Feasibility study of the $K^{+} d \to K^{0} p p$ reaction in killing/saving the "$\Theta ^{+}$" pentaquark.

We investigate theoretically the $K^{0} p$ invariant mass spectrum of the $K^{+} d \to K^{0} p p$ reaction and scrutinize how the signal of the "$\Theta ^{+}$" pentaquark, if it exists, emerges in the $K^{0} p$ spectrum. The most prominent advantage of this reaction is that we can clearly judge whether the "$\Theta ^{+}$" exists or not as a direct-formation production without significant backgrounds, in contrast to other reactions such as photoproduction and $\pi$-induced productions. We show that while the impulse or single-step scattering process can cover the "$\Theta ^{+}$" energy region with an initial kaon momentum $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ in the laboratory frame, the contributions from double-step processes may have a potential possibility to reach the "$\Theta ^{+}$" energy region with a higher kaon momentum $k_{\mathrm{lab}} \sim 1 \, \mathrm{GeV} / c$. Assuming that the full decay width of the "$\Theta ^{+}$" is around $0.5\, \mathrm{MeV}$, we predict that the magnitude of the peak corresponding to the "$\Theta^+$" is around a few hundred $\mu \mathrm{b}$ to $1\, \mathrm{mb}$ with the momentum of the kaon beam $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ while it is around $\lesssim 1\, \mu \mathrm{b}$ with $k_{\mathrm{lab}} \approx 0.85\, \mathrm{GeV} / c$. Thus, the "$\Theta^+$" peak is more likely to be seen at $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ than at $k_{\mathrm{lab}} \approx 0.85 \,\mathrm{GeV} / c$.