Faddeev calculation of pentaquark ${\mathrm{\Theta }}^{+}$ in the Nambu-Jona-Lasinio model-based diquark picture

A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark ${\mathrm{\Theta }}^{+}$ in the diquark picture of Jaffe and Wilczek in which ${\mathrm{\Theta }}^{+}$ is a diquark-diquark-$\bar{s}$ three-body system. Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two-body scatterings of $\bar{s}D$ and $DD$. With the use of coupling constants determined from the meson sector, we find that $\bar{s}D$ interaction is attractive in s-wave while $DD$ interaction is repulsive in p-wave. With only the lowest three-body channel considered, we do not find a bound ${\frac{1}{2}}^{+}$ pentaquark state. Instead, a bound pentaquark ${\mathrm{\Theta }}^{+}$ with ${\frac{1}{2}}^{-}$ is obtained with a unphysically strong vector mesonic coupling constants.