Decay Probability Ratio of Pentaquark Theta^+ State

The pentaquark state of $\Theta^{+}(uudd\bar{s})$ has been observed to decay with two decay modes: $\Theta^+\to n K^+$ and $\Theta^+ \to p K^0$. The decay probability ratio of the two decay modes is studied with general symmetry consideration of isospin, spin, and parity. We arrive at a result of the ratio $\frac{\Gamma(\Theta^+\to nK^+)}{\Gamma(\Theta^+\to pK^0)} =\frac{(\alpha-\beta)^2}{(\alpha+\beta)^2}(\frac{k_1}{k_2})^{2L+1}$, which is valid for the $\Theta^+$ state to be a pure isoscalar or isovector state, or an isoscalar and isovector mixing state, or an isotensor state with mixture of isoscalar and isovector components with coefficients $\alpha$ and $\beta$. The dependence on spin and parity of the pentaquark $\Theta^+$ state is found to be small due to small difference between the center of mass decay momenta $k_1$ and $k_2$ of the two decay modes. We also provide an analysis on the constraint of the isospin of $\Theta^+$ from the absence of a peak in the $pK^+$ invariant mass distribution in the $\gamma p \to pK^+K^-$ process. Future experimental results about the decay probability ratio may provide information about the properties of the pentaquark $\Theta^+$ state.