${\Theta }^{++}$ from the QCD sum rule

We study the pentaquark $uudd\overline{s}$ with $J=3/2$ and $I=1$ (${\Theta }^{++}$) in the QCD sum rule approach. We derive the QCD sum rules for positive and negative parity states of the pentaquark. The QCD sum rule predicts that there exists ${\Theta }^{++}$ with negative parity and its mass is $1.5\sim 1.6$ GeV. The negative parity ${\Theta }^{++}$ can be extremely narrow, since it lies much below the $\Delta K$ threshold and the decay into $NK$ state is strongly suppressed due to the $D$-wave centrifugal barrier. Also, the possibility of the existence of the ${\Theta }^{++}$ with positive parity is not excluded. Although it nearly degenerates with the negative parity state, it may be broader than the negative parity state.

Figure 1

$|{\lambda }_{-}{|}^2\exp (-\frac{m_{-}^2}{M^2})$ as functions of Borel mass, $M$ with the continuum threshold parameter ${\omega }_{-}=1.8\mathrm{G}\mathrm{e}\mathrm{V}$.

Figure 2

Mass of $J^P=3/2^{-}$, $I=1$ pentaquark as a function of Borel mass, $M$, with the continuum threshold parameters ${\omega }_{-}=1.73\mathrm{G}\mathrm{e}\mathrm{V}$ (solid line), $1.8\mathrm{G}\mathrm{e}\mathrm{V}$ (dashed), $1.9\mathrm{G}\mathrm{e}\mathrm{V}$ (dotted).

Figure 3

$|{\lambda }_{+}{|}^2\exp (-\frac{m_{+}^2}{M^2})$ as functions of Borel mass, $M$ with the continuum threshold parameter ${\omega }_{+}=1.8\mathrm{G}\mathrm{e}\mathrm{V}$.

Figure 4

Mass of $J^P=3/2^{+}$, $I=1$ pentaquark as a function of Borel mass, $M$ with the continuum threshold parameters ${\omega }_{+}=1.7\mathrm{G}\mathrm{e}\mathrm{V}$ (solid line), $1.8\mathrm{G}\mathrm{e}\mathrm{V}$ (dashed), $1.9\mathrm{G}\mathrm{e}\mathrm{V}$ (dotted).