Five-body calculation of resonance and scattering states of pentaquark system

Scattering problem of the $uudd\bar{s}$ system, in the standard non-relativistic quark model of Isgur-Karl, is solved for the first time, by treating the large five-body modelspace including the $NK$ scattering channel accurately with the Gaussian expansion method and the Kohn-type coupled-channel variational method. The two-body interaction that reproduces observed properties of ordinary baryons and mesons is applied to the pentaquark system with no additional adjustable parameter. The five-body wave function calculated has the correct asymptotic form in its the scattering-channel component and describes $qq$ and $q\bar{q}$ correlations properly. The $NK$ scattering phase shift calculated shows no resonance in the energy region of the reported pentaquark ${\mathrm{\Theta }}^{+}(1540)$, that is, at 0-500 MeV above the $NK$ threshold ($1.4-1.9$ GeV in mass). The phase shift does show two resonances just above 500 MeV: a broad ${1\mathrm{/}2}^{+}$ resonance with a width of $\mathrm{\Gamma }\sim 110$ MeV located at $\sim 520$ MeV ($\sim 2.0$ GeV in mass) and a sharp ${1\mathrm{/}2}^{-}$ resonance with $\mathrm{\Gamma }=$0.12 MeV at 540 MeV. Properties of these states are discussed.