  We systematically construct $I(J^P)=0(3^+)$ six-quark local interpolating
currents without derivative operators. We discuss the best choice of operator,
and select three $\Delta$-$\Delta$ like operators to perform QCD sum rule
analyses to calculate the mass of the $d^*(2380)$. The mass extracted from this
analysis is $M_{d^*} = 2.4\pm0.2$ GeV, consistent with the $d^*(2380)$ mass
observed by the WASA detector at COSY. We also obtain a sum-rule lower mass
bound $M_{d^*} > 2.25$ GeV. We also consider the effect of mixing of singlet
dibaryon fields with the same quantum numbers, and perform the QCD sum rule
analysis of the mixed interpolating current and extract the mass of the
$d^*(2380)$ and its lower mass bound. With optimized mixing parameters, we find
that the mixed current does not change the numerical result significantly.
