QCD sum rule for spin-$3/2$ pentaquarks

Most QCD-based-approaches, i.e, sum rules and lattice simulations, for pentaquark baryons have been done under the assumption that the pentaquarks have spin-$1/2$. But, the quark model calculations indicate possibility of spin- $3/2$. Because the $\Theta^+$ with $J^{\pi}=3/2^-$ decays into $D$- wave $NK$ states, this scenario may explain the narrow decay width of $\Theta^+$. Thus, we study the spin-$3/2$ pentaquarks using QCD sum rule technique. The spin-$3/2$ field is treated as a Rarita-Schwinger field. We consider two kinds of the diquark-type interpolating field operators and analyze which one is preferable. We perform parity projection and explore the existence of the pentaquark with $J^{\pi}=3/2^+$ and $J^{\pi}=3/2^-$. We find that $\Theta^+$ both of $3/2^+$ and $3/2^-$ are possible to exist, the $3/2^-$ state comes lower in energy than $3/2^+$ by about $60\mathrm{MeV}$ and their masses are around $1.5\mathrm{GeV}$, but they depend on the threshold parameters. We will report the results of the other pentaquark baryon.