^{\sigma f}$ are added, which are unnegligible in the $J^{\pi}={1/2}^+$ case and were not considered in our previous paper, further the mixing between configurations $[31]_{orb}(0s^30p)[4]^{\sigma f}$ and $[4]_{orb}(0s^30p)[31]^{\sigma f}$ is also investigated. The results show that the T=0 state is still always the lowest one for both $J^{\pi}={1/2}^-$ and $J^{\pi}={1/2}^+$ states, and $J^{\pi}={1/2}^-, T=0$ state is always lower than that of $J^{\pi}={1/2}^+$. All of these modifications can only offer several tens to hundred MeV effect, and the theoretical value of the lowest state is still about 245 MeV higher than the experimental mass of $\Theta^+$. It seems to be difficult to get the calculated mass close to the observed one with the reasonable parameters in the framework of the chiral SU(3) quark model when the model space is chosen as a $5q$ cluster.