Hexaquark picture for $d^*(2380)$

Hexaquark wave function with the quantum numbers $I(J^P)=0(3^+)$, which might be relevant for $d^*(2380)$, is constructed under an assumption that this is composed only of $u,d$ quarks in an $S$-wave. By combining three diquarks of either type, ($\bm{\bar{3}}_c, I=1$) or ($\bm{6}_c, I=0$), we demonstrate that there are five possible configurations for the six-quark state. The fully antisymmetric wave function is constructed by linearly combining the five configurations on an equal footing. We then take this wave function as well as the five configurations to calculate the hexaquark mass using the contact type effective potential consisting of the color-spin, color electric and constant shift. The mass is found to be the same regardless of the configurations being used including the fully antisymmetric one. This result can be traced to the fact that the hexaquark system has a freedom in choosing three diquarks in the construction of its wave function. The calculated hexaquark mass using the empirical parameters independently fixed from the baryon spectroscopy is found to be around $2342$ MeV, which is indeed very close to the experimental mass of $d^*(2380)$. Therefore, the hexaquark picture is promising for $d^*(2380)$ as far as the mass is concerned.