Relativistic calculation of the width of the ${\Theta }^{+}(1540)$ pentaquark

We calculate the width of the ${\Theta }^{+}(1540)$ pentaquark in a relativistic model in which the pentaquark is considered to be composed of a scalar diquark and a spin 1/2 triquark. We consider both positive and negative parity for the pentaquark. There is a single parameter in our model which we vary and which describes the size of the pentaquark. If the pentaquark size is somewhat smaller than that of the nucleon, we find quite small widths for the pentaquark of about 1 MeV or less. Our model of confinement plays an important role in our analysis and makes it possible to use Feynman diagrams to describe the decay of the pentaquark.

Figure 1

Heavy line denotes the pentaquark; line of momentum $-k+P_N$ denotes an on-mass-shell diquark; line of momentum k represents the quark; and $P+k-P_N$ is the momentum of the triquark. In the final state, we have a nucleon of momentum $P_N$ and a kaon of momentum $P-P_N$.

Figure 2

Width of negative-parity pentaquark as a function of the parameter $w_{\Theta }$. (The quite small values in the vicinity of the minimum have rather large uncertainties because of the limitation of the number of points used in our five-dimensional integral which determines the width.) The triquark had negative parity and mass of $m_T=0.800$ GeV. The diquark mass was 0.400 GeV.

Figure 3

Width of the positive-parity pentaquark as a function of the parameter, $w_{\Theta }$, which governs the extent of the pentaquark wave function in momentum space. (In the case of the nucleon we found $w=0.66$ GeV for the wave function calculated in Ref. [25].) In this calculation, the triquark had positive parity with mass $m_T=0.800$ GeV. The diquark mass was 0.400 GeV.